From fcbdc4d3a54f2c110f5e7433d620f331ef0e356a Mon Sep 17 00:00:00 2001
From: Aaron Meurer <asmeurer@gmail.com>
Date: Thu, 11 Feb 2016 11:41:25 -0500
Subject: [PATCH 1/2] Remove the dev-py3k docs

We only have one set of dev docs (preferably built with Python 3, but it
shouldn't even matter, since we don't use 2to3 anymore). The dev-py3k docs
here were not at all up-to-date.
---
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diff --git a/0.7.6/index.html b/0.7.6/index.html
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diff --git a/dev-py3k/_modules/mpmath.html b/dev-py3k/_modules/mpmath.html
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-            
-  <h1>Source code for mpmath</h1><div class="highlight"><pre>
-<span class="n">__version__</span> <span class="o">=</span> <span class="s">&#39;0.17&#39;</span>
-
-<span class="kn">from</span> <span class="nn">.usertools</span> <span class="kn">import</span> <span class="n">monitor</span><span class="p">,</span> <span class="n">timing</span>
-
-<span class="kn">from</span> <span class="nn">.ctx_fp</span> <span class="kn">import</span> <span class="n">FPContext</span>
-<span class="kn">from</span> <span class="nn">.ctx_mp</span> <span class="kn">import</span> <span class="n">MPContext</span>
-<span class="kn">from</span> <span class="nn">.ctx_iv</span> <span class="kn">import</span> <span class="n">MPIntervalContext</span>
-
-<span class="n">fp</span> <span class="o">=</span> <span class="n">FPContext</span><span class="p">()</span>
-<span class="n">mp</span> <span class="o">=</span> <span class="n">MPContext</span><span class="p">()</span>
-<span class="n">iv</span> <span class="o">=</span> <span class="n">MPIntervalContext</span><span class="p">()</span>
-
-<span class="n">fp</span><span class="o">.</span><span class="n">_mp</span> <span class="o">=</span> <span class="n">mp</span>
-<span class="n">mp</span><span class="o">.</span><span class="n">_mp</span> <span class="o">=</span> <span class="n">mp</span>
-<span class="n">iv</span><span class="o">.</span><span class="n">_mp</span> <span class="o">=</span> <span class="n">mp</span>
-<span class="n">mp</span><span class="o">.</span><span class="n">_fp</span> <span class="o">=</span> <span class="n">fp</span>
-<span class="n">fp</span><span class="o">.</span><span class="n">_fp</span> <span class="o">=</span> <span class="n">fp</span>
-<span class="n">mp</span><span class="o">.</span><span class="n">_iv</span> <span class="o">=</span> <span class="n">iv</span>
-<span class="n">fp</span><span class="o">.</span><span class="n">_iv</span> <span class="o">=</span> <span class="n">iv</span>
-<span class="n">iv</span><span class="o">.</span><span class="n">_iv</span> <span class="o">=</span> <span class="n">iv</span>
-
-<span class="c"># XXX: extremely bad pickle hack</span>
-<span class="kn">from</span> <span class="nn">.</span> <span class="kn">import</span> <span class="n">ctx_mp</span> <span class="k">as</span> <span class="n">_ctx_mp</span>
-<span class="n">_ctx_mp</span><span class="o">.</span><span class="n">_mpf_module</span><span class="o">.</span><span class="n">mpf</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">mpf</span>
-<span class="n">_ctx_mp</span><span class="o">.</span><span class="n">_mpf_module</span><span class="o">.</span><span class="n">mpc</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">mpc</span>
-
-<span class="n">make_mpf</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">make_mpf</span>
-<span class="n">make_mpc</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">make_mpc</span>
-
-<span class="n">extraprec</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">extraprec</span>
-<span class="n">extradps</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">extradps</span>
-<span class="n">workprec</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">workprec</span>
-<span class="n">workdps</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">workdps</span>
-<span class="n">autoprec</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">autoprec</span>
-<span class="n">maxcalls</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">maxcalls</span>
-<span class="n">memoize</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">memoize</span>
-
-<span class="n">mag</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">mag</span>
-
-<span class="n">bernfrac</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">bernfrac</span>
-
-<span class="n">qfrom</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">qfrom</span>
-<span class="n">mfrom</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">mfrom</span>
-<span class="n">kfrom</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">kfrom</span>
-<span class="n">taufrom</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">taufrom</span>
-<span class="n">qbarfrom</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">qbarfrom</span>
-<span class="n">ellipfun</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">ellipfun</span>
-<span class="n">jtheta</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">jtheta</span>
-<span class="n">kleinj</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">kleinj</span>
-
-<span class="n">qp</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">qp</span>
-<span class="n">qhyper</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">qhyper</span>
-<span class="n">qgamma</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">qgamma</span>
-<span class="n">qfac</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">qfac</span>
-
-<span class="n">nint_distance</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">nint_distance</span>
-
-<span class="n">plot</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">plot</span>
-<span class="n">cplot</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">cplot</span>
-<span class="n">splot</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">splot</span>
-
-<span class="n">odefun</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">odefun</span>
-
-<span class="n">jacobian</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">jacobian</span>
-<span class="n">findroot</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">findroot</span>
-<span class="n">multiplicity</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">multiplicity</span>
-
-<span class="n">isinf</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">isinf</span>
-<span class="n">isnan</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">isnan</span>
-<span class="n">isnormal</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">isnormal</span>
-<span class="n">isint</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">isint</span>
-<span class="n">almosteq</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">almosteq</span>
-<span class="n">nan</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">nan</span>
-<span class="n">rand</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">rand</span>
-
-<span class="n">absmin</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">absmin</span>
-<span class="n">absmax</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">absmax</span>
-
-<span class="n">fraction</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">fraction</span>
-
-<span class="n">linspace</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">linspace</span>
-<span class="n">arange</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">arange</span>
-
-<span class="n">mpmathify</span> <span class="o">=</span> <span class="n">convert</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">convert</span>
-<span class="n">mpc</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">mpc</span>
-
-<span class="n">mpi</span> <span class="o">=</span> <span class="n">iv</span><span class="o">.</span><span class="n">_mpi</span>
-
-<span class="n">nstr</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">nstr</span>
-<span class="n">nprint</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">nprint</span>
-<span class="n">chop</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">chop</span>
-
-<span class="n">fneg</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">fneg</span>
-<span class="n">fadd</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">fadd</span>
-<span class="n">fsub</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">fsub</span>
-<span class="n">fmul</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">fmul</span>
-<span class="n">fdiv</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">fdiv</span>
-<span class="n">fprod</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">fprod</span>
-
-<span class="n">quad</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">quad</span>
-<span class="n">quadgl</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">quadgl</span>
-<span class="n">quadts</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">quadts</span>
-<span class="n">quadosc</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">quadosc</span>
-
-<span class="n">pslq</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">pslq</span>
-<span class="n">identify</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">identify</span>
-<span class="n">findpoly</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">findpoly</span>
-
-<span class="n">richardson</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">richardson</span>
-<span class="n">shanks</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">shanks</span>
-<span class="n">nsum</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">nsum</span>
-<span class="n">nprod</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">nprod</span>
-<span class="n">difference</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">difference</span>
-<span class="n">diff</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">diff</span>
-<span class="n">diffs</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">diffs</span>
-<span class="n">diffs_prod</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">diffs_prod</span>
-<span class="n">diffs_exp</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">diffs_exp</span>
-<span class="n">diffun</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">diffun</span>
-<span class="n">differint</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">differint</span>
-<span class="n">taylor</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">taylor</span>
-<span class="n">pade</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">pade</span>
-<span class="n">polyval</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">polyval</span>
-<span class="n">polyroots</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">polyroots</span>
-<span class="n">fourier</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">fourier</span>
-<span class="n">fourierval</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">fourierval</span>
-<span class="n">sumem</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">sumem</span>
-<span class="n">sumap</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">sumap</span>
-<span class="n">chebyfit</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">chebyfit</span>
-<span class="n">limit</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">limit</span>
-
-<span class="n">matrix</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">matrix</span>
-<span class="n">eye</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">eye</span>
-<span class="n">diag</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">diag</span>
-<span class="n">zeros</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">zeros</span>
-<span class="n">ones</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">ones</span>
-<span class="n">hilbert</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">hilbert</span>
-<span class="n">randmatrix</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">randmatrix</span>
-<span class="n">swap_row</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">swap_row</span>
-<span class="n">extend</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">extend</span>
-<span class="n">norm</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">norm</span>
-<span class="n">mnorm</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">mnorm</span>
-
-<span class="n">lu_solve</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">lu_solve</span>
-<span class="n">lu</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">lu</span>
-<span class="n">unitvector</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">unitvector</span>
-<span class="n">inverse</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">inverse</span>
-<span class="n">residual</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">residual</span>
-<span class="n">qr_solve</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">qr_solve</span>
-<span class="n">cholesky</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">cholesky</span>
-<span class="n">cholesky_solve</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">cholesky_solve</span>
-<span class="n">det</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">det</span>
-<span class="n">cond</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">cond</span>
-
-<span class="n">expm</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">expm</span>
-<span class="n">sqrtm</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">sqrtm</span>
-<span class="n">powm</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">powm</span>
-<span class="n">logm</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">logm</span>
-<span class="n">sinm</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">sinm</span>
-<span class="n">cosm</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">cosm</span>
-
-<span class="n">mpf</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">mpf</span>
-<span class="n">j</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">j</span>
-<span class="n">exp</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">exp</span>
-<span class="n">expj</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">expj</span>
-<span class="n">expjpi</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">expjpi</span>
-<span class="n">ln</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">ln</span>
-<span class="n">im</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">im</span>
-<span class="n">re</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">re</span>
-<span class="n">inf</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">inf</span>
-<span class="n">ninf</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">ninf</span>
-<span class="n">sign</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">sign</span>
-
-<span class="n">eps</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">eps</span>
-<span class="n">pi</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">pi</span>
-<span class="n">ln2</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">ln2</span>
-<span class="n">ln10</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">ln10</span>
-<span class="n">phi</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">phi</span>
-<span class="n">e</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">e</span>
-<span class="n">euler</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">euler</span>
-<span class="n">catalan</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">catalan</span>
-<span class="n">khinchin</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">khinchin</span>
-<span class="n">glaisher</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">glaisher</span>
-<span class="n">apery</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">apery</span>
-<span class="n">degree</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">degree</span>
-<span class="n">twinprime</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">twinprime</span>
-<span class="n">mertens</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">mertens</span>
-
-<span class="n">ldexp</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">ldexp</span>
-<span class="n">frexp</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">frexp</span>
-
-<span class="n">fsum</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">fsum</span>
-<span class="n">fdot</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">fdot</span>
-
-<span class="n">sqrt</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">sqrt</span>
-<span class="n">cbrt</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">cbrt</span>
-<span class="n">exp</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">exp</span>
-<span class="n">ln</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">ln</span>
-<span class="n">log</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">log</span>
-<span class="n">log10</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">log10</span>
-<span class="n">power</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">power</span>
-<span class="n">cos</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">cos</span>
-<span class="n">sin</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">sin</span>
-<span class="n">tan</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">tan</span>
-<span class="n">cosh</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">cosh</span>
-<span class="n">sinh</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">sinh</span>
-<span class="n">tanh</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">tanh</span>
-<span class="n">acos</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">acos</span>
-<span class="n">asin</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">asin</span>
-<span class="n">atan</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">atan</span>
-<span class="n">asinh</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">asinh</span>
-<span class="n">acosh</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">acosh</span>
-<span class="n">atanh</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">atanh</span>
-<span class="n">sec</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">sec</span>
-<span class="n">csc</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">csc</span>
-<span class="n">cot</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">cot</span>
-<span class="n">sech</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">sech</span>
-<span class="n">csch</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">csch</span>
-<span class="n">coth</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">coth</span>
-<span class="n">asec</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">asec</span>
-<span class="n">acsc</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">acsc</span>
-<span class="n">acot</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">acot</span>
-<span class="n">asech</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">asech</span>
-<span class="n">acsch</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">acsch</span>
-<span class="n">acoth</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">acoth</span>
-<span class="n">cospi</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">cospi</span>
-<span class="n">sinpi</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">sinpi</span>
-<span class="n">sinc</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">sinc</span>
-<span class="n">sincpi</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">sincpi</span>
-<span class="n">cos_sin</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">cos_sin</span>
-<span class="n">cospi_sinpi</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">cospi_sinpi</span>
-<span class="n">fabs</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">fabs</span>
-<span class="n">re</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">re</span>
-<span class="n">im</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">im</span>
-<span class="n">conj</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">conj</span>
-<span class="n">floor</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">floor</span>
-<span class="n">ceil</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">ceil</span>
-<span class="n">nint</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">nint</span>
-<span class="n">frac</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">frac</span>
-<span class="n">root</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">root</span>
-<span class="n">nthroot</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">nthroot</span>
-<span class="n">hypot</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">hypot</span>
-<span class="n">fmod</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">fmod</span>
-<span class="n">ldexp</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">ldexp</span>
-<span class="n">frexp</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">frexp</span>
-<span class="n">sign</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">sign</span>
-<span class="n">arg</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">arg</span>
-<span class="n">phase</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">phase</span>
-<span class="n">polar</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">polar</span>
-<span class="n">rect</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">rect</span>
-<span class="n">degrees</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">degrees</span>
-<span class="n">radians</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">radians</span>
-<span class="n">atan2</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">atan2</span>
-<span class="n">fib</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">fib</span>
-<span class="n">fibonacci</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">fibonacci</span>
-<span class="n">lambertw</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">lambertw</span>
-<span class="n">zeta</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">zeta</span>
-<span class="n">altzeta</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">altzeta</span>
-<span class="n">gamma</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">gamma</span>
-<span class="n">rgamma</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">rgamma</span>
-<span class="n">factorial</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">factorial</span>
-<span class="n">fac</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">fac</span>
-<span class="n">fac2</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">fac2</span>
-<span class="n">beta</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">beta</span>
-<span class="n">betainc</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">betainc</span>
-<span class="n">psi</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">psi</span>
-<span class="c">#psi0 = mp.psi0</span>
-<span class="c">#psi1 = mp.psi1</span>
-<span class="c">#psi2 = mp.psi2</span>
-<span class="c">#psi3 = mp.psi3</span>
-<span class="n">polygamma</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">polygamma</span>
-<span class="n">digamma</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">digamma</span>
-<span class="c">#trigamma = mp.trigamma</span>
-<span class="c">#tetragamma = mp.tetragamma</span>
-<span class="c">#pentagamma = mp.pentagamma</span>
-<span class="n">harmonic</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">harmonic</span>
-<span class="n">bernoulli</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">bernoulli</span>
-<span class="n">bernfrac</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">bernfrac</span>
-<span class="n">stieltjes</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">stieltjes</span>
-<span class="n">hurwitz</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">hurwitz</span>
-<span class="n">dirichlet</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">dirichlet</span>
-<span class="n">bernpoly</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">bernpoly</span>
-<span class="n">eulerpoly</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">eulerpoly</span>
-<span class="n">eulernum</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">eulernum</span>
-<span class="n">polylog</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">polylog</span>
-<span class="n">clsin</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">clsin</span>
-<span class="n">clcos</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">clcos</span>
-<span class="n">gammainc</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">gammainc</span>
-<span class="n">gammaprod</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">gammaprod</span>
-<span class="n">binomial</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">binomial</span>
-<span class="n">rf</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">rf</span>
-<span class="n">ff</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">ff</span>
-<span class="n">hyper</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">hyper</span>
-<span class="n">hyp0f1</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">hyp0f1</span>
-<span class="n">hyp1f1</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">hyp1f1</span>
-<span class="n">hyp1f2</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">hyp1f2</span>
-<span class="n">hyp2f1</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">hyp2f1</span>
-<span class="n">hyp2f2</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">hyp2f2</span>
-<span class="n">hyp2f0</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">hyp2f0</span>
-<span class="n">hyp2f3</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">hyp2f3</span>
-<span class="n">hyp3f2</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">hyp3f2</span>
-<span class="n">hyperu</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">hyperu</span>
-<span class="n">hypercomb</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">hypercomb</span>
-<span class="n">meijerg</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">meijerg</span>
-<span class="n">appellf1</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">appellf1</span>
-<span class="n">appellf2</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">appellf2</span>
-<span class="n">appellf3</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">appellf3</span>
-<span class="n">appellf4</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">appellf4</span>
-<span class="n">hyper2d</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">hyper2d</span>
-<span class="n">bihyper</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">bihyper</span>
-<span class="n">erf</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">erf</span>
-<span class="n">erfc</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">erfc</span>
-<span class="n">erfi</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">erfi</span>
-<span class="n">erfinv</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">erfinv</span>
-<span class="n">npdf</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">npdf</span>
-<span class="n">ncdf</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">ncdf</span>
-<span class="n">expint</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">expint</span>
-<span class="n">e1</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">e1</span>
-<span class="n">ei</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">ei</span>
-<span class="n">li</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">li</span>
-<span class="n">ci</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">ci</span>
-<span class="n">si</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">si</span>
-<span class="n">chi</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">chi</span>
-<span class="n">shi</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">shi</span>
-<span class="n">fresnels</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">fresnels</span>
-<span class="n">fresnelc</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">fresnelc</span>
-<span class="n">airyai</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">airyai</span>
-<span class="n">airybi</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">airybi</span>
-<span class="n">airyaizero</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">airyaizero</span>
-<span class="n">airybizero</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">airybizero</span>
-<span class="n">scorergi</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">scorergi</span>
-<span class="n">scorerhi</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">scorerhi</span>
-<span class="n">ellipk</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">ellipk</span>
-<span class="n">ellipe</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">ellipe</span>
-<span class="n">ellipf</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">ellipf</span>
-<span class="n">ellippi</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">ellippi</span>
-<span class="n">elliprc</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">elliprc</span>
-<span class="n">elliprj</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">elliprj</span>
-<span class="n">elliprf</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">elliprf</span>
-<span class="n">elliprd</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">elliprd</span>
-<span class="n">elliprg</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">elliprg</span>
-<span class="n">agm</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">agm</span>
-<span class="n">jacobi</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">jacobi</span>
-<span class="n">chebyt</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">chebyt</span>
-<span class="n">chebyu</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">chebyu</span>
-<span class="n">legendre</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">legendre</span>
-<span class="n">legenp</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">legenp</span>
-<span class="n">legenq</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">legenq</span>
-<span class="n">hermite</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">hermite</span>
-<span class="n">pcfd</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">pcfd</span>
-<span class="n">pcfu</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">pcfu</span>
-<span class="n">pcfv</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">pcfv</span>
-<span class="n">pcfw</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">pcfw</span>
-<span class="n">gegenbauer</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">gegenbauer</span>
-<span class="n">laguerre</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">laguerre</span>
-<span class="n">spherharm</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">spherharm</span>
-<span class="n">besselj</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">besselj</span>
-<span class="n">j0</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">j0</span>
-<span class="n">j1</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">j1</span>
-<span class="n">besseli</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">besseli</span>
-<span class="n">bessely</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">bessely</span>
-<span class="n">besselk</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">besselk</span>
-<span class="n">besseljzero</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">besseljzero</span>
-<span class="n">besselyzero</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">besselyzero</span>
-<span class="n">hankel1</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">hankel1</span>
-<span class="n">hankel2</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">hankel2</span>
-<span class="n">struveh</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">struveh</span>
-<span class="n">struvel</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">struvel</span>
-<span class="n">angerj</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">angerj</span>
-<span class="n">webere</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">webere</span>
-<span class="n">lommels1</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">lommels1</span>
-<span class="n">lommels2</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">lommels2</span>
-<span class="n">whitm</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">whitm</span>
-<span class="n">whitw</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">whitw</span>
-<span class="n">ber</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">ber</span>
-<span class="n">bei</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">bei</span>
-<span class="n">ker</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">ker</span>
-<span class="n">kei</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">kei</span>
-<span class="n">coulombc</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">coulombc</span>
-<span class="n">coulombf</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">coulombf</span>
-<span class="n">coulombg</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">coulombg</span>
-<span class="n">lambertw</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">lambertw</span>
-<span class="n">barnesg</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">barnesg</span>
-<span class="n">superfac</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">superfac</span>
-<span class="n">hyperfac</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">hyperfac</span>
-<span class="n">loggamma</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">loggamma</span>
-<span class="n">siegeltheta</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">siegeltheta</span>
-<span class="n">siegelz</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">siegelz</span>
-<span class="n">grampoint</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">grampoint</span>
-<span class="n">zetazero</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">zetazero</span>
-<span class="n">riemannr</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">riemannr</span>
-<span class="n">primepi</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">primepi</span>
-<span class="n">primepi2</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">primepi2</span>
-<span class="n">primezeta</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">primezeta</span>
-<span class="n">bell</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">bell</span>
-<span class="n">polyexp</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">polyexp</span>
-<span class="n">expm1</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">expm1</span>
-<span class="n">powm1</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">powm1</span>
-<span class="n">unitroots</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">unitroots</span>
-<span class="n">cyclotomic</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">cyclotomic</span>
-<span class="n">mangoldt</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">mangoldt</span>
-<span class="n">secondzeta</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">secondzeta</span>
-<span class="n">nzeros</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">nzeros</span>
-<span class="n">backlunds</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">backlunds</span>
-<span class="n">lerchphi</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">lerchphi</span>
-
-<span class="c"># be careful when changing this name, don&#39;t use test*!</span>
-<span class="k">def</span> <span class="nf">runtests</span><span class="p">():</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Run all mpmath tests and print output.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">import</span> <span class="nn">os.path</span>
-    <span class="kn">from</span> <span class="nn">inspect</span> <span class="kn">import</span> <span class="n">getsourcefile</span>
-    <span class="kn">from</span> <span class="nn">.tests</span> <span class="kn">import</span> <span class="n">runtests</span> <span class="k">as</span> <span class="n">tests</span>
-    <span class="n">testdir</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">dirname</span><span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">abspath</span><span class="p">(</span><span class="n">getsourcefile</span><span class="p">(</span><span class="n">tests</span><span class="p">)))</span>
-    <span class="n">importdir</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">abspath</span><span class="p">(</span><span class="n">testdir</span> <span class="o">+</span> <span class="s">&#39;/../..&#39;</span><span class="p">)</span>
-    <span class="n">tests</span><span class="o">.</span><span class="n">testit</span><span class="p">(</span><span class="n">importdir</span><span class="p">,</span> <span class="n">testdir</span><span class="p">)</span>
-
-<span class="k">def</span> <span class="nf">doctests</span><span class="p">():</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="kn">import</span> <span class="nn">psyco</span>
-        <span class="n">psyco</span><span class="o">.</span><span class="n">full</span><span class="p">()</span>
-    <span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>
-        <span class="k">pass</span>
-    <span class="kn">import</span> <span class="nn">sys</span>
-    <span class="kn">from</span> <span class="nn">timeit</span> <span class="kn">import</span> <span class="n">default_timer</span> <span class="k">as</span> <span class="n">clock</span>
-    <span class="nb">filter</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">arg</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">sys</span><span class="o">.</span><span class="n">argv</span><span class="p">):</span>
-        <span class="k">if</span> <span class="s">&#39;__init__.py&#39;</span> <span class="ow">in</span> <span class="n">arg</span><span class="p">:</span>
-            <span class="nb">filter</span> <span class="o">=</span> <span class="p">[</span><span class="n">sn</span> <span class="k">for</span> <span class="n">sn</span> <span class="ow">in</span> <span class="n">sys</span><span class="o">.</span><span class="n">argv</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">:]</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">sn</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&quot;-&quot;</span><span class="p">)]</span>
-            <span class="k">break</span>
-    <span class="kn">import</span> <span class="nn">doctest</span>
-    <span class="n">globs</span> <span class="o">=</span> <span class="nb">globals</span><span class="p">()</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-    <span class="k">for</span> <span class="n">obj</span> <span class="ow">in</span> <span class="n">globs</span><span class="p">:</span> <span class="c">#sorted(globs.keys()):</span>
-        <span class="k">if</span> <span class="nb">filter</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">sum</span><span class="p">([</span><span class="n">pat</span> <span class="ow">in</span> <span class="n">obj</span> <span class="k">for</span> <span class="n">pat</span> <span class="ow">in</span> <span class="nb">filter</span><span class="p">]):</span>
-                <span class="k">continue</span>
-        <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot; &quot;</span><span class="p">)</span>
-        <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span><span class="o">.</span><span class="n">flush</span><span class="p">()</span>
-        <span class="n">t1</span> <span class="o">=</span> <span class="n">clock</span><span class="p">()</span>
-        <span class="n">doctest</span><span class="o">.</span><span class="n">run_docstring_examples</span><span class="p">(</span><span class="n">globs</span><span class="p">[</span><span class="n">obj</span><span class="p">],</span> <span class="p">{},</span> <span class="n">verbose</span><span class="o">=</span><span class="p">(</span><span class="s">&quot;-v&quot;</span> <span class="ow">in</span> <span class="n">sys</span><span class="o">.</span><span class="n">argv</span><span class="p">))</span>
-        <span class="n">t2</span> <span class="o">=</span> <span class="n">clock</span><span class="p">()</span>
-        <span class="k">print</span><span class="p">(</span><span class="nb">round</span><span class="p">(</span><span class="n">t2</span><span class="o">-</span><span class="n">t1</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
-
-<span class="k">if</span> <span class="n">__name__</span> <span class="o">==</span> <span class="s">&#39;__main__&#39;</span><span class="p">:</span>
-    <span class="n">doctests</span><span class="p">()</span>
-</pre></div>
-
-          </div>
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-            
-  <h1>Source code for mpmath.calculus.optimization</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">copy</span> <span class="kn">import</span> <span class="n">copy</span>
-
-<span class="kn">from</span> <span class="nn">..libmp.backend</span> <span class="kn">import</span> <span class="nb">xrange</span>
-
-<span class="k">class</span> <span class="nc">OptimizationMethods</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="n">ctx</span><span class="p">):</span>
-        <span class="k">pass</span>
-
-<span class="c">##############</span>
-<span class="c"># 1D-SOLVERS #</span>
-<span class="c">##############</span>
-
-<div class="viewcode-block" id="Newton"><a class="viewcode-back" href="../../../modules/mpmath/calculus/optimization.html#mpmath.calculus.optimization.Newton">[docs]</a><span class="k">class</span> <span class="nc">Newton</span><span class="p">:</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    1d-solver generating pairs of approximative root and error.</span>
-
-<span class="sd">    Needs starting points x0 close to the root.</span>
-
-<span class="sd">    Pro:</span>
-
-<span class="sd">    * converges fast</span>
-<span class="sd">    * sometimes more robust than secant with bad second starting point</span>
-
-<span class="sd">    Contra:</span>
-
-<span class="sd">    * converges slowly for multiple roots</span>
-<span class="sd">    * needs first derivative</span>
-<span class="sd">    * 2 function evaluations per iteration</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">maxsteps</span> <span class="o">=</span> <span class="mi">20</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ctx</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span> <span class="o">=</span> <span class="n">ctx</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">x0</span> <span class="o">=</span> <span class="n">x0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;expected 1 starting point, got </span><span class="si">%i</span><span class="s">&#39;</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">f</span> <span class="o">=</span> <span class="n">f</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="s">&#39;df&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
-            <span class="k">def</span> <span class="nf">df</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">df</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;df&#39;</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">df</span> <span class="o">=</span> <span class="n">df</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">f</span>
-        <span class="n">df</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">df</span>
-        <span class="n">x0</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">x0</span>
-        <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">x1</span> <span class="o">=</span> <span class="n">x0</span> <span class="o">-</span> <span class="n">f</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span> <span class="o">/</span> <span class="n">df</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span>
-            <span class="n">error</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">x1</span> <span class="o">-</span> <span class="n">x0</span><span class="p">)</span>
-            <span class="n">x0</span> <span class="o">=</span> <span class="n">x1</span>
-            <span class="k">yield</span> <span class="p">(</span><span class="n">x1</span><span class="p">,</span> <span class="n">error</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Secant"><a class="viewcode-back" href="../../../modules/mpmath/calculus/optimization.html#mpmath.calculus.optimization.Secant">[docs]</a><span class="k">class</span> <span class="nc">Secant</span><span class="p">:</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    1d-solver generating pairs of approximative root and error.</span>
-
-<span class="sd">    Needs starting points x0 and x1 close to the root.</span>
-<span class="sd">    x1 defaults to x0 + 0.25.</span>
-
-<span class="sd">    Pro:</span>
-
-<span class="sd">    * converges fast</span>
-
-<span class="sd">    Contra:</span>
-
-<span class="sd">    * converges slowly for multiple roots</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">maxsteps</span> <span class="o">=</span> <span class="mi">30</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ctx</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span> <span class="o">=</span> <span class="n">ctx</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">x0</span> <span class="o">=</span> <span class="n">x0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">x1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">x0</span> <span class="o">+</span> <span class="mf">0.25</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">x0</span> <span class="o">=</span> <span class="n">x0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">x1</span> <span class="o">=</span> <span class="n">x0</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;expected 1 or 2 starting points, got </span><span class="si">%i</span><span class="s">&#39;</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">f</span> <span class="o">=</span> <span class="n">f</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">f</span>
-        <span class="n">x0</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">x0</span>
-        <span class="n">x1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">x1</span>
-        <span class="n">f0</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span>
-        <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">f1</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x1</span><span class="p">)</span>
-            <span class="n">l</span> <span class="o">=</span> <span class="n">x1</span> <span class="o">-</span> <span class="n">x0</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">l</span><span class="p">:</span>
-                <span class="k">break</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="p">(</span><span class="n">f1</span> <span class="o">-</span> <span class="n">f0</span><span class="p">)</span> <span class="o">/</span> <span class="n">l</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">s</span><span class="p">:</span>
-                <span class="k">break</span>
-            <span class="n">x0</span><span class="p">,</span> <span class="n">x1</span> <span class="o">=</span> <span class="n">x1</span><span class="p">,</span> <span class="n">x1</span> <span class="o">-</span> <span class="n">f1</span><span class="o">/</span><span class="n">s</span>
-            <span class="n">f0</span> <span class="o">=</span> <span class="n">f1</span>
-            <span class="k">yield</span> <span class="n">x1</span><span class="p">,</span> <span class="nb">abs</span><span class="p">(</span><span class="n">l</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MNewton"><a class="viewcode-back" href="../../../modules/mpmath/calculus/optimization.html#mpmath.calculus.optimization.MNewton">[docs]</a><span class="k">class</span> <span class="nc">MNewton</span><span class="p">:</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    1d-solver generating pairs of approximative root and error.</span>
-
-<span class="sd">    Needs starting point x0 close to the root.</span>
-<span class="sd">    Uses modified Newton&#39;s method that converges fast regardless of the</span>
-<span class="sd">    multiplicity of the root.</span>
-
-<span class="sd">    Pro:</span>
-
-<span class="sd">    * converges fast for multiple roots</span>
-
-<span class="sd">    Contra:</span>
-
-<span class="sd">    * needs first and second derivative of f</span>
-<span class="sd">    * 3 function evaluations per iteration</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">maxsteps</span> <span class="o">=</span> <span class="mi">20</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ctx</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span> <span class="o">=</span> <span class="n">ctx</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;expected 1 starting point, got </span><span class="si">%i</span><span class="s">&#39;</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">x0</span> <span class="o">=</span> <span class="n">x0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">f</span> <span class="o">=</span> <span class="n">f</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="s">&#39;df&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
-            <span class="k">def</span> <span class="nf">df</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">df</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;df&#39;</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">df</span> <span class="o">=</span> <span class="n">df</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="s">&#39;d2f&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
-            <span class="k">def</span> <span class="nf">d2f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">df</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">d2f</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;df&#39;</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">d2f</span> <span class="o">=</span> <span class="n">d2f</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">x0</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">f</span>
-        <span class="n">df</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">df</span>
-        <span class="n">d2f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">d2f</span>
-        <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">prevx</span> <span class="o">=</span> <span class="n">x</span>
-            <span class="n">fx</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">fx</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">break</span>
-            <span class="n">dfx</span> <span class="o">=</span> <span class="n">df</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="n">d2fx</span> <span class="o">=</span> <span class="n">d2f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="c"># x = x - F(x)/F&#39;(x) with F(x) = f(x)/f&#39;(x)</span>
-            <span class="n">x</span> <span class="o">-=</span> <span class="n">fx</span> <span class="o">/</span> <span class="p">(</span><span class="n">dfx</span> <span class="o">-</span> <span class="n">fx</span> <span class="o">*</span> <span class="n">d2fx</span> <span class="o">/</span> <span class="n">dfx</span><span class="p">)</span>
-            <span class="n">error</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">prevx</span><span class="p">)</span>
-            <span class="k">yield</span> <span class="n">x</span><span class="p">,</span> <span class="n">error</span>
-</div>
-<div class="viewcode-block" id="Halley"><a class="viewcode-back" href="../../../modules/mpmath/calculus/optimization.html#mpmath.calculus.optimization.Halley">[docs]</a><span class="k">class</span> <span class="nc">Halley</span><span class="p">:</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    1d-solver generating pairs of approximative root and error.</span>
-
-<span class="sd">    Needs a starting point x0 close to the root.</span>
-<span class="sd">    Uses Halley&#39;s method with cubic convergence rate.</span>
-
-<span class="sd">    Pro:</span>
-
-<span class="sd">    * converges even faster the Newton&#39;s method</span>
-<span class="sd">    * useful when computing with *many* digits</span>
-
-<span class="sd">    Contra:</span>
-
-<span class="sd">    * needs first and second derivative of f</span>
-<span class="sd">    * 3 function evaluations per iteration</span>
-<span class="sd">    * converges slowly for multiple roots</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">maxsteps</span> <span class="o">=</span> <span class="mi">20</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ctx</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span> <span class="o">=</span> <span class="n">ctx</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;expected 1 starting point, got </span><span class="si">%i</span><span class="s">&#39;</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">x0</span> <span class="o">=</span> <span class="n">x0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">f</span> <span class="o">=</span> <span class="n">f</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="s">&#39;df&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
-            <span class="k">def</span> <span class="nf">df</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">df</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;df&#39;</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">df</span> <span class="o">=</span> <span class="n">df</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="s">&#39;d2f&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
-            <span class="k">def</span> <span class="nf">d2f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">df</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">d2f</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;df&#39;</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">d2f</span> <span class="o">=</span> <span class="n">d2f</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">x0</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">f</span>
-        <span class="n">df</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">df</span>
-        <span class="n">d2f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">d2f</span>
-        <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">prevx</span> <span class="o">=</span> <span class="n">x</span>
-            <span class="n">fx</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="n">dfx</span> <span class="o">=</span> <span class="n">df</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="n">d2fx</span> <span class="o">=</span> <span class="n">d2f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="n">x</span> <span class="o">-=</span>  <span class="mi">2</span><span class="o">*</span><span class="n">fx</span><span class="o">*</span><span class="n">dfx</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">dfx</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">fx</span><span class="o">*</span><span class="n">d2fx</span><span class="p">)</span>
-            <span class="n">error</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">prevx</span><span class="p">)</span>
-            <span class="k">yield</span> <span class="n">x</span><span class="p">,</span> <span class="n">error</span>
-</div>
-<div class="viewcode-block" id="Muller"><a class="viewcode-back" href="../../../modules/mpmath/calculus/optimization.html#mpmath.calculus.optimization.Muller">[docs]</a><span class="k">class</span> <span class="nc">Muller</span><span class="p">:</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    1d-solver generating pairs of approximative root and error.</span>
-
-<span class="sd">    Needs starting points x0, x1 and x2 close to the root.</span>
-<span class="sd">    x1 defaults to x0 + 0.25; x2 to x1 + 0.25.</span>
-<span class="sd">    Uses Muller&#39;s method that converges towards complex roots.</span>
-
-<span class="sd">    Pro:</span>
-
-<span class="sd">    * converges fast (somewhat faster than secant)</span>
-<span class="sd">    * can find complex roots</span>
-
-<span class="sd">    Contra:</span>
-
-<span class="sd">    * converges slowly for multiple roots</span>
-<span class="sd">    * may have complex values for real starting points and real roots</span>
-
-<span class="sd">    http://en.wikipedia.org/wiki/Muller&#39;s_method</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">maxsteps</span> <span class="o">=</span> <span class="mi">30</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ctx</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span> <span class="o">=</span> <span class="n">ctx</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">x0</span> <span class="o">=</span> <span class="n">x0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">x1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">x0</span> <span class="o">+</span> <span class="mf">0.25</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">x2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">x1</span> <span class="o">+</span> <span class="mf">0.25</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">x0</span> <span class="o">=</span> <span class="n">x0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">x1</span> <span class="o">=</span> <span class="n">x0</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">x2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">x1</span> <span class="o">+</span> <span class="mf">0.25</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">x0</span> <span class="o">=</span> <span class="n">x0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">x1</span> <span class="o">=</span> <span class="n">x0</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">x2</span> <span class="o">=</span> <span class="n">x0</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;expected 1, 2 or 3 starting points, got </span><span class="si">%i</span><span class="s">&#39;</span>
-                             <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">f</span> <span class="o">=</span> <span class="n">f</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;verbose&#39;</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">f</span>
-        <span class="n">x0</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">x0</span>
-        <span class="n">x1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">x1</span>
-        <span class="n">x2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">x2</span>
-        <span class="n">fx0</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span>
-        <span class="n">fx1</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x1</span><span class="p">)</span>
-        <span class="n">fx2</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x2</span><span class="p">)</span>
-        <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="c"># TODO: maybe refactoring with function for divided differences</span>
-            <span class="c"># calculate divided differences</span>
-            <span class="n">fx2x1</span> <span class="o">=</span> <span class="p">(</span><span class="n">fx1</span> <span class="o">-</span> <span class="n">fx2</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">x1</span> <span class="o">-</span> <span class="n">x2</span><span class="p">)</span>
-            <span class="n">fx2x0</span> <span class="o">=</span> <span class="p">(</span><span class="n">fx0</span> <span class="o">-</span> <span class="n">fx2</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">x0</span> <span class="o">-</span> <span class="n">x2</span><span class="p">)</span>
-            <span class="n">fx1x0</span> <span class="o">=</span> <span class="p">(</span><span class="n">fx0</span> <span class="o">-</span> <span class="n">fx1</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">x0</span> <span class="o">-</span> <span class="n">x1</span><span class="p">)</span>
-            <span class="n">w</span> <span class="o">=</span> <span class="n">fx2x1</span> <span class="o">+</span> <span class="n">fx2x0</span> <span class="o">-</span> <span class="n">fx1x0</span>
-            <span class="n">fx2x1x0</span> <span class="o">=</span> <span class="p">(</span><span class="n">fx1x0</span> <span class="o">-</span> <span class="n">fx2x1</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">x0</span> <span class="o">-</span> <span class="n">x2</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">w</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">fx2x1x0</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span><span class="p">:</span>
-                    <span class="k">print</span><span class="p">(</span><span class="s">&#39;canceled with&#39;</span><span class="p">)</span>
-                    <span class="k">print</span><span class="p">(</span><span class="s">&#39;x0 =&#39;</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="s">&#39;, x1 =&#39;</span><span class="p">,</span> <span class="n">x1</span><span class="p">,</span> <span class="s">&#39;and x2 =&#39;</span><span class="p">,</span> <span class="n">x2</span><span class="p">)</span>
-                <span class="k">break</span>
-            <span class="n">x0</span> <span class="o">=</span> <span class="n">x1</span>
-            <span class="n">fx0</span> <span class="o">=</span> <span class="n">fx1</span>
-            <span class="n">x1</span> <span class="o">=</span> <span class="n">x2</span>
-            <span class="n">fx1</span> <span class="o">=</span> <span class="n">fx2</span>
-            <span class="c"># denominator should be as large as possible =&gt; choose sign</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">w</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">4</span><span class="o">*</span><span class="n">fx2</span><span class="o">*</span><span class="n">fx2x1x0</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="n">w</span> <span class="o">-</span> <span class="n">r</span><span class="p">)</span> <span class="o">&gt;</span> <span class="nb">abs</span><span class="p">(</span><span class="n">w</span> <span class="o">+</span> <span class="n">r</span><span class="p">):</span>
-                <span class="n">r</span> <span class="o">=</span> <span class="o">-</span><span class="n">r</span>
-            <span class="n">x2</span> <span class="o">-=</span> <span class="mi">2</span><span class="o">*</span><span class="n">fx2</span> <span class="o">/</span> <span class="p">(</span><span class="n">w</span> <span class="o">+</span> <span class="n">r</span><span class="p">)</span>
-            <span class="n">fx2</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x2</span><span class="p">)</span>
-            <span class="n">error</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">x2</span> <span class="o">-</span> <span class="n">x1</span><span class="p">)</span>
-            <span class="k">yield</span> <span class="n">x2</span><span class="p">,</span> <span class="n">error</span>
-
-<span class="c"># TODO: consider raising a ValueError when there&#39;s no sign change in a and b</span></div>
-<div class="viewcode-block" id="Bisection"><a class="viewcode-back" href="../../../modules/mpmath/calculus/optimization.html#mpmath.calculus.optimization.Bisection">[docs]</a><span class="k">class</span> <span class="nc">Bisection</span><span class="p">:</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    1d-solver generating pairs of approximative root and error.</span>
-
-<span class="sd">    Uses bisection method to find a root of f in [a, b].</span>
-<span class="sd">    Might fail for multiple roots (needs sign change).</span>
-
-<span class="sd">    Pro:</span>
-
-<span class="sd">    * robust and reliable</span>
-
-<span class="sd">    Contra:</span>
-
-<span class="sd">    * converges slowly</span>
-<span class="sd">    * needs sign change</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">maxsteps</span> <span class="o">=</span> <span class="mi">100</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ctx</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span> <span class="o">=</span> <span class="n">ctx</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;expected interval of 2 points, got </span><span class="si">%i</span><span class="s">&#39;</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">f</span> <span class="o">=</span> <span class="n">f</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">a</span> <span class="o">=</span> <span class="n">x0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">b</span> <span class="o">=</span> <span class="n">x0</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">f</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">a</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span>
-        <span class="n">l</span> <span class="o">=</span> <span class="n">b</span> <span class="o">-</span> <span class="n">a</span>
-        <span class="n">fb</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">ldexp</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-            <span class="n">fm</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">fm</span> <span class="o">*</span> <span class="n">fb</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">a</span> <span class="o">=</span> <span class="n">m</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="n">m</span>
-                <span class="n">fb</span> <span class="o">=</span> <span class="n">fm</span>
-            <span class="n">l</span> <span class="o">/=</span> <span class="mi">2</span>
-            <span class="k">yield</span> <span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="nb">abs</span><span class="p">(</span><span class="n">l</span><span class="p">)</span>
-</div>
-<span class="k">def</span> <span class="nf">_getm</span><span class="p">(</span><span class="n">method</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a function to calculate m for Illinois-like methods.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">method</span> <span class="o">==</span> <span class="s">&#39;illinois&#39;</span><span class="p">:</span>
-        <span class="k">def</span> <span class="nf">getm</span><span class="p">(</span><span class="n">fz</span><span class="p">,</span> <span class="n">fb</span><span class="p">):</span>
-            <span class="k">return</span> <span class="mf">0.5</span>
-    <span class="k">elif</span> <span class="n">method</span> <span class="o">==</span> <span class="s">&#39;pegasus&#39;</span><span class="p">:</span>
-        <span class="k">def</span> <span class="nf">getm</span><span class="p">(</span><span class="n">fz</span><span class="p">,</span> <span class="n">fb</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">fb</span><span class="o">/</span><span class="p">(</span><span class="n">fb</span> <span class="o">+</span> <span class="n">fz</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">method</span> <span class="o">==</span> <span class="s">&#39;anderson&#39;</span><span class="p">:</span>
-        <span class="k">def</span> <span class="nf">getm</span><span class="p">(</span><span class="n">fz</span><span class="p">,</span> <span class="n">fb</span><span class="p">):</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">fz</span><span class="o">/</span><span class="n">fb</span>
-            <span class="k">if</span> <span class="n">m</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">m</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="mf">0.5</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;method &#39;</span><span class="si">%s</span><span class="s">&#39; not recognized&quot;</span> <span class="o">%</span> <span class="n">method</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">getm</span>
-
-<div class="viewcode-block" id="Illinois"><a class="viewcode-back" href="../../../modules/mpmath/calculus/optimization.html#mpmath.calculus.optimization.Illinois">[docs]</a><span class="k">class</span> <span class="nc">Illinois</span><span class="p">:</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    1d-solver generating pairs of approximative root and error.</span>
-
-<span class="sd">    Uses Illinois method or similar to find a root of f in [a, b].</span>
-<span class="sd">    Might fail for multiple roots (needs sign change).</span>
-<span class="sd">    Combines bisect with secant (improved regula falsi).</span>
-
-<span class="sd">    The only difference between the methods is the scaling factor m, which is</span>
-<span class="sd">    used to ensure convergence (you can choose one using the &#39;method&#39; keyword):</span>
-
-<span class="sd">    Illinois method (&#39;illinois&#39;):</span>
-<span class="sd">        m = 0.5</span>
-
-<span class="sd">    Pegasus method (&#39;pegasus&#39;):</span>
-<span class="sd">        m = fb/(fb + fz)</span>
-
-<span class="sd">    Anderson-Bjoerk method (&#39;anderson&#39;):</span>
-<span class="sd">        m = 1 - fz/fb if positive else 0.5</span>
-
-<span class="sd">    Pro:</span>
-
-<span class="sd">    * converges very fast</span>
-
-<span class="sd">    Contra:</span>
-
-<span class="sd">    * has problems with multiple roots</span>
-<span class="sd">    * needs sign change</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">maxsteps</span> <span class="o">=</span> <span class="mi">30</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ctx</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span> <span class="o">=</span> <span class="n">ctx</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;expected interval of 2 points, got </span><span class="si">%i</span><span class="s">&#39;</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">a</span> <span class="o">=</span> <span class="n">x0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">b</span> <span class="o">=</span> <span class="n">x0</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">f</span> <span class="o">=</span> <span class="n">f</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">tol</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;tol&#39;</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;verbose&#39;</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">method</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;method&#39;</span><span class="p">,</span> <span class="s">&#39;illinois&#39;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">getm</span> <span class="o">=</span> <span class="n">_getm</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">method</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;using </span><span class="si">%s</span><span class="s"> method&#39;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">method</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">method</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">method</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">f</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">a</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span>
-        <span class="n">fa</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="n">fb</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">l</span> <span class="o">=</span> <span class="n">b</span> <span class="o">-</span> <span class="n">a</span>
-            <span class="k">if</span> <span class="n">l</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">break</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="p">(</span><span class="n">fb</span> <span class="o">-</span> <span class="n">fa</span><span class="p">)</span> <span class="o">/</span> <span class="n">l</span>
-            <span class="n">z</span> <span class="o">=</span> <span class="n">a</span> <span class="o">-</span> <span class="n">fa</span><span class="o">/</span><span class="n">s</span>
-            <span class="n">fz</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="n">fz</span><span class="p">)</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">tol</span><span class="p">:</span>
-                <span class="c"># TODO: better condition (when f is very flat)</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span><span class="p">:</span>
-                    <span class="k">print</span><span class="p">(</span><span class="s">&#39;canceled with z =&#39;</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-                <span class="k">yield</span> <span class="n">z</span><span class="p">,</span> <span class="n">l</span>
-                <span class="k">break</span>
-            <span class="k">if</span> <span class="n">fz</span> <span class="o">*</span> <span class="n">fb</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span> <span class="c"># root in [z, b]</span>
-                <span class="n">a</span> <span class="o">=</span> <span class="n">b</span>
-                <span class="n">fa</span> <span class="o">=</span> <span class="n">fb</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="n">z</span>
-                <span class="n">fb</span> <span class="o">=</span> <span class="n">fz</span>
-            <span class="k">else</span><span class="p">:</span> <span class="c"># root in [a, z]</span>
-                <span class="n">m</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">getm</span><span class="p">(</span><span class="n">fz</span><span class="p">,</span> <span class="n">fb</span><span class="p">)</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="n">z</span>
-                <span class="n">fb</span> <span class="o">=</span> <span class="n">fz</span>
-                <span class="n">fa</span> <span class="o">=</span> <span class="n">m</span><span class="o">*</span><span class="n">fa</span> <span class="c"># scale down to ensure convergence</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span> <span class="ow">and</span> <span class="n">m</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">method</span> <span class="o">==</span> <span class="s">&#39;illinois&#39;</span><span class="p">:</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&#39;m:&#39;</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-            <span class="k">yield</span> <span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="nb">abs</span><span class="p">(</span><span class="n">l</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Pegasus"><a class="viewcode-back" href="../../../modules/mpmath/calculus/optimization.html#mpmath.calculus.optimization.Pegasus">[docs]</a><span class="k">def</span> <span class="nf">Pegasus</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    1d-solver generating pairs of approximative root and error.</span>
-
-<span class="sd">    Uses Pegasus method to find a root of f in [a, b].</span>
-<span class="sd">    Wrapper for illinois to use method=&#39;pegasus&#39;.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;method&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="s">&#39;pegasus&#39;</span>
-    <span class="k">return</span> <span class="n">Illinois</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Anderson"><a class="viewcode-back" href="../../../modules/mpmath/calculus/optimization.html#mpmath.calculus.optimization.Anderson">[docs]</a><span class="k">def</span> <span class="nf">Anderson</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    1d-solver generating pairs of approximative root and error.</span>
-
-<span class="sd">    Uses Anderson-Bjoerk method to find a root of f in [a, b].</span>
-<span class="sd">    Wrapper for illinois to use method=&#39;pegasus&#39;.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;method&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="s">&#39;anderson&#39;</span>
-    <span class="k">return</span> <span class="n">Illinois</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-<span class="c"># TODO: check whether it&#39;s possible to combine it with Illinois stuff</span></div>
-<div class="viewcode-block" id="Ridder"><a class="viewcode-back" href="../../../modules/mpmath/calculus/optimization.html#mpmath.calculus.optimization.Ridder">[docs]</a><span class="k">class</span> <span class="nc">Ridder</span><span class="p">:</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    1d-solver generating pairs of approximative root and error.</span>
-
-<span class="sd">    Ridders&#39; method to find a root of f in [a, b].</span>
-<span class="sd">    Is told to perform as well as Brent&#39;s method while being simpler.</span>
-
-<span class="sd">    Pro:</span>
-
-<span class="sd">    * very fast</span>
-<span class="sd">    * simpler than Brent&#39;s method</span>
-
-<span class="sd">    Contra:</span>
-
-<span class="sd">    * two function evaluations per step</span>
-<span class="sd">    * has problems with multiple roots</span>
-<span class="sd">    * needs sign change</span>
-
-<span class="sd">    http://en.wikipedia.org/wiki/Ridders&#39;_method</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">maxsteps</span> <span class="o">=</span> <span class="mi">30</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ctx</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span> <span class="o">=</span> <span class="n">ctx</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">f</span> <span class="o">=</span> <span class="n">f</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;expected interval of 2 points, got </span><span class="si">%i</span><span class="s">&#39;</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">x1</span> <span class="o">=</span> <span class="n">x0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">x2</span> <span class="o">=</span> <span class="n">x0</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;verbose&#39;</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">tol</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;tol&#39;</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">ctx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">f</span>
-        <span class="n">x1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">x1</span>
-        <span class="n">fx1</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x1</span><span class="p">)</span>
-        <span class="n">x2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">x2</span>
-        <span class="n">fx2</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x2</span><span class="p">)</span>
-        <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">x3</span> <span class="o">=</span> <span class="mf">0.5</span><span class="o">*</span><span class="p">(</span><span class="n">x1</span> <span class="o">+</span> <span class="n">x2</span><span class="p">)</span>
-            <span class="n">fx3</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x3</span><span class="p">)</span>
-            <span class="n">x4</span> <span class="o">=</span> <span class="n">x3</span> <span class="o">+</span> <span class="p">(</span><span class="n">x3</span> <span class="o">-</span> <span class="n">x1</span><span class="p">)</span> <span class="o">*</span> <span class="n">ctx</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">fx1</span> <span class="o">-</span> <span class="n">fx2</span><span class="p">)</span> <span class="o">*</span> <span class="n">fx3</span> <span class="o">/</span> <span class="n">ctx</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">fx3</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">fx1</span><span class="o">*</span><span class="n">fx2</span><span class="p">)</span>
-            <span class="n">fx4</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x4</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="n">fx4</span><span class="p">)</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">tol</span><span class="p">:</span>
-                <span class="c"># TODO: better condition (when f is very flat)</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span><span class="p">:</span>
-                    <span class="k">print</span><span class="p">(</span><span class="s">&#39;canceled with f(x4) =&#39;</span><span class="p">,</span> <span class="n">fx4</span><span class="p">)</span>
-                <span class="k">yield</span> <span class="n">x4</span><span class="p">,</span> <span class="nb">abs</span><span class="p">(</span><span class="n">x1</span> <span class="o">-</span> <span class="n">x2</span><span class="p">)</span>
-                <span class="k">break</span>
-            <span class="k">if</span> <span class="n">fx4</span> <span class="o">*</span> <span class="n">fx2</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span> <span class="c"># root in [x4, x2]</span>
-                <span class="n">x1</span> <span class="o">=</span> <span class="n">x4</span>
-                <span class="n">fx1</span> <span class="o">=</span> <span class="n">fx4</span>
-            <span class="k">else</span><span class="p">:</span> <span class="c"># root in [x1, x4]</span>
-                <span class="n">x2</span> <span class="o">=</span> <span class="n">x4</span>
-                <span class="n">fx2</span> <span class="o">=</span> <span class="n">fx4</span>
-            <span class="n">error</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">x1</span> <span class="o">-</span> <span class="n">x2</span><span class="p">)</span>
-            <span class="k">yield</span> <span class="p">(</span><span class="n">x1</span> <span class="o">+</span> <span class="n">x2</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">error</span>
-</div>
-<div class="viewcode-block" id="ANewton"><a class="viewcode-back" href="../../../modules/mpmath/calculus/optimization.html#mpmath.calculus.optimization.ANewton">[docs]</a><span class="k">class</span> <span class="nc">ANewton</span><span class="p">:</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    EXPERIMENTAL 1d-solver generating pairs of approximative root and error.</span>
-
-<span class="sd">    Uses Newton&#39;s method modified to use Steffensens method when convergence is</span>
-<span class="sd">    slow. (I.e. for multiple roots.)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">maxsteps</span> <span class="o">=</span> <span class="mi">20</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ctx</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span> <span class="o">=</span> <span class="n">ctx</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;expected 1 starting point, got </span><span class="si">%i</span><span class="s">&#39;</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">x0</span> <span class="o">=</span> <span class="n">x0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">f</span> <span class="o">=</span> <span class="n">f</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="s">&#39;df&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
-            <span class="k">def</span> <span class="nf">df</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">df</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;df&#39;</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">df</span> <span class="o">=</span> <span class="n">df</span>
-        <span class="k">def</span> <span class="nf">phi</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">x</span> <span class="o">-</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">/</span> <span class="n">df</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">phi</span> <span class="o">=</span> <span class="n">phi</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;verbose&#39;</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">x0</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">x0</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">f</span>
-        <span class="n">df</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">df</span>
-        <span class="n">phi</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">phi</span>
-        <span class="n">error</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">counter</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">prevx</span> <span class="o">=</span> <span class="n">x0</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">x0</span> <span class="o">=</span> <span class="n">phi</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">ZeroDivisionError</span><span class="p">:</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span><span class="p">:</span>
-                    <span class="s">&#39;ZeroDivisionError: canceled with x =&#39;</span><span class="p">,</span> <span class="n">x0</span>
-                <span class="k">break</span>
-            <span class="n">preverror</span> <span class="o">=</span> <span class="n">error</span>
-            <span class="n">error</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">prevx</span> <span class="o">-</span> <span class="n">x0</span><span class="p">)</span>
-            <span class="c"># TODO: decide not to use convergence acceleration</span>
-            <span class="k">if</span> <span class="n">error</span> <span class="ow">and</span> <span class="nb">abs</span><span class="p">(</span><span class="n">error</span> <span class="o">-</span> <span class="n">preverror</span><span class="p">)</span> <span class="o">/</span> <span class="n">error</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span><span class="p">:</span>
-                    <span class="k">print</span><span class="p">(</span><span class="s">&#39;converging slowly&#39;</span><span class="p">)</span>
-                <span class="n">counter</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="n">counter</span> <span class="o">&gt;=</span> <span class="mi">3</span><span class="p">:</span>
-                <span class="c"># accelerate convergence</span>
-                <span class="n">phi</span> <span class="o">=</span> <span class="n">steffensen</span><span class="p">(</span><span class="n">phi</span><span class="p">)</span>
-                <span class="n">counter</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span><span class="p">:</span>
-                    <span class="k">print</span><span class="p">(</span><span class="s">&#39;accelerating convergence&#39;</span><span class="p">)</span>
-            <span class="k">yield</span> <span class="n">x0</span><span class="p">,</span> <span class="n">error</span>
-
-<span class="c"># TODO: add Brent</span>
-
-<span class="c">############################</span>
-<span class="c"># MULTIDIMENSIONAL SOLVERS #</span>
-<span class="c">############################</span>
-</div>
-<span class="k">def</span> <span class="nf">jacobian</span><span class="p">(</span><span class="n">ctx</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Calculate the Jacobian matrix of a function at the point x0.</span>
-
-<span class="sd">    This is the first derivative of a vectorial function:</span>
-
-<span class="sd">        f : R^m -&gt; R^n with m &gt;= n</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">matrix</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">ctx</span><span class="o">.</span><span class="n">eps</span><span class="p">)</span>
-    <span class="n">fx</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">matrix</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="o">*</span><span class="n">x</span><span class="p">))</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">fx</span><span class="p">)</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="n">J</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">matrix</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-        <span class="n">xj</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="n">xj</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="n">h</span>
-        <span class="n">Jj</span> <span class="o">=</span> <span class="p">(</span><span class="n">ctx</span><span class="o">.</span><span class="n">matrix</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="o">*</span><span class="n">xj</span><span class="p">))</span> <span class="o">-</span> <span class="n">fx</span><span class="p">)</span> <span class="o">/</span> <span class="n">h</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
-            <span class="n">J</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">Jj</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">J</span>
-
-<span class="c"># TODO: test with user-specified jacobian matrix, support force_type</span>
-<div class="viewcode-block" id="MDNewton"><a class="viewcode-back" href="../../../modules/mpmath/calculus/optimization.html#mpmath.calculus.optimization.MDNewton">[docs]</a><span class="k">class</span> <span class="nc">MDNewton</span><span class="p">:</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Find the root of a vector function numerically using Newton&#39;s method.</span>
-
-<span class="sd">    f is a vector function representing a nonlinear equation system.</span>
-
-<span class="sd">    x0 is the starting point close to the root.</span>
-
-<span class="sd">    J is a function returning the Jacobian matrix for a point.</span>
-
-<span class="sd">    Supports overdetermined systems.</span>
-
-<span class="sd">    Use the &#39;norm&#39; keyword to specify which norm to use. Defaults to max-norm.</span>
-<span class="sd">    The function to calculate the Jacobian matrix can be given using the</span>
-<span class="sd">    keyword &#39;J&#39;. Otherwise it will be calculated numerically.</span>
-
-<span class="sd">    Please note that this method converges only locally. Especially for high-</span>
-<span class="sd">    dimensional systems it is not trivial to find a good starting point being</span>
-<span class="sd">    close enough to the root.</span>
-
-<span class="sd">    It is recommended to use a faster, low-precision solver from SciPy [1] or</span>
-<span class="sd">    OpenOpt [2] to get an initial guess. Afterwards you can use this method for</span>
-<span class="sd">    root-polishing to any precision.</span>
-
-<span class="sd">    [1] http://scipy.org</span>
-
-<span class="sd">    [2] http://openopt.org</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">maxsteps</span> <span class="o">=</span> <span class="mi">10</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ctx</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span> <span class="o">=</span> <span class="n">ctx</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">f</span> <span class="o">=</span> <span class="n">f</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x0</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="nb">list</span><span class="p">)):</span>
-            <span class="n">x0</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">matrix</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span>
-        <span class="k">assert</span> <span class="n">x0</span><span class="o">.</span><span class="n">cols</span> <span class="o">==</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;need a vector&#39;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">x0</span> <span class="o">=</span> <span class="n">x0</span>
-        <span class="k">if</span> <span class="s">&#39;J&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">J</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;J&#39;</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">def</span> <span class="nf">J</span><span class="p">(</span><span class="o">*</span><span class="n">x</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">ctx</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">J</span> <span class="o">=</span> <span class="n">J</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">norm</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;norm&#39;</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;verbose&#39;</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">f</span>
-        <span class="n">x0</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">x0</span>
-        <span class="n">norm</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">norm</span>
-        <span class="n">J</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">J</span>
-        <span class="n">fx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">matrix</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="o">*</span><span class="n">x0</span><span class="p">))</span>
-        <span class="n">fxnorm</span> <span class="o">=</span> <span class="n">norm</span><span class="p">(</span><span class="n">fx</span><span class="p">)</span>
-        <span class="n">cancel</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">while</span> <span class="ow">not</span> <span class="n">cancel</span><span class="p">:</span>
-            <span class="c"># get direction of descent</span>
-            <span class="n">fxn</span> <span class="o">=</span> <span class="o">-</span><span class="n">fx</span>
-            <span class="n">Jx</span> <span class="o">=</span> <span class="n">J</span><span class="p">(</span><span class="o">*</span><span class="n">x0</span><span class="p">)</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">lu_solve</span><span class="p">(</span><span class="n">Jx</span><span class="p">,</span> <span class="n">fxn</span><span class="p">)</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span><span class="p">:</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&#39;Jx:&#39;</span><span class="p">)</span>
-                <span class="k">print</span><span class="p">(</span><span class="n">Jx</span><span class="p">)</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&#39;s:&#39;</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span>
-            <span class="c"># damping step size TODO: better strategy (hard task)</span>
-            <span class="n">l</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">one</span>
-            <span class="n">x1</span> <span class="o">=</span> <span class="n">x0</span> <span class="o">+</span> <span class="n">s</span>
-            <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">x1</span> <span class="o">==</span> <span class="n">x0</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">verbose</span><span class="p">:</span>
-                        <span class="k">print</span><span class="p">(</span><span class="s">&quot;canceled, won&#39;t get more excact&quot;</span><span class="p">)</span>
-                    <span class="n">cancel</span> <span class="o">=</span> <span class="bp">True</span>
-                    <span class="k">break</span>
-                <span class="n">fx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">matrix</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="o">*</span><span class="n">x1</span><span class="p">))</span>
-                <span class="n">newnorm</span> <span class="o">=</span> <span class="n">norm</span><span class="p">(</span><span class="n">fx</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">newnorm</span> <span class="o">&lt;</span> <span class="n">fxnorm</span><span class="p">:</span>
-                    <span class="c"># new x accepted</span>
-                    <span class="n">fxnorm</span> <span class="o">=</span> <span class="n">newnorm</span>
-                    <span class="n">x0</span> <span class="o">=</span> <span class="n">x1</span>
-                    <span class="k">break</span>
-                <span class="n">l</span> <span class="o">/=</span> <span class="mi">2</span>
-                <span class="n">x1</span> <span class="o">=</span> <span class="n">x0</span> <span class="o">+</span> <span class="n">l</span><span class="o">*</span><span class="n">s</span>
-            <span class="k">yield</span> <span class="p">(</span><span class="n">x0</span><span class="p">,</span> <span class="n">fxnorm</span><span class="p">)</span>
-
-<span class="c">#############</span>
-<span class="c"># UTILITIES #</span>
-<span class="c">#############</span>
-</div>
-<span class="n">str2solver</span> <span class="o">=</span> <span class="p">{</span><span class="s">&#39;newton&#39;</span><span class="p">:</span><span class="n">Newton</span><span class="p">,</span> <span class="s">&#39;secant&#39;</span><span class="p">:</span><span class="n">Secant</span><span class="p">,</span><span class="s">&#39;mnewton&#39;</span><span class="p">:</span><span class="n">MNewton</span><span class="p">,</span>
-              <span class="s">&#39;halley&#39;</span><span class="p">:</span><span class="n">Halley</span><span class="p">,</span> <span class="s">&#39;muller&#39;</span><span class="p">:</span><span class="n">Muller</span><span class="p">,</span> <span class="s">&#39;bisect&#39;</span><span class="p">:</span><span class="n">Bisection</span><span class="p">,</span>
-              <span class="s">&#39;illinois&#39;</span><span class="p">:</span><span class="n">Illinois</span><span class="p">,</span> <span class="s">&#39;pegasus&#39;</span><span class="p">:</span><span class="n">Pegasus</span><span class="p">,</span> <span class="s">&#39;anderson&#39;</span><span class="p">:</span><span class="n">Anderson</span><span class="p">,</span>
-              <span class="s">&#39;ridder&#39;</span><span class="p">:</span><span class="n">Ridder</span><span class="p">,</span> <span class="s">&#39;anewton&#39;</span><span class="p">:</span><span class="n">ANewton</span><span class="p">,</span> <span class="s">&#39;mdnewton&#39;</span><span class="p">:</span><span class="n">MDNewton</span><span class="p">}</span>
-
-<span class="k">def</span> <span class="nf">findroot</span><span class="p">(</span><span class="n">ctx</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">solver</span><span class="o">=</span><span class="n">Secant</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">verify</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Find a solution to `f(x) = 0`, using *x0* as starting point or</span>
-<span class="sd">    interval for *x*.</span>
-
-<span class="sd">    Multidimensional overdetermined systems are supported.</span>
-<span class="sd">    You can specify them using a function or a list of functions.</span>
-
-<span class="sd">    If the found root does not satisfy `|f(x)^2 &lt; \mathrm{tol}|`,</span>
-<span class="sd">    an exception is raised (this can be disabled with *verify=False*).</span>
-
-<span class="sd">    **Arguments**</span>
-
-<span class="sd">    *f*</span>
-<span class="sd">        one dimensional function</span>
-<span class="sd">    *x0*</span>
-<span class="sd">        starting point, several starting points or interval (depends on solver)</span>
-<span class="sd">    *tol*</span>
-<span class="sd">        the returned solution has an error smaller than this</span>
-<span class="sd">    *verbose*</span>
-<span class="sd">        print additional information for each iteration if true</span>
-<span class="sd">    *verify*</span>
-<span class="sd">        verify the solution and raise a ValueError if `|f(x) &gt; \mathrm{tol}|`</span>
-<span class="sd">    *solver*</span>
-<span class="sd">        a generator for *f* and *x0* returning approximative solution and error</span>
-<span class="sd">    *maxsteps*</span>
-<span class="sd">        after how many steps the solver will cancel</span>
-<span class="sd">    *df*</span>
-<span class="sd">        first derivative of *f* (used by some solvers)</span>
-<span class="sd">    *d2f*</span>
-<span class="sd">        second derivative of *f* (used by some solvers)</span>
-<span class="sd">    *multidimensional*</span>
-<span class="sd">        force multidimensional solving</span>
-<span class="sd">    *J*</span>
-<span class="sd">        Jacobian matrix of *f* (used by multidimensional solvers)</span>
-<span class="sd">    *norm*</span>
-<span class="sd">        used vector norm (used by multidimensional solvers)</span>
-
-<span class="sd">    solver has to be callable with ``(f, x0, **kwargs)`` and return an generator</span>
-<span class="sd">    yielding pairs of approximative solution and estimated error (which is</span>
-<span class="sd">    expected to be positive).</span>
-<span class="sd">    You can use the following string aliases:</span>
-<span class="sd">    &#39;secant&#39;, &#39;mnewton&#39;, &#39;halley&#39;, &#39;muller&#39;, &#39;illinois&#39;, &#39;pegasus&#39;, &#39;anderson&#39;,</span>
-<span class="sd">    &#39;ridder&#39;, &#39;anewton&#39;, &#39;bisect&#39;</span>
-
-<span class="sd">    See mpmath.optimization for their documentation.</span>
-
-<span class="sd">    **Examples**</span>
-
-<span class="sd">    The function :func:`~mpmath.findroot` locates a root of a given function using the</span>
-<span class="sd">    secant method by default. A simple example use of the secant method is to</span>
-<span class="sd">    compute `\pi` as the root of `\sin x` closest to `x_0 = 3`::</span>
-
-<span class="sd">        &gt;&gt;&gt; from mpmath import *</span>
-<span class="sd">        &gt;&gt;&gt; mp.dps = 30; mp.pretty = True</span>
-<span class="sd">        &gt;&gt;&gt; findroot(sin, 3)</span>
-<span class="sd">        3.14159265358979323846264338328</span>
-
-<span class="sd">    The secant method can be used to find complex roots of analytic functions,</span>
-<span class="sd">    although it must in that case generally be given a nonreal starting value</span>
-<span class="sd">    (or else it will never leave the real line)::</span>
-
-<span class="sd">        &gt;&gt;&gt; mp.dps = 15</span>
-<span class="sd">        &gt;&gt;&gt; findroot(lambda x: x**3 + 2*x + 1, j)</span>
-<span class="sd">        (0.226698825758202 + 1.46771150871022j)</span>
-
-<span class="sd">    A nice application is to compute nontrivial roots of the Riemann zeta</span>
-<span class="sd">    function with many digits (good initial values are needed for convergence)::</span>
-
-<span class="sd">        &gt;&gt;&gt; mp.dps = 30</span>
-<span class="sd">        &gt;&gt;&gt; findroot(zeta, 0.5+14j)</span>
-<span class="sd">        (0.5 + 14.1347251417346937904572519836j)</span>
-
-<span class="sd">    The secant method can also be used as an optimization algorithm, by passing</span>
-<span class="sd">    it a derivative of a function. The following example locates the positive</span>
-<span class="sd">    minimum of the gamma function::</span>
-
-<span class="sd">        &gt;&gt;&gt; mp.dps = 20</span>
-<span class="sd">        &gt;&gt;&gt; findroot(lambda x: diff(gamma, x), 1)</span>
-<span class="sd">        1.4616321449683623413</span>
-
-<span class="sd">    Finally, a useful application is to compute inverse functions, such as the</span>
-<span class="sd">    Lambert W function which is the inverse of `w e^w`, given the first</span>
-<span class="sd">    term of the solution&#39;s asymptotic expansion as the initial value. In basic</span>
-<span class="sd">    cases, this gives identical results to mpmath&#39;s built-in ``lambertw``</span>
-<span class="sd">    function::</span>
-
-<span class="sd">        &gt;&gt;&gt; def lambert(x):</span>
-<span class="sd">        ...     return findroot(lambda w: w*exp(w) - x, log(1+x))</span>
-<span class="sd">        ...</span>
-<span class="sd">        &gt;&gt;&gt; mp.dps = 15</span>
-<span class="sd">        &gt;&gt;&gt; lambert(1); lambertw(1)</span>
-<span class="sd">        0.567143290409784</span>
-<span class="sd">        0.567143290409784</span>
-<span class="sd">        &gt;&gt;&gt; lambert(1000); lambert(1000)</span>
-<span class="sd">        5.2496028524016</span>
-<span class="sd">        5.2496028524016</span>
-
-<span class="sd">    Multidimensional functions are also supported::</span>
-
-<span class="sd">        &gt;&gt;&gt; f = [lambda x1, x2: x1**2 + x2,</span>
-<span class="sd">        ...      lambda x1, x2: 5*x1**2 - 3*x1 + 2*x2 - 3]</span>
-<span class="sd">        &gt;&gt;&gt; findroot(f, (0, 0))</span>
-<span class="sd">        [-0.618033988749895]</span>
-<span class="sd">        [-0.381966011250105]</span>
-<span class="sd">        &gt;&gt;&gt; findroot(f, (10, 10))</span>
-<span class="sd">        [ 1.61803398874989]</span>
-<span class="sd">        [-2.61803398874989]</span>
-
-<span class="sd">    You can verify this by solving the system manually.</span>
-
-<span class="sd">    Please note that the following (more general) syntax also works::</span>
-
-<span class="sd">        &gt;&gt;&gt; def f(x1, x2):</span>
-<span class="sd">        ...     return x1**2 + x2, 5*x1**2 - 3*x1 + 2*x2 - 3</span>
-<span class="sd">        ...</span>
-<span class="sd">        &gt;&gt;&gt; findroot(f, (0, 0))</span>
-<span class="sd">        [-0.618033988749895]</span>
-<span class="sd">        [-0.381966011250105]</span>
-
-
-<span class="sd">    **Multiple roots**</span>
-
-<span class="sd">    For multiple roots all methods of the Newtonian family (including secant)</span>
-<span class="sd">    converge slowly. Consider this example::</span>
-
-<span class="sd">        &gt;&gt;&gt; f = lambda x: (x - 1)**99</span>
-<span class="sd">        &gt;&gt;&gt; findroot(f, 0.9, verify=False)</span>
-<span class="sd">        0.918073542444929</span>
-
-<span class="sd">    Even for a very close starting point the secant method converges very</span>
-<span class="sd">    slowly. Use ``verbose=True`` to illustrate this.</span>
-
-<span class="sd">    It is possible to modify Newton&#39;s method to make it converge regardless of</span>
-<span class="sd">    the root&#39;s multiplicity::</span>
-
-<span class="sd">        &gt;&gt;&gt; findroot(f, -10, solver=&#39;mnewton&#39;)</span>
-<span class="sd">        1.0</span>
-
-<span class="sd">    This variant uses the first and second derivative of the function, which is</span>
-<span class="sd">    not very efficient.</span>
-
-<span class="sd">    Alternatively you can use an experimental Newtonian solver that keeps track</span>
-<span class="sd">    of the speed of convergence and accelerates it using Steffensen&#39;s method if</span>
-<span class="sd">    necessary::</span>
-
-<span class="sd">        &gt;&gt;&gt; findroot(f, -10, solver=&#39;anewton&#39;, verbose=True)</span>
-<span class="sd">        x:     -9.88888888888888888889</span>
-<span class="sd">        error: 0.111111111111111111111</span>
-<span class="sd">        converging slowly</span>
-<span class="sd">        x:     -9.77890011223344556678</span>
-<span class="sd">        error: 0.10998877665544332211</span>
-<span class="sd">        converging slowly</span>
-<span class="sd">        x:     -9.67002233332199662166</span>
-<span class="sd">        error: 0.108877778911448945119</span>
-<span class="sd">        converging slowly</span>
-<span class="sd">        accelerating convergence</span>
-<span class="sd">        x:     -9.5622443299551077669</span>
-<span class="sd">        error: 0.107778003366888854764</span>
-<span class="sd">        converging slowly</span>
-<span class="sd">        x:     0.99999999999999999214</span>
-<span class="sd">        error: 10.562244329955107759</span>
-<span class="sd">        x:     1.0</span>
-<span class="sd">        error: 7.8598304758094664213e-18</span>
-<span class="sd">        1.0</span>
-
-
-<span class="sd">    **Complex roots**</span>
-
-<span class="sd">    For complex roots it&#39;s recommended to use Muller&#39;s method as it converges</span>
-<span class="sd">    even for real starting points very fast::</span>
-
-<span class="sd">        &gt;&gt;&gt; findroot(lambda x: x**4 + x + 1, (0, 1, 2), solver=&#39;muller&#39;)</span>
-<span class="sd">        (0.727136084491197 + 0.934099289460529j)</span>
-
-
-<span class="sd">    **Intersection methods**</span>
-
-<span class="sd">    When you need to find a root in a known interval, it&#39;s highly recommended to</span>
-<span class="sd">    use an intersection-based solver like ``&#39;anderson&#39;`` or ``&#39;ridder&#39;``.</span>
-<span class="sd">    Usually they converge faster and more reliable. They have however problems</span>
-<span class="sd">    with multiple roots and usually need a sign change to find a root::</span>
-
-<span class="sd">        &gt;&gt;&gt; findroot(lambda x: x**3, (-1, 1), solver=&#39;anderson&#39;)</span>
-<span class="sd">        0.0</span>
-
-<span class="sd">    Be careful with symmetric functions::</span>
-
-<span class="sd">        &gt;&gt;&gt; findroot(lambda x: x**2, (-1, 1), solver=&#39;anderson&#39;) #doctest:+ELLIPSIS</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">          ...</span>
-<span class="sd">        ZeroDivisionError</span>
-
-<span class="sd">    It fails even for better starting points, because there is no sign change::</span>
-
-<span class="sd">        &gt;&gt;&gt; findroot(lambda x: x**2, (-1, .5), solver=&#39;anderson&#39;)</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">          ...</span>
-<span class="sd">        ValueError: Could not find root within given tolerance. (1 &gt; 2.1684e-19)</span>
-<span class="sd">        Try another starting point or tweak arguments.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">prec</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">prec</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">ctx</span><span class="o">.</span><span class="n">prec</span> <span class="o">+=</span> <span class="mi">20</span>
-
-        <span class="c"># initialize arguments</span>
-        <span class="k">if</span> <span class="n">tol</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">tol</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">eps</span> <span class="o">*</span> <span class="mi">2</span><span class="o">**</span><span class="mi">10</span>
-
-        <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;verbose&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;verbose&#39;</span><span class="p">,</span> <span class="n">verbose</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="s">&#39;d1f&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
-            <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;df&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;d1f&#39;</span><span class="p">]</span>
-
-        <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;tol&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">tol</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x0</span><span class="p">,</span> <span class="p">(</span><span class="nb">list</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">)):</span>
-            <span class="n">x0</span> <span class="o">=</span> <span class="p">[</span><span class="n">ctx</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">x0</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">x0</span> <span class="o">=</span> <span class="p">[</span><span class="n">ctx</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">x0</span><span class="p">)]</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">solver</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">solver</span> <span class="o">=</span> <span class="n">str2solver</span><span class="p">[</span><span class="n">solver</span><span class="p">]</span>
-            <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;could not recognize solver&#39;</span><span class="p">)</span>
-
-        <span class="c"># accept list of functions</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="nb">list</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">)):</span>
-            <span class="n">f2</span> <span class="o">=</span> <span class="n">copy</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-            <span class="k">def</span> <span class="nf">tmp</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">):</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">fn</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span> <span class="k">for</span> <span class="n">fn</span> <span class="ow">in</span> <span class="n">f2</span><span class="p">]</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">tmp</span>
-
-        <span class="c"># detect multidimensional functions</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">fx</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="o">*</span><span class="n">x0</span><span class="p">)</span>
-            <span class="n">multidimensional</span> <span class="o">=</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">fx</span><span class="p">,</span> <span class="p">(</span><span class="nb">list</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">,</span> <span class="n">ctx</span><span class="o">.</span><span class="n">matrix</span><span class="p">))</span>
-        <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-            <span class="n">fx</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x0</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="n">multidimensional</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="s">&#39;multidimensional&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
-            <span class="n">multidimensional</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;multidimensional&#39;</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">multidimensional</span><span class="p">:</span>
-            <span class="c"># only one multidimensional solver available at the moment</span>
-            <span class="n">solver</span> <span class="o">=</span> <span class="n">MDNewton</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="s">&#39;norm&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
-                <span class="n">norm</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">ctx</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="s">&#39;inf&#39;</span><span class="p">)</span>
-                <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;norm&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">norm</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">norm</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;norm&#39;</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">norm</span> <span class="o">=</span> <span class="nb">abs</span>
-
-        <span class="c"># happily return starting point if it&#39;s a root</span>
-        <span class="k">if</span> <span class="n">norm</span><span class="p">(</span><span class="n">fx</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">multidimensional</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">ctx</span><span class="o">.</span><span class="n">matrix</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">x0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="c"># use solver</span>
-        <span class="n">iterations</span> <span class="o">=</span> <span class="n">solver</span><span class="p">(</span><span class="n">ctx</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">if</span> <span class="s">&#39;maxsteps&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
-            <span class="n">maxsteps</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;maxsteps&#39;</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">maxsteps</span> <span class="o">=</span> <span class="n">iterations</span><span class="o">.</span><span class="n">maxsteps</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="n">error</span> <span class="ow">in</span> <span class="n">iterations</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&#39;x:    &#39;</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&#39;error:&#39;</span><span class="p">,</span> <span class="n">error</span><span class="p">)</span>
-            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="n">error</span> <span class="o">&lt;</span> <span class="n">tol</span> <span class="o">*</span> <span class="nb">max</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">norm</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="ow">or</span> <span class="n">i</span> <span class="o">&gt;=</span> <span class="n">maxsteps</span><span class="p">:</span>
-                <span class="k">break</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="nb">list</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">,</span> <span class="n">ctx</span><span class="o">.</span><span class="n">matrix</span><span class="p">)):</span>
-            <span class="n">xl</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">xl</span> <span class="o">=</span> <span class="n">x</span>
-        <span class="k">if</span> <span class="n">verify</span> <span class="ow">and</span> <span class="n">norm</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="o">*</span><span class="n">xl</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span> <span class="o">&gt;</span> <span class="n">tol</span><span class="p">:</span> <span class="c"># TODO: better condition?</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Could not find root within given tolerance. &#39;</span>
-                             <span class="s">&#39;(</span><span class="si">%g</span><span class="s"> &gt; </span><span class="si">%g</span><span class="s">)</span><span class="se">\n</span><span class="s">&#39;</span>
-                             <span class="s">&#39;Try another starting point or tweak arguments.&#39;</span>
-                             <span class="o">%</span> <span class="p">(</span><span class="n">norm</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="o">*</span><span class="n">xl</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">tol</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">x</span>
-    <span class="k">finally</span><span class="p">:</span>
-        <span class="n">ctx</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="n">prec</span>
-
-
-<span class="k">def</span> <span class="nf">multiplicity</span><span class="p">(</span><span class="n">ctx</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">root</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">maxsteps</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the multiplicity of a given root of f.</span>
-
-<span class="sd">    Internally, numerical derivatives are used. This might be inefficient for</span>
-<span class="sd">    higher order derviatives. Due to this, ``multiplicity`` cancels after</span>
-<span class="sd">    evaluating 10 derivatives by default. You can be specify the n-th derivative</span>
-<span class="sd">    using the dnf keyword.</span>
-
-<span class="sd">    &gt;&gt;&gt; from mpmath import *</span>
-<span class="sd">    &gt;&gt;&gt; multiplicity(lambda x: sin(x) - 1, pi/2)</span>
-<span class="sd">    2</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">tol</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">tol</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">eps</span> <span class="o">**</span> <span class="mf">0.8</span>
-    <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;d0f&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="n">maxsteps</span><span class="p">):</span>
-        <span class="n">dfstr</span> <span class="o">=</span> <span class="s">&#39;d&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39;f&#39;</span>
-        <span class="k">if</span> <span class="n">dfstr</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
-            <span class="n">df</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="n">dfstr</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">df</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">ctx</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">abs</span><span class="p">(</span><span class="n">df</span><span class="p">(</span><span class="n">root</span><span class="p">))</span> <span class="o">&lt;</span> <span class="n">tol</span><span class="p">:</span>
-            <span class="k">break</span>
-    <span class="k">return</span> <span class="n">i</span>
-
-<span class="k">def</span> <span class="nf">steffensen</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    linear convergent function -&gt; quadratic convergent function</span>
-
-<span class="sd">    Steffensen&#39;s method for quadratic convergence of a linear converging</span>
-<span class="sd">    sequence.</span>
-<span class="sd">    Don not use it for higher rates of convergence.</span>
-<span class="sd">    It may even work for divergent sequences.</span>
-
-<span class="sd">    Definition:</span>
-<span class="sd">    F(x) = (x*f(f(x)) - f(x)**2) / (f(f(x)) - 2*f(x) + x)</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    --------</span>
-
-<span class="sd">    You can use Steffensen&#39;s method to accelerate a fixpoint iteration of linear</span>
-<span class="sd">    (or less) convergence.</span>
-
-<span class="sd">    x* is a fixpoint of the iteration x_{k+1} = phi(x_k) if x* = phi(x*). For</span>
-<span class="sd">    phi(x) = x**2 there are two fixpoints: 0 and 1.</span>
-
-<span class="sd">    Let&#39;s try Steffensen&#39;s method:</span>
-
-<span class="sd">    &gt;&gt;&gt; f = lambda x: x**2</span>
-<span class="sd">    &gt;&gt;&gt; from mpmath.optimization import steffensen</span>
-<span class="sd">    &gt;&gt;&gt; F = steffensen(f)</span>
-<span class="sd">    &gt;&gt;&gt; for x in [0.5, 0.9, 2.0]:</span>
-<span class="sd">    ...     fx = Fx = x</span>
-<span class="sd">    ...     for i in xrange(10):</span>
-<span class="sd">    ...         try:</span>
-<span class="sd">    ...             fx = f(fx)</span>
-<span class="sd">    ...         except OverflowError:</span>
-<span class="sd">    ...             pass</span>
-<span class="sd">    ...         try:</span>
-<span class="sd">    ...             Fx = F(Fx)</span>
-<span class="sd">    ...         except ZeroDivisionError:</span>
-<span class="sd">    ...             pass</span>
-<span class="sd">    ...         print &#39;%20g  %20g&#39; % (fx, Fx)</span>
-<span class="sd">                    0.25                  -0.5</span>
-<span class="sd">                  0.0625                   0.1</span>
-<span class="sd">              0.00390625            -0.0011236</span>
-<span class="sd">            1.52588e-005          1.41691e-009</span>
-<span class="sd">            2.32831e-010         -2.84465e-027</span>
-<span class="sd">            5.42101e-020          2.30189e-080</span>
-<span class="sd">            2.93874e-039          -1.2197e-239</span>
-<span class="sd">            8.63617e-078                     0</span>
-<span class="sd">            7.45834e-155                     0</span>
-<span class="sd">            5.56268e-309                     0</span>
-<span class="sd">                    0.81               1.02676</span>
-<span class="sd">                  0.6561               1.00134</span>
-<span class="sd">                0.430467                     1</span>
-<span class="sd">                0.185302                     1</span>
-<span class="sd">               0.0343368                     1</span>
-<span class="sd">              0.00117902                     1</span>
-<span class="sd">            1.39008e-006                     1</span>
-<span class="sd">            1.93233e-012                     1</span>
-<span class="sd">            3.73392e-024                     1</span>
-<span class="sd">            1.39421e-047                     1</span>
-<span class="sd">                       4                   1.6</span>
-<span class="sd">                      16                1.2962</span>
-<span class="sd">                     256               1.10194</span>
-<span class="sd">                   65536               1.01659</span>
-<span class="sd">            4.29497e+009               1.00053</span>
-<span class="sd">            1.84467e+019                     1</span>
-<span class="sd">            3.40282e+038                     1</span>
-<span class="sd">            1.15792e+077                     1</span>
-<span class="sd">            1.34078e+154                     1</span>
-<span class="sd">            1.34078e+154                     1</span>
-
-<span class="sd">    Unmodified, the iteration converges only towards 0. Modified it converges</span>
-<span class="sd">    not only much faster, it converges even to the repelling fixpoint 1.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">F</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-        <span class="n">fx</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="n">ffx</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">fx</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">ffx</span> <span class="o">-</span> <span class="n">fx</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">ffx</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">fx</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">F</span>
-
-<span class="n">OptimizationMethods</span><span class="o">.</span><span class="n">jacobian</span> <span class="o">=</span> <span class="n">jacobian</span>
-<span class="n">OptimizationMethods</span><span class="o">.</span><span class="n">findroot</span> <span class="o">=</span> <span class="n">findroot</span>
-<span class="n">OptimizationMethods</span><span class="o">.</span><span class="n">multiplicity</span> <span class="o">=</span> <span class="n">multiplicity</span>
-
-<span class="k">if</span> <span class="n">__name__</span> <span class="o">==</span> <span class="s">&#39;__main__&#39;</span><span class="p">:</span>
-    <span class="kn">import</span> <span class="nn">doctest</span>
-    <span class="n">doctest</span><span class="o">.</span><span class="n">testmod</span><span class="p">()</span>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
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-  <h3>Quick search</h3>
-    <form class="search" action="../../../search.html" method="get">
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-      <input type="submit" value="Go" />
-      <input type="hidden" name="check_keywords" value="yes" />
-      <input type="hidden" name="area" value="default" />
-    </form>
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-    Enter search terms or a module, class or function name.
-    </p>
-</div>
-<script type="text/javascript">$('#searchbox').show(0);</script>
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-          <a href="../../../genindex.html" title="General Index"
-             >index</a></li>
-        <li class="right" >
-          <a href="../../../py-modindex.html" title="Python Module Index"
-             >modules</a> |</li>
-        <li class="right" >
-          <a href="../../../np-modindex.html" title="Python Module Index"
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-        <li><a href="../../../index.html">SymPy 0.7.2-git documentation</a> &raquo;</li>
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-      Last updated on Jan 20, 2013.
-      Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.1.3.
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-  <h1>Source code for mpmath.calculus.quadrature</h1><div class="highlight"><pre>
-<span class="kn">import</span> <span class="nn">math</span>
-
-<span class="kn">from</span> <span class="nn">..libmp.backend</span> <span class="kn">import</span> <span class="nb">xrange</span>
-
-<div class="viewcode-block" id="QuadratureRule"><a class="viewcode-back" href="../../../modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.QuadratureRule">[docs]</a><span class="k">class</span> <span class="nc">QuadratureRule</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Quadrature rules are implemented using this class, in order to</span>
-<span class="sd">    simplify the code and provide a common infrastructure</span>
-<span class="sd">    for tasks such as error estimation and node caching.</span>
-
-<span class="sd">    You can implement a custom quadrature rule by subclassing</span>
-<span class="sd">    :class:`QuadratureRule` and implementing the appropriate</span>
-<span class="sd">    methods. The subclass can then be used by :func:`~mpmath.quad` by</span>
-<span class="sd">    passing it as the *method* argument.</span>
-
-<span class="sd">    :class:`QuadratureRule` instances are supposed to be singletons.</span>
-<span class="sd">    :class:`QuadratureRule` therefore implements instance caching</span>
-<span class="sd">    in :func:`~mpmath.__new__`.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ctx</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span> <span class="o">=</span> <span class="n">ctx</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">standard_cache</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">transformed_cache</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">interval_count</span> <span class="o">=</span> <span class="p">{}</span>
-
-<div class="viewcode-block" id="QuadratureRule.clear"><a class="viewcode-back" href="../../../modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.QuadratureRule.clear">[docs]</a>    <span class="k">def</span> <span class="nf">clear</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Delete cached node data.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">standard_cache</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">transformed_cache</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">interval_count</span> <span class="o">=</span> <span class="p">{}</span>
-</div>
-<div class="viewcode-block" id="QuadratureRule.calc_nodes"><a class="viewcode-back" href="../../../modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.QuadratureRule.calc_nodes">[docs]</a>    <span class="k">def</span> <span class="nf">calc_nodes</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">        Compute nodes for the standard interval `[-1, 1]`. Subclasses</span>
-<span class="sd">        should probably implement only this method, and use</span>
-<span class="sd">        :func:`~mpmath.get_nodes` method to retrieve the nodes.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-</div>
-<div class="viewcode-block" id="QuadratureRule.get_nodes"><a class="viewcode-back" href="../../../modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.QuadratureRule.get_nodes">[docs]</a>    <span class="k">def</span> <span class="nf">get_nodes</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return nodes for given interval, degree and precision. The</span>
-<span class="sd">        nodes are retrieved from a cache if already computed;</span>
-<span class="sd">        otherwise they are computed by calling :func:`~mpmath.calc_nodes`</span>
-<span class="sd">        and are then cached.</span>
-
-<span class="sd">        Subclasses should probably not implement this method,</span>
-<span class="sd">        but just implement :func:`~mpmath.calc_nodes` for the actual</span>
-<span class="sd">        node computation.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">key</span> <span class="o">=</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">key</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">transformed_cache</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">transformed_cache</span><span class="p">[</span><span class="n">key</span><span class="p">]</span>
-        <span class="n">orig</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">prec</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="n">prec</span><span class="o">+</span><span class="mi">20</span>
-            <span class="c"># Get nodes on standard interval</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">standard_cache</span><span class="p">:</span>
-                <span class="n">nodes</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">standard_cache</span><span class="p">[</span><span class="n">degree</span><span class="p">,</span> <span class="n">prec</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">nodes</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">calc_nodes</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">verbose</span><span class="p">)</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">standard_cache</span><span class="p">[</span><span class="n">degree</span><span class="p">,</span> <span class="n">prec</span><span class="p">]</span> <span class="o">=</span> <span class="n">nodes</span>
-            <span class="c"># Transform to general interval</span>
-            <span class="n">nodes</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform_nodes</span><span class="p">(</span><span class="n">nodes</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">verbose</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">key</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">interval_count</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">transformed_cache</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">=</span> <span class="n">nodes</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">interval_count</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">finally</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="n">orig</span>
-        <span class="k">return</span> <span class="n">nodes</span>
-</div>
-<div class="viewcode-block" id="QuadratureRule.transform_nodes"><a class="viewcode-back" href="../../../modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.QuadratureRule.transform_nodes">[docs]</a>    <span class="k">def</span> <span class="nf">transform_nodes</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">nodes</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">        Rescale standardized nodes (for `[-1, 1]`) to a general</span>
-<span class="sd">        interval `[a, b]`. For a finite interval, a simple linear</span>
-<span class="sd">        change of variables is used. Otherwise, the following</span>
-<span class="sd">        transformations are used:</span>
-
-<span class="sd">        .. math ::</span>
-
-<span class="sd">            [a, \infty] : t = \frac{1}{x} + (a-1)</span>
-
-<span class="sd">            [-\infty, b] : t = (b+1) - \frac{1}{x}</span>
-
-<span class="sd">            [-\infty, \infty] : t = \frac{x}{\sqrt{1-x^2}}</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">ctx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="n">one</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">one</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">==</span> <span class="p">(</span><span class="o">-</span><span class="n">one</span><span class="p">,</span> <span class="n">one</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">nodes</span>
-        <span class="n">half</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>
-        <span class="n">new_nodes</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">if</span> <span class="n">ctx</span><span class="o">.</span><span class="n">isinf</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="ow">or</span> <span class="n">ctx</span><span class="o">.</span><span class="n">isinf</span><span class="p">(</span><span class="n">b</span><span class="p">):</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">==</span> <span class="p">(</span><span class="n">ctx</span><span class="o">.</span><span class="n">ninf</span><span class="p">,</span> <span class="n">ctx</span><span class="o">.</span><span class="n">inf</span><span class="p">):</span>
-                <span class="n">p05</span> <span class="o">=</span> <span class="o">-</span><span class="n">half</span>
-                <span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">nodes</span><span class="p">:</span>
-                    <span class="n">x2</span> <span class="o">=</span> <span class="n">x</span><span class="o">*</span><span class="n">x</span>
-                    <span class="n">px1</span> <span class="o">=</span> <span class="n">one</span><span class="o">-</span><span class="n">x2</span>
-                    <span class="n">spx1</span> <span class="o">=</span> <span class="n">px1</span><span class="o">**</span><span class="n">p05</span>
-                    <span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">*</span><span class="n">spx1</span>
-                    <span class="n">w</span> <span class="o">*=</span> <span class="n">spx1</span><span class="o">/</span><span class="n">px1</span>
-                    <span class="n">new_nodes</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">w</span><span class="p">))</span>
-            <span class="k">elif</span> <span class="n">a</span> <span class="o">==</span> <span class="n">ctx</span><span class="o">.</span><span class="n">ninf</span><span class="p">:</span>
-                <span class="n">b1</span> <span class="o">=</span> <span class="n">b</span><span class="o">+</span><span class="mi">1</span>
-                <span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">nodes</span><span class="p">:</span>
-                    <span class="n">u</span> <span class="o">=</span> <span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">one</span><span class="p">)</span>
-                    <span class="n">x</span> <span class="o">=</span> <span class="n">b1</span><span class="o">-</span><span class="n">u</span>
-                    <span class="n">w</span> <span class="o">*=</span> <span class="n">half</span><span class="o">*</span><span class="n">u</span><span class="o">**</span><span class="mi">2</span>
-                    <span class="n">new_nodes</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">w</span><span class="p">))</span>
-            <span class="k">elif</span> <span class="n">b</span> <span class="o">==</span> <span class="n">ctx</span><span class="o">.</span><span class="n">inf</span><span class="p">:</span>
-                <span class="n">a1</span> <span class="o">=</span> <span class="n">a</span><span class="o">-</span><span class="mi">1</span>
-                <span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">nodes</span><span class="p">:</span>
-                    <span class="n">u</span> <span class="o">=</span> <span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">one</span><span class="p">)</span>
-                    <span class="n">x</span> <span class="o">=</span> <span class="n">a1</span><span class="o">+</span><span class="n">u</span>
-                    <span class="n">w</span> <span class="o">*=</span> <span class="n">half</span><span class="o">*</span><span class="n">u</span><span class="o">**</span><span class="mi">2</span>
-                    <span class="n">new_nodes</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">w</span><span class="p">))</span>
-            <span class="k">elif</span> <span class="n">a</span> <span class="o">==</span> <span class="n">ctx</span><span class="o">.</span><span class="n">inf</span> <span class="ow">or</span> <span class="n">b</span> <span class="o">==</span> <span class="n">ctx</span><span class="o">.</span><span class="n">ninf</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[(</span><span class="n">x</span><span class="p">,</span><span class="o">-</span><span class="n">w</span><span class="p">)</span> <span class="k">for</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">w</span><span class="p">)</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform_nodes</span><span class="p">(</span><span class="n">nodes</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">verbose</span><span class="p">)]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># Simple linear change of variables</span>
-            <span class="n">C</span> <span class="o">=</span> <span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-            <span class="n">D</span> <span class="o">=</span> <span class="p">(</span><span class="n">b</span><span class="o">+</span><span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-            <span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">nodes</span><span class="p">:</span>
-                <span class="n">new_nodes</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">D</span><span class="o">+</span><span class="n">C</span><span class="o">*</span><span class="n">x</span><span class="p">,</span> <span class="n">C</span><span class="o">*</span><span class="n">w</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">new_nodes</span>
-</div>
-<div class="viewcode-block" id="QuadratureRule.guess_degree"><a class="viewcode-back" href="../../../modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.QuadratureRule.guess_degree">[docs]</a>    <span class="k">def</span> <span class="nf">guess_degree</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Given a desired precision `p` in bits, estimate the degree `m`</span>
-<span class="sd">        of the quadrature required to accomplish full accuracy for</span>
-<span class="sd">        typical integrals. By default, :func:`~mpmath.quad` will perform up</span>
-<span class="sd">        to `m` iterations. The value of `m` should be a slight</span>
-<span class="sd">        overestimate, so that &quot;slightly bad&quot; integrals can be dealt</span>
-<span class="sd">        with automatically using a few extra iterations. On the</span>
-<span class="sd">        other hand, it should not be too big, so :func:`~mpmath.quad` can</span>
-<span class="sd">        quit within a reasonable amount of time when it is given</span>
-<span class="sd">        an &quot;unsolvable&quot; integral.</span>
-
-<span class="sd">        The default formula used by :func:`~mpmath.guess_degree` is tuned</span>
-<span class="sd">        for both :class:`TanhSinh` and :class:`GaussLegendre`.</span>
-<span class="sd">        The output is roughly as follows:</span>
-
-<span class="sd">            +---------+---------+</span>
-<span class="sd">            | `p`     | `m`     |</span>
-<span class="sd">            +=========+=========+</span>
-<span class="sd">            | 50      | 6       |</span>
-<span class="sd">            +---------+---------+</span>
-<span class="sd">            | 100     | 7       |</span>
-<span class="sd">            +---------+---------+</span>
-<span class="sd">            | 500     | 10      |</span>
-<span class="sd">            +---------+---------+</span>
-<span class="sd">            | 3000    | 12      |</span>
-<span class="sd">            +---------+---------+</span>
-
-<span class="sd">        This formula is based purely on a limited amount of</span>
-<span class="sd">        experimentation and will sometimes be wrong.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># Expected degree</span>
-        <span class="c"># XXX: use mag</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="mi">4</span> <span class="o">+</span> <span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">prec</span><span class="o">/</span><span class="mf">30.0</span><span class="p">,</span> <span class="mi">2</span><span class="p">)))</span>
-        <span class="c"># Reasonable &quot;worst case&quot;</span>
-        <span class="n">g</span> <span class="o">+=</span> <span class="mi">2</span>
-        <span class="k">return</span> <span class="n">g</span>
-</div>
-<div class="viewcode-block" id="QuadratureRule.estimate_error"><a class="viewcode-back" href="../../../modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.QuadratureRule.estimate_error">[docs]</a>    <span class="k">def</span> <span class="nf">estimate_error</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">results</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">epsilon</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">        Given results from integrations `[I_1, I_2, \ldots, I_k]` done</span>
-<span class="sd">        with a quadrature of rule of degree `1, 2, \ldots, k`, estimate</span>
-<span class="sd">        the error of `I_k`.</span>
-
-<span class="sd">        For `k = 2`, we estimate  `|I_{\infty}-I_2|` as `|I_2-I_1|`.</span>
-
-<span class="sd">        For `k &gt; 2`, we extrapolate `|I_{\infty}-I_k| \approx |I_{k+1}-I_k|`</span>
-<span class="sd">        from `|I_k-I_{k-1}|` and `|I_k-I_{k-2}|` under the assumption</span>
-<span class="sd">        that each degree increment roughly doubles the accuracy of</span>
-<span class="sd">        the quadrature rule (this is true for both :class:`TanhSinh`</span>
-<span class="sd">        and :class:`GaussLegendre`). The extrapolation formula is given</span>
-<span class="sd">        by Borwein, Bailey &amp; Girgensohn. Although not very conservative,</span>
-<span class="sd">        this method seems to be very robust in practice.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">results</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">abs</span><span class="p">(</span><span class="n">results</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">-</span><span class="n">results</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">results</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">results</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="n">results</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">zero</span>
-            <span class="n">D1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">results</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">results</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]),</span> <span class="mi">10</span><span class="p">)</span>
-            <span class="n">D2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">results</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">results</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]),</span> <span class="mi">10</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">epsilon</span>
-        <span class="n">D3</span> <span class="o">=</span> <span class="o">-</span><span class="n">prec</span>
-        <span class="n">D4</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">max</span><span class="p">(</span><span class="n">D1</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="n">D2</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">D1</span><span class="p">,</span> <span class="n">D3</span><span class="p">))</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span> <span class="o">**</span> <span class="nb">int</span><span class="p">(</span><span class="n">D4</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="QuadratureRule.summation"><a class="viewcode-back" href="../../../modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.QuadratureRule.summation">[docs]</a>    <span class="k">def</span> <span class="nf">summation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">points</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">epsilon</span><span class="p">,</span> <span class="n">max_degree</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Main integration function. Computes the 1D integral over</span>
-<span class="sd">        the interval specified by *points*. For each subinterval,</span>
-<span class="sd">        performs quadrature of degree from 1 up to *max_degree*</span>
-<span class="sd">        until :func:`~mpmath.estimate_error` signals convergence.</span>
-
-<span class="sd">        :func:`~mpmath.summation` transforms each subintegration to</span>
-<span class="sd">        the standard interval and then calls :func:`~mpmath.sum_next`.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">ctx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span>
-        <span class="n">I</span> <span class="o">=</span> <span class="n">err</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">zero</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">points</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">points</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">points</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="n">b</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="c"># XXX: we could use a single variable transformation,</span>
-            <span class="c"># but this is not good in practice. We get better accuracy</span>
-            <span class="c"># by having 0 as an endpoint.</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">==</span> <span class="p">(</span><span class="n">ctx</span><span class="o">.</span><span class="n">ninf</span><span class="p">,</span> <span class="n">ctx</span><span class="o">.</span><span class="n">inf</span><span class="p">):</span>
-                <span class="n">_f</span> <span class="o">=</span> <span class="n">f</span>
-                <span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">_f</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">_f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-                <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="p">(</span><span class="n">ctx</span><span class="o">.</span><span class="n">zero</span><span class="p">,</span> <span class="n">ctx</span><span class="o">.</span><span class="n">inf</span><span class="p">)</span>
-            <span class="n">results</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">degree</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">max_degree</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
-                <span class="n">nodes</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_nodes</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">verbose</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-                    <span class="k">print</span><span class="p">(</span><span class="s">&quot;Integrating from </span><span class="si">%s</span><span class="s"> to </span><span class="si">%s</span><span class="s"> (degree </span><span class="si">%s</span><span class="s"> of </span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> \
-                        <span class="p">(</span><span class="n">ctx</span><span class="o">.</span><span class="n">nstr</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">ctx</span><span class="o">.</span><span class="n">nstr</span><span class="p">(</span><span class="n">b</span><span class="p">),</span> <span class="n">degree</span><span class="p">,</span> <span class="n">max_degree</span><span class="p">))</span>
-                <span class="n">results</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">sum_next</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">nodes</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">results</span><span class="p">,</span> <span class="n">verbose</span><span class="p">))</span>
-                <span class="k">if</span> <span class="n">degree</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">err</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">estimate_error</span><span class="p">(</span><span class="n">results</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">epsilon</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">err</span> <span class="o">&lt;=</span> <span class="n">epsilon</span><span class="p">:</span>
-                        <span class="k">break</span>
-                    <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-                        <span class="k">print</span><span class="p">(</span><span class="s">&quot;Estimated error:&quot;</span><span class="p">,</span> <span class="n">ctx</span><span class="o">.</span><span class="n">nstr</span><span class="p">(</span><span class="n">err</span><span class="p">))</span>
-            <span class="n">I</span> <span class="o">+=</span> <span class="n">results</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">err</span> <span class="o">&gt;</span> <span class="n">epsilon</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&quot;Failed to reach full accuracy. Estimated error:&quot;</span><span class="p">,</span> <span class="n">ctx</span><span class="o">.</span><span class="n">nstr</span><span class="p">(</span><span class="n">err</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">I</span><span class="p">,</span> <span class="n">err</span>
-</div>
-<div class="viewcode-block" id="QuadratureRule.sum_next"><a class="viewcode-back" href="../../../modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.QuadratureRule.sum_next">[docs]</a>    <span class="k">def</span> <span class="nf">sum_next</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">nodes</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">previous</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">        Evaluates the step sum `\sum w_k f(x_k)` where the *nodes* list</span>
-<span class="sd">        contains the `(w_k, x_k)` pairs.</span>
-
-<span class="sd">        :func:`~mpmath.summation` will supply the list *results* of</span>
-<span class="sd">        values computed by :func:`~mpmath.sum_next` at previous degrees, in</span>
-<span class="sd">        case the quadrature rule is able to reuse them.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">fdot</span><span class="p">((</span><span class="n">w</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="k">for</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">w</span><span class="p">)</span> <span class="ow">in</span> <span class="n">nodes</span><span class="p">)</span>
-
-</div></div>
-<div class="viewcode-block" id="TanhSinh"><a class="viewcode-back" href="../../../modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.TanhSinh">[docs]</a><span class="k">class</span> <span class="nc">TanhSinh</span><span class="p">(</span><span class="n">QuadratureRule</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    This class implements &quot;tanh-sinh&quot; or &quot;doubly exponential&quot;</span>
-<span class="sd">    quadrature. This quadrature rule is based on the Euler-Maclaurin</span>
-<span class="sd">    integral formula. By performing a change of variables involving</span>
-<span class="sd">    nested exponentials / hyperbolic functions (hence the name), the</span>
-<span class="sd">    derivatives at the endpoints vanish rapidly. Since the error term</span>
-<span class="sd">    in the Euler-Maclaurin formula depends on the derivatives at the</span>
-<span class="sd">    endpoints, a simple step sum becomes extremely accurate. In</span>
-<span class="sd">    practice, this means that doubling the number of evaluation</span>
-<span class="sd">    points roughly doubles the number of accurate digits.</span>
-
-<span class="sd">    Comparison to Gauss-Legendre:</span>
-<span class="sd">      * Initial computation of nodes is usually faster</span>
-<span class="sd">      * Handles endpoint singularities better</span>
-<span class="sd">      * Handles infinite integration intervals better</span>
-<span class="sd">      * Is slower for smooth integrands once nodes have been computed</span>
-
-<span class="sd">    The implementation of the tanh-sinh algorithm is based on the</span>
-<span class="sd">    description given in Borwein, Bailey &amp; Girgensohn, &quot;Experimentation</span>
-<span class="sd">    in Mathematics - Computational Paths to Discovery&quot;, A K Peters,</span>
-<span class="sd">    2003, pages 312-313. In the present implementation, a few</span>
-<span class="sd">    improvements have been made:</span>
-
-<span class="sd">      * A more efficient scheme is used to compute nodes (exploiting</span>
-<span class="sd">        recurrence for the exponential function)</span>
-<span class="sd">      * The nodes are computed successively instead of all at once</span>
-
-<span class="sd">    Various documents describing the algorithm are available online, e.g.:</span>
-
-<span class="sd">      * http://crd.lbl.gov/~dhbailey/dhbpapers/dhb-tanh-sinh.pdf</span>
-<span class="sd">      * http://users.cs.dal.ca/~jborwein/tanh-sinh.pdf</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-<div class="viewcode-block" id="TanhSinh.sum_next"><a class="viewcode-back" href="../../../modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.TanhSinh.sum_next">[docs]</a>    <span class="k">def</span> <span class="nf">sum_next</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">nodes</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">previous</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Step sum for tanh-sinh quadrature of degree `m`. We exploit the</span>
-<span class="sd">        fact that half of the abscissas at degree `m` are precisely the</span>
-<span class="sd">        abscissas from degree `m-1`. Thus reusing the result from</span>
-<span class="sd">        the previous level allows a 2x speedup.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">degree</span><span class="p">)</span>
-        <span class="c"># Abscissas overlap, so reusing saves half of the time</span>
-        <span class="k">if</span> <span class="n">previous</span><span class="p">:</span>
-            <span class="n">S</span> <span class="o">=</span> <span class="n">previous</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">/</span><span class="p">(</span><span class="n">h</span><span class="o">*</span><span class="mi">2</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">S</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">zero</span>
-        <span class="n">S</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span><span class="o">.</span><span class="n">fdot</span><span class="p">((</span><span class="n">w</span><span class="p">,</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="k">for</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">w</span><span class="p">)</span> <span class="ow">in</span> <span class="n">nodes</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">h</span><span class="o">*</span><span class="n">S</span>
-</div>
-<div class="viewcode-block" id="TanhSinh.calc_nodes"><a class="viewcode-back" href="../../../modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.TanhSinh.calc_nodes">[docs]</a>    <span class="k">def</span> <span class="nf">calc_nodes</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">        The abscissas and weights for tanh-sinh quadrature of degree</span>
-<span class="sd">        `m` are given by</span>
-
-<span class="sd">        .. math::</span>
-
-<span class="sd">            x_k = \tanh(\pi/2 \sinh(t_k))</span>
-
-<span class="sd">            w_k = \pi/2 \cosh(t_k) / \cosh(\pi/2 \sinh(t_k))^2</span>
-
-<span class="sd">        where `t_k = t_0 + hk` for a step length `h \sim 2^{-m}`. The</span>
-<span class="sd">        list of nodes is actually infinite, but the weights die off so</span>
-<span class="sd">        rapidly that only a few are needed.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">ctx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span>
-        <span class="n">nodes</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="n">extra</span> <span class="o">=</span> <span class="mi">20</span>
-        <span class="n">ctx</span><span class="o">.</span><span class="n">prec</span> <span class="o">+=</span> <span class="n">extra</span>
-        <span class="n">tol</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">ldexp</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="n">prec</span><span class="o">-</span><span class="mi">10</span><span class="p">)</span>
-        <span class="n">pi4</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span>
-
-        <span class="c"># For simplicity, we work in steps h = 1/2^n, with the first point</span>
-        <span class="c"># offset so that we can reuse the sum from the previous degree</span>
-
-        <span class="c"># We define degree 1 to include the &quot;degree 0&quot; steps, including</span>
-        <span class="c"># the point x = 0. (It doesn&#39;t work well otherwise; not sure why.)</span>
-        <span class="n">t0</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">ldexp</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="n">degree</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">degree</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c">#nodes.append((mpf(0), pi4))</span>
-            <span class="c">#nodes.append((-mpf(0), pi4))</span>
-            <span class="n">nodes</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">ctx</span><span class="o">.</span><span class="n">zero</span><span class="p">,</span> <span class="n">ctx</span><span class="o">.</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">t0</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">t0</span><span class="o">*</span><span class="mi">2</span>
-
-        <span class="c"># Since h is fixed, we can compute the next exponential</span>
-        <span class="c"># by simply multiplying by exp(h)</span>
-        <span class="n">expt0</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">t0</span><span class="p">)</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">pi4</span> <span class="o">*</span> <span class="n">expt0</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">pi4</span> <span class="o">/</span> <span class="n">expt0</span>
-        <span class="n">udelta</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-        <span class="n">urdelta</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">udelta</span>
-
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">20</span><span class="o">*</span><span class="mi">2</span><span class="o">**</span><span class="n">degree</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
-            <span class="c"># Reference implementation:</span>
-            <span class="c"># t = t0 + k*h</span>
-            <span class="c"># x = tanh(pi/2 * sinh(t))</span>
-            <span class="c"># w = pi/2 * cosh(t) / cosh(pi/2 * sinh(t))**2</span>
-
-            <span class="c"># Fast implementation. Note that c = exp(pi/2 * sinh(t))</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">a</span><span class="o">-</span><span class="n">b</span><span class="p">)</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">c</span>
-            <span class="n">co</span> <span class="o">=</span> <span class="p">(</span><span class="n">c</span><span class="o">+</span><span class="n">d</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-            <span class="n">si</span> <span class="o">=</span> <span class="p">(</span><span class="n">c</span><span class="o">-</span><span class="n">d</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">si</span> <span class="o">/</span> <span class="n">co</span>
-            <span class="n">w</span> <span class="o">=</span> <span class="p">(</span><span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="p">)</span> <span class="o">/</span> <span class="n">co</span><span class="o">**</span><span class="mi">2</span>
-            <span class="n">diff</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">diff</span> <span class="o">&lt;=</span> <span class="n">tol</span><span class="p">:</span>
-                <span class="k">break</span>
-
-            <span class="n">nodes</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">w</span><span class="p">))</span>
-            <span class="n">nodes</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="o">-</span><span class="n">x</span><span class="p">,</span> <span class="n">w</span><span class="p">))</span>
-
-            <span class="n">a</span> <span class="o">*=</span> <span class="n">udelta</span>
-            <span class="n">b</span> <span class="o">*=</span> <span class="n">urdelta</span>
-
-            <span class="k">if</span> <span class="n">verbose</span> <span class="ow">and</span> <span class="n">k</span> <span class="o">%</span> <span class="mi">300</span> <span class="o">==</span> <span class="mi">150</span><span class="p">:</span>
-                <span class="c"># Note: the number displayed is rather arbitrary. Should</span>
-                <span class="c"># figure out how to print something that looks more like a</span>
-                <span class="c"># percentage</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&quot;Calculating nodes:&quot;</span><span class="p">,</span> <span class="n">ctx</span><span class="o">.</span><span class="n">nstr</span><span class="p">(</span><span class="o">-</span><span class="n">ctx</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">diff</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span> <span class="o">/</span> <span class="n">prec</span><span class="p">))</span>
-
-        <span class="n">ctx</span><span class="o">.</span><span class="n">prec</span> <span class="o">-=</span> <span class="n">extra</span>
-        <span class="k">return</span> <span class="n">nodes</span>
-
-</div></div>
-<div class="viewcode-block" id="GaussLegendre"><a class="viewcode-back" href="../../../modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.GaussLegendre">[docs]</a><span class="k">class</span> <span class="nc">GaussLegendre</span><span class="p">(</span><span class="n">QuadratureRule</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    This class implements Gauss-Legendre quadrature, which is</span>
-<span class="sd">    exceptionally efficient for polynomials and polynomial-like (i.e.</span>
-<span class="sd">    very smooth) integrands.</span>
-
-<span class="sd">    The abscissas and weights are given by roots and values of</span>
-<span class="sd">    Legendre polynomials, which are the orthogonal polynomials</span>
-<span class="sd">    on `[-1, 1]` with respect to the unit weight</span>
-<span class="sd">    (see :func:`~mpmath.legendre`).</span>
-
-<span class="sd">    In this implementation, we take the &quot;degree&quot; `m` of the quadrature</span>
-<span class="sd">    to denote a Gauss-Legendre rule of degree `3 \cdot 2^m` (following</span>
-<span class="sd">    Borwein, Bailey &amp; Girgensohn). This way we get quadratic, rather</span>
-<span class="sd">    than linear, convergence as the degree is incremented.</span>
-
-<span class="sd">    Comparison to tanh-sinh quadrature:</span>
-<span class="sd">      * Is faster for smooth integrands once nodes have been computed</span>
-<span class="sd">      * Initial computation of nodes is usually slower</span>
-<span class="sd">      * Handles endpoint singularities worse</span>
-<span class="sd">      * Handles infinite integration intervals worse</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-<div class="viewcode-block" id="GaussLegendre.calc_nodes"><a class="viewcode-back" href="../../../modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.GaussLegendre.calc_nodes">[docs]</a>    <span class="k">def</span> <span class="nf">calc_nodes</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Calculates the abscissas and weights for Gauss-Legendre</span>
-<span class="sd">        quadrature of degree of given degree (actually `3 \cdot 2^m`).</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">ctx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ctx</span>
-        <span class="c"># It is important that the epsilon is set lower than the</span>
-        <span class="c"># &quot;real&quot; epsilon</span>
-        <span class="n">epsilon</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">ldexp</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="n">prec</span><span class="o">-</span><span class="mi">8</span><span class="p">)</span>
-        <span class="c"># Fairly high precision might be required for accurate</span>
-        <span class="c"># evaluation of the roots</span>
-        <span class="n">orig</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">prec</span>
-        <span class="n">ctx</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">prec</span><span class="o">*</span><span class="mf">1.5</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">degree</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">5</span>
-            <span class="n">w</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span><span class="o">/</span><span class="mi">9</span>
-            <span class="n">nodes</span> <span class="o">=</span> <span class="p">[(</span><span class="o">-</span><span class="n">x</span><span class="p">,</span><span class="n">w</span><span class="p">),(</span><span class="n">ctx</span><span class="o">.</span><span class="n">zero</span><span class="p">,</span><span class="n">ctx</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span><span class="mi">8</span><span class="p">)</span><span class="o">/</span><span class="mi">9</span><span class="p">),(</span><span class="n">x</span><span class="p">,</span><span class="n">w</span><span class="p">)]</span>
-            <span class="n">ctx</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="n">orig</span>
-            <span class="k">return</span> <span class="n">nodes</span>
-        <span class="n">nodes</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="mi">3</span><span class="o">*</span><span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="n">degree</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-        <span class="n">upto</span> <span class="o">=</span> <span class="n">n</span><span class="o">//</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">upto</span><span class="p">):</span>
-            <span class="c"># Asymptotic formula for the roots</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="p">(</span><span class="n">j</span><span class="o">-</span><span class="mf">0.25</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mf">0.5</span><span class="p">)))</span>
-            <span class="c"># Newton iteration</span>
-            <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">t1</span><span class="p">,</span> <span class="n">t2</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span>
-                <span class="c"># Evaluates the Legendre polynomial using its defining</span>
-                <span class="c"># recurrence relation</span>
-                <span class="k">for</span> <span class="n">j1</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
-                    <span class="n">t3</span><span class="p">,</span> <span class="n">t2</span><span class="p">,</span> <span class="n">t1</span> <span class="o">=</span> <span class="n">t2</span><span class="p">,</span> <span class="n">t1</span><span class="p">,</span> <span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">j1</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">r</span><span class="o">*</span><span class="n">t1</span> <span class="o">-</span> <span class="p">(</span><span class="n">j1</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">t2</span><span class="p">)</span><span class="o">/</span><span class="n">j1</span>
-                <span class="n">t4</span> <span class="o">=</span> <span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">r</span><span class="o">*</span><span class="n">t1</span><span class="o">-</span> <span class="n">t2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-                <span class="n">t5</span> <span class="o">=</span> <span class="n">r</span>
-                <span class="n">a</span> <span class="o">=</span> <span class="n">t1</span><span class="o">/</span><span class="n">t4</span>
-                <span class="n">r</span> <span class="o">=</span> <span class="n">r</span> <span class="o">-</span> <span class="n">a</span>
-                <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">epsilon</span><span class="p">:</span>
-                    <span class="k">break</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">r</span>
-            <span class="n">w</span> <span class="o">=</span> <span class="mi">2</span><span class="o">/</span><span class="p">((</span><span class="mi">1</span><span class="o">-</span><span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">t4</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">verbose</span>  <span class="ow">and</span> <span class="n">j</span> <span class="o">%</span> <span class="mi">30</span> <span class="o">==</span> <span class="mi">15</span><span class="p">:</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&quot;Computing nodes (</span><span class="si">%i</span><span class="s"> of </span><span class="si">%i</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">upto</span><span class="p">))</span>
-            <span class="n">nodes</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">w</span><span class="p">))</span>
-            <span class="n">nodes</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="o">-</span><span class="n">x</span><span class="p">,</span> <span class="n">w</span><span class="p">))</span>
-        <span class="n">ctx</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="n">orig</span>
-        <span class="k">return</span> <span class="n">nodes</span>
-</div></div>
-<span class="k">class</span> <span class="nc">QuadratureMethods</span><span class="p">:</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="n">ctx</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">ctx</span><span class="o">.</span><span class="n">_gauss_legendre</span> <span class="o">=</span> <span class="n">GaussLegendre</span><span class="p">(</span><span class="n">ctx</span><span class="p">)</span>
-        <span class="n">ctx</span><span class="o">.</span><span class="n">_tanh_sinh</span> <span class="o">=</span> <span class="n">TanhSinh</span><span class="p">(</span><span class="n">ctx</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">quad</span><span class="p">(</span><span class="n">ctx</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">points</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">        Computes a single, double or triple integral over a given</span>
-<span class="sd">        1D interval, 2D rectangle, or 3D cuboid. A basic example::</span>
-
-<span class="sd">            &gt;&gt;&gt; from mpmath import *</span>
-<span class="sd">            &gt;&gt;&gt; mp.dps = 15; mp.pretty = True</span>
-<span class="sd">            &gt;&gt;&gt; quad(sin, [0, pi])</span>
-<span class="sd">            2.0</span>
-
-<span class="sd">        A basic 2D integral::</span>
-
-<span class="sd">            &gt;&gt;&gt; f = lambda x, y: cos(x+y/2)</span>
-<span class="sd">            &gt;&gt;&gt; quad(f, [-pi/2, pi/2], [0, pi])</span>
-<span class="sd">            4.0</span>
-
-<span class="sd">        **Interval format**</span>
-
-<span class="sd">        The integration range for each dimension may be specified</span>
-<span class="sd">        using a list or tuple. Arguments are interpreted as follows:</span>
-
-<span class="sd">        ``quad(f, [x1, x2])`` -- calculates</span>
-<span class="sd">        `\int_{x_1}^{x_2} f(x) \, dx`</span>
-
-<span class="sd">        ``quad(f, [x1, x2], [y1, y2])`` -- calculates</span>
-<span class="sd">        `\int_{x_1}^{x_2} \int_{y_1}^{y_2} f(x,y) \, dy \, dx`</span>
-
-<span class="sd">        ``quad(f, [x1, x2], [y1, y2], [z1, z2])`` -- calculates</span>
-<span class="sd">        `\int_{x_1}^{x_2} \int_{y_1}^{y_2} \int_{z_1}^{z_2} f(x,y,z)</span>
-<span class="sd">        \, dz \, dy \, dx`</span>
-
-<span class="sd">        Endpoints may be finite or infinite. An interval descriptor</span>
-<span class="sd">        may also contain more than two points. In this</span>
-<span class="sd">        case, the integration is split into subintervals, between</span>
-<span class="sd">        each pair of consecutive points. This is useful for</span>
-<span class="sd">        dealing with mid-interval discontinuities, or integrating</span>
-<span class="sd">        over large intervals where the function is irregular or</span>
-<span class="sd">        oscillates.</span>
-
-<span class="sd">        **Options**</span>
-
-<span class="sd">        :func:`~mpmath.quad` recognizes the following keyword arguments:</span>
-
-<span class="sd">        *method*</span>
-<span class="sd">            Chooses integration algorithm (described below).</span>
-<span class="sd">        *error*</span>
-<span class="sd">            If set to true, :func:`~mpmath.quad` returns `(v, e)` where `v` is the</span>
-<span class="sd">            integral and `e` is the estimated error.</span>
-<span class="sd">        *maxdegree*</span>
-<span class="sd">            Maximum degree of the quadrature rule to try before</span>
-<span class="sd">            quitting.</span>
-<span class="sd">        *verbose*</span>
-<span class="sd">            Print details about progress.</span>
-
-<span class="sd">        **Algorithms**</span>
-
-<span class="sd">        Mpmath presently implements two integration algorithms: tanh-sinh</span>
-<span class="sd">        quadrature and Gauss-Legendre quadrature. These can be selected</span>
-<span class="sd">        using *method=&#39;tanh-sinh&#39;* or *method=&#39;gauss-legendre&#39;* or by</span>
-<span class="sd">        passing the classes *method=TanhSinh*, *method=GaussLegendre*.</span>
-<span class="sd">        The functions :func:`~mpmath.quadts` and :func:`~mpmath.quadgl` are also available</span>
-<span class="sd">        as shortcuts.</span>
-
-<span class="sd">        Both algorithms have the property that doubling the number of</span>
-<span class="sd">        evaluation points roughly doubles the accuracy, so both are ideal</span>
-<span class="sd">        for high precision quadrature (hundreds or thousands of digits).</span>
-
-<span class="sd">        At high precision, computing the nodes and weights for the</span>
-<span class="sd">        integration can be expensive (more expensive than computing the</span>
-<span class="sd">        function values). To make repeated integrations fast, nodes</span>
-<span class="sd">        are automatically cached.</span>
-
-<span class="sd">        The advantages of the tanh-sinh algorithm are that it tends to</span>
-<span class="sd">        handle endpoint singularities well, and that the nodes are cheap</span>
-<span class="sd">        to compute on the first run. For these reasons, it is used by</span>
-<span class="sd">        :func:`~mpmath.quad` as the default algorithm.</span>
-
-<span class="sd">        Gauss-Legendre quadrature often requires fewer function</span>
-<span class="sd">        evaluations, and is therefore often faster for repeated use, but</span>
-<span class="sd">        the algorithm does not handle endpoint singularities as well and</span>
-<span class="sd">        the nodes are more expensive to compute. Gauss-Legendre quadrature</span>
-<span class="sd">        can be a better choice if the integrand is smooth and repeated</span>
-<span class="sd">        integrations are required (e.g. for multiple integrals).</span>
-
-<span class="sd">        See the documentation for :class:`TanhSinh` and</span>
-<span class="sd">        :class:`GaussLegendre` for additional details.</span>
-
-<span class="sd">        **Examples of 1D integrals**</span>
-
-<span class="sd">        Intervals may be infinite or half-infinite. The following two</span>
-<span class="sd">        examples evaluate the limits of the inverse tangent function</span>
-<span class="sd">        (`\int 1/(1+x^2) = \tan^{-1} x`), and the Gaussian integral</span>
-<span class="sd">        `\int_{\infty}^{\infty} \exp(-x^2)\,dx = \sqrt{\pi}`::</span>
-
-<span class="sd">            &gt;&gt;&gt; mp.dps = 15</span>
-<span class="sd">            &gt;&gt;&gt; quad(lambda x: 2/(x**2+1), [0, inf])</span>
-<span class="sd">            3.14159265358979</span>
-<span class="sd">            &gt;&gt;&gt; quad(lambda x: exp(-x**2), [-inf, inf])**2</span>
-<span class="sd">            3.14159265358979</span>
-
-<span class="sd">        Integrals can typically be resolved to high precision.</span>
-<span class="sd">        The following computes 50 digits of `\pi` by integrating the</span>
-<span class="sd">        area of the half-circle defined by `x^2 + y^2 \le 1`,</span>
-<span class="sd">        `-1 \le x \le 1`, `y \ge 0`::</span>
-
-<span class="sd">            &gt;&gt;&gt; mp.dps = 50</span>
-<span class="sd">            &gt;&gt;&gt; 2*quad(lambda x: sqrt(1-x**2), [-1, 1])</span>
-<span class="sd">            3.1415926535897932384626433832795028841971693993751</span>
-
-<span class="sd">        One can just as well compute 1000 digits (output truncated)::</span>
-
-<span class="sd">            &gt;&gt;&gt; mp.dps = 1000</span>
-<span class="sd">            &gt;&gt;&gt; 2*quad(lambda x: sqrt(1-x**2), [-1, 1])  #doctest:+ELLIPSIS</span>
-<span class="sd">            3.141592653589793238462643383279502884...216420198</span>
-
-<span class="sd">        Complex integrals are supported. The following computes</span>
-<span class="sd">        a residue at `z = 0` by integrating counterclockwise along the</span>
-<span class="sd">        diamond-shaped path from `1` to `+i` to `-1` to `-i` to `1`::</span>
-
-<span class="sd">            &gt;&gt;&gt; mp.dps = 15</span>
-<span class="sd">            &gt;&gt;&gt; chop(quad(lambda z: 1/z, [1,j,-1,-j,1]))</span>
-<span class="sd">            (0.0 + 6.28318530717959j)</span>
-
-<span class="sd">        **Examples of 2D and 3D integrals**</span>
-
-<span class="sd">        Here are several nice examples of analytically solvable</span>
-<span class="sd">        2D integrals (taken from MathWorld [1]) that can be evaluated</span>
-<span class="sd">        to high precision fairly rapidly by :func:`~mpmath.quad`::</span>
-
-<span class="sd">            &gt;&gt;&gt; mp.dps = 30</span>
-<span class="sd">            &gt;&gt;&gt; f = lambda x, y: (x-1)/((1-x*y)*log(x*y))</span>
-<span class="sd">            &gt;&gt;&gt; quad(f, [0, 1], [0, 1])</span>
-<span class="sd">            0.577215664901532860606512090082</span>
-<span class="sd">            &gt;&gt;&gt; +euler</span>
-<span class="sd">            0.577215664901532860606512090082</span>
-
-<span class="sd">            &gt;&gt;&gt; f = lambda x, y: 1/sqrt(1+x**2+y**2)</span>
-<span class="sd">            &gt;&gt;&gt; quad(f, [-1, 1], [-1, 1])</span>
-<span class="sd">            3.17343648530607134219175646705</span>
-<span class="sd">            &gt;&gt;&gt; 4*log(2+sqrt(3))-2*pi/3</span>
-<span class="sd">            3.17343648530607134219175646705</span>
-
-<span class="sd">            &gt;&gt;&gt; f = lambda x, y: 1/(1-x**2 * y**2)</span>
-<span class="sd">            &gt;&gt;&gt; quad(f, [0, 1], [0, 1])</span>
-<span class="sd">            1.23370055013616982735431137498</span>
-<span class="sd">            &gt;&gt;&gt; pi**2 / 8</span>
-<span class="sd">            1.23370055013616982735431137498</span>
-
-<span class="sd">            &gt;&gt;&gt; quad(lambda x, y: 1/(1-x*y), [0, 1], [0, 1])</span>
-<span class="sd">            1.64493406684822643647241516665</span>
-<span class="sd">            &gt;&gt;&gt; pi**2 / 6</span>
-<span class="sd">            1.64493406684822643647241516665</span>
-
-<span class="sd">        Multiple integrals may be done over infinite ranges::</span>
-
-<span class="sd">            &gt;&gt;&gt; mp.dps = 15</span>
-<span class="sd">            &gt;&gt;&gt; print(quad(lambda x,y: exp(-x-y), [0, inf], [1, inf]))</span>
-<span class="sd">            0.367879441171442</span>
-<span class="sd">            &gt;&gt;&gt; print(1/e)</span>
-<span class="sd">            0.367879441171442</span>
-
-<span class="sd">        For nonrectangular areas, one can call :func:`~mpmath.quad` recursively.</span>
-<span class="sd">        For example, we can replicate the earlier example of calculating</span>
-<span class="sd">        `\pi` by integrating over the unit-circle, and actually use double</span>
-<span class="sd">        quadrature to actually measure the area circle::</span>
-
-<span class="sd">            &gt;&gt;&gt; f = lambda x: quad(lambda y: 1, [-sqrt(1-x**2), sqrt(1-x**2)])</span>
-<span class="sd">            &gt;&gt;&gt; quad(f, [-1, 1])</span>
-<span class="sd">            3.14159265358979</span>
-
-<span class="sd">        Here is a simple triple integral::</span>
-
-<span class="sd">            &gt;&gt;&gt; mp.dps = 15</span>
-<span class="sd">            &gt;&gt;&gt; f = lambda x,y,z: x*y/(1+z)</span>
-<span class="sd">            &gt;&gt;&gt; quad(f, [0,1], [0,1], [1,2], method=&#39;gauss-legendre&#39;)</span>
-<span class="sd">            0.101366277027041</span>
-<span class="sd">            &gt;&gt;&gt; (log(3)-log(2))/4</span>
-<span class="sd">            0.101366277027041</span>
-
-<span class="sd">        **Singularities**</span>
-
-<span class="sd">        Both tanh-sinh and Gauss-Legendre quadrature are designed to</span>
-<span class="sd">        integrate smooth (infinitely differentiable) functions. Neither</span>
-<span class="sd">        algorithm copes well with mid-interval singularities (such as</span>
-<span class="sd">        mid-interval discontinuities in `f(x)` or `f&#39;(x)`).</span>
-<span class="sd">        The best solution is to split the integral into parts::</span>
-
-<span class="sd">            &gt;&gt;&gt; mp.dps = 15</span>
-<span class="sd">            &gt;&gt;&gt; quad(lambda x: abs(sin(x)), [0, 2*pi])   # Bad</span>
-<span class="sd">            3.99900894176779</span>
-<span class="sd">            &gt;&gt;&gt; quad(lambda x: abs(sin(x)), [0, pi, 2*pi])  # Good</span>
-<span class="sd">            4.0</span>
-
-<span class="sd">        The tanh-sinh rule often works well for integrands having a</span>
-<span class="sd">        singularity at one or both endpoints::</span>
-
-<span class="sd">            &gt;&gt;&gt; mp.dps = 15</span>
-<span class="sd">            &gt;&gt;&gt; quad(log, [0, 1], method=&#39;tanh-sinh&#39;)  # Good</span>
-<span class="sd">            -1.0</span>
-<span class="sd">            &gt;&gt;&gt; quad(log, [0, 1], method=&#39;gauss-legendre&#39;)  # Bad</span>
-<span class="sd">            -0.999932197413801</span>
-
-<span class="sd">        However, the result may still be inaccurate for some functions::</span>
-
-<span class="sd">            &gt;&gt;&gt; quad(lambda x: 1/sqrt(x), [0, 1], method=&#39;tanh-sinh&#39;)</span>
-<span class="sd">            1.99999999946942</span>
-
-<span class="sd">        This problem is not due to the quadrature rule per se, but to</span>
-<span class="sd">        numerical amplification of errors in the nodes. The problem can be</span>
-<span class="sd">        circumvented by temporarily increasing the precision::</span>
-
-<span class="sd">            &gt;&gt;&gt; mp.dps = 30</span>
-<span class="sd">            &gt;&gt;&gt; a = quad(lambda x: 1/sqrt(x), [0, 1], method=&#39;tanh-sinh&#39;)</span>
-<span class="sd">            &gt;&gt;&gt; mp.dps = 15</span>
-<span class="sd">            &gt;&gt;&gt; +a</span>
-<span class="sd">            2.0</span>
-
-<span class="sd">        **Highly variable functions**</span>
-
-<span class="sd">        For functions that are smooth (in the sense of being infinitely</span>
-<span class="sd">        differentiable) but contain sharp mid-interval peaks or many</span>
-<span class="sd">        &quot;bumps&quot;, :func:`~mpmath.quad` may fail to provide full accuracy. For</span>
-<span class="sd">        example, with default settings, :func:`~mpmath.quad` is able to integrate</span>
-<span class="sd">        `\sin(x)` accurately over an interval of length 100 but not over</span>
-<span class="sd">        length 1000::</span>
-
-<span class="sd">            &gt;&gt;&gt; quad(sin, [0, 100]); 1-cos(100)   # Good</span>
-<span class="sd">            0.137681127712316</span>
-<span class="sd">            0.137681127712316</span>
-<span class="sd">            &gt;&gt;&gt; quad(sin, [0, 1000]); 1-cos(1000)   # Bad</span>
-<span class="sd">            -37.8587612408485</span>
-<span class="sd">            0.437620923709297</span>
-
-<span class="sd">        One solution is to break the integration into 10 intervals of</span>
-<span class="sd">        length 100::</span>
-
-<span class="sd">            &gt;&gt;&gt; quad(sin, linspace(0, 1000, 10))   # Good</span>
-<span class="sd">            0.437620923709297</span>
-
-<span class="sd">        Another is to increase the degree of the quadrature::</span>
-
-<span class="sd">            &gt;&gt;&gt; quad(sin, [0, 1000], maxdegree=10)   # Also good</span>
-<span class="sd">            0.437620923709297</span>
-
-<span class="sd">        Whether splitting the interval or increasing the degree is</span>
-<span class="sd">        more efficient differs from case to case. Another example is the</span>
-<span class="sd">        function `1/(1+x^2)`, which has a sharp peak centered around</span>
-<span class="sd">        `x = 0`::</span>
-
-<span class="sd">            &gt;&gt;&gt; f = lambda x: 1/(1+x**2)</span>
-<span class="sd">            &gt;&gt;&gt; quad(f, [-100, 100])   # Bad</span>
-<span class="sd">            3.64804647105268</span>
-<span class="sd">            &gt;&gt;&gt; quad(f, [-100, 100], maxdegree=10)   # Good</span>
-<span class="sd">            3.12159332021646</span>
-<span class="sd">            &gt;&gt;&gt; quad(f, [-100, 0, 100])   # Also good</span>
-<span class="sd">            3.12159332021646</span>
-
-<span class="sd">        **References**</span>
-
-<span class="sd">        1. http://mathworld.wolfram.com/DoubleIntegral.html</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">rule</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;method&#39;</span><span class="p">,</span> <span class="s">&#39;tanh-sinh&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">rule</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">str</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">rule</span> <span class="o">==</span> <span class="s">&#39;tanh-sinh&#39;</span><span class="p">:</span>
-                <span class="n">rule</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">_tanh_sinh</span>
-            <span class="k">elif</span> <span class="n">rule</span> <span class="o">==</span> <span class="s">&#39;gauss-legendre&#39;</span><span class="p">:</span>
-                <span class="n">rule</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">_gauss_legendre</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;unknown quadrature rule: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">rule</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">rule</span> <span class="o">=</span> <span class="n">rule</span><span class="p">(</span><span class="n">ctx</span><span class="p">)</span>
-        <span class="n">verbose</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;verbose&#39;</span><span class="p">)</span>
-        <span class="n">dim</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">points</span><span class="p">)</span>
-        <span class="n">orig</span> <span class="o">=</span> <span class="n">prec</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">prec</span>
-        <span class="n">epsilon</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">eps</span><span class="o">/</span><span class="mi">8</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;maxdegree&#39;</span><span class="p">)</span> <span class="ow">or</span> <span class="n">rule</span><span class="o">.</span><span class="n">guess_degree</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-        <span class="n">points</span> <span class="o">=</span> <span class="p">[</span><span class="n">ctx</span><span class="o">.</span><span class="n">_as_points</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">points</span><span class="p">]</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">ctx</span><span class="o">.</span><span class="n">prec</span> <span class="o">+=</span> <span class="mi">20</span>
-            <span class="k">if</span> <span class="n">dim</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">v</span><span class="p">,</span> <span class="n">err</span> <span class="o">=</span> <span class="n">rule</span><span class="o">.</span><span class="n">summation</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">points</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">prec</span><span class="p">,</span> <span class="n">epsilon</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">verbose</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">dim</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">v</span><span class="p">,</span> <span class="n">err</span> <span class="o">=</span> <span class="n">rule</span><span class="o">.</span><span class="n">summation</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> \
-                        <span class="n">rule</span><span class="o">.</span><span class="n">summation</span><span class="p">(</span><span class="k">lambda</span> <span class="n">y</span><span class="p">:</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),</span> \
-                        <span class="n">points</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">prec</span><span class="p">,</span> <span class="n">epsilon</span><span class="p">,</span> <span class="n">m</span><span class="p">)[</span><span class="mi">0</span><span class="p">],</span>
-                    <span class="n">points</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">prec</span><span class="p">,</span> <span class="n">epsilon</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">verbose</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">dim</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-                <span class="n">v</span><span class="p">,</span> <span class="n">err</span> <span class="o">=</span> <span class="n">rule</span><span class="o">.</span><span class="n">summation</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> \
-                        <span class="n">rule</span><span class="o">.</span><span class="n">summation</span><span class="p">(</span><span class="k">lambda</span> <span class="n">y</span><span class="p">:</span> \
-                            <span class="n">rule</span><span class="o">.</span><span class="n">summation</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span><span class="p">),</span> \
-                            <span class="n">points</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">prec</span><span class="p">,</span> <span class="n">epsilon</span><span class="p">,</span> <span class="n">m</span><span class="p">)[</span><span class="mi">0</span><span class="p">],</span>
-                        <span class="n">points</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">prec</span><span class="p">,</span> <span class="n">epsilon</span><span class="p">,</span> <span class="n">m</span><span class="p">)[</span><span class="mi">0</span><span class="p">],</span>
-                    <span class="n">points</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">prec</span><span class="p">,</span> <span class="n">epsilon</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">verbose</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;quadrature must have dim 1, 2 or 3&quot;</span><span class="p">)</span>
-        <span class="k">finally</span><span class="p">:</span>
-            <span class="n">ctx</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="n">orig</span>
-        <span class="k">if</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;error&quot;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="o">+</span><span class="n">v</span><span class="p">,</span> <span class="n">err</span>
-        <span class="k">return</span> <span class="o">+</span><span class="n">v</span>
-
-    <span class="k">def</span> <span class="nf">quadts</span><span class="p">(</span><span class="n">ctx</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Performs tanh-sinh quadrature. The call</span>
-
-<span class="sd">            quadts(func, *points, ...)</span>
-
-<span class="sd">        is simply a shortcut for:</span>
-
-<span class="sd">            quad(func, *points, ..., method=TanhSinh)</span>
-
-<span class="sd">        For example, a single integral and a double integral:</span>
-
-<span class="sd">            quadts(lambda x: exp(cos(x)), [0, 1])</span>
-<span class="sd">            quadts(lambda x, y: exp(cos(x+y)), [0, 1], [0, 1])</span>
-
-<span class="sd">        See the documentation for quad for information about how points</span>
-<span class="sd">        arguments and keyword arguments are parsed.</span>
-
-<span class="sd">        See documentation for TanhSinh for algorithmic information about</span>
-<span class="sd">        tanh-sinh quadrature.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;method&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="s">&#39;tanh-sinh&#39;</span>
-        <span class="k">return</span> <span class="n">ctx</span><span class="o">.</span><span class="n">quad</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">quadgl</span><span class="p">(</span><span class="n">ctx</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Performs Gauss-Legendre quadrature. The call</span>
-
-<span class="sd">            quadgl(func, *points, ...)</span>
-
-<span class="sd">        is simply a shortcut for:</span>
-
-<span class="sd">            quad(func, *points, ..., method=GaussLegendre)</span>
-
-<span class="sd">        For example, a single integral and a double integral:</span>
-
-<span class="sd">            quadgl(lambda x: exp(cos(x)), [0, 1])</span>
-<span class="sd">            quadgl(lambda x, y: exp(cos(x+y)), [0, 1], [0, 1])</span>
-
-<span class="sd">        See the documentation for quad for information about how points</span>
-<span class="sd">        arguments and keyword arguments are parsed.</span>
-
-<span class="sd">        See documentation for TanhSinh for algorithmic information about</span>
-<span class="sd">        tanh-sinh quadrature.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;method&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="s">&#39;gauss-legendre&#39;</span>
-        <span class="k">return</span> <span class="n">ctx</span><span class="o">.</span><span class="n">quad</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">quadosc</span><span class="p">(</span><span class="n">ctx</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">interval</span><span class="p">,</span> <span class="n">omega</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">period</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">zeros</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">        Calculates</span>
-
-<span class="sd">        .. math ::</span>
-
-<span class="sd">            I = \int_a^b f(x) dx</span>
-
-<span class="sd">        where at least one of `a` and `b` is infinite and where</span>
-<span class="sd">        `f(x) = g(x) \cos(\omega x  + \phi)` for some slowly</span>
-<span class="sd">        decreasing function `g(x)`. With proper input, :func:`~mpmath.quadosc`</span>
-<span class="sd">        can also handle oscillatory integrals where the oscillation</span>
-<span class="sd">        rate is different from a pure sine or cosine wave.</span>
-
-<span class="sd">        In the standard case when `|a| &lt; \infty, b = \infty`,</span>
-<span class="sd">        :func:`~mpmath.quadosc` works by evaluating the infinite series</span>
-
-<span class="sd">        .. math ::</span>
-
-<span class="sd">            I = \int_a^{x_1} f(x) dx +</span>
-<span class="sd">            \sum_{k=1}^{\infty} \int_{x_k}^{x_{k+1}} f(x) dx</span>
-
-<span class="sd">        where `x_k` are consecutive zeros (alternatively</span>
-<span class="sd">        some other periodic reference point) of `f(x)`.</span>
-<span class="sd">        Accordingly, :func:`~mpmath.quadosc` requires information about the</span>
-<span class="sd">        zeros of `f(x)`. For a periodic function, you can specify</span>
-<span class="sd">        the zeros by either providing the angular frequency `\omega`</span>
-<span class="sd">        (*omega*) or the *period* `2 \pi/\omega`. In general, you can</span>
-<span class="sd">        specify the `n`-th zero by providing the *zeros* arguments.</span>
-<span class="sd">        Below is an example of each::</span>
-
-<span class="sd">            &gt;&gt;&gt; from mpmath import *</span>
-<span class="sd">            &gt;&gt;&gt; mp.dps = 15; mp.pretty = True</span>
-<span class="sd">            &gt;&gt;&gt; f = lambda x: sin(3*x)/(x**2+1)</span>
-<span class="sd">            &gt;&gt;&gt; quadosc(f, [0,inf], omega=3)</span>
-<span class="sd">            0.37833007080198</span>
-<span class="sd">            &gt;&gt;&gt; quadosc(f, [0,inf], period=2*pi/3)</span>
-<span class="sd">            0.37833007080198</span>
-<span class="sd">            &gt;&gt;&gt; quadosc(f, [0,inf], zeros=lambda n: pi*n/3)</span>
-<span class="sd">            0.37833007080198</span>
-<span class="sd">            &gt;&gt;&gt; (ei(3)*exp(-3)-exp(3)*ei(-3))/2  # Computed by Mathematica</span>
-<span class="sd">            0.37833007080198</span>
-
-<span class="sd">        Note that *zeros* was specified to multiply `n` by the</span>
-<span class="sd">        *half-period*, not the full period. In theory, it does not matter</span>
-<span class="sd">        whether each partial integral is done over a half period or a full</span>
-<span class="sd">        period. However, if done over half-periods, the infinite series</span>
-<span class="sd">        passed to :func:`~mpmath.nsum` becomes an *alternating series* and this</span>
-<span class="sd">        typically makes the extrapolation much more efficient.</span>
-
-<span class="sd">        Here is an example of an integration over the entire real line,</span>
-<span class="sd">        and a half-infinite integration starting at `-\infty`::</span>
-
-<span class="sd">            &gt;&gt;&gt; quadosc(lambda x: cos(x)/(1+x**2), [-inf, inf], omega=1)</span>
-<span class="sd">            1.15572734979092</span>
-<span class="sd">            &gt;&gt;&gt; pi/e</span>
-<span class="sd">            1.15572734979092</span>
-<span class="sd">            &gt;&gt;&gt; quadosc(lambda x: cos(x)/x**2, [-inf, -1], period=2*pi)</span>
-<span class="sd">            -0.0844109505595739</span>
-<span class="sd">            &gt;&gt;&gt; cos(1)+si(1)-pi/2</span>
-<span class="sd">            -0.0844109505595738</span>
-
-<span class="sd">        Of course, the integrand may contain a complex exponential just as</span>
-<span class="sd">        well as a real sine or cosine::</span>
-
-<span class="sd">            &gt;&gt;&gt; quadosc(lambda x: exp(3*j*x)/(1+x**2), [-inf,inf], omega=3)</span>
-<span class="sd">            (0.156410688228254 + 0.0j)</span>
-<span class="sd">            &gt;&gt;&gt; pi/e**3</span>
-<span class="sd">            0.156410688228254</span>
-<span class="sd">            &gt;&gt;&gt; quadosc(lambda x: exp(3*j*x)/(2+x+x**2), [-inf,inf], omega=3)</span>
-<span class="sd">            (0.00317486988463794 - 0.0447701735209082j)</span>
-<span class="sd">            &gt;&gt;&gt; 2*pi/sqrt(7)/exp(3*(j+sqrt(7))/2)</span>
-<span class="sd">            (0.00317486988463794 - 0.0447701735209082j)</span>
-
-<span class="sd">        **Non-periodic functions**</span>
-
-<span class="sd">        If `f(x) = g(x) h(x)` for some function `h(x)` that is not</span>
-<span class="sd">        strictly periodic, *omega* or *period* might not work, and it might</span>
-<span class="sd">        be necessary to use *zeros*.</span>
-
-<span class="sd">        A notable exception can be made for Bessel functions which, though not</span>
-<span class="sd">        periodic, are &quot;asymptotically periodic&quot; in a sufficiently strong sense</span>
-<span class="sd">        that the sum extrapolation will work out::</span>
-
-<span class="sd">            &gt;&gt;&gt; quadosc(j0, [0, inf], period=2*pi)</span>
-<span class="sd">            1.0</span>
-<span class="sd">            &gt;&gt;&gt; quadosc(j1, [0, inf], period=2*pi)</span>
-<span class="sd">            1.0</span>
-
-<span class="sd">        More properly, one should provide the exact Bessel function zeros::</span>
-
-<span class="sd">            &gt;&gt;&gt; j0zero = lambda n: findroot(j0, pi*(n-0.25))</span>
-<span class="sd">            &gt;&gt;&gt; quadosc(j0, [0, inf], zeros=j0zero)</span>
-<span class="sd">            1.0</span>
-
-<span class="sd">        For an example where *zeros* becomes necessary, consider the</span>
-<span class="sd">        complete Fresnel integrals</span>
-
-<span class="sd">        .. math ::</span>
-
-<span class="sd">            \int_0^{\infty} \cos x^2\,dx = \int_0^{\infty} \sin x^2\,dx</span>
-<span class="sd">            = \sqrt{\frac{\pi}{8}}.</span>
-
-<span class="sd">        Although the integrands do not decrease in magnitude as</span>
-<span class="sd">        `x \to \infty`, the integrals are convergent since the oscillation</span>
-<span class="sd">        rate increases (causing consecutive periods to asymptotically</span>
-<span class="sd">        cancel out). These integrals are virtually impossible to calculate</span>
-<span class="sd">        to any kind of accuracy using standard quadrature rules. However,</span>
-<span class="sd">        if one provides the correct asymptotic distribution of zeros</span>
-<span class="sd">        (`x_n \sim \sqrt{n}`), :func:`~mpmath.quadosc` works::</span>
-
-<span class="sd">            &gt;&gt;&gt; mp.dps = 30</span>
-<span class="sd">            &gt;&gt;&gt; f = lambda x: cos(x**2)</span>
-<span class="sd">            &gt;&gt;&gt; quadosc(f, [0,inf], zeros=lambda n:sqrt(pi*n))</span>
-<span class="sd">            0.626657068657750125603941321203</span>
-<span class="sd">            &gt;&gt;&gt; f = lambda x: sin(x**2)</span>
-<span class="sd">            &gt;&gt;&gt; quadosc(f, [0,inf], zeros=lambda n:sqrt(pi*n))</span>
-<span class="sd">            0.626657068657750125603941321203</span>
-<span class="sd">            &gt;&gt;&gt; sqrt(pi/8)</span>
-<span class="sd">            0.626657068657750125603941321203</span>
-
-<span class="sd">        (Interestingly, these integrals can still be evaluated if one</span>
-<span class="sd">        places some other constant than `\pi` in the square root sign.)</span>
-
-<span class="sd">        In general, if `f(x) \sim g(x) \cos(h(x))`, the zeros follow</span>
-<span class="sd">        the inverse-function distribution `h^{-1}(x)`::</span>
-
-<span class="sd">            &gt;&gt;&gt; mp.dps = 15</span>
-<span class="sd">            &gt;&gt;&gt; f = lambda x: sin(exp(x))</span>
-<span class="sd">            &gt;&gt;&gt; quadosc(f, [1,inf], zeros=lambda n: log(n))</span>
-<span class="sd">            -0.25024394235267</span>
-<span class="sd">            &gt;&gt;&gt; pi/2-si(e)</span>
-<span class="sd">            -0.250243942352671</span>
-
-<span class="sd">        **Non-alternating functions**</span>
-
-<span class="sd">        If the integrand oscillates around a positive value, without</span>
-<span class="sd">        alternating signs, the extrapolation might fail. A simple trick</span>
-<span class="sd">        that sometimes works is to multiply or divide the frequency by 2::</span>
-
-<span class="sd">            &gt;&gt;&gt; f = lambda x: 1/x**2+sin(x)/x**4</span>
-<span class="sd">            &gt;&gt;&gt; quadosc(f, [1,inf], omega=1)  # Bad</span>
-<span class="sd">            1.28642190869861</span>
-<span class="sd">            &gt;&gt;&gt; quadosc(f, [1,inf], omega=0.5)  # Perfect</span>
-<span class="sd">            1.28652953559617</span>
-<span class="sd">            &gt;&gt;&gt; 1+(cos(1)+ci(1)+sin(1))/6</span>
-<span class="sd">            1.28652953559617</span>
-
-<span class="sd">        **Fast decay**</span>
-
-<span class="sd">        :func:`~mpmath.quadosc` is primarily useful for slowly decaying</span>
-<span class="sd">        integrands. If the integrand decreases exponentially or faster,</span>
-<span class="sd">        :func:`~mpmath.quad` will likely handle it without trouble (and generally be</span>
-<span class="sd">        much faster than :func:`~mpmath.quadosc`)::</span>
-
-<span class="sd">            &gt;&gt;&gt; quadosc(lambda x: cos(x)/exp(x), [0, inf], omega=1)</span>
-<span class="sd">            0.5</span>
-<span class="sd">            &gt;&gt;&gt; quad(lambda x: cos(x)/exp(x), [0, inf])</span>
-<span class="sd">            0.5</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">_as_points</span><span class="p">(</span><span class="n">interval</span><span class="p">)</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="k">if</span> <span class="p">[</span><span class="n">omega</span><span class="p">,</span> <span class="n">period</span><span class="p">,</span> <span class="n">zeros</span><span class="p">]</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="bp">None</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span> \
-                <span class="s">&quot;must specify exactly one of omega, period, zeros&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="n">ctx</span><span class="o">.</span><span class="n">ninf</span> <span class="ow">and</span> <span class="n">b</span> <span class="o">==</span> <span class="n">ctx</span><span class="o">.</span><span class="n">inf</span><span class="p">:</span>
-            <span class="n">s1</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">quadosc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="n">a</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">omega</span><span class="o">=</span><span class="n">omega</span><span class="p">,</span> <span class="n">zeros</span><span class="o">=</span><span class="n">zeros</span><span class="p">,</span> <span class="n">period</span><span class="o">=</span><span class="n">period</span><span class="p">)</span>
-            <span class="n">s2</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">quadosc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">b</span><span class="p">],</span> <span class="n">omega</span><span class="o">=</span><span class="n">omega</span><span class="p">,</span> <span class="n">zeros</span><span class="o">=</span><span class="n">zeros</span><span class="p">,</span> <span class="n">period</span><span class="o">=</span><span class="n">period</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">s1</span> <span class="o">+</span> <span class="n">s2</span>
-        <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="n">ctx</span><span class="o">.</span><span class="n">ninf</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">zeros</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">ctx</span><span class="o">.</span><span class="n">quadosc</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span><span class="n">f</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="n">b</span><span class="p">,</span><span class="o">-</span><span class="n">a</span><span class="p">],</span> <span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="n">zeros</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">ctx</span><span class="o">.</span><span class="n">quadosc</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span><span class="n">f</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="n">b</span><span class="p">,</span><span class="o">-</span><span class="n">a</span><span class="p">],</span> <span class="n">omega</span><span class="o">=</span><span class="n">omega</span><span class="p">,</span> <span class="n">period</span><span class="o">=</span><span class="n">period</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">b</span> <span class="o">!=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">inf</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;quadosc requires an infinite integration interval&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">zeros</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">omega</span><span class="p">:</span>
-                <span class="n">period</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">ctx</span><span class="o">.</span><span class="n">pi</span><span class="o">/</span><span class="n">omega</span>
-            <span class="n">zeros</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="n">n</span><span class="o">*</span><span class="n">period</span><span class="o">/</span><span class="mi">2</span>
-        <span class="c">#for n in range(1,10):</span>
-        <span class="c">#    p = zeros(n)</span>
-        <span class="c">#    if p &gt; a:</span>
-        <span class="c">#        break</span>
-        <span class="c">#if n &gt;= 9:</span>
-        <span class="c">#    raise ValueError(&quot;zeros do not appear to be correctly indexed&quot;)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">quadgl</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="n">a</span><span class="p">,</span> <span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">)])</span>
-        <span class="k">def</span> <span class="nf">term</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">ctx</span><span class="o">.</span><span class="n">quadgl</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="n">zeros</span><span class="p">(</span><span class="n">k</span><span class="p">),</span> <span class="n">zeros</span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)])</span>
-        <span class="n">s</span> <span class="o">+=</span> <span class="n">ctx</span><span class="o">.</span><span class="n">nsum</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="n">ctx</span><span class="o">.</span><span class="n">inf</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">s</span>
-
-<span class="k">if</span> <span class="n">__name__</span> <span class="o">==</span> <span class="s">&#39;__main__&#39;</span><span class="p">:</span>
-    <span class="kn">import</span> <span class="nn">doctest</span>
-    <span class="n">doctest</span><span class="o">.</span><span class="n">testmod</span><span class="p">()</span>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
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-      Last updated on Jan 20, 2013.
-      Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.1.3.
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diff --git a/dev-py3k/_modules/sympy.html b/dev-py3k/_modules/sympy.html
deleted file mode 100644
index 01252b83f8a..00000000000
--- a/dev-py3k/_modules/sympy.html
+++ /dev/null
@@ -1,196 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
-
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-<html xmlns="http://www.w3.org/1999/xhtml">
-  <head>
-    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
-    
-    <title>sympy &mdash; SymPy 0.7.2-git documentation</title>
-    
-    <link rel="stylesheet" href="../_static/default.css" type="text/css" />
-    <link rel="stylesheet" href="../_static/pygments.css" type="text/css" />
-    <link rel="stylesheet" href="http://live.sympy.org/static/live-core.css" type="text/css" />
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-    
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-      var DOCUMENTATION_OPTIONS = {
-        URL_ROOT:    '../',
-        VERSION:     '0.7.2-git',
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-      };
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-        <div class="bodywrapper">
-          <div class="body">
-            
-  <h1>Source code for sympy</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">SymPy is a Python library for symbolic mathematics. It aims to become a</span>
-<span class="sd">full-featured computer algebra system (CAS) while keeping the code as</span>
-<span class="sd">simple as possible in order to be comprehensible and easily extensible.</span>
-<span class="sd">SymPy is written entirely in Python and does not require any external</span>
-<span class="sd">libraries, except optionally for plotting support.</span>
-
-<span class="sd">See the webpage for more information and documentation:</span>
-
-<span class="sd">    http://code.google.com/p/sympy/&quot;&quot;&quot;</span>
-
-<span class="n">__version__</span> <span class="o">=</span> <span class="s">&quot;0.7.2-git&quot;</span>
-
-<span class="c"># Try to determine if 2to3 has been run. To do this, we look at long.__name__.</span>
-<span class="c"># If 2to3 has been run, it should convert long to int.</span>
-<span class="kn">import</span> <span class="nn">sys</span>
-<span class="k">if</span> <span class="n">sys</span><span class="o">.</span><span class="n">version_info</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">HAS_2TO3</span> <span class="o">=</span> <span class="nb">int</span><span class="o">.</span><span class="n">__name__</span> <span class="o">==</span> <span class="s">&quot;int&quot;</span>
-    <span class="k">except</span> <span class="ne">NameError</span><span class="p">:</span>  <span class="c"># it tries to see long but long doesn&#39;t exist in Python 3</span>
-        <span class="n">HAS_2TO3</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="k">else</span><span class="p">:</span>
-    <span class="n">HAS_2TO3</span> <span class="o">=</span> <span class="nb">int</span><span class="o">.</span><span class="n">__name__</span> <span class="o">==</span> <span class="s">&quot;int&quot;</span>
-
-<span class="k">if</span> <span class="n">sys</span><span class="o">.</span><span class="n">version_info</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-    <span class="k">if</span> <span class="n">HAS_2TO3</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ImportError</span><span class="p">(</span><span class="s">&quot;It appears 2to3 has been run on the codebase. Use &quot;</span>
-                          <span class="s">&quot;Python 3 or get the original source code.&quot;</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">sys</span><span class="o">.</span><span class="n">version_info</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">&lt;</span> <span class="mi">5</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ImportError</span><span class="p">(</span>
-                <span class="s">&quot;Python Version 2.5 or above is required for SymPy.&quot;</span><span class="p">)</span>
-<span class="k">else</span><span class="p">:</span>  <span class="c"># Python 3</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">HAS_2TO3</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ImportError</span><span class="p">(</span><span class="s">&quot;This is the Python 2 version of SymPy. To use SymPy &quot;</span>
-    <span class="s">&quot;with Python 3, please obtain a Python 3 version from http://sympy.org, &quot;</span>
-    <span class="s">&quot;or use the bin/use2to3 script if you are using the git version.&quot;</span><span class="p">)</span>
-    <span class="c"># Here we can also check for specific Python 3 versions, if needed</span>
-
-<span class="k">del</span> <span class="n">sys</span>
-<span class="k">del</span> <span class="n">HAS_2TO3</span>
-
-
-<span class="k">def</span> <span class="nf">__sympy_debug</span><span class="p">():</span>
-    <span class="c"># helper function so we don&#39;t import os globally</span>
-    <span class="kn">import</span> <span class="nn">os</span>
-    <span class="k">return</span> <span class="nb">eval</span><span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">getenv</span><span class="p">(</span><span class="s">&#39;SYMPY_DEBUG&#39;</span><span class="p">,</span> <span class="s">&#39;False&#39;</span><span class="p">))</span>
-<span class="n">SYMPY_DEBUG</span> <span class="o">=</span> <span class="n">__sympy_debug</span><span class="p">()</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="kn">from</span> <span class="nn">.logic</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="kn">from</span> <span class="nn">.assumptions</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="kn">from</span> <span class="nn">.polys</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="kn">from</span> <span class="nn">.series</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="kn">from</span> <span class="nn">.functions</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="kn">from</span> <span class="nn">.ntheory</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="kn">from</span> <span class="nn">.concrete</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="kn">from</span> <span class="nn">.simplify</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="kn">from</span> <span class="nn">.sets</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="kn">from</span> <span class="nn">.solvers</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="kn">from</span> <span class="nn">.matrices</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="kn">from</span> <span class="nn">.geometry</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="kn">from</span> <span class="nn">.utilities</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="kn">from</span> <span class="nn">.integrals</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="kn">from</span> <span class="nn">.tensor</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="kn">from</span> <span class="nn">.parsing</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="c"># Adds about .04-.05 seconds of import time</span>
-<span class="c"># from combinatorics import *</span>
-<span class="c"># This module is slow to import:</span>
-<span class="c">#from physics import units</span>
-<span class="kn">from</span> <span class="nn">.plotting</span> <span class="kn">import</span> <span class="n">plot</span><span class="p">,</span> <span class="n">Plot</span><span class="p">,</span> <span class="n">textplot</span><span class="p">,</span> <span class="n">plot_backends</span><span class="p">,</span> <span class="n">plot_implicit</span>
-<span class="kn">from</span> <span class="nn">.printing</span> <span class="kn">import</span> <span class="n">pretty</span><span class="p">,</span> <span class="n">pretty_print</span><span class="p">,</span> <span class="n">pprint</span><span class="p">,</span> <span class="n">pprint_use_unicode</span><span class="p">,</span> \
-    <span class="n">pprint_try_use_unicode</span><span class="p">,</span> <span class="n">print_gtk</span><span class="p">,</span> <span class="n">print_tree</span><span class="p">,</span> <span class="n">pager_print</span><span class="p">,</span> <span class="n">TableForm</span>
-<span class="kn">from</span> <span class="nn">.printing</span> <span class="kn">import</span> <span class="n">ccode</span><span class="p">,</span> <span class="n">fcode</span><span class="p">,</span> <span class="n">jscode</span><span class="p">,</span> <span class="n">latex</span><span class="p">,</span> <span class="n">preview</span>
-<span class="kn">from</span> <span class="nn">.printing</span> <span class="kn">import</span> <span class="n">python</span><span class="p">,</span> <span class="n">print_python</span><span class="p">,</span> <span class="n">srepr</span><span class="p">,</span> <span class="n">sstr</span><span class="p">,</span> <span class="n">sstrrepr</span>
-<span class="kn">from</span> <span class="nn">.interactive</span> <span class="kn">import</span> <span class="n">init_session</span><span class="p">,</span> <span class="n">init_printing</span>
-
-<span class="n">evalf</span><span class="o">.</span><span class="n">_create_evalf_table</span><span class="p">()</span>
-
-<span class="c"># This is slow to import:</span>
-<span class="c">#import abc</span>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../index.html">
-              <img class="logo" src="../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-<div id="searchbox" style="display: none">
-  <h3>Quick search</h3>
-    <form class="search" action="../search.html" method="get">
-      <input type="text" name="q" />
-      <input type="submit" value="Go" />
-      <input type="hidden" name="check_keywords" value="yes" />
-      <input type="hidden" name="area" value="default" />
-    </form>
-    <p class="searchtip" style="font-size: 90%">
-    Enter search terms or a module, class or function name.
-    </p>
-</div>
-<script type="text/javascript">$('#searchbox').show(0);</script>
-        </div>
-      </div>
-      <div class="clearer"></div>
-    </div>
-    <div class="related">
-      <h3>Navigation</h3>
-      <ul>
-        <li class="right" style="margin-right: 10px">
-          <a href="../genindex.html" title="General Index"
-             >index</a></li>
-        <li class="right" >
-          <a href="../py-modindex.html" title="Python Module Index"
-             >modules</a> |</li>
-        <li class="right" >
-          <a href="../np-modindex.html" title="Python Module Index"
-             >modules</a> |</li>
-        <li><a href="../index.html">SymPy 0.7.2-git documentation</a> &raquo;</li>
-          <li><a href="index.html" >Module code</a> &raquo;</li> 
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-    <div class="footer">
-        &copy; Copyright 2013 SymPy Development Team.
-      Last updated on Jan 20, 2013.
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-</html>
diff --git a/dev-py3k/_modules/sympy/assumptions/ask.html b/dev-py3k/_modules/sympy/assumptions/ask.html
deleted file mode 100644
index 3cd347d90bb..00000000000
--- a/dev-py3k/_modules/sympy/assumptions/ask.html
+++ /dev/null
@@ -1,420 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
-
-
-<html xmlns="http://www.w3.org/1999/xhtml">
-  <head>
-    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
-    
-    <title>sympy.assumptions.ask &mdash; SymPy 0.7.2-git documentation</title>
-    
-    <link rel="stylesheet" href="../../../_static/default.css" type="text/css" />
-    <link rel="stylesheet" href="../../../_static/pygments.css" type="text/css" />
-    <link rel="stylesheet" href="http://live.sympy.org/static/live-core.css" type="text/css" />
-    <link rel="stylesheet" href="http://live.sympy.org/static/live-autocomplete.css" type="text/css" />
-    <link rel="stylesheet" href="http://live.sympy.org/static/live-sphinx.css" type="text/css" />
-    
-    <script type="text/javascript">
-      var DOCUMENTATION_OPTIONS = {
-        URL_ROOT:    '../../../',
-        VERSION:     '0.7.2-git',
-        COLLAPSE_INDEX: false,
-        FILE_SUFFIX: '.html',
-        HAS_SOURCE:  true
-      };
-    </script>
-    <script type="text/javascript" src="../../../_static/jquery.js"></script>
-    <script type="text/javascript" src="../../../_static/underscore.js"></script>
-    <script type="text/javascript" src="../../../_static/doctools.js"></script>
-    <script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_HTML-full"></script>
-    <script type="text/javascript" src="http://live.sympy.org/static/utilities.js"></script>
-    <script type="text/javascript" src="http://live.sympy.org/static/external/classy.js"></script>
-    <script type="text/javascript" src="http://live.sympy.org/static/live-core.js"></script>
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-    <script type="text/javascript" src="http://live.sympy.org/static/live-sphinx.js"></script>
-    <script type="text/javascript" src="../../../_static/sidebar.js"></script>
-    <link rel="shortcut icon" href="../../../_static/SymPy-Favicon.ico"/>
-    <link rel="top" title="SymPy 0.7.2-git documentation" href="../../../index.html" />
-    <link rel="up" title="sympy" href="../../sympy.html" /> 
-  </head>
-  <body>
-    <div class="related">
-      <h3>Navigation</h3>
-      <ul>
-        <li class="right" style="margin-right: 10px">
-          <a href="../../../genindex.html" title="General Index"
-             accesskey="I">index</a></li>
-        <li class="right" >
-          <a href="../../../py-modindex.html" title="Python Module Index"
-             >modules</a> |</li>
-        <li class="right" >
-          <a href="../../../np-modindex.html" title="Python Module Index"
-             >modules</a> |</li>
-        <li><a href="../../../index.html">SymPy 0.7.2-git documentation</a> &raquo;</li>
-          <li><a href="../../index.html" >Module code</a> &raquo;</li>
-          <li><a href="../../sympy.html" accesskey="U">sympy</a> &raquo;</li> 
-      </ul>
-    </div>  
-
-    <div class="document">
-      <div class="documentwrapper">
-        <div class="bodywrapper">
-          <div class="body">
-            
-  <h1>Source code for sympy.assumptions.ask</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Module for querying SymPy objects about assumptions.&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.logic.boolalg</span> <span class="kn">import</span> <span class="n">to_cnf</span><span class="p">,</span> <span class="n">And</span><span class="p">,</span> <span class="n">Not</span><span class="p">,</span> <span class="n">Or</span><span class="p">,</span> <span class="n">Implies</span><span class="p">,</span> <span class="n">Equivalent</span>
-<span class="kn">from</span> <span class="nn">sympy.logic.inference</span> <span class="kn">import</span> <span class="n">satisfiable</span>
-<span class="kn">from</span> <span class="nn">sympy.assumptions.assume</span> <span class="kn">import</span> <span class="p">(</span><span class="n">global_assumptions</span><span class="p">,</span> <span class="n">Predicate</span><span class="p">,</span>
-        <span class="n">AppliedPredicate</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="Q"><a class="viewcode-back" href="../../../modules/assumptions/ask.html#sympy.assumptions.ask.Q">[docs]</a><span class="k">class</span> <span class="nc">Q</span><span class="p">:</span>
-    <span class="sd">&quot;&quot;&quot;Supported ask keys.&quot;&quot;&quot;</span>
-    <span class="n">antihermitian</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;antihermitian&#39;</span><span class="p">)</span>
-    <span class="n">bounded</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;bounded&#39;</span><span class="p">)</span>
-    <span class="n">commutative</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;commutative&#39;</span><span class="p">)</span>
-    <span class="nb">complex</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;complex&#39;</span><span class="p">)</span>
-    <span class="n">composite</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;composite&#39;</span><span class="p">)</span>
-    <span class="n">even</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;even&#39;</span><span class="p">)</span>
-    <span class="n">extended_real</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;extended_real&#39;</span><span class="p">)</span>
-    <span class="n">hermitian</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;hermitian&#39;</span><span class="p">)</span>
-    <span class="n">imaginary</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;imaginary&#39;</span><span class="p">)</span>
-    <span class="n">infinitesimal</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;infinitesimal&#39;</span><span class="p">)</span>
-    <span class="n">infinity</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;infinity&#39;</span><span class="p">)</span>
-    <span class="n">integer</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;integer&#39;</span><span class="p">)</span>
-    <span class="n">irrational</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;irrational&#39;</span><span class="p">)</span>
-    <span class="n">rational</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;rational&#39;</span><span class="p">)</span>
-    <span class="n">negative</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;negative&#39;</span><span class="p">)</span>
-    <span class="n">nonzero</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;nonzero&#39;</span><span class="p">)</span>
-    <span class="n">positive</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;positive&#39;</span><span class="p">)</span>
-    <span class="n">prime</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;prime&#39;</span><span class="p">)</span>
-    <span class="n">real</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;real&#39;</span><span class="p">)</span>
-    <span class="n">odd</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;odd&#39;</span><span class="p">)</span>
-    <span class="n">is_true</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;is_true&#39;</span><span class="p">)</span>
-    <span class="n">symmetric</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;symmetric&#39;</span><span class="p">)</span>
-    <span class="n">invertible</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;invertible&#39;</span><span class="p">)</span>
-    <span class="n">orthogonal</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;orthogonal&#39;</span><span class="p">)</span>
-    <span class="n">positive_definite</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;positive_definite&#39;</span><span class="p">)</span>
-    <span class="n">upper_triangular</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;upper_triangular&#39;</span><span class="p">)</span>
-    <span class="n">lower_triangular</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;lower_triangular&#39;</span><span class="p">)</span>
-    <span class="n">diagonal</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;diagonal&#39;</span><span class="p">)</span>
-    <span class="n">triangular</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;triangular&#39;</span><span class="p">)</span>
-    <span class="n">unit_triangular</span> <span class="o">=</span> <span class="n">Predicate</span><span class="p">(</span><span class="s">&#39;unit_triangular&#39;</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_extract_facts</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">symbol</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Helper for ask().</span>
-
-<span class="sd">    Extracts the facts relevant to the symbol from an assumption.</span>
-<span class="sd">    Returns None if there is nothing to extract.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">symbol</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">None</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">AppliedPredicate</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span>
-    <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="p">[</span><span class="n">_extract_facts</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">symbol</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">]</span> <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">])</span>
-
-
-<div class="viewcode-block" id="ask"><a class="viewcode-back" href="../../../modules/assumptions/index.html#sympy.assumptions.ask.ask">[docs]</a><span class="k">def</span> <span class="nf">ask</span><span class="p">(</span><span class="n">proposition</span><span class="p">,</span> <span class="n">assumptions</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">context</span><span class="o">=</span><span class="n">global_assumptions</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Method for inferring properties about objects.</span>
-
-<span class="sd">    **Syntax**</span>
-
-<span class="sd">        * ask(proposition)</span>
-
-<span class="sd">        * ask(proposition, assumptions)</span>
-
-<span class="sd">            where ``proposition`` is any boolean expression</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import ask, Q, pi</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; ask(Q.rational(pi))</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; ask(Q.even(x*y), Q.even(x) &amp; Q.integer(y))</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; ask(Q.prime(x*y), Q.integer(x) &amp;  Q.integer(y))</span>
-<span class="sd">    False</span>
-
-<span class="sd">    **Remarks**</span>
-<span class="sd">        Relations in assumptions are not implemented (yet), so the following</span>
-<span class="sd">        will not give a meaningful result.</span>
-
-<span class="sd">        &gt;&gt;&gt; ask(Q.positive(x), Q.is_true(x &gt; 0)) # doctest: +SKIP</span>
-
-<span class="sd">        It is however a work in progress.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">assumptions</span> <span class="o">=</span> <span class="n">And</span><span class="p">(</span><span class="n">assumptions</span><span class="p">,</span> <span class="n">And</span><span class="p">(</span><span class="o">*</span><span class="n">context</span><span class="p">))</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">proposition</span><span class="p">,</span> <span class="n">AppliedPredicate</span><span class="p">):</span>
-        <span class="n">key</span><span class="p">,</span> <span class="n">expr</span> <span class="o">=</span> <span class="n">proposition</span><span class="o">.</span><span class="n">func</span><span class="p">,</span> <span class="n">sympify</span><span class="p">(</span><span class="n">proposition</span><span class="o">.</span><span class="n">arg</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">key</span><span class="p">,</span> <span class="n">expr</span> <span class="o">=</span> <span class="n">Q</span><span class="o">.</span><span class="n">is_true</span><span class="p">,</span> <span class="n">sympify</span><span class="p">(</span><span class="n">proposition</span><span class="p">)</span>
-
-    <span class="c"># direct resolution method, no logic</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="n">key</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_ask</span><span class="p">(</span><span class="n">assumptions</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">res</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">res</span>
-
-    <span class="k">if</span> <span class="n">assumptions</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="k">return</span>
-
-    <span class="n">local_facts</span> <span class="o">=</span> <span class="n">_extract_facts</span><span class="p">(</span><span class="n">assumptions</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">local_facts</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">local_facts</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="k">return</span>
-
-    <span class="c"># See if there&#39;s a straight-forward conclusion we can make for the inference</span>
-    <span class="k">if</span> <span class="n">local_facts</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">key</span> <span class="ow">in</span> <span class="n">known_facts_dict</span><span class="p">[</span><span class="n">local_facts</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="n">Not</span><span class="p">(</span><span class="n">key</span><span class="p">)</span> <span class="ow">in</span> <span class="n">known_facts_dict</span><span class="p">[</span><span class="n">local_facts</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-    <span class="k">elif</span> <span class="n">local_facts</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">And</span> <span class="ow">and</span> <span class="nb">all</span><span class="p">(</span><span class="n">k</span> <span class="ow">in</span> <span class="n">known_facts_dict</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">local_facts</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">assum</span> <span class="ow">in</span> <span class="n">local_facts</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">assum</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">key</span> <span class="ow">in</span> <span class="n">known_facts_dict</span><span class="p">[</span><span class="n">assum</span><span class="p">]:</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-                <span class="k">if</span> <span class="n">Not</span><span class="p">(</span><span class="n">key</span><span class="p">)</span> <span class="ow">in</span> <span class="n">known_facts_dict</span><span class="p">[</span><span class="n">assum</span><span class="p">]:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">elif</span> <span class="n">assum</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Not</span> <span class="ow">and</span> <span class="n">assum</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">key</span> <span class="ow">in</span> <span class="n">known_facts_dict</span><span class="p">[</span><span class="n">assum</span><span class="p">]:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-                <span class="k">if</span> <span class="n">Not</span><span class="p">(</span><span class="n">key</span><span class="p">)</span> <span class="ow">in</span> <span class="n">known_facts_dict</span><span class="p">[</span><span class="n">assum</span><span class="p">]:</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-    <span class="k">elif</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">Predicate</span><span class="p">)</span> <span class="ow">and</span>
-            <span class="n">local_facts</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Not</span> <span class="ow">and</span> <span class="n">local_facts</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">local_facts</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">in</span> <span class="n">known_facts_dict</span><span class="p">[</span><span class="n">key</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="c"># Failing all else, we do a full logical inference</span>
-    <span class="k">return</span> <span class="n">ask_full_inference</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">local_facts</span><span class="p">,</span> <span class="n">known_facts_cnf</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="ask_full_inference"><a class="viewcode-back" href="../../../modules/assumptions/ask.html#sympy.assumptions.ask.ask_full_inference">[docs]</a><span class="k">def</span> <span class="nf">ask_full_inference</span><span class="p">(</span><span class="n">proposition</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">,</span> <span class="n">known_facts_cnf</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Method for inferring properties about objects.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">satisfiable</span><span class="p">(</span><span class="n">And</span><span class="p">(</span><span class="n">known_facts_cnf</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">,</span> <span class="n">proposition</span><span class="p">)):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">satisfiable</span><span class="p">(</span><span class="n">And</span><span class="p">(</span><span class="n">known_facts_cnf</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">,</span> <span class="n">Not</span><span class="p">(</span><span class="n">proposition</span><span class="p">))):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-    <span class="k">return</span> <span class="bp">None</span>
-
-</div>
-<div class="viewcode-block" id="register_handler"><a class="viewcode-back" href="../../../modules/assumptions/index.html#sympy.assumptions.ask.register_handler">[docs]</a><span class="k">def</span> <span class="nf">register_handler</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">handler</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Register a handler in the ask system. key must be a string and handler a</span>
-<span class="sd">    class inheriting from AskHandler::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.assumptions import register_handler, ask, Q</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.assumptions.handlers import AskHandler</span>
-<span class="sd">        &gt;&gt;&gt; class MersenneHandler(AskHandler):</span>
-<span class="sd">        ...     # Mersenne numbers are in the form 2**n + 1, n integer</span>
-<span class="sd">        ...     @staticmethod</span>
-<span class="sd">        ...     def Integer(expr, assumptions):</span>
-<span class="sd">        ...         import math</span>
-<span class="sd">        ...         return ask(Q.integer(math.log(expr + 1, 2)))</span>
-<span class="sd">        &gt;&gt;&gt; register_handler(&#39;mersenne&#39;, MersenneHandler)</span>
-<span class="sd">        &gt;&gt;&gt; ask(Q.mersenne(7))</span>
-<span class="sd">        True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">key</span><span class="p">)</span> <span class="ow">is</span> <span class="n">Predicate</span><span class="p">:</span>
-        <span class="n">key</span> <span class="o">=</span> <span class="n">key</span><span class="o">.</span><span class="n">name</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="nb">getattr</span><span class="p">(</span><span class="n">Q</span><span class="p">,</span> <span class="n">key</span><span class="p">)</span><span class="o">.</span><span class="n">add_handler</span><span class="p">(</span><span class="n">handler</span><span class="p">)</span>
-    <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-        <span class="nb">setattr</span><span class="p">(</span><span class="n">Q</span><span class="p">,</span> <span class="n">key</span><span class="p">,</span> <span class="n">Predicate</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">handlers</span><span class="o">=</span><span class="p">[</span><span class="n">handler</span><span class="p">]))</span>
-
-</div>
-<div class="viewcode-block" id="remove_handler"><a class="viewcode-back" href="../../../modules/assumptions/index.html#sympy.assumptions.ask.remove_handler">[docs]</a><span class="k">def</span> <span class="nf">remove_handler</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">handler</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Removes a handler from the ask system. Same syntax as register_handler&quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">key</span><span class="p">)</span> <span class="ow">is</span> <span class="n">Predicate</span><span class="p">:</span>
-        <span class="n">key</span> <span class="o">=</span> <span class="n">key</span><span class="o">.</span><span class="n">name</span>
-    <span class="nb">getattr</span><span class="p">(</span><span class="n">Q</span><span class="p">,</span> <span class="n">key</span><span class="p">)</span><span class="o">.</span><span class="n">remove_handler</span><span class="p">(</span><span class="n">handler</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">single_fact_lookup</span><span class="p">(</span><span class="n">known_facts_keys</span><span class="p">,</span> <span class="n">known_facts_cnf</span><span class="p">):</span>
-    <span class="c"># Compute the quick lookup for single facts</span>
-    <span class="n">mapping</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">for</span> <span class="n">key</span> <span class="ow">in</span> <span class="n">known_facts_keys</span><span class="p">:</span>
-        <span class="n">mapping</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span><span class="n">key</span><span class="p">])</span>
-        <span class="k">for</span> <span class="n">other_key</span> <span class="ow">in</span> <span class="n">known_facts_keys</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">other_key</span> <span class="o">!=</span> <span class="n">key</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">ask_full_inference</span><span class="p">(</span><span class="n">other_key</span><span class="p">,</span> <span class="n">key</span><span class="p">,</span> <span class="n">known_facts_cnf</span><span class="p">):</span>
-                    <span class="n">mapping</span><span class="p">[</span><span class="n">key</span><span class="p">]</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">other_key</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">mapping</span>
-
-
-<div class="viewcode-block" id="compute_known_facts"><a class="viewcode-back" href="../../../modules/assumptions/ask.html#sympy.assumptions.ask.compute_known_facts">[docs]</a><span class="k">def</span> <span class="nf">compute_known_facts</span><span class="p">(</span><span class="n">known_facts</span><span class="p">,</span> <span class="n">known_facts_keys</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Compute the various forms of knowledge compilation used by the</span>
-<span class="sd">    assumptions system.</span>
-
-<span class="sd">    This function is typically applied to the variables</span>
-<span class="sd">    ``known_facts`` and ``known_facts_keys`` defined at the bottom of</span>
-<span class="sd">    this file.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">textwrap</span> <span class="kn">import</span> <span class="n">dedent</span><span class="p">,</span> <span class="n">wrap</span>
-
-    <span class="n">fact_string</span> <span class="o">=</span> <span class="n">dedent</span><span class="p">(</span><span class="s">&#39;&#39;&#39;</span><span class="se">\</span>
-<span class="s">    &quot;&quot;&quot;</span>
-<span class="s">    The contents of this file are the return value of</span>
-<span class="s">    ``sympy.assumptions.ask.compute_known_facts``.  Do NOT manually</span>
-<span class="s">    edit this file.</span>
-<span class="s">    &quot;&quot;&quot;</span>
-
-<span class="s">    from sympy.logic.boolalg import And, Not, Or</span>
-<span class="s">    from sympy.assumptions.ask import Q</span>
-
-<span class="s">    # -{ Known facts in CNF }-</span>
-<span class="s">    known_facts_cnf = And(</span>
-<span class="s">        </span><span class="si">%s</span><span class="s"></span>
-<span class="s">    )</span>
-
-<span class="s">    # -{ Known facts in compressed sets }-</span>
-<span class="s">    known_facts_dict = {</span>
-<span class="s">        </span><span class="si">%s</span><span class="s"></span>
-<span class="s">    }</span>
-<span class="s">    &#39;&#39;&#39;</span><span class="p">)</span>
-    <span class="c"># Compute the known facts in CNF form for logical inference</span>
-    <span class="n">LINE</span> <span class="o">=</span> <span class="s">&quot;,</span><span class="se">\n</span><span class="s">    &quot;</span>
-    <span class="n">HANG</span> <span class="o">=</span> <span class="s">&#39; &#39;</span><span class="o">*</span><span class="mi">8</span>
-    <span class="n">cnf</span> <span class="o">=</span> <span class="n">to_cnf</span><span class="p">(</span><span class="n">known_facts</span><span class="p">)</span>
-    <span class="n">c</span> <span class="o">=</span> <span class="n">LINE</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="nb">str</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">cnf</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-    <span class="n">mapping</span> <span class="o">=</span> <span class="n">single_fact_lookup</span><span class="p">(</span><span class="n">known_facts_keys</span><span class="p">,</span> <span class="n">cnf</span><span class="p">)</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">LINE</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span>
-        <span class="n">wrap</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s">: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">item</span><span class="p">,</span>
-            <span class="n">subsequent_indent</span><span class="o">=</span><span class="n">HANG</span><span class="p">,</span>
-            <span class="n">break_long_words</span><span class="o">=</span><span class="bp">False</span><span class="p">))</span>
-        <span class="k">for</span> <span class="n">item</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">mapping</span><span class="o">.</span><span class="n">items</span><span class="p">())])</span> <span class="o">+</span> <span class="s">&#39;,&#39;</span>
-    <span class="k">return</span> <span class="n">fact_string</span> <span class="o">%</span> <span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-
-<span class="c"># handlers_dict tells us what ask handler we should use</span>
-<span class="c"># for a particular key</span></div>
-<span class="n">_val_template</span> <span class="o">=</span> <span class="s">&#39;sympy.assumptions.handlers.</span><span class="si">%s</span><span class="s">&#39;</span>
-<span class="n">_handlers</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="p">(</span><span class="s">&quot;antihermitian&quot;</span><span class="p">,</span>     <span class="s">&quot;sets.AskAntiHermitianHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;bounded&quot;</span><span class="p">,</span>           <span class="s">&quot;calculus.AskBoundedHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;commutative&quot;</span><span class="p">,</span>       <span class="s">&quot;AskCommutativeHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;complex&quot;</span><span class="p">,</span>           <span class="s">&quot;sets.AskComplexHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;composite&quot;</span><span class="p">,</span>         <span class="s">&quot;ntheory.AskCompositeHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;even&quot;</span><span class="p">,</span>              <span class="s">&quot;ntheory.AskEvenHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;extended_real&quot;</span><span class="p">,</span>     <span class="s">&quot;sets.AskExtendedRealHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;hermitian&quot;</span><span class="p">,</span>         <span class="s">&quot;sets.AskHermitianHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;imaginary&quot;</span><span class="p">,</span>         <span class="s">&quot;sets.AskImaginaryHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;infinitesimal&quot;</span><span class="p">,</span>     <span class="s">&quot;calculus.AskInfinitesimalHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;integer&quot;</span><span class="p">,</span>           <span class="s">&quot;sets.AskIntegerHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;irrational&quot;</span><span class="p">,</span>        <span class="s">&quot;sets.AskIrrationalHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;rational&quot;</span><span class="p">,</span>          <span class="s">&quot;sets.AskRationalHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;negative&quot;</span><span class="p">,</span>          <span class="s">&quot;order.AskNegativeHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;nonzero&quot;</span><span class="p">,</span>           <span class="s">&quot;order.AskNonZeroHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;positive&quot;</span><span class="p">,</span>          <span class="s">&quot;order.AskPositiveHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;prime&quot;</span><span class="p">,</span>             <span class="s">&quot;ntheory.AskPrimeHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;real&quot;</span><span class="p">,</span>              <span class="s">&quot;sets.AskRealHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;odd&quot;</span><span class="p">,</span>               <span class="s">&quot;ntheory.AskOddHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;algebraic&quot;</span><span class="p">,</span>         <span class="s">&quot;sets.AskAlgebraicHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;is_true&quot;</span><span class="p">,</span>           <span class="s">&quot;TautologicalHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;symmetric&quot;</span><span class="p">,</span>         <span class="s">&quot;matrices.AskSymmetricHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;invertible&quot;</span><span class="p">,</span>        <span class="s">&quot;matrices.AskInvertibleHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;orthogonal&quot;</span><span class="p">,</span>        <span class="s">&quot;matrices.AskOrthogonalHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;positive_definite&quot;</span><span class="p">,</span> <span class="s">&quot;matrices.AskPositiveDefiniteHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;upper_triangular&quot;</span><span class="p">,</span>  <span class="s">&quot;matrices.AskUpperTriangularHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;lower_triangular&quot;</span><span class="p">,</span>  <span class="s">&quot;matrices.AskLowerTriangularHandler&quot;</span><span class="p">),</span>
-    <span class="p">(</span><span class="s">&quot;diagonal&quot;</span><span class="p">,</span>          <span class="s">&quot;matrices.AskDiagonalHandler&quot;</span><span class="p">),</span>
-<span class="p">]</span>
-<span class="k">for</span> <span class="n">name</span><span class="p">,</span> <span class="n">value</span> <span class="ow">in</span> <span class="n">_handlers</span><span class="p">:</span>
-    <span class="n">register_handler</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">_val_template</span> <span class="o">%</span> <span class="n">value</span><span class="p">)</span>
-
-
-<span class="n">known_facts_keys</span> <span class="o">=</span> <span class="p">[</span><span class="nb">getattr</span><span class="p">(</span><span class="n">Q</span><span class="p">,</span> <span class="n">attr</span><span class="p">)</span> <span class="k">for</span> <span class="n">attr</span> <span class="ow">in</span> <span class="n">Q</span><span class="o">.</span><span class="n">__dict__</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">attr</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39;__&#39;</span><span class="p">)]</span>
-<span class="n">known_facts</span> <span class="o">=</span> <span class="n">And</span><span class="p">(</span>
-    <span class="n">Implies</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">complex</span><span class="p">),</span>
-    <span class="n">Implies</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">hermitian</span><span class="p">),</span>
-    <span class="n">Equivalent</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">integer</span> <span class="o">&amp;</span> <span class="o">~</span><span class="n">Q</span><span class="o">.</span><span class="n">odd</span><span class="p">),</span>
-    <span class="n">Equivalent</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">extended_real</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">real</span> <span class="o">|</span> <span class="n">Q</span><span class="o">.</span><span class="n">infinity</span><span class="p">),</span>
-    <span class="n">Equivalent</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">odd</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">integer</span> <span class="o">&amp;</span> <span class="o">~</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">),</span>
-    <span class="n">Equivalent</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">prime</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">integer</span> <span class="o">&amp;</span> <span class="n">Q</span><span class="o">.</span><span class="n">positive</span> <span class="o">&amp;</span> <span class="o">~</span><span class="n">Q</span><span class="o">.</span><span class="n">composite</span><span class="p">),</span>
-    <span class="n">Implies</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">rational</span><span class="p">),</span>
-    <span class="n">Implies</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">imaginary</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">complex</span> <span class="o">&amp;</span> <span class="o">~</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">),</span>
-    <span class="n">Implies</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">imaginary</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">antihermitian</span><span class="p">),</span>
-    <span class="n">Implies</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">antihermitian</span><span class="p">,</span> <span class="o">~</span><span class="n">Q</span><span class="o">.</span><span class="n">hermitian</span><span class="p">),</span>
-    <span class="n">Equivalent</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">nonzero</span> <span class="o">&amp;</span> <span class="o">~</span><span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">),</span>
-    <span class="n">Equivalent</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">nonzero</span> <span class="o">&amp;</span> <span class="o">~</span><span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">),</span>
-    <span class="n">Equivalent</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">rational</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">real</span> <span class="o">&amp;</span> <span class="o">~</span><span class="n">Q</span><span class="o">.</span><span class="n">irrational</span><span class="p">),</span>
-    <span class="n">Equivalent</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">rational</span> <span class="o">|</span> <span class="n">Q</span><span class="o">.</span><span class="n">irrational</span><span class="p">),</span>
-    <span class="n">Implies</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">nonzero</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">),</span>
-    <span class="n">Equivalent</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">nonzero</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">positive</span> <span class="o">|</span> <span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">),</span>
-    <span class="n">Implies</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">orthogonal</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">positive_definite</span><span class="p">),</span>
-    <span class="n">Implies</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">positive_definite</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">invertible</span><span class="p">),</span>
-    <span class="n">Implies</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">diagonal</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">upper_triangular</span><span class="p">),</span>
-    <span class="n">Implies</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">diagonal</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">lower_triangular</span><span class="p">),</span>
-    <span class="n">Implies</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">lower_triangular</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">triangular</span><span class="p">),</span>
-    <span class="n">Implies</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">upper_triangular</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">triangular</span><span class="p">),</span>
-    <span class="n">Implies</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">triangular</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">upper_triangular</span> <span class="o">|</span> <span class="n">Q</span><span class="o">.</span><span class="n">lower_triangular</span><span class="p">),</span>
-    <span class="n">Implies</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">upper_triangular</span> <span class="o">&amp;</span> <span class="n">Q</span><span class="o">.</span><span class="n">lower_triangular</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">diagonal</span><span class="p">),</span>
-    <span class="n">Implies</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">diagonal</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">symmetric</span><span class="p">),</span>
-    <span class="n">Implies</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">unit_triangular</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">triangular</span><span class="p">),</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.assumptions.ask_generated</span> <span class="kn">import</span> <span class="n">known_facts_dict</span><span class="p">,</span> <span class="n">known_facts_cnf</span>
-</pre></div>
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+++ /dev/null
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-            
-  <h1>Source code for sympy.assumptions.assume</h1><div class="highlight"><pre>
-<span class="kn">import</span> <span class="nn">inspect</span>
-<span class="kn">from</span> <span class="nn">sympy.core.cache</span> <span class="kn">import</span> <span class="n">cacheit</span>
-<span class="kn">from</span> <span class="nn">sympy.core.singleton</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="kn">from</span> <span class="nn">sympy.logic.boolalg</span> <span class="kn">import</span> <span class="n">Boolean</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.source</span> <span class="kn">import</span> <span class="n">get_class</span>
-
-
-<div class="viewcode-block" id="AssumptionsContext"><a class="viewcode-back" href="../../../modules/assumptions/assume.html#sympy.assumptions.assume.AssumptionsContext">[docs]</a><span class="k">class</span> <span class="nc">AssumptionsContext</span><span class="p">(</span><span class="nb">set</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Set representing assumptions.</span>
-
-<span class="sd">    This is used to represent global assumptions, but you can also use this</span>
-<span class="sd">    class to create your own local assumptions contexts. It is basically a thin</span>
-<span class="sd">    wrapper to Python&#39;s set, so see its documentation for advanced usage.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import global_assumptions, AppliedPredicate, Q</span>
-<span class="sd">        &gt;&gt;&gt; global_assumptions</span>
-<span class="sd">        AssumptionsContext()</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; global_assumptions.add(Q.real(x))</span>
-<span class="sd">        &gt;&gt;&gt; global_assumptions</span>
-<span class="sd">        AssumptionsContext([Q.real(x)])</span>
-<span class="sd">        &gt;&gt;&gt; global_assumptions.remove(Q.real(x))</span>
-<span class="sd">        &gt;&gt;&gt; global_assumptions</span>
-<span class="sd">        AssumptionsContext()</span>
-<span class="sd">        &gt;&gt;&gt; global_assumptions.clear()</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-<div class="viewcode-block" id="AssumptionsContext.add"><a class="viewcode-back" href="../../../modules/assumptions/assume.html#sympy.assumptions.assume.AssumptionsContext.add">[docs]</a>    <span class="k">def</span> <span class="nf">add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Add an assumption.&quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">assumptions</span><span class="p">:</span>
-            <span class="nb">super</span><span class="p">(</span><span class="n">AssumptionsContext</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-</div></div>
-<span class="n">global_assumptions</span> <span class="o">=</span> <span class="n">AssumptionsContext</span><span class="p">()</span>
-
-
-<div class="viewcode-block" id="AppliedPredicate"><a class="viewcode-back" href="../../../modules/assumptions/assume.html#sympy.assumptions.assume.AppliedPredicate">[docs]</a><span class="k">class</span> <span class="nc">AppliedPredicate</span><span class="p">(</span><span class="n">Boolean</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The class of expressions resulting from applying a Predicate.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Q, Symbol</span>
-<span class="sd">    &gt;&gt;&gt; x = Symbol(&#39;x&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; Q.integer(x)</span>
-<span class="sd">    Q.integer(x)</span>
-<span class="sd">    &gt;&gt;&gt; type(Q.integer(x))</span>
-<span class="sd">    &lt;class &#39;sympy.assumptions.assume.AppliedPredicate&#39;&gt;</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">predicate</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Boolean</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">predicate</span><span class="p">,</span> <span class="n">arg</span><span class="p">)</span>
-
-    <span class="n">is_Atom</span> <span class="o">=</span> <span class="bp">True</span>  <span class="c"># do not attempt to decompose this</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="AppliedPredicate.arg"><a class="viewcode-back" href="../../../modules/assumptions/assume.html#sympy.assumptions.assume.AppliedPredicate.arg">[docs]</a>    <span class="k">def</span> <span class="nf">arg</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the expression used by this assumption.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy import Q, Symbol</span>
-<span class="sd">            &gt;&gt;&gt; x = Symbol(&#39;x&#39;)</span>
-<span class="sd">            &gt;&gt;&gt; a = Q.integer(x + 1)</span>
-<span class="sd">            &gt;&gt;&gt; a.arg</span>
-<span class="sd">            x + 1</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">sort_key</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">class_key</span><span class="p">(),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">arg</span><span class="o">.</span><span class="n">sort_key</span><span class="p">())),</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="o">.</span><span class="n">sort_key</span><span class="p">(),</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">other</span><span class="p">)</span> <span class="ow">is</span> <span class="n">AppliedPredicate</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">_args</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">AppliedPredicate</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">_eval_ask</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">arg</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="Predicate"><a class="viewcode-back" href="../../../modules/assumptions/assume.html#sympy.assumptions.assume.Predicate">[docs]</a><span class="k">class</span> <span class="nc">Predicate</span><span class="p">(</span><span class="n">Boolean</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A predicate is a function that returns a boolean value.</span>
-
-<span class="sd">    Predicates merely wrap their argument and remain unevaluated:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Q, ask, Symbol, S</span>
-<span class="sd">        &gt;&gt;&gt; x = Symbol(&#39;x&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; Q.prime(7)</span>
-<span class="sd">        Q.prime(7)</span>
-
-<span class="sd">    To obtain the truth value of an expression containing predicates, use</span>
-<span class="sd">    the function `ask`:</span>
-
-<span class="sd">        &gt;&gt;&gt; ask(Q.prime(7))</span>
-<span class="sd">        True</span>
-
-<span class="sd">    The tautological predicate `Q.is_true` can be used to wrap other objects:</span>
-
-<span class="sd">        &gt;&gt;&gt; Q.is_true(x &gt; 1)</span>
-<span class="sd">        Q.is_true(x &gt; 1)</span>
-<span class="sd">        &gt;&gt;&gt; Q.is_true(S(1) &lt; x)</span>
-<span class="sd">        Q.is_true(1 &lt; x)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_Atom</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">handlers</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Boolean</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="n">name</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">handlers</span> <span class="o">=</span> <span class="n">handlers</span> <span class="ow">or</span> <span class="p">[]</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">_hashable_content</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">,)</span>
-
-    <span class="k">def</span> <span class="nf">__getnewargs__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">,)</span>
-
-    <span class="k">def</span> <span class="nf">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">AppliedPredicate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">add_handler</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">handler</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">handlers</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">handler</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">remove_handler</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">handler</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">handlers</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">handler</span><span class="p">)</span>
-
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">sort_key</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">class_key</span><span class="p">(),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">,)),</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="o">.</span><span class="n">sort_key</span><span class="p">(),</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-<div class="viewcode-block" id="Predicate.eval"><a class="viewcode-back" href="../../../modules/assumptions/assume.html#sympy.assumptions.assume.Predicate.eval">[docs]</a>    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Evaluate self(expr) under the given assumptions.</span>
-
-<span class="sd">        This uses only direct resolution methods, not logical inference.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">res</span><span class="p">,</span> <span class="n">_res</span> <span class="o">=</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-        <span class="n">mro</span> <span class="o">=</span> <span class="n">inspect</span><span class="o">.</span><span class="n">getmro</span><span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-        <span class="k">for</span> <span class="n">handler</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">handlers</span><span class="p">:</span>
-            <span class="n">cls</span> <span class="o">=</span> <span class="n">get_class</span><span class="p">(</span><span class="n">handler</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">subclass</span> <span class="ow">in</span> <span class="n">mro</span><span class="p">:</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="nb">eval</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">subclass</span><span class="o">.</span><span class="n">__name__</span><span class="p">)</span>
-                <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-                    <span class="k">continue</span>
-                <span class="n">res</span> <span class="o">=</span> <span class="nb">eval</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">_res</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">_res</span> <span class="o">=</span> <span class="n">res</span>
-                <span class="k">elif</span> <span class="n">res</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="c"># since first resolutor was conclusive, we keep that value</span>
-                    <span class="n">res</span> <span class="o">=</span> <span class="n">_res</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="c"># only check consistency if both resolutors have concluded</span>
-                    <span class="k">if</span> <span class="n">_res</span> <span class="o">!=</span> <span class="n">res</span><span class="p">:</span>
-                        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;incompatible resolutors&#39;</span><span class="p">)</span>
-                <span class="k">break</span>
-        <span class="k">return</span> <span class="n">res</span></div></div>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.assumptions.handlers.calculus</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">This module contains query handlers responsible for calculus queries:</span>
-<span class="sd">infinitesimal, bounded, etc.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">sympy.logic.boolalg</span> <span class="kn">import</span> <span class="n">conjuncts</span>
-<span class="kn">from</span> <span class="nn">sympy.assumptions</span> <span class="kn">import</span> <span class="n">Q</span><span class="p">,</span> <span class="n">ask</span>
-<span class="kn">from</span> <span class="nn">sympy.assumptions.handlers</span> <span class="kn">import</span> <span class="n">CommonHandler</span>
-
-
-<div class="viewcode-block" id="AskInfinitesimalHandler"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/calculus.html#sympy.assumptions.handlers.calculus.AskInfinitesimalHandler">[docs]</a><span class="k">class</span> <span class="nc">AskInfinitesimalHandler</span><span class="p">(</span><span class="n">CommonHandler</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Handler for key &#39;infinitesimal&#39;</span>
-<span class="sd">    Test that a given expression is equivalent to an infinitesimal</span>
-<span class="sd">    number</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="c"># helper method</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span> <span class="o">==</span> <span class="mi">0</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Basic</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskInfinitesimalHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskInfinitesimalHandler.Mul"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/calculus.html#sympy.assumptions.handlers.calculus.AskInfinitesimalHandler.Mul">[docs]</a>    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Infinitesimal*Bounded -&gt; Infinitesimal</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskInfinitesimalHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">infinitesimal</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">bounded</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">continue</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">result</span>
-</div>
-    <span class="n">Add</span><span class="p">,</span> <span class="n">Pow</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Mul</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span> <span class="o">==</span> <span class="mi">0</span>
-
-    <span class="n">NumberSymbol</span> <span class="o">=</span> <span class="n">Number</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">ImaginaryUnit</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-</div>
-<div class="viewcode-block" id="AskBoundedHandler"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/calculus.html#sympy.assumptions.handlers.calculus.AskBoundedHandler">[docs]</a><span class="k">class</span> <span class="nc">AskBoundedHandler</span><span class="p">(</span><span class="n">CommonHandler</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Handler for key &#39;bounded&#39;.</span>
-
-<span class="sd">    Test that an expression is bounded respect to all its variables.</span>
-
-<span class="sd">    Examples of usage:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, Q</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.assumptions.handlers.calculus import AskBoundedHandler</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; a = AskBoundedHandler()</span>
-<span class="sd">    &gt;&gt;&gt; a.Symbol(x, Q.positive(x)) == None</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; a.Symbol(x, Q.bounded(x))</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskBoundedHandler.Symbol"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/calculus.html#sympy.assumptions.handlers.calculus.AskBoundedHandler.Symbol">[docs]</a>    <span class="k">def</span> <span class="nf">Symbol</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Handles Symbol.</span>
-
-<span class="sd">        Examples:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol, Q</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.assumptions.handlers.calculus import AskBoundedHandler</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; a = AskBoundedHandler()</span>
-<span class="sd">        &gt;&gt;&gt; a.Symbol(x, Q.positive(x)) == None</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; a.Symbol(x, Q.bounded(x))</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">Q</span><span class="o">.</span><span class="n">bounded</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span> <span class="ow">in</span> <span class="n">conjuncts</span><span class="p">(</span><span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="bp">None</span>
-</div>
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskBoundedHandler.Add"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/calculus.html#sympy.assumptions.handlers.calculus.AskBoundedHandler.Add">[docs]</a>    <span class="k">def</span> <span class="nf">Add</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if expr is bounded, False if not and None if unknown.</span>
-
-<span class="sd">        Truth Table:</span>
-
-<span class="sd">        +-------+-----+-----------+-----------+</span>
-<span class="sd">        |       |     |           |           |</span>
-<span class="sd">        |       |  B  |     U     |     ?     |</span>
-<span class="sd">        |       |     |           |           |</span>
-<span class="sd">        +-------+-----+---+---+---+---+---+---+</span>
-<span class="sd">        |       |     |   |   |   |   |   |   |</span>
-<span class="sd">        |       |     |&#39;+&#39;|&#39;-&#39;|&#39;x&#39;|&#39;+&#39;|&#39;-&#39;|&#39;x&#39;|</span>
-<span class="sd">        |       |     |   |   |   |   |   |   |</span>
-<span class="sd">        +-------+-----+---+---+---+---+---+---+</span>
-<span class="sd">        |       |     |           |           |</span>
-<span class="sd">        |   B   |  B  |     U     |     ?     |</span>
-<span class="sd">        |       |     |           |           |</span>
-<span class="sd">        +---+---+-----+---+---+---+---+---+---+</span>
-<span class="sd">        |   |   |     |   |   |   |   |   |   |</span>
-<span class="sd">        |   |&#39;+&#39;|     | U | ? | ? | U | ? | ? |</span>
-<span class="sd">        |   |   |     |   |   |   |   |   |   |</span>
-<span class="sd">        |   +---+-----+---+---+---+---+---+---+</span>
-<span class="sd">        |   |   |     |   |   |   |   |   |   |</span>
-<span class="sd">        | U |&#39;-&#39;|     | ? | U | ? | ? | U | ? |</span>
-<span class="sd">        |   |   |     |   |   |   |   |   |   |</span>
-<span class="sd">        |   +---+-----+---+---+---+---+---+---+</span>
-<span class="sd">        |   |   |     |           |           |</span>
-<span class="sd">        |   |&#39;x&#39;|     |     ?     |     ?     |</span>
-<span class="sd">        |   |   |     |           |           |</span>
-<span class="sd">        +---+---+-----+---+---+---+---+---+---+</span>
-<span class="sd">        |       |     |           |           |</span>
-<span class="sd">        |   ?   |     |           |     ?     |</span>
-<span class="sd">        |       |     |           |           |</span>
-<span class="sd">        +-------+-----+-----------+---+---+---+</span>
-
-<span class="sd">            * &#39;B&#39; = Bounded</span>
-
-<span class="sd">            * &#39;U&#39; = Unbounded</span>
-
-<span class="sd">            * &#39;?&#39; = unknown boundedness</span>
-
-<span class="sd">            * &#39;+&#39; = positive sign</span>
-
-<span class="sd">            * &#39;-&#39; = negative sign</span>
-
-<span class="sd">            * &#39;x&#39; = sign unknown</span>
-
-<span class="sd">|</span>
-
-<span class="sd">            * All Bounded -&gt; True</span>
-
-<span class="sd">            * 1 Unbounded and the rest Bounded -&gt; False</span>
-
-<span class="sd">            * &gt;1 Unbounded, all with same known sign -&gt; False</span>
-
-<span class="sd">            * Any Unknown and unknown sign -&gt; None</span>
-
-<span class="sd">            * Else -&gt; None</span>
-
-<span class="sd">        When the signs are not the same you can have an undefined</span>
-<span class="sd">        result as in oo - oo, hence &#39;bounded&#39; is also undefined.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">sign</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>  <span class="c"># sign of unknown or unbounded</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">_bounded</span> <span class="o">=</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">bounded</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">_bounded</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span>
-            <span class="c"># if there has been more than one sign or if the sign of this arg</span>
-            <span class="c"># is None and Bounded is None or there was already</span>
-            <span class="c"># an unknown sign, return None</span>
-            <span class="k">if</span> <span class="n">sign</span> <span class="o">!=</span> <span class="o">-</span><span class="mi">1</span> <span class="ow">and</span> <span class="n">s</span> <span class="o">!=</span> <span class="n">sign</span> <span class="ow">or</span> \
-                    <span class="n">s</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">and</span> <span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="n">_bounded</span> <span class="ow">or</span> <span class="n">s</span> <span class="o">==</span> <span class="n">sign</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">sign</span> <span class="o">=</span> <span class="n">s</span>
-            <span class="c"># once False, do not change</span>
-            <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">_bounded</span>
-        <span class="k">return</span> <span class="n">result</span>
-</div>
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskBoundedHandler.Mul"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/calculus.html#sympy.assumptions.handlers.calculus.AskBoundedHandler.Mul">[docs]</a>    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if expr is bounded, False if not and None if unknown.</span>
-
-<span class="sd">        Truth Table:</span>
-
-<span class="sd">        +---+---+---+--------+</span>
-<span class="sd">        |   |   |   |        |</span>
-<span class="sd">        |   | B | U |   ?    |</span>
-<span class="sd">        |   |   |   |        |</span>
-<span class="sd">        +---+---+---+---+----+</span>
-<span class="sd">        |   |   |   |   |    |</span>
-<span class="sd">        |   |   |   | s | /s |</span>
-<span class="sd">        |   |   |   |   |    |</span>
-<span class="sd">        +---+---+---+---+----+</span>
-<span class="sd">        |   |   |   |        |</span>
-<span class="sd">        | B | B | U |   ?    |</span>
-<span class="sd">        |   |   |   |        |</span>
-<span class="sd">        +---+---+---+---+----+</span>
-<span class="sd">        |   |   |   |   |    |</span>
-<span class="sd">        | U |   | U | U | ?  |</span>
-<span class="sd">        |   |   |   |   |    |</span>
-<span class="sd">        +---+---+---+---+----+</span>
-<span class="sd">        |   |   |   |        |</span>
-<span class="sd">        | ? |   |   |   ?    |</span>
-<span class="sd">        |   |   |   |        |</span>
-<span class="sd">        +---+---+---+---+----+</span>
-
-<span class="sd">            * B = Bounded</span>
-
-<span class="sd">            * U = Unbounded</span>
-
-<span class="sd">            * ? = unknown boundedness</span>
-
-<span class="sd">            * s = signed (hence nonzero)</span>
-
-<span class="sd">            * /s = not signed</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">_bounded</span> <span class="o">=</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">bounded</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">_bounded</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="n">_bounded</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-                <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">nonzero</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-                <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">:</span>
-                    <span class="n">result</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="n">result</span>
-</div>
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskBoundedHandler.Pow"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/calculus.html#sympy.assumptions.handlers.calculus.AskBoundedHandler.Pow">[docs]</a>    <span class="k">def</span> <span class="nf">Pow</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Unbounded ** NonZero -&gt; Unbounded</span>
-<span class="sd">        Bounded ** Bounded -&gt; Bounded</span>
-<span class="sd">        Abs()&lt;=1 ** Positive -&gt; Bounded</span>
-<span class="sd">        Abs()&gt;=1 ** Negative -&gt; Bounded</span>
-<span class="sd">        Otherwise unknown</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">base_bounded</span> <span class="o">=</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">bounded</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="n">exp_bounded</span> <span class="o">=</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">bounded</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">base_bounded</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">exp_bounded</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>  <span class="c"># Common Case</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">base_bounded</span> <span class="ow">is</span> <span class="bp">False</span> <span class="ow">and</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">nonzero</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">base_bounded</span> <span class="ow">and</span> <span class="n">exp_bounded</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">exp_bounded</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="bp">None</span>
-</div>
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">log</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">bounded</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">assumptions</span><span class="p">)</span>
-
-    <span class="n">exp</span> <span class="o">=</span> <span class="n">log</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">sin</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="n">cos</span> <span class="o">=</span> <span class="n">sin</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Infinity</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">NegativeInfinity</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Pi</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Exp1</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">ImaginaryUnit</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">sign</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span></div>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../../index.html">
-              <img class="logo" src="../../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-<div id="searchbox" style="display: none">
-  <h3>Quick search</h3>
-    <form class="search" action="../../../../search.html" method="get">
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diff --git a/dev-py3k/_modules/sympy/assumptions/handlers/ntheory.html b/dev-py3k/_modules/sympy/assumptions/handlers/ntheory.html
deleted file mode 100644
index a1ecbdb17eb..00000000000
--- a/dev-py3k/_modules/sympy/assumptions/handlers/ntheory.html
+++ /dev/null
@@ -1,355 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
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-    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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-    
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-
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-      <div class="documentwrapper">
-        <div class="bodywrapper">
-          <div class="body">
-            
-  <h1>Source code for sympy.assumptions.handlers.ntheory</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Handlers for keys related to number theory: prime, even, odd, etc.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">sympy.assumptions</span> <span class="kn">import</span> <span class="n">Q</span><span class="p">,</span> <span class="n">ask</span>
-<span class="kn">from</span> <span class="nn">sympy.assumptions.handlers</span> <span class="kn">import</span> <span class="n">CommonHandler</span>
-<span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">isprime</span>
-
-
-<div class="viewcode-block" id="AskPrimeHandler"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/ntheory.html#sympy.assumptions.handlers.ntheory.AskPrimeHandler">[docs]</a><span class="k">class</span> <span class="nc">AskPrimeHandler</span><span class="p">(</span><span class="n">CommonHandler</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Handler for key &#39;prime&#39;</span>
-<span class="sd">    Test that an expression represents a prime number. When the</span>
-<span class="sd">    expression is a number the result, when True, is subject to</span>
-<span class="sd">    the limitations of isprime() which is used to return the result.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="c"># helper method</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">round</span><span class="p">())</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">expr</span> <span class="o">-</span> <span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="mi">0</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span>
-        <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="n">isprime</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Basic</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="c"># Just use int(expr) once</span>
-        <span class="c"># http://code.google.com/p/sympy/issues/detail?id=1462</span>
-        <span class="c"># is solved</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskPrimeHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskPrimeHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">pass</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># a product of integers can&#39;t be a prime</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskPrimeHandler.Pow"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/ntheory.html#sympy.assumptions.handlers.ntheory.AskPrimeHandler.Pow">[docs]</a>    <span class="k">def</span> <span class="nf">Pow</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Integer**Integer     -&gt; !Prime</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskPrimeHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span> <span class="ow">and</span> \
-                <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-</div>
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Integer</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">isprime</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Rational</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Float</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">AskPrimeHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Infinity</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">NegativeInfinity</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">ImaginaryUnit</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">NumberSymbol</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">AskPrimeHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-
-</div>
-<span class="k">class</span> <span class="nc">AskCompositeHandler</span><span class="p">(</span><span class="n">CommonHandler</span><span class="p">):</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Basic</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="n">_positive</span> <span class="o">=</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">expr</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">_positive</span><span class="p">:</span>
-            <span class="n">_integer</span> <span class="o">=</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">expr</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">_integer</span><span class="p">:</span>
-                <span class="n">_prime</span> <span class="o">=</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">prime</span><span class="p">(</span><span class="n">expr</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">_prime</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">return</span>
-                <span class="k">return</span> <span class="ow">not</span> <span class="n">_prime</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">_integer</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_positive</span>
-
-
-<span class="k">class</span> <span class="nc">AskEvenHandler</span><span class="p">(</span><span class="n">CommonHandler</span><span class="p">):</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="c"># helper method</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">round</span><span class="p">())</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">expr</span> <span class="o">-</span> <span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="mi">0</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span>
-        <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="n">i</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Basic</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskEvenHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Even * Integer -&gt; Even</span>
-<span class="sd">        Even * Odd     -&gt; Even</span>
-<span class="sd">        Integer * Odd  -&gt; ?</span>
-<span class="sd">        Odd * Odd      -&gt; Odd</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskEvenHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="n">even</span><span class="p">,</span> <span class="n">odd</span><span class="p">,</span> <span class="n">irrational</span> <span class="o">=</span> <span class="bp">False</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="c"># check for all integers and at least one even</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                    <span class="n">even</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">odd</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                    <span class="n">odd</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">irrational</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="c"># one irrational makes the result False</span>
-                <span class="c"># two makes it undefined</span>
-                <span class="k">if</span> <span class="n">irrational</span><span class="p">:</span>
-                    <span class="k">break</span>
-                <span class="n">irrational</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">irrational</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">if</span> <span class="n">even</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">if</span> <span class="n">odd</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Add</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Even + Odd  -&gt; Odd</span>
-<span class="sd">        Even + Even -&gt; Even</span>
-<span class="sd">        Odd  + Odd  -&gt; Even</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskEvenHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="n">_result</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">pass</span>
-            <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">odd</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="n">_result</span> <span class="o">=</span> <span class="ow">not</span> <span class="n">_result</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_result</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Integer</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="nb">bool</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">p</span> <span class="o">&amp;</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Rational</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Float</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Infinity</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">NegativeInfinity</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">NumberSymbol</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">AskEvenHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">ImaginaryUnit</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Abs</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">assumptions</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">re</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">assumptions</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">im</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-
-<div class="viewcode-block" id="AskOddHandler"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/ntheory.html#sympy.assumptions.handlers.ntheory.AskOddHandler">[docs]</a><span class="k">class</span> <span class="nc">AskOddHandler</span><span class="p">(</span><span class="n">CommonHandler</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Handler for key &#39;odd&#39;</span>
-<span class="sd">    Test that an expression represents an odd number</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Basic</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="n">_integer</span> <span class="o">=</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">expr</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">_integer</span><span class="p">:</span>
-            <span class="n">_even</span> <span class="o">=</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">expr</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">_even</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="k">return</span> <span class="ow">not</span> <span class="n">_even</span>
-        <span class="k">return</span> <span class="n">_integer</span></div>
-</pre></div>
-
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-        </div>
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diff --git a/dev-py3k/_modules/sympy/assumptions/handlers/order.html b/dev-py3k/_modules/sympy/assumptions/handlers/order.html
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-            
-  <h1>Source code for sympy.assumptions.handlers.order</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">AskHandlers related to order relations: positive, negative, etc.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">sympy.assumptions</span> <span class="kn">import</span> <span class="n">Q</span><span class="p">,</span> <span class="n">ask</span>
-<span class="kn">from</span> <span class="nn">sympy.assumptions.handlers</span> <span class="kn">import</span> <span class="n">CommonHandler</span>
-
-
-<div class="viewcode-block" id="AskNegativeHandler"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/order.html#sympy.assumptions.handlers.order.AskNegativeHandler">[docs]</a><span class="k">class</span> <span class="nc">AskNegativeHandler</span><span class="p">(</span><span class="n">CommonHandler</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    This is called by ask() when key=&#39;negative&#39;</span>
-
-<span class="sd">    Test that an expression is less (strict) than zero.</span>
-
-<span class="sd">    Examples:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import ask, Q, pi</span>
-<span class="sd">    &gt;&gt;&gt; ask(Q.negative(pi+1)) # this calls AskNegativeHandler.Add</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; ask(Q.negative(pi**2)) # this calls AskNegativeHandler.Pow</span>
-<span class="sd">    False</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()[</span><span class="mi">1</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span> <span class="o">&lt;</span> <span class="mi">0</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Basic</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskNegativeHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskNegativeHandler.Add"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/order.html#sympy.assumptions.handlers.order.AskNegativeHandler.Add">[docs]</a>    <span class="k">def</span> <span class="nf">Add</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Positive + Positive -&gt; Positive,</span>
-<span class="sd">        Negative + Negative -&gt; Negative</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskNegativeHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># if all argument&#39;s are negative</span>
-            <span class="k">return</span> <span class="bp">True</span>
-</div>
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskNegativeHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="ow">not</span> <span class="n">result</span>
-            <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">pass</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskNegativeHandler.Pow"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/order.html#sympy.assumptions.handlers.order.AskNegativeHandler.Pow">[docs]</a>    <span class="k">def</span> <span class="nf">Pow</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Real ** Even -&gt; NonNegative</span>
-<span class="sd">        Real ** Odd  -&gt; same_as_base</span>
-<span class="sd">        NonNegative ** Positive -&gt; NonNegative</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskNegativeHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">odd</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span>
-</div>
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">ImaginaryUnit</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Abs</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-</div>
-<div class="viewcode-block" id="AskNonZeroHandler"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/order.html#sympy.assumptions.handlers.order.AskNonZeroHandler">[docs]</a><span class="k">class</span> <span class="nc">AskNonZeroHandler</span><span class="p">(</span><span class="n">CommonHandler</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Handler for key &#39;zero&#39;</span>
-<span class="sd">    Test that an expression is not identically zero</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Basic</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="c"># if there are no symbols just evalf</span>
-            <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Add</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> \
-                <span class="ow">or</span> <span class="nb">all</span><span class="p">(</span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">nonzero</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">result</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">return</span> <span class="n">result</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Pow</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">nonzero</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">NaN</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Abs</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">nonzero</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">assumptions</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="AskPositiveHandler"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/order.html#sympy.assumptions.handlers.order.AskPositiveHandler">[docs]</a><span class="k">class</span> <span class="nc">AskPositiveHandler</span><span class="p">(</span><span class="n">CommonHandler</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Handler for key &#39;positive&#39;</span>
-<span class="sd">    Test that an expression is greater (strict) than zero</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()[</span><span class="mi">1</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span> <span class="o">&gt;</span> <span class="mi">0</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Basic</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskPositiveHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskPositiveHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">result</span> <span class="o">^</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Add</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskPositiveHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">True</span><span class="p">:</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># if all argument&#39;s are positive</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Pow</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span> <span class="o">&gt;</span> <span class="mi">0</span>
-        <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">exp</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">ImaginaryUnit</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Abs</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">nonzero</span><span class="p">(</span><span class="n">expr</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span></div>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
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-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../../index.html">
-              <img class="logo" src="../../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-<div id="searchbox" style="display: none">
-  <h3>Quick search</h3>
-    <form class="search" action="../../../../search.html" method="get">
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diff --git a/dev-py3k/_modules/sympy/assumptions/handlers/sets.html b/dev-py3k/_modules/sympy/assumptions/handlers/sets.html
deleted file mode 100644
index a8a49bec97e..00000000000
--- a/dev-py3k/_modules/sympy/assumptions/handlers/sets.html
+++ /dev/null
@@ -1,755 +0,0 @@
-
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-            
-  <h1>Source code for sympy.assumptions.handlers.sets</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Handlers for predicates related to set membership: integer, rational, etc.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">sympy.assumptions</span> <span class="kn">import</span> <span class="n">Q</span><span class="p">,</span> <span class="n">ask</span>
-<span class="kn">from</span> <span class="nn">sympy.assumptions.handlers</span> <span class="kn">import</span> <span class="n">CommonHandler</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">I</span>
-
-
-<div class="viewcode-block" id="AskIntegerHandler"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskIntegerHandler">[docs]</a><span class="k">class</span> <span class="nc">AskIntegerHandler</span><span class="p">(</span><span class="n">CommonHandler</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Handler for Q.integer</span>
-<span class="sd">    Test that an expression belongs to the field of integer numbers</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="c"># helper method</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">round</span><span class="p">())</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">expr</span> <span class="o">-</span> <span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="mi">0</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskIntegerHandler.Add"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskIntegerHandler.Add">[docs]</a>    <span class="k">def</span> <span class="nf">Add</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Integer + Integer       -&gt; Integer</span>
-<span class="sd">        Integer + !Integer      -&gt; !Integer</span>
-<span class="sd">        !Integer + !Integer -&gt; ?</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskIntegerHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">test_closed_group</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">)</span>
-</div>
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskIntegerHandler.Mul"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskIntegerHandler.Mul">[docs]</a>    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Integer*Integer      -&gt; Integer</span>
-<span class="sd">        Integer*Irrational   -&gt; !Integer</span>
-<span class="sd">        Odd/Even             -&gt; !Integer</span>
-<span class="sd">        Integer*Rational     -&gt; ?</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskIntegerHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="n">_output</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">q</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">expr</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="o">~</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">q</span> <span class="o">&amp;</span> <span class="mi">1</span><span class="p">):</span>
-                        <span class="k">return</span> <span class="bp">None</span>
-                <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">irrational</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">_output</span><span class="p">:</span>
-                        <span class="n">_output</span> <span class="o">=</span> <span class="bp">False</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">return</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_output</span>
-</div>
-    <span class="n">Pow</span> <span class="o">=</span> <span class="n">Add</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">int</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Integer</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Rational</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="c"># rationals with denominator one get</span>
-        <span class="c"># evaluated to Integers</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Float</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span> <span class="o">==</span> <span class="n">expr</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Pi</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Exp1</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Infinity</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">NegativeInfinity</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">ImaginaryUnit</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Abs</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">assumptions</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="AskRationalHandler"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskRationalHandler">[docs]</a><span class="k">class</span> <span class="nc">AskRationalHandler</span><span class="p">(</span><span class="n">CommonHandler</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Handler for Q.rational</span>
-<span class="sd">    Test that an expression belongs to the field of rational numbers</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskRationalHandler.Add"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskRationalHandler.Add">[docs]</a>    <span class="k">def</span> <span class="nf">Add</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Rational + Rational     -&gt; Rational</span>
-<span class="sd">        Rational + !Rational    -&gt; !Rational</span>
-<span class="sd">        !Rational + !Rational   -&gt; ?</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()[</span><span class="mi">1</span><span class="p">]:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="n">test_closed_group</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">rational</span><span class="p">)</span>
-</div>
-    <span class="n">Mul</span> <span class="o">=</span> <span class="n">Add</span>
-
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskRationalHandler.Pow"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskRationalHandler.Pow">[docs]</a>    <span class="k">def</span> <span class="nf">Pow</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Rational ** Integer      -&gt; Rational</span>
-<span class="sd">        Irrational ** Rational   -&gt; Irrational</span>
-<span class="sd">        Rational ** Irrational   -&gt; ?</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">rational</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">rational</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">prime</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">False</span>
-</div>
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Rational</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Float</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="c"># it&#39;s finite-precision</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">ImaginaryUnit</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Infinity</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">NegativeInfinity</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Pi</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Exp1</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-</div>
-<span class="k">class</span> <span class="nc">AskIrrationalHandler</span><span class="p">(</span><span class="n">CommonHandler</span><span class="p">):</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Basic</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="n">_real</span> <span class="o">=</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">expr</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">_real</span><span class="p">:</span>
-            <span class="n">_rational</span> <span class="o">=</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">rational</span><span class="p">(</span><span class="n">expr</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">_rational</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="k">return</span> <span class="ow">not</span> <span class="n">_rational</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_real</span>
-
-
-<div class="viewcode-block" id="AskRealHandler"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskRealHandler">[docs]</a><span class="k">class</span> <span class="nc">AskRealHandler</span><span class="p">(</span><span class="n">CommonHandler</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Handler for Q.real</span>
-<span class="sd">    Test that an expression belongs to the field of real numbers</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="c"># let as_real_imag() work first since the expression may</span>
-        <span class="c"># be simpler to evaluate</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">_prec</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="ow">not</span> <span class="n">i</span>
-        <span class="c"># allow None to be returned if we couldn&#39;t show for sure</span>
-        <span class="c"># that i was 0</span>
-
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskRealHandler.Add"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskRealHandler.Add">[docs]</a>    <span class="k">def</span> <span class="nf">Add</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Real + Real              -&gt; Real</span>
-<span class="sd">        Real + (Complex &amp; !Real) -&gt; !Real</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskRealHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">test_closed_group</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">)</span>
-</div>
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskRealHandler.Mul"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskRealHandler.Mul">[docs]</a>    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Real*Real               -&gt; Real</span>
-<span class="sd">        Real*Imaginary          -&gt; !Real</span>
-<span class="sd">        Imaginary*Imaginary     -&gt; Real</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskRealHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">pass</span>
-            <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">imaginary</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">result</span> <span class="o">^</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">result</span>
-</div>
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskRealHandler.Pow"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskRealHandler.Pow">[docs]</a>    <span class="k">def</span> <span class="nf">Pow</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Real**Integer         -&gt; Real</span>
-<span class="sd">        Positive**Real        -&gt; Real</span>
-<span class="sd">        Real**(Integer/Even)  -&gt; Real if base is nonnegative</span>
-<span class="sd">        Real**(Integer/Odd)   -&gt; Real</span>
-<span class="sd">        Real**Imaginary       -&gt; ?</span>
-<span class="sd">        Imaginary**Real       -&gt; ?</span>
-<span class="sd">        Real**Real            -&gt; ?</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskRealHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">imaginary</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">odd</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-                <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> \
-                   <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">q</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-                <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-                <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-</div>
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Rational</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Float</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Pi</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Exp1</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Abs</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">re</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="n">im</span> <span class="o">=</span> <span class="n">re</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">ImaginaryUnit</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Infinity</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">NegativeInfinity</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">sin</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="n">cos</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="n">sin</span><span class="p">,</span> <span class="n">sin</span>
-
-</div>
-<div class="viewcode-block" id="AskExtendedRealHandler"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskExtendedRealHandler">[docs]</a><span class="k">class</span> <span class="nc">AskExtendedRealHandler</span><span class="p">(</span><span class="n">AskRealHandler</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Handler for Q.extended_real</span>
-<span class="sd">    Test that an expression belongs to the field of extended real numbers,</span>
-<span class="sd">    that is real numbers union {Infinity, -Infinity}</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Add</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">test_closed_group</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">extended_real</span><span class="p">)</span>
-
-    <span class="n">Mul</span><span class="p">,</span> <span class="n">Pow</span> <span class="o">=</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Add</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Infinity</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">NegativeInfinity</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-</div>
-<div class="viewcode-block" id="AskHermitianHandler"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskHermitianHandler">[docs]</a><span class="k">class</span> <span class="nc">AskHermitianHandler</span><span class="p">(</span><span class="n">AskRealHandler</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Handler for Q.hermitian</span>
-<span class="sd">    Test that an expression belongs to the field of Hermitian operators</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskHermitianHandler.Add"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskHermitianHandler.Add">[docs]</a>    <span class="k">def</span> <span class="nf">Add</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Hermitian + Hermitian  -&gt; Hermitian</span>
-<span class="sd">        Hermitian + !Hermitian -&gt; !Hermitian</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskRealHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">test_closed_group</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">hermitian</span><span class="p">)</span>
-</div>
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskHermitianHandler.Mul"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskHermitianHandler.Mul">[docs]</a>    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        As long as there is at most only one noncommutative term:</span>
-<span class="sd">        Hermitian*Hermitian         -&gt; Hermitian</span>
-<span class="sd">        Hermitian*Antihermitian     -&gt; !Hermitian</span>
-<span class="sd">        Antihermitian*Antihermitian -&gt; Hermitian</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskRealHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="n">nccount</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">antihermitian</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">result</span> <span class="o">^</span> <span class="bp">True</span>
-            <span class="k">elif</span> <span class="ow">not</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">hermitian</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">break</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="o">~</span><span class="n">Q</span><span class="o">.</span><span class="n">commutative</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="n">nccount</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">if</span> <span class="n">nccount</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">result</span>
-</div>
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskHermitianHandler.Pow"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskHermitianHandler.Pow">[docs]</a>    <span class="k">def</span> <span class="nf">Pow</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Hermitian**Integer -&gt; Hermitian</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskRealHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">hermitian</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">True</span>
-</div>
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">sin</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">hermitian</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="n">cos</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="n">sin</span><span class="p">,</span> <span class="n">sin</span>
-
-</div>
-<div class="viewcode-block" id="AskComplexHandler"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskComplexHandler">[docs]</a><span class="k">class</span> <span class="nc">AskComplexHandler</span><span class="p">(</span><span class="n">CommonHandler</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Handler for Q.complex</span>
-<span class="sd">    Test that an expression belongs to the field of complex numbers</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Add</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">test_closed_group</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">complex</span><span class="p">)</span>
-
-    <span class="n">Mul</span><span class="p">,</span> <span class="n">Pow</span> <span class="o">=</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Add</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">NumberSymbol</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Abs</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">ImaginaryUnit</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Infinity</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">NegativeInfinity</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="p">[</span><span class="n">Abs</span><span class="p">]</span><span class="o">*</span><span class="mi">5</span>  <span class="c"># they are all complex functions</span>
-
-</div>
-<div class="viewcode-block" id="AskImaginaryHandler"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskImaginaryHandler">[docs]</a><span class="k">class</span> <span class="nc">AskImaginaryHandler</span><span class="p">(</span><span class="n">CommonHandler</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Handler for Q.imaginary</span>
-<span class="sd">    Test that an expression belongs to the field of imaginary numbers,</span>
-<span class="sd">    that is, numbers in the form x*I, where x is real</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="c"># helper method</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskImaginaryHandler.Add"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskImaginaryHandler.Add">[docs]</a>    <span class="k">def</span> <span class="nf">Add</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Imaginary + Imaginary -&gt; Imaginary</span>
-<span class="sd">        Imaginary + Complex   -&gt; ?</span>
-<span class="sd">        Imaginary + Real      -&gt; !Imaginary</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskImaginaryHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="n">reals</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">imaginary</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">pass</span>
-            <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="n">reals</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">reals</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">if</span> <span class="n">reals</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="n">reals</span><span class="p">):</span>
-                <span class="c"># two reals could sum 0 thus giving an imaginary</span>
-                <span class="k">return</span> <span class="bp">False</span>
-</div>
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskImaginaryHandler.Mul"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskImaginaryHandler.Mul">[docs]</a>    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Real*Imaginary      -&gt; Imaginary</span>
-<span class="sd">        Imaginary*Imaginary -&gt; Real</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskImaginaryHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="n">reals</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">imaginary</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">result</span> <span class="o">^</span> <span class="bp">True</span>
-            <span class="k">elif</span> <span class="ow">not</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">reals</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">return</span> <span class="n">result</span>
-</div>
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskImaginaryHandler.Pow"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskImaginaryHandler.Pow">[docs]</a>    <span class="k">def</span> <span class="nf">Pow</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Imaginary**integer -&gt; Imaginary if integer % 2 == 1</span>
-<span class="sd">        Imaginary**integer -&gt; real if integer % 2 == 0</span>
-<span class="sd">        Imaginary**Imaginary    -&gt; ?</span>
-<span class="sd">        Imaginary**Real         -&gt; ?</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskImaginaryHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">imaginary</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">odd</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-                <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> \
-                   <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">q</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span><span class="n">assumptions</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-                <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-                <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-
-
-</div>
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span>
-
-    <span class="n">NumberSymbol</span> <span class="o">=</span> <span class="n">Number</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">ImaginaryUnit</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-</div>
-<div class="viewcode-block" id="AskAntiHermitianHandler"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskAntiHermitianHandler">[docs]</a><span class="k">class</span> <span class="nc">AskAntiHermitianHandler</span><span class="p">(</span><span class="n">AskImaginaryHandler</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Handler for Q.antihermitian</span>
-<span class="sd">    Test that an expression belongs to the field of anti-Hermitian operators,</span>
-<span class="sd">    that is, operators in the form x*I, where x is Hermitian</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskAntiHermitianHandler.Add"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskAntiHermitianHandler.Add">[docs]</a>    <span class="k">def</span> <span class="nf">Add</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Antihermitian + Antihermitian  -&gt; Antihermitian</span>
-<span class="sd">        Antihermitian + !Antihermitian -&gt; !Antihermitian</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskImaginaryHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">test_closed_group</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">antihermitian</span><span class="p">)</span>
-</div>
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskAntiHermitianHandler.Mul"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskAntiHermitianHandler.Mul">[docs]</a>    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        As long as there is at most only one noncommutative term:</span>
-<span class="sd">        Hermitian*Hermitian         -&gt; !Antihermitian</span>
-<span class="sd">        Hermitian*Antihermitian     -&gt; Antihermitian</span>
-<span class="sd">        Antihermitian*Antihermitian -&gt; !Antihermitian</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskImaginaryHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="n">nccount</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">antihermitian</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">result</span> <span class="o">^</span> <span class="bp">True</span>
-            <span class="k">elif</span> <span class="ow">not</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">hermitian</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">break</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="o">~</span><span class="n">Q</span><span class="o">.</span><span class="n">commutative</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="n">nccount</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">if</span> <span class="n">nccount</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">result</span>
-</div>
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="AskAntiHermitianHandler.Pow"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskAntiHermitianHandler.Pow">[docs]</a>    <span class="k">def</span> <span class="nf">Pow</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Hermitian**Integer  -&gt; !Antihermitian</span>
-<span class="sd">        Antihermitian**Even -&gt; !Antihermitian</span>
-<span class="sd">        Antihermitian**Odd  -&gt; Antihermitian</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AskImaginaryHandler</span><span class="o">.</span><span class="n">_number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">hermitian</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">antihermitian</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">odd</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">True</span>
-
-</div></div>
-<div class="viewcode-block" id="AskAlgebraicHandler"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskAlgebraicHandler">[docs]</a><span class="k">class</span> <span class="nc">AskAlgebraicHandler</span><span class="p">(</span><span class="n">CommonHandler</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Handler for Q.algebraic key. &quot;&quot;&quot;</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Add</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">test_closed_group</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">algebraic</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">test_closed_group</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">algebraic</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Pow</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">ask</span><span class="p">(</span>
-            <span class="n">Q</span><span class="o">.</span><span class="n">algebraic</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Number</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">Rational</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">q</span> <span class="o">!=</span> <span class="mi">0</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">ImaginaryUnit</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">AlgebraicNumber</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-<span class="c">#### Helper methods</span>
-
-</div>
-<div class="viewcode-block" id="test_closed_group"><a class="viewcode-back" href="../../../../modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.test_closed_group">[docs]</a><span class="k">def</span> <span class="nf">test_closed_group</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">,</span> <span class="n">key</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Test for membership in a group with respect</span>
-<span class="sd">    to the current operation</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-        <span class="n">_out</span> <span class="o">=</span> <span class="n">ask</span><span class="p">(</span><span class="n">key</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">_out</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">break</span>
-        <span class="k">elif</span> <span class="n">_out</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">result</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">break</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span></div>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/assumptions/refine.html b/dev-py3k/_modules/sympy/assumptions/refine.html
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-            
-  <h1>Source code for sympy.assumptions.refine</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">Add</span>
-<span class="kn">from</span> <span class="nn">sympy.assumptions</span> <span class="kn">import</span> <span class="n">Q</span><span class="p">,</span> <span class="n">ask</span>
-<span class="kn">from</span> <span class="nn">sympy.core.logic</span> <span class="kn">import</span> <span class="n">fuzzy_not</span>
-
-
-<div class="viewcode-block" id="refine"><a class="viewcode-back" href="../../../modules/assumptions/refine.html#sympy.assumptions.refine.refine">[docs]</a><span class="k">def</span> <span class="nf">refine</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Simplify an expression using assumptions.</span>
-
-<span class="sd">    Gives the form of expr that would be obtained if symbols</span>
-<span class="sd">    in it were replaced by explicit numerical expressions satisfying</span>
-<span class="sd">    the assumptions.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import refine, sqrt, Q</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; refine(sqrt(x**2), Q.real(x))</span>
-<span class="sd">        Abs(x)</span>
-<span class="sd">        &gt;&gt;&gt; refine(sqrt(x**2), Q.positive(x))</span>
-<span class="sd">        x</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">refine</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="c"># TODO: this will probably not work with Integral or Polynomial</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-    <span class="n">name</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span>
-    <span class="n">handler</span> <span class="o">=</span> <span class="n">handlers_dict</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">handler</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span>
-    <span class="n">new_expr</span> <span class="o">=</span> <span class="n">handler</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-    <span class="k">if</span> <span class="p">(</span><span class="n">new_expr</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="n">expr</span> <span class="o">==</span> <span class="n">new_expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span>
-    <span class="k">return</span> <span class="n">refine</span><span class="p">(</span><span class="n">new_expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="refine_abs"><a class="viewcode-back" href="../../../modules/assumptions/refine.html#sympy.assumptions.refine.refine_abs">[docs]</a><span class="k">def</span> <span class="nf">refine_abs</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Handler for the absolute value.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, Q, refine, Abs</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.assumptions.refine import refine_abs</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; refine_abs(Abs(x), Q.real(x))</span>
-<span class="sd">    &gt;&gt;&gt; refine_abs(Abs(x), Q.positive(x))</span>
-<span class="sd">    x</span>
-<span class="sd">    &gt;&gt;&gt; refine_abs(Abs(x), Q.negative(x))</span>
-<span class="sd">    -x</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">arg</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)</span> <span class="ow">and</span> \
-            <span class="n">fuzzy_not</span><span class="p">(</span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">)):</span>
-        <span class="c"># if it&#39;s nonnegative</span>
-        <span class="k">return</span> <span class="n">arg</span>
-    <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">arg</span>
-
-</div>
-<div class="viewcode-block" id="refine_Pow"><a class="viewcode-back" href="../../../modules/assumptions/refine.html#sympy.assumptions.refine.refine_Pow">[docs]</a><span class="k">def</span> <span class="nf">refine_Pow</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Handler for instances of Pow.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, Q</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.assumptions.refine import refine_Pow</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x,y,z</span>
-<span class="sd">    &gt;&gt;&gt; refine_Pow((-1)**x, Q.real(x))</span>
-<span class="sd">    &gt;&gt;&gt; refine_Pow((-1)**x, Q.even(x))</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; refine_Pow((-1)**x, Q.odd(x))</span>
-<span class="sd">    -1</span>
-
-<span class="sd">    For powers of -1, even parts of the exponent can be simplified:</span>
-
-<span class="sd">    &gt;&gt;&gt; refine_Pow((-1)**(x+y), Q.even(x))</span>
-<span class="sd">    (-1)**y</span>
-<span class="sd">    &gt;&gt;&gt; refine_Pow((-1)**(x+y+z), Q.odd(x) &amp; Q.odd(z))</span>
-<span class="sd">    (-1)**y</span>
-<span class="sd">    &gt;&gt;&gt; refine_Pow((-1)**(x+y+2), Q.odd(x))</span>
-<span class="sd">    (-1)**(y + 1)</span>
-<span class="sd">    &gt;&gt;&gt; refine_Pow((-1)**(x+3), True)</span>
-<span class="sd">    (-1)**(x + 1)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Pow</span><span class="p">,</span> <span class="n">Rational</span>
-    <span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">sign</span>
-    <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">return</span> <span class="nb">abs</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">)</span> <span class="o">**</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">odd</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">sign</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">)</span> <span class="o">*</span> <span class="nb">abs</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">)</span> <span class="o">**</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">)</span> <span class="ow">is</span> <span class="n">Pow</span><span class="p">:</span>
-                <span class="k">return</span> <span class="nb">abs</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">base</span><span class="p">)</span> <span class="o">**</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">exp</span> <span class="o">*</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">base</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-
-                <span class="c"># For powers of (-1) we can remove</span>
-                <span class="c">#  - even terms</span>
-                <span class="c">#  - pairs of odd terms</span>
-                <span class="c">#  - a single odd term + 1</span>
-                <span class="c">#  - A numerical constant N can be replaced with mod(N,2)</span>
-
-                <span class="n">coeff</span><span class="p">,</span> <span class="n">terms</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">as_coeff_add</span><span class="p">()</span>
-                <span class="n">terms</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">terms</span><span class="p">)</span>
-                <span class="n">even_terms</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([])</span>
-                <span class="n">odd_terms</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([])</span>
-                <span class="n">initial_number_of_terms</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">terms</span><span class="p">)</span>
-
-                <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                        <span class="n">even_terms</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-                    <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">odd</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                        <span class="n">odd_terms</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-
-                <span class="n">terms</span> <span class="o">-=</span> <span class="n">even_terms</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">odd_terms</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span><span class="p">:</span>
-                    <span class="n">terms</span> <span class="o">-=</span> <span class="n">odd_terms</span>
-                    <span class="n">new_coeff</span> <span class="o">=</span> <span class="p">(</span><span class="n">coeff</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">terms</span> <span class="o">-=</span> <span class="n">odd_terms</span>
-                    <span class="n">new_coeff</span> <span class="o">=</span> <span class="n">coeff</span> <span class="o">%</span> <span class="mi">2</span>
-
-                <span class="k">if</span> <span class="n">new_coeff</span> <span class="o">!=</span> <span class="n">coeff</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">terms</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">initial_number_of_terms</span><span class="p">:</span>
-                    <span class="n">terms</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">new_coeff</span><span class="p">)</span>
-                    <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="o">**</span><span class="p">(</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">terms</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="refine_exp"><a class="viewcode-back" href="../../../modules/assumptions/refine.html#sympy.assumptions.refine.refine_exp">[docs]</a><span class="k">def</span> <span class="nf">refine_exp</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Handler for exponential function.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, Q, exp, I, pi</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.assumptions.refine import refine_exp</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; refine_exp(exp(pi*I*2*x), Q.real(x))</span>
-<span class="sd">    &gt;&gt;&gt; refine_exp(exp(pi*I*2*x), Q.integer(x))</span>
-<span class="sd">    1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">arg</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-        <span class="n">coeff</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">coeff</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">coeff</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">coeff</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-                <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">odd</span><span class="p">(</span><span class="n">coeff</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span>
-                <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">coeff</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-                <span class="k">elif</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">odd</span><span class="p">(</span><span class="n">coeff</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">),</span> <span class="n">assumptions</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-</div>
-<span class="n">handlers_dict</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="s">&#39;Abs&#39;</span><span class="p">:</span> <span class="n">refine_abs</span><span class="p">,</span>
-    <span class="s">&#39;Pow&#39;</span><span class="p">:</span> <span class="n">refine_Pow</span><span class="p">,</span>
-    <span class="s">&#39;exp&#39;</span><span class="p">:</span> <span class="n">refine_exp</span><span class="p">,</span>
-<span class="p">}</span>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/categories.html b/dev-py3k/_modules/sympy/categories.html
deleted file mode 100644
index 040239d1271..00000000000
--- a/dev-py3k/_modules/sympy/categories.html
+++ /dev/null
@@ -1,142 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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-    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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-             >modules</a> |</li>
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-             >modules</a> |</li>
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-      <div class="documentwrapper">
-        <div class="bodywrapper">
-          <div class="body">
-            
-  <h1>Source code for sympy.categories</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Category Theory module.</span>
-
-<span class="sd">Provides some of the fundamental category-theory-related classes,</span>
-<span class="sd">including categories, morphisms, diagrams.  Functors are not</span>
-<span class="sd">implemented yet.</span>
-
-<span class="sd">The general reference work this module tries to follow is</span>
-
-<span class="sd">  [JoyOfCats] J. Adamek, H. Herrlich. G. E. Strecker: Abstract and</span>
-<span class="sd">              Concrete Categories. The Joy of Cats.</span>
-
-<span class="sd">The latest version of this book should be available for free download</span>
-<span class="sd">from</span>
-
-<span class="sd">   katmat.math.uni-bremen.de/acc/acc.pdf</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">.baseclasses</span> <span class="kn">import</span> <span class="p">(</span><span class="n">Object</span><span class="p">,</span> <span class="n">Morphism</span><span class="p">,</span> <span class="n">IdentityMorphism</span><span class="p">,</span>
-                         <span class="n">NamedMorphism</span><span class="p">,</span> <span class="n">CompositeMorphism</span><span class="p">,</span> <span class="n">Category</span><span class="p">,</span>
-                         <span class="n">Diagram</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">.diagram_drawing</span> <span class="kn">import</span> <span class="p">(</span><span class="n">DiagramGrid</span><span class="p">,</span> <span class="n">XypicDiagramDrawer</span><span class="p">,</span>
-                             <span class="n">xypic_draw_diagram</span><span class="p">,</span> <span class="n">preview_diagram</span><span class="p">)</span>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../index.html">
-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
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-  <h3>Quick search</h3>
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-    </p>
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-    </div>
-  </body>
-</html>
diff --git a/dev-py3k/_modules/sympy/categories/diagram_drawing.html b/dev-py3k/_modules/sympy/categories/diagram_drawing.html
deleted file mode 100644
index 2eedfdd73be..00000000000
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+++ /dev/null
@@ -1,2699 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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-            
-  <h1>Source code for sympy.categories.diagram_drawing</h1><div class="highlight"><pre>
-<span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">This module contains the functionality to arrange the nodes of a</span>
-<span class="sd">diagram on an abstract grid, and then to produce a graphical</span>
-<span class="sd">representation of the grid.</span>
-
-<span class="sd">The currently supported back-ends are Xy-pic [Xypic].</span>
-
-<span class="sd">Layout Algorithm</span>
-<span class="sd">================</span>
-
-<span class="sd">This section provides an overview of the algorithms implemented in</span>
-<span class="sd">:class:`DiagramGrid` to lay out diagrams.</span>
-
-<span class="sd">The first step of the algorithm is the removal composite and identity</span>
-<span class="sd">morphisms which do not have properties in the supplied diagram.  The</span>
-<span class="sd">premises and conclusions of the diagram are then merged.</span>
-
-<span class="sd">The generic layout algorithm begins with the construction of the</span>
-<span class="sd">&quot;skeleton&quot; of the diagram.  The skeleton is an undirected graph which</span>
-<span class="sd">has the objects of the diagram as vertices and has an (undirected)</span>
-<span class="sd">edge between each pair of objects between which there exist morphisms.</span>
-<span class="sd">The direction of the morphisms does not matter at this stage.  The</span>
-<span class="sd">skeleton also includes an edge between each pair of vertices `A` and</span>
-<span class="sd">`C` such that there exists an object `B` which is connected via</span>
-<span class="sd">a morphism to `A`, and via a morphism to `C`.</span>
-
-<span class="sd">The skeleton constructed in this way has the property that every</span>
-<span class="sd">object is a vertex of a triangle formed by three edges of the</span>
-<span class="sd">skeleton.  This property lies at the base of the generic layout</span>
-<span class="sd">algorithm.</span>
-
-<span class="sd">After the skeleton has been constructed, the algorithm lists all</span>
-<span class="sd">triangles which can be formed.  Note that some triangles will not have</span>
-<span class="sd">all edges corresponding to morphisms which will actually be drawn.</span>
-<span class="sd">Triangles which have only one edge or less which will actually be</span>
-<span class="sd">drawn are immediately discarded.</span>
-
-<span class="sd">The list of triangles is sorted according to the number of edges which</span>
-<span class="sd">correspond to morphisms, then the triangle with the least number of such</span>
-<span class="sd">edges is selected.  One of such edges is picked and the corresponding</span>
-<span class="sd">objects are placed horizontally, on a grid.  This edge is recorded to</span>
-<span class="sd">be in the fringe.  The algorithm then finds a &quot;welding&quot; of a triangle</span>
-<span class="sd">to the fringe.  A welding is an edge in the fringe where a triangle</span>
-<span class="sd">could be attached.  If the algorithm succeeds in finding such a</span>
-<span class="sd">welding, it adds to the grid that vertex of the triangle which was not</span>
-<span class="sd">yet included in any edge in the fringe and records the two new edges in</span>
-<span class="sd">the fringe.  This process continues iteratively until all objects of</span>
-<span class="sd">the diagram has been placed or until no more weldings can be found.</span>
-
-<span class="sd">An edge is only removed from the fringe when a welding to this edge</span>
-<span class="sd">has been found, and there is no room around this edge to place</span>
-<span class="sd">another vertex.</span>
-
-<span class="sd">When no more weldings can be found, but there are still triangles</span>
-<span class="sd">left, the algorithm searches for a possibility of attaching one of the</span>
-<span class="sd">remaining triangles to the existing structure by a vertex.  If such a</span>
-<span class="sd">possibility is found, the corresponding edge of the found triangle is</span>
-<span class="sd">placed in the found space and the iterative process of welding</span>
-<span class="sd">triangles restarts.</span>
-
-<span class="sd">When logical groups are supplied, each of these groups is laid out</span>
-<span class="sd">independently.  Then a diagram is constructed in which groups are</span>
-<span class="sd">objects and any two logical groups between which there exist morphisms</span>
-<span class="sd">are connected via a morphism.  This diagram is laid out.  Finally,</span>
-<span class="sd">the grid which includes all objects of the initial diagram is</span>
-<span class="sd">constructed by replacing the cells which contain logical groups with</span>
-<span class="sd">the corresponding laid out grids, and by correspondingly expanding the</span>
-<span class="sd">rows and columns.</span>
-
-<span class="sd">The sequential layout algorithm begins by constructing the</span>
-<span class="sd">underlying undirected graph defined by the morphisms obtained after</span>
-<span class="sd">simplifying premises and conclusions and merging them (see above).</span>
-<span class="sd">The vertex with the minimal degree is then picked up and depth-first</span>
-<span class="sd">search is started from it.  All objects which are located at distance</span>
-<span class="sd">`n` from the root in the depth-first search tree, are positioned in</span>
-<span class="sd">the `n`-th column of the resulting grid.  The sequential layout will</span>
-<span class="sd">therefore attempt to lay the objects out along a line.</span>
-
-<span class="sd">References</span>
-<span class="sd">==========</span>
-
-<span class="sd">[Xypic] http://www.tug.org/applications/Xy-pic/</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">FiniteSet</span><span class="p">,</span> <span class="n">Dict</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="p">(</span><span class="n">CompositeMorphism</span><span class="p">,</span> <span class="n">IdentityMorphism</span><span class="p">,</span>
-                              <span class="n">NamedMorphism</span><span class="p">,</span> <span class="n">Diagram</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">default_sort_key</span>
-<span class="kn">from</span> <span class="nn">itertools</span> <span class="kn">import</span> <span class="n">chain</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">iterable</span>
-<span class="kn">from</span> <span class="nn">sympy.printing</span> <span class="kn">import</span> <span class="n">latex</span>
-
-
-<span class="k">class</span> <span class="nc">_GrowableGrid</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Holds a growable grid of objects.</span>
-
-<span class="sd">    It is possible to append or prepend a row or a column to the grid</span>
-<span class="sd">    using the corresponding methods.  Prepending rows or columns has</span>
-<span class="sd">    the effect of changing the coordinates of the already existing</span>
-<span class="sd">    elements.</span>
-
-<span class="sd">    This class currently represents a naive implementation of the</span>
-<span class="sd">    functionality with little attempt at optimisation.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">width</span><span class="p">,</span> <span class="n">height</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_width</span> <span class="o">=</span> <span class="n">width</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_height</span> <span class="o">=</span> <span class="n">height</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_array</span> <span class="o">=</span> <span class="p">[[</span><span class="bp">None</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">width</span><span class="p">)]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">height</span><span class="p">)]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">width</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_width</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">height</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_height</span>
-
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">xxx_todo_changeme</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the element located at in the i-th line and j-th</span>
-<span class="sd">        column.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="o">=</span> <span class="n">xxx_todo_changeme</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_array</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__setitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">xxx_todo_changeme1</span><span class="p">,</span> <span class="n">newvalue</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Sets the element located at in the i-th line and j-th</span>
-<span class="sd">        column.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="o">=</span> <span class="n">xxx_todo_changeme1</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_array</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">newvalue</span>
-
-    <span class="k">def</span> <span class="nf">append_row</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Appends an empty row to the grid.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_height</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_array</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="bp">None</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_width</span><span class="p">)])</span>
-
-    <span class="k">def</span> <span class="nf">append_column</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Appends an empty column to the grid.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_width</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_height</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_array</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">None</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">prepend_row</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Prepends the grid with an empty row.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_height</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_array</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="p">[</span><span class="bp">None</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_width</span><span class="p">)])</span>
-
-    <span class="k">def</span> <span class="nf">prepend_column</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Prepends the grid with an empty column.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_width</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_height</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_array</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="DiagramGrid"><a class="viewcode-back" href="../../../modules/categories.html#sympy.categories.diagram_drawing.DiagramGrid">[docs]</a><span class="k">class</span> <span class="nc">DiagramGrid</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Constructs and holds the fitting of the diagram into a grid.</span>
-
-<span class="sd">    The mission of this class is to analyse the structure of the</span>
-<span class="sd">    supplied diagram and to place its objects on a grid such that,</span>
-<span class="sd">    when the objects and the morphisms are actually drawn, the diagram</span>
-<span class="sd">    would be &quot;readable&quot;, in the sense that there will not be many</span>
-<span class="sd">    intersections of moprhisms.  This class does not perform any</span>
-<span class="sd">    actual drawing.  It does strive nevertheless to offer sufficient</span>
-<span class="sd">    metadata to draw a diagram.</span>
-
-<span class="sd">    Consider the following simple diagram.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.categories import Object, NamedMorphism</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.categories import Diagram, DiagramGrid</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import pprint</span>
-<span class="sd">    &gt;&gt;&gt; A = Object(&quot;A&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; B = Object(&quot;B&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; C = Object(&quot;C&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; f = NamedMorphism(A, B, &quot;f&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; g = NamedMorphism(B, C, &quot;g&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; diagram = Diagram([f, g])</span>
-
-<span class="sd">    The simplest way to have a diagram laid out is the following:</span>
-
-<span class="sd">    &gt;&gt;&gt; grid = DiagramGrid(diagram)</span>
-<span class="sd">    &gt;&gt;&gt; (grid.width, grid.height)</span>
-<span class="sd">    (2, 2)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(grid)</span>
-<span class="sd">    A  B</span>
-<span class="sd">    &lt;BLANKLINE&gt;</span>
-<span class="sd">       C</span>
-
-<span class="sd">    Sometimes one sees the diagram as consisting of logical groups.</span>
-<span class="sd">    One can advise ``DiagramGrid`` as to such groups by employing the</span>
-<span class="sd">    ``groups`` keyword argument.</span>
-
-<span class="sd">    Consider the following diagram:</span>
-
-<span class="sd">    &gt;&gt;&gt; D = Object(&quot;D&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; f = NamedMorphism(A, B, &quot;f&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; g = NamedMorphism(B, C, &quot;g&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; h = NamedMorphism(D, A, &quot;h&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; k = NamedMorphism(D, B, &quot;k&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; diagram = Diagram([f, g, h, k])</span>
-
-<span class="sd">    Lay it out with generic layout:</span>
-
-<span class="sd">    &gt;&gt;&gt; grid = DiagramGrid(diagram)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(grid)</span>
-<span class="sd">    A  B  D</span>
-<span class="sd">    &lt;BLANKLINE&gt;</span>
-<span class="sd">       C</span>
-
-<span class="sd">    Now, we can group the objects `A` and `D` to have them near one</span>
-<span class="sd">    another:</span>
-
-<span class="sd">    &gt;&gt;&gt; grid = DiagramGrid(diagram, groups=[[A, D], B, C])</span>
-<span class="sd">    &gt;&gt;&gt; pprint(grid)</span>
-<span class="sd">    B     C</span>
-<span class="sd">    &lt;BLANKLINE&gt;</span>
-<span class="sd">    A  D</span>
-
-<span class="sd">    Note how the positioning of the other objects changes.</span>
-
-<span class="sd">    Further indications can be supplied to the constructor of</span>
-<span class="sd">    :class:`DiagramGrid` using keyword arguments.  The currently</span>
-<span class="sd">    supported hints are explained in the following paragraphs.</span>
-
-<span class="sd">    :class:`DiagramGrid` does not automatically guess which layout</span>
-<span class="sd">    would suit the supplied diagram better.  Consider, for example,</span>
-<span class="sd">    the following linear diagram:</span>
-
-<span class="sd">    &gt;&gt;&gt; E = Object(&quot;E&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; f = NamedMorphism(A, B, &quot;f&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; g = NamedMorphism(B, C, &quot;g&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; h = NamedMorphism(C, D, &quot;h&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; i = NamedMorphism(D, E, &quot;i&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; diagram = Diagram([f, g, h, i])</span>
-
-<span class="sd">    When laid out with the generic layout, it does not get to look</span>
-<span class="sd">    linear:</span>
-
-<span class="sd">    &gt;&gt;&gt; grid = DiagramGrid(diagram)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(grid)</span>
-<span class="sd">    A  B</span>
-<span class="sd">    &lt;BLANKLINE&gt;</span>
-<span class="sd">       C  D</span>
-<span class="sd">    &lt;BLANKLINE&gt;</span>
-<span class="sd">          E</span>
-
-<span class="sd">    To get it laid out in a line, use ``layout=&quot;sequential&quot;``:</span>
-
-<span class="sd">    &gt;&gt;&gt; grid = DiagramGrid(diagram, layout=&quot;sequential&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(grid)</span>
-<span class="sd">    A  B  C  D  E</span>
-
-<span class="sd">    One may sometimes need to transpose the resulting layout.  While</span>
-<span class="sd">    this can always be done by hand, :class:`DiagramGrid` provides a</span>
-<span class="sd">    hint for that purpose:</span>
-
-<span class="sd">    &gt;&gt;&gt; grid = DiagramGrid(diagram, layout=&quot;sequential&quot;, transpose=True)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(grid)</span>
-<span class="sd">    A</span>
-<span class="sd">    &lt;BLANKLINE&gt;</span>
-<span class="sd">    B</span>
-<span class="sd">    &lt;BLANKLINE&gt;</span>
-<span class="sd">    C</span>
-<span class="sd">    &lt;BLANKLINE&gt;</span>
-<span class="sd">    D</span>
-<span class="sd">    &lt;BLANKLINE&gt;</span>
-<span class="sd">    E</span>
-
-<span class="sd">    Separate hints can also be provided for each group.  For an</span>
-<span class="sd">    example, refer to ``tests/test_drawing.py``, and see the different</span>
-<span class="sd">    ways in which the five lemma [FiveLemma] can be laid out.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Diagram</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [FiveLemma] http://en.wikipedia.org/wiki/Five_lemma</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_simplify_morphisms</span><span class="p">(</span><span class="n">morphisms</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Given a dictionary mapping morphisms to their properties,</span>
-<span class="sd">        returns a new dictionary in which there are no morphisms which</span>
-<span class="sd">        do not have properties, and which are compositions of other</span>
-<span class="sd">        morphisms included in the dictionary.  Identities are dropped</span>
-<span class="sd">        as well.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">newmorphisms</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">morphism</span><span class="p">,</span> <span class="n">props</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">morphisms</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">morphism</span><span class="p">,</span> <span class="n">CompositeMorphism</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">props</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">morphism</span><span class="p">,</span> <span class="n">IdentityMorphism</span><span class="p">):</span>
-                <span class="k">continue</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">newmorphisms</span><span class="p">[</span><span class="n">morphism</span><span class="p">]</span> <span class="o">=</span> <span class="n">props</span>
-        <span class="k">return</span> <span class="n">newmorphisms</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_merge_premises_conclusions</span><span class="p">(</span><span class="n">premises</span><span class="p">,</span> <span class="n">conclusions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Given two dictionaries of morphisms and their properties,</span>
-<span class="sd">        produces a single dictionary which includes elements from both</span>
-<span class="sd">        dictionaries.  If a morphism has some properties in premises</span>
-<span class="sd">        and also in conclusions, the properties in conclusions take</span>
-<span class="sd">        priority.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">dict</span><span class="p">(</span><span class="n">chain</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">premises</span><span class="o">.</span><span class="n">items</span><span class="p">()),</span> <span class="nb">list</span><span class="p">(</span><span class="n">conclusions</span><span class="o">.</span><span class="n">items</span><span class="p">())))</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_juxtapose_edges</span><span class="p">(</span><span class="n">edge1</span><span class="p">,</span> <span class="n">edge2</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        If ``edge1`` and ``edge2`` have precisely one common endpoint,</span>
-<span class="sd">        returns an edge which would form a triangle with ``edge1`` and</span>
-<span class="sd">        ``edge2``.</span>
-
-<span class="sd">        If ``edge1`` and ``edge2`` don&#39;t have a common endpoint,</span>
-<span class="sd">        returns ``None``.</span>
-
-<span class="sd">        If ``edge1`` and ``edge`` are the same edge, returns ``None``.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">intersection</span> <span class="o">=</span> <span class="n">edge1</span> <span class="o">&amp;</span> <span class="n">edge2</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">intersection</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># The edges either have no common points or are equal.</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="c"># The edges have a common endpoint.  Extract the different</span>
-        <span class="c"># endpoints and set up the new edge.</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">edge1</span> <span class="o">-</span> <span class="n">intersection</span><span class="p">)</span> <span class="o">|</span> <span class="p">(</span><span class="n">edge2</span> <span class="o">-</span> <span class="n">intersection</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_add_edge_append</span><span class="p">(</span><span class="n">dictionary</span><span class="p">,</span> <span class="n">edge</span><span class="p">,</span> <span class="n">elem</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        If ``edge`` is not in ``dictionary``, adds ``edge`` to the</span>
-<span class="sd">        dictionary and sets its value to ``[elem]``.  Otherwise</span>
-<span class="sd">        appends ``elem`` to the value of existing entry.</span>
-
-<span class="sd">        Note that edges are undirected, thus `(A, B) = (B, A)`.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">edge</span> <span class="ow">in</span> <span class="n">dictionary</span><span class="p">:</span>
-            <span class="n">dictionary</span><span class="p">[</span><span class="n">edge</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">elem</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">dictionary</span><span class="p">[</span><span class="n">edge</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">elem</span><span class="p">]</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_build_skeleton</span><span class="p">(</span><span class="n">morphisms</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Creates a dictionary which maps edges to corresponding</span>
-<span class="sd">        morphisms.  Thus for a morphism `f:A\rightarrow B`, the edge</span>
-<span class="sd">        `(A, B)` will be associated with `f`.  This function also adds</span>
-<span class="sd">        to the list those edges which are formed by juxtaposition of</span>
-<span class="sd">        two edges already in the list.  These new edges are not</span>
-<span class="sd">        associated with any morphism and are only added to assure that</span>
-<span class="sd">        the diagram can be decomposed into triangles.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">edges</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="c"># Create edges for morphisms.</span>
-        <span class="k">for</span> <span class="n">morphism</span> <span class="ow">in</span> <span class="n">morphisms</span><span class="p">:</span>
-            <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_add_edge_append</span><span class="p">(</span>
-                <span class="n">edges</span><span class="p">,</span> <span class="nb">frozenset</span><span class="p">([</span><span class="n">morphism</span><span class="o">.</span><span class="n">domain</span><span class="p">,</span> <span class="n">morphism</span><span class="o">.</span><span class="n">codomain</span><span class="p">]),</span> <span class="n">morphism</span><span class="p">)</span>
-
-        <span class="c"># Create new edges by juxtaposing existing edges.</span>
-        <span class="n">edges1</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">edges</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">edges1</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">edges1</span><span class="p">:</span>
-                <span class="n">wv</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_juxtapose_edges</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">wv</span> <span class="ow">and</span> <span class="n">wv</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">edges</span><span class="p">:</span>
-                    <span class="n">edges</span><span class="p">[</span><span class="n">wv</span><span class="p">]</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">return</span> <span class="n">edges</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_list_triangles</span><span class="p">(</span><span class="n">edges</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Builds the set of triangles formed by the supplied edges.  The</span>
-<span class="sd">        triangles are arbitrary and need not be commutative.  A</span>
-<span class="sd">        triangle is a set that contains all three of its sides.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">triangles</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-
-        <span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">edges</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">edges</span><span class="p">:</span>
-                <span class="n">wv</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_juxtapose_edges</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">wv</span> <span class="ow">and</span> <span class="n">wv</span> <span class="ow">in</span> <span class="n">edges</span><span class="p">:</span>
-                    <span class="n">triangles</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="nb">frozenset</span><span class="p">([</span><span class="n">w</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">wv</span><span class="p">]))</span>
-
-        <span class="k">return</span> <span class="n">triangles</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_drop_redundant_triangles</span><span class="p">(</span><span class="n">triangles</span><span class="p">,</span> <span class="n">skeleton</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns a list which contains only those triangles who have</span>
-<span class="sd">        morphisms associated with at least two edges.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">tri</span> <span class="k">for</span> <span class="n">tri</span> <span class="ow">in</span> <span class="n">triangles</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">([</span><span class="n">e</span> <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">tri</span> <span class="k">if</span> <span class="n">skeleton</span><span class="p">[</span><span class="n">e</span><span class="p">]])</span> <span class="o">&gt;=</span> <span class="mi">2</span><span class="p">]</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_morphism_length</span><span class="p">(</span><span class="n">morphism</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the length of a morphism.  The length of a morphism is</span>
-<span class="sd">        the number of components it consists of.  A non-composite</span>
-<span class="sd">        morphism is of length 1.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">morphism</span><span class="p">,</span> <span class="n">CompositeMorphism</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="n">morphism</span><span class="o">.</span><span class="n">components</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">1</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_compute_triangle_min_sizes</span><span class="p">(</span><span class="n">triangles</span><span class="p">,</span> <span class="n">edges</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">        Returns a dictionary mapping triangles to their minimal sizes.</span>
-<span class="sd">        The minimal size of a triangle is the sum of maximal lengths</span>
-<span class="sd">        of morphisms associated to the sides of the triangle.  The</span>
-<span class="sd">        length of a morphism is the number of components it consists</span>
-<span class="sd">        of.  A non-composite morphism is of length 1.</span>
-
-<span class="sd">        Sorting triangles by this metric attempts to address two</span>
-<span class="sd">        aspects of layout.  For triangles with only simple morphisms</span>
-<span class="sd">        in the edge, this assures that triangles with all three edges</span>
-<span class="sd">        visible will get typeset after triangles with less visible</span>
-<span class="sd">        edges, which sometimes minimises the necessity in diagonal</span>
-<span class="sd">        arrows.  For triangles with composite morphisms in the edges,</span>
-<span class="sd">        this assures that objects connected with shorter morphisms</span>
-<span class="sd">        will be laid out first, resulting the visual proximity of</span>
-<span class="sd">        those objects which are connected by shorter morphisms.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">triangle_sizes</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">triangle</span> <span class="ow">in</span> <span class="n">triangles</span><span class="p">:</span>
-            <span class="n">size</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">triangle</span><span class="p">:</span>
-                <span class="n">morphisms</span> <span class="o">=</span> <span class="n">edges</span><span class="p">[</span><span class="n">e</span><span class="p">]</span>
-                <span class="k">if</span> <span class="n">morphisms</span><span class="p">:</span>
-                    <span class="n">size</span> <span class="o">+=</span> <span class="nb">max</span><span class="p">(</span><span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_morphism_length</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-                                <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">morphisms</span><span class="p">)</span>
-            <span class="n">triangle_sizes</span><span class="p">[</span><span class="n">triangle</span><span class="p">]</span> <span class="o">=</span> <span class="n">size</span>
-        <span class="k">return</span> <span class="n">triangle_sizes</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_triangle_objects</span><span class="p">(</span><span class="n">triangle</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Given a triangle, returns the objects included in it.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># A triangle is a frozenset of three two-element frozensets</span>
-        <span class="c"># (the edges).  This chains the three edges together and</span>
-        <span class="c"># creates a frozenset from the iterator, thus producing a</span>
-        <span class="c"># frozenset of objects of the triangle.</span>
-        <span class="k">return</span> <span class="nb">frozenset</span><span class="p">(</span><span class="n">chain</span><span class="p">(</span><span class="o">*</span><span class="nb">tuple</span><span class="p">(</span><span class="n">triangle</span><span class="p">)))</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_other_vertex</span><span class="p">(</span><span class="n">triangle</span><span class="p">,</span> <span class="n">edge</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Given a triangle and an edge of it, returns the vertex which</span>
-<span class="sd">        opposes the edge.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># This gets the set of objects of the triangle and then</span>
-        <span class="c"># subtracts the set of objects employed in ``edge`` to get the</span>
-        <span class="c"># vertex opposite to ``edge``.</span>
-        <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_triangle_objects</span><span class="p">(</span><span class="n">triangle</span><span class="p">)</span> <span class="o">-</span> <span class="nb">set</span><span class="p">(</span><span class="n">edge</span><span class="p">))[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_empty_point</span><span class="p">(</span><span class="n">pt</span><span class="p">,</span> <span class="n">grid</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Checks if the cell at coordinates ``pt`` is either empty or</span>
-<span class="sd">        out of the bounds of the grid.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">pt</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="n">pt</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">)</span> <span class="ow">or</span> \
-           <span class="p">(</span><span class="n">pt</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&gt;=</span> <span class="n">grid</span><span class="o">.</span><span class="n">height</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="n">pt</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">&gt;=</span> <span class="n">grid</span><span class="o">.</span><span class="n">width</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="n">grid</span><span class="p">[</span><span class="n">pt</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_put_object</span><span class="p">(</span><span class="n">coords</span><span class="p">,</span> <span class="n">obj</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">fringe</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Places an object at the coordinate ``cords`` in ``grid``,</span>
-<span class="sd">        growing the grid and updating ``fringe``, if necessary.</span>
-<span class="sd">        Returns (0, 0) if no row or column has been prepended, (1, 0)</span>
-<span class="sd">        if a row was prepended, (0, 1) if a column was prepended and</span>
-<span class="sd">        (1, 1) if both a column and a row were prepended.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="o">=</span> <span class="n">coords</span>
-        <span class="n">offset</span> <span class="o">=</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-            <span class="n">grid</span><span class="o">.</span><span class="n">prepend_row</span><span class="p">()</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="n">offset</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">fringe</span><span class="p">)):</span>
-                <span class="p">((</span><span class="n">i1</span><span class="p">,</span> <span class="n">j1</span><span class="p">),</span> <span class="p">(</span><span class="n">i2</span><span class="p">,</span> <span class="n">j2</span><span class="p">))</span> <span class="o">=</span> <span class="n">fringe</span><span class="p">[</span><span class="n">k</span><span class="p">]</span>
-                <span class="n">fringe</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="p">((</span><span class="n">i1</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j1</span><span class="p">),</span> <span class="p">(</span><span class="n">i2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j2</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="n">i</span> <span class="o">==</span> <span class="n">grid</span><span class="o">.</span><span class="n">height</span><span class="p">:</span>
-            <span class="n">grid</span><span class="o">.</span><span class="n">append_row</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="n">j</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-            <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="n">offset</span> <span class="o">=</span> <span class="p">(</span><span class="n">offset</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="n">grid</span><span class="o">.</span><span class="n">prepend_column</span><span class="p">()</span>
-            <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">fringe</span><span class="p">)):</span>
-                <span class="p">((</span><span class="n">i1</span><span class="p">,</span> <span class="n">j1</span><span class="p">),</span> <span class="p">(</span><span class="n">i2</span><span class="p">,</span> <span class="n">j2</span><span class="p">))</span> <span class="o">=</span> <span class="n">fringe</span><span class="p">[</span><span class="n">k</span><span class="p">]</span>
-                <span class="n">fringe</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="p">((</span><span class="n">i1</span><span class="p">,</span> <span class="n">j1</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="n">i2</span><span class="p">,</span> <span class="n">j2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="n">j</span> <span class="o">==</span> <span class="n">grid</span><span class="o">.</span><span class="n">width</span><span class="p">:</span>
-            <span class="n">grid</span><span class="o">.</span><span class="n">append_column</span><span class="p">()</span>
-
-        <span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">obj</span>
-        <span class="k">return</span> <span class="n">offset</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_choose_target_cell</span><span class="p">(</span><span class="n">pt1</span><span class="p">,</span> <span class="n">pt2</span><span class="p">,</span> <span class="n">edge</span><span class="p">,</span> <span class="n">obj</span><span class="p">,</span> <span class="n">skeleton</span><span class="p">,</span> <span class="n">grid</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Given two points, ``pt1`` and ``pt2``, and the welding edge</span>
-<span class="sd">        ``edge``, chooses one of the two points to place the opposing</span>
-<span class="sd">        vertex ``obj`` of the triangle.  If neither of this points</span>
-<span class="sd">        fits, returns ``None``.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">pt1_empty</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_empty_point</span><span class="p">(</span><span class="n">pt1</span><span class="p">,</span> <span class="n">grid</span><span class="p">)</span>
-        <span class="n">pt2_empty</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_empty_point</span><span class="p">(</span><span class="n">pt2</span><span class="p">,</span> <span class="n">grid</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">pt1_empty</span> <span class="ow">and</span> <span class="n">pt2_empty</span><span class="p">:</span>
-            <span class="c"># Both cells are empty.  Of these two, choose that cell</span>
-            <span class="c"># which will assure that a visible edge of the triangle</span>
-            <span class="c"># will be drawn perpendicularly to the current welding</span>
-            <span class="c"># edge.</span>
-
-            <span class="n">A</span> <span class="o">=</span> <span class="n">grid</span><span class="p">[</span><span class="n">edge</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span>
-            <span class="n">B</span> <span class="o">=</span> <span class="n">grid</span><span class="p">[</span><span class="n">edge</span><span class="p">[</span><span class="mi">1</span><span class="p">]]</span>
-
-            <span class="k">if</span> <span class="n">skeleton</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="nb">frozenset</span><span class="p">([</span><span class="n">A</span><span class="p">,</span> <span class="n">obj</span><span class="p">])):</span>
-                <span class="k">return</span> <span class="n">pt1</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">pt2</span>
-        <span class="k">if</span> <span class="n">pt1_empty</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">pt1</span>
-        <span class="k">elif</span> <span class="n">pt2_empty</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">pt2</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_find_triangle_to_weld</span><span class="p">(</span><span class="n">triangles</span><span class="p">,</span> <span class="n">fringe</span><span class="p">,</span> <span class="n">grid</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Finds, if possible, a triangle and an edge in the fringe to</span>
-<span class="sd">        which the triangle could be attached.  Returns the tuple</span>
-<span class="sd">        containing the triangle and the index of the corresponding</span>
-<span class="sd">        edge in the fringe.</span>
-
-<span class="sd">        This function relies on the fact that objects are unique in</span>
-<span class="sd">        the diagram.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">triangle</span> <span class="ow">in</span> <span class="n">triangles</span><span class="p">:</span>
-            <span class="k">for</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="ow">in</span> <span class="n">fringe</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">frozenset</span><span class="p">([</span><span class="n">grid</span><span class="p">[</span><span class="n">a</span><span class="p">],</span> <span class="n">grid</span><span class="p">[</span><span class="n">b</span><span class="p">]])</span> <span class="ow">in</span> <span class="n">triangle</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="p">(</span><span class="n">triangle</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_weld_triangle</span><span class="p">(</span><span class="n">tri</span><span class="p">,</span> <span class="n">welding_edge</span><span class="p">,</span> <span class="n">fringe</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">skeleton</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        If possible, welds the triangle ``tri`` to ``fringe`` and</span>
-<span class="sd">        returns ``False``.  If this method encounters a degenerate</span>
-<span class="sd">        situation in the fringe and corrects it such that a restart of</span>
-<span class="sd">        the search is required, it returns ``True`` (which means that</span>
-<span class="sd">        a restart in finding triangle weldings is required).</span>
-
-<span class="sd">        A degenerate situation is a situation when an edge listed in</span>
-<span class="sd">        the fringe does not belong to the visual boundary of the</span>
-<span class="sd">        diagram.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">welding_edge</span>
-        <span class="n">target_cell</span> <span class="o">=</span> <span class="bp">None</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_other_vertex</span><span class="p">(</span><span class="n">tri</span><span class="p">,</span> <span class="p">(</span><span class="n">grid</span><span class="p">[</span><span class="n">a</span><span class="p">],</span> <span class="n">grid</span><span class="p">[</span><span class="n">b</span><span class="p">]))</span>
-
-        <span class="c"># We now have a triangle and an edge where it can be welded to</span>
-        <span class="c"># the fringe.  Decide where to place the other vertex of the</span>
-        <span class="c"># triangle and check for degenerate situations en route.</span>
-
-        <span class="k">if</span> <span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">b</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="o">==</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="c"># A diagonal edge.</span>
-            <span class="n">target_cell</span> <span class="o">=</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">b</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-            <span class="k">if</span> <span class="n">grid</span><span class="p">[</span><span class="n">target_cell</span><span class="p">]:</span>
-                <span class="c"># That cell is already occupied.</span>
-                <span class="n">target_cell</span> <span class="o">=</span> <span class="p">(</span><span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-
-                <span class="k">if</span> <span class="n">grid</span><span class="p">[</span><span class="n">target_cell</span><span class="p">]:</span>
-                    <span class="c"># Degenerate situation, this edge is not</span>
-                    <span class="c"># on the actual fringe.  Correct the</span>
-                    <span class="c"># fringe and go on.</span>
-                    <span class="n">fringe</span><span class="o">.</span><span class="n">remove</span><span class="p">((</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">elif</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-            <span class="c"># A horizontal edge.  We first attempt to build the</span>
-            <span class="c"># triangle in the downward direction.</span>
-
-            <span class="n">down_left</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-            <span class="n">down_right</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">b</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-            <span class="n">target_cell</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_choose_target_cell</span><span class="p">(</span>
-                <span class="n">down_left</span><span class="p">,</span> <span class="n">down_right</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">obj</span><span class="p">,</span> <span class="n">skeleton</span><span class="p">,</span> <span class="n">grid</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">target_cell</span><span class="p">:</span>
-                <span class="c"># No room below this edge.  Check above.</span>
-                <span class="n">up_left</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-                <span class="n">up_right</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">b</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-                <span class="n">target_cell</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_choose_target_cell</span><span class="p">(</span>
-                    <span class="n">up_left</span><span class="p">,</span> <span class="n">up_right</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">obj</span><span class="p">,</span> <span class="n">skeleton</span><span class="p">,</span> <span class="n">grid</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">target_cell</span><span class="p">:</span>
-                    <span class="c"># This edge is not in the fringe, remove it</span>
-                    <span class="c"># and restart.</span>
-                    <span class="n">fringe</span><span class="o">.</span><span class="n">remove</span><span class="p">((</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">elif</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">b</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span>
-            <span class="c"># A vertical edge.  We will attempt to place the other</span>
-            <span class="c"># vertex of the triangle to the right of this edge.</span>
-            <span class="n">right_up</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span>
-            <span class="n">right_down</span> <span class="o">=</span> <span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span>
-
-            <span class="n">target_cell</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_choose_target_cell</span><span class="p">(</span>
-                <span class="n">right_up</span><span class="p">,</span> <span class="n">right_down</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">obj</span><span class="p">,</span> <span class="n">skeleton</span><span class="p">,</span> <span class="n">grid</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">target_cell</span><span class="p">:</span>
-                <span class="c"># No room to the left.  See what&#39;s to the right.</span>
-                <span class="n">left_up</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span>
-                <span class="n">left_down</span> <span class="o">=</span> <span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span>
-
-                <span class="n">target_cell</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_choose_target_cell</span><span class="p">(</span>
-                    <span class="n">left_up</span><span class="p">,</span> <span class="n">left_down</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">obj</span><span class="p">,</span> <span class="n">skeleton</span><span class="p">,</span> <span class="n">grid</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">target_cell</span><span class="p">:</span>
-                    <span class="c"># This edge is not in the fringe, remove it</span>
-                    <span class="c"># and restart.</span>
-                    <span class="n">fringe</span><span class="o">.</span><span class="n">remove</span><span class="p">((</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-
-        <span class="c"># We now know where to place the other vertex of the</span>
-        <span class="c"># triangle.</span>
-        <span class="n">offset</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_put_object</span><span class="p">(</span><span class="n">target_cell</span><span class="p">,</span> <span class="n">obj</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">fringe</span><span class="p">)</span>
-
-        <span class="c"># Take care of the displacement of coordinates if a row or</span>
-        <span class="c"># a column was prepended.</span>
-        <span class="n">target_cell</span> <span class="o">=</span> <span class="p">(</span><span class="n">target_cell</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">offset</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span>
-                       <span class="n">target_cell</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">offset</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">offset</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">offset</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="p">(</span><span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">offset</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">b</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">offset</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-
-        <span class="n">fringe</span><span class="o">.</span><span class="n">extend</span><span class="p">([(</span><span class="n">a</span><span class="p">,</span> <span class="n">target_cell</span><span class="p">),</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">target_cell</span><span class="p">)])</span>
-
-        <span class="c"># No restart is required.</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_triangle_key</span><span class="p">(</span><span class="n">tri</span><span class="p">,</span> <span class="n">triangle_sizes</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns a key for the supplied triangle.  It should be the</span>
-<span class="sd">        same independently of the hash randomisation.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">objects</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span>
-            <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_triangle_objects</span><span class="p">(</span><span class="n">tri</span><span class="p">),</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">triangle_sizes</span><span class="p">[</span><span class="n">tri</span><span class="p">],</span> <span class="n">default_sort_key</span><span class="p">(</span><span class="n">objects</span><span class="p">))</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_pick_root_edge</span><span class="p">(</span><span class="n">tri</span><span class="p">,</span> <span class="n">skeleton</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        For a given triangle always picks the same root edge.  The</span>
-<span class="sd">        root edge is the edge that will be placed first on the grid.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">candidates</span> <span class="o">=</span> <span class="p">[</span><span class="nb">sorted</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-                      <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">tri</span> <span class="k">if</span> <span class="n">skeleton</span><span class="p">[</span><span class="n">e</span><span class="p">]]</span>
-        <span class="n">sorted_candidates</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">candidates</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="c"># Don&#39;t forget to assure the proper ordering of the vertices</span>
-        <span class="c"># in this edge.</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="n">sorted_candidates</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">))</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_drop_irrelevant_triangles</span><span class="p">(</span><span class="n">triangles</span><span class="p">,</span> <span class="n">placed_objects</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns only those triangles whose set of objects is not</span>
-<span class="sd">        completely included in ``placed_objects``.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">tri</span> <span class="k">for</span> <span class="n">tri</span> <span class="ow">in</span> <span class="n">triangles</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">placed_objects</span><span class="o">.</span><span class="n">issuperset</span><span class="p">(</span>
-            <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_triangle_objects</span><span class="p">(</span><span class="n">tri</span><span class="p">))]</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_grow_pseudopod</span><span class="p">(</span><span class="n">triangles</span><span class="p">,</span> <span class="n">fringe</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">skeleton</span><span class="p">,</span> <span class="n">placed_objects</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Starting from an object in the existing structure on the grid,</span>
-<span class="sd">        adds an edge to which a triangle from ``triangles`` could be</span>
-<span class="sd">        welded.  If this method has found a way to do so, it returns</span>
-<span class="sd">        the object it has just added.</span>
-
-<span class="sd">        This method should be applied when ``_weld_triangle`` cannot</span>
-<span class="sd">        find weldings any more.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid</span><span class="o">.</span><span class="n">height</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid</span><span class="o">.</span><span class="n">width</span><span class="p">):</span>
-                <span class="n">obj</span> <span class="o">=</span> <span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">obj</span><span class="p">:</span>
-                    <span class="k">continue</span>
-
-                <span class="c"># Here we need to choose a triangle which has only</span>
-                <span class="c"># ``obj`` in common with the existing structure.  The</span>
-                <span class="c"># situations when this is not possible should be</span>
-                <span class="c"># handled elsewhere.</span>
-
-                <span class="k">def</span> <span class="nf">good_triangle</span><span class="p">(</span><span class="n">tri</span><span class="p">):</span>
-                    <span class="n">objs</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_triangle_objects</span><span class="p">(</span><span class="n">tri</span><span class="p">)</span>
-                    <span class="k">return</span> <span class="n">obj</span> <span class="ow">in</span> <span class="n">objs</span> <span class="ow">and</span> \
-                        <span class="n">placed_objects</span> <span class="o">&amp;</span> <span class="p">(</span><span class="n">objs</span> <span class="o">-</span> <span class="nb">set</span><span class="p">([</span><span class="n">obj</span><span class="p">]))</span> <span class="o">==</span> <span class="nb">set</span><span class="p">()</span>
-
-                <span class="n">tris</span> <span class="o">=</span> <span class="p">[</span><span class="n">tri</span> <span class="k">for</span> <span class="n">tri</span> <span class="ow">in</span> <span class="n">triangles</span> <span class="k">if</span> <span class="n">good_triangle</span><span class="p">(</span><span class="n">tri</span><span class="p">)]</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">tris</span><span class="p">:</span>
-                    <span class="c"># This object is not interesting.</span>
-                    <span class="k">continue</span>
-
-                <span class="c"># Pick the &quot;simplest&quot; of the triangles which could be</span>
-                <span class="c"># attached.  Remember that the list of triangles is</span>
-                <span class="c"># sorted according to their &quot;simplicity&quot; (see</span>
-                <span class="c"># _compute_triangle_min_sizes for the metric).</span>
-                <span class="c">#</span>
-                <span class="c"># Note that ``tris`` are sequentially built from</span>
-                <span class="c"># ``triangles``, so we don&#39;t have to worry about hash</span>
-                <span class="c"># randomisation.</span>
-                <span class="n">tri</span> <span class="o">=</span> <span class="n">tris</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-                <span class="c"># We have found a triangle which could be attached to</span>
-                <span class="c"># the existing structure by a vertex.</span>
-
-                <span class="n">candidates</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">([</span><span class="n">e</span> <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">tri</span> <span class="k">if</span> <span class="n">skeleton</span><span class="p">[</span><span class="n">e</span><span class="p">]],</span>
-                                    <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">e</span><span class="p">:</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="n">e</span><span class="p">)</span><span class="o">.</span><span class="n">sort_key</span><span class="p">())</span>
-                <span class="n">edges</span> <span class="o">=</span> <span class="p">[</span><span class="n">e</span> <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">candidates</span> <span class="k">if</span> <span class="n">obj</span> <span class="ow">in</span> <span class="n">e</span><span class="p">]</span>
-
-                <span class="c"># Note that a meaningful edge (i.e., and edge that is</span>
-                <span class="c"># associated with a morphism) containing ``obj``</span>
-                <span class="c"># always exists.  That&#39;s because all triangles are</span>
-                <span class="c"># guaranteed to have at least two meaningful edges.</span>
-                <span class="c"># See _drop_redundant_triangles.</span>
-
-                <span class="c"># Get the object at the other end of the edge.</span>
-                <span class="n">edge</span> <span class="o">=</span> <span class="n">edges</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="n">other_obj</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">edge</span> <span class="o">-</span> <span class="nb">frozenset</span><span class="p">([</span><span class="n">obj</span><span class="p">]))[</span><span class="mi">0</span><span class="p">]</span>
-
-                <span class="c"># Now check for free directions.  When checking for</span>
-                <span class="c"># free directions, prefer the horizontal and vertical</span>
-                <span class="c"># directions.</span>
-                <span class="n">neighbours</span> <span class="o">=</span> <span class="p">[(</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j</span><span class="p">),</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j</span><span class="p">),</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span>
-                              <span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
-
-                <span class="k">for</span> <span class="n">pt</span> <span class="ow">in</span> <span class="n">neighbours</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_empty_point</span><span class="p">(</span><span class="n">pt</span><span class="p">,</span> <span class="n">grid</span><span class="p">):</span>
-                        <span class="c"># We have a found a place to grow the</span>
-                        <span class="c"># pseudopod into.</span>
-                        <span class="n">offset</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_put_object</span><span class="p">(</span>
-                            <span class="n">pt</span><span class="p">,</span> <span class="n">other_obj</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">fringe</span><span class="p">)</span>
-
-                        <span class="n">i</span> <span class="o">+=</span> <span class="n">offset</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                        <span class="n">j</span> <span class="o">+=</span> <span class="n">offset</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-                        <span class="n">pt</span> <span class="o">=</span> <span class="p">(</span><span class="n">pt</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">offset</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">pt</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">offset</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-                        <span class="n">fringe</span><span class="o">.</span><span class="n">append</span><span class="p">(((</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">),</span> <span class="n">pt</span><span class="p">))</span>
-
-                        <span class="k">return</span> <span class="n">other_obj</span>
-
-        <span class="c"># This diagram is actually cooler that I can handle.  Fail cowardly.</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_handle_groups</span><span class="p">(</span><span class="n">diagram</span><span class="p">,</span> <span class="n">groups</span><span class="p">,</span> <span class="n">merged_morphisms</span><span class="p">,</span> <span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Given the slightly preprocessed morphisms of the diagram,</span>
-<span class="sd">        produces a grid laid out according to ``groups``.</span>
-
-<span class="sd">        If a group has hints, it is laid out with those hints only,</span>
-<span class="sd">        without any influence from ``hints``.  Otherwise, it is laid</span>
-<span class="sd">        out with ``hints``.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">def</span> <span class="nf">lay_out_group</span><span class="p">(</span><span class="n">group</span><span class="p">,</span> <span class="n">local_hints</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            If ``group`` is a set of objects, uses a ``DiagramGrid``</span>
-<span class="sd">            to lay it out and returns the grid.  Otherwise returns the</span>
-<span class="sd">            object (i.e., ``group``).  If ``local_hints`` is not</span>
-<span class="sd">            empty, it is supplied to ``DiagramGrid`` as the dictionary</span>
-<span class="sd">            of hints.  Otherwise, the ``hints`` argument of</span>
-<span class="sd">            ``_handle_groups`` is used.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">group</span><span class="p">,</span> <span class="n">FiniteSet</span><span class="p">):</span>
-                <span class="c"># Set up the corresponding object-to-group</span>
-                <span class="c"># mappings.</span>
-                <span class="k">for</span> <span class="n">obj</span> <span class="ow">in</span> <span class="n">group</span><span class="p">:</span>
-                    <span class="n">obj_groups</span><span class="p">[</span><span class="n">obj</span><span class="p">]</span> <span class="o">=</span> <span class="n">group</span>
-
-                <span class="c"># Lay out the current group.</span>
-                <span class="k">if</span> <span class="n">local_hints</span><span class="p">:</span>
-                    <span class="n">groups_grids</span><span class="p">[</span><span class="n">group</span><span class="p">]</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="p">(</span>
-                        <span class="n">diagram</span><span class="o">.</span><span class="n">subdiagram_from_objects</span><span class="p">(</span><span class="n">group</span><span class="p">),</span> <span class="o">**</span><span class="n">local_hints</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">groups_grids</span><span class="p">[</span><span class="n">group</span><span class="p">]</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="p">(</span>
-                        <span class="n">diagram</span><span class="o">.</span><span class="n">subdiagram_from_objects</span><span class="p">(</span><span class="n">group</span><span class="p">),</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">obj_groups</span><span class="p">[</span><span class="n">group</span><span class="p">]</span> <span class="o">=</span> <span class="n">group</span>
-
-        <span class="k">def</span> <span class="nf">group_to_finiteset</span><span class="p">(</span><span class="n">group</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            Converts ``group`` to a :class:``FiniteSet`` if it is an</span>
-<span class="sd">            iterable.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">if</span> <span class="n">iterable</span><span class="p">(</span><span class="n">group</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="n">group</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">group</span>
-
-        <span class="n">obj_groups</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="n">groups_grids</span> <span class="o">=</span> <span class="p">{}</span>
-
-        <span class="c"># We would like to support various containers to represent</span>
-        <span class="c"># groups.  To achieve that, before laying each group out, it</span>
-        <span class="c"># should be converted to a FiniteSet, because that is what the</span>
-        <span class="c"># following code expects.</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">groups</span><span class="p">,</span> <span class="nb">dict</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">groups</span><span class="p">,</span> <span class="n">Dict</span><span class="p">):</span>
-            <span class="n">finiteset_groups</span> <span class="o">=</span> <span class="p">{}</span>
-            <span class="k">for</span> <span class="n">group</span><span class="p">,</span> <span class="n">local_hints</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">groups</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-                <span class="n">finiteset_group</span> <span class="o">=</span> <span class="n">group_to_finiteset</span><span class="p">(</span><span class="n">group</span><span class="p">)</span>
-                <span class="n">finiteset_groups</span><span class="p">[</span><span class="n">finiteset_group</span><span class="p">]</span> <span class="o">=</span> <span class="n">local_hints</span>
-                <span class="n">lay_out_group</span><span class="p">(</span><span class="n">group</span><span class="p">,</span> <span class="n">local_hints</span><span class="p">)</span>
-            <span class="n">groups</span> <span class="o">=</span> <span class="n">finiteset_groups</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">finiteset_groups</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">group</span> <span class="ow">in</span> <span class="n">groups</span><span class="p">:</span>
-                <span class="n">finiteset_group</span> <span class="o">=</span> <span class="n">group_to_finiteset</span><span class="p">(</span><span class="n">group</span><span class="p">)</span>
-                <span class="n">finiteset_groups</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">finiteset_group</span><span class="p">)</span>
-                <span class="n">lay_out_group</span><span class="p">(</span><span class="n">finiteset_group</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-            <span class="n">groups</span> <span class="o">=</span> <span class="n">finiteset_groups</span>
-
-        <span class="n">new_morphisms</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">morphism</span> <span class="ow">in</span> <span class="n">merged_morphisms</span><span class="p">:</span>
-            <span class="n">dom</span> <span class="o">=</span> <span class="n">obj_groups</span><span class="p">[</span><span class="n">morphism</span><span class="o">.</span><span class="n">domain</span><span class="p">]</span>
-            <span class="n">cod</span> <span class="o">=</span> <span class="n">obj_groups</span><span class="p">[</span><span class="n">morphism</span><span class="o">.</span><span class="n">codomain</span><span class="p">]</span>
-            <span class="c"># Note that we are not really interested in morphisms</span>
-            <span class="c"># which do not employ two different groups, because</span>
-            <span class="c"># these do not influence the layout.</span>
-            <span class="k">if</span> <span class="n">dom</span> <span class="o">!=</span> <span class="n">cod</span><span class="p">:</span>
-                <span class="c"># These are essentially unnamed morphisms; they are</span>
-                <span class="c"># not going to mess in the final layout.  By giving</span>
-                <span class="c"># them the same names, we avoid unnecessary</span>
-                <span class="c"># duplicates.</span>
-                <span class="n">new_morphisms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">NamedMorphism</span><span class="p">(</span><span class="n">dom</span><span class="p">,</span> <span class="n">cod</span><span class="p">,</span> <span class="s">&quot;dummy&quot;</span><span class="p">))</span>
-
-        <span class="c"># Lay out the new diagram.  Since these are dummy morphisms,</span>
-        <span class="c"># properties and conclusions are irrelevant.</span>
-        <span class="n">top_grid</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="p">(</span><span class="n">Diagram</span><span class="p">(</span><span class="n">new_morphisms</span><span class="p">))</span>
-
-        <span class="c"># We now have to substitute the groups with the corresponding</span>
-        <span class="c"># grids, laid out at the beginning of this function.  Compute</span>
-        <span class="c"># the size of each row and column in the grid, so that all</span>
-        <span class="c"># nested grids fit.</span>
-
-        <span class="k">def</span> <span class="nf">group_size</span><span class="p">(</span><span class="n">group</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            For the supplied group (or object, eventually), returns</span>
-<span class="sd">            the size of the cell that will hold this group (object).</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">if</span> <span class="n">group</span> <span class="ow">in</span> <span class="n">groups_grids</span><span class="p">:</span>
-                <span class="n">grid</span> <span class="o">=</span> <span class="n">groups_grids</span><span class="p">[</span><span class="n">group</span><span class="p">]</span>
-                <span class="k">return</span> <span class="p">(</span><span class="n">grid</span><span class="o">.</span><span class="n">height</span><span class="p">,</span> <span class="n">grid</span><span class="o">.</span><span class="n">width</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-
-        <span class="n">row_heights</span> <span class="o">=</span> <span class="p">[</span><span class="nb">max</span><span class="p">(</span><span class="n">group_size</span><span class="p">(</span><span class="n">top_grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">])[</span><span class="mi">0</span><span class="p">]</span>
-                           <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">top_grid</span><span class="o">.</span><span class="n">width</span><span class="p">))</span>
-                       <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">top_grid</span><span class="o">.</span><span class="n">height</span><span class="p">)]</span>
-
-        <span class="n">column_widths</span> <span class="o">=</span> <span class="p">[</span><span class="nb">max</span><span class="p">(</span><span class="n">group_size</span><span class="p">(</span><span class="n">top_grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">])[</span><span class="mi">1</span><span class="p">]</span>
-                             <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">top_grid</span><span class="o">.</span><span class="n">height</span><span class="p">))</span>
-                         <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">top_grid</span><span class="o">.</span><span class="n">width</span><span class="p">)]</span>
-
-        <span class="n">grid</span> <span class="o">=</span> <span class="n">_GrowableGrid</span><span class="p">(</span><span class="nb">sum</span><span class="p">(</span><span class="n">column_widths</span><span class="p">),</span> <span class="nb">sum</span><span class="p">(</span><span class="n">row_heights</span><span class="p">))</span>
-
-        <span class="n">real_row</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">real_column</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">logical_row</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">top_grid</span><span class="o">.</span><span class="n">height</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">logical_column</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">top_grid</span><span class="o">.</span><span class="n">width</span><span class="p">):</span>
-                <span class="n">obj</span> <span class="o">=</span> <span class="n">top_grid</span><span class="p">[</span><span class="n">logical_row</span><span class="p">,</span> <span class="n">logical_column</span><span class="p">]</span>
-
-                <span class="k">if</span> <span class="n">obj</span> <span class="ow">in</span> <span class="n">groups_grids</span><span class="p">:</span>
-                    <span class="c"># This is a group.  Copy the corresponding grid in</span>
-                    <span class="c"># place.</span>
-                    <span class="n">local_grid</span> <span class="o">=</span> <span class="n">groups_grids</span><span class="p">[</span><span class="n">obj</span><span class="p">]</span>
-                    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">local_grid</span><span class="o">.</span><span class="n">height</span><span class="p">):</span>
-                        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">local_grid</span><span class="o">.</span><span class="n">width</span><span class="p">):</span>
-                            <span class="n">grid</span><span class="p">[</span><span class="n">real_row</span> <span class="o">+</span> <span class="n">i</span><span class="p">,</span>
-                                <span class="n">real_column</span> <span class="o">+</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">local_grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="c"># This is an object.  Just put it there.</span>
-                    <span class="n">grid</span><span class="p">[</span><span class="n">real_row</span><span class="p">,</span> <span class="n">real_column</span><span class="p">]</span> <span class="o">=</span> <span class="n">obj</span>
-
-                <span class="n">real_column</span> <span class="o">+=</span> <span class="n">column_widths</span><span class="p">[</span><span class="n">logical_column</span><span class="p">]</span>
-            <span class="n">real_column</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="n">real_row</span> <span class="o">+=</span> <span class="n">row_heights</span><span class="p">[</span><span class="n">logical_row</span><span class="p">]</span>
-
-        <span class="k">return</span> <span class="n">grid</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_generic_layout</span><span class="p">(</span><span class="n">diagram</span><span class="p">,</span> <span class="n">merged_morphisms</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Produces the generic layout for the supplied diagram.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">all_objects</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">diagram</span><span class="o">.</span><span class="n">objects</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">all_objects</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># There only one object in the diagram, just put in on 1x1</span>
-            <span class="c"># grid.</span>
-            <span class="n">grid</span> <span class="o">=</span> <span class="n">_GrowableGrid</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">all_objects</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">grid</span>
-
-        <span class="n">skeleton</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_build_skeleton</span><span class="p">(</span><span class="n">merged_morphisms</span><span class="p">)</span>
-
-        <span class="n">grid</span> <span class="o">=</span> <span class="n">_GrowableGrid</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">skeleton</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># This diagram contains only one morphism.  Draw it</span>
-            <span class="c"># horizontally.</span>
-            <span class="n">objects</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">all_objects</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-            <span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">objects</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">objects</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-            <span class="k">return</span> <span class="n">grid</span>
-
-        <span class="n">triangles</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_list_triangles</span><span class="p">(</span><span class="n">skeleton</span><span class="p">)</span>
-        <span class="n">triangles</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_drop_redundant_triangles</span><span class="p">(</span><span class="n">triangles</span><span class="p">,</span> <span class="n">skeleton</span><span class="p">)</span>
-        <span class="n">triangle_sizes</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_compute_triangle_min_sizes</span><span class="p">(</span>
-            <span class="n">triangles</span><span class="p">,</span> <span class="n">skeleton</span><span class="p">)</span>
-
-        <span class="n">triangles</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">triangles</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">tri</span><span class="p">:</span>
-                           <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_triangle_key</span><span class="p">(</span><span class="n">tri</span><span class="p">,</span> <span class="n">triangle_sizes</span><span class="p">))</span>
-
-        <span class="c"># Place the first edge on the grid.</span>
-        <span class="n">root_edge</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_pick_root_edge</span><span class="p">(</span><span class="n">triangles</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">skeleton</span><span class="p">)</span>
-        <span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">root_edge</span>
-        <span class="n">fringe</span> <span class="o">=</span> <span class="p">[((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))]</span>
-
-        <span class="c"># Record which objects we now have on the grid.</span>
-        <span class="n">placed_objects</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">root_edge</span><span class="p">)</span>
-
-        <span class="k">while</span> <span class="n">placed_objects</span> <span class="o">!=</span> <span class="n">all_objects</span><span class="p">:</span>
-            <span class="n">welding</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_find_triangle_to_weld</span><span class="p">(</span>
-                <span class="n">triangles</span><span class="p">,</span> <span class="n">fringe</span><span class="p">,</span> <span class="n">grid</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">welding</span><span class="p">:</span>
-                <span class="p">(</span><span class="n">triangle</span><span class="p">,</span> <span class="n">welding_edge</span><span class="p">)</span> <span class="o">=</span> <span class="n">welding</span>
-
-                <span class="n">restart_required</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_weld_triangle</span><span class="p">(</span>
-                    <span class="n">triangle</span><span class="p">,</span> <span class="n">welding_edge</span><span class="p">,</span> <span class="n">fringe</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">skeleton</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">restart_required</span><span class="p">:</span>
-                    <span class="k">continue</span>
-
-                <span class="n">placed_objects</span><span class="o">.</span><span class="n">update</span><span class="p">(</span>
-                    <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_triangle_objects</span><span class="p">(</span><span class="n">triangle</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># No more weldings found.  Try to attach triangles by</span>
-                <span class="c"># vertices.</span>
-                <span class="n">new_obj</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_grow_pseudopod</span><span class="p">(</span>
-                    <span class="n">triangles</span><span class="p">,</span> <span class="n">fringe</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">skeleton</span><span class="p">,</span> <span class="n">placed_objects</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">new_obj</span><span class="p">:</span>
-                    <span class="c"># No more triangles can be attached, not even by</span>
-                    <span class="c"># the edge.  We will set up a new diagram out of</span>
-                    <span class="c"># what has been left, laid it out independently,</span>
-                    <span class="c"># and then attach it to this one.</span>
-
-                    <span class="n">remaining_objects</span> <span class="o">=</span> <span class="n">all_objects</span> <span class="o">-</span> <span class="n">placed_objects</span>
-
-                    <span class="n">remaining_diagram</span> <span class="o">=</span> <span class="n">diagram</span><span class="o">.</span><span class="n">subdiagram_from_objects</span><span class="p">(</span>
-                        <span class="n">FiniteSet</span><span class="p">(</span><span class="n">remaining_objects</span><span class="p">))</span>
-                    <span class="n">remaining_grid</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="p">(</span><span class="n">remaining_diagram</span><span class="p">)</span>
-
-                    <span class="c"># Now, let&#39;s glue ``remaining_grid`` to ``grid``.</span>
-                    <span class="n">final_width</span> <span class="o">=</span> <span class="n">grid</span><span class="o">.</span><span class="n">width</span> <span class="o">+</span> <span class="n">remaining_grid</span><span class="o">.</span><span class="n">width</span>
-                    <span class="n">final_height</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">grid</span><span class="o">.</span><span class="n">height</span><span class="p">,</span> <span class="n">remaining_grid</span><span class="o">.</span><span class="n">height</span><span class="p">)</span>
-                    <span class="n">final_grid</span> <span class="o">=</span> <span class="n">_GrowableGrid</span><span class="p">(</span><span class="n">final_width</span><span class="p">,</span> <span class="n">final_height</span><span class="p">)</span>
-
-                    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid</span><span class="o">.</span><span class="n">width</span><span class="p">):</span>
-                        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid</span><span class="o">.</span><span class="n">height</span><span class="p">):</span>
-                            <span class="n">final_grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-
-                    <span class="n">start_j</span> <span class="o">=</span> <span class="n">grid</span><span class="o">.</span><span class="n">width</span>
-                    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">remaining_grid</span><span class="o">.</span><span class="n">height</span><span class="p">):</span>
-                        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">remaining_grid</span><span class="o">.</span><span class="n">width</span><span class="p">):</span>
-                            <span class="n">final_grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">start_j</span> <span class="o">+</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">remaining_grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-
-                    <span class="k">return</span> <span class="n">final_grid</span>
-
-                <span class="n">placed_objects</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">new_obj</span><span class="p">)</span>
-
-            <span class="n">triangles</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_drop_irrelevant_triangles</span><span class="p">(</span>
-                <span class="n">triangles</span><span class="p">,</span> <span class="n">placed_objects</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">grid</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_get_undirected_graph</span><span class="p">(</span><span class="n">objects</span><span class="p">,</span> <span class="n">merged_morphisms</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Given the objects and the relevant morphisms of a diagram,</span>
-<span class="sd">        returns the adjacency lists of the underlying undirected</span>
-<span class="sd">        graph.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">adjlists</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">obj</span> <span class="ow">in</span> <span class="n">objects</span><span class="p">:</span>
-            <span class="n">adjlists</span><span class="p">[</span><span class="n">obj</span><span class="p">]</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">morphism</span> <span class="ow">in</span> <span class="n">merged_morphisms</span><span class="p">:</span>
-            <span class="n">adjlists</span><span class="p">[</span><span class="n">morphism</span><span class="o">.</span><span class="n">domain</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">morphism</span><span class="o">.</span><span class="n">codomain</span><span class="p">)</span>
-            <span class="n">adjlists</span><span class="p">[</span><span class="n">morphism</span><span class="o">.</span><span class="n">codomain</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">morphism</span><span class="o">.</span><span class="n">domain</span><span class="p">)</span>
-
-        <span class="c"># Assure that the objects in the adjacency list are always in</span>
-        <span class="c"># the same order.</span>
-        <span class="k">for</span> <span class="n">obj</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">adjlists</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
-            <span class="n">adjlists</span><span class="p">[</span><span class="n">obj</span><span class="p">]</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">adjlists</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_sequential_layout</span><span class="p">(</span><span class="n">diagram</span><span class="p">,</span> <span class="n">merged_morphisms</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">        Lays out the diagram in &quot;sequential&quot; layout.  This method</span>
-<span class="sd">        will attempt to produce a result as close to a line as</span>
-<span class="sd">        possible.  For linear diagrams, the result will actually be a</span>
-<span class="sd">        line.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">objects</span> <span class="o">=</span> <span class="n">diagram</span><span class="o">.</span><span class="n">objects</span>
-        <span class="n">sorted_objects</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">objects</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-
-        <span class="c"># Set up the adjacency lists of the underlying undirected</span>
-        <span class="c"># graph of ``merged_morphisms``.</span>
-        <span class="n">adjlists</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_get_undirected_graph</span><span class="p">(</span><span class="n">objects</span><span class="p">,</span> <span class="n">merged_morphisms</span><span class="p">)</span>
-
-        <span class="c"># Find an object with the minimal degree.  This is going to be</span>
-        <span class="c"># the root.</span>
-        <span class="n">root</span> <span class="o">=</span> <span class="n">sorted_objects</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">mindegree</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">adjlists</span><span class="p">[</span><span class="n">root</span><span class="p">])</span>
-        <span class="k">for</span> <span class="n">obj</span> <span class="ow">in</span> <span class="n">sorted_objects</span><span class="p">:</span>
-            <span class="n">current_degree</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">adjlists</span><span class="p">[</span><span class="n">obj</span><span class="p">])</span>
-            <span class="k">if</span> <span class="n">current_degree</span> <span class="o">&lt;</span> <span class="n">mindegree</span><span class="p">:</span>
-                <span class="n">root</span> <span class="o">=</span> <span class="n">obj</span>
-                <span class="n">mindegree</span> <span class="o">=</span> <span class="n">current_degree</span>
-
-        <span class="n">grid</span> <span class="o">=</span> <span class="n">_GrowableGrid</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">root</span>
-
-        <span class="n">placed_objects</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span><span class="n">root</span><span class="p">])</span>
-
-        <span class="k">def</span> <span class="nf">place_objects</span><span class="p">(</span><span class="n">pt</span><span class="p">,</span> <span class="n">placed_objects</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            Does depth-first search in the underlying graph of the</span>
-<span class="sd">            diagram and places the objects en route.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="c"># We will start placing new objects from here.</span>
-            <span class="n">new_pt</span> <span class="o">=</span> <span class="p">(</span><span class="n">pt</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">pt</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">adjacent_obj</span> <span class="ow">in</span> <span class="n">adjlists</span><span class="p">[</span><span class="n">grid</span><span class="p">[</span><span class="n">pt</span><span class="p">]]:</span>
-                <span class="k">if</span> <span class="n">adjacent_obj</span> <span class="ow">in</span> <span class="n">placed_objects</span><span class="p">:</span>
-                    <span class="c"># This object has already been placed.</span>
-                    <span class="k">continue</span>
-
-                <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_put_object</span><span class="p">(</span><span class="n">new_pt</span><span class="p">,</span> <span class="n">adjacent_obj</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="p">[])</span>
-                <span class="n">placed_objects</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">adjacent_obj</span><span class="p">)</span>
-                <span class="n">placed_objects</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">place_objects</span><span class="p">(</span><span class="n">new_pt</span><span class="p">,</span> <span class="n">placed_objects</span><span class="p">))</span>
-
-                <span class="n">new_pt</span> <span class="o">=</span> <span class="p">(</span><span class="n">new_pt</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">new_pt</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-
-            <span class="k">return</span> <span class="n">placed_objects</span>
-
-        <span class="n">place_objects</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">placed_objects</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">grid</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_drop_inessential_morphisms</span><span class="p">(</span><span class="n">merged_morphisms</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">        Removes those morphisms which should appear in the diagram,</span>
-<span class="sd">        but which have no relevance to object layout.</span>
-
-<span class="sd">        Currently this removes &quot;loop&quot; morphisms: the non-identity</span>
-<span class="sd">        morphisms with the same domains and codomains.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">morphisms</span> <span class="o">=</span> <span class="p">[</span><span class="n">m</span> <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">merged_morphisms</span> <span class="k">if</span> <span class="n">m</span><span class="o">.</span><span class="n">domain</span> <span class="o">!=</span> <span class="n">m</span><span class="o">.</span><span class="n">codomain</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">morphisms</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_get_connected_components</span><span class="p">(</span><span class="n">objects</span><span class="p">,</span> <span class="n">merged_morphisms</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Given a container of morphisms, returns a list of connected</span>
-<span class="sd">        components formed by these morphisms.  A connected component</span>
-<span class="sd">        is represented by a diagram consisting of the corresponding</span>
-<span class="sd">        morphisms.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">component_index</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">objects</span><span class="p">:</span>
-            <span class="n">component_index</span><span class="p">[</span><span class="n">o</span><span class="p">]</span> <span class="o">=</span> <span class="bp">None</span>
-
-        <span class="c"># Get the underlying undirected graph of the diagram.</span>
-        <span class="n">adjlist</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_get_undirected_graph</span><span class="p">(</span><span class="n">objects</span><span class="p">,</span> <span class="n">merged_morphisms</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">traverse_component</span><span class="p">(</span><span class="nb">object</span><span class="p">,</span> <span class="n">current_index</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            Does a depth-first search traversal of the component</span>
-<span class="sd">            containing ``object``.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="n">component_index</span><span class="p">[</span><span class="nb">object</span><span class="p">]</span> <span class="o">=</span> <span class="n">current_index</span>
-            <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">adjlist</span><span class="p">[</span><span class="nb">object</span><span class="p">]:</span>
-                <span class="k">if</span> <span class="n">component_index</span><span class="p">[</span><span class="n">o</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">traverse_component</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">current_index</span><span class="p">)</span>
-
-        <span class="c"># Traverse all components.</span>
-        <span class="n">current_index</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">adjlist</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">component_index</span><span class="p">[</span><span class="n">o</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">traverse_component</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">current_index</span><span class="p">)</span>
-                <span class="n">current_index</span> <span class="o">+=</span> <span class="mi">1</span>
-
-        <span class="c"># List the objects of the components.</span>
-        <span class="n">component_objects</span> <span class="o">=</span> <span class="p">[[]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">current_index</span><span class="p">)]</span>
-        <span class="k">for</span> <span class="n">o</span><span class="p">,</span> <span class="n">idx</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">component_index</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-            <span class="n">component_objects</span><span class="p">[</span><span class="n">idx</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-
-        <span class="c"># Finally, list the morphisms belonging to each component.</span>
-        <span class="c">#</span>
-        <span class="c"># Note: If some objects are isolated, they will not get any</span>
-        <span class="c"># morphisms at this stage, and since the layout algorithm</span>
-        <span class="c"># relies, we are essentially going to lose this object.</span>
-        <span class="c"># Therefore, check if there are isolated objects and, for each</span>
-        <span class="c"># of them, provide the trivial identity morphism.  It will get</span>
-        <span class="c"># discarded later, but the object will be there.</span>
-
-        <span class="n">component_morphisms</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">component</span> <span class="ow">in</span> <span class="n">component_objects</span><span class="p">:</span>
-            <span class="n">current_morphisms</span> <span class="o">=</span> <span class="p">{}</span>
-            <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">merged_morphisms</span><span class="p">:</span>
-                <span class="k">if</span> <span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">domain</span> <span class="ow">in</span> <span class="n">component</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">codomain</span> <span class="ow">in</span> <span class="n">component</span><span class="p">):</span>
-                    <span class="n">current_morphisms</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">=</span> <span class="n">merged_morphisms</span><span class="p">[</span><span class="n">m</span><span class="p">]</span>
-
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">component</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="c"># Let&#39;s add an identity morphism, for the sake of</span>
-                <span class="c"># surely having morphisms in this component.</span>
-                <span class="n">current_morphisms</span><span class="p">[</span><span class="n">IdentityMorphism</span><span class="p">(</span><span class="n">component</span><span class="p">[</span><span class="mi">0</span><span class="p">])]</span> <span class="o">=</span> <span class="n">FiniteSet</span><span class="p">()</span>
-
-            <span class="n">component_morphisms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Diagram</span><span class="p">(</span><span class="n">current_morphisms</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">component_morphisms</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">diagram</span><span class="p">,</span> <span class="n">groups</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="n">premises</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_simplify_morphisms</span><span class="p">(</span><span class="n">diagram</span><span class="o">.</span><span class="n">premises</span><span class="p">)</span>
-        <span class="n">conclusions</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_simplify_morphisms</span><span class="p">(</span><span class="n">diagram</span><span class="o">.</span><span class="n">conclusions</span><span class="p">)</span>
-        <span class="n">all_merged_morphisms</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_merge_premises_conclusions</span><span class="p">(</span>
-            <span class="n">premises</span><span class="p">,</span> <span class="n">conclusions</span><span class="p">)</span>
-        <span class="n">merged_morphisms</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_drop_inessential_morphisms</span><span class="p">(</span>
-            <span class="n">all_merged_morphisms</span><span class="p">)</span>
-
-        <span class="c"># Store the merged morphisms for later use.</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_morphisms</span> <span class="o">=</span> <span class="n">all_merged_morphisms</span>
-
-        <span class="n">components</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_get_connected_components</span><span class="p">(</span>
-            <span class="n">diagram</span><span class="o">.</span><span class="n">objects</span><span class="p">,</span> <span class="n">all_merged_morphisms</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">groups</span> <span class="ow">and</span> <span class="p">(</span><span class="n">groups</span> <span class="o">!=</span> <span class="n">diagram</span><span class="o">.</span><span class="n">objects</span><span class="p">):</span>
-            <span class="c"># Lay out the diagram according to the groups.</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_grid</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_handle_groups</span><span class="p">(</span>
-                <span class="n">diagram</span><span class="p">,</span> <span class="n">groups</span><span class="p">,</span> <span class="n">merged_morphisms</span><span class="p">,</span> <span class="n">hints</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">components</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># Note that we check for connectedness _before_ checking</span>
-            <span class="c"># the layout hints because the layout strategies don&#39;t</span>
-            <span class="c"># know how to deal with disconnected diagrams.</span>
-
-            <span class="c"># The diagram is disconnected.  Lay out the components</span>
-            <span class="c"># independently.</span>
-            <span class="n">grids</span> <span class="o">=</span> <span class="p">[]</span>
-
-            <span class="c"># Sort the components to eventually get the grids arranged</span>
-            <span class="c"># in a fixed, hash-independent order.</span>
-            <span class="n">components</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">components</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">component</span> <span class="ow">in</span> <span class="n">components</span><span class="p">:</span>
-                <span class="n">grid</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="p">(</span><span class="n">component</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-                <span class="n">grids</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">grid</span><span class="p">)</span>
-
-            <span class="c"># Throw the grids together, in a line.</span>
-            <span class="n">total_width</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">width</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">grids</span><span class="p">)</span>
-            <span class="n">total_height</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">height</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">grids</span><span class="p">)</span>
-
-            <span class="n">grid</span> <span class="o">=</span> <span class="n">_GrowableGrid</span><span class="p">(</span><span class="n">total_width</span><span class="p">,</span> <span class="n">total_height</span><span class="p">)</span>
-            <span class="n">start_j</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">grids</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">height</span><span class="p">):</span>
-                    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">width</span><span class="p">):</span>
-                        <span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">start_j</span> <span class="o">+</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-
-                <span class="n">start_j</span> <span class="o">+=</span> <span class="n">g</span><span class="o">.</span><span class="n">width</span>
-
-            <span class="bp">self</span><span class="o">.</span><span class="n">_grid</span> <span class="o">=</span> <span class="n">grid</span>
-        <span class="k">elif</span> <span class="s">&quot;layout&quot;</span> <span class="ow">in</span> <span class="n">hints</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">hints</span><span class="p">[</span><span class="s">&quot;layout&quot;</span><span class="p">]</span> <span class="o">==</span> <span class="s">&quot;sequential&quot;</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_grid</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_sequential_layout</span><span class="p">(</span>
-                    <span class="n">diagram</span><span class="p">,</span> <span class="n">merged_morphisms</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_grid</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="o">.</span><span class="n">_generic_layout</span><span class="p">(</span><span class="n">diagram</span><span class="p">,</span> <span class="n">merged_morphisms</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;transpose&quot;</span><span class="p">):</span>
-            <span class="c"># Transpose the resulting grid.</span>
-            <span class="n">grid</span> <span class="o">=</span> <span class="n">_GrowableGrid</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_grid</span><span class="o">.</span><span class="n">height</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_grid</span><span class="o">.</span><span class="n">width</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_grid</span><span class="o">.</span><span class="n">height</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_grid</span><span class="o">.</span><span class="n">width</span><span class="p">):</span>
-                    <span class="n">grid</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_grid</span> <span class="o">=</span> <span class="n">grid</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="DiagramGrid.width"><a class="viewcode-back" href="../../../modules/categories.html#sympy.categories.diagram_drawing.DiagramGrid.width">[docs]</a>    <span class="k">def</span> <span class="nf">width</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the number of columns in this diagram layout.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.categories import Object, NamedMorphism</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.categories import Diagram, DiagramGrid</span>
-<span class="sd">        &gt;&gt;&gt; A = Object(&quot;A&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; B = Object(&quot;B&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; C = Object(&quot;C&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; f = NamedMorphism(A, B, &quot;f&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; g = NamedMorphism(B, C, &quot;g&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; diagram = Diagram([f, g])</span>
-<span class="sd">        &gt;&gt;&gt; grid = DiagramGrid(diagram)</span>
-<span class="sd">        &gt;&gt;&gt; grid.width</span>
-<span class="sd">        2</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_grid</span><span class="o">.</span><span class="n">width</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="DiagramGrid.height"><a class="viewcode-back" href="../../../modules/categories.html#sympy.categories.diagram_drawing.DiagramGrid.height">[docs]</a>    <span class="k">def</span> <span class="nf">height</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the number of rows in this diagram layout.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.categories import Object, NamedMorphism</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.categories import Diagram, DiagramGrid</span>
-<span class="sd">        &gt;&gt;&gt; A = Object(&quot;A&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; B = Object(&quot;B&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; C = Object(&quot;C&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; f = NamedMorphism(A, B, &quot;f&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; g = NamedMorphism(B, C, &quot;g&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; diagram = Diagram([f, g])</span>
-<span class="sd">        &gt;&gt;&gt; grid = DiagramGrid(diagram)</span>
-<span class="sd">        &gt;&gt;&gt; grid.height</span>
-<span class="sd">        2</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_grid</span><span class="o">.</span><span class="n">height</span>
-</div>
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">xxx_todo_changeme2</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the object placed in the row ``i`` and column ``j``.</span>
-<span class="sd">        The indices are 0-based.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.categories import Object, NamedMorphism</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.categories import Diagram, DiagramGrid</span>
-<span class="sd">        &gt;&gt;&gt; A = Object(&quot;A&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; B = Object(&quot;B&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; C = Object(&quot;C&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; f = NamedMorphism(A, B, &quot;f&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; g = NamedMorphism(B, C, &quot;g&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; diagram = Diagram([f, g])</span>
-<span class="sd">        &gt;&gt;&gt; grid = DiagramGrid(diagram)</span>
-<span class="sd">        &gt;&gt;&gt; (grid[0, 0], grid[0, 1])</span>
-<span class="sd">        (Object(&quot;A&quot;), Object(&quot;B&quot;))</span>
-<span class="sd">        &gt;&gt;&gt; (grid[1, 0], grid[1, 1])</span>
-<span class="sd">        (None, Object(&quot;C&quot;))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="o">=</span> <span class="n">xxx_todo_changeme2</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="DiagramGrid.morphisms"><a class="viewcode-back" href="../../../modules/categories.html#sympy.categories.diagram_drawing.DiagramGrid.morphisms">[docs]</a>    <span class="k">def</span> <span class="nf">morphisms</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns those morphisms (and their properties) which are</span>
-<span class="sd">        sufficiently meaningful to be drawn.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.categories import Object, NamedMorphism</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.categories import Diagram, DiagramGrid</span>
-<span class="sd">        &gt;&gt;&gt; A = Object(&quot;A&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; B = Object(&quot;B&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; C = Object(&quot;C&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; f = NamedMorphism(A, B, &quot;f&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; g = NamedMorphism(B, C, &quot;g&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; diagram = Diagram([f, g])</span>
-<span class="sd">        &gt;&gt;&gt; grid = DiagramGrid(diagram)</span>
-<span class="sd">        &gt;&gt;&gt; grid.morphisms</span>
-<span class="sd">        {NamedMorphism(Object(&quot;A&quot;), Object(&quot;B&quot;), &quot;f&quot;): EmptySet(),</span>
-<span class="sd">        NamedMorphism(Object(&quot;B&quot;), Object(&quot;C&quot;), &quot;g&quot;): EmptySet()}</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_morphisms</span>
-</div>
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Produces a string representation of this class.</span>
-
-<span class="sd">        This method returns a string representation of the underlying</span>
-<span class="sd">        list of lists of objects.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.categories import Object, NamedMorphism</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.categories import Diagram, DiagramGrid</span>
-<span class="sd">        &gt;&gt;&gt; A = Object(&quot;A&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; B = Object(&quot;B&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; C = Object(&quot;C&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; f = NamedMorphism(A, B, &quot;f&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; g = NamedMorphism(B, C, &quot;g&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; diagram = Diagram([f, g])</span>
-<span class="sd">        &gt;&gt;&gt; grid = DiagramGrid(diagram)</span>
-<span class="sd">        &gt;&gt;&gt; print(grid)</span>
-<span class="sd">        [[Object(&quot;A&quot;), Object(&quot;B&quot;)],</span>
-<span class="sd">        [None, Object(&quot;C&quot;)]]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">repr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_grid</span><span class="o">.</span><span class="n">_array</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="ArrowStringDescription"><a class="viewcode-back" href="../../../modules/categories.html#sympy.categories.diagram_drawing.ArrowStringDescription">[docs]</a><span class="k">class</span> <span class="nc">ArrowStringDescription</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Stores the information necessary for producing an Xy-pic</span>
-<span class="sd">    description of an arrow.</span>
-
-<span class="sd">    The principal goal of this class is to abstract away the string</span>
-<span class="sd">    representation of an arrow and to also provide the functionality</span>
-<span class="sd">    to produce the actual Xy-pic string.</span>
-
-<span class="sd">    ``unit`` sets the unit which will be used to specify the amount of</span>
-<span class="sd">    curving and other distances.  ``horizontal_direction`` should be a</span>
-<span class="sd">    string of ``&quot;r&quot;`` or ``&quot;l&quot;`` specifying the horizontal offset of the</span>
-<span class="sd">    target cell of the arrow relatively to the current one.</span>
-<span class="sd">    ``vertical_direction`` should  specify the vertical offset using a</span>
-<span class="sd">    series of either ``&quot;d&quot;`` or ``&quot;u&quot;``.  ``label_position`` should be</span>
-<span class="sd">    either ``&quot;^&quot;``, ``&quot;_&quot;``,  or ``&quot;|&quot;`` to specify that the label should</span>
-<span class="sd">    be positioned above the arrow, below the arrow or just over the arrow,</span>
-<span class="sd">    in a break.  Note that the notions &quot;above&quot; and &quot;below&quot; are relative</span>
-<span class="sd">    to arrow direction.  ``label`` stores the morphism label.</span>
-
-<span class="sd">    This works as follows (disregard the yet unexplained arguments):</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.categories.diagram_drawing import ArrowStringDescription</span>
-<span class="sd">    &gt;&gt;&gt; astr = ArrowStringDescription(</span>
-<span class="sd">    ... unit=&quot;mm&quot;, curving=None, curving_amount=None,</span>
-<span class="sd">    ... looping_start=None, looping_end=None, horizontal_direction=&quot;d&quot;,</span>
-<span class="sd">    ... vertical_direction=&quot;r&quot;, label_position=&quot;_&quot;, label=&quot;f&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; print(str(astr))</span>
-<span class="sd">    \ar[dr]_{f}</span>
-
-<span class="sd">    ``curving`` should be one of ``&quot;^&quot;``, ``&quot;_&quot;`` to specify in which</span>
-<span class="sd">    direction the arrow is going to curve. ``curving_amount`` is a number</span>
-<span class="sd">    describing how many ``unit``&#39;s the morphism is going to curve:</span>
-
-<span class="sd">    &gt;&gt;&gt; astr = ArrowStringDescription(</span>
-<span class="sd">    ... unit=&quot;mm&quot;, curving=&quot;^&quot;, curving_amount=12,</span>
-<span class="sd">    ... looping_start=None, looping_end=None, horizontal_direction=&quot;d&quot;,</span>
-<span class="sd">    ... vertical_direction=&quot;r&quot;, label_position=&quot;_&quot;, label=&quot;f&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; print(str(astr))</span>
-<span class="sd">    \ar@/^12mm/[dr]_{f}</span>
-
-<span class="sd">    ``looping_start`` and ``looping_end`` are currently only used for</span>
-<span class="sd">    loop morphisms, those which have the same domain and codomain.</span>
-<span class="sd">    These two attributes should store a valid Xy-pic direction and</span>
-<span class="sd">    specify, correspondingly, the direction the arrow gets out into</span>
-<span class="sd">    and the direction the arrow gets back from:</span>
-
-<span class="sd">    &gt;&gt;&gt; astr = ArrowStringDescription(</span>
-<span class="sd">    ... unit=&quot;mm&quot;, curving=None, curving_amount=None,</span>
-<span class="sd">    ... looping_start=&quot;u&quot;, looping_end=&quot;l&quot;, horizontal_direction=&quot;&quot;,</span>
-<span class="sd">    ... vertical_direction=&quot;&quot;, label_position=&quot;_&quot;, label=&quot;f&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; print(str(astr))</span>
-<span class="sd">    \ar@(u,l)[]_{f}</span>
-
-<span class="sd">    ``label_displacement`` controls how far the arrow label is from</span>
-<span class="sd">    the ends of the arrow.  For example, to position the arrow label</span>
-<span class="sd">    near the arrow head, use &quot;&gt;&quot;:</span>
-
-<span class="sd">    &gt;&gt;&gt; astr = ArrowStringDescription(</span>
-<span class="sd">    ... unit=&quot;mm&quot;, curving=&quot;^&quot;, curving_amount=12,</span>
-<span class="sd">    ... looping_start=None, looping_end=None, horizontal_direction=&quot;d&quot;,</span>
-<span class="sd">    ... vertical_direction=&quot;r&quot;, label_position=&quot;_&quot;, label=&quot;f&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; astr.label_displacement = &quot;&gt;&quot;</span>
-<span class="sd">    &gt;&gt;&gt; print(str(astr))</span>
-<span class="sd">    \ar@/^12mm/[dr]_&gt;{f}</span>
-
-<span class="sd">    Finally, ``arrow_style`` is used to specify the arrow style.  To</span>
-<span class="sd">    get a dashed arrow, for example, use &quot;{--&gt;}&quot; as arrow style:</span>
-
-<span class="sd">    &gt;&gt;&gt; astr = ArrowStringDescription(</span>
-<span class="sd">    ... unit=&quot;mm&quot;, curving=&quot;^&quot;, curving_amount=12,</span>
-<span class="sd">    ... looping_start=None, looping_end=None, horizontal_direction=&quot;d&quot;,</span>
-<span class="sd">    ... vertical_direction=&quot;r&quot;, label_position=&quot;_&quot;, label=&quot;f&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; astr.arrow_style = &quot;{--&gt;}&quot;</span>
-<span class="sd">    &gt;&gt;&gt; print(str(astr))</span>
-<span class="sd">    \ar@/^12mm/@{--&gt;}[dr]_{f}</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    Instances of :class:`ArrowStringDescription` will be constructed</span>
-<span class="sd">    by :class:`XypicDiagramDrawer` and provided for further use in</span>
-<span class="sd">    formatters.  The user is not expected to construct instances of</span>
-<span class="sd">    :class:`ArrowStringDescription` themselves.</span>
-
-<span class="sd">    To be able to properly utilise this class, the reader is encouraged</span>
-<span class="sd">    to checkout the Xy-pic user guide, available at [Xypic].</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    XypicDiagramDrawer</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [Xypic] http://www.tug.org/applications/Xy-pic/</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">unit</span><span class="p">,</span> <span class="n">curving</span><span class="p">,</span> <span class="n">curving_amount</span><span class="p">,</span> <span class="n">looping_start</span><span class="p">,</span>
-                 <span class="n">looping_end</span><span class="p">,</span> <span class="n">horizontal_direction</span><span class="p">,</span> <span class="n">vertical_direction</span><span class="p">,</span>
-                 <span class="n">label_position</span><span class="p">,</span> <span class="n">label</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">unit</span> <span class="o">=</span> <span class="n">unit</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">curving</span> <span class="o">=</span> <span class="n">curving</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">curving_amount</span> <span class="o">=</span> <span class="n">curving_amount</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">looping_start</span> <span class="o">=</span> <span class="n">looping_start</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">looping_end</span> <span class="o">=</span> <span class="n">looping_end</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">horizontal_direction</span> <span class="o">=</span> <span class="n">horizontal_direction</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">vertical_direction</span> <span class="o">=</span> <span class="n">vertical_direction</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">label_position</span> <span class="o">=</span> <span class="n">label_position</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">label</span> <span class="o">=</span> <span class="n">label</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">label_displacement</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">arrow_style</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-
-        <span class="c"># This flag shows that the position of the label of this</span>
-        <span class="c"># morphism was set while typesetting a curved morphism and</span>
-        <span class="c"># should not be modified later.</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">forced_label_position</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">curving</span><span class="p">:</span>
-            <span class="n">curving_str</span> <span class="o">=</span> <span class="s">&quot;@/</span><span class="si">%s%d%s</span><span class="s">/&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">curving</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">curving_amount</span><span class="p">,</span>
-                                         <span class="bp">self</span><span class="o">.</span><span class="n">unit</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">curving_str</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">looping_start</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">looping_end</span><span class="p">:</span>
-            <span class="n">looping_str</span> <span class="o">=</span> <span class="s">&quot;@(</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">looping_start</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">looping_end</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">looping_str</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">arrow_style</span><span class="p">:</span>
-
-            <span class="n">style_str</span> <span class="o">=</span> <span class="s">&quot;@&quot;</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">arrow_style</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">style_str</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-
-        <span class="k">return</span> <span class="s">&quot;</span><span class="se">\\</span><span class="s">ar</span><span class="si">%s%s%s</span><span class="s">[</span><span class="si">%s%s</span><span class="s">]</span><span class="si">%s%s</span><span class="s">{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> \
-               <span class="p">(</span><span class="n">curving_str</span><span class="p">,</span> <span class="n">looping_str</span><span class="p">,</span> <span class="n">style_str</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">horizontal_direction</span><span class="p">,</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">vertical_direction</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">label_position</span><span class="p">,</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">label_displacement</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="XypicDiagramDrawer"><a class="viewcode-back" href="../../../modules/categories.html#sympy.categories.diagram_drawing.XypicDiagramDrawer">[docs]</a><span class="k">class</span> <span class="nc">XypicDiagramDrawer</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Given a :class:`Diagram` and the corresponding</span>
-<span class="sd">    :class:`DiagramGrid`, produces the Xy-pic representation of the</span>
-<span class="sd">    diagram.</span>
-
-<span class="sd">    The most important method in this class is ``draw``.  Consider the</span>
-<span class="sd">    following triangle diagram:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.categories import Object, NamedMorphism, Diagram</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.categories import DiagramGrid, XypicDiagramDrawer</span>
-<span class="sd">    &gt;&gt;&gt; A = Object(&quot;A&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; B = Object(&quot;B&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; C = Object(&quot;C&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; f = NamedMorphism(A, B, &quot;f&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; g = NamedMorphism(B, C, &quot;g&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; diagram = Diagram([f, g], {g * f: &quot;unique&quot;})</span>
-
-<span class="sd">    To draw this diagram, its objects need to be laid out with a</span>
-<span class="sd">    :class:`DiagramGrid`::</span>
-
-<span class="sd">    &gt;&gt;&gt; grid = DiagramGrid(diagram)</span>
-
-<span class="sd">    Finally, the drawing:</span>
-
-<span class="sd">    &gt;&gt;&gt; drawer = XypicDiagramDrawer()</span>
-<span class="sd">    &gt;&gt;&gt; print(drawer.draw(diagram, grid))</span>
-<span class="sd">    \xymatrix{</span>
-<span class="sd">    A \ar[d]_{g\circ f} \ar[r]^{f} &amp; B \ar[ld]^{g} \\</span>
-<span class="sd">    C &amp;</span>
-<span class="sd">    }</span>
-
-<span class="sd">    For further details see the docstring of this method.</span>
-
-<span class="sd">    To control the appearance of the arrows, formatters are used.  The</span>
-<span class="sd">    dictionary ``arrow_formatters`` maps morphisms to formatter</span>
-<span class="sd">    functions.  A formatter is accepts an</span>
-<span class="sd">    :class:`ArrowStringDescription` and is allowed to modify any of</span>
-<span class="sd">    the arrow properties exposed thereby.  For example, to have all</span>
-<span class="sd">    morphisms with the property ``unique`` appear as dashed arrows,</span>
-<span class="sd">    and to have their names prepended with `\exists !`, the following</span>
-<span class="sd">    should be done:</span>
-
-<span class="sd">    &gt;&gt;&gt; def formatter(astr):</span>
-<span class="sd">    ...   astr.label = &quot;\exists !&quot; + astr.label</span>
-<span class="sd">    ...   astr.arrow_style = &quot;{--&gt;}&quot;</span>
-<span class="sd">    &gt;&gt;&gt; drawer.arrow_formatters[&quot;unique&quot;] = formatter</span>
-<span class="sd">    &gt;&gt;&gt; print(drawer.draw(diagram, grid))</span>
-<span class="sd">    \xymatrix{</span>
-<span class="sd">    A \ar@{--&gt;}[d]_{\exists !g\circ f} \ar[r]^{f} &amp; B \ar[ld]^{g} \\</span>
-<span class="sd">    C &amp;</span>
-<span class="sd">    }</span>
-
-<span class="sd">    To modify the appearance of all arrows in the diagram, set</span>
-<span class="sd">    ``default_arrow_formatter``.  For example, to place all morphism</span>
-<span class="sd">    labels a little bit farther from the arrow head so that they look</span>
-<span class="sd">    more centred, do as follows:</span>
-
-<span class="sd">    &gt;&gt;&gt; def default_formatter(astr):</span>
-<span class="sd">    ...   astr.label_displacement = &quot;(0.45)&quot;</span>
-<span class="sd">    &gt;&gt;&gt; drawer.default_arrow_formatter = default_formatter</span>
-<span class="sd">    &gt;&gt;&gt; print(drawer.draw(diagram, grid))</span>
-<span class="sd">    \xymatrix{</span>
-<span class="sd">    A \ar@{--&gt;}[d]_(0.45){\exists !g\circ f} \ar[r]^(0.45){f} &amp; B \ar[ld]^(0.45){g} \\</span>
-<span class="sd">    C &amp;</span>
-<span class="sd">    }</span>
-
-<span class="sd">    In some diagrams some morphisms are drawn as curved arrows.</span>
-<span class="sd">    Consider the following diagram:</span>
-
-<span class="sd">    &gt;&gt;&gt; D = Object(&quot;D&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; E = Object(&quot;E&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; h = NamedMorphism(D, A, &quot;h&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; k = NamedMorphism(D, B, &quot;k&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; diagram = Diagram([f, g, h, k])</span>
-<span class="sd">    &gt;&gt;&gt; grid = DiagramGrid(diagram)</span>
-<span class="sd">    &gt;&gt;&gt; drawer = XypicDiagramDrawer()</span>
-<span class="sd">    &gt;&gt;&gt; print(drawer.draw(diagram, grid))</span>
-<span class="sd">    \xymatrix{</span>
-<span class="sd">    A \ar[r]_{f} &amp; B \ar[d]^{g} &amp; D \ar[l]^{k} \ar@/_3mm/[ll]_{h} \\</span>
-<span class="sd">    &amp; C &amp;</span>
-<span class="sd">    }</span>
-
-<span class="sd">    To control how far the morphisms are curved by default, one can</span>
-<span class="sd">    use the ``unit`` and ``default_curving_amount`` attributes:</span>
-
-<span class="sd">    &gt;&gt;&gt; drawer.unit = &quot;cm&quot;</span>
-<span class="sd">    &gt;&gt;&gt; drawer.default_curving_amount = 1</span>
-<span class="sd">    &gt;&gt;&gt; print(drawer.draw(diagram, grid))</span>
-<span class="sd">    \xymatrix{</span>
-<span class="sd">    A \ar[r]_{f} &amp; B \ar[d]^{g} &amp; D \ar[l]^{k} \ar@/_1cm/[ll]_{h} \\</span>
-<span class="sd">    &amp; C &amp;</span>
-<span class="sd">    }</span>
-
-<span class="sd">    In some diagrams, there are multiple curved morphisms between the</span>
-<span class="sd">    same two objects.  To control by how much the curving changes</span>
-<span class="sd">    between two such successive morphisms, use</span>
-<span class="sd">    ``default_curving_step``:</span>
-
-<span class="sd">    &gt;&gt;&gt; drawer.default_curving_step = 1</span>
-<span class="sd">    &gt;&gt;&gt; h1 = NamedMorphism(A, D, &quot;h1&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; diagram = Diagram([f, g, h, k, h1])</span>
-<span class="sd">    &gt;&gt;&gt; grid = DiagramGrid(diagram)</span>
-<span class="sd">    &gt;&gt;&gt; print(drawer.draw(diagram, grid))</span>
-<span class="sd">    \xymatrix{</span>
-<span class="sd">    A \ar[r]_{f} \ar@/^1cm/[rr]^{h_{1}} &amp; B \ar[d]^{g} &amp; D \ar[l]^{k} \ar@/_2cm/[ll]_{h} \\</span>
-<span class="sd">    &amp; C &amp;</span>
-<span class="sd">    }</span>
-
-<span class="sd">    The default value of ``default_curving_step`` is 4 units.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    draw, ArrowStringDescription</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">unit</span> <span class="o">=</span> <span class="s">&quot;mm&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">default_curving_amount</span> <span class="o">=</span> <span class="mi">3</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">default_curving_step</span> <span class="o">=</span> <span class="mi">4</span>
-
-        <span class="c"># This dictionary maps properties to the corresponding arrow</span>
-        <span class="c"># formatters.</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">arrow_formatters</span> <span class="o">=</span> <span class="p">{}</span>
-
-        <span class="c"># This is the default arrow formatter which will be applied to</span>
-        <span class="c"># each arrow independently of its properties.</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">default_arrow_formatter</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_process_loop_morphism</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">morphisms_str_info</span><span class="p">,</span> <span class="n">object_coords</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Produces the information required for constructing the string</span>
-<span class="sd">        representation of a loop morphism.  This function is invoked</span>
-<span class="sd">        from ``_process_morphism``.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        _process_morphism</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">curving</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-        <span class="n">label_pos</span> <span class="o">=</span> <span class="s">&quot;^&quot;</span>
-        <span class="n">looping_start</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-        <span class="n">looping_end</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-
-        <span class="c"># This is a loop morphism.  Count how many morphisms stick</span>
-        <span class="c"># in each of the four quadrants.  Note that straight</span>
-        <span class="c"># vertical and horizontal morphisms count in two quadrants</span>
-        <span class="c"># at the same time (i.e., a morphism going up counts both</span>
-        <span class="c"># in the first and the second quadrants).</span>
-
-        <span class="c"># The usual numbering (counterclockwise) of quadrants</span>
-        <span class="c"># applies.</span>
-        <span class="n">quadrant</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-
-        <span class="k">for</span> <span class="n">m</span><span class="p">,</span> <span class="n">m_str_info</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">morphisms_str_info</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">domain</span> <span class="o">==</span> <span class="n">obj</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">codomain</span> <span class="o">==</span> <span class="n">obj</span><span class="p">):</span>
-                <span class="c"># That&#39;s another loop morphism.  Check how it</span>
-                <span class="c"># loops and mark the corresponding quadrants as</span>
-                <span class="c"># busy.</span>
-                <span class="p">(</span><span class="n">l_s</span><span class="p">,</span> <span class="n">l_e</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">m_str_info</span><span class="o">.</span><span class="n">looping_start</span><span class="p">,</span> <span class="n">m_str_info</span><span class="o">.</span><span class="n">looping_end</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="p">(</span><span class="n">l_s</span><span class="p">,</span> <span class="n">l_e</span><span class="p">)</span> <span class="o">==</span> <span class="p">(</span><span class="s">&quot;r&quot;</span><span class="p">,</span> <span class="s">&quot;u&quot;</span><span class="p">):</span>
-                    <span class="n">quadrant</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">elif</span> <span class="p">(</span><span class="n">l_s</span><span class="p">,</span> <span class="n">l_e</span><span class="p">)</span> <span class="o">==</span> <span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">,</span> <span class="s">&quot;l&quot;</span><span class="p">):</span>
-                    <span class="n">quadrant</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">elif</span> <span class="p">(</span><span class="n">l_s</span><span class="p">,</span> <span class="n">l_e</span><span class="p">)</span> <span class="o">==</span> <span class="p">(</span><span class="s">&quot;l&quot;</span><span class="p">,</span> <span class="s">&quot;d&quot;</span><span class="p">):</span>
-                    <span class="n">quadrant</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">elif</span> <span class="p">(</span><span class="n">l_s</span><span class="p">,</span> <span class="n">l_e</span><span class="p">)</span> <span class="o">==</span> <span class="p">(</span><span class="s">&quot;d&quot;</span><span class="p">,</span> <span class="s">&quot;r&quot;</span><span class="p">):</span>
-                    <span class="n">quadrant</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-
-                <span class="k">continue</span>
-            <span class="k">if</span> <span class="n">m</span><span class="o">.</span><span class="n">domain</span> <span class="o">==</span> <span class="n">obj</span><span class="p">:</span>
-                <span class="p">(</span><span class="n">end_i</span><span class="p">,</span> <span class="n">end_j</span><span class="p">)</span> <span class="o">=</span> <span class="n">object_coords</span><span class="p">[</span><span class="n">m</span><span class="o">.</span><span class="n">codomain</span><span class="p">]</span>
-                <span class="n">goes_out</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">elif</span> <span class="n">m</span><span class="o">.</span><span class="n">codomain</span> <span class="o">==</span> <span class="n">obj</span><span class="p">:</span>
-                <span class="p">(</span><span class="n">end_i</span><span class="p">,</span> <span class="n">end_j</span><span class="p">)</span> <span class="o">=</span> <span class="n">object_coords</span><span class="p">[</span><span class="n">m</span><span class="o">.</span><span class="n">domain</span><span class="p">]</span>
-                <span class="n">goes_out</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">continue</span>
-
-            <span class="n">d_i</span> <span class="o">=</span> <span class="n">end_i</span> <span class="o">-</span> <span class="n">i</span>
-            <span class="n">d_j</span> <span class="o">=</span> <span class="n">end_j</span> <span class="o">-</span> <span class="n">j</span>
-            <span class="n">m_curving</span> <span class="o">=</span> <span class="n">m_str_info</span><span class="o">.</span><span class="n">curving</span>
-
-            <span class="k">if</span> <span class="p">(</span><span class="n">d_i</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">d_j</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">):</span>
-                <span class="c"># This is really a diagonal morphism.  Detect the</span>
-                <span class="c"># quadrant.</span>
-                <span class="k">if</span> <span class="p">(</span><span class="n">d_i</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">d_j</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">):</span>
-                    <span class="n">quadrant</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">elif</span> <span class="p">(</span><span class="n">d_i</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">d_j</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">):</span>
-                    <span class="n">quadrant</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">elif</span> <span class="p">(</span><span class="n">d_i</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">d_j</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">):</span>
-                    <span class="n">quadrant</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">elif</span> <span class="p">(</span><span class="n">d_i</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">d_j</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">):</span>
-                    <span class="n">quadrant</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">elif</span> <span class="n">d_i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="c"># Knowing where the other end of the morphism is</span>
-                <span class="c"># and which way it goes, we now have to decide</span>
-                <span class="c"># which quadrant is now the upper one and which is</span>
-                <span class="c"># the lower one.</span>
-                <span class="k">if</span> <span class="n">d_j</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">goes_out</span><span class="p">:</span>
-                        <span class="n">upper_quadrant</span> <span class="o">=</span> <span class="mi">0</span>
-                        <span class="n">lower_quadrant</span> <span class="o">=</span> <span class="mi">3</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">upper_quadrant</span> <span class="o">=</span> <span class="mi">3</span>
-                        <span class="n">lower_quadrant</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">goes_out</span><span class="p">:</span>
-                        <span class="n">upper_quadrant</span> <span class="o">=</span> <span class="mi">2</span>
-                        <span class="n">lower_quadrant</span> <span class="o">=</span> <span class="mi">1</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">upper_quadrant</span> <span class="o">=</span> <span class="mi">1</span>
-                        <span class="n">lower_quadrant</span> <span class="o">=</span> <span class="mi">2</span>
-
-                <span class="k">if</span> <span class="n">m_curving</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">m_curving</span> <span class="o">==</span> <span class="s">&quot;^&quot;</span><span class="p">:</span>
-                        <span class="n">quadrant</span><span class="p">[</span><span class="n">upper_quadrant</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-                    <span class="k">elif</span> <span class="n">m_curving</span> <span class="o">==</span> <span class="s">&quot;_&quot;</span><span class="p">:</span>
-                        <span class="n">quadrant</span><span class="p">[</span><span class="n">lower_quadrant</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="c"># This morphism counts in both upper and lower</span>
-                    <span class="c"># quadrants.</span>
-                    <span class="n">quadrant</span><span class="p">[</span><span class="n">upper_quadrant</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-                    <span class="n">quadrant</span><span class="p">[</span><span class="n">lower_quadrant</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">elif</span> <span class="n">d_j</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="c"># Knowing where the other end of the morphism is</span>
-                <span class="c"># and which way it goes, we now have to decide</span>
-                <span class="c"># which quadrant is now the left one and which is</span>
-                <span class="c"># the right one.</span>
-                <span class="k">if</span> <span class="n">d_i</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">goes_out</span><span class="p">:</span>
-                        <span class="n">left_quadrant</span> <span class="o">=</span> <span class="mi">1</span>
-                        <span class="n">right_quadrant</span> <span class="o">=</span> <span class="mi">0</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">left_quadrant</span> <span class="o">=</span> <span class="mi">0</span>
-                        <span class="n">right_quadrant</span> <span class="o">=</span> <span class="mi">1</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">goes_out</span><span class="p">:</span>
-                        <span class="n">left_quadrant</span> <span class="o">=</span> <span class="mi">3</span>
-                        <span class="n">right_quadrant</span> <span class="o">=</span> <span class="mi">2</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">left_quadrant</span> <span class="o">=</span> <span class="mi">2</span>
-                        <span class="n">right_quadrant</span> <span class="o">=</span> <span class="mi">3</span>
-
-                <span class="k">if</span> <span class="n">m_curving</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">m_curving</span> <span class="o">==</span> <span class="s">&quot;^&quot;</span><span class="p">:</span>
-                        <span class="n">quadrant</span><span class="p">[</span><span class="n">left_quadrant</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-                    <span class="k">elif</span> <span class="n">m_curving</span> <span class="o">==</span> <span class="s">&quot;_&quot;</span><span class="p">:</span>
-                        <span class="n">quadrant</span><span class="p">[</span><span class="n">right_quadrant</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="c"># This morphism counts in both upper and lower</span>
-                    <span class="c"># quadrants.</span>
-                    <span class="n">quadrant</span><span class="p">[</span><span class="n">left_quadrant</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-                    <span class="n">quadrant</span><span class="p">[</span><span class="n">right_quadrant</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-
-        <span class="c"># Pick the freest quadrant to curve our morphism into.</span>
-        <span class="n">freest_quadrant</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">quadrant</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">quadrant</span><span class="p">[</span><span class="n">freest_quadrant</span><span class="p">]:</span>
-                <span class="n">freest_quadrant</span> <span class="o">=</span> <span class="n">i</span>
-
-        <span class="c"># Now set up proper looping.</span>
-        <span class="p">(</span><span class="n">looping_start</span><span class="p">,</span> <span class="n">looping_end</span><span class="p">)</span> <span class="o">=</span> <span class="p">[(</span><span class="s">&quot;r&quot;</span><span class="p">,</span> <span class="s">&quot;u&quot;</span><span class="p">),</span> <span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">,</span> <span class="s">&quot;l&quot;</span><span class="p">),</span> <span class="p">(</span><span class="s">&quot;l&quot;</span><span class="p">,</span> <span class="s">&quot;d&quot;</span><span class="p">),</span>
-                                        <span class="p">(</span><span class="s">&quot;d&quot;</span><span class="p">,</span> <span class="s">&quot;r&quot;</span><span class="p">)][</span><span class="n">freest_quadrant</span><span class="p">]</span>
-
-        <span class="k">return</span> <span class="p">(</span><span class="n">curving</span><span class="p">,</span> <span class="n">label_pos</span><span class="p">,</span> <span class="n">looping_start</span><span class="p">,</span> <span class="n">looping_end</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_process_horizontal_morphism</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">target_j</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">morphisms_str_info</span><span class="p">,</span>
-                                     <span class="n">object_coords</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Produces the information required for constructing the string</span>
-<span class="sd">        representation of a horizontal morphism.  This function is</span>
-<span class="sd">        invoked from ``_process_morphism``.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        _process_morphism</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># The arrow is horizontal.  Check if it goes from left to</span>
-        <span class="c"># right (``backwards == False``) or from right to left</span>
-        <span class="c"># (``backwards == True``).</span>
-        <span class="n">backwards</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="n">start</span> <span class="o">=</span> <span class="n">j</span>
-        <span class="n">end</span> <span class="o">=</span> <span class="n">target_j</span>
-        <span class="k">if</span> <span class="n">end</span> <span class="o">&lt;</span> <span class="n">start</span><span class="p">:</span>
-            <span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="n">end</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">end</span><span class="p">,</span> <span class="n">start</span><span class="p">)</span>
-            <span class="n">backwards</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="c"># Let&#39;s see which objects are there between ``start`` and</span>
-        <span class="c"># ``end``, and then count how many morphisms stick out</span>
-        <span class="c"># upwards, and how many stick out downwards.</span>
-        <span class="c">#</span>
-        <span class="c"># For example, consider the situation:</span>
-        <span class="c">#</span>
-        <span class="c">#    B1 C1</span>
-        <span class="c">#    |  |</span>
-        <span class="c"># A--B--C--D</span>
-        <span class="c">#    |</span>
-        <span class="c">#    B2</span>
-        <span class="c">#</span>
-        <span class="c"># Between the objects `A` and `D` there are two objects:</span>
-        <span class="c"># `B` and `C`.  Further, there are two morphisms which</span>
-        <span class="c"># stick out upward (the ones between `B1` and `B` and</span>
-        <span class="c"># between `C` and `C1`) and one morphism which sticks out</span>
-        <span class="c"># downward (the one between `B and `B2`).</span>
-        <span class="c">#</span>
-        <span class="c"># We need this information to decide how to curve the</span>
-        <span class="c"># arrow between `A` and `D`.  First of all, since there</span>
-        <span class="c"># are two objects between `A` and `D``, we must curve the</span>
-        <span class="c"># arrow.  Then, we will have it curve downward, because</span>
-        <span class="c"># there is more space (less morphisms stick out downward</span>
-        <span class="c"># than upward).</span>
-        <span class="n">up</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">down</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">straight_horizontal</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">start</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">end</span><span class="p">):</span>
-            <span class="n">obj</span> <span class="o">=</span> <span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">obj</span><span class="p">:</span>
-                <span class="k">continue</span>
-
-            <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">morphisms_str_info</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">m</span><span class="o">.</span><span class="n">domain</span> <span class="o">==</span> <span class="n">obj</span><span class="p">:</span>
-                    <span class="p">(</span><span class="n">end_i</span><span class="p">,</span> <span class="n">end_j</span><span class="p">)</span> <span class="o">=</span> <span class="n">object_coords</span><span class="p">[</span><span class="n">m</span><span class="o">.</span><span class="n">codomain</span><span class="p">]</span>
-                <span class="k">elif</span> <span class="n">m</span><span class="o">.</span><span class="n">codomain</span> <span class="o">==</span> <span class="n">obj</span><span class="p">:</span>
-                    <span class="p">(</span><span class="n">end_i</span><span class="p">,</span> <span class="n">end_j</span><span class="p">)</span> <span class="o">=</span> <span class="n">object_coords</span><span class="p">[</span><span class="n">m</span><span class="o">.</span><span class="n">domain</span><span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">continue</span>
-
-                <span class="k">if</span> <span class="n">end_i</span> <span class="o">&gt;</span> <span class="n">i</span><span class="p">:</span>
-                    <span class="n">down</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="n">end_i</span> <span class="o">&lt;</span> <span class="n">i</span><span class="p">:</span>
-                    <span class="n">up</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="ow">not</span> <span class="n">morphisms_str_info</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="o">.</span><span class="n">curving</span><span class="p">:</span>
-                    <span class="c"># This is a straight horizontal morphism,</span>
-                    <span class="c"># because it has no curving.</span>
-                    <span class="n">straight_horizontal</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">up</span><span class="p">)</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">down</span><span class="p">):</span>
-            <span class="c"># More morphisms stick out downward than upward, let&#39;s</span>
-            <span class="c"># curve the morphism up.</span>
-            <span class="k">if</span> <span class="n">backwards</span><span class="p">:</span>
-                <span class="n">curving</span> <span class="o">=</span> <span class="s">&quot;_&quot;</span>
-                <span class="n">label_pos</span> <span class="o">=</span> <span class="s">&quot;_&quot;</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">curving</span> <span class="o">=</span> <span class="s">&quot;^&quot;</span>
-                <span class="n">label_pos</span> <span class="o">=</span> <span class="s">&quot;^&quot;</span>
-
-            <span class="c"># Assure that the straight horizontal morphisms have</span>
-            <span class="c"># their labels on the lower side of the arrow.</span>
-            <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">straight_horizontal</span><span class="p">:</span>
-                <span class="p">(</span><span class="n">i1</span><span class="p">,</span> <span class="n">j1</span><span class="p">)</span> <span class="o">=</span> <span class="n">object_coords</span><span class="p">[</span><span class="n">m</span><span class="o">.</span><span class="n">domain</span><span class="p">]</span>
-                <span class="p">(</span><span class="n">i2</span><span class="p">,</span> <span class="n">j2</span><span class="p">)</span> <span class="o">=</span> <span class="n">object_coords</span><span class="p">[</span><span class="n">m</span><span class="o">.</span><span class="n">codomain</span><span class="p">]</span>
-
-                <span class="n">m_str_info</span> <span class="o">=</span> <span class="n">morphisms_str_info</span><span class="p">[</span><span class="n">m</span><span class="p">]</span>
-                <span class="k">if</span> <span class="n">j1</span> <span class="o">&lt;</span> <span class="n">j2</span><span class="p">:</span>
-                    <span class="n">m_str_info</span><span class="o">.</span><span class="n">label_position</span> <span class="o">=</span> <span class="s">&quot;_&quot;</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">m_str_info</span><span class="o">.</span><span class="n">label_position</span> <span class="o">=</span> <span class="s">&quot;^&quot;</span>
-
-                <span class="c"># Don&#39;t allow any further modifications of the</span>
-                <span class="c"># position of this label.</span>
-                <span class="n">m_str_info</span><span class="o">.</span><span class="n">forced_label_position</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># More morphisms stick out downward than upward, let&#39;s</span>
-            <span class="c"># curve the morphism up.</span>
-            <span class="k">if</span> <span class="n">backwards</span><span class="p">:</span>
-                <span class="n">curving</span> <span class="o">=</span> <span class="s">&quot;^&quot;</span>
-                <span class="n">label_pos</span> <span class="o">=</span> <span class="s">&quot;^&quot;</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">curving</span> <span class="o">=</span> <span class="s">&quot;_&quot;</span>
-                <span class="n">label_pos</span> <span class="o">=</span> <span class="s">&quot;_&quot;</span>
-
-            <span class="c"># Assure that the straight horizontal morphisms have</span>
-            <span class="c"># their labels on the upper side of the arrow.</span>
-            <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">straight_horizontal</span><span class="p">:</span>
-                <span class="p">(</span><span class="n">i1</span><span class="p">,</span> <span class="n">j1</span><span class="p">)</span> <span class="o">=</span> <span class="n">object_coords</span><span class="p">[</span><span class="n">m</span><span class="o">.</span><span class="n">domain</span><span class="p">]</span>
-                <span class="p">(</span><span class="n">i2</span><span class="p">,</span> <span class="n">j2</span><span class="p">)</span> <span class="o">=</span> <span class="n">object_coords</span><span class="p">[</span><span class="n">m</span><span class="o">.</span><span class="n">codomain</span><span class="p">]</span>
-
-                <span class="n">m_str_info</span> <span class="o">=</span> <span class="n">morphisms_str_info</span><span class="p">[</span><span class="n">m</span><span class="p">]</span>
-                <span class="k">if</span> <span class="n">j1</span> <span class="o">&lt;</span> <span class="n">j2</span><span class="p">:</span>
-                    <span class="n">m_str_info</span><span class="o">.</span><span class="n">label_position</span> <span class="o">=</span> <span class="s">&quot;^&quot;</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">m_str_info</span><span class="o">.</span><span class="n">label_position</span> <span class="o">=</span> <span class="s">&quot;_&quot;</span>
-
-                <span class="c"># Don&#39;t allow any further modifications of the</span>
-                <span class="c"># position of this label.</span>
-                <span class="n">m_str_info</span><span class="o">.</span><span class="n">forced_label_position</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="k">return</span> <span class="p">(</span><span class="n">curving</span><span class="p">,</span> <span class="n">label_pos</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_process_vertical_morphism</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">target_i</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">morphisms_str_info</span><span class="p">,</span>
-                                   <span class="n">object_coords</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Produces the information required for constructing the string</span>
-<span class="sd">        representation of a vertical morphism.  This function is</span>
-<span class="sd">        invoked from ``_process_morphism``.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        _process_morphism</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># This arrow is vertical.  Check if it goes from top to</span>
-        <span class="c"># bottom (``backwards == False``) or from bottom to top</span>
-        <span class="c"># (``backwards == True``).</span>
-        <span class="n">backwards</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="n">start</span> <span class="o">=</span> <span class="n">i</span>
-        <span class="n">end</span> <span class="o">=</span> <span class="n">target_i</span>
-        <span class="k">if</span> <span class="n">end</span> <span class="o">&lt;</span> <span class="n">start</span><span class="p">:</span>
-            <span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="n">end</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">end</span><span class="p">,</span> <span class="n">start</span><span class="p">)</span>
-            <span class="n">backwards</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="c"># Let&#39;s see which objects are there between ``start`` and</span>
-        <span class="c"># ``end``, and then count how many morphisms stick out to</span>
-        <span class="c"># the left, and how many stick out to the right.</span>
-        <span class="c">#</span>
-        <span class="c"># See the corresponding comment in the previous branch of</span>
-        <span class="c"># this if-statement for more details.</span>
-        <span class="n">left</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">right</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">straight_vertical</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">start</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">end</span><span class="p">):</span>
-            <span class="n">obj</span> <span class="o">=</span> <span class="n">grid</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">obj</span><span class="p">:</span>
-                <span class="k">continue</span>
-
-            <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">morphisms_str_info</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">m</span><span class="o">.</span><span class="n">domain</span> <span class="o">==</span> <span class="n">obj</span><span class="p">:</span>
-                    <span class="p">(</span><span class="n">end_i</span><span class="p">,</span> <span class="n">end_j</span><span class="p">)</span> <span class="o">=</span> <span class="n">object_coords</span><span class="p">[</span><span class="n">m</span><span class="o">.</span><span class="n">codomain</span><span class="p">]</span>
-                <span class="k">elif</span> <span class="n">m</span><span class="o">.</span><span class="n">codomain</span> <span class="o">==</span> <span class="n">obj</span><span class="p">:</span>
-                    <span class="p">(</span><span class="n">end_i</span><span class="p">,</span> <span class="n">end_j</span><span class="p">)</span> <span class="o">=</span> <span class="n">object_coords</span><span class="p">[</span><span class="n">m</span><span class="o">.</span><span class="n">domain</span><span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">continue</span>
-
-                <span class="k">if</span> <span class="n">end_j</span> <span class="o">&gt;</span> <span class="n">j</span><span class="p">:</span>
-                    <span class="n">right</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="n">end_j</span> <span class="o">&lt;</span> <span class="n">j</span><span class="p">:</span>
-                    <span class="n">left</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="ow">not</span> <span class="n">morphisms_str_info</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="o">.</span><span class="n">curving</span><span class="p">:</span>
-                    <span class="c"># This is a straight vertical morphism,</span>
-                    <span class="c"># because it has no curving.</span>
-                    <span class="n">straight_vertical</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">left</span><span class="p">)</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">right</span><span class="p">):</span>
-            <span class="c"># More morphisms stick out to the left than to the</span>
-            <span class="c"># right, let&#39;s curve the morphism to the right.</span>
-            <span class="k">if</span> <span class="n">backwards</span><span class="p">:</span>
-                <span class="n">curving</span> <span class="o">=</span> <span class="s">&quot;^&quot;</span>
-                <span class="n">label_pos</span> <span class="o">=</span> <span class="s">&quot;^&quot;</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">curving</span> <span class="o">=</span> <span class="s">&quot;_&quot;</span>
-                <span class="n">label_pos</span> <span class="o">=</span> <span class="s">&quot;_&quot;</span>
-
-            <span class="c"># Assure that the straight vertical morphisms have</span>
-            <span class="c"># their labels on the left side of the arrow.</span>
-            <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">straight_vertical</span><span class="p">:</span>
-                <span class="p">(</span><span class="n">i1</span><span class="p">,</span> <span class="n">j1</span><span class="p">)</span> <span class="o">=</span> <span class="n">object_coords</span><span class="p">[</span><span class="n">m</span><span class="o">.</span><span class="n">domain</span><span class="p">]</span>
-                <span class="p">(</span><span class="n">i2</span><span class="p">,</span> <span class="n">j2</span><span class="p">)</span> <span class="o">=</span> <span class="n">object_coords</span><span class="p">[</span><span class="n">m</span><span class="o">.</span><span class="n">codomain</span><span class="p">]</span>
-
-                <span class="n">m_str_info</span> <span class="o">=</span> <span class="n">morphisms_str_info</span><span class="p">[</span><span class="n">m</span><span class="p">]</span>
-                <span class="k">if</span> <span class="n">i1</span> <span class="o">&lt;</span> <span class="n">i2</span><span class="p">:</span>
-                    <span class="n">m_str_info</span><span class="o">.</span><span class="n">label_position</span> <span class="o">=</span> <span class="s">&quot;^&quot;</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">m_str_info</span><span class="o">.</span><span class="n">label_position</span> <span class="o">=</span> <span class="s">&quot;_&quot;</span>
-
-                <span class="c"># Don&#39;t allow any further modifications of the</span>
-                <span class="c"># position of this label.</span>
-                <span class="n">m_str_info</span><span class="o">.</span><span class="n">forced_label_position</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># More morphisms stick out to the right than to the</span>
-            <span class="c"># left, let&#39;s curve the morphism to the left.</span>
-            <span class="k">if</span> <span class="n">backwards</span><span class="p">:</span>
-                <span class="n">curving</span> <span class="o">=</span> <span class="s">&quot;_&quot;</span>
-                <span class="n">label_pos</span> <span class="o">=</span> <span class="s">&quot;_&quot;</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">curving</span> <span class="o">=</span> <span class="s">&quot;^&quot;</span>
-                <span class="n">label_pos</span> <span class="o">=</span> <span class="s">&quot;^&quot;</span>
-
-            <span class="c"># Assure that the straight vertical morphisms have</span>
-            <span class="c"># their labels on the right side of the arrow.</span>
-            <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">straight_vertical</span><span class="p">:</span>
-                <span class="p">(</span><span class="n">i1</span><span class="p">,</span> <span class="n">j1</span><span class="p">)</span> <span class="o">=</span> <span class="n">object_coords</span><span class="p">[</span><span class="n">m</span><span class="o">.</span><span class="n">domain</span><span class="p">]</span>
-                <span class="p">(</span><span class="n">i2</span><span class="p">,</span> <span class="n">j2</span><span class="p">)</span> <span class="o">=</span> <span class="n">object_coords</span><span class="p">[</span><span class="n">m</span><span class="o">.</span><span class="n">codomain</span><span class="p">]</span>
-
-                <span class="n">m_str_info</span> <span class="o">=</span> <span class="n">morphisms_str_info</span><span class="p">[</span><span class="n">m</span><span class="p">]</span>
-                <span class="k">if</span> <span class="n">i1</span> <span class="o">&lt;</span> <span class="n">i2</span><span class="p">:</span>
-                    <span class="n">m_str_info</span><span class="o">.</span><span class="n">label_position</span> <span class="o">=</span> <span class="s">&quot;_&quot;</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">m_str_info</span><span class="o">.</span><span class="n">label_position</span> <span class="o">=</span> <span class="s">&quot;^&quot;</span>
-
-                <span class="c"># Don&#39;t allow any further modifications of the</span>
-                <span class="c"># position of this label.</span>
-                <span class="n">m_str_info</span><span class="o">.</span><span class="n">forced_label_position</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="k">return</span> <span class="p">(</span><span class="n">curving</span><span class="p">,</span> <span class="n">label_pos</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_process_morphism</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">diagram</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">morphism</span><span class="p">,</span> <span class="n">object_coords</span><span class="p">,</span>
-                          <span class="n">morphisms</span><span class="p">,</span> <span class="n">morphisms_str_info</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Given the required information, produces the string</span>
-<span class="sd">        representation of ``morphism``.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">def</span> <span class="nf">repeat_string_cond</span><span class="p">(</span><span class="n">times</span><span class="p">,</span> <span class="n">str_gt</span><span class="p">,</span> <span class="n">str_lt</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            If ``times &gt; 0``, repeats ``str_gt`` ``times`` times.</span>
-<span class="sd">            Otherwise, repeats ``str_lt`` ``-times`` times.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">if</span> <span class="n">times</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">str_gt</span> <span class="o">*</span> <span class="n">times</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">str_lt</span> <span class="o">*</span> <span class="p">(</span><span class="o">-</span><span class="n">times</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">count_morphisms_undirected</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            Counts how many processed morphisms there are between the</span>
-<span class="sd">            two supplied objects.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">return</span> <span class="nb">len</span><span class="p">([</span><span class="n">m</span> <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">morphisms_str_info</span>
-                        <span class="k">if</span> <span class="nb">set</span><span class="p">([</span><span class="n">m</span><span class="o">.</span><span class="n">domain</span><span class="p">,</span> <span class="n">m</span><span class="o">.</span><span class="n">codomain</span><span class="p">])</span> <span class="o">==</span> <span class="nb">set</span><span class="p">([</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">])])</span>
-
-        <span class="k">def</span> <span class="nf">count_morphisms_filtered</span><span class="p">(</span><span class="n">dom</span><span class="p">,</span> <span class="n">cod</span><span class="p">,</span> <span class="n">curving</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            Counts the processed morphisms which go out of ``dom``</span>
-<span class="sd">            into ``cod`` with curving ``curving``.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">return</span> <span class="nb">len</span><span class="p">([</span><span class="n">m</span> <span class="k">for</span> <span class="n">m</span><span class="p">,</span> <span class="n">m_str_info</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">morphisms_str_info</span><span class="o">.</span><span class="n">items</span><span class="p">())</span>
-                        <span class="k">if</span> <span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">domain</span><span class="p">,</span> <span class="n">m</span><span class="o">.</span><span class="n">codomain</span><span class="p">)</span> <span class="o">==</span> <span class="p">(</span><span class="n">dom</span><span class="p">,</span> <span class="n">cod</span><span class="p">)</span> <span class="ow">and</span>
-                        <span class="p">(</span><span class="n">m_str_info</span><span class="o">.</span><span class="n">curving</span> <span class="o">==</span> <span class="n">curving</span><span class="p">)])</span>
-
-        <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="o">=</span> <span class="n">object_coords</span><span class="p">[</span><span class="n">morphism</span><span class="o">.</span><span class="n">domain</span><span class="p">]</span>
-        <span class="p">(</span><span class="n">target_i</span><span class="p">,</span> <span class="n">target_j</span><span class="p">)</span> <span class="o">=</span> <span class="n">object_coords</span><span class="p">[</span><span class="n">morphism</span><span class="o">.</span><span class="n">codomain</span><span class="p">]</span>
-
-        <span class="c"># We now need to determine the direction of</span>
-        <span class="c"># the arrow.</span>
-        <span class="n">delta_i</span> <span class="o">=</span> <span class="n">target_i</span> <span class="o">-</span> <span class="n">i</span>
-        <span class="n">delta_j</span> <span class="o">=</span> <span class="n">target_j</span> <span class="o">-</span> <span class="n">j</span>
-        <span class="n">vertical_direction</span> <span class="o">=</span> <span class="n">repeat_string_cond</span><span class="p">(</span><span class="n">delta_i</span><span class="p">,</span>
-                                                <span class="s">&quot;d&quot;</span><span class="p">,</span> <span class="s">&quot;u&quot;</span><span class="p">)</span>
-        <span class="n">horizontal_direction</span> <span class="o">=</span> <span class="n">repeat_string_cond</span><span class="p">(</span><span class="n">delta_j</span><span class="p">,</span>
-                                                  <span class="s">&quot;r&quot;</span><span class="p">,</span> <span class="s">&quot;l&quot;</span><span class="p">)</span>
-
-        <span class="n">curving</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-        <span class="n">label_pos</span> <span class="o">=</span> <span class="s">&quot;^&quot;</span>
-        <span class="n">looping_start</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-        <span class="n">looping_end</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="p">(</span><span class="n">delta_i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">delta_j</span> <span class="o">==</span> <span class="mi">0</span><span class="p">):</span>
-            <span class="c"># This is a loop morphism.</span>
-            <span class="p">(</span><span class="n">curving</span><span class="p">,</span> <span class="n">label_pos</span><span class="p">,</span> <span class="n">looping_start</span><span class="p">,</span>
-             <span class="n">looping_end</span><span class="p">)</span> <span class="o">=</span> <span class="n">XypicDiagramDrawer</span><span class="o">.</span><span class="n">_process_loop_morphism</span><span class="p">(</span>
-                 <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">morphisms_str_info</span><span class="p">,</span> <span class="n">object_coords</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="p">(</span><span class="n">delta_i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">j</span> <span class="o">-</span> <span class="n">target_j</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="c"># This is a horizontal morphism.</span>
-            <span class="p">(</span><span class="n">curving</span><span class="p">,</span> <span class="n">label_pos</span><span class="p">)</span> <span class="o">=</span> <span class="n">XypicDiagramDrawer</span><span class="o">.</span><span class="n">_process_horizontal_morphism</span><span class="p">(</span>
-                <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">target_j</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">morphisms_str_info</span><span class="p">,</span> <span class="n">object_coords</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="p">(</span><span class="n">delta_j</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="n">target_i</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="c"># This is a vertical morphism.</span>
-            <span class="p">(</span><span class="n">curving</span><span class="p">,</span> <span class="n">label_pos</span><span class="p">)</span> <span class="o">=</span> <span class="n">XypicDiagramDrawer</span><span class="o">.</span><span class="n">_process_vertical_morphism</span><span class="p">(</span>
-                <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">target_i</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">morphisms_str_info</span><span class="p">,</span> <span class="n">object_coords</span><span class="p">)</span>
-
-        <span class="n">count</span> <span class="o">=</span> <span class="n">count_morphisms_undirected</span><span class="p">(</span><span class="n">morphism</span><span class="o">.</span><span class="n">domain</span><span class="p">,</span> <span class="n">morphism</span><span class="o">.</span><span class="n">codomain</span><span class="p">)</span>
-        <span class="n">curving_amount</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">curving</span><span class="p">:</span>
-            <span class="c"># This morphisms should be curved anyway.</span>
-            <span class="n">curving_amount</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">default_curving_amount</span> <span class="o">+</span> <span class="n">count</span> <span class="o">*</span> \
-                <span class="bp">self</span><span class="o">.</span><span class="n">default_curving_step</span>
-        <span class="k">elif</span> <span class="n">count</span><span class="p">:</span>
-            <span class="c"># There are no objects between the domain and codomain of</span>
-            <span class="c"># the current morphism, but this is not there already are</span>
-            <span class="c"># some morphisms with the same domain and codomain, so we</span>
-            <span class="c"># have to curve this one.</span>
-            <span class="n">curving</span> <span class="o">=</span> <span class="s">&quot;^&quot;</span>
-            <span class="n">filtered_morphisms</span> <span class="o">=</span> <span class="n">count_morphisms_filtered</span><span class="p">(</span>
-                <span class="n">morphism</span><span class="o">.</span><span class="n">domain</span><span class="p">,</span> <span class="n">morphism</span><span class="o">.</span><span class="n">codomain</span><span class="p">,</span> <span class="n">curving</span><span class="p">)</span>
-            <span class="n">curving_amount</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">default_curving_amount</span> <span class="o">+</span> \
-                <span class="n">filtered_morphisms</span> <span class="o">*</span> \
-                <span class="bp">self</span><span class="o">.</span><span class="n">default_curving_step</span>
-
-        <span class="c"># Let&#39;s now get the name of the morphism.</span>
-        <span class="n">morphism_name</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">morphism</span><span class="p">,</span> <span class="n">IdentityMorphism</span><span class="p">):</span>
-            <span class="n">morphism_name</span> <span class="o">=</span> <span class="s">&quot;id_{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">+</span> <span class="n">latex</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">morphism</span><span class="p">,</span> <span class="n">CompositeMorphism</span><span class="p">):</span>
-            <span class="n">component_names</span> <span class="o">=</span> <span class="p">[</span><span class="n">latex</span><span class="p">(</span><span class="n">Symbol</span><span class="p">(</span><span class="n">component</span><span class="o">.</span><span class="n">name</span><span class="p">))</span> <span class="k">for</span>
-                               <span class="n">component</span> <span class="ow">in</span> <span class="n">morphism</span><span class="o">.</span><span class="n">components</span><span class="p">]</span>
-            <span class="n">component_names</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-            <span class="n">morphism_name</span> <span class="o">=</span> <span class="s">&quot;</span><span class="se">\\</span><span class="s">circ &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">component_names</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">morphism</span><span class="p">,</span> <span class="n">NamedMorphism</span><span class="p">):</span>
-            <span class="n">morphism_name</span> <span class="o">=</span> <span class="n">latex</span><span class="p">(</span><span class="n">Symbol</span><span class="p">(</span><span class="n">morphism</span><span class="o">.</span><span class="n">name</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">ArrowStringDescription</span><span class="p">(</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">unit</span><span class="p">,</span> <span class="n">curving</span><span class="p">,</span> <span class="n">curving_amount</span><span class="p">,</span> <span class="n">looping_start</span><span class="p">,</span>
-            <span class="n">looping_end</span><span class="p">,</span> <span class="n">horizontal_direction</span><span class="p">,</span> <span class="n">vertical_direction</span><span class="p">,</span>
-            <span class="n">label_pos</span><span class="p">,</span> <span class="n">morphism_name</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_check_free_space_horizontal</span><span class="p">(</span><span class="n">dom_i</span><span class="p">,</span> <span class="n">dom_j</span><span class="p">,</span> <span class="n">cod_j</span><span class="p">,</span> <span class="n">grid</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        For a horizontal morphism, checks whether there is free space</span>
-<span class="sd">        (i.e., space not occupied by any objects) above the morphism</span>
-<span class="sd">        or below it.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">dom_j</span> <span class="o">&lt;</span> <span class="n">cod_j</span><span class="p">:</span>
-            <span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="n">end</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">dom_j</span><span class="p">,</span> <span class="n">cod_j</span><span class="p">)</span>
-            <span class="n">backwards</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="n">end</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">cod_j</span><span class="p">,</span> <span class="n">dom_j</span><span class="p">)</span>
-            <span class="n">backwards</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="c"># Check for free space above.</span>
-        <span class="k">if</span> <span class="n">dom_i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">free_up</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">free_up</span> <span class="o">=</span> <span class="nb">all</span><span class="p">([</span><span class="n">grid</span><span class="p">[</span><span class="n">dom_i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span>
-                           <span class="nb">range</span><span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="n">end</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)])</span>
-
-        <span class="c"># Check for free space below.</span>
-        <span class="k">if</span> <span class="n">dom_i</span> <span class="o">==</span> <span class="n">grid</span><span class="o">.</span><span class="n">height</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">free_down</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">free_down</span> <span class="o">=</span> <span class="nb">all</span><span class="p">([</span><span class="ow">not</span> <span class="n">grid</span><span class="p">[</span><span class="n">dom_i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span>
-                             <span class="nb">range</span><span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="n">end</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)])</span>
-
-        <span class="k">return</span> <span class="p">(</span><span class="n">free_up</span><span class="p">,</span> <span class="n">free_down</span><span class="p">,</span> <span class="n">backwards</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_check_free_space_vertical</span><span class="p">(</span><span class="n">dom_i</span><span class="p">,</span> <span class="n">cod_i</span><span class="p">,</span> <span class="n">dom_j</span><span class="p">,</span> <span class="n">grid</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        For a vertical morphism, checks whether there is free space</span>
-<span class="sd">        (i.e., space not occupied by any objects) to the left of the</span>
-<span class="sd">        morphism or to the right of it.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">dom_i</span> <span class="o">&lt;</span> <span class="n">cod_i</span><span class="p">:</span>
-            <span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="n">end</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">dom_i</span><span class="p">,</span> <span class="n">cod_i</span><span class="p">)</span>
-            <span class="n">backwards</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="n">end</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">cod_i</span><span class="p">,</span> <span class="n">dom_i</span><span class="p">)</span>
-            <span class="n">backwards</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="c"># Check if there&#39;s space to the left.</span>
-        <span class="k">if</span> <span class="n">dom_j</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">free_left</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">free_left</span> <span class="o">=</span> <span class="nb">all</span><span class="p">([</span><span class="ow">not</span> <span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">dom_j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span>
-                             <span class="nb">range</span><span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="n">end</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)])</span>
-
-        <span class="k">if</span> <span class="n">dom_j</span> <span class="o">==</span> <span class="n">grid</span><span class="o">.</span><span class="n">width</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">free_right</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">free_right</span> <span class="o">=</span> <span class="nb">all</span><span class="p">([</span><span class="ow">not</span> <span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">dom_j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span>
-                              <span class="nb">range</span><span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="n">end</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)])</span>
-
-        <span class="k">return</span> <span class="p">(</span><span class="n">free_left</span><span class="p">,</span> <span class="n">free_right</span><span class="p">,</span> <span class="n">backwards</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_check_free_space_diagonal</span><span class="p">(</span><span class="n">dom_i</span><span class="p">,</span> <span class="n">cod_i</span><span class="p">,</span> <span class="n">dom_j</span><span class="p">,</span> <span class="n">cod_j</span><span class="p">,</span> <span class="n">grid</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        For a diagonal morphism, checks whether there is free space</span>
-<span class="sd">        (i.e., space not occupied by any objects) above the morphism</span>
-<span class="sd">        or below it.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">def</span> <span class="nf">abs_xrange</span><span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="n">end</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">start</span> <span class="o">&lt;</span> <span class="n">end</span><span class="p">:</span>
-                <span class="k">return</span> <span class="nb">range</span><span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="n">end</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="nb">range</span><span class="p">(</span><span class="n">end</span><span class="p">,</span> <span class="n">start</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">dom_i</span> <span class="o">&lt;</span> <span class="n">cod_i</span> <span class="ow">and</span> <span class="n">dom_j</span> <span class="o">&lt;</span> <span class="n">cod_j</span><span class="p">:</span>
-            <span class="c"># This morphism goes from top-left to</span>
-            <span class="c"># bottom-right.</span>
-            <span class="p">(</span><span class="n">start_i</span><span class="p">,</span> <span class="n">start_j</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">dom_i</span><span class="p">,</span> <span class="n">dom_j</span><span class="p">)</span>
-            <span class="p">(</span><span class="n">end_i</span><span class="p">,</span> <span class="n">end_j</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">cod_i</span><span class="p">,</span> <span class="n">cod_j</span><span class="p">)</span>
-            <span class="n">backwards</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="n">dom_i</span> <span class="o">&gt;</span> <span class="n">cod_i</span> <span class="ow">and</span> <span class="n">dom_j</span> <span class="o">&gt;</span> <span class="n">cod_j</span><span class="p">:</span>
-            <span class="c"># This morphism goes from bottom-right to</span>
-            <span class="c"># top-left.</span>
-            <span class="p">(</span><span class="n">start_i</span><span class="p">,</span> <span class="n">start_j</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">cod_i</span><span class="p">,</span> <span class="n">cod_j</span><span class="p">)</span>
-            <span class="p">(</span><span class="n">end_i</span><span class="p">,</span> <span class="n">end_j</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">dom_i</span><span class="p">,</span> <span class="n">dom_j</span><span class="p">)</span>
-            <span class="n">backwards</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="n">dom_i</span> <span class="o">&lt;</span> <span class="n">cod_i</span> <span class="ow">and</span> <span class="n">dom_j</span> <span class="o">&gt;</span> <span class="n">cod_j</span><span class="p">:</span>
-            <span class="c"># This morphism goes from top-right to</span>
-            <span class="c"># bottom-left.</span>
-            <span class="p">(</span><span class="n">start_i</span><span class="p">,</span> <span class="n">start_j</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">dom_i</span><span class="p">,</span> <span class="n">dom_j</span><span class="p">)</span>
-            <span class="p">(</span><span class="n">end_i</span><span class="p">,</span> <span class="n">end_j</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">cod_i</span><span class="p">,</span> <span class="n">cod_j</span><span class="p">)</span>
-            <span class="n">backwards</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">elif</span> <span class="n">dom_i</span> <span class="o">&gt;</span> <span class="n">cod_i</span> <span class="ow">and</span> <span class="n">dom_j</span> <span class="o">&lt;</span> <span class="n">cod_j</span><span class="p">:</span>
-            <span class="c"># This morphism goes from bottom-left to</span>
-            <span class="c"># top-right.</span>
-            <span class="p">(</span><span class="n">start_i</span><span class="p">,</span> <span class="n">start_j</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">cod_i</span><span class="p">,</span> <span class="n">cod_j</span><span class="p">)</span>
-            <span class="p">(</span><span class="n">end_i</span><span class="p">,</span> <span class="n">end_j</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">dom_i</span><span class="p">,</span> <span class="n">dom_j</span><span class="p">)</span>
-            <span class="n">backwards</span> <span class="o">=</span> <span class="bp">False</span>
-
-        <span class="c"># This is an attempt at a fast and furious strategy to</span>
-        <span class="c"># decide where there is free space on the two sides of</span>
-        <span class="c"># a diagonal morphism.  For a diagonal morphism</span>
-        <span class="c"># starting at ``(start_i, start_j)`` and ending at</span>
-        <span class="c"># ``(end_i, end_j)`` the rectangle defined by these</span>
-        <span class="c"># two points is considered.  The slope of the diagonal</span>
-        <span class="c"># ``alpha`` is then computed.  Then, for every cell</span>
-        <span class="c"># ``(i, j)`` within the rectangle, the slope</span>
-        <span class="c"># ``alpha1`` of the line through ``(start_i,</span>
-        <span class="c"># start_j)`` and ``(i, j)`` is considered.  If</span>
-        <span class="c"># ``alpha1`` is between 0 and ``alpha``, the point</span>
-        <span class="c"># ``(i, j)`` is above the diagonal, if ``alpha1`` is</span>
-        <span class="c"># between ``alpha`` and infinity, the point is below</span>
-        <span class="c"># the diagonal.  Also note that, with some beforehand</span>
-        <span class="c"># precautions, this trick works for both the main and</span>
-        <span class="c"># the secondary diagonals of the rectangle.</span>
-
-        <span class="c"># I have considered the possibility to only follow the</span>
-        <span class="c"># shorter diagonals immediately above and below the</span>
-        <span class="c"># main (or secondary) diagonal.  This, however,</span>
-        <span class="c"># wouldn&#39;t have resulted in much performance gain or</span>
-        <span class="c"># better detection of outer edges, because of</span>
-        <span class="c"># relatively small sizes of diagram grids, while the</span>
-        <span class="c"># code would have become harder to understand.</span>
-
-        <span class="n">alpha</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">end_i</span> <span class="o">-</span> <span class="n">start_i</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">end_j</span> <span class="o">-</span> <span class="n">start_j</span><span class="p">)</span>
-        <span class="n">free_up</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="n">free_down</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">abs_xrange</span><span class="p">(</span><span class="n">start_i</span><span class="p">,</span> <span class="n">end_i</span><span class="p">):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">free_up</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">free_down</span><span class="p">:</span>
-                <span class="k">break</span>
-
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">abs_xrange</span><span class="p">(</span><span class="n">start_j</span><span class="p">,</span> <span class="n">end_j</span><span class="p">):</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">free_up</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">free_down</span><span class="p">:</span>
-                    <span class="k">break</span>
-
-                <span class="k">if</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="o">==</span> <span class="p">(</span><span class="n">start_i</span><span class="p">,</span> <span class="n">start_j</span><span class="p">):</span>
-                    <span class="k">continue</span>
-
-                <span class="k">if</span> <span class="n">j</span> <span class="o">==</span> <span class="n">start_j</span><span class="p">:</span>
-                    <span class="n">alpha1</span> <span class="o">=</span> <span class="s">&quot;inf&quot;</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">alpha1</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="n">start_i</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">j</span> <span class="o">-</span> <span class="n">start_j</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]:</span>
-                    <span class="k">if</span> <span class="p">(</span><span class="n">alpha1</span> <span class="o">==</span> <span class="s">&quot;inf&quot;</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">alpha1</span><span class="p">)</span> <span class="o">&gt;</span> <span class="nb">abs</span><span class="p">(</span><span class="n">alpha</span><span class="p">)):</span>
-                        <span class="n">free_down</span> <span class="o">=</span> <span class="bp">False</span>
-                    <span class="k">elif</span> <span class="nb">abs</span><span class="p">(</span><span class="n">alpha1</span><span class="p">)</span> <span class="o">&lt;</span> <span class="nb">abs</span><span class="p">(</span><span class="n">alpha</span><span class="p">):</span>
-                        <span class="n">free_up</span> <span class="o">=</span> <span class="bp">False</span>
-
-        <span class="k">return</span> <span class="p">(</span><span class="n">free_up</span><span class="p">,</span> <span class="n">free_down</span><span class="p">,</span> <span class="n">backwards</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_push_labels_out</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">morphisms_str_info</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">object_coords</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        For all straight morphisms which form the visual boundary of</span>
-<span class="sd">        the laid out diagram, puts their labels on their outer sides.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">def</span> <span class="nf">set_label_position</span><span class="p">(</span><span class="n">free1</span><span class="p">,</span> <span class="n">free2</span><span class="p">,</span> <span class="n">pos1</span><span class="p">,</span> <span class="n">pos2</span><span class="p">,</span> <span class="n">backwards</span><span class="p">,</span> <span class="n">m_str_info</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            Given the information about room available to one side and</span>
-<span class="sd">            to the other side of a morphism (``free1`` and ``free2``),</span>
-<span class="sd">            sets the position of the morphism label in such a way that</span>
-<span class="sd">            it is on the freer side.  This latter operations involves</span>
-<span class="sd">            choice between ``pos1`` and ``pos2``, taking ``backwards``</span>
-<span class="sd">            in consideration.</span>
-
-<span class="sd">            Thus this function will do nothing if either both ``free1</span>
-<span class="sd">            == True`` and ``free2 == True`` or both ``free1 == False``</span>
-<span class="sd">            and ``free2 == False``.  In either case, choosing one side</span>
-<span class="sd">            over the other presents no advantage.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">if</span> <span class="n">backwards</span><span class="p">:</span>
-                <span class="p">(</span><span class="n">pos1</span><span class="p">,</span> <span class="n">pos2</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">pos2</span><span class="p">,</span> <span class="n">pos1</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">free1</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">free2</span><span class="p">:</span>
-                <span class="n">m_str_info</span><span class="o">.</span><span class="n">label_position</span> <span class="o">=</span> <span class="n">pos1</span>
-            <span class="k">elif</span> <span class="n">free2</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">free1</span><span class="p">:</span>
-                <span class="n">m_str_info</span><span class="o">.</span><span class="n">label_position</span> <span class="o">=</span> <span class="n">pos2</span>
-
-        <span class="k">for</span> <span class="n">m</span><span class="p">,</span> <span class="n">m_str_info</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">morphisms_str_info</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-            <span class="k">if</span> <span class="n">m_str_info</span><span class="o">.</span><span class="n">curving</span> <span class="ow">or</span> <span class="n">m_str_info</span><span class="o">.</span><span class="n">forced_label_position</span><span class="p">:</span>
-                <span class="c"># This is either a curved morphism, and curved</span>
-                <span class="c"># morphisms have other magic, or the position of this</span>
-                <span class="c"># label has already been fixed.</span>
-                <span class="k">continue</span>
-
-            <span class="k">if</span> <span class="n">m</span><span class="o">.</span><span class="n">domain</span> <span class="o">==</span> <span class="n">m</span><span class="o">.</span><span class="n">codomain</span><span class="p">:</span>
-                <span class="c"># This is a loop morphism, their labels, again have a</span>
-                <span class="c"># different magic.</span>
-                <span class="k">continue</span>
-
-            <span class="p">(</span><span class="n">dom_i</span><span class="p">,</span> <span class="n">dom_j</span><span class="p">)</span> <span class="o">=</span> <span class="n">object_coords</span><span class="p">[</span><span class="n">m</span><span class="o">.</span><span class="n">domain</span><span class="p">]</span>
-            <span class="p">(</span><span class="n">cod_i</span><span class="p">,</span> <span class="n">cod_j</span><span class="p">)</span> <span class="o">=</span> <span class="n">object_coords</span><span class="p">[</span><span class="n">m</span><span class="o">.</span><span class="n">codomain</span><span class="p">]</span>
-
-            <span class="k">if</span> <span class="n">dom_i</span> <span class="o">==</span> <span class="n">cod_i</span><span class="p">:</span>
-                <span class="c"># Horizontal morphism.</span>
-                <span class="p">(</span><span class="n">free_up</span><span class="p">,</span> <span class="n">free_down</span><span class="p">,</span>
-                 <span class="n">backwards</span><span class="p">)</span> <span class="o">=</span> <span class="n">XypicDiagramDrawer</span><span class="o">.</span><span class="n">_check_free_space_horizontal</span><span class="p">(</span>
-                     <span class="n">dom_i</span><span class="p">,</span> <span class="n">dom_j</span><span class="p">,</span> <span class="n">cod_j</span><span class="p">,</span> <span class="n">grid</span><span class="p">)</span>
-
-                <span class="n">set_label_position</span><span class="p">(</span><span class="n">free_up</span><span class="p">,</span> <span class="n">free_down</span><span class="p">,</span> <span class="s">&quot;^&quot;</span><span class="p">,</span> <span class="s">&quot;_&quot;</span><span class="p">,</span>
-                                   <span class="n">backwards</span><span class="p">,</span> <span class="n">m_str_info</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">dom_j</span> <span class="o">==</span> <span class="n">cod_j</span><span class="p">:</span>
-                <span class="c"># Vertical morphism.</span>
-                <span class="p">(</span><span class="n">free_left</span><span class="p">,</span> <span class="n">free_right</span><span class="p">,</span>
-                 <span class="n">backwards</span><span class="p">)</span> <span class="o">=</span> <span class="n">XypicDiagramDrawer</span><span class="o">.</span><span class="n">_check_free_space_vertical</span><span class="p">(</span>
-                     <span class="n">dom_i</span><span class="p">,</span> <span class="n">cod_i</span><span class="p">,</span> <span class="n">dom_j</span><span class="p">,</span> <span class="n">grid</span><span class="p">)</span>
-
-                <span class="n">set_label_position</span><span class="p">(</span><span class="n">free_left</span><span class="p">,</span> <span class="n">free_right</span><span class="p">,</span> <span class="s">&quot;_&quot;</span><span class="p">,</span> <span class="s">&quot;^&quot;</span><span class="p">,</span>
-                                   <span class="n">backwards</span><span class="p">,</span> <span class="n">m_str_info</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># A diagonal morphism.</span>
-                <span class="p">(</span><span class="n">free_up</span><span class="p">,</span> <span class="n">free_down</span><span class="p">,</span>
-                 <span class="n">backwards</span><span class="p">)</span> <span class="o">=</span> <span class="n">XypicDiagramDrawer</span><span class="o">.</span><span class="n">_check_free_space_diagonal</span><span class="p">(</span>
-                     <span class="n">dom_i</span><span class="p">,</span> <span class="n">cod_i</span><span class="p">,</span> <span class="n">dom_j</span><span class="p">,</span> <span class="n">cod_j</span><span class="p">,</span> <span class="n">grid</span><span class="p">)</span>
-
-                <span class="n">set_label_position</span><span class="p">(</span><span class="n">free_up</span><span class="p">,</span> <span class="n">free_down</span><span class="p">,</span> <span class="s">&quot;^&quot;</span><span class="p">,</span> <span class="s">&quot;_&quot;</span><span class="p">,</span>
-                                   <span class="n">backwards</span><span class="p">,</span> <span class="n">m_str_info</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_morphism_sort_key</span><span class="p">(</span><span class="n">morphism</span><span class="p">,</span> <span class="n">object_coords</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Provides a morphism sorting key such that horizontal or</span>
-<span class="sd">        vertical morphisms between neighbouring objects come</span>
-<span class="sd">        first, then horizontal or vertical morphisms between more</span>
-<span class="sd">        far away objects, and finally, all other morphisms.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="o">=</span> <span class="n">object_coords</span><span class="p">[</span><span class="n">morphism</span><span class="o">.</span><span class="n">domain</span><span class="p">]</span>
-        <span class="p">(</span><span class="n">target_i</span><span class="p">,</span> <span class="n">target_j</span><span class="p">)</span> <span class="o">=</span> <span class="n">object_coords</span><span class="p">[</span><span class="n">morphism</span><span class="o">.</span><span class="n">codomain</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">morphism</span><span class="o">.</span><span class="n">domain</span> <span class="o">==</span> <span class="n">morphism</span><span class="o">.</span><span class="n">codomain</span><span class="p">:</span>
-            <span class="c"># Loop morphisms should get after diagonal morphisms</span>
-            <span class="c"># so that the proper direction in which to curve the</span>
-            <span class="c"># loop can be determined.</span>
-            <span class="k">return</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">default_sort_key</span><span class="p">(</span><span class="n">morphism</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">target_i</span> <span class="o">==</span> <span class="n">i</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">abs</span><span class="p">(</span><span class="n">target_j</span> <span class="o">-</span> <span class="n">j</span><span class="p">),</span> <span class="n">default_sort_key</span><span class="p">(</span><span class="n">morphism</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">target_j</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">abs</span><span class="p">(</span><span class="n">target_i</span> <span class="o">-</span> <span class="n">i</span><span class="p">),</span> <span class="n">default_sort_key</span><span class="p">(</span><span class="n">morphism</span><span class="p">))</span>
-
-        <span class="c"># Diagonal morphism.</span>
-        <span class="k">return</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">default_sort_key</span><span class="p">(</span><span class="n">morphism</span><span class="p">))</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_build_xypic_string</span><span class="p">(</span><span class="n">diagram</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">morphisms</span><span class="p">,</span>
-                            <span class="n">morphisms_str_info</span><span class="p">,</span> <span class="n">diagram_format</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Given a collection of :class:`ArrowStringDescription`</span>
-<span class="sd">        describing the morphisms of a diagram and the object layout</span>
-<span class="sd">        information of a diagram, produces the final Xy-pic picture.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># Build the mapping between objects and morphisms which have</span>
-        <span class="c"># them as domains.</span>
-        <span class="n">object_morphisms</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">obj</span> <span class="ow">in</span> <span class="n">diagram</span><span class="o">.</span><span class="n">objects</span><span class="p">:</span>
-            <span class="n">object_morphisms</span><span class="p">[</span><span class="n">obj</span><span class="p">]</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">morphism</span> <span class="ow">in</span> <span class="n">morphisms</span><span class="p">:</span>
-            <span class="n">object_morphisms</span><span class="p">[</span><span class="n">morphism</span><span class="o">.</span><span class="n">domain</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">morphism</span><span class="p">)</span>
-
-        <span class="n">result</span> <span class="o">=</span> <span class="s">&quot;</span><span class="se">\\</span><span class="s">xymatrix</span><span class="si">%s</span><span class="s">{</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">diagram_format</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid</span><span class="o">.</span><span class="n">height</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid</span><span class="o">.</span><span class="n">width</span><span class="p">):</span>
-                <span class="n">obj</span> <span class="o">=</span> <span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-                <span class="k">if</span> <span class="n">obj</span><span class="p">:</span>
-                    <span class="n">result</span> <span class="o">+=</span> <span class="n">latex</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot; &quot;</span>
-
-                    <span class="n">morphisms_to_draw</span> <span class="o">=</span> <span class="n">object_morphisms</span><span class="p">[</span><span class="n">obj</span><span class="p">]</span>
-                    <span class="k">for</span> <span class="n">morphism</span> <span class="ow">in</span> <span class="n">morphisms_to_draw</span><span class="p">:</span>
-                        <span class="n">result</span> <span class="o">+=</span> <span class="nb">str</span><span class="p">(</span><span class="n">morphisms_str_info</span><span class="p">[</span><span class="n">morphism</span><span class="p">])</span> <span class="o">+</span> <span class="s">&quot; &quot;</span>
-
-                <span class="c"># Don&#39;t put the &amp; after the last column.</span>
-                <span class="k">if</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="n">grid</span><span class="o">.</span><span class="n">width</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">result</span> <span class="o">+=</span> <span class="s">&quot;&amp; &quot;</span>
-
-            <span class="c"># Don&#39;t put the line break after the last row.</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">grid</span><span class="o">.</span><span class="n">height</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">+=</span> <span class="s">&quot;</span><span class="se">\\\\</span><span class="s">&quot;</span>
-            <span class="n">result</span> <span class="o">+=</span> <span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span>
-
-        <span class="n">result</span> <span class="o">+=</span> <span class="s">&quot;}</span><span class="se">\n</span><span class="s">&quot;</span>
-
-        <span class="k">return</span> <span class="n">result</span>
-
-<div class="viewcode-block" id="XypicDiagramDrawer.draw"><a class="viewcode-back" href="../../../modules/categories.html#sympy.categories.diagram_drawing.XypicDiagramDrawer.draw">[docs]</a>    <span class="k">def</span> <span class="nf">draw</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">diagram</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">masked</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">diagram_format</span><span class="o">=</span><span class="s">&quot;&quot;</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">        Returns the Xy-pic representation of ``diagram`` laid out in</span>
-<span class="sd">        ``grid``.</span>
-
-<span class="sd">        Consider the following simple triangle diagram.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.categories import Object, NamedMorphism, Diagram</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.categories import DiagramGrid, XypicDiagramDrawer</span>
-<span class="sd">        &gt;&gt;&gt; A = Object(&quot;A&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; B = Object(&quot;B&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; C = Object(&quot;C&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; f = NamedMorphism(A, B, &quot;f&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; g = NamedMorphism(B, C, &quot;g&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; diagram = Diagram([f, g], {g * f: &quot;unique&quot;})</span>
-
-<span class="sd">        To draw this diagram, its objects need to be laid out with a</span>
-<span class="sd">        :class:`DiagramGrid`::</span>
-
-<span class="sd">        &gt;&gt;&gt; grid = DiagramGrid(diagram)</span>
-
-<span class="sd">        Finally, the drawing:</span>
-
-<span class="sd">        &gt;&gt;&gt; drawer = XypicDiagramDrawer()</span>
-<span class="sd">        &gt;&gt;&gt; print(drawer.draw(diagram, grid))</span>
-<span class="sd">        \xymatrix{</span>
-<span class="sd">        A \ar[d]_{g\circ f} \ar[r]^{f} &amp; B \ar[ld]^{g} \\</span>
-<span class="sd">        C &amp;</span>
-<span class="sd">        }</span>
-
-<span class="sd">        The argument ``masked`` can be used to skip morphisms in the</span>
-<span class="sd">        presentation of the diagram:</span>
-
-<span class="sd">        &gt;&gt;&gt; print(drawer.draw(diagram, grid, masked=[g * f]))</span>
-<span class="sd">        \xymatrix{</span>
-<span class="sd">        A \ar[r]^{f} &amp; B \ar[ld]^{g} \\</span>
-<span class="sd">        C &amp;</span>
-<span class="sd">        }</span>
-
-<span class="sd">        Finally, the ``diagram_format`` argument can be used to</span>
-<span class="sd">        specify the format string of the diagram.  For example, to</span>
-<span class="sd">        increase the spacing by 1 cm, proceeding as follows:</span>
-
-<span class="sd">        &gt;&gt;&gt; print(drawer.draw(diagram, grid, diagram_format=&quot;@+1cm&quot;))</span>
-<span class="sd">        \xymatrix@+1cm{</span>
-<span class="sd">        A \ar[d]_{g\circ f} \ar[r]^{f} &amp; B \ar[ld]^{g} \\</span>
-<span class="sd">        C &amp;</span>
-<span class="sd">        }</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># This method works in several steps.  It starts by removing</span>
-        <span class="c"># the masked morphisms, if necessary, and then maps objects to</span>
-        <span class="c"># their positions in the grid (coordinate tuples).  Remember</span>
-        <span class="c"># that objects are unique in ``Diagram`` and in the layout</span>
-        <span class="c"># produced by ``DiagramGrid``, so every object is mapped to a</span>
-        <span class="c"># single coordinate pair.</span>
-        <span class="c">#</span>
-        <span class="c"># The next step is the central step and is concerned with</span>
-        <span class="c"># analysing the morphisms of the diagram and deciding how to</span>
-        <span class="c"># draw them.  For example, how to curve the arrows is decided</span>
-        <span class="c"># at this step.  The bulk of the analysis is implemented in</span>
-        <span class="c"># ``_process_morphism``, to the result of which the</span>
-        <span class="c"># appropriate formatters are applied.</span>
-        <span class="c">#</span>
-        <span class="c"># The result of the previous step is a list of</span>
-        <span class="c"># ``ArrowStringDescription``.  After the analysis and</span>
-        <span class="c"># application of formatters, some extra logic tries to assure</span>
-        <span class="c"># better positioning of morphism labels (for example, an</span>
-        <span class="c"># attempt is made to avoid the situations when arrows cross</span>
-        <span class="c"># labels).  This functionality constitutes the next step and</span>
-        <span class="c"># is implemented in ``_push_labels_out``.  Note that label</span>
-        <span class="c"># positions which have been set via a formatter are not</span>
-        <span class="c"># affected in this step.</span>
-        <span class="c">#</span>
-        <span class="c"># Finally, at the closing step, the array of</span>
-        <span class="c"># ``ArrowStringDescription`` and the layout information</span>
-        <span class="c"># incorporated in ``DiagramGrid`` are combined to produce the</span>
-        <span class="c"># resulting Xy-pic picture.  This part of code lies in</span>
-        <span class="c"># ``_build_xypic_string``.</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">masked</span><span class="p">:</span>
-            <span class="n">morphisms_props</span> <span class="o">=</span> <span class="n">grid</span><span class="o">.</span><span class="n">morphisms</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">morphisms_props</span> <span class="o">=</span> <span class="p">{}</span>
-            <span class="k">for</span> <span class="n">m</span><span class="p">,</span> <span class="n">props</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">grid</span><span class="o">.</span><span class="n">morphisms</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-                <span class="k">if</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">masked</span><span class="p">:</span>
-                    <span class="k">continue</span>
-                <span class="n">morphisms_props</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">=</span> <span class="n">props</span>
-
-        <span class="c"># Build the mapping between objects and their position in the</span>
-        <span class="c"># grid.</span>
-        <span class="n">object_coords</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid</span><span class="o">.</span><span class="n">height</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid</span><span class="o">.</span><span class="n">width</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]:</span>
-                    <span class="n">object_coords</span><span class="p">[</span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]]</span> <span class="o">=</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
-
-        <span class="n">morphisms</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">morphisms_props</span><span class="p">,</span>
-                           <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">m</span><span class="p">:</span> <span class="n">XypicDiagramDrawer</span><span class="o">.</span><span class="n">_morphism_sort_key</span><span class="p">(</span>
-                               <span class="n">m</span><span class="p">,</span> <span class="n">object_coords</span><span class="p">))</span>
-
-        <span class="c"># Build the tuples defining the string representations of</span>
-        <span class="c"># morphisms.</span>
-        <span class="n">morphisms_str_info</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">morphism</span> <span class="ow">in</span> <span class="n">morphisms</span><span class="p">:</span>
-            <span class="n">string_description</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_process_morphism</span><span class="p">(</span>
-                <span class="n">diagram</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">morphism</span><span class="p">,</span> <span class="n">object_coords</span><span class="p">,</span> <span class="n">morphisms</span><span class="p">,</span>
-                <span class="n">morphisms_str_info</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">default_arrow_formatter</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">default_arrow_formatter</span><span class="p">(</span><span class="n">string_description</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">prop</span> <span class="ow">in</span> <span class="n">morphisms_props</span><span class="p">[</span><span class="n">morphism</span><span class="p">]:</span>
-                <span class="c"># prop is a Symbol.  TODO: Find out why.</span>
-                <span class="k">if</span> <span class="n">prop</span><span class="o">.</span><span class="n">name</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">arrow_formatters</span><span class="p">:</span>
-                    <span class="n">formatter</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">arrow_formatters</span><span class="p">[</span><span class="n">prop</span><span class="o">.</span><span class="n">name</span><span class="p">]</span>
-                    <span class="n">formatter</span><span class="p">(</span><span class="n">string_description</span><span class="p">)</span>
-
-            <span class="n">morphisms_str_info</span><span class="p">[</span><span class="n">morphism</span><span class="p">]</span> <span class="o">=</span> <span class="n">string_description</span>
-
-        <span class="c"># Reposition the labels a bit.</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_push_labels_out</span><span class="p">(</span><span class="n">morphisms_str_info</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">object_coords</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">XypicDiagramDrawer</span><span class="o">.</span><span class="n">_build_xypic_string</span><span class="p">(</span>
-            <span class="n">diagram</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">morphisms</span><span class="p">,</span> <span class="n">morphisms_str_info</span><span class="p">,</span> <span class="n">diagram_format</span><span class="p">)</span>
-
-</div></div>
-<div class="viewcode-block" id="xypic_draw_diagram"><a class="viewcode-back" href="../../../modules/categories.html#sympy.categories.diagram_drawing.xypic_draw_diagram">[docs]</a><span class="k">def</span> <span class="nf">xypic_draw_diagram</span><span class="p">(</span><span class="n">diagram</span><span class="p">,</span> <span class="n">masked</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">diagram_format</span><span class="o">=</span><span class="s">&quot;&quot;</span><span class="p">,</span>
-                       <span class="n">groups</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Provides a shortcut combining :class:`DiagramGrid` and</span>
-<span class="sd">    :class:`XypicDiagramDrawer`.  Returns an Xy-pic presentation of</span>
-<span class="sd">    ``diagram``.  The argument ``masked`` is a list of morphisms which</span>
-<span class="sd">    will be not be drawn.  The argument ``diagram_format`` is the</span>
-<span class="sd">    format string inserted after &quot;\xymatrix&quot;.  ``groups`` should be a</span>
-<span class="sd">    set of logical groups.  The ``hints`` will be passed directly to</span>
-<span class="sd">    the constructor of :class:`DiagramGrid`.</span>
-
-<span class="sd">    For more information about the arguments, see the docstrings of</span>
-<span class="sd">    :class:`DiagramGrid` and ``XypicDiagramDrawer.draw``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.categories import Object, NamedMorphism, Diagram</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.categories import xypic_draw_diagram</span>
-<span class="sd">    &gt;&gt;&gt; A = Object(&quot;A&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; B = Object(&quot;B&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; C = Object(&quot;C&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; f = NamedMorphism(A, B, &quot;f&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; g = NamedMorphism(B, C, &quot;g&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; diagram = Diagram([f, g], {g * f: &quot;unique&quot;})</span>
-<span class="sd">    &gt;&gt;&gt; print(xypic_draw_diagram(diagram))</span>
-<span class="sd">    \xymatrix{</span>
-<span class="sd">    A \ar[d]_{g\circ f} \ar[r]^{f} &amp; B \ar[ld]^{g} \\</span>
-<span class="sd">    C &amp;</span>
-<span class="sd">    }</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    XypicDiagramDrawer, DiagramGrid</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">grid</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="p">(</span><span class="n">diagram</span><span class="p">,</span> <span class="n">groups</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-    <span class="n">drawer</span> <span class="o">=</span> <span class="n">XypicDiagramDrawer</span><span class="p">()</span>
-    <span class="k">return</span> <span class="n">drawer</span><span class="o">.</span><span class="n">draw</span><span class="p">(</span><span class="n">diagram</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">masked</span><span class="p">,</span> <span class="n">diagram_format</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="preview_diagram"><a class="viewcode-back" href="../../../modules/categories.html#sympy.categories.diagram_drawing.preview_diagram">[docs]</a><span class="k">def</span> <span class="nf">preview_diagram</span><span class="p">(</span><span class="n">diagram</span><span class="p">,</span> <span class="n">masked</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">diagram_format</span><span class="o">=</span><span class="s">&quot;&quot;</span><span class="p">,</span> <span class="n">groups</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span>
-                    <span class="n">output</span><span class="o">=</span><span class="s">&#39;png&#39;</span><span class="p">,</span> <span class="n">viewer</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">euler</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Combines the functionality of ``xypic_draw_diagram`` and</span>
-<span class="sd">    ``sympy.printing.preview``.  The arguments ``masked``,</span>
-<span class="sd">    ``diagram_format``, ``groups``, and ``hints`` are passed to</span>
-<span class="sd">    ``xypic_draw_diagram``, while ``output``, ``viewer, and ``euler``</span>
-<span class="sd">    are passed to ``preview``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.categories import Object, NamedMorphism, Diagram</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.categories import preview_diagram</span>
-<span class="sd">    &gt;&gt;&gt; A = Object(&quot;A&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; B = Object(&quot;B&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; C = Object(&quot;C&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; f = NamedMorphism(A, B, &quot;f&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; g = NamedMorphism(B, C, &quot;g&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; d = Diagram([f, g], {g * f: &quot;unique&quot;})</span>
-<span class="sd">    &gt;&gt;&gt; preview_diagram(d)  #doctest: +SKIP</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    xypic_diagram_drawer</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.printing</span> <span class="kn">import</span> <span class="n">preview</span>
-    <span class="n">latex_output</span> <span class="o">=</span> <span class="n">xypic_draw_diagram</span><span class="p">(</span><span class="n">diagram</span><span class="p">,</span> <span class="n">masked</span><span class="p">,</span> <span class="n">diagram_format</span><span class="p">,</span>
-                                      <span class="n">groups</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-    <span class="n">preview</span><span class="p">(</span><span class="n">latex_output</span><span class="p">,</span> <span class="n">output</span><span class="p">,</span> <span class="n">viewer</span><span class="p">,</span> <span class="n">euler</span><span class="p">,</span> <span class="p">(</span><span class="s">&quot;xypic&quot;</span><span class="p">,))</span></div>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.combinatorics</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span><span class="p">,</span> <span class="n">Cycle</span>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.prufer</span> <span class="kn">import</span> <span class="n">Prufer</span>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.generators</span> <span class="kn">import</span> <span class="n">cyclic</span><span class="p">,</span> <span class="n">alternating</span><span class="p">,</span> <span class="n">symmetric</span><span class="p">,</span> <span class="n">dihedral</span>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.subsets</span> <span class="kn">import</span> <span class="n">Subset</span>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.partitions</span> <span class="kn">import</span> <span class="p">(</span><span class="n">Partition</span><span class="p">,</span> <span class="n">IntegerPartition</span><span class="p">,</span>
-    <span class="n">RGS_rank</span><span class="p">,</span> <span class="n">RGS_unrank</span><span class="p">,</span> <span class="n">RGS_enum</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.polyhedron</span> <span class="kn">import</span> <span class="p">(</span><span class="n">Polyhedron</span><span class="p">,</span> <span class="n">tetrahedron</span><span class="p">,</span> <span class="n">cube</span><span class="p">,</span>
-    <span class="n">octahedron</span><span class="p">,</span> <span class="n">dodecahedron</span><span class="p">,</span> <span class="n">icosahedron</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.group_constructs</span> <span class="kn">import</span> <span class="n">DirectProduct</span>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.graycode</span> <span class="kn">import</span> <span class="n">GrayCode</span>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="p">(</span><span class="n">SymmetricGroup</span><span class="p">,</span> <span class="n">DihedralGroup</span><span class="p">,</span>
-    <span class="n">CyclicGroup</span><span class="p">,</span> <span class="n">AlternatingGroup</span><span class="p">,</span> <span class="n">AbelianGroup</span><span class="p">,</span> <span class="n">RubikGroup</span><span class="p">)</span>
-</pre></div>
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-            
-  <h1>Source code for sympy.combinatorics.graycode</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Basic</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="nb">bin</span>
-
-<span class="kn">import</span> <span class="nn">random</span>
-
-
-<div class="viewcode-block" id="GrayCode"><a class="viewcode-back" href="../../../modules/combinatorics/graycode.html#sympy.combinatorics.graycode.GrayCode">[docs]</a><span class="k">class</span> <span class="nc">GrayCode</span><span class="p">(</span><span class="n">Basic</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A Gray code is essentially a Hamiltonian walk on</span>
-<span class="sd">    a n-dimensional cube with edge length of one.</span>
-<span class="sd">    The vertices of the cube are represented by vectors</span>
-<span class="sd">    whose values are binary. The Hamilton walk visits</span>
-<span class="sd">    each vertex exactly once. The Gray code for a 3d</span>
-<span class="sd">    cube is [&#39;000&#39;,&#39;100&#39;,&#39;110&#39;,&#39;010&#39;,&#39;011&#39;,&#39;111&#39;,&#39;101&#39;,</span>
-<span class="sd">    &#39;001&#39;].</span>
-
-<span class="sd">    A Gray code solves the problem of sequentially</span>
-<span class="sd">    generating all possible subsets of n objects in such</span>
-<span class="sd">    a way that each subset is obtained from the previous</span>
-<span class="sd">    one by either deleting or adding a single object.</span>
-<span class="sd">    In the above example, 1 indicates that the object is</span>
-<span class="sd">    present, and 0 indicates that its absent.</span>
-
-<span class="sd">    Gray codes have applications in statistics as well when</span>
-<span class="sd">    we want to compute various statistics related to subsets</span>
-<span class="sd">    in an efficient manner.</span>
-
-<span class="sd">    References:</span>
-<span class="sd">    [1] Nijenhuis,A. and Wilf,H.S.(1978).</span>
-<span class="sd">    Combinatorial Algorithms. Academic Press.</span>
-<span class="sd">    [2] Knuth, D. (2011). The Art of Computer Programming, Vol 4</span>
-<span class="sd">    Addison Wesley</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.graycode import GrayCode</span>
-<span class="sd">    &gt;&gt;&gt; a = GrayCode(3)</span>
-<span class="sd">    &gt;&gt;&gt; list(a.generate_gray())</span>
-<span class="sd">    [&#39;000&#39;, &#39;001&#39;, &#39;011&#39;, &#39;010&#39;, &#39;110&#39;, &#39;111&#39;, &#39;101&#39;, &#39;100&#39;]</span>
-<span class="sd">    &gt;&gt;&gt; a = GrayCode(4)</span>
-<span class="sd">    &gt;&gt;&gt; list(a.generate_gray())</span>
-<span class="sd">    [&#39;0000&#39;, &#39;0001&#39;, &#39;0011&#39;, &#39;0010&#39;, &#39;0110&#39;, &#39;0111&#39;, &#39;0101&#39;, &#39;0100&#39;, \</span>
-<span class="sd">    &#39;1100&#39;, &#39;1101&#39;, &#39;1111&#39;, &#39;1110&#39;, &#39;1010&#39;, &#39;1011&#39;, &#39;1001&#39;, &#39;1000&#39;]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_skip</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">_current</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">_rank</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Default constructor.</span>
-
-<span class="sd">        It takes a single argument ``n`` which gives the dimension of the Gray</span>
-<span class="sd">        code. The starting Gray code string (``start``) or the starting ``rank``</span>
-<span class="sd">        may also be given; the default is to start at rank = 0 (&#39;0...0&#39;).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.graycode import GrayCode</span>
-<span class="sd">        &gt;&gt;&gt; a = GrayCode(3)</span>
-<span class="sd">        &gt;&gt;&gt; a</span>
-<span class="sd">        GrayCode(3)</span>
-<span class="sd">        &gt;&gt;&gt; a.n</span>
-<span class="sd">        3</span>
-
-<span class="sd">        &gt;&gt;&gt; a = GrayCode(3, start=&#39;100&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; a.current</span>
-<span class="sd">        &#39;100&#39;</span>
-
-<span class="sd">        &gt;&gt;&gt; a = GrayCode(4, rank=4)</span>
-<span class="sd">        &gt;&gt;&gt; a.current</span>
-<span class="sd">        &#39;0110&#39;</span>
-<span class="sd">        &gt;&gt;&gt; a.rank</span>
-<span class="sd">        4</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">1</span> <span class="ow">or</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">!=</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&#39;Gray code dimension must be a positive integer, not </span><span class="si">%i</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">(</span><span class="n">n</span><span class="p">,)</span> <span class="o">+</span> <span class="n">args</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="s">&#39;start&#39;</span> <span class="ow">in</span> <span class="n">kw_args</span><span class="p">:</span>
-            <span class="n">obj</span><span class="o">.</span><span class="n">_current</span> <span class="o">=</span> <span class="n">kw_args</span><span class="p">[</span><span class="s">&quot;start&quot;</span><span class="p">]</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">obj</span><span class="o">.</span><span class="n">_current</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">n</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Gray code start has length </span><span class="si">%i</span><span class="s"> but &#39;</span>
-                <span class="s">&#39;should not be greater than </span><span class="si">%i</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">obj</span><span class="o">.</span><span class="n">_current</span><span class="p">),</span> <span class="n">n</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="s">&#39;rank&#39;</span> <span class="ow">in</span> <span class="n">kw_args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">int</span><span class="p">(</span><span class="n">kw_args</span><span class="p">[</span><span class="s">&quot;rank&quot;</span><span class="p">])</span> <span class="o">!=</span> <span class="n">kw_args</span><span class="p">[</span><span class="s">&quot;rank&quot;</span><span class="p">]:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Gray code rank must be a positive integer, &#39;</span>
-                <span class="s">&#39;not </span><span class="si">%i</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">kw_args</span><span class="p">[</span><span class="s">&quot;rank&quot;</span><span class="p">])</span>
-            <span class="n">obj</span><span class="o">.</span><span class="n">_rank</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">kw_args</span><span class="p">[</span><span class="s">&quot;rank&quot;</span><span class="p">])</span> <span class="o">%</span> <span class="n">obj</span><span class="o">.</span><span class="n">selections</span>
-            <span class="n">obj</span><span class="o">.</span><span class="n">_current</span> <span class="o">=</span> <span class="n">obj</span><span class="o">.</span><span class="n">unrank</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">obj</span><span class="o">.</span><span class="n">_rank</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-<div class="viewcode-block" id="GrayCode.next"><a class="viewcode-back" href="../../../modules/combinatorics/graycode.html#sympy.combinatorics.graycode.GrayCode.next">[docs]</a>    <span class="k">def</span> <span class="nf">next</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">delta</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the Gray code a distance ``delta`` (default = 1) from the</span>
-<span class="sd">        current value in canonical order.</span>
-
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.graycode import GrayCode</span>
-<span class="sd">        &gt;&gt;&gt; a = GrayCode(3, start=&#39;110&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; a.next().current</span>
-<span class="sd">        &#39;111&#39;</span>
-<span class="sd">        &gt;&gt;&gt; a.next(-1).current</span>
-<span class="sd">        &#39;010&#39;</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">GrayCode</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n</span><span class="p">,</span> <span class="n">rank</span><span class="o">=</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rank</span> <span class="o">+</span> <span class="n">delta</span><span class="p">)</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">selections</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="GrayCode.selections"><a class="viewcode-back" href="../../../modules/combinatorics/graycode.html#sympy.combinatorics.graycode.GrayCode.selections">[docs]</a>    <span class="k">def</span> <span class="nf">selections</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the number of bit vectors in the Gray code.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.graycode import GrayCode</span>
-<span class="sd">        &gt;&gt;&gt; a = GrayCode(3)</span>
-<span class="sd">        &gt;&gt;&gt; a.selections</span>
-<span class="sd">        8</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="mi">2</span><span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">n</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="GrayCode.n"><a class="viewcode-back" href="../../../modules/combinatorics/graycode.html#sympy.combinatorics.graycode.GrayCode.n">[docs]</a>    <span class="k">def</span> <span class="nf">n</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the dimension of the Gray code.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.graycode import GrayCode</span>
-<span class="sd">        &gt;&gt;&gt; a = GrayCode(5)</span>
-<span class="sd">        &gt;&gt;&gt; a.n</span>
-<span class="sd">        5</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-<div class="viewcode-block" id="GrayCode.generate_gray"><a class="viewcode-back" href="../../../modules/combinatorics/graycode.html#sympy.combinatorics.graycode.GrayCode.generate_gray">[docs]</a>    <span class="k">def</span> <span class="nf">generate_gray</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Generates the sequence of bit vectors of a Gray Code.</span>
-
-<span class="sd">        [1] Knuth, D. (2011). The Art of Computer Programming,</span>
-<span class="sd">        Vol 4, Addison Wesley</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.graycode import GrayCode</span>
-<span class="sd">        &gt;&gt;&gt; a = GrayCode(3)</span>
-<span class="sd">        &gt;&gt;&gt; list(a.generate_gray())</span>
-<span class="sd">        [&#39;000&#39;, &#39;001&#39;, &#39;011&#39;, &#39;010&#39;, &#39;110&#39;, &#39;111&#39;, &#39;101&#39;, &#39;100&#39;]</span>
-<span class="sd">        &gt;&gt;&gt; list(a.generate_gray(start=&#39;011&#39;))</span>
-<span class="sd">        [&#39;011&#39;, &#39;010&#39;, &#39;110&#39;, &#39;111&#39;, &#39;101&#39;, &#39;100&#39;]</span>
-<span class="sd">        &gt;&gt;&gt; list(a.generate_gray(rank=4))</span>
-<span class="sd">        [&#39;110&#39;, &#39;111&#39;, &#39;101&#39;, &#39;100&#39;]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        skip</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">bits</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">n</span>
-        <span class="n">start</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="s">&quot;start&quot;</span> <span class="ow">in</span> <span class="n">hints</span><span class="p">:</span>
-            <span class="n">start</span> <span class="o">=</span> <span class="n">hints</span><span class="p">[</span><span class="s">&quot;start&quot;</span><span class="p">]</span>
-        <span class="k">elif</span> <span class="s">&quot;rank&quot;</span> <span class="ow">in</span> <span class="n">hints</span><span class="p">:</span>
-            <span class="n">start</span> <span class="o">=</span> <span class="n">GrayCode</span><span class="o">.</span><span class="n">unrank</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n</span><span class="p">,</span> <span class="n">hints</span><span class="p">[</span><span class="s">&quot;rank&quot;</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">start</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_current</span> <span class="o">=</span> <span class="n">start</span>
-        <span class="n">current</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">current</span>
-        <span class="n">graycode_bin</span> <span class="o">=</span> <span class="n">gray_to_bin</span><span class="p">(</span><span class="n">current</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">graycode_bin</span><span class="p">)</span> <span class="o">&gt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">n</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Gray code start has length </span><span class="si">%i</span><span class="s"> but should &#39;</span>
-            <span class="s">&#39;not be greater than </span><span class="si">%i</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">graycode_bin</span><span class="p">),</span> <span class="n">bits</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_current</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">current</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-        <span class="n">graycode_int</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">graycode_bin</span><span class="p">),</span> <span class="mi">2</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">graycode_int</span><span class="p">,</span> <span class="mi">1</span> <span class="o">&lt;&lt;</span> <span class="n">bits</span><span class="p">):</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_skip</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_skip</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">yield</span> <span class="bp">self</span><span class="o">.</span><span class="n">current</span>
-            <span class="n">bbtc</span> <span class="o">=</span> <span class="p">(</span><span class="n">i</span> <span class="o">^</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-            <span class="n">gbtc</span> <span class="o">=</span> <span class="p">(</span><span class="n">bbtc</span> <span class="o">^</span> <span class="p">(</span><span class="n">bbtc</span> <span class="o">&gt;&gt;</span> <span class="mi">1</span><span class="p">))</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_current</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_current</span> <span class="o">^</span> <span class="n">gbtc</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_current</span> <span class="o">=</span> <span class="mi">0</span>
-</div>
-<div class="viewcode-block" id="GrayCode.skip"><a class="viewcode-back" href="../../../modules/combinatorics/graycode.html#sympy.combinatorics.graycode.GrayCode.skip">[docs]</a>    <span class="k">def</span> <span class="nf">skip</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Skips the bit generation.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.graycode import GrayCode</span>
-<span class="sd">        &gt;&gt;&gt; a = GrayCode(3)</span>
-<span class="sd">        &gt;&gt;&gt; for i in a.generate_gray():</span>
-<span class="sd">        ...     if i == &#39;010&#39;:</span>
-<span class="sd">        ...         a.skip()</span>
-<span class="sd">        ...     print(i)</span>
-<span class="sd">        ...</span>
-<span class="sd">        000</span>
-<span class="sd">        001</span>
-<span class="sd">        011</span>
-<span class="sd">        010</span>
-<span class="sd">        111</span>
-<span class="sd">        101</span>
-<span class="sd">        100</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        generate_gray</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_skip</span> <span class="o">=</span> <span class="bp">True</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="GrayCode.rank"><a class="viewcode-back" href="../../../modules/combinatorics/graycode.html#sympy.combinatorics.graycode.GrayCode.rank">[docs]</a>    <span class="k">def</span> <span class="nf">rank</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Ranks the Gray code.</span>
-
-<span class="sd">        A ranking algorithm determines the position (or rank)</span>
-<span class="sd">        of a combinatorial object among all the objects w.r.t.</span>
-<span class="sd">        a given order. For example, the 4 bit binary reflected</span>
-<span class="sd">        Gray code (BRGC) &#39;0101&#39; has a rank of 6 as it appears in</span>
-<span class="sd">        the 6th position in the canonical ordering of the family</span>
-<span class="sd">        of 4 bit Gray codes.</span>
-
-<span class="sd">        References:</span>
-<span class="sd">        [1] http://www-stat.stanford.edu/~susan/courses/s208/node12.html</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.graycode import GrayCode</span>
-<span class="sd">        &gt;&gt;&gt; a = GrayCode(3)</span>
-<span class="sd">        &gt;&gt;&gt; list(a.generate_gray())</span>
-<span class="sd">        [&#39;000&#39;, &#39;001&#39;, &#39;011&#39;, &#39;010&#39;, &#39;110&#39;, &#39;111&#39;, &#39;101&#39;, &#39;100&#39;]</span>
-<span class="sd">        &gt;&gt;&gt; GrayCode(3, start=&#39;100&#39;).rank</span>
-<span class="sd">        7</span>
-<span class="sd">        &gt;&gt;&gt; GrayCode(3, rank=7).current</span>
-<span class="sd">        &#39;100&#39;</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        unrank</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rank</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_rank</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">gray_to_bin</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">current</span><span class="p">),</span> <span class="mi">2</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rank</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="GrayCode.current"><a class="viewcode-back" href="../../../modules/combinatorics/graycode.html#sympy.combinatorics.graycode.GrayCode.current">[docs]</a>    <span class="k">def</span> <span class="nf">current</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the currently referenced Gray code as a bit string.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.graycode import GrayCode</span>
-<span class="sd">        &gt;&gt;&gt; GrayCode(3, start=&#39;100&#39;).current</span>
-<span class="sd">        &#39;100&#39;</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_current</span> <span class="ow">or</span> <span class="s">&#39;0&#39;</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">rv</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="nb">str</span><span class="p">:</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="nb">bin</span><span class="p">(</span><span class="n">rv</span><span class="p">)[</span><span class="mi">2</span><span class="p">:]</span>
-        <span class="k">return</span> <span class="n">rv</span><span class="o">.</span><span class="n">rjust</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n</span><span class="p">,</span> <span class="s">&#39;0&#39;</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="GrayCode.unrank"><a class="viewcode-back" href="../../../modules/combinatorics/graycode.html#sympy.combinatorics.graycode.GrayCode.unrank">[docs]</a>    <span class="k">def</span> <span class="nf">unrank</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">rank</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Unranks an n-bit sized Gray code of rank k. This method exists</span>
-<span class="sd">        so that a derivative GrayCode class can define its own code of</span>
-<span class="sd">        a given rank.</span>
-
-<span class="sd">        The string here is generated in reverse order to allow for tail-call</span>
-<span class="sd">        optimization.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.graycode import GrayCode</span>
-<span class="sd">        &gt;&gt;&gt; GrayCode(5, rank=3).current</span>
-<span class="sd">        &#39;00010&#39;</span>
-<span class="sd">        &gt;&gt;&gt; GrayCode.unrank(5, 3)</span>
-<span class="sd">        &#39;00010&#39;</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        rank</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">def</span> <span class="nf">_unrank</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="n">k</span> <span class="o">%</span> <span class="mi">2</span><span class="p">)</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">k</span> <span class="o">&lt;</span> <span class="n">m</span><span class="p">:</span>
-                <span class="k">return</span> <span class="s">&#39;0&#39;</span> <span class="o">+</span> <span class="n">_unrank</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="k">return</span> <span class="s">&#39;1&#39;</span> <span class="o">+</span> <span class="n">_unrank</span><span class="p">(</span><span class="n">m</span> <span class="o">-</span> <span class="p">(</span><span class="n">k</span> <span class="o">%</span> <span class="n">m</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">_unrank</span><span class="p">(</span><span class="n">rank</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-
-</div></div>
-<span class="k">def</span> <span class="nf">random_bitstring</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Generates a random bitlist of length n.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.graycode import random_bitstring</span>
-<span class="sd">    &gt;&gt;&gt; random_bitstring(3) # doctest: +SKIP</span>
-<span class="sd">    100</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="s">&#39;01&#39;</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)])</span>
-
-
-<span class="k">def</span> <span class="nf">gray_to_bin</span><span class="p">(</span><span class="n">bin_list</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Convert from Gray coding to binary coding.</span>
-
-<span class="sd">    We assume big endian encoding.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.graycode import gray_to_bin</span>
-<span class="sd">    &gt;&gt;&gt; gray_to_bin(&#39;100&#39;)</span>
-<span class="sd">    &#39;111&#39;</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    bin_to_gray</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">b</span> <span class="o">=</span> <span class="p">[</span><span class="n">bin_list</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">bin_list</span><span class="p">)):</span>
-        <span class="n">b</span> <span class="o">+=</span> <span class="nb">str</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">b</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">!=</span> <span class="n">bin_list</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span>
-    <span class="k">return</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">bin_to_gray</span><span class="p">(</span><span class="n">bin_list</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Convert from binary coding to gray coding.</span>
-
-<span class="sd">    We assume big endian encoding.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.graycode import bin_to_gray</span>
-<span class="sd">    &gt;&gt;&gt; bin_to_gray(&#39;111&#39;)</span>
-<span class="sd">    &#39;100&#39;</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    gray_to_bin</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">b</span> <span class="o">=</span> <span class="p">[</span><span class="n">bin_list</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">bin_list</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">b</span> <span class="o">+=</span> <span class="nb">str</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">bin_list</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="o">^</span> <span class="nb">int</span><span class="p">(</span><span class="n">b</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]))</span>
-    <span class="k">return</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">get_subset_from_bitstring</span><span class="p">(</span><span class="n">super_set</span><span class="p">,</span> <span class="n">bitstring</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Gets the subset defined by the bitstring.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.graycode import get_subset_from_bitstring</span>
-<span class="sd">    &gt;&gt;&gt; get_subset_from_bitstring([&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;d&#39;], &#39;0011&#39;)</span>
-<span class="sd">    [&#39;c&#39;, &#39;d&#39;]</span>
-<span class="sd">    &gt;&gt;&gt; get_subset_from_bitstring([&#39;c&#39;,&#39;a&#39;,&#39;c&#39;,&#39;c&#39;], &#39;1100&#39;)</span>
-<span class="sd">    [&#39;c&#39;, &#39;a&#39;]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    graycode_subsets</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">super_set</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">bitstring</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;The sizes of the lists are not equal&quot;</span><span class="p">)</span>
-    <span class="k">return</span> <span class="p">[</span><span class="n">super_set</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">bitstring</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">bitstring</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;1&#39;</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">graycode_subsets</span><span class="p">(</span><span class="n">gray_code_set</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Generates the subsets as enumerated by a Gray code.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.graycode import graycode_subsets</span>
-<span class="sd">    &gt;&gt;&gt; list(graycode_subsets([&#39;a&#39;,&#39;b&#39;,&#39;c&#39;]))</span>
-<span class="sd">    [[], [&#39;c&#39;], [&#39;b&#39;, &#39;c&#39;], [&#39;b&#39;], [&#39;a&#39;, &#39;b&#39;], [&#39;a&#39;, &#39;b&#39;, &#39;c&#39;], \</span>
-<span class="sd">    [&#39;a&#39;, &#39;c&#39;], [&#39;a&#39;]]</span>
-<span class="sd">    &gt;&gt;&gt; list(graycode_subsets([&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;c&#39;]))</span>
-<span class="sd">    [[], [&#39;c&#39;], [&#39;c&#39;, &#39;c&#39;], [&#39;c&#39;], [&#39;b&#39;, &#39;c&#39;], [&#39;b&#39;, &#39;c&#39;, &#39;c&#39;], \</span>
-<span class="sd">    [&#39;b&#39;, &#39;c&#39;], [&#39;b&#39;], [&#39;a&#39;, &#39;b&#39;], [&#39;a&#39;, &#39;b&#39;, &#39;c&#39;], [&#39;a&#39;, &#39;b&#39;, &#39;c&#39;, &#39;c&#39;], \</span>
-<span class="sd">    [&#39;a&#39;, &#39;b&#39;, &#39;c&#39;], [&#39;a&#39;, &#39;c&#39;], [&#39;a&#39;, &#39;c&#39;, &#39;c&#39;], [&#39;a&#39;, &#39;c&#39;], [&#39;a&#39;]]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    get_subset_from_bitstring</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">for</span> <span class="n">bitstring</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">GrayCode</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">gray_code_set</span><span class="p">))</span><span class="o">.</span><span class="n">generate_gray</span><span class="p">()):</span>
-        <span class="k">yield</span> <span class="n">get_subset_from_bitstring</span><span class="p">(</span><span class="n">gray_code_set</span><span class="p">,</span> <span class="n">bitstring</span><span class="p">)</span>
-</pre></div>
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-            
-  <h1>Source code for sympy.combinatorics.group_constructs</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">uniq</span>
-
-<span class="n">_af_new</span> <span class="o">=</span> <span class="n">Permutation</span><span class="o">.</span><span class="n">_af_new</span>
-
-
-<div class="viewcode-block" id="DirectProduct"><a class="viewcode-back" href="../../../modules/combinatorics/group_constructs.html#sympy.combinatorics.group_constructs.DirectProduct">[docs]</a><span class="k">def</span> <span class="nf">DirectProduct</span><span class="p">(</span><span class="o">*</span><span class="n">groups</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the direct product of several groups as a permutation group.</span>
-
-<span class="sd">    This is implemented much like the __mul__ procedure for taking the direct</span>
-<span class="sd">    product of two permutation groups, but the idea of shifting the</span>
-<span class="sd">    generators is realized in the case of an arbitrary number of groups.</span>
-<span class="sd">    A call to DirectProduct(G1, G2, ..., Gn) is generally expected to be faster</span>
-<span class="sd">    than a call to G1*G2*...*Gn (and thus the need for this algorithm).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.group_constructs import DirectProduct</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.named_groups import CyclicGroup</span>
-<span class="sd">    &gt;&gt;&gt; C = CyclicGroup(4)</span>
-<span class="sd">    &gt;&gt;&gt; G = DirectProduct(C,C,C)</span>
-<span class="sd">    &gt;&gt;&gt; G.order()</span>
-<span class="sd">    64</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    __mul__</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">degrees</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">gens_count</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">total_degree</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">total_gens</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">group</span> <span class="ow">in</span> <span class="n">groups</span><span class="p">:</span>
-        <span class="n">current_deg</span> <span class="o">=</span> <span class="n">group</span><span class="o">.</span><span class="n">degree</span>
-        <span class="n">current_num_gens</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">group</span><span class="o">.</span><span class="n">generators</span><span class="p">)</span>
-        <span class="n">degrees</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">current_deg</span><span class="p">)</span>
-        <span class="n">total_degree</span> <span class="o">+=</span> <span class="n">current_deg</span>
-        <span class="n">gens_count</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">current_num_gens</span><span class="p">)</span>
-        <span class="n">total_gens</span> <span class="o">+=</span> <span class="n">current_num_gens</span>
-    <span class="n">array_gens</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">total_gens</span><span class="p">):</span>
-        <span class="n">array_gens</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">total_degree</span><span class="p">)))</span>
-    <span class="n">current_gen</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">current_deg</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">gens_count</span><span class="p">)):</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">current_gen</span><span class="p">,</span> <span class="n">current_gen</span> <span class="o">+</span> <span class="n">gens_count</span><span class="p">[</span><span class="n">i</span><span class="p">]):</span>
-            <span class="n">gen</span> <span class="o">=</span> <span class="p">((</span><span class="n">groups</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">generators</span><span class="p">)[</span><span class="n">j</span> <span class="o">-</span> <span class="n">current_gen</span><span class="p">])</span><span class="o">.</span><span class="n">array_form</span>
-            <span class="n">array_gens</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">current_deg</span><span class="p">:</span><span class="n">current_deg</span> <span class="o">+</span> <span class="n">degrees</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span> <span class="o">=</span> \
-                <span class="p">[</span> <span class="n">x</span> <span class="o">+</span> <span class="n">current_deg</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">gen</span><span class="p">]</span>
-        <span class="n">current_gen</span> <span class="o">+=</span> <span class="n">gens_count</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-        <span class="n">current_deg</span> <span class="o">+=</span> <span class="n">degrees</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-    <span class="n">perm_gens</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">uniq</span><span class="p">([</span><span class="n">_af_new</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">array_gens</span><span class="p">]))</span>
-    <span class="k">return</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="n">perm_gens</span><span class="p">,</span> <span class="n">dups</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span></div>
-</pre></div>
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-  <h1>Source code for sympy.combinatorics.named_groups</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.group_constructs</span> <span class="kn">import</span> <span class="n">DirectProduct</span>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-
-<span class="n">_af_new</span> <span class="o">=</span> <span class="n">Permutation</span><span class="o">.</span><span class="n">_af_new</span>
-
-
-<div class="viewcode-block" id="AbelianGroup"><a class="viewcode-back" href="../../../modules/combinatorics/named_groups.html#sympy.combinatorics.named_groups.AbelianGroup">[docs]</a><span class="k">def</span> <span class="nf">AbelianGroup</span><span class="p">(</span><span class="o">*</span><span class="n">cyclic_orders</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the direct product of cyclic groups with the given orders.</span>
-
-<span class="sd">    According to the structure theorem for finite abelian groups ([1]),</span>
-<span class="sd">    every finite abelian group can be written as the direct product of finitely</span>
-<span class="sd">    many cyclic groups.</span>
-<span class="sd">    [1] http://groupprops.subwiki.org/wiki/Structure_theorem_for_finitely_generated_abelian_groups</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.named_groups import AbelianGroup</span>
-<span class="sd">    &gt;&gt;&gt; AbelianGroup(3, 4)</span>
-<span class="sd">    PermutationGroup([</span>
-<span class="sd">            Permutation(6)(0, 1, 2),</span>
-<span class="sd">            Permutation(3, 4, 5, 6)])</span>
-<span class="sd">    &gt;&gt;&gt; _.is_group()</span>
-<span class="sd">    False</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    DirectProduct</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">groups</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">degree</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">order</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="k">for</span> <span class="n">size</span> <span class="ow">in</span> <span class="n">cyclic_orders</span><span class="p">:</span>
-        <span class="n">degree</span> <span class="o">+=</span> <span class="n">size</span>
-        <span class="n">order</span> <span class="o">*=</span> <span class="n">size</span>
-        <span class="n">groups</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">CyclicGroup</span><span class="p">(</span><span class="n">size</span><span class="p">))</span>
-    <span class="n">G</span> <span class="o">=</span> <span class="n">DirectProduct</span><span class="p">(</span><span class="o">*</span><span class="n">groups</span><span class="p">)</span>
-    <span class="n">G</span><span class="o">.</span><span class="n">_is_abelian</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">G</span><span class="o">.</span><span class="n">_degree</span> <span class="o">=</span> <span class="n">degree</span>
-    <span class="n">G</span><span class="o">.</span><span class="n">_order</span> <span class="o">=</span> <span class="n">order</span>
-
-    <span class="k">return</span> <span class="n">G</span>
-
-</div>
-<div class="viewcode-block" id="AlternatingGroup"><a class="viewcode-back" href="../../../modules/combinatorics/named_groups.html#sympy.combinatorics.named_groups.AlternatingGroup">[docs]</a><span class="k">def</span> <span class="nf">AlternatingGroup</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Generates the alternating group on ``n`` elements as a permutation group.</span>
-
-<span class="sd">    For ``n &gt; 2``, the generators taken are ``(0 1 2), (0 1 2 ... n-1)`` for</span>
-<span class="sd">    ``n`` odd</span>
-<span class="sd">    and ``(0 1 2), (1 2 ... n-1)`` for ``n`` even (See [1], p.31, ex.6.9.).</span>
-<span class="sd">    After the group is generated, some of its basic properties are set.</span>
-<span class="sd">    The cases ``n = 1, 2`` are handled separately.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.named_groups import AlternatingGroup</span>
-<span class="sd">    &gt;&gt;&gt; G = AlternatingGroup(4)</span>
-<span class="sd">    &gt;&gt;&gt; G.is_group()</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; a = list(G.generate_dimino())</span>
-<span class="sd">    &gt;&gt;&gt; len(a)</span>
-<span class="sd">    12</span>
-<span class="sd">    &gt;&gt;&gt; all(perm.is_even for perm in a)</span>
-<span class="sd">    True</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    SymmetricGroup, CyclicGroup, DihedralGroup</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] Armstrong, M. &quot;Groups and Symmetry&quot;</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># small cases are special</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="ow">in</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">])])</span>
-
-    <span class="n">a</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-    <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">a</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">a</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">gen1</span> <span class="o">=</span> <span class="n">a</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-        <span class="n">a</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="n">gen2</span> <span class="o">=</span> <span class="n">a</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-        <span class="n">a</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-        <span class="n">a</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="n">gen2</span> <span class="o">=</span> <span class="n">a</span>
-    <span class="n">gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">gen1</span><span class="p">,</span> <span class="n">gen2</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">gen1</span> <span class="o">==</span> <span class="n">gen2</span><span class="p">:</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="n">gens</span><span class="p">[:</span><span class="mi">1</span><span class="p">]</span>
-    <span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">_af_new</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">],</span> <span class="n">dups</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">4</span><span class="p">:</span>
-        <span class="n">G</span><span class="o">.</span><span class="n">_is_abelian</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">G</span><span class="o">.</span><span class="n">_is_abelian</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">G</span><span class="o">.</span><span class="n">_degree</span> <span class="o">=</span> <span class="n">n</span>
-    <span class="n">G</span><span class="o">.</span><span class="n">_is_transitive</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">G</span><span class="o">.</span><span class="n">_is_alt</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="k">return</span> <span class="n">G</span>
-
-</div>
-<div class="viewcode-block" id="CyclicGroup"><a class="viewcode-back" href="../../../modules/combinatorics/named_groups.html#sympy.combinatorics.named_groups.CyclicGroup">[docs]</a><span class="k">def</span> <span class="nf">CyclicGroup</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Generates the cyclic group of order ``n`` as a permutation group.</span>
-
-<span class="sd">    The generator taken is the ``n``-cycle ``(0 1 2 ... n-1)``</span>
-<span class="sd">    (in cycle notation). After the group is generated, some of its basic</span>
-<span class="sd">    properties are set.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.named_groups import CyclicGroup</span>
-<span class="sd">    &gt;&gt;&gt; G = CyclicGroup(6)</span>
-<span class="sd">    &gt;&gt;&gt; G.is_group()</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; G.order()</span>
-<span class="sd">    6</span>
-<span class="sd">    &gt;&gt;&gt; list(G.generate_schreier_sims(af=True))</span>
-<span class="sd">    [[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1],</span>
-<span class="sd">    [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    SymmetricGroup, DihedralGroup, AlternatingGroup</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-    <span class="n">a</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-    <span class="n">gen</span> <span class="o">=</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-    <span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">gen</span><span class="p">])</span>
-
-    <span class="n">G</span><span class="o">.</span><span class="n">_is_abelian</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">G</span><span class="o">.</span><span class="n">_degree</span> <span class="o">=</span> <span class="n">n</span>
-    <span class="n">G</span><span class="o">.</span><span class="n">_is_transitive</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">G</span><span class="o">.</span><span class="n">_order</span> <span class="o">=</span> <span class="n">n</span>
-    <span class="k">return</span> <span class="n">G</span>
-
-</div>
-<div class="viewcode-block" id="DihedralGroup"><a class="viewcode-back" href="../../../modules/combinatorics/named_groups.html#sympy.combinatorics.named_groups.DihedralGroup">[docs]</a><span class="k">def</span> <span class="nf">DihedralGroup</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Generates the dihedral group `D_n` as a permutation group.</span>
-
-<span class="sd">    The dihedral group `D_n` is the group of symmetries of the regular</span>
-<span class="sd">    ``n``-gon. The generators taken are the ``n``-cycle ``a = (0 1 2 ... n-1)``</span>
-<span class="sd">    (a rotation of the ``n``-gon) and ``b = (0 n-1)(1 n-2)...``</span>
-<span class="sd">    (a reflection of the ``n``-gon) in cycle rotation. It is easy to see that</span>
-<span class="sd">    these satisfy ``a**n = b**2 = 1`` and ``bab = ~a`` so they indeed generate</span>
-<span class="sd">    `D_n` (See [1]). After the group is generated, some of its basic properties</span>
-<span class="sd">    are set.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.named_groups import DihedralGroup</span>
-<span class="sd">    &gt;&gt;&gt; G = DihedralGroup(5)</span>
-<span class="sd">    &gt;&gt;&gt; G.is_group()</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; a = list(G.generate_dimino())</span>
-<span class="sd">    &gt;&gt;&gt; [perm.cyclic_form for perm in a]</span>
-<span class="sd">    [[], [[0, 1, 2, 3, 4]], [[0, 2, 4, 1, 3]],</span>
-<span class="sd">    [[0, 3, 1, 4, 2]], [[0, 4, 3, 2, 1]], [[0, 4], [1, 3]],</span>
-<span class="sd">    [[1, 4], [2, 3]], [[0, 1], [2, 4]], [[0, 2], [3, 4]],</span>
-<span class="sd">    [[0, 3], [1, 2]]]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    SymmetricGroup, CyclicGroup, AlternatingGroup</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Dihedral_group</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># small cases are special</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])])</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">]),</span>
-               <span class="n">Permutation</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]),</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])])</span>
-
-    <span class="n">a</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-    <span class="n">a</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-    <span class="n">gen1</span> <span class="o">=</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-    <span class="n">a</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-    <span class="n">gen2</span> <span class="o">=</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-    <span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">gen1</span><span class="p">,</span> <span class="n">gen2</span><span class="p">])</span>
-
-    <span class="n">G</span><span class="o">.</span><span class="n">_is_abelian</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">G</span><span class="o">.</span><span class="n">_degree</span> <span class="o">=</span> <span class="n">n</span>
-    <span class="n">G</span><span class="o">.</span><span class="n">_is_transitive</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">G</span><span class="o">.</span><span class="n">_order</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">n</span>
-    <span class="k">return</span> <span class="n">G</span>
-
-</div>
-<div class="viewcode-block" id="SymmetricGroup"><a class="viewcode-back" href="../../../modules/combinatorics/named_groups.html#sympy.combinatorics.named_groups.SymmetricGroup">[docs]</a><span class="k">def</span> <span class="nf">SymmetricGroup</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Generates the symmetric group on ``n`` elements as a permutation group.</span>
-
-<span class="sd">    The generators taken are the ``n``-cycle</span>
-<span class="sd">    ``(0 1 2 ... n-1)`` and the transposition ``(0 1)`` (in cycle notation).</span>
-<span class="sd">    (See [1]). After the group is generated, some of its basic properties</span>
-<span class="sd">    are set.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.named_groups import SymmetricGroup</span>
-<span class="sd">    &gt;&gt;&gt; G = SymmetricGroup(4)</span>
-<span class="sd">    &gt;&gt;&gt; G.is_group()</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; G.order()</span>
-<span class="sd">    24</span>
-<span class="sd">    &gt;&gt;&gt; list(G.generate_schreier_sims(af=True))</span>
-<span class="sd">    [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 1, 2, 0], [0, 2, 3, 1],</span>
-<span class="sd">    [1, 3, 0, 2], [2, 0, 1, 3], [3, 2, 0, 1], [0, 3, 1, 2], [1, 0, 2, 3],</span>
-<span class="sd">    [2, 1, 3, 0], [3, 0, 1, 2], [0, 1, 3, 2], [1, 2, 0, 3], [2, 3, 1, 0],</span>
-<span class="sd">    [3, 1, 0, 2], [0, 2, 1, 3], [1, 3, 2, 0], [2, 0, 3, 1], [3, 2, 1, 0],</span>
-<span class="sd">    [0, 3, 2, 1], [1, 0, 3, 2], [2, 1, 0, 3], [3, 0, 2, 1]]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    CyclicGroup, DihedralGroup, AlternatingGroup</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Symmetric_group#Generators_and_relations</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">])])</span>
-    <span class="k">elif</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])])</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-        <span class="n">a</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="n">gen1</span> <span class="o">=</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-        <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">gen2</span> <span class="o">=</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">gen1</span><span class="p">,</span> <span class="n">gen2</span><span class="p">])</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">3</span><span class="p">:</span>
-        <span class="n">G</span><span class="o">.</span><span class="n">_is_abelian</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">G</span><span class="o">.</span><span class="n">_is_abelian</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">G</span><span class="o">.</span><span class="n">_degree</span> <span class="o">=</span> <span class="n">n</span>
-    <span class="n">G</span><span class="o">.</span><span class="n">_is_transitive</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">G</span><span class="o">.</span><span class="n">_is_sym</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="k">return</span> <span class="n">G</span>
-
-</div>
-<span class="k">def</span> <span class="nf">RubikGroup</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return a group of Rubik&#39;s cube generators.</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.named_groups import RubikGroup</span>
-<span class="sd">    &gt;&gt;&gt; RubikGroup(2).is_group()</span>
-<span class="sd">    False</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.combinatorics.generators</span> <span class="kn">import</span> <span class="n">rubik</span>
-    <span class="k">assert</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">1</span>
-    <span class="k">return</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="n">rubik</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.combinatorics.partitions</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">Dict</span><span class="p">,</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">as_int</span><span class="p">,</span> <span class="n">default_sort_key</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.combinatorial.numbers</span> <span class="kn">import</span> <span class="n">bell</span>
-<span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">zeros</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">has_dups</span><span class="p">,</span> <span class="n">flatten</span><span class="p">,</span> <span class="n">group</span>
-
-<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</span>
-
-
-<div class="viewcode-block" id="Partition"><a class="viewcode-back" href="../../../modules/combinatorics/partitions.html#sympy.combinatorics.partitions.Partition">[docs]</a><span class="k">class</span> <span class="nc">Partition</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">FiniteSet</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    This class represents an abstract partition.</span>
-
-<span class="sd">    A partition is a set of disjoint sets whose union equals a given set.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.utilities.iterables.partitions,</span>
-<span class="sd">    sympy.utilities.iterables.multiset_partitions</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_rank</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">_partition</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">partition</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Generates a new partition object.</span>
-
-<span class="sd">        This method also verifies if the arguments passed are</span>
-<span class="sd">        valid and raises a ValueError if they are not.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.partitions import Partition</span>
-<span class="sd">        &gt;&gt;&gt; a = Partition([[1, 2], [3]])</span>
-<span class="sd">        &gt;&gt;&gt; a</span>
-<span class="sd">        {{1, 2}, {3}}</span>
-<span class="sd">        &gt;&gt;&gt; a.partition</span>
-<span class="sd">        [[1, 2], [3]]</span>
-<span class="sd">        &gt;&gt;&gt; len(a)</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; a.members</span>
-<span class="sd">        (1, 2, 3)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="n">partition</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">part</span><span class="p">,</span> <span class="nb">list</span><span class="p">)</span> <span class="k">for</span> <span class="n">part</span> <span class="ow">in</span> <span class="n">args</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Partition should be a list of lists.&quot;</span><span class="p">)</span>
-
-        <span class="c"># sort so we have a canonical reference for RGS</span>
-        <span class="n">partition</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="nb">sum</span><span class="p">(</span><span class="n">partition</span><span class="p">,</span> <span class="p">[]),</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">has_dups</span><span class="p">(</span><span class="n">partition</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Partition contained duplicated elements.&quot;</span><span class="p">)</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">FiniteSet</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">FiniteSet</span><span class="p">,</span> <span class="n">args</span><span class="p">)))</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">members</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">partition</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">size</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">partition</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-<div class="viewcode-block" id="Partition.sort_key"><a class="viewcode-back" href="../../../modules/combinatorics/partitions.html#sympy.combinatorics.partitions.Partition.sort_key">[docs]</a>    <span class="k">def</span> <span class="nf">sort_key</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return a canonical key that can be used for sorting.</span>
-
-<span class="sd">        Ordering is based on the size and sorted elements of the partition</span>
-<span class="sd">        and ties are broken with the rank.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.utilities.iterables import default_sort_key</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.partitions import Partition</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; a = Partition([[1, 2]])</span>
-<span class="sd">        &gt;&gt;&gt; b = Partition([[3, 4]])</span>
-<span class="sd">        &gt;&gt;&gt; c = Partition([[1, x]])</span>
-<span class="sd">        &gt;&gt;&gt; d = Partition([list(range(4))])</span>
-<span class="sd">        &gt;&gt;&gt; l = [d, b, a + 1, a, c]</span>
-<span class="sd">        &gt;&gt;&gt; l.sort(key=default_sort_key); l</span>
-<span class="sd">        [{{1, 2}}, {{1}, {2}}, {{1, x}}, {{3, 4}}, {{0, 1, 2, 3}}]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">order</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">members</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">members</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">members</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">members</span><span class="p">,</span>
-                             <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">w</span><span class="p">:</span> <span class="n">default_sort_key</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">order</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">,</span> <span class="n">members</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rank</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Partition.partition"><a class="viewcode-back" href="../../../modules/combinatorics/partitions.html#sympy.combinatorics.partitions.Partition.partition">[docs]</a>    <span class="k">def</span> <span class="nf">partition</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return partition as a sorted list of lists.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.partitions import Partition</span>
-<span class="sd">        &gt;&gt;&gt; Partition([[1], [2, 3]]).partition</span>
-<span class="sd">        [[1], [2, 3]]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_partition</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_partition</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_partition</span>
-</div>
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return permutation whose rank is ``other`` greater than current rank,</span>
-<span class="sd">        (mod the maximum rank for the set).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.partitions import Partition</span>
-<span class="sd">        &gt;&gt;&gt; a = Partition([[1, 2], [3]])</span>
-<span class="sd">        &gt;&gt;&gt; a.rank</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; (a + 1).rank</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; (a + 100).rank</span>
-<span class="sd">        1</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">other</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="n">offset</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rank</span> <span class="o">+</span> <span class="n">other</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">RGS_unrank</span><span class="p">((</span><span class="n">offset</span><span class="p">)</span> <span class="o">%</span>
-                            <span class="n">RGS_enum</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">),</span>
-                            <span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Partition</span><span class="o">.</span><span class="n">from_rgs</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">members</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return permutation whose rank is ``other`` less than current rank,</span>
-<span class="sd">        (mod the maximum rank for the set).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.partitions import Partition</span>
-<span class="sd">        &gt;&gt;&gt; a = Partition([[1, 2], [3]])</span>
-<span class="sd">        &gt;&gt;&gt; a.rank</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; (a - 1).rank</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; (a - 100).rank</span>
-<span class="sd">        1</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__add__</span><span class="p">(</span><span class="o">-</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__le__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Checks if a partition is less than or equal to</span>
-<span class="sd">        the other based on rank.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.partitions import Partition</span>
-<span class="sd">        &gt;&gt;&gt; a = Partition([[1, 2], [3, 4, 5]])</span>
-<span class="sd">        &gt;&gt;&gt; b = Partition([[1], [2, 3], [4], [5]])</span>
-<span class="sd">        &gt;&gt;&gt; a.rank, b.rank</span>
-<span class="sd">        (9, 34)</span>
-<span class="sd">        &gt;&gt;&gt; a &lt;= a</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; a &lt;= b</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">sort_key</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span><span class="o">.</span><span class="n">sort_key</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Checks if a partition is less than the other.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.partitions import Partition</span>
-<span class="sd">        &gt;&gt;&gt; a = Partition([[1, 2], [3, 4, 5]])</span>
-<span class="sd">        &gt;&gt;&gt; b = Partition([[1], [2, 3], [4], [5]])</span>
-<span class="sd">        &gt;&gt;&gt; a.rank, b.rank</span>
-<span class="sd">        (9, 34)</span>
-<span class="sd">        &gt;&gt;&gt; a &lt; b</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">sort_key</span><span class="p">()</span> <span class="o">&lt;</span> <span class="n">sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span><span class="o">.</span><span class="n">sort_key</span><span class="p">()</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Partition.rank"><a class="viewcode-back" href="../../../modules/combinatorics/partitions.html#sympy.combinatorics.partitions.Partition.rank">[docs]</a>    <span class="k">def</span> <span class="nf">rank</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Gets the rank of a partition.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.partitions import Partition</span>
-<span class="sd">        &gt;&gt;&gt; a = Partition([[1, 2], [3], [4, 5]])</span>
-<span class="sd">        &gt;&gt;&gt; a.rank</span>
-<span class="sd">        13</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rank</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rank</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_rank</span> <span class="o">=</span> <span class="n">RGS_rank</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">RGS</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rank</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Partition.RGS"><a class="viewcode-back" href="../../../modules/combinatorics/partitions.html#sympy.combinatorics.partitions.Partition.RGS">[docs]</a>    <span class="k">def</span> <span class="nf">RGS</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the &quot;restricted growth string&quot; of the partition.</span>
-
-<span class="sd">        The RGS is returned as a list of indices, L, where L[i] indicates</span>
-<span class="sd">        the block in which element i appears. For example, in a partition</span>
-<span class="sd">        of 3 elements (a, b, c) into 2 blocks ([c], [a, b]) the RGS is</span>
-<span class="sd">        [1, 1, 0]: &quot;a&quot; is in block 1, &quot;b&quot; is in block 1 and &quot;c&quot; is in block 0.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.partitions import Partition</span>
-<span class="sd">        &gt;&gt;&gt; a = Partition([[1, 2], [3], [4, 5]])</span>
-<span class="sd">        &gt;&gt;&gt; a.members</span>
-<span class="sd">        (1, 2, 3, 4, 5)</span>
-<span class="sd">        &gt;&gt;&gt; a.RGS</span>
-<span class="sd">        (0, 0, 1, 2, 2)</span>
-<span class="sd">        &gt;&gt;&gt; a + 1</span>
-<span class="sd">        {{1, 2}, {3}, {4}, {5}}</span>
-<span class="sd">        &gt;&gt;&gt; _.RGS</span>
-<span class="sd">        (0, 0, 1, 2, 3)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">rgs</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="n">partition</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">partition</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">part</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">partition</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">part</span><span class="p">:</span>
-                <span class="n">rgs</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">i</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">([</span><span class="n">rgs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">i</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">partition</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">p</span><span class="p">)])</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Partition.from_rgs"><a class="viewcode-back" href="../../../modules/combinatorics/partitions.html#sympy.combinatorics.partitions.Partition.from_rgs">[docs]</a>    <span class="k">def</span> <span class="nf">from_rgs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rgs</span><span class="p">,</span> <span class="n">elements</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Creates a set partition from a restricted growth string.</span>
-
-<span class="sd">        The indices given in rgs are assumed to be the index</span>
-<span class="sd">        of the element as given in elements *as provided* (the</span>
-<span class="sd">        elements are not sorted by this routine). Block numbering</span>
-<span class="sd">        starts from 0. If any block was not referenced in ``rgs``</span>
-<span class="sd">        an error will be raised.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.partitions import Partition</span>
-<span class="sd">        &gt;&gt;&gt; Partition.from_rgs([0, 1, 2, 0, 1], list(&#39;abcde&#39;))</span>
-<span class="sd">        {{c}, {a, d}, {b, e}}</span>
-<span class="sd">        &gt;&gt;&gt; Partition.from_rgs([0, 1, 2, 0, 1], list(&#39;cbead&#39;))</span>
-<span class="sd">        {{e}, {a, c}, {b, d}}</span>
-<span class="sd">        &gt;&gt;&gt; a = Partition([[1, 4], [2], [3, 5]])</span>
-<span class="sd">        &gt;&gt;&gt; Partition.from_rgs(a.RGS, a.members)</span>
-<span class="sd">        {{1, 4}, {2}, {3, 5}}</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">rgs</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">elements</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;mismatch in rgs and element lengths&#39;</span><span class="p">)</span>
-        <span class="n">max_elem</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">rgs</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="n">partition</span> <span class="o">=</span> <span class="p">[[]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">max_elem</span><span class="p">)]</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">rgs</span><span class="p">:</span>
-            <span class="n">partition</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">elements</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-            <span class="n">j</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">(</span><span class="n">p</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">partition</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;some blocks of the partition were empty.&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Partition</span><span class="p">(</span><span class="n">partition</span><span class="p">)</span>
-
-</div></div>
-<div class="viewcode-block" id="IntegerPartition"><a class="viewcode-back" href="../../../modules/combinatorics/partitions.html#sympy.combinatorics.partitions.IntegerPartition">[docs]</a><span class="k">class</span> <span class="nc">IntegerPartition</span><span class="p">(</span><span class="n">Basic</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    This class represents an integer partition.</span>
-
-<span class="sd">    In number theory and combinatorics, a partition of a positive integer,</span>
-<span class="sd">    ``n``, also called an integer partition, is a way of writing ``n`` as a</span>
-<span class="sd">    list of positive integers that sum to n. Two partitions that differ only</span>
-<span class="sd">    in the order of summands are considered to be the same partition; if order</span>
-<span class="sd">    matters then the partitions are referred to as compositions. For example,</span>
-<span class="sd">    4 has five partitions: [4], [3, 1], [2, 2], [2, 1, 1], and [1, 1, 1, 1];</span>
-<span class="sd">    the compositions [1, 2, 1] and [1, 1, 2] are the same as partition</span>
-<span class="sd">    [2, 1, 1].</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.utilities.iterables.partitions,</span>
-<span class="sd">    sympy.utilities.iterables.multiset_partitions</span>
-
-<span class="sd">    Reference: http://en.wikipedia.org/wiki/Partition_%28number_theory%29</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_dict</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">_keys</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">partition</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Generates a new IntegerPartition object from a list or dictionary.</span>
-
-<span class="sd">        The partition can be given as a list of positive integers or a</span>
-<span class="sd">        dictionary of (integer, multiplicity) items. If the partition is</span>
-<span class="sd">        preceeded by an integer an error will be raised if the partition</span>
-<span class="sd">        does not sum to that given integer.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.partitions import IntegerPartition</span>
-<span class="sd">        &gt;&gt;&gt; a = IntegerPartition([5, 4, 3, 1, 1])</span>
-<span class="sd">        &gt;&gt;&gt; a</span>
-<span class="sd">        IntegerPartition(14, (5, 4, 3, 1, 1))</span>
-<span class="sd">        &gt;&gt;&gt; print(a)</span>
-<span class="sd">        [5, 4, 3, 1, 1]</span>
-<span class="sd">        &gt;&gt;&gt; IntegerPartition({1:3, 2:1})</span>
-<span class="sd">        IntegerPartition(5, (2, 1, 1, 1))</span>
-
-<span class="sd">        If the value that the partion should sum to is given first, a check</span>
-<span class="sd">        will be made to see n error will be raised if there is a discrepancy:</span>
-<span class="sd">        &gt;&gt;&gt; IntegerPartition(10, [5, 4, 3, 1])</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">        ...</span>
-<span class="sd">        ValueError: The partition is not valid</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">integer</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">integer</span><span class="p">,</span> <span class="n">partition</span> <span class="o">=</span> <span class="n">partition</span><span class="p">,</span> <span class="n">integer</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">partition</span><span class="p">,</span> <span class="p">(</span><span class="nb">dict</span><span class="p">,</span> <span class="n">Dict</span><span class="p">)):</span>
-            <span class="n">_</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">partition</span><span class="o">.</span><span class="n">items</span><span class="p">()),</span> <span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">v</span><span class="p">:</span>
-                    <span class="k">continue</span>
-                <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">k</span><span class="p">),</span> <span class="n">as_int</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-                <span class="n">_</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">v</span><span class="p">)</span>
-            <span class="n">partition</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">_</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">partition</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">as_int</span><span class="p">,</span> <span class="n">partition</span><span class="p">),</span> <span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-        <span class="n">sum_ok</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">integer</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">integer</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">partition</span><span class="p">)</span>
-            <span class="n">sum_ok</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">integer</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">integer</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">sum_ok</span> <span class="ow">and</span> <span class="nb">sum</span><span class="p">(</span><span class="n">partition</span><span class="p">)</span> <span class="o">!=</span> <span class="n">integer</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Partition did not add to </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">integer</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">i</span> <span class="o">&lt;</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">partition</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;The summands must all be positive.&quot;</span><span class="p">)</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">integer</span><span class="p">,</span> <span class="n">partition</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">partition</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">partition</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">integer</span> <span class="o">=</span> <span class="n">integer</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-<div class="viewcode-block" id="IntegerPartition.prev_lex"><a class="viewcode-back" href="../../../modules/combinatorics/partitions.html#sympy.combinatorics.partitions.IntegerPartition.prev_lex">[docs]</a>    <span class="k">def</span> <span class="nf">prev_lex</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the previous partition of the integer, n, in lexical order,</span>
-<span class="sd">        wrapping around to [1, ..., 1] if the partition is [n].</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.partitions import IntegerPartition</span>
-<span class="sd">        &gt;&gt;&gt; p = IntegerPartition([4])</span>
-<span class="sd">        &gt;&gt;&gt; print(p.prev_lex())</span>
-<span class="sd">        [3, 1]</span>
-<span class="sd">        &gt;&gt;&gt; p.partition &gt; p.prev_lex().partition</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
-        <span class="n">d</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">as_dict</span><span class="p">())</span>
-        <span class="n">keys</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_keys</span>
-        <span class="k">if</span> <span class="n">keys</span> <span class="o">==</span> <span class="p">[</span><span class="mi">1</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="n">IntegerPartition</span><span class="p">({</span><span class="bp">self</span><span class="o">.</span><span class="n">integer</span><span class="p">:</span> <span class="mi">1</span><span class="p">})</span>
-        <span class="k">if</span> <span class="n">keys</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">d</span><span class="p">[</span><span class="n">keys</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="n">keys</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">d</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">2</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">d</span><span class="p">[</span><span class="n">keys</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">d</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">d</span><span class="p">[</span><span class="n">keys</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]]</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="n">left</span> <span class="o">=</span> <span class="n">d</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">keys</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span>
-            <span class="n">new</span> <span class="o">=</span> <span class="n">keys</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span>
-            <span class="n">d</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">while</span> <span class="n">left</span><span class="p">:</span>
-                <span class="n">new</span> <span class="o">-=</span> <span class="mi">1</span>
-                <span class="k">if</span> <span class="n">left</span> <span class="o">-</span> <span class="n">new</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">d</span><span class="p">[</span><span class="n">new</span><span class="p">]</span> <span class="o">+=</span> <span class="n">left</span><span class="o">//</span><span class="n">new</span>
-                    <span class="n">left</span> <span class="o">-=</span> <span class="n">d</span><span class="p">[</span><span class="n">new</span><span class="p">]</span><span class="o">*</span><span class="n">new</span>
-        <span class="k">return</span> <span class="n">IntegerPartition</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">integer</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="IntegerPartition.next_lex"><a class="viewcode-back" href="../../../modules/combinatorics/partitions.html#sympy.combinatorics.partitions.IntegerPartition.next_lex">[docs]</a>    <span class="k">def</span> <span class="nf">next_lex</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the next partition of the integer, n, in lexical order,</span>
-<span class="sd">        wrapping around to [n] if the partition is [1, ..., 1].</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.partitions import IntegerPartition</span>
-<span class="sd">        &gt;&gt;&gt; p = IntegerPartition([3, 1])</span>
-<span class="sd">        &gt;&gt;&gt; print(p.next_lex())</span>
-<span class="sd">        [4]</span>
-<span class="sd">        &gt;&gt;&gt; p.partition &lt; p.next_lex().partition</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
-        <span class="n">d</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">as_dict</span><span class="p">())</span>
-        <span class="n">key</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_keys</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">key</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">integer</span><span class="p">:</span>
-            <span class="n">d</span><span class="o">.</span><span class="n">clear</span><span class="p">()</span>
-            <span class="n">d</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">integer</span>
-        <span class="k">elif</span> <span class="n">a</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">d</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">d</span><span class="p">[</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="n">d</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">2</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="n">key</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span>
-                <span class="n">d</span><span class="p">[</span><span class="n">b</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="n">d</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">d</span><span class="p">[</span><span class="n">b</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">b</span>
-                <span class="n">d</span><span class="p">[</span><span class="n">b</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">d</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">key</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">d</span><span class="o">.</span><span class="n">clear</span><span class="p">()</span>
-                    <span class="n">d</span><span class="p">[</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-                    <span class="n">d</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">integer</span> <span class="o">-</span> <span class="n">a</span> <span class="o">-</span> <span class="mi">1</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">a1</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">1</span>
-                    <span class="n">d</span><span class="p">[</span><span class="n">a1</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-                    <span class="n">d</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">d</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="n">a1</span>
-                    <span class="n">d</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="n">key</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span>
-                <span class="n">b1</span> <span class="o">=</span> <span class="n">b</span> <span class="o">+</span> <span class="mi">1</span>
-                <span class="n">d</span><span class="p">[</span><span class="n">b1</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="n">need</span> <span class="o">=</span> <span class="n">d</span><span class="p">[</span><span class="n">b</span><span class="p">]</span><span class="o">*</span><span class="n">b</span> <span class="o">+</span> <span class="n">d</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="n">b1</span>
-                <span class="n">d</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="n">d</span><span class="p">[</span><span class="n">b</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="n">d</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">need</span>
-        <span class="k">return</span> <span class="n">IntegerPartition</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">integer</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="IntegerPartition.as_dict"><a class="viewcode-back" href="../../../modules/combinatorics/partitions.html#sympy.combinatorics.partitions.IntegerPartition.as_dict">[docs]</a>    <span class="k">def</span> <span class="nf">as_dict</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the partition as a dictionary whose keys are the</span>
-<span class="sd">        partition integers and the values are the multiplicity of that</span>
-<span class="sd">        integer.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.partitions import IntegerPartition</span>
-<span class="sd">        &gt;&gt;&gt; IntegerPartition([1]*3 + [2] + [3]*4).as_dict()</span>
-<span class="sd">        {1: 3, 2: 1, 3: 4}</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_dict</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">groups</span> <span class="o">=</span> <span class="n">group</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">partition</span><span class="p">,</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_keys</span> <span class="o">=</span> <span class="p">[</span><span class="n">g</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">groups</span><span class="p">]</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_dict</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">groups</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_dict</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="IntegerPartition.conjugate"><a class="viewcode-back" href="../../../modules/combinatorics/partitions.html#sympy.combinatorics.partitions.IntegerPartition.conjugate">[docs]</a>    <span class="k">def</span> <span class="nf">conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes the conjugate partition of itself.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.partitions import IntegerPartition</span>
-<span class="sd">        &gt;&gt;&gt; a = IntegerPartition([6, 3, 3, 2, 1])</span>
-<span class="sd">        &gt;&gt;&gt; a.conjugate</span>
-<span class="sd">        [5, 4, 3, 1, 1, 1]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">temp_arr</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">partition</span><span class="p">)</span> <span class="o">+</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">temp_arr</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">k</span>
-        <span class="k">while</span> <span class="n">k</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">while</span> <span class="n">k</span> <span class="o">&gt;</span> <span class="n">temp_arr</span><span class="p">[</span><span class="n">j</span><span class="p">]:</span>
-                <span class="n">b</span><span class="p">[</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">j</span>
-                <span class="n">k</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="n">j</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="n">b</span>
-</div>
-    <span class="k">def</span> <span class="nf">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return True if self is less than other when the partition</span>
-<span class="sd">        is listed from smallest to biggest.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.partitions import IntegerPartition</span>
-<span class="sd">        &gt;&gt;&gt; a = IntegerPartition([3, 1])</span>
-<span class="sd">        &gt;&gt;&gt; a &lt; a</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; b = a.next_lex()</span>
-<span class="sd">        &gt;&gt;&gt; a &lt; b</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; a == b</span>
-<span class="sd">        False</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">partition</span><span class="p">))</span> <span class="o">&lt;</span> <span class="nb">list</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">partition</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__le__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return True if self is less than other when the partition</span>
-<span class="sd">        is listed from smallest to biggest.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.partitions import IntegerPartition</span>
-<span class="sd">        &gt;&gt;&gt; a = IntegerPartition([4])</span>
-<span class="sd">        &gt;&gt;&gt; a &lt;= a</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">partition</span><span class="p">))</span> <span class="o">&lt;=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">partition</span><span class="p">))</span>
-
-<div class="viewcode-block" id="IntegerPartition.as_ferrers"><a class="viewcode-back" href="../../../modules/combinatorics/partitions.html#sympy.combinatorics.partitions.IntegerPartition.as_ferrers">[docs]</a>    <span class="k">def</span> <span class="nf">as_ferrers</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">char</span><span class="o">=</span><span class="s">&#39;#&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Prints the ferrer diagram of a partition.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.partitions import IntegerPartition</span>
-<span class="sd">        &gt;&gt;&gt; print(IntegerPartition([1, 1, 5]).as_ferrers())</span>
-<span class="sd">        #####</span>
-<span class="sd">        #</span>
-<span class="sd">        #</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="n">char</span><span class="o">*</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">partition</span><span class="p">])</span>
-</div>
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">partition</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="random_integer_partition"><a class="viewcode-back" href="../../../modules/combinatorics/partitions.html#sympy.combinatorics.partitions.random_integer_partition">[docs]</a><span class="k">def</span> <span class="nf">random_integer_partition</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Generates a random integer partition summing to ``n`` as a list</span>
-<span class="sd">    of reverse-sorted integers.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.partitions import random_integer_partition</span>
-
-<span class="sd">    For the following, a seed is given so a known value can be shown; in</span>
-<span class="sd">    practice, the seed would not be given.</span>
-
-<span class="sd">    &gt;&gt;&gt; random_integer_partition(100, seed=[1, 1, 12, 1, 2, 1, 85, 1])</span>
-<span class="sd">    [85, 12, 2, 1]</span>
-<span class="sd">    &gt;&gt;&gt; random_integer_partition(10, seed=[1, 2, 3, 1, 5, 1])</span>
-<span class="sd">    [5, 3, 1, 1]</span>
-<span class="sd">    &gt;&gt;&gt; random_integer_partition(1)</span>
-<span class="sd">    [1]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.utilities.randtest</span> <span class="kn">import</span> <span class="n">_randint</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;n must be a positive integer&#39;</span><span class="p">)</span>
-
-    <span class="n">randint</span> <span class="o">=</span> <span class="n">_randint</span><span class="p">(</span><span class="n">seed</span><span class="p">)</span>
-
-    <span class="n">partition</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">while</span> <span class="p">(</span><span class="n">n</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">):</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">randint</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-        <span class="n">mult</span> <span class="o">=</span> <span class="n">randint</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">//</span><span class="n">k</span><span class="p">)</span>
-        <span class="n">partition</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">k</span><span class="p">,</span> <span class="n">mult</span><span class="p">))</span>
-        <span class="n">n</span> <span class="o">-=</span> <span class="n">k</span><span class="o">*</span><span class="n">mult</span>
-    <span class="n">partition</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">partition</span> <span class="o">=</span> <span class="n">flatten</span><span class="p">([[</span><span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">m</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">partition</span><span class="p">])</span>
-    <span class="k">return</span> <span class="n">partition</span>
-
-</div>
-<div class="viewcode-block" id="RGS_generalized"><a class="viewcode-back" href="../../../modules/combinatorics/partitions.html#sympy.combinatorics.partitions.RGS_generalized">[docs]</a><span class="k">def</span> <span class="nf">RGS_generalized</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes the m + 1 generalized unrestricted growth strings</span>
-<span class="sd">    and returns them as rows in matrix.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.partitions import RGS_generalized</span>
-<span class="sd">    &gt;&gt;&gt; RGS_generalized(6)</span>
-<span class="sd">    [  1,   1,   1,  1,  1, 1, 1]</span>
-<span class="sd">    [  1,   2,   3,  4,  5, 6, 0]</span>
-<span class="sd">    [  2,   5,  10, 17, 26, 0, 0]</span>
-<span class="sd">    [  5,  15,  37, 77,  0, 0, 0]</span>
-<span class="sd">    [ 15,  52, 151,  0,  0, 0, 0]</span>
-<span class="sd">    [ 52, 203,   0,  0,  0, 0, 0]</span>
-<span class="sd">    [203,   0,   0,  0,  0, 0, 0]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">d</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">d</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">j</span> <span class="o">&lt;=</span> <span class="n">m</span> <span class="o">-</span> <span class="n">i</span><span class="p">:</span>
-                <span class="n">d</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">j</span> <span class="o">*</span> <span class="n">d</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">+</span> <span class="n">d</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">d</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">return</span> <span class="n">d</span>
-
-</div>
-<div class="viewcode-block" id="RGS_enum"><a class="viewcode-back" href="../../../modules/combinatorics/partitions.html#sympy.combinatorics.partitions.RGS_enum">[docs]</a><span class="k">def</span> <span class="nf">RGS_enum</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    RGS_enum computes the total number of restricted growth strings</span>
-<span class="sd">    possible for a superset of size m.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.partitions import RGS_enum</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.partitions import Partition</span>
-<span class="sd">    &gt;&gt;&gt; RGS_enum(4)</span>
-<span class="sd">    15</span>
-<span class="sd">    &gt;&gt;&gt; RGS_enum(5)</span>
-<span class="sd">    52</span>
-<span class="sd">    &gt;&gt;&gt; RGS_enum(6)</span>
-<span class="sd">    203</span>
-
-<span class="sd">    We can check that the enumeration is correct by actually generating</span>
-<span class="sd">    the partitions. Here, the 15 partitions of 4 items are generated:</span>
-
-<span class="sd">    &gt;&gt;&gt; a = Partition([list(range(4))])</span>
-<span class="sd">    &gt;&gt;&gt; s = set()</span>
-<span class="sd">    &gt;&gt;&gt; for i in range(20):</span>
-<span class="sd">    ...     s.add(a)</span>
-<span class="sd">    ...     a += 1</span>
-<span class="sd">    ...</span>
-<span class="sd">    &gt;&gt;&gt; assert len(s) == 15</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="p">(</span><span class="n">m</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="k">elif</span> <span class="p">(</span><span class="n">m</span> <span class="o">==</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">1</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">bell</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="RGS_unrank"><a class="viewcode-back" href="../../../modules/combinatorics/partitions.html#sympy.combinatorics.partitions.RGS_unrank">[docs]</a><span class="k">def</span> <span class="nf">RGS_unrank</span><span class="p">(</span><span class="n">rank</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Gives the unranked restricted growth string for a given</span>
-<span class="sd">    superset size.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.partitions import RGS_unrank</span>
-<span class="sd">    &gt;&gt;&gt; RGS_unrank(14, 4)</span>
-<span class="sd">    [0, 1, 2, 3]</span>
-<span class="sd">    &gt;&gt;&gt; RGS_unrank(0, 4)</span>
-<span class="sd">    [0, 0, 0, 0]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;The superset size must be &gt;= 1&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">rank</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">RGS_enum</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="n">rank</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Invalid arguments&quot;</span><span class="p">)</span>
-
-    <span class="n">L</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="n">j</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">D</span> <span class="o">=</span> <span class="n">RGS_generalized</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="n">D</span><span class="p">[</span><span class="n">m</span> <span class="o">-</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-        <span class="n">cr</span> <span class="o">=</span> <span class="n">j</span><span class="o">*</span><span class="n">v</span>
-        <span class="k">if</span> <span class="n">cr</span> <span class="o">&lt;=</span> <span class="n">rank</span><span class="p">:</span>
-            <span class="n">L</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">1</span>
-            <span class="n">rank</span> <span class="o">-=</span> <span class="n">cr</span>
-            <span class="n">j</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">L</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">rank</span> <span class="o">/</span> <span class="n">v</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="n">rank</span> <span class="o">%=</span> <span class="n">v</span>
-    <span class="k">return</span> <span class="p">[</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">L</span><span class="p">[</span><span class="mi">1</span><span class="p">:]]</span>
-
-</div>
-<div class="viewcode-block" id="RGS_rank"><a class="viewcode-back" href="../../../modules/combinatorics/partitions.html#sympy.combinatorics.partitions.RGS_rank">[docs]</a><span class="k">def</span> <span class="nf">RGS_rank</span><span class="p">(</span><span class="n">rgs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes the rank of a restricted growth string.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.partitions import RGS_rank, RGS_unrank</span>
-<span class="sd">    &gt;&gt;&gt; RGS_rank([0, 1, 2, 1, 3])</span>
-<span class="sd">    42</span>
-<span class="sd">    &gt;&gt;&gt; RGS_rank(RGS_unrank(4, 7))</span>
-<span class="sd">    4</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">rgs_size</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">rgs</span><span class="p">)</span>
-    <span class="n">rank</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">D</span> <span class="o">=</span> <span class="n">RGS_generalized</span><span class="p">(</span><span class="n">rgs_size</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">rgs_size</span><span class="p">):</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">rgs</span><span class="p">[(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):])</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">rgs</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="n">i</span><span class="p">])</span>
-        <span class="n">rank</span> <span class="o">+=</span> <span class="n">D</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">rgs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">rank</span></div>
-</pre></div>
-
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-  <h1>Source code for sympy.combinatorics.perm_groups</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">randrange</span><span class="p">,</span> <span class="n">choice</span>
-<span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">log</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Basic</span>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="p">(</span><span class="n">_af_commutes_with</span><span class="p">,</span> <span class="n">_af_invert</span><span class="p">,</span>
-    <span class="n">_af_rmul</span><span class="p">,</span> <span class="n">_af_rmuln</span><span class="p">,</span> <span class="n">_af_pow</span><span class="p">,</span> <span class="n">Cycle</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.util</span> <span class="kn">import</span> <span class="p">(</span><span class="n">_check_cycles_alt_sym</span><span class="p">,</span>
-    <span class="n">_distribute_gens_by_base</span><span class="p">,</span> <span class="n">_orbits_transversals_from_bsgs</span><span class="p">,</span>
-    <span class="n">_handle_precomputed_bsgs</span><span class="p">,</span> <span class="n">_base_ordering</span><span class="p">,</span> <span class="n">_strong_gens_from_distr</span><span class="p">,</span>
-    <span class="n">_strip</span><span class="p">,</span> <span class="n">_strip_af</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.combinatorial.factorials</span> <span class="kn">import</span> <span class="n">factorial</span>
-<span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">sieve</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">has_variety</span><span class="p">,</span> <span class="n">is_sequence</span><span class="p">,</span> <span class="n">uniq</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.randtest</span> <span class="kn">import</span> <span class="n">_randrange</span>
-
-<span class="n">rmul</span> <span class="o">=</span> <span class="n">Permutation</span><span class="o">.</span><span class="n">rmul_with_af</span>
-<span class="n">_af_new</span> <span class="o">=</span> <span class="n">Permutation</span><span class="o">.</span><span class="n">_af_new</span>
-
-
-<div class="viewcode-block" id="PermutationGroup"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup">[docs]</a><span class="k">class</span> <span class="nc">PermutationGroup</span><span class="p">(</span><span class="n">Basic</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The class defining a Permutation group.</span>
-
-<span class="sd">    PermutationGroup([p1, p2, ..., pn]) returns the permutation group</span>
-<span class="sd">    generated by the list of permutations. This group can be supplied</span>
-<span class="sd">    to Polyhedron if one desires to decorate the elements to which the</span>
-<span class="sd">    indices of the permutation refer.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.permutations import Cycle</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.polyhedron import Polyhedron</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-
-<span class="sd">    The permutations corresponding to motion of the front, right and</span>
-<span class="sd">    bottom face of a 2x2 Rubik&#39;s cube are defined:</span>
-
-<span class="sd">    &gt;&gt;&gt; F = Permutation(2, 19, 21, 8)(3, 17, 20, 10)(4, 6, 7, 5)</span>
-<span class="sd">    &gt;&gt;&gt; R = Permutation(1, 5, 21, 14)(3, 7, 23, 12)(8, 10, 11, 9)</span>
-<span class="sd">    &gt;&gt;&gt; D = Permutation(6, 18, 14, 10)(7, 19, 15, 11)(20, 22, 23, 21)</span>
-
-<span class="sd">    These are passed as permutations to PermutationGroup:</span>
-
-<span class="sd">    &gt;&gt;&gt; G = PermutationGroup(F, R, D)</span>
-<span class="sd">    &gt;&gt;&gt; G.order()</span>
-<span class="sd">    3674160</span>
-
-<span class="sd">    The group can be supplied to a Polyhedron in order to track the</span>
-<span class="sd">    objects being moved. An example involving the 2x2 Rubik&#39;s cube is</span>
-<span class="sd">    given there, but here is a simple demonstration:</span>
-
-<span class="sd">    &gt;&gt;&gt; a = Permutation(2, 1)</span>
-<span class="sd">    &gt;&gt;&gt; b = Permutation(1, 0)</span>
-<span class="sd">    &gt;&gt;&gt; G = PermutationGroup(a, b)</span>
-<span class="sd">    &gt;&gt;&gt; P = Polyhedron(list(&#39;ABC&#39;), pgroup=G)</span>
-<span class="sd">    &gt;&gt;&gt; P.corners</span>
-<span class="sd">    (A, B, C)</span>
-<span class="sd">    &gt;&gt;&gt; P.rotate(0) # apply permutation 0</span>
-<span class="sd">    &gt;&gt;&gt; P.corners</span>
-<span class="sd">    (A, C, B)</span>
-<span class="sd">    &gt;&gt;&gt; P.reset()</span>
-<span class="sd">    &gt;&gt;&gt; P.corners</span>
-<span class="sd">    (A, B, C)</span>
-
-<span class="sd">    Or one can make a permutation as a product of selected permutations</span>
-<span class="sd">    and apply them to an iterable directly:</span>
-
-<span class="sd">    &gt;&gt;&gt; P10 = G.make_perm([0, 1])</span>
-<span class="sd">    &gt;&gt;&gt; P10(&#39;ABC&#39;)</span>
-<span class="sd">    [&#39;C&#39;, &#39;A&#39;, &#39;B&#39;]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.combinatorics.polyhedron.Polyhedron,</span>
-<span class="sd">    sympy.combinatorics.permutations.Permutation</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] Holt, D., Eick, B., O&#39;Brien, E.</span>
-<span class="sd">    &quot;Handbook of Computational Group Theory&quot;</span>
-
-<span class="sd">    [2] Seress, A.</span>
-<span class="sd">    &quot;Permutation Group Algorithms&quot;</span>
-
-<span class="sd">    [3] http://en.wikipedia.org/wiki/Schreier_vector</span>
-
-<span class="sd">    [4] http://en.wikipedia.org/wiki/Nielsen_transformation</span>
-<span class="sd">    #Product_replacement_algorithm</span>
-
-<span class="sd">    [5] Frank Celler, Charles R.Leedham-Green, Scott H.Murray,</span>
-<span class="sd">    Alice C.Niemeyer, and E.A.O&#39;Brien. &quot;Generating Random</span>
-<span class="sd">    Elements of a Finite Group&quot;</span>
-
-<span class="sd">    [6] http://en.wikipedia.org/wiki/Block_%28permutation_group_theory%29</span>
-
-<span class="sd">    [7] http://www.algorithmist.com/index.php/Union_Find</span>
-
-<span class="sd">    [8] http://en.wikipedia.org/wiki/Multiply_transitive_group#Multiply_transitive_groups</span>
-
-<span class="sd">    [9] http://en.wikipedia.org/wiki/Center_%28group_theory%29</span>
-
-<span class="sd">    [10] http://en.wikipedia.org/wiki/Centralizer_and_normalizer</span>
-
-<span class="sd">    [11] http://groupprops.subwiki.org/wiki/Derived_subgroup</span>
-
-<span class="sd">    [12] http://en.wikipedia.org/wiki/Nilpotent_group</span>
-
-<span class="sd">    [13] http://www.math.colostate.edu/~hulpke/CGT/cgtnotes.pdf</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The default constructor. Accepts Cycle and Permutation forms.</span>
-<span class="sd">        Removes duplicates unless ``dups`` keyword is False.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">if</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="k">else</span> <span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;must supply one or more permutations &#39;</span>
-            <span class="s">&#39;to define the group&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Cycle</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">args</span><span class="p">):</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">Permutation</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">has_variety</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">size</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">args</span><span class="p">):</span>
-            <span class="n">degree</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;degree&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">degree</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">degree</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">size</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">args</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)):</span>
-                <span class="k">if</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">size</span> <span class="o">!=</span> <span class="n">degree</span><span class="p">:</span>
-                    <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">size</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;dups&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">uniq</span><span class="p">([</span><span class="n">_af_new</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]))</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_generators</span> <span class="o">=</span> <span class="n">args</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_order</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_center</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_is_abelian</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_is_transitive</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_is_sym</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_is_alt</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_is_primitive</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_is_nilpotent</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_is_solvable</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_is_trivial</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_transitivity_degree</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_max_div</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_r</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">obj</span><span class="o">.</span><span class="n">_generators</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_degree</span> <span class="o">=</span> <span class="n">obj</span><span class="o">.</span><span class="n">_generators</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">size</span>
-
-        <span class="c"># these attributes are assigned after running schreier_sims</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_base</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_strong_gens</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_basic_orbits</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_transversals</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="c"># these attributes are assigned after running _random_pr_init</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_random_gens</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_generators</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__len__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_generators</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return True if self and other have the same generators.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation(0, 1, 2, 3, 4, 5)</span>
-<span class="sd">        &gt;&gt;&gt; G = PermutationGroup([p, p**2])</span>
-<span class="sd">        &gt;&gt;&gt; H = PermutationGroup([p**2, p])</span>
-<span class="sd">        &gt;&gt;&gt; G.generators == H.generators</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; G == H</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">PermutationGroup</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="nb">set</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">generators</span><span class="p">)</span> <span class="o">==</span> <span class="nb">set</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">generators</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">PermutationGroup</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the direct product of two permutation groups as a permutation</span>
-<span class="sd">        group.</span>
-
-<span class="sd">        This implementation realizes the direct product by shifting</span>
-<span class="sd">        the index set for the generators of the second group: so if we have</span>
-<span class="sd">        G acting on n1 points and H acting on n2 points, G*H acts on n1 + n2</span>
-<span class="sd">        points.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import CyclicGroup</span>
-<span class="sd">        &gt;&gt;&gt; G = CyclicGroup(5)</span>
-<span class="sd">        &gt;&gt;&gt; H = G*G</span>
-<span class="sd">        &gt;&gt;&gt; H</span>
-<span class="sd">        PermutationGroup([</span>
-<span class="sd">            Permutation(9)(0, 1, 2, 3, 4),</span>
-<span class="sd">            Permutation(5, 6, 7, 8, 9)])</span>
-<span class="sd">        &gt;&gt;&gt; H.order()</span>
-<span class="sd">        25</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">gens1</span> <span class="o">=</span> <span class="p">[</span><span class="n">perm</span><span class="o">.</span><span class="n">_array_form</span> <span class="k">for</span> <span class="n">perm</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">generators</span><span class="p">]</span>
-        <span class="n">gens2</span> <span class="o">=</span> <span class="p">[</span><span class="n">perm</span><span class="o">.</span><span class="n">_array_form</span> <span class="k">for</span> <span class="n">perm</span> <span class="ow">in</span> <span class="n">other</span><span class="o">.</span><span class="n">generators</span><span class="p">]</span>
-        <span class="n">n1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_degree</span>
-        <span class="n">n2</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">_degree</span>
-        <span class="n">start</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n1</span><span class="p">))</span>
-        <span class="n">end</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n1</span><span class="p">,</span> <span class="n">n1</span> <span class="o">+</span> <span class="n">n2</span><span class="p">))</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">gens2</span><span class="p">)):</span>
-            <span class="n">gens2</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="o">+</span> <span class="n">n1</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">gens2</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span>
-        <span class="n">gens2</span> <span class="o">=</span> <span class="p">[</span><span class="n">start</span> <span class="o">+</span> <span class="n">gen</span> <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">gens2</span><span class="p">]</span>
-        <span class="n">gens1</span> <span class="o">=</span> <span class="p">[</span><span class="n">gen</span> <span class="o">+</span> <span class="n">end</span> <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">gens1</span><span class="p">]</span>
-        <span class="n">together</span> <span class="o">=</span> <span class="n">gens1</span> <span class="o">+</span> <span class="n">gens2</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">_af_new</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">together</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_random_pr_init</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">_random_prec_n</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;Initialize random generators for the product replacement algorithm.</span>
-
-<span class="sd">        The implementation uses a modification of the original product</span>
-<span class="sd">        replacement algorithm due to Leedham-Green, as described in [1],</span>
-<span class="sd">        pp. 69-71; also, see [2], pp. 27-29 for a detailed theoretical</span>
-<span class="sd">        analysis of the original product replacement algorithm, and [4].</span>
-
-<span class="sd">        The product replacement algorithm is used for producing random,</span>
-<span class="sd">        uniformly distributed elements of a group ``G`` with a set of generators</span>
-<span class="sd">        ``S``. For the initialization ``_random_pr_init``, a list ``R`` of</span>
-<span class="sd">        ``\max\{r, |S|\}`` group generators is created as the attribute</span>
-<span class="sd">        ``G._random_gens``, repeating elements of ``S`` if necessary, and the</span>
-<span class="sd">        identity element of ``G`` is appended to ``R`` - we shall refer to this</span>
-<span class="sd">        last element as the accumulator. Then the function ``random_pr()``</span>
-<span class="sd">        is called ``n`` times, randomizing the list ``R`` while preserving</span>
-<span class="sd">        the generation of ``G`` by ``R``. The function ``random_pr()`` itself</span>
-<span class="sd">        takes two random elements ``g, h`` among all elements of ``R`` but</span>
-<span class="sd">        the accumulator and replaces ``g`` with a randomly chosen element</span>
-<span class="sd">        from ``\{gh, g(~h), hg, (~h)g\}``. Then the accumulator is multiplied</span>
-<span class="sd">        by whatever ``g`` was replaced by. The new value of the accumulator is</span>
-<span class="sd">        then returned by ``random_pr()``.</span>
-
-<span class="sd">        The elements returned will eventually (for ``n`` large enough) become</span>
-<span class="sd">        uniformly distributed across ``G`` ([5]). For practical purposes however,</span>
-<span class="sd">        the values ``n = 50, r = 11`` are suggested in [1].</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        THIS FUNCTION HAS SIDE EFFECTS: it changes the attribute</span>
-<span class="sd">        self._random_gens</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        random_pr</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">deg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">degree</span>
-        <span class="n">random_gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="o">.</span><span class="n">_array_form</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">generators</span><span class="p">]</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">random_gens</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">k</span> <span class="o">&lt;</span> <span class="n">r</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">r</span><span class="p">):</span>
-                <span class="n">random_gens</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">random_gens</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="n">k</span><span class="p">])</span>
-        <span class="n">acc</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">deg</span><span class="p">))</span>
-        <span class="n">random_gens</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">acc</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_random_gens</span> <span class="o">=</span> <span class="n">random_gens</span>
-
-        <span class="c"># handle randomized input for testing purposes</span>
-        <span class="k">if</span> <span class="n">_random_prec_n</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">random_pr</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">random_pr</span><span class="p">(</span><span class="n">_random_prec</span><span class="o">=</span><span class="n">_random_prec_n</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_union_find_merge</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">first</span><span class="p">,</span> <span class="n">second</span><span class="p">,</span> <span class="n">ranks</span><span class="p">,</span> <span class="n">parents</span><span class="p">,</span> <span class="n">not_rep</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Merges two classes in a union-find data structure.</span>
-
-<span class="sd">        Used in the implementation of Atkinson&#39;s algorithm as suggested in [1],</span>
-<span class="sd">        pp. 83-87. The class merging process uses union by rank as an</span>
-<span class="sd">        optimization. ([7])</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        THIS FUNCTION HAS SIDE EFFECTS: the list of class representatives,</span>
-<span class="sd">        ``parents``, the list of class sizes, ``ranks``, and the list of</span>
-<span class="sd">        elements that are not representatives, ``not_rep``, are changed due to</span>
-<span class="sd">        class merging.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        minimal_block, _union_find_rep</span>
-
-<span class="sd">        References</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        [1] Holt, D., Eick, B., O&#39;Brien, E.</span>
-<span class="sd">        &quot;Handbook of computational group theory&quot;</span>
-
-<span class="sd">        [7] http://www.algorithmist.com/index.php/Union_Find</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">rep_first</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_union_find_rep</span><span class="p">(</span><span class="n">first</span><span class="p">,</span> <span class="n">parents</span><span class="p">)</span>
-        <span class="n">rep_second</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_union_find_rep</span><span class="p">(</span><span class="n">second</span><span class="p">,</span> <span class="n">parents</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">rep_first</span> <span class="o">!=</span> <span class="n">rep_second</span><span class="p">:</span>
-            <span class="c"># union by rank</span>
-            <span class="k">if</span> <span class="n">ranks</span><span class="p">[</span><span class="n">rep_first</span><span class="p">]</span> <span class="o">&gt;=</span> <span class="n">ranks</span><span class="p">[</span><span class="n">rep_second</span><span class="p">]:</span>
-                <span class="n">new_1</span><span class="p">,</span> <span class="n">new_2</span> <span class="o">=</span> <span class="n">rep_first</span><span class="p">,</span> <span class="n">rep_second</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">new_1</span><span class="p">,</span> <span class="n">new_2</span> <span class="o">=</span> <span class="n">rep_second</span><span class="p">,</span> <span class="n">rep_first</span>
-            <span class="n">total_rank</span> <span class="o">=</span> <span class="n">ranks</span><span class="p">[</span><span class="n">new_1</span><span class="p">]</span> <span class="o">+</span> <span class="n">ranks</span><span class="p">[</span><span class="n">new_2</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">total_rank</span> <span class="o">&gt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">max_div</span><span class="p">:</span>
-                <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-            <span class="n">parents</span><span class="p">[</span><span class="n">new_2</span><span class="p">]</span> <span class="o">=</span> <span class="n">new_1</span>
-            <span class="n">ranks</span><span class="p">[</span><span class="n">new_1</span><span class="p">]</span> <span class="o">=</span> <span class="n">total_rank</span>
-            <span class="n">not_rep</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">new_2</span><span class="p">)</span>
-            <span class="k">return</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="mi">0</span>
-
-    <span class="k">def</span> <span class="nf">_union_find_rep</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">num</span><span class="p">,</span> <span class="n">parents</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Find representative of a class in a union-find data structure.</span>
-
-<span class="sd">        Used in the implementation of Atkinson&#39;s algorithm as suggested in [1],</span>
-<span class="sd">        pp. 83-87. After the representative of the class to which ``num``</span>
-<span class="sd">        belongs is found, path compression is performed as an optimization</span>
-<span class="sd">        ([7]).</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        THIS FUNCTION HAS SIDE EFFECTS: the list of class representatives,</span>
-<span class="sd">        ``parents``, is altered due to path compression.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        minimal_block, _union_find_merge</span>
-
-<span class="sd">        References</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        [1] Holt, D., Eick, B., O&#39;Brien, E.</span>
-<span class="sd">        &quot;Handbook of computational group theory&quot;</span>
-
-<span class="sd">        [7] http://www.algorithmist.com/index.php/Union_Find</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">rep</span><span class="p">,</span> <span class="n">parent</span> <span class="o">=</span> <span class="n">num</span><span class="p">,</span> <span class="n">parents</span><span class="p">[</span><span class="n">num</span><span class="p">]</span>
-        <span class="k">while</span> <span class="n">parent</span> <span class="o">!=</span> <span class="n">rep</span><span class="p">:</span>
-            <span class="n">rep</span> <span class="o">=</span> <span class="n">parent</span>
-            <span class="n">parent</span> <span class="o">=</span> <span class="n">parents</span><span class="p">[</span><span class="n">rep</span><span class="p">]</span>
-        <span class="c"># path compression</span>
-        <span class="n">temp</span><span class="p">,</span> <span class="n">parent</span> <span class="o">=</span> <span class="n">num</span><span class="p">,</span> <span class="n">parents</span><span class="p">[</span><span class="n">num</span><span class="p">]</span>
-        <span class="k">while</span> <span class="n">parent</span> <span class="o">!=</span> <span class="n">rep</span><span class="p">:</span>
-            <span class="n">parents</span><span class="p">[</span><span class="n">temp</span><span class="p">]</span> <span class="o">=</span> <span class="n">rep</span>
-            <span class="n">temp</span> <span class="o">=</span> <span class="n">parent</span>
-            <span class="n">parent</span> <span class="o">=</span> <span class="n">parents</span><span class="p">[</span><span class="n">temp</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">rep</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="PermutationGroup.base"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.base">[docs]</a>    <span class="k">def</span> <span class="nf">base</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return a base from the Schreier-Sims algorithm.</span>
-
-<span class="sd">        For a permutation group ``G``, a base is a sequence of points</span>
-<span class="sd">        ``B = (b_1, b_2, ..., b_k)`` such that no element of ``G`` apart</span>
-<span class="sd">        from the identity fixes all the points in ``B``. The concepts of</span>
-<span class="sd">        a base and strong generating set and their applications are</span>
-<span class="sd">        discussed in depth in [1], pp. 87-89 and [2], pp. 55-57.</span>
-
-<span class="sd">        An alternative way to think of ``B`` is that it gives the</span>
-<span class="sd">        indices of the stabilizer cosets that contain more than the</span>
-<span class="sd">        identity permutation.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation, PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; G = PermutationGroup([Permutation(0, 1, 3)(2, 4)])</span>
-<span class="sd">        &gt;&gt;&gt; G.base</span>
-<span class="sd">        [0, 2]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        strong_gens, basic_transversals, basic_orbits, basic_stabilizers</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_base</span> <span class="o">==</span> <span class="p">[]:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">schreier_sims</span><span class="p">()</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_base</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.baseswap"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.baseswap">[docs]</a>    <span class="k">def</span> <span class="nf">baseswap</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span><span class="p">,</span> <span class="n">pos</span><span class="p">,</span> <span class="n">randomized</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span>
-                 <span class="n">transversals</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">basic_orbits</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">strong_gens_distr</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;Swap two consecutive base points in base and strong generating set.</span>
-
-<span class="sd">        If a base for a group ``G`` is given by ``(b_1, b_2, ..., b_k)``, this</span>
-<span class="sd">        function returns a base ``(b_1, b_2, ..., b_{i+1}, b_i, ..., b_k)``,</span>
-<span class="sd">        where ``i`` is given by ``pos``, and a strong generating set relative</span>
-<span class="sd">        to that base. The original base and strong generating set are not</span>
-<span class="sd">        modified.</span>
-
-<span class="sd">        The randomized version (default) is of Las Vegas type.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        base, strong_gens</span>
-<span class="sd">            The base and strong generating set.</span>
-<span class="sd">        pos</span>
-<span class="sd">            The position at which swapping is performed.</span>
-<span class="sd">        randomized</span>
-<span class="sd">            A switch between randomized and deterministic version.</span>
-<span class="sd">        transversals</span>
-<span class="sd">            The transversals for the basic orbits, if known.</span>
-<span class="sd">        basic_orbits</span>
-<span class="sd">            The basic orbits, if known.</span>
-<span class="sd">        strong_gens_distr</span>
-<span class="sd">            The strong generators distributed by basic stabilizers, if known.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        (base, strong_gens)</span>
-<span class="sd">            ``base`` is the new base, and ``strong_gens`` is a generating set</span>
-<span class="sd">            relative to it.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import SymmetricGroup</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.testutil import _verify_bsgs</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; S = SymmetricGroup(4)</span>
-<span class="sd">        &gt;&gt;&gt; S.schreier_sims()</span>
-<span class="sd">        &gt;&gt;&gt; S.base</span>
-<span class="sd">        [0, 1, 2]</span>
-<span class="sd">        &gt;&gt;&gt; base, gens = S.baseswap(S.base, S.strong_gens, 1, randomized=False)</span>
-<span class="sd">        &gt;&gt;&gt; base, gens</span>
-<span class="sd">        ([0, 2, 1],</span>
-<span class="sd">        [Permutation(0, 1, 2, 3), Permutation(3)(0, 1), Permutation(1, 3, 2),</span>
-<span class="sd">         Permutation(2, 3), Permutation(1, 3)])</span>
-
-<span class="sd">        check that base, gens is a BSGS</span>
-
-<span class="sd">        &gt;&gt;&gt; S1 = PermutationGroup(gens)</span>
-<span class="sd">        &gt;&gt;&gt; _verify_bsgs(S1, base, gens)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        schreier_sims</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        The deterministic version of the algorithm is discussed in</span>
-<span class="sd">        [1], pp. 102-103; the randomized version is discussed in [1], p.103, and</span>
-<span class="sd">        [2], p.98. It is of Las Vegas type.</span>
-<span class="sd">        Notice that [1] contains a mistake in the pseudocode and</span>
-<span class="sd">        discussion of BASESWAP: on line 3 of the pseudocode,</span>
-<span class="sd">        ``|\beta_{i+1}^{\left\langle T\right\rangle}|`` should be replaced by</span>
-<span class="sd">        ``|\beta_{i}^{\left\langle T\right\rangle}|``, and the same for the</span>
-<span class="sd">        discussion of the algorithm.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># construct the basic orbits, generators for the stabilizer chain</span>
-        <span class="c"># and transversal elements from whatever was provided</span>
-        <span class="n">transversals</span><span class="p">,</span> <span class="n">basic_orbits</span><span class="p">,</span> <span class="n">strong_gens_distr</span> <span class="o">=</span> \
-            <span class="n">_handle_precomputed_bsgs</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span><span class="p">,</span> <span class="n">transversals</span><span class="p">,</span>
-                                 <span class="n">basic_orbits</span><span class="p">,</span> <span class="n">strong_gens_distr</span><span class="p">)</span>
-        <span class="n">base_len</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-        <span class="n">degree</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">degree</span>
-        <span class="c"># size of orbit of base[pos] under the stabilizer we seek to insert</span>
-        <span class="c"># in the stabilizer chain at position pos + 1</span>
-        <span class="n">size</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">basic_orbits</span><span class="p">[</span><span class="n">pos</span><span class="p">])</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="n">basic_orbits</span><span class="p">[</span><span class="n">pos</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span> \
-            <span class="o">//</span><span class="nb">len</span><span class="p">(</span><span class="n">_orbit</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">strong_gens_distr</span><span class="p">[</span><span class="n">pos</span><span class="p">],</span> <span class="n">base</span><span class="p">[</span><span class="n">pos</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]))</span>
-        <span class="c"># initialize the wanted stabilizer by a subgroup</span>
-        <span class="k">if</span> <span class="n">pos</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">&gt;</span> <span class="n">base_len</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">T</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">T</span> <span class="o">=</span> <span class="n">strong_gens_distr</span><span class="p">[</span><span class="n">pos</span> <span class="o">+</span> <span class="mi">2</span><span class="p">][:]</span>
-        <span class="c"># randomized version</span>
-        <span class="k">if</span> <span class="n">randomized</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">stab_pos</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="n">strong_gens_distr</span><span class="p">[</span><span class="n">pos</span><span class="p">])</span>
-            <span class="n">schreier_vector</span> <span class="o">=</span> <span class="n">stab_pos</span><span class="o">.</span><span class="n">schreier_vector</span><span class="p">(</span><span class="n">base</span><span class="p">[</span><span class="n">pos</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span>
-            <span class="c"># add random elements of the stabilizer until they generate it</span>
-            <span class="k">while</span> <span class="nb">len</span><span class="p">(</span><span class="n">_orbit</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">base</span><span class="p">[</span><span class="n">pos</span><span class="p">]))</span> <span class="o">!=</span> <span class="n">size</span><span class="p">:</span>
-                <span class="n">new</span> <span class="o">=</span> <span class="n">stab_pos</span><span class="o">.</span><span class="n">random_stab</span><span class="p">(</span><span class="n">base</span><span class="p">[</span><span class="n">pos</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span>
-                                           <span class="n">schreier_vector</span><span class="o">=</span><span class="n">schreier_vector</span><span class="p">)</span>
-                <span class="n">T</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">new</span><span class="p">)</span>
-        <span class="c"># deterministic version</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">Gamma</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">basic_orbits</span><span class="p">[</span><span class="n">pos</span><span class="p">])</span>
-            <span class="n">Gamma</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">base</span><span class="p">[</span><span class="n">pos</span><span class="p">])</span>
-            <span class="k">if</span> <span class="n">base</span><span class="p">[</span><span class="n">pos</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="ow">in</span> <span class="n">Gamma</span><span class="p">:</span>
-                <span class="n">Gamma</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">base</span><span class="p">[</span><span class="n">pos</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span>
-            <span class="c"># add elements of the stabilizer until they generate it by</span>
-            <span class="c"># ruling out member of the basic orbit of base[pos] along the way</span>
-            <span class="k">while</span> <span class="nb">len</span><span class="p">(</span><span class="n">_orbit</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">base</span><span class="p">[</span><span class="n">pos</span><span class="p">]))</span> <span class="o">!=</span> <span class="n">size</span><span class="p">:</span>
-                <span class="n">gamma</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="nb">iter</span><span class="p">(</span><span class="n">Gamma</span><span class="p">))</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="n">transversals</span><span class="p">[</span><span class="n">pos</span><span class="p">][</span><span class="n">gamma</span><span class="p">]</span>
-                <span class="n">temp</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">_array_form</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">base</span><span class="p">[</span><span class="n">pos</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span> <span class="c"># (~x)(base[pos + 1])</span>
-                <span class="k">if</span> <span class="n">temp</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">basic_orbits</span><span class="p">[</span><span class="n">pos</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]:</span>
-                    <span class="n">Gamma</span> <span class="o">=</span> <span class="n">Gamma</span> <span class="o">-</span> <span class="n">_orbit</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">gamma</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">y</span> <span class="o">=</span> <span class="n">transversals</span><span class="p">[</span><span class="n">pos</span> <span class="o">+</span> <span class="mi">1</span><span class="p">][</span><span class="n">temp</span><span class="p">]</span>
-                    <span class="n">el</span> <span class="o">=</span> <span class="n">rmul</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">el</span><span class="p">(</span><span class="n">base</span><span class="p">[</span><span class="n">pos</span><span class="p">])</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">_orbit</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">base</span><span class="p">[</span><span class="n">pos</span><span class="p">]):</span>
-                        <span class="n">T</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">el</span><span class="p">)</span>
-                        <span class="n">Gamma</span> <span class="o">=</span> <span class="n">Gamma</span> <span class="o">-</span> <span class="n">_orbit</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">base</span><span class="p">[</span><span class="n">pos</span><span class="p">])</span>
-        <span class="c"># build the new base and strong generating set</span>
-        <span class="n">strong_gens_new_distr</span> <span class="o">=</span> <span class="n">strong_gens_distr</span><span class="p">[:]</span>
-        <span class="n">strong_gens_new_distr</span><span class="p">[</span><span class="n">pos</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">T</span>
-        <span class="n">base_new</span> <span class="o">=</span> <span class="n">base</span><span class="p">[:]</span>
-        <span class="n">base_new</span><span class="p">[</span><span class="n">pos</span><span class="p">],</span> <span class="n">base_new</span><span class="p">[</span><span class="n">pos</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">base_new</span><span class="p">[</span><span class="n">pos</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">base_new</span><span class="p">[</span><span class="n">pos</span><span class="p">]</span>
-        <span class="n">strong_gens_new</span> <span class="o">=</span> <span class="n">_strong_gens_from_distr</span><span class="p">(</span><span class="n">strong_gens_new_distr</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">T</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">gen</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">strong_gens_new</span><span class="p">:</span>
-                <span class="n">strong_gens_new</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">gen</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">base_new</span><span class="p">,</span> <span class="n">strong_gens_new</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="PermutationGroup.basic_orbits"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.basic_orbits">[docs]</a>    <span class="k">def</span> <span class="nf">basic_orbits</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the basic orbits relative to a base and strong generating set.</span>
-
-<span class="sd">        If ``(b_1, b_2, ..., b_k)`` is a base for a group ``G``, and</span>
-<span class="sd">        ``G^{(i)} = G_{b_1, b_2, ..., b_{i-1}}`` is the ``i``-th basic stabilizer</span>
-<span class="sd">        (so that ``G^{(1)} = G``), the ``i``-th basic orbit relative to this base</span>
-<span class="sd">        is the orbit of ``b_i`` under ``G^{(i)}``. See [1], pp. 87-89 for more</span>
-<span class="sd">        information.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import SymmetricGroup</span>
-<span class="sd">        &gt;&gt;&gt; S = SymmetricGroup(4)</span>
-<span class="sd">        &gt;&gt;&gt; S.basic_orbits</span>
-<span class="sd">        [[0, 1, 2, 3], [1, 2, 3], [2, 3]]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        base, strong_gens, basic_transversals, basic_stabilizers</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_basic_orbits</span> <span class="o">==</span> <span class="p">[]:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">schreier_sims</span><span class="p">()</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_basic_orbits</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="PermutationGroup.basic_stabilizers"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.basic_stabilizers">[docs]</a>    <span class="k">def</span> <span class="nf">basic_stabilizers</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return a chain of stabilizers relative to a base and strong generating</span>
-<span class="sd">        set.</span>
-
-<span class="sd">        The ``i``-th basic stabilizer ``G^{(i)}`` relative to a base</span>
-<span class="sd">        ``(b_1, b_2, ..., b_k)`` is ``G_{b_1, b_2, ..., b_{i-1}}``. For more</span>
-<span class="sd">        information, see [1], pp. 87-89.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import AlternatingGroup</span>
-<span class="sd">        &gt;&gt;&gt; A = AlternatingGroup(4)</span>
-<span class="sd">        &gt;&gt;&gt; A.schreier_sims()</span>
-<span class="sd">        &gt;&gt;&gt; A.base</span>
-<span class="sd">        [0, 1]</span>
-<span class="sd">        &gt;&gt;&gt; for g in A.basic_stabilizers:</span>
-<span class="sd">        ...     print(g)</span>
-<span class="sd">        ...</span>
-<span class="sd">        PermutationGroup([</span>
-<span class="sd">            Permutation(3)(0, 1, 2),</span>
-<span class="sd">            Permutation(1, 2, 3)])</span>
-<span class="sd">        PermutationGroup([</span>
-<span class="sd">            Permutation(1, 2, 3)])</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        base, strong_gens, basic_orbits, basic_transversals</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_transversals</span> <span class="o">==</span> <span class="p">[]:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">schreier_sims</span><span class="p">()</span>
-        <span class="n">strong_gens</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_strong_gens</span>
-        <span class="n">base</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_base</span>
-        <span class="n">strong_gens_distr</span> <span class="o">=</span> <span class="n">_distribute_gens_by_base</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span><span class="p">)</span>
-        <span class="n">basic_stabilizers</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">gens</span> <span class="ow">in</span> <span class="n">strong_gens_distr</span><span class="p">:</span>
-            <span class="n">basic_stabilizers</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">PermutationGroup</span><span class="p">(</span><span class="n">gens</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">basic_stabilizers</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="PermutationGroup.basic_transversals"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.basic_transversals">[docs]</a>    <span class="k">def</span> <span class="nf">basic_transversals</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return basic transversals relative to a base and strong generating set.</span>
-
-<span class="sd">        The basic transversals are transversals of the basic orbits. They</span>
-<span class="sd">        are provided as a list of dictionaries, each dictionary having</span>
-<span class="sd">        keys - the elements of one of the basic orbits, and values - the</span>
-<span class="sd">        corresponding transversal elements. See [1], pp. 87-89 for more</span>
-<span class="sd">        information.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import AlternatingGroup</span>
-<span class="sd">        &gt;&gt;&gt; A = AlternatingGroup(4)</span>
-<span class="sd">        &gt;&gt;&gt; A.basic_transversals</span>
-<span class="sd">        [{0: Permutation(3),</span>
-<span class="sd">          1: Permutation(3)(0, 1, 2),</span>
-<span class="sd">          2: Permutation(3)(0, 2, 1),</span>
-<span class="sd">          3: Permutation(0, 3, 1)},</span>
-<span class="sd">         {1: Permutation(3),</span>
-<span class="sd">          2: Permutation(1, 2, 3),</span>
-<span class="sd">          3: Permutation(1, 3, 2)}]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        strong_gens, base, basic_orbits, basic_stabilizers</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_transversals</span> <span class="o">==</span> <span class="p">[]:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">schreier_sims</span><span class="p">()</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_transversals</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.center"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.center">[docs]</a>    <span class="k">def</span> <span class="nf">center</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">        Return the center of a permutation group.</span>
-
-<span class="sd">        The center for a group ``G`` is defined as</span>
-<span class="sd">        ``Z(G) = \{z\in G | \forall g\in G, zg = gz \}``,</span>
-<span class="sd">        the set of elements of ``G`` that commute with all elements of ``G``.</span>
-<span class="sd">        It is equal to the centralizer of ``G`` inside ``G``, and is naturally a</span>
-<span class="sd">        subgroup of ``G`` ([9]).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import DihedralGroup</span>
-<span class="sd">        &gt;&gt;&gt; D = DihedralGroup(4)</span>
-<span class="sd">        &gt;&gt;&gt; G = D.center()</span>
-<span class="sd">        &gt;&gt;&gt; G.order()</span>
-<span class="sd">        2</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        centralizer</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        This is a naive implementation that is a straightforward application</span>
-<span class="sd">        of ``.centralizer()``</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">centralizer</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.centralizer"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.centralizer">[docs]</a>    <span class="k">def</span> <span class="nf">centralizer</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">        Return the centralizer of a group/set/element.</span>
-
-<span class="sd">        The centralizer of a set of permutations ``S`` inside</span>
-<span class="sd">        a group ``G`` is the set of elements of ``G`` that commute with all</span>
-<span class="sd">        elements of ``S``::</span>
-
-<span class="sd">            ``C_G(S) = \{ g \in G | gs = sg \forall s \in S\}`` ([10])</span>
-
-<span class="sd">        Usually, ``S`` is a subset of ``G``, but if ``G`` is a proper subgroup of</span>
-<span class="sd">        the full symmetric group, we allow for ``S`` to have elements outside</span>
-<span class="sd">        ``G``.</span>
-
-<span class="sd">        It is naturally a subgroup of ``G``; the centralizer of a permutation</span>
-<span class="sd">        group is equal to the centralizer of any set of generators for that</span>
-<span class="sd">        group, since any element commuting with the generators commutes with</span>
-<span class="sd">        any product of the  generators.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        other</span>
-<span class="sd">            a permutation group/list of permutations/single permutation</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import (SymmetricGroup,</span>
-<span class="sd">        ... CyclicGroup)</span>
-<span class="sd">        &gt;&gt;&gt; S = SymmetricGroup(6)</span>
-<span class="sd">        &gt;&gt;&gt; C = CyclicGroup(6)</span>
-<span class="sd">        &gt;&gt;&gt; H = S.centralizer(C)</span>
-<span class="sd">        &gt;&gt;&gt; H.is_subgroup(C)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        subgroup_search</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        The implementation is an application of ``.subgroup_search()`` with</span>
-<span class="sd">        tests using a specific base for the group ``G``.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="s">&#39;generators&#39;</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">is_trivial</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_trivial</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span>
-            <span class="n">degree</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">degree</span>
-            <span class="n">identity</span> <span class="o">=</span> <span class="n">_af_new</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">degree</span><span class="p">)))</span>
-            <span class="n">orbits</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">orbits</span><span class="p">()</span>
-            <span class="n">num_orbits</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">orbits</span><span class="p">)</span>
-            <span class="n">orbits</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="o">-</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-            <span class="n">long_base</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">orbit_reps</span> <span class="o">=</span> <span class="p">[</span><span class="bp">None</span><span class="p">]</span><span class="o">*</span><span class="n">num_orbits</span>
-            <span class="n">orbit_reps_indices</span> <span class="o">=</span> <span class="p">[</span><span class="bp">None</span><span class="p">]</span><span class="o">*</span><span class="n">num_orbits</span>
-            <span class="n">orbit_descr</span> <span class="o">=</span> <span class="p">[</span><span class="bp">None</span><span class="p">]</span><span class="o">*</span><span class="n">degree</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_orbits</span><span class="p">):</span>
-                <span class="n">orbit</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">orbits</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-                <span class="n">orbit_reps</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">orbit</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="n">orbit_reps_indices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">long_base</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">point</span> <span class="ow">in</span> <span class="n">orbit</span><span class="p">:</span>
-                    <span class="n">orbit_descr</span><span class="p">[</span><span class="n">point</span><span class="p">]</span> <span class="o">=</span> <span class="n">i</span>
-                <span class="n">long_base</span> <span class="o">=</span> <span class="n">long_base</span> <span class="o">+</span> <span class="n">orbit</span>
-            <span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">schreier_sims_incremental</span><span class="p">(</span><span class="n">base</span><span class="o">=</span><span class="n">long_base</span><span class="p">)</span>
-            <span class="n">strong_gens_distr</span> <span class="o">=</span> <span class="n">_distribute_gens_by_base</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span><span class="p">)</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">)):</span>
-                <span class="k">if</span> <span class="n">strong_gens_distr</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="p">[</span><span class="n">identity</span><span class="p">]:</span>
-                    <span class="k">break</span>
-            <span class="n">base</span> <span class="o">=</span> <span class="n">base</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span>
-            <span class="n">base_len</span> <span class="o">=</span> <span class="n">i</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_orbits</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">base</span><span class="p">[</span><span class="n">base_len</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="ow">in</span> <span class="n">orbits</span><span class="p">[</span><span class="n">j</span><span class="p">]:</span>
-                    <span class="k">break</span>
-            <span class="n">rel_orbits</span> <span class="o">=</span> <span class="n">orbits</span><span class="p">[:</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
-            <span class="n">num_rel_orbits</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">rel_orbits</span><span class="p">)</span>
-            <span class="n">transversals</span> <span class="o">=</span> <span class="p">[</span><span class="bp">None</span><span class="p">]</span><span class="o">*</span><span class="n">num_rel_orbits</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_rel_orbits</span><span class="p">):</span>
-                <span class="n">rep</span> <span class="o">=</span> <span class="n">orbit_reps</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-                <span class="n">transversals</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span>
-                    <span class="n">other</span><span class="o">.</span><span class="n">orbit_transversal</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">pairs</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-            <span class="n">trivial_test</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="bp">True</span>
-            <span class="n">tests</span> <span class="o">=</span> <span class="p">[</span><span class="bp">None</span><span class="p">]</span><span class="o">*</span><span class="n">base_len</span>
-            <span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">base_len</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">base</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="ow">in</span> <span class="n">orbit_reps</span><span class="p">:</span>
-                    <span class="n">tests</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">trivial_test</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">def</span> <span class="nf">test</span><span class="p">(</span><span class="n">computed_words</span><span class="p">,</span> <span class="n">l</span><span class="o">=</span><span class="n">l</span><span class="p">):</span>
-                        <span class="n">g</span> <span class="o">=</span> <span class="n">computed_words</span><span class="p">[</span><span class="n">l</span><span class="p">]</span>
-                        <span class="n">rep_orb_index</span> <span class="o">=</span> <span class="n">orbit_descr</span><span class="p">[</span><span class="n">base</span><span class="p">[</span><span class="n">l</span><span class="p">]]</span>
-                        <span class="n">rep</span> <span class="o">=</span> <span class="n">orbit_reps</span><span class="p">[</span><span class="n">rep_orb_index</span><span class="p">]</span>
-                        <span class="n">im</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">_array_form</span><span class="p">[</span><span class="n">base</span><span class="p">[</span><span class="n">l</span><span class="p">]]</span>
-                        <span class="n">im_rep</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">_array_form</span><span class="p">[</span><span class="n">rep</span><span class="p">]</span>
-                        <span class="n">tr_el</span> <span class="o">=</span> <span class="n">transversals</span><span class="p">[</span><span class="n">rep_orb_index</span><span class="p">][</span><span class="n">base</span><span class="p">[</span><span class="n">l</span><span class="p">]]</span>
-                        <span class="c"># using the definition of transversal,</span>
-                        <span class="c"># base[l]^g = rep^(tr_el*g);</span>
-                        <span class="c"># if g belongs to the centralizer, then</span>
-                        <span class="c"># base[l]^g = (rep^g)^tr_el</span>
-                        <span class="k">return</span> <span class="n">im</span> <span class="o">==</span> <span class="n">tr_el</span><span class="o">.</span><span class="n">_array_form</span><span class="p">[</span><span class="n">im_rep</span><span class="p">]</span>
-                    <span class="n">tests</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">test</span>
-
-            <span class="k">def</span> <span class="nf">prop</span><span class="p">(</span><span class="n">g</span><span class="p">):</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">rmul</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">gen</span><span class="p">)</span> <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">other</span><span class="o">.</span><span class="n">generators</span><span class="p">]</span> <span class="o">==</span> \
-                       <span class="p">[</span><span class="n">rmul</span><span class="p">(</span><span class="n">gen</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span> <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">other</span><span class="o">.</span><span class="n">generators</span><span class="p">]</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">subgroup_search</span><span class="p">(</span><span class="n">prop</span><span class="p">,</span> <span class="n">base</span><span class="o">=</span><span class="n">base</span><span class="p">,</span>
-                                        <span class="n">strong_gens</span><span class="o">=</span><span class="n">strong_gens</span><span class="p">,</span> <span class="n">tests</span><span class="o">=</span><span class="n">tests</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="s">&#39;__getitem__&#39;</span><span class="p">):</span>
-            <span class="n">gens</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">centralizer</span><span class="p">(</span><span class="n">PermutationGroup</span><span class="p">(</span><span class="n">gens</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="s">&#39;array_form&#39;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">centralizer</span><span class="p">(</span><span class="n">PermutationGroup</span><span class="p">([</span><span class="n">other</span><span class="p">]))</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.commutator"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.commutator">[docs]</a>    <span class="k">def</span> <span class="nf">commutator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">H</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the commutator of two subgroups.</span>
-
-<span class="sd">        For a permutation group ``K`` and subgroups ``G``, ``H``, the</span>
-<span class="sd">        commutator of ``G`` and ``H`` is defined as the group generated</span>
-<span class="sd">        by all the commutators ``[g, h] = hgh^{-1}g^{-1}`` for ``g`` in ``G`` and</span>
-<span class="sd">        ``h`` in ``H``. It is naturally a subgroup of ``K`` ([1], p.27).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import (SymmetricGroup,</span>
-<span class="sd">        ... AlternatingGroup)</span>
-<span class="sd">        &gt;&gt;&gt; S = SymmetricGroup(5)</span>
-<span class="sd">        &gt;&gt;&gt; A = AlternatingGroup(5)</span>
-<span class="sd">        &gt;&gt;&gt; G = S.commutator(S, A)</span>
-<span class="sd">        &gt;&gt;&gt; G.is_subgroup(A)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        derived_subgroup</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        The commutator of two subgroups ``H, G`` is equal to the normal closure</span>
-<span class="sd">        of the commutators of all the generators, i.e. ``hgh^{-1}g^{-1}`` for ``h``</span>
-<span class="sd">        a generator of ``H`` and ``g`` a generator of ``G`` ([1], p.28)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">ggens</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">generators</span>
-        <span class="n">hgens</span> <span class="o">=</span> <span class="n">H</span><span class="o">.</span><span class="n">generators</span>
-        <span class="n">commutators</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">ggen</span> <span class="ow">in</span> <span class="n">ggens</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">hgen</span> <span class="ow">in</span> <span class="n">hgens</span><span class="p">:</span>
-                <span class="n">commutator</span> <span class="o">=</span> <span class="n">rmul</span><span class="p">(</span><span class="n">hgen</span><span class="p">,</span> <span class="n">ggen</span><span class="p">,</span> <span class="o">~</span><span class="n">hgen</span><span class="p">,</span> <span class="o">~</span><span class="n">ggen</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">commutator</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">commutators</span><span class="p">:</span>
-                    <span class="n">commutators</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">commutator</span><span class="p">)</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">normal_closure</span><span class="p">(</span><span class="n">commutators</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">res</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.coset_factor"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.coset_factor">[docs]</a>    <span class="k">def</span> <span class="nf">coset_factor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">factor_index</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return ``G``&#39;s (self&#39;s) coset factorization of ``g``</span>
-
-<span class="sd">        If ``g`` is an element of ``G`` then it can be written as the product</span>
-<span class="sd">        of permutations drawn from the Schreier-Sims coset decomposition,</span>
-
-<span class="sd">        The permutations returned in ``f`` are those for which</span>
-<span class="sd">        the product gives ``g``: ``g = f[n]*...f[1]*f[0]`` where ``n = len(B)``</span>
-<span class="sd">        and ``B = G.base``. f[i] is one of the permutations in</span>
-<span class="sd">        ``self._basic_orbits[i]``.</span>
-
-<span class="sd">        If factor_index==True,</span>
-<span class="sd">        returns a tuple ``[b[0],..,b[n]]``, where ``b[i]``</span>
-<span class="sd">        belongs to ``self._basic_orbits[i]``</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation, PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">        &gt;&gt;&gt; a = Permutation(0, 1, 3, 7, 6, 4)(2, 5)</span>
-<span class="sd">        &gt;&gt;&gt; b = Permutation(0, 1, 3, 2)(4, 5, 7, 6)</span>
-<span class="sd">        &gt;&gt;&gt; G = PermutationGroup([a, b])</span>
-
-<span class="sd">        Define g:</span>
-
-<span class="sd">        &gt;&gt;&gt; g = Permutation(7)(1, 2, 4)(3, 6, 5)</span>
-
-<span class="sd">        Confirm that it is an element of G:</span>
-
-<span class="sd">        &gt;&gt;&gt; G.contains(g)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        Thus, it can be written as a product of factors (up to</span>
-<span class="sd">        3) drawn from u. See below that a factor from u1 and u2</span>
-<span class="sd">        and the Identity permutation have been used:</span>
-
-<span class="sd">        &gt;&gt;&gt; f = G.coset_factor(g)</span>
-<span class="sd">        &gt;&gt;&gt; f[2]*f[1]*f[0] == g</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; f1 = G.coset_factor(g, True); f1</span>
-<span class="sd">        [0, 4, 4]</span>
-<span class="sd">        &gt;&gt;&gt; tr = G.basic_transversals</span>
-<span class="sd">        &gt;&gt;&gt; f[0] == tr[0][f1[0]]</span>
-<span class="sd">        True</span>
-
-<span class="sd">        If g is not an element of G then [] is returned:</span>
-
-<span class="sd">        &gt;&gt;&gt; c = Permutation(5, 6, 7)</span>
-<span class="sd">        &gt;&gt;&gt; G.coset_factor(c)</span>
-<span class="sd">        []</span>
-
-<span class="sd">        see util._strip</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="p">(</span><span class="n">Cycle</span><span class="p">,</span> <span class="n">Permutation</span><span class="p">)):</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">list</span><span class="p">()</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">g</span><span class="p">)</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_degree</span><span class="p">:</span>
-            <span class="c"># this could either adjust the size or return [] immediately</span>
-            <span class="c"># but we don&#39;t choose between the two and just signal a possible</span>
-            <span class="c"># error</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;g should be the same size as permutations of G&#39;</span><span class="p">)</span>
-        <span class="n">I</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_degree</span><span class="p">))</span>
-        <span class="n">basic_orbits</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">basic_orbits</span>
-        <span class="n">transversals</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_transversals</span>
-        <span class="n">factors</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">base</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">g</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">)):</span>
-            <span class="n">beta</span> <span class="o">=</span> <span class="n">h</span><span class="p">[</span><span class="n">base</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span>
-            <span class="k">if</span> <span class="n">beta</span> <span class="o">==</span> <span class="n">base</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                <span class="n">factors</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span>
-                <span class="k">continue</span>
-            <span class="k">if</span> <span class="n">beta</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">basic_orbits</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                <span class="k">return</span> <span class="p">[]</span>
-            <span class="n">u</span> <span class="o">=</span> <span class="n">transversals</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">beta</span><span class="p">]</span><span class="o">.</span><span class="n">_array_form</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">_af_rmul</span><span class="p">(</span><span class="n">_af_invert</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">h</span><span class="p">)</span>
-            <span class="n">factors</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">h</span> <span class="o">!=</span> <span class="n">I</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[]</span>
-        <span class="k">if</span> <span class="n">factor_index</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">factors</span>
-        <span class="n">tr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">basic_transversals</span>
-        <span class="n">factors</span> <span class="o">=</span> <span class="p">[</span><span class="n">tr</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">factors</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">))]</span>
-        <span class="k">return</span> <span class="n">factors</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.coset_rank"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.coset_rank">[docs]</a>    <span class="k">def</span> <span class="nf">coset_rank</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;rank using Schreier-Sims representation</span>
-
-<span class="sd">        The coset rank of ``g`` is the ordering number in which</span>
-<span class="sd">        it appears in the lexicographic listing according to the</span>
-<span class="sd">        coset decomposition</span>
-
-<span class="sd">        The ordering is the same as in G.generate(method=&#39;coset&#39;).</span>
-<span class="sd">        If ``g`` does not belong to the group it returns None.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; a = Permutation(0, 1, 3, 7, 6, 4)(2, 5)</span>
-<span class="sd">        &gt;&gt;&gt; b = Permutation(0, 1, 3, 2)(4, 5, 7, 6)</span>
-<span class="sd">        &gt;&gt;&gt; G = PermutationGroup([a, b])</span>
-<span class="sd">        &gt;&gt;&gt; c = Permutation(7)(2, 4)(3, 5)</span>
-<span class="sd">        &gt;&gt;&gt; G.coset_rank(c)</span>
-<span class="sd">        16</span>
-<span class="sd">        &gt;&gt;&gt; G.coset_unrank(16)</span>
-<span class="sd">        Permutation(7)(2, 4)(3, 5)</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        coset_factor</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">factors</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">coset_factor</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">factors</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="n">rank</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">transversals</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_transversals</span>
-        <span class="n">base</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_base</span>
-        <span class="n">basic_orbits</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_basic_orbits</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">)):</span>
-            <span class="n">k</span> <span class="o">=</span> <span class="n">factors</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="n">j</span> <span class="o">=</span> <span class="n">basic_orbits</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-            <span class="n">rank</span> <span class="o">+=</span> <span class="n">b</span><span class="o">*</span><span class="n">j</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="n">transversals</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">rank</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.coset_unrank"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.coset_unrank">[docs]</a>    <span class="k">def</span> <span class="nf">coset_unrank</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rank</span><span class="p">,</span> <span class="n">af</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;unrank using Schreier-Sims representation</span>
-
-<span class="sd">        coset_unrank is the inverse operation of coset_rank</span>
-<span class="sd">        if 0 &lt;= rank &lt; order; otherwise it returns None.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">rank</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">rank</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span><span class="p">():</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="n">base</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_base</span>
-        <span class="n">transversals</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_transversals</span>
-        <span class="n">basic_orbits</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_basic_orbits</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">m</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
-            <span class="n">rank</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="nb">divmod</span><span class="p">(</span><span class="n">rank</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">transversals</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span>
-            <span class="n">v</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">basic_orbits</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="p">[</span><span class="n">transversals</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">v</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span><span class="o">.</span><span class="n">_array_form</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="p">)]</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">_af_rmuln</span><span class="p">(</span><span class="o">*</span><span class="n">a</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">af</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">h</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="PermutationGroup.degree"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.degree">[docs]</a>    <span class="k">def</span> <span class="nf">degree</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the size of the permutations in the group.</span>
-
-<span class="sd">        The number of permutations comprising the group is given by</span>
-<span class="sd">        len(group); the number of permutations that can be generated</span>
-<span class="sd">        by the group is given by group.order().</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; a = Permutation([1, 0, 2])</span>
-<span class="sd">        &gt;&gt;&gt; G = PermutationGroup([a])</span>
-<span class="sd">        &gt;&gt;&gt; G.degree</span>
-<span class="sd">        3</span>
-<span class="sd">        &gt;&gt;&gt; len(G)</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; G.order()</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; list(G.generate())</span>
-<span class="sd">        [Permutation(2), Permutation(2)(0, 1)]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        order</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_degree</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.derived_series"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.derived_series">[docs]</a>    <span class="k">def</span> <span class="nf">derived_series</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;Return the derived series for the group.</span>
-
-<span class="sd">        The derived series for a group ``G`` is defined as</span>
-<span class="sd">        ``G = G_0 &gt; G_1 &gt; G_2 &gt; \ldots`` where ``G_i = [G_{i-1}, G_{i-1}]``,</span>
-<span class="sd">        i.e. ``G_i`` is the derived subgroup of ``G_{i-1}``, for</span>
-<span class="sd">        ``i\in\mathbb{N}``. When we have ``G_k = G_{k-1}`` for some</span>
-<span class="sd">        ``k\in\mathbb{N}``, the series terminates.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        A list of permutation groups containing the members of the derived</span>
-<span class="sd">        series in the order ``G = G_0, G_1, G_2, \ldots``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import (SymmetricGroup,</span>
-<span class="sd">        ... AlternatingGroup, DihedralGroup)</span>
-<span class="sd">        &gt;&gt;&gt; A = AlternatingGroup(5)</span>
-<span class="sd">        &gt;&gt;&gt; len(A.derived_series())</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; S = SymmetricGroup(4)</span>
-<span class="sd">        &gt;&gt;&gt; len(S.derived_series())</span>
-<span class="sd">        4</span>
-<span class="sd">        &gt;&gt;&gt; S.derived_series()[1].is_subgroup(AlternatingGroup(4))</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; S.derived_series()[2].is_subgroup(DihedralGroup(2))</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        derived_subgroup</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="p">]</span>
-        <span class="n">current</span> <span class="o">=</span> <span class="bp">self</span>
-        <span class="nb">next</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">derived_subgroup</span><span class="p">()</span>
-        <span class="k">while</span> <span class="ow">not</span> <span class="n">current</span><span class="o">.</span><span class="n">is_subgroup</span><span class="p">(</span><span class="nb">next</span><span class="p">):</span>
-            <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">next</span><span class="p">)</span>
-            <span class="n">current</span> <span class="o">=</span> <span class="nb">next</span>
-            <span class="nb">next</span> <span class="o">=</span> <span class="nb">next</span><span class="o">.</span><span class="n">derived_subgroup</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">res</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.derived_subgroup"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.derived_subgroup">[docs]</a>    <span class="k">def</span> <span class="nf">derived_subgroup</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute the derived subgroup.</span>
-
-<span class="sd">        The derived subgroup, or commutator subgroup is the subgroup generated</span>
-<span class="sd">        by all commutators ``[g, h] = hgh^{-1}g^{-1}`` for ``g, h\in G`` ; it is</span>
-<span class="sd">        equal to the normal closure of the set of commutators of the generators</span>
-<span class="sd">        ([1], p.28, [11]).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; a = Permutation([1, 0, 2, 4, 3])</span>
-<span class="sd">        &gt;&gt;&gt; b = Permutation([0, 1, 3, 2, 4])</span>
-<span class="sd">        &gt;&gt;&gt; G = PermutationGroup([a, b])</span>
-<span class="sd">        &gt;&gt;&gt; C = G.derived_subgroup()</span>
-<span class="sd">        &gt;&gt;&gt; list(C.generate(af=True))</span>
-<span class="sd">        [[0, 1, 2, 3, 4], [0, 1, 3, 4, 2], [0, 1, 4, 2, 3]]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        derived_series</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_r</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">p</span><span class="o">.</span><span class="n">_array_form</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">generators</span><span class="p">]</span>
-        <span class="n">gens_inv</span> <span class="o">=</span> <span class="p">[</span><span class="n">_af_invert</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">]</span>
-        <span class="n">set_commutators</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="n">degree</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_degree</span>
-        <span class="n">rng</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">degree</span><span class="p">))</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">r</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">r</span><span class="p">):</span>
-                <span class="n">p1</span> <span class="o">=</span> <span class="n">gens</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-                <span class="n">p2</span> <span class="o">=</span> <span class="n">gens</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">degree</span><span class="p">))</span>
-                <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">rng</span><span class="p">:</span>
-                    <span class="n">c</span><span class="p">[</span><span class="n">p2</span><span class="p">[</span><span class="n">p1</span><span class="p">[</span><span class="n">k</span><span class="p">]]]</span> <span class="o">=</span> <span class="n">p1</span><span class="p">[</span><span class="n">p2</span><span class="p">[</span><span class="n">k</span><span class="p">]]</span>
-                <span class="n">ct</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">ct</span> <span class="ow">in</span> <span class="n">set_commutators</span><span class="p">:</span>
-                    <span class="n">set_commutators</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">ct</span><span class="p">)</span>
-        <span class="n">cms</span> <span class="o">=</span> <span class="p">[</span><span class="n">_af_new</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">set_commutators</span><span class="p">]</span>
-        <span class="n">G2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">normal_closure</span><span class="p">(</span><span class="n">cms</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">G2</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.generate"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.generate">[docs]</a>    <span class="k">def</span> <span class="nf">generate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">&quot;coset&quot;</span><span class="p">,</span> <span class="n">af</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return iterator to generate the elements of the group</span>
-
-<span class="sd">        Iteration is done with one of these methods::</span>
-
-<span class="sd">          method=&#39;coset&#39;  using the Schreier-Sims coset representation</span>
-<span class="sd">          method=&#39;dimino&#39; using the Dimino method</span>
-
-<span class="sd">        If af = True it yields the array form of the permutations</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.polyhedron import tetrahedron</span>
-
-<span class="sd">        The permutation group given in the tetrahedron object is not</span>
-<span class="sd">        true groups:</span>
-
-<span class="sd">        &gt;&gt;&gt; G = tetrahedron.pgroup</span>
-<span class="sd">        &gt;&gt;&gt; G.is_group()</span>
-<span class="sd">        False</span>
-
-<span class="sd">        But the group generated by the permutations in the tetrahedron</span>
-<span class="sd">        pgroup -- even the first two -- is a proper group:</span>
-
-<span class="sd">        &gt;&gt;&gt; H = PermutationGroup(G[0], G[1])</span>
-<span class="sd">        &gt;&gt;&gt; J = PermutationGroup(list(H.generate())); J</span>
-<span class="sd">        PermutationGroup([</span>
-<span class="sd">            Permutation(0, 1)(2, 3),</span>
-<span class="sd">            Permutation(3),</span>
-<span class="sd">            Permutation(1, 2, 3),</span>
-<span class="sd">            Permutation(1, 3, 2),</span>
-<span class="sd">            Permutation(0, 3, 1),</span>
-<span class="sd">            Permutation(0, 2, 3),</span>
-<span class="sd">            Permutation(0, 3)(1, 2),</span>
-<span class="sd">            Permutation(0, 1, 3),</span>
-<span class="sd">            Permutation(3)(0, 2, 1),</span>
-<span class="sd">            Permutation(0, 3, 2),</span>
-<span class="sd">            Permutation(3)(0, 1, 2),</span>
-<span class="sd">            Permutation(0, 2)(1, 3)])</span>
-<span class="sd">        &gt;&gt;&gt; _.is_group()</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">method</span> <span class="o">==</span> <span class="s">&quot;coset&quot;</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">generate_schreier_sims</span><span class="p">(</span><span class="n">af</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">method</span> <span class="o">==</span> <span class="s">&quot;dimino&quot;</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">generate_dimino</span><span class="p">(</span><span class="n">af</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;No generation defined for </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">method</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.generate_dimino"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.generate_dimino">[docs]</a>    <span class="k">def</span> <span class="nf">generate_dimino</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">af</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Yield group elements using Dimino&#39;s algorithm</span>
-
-<span class="sd">        If af == True it yields the array form of the permutations</span>
-
-<span class="sd">        References</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        [1] The Implementation of Various Algorithms for Permutation Groups in</span>
-<span class="sd">        the Computer Algebra System: AXIOM, N.J. Doye, M.Sc. Thesis</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; a = Permutation([0, 2, 1, 3])</span>
-<span class="sd">        &gt;&gt;&gt; b = Permutation([0, 2, 3, 1])</span>
-<span class="sd">        &gt;&gt;&gt; g = PermutationGroup([a, b])</span>
-<span class="sd">        &gt;&gt;&gt; list(g.generate_dimino(af=True))</span>
-<span class="sd">        [[0, 1, 2, 3], [0, 2, 1, 3], [0, 2, 3, 1],</span>
-<span class="sd">         [0, 1, 3, 2], [0, 3, 2, 1], [0, 3, 1, 2]]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">idn</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">degree</span><span class="p">))</span>
-        <span class="n">order</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">element_list</span> <span class="o">=</span> <span class="p">[</span><span class="n">idn</span><span class="p">]</span>
-        <span class="n">set_element_list</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span><span class="nb">tuple</span><span class="p">(</span><span class="n">idn</span><span class="p">)])</span>
-        <span class="k">if</span> <span class="n">af</span><span class="p">:</span>
-            <span class="k">yield</span> <span class="n">idn</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">yield</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">idn</span><span class="p">)</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">p</span><span class="o">.</span><span class="n">_array_form</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">generators</span><span class="p">]</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">)):</span>
-            <span class="c"># D elements of the subgroup G_i generated by gens[:i]</span>
-            <span class="n">D</span> <span class="o">=</span> <span class="n">element_list</span><span class="p">[:]</span>
-            <span class="n">N</span> <span class="o">=</span> <span class="p">[</span><span class="n">idn</span><span class="p">]</span>
-            <span class="k">while</span> <span class="n">N</span><span class="p">:</span>
-                <span class="n">A</span> <span class="o">=</span> <span class="n">N</span>
-                <span class="n">N</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">A</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">[:</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]:</span>
-                        <span class="n">ag</span> <span class="o">=</span> <span class="n">_af_rmul</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">ag</span><span class="p">)</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">set_element_list</span><span class="p">:</span>
-                            <span class="c"># produce G_i*g</span>
-                            <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">D</span><span class="p">:</span>
-                                <span class="n">order</span> <span class="o">+=</span> <span class="mi">1</span>
-                                <span class="n">ap</span> <span class="o">=</span> <span class="n">_af_rmul</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">ag</span><span class="p">)</span>
-                                <span class="k">if</span> <span class="n">af</span><span class="p">:</span>
-                                    <span class="k">yield</span> <span class="n">ap</span>
-                                <span class="k">else</span><span class="p">:</span>
-                                    <span class="n">p</span> <span class="o">=</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">ap</span><span class="p">)</span>
-                                    <span class="k">yield</span> <span class="n">p</span>
-                                <span class="n">element_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ap</span><span class="p">)</span>
-                                <span class="n">set_element_list</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="nb">tuple</span><span class="p">(</span><span class="n">ap</span><span class="p">))</span>
-                                <span class="n">N</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ap</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_order</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">element_list</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.generate_schreier_sims"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.generate_schreier_sims">[docs]</a>    <span class="k">def</span> <span class="nf">generate_schreier_sims</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">af</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Yield group elements using the Schreier-Sims representation</span>
-<span class="sd">        in coset_rank order</span>
-
-<span class="sd">        If af = True it yields the array form of the permutations</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; a = Permutation([0, 2, 1, 3])</span>
-<span class="sd">        &gt;&gt;&gt; b = Permutation([0, 2, 3, 1])</span>
-<span class="sd">        &gt;&gt;&gt; g = PermutationGroup([a, b])</span>
-<span class="sd">        &gt;&gt;&gt; list(g.generate_schreier_sims(af=True))</span>
-<span class="sd">        [[0, 1, 2, 3], [0, 2, 1, 3], [0, 3, 2, 1],</span>
-<span class="sd">         [0, 1, 3, 2], [0, 2, 3, 1], [0, 3, 1, 2]]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_degree</span>
-        <span class="n">u</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">basic_transversals</span>
-        <span class="n">basic_orbits</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_basic_orbits</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">u</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">generators</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">af</span><span class="p">:</span>
-                    <span class="k">yield</span> <span class="n">x</span><span class="o">.</span><span class="n">_array_form</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">yield</span> <span class="n">x</span>
-            <span class="k">raise</span> <span class="ne">StopIteration</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">u</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">basic_orbits</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-                <span class="k">if</span> <span class="n">af</span><span class="p">:</span>
-                    <span class="k">yield</span> <span class="n">u</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">_array_form</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">yield</span> <span class="n">u</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">i</span><span class="p">]</span>
-            <span class="k">raise</span> <span class="ne">StopIteration</span>
-
-        <span class="n">u</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">u</span><span class="p">))</span>
-        <span class="n">basic_orbits</span> <span class="o">=</span> <span class="n">basic_orbits</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-        <span class="c"># stg stack of group elements</span>
-        <span class="n">stg</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">))]</span>
-        <span class="n">posmax</span> <span class="o">=</span> <span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">u</span><span class="p">]</span>
-        <span class="n">n1</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">posmax</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="n">pos</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">n1</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># backtrack when finished iterating over coset</span>
-            <span class="k">if</span> <span class="n">pos</span><span class="p">[</span><span class="n">h</span><span class="p">]</span> <span class="o">&gt;=</span> <span class="n">posmax</span><span class="p">[</span><span class="n">h</span><span class="p">]:</span>
-                <span class="c">#count_b += 1</span>
-                <span class="k">if</span> <span class="n">h</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">StopIteration</span>
-                <span class="n">pos</span><span class="p">[</span><span class="n">h</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="n">h</span> <span class="o">-=</span> <span class="mi">1</span>
-                <span class="n">stg</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-                <span class="k">continue</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">_af_rmul</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="n">h</span><span class="p">][</span><span class="n">basic_orbits</span><span class="p">[</span><span class="n">h</span><span class="p">][</span><span class="n">pos</span><span class="p">[</span><span class="n">h</span><span class="p">]]]</span><span class="o">.</span><span class="n">_array_form</span><span class="p">,</span> <span class="n">stg</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-            <span class="n">pos</span><span class="p">[</span><span class="n">h</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">stg</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-            <span class="n">h</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="n">h</span> <span class="o">==</span> <span class="n">n1</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">af</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">basic_orbits</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-                        <span class="n">p</span> <span class="o">=</span> <span class="n">_af_rmul</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">_array_form</span><span class="p">,</span> <span class="n">stg</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-                        <span class="k">yield</span> <span class="n">p</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">basic_orbits</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-                        <span class="n">p</span> <span class="o">=</span> <span class="n">_af_rmul</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">_array_form</span><span class="p">,</span> <span class="n">stg</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-                        <span class="n">p1</span> <span class="o">=</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-                        <span class="k">yield</span> <span class="n">p1</span>
-                <span class="n">stg</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-                <span class="n">h</span> <span class="o">-=</span> <span class="mi">1</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="PermutationGroup.generators"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.generators">[docs]</a>    <span class="k">def</span> <span class="nf">generators</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the generators of the group.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; a = Permutation([0, 2, 1])</span>
-<span class="sd">        &gt;&gt;&gt; b = Permutation([1, 0, 2])</span>
-<span class="sd">        &gt;&gt;&gt; G = PermutationGroup([a, b])</span>
-<span class="sd">        &gt;&gt;&gt; G.generators</span>
-<span class="sd">        [Permutation(1, 2), Permutation(2)(0, 1)]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_generators</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.contains"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.contains">[docs]</a>    <span class="k">def</span> <span class="nf">contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Test if permutation ``g`` belong to self, ``G``.</span>
-
-<span class="sd">        If ``g`` is an element of ``G`` it can be written as a product</span>
-<span class="sd">        of factors drawn from the cosets of ``G``&#39;s stabilizers. To see</span>
-<span class="sd">        if ``g`` is one of the actual generators defining the group use</span>
-<span class="sd">        ``G.has(g)``.</span>
-
-<span class="sd">        If ``strict`` is not True, ``g`` will be resized, if necessary,</span>
-<span class="sd">        to match the size of permutations in ``self``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-
-<span class="sd">        &gt;&gt;&gt; a = Permutation(1, 2)</span>
-<span class="sd">        &gt;&gt;&gt; b = Permutation(2, 3, 1)</span>
-<span class="sd">        &gt;&gt;&gt; G = PermutationGroup(a, b, degree=5)</span>
-<span class="sd">        &gt;&gt;&gt; G.contains(G[0]) # trivial check</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; elem = Permutation([[2, 3]], size=5)</span>
-<span class="sd">        &gt;&gt;&gt; G.contains(elem)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; G.contains(Permutation(4)(0, 1, 2, 3))</span>
-<span class="sd">        False</span>
-
-<span class="sd">        If strict is False, a permutation will be resized, if</span>
-<span class="sd">        necessary:</span>
-
-<span class="sd">        &gt;&gt;&gt; H = PermutationGroup(Permutation(5))</span>
-<span class="sd">        &gt;&gt;&gt; H.contains(Permutation(3))</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; H.contains(Permutation(3), strict=False)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        To test if a given permutation is present in the group:</span>
-
-<span class="sd">        &gt;&gt;&gt; elem in G.generators</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; G.has(elem)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        coset_factor, has, in</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">Permutation</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">g</span><span class="o">.</span><span class="n">size</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">degree</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">strict</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">degree</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">g</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">generators</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">coset_factor</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">array_form</span><span class="p">,</span> <span class="bp">True</span><span class="p">))</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="PermutationGroup.is_abelian"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.is_abelian">[docs]</a>    <span class="k">def</span> <span class="nf">is_abelian</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Test if the group is Abelian.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; a = Permutation([0, 2, 1])</span>
-<span class="sd">        &gt;&gt;&gt; b = Permutation([1, 0, 2])</span>
-<span class="sd">        &gt;&gt;&gt; G = PermutationGroup([a, b])</span>
-<span class="sd">        &gt;&gt;&gt; G.is_abelian</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; a = Permutation([0, 2, 1])</span>
-<span class="sd">        &gt;&gt;&gt; G = PermutationGroup([a])</span>
-<span class="sd">        &gt;&gt;&gt; G.is_abelian</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_is_abelian</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_is_abelian</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_is_abelian</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">p</span><span class="o">.</span><span class="n">_array_form</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">generators</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">y</span> <span class="o">&lt;=</span> <span class="n">x</span><span class="p">:</span>
-                    <span class="k">continue</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">_af_commutes_with</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">_is_abelian</span> <span class="o">=</span> <span class="bp">False</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="bp">True</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.is_alt_sym"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.is_alt_sym">[docs]</a>    <span class="k">def</span> <span class="nf">is_alt_sym</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="mf">0.05</span><span class="p">,</span> <span class="n">_random_prec</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;Monte Carlo test for the symmetric/alternating group for degrees</span>
-<span class="sd">        &gt;= 8.</span>
-
-<span class="sd">        More specifically, it is one-sided Monte Carlo with the</span>
-<span class="sd">        answer True (i.e., G is symmetric/alternating) guaranteed to be</span>
-<span class="sd">        correct, and the answer False being incorrect with probability eps.</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        The algorithm itself uses some nontrivial results from group theory and</span>
-<span class="sd">        number theory:</span>
-<span class="sd">        1) If a transitive group ``G`` of degree ``n`` contains an element</span>
-<span class="sd">        with a cycle of length ``n/2 &lt; p &lt; n-2`` for ``p`` a prime, ``G`` is the</span>
-<span class="sd">        symmetric or alternating group ([1], pp. 81-82)</span>
-<span class="sd">        2) The proportion of elements in the symmetric/alternating group having</span>
-<span class="sd">        the property described in 1) is approximately ``\log(2)/\log(n)``</span>
-<span class="sd">        ([1], p.82; [2], pp. 226-227).</span>
-<span class="sd">        The helper function ``_check_cycles_alt_sym`` is used to</span>
-<span class="sd">        go over the cycles in a permutation and look for ones satisfying 1).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import DihedralGroup</span>
-<span class="sd">        &gt;&gt;&gt; D = DihedralGroup(10)</span>
-<span class="sd">        &gt;&gt;&gt; D.is_alt_sym()</span>
-<span class="sd">        False</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        _check_cycles_alt_sym</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">_random_prec</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">degree</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">8</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_transitive</span><span class="p">():</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">17</span><span class="p">:</span>
-                <span class="n">c_n</span> <span class="o">=</span> <span class="mf">0.34</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">c_n</span> <span class="o">=</span> <span class="mf">0.57</span>
-            <span class="n">d_n</span> <span class="o">=</span> <span class="p">(</span><span class="n">c_n</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-            <span class="n">N_eps</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="o">-</span><span class="n">log</span><span class="p">(</span><span class="n">eps</span><span class="p">)</span><span class="o">/</span><span class="n">d_n</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N_eps</span><span class="p">):</span>
-                <span class="n">perm</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">random_pr</span><span class="p">()</span>
-                <span class="k">if</span> <span class="n">_check_cycles_alt_sym</span><span class="p">(</span><span class="n">perm</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">_random_prec</span><span class="p">[</span><span class="s">&#39;N_eps&#39;</span><span class="p">]):</span>
-                <span class="n">perm</span> <span class="o">=</span> <span class="n">_random_prec</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-                <span class="k">if</span> <span class="n">_check_cycles_alt_sym</span><span class="p">(</span><span class="n">perm</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">return</span> <span class="bp">False</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="PermutationGroup.is_nilpotent"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.is_nilpotent">[docs]</a>    <span class="k">def</span> <span class="nf">is_nilpotent</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Test if the group is nilpotent.</span>
-
-<span class="sd">        A group ``G`` is nilpotent if it has a central series of finite length.</span>
-<span class="sd">        Alternatively, ``G`` is nilpotent if its lower central series terminates</span>
-<span class="sd">        with the trivial group. Every nilpotent group is also solvable</span>
-<span class="sd">        ([1], p.29, [12]).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import (SymmetricGroup,</span>
-<span class="sd">        ... CyclicGroup)</span>
-<span class="sd">        &gt;&gt;&gt; C = CyclicGroup(6)</span>
-<span class="sd">        &gt;&gt;&gt; C.is_nilpotent</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; S = SymmetricGroup(5)</span>
-<span class="sd">        &gt;&gt;&gt; S.is_nilpotent</span>
-<span class="sd">        False</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        lower_central_series, is_solvable</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_is_nilpotent</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">lcs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lower_central_series</span><span class="p">()</span>
-            <span class="n">terminator</span> <span class="o">=</span> <span class="n">lcs</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="n">lcs</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-            <span class="n">gens</span> <span class="o">=</span> <span class="n">terminator</span><span class="o">.</span><span class="n">generators</span>
-            <span class="n">degree</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">degree</span>
-            <span class="n">identity</span> <span class="o">=</span> <span class="n">_af_new</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">degree</span><span class="p">)))</span>
-            <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">g</span> <span class="o">==</span> <span class="n">identity</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">):</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_is_solvable</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_is_nilpotent</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_is_nilpotent</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_is_nilpotent</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.is_normal"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.is_normal">[docs]</a>    <span class="k">def</span> <span class="nf">is_normal</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">gr</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Test if G=self is a normal subgroup of gr.</span>
-
-<span class="sd">        G is normal in gr if</span>
-<span class="sd">        for each g2 in G, g1 in gr, g = g1*g2*g1**-1 belongs to G</span>
-<span class="sd">        It is sufficient to check this for each g1 in gr.generator and</span>
-<span class="sd">        g2 g2 in G.generator</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; a = Permutation([1, 2, 0])</span>
-<span class="sd">        &gt;&gt;&gt; b = Permutation([1, 0, 2])</span>
-<span class="sd">        &gt;&gt;&gt; G = PermutationGroup([a, b])</span>
-<span class="sd">        &gt;&gt;&gt; G1 = PermutationGroup([a, Permutation([2, 0, 1])])</span>
-<span class="sd">        &gt;&gt;&gt; G1.is_normal(G)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">gens2</span> <span class="o">=</span> <span class="p">[</span><span class="n">p</span><span class="o">.</span><span class="n">_array_form</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">generators</span><span class="p">]</span>
-        <span class="n">gens1</span> <span class="o">=</span> <span class="p">[</span><span class="n">p</span><span class="o">.</span><span class="n">_array_form</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">gr</span><span class="o">.</span><span class="n">generators</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">g1</span> <span class="ow">in</span> <span class="n">gens1</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">g2</span> <span class="ow">in</span> <span class="n">gens2</span><span class="p">:</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="n">_af_rmuln</span><span class="p">(</span><span class="n">g1</span><span class="p">,</span> <span class="n">g2</span><span class="p">,</span> <span class="n">_af_invert</span><span class="p">(</span><span class="n">g1</span><span class="p">))</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">coset_factor</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="bp">True</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.is_primitive"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.is_primitive">[docs]</a>    <span class="k">def</span> <span class="nf">is_primitive</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">randomized</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Test if a group is primitive.</span>
-
-<span class="sd">        A permutation group ``G`` acting on a set ``S`` is called primitive if</span>
-<span class="sd">        ``S`` contains no nontrivial block under the action of ``G``</span>
-<span class="sd">        (a block is nontrivial if its cardinality is more than ``1``).</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        The algorithm is described in [1], p.83, and uses the function</span>
-<span class="sd">        minimal_block to search for blocks of the form ``\{0, k\}`` for ``k``</span>
-<span class="sd">        ranging over representatives for the orbits of ``G_0``, the stabilizer of</span>
-<span class="sd">        ``0``. This algorithm has complexity ``O(n^2)`` where ``n`` is the degree</span>
-<span class="sd">        of the group, and will perform badly if ``G_0`` is small.</span>
-
-<span class="sd">        There are two implementations offered: one finds ``G_0``</span>
-<span class="sd">        deterministically using the function ``stabilizer``, and the other</span>
-<span class="sd">        (default) produces random elements of ``G_0`` using ``random_stab``,</span>
-<span class="sd">        hoping that they generate a subgroup of ``G_0`` with not too many more</span>
-<span class="sd">        orbits than G_0 (this is suggested in [1], p.83). Behavior is changed</span>
-<span class="sd">        by the ``randomized`` flag.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import DihedralGroup</span>
-<span class="sd">        &gt;&gt;&gt; D = DihedralGroup(10)</span>
-<span class="sd">        &gt;&gt;&gt; D.is_primitive()</span>
-<span class="sd">        False</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        minimal_block, random_stab</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_is_primitive</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_is_primitive</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">degree</span>
-        <span class="k">if</span> <span class="n">randomized</span><span class="p">:</span>
-            <span class="n">random_stab_gens</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">v</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">schreier_vector</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="p">)):</span>
-                <span class="n">random_stab_gens</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">random_stab</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">v</span><span class="p">))</span>
-            <span class="n">stab</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="n">random_stab_gens</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">stab</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">stabilizer</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="n">orbits</span> <span class="o">=</span> <span class="n">stab</span><span class="o">.</span><span class="n">orbits</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">orb</span> <span class="ow">in</span> <span class="n">orbits</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">orb</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">x</span> <span class="o">!=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">minimal_block</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span> <span class="o">!=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">n</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_is_primitive</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_is_primitive</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="bp">True</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="PermutationGroup.is_solvable"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.is_solvable">[docs]</a>    <span class="k">def</span> <span class="nf">is_solvable</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Test if the group is solvable.</span>
-
-<span class="sd">        ``G`` is solvable if its derived series terminates with the trivial</span>
-<span class="sd">        group ([1], p.29).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import SymmetricGroup</span>
-<span class="sd">        &gt;&gt;&gt; S = SymmetricGroup(3)</span>
-<span class="sd">        &gt;&gt;&gt; S.is_solvable</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        is_nilpotent, derived_series</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_is_solvable</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">ds</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">derived_series</span><span class="p">()</span>
-            <span class="n">terminator</span> <span class="o">=</span> <span class="n">ds</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="n">ds</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-            <span class="n">gens</span> <span class="o">=</span> <span class="n">terminator</span><span class="o">.</span><span class="n">generators</span>
-            <span class="n">degree</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">degree</span>
-            <span class="n">identity</span> <span class="o">=</span> <span class="n">_af_new</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">degree</span><span class="p">)))</span>
-            <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">g</span> <span class="o">==</span> <span class="n">identity</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">):</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_is_solvable</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_is_solvable</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_is_solvable</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.is_subgroup"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.is_subgroup">[docs]</a>    <span class="k">def</span> <span class="nf">is_subgroup</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return True if all elements of self belong to G.</span>
-
-<span class="sd">        If ``strict`` is False then if ``self``&#39;s degree is smaller</span>
-<span class="sd">        than ``G``&#39;s, the elements will be resized to have the same degree.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation, PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import (SymmetricGroup,</span>
-<span class="sd">        ...    CyclicGroup)</span>
-
-<span class="sd">        Testing is strict by default: the degree of each group must be the</span>
-<span class="sd">        same:</span>
-
-<span class="sd">        &gt;&gt;&gt; p = Permutation(0, 1, 2, 3, 4, 5)</span>
-<span class="sd">        &gt;&gt;&gt; G1 = PermutationGroup([Permutation(0, 1, 2), Permutation(0, 1)])</span>
-<span class="sd">        &gt;&gt;&gt; G2 = PermutationGroup([Permutation(0, 2), Permutation(0, 1, 2)])</span>
-<span class="sd">        &gt;&gt;&gt; G3 = PermutationGroup([p, p**2])</span>
-<span class="sd">        &gt;&gt;&gt; assert G1.order() == G2.order() == G3.order() == 6</span>
-<span class="sd">        &gt;&gt;&gt; G1.is_subgroup(G2)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; G1.is_subgroup(G3)</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; G3.is_subgroup(PermutationGroup(G3[1]))</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; G3.is_subgroup(PermutationGroup(G3[0]))</span>
-<span class="sd">        True</span>
-
-<span class="sd">        To ignore the size, set ``strict`` to False:</span>
-
-<span class="sd">        &gt;&gt;&gt; S3 = SymmetricGroup(3)</span>
-<span class="sd">        &gt;&gt;&gt; S5 = SymmetricGroup(5)</span>
-<span class="sd">        &gt;&gt;&gt; S3.is_subgroup(S5, strict=False)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; C7 = CyclicGroup(7)</span>
-<span class="sd">        &gt;&gt;&gt; G = S5*C7</span>
-<span class="sd">        &gt;&gt;&gt; S5.is_subgroup(G, False)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; C7.is_subgroup(G, 0)</span>
-<span class="sd">        False</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">PermutationGroup</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="bp">self</span> <span class="o">==</span> <span class="n">G</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">order</span><span class="p">()</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">degree</span> <span class="o">==</span> <span class="n">G</span><span class="o">.</span><span class="n">degree</span> <span class="ow">or</span> \
-                <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">degree</span> <span class="o">&lt;</span> <span class="n">G</span><span class="o">.</span><span class="n">degree</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">strict</span><span class="p">):</span>
-            <span class="n">gens</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">generators</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="n">strict</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.is_transitive"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.is_transitive">[docs]</a>    <span class="k">def</span> <span class="nf">is_transitive</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Test if the group is transitive.</span>
-
-<span class="sd">        A group is transitive if it has a single orbit.</span>
-
-<span class="sd">        If ``strict`` is False the group is transitive if it has</span>
-<span class="sd">        a single orbit of length different from 1.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; a = Permutation([0, 2, 1, 3])</span>
-<span class="sd">        &gt;&gt;&gt; b = Permutation([2, 0, 1, 3])</span>
-<span class="sd">        &gt;&gt;&gt; G1 = PermutationGroup([a, b])</span>
-<span class="sd">        &gt;&gt;&gt; G1.is_transitive()</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; G1.is_transitive(strict=False)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; c = Permutation([2, 3, 0, 1])</span>
-<span class="sd">        &gt;&gt;&gt; G2 = PermutationGroup([a, c])</span>
-<span class="sd">        &gt;&gt;&gt; G2.is_transitive()</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; d = Permutation([1,0,2,3])</span>
-<span class="sd">        &gt;&gt;&gt; e = Permutation([0,1,3,2])</span>
-<span class="sd">        &gt;&gt;&gt; G3 = PermutationGroup([d, e])</span>
-<span class="sd">        &gt;&gt;&gt; G3.is_transitive() or G3.is_transitive(strict=False)</span>
-<span class="sd">        False</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_is_transitive</span><span class="p">:</span>  <span class="c"># strict or not, if True then True</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_is_transitive</span>
-        <span class="k">if</span> <span class="n">strict</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_is_transitive</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>  <span class="c"># we only store strict=True</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_is_transitive</span>
-
-            <span class="n">ans</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">orbit</span><span class="p">(</span><span class="mi">0</span><span class="p">))</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">degree</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_is_transitive</span> <span class="o">=</span> <span class="n">ans</span>
-            <span class="k">return</span> <span class="n">ans</span>
-
-        <span class="n">got_orb</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">orbits</span><span class="p">():</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">got_orb</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-                <span class="n">got_orb</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="n">got_orb</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="PermutationGroup.is_trivial"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.is_trivial">[docs]</a>    <span class="k">def</span> <span class="nf">is_trivial</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Test if the group is the trivial group.</span>
-
-<span class="sd">        This is true if the group contains only the identity permutation.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; G = PermutationGroup([Permutation([0, 1, 2])])</span>
-<span class="sd">        &gt;&gt;&gt; G.is_trivial</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_is_trivial</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_is_trivial</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="bp">self</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Identity</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_is_trivial</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.lower_central_series"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.lower_central_series">[docs]</a>    <span class="k">def</span> <span class="nf">lower_central_series</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;Return the lower central series for the group.</span>
-
-<span class="sd">        The lower central series for a group ``G`` is the series</span>
-<span class="sd">        ``G = G_0 &gt; G_1 &gt; G_2 &gt; \ldots`` where</span>
-<span class="sd">        ``G_k = [G, G_{k-1}]``, i.e. every term after the first is equal to the</span>
-<span class="sd">        commutator of ``G`` and the previous term in ``G1`` ([1], p.29).</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        A list of permutation groups in the order</span>
-<span class="sd">        ``G = G_0, G_1, G_2, \ldots``</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import (AlternatingGroup,</span>
-<span class="sd">        ... DihedralGroup)</span>
-<span class="sd">        &gt;&gt;&gt; A = AlternatingGroup(4)</span>
-<span class="sd">        &gt;&gt;&gt; len(A.lower_central_series())</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; A.lower_central_series()[1].is_subgroup(DihedralGroup(2))</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        commutator, derived_series</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="p">]</span>
-        <span class="n">current</span> <span class="o">=</span> <span class="bp">self</span>
-        <span class="nb">next</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">commutator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">current</span><span class="p">)</span>
-        <span class="k">while</span> <span class="ow">not</span> <span class="n">current</span><span class="o">.</span><span class="n">is_subgroup</span><span class="p">(</span><span class="nb">next</span><span class="p">):</span>
-            <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">next</span><span class="p">)</span>
-            <span class="n">current</span> <span class="o">=</span> <span class="nb">next</span>
-            <span class="nb">next</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">commutator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">current</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">res</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="PermutationGroup.max_div"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.max_div">[docs]</a>    <span class="k">def</span> <span class="nf">max_div</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Maximum proper divisor of the degree of a permutation group.</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        Obviously, this is the degree divided by its minimal proper divisor</span>
-<span class="sd">        (larger than ``1``, if one exists). As it is guaranteed to be prime,</span>
-<span class="sd">        the ``sieve`` from ``sympy.ntheory`` is used.</span>
-<span class="sd">        This function is also used as an optimization tool for the functions</span>
-<span class="sd">        ``minimal_block`` and ``_union_find_merge``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; G = PermutationGroup([Permutation([0,2,1,3])])</span>
-<span class="sd">        &gt;&gt;&gt; G.max_div</span>
-<span class="sd">        2</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        minimal_block, _union_find_merge</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_max_div</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_max_div</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">degree</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">1</span>
-        <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">sieve</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">%</span> <span class="n">x</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">d</span> <span class="o">=</span> <span class="n">n</span><span class="o">//</span><span class="n">x</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_max_div</span> <span class="o">=</span> <span class="n">d</span>
-                <span class="k">return</span> <span class="n">d</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.minimal_block"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.minimal_block">[docs]</a>    <span class="k">def</span> <span class="nf">minimal_block</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">points</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;For a transitive group, finds the block system generated by</span>
-<span class="sd">        ``points``.</span>
-
-<span class="sd">        If a group ``G`` acts on a set ``S``, a nonempty subset ``B`` of ``S``</span>
-<span class="sd">        is called a block under the action of ``G`` if for all ``g`` in ``G``</span>
-<span class="sd">        we have ``gB = B`` (``g`` fixes ``B``) or ``gB`` and ``B`` have no</span>
-<span class="sd">        common points (``g`` moves ``B`` entirely). ([1], p.23; [6]).</span>
-
-<span class="sd">        The distinct translates ``gB`` of a block ``B`` for ``g`` in ``G``</span>
-<span class="sd">        partition the set ``S`` and this set of translates is known as a block</span>
-<span class="sd">        system. Moreover, we obviously have that all blocks in the partition</span>
-<span class="sd">        have the same size, hence the block size divides ``|S|`` ([1], p.23).</span>
-<span class="sd">        A ``G``-congruence is an equivalence relation ``~`` on the set ``S``</span>
-<span class="sd">        such that ``a ~ b`` implies ``g(a) ~ g(b)`` for all ``g`` in ``G``.</span>
-<span class="sd">        For a transitive group, the equivalence classes of a ``G``-congruence</span>
-<span class="sd">        and the blocks of a block system are the same thing ([1], p.23).</span>
-
-<span class="sd">        The algorithm below checks the group for transitivity, and then finds</span>
-<span class="sd">        the ``G``-congruence generated by the pairs ``(p_0, p_1), (p_0, p_2),</span>
-<span class="sd">        ..., (p_0,p_{k-1})`` which is the same as finding the maximal block</span>
-<span class="sd">        system (i.e., the one with minimum block size) such that</span>
-<span class="sd">        ``p_0, ..., p_{k-1}`` are in the same block ([1], p.83).</span>
-
-<span class="sd">        It is an implementation of Atkinson&#39;s algorithm, as suggested in [1],</span>
-<span class="sd">        and manipulates an equivalence relation on the set ``S`` using a</span>
-<span class="sd">        union-find data structure. The running time is just above</span>
-<span class="sd">        ``O(|points||S|)``. ([1], pp. 83-87; [7]).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import DihedralGroup</span>
-<span class="sd">        &gt;&gt;&gt; D = DihedralGroup(10)</span>
-<span class="sd">        &gt;&gt;&gt; D.minimal_block([0,5])</span>
-<span class="sd">        [0, 6, 2, 8, 4, 0, 6, 2, 8, 4]</span>
-<span class="sd">        &gt;&gt;&gt; D.minimal_block([0,1])</span>
-<span class="sd">        [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        _union_find_rep, _union_find_merge, is_transitive, is_primitive</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_transitive</span><span class="p">():</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">degree</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">generators</span>
-        <span class="c"># initialize the list of equivalence class representatives</span>
-        <span class="n">parents</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-        <span class="n">ranks</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">n</span>
-        <span class="n">not_rep</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">points</span><span class="p">)</span>
-        <span class="c"># the block size must divide the degree of the group</span>
-        <span class="k">if</span> <span class="n">k</span> <span class="o">&gt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">max_div</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">n</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">parents</span><span class="p">[</span><span class="n">points</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]]</span> <span class="o">=</span> <span class="n">points</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">not_rep</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">points</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span>
-        <span class="n">ranks</span><span class="p">[</span><span class="n">points</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span> <span class="o">=</span> <span class="n">k</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">len_not_rep</span> <span class="o">=</span> <span class="n">k</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">len_not_rep</span><span class="p">:</span>
-            <span class="n">temp</span> <span class="o">=</span> <span class="n">not_rep</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">:</span>
-                <span class="c"># find has side effects: performs path compression on the list</span>
-                <span class="c"># of representatives</span>
-                <span class="n">delta</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_union_find_rep</span><span class="p">(</span><span class="n">temp</span><span class="p">,</span> <span class="n">parents</span><span class="p">)</span>
-                <span class="c"># union has side effects: performs union by rank on the list</span>
-                <span class="c"># of representatives</span>
-                <span class="n">temp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_union_find_merge</span><span class="p">(</span><span class="n">gen</span><span class="p">(</span><span class="n">temp</span><span class="p">),</span> <span class="n">gen</span><span class="p">(</span><span class="n">delta</span><span class="p">),</span> <span class="n">ranks</span><span class="p">,</span>
-                                              <span class="n">parents</span><span class="p">,</span> <span class="n">not_rep</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">temp</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">n</span>
-                <span class="n">len_not_rep</span> <span class="o">+=</span> <span class="n">temp</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="c"># force path compression to get the final state of the equivalence</span>
-            <span class="c"># relation</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_union_find_rep</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">parents</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">parents</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.normal_closure"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.normal_closure">[docs]</a>    <span class="k">def</span> <span class="nf">normal_closure</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">10</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;Return the normal closure of a subgroup/set of permutations.</span>
-
-<span class="sd">        If ``S`` is a subset of a group ``G``, the normal closure of ``A`` in ``G``</span>
-<span class="sd">        is defined as the intersection of all normal subgroups of ``G`` that</span>
-<span class="sd">        contain ``A`` ([1], p.14). Alternatively, it is the group generated by</span>
-<span class="sd">        the conjugates ``x^{-1}yx`` for ``x`` a generator of ``G`` and ``y`` a</span>
-<span class="sd">        generator of the subgroup ``\left\langle S\right\rangle`` generated by</span>
-<span class="sd">        ``S`` (for some chosen generating set for ``\left\langle S\right\rangle``)</span>
-<span class="sd">        ([1], p.73).</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        other</span>
-<span class="sd">            a subgroup/list of permutations/single permutation</span>
-<span class="sd">        k</span>
-<span class="sd">            an implementation-specific parameter that determines the number</span>
-<span class="sd">            of conjugates that are adjoined to ``other`` at once</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import (SymmetricGroup,</span>
-<span class="sd">        ... CyclicGroup, AlternatingGroup)</span>
-<span class="sd">        &gt;&gt;&gt; S = SymmetricGroup(5)</span>
-<span class="sd">        &gt;&gt;&gt; C = CyclicGroup(5)</span>
-<span class="sd">        &gt;&gt;&gt; G = S.normal_closure(C)</span>
-<span class="sd">        &gt;&gt;&gt; G.order()</span>
-<span class="sd">        60</span>
-<span class="sd">        &gt;&gt;&gt; G.is_subgroup(AlternatingGroup(5))</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        commutator, derived_subgroup, random_pr</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        The algorithm is described in [1], pp. 73-74; it makes use of the</span>
-<span class="sd">        generation of random elements for permutation groups by the product</span>
-<span class="sd">        replacement algorithm.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="s">&#39;generators&#39;</span><span class="p">):</span>
-            <span class="n">degree</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">degree</span>
-            <span class="n">identity</span> <span class="o">=</span> <span class="n">_af_new</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">degree</span><span class="p">)))</span>
-
-            <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">g</span> <span class="o">==</span> <span class="n">identity</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">other</span><span class="o">.</span><span class="n">generators</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">other</span>
-            <span class="n">Z</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">generators</span><span class="p">[:])</span>
-            <span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span> <span class="o">=</span> <span class="n">Z</span><span class="o">.</span><span class="n">schreier_sims_incremental</span><span class="p">()</span>
-            <span class="n">strong_gens_distr</span> <span class="o">=</span> <span class="n">_distribute_gens_by_base</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span><span class="p">)</span>
-            <span class="n">basic_orbits</span><span class="p">,</span> <span class="n">basic_transversals</span> <span class="o">=</span> \
-                <span class="n">_orbits_transversals_from_bsgs</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens_distr</span><span class="p">)</span>
-
-            <span class="bp">self</span><span class="o">.</span><span class="n">_random_pr_init</span><span class="p">(</span><span class="n">r</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
-
-            <span class="n">_loop</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">while</span> <span class="n">_loop</span><span class="p">:</span>
-                <span class="n">Z</span><span class="o">.</span><span class="n">_random_pr_init</span><span class="p">(</span><span class="n">r</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
-                    <span class="n">g</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">random_pr</span><span class="p">()</span>
-                    <span class="n">h</span> <span class="o">=</span> <span class="n">Z</span><span class="o">.</span><span class="n">random_pr</span><span class="p">()</span>
-                    <span class="n">conj</span> <span class="o">=</span> <span class="n">h</span><span class="o">^</span><span class="n">g</span>
-                    <span class="n">res</span> <span class="o">=</span> <span class="n">_strip</span><span class="p">(</span><span class="n">conj</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="n">basic_orbits</span><span class="p">,</span> <span class="n">basic_transversals</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">res</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="n">identity</span> <span class="ow">or</span> <span class="n">res</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="n">gens</span> <span class="o">=</span> <span class="n">Z</span><span class="o">.</span><span class="n">generators</span>
-                        <span class="n">gens</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">conj</span><span class="p">)</span>
-                        <span class="n">Z</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span>
-                        <span class="n">strong_gens</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">conj</span><span class="p">)</span>
-                        <span class="n">temp_base</span><span class="p">,</span> <span class="n">temp_strong_gens</span> <span class="o">=</span> \
-                            <span class="n">Z</span><span class="o">.</span><span class="n">schreier_sims_incremental</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span><span class="p">)</span>
-                        <span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span> <span class="o">=</span> <span class="n">temp_base</span><span class="p">,</span> <span class="n">temp_strong_gens</span>
-                        <span class="n">strong_gens_distr</span> <span class="o">=</span> \
-                            <span class="n">_distribute_gens_by_base</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span><span class="p">)</span>
-                        <span class="n">basic_orbits</span><span class="p">,</span> <span class="n">basic_transversals</span> <span class="o">=</span> \
-                            <span class="n">_orbits_transversals_from_bsgs</span><span class="p">(</span><span class="n">base</span><span class="p">,</span>
-                                <span class="n">strong_gens_distr</span><span class="p">)</span>
-                <span class="n">_loop</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">generators</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">Z</span><span class="o">.</span><span class="n">generators</span><span class="p">:</span>
-                        <span class="n">conj</span> <span class="o">=</span> <span class="n">h</span><span class="o">^</span><span class="n">g</span>
-                        <span class="n">res</span> <span class="o">=</span> <span class="n">_strip</span><span class="p">(</span><span class="n">conj</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="n">basic_orbits</span><span class="p">,</span>
-                                     <span class="n">basic_transversals</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">res</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="n">identity</span> <span class="ow">or</span> <span class="n">res</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:</span>
-                            <span class="n">_loop</span> <span class="o">=</span> <span class="bp">True</span>
-                            <span class="k">break</span>
-                    <span class="k">if</span> <span class="n">_loop</span><span class="p">:</span>
-                        <span class="k">break</span>
-            <span class="k">return</span> <span class="n">Z</span>
-        <span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="s">&#39;__getitem__&#39;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">normal_closure</span><span class="p">(</span><span class="n">PermutationGroup</span><span class="p">(</span><span class="n">other</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="s">&#39;array_form&#39;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">normal_closure</span><span class="p">(</span><span class="n">PermutationGroup</span><span class="p">([</span><span class="n">other</span><span class="p">]))</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.orbit"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.orbit">[docs]</a>    <span class="k">def</span> <span class="nf">orbit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">action</span><span class="o">=</span><span class="s">&#39;tuples&#39;</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;Compute the orbit of alpha ``\{g(\alpha) | g \in G\}`` as a set.</span>
-
-<span class="sd">        The time complexity of the algorithm used here is ``O(|Orb|*r)`` where</span>
-<span class="sd">        ``|Orb|`` is the size of the orbit and ``r`` is the number of generators of</span>
-<span class="sd">        the group. For a more detailed analysis, see [1], p.78, [2], pp. 19-21.</span>
-<span class="sd">        Here alpha can be a single point, or a list of points.</span>
-
-<span class="sd">        If alpha is a single point, the ordinary orbit is computed.</span>
-<span class="sd">        if alpha is a list of points, there are three available options:</span>
-
-<span class="sd">        &#39;union&#39; - computes the union of the orbits of the points in the list</span>
-<span class="sd">        &#39;tuples&#39; - computes the orbit of the list interpreted as an ordered</span>
-<span class="sd">        tuple under the group action ( i.e., g((1,2,3)) = (g(1), g(2), g(3)) )</span>
-<span class="sd">        &#39;sets&#39; - computes the orbit of the list interpreted as a sets</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; a = Permutation([1,2,0,4,5,6,3])</span>
-<span class="sd">        &gt;&gt;&gt; G = PermutationGroup([a])</span>
-<span class="sd">        &gt;&gt;&gt; G.orbit(0)</span>
-<span class="sd">        set([0, 1, 2])</span>
-<span class="sd">        &gt;&gt;&gt; G.orbit([0,4], &#39;union&#39;)</span>
-<span class="sd">        set([0, 1, 2, 3, 4, 5, 6])</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        orbit_transversal</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">_orbit</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">degree</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">generators</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">action</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.orbit_rep"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.orbit_rep">[docs]</a>    <span class="k">def</span> <span class="nf">orbit_rep</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">schreier_vector</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return a group element which sends ``alpha`` to ``beta``.</span>
-
-<span class="sd">        If ``beta`` is not in the orbit of ``alpha``, the function returns</span>
-<span class="sd">        ``False``. This implementation makes use of the schreier vector.</span>
-<span class="sd">        For a proof of correctness, see [1], p.80</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import AlternatingGroup</span>
-<span class="sd">        &gt;&gt;&gt; G = AlternatingGroup(5)</span>
-<span class="sd">        &gt;&gt;&gt; G.orbit_rep(0, 4)</span>
-<span class="sd">        Permutation(0, 4, 1, 2, 3)</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        schreier_vector</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">schreier_vector</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">schreier_vector</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">schreier_vector</span><span class="p">(</span><span class="n">alpha</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">schreier_vector</span><span class="p">[</span><span class="n">beta</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">schreier_vector</span><span class="p">[</span><span class="n">beta</span><span class="p">]</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="o">.</span><span class="n">_array_form</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">generators</span><span class="p">]</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">while</span> <span class="n">k</span> <span class="o">!=</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-            <span class="n">a</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">gens</span><span class="p">[</span><span class="n">k</span><span class="p">])</span>
-            <span class="n">beta</span> <span class="o">=</span> <span class="n">gens</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span> <span class="c"># beta = (~gens[k])(beta)</span>
-            <span class="n">k</span> <span class="o">=</span> <span class="n">schreier_vector</span><span class="p">[</span><span class="n">beta</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">a</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">_af_rmuln</span><span class="p">(</span><span class="o">*</span><span class="n">a</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_af_new</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_degree</span><span class="p">)))</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.orbit_transversal"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.orbit_transversal">[docs]</a>    <span class="k">def</span> <span class="nf">orbit_transversal</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">pairs</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;Computes a transversal for the orbit of ``alpha`` as a set.</span>
-
-<span class="sd">        For a permutation group ``G``, a transversal for the orbit</span>
-<span class="sd">        ``Orb = \{g(\alpha) | g \in G\}`` is a set</span>
-<span class="sd">        ``\{g_\beta | g_\beta(\alpha) = \beta\}`` for ``\beta \in Orb``.</span>
-<span class="sd">        Note that there may be more than one possible transversal.</span>
-<span class="sd">        If ``pairs`` is set to ``True``, it returns the list of pairs</span>
-<span class="sd">        ``(\beta, g_\beta)``. For a proof of correctness, see [1], p.79</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import DihedralGroup</span>
-<span class="sd">        &gt;&gt;&gt; G = DihedralGroup(6)</span>
-<span class="sd">        &gt;&gt;&gt; G.orbit_transversal(0)</span>
-<span class="sd">        [Permutation(5),</span>
-<span class="sd">         Permutation(0, 1, 2, 3, 4, 5),</span>
-<span class="sd">         Permutation(0, 5)(1, 4)(2, 3),</span>
-<span class="sd">         Permutation(0, 2, 4)(1, 3, 5),</span>
-<span class="sd">         Permutation(5)(0, 4)(1, 3),</span>
-<span class="sd">         Permutation(0, 3)(1, 4)(2, 5)]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        orbit</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">_orbit_transversal</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_degree</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">generators</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">pairs</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.orbits"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.orbits">[docs]</a>    <span class="k">def</span> <span class="nf">orbits</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rep</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the orbits of self, ordered according to lowest element</span>
-<span class="sd">        in each orbit.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; a = Permutation(1,5)(2,3)(4,0,6)</span>
-<span class="sd">        &gt;&gt;&gt; b = Permutation(1,5)(3,4)(2,6,0)</span>
-<span class="sd">        &gt;&gt;&gt; G = PermutationGroup([a, b])</span>
-<span class="sd">        &gt;&gt;&gt; G.orbits()</span>
-<span class="sd">        [set([0, 2, 3, 4, 6]), set([1, 5])]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">_orbits</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_degree</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_generators</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.order"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.order">[docs]</a>    <span class="k">def</span> <span class="nf">order</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the order of the group: the number of permutations that</span>
-<span class="sd">        can be generated from elements of the group.</span>
-
-<span class="sd">        The number of permutations comprising the group is given by</span>
-<span class="sd">        len(group); the length of each permutation in the group is</span>
-<span class="sd">        given by group.size.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-
-<span class="sd">        &gt;&gt;&gt; a = Permutation([1, 0, 2])</span>
-<span class="sd">        &gt;&gt;&gt; G = PermutationGroup([a])</span>
-<span class="sd">        &gt;&gt;&gt; G.degree</span>
-<span class="sd">        3</span>
-<span class="sd">        &gt;&gt;&gt; len(G)</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; G.order()</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; list(G.generate())</span>
-<span class="sd">        [Permutation(2), Permutation(2)(0, 1)]</span>
-
-<span class="sd">        &gt;&gt;&gt; a = Permutation([0, 2, 1])</span>
-<span class="sd">        &gt;&gt;&gt; b = Permutation([1, 0, 2])</span>
-<span class="sd">        &gt;&gt;&gt; G = PermutationGroup([a, b])</span>
-<span class="sd">        &gt;&gt;&gt; G.order()</span>
-<span class="sd">        6</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        degree</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_order</span> <span class="o">!=</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_order</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_is_sym</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_degree</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_order</span> <span class="o">=</span> <span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_order</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_is_alt</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_degree</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_order</span> <span class="o">=</span> <span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_order</span>
-
-        <span class="n">basic_transversals</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">basic_transversals</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">basic_transversals</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">*=</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_order</span> <span class="o">=</span> <span class="n">m</span>
-        <span class="k">return</span> <span class="n">m</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.pointwise_stabilizer"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.pointwise_stabilizer">[docs]</a>    <span class="k">def</span> <span class="nf">pointwise_stabilizer</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">points</span><span class="p">,</span> <span class="n">incremental</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;Return the pointwise stabilizer for a set of points.</span>
-
-<span class="sd">        For a permutation group ``G`` and a set of points</span>
-<span class="sd">        ``\{p_1, p_2,\ldots, p_k\}``, the pointwise stabilizer of</span>
-<span class="sd">        ``p_1, p_2, \ldots, p_k`` is defined as</span>
-<span class="sd">        ``G_{p_1,\ldots, p_k} =</span>
-<span class="sd">        \{g\in G | g(p_i) = p_i \forall i\in\{1, 2,\ldots,k\}\} ([1],p20).</span>
-<span class="sd">        It is a subgroup of ``G``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import SymmetricGroup</span>
-<span class="sd">        &gt;&gt;&gt; S = SymmetricGroup(7)</span>
-<span class="sd">        &gt;&gt;&gt; Stab = S.pointwise_stabilizer([2, 3, 5])</span>
-<span class="sd">        &gt;&gt;&gt; Stab.is_subgroup(S.stabilizer(2).stabilizer(3).stabilizer(5))</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        stabilizer, schreier_sims_incremental</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        When incremental == True,</span>
-<span class="sd">        rather than the obvious implementation using successive calls to</span>
-<span class="sd">        .stabilizer(), this uses the incremental Schreier-Sims algorithm</span>
-<span class="sd">        to obtain a base with starting segment - the given points.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">incremental</span><span class="p">:</span>
-            <span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">schreier_sims_incremental</span><span class="p">(</span><span class="n">base</span><span class="o">=</span><span class="n">points</span><span class="p">)</span>
-            <span class="n">stab_gens</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">degree</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">degree</span>
-            <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">strong_gens</span><span class="p">:</span>
-                <span class="k">if</span> <span class="p">[</span><span class="n">gen</span><span class="p">(</span><span class="n">point</span><span class="p">)</span> <span class="k">for</span> <span class="n">point</span> <span class="ow">in</span> <span class="n">points</span><span class="p">]</span> <span class="o">==</span> <span class="n">points</span><span class="p">:</span>
-                    <span class="n">stab_gens</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">gen</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">stab_gens</span><span class="p">:</span>
-                <span class="n">stab_gens</span> <span class="o">=</span> <span class="n">_af_new</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">degree</span><span class="p">)))</span>
-            <span class="k">return</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="n">stab_gens</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">gens</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_generators</span>
-            <span class="n">degree</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">degree</span>
-            <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">points</span><span class="p">:</span>
-                <span class="n">gens</span> <span class="o">=</span> <span class="n">_stabilizer</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.make_perm"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.make_perm">[docs]</a>    <span class="k">def</span> <span class="nf">make_perm</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Multiply ``n`` randomly selected permutations from</span>
-<span class="sd">        pgroup together, starting with the identity</span>
-<span class="sd">        permutation. If ``n`` is a list of integers, those</span>
-<span class="sd">        integers will be used to select the permutations and they</span>
-<span class="sd">        will be applied in L to R order: make_perm((A, B, C)) will</span>
-<span class="sd">        give CBA(I) where I is the identity permutation.</span>
-
-<span class="sd">        ``seed`` is used to set the seed for the random selection</span>
-<span class="sd">        of permutations from pgroup. If this is a list of integers,</span>
-<span class="sd">        the corresponding permutations from pgroup will be selected</span>
-<span class="sd">        in the order give. This is mainly used for testing purposes.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; a, b = [Permutation([1, 0, 3, 2]), Permutation([1, 3, 0, 2])]</span>
-<span class="sd">        &gt;&gt;&gt; G = PermutationGroup([a, b])</span>
-<span class="sd">        &gt;&gt;&gt; G.make_perm(1, [0])</span>
-<span class="sd">        Permutation(0, 1)(2, 3)</span>
-<span class="sd">        &gt;&gt;&gt; G.make_perm(3, [0, 1, 0])</span>
-<span class="sd">        Permutation(0, 2, 3, 1)</span>
-<span class="sd">        &gt;&gt;&gt; G.make_perm([0, 1, 0])</span>
-<span class="sd">        Permutation(0, 2, 3, 1)</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        random</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">seed</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;If n is a sequence, seed should be None&#39;</span><span class="p">)</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">seed</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">n</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;n must be an integer or a sequence.&#39;</span><span class="p">)</span>
-        <span class="n">randrange</span> <span class="o">=</span> <span class="n">_randrange</span><span class="p">(</span><span class="n">seed</span><span class="p">)</span>
-
-        <span class="c"># start with the identity permutation</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">degree</span><span class="p">)))</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="bp">self</span><span class="p">[</span><span class="n">randrange</span><span class="p">(</span><span class="n">m</span><span class="p">)]</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">rmul</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">result</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.random"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.random">[docs]</a>    <span class="k">def</span> <span class="nf">random</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">af</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return a random group element</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">rank</span> <span class="o">=</span> <span class="n">randrange</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">order</span><span class="p">())</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">coset_unrank</span><span class="p">(</span><span class="n">rank</span><span class="p">,</span> <span class="n">af</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.random_pr"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.random_pr">[docs]</a>    <span class="k">def</span> <span class="nf">random_pr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">gen_count</span><span class="o">=</span><span class="mi">11</span><span class="p">,</span> <span class="n">iterations</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span> <span class="n">_random_prec</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return a random group element using product replacement.</span>
-
-<span class="sd">        For the details of the product replacement algorithm, see</span>
-<span class="sd">        ``_random_pr_init`` In ``random_pr`` the actual &#39;product replacement&#39;</span>
-<span class="sd">        is performed. Notice that if the attribute ``_random_gens``</span>
-<span class="sd">        is empty, it needs to be initialized by ``_random_pr_init``.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        _random_pr_init</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_random_gens</span> <span class="o">==</span> <span class="p">[]:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_random_pr_init</span><span class="p">(</span><span class="n">gen_count</span><span class="p">,</span> <span class="n">iterations</span><span class="p">)</span>
-        <span class="n">random_gens</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_random_gens</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">random_gens</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="c"># handle randomized input for testing purposes</span>
-        <span class="k">if</span> <span class="n">_random_prec</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">randrange</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-            <span class="n">t</span> <span class="o">=</span> <span class="n">randrange</span><span class="p">(</span><span class="n">r</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">t</span> <span class="o">==</span> <span class="n">s</span><span class="p">:</span>
-                <span class="n">t</span> <span class="o">=</span> <span class="n">r</span> <span class="o">-</span> <span class="mi">1</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">choice</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-            <span class="n">e</span> <span class="o">=</span> <span class="n">choice</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">_random_prec</span><span class="p">[</span><span class="s">&#39;s&#39;</span><span class="p">]</span>
-            <span class="n">t</span> <span class="o">=</span> <span class="n">_random_prec</span><span class="p">[</span><span class="s">&#39;t&#39;</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">t</span> <span class="o">==</span> <span class="n">s</span><span class="p">:</span>
-                <span class="n">t</span> <span class="o">=</span> <span class="n">r</span> <span class="o">-</span> <span class="mi">1</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">_random_prec</span><span class="p">[</span><span class="s">&#39;x&#39;</span><span class="p">]</span>
-            <span class="n">e</span> <span class="o">=</span> <span class="n">_random_prec</span><span class="p">[</span><span class="s">&#39;e&#39;</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">random_gens</span><span class="p">[</span><span class="n">s</span><span class="p">]</span> <span class="o">=</span> <span class="n">_af_rmul</span><span class="p">(</span><span class="n">random_gens</span><span class="p">[</span><span class="n">s</span><span class="p">],</span> <span class="n">_af_pow</span><span class="p">(</span><span class="n">random_gens</span><span class="p">[</span><span class="n">t</span><span class="p">],</span> <span class="n">e</span><span class="p">))</span>
-            <span class="n">random_gens</span><span class="p">[</span><span class="n">r</span><span class="p">]</span> <span class="o">=</span> <span class="n">_af_rmul</span><span class="p">(</span><span class="n">random_gens</span><span class="p">[</span><span class="n">r</span><span class="p">],</span> <span class="n">random_gens</span><span class="p">[</span><span class="n">s</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">random_gens</span><span class="p">[</span><span class="n">s</span><span class="p">]</span> <span class="o">=</span> <span class="n">_af_rmul</span><span class="p">(</span><span class="n">_af_pow</span><span class="p">(</span><span class="n">random_gens</span><span class="p">[</span><span class="n">t</span><span class="p">],</span> <span class="n">e</span><span class="p">),</span> <span class="n">random_gens</span><span class="p">[</span><span class="n">s</span><span class="p">])</span>
-            <span class="n">random_gens</span><span class="p">[</span><span class="n">r</span><span class="p">]</span> <span class="o">=</span> <span class="n">_af_rmul</span><span class="p">(</span><span class="n">random_gens</span><span class="p">[</span><span class="n">s</span><span class="p">],</span> <span class="n">random_gens</span><span class="p">[</span><span class="n">r</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">random_gens</span><span class="p">[</span><span class="n">r</span><span class="p">])</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.random_stab"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.random_stab">[docs]</a>    <span class="k">def</span> <span class="nf">random_stab</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">schreier_vector</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">_random_prec</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Random element from the stabilizer of ``alpha``.</span>
-
-<span class="sd">        The schreier vector for ``alpha`` is an optional argument used</span>
-<span class="sd">        for speeding up repeated calls. The algorithm is described in [1], p.81</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        random_pr, orbit_rep</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">schreier_vector</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">schreier_vector</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">schreier_vector</span><span class="p">(</span><span class="n">alpha</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">_random_prec</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">rand</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">random_pr</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">rand</span> <span class="o">=</span> <span class="n">_random_prec</span><span class="p">[</span><span class="s">&#39;rand&#39;</span><span class="p">]</span>
-        <span class="n">beta</span> <span class="o">=</span> <span class="n">rand</span><span class="p">(</span><span class="n">alpha</span><span class="p">)</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">orbit_rep</span><span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">schreier_vector</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">rmul</span><span class="p">(</span><span class="o">~</span><span class="n">h</span><span class="p">,</span> <span class="n">rand</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.schreier_sims"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims">[docs]</a>    <span class="k">def</span> <span class="nf">schreier_sims</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Schreier-Sims algorithm.</span>
-
-<span class="sd">        It computes the generators of the chain of stabilizers</span>
-<span class="sd">        G &gt; G_{b_1} &gt; .. &gt; G_{b1,..,b_r} &gt; 1</span>
-<span class="sd">        in which G_{b_1,..,b_i} stabilizes b_1,..,b_i,</span>
-<span class="sd">        and the corresponding ``s`` cosets.</span>
-<span class="sd">        An element of the group can be written as the product</span>
-<span class="sd">        h_1*..*h_s.</span>
-
-<span class="sd">        We use the incremental Schreier-Sims algorithm.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; a = Permutation([0, 2, 1])</span>
-<span class="sd">        &gt;&gt;&gt; b = Permutation([1, 0, 2])</span>
-<span class="sd">        &gt;&gt;&gt; G = PermutationGroup([a, b])</span>
-<span class="sd">        &gt;&gt;&gt; G.schreier_sims()</span>
-<span class="sd">        &gt;&gt;&gt; G.basic_transversals</span>
-<span class="sd">        [{0: Permutation(2)(0, 1), 1: Permutation(2), 2: Permutation(1, 2)},</span>
-<span class="sd">         {0: Permutation(2), 2: Permutation(0, 2)}]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_transversals</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">schreier_sims_incremental</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_base</span> <span class="o">=</span> <span class="n">base</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_strong_gens</span> <span class="o">=</span> <span class="n">strong_gens</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">base</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_transversals</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_basic_orbits</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">return</span>
-
-        <span class="n">strong_gens_distr</span> <span class="o">=</span> <span class="n">_distribute_gens_by_base</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span><span class="p">)</span>
-        <span class="n">basic_orbits</span><span class="p">,</span> <span class="n">transversals</span> <span class="o">=</span> <span class="n">_orbits_transversals_from_bsgs</span><span class="p">(</span><span class="n">base</span><span class="p">,</span>\
-                <span class="n">strong_gens_distr</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_transversals</span> <span class="o">=</span> <span class="n">transversals</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_basic_orbits</span> <span class="o">=</span> <span class="p">[</span><span class="nb">sorted</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">basic_orbits</span><span class="p">]</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.schreier_sims_incremental"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims_incremental">[docs]</a>    <span class="k">def</span> <span class="nf">schreier_sims_incremental</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">base</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">gens</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Extend a sequence of points and generating set to a base and strong</span>
-<span class="sd">        generating set.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        base</span>
-<span class="sd">            The sequence of points to be extended to a base. Optional</span>
-<span class="sd">            parameter with default value ``[]``.</span>
-<span class="sd">        gens</span>
-<span class="sd">            The generating set to be extended to a strong generating set</span>
-<span class="sd">            relative to the base obtained. Optional parameter with default</span>
-<span class="sd">            value ``self.generators``.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        (base, strong_gens)</span>
-<span class="sd">            ``base`` is the base obtained, and ``strong_gens`` is the strong</span>
-<span class="sd">            generating set relative to it. The original parameters ``base``,</span>
-<span class="sd">            ``gens`` remain unchanged.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import AlternatingGroup</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.testutil import _verify_bsgs</span>
-<span class="sd">        &gt;&gt;&gt; A = AlternatingGroup(7)</span>
-<span class="sd">        &gt;&gt;&gt; base = [2, 3]</span>
-<span class="sd">        &gt;&gt;&gt; seq = [2, 3]</span>
-<span class="sd">        &gt;&gt;&gt; base, strong_gens = A.schreier_sims_incremental(base=seq)</span>
-<span class="sd">        &gt;&gt;&gt; _verify_bsgs(A, base, strong_gens)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; base[:2]</span>
-<span class="sd">        [2, 3]</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        This version of the Schreier-Sims algorithm runs in polynomial time.</span>
-<span class="sd">        There are certain assumptions in the implementation - if the trivial</span>
-<span class="sd">        group is provided, ``base`` and ``gens`` are returned immediately,</span>
-<span class="sd">        as any sequence of points is a base for the trivial group. If the</span>
-<span class="sd">        identity is present in the generators ``gens``, it is removed as</span>
-<span class="sd">        it is a redundant generator.</span>
-<span class="sd">        The implementation is described in [1], pp. 90-93.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        schreier_sims, schreier_sims_random</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">base</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">base</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">if</span> <span class="n">gens</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">gens</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">generators</span><span class="p">[:]</span>
-        <span class="n">degree</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">degree</span>
-        <span class="n">id_af</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">degree</span><span class="p">))</span>
-        <span class="c"># handle the trivial group</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">gens</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Identity</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">base</span><span class="p">,</span> <span class="n">gens</span>
-        <span class="c"># prevent side effects</span>
-        <span class="n">_base</span><span class="p">,</span> <span class="n">_gens</span> <span class="o">=</span> <span class="n">base</span><span class="p">[:],</span> <span class="n">gens</span><span class="p">[:]</span>
-        <span class="c"># remove the identity as a generator</span>
-        <span class="n">_gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">_gens</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">x</span><span class="o">.</span><span class="n">is_Identity</span><span class="p">]</span>
-        <span class="c"># make sure no generator fixes all base points</span>
-        <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">_gens</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">x</span> <span class="o">==</span> <span class="n">gen</span><span class="o">.</span><span class="n">_array_form</span><span class="p">[</span><span class="n">x</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">_base</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">new</span> <span class="ow">in</span> <span class="n">id_af</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">gen</span><span class="o">.</span><span class="n">_array_form</span><span class="p">[</span><span class="n">new</span><span class="p">]</span> <span class="o">!=</span> <span class="n">new</span><span class="p">:</span>
-                        <span class="k">break</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">assert</span> <span class="bp">None</span>  <span class="c"># can this ever happen?</span>
-                <span class="n">_base</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">new</span><span class="p">)</span>
-        <span class="c"># distribute generators according to basic stabilizers</span>
-        <span class="n">strong_gens_distr</span> <span class="o">=</span> <span class="n">_distribute_gens_by_base</span><span class="p">(</span><span class="n">_base</span><span class="p">,</span> <span class="n">_gens</span><span class="p">)</span>
-        <span class="c"># initialize the basic stabilizers, basic orbits and basic transversals</span>
-        <span class="n">orbs</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="n">transversals</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="n">base_len</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">_base</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">base_len</span><span class="p">):</span>
-            <span class="n">transversals</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">_orbit_transversal</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">strong_gens_distr</span><span class="p">[</span><span class="n">i</span><span class="p">],</span>
-                <span class="n">_base</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">pairs</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">af</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-            <span class="n">orbs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">transversals</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-        <span class="c"># main loop: amend the stabilizer chain until we have generators</span>
-        <span class="c"># for all stabilizers</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="n">base_len</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="k">while</span> <span class="n">i</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="c"># this flag is used to continue with the main loop from inside</span>
-            <span class="c"># a nested loop</span>
-            <span class="n">continue_i</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="c"># test the generators for being a strong generating set</span>
-            <span class="n">db</span> <span class="o">=</span> <span class="p">{}</span>
-            <span class="k">for</span> <span class="n">beta</span><span class="p">,</span> <span class="n">u_beta</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">transversals</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-                <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">strong_gens_distr</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                    <span class="n">gb</span> <span class="o">=</span> <span class="n">gen</span><span class="o">.</span><span class="n">_array_form</span><span class="p">[</span><span class="n">beta</span><span class="p">]</span>
-                    <span class="n">u1</span> <span class="o">=</span> <span class="n">transversals</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">gb</span><span class="p">]</span>
-                    <span class="n">g1</span> <span class="o">=</span> <span class="n">_af_rmul</span><span class="p">(</span><span class="n">gen</span><span class="o">.</span><span class="n">_array_form</span><span class="p">,</span> <span class="n">u_beta</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">g1</span> <span class="o">!=</span> <span class="n">u1</span><span class="p">:</span>
-                        <span class="c"># test if the schreier generator is in the i+1-th</span>
-                        <span class="c"># would-be basic stabilizer</span>
-                        <span class="n">y</span> <span class="o">=</span> <span class="bp">True</span>
-                        <span class="k">try</span><span class="p">:</span>
-                            <span class="n">u1_inv</span> <span class="o">=</span> <span class="n">db</span><span class="p">[</span><span class="n">gb</span><span class="p">]</span>
-                        <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-                            <span class="n">u1_inv</span> <span class="o">=</span> <span class="n">db</span><span class="p">[</span><span class="n">gb</span><span class="p">]</span> <span class="o">=</span> <span class="n">_af_invert</span><span class="p">(</span><span class="n">u1</span><span class="p">)</span>
-                        <span class="n">schreier_gen</span> <span class="o">=</span> <span class="n">_af_rmul</span><span class="p">(</span><span class="n">u1_inv</span><span class="p">,</span> <span class="n">g1</span><span class="p">)</span>
-                        <span class="n">h</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">_strip_af</span><span class="p">(</span><span class="n">schreier_gen</span><span class="p">,</span> <span class="n">_base</span><span class="p">,</span> <span class="n">orbs</span><span class="p">,</span> <span class="n">transversals</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">j</span> <span class="o">&lt;=</span> <span class="n">base_len</span><span class="p">:</span>
-                            <span class="c"># new strong generator h at level j</span>
-                            <span class="n">y</span> <span class="o">=</span> <span class="bp">False</span>
-                        <span class="k">elif</span> <span class="n">h</span><span class="p">:</span>
-                            <span class="c"># h fixes all base points</span>
-                            <span class="n">y</span> <span class="o">=</span> <span class="bp">False</span>
-                            <span class="n">moved</span> <span class="o">=</span> <span class="mi">0</span>
-                            <span class="k">while</span> <span class="n">h</span><span class="p">[</span><span class="n">moved</span><span class="p">]</span> <span class="o">==</span> <span class="n">moved</span><span class="p">:</span>
-                                <span class="n">moved</span> <span class="o">+=</span> <span class="mi">1</span>
-                            <span class="n">_base</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">moved</span><span class="p">)</span>
-                            <span class="n">base_len</span> <span class="o">+=</span> <span class="mi">1</span>
-                            <span class="n">strong_gens_distr</span><span class="o">.</span><span class="n">append</span><span class="p">([])</span>
-                        <span class="k">if</span> <span class="n">y</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                            <span class="c"># if a new strong generator is found, update the</span>
-                            <span class="c"># data structures and start over</span>
-                            <span class="n">h</span> <span class="o">=</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-                            <span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j</span><span class="p">):</span>
-                                <span class="n">strong_gens_distr</span><span class="p">[</span><span class="n">l</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-                                <span class="n">transversals</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span>\
-                                <span class="nb">dict</span><span class="p">(</span><span class="n">_orbit_transversal</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">strong_gens_distr</span><span class="p">[</span><span class="n">l</span><span class="p">],</span>
-                                    <span class="n">_base</span><span class="p">[</span><span class="n">l</span><span class="p">],</span> <span class="n">pairs</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">af</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-                                <span class="n">orbs</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">transversals</span><span class="p">[</span><span class="n">l</span><span class="p">]</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-                            <span class="n">i</span> <span class="o">=</span> <span class="n">j</span> <span class="o">-</span> <span class="mi">1</span>
-                            <span class="c"># continue main loop using the flag</span>
-                            <span class="n">continue_i</span> <span class="o">=</span> <span class="bp">True</span>
-                    <span class="k">if</span> <span class="n">continue_i</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                        <span class="k">break</span>
-                <span class="k">if</span> <span class="n">continue_i</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                    <span class="k">break</span>
-            <span class="k">if</span> <span class="n">continue_i</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="n">i</span> <span class="o">-=</span> <span class="mi">1</span>
-        <span class="c"># build the strong generating set</span>
-        <span class="n">strong_gens</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">gens</span> <span class="ow">in</span> <span class="n">strong_gens_distr</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">gen</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">strong_gens</span><span class="p">:</span>
-                    <span class="n">strong_gens</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">gen</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">_base</span><span class="p">,</span> <span class="n">strong_gens</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.schreier_sims_random"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims_random">[docs]</a>    <span class="k">def</span> <span class="nf">schreier_sims_random</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">base</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">gens</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">consec_succ</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span>
-                             <span class="n">_random_prec</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;Randomized Schreier-Sims algorithm.</span>
-
-<span class="sd">        The randomized Schreier-Sims algorithm takes the sequence ``base``</span>
-<span class="sd">        and the generating set ``gens``, and extends ``base`` to a base, and</span>
-<span class="sd">        ``gens`` to a strong generating set relative to that base with</span>
-<span class="sd">        probability of a wrong answer at most ``2^{-consec\_succ}``,</span>
-<span class="sd">        provided the random generators are sufficiently random.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        base</span>
-<span class="sd">            The sequence to be extended to a base.</span>
-<span class="sd">        gens</span>
-<span class="sd">            The generating set to be extended to a strong generating set.</span>
-<span class="sd">        consec_succ</span>
-<span class="sd">            The parameter defining the probability of a wrong answer.</span>
-<span class="sd">        _random_prec</span>
-<span class="sd">            An internal parameter used for testing purposes.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        (base, strong_gens)</span>
-<span class="sd">            ``base`` is the base and ``strong_gens`` is the strong generating</span>
-<span class="sd">            set relative to it.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.testutil import _verify_bsgs</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import SymmetricGroup</span>
-<span class="sd">        &gt;&gt;&gt; S = SymmetricGroup(5)</span>
-<span class="sd">        &gt;&gt;&gt; base, strong_gens = S.schreier_sims_random(consec_succ=5)</span>
-<span class="sd">        &gt;&gt;&gt; _verify_bsgs(S, base, strong_gens) #doctest: +SKIP</span>
-<span class="sd">        True</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        The algorithm is described in detail in [1], pp. 97-98. It extends</span>
-<span class="sd">        the orbits ``orbs`` and the permutation groups ``stabs`` to</span>
-<span class="sd">        basic orbits and basic stabilizers for the base and strong generating</span>
-<span class="sd">        set produced in the end.</span>
-<span class="sd">        The idea of the extension process</span>
-<span class="sd">        is to &quot;sift&quot; random group elements through the stabilizer chain</span>
-<span class="sd">        and amend the stabilizers/orbits along the way when a sift</span>
-<span class="sd">        is not successful.</span>
-<span class="sd">        The helper function ``_strip`` is used to attempt</span>
-<span class="sd">        to decompose a random group element according to the current</span>
-<span class="sd">        state of the stabilizer chain and report whether the element was</span>
-<span class="sd">        fully decomposed (successful sift) or not (unsuccessful sift). In</span>
-<span class="sd">        the latter case, the level at which the sift failed is reported and</span>
-<span class="sd">        used to amend ``stabs``, ``base``, ``gens`` and ``orbs`` accordingly.</span>
-<span class="sd">        The halting condition is for ``consec_succ`` consecutive successful</span>
-<span class="sd">        sifts to pass. This makes sure that the current ``base`` and ``gens``</span>
-<span class="sd">        form a BSGS with probability at least ``1 - 1/\text{consec\_succ}``.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        schreier_sims</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">base</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">base</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">if</span> <span class="n">gens</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">gens</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">generators</span>
-        <span class="n">base_len</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">degree</span>
-        <span class="c"># make sure no generator fixes all base points</span>
-        <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">gen</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">==</span> <span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">base</span><span class="p">):</span>
-                <span class="n">new</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="k">while</span> <span class="n">gen</span><span class="o">.</span><span class="n">_array_form</span><span class="p">[</span><span class="n">new</span><span class="p">]</span> <span class="o">==</span> <span class="n">new</span><span class="p">:</span>
-                    <span class="n">new</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="n">base</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">new</span><span class="p">)</span>
-                <span class="n">base_len</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="c"># distribute generators according to basic stabilizers</span>
-        <span class="n">strong_gens_distr</span> <span class="o">=</span> <span class="n">_distribute_gens_by_base</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">gens</span><span class="p">)</span>
-        <span class="c"># initialize the basic stabilizers, basic transversals and basic orbits</span>
-        <span class="n">transversals</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="n">orbs</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">base_len</span><span class="p">):</span>
-            <span class="n">transversals</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">_orbit_transversal</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">strong_gens_distr</span><span class="p">[</span><span class="n">i</span><span class="p">],</span>
-                <span class="n">base</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">pairs</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-            <span class="n">orbs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">transversals</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-        <span class="c"># initialize the number of consecutive elements sifted</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="c"># start sifting random elements while the number of consecutive sifts</span>
-        <span class="c"># is less than consec_succ</span>
-        <span class="k">while</span> <span class="n">c</span> <span class="o">&lt;</span> <span class="n">consec_succ</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">_random_prec</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">g</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">random_pr</span><span class="p">()</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">g</span> <span class="o">=</span> <span class="n">_random_prec</span><span class="p">[</span><span class="s">&#39;g&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">_strip</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="n">orbs</span><span class="p">,</span> <span class="n">transversals</span><span class="p">)</span>
-            <span class="n">y</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="c"># determine whether a new base point is needed</span>
-            <span class="k">if</span> <span class="n">j</span> <span class="o">&lt;=</span> <span class="n">base_len</span><span class="p">:</span>
-                <span class="n">y</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="k">elif</span> <span class="ow">not</span> <span class="n">h</span><span class="o">.</span><span class="n">is_Identity</span><span class="p">:</span>
-                <span class="n">y</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="n">moved</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="k">while</span> <span class="n">h</span><span class="p">(</span><span class="n">moved</span><span class="p">)</span> <span class="o">==</span> <span class="n">moved</span><span class="p">:</span>
-                    <span class="n">moved</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="n">base</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">moved</span><span class="p">)</span>
-                <span class="n">base_len</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="n">strong_gens_distr</span><span class="o">.</span><span class="n">append</span><span class="p">([])</span>
-            <span class="c"># if the element doesn&#39;t sift, amend the strong generators and</span>
-            <span class="c"># associated stabilizers and orbits</span>
-            <span class="k">if</span> <span class="n">y</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">j</span><span class="p">):</span>
-                    <span class="n">strong_gens_distr</span><span class="p">[</span><span class="n">l</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-                    <span class="n">transversals</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">_orbit_transversal</span><span class="p">(</span><span class="n">n</span><span class="p">,</span>
-                        <span class="n">strong_gens_distr</span><span class="p">[</span><span class="n">l</span><span class="p">],</span> <span class="n">base</span><span class="p">[</span><span class="n">l</span><span class="p">],</span> <span class="n">pairs</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-                    <span class="n">orbs</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">transversals</span><span class="p">[</span><span class="n">l</span><span class="p">]</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">c</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="c"># build the strong generating set</span>
-        <span class="n">strong_gens</span> <span class="o">=</span> <span class="n">strong_gens_distr</span><span class="p">[</span><span class="mi">0</span><span class="p">][:]</span>
-        <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">strong_gens_distr</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span>
-            <span class="k">if</span> <span class="n">gen</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">strong_gens</span><span class="p">:</span>
-                <span class="n">strong_gens</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">gen</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.schreier_vector"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.schreier_vector">[docs]</a>    <span class="k">def</span> <span class="nf">schreier_vector</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">alpha</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes the schreier vector for ``alpha``.</span>
-
-<span class="sd">        The Schreier vector efficiently stores information</span>
-<span class="sd">        about the orbit of ``alpha``. It can later be used to quickly obtain</span>
-<span class="sd">        elements of the group that send ``alpha`` to a particular element</span>
-<span class="sd">        in the orbit. Notice that the Schreier vector depends on the order</span>
-<span class="sd">        in which the group generators are listed. For a definition, see [3].</span>
-<span class="sd">        Since list indices start from zero, we adopt the convention to use</span>
-<span class="sd">        &quot;None&quot; instead of 0 to signify that an element doesn&#39;t belong</span>
-<span class="sd">        to the orbit.</span>
-<span class="sd">        For the algorithm and its correctness, see [2], pp.78-80.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; a = Permutation([2,4,6,3,1,5,0])</span>
-<span class="sd">        &gt;&gt;&gt; b = Permutation([0,1,3,5,4,6,2])</span>
-<span class="sd">        &gt;&gt;&gt; G = PermutationGroup([a,b])</span>
-<span class="sd">        &gt;&gt;&gt; G.schreier_vector(0)</span>
-<span class="sd">        [-1, None, 0, 1, None, 1, 0]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        orbit</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">degree</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="bp">None</span><span class="p">]</span><span class="o">*</span><span class="n">n</span>
-        <span class="n">v</span><span class="p">[</span><span class="n">alpha</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-        <span class="n">orb</span> <span class="o">=</span> <span class="p">[</span><span class="n">alpha</span><span class="p">]</span>
-        <span class="n">used</span> <span class="o">=</span> <span class="p">[</span><span class="bp">False</span><span class="p">]</span><span class="o">*</span><span class="n">n</span>
-        <span class="n">used</span><span class="p">[</span><span class="n">alpha</span><span class="p">]</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">generators</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">orb</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">r</span><span class="p">):</span>
-                <span class="n">temp</span> <span class="o">=</span> <span class="n">gens</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">_array_form</span><span class="p">[</span><span class="n">b</span><span class="p">]</span>
-                <span class="k">if</span> <span class="n">used</span><span class="p">[</span><span class="n">temp</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                    <span class="n">orb</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">temp</span><span class="p">)</span>
-                    <span class="n">used</span><span class="p">[</span><span class="n">temp</span><span class="p">]</span> <span class="o">=</span> <span class="bp">True</span>
-                    <span class="n">v</span><span class="p">[</span><span class="n">temp</span><span class="p">]</span> <span class="o">=</span> <span class="n">i</span>
-        <span class="k">return</span> <span class="n">v</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.stabilizer"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.stabilizer">[docs]</a>    <span class="k">def</span> <span class="nf">stabilizer</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">alpha</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;Return the stabilizer subgroup of ``alpha``.</span>
-
-<span class="sd">        The stabilizer of ``\alpha`` is the group ``G_\alpha =</span>
-<span class="sd">        \{g \in G | g(\alpha) = \alpha\}``.</span>
-<span class="sd">        For a proof of correctness, see [1], p.79.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import DihedralGroup</span>
-<span class="sd">        &gt;&gt;&gt; G = DihedralGroup(6)</span>
-<span class="sd">        &gt;&gt;&gt; G.stabilizer(5)</span>
-<span class="sd">        PermutationGroup([</span>
-<span class="sd">            Permutation(5)(0, 4)(1, 3),</span>
-<span class="sd">            Permutation(5)])</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        orbit</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">PermGroup</span><span class="p">(</span><span class="n">_stabilizer</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_degree</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_generators</span><span class="p">,</span> <span class="n">alpha</span><span class="p">))</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="PermutationGroup.strong_gens"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.strong_gens">[docs]</a>    <span class="k">def</span> <span class="nf">strong_gens</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return a strong generating set from the Schreier-Sims algorithm.</span>
-
-<span class="sd">        A generating set ``S = \{g_1, g_2, ..., g_t\}`` for a permutation group</span>
-<span class="sd">        ``G`` is a strong generating set relative to the sequence of points</span>
-<span class="sd">        (referred to as a &quot;base&quot;) ``(b_1, b_2, ..., b_k)`` if, for</span>
-<span class="sd">        ``1 \leq i \leq k`` we have that the intersection of the pointwise</span>
-<span class="sd">        stabilizer ``G^{(i+1)} := G_{b_1, b_2, ..., b_i}`` with ``S`` generates</span>
-<span class="sd">        the pointwise stabilizer ``G^{(i+1)}``. The concepts of a base and</span>
-<span class="sd">        strong generating set and their applications are discussed in depth</span>
-<span class="sd">        in [1], pp. 87-89 and [2], pp. 55-57.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import DihedralGroup</span>
-<span class="sd">        &gt;&gt;&gt; D = DihedralGroup(4)</span>
-<span class="sd">        &gt;&gt;&gt; D.strong_gens</span>
-<span class="sd">        [Permutation(0, 1, 2, 3), Permutation(0, 3)(1, 2), Permutation(1, 3)]</span>
-<span class="sd">        &gt;&gt;&gt; D.base</span>
-<span class="sd">        [0, 1]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        base, basic_transversals, basic_orbits, basic_stabilizers</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_strong_gens</span> <span class="o">==</span> <span class="p">[]:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">schreier_sims</span><span class="p">()</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_strong_gens</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.subgroup_search"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.subgroup_search">[docs]</a>    <span class="k">def</span> <span class="nf">subgroup_search</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prop</span><span class="p">,</span> <span class="n">base</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">strong_gens</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">tests</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span>
-                        <span class="n">init_subgroup</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Find the subgroup of all elements satisfying the property ``prop``.</span>
-
-<span class="sd">        This is done by a depth-first search with respect to base images that</span>
-<span class="sd">        uses several tests to prune the search tree.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        prop</span>
-<span class="sd">            The property to be used. Has to be callable on group elements</span>
-<span class="sd">            and always return ``True`` or ``False``. It is assumed that</span>
-<span class="sd">            all group elements satisfying ``prop`` indeed form a subgroup.</span>
-<span class="sd">        base</span>
-<span class="sd">            A base for the supergroup.</span>
-<span class="sd">        strong_gens</span>
-<span class="sd">            A strong generating set for the supergroup.</span>
-<span class="sd">        tests</span>
-<span class="sd">            A list of callables of length equal to the length of ``base``.</span>
-<span class="sd">            These are used to rule out group elements by partial base images,</span>
-<span class="sd">            so that ``tests[l](g)`` returns False if the element ``g`` is known</span>
-<span class="sd">            not to satisfy prop base on where g sends the first ``l + 1`` base</span>
-<span class="sd">            points.</span>
-<span class="sd">        init_subgroup</span>
-<span class="sd">            if a subgroup of the sought group is</span>
-<span class="sd">            known in advance, it can be passed to the function as this</span>
-<span class="sd">            parameter.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        res</span>
-<span class="sd">            The subgroup of all elements satisfying ``prop``. The generating</span>
-<span class="sd">            set for this group is guaranteed to be a strong generating set</span>
-<span class="sd">            relative to the base ``base``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.named_groups import (SymmetricGroup,</span>
-<span class="sd">        ... AlternatingGroup)</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.testutil import _verify_bsgs</span>
-<span class="sd">        &gt;&gt;&gt; S = SymmetricGroup(7)</span>
-<span class="sd">        &gt;&gt;&gt; prop_even = lambda x: x.is_even</span>
-<span class="sd">        &gt;&gt;&gt; base, strong_gens = S.schreier_sims_incremental()</span>
-<span class="sd">        &gt;&gt;&gt; G = S.subgroup_search(prop_even, base=base, strong_gens=strong_gens)</span>
-<span class="sd">        &gt;&gt;&gt; G.is_subgroup(AlternatingGroup(7))</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; _verify_bsgs(G, base, G.generators)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        This function is extremely lenghty and complicated and will require</span>
-<span class="sd">        some careful attention. The implementation is described in</span>
-<span class="sd">        [1], pp. 114-117, and the comments for the code here follow the lines</span>
-<span class="sd">        of the pseudocode in the book for clarity.</span>
-
-<span class="sd">        The complexity is exponential in general, since the search process by</span>
-<span class="sd">        itself visits all members of the supergroup. However, there are a lot</span>
-<span class="sd">        of tests which are used to prune the search tree, and users can define</span>
-<span class="sd">        their own tests via the ``tests`` parameter, so in practice, and for</span>
-<span class="sd">        some computations, it&#39;s not terrible.</span>
-
-<span class="sd">        A crucial part in the procedure is the frequent base change performed</span>
-<span class="sd">        (this is line 11 in the pseudocode) in order to obtain a new basic</span>
-<span class="sd">        stabilizer. The book mentiones that this can be done by using</span>
-<span class="sd">        ``.baseswap(...)``, however the current imlementation uses a more</span>
-<span class="sd">        straightforward way to find the next basic stabilizer - calling the</span>
-<span class="sd">        function ``.stabilizer(...)`` on the previous basic stabilizer.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># initialize BSGS and basic group properties</span>
-        <span class="k">def</span> <span class="nf">get_reps</span><span class="p">(</span><span class="n">orbits</span><span class="p">):</span>
-            <span class="c"># get the minimal element in the base ordering</span>
-            <span class="k">return</span> <span class="p">[</span><span class="nb">min</span><span class="p">(</span><span class="n">orbit</span><span class="p">,</span> <span class="n">key</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">base_ordering</span><span class="p">[</span><span class="n">x</span><span class="p">])</span> \
-              <span class="k">for</span> <span class="n">orbit</span> <span class="ow">in</span> <span class="n">orbits</span><span class="p">]</span>
-
-        <span class="k">def</span> <span class="nf">update_nu</span><span class="p">(</span><span class="n">l</span><span class="p">):</span>
-            <span class="n">temp_index</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">basic_orbits</span><span class="p">[</span><span class="n">l</span><span class="p">])</span> <span class="o">+</span> <span class="mi">1</span> <span class="o">-</span>\
-                         <span class="nb">len</span><span class="p">(</span><span class="n">res_basic_orbits_init_base</span><span class="p">[</span><span class="n">l</span><span class="p">])</span>
-            <span class="c"># this corresponds to the element larger than all points</span>
-            <span class="k">if</span> <span class="n">temp_index</span> <span class="o">&gt;=</span> <span class="nb">len</span><span class="p">(</span><span class="n">sorted_orbits</span><span class="p">[</span><span class="n">l</span><span class="p">]):</span>
-                <span class="n">nu</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">base_ordering</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">nu</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">sorted_orbits</span><span class="p">[</span><span class="n">l</span><span class="p">][</span><span class="n">temp_index</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">base</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">schreier_sims_incremental</span><span class="p">()</span>
-        <span class="n">base_len</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-        <span class="n">degree</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">degree</span>
-        <span class="n">identity</span> <span class="o">=</span> <span class="n">_af_new</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">degree</span><span class="p">)))</span>
-        <span class="n">base_ordering</span> <span class="o">=</span> <span class="n">_base_ordering</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">degree</span><span class="p">)</span>
-        <span class="c"># add an element larger than all points</span>
-        <span class="n">base_ordering</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">degree</span><span class="p">)</span>
-        <span class="c"># add an element smaller than all points</span>
-        <span class="n">base_ordering</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-        <span class="c"># compute BSGS-related structures</span>
-        <span class="n">strong_gens_distr</span> <span class="o">=</span> <span class="n">_distribute_gens_by_base</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span><span class="p">)</span>
-        <span class="n">basic_orbits</span><span class="p">,</span> <span class="n">transversals</span> <span class="o">=</span> <span class="n">_orbits_transversals_from_bsgs</span><span class="p">(</span><span class="n">base</span><span class="p">,</span>
-                                     <span class="n">strong_gens_distr</span><span class="p">)</span>
-        <span class="c"># handle subgroup initialization and tests</span>
-        <span class="k">if</span> <span class="n">init_subgroup</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">init_subgroup</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">identity</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">tests</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">trivial_test</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="bp">True</span>
-            <span class="n">tests</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">base_len</span><span class="p">):</span>
-                <span class="n">tests</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">trivial_test</span><span class="p">)</span>
-        <span class="c"># line 1: more initializations.</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="n">init_subgroup</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">base_len</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="n">l</span> <span class="o">=</span> <span class="n">base_len</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="c"># line 2: set the base for K to the base for G</span>
-        <span class="n">res_base</span> <span class="o">=</span> <span class="n">base</span><span class="p">[:]</span>
-        <span class="c"># line 3: compute BSGS and related structures for K</span>
-        <span class="n">res_base</span><span class="p">,</span> <span class="n">res_strong_gens</span> <span class="o">=</span> <span class="n">res</span><span class="o">.</span><span class="n">schreier_sims_incremental</span><span class="p">(</span>
-            <span class="n">base</span><span class="o">=</span><span class="n">res_base</span><span class="p">)</span>
-        <span class="n">res_strong_gens_distr</span> <span class="o">=</span> <span class="n">_distribute_gens_by_base</span><span class="p">(</span><span class="n">res_base</span><span class="p">,</span>
-                                <span class="n">res_strong_gens</span><span class="p">)</span>
-        <span class="n">res_generators</span> <span class="o">=</span> <span class="n">res</span><span class="o">.</span><span class="n">generators</span>
-        <span class="n">res_basic_orbits_init_base</span> <span class="o">=</span> \
-        <span class="p">[</span><span class="n">_orbit</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">res_strong_gens_distr</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">res_base</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>\
-         <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">base_len</span><span class="p">)]</span>
-        <span class="c"># initialize orbit representatives</span>
-        <span class="n">orbit_reps</span> <span class="o">=</span> <span class="p">[</span><span class="bp">None</span><span class="p">]</span><span class="o">*</span><span class="n">base_len</span>
-        <span class="c"># line 4: orbit representatives for f-th basic stabilizer of K</span>
-        <span class="n">orbits</span> <span class="o">=</span> <span class="n">_orbits</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">res_strong_gens_distr</span><span class="p">[</span><span class="n">f</span><span class="p">])</span>
-        <span class="n">orbit_reps</span><span class="p">[</span><span class="n">f</span><span class="p">]</span> <span class="o">=</span> <span class="n">get_reps</span><span class="p">(</span><span class="n">orbits</span><span class="p">)</span>
-        <span class="c"># line 5: remove the base point from the representatives to avoid</span>
-        <span class="c"># getting the identity element as a generator for K</span>
-        <span class="n">orbit_reps</span><span class="p">[</span><span class="n">f</span><span class="p">]</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">base</span><span class="p">[</span><span class="n">f</span><span class="p">])</span>
-        <span class="c"># line 6: more initializations</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">base_len</span>
-        <span class="n">u</span> <span class="o">=</span> <span class="p">[</span><span class="n">identity</span><span class="p">]</span><span class="o">*</span><span class="n">base_len</span>
-        <span class="n">sorted_orbits</span> <span class="o">=</span> <span class="p">[</span><span class="bp">None</span><span class="p">]</span><span class="o">*</span><span class="n">base_len</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">base_len</span><span class="p">):</span>
-            <span class="n">sorted_orbits</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">basic_orbits</span><span class="p">[</span><span class="n">i</span><span class="p">][:]</span>
-            <span class="n">sorted_orbits</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">point</span><span class="p">:</span> <span class="n">base_ordering</span><span class="p">[</span><span class="n">point</span><span class="p">])</span>
-        <span class="c"># line 7: initializations</span>
-        <span class="n">mu</span> <span class="o">=</span> <span class="p">[</span><span class="bp">None</span><span class="p">]</span><span class="o">*</span><span class="n">base_len</span>
-        <span class="n">nu</span> <span class="o">=</span> <span class="p">[</span><span class="bp">None</span><span class="p">]</span><span class="o">*</span><span class="n">base_len</span>
-        <span class="c"># this corresponds to the element smaller than all points</span>
-        <span class="n">mu</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">degree</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="n">update_nu</span><span class="p">(</span><span class="n">l</span><span class="p">)</span>
-        <span class="c"># initialize computed words</span>
-        <span class="n">computed_words</span> <span class="o">=</span> <span class="p">[</span><span class="n">identity</span><span class="p">]</span><span class="o">*</span><span class="n">base_len</span>
-        <span class="c"># line 8: main loop</span>
-        <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="c"># apply all the tests</span>
-            <span class="k">while</span> <span class="n">l</span> <span class="o">&lt;</span> <span class="n">base_len</span> <span class="o">-</span> <span class="mi">1</span> <span class="ow">and</span> \
-                <span class="n">computed_words</span><span class="p">[</span><span class="n">l</span><span class="p">](</span><span class="n">base</span><span class="p">[</span><span class="n">l</span><span class="p">])</span> <span class="ow">in</span> <span class="n">orbit_reps</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="ow">and</span> \
-                <span class="n">base_ordering</span><span class="p">[</span><span class="n">mu</span><span class="p">[</span><span class="n">l</span><span class="p">]]</span> <span class="o">&lt;</span> \
-                <span class="n">base_ordering</span><span class="p">[</span><span class="n">computed_words</span><span class="p">[</span><span class="n">l</span><span class="p">](</span><span class="n">base</span><span class="p">[</span><span class="n">l</span><span class="p">])]</span> <span class="o">&lt;</span> \
-                <span class="n">base_ordering</span><span class="p">[</span><span class="n">nu</span><span class="p">[</span><span class="n">l</span><span class="p">]]</span> <span class="ow">and</span> \
-                    <span class="n">tests</span><span class="p">[</span><span class="n">l</span><span class="p">](</span><span class="n">computed_words</span><span class="p">):</span>
-                <span class="c"># line 11: change the (partial) base of K</span>
-                <span class="n">new_point</span> <span class="o">=</span> <span class="n">computed_words</span><span class="p">[</span><span class="n">l</span><span class="p">](</span><span class="n">base</span><span class="p">[</span><span class="n">l</span><span class="p">])</span>
-                <span class="n">res_base</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">new_point</span>
-                <span class="n">new_stab_gens</span> <span class="o">=</span> <span class="n">_stabilizer</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">res_strong_gens_distr</span><span class="p">[</span><span class="n">l</span><span class="p">],</span>
-                        <span class="n">new_point</span><span class="p">)</span>
-                <span class="n">res_strong_gens_distr</span><span class="p">[</span><span class="n">l</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">new_stab_gens</span>
-                <span class="c"># line 12: calculate minimal orbit representatives for the</span>
-                <span class="c"># l+1-th basic stabilizer</span>
-                <span class="n">orbits</span> <span class="o">=</span> <span class="n">_orbits</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">new_stab_gens</span><span class="p">)</span>
-                <span class="n">orbit_reps</span><span class="p">[</span><span class="n">l</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">get_reps</span><span class="p">(</span><span class="n">orbits</span><span class="p">)</span>
-                <span class="c"># line 13: amend sorted orbits</span>
-                <span class="n">l</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="n">temp_orbit</span> <span class="o">=</span> <span class="p">[</span><span class="n">computed_words</span><span class="p">[</span><span class="n">l</span> <span class="o">-</span> <span class="mi">1</span><span class="p">](</span><span class="n">point</span><span class="p">)</span> <span class="k">for</span> <span class="n">point</span>
-                             <span class="ow">in</span> <span class="n">basic_orbits</span><span class="p">[</span><span class="n">l</span><span class="p">]]</span>
-                <span class="n">temp_orbit</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">point</span><span class="p">:</span> <span class="n">base_ordering</span><span class="p">[</span><span class="n">point</span><span class="p">])</span>
-                <span class="n">sorted_orbits</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">temp_orbit</span>
-                <span class="c"># lines 14 and 15: update variables used minimality tests</span>
-                <span class="n">new_mu</span> <span class="o">=</span> <span class="n">degree</span> <span class="o">+</span> <span class="mi">1</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">l</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">base</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="ow">in</span> <span class="n">res_basic_orbits_init_base</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                        <span class="n">candidate</span> <span class="o">=</span> <span class="n">computed_words</span><span class="p">[</span><span class="n">i</span><span class="p">](</span><span class="n">base</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-                        <span class="k">if</span> <span class="n">base_ordering</span><span class="p">[</span><span class="n">candidate</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">base_ordering</span><span class="p">[</span><span class="n">new_mu</span><span class="p">]:</span>
-                            <span class="n">new_mu</span> <span class="o">=</span> <span class="n">candidate</span>
-                <span class="n">mu</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">new_mu</span>
-                <span class="n">update_nu</span><span class="p">(</span><span class="n">l</span><span class="p">)</span>
-                <span class="c"># line 16: determine the new transversal element</span>
-                <span class="n">c</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="n">temp_point</span> <span class="o">=</span> <span class="n">sorted_orbits</span><span class="p">[</span><span class="n">l</span><span class="p">][</span><span class="n">c</span><span class="p">[</span><span class="n">l</span><span class="p">]]</span>
-                <span class="n">gamma</span> <span class="o">=</span> <span class="n">computed_words</span><span class="p">[</span><span class="n">l</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">_array_form</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">temp_point</span><span class="p">)</span>
-                <span class="n">u</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">transversals</span><span class="p">[</span><span class="n">l</span><span class="p">][</span><span class="n">gamma</span><span class="p">]</span>
-                <span class="c"># update computed words</span>
-                <span class="n">computed_words</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">rmul</span><span class="p">(</span><span class="n">computed_words</span><span class="p">[</span><span class="n">l</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">u</span><span class="p">[</span><span class="n">l</span><span class="p">])</span>
-            <span class="c"># lines 17 &amp; 18: apply the tests to the group element found</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">computed_words</span><span class="p">[</span><span class="n">l</span><span class="p">]</span>
-            <span class="n">temp_point</span> <span class="o">=</span> <span class="n">g</span><span class="p">(</span><span class="n">base</span><span class="p">[</span><span class="n">l</span><span class="p">])</span>
-            <span class="k">if</span> <span class="n">l</span> <span class="o">==</span> <span class="n">base_len</span> <span class="o">-</span> <span class="mi">1</span> <span class="ow">and</span> \
-                <span class="n">base_ordering</span><span class="p">[</span><span class="n">mu</span><span class="p">[</span><span class="n">l</span><span class="p">]]</span> <span class="o">&lt;</span> \
-                <span class="n">base_ordering</span><span class="p">[</span><span class="n">temp_point</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">base_ordering</span><span class="p">[</span><span class="n">nu</span><span class="p">[</span><span class="n">l</span><span class="p">]]</span> <span class="ow">and</span> \
-                <span class="n">temp_point</span> <span class="ow">in</span> <span class="n">orbit_reps</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="ow">and</span> \
-                <span class="n">tests</span><span class="p">[</span><span class="n">l</span><span class="p">](</span><span class="n">computed_words</span><span class="p">)</span> <span class="ow">and</span> \
-                    <span class="n">prop</span><span class="p">(</span><span class="n">g</span><span class="p">):</span>
-                <span class="c"># line 19: reset the base of K</span>
-                <span class="n">res_generators</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-                <span class="n">res_base</span> <span class="o">=</span> <span class="n">base</span><span class="p">[:]</span>
-                <span class="c"># line 20: recalculate basic orbits (and transversals)</span>
-                <span class="n">res_strong_gens</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-                <span class="n">res_strong_gens_distr</span> <span class="o">=</span> <span class="n">_distribute_gens_by_base</span><span class="p">(</span><span class="n">res_base</span><span class="p">,</span>
-                                                          <span class="n">res_strong_gens</span><span class="p">)</span>
-                <span class="n">res_basic_orbits_init_base</span> <span class="o">=</span> \
-                <span class="p">[</span><span class="n">_orbit</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">res_strong_gens_distr</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">res_base</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> \
-                 <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">base_len</span><span class="p">)]</span>
-                <span class="c"># line 21: recalculate orbit representatives</span>
-                <span class="c"># line 22: reset the search depth</span>
-                <span class="n">orbit_reps</span><span class="p">[</span><span class="n">f</span><span class="p">]</span> <span class="o">=</span> <span class="n">get_reps</span><span class="p">(</span><span class="n">orbits</span><span class="p">)</span>
-                <span class="n">l</span> <span class="o">=</span> <span class="n">f</span>
-            <span class="c"># line 23: go up the tree until in the first branch not fully</span>
-            <span class="c"># searched</span>
-            <span class="k">while</span> <span class="n">l</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">c</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">basic_orbits</span><span class="p">[</span><span class="n">l</span><span class="p">])</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">l</span> <span class="o">=</span> <span class="n">l</span> <span class="o">-</span> <span class="mi">1</span>
-            <span class="c"># line 24: if the entire tree is traversed, return K</span>
-            <span class="k">if</span> <span class="n">l</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="n">res_generators</span><span class="p">)</span>
-            <span class="c"># lines 25-27: update orbit representatives</span>
-            <span class="k">if</span> <span class="n">l</span> <span class="o">&lt;</span> <span class="n">f</span><span class="p">:</span>
-                <span class="c"># line 26</span>
-                <span class="n">f</span> <span class="o">=</span> <span class="n">l</span>
-                <span class="n">c</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="c"># line 27</span>
-                <span class="n">temp_orbits</span> <span class="o">=</span> <span class="n">_orbits</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">res_strong_gens_distr</span><span class="p">[</span><span class="n">f</span><span class="p">])</span>
-                <span class="n">orbit_reps</span><span class="p">[</span><span class="n">f</span><span class="p">]</span> <span class="o">=</span> <span class="n">get_reps</span><span class="p">(</span><span class="n">temp_orbits</span><span class="p">)</span>
-                <span class="c"># line 28: update variables used for minimality testing</span>
-                <span class="n">mu</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">degree</span> <span class="o">+</span> <span class="mi">1</span>
-                <span class="n">temp_index</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">basic_orbits</span><span class="p">[</span><span class="n">l</span><span class="p">])</span> <span class="o">+</span> <span class="mi">1</span> <span class="o">-</span> \
-                    <span class="nb">len</span><span class="p">(</span><span class="n">res_basic_orbits_init_base</span><span class="p">[</span><span class="n">l</span><span class="p">])</span>
-                <span class="k">if</span> <span class="n">temp_index</span> <span class="o">&gt;=</span> <span class="nb">len</span><span class="p">(</span><span class="n">sorted_orbits</span><span class="p">[</span><span class="n">l</span><span class="p">]):</span>
-                    <span class="n">nu</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">base_ordering</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">nu</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">sorted_orbits</span><span class="p">[</span><span class="n">l</span><span class="p">][</span><span class="n">temp_index</span><span class="p">]</span>
-            <span class="c"># line 29: set the next element from the current branch and update</span>
-            <span class="c"># accorndingly</span>
-            <span class="n">c</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="n">l</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">gamma</span>  <span class="o">=</span> <span class="n">sorted_orbits</span><span class="p">[</span><span class="n">l</span><span class="p">][</span><span class="n">c</span><span class="p">[</span><span class="n">l</span><span class="p">]]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">gamma</span> <span class="o">=</span> <span class="n">computed_words</span><span class="p">[</span><span class="n">l</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">_array_form</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">sorted_orbits</span><span class="p">[</span><span class="n">l</span><span class="p">][</span><span class="n">c</span><span class="p">[</span><span class="n">l</span><span class="p">]])</span>
-
-            <span class="n">u</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">transversals</span><span class="p">[</span><span class="n">l</span><span class="p">][</span><span class="n">gamma</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">l</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">computed_words</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">u</span><span class="p">[</span><span class="n">l</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">computed_words</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">rmul</span><span class="p">(</span><span class="n">computed_words</span><span class="p">[</span><span class="n">l</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">u</span><span class="p">[</span><span class="n">l</span><span class="p">])</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="PermutationGroup.transitivity_degree"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.transitivity_degree">[docs]</a>    <span class="k">def</span> <span class="nf">transitivity_degree</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute the degree of transitivity of the group.</span>
-
-<span class="sd">        A permutation group ``G`` acting on ``\Omega = \{0, 1, ..., n-1\}`` is</span>
-<span class="sd">        ``k``-fold transitive, if, for any k points</span>
-<span class="sd">        ``(a_1, a_2, ..., a_k)\in\Omega`` and any k points</span>
-<span class="sd">        ``(b_1, b_2, ..., b_k)\in\Omega`` there exists ``g\in G`` such that</span>
-<span class="sd">        ``g(a_1)=b_1, g(a_2)=b_2, ..., g(a_k)=b_k``</span>
-<span class="sd">        The degree of transitivity of ``G`` is the maximum ``k`` such that</span>
-<span class="sd">        ``G`` is ``k``-fold transitive. ([8])</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; a = Permutation([1, 2, 0])</span>
-<span class="sd">        &gt;&gt;&gt; b = Permutation([1, 0, 2])</span>
-<span class="sd">        &gt;&gt;&gt; G = PermutationGroup([a,b])</span>
-<span class="sd">        &gt;&gt;&gt; G.transitivity_degree</span>
-<span class="sd">        3</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        is_transitive, orbit</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_transitivity_degree</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">degree</span>
-            <span class="n">G</span> <span class="o">=</span> <span class="bp">self</span>
-            <span class="c"># if G is k-transitive, a tuple (a_0,..,a_k)</span>
-            <span class="c"># can be brought to (b_0,...,b_(k-1), b_k)</span>
-            <span class="c"># where b_0,...,b_(k-1) are fixed points;</span>
-            <span class="c"># consider the group G_k which stabilizes b_0,...,b_(k-1)</span>
-            <span class="c"># if G_k is transitive on the subset excluding b_0,...,b_(k-1)</span>
-            <span class="c"># then G is (k+1)-transitive</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-                <span class="n">orb</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">orbit</span><span class="p">((</span><span class="n">i</span><span class="p">))</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">orb</span><span class="p">)</span> <span class="o">!=</span> <span class="n">n</span> <span class="o">-</span> <span class="n">i</span><span class="p">:</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">_transitivity_degree</span> <span class="o">=</span> <span class="n">i</span>
-                    <span class="k">return</span> <span class="n">i</span>
-                <span class="n">G</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">stabilizer</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_transitivity_degree</span> <span class="o">=</span> <span class="n">n</span>
-            <span class="k">return</span> <span class="n">n</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_transitivity_degree</span>
-</div>
-<div class="viewcode-block" id="PermutationGroup.is_group"><a class="viewcode-back" href="../../../modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.is_group">[docs]</a>    <span class="k">def</span> <span class="nf">is_group</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return True if the group meets three criteria: identity is present,</span>
-<span class="sd">        the inverse of every element is also an element, and the product of</span>
-<span class="sd">        any two elements is also an element. If any of the tests fail, False</span>
-<span class="sd">        is returned.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import PermutationGroup</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.polyhedron import tetrahedron</span>
-
-<span class="sd">        The permutation group given in the tetrahedron object is not</span>
-<span class="sd">        a true group:</span>
-
-<span class="sd">        &gt;&gt;&gt; G = tetrahedron.pgroup</span>
-<span class="sd">        &gt;&gt;&gt; G.is_group()</span>
-<span class="sd">        False</span>
-
-<span class="sd">        But the group generated by the permutations in the tetrahedron</span>
-<span class="sd">        pgroup is a proper group:</span>
-
-<span class="sd">        &gt;&gt;&gt; H = PermutationGroup(list(G.generate()))</span>
-<span class="sd">        &gt;&gt;&gt; H.is_group()</span>
-<span class="sd">        True</span>
-
-<span class="sd">        The identity permutation is present:</span>
-
-<span class="sd">        &gt;&gt;&gt; H.has(Permutation(G.degree - 1))</span>
-<span class="sd">        True</span>
-
-<span class="sd">        The product of any two elements from the group is also in the group:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import TableForm</span>
-<span class="sd">        &gt;&gt;&gt; g = list(H)</span>
-<span class="sd">        &gt;&gt;&gt; n = len(g)</span>
-<span class="sd">        &gt;&gt;&gt; m = []</span>
-<span class="sd">        &gt;&gt;&gt; for i in g:</span>
-<span class="sd">        ...     m.append([g.index(i*H) for H in g])</span>
-<span class="sd">        ...</span>
-<span class="sd">        &gt;&gt;&gt; TableForm(m, headings=[list(range(n)), list(range(n))], wipe_zeros=False)</span>
-<span class="sd">           | 0  1  2  3  4  5  6  7  8  9  10 11</span>
-<span class="sd">        ----------------------------------------</span>
-<span class="sd">         0 | 11 0  8  10 6  2  7  4  5  3  9  1</span>
-<span class="sd">         1 | 0  1  2  3  4  5  6  7  8  9  10 11</span>
-<span class="sd">         2 | 6  2  7  4  5  3  9  1  11 0  8  10</span>
-<span class="sd">         3 | 5  3  9  1  11 0  8  10 6  2  7  4</span>
-<span class="sd">         4 | 3  4  0  2  10 6  11 8  9  7  1  5</span>
-<span class="sd">         5 | 4  5  6  7  8  9  10 11 0  1  2  3</span>
-<span class="sd">         6 | 10 6  11 8  9  7  1  5  3  4  0  2</span>
-<span class="sd">         7 | 9  7  1  5  3  4  0  2  10 6  11 8</span>
-<span class="sd">         8 | 7  8  4  6  2  10 3  0  1  11 5  9</span>
-<span class="sd">         9 | 8  9  10 11 0  1  2  3  4  5  6  7</span>
-<span class="sd">        10 | 2  10 3  0  1  11 5  9  7  8  4  6</span>
-<span class="sd">        11 | 1  11 5  9  7  8  4  6  2  10 3  0</span>
-<span class="sd">        &gt;&gt;&gt;</span>
-<span class="sd">        The entries in the table give the element in the group corresponding</span>
-<span class="sd">        to the product of a given column element and row element:</span>
-
-<span class="sd">        &gt;&gt;&gt; g[3]*g[2] == g[9]</span>
-<span class="sd">        True</span>
-
-<span class="sd">        The inverse of every element is also in the group:</span>
-
-<span class="sd">        &gt;&gt;&gt; TableForm([[g.index(~gi) for gi in g]], headings=[[], list(range(n))],</span>
-<span class="sd">        ...    wipe_zeros=False)</span>
-<span class="sd">        0  1 2 3 4  5 6 7 8 9 10 11</span>
-<span class="sd">        ---------------------------</span>
-<span class="sd">        11 1 7 3 10 9 6 2 8 5 4  0</span>
-
-<span class="sd">        So we see that g[1] and g[3] are equivalent to their inverse while</span>
-<span class="sd">        g[7] == ~g[2].</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># identity present</span>
-        <span class="n">I</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">degree</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">g</span> <span class="o">==</span> <span class="n">I</span><span class="p">:</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="c"># associativity already holds: a*(b*c) == (a*b)*c for permutations</span>
-
-        <span class="c"># inverse of each is present</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">~</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="c"># closure</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">b</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="bp">True</span>
-</div></div>
-<span class="k">def</span> <span class="nf">_orbit</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">generators</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">action</span><span class="o">=</span><span class="s">&#39;tuples&#39;</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;Compute the orbit of alpha ``\{g(\alpha) | g \in G\}`` as a set.</span>
-
-<span class="sd">    The time complexity of the algorithm used here is ``O(|Orb|*r)`` where</span>
-<span class="sd">    ``|Orb|`` is the size of the orbit and ``r`` is the number of generators of</span>
-<span class="sd">    the group. For a more detailed analysis, see [1], p.78, [2], pp. 19-21.</span>
-<span class="sd">    Here alpha can be a single point, or a list of points.</span>
-
-<span class="sd">    If alpha is a single point, the ordinary orbit is computed.</span>
-<span class="sd">    if alpha is a list of points, there are three available options:</span>
-
-<span class="sd">    &#39;union&#39; - computes the union of the orbits of the points in the list</span>
-<span class="sd">    &#39;tuples&#39; - computes the orbit of the list interpreted as an ordered</span>
-<span class="sd">    tuple under the group action ( i.e., g((1,2,3)) = (g(1), g(2), g(3)) )</span>
-<span class="sd">    &#39;sets&#39; - computes the orbit of the list interpreted as a sets</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup, _orbit</span>
-<span class="sd">    &gt;&gt;&gt; a = Permutation([1,2,0,4,5,6,3])</span>
-<span class="sd">    &gt;&gt;&gt; G = PermutationGroup([a])</span>
-<span class="sd">    &gt;&gt;&gt; _orbit(G.degree, G.generators, 0)</span>
-<span class="sd">    set([0, 1, 2])</span>
-<span class="sd">    &gt;&gt;&gt; _orbit(G.degree, G.generators, [0, 4], &#39;union&#39;)</span>
-<span class="sd">    set([0, 1, 2, 3, 4, 5, 6])</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    orbit, orbit_transversal</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="s">&#39;__getitem__&#39;</span><span class="p">):</span>
-        <span class="n">alpha</span> <span class="o">=</span> <span class="p">[</span><span class="n">alpha</span><span class="p">]</span>
-
-    <span class="n">gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="o">.</span><span class="n">_array_form</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">generators</span><span class="p">]</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">alpha</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">action</span> <span class="o">==</span> <span class="s">&#39;union&#39;</span><span class="p">:</span>
-        <span class="n">orb</span> <span class="o">=</span> <span class="n">alpha</span>
-        <span class="n">used</span> <span class="o">=</span> <span class="p">[</span><span class="bp">False</span><span class="p">]</span><span class="o">*</span><span class="n">degree</span>
-        <span class="k">for</span> <span class="n">el</span> <span class="ow">in</span> <span class="n">alpha</span><span class="p">:</span>
-            <span class="n">used</span><span class="p">[</span><span class="n">el</span><span class="p">]</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">orb</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">:</span>
-                <span class="n">temp</span> <span class="o">=</span> <span class="n">gen</span><span class="p">[</span><span class="n">b</span><span class="p">]</span>
-                <span class="k">if</span> <span class="n">used</span><span class="p">[</span><span class="n">temp</span><span class="p">]</span> <span class="o">==</span> <span class="bp">False</span><span class="p">:</span>
-                    <span class="n">orb</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">temp</span><span class="p">)</span>
-                    <span class="n">used</span><span class="p">[</span><span class="n">temp</span><span class="p">]</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="nb">set</span><span class="p">(</span><span class="n">orb</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">action</span> <span class="o">==</span> <span class="s">&#39;tuples&#39;</span><span class="p">:</span>
-        <span class="n">alpha</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">alpha</span><span class="p">)</span>
-        <span class="n">orb</span> <span class="o">=</span> <span class="p">[</span><span class="n">alpha</span><span class="p">]</span>
-        <span class="n">used</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span><span class="n">alpha</span><span class="p">])</span>
-        <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">orb</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">:</span>
-                <span class="n">temp</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">([</span><span class="n">gen</span><span class="p">[</span><span class="n">x</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">b</span><span class="p">])</span>
-                <span class="k">if</span> <span class="n">temp</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">used</span><span class="p">:</span>
-                    <span class="n">orb</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">temp</span><span class="p">)</span>
-                    <span class="n">used</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">temp</span><span class="p">)</span>
-        <span class="k">return</span> <span class="nb">set</span><span class="p">(</span><span class="n">orb</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">action</span> <span class="o">==</span> <span class="s">&#39;sets&#39;</span><span class="p">:</span>
-        <span class="n">alpha</span> <span class="o">=</span> <span class="nb">frozenset</span><span class="p">(</span><span class="n">alpha</span><span class="p">)</span>
-        <span class="n">orb</span> <span class="o">=</span> <span class="p">[</span><span class="n">alpha</span><span class="p">]</span>
-        <span class="n">used</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span><span class="n">alpha</span><span class="p">])</span>
-        <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">orb</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">:</span>
-                <span class="n">temp</span> <span class="o">=</span> <span class="nb">frozenset</span><span class="p">([</span><span class="n">gen</span><span class="p">[</span><span class="n">x</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">b</span><span class="p">])</span>
-                <span class="k">if</span> <span class="n">temp</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">used</span><span class="p">:</span>
-                    <span class="n">orb</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">temp</span><span class="p">)</span>
-                    <span class="n">used</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">temp</span><span class="p">)</span>
-        <span class="k">return</span> <span class="nb">set</span><span class="p">([</span><span class="nb">tuple</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">orb</span><span class="p">])</span>
-
-<span class="k">def</span> <span class="nf">_orbits</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">generators</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Compute the orbits of G.</span>
-
-<span class="sd">    If rep=False it returns a list of sets else it returns a list of</span>
-<span class="sd">    representatives of the orbits</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup, _orbits</span>
-<span class="sd">    &gt;&gt;&gt; a = Permutation([0, 2, 1])</span>
-<span class="sd">    &gt;&gt;&gt; b = Permutation([1, 0, 2])</span>
-<span class="sd">    &gt;&gt;&gt; _orbits(a.size, [a, b])</span>
-<span class="sd">    [set([0, 1, 2])]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">seen</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>  <span class="c"># elements that have already appeared in orbits</span>
-    <span class="n">orbs</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">sorted_I</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">degree</span><span class="p">))</span>
-    <span class="n">I</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">sorted_I</span><span class="p">)</span>
-    <span class="k">while</span> <span class="n">I</span><span class="p">:</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="n">sorted_I</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">orb</span> <span class="o">=</span> <span class="n">_orbit</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">generators</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-        <span class="n">orbs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">orb</span><span class="p">)</span>
-        <span class="c"># remove all indices that are in this orbit</span>
-        <span class="n">I</span> <span class="o">-=</span> <span class="n">orb</span>
-        <span class="n">sorted_I</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">sorted_I</span> <span class="k">if</span> <span class="n">i</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">orb</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">orbs</span>
-
-<span class="k">def</span> <span class="nf">_orbit_transversal</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">generators</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">pairs</span><span class="p">,</span> <span class="n">af</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;Computes a transversal for the orbit of ``alpha`` as a set.</span>
-
-<span class="sd">    generators   generators of the group ``G``</span>
-
-<span class="sd">    For a permutation group ``G``, a transversal for the orbit</span>
-<span class="sd">    ``Orb = \{g(\alpha) | g \in G\}`` is a set</span>
-<span class="sd">    ``\{g_\beta | g_\beta(\alpha) = \beta\}`` for ``\beta \in Orb``.</span>
-<span class="sd">    Note that there may be more than one possible transversal.</span>
-<span class="sd">    If ``pairs`` is set to ``True``, it returns the list of pairs</span>
-<span class="sd">    ``(\beta, g_\beta)``. For a proof of correctness, see [1], p.79</span>
-
-<span class="sd">    if af is True, the transversal elements are given in array form</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.named_groups import DihedralGroup</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.perm_groups import _orbit_transversal</span>
-<span class="sd">    &gt;&gt;&gt; G = DihedralGroup(6)</span>
-<span class="sd">    &gt;&gt;&gt; _orbit_transversal(G.degree, G.generators, 0, False)</span>
-<span class="sd">        [Permutation(5),</span>
-<span class="sd">         Permutation(0, 1, 2, 3, 4, 5),</span>
-<span class="sd">         Permutation(0, 5)(1, 4)(2, 3),</span>
-<span class="sd">         Permutation(0, 2, 4)(1, 3, 5),</span>
-<span class="sd">         Permutation(5)(0, 4)(1, 3),</span>
-<span class="sd">         Permutation(0, 3)(1, 4)(2, 5)]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">tr</span> <span class="o">=</span> <span class="p">[(</span><span class="n">alpha</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">degree</span><span class="p">)))]</span>
-    <span class="n">used</span> <span class="o">=</span> <span class="p">[</span><span class="bp">False</span><span class="p">]</span><span class="o">*</span><span class="n">degree</span>
-    <span class="n">used</span><span class="p">[</span><span class="n">alpha</span><span class="p">]</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="o">.</span><span class="n">_array_form</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">generators</span><span class="p">]</span>
-    <span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="n">px</span> <span class="ow">in</span> <span class="n">tr</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">:</span>
-            <span class="n">temp</span> <span class="o">=</span> <span class="n">gen</span><span class="p">[</span><span class="n">x</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">used</span><span class="p">[</span><span class="n">temp</span><span class="p">]</span> <span class="o">==</span> <span class="bp">False</span><span class="p">:</span>
-                <span class="n">tr</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">temp</span><span class="p">,</span> <span class="n">_af_rmul</span><span class="p">(</span><span class="n">gen</span><span class="p">,</span> <span class="n">px</span><span class="p">)))</span>
-                <span class="n">used</span><span class="p">[</span><span class="n">temp</span><span class="p">]</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="k">if</span> <span class="n">pairs</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">af</span><span class="p">:</span>
-            <span class="n">tr</span> <span class="o">=</span> <span class="p">[(</span><span class="n">x</span><span class="p">,</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">y</span><span class="p">))</span> <span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">tr</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">tr</span>
-
-    <span class="k">if</span> <span class="n">af</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">y</span> <span class="k">for</span> <span class="n">_</span><span class="p">,</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">tr</span><span class="p">]</span>
-
-    <span class="k">return</span> <span class="p">[</span><span class="n">_af_new</span><span class="p">(</span><span class="n">y</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span><span class="p">,</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">tr</span><span class="p">]</span>
-
-<span class="k">def</span> <span class="nf">_stabilizer</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">generators</span><span class="p">,</span> <span class="n">alpha</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;Return the stabilizer subgroup of ``alpha``.</span>
-
-<span class="sd">    The stabilizer of ``\alpha`` is the group ``G_\alpha =</span>
-<span class="sd">    \{g \in G | g(\alpha) = \alpha\}``.</span>
-<span class="sd">    For a proof of correctness, see [1], p.79.</span>
-
-<span class="sd">    degree       degree of G</span>
-<span class="sd">    generators   generators of G</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.perm_groups import _stabilizer</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.named_groups import DihedralGroup</span>
-<span class="sd">    &gt;&gt;&gt; G = DihedralGroup(6)</span>
-<span class="sd">    &gt;&gt;&gt; _stabilizer(G.degree, G.generators, 5)</span>
-<span class="sd">    [Permutation(5)(0, 4)(1, 3), Permutation(5)]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    orbit</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">orb</span> <span class="o">=</span> <span class="p">[</span><span class="n">alpha</span><span class="p">]</span>
-    <span class="n">table</span> <span class="o">=</span> <span class="p">{</span><span class="n">alpha</span><span class="p">:</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">degree</span><span class="p">))}</span>
-    <span class="n">table_inv</span> <span class="o">=</span> <span class="p">{</span><span class="n">alpha</span><span class="p">:</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">degree</span><span class="p">))}</span>
-    <span class="n">used</span> <span class="o">=</span> <span class="p">[</span><span class="bp">False</span><span class="p">]</span><span class="o">*</span><span class="n">degree</span>
-    <span class="n">used</span><span class="p">[</span><span class="n">alpha</span><span class="p">]</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="o">.</span><span class="n">_array_form</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">generators</span><span class="p">]</span>
-    <span class="n">stab_gens</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">orb</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">:</span>
-            <span class="n">temp</span> <span class="o">=</span> <span class="n">gen</span><span class="p">[</span><span class="n">b</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">used</span><span class="p">[</span><span class="n">temp</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                <span class="n">gen_temp</span> <span class="o">=</span> <span class="n">_af_rmul</span><span class="p">(</span><span class="n">gen</span><span class="p">,</span> <span class="n">table</span><span class="p">[</span><span class="n">b</span><span class="p">])</span>
-                <span class="n">orb</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">temp</span><span class="p">)</span>
-                <span class="n">table</span><span class="p">[</span><span class="n">temp</span><span class="p">]</span> <span class="o">=</span> <span class="n">gen_temp</span>
-                <span class="n">table_inv</span><span class="p">[</span><span class="n">temp</span><span class="p">]</span> <span class="o">=</span> <span class="n">_af_invert</span><span class="p">(</span><span class="n">gen_temp</span><span class="p">)</span>
-                <span class="n">used</span><span class="p">[</span><span class="n">temp</span><span class="p">]</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">schreier_gen</span> <span class="o">=</span> <span class="n">_af_rmuln</span><span class="p">(</span><span class="n">table_inv</span><span class="p">[</span><span class="n">temp</span><span class="p">],</span> <span class="n">gen</span><span class="p">,</span> <span class="n">table</span><span class="p">[</span><span class="n">b</span><span class="p">])</span>
-                <span class="k">if</span> <span class="n">schreier_gen</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">stab_gens</span><span class="p">:</span>
-                    <span class="n">stab_gens</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">schreier_gen</span><span class="p">)</span>
-    <span class="k">return</span> <span class="p">[</span><span class="n">_af_new</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">stab_gens</span><span class="p">]</span>
-
-<span class="n">PermGroup</span> <span class="o">=</span> <span class="n">PermutationGroup</span>
-</pre></div>
-
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-          <div class="body">
-            
-  <h1>Source code for sympy.combinatorics.permutations</h1><div class="highlight"><pre>
-<span class="kn">import</span> <span class="nn">random</span>
-<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Basic</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">is_sequence</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="p">(</span><span class="n">flatten</span><span class="p">,</span> <span class="n">has_variety</span><span class="p">,</span> <span class="n">minlex</span><span class="p">,</span>
-    <span class="n">has_dups</span><span class="p">,</span> <span class="n">runs</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polytools</span> <span class="kn">import</span> <span class="n">lcm</span>
-<span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">zeros</span>
-<span class="kn">from</span> <span class="nn">sympy.mpmath.libmp.libintmath</span> <span class="kn">import</span> <span class="n">ifac</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-
-<span class="k">def</span> <span class="nf">_af_rmul</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the product b*a; input and output are array forms. The ith value</span>
-<span class="sd">    is a[b[i]].</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.permutations import _af_rmul, Permutation</span>
-<span class="sd">    &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-
-<span class="sd">    &gt;&gt;&gt; a, b = [1, 0, 2], [0, 2, 1]</span>
-<span class="sd">    &gt;&gt;&gt; _af_rmul(a, b)</span>
-<span class="sd">    [1, 2, 0]</span>
-<span class="sd">    &gt;&gt;&gt; [a[b[i]] for i in range(3)]</span>
-<span class="sd">    [1, 2, 0]</span>
-
-<span class="sd">    This handles the operands in reverse order compared to the ``*`` operator:</span>
-
-<span class="sd">    &gt;&gt;&gt; a = Permutation(a); b = Permutation(b)</span>
-<span class="sd">    &gt;&gt;&gt; list(a*b)</span>
-<span class="sd">    [2, 0, 1]</span>
-<span class="sd">    &gt;&gt;&gt; [b(a(i)) for i in range(3)]</span>
-<span class="sd">    [2, 0, 1]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    rmul, _af_rmuln</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">[</span><span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">b</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">_af_rmuln</span><span class="p">(</span><span class="o">*</span><span class="n">abc</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Given [a, b, c, ...] return the product of ...*c*b*a using array forms.</span>
-<span class="sd">    The ith value is a[b[c[i]]].</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.permutations import _af_rmul, Permutation</span>
-<span class="sd">    &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-
-<span class="sd">    &gt;&gt;&gt; a, b = [1, 0, 2], [0, 2, 1]</span>
-<span class="sd">    &gt;&gt;&gt; _af_rmul(a, b)</span>
-<span class="sd">    [1, 2, 0]</span>
-<span class="sd">    &gt;&gt;&gt; [a[b[i]] for i in range(3)]</span>
-<span class="sd">    [1, 2, 0]</span>
-
-<span class="sd">    This handles the operands in reverse order compared to the ``*`` operator:</span>
-
-<span class="sd">    &gt;&gt;&gt; a = Permutation(a); b = Permutation(b)</span>
-<span class="sd">    &gt;&gt;&gt; list(a*b)</span>
-<span class="sd">    [2, 0, 1]</span>
-<span class="sd">    &gt;&gt;&gt; [b(a(i)) for i in range(3)]</span>
-<span class="sd">    [2, 0, 1]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    rmul, _af_rmul</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">abc</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-        <span class="n">p0</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">a</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">p0</span><span class="p">[</span><span class="n">p1</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">p2</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">4</span><span class="p">:</span>
-        <span class="n">p0</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">a</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">p0</span><span class="p">[</span><span class="n">p1</span><span class="p">[</span><span class="n">p2</span><span class="p">[</span><span class="n">i</span><span class="p">]]]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">p3</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">5</span><span class="p">:</span>
-        <span class="n">p0</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span> <span class="o">=</span> <span class="n">a</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">p0</span><span class="p">[</span><span class="n">p1</span><span class="p">[</span><span class="n">p2</span><span class="p">[</span><span class="n">p3</span><span class="p">[</span><span class="n">i</span><span class="p">]]]]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">p4</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">6</span><span class="p">:</span>
-        <span class="n">p0</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span><span class="p">,</span> <span class="n">p5</span> <span class="o">=</span> <span class="n">a</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">p0</span><span class="p">[</span><span class="n">p1</span><span class="p">[</span><span class="n">p2</span><span class="p">[</span><span class="n">p3</span><span class="p">[</span><span class="n">p4</span><span class="p">[</span><span class="n">i</span><span class="p">]]]]]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">p5</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">7</span><span class="p">:</span>
-        <span class="n">p0</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span><span class="p">,</span> <span class="n">p5</span><span class="p">,</span> <span class="n">p6</span> <span class="o">=</span> <span class="n">a</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">p0</span><span class="p">[</span><span class="n">p1</span><span class="p">[</span><span class="n">p2</span><span class="p">[</span><span class="n">p3</span><span class="p">[</span><span class="n">p4</span><span class="p">[</span><span class="n">p5</span><span class="p">[</span><span class="n">i</span><span class="p">]]]]]]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">p6</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">8</span><span class="p">:</span>
-        <span class="n">p0</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span><span class="p">,</span> <span class="n">p5</span><span class="p">,</span> <span class="n">p6</span><span class="p">,</span> <span class="n">p7</span> <span class="o">=</span> <span class="n">a</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">p0</span><span class="p">[</span><span class="n">p1</span><span class="p">[</span><span class="n">p2</span><span class="p">[</span><span class="n">p3</span><span class="p">[</span><span class="n">p4</span><span class="p">[</span><span class="n">p5</span><span class="p">[</span><span class="n">p6</span><span class="p">[</span><span class="n">i</span><span class="p">]]]]]]]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">p7</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">][:]</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">a</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">b</span><span class="p">]</span>
-    <span class="k">assert</span> <span class="n">m</span> <span class="o">!=</span> <span class="mi">0</span>
-    <span class="n">p0</span> <span class="o">=</span> <span class="n">_af_rmuln</span><span class="p">(</span><span class="o">*</span><span class="n">a</span><span class="p">[:</span><span class="n">m</span><span class="o">//</span><span class="mi">2</span><span class="p">])</span>
-    <span class="n">p1</span> <span class="o">=</span> <span class="n">_af_rmuln</span><span class="p">(</span><span class="o">*</span><span class="n">a</span><span class="p">[</span><span class="n">m</span><span class="o">//</span><span class="mi">2</span><span class="p">:])</span>
-    <span class="k">return</span> <span class="p">[</span><span class="n">p0</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">p1</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">_af_parity</span><span class="p">(</span><span class="n">pi</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes the parity of a permutation in array form.</span>
-
-<span class="sd">    The parity of a permutation reflects the parity of the</span>
-<span class="sd">    number of inversions in the permutation, i.e., the</span>
-<span class="sd">    number of pairs of x and y such that x &gt; y but p[x] &lt; p[y].</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.permutations import _af_parity</span>
-<span class="sd">    &gt;&gt;&gt; _af_parity([0,1,2,3])</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; _af_parity([3,2,0,1])</span>
-<span class="sd">    1</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Permutation</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">n</span>
-    <span class="n">c</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">a</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">c</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">a</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="n">j</span>
-            <span class="k">while</span> <span class="n">pi</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">!=</span> <span class="n">j</span><span class="p">:</span>
-                <span class="n">i</span> <span class="o">=</span> <span class="n">pi</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-                <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="k">return</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">c</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span>
-
-
-<span class="k">def</span> <span class="nf">_af_invert</span><span class="p">(</span><span class="n">a</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Finds the inverse, ~A, of a permutation, A, given in array form.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.permutations import _af_invert, _af_rmul</span>
-<span class="sd">    &gt;&gt;&gt; A = [1, 2, 0, 3]</span>
-<span class="sd">    &gt;&gt;&gt; _af_invert(A)</span>
-<span class="sd">    [2, 0, 1, 3]</span>
-<span class="sd">    &gt;&gt;&gt; _af_rmul(_, A)</span>
-<span class="sd">    [0, 1, 2, 3]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Permutation, __invert__</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">inv_form</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">ai</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">a</span><span class="p">):</span>
-        <span class="n">inv_form</span><span class="p">[</span><span class="n">ai</span><span class="p">]</span> <span class="o">=</span> <span class="n">i</span>
-    <span class="k">return</span> <span class="n">inv_form</span>
-
-<span class="k">def</span> <span class="nf">_af_pow</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Routine for finding powers of a permutation.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation, _af_pow</span>
-<span class="sd">    &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-<span class="sd">    &gt;&gt;&gt; p = Permutation([2,0,3,1])</span>
-<span class="sd">    &gt;&gt;&gt; p.order()</span>
-<span class="sd">    4</span>
-<span class="sd">    &gt;&gt;&gt; _af_pow(p._array_form, 4)</span>
-<span class="sd">    [0, 1, 2, 3]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)))</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">_af_pow</span><span class="p">(</span><span class="n">_af_invert</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="o">-</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">a</span><span class="p">[:]</span>
-    <span class="k">elif</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">a</span><span class="p">]</span>
-    <span class="k">elif</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span><span class="p">[</span><span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">a</span><span class="p">]</span>
-    <span class="k">elif</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">4</span><span class="p">:</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span><span class="p">[</span><span class="n">a</span><span class="p">[</span><span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]]]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">a</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="c"># use binary multiplication</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)))</span>
-        <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">&amp;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="p">[</span><span class="n">b</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">a</span><span class="p">]</span>
-                <span class="n">n</span> <span class="o">-=</span> <span class="mi">1</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-                    <span class="k">break</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">4</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">a</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span><span class="p">[</span><span class="n">a</span><span class="p">[</span><span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]]]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">a</span><span class="p">]</span>
-                <span class="n">n</span> <span class="o">=</span> <span class="n">n</span> <span class="o">//</span> <span class="mi">4</span>
-            <span class="k">elif</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">a</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">a</span><span class="p">]</span>
-                <span class="n">n</span> <span class="o">=</span> <span class="n">n</span> <span class="o">//</span> <span class="mi">2</span>
-    <span class="k">return</span> <span class="n">b</span>
-
-<span class="k">def</span> <span class="nf">_af_commutes_with</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Checks if the two permutations with array forms</span>
-<span class="sd">    given by ``a`` and ``b`` commute.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.permutations import _af_commutes_with</span>
-<span class="sd">    &gt;&gt;&gt; _af_commutes_with([1,2,0], [0,2,1])</span>
-<span class="sd">    False</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Permutation, commutes_with</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="ow">not</span> <span class="nb">any</span><span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="n">b</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span> <span class="o">!=</span> <span class="n">b</span><span class="p">[</span><span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-
-
-<div class="viewcode-block" id="Cycle"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Cycle">[docs]</a><span class="k">class</span> <span class="nc">Cycle</span><span class="p">(</span><span class="nb">dict</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Wrapper around dict which provides the functionality of a disjoint cycle.</span>
-
-<span class="sd">    A cycle shows the rule to use to move subsets of elements to obtain</span>
-<span class="sd">    a permutation. The Cycle class is more flexible that Permutation in</span>
-<span class="sd">    that 1) all elements need not be present in order to investigate how</span>
-<span class="sd">    multiple cycles act in sequence and 2) it can contain singletons:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.permutations import Perm, Cycle</span>
-
-<span class="sd">    A Cycle will automatically parse a cycle given as a tuple on the rhs:</span>
-
-<span class="sd">    &gt;&gt;&gt; Cycle(1, 2)(2, 3)</span>
-<span class="sd">    Cycle(1, 3, 2)</span>
-
-<span class="sd">    The identity cycle, Cycle(), can be used to start a product:</span>
-
-<span class="sd">    &gt;&gt;&gt; Cycle()(1, 2)(2,3)</span>
-<span class="sd">    Cycle(1, 3, 2)</span>
-
-<span class="sd">    The array form of a Cycle can be obtained by calling the list</span>
-<span class="sd">    method (or passing it to the list function) and all elements from</span>
-<span class="sd">    0 will be shown:</span>
-
-<span class="sd">    &gt;&gt;&gt; a = Cycle(1, 2)</span>
-<span class="sd">    &gt;&gt;&gt; a.list()</span>
-<span class="sd">    [0, 2, 1]</span>
-<span class="sd">    &gt;&gt;&gt; list(a)</span>
-<span class="sd">    [0, 2, 1]</span>
-
-<span class="sd">    If a larger (or smaller) range is desired use the list method and</span>
-<span class="sd">    provide the desired size -- but the Cycle cannot be truncated to</span>
-<span class="sd">    a size smaller than the largest element that is out of place:</span>
-
-<span class="sd">    &gt;&gt;&gt; b = Cycle(2,4)(1,2)(3,1,4)(1,3)</span>
-<span class="sd">    &gt;&gt;&gt; b.list()</span>
-<span class="sd">    [0, 2, 1, 3, 4]</span>
-<span class="sd">    &gt;&gt;&gt; b.list(b.size + 1)</span>
-<span class="sd">    [0, 2, 1, 3, 4, 5]</span>
-<span class="sd">    &gt;&gt;&gt; b.list(-1)</span>
-<span class="sd">    [0, 2, 1]</span>
-
-<span class="sd">    Singletons are not shown when printing with one exception: the largest</span>
-<span class="sd">    element is always shown -- as a singleton if necessary:</span>
-
-<span class="sd">    &gt;&gt;&gt; Cycle(1, 4, 10)(4, 5)</span>
-<span class="sd">    Cycle(1, 5, 4, 10)</span>
-<span class="sd">    &gt;&gt;&gt; Cycle(1, 2)(4)(5)(10)</span>
-<span class="sd">    Cycle(1, 2)(10)</span>
-
-<span class="sd">    The array form can be used to instantiate a Permutation so other</span>
-<span class="sd">    properties of the permutation can be investigated:</span>
-
-<span class="sd">    &gt;&gt;&gt; Perm(Cycle(1,2)(3,4).list()).transpositions()</span>
-<span class="sd">    [(1, 2), (3, 4)]</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    The underlying structure of the Cycle is a dictionary and although</span>
-<span class="sd">    the __iter__ method has been redefiend to give the array form of the</span>
-<span class="sd">    cycle, the underlying dictionary items are still available with the</span>
-<span class="sd">    such methods as items():</span>
-
-<span class="sd">    &gt;&gt;&gt; list(Cycle(1, 2).items())</span>
-<span class="sd">    [(1, 2), (2, 1)]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Permutation</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__missing__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Enter arg into dictionary and return arg.&quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="p">[</span><span class="n">arg</span><span class="p">]</span> <span class="o">=</span> <span class="n">arg</span>
-        <span class="k">return</span> <span class="n">arg</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">list</span><span class="p">():</span>
-            <span class="k">yield</span> <span class="n">i</span>
-
-    <span class="k">def</span> <span class="nf">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return product of cycles processed from R to L.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Cycle as C</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation as Perm</span>
-<span class="sd">        &gt;&gt;&gt; C(1, 2)(2, 3)</span>
-<span class="sd">        Cycle(1, 3, 2)</span>
-
-<span class="sd">        An instance of a Cycle will automatically parse list-like</span>
-<span class="sd">        objects and Permutations that are on the right. It is more</span>
-<span class="sd">        flexible than the Permutation in that all elements need not</span>
-<span class="sd">        be present:</span>
-
-<span class="sd">        &gt;&gt;&gt; a = C(1, 2)</span>
-<span class="sd">        &gt;&gt;&gt; a(2, 3)</span>
-<span class="sd">        Cycle(1, 3, 2)</span>
-<span class="sd">        &gt;&gt;&gt; a(2, 3)(4, 5)</span>
-<span class="sd">        Cycle(1, 3, 2)(4, 5)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">Cycle</span><span class="p">(</span><span class="o">*</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">keys</span><span class="p">()),</span> <span class="p">[</span><span class="n">rv</span><span class="p">[</span><span class="bp">self</span><span class="p">[</span><span class="n">k</span><span class="p">]]</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">keys</span><span class="p">())]):</span>
-            <span class="n">rv</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">v</span>
-        <span class="k">return</span> <span class="n">rv</span>
-
-<div class="viewcode-block" id="Cycle.list"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Cycle.list">[docs]</a>    <span class="k">def</span> <span class="nf">list</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the cycles as an explicit list starting from 0 up</span>
-<span class="sd">        to the greater of the largest value in the cycles and size.</span>
-
-<span class="sd">        Truncation of trailing unmoved items will occur when size</span>
-<span class="sd">        is less than the maximum element in the cycle; if this is</span>
-<span class="sd">        desired, setting ``size=-1`` will guarantee such trimming.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Cycle</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-<span class="sd">        &gt;&gt;&gt; p = Cycle(2, 3)(4, 5)</span>
-<span class="sd">        &gt;&gt;&gt; p.list()</span>
-<span class="sd">        [0, 1, 3, 2, 5, 4]</span>
-<span class="sd">        &gt;&gt;&gt; p.list(10)</span>
-<span class="sd">        [0, 1, 3, 2, 5, 4, 6, 7, 8, 9]</span>
-
-<span class="sd">        Passing a length too small will trim trailing, unchanged elements</span>
-<span class="sd">        in the permutation:</span>
-
-<span class="sd">        &gt;&gt;&gt; Cycle(2, 4)(1, 2, 4).list(-1)</span>
-<span class="sd">        [0, 2, 1]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span> <span class="ow">and</span> <span class="n">size</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;must give size for empty Cycle&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">size</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">big</span> <span class="o">=</span> <span class="nb">max</span><span class="p">([</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span> <span class="k">if</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">!=</span> <span class="n">i</span><span class="p">])</span>
-            <span class="n">size</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">size</span><span class="p">,</span> <span class="n">big</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">size</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span>
-        <span class="k">return</span> <span class="p">[</span><span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">)]</span>
-</div>
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;We want it to print as a Cycle, not as a dict.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Cycle</span>
-<span class="sd">        &gt;&gt;&gt; Cycle(1, 2)</span>
-<span class="sd">        Cycle(1, 2)</span>
-<span class="sd">        &gt;&gt;&gt; print(_)</span>
-<span class="sd">        Cycle(1, 2)</span>
-<span class="sd">        &gt;&gt;&gt; list(Cycle(1, 2).items())</span>
-<span class="sd">        [(1, 2), (2, 1)]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&#39;Cycle()&#39;</span>
-        <span class="n">cycles</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">cyclic_form</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="nb">tuple</span><span class="p">(</span><span class="n">c</span><span class="p">))</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">cycles</span><span class="p">)</span>
-        <span class="n">big</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">any</span><span class="p">(</span><span class="n">i</span> <span class="o">==</span> <span class="n">big</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">cycles</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">c</span><span class="p">):</span>
-            <span class="n">s</span> <span class="o">+=</span> <span class="s">&#39;(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="n">big</span>
-        <span class="k">return</span> <span class="s">&#39;Cycle</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">s</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Load up a Cycle instance with the values for the cycle.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Cycle</span>
-<span class="sd">        &gt;&gt;&gt; Cycle(1, 2, 6)</span>
-<span class="sd">        Cycle(1, 2, 6)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Permutation</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">cyclic_form</span><span class="p">:</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="bp">self</span><span class="p">(</span><span class="o">*</span><span class="n">c</span><span class="p">))</span>
-                <span class="k">return</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Cycle</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                    <span class="bp">self</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">v</span>
-                <span class="k">return</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">has_dups</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;All elements must be unique in a cycle.&#39;</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">),</span> <span class="mi">0</span><span class="p">):</span>
-            <span class="bp">self</span><span class="p">[</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">size</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">max</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span> <span class="o">+</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">copy</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Cycle</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Permutation"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation">[docs]</a><span class="k">class</span> <span class="nc">Permutation</span><span class="p">(</span><span class="n">Basic</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A permutation, alternatively known as an &#39;arrangement number&#39; or &#39;ordering&#39;</span>
-<span class="sd">    is an arrangement of the elements of an ordered list into a one-to-one</span>
-<span class="sd">    mapping with itself. The permutation of a given arrangement is given by</span>
-<span class="sd">    indicating the positions of the elements after re-arrangment [2]_. For</span>
-<span class="sd">    example, if one started with elements [x, y, a, b] (in that order) and</span>
-<span class="sd">    they were reordered as [x, y, b, a] then the permutation would be</span>
-<span class="sd">    [0, 1, 3, 2]. Notice that (in SymPy) the first element is always referred</span>
-<span class="sd">    to as 0 and the permutation uses the indices of the elements in the</span>
-<span class="sd">    original ordering, not the elements (a, b, etc...) themselves.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-
-<span class="sd">    Permutations Notation</span>
-<span class="sd">    =====================</span>
-
-<span class="sd">    Permutations are commonly represented in disjoint cycle or array forms.</span>
-
-<span class="sd">    Array Notation and 2-line Form</span>
-<span class="sd">    ------------------------------------</span>
-
-<span class="sd">    In the 2-line form, the elements and their final positions are shown</span>
-<span class="sd">    as a matrix with 2 rows:</span>
-
-<span class="sd">    [0    1    2     ... n-1]</span>
-<span class="sd">    [p(0) p(1) p(2)  ... p(n-1)]</span>
-
-<span class="sd">    Since the first line is always range(n), where n is the size of p,</span>
-<span class="sd">    it is sufficient to represent the permutation by the second line,</span>
-<span class="sd">    referred to as the &quot;array form&quot; of the permutation. This is entered</span>
-<span class="sd">    in brackets as the argument to the Permutation class:</span>
-
-<span class="sd">    &gt;&gt;&gt; p = Permutation([0, 2, 1]); p</span>
-<span class="sd">    Permutation([0, 2, 1])</span>
-
-<span class="sd">    Given i in range(p.size), the permutation maps i to i^p</span>
-
-<span class="sd">    &gt;&gt;&gt; [i^p for i in range(p.size)]</span>
-<span class="sd">    [0, 2, 1]</span>
-
-<span class="sd">    The composite of two permutations p*q means first apply p, then q, so</span>
-<span class="sd">    i^(p*q) = (i^p)^q which is i^p^q according to Python precedence rules:</span>
-
-<span class="sd">    &gt;&gt;&gt; q = Permutation([2, 1, 0])</span>
-<span class="sd">    &gt;&gt;&gt; [i^p^q for i in range(3)]</span>
-<span class="sd">    [2, 0, 1]</span>
-<span class="sd">    &gt;&gt;&gt; [i^(p*q) for i in range(3)]</span>
-<span class="sd">    [2, 0, 1]</span>
-
-<span class="sd">    One can use also the notation p(i) = i^p, but then the composition</span>
-<span class="sd">    rule is (p*q)(i) = q(p(i)), not p(q(i)):</span>
-
-<span class="sd">    &gt;&gt;&gt; [(p*q)(i) for i in range(p.size)]</span>
-<span class="sd">    [2, 0, 1]</span>
-<span class="sd">    &gt;&gt;&gt; [q(p(i)) for i in range(p.size)]</span>
-<span class="sd">    [2, 0, 1]</span>
-<span class="sd">    &gt;&gt;&gt; [p(q(i)) for i in range(p.size)]</span>
-<span class="sd">    [1, 2, 0]</span>
-
-<span class="sd">    Disjoint Cycle Notation</span>
-<span class="sd">    -----------------------</span>
-
-<span class="sd">    In disjoint cycle notation, only the elements that have shifted are</span>
-<span class="sd">    indicated. In the above case, the 2 and 1 switched places. This can</span>
-<span class="sd">    be entered in two ways:</span>
-
-<span class="sd">    &gt;&gt;&gt; Permutation(1, 2) == Permutation([[1, 2]]) == p</span>
-<span class="sd">    True</span>
-
-<span class="sd">    Only the relative ordering of elements in a cycle matter:</span>
-
-<span class="sd">    &gt;&gt;&gt; Permutation(1,2,3) == Permutation(2,3,1) == Permutation(3,1,2)</span>
-<span class="sd">    True</span>
-
-<span class="sd">    The disjoint cycle notation is convenient when representing permutations</span>
-<span class="sd">    that have several cycles in them:</span>
-
-<span class="sd">    &gt;&gt;&gt; Permutation(1, 2)(3, 5) == Permutation([[1, 2], [3, 5]])</span>
-<span class="sd">    True</span>
-
-<span class="sd">    It also provides some economy in entry when computing products of</span>
-<span class="sd">    permutations that are written in disjoint cycle notation:</span>
-
-<span class="sd">    &gt;&gt;&gt; Permutation(1, 2)(1, 3)(2, 3)</span>
-<span class="sd">    Permutation([0, 3, 2, 1])</span>
-<span class="sd">    &gt;&gt;&gt; _ == Permutation([[1, 2]])*Permutation([[1, 3]])*Permutation([[2, 3]])</span>
-<span class="sd">    True</span>
-
-<span class="sd">    Entering a singleton in a permutation is a way to indicate the size of the</span>
-<span class="sd">    permutation. The ``size`` keyword can also be used.</span>
-
-<span class="sd">    Array-form entry:</span>
-
-<span class="sd">    &gt;&gt;&gt; Permutation([[1, 2], [9]])</span>
-<span class="sd">    Permutation([0, 2, 1], size=10)</span>
-<span class="sd">    &gt;&gt;&gt; Permutation([[1, 2]], size=10)</span>
-<span class="sd">    Permutation([0, 2, 1], size=10)</span>
-
-<span class="sd">    Cyclic-form entry:</span>
-
-<span class="sd">    &gt;&gt;&gt; Permutation(1, 2, size=10)</span>
-<span class="sd">    Permutation([0, 2, 1], size=10)</span>
-<span class="sd">    &gt;&gt;&gt; Permutation(9)(1, 2)</span>
-<span class="sd">    Permutation([0, 2, 1], size=10)</span>
-
-<span class="sd">    Caution: no singleton containing an element larger than the largest</span>
-<span class="sd">    in any previous cycle can be entered. This is an important difference</span>
-<span class="sd">    in how Permutation and Cycle handle the __call__ syntax. A singleton</span>
-<span class="sd">    argument at the start of a Permutation performs instantiation of the</span>
-<span class="sd">    Permutation and is permitted:</span>
-
-<span class="sd">    &gt;&gt;&gt; Permutation(5)</span>
-<span class="sd">    Permutation([], size=6)</span>
-
-<span class="sd">    A singleton entered after instantiation is a call to the permutation</span>
-<span class="sd">    -- a function call -- and if the argument is out of range it will</span>
-<span class="sd">    trigger an error. For this reason, it is better to start the cycle</span>
-<span class="sd">    with the singleton:</span>
-
-<span class="sd">    The following fails because there is is no element 3:</span>
-
-<span class="sd">    &gt;&gt;&gt; Permutation(1, 2)(3)</span>
-<span class="sd">    Traceback (most recent call last):</span>
-<span class="sd">    ...</span>
-<span class="sd">    IndexError: list index out of range</span>
-
-<span class="sd">    This is ok: only the call to an out of range singleton is prohibited;</span>
-<span class="sd">    otherwise the permutation autosizes:</span>
-
-<span class="sd">    &gt;&gt;&gt; Permutation(3)(1, 2)</span>
-<span class="sd">    Permutation([0, 2, 1, 3])</span>
-<span class="sd">    &gt;&gt;&gt; Permutation(1, 2)(3, 4) == Permutation(3, 4)(1, 2)</span>
-<span class="sd">    True</span>
-
-
-<span class="sd">    Equality testing</span>
-<span class="sd">    ----------------</span>
-
-<span class="sd">    The array forms must be the same in order for permutations to be equal:</span>
-
-<span class="sd">    &gt;&gt;&gt; Permutation([1, 0, 2, 3]) == Permutation([1, 0])</span>
-<span class="sd">    False</span>
-
-
-<span class="sd">    Identity Permutation</span>
-<span class="sd">    --------------------</span>
-
-<span class="sd">    The identity permutation is a permutation in which no element is out of</span>
-<span class="sd">    place. It can be entered in a variety of ways. All the following create</span>
-<span class="sd">    an identity permutation of size 4:</span>
-
-<span class="sd">    &gt;&gt;&gt; I = Permutation([0, 1, 2, 3])</span>
-<span class="sd">    &gt;&gt;&gt; all(p == I for p in [</span>
-<span class="sd">    ... Permutation(3),</span>
-<span class="sd">    ... Permutation(list(range(4))),</span>
-<span class="sd">    ... Permutation([], size=4),</span>
-<span class="sd">    ... Permutation(size=4)])</span>
-<span class="sd">    True</span>
-
-<span class="sd">    Watch out for entering the range *inside* a set of brackets (which is</span>
-<span class="sd">    cycle notation):</span>
-
-<span class="sd">    &gt;&gt;&gt; I == Permutation([list(range(4))])</span>
-<span class="sd">    False</span>
-
-
-<span class="sd">    Permutation Printing</span>
-<span class="sd">    ====================</span>
-
-<span class="sd">    There are a few things to note about how Permutations are printed.</span>
-
-<span class="sd">    1) If you prefer one form (array or cycle) over another, you can set that</span>
-<span class="sd">    with the print_cyclic flag.</span>
-
-<span class="sd">    &gt;&gt;&gt; Permutation(1, 2)(4, 5)(3, 4)</span>
-<span class="sd">    Permutation([0, 2, 1, 4, 5, 3])</span>
-<span class="sd">    &gt;&gt;&gt; p = _</span>
-
-<span class="sd">    &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">    &gt;&gt;&gt; p</span>
-<span class="sd">    Permutation(1, 2)(3, 4, 5)</span>
-<span class="sd">    &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-
-<span class="sd">    2) Regardless of the setting, a list of elements in the array for cyclic</span>
-<span class="sd">    form can be obtained and either of those can be copied and supplied as</span>
-<span class="sd">    the argument to Permutation:</span>
-
-<span class="sd">    &gt;&gt;&gt; p.array_form</span>
-<span class="sd">    [0, 2, 1, 4, 5, 3]</span>
-<span class="sd">    &gt;&gt;&gt; p.cyclic_form</span>
-<span class="sd">    [[1, 2], [3, 4, 5]]</span>
-<span class="sd">    &gt;&gt;&gt; Permutation(_) == p</span>
-<span class="sd">    True</span>
-
-<span class="sd">    3) Printing is economical in that as little as possible is printed while</span>
-<span class="sd">    retaining all information about the size of the permutation:</span>
-
-<span class="sd">    &gt;&gt;&gt; Permutation([1, 0, 2, 3])</span>
-<span class="sd">    Permutation([1, 0, 2, 3])</span>
-<span class="sd">    &gt;&gt;&gt; Permutation([1, 0, 2, 3], size=20)</span>
-<span class="sd">    Permutation([1, 0], size=20)</span>
-<span class="sd">    &gt;&gt;&gt; Permutation([1, 0, 2, 4, 3, 5, 6], size=20)</span>
-<span class="sd">    Permutation([1, 0, 2, 4, 3], size=20)</span>
-
-<span class="sd">    &gt;&gt;&gt; p = Permutation([1, 0, 2, 3])</span>
-<span class="sd">    &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">    &gt;&gt;&gt; p</span>
-<span class="sd">    Permutation(3)(0, 1)</span>
-<span class="sd">    &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-
-<span class="sd">    The 2 was not printed but it is still there as can be seen with the</span>
-<span class="sd">    array_form and size methods:</span>
-
-<span class="sd">    &gt;&gt;&gt; p.array_form</span>
-<span class="sd">    [1, 0, 2, 3]</span>
-<span class="sd">    &gt;&gt;&gt; p.size</span>
-<span class="sd">    4</span>
-
-<span class="sd">    Short introduction to other methods</span>
-<span class="sd">    ===================================</span>
-
-<span class="sd">    The permutation can act as a bijective function, telling what element is</span>
-<span class="sd">    located at a given position</span>
-
-<span class="sd">    &gt;&gt;&gt; q = Permutation([5, 2, 3, 4, 1, 0])</span>
-<span class="sd">    &gt;&gt;&gt; q.array_form[1] # the hard way</span>
-<span class="sd">    2</span>
-<span class="sd">    &gt;&gt;&gt; q(1) # the easy way</span>
-<span class="sd">    2</span>
-<span class="sd">    &gt;&gt;&gt; dict([(i, q(i)) for i in range(q.size)]) # showing the bijection</span>
-<span class="sd">    {0: 5, 1: 2, 2: 3, 3: 4, 4: 1, 5: 0}</span>
-
-<span class="sd">    The full cyclic form (including singletons) can be obtained:</span>
-
-<span class="sd">    &gt;&gt;&gt; p.full_cyclic_form</span>
-<span class="sd">    [[0, 1], [2], [3]]</span>
-
-<span class="sd">    Any permutation can be factored into transpositions of pairs of elements:</span>
-
-<span class="sd">    &gt;&gt;&gt; Permutation([[1, 2], [3, 4, 5]]).transpositions()</span>
-<span class="sd">    [(1, 2), (3, 5), (3, 4)]</span>
-<span class="sd">    &gt;&gt;&gt; Permutation.rmul(*[Permutation([ti], size=6) for ti in _]).cyclic_form</span>
-<span class="sd">    [[1, 2], [3, 4, 5]]</span>
-
-<span class="sd">    The number of permutations on a set of n elements is given by n! and is</span>
-<span class="sd">    called the cardinality.</span>
-
-<span class="sd">    &gt;&gt;&gt; p.size</span>
-<span class="sd">    4</span>
-<span class="sd">    &gt;&gt;&gt; p.cardinality</span>
-<span class="sd">    24</span>
-
-<span class="sd">    A given permutation has a rank among all the possible permutations of the</span>
-<span class="sd">    same elements, but what that rank is depends on how the permutations are</span>
-<span class="sd">    enumerated. (There are a number of different methods of doing so.) The</span>
-<span class="sd">    lexicographic rank is given by the rank method and this rank is used to</span>
-<span class="sd">    increment a partion with addition/subtraction:</span>
-
-<span class="sd">    &gt;&gt;&gt; p.rank()</span>
-<span class="sd">    6</span>
-<span class="sd">    &gt;&gt;&gt; p + 1</span>
-<span class="sd">    Permutation([1, 0, 3, 2])</span>
-<span class="sd">    &gt;&gt;&gt; p.next_lex()</span>
-<span class="sd">    Permutation([1, 0, 3, 2])</span>
-<span class="sd">    &gt;&gt;&gt; _.rank()</span>
-<span class="sd">    7</span>
-<span class="sd">    &gt;&gt;&gt; p.unrank_lex(p.size, rank=7)</span>
-<span class="sd">    Permutation([1, 0, 3, 2])</span>
-
-<span class="sd">    The product of two permutations p and q is defined as their composition as</span>
-<span class="sd">    functions, (p*q)(i) = q(p(i)) [6]_.</span>
-
-<span class="sd">    &gt;&gt;&gt; p = Permutation([1, 0, 2, 3])</span>
-<span class="sd">    &gt;&gt;&gt; q = Permutation([2, 3, 1, 0])</span>
-<span class="sd">    &gt;&gt;&gt; list(q*p)</span>
-<span class="sd">    [2, 3, 0, 1]</span>
-<span class="sd">    &gt;&gt;&gt; list(p*q)</span>
-<span class="sd">    [3, 2, 1, 0]</span>
-<span class="sd">    &gt;&gt;&gt; [q(p(i)) for i in range(p.size)]</span>
-<span class="sd">    [3, 2, 1, 0]</span>
-
-<span class="sd">    The permutation can be &#39;applied&#39; to any list-like object, not only</span>
-<span class="sd">    Permutations:</span>
-
-<span class="sd">    &gt;&gt;&gt; p([&#39;zero&#39;, &#39;one&#39;, &#39;four&#39;, &#39;two&#39;])</span>
-<span class="sd">     [&#39;one&#39;, &#39;zero&#39;, &#39;four&#39;, &#39;two&#39;]</span>
-<span class="sd">    &gt;&gt;&gt; p(&#39;zo42&#39;)</span>
-<span class="sd">    [&#39;o&#39;, &#39;z&#39;, &#39;4&#39;, &#39;2&#39;]</span>
-
-<span class="sd">    If you have a list of arbitrary elements, the corresponding permutation</span>
-<span class="sd">    can be found with the from_sequence method:</span>
-
-<span class="sd">    &gt;&gt;&gt; Permutation.from_sequence(&#39;SymPy&#39;)</span>
-<span class="sd">    Permutation([1, 3, 2, 0, 4])</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Cycle</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] Skiena, S. &#39;Permutations.&#39; 1.1 in Implementing Discrete Mathematics</span>
-<span class="sd">           Combinatorics and Graph Theory with Mathematica.  Reading, MA:</span>
-<span class="sd">           Addison-Wesley, pp. 3-16, 1990.</span>
-
-<span class="sd">    .. [2] Knuth, D. E. The Art of Computer Programming, Vol. 4: Combinatorial</span>
-<span class="sd">           Algorithms, 1st ed. Reading, MA: Addison-Wesley, 2011.</span>
-
-<span class="sd">    .. [3] Wendy Myrvold and Frank Ruskey. 2001. Ranking and unranking</span>
-<span class="sd">           permutations in linear time. Inf. Process. Lett. 79, 6 (September 2001),</span>
-<span class="sd">           281-284. DOI=10.1016/S0020-0190(01)00141-7</span>
-
-<span class="sd">    .. [4] D. L. Kreher, D. R. Stinson &#39;Combinatorial Algorithms&#39;</span>
-<span class="sd">           CRC Press, 1999</span>
-
-<span class="sd">    .. [5] Graham, R. L.; Knuth, D. E.; and Patashnik, O.</span>
-<span class="sd">           Concrete Mathematics: A Foundation for Computer Science, 2nd ed.</span>
-<span class="sd">           Reading, MA: Addison-Wesley, 1994.</span>
-
-<span class="sd">    .. [6] http://en.wikipedia.org/wiki/Permutation#Product_and_inverse</span>
-
-<span class="sd">    .. [7] http://en.wikipedia.org/wiki/Lehmer_code</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_Permutation</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="n">_array_form</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">_cyclic_form</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">_cycle_structure</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">_size</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">_rank</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Constructor for the Permutation object from a list or a</span>
-<span class="sd">        list of lists in which all elements of the permutation may</span>
-<span class="sd">        appear only once.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-
-<span class="sd">        Permutations entered in array-form are left unaltered:</span>
-
-<span class="sd">        &gt;&gt;&gt; Permutation([0, 2, 1])</span>
-<span class="sd">        Permutation([0, 2, 1])</span>
-
-<span class="sd">        Permutations entered in cyclic form are converted to array form;</span>
-<span class="sd">        singletons need not be entered, but can be entered to indicate the</span>
-<span class="sd">        largest element:</span>
-
-<span class="sd">        &gt;&gt;&gt; Permutation([[4, 5, 6], [0, 1]])</span>
-<span class="sd">        Permutation([1, 0, 2, 3, 5, 6, 4])</span>
-<span class="sd">        &gt;&gt;&gt; Permutation([[4, 5, 6], [0, 1], [19]])</span>
-<span class="sd">        Permutation([1, 0, 2, 3, 5, 6, 4], size=20)</span>
-
-<span class="sd">        All manipulation of permutations assumes that the smallest element</span>
-<span class="sd">        is 0 (in keeping with 0-based indexing in Python) so if the 0 is</span>
-<span class="sd">        missing when entering a permutation in array form, an error will be</span>
-<span class="sd">        raised:</span>
-
-<span class="sd">        &gt;&gt;&gt; Permutation([2, 1])</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">        ...</span>
-<span class="sd">        ValueError: Integers 0 through 2 must be present.</span>
-
-<span class="sd">        If a permutation is entered in cyclic form, it can be entered without</span>
-<span class="sd">        singletons and the ``size`` specified so those values can be filled</span>
-<span class="sd">        in, otherwise the array form will only extend to the maximum value</span>
-<span class="sd">        in the cycles:</span>
-
-<span class="sd">        &gt;&gt;&gt; Permutation([[1, 4], [3, 5, 2]], size=10)</span>
-<span class="sd">        Permutation([0, 4, 3, 5, 1, 2], size=10)</span>
-<span class="sd">        &gt;&gt;&gt; _.array_form</span>
-<span class="sd">        [0, 4, 3, 5, 1, 2, 6, 7, 8, 9]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">size</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;size&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">size</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">size</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">size</span><span class="p">)</span>
-
-        <span class="c">#a) ()</span>
-        <span class="c">#b) (1) = identity</span>
-        <span class="c">#c) (1, 2) = cycle</span>
-        <span class="c">#d) ([1, 2, 3]) = array form</span>
-        <span class="c">#e) ([[1, 2]]) = cyclic form</span>
-        <span class="c">#f) (Cycle) = conversion to permutation</span>
-        <span class="c">#g) (Permutation) = adjust size or return copy</span>
-        <span class="n">ok</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="p">:</span>  <span class="c"># a</span>
-            <span class="k">return</span> <span class="n">_af_new</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">size</span> <span class="ow">or</span> <span class="mi">0</span><span class="p">)))</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>  <span class="c"># c</span>
-            <span class="k">return</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">Cycle</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span><span class="o">.</span><span class="n">list</span><span class="p">(</span><span class="n">size</span><span class="p">))</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Perm</span><span class="p">):</span>  <span class="c"># g</span>
-                <span class="k">if</span> <span class="n">size</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">size</span> <span class="o">==</span> <span class="n">a</span><span class="o">.</span><span class="n">size</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">a</span>
-                <span class="k">return</span> <span class="n">Perm</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">array_form</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">size</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Cycle</span><span class="p">):</span>  <span class="c"># f</span>
-                <span class="k">return</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">list</span><span class="p">(</span><span class="n">size</span><span class="p">))</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">a</span><span class="p">):</span>  <span class="c"># b</span>
-                <span class="k">return</span> <span class="n">_af_new</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)))</span>
-            <span class="k">if</span> <span class="n">has_variety</span><span class="p">(</span><span class="n">is_sequence</span><span class="p">(</span><span class="n">ai</span><span class="p">)</span> <span class="k">for</span> <span class="n">ai</span> <span class="ow">in</span> <span class="n">a</span><span class="p">):</span>
-                <span class="n">ok</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">ok</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">ok</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Permutation argument must be a list of ints, &quot;</span>
-                             <span class="s">&quot;a list of lists, Permutation or Cycle.&quot;</span><span class="p">)</span>
-
-
-        <span class="c"># safe to assume args are valid; this also makes a copy</span>
-        <span class="c"># of the args</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="n">is_cycle</span> <span class="o">=</span> <span class="n">args</span> <span class="ow">and</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">is_cycle</span><span class="p">:</span>  <span class="c"># e</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="p">[[</span><span class="nb">int</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">c</span><span class="p">]</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># d</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span>
-
-        <span class="c"># if there are n elements present, 0, 1, ..., n-1 should be present</span>
-        <span class="c"># unless a cycle notation has been provided. A 0 will be added</span>
-        <span class="c"># for convenience in case one wants to enter permutations where</span>
-        <span class="c"># counting starts from 1.</span>
-
-        <span class="n">temp</span> <span class="o">=</span> <span class="n">flatten</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">has_dups</span><span class="p">(</span><span class="n">temp</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">is_cycle</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;there were repeated elements; to resolve &#39;</span>
-                <span class="s">&#39;cycles use Cycle</span><span class="si">%s</span><span class="s">.&#39;</span> <span class="o">%</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="nb">str</span><span class="p">(</span><span class="nb">tuple</span><span class="p">(</span><span class="n">c</span><span class="p">))</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;there were repeated elements.&#39;</span><span class="p">)</span>
-        <span class="n">temp</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">temp</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">is_cycle</span> <span class="ow">and</span> \
-                <span class="nb">any</span><span class="p">(</span><span class="n">i</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">temp</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">temp</span><span class="p">))):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Integers 0 through </span><span class="si">%s</span><span class="s"> must be present.&quot;</span> <span class="o">%</span>
-                             <span class="nb">max</span><span class="p">(</span><span class="n">temp</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">is_cycle</span><span class="p">:</span>
-            <span class="c"># it&#39;s not necessarily canonical so we won&#39;t store</span>
-            <span class="c"># it -- use the array form instead</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">Cycle</span><span class="p">()</span>
-            <span class="k">for</span> <span class="n">ci</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="n">c</span><span class="p">(</span><span class="o">*</span><span class="n">ci</span><span class="p">)</span>
-            <span class="n">aform</span> <span class="o">=</span> <span class="n">c</span><span class="o">.</span><span class="n">list</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">aform</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">size</span> <span class="ow">and</span> <span class="n">size</span> <span class="o">&gt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">aform</span><span class="p">):</span>
-            <span class="c"># don&#39;t allow for truncation of permutation which</span>
-            <span class="c"># might split a cycle and lead to an invalid aform</span>
-            <span class="c"># but do allow the permutation size to be increased</span>
-            <span class="n">aform</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">aform</span><span class="p">),</span> <span class="n">size</span><span class="p">)))</span>
-        <span class="n">size</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">aform</span><span class="p">)</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">aform</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_array_form</span> <span class="o">=</span> <span class="n">aform</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_size</span> <span class="o">=</span> <span class="n">size</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_af_new</span><span class="p">(</span><span class="n">perm</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A method to produce a Permutation object from a list;</span>
-<span class="sd">        the list is bound to the _array_form attribute, so it must</span>
-<span class="sd">        not be modified; this method is meant for internal use only;</span>
-<span class="sd">        the list ``a`` is supposed to be generated as a temporary value</span>
-<span class="sd">        in a method, so p = Perm._af_new(a) is the only object</span>
-<span class="sd">        to hold a reference to ``a``::</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Perm</span>
-<span class="sd">        &gt;&gt;&gt; Perm.print_cyclic = False</span>
-<span class="sd">        &gt;&gt;&gt; a = [2,1,3,0]</span>
-<span class="sd">        &gt;&gt;&gt; p = Perm._af_new(a)</span>
-<span class="sd">        &gt;&gt;&gt; p</span>
-<span class="sd">        Permutation([2, 1, 3, 0])</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">Perm</span><span class="p">,</span> <span class="n">perm</span><span class="p">)</span>
-        <span class="n">p</span><span class="o">.</span><span class="n">_array_form</span> <span class="o">=</span> <span class="n">perm</span>
-        <span class="n">p</span><span class="o">.</span><span class="n">_size</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">perm</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">p</span>
-
-    <span class="k">def</span> <span class="nf">_hashable_content</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># the array_form (a list) is the Permutation arg, so we need to</span>
-        <span class="c"># return a tuple, instead</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">array_form</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Permutation.array_form"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.array_form">[docs]</a>    <span class="k">def</span> <span class="nf">array_form</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return a copy of the attribute _array_form</span>
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([[2,0], [3,1]])</span>
-<span class="sd">        &gt;&gt;&gt; p.array_form</span>
-<span class="sd">        [2, 3, 0, 1]</span>
-<span class="sd">        &gt;&gt;&gt; Permutation([[2,0,3,1]]).array_form</span>
-<span class="sd">        [3, 2, 0, 1]</span>
-<span class="sd">        &gt;&gt;&gt; Permutation([2,0,3,1]).array_form</span>
-<span class="sd">        [2, 0, 3, 1]</span>
-<span class="sd">        &gt;&gt;&gt; Permutation([[1, 2], [4, 5]]).array_form</span>
-<span class="sd">        [0, 2, 1, 3, 5, 4]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_array_form</span><span class="p">[:]</span>
-</div>
-<div class="viewcode-block" id="Permutation.list"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.list">[docs]</a>    <span class="k">def</span> <span class="nf">list</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the permutation as an explicit list, possibly</span>
-<span class="sd">        trimming unmoved elements if size is less than the maximum</span>
-<span class="sd">        element in the permutation; if this is desired, setting</span>
-<span class="sd">        ``size=-1`` will guarantee such trimming.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation(2, 3)(4, 5)</span>
-<span class="sd">        &gt;&gt;&gt; p.list()</span>
-<span class="sd">        [0, 1, 3, 2, 5, 4]</span>
-<span class="sd">        &gt;&gt;&gt; p.list(10)</span>
-<span class="sd">        [0, 1, 3, 2, 5, 4, 6, 7, 8, 9]</span>
-
-<span class="sd">        Passing a length too small will trim trailing, unchanged elements</span>
-<span class="sd">        in the permutation:</span>
-
-<span class="sd">        &gt;&gt;&gt; Permutation(2, 4)(1, 2, 4).list(-1)</span>
-<span class="sd">        [0, 2, 1]</span>
-<span class="sd">        &gt;&gt;&gt; Permutation(3).list(-1)</span>
-<span class="sd">        []</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span> <span class="ow">and</span> <span class="n">size</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;must give size for empty Cycle&#39;</span><span class="p">)</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="k">if</span> <span class="n">size</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">size</span> <span class="o">&gt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">:</span>
-                <span class="n">rv</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">,</span> <span class="n">size</span><span class="p">)))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># find first value from rhs where rv[i] != i</span>
-                <span class="n">i</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">-</span> <span class="mi">1</span>
-                <span class="k">while</span> <span class="n">rv</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">rv</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">!=</span> <span class="n">i</span><span class="p">:</span>
-                        <span class="k">break</span>
-                    <span class="n">rv</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-                    <span class="n">i</span> <span class="o">-=</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="n">rv</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Permutation.cyclic_form"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.cyclic_form">[docs]</a>    <span class="k">def</span> <span class="nf">cyclic_form</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        This is used to convert to the cyclic notation</span>
-<span class="sd">        from the canonical notation. Singletons are omitted.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([0, 3, 1, 2])</span>
-<span class="sd">        &gt;&gt;&gt; p.cyclic_form</span>
-<span class="sd">        [[1, 3, 2]]</span>
-<span class="sd">        &gt;&gt;&gt; Permutation([1, 0, 2, 4, 3, 5]).cyclic_form</span>
-<span class="sd">        [[0, 1], [3, 4]]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        array_form, full_cyclic_form</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_cyclic_form</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_cyclic_form</span><span class="p">)</span>
-        <span class="n">array_form</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="n">unchecked</span> <span class="o">=</span> <span class="p">[</span><span class="bp">True</span><span class="p">]</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">array_form</span><span class="p">)</span>
-        <span class="n">cyclic_form</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">array_form</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="n">unchecked</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                <span class="n">cycle</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="n">cycle</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                <span class="n">unchecked</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="n">j</span> <span class="o">=</span> <span class="n">i</span>
-                <span class="k">while</span> <span class="n">unchecked</span><span class="p">[</span><span class="n">array_form</span><span class="p">[</span><span class="n">j</span><span class="p">]]:</span>
-                    <span class="n">j</span> <span class="o">=</span> <span class="n">array_form</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-                    <span class="n">cycle</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-                    <span class="n">unchecked</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">cycle</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">cyclic_form</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">cycle</span><span class="p">)</span>
-                    <span class="k">assert</span> <span class="n">cycle</span> <span class="o">==</span> <span class="nb">list</span><span class="p">(</span><span class="n">minlex</span><span class="p">(</span><span class="n">cycle</span><span class="p">,</span> <span class="n">is_set</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-        <span class="n">cyclic_form</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_cyclic_form</span> <span class="o">=</span> <span class="n">cyclic_form</span><span class="p">[:]</span>
-        <span class="k">return</span> <span class="n">cyclic_form</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Permutation.full_cyclic_form"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.full_cyclic_form">[docs]</a>    <span class="k">def</span> <span class="nf">full_cyclic_form</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return permutation in cyclic form including singletons.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation([0, 2, 1]).full_cyclic_form</span>
-<span class="sd">        [[0], [1, 2]]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">need</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">))</span> <span class="o">-</span> <span class="nb">set</span><span class="p">(</span><span class="n">flatten</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cyclic_form</span><span class="p">))</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">cyclic_form</span>
-        <span class="n">rv</span><span class="o">.</span><span class="n">extend</span><span class="p">([[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">need</span><span class="p">])</span>
-        <span class="n">rv</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">rv</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Permutation.size"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.size">[docs]</a>    <span class="k">def</span> <span class="nf">size</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the number of elements in the permutation.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation([[3, 2], [0, 1]]).size</span>
-<span class="sd">        4</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        cardinality, length, order, rank</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span>
-</div>
-<div class="viewcode-block" id="Permutation.support"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.support">[docs]</a>    <span class="k">def</span> <span class="nf">support</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the elements in permutation, P, for which P[i] != i.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([[3, 2], [0, 1], [4]])</span>
-<span class="sd">        &gt;&gt;&gt; p.array_form</span>
-<span class="sd">        [1, 0, 3, 2, 4]</span>
-<span class="sd">        &gt;&gt;&gt; p.support()</span>
-<span class="sd">        [0, 1, 2, 3]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">if</span> <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">!=</span> <span class="n">i</span><span class="p">]</span>
-</div>
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return permutation that is other higher in rank than self.</span>
-
-<span class="sd">        The rank is the lexicographical rank, with the identity permutation</span>
-<span class="sd">        having rank of 0.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-<span class="sd">        &gt;&gt;&gt; I = Permutation([0, 1, 2, 3])</span>
-<span class="sd">        &gt;&gt;&gt; a = Permutation([2, 1, 3, 0])</span>
-<span class="sd">        &gt;&gt;&gt; I + a.rank() == a</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        __sub__, inversion_vector</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">rank</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rank</span><span class="p">()</span> <span class="o">+</span> <span class="n">other</span><span class="p">)</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">cardinality</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">Perm</span><span class="o">.</span><span class="n">unrank_lex</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">,</span> <span class="n">rank</span><span class="p">)</span>
-        <span class="n">rv</span><span class="o">.</span><span class="n">_rank</span> <span class="o">=</span> <span class="n">rank</span>
-        <span class="k">return</span> <span class="n">rv</span>
-
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the permutation that is other lower in rank than self.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        __add__</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__add__</span><span class="p">(</span><span class="o">-</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="Permutation.rmul"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.rmul">[docs]</a>    <span class="k">def</span> <span class="nf">rmul</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return product of Permutations [a, b, c, ...] as the Permutation whose</span>
-<span class="sd">        ith value is a(b(c(i))).</span>
-
-<span class="sd">        a, b, c, ... can be Permutation objects or tuples.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import _af_rmul, Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-
-<span class="sd">        &gt;&gt;&gt; a, b = [1, 0, 2], [0, 2, 1]</span>
-<span class="sd">        &gt;&gt;&gt; a = Permutation(a); b = Permutation(b)</span>
-<span class="sd">        &gt;&gt;&gt; list(Permutation.rmul(a, b))</span>
-<span class="sd">        [1, 2, 0]</span>
-<span class="sd">        &gt;&gt;&gt; [a(b(i)) for i in range(3)]</span>
-<span class="sd">        [1, 2, 0]</span>
-
-<span class="sd">        This handles the operands in reverse order compared to the ``*`` operator:</span>
-
-<span class="sd">        &gt;&gt;&gt; a = Permutation(a); b = Permutation(b)</span>
-<span class="sd">        &gt;&gt;&gt; list(a*b)</span>
-<span class="sd">        [2, 0, 1]</span>
-<span class="sd">        &gt;&gt;&gt; [b(a(i)) for i in range(3)]</span>
-<span class="sd">        [2, 0, 1]</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        All items in the sequence will be parsed by Permutation as</span>
-<span class="sd">        necessary as long as the first item is a Permutation:</span>
-
-<span class="sd">        &gt;&gt;&gt; Permutation.rmul(a, [0, 2, 1]) == Permutation.rmul(a, b)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        The reverse order of arguments will raise a TypeError.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)):</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">rv</span>
-        <span class="k">return</span> <span class="n">rv</span>
-</div>
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="Permutation.rmul_with_af"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.rmul_with_af">[docs]</a>    <span class="k">def</span> <span class="nf">rmul_with_af</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        same as rmul, but the elements of args are Permutation objects</span>
-<span class="sd">        which have _array_form</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="o">.</span><span class="n">_array_form</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">_af_rmuln</span><span class="p">(</span><span class="o">*</span><span class="n">a</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">rv</span>
-</div>
-<div class="viewcode-block" id="Permutation.mul_inv"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.mul_inv">[docs]</a>    <span class="k">def</span> <span class="nf">mul_inv</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        other*~self, self and other have _array_form</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">_af_invert</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_array_form</span><span class="p">)</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">_array_form</span>
-        <span class="k">return</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">_af_rmul</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-</div>
-    <span class="k">def</span> <span class="nf">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;This is needed to coerse other to Permutation in rmul.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Perm</span><span class="p">(</span><span class="n">other</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the product a*b as a Permutation; the ith value is b(a(i)).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import _af_rmul, Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-
-<span class="sd">        &gt;&gt;&gt; a, b = [1, 0, 2], [0, 2, 1]</span>
-<span class="sd">        &gt;&gt;&gt; a = Permutation(a); b = Permutation(b)</span>
-<span class="sd">        &gt;&gt;&gt; list(a*b)</span>
-<span class="sd">        [2, 0, 1]</span>
-<span class="sd">        &gt;&gt;&gt; [b(a(i)) for i in range(3)]</span>
-<span class="sd">        [2, 0, 1]</span>
-
-<span class="sd">        This handles operands in reverse order compared to _af_rmul and rmul:</span>
-
-<span class="sd">        &gt;&gt;&gt; al = list(a); bl = list(b)</span>
-<span class="sd">        &gt;&gt;&gt; _af_rmul(al, bl)</span>
-<span class="sd">        [1, 2, 0]</span>
-<span class="sd">        &gt;&gt;&gt; [al[bl[i]] for i in range(3)]</span>
-<span class="sd">        [1, 2, 0]</span>
-
-<span class="sd">        It is acceptable for the arrays to have different lengths; the shorter</span>
-<span class="sd">        one will be padded to match the longer one:</span>
-
-<span class="sd">        &gt;&gt;&gt; b*Permutation([1, 0])</span>
-<span class="sd">        Permutation([1, 2, 0])</span>
-<span class="sd">        &gt;&gt;&gt; Permutation([1, 0])*b</span>
-<span class="sd">        Permutation([2, 0, 1])</span>
-
-<span class="sd">        It is also acceptable to allow coercion to handle conversion of a</span>
-<span class="sd">        single list to the left of a Permutation:</span>
-
-<span class="sd">        &gt;&gt;&gt; [0, 1]*a # no change: 2-element identity</span>
-<span class="sd">        Permutation([1, 0, 2])</span>
-<span class="sd">        &gt;&gt;&gt; [[0, 1]]*a # exchange first two elements</span>
-<span class="sd">        Permutation([0, 1, 2])</span>
-
-<span class="sd">        You cannot use more than 1 cycle notation in a product of cycles</span>
-<span class="sd">        since coercion can only handle one argument to the left. To handle</span>
-<span class="sd">        multiple cycles it is convenient to use Cycle instead of Permutation:</span>
-
-<span class="sd">        &gt;&gt;&gt; [[1, 2]]*[[2, 3]]*Permutation([]) # doctest: +SKIP</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Cycle</span>
-<span class="sd">        &gt;&gt;&gt; Cycle(1, 2)(2, 3)</span>
-<span class="sd">        Cycle(1, 3, 2)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="c"># __rmul__ makes sure the other is a Permutation</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">b</span><span class="p">:</span>
-            <span class="n">perm</span> <span class="o">=</span> <span class="n">a</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">b</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">b</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">))))</span>
-            <span class="n">perm</span> <span class="o">=</span> <span class="p">[</span><span class="n">b</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">a</span><span class="p">]</span> <span class="o">+</span> <span class="n">b</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">):]</span>
-        <span class="k">return</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">perm</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Permutation.commutes_with"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.commutes_with">[docs]</a>    <span class="k">def</span> <span class="nf">commutes_with</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Checks if the elements are commuting.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; a = Permutation([1,4,3,0,2,5])</span>
-<span class="sd">        &gt;&gt;&gt; b = Permutation([0,1,2,3,4,5])</span>
-<span class="sd">        &gt;&gt;&gt; a.commutes_with(b)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; b = Permutation([2,3,5,4,1,0])</span>
-<span class="sd">        &gt;&gt;&gt; a.commutes_with(b)</span>
-<span class="sd">        False</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="k">return</span> <span class="n">_af_commutes_with</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">__pow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Routine for finding powers of a permutation.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([2,0,3,1])</span>
-<span class="sd">        &gt;&gt;&gt; p.order()</span>
-<span class="sd">        4</span>
-<span class="sd">        &gt;&gt;&gt; p**4</span>
-<span class="sd">        Permutation([0, 1, 2, 3])</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="n">Perm</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                <span class="s">&#39;p**p is not defined; do you mean p^p (conjugate)?&#39;</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">_af_pow</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">array_form</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__rxor__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return self(i) when ``i`` is an int.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation(1, 2, 9)</span>
-<span class="sd">        &gt;&gt;&gt; 2^p == p(2) == 9</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">int</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">==</span> <span class="n">i</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                <span class="s">&quot;i^p = p(i) when i is an integer, not </span><span class="si">%s</span><span class="s">.&quot;</span> <span class="o">%</span> <span class="n">i</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__xor__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">h</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the conjugate permutation ``~h*self*h` `.</span>
-
-<span class="sd">        If ``a`` and ``b`` are conjugates, ``a = h*b*~h`` and</span>
-<span class="sd">        ``b = ~h*a*h`` and both have the same cycle structure.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation(1, 2, 9)</span>
-<span class="sd">        &gt;&gt;&gt; q = Permutation(6, 9, 8)</span>
-<span class="sd">        &gt;&gt;&gt; p*q != q*p</span>
-<span class="sd">        True</span>
-
-<span class="sd">        Calculate and check properties of the conjugate:</span>
-
-<span class="sd">        &gt;&gt;&gt; c = p^q</span>
-<span class="sd">        &gt;&gt;&gt; c == ~q*p*q and p == q*c*~q</span>
-<span class="sd">        True</span>
-
-<span class="sd">        The expression q^p^r is equivalent to q^(p*r):</span>
-
-<span class="sd">        &gt;&gt;&gt; r = Permutation(9)(4,6,8)</span>
-<span class="sd">        &gt;&gt;&gt; q^p^r == q^(p*r)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        If the term to the left of the conjugate operator, i, is an integer</span>
-<span class="sd">        then this is interpreted as selecting the ith element from the</span>
-<span class="sd">        permutation to the right:</span>
-
-<span class="sd">        &gt;&gt;&gt; all(i^p == p(i) for i in range(p.size))</span>
-<span class="sd">        True</span>
-
-<span class="sd">        Note that the * operator as higher precedence than the ^ operator:</span>
-
-<span class="sd">        &gt;&gt;&gt; q^r*p^r == q^(r*p)^r == Permutation(9)(1, 6, 4)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        In Python the precedence rule is p^q^r = (p^q)^r which differs</span>
-<span class="sd">        in general from p^(q^r)</span>
-
-<span class="sd">        &gt;&gt;&gt; q^p^r</span>
-<span class="sd">        Permutation(9)(1, 4, 8)</span>
-<span class="sd">        &gt;&gt;&gt; q^(p^r)</span>
-<span class="sd">        Permutation(9)(1, 8, 6)</span>
-
-<span class="sd">        For a given r and p, both of the following are conjugates of p:</span>
-<span class="sd">        ~r*p*r and r*p*~r. But these are not necessarily the same:</span>
-
-<span class="sd">        &gt;&gt;&gt; ~r*p*r == r*p*~r</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &gt;&gt;&gt; p = Permutation(1, 2, 9)(5, 6)</span>
-<span class="sd">        &gt;&gt;&gt; ~r*p*r == r*p*~r</span>
-<span class="sd">        False</span>
-
-<span class="sd">        The conjugate ~r*p*r was chosen so that ``p^q^r`` would be equivalent</span>
-<span class="sd">        to ``p^(q*r)`` rather than ``p^(r*q)``. To obtain r*p*~r, pass ~r to</span>
-<span class="sd">        this method:</span>
-
-<span class="sd">        &gt;&gt;&gt; p^~r == r*p*~r</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">!=</span> <span class="n">h</span><span class="o">.</span><span class="n">size</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;The permutations must be of equal size.&quot;</span><span class="p">)</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="p">[</span><span class="bp">None</span><span class="p">]</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">h</span><span class="o">.</span><span class="n">_array_form</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_array_form</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">):</span>
-            <span class="n">a</span><span class="p">[</span><span class="n">h</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span> <span class="o">=</span> <span class="n">h</span><span class="p">[</span><span class="n">p</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span>
-        <span class="k">return</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Permutation.transpositions"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.transpositions">[docs]</a>    <span class="k">def</span> <span class="nf">transpositions</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the permutation decomposed into a list of transpositions.</span>
-
-<span class="sd">        It is always possible to express a permutation as the product of</span>
-<span class="sd">        transpositions, see [1]</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([[1, 2, 3], [0, 4, 5, 6, 7]])</span>
-<span class="sd">        &gt;&gt;&gt; t = p.transpositions()</span>
-<span class="sd">        &gt;&gt;&gt; t</span>
-<span class="sd">        [(0, 7), (0, 6), (0, 5), (0, 4), (1, 3), (1, 2)]</span>
-<span class="sd">        &gt;&gt;&gt; print(&#39;&#39;.join(str(c) for c in t))</span>
-<span class="sd">        (0, 7)(0, 6)(0, 5)(0, 4)(1, 3)(1, 2)</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.rmul(*[Permutation([ti], size=p.size) for ti in t]) == p</span>
-<span class="sd">        True</span>
-
-<span class="sd">        References</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        1. http://en.wikipedia.org/wiki/Transposition_%28mathematics%29#Properties</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">cyclic_form</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">a</span><span class="p">:</span>
-            <span class="n">nx</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">nx</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">tuple</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-            <span class="k">elif</span> <span class="n">nx</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">first</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">x</span><span class="p">[</span><span class="n">nx</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span><span class="mi">0</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-                    <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">first</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">res</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Permutation.from_sequence"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.from_sequence">[docs]</a>    <span class="k">def</span> <span class="nf">from_sequence</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the permutation needed to obtain ``i`` from the sorted</span>
-<span class="sd">        elements of ``i``. If custom sorting is desired, a key can be given.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-
-<span class="sd">        &gt;&gt;&gt; Permutation.from_sequence(&#39;SymPy&#39;)</span>
-<span class="sd">        Permutation(4)(0, 1, 3)</span>
-<span class="sd">        &gt;&gt;&gt; _(sorted(&quot;SymPy&quot;))</span>
-<span class="sd">        [&#39;S&#39;, &#39;y&#39;, &#39;m&#39;, &#39;P&#39;, &#39;y&#39;]</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.from_sequence(&#39;SymPy&#39;, key=lambda x: x.lower())</span>
-<span class="sd">        Permutation(4)(0, 2)(1, 3)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">ic</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">i</span><span class="p">)))))</span>
-        <span class="k">if</span> <span class="n">key</span><span class="p">:</span>
-            <span class="n">ic</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">key</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">ic</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-        <span class="k">return</span> <span class="o">~</span><span class="n">Permutation</span><span class="p">([</span><span class="n">i</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">ic</span><span class="p">])</span>
-</div>
-    <span class="k">def</span> <span class="nf">__invert__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the inverse of the permutation.</span>
-
-<span class="sd">        A permutation multiplied by its inverse is the identity permutation.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([[2,0], [3,1]])</span>
-<span class="sd">        &gt;&gt;&gt; ~p</span>
-<span class="sd">        Permutation([2, 3, 0, 1])</span>
-<span class="sd">        &gt;&gt;&gt; _ == p**-1</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; p*~p == ~p*p == Permutation([0, 1, 2, 3])</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">_af_invert</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_array_form</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Yield elements from array form.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; list(Permutation(list(range(3))))</span>
-<span class="sd">        [0, 1, 2]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span><span class="p">:</span>
-            <span class="k">yield</span> <span class="n">i</span>
-
-    <span class="k">def</span> <span class="nf">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">i</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Allows applying a permutation instance as a bijective function.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([[2,0], [3,1]])</span>
-<span class="sd">        &gt;&gt;&gt; p.array_form</span>
-<span class="sd">        [2, 3, 0, 1]</span>
-<span class="sd">        &gt;&gt;&gt; [p(i) for i in range(4)]</span>
-<span class="sd">        [2, 3, 0, 1]</span>
-
-<span class="sd">        If an array is given then the permutation selects the items</span>
-<span class="sd">        from the array (i.e. the permutation is applied to the array):</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; p([x, 1, 0, x**2])</span>
-<span class="sd">        [0, x**2, x, 1]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># list indices can be Integer or int; leave this</span>
-        <span class="c"># as it is (don&#39;t test or convert it) because this</span>
-        <span class="c"># gets called a lot and should be fast</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="n">i</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="c"># P(1)</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_array_form</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="c"># P([a, b, c])</span>
-                    <span class="k">return</span> <span class="p">[</span><span class="n">i</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_array_form</span><span class="p">]</span>
-                <span class="k">except</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;unrecognized argument&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># P(1, 2, 3)</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">*</span><span class="n">Permutation</span><span class="p">(</span><span class="n">Cycle</span><span class="p">(</span><span class="o">*</span><span class="n">i</span><span class="p">),</span> <span class="n">size</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Permutation.atoms"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.atoms">[docs]</a>    <span class="k">def</span> <span class="nf">atoms</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns all the elements of a permutation</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation([0, 1, 2, 3, 4, 5]).atoms()</span>
-<span class="sd">        set([0, 1, 2, 3, 4, 5])</span>
-<span class="sd">        &gt;&gt;&gt; Permutation([[0, 1], [2, 3], [4, 5]]).atoms()</span>
-<span class="sd">        set([0, 1, 2, 3, 4, 5])</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">set</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">array_form</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Permutation.next_lex"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.next_lex">[docs]</a>    <span class="k">def</span> <span class="nf">next_lex</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the next permutation in lexicographical order.</span>
-<span class="sd">        If self is the last permutation in lexicographical order</span>
-<span class="sd">        it returns None.</span>
-<span class="sd">        See [4] section 2.4.</span>
-
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([2, 3, 1, 0])</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([2, 3, 1, 0]); p.rank()</span>
-<span class="sd">        17</span>
-<span class="sd">        &gt;&gt;&gt; p = p.next_lex(); p.rank()</span>
-<span class="sd">        18</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        rank, unrank_lex</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">perm</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span><span class="p">[:]</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">perm</span><span class="p">)</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">2</span>
-        <span class="k">while</span> <span class="n">perm</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">perm</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-            <span class="n">i</span> <span class="o">-=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">j</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span>
-            <span class="k">while</span> <span class="n">perm</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">perm</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                <span class="n">j</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="n">perm</span><span class="p">[</span><span class="n">j</span><span class="p">],</span> <span class="n">perm</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">perm</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">perm</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">j</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span>
-            <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">j</span><span class="p">:</span>
-                <span class="n">perm</span><span class="p">[</span><span class="n">j</span><span class="p">],</span> <span class="n">perm</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">perm</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">perm</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-                <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="n">j</span> <span class="o">-=</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">perm</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Permutation.unrank_nonlex"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.unrank_nonlex">[docs]</a>    <span class="k">def</span> <span class="nf">unrank_nonlex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">r</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        This is a linear time unranking algorithm that does not</span>
-<span class="sd">        respect lexicographic order [3].</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.unrank_nonlex(4, 5)</span>
-<span class="sd">        Permutation([2, 0, 3, 1])</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.unrank_nonlex(4, -1)</span>
-<span class="sd">        Permutation([0, 1, 2, 3])</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        next_nonlex, rank_nonlex</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">def</span> <span class="nf">_unrank1</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">a</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">a</span><span class="p">[</span><span class="n">r</span> <span class="o">%</span> <span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="n">r</span> <span class="o">%</span> <span class="n">n</span><span class="p">],</span> <span class="n">a</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-                <span class="n">_unrank1</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">r</span><span class="o">//</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-
-        <span class="n">id_perm</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">r</span> <span class="o">%</span> <span class="n">ifac</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="n">_unrank1</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">id_perm</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">id_perm</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Permutation.rank_nonlex"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.rank_nonlex">[docs]</a>    <span class="k">def</span> <span class="nf">rank_nonlex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">inv_perm</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        This is a linear time ranking algorithm that does not</span>
-<span class="sd">        enforce lexicographic order [3].</span>
-
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([0,1,2,3])</span>
-<span class="sd">        &gt;&gt;&gt; p.rank_nonlex()</span>
-<span class="sd">        23</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        next_nonlex, unrank_nonlex</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">def</span> <span class="nf">_rank1</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">perm</span><span class="p">,</span> <span class="n">inv_perm</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="mi">0</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">perm</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-            <span class="n">t</span> <span class="o">=</span> <span class="n">inv_perm</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-            <span class="n">perm</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">perm</span><span class="p">[</span><span class="n">t</span><span class="p">]</span> <span class="o">=</span> <span class="n">perm</span><span class="p">[</span><span class="n">t</span><span class="p">],</span> <span class="n">s</span>
-            <span class="n">inv_perm</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">inv_perm</span><span class="p">[</span><span class="n">s</span><span class="p">]</span> <span class="o">=</span> <span class="n">inv_perm</span><span class="p">[</span><span class="n">s</span><span class="p">],</span> <span class="n">t</span>
-            <span class="k">return</span> <span class="n">s</span> <span class="o">+</span> <span class="n">n</span><span class="o">*</span><span class="n">_rank1</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">perm</span><span class="p">,</span> <span class="n">inv_perm</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">inv_perm</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">inv_perm</span> <span class="o">=</span> <span class="p">(</span><span class="o">~</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">inv_perm</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">0</span>
-        <span class="n">perm</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span><span class="p">[:]</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">_rank1</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">perm</span><span class="p">),</span> <span class="n">perm</span><span class="p">,</span> <span class="n">inv_perm</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">r</span>
-</div>
-<div class="viewcode-block" id="Permutation.next_nonlex"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.next_nonlex">[docs]</a>    <span class="k">def</span> <span class="nf">next_nonlex</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the next permutation in nonlex order [3].</span>
-<span class="sd">        If self is the last permutation in this order it returns None.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([2, 0, 3, 1]); p.rank_nonlex()</span>
-<span class="sd">        5</span>
-<span class="sd">        &gt;&gt;&gt; p = p.next_nonlex(); p</span>
-<span class="sd">        Permutation([3, 0, 1, 2])</span>
-<span class="sd">        &gt;&gt;&gt; p.rank_nonlex()</span>
-<span class="sd">        6</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        rank_nonlex, unrank_nonlex</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rank_nonlex</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">r</span> <span class="o">==</span> <span class="n">ifac</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">return</span> <span class="n">Perm</span><span class="o">.</span><span class="n">unrank_nonlex</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">,</span> <span class="n">r</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Permutation.rank"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.rank">[docs]</a>    <span class="k">def</span> <span class="nf">rank</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the lexicographic rank of the permutation.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([0, 1, 2, 3])</span>
-<span class="sd">        &gt;&gt;&gt; p.rank()</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([3, 2, 1, 0])</span>
-<span class="sd">        &gt;&gt;&gt; p.rank()</span>
-<span class="sd">        23</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        next_lex, unrank_lex, cardinality, length, order, size</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rank</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rank</span>
-        <span class="n">rank</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">rho</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span><span class="p">[:]</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="n">size</span> <span class="o">=</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="n">psize</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">ifac</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">rank</span> <span class="o">+=</span> <span class="n">rho</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">*</span><span class="n">psize</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">size</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">rho</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">rho</span><span class="p">[</span><span class="n">j</span><span class="p">]:</span>
-                    <span class="n">rho</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="n">psize</span> <span class="o">//=</span> <span class="n">n</span>
-            <span class="n">n</span> <span class="o">-=</span> <span class="mi">1</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_rank</span> <span class="o">=</span> <span class="n">rank</span>
-        <span class="k">return</span> <span class="n">rank</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Permutation.cardinality"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.cardinality">[docs]</a>    <span class="k">def</span> <span class="nf">cardinality</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the number of all possible permutations.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([0,1,2,3])</span>
-<span class="sd">        &gt;&gt;&gt; p.cardinality</span>
-<span class="sd">        24</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        length, order, rank, size</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">ifac</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="Permutation.parity"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.parity">[docs]</a>    <span class="k">def</span> <span class="nf">parity</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes the parity of a permutation.</span>
-
-<span class="sd">        The parity of a permutation reflects the parity of the</span>
-<span class="sd">        number of inversions in the permutation, i.e., the</span>
-<span class="sd">        number of pairs of x and y such that ``x &gt; y`` but ``p[x] &lt; p[y]``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([0,1,2,3])</span>
-<span class="sd">        &gt;&gt;&gt; p.parity()</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([3,2,0,1])</span>
-<span class="sd">        &gt;&gt;&gt; p.parity()</span>
-<span class="sd">        1</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        _af_parity</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_cyclic_form</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">cycles</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span>
-
-        <span class="k">return</span> <span class="n">_af_parity</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">array_form</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Permutation.is_even"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.is_even">[docs]</a>    <span class="k">def</span> <span class="nf">is_even</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Checks if a permutation is even.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([0,1,2,3])</span>
-<span class="sd">        &gt;&gt;&gt; p.is_even</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([3,2,1,0])</span>
-<span class="sd">        &gt;&gt;&gt; p.is_even</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        is_odd</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_odd</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Permutation.is_odd"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.is_odd">[docs]</a>    <span class="k">def</span> <span class="nf">is_odd</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Checks if a permutation is odd.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([0,1,2,3])</span>
-<span class="sd">        &gt;&gt;&gt; p.is_odd</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([3,2,0,1])</span>
-<span class="sd">        &gt;&gt;&gt; p.is_odd</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        is_even</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">parity</span><span class="p">()</span> <span class="o">%</span> <span class="mi">2</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Permutation.is_Singleton"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.is_Singleton">[docs]</a>    <span class="k">def</span> <span class="nf">is_Singleton</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Checks to see if the permutation contains only one number and is</span>
-<span class="sd">        thus the only possible permutation of this set of numbers</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation([0]).is_Singleton</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Permutation([0, 1]).is_Singleton</span>
-<span class="sd">        False</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        is_Empty</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">==</span> <span class="mi">1</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Permutation.is_Empty"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.is_Empty">[docs]</a>    <span class="k">def</span> <span class="nf">is_Empty</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Checks to see if the permutation is a set with zero elements</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation([]).is_Empty</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Permutation([0]).is_Empty</span>
-<span class="sd">        False</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        is_Singleton</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">==</span> <span class="mi">0</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Permutation.is_Identity"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.is_Identity">[docs]</a>    <span class="k">def</span> <span class="nf">is_Identity</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns True if the Permutation is an identity permutation.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([])</span>
-<span class="sd">        &gt;&gt;&gt; p.is_Identity</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([[0], [1], [2]])</span>
-<span class="sd">        &gt;&gt;&gt; p.is_Identity</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([0, 1, 2])</span>
-<span class="sd">        &gt;&gt;&gt; p.is_Identity</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([0, 2, 1])</span>
-<span class="sd">        &gt;&gt;&gt; p.is_Identity</span>
-<span class="sd">        False</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        order</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">af</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="n">af</span> <span class="ow">or</span> <span class="nb">all</span><span class="p">(</span><span class="n">i</span> <span class="o">==</span> <span class="n">af</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="Permutation.ascents"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.ascents">[docs]</a>    <span class="k">def</span> <span class="nf">ascents</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the positions of ascents in a permutation, ie, the location</span>
-<span class="sd">        where p[i] &lt; p[i+1]</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([4,0,1,3,2])</span>
-<span class="sd">        &gt;&gt;&gt; p.ascents()</span>
-<span class="sd">        [1, 2]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        descents, inversions, min, max</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="n">pos</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="k">if</span> <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">a</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]]</span>
-        <span class="k">return</span> <span class="n">pos</span>
-</div>
-<div class="viewcode-block" id="Permutation.descents"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.descents">[docs]</a>    <span class="k">def</span> <span class="nf">descents</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the positions of descents in a permutation, ie, the location</span>
-<span class="sd">        where p[i] &gt; p[i+1]</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([4,0,1,3,2])</span>
-<span class="sd">        &gt;&gt;&gt; p.descents()</span>
-<span class="sd">        [0, 3]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        ascents, inversions, min, max</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="n">pos</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="k">if</span> <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">a</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]]</span>
-        <span class="k">return</span> <span class="n">pos</span>
-</div>
-<div class="viewcode-block" id="Permutation.max"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.max">[docs]</a>    <span class="k">def</span> <span class="nf">max</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The maximum element moved by the permutation.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([1,0,2,3,4])</span>
-<span class="sd">        &gt;&gt;&gt; p.max()</span>
-<span class="sd">        1</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        min, descents, ascents, inversions</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="nb">max</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">!=</span> <span class="n">i</span> <span class="ow">and</span> <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&gt;</span> <span class="nb">max</span><span class="p">:</span>
-                <span class="nb">max</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-        <span class="k">return</span> <span class="nb">max</span>
-</div>
-<div class="viewcode-block" id="Permutation.min"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.min">[docs]</a>    <span class="k">def</span> <span class="nf">min</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The minimum element moved by the permutation.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([0,1,4,3,2])</span>
-<span class="sd">        &gt;&gt;&gt; p.min()</span>
-<span class="sd">        2</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        max, descents, ascents, inversions</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="nb">min</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">!=</span> <span class="n">i</span> <span class="ow">and</span> <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&lt;</span> <span class="nb">min</span><span class="p">:</span>
-                <span class="nb">min</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-        <span class="k">return</span> <span class="nb">min</span>
-</div>
-<div class="viewcode-block" id="Permutation.inversions"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.inversions">[docs]</a>    <span class="k">def</span> <span class="nf">inversions</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes the number of inversions of a permutation.</span>
-
-<span class="sd">        An inversion is where i &gt; j but p[i] &lt; p[j].</span>
-
-<span class="sd">        For small length of p, it iterates over all i and j</span>
-<span class="sd">        values and calculates the number of inversions.</span>
-<span class="sd">        For large length of p, it uses a variation of merge</span>
-<span class="sd">        sort to calculate the number of inversions.</span>
-
-<span class="sd">        References</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        [1] http://www.cp.eng.chula.ac.th/~piak/teaching/algo/algo2008/count-inv.htm</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([0,1,2,3,4,5])</span>
-<span class="sd">        &gt;&gt;&gt; p.inversions()</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; Permutation([3,2,1,0]).inversions()</span>
-<span class="sd">        6</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        descents, ascents, min, max</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">inversions</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">130</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-                <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">a</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]:</span>
-                    <span class="k">if</span> <span class="n">b</span> <span class="o">&gt;</span> <span class="n">c</span><span class="p">:</span>
-                        <span class="n">inversions</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">k</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="n">right</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="n">arr</span> <span class="o">=</span> <span class="n">a</span><span class="p">[:]</span>
-            <span class="n">temp</span> <span class="o">=</span> <span class="n">a</span><span class="p">[:]</span>
-            <span class="k">while</span> <span class="n">k</span> <span class="o">&lt;</span> <span class="n">n</span><span class="p">:</span>
-                <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="k">while</span> <span class="n">i</span> <span class="o">+</span> <span class="n">k</span> <span class="o">&lt;</span> <span class="n">n</span><span class="p">:</span>
-                    <span class="n">right</span> <span class="o">=</span> <span class="n">i</span> <span class="o">+</span> <span class="n">k</span> <span class="o">*</span> <span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span>
-                    <span class="k">if</span> <span class="n">right</span> <span class="o">&gt;=</span> <span class="n">n</span><span class="p">:</span>
-                        <span class="n">right</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span>
-                    <span class="n">inversions</span> <span class="o">+=</span> <span class="n">_merge</span><span class="p">(</span><span class="n">arr</span><span class="p">,</span> <span class="n">temp</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="n">k</span><span class="p">,</span> <span class="n">right</span><span class="p">)</span>
-                    <span class="n">i</span> <span class="o">=</span> <span class="n">i</span> <span class="o">+</span> <span class="n">k</span> <span class="o">*</span> <span class="mi">2</span>
-                <span class="n">k</span> <span class="o">=</span> <span class="n">k</span> <span class="o">*</span> <span class="mi">2</span>
-        <span class="k">return</span> <span class="n">inversions</span>
-</div>
-<div class="viewcode-block" id="Permutation.commutator"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.commutator">[docs]</a>    <span class="k">def</span> <span class="nf">commutator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the commutator of self and x: ``~x*~self*x*self``</span>
-
-<span class="sd">        If f and g are part of a group, G, then the commutator of f and g</span>
-<span class="sd">        is the group identity iff f and g commute, i.e. fg == gf.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([0, 2, 3, 1])</span>
-<span class="sd">        &gt;&gt;&gt; x = Permutation([2, 0, 3, 1])</span>
-<span class="sd">        &gt;&gt;&gt; c = p.commutator(x); c</span>
-<span class="sd">        Permutation([2, 1, 3, 0])</span>
-<span class="sd">        &gt;&gt;&gt; c == ~x*~p*x*p</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &gt;&gt;&gt; I = Permutation(3)</span>
-<span class="sd">        &gt;&gt;&gt; p = [I + i for i in range(6)]</span>
-<span class="sd">        &gt;&gt;&gt; for i in range(len(p)):</span>
-<span class="sd">        ...     for j in range(len(p)):</span>
-<span class="sd">        ...         c = p[i].commutator(p[j])</span>
-<span class="sd">        ...         if p[i]*p[j] == p[j]*p[i]:</span>
-<span class="sd">        ...             assert c == I</span>
-<span class="sd">        ...         else:</span>
-<span class="sd">        ...             assert c != I</span>
-<span class="sd">        ...</span>
-
-<span class="sd">        References</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        http://en.wikipedia.org/wiki/Commutator</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">!=</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;The permutations must be of equal size.&quot;</span><span class="p">)</span>
-        <span class="n">inva</span> <span class="o">=</span> <span class="p">[</span><span class="bp">None</span><span class="p">]</span><span class="o">*</span><span class="n">n</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="n">inva</span><span class="p">[</span><span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span> <span class="o">=</span> <span class="n">i</span>
-        <span class="n">invb</span> <span class="o">=</span> <span class="p">[</span><span class="bp">None</span><span class="p">]</span><span class="o">*</span><span class="n">n</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="n">invb</span><span class="p">[</span><span class="n">b</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span> <span class="o">=</span> <span class="n">i</span>
-        <span class="k">return</span> <span class="n">_af_new</span><span class="p">([</span><span class="n">a</span><span class="p">[</span><span class="n">b</span><span class="p">[</span><span class="n">inva</span><span class="p">[</span><span class="n">i</span><span class="p">]]]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">invb</span><span class="p">])</span>
-</div>
-<div class="viewcode-block" id="Permutation.signature"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.signature">[docs]</a>    <span class="k">def</span> <span class="nf">signature</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Gives the signature of the permutation needed to place the</span>
-<span class="sd">        elements of the permutation in canonical order.</span>
-
-<span class="sd">        The signature is calculated as (-1)^&lt;number of inversions&gt;</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([0,1,2])</span>
-<span class="sd">        &gt;&gt;&gt; p.inversions()</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; p.signature()</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; q = Permutation([0,2,1])</span>
-<span class="sd">        &gt;&gt;&gt; q.inversions()</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; q.signature()</span>
-<span class="sd">        -1</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        inversions</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-</div>
-<div class="viewcode-block" id="Permutation.order"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.order">[docs]</a>    <span class="k">def</span> <span class="nf">order</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes the order of a permutation.</span>
-
-<span class="sd">        When the permutation is raised to the power of its</span>
-<span class="sd">        order it equals the identity permutation.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([3, 1, 5, 2, 4, 0])</span>
-<span class="sd">        &gt;&gt;&gt; p.order()</span>
-<span class="sd">        4</span>
-<span class="sd">        &gt;&gt;&gt; (p**(p.order()))</span>
-<span class="sd">        Permutation([], size=6)</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        identity, cardinality, length, rank, size</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">return</span> <span class="nb">reduce</span><span class="p">(</span><span class="n">lcm</span><span class="p">,</span> <span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="n">cycle</span><span class="p">)</span> <span class="k">for</span> <span class="n">cycle</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">cyclic_form</span><span class="p">],</span> <span class="mi">1</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Permutation.length"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.length">[docs]</a>    <span class="k">def</span> <span class="nf">length</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the number of integers moved by a permutation.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation([0, 3, 2, 1]).length()</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; Permutation([[0, 1], [2, 3]]).length()</span>
-<span class="sd">        4</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        min, max, suppport, cardinality, order, rank, size</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">support</span><span class="p">())</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Permutation.cycle_structure"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.cycle_structure">[docs]</a>    <span class="k">def</span> <span class="nf">cycle_structure</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the cycle structure of the permutation as a dictionary</span>
-<span class="sd">        indicating the multiplicity of each cycle length.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">        &gt;&gt;&gt; Permutation(3).cycle_structure</span>
-<span class="sd">        {1: 4}</span>
-<span class="sd">        &gt;&gt;&gt; Permutation(0, 4, 3)(1, 2)(5, 6).cycle_structure</span>
-<span class="sd">        {2: 2, 3: 1}</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_cycle_structure</span><span class="p">:</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_cycle_structure</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
-            <span class="n">singletons</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span>
-            <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">cyclic_form</span><span class="p">:</span>
-                <span class="n">rv</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="n">c</span><span class="p">)]</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="n">singletons</span> <span class="o">-=</span> <span class="nb">len</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">singletons</span><span class="p">:</span>
-                <span class="n">rv</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">singletons</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_cycle_structure</span> <span class="o">=</span> <span class="n">rv</span>
-        <span class="k">return</span> <span class="nb">dict</span><span class="p">(</span><span class="n">rv</span><span class="p">)</span>  <span class="c"># make a copy</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Permutation.cycles"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.cycles">[docs]</a>    <span class="k">def</span> <span class="nf">cycles</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the number of cycles contained in the permutation</span>
-<span class="sd">        (including singletons).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation([0, 1, 2]).cycles</span>
-<span class="sd">        3</span>
-<span class="sd">        &gt;&gt;&gt; Permutation([0, 1, 2]).full_cyclic_form</span>
-<span class="sd">        [[0], [1], [2]]</span>
-<span class="sd">        &gt;&gt;&gt; Permutation(0, 1)(2, 3).cycles</span>
-<span class="sd">        2</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">full_cyclic_form</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Permutation.index"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.index">[docs]</a>    <span class="k">def</span> <span class="nf">index</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the index of a permutation.</span>
-
-<span class="sd">        The index of a permutation is the sum of all subscripts j such</span>
-<span class="sd">        that p[j] is greater than p[j+1].</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([3, 0, 2, 1, 4])</span>
-<span class="sd">        &gt;&gt;&gt; p.index()</span>
-<span class="sd">        2</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span>
-
-        <span class="k">return</span> <span class="nb">sum</span><span class="p">([</span><span class="n">j</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="k">if</span> <span class="n">a</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">a</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]])</span>
-</div>
-<div class="viewcode-block" id="Permutation.runs"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.runs">[docs]</a>    <span class="k">def</span> <span class="nf">runs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the runs of a permutation.</span>
-
-<span class="sd">        An ascending sequence in a permutation is called a run [5].</span>
-
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([2,5,7,3,6,0,1,4,8])</span>
-<span class="sd">        &gt;&gt;&gt; p.runs()</span>
-<span class="sd">        [[2, 5, 7], [3, 6], [0, 1, 4, 8]]</span>
-<span class="sd">        &gt;&gt;&gt; q = Permutation([1,3,2,0])</span>
-<span class="sd">        &gt;&gt;&gt; q.runs()</span>
-<span class="sd">        [[1, 3], [2], [0]]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">runs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">array_form</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Permutation.inversion_vector"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.inversion_vector">[docs]</a>    <span class="k">def</span> <span class="nf">inversion_vector</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the inversion vector of the permutation.</span>
-
-<span class="sd">        The inversion vector consists of elements whose value</span>
-<span class="sd">        indicates the number of elements in the permutation</span>
-<span class="sd">        that are lesser than it and lie on its right hand side.</span>
-
-<span class="sd">        The inversion vector is the same as the Lehmer encoding of a</span>
-<span class="sd">        permutation.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([4, 8, 0, 7, 1, 5, 3, 6, 2])</span>
-<span class="sd">        &gt;&gt;&gt; p.inversion_vector()</span>
-<span class="sd">        [4, 7, 0, 5, 0, 2, 1, 1]</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([3, 2, 1, 0])</span>
-<span class="sd">        &gt;&gt;&gt; p.inversion_vector()</span>
-<span class="sd">        [3, 2, 1]</span>
-
-<span class="sd">        The inversion vector increases lexicographically with the rank</span>
-<span class="sd">        of the permutation, the -ith element cycling through 0..i.</span>
-
-<span class="sd">        &gt;&gt;&gt; p = Permutation(2)</span>
-<span class="sd">        &gt;&gt;&gt; while p:</span>
-<span class="sd">        ...     print(p, p.inversion_vector(), p.rank())</span>
-<span class="sd">        ...     p = p.next_lex()</span>
-<span class="sd">        ...</span>
-<span class="sd">        Permutation([0, 1, 2]) [0, 0] 0</span>
-<span class="sd">        Permutation([0, 2, 1]) [0, 1] 1</span>
-<span class="sd">        Permutation([1, 0, 2]) [1, 0] 2</span>
-<span class="sd">        Permutation([1, 2, 0]) [1, 1] 3</span>
-<span class="sd">        Permutation([2, 0, 1]) [2, 0] 4</span>
-<span class="sd">        Permutation([2, 1, 0]) [2, 1] 5</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        from_inversion_vector</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">self_array_form</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">self_array_form</span><span class="p">)</span>
-        <span class="n">inversion_vector</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">val</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">self_array_form</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">self_array_form</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                    <span class="n">val</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">inversion_vector</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">val</span>
-        <span class="k">return</span> <span class="n">inversion_vector</span>
-</div>
-<div class="viewcode-block" id="Permutation.rank_trotterjohnson"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.rank_trotterjohnson">[docs]</a>    <span class="k">def</span> <span class="nf">rank_trotterjohnson</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the Trotter Johnson rank, which we get from the minimal</span>
-<span class="sd">        change algorithm. See [4] section 2.4.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([0,1,2,3])</span>
-<span class="sd">        &gt;&gt;&gt; p.rank_trotterjohnson()</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([0,2,1,3])</span>
-<span class="sd">        &gt;&gt;&gt; p.rank_trotterjohnson()</span>
-<span class="sd">        7</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        unrank_trotterjohnson, next_trotterjohnson</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span> <span class="o">==</span> <span class="p">[]</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Identity</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">0</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span> <span class="o">==</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="mi">1</span>
-        <span class="n">perm</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span>
-        <span class="n">rank</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-            <span class="n">k</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">while</span> <span class="n">perm</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">!=</span> <span class="n">j</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">perm</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">j</span><span class="p">:</span>
-                    <span class="n">k</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">j1</span> <span class="o">=</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="n">rank</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">rank</span> <span class="o">=</span> <span class="n">j1</span><span class="o">*</span><span class="n">rank</span> <span class="o">+</span> <span class="n">j1</span> <span class="o">-</span> <span class="n">k</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">rank</span> <span class="o">=</span> <span class="n">j1</span><span class="o">*</span><span class="n">rank</span> <span class="o">+</span> <span class="n">k</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="n">rank</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Permutation.unrank_trotterjohnson"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.unrank_trotterjohnson">[docs]</a>    <span class="k">def</span> <span class="nf">unrank_trotterjohnson</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">size</span><span class="p">,</span> <span class="n">rank</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Trotter Johnson permutation unranking. See [4] section 2.4.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.unrank_trotterjohnson(5, 10)</span>
-<span class="sd">        Permutation([0, 3, 1, 2, 4])</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        rank_trotterjohnson, next_trotterjohnson</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">perm</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">size</span>
-        <span class="n">r2</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">ifac</span><span class="p">(</span><span class="n">size</span><span class="p">)</span>
-        <span class="n">pj</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">size</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">pj</span> <span class="o">*=</span> <span class="n">j</span>
-            <span class="n">r1</span> <span class="o">=</span> <span class="p">(</span><span class="n">rank</span> <span class="o">*</span> <span class="n">pj</span><span class="p">)</span> <span class="o">//</span> <span class="n">n</span>
-            <span class="n">k</span> <span class="o">=</span> <span class="n">r1</span> <span class="o">-</span> <span class="n">j</span><span class="o">*</span><span class="n">r2</span>
-            <span class="k">if</span> <span class="n">r2</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j</span> <span class="o">-</span> <span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-                    <span class="n">perm</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">perm</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-                <span class="n">perm</span><span class="p">[</span><span class="n">j</span> <span class="o">-</span> <span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">j</span> <span class="o">-</span> <span class="mi">1</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-                    <span class="n">perm</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">perm</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-                <span class="n">perm</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">j</span> <span class="o">-</span> <span class="mi">1</span>
-            <span class="n">r2</span> <span class="o">=</span> <span class="n">r1</span>
-        <span class="k">return</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">perm</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Permutation.next_trotterjohnson"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.next_trotterjohnson">[docs]</a>    <span class="k">def</span> <span class="nf">next_trotterjohnson</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the next permutation in Trotter-Johnson order.</span>
-<span class="sd">        If self is the last permutation it returns None.</span>
-<span class="sd">        See [4] section 2.4.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([3, 0, 2, 1])</span>
-<span class="sd">        &gt;&gt;&gt; p.rank_trotterjohnson()</span>
-<span class="sd">        4</span>
-<span class="sd">        &gt;&gt;&gt; p = p.next_trotterjohnson(); p</span>
-<span class="sd">        Permutation([0, 3, 2, 1])</span>
-<span class="sd">        &gt;&gt;&gt; p.rank_trotterjohnson()</span>
-<span class="sd">        5</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        rank_trotterjohnson, unrank_trotterjohnson</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">pi</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span><span class="p">[:]</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-        <span class="n">st</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">rho</span> <span class="o">=</span> <span class="n">pi</span><span class="p">[:]</span>
-        <span class="n">done</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span>
-        <span class="k">while</span> <span class="n">m</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">done</span><span class="p">:</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="n">rho</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-                <span class="n">rho</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">rho</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
-            <span class="n">par</span> <span class="o">=</span> <span class="n">_af_parity</span><span class="p">(</span><span class="n">rho</span><span class="p">[:</span><span class="n">m</span><span class="p">])</span>
-            <span class="k">if</span> <span class="n">par</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">d</span> <span class="o">==</span> <span class="n">m</span><span class="p">:</span>
-                    <span class="n">m</span> <span class="o">-=</span> <span class="mi">1</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">pi</span><span class="p">[</span><span class="n">st</span> <span class="o">+</span> <span class="n">d</span><span class="p">],</span> <span class="n">pi</span><span class="p">[</span><span class="n">st</span> <span class="o">+</span> <span class="n">d</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">pi</span><span class="p">[</span><span class="n">st</span> <span class="o">+</span> <span class="n">d</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">pi</span><span class="p">[</span><span class="n">st</span> <span class="o">+</span> <span class="n">d</span><span class="p">]</span>
-                    <span class="n">done</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">d</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">m</span> <span class="o">-=</span> <span class="mi">1</span>
-                    <span class="n">st</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">pi</span><span class="p">[</span><span class="n">st</span> <span class="o">+</span> <span class="n">d</span><span class="p">],</span> <span class="n">pi</span><span class="p">[</span><span class="n">st</span> <span class="o">+</span> <span class="n">d</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">pi</span><span class="p">[</span><span class="n">st</span> <span class="o">+</span> <span class="n">d</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">pi</span><span class="p">[</span><span class="n">st</span> <span class="o">+</span> <span class="n">d</span><span class="p">]</span>
-                    <span class="n">done</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">return</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Permutation.get_precedence_matrix"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.get_precedence_matrix">[docs]</a>    <span class="k">def</span> <span class="nf">get_precedence_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Gets the precedence matrix. This is used for computing the</span>
-<span class="sd">        distance between two permutations.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation.josephus(3,6,1)</span>
-<span class="sd">        &gt;&gt;&gt; p</span>
-<span class="sd">        Permutation([2, 5, 3, 1, 4, 0])</span>
-<span class="sd">        &gt;&gt;&gt; p.get_precedence_matrix()</span>
-<span class="sd">        [0, 0, 0, 0, 0, 0]</span>
-<span class="sd">        [1, 0, 0, 0, 1, 0]</span>
-<span class="sd">        [1, 1, 0, 1, 1, 1]</span>
-<span class="sd">        [1, 1, 0, 0, 1, 0]</span>
-<span class="sd">        [1, 0, 0, 0, 0, 0]</span>
-<span class="sd">        [1, 1, 0, 1, 1, 0]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        get_precedence_distance, get_adjacency_matrix, get_adjacency_distance</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">)</span>
-        <span class="n">perm</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">m</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-                <span class="n">m</span><span class="p">[</span><span class="n">perm</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">perm</span><span class="p">[</span><span class="n">j</span><span class="p">]]</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="n">m</span>
-</div>
-<div class="viewcode-block" id="Permutation.get_precedence_distance"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.get_precedence_distance">[docs]</a>    <span class="k">def</span> <span class="nf">get_precedence_distance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes the precedence distance between two permutations.</span>
-
-<span class="sd">        Suppose p and p&#39; represent n jobs. The precedence metric</span>
-<span class="sd">        counts the number of times a job j is prededed by job i</span>
-<span class="sd">        in both p and p&#39;. This metric is commutative.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([2, 0, 4, 3, 1])</span>
-<span class="sd">        &gt;&gt;&gt; q = Permutation([3, 1, 2, 4, 0])</span>
-<span class="sd">        &gt;&gt;&gt; p.get_precedence_distance(q)</span>
-<span class="sd">        7</span>
-<span class="sd">        &gt;&gt;&gt; q.get_precedence_distance(p)</span>
-<span class="sd">        7</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        get_precedence_matrix, get_adjacency_matrix, get_adjacency_distance</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">!=</span> <span class="n">other</span><span class="o">.</span><span class="n">size</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;The permutations must be of equal size.&quot;</span><span class="p">)</span>
-        <span class="n">self_prec_mat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_precedence_matrix</span><span class="p">()</span>
-        <span class="n">other_prec_mat</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">get_precedence_matrix</span><span class="p">()</span>
-        <span class="n">n_prec</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
-                    <span class="k">continue</span>
-                <span class="k">if</span> <span class="n">self_prec_mat</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">*</span> <span class="n">other_prec_mat</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">n_prec</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span> <span class="o">-</span> <span class="n">n_prec</span>
-        <span class="k">return</span> <span class="n">d</span>
-</div>
-<div class="viewcode-block" id="Permutation.get_adjacency_matrix"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.get_adjacency_matrix">[docs]</a>    <span class="k">def</span> <span class="nf">get_adjacency_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes the adjacency matrix of a permutation.</span>
-
-<span class="sd">        If job i is adjacent to job j in a permutation p</span>
-<span class="sd">        then we set m[i, j] = 1 where m is the adjacency</span>
-<span class="sd">        matrix of p.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation.josephus(3,6,1)</span>
-<span class="sd">        &gt;&gt;&gt; p.get_adjacency_matrix()</span>
-<span class="sd">        [0, 0, 0, 0, 0, 0]</span>
-<span class="sd">        [0, 0, 0, 0, 1, 0]</span>
-<span class="sd">        [0, 0, 0, 0, 0, 1]</span>
-<span class="sd">        [0, 1, 0, 0, 0, 0]</span>
-<span class="sd">        [1, 0, 0, 0, 0, 0]</span>
-<span class="sd">        [0, 0, 0, 1, 0, 0]</span>
-
-<span class="sd">        &gt;&gt;&gt; q = Permutation([0, 1, 2, 3])</span>
-<span class="sd">        &gt;&gt;&gt; q.get_adjacency_matrix()</span>
-<span class="sd">        [0, 1, 0, 0]</span>
-<span class="sd">        [0, 0, 1, 0]</span>
-<span class="sd">        [0, 0, 0, 1]</span>
-<span class="sd">        [0, 0, 0, 0]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        get_precedence_matrix, get_precedence_distance, get_adjacency_distance</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">)</span>
-        <span class="n">perm</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">m</span><span class="p">[</span><span class="n">perm</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">perm</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]]</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="n">m</span>
-</div>
-<div class="viewcode-block" id="Permutation.get_adjacency_distance"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.get_adjacency_distance">[docs]</a>    <span class="k">def</span> <span class="nf">get_adjacency_distance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes the adjacency distance between two permutations.</span>
-
-<span class="sd">        This metric counts the number of times a pair i,j of jobs is</span>
-<span class="sd">        adjacent in both p and p&#39;. If n_adj is this quantity then</span>
-<span class="sd">        the adjacency distance is n - n_adj - 1 [1]</span>
-
-<span class="sd">        [1] Reeves, Colin R. Landscapes, Operators and Heuristic search, Annals</span>
-<span class="sd">        of Operational Research, 86, pp 473-490. (1999)</span>
-
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([0, 3, 1, 2, 4])</span>
-<span class="sd">        &gt;&gt;&gt; q = Permutation.josephus(4, 5, 2)</span>
-<span class="sd">        &gt;&gt;&gt; p.get_adjacency_distance(q)</span>
-<span class="sd">        3</span>
-<span class="sd">        &gt;&gt;&gt; r = Permutation([0, 2, 1, 4, 3])</span>
-<span class="sd">        &gt;&gt;&gt; p.get_adjacency_distance(r)</span>
-<span class="sd">        4</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        get_precedence_matrix, get_precedence_distance, get_adjacency_matrix</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">!=</span> <span class="n">other</span><span class="o">.</span><span class="n">size</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;The permutations must be of the same size.&quot;</span><span class="p">)</span>
-        <span class="n">self_adj_mat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_adjacency_matrix</span><span class="p">()</span>
-        <span class="n">other_adj_mat</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">get_adjacency_matrix</span><span class="p">()</span>
-        <span class="n">n_adj</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
-                    <span class="k">continue</span>
-                <span class="k">if</span> <span class="n">self_adj_mat</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">*</span> <span class="n">other_adj_mat</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">n_adj</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">-</span> <span class="n">n_adj</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="n">d</span>
-</div>
-<div class="viewcode-block" id="Permutation.get_positional_distance"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.get_positional_distance">[docs]</a>    <span class="k">def</span> <span class="nf">get_positional_distance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes the positional distance between two permutations.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([0, 3, 1, 2, 4])</span>
-<span class="sd">        &gt;&gt;&gt; q = Permutation.josephus(4, 5, 2)</span>
-<span class="sd">        &gt;&gt;&gt; r = Permutation([3, 1, 4, 0, 2])</span>
-<span class="sd">        &gt;&gt;&gt; p.get_positional_distance(q)</span>
-<span class="sd">        12</span>
-<span class="sd">        &gt;&gt;&gt; p.get_positional_distance(r)</span>
-<span class="sd">        12</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        get_precedence_distance, get_adjacency_distance</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">b</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;The permutations must be of the same size.&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="nb">sum</span><span class="p">([</span><span class="nb">abs</span><span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">b</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">))])</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Permutation.josephus"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.josephus">[docs]</a>    <span class="k">def</span> <span class="nf">josephus</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return as a permutation the shuffling of range(n) using the Josephus</span>
-<span class="sd">        scheme in which every m-th item is selected until all have been chosen.</span>
-<span class="sd">        The returned permutation has elements listed by the order in which they</span>
-<span class="sd">        were selected.</span>
-
-<span class="sd">        The parameter ``s`` stops the selection process when there are ``s``</span>
-<span class="sd">        items remaining and these are selected by countinuing the selection,</span>
-<span class="sd">        counting by 1 rather than by ``m``.</span>
-
-<span class="sd">        Consider selecting every 3rd item from 6 until only 2 remain::</span>
-
-<span class="sd">            choices    chosen</span>
-<span class="sd">            ========   ======</span>
-<span class="sd">              012345</span>
-<span class="sd">              01 345   2</span>
-<span class="sd">              01 34    25</span>
-<span class="sd">              01  4    253</span>
-<span class="sd">              0   4    2531</span>
-<span class="sd">              0        25314</span>
-<span class="sd">                       253140</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.josephus(3, 6, 2).array_form</span>
-<span class="sd">        [2, 5, 3, 1, 4, 0]</span>
-
-<span class="sd">        References</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        1. http://en.wikipedia.org/wiki/Flavius_Josephus</span>
-<span class="sd">        2. http://en.wikipedia.org/wiki/Josephus_problem</span>
-<span class="sd">        3. http://www.wou.edu/~burtonl/josephus.html</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">deque</span>
-        <span class="n">m</span> <span class="o">-=</span> <span class="mi">1</span>
-        <span class="n">Q</span> <span class="o">=</span> <span class="n">deque</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)))</span>
-        <span class="n">perm</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">while</span> <span class="nb">len</span><span class="p">(</span><span class="n">Q</span><span class="p">)</span> <span class="o">&gt;</span> <span class="nb">max</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">dp</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
-                <span class="n">Q</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">popleft</span><span class="p">())</span>
-            <span class="n">perm</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">popleft</span><span class="p">())</span>
-        <span class="n">perm</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">Q</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">Perm</span><span class="p">(</span><span class="n">perm</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Permutation.from_inversion_vector"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.from_inversion_vector">[docs]</a>    <span class="k">def</span> <span class="nf">from_inversion_vector</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">inversion</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Calculates the permutation from the inversion vector.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.from_inversion_vector([3, 2, 1, 0, 0])</span>
-<span class="sd">        Permutation([3, 2, 1, 0, 4, 5])</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">size</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">inversion</span><span class="p">)</span>
-        <span class="n">N</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">size</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-        <span class="n">perm</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">):</span>
-                <span class="n">val</span> <span class="o">=</span> <span class="n">N</span><span class="p">[</span><span class="n">inversion</span><span class="p">[</span><span class="n">k</span><span class="p">]]</span>
-                <span class="n">perm</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
-                <span class="n">N</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;The inversion vector is not valid.&quot;</span><span class="p">)</span>
-        <span class="n">perm</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">perm</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Permutation.random"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.random">[docs]</a>    <span class="k">def</span> <span class="nf">random</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Generates a random permutation of length ``n``.</span>
-
-<span class="sd">        Uses the underlying Python psuedo-random number generator.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.random(2) in (Permutation([1, 0]), Permutation([0, 1]))</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">perm_array</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-        <span class="n">random</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="n">perm_array</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">perm_array</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Permutation.unrank_lex"><a class="viewcode-back" href="../../../modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.unrank_lex">[docs]</a>    <span class="k">def</span> <span class="nf">unrank_lex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">size</span><span class="p">,</span> <span class="n">rank</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Lexicographic permutation unranking.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-<span class="sd">        &gt;&gt;&gt; a = Permutation.unrank_lex(5, 10)</span>
-<span class="sd">        &gt;&gt;&gt; a.rank()</span>
-<span class="sd">        10</span>
-<span class="sd">        &gt;&gt;&gt; a</span>
-<span class="sd">        Permutation([0, 2, 4, 1, 3])</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        rank, next_lex</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">perm_array</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">size</span>
-        <span class="n">psize</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">):</span>
-            <span class="n">new_psize</span> <span class="o">=</span> <span class="n">psize</span><span class="o">*</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="p">(</span><span class="n">rank</span> <span class="o">%</span> <span class="n">new_psize</span><span class="p">)</span> <span class="o">//</span> <span class="n">psize</span>
-            <span class="n">rank</span> <span class="o">-=</span> <span class="n">d</span><span class="o">*</span><span class="n">psize</span>
-            <span class="n">perm_array</span><span class="p">[</span><span class="n">size</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">d</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span> <span class="o">-</span> <span class="n">i</span><span class="p">,</span> <span class="n">size</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">perm_array</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">d</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">perm_array</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">psize</span> <span class="o">=</span> <span class="n">new_psize</span>
-        <span class="k">return</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">perm_array</span><span class="p">)</span>
-
-    <span class="c"># global flag to control how permutations are printed</span>
-    <span class="c"># when True, Permutation([0, 2, 1, 3]) -&gt; Cycle(1, 2)</span>
-    <span class="c"># when False, Permutation([0, 2, 1, 3]) -&gt; Permutation([0, 2, 1])</span></div>
-    <span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_merge</span><span class="p">(</span><span class="n">arr</span><span class="p">,</span> <span class="n">temp</span><span class="p">,</span> <span class="n">left</span><span class="p">,</span> <span class="n">mid</span><span class="p">,</span> <span class="n">right</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Merges two sorted arrays and calculates the inversion count.</span>
-
-<span class="sd">    Helper function for calculating inversions. This method is</span>
-<span class="sd">    for internal use only.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">i</span> <span class="o">=</span> <span class="n">k</span> <span class="o">=</span> <span class="n">left</span>
-    <span class="n">j</span> <span class="o">=</span> <span class="n">mid</span>
-    <span class="n">inv_count</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">mid</span> <span class="ow">and</span> <span class="n">j</span> <span class="o">&lt;=</span> <span class="n">right</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">arr</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">arr</span><span class="p">[</span><span class="n">j</span><span class="p">]:</span>
-            <span class="n">temp</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">arr</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="n">k</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">temp</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">arr</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-            <span class="n">k</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">j</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">inv_count</span> <span class="o">+=</span> <span class="p">(</span><span class="n">mid</span> <span class="o">-</span><span class="n">i</span><span class="p">)</span>
-    <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">mid</span><span class="p">:</span>
-        <span class="n">temp</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">arr</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-        <span class="n">k</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-    <span class="k">if</span> <span class="n">j</span> <span class="o">&lt;=</span> <span class="n">right</span><span class="p">:</span>
-        <span class="n">k</span> <span class="o">+=</span> <span class="n">right</span> <span class="o">-</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="n">j</span> <span class="o">+=</span> <span class="n">right</span> <span class="o">-</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="n">arr</span><span class="p">[</span><span class="n">left</span><span class="p">:</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">temp</span><span class="p">[</span><span class="n">left</span><span class="p">:</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">arr</span><span class="p">[</span><span class="n">left</span><span class="p">:</span><span class="n">right</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">temp</span><span class="p">[</span><span class="n">left</span><span class="p">:</span><span class="n">right</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">inv_count</span>
-
-<span class="n">Perm</span> <span class="o">=</span> <span class="n">Permutation</span>
-<span class="n">_af_new</span> <span class="o">=</span> <span class="n">Perm</span><span class="o">.</span><span class="n">_af_new</span>
-</pre></div>
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-  <h1>Source code for sympy.combinatorics.polyhedron</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">,</span> <span class="n">FiniteSet</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">as_int</span>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span> <span class="k">as</span> <span class="n">Perm</span>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="p">(</span><span class="n">minlex</span><span class="p">,</span> <span class="n">unflatten</span><span class="p">,</span> <span class="n">flatten</span><span class="p">)</span>
-
-<span class="n">rmul</span> <span class="o">=</span> <span class="n">Perm</span><span class="o">.</span><span class="n">rmul</span>
-
-
-<div class="viewcode-block" id="Polyhedron"><a class="viewcode-back" href="../../../modules/combinatorics/polyhedron.html#sympy.combinatorics.polyhedron.Polyhedron">[docs]</a><span class="k">class</span> <span class="nc">Polyhedron</span><span class="p">(</span><span class="n">Basic</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Represents the polyhedral symmetry group (PSG).</span>
-
-<span class="sd">    The PSG is one of the symmetry groups of the Platonic solids.</span>
-<span class="sd">    There are three polyhedral groups: the tetrahedral group</span>
-<span class="sd">    of order 12, the octahedral group of order 24, and the</span>
-<span class="sd">    icosahedral group of order 60.</span>
-
-<span class="sd">    All doctests have been given in the docstring of the</span>
-<span class="sd">    constructor of the object.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    http://mathworld.wolfram.com/PolyhedralGroup.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">_edges</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">corners</span><span class="p">,</span> <span class="n">faces</span><span class="o">=</span><span class="p">[],</span> <span class="n">pgroup</span><span class="o">=</span><span class="p">[]):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The constructor of the Polyhedron group object.</span>
-
-<span class="sd">        It takes up to three parameters: the corners, faces, and</span>
-<span class="sd">        allowed transformations.</span>
-
-<span class="sd">        The corners/vertices are entered as a list of arbitrary</span>
-<span class="sd">        expressions that are used to identify each vertex.</span>
-
-<span class="sd">        The faces are entered as a list of tuples of indices; a tuple</span>
-<span class="sd">        of indices identifies the vertices which define the face. They</span>
-<span class="sd">        should be entered in a cw or ccw order; they will be standardized</span>
-<span class="sd">        by reversal and rotation to be give the lowest lexical ordering.</span>
-<span class="sd">        If no faces are given then no edges will be computed.</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy.combinatorics.polyhedron import Polyhedron</span>
-<span class="sd">            &gt;&gt;&gt; Polyhedron(list(&#39;abc&#39;), [(1, 2, 0)]).faces</span>
-<span class="sd">            {(0, 1, 2)}</span>
-<span class="sd">            &gt;&gt;&gt; Polyhedron(list(&#39;abc&#39;), [(1, 0, 2)]).faces</span>
-<span class="sd">            {(0, 1, 2)}</span>
-
-<span class="sd">        The allowed transformations are entered as allowable permutations</span>
-<span class="sd">        of the vertices for the polyhedron. Instance of Permutations</span>
-<span class="sd">        (as with faces) should refer to the supplied vertices by index.</span>
-<span class="sd">        These permutation are stored as a PermutationGroup.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">        &gt;&gt;&gt; Permutation.print_cyclic = False</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import w, x, y, z</span>
-
-<span class="sd">        Here we construct the Polyhedron object for a tetrahedron.</span>
-
-<span class="sd">        &gt;&gt;&gt; corners = [w, x, y, z]</span>
-<span class="sd">        &gt;&gt;&gt; faces = [(0,1,2), (0,2,3), (0,3,1), (1,2,3)]</span>
-
-<span class="sd">        Next, allowed transformations of the polyhedron must be given. This</span>
-<span class="sd">        is given as permutations of vertices.</span>
-
-<span class="sd">        Although the vertices of a tetrahedron can be numbered in 24 (4!)</span>
-<span class="sd">        different ways, there are only 12 different orientations for a</span>
-<span class="sd">        physical tetrahedron. The following permutations, applied once or</span>
-<span class="sd">        twice, will generate all 12 of the orientations. (The identity</span>
-<span class="sd">        permutation, Permutation(range(4)), is not included since it does</span>
-<span class="sd">        not change the orientation of the vertices.)</span>
-
-<span class="sd">        &gt;&gt;&gt; pgroup = [Permutation([[0,1,2], [3]]), \</span>
-<span class="sd">                      Permutation([[0,1,3], [2]]), \</span>
-<span class="sd">                      Permutation([[0,2,3], [1]]), \</span>
-<span class="sd">                      Permutation([[1,2,3], [0]]), \</span>
-<span class="sd">                      Permutation([[0,1], [2,3]]), \</span>
-<span class="sd">                      Permutation([[0,2], [1,3]]), \</span>
-<span class="sd">                      Permutation([[0,3], [1,2]])]</span>
-
-<span class="sd">        The Polyhedron is now constructed and demonstrated:</span>
-
-<span class="sd">        &gt;&gt;&gt; tetra = Polyhedron(corners, faces, pgroup)</span>
-<span class="sd">        &gt;&gt;&gt; tetra.size</span>
-<span class="sd">        4</span>
-<span class="sd">        &gt;&gt;&gt; tetra.edges</span>
-<span class="sd">        {(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)}</span>
-<span class="sd">        &gt;&gt;&gt; tetra.corners</span>
-<span class="sd">        (w, x, y, z)</span>
-
-<span class="sd">        It can be rotated with an arbitrary permutation of vertices, e.g.</span>
-<span class="sd">        the following permutation is not in the pgroup:</span>
-
-<span class="sd">        &gt;&gt;&gt; tetra.rotate(Permutation([0, 1, 3, 2]))</span>
-<span class="sd">        &gt;&gt;&gt; tetra.corners</span>
-<span class="sd">        (w, x, z, y)</span>
-
-<span class="sd">        An allowed permutation of the vertices can be constructed by</span>
-<span class="sd">        repeatedly applying permutations from the pgroup to the vertices.</span>
-<span class="sd">        Here is a demonstration that applying p and p**2 for every p in</span>
-<span class="sd">        pgroup generates all the orientations of a tetrahedron and no others:</span>
-
-<span class="sd">        &gt;&gt;&gt; all = ( (w, x, y, z), \</span>
-<span class="sd">                    (x, y, w, z), \</span>
-<span class="sd">                    (y, w, x, z), \</span>
-<span class="sd">                    (w, z, x, y), \</span>
-<span class="sd">                    (z, w, y, x), \</span>
-<span class="sd">                    (w, y, z, x), \</span>
-<span class="sd">                    (y, z, w, x), \</span>
-<span class="sd">                    (x, z, y, w), \</span>
-<span class="sd">                    (z, y, x, w), \</span>
-<span class="sd">                    (y, x, z, w), \</span>
-<span class="sd">                    (x, w, z, y), \</span>
-<span class="sd">                    (z, x, w, y) )</span>
-
-<span class="sd">        &gt;&gt;&gt; got = []</span>
-<span class="sd">        &gt;&gt;&gt; for p in (pgroup + [p**2 for p in pgroup]):</span>
-<span class="sd">        ...     h = Polyhedron(corners)</span>
-<span class="sd">        ...     h.rotate(p)</span>
-<span class="sd">        ...     got.append(h.corners)</span>
-<span class="sd">        ...</span>
-<span class="sd">        &gt;&gt;&gt; set(got) == set(all)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        The make_perm method of a PermutationGroup will randomly pick</span>
-<span class="sd">        permutations, multiply them together, and return the permutation that</span>
-<span class="sd">        can be applied to the polyhedron to give the orientation produced</span>
-<span class="sd">        by those individual permutations.</span>
-
-<span class="sd">        Here, 3 permutations are used:</span>
-
-<span class="sd">        &gt;&gt;&gt; tetra.pgroup.make_perm(3) # doctest: +SKIP</span>
-<span class="sd">        Permutation([0, 3, 1, 2])</span>
-
-<span class="sd">        To select the permutations that should be used, supply a list</span>
-<span class="sd">        of indices to the permutations in pgroup in the order they should</span>
-<span class="sd">        be applied:</span>
-
-<span class="sd">        &gt;&gt;&gt; use = [0, 0, 2]</span>
-<span class="sd">        &gt;&gt;&gt; p002 = tetra.pgroup.make_perm(3, use)</span>
-<span class="sd">        &gt;&gt;&gt; p002</span>
-<span class="sd">        Permutation([1, 0, 3, 2])</span>
-
-
-<span class="sd">        Apply them one at a time:</span>
-
-<span class="sd">        &gt;&gt;&gt; tetra.reset()</span>
-<span class="sd">        &gt;&gt;&gt; for i in use:</span>
-<span class="sd">        ...     tetra.rotate(pgroup[i])</span>
-<span class="sd">        ...</span>
-<span class="sd">        &gt;&gt;&gt; tetra.vertices</span>
-<span class="sd">        (x, w, z, y)</span>
-<span class="sd">        &gt;&gt;&gt; sequentially = tetra.vertices</span>
-
-<span class="sd">        Apply the composite permutation:</span>
-
-<span class="sd">        &gt;&gt;&gt; tetra.reset()</span>
-<span class="sd">        &gt;&gt;&gt; tetra.rotate(p002)</span>
-<span class="sd">        &gt;&gt;&gt; tetra.corners</span>
-<span class="sd">        (x, w, z, y)</span>
-<span class="sd">        &gt;&gt;&gt; tetra.corners in all and tetra.corners == sequentially</span>
-<span class="sd">        True</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        Defining permutation groups</span>
-<span class="sd">        ---------------------------</span>
-
-<span class="sd">        It is not necessary to enter any permutations, nor is necessary to</span>
-<span class="sd">        enter a complete set of transforations. In fact, for a polyhedron,</span>
-<span class="sd">        all configurations can be constructed from just two permutations.</span>
-<span class="sd">        For example, the orientations of a tetrahedron can be generated from</span>
-<span class="sd">        an axis passing through a vertex and face and another axis passing</span>
-<span class="sd">        through a different vertex or from an axis passing through the</span>
-<span class="sd">        midpoints of two edges opposite of each other.</span>
-
-<span class="sd">        For simplicity of presentation, consider a square --</span>
-<span class="sd">        not a cube -- with vertices 1, 2, 3, and 4:</span>
-
-<span class="sd">        1-----2  We could think of axes of rotation being:</span>
-<span class="sd">        |     |  1) through the face</span>
-<span class="sd">        |     |  2) from midpoint 1-2 to 3-4 or 1-3 to 2-4</span>
-<span class="sd">        3-----4  3) lines 1-4 or 2-3</span>
-
-
-<span class="sd">        To determine how to write the permutations, imagine 4 cameras,</span>
-<span class="sd">        one at each corner, labeled A-D:</span>
-
-<span class="sd">        A       B          A       B</span>
-<span class="sd">         1-----2            1-----3             vertex index:</span>
-<span class="sd">         |     |            |     |                 1   0</span>
-<span class="sd">         |     |            |     |                 2   1</span>
-<span class="sd">         3-----4            2-----4                 3   2</span>
-<span class="sd">        C       D          C       D                4   3</span>
-
-<span class="sd">        original           after rotation</span>
-<span class="sd">                           along 1-4</span>
-
-<span class="sd">        A diagonal and a face axis will be chosen for the &quot;permutation group&quot;</span>
-<span class="sd">        from which any orientation can be constructed.</span>
-
-<span class="sd">        &gt;&gt;&gt; pgroup = []</span>
-
-<span class="sd">        Imagine a clockwise rotation when viewing 1-4 from camera A. The new</span>
-<span class="sd">        orientation is (in camera-order): 1, 3, 2, 4 so the permutation is</span>
-<span class="sd">        given using the *indices* of the vertices as:</span>
-
-<span class="sd">        &gt;&gt;&gt; pgroup.append(Permutation((0, 2, 1, 3)))</span>
-
-<span class="sd">        Now imagine rotating clockwise when looking down an axis entering the</span>
-<span class="sd">        center of the square as viewed. The new camera-order would be</span>
-<span class="sd">        3, 1, 4, 2 so the permutation is (using indices):</span>
-
-<span class="sd">        &gt;&gt;&gt; pgroup.append(Permutation((2, 0, 3, 1)))</span>
-
-<span class="sd">        The square can now be constructed:</span>
-<span class="sd">            ** use real-world labels for the vertices, entering them in</span>
-<span class="sd">               camera order</span>
-<span class="sd">            ** for the faces we use zero-based indices of the vertices</span>
-<span class="sd">               in *edge-order* as the face is traversed; neither the</span>
-<span class="sd">               direction nor the starting point matter -- the faces are</span>
-<span class="sd">               only used to define edges (if so desired).</span>
-
-<span class="sd">        &gt;&gt;&gt; square = Polyhedron((1, 2, 3, 4), [(0, 1, 3, 2)], pgroup)</span>
-
-<span class="sd">        To rotate the square with a single permutation we can do:</span>
-
-<span class="sd">        &gt;&gt;&gt; square.rotate(square.pgroup[0]); square.corners</span>
-<span class="sd">        (1, 3, 2, 4)</span>
-
-<span class="sd">        To use more than one permutation (or to use one permutation more</span>
-<span class="sd">        than once) it is more convenient to use the make_perm method:</span>
-
-<span class="sd">        &gt;&gt;&gt; p011 = square.pgroup.make_perm([0,1,1]) # diag flip + 2 rotations</span>
-<span class="sd">        &gt;&gt;&gt; square.reset() # return to initial orientation</span>
-<span class="sd">        &gt;&gt;&gt; square.rotate(p011); square.corners</span>
-<span class="sd">        (4, 2, 3, 1)</span>
-
-<span class="sd">        Thinking outside the box</span>
-<span class="sd">        ------------------------</span>
-
-<span class="sd">        Although the Polyhedron object has a direct physical meaning, it</span>
-<span class="sd">        actually has broader application. In the most general sense it is</span>
-<span class="sd">        just a decorated PermutationGroup, allowing one to connect the</span>
-<span class="sd">        permutations to something physical. For example, a Rubik&#39;s cube is</span>
-<span class="sd">        not a proper polyhedron, but the Polyhedron class can be used to</span>
-<span class="sd">        represent it in a way that helps to visualize the Rubik&#39;s cube.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.utilities.iterables import flatten, unflatten</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import RubikGroup</span>
-<span class="sd">        &gt;&gt;&gt; facelets = flatten([symbols(s+&#39;1:5&#39;) for s in &#39;UFRBLD&#39;])</span>
-<span class="sd">        &gt;&gt;&gt; def show():</span>
-<span class="sd">        ...     pairs = unflatten(r2.corners, 2)</span>
-<span class="sd">        ...     print(pairs[::2])</span>
-<span class="sd">        ...     print(pairs[1::2])</span>
-<span class="sd">        ...</span>
-<span class="sd">        &gt;&gt;&gt; r2 = Polyhedron(facelets, pgroup=RubikGroup(2))</span>
-<span class="sd">        &gt;&gt;&gt; show()</span>
-<span class="sd">        [(U1, U2), (F1, F2), (R1, R2), (B1, B2), (L1, L2), (D1, D2)]</span>
-<span class="sd">        [(U3, U4), (F3, F4), (R3, R4), (B3, B4), (L3, L4), (D3, D4)]</span>
-<span class="sd">        &gt;&gt;&gt; r2.rotate(0) # cw rotation of F</span>
-<span class="sd">        &gt;&gt;&gt; show()</span>
-<span class="sd">        [(U1, U2), (F3, F1), (U3, R2), (B1, B2), (L1, D1), (R3, R1)]</span>
-<span class="sd">        [(L4, L2), (F4, F2), (U4, R4), (B3, B4), (L3, D2), (D3, D4)]</span>
-
-<span class="sd">        Predefined Polyhedra</span>
-<span class="sd">        ====================</span>
-
-<span class="sd">        For convenience, the vertices and faces are defined for the following</span>
-<span class="sd">        standard solids along with a permutation group for transformations.</span>
-<span class="sd">        When the polyhedron is oriented as indicated below, the vertices in</span>
-<span class="sd">        a given horizontal plane are numbered in ccw direction, starting from</span>
-<span class="sd">        the vertex that will give the lowest indices in a given face. (In the</span>
-<span class="sd">        net of the vertices, indices preceded by &quot;-&quot; indicate replication of</span>
-<span class="sd">        the lhs index in the net.)</span>
-
-<span class="sd">        tetrahedron, tetrahedron_faces</span>
-<span class="sd">        ------------------------------</span>
-
-<span class="sd">            4 vertices (vertex up) net:</span>
-
-<span class="sd">                 0 0-0</span>
-<span class="sd">                1 2 3-1</span>
-
-<span class="sd">            4 faces:</span>
-
-<span class="sd">            (0,1,2) (0,2,3) (0,3,1) (1,2,3)</span>
-
-<span class="sd">        cube, cube_faces</span>
-<span class="sd">        ----------------</span>
-
-<span class="sd">            8 vertices (face up) net:</span>
-
-<span class="sd">                0 1 2 3-0</span>
-<span class="sd">                4 5 6 7-4</span>
-
-<span class="sd">            6 faces:</span>
-
-<span class="sd">            (0,1,2,3)</span>
-<span class="sd">            (0,1,5,4) (1,2,6,5) (2,3,7,6) (0,3,7,4)</span>
-<span class="sd">            (4,5,6,7)</span>
-
-<span class="sd">        octahedron, octahedron_faces</span>
-<span class="sd">        ----------------------------</span>
-
-<span class="sd">            6 vertices (vertex up) net:</span>
-
-<span class="sd">                 0 0 0-0</span>
-<span class="sd">                1 2 3 4-1</span>
-<span class="sd">                 5 5 5-5</span>
-
-<span class="sd">            8 faces:</span>
-
-<span class="sd">            (0,1,2) (0,2,3) (0,3,4) (0,1,4)</span>
-<span class="sd">            (1,2,5) (2,3,5) (3,4,5) (1,4,5)</span>
-
-<span class="sd">        dodecahedron, dodecahedron_faces</span>
-<span class="sd">        --------------------------------</span>
-
-<span class="sd">            20 vertices (vertex up) net:</span>
-
-<span class="sd">                  0  1  2  3  4 -0</span>
-<span class="sd">                  5  6  7  8  9 -5</span>
-<span class="sd">                14 10 11 12 13-14</span>
-<span class="sd">                15 16 17 18 19-15</span>
-
-<span class="sd">            12 faces:</span>
-
-<span class="sd">            (0,1,2,3,4)</span>
-<span class="sd">            (0,1,6,10,5) (1,2,7,11,6) (2,3,8,12,7) (3,4,9,13,8) (0,4,9,14,5)</span>
-<span class="sd">            (5,10,16,15,14) (</span>
-<span class="sd">                6,10,16,17,11) (7,11,17,18,12) (8,12,18,19,13) (9,13,19,15,14)</span>
-<span class="sd">            (15,16,17,18,19)</span>
-
-<span class="sd">        icosahedron, icosahedron_faces</span>
-<span class="sd">        ------------------------------</span>
-
-<span class="sd">            12 vertices (face up) net:</span>
-
-<span class="sd">                 0  0  0  0 -0</span>
-<span class="sd">                1  2  3  4  5 -1</span>
-<span class="sd">                 6  7  8  9  10 -6</span>
-<span class="sd">                  11 11 11 11 -11</span>
-
-<span class="sd">            20 faces:</span>
-
-<span class="sd">            (0,1,2) (0,2,3) (0,3,4) (0,4,5) (0,1,5)</span>
-<span class="sd">            (1,2,6) (2,3,7) (3,4,8) (4,5,9) (1,5,10)</span>
-<span class="sd">            (2,6,7) (3,7,8) (4,8,9) (5,9,10) (1,6,10)</span>
-<span class="sd">            (6,7,11,) (7,8,11) (8,9,11) (9,10,11) (6,10,11)</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.polyhedron import cube</span>
-<span class="sd">        &gt;&gt;&gt; cube.edges</span>
-<span class="sd">        {(0, 1), (0, 3), (0, 4), &#39;...&#39;, (4, 7), (5, 6), (6, 7)}</span>
-
-<span class="sd">        If you want to use letters or other names for the corners you</span>
-<span class="sd">        can still use the pre-calculated faces:</span>
-
-<span class="sd">        &gt;&gt;&gt; corners = list(&#39;abcdefgh&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; Polyhedron(corners, cube.faces).corners</span>
-<span class="sd">        (a, b, c, d, e, f, g, h)</span>
-
-<span class="sd">        References</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        [1] www.ocf.berkeley.edu/~wwu/articles/platonicsolids.pdf</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">faces</span> <span class="o">=</span> <span class="p">[</span><span class="n">minlex</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">directed</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">is_set</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">faces</span><span class="p">]</span>
-        <span class="n">corners</span><span class="p">,</span> <span class="n">faces</span><span class="p">,</span> <span class="n">pgroup</span> <span class="o">=</span> <span class="n">args</span> <span class="o">=</span> \
-            <span class="p">[</span><span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="p">(</span><span class="n">corners</span><span class="p">,</span> <span class="n">faces</span><span class="p">,</span> <span class="n">pgroup</span><span class="p">)]</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_corners</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">corners</span><span class="p">)</span>  <span class="c"># in order given</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_faces</span> <span class="o">=</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="n">faces</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">pgroup</span> <span class="ow">and</span> <span class="n">pgroup</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">size</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">corners</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Permutation size unequal to number of corners.&quot;</span><span class="p">)</span>
-        <span class="c"># use the identity permutation if none are given</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_pgroup</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">((</span>
-            <span class="n">pgroup</span> <span class="ow">or</span> <span class="p">[</span><span class="n">Perm</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">corners</span><span class="p">))))]</span> <span class="p">))</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Polyhedron.corners"><a class="viewcode-back" href="../../../modules/combinatorics/polyhedron.html#sympy.combinatorics.polyhedron.Polyhedron.corners">[docs]</a>    <span class="k">def</span> <span class="nf">corners</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Get the corners of the Polyhedron.</span>
-
-<span class="sd">        The method ``vertices`` is an alias for ``corners``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Polyhedron</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import a, b, c, d</span>
-<span class="sd">        &gt;&gt;&gt; p = Polyhedron(list(&#39;abcd&#39;))</span>
-<span class="sd">        &gt;&gt;&gt; p.corners == p.vertices == (a, b, c, d)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        array_form, cyclic_form</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_corners</span></div>
-    <span class="n">vertices</span> <span class="o">=</span> <span class="n">corners</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Polyhedron.array_form"><a class="viewcode-back" href="../../../modules/combinatorics/polyhedron.html#sympy.combinatorics.polyhedron.Polyhedron.array_form">[docs]</a>    <span class="k">def</span> <span class="nf">array_form</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the indices of the corners.</span>
-
-<span class="sd">        The indices are given relative to the original position of corners.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Permutation, Cycle</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.polyhedron import tetrahedron</span>
-<span class="sd">        &gt;&gt;&gt; tetrahedron.array_form</span>
-<span class="sd">        [0, 1, 2, 3]</span>
-
-<span class="sd">        &gt;&gt;&gt; tetrahedron.rotate(0)</span>
-<span class="sd">        &gt;&gt;&gt; tetrahedron.array_form</span>
-<span class="sd">        [0, 2, 3, 1]</span>
-<span class="sd">        &gt;&gt;&gt; tetrahedron.pgroup[0].array_form</span>
-<span class="sd">        [0, 2, 3, 1]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        corners, cyclic_form</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">corners</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">corners</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">corners</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Polyhedron.cyclic_form"><a class="viewcode-back" href="../../../modules/combinatorics/polyhedron.html#sympy.combinatorics.polyhedron.Polyhedron.cyclic_form">[docs]</a>    <span class="k">def</span> <span class="nf">cyclic_form</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the indices of the corners in cyclic notation.</span>
-
-<span class="sd">        The indices are given relative to the original position of corners.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        corners, array_form</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Perm</span><span class="o">.</span><span class="n">_af_new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">array_form</span><span class="p">)</span><span class="o">.</span><span class="n">cyclic_form</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Polyhedron.size"><a class="viewcode-back" href="../../../modules/combinatorics/polyhedron.html#sympy.combinatorics.polyhedron.Polyhedron.size">[docs]</a>    <span class="k">def</span> <span class="nf">size</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Get the number of corners of the Polyhedron.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_corners</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Polyhedron.faces"><a class="viewcode-back" href="../../../modules/combinatorics/polyhedron.html#sympy.combinatorics.polyhedron.Polyhedron.faces">[docs]</a>    <span class="k">def</span> <span class="nf">faces</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Get the faces of the Polyhedron.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_faces</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Polyhedron.pgroup"><a class="viewcode-back" href="../../../modules/combinatorics/polyhedron.html#sympy.combinatorics.polyhedron.Polyhedron.pgroup">[docs]</a>    <span class="k">def</span> <span class="nf">pgroup</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Get the permutations of the Polyhedron.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_pgroup</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Polyhedron.edges"><a class="viewcode-back" href="../../../modules/combinatorics/polyhedron.html#sympy.combinatorics.polyhedron.Polyhedron.edges">[docs]</a>    <span class="k">def</span> <span class="nf">edges</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Given the faces of the polyhedra we can get the edges.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Polyhedron</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import a, b, c</span>
-<span class="sd">        &gt;&gt;&gt; corners = (a, b, c)</span>
-<span class="sd">        &gt;&gt;&gt; faces = [(0, 1, 2)]</span>
-<span class="sd">        &gt;&gt;&gt; Polyhedron(corners, faces).edges</span>
-<span class="sd">        {(0, 1), (0, 2), (1, 2)}</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_edges</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">output</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-            <span class="k">for</span> <span class="n">face</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">faces</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">face</span><span class="p">)):</span>
-                    <span class="n">edge</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="nb">sorted</span><span class="p">([</span><span class="n">face</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">face</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]]))</span>
-                    <span class="n">output</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">edge</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_edges</span> <span class="o">=</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="o">*</span><span class="n">output</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_edges</span>
-</div>
-<div class="viewcode-block" id="Polyhedron.rotate"><a class="viewcode-back" href="../../../modules/combinatorics/polyhedron.html#sympy.combinatorics.polyhedron.Polyhedron.rotate">[docs]</a>    <span class="k">def</span> <span class="nf">rotate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">perm</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Apply a permutation to the polyhedron *in place*. The permutation</span>
-<span class="sd">        may be given as a Permutation instance or an integer indicating</span>
-<span class="sd">        which permutation from pgroup of the Polyhedron should be</span>
-<span class="sd">        applied.</span>
-
-<span class="sd">        This is an operation that is analogous to rotation about</span>
-<span class="sd">        an axis by a fixed increment.</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        When a Permutation is applied, no check is done to see if that</span>
-<span class="sd">        is a valid permutation for the Polyhedron. For example, a cube</span>
-<span class="sd">        could be given a permutation which effectively swaps only 2</span>
-<span class="sd">        vertices. A valid permutation (that rotates the object in a</span>
-<span class="sd">        physical way) will be obtained if one only uses</span>
-<span class="sd">        permutations from the ``pgroup`` of the Polyhedron. On the other</span>
-<span class="sd">        hand, allowing arbitrary rotations (applications of permutations)</span>
-<span class="sd">        gives a way to follow named elements rather than indices since</span>
-<span class="sd">        Polyhedron allows vertices to be named while Permutation works</span>
-<span class="sd">        only with indices.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Polyhedron, Permutation</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.polyhedron import cube</span>
-<span class="sd">        &gt;&gt;&gt; cube.corners</span>
-<span class="sd">        (0, 1, 2, 3, 4, 5, 6, 7)</span>
-<span class="sd">        &gt;&gt;&gt; cube.rotate(0)</span>
-<span class="sd">        &gt;&gt;&gt; cube.corners</span>
-<span class="sd">        (1, 2, 3, 0, 5, 6, 7, 4)</span>
-
-<span class="sd">        A non-physical &quot;rotation&quot; that is not prohibited by this method:</span>
-
-<span class="sd">        &gt;&gt;&gt; cube.reset()</span>
-<span class="sd">        &gt;&gt;&gt; cube.rotate(Permutation([[1,2]], size=8))</span>
-<span class="sd">        &gt;&gt;&gt; cube.corners</span>
-<span class="sd">        (0, 2, 1, 3, 4, 5, 6, 7)</span>
-
-<span class="sd">        Polyhedron can be used to follow elements of set that are</span>
-<span class="sd">        identified by letters instead of integers:</span>
-
-<span class="sd">        &gt;&gt;&gt; shadow = h5 = Polyhedron(list(&#39;abcde&#39;))</span>
-<span class="sd">        &gt;&gt;&gt; p = Permutation([3, 0, 1, 2, 4])</span>
-<span class="sd">        &gt;&gt;&gt; h5.rotate(p)</span>
-<span class="sd">        &gt;&gt;&gt; h5.corners</span>
-<span class="sd">        (d, a, b, c, e)</span>
-<span class="sd">        &gt;&gt;&gt; _ == shadow.corners</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; copy = h5.copy()</span>
-<span class="sd">        &gt;&gt;&gt; h5.rotate(p)</span>
-<span class="sd">        &gt;&gt;&gt; h5.corners == copy.corners</span>
-<span class="sd">        False</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">perm</span><span class="p">,</span> <span class="n">Perm</span><span class="p">):</span>
-            <span class="n">perm</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">pgroup</span><span class="p">[</span><span class="n">perm</span><span class="p">]</span>
-            <span class="c"># and we know it&#39;s valid</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">perm</span><span class="o">.</span><span class="n">size</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Polyhedron and Permutation sizes differ.&#39;</span><span class="p">)</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">perm</span><span class="o">.</span><span class="n">array_form</span>
-        <span class="n">corners</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">corners</span><span class="p">[</span><span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">corners</span><span class="p">))]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_corners</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">corners</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Polyhedron.reset"><a class="viewcode-back" href="../../../modules/combinatorics/polyhedron.html#sympy.combinatorics.polyhedron.Polyhedron.reset">[docs]</a>    <span class="k">def</span> <span class="nf">reset</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return corners to their original positions.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.polyhedron import tetrahedron as T</span>
-<span class="sd">        &gt;&gt;&gt; T.corners</span>
-<span class="sd">        (0, 1, 2, 3)</span>
-<span class="sd">        &gt;&gt;&gt; T.rotate(0)</span>
-<span class="sd">        &gt;&gt;&gt; T.corners</span>
-<span class="sd">        (0, 2, 3, 1)</span>
-<span class="sd">        &gt;&gt;&gt; T.reset()</span>
-<span class="sd">        &gt;&gt;&gt; T.corners</span>
-<span class="sd">        (0, 1, 2, 3)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_corners</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-</div></div>
-<span class="k">def</span> <span class="nf">_pgroup_calcs</span><span class="p">():</span>
-    <span class="sd">&quot;&quot;&quot;Return the permutation groups for each of the polyhedra and the face</span>
-<span class="sd">    definitions: tetrahedron, cube, octahedron, dodecahedron, icosahedron,</span>
-<span class="sd">    tetrahedron_faces, cube_faces, octahedron_faces, dodecahedron_faces,</span>
-<span class="sd">    icosahedron_faces</span>
-
-<span class="sd">    (This author didn&#39;t find and didn&#39;t know of a better way to do it though</span>
-<span class="sd">    there likely is such a way.)</span>
-
-<span class="sd">    Although only 2 permutations are needed for a polyhedron in order to</span>
-<span class="sd">    generate all the possible orientations, a group of permutations is</span>
-<span class="sd">    provided instead. A set of permutations is called a &quot;group&quot; if::</span>
-
-<span class="sd">    a*b = c (for any pair of permutations in the group, a and b, their</span>
-<span class="sd">    product, c, is in the group)</span>
-
-<span class="sd">    a*(b*c) = (a*b)*c (for any 3 permutations in the group associativity holds)</span>
-
-<span class="sd">    there is an identity permutation, I, such that I*a = a*I for all elements</span>
-<span class="sd">    in the group</span>
-
-<span class="sd">    a*b = I (the inverse of each permutation is also in the group)</span>
-
-<span class="sd">    None of the polyhedron groups defined follow these definitions of a group.</span>
-<span class="sd">    Instead, they are selected to contain those permutations whose powers</span>
-<span class="sd">    alone will construct all orientations of the polyhedron, i.e. for</span>
-<span class="sd">    permutations ``a``, ``b``, etc... in the group, ``a, a**2, ..., a**o_a``,</span>
-<span class="sd">    ``b, b**2, ..., b**o_b``, etc... (where ``o_i`` is the order of</span>
-<span class="sd">    permutation ``i``) generate all permutations of the polyhedron instead of</span>
-<span class="sd">    mixed products like ``a*b``, ``a*b**2``, etc....</span>
-
-<span class="sd">    Note that for a polyhedron with n vertices, the valid permutations of the</span>
-<span class="sd">    vertices exclude those that do not maintain its faces. e.g. the</span>
-<span class="sd">    permutation BCDE of a square&#39;s four corners, ABCD, is a valid</span>
-<span class="sd">    permutation while CBDE is not (because this would twist the square).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    The is_group checks for: closure, the presence of the Identity permutation,</span>
-<span class="sd">    and the presence of the inverse for each of the elements in the group. This</span>
-<span class="sd">    confirms that none of the polyhedra are true groups:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.polyhedron import (</span>
-<span class="sd">    ... tetrahedron, cube, octahedron, dodecahedron, icosahedron)</span>
-<span class="sd">    ...</span>
-<span class="sd">    &gt;&gt;&gt; polyhedra = (tetrahedron, cube, octahedron, dodecahedron, icosahedron)</span>
-<span class="sd">    &gt;&gt;&gt; [h.pgroup.is_group() for h in polyhedra]</span>
-<span class="sd">    ...</span>
-<span class="sd">    [False, False, False, False, False]</span>
-
-<span class="sd">    Although tests in polyhedron&#39;s test suite check that powers of the</span>
-<span class="sd">    permutations in the groups generate all permutations of the vertices</span>
-<span class="sd">    of the polyhedron, here we also demonstrate the powers of the given</span>
-<span class="sd">    permutations create a complete group for the tetrahedron:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics import Permutation, PermutationGroup</span>
-<span class="sd">    &gt;&gt;&gt; for h in polyhedra[:1]:</span>
-<span class="sd">    ...     G = h.pgroup</span>
-<span class="sd">    ...     perms = set()</span>
-<span class="sd">    ...     for g in G:</span>
-<span class="sd">    ...         for e in range(g.order()):</span>
-<span class="sd">    ...             p = tuple((g**e).array_form)</span>
-<span class="sd">    ...             perms.add(p)</span>
-<span class="sd">    ...</span>
-<span class="sd">    ...     perms = [Permutation(p) for p in perms]</span>
-<span class="sd">    ...     assert PermutationGroup(perms).is_group()</span>
-
-<span class="sd">    In addition to doing the above, the tests in the suite confirm that the</span>
-<span class="sd">    faces are all present after the application of each permutation.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    http://dogschool.tripod.com/trianglegroup.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">_pgroup_of_double</span><span class="p">(</span><span class="n">polyh</span><span class="p">,</span> <span class="n">ordered_faces</span><span class="p">,</span> <span class="n">pgroup</span><span class="p">):</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">ordered_faces</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="c"># the vertices of the double which sits inside a give polyhedron</span>
-        <span class="c"># can be found by tracking the faces of the outer polyhedron.</span>
-        <span class="c"># A map between face and the vertex of the double is made so that</span>
-        <span class="c"># after rotation the position of the vertices can be located</span>
-        <span class="n">fmap</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">ordered_faces</span><span class="p">,</span>
-                        <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">ordered_faces</span><span class="p">))))))</span>
-        <span class="n">flat_faces</span> <span class="o">=</span> <span class="n">flatten</span><span class="p">(</span><span class="n">ordered_faces</span><span class="p">)</span>
-        <span class="n">new_pgroup</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">pgroup</span><span class="p">):</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">polyh</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-            <span class="n">h</span><span class="o">.</span><span class="n">rotate</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">h</span><span class="o">.</span><span class="n">corners</span>
-            <span class="c"># reorder corners in the order they should appear when</span>
-            <span class="c"># enumerating the faces</span>
-            <span class="n">reorder</span> <span class="o">=</span> <span class="n">unflatten</span><span class="p">([</span><span class="n">c</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">flat_faces</span><span class="p">],</span> <span class="n">n</span><span class="p">)</span>
-            <span class="c"># make them canonical</span>
-            <span class="n">reorder</span> <span class="o">=</span> <span class="p">[</span><span class="nb">tuple</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">as_int</span><span class="p">,</span>
-                       <span class="n">minlex</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">directed</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">is_set</span><span class="o">=</span><span class="bp">True</span><span class="p">)))</span>
-                       <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">reorder</span><span class="p">]</span>
-            <span class="c"># map face to vertex: the resulting list of vertices are the</span>
-            <span class="c"># permutation that we seek for the double</span>
-            <span class="n">new_pgroup</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Perm</span><span class="p">([</span><span class="n">fmap</span><span class="p">[</span><span class="n">f</span><span class="p">]</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">reorder</span><span class="p">]))</span>
-        <span class="k">return</span> <span class="n">new_pgroup</span>
-
-    <span class="n">tetrahedron_faces</span> <span class="o">=</span> <span class="p">[</span>
-        <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span>  <span class="c"># upper 3</span>
-        <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span>  <span class="c"># bottom</span>
-    <span class="p">]</span>
-
-    <span class="c"># cw from top</span>
-    <span class="c">#</span>
-    <span class="n">_t_pgroup</span> <span class="o">=</span> <span class="p">[</span>
-        <span class="n">Perm</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">]]),</span>  <span class="c"># cw from top</span>
-        <span class="n">Perm</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">]]),</span>  <span class="c"># cw from front face</span>
-        <span class="n">Perm</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">]]),</span>  <span class="c"># cw from back right face</span>
-        <span class="n">Perm</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]]),</span>  <span class="c"># cw from back left face</span>
-        <span class="n">Perm</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]]),</span>  <span class="c"># through front left edge</span>
-        <span class="n">Perm</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">]]),</span>  <span class="c"># through front right edge</span>
-        <span class="n">Perm</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]]),</span>  <span class="c"># through back edge</span>
-    <span class="p">]</span>
-
-    <span class="n">tetrahedron</span> <span class="o">=</span> <span class="n">Polyhedron</span><span class="p">(</span>
-        <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">)),</span>
-        <span class="n">tetrahedron_faces</span><span class="p">,</span>
-        <span class="n">_t_pgroup</span><span class="p">)</span>
-
-    <span class="n">cube_faces</span> <span class="o">=</span> <span class="p">[</span>
-        <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span>  <span class="c"># upper</span>
-        <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span>  <span class="c"># middle 4</span>
-        <span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">),</span>  <span class="c"># lower</span>
-    <span class="p">]</span>
-
-    <span class="c"># U, D, F, B, L, R = up, down, front, back, left, right</span>
-    <span class="n">_c_pgroup</span> <span class="o">=</span> <span class="p">[</span><span class="n">Perm</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span>
-        <span class="p">[</span>
-        <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span>  <span class="c"># cw from top, U</span>
-        <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span>  <span class="c"># cw from F face</span>
-        <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span>  <span class="c"># cw from R face</span>
-
-        <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span>  <span class="c"># cw through UF edge</span>
-        <span class="p">[</span><span class="mi">6</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span>  <span class="c"># cw through UR edge</span>
-        <span class="p">[</span><span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>  <span class="c"># cw through UB edge</span>
-        <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>  <span class="c"># cw through UL edge</span>
-        <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>  <span class="c"># cw through FL edge</span>
-        <span class="p">[</span><span class="mi">6</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span>  <span class="c"># cw through FR edge</span>
-
-        <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</span>  <span class="c"># cw through UFL vertex</span>
-        <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">],</span>  <span class="c"># cw through UFR vertex</span>
-        <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>  <span class="c"># cw through UBR vertex</span>
-        <span class="p">[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span>  <span class="c"># cw through UBL</span>
-        <span class="p">]]</span>
-
-    <span class="n">cube</span> <span class="o">=</span> <span class="n">Polyhedron</span><span class="p">(</span>
-        <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">8</span><span class="p">)),</span>
-        <span class="n">cube_faces</span><span class="p">,</span>
-        <span class="n">_c_pgroup</span><span class="p">)</span>
-
-    <span class="n">octahedron_faces</span> <span class="o">=</span> <span class="p">[</span>
-        <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span>  <span class="c"># top 4</span>
-        <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span>  <span class="c"># bottom 4</span>
-    <span class="p">]</span>
-
-    <span class="n">octahedron</span> <span class="o">=</span> <span class="n">Polyhedron</span><span class="p">(</span>
-        <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">6</span><span class="p">)),</span>
-        <span class="n">octahedron_faces</span><span class="p">,</span>
-        <span class="n">_pgroup_of_double</span><span class="p">(</span><span class="n">cube</span><span class="p">,</span> <span class="n">cube_faces</span><span class="p">,</span> <span class="n">_c_pgroup</span><span class="p">))</span>
-
-    <span class="n">dodecahedron_faces</span> <span class="o">=</span> <span class="p">[</span>
-        <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span>  <span class="c"># top</span>
-        <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">7</span><span class="p">),</span>  <span class="c"># upper 5</span>
-        <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">8</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">14</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span>
-        <span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">16</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">14</span><span class="p">),</span> <span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">16</span><span class="p">,</span> <span class="mi">17</span><span class="p">,</span> <span class="mi">11</span><span class="p">),</span> <span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">17</span><span class="p">,</span> <span class="mi">18</span><span class="p">,</span>
-          <span class="mi">12</span><span class="p">),</span>  <span class="c"># lower 5</span>
-        <span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">18</span><span class="p">,</span> <span class="mi">19</span><span class="p">,</span> <span class="mi">13</span><span class="p">),</span> <span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">19</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">14</span><span class="p">),</span>
-        <span class="p">(</span><span class="mi">15</span><span class="p">,</span> <span class="mi">16</span><span class="p">,</span> <span class="mi">17</span><span class="p">,</span> <span class="mi">18</span><span class="p">,</span> <span class="mi">19</span><span class="p">)</span>  <span class="c"># bottom</span>
-    <span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">_string_to_perm</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="p">[</span><span class="n">Perm</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">20</span><span class="p">)))]</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">for</span> <span class="n">si</span> <span class="ow">in</span> <span class="n">s</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">si</span> <span class="ow">not</span> <span class="ow">in</span> <span class="s">&#39;01&#39;</span><span class="p">:</span>
-                <span class="n">count</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">si</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">count</span> <span class="o">=</span> <span class="mi">1</span>
-                <span class="k">if</span> <span class="n">si</span> <span class="o">==</span> <span class="s">&#39;0&#39;</span><span class="p">:</span>
-                    <span class="n">p</span> <span class="o">=</span> <span class="n">_f0</span>
-                <span class="k">elif</span> <span class="n">si</span> <span class="o">==</span> <span class="s">&#39;1&#39;</span><span class="p">:</span>
-                    <span class="n">p</span> <span class="o">=</span> <span class="n">_f1</span>
-            <span class="n">rv</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">p</span><span class="p">]</span><span class="o">*</span><span class="n">count</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Perm</span><span class="o">.</span><span class="n">rmul</span><span class="p">(</span><span class="o">*</span><span class="n">rv</span><span class="p">)</span>
-
-    <span class="c"># top face cw</span>
-    <span class="n">_f0</span> <span class="o">=</span> <span class="n">Perm</span><span class="p">([</span>
-        <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span>
-        <span class="mi">12</span><span class="p">,</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">14</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">16</span><span class="p">,</span> <span class="mi">17</span><span class="p">,</span> <span class="mi">18</span><span class="p">,</span> <span class="mi">19</span><span class="p">,</span> <span class="mi">15</span><span class="p">])</span>
-    <span class="c"># front face cw</span>
-    <span class="n">_f1</span> <span class="o">=</span> <span class="n">Perm</span><span class="p">([</span>
-        <span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">14</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span>
-        <span class="mi">6</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">19</span><span class="p">,</span> <span class="mi">16</span><span class="p">,</span> <span class="mi">17</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">18</span><span class="p">])</span>
-    <span class="c"># the strings below, like 0104 are shorthand for F0*F1*F0**4 and are</span>
-    <span class="c"># the remaining 4 face rotations, 15 edge permutations, and the</span>
-    <span class="c"># 10 vertex rotations.</span>
-    <span class="n">_dodeca_pgroup</span> <span class="o">=</span> <span class="p">[</span><span class="n">_f0</span><span class="p">,</span> <span class="n">_f1</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">_string_to_perm</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="s">&#39;&#39;&#39;</span>
-<span class="s">    0104 140 014 0410</span>
-<span class="s">    010 1403 03104 04103 102</span>
-<span class="s">    120 1304 01303 021302 03130</span>
-<span class="s">    0412041 041204103 04120410 041204104 041204102</span>
-<span class="s">    10 01 1402 0140 04102 0412 1204 1302 0130 03120&#39;&#39;&#39;</span><span class="o">.</span><span class="n">strip</span><span class="p">()</span><span class="o">.</span><span class="n">split</span><span class="p">()]</span>
-
-    <span class="n">dodecahedron</span> <span class="o">=</span> <span class="n">Polyhedron</span><span class="p">(</span>
-        <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">20</span><span class="p">)),</span>
-        <span class="n">dodecahedron_faces</span><span class="p">,</span>
-        <span class="n">_dodeca_pgroup</span><span class="p">)</span>
-
-    <span class="n">icosahedron_faces</span> <span class="o">=</span> <span class="p">[</span>
-        <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</span>
-        <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">7</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">8</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span> <span class="p">],</span>
-        <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">9</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">10</span> <span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">10</span><span class="p">],</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">10</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span> <span class="p">],</span>
-        <span class="p">[</span><span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">11</span><span class="p">],</span> <span class="p">[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">11</span><span class="p">],</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">11</span><span class="p">],</span> <span class="p">[</span><span class="mi">9</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">11</span><span class="p">],</span> <span class="p">[</span><span class="mi">6</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">11</span><span class="p">]]</span>
-
-    <span class="n">icosahedron</span> <span class="o">=</span> <span class="n">Polyhedron</span><span class="p">(</span>
-        <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">12</span><span class="p">)),</span>
-        <span class="n">icosahedron_faces</span><span class="p">,</span>
-        <span class="n">_pgroup_of_double</span><span class="p">(</span>
-            <span class="n">dodecahedron</span><span class="p">,</span> <span class="n">dodecahedron_faces</span><span class="p">,</span> <span class="n">_dodeca_pgroup</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="p">(</span><span class="n">tetrahedron</span><span class="p">,</span> <span class="n">cube</span><span class="p">,</span> <span class="n">octahedron</span><span class="p">,</span> <span class="n">dodecahedron</span><span class="p">,</span> <span class="n">icosahedron</span><span class="p">,</span>
-        <span class="n">tetrahedron_faces</span><span class="p">,</span> <span class="n">cube_faces</span><span class="p">,</span> <span class="n">octahedron_faces</span><span class="p">,</span>
-        <span class="n">dodecahedron_faces</span><span class="p">,</span> <span class="n">icosahedron_faces</span><span class="p">)</span>
-
-<span class="p">(</span><span class="n">tetrahedron</span><span class="p">,</span> <span class="n">cube</span><span class="p">,</span> <span class="n">octahedron</span><span class="p">,</span> <span class="n">dodecahedron</span><span class="p">,</span> <span class="n">icosahedron</span><span class="p">,</span>
-<span class="n">tetrahedron_faces</span><span class="p">,</span> <span class="n">cube_faces</span><span class="p">,</span> <span class="n">octahedron_faces</span><span class="p">,</span>
-<span class="n">dodecahedron_faces</span><span class="p">,</span> <span class="n">icosahedron_faces</span><span class="p">)</span> <span class="o">=</span> <span class="n">_pgroup_calcs</span><span class="p">()</span>
-</pre></div>
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-            
-  <h1>Source code for sympy.combinatorics.prufer</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Basic</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">iterable</span><span class="p">,</span> <span class="n">as_int</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">flatten</span>
-
-<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</span>
-
-
-<div class="viewcode-block" id="Prufer"><a class="viewcode-back" href="../../../modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer">[docs]</a><span class="k">class</span> <span class="nc">Prufer</span><span class="p">(</span><span class="n">Basic</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The Prufer correspondence is an algorithm that describes the</span>
-<span class="sd">    bijection between labeled trees and the Prufer code. A Prufer</span>
-<span class="sd">    code of a labeled tree is unique up to isomorphism and has</span>
-<span class="sd">    a length of n - 2.</span>
-
-<span class="sd">    Prufer sequences were first used by Heinz Prufer to give a</span>
-<span class="sd">    proof of Cayley&#39;s formula.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://mathworld.wolfram.com/LabeledTree.html</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">_prufer_repr</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">_tree_repr</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">_nodes</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">_rank</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Prufer.prufer_repr"><a class="viewcode-back" href="../../../modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.prufer_repr">[docs]</a>    <span class="k">def</span> <span class="nf">prufer_repr</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns Prufer sequence for the Prufer object.</span>
-
-<span class="sd">        This sequence is found by removing the highest numbered vertex,</span>
-<span class="sd">        recording the node it was attached to, and continuuing until only</span>
-<span class="sd">        two verices remain. The Prufer sequence is the list of recorded nodes.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.prufer import Prufer</span>
-<span class="sd">        &gt;&gt;&gt; Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]).prufer_repr</span>
-<span class="sd">        [3, 3, 3, 4]</span>
-<span class="sd">        &gt;&gt;&gt; Prufer([1, 0, 0]).prufer_repr</span>
-<span class="sd">        [1, 0, 0]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        to_prufer</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_prufer_repr</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_prufer_repr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">to_prufer</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_tree_repr</span><span class="p">[:],</span> <span class="bp">self</span><span class="o">.</span><span class="n">nodes</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_prufer_repr</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Prufer.tree_repr"><a class="viewcode-back" href="../../../modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.tree_repr">[docs]</a>    <span class="k">def</span> <span class="nf">tree_repr</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the tree representation of the Prufer object.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.prufer import Prufer</span>
-<span class="sd">        &gt;&gt;&gt; Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]).tree_repr</span>
-<span class="sd">        [[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]</span>
-<span class="sd">        &gt;&gt;&gt; Prufer([1, 0, 0]).tree_repr</span>
-<span class="sd">        [[1, 2], [0, 1], [0, 3], [0, 4]]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        to_tree</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_tree_repr</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_tree_repr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">to_tree</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_prufer_repr</span><span class="p">[:])</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_tree_repr</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Prufer.nodes"><a class="viewcode-back" href="../../../modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.nodes">[docs]</a>    <span class="k">def</span> <span class="nf">nodes</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the number of nodes in the tree.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.prufer import Prufer</span>
-<span class="sd">        &gt;&gt;&gt; Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]).nodes</span>
-<span class="sd">        6</span>
-<span class="sd">        &gt;&gt;&gt; Prufer([1, 0, 0]).nodes</span>
-<span class="sd">        5</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_nodes</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Prufer.rank"><a class="viewcode-back" href="../../../modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.rank">[docs]</a>    <span class="k">def</span> <span class="nf">rank</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the rank of the Prufer sequence.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.prufer import Prufer</span>
-<span class="sd">        &gt;&gt;&gt; p = Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]])</span>
-<span class="sd">        &gt;&gt;&gt; p.rank</span>
-<span class="sd">        778</span>
-<span class="sd">        &gt;&gt;&gt; p.next(1).rank</span>
-<span class="sd">        779</span>
-<span class="sd">        &gt;&gt;&gt; p.prev().rank</span>
-<span class="sd">        777</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        prufer_rank, next, prev, size</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rank</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_rank</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">prufer_rank</span><span class="p">()</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rank</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Prufer.size"><a class="viewcode-back" href="../../../modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.size">[docs]</a>    <span class="k">def</span> <span class="nf">size</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the number of possible trees of this Prufer object.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.prufer import Prufer</span>
-<span class="sd">        &gt;&gt;&gt; Prufer([0]*4).size == Prufer([6]*4).size == 1296</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        prufer_rank, rank, next, prev</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">prev</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rank</span><span class="p">)</span><span class="o">.</span><span class="n">prev</span><span class="p">()</span><span class="o">.</span><span class="n">rank</span> <span class="o">+</span> <span class="mi">1</span>
-</div>
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="Prufer.to_prufer"><a class="viewcode-back" href="../../../modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.to_prufer">[docs]</a>    <span class="k">def</span> <span class="nf">to_prufer</span><span class="p">(</span><span class="n">tree</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the Prufer sequence for a tree given as a list of edges where</span>
-<span class="sd">        ``n`` is the number of nodes in the tree.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.prufer import Prufer</span>
-<span class="sd">        &gt;&gt;&gt; a = Prufer([[0, 1], [0, 2], [0, 3]])</span>
-<span class="sd">        &gt;&gt;&gt; a.prufer_repr</span>
-<span class="sd">        [0, 0]</span>
-<span class="sd">        &gt;&gt;&gt; Prufer.to_prufer([[0, 1], [0, 2], [0, 3]], 4)</span>
-<span class="sd">        [0, 0]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        prufer_repr: returns Prufer sequence of a Prufer object.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
-        <span class="n">L</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">edge</span> <span class="ow">in</span> <span class="n">tree</span><span class="p">:</span>
-            <span class="c"># Increment the value of the corresponding</span>
-            <span class="c"># node in the degree list as we encounter an</span>
-            <span class="c"># edge involving it.</span>
-            <span class="n">d</span><span class="p">[</span><span class="n">edge</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">d</span><span class="p">[</span><span class="n">edge</span><span class="p">[</span><span class="mi">1</span><span class="p">]]</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="p">):</span>
-            <span class="c"># find the smallest leaf</span>
-            <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">d</span><span class="p">[</span><span class="n">x</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">break</span>
-            <span class="c"># find the node it was connected to</span>
-            <span class="n">y</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">for</span> <span class="n">edge</span> <span class="ow">in</span> <span class="n">tree</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="n">edge</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-                    <span class="n">y</span> <span class="o">=</span> <span class="n">edge</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-                <span class="k">elif</span> <span class="n">x</span> <span class="o">==</span> <span class="n">edge</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span>
-                    <span class="n">y</span> <span class="o">=</span> <span class="n">edge</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="k">if</span> <span class="n">y</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">break</span>
-            <span class="c"># record and update</span>
-            <span class="n">L</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
-                <span class="n">d</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">1</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">d</span><span class="p">[</span><span class="n">j</span><span class="p">]:</span>
-                    <span class="n">d</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-            <span class="n">tree</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">edge</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">L</span>
-</div>
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="Prufer.to_tree"><a class="viewcode-back" href="../../../modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.to_tree">[docs]</a>    <span class="k">def</span> <span class="nf">to_tree</span><span class="p">(</span><span class="n">prufer</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the tree (as a list of edges) of the given Prufer sequence.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.prufer import Prufer</span>
-<span class="sd">        &gt;&gt;&gt; a = Prufer([0, 2], 4)</span>
-<span class="sd">        &gt;&gt;&gt; a.tree_repr</span>
-<span class="sd">        [[0, 1], [0, 2], [2, 3]]</span>
-<span class="sd">        &gt;&gt;&gt; Prufer.to_tree([0, 2])</span>
-<span class="sd">        [[0, 1], [0, 2], [2, 3]]</span>
-
-<span class="sd">        References</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        - http://hamberg.no/erlend/2010/11/06/prufer-sequence/</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        tree_repr: returns tree representation of a Prufer object.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">tree</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">last</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">prufer</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="k">lambda</span><span class="p">:</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">prufer</span><span class="p">:</span>
-            <span class="n">d</span><span class="p">[</span><span class="n">p</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">prufer</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="c"># find the smallest leaf (degree = 1)</span>
-                <span class="k">if</span> <span class="n">d</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">break</span>
-            <span class="c"># (i, j) is the new edge that we append to the tree</span>
-            <span class="c"># and remove from the degree dictionary</span>
-            <span class="n">d</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="n">d</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="n">tree</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">sorted</span><span class="p">([</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]))</span>
-        <span class="n">last</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">if</span> <span class="n">d</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">]</span> <span class="ow">or</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
-        <span class="n">tree</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">last</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">tree</span>
-</div>
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="Prufer.edges"><a class="viewcode-back" href="../../../modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.edges">[docs]</a>    <span class="k">def</span> <span class="nf">edges</span><span class="p">(</span><span class="o">*</span><span class="n">runs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return a list of edges and the number of nodes from the given runs</span>
-<span class="sd">        that connect nodes in an integer-labelled tree.</span>
-
-<span class="sd">        All node numbers will be shifted so that the minimum node is 0. It is</span>
-<span class="sd">        not a problem if edges are repeated in the runs; only unique edges are</span>
-<span class="sd">        returned. There is no assumption made about what the range of the node</span>
-<span class="sd">        labels should be, but all nodes from the smallest through the largest</span>
-<span class="sd">        must be present.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.prufer import Prufer</span>
-<span class="sd">        &gt;&gt;&gt; Prufer.edges([1, 2, 3], [2, 4, 5]) # a T</span>
-<span class="sd">        ([[0, 1], [3, 4], [1, 2], [1, 3]], 5)</span>
-
-<span class="sd">        Duplicate edges are removed:</span>
-
-<span class="sd">        &gt;&gt;&gt; Prufer.edges([0, 1, 2, 3], [1, 4, 5], [1, 4, 6]) # a K</span>
-<span class="sd">        ([[0, 1], [1, 2], [4, 6], [4, 5], [1, 4], [2, 3]], 7)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">e</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="n">nmin</span> <span class="o">=</span> <span class="n">runs</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">runs</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">r</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">r</span><span class="p">[</span><span class="n">i</span><span class="p">:</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">2</span><span class="p">]</span>
-                <span class="k">if</span> <span class="n">b</span> <span class="o">&lt;</span> <span class="n">a</span><span class="p">:</span>
-                    <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="p">,</span> <span class="n">a</span>
-                <span class="n">e</span><span class="o">.</span><span class="n">add</span><span class="p">((</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">got</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="n">nmin</span> <span class="o">=</span> <span class="n">nmax</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">while</span> <span class="n">e</span><span class="p">:</span>
-            <span class="n">ei</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">ei</span><span class="p">:</span>
-                <span class="n">got</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="n">nmin</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">ei</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">nmin</span><span class="p">)</span> <span class="k">if</span> <span class="n">nmin</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="k">else</span> <span class="n">ei</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">nmax</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">ei</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">nmax</span><span class="p">)</span> <span class="k">if</span> <span class="n">nmax</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="k">else</span> <span class="n">ei</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-            <span class="n">rv</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">ei</span><span class="p">))</span>
-        <span class="n">missing</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">nmin</span><span class="p">,</span> <span class="n">nmax</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span> <span class="o">-</span> <span class="n">got</span>
-        <span class="k">if</span> <span class="n">missing</span><span class="p">:</span>
-            <span class="n">missing</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="n">nmin</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">missing</span><span class="p">]</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">missing</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">msg</span> <span class="o">=</span> <span class="s">&#39;Node </span><span class="si">%s</span><span class="s"> is missing.&#39;</span> <span class="o">%</span> <span class="n">missing</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">msg</span> <span class="o">=</span> <span class="s">&#39;Nodes </span><span class="si">%s</span><span class="s"> are missing.&#39;</span> <span class="o">%</span> <span class="nb">list</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="n">missing</span><span class="p">))</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="n">msg</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">nmin</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">ei</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">rv</span><span class="p">):</span>
-                <span class="n">rv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="n">nmin</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">ei</span><span class="p">]</span>
-            <span class="n">nmax</span> <span class="o">-=</span> <span class="n">nmin</span>
-        <span class="k">return</span> <span class="n">rv</span><span class="p">,</span> <span class="n">nmax</span> <span class="o">+</span> <span class="mi">1</span>
-</div>
-<div class="viewcode-block" id="Prufer.prufer_rank"><a class="viewcode-back" href="../../../modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.prufer_rank">[docs]</a>    <span class="k">def</span> <span class="nf">prufer_rank</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes the rank of a Prufer sequence.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.prufer import Prufer</span>
-<span class="sd">        &gt;&gt;&gt; a = Prufer([[0, 1], [0, 2], [0, 3]])</span>
-<span class="sd">        &gt;&gt;&gt; a.prufer_rank()</span>
-<span class="sd">        0</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        rank, next, prev, size</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">nodes</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-            <span class="n">r</span> <span class="o">+=</span> <span class="n">p</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">prufer_repr</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="n">p</span> <span class="o">*=</span> <span class="bp">self</span><span class="o">.</span><span class="n">nodes</span>
-        <span class="k">return</span> <span class="n">r</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Prufer.unrank"><a class="viewcode-back" href="../../../modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.unrank">[docs]</a>    <span class="k">def</span> <span class="nf">unrank</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rank</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Finds the unranked Prufer sequence.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.prufer import Prufer</span>
-<span class="sd">        &gt;&gt;&gt; Prufer.unrank(0, 4)</span>
-<span class="sd">        Prufer([0, 0])</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">rank</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">as_int</span><span class="p">(</span><span class="n">rank</span><span class="p">)</span>
-        <span class="n">L</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-            <span class="n">L</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">rank</span> <span class="o">%</span> <span class="n">n</span>
-            <span class="n">rank</span> <span class="o">=</span> <span class="p">(</span><span class="n">rank</span> <span class="o">-</span> <span class="n">L</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="o">//</span><span class="n">n</span>
-        <span class="k">return</span> <span class="n">Prufer</span><span class="p">([</span><span class="n">L</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">L</span><span class="p">))])</span>
-</div>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The constructor for the Prufer object.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.prufer import Prufer</span>
-
-<span class="sd">        A Prufer object can be constructed from a list of edges:</span>
-
-<span class="sd">        &gt;&gt;&gt; a = Prufer([[0, 1], [0, 2], [0, 3]])</span>
-<span class="sd">        &gt;&gt;&gt; a.prufer_repr</span>
-<span class="sd">        [0, 0]</span>
-
-<span class="sd">        If the number of nodes is given, no checking of the nodes will</span>
-<span class="sd">        be performed; it will be assumed that nodes 0 through n - 1 are</span>
-<span class="sd">        present:</span>
-
-<span class="sd">        &gt;&gt;&gt; Prufer([[0, 1], [0, 2], [0, 3]], 4)</span>
-<span class="sd">        Prufer([[0, 1], [0, 2], [0, 3]], 4)</span>
-
-<span class="sd">        A Prufer object can be constructed from a Prufer sequence:</span>
-
-<span class="sd">        &gt;&gt;&gt; b = Prufer([1, 3])</span>
-<span class="sd">        &gt;&gt;&gt; b.tree_repr</span>
-<span class="sd">        [[0, 1], [1, 3], [2, 3]]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">ret_obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">)</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])]</span>
-        <span class="k">if</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">and</span> <span class="n">iterable</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                    <span class="s">&#39;Prufer expects at least one edge in the tree.&#39;</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">nnodes</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">nodes</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">flatten</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-                <span class="n">nnodes</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">nodes</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
-                <span class="k">if</span> <span class="n">nnodes</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">nodes</span><span class="p">):</span>
-                    <span class="n">missing</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">nnodes</span><span class="p">))</span> <span class="o">-</span> <span class="n">nodes</span>
-                    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">missing</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="n">msg</span> <span class="o">=</span> <span class="s">&#39;Node </span><span class="si">%s</span><span class="s"> is missing.&#39;</span> <span class="o">%</span> <span class="n">missing</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">msg</span> <span class="o">=</span> <span class="s">&#39;Nodes </span><span class="si">%s</span><span class="s"> are missing.&#39;</span> <span class="o">%</span> <span class="nb">list</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="n">missing</span><span class="p">))</span>
-                    <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="n">msg</span><span class="p">)</span>
-            <span class="n">ret_obj</span><span class="o">.</span><span class="n">_tree_repr</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span>
-            <span class="n">ret_obj</span><span class="o">.</span><span class="n">_nodes</span> <span class="o">=</span> <span class="n">nnodes</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">ret_obj</span><span class="o">.</span><span class="n">_prufer_repr</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">ret_obj</span><span class="o">.</span><span class="n">_nodes</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">ret_obj</span><span class="o">.</span><span class="n">_prufer_repr</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span>
-        <span class="k">return</span> <span class="n">ret_obj</span>
-
-<div class="viewcode-block" id="Prufer.next"><a class="viewcode-back" href="../../../modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.next">[docs]</a>    <span class="k">def</span> <span class="nf">next</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">delta</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Generates the Prufer sequence that is delta beyond the current one.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.prufer import Prufer</span>
-<span class="sd">        &gt;&gt;&gt; a = Prufer([[0, 1], [0, 2], [0, 3]])</span>
-<span class="sd">        &gt;&gt;&gt; b = a.next(1) # == a.next()</span>
-<span class="sd">        &gt;&gt;&gt; b.tree_repr</span>
-<span class="sd">        [[0, 2], [0, 1], [1, 3]]</span>
-<span class="sd">        &gt;&gt;&gt; b.rank</span>
-<span class="sd">        1</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        prufer_rank, rank, prev, size</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Prufer</span><span class="o">.</span><span class="n">unrank</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rank</span> <span class="o">+</span> <span class="n">delta</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">nodes</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Prufer.prev"><a class="viewcode-back" href="../../../modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.prev">[docs]</a>    <span class="k">def</span> <span class="nf">prev</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">delta</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Generates the Prufer sequence that is -delta before the current one.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.prufer import Prufer</span>
-<span class="sd">        &gt;&gt;&gt; a = Prufer([[0, 1], [1, 2], [2, 3], [1, 4]])</span>
-<span class="sd">        &gt;&gt;&gt; a.rank</span>
-<span class="sd">        36</span>
-<span class="sd">        &gt;&gt;&gt; b = a.prev()</span>
-<span class="sd">        &gt;&gt;&gt; b</span>
-<span class="sd">        Prufer([1, 2, 0])</span>
-<span class="sd">        &gt;&gt;&gt; b.rank</span>
-<span class="sd">        35</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        prufer_rank, rank, next, size</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Prufer</span><span class="o">.</span><span class="n">unrank</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rank</span> <span class="o">-</span> <span class="n">delta</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">nodes</span><span class="p">)</span></div></div>
-</pre></div>
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-            
-  <h1>Source code for sympy.combinatorics.subsets</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Basic</span>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.graycode</span> <span class="kn">import</span> <span class="n">GrayCode</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="nb">bin</span><span class="p">,</span> <span class="n">combinations</span>
-
-
-<div class="viewcode-block" id="Subset"><a class="viewcode-back" href="../../../modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset">[docs]</a><span class="k">class</span> <span class="nc">Subset</span><span class="p">(</span><span class="n">Basic</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Represents a basic subset object.</span>
-
-<span class="sd">    We generate subsets using essentially two techniques,</span>
-<span class="sd">    binary enumeration and lexicographic enumeration.</span>
-<span class="sd">    The Subset class takes two arguments, the first one</span>
-<span class="sd">    describes the initial subset to consider and the second</span>
-<span class="sd">    describes the superset.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.subsets import Subset</span>
-<span class="sd">    &gt;&gt;&gt; a = Subset([&#39;c&#39;,&#39;d&#39;], [&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;d&#39;])</span>
-<span class="sd">    &gt;&gt;&gt; a.next_binary().subset</span>
-<span class="sd">    [&#39;b&#39;]</span>
-<span class="sd">    &gt;&gt;&gt; a.prev_binary().subset</span>
-<span class="sd">    [&#39;c&#39;]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_rank_binary</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">_rank_lex</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">_rank_graycode</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">_subset</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">_superset</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">subset</span><span class="p">,</span> <span class="n">superset</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Default constructor.</span>
-
-<span class="sd">        It takes the subset and its superset as its parameters.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.subsets import Subset</span>
-<span class="sd">        &gt;&gt;&gt; a = Subset([&#39;c&#39;,&#39;d&#39;], [&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;d&#39;])</span>
-<span class="sd">        &gt;&gt;&gt; a.subset</span>
-<span class="sd">        [&#39;c&#39;, &#39;d&#39;]</span>
-<span class="sd">        &gt;&gt;&gt; a.superset</span>
-<span class="sd">        [&#39;a&#39;, &#39;b&#39;, &#39;c&#39;, &#39;d&#39;]</span>
-<span class="sd">        &gt;&gt;&gt; a.size</span>
-<span class="sd">        2</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">subset</span><span class="p">)</span> <span class="o">&gt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">superset</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Invalid arguments have been provided. The superset must be larger than the subset.&#39;</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">elem</span> <span class="ow">in</span> <span class="n">subset</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">elem</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">superset</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;The superset provided is invalid as it does not contain the element </span><span class="si">%i</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">elem</span><span class="p">)</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_subset</span> <span class="o">=</span> <span class="n">subset</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_superset</span> <span class="o">=</span> <span class="n">superset</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-<div class="viewcode-block" id="Subset.iterate_binary"><a class="viewcode-back" href="../../../modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.iterate_binary">[docs]</a>    <span class="k">def</span> <span class="nf">iterate_binary</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        This is a helper function. It iterates over the</span>
-<span class="sd">        binary subsets by k steps. This variable can be</span>
-<span class="sd">        both positive or negative.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.subsets import Subset</span>
-<span class="sd">        &gt;&gt;&gt; a = Subset([&#39;c&#39;,&#39;d&#39;], [&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;d&#39;])</span>
-<span class="sd">        &gt;&gt;&gt; a.iterate_binary(-2).subset</span>
-<span class="sd">        [&#39;d&#39;]</span>
-<span class="sd">        &gt;&gt;&gt; a = Subset([&#39;a&#39;,&#39;b&#39;,&#39;c&#39;], [&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;d&#39;])</span>
-<span class="sd">        &gt;&gt;&gt; a.iterate_binary(2).subset</span>
-<span class="sd">        []</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        next_binary, prev_binary</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">bin_list</span> <span class="o">=</span> <span class="n">Subset</span><span class="o">.</span><span class="n">bitlist_from_subset</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">subset</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">superset</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">bin_list</span><span class="p">),</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span><span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">superset_size</span>
-        <span class="n">bits</span> <span class="o">=</span> <span class="nb">bin</span><span class="p">(</span><span class="n">n</span><span class="p">)[</span><span class="mi">2</span><span class="p">:]</span><span class="o">.</span><span class="n">rjust</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">superset_size</span><span class="p">,</span> <span class="s">&#39;0&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Subset</span><span class="o">.</span><span class="n">subset_from_bitlist</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">superset</span><span class="p">,</span> <span class="n">bits</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Subset.next_binary"><a class="viewcode-back" href="../../../modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.next_binary">[docs]</a>    <span class="k">def</span> <span class="nf">next_binary</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Generates the next binary ordered subset.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.subsets import Subset</span>
-<span class="sd">        &gt;&gt;&gt; a = Subset([&#39;c&#39;,&#39;d&#39;], [&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;d&#39;])</span>
-<span class="sd">        &gt;&gt;&gt; a.next_binary().subset</span>
-<span class="sd">        [&#39;b&#39;]</span>
-<span class="sd">        &gt;&gt;&gt; a = Subset([&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;d&#39;], [&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;d&#39;])</span>
-<span class="sd">        &gt;&gt;&gt; a.next_binary().subset</span>
-<span class="sd">        []</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        prev_binary, iterate_binary</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">iterate_binary</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Subset.prev_binary"><a class="viewcode-back" href="../../../modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.prev_binary">[docs]</a>    <span class="k">def</span> <span class="nf">prev_binary</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Generates the previous binary ordered subset.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.subsets import Subset</span>
-<span class="sd">        &gt;&gt;&gt; a = Subset([], [&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;d&#39;])</span>
-<span class="sd">        &gt;&gt;&gt; a.prev_binary().subset</span>
-<span class="sd">        [&#39;a&#39;, &#39;b&#39;, &#39;c&#39;, &#39;d&#39;]</span>
-<span class="sd">        &gt;&gt;&gt; a = Subset([&#39;c&#39;,&#39;d&#39;], [&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;d&#39;])</span>
-<span class="sd">        &gt;&gt;&gt; a.prev_binary().subset</span>
-<span class="sd">        [&#39;c&#39;]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        next_binary, iterate_binary</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">iterate_binary</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Subset.next_lexicographic"><a class="viewcode-back" href="../../../modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.next_lexicographic">[docs]</a>    <span class="k">def</span> <span class="nf">next_lexicographic</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Generates the next lexicographically ordered subset.</span>
-
-<span class="sd">        NOT IMPLEMENTED</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-</div>
-<div class="viewcode-block" id="Subset.prev_lexicographic"><a class="viewcode-back" href="../../../modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.prev_lexicographic">[docs]</a>    <span class="k">def</span> <span class="nf">prev_lexicographic</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Generates the previous lexicographically ordered subset.</span>
-
-<span class="sd">        NOT YET IMPLEMENTED</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-</div>
-<div class="viewcode-block" id="Subset.iterate_graycode"><a class="viewcode-back" href="../../../modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.iterate_graycode">[docs]</a>    <span class="k">def</span> <span class="nf">iterate_graycode</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Helper function used for prev_gray and next_gray.</span>
-<span class="sd">        It performs k step overs to get the respective Gray codes.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.subsets import Subset</span>
-<span class="sd">        &gt;&gt;&gt; a = Subset([1,2,3], [1,2,3,4])</span>
-<span class="sd">        &gt;&gt;&gt; a.iterate_graycode(3).subset</span>
-<span class="sd">        [1, 4]</span>
-<span class="sd">        &gt;&gt;&gt; a.iterate_graycode(-2).subset</span>
-<span class="sd">        [1, 2, 4]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        next_gray, prev_gray</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">unranked_code</span> <span class="o">=</span> <span class="n">GrayCode</span><span class="o">.</span><span class="n">unrank</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">superset_size</span><span class="p">,</span>
-                                       <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rank_gray</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">cardinality</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Subset</span><span class="o">.</span><span class="n">subset_from_bitlist</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">superset</span><span class="p">,</span>
-                                          <span class="n">unranked_code</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Subset.next_gray"><a class="viewcode-back" href="../../../modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.next_gray">[docs]</a>    <span class="k">def</span> <span class="nf">next_gray</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Generates the next Gray code ordered subset.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.subsets import Subset</span>
-<span class="sd">        &gt;&gt;&gt; a = Subset([1,2,3], [1,2,3,4])</span>
-<span class="sd">        &gt;&gt;&gt; a.next_gray().subset</span>
-<span class="sd">        [1, 3]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        iterate_graycode, prev_gray</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">iterate_graycode</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Subset.prev_gray"><a class="viewcode-back" href="../../../modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.prev_gray">[docs]</a>    <span class="k">def</span> <span class="nf">prev_gray</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Generates the previous Gray code ordered subset.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.subsets import Subset</span>
-<span class="sd">        &gt;&gt;&gt; a = Subset([2,3,4], [1,2,3,4,5])</span>
-<span class="sd">        &gt;&gt;&gt; a.prev_gray().subset</span>
-<span class="sd">        [2, 3, 4, 5]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        iterate_graycode, next_gray</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">iterate_graycode</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Subset.rank_binary"><a class="viewcode-back" href="../../../modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.rank_binary">[docs]</a>    <span class="k">def</span> <span class="nf">rank_binary</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes the binary ordered rank.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.subsets import Subset</span>
-<span class="sd">        &gt;&gt;&gt; a = Subset([], [&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;d&#39;])</span>
-<span class="sd">        &gt;&gt;&gt; a.rank_binary</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; a = Subset([&#39;c&#39;,&#39;d&#39;], [&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;d&#39;])</span>
-<span class="sd">        &gt;&gt;&gt; a.rank_binary</span>
-<span class="sd">        3</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        iterate_binary, unrank_binary</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rank_binary</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_rank_binary</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="s">&quot;&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span>
-                <span class="n">Subset</span><span class="o">.</span><span class="n">bitlist_from_subset</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">subset</span><span class="p">,</span>
-                                           <span class="bp">self</span><span class="o">.</span><span class="n">superset</span><span class="p">)),</span> <span class="mi">2</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rank_binary</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Subset.rank_lexicographic"><a class="viewcode-back" href="../../../modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.rank_lexicographic">[docs]</a>    <span class="k">def</span> <span class="nf">rank_lexicographic</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes the lexicographic ranking of the subset.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.subsets import Subset</span>
-<span class="sd">        &gt;&gt;&gt; a = Subset([&#39;c&#39;,&#39;d&#39;], [&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;d&#39;])</span>
-<span class="sd">        &gt;&gt;&gt; a.rank_lexicographic</span>
-<span class="sd">        14</span>
-<span class="sd">        &gt;&gt;&gt; a = Subset([2,4,5], [1,2,3,4,5,6])</span>
-<span class="sd">        &gt;&gt;&gt; a.rank_lexicographic</span>
-<span class="sd">        43</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rank_lex</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">def</span> <span class="nf">_ranklex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">subset_index</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">subset_index</span> <span class="o">==</span> <span class="p">[]</span> <span class="ow">or</span> <span class="n">i</span> <span class="o">&gt;</span> <span class="n">n</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="mi">0</span>
-                <span class="k">if</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">subset_index</span><span class="p">:</span>
-                    <span class="n">subset_index</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                    <span class="k">return</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">_ranklex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">subset_index</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-                <span class="k">return</span> <span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">_ranklex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">subset_index</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-            <span class="n">indices</span> <span class="o">=</span> <span class="n">Subset</span><span class="o">.</span><span class="n">subset_indices</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">subset</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">superset</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_rank_lex</span> <span class="o">=</span> <span class="n">_ranklex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">indices</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">superset_size</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rank_lex</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Subset.rank_gray"><a class="viewcode-back" href="../../../modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.rank_gray">[docs]</a>    <span class="k">def</span> <span class="nf">rank_gray</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes the Gray code ranking of the subset.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.subsets import Subset</span>
-<span class="sd">        &gt;&gt;&gt; a = Subset([&#39;c&#39;,&#39;d&#39;], [&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;d&#39;])</span>
-<span class="sd">        &gt;&gt;&gt; a.rank_gray</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; a = Subset([2,4,5], [1,2,3,4,5,6])</span>
-<span class="sd">        &gt;&gt;&gt; a.rank_gray</span>
-<span class="sd">        27</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        iterate_graycode, unrank_gray</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rank_graycode</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">bits</span> <span class="o">=</span> <span class="n">Subset</span><span class="o">.</span><span class="n">bitlist_from_subset</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">subset</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">superset</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_rank_graycode</span> <span class="o">=</span> <span class="n">GrayCode</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">bits</span><span class="p">),</span> <span class="n">start</span><span class="o">=</span><span class="n">bits</span><span class="p">)</span><span class="o">.</span><span class="n">rank</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rank_graycode</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Subset.subset"><a class="viewcode-back" href="../../../modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.subset">[docs]</a>    <span class="k">def</span> <span class="nf">subset</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Gets the subset represented by the current instance.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.subsets import Subset</span>
-<span class="sd">        &gt;&gt;&gt; a = Subset([&#39;c&#39;,&#39;d&#39;], [&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;d&#39;])</span>
-<span class="sd">        &gt;&gt;&gt; a.subset</span>
-<span class="sd">        [&#39;c&#39;, &#39;d&#39;]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        superset, size, superset_size, cardinality</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_subset</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Subset.size"><a class="viewcode-back" href="../../../modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.size">[docs]</a>    <span class="k">def</span> <span class="nf">size</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Gets the size of the subset.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.subsets import Subset</span>
-<span class="sd">        &gt;&gt;&gt; a = Subset([&#39;c&#39;,&#39;d&#39;], [&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;d&#39;])</span>
-<span class="sd">        &gt;&gt;&gt; a.size</span>
-<span class="sd">        2</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        subset, superset, superset_size, cardinality</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">subset</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Subset.superset"><a class="viewcode-back" href="../../../modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.superset">[docs]</a>    <span class="k">def</span> <span class="nf">superset</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Gets the superset of the subset.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.subsets import Subset</span>
-<span class="sd">        &gt;&gt;&gt; a = Subset([&#39;c&#39;,&#39;d&#39;], [&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;d&#39;])</span>
-<span class="sd">        &gt;&gt;&gt; a.superset</span>
-<span class="sd">        [&#39;a&#39;, &#39;b&#39;, &#39;c&#39;, &#39;d&#39;]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        subset, size, superset_size, cardinality</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_superset</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Subset.superset_size"><a class="viewcode-back" href="../../../modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.superset_size">[docs]</a>    <span class="k">def</span> <span class="nf">superset_size</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the size of the superset.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.subsets import Subset</span>
-<span class="sd">        &gt;&gt;&gt; a = Subset([&#39;c&#39;,&#39;d&#39;], [&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;d&#39;])</span>
-<span class="sd">        &gt;&gt;&gt; a.superset_size</span>
-<span class="sd">        4</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        subset, superset, size, cardinality</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">superset</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Subset.cardinality"><a class="viewcode-back" href="../../../modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.cardinality">[docs]</a>    <span class="k">def</span> <span class="nf">cardinality</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the number of all possible subsets.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.subsets import Subset</span>
-<span class="sd">        &gt;&gt;&gt; a = Subset([&#39;c&#39;,&#39;d&#39;], [&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;d&#39;])</span>
-<span class="sd">        &gt;&gt;&gt; a.cardinality</span>
-<span class="sd">        16</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        subset, superset, size, superset_size</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">superset_size</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Subset.subset_from_bitlist"><a class="viewcode-back" href="../../../modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.subset_from_bitlist">[docs]</a>    <span class="k">def</span> <span class="nf">subset_from_bitlist</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">super_set</span><span class="p">,</span> <span class="n">bitlist</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Gets the subset defined by the bitlist.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.subsets import Subset</span>
-<span class="sd">        &gt;&gt;&gt; Subset.subset_from_bitlist([&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;d&#39;], &#39;0011&#39;).subset</span>
-<span class="sd">        [&#39;c&#39;, &#39;d&#39;]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        bitlist_from_subset</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">super_set</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">bitlist</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;The sizes of the lists are not equal&quot;</span><span class="p">)</span>
-        <span class="n">ret_set</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">bitlist</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="n">bitlist</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;1&#39;</span><span class="p">:</span>
-                <span class="n">ret_set</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">super_set</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">Subset</span><span class="p">(</span><span class="n">ret_set</span><span class="p">,</span> <span class="n">super_set</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Subset.bitlist_from_subset"><a class="viewcode-back" href="../../../modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.bitlist_from_subset">[docs]</a>    <span class="k">def</span> <span class="nf">bitlist_from_subset</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">subset</span><span class="p">,</span> <span class="n">superset</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Gets the bitlist corresponding to a subset.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.subsets import Subset</span>
-<span class="sd">        &gt;&gt;&gt; Subset.bitlist_from_subset([&#39;c&#39;,&#39;d&#39;], [&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;d&#39;])</span>
-<span class="sd">        &#39;0011&#39;</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        subset_from_bitlist</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">bitlist</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;0&#39;</span><span class="p">]</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">superset</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">subset</span><span class="p">)</span> <span class="ow">is</span> <span class="n">Subset</span><span class="p">:</span>
-            <span class="n">subset</span> <span class="o">=</span> <span class="n">subset</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">Subset</span><span class="o">.</span><span class="n">subset_indices</span><span class="p">(</span><span class="n">subset</span><span class="p">,</span> <span class="n">superset</span><span class="p">):</span>
-            <span class="n">bitlist</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="s">&#39;1&#39;</span>
-        <span class="k">return</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">bitlist</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Subset.unrank_binary"><a class="viewcode-back" href="../../../modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.unrank_binary">[docs]</a>    <span class="k">def</span> <span class="nf">unrank_binary</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rank</span><span class="p">,</span> <span class="n">superset</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Gets the binary ordered subset of the specified rank.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.subsets import Subset</span>
-<span class="sd">        &gt;&gt;&gt; Subset.unrank_binary(4, [&#39;a&#39;,&#39;b&#39;,&#39;c&#39;,&#39;d&#39;]).subset</span>
-<span class="sd">        [&#39;b&#39;]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        iterate_binary, rank_binary</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">bits</span> <span class="o">=</span> <span class="nb">bin</span><span class="p">(</span><span class="n">rank</span><span class="p">)[</span><span class="mi">2</span><span class="p">:]</span><span class="o">.</span><span class="n">rjust</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">superset</span><span class="p">),</span> <span class="s">&#39;0&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Subset</span><span class="o">.</span><span class="n">subset_from_bitlist</span><span class="p">(</span><span class="n">superset</span><span class="p">,</span> <span class="n">bits</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Subset.unrank_gray"><a class="viewcode-back" href="../../../modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.unrank_gray">[docs]</a>    <span class="k">def</span> <span class="nf">unrank_gray</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rank</span><span class="p">,</span> <span class="n">superset</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Gets the Gray code ordered subset of the specified rank.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics.subsets import Subset</span>
-<span class="sd">        &gt;&gt;&gt; Subset.unrank_gray(4, [&#39;a&#39;,&#39;b&#39;,&#39;c&#39;]).subset</span>
-<span class="sd">        [&#39;a&#39;, &#39;b&#39;]</span>
-<span class="sd">        &gt;&gt;&gt; Subset.unrank_gray(0, [&#39;a&#39;,&#39;b&#39;,&#39;c&#39;]).subset</span>
-<span class="sd">        []</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        iterate_graycode, rank_gray</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">graycode_bitlist</span> <span class="o">=</span> <span class="n">GrayCode</span><span class="o">.</span><span class="n">unrank</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">superset</span><span class="p">),</span> <span class="n">rank</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Subset</span><span class="o">.</span><span class="n">subset_from_bitlist</span><span class="p">(</span><span class="n">superset</span><span class="p">,</span> <span class="n">graycode_bitlist</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Subset.subset_indices"><a class="viewcode-back" href="../../../modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.subset_indices">[docs]</a>    <span class="k">def</span> <span class="nf">subset_indices</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">subset</span><span class="p">,</span> <span class="n">superset</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return indices of subset in superset in a list; the list is empty</span>
-<span class="sd">        if all elements of subset are not in superset.</span>
-
-<span class="sd">        Examples::</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy.combinatorics import Subset</span>
-<span class="sd">            &gt;&gt;&gt; superset = [1, 3, 2, 5, 4]</span>
-<span class="sd">            &gt;&gt;&gt; Subset.subset_indices([3, 2, 1], superset)</span>
-<span class="sd">            [1, 2, 0]</span>
-<span class="sd">            &gt;&gt;&gt; Subset.subset_indices([1, 6], superset)</span>
-<span class="sd">            []</span>
-<span class="sd">            &gt;&gt;&gt; Subset.subset_indices([], superset)</span>
-<span class="sd">            []</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">superset</span><span class="p">,</span> <span class="n">subset</span>
-        <span class="n">sb</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">ai</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">a</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">ai</span> <span class="ow">in</span> <span class="n">sb</span><span class="p">:</span>
-                <span class="n">d</span><span class="p">[</span><span class="n">ai</span><span class="p">]</span> <span class="o">=</span> <span class="n">i</span>
-                <span class="n">sb</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">ai</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">sb</span><span class="p">:</span>
-                    <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">list</span><span class="p">()</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">d</span><span class="p">[</span><span class="n">bi</span><span class="p">]</span> <span class="k">for</span> <span class="n">bi</span> <span class="ow">in</span> <span class="n">b</span><span class="p">]</span>
-
-</div></div>
-<span class="k">def</span> <span class="nf">ksubsets</span><span class="p">(</span><span class="n">superset</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Finds the subsets of size k in lexicographic order.</span>
-
-<span class="sd">    This uses the itertools generator.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.subsets import ksubsets</span>
-<span class="sd">    &gt;&gt;&gt; list(ksubsets([1,2,3], 2))</span>
-<span class="sd">    [(1, 2), (1, 3), (2, 3)]</span>
-<span class="sd">    &gt;&gt;&gt; list(ksubsets([1,2,3,4,5], 2))</span>
-<span class="sd">    [(1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), \</span>
-<span class="sd">    (2, 5), (3, 4), (3, 5), (4, 5)]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    class:Subset</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">combinations</span><span class="p">(</span><span class="n">superset</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.combinatorics.tensor_can</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span><span class="p">,</span> <span class="n">_af_rmul</span><span class="p">,</span> <span class="n">_af_rmuln</span><span class="p">,</span>\
-  <span class="n">_af_invert</span><span class="p">,</span> <span class="n">_af_new</span>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span><span class="p">,</span> <span class="n">_orbit</span><span class="p">,</span> \
-  <span class="n">_orbit_transversal</span>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.util</span> <span class="kn">import</span> <span class="n">_distribute_gens_by_base</span><span class="p">,</span> \
-  <span class="n">_orbits_transversals_from_bsgs</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    References for tensor canonicalization:</span>
-
-<span class="sd">    [1] R. Portugal &quot;Algorithmic simplification of tensor expressions&quot;,</span>
-<span class="sd">        J. Phys. A 32 (1999) 7779-7789</span>
-
-<span class="sd">    [2] R. Portugal, B.F. Svaiter &quot;Group-theoretic Approach for Symbolic</span>
-<span class="sd">        Tensor Manipulation: I. Free Indices&quot;</span>
-<span class="sd">        arXiv:math-ph/0107031v1</span>
-
-<span class="sd">    [3] L.R.U. Manssur, R. Portugal &quot;Group-theoretic Approach for Symbolic</span>
-<span class="sd">        Tensor Manipulation: II. Dummy Indices&quot;</span>
-<span class="sd">        arXiv:math-ph/0107032v1</span>
-
-<span class="sd">    [4] xperm.c part of XPerm written by J. M. Martin-Garcia</span>
-<span class="sd">        http://www.xact.es/index.html</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-
-<span class="k">def</span> <span class="nf">dummy_sgs</span><span class="p">(</span><span class="n">dummies</span><span class="p">,</span> <span class="n">sym</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the strong generators for dummy indices</span>
-
-<span class="sd">    dummies   list of dummy indices,</span>
-<span class="sd">              `dummies[2k], dummies[2k+1]` are paired indices</span>
-<span class="sd">    sym       symmetry under interchange of contracted dummies</span>
-<span class="sd">    sym =     None  no symmetry</span>
-<span class="sd">              0 symmetric</span>
-<span class="sd">              1 antisymmetric</span>
-<span class="sd">    n         number of indices</span>
-
-<span class="sd">    in base form the dummy indices are always in consecutive positions</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.tensor_can import dummy_sgs</span>
-<span class="sd">    &gt;&gt;&gt; dummy_sgs([2,3,4,5,6,7], 0, 8)</span>
-<span class="sd">    [[0, 1, 3, 2, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 5, 4, 6, 7, 8, 9], [0, 1, 2, 3, 4, 5, 7, 6, 8, 9], [0, 1, 4, 5, 2, 3, 6, 7, 8, 9], [0, 1, 2, 3, 6, 7, 4, 5, 8, 9]]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">dummies</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="n">n</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="c"># exchange of contravariant and covariant indices</span>
-    <span class="k">if</span> <span class="n">sym</span> <span class="o">!=</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">dummies</span><span class="p">[::</span><span class="mi">2</span><span class="p">]:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">2</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">sym</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">a</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">n</span><span class="o">+</span><span class="mi">1</span>
-                <span class="n">a</span><span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">n</span>
-            <span class="n">a</span><span class="p">[</span><span class="n">j</span><span class="p">],</span> <span class="n">a</span><span class="p">[</span><span class="n">j</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="n">j</span><span class="o">+</span><span class="mi">1</span><span class="p">],</span> <span class="n">a</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-            <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-    <span class="c"># rename dummy indices</span>
-    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">dummies</span><span class="p">[:</span><span class="o">-</span><span class="mi">3</span><span class="p">:</span><span class="mi">2</span><span class="p">]:</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">2</span><span class="p">))</span>
-        <span class="n">a</span><span class="p">[</span><span class="n">j</span><span class="p">:</span><span class="n">j</span><span class="o">+</span><span class="mi">4</span><span class="p">]</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="n">j</span><span class="o">+</span><span class="mi">2</span><span class="p">],</span><span class="n">a</span><span class="p">[</span><span class="n">j</span><span class="o">+</span><span class="mi">3</span><span class="p">],</span><span class="n">a</span><span class="p">[</span><span class="n">j</span><span class="p">],</span><span class="n">a</span><span class="p">[</span><span class="n">j</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">res</span>
-
-<span class="k">def</span> <span class="nf">_min_dummies</span><span class="p">(</span><span class="n">dummies</span><span class="p">,</span> <span class="n">sym</span><span class="p">,</span> <span class="n">indices</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return list of minima of the orbits of indices in group of dummies</span>
-<span class="sd">    see `double_coset_can_rep` for the description of `dummies` and `sym`</span>
-<span class="sd">    indices is the initial list of dummy indices</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.tensor_can import _min_dummies</span>
-<span class="sd">    &gt;&gt;&gt; _min_dummies([[2,3,4,5,6,7]], [0], list(range(10)))</span>
-<span class="sd">    [0, 1, 2, 2, 2, 2, 2, 2, 8, 9]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">num_types</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">sym</span><span class="p">)</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">dx</span> <span class="ow">in</span> <span class="n">dummies</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">dx</span><span class="p">:</span>
-            <span class="n">m</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">min</span><span class="p">(</span><span class="n">dx</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">m</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">None</span><span class="p">)</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="n">indices</span><span class="p">[:]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_types</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">c</span><span class="p">,</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">indices</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_types</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">dummies</span><span class="p">[</span><span class="n">j</span><span class="p">]:</span>
-                    <span class="n">res</span><span class="p">[</span><span class="n">c</span><span class="p">]</span> <span class="o">=</span> <span class="n">m</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-                    <span class="k">break</span>
-    <span class="k">return</span> <span class="n">res</span>
-
-
-<span class="k">def</span> <span class="nf">_trace_S</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">S_cosets</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the representative h satisfying s[h[b]] == j</span>
-
-<span class="sd">    If there is not such a representative return None</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">S_cosets</span><span class="p">[</span><span class="n">b</span><span class="p">]:</span>
-        <span class="k">if</span> <span class="n">s</span><span class="p">[</span><span class="n">h</span><span class="p">[</span><span class="n">b</span><span class="p">]]</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">h</span>
-    <span class="k">return</span> <span class="bp">None</span>
-
-<span class="k">def</span> <span class="nf">_trace_D</span><span class="p">(</span><span class="n">gj</span><span class="p">,</span> <span class="n">p_i</span><span class="p">,</span> <span class="n">Dxtrav</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the representative h satisfying h[gj] == p_i</span>
-
-<span class="sd">    If there is not such a representative return None</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">Dxtrav</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">h</span><span class="p">[</span><span class="n">gj</span><span class="p">]</span> <span class="o">==</span> <span class="n">p_i</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">h</span>
-    <span class="k">return</span> <span class="bp">None</span>
-
-<span class="k">def</span> <span class="nf">_dumx_remove</span><span class="p">(</span><span class="n">dumx</span><span class="p">,</span> <span class="n">dumx_flat</span><span class="p">,</span> <span class="n">p0</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    remove p0 from dumx</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">dx</span> <span class="ow">in</span> <span class="n">dumx</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">p0</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">dx</span><span class="p">:</span>
-            <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dx</span><span class="p">)</span>
-            <span class="k">continue</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">dx</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">p0</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">k</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">p0_paired</span> <span class="o">=</span> <span class="n">dx</span><span class="p">[</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">p0_paired</span> <span class="o">=</span> <span class="n">dx</span><span class="p">[</span><span class="n">k</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">dx</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">p0</span><span class="p">)</span>
-        <span class="n">dx</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">p0_paired</span><span class="p">)</span>
-        <span class="n">dumx_flat</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">p0</span><span class="p">)</span>
-        <span class="n">dumx_flat</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">p0_paired</span><span class="p">)</span>
-        <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dx</span><span class="p">)</span>
-
-<span class="k">def</span> <span class="nf">transversal2coset</span><span class="p">(</span><span class="n">size</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="n">transversal</span><span class="p">):</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">base</span><span class="p">:</span>
-            <span class="n">a</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="n">transversal</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">values</span><span class="p">()))</span>
-            <span class="n">j</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">a</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">))])</span>
-    <span class="n">j</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span>
-    <span class="k">while</span> <span class="n">a</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">))]:</span>
-        <span class="n">j</span> <span class="o">-=</span> <span class="mi">1</span>
-    <span class="k">return</span> <span class="n">a</span><span class="p">[:</span><span class="n">j</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span>
-
-
-<div class="viewcode-block" id="double_coset_can_rep"><a class="viewcode-back" href="../../../modules/combinatorics/tensor_can.html#sympy.combinatorics.tensor_can.double_coset_can_rep">[docs]</a><span class="k">def</span> <span class="nf">double_coset_can_rep</span><span class="p">(</span><span class="n">dummies</span><span class="p">,</span> <span class="n">sym</span><span class="p">,</span> <span class="n">b_S</span><span class="p">,</span> <span class="n">sgens</span><span class="p">,</span> <span class="n">S_transversals</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Butler-Portugal algorithm for tensor canonicalization with dummy indices</span>
-
-<span class="sd">      dummies</span>
-<span class="sd">        list of lists of dummy indices,</span>
-<span class="sd">        one list for each type of index;</span>
-<span class="sd">        the dummy indices are put in order contravariant, covariant</span>
-<span class="sd">        [d0, -d0, d1,-d1,...].</span>
-
-<span class="sd">      sym</span>
-<span class="sd">        list of the symmetries of the index metric for each type.</span>
-
-<span class="sd">      possible symmetries of the metrics</span>
-<span class="sd">              * 0     symmetric</span>
-<span class="sd">              * 1     antisymmetric</span>
-<span class="sd">              * None  no symmetry</span>
-
-<span class="sd">      b_S</span>
-<span class="sd">        base of a minimal slot symmetry BSGS.</span>
-
-<span class="sd">      sgens</span>
-<span class="sd">        generators of the slot symmetry BSGS.</span>
-
-<span class="sd">      S_transversals</span>
-<span class="sd">        transversals for the slot BSGS.</span>
-
-<span class="sd">      g</span>
-<span class="sd">        permutation representing the tensor.</span>
-
-<span class="sd">    Return 0 if the tensor is zero, else return the array form of</span>
-<span class="sd">    the permutation representing the canonical form of the tensor.</span>
-
-
-<span class="sd">    A tensor with dummy indices can be represented in a number</span>
-<span class="sd">    of equivalent ways which typically grows exponentially with</span>
-<span class="sd">    the number of indices. To be able to establish if two tensors</span>
-<span class="sd">    with many indices are equal becomes computationally very slow</span>
-<span class="sd">    in absence of an efficient algorithm.</span>
-
-<span class="sd">    The Butler-Portugal algorithm [3] is an efficient algorithm to</span>
-<span class="sd">    put tensors in canonical form, solving the above problem.</span>
-
-<span class="sd">    Portugal observed that a tensor can be represented by a permutation,</span>
-<span class="sd">    and that the class of tensors equivalent to it under slot and dummy</span>
-<span class="sd">    symmetries is equivalent to the double coset `D*g*S`</span>
-<span class="sd">    (Note: in this documentation we use the conventions for multiplication</span>
-<span class="sd">    of permutations p, q with (p*q)(i) = p[q[i]] which is opposite</span>
-<span class="sd">    to the one used in the Permutation class)</span>
-
-<span class="sd">    Using the algorithm by Butler to find a representative of the</span>
-<span class="sd">    double coset one can find a canonical form for the tensor.</span>
-
-<span class="sd">    To see this correspondence,</span>
-<span class="sd">    let `g` be a permutation in array form; a tensor with indices `ind`</span>
-<span class="sd">    (the indices including both the contravariant and the covariant ones)</span>
-<span class="sd">    can be written as</span>
-
-<span class="sd">    `t = T(ind[g[0],..., ind[g[n-1]])`,</span>
-
-<span class="sd">    where `n= len(ind)`;</span>
-<span class="sd">    `g` has size `n + 2`, the last two indices for the sign of the tensor</span>
-<span class="sd">    (trick introduced in [4]).</span>
-
-<span class="sd">    A slot symmetry transformation `s` is a permutation acting on the slots</span>
-<span class="sd">    `t -&gt; T(ind[(g*s)[0]],..., ind[(g*s)[n-1]])`</span>
-
-<span class="sd">    A dummy symmetry transformation acts on `ind`</span>
-<span class="sd">    `t -&gt; T(ind[(d*g)[0]],..., ind[(d*g)[n-1]])`</span>
-
-<span class="sd">    Being interested only in the transformations of the tensor under</span>
-<span class="sd">    these symmetries, one can represent the tensor by `g`, which transforms</span>
-<span class="sd">    as</span>
-
-<span class="sd">    `g -&gt; d*g*s`, so it belongs to the coset `D*g*S`.</span>
-
-<span class="sd">    Let us explain the conventions by an example.</span>
-
-<span class="sd">    Given a tensor `T^{d3 d2 d1}{}_{d1 d2 d3}` with the slot symmetries</span>
-<span class="sd">          `T^{a0 a1 a2 a3 a4 a5} = -T^{a2 a1 a0 a3 a4 a5}`</span>
-
-<span class="sd">          `T^{a0 a1 a2 a3 a4 a5} = -T^{a4 a1 a2 a3 a0 a5}`</span>
-
-<span class="sd">    and symmetric metric, find the tensor equivalent to it which</span>
-<span class="sd">    is the lowest under the ordering of indices:</span>
-<span class="sd">    lexicographic ordering `d1, d2, d3` then and contravariant index</span>
-<span class="sd">    before covariant index; that is the canonical form of the tensor.</span>
-
-<span class="sd">    The canonical form is `-T^{d1 d2 d3}{}_{d1 d2 d3}`</span>
-<span class="sd">    obtained using `T^{a0 a1 a2 a3 a4 a5} = -T^{a2 a1 a0 a3 a4 a5}`.</span>
-
-<span class="sd">    To convert this problem in the input for this function,</span>
-<span class="sd">    use the following labelling of the index names</span>
-<span class="sd">    (- for covariant for short) `d1, -d1, d2, -d2, d3, -d3`</span>
-
-<span class="sd">    `T^{d3 d2 d1}{}_{d1 d2 d3}` corresponds to `g = [4,2,0,1,3,5,6,7]`</span>
-<span class="sd">    where the last two indices are for the sign</span>
-
-<span class="sd">    `sgens = [Permutation(0,2)(6,7), Permutation(0,4)(6,7)]`</span>
-
-<span class="sd">    sgens[0] is the slot symmetry `-(0,2)`</span>
-<span class="sd">    `T^{a0 a1 a2 a3 a4 a5} = -T^{a2 a1 a0 a3 a4 a5}`</span>
-
-<span class="sd">    sgens[1] is the slot symmetry `-(0,4)`</span>
-<span class="sd">    `T^{a0 a1 a2 a3 a4 a5} = -T^{a4 a1 a2 a3 a0 a5}`</span>
-
-<span class="sd">    The dummy symmetry group D is generated by the strong base generators</span>
-<span class="sd">    `[(0,1),(2,3),(4,5),(0,1)(2,3),(2,3)(4,5)]`</span>
-
-<span class="sd">    The dummy symmetry acts from the left</span>
-<span class="sd">    `d = [1,0,2,3,4,5,6,7]`  exchange `d1 -&gt; -d1`</span>
-<span class="sd">    `T^{d3 d2 d1}{}_{d1 d2 d3} == T^{d3 d2}{}_{d1}{}^{d1}{}_{d2 d3}`</span>
-
-<span class="sd">    `g=[4,2,0,1,3,5,6,7]  -&gt; [4,2,1,0,3,5,6,7] = _af_rmul(d, g)`</span>
-<span class="sd">    which differs from `_af_rmul(g, d)`.</span>
-
-<span class="sd">    The slot symmetry acts from the right</span>
-<span class="sd">    `s = [2,1,0,3,4,5,7,6]`  exchanges slots 0 and 2 and changes sign</span>
-<span class="sd">    `T^{d3 d2 d1}{}_{d1 d2 d3} == -T^{d1 d2 d3}{}_{d1 d2 d3}`</span>
-
-<span class="sd">    `g=[4,2,0,1,3,5,6,7]  -&gt; [0,2,4,1,3,5,7,6] = _af_rmul(g, s)`</span>
-
-<span class="sd">    Example in which the tensor is zero, same slot symmetries as above:</span>
-<span class="sd">    `T^{d3}{}_{d1,d2}{}^{d1}{}_{d3}{}^{d2}`</span>
-
-<span class="sd">    `= -T^{d3}{}_{d1,d3}{}^{d1}{}_{d2}{}^{d2}`   under slot symmetry `-(2,4)`;</span>
-
-<span class="sd">    `= T_{d3 d1}{}^{d3}{}^{d1}{}_{d2}{}^{d2}`    under slot symmetry `-(0,2)`;</span>
-
-<span class="sd">    `= T^{d3}{}_{d1 d3}{}^{d1}{}_{d2}{}^{d2}`    symmetric metric;</span>
-
-<span class="sd">    `= 0`  since two of these lines have tensors differ only for the sign.</span>
-
-<span class="sd">    The double coset D*g*S consists of permutations `h = d*g*s` corresponding</span>
-<span class="sd">    to equivalent tensors; if there are two `h` which are the same apart</span>
-<span class="sd">    from the sign, return zero; otherwise</span>
-<span class="sd">    choose as representative the tensor with indices</span>
-<span class="sd">    ordered lexicographically according to `[d1, -d1, d2, -d2, d3, -d3]`</span>
-<span class="sd">    that is `rep = min(D*g*S) = min([d*g*s for d in D for s in S])`</span>
-
-<span class="sd">    The indices are fixed one by one; first choose the lowest index</span>
-<span class="sd">    for slot 0, then the lowest remaining index for slot 1, etc.</span>
-<span class="sd">    Doing this one obtains a chain of stabilizers</span>
-
-<span class="sd">    `S -&gt; S_{b0} -&gt; S_{b0,b1} -&gt; ...` and</span>
-<span class="sd">    `D -&gt; D_{p0} -&gt; D_{p0,p1} -&gt; ...`</span>
-
-<span class="sd">    where `[b0, b1, ...] = range(b)` is a base of the symmetric group;</span>
-<span class="sd">    the strong base `b_S` of S is an ordered sublist of it;</span>
-<span class="sd">    therefore it is sufficient to compute once the</span>
-<span class="sd">    strong base generators of S using the Schreier-Sims algorithm;</span>
-<span class="sd">    the stabilizers of the strong base generators are the</span>
-<span class="sd">    strong base generators of the stabilizer subgroup.</span>
-
-<span class="sd">    `dbase = [p0,p1,...]` is not in general in lexicographic order,</span>
-<span class="sd">    so that one must recompute the strong base generators each time;</span>
-<span class="sd">    however this is trivial, there is no need to use the Schreier-Sims</span>
-<span class="sd">    algorithm for D.</span>
-
-<span class="sd">    The algorithm keeps a TAB of elements `(s_i, d_i, h_i)`</span>
-<span class="sd">    where `h_i = d_i*g*s_i` satisfying `h_i[j] = p_j` for `0 &lt;= j &lt; i`</span>
-<span class="sd">    starting from `s_0 = id, d_0 = id, h_0 = g`.</span>
-
-<span class="sd">    The equations `h_0[0] = p_0, h_1[1] = p_1,...` are solved in this order,</span>
-<span class="sd">    choosing each time the lowest possible value of p_i</span>
-
-<span class="sd">    For `j &lt; i`</span>
-<span class="sd">    `d_i*g*s_i*S_{b_0,...,b_{i-1}}*b_j = D_{p_0,...,p_{i-1}}*p_j`</span>
-<span class="sd">    so that for dx in `D_{p_0,...,p_{i-1}}` and sx in</span>
-<span class="sd">    `S_{base[0],...,base[i-1]}` one has `dx*d_i*g*s_i*sx*b_j = p_j`</span>
-
-<span class="sd">    Search for dx, sx such that this equation holds for `j = i`;</span>
-<span class="sd">    it can be written as `s_i*sx*b_j = J, dx*d_i*g*J = p_j`</span>
-<span class="sd">    `sx*b_j = s_i**-1*J; sx = trace(s_i**-1, S_{b_0,...,b_{i-1}})`</span>
-<span class="sd">    `dx**-1*p_j = d_i*g*J; dx = trace(d_i*g*J, D_{p_0,...,p_{i-1}})`</span>
-
-<span class="sd">    `s_{i+1} = s_i*trace(s_i**-1*J, S_{b_0,...,b_{i-1}})`</span>
-<span class="sd">    `d_{i+1} = trace(d_i*g*J, D_{p_0,...,p_{i-1}})**-1*d_i`</span>
-<span class="sd">    `h_{i+1}*b_i = d_{i+1}*g*s_{i+1}*b_i = p_i`</span>
-
-<span class="sd">    `h_n*b_j = p_j` for all j, so that `h_n` is the solution.</span>
-
-<span class="sd">    Add the found `(s, d, h)` to TAB1.</span>
-
-<span class="sd">    At the end of the iteration sort TAB1 with respect to the `h`;</span>
-<span class="sd">    if there are two consecutive `h` in TAB1 which differ only for the</span>
-<span class="sd">    sign, the tensor is zero, so return 0;</span>
-<span class="sd">    if there are two consecutive `h` which are equal, keep only one.</span>
-
-<span class="sd">    Then stabilize the slot generators under `i` and the dummy generators</span>
-<span class="sd">    under `p_i`.</span>
-
-<span class="sd">    Assign `TAB = TAB1` at the end of the iteration step.</span>
-
-<span class="sd">    At the end `TAB` contains a unique `(s, d, h)`, since all the slots</span>
-<span class="sd">    of the tensor `h` have been fixed to have the minimum value according</span>
-<span class="sd">    to the symmetries. The algorithm returns `h`.</span>
-
-<span class="sd">    It is important that the slot BSGS has lexicographic minimal base,</span>
-<span class="sd">    otherwise there is an `i` which does not belong to the slot base</span>
-<span class="sd">    for which `p_i` is fixed by the dummy symmetry only, while `i`</span>
-<span class="sd">    is not invariant from the slot stabilizer, so `p_i` is not in</span>
-<span class="sd">    general the minimal value.</span>
-
-<span class="sd">    This algorithm differs slightly from the original algorithm [3]:</span>
-<span class="sd">      the canonical form is minimal lexicographically, and</span>
-<span class="sd">      the BSGS has minimal base under lexicographic order.</span>
-<span class="sd">      Equal tensors `h` are eliminated from TAB.</span>
-
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.tensor_can import double_coset_can_rep, get_transversals</span>
-<span class="sd">    &gt;&gt;&gt; gens = [Permutation(x) for x in [[2,1,0,3,4,5,7,6], [4,1,2,3,0,5,7,6]]]</span>
-<span class="sd">    &gt;&gt;&gt; base = [0, 2]</span>
-<span class="sd">    &gt;&gt;&gt; g = Permutation([4,2,0,1,3,5,6,7])</span>
-<span class="sd">    &gt;&gt;&gt; transversals = get_transversals(base, gens)</span>
-<span class="sd">    &gt;&gt;&gt; double_coset_can_rep([list(range(6))], [0], base, gens, transversals, g)</span>
-<span class="sd">    [0, 1, 2, 3, 4, 5, 7, 6]</span>
-
-<span class="sd">    &gt;&gt;&gt; g = Permutation([4,1,3,0,5,2,6,7])</span>
-<span class="sd">    &gt;&gt;&gt; double_coset_can_rep([list(range(6))], [0], base, gens, transversals, g)</span>
-<span class="sd">    0</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">size</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">size</span>
-    <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">array_form</span>
-    <span class="n">num_dummies</span> <span class="o">=</span> <span class="n">size</span> <span class="o">-</span> <span class="mi">2</span>
-    <span class="n">indices</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">num_dummies</span><span class="p">))</span>
-    <span class="n">all_metrics_with_sym</span> <span class="o">=</span> <span class="nb">all</span><span class="p">([</span><span class="n">_</span> <span class="o">!=</span> <span class="bp">None</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">sym</span><span class="p">])</span>
-    <span class="n">num_types</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">sym</span><span class="p">)</span>
-    <span class="n">dumx</span> <span class="o">=</span> <span class="n">dummies</span><span class="p">[:]</span>
-    <span class="n">dumx_flat</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">dx</span> <span class="ow">in</span> <span class="n">dumx</span><span class="p">:</span>
-        <span class="n">dumx_flat</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">dx</span><span class="p">)</span>
-    <span class="n">b_S</span> <span class="o">=</span> <span class="n">b_S</span><span class="p">[:]</span>
-    <span class="n">sgensx</span> <span class="o">=</span> <span class="p">[</span><span class="n">h</span><span class="o">.</span><span class="n">_array_form</span> <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">sgens</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">b_S</span><span class="p">:</span>
-        <span class="n">S_transversals</span> <span class="o">=</span> <span class="n">transversal2coset</span><span class="p">(</span><span class="n">size</span><span class="p">,</span> <span class="n">b_S</span><span class="p">,</span> <span class="n">S_transversals</span><span class="p">)</span>
-    <span class="c"># strong generating set for D</span>
-    <span class="n">dsgsx</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_types</span><span class="p">):</span>
-       <span class="n">dsgsx</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">dummy_sgs</span><span class="p">(</span><span class="n">dumx</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">sym</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">num_dummies</span><span class="p">))</span>
-    <span class="n">ginv</span> <span class="o">=</span> <span class="n">_af_invert</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-    <span class="n">idn</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">))</span>
-    <span class="c"># TAB = list of entries (s, d, h) where h = _af_rmuln(d,g,s)</span>
-    <span class="c"># for short, in the following d*g*s means _af_rmuln(d,g,s)</span>
-    <span class="n">TAB</span> <span class="o">=</span> <span class="p">[(</span><span class="n">idn</span><span class="p">,</span> <span class="n">idn</span><span class="p">,</span> <span class="n">g</span><span class="p">)]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span> <span class="o">-</span> <span class="mi">2</span><span class="p">):</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">i</span>
-        <span class="n">testb</span> <span class="o">=</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">b_S</span> <span class="ow">and</span> <span class="n">sgensx</span>
-        <span class="k">if</span> <span class="n">testb</span><span class="p">:</span>
-            <span class="n">sgensx1</span> <span class="o">=</span> <span class="p">[</span><span class="n">_af_new</span><span class="p">(</span><span class="n">_</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">sgensx</span><span class="p">]</span>
-            <span class="n">deltab</span> <span class="o">=</span> <span class="n">_orbit</span><span class="p">(</span><span class="n">size</span><span class="p">,</span> <span class="n">sgensx1</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">deltab</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span><span class="n">b</span><span class="p">])</span>
-        <span class="c"># p1 = min(IMAGES) = min(Union D_p*h*deltab for h in TAB)</span>
-        <span class="k">if</span> <span class="n">all_metrics_with_sym</span><span class="p">:</span>
-            <span class="n">md</span> <span class="o">=</span> <span class="n">_min_dummies</span><span class="p">(</span><span class="n">dumx</span><span class="p">,</span> <span class="n">sym</span><span class="p">,</span> <span class="n">indices</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">md</span> <span class="o">=</span> <span class="p">[</span><span class="nb">min</span><span class="p">(</span><span class="n">_orbit</span><span class="p">(</span><span class="n">size</span><span class="p">,</span> <span class="p">[</span><span class="n">_af_new</span><span class="p">(</span><span class="n">ddx</span><span class="p">)</span> <span class="k">for</span> <span class="n">ddx</span> <span class="ow">in</span> <span class="n">dsgsx</span><span class="p">],</span> <span class="n">ii</span><span class="p">))</span> <span class="k">for</span> <span class="n">ii</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="o">-</span><span class="mi">2</span><span class="p">)]</span>
-
-        <span class="n">p_i</span> <span class="o">=</span> <span class="nb">min</span><span class="p">([</span><span class="nb">min</span><span class="p">([</span><span class="n">md</span><span class="p">[</span><span class="n">h</span><span class="p">[</span><span class="n">x</span><span class="p">]]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">deltab</span><span class="p">])</span> <span class="k">for</span> <span class="n">s</span><span class="p">,</span><span class="n">d</span><span class="p">,</span><span class="n">h</span> <span class="ow">in</span> <span class="n">TAB</span><span class="p">])</span>
-        <span class="n">dsgsx1</span> <span class="o">=</span> <span class="p">[</span><span class="n">_af_new</span><span class="p">(</span><span class="n">_</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">dsgsx</span><span class="p">]</span>
-        <span class="n">Dxtrav</span> <span class="o">=</span> <span class="n">_orbit_transversal</span><span class="p">(</span><span class="n">size</span><span class="p">,</span> <span class="n">dsgsx1</span><span class="p">,</span> <span class="n">p_i</span><span class="p">,</span> <span class="bp">False</span><span class="p">,</span> <span class="n">af</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> \
-                <span class="k">if</span> <span class="n">dsgsx</span> <span class="k">else</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">Dxtrav</span><span class="p">:</span>
-            <span class="n">Dxtrav</span> <span class="o">=</span> <span class="p">[</span><span class="n">_af_invert</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">Dxtrav</span><span class="p">]</span>
-        <span class="c"># compute the orbit of p_i</span>
-        <span class="k">for</span> <span class="n">ii</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_types</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">p_i</span> <span class="ow">in</span> <span class="n">dumx</span><span class="p">[</span><span class="n">ii</span><span class="p">]:</span>
-                <span class="c"># the orbit is made by all the indices in dum[ii]</span>
-                <span class="k">if</span> <span class="n">sym</span><span class="p">[</span><span class="n">ii</span><span class="p">]</span> <span class="o">!=</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">deltap</span> <span class="o">=</span> <span class="n">dumx</span><span class="p">[</span><span class="n">ii</span><span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="c"># the orbit is made by all the even indices if p_i</span>
-                    <span class="c"># is even, by all the odd indices if p_i is odd</span>
-                    <span class="n">p_i_index</span> <span class="o">=</span> <span class="n">dumx</span><span class="p">[</span><span class="n">ii</span><span class="p">]</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">p_i</span><span class="p">)</span><span class="o">%</span><span class="mi">2</span>
-                    <span class="n">deltap</span> <span class="o">=</span> <span class="n">dumx</span><span class="p">[</span><span class="n">ii</span><span class="p">][</span><span class="n">p_i_index</span><span class="p">::</span><span class="mi">2</span><span class="p">]</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">deltap</span> <span class="o">=</span> <span class="p">[</span><span class="n">p_i</span><span class="p">]</span>
-        <span class="n">TAB1</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">nTAB</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">TAB</span><span class="p">)</span>
-        <span class="k">while</span> <span class="n">TAB</span><span class="p">:</span>
-            <span class="n">s</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">TAB</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-            <span class="k">if</span> <span class="nb">min</span><span class="p">([</span><span class="n">md</span><span class="p">[</span><span class="n">h</span><span class="p">[</span><span class="n">x</span><span class="p">]]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">deltab</span><span class="p">])</span> <span class="o">!=</span> <span class="n">p_i</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="n">deltab1</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">deltab</span> <span class="k">if</span> <span class="n">md</span><span class="p">[</span><span class="n">h</span><span class="p">[</span><span class="n">x</span><span class="p">]]</span> <span class="o">==</span> <span class="n">p_i</span><span class="p">]</span>
-            <span class="c"># NEXT = s*deltab1 intersection (d*g)**-1*deltap</span>
-            <span class="n">dg</span> <span class="o">=</span> <span class="n">_af_rmul</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-            <span class="n">dginv</span> <span class="o">=</span> <span class="n">_af_invert</span><span class="p">(</span><span class="n">dg</span><span class="p">)</span>
-            <span class="n">sdeltab</span> <span class="o">=</span> <span class="p">[</span><span class="n">s</span><span class="p">[</span><span class="n">x</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">deltab1</span><span class="p">]</span>
-            <span class="n">gdeltap</span> <span class="o">=</span> <span class="p">[</span><span class="n">dginv</span><span class="p">[</span><span class="n">x</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">deltap</span><span class="p">]</span>
-            <span class="n">NEXT</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">sdeltab</span> <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">gdeltap</span><span class="p">]</span>
-            <span class="c"># d, s satisfy</span>
-            <span class="c"># d*g*s*base[i-1] = p_{i-1}; using the stabilizers</span>
-            <span class="c"># d*g*s*S_{base[0],...,base[i-1]}*base[i-1] =</span>
-            <span class="c"># D_{p_0,...,p_{i-1}}*p_{i-1}</span>
-            <span class="c"># so that to find d1, s1 satisfying d1*g*s1*b = p_i</span>
-            <span class="c"># one can look for dx in D_{p_0,...,p_{i-1}} and</span>
-            <span class="c"># sx in S_{base[0],...,base[i-1]}</span>
-            <span class="c"># d1 = dx*d; s1 = s*sx</span>
-            <span class="c"># d1*g*s1*b = dx*d*g*s*sx*b = p_i</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">NEXT</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">testb</span><span class="p">:</span>
-                    <span class="c"># solve s1*b = j with s1 = s*sx for some element sx</span>
-                    <span class="c"># of the stabilizer of ..., base[i-1]</span>
-                    <span class="c"># sx*b = s**-1*j; sx = _trace_S(s, j,...)</span>
-                    <span class="c"># s1 = s*trace_S(s**-1*j,...)</span>
-                    <span class="n">s1</span> <span class="o">=</span> <span class="n">_trace_S</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">S_transversals</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">s1</span><span class="p">:</span>
-                        <span class="k">continue</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">s1</span> <span class="o">=</span> <span class="p">[</span><span class="n">s</span><span class="p">[</span><span class="n">ix</span><span class="p">]</span> <span class="k">for</span> <span class="n">ix</span> <span class="ow">in</span> <span class="n">s1</span><span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">s1</span> <span class="o">=</span> <span class="n">s</span>
-                <span class="c">#assert s1[b] == j  # invariant</span>
-                <span class="c"># solve d1*g*j = p_i with d1 = dx*d for some element dg</span>
-                <span class="c"># of the stabilizer of ..., p_{i-1}</span>
-                <span class="c"># dx**-1*p_i = d*g*j; dx**-1 = trace_D(d*g*j,...)</span>
-                <span class="c"># d1 = trace_D(d*g*j,...)**-1*d</span>
-                <span class="c"># to save an inversion in the inner loop; notice we did</span>
-                <span class="c"># Dxtrav = [perm_af_invert(x) for x in Dxtrav] out of the loop</span>
-                <span class="k">if</span> <span class="n">Dxtrav</span><span class="p">:</span>
-                    <span class="n">d1</span> <span class="o">=</span> <span class="n">_trace_D</span><span class="p">(</span><span class="n">dg</span><span class="p">[</span><span class="n">j</span><span class="p">],</span> <span class="n">p_i</span><span class="p">,</span> <span class="n">Dxtrav</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">d1</span><span class="p">:</span>
-                        <span class="k">continue</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">p_i</span> <span class="o">!=</span> <span class="n">dg</span><span class="p">[</span><span class="n">j</span><span class="p">]:</span>
-                        <span class="k">continue</span>
-                    <span class="n">d1</span> <span class="o">=</span> <span class="n">idn</span>
-                <span class="k">assert</span> <span class="n">d1</span><span class="p">[</span><span class="n">dg</span><span class="p">[</span><span class="n">j</span><span class="p">]]</span> <span class="o">==</span> <span class="n">p_i</span>  <span class="c"># invariant</span>
-                <span class="n">d1</span> <span class="o">=</span> <span class="p">[</span><span class="n">d1</span><span class="p">[</span><span class="n">ix</span><span class="p">]</span> <span class="k">for</span> <span class="n">ix</span> <span class="ow">in</span> <span class="n">d</span><span class="p">]</span>
-                <span class="n">h1</span> <span class="o">=</span> <span class="p">[</span><span class="n">d1</span><span class="p">[</span><span class="n">g</span><span class="p">[</span><span class="n">ix</span><span class="p">]]</span> <span class="k">for</span> <span class="n">ix</span> <span class="ow">in</span> <span class="n">s1</span><span class="p">]</span>
-                <span class="c">#assert h1[b] == p_i  # invariant</span>
-                <span class="n">TAB1</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">s1</span><span class="p">,</span> <span class="n">d1</span><span class="p">,</span> <span class="n">h1</span><span class="p">))</span>
-
-        <span class="c"># if TAB contains equal permutations, keep only one of them;</span>
-        <span class="c"># if TAB contains equal permutations up to the sign, return 0</span>
-        <span class="n">TAB1</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-        <span class="n">nTAB1</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">TAB1</span><span class="p">)</span>
-        <span class="n">prev</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">size</span>
-        <span class="k">while</span> <span class="n">TAB1</span><span class="p">:</span>
-            <span class="n">s</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">TAB1</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">h</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="n">prev</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">]:</span>
-                <span class="k">if</span> <span class="n">h</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">!=</span> <span class="n">prev</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-                    <span class="k">return</span> <span class="mi">0</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">TAB</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">s</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">h</span><span class="p">))</span>
-            <span class="n">prev</span> <span class="o">=</span> <span class="n">h</span>
-
-        <span class="c"># stabilize the SGS</span>
-        <span class="n">sgensx</span> <span class="o">=</span> <span class="p">[</span><span class="n">h</span> <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">sgensx</span> <span class="k">if</span> <span class="n">h</span><span class="p">[</span><span class="n">b</span><span class="p">]</span> <span class="o">==</span> <span class="n">b</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">b_S</span><span class="p">:</span>
-            <span class="n">b_S</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="n">_dumx_remove</span><span class="p">(</span><span class="n">dumx</span><span class="p">,</span> <span class="n">dumx_flat</span><span class="p">,</span> <span class="n">p_i</span><span class="p">)</span>
-        <span class="n">dsgsx</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_types</span><span class="p">):</span>
-            <span class="n">dsgsx</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">dummy_sgs</span><span class="p">(</span><span class="n">dumx</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">sym</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">num_dummies</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">TAB</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-
-</div>
-<span class="k">def</span> <span class="nf">canonical_free</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">num_free</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    canonicalization of a tensor with respect to free indices</span>
-<span class="sd">    choosing the minimum with respect to lexicographical ordering</span>
-
-<span class="sd">    base, gens  BSGS for slot permutation group</span>
-<span class="sd">    g           permutation representing the tensor</span>
-<span class="sd">    num_free    number of free indices</span>
-<span class="sd">    The indices must be ordered with first the free indices</span>
-
-<span class="sd">    see explanation in double_coset_can_rep</span>
-<span class="sd">    The algorithm is given in [2]</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.tensor_can import canonical_free</span>
-<span class="sd">    &gt;&gt;&gt; gens = [[1,0,2,3,5,4], [2,3,0,1,4,5],[0,1,3,2,5,4]]</span>
-<span class="sd">    &gt;&gt;&gt; gens = [Permutation(h) for h in gens]</span>
-<span class="sd">    &gt;&gt;&gt; base = [0, 2]</span>
-<span class="sd">    &gt;&gt;&gt; g = Permutation([2, 1, 0, 3, 4, 5])</span>
-<span class="sd">    &gt;&gt;&gt; canonical_free(base, gens, g, 4)</span>
-<span class="sd">    [0, 3, 1, 2, 5, 4]</span>
-
-<span class="sd">    Consider the product of Riemann tensors</span>
-<span class="sd">    `T = R^{a}_{d0}^{d1,d2}*R_{d2,d1}^{d0,b}`</span>
-<span class="sd">    The order of the indices is [a,b,d0,-d0,d1,-d1,d2,-d2]</span>
-<span class="sd">    The permutation corresponding to the tensor is</span>
-<span class="sd">    g = [0,3,4,6,7,5,2,1,8,9]</span>
-<span class="sd">    Use the slot symmetries to get `T` is the form which is the minimum</span>
-<span class="sd">    in lexicographic order</span>
-<span class="sd">    `R^{a}_{d0}^{d1,d2}*R^{b,d0}_{d1,d2}` corresponding to</span>
-<span class="sd">    `[0, 3, 4, 6, 1, 2, 5, 7, 8, 9]`</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.tensor_can import riemann_bsgs, tensor_gens</span>
-<span class="sd">    &gt;&gt;&gt; base, gens = riemann_bsgs</span>
-<span class="sd">    &gt;&gt;&gt; size, sbase, sgens = tensor_gens(base, gens, [[],[]], 0)</span>
-<span class="sd">    &gt;&gt;&gt; g = Permutation([0,3,4,6,7,5,2,1,8,9])</span>
-<span class="sd">    &gt;&gt;&gt; canonical_free(sbase, [Permutation(h) for h in sgens], g, 2)</span>
-<span class="sd">    [0, 3, 4, 6, 1, 2, 5, 7, 8, 9]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">array_form</span>
-    <span class="n">size</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">base</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">g</span><span class="p">[:]</span>
-
-    <span class="n">transversals</span> <span class="o">=</span> <span class="n">get_transversals</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">gens</span><span class="p">)</span>
-    <span class="n">cosets</span> <span class="o">=</span> <span class="n">transversal2coset</span><span class="p">(</span><span class="n">size</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="n">transversals</span><span class="p">)</span>
-    <span class="n">K</span> <span class="o">=</span> <span class="p">[</span><span class="n">h</span><span class="o">.</span><span class="n">_array_form</span> <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">]</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">g</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">]):</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">base</span><span class="p">:</span>
-            <span class="n">base</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="n">lambd</span> <span class="o">=</span> <span class="n">g</span>
-    <span class="n">K1</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">base</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-        <span class="n">delta</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="p">[</span><span class="n">b</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">cosets</span><span class="p">[</span><span class="n">b</span><span class="p">]]</span>
-        <span class="n">delta1</span> <span class="o">=</span> <span class="p">[</span><span class="n">lambd</span><span class="p">[</span><span class="n">x</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">delta</span><span class="p">]</span>
-        <span class="n">delta1min</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">delta1</span><span class="p">)</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">delta1</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">delta1min</span><span class="p">)</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">delta</span><span class="p">[</span><span class="n">k</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">omega</span> <span class="ow">in</span> <span class="n">cosets</span><span class="p">[</span><span class="n">b</span><span class="p">]:</span>
-            <span class="k">if</span> <span class="n">omega</span><span class="p">[</span><span class="n">b</span><span class="p">]</span> <span class="o">==</span> <span class="n">p</span><span class="p">:</span>
-                <span class="k">break</span>
-        <span class="n">lambd</span> <span class="o">=</span> <span class="n">_af_rmul</span><span class="p">(</span><span class="n">lambd</span><span class="p">,</span> <span class="n">omega</span><span class="p">)</span>
-        <span class="n">K</span> <span class="o">=</span> <span class="p">[</span><span class="n">px</span> <span class="k">for</span> <span class="n">px</span> <span class="ow">in</span> <span class="n">K</span> <span class="k">if</span> <span class="n">px</span><span class="p">[</span><span class="n">b</span><span class="p">]</span> <span class="o">==</span> <span class="n">b</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">lambd</span>
-
-
-<span class="k">def</span> <span class="nf">_get_map_slots</span><span class="p">(</span><span class="n">size</span><span class="p">,</span> <span class="n">fixed_slots</span><span class="p">):</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">))</span>
-    <span class="n">pos</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">):</span>
-      <span class="k">if</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">fixed_slots</span><span class="p">:</span>
-          <span class="k">continue</span>
-      <span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">pos</span>
-      <span class="n">pos</span> <span class="o">+=</span> <span class="mi">1</span>
-    <span class="k">return</span> <span class="n">res</span>
-
-<span class="k">def</span> <span class="nf">_lift_sgens</span><span class="p">(</span><span class="n">size</span><span class="p">,</span> <span class="n">fixed_slots</span><span class="p">,</span> <span class="n">free</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">j</span> <span class="o">=</span> <span class="n">k</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">fd</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">fixed_slots</span><span class="p">,</span> <span class="n">free</span><span class="p">))</span>
-    <span class="n">fd</span> <span class="o">=</span> <span class="p">[</span><span class="n">y</span> <span class="k">for</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">fd</span><span class="p">)]</span>
-    <span class="n">num_free</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">free</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">fixed_slots</span><span class="p">:</span>
-            <span class="n">a</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">fd</span><span class="p">[</span><span class="n">k</span><span class="p">])</span>
-            <span class="n">k</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">a</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+</span> <span class="n">num_free</span><span class="p">)</span>
-            <span class="n">j</span> <span class="o">+=</span> <span class="mi">1</span>
-    <span class="k">return</span> <span class="n">a</span>
-
-<div class="viewcode-block" id="canonicalize"><a class="viewcode-back" href="../../../modules/combinatorics/tensor_can.html#sympy.combinatorics.tensor_can.canonicalize">[docs]</a><span class="k">def</span> <span class="nf">canonicalize</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">dummies</span><span class="p">,</span> <span class="n">msym</span><span class="p">,</span> <span class="o">*</span><span class="n">v</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    canonicalize tensor formed by tensors</span>
-
-<span class="sd">      g     permutation representing the tensor</span>
-
-<span class="sd">      dummies list representing the dummy indices</span>
-<span class="sd">              it can be a list of dummy indices of the same type</span>
-<span class="sd">              or a list of lists of dummy indices, one list for each</span>
-<span class="sd">              type of index;</span>
-<span class="sd">              the dummy indices must come after the free indices,</span>
-<span class="sd">              and put in order contravariant, covariant</span>
-<span class="sd">              [d0, -d0, d1,-d1,...]</span>
-
-<span class="sd">      msym  symmetry of the metric(s)</span>
-<span class="sd">            it can be an integer or a list;</span>
-<span class="sd">            in the first case it is the symmetry of the dummy index metric;</span>
-<span class="sd">            in the second case it is the list of the symmetries of the</span>
-<span class="sd">            index metric for each type</span>
-
-<span class="sd">      v     is a list of (base_i, gens_i, n_i, sym_i) for tensors of type `i`</span>
-<span class="sd">            base_i, gens_i BSGS for tensors of this type.</span>
-
-<span class="sd">            The BSGS should have minimal base under lexicographic ordering;</span>
-<span class="sd">            if not, an attempt is made do get the minimal BSGS;</span>
-<span class="sd">            in case of failure,</span>
-<span class="sd">            canonicalize_naive is used, which is much slower.</span>
-
-<span class="sd">            n_i     number ot tensors of type `i`.</span>
-
-<span class="sd">            sym_i   symmetry under exchange of component tensors of type `i`.</span>
-
-<span class="sd">      Both for msym and sym_i the cases are</span>
-<span class="sd">            * None  no symmetry</span>
-<span class="sd">            * 0     commuting</span>
-<span class="sd">            * 1     anticommuting</span>
-
-<span class="sd">    Return 0 if the tensor is zero, else return the array form of</span>
-<span class="sd">    the permutation representing the canonical form of the tensor.</span>
-
-<span class="sd">    Algorithm</span>
-<span class="sd">    =========</span>
-
-<span class="sd">    First one uses canonical_free to get the minimum tensor under</span>
-<span class="sd">    lexicographic order, using only the slot symmetries.</span>
-<span class="sd">    If the component tensors have not minimal BSGS, it is attempted</span>
-<span class="sd">    to find it; if the attempt fails canonicalize_naive</span>
-<span class="sd">    is used instead.</span>
-
-<span class="sd">    Compute the residual slot symmetry keeping fixed the free indices</span>
-<span class="sd">    using tensor_gens(base, gens, list_free_indices, sym).</span>
-
-<span class="sd">    Reduce the problem eliminating the free indices.</span>
-
-<span class="sd">    Then use double_coset_can_rep and lift back the result reintroducing</span>
-<span class="sd">    the free indices.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    one type of index with commuting metric;</span>
-
-<span class="sd">    `A_{a b}` and `B_{a b}` antisymmetric and commuting</span>
-
-<span class="sd">    `T = A_{d0 d1} * B^{d0}{}_{d2} * B^{d2 d1}`</span>
-
-<span class="sd">    `ord = [d0,-d0,d1,-d1,d2,-d2]` order of the indices</span>
-
-<span class="sd">    g = [1,3,0,5,4,2,6,7]</span>
-
-<span class="sd">    T_c = 0</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.tensor_can import get_symmetric_group_sgs, canonicalize, bsgs_direct_product</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; base2a, gens2a = get_symmetric_group_sgs(2, 1)</span>
-<span class="sd">    &gt;&gt;&gt; t0 = (base2a, gens2a, 1, 0)</span>
-<span class="sd">    &gt;&gt;&gt; t1 = (base2a, gens2a, 2, 0)</span>
-<span class="sd">    &gt;&gt;&gt; g = Permutation([1,3,0,5,4,2,6,7])</span>
-<span class="sd">    &gt;&gt;&gt; canonicalize(g, list(range(6)), 0, t0, t1)</span>
-<span class="sd">    0</span>
-
-<span class="sd">    same as above, but with `B_{a b}` anticommuting</span>
-
-<span class="sd">    `T_c = -A^{d0 d1} * B_{d0}{}^{d2} * B_{d1 d2}`</span>
-
-<span class="sd">    `can = [0,2,1,4,3,5,7,6]`</span>
-
-<span class="sd">    &gt;&gt;&gt; t1 = (base2a, gens2a, 2, 1)</span>
-<span class="sd">    &gt;&gt;&gt; canonicalize(g, list(range(6)), 0, t0, t1)</span>
-<span class="sd">    [0, 2, 1, 4, 3, 5, 7, 6]</span>
-
-<span class="sd">    two types of indices `[a,b,c,d,e,f]` and `[m,n]`, in this order,</span>
-<span class="sd">    both with commuting metric</span>
-
-<span class="sd">    `f^{a b c}` antisymmetric, commuting</span>
-
-<span class="sd">    `A_{m a}` no symmetry, commuting</span>
-
-<span class="sd">    `T = f^c{}_{d a} * f^f{}_{e b} * A_m{}^d * A^{m b} * A_n{}^a * A^{n e}`</span>
-
-<span class="sd">    ord = [c,f,a,-a,b,-b,d,-d,e,-e,m,-m,n,-n]</span>
-
-<span class="sd">    g = [0,7,3, 1,9,5, 11,6, 10,4, 13,2, 12,8, 14,15]</span>
-
-<span class="sd">    The canonical tensor is</span>
-<span class="sd">    `T_c = -f^{c a b} * f^{f d e} * A^m{}_a * A_{m d} * A^n{}_b * A_{n e}`</span>
-
-<span class="sd">    can = [0,2,4, 1,6,8, 10,3, 11,7, 12,5, 13,9, 15,14]</span>
-
-<span class="sd">    &gt;&gt;&gt; base_f, gens_f = get_symmetric_group_sgs(3, 1)</span>
-<span class="sd">    &gt;&gt;&gt; base1, gens1 = get_symmetric_group_sgs(1)</span>
-<span class="sd">    &gt;&gt;&gt; base_A, gens_A = bsgs_direct_product(base1, gens1, base1, gens1)</span>
-<span class="sd">    &gt;&gt;&gt; t0 = (base_f, gens_f, 2, 0)</span>
-<span class="sd">    &gt;&gt;&gt; t1 = (base_A, gens_A, 4, 0)</span>
-<span class="sd">    &gt;&gt;&gt; dummies = [list(range(2, 10)), list(range(10, 14))]</span>
-<span class="sd">    &gt;&gt;&gt; g = Permutation([0,7,3,1,9,5,11,6,10,4,13,2,12,8,14,15])</span>
-<span class="sd">    &gt;&gt;&gt; canonicalize(g, dummies, [0, 0], t0, t1)</span>
-<span class="sd">    [0, 2, 4, 1, 6, 8, 10, 3, 11, 7, 12, 5, 13, 9, 15, 14]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.combinatorics.testutil</span> <span class="kn">import</span> <span class="n">canonicalize_naive</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">msym</span><span class="p">,</span> <span class="nb">list</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">msym</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="bp">None</span><span class="p">]:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;msym must be 0, 1 or None&#39;</span><span class="p">)</span>
-        <span class="n">num_types</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">num_types</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">msym</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">(</span><span class="n">msymx</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="bp">None</span><span class="p">]</span> <span class="k">for</span> <span class="n">msymx</span> <span class="ow">in</span> <span class="n">msym</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;msym entries must be 0, 1 or None&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">dummies</span><span class="p">)</span> <span class="o">!=</span> <span class="n">num_types</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;dummies and msym must have the same number of elements&#39;</span><span class="p">)</span>
-    <span class="n">size</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">size</span>
-    <span class="n">num_tensors</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">v1</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">v</span><span class="p">)):</span>
-        <span class="n">base_i</span><span class="p">,</span> <span class="n">gens_i</span><span class="p">,</span> <span class="n">n_i</span><span class="p">,</span> <span class="n">sym_i</span> <span class="o">=</span> <span class="n">v</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-        <span class="c"># check that the BSGS is minimal;</span>
-        <span class="c"># this property is used in double_coset_can_rep;</span>
-        <span class="c"># if it is not minimal use canonicalize_naive</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">_is_minimal_bsgs</span><span class="p">(</span><span class="n">base_i</span><span class="p">,</span> <span class="n">gens_i</span><span class="p">):</span>
-            <span class="n">mbsgs</span> <span class="o">=</span> <span class="n">get_minimal_bsgs</span><span class="p">(</span><span class="n">base_i</span><span class="p">,</span> <span class="n">gens_i</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">mbsgs</span><span class="p">:</span>
-                <span class="n">can</span> <span class="o">=</span> <span class="n">canonicalize_naive</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">dummies</span><span class="p">,</span> <span class="n">msym</span><span class="p">,</span> <span class="o">*</span><span class="n">v</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">can</span>
-            <span class="n">base_i</span><span class="p">,</span> <span class="n">gens_i</span> <span class="o">=</span> <span class="n">mbsgs</span>
-        <span class="n">v1</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">base_i</span><span class="p">,</span> <span class="n">gens_i</span><span class="p">,</span> <span class="p">[[]]</span><span class="o">*</span><span class="n">n_i</span><span class="p">,</span> <span class="n">sym_i</span><span class="p">))</span>
-        <span class="n">num_tensors</span> <span class="o">+=</span> <span class="n">n_i</span>
-
-    <span class="k">if</span> <span class="n">num_types</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">dummies</span> <span class="o">=</span> <span class="p">[</span><span class="n">dummies</span><span class="p">]</span>
-        <span class="n">msym</span> <span class="o">=</span> <span class="p">[</span><span class="n">msym</span><span class="p">]</span>
-    <span class="n">flat_dummies</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">dumx</span> <span class="ow">in</span> <span class="n">dummies</span><span class="p">:</span>
-        <span class="n">flat_dummies</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">dumx</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">flat_dummies</span> <span class="ow">and</span> <span class="n">flat_dummies</span> <span class="o">!=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">flat_dummies</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">flat_dummies</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)):</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;dummies is not valid&#39;</span><span class="p">)</span>
-
-    <span class="c"># slot symmetry of the tensor</span>
-    <span class="n">size1</span><span class="p">,</span> <span class="n">sbase</span><span class="p">,</span> <span class="n">sgens</span> <span class="o">=</span> <span class="n">gens_products</span><span class="p">(</span><span class="o">*</span><span class="n">v1</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">size</span> <span class="o">!=</span> <span class="n">size1</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;g has size </span><span class="si">%d</span><span class="s">, generators have size </span><span class="si">%d</span><span class="s">&#39;</span> <span class="o">%</span><span class="p">(</span><span class="n">size</span><span class="p">,</span> <span class="n">size1</span><span class="p">))</span>
-    <span class="n">free</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span> <span class="k">if</span> <span class="n">i</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">flat_dummies</span><span class="p">]</span>
-    <span class="n">num_free</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">free</span><span class="p">)</span>
-
-    <span class="c"># g1 minimal tensor under slot symmetry</span>
-    <span class="n">g1</span> <span class="o">=</span> <span class="n">canonical_free</span><span class="p">(</span><span class="n">sbase</span><span class="p">,</span> <span class="n">sgens</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">num_free</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">flat_dummies</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">g1</span>
-    <span class="c"># save the sign of g1</span>
-    <span class="n">sign</span> <span class="o">=</span> <span class="mi">0</span> <span class="k">if</span> <span class="n">g1</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">size</span><span class="o">-</span><span class="mi">1</span> <span class="k">else</span> <span class="mi">1</span>
-
-    <span class="c"># the free indices are kept fixed.</span>
-    <span class="c"># Determine free_i, the list of slots of tensors which are fixed</span>
-    <span class="c"># since they are occupied by free indices, which are fixed.</span>
-    <span class="n">start</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">v</span><span class="p">)):</span>
-        <span class="n">free_i</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">base_i</span><span class="p">,</span> <span class="n">gens_i</span><span class="p">,</span> <span class="n">n_i</span><span class="p">,</span> <span class="n">sym_i</span> <span class="o">=</span> <span class="n">v</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-        <span class="n">len_tens</span> <span class="o">=</span> <span class="n">gens_i</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">size</span> <span class="o">-</span> <span class="mi">2</span>
-        <span class="c"># for each component tensor get a list od fixed islots</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_i</span><span class="p">):</span>
-            <span class="c"># get the elements corresponding to the component tensor</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">g1</span><span class="p">[</span><span class="n">start</span><span class="p">:(</span><span class="n">start</span> <span class="o">+</span> <span class="n">len_tens</span><span class="p">)]</span>
-            <span class="n">fr</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="c"># get the positions of the fixed elements in h</span>
-            <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">free</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">h</span><span class="p">:</span>
-                    <span class="n">fr</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">h</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">k</span><span class="p">))</span>
-            <span class="n">free_i</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">fr</span><span class="p">)</span>
-            <span class="n">start</span> <span class="o">+=</span> <span class="n">len_tens</span>
-        <span class="n">v1</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">base_i</span><span class="p">,</span> <span class="n">gens_i</span><span class="p">,</span> <span class="n">free_i</span><span class="p">,</span> <span class="n">sym_i</span><span class="p">)</span>
-    <span class="c"># BSGS of the tensor with fixed free indices</span>
-    <span class="c"># if tensor_gens fails in gens_product, use canonicalize_naive</span>
-    <span class="n">size</span><span class="p">,</span> <span class="n">sbase</span><span class="p">,</span> <span class="n">sgens</span> <span class="o">=</span> <span class="n">gens_products</span><span class="p">(</span><span class="o">*</span><span class="n">v1</span><span class="p">)</span>
-
-    <span class="c"># reduce the permutations getting rid of the free indices</span>
-    <span class="n">pos_dummies</span> <span class="o">=</span> <span class="p">[</span><span class="n">g1</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">flat_dummies</span><span class="p">]</span>
-    <span class="n">pos_free</span> <span class="o">=</span> <span class="p">[</span><span class="n">g1</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_free</span><span class="p">)]</span>
-    <span class="n">size_red</span> <span class="o">=</span> <span class="n">size</span> <span class="o">-</span> <span class="n">num_free</span>
-    <span class="n">g1_red</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="o">-</span> <span class="n">num_free</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">g1</span> <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">flat_dummies</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">sign</span><span class="p">:</span>
-        <span class="n">g1_red</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">size_red</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">size_red</span> <span class="o">-</span> <span class="mi">2</span><span class="p">])</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">g1_red</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">size_red</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">size_red</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span>
-    <span class="n">map_slots</span> <span class="o">=</span> <span class="n">_get_map_slots</span><span class="p">(</span><span class="n">size</span><span class="p">,</span> <span class="n">pos_free</span><span class="p">)</span>
-    <span class="n">sbase_red</span> <span class="o">=</span> <span class="p">[</span><span class="n">map_slots</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">sbase</span> <span class="k">if</span> <span class="n">i</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">pos_free</span><span class="p">]</span>
-    <span class="n">sgens_red</span> <span class="o">=</span> <span class="p">[</span><span class="n">_af_new</span><span class="p">([</span><span class="n">map_slots</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">y</span><span class="o">.</span><span class="n">_array_form</span> <span class="k">if</span> <span class="n">i</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">pos_free</span><span class="p">])</span> <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">sgens</span><span class="p">]</span>
-    <span class="n">dummies_red</span> <span class="o">=</span> <span class="p">[[</span><span class="n">x</span> <span class="o">-</span> <span class="n">num_free</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">y</span><span class="p">]</span> <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">dummies</span><span class="p">]</span>
-    <span class="n">transv_red</span> <span class="o">=</span> <span class="n">get_transversals</span><span class="p">(</span><span class="n">sbase_red</span><span class="p">,</span> <span class="n">sgens_red</span><span class="p">)</span>
-    <span class="n">g1_red</span> <span class="o">=</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">g1_red</span><span class="p">)</span>
-    <span class="n">g2</span> <span class="o">=</span> <span class="n">double_coset_can_rep</span><span class="p">(</span><span class="n">dummies_red</span><span class="p">,</span> <span class="n">msym</span><span class="p">,</span> <span class="n">sbase_red</span><span class="p">,</span> <span class="n">sgens_red</span><span class="p">,</span> <span class="n">transv_red</span><span class="p">,</span> <span class="n">g1_red</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">g2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="c"># lift to the case with the free indices</span>
-    <span class="n">g3</span> <span class="o">=</span> <span class="n">_lift_sgens</span><span class="p">(</span><span class="n">size</span><span class="p">,</span> <span class="n">pos_free</span><span class="p">,</span> <span class="n">free</span><span class="p">,</span> <span class="n">g2</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">g3</span>
-
-</div>
-<span class="k">def</span> <span class="nf">perm_af_direct_product</span><span class="p">(</span><span class="n">gens1</span><span class="p">,</span> <span class="n">gens2</span><span class="p">,</span> <span class="n">signed</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    direct products of the generators gens1 and gens2</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.tensor_can import perm_af_direct_product</span>
-<span class="sd">    &gt;&gt;&gt; gens1 = [[1,0,2,3], [0,1,3,2]]</span>
-<span class="sd">    &gt;&gt;&gt; gens2 = [[1,0]]</span>
-<span class="sd">    &gt;&gt;&gt; perm_af_direct_product(gens1, gens2, False)</span>
-<span class="sd">    [[1, 0, 2, 3, 4, 5], [0, 1, 3, 2, 4, 5], [0, 1, 2, 3, 5, 4]]</span>
-<span class="sd">    &gt;&gt;&gt; gens1 = [[1,0,2,3,5,4], [0,1,3,2,4,5]]</span>
-<span class="sd">    &gt;&gt;&gt; gens2 = [[1,0,2,3]]</span>
-<span class="sd">    &gt;&gt;&gt; perm_af_direct_product(gens1, gens2, True)</span>
-<span class="sd">    [[1, 0, 2, 3, 4, 5, 7, 6], [0, 1, 3, 2, 4, 5, 6, 7], [0, 1, 2, 3, 5, 4, 6, 7]]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">gens1</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">gens1</span><span class="p">]</span>
-    <span class="n">gens2</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">gens2</span><span class="p">]</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="mi">2</span> <span class="k">if</span> <span class="n">signed</span> <span class="k">else</span> <span class="mi">0</span>
-    <span class="n">n1</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens1</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">-</span> <span class="n">s</span>
-    <span class="n">n2</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens2</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">-</span> <span class="n">s</span>
-    <span class="n">start</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n1</span><span class="p">))</span>
-    <span class="n">end</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n1</span><span class="p">,</span> <span class="n">n1</span> <span class="o">+</span> <span class="n">n2</span><span class="p">))</span>
-    <span class="k">if</span> <span class="n">signed</span><span class="p">:</span>
-        <span class="n">gens1</span> <span class="o">=</span> <span class="p">[</span><span class="n">gen</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="n">end</span> <span class="o">+</span> <span class="p">[</span><span class="n">gen</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span><span class="o">+</span><span class="n">n2</span><span class="p">,</span> <span class="n">gen</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="n">n2</span><span class="p">]</span> <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">gens1</span><span class="p">]</span>
-        <span class="n">gens2</span> <span class="o">=</span> <span class="p">[</span><span class="n">start</span> <span class="o">+</span> <span class="p">[</span><span class="n">x</span> <span class="o">+</span> <span class="n">n1</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">gen</span><span class="p">]</span> <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">gens2</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">gens1</span> <span class="o">=</span> <span class="p">[</span><span class="n">gen</span> <span class="o">+</span> <span class="n">end</span> <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">gens1</span><span class="p">]</span>
-        <span class="n">gens2</span> <span class="o">=</span> <span class="p">[</span><span class="n">start</span> <span class="o">+</span> <span class="p">[</span><span class="n">x</span> <span class="o">+</span> <span class="n">n1</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">gen</span><span class="p">]</span> <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">gens2</span><span class="p">]</span>
-
-    <span class="n">res</span> <span class="o">=</span> <span class="n">gens1</span> <span class="o">+</span> <span class="n">gens2</span>
-
-    <span class="k">return</span> <span class="n">res</span>
-
-<div class="viewcode-block" id="bsgs_direct_product"><a class="viewcode-back" href="../../../modules/combinatorics/tensor_can.html#sympy.combinatorics.tensor_can.bsgs_direct_product">[docs]</a><span class="k">def</span> <span class="nf">bsgs_direct_product</span><span class="p">(</span><span class="n">base1</span><span class="p">,</span> <span class="n">gens1</span><span class="p">,</span> <span class="n">base2</span><span class="p">,</span> <span class="n">gens2</span><span class="p">,</span> <span class="n">signed</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    direct product of two BSGS</span>
-
-<span class="sd">    base1    base of the first BSGS.</span>
-
-<span class="sd">    gens1    strong generating sequence of the first BSGS.</span>
-
-<span class="sd">    base2, gens2   similarly for the second BSGS.</span>
-
-<span class="sd">    signed   flag for signed permutations.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.tensor_can import (get_symmetric_group_sgs, bsgs_direct_product)</span>
-<span class="sd">    &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">    &gt;&gt;&gt; base1, gens1 = get_symmetric_group_sgs(1)</span>
-<span class="sd">    &gt;&gt;&gt; base2, gens2 = get_symmetric_group_sgs(2)</span>
-<span class="sd">    &gt;&gt;&gt; bsgs_direct_product(base1, gens1, base2, gens2)</span>
-<span class="sd">    ([1], [Permutation(4)(1, 2)])</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="mi">2</span> <span class="k">if</span> <span class="n">signed</span> <span class="k">else</span> <span class="mi">0</span>
-    <span class="n">n1</span> <span class="o">=</span> <span class="n">gens1</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">size</span> <span class="o">-</span> <span class="n">s</span>
-    <span class="n">base</span> <span class="o">=</span> <span class="n">base1</span><span class="p">[:]</span>
-    <span class="n">base</span> <span class="o">+=</span> <span class="p">[</span><span class="n">x</span> <span class="o">+</span> <span class="n">n1</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">base2</span><span class="p">]</span>
-    <span class="n">gens1</span> <span class="o">=</span> <span class="p">[</span><span class="n">h</span><span class="o">.</span><span class="n">_array_form</span> <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">gens1</span><span class="p">]</span>
-    <span class="n">gens2</span> <span class="o">=</span> <span class="p">[</span><span class="n">h</span><span class="o">.</span><span class="n">_array_form</span> <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">gens2</span><span class="p">]</span>
-    <span class="n">gens</span> <span class="o">=</span> <span class="n">perm_af_direct_product</span><span class="p">(</span><span class="n">gens1</span><span class="p">,</span> <span class="n">gens2</span><span class="p">,</span> <span class="n">signed</span><span class="p">)</span>
-    <span class="n">size</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-    <span class="n">id_af</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">))</span>
-    <span class="n">gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">h</span> <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">gens</span> <span class="k">if</span> <span class="n">h</span> <span class="o">!=</span> <span class="n">id_af</span><span class="p">]</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">gens</span><span class="p">:</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">id_af</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">base</span><span class="p">,</span> <span class="p">[</span><span class="n">_af_new</span><span class="p">(</span><span class="n">h</span><span class="p">)</span> <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">]</span>
-</div>
-<div class="viewcode-block" id="get_symmetric_group_sgs"><a class="viewcode-back" href="../../../modules/combinatorics/tensor_can.html#sympy.combinatorics.tensor_can.get_symmetric_group_sgs">[docs]</a><span class="k">def</span> <span class="nf">get_symmetric_group_sgs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">sym</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return base, gens of the minimal BSGS for (anti)symmetric group</span>
-<span class="sd">    with n elements</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.tensor_can import get_symmetric_group_sgs</span>
-<span class="sd">    &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">    &gt;&gt;&gt; get_symmetric_group_sgs(3)</span>
-<span class="sd">    ([0, 1], [Permutation(4)(0, 1), Permutation(4)(1, 2)])</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[],</span> <span class="p">[</span><span class="n">_af_new</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">)))]</span>
-    <span class="n">gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">Permutation</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)(</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">_array_form</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)]</span>
-    <span class="k">if</span> <span class="n">sym</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="o">+</span> <span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="o">+</span> <span class="p">[</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">]</span>
-    <span class="n">base</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">base</span><span class="p">,</span> <span class="p">[</span><span class="n">_af_new</span><span class="p">(</span><span class="n">h</span><span class="p">)</span> <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">]</span>
-</div>
-<span class="n">riemann_bsgs</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="n">Permutation</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">)(</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">),</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)(</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">),</span> \
-               <span class="n">Permutation</span><span class="p">(</span><span class="mi">5</span><span class="p">)(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">)(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">)]</span>
-
-<span class="k">def</span> <span class="nf">get_transversals</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">gens</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return transversals for the group with BSGS base, gens</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">base</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-    <span class="n">stabs</span> <span class="o">=</span>  <span class="n">_distribute_gens_by_base</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">gens</span><span class="p">)</span>
-    <span class="n">orbits</span><span class="p">,</span> <span class="n">transversals</span> <span class="o">=</span> <span class="n">_orbits_transversals_from_bsgs</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">stabs</span><span class="p">)</span>
-    <span class="n">transversals</span> <span class="o">=</span> <span class="p">[</span><span class="nb">dict</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">h</span><span class="o">.</span><span class="n">_array_form</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="n">h</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">y</span><span class="o">.</span><span class="n">items</span><span class="p">()))</span> <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> \
-            <span class="n">transversals</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">transversals</span>
-
-
-<span class="k">def</span> <span class="nf">_is_minimal_bsgs</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">gens</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Check if the BSGS has minimal base under lexigographic order.</span>
-
-<span class="sd">    base, gens BSGS</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.tensor_can import riemann_bsgs, _is_minimal_bsgs</span>
-<span class="sd">    &gt;&gt;&gt; _is_minimal_bsgs(*riemann_bsgs)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; riemann_bsgs1 = ([2, 0], ([Permutation(5)(0,1)(4,5), Permutation(5)(0,2)(1,3)]))</span>
-<span class="sd">    &gt;&gt;&gt; _is_minimal_bsgs(*riemann_bsgs1)</span>
-<span class="sd">    False</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">base1</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">sgs1</span> <span class="o">=</span> <span class="n">gens</span><span class="p">[:]</span>
-    <span class="n">size</span> <span class="o">=</span> <span class="n">gens</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">size</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">(</span><span class="n">h</span><span class="o">.</span><span class="n">_array_form</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="n">i</span> <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">sgs1</span><span class="p">):</span>
-            <span class="n">base1</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="n">sgs1</span> <span class="o">=</span> <span class="p">[</span><span class="n">h</span> <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">sgs1</span> <span class="k">if</span> <span class="n">h</span><span class="o">.</span><span class="n">_array_form</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="n">i</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">base1</span> <span class="o">==</span> <span class="n">base</span>
-
-<span class="k">def</span> <span class="nf">get_minimal_bsgs</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">gens</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute a minimal GSGS</span>
-
-<span class="sd">    base, gens BSGS</span>
-
-<span class="sd">    If base, gens is a minimal BSGS return it; else return a minimal BSGS</span>
-<span class="sd">    if it fails in finding one, it returns None</span>
-
-<span class="sd">    TODO: use baseswap in the case in which if it fails in finding a</span>
-<span class="sd">    minimal BSGS</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.tensor_can import get_minimal_bsgs</span>
-<span class="sd">    &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">    &gt;&gt;&gt; riemann_bsgs1 = ([2, 0], ([Permutation(5)(0,1)(4,5), Permutation(5)(0,2)(1,3)]))</span>
-<span class="sd">    &gt;&gt;&gt; get_minimal_bsgs(*riemann_bsgs1)</span>
-<span class="sd">    ([0, 2], [Permutation(0, 1)(4, 5), Permutation(5)(0, 2)(1, 3), Permutation(2, 3)(4, 5)])</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span>
-    <span class="n">base</span><span class="p">,</span> <span class="n">gens</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">schreier_sims_incremental</span><span class="p">()</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">_is_minimal_bsgs</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">gens</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">None</span>
-    <span class="k">return</span> <span class="n">base</span><span class="p">,</span> <span class="n">gens</span>
-
-<span class="k">def</span> <span class="nf">tensor_gens</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">list_free_indices</span><span class="p">,</span> <span class="n">sym</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns size, res_base, res_gens BSGS for n tensors of the same type</span>
-
-<span class="sd">    base, gens BSGS for tensors of this type</span>
-<span class="sd">    list_free_indices  list of the slots occupied by fixed indices</span>
-<span class="sd">                       for each of the tensors</span>
-
-<span class="sd">    sym symmetry under commutation of two tensors</span>
-<span class="sd">    sym   None  no symmetry</span>
-<span class="sd">    sym   0     commuting</span>
-<span class="sd">    sym   1     anticommuting</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.tensor_can import tensor_gens, get_symmetric_group_sgs</span>
-<span class="sd">    &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-
-<span class="sd">    two symmetric tensors with 3 indices without free indices</span>
-<span class="sd">    &gt;&gt;&gt; base, gens = get_symmetric_group_sgs(3)</span>
-<span class="sd">    &gt;&gt;&gt; tensor_gens(base, gens, [[], []])</span>
-<span class="sd">    (8, [0, 1, 3, 4], [Permutation(7)(0, 1), Permutation(7)(1, 2), Permutation(7)(3, 4), Permutation(7)(4, 5), Permutation(7)(0, 3)(1, 4)(2, 5)])</span>
-
-<span class="sd">    two symmetric tensors with 3 indices with free indices in slot 1 and 0</span>
-<span class="sd">    &gt;&gt;&gt; tensor_gens(base, gens, [[1],[0]])</span>
-<span class="sd">    (8, [0, 4], [Permutation(7)(0, 2), Permutation(7)(4, 5)])</span>
-
-<span class="sd">    four symmetric tensors with 3 indices, two of which with free indices</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">_get_bsgs</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">free_indices</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        return the BSGS for G.pointwise_stabilizer(free_indices)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">free_indices</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">base</span><span class="p">[:],</span> <span class="n">gens</span><span class="p">[:]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">H</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">pointwise_stabilizer</span><span class="p">(</span><span class="n">free_indices</span><span class="p">)</span>
-            <span class="n">base</span><span class="p">,</span> <span class="n">sgs</span> <span class="o">=</span> <span class="n">H</span><span class="o">.</span><span class="n">schreier_sims_incremental</span><span class="p">()</span>
-            <span class="k">return</span> <span class="n">base</span><span class="p">,</span> <span class="n">sgs</span>
-
-    <span class="c"># if not base there is no slot symmetry for the component tensors</span>
-    <span class="c"># if list_free_indices.count([]) &lt; 2 there is no commutation symmetry</span>
-    <span class="c"># so there is no resulting slot symmetry</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">base</span> <span class="ow">and</span> <span class="n">list_free_indices</span><span class="o">.</span><span class="n">count</span><span class="p">([])</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">list_free_indices</span><span class="p">)</span>
-        <span class="n">size</span> <span class="o">=</span> <span class="n">gens</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">size</span>
-        <span class="n">size</span> <span class="o">=</span> <span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">gens</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">size</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span>
-        <span class="k">return</span> <span class="n">size</span><span class="p">,</span> <span class="p">[],</span> <span class="p">[</span><span class="n">_af_new</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">)))]</span>
-
-    <span class="c"># if any(list_free_indices) one needs to compute the pointwise</span>
-    <span class="c"># stabilizer, so G is needed</span>
-    <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">list_free_indices</span><span class="p">):</span>
-        <span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">G</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="c"># no_free list of lists of indices for component tensors without fixed</span>
-    <span class="c"># indices</span>
-    <span class="n">no_free</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">size</span> <span class="o">=</span> <span class="n">gens</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">size</span>
-    <span class="n">id_af</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">))</span>
-    <span class="n">num_indices</span> <span class="o">=</span> <span class="n">size</span> <span class="o">-</span> <span class="mi">2</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">list_free_indices</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-        <span class="n">no_free</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">num_indices</span><span class="p">)))</span>
-    <span class="n">res_base</span><span class="p">,</span> <span class="n">res_gens</span> <span class="o">=</span> <span class="n">_get_bsgs</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">list_free_indices</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">list_free_indices</span><span class="p">)):</span>
-        <span class="n">base1</span><span class="p">,</span> <span class="n">gens1</span> <span class="o">=</span> <span class="n">_get_bsgs</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">list_free_indices</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-        <span class="n">res_base</span><span class="p">,</span> <span class="n">res_gens</span> <span class="o">=</span> <span class="n">bsgs_direct_product</span><span class="p">(</span><span class="n">res_base</span><span class="p">,</span> <span class="n">res_gens</span><span class="p">,</span>
-            <span class="n">base1</span><span class="p">,</span> <span class="n">gens1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">list_free_indices</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-            <span class="n">no_free</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="n">size</span> <span class="o">-</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">num_indices</span><span class="p">)))</span>
-        <span class="n">size</span> <span class="o">+=</span> <span class="n">num_indices</span>
-    <span class="n">nr</span> <span class="o">=</span> <span class="n">size</span> <span class="o">-</span> <span class="mi">2</span>
-    <span class="n">res_gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">h</span> <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">res_gens</span> <span class="k">if</span> <span class="n">h</span><span class="o">.</span><span class="n">_array_form</span> <span class="o">!=</span> <span class="n">id_af</span><span class="p">]</span>
-    <span class="c"># if sym there are no commuting tensors stop here</span>
-    <span class="k">if</span> <span class="n">sym</span> <span class="o">==</span> <span class="bp">None</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">no_free</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">res_gens</span><span class="p">:</span>
-            <span class="n">res_gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">_af_new</span><span class="p">(</span><span class="n">id_af</span><span class="p">)]</span>
-        <span class="k">return</span> <span class="n">size</span><span class="p">,</span> <span class="n">res_base</span><span class="p">,</span> <span class="n">res_gens</span>
-
-    <span class="c"># if the component tensors have moinimal BSGS, so is their direct</span>
-    <span class="c"># product P; the slot symmetry group is S = P*C, where C is the group</span>
-    <span class="c"># to (anti)commute the component tensors with no free indices</span>
-    <span class="c"># a stabilizer has the property S_i = P_i*C_i;</span>
-    <span class="c"># the BSGS of P*C has SGS_P + SGS_C and the base is</span>
-    <span class="c"># the ordered union of the bases of P and C.</span>
-    <span class="c"># If P has minimal BSGS, so has S with this base.</span>
-    <span class="n">base_comm</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">no_free</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">ind1</span> <span class="o">=</span> <span class="n">no_free</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-        <span class="n">ind2</span> <span class="o">=</span> <span class="n">no_free</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">ind1</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-        <span class="n">a</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">ind2</span><span class="p">)</span>
-        <span class="n">a</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">ind1</span><span class="p">)</span>
-        <span class="n">base_comm</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ind1</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="n">a</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">ind2</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">nr</span><span class="p">)))</span>
-        <span class="k">if</span> <span class="n">sym</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">a</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">nr</span><span class="p">,</span> <span class="n">nr</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">a</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">nr</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">nr</span><span class="p">])</span>
-        <span class="n">res_gens</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">_af_new</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
-    <span class="c"># each base is ordered; order the union of the two bases</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">base_comm</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">res_base</span><span class="p">:</span>
-            <span class="n">res_base</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-    <span class="n">res_base</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">res_gens</span><span class="p">:</span>
-        <span class="n">res_gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">_af_new</span><span class="p">(</span><span class="n">id_af</span><span class="p">)]</span>
-
-    <span class="k">return</span> <span class="n">size</span><span class="p">,</span> <span class="n">res_base</span><span class="p">,</span> <span class="n">res_gens</span>
-
-
-<span class="k">def</span> <span class="nf">gens_products</span><span class="p">(</span><span class="o">*</span><span class="n">v</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns size, res_base, res_gens BSGS for n tensors of different types</span>
-
-<span class="sd">    v is a sequence of (base_i, gens_i, free_i, sym_i)</span>
-<span class="sd">    where</span>
-<span class="sd">    base_i, gens_i  BSGS of tensor of type `i`</span>
-<span class="sd">    free_i          list of the fixed slots for each of the tensors</span>
-<span class="sd">                    of type `i`; if there are `n_i` tensors of type `i`</span>
-<span class="sd">                    and none of them have fixed slots, `free = [[]]*n_i`</span>
-<span class="sd">    sym   0 (1) if the tensors of type `i` (anti)commute among themselves</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.tensor_can import get_symmetric_group_sgs, gens_products</span>
-<span class="sd">    &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">    &gt;&gt;&gt; base, gens = get_symmetric_group_sgs(2)</span>
-<span class="sd">    &gt;&gt;&gt; gens_products((base,gens,[[],[]],0))</span>
-<span class="sd">    (6, [0, 2], [Permutation(5)(0, 1), Permutation(5)(2, 3), Permutation(5)(0, 2)(1, 3)])</span>
-<span class="sd">    &gt;&gt;&gt; gens_products((base,gens,[[1],[]],0))</span>
-<span class="sd">    (6, [2], [Permutation(5)(2, 3)])</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">res_size</span><span class="p">,</span> <span class="n">res_base</span><span class="p">,</span> <span class="n">res_gens</span> <span class="o">=</span> <span class="n">tensor_gens</span><span class="p">(</span><span class="o">*</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">v</span><span class="p">)):</span>
-        <span class="n">size</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="n">gens</span> <span class="o">=</span> <span class="n">tensor_gens</span><span class="p">(</span><span class="o">*</span><span class="n">v</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-        <span class="n">res_base</span><span class="p">,</span> <span class="n">res_gens</span> <span class="o">=</span> <span class="n">bsgs_direct_product</span><span class="p">(</span><span class="n">res_base</span><span class="p">,</span> <span class="n">res_gens</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span>
-                                               <span class="n">gens</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="n">res_size</span> <span class="o">=</span> <span class="n">res_gens</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">size</span>
-    <span class="n">id_af</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">res_size</span><span class="p">))</span>
-    <span class="n">res_gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">h</span> <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">res_gens</span> <span class="k">if</span> <span class="n">h</span> <span class="o">!=</span> <span class="n">id_af</span><span class="p">]</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">res_gens</span><span class="p">:</span>
-        <span class="n">res_gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">id_af</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">res_size</span><span class="p">,</span> <span class="n">res_base</span><span class="p">,</span> <span class="n">res_gens</span>
-</pre></div>
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-            
-  <h1>Source code for sympy.combinatorics.testutil</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.util</span> <span class="kn">import</span> <span class="n">_distribute_gens_by_base</span>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-
-<span class="n">rmul</span> <span class="o">=</span> <span class="n">Permutation</span><span class="o">.</span><span class="n">rmul</span>
-
-
-<div class="viewcode-block" id="_cmp_perm_lists"><a class="viewcode-back" href="../../../modules/combinatorics/testutil.html#sympy.combinatorics.testutil._cmp_perm_lists">[docs]</a><span class="k">def</span> <span class="nf">_cmp_perm_lists</span><span class="p">(</span><span class="n">first</span><span class="p">,</span> <span class="n">second</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compare two lists of permutations as sets.</span>
-
-<span class="sd">    This is used for testing purposes. Since the array form of a</span>
-<span class="sd">    permutation is currently a list, Permutation is not hashable</span>
-<span class="sd">    and cannot be put into a set.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.testutil import _cmp_perm_lists</span>
-<span class="sd">    &gt;&gt;&gt; a = Permutation([0, 2, 3, 4, 1])</span>
-<span class="sd">    &gt;&gt;&gt; b = Permutation([1, 2, 0, 4, 3])</span>
-<span class="sd">    &gt;&gt;&gt; c = Permutation([3, 4, 0, 1, 2])</span>
-<span class="sd">    &gt;&gt;&gt; ls1 = [a, b, c]</span>
-<span class="sd">    &gt;&gt;&gt; ls2 = [b, c, a]</span>
-<span class="sd">    &gt;&gt;&gt; _cmp_perm_lists(ls1, ls2)</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="nb">set</span><span class="p">([</span><span class="nb">tuple</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">first</span><span class="p">])</span> <span class="o">==</span> \
-        <span class="nb">set</span><span class="p">([</span><span class="nb">tuple</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">second</span><span class="p">])</span>
-
-</div>
-<div class="viewcode-block" id="_naive_list_centralizer"><a class="viewcode-back" href="../../../modules/combinatorics/testutil.html#sympy.combinatorics.testutil._naive_list_centralizer">[docs]</a><span class="k">def</span> <span class="nf">_naive_list_centralizer</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="n">af</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a list of elements for the centralizer of a subgroup/set/element.</span>
-
-<span class="sd">    This is a brute-force implementation that goes over all elements of the</span>
-<span class="sd">    group and checks for membership in the centralizer. It is used to</span>
-<span class="sd">    test ``.centralizer()`` from ``sympy.combinatorics.perm_groups``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.testutil import _naive_list_centralizer</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.named_groups import DihedralGroup</span>
-<span class="sd">    &gt;&gt;&gt; D = DihedralGroup(4)</span>
-<span class="sd">    &gt;&gt;&gt; _naive_list_centralizer(D, D)</span>
-<span class="sd">    [Permutation([0, 1, 2, 3]), Permutation([2, 3, 0, 1])]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.combinatorics.perm_groups.centralizer</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">_af_commutes_with</span>
-    <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="s">&#39;generators&#39;</span><span class="p">):</span>
-        <span class="n">elements</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">generate_dimino</span><span class="p">(</span><span class="n">af</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="o">.</span><span class="n">_array_form</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">other</span><span class="o">.</span><span class="n">generators</span><span class="p">]</span>
-        <span class="n">commutes_with_gens</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="nb">all</span><span class="p">(</span><span class="n">_af_commutes_with</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">gen</span><span class="p">)</span> <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">)</span>
-        <span class="n">centralizer_list</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">af</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">element</span> <span class="ow">in</span> <span class="n">elements</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">commutes_with_gens</span><span class="p">(</span><span class="n">element</span><span class="p">):</span>
-                    <span class="n">centralizer_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Permutation</span><span class="o">.</span><span class="n">_af_new</span><span class="p">(</span><span class="n">element</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">element</span> <span class="ow">in</span> <span class="n">elements</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">commutes_with_gens</span><span class="p">(</span><span class="n">element</span><span class="p">):</span>
-                    <span class="n">centralizer_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">element</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">centralizer_list</span>
-    <span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="s">&#39;getitem&#39;</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_naive_list_centralizer</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="n">other</span><span class="p">),</span> <span class="n">af</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="s">&#39;array_form&#39;</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_naive_list_centralizer</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">other</span><span class="p">]),</span> <span class="n">af</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="_verify_bsgs"><a class="viewcode-back" href="../../../modules/combinatorics/testutil.html#sympy.combinatorics.testutil._verify_bsgs">[docs]</a><span class="k">def</span> <span class="nf">_verify_bsgs</span><span class="p">(</span><span class="n">group</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="n">gens</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Verify the correctness of a base and strong generating set.</span>
-
-<span class="sd">    This is a naive implementation using the definition of a base and a strong</span>
-<span class="sd">    generating set relative to it. There are other procedures for</span>
-<span class="sd">    verifying a base and strong generating set, but this one will</span>
-<span class="sd">    serve for more robust testing.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.named_groups import AlternatingGroup</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.testutil import _verify_bsgs</span>
-<span class="sd">    &gt;&gt;&gt; A = AlternatingGroup(4)</span>
-<span class="sd">    &gt;&gt;&gt; A.schreier_sims()</span>
-<span class="sd">    &gt;&gt;&gt; _verify_bsgs(A, A.base, A.strong_gens)</span>
-<span class="sd">    True</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-    <span class="n">strong_gens_distr</span> <span class="o">=</span> <span class="n">_distribute_gens_by_base</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">gens</span><span class="p">)</span>
-    <span class="n">current_stabilizer</span> <span class="o">=</span> <span class="n">group</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">)):</span>
-        <span class="n">candidate</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="n">strong_gens_distr</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">current_stabilizer</span><span class="o">.</span><span class="n">order</span><span class="p">()</span> <span class="o">!=</span> <span class="n">candidate</span><span class="o">.</span><span class="n">order</span><span class="p">():</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="n">current_stabilizer</span> <span class="o">=</span> <span class="n">current_stabilizer</span><span class="o">.</span><span class="n">stabilizer</span><span class="p">(</span><span class="n">base</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-    <span class="k">if</span> <span class="n">current_stabilizer</span><span class="o">.</span><span class="n">order</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">False</span>
-    <span class="k">return</span> <span class="bp">True</span>
-
-</div>
-<div class="viewcode-block" id="_verify_centralizer"><a class="viewcode-back" href="../../../modules/combinatorics/testutil.html#sympy.combinatorics.testutil._verify_centralizer">[docs]</a><span class="k">def</span> <span class="nf">_verify_centralizer</span><span class="p">(</span><span class="n">group</span><span class="p">,</span> <span class="n">arg</span><span class="p">,</span> <span class="n">centr</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Verify the centralizer of a group/set/element inside another group.</span>
-
-<span class="sd">    This is used for testing ``.centralizer()`` from</span>
-<span class="sd">    ``sympy.combinatorics.perm_groups``</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.named_groups import (SymmetricGroup,</span>
-<span class="sd">    ... AlternatingGroup)</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.testutil import _verify_centralizer</span>
-<span class="sd">    &gt;&gt;&gt; S = SymmetricGroup(5)</span>
-<span class="sd">    &gt;&gt;&gt; A = AlternatingGroup(5)</span>
-<span class="sd">    &gt;&gt;&gt; centr = PermutationGroup([Permutation([0, 1, 2, 3, 4])])</span>
-<span class="sd">    &gt;&gt;&gt; _verify_centralizer(S, A, centr)</span>
-<span class="sd">    True</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    _naive_list_centralizer,</span>
-<span class="sd">    sympy.combinatorics.perm_groups.PermutationGroup.centralizer,</span>
-<span class="sd">    _cmp_perm_lists</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">centr</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">centr</span> <span class="o">=</span> <span class="n">group</span><span class="o">.</span><span class="n">centralizer</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-    <span class="n">centr_list</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">centr</span><span class="o">.</span><span class="n">generate_dimino</span><span class="p">(</span><span class="n">af</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-    <span class="n">centr_list_naive</span> <span class="o">=</span> <span class="n">_naive_list_centralizer</span><span class="p">(</span><span class="n">group</span><span class="p">,</span> <span class="n">arg</span><span class="p">,</span> <span class="n">af</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">_cmp_perm_lists</span><span class="p">(</span><span class="n">centr_list</span><span class="p">,</span> <span class="n">centr_list_naive</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="_verify_normal_closure"><a class="viewcode-back" href="../../../modules/combinatorics/testutil.html#sympy.combinatorics.testutil._verify_normal_closure">[docs]</a><span class="k">def</span> <span class="nf">_verify_normal_closure</span><span class="p">(</span><span class="n">group</span><span class="p">,</span> <span class="n">arg</span><span class="p">,</span> <span class="n">closure</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Verify the normal closure of a subgroup/subset/element in a group.</span>
-
-<span class="sd">    This is used to test</span>
-<span class="sd">    sympy.combinatorics.perm_groups.PermutationGroup.normal_closure</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.named_groups import (SymmetricGroup,</span>
-<span class="sd">    ... AlternatingGroup)</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.testutil import _verify_normal_closure</span>
-<span class="sd">    &gt;&gt;&gt; S = SymmetricGroup(3)</span>
-<span class="sd">    &gt;&gt;&gt; A = AlternatingGroup(3)</span>
-<span class="sd">    &gt;&gt;&gt; _verify_normal_closure(S, A, closure=A)</span>
-<span class="sd">    True</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.combinatorics.perm_groups.PermutationGroup.normal_closure</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">closure</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">closure</span> <span class="o">=</span> <span class="n">group</span><span class="o">.</span><span class="n">normal_closure</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-    <span class="n">conjugates</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="s">&#39;generators&#39;</span><span class="p">):</span>
-        <span class="n">subgr_gens</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">generators</span>
-    <span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="s">&#39;__getitem__&#39;</span><span class="p">):</span>
-        <span class="n">subgr_gens</span> <span class="o">=</span> <span class="n">arg</span>
-    <span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="s">&#39;array_form&#39;</span><span class="p">):</span>
-        <span class="n">subgr_gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">arg</span><span class="p">]</span>
-    <span class="k">for</span> <span class="n">el</span> <span class="ow">in</span> <span class="n">group</span><span class="o">.</span><span class="n">generate_dimino</span><span class="p">():</span>
-        <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">subgr_gens</span><span class="p">:</span>
-            <span class="n">conjugates</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">gen</span><span class="o">^</span><span class="n">el</span><span class="p">)</span>
-    <span class="n">naive_closure</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">conjugates</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">closure</span><span class="o">.</span><span class="n">is_subgroup</span><span class="p">(</span><span class="n">naive_closure</span><span class="p">)</span>
-</div>
-<span class="k">def</span> <span class="nf">canonicalize_naive</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">dummies</span><span class="p">,</span> <span class="n">sym</span><span class="p">,</span> <span class="o">*</span><span class="n">v</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    canonicalize tensor formed by tensors of the different types</span>
-
-<span class="sd">    g  permutation representing the tensor</span>
-<span class="sd">    dummies  list of dummy indices</span>
-<span class="sd">    msym symmetry of the metric</span>
-
-<span class="sd">    v is a list of (base_i, gens_i, n_i, sym_i) for tensors of type `i`</span>
-<span class="sd">    base_i, gens_i BSGS for tensors of this type</span>
-<span class="sd">    n_i  number ot tensors of type `i`</span>
-
-<span class="sd">    sym_i symmetry under exchange of two component tensors of type `i`</span>
-<span class="sd">          None  no symmetry</span>
-<span class="sd">          0     commuting</span>
-<span class="sd">          1     anticommuting</span>
-
-<span class="sd">    Return 0 if the tensor is zero, else return the array form of</span>
-<span class="sd">    the permutation representing the canonical form of the tensor.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.testutil import canonicalize_naive</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.tensor_can import get_symmetric_group_sgs</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics import Permutation, PermutationGroup</span>
-<span class="sd">    &gt;&gt;&gt; g = Permutation([1,3,2,0,4,5])</span>
-<span class="sd">    &gt;&gt;&gt; base2, gens2 = get_symmetric_group_sgs(2)</span>
-<span class="sd">    &gt;&gt;&gt; canonicalize_naive(g, [2, 3], 0, (base2, gens2, 2, 0))</span>
-<span class="sd">    [0, 2, 1, 3, 4, 5]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-    <span class="kn">from</span> <span class="nn">sympy.combinatorics.tensor_can</span> <span class="kn">import</span> <span class="n">gens_products</span><span class="p">,</span> <span class="n">dummy_sgs</span>
-    <span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span><span class="p">,</span> <span class="n">_af_rmul</span>
-    <span class="n">v1</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">v</span><span class="p">)):</span>
-        <span class="n">base_i</span><span class="p">,</span> <span class="n">gens_i</span><span class="p">,</span> <span class="n">n_i</span><span class="p">,</span> <span class="n">sym_i</span> <span class="o">=</span> <span class="n">v</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-        <span class="n">v1</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">base_i</span><span class="p">,</span> <span class="n">gens_i</span><span class="p">,</span> <span class="p">[[]]</span><span class="o">*</span><span class="n">n_i</span><span class="p">,</span> <span class="n">sym_i</span><span class="p">))</span>
-    <span class="n">size</span><span class="p">,</span> <span class="n">sbase</span><span class="p">,</span> <span class="n">sgens</span> <span class="o">=</span> <span class="n">gens_products</span><span class="p">(</span><span class="o">*</span><span class="n">v1</span><span class="p">)</span>
-    <span class="n">dgens</span> <span class="o">=</span> <span class="n">dummy_sgs</span><span class="p">(</span><span class="n">dummies</span><span class="p">,</span> <span class="n">sym</span><span class="p">,</span> <span class="n">size</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-        <span class="n">num_types</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">dummies</span> <span class="o">=</span> <span class="p">[</span><span class="n">dummies</span><span class="p">]</span>
-        <span class="n">sym</span> <span class="o">=</span> <span class="p">[</span><span class="n">sym</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">num_types</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">sym</span><span class="p">)</span>
-    <span class="n">dgens</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">num_types</span><span class="p">):</span>
-        <span class="n">dgens</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">dummy_sgs</span><span class="p">(</span><span class="n">dummies</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">sym</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">size</span> <span class="o">-</span> <span class="mi">2</span><span class="p">))</span>
-    <span class="n">S</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="n">sgens</span><span class="p">)</span>
-    <span class="n">D</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">Permutation</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">dgens</span><span class="p">])</span>
-    <span class="n">dlist</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">D</span><span class="o">.</span><span class="n">generate</span><span class="p">(</span><span class="n">af</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-    <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">array_form</span>
-    <span class="n">st</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">S</span><span class="o">.</span><span class="n">generate</span><span class="p">(</span><span class="n">af</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">_af_rmul</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">dlist</span><span class="p">:</span>
-            <span class="n">q</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">_af_rmul</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">h</span><span class="p">))</span>
-            <span class="n">st</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">st</span><span class="p">)</span>
-    <span class="n">a</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-    <span class="n">prev</span> <span class="o">=</span> <span class="p">(</span><span class="mi">0</span><span class="p">,)</span><span class="o">*</span><span class="n">size</span>
-    <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">a</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">h</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="n">prev</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">]:</span>
-            <span class="k">if</span> <span class="n">h</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">!=</span> <span class="n">prev</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-                <span class="k">return</span> <span class="mi">0</span>
-        <span class="n">prev</span> <span class="o">=</span> <span class="n">h</span>
-    <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-<span class="k">def</span> <span class="nf">graph_certificate</span><span class="p">(</span><span class="n">gr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a certificate for the graph</span>
-
-<span class="sd">    gr adjacency list</span>
-
-<span class="sd">    The graph is assumed to be unoriented and without</span>
-<span class="sd">    external lines.</span>
-
-<span class="sd">    Associate to each vertex of the graph a symmetric tensor with</span>
-<span class="sd">    number of indices equal to the degree of the vertex; indices</span>
-<span class="sd">    are contracted when they correspond to the same line of the graph.</span>
-<span class="sd">    The canonical form of the tensor gives a certificate for the graph.</span>
-
-<span class="sd">    This is not an efficient algorithm to get the certificate of a graph.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.testutil import graph_certificate</span>
-<span class="sd">    &gt;&gt;&gt; gr1 = {0:[1,2,3,5], 1:[0,2,4], 2:[0,1,3,4], 3:[0,2,4], 4:[1,2,3,5], 5:[0,4]}</span>
-<span class="sd">    &gt;&gt;&gt; gr2 = {0:[1,5], 1:[0,2,3,4], 2:[1,3,5], 3:[1,2,4,5], 4:[1,3,5], 5:[0,2,3,4]}</span>
-<span class="sd">    &gt;&gt;&gt; c1 = graph_certificate(gr1)</span>
-<span class="sd">    &gt;&gt;&gt; c2 = graph_certificate(gr2)</span>
-<span class="sd">    &gt;&gt;&gt; c1</span>
-<span class="sd">    [0, 2, 4, 6, 1, 8, 10, 12, 3, 14, 16, 18, 5, 9, 15, 7, 11, 17, 13, 19, 20, 21]</span>
-<span class="sd">    &gt;&gt;&gt; c1 == c2</span>
-<span class="sd">    True</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">_af_invert</span>
-    <span class="kn">from</span> <span class="nn">sympy.combinatorics.tensor_can</span> <span class="kn">import</span> <span class="n">get_symmetric_group_sgs</span><span class="p">,</span> <span class="n">canonicalize</span>
-    <span class="n">items</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">gr</span><span class="o">.</span><span class="n">items</span><span class="p">())</span>
-    <span class="n">items</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">pvert</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">items</span><span class="p">]</span>
-    <span class="n">pvert</span> <span class="o">=</span> <span class="n">_af_invert</span><span class="p">(</span><span class="n">pvert</span><span class="p">)</span>
-
-    <span class="c"># the indices of the tensor are twice the number of lines of the graph</span>
-    <span class="n">num_indices</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">neigh</span> <span class="ow">in</span> <span class="n">items</span><span class="p">:</span>
-      <span class="n">num_indices</span> <span class="o">+=</span> <span class="nb">len</span><span class="p">(</span><span class="n">neigh</span><span class="p">)</span>
-    <span class="c"># associate to each vertex its indices; for each line</span>
-    <span class="c"># between two vertices assign the</span>
-    <span class="c"># even index to the vertex which comes first in items,</span>
-    <span class="c"># the odd index to the other vertex</span>
-    <span class="n">vertices</span> <span class="o">=</span> <span class="p">[[]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">items</span><span class="p">]</span>
-    <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">neigh</span> <span class="ow">in</span> <span class="n">items</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">v2</span> <span class="ow">in</span> <span class="n">neigh</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">pvert</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">pvert</span><span class="p">[</span><span class="n">v2</span><span class="p">]:</span>
-                <span class="n">vertices</span><span class="p">[</span><span class="n">pvert</span><span class="p">[</span><span class="n">v</span><span class="p">]]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                <span class="n">vertices</span><span class="p">[</span><span class="n">pvert</span><span class="p">[</span><span class="n">v2</span><span class="p">]]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-                <span class="n">i</span> <span class="o">+=</span> <span class="mi">2</span>
-    <span class="n">g</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">vertices</span><span class="p">:</span>
-      <span class="n">g</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-    <span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">g</span><span class="p">)</span> <span class="o">==</span> <span class="n">num_indices</span>
-    <span class="n">g</span> <span class="o">+=</span> <span class="p">[</span><span class="n">num_indices</span><span class="p">,</span> <span class="n">num_indices</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
-    <span class="n">size</span> <span class="o">=</span> <span class="n">num_indices</span> <span class="o">+</span> <span class="mi">2</span>
-    <span class="k">assert</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">g</span><span class="p">)</span> <span class="o">==</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">))</span>
-    <span class="n">g</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-    <span class="n">vlen</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">vertices</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">neigh</span> <span class="ow">in</span> <span class="n">vertices</span><span class="p">:</span>
-        <span class="n">vlen</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="n">neigh</span><span class="p">)]</span> <span class="o">+=</span> <span class="mi">1</span>
-    <span class="n">v</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">vlen</span><span class="p">)):</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">vlen</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">n</span><span class="p">:</span>
-            <span class="n">base</span><span class="p">,</span> <span class="n">gens</span> <span class="o">=</span> <span class="n">get_symmetric_group_sgs</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="n">v</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">base</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-    <span class="n">v</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-    <span class="n">dummies</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">num_indices</span><span class="p">))</span>
-    <span class="n">can</span> <span class="o">=</span> <span class="n">canonicalize</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">dummies</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">*</span><span class="n">v</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">can</span>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.combinatorics.util</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">isprime</span>
-<span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span><span class="p">,</span> <span class="n">_af_invert</span><span class="p">,</span> <span class="n">_af_rmul</span>
-
-<span class="n">rmul</span> <span class="o">=</span> <span class="n">Permutation</span><span class="o">.</span><span class="n">rmul</span>
-<span class="n">_af_new</span> <span class="o">=</span> <span class="n">Permutation</span><span class="o">.</span><span class="n">_af_new</span>
-
-<span class="c">############################################</span>
-<span class="c">###</span>
-<span class="c">### Utilities for computational group theory</span>
-<span class="c">###</span>
-<span class="c">############################################</span>
-
-
-<div class="viewcode-block" id="_base_ordering"><a class="viewcode-back" href="../../../modules/combinatorics/util.html#sympy.combinatorics.util._base_ordering">[docs]</a><span class="k">def</span> <span class="nf">_base_ordering</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">degree</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Order `\{0, 1, ..., n-1\}` so that base points come first and in order.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    ``base`` - the base</span>
-<span class="sd">    ``degree`` - the degree of the associated permutation group</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A list ``base_ordering`` such that ``base_ordering[point]`` is the</span>
-<span class="sd">    number of ``point`` in the ordering.</span>
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.named_groups import SymmetricGroup</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.util import _base_ordering</span>
-<span class="sd">    &gt;&gt;&gt; S = SymmetricGroup(4)</span>
-<span class="sd">    &gt;&gt;&gt; S.schreier_sims()</span>
-<span class="sd">    &gt;&gt;&gt; _base_ordering(S.base, S.degree)</span>
-<span class="sd">    [0, 1, 2, 3]</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    This is used in backtrack searches, when we define a relation `&lt;&lt;` on</span>
-<span class="sd">    the underlying set for a permutation group of degree `n`,</span>
-<span class="sd">    `\{0, 1, ..., n-1\}`, so that if `(b_1, b_2, ..., b_k)` is a base we</span>
-<span class="sd">    have `b_i &lt;&lt; b_j` whenever `i&lt;j` and `b_i &lt;&lt; a` for all</span>
-<span class="sd">    `i\in\{1,2, ..., k\}` and `a` is not in the base. The idea is developed</span>
-<span class="sd">    and applied to backtracking algorithms in [1], pp.108-132. The points</span>
-<span class="sd">    that are not in the base are taken in increasing order.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] Holt, D., Eick, B., O&#39;Brien, E.</span>
-<span class="sd">    &quot;Handbook of computational group theory&quot;</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">base_len</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-    <span class="n">ordering</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">degree</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">base_len</span><span class="p">):</span>
-        <span class="n">ordering</span><span class="p">[</span><span class="n">base</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span> <span class="o">=</span> <span class="n">i</span>
-    <span class="n">current</span> <span class="o">=</span> <span class="n">base_len</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">degree</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">base</span><span class="p">:</span>
-            <span class="n">ordering</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">current</span>
-            <span class="n">current</span> <span class="o">+=</span> <span class="mi">1</span>
-    <span class="k">return</span> <span class="n">ordering</span>
-
-</div>
-<div class="viewcode-block" id="_check_cycles_alt_sym"><a class="viewcode-back" href="../../../modules/combinatorics/util.html#sympy.combinatorics.util._check_cycles_alt_sym">[docs]</a><span class="k">def</span> <span class="nf">_check_cycles_alt_sym</span><span class="p">(</span><span class="n">perm</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Checks for cycles of prime length p with n/2 &lt; p &lt; n-2.</span>
-
-<span class="sd">    Here `n` is the degree of the permutation. This is a helper function for</span>
-<span class="sd">    the function is_alt_sym from sympy.combinatorics.perm_groups.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.util import _check_cycles_alt_sym</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; a = Permutation([[0,1,2,3,4,5,6,7,8,9,10], [11, 12]])</span>
-<span class="sd">    &gt;&gt;&gt; _check_cycles_alt_sym(a)</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; b = Permutation([[0,1,2,3,4,5,6], [7,8,9,10]])</span>
-<span class="sd">    &gt;&gt;&gt; _check_cycles_alt_sym(b)</span>
-<span class="sd">    True</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.combinatorics.perm_groups.PermutationGroup.is_alt_sym</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">perm</span><span class="o">.</span><span class="n">size</span>
-    <span class="n">af</span> <span class="o">=</span> <span class="n">perm</span><span class="o">.</span><span class="n">array_form</span>
-    <span class="n">current_len</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">total_len</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">used</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="o">//</span><span class="mi">2</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">used</span> <span class="ow">and</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">n</span><span class="o">//</span><span class="mi">2</span> <span class="o">-</span> <span class="n">total_len</span><span class="p">:</span>
-            <span class="n">current_len</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="n">used</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="n">j</span> <span class="o">=</span> <span class="n">i</span>
-            <span class="k">while</span><span class="p">(</span><span class="n">af</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">!=</span> <span class="n">i</span><span class="p">):</span>
-                <span class="n">current_len</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="n">j</span> <span class="o">=</span> <span class="n">af</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-                <span class="n">used</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-            <span class="n">total_len</span> <span class="o">+=</span> <span class="n">current_len</span>
-            <span class="k">if</span> <span class="n">current_len</span> <span class="o">&gt;</span> <span class="n">n</span><span class="o">//</span><span class="mi">2</span> <span class="ow">and</span> <span class="n">current_len</span> <span class="o">&lt;</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">isprime</span><span class="p">(</span><span class="n">current_len</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">True</span>
-    <span class="k">return</span> <span class="bp">False</span>
-
-</div>
-<div class="viewcode-block" id="_distribute_gens_by_base"><a class="viewcode-back" href="../../../modules/combinatorics/util.html#sympy.combinatorics.util._distribute_gens_by_base">[docs]</a><span class="k">def</span> <span class="nf">_distribute_gens_by_base</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">gens</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Distribute the group elements ``gens`` by membership in basic stabilizers.</span>
-
-<span class="sd">    Notice that for a base `(b_1, b_2, ..., b_k)`, the basic stabilizers</span>
-<span class="sd">    are defined as `G^{(i)} = G_{b_1, ..., b_{i-1}}` for</span>
-<span class="sd">    `i \in\{1, 2, ..., k\}`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    ``base`` - a sequence of points in `\{0, 1, ..., n-1\}`</span>
-<span class="sd">    ``gens`` - a list of elements of a permutation group of degree `n`.</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    List of length `k`, where `k` is</span>
-<span class="sd">    the length of ``base``. The `i`-th entry contains those elements in</span>
-<span class="sd">    ``gens`` which fix the first `i` elements of ``base`` (so that the</span>
-<span class="sd">    `0`-th entry is equal to ``gens`` itself). If no element fixes the first</span>
-<span class="sd">    `i` elements of ``base``, the `i`-th element is set to a list containing</span>
-<span class="sd">    the identity element.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.named_groups import DihedralGroup</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.util import _distribute_gens_by_base</span>
-<span class="sd">    &gt;&gt;&gt; D = DihedralGroup(3)</span>
-<span class="sd">    &gt;&gt;&gt; D.schreier_sims()</span>
-<span class="sd">    &gt;&gt;&gt; D.strong_gens</span>
-<span class="sd">    [Permutation(0, 1, 2), Permutation(0, 2), Permutation(1, 2)]</span>
-<span class="sd">    &gt;&gt;&gt; D.base</span>
-<span class="sd">    [0, 1]</span>
-<span class="sd">    &gt;&gt;&gt; _distribute_gens_by_base(D.base, D.strong_gens)</span>
-<span class="sd">    [[Permutation(0, 1, 2), Permutation(0, 2), Permutation(1, 2)],</span>
-<span class="sd">     [Permutation(1, 2)]]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    _strong_gens_from_distr, _orbits_transversals_from_bsgs,</span>
-<span class="sd">    _handle_precomputed_bsgs</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">base_len</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-    <span class="n">degree</span> <span class="o">=</span> <span class="n">gens</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">size</span>
-    <span class="n">stabs</span> <span class="o">=</span> <span class="p">[[]</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">base_len</span><span class="p">)]</span>
-    <span class="n">max_stab_index</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">:</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">while</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="n">base_len</span> <span class="o">-</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">gen</span><span class="o">.</span><span class="n">_array_form</span><span class="p">[</span><span class="n">base</span><span class="p">[</span><span class="n">j</span><span class="p">]]</span> <span class="o">==</span> <span class="n">base</span><span class="p">[</span><span class="n">j</span><span class="p">]:</span>
-            <span class="n">j</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">j</span> <span class="o">&gt;</span> <span class="n">max_stab_index</span><span class="p">:</span>
-            <span class="n">max_stab_index</span> <span class="o">=</span> <span class="n">j</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">stabs</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">gen</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">max_stab_index</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">base_len</span><span class="p">):</span>
-        <span class="n">stabs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">_af_new</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">degree</span><span class="p">))))</span>
-    <span class="k">return</span> <span class="n">stabs</span>
-</div>
-<div class="viewcode-block" id="_handle_precomputed_bsgs"><a class="viewcode-back" href="../../../modules/combinatorics/util.html#sympy.combinatorics.util._handle_precomputed_bsgs">[docs]</a><span class="k">def</span> <span class="nf">_handle_precomputed_bsgs</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span><span class="p">,</span> <span class="n">transversals</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span>
-                             <span class="n">basic_orbits</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">strong_gens_distr</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Calculate BSGS-related structures from those present.</span>
-
-<span class="sd">    The base and strong generating set must be provided; if any of the</span>
-<span class="sd">    transversals, basic orbits or distributed strong generators are not</span>
-<span class="sd">    provided, they will be calculated from the base and strong generating set.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    ``base`` - the base</span>
-<span class="sd">    ``strong_gens`` - the strong generators</span>
-<span class="sd">    ``transversals`` - basic transversals</span>
-<span class="sd">    ``basic_orbits`` - basic orbits</span>
-<span class="sd">    ``strong_gens_distr`` - strong generators distributed by membership in basic</span>
-<span class="sd">    stabilizers</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    ``(transversals, basic_orbits, strong_gens_distr)`` where ``transversals``</span>
-<span class="sd">    are the basic transversals, ``basic_orbits`` are the basic orbits, and</span>
-<span class="sd">    ``strong_gens_distr`` are the strong generators distributed by membership</span>
-<span class="sd">    in basic stabilizers.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.named_groups import DihedralGroup</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.util import _handle_precomputed_bsgs</span>
-<span class="sd">    &gt;&gt;&gt; D = DihedralGroup(3)</span>
-<span class="sd">    &gt;&gt;&gt; D.schreier_sims()</span>
-<span class="sd">    &gt;&gt;&gt; _handle_precomputed_bsgs(D.base, D.strong_gens,</span>
-<span class="sd">    ... basic_orbits=D.basic_orbits)</span>
-<span class="sd">    ([{0: Permutation(2), 1: Permutation(0, 1, 2), 2: Permutation(0, 2)},</span>
-<span class="sd">    {1: Permutation(2), 2: Permutation(1, 2)}],</span>
-<span class="sd">    [[0, 1, 2], [1, 2]], [[Permutation(0, 1, 2),</span>
-<span class="sd">                           Permutation(0, 2),</span>
-<span class="sd">                           Permutation(1, 2)],</span>
-<span class="sd">                          [Permutation(1, 2)]])</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    _orbits_transversals_from_bsgs, distribute_gens_by_base</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">strong_gens_distr</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">strong_gens_distr</span> <span class="o">=</span> <span class="n">_distribute_gens_by_base</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">transversals</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">basic_orbits</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">basic_orbits</span><span class="p">,</span> <span class="n">transversals</span> <span class="o">=</span> \
-                <span class="n">_orbits_transversals_from_bsgs</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens_distr</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">transversals</span> <span class="o">=</span> \
-                <span class="n">_orbits_transversals_from_bsgs</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens_distr</span><span class="p">,</span>
-                                           <span class="n">transversals_only</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">basic_orbits</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">base_len</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-            <span class="n">basic_orbits</span> <span class="o">=</span> <span class="p">[</span><span class="bp">None</span><span class="p">]</span><span class="o">*</span><span class="n">base_len</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">base_len</span><span class="p">):</span>
-                <span class="n">basic_orbits</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">transversals</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-    <span class="k">return</span> <span class="n">transversals</span><span class="p">,</span> <span class="n">basic_orbits</span><span class="p">,</span> <span class="n">strong_gens_distr</span>
-
-</div>
-<div class="viewcode-block" id="_orbits_transversals_from_bsgs"><a class="viewcode-back" href="../../../modules/combinatorics/util.html#sympy.combinatorics.util._orbits_transversals_from_bsgs">[docs]</a><span class="k">def</span> <span class="nf">_orbits_transversals_from_bsgs</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens_distr</span><span class="p">,</span>
-                                   <span class="n">transversals_only</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute basic orbits and transversals from a base and strong generating set.</span>
-
-<span class="sd">    The generators are provided as distributed across the basic stabilizers.</span>
-<span class="sd">    If the optional argument ``transversals_only`` is set to True, only the</span>
-<span class="sd">    transversals are returned.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    ``base`` - the base</span>
-<span class="sd">    ``strong_gens_distr`` - strong generators distributed by membership in basic</span>
-<span class="sd">    stabilizers</span>
-<span class="sd">    ``transversals_only`` - a flag swithing between returning only the</span>
-<span class="sd">    transversals/ both orbits and transversals</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.named_groups import SymmetricGroup</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.util import _orbits_transversals_from_bsgs</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.util import (_orbits_transversals_from_bsgs,</span>
-<span class="sd">    ... _distribute_gens_by_base)</span>
-<span class="sd">    &gt;&gt;&gt; S = SymmetricGroup(3)</span>
-<span class="sd">    &gt;&gt;&gt; S.schreier_sims()</span>
-<span class="sd">    &gt;&gt;&gt; strong_gens_distr = _distribute_gens_by_base(S.base, S.strong_gens)</span>
-<span class="sd">    &gt;&gt;&gt; _orbits_transversals_from_bsgs(S.base, strong_gens_distr)</span>
-<span class="sd">    ([[0, 1, 2], [1, 2]],</span>
-<span class="sd">    [{0: Permutation(2), 1: Permutation(0, 1, 2), 2: Permutation(0, 2, 1)},</span>
-<span class="sd">    {1: Permutation(2), 2: Permutation(1, 2)}])</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    _distribute_gens_by_base, _handle_precomputed_bsgs</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">_orbit_transversal</span>
-    <span class="n">base_len</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-    <span class="n">degree</span> <span class="o">=</span> <span class="n">strong_gens_distr</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">size</span>
-    <span class="n">transversals</span> <span class="o">=</span> <span class="p">[</span><span class="bp">None</span><span class="p">]</span><span class="o">*</span><span class="n">base_len</span>
-    <span class="k">if</span> <span class="n">transversals_only</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-        <span class="n">basic_orbits</span> <span class="o">=</span> <span class="p">[</span><span class="bp">None</span><span class="p">]</span><span class="o">*</span><span class="n">base_len</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">base_len</span><span class="p">):</span>
-        <span class="n">transversals</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">_orbit_transversal</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">strong_gens_distr</span><span class="p">[</span><span class="n">i</span><span class="p">],</span>
-                                 <span class="n">base</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">pairs</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">transversals_only</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="n">basic_orbits</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">transversals</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-    <span class="k">if</span> <span class="n">transversals_only</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">transversals</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">basic_orbits</span><span class="p">,</span> <span class="n">transversals</span>
-
-</div>
-<div class="viewcode-block" id="_remove_gens"><a class="viewcode-back" href="../../../modules/combinatorics/util.html#sympy.combinatorics.util._remove_gens">[docs]</a><span class="k">def</span> <span class="nf">_remove_gens</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span><span class="p">,</span> <span class="n">basic_orbits</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">strong_gens_distr</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Remove redundant generators from a strong generating set.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    ``base`` - a base</span>
-<span class="sd">    ``strong_gens`` - a strong generating set relative to ``base``</span>
-<span class="sd">    ``basic_orbits`` - basic orbits</span>
-<span class="sd">    ``strong_gens_distr`` - strong generators distributed by membership in basic</span>
-<span class="sd">    stabilizers</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A strong generating set with respect to ``base`` which is a subset of</span>
-<span class="sd">    ``strong_gens``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.named_groups import SymmetricGroup</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.perm_groups import PermutationGroup</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.util import _remove_gens</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.testutil import _verify_bsgs</span>
-<span class="sd">    &gt;&gt;&gt; S = SymmetricGroup(15)</span>
-<span class="sd">    &gt;&gt;&gt; base, strong_gens = S.schreier_sims_incremental()</span>
-<span class="sd">    &gt;&gt;&gt; len(strong_gens)</span>
-<span class="sd">    26</span>
-<span class="sd">    &gt;&gt;&gt; new_gens = _remove_gens(base, strong_gens)</span>
-<span class="sd">    &gt;&gt;&gt; len(new_gens)</span>
-<span class="sd">    14</span>
-<span class="sd">    &gt;&gt;&gt; _verify_bsgs(S, base, new_gens)</span>
-<span class="sd">    True</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    This procedure is outlined in [1],p.95.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] Holt, D., Eick, B., O&#39;Brien, E.</span>
-<span class="sd">    &quot;Handbook of computational group theory&quot;</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span><span class="p">,</span> <span class="n">_orbit</span>
-    <span class="n">base_len</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-    <span class="n">degree</span> <span class="o">=</span> <span class="n">strong_gens</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">size</span>
-    <span class="k">if</span> <span class="n">strong_gens_distr</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">strong_gens_distr</span> <span class="o">=</span> <span class="n">_distribute_gens_by_base</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span><span class="p">)</span>
-    <span class="n">temp</span> <span class="o">=</span> <span class="n">strong_gens_distr</span><span class="p">[:]</span>
-    <span class="k">if</span> <span class="n">basic_orbits</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">basic_orbits</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">base_len</span><span class="p">):</span>
-            <span class="n">basic_orbit</span> <span class="o">=</span> <span class="n">_orbit</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">strong_gens_distr</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">base</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-            <span class="n">basic_orbits</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">basic_orbit</span><span class="p">)</span>
-    <span class="n">strong_gens_distr</span><span class="o">.</span><span class="n">append</span><span class="p">([])</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="n">strong_gens</span><span class="p">[:]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">base_len</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-        <span class="n">gens_copy</span> <span class="o">=</span> <span class="n">strong_gens_distr</span><span class="p">[</span><span class="n">i</span><span class="p">][:]</span>
-        <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">strong_gens_distr</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-            <span class="k">if</span> <span class="n">gen</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">strong_gens_distr</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]:</span>
-                <span class="n">temp_gens</span> <span class="o">=</span> <span class="n">gens_copy</span><span class="p">[:]</span>
-                <span class="n">temp_gens</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">gen</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">temp_gens</span> <span class="o">==</span> <span class="p">[]:</span>
-                    <span class="k">continue</span>
-                <span class="n">temp_orbit</span> <span class="o">=</span> <span class="n">_orbit</span><span class="p">(</span><span class="n">degree</span><span class="p">,</span> <span class="n">temp_gens</span><span class="p">,</span> <span class="n">base</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-                <span class="k">if</span> <span class="n">temp_orbit</span> <span class="o">==</span> <span class="n">basic_orbits</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                    <span class="n">gens_copy</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">gen</span><span class="p">)</span>
-                    <span class="n">res</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">gen</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">res</span>
-</div>
-<div class="viewcode-block" id="_strip"><a class="viewcode-back" href="../../../modules/combinatorics/util.html#sympy.combinatorics.util._strip">[docs]</a><span class="k">def</span> <span class="nf">_strip</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="n">orbits</span><span class="p">,</span> <span class="n">transversals</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Attempt to decompose a permutation using a (possibly partial) BSGS</span>
-<span class="sd">    structure.</span>
-
-<span class="sd">    This is done by treating the sequence ``base`` as an actual base, and</span>
-<span class="sd">    the orbits ``orbits`` and transversals ``transversals`` as basic orbits and</span>
-<span class="sd">    transversals relative to it.</span>
-
-<span class="sd">    This process is called &quot;sifting&quot;. A sift is unsuccessful when a certain</span>
-<span class="sd">    orbit element is not found or when after the sift the decomposition</span>
-<span class="sd">    doesn&#39;t end with the identity element.</span>
-
-<span class="sd">    The argument ``transversals`` is a list of dictionaries that provides</span>
-<span class="sd">    transversal elements for the orbits ``orbits``.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    ``g`` - permutation to be decomposed</span>
-<span class="sd">    ``base`` - sequence of points</span>
-<span class="sd">    ``orbits`` - a list in which the ``i``-th entry is an orbit of ``base[i]``</span>
-<span class="sd">    under some subgroup of the pointwise stabilizer of `</span>
-<span class="sd">    `base[0], base[1], ..., base[i - 1]``. The groups themselves are implicit</span>
-<span class="sd">    in this function since the only infromation we need is encoded in the orbits</span>
-<span class="sd">    and transversals</span>
-<span class="sd">    ``transversals`` - a list of orbit transversals associated with the orbits</span>
-<span class="sd">    ``orbits``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.named_groups import SymmetricGroup</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.permutations import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.util import _strip</span>
-<span class="sd">    &gt;&gt;&gt; S = SymmetricGroup(5)</span>
-<span class="sd">    &gt;&gt;&gt; S.schreier_sims()</span>
-<span class="sd">    &gt;&gt;&gt; g = Permutation([0, 2, 3, 1, 4])</span>
-<span class="sd">    &gt;&gt;&gt; _strip(g, S.base, S.basic_orbits, S.basic_transversals)</span>
-<span class="sd">    (Permutation(4), 5)</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    The algorithm is described in [1],pp.89-90. The reason for returning</span>
-<span class="sd">    both the current state of the element being decomposed and the level</span>
-<span class="sd">    at which the sifting ends is that they provide important information for</span>
-<span class="sd">    the randomized version of the Schreier-Sims algorithm.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] Holt, D., Eick, B., O&#39;Brien, E.</span>
-<span class="sd">    &quot;Handbook of computational group theory&quot;</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims</span>
-
-<span class="sd">    sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims_random</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">_array_form</span>
-    <span class="n">base_len</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">base_len</span><span class="p">):</span>
-        <span class="n">beta</span> <span class="o">=</span> <span class="n">h</span><span class="p">[</span><span class="n">base</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span>
-        <span class="k">if</span> <span class="n">beta</span> <span class="o">==</span> <span class="n">base</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-            <span class="k">continue</span>
-        <span class="k">if</span> <span class="n">beta</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">orbits</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">h</span><span class="p">),</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="n">u</span> <span class="o">=</span> <span class="n">transversals</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">beta</span><span class="p">]</span><span class="o">.</span><span class="n">_array_form</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">_af_rmul</span><span class="p">(</span><span class="n">_af_invert</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">h</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">_af_new</span><span class="p">(</span><span class="n">h</span><span class="p">),</span> <span class="n">base_len</span> <span class="o">+</span> <span class="mi">1</span>
-</div>
-<span class="k">def</span> <span class="nf">_strip_af</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="n">orbits</span><span class="p">,</span> <span class="n">transversals</span><span class="p">,</span> <span class="n">j</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    optimized _strip, with h, transversals and result in array form</span>
-<span class="sd">    if the stripped elements is the identity, it returns False, base_len + 1</span>
-
-<span class="sd">    j    h[base[i]] == base[i] for i &lt;= j</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">base_len</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">j</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">base_len</span><span class="p">):</span>
-        <span class="n">beta</span> <span class="o">=</span> <span class="n">h</span><span class="p">[</span><span class="n">base</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span>
-        <span class="k">if</span> <span class="n">beta</span> <span class="o">==</span> <span class="n">base</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-            <span class="k">continue</span>
-        <span class="k">if</span> <span class="n">beta</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">orbits</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="n">h</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="n">u</span> <span class="o">=</span> <span class="n">transversals</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">beta</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">h</span> <span class="o">==</span> <span class="n">u</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span><span class="p">,</span> <span class="n">base_len</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">_af_rmul</span><span class="p">(</span><span class="n">_af_invert</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">h</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">h</span><span class="p">,</span> <span class="n">base_len</span> <span class="o">+</span> <span class="mi">1</span>
-
-<div class="viewcode-block" id="_strong_gens_from_distr"><a class="viewcode-back" href="../../../modules/combinatorics/util.html#sympy.combinatorics.util._strong_gens_from_distr">[docs]</a><span class="k">def</span> <span class="nf">_strong_gens_from_distr</span><span class="p">(</span><span class="n">strong_gens_distr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Retrieve strong generating set from generators of basic stabilizers.</span>
-
-<span class="sd">    This is just the union of the generators of the first and second basic</span>
-<span class="sd">    stabilizers.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    ``strong_gens_distr`` - strong generators distributed by membership in basic</span>
-<span class="sd">    stabilizers</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics import Permutation</span>
-<span class="sd">    &gt;&gt;&gt; Permutation.print_cyclic = True</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.named_groups import SymmetricGroup</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.util import (_strong_gens_from_distr,</span>
-<span class="sd">    ... _distribute_gens_by_base)</span>
-<span class="sd">    &gt;&gt;&gt; S = SymmetricGroup(3)</span>
-<span class="sd">    &gt;&gt;&gt; S.schreier_sims()</span>
-<span class="sd">    &gt;&gt;&gt; S.strong_gens</span>
-<span class="sd">    [Permutation(0, 1, 2), Permutation(2)(0, 1), Permutation(1, 2)]</span>
-<span class="sd">    &gt;&gt;&gt; strong_gens_distr = _distribute_gens_by_base(S.base, S.strong_gens)</span>
-<span class="sd">    &gt;&gt;&gt; _strong_gens_from_distr(strong_gens_distr)</span>
-<span class="sd">    [Permutation(0, 1, 2), Permutation(2)(0, 1), Permutation(1, 2)]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    _distribute_gens_by_base</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">strong_gens_distr</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">strong_gens_distr</span><span class="p">[</span><span class="mi">0</span><span class="p">][:]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">strong_gens_distr</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">strong_gens_distr</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span>
-            <span class="k">if</span> <span class="n">gen</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">result</span><span class="p">:</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">gen</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">result</span></div>
-</pre></div>
-
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-  <h1>Source code for sympy.concrete.gosper</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Gosper&#39;s algorithm for hypergeometric summation. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">is_sequence</span>
-<span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">parallel_poly_from_expr</span><span class="p">,</span> <span class="n">factor</span>
-<span class="kn">from</span> <span class="nn">sympy.solvers</span> <span class="kn">import</span> <span class="n">solve</span>
-<span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">hypersimp</span>
-
-
-<div class="viewcode-block" id="gosper_normal"><a class="viewcode-back" href="../../../modules/concrete.html#sympy.concrete.gosper.gosper_normal">[docs]</a><span class="k">def</span> <span class="nf">gosper_normal</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">polys</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Compute the Gosper&#39;s normal form of ``f`` and ``g``.</span>
-
-<span class="sd">    Given relatively prime univariate polynomials ``f`` and ``g``,</span>
-<span class="sd">    rewrite their quotient to a normal form defined as follows:</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        \frac{f(n)}{g(n)} = Z \cdot \frac{A(n) C(n+1)}{B(n) C(n)}</span>
-
-<span class="sd">    where ``Z`` is an arbitrary constant and ``A``, ``B``, ``C`` are</span>
-<span class="sd">    monic polynomials in ``n`` with the following properties:</span>
-
-<span class="sd">    1. `\gcd(A(n), B(n+h)) = 1 \forall h \in \mathbb{N}`</span>
-<span class="sd">    2. `\gcd(B(n), C(n+1)) = 1`</span>
-<span class="sd">    3. `\gcd(A(n), C(n)) = 1`</span>
-
-<span class="sd">    This normal form, or rational factorization in other words, is a</span>
-<span class="sd">    crucial step in Gosper&#39;s algorithm and in solving of difference</span>
-<span class="sd">    equations. It can be also used to decide if two hypergeometric</span>
-<span class="sd">    terms are similar or not.</span>
-
-<span class="sd">    This procedure will return a tuple containing elements of this</span>
-<span class="sd">    factorization in the form ``(Z*A, B, C)``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.concrete.gosper import gosper_normal</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import n</span>
-
-<span class="sd">    &gt;&gt;&gt; gosper_normal(4*n+5, 2*(4*n+1)*(2*n+3), n, polys=False)</span>
-<span class="sd">    (1/4, n + 3/2, n + 1/4)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">),</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">(</span>
-        <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">),</span> <span class="n">n</span><span class="p">,</span> <span class="n">field</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">extension</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="n">a</span><span class="p">,</span> <span class="n">A</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">LC</span><span class="p">(),</span> <span class="n">p</span><span class="o">.</span><span class="n">monic</span><span class="p">()</span>
-    <span class="n">b</span><span class="p">,</span> <span class="n">B</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">LC</span><span class="p">(),</span> <span class="n">q</span><span class="o">.</span><span class="n">monic</span><span class="p">()</span>
-
-    <span class="n">C</span><span class="p">,</span> <span class="n">Z</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">a</span><span class="o">/</span><span class="n">b</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;h&#39;</span><span class="p">)</span>
-
-    <span class="n">D</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="n">h</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="n">opt</span><span class="o">.</span><span class="n">domain</span><span class="p">)</span>
-
-    <span class="n">R</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">resultant</span><span class="p">(</span><span class="n">B</span><span class="o">.</span><span class="n">compose</span><span class="p">(</span><span class="n">D</span><span class="p">))</span>
-    <span class="n">roots</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">R</span><span class="o">.</span><span class="n">ground_roots</span><span class="p">()</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-
-    <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="nb">set</span><span class="p">(</span><span class="n">roots</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">or</span> <span class="n">r</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">roots</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">roots</span><span class="p">):</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">B</span><span class="o">.</span><span class="n">shift</span><span class="p">(</span><span class="o">+</span><span class="n">i</span><span class="p">))</span>
-
-        <span class="n">A</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-        <span class="n">B</span> <span class="o">=</span> <span class="n">B</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">shift</span><span class="p">(</span><span class="o">-</span><span class="n">i</span><span class="p">))</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">C</span> <span class="o">*=</span> <span class="n">d</span><span class="o">.</span><span class="n">shift</span><span class="p">(</span><span class="o">-</span><span class="n">j</span><span class="p">)</span>
-
-    <span class="n">A</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">mul_ground</span><span class="p">(</span><span class="n">Z</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">polys</span><span class="p">:</span>
-        <span class="n">A</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-        <span class="n">B</span> <span class="o">=</span> <span class="n">B</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-        <span class="n">C</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-
-    <span class="k">return</span> <span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span>
-
-</div>
-<div class="viewcode-block" id="gosper_term"><a class="viewcode-back" href="../../../modules/concrete.html#sympy.concrete.gosper.gosper_term">[docs]</a><span class="k">def</span> <span class="nf">gosper_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Compute Gosper&#39;s hypergeometric term for ``f``.</span>
-
-<span class="sd">    Suppose ``f`` is a hypergeometric term such that:</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        s_n = \sum_{k=0}^{n-1} f_k</span>
-
-<span class="sd">    and `f_k` doesn&#39;t depend on `n`. Returns a hypergeometric</span>
-<span class="sd">    term `g_n` such that `g_{n+1} - g_n = f_n`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.concrete.gosper import gosper_term</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.functions import factorial</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import n</span>
-
-<span class="sd">    &gt;&gt;&gt; gosper_term((4*n + 1)*factorial(n)/factorial(2*n + 1), n)</span>
-<span class="sd">    (-n - 1/2)/(n + 1/4)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">hypersimp</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">r</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>    <span class="c"># &#39;f&#39; is *not* a hypergeometric term</span>
-
-    <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">r</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-
-    <span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span> <span class="o">=</span> <span class="n">gosper_normal</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-    <span class="n">B</span> <span class="o">=</span> <span class="n">B</span><span class="o">.</span><span class="n">shift</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-
-    <span class="n">N</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">degree</span><span class="p">())</span>
-    <span class="n">M</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="n">B</span><span class="o">.</span><span class="n">degree</span><span class="p">())</span>
-    <span class="n">K</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">degree</span><span class="p">())</span>
-
-    <span class="k">if</span> <span class="p">(</span><span class="n">N</span> <span class="o">!=</span> <span class="n">M</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">LC</span><span class="p">()</span> <span class="o">!=</span> <span class="n">B</span><span class="o">.</span><span class="n">LC</span><span class="p">()):</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span><span class="n">K</span> <span class="o">-</span> <span class="nb">max</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">M</span><span class="p">)])</span>
-    <span class="k">elif</span> <span class="ow">not</span> <span class="n">N</span><span class="p">:</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span><span class="n">K</span> <span class="o">-</span> <span class="n">N</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)])</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span><span class="n">K</span> <span class="o">-</span> <span class="n">N</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="n">B</span><span class="o">.</span><span class="n">nth</span><span class="p">(</span><span class="n">N</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">A</span><span class="o">.</span><span class="n">nth</span><span class="p">(</span><span class="n">N</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span><span class="o">/</span><span class="n">A</span><span class="o">.</span><span class="n">LC</span><span class="p">()])</span>
-
-    <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="nb">set</span><span class="p">(</span><span class="n">D</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">d</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">or</span> <span class="n">d</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">D</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">D</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>    <span class="c"># &#39;f(n)&#39; is *not* Gosper-summable</span>
-
-    <span class="n">d</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">D</span><span class="p">)</span>
-
-    <span class="n">coeffs</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;c:</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">d</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">cls</span><span class="o">=</span><span class="n">Dummy</span><span class="p">)</span>
-    <span class="n">domain</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">get_domain</span><span class="p">()</span><span class="o">.</span><span class="n">inject</span><span class="p">(</span><span class="o">*</span><span class="n">coeffs</span><span class="p">)</span>
-
-    <span class="n">x</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="n">domain</span><span class="p">)</span>
-    <span class="n">H</span> <span class="o">=</span> <span class="n">A</span><span class="o">*</span><span class="n">x</span><span class="o">.</span><span class="n">shift</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">B</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="n">C</span>
-
-    <span class="n">solution</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">H</span><span class="o">.</span><span class="n">coeffs</span><span class="p">(),</span> <span class="n">coeffs</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">solution</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>    <span class="c"># &#39;f(n)&#39; is *not* Gosper-summable</span>
-
-    <span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">solution</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">coeffs</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">coeff</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">solution</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>    <span class="c"># &#39;f(n)&#39; is *not* Gosper-summable</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">B</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span><span class="o">*</span><span class="n">x</span><span class="o">/</span><span class="n">C</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-
-</div>
-<div class="viewcode-block" id="gosper_sum"><a class="viewcode-back" href="../../../modules/concrete.html#sympy.concrete.gosper.gosper_sum">[docs]</a><span class="k">def</span> <span class="nf">gosper_sum</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Gosper&#39;s hypergeometric summation algorithm.</span>
-
-<span class="sd">    Given a hypergeometric term ``f`` such that:</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        s_n = \sum_{k=0}^{n-1} f_k</span>
-
-<span class="sd">    and `f(n)` doesn&#39;t depend on `n`, returns `g_{n} - g(0)` where</span>
-<span class="sd">    `g_{n+1} - g_n = f_n`, or ``None`` if `s_n` can not be expressed</span>
-<span class="sd">    in closed form as a sum of hypergeometric terms.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.concrete.gosper import gosper_sum</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.functions import factorial</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import i, n, k</span>
-
-<span class="sd">    &gt;&gt;&gt; f = (4*k + 1)*factorial(k)/factorial(2*k + 1)</span>
-<span class="sd">    &gt;&gt;&gt; gosper_sum(f, (k, 0, n))</span>
-<span class="sd">    (-n! + 2*(2*n + 1)!)/(2*n + 1)!</span>
-<span class="sd">    &gt;&gt;&gt; _.subs(n, 2) == sum(f.subs(k, i) for i in [0, 1, 2])</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; gosper_sum(f, (k, 3, n))</span>
-<span class="sd">    (-60*n! + (2*n + 1)!)/(60*(2*n + 1)!)</span>
-<span class="sd">    &gt;&gt;&gt; _.subs(n, 5) == sum(f.subs(k, i) for i in [3, 4, 5])</span>
-<span class="sd">    True</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger, A = B,</span>
-<span class="sd">           AK Peters, Ltd., Wellesley, MA, USA, 1997, pp. 73--100</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">indefinite</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="k">if</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
-        <span class="n">k</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">k</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">indefinite</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="n">g</span> <span class="o">=</span> <span class="n">gosper_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">g</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">if</span> <span class="n">indefinite</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">*</span><span class="n">g</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="p">(</span><span class="n">g</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="n">g</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">factor</span><span class="p">(</span><span class="n">result</span><span class="p">)</span></div>
-</pre></div>
-
-          </div>
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-            
-  <h1>Source code for sympy.concrete.products</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">C</span><span class="p">,</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.piecewise</span> <span class="kn">import</span> <span class="n">piecewise_fold</span>
-<span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">quo</span><span class="p">,</span> <span class="n">roots</span>
-<span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">powsimp</span>
-
-
-<div class="viewcode-block" id="Product"><a class="viewcode-back" href="../../../modules/concrete.html#sympy.concrete.products.Product">[docs]</a><span class="k">class</span> <span class="nc">Product</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Represents unevaluated product.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;is_commutative&#39;</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">function</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.integrals.integrals</span> <span class="kn">import</span> <span class="n">_process_limits</span>
-
-        <span class="c"># Any embedded piecewise functions need to be brought out to the</span>
-        <span class="c"># top level so that integration can go into piecewise mode at the</span>
-        <span class="c"># earliest possible moment.</span>
-        <span class="n">function</span> <span class="o">=</span> <span class="n">piecewise_fold</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">function</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">function</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">symbols</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Product variables must be given&quot;</span><span class="p">)</span>
-
-        <span class="n">limits</span><span class="p">,</span> <span class="n">sign</span> <span class="o">=</span> <span class="n">_process_limits</span><span class="p">(</span><span class="o">*</span><span class="n">symbols</span><span class="p">)</span>
-
-        <span class="c"># Only limits with lower and upper bounds are supported; the indefinite</span>
-        <span class="c"># Product is not supported</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">l</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">3</span> <span class="ow">or</span> <span class="bp">None</span> <span class="ow">in</span> <span class="n">l</span> <span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="n">limits</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&#39;Product requires values for lower and upper bounds.&#39;</span><span class="p">)</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">)</span>
-        <span class="n">arglist</span> <span class="o">=</span> <span class="p">[</span><span class="n">sign</span><span class="o">*</span><span class="n">function</span><span class="p">]</span>
-        <span class="n">arglist</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">limits</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_args</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">arglist</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">is_commutative</span> <span class="o">=</span> <span class="n">function</span><span class="o">.</span><span class="n">is_commutative</span>  <span class="c"># limits already checked</span>
-
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">term</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">function</span> <span class="o">=</span> <span class="n">term</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">limits</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Product.variables"><a class="viewcode-back" href="../../../modules/concrete.html#sympy.concrete.products.Product.variables">[docs]</a>    <span class="k">def</span> <span class="nf">variables</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return a list of the product variables</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Product</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, i</span>
-<span class="sd">        &gt;&gt;&gt; Product(x**i, (i, 1, 3)).variables</span>
-<span class="sd">        [i]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">l</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">limits</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Product.free_symbols"><a class="viewcode-back" href="../../../modules/concrete.html#sympy.concrete.products.Product.free_symbols">[docs]</a>    <span class="k">def</span> <span class="nf">free_symbols</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        This method returns the symbols that will affect the value of</span>
-<span class="sd">        the Product when evaluated. This is useful if one is trying to</span>
-<span class="sd">        determine whether a product depends on a certain symbol or not.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Product</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; Product(x, (x, y, 1)).free_symbols</span>
-<span class="sd">        set([y])</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.concrete.summations</span> <span class="kn">import</span> <span class="n">_free_symbols</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="o">.</span><span class="n">is_zero</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">function</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">_free_symbols</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">limits</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Product.is_zero"><a class="viewcode-back" href="../../../modules/concrete.html#sympy.concrete.products.Product.is_zero">[docs]</a>    <span class="k">def</span> <span class="nf">is_zero</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A Product is zero only if its term is zero.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">term</span><span class="o">.</span><span class="n">is_zero</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Product.is_number"><a class="viewcode-back" href="../../../modules/concrete.html#sympy.concrete.products.Product.is_number">[docs]</a>    <span class="k">def</span> <span class="nf">is_number</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if the Product will result in a number, else False.</span>
-
-<span class="sd">        sympy considers anything that will result in a number to have</span>
-<span class="sd">        is_number == True.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import log, Product</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">        &gt;&gt;&gt; log(2).is_number</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Product(x, (x, 1, 2)).is_number</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Product(y, (x, 1, 2)).is_number</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Product(1, (x, y, z)).is_number</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Product(2, (x, y, z)).is_number</span>
-<span class="sd">        False</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="o">.</span><span class="n">is_zero</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">function</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span>
-</div>
-    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">g</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">function</span>
-        <span class="k">for</span> <span class="n">index</span><span class="p">,</span> <span class="n">limit</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">limits</span><span class="p">):</span>
-            <span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">limit</span>
-            <span class="n">dif</span> <span class="o">=</span> <span class="n">b</span> <span class="o">-</span> <span class="n">a</span>
-            <span class="k">if</span> <span class="n">dif</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">dif</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="p">,</span> <span class="n">a</span>
-
-            <span class="n">g</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_product</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">g</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Product</span><span class="p">(</span><span class="n">powsimp</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">limits</span><span class="p">[</span><span class="n">index</span><span class="p">:])</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">f</span> <span class="o">=</span> <span class="n">g</span>
-
-        <span class="k">if</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;deep&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">powsimp</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_adjoint</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Product</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="o">.</span><span class="n">adjoint</span><span class="p">(),</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">limits</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Product</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="o">.</span><span class="n">conjugate</span><span class="p">(),</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">limits</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_product</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">term</span><span class="p">,</span> <span class="n">limits</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">summation</span>
-
-        <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">=</span> <span class="n">limits</span>
-
-        <span class="k">if</span> <span class="n">k</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">term</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">term</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">term</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-
-        <span class="n">dif</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="n">a</span>
-        <span class="k">if</span> <span class="n">dif</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">term</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">dif</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)])</span>
-
-        <span class="k">elif</span> <span class="n">term</span><span class="o">.</span><span class="n">is_polynomial</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
-            <span class="n">poly</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">as_poly</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-
-            <span class="n">A</span> <span class="o">=</span> <span class="n">B</span> <span class="o">=</span> <span class="n">Q</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-            <span class="n">all_roots</span> <span class="o">=</span> <span class="n">roots</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">all_roots</span><span class="p">:</span>
-                <span class="n">A</span> <span class="o">*=</span> <span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">r</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-                <span class="n">Q</span> <span class="o">*=</span> <span class="n">n</span> <span class="o">-</span> <span class="n">r</span>
-
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">all_roots</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">poly</span><span class="o">.</span><span class="n">degree</span><span class="p">():</span>
-                <span class="n">arg</span> <span class="o">=</span> <span class="n">quo</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">as_poly</span><span class="p">(</span><span class="n">k</span><span class="p">))</span>
-                <span class="n">B</span> <span class="o">=</span> <span class="n">Product</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-
-            <span class="k">return</span> <span class="n">poly</span><span class="o">.</span><span class="n">LC</span><span class="p">()</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">A</span> <span class="o">*</span> <span class="n">B</span>
-
-        <span class="k">elif</span> <span class="n">term</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-
-            <span class="n">p</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_product</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-            <span class="n">q</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_product</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-
-            <span class="k">return</span> <span class="n">p</span> <span class="o">/</span> <span class="n">q</span>
-
-        <span class="k">elif</span> <span class="n">term</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="n">exclude</span><span class="p">,</span> <span class="n">include</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
-
-            <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">term</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_product</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-
-                <span class="k">if</span> <span class="n">p</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">exclude</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">include</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">exclude</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">arg</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">_new_rawargs</span><span class="p">(</span><span class="o">*</span><span class="n">include</span><span class="p">)</span>
-                <span class="n">A</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">exclude</span><span class="p">)</span>
-                <span class="n">B</span> <span class="o">=</span> <span class="n">Product</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-                <span class="k">return</span> <span class="n">A</span> <span class="o">*</span> <span class="n">B</span>
-
-        <span class="k">elif</span> <span class="n">term</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">term</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">summation</span><span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">exp</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-
-                <span class="k">return</span> <span class="n">term</span><span class="o">.</span><span class="n">base</span><span class="o">**</span><span class="n">s</span>
-            <span class="k">elif</span> <span class="ow">not</span> <span class="n">term</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_product</span><span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-
-                <span class="k">if</span> <span class="n">p</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">p</span><span class="o">**</span><span class="n">term</span><span class="o">.</span><span class="n">exp</span>
-
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">Product</span><span class="p">):</span>
-            <span class="n">evaluated</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_product</span><span class="p">(</span><span class="n">evaluated</span><span class="p">,</span> <span class="n">limits</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">f</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Product</span><span class="p">(</span><span class="n">evaluated</span><span class="p">,</span> <span class="n">limits</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span>
-
-    <span class="k">def</span> <span class="nf">_eval_transpose</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Product</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="o">.</span><span class="n">transpose</span><span class="p">(),</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">limits</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-</div>
-<div class="viewcode-block" id="product"><a class="viewcode-back" href="../../../modules/concrete.html#sympy.concrete.products.product">[docs]</a><span class="k">def</span> <span class="nf">product</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Compute the product.</span>
-
-<span class="sd">    The notation for symbols is similiar to the notation used in Sum or</span>
-<span class="sd">    Integral. product(f, (i, a, b)) computes the product of f with</span>
-<span class="sd">    respect to i from a to b, i.e.,</span>
-
-<span class="sd">    ::</span>
-
-<span class="sd">                                     b</span>
-<span class="sd">                                   _____</span>
-<span class="sd">        product(f(n), (i, a, b)) = |   | f(n)</span>
-<span class="sd">                                   |   |</span>
-<span class="sd">                                   i = a</span>
-
-<span class="sd">    If it cannot compute the product, it returns an unevaluated Product object.</span>
-<span class="sd">    Repeated products can be computed by introducing additional symbols tuples::</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import product, symbols</span>
-<span class="sd">    &gt;&gt;&gt; i, n, m, k = symbols(&#39;i n m k&#39;, integer=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; product(i, (i, 1, k))</span>
-<span class="sd">    k!</span>
-<span class="sd">    &gt;&gt;&gt; product(m, (i, 1, k))</span>
-<span class="sd">    m**k</span>
-<span class="sd">    &gt;&gt;&gt; product(i, (i, 1, k), (k, 1, n))</span>
-<span class="sd">    Product(k!, (k, 1, n))</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">prod</span> <span class="o">=</span> <span class="n">Product</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">prod</span><span class="p">,</span> <span class="n">Product</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">prod</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">prod</span></div>
-</pre></div>
-
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-  <h1>Source code for sympy.concrete.summations</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Add</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">,</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">Wild</span>
-<span class="kn">from</span> <span class="nn">sympy.concrete.gosper</span> <span class="kn">import</span> <span class="n">gosper_sum</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.piecewise</span> <span class="kn">import</span> <span class="n">piecewise_fold</span>
-<span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">apart</span><span class="p">,</span> <span class="n">PolynomialError</span>
-<span class="kn">from</span> <span class="nn">sympy.solvers</span> <span class="kn">import</span> <span class="n">solve</span>
-
-
-<span class="k">def</span> <span class="nf">_free_symbols</span><span class="p">(</span><span class="n">function</span><span class="p">,</span> <span class="n">limits</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Helper function to return the symbols that appear in a sum-like object</span>
-<span class="sd">    once it is evaluated.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">isyms</span> <span class="o">=</span> <span class="n">function</span><span class="o">.</span><span class="n">free_symbols</span>
-    <span class="k">for</span> <span class="n">xab</span> <span class="ow">in</span> <span class="n">limits</span><span class="p">:</span>
-        <span class="c"># take out the target symbol</span>
-        <span class="k">if</span> <span class="n">xab</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">in</span> <span class="n">isyms</span><span class="p">:</span>
-            <span class="n">isyms</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">xab</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="c"># add in the new symbols</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">xab</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="n">isyms</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">isyms</span>
-
-
-<div class="viewcode-block" id="Sum"><a class="viewcode-back" href="../../../modules/concrete.html#sympy.concrete.summations.Sum">[docs]</a><span class="k">class</span> <span class="nc">Sum</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Represents unevaluated summation.&quot;&quot;&quot;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;is_commutative&#39;</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">function</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.integrals.integrals</span> <span class="kn">import</span> <span class="n">_process_limits</span>
-
-        <span class="c"># Any embedded piecewise functions need to be brought out to the</span>
-        <span class="c"># top level so that integration can go into piecewise mode at the</span>
-        <span class="c"># earliest possible moment.</span>
-        <span class="n">function</span> <span class="o">=</span> <span class="n">piecewise_fold</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">function</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">function</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">symbols</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Summation variables must be given&quot;</span><span class="p">)</span>
-
-        <span class="n">limits</span><span class="p">,</span> <span class="n">sign</span> <span class="o">=</span> <span class="n">_process_limits</span><span class="p">(</span><span class="o">*</span><span class="n">symbols</span><span class="p">)</span>
-
-        <span class="c"># Only limits with lower and upper bounds are supported; the indefinite Sum</span>
-        <span class="c"># is not supported</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">l</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">3</span> <span class="ow">or</span> <span class="bp">None</span> <span class="ow">in</span> <span class="n">l</span> <span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="n">limits</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Sum requires values for lower and upper bounds.&#39;</span><span class="p">)</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">)</span>
-        <span class="n">arglist</span> <span class="o">=</span> <span class="p">[</span><span class="n">sign</span><span class="o">*</span><span class="n">function</span><span class="p">]</span>
-        <span class="n">arglist</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">limits</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_args</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">arglist</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">is_commutative</span> <span class="o">=</span> <span class="n">function</span><span class="o">.</span><span class="n">is_commutative</span>  <span class="c"># limits already checked</span>
-
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">function</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">limits</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Sum.variables"><a class="viewcode-back" href="../../../modules/concrete.html#sympy.concrete.summations.Sum.variables">[docs]</a>    <span class="k">def</span> <span class="nf">variables</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return a list of the summation variables</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Sum</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, i</span>
-<span class="sd">        &gt;&gt;&gt; Sum(x**i, (i, 1, 3)).variables</span>
-<span class="sd">        [i]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">l</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">limits</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Sum.free_symbols"><a class="viewcode-back" href="../../../modules/concrete.html#sympy.concrete.summations.Sum.free_symbols">[docs]</a>    <span class="k">def</span> <span class="nf">free_symbols</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        This method returns the symbols that will exist when the</span>
-<span class="sd">        summation is evaluated. This is useful if one is trying to</span>
-<span class="sd">        determine whether a sum depends on a certain symbol or not.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Sum</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; Sum(x, (x, y, 1)).free_symbols</span>
-<span class="sd">        set([y])</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">_free_symbols</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">limits</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Sum.is_zero"><a class="viewcode-back" href="../../../modules/concrete.html#sympy.concrete.summations.Sum.is_zero">[docs]</a>    <span class="k">def</span> <span class="nf">is_zero</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A Sum is only zero if its function is zero or if all terms</span>
-<span class="sd">        cancel out. This only answers whether the summand zero.&quot;&quot;&quot;</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="o">.</span><span class="n">is_zero</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Sum.is_number"><a class="viewcode-back" href="../../../modules/concrete.html#sympy.concrete.summations.Sum.is_number">[docs]</a>    <span class="k">def</span> <span class="nf">is_number</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if the Sum will result in a number, else False.</span>
-
-<span class="sd">        sympy considers anything that will result in a number to have</span>
-<span class="sd">        is_number == True.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import log</span>
-<span class="sd">        &gt;&gt;&gt; log(2).is_number</span>
-<span class="sd">        True</span>
-
-<span class="sd">        Sums are a special case since they contain symbols that can</span>
-<span class="sd">        be replaced with numbers. Whether the integral can be done or not is</span>
-<span class="sd">        another issue. But answering whether the final result is a number is</span>
-<span class="sd">        not difficult.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Sum</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; Sum(x, (y, 1, x)).is_number</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Sum(1, (y, 1, x)).is_number</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Sum(0, (y, 1, x)).is_number</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Sum(x, (y, 1, 2)).is_number</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Sum(x, (y, 1, 1)).is_number</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Sum(x, (x, 1, 2)).is_number</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Sum(x*y, (x, 1, 2), (y, 1, 3)).is_number</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="o">.</span><span class="n">is_zero</span> <span class="ow">or</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span>
-</div>
-    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="c">#if not hints.get(&#39;sums&#39;, True):</span>
-        <span class="c">#    return self</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">function</span>
-        <span class="k">for</span> <span class="n">limit</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">limits</span><span class="p">:</span>
-            <span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">limit</span>
-            <span class="n">dif</span> <span class="o">=</span> <span class="n">b</span> <span class="o">-</span> <span class="n">a</span>
-            <span class="k">if</span> <span class="n">dif</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">dif</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="p">,</span> <span class="n">a</span>
-
-            <span class="n">f</span> <span class="o">=</span> <span class="n">eval_sum</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">f</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span>
-
-        <span class="k">if</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;deep&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span>
-
-    <span class="k">def</span> <span class="nf">_eval_adjoint</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Sum</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="o">.</span><span class="n">adjoint</span><span class="p">(),</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">limits</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Sum</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="o">.</span><span class="n">conjugate</span><span class="p">(),</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">limits</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Differentiate wrt x as long as x is not in the free symbols of any of</span>
-<span class="sd">        the upper or lower limits.</span>
-
-<span class="sd">        Sum(a*b*x, (x, 1, a)) can be differentiated wrt x or b but not `a`</span>
-<span class="sd">        since the value of the sum is discontinuous in `a`. In a case</span>
-<span class="sd">        involving a limit variable, the unevaluated derivative is returned.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="c"># diff already confirmed that x is in the free symbols of self, but we</span>
-        <span class="c"># don&#39;t want to differentiate wrt any free symbol in the upper or lower</span>
-        <span class="c"># limits</span>
-        <span class="c"># XXX remove this test for free_symbols when the default _eval_derivative is in</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">not</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-        <span class="c"># get limits and the function</span>
-        <span class="n">f</span><span class="p">,</span> <span class="n">limits</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">limits</span><span class="p">)</span>
-
-        <span class="n">limit</span> <span class="o">=</span> <span class="n">limits</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">limits</span><span class="p">:</span>  <span class="c"># f is the argument to a Sum</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">Sum</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">limits</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">limit</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="n">_</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">limit</span>
-            <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">a</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">or</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">b</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="n">df</span> <span class="o">=</span> <span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;evaluate&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">})</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="n">Sum</span><span class="p">(</span><span class="n">df</span><span class="p">,</span> <span class="n">limit</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">limit</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">df</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-                <span class="n">rv</span> <span class="o">=</span> <span class="n">rv</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-            <span class="k">return</span> <span class="n">rv</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;Lower and upper bound expected.&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_summation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">_eval_transpose</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Sum</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="o">.</span><span class="n">transpose</span><span class="p">(),</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">limits</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Sum.euler_maclaurin"><a class="viewcode-back" href="../../../modules/concrete.html#sympy.concrete.summations.Sum.euler_maclaurin">[docs]</a>    <span class="k">def</span> <span class="nf">euler_maclaurin</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">m</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">eval_integral</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return an Euler-Maclaurin approximation of self, where m is the</span>
-<span class="sd">        number of leading terms to sum directly and n is the number of</span>
-<span class="sd">        terms in the tail.</span>
-
-<span class="sd">        With m = n = 0, this is simply the corresponding integral</span>
-<span class="sd">        plus a first-order endpoint correction.</span>
-
-<span class="sd">        Returns (s, e) where s is the Euler-Maclaurin approximation</span>
-<span class="sd">        and e is the estimated error (taken to be the magnitude of</span>
-<span class="sd">        the first omitted term in the tail):</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy.abc import k, a, b</span>
-<span class="sd">            &gt;&gt;&gt; from sympy import Sum</span>
-<span class="sd">            &gt;&gt;&gt; Sum(1/k, (k, 2, 5)).doit().evalf()</span>
-<span class="sd">            1.28333333333333</span>
-<span class="sd">            &gt;&gt;&gt; s, e = Sum(1/k, (k, 2, 5)).euler_maclaurin()</span>
-<span class="sd">            &gt;&gt;&gt; s</span>
-<span class="sd">            -log(2) + 7/20 + log(5)</span>
-<span class="sd">            &gt;&gt;&gt; from sympy import sstr</span>
-<span class="sd">            &gt;&gt;&gt; print(sstr((s.evalf(), e.evalf()), full_prec=True))</span>
-<span class="sd">            (1.26629073187415, 0.0175000000000000)</span>
-
-<span class="sd">        The endpoints may be symbolic:</span>
-
-<span class="sd">            &gt;&gt;&gt; s, e = Sum(1/k, (k, a, b)).euler_maclaurin()</span>
-<span class="sd">            &gt;&gt;&gt; s</span>
-<span class="sd">            -log(a) + log(b) + 1/(2*b) + 1/(2*a)</span>
-<span class="sd">            &gt;&gt;&gt; e</span>
-<span class="sd">            Abs(-1/(12*b**2) + 1/(12*a**2))</span>
-
-<span class="sd">        If the function is a polynomial of degree at most 2n+1, the</span>
-<span class="sd">        Euler-Maclaurin formula becomes exact (and e = 0 is returned):</span>
-
-<span class="sd">            &gt;&gt;&gt; Sum(k, (k, 2, b)).euler_maclaurin()</span>
-<span class="sd">            (b**2/2 + b/2 - 1, 0)</span>
-<span class="sd">            &gt;&gt;&gt; Sum(k, (k, 2, b)).doit()</span>
-<span class="sd">            b**2/2 + b/2 - 1</span>
-
-<span class="sd">        With a nonzero eps specified, the summation is ended</span>
-<span class="sd">        as soon as the remainder term is less than the epsilon.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">function</span>
-        <span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">limits</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span>
-        <span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">limits</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">if</span> <span class="n">m</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
-                <span class="n">term</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span>
-                <span class="k">if</span> <span class="p">(</span><span class="n">eps</span> <span class="ow">and</span> <span class="n">term</span> <span class="ow">and</span> <span class="nb">abs</span><span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span> <span class="o">&lt;</span> <span class="n">eps</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">s</span><span class="p">,</span> <span class="nb">abs</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-                <span class="n">s</span> <span class="o">+=</span> <span class="n">term</span>
-            <span class="n">a</span> <span class="o">+=</span> <span class="n">m</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-        <span class="n">I</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Integral</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">eval_integral</span><span class="p">:</span>
-            <span class="n">I</span> <span class="o">=</span> <span class="n">I</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-        <span class="n">s</span> <span class="o">+=</span> <span class="n">I</span>
-
-        <span class="k">def</span> <span class="nf">fpoint</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">b</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">),</span> <span class="mi">0</span>
-            <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">),</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-        <span class="n">fa</span><span class="p">,</span> <span class="n">fb</span> <span class="o">=</span> <span class="n">fpoint</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-        <span class="n">iterm</span> <span class="o">=</span> <span class="p">(</span><span class="n">fa</span> <span class="o">+</span> <span class="n">fb</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">2</span><span class="p">):</span>
-            <span class="n">ga</span><span class="p">,</span> <span class="n">gb</span> <span class="o">=</span> <span class="n">fpoint</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-            <span class="n">term</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">bernoulli</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">)</span><span class="o">/</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">gb</span> <span class="o">-</span> <span class="n">ga</span><span class="p">)</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">eps</span> <span class="ow">and</span> <span class="n">term</span> <span class="ow">and</span> <span class="nb">abs</span><span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span> <span class="o">&lt;</span> <span class="n">eps</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="n">k</span> <span class="o">&gt;</span> <span class="n">n</span><span class="p">):</span>
-                <span class="k">break</span>
-            <span class="n">s</span> <span class="o">+=</span> <span class="n">term</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">s</span> <span class="o">+</span> <span class="n">iterm</span><span class="p">,</span> <span class="nb">abs</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">):</span>  <span class="c"># XXX this should be the same as Integral&#39;s</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">old</span> <span class="o">==</span> <span class="n">v</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span>
-
-</div>
-<div class="viewcode-block" id="summation"><a class="viewcode-back" href="../../../modules/concrete.html#sympy.concrete.summations.summation">[docs]</a><span class="k">def</span> <span class="nf">summation</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Compute the summation of f with respect to symbols.</span>
-
-<span class="sd">    The notation for symbols is similar to the notation used in Integral.</span>
-<span class="sd">    summation(f, (i, a, b)) computes the sum of f with respect to i from a to b,</span>
-<span class="sd">    i.e.,</span>
-
-<span class="sd">    ::</span>
-
-<span class="sd">                                    b</span>
-<span class="sd">                                  ____</span>
-<span class="sd">                                  \   `</span>
-<span class="sd">        summation(f, (i, a, b)) =  )    f</span>
-<span class="sd">                                  /___,</span>
-<span class="sd">                                  i = a</span>
-
-<span class="sd">    If it cannot compute the sum, it returns an unevaluated Sum object.</span>
-<span class="sd">    Repeated sums can be computed by introducing additional symbols tuples::</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import summation, oo, symbols, log</span>
-<span class="sd">    &gt;&gt;&gt; i, n, m = symbols(&#39;i n m&#39;, integer=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; summation(2*i - 1, (i, 1, n))</span>
-<span class="sd">    n**2</span>
-<span class="sd">    &gt;&gt;&gt; summation(1/2**i, (i, 0, oo))</span>
-<span class="sd">    2</span>
-<span class="sd">    &gt;&gt;&gt; summation(1/log(n)**n, (n, 2, oo))</span>
-<span class="sd">    Sum(log(n)**(-n), (n, 2, oo))</span>
-<span class="sd">    &gt;&gt;&gt; summation(i, (i, 0, n), (n, 0, m))</span>
-<span class="sd">    m**3/6 + m**2/2 + m/3</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import factorial</span>
-<span class="sd">    &gt;&gt;&gt; summation(x**n/factorial(n), (n, 0, oo))</span>
-<span class="sd">    exp(x)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">Sum</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">telescopic_direct</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">limits</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns the direct summation of the terms of a telescopic sum</span>
-
-<span class="sd">    L is the term with lower index</span>
-<span class="sd">    R is the term with higher index</span>
-<span class="sd">    n difference between the indexes of L and R</span>
-
-<span class="sd">    For example:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.concrete.summations import telescopic_direct</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import k, a, b</span>
-<span class="sd">    &gt;&gt;&gt; telescopic_direct(1/k, -1/(k+2), 2, (k, a, b))</span>
-<span class="sd">    -1/(b + 2) - 1/(b + 1) + 1/(a + 1) + 1/a</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">=</span> <span class="n">limits</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-        <span class="n">s</span> <span class="o">+=</span> <span class="n">L</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="n">m</span><span class="p">)</span> <span class="o">+</span> <span class="n">R</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">b</span> <span class="o">-</span> <span class="n">m</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">s</span>
-
-
-<span class="k">def</span> <span class="nf">telescopic</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="n">limits</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;Tries to perform the summation using the telescopic property</span>
-
-<span class="sd">    return None if not possible</span>
-<span class="sd">    &#39;&#39;&#39;</span>
-    <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">=</span> <span class="n">limits</span>
-    <span class="k">if</span> <span class="n">L</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">or</span> <span class="n">R</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="c"># We want to solve(L.subs(i, i + m) + R, m)</span>
-    <span class="c"># First we try a simple match since this does things that</span>
-    <span class="c"># solve doesn&#39;t do, e.g. solve(f(k+m)-f(k), m) fails</span>
-
-    <span class="n">k</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">)</span>
-    <span class="n">sol</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="n">R</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">L</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="n">k</span><span class="p">))</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="k">if</span> <span class="n">sol</span> <span class="ow">and</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">sol</span><span class="p">:</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">sol</span><span class="p">[</span><span class="n">k</span><span class="p">]</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">L</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="n">s</span><span class="p">)</span> <span class="o">==</span> <span class="o">-</span><span class="n">R</span><span class="p">):</span>
-            <span class="c">#sometimes match fail(f(x+2).match(-f(x+k))-&gt;{k: -2 - 2x}))</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="c"># But there are things that match doesn&#39;t do that solve</span>
-    <span class="c"># can do, e.g. determine that 1/(x + m) = 1/(1 - x) when m = 1</span>
-
-    <span class="k">if</span> <span class="n">s</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;m&#39;</span><span class="p">)</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">sol</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">L</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="n">m</span><span class="p">)</span> <span class="o">+</span> <span class="n">R</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span> <span class="ow">or</span> <span class="p">[]</span>
-        <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="n">sol</span> <span class="o">=</span> <span class="p">[</span><span class="n">si</span> <span class="k">for</span> <span class="n">si</span> <span class="ow">in</span> <span class="n">sol</span> <span class="k">if</span> <span class="n">si</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span>
-               <span class="p">(</span><span class="n">L</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="n">si</span><span class="p">)</span> <span class="o">+</span> <span class="n">R</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span><span class="o">.</span><span class="n">is_zero</span><span class="p">]</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">sol</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">sol</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">if</span> <span class="n">s</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">telescopic_direct</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">L</span><span class="p">,</span> <span class="nb">abs</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-    <span class="k">elif</span> <span class="n">s</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">telescopic_direct</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-
-
-<span class="k">def</span> <span class="nf">eval_sum</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">limits</span><span class="p">):</span>
-    <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">=</span> <span class="n">limits</span>
-    <span class="k">if</span> <span class="n">f</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-    <span class="k">if</span> <span class="n">i</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">*</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="n">b</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-
-    <span class="n">dif</span> <span class="o">=</span> <span class="n">b</span> <span class="o">-</span> <span class="n">a</span>
-    <span class="n">definite</span> <span class="o">=</span> <span class="n">dif</span><span class="o">.</span><span class="n">is_Integer</span>
-    <span class="c"># Doing it directly may be faster if there are very few terms.</span>
-    <span class="k">if</span> <span class="n">definite</span> <span class="ow">and</span> <span class="p">(</span><span class="n">dif</span> <span class="o">&lt;</span> <span class="mi">100</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">eval_sum_direct</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-    <span class="c"># Try to do it symbolically. Even when the number of terms is known,</span>
-    <span class="c"># this can save time when b-a is big.</span>
-    <span class="c"># We should try to transform to partial fractions</span>
-    <span class="n">value</span> <span class="o">=</span> <span class="n">eval_sum_symbolic</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">expand</span><span class="p">(),</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-    <span class="k">if</span> <span class="n">value</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">value</span>
-    <span class="c"># Do it directly</span>
-    <span class="k">if</span> <span class="n">definite</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">eval_sum_direct</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-
-
-<span class="k">def</span> <span class="nf">eval_sum_direct</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">limits</span><span class="p">):</span>
-    <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">=</span> <span class="n">limits</span>
-
-    <span class="n">dif</span> <span class="o">=</span> <span class="n">b</span> <span class="o">-</span> <span class="n">a</span>
-    <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="n">j</span><span class="p">)</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">dif</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)])</span>
-
-
-<span class="k">def</span> <span class="nf">eval_sum_symbolic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">limits</span><span class="p">):</span>
-    <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">=</span> <span class="n">limits</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">*</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="c"># Linearity</span>
-    <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-        <span class="n">L</span><span class="p">,</span> <span class="n">R</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">as_two_terms</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">L</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
-            <span class="n">sR</span> <span class="o">=</span> <span class="n">eval_sum_symbolic</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">sR</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">L</span><span class="o">*</span><span class="n">sR</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">R</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
-            <span class="n">sL</span> <span class="o">=</span> <span class="n">eval_sum_symbolic</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">sL</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">R</span><span class="o">*</span><span class="n">sL</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">apart</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>  <span class="c"># see if it becomes an Add</span>
-        <span class="k">except</span> <span class="n">PolynomialError</span><span class="p">:</span>
-            <span class="k">pass</span>
-
-    <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-        <span class="n">L</span><span class="p">,</span> <span class="n">R</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">as_two_terms</span><span class="p">()</span>
-        <span class="n">lrsum</span> <span class="o">=</span> <span class="n">telescopic</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">lrsum</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">lrsum</span>
-
-        <span class="n">lsum</span> <span class="o">=</span> <span class="n">eval_sum_symbolic</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-        <span class="n">rsum</span> <span class="o">=</span> <span class="n">eval_sum_symbolic</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="bp">None</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">(</span><span class="n">lsum</span><span class="p">,</span> <span class="n">rsum</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">lsum</span> <span class="o">+</span> <span class="n">rsum</span>
-
-    <span class="c"># Polynomial terms with Faulhaber&#39;s formula</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">)</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">i</span><span class="o">**</span><span class="n">n</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">result</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">((</span><span class="n">C</span><span class="o">.</span><span class="n">bernoulli</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">b</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">C</span><span class="o">.</span><span class="n">bernoulli</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-            <span class="k">elif</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">a</span> <span class="o">&gt;=</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">harmonic</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">-</span> <span class="n">C</span><span class="o">.</span><span class="n">harmonic</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">harmonic</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="nb">abs</span><span class="p">(</span><span class="n">n</span><span class="p">))</span> <span class="o">-</span> <span class="n">C</span><span class="o">.</span><span class="n">harmonic</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">abs</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-
-    <span class="c"># Geometric terms</span>
-    <span class="n">c1</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Wild</span><span class="p">(</span><span class="s">&#39;c1&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-    <span class="n">c2</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Wild</span><span class="p">(</span><span class="s">&#39;c2&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-    <span class="n">c3</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Wild</span><span class="p">(</span><span class="s">&#39;c3&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-
-    <span class="n">e</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">c1</span><span class="o">**</span><span class="p">(</span><span class="n">c2</span><span class="o">*</span><span class="n">i</span> <span class="o">+</span> <span class="n">c3</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="n">e</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">c1</span> <span class="o">=</span> <span class="n">c1</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-        <span class="n">c2</span> <span class="o">=</span> <span class="n">c2</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-        <span class="n">c3</span> <span class="o">=</span> <span class="n">c3</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-
-        <span class="c"># TODO: more general limit handling</span>
-        <span class="k">return</span> <span class="n">c1</span><span class="o">**</span><span class="n">c3</span> <span class="o">*</span> <span class="p">(</span><span class="n">c1</span><span class="o">**</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">c2</span><span class="p">)</span> <span class="o">-</span> <span class="n">c1</span><span class="o">**</span><span class="p">(</span><span class="n">c2</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">c2</span><span class="p">))</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">c1</span><span class="o">**</span><span class="n">c2</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">)</span> <span class="ow">or</span>
-            <span class="n">b</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">)):</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">gosper_sum</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span> <span class="ow">in</span> <span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">r</span>
-
-    <span class="k">return</span> <span class="n">eval_sum_hyper</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-
-
-<span class="k">def</span> <span class="nf">_eval_sum_hyper</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Returns (res, cond). Sums from a to oo. &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">hyper</span>
-    <span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">hyperexpand</span><span class="p">,</span> <span class="n">hypersimp</span><span class="p">,</span> <span class="n">fraction</span><span class="p">,</span> <span class="n">simplify</span>
-    <span class="kn">from</span> <span class="nn">sympy.polys.polytools</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">factor</span>
-
-    <span class="k">if</span> <span class="n">a</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">_eval_sum_hyper</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="n">a</span><span class="p">),</span> <span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">simplify</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)))</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="n">_eval_sum_hyper</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-
-    <span class="n">hs</span> <span class="o">=</span> <span class="n">hypersimp</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">hs</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="o">=</span> <span class="n">fraction</span><span class="p">(</span><span class="n">factor</span><span class="p">(</span><span class="n">hs</span><span class="p">))</span>
-    <span class="n">top</span><span class="p">,</span> <span class="n">topl</span> <span class="o">=</span> <span class="n">numer</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-    <span class="n">bot</span><span class="p">,</span> <span class="n">botl</span> <span class="o">=</span> <span class="n">denom</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-    <span class="n">ab</span> <span class="o">=</span> <span class="p">[</span><span class="n">top</span><span class="p">,</span> <span class="n">bot</span><span class="p">]</span>
-    <span class="n">factors</span> <span class="o">=</span> <span class="p">[</span><span class="n">topl</span><span class="p">,</span> <span class="n">botl</span><span class="p">]</span>
-    <span class="n">params</span> <span class="o">=</span> <span class="p">[[],</span> <span class="p">[]]</span>
-    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">fac</span> <span class="ow">in</span> <span class="n">factors</span><span class="p">[</span><span class="n">k</span><span class="p">]:</span>
-            <span class="n">mul</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="n">fac</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-                <span class="n">mul</span> <span class="o">=</span> <span class="n">fac</span><span class="o">.</span><span class="n">exp</span>
-                <span class="n">fac</span> <span class="o">=</span> <span class="n">fac</span><span class="o">.</span><span class="n">base</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">mul</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">fac</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()</span>
-            <span class="n">ab</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">*=</span> <span class="n">m</span><span class="o">**</span><span class="n">mul</span>
-            <span class="n">params</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">+=</span> <span class="p">[</span><span class="n">n</span><span class="o">/</span><span class="n">m</span><span class="p">]</span><span class="o">*</span><span class="n">mul</span>
-
-    <span class="c"># Add &quot;1&quot; to numerator parameters, to account for implicit n! in</span>
-    <span class="c"># hypergeometric series.</span>
-    <span class="n">ap</span> <span class="o">=</span> <span class="n">params</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-    <span class="n">bq</span> <span class="o">=</span> <span class="n">params</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">ab</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="n">ab</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">hyper</span><span class="p">(</span><span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span><span class="o">*</span><span class="n">hyperexpand</span><span class="p">(</span><span class="n">h</span><span class="p">),</span> <span class="n">h</span><span class="o">.</span><span class="n">convergence_statement</span>
-
-
-<span class="k">def</span> <span class="nf">eval_sum_hyper</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">xxx_todo_changeme</span><span class="p">):</span>
-    <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">=</span> <span class="n">xxx_todo_changeme</span>
-    <span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">Piecewise</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">oo</span><span class="p">,</span> <span class="n">And</span>
-
-    <span class="k">if</span> <span class="n">b</span> <span class="o">!=</span> <span class="n">oo</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="o">-</span><span class="n">oo</span><span class="p">:</span>
-            <span class="n">res</span> <span class="o">=</span> <span class="n">_eval_sum_hyper</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="o">-</span><span class="n">i</span><span class="p">),</span> <span class="n">i</span><span class="p">,</span> <span class="o">-</span><span class="n">b</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">res</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Piecewise</span><span class="p">(</span><span class="n">res</span><span class="p">,</span> <span class="p">(</span><span class="n">Sum</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)),</span> <span class="bp">True</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">res1</span> <span class="o">=</span> <span class="n">_eval_sum_hyper</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-            <span class="n">res2</span> <span class="o">=</span> <span class="n">_eval_sum_hyper</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">b</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">res1</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">res2</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="p">(</span><span class="n">res1</span><span class="p">,</span> <span class="n">cond1</span><span class="p">),</span> <span class="p">(</span><span class="n">res2</span><span class="p">,</span> <span class="n">cond2</span><span class="p">)</span> <span class="o">=</span> <span class="n">res1</span><span class="p">,</span> <span class="n">res2</span>
-            <span class="n">cond</span> <span class="o">=</span> <span class="n">And</span><span class="p">(</span><span class="n">cond1</span><span class="p">,</span> <span class="n">cond2</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">cond</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">return</span> <span class="n">Piecewise</span><span class="p">((</span><span class="n">res1</span> <span class="o">-</span> <span class="n">res2</span><span class="p">,</span> <span class="n">cond</span><span class="p">),</span> <span class="p">(</span><span class="n">Sum</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)),</span> <span class="bp">True</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="o">-</span><span class="n">oo</span><span class="p">:</span>
-        <span class="n">res1</span> <span class="o">=</span> <span class="n">_eval_sum_hyper</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="o">-</span><span class="n">i</span><span class="p">),</span> <span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">res2</span> <span class="o">=</span> <span class="n">_eval_sum_hyper</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">res1</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">res2</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="n">res1</span><span class="p">,</span> <span class="n">cond1</span> <span class="o">=</span> <span class="n">res1</span>
-        <span class="n">res2</span><span class="p">,</span> <span class="n">cond2</span> <span class="o">=</span> <span class="n">res2</span>
-        <span class="n">cond</span> <span class="o">=</span> <span class="n">And</span><span class="p">(</span><span class="n">cond1</span><span class="p">,</span> <span class="n">cond2</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">cond</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">return</span> <span class="n">Piecewise</span><span class="p">((</span><span class="n">res1</span> <span class="o">+</span> <span class="n">res2</span><span class="p">,</span> <span class="n">cond</span><span class="p">),</span> <span class="p">(</span><span class="n">Sum</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)),</span> <span class="bp">True</span><span class="p">))</span>
-
-    <span class="c"># Now b == oo, a != -oo</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="n">_eval_sum_hyper</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">res</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">Piecewise</span><span class="p">(</span><span class="n">res</span><span class="p">,</span> <span class="p">(</span><span class="n">Sum</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)),</span> <span class="bp">True</span><span class="p">))</span>
-</pre></div>
-
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-  <h1>Source code for sympy.core.add</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.core</span> <span class="kn">import</span> <span class="n">C</span>
-<span class="kn">from</span> <span class="nn">sympy.core.singleton</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="kn">from</span> <span class="nn">sympy.core.operations</span> <span class="kn">import</span> <span class="n">AssocOp</span>
-<span class="kn">from</span> <span class="nn">sympy.core.cache</span> <span class="kn">import</span> <span class="n">cacheit</span>
-<span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">ilcm</span><span class="p">,</span> <span class="n">igcd</span>
-<span class="kn">from</span> <span class="nn">sympy.core.expr</span> <span class="kn">import</span> <span class="n">Expr</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-
-<div class="viewcode-block" id="Add"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.add.Add">[docs]</a><span class="k">class</span> <span class="nc">Add</span><span class="p">(</span><span class="n">Expr</span><span class="p">,</span> <span class="n">AssocOp</span><span class="p">):</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="n">is_Add</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="c">#identity = S.Zero</span>
-    <span class="c"># cyclic import, so defined in numbers.py</span>
-
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Add.flatten"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.add.Add.flatten">[docs]</a>    <span class="k">def</span> <span class="nf">flatten</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">seq</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Takes the sequence &quot;seq&quot; of nested Adds and returns a flatten list.</span>
-
-<span class="sd">        Returns: (commutative_part, noncommutative_part, order_symbols)</span>
-
-<span class="sd">        Applies associativity, all terms are commutable with respect to</span>
-<span class="sd">        addition.</span>
-
-<span class="sd">        NB: the removal of 0 is already handled by AssocOp.__new__</span>
-
-<span class="sd">        See also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.core.mul.Mul.flatten</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">seq</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">seq</span>
-            <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="p">,</span> <span class="n">a</span>
-            <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                    <span class="c"># if it&#39;s an unevaluated 2-arg, expand it</span>
-                    <span class="n">c</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-                    <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                        <span class="n">h</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">t</span><span class="o">.</span><span class="n">as_coeff_Add</span><span class="p">()</span>
-                        <span class="n">bargs</span> <span class="o">=</span> <span class="p">[</span><span class="n">c</span><span class="o">*</span><span class="n">ti</span> <span class="k">for</span> <span class="n">ti</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">t</span><span class="p">)]</span>
-                        <span class="n">bargs</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="nb">hash</span><span class="p">)</span>
-                        <span class="n">ch</span> <span class="o">=</span> <span class="n">c</span><span class="o">*</span><span class="n">h</span>
-                        <span class="k">if</span> <span class="n">ch</span><span class="p">:</span>
-                            <span class="n">bargs</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">ch</span><span class="p">)</span>
-                        <span class="n">b</span> <span class="o">=</span> <span class="n">Add</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">bargs</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                    <span class="n">bargs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">bargs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                        <span class="n">bargs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+=</span> <span class="n">a</span>
-                        <span class="k">if</span> <span class="ow">not</span> <span class="n">bargs</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-                            <span class="n">bargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">bargs</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-                    <span class="n">rv</span> <span class="o">=</span> <span class="n">bargs</span><span class="p">,</span> <span class="p">[],</span> <span class="bp">None</span>
-                <span class="k">elif</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                    <span class="n">rv</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">],</span> <span class="p">[],</span> <span class="bp">None</span>
-            <span class="k">if</span> <span class="n">rv</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">is_commutative</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">rv</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
-                    <span class="k">return</span> <span class="n">rv</span>
-                <span class="k">return</span> <span class="p">[],</span> <span class="n">rv</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">None</span>
-
-        <span class="n">terms</span> <span class="o">=</span> <span class="p">{}</span>      <span class="c"># term -&gt; coeff</span>
-                        <span class="c"># e.g. x**2 -&gt; 5   for ... + 5*x**2 + ...</span>
-
-        <span class="n">coeff</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>  <span class="c"># coefficient (Number or zoo) to always be in slot 0</span>
-                        <span class="c"># e.g. 3 + ...</span>
-        <span class="n">order_factors</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">seq</span><span class="p">:</span>
-
-            <span class="c"># O(x)</span>
-            <span class="k">if</span> <span class="n">o</span><span class="o">.</span><span class="n">is_Order</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">o1</span> <span class="ow">in</span> <span class="n">order_factors</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">o1</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">o</span><span class="p">):</span>
-                        <span class="n">o</span> <span class="o">=</span> <span class="bp">None</span>
-                        <span class="k">break</span>
-                <span class="k">if</span> <span class="n">o</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">continue</span>
-                <span class="n">order_factors</span> <span class="o">=</span> <span class="p">[</span><span class="n">o</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span>
-                    <span class="n">o1</span> <span class="k">for</span> <span class="n">o1</span> <span class="ow">in</span> <span class="n">order_factors</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">o</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">o1</span><span class="p">)]</span>
-                <span class="k">continue</span>
-
-            <span class="c"># 3 or NaN</span>
-            <span class="k">elif</span> <span class="n">o</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                <span class="k">if</span> <span class="p">(</span><span class="n">o</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span> <span class="ow">or</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span> <span class="ow">and</span>
-                        <span class="n">o</span><span class="o">.</span><span class="n">is_bounded</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">):</span>
-                    <span class="c"># we know for sure the result will be nan</span>
-                    <span class="k">return</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">],</span> <span class="p">[],</span> <span class="bp">None</span>
-                <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                    <span class="n">coeff</span> <span class="o">+=</span> <span class="n">o</span>
-                    <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                        <span class="c"># we know for sure the result will be nan</span>
-                        <span class="k">return</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">],</span> <span class="p">[],</span> <span class="bp">None</span>
-                <span class="k">continue</span>
-
-            <span class="k">elif</span> <span class="n">o</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_bounded</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                    <span class="c"># we know for sure the result will be nan</span>
-                    <span class="k">return</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">],</span> <span class="p">[],</span> <span class="bp">None</span>
-                <span class="n">coeff</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span>
-                <span class="k">continue</span>
-
-            <span class="c"># Add([...])</span>
-            <span class="k">elif</span> <span class="n">o</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="c"># NB: here we assume Add is always commutative</span>
-                <span class="n">seq</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">o</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>  <span class="c"># TODO zerocopy?</span>
-                <span class="k">continue</span>
-
-            <span class="c"># Mul([...])</span>
-            <span class="k">elif</span> <span class="n">o</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                <span class="n">c</span><span class="p">,</span> <span class="n">s</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-
-                <span class="c"># 3*...</span>
-                <span class="c"># unevaluated 2-arg Mul, but we always unfold it so</span>
-                <span class="c"># it can combine with other terms (just like is done</span>
-                <span class="c"># with the Pow below)</span>
-                <span class="k">if</span> <span class="n">c</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">and</span> <span class="n">s</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                    <span class="n">seq</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">c</span><span class="o">*</span><span class="n">a</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">s</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-                    <span class="k">continue</span>
-
-            <span class="c"># check for unevaluated Pow, e.g. 2**3 or 2**(-1/2)</span>
-            <span class="k">elif</span> <span class="n">o</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-                <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">and</span> <span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">or</span>
-                                   <span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">e</span><span class="o">.</span><span class="n">is_negative</span><span class="p">)):</span>
-                    <span class="n">seq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">b</span><span class="o">**</span><span class="n">e</span><span class="p">)</span>
-                    <span class="k">continue</span>
-                <span class="n">c</span><span class="p">,</span> <span class="n">s</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">o</span>
-
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># everything else</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">o</span>
-
-            <span class="c"># now we have:</span>
-            <span class="c"># o = c*s, where</span>
-            <span class="c">#</span>
-            <span class="c"># c is a Number</span>
-            <span class="c"># s is an expression with number factor extracted</span>
-            <span class="c"># let&#39;s collect terms with the same s, so e.g.</span>
-            <span class="c"># 2*x**2 + 3*x**2  -&gt;  5*x**2</span>
-            <span class="k">if</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">:</span>
-                <span class="n">terms</span><span class="p">[</span><span class="n">s</span><span class="p">]</span> <span class="o">+=</span> <span class="n">c</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">terms</span><span class="p">[</span><span class="n">s</span><span class="p">]</span> <span class="o">=</span> <span class="n">c</span>
-
-        <span class="c"># now let&#39;s construct new args:</span>
-        <span class="c"># [2*x**2, x**3, 7*x**4, pi, ...]</span>
-        <span class="n">newseq</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">noncommutative</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">s</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">terms</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-            <span class="c"># 0*s</span>
-            <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="c"># 1*s</span>
-            <span class="k">elif</span> <span class="n">c</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="n">newseq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-            <span class="c"># c*s</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">s</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                    <span class="c"># Mul, already keeps its arguments in perfect order.</span>
-                    <span class="c"># so we can simply put c in slot0 and go the fast way.</span>
-                    <span class="n">cs</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">_new_rawargs</span><span class="p">(</span><span class="o">*</span><span class="p">((</span><span class="n">c</span><span class="p">,)</span> <span class="o">+</span> <span class="n">s</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-                    <span class="n">newseq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">cs</span><span class="p">)</span>
-
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="c"># alternatively we have to call all Mul&#39;s machinery (slow)</span>
-                    <span class="n">newseq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">s</span><span class="p">))</span>
-
-            <span class="n">noncommutative</span> <span class="o">=</span> <span class="n">noncommutative</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">s</span><span class="o">.</span><span class="n">is_commutative</span>
-
-        <span class="c"># oo, -oo</span>
-        <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-            <span class="n">newseq</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">newseq</span> <span class="k">if</span> <span class="ow">not</span>
-                      <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">is_nonnegative</span> <span class="ow">or</span> <span class="n">f</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span>
-                       <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">is_bounded</span> <span class="ow">or</span> <span class="n">f</span><span class="o">.</span><span class="n">is_finite</span> <span class="ow">or</span> <span class="n">f</span><span class="o">.</span><span class="n">is_infinitesimal</span><span class="p">))]</span>
-
-        <span class="k">elif</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-            <span class="n">newseq</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">newseq</span> <span class="k">if</span> <span class="ow">not</span>
-                      <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">is_nonpositive</span> <span class="ow">or</span> <span class="n">f</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span>
-                       <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">is_bounded</span> <span class="ow">or</span> <span class="n">f</span><span class="o">.</span><span class="n">is_finite</span> <span class="ow">or</span> <span class="n">f</span><span class="o">.</span><span class="n">is_infinitesimal</span><span class="p">))]</span>
-
-        <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">:</span>
-            <span class="c"># zoo might be</span>
-            <span class="c">#   unbounded_real + bounded_im</span>
-            <span class="c">#   bounded_real + unbounded_im</span>
-            <span class="c">#   unbounded_real + unbounded_im</span>
-            <span class="c"># addition of a bounded real or imaginary number won&#39;t be able to</span>
-            <span class="c"># change the zoo nature; if unbounded a NaN condition could result</span>
-            <span class="c"># if the unbounded symbol had sign opposite of the unbounded</span>
-            <span class="c"># portion of zoo, e.g., unbounded_real - unbounded_real.</span>
-            <span class="n">newseq</span> <span class="o">=</span> <span class="p">[</span><span class="n">c</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">newseq</span> <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">c</span><span class="o">.</span><span class="n">is_bounded</span> <span class="ow">and</span>
-                                                <span class="n">c</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">)]</span>
-
-        <span class="c"># process O(x)</span>
-        <span class="k">if</span> <span class="n">order_factors</span><span class="p">:</span>
-            <span class="n">newseq2</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">newseq</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">order_factors</span><span class="p">:</span>
-                    <span class="c"># x + O(x) -&gt; O(x)</span>
-                    <span class="k">if</span> <span class="n">o</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
-                        <span class="n">t</span> <span class="o">=</span> <span class="bp">None</span>
-                        <span class="k">break</span>
-                <span class="c"># x + O(x**2) -&gt; x + O(x**2)</span>
-                <span class="k">if</span> <span class="n">t</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">newseq2</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-            <span class="n">newseq</span> <span class="o">=</span> <span class="n">newseq2</span> <span class="o">+</span> <span class="n">order_factors</span>
-            <span class="c"># 1 + O(1) -&gt; O(1)</span>
-            <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">order_factors</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">o</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">coeff</span><span class="p">):</span>
-                    <span class="n">coeff</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-                    <span class="k">break</span>
-
-        <span class="c"># order args canonically</span>
-        <span class="c"># Currently we sort things using hashes, as it is quite fast. A better</span>
-        <span class="c"># solution is not to sort things at all - but this needs some more</span>
-        <span class="c"># fixing. NOTE: this is used in primitive and Mul.flattten, too, so if</span>
-        <span class="c"># it changes here it should be changed there.</span>
-        <span class="n">newseq</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="nb">hash</span><span class="p">)</span>
-
-        <span class="c"># current code expects coeff to be first</span>
-        <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-            <span class="n">newseq</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">coeff</span><span class="p">)</span>
-
-        <span class="c"># we are done</span>
-        <span class="k">if</span> <span class="n">noncommutative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[],</span> <span class="n">newseq</span><span class="p">,</span> <span class="bp">None</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">newseq</span><span class="p">,</span> <span class="p">[],</span> <span class="bp">None</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Add.class_key"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.add.Add.class_key">[docs]</a>    <span class="k">def</span> <span class="nf">class_key</span><span class="p">(</span><span class="n">cls</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Nice order of classes&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">cls</span><span class="o">.</span><span class="n">__name__</span>
-</div>
-<div class="viewcode-block" id="Add.as_coefficients_dict"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.add.Add.as_coefficients_dict">[docs]</a>    <span class="k">def</span> <span class="nf">as_coefficients_dict</span><span class="p">(</span><span class="n">a</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return a dictionary mapping terms to their Rational coefficient.</span>
-<span class="sd">        Since the dictionary is a defaultdict, inquiries about terms which</span>
-<span class="sd">        were not present will return a coefficient of 0. If an expression is</span>
-<span class="sd">        not an Add it is considered to have a single term.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import a, x</span>
-<span class="sd">        &gt;&gt;&gt; (3*x + a*x + 4).as_coefficients_dict()</span>
-<span class="sd">        {1: 4, x: 3, a*x: 1}</span>
-<span class="sd">        &gt;&gt;&gt; _[a]</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; (3*a*x).as_coefficients_dict()</span>
-<span class="sd">        {a*x: 3}</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">d</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">list</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">ai</span> <span class="ow">in</span> <span class="n">a</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">c</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">ai</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-            <span class="n">d</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">d</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">d</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">d</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">v</span><span class="p">)</span>
-        <span class="n">di</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
-        <span class="n">di</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">di</span>
-</div>
-    <span class="nd">@cacheit</span>
-<div class="viewcode-block" id="Add.as_coeff_add"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.add.Add.as_coeff_add">[docs]</a>    <span class="k">def</span> <span class="nf">as_coeff_add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">deps</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns a tuple (coeff, args) where self is treated as an Add and coeff</span>
-<span class="sd">        is the Number term and args is a tuple of all other terms.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; (7 + 3*x).as_coeff_add()</span>
-<span class="sd">        (7, (3*x,))</span>
-<span class="sd">        &gt;&gt;&gt; (7*x).as_coeff_add()</span>
-<span class="sd">        (0, (7*x,))</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">deps</span><span class="p">:</span>
-            <span class="n">l1</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">l2</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">deps</span><span class="p">):</span>
-                    <span class="n">l2</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">l1</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new_rawargs</span><span class="p">(</span><span class="o">*</span><span class="n">l1</span><span class="p">),</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">l2</span><span class="p">)</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">notrat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_coeff_add</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">notrat</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-</div>
-<div class="viewcode-block" id="Add.as_coeff_Add"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.add.Add.as_coeff_Add">[docs]</a>    <span class="k">def</span> <span class="nf">as_coeff_Add</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Efficiently extract the coefficient of a summation. &quot;&quot;&quot;</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-        <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">coeff</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new_rawargs</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="bp">self</span>
-
-    <span class="c"># Note, we intentionally do not implement Add.as_coeff_mul().  Rather, we</span>
-    <span class="c"># let Expr.as_coeff_mul() just always return (S.One, self) for an Add.  See</span>
-    <span class="c"># issue 2425.</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">f</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_nseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="n">terms</span> <span class="o">=</span> <span class="p">[</span><span class="n">t</span><span class="o">.</span><span class="n">nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">terms</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_matches_simple</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">repl_dict</span><span class="p">):</span>
-        <span class="c"># handle (w+3).matches(&#39;x+5&#39;) -&gt; {w: x+2}</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">terms</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_coeff_add</span><span class="p">()</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">terms</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">terms</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">matches</span><span class="p">(</span><span class="n">expr</span> <span class="o">-</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">repl_dict</span><span class="p">)</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">matches</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">repl_dict</span><span class="o">=</span><span class="p">{},</span> <span class="n">old</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">AssocOp</span><span class="o">.</span><span class="n">_matches_commutative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">repl_dict</span><span class="p">,</span> <span class="n">old</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_combine_inverse</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns lhs - rhs, but treats arguments like symbols, so things like</span>
-<span class="sd">        oo - oo return 0, instead of a nan.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">oo</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">expand_mul</span>
-        <span class="k">if</span> <span class="n">lhs</span> <span class="o">==</span> <span class="n">oo</span> <span class="ow">and</span> <span class="n">rhs</span> <span class="o">==</span> <span class="n">oo</span> <span class="ow">or</span> <span class="n">lhs</span> <span class="o">==</span> <span class="n">oo</span><span class="o">*</span><span class="n">I</span> <span class="ow">and</span> <span class="n">rhs</span> <span class="o">==</span> <span class="n">oo</span><span class="o">*</span><span class="n">I</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">return</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">lhs</span> <span class="o">-</span> <span class="n">rhs</span><span class="p">)</span>
-
-    <span class="nd">@cacheit</span>
-<div class="viewcode-block" id="Add.as_two_terms"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.add.Add.as_two_terms">[docs]</a>    <span class="k">def</span> <span class="nf">as_two_terms</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return head and tail of self.</span>
-
-<span class="sd">        This is the most efficient way to get the head and tail of an</span>
-<span class="sd">        expression.</span>
-
-<span class="sd">        - if you want only the head, use self.args[0];</span>
-<span class="sd">        - if you want to process the arguments of the tail then use</span>
-<span class="sd">          self.as_coef_add() which gives the head and a tuple containing</span>
-<span class="sd">          the arguments of the tail when treated as an Add.</span>
-<span class="sd">        - if you want the coefficient when self is treated as a Mul</span>
-<span class="sd">          then use self.as_coeff_mul()[0]</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; (3*x*y).as_two_terms()</span>
-<span class="sd">        (3, x*y)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="bp">self</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new_rawargs</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-</div>
-    <span class="k">def</span> <span class="nf">as_numer_denom</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-
-        <span class="c"># clear rational denominator</span>
-        <span class="n">content</span><span class="p">,</span> <span class="n">expr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">primitive</span><span class="p">()</span>
-        <span class="n">ncon</span><span class="p">,</span> <span class="n">dcon</span> <span class="o">=</span> <span class="n">content</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-
-        <span class="c"># collect numerators and denominators of the terms</span>
-        <span class="n">nd</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">list</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">ni</span><span class="p">,</span> <span class="n">di</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-            <span class="n">nd</span><span class="p">[</span><span class="n">di</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ni</span><span class="p">)</span>
-        <span class="c"># put infinity in the numerator</span>
-        <span class="k">if</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span> <span class="ow">in</span> <span class="n">nd</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">nd</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-            <span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">n</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">nd</span><span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-
-        <span class="c"># check for quick exit</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">nd</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">d</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">nd</span><span class="o">.</span><span class="n">popitem</span><span class="p">()</span>
-            <span class="k">return</span> <span class="n">Add</span><span class="p">(</span>
-                <span class="o">*</span><span class="p">[</span><span class="n">_keep_coeff</span><span class="p">(</span><span class="n">ncon</span><span class="p">,</span> <span class="n">ni</span><span class="p">)</span> <span class="k">for</span> <span class="n">ni</span> <span class="ow">in</span> <span class="n">n</span><span class="p">]),</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">dcon</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
-
-        <span class="c"># sum up the terms having a common denominator</span>
-        <span class="k">for</span> <span class="n">d</span><span class="p">,</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">nd</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">nd</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">=</span> <span class="n">n</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">nd</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">n</span><span class="p">)</span>
-
-        <span class="c"># assemble single numerator and denominator</span>
-        <span class="n">denoms</span><span class="p">,</span> <span class="n">numers</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="nb">iter</span><span class="p">(</span><span class="n">nd</span><span class="o">.</span><span class="n">items</span><span class="p">()))]</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">denoms</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">numers</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span> <span class="o">+</span> <span class="n">denoms</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]))</span>
-                   <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">numers</span><span class="p">))]),</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">denoms</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">ncon</span><span class="p">,</span> <span class="n">n</span><span class="p">),</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">dcon</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_polynomial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">syms</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">_eval_is_polynomial</span><span class="p">(</span><span class="n">syms</span><span class="p">)</span> <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_rational_function</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">syms</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">_eval_is_rational_function</span><span class="p">(</span><span class="n">syms</span><span class="p">)</span> <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="c"># assumption methods</span>
-    <span class="n">_eval_is_real</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span>
-        <span class="s">&#39;is_real&#39;</span><span class="p">,</span> <span class="n">when_multiple</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span>
-    <span class="n">_eval_is_antihermitian</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span>
-        <span class="s">&#39;is_antihermitian&#39;</span><span class="p">,</span> <span class="n">when_multiple</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span>
-    <span class="n">_eval_is_bounded</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span>
-        <span class="s">&#39;is_bounded&#39;</span><span class="p">,</span> <span class="n">when_multiple</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span>
-    <span class="n">_eval_is_hermitian</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span>
-        <span class="s">&#39;is_hermitian&#39;</span><span class="p">,</span> <span class="n">when_multiple</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span>
-    <span class="n">_eval_is_imaginary</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span>
-        <span class="s">&#39;is_imaginary&#39;</span><span class="p">,</span> <span class="n">when_multiple</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span>
-    <span class="n">_eval_is_integer</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span>
-        <span class="s">&#39;is_integer&#39;</span><span class="p">,</span> <span class="n">when_multiple</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span>
-    <span class="n">_eval_is_commutative</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span>
-        <span class="s">&#39;is_commutative&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_odd</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">l</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">is_even</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">)]</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">l</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">l</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_odd</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new_rawargs</span><span class="p">(</span><span class="o">*</span><span class="n">l</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span><span class="o">.</span><span class="n">is_even</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_irrational</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">t</span><span class="o">.</span><span class="n">is_irrational</span>
-            <span class="k">if</span> <span class="n">a</span><span class="p">:</span>
-                <span class="n">others</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-                <span class="n">others</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">is_rational</span> <span class="ow">is</span> <span class="bp">True</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">others</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_positive</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Add</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_is_positive</span><span class="p">()</span>
-        <span class="n">pos</span> <span class="o">=</span> <span class="n">nonneg</span> <span class="o">=</span> <span class="n">nonpos</span> <span class="o">=</span> <span class="n">unknown_sign</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="n">unbounded</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">a</span><span class="o">.</span><span class="n">is_zero</span><span class="p">]</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="n">ispos</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">is_positive</span>
-            <span class="n">ubound</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">is_unbounded</span>
-            <span class="k">if</span> <span class="n">ubound</span><span class="p">:</span>
-                <span class="n">unbounded</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">ispos</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">unbounded</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-            <span class="k">if</span> <span class="n">ispos</span><span class="p">:</span>
-                <span class="n">pos</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="n">a</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">:</span>
-                <span class="n">nonneg</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="n">a</span><span class="o">.</span><span class="n">is_nonpositive</span><span class="p">:</span>
-                <span class="n">nonpos</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="n">a</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-                <span class="k">continue</span>
-
-            <span class="k">if</span> <span class="n">ubound</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="c"># sign is unknown; if we don&#39;t know the boundedness</span>
-                <span class="c"># we&#39;re done: we don&#39;t know. That is technically true,</span>
-                <span class="c"># but the only option is that we have something like</span>
-                <span class="c"># oo - oo which is NaN and it really doesn&#39;t matter</span>
-                <span class="c"># what sign we apply to that because it (when finally</span>
-                <span class="c"># computed) will trump any sign. So instead of returning</span>
-                <span class="c"># None, we pass.</span>
-                <span class="k">pass</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="n">unknown_sign</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="k">if</span> <span class="n">unbounded</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">unbounded</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-        <span class="k">elif</span> <span class="n">unknown_sign</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="n">nonpos</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">nonneg</span> <span class="ow">and</span> <span class="n">pos</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="n">nonpos</span> <span class="ow">and</span> <span class="n">pos</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="n">pos</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">nonneg</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_negative</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Add</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_is_negative</span><span class="p">()</span>
-        <span class="n">neg</span> <span class="o">=</span> <span class="n">nonpos</span> <span class="o">=</span> <span class="n">nonneg</span> <span class="o">=</span> <span class="n">unknown_sign</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="n">unbounded</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">a</span><span class="o">.</span><span class="n">is_zero</span><span class="p">]</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="n">isneg</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">is_negative</span>
-            <span class="n">ubound</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">is_unbounded</span>
-            <span class="k">if</span> <span class="n">ubound</span><span class="p">:</span>
-                <span class="n">unbounded</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">isneg</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">unbounded</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-            <span class="k">if</span> <span class="n">isneg</span><span class="p">:</span>
-                <span class="n">neg</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="n">a</span><span class="o">.</span><span class="n">is_nonpositive</span><span class="p">:</span>
-                <span class="n">nonpos</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="n">a</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">:</span>
-                <span class="n">nonneg</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="n">a</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-                <span class="k">continue</span>
-
-            <span class="k">if</span> <span class="n">ubound</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="c"># sign is unknown; if we don&#39;t know the boundedness</span>
-                <span class="c"># we&#39;re done: we don&#39;t know. That is technically true,</span>
-                <span class="c"># but the only option is that we have something like</span>
-                <span class="c"># oo - oo which is NaN and it really doesn&#39;t matter</span>
-                <span class="c"># what sign we apply to that because it (when finally</span>
-                <span class="c"># computed) will trump any sign. So instead of returning</span>
-                <span class="c"># None, we pass.</span>
-                <span class="k">pass</span>
-            <span class="n">unknown_sign</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="k">if</span> <span class="n">unbounded</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">unbounded</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-        <span class="k">elif</span> <span class="n">unknown_sign</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="n">nonneg</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">nonpos</span> <span class="ow">and</span> <span class="n">neg</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="n">nonneg</span> <span class="ow">and</span> <span class="n">neg</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="n">neg</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">nonpos</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">_eval_subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">old</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="n">coeff_self</span><span class="p">,</span> <span class="n">terms_self</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_coeff_Add</span><span class="p">()</span>
-        <span class="n">coeff_old</span><span class="p">,</span> <span class="n">terms_old</span> <span class="o">=</span> <span class="n">old</span><span class="o">.</span><span class="n">as_coeff_Add</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="n">coeff_self</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">coeff_old</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">terms_self</span> <span class="o">==</span> <span class="n">terms_old</span><span class="p">:</span>   <span class="c"># (2 + a).subs( 3 + a, y) -&gt; -1 + y</span>
-                <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="n">new</span><span class="p">,</span> <span class="n">coeff_self</span><span class="p">,</span> <span class="o">-</span><span class="n">coeff_old</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">terms_self</span> <span class="o">==</span> <span class="o">-</span><span class="n">terms_old</span><span class="p">:</span>  <span class="c"># (2 + a).subs(-3 - a, y) -&gt; -1 - y</span>
-                <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">-</span><span class="n">new</span><span class="p">,</span> <span class="n">coeff_self</span><span class="p">,</span> <span class="n">coeff_old</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">coeff_self</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">coeff_old</span><span class="o">.</span><span class="n">is_Rational</span> \
-                <span class="ow">or</span> <span class="n">coeff_self</span> <span class="o">==</span> <span class="n">coeff_old</span><span class="p">:</span>
-            <span class="n">args_old</span><span class="p">,</span> <span class="n">args_self</span> <span class="o">=</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span>
-                <span class="n">terms_old</span><span class="p">),</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">terms_self</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args_old</span><span class="p">)</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">args_self</span><span class="p">):</span>  <span class="c"># (a+b+c).subs(b+c,x) -&gt; a+x</span>
-                <span class="n">self_set</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">args_self</span><span class="p">)</span>
-                <span class="n">old_set</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">args_old</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">old_set</span> <span class="o">&lt;</span> <span class="n">self_set</span><span class="p">:</span>
-                    <span class="n">ret_set</span> <span class="o">=</span> <span class="n">self_set</span> <span class="o">-</span> <span class="n">old_set</span>
-                    <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="n">new</span><span class="p">,</span> <span class="n">coeff_self</span><span class="p">,</span> <span class="o">-</span><span class="n">coeff_old</span><span class="p">,</span>
-                               <span class="o">*</span><span class="p">[</span><span class="n">s</span><span class="o">.</span><span class="n">_subs</span><span class="p">(</span><span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">ret_set</span><span class="p">])</span>
-
-                <span class="n">args_old</span> <span class="o">=</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span>
-                    <span class="o">-</span><span class="n">terms_old</span><span class="p">)</span>     <span class="c"># (a+b+c+d).subs(-b-c,x) -&gt; a-x+d</span>
-                <span class="n">old_set</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">args_old</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">old_set</span> <span class="o">&lt;</span> <span class="n">self_set</span><span class="p">:</span>
-                    <span class="n">ret_set</span> <span class="o">=</span> <span class="n">self_set</span> <span class="o">-</span> <span class="n">old_set</span>
-                    <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">-</span><span class="n">new</span><span class="p">,</span> <span class="n">coeff_self</span><span class="p">,</span> <span class="n">coeff_old</span><span class="p">,</span>
-                               <span class="o">*</span><span class="p">[</span><span class="n">s</span><span class="o">.</span><span class="n">_subs</span><span class="p">(</span><span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">ret_set</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">removeO</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Order</span><span class="p">]</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new_rawargs</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">getO</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Order</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new_rawargs</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="nd">@cacheit</span>
-<div class="viewcode-block" id="Add.extract_leading_order"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.add.Add.extract_leading_order">[docs]</a>    <span class="k">def</span> <span class="nf">extract_leading_order</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the leading term and it&#39;s order.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; (x + 1 + 1/x**5).extract_leading_order(x)</span>
-<span class="sd">        ((x**(-5), O(x**(-5))),)</span>
-<span class="sd">        &gt;&gt;&gt; (1 + x).extract_leading_order(x)</span>
-<span class="sd">        ((1, O(1)),)</span>
-<span class="sd">        &gt;&gt;&gt; (x + x**2).extract_leading_order(x)</span>
-<span class="sd">        ((x, O(x)),)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">lst</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">seq</span> <span class="o">=</span> <span class="p">[(</span><span class="n">f</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">))</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">ef</span><span class="p">,</span> <span class="n">of</span> <span class="ow">in</span> <span class="n">seq</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">lst</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">o</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">of</span><span class="p">)</span> <span class="ow">and</span> <span class="n">o</span> <span class="o">!=</span> <span class="n">of</span><span class="p">:</span>
-                    <span class="n">of</span> <span class="o">=</span> <span class="bp">None</span>
-                    <span class="k">break</span>
-            <span class="k">if</span> <span class="n">of</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="n">new_lst</span> <span class="o">=</span> <span class="p">[(</span><span class="n">ef</span><span class="p">,</span> <span class="n">of</span><span class="p">)]</span>
-            <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">lst</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">of</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">o</span><span class="p">)</span> <span class="ow">and</span> <span class="n">o</span> <span class="o">!=</span> <span class="n">of</span><span class="p">:</span>
-                    <span class="k">continue</span>
-                <span class="n">new_lst</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">e</span><span class="p">,</span> <span class="n">o</span><span class="p">))</span>
-            <span class="n">lst</span> <span class="o">=</span> <span class="n">new_lst</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">lst</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Add.as_real_imag"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.add.Add.as_real_imag">[docs]</a>    <span class="k">def</span> <span class="nf">as_real_imag</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        returns a tuple represeting a complex numbers</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import I</span>
-<span class="sd">        &gt;&gt;&gt; (7 + 9*I).as_real_imag()</span>
-<span class="sd">        (7, 9)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">sargs</span><span class="p">,</span> <span class="n">terms</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="p">[]</span>
-        <span class="n">re_part</span><span class="p">,</span> <span class="n">im_part</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">sargs</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">(</span><span class="n">deep</span><span class="o">=</span><span class="n">deep</span><span class="p">)</span>
-            <span class="n">re_part</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">re</span><span class="p">)</span>
-            <span class="n">im_part</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">im</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">re_part</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">im_part</span><span class="p">))</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expand_mul</span><span class="p">,</span> <span class="n">factor_terms</span>
-
-        <span class="n">old</span> <span class="o">=</span> <span class="bp">self</span>
-
-        <span class="bp">self</span> <span class="o">=</span> <span class="n">expand_mul</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="n">unbounded</span> <span class="o">=</span> <span class="p">[</span><span class="n">t</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">is_unbounded</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">unbounded</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Add</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">unbounded</span><span class="p">)</span>
-
-        <span class="bp">self</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">t</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span><span class="o">.</span><span class="n">removeO</span><span class="p">()</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="p">:</span>
-            <span class="c"># simple leading term analysis gave us 0 but we have to send</span>
-            <span class="c"># back a term, so compute the leading term (via series)</span>
-            <span class="k">return</span> <span class="n">old</span><span class="o">.</span><span class="n">compute_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">plain</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">s</span> <span class="k">for</span> <span class="n">s</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">extract_leading_order</span><span class="p">(</span><span class="n">x</span><span class="p">)])</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="n">factor_terms</span><span class="p">(</span><span class="n">plain</span><span class="p">,</span> <span class="n">fraction</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-            <span class="n">rv_fraction</span> <span class="o">=</span> <span class="n">factor_terms</span><span class="p">(</span><span class="n">rv</span><span class="p">,</span> <span class="n">fraction</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="c"># if it simplifies to an x-free expression, return that;</span>
-            <span class="c"># tests don&#39;t fail if we don&#39;t but it seems nicer to do this</span>
-            <span class="k">if</span> <span class="n">x</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">rv_fraction</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">rv_fraction</span>
-            <span class="k">return</span> <span class="n">rv</span>
-
-    <span class="k">def</span> <span class="nf">_eval_adjoint</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">t</span><span class="o">.</span><span class="n">adjoint</span><span class="p">()</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">t</span><span class="o">.</span><span class="n">conjugate</span><span class="p">()</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_transpose</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">t</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="o">-</span><span class="n">t</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">s</span> <span class="o">+=</span> <span class="n">x</span><span class="o">.</span><span class="n">_sage_</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">s</span>
-
-<div class="viewcode-block" id="Add.primitive"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.add.Add.primitive">[docs]</a>    <span class="k">def</span> <span class="nf">primitive</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return ``(R, self/R)`` where ``R``` is the Rational GCD of ``self```.</span>
-
-<span class="sd">        ``R`` is collected only from the leading coefficient of each term.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; (2*x + 4*y).primitive()</span>
-<span class="sd">        (2, x + 2*y)</span>
-
-<span class="sd">        &gt;&gt;&gt; (2*x/3 + 4*y/9).primitive()</span>
-<span class="sd">        (2/9, 3*x + 2*y)</span>
-
-<span class="sd">        &gt;&gt;&gt; (2*x/3 + 4.2*y).primitive()</span>
-<span class="sd">        (1/3, 2*x + 12.6*y)</span>
-
-<span class="sd">        No subprocessing of term factors is performed:</span>
-
-<span class="sd">        &gt;&gt;&gt; ((2 + 2*x)*x + 2).primitive()</span>
-<span class="sd">        (1, x*(2*x + 2) + 2)</span>
-
-<span class="sd">        Recursive subprocessing can be done with the as_content_primitive()</span>
-<span class="sd">        method:</span>
-
-<span class="sd">        &gt;&gt;&gt; ((2 + 2*x)*x + 2).as_content_primitive()</span>
-<span class="sd">        (2, x*(x + 1) + 1)</span>
-
-<span class="sd">        See also: primitive() function in polytools.py</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">terms</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">inf</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">c</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">c</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-                <span class="n">m</span> <span class="o">=</span> <span class="n">a</span>
-            <span class="n">inf</span> <span class="o">=</span> <span class="n">inf</span> <span class="ow">or</span> <span class="n">m</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span>
-            <span class="n">terms</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">c</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="n">c</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="n">m</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">inf</span><span class="p">:</span>
-            <span class="n">ngcd</span> <span class="o">=</span> <span class="nb">reduce</span><span class="p">(</span><span class="n">igcd</span><span class="p">,</span> <span class="p">[</span><span class="n">t</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
-            <span class="n">dlcm</span> <span class="o">=</span> <span class="nb">reduce</span><span class="p">(</span><span class="n">ilcm</span><span class="p">,</span> <span class="p">[</span><span class="n">t</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">],</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">ngcd</span> <span class="o">=</span> <span class="nb">reduce</span><span class="p">(</span><span class="n">igcd</span><span class="p">,</span> <span class="p">[</span><span class="n">t</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">terms</span> <span class="k">if</span> <span class="n">t</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span> <span class="mi">0</span><span class="p">)</span>
-            <span class="n">dlcm</span> <span class="o">=</span> <span class="nb">reduce</span><span class="p">(</span><span class="n">ilcm</span><span class="p">,</span> <span class="p">[</span><span class="n">t</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">terms</span> <span class="k">if</span> <span class="n">t</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span> <span class="mi">1</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">ngcd</span> <span class="o">==</span> <span class="n">dlcm</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="bp">self</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">inf</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">term</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">terms</span><span class="p">):</span>
-                <span class="n">terms</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">Rational</span><span class="p">((</span><span class="n">p</span><span class="o">//</span><span class="n">ngcd</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">dlcm</span><span class="o">//</span><span class="n">q</span><span class="p">)),</span> <span class="n">term</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">term</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">terms</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">q</span><span class="p">:</span>
-                    <span class="n">terms</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">Rational</span><span class="p">((</span><span class="n">p</span><span class="o">//</span><span class="n">ngcd</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">dlcm</span><span class="o">//</span><span class="n">q</span><span class="p">)),</span> <span class="n">term</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">terms</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">),</span> <span class="n">term</span><span class="p">)</span>
-
-        <span class="c"># we don&#39;t need a complete re-flattening since no new terms will join</span>
-        <span class="c"># so we just use the same sort as is used in Add.flatten. When the</span>
-        <span class="c"># coefficient changes, the ordering of terms may change, e.g.</span>
-        <span class="c">#     (3*x, 6*y) -&gt; (2*y, x)</span>
-        <span class="c">#</span>
-        <span class="c"># We do need to make sure that term[0] stays in position 0, however.</span>
-        <span class="c">#</span>
-        <span class="k">if</span> <span class="n">terms</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">or</span> <span class="n">terms</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">:</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">terms</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="n">terms</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="nb">hash</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">c</span><span class="p">:</span>
-            <span class="n">terms</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="n">ngcd</span><span class="p">,</span> <span class="n">dlcm</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new_rawargs</span><span class="p">(</span><span class="o">*</span><span class="n">terms</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Add.as_content_primitive"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.add.Add.as_content_primitive">[docs]</a>    <span class="k">def</span> <span class="nf">as_content_primitive</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">radical</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the tuple (R, self/R) where R is the positive Rational</span>
-<span class="sd">        extracted from self. If radical is True (default is False) then</span>
-<span class="sd">        common radicals will be removed and included as a factor of the</span>
-<span class="sd">        primitive expression.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sqrt</span>
-<span class="sd">        &gt;&gt;&gt; (3 + 3*sqrt(2)).as_content_primitive()</span>
-<span class="sd">        (3, 1 + sqrt(2))</span>
-
-<span class="sd">        Radical content can also be factored out of the primitive:</span>
-
-<span class="sd">        &gt;&gt;&gt; (2*sqrt(2) + 4*sqrt(10)).as_content_primitive(radical=True)</span>
-<span class="sd">        (2, sqrt(2)*(1 + 2*sqrt(5)))</span>
-
-<span class="sd">        See docstring of Expr.as_content_primitive for more examples.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">con</span><span class="p">,</span> <span class="n">prim</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">_keep_coeff</span><span class="p">(</span><span class="o">*</span><span class="n">a</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">(</span>
-            <span class="n">radical</span><span class="o">=</span><span class="n">radical</span><span class="p">))</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span><span class="o">.</span><span class="n">primitive</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">radical</span> <span class="ow">and</span> <span class="n">prim</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="c"># look for common radicals that can be removed</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="n">prim</span><span class="o">.</span><span class="n">args</span>
-            <span class="n">rads</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">common_q</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-                <span class="n">term_rads</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">list</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">ai</span> <span class="ow">in</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">ai</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-                        <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">ai</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">b</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                            <span class="n">term_rads</span><span class="p">[</span><span class="n">e</span><span class="o">.</span><span class="n">q</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">e</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">term_rads</span><span class="p">:</span>
-                    <span class="k">break</span>
-                <span class="k">if</span> <span class="n">common_q</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">common_q</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">term_rads</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">common_q</span> <span class="o">=</span> <span class="n">common_q</span> <span class="o">&amp;</span> <span class="nb">set</span><span class="p">(</span><span class="n">term_rads</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">common_q</span><span class="p">:</span>
-                        <span class="k">break</span>
-                <span class="n">rads</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">term_rads</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># process rads</span>
-                <span class="c"># keep only those in common_q</span>
-                <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">rads</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">r</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
-                        <span class="k">if</span> <span class="n">q</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">common_q</span><span class="p">:</span>
-                            <span class="n">r</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-                    <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">r</span><span class="p">:</span>
-                        <span class="n">r</span><span class="p">[</span><span class="n">q</span><span class="p">]</span> <span class="o">=</span> <span class="n">prod</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">q</span><span class="p">])</span>
-                <span class="c"># find the gcd of bases for each q</span>
-                <span class="n">G</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">common_q</span><span class="p">:</span>
-                    <span class="n">g</span> <span class="o">=</span> <span class="nb">reduce</span><span class="p">(</span><span class="n">igcd</span><span class="p">,</span> <span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="n">q</span><span class="p">]</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">rads</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">g</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="n">G</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">g</span><span class="o">**</span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">q</span><span class="p">))</span>
-                <span class="k">if</span> <span class="n">G</span><span class="p">:</span>
-                    <span class="n">G</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">G</span><span class="p">)</span>
-                    <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">ai</span><span class="o">/</span><span class="n">G</span> <span class="k">for</span> <span class="n">ai</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span>
-                    <span class="n">prim</span> <span class="o">=</span> <span class="n">G</span><span class="o">*</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">con</span><span class="p">,</span> <span class="n">prim</span>
-</div>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_sorted_args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">default_sort_key</span>
-        <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">w</span><span class="p">:</span> <span class="n">default_sort_key</span><span class="p">(</span><span class="n">w</span><span class="p">))</span>
-</div>
-<span class="kn">from</span> <span class="nn">.mul</span> <span class="kn">import</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">_keep_coeff</span><span class="p">,</span> <span class="n">prod</span>
-<span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">Rational</span>
-</pre></div>
-
-          </div>
-        </div>
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-            <p class="logo"><a href="../../../index.html">
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-            
-  <h1>Source code for sympy.core.basic</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Base class for all the objects in SymPy&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">copy</span> <span class="kn">import</span> <span class="n">copy</span>
-<span class="kn">from</span> <span class="nn">sympy.core.assumptions</span> <span class="kn">import</span> <span class="n">ManagedProperties</span>
-<span class="kn">from</span> <span class="nn">sympy.core.cache</span> <span class="kn">import</span> <span class="n">cacheit</span>
-<span class="kn">from</span> <span class="nn">sympy.core.core</span> <span class="kn">import</span> <span class="n">BasicType</span><span class="p">,</span> <span class="n">C</span>
-<span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="n">_sympify</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">SympifyError</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="p">(</span><span class="nb">callable</span><span class="p">,</span> <span class="nb">reduce</span><span class="p">,</span> <span class="nb">cmp</span><span class="p">,</span> <span class="n">iterable</span><span class="p">,</span>
-    <span class="n">ordered</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.core.decorators</span> <span class="kn">import</span> <span class="n">deprecated</span>
-<span class="kn">from</span> <span class="nn">sympy.core.singleton</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="kn">import</span> <span class="nn">collections</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-
-<div class="viewcode-block" id="Basic"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic">[docs]</a><span class="k">class</span> <span class="nc">Basic</span><span class="p">(</span><span class="nb">object</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">ManagedProperties</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Base class for all objects in sympy.</span>
-
-<span class="sd">    Conventions:</span>
-
-<span class="sd">    1) Always use ``.args``, when accessing parameters of some instance:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols, cot</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; cot(x).args</span>
-<span class="sd">        (x,)</span>
-
-<span class="sd">        &gt;&gt;&gt; cot(x).args[0]</span>
-<span class="sd">        x</span>
-
-<span class="sd">        &gt;&gt;&gt; (x*y).args</span>
-<span class="sd">        (x, y)</span>
-
-<span class="sd">        &gt;&gt;&gt; (x*y).args[1]</span>
-<span class="sd">        y</span>
-
-
-<span class="sd">    2) Never use internal methods or variables (the ones prefixed with ``_``):</span>
-
-<span class="sd">        &gt;&gt;&gt; cot(x)._args    # do not use this, use cot(x).args instead</span>
-<span class="sd">        (x,)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;_mhash&#39;</span><span class="p">,</span>              <span class="c"># hash value</span>
-                 <span class="s">&#39;_args&#39;</span><span class="p">,</span>               <span class="c"># arguments</span>
-                 <span class="s">&#39;_assumptions&#39;</span>
-                <span class="p">]</span>
-
-    <span class="c"># To be overridden with True in the appropriate subclasses</span>
-    <span class="n">is_Atom</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Symbol</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Dummy</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Wild</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Function</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Add</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Mul</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Pow</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Number</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Float</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Rational</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Integer</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_NumberSymbol</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Order</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Derivative</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Piecewise</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Poly</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_AlgebraicNumber</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Relational</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Equality</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Boolean</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Not</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Matrix</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="nd">@property</span>
-    <span class="nd">@deprecated</span><span class="p">(</span><span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;is_Float&quot;</span><span class="p">,</span> <span class="n">issue</span><span class="o">=</span><span class="mi">1721</span><span class="p">,</span> <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.0&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="Basic.is_Real"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.is_Real">[docs]</a>    <span class="k">def</span> <span class="nf">is_Real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>  <span class="c"># pragma: no cover</span>
-        <span class="sd">&quot;&quot;&quot;Deprecated alias for ``is_Float``&quot;&quot;&quot;</span>
-        <span class="c"># When this is removed, remove the piece of code disabling the warning</span>
-        <span class="c"># from test_pickling.py</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Float</span>
-</div>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="nb">object</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_assumptions</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">default_assumptions</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_mhash</span> <span class="o">=</span> <span class="bp">None</span>  <span class="c"># will be set by __hash__ method.</span>
-
-        <span class="n">obj</span><span class="o">.</span><span class="n">_args</span> <span class="o">=</span> <span class="n">args</span>  <span class="c"># all items in args must be Basic objects</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">copy</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__reduce_ex__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">proto</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Pickling support.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">__getnewargs__</span><span class="p">(),</span> <span class="bp">self</span><span class="o">.</span><span class="n">__getstate__</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__getnewargs__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-
-    <span class="k">def</span> <span class="nf">__getstate__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">{}</span>
-
-    <span class="k">def</span> <span class="nf">__setstate__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">state</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="nb">setattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># hash cannot be cached using cache_it because infinite recurrence</span>
-        <span class="c"># occurs as hash is needed for setting cache dictionary keys</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mhash</span>
-        <span class="k">if</span> <span class="n">h</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="nb">hash</span><span class="p">((</span><span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__name__</span><span class="p">,)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_hashable_content</span><span class="p">())</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_mhash</span> <span class="o">=</span> <span class="n">h</span>
-        <span class="k">return</span> <span class="n">h</span>
-
-    <span class="k">def</span> <span class="nf">_hashable_content</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return a tuple of information about self that can be used to</span>
-<span class="sd">        compute the hash. If a class defines additional attributes,</span>
-<span class="sd">        like ``name`` in Symbol, then this method should be updated</span>
-<span class="sd">        accordingly to return such relevent attributes.</span>
-
-<span class="sd">        Defining more than _hashable_content is necessary if __eq__ has</span>
-<span class="sd">        been defined by a class. See note about this in Basic.__eq__.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Basic.assumptions0"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.assumptions0">[docs]</a>    <span class="k">def</span> <span class="nf">assumptions0</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return object `type` assumptions.</span>
-
-<span class="sd">        For example:</span>
-
-<span class="sd">          Symbol(&#39;x&#39;, real=True)</span>
-<span class="sd">          Symbol(&#39;x&#39;, integer=True)</span>
-
-<span class="sd">        are different objects. In other words, besides Python type (Symbol in</span>
-<span class="sd">        this case), the initial assumptions are also forming their typeinfo.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; x.assumptions0</span>
-<span class="sd">        {&#39;commutative&#39;: True}</span>
-<span class="sd">        &gt;&gt;&gt; x = Symbol(&quot;x&quot;, positive=True)</span>
-<span class="sd">        &gt;&gt;&gt; x.assumptions0</span>
-<span class="sd">        {&#39;commutative&#39;: True, &#39;complex&#39;: True, &#39;hermitian&#39;: True,</span>
-<span class="sd">        &#39;imaginary&#39;: False, &#39;negative&#39;: False, &#39;nonnegative&#39;: True,</span>
-<span class="sd">        &#39;nonpositive&#39;: False, &#39;nonzero&#39;: True, &#39;positive&#39;: True, &#39;real&#39;: True,</span>
-<span class="sd">        &#39;zero&#39;: False}</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">{}</span>
-</div>
-<div class="viewcode-block" id="Basic.compare"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.compare">[docs]</a>    <span class="k">def</span> <span class="nf">compare</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return -1, 0, 1 if the object is smaller, equal, or greater than other.</span>
-
-<span class="sd">        Not in the mathematical sense. If the object is of a different type</span>
-<span class="sd">        from the &quot;other&quot; then their classes are ordered according to</span>
-<span class="sd">        the sorted_classes list.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; x.compare(y)</span>
-<span class="sd">        -1</span>
-<span class="sd">        &gt;&gt;&gt; x.compare(x)</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; y.compare(x)</span>
-<span class="sd">        1</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># all redefinitions of __cmp__ method should start with the</span>
-        <span class="c"># following three lines:</span>
-        <span class="k">if</span> <span class="bp">self</span> <span class="ow">is</span> <span class="n">other</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">0</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="nb">cmp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">__class__</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">c</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">c</span>
-        <span class="c">#</span>
-        <span class="n">st</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_hashable_content</span><span class="p">()</span>
-        <span class="n">ot</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">_hashable_content</span><span class="p">()</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="nb">cmp</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">st</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">ot</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">c</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">c</span>
-        <span class="k">for</span> <span class="n">l</span><span class="p">,</span> <span class="n">r</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">st</span><span class="p">,</span> <span class="n">ot</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="n">l</span><span class="o">.</span><span class="n">compare</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="nb">frozenset</span><span class="p">):</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="nb">cmp</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">c</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">c</span>
-        <span class="k">return</span> <span class="mi">0</span>
-</div>
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_compare_pretty</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.series.order</span> <span class="kn">import</span> <span class="n">Order</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Order</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Order</span><span class="p">):</span>
-            <span class="k">return</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Order</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Order</span><span class="p">):</span>
-            <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-
-        <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">cmp</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">p</span><span class="o">*</span><span class="n">b</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">p</span><span class="o">*</span><span class="n">a</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">sympy.core.symbol</span> <span class="kn">import</span> <span class="n">Wild</span>
-            <span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&quot;p1&quot;</span><span class="p">),</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&quot;p2&quot;</span><span class="p">),</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&quot;p3&quot;</span><span class="p">)</span>
-            <span class="n">r_a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p1</span> <span class="o">*</span> <span class="n">p2</span><span class="o">**</span><span class="n">p3</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">r_a</span> <span class="ow">and</span> <span class="n">p3</span> <span class="ow">in</span> <span class="n">r_a</span><span class="p">:</span>
-                <span class="n">a3</span> <span class="o">=</span> <span class="n">r_a</span><span class="p">[</span><span class="n">p3</span><span class="p">]</span>
-                <span class="n">r_b</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p1</span> <span class="o">*</span> <span class="n">p2</span><span class="o">**</span><span class="n">p3</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">r_b</span> <span class="ow">and</span> <span class="n">p3</span> <span class="ow">in</span> <span class="n">r_b</span><span class="p">:</span>
-                    <span class="n">b3</span> <span class="o">=</span> <span class="n">r_b</span><span class="p">[</span><span class="n">p3</span><span class="p">]</span>
-                    <span class="n">c</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">compare</span><span class="p">(</span><span class="n">a3</span><span class="p">,</span> <span class="n">b3</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">c</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">c</span>
-
-        <span class="k">return</span> <span class="n">Basic</span><span class="o">.</span><span class="n">compare</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@deprecated</span><span class="p">(</span><span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;default_sort_key&quot;</span><span class="p">,</span> <span class="n">issue</span><span class="o">=</span><span class="mi">1491</span><span class="p">,</span> <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.2&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="Basic.compare_pretty"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.compare_pretty">[docs]</a>    <span class="k">def</span> <span class="nf">compare_pretty</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Is a &gt; b in the sense of ordering in printing?</span>
-
-<span class="sd">        THIS FUNCTION IS DEPRECATED.  Use ``default_sort_key`` instead.</span>
-
-<span class="sd">        ::</span>
-
-<span class="sd">          yes ..... return 1</span>
-<span class="sd">          no ...... return -1</span>
-<span class="sd">          equal ... return 0</span>
-
-<span class="sd">        Strategy:</span>
-
-<span class="sd">        It uses Basic.compare as a fallback, but improves it in many cases,</span>
-<span class="sd">        like ``x**3``, ``x**4``, ``O(x**3)`` etc. In those simple cases, it just parses the</span>
-<span class="sd">        expression and returns the &quot;sane&quot; ordering such as::</span>
-
-<span class="sd">          1 &lt; x &lt; x**2 &lt; x**3 &lt; O(x**4) etc.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import Basic, Number</span>
-<span class="sd">        &gt;&gt;&gt; Basic._compare_pretty(x, x**2)</span>
-<span class="sd">        -1</span>
-<span class="sd">        &gt;&gt;&gt; Basic._compare_pretty(x**2, x**2)</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; Basic._compare_pretty(x**3, x**2)</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; Basic._compare_pretty(Number(1, 2), Number(1, 3))</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; Basic._compare_pretty(Number(0), Number(-1))</span>
-<span class="sd">        1</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">SympifyError</span><span class="p">:</span>
-            <span class="k">pass</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">SympifyError</span><span class="p">:</span>
-            <span class="k">pass</span>
-
-        <span class="c"># both objects are non-SymPy</span>
-        <span class="k">if</span> <span class="p">(</span><span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Basic</span><span class="p">))</span> <span class="ow">and</span> <span class="p">(</span><span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Basic</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="nb">cmp</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-            <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>   <span class="c"># other &lt; sympy</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-            <span class="k">return</span> <span class="o">+</span><span class="mi">1</span>   <span class="c"># sympy &gt; other</span>
-
-        <span class="c"># now both objects are from SymPy, so we can proceed to usual comparison</span>
-        <span class="k">return</span> <span class="nb">cmp</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">sort_key</span><span class="p">(),</span> <span class="n">b</span><span class="o">.</span><span class="n">sort_key</span><span class="p">())</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Basic.fromiter"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.fromiter">[docs]</a>    <span class="k">def</span> <span class="nf">fromiter</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Create a new object from an iterable.</span>
-
-<span class="sd">        This is a convenience function that allows one to create objects from</span>
-<span class="sd">        any iterable, without having to convert to a list or tuple first.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Tuple</span>
-<span class="sd">        &gt;&gt;&gt; Tuple.fromiter(i for i in range(5))</span>
-<span class="sd">        (0, 1, 2, 3, 4)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="o">*</span><span class="nb">tuple</span><span class="p">(</span><span class="n">args</span><span class="p">),</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Basic.class_key"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.class_key">[docs]</a>    <span class="k">def</span> <span class="nf">class_key</span><span class="p">(</span><span class="n">cls</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Nice order of classes. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">cls</span><span class="o">.</span><span class="n">__name__</span>
-</div>
-    <span class="nd">@cacheit</span>
-<div class="viewcode-block" id="Basic.sort_key"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.sort_key">[docs]</a>    <span class="k">def</span> <span class="nf">sort_key</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return a sort key.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.core import Basic, S, I</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; sorted([S(1)/2, I, -I], key=lambda x: x.sort_key())</span>
-<span class="sd">        [1/2, -I, I]</span>
-
-<span class="sd">        &gt;&gt;&gt; S(&quot;[x, 1/x, 1/x**2, x**2, x**(1/2), x**(1/4), x**(3/2)]&quot;)</span>
-<span class="sd">        [x, 1/x, x**(-2), x**2, sqrt(x), x**(1/4), x**(3/2)]</span>
-<span class="sd">        &gt;&gt;&gt; sorted(_, key=lambda x: x.sort_key())</span>
-<span class="sd">        [x**(-2), 1/x, x**(1/4), sqrt(x), x, x**(3/2), x**2]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="c"># XXX: remove this when issue #2070 is fixed</span>
-        <span class="k">def</span> <span class="nf">inner_key</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">arg</span><span class="o">.</span><span class="n">sort_key</span><span class="p">(</span><span class="n">order</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">arg</span>
-
-        <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sorted_args</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">),</span> <span class="nb">tuple</span><span class="p">([</span> <span class="n">inner_key</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span> <span class="p">])</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">class_key</span><span class="p">(),</span> <span class="n">args</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="o">.</span><span class="n">sort_key</span><span class="p">(),</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-</div>
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return a boolean indicating whether a == b on the basis of</span>
-<span class="sd">        their symbolic trees.</span>
-
-<span class="sd">        This is the same as a.compare(b) == 0 but faster.</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        If a class that overrides __eq__() needs to retain the</span>
-<span class="sd">        implementation of __hash__() from a parent class, the</span>
-<span class="sd">        interpreter must be told this explicitly by setting __hash__ =</span>
-<span class="sd">        &lt;ParentClass&gt;.__hash__. Otherwise the inheritance of __hash__()</span>
-<span class="sd">        will be blocked, just as if __hash__ had been explicitly set to</span>
-<span class="sd">        None.</span>
-
-<span class="sd">        References</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        from http://docs.python.org/dev/reference/datamodel.html#object.__hash__</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="nb">type</span><span class="p">(</span><span class="n">other</span><span class="p">):</span>
-            <span class="c"># issue 3001 a**1.0 == a like a**2.0 == a**2</span>
-            <span class="k">while</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Pow</span><span class="p">)</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="bp">self</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span>
-            <span class="k">while</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Pow</span><span class="p">)</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">exp</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">other</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">base</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">other</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">SympifyError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>    <span class="c"># sympy != other</span>
-
-            <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="nb">type</span><span class="p">(</span><span class="n">other</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_hashable_content</span><span class="p">()</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">_hashable_content</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;a != b  -&gt; Compare two symbolic trees and see whether they are different</span>
-
-<span class="sd">           this is the same as:</span>
-
-<span class="sd">             a.compare(b) != 0</span>
-
-<span class="sd">           but faster</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="nb">type</span><span class="p">(</span><span class="n">other</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">other</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">SympifyError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>     <span class="c"># sympy != other</span>
-
-            <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="nb">type</span><span class="p">(</span><span class="n">other</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">True</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_hashable_content</span><span class="p">()</span> <span class="o">!=</span> <span class="n">other</span><span class="o">.</span><span class="n">_hashable_content</span><span class="p">()</span>
-
-<div class="viewcode-block" id="Basic.dummy_eq"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.dummy_eq">[docs]</a>    <span class="k">def</span> <span class="nf">dummy_eq</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="n">symbol</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Compare two expressions and handle dummy symbols.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Dummy</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; u = Dummy(&#39;u&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; (u**2 + 1).dummy_eq(x**2 + 1)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; (u**2 + 1) == (x**2 + 1)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &gt;&gt;&gt; (u**2 + y).dummy_eq(x**2 + y, x)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; (u**2 + y).dummy_eq(x**2 + y, y)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">dummy_symbols</span> <span class="o">=</span> <span class="p">[</span> <span class="n">s</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span> <span class="k">if</span> <span class="n">s</span><span class="o">.</span><span class="n">is_Dummy</span> <span class="p">]</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">dummy_symbols</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span> <span class="o">==</span> <span class="n">other</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">dummy_symbols</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">dummy</span> <span class="o">=</span> <span class="n">dummy_symbols</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&quot;only one dummy symbol allowed on the left-hand side&quot;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">symbol</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">symbols</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">free_symbols</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">symbols</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span> <span class="o">==</span> <span class="n">other</span>
-            <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">symbol</span> <span class="o">=</span> <span class="n">symbols</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;specify a symbol in which expressions should be compared&quot;</span><span class="p">)</span>
-
-        <span class="n">tmp</span> <span class="o">=</span> <span class="n">dummy</span><span class="o">.</span><span class="n">__class__</span><span class="p">()</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">dummy</span><span class="p">,</span> <span class="n">tmp</span><span class="p">)</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">symbol</span><span class="p">,</span> <span class="n">tmp</span><span class="p">)</span>
-
-    <span class="c"># Note, we always use the default ordering (lex) in __str__ and __repr__,</span>
-    <span class="c"># regardless of the global setting.  See issue 2388.</span></div>
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.printing</span> <span class="kn">import</span> <span class="n">sstr</span>
-        <span class="k">return</span> <span class="n">sstr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.printing</span> <span class="kn">import</span> <span class="n">sstr</span>
-        <span class="k">return</span> <span class="n">sstr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Basic.atoms"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.atoms">[docs]</a>    <span class="k">def</span> <span class="nf">atoms</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">types</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the atoms that form the current object.</span>
-
-<span class="sd">           By default, only objects that are truly atomic and can&#39;t</span>
-<span class="sd">           be divided into smaller pieces are returned: symbols, numbers,</span>
-<span class="sd">           and number symbols like I and pi. It is possible to request</span>
-<span class="sd">           atoms of any type, however, as demonstrated below.</span>
-
-<span class="sd">           Examples</span>
-<span class="sd">           ========</span>
-
-<span class="sd">           &gt;&gt;&gt; from sympy import I, pi, sin</span>
-<span class="sd">           &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">           &gt;&gt;&gt; (1 + x + 2*sin(y + I*pi)).atoms()</span>
-<span class="sd">           set([1, 2, I, pi, x, y])</span>
-
-<span class="sd">           If one or more types are given, the results will contain only</span>
-<span class="sd">           those types of atoms.</span>
-
-<span class="sd">           Examples</span>
-<span class="sd">           ========</span>
-
-<span class="sd">           &gt;&gt;&gt; from sympy import Number, NumberSymbol, Symbol</span>
-<span class="sd">           &gt;&gt;&gt; (1 + x + 2*sin(y + I*pi)).atoms(Symbol)</span>
-<span class="sd">           set([x, y])</span>
-
-<span class="sd">           &gt;&gt;&gt; (1 + x + 2*sin(y + I*pi)).atoms(Number)</span>
-<span class="sd">           set([1, 2])</span>
-
-<span class="sd">           &gt;&gt;&gt; (1 + x + 2*sin(y + I*pi)).atoms(Number, NumberSymbol)</span>
-<span class="sd">           set([1, 2, pi])</span>
-
-<span class="sd">           &gt;&gt;&gt; (1 + x + 2*sin(y + I*pi)).atoms(Number, NumberSymbol, I)</span>
-<span class="sd">           set([1, 2, I, pi])</span>
-
-<span class="sd">           Note that I (imaginary unit) and zoo (complex infinity) are special</span>
-<span class="sd">           types of number symbols and are not part of the NumberSymbol class.</span>
-
-<span class="sd">           The type can be given implicitly, too:</span>
-
-<span class="sd">           &gt;&gt;&gt; (1 + x + 2*sin(y + I*pi)).atoms(x) # x is a Symbol</span>
-<span class="sd">           set([x, y])</span>
-
-<span class="sd">           Be careful to check your assumptions when using the implicit option</span>
-<span class="sd">           since ``S(1).is_Integer = True`` but ``type(S(1))`` is ``One``, a special type</span>
-<span class="sd">           of sympy atom, while ``type(S(2))`` is type ``Integer`` and will find all</span>
-<span class="sd">           integers in an expression:</span>
-
-<span class="sd">           &gt;&gt;&gt; from sympy import S</span>
-<span class="sd">           &gt;&gt;&gt; (1 + x + 2*sin(y + I*pi)).atoms(S(1))</span>
-<span class="sd">           set([1])</span>
-
-<span class="sd">           &gt;&gt;&gt; (1 + x + 2*sin(y + I*pi)).atoms(S(2))</span>
-<span class="sd">           set([1, 2])</span>
-
-<span class="sd">           Finally, arguments to atoms() can select more than atomic atoms: any</span>
-<span class="sd">           sympy type (loaded in core/__init__.py) can be listed as an argument</span>
-<span class="sd">           and those types of &quot;atoms&quot; as found in scanning the arguments of the</span>
-<span class="sd">           expression recursively:</span>
-
-<span class="sd">           &gt;&gt;&gt; from sympy import Function, Mul</span>
-<span class="sd">           &gt;&gt;&gt; from sympy.core.function import AppliedUndef</span>
-<span class="sd">           &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">           &gt;&gt;&gt; (1 + f(x) + 2*sin(y + I*pi)).atoms(Function)</span>
-<span class="sd">           set([f(x), sin(y + I*pi)])</span>
-<span class="sd">           &gt;&gt;&gt; (1 + f(x) + 2*sin(y + I*pi)).atoms(AppliedUndef)</span>
-<span class="sd">           set([f(x)])</span>
-
-<span class="sd">           &gt;&gt;&gt; (1 + x + 2*sin(y + I*pi)).atoms(Mul)</span>
-<span class="sd">           set([I*pi, 2*sin(y + I*pi)])</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">types</span><span class="p">:</span>
-            <span class="n">types</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span>
-                <span class="p">[</span><span class="n">t</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="nb">type</span><span class="p">)</span> <span class="k">else</span> <span class="nb">type</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">types</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">types</span> <span class="o">=</span> <span class="p">(</span><span class="n">Atom</span><span class="p">,)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">expr</span> <span class="ow">in</span> <span class="n">preorder_traversal</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">types</span><span class="p">):</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">result</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Basic.free_symbols"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.free_symbols">[docs]</a>    <span class="k">def</span> <span class="nf">free_symbols</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return from the atoms of self those which are free symbols.</span>
-
-<span class="sd">        For most expressions, all symbols are free symbols. For some classes</span>
-<span class="sd">        this is not true. e.g. Integrals use Symbols for the dummy variables</span>
-<span class="sd">        which are bound variables, so Integral has a method to return all symbols</span>
-<span class="sd">        except those. Derivative keeps track of symbols with respect to which it</span>
-<span class="sd">        will perform a derivative; those are bound variables, too, so it has</span>
-<span class="sd">        its own symbols method.</span>
-
-<span class="sd">        Any other method that uses bound variables should implement a symbols</span>
-<span class="sd">        method.&quot;&quot;&quot;</span>
-        <span class="n">union</span> <span class="o">=</span> <span class="nb">set</span><span class="o">.</span><span class="n">union</span>
-        <span class="k">return</span> <span class="nb">reduce</span><span class="p">(</span><span class="n">union</span><span class="p">,</span> <span class="p">[</span><span class="n">arg</span><span class="o">.</span><span class="n">free_symbols</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">],</span> <span class="nb">set</span><span class="p">())</span>
-</div>
-    <span class="k">def</span> <span class="nf">is_hypergeometric</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">hypersimp</span>
-        <span class="k">return</span> <span class="n">hypersimp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Basic.is_number"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.is_number">[docs]</a>    <span class="k">def</span> <span class="nf">is_number</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns ``True`` if &#39;self&#39; is a number.</span>
-
-<span class="sd">           &gt;&gt;&gt; from sympy import log, Integral</span>
-<span class="sd">           &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">           &gt;&gt;&gt; x.is_number</span>
-<span class="sd">           False</span>
-<span class="sd">           &gt;&gt;&gt; (2*x).is_number</span>
-<span class="sd">           False</span>
-<span class="sd">           &gt;&gt;&gt; (2 + log(2)).is_number</span>
-<span class="sd">           True</span>
-<span class="sd">           &gt;&gt;&gt; (2 + Integral(2, x)).is_number</span>
-<span class="sd">           False</span>
-<span class="sd">           &gt;&gt;&gt; (2 + Integral(2, (x, 1, 2))).is_number</span>
-<span class="sd">           True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># should be overriden by subclasses</span>
-        <span class="k">return</span> <span class="bp">False</span>
-</div>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_comparable</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">is_real</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_real</span>
-        <span class="k">if</span> <span class="n">is_real</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="n">is_number</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_number</span>
-        <span class="k">if</span> <span class="n">is_number</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">is_real</span> <span class="ow">and</span> <span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="p">[</span><span class="n">p</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()]</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">i</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">i</span><span class="p">:</span>
-            <span class="c"># if _prec = 1 we can&#39;t decide and if not,</span>
-            <span class="c"># the answer is False so return False</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">n</span><span class="o">.</span><span class="n">_prec</span> <span class="o">!=</span> <span class="mi">1</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Basic.func"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.func">[docs]</a>    <span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The top-level function in an expression.</span>
-
-<span class="sd">        The following should hold for all objects::</span>
-
-<span class="sd">            &gt;&gt; x == x.func(*x.args)</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; a = 2*x</span>
-<span class="sd">        &gt;&gt;&gt; a.func</span>
-<span class="sd">        &lt;class &#39;sympy.core.mul.Mul&#39;&gt;</span>
-<span class="sd">        &gt;&gt;&gt; a.args</span>
-<span class="sd">        (2, x)</span>
-<span class="sd">        &gt;&gt;&gt; a.func(*a.args)</span>
-<span class="sd">        2*x</span>
-<span class="sd">        &gt;&gt;&gt; a == a.func(*a.args)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Basic.args"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.args">[docs]</a>    <span class="k">def</span> <span class="nf">args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a tuple of arguments of &#39;self&#39;.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols, cot</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; cot(x).args</span>
-<span class="sd">        (x,)</span>
-
-<span class="sd">        &gt;&gt;&gt; cot(x).args[0]</span>
-<span class="sd">        x</span>
-
-<span class="sd">        &gt;&gt;&gt; (x*y).args</span>
-<span class="sd">        (x, y)</span>
-
-<span class="sd">        &gt;&gt;&gt; (x*y).args[1]</span>
-<span class="sd">        y</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        Never use self._args, always use self.args.</span>
-<span class="sd">        Only use _args in __new__ when creating a new function.</span>
-<span class="sd">        Don&#39;t override .args() from Basic (so that it&#39;s easy to</span>
-<span class="sd">        change the interface in the future if needed).</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span>
-</div>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_sorted_args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The same as ``args``.  Derived classes which don&#39;t fix an</span>
-<span class="sd">        order on their arguments should override this method to</span>
-<span class="sd">        produce the sorted representation.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-
-<div class="viewcode-block" id="Basic.iter_basic_args"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.iter_basic_args">[docs]</a>    <span class="k">def</span> <span class="nf">iter_basic_args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Iterates arguments of ``self``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; a = 2*x</span>
-<span class="sd">        &gt;&gt;&gt; a.iter_basic_args()</span>
-<span class="sd">        &lt;...iterator object at 0x...&gt;</span>
-<span class="sd">        &gt;&gt;&gt; list(a.iter_basic_args())</span>
-<span class="sd">        [2, x]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">iter</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Basic.as_poly"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.as_poly">[docs]</a>    <span class="k">def</span> <span class="nf">as_poly</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Converts ``self`` to a polynomial or returns ``None``.</span>
-
-<span class="sd">           &gt;&gt;&gt; from sympy import Poly, sin</span>
-<span class="sd">           &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">           &gt;&gt;&gt; print((x**2 + x*y).as_poly())</span>
-<span class="sd">           Poly(x**2 + x*y, x, y, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">           &gt;&gt;&gt; print((x**2 + x*y).as_poly(x, y))</span>
-<span class="sd">           Poly(x**2 + x*y, x, y, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">           &gt;&gt;&gt; print((x**2 + sin(y)).as_poly(x, y))</span>
-<span class="sd">           None</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">PolynomialError</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">poly</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">poly</span>
-        <span class="k">except</span> <span class="n">PolynomialError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-</div>
-<div class="viewcode-block" id="Basic.as_content_primitive"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.as_content_primitive">[docs]</a>    <span class="k">def</span> <span class="nf">as_content_primitive</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">radical</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A stub to allow Basic args (like Tuple) to be skipped when computing</span>
-<span class="sd">        the content and primitive components of an expression.</span>
-
-<span class="sd">        See docstring of Expr.as_content_primitive</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="bp">self</span>
-</div>
-<div class="viewcode-block" id="Basic.subs"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.subs">[docs]</a>    <span class="k">def</span> <span class="nf">subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Substitutes old for new in an expression after sympifying args.</span>
-
-<span class="sd">        `args` is either:</span>
-<span class="sd">          - two arguments, e.g. foo.subs(old, new)</span>
-<span class="sd">          - one iterable argument, e.g. foo.subs(iterable). The iterable may be</span>
-<span class="sd">             o an iterable container with (old, new) pairs. In this case the</span>
-<span class="sd">               replacements are processed in the order given with successive</span>
-<span class="sd">               patterns possibly affecting replacements already made.</span>
-<span class="sd">             o a dict or set whose key/value items correspond to old/new pairs.</span>
-<span class="sd">               In this case the old/new pairs will be sorted by op count and in</span>
-<span class="sd">               case of a tie, by number of args and the default_sort_key. The</span>
-<span class="sd">               resulting sorted list is then processed as an iterable container</span>
-<span class="sd">               (see previous).</span>
-
-<span class="sd">        If the keyword ``simultaneous`` is True, the subexpressions will not be</span>
-<span class="sd">        evaluated until all the substitutions have been made.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import pi, exp</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; (1 + x*y).subs(x, pi)</span>
-<span class="sd">        pi*y + 1</span>
-<span class="sd">        &gt;&gt;&gt; (1 + x*y).subs({x:pi, y:2})</span>
-<span class="sd">        1 + 2*pi</span>
-<span class="sd">        &gt;&gt;&gt; (1 + x*y).subs([(x, pi), (y, 2)])</span>
-<span class="sd">        1 + 2*pi</span>
-<span class="sd">        &gt;&gt;&gt; reps = [(y, x**2), (x, 2)]</span>
-<span class="sd">        &gt;&gt;&gt; (x + y).subs(reps)</span>
-<span class="sd">        6</span>
-<span class="sd">        &gt;&gt;&gt; (x + y).subs(reversed(reps))</span>
-<span class="sd">        x**2 + 2</span>
-
-<span class="sd">        &gt;&gt;&gt; (x**2 + x**4).subs(x**2, y)</span>
-<span class="sd">        y**2 + y</span>
-
-<span class="sd">        To replace only the x**2 but not the x**4, use xreplace:</span>
-
-<span class="sd">        &gt;&gt;&gt; (x**2 + x**4).xreplace({x**2: y})</span>
-<span class="sd">        x**4 + y</span>
-
-<span class="sd">        To delay evaluation until all substitutions have been made,</span>
-<span class="sd">        set the keyword ``simultaneous`` to True:</span>
-
-<span class="sd">        &gt;&gt;&gt; (x/y).subs([(x, 0), (y, 0)])</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; (x/y).subs([(x, 0), (y, 0)], simultaneous=True)</span>
-<span class="sd">        nan</span>
-
-<span class="sd">        This has the added feature of not allowing subsequent substitutions</span>
-<span class="sd">        to affect those already made:</span>
-
-<span class="sd">        &gt;&gt;&gt; ((x + y)/y).subs({x + y: y, y: x + y})</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True)</span>
-<span class="sd">        y/(x + y)</span>
-
-<span class="sd">        In order to obtain a canonical result, unordered iterables are</span>
-<span class="sd">        sorted by count_op length, number of arguments and by the</span>
-<span class="sd">        default_sort_key to break any ties. All other iterables are left</span>
-<span class="sd">        unsorted.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sqrt, sin, cos, exp</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import a, b, c, d, e</span>
-
-<span class="sd">        &gt;&gt;&gt; A = (sqrt(sin(2*x)), a)</span>
-<span class="sd">        &gt;&gt;&gt; B = (sin(2*x), b)</span>
-<span class="sd">        &gt;&gt;&gt; C = (cos(2*x), c)</span>
-<span class="sd">        &gt;&gt;&gt; D = (x, d)</span>
-<span class="sd">        &gt;&gt;&gt; E = (exp(x), e)</span>
-
-<span class="sd">        &gt;&gt;&gt; expr = sqrt(sin(2*x))*sin(exp(x)*x)*cos(2*x) + sin(2*x)</span>
-
-<span class="sd">        &gt;&gt;&gt; expr.subs(dict([A,B,C,D,E]))</span>
-<span class="sd">        a*c*sin(d*e) + b</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        replace: replacement capable of doing wildcard-like matching,</span>
-<span class="sd">                 parsing of match, and conditional replacements</span>
-<span class="sd">        xreplace: exact node replacement in expr tree; also capable of</span>
-<span class="sd">                  using matching rules</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.containers</span> <span class="kn">import</span> <span class="n">Dict</span>
-        <span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">default_sort_key</span>
-
-        <span class="n">unordered</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">sequence</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">sequence</span><span class="p">,</span> <span class="nb">set</span><span class="p">):</span>
-                <span class="n">unordered</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">sequence</span><span class="p">,</span> <span class="p">(</span><span class="n">Dict</span><span class="p">,</span> <span class="nb">dict</span><span class="p">)):</span>
-                <span class="n">unordered</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="n">sequence</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">sequence</span><span class="o">.</span><span class="n">items</span><span class="p">())</span>
-            <span class="k">elif</span> <span class="ow">not</span> <span class="n">iterable</span><span class="p">(</span><span class="n">sequence</span><span class="p">):</span>
-                <span class="kn">from</span> <span class="nn">sympy.utilities.misc</span> <span class="kn">import</span> <span class="n">filldedent</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&quot;&quot;&quot;</span>
-<span class="s">                   When a single argument is passed to subs</span>
-<span class="s">                   it should be a dictionary of old: new pairs or an iterable</span>
-<span class="s">                   of (old, new) tuples.&quot;&quot;&quot;</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">sequence</span> <span class="o">=</span> <span class="p">[</span><span class="n">args</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;subs accepts either 1 or 2 arguments&quot;</span><span class="p">)</span>
-
-        <span class="n">sequence</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">sequence</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">sequence</span><span class="p">)):</span>
-            <span class="n">o</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">sequence</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="n">so</span><span class="p">,</span> <span class="n">sn</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">o</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">so</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-                <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">o</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">str</span><span class="p">:</span>
-                    <span class="n">so</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-            <span class="n">sequence</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">so</span><span class="p">,</span> <span class="n">sn</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">_aresame</span><span class="p">(</span><span class="n">so</span><span class="p">,</span> <span class="n">sn</span><span class="p">):</span>
-                <span class="n">sequence</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="bp">None</span>
-                <span class="k">continue</span>
-        <span class="n">sequence</span> <span class="o">=</span> <span class="p">[</span><span class="n">_f</span> <span class="k">for</span> <span class="n">_f</span> <span class="ow">in</span> <span class="n">sequence</span> <span class="k">if</span> <span class="n">_f</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">unordered</span><span class="p">:</span>
-            <span class="n">sequence</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">sequence</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">(</span><span class="n">k</span><span class="o">.</span><span class="n">is_Atom</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">sequence</span><span class="p">):</span>
-                <span class="n">d</span> <span class="o">=</span> <span class="p">{}</span>
-                <span class="k">for</span> <span class="n">o</span><span class="p">,</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">sequence</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                    <span class="k">try</span><span class="p">:</span>
-                        <span class="n">ops</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">count_ops</span><span class="p">(),</span> <span class="nb">len</span><span class="p">(</span><span class="n">o</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-                    <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                        <span class="n">ops</span> <span class="o">=</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-                    <span class="n">d</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="n">ops</span><span class="p">,</span> <span class="p">[])</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">o</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-                <span class="n">newseq</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">keys</span><span class="p">()),</span> <span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-                    <span class="n">newseq</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span>
-                        <span class="nb">sorted</span><span class="p">([</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">d</span><span class="p">[</span><span class="n">k</span><span class="p">]],</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">))</span>
-                <span class="n">sequence</span> <span class="o">=</span> <span class="p">[(</span><span class="n">k</span><span class="p">,</span> <span class="n">sequence</span><span class="p">[</span><span class="n">k</span><span class="p">])</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">newseq</span><span class="p">]</span>
-                <span class="k">del</span> <span class="n">newseq</span><span class="p">,</span> <span class="n">d</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">sequence</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">([(</span><span class="n">k</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="k">for</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="ow">in</span> <span class="n">sequence</span><span class="o">.</span><span class="n">items</span><span class="p">()],</span>
-                                  <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;simultaneous&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>  <span class="c"># XXX should this be the default for dict subs?</span>
-            <span class="n">reps</span> <span class="o">=</span> <span class="p">{}</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="bp">self</span>
-            <span class="k">for</span> <span class="n">old</span><span class="p">,</span> <span class="n">new</span> <span class="ow">in</span> <span class="n">sequence</span><span class="p">:</span>
-                <span class="n">d</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">()</span>
-                <span class="n">rv</span> <span class="o">=</span> <span class="n">rv</span><span class="o">.</span><span class="n">_subs</span><span class="p">(</span><span class="n">old</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
-                <span class="n">reps</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">=</span> <span class="n">new</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">rv</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-                    <span class="k">break</span>
-            <span class="k">return</span> <span class="n">rv</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="n">reps</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="bp">self</span>
-            <span class="k">for</span> <span class="n">old</span><span class="p">,</span> <span class="n">new</span> <span class="ow">in</span> <span class="n">sequence</span><span class="p">:</span>
-                <span class="n">rv</span> <span class="o">=</span> <span class="n">rv</span><span class="o">.</span><span class="n">_subs</span><span class="p">(</span><span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">rv</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-                    <span class="k">break</span>
-            <span class="k">return</span> <span class="n">rv</span>
-</div>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">_subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Substitutes an expression old -&gt; new.</span>
-
-<span class="sd">        If self is not equal to old then _eval_subs is called.</span>
-<span class="sd">        If _eval_subs doesn&#39;t want to make any special replacement</span>
-<span class="sd">        then a None is received which indicates that the fallback</span>
-<span class="sd">        should be applied wherein a search for replacements is made</span>
-<span class="sd">        amongst the arguments of self.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Basic, Add, Mul</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        Add&#39;s _eval_subs knows how to target x + y in the following</span>
-<span class="sd">        so it makes the change:</span>
-
-<span class="sd">            &gt;&gt;&gt; (x + y + z).subs(x + y, 1)</span>
-<span class="sd">            z + 1</span>
-
-<span class="sd">        Add&#39;s _eval_subs doesn&#39;t need to know how to find x + y in</span>
-<span class="sd">        the following:</span>
-
-<span class="sd">            &gt;&gt;&gt; Add._eval_subs(z*(x + y) + 3, x + y, 1) is None</span>
-<span class="sd">            True</span>
-
-<span class="sd">        The returned None will cause the fallback routine to traverse the args and</span>
-<span class="sd">        pass the z*(x + y) arg to Mul where the change will take place and the</span>
-<span class="sd">        substitution will succeed:</span>
-
-<span class="sd">            &gt;&gt;&gt; (z*(x + y) + 3).subs(x + y, 1)</span>
-<span class="sd">            z + 3</span>
-
-<span class="sd">        ** Developers Notes **</span>
-
-<span class="sd">        An _eval_subs routine for a class should be written if:</span>
-
-<span class="sd">            1) any arguments are not instances of Basic (e.g. bool, tuple);</span>
-
-<span class="sd">            2) some arguments should not be targeted (as in integration</span>
-<span class="sd">               variables);</span>
-
-<span class="sd">            3) if there is something other than a literal replacement</span>
-<span class="sd">               that should be attempted (as in Piecewise where the condition</span>
-<span class="sd">               may be updated without doing a replacement).</span>
-
-<span class="sd">        If it is overridden, here are some special cases that might arise:</span>
-
-<span class="sd">            1) If it turns out that no special change was made and all</span>
-<span class="sd">               the original sub-arguments should be checked for</span>
-<span class="sd">               replacements then None should be returned.</span>
-
-<span class="sd">            2) If it is necessary to do substitutions on a portion of</span>
-<span class="sd">               the expression then _subs should be called. _subs will</span>
-<span class="sd">               handle the case of any sub-expression being equal to old</span>
-<span class="sd">               (which usually would not be the case) while its fallback</span>
-<span class="sd">               will handle the recursion into the sub-arguments. For</span>
-<span class="sd">               example, after Add&#39;s _eval_subs removes some matching terms</span>
-<span class="sd">               it must process the remaining terms so it calls _subs</span>
-<span class="sd">               on each of the un-matched terms and then adds them</span>
-<span class="sd">               onto the terms previously obtained.</span>
-
-<span class="sd">           3) If the initial expression should remain unchanged then</span>
-<span class="sd">              the original expression should be returned. (Whenever an</span>
-<span class="sd">              expression is returned, modified or not, no further</span>
-<span class="sd">              substitution of old -&gt; new is attempted.) Sum&#39;s _eval_subs</span>
-<span class="sd">              routine uses this strategy when a substitution is attempted</span>
-<span class="sd">              on any of its summation variables.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">def</span> <span class="nf">fallback</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            Try to replace old with new in any of self&#39;s arguments.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="n">hit</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">arg</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="s">&#39;_eval_subs&#39;</span><span class="p">):</span>
-                    <span class="k">continue</span>
-                <span class="n">arg</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">_subs</span><span class="p">(</span><span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                    <span class="n">hit</span> <span class="o">=</span> <span class="bp">True</span>
-                    <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">arg</span>
-            <span class="k">if</span> <span class="n">hit</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-            <span class="k">return</span> <span class="bp">self</span>
-
-        <span class="k">if</span> <span class="n">_aresame</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">old</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">new</span>
-
-        <span class="n">rv</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_subs</span><span class="p">(</span><span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">rv</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="n">fallback</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">rv</span>
-
-    <span class="k">def</span> <span class="nf">_eval_subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Override this stub if you want to do anything more than</span>
-<span class="sd">        attempt a replacement of old with new in the arguments of self.</span>
-
-<span class="sd">        See also: _subs</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-<div class="viewcode-block" id="Basic.xreplace"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.xreplace">[docs]</a>    <span class="k">def</span> <span class="nf">xreplace</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rule</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Replace occurrences of objects within the expression.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-<span class="sd">        rule : dict-like</span>
-<span class="sd">            Expresses a replacement rule</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-<span class="sd">        xreplace : the result of the replacement</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols, pi, exp</span>
-<span class="sd">        &gt;&gt;&gt; x, y, z = symbols(&#39;x y z&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; (1 + x*y).xreplace({x: pi})</span>
-<span class="sd">        pi*y + 1</span>
-<span class="sd">        &gt;&gt;&gt; (1 + x*y).xreplace({x:pi, y:2})</span>
-<span class="sd">        1 + 2*pi</span>
-
-<span class="sd">        Replacements occur only if an entire node in the expression tree is</span>
-<span class="sd">        matched:</span>
-
-<span class="sd">        &gt;&gt;&gt; (x*y + z).xreplace({x*y: pi})</span>
-<span class="sd">        z + pi</span>
-<span class="sd">        &gt;&gt;&gt; (x*y*z).xreplace({x*y: pi})</span>
-<span class="sd">        x*y*z</span>
-<span class="sd">        &gt;&gt;&gt; (2*x).xreplace({2*x: y, x: z})</span>
-<span class="sd">        y</span>
-<span class="sd">        &gt;&gt;&gt; (2*2*x).xreplace({2*x: y, x: z})</span>
-<span class="sd">        4*z</span>
-<span class="sd">        &gt;&gt;&gt; (x + y + 2).xreplace({x + y: 2})</span>
-<span class="sd">        x + y + 2</span>
-<span class="sd">        &gt;&gt;&gt; (x + 2 + exp(x + 2)).xreplace({x + 2: y})</span>
-<span class="sd">        x + exp(y) + 2</span>
-
-<span class="sd">        xreplace doesn&#39;t differentiate between free and bound symbols. In the</span>
-<span class="sd">        following, subs(x, y) would not change x since it is a bound symbol,</span>
-<span class="sd">        but xreplace does:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Integral</span>
-<span class="sd">        &gt;&gt;&gt; Integral(x, (x, 1, 2*x)).xreplace({x: y})</span>
-<span class="sd">        Integral(y, (y, 1, 2*y))</span>
-
-<span class="sd">        Trying to replace x with an expression raises an error:</span>
-
-<span class="sd">        &gt;&gt;&gt; Integral(x, (x, 1, 2*x)).xreplace({x: 2*y}) #doctest: +SKIP</span>
-<span class="sd">        ValueError: Invalid limits given: ((2*y, 1, 4*y),)</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        replace: replacement capable of doing wildcard-like matching,</span>
-<span class="sd">                 parsing of match, and conditional replacements</span>
-<span class="sd">        subs: substitution of subexpressions as defined by the objects</span>
-<span class="sd">              themselves.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span> <span class="ow">in</span> <span class="n">rule</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">rule</span><span class="p">[</span><span class="bp">self</span><span class="p">]</span>
-        <span class="k">elif</span> <span class="n">rule</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">([</span><span class="n">arg</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="n">rule</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">_aresame</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span>
-</div>
-    <span class="nd">@deprecated</span><span class="p">(</span><span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;has&quot;</span><span class="p">,</span> <span class="n">issue</span><span class="o">=</span><span class="mi">2389</span><span class="p">,</span> <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.2&quot;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__contains__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">obj</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span> <span class="o">==</span> <span class="n">obj</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">obj</span> <span class="ow">in</span> <span class="n">arg</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">obj</span> <span class="o">==</span> <span class="n">arg</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@cacheit</span>
-<div class="viewcode-block" id="Basic.has"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.has">[docs]</a>    <span class="k">def</span> <span class="nf">has</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">patterns</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Test whether any subexpression matches any of the patterns.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sin, S</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">        &gt;&gt;&gt; (x**2 + sin(x*y)).has(z)</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; (x**2 + sin(x*y)).has(x, y, z)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; x.has(x)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        Note that ``expr.has(*patterns)`` is exactly equivalent to</span>
-<span class="sd">        ``any(expr.has(p) for p in patterns)``. In particular, ``False`` is</span>
-<span class="sd">        returned when the list of patterns is empty.</span>
-
-<span class="sd">        &gt;&gt;&gt; x.has()</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">any</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_has</span><span class="p">(</span><span class="n">pattern</span><span class="p">)</span> <span class="k">for</span> <span class="n">pattern</span> <span class="ow">in</span> <span class="n">patterns</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_has</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">pattern</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Helper for .has()&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">UndefinedFunction</span><span class="p">,</span> <span class="n">Function</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">pattern</span><span class="p">,</span> <span class="n">UndefinedFunction</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">any</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">pattern</span> <span class="ow">or</span> <span class="n">f</span> <span class="o">==</span> <span class="n">pattern</span>
-            <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Function</span><span class="p">,</span> <span class="n">UndefinedFunction</span><span class="p">))</span>
-
-        <span class="n">pattern</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">pattern</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">pattern</span><span class="p">,</span> <span class="n">BasicType</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">any</span><span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">pattern</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">preorder_traversal</span><span class="p">(</span><span class="bp">self</span><span class="p">))</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">match</span> <span class="o">=</span> <span class="n">pattern</span><span class="o">.</span><span class="n">_has_matcher</span><span class="p">()</span>
-            <span class="k">return</span> <span class="nb">any</span><span class="p">(</span><span class="n">match</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">preorder_traversal</span><span class="p">(</span><span class="bp">self</span><span class="p">))</span>
-        <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">any</span><span class="p">(</span><span class="n">arg</span> <span class="o">==</span> <span class="n">pattern</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">preorder_traversal</span><span class="p">(</span><span class="bp">self</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_has_matcher</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Helper for .has()&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__eq__</span>
-
-<div class="viewcode-block" id="Basic.replace"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.replace">[docs]</a>    <span class="k">def</span> <span class="nf">replace</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">query</span><span class="p">,</span> <span class="n">value</span><span class="p">,</span> <span class="nb">map</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Replace matching subexpressions of ``self`` with ``value``.</span>
-
-<span class="sd">        If ``map = True`` then also return the mapping {old: new} where ``old``</span>
-<span class="sd">        was a sub-expression found with query and ``new`` is the replacement</span>
-<span class="sd">        value for it.</span>
-
-<span class="sd">        Traverses an expression tree and performs replacement of matching</span>
-<span class="sd">        subexpressions from the bottom to the top of the tree. The list of</span>
-<span class="sd">        possible combinations of queries and replacement values is listed</span>
-<span class="sd">        below:</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        Initial setup</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy import log, sin, cos, tan, Wild</span>
-<span class="sd">            &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">            &gt;&gt;&gt; f = log(sin(x)) + tan(sin(x**2))</span>
-
-<span class="sd">        1.1. type -&gt; type</span>
-<span class="sd">            obj.replace(sin, tan)</span>
-
-<span class="sd">            &gt;&gt;&gt; f.replace(sin, cos)</span>
-<span class="sd">            log(cos(x)) + tan(cos(x**2))</span>
-<span class="sd">            &gt;&gt;&gt; sin(x).replace(sin, cos, map=True)</span>
-<span class="sd">            (cos(x), {sin(x): cos(x)})</span>
-
-<span class="sd">        1.2. type -&gt; func</span>
-<span class="sd">            obj.replace(sin, lambda arg: ...)</span>
-
-<span class="sd">            &gt;&gt;&gt; f.replace(sin, lambda arg: sin(2*arg))</span>
-<span class="sd">            log(sin(2*x)) + tan(sin(2*x**2))</span>
-
-<span class="sd">        2.1. expr -&gt; expr</span>
-<span class="sd">            obj.replace(sin(a), tan(a))</span>
-
-<span class="sd">            &gt;&gt;&gt; a = Wild(&#39;a&#39;)</span>
-<span class="sd">            &gt;&gt;&gt; f.replace(sin(a), tan(a))</span>
-<span class="sd">            log(tan(x)) + tan(tan(x**2))</span>
-
-<span class="sd">        2.2. expr -&gt; func</span>
-<span class="sd">            obj.replace(sin(a), lambda a: ...)</span>
-
-<span class="sd">            &gt;&gt;&gt; f.replace(sin(a), cos(a))</span>
-<span class="sd">            log(cos(x)) + tan(cos(x**2))</span>
-<span class="sd">            &gt;&gt;&gt; f.replace(sin(a), lambda a: sin(2*a))</span>
-<span class="sd">            log(sin(2*x)) + tan(sin(2*x**2))</span>
-
-<span class="sd">        3.1. func -&gt; func</span>
-<span class="sd">            obj.replace(lambda expr: ..., lambda expr: ...)</span>
-
-<span class="sd">            &gt;&gt;&gt; g = 2*sin(x**3)</span>
-<span class="sd">            &gt;&gt;&gt; g.replace(lambda expr: expr.is_Number, lambda expr: expr**2)</span>
-<span class="sd">            4*sin(x**9)</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        subs: substitution of subexpressions as defined by the objects</span>
-<span class="sd">              themselves.</span>
-<span class="sd">        xreplace: exact node replacement in expr tree; also capable of</span>
-<span class="sd">                  using matching rules</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">query</span><span class="p">,</span> <span class="nb">type</span><span class="p">):</span>
-            <span class="n">_query</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">expr</span><span class="p">:</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">query</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="nb">type</span><span class="p">):</span>
-                <span class="n">_value</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">expr</span><span class="p">,</span> <span class="n">result</span><span class="p">:</span> <span class="n">value</span><span class="p">(</span><span class="o">*</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">collections</span><span class="o">.</span><span class="n">Callable</span><span class="p">):</span>
-                <span class="n">_value</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">expr</span><span class="p">,</span> <span class="n">result</span><span class="p">:</span> <span class="n">value</span><span class="p">(</span><span class="o">*</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span>
-                    <span class="s">&quot;given a type, replace() expects another &quot;</span>
-                    <span class="s">&quot;type or a callable&quot;</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">query</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-            <span class="n">_query</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">expr</span><span class="p">:</span> <span class="n">expr</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">query</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-                <span class="n">_value</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">expr</span><span class="p">,</span> <span class="n">result</span><span class="p">:</span> <span class="n">value</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">collections</span><span class="o">.</span><span class="n">Callable</span><span class="p">):</span>
-                <span class="n">_value</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">expr</span><span class="p">,</span> <span class="n">result</span><span class="p">:</span> <span class="n">value</span><span class="p">(</span><span class="o">**</span><span class="nb">dict</span><span class="p">([</span> <span class="p">(</span>
-                    <span class="nb">str</span><span class="p">(</span><span class="n">key</span><span class="p">)[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">val</span><span class="p">)</span> <span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="n">val</span> <span class="ow">in</span> <span class="n">result</span><span class="o">.</span><span class="n">items</span><span class="p">()</span> <span class="p">]))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span>
-                    <span class="s">&quot;given an expression, replace() expects &quot;</span>
-                    <span class="s">&quot;another expression or a callable&quot;</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">query</span><span class="p">,</span> <span class="n">collections</span><span class="o">.</span><span class="n">Callable</span><span class="p">):</span>
-            <span class="n">_query</span> <span class="o">=</span> <span class="n">query</span>
-
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">collections</span><span class="o">.</span><span class="n">Callable</span><span class="p">):</span>
-                <span class="n">_value</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">expr</span><span class="p">,</span> <span class="n">result</span><span class="p">:</span> <span class="n">value</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span>
-                    <span class="s">&quot;given a callable, replace() expects &quot;</span>
-                    <span class="s">&quot;another callable&quot;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span>
-                <span class="s">&quot;first argument to replace() must be a &quot;</span>
-                <span class="s">&quot;type, an expression or a callable&quot;</span><span class="p">)</span>
-
-        <span class="n">mapping</span> <span class="o">=</span> <span class="p">{}</span>
-
-        <span class="k">def</span> <span class="nf">rec_replace</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-            <span class="n">args</span><span class="p">,</span> <span class="n">construct</span> <span class="o">=</span> <span class="p">[],</span> <span class="bp">False</span>
-
-            <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">rec_replace</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">construct</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">result</span> <span class="o">=</span> <span class="n">arg</span>
-
-                <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">construct</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">_query</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">result</span><span class="p">:</span>
-                    <span class="n">value</span> <span class="o">=</span> <span class="n">_value</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">result</span><span class="p">)</span>
-
-                    <span class="k">if</span> <span class="nb">map</span><span class="p">:</span>
-                        <span class="n">mapping</span><span class="p">[</span><span class="n">expr</span><span class="p">]</span> <span class="o">=</span> <span class="n">value</span>
-
-                    <span class="k">return</span> <span class="n">value</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="n">result</span> <span class="o">=</span> <span class="n">rec_replace</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="bp">self</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">map</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">result</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">result</span><span class="p">,</span> <span class="n">mapping</span>
-</div>
-<div class="viewcode-block" id="Basic.find"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.find">[docs]</a>    <span class="k">def</span> <span class="nf">find</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">query</span><span class="p">,</span> <span class="n">group</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Find all subexpressions matching a query. &quot;&quot;&quot;</span>
-        <span class="n">query</span> <span class="o">=</span> <span class="n">_make_find_query</span><span class="p">(</span><span class="n">query</span><span class="p">)</span>
-        <span class="n">results</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">filter</span><span class="p">(</span><span class="n">query</span><span class="p">,</span> <span class="n">preorder_traversal</span><span class="p">(</span><span class="bp">self</span><span class="p">)))</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">group</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">set</span><span class="p">(</span><span class="n">results</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">groups</span> <span class="o">=</span> <span class="p">{}</span>
-
-            <span class="k">for</span> <span class="n">result</span> <span class="ow">in</span> <span class="n">results</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">result</span> <span class="ow">in</span> <span class="n">groups</span><span class="p">:</span>
-                    <span class="n">groups</span><span class="p">[</span><span class="n">result</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">groups</span><span class="p">[</span><span class="n">result</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-
-            <span class="k">return</span> <span class="n">groups</span>
-</div>
-<div class="viewcode-block" id="Basic.count"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.count">[docs]</a>    <span class="k">def</span> <span class="nf">count</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">query</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Count the number of matching subexpressions. &quot;&quot;&quot;</span>
-        <span class="n">query</span> <span class="o">=</span> <span class="n">_make_find_query</span><span class="p">(</span><span class="n">query</span><span class="p">)</span>
-        <span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">bool</span><span class="p">(</span><span class="n">query</span><span class="p">(</span><span class="n">sub</span><span class="p">))</span> <span class="k">for</span> <span class="n">sub</span> <span class="ow">in</span> <span class="n">preorder_traversal</span><span class="p">(</span><span class="bp">self</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="Basic.matches"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.matches">[docs]</a>    <span class="k">def</span> <span class="nf">matches</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">repl_dict</span><span class="o">=</span><span class="p">{},</span> <span class="n">old</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Helper method for match() that looks for a match between wild symbols</span>
-<span class="sd">        in self and expressions in expr.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols, Wild, Integer, Basic</span>
-<span class="sd">        &gt;&gt;&gt; a, b, c = symbols(&#39;a b c&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; x = Wild(&#39;x&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; Basic(a + x, x).matches(Basic(a + b, c)) is None</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Basic(a + x, x).matches(Basic(a + b + c, b + c))</span>
-<span class="sd">        {x_: b + c}</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="k">if</span> <span class="bp">self</span> <span class="o">==</span> <span class="n">expr</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">repl_dict</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="n">d</span> <span class="o">=</span> <span class="n">repl_dict</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">arg</span><span class="p">,</span> <span class="n">other_arg</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="o">==</span> <span class="n">other_arg</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="n">d</span><span class="p">)</span><span class="o">.</span><span class="n">matches</span><span class="p">(</span><span class="n">other_arg</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">old</span><span class="o">=</span><span class="n">old</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">d</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">return</span> <span class="n">d</span>
-</div>
-<div class="viewcode-block" id="Basic.match"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.match">[docs]</a>    <span class="k">def</span> <span class="nf">match</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">pattern</span><span class="p">,</span> <span class="n">old</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Pattern matching.</span>
-
-<span class="sd">        Wild symbols match all.</span>
-
-<span class="sd">        Return ``None`` when expression (self) does not match</span>
-<span class="sd">        with pattern. Otherwise return a dictionary such that::</span>
-
-<span class="sd">          pattern.xreplace(self.match(pattern)) == self</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols, Wild</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; p = Wild(&quot;p&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; q = Wild(&quot;q&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; r = Wild(&quot;r&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; e = (x+y)**(x+y)</span>
-<span class="sd">        &gt;&gt;&gt; e.match(p**p)</span>
-<span class="sd">        {p_: x + y}</span>
-<span class="sd">        &gt;&gt;&gt; e.match(p**q)</span>
-<span class="sd">        {p_: x + y, q_: x + y}</span>
-<span class="sd">        &gt;&gt;&gt; e = (2*x)**2</span>
-<span class="sd">        &gt;&gt;&gt; e.match(p*q**r)</span>
-<span class="sd">        {p_: 4, q_: x, r_: 2}</span>
-<span class="sd">        &gt;&gt;&gt; (p*q**r).xreplace(e.match(p*q**r))</span>
-<span class="sd">        4*x**2</span>
-
-<span class="sd">        The ``old`` flag will give the old-style pattern matching where</span>
-<span class="sd">        expressions and patterns are essentially solved to give the</span>
-<span class="sd">        match. Both of the following give None unless ``old=True``:</span>
-
-<span class="sd">        &gt;&gt;&gt; (x - 2).match(p - x, old=True)</span>
-<span class="sd">        {p_: 2*x - 2}</span>
-<span class="sd">        &gt;&gt;&gt; (2/x).match(p*x, old=True)</span>
-<span class="sd">        {p_: 2/x**2}</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">signsimp</span><span class="p">,</span> <span class="n">count_ops</span>
-        <span class="n">pattern</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">pattern</span><span class="p">)</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">signsimp</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">signsimp</span><span class="p">(</span><span class="n">pattern</span><span class="p">)</span>
-        <span class="c"># if we still have the same relationship between the types of</span>
-        <span class="c"># input, then use the sign simplified forms</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">pattern</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">p</span><span class="o">.</span><span class="n">func</span><span class="p">):</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">matches</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">old</span><span class="o">=</span><span class="n">old</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="n">pattern</span><span class="o">.</span><span class="n">matches</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">old</span><span class="o">=</span><span class="n">old</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">rv</span>
-</div>
-<div class="viewcode-block" id="Basic.count_ops"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.count_ops">[docs]</a>    <span class="k">def</span> <span class="nf">count_ops</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;wrapper for count_ops that returns the operation count.&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">count_ops</span>
-        <span class="k">return</span> <span class="n">count_ops</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">visual</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Basic.doit"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.doit">[docs]</a>    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Evaluate objects that are not evaluated by default like limits,</span>
-<span class="sd">           integrals, sums and products. All objects of this kind will be</span>
-<span class="sd">           evaluated recursively, unless some species were excluded via &#39;hints&#39;</span>
-<span class="sd">           or unless the &#39;deep&#39; hint was set to &#39;False&#39;.</span>
-
-<span class="sd">           &gt;&gt;&gt; from sympy import Integral</span>
-<span class="sd">           &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">           &gt;&gt;&gt; 2*Integral(x, x)</span>
-<span class="sd">           2*Integral(x, x)</span>
-
-<span class="sd">           &gt;&gt;&gt; (2*Integral(x, x)).doit()</span>
-<span class="sd">           x**2</span>
-
-<span class="sd">           &gt;&gt;&gt; (2*Integral(x, x)).doit(deep = False)</span>
-<span class="sd">           2*Integral(x, x)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;deep&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-            <span class="n">terms</span> <span class="o">=</span> <span class="p">[</span> <span class="n">term</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span> <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="p">]</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">terms</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_rewrite</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">pattern</span><span class="p">,</span> <span class="n">rule</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="n">sargs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="n">terms</span> <span class="o">=</span> <span class="p">[</span> <span class="n">t</span><span class="o">.</span><span class="n">_eval_rewrite</span><span class="p">(</span><span class="n">pattern</span><span class="p">,</span> <span class="n">rule</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">sargs</span> <span class="p">]</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">terms</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Basic.rewrite"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Basic.rewrite">[docs]</a>    <span class="k">def</span> <span class="nf">rewrite</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Rewrite functions in terms of other functions.</span>
-
-<span class="sd">        Rewrites expression containing applications of functions</span>
-<span class="sd">        of one kind in terms of functions of different kind. For</span>
-<span class="sd">        example you can rewrite trigonometric functions as complex</span>
-<span class="sd">        exponentials or combinatorial functions as gamma function.</span>
-
-<span class="sd">        As a pattern this function accepts a list of functions to</span>
-<span class="sd">        to rewrite (instances of DefinedFunction class). As rule</span>
-<span class="sd">        you can use string or a destination function instance (in</span>
-<span class="sd">        this case rewrite() will use the str() function).</span>
-
-<span class="sd">        There is also possibility to pass hints on how to rewrite</span>
-<span class="sd">        the given expressions. For now there is only one such hint</span>
-<span class="sd">        defined called &#39;deep&#39;. When &#39;deep&#39; is set to False it will</span>
-<span class="sd">        forbid functions to rewrite their contents.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sin, exp, I</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        Unspecified pattern:</span>
-<span class="sd">        &gt;&gt;&gt; sin(x).rewrite(exp)</span>
-<span class="sd">        -I*(exp(I*x) - exp(-I*x))/2</span>
-
-<span class="sd">        Pattern as a single function:</span>
-<span class="sd">        &gt;&gt;&gt; sin(x).rewrite(sin, exp)</span>
-<span class="sd">        -I*(exp(I*x) - exp(-I*x))/2</span>
-
-<span class="sd">        Pattern as a list of functions:</span>
-<span class="sd">        &gt;&gt;&gt; sin(x).rewrite([sin, ], exp)</span>
-<span class="sd">        -I*(exp(I*x) - exp(-I*x))/2</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Atom</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">pattern</span> <span class="o">=</span> <span class="n">args</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="nb">str</span><span class="p">):</span>
-                <span class="n">rule</span> <span class="o">=</span> <span class="s">&#39;_eval_rewrite_as_&#39;</span> <span class="o">+</span> <span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">rule</span> <span class="o">=</span> <span class="s">&#39;_eval_rewrite_as_&#39;</span> <span class="o">+</span> <span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">__name__</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">pattern</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_rewrite</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="n">rule</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">iterable</span><span class="p">(</span><span class="n">pattern</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
-                    <span class="n">pattern</span> <span class="o">=</span> <span class="n">pattern</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-                <span class="n">pattern</span> <span class="o">=</span> <span class="p">[</span> <span class="n">p</span><span class="o">.</span><span class="n">__class__</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">pattern</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="p">]</span>
-
-                <span class="k">if</span> <span class="n">pattern</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_rewrite</span><span class="p">(</span><span class="nb">tuple</span><span class="p">(</span><span class="n">pattern</span><span class="p">),</span> <span class="n">rule</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">self</span>
-
-</div></div>
-<div class="viewcode-block" id="Atom"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.basic.Atom">[docs]</a><span class="k">class</span> <span class="nc">Atom</span><span class="p">(</span><span class="n">Basic</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A parent class for atomic things. An atom is an expression with no subexpressions.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Symbol, Number, Rational, Integer, ...</span>
-<span class="sd">    But not: Add, Mul, Pow, ...</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_Atom</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">matches</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">repl_dict</span><span class="o">=</span><span class="p">{},</span> <span class="n">old</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span> <span class="o">==</span> <span class="n">expr</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">repl_dict</span>
-
-    <span class="k">def</span> <span class="nf">xreplace</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rule</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">rule</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">class_key</span><span class="p">(</span><span class="n">cls</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">cls</span><span class="o">.</span><span class="n">__name__</span>
-
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">sort_key</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">class_key</span><span class="p">(),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="p">),)),</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="o">.</span><span class="n">sort_key</span><span class="p">(),</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-    <span class="k">def</span> <span class="nf">_eval_simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ratio</span><span class="p">,</span> <span class="n">measure</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_sorted_args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># this is here as a safeguard against accidentally using _sorted_args</span>
-        <span class="c"># on Atoms -- they cannot be rebuilt as atom.func(*atom._sorted_args)</span>
-        <span class="c"># since there are no args. So the calling routine should be checking</span>
-        <span class="c"># to see that this property is not called for Atoms.</span>
-        <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s">&#39;Atoms have no args. It might be necessary&#39;</span>
-        <span class="s">&#39; to make a check for Atoms in the calling code.&#39;</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_aresame</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return True if a and b are structurally the same, else False.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    To SymPy, 2.0 == 2:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import S, Symbol, cos, sin</span>
-<span class="sd">    &gt;&gt;&gt; 2.0 == S(2)</span>
-<span class="sd">    True</span>
-
-<span class="sd">    Since a simple &#39;same or not&#39; result is sometimes useful, this routine was</span>
-<span class="sd">    written to provide that query:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.core.basic import _aresame</span>
-<span class="sd">    &gt;&gt;&gt; _aresame(S(2.0), S(2))</span>
-<span class="sd">    False</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">preorder_traversal</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">preorder_traversal</span><span class="p">(</span><span class="n">b</span><span class="p">)):</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="n">j</span> <span class="ow">or</span> <span class="nb">type</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">type</span><span class="p">(</span><span class="n">j</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-
-<span class="k">def</span> <span class="nf">_atomic</span><span class="p">(</span><span class="n">e</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return atom-like quantities as far as substitution is</span>
-<span class="sd">    concerned: Derivatives, Functions and Symbols. Don&#39;t</span>
-<span class="sd">    return any &#39;atoms&#39; that are inside such quantities unless</span>
-<span class="sd">    they also appear outside, too.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Derivative, Function, cos</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.core.basic import _atomic</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; _atomic(x + y)</span>
-<span class="sd">    set([x, y])</span>
-<span class="sd">    &gt;&gt;&gt; _atomic(x + f(y))</span>
-<span class="sd">    set([x, f(y)])</span>
-<span class="sd">    &gt;&gt;&gt; _atomic(Derivative(f(x), x) + cos(x) + y)</span>
-<span class="sd">    set([y, cos(x), Derivative(f(x), x)])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Derivative</span><span class="p">,</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Symbol</span>
-    <span class="n">pot</span> <span class="o">=</span> <span class="n">preorder_traversal</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-    <span class="n">seen</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">free</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">free_symbols</span>
-    <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">set</span><span class="p">([</span><span class="n">e</span><span class="p">])</span>
-    <span class="n">atoms</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">pot</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">seen</span><span class="p">:</span>
-            <span class="n">pot</span><span class="o">.</span><span class="n">skip</span><span class="p">()</span>
-            <span class="k">continue</span>
-        <span class="n">seen</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">)</span> <span class="ow">and</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">free</span><span class="p">:</span>
-            <span class="n">atoms</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="p">(</span><span class="n">Derivative</span><span class="p">,</span> <span class="n">Function</span><span class="p">)):</span>
-            <span class="n">pot</span><span class="o">.</span><span class="n">skip</span><span class="p">()</span>
-            <span class="n">atoms</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">atoms</span>
-
-
-<span class="k">class</span> <span class="nc">preorder_traversal</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Do a pre-order traversal of a tree.</span>
-
-<span class="sd">    This iterator recursively yields nodes that it has visited in a pre-order</span>
-<span class="sd">    fashion. That is, it yields the current node then descends through the</span>
-<span class="sd">    tree breadth-first to yield all of a node&#39;s children&#39;s pre-order</span>
-<span class="sd">    traversal.</span>
-
-
-<span class="sd">    For an expression, the order of the traversal depends on the order of</span>
-<span class="sd">    .args, which in many cases can be arbitrary.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-<span class="sd">    node : sympy expression</span>
-<span class="sd">        The expression to traverse.</span>
-<span class="sd">    keys : (default None) sort key(s)</span>
-<span class="sd">        The key(s) used to sort args of Basic objects. When None, args of Basic</span>
-<span class="sd">        objects are processed in arbitrary order. If key is defined, it will</span>
-<span class="sd">        be passed along to ordered() as the only key(s) to use to sort the</span>
-<span class="sd">        arguments; if ``key`` is simply True then the default keys of ordered</span>
-<span class="sd">        will be used.</span>
-
-<span class="sd">    Yields</span>
-<span class="sd">    ======</span>
-<span class="sd">    subtree : sympy expression</span>
-<span class="sd">        All of the subtrees in the tree.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, default_sort_key</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.core.basic import preorder_traversal</span>
-<span class="sd">    &gt;&gt;&gt; x, y, z = symbols(&#39;x y z&#39;)</span>
-
-<span class="sd">    The nodes are returned in the order that they are encountered unless key</span>
-<span class="sd">    is given; simply passing key=True will guarantee that the traversal is</span>
-<span class="sd">    unique.</span>
-
-<span class="sd">    &gt;&gt;&gt; list(preorder_traversal((x + y)*z, keys=None)) # doctest: +SKIP</span>
-<span class="sd">    [z*(x + y), z, x + y, y, x]</span>
-<span class="sd">    &gt;&gt;&gt; list(preorder_traversal((x + y)*z, keys=True))</span>
-<span class="sd">    [z*(x + y), z, x + y, x, y]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">node</span><span class="p">,</span> <span class="n">keys</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_skip_flag</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_pt</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_preorder_traversal</span><span class="p">(</span><span class="n">node</span><span class="p">,</span> <span class="n">keys</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_preorder_traversal</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">node</span><span class="p">,</span> <span class="n">keys</span><span class="p">):</span>
-        <span class="k">yield</span> <span class="n">node</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_skip_flag</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_skip_flag</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="k">return</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">node</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="n">node</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">if</span> <span class="n">keys</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">keys</span> <span class="o">!=</span> <span class="bp">True</span><span class="p">:</span>
-                    <span class="n">args</span> <span class="o">=</span> <span class="n">ordered</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="n">keys</span><span class="p">,</span> <span class="n">default</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">args</span> <span class="o">=</span> <span class="n">ordered</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">subtree</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_preorder_traversal</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">keys</span><span class="p">):</span>
-                    <span class="k">yield</span> <span class="n">subtree</span>
-        <span class="k">elif</span> <span class="n">iterable</span><span class="p">(</span><span class="n">node</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">item</span> <span class="ow">in</span> <span class="n">node</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">subtree</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_preorder_traversal</span><span class="p">(</span><span class="n">item</span><span class="p">,</span> <span class="n">keys</span><span class="p">):</span>
-                    <span class="k">yield</span> <span class="n">subtree</span>
-
-    <span class="k">def</span> <span class="nf">skip</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Skip yielding current node&#39;s (last yielded node&#39;s) subtrees.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        --------</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.core import symbols</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.core.basic import preorder_traversal</span>
-<span class="sd">        &gt;&gt;&gt; x, y, z = symbols(&#39;x y z&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; pt = preorder_traversal((x+y*z)*z)</span>
-<span class="sd">        &gt;&gt;&gt; for i in pt:</span>
-<span class="sd">        ...     print(i)</span>
-<span class="sd">        ...     if i == x+y*z:</span>
-<span class="sd">        ...             pt.skip()</span>
-<span class="sd">        z*(x + y*z)</span>
-<span class="sd">        z</span>
-<span class="sd">        x + y*z</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_skip_flag</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__next__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">next</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_pt</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-
-<span class="k">def</span> <span class="nf">_make_find_query</span><span class="p">(</span><span class="n">query</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Convert the argument of Basic.find() into a callable&quot;&quot;&quot;</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">query</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">query</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">SympifyError</span><span class="p">:</span>
-        <span class="k">pass</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">query</span><span class="p">,</span> <span class="nb">type</span><span class="p">):</span>
-        <span class="k">return</span> <span class="k">lambda</span> <span class="n">expr</span><span class="p">:</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">query</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">query</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-        <span class="k">return</span> <span class="k">lambda</span> <span class="n">expr</span><span class="p">:</span> <span class="n">expr</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">query</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span>
-    <span class="k">return</span> <span class="n">query</span>
-</pre></div>
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-            
-  <h1>Source code for sympy.core.cache</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot; Caching facility for SymPy &quot;&quot;&quot;</span>
-
-<span class="c"># TODO: refactor CACHE &amp; friends into class?</span>
-
-<span class="c"># global cache registry:</span>
-<span class="n">CACHE</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c"># [] of</span>
-            <span class="c">#    (item, {} or tuple of {})</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.decorators</span> <span class="kn">import</span> <span class="n">wraps</span>
-
-
-<span class="k">def</span> <span class="nf">print_cache</span><span class="p">():</span>
-    <span class="sd">&quot;&quot;&quot;print cache content&quot;&quot;&quot;</span>
-
-    <span class="k">for</span> <span class="n">item</span><span class="p">,</span> <span class="n">cache</span> <span class="ow">in</span> <span class="n">CACHE</span><span class="p">:</span>
-        <span class="n">item</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">item</span><span class="p">)</span>
-        <span class="n">head</span> <span class="o">=</span> <span class="s">&#39;=&#39;</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="n">item</span><span class="p">)</span>
-
-        <span class="k">print</span><span class="p">(</span><span class="n">head</span><span class="p">)</span>
-        <span class="k">print</span><span class="p">(</span><span class="n">item</span><span class="p">)</span>
-        <span class="k">print</span><span class="p">(</span><span class="n">head</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">cache</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span>
-            <span class="n">cache</span> <span class="o">=</span> <span class="p">(</span><span class="n">cache</span><span class="p">,)</span>
-            <span class="n">shown</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">shown</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">kv</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">cache</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">shown</span><span class="p">:</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\n</span><span class="s">*** </span><span class="si">%i</span><span class="s"> ***</span><span class="se">\n</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">i</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">kv</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&#39;  </span><span class="si">%s</span><span class="s"> :</span><span class="se">\t</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">v</span><span class="p">))</span>
-
-
-<span class="k">def</span> <span class="nf">clear_cache</span><span class="p">():</span>
-    <span class="sd">&quot;&quot;&quot;clear cache content&quot;&quot;&quot;</span>
-    <span class="k">for</span> <span class="n">item</span><span class="p">,</span> <span class="n">cache</span> <span class="ow">in</span> <span class="n">CACHE</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">cache</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span>
-            <span class="n">cache</span> <span class="o">=</span> <span class="p">(</span><span class="n">cache</span><span class="p">,)</span>
-
-        <span class="k">for</span> <span class="n">kv</span> <span class="ow">in</span> <span class="n">cache</span><span class="p">:</span>
-            <span class="n">kv</span><span class="o">.</span><span class="n">clear</span><span class="p">()</span>
-
-<span class="c">########################################</span>
-
-
-<span class="k">def</span> <span class="nf">__cacheit_nocache</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">func</span>
-
-
-<span class="k">def</span> <span class="nf">__cacheit</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;caching decorator.</span>
-
-<span class="sd">       important: the result of cached function must be *immutable*</span>
-
-
-<span class="sd">       Examples</span>
-<span class="sd">       ========</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy.core.cache import cacheit</span>
-<span class="sd">       &gt;&gt;&gt; @cacheit</span>
-<span class="sd">       ... def f(a,b):</span>
-<span class="sd">       ...    return a+b</span>
-
-<span class="sd">       &gt;&gt;&gt; @cacheit</span>
-<span class="sd">       ... def f(a,b):</span>
-<span class="sd">       ...    return [a,b] # &lt;-- WRONG, returns mutable object</span>
-
-<span class="sd">       to force cacheit to check returned results mutability and consistency,</span>
-<span class="sd">       set environment variable SYMPY_USE_CACHE to &#39;debug&#39;</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">func</span><span class="o">.</span><span class="n">_cache_it_cache</span> <span class="o">=</span> <span class="n">func_cache_it_cache</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="n">CACHE</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">func</span><span class="p">,</span> <span class="n">func_cache_it_cache</span><span class="p">))</span>
-
-    <span class="nd">@wraps</span><span class="p">(</span><span class="n">func</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">wrapper</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Assemble the args and kw_args to compute the hash.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="p">[(</span><span class="n">x</span><span class="p">,</span> <span class="nb">type</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">kw_args</span><span class="p">:</span>
-            <span class="n">keys</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">kw_args</span><span class="p">)</span>
-            <span class="n">k</span><span class="o">.</span><span class="n">extend</span><span class="p">([(</span><span class="n">x</span><span class="p">,</span> <span class="n">kw_args</span><span class="p">[</span><span class="n">x</span><span class="p">],</span> <span class="nb">type</span><span class="p">(</span><span class="n">kw_args</span><span class="p">[</span><span class="n">x</span><span class="p">]))</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">keys</span><span class="p">])</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">func_cache_it_cache</span><span class="p">[</span><span class="n">k</span><span class="p">]</span>
-        <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-            <span class="k">pass</span>
-        <span class="n">func_cache_it_cache</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">r</span>
-    <span class="k">return</span> <span class="n">wrapper</span>
-
-
-<span class="k">def</span> <span class="nf">__cacheit_debug</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;cacheit + code to check cache consistency&quot;&quot;&quot;</span>
-    <span class="n">cfunc</span> <span class="o">=</span> <span class="n">__cacheit</span><span class="p">(</span><span class="n">func</span><span class="p">)</span>
-
-    <span class="nd">@wraps</span><span class="p">(</span><span class="n">func</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">wrapper</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">):</span>
-        <span class="c"># always call function itself and compare it with cached version</span>
-        <span class="n">r1</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">)</span>
-        <span class="n">r2</span> <span class="o">=</span> <span class="n">cfunc</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">)</span>
-
-        <span class="c"># try to see if the result is immutable</span>
-        <span class="c">#</span>
-        <span class="c"># this works because:</span>
-        <span class="c">#</span>
-        <span class="c"># hash([1,2,3])         -&gt; raise TypeError</span>
-        <span class="c"># hash({&#39;a&#39;:1, &#39;b&#39;:2})  -&gt; raise TypeError</span>
-        <span class="c"># hash((1,[2,3]))       -&gt; raise TypeError</span>
-        <span class="c">#</span>
-        <span class="c"># hash((1,2,3))         -&gt; just computes the hash</span>
-        <span class="nb">hash</span><span class="p">(</span><span class="n">r1</span><span class="p">),</span> <span class="nb">hash</span><span class="p">(</span><span class="n">r2</span><span class="p">)</span>
-
-        <span class="c"># also see if returned values are the same</span>
-        <span class="k">assert</span> <span class="n">r1</span> <span class="o">==</span> <span class="n">r2</span>
-
-        <span class="k">return</span> <span class="n">r1</span>
-    <span class="k">return</span> <span class="n">wrapper</span>
-
-
-<span class="k">def</span> <span class="nf">_getenv</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">default</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="kn">from</span> <span class="nn">os</span> <span class="kn">import</span> <span class="n">getenv</span>
-    <span class="k">return</span> <span class="n">getenv</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">default</span><span class="p">)</span>
-
-<span class="c"># SYMPY_USE_CACHE=yes/no/debug</span>
-<span class="n">USE_CACHE</span> <span class="o">=</span> <span class="n">_getenv</span><span class="p">(</span><span class="s">&#39;SYMPY_USE_CACHE&#39;</span><span class="p">,</span> <span class="s">&#39;yes&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span>
-
-<span class="k">if</span> <span class="n">USE_CACHE</span> <span class="o">==</span> <span class="s">&#39;no&#39;</span><span class="p">:</span>
-    <span class="n">cacheit</span> <span class="o">=</span> <span class="n">__cacheit_nocache</span>
-<span class="k">elif</span> <span class="n">USE_CACHE</span> <span class="o">==</span> <span class="s">&#39;yes&#39;</span><span class="p">:</span>
-    <span class="n">cacheit</span> <span class="o">=</span> <span class="n">__cacheit</span>
-<span class="k">elif</span> <span class="n">USE_CACHE</span> <span class="o">==</span> <span class="s">&#39;debug&#39;</span><span class="p">:</span>
-    <span class="n">cacheit</span> <span class="o">=</span> <span class="n">__cacheit_debug</span>   <span class="c"># a lot slower</span>
-<span class="k">else</span><span class="p">:</span>
-    <span class="k">raise</span> <span class="ne">RuntimeError</span><span class="p">(</span>
-        <span class="s">&#39;unrecognized value for SYMPY_USE_CACHE: </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">USE_CACHE</span><span class="p">)</span>
-</pre></div>
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-            
-  <h1>Source code for sympy.core.compatibility</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Reimplementations of constructs introduced in later versions of Python than</span>
-<span class="sd">we support. Also some functions that are needed SymPy-wide and are located</span>
-<span class="sd">here for easy import.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</span>
-
-<span class="c"># These are in here because telling if something is an iterable just by calling</span>
-<span class="c"># hasattr(obj, &quot;__iter__&quot;) behaves differently in Python 2 and Python 3.  In</span>
-<span class="c"># particular, hasattr(str, &quot;__iter__&quot;) is False in Python 2 and True in Python 3.</span>
-<span class="c"># I think putting them here also makes it easier to use them in the core.</span>
-
-
-<div class="viewcode-block" id="iterable"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.compatibility.iterable">[docs]</a><span class="k">def</span> <span class="nf">iterable</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">(</span><span class="nb">str</span><span class="p">,</span> <span class="nb">dict</span><span class="p">)):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a boolean indicating whether ``i`` is SymPy iterable.</span>
-
-<span class="sd">    When SymPy is working with iterables, it is almost always assuming</span>
-<span class="sd">    that the iterable is not a string or a mapping, so those are excluded</span>
-<span class="sd">    by default. If you want a pure Python definition, make exclude=None. To</span>
-<span class="sd">    exclude multiple items, pass them as a tuple.</span>
-
-<span class="sd">    See also: is_sequence</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import iterable</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Tuple</span>
-<span class="sd">    &gt;&gt;&gt; things = [[1], (1,), set([1]), Tuple(1), (j for j in [1, 2]), {1:2}, &#39;1&#39;, 1]</span>
-<span class="sd">    &gt;&gt;&gt; for i in things:</span>
-<span class="sd">    ...     print(iterable(i), type(i))</span>
-<span class="sd">    True &lt;... &#39;list&#39;&gt;</span>
-<span class="sd">    True &lt;... &#39;tuple&#39;&gt;</span>
-<span class="sd">    True &lt;... &#39;set&#39;&gt;</span>
-<span class="sd">    True &lt;class &#39;sympy.core.containers.Tuple&#39;&gt;</span>
-<span class="sd">    True &lt;... &#39;generator&#39;&gt;</span>
-<span class="sd">    False &lt;... &#39;dict&#39;&gt;</span>
-<span class="sd">    False &lt;... &#39;str&#39;&gt;</span>
-<span class="sd">    False &lt;... &#39;int&#39;&gt;</span>
-
-<span class="sd">    &gt;&gt;&gt; iterable({}, exclude=None)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; iterable({}, exclude=str)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; iterable(&quot;no&quot;, exclude=str)</span>
-<span class="sd">    False</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="nb">iter</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-    <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">False</span>
-    <span class="k">if</span> <span class="n">exclude</span><span class="p">:</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">exclude</span><span class="p">)</span>
-    <span class="k">return</span> <span class="bp">True</span>
-
-</div>
-<div class="viewcode-block" id="is_sequence"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.compatibility.is_sequence">[docs]</a><span class="k">def</span> <span class="nf">is_sequence</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">include</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a boolean indicating whether ``i`` is a sequence in the SymPy</span>
-<span class="sd">    sense. If anything that fails the test below should be included as</span>
-<span class="sd">    being a sequence for your application, set &#39;include&#39; to that object&#39;s</span>
-<span class="sd">    type; multiple types should be passed as a tuple of types.</span>
-
-<span class="sd">    Note: although generators can generate a sequence, they often need special</span>
-<span class="sd">    handling to make sure their elements are captured before the generator is</span>
-<span class="sd">    exhausted, so these are not included by default in the definition of a</span>
-<span class="sd">    sequence.</span>
-
-<span class="sd">    See also: iterable</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import is_sequence</span>
-<span class="sd">    &gt;&gt;&gt; from types import GeneratorType</span>
-<span class="sd">    &gt;&gt;&gt; is_sequence([])</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; is_sequence(set())</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; is_sequence(&#39;abc&#39;)</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; is_sequence(&#39;abc&#39;, include=str)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; generator = (c for c in &#39;abc&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; is_sequence(generator)</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; is_sequence(generator, include=(str, GeneratorType))</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">(</span><span class="nb">hasattr</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="s">&#39;__getitem__&#39;</span><span class="p">)</span> <span class="ow">and</span>
-            <span class="n">iterable</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="ow">or</span>
-            <span class="nb">bool</span><span class="p">(</span><span class="n">include</span><span class="p">)</span> <span class="ow">and</span>
-            <span class="nb">isinstance</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">include</span><span class="p">))</span>
-</div>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Wrapping some imports in try/except statements to allow the same code to</span>
-<span class="sd">be used in Python 3+ as well.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="k">try</span><span class="p">:</span>
-    <span class="nb">callable</span> <span class="o">=</span> <span class="nb">callable</span>
-<span class="k">except</span> <span class="ne">NameError</span><span class="p">:</span>
-    <span class="kn">import</span> <span class="nn">collections</span>
-
-    <span class="k">def</span> <span class="nf">callable</span><span class="p">(</span><span class="n">obj</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="n">collections</span><span class="o">.</span><span class="n">Callable</span><span class="p">)</span>
-
-<span class="k">try</span><span class="p">:</span>
-    <span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-<span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>
-    <span class="nb">reduce</span> <span class="o">=</span> <span class="nb">reduce</span>
-
-
-<span class="k">def</span> <span class="nf">cmp_to_key</span><span class="p">(</span><span class="n">mycmp</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Convert a cmp= function into a key= function</span>
-
-<span class="sd">    This code is included in Python 2.7 and 3.2 in functools.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">class</span> <span class="nc">K</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-        <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">obj</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">obj</span> <span class="o">=</span> <span class="n">obj</span>
-
-        <span class="k">def</span> <span class="nf">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">mycmp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">obj</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">obj</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">0</span>
-
-        <span class="k">def</span> <span class="nf">__gt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">mycmp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">obj</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">obj</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span>
-
-        <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">mycmp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">obj</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">obj</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span>
-
-        <span class="k">def</span> <span class="nf">__le__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">mycmp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">obj</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">obj</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">0</span>
-
-        <span class="k">def</span> <span class="nf">__ge__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">mycmp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">obj</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">obj</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mi">0</span>
-
-        <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">mycmp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">obj</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">obj</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span>
-    <span class="k">return</span> <span class="n">K</span>
-
-<span class="k">try</span><span class="p">:</span>
-    <span class="kn">import</span> <span class="nn">builtins</span>
-    <span class="nb">cmp</span> <span class="o">=</span> <span class="n">builtins</span><span class="o">.</span><span class="n">cmp</span>
-<span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-    <span class="k">def</span> <span class="nf">cmp</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">a</span> <span class="o">&gt;</span> <span class="n">b</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="n">a</span> <span class="o">&lt;</span> <span class="n">b</span><span class="p">)</span>
-
-<span class="k">try</span><span class="p">:</span>
-    <span class="kn">from</span> <span class="nn">itertools</span> <span class="kn">import</span> <span class="n">product</span>
-<span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>  <span class="c"># Python 2.5</span>
-    <span class="k">def</span> <span class="nf">product</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Cartesian product of input iterables.</span>
-
-<span class="sd">        Equivalent to nested for-loops in a generator expression. For example,</span>
-<span class="sd">        cartes(A, B) returns the same as ((x,y) for x in A for y in B).</span>
-
-<span class="sd">        The nested loops cycle like an odometer with the rightmost element</span>
-<span class="sd">        advancing on every iteration. This pattern creates a lexicographic</span>
-<span class="sd">        ordering so that if the input&#39;s iterables are sorted, the product</span>
-<span class="sd">        tuples are emitted in sorted order.</span>
-
-<span class="sd">        To compute the product of an iterable with itself, specify the number</span>
-<span class="sd">        of repetitions with the optional repeat keyword argument. For example,</span>
-<span class="sd">        product(A, repeat=4) means the same as product(A, A, A, A).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.utilities.iterables import cartes</span>
-<span class="sd">        &gt;&gt;&gt; [&#39;&#39;.join(p) for p in list(cartes(&#39;ABC&#39;, &#39;xy&#39;))]</span>
-<span class="sd">        [&#39;Ax&#39;, &#39;Ay&#39;, &#39;Bx&#39;, &#39;By&#39;, &#39;Cx&#39;, &#39;Cy&#39;]</span>
-<span class="sd">        &gt;&gt;&gt; list(cartes(list(range(2)), repeat=2))</span>
-<span class="sd">        [(0, 0), (0, 1), (1, 0), (1, 1)]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        variations</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">pools</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="n">args</span><span class="p">))</span> <span class="o">*</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;repeat&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[[]]</span>
-        <span class="k">for</span> <span class="n">pool</span> <span class="ow">in</span> <span class="n">pools</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="o">+</span> <span class="p">[</span><span class="n">y</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">result</span> <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">pool</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">prod</span> <span class="ow">in</span> <span class="n">result</span><span class="p">:</span>
-            <span class="k">yield</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">prod</span><span class="p">)</span>
-
-<span class="k">try</span><span class="p">:</span>
-    <span class="kn">from</span> <span class="nn">itertools</span> <span class="kn">import</span> <span class="n">permutations</span>
-<span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>  <span class="c"># Python 2.5</span>
-    <span class="k">def</span> <span class="nf">permutations</span><span class="p">(</span><span class="n">iterable</span><span class="p">,</span> <span class="n">r</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return successive r length permutations of elements in the iterable.</span>
-
-<span class="sd">        If r is not specified or is None, then r defaults to the length of</span>
-<span class="sd">        the iterable and all possible full-length permutations are generated.</span>
-
-<span class="sd">        Permutations are emitted in lexicographic sort order. So, if the input</span>
-<span class="sd">        iterable is sorted, the permutation tuples will be produced in sorted</span>
-<span class="sd">        order.</span>
-
-<span class="sd">        Elements are treated as unique based on their position, not on their</span>
-<span class="sd">        value. So if the input elements are unique, there will be no repeat</span>
-<span class="sd">        values in each permutation.</span>
-
-<span class="sd">        Examples;</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.core.compatibility import permutations</span>
-<span class="sd">        &gt;&gt;&gt; [&#39;&#39;.join(p) for p in list(permutations(&#39;ABC&#39;, 2))]</span>
-<span class="sd">        [&#39;AB&#39;, &#39;AC&#39;, &#39;BA&#39;, &#39;BC&#39;, &#39;CA&#39;, &#39;CB&#39;]</span>
-<span class="sd">        &gt;&gt;&gt; list(permutations(list(range(3))))</span>
-<span class="sd">        [(0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0)]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">pool</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">iterable</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">pool</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">n</span> <span class="k">if</span> <span class="n">r</span> <span class="ow">is</span> <span class="bp">None</span> <span class="k">else</span> <span class="n">r</span>
-        <span class="k">if</span> <span class="n">r</span> <span class="o">&gt;</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="n">indices</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-        <span class="n">cycles</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">r</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">))</span>
-        <span class="k">yield</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">pool</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">indices</span><span class="p">[:</span><span class="n">r</span><span class="p">])</span>
-        <span class="k">while</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">r</span><span class="p">))):</span>
-                <span class="n">cycles</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">1</span>
-                <span class="k">if</span> <span class="n">cycles</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">indices</span><span class="p">[</span><span class="n">i</span><span class="p">:]</span> <span class="o">=</span> <span class="n">indices</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span> <span class="o">+</span> <span class="n">indices</span><span class="p">[</span><span class="n">i</span><span class="p">:</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
-                    <span class="n">cycles</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="n">i</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">j</span> <span class="o">=</span> <span class="n">cycles</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-                    <span class="n">indices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">indices</span><span class="p">[</span><span class="o">-</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">indices</span><span class="p">[</span><span class="o">-</span><span class="n">j</span><span class="p">],</span> <span class="n">indices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-                    <span class="k">yield</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">pool</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">indices</span><span class="p">[:</span><span class="n">r</span><span class="p">])</span>
-                    <span class="k">break</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span>
-
-<span class="k">try</span><span class="p">:</span>
-    <span class="kn">from</span> <span class="nn">itertools</span> <span class="kn">import</span> <span class="n">combinations</span><span class="p">,</span> <span class="n">combinations_with_replacement</span>
-<span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>  <span class="c"># &lt; python 2.6</span>
-    <span class="k">def</span> <span class="nf">combinations</span><span class="p">(</span><span class="n">iterable</span><span class="p">,</span> <span class="n">r</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return r length subsequences of elements from the input iterable.</span>
-
-<span class="sd">        Combinations are emitted in lexicographic sort order. So, if the</span>
-<span class="sd">        input iterable is sorted, the combination tuples will be produced</span>
-<span class="sd">        in sorted order.</span>
-
-<span class="sd">        Elements are treated as unique based on their position, not on their</span>
-<span class="sd">        value. So if the input elements are unique, there will be no repeat</span>
-<span class="sd">        values in each combination.</span>
-
-<span class="sd">        See also: combinations_with_replacement</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.core.compatibility import combinations</span>
-<span class="sd">        &gt;&gt;&gt; list(combinations(&#39;ABC&#39;, 2))</span>
-<span class="sd">        [(&#39;A&#39;, &#39;B&#39;), (&#39;A&#39;, &#39;C&#39;), (&#39;B&#39;, &#39;C&#39;)]</span>
-<span class="sd">        &gt;&gt;&gt; list(combinations(list(range(4)), 3))</span>
-<span class="sd">        [(0, 1, 2), (0, 1, 3), (0, 2, 3), (1, 2, 3)]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">pool</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">iterable</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">pool</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">r</span> <span class="o">&gt;</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="n">indices</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">r</span><span class="p">))</span>
-        <span class="k">yield</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">pool</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">indices</span><span class="p">)</span>
-        <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">r</span><span class="p">))):</span>
-                <span class="k">if</span> <span class="n">indices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">!=</span> <span class="n">i</span> <span class="o">+</span> <span class="n">n</span> <span class="o">-</span> <span class="n">r</span><span class="p">:</span>
-                    <span class="k">break</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span>
-            <span class="n">indices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">r</span><span class="p">):</span>
-                <span class="n">indices</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">indices</span><span class="p">[</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span>
-            <span class="k">yield</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">pool</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">indices</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">combinations_with_replacement</span><span class="p">(</span><span class="n">iterable</span><span class="p">,</span> <span class="n">r</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return r length subsequences of elements from the input iterable</span>
-<span class="sd">        allowing individual elements to be repeated more than once.</span>
-
-<span class="sd">        Combinations are emitted in lexicographic sort order. So, if the</span>
-<span class="sd">        input iterable is sorted, the combination tuples will be produced</span>
-<span class="sd">        in sorted order.</span>
-
-<span class="sd">        Elements are treated as unique based on their position, not on their</span>
-<span class="sd">        value. So if the input elements are unique, the generated combinations</span>
-<span class="sd">        will also be unique.</span>
-
-<span class="sd">        See also: combinations</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.core.compatibility import combinations_with_replacement</span>
-<span class="sd">        &gt;&gt;&gt; list(combinations_with_replacement(&#39;AB&#39;, 2))</span>
-<span class="sd">        [(&#39;A&#39;, &#39;A&#39;), (&#39;A&#39;, &#39;B&#39;), (&#39;B&#39;, &#39;B&#39;)]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">pool</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">iterable</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">pool</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span> <span class="ow">and</span> <span class="n">r</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="n">indices</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">r</span>
-        <span class="k">yield</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">pool</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">indices</span><span class="p">)</span>
-        <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">r</span><span class="p">))):</span>
-                <span class="k">if</span> <span class="n">indices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">!=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">break</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span>
-            <span class="n">indices</span><span class="p">[</span><span class="n">i</span><span class="p">:]</span> <span class="o">=</span> <span class="p">[</span><span class="n">indices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">r</span> <span class="o">-</span> <span class="n">i</span><span class="p">)</span>
-            <span class="k">yield</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">pool</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">indices</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="set_intersection"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.compatibility.set_intersection">[docs]</a><span class="k">def</span> <span class="nf">set_intersection</span><span class="p">(</span><span class="o">*</span><span class="n">sets</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return the intersection of all the given sets.</span>
-
-<span class="sd">    As of Python 2.6 you can write ``set.intersection(*sets)``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.core.compatibility import set_intersection</span>
-<span class="sd">    &gt;&gt;&gt; set_intersection(set([1, 2]), set([2, 3]))</span>
-<span class="sd">    set([2])</span>
-<span class="sd">    &gt;&gt;&gt; set_intersection()</span>
-<span class="sd">    set()</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">sets</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="n">rv</span> <span class="o">=</span> <span class="n">sets</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">sets</span><span class="p">:</span>
-        <span class="n">rv</span> <span class="o">&amp;=</span> <span class="n">s</span>
-    <span class="k">return</span> <span class="n">rv</span>
-
-</div>
-<div class="viewcode-block" id="set_union"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.compatibility.set_union">[docs]</a><span class="k">def</span> <span class="nf">set_union</span><span class="p">(</span><span class="o">*</span><span class="n">sets</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return the union of all the given sets.</span>
-
-<span class="sd">    As of Python 2.6 you can write ``set.union(*sets)``.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.core.compatibility import set_union</span>
-<span class="sd">    &gt;&gt;&gt; set_union(set([1, 2]), set([2, 3]))</span>
-<span class="sd">    set([1, 2, 3])</span>
-<span class="sd">    &gt;&gt;&gt; set_union()</span>
-<span class="sd">    set()</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">rv</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">sets</span><span class="p">:</span>
-        <span class="n">rv</span> <span class="o">|=</span> <span class="n">s</span>
-    <span class="k">return</span> <span class="n">rv</span>
-</div>
-<span class="k">try</span><span class="p">:</span>
-    <span class="nb">bin</span> <span class="o">=</span> <span class="nb">bin</span>
-<span class="k">except</span> <span class="ne">NameError</span><span class="p">:</span>  <span class="c"># Python 2.5</span>
-    <span class="k">def</span> <span class="nf">bin</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        bin(number) -&gt; string</span>
-
-<span class="sd">        Stringifies an int or long in base 2.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&#39;-&#39;</span> <span class="o">+</span> <span class="nb">bin</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">)</span>
-        <span class="n">out</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">out</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;0&#39;</span><span class="p">)</span>
-        <span class="k">while</span> <span class="n">x</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">out</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;01&#39;</span><span class="p">[</span><span class="n">x</span> <span class="o">&amp;</span> <span class="mi">1</span><span class="p">])</span>
-            <span class="n">x</span> <span class="o">&gt;&gt;=</span> <span class="mi">1</span>
-            <span class="k">pass</span>
-        <span class="k">return</span> <span class="s">&#39;0b&#39;</span> <span class="o">+</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">out</span><span class="p">))</span>
-
-<span class="k">try</span><span class="p">:</span>
-    <span class="nb">next</span> <span class="o">=</span> <span class="nb">next</span>
-<span class="k">except</span> <span class="ne">NameError</span><span class="p">:</span>  <span class="c"># Python 2.5</span>
-    <span class="k">def</span> <span class="nf">next</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        next(iterator[, default])</span>
-
-<span class="sd">        Return the next item from the iterator. If default is given and the</span>
-<span class="sd">        iterator is exhausted, it is returned instead of raising StopIteration.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">__next__</span><span class="p">()</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">__next__</span><span class="p">()</span>
-            <span class="k">except</span> <span class="ne">StopIteration</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Expected 1 or 2 arguments, got </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">))</span>
-
-<span class="k">try</span><span class="p">:</span>
-    <span class="kn">from</span> <span class="nn">builtins</span> <span class="kn">import</span> <span class="nb">bin</span>
-<span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>  <span class="c"># Python 2.5</span>
-    <span class="n">_hexDict</span> <span class="o">=</span> <span class="p">{</span>
-        <span class="s">&#39;0&#39;</span><span class="p">:</span> <span class="s">&#39;0000&#39;</span><span class="p">,</span> <span class="s">&#39;1&#39;</span><span class="p">:</span> <span class="s">&#39;0001&#39;</span><span class="p">,</span> <span class="s">&#39;2&#39;</span><span class="p">:</span> <span class="s">&#39;0010&#39;</span><span class="p">,</span> <span class="s">&#39;3&#39;</span><span class="p">:</span> <span class="s">&#39;0011&#39;</span><span class="p">,</span> <span class="s">&#39;4&#39;</span><span class="p">:</span> <span class="s">&#39;0100&#39;</span><span class="p">,</span> <span class="s">&#39;5&#39;</span><span class="p">:</span> <span class="s">&#39;0101&#39;</span><span class="p">,</span>
-        <span class="s">&#39;6&#39;</span><span class="p">:</span> <span class="s">&#39;0110&#39;</span><span class="p">,</span> <span class="s">&#39;7&#39;</span><span class="p">:</span> <span class="s">&#39;0111&#39;</span><span class="p">,</span> <span class="s">&#39;8&#39;</span><span class="p">:</span> <span class="s">&#39;1000&#39;</span><span class="p">,</span> <span class="s">&#39;9&#39;</span><span class="p">:</span> <span class="s">&#39;1001&#39;</span><span class="p">,</span> <span class="s">&#39;a&#39;</span><span class="p">:</span> <span class="s">&#39;1010&#39;</span><span class="p">,</span> <span class="s">&#39;b&#39;</span><span class="p">:</span> <span class="s">&#39;1011&#39;</span><span class="p">,</span>
-        <span class="s">&#39;c&#39;</span><span class="p">:</span> <span class="s">&#39;1100&#39;</span><span class="p">,</span> <span class="s">&#39;d&#39;</span><span class="p">:</span> <span class="s">&#39;1101&#39;</span><span class="p">,</span> <span class="s">&#39;e&#39;</span><span class="p">:</span> <span class="s">&#39;1110&#39;</span><span class="p">,</span> <span class="s">&#39;f&#39;</span><span class="p">:</span> <span class="s">&#39;1111&#39;</span><span class="p">,</span> <span class="s">&#39;L&#39;</span><span class="p">:</span> <span class="s">&#39;&#39;</span><span class="p">}</span>
-
-    <span class="k">def</span> <span class="nf">bin</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the equivalent to Python 2.6&#39;s bin function.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.core.compatibility import bin</span>
-<span class="sd">        &gt;&gt;&gt; bin(-123)</span>
-<span class="sd">        &#39;-0b1111011&#39;</span>
-<span class="sd">        &gt;&gt;&gt; bin(0) # this is the only time a 0 will be to the right of &#39;b&#39;</span>
-<span class="sd">        &#39;0b0&#39;</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        sympy.physics.quantum.shor.arr</span>
-
-<span class="sd">        Modified from http://code.activestate.com/recipes/576847/</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># =========================================================</span>
-        <span class="c"># create hex of int, remove &#39;0x&#39;. now for each hex char,</span>
-        <span class="c"># look up binary string, append in list and join at the end.</span>
-        <span class="c"># =========================================================</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&#39;-</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="nb">bin</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&#39;0b</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="n">_hexDict</span><span class="p">[</span><span class="n">hstr</span><span class="p">]</span> <span class="k">for</span> <span class="n">hstr</span> <span class="ow">in</span> <span class="nb">hex</span><span class="p">(</span><span class="n">n</span><span class="p">)[</span><span class="mi">2</span><span class="p">:]</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span>
-                                  <span class="p">])</span><span class="o">.</span><span class="n">lstrip</span><span class="p">(</span><span class="s">&#39;0&#39;</span><span class="p">)</span> <span class="ow">or</span> <span class="s">&#39;0&#39;</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="as_int"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.compatibility.as_int">[docs]</a><span class="k">def</span> <span class="nf">as_int</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Convert the argument to a builtin integer.</span>
-
-<span class="sd">    The return value is guaranteed to be equal to the input. ValueError is</span>
-<span class="sd">    raised if the input has a non-integral value.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.core.compatibility import as_int</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import sqrt</span>
-<span class="sd">    &gt;&gt;&gt; 3.0</span>
-<span class="sd">    3.0</span>
-<span class="sd">    &gt;&gt;&gt; as_int(3.0) # convert to int and test for equality</span>
-<span class="sd">    3</span>
-<span class="sd">    &gt;&gt;&gt; int(sqrt(10))</span>
-<span class="sd">    3</span>
-<span class="sd">    &gt;&gt;&gt; as_int(sqrt(10))</span>
-<span class="sd">    Traceback (most recent call last):</span>
-<span class="sd">    ...</span>
-<span class="sd">    ValueError: ... is not an integer</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">result</span> <span class="o">!=</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span>
-    <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> is not an integer&#39;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<span class="k">def</span> <span class="nf">default_sort_key</span><span class="p">(</span><span class="n">item</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return a key that can be used for sorting.</span>
-
-<span class="sd">    The key has the structure:</span>
-
-<span class="sd">    (class_key, (len(args), args), exponent.sort_key(), coefficient)</span>
-
-<span class="sd">    This key is supplied by the sort_key routine of Basic objects when</span>
-<span class="sd">    ``item`` is a Basic object or an object (other than a string) that</span>
-<span class="sd">    sympifies to a Basic object. Otherwise, this function produces the</span>
-<span class="sd">    key.</span>
-
-<span class="sd">    The ``order`` argument is passed along to the sort_key routine and is</span>
-<span class="sd">    used to determine how the terms *within* an expression are ordered.</span>
-<span class="sd">    (See examples below) ``order`` options are: &#39;lex&#39;, &#39;grlex&#39;, &#39;grevlex&#39;,</span>
-<span class="sd">    and reversed values of the same (e.g. &#39;rev-lex&#39;). The default order</span>
-<span class="sd">    value is None (which translates to &#39;lex&#39;).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Basic, S, I, default_sort_key</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.core.function import UndefinedFunction</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    The following are eqivalent ways of getting the key for an object:</span>
-
-<span class="sd">    &gt;&gt;&gt; x.sort_key() == default_sort_key(x)</span>
-<span class="sd">    True</span>
-
-<span class="sd">    Here are some examples of the key that is produced:</span>
-
-<span class="sd">    &gt;&gt;&gt; default_sort_key(UndefinedFunction(&#39;f&#39;))</span>
-<span class="sd">    ((0, 0, &#39;UndefinedFunction&#39;), (1, (&#39;f&#39;,)), ((1, 0, &#39;Number&#39;),</span>
-<span class="sd">        (0, ()), (), 1), 1)</span>
-<span class="sd">    &gt;&gt;&gt; default_sort_key(&#39;1&#39;)</span>
-<span class="sd">    ((0, 0, &#39;str&#39;), (1, (&#39;1&#39;,)), ((1, 0, &#39;Number&#39;), (0, ()), (), 1), 1)</span>
-<span class="sd">    &gt;&gt;&gt; default_sort_key(S.One)</span>
-<span class="sd">    ((1, 0, &#39;Number&#39;), (0, ()), (), 1)</span>
-<span class="sd">    &gt;&gt;&gt; default_sort_key(2)</span>
-<span class="sd">    ((1, 0, &#39;Number&#39;), (0, ()), (), 2)</span>
-
-
-<span class="sd">    While sort_key is a method only defined for SymPy objects,</span>
-<span class="sd">    default_sort_key will accept anything as an argument so it is</span>
-<span class="sd">    more robust as a sorting key. For the following, using key=</span>
-<span class="sd">    lambda i: i.sort_key() would fail because 2 doesn&#39;t have a sort_key</span>
-<span class="sd">    method; that&#39;s why default_sort_key is used. Note, that it also</span>
-<span class="sd">    handles sympification of non-string items likes ints:</span>
-
-<span class="sd">    &gt;&gt;&gt; a = [2, I, -I]</span>
-<span class="sd">    &gt;&gt;&gt; sorted(a, key=default_sort_key)</span>
-<span class="sd">    [2, -I, I]</span>
-
-<span class="sd">    The returned key can be used anywhere that a key can be specified for</span>
-<span class="sd">    a function, e.g. sort, min, max, etc...:</span>
-
-<span class="sd">    &gt;&gt;&gt; a.sort(key=default_sort_key); a[0]</span>
-<span class="sd">    2</span>
-<span class="sd">    &gt;&gt;&gt; min(a, key=default_sort_key)</span>
-<span class="sd">    2</span>
-
-<span class="sd">    Note</span>
-<span class="sd">    ----</span>
-
-<span class="sd">    The key returned is useful for getting items into a canonical order</span>
-<span class="sd">    that will be the same across platforms. It is not directly useful for</span>
-<span class="sd">    sorting lists of expressions:</span>
-
-<span class="sd">    &gt;&gt;&gt; a, b = x, 1/x</span>
-
-<span class="sd">    Since ``a`` has only 1 term, its value of sort_key is unaffected by</span>
-<span class="sd">    ``order``:</span>
-
-<span class="sd">    &gt;&gt;&gt; a.sort_key() == a.sort_key(&#39;rev-lex&#39;)</span>
-<span class="sd">    True</span>
-
-<span class="sd">    If ``a`` and ``b`` are combined then the key will differ because there</span>
-<span class="sd">    are terms that can be ordered:</span>
-
-<span class="sd">    &gt;&gt;&gt; eq = a + b</span>
-<span class="sd">    &gt;&gt;&gt; eq.sort_key() == eq.sort_key(&#39;rev-lex&#39;)</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; eq.as_ordered_terms()</span>
-<span class="sd">    [x, 1/x]</span>
-<span class="sd">    &gt;&gt;&gt; eq.as_ordered_terms(&#39;rev-lex&#39;)</span>
-<span class="sd">    [1/x, x]</span>
-
-<span class="sd">    But since the keys for each of these terms are independent of ``order``&#39;s</span>
-<span class="sd">    value, they don&#39;t sort differently when they appear separately in a list:</span>
-
-<span class="sd">    &gt;&gt;&gt; sorted(eq.args, key=default_sort_key)</span>
-<span class="sd">    [1/x, x]</span>
-<span class="sd">    &gt;&gt;&gt; sorted(eq.args, key=lambda i: default_sort_key(i, order=&#39;rev-lex&#39;))</span>
-<span class="sd">    [1/x, x]</span>
-
-<span class="sd">    The order of terms obtained when using these keys is the order that would</span>
-<span class="sd">    be obtained if those terms were *factors* in a product.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.core.expr.as_ordered_factors, sympy.core.expr.as_ordered_terms</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">Basic</span>
-    <span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">SympifyError</span>
-    <span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">iterable</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">item</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">item</span><span class="o">.</span><span class="n">sort_key</span><span class="p">(</span><span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">iterable</span><span class="p">(</span><span class="n">item</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="nb">str</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">item</span><span class="p">,</span> <span class="nb">dict</span><span class="p">):</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">item</span><span class="o">.</span><span class="n">items</span><span class="p">())</span>
-            <span class="n">unordered</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">item</span><span class="p">,</span> <span class="nb">set</span><span class="p">):</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="n">item</span>
-            <span class="n">unordered</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># e.g. tuple, list</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">item</span><span class="p">)</span>
-            <span class="n">unordered</span> <span class="o">=</span> <span class="bp">False</span>
-
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">default_sort_key</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">unordered</span><span class="p">:</span>
-            <span class="c"># e.g. dict, set</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="n">cls_index</span><span class="p">,</span> <span class="n">args</span> <span class="o">=</span> <span class="mi">10</span><span class="p">,</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">),</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">args</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">item</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">item</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">item</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">SympifyError</span><span class="p">:</span>
-                <span class="c"># e.g. lambda x: x</span>
-                <span class="k">pass</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">item</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-                    <span class="c"># e.g int -&gt; Integer</span>
-                    <span class="k">return</span> <span class="n">default_sort_key</span><span class="p">(</span><span class="n">item</span><span class="p">)</span>
-                <span class="c"># e.g. UndefinedFunction</span>
-
-        <span class="c"># e.g. str</span>
-        <span class="n">cls_index</span><span class="p">,</span> <span class="n">args</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">item</span><span class="p">),))</span>
-
-    <span class="k">return</span> <span class="p">(</span><span class="n">cls_index</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">item</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span>
-            <span class="p">),</span> <span class="n">args</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="o">.</span><span class="n">sort_key</span><span class="p">(),</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-
-<span class="k">def</span> <span class="nf">_nodes</span><span class="p">(</span><span class="n">e</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A helper for ordered() which returns the node count of ``e`` which</span>
-<span class="sd">    for Basic object is the number of Basic nodes in the expression tree</span>
-<span class="sd">    but for other object is 1 (unless the object is an iterable or dict</span>
-<span class="sd">    for which the sum of nodes is returned).</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">.basic</span> <span class="kn">import</span> <span class="n">Basic</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">e</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="n">Basic</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">iterable</span><span class="p">(</span><span class="n">e</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">1</span> <span class="o">+</span> <span class="nb">sum</span><span class="p">(</span><span class="n">_nodes</span><span class="p">(</span><span class="n">ei</span><span class="p">)</span> <span class="k">for</span> <span class="n">ei</span> <span class="ow">in</span> <span class="n">e</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="nb">dict</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">1</span> <span class="o">+</span> <span class="nb">sum</span><span class="p">(</span><span class="n">_nodes</span><span class="p">(</span><span class="n">k</span><span class="p">)</span> <span class="o">+</span> <span class="n">_nodes</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">items</span><span class="p">())</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">1</span>
-
-
-<span class="k">def</span> <span class="nf">ordered</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="n">keys</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">default</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">warn</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return an iterator of the seq where keys are used to break ties.</span>
-<span class="sd">    Two default keys will be applied after and provided unless ``default``</span>
-<span class="sd">    is False. The two keys are _nodes and default_sort_key which will</span>
-<span class="sd">    place smaller expressions before larger ones (in terms of Basic nodes)</span>
-<span class="sd">    and where there are ties, they will be broken by the default_sort_key.</span>
-
-<span class="sd">    If ``warn`` is True then an error will be raised if there were no</span>
-<span class="sd">    keys remaining to break ties. This can be used if it was expected that</span>
-<span class="sd">    there should be no ties.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import ordered, default_sort_key</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import count_ops</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">    The count_ops is not sufficient to break ties in this list and the first</span>
-<span class="sd">    two items appear in their original order (i.e. the sorting is stable):</span>
-
-<span class="sd">    &gt;&gt;&gt; list(ordered([y + 2, x + 2, x**2 + y + 3],</span>
-<span class="sd">    ...    count_ops, default=False, warn=False))</span>
-<span class="sd">    ...</span>
-<span class="sd">    [y + 2, x + 2, x**2 + y + 3]</span>
-
-<span class="sd">    The default_sort_key allows the tie to be broken:</span>
-
-<span class="sd">    &gt;&gt;&gt; list(ordered([y + 2, x + 2, x**2 + y + 3]))</span>
-<span class="sd">    ...</span>
-<span class="sd">    [x + 2, y + 2, x**2 + y + 3]</span>
-
-<span class="sd">    Here, sequences are sorted by length, then sum:</span>
-
-<span class="sd">    &gt;&gt;&gt; seq, keys = [[[1, 2, 1], [0, 3, 1], [1, 1, 3], [2], [1]], [</span>
-<span class="sd">    ...    lambda x: len(x),</span>
-<span class="sd">    ...    lambda x: sum(x)]]</span>
-<span class="sd">    ...</span>
-<span class="sd">    &gt;&gt;&gt; list(ordered(seq, keys, default=False, warn=False))</span>
-<span class="sd">    [[1], [2], [1, 2, 1], [0, 3, 1], [1, 1, 3]]</span>
-
-<span class="sd">    If ``warn`` is True, an error will be raised if there were not</span>
-<span class="sd">    enough keys to break ties:</span>
-
-<span class="sd">    &gt;&gt;&gt; list(ordered(seq, keys, default=False, warn=True))</span>
-<span class="sd">    Traceback (most recent call last):</span>
-<span class="sd">    ...</span>
-<span class="sd">    ValueError: not enough keys to break ties</span>
-
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    The decorated sort is one of the fastest ways to sort a sequence for</span>
-<span class="sd">    which special item comparison is desired: the sequence is decorated,</span>
-<span class="sd">    sorted on the basis of the decoration (e.g. making all letters lower</span>
-<span class="sd">    case) and then undecorated. If one wants to break ties for items that</span>
-<span class="sd">    have the same decorated value, a second key can be used. But if the</span>
-<span class="sd">    second key is expensive to compute then it is inefficient to decorate</span>
-<span class="sd">    all items with both keys: only those items having identical first key</span>
-<span class="sd">    values need to be decorated. This function applies keys successively</span>
-<span class="sd">    only when needed to break ties. By yielding an iterator, use of the</span>
-<span class="sd">    tie-breaker is delayed as long as possible.</span>
-
-<span class="sd">    This function is best used in cases when use of the first key is</span>
-<span class="sd">    expected to be a good hashing function; if there are no unique hashes</span>
-<span class="sd">    from application of a key then that key should not have been used. The</span>
-<span class="sd">    exception, however, is that even if there are many collisions, if the</span>
-<span class="sd">    first group is small and one does not need to process all items in the</span>
-<span class="sd">    list then time will not be wasted sorting what one was not interested</span>
-<span class="sd">    in. For example, if one were looking for the minimum in a list and</span>
-<span class="sd">    there were several criteria used to define the sort order, then this</span>
-<span class="sd">    function would be good at returning that quickly if the first group</span>
-<span class="sd">    of candidates is small relative to the number of items being processed.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">d</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">list</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">keys</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">keys</span><span class="p">,</span> <span class="p">(</span><span class="nb">list</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">)):</span>
-            <span class="n">keys</span> <span class="o">=</span> <span class="p">[</span><span class="n">keys</span><span class="p">]</span>
-        <span class="n">keys</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">keys</span><span class="p">)</span>
-
-        <span class="n">f</span> <span class="o">=</span> <span class="n">keys</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">seq</span><span class="p">:</span>
-            <span class="n">d</span><span class="p">[</span><span class="n">f</span><span class="p">(</span><span class="n">a</span><span class="p">)]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">default</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;if default=False then keys must be provided&#39;</span><span class="p">)</span>
-        <span class="n">d</span><span class="p">[</span><span class="bp">None</span><span class="p">]</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">seq</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">d</span><span class="p">[</span><span class="n">k</span><span class="p">])</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">keys</span><span class="p">:</span>
-                <span class="n">d</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">ordered</span><span class="p">(</span><span class="n">d</span><span class="p">[</span><span class="n">k</span><span class="p">],</span> <span class="n">keys</span><span class="p">,</span> <span class="n">default</span><span class="p">,</span> <span class="n">warn</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">default</span><span class="p">:</span>
-                <span class="n">d</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">ordered</span><span class="p">(</span><span class="n">d</span><span class="p">[</span><span class="n">k</span><span class="p">],</span> <span class="p">(</span><span class="n">_nodes</span><span class="p">,</span> <span class="n">default_sort_key</span><span class="p">,),</span>
-                               <span class="n">default</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">warn</span><span class="o">=</span><span class="n">warn</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">warn</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;not enough keys to break ties&#39;</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">d</span><span class="p">[</span><span class="n">k</span><span class="p">]:</span>
-            <span class="k">yield</span> <span class="n">v</span>
-        <span class="n">d</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-
-<span class="k">try</span><span class="p">:</span>
-    <span class="nb">next</span> <span class="o">=</span> <span class="nb">next</span>
-<span class="k">except</span> <span class="ne">NameError</span><span class="p">:</span>
-    <span class="k">def</span> <span class="nf">next</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">x</span><span class="o">.</span><span class="n">__next__</span><span class="p">()</span>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.core.containers</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Module for SymPy containers</span>
-
-<span class="sd">    (SymPy objects that store other SymPy objects)</span>
-
-<span class="sd">    The containers implemented in this module are subclassed to Basic.</span>
-<span class="sd">    They are supposed to work seamlessly within the SymPy framework.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.basic</span> <span class="kn">import</span> <span class="n">Basic</span>
-<span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">converter</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">iterable</span>
-
-
-<div class="viewcode-block" id="Tuple"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.containers.Tuple">[docs]</a><span class="k">class</span> <span class="nc">Tuple</span><span class="p">(</span><span class="n">Basic</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Wrapper around the builtin tuple object</span>
-
-<span class="sd">    The Tuple is a subclass of Basic, so that it works well in the</span>
-<span class="sd">    SymPy framework.  The wrapped tuple is available as self.args, but</span>
-<span class="sd">    you can also access elements or slices with [:] syntax.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.core.containers import Tuple</span>
-<span class="sd">    &gt;&gt;&gt; a, b, c, d = symbols(&#39;a b c d&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; Tuple(a, b, c)[1:]</span>
-<span class="sd">    (b, c)</span>
-<span class="sd">    &gt;&gt;&gt; Tuple(a, b, c).subs(a, d)</span>
-<span class="sd">    (d, b, c)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[</span> <span class="n">sympify</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span> <span class="p">]</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="nb">slice</span><span class="p">):</span>
-            <span class="n">indices</span> <span class="o">=</span> <span class="n">i</span><span class="o">.</span><span class="n">indices</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="o">*</span><span class="n">indices</span><span class="p">)])</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__len__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__contains__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">item</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">item</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">iter</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">+</span> <span class="n">other</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">+</span> <span class="n">other</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="nf">__radd__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">args</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">other</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Tuple</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">==</span> <span class="n">other</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Tuple</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__ne__</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">!=</span> <span class="n">other</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">hash</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_to_mpmath</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">([</span><span class="n">a</span><span class="o">.</span><span class="n">_to_mpmath</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">&lt;</span> <span class="n">other</span><span class="o">.</span><span class="n">args</span>
-
-    <span class="k">def</span> <span class="nf">__le__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">&lt;=</span> <span class="n">other</span><span class="o">.</span><span class="n">args</span>
-</div>
-<span class="n">converter</span><span class="p">[</span><span class="nb">tuple</span><span class="p">]</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">tup</span><span class="p">:</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">tup</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">tuple_wrapper</span><span class="p">(</span><span class="n">method</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Decorator that converts any tuple in the function arguments into a Tuple.</span>
-
-<span class="sd">    The motivation for this is to provide simple user interfaces.  The user can</span>
-<span class="sd">    call a function with regular tuples in the argument, and the wrapper will</span>
-<span class="sd">    convert them to Tuples before handing them to the function.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.core.containers import tuple_wrapper, Tuple</span>
-<span class="sd">    &gt;&gt;&gt; def f(*args):</span>
-<span class="sd">    ...    return args</span>
-<span class="sd">    &gt;&gt;&gt; g = tuple_wrapper(f)</span>
-
-<span class="sd">    The decorated function g sees only the Tuple argument:</span>
-
-<span class="sd">    &gt;&gt;&gt; g(0, (1, 2), 3)</span>
-<span class="sd">    (0, (1, 2), 3)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">wrap_tuples</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">):</span>
-        <span class="n">newargs</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">tuple</span><span class="p">:</span>
-                <span class="n">newargs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">arg</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">newargs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">method</span><span class="p">(</span><span class="o">*</span><span class="n">newargs</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">wrap_tuples</span>
-
-
-<div class="viewcode-block" id="Dict"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.containers.Dict">[docs]</a><span class="k">class</span> <span class="nc">Dict</span><span class="p">(</span><span class="n">Basic</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Wrapper around the builtin dict object</span>
-
-<span class="sd">    The Dict is a subclass of Basic, so that it works well in the</span>
-<span class="sd">    SymPy framework.  Because it is immutable, it may be included</span>
-<span class="sd">    in sets, but its values must all be given at instantiation and</span>
-<span class="sd">    cannot be changed afterwards.  Otherwise it behaves identically</span>
-<span class="sd">    to the Python dict.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import S</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.core.containers import Dict</span>
-
-<span class="sd">    &gt;&gt;&gt; D = Dict({1: &#39;one&#39;, 2: &#39;two&#39;})</span>
-<span class="sd">    &gt;&gt;&gt; for key in D:</span>
-<span class="sd">    ...    if key == 1:</span>
-<span class="sd">    ...        print(key, D[key])</span>
-<span class="sd">    1 one</span>
-
-<span class="sd">    The args are sympified so the 1 and 2 are Integers and the values</span>
-<span class="sd">    are Symbols. Queries automatically sympify args so the following work:</span>
-
-<span class="sd">    &gt;&gt;&gt; 1 in D</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; D.has(&#39;one&#39;) # searches keys and values</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; &#39;one&#39; in D # not in the keys</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; D[1]</span>
-<span class="sd">    one</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="p">((</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">__class__</span> <span class="ow">is</span> <span class="nb">dict</span><span class="p">)</span> <span class="ow">or</span>
-                             <span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">__class__</span> <span class="ow">is</span> <span class="n">Dict</span><span class="p">)):</span>
-            <span class="n">items</span> <span class="o">=</span> <span class="p">[</span><span class="n">Tuple</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">items</span><span class="p">())]</span>
-        <span class="k">elif</span> <span class="n">iterable</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">all</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">):</span>
-            <span class="n">items</span> <span class="o">=</span> <span class="p">[</span><span class="n">Tuple</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Pass Dict args as Dict((k1, v1), ...) or Dict({k1: v1, ...})&#39;</span><span class="p">)</span>
-        <span class="n">elements</span> <span class="o">=</span> <span class="nb">frozenset</span><span class="p">(</span><span class="n">items</span><span class="p">)</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">elements</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">elements</span> <span class="o">=</span> <span class="n">elements</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_dict</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">items</span><span class="p">)</span>  <span class="c"># In case Tuple decides it wants to sympify</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;x.__getitem__(y) &lt;==&gt; x[y]&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_dict</span><span class="p">[</span><span class="n">sympify</span><span class="p">(</span><span class="n">key</span><span class="p">)]</span>
-
-    <span class="k">def</span> <span class="nf">__setitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;SymPy Dicts are Immutable&quot;</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">elements</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Dict.items"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.containers.Dict.items">[docs]</a>    <span class="k">def</span> <span class="nf">items</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&#39;&#39;&#39;D.items() -&gt; list of D&#39;s (key, value) pairs, as 2-tuples&#39;&#39;&#39;</span>
-        <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_dict</span><span class="o">.</span><span class="n">items</span><span class="p">())</span>
-</div>
-<div class="viewcode-block" id="Dict.keys"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.containers.Dict.keys">[docs]</a>    <span class="k">def</span> <span class="nf">keys</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&#39;&#39;&#39;D.keys() -&gt; list of D&#39;s keys&#39;&#39;&#39;</span>
-        <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_dict</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-</div>
-<div class="viewcode-block" id="Dict.values"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.containers.Dict.values">[docs]</a>    <span class="k">def</span> <span class="nf">values</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&#39;&#39;&#39;D.values() -&gt; list of D&#39;s values&#39;&#39;&#39;</span>
-        <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_dict</span><span class="o">.</span><span class="n">values</span><span class="p">())</span>
-</div>
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&#39;&#39;&#39;x.__iter__() &lt;==&gt; iter(x)&#39;&#39;&#39;</span>
-        <span class="k">return</span> <span class="nb">iter</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_dict</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__len__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&#39;&#39;&#39;x.__len__() &lt;==&gt; len(x)&#39;&#39;&#39;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_dict</span><span class="o">.</span><span class="n">__len__</span><span class="p">()</span>
-
-<div class="viewcode-block" id="Dict.get"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.containers.Dict.get">[docs]</a>    <span class="k">def</span> <span class="nf">get</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">,</span> <span class="n">default</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&#39;&#39;&#39;D.get(k[,d]) -&gt; D[k] if k in D, else d.  d defaults to None.&#39;&#39;&#39;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_dict</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">key</span><span class="p">),</span> <span class="n">default</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">__contains__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">):</span>
-        <span class="sd">&#39;&#39;&#39;D.__contains__(k) -&gt; True if D has a key k, else False&#39;&#39;&#39;</span>
-        <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="n">key</span><span class="p">)</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_dict</span>
-
-    <span class="k">def</span> <span class="nf">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">&lt;</span> <span class="n">other</span><span class="o">.</span><span class="n">args</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_sorted_args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">default_sort_key</span>
-        <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span></div>
-</pre></div>
-
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-  <h1>Source code for sympy.core.evalf</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Adaptive numerical evaluation of SymPy expressions, using mpmath</span>
-<span class="sd">for mathematical functions.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">import</span> <span class="nn">sympy.mpmath.libmp</span> <span class="kn">as</span> <span class="nn">libmp</span>
-<span class="kn">from</span> <span class="nn">sympy.mpmath</span> <span class="kn">import</span> <span class="n">make_mpc</span><span class="p">,</span> <span class="n">make_mpf</span><span class="p">,</span> <span class="n">mp</span><span class="p">,</span> <span class="n">mpc</span><span class="p">,</span> <span class="n">mpf</span><span class="p">,</span> <span class="n">nsum</span><span class="p">,</span> <span class="n">quadts</span><span class="p">,</span> <span class="n">quadosc</span>
-<span class="kn">from</span> <span class="nn">sympy.mpmath</span> <span class="kn">import</span> <span class="n">inf</span> <span class="k">as</span> <span class="n">mpmath_inf</span>
-<span class="kn">from</span> <span class="nn">sympy.mpmath.libmp</span> <span class="kn">import</span> <span class="p">(</span><span class="n">bitcount</span><span class="p">,</span> <span class="n">from_int</span><span class="p">,</span> <span class="n">from_man_exp</span><span class="p">,</span>
-        <span class="n">from_rational</span><span class="p">,</span> <span class="n">fhalf</span><span class="p">,</span> <span class="n">fnan</span><span class="p">,</span> <span class="n">fnone</span><span class="p">,</span> <span class="n">fone</span><span class="p">,</span> <span class="n">fzero</span><span class="p">,</span> <span class="n">mpf_abs</span><span class="p">,</span> <span class="n">mpf_add</span><span class="p">,</span>
-        <span class="n">mpf_atan</span><span class="p">,</span> <span class="n">mpf_atan2</span><span class="p">,</span> <span class="n">mpf_cmp</span><span class="p">,</span> <span class="n">mpf_cos</span><span class="p">,</span> <span class="n">mpf_e</span><span class="p">,</span> <span class="n">mpf_exp</span><span class="p">,</span> <span class="n">mpf_log</span><span class="p">,</span> <span class="n">mpf_lt</span><span class="p">,</span>
-        <span class="n">mpf_mul</span><span class="p">,</span> <span class="n">mpf_neg</span><span class="p">,</span> <span class="n">mpf_pi</span><span class="p">,</span> <span class="n">mpf_pow</span><span class="p">,</span> <span class="n">mpf_pow_int</span><span class="p">,</span> <span class="n">mpf_shift</span><span class="p">,</span> <span class="n">mpf_sin</span><span class="p">,</span>
-        <span class="n">mpf_sqrt</span><span class="p">,</span> <span class="n">normalize</span><span class="p">,</span> <span class="n">round_nearest</span><span class="p">,</span> <span class="n">to_int</span><span class="p">,</span> <span class="n">to_str</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.mpmath.libmp.backend</span> <span class="kn">import</span> <span class="n">MPZ</span>
-<span class="kn">from</span> <span class="nn">sympy.mpmath.libmp.libmpc</span> <span class="kn">import</span> <span class="n">_infs_nan</span>
-<span class="kn">from</span> <span class="nn">sympy.mpmath.libmp.libmpf</span> <span class="kn">import</span> <span class="n">dps_to_prec</span><span class="p">,</span> <span class="n">prec_to_dps</span>
-
-<span class="kn">from</span> <span class="nn">sympy.mpmath.libmp.gammazeta</span> <span class="kn">import</span> <span class="n">mpf_bernoulli</span>
-
-<span class="kn">import</span> <span class="nn">math</span>
-
-<span class="kn">from</span> <span class="nn">.sympify</span> <span class="kn">import</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">.core</span> <span class="kn">import</span> <span class="n">C</span>
-<span class="kn">from</span> <span class="nn">.singleton</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="kn">from</span> <span class="nn">.containers</span> <span class="kn">import</span> <span class="n">Tuple</span>
-
-<span class="n">LG10</span> <span class="o">=</span> <span class="n">math</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="n">rnd</span> <span class="o">=</span> <span class="n">round_nearest</span>
-
-<span class="c"># Used in a few places as placeholder values to denote exponents and</span>
-<span class="c"># precision levels, e.g. of exact numbers. Must be careful to avoid</span>
-<span class="c"># passing these to mpmath functions or returning them in final results.</span>
-<span class="n">INF</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">mpmath_inf</span><span class="p">)</span>
-<span class="n">MINUS_INF</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="o">-</span><span class="n">mpmath_inf</span><span class="p">)</span>
-
-<span class="c"># ~= 100 digits. Real men set this to INF.</span>
-<span class="n">DEFAULT_MAXPREC</span> <span class="o">=</span> <span class="mi">333</span>
-
-
-<div class="viewcode-block" id="PrecisionExhausted"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.evalf.PrecisionExhausted">[docs]</a><span class="k">class</span> <span class="nc">PrecisionExhausted</span><span class="p">(</span><span class="ne">ArithmeticError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-<span class="c">#----------------------------------------------------------------------------#</span>
-<span class="c">#                                                                            #</span>
-<span class="c">#              Helper functions for arithmetic and complex parts             #</span>
-<span class="c">#                                                                            #</span>
-<span class="c">#----------------------------------------------------------------------------#</span>
-</div>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">An mpf value tuple is a tuple of integers (sign, man, exp, bc)</span>
-<span class="sd">representing a floating-point number: [1, -1][sign]*man*2**exp where</span>
-<span class="sd">sign is 0 or 1 and bc should correspond to the number of bits used to</span>
-<span class="sd">represent the mantissa (man) in binary notation, e.g.</span>
-
-<span class="sd">&gt;&gt;&gt; from sympy.core.evalf import bitcount</span>
-<span class="sd">&gt;&gt;&gt; sign, man, exp, bc = 0, 5, 1, 3</span>
-<span class="sd">&gt;&gt;&gt; n = [1, -1][sign]*man*2**exp</span>
-<span class="sd">&gt;&gt;&gt; n, bitcount(man)</span>
-<span class="sd">(10, 3)</span>
-
-<span class="sd">A temporary result is a tuple (re, im, re_acc, im_acc) where</span>
-<span class="sd">re and im are nonzero mpf value tuples representing approximate</span>
-<span class="sd">numbers, or None to denote exact zeros.</span>
-
-<span class="sd">re_acc, im_acc are integers denoting log2(e) where e is the estimated</span>
-<span class="sd">relative accuracy of the respective complex part, but may be anything</span>
-<span class="sd">if the corresponding complex part is None.</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-
-
-<span class="k">def</span> <span class="nf">fastlog</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Fast approximation of log2(x) for an mpf value tuple x.</span>
-
-<span class="sd">    Notes: Calculated as exponent + width of mantissa. This is an</span>
-<span class="sd">    approximation for two reasons: 1) it gives the ceil(log2(abs(x)))</span>
-<span class="sd">    value and 2) it is too high by 1 in the case that x is an exact</span>
-<span class="sd">    power of 2. Although this is easy to remedy by testing to see if</span>
-<span class="sd">    the odd mpf mantissa is 1 (indicating that one was dealing with</span>
-<span class="sd">    an exact power of 2) that would decrease the speed and is not</span>
-<span class="sd">    necessary as this is only being used as an approximation for the</span>
-<span class="sd">    number of bits in x. The correct return value could be written as</span>
-<span class="sd">    &quot;x[2] + (x[3] if x[1] != 1 else 0)&quot;.</span>
-<span class="sd">        Since mpf tuples always have an odd mantissa, no check is done</span>
-<span class="sd">    to see if the mantissa is a multiple of 2 (in which case the</span>
-<span class="sd">    result would be too large by 1).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import log</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.core.evalf import fastlog, bitcount</span>
-<span class="sd">    &gt;&gt;&gt; s, m, e = 0, 5, 1</span>
-<span class="sd">    &gt;&gt;&gt; bc = bitcount(m)</span>
-<span class="sd">    &gt;&gt;&gt; n = [1, -1][s]*m*2**e</span>
-<span class="sd">    &gt;&gt;&gt; n, (log(n)/log(2)).evalf(2), fastlog((s, m, e, bc))</span>
-<span class="sd">    (10, 3.3, 4)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">x</span> <span class="ow">or</span> <span class="n">x</span> <span class="o">==</span> <span class="n">fzero</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">MINUS_INF</span>
-    <span class="k">return</span> <span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="n">x</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">pure_complex</span><span class="p">(</span><span class="n">v</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return a and b if v matches a + I*b where b is not zero and</span>
-<span class="sd">    a and b are Numbers, else None.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.core.evalf import pure_complex</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Tuple, I</span>
-<span class="sd">    &gt;&gt;&gt; a, b = Tuple(2, 3)</span>
-<span class="sd">    &gt;&gt;&gt; pure_complex(a)</span>
-<span class="sd">    &gt;&gt;&gt; pure_complex(a + b*I)</span>
-<span class="sd">    (2, 3)</span>
-<span class="sd">    &gt;&gt;&gt; pure_complex(I)</span>
-<span class="sd">    (0, 1)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">h</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">v</span><span class="o">.</span><span class="n">as_coeff_Add</span><span class="p">()</span>
-    <span class="n">c</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="n">t</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-    <span class="k">if</span> <span class="n">i</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">h</span><span class="p">,</span> <span class="n">c</span>
-
-
-<span class="k">def</span> <span class="nf">scaled_zero</span><span class="p">(</span><span class="n">mag</span><span class="p">,</span> <span class="n">sign</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return an mpf representing a power of two with magnitude ``mag``</span>
-<span class="sd">    and -1 for precision. Or, if ``mag`` is a scaled_zero tuple, then just</span>
-<span class="sd">    remove the sign from within the list that it was initially wrapped</span>
-<span class="sd">    in.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.core.evalf import scaled_zero</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Float</span>
-<span class="sd">    &gt;&gt;&gt; z, p = scaled_zero(100)</span>
-<span class="sd">    &gt;&gt;&gt; z, p</span>
-<span class="sd">    (([0], 1, 100, 1), -1)</span>
-<span class="sd">    &gt;&gt;&gt; ok = scaled_zero(z)</span>
-<span class="sd">    &gt;&gt;&gt; ok</span>
-<span class="sd">    (0, 1, 100, 1)</span>
-<span class="sd">    &gt;&gt;&gt; Float(ok)</span>
-<span class="sd">    1.26765060022823e+30</span>
-<span class="sd">    &gt;&gt;&gt; Float(ok, p)</span>
-<span class="sd">    0.e+30</span>
-<span class="sd">    &gt;&gt;&gt; ok, p = scaled_zero(100, -1)</span>
-<span class="sd">    &gt;&gt;&gt; Float(scaled_zero(ok), p)</span>
-<span class="sd">    -0.e+30</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">mag</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">tuple</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">mag</span><span class="p">)</span> <span class="o">==</span> <span class="mi">4</span> <span class="ow">and</span> <span class="n">iszero</span><span class="p">(</span><span class="n">mag</span><span class="p">,</span> <span class="n">scaled</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">mag</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">],)</span> <span class="o">+</span> <span class="n">mag</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-    <span class="k">elif</span> <span class="nb">type</span><span class="p">(</span><span class="n">mag</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">int</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">sign</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;sign must be +/-1&#39;</span><span class="p">)</span>
-        <span class="n">rv</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="n">mpf_shift</span><span class="p">(</span><span class="n">fone</span><span class="p">,</span> <span class="n">mag</span><span class="p">),</span> <span class="o">-</span><span class="mi">1</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="mi">0</span> <span class="k">if</span> <span class="n">sign</span> <span class="o">==</span> <span class="mi">1</span> <span class="k">else</span> <span class="mi">1</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="p">([</span><span class="n">s</span><span class="p">],)</span> <span class="o">+</span> <span class="n">rv</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-        <span class="k">return</span> <span class="n">rv</span><span class="p">,</span> <span class="n">p</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;scaled zero expects int or scaled_zero tuple.&#39;</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">iszero</span><span class="p">(</span><span class="n">mpf</span><span class="p">,</span> <span class="n">scaled</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">scaled</span><span class="p">:</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="n">mpf</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">mpf</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">mpf</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">mpf</span> <span class="ow">and</span> <span class="nb">type</span><span class="p">(</span><span class="n">mpf</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="ow">is</span> <span class="nb">list</span> <span class="ow">and</span> <span class="n">mpf</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">mpf</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span>
-
-
-<span class="k">def</span> <span class="nf">complex_accuracy</span><span class="p">(</span><span class="n">result</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns relative accuracy of a complex number with given accuracies</span>
-<span class="sd">    for the real and imaginary parts. The relative accuracy is defined</span>
-<span class="sd">    in the complex norm sense as ||z|+|error|| / |z| where error</span>
-<span class="sd">    is equal to (real absolute error) + (imag absolute error)*i.</span>
-
-<span class="sd">    The full expression for the (logarithmic) error can be approximated</span>
-<span class="sd">    easily by using the max norm to approximate the complex norm.</span>
-
-<span class="sd">    In the worst case (re and im equal), this is wrong by a factor</span>
-<span class="sd">    sqrt(2), or by log2(sqrt(2)) = 0.5 bit.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_acc</span> <span class="o">=</span> <span class="n">result</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">im</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">re</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">INF</span>
-        <span class="k">return</span> <span class="n">re_acc</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">re</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">im_acc</span>
-    <span class="n">re_size</span> <span class="o">=</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">re</span><span class="p">)</span>
-    <span class="n">im_size</span> <span class="o">=</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">im</span><span class="p">)</span>
-    <span class="n">absolute_error</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">re_size</span> <span class="o">-</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_size</span> <span class="o">-</span> <span class="n">im_acc</span><span class="p">)</span>
-    <span class="n">relative_error</span> <span class="o">=</span> <span class="n">absolute_error</span> <span class="o">-</span> <span class="nb">max</span><span class="p">(</span><span class="n">re_size</span><span class="p">,</span> <span class="n">im_size</span><span class="p">)</span>
-    <span class="k">return</span> <span class="o">-</span><span class="n">relative_error</span>
-
-
-<span class="k">def</span> <span class="nf">get_abs</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">):</span>
-    <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_acc</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">prec</span> <span class="o">+</span> <span class="mi">2</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">re</span><span class="p">:</span>
-        <span class="n">re</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">im_acc</span> <span class="o">=</span> <span class="n">im</span><span class="p">,</span> <span class="n">im_acc</span><span class="p">,</span> <span class="n">re</span><span class="p">,</span> <span class="n">re_acc</span>
-    <span class="k">if</span> <span class="n">im</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">libmp</span><span class="o">.</span><span class="n">mpc_abs</span><span class="p">((</span><span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">),</span> <span class="n">prec</span><span class="p">),</span> <span class="bp">None</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="bp">None</span>
-    <span class="k">elif</span> <span class="n">re</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">mpf_abs</span><span class="p">(</span><span class="n">re</span><span class="p">),</span> <span class="bp">None</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="bp">None</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-
-
-<span class="k">def</span> <span class="nf">get_complex_part</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">no</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;no = 0 for real part, no = 1 for imaginary part&quot;&quot;&quot;</span>
-    <span class="n">workprec</span> <span class="o">=</span> <span class="n">prec</span>
-    <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">workprec</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-        <span class="n">value</span><span class="p">,</span> <span class="n">accuracy</span> <span class="o">=</span> <span class="n">res</span><span class="p">[</span><span class="n">no</span><span class="p">::</span><span class="mi">2</span><span class="p">]</span>
-        <span class="c"># XXX is the last one correct? Consider re((1+I)**2).n()</span>
-        <span class="k">if</span> <span class="p">(</span><span class="ow">not</span> <span class="n">value</span><span class="p">)</span> <span class="ow">or</span> <span class="n">accuracy</span> <span class="o">&gt;=</span> <span class="n">prec</span> <span class="ow">or</span> <span class="o">-</span><span class="n">value</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">prec</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">value</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">accuracy</span><span class="p">,</span> <span class="bp">None</span>
-        <span class="n">workprec</span> <span class="o">+=</span> <span class="nb">max</span><span class="p">(</span><span class="mi">30</span><span class="p">,</span> <span class="mi">2</span><span class="o">**</span><span class="n">i</span><span class="p">)</span>
-        <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-
-
-<span class="k">def</span> <span class="nf">evalf_abs</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">get_abs</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">evalf_re</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">get_complex_part</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">0</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">evalf_im</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">get_complex_part</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">finalize_complex</span><span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-    <span class="k">if</span> <span class="n">re</span> <span class="o">==</span> <span class="n">fzero</span> <span class="ow">and</span> <span class="n">im</span> <span class="o">==</span> <span class="n">fzero</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;got complex zero with unknown accuracy&quot;</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">re</span> <span class="o">==</span> <span class="n">fzero</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span>
-    <span class="k">elif</span> <span class="n">im</span> <span class="o">==</span> <span class="n">fzero</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">re</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="bp">None</span>
-
-    <span class="n">size_re</span> <span class="o">=</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">re</span><span class="p">)</span>
-    <span class="n">size_im</span> <span class="o">=</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">im</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">size_re</span> <span class="o">&gt;</span> <span class="n">size_im</span><span class="p">:</span>
-        <span class="n">re_acc</span> <span class="o">=</span> <span class="n">prec</span>
-        <span class="n">im_acc</span> <span class="o">=</span> <span class="n">prec</span> <span class="o">+</span> <span class="nb">min</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">size_re</span> <span class="o">-</span> <span class="n">size_im</span><span class="p">),</span> <span class="mi">0</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">im_acc</span> <span class="o">=</span> <span class="n">prec</span>
-        <span class="n">re_acc</span> <span class="o">=</span> <span class="n">prec</span> <span class="o">+</span> <span class="nb">min</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">size_im</span> <span class="o">-</span> <span class="n">size_re</span><span class="p">),</span> <span class="mi">0</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_acc</span>
-
-
-<span class="k">def</span> <span class="nf">chop_parts</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Chop off tiny real or complex parts.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_acc</span> <span class="o">=</span> <span class="n">value</span>
-    <span class="c"># Method 1: chop based on absolute value</span>
-    <span class="k">if</span> <span class="n">re</span> <span class="ow">and</span> <span class="n">re</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">_infs_nan</span> <span class="ow">and</span> <span class="p">(</span><span class="n">fastlog</span><span class="p">(</span><span class="n">re</span><span class="p">)</span> <span class="o">&lt;</span> <span class="o">-</span><span class="n">prec</span> <span class="o">+</span> <span class="mi">4</span><span class="p">):</span>
-        <span class="n">re</span><span class="p">,</span> <span class="n">re_acc</span> <span class="o">=</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-    <span class="k">if</span> <span class="n">im</span> <span class="ow">and</span> <span class="n">im</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">_infs_nan</span> <span class="ow">and</span> <span class="p">(</span><span class="n">fastlog</span><span class="p">(</span><span class="n">im</span><span class="p">)</span> <span class="o">&lt;</span> <span class="o">-</span><span class="n">prec</span> <span class="o">+</span> <span class="mi">4</span><span class="p">):</span>
-        <span class="n">im</span><span class="p">,</span> <span class="n">im_acc</span> <span class="o">=</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-    <span class="c"># Method 2: chop if inaccurate and relatively small</span>
-    <span class="k">if</span> <span class="n">re</span> <span class="ow">and</span> <span class="n">im</span><span class="p">:</span>
-        <span class="n">delta</span> <span class="o">=</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">re</span><span class="p">)</span> <span class="o">-</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">im</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">re_acc</span> <span class="o">&lt;</span> <span class="mi">2</span> <span class="ow">and</span> <span class="p">(</span><span class="n">delta</span> <span class="o">-</span> <span class="n">re_acc</span> <span class="o">&lt;=</span> <span class="o">-</span><span class="n">prec</span> <span class="o">+</span> <span class="mi">4</span><span class="p">):</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">re_acc</span> <span class="o">=</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">im_acc</span> <span class="o">&lt;</span> <span class="mi">2</span> <span class="ow">and</span> <span class="p">(</span><span class="n">delta</span> <span class="o">-</span> <span class="n">im_acc</span> <span class="o">&gt;=</span> <span class="n">prec</span> <span class="o">-</span> <span class="mi">4</span><span class="p">):</span>
-            <span class="n">im</span><span class="p">,</span> <span class="n">im_acc</span> <span class="o">=</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-    <span class="k">return</span> <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_acc</span>
-
-
-<span class="k">def</span> <span class="nf">check_target</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">result</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">complex_accuracy</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">a</span> <span class="o">&lt;</span> <span class="n">prec</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">PrecisionExhausted</span><span class="p">(</span><span class="s">&quot;Failed to distinguish the expression: </span><span class="se">\n\n</span><span class="si">%s</span><span class="se">\n\n</span><span class="s">&quot;</span>
-            <span class="s">&quot;from zero. Try simplifying the input, using chop=True, or providing &quot;</span>
-            <span class="s">&quot;a higher maxn for evalf&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-
-
-<span class="k">def</span> <span class="nf">get_integer_part</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">no</span><span class="p">,</span> <span class="n">options</span><span class="p">,</span> <span class="n">return_ints</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    With no = 1, computes ceiling(expr)</span>
-<span class="sd">    With no = -1, computes floor(expr)</span>
-
-<span class="sd">    Note: this function either gives the exact result or signals failure.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="c"># The expression is likely less than 2^30 or so</span>
-    <span class="n">assumed_size</span> <span class="o">=</span> <span class="mi">30</span>
-    <span class="n">ire</span><span class="p">,</span> <span class="n">iim</span><span class="p">,</span> <span class="n">ire_acc</span><span class="p">,</span> <span class="n">iim_acc</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumed_size</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-
-    <span class="c"># We now know the size, so we can calculate how much extra precision</span>
-    <span class="c"># (if any) is needed to get within the nearest integer</span>
-    <span class="k">if</span> <span class="n">ire</span> <span class="ow">and</span> <span class="n">iim</span><span class="p">:</span>
-        <span class="n">gap</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">fastlog</span><span class="p">(</span><span class="n">ire</span><span class="p">)</span> <span class="o">-</span> <span class="n">ire_acc</span><span class="p">,</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">iim</span><span class="p">)</span> <span class="o">-</span> <span class="n">iim_acc</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">ire</span><span class="p">:</span>
-        <span class="n">gap</span> <span class="o">=</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">ire</span><span class="p">)</span> <span class="o">-</span> <span class="n">ire_acc</span>
-    <span class="k">elif</span> <span class="n">iim</span><span class="p">:</span>
-        <span class="n">gap</span> <span class="o">=</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">iim</span><span class="p">)</span> <span class="o">-</span> <span class="n">iim_acc</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="c"># ... or maybe the expression was exactly zero</span>
-        <span class="k">return</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-
-    <span class="n">margin</span> <span class="o">=</span> <span class="mi">10</span>
-
-    <span class="k">if</span> <span class="n">gap</span> <span class="o">&gt;=</span> <span class="o">-</span><span class="n">margin</span><span class="p">:</span>
-        <span class="n">ire</span><span class="p">,</span> <span class="n">iim</span><span class="p">,</span> <span class="n">ire_acc</span><span class="p">,</span> <span class="n">iim_acc</span> <span class="o">=</span> \
-            <span class="n">evalf</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">margin</span> <span class="o">+</span> <span class="n">assumed_size</span> <span class="o">+</span> <span class="n">gap</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-
-    <span class="c"># We can now easily find the nearest integer, but to find floor/ceil, we</span>
-    <span class="c"># must also calculate whether the difference to the nearest integer is</span>
-    <span class="c"># positive or negative (which may fail if very close).</span>
-    <span class="k">def</span> <span class="nf">calc_part</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">nexpr</span><span class="p">):</span>
-        <span class="n">nint</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">to_int</span><span class="p">(</span><span class="n">nexpr</span><span class="p">,</span> <span class="n">rnd</span><span class="p">))</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Add</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">-</span><span class="n">nint</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">x_acc</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">check_target</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">x_acc</span><span class="p">,</span> <span class="bp">None</span><span class="p">),</span> <span class="mi">3</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">PrecisionExhausted</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="mi">0</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="n">PrecisionExhausted</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">fzero</span>
-        <span class="n">nint</span> <span class="o">+=</span> <span class="nb">int</span><span class="p">(</span><span class="n">no</span><span class="o">*</span><span class="p">(</span><span class="n">mpf_cmp</span><span class="p">(</span><span class="n">x</span> <span class="ow">or</span> <span class="n">fzero</span><span class="p">,</span> <span class="n">fzero</span><span class="p">)</span> <span class="o">==</span> <span class="n">no</span><span class="p">))</span>
-        <span class="n">nint</span> <span class="o">=</span> <span class="n">from_int</span><span class="p">(</span><span class="n">nint</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">nint</span><span class="p">,</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">nint</span><span class="p">)</span> <span class="o">+</span> <span class="mi">10</span>
-
-    <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_acc</span> <span class="o">=</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-
-    <span class="k">if</span> <span class="n">ire</span><span class="p">:</span>
-        <span class="n">re</span><span class="p">,</span> <span class="n">re_acc</span> <span class="o">=</span> <span class="n">calc_part</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">re</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">),</span> <span class="n">ire</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">iim</span><span class="p">:</span>
-        <span class="n">im</span><span class="p">,</span> <span class="n">im_acc</span> <span class="o">=</span> <span class="n">calc_part</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">im</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">),</span> <span class="n">iim</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">return_ints</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">to_int</span><span class="p">(</span><span class="n">re</span> <span class="ow">or</span> <span class="n">fzero</span><span class="p">)),</span> <span class="nb">int</span><span class="p">(</span><span class="n">to_int</span><span class="p">(</span><span class="n">im</span> <span class="ow">or</span> <span class="n">fzero</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_acc</span>
-
-
-<span class="k">def</span> <span class="nf">evalf_ceiling</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">get_integer_part</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">evalf_floor</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">get_integer_part</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-
-<span class="c">#----------------------------------------------------------------------------#</span>
-<span class="c">#                                                                            #</span>
-<span class="c">#                            Arithmetic operations                           #</span>
-<span class="c">#                                                                            #</span>
-<span class="c">#----------------------------------------------------------------------------#</span>
-
-
-<span class="k">def</span> <span class="nf">add_terms</span><span class="p">(</span><span class="n">terms</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">target_prec</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Helper for evalf_add. Adds a list of (mpfval, accuracy) terms.</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    -------</span>
-
-<span class="sd">    - None, None if there are no non-zero terms;</span>
-<span class="sd">    - terms[0] if there is only 1 term;</span>
-<span class="sd">    - scaled_zero if the sum of the terms produces a zero by cancellation</span>
-<span class="sd">      e.g. mpfs representing 1 and -1 would produce a scaled zero which need</span>
-<span class="sd">      special handling since they are not actually zero and they are purposely</span>
-<span class="sd">      malformed to ensure that they can&#39;t be used in anything but accuracy</span>
-<span class="sd">      calculations;</span>
-<span class="sd">    - a tuple that is scaled to target_prec that corresponds to the</span>
-<span class="sd">      sum of the terms.</span>
-
-<span class="sd">    The returned mpf tuple will be normalized to target_prec; the input</span>
-<span class="sd">    prec is used to define the working precision.</span>
-
-<span class="sd">    XXX explain why this is needed and why one can&#39;t just loop using mpf_add</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">terms</span> <span class="o">=</span> <span class="p">[</span><span class="n">t</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">terms</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">iszero</span><span class="p">(</span><span class="n">t</span><span class="p">)]</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">terms</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-    <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">terms</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">terms</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">working_prec</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">prec</span>
-    <span class="n">sum_man</span><span class="p">,</span> <span class="n">sum_exp</span><span class="p">,</span> <span class="n">absolute_error</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">MINUS_INF</span>
-    <span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="n">accuracy</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">:</span>
-        <span class="n">sign</span><span class="p">,</span> <span class="n">man</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">bc</span> <span class="o">=</span> <span class="n">x</span>
-        <span class="k">if</span> <span class="n">sign</span><span class="p">:</span>
-            <span class="n">man</span> <span class="o">=</span> <span class="o">-</span><span class="n">man</span>
-        <span class="n">absolute_error</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">absolute_error</span><span class="p">,</span> <span class="n">bc</span> <span class="o">+</span> <span class="n">exp</span> <span class="o">-</span> <span class="n">accuracy</span><span class="p">)</span>
-        <span class="n">delta</span> <span class="o">=</span> <span class="n">exp</span> <span class="o">-</span> <span class="n">sum_exp</span>
-        <span class="k">if</span> <span class="n">exp</span> <span class="o">&gt;=</span> <span class="n">sum_exp</span><span class="p">:</span>
-            <span class="c"># x much larger than existing sum?</span>
-            <span class="c"># first: quick test</span>
-            <span class="k">if</span> <span class="p">((</span><span class="n">delta</span> <span class="o">&gt;</span> <span class="n">working_prec</span><span class="p">)</span> <span class="ow">and</span>
-                <span class="p">((</span><span class="ow">not</span> <span class="n">sum_man</span><span class="p">)</span> <span class="ow">or</span>
-                 <span class="n">delta</span> <span class="o">-</span> <span class="n">bitcount</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">sum_man</span><span class="p">))</span> <span class="o">&gt;</span> <span class="n">working_prec</span><span class="p">)):</span>
-                <span class="n">sum_man</span> <span class="o">=</span> <span class="n">man</span>
-                <span class="n">sum_exp</span> <span class="o">=</span> <span class="n">exp</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">sum_man</span> <span class="o">+=</span> <span class="p">(</span><span class="n">man</span> <span class="o">&lt;&lt;</span> <span class="n">delta</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">delta</span> <span class="o">=</span> <span class="o">-</span><span class="n">delta</span>
-            <span class="c"># x much smaller than existing sum?</span>
-            <span class="k">if</span> <span class="n">delta</span> <span class="o">-</span> <span class="n">bc</span> <span class="o">&gt;</span> <span class="n">working_prec</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">sum_man</span><span class="p">:</span>
-                    <span class="n">sum_man</span><span class="p">,</span> <span class="n">sum_exp</span> <span class="o">=</span> <span class="n">man</span><span class="p">,</span> <span class="n">exp</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">sum_man</span> <span class="o">=</span> <span class="p">(</span><span class="n">sum_man</span> <span class="o">&lt;&lt;</span> <span class="n">delta</span><span class="p">)</span> <span class="o">+</span> <span class="n">man</span>
-                <span class="n">sum_exp</span> <span class="o">=</span> <span class="n">exp</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">sum_man</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">scaled_zero</span><span class="p">(</span><span class="n">absolute_error</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">sum_man</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">sum_sign</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">sum_man</span> <span class="o">=</span> <span class="o">-</span><span class="n">sum_man</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">sum_sign</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">sum_bc</span> <span class="o">=</span> <span class="n">bitcount</span><span class="p">(</span><span class="n">sum_man</span><span class="p">)</span>
-    <span class="n">sum_accuracy</span> <span class="o">=</span> <span class="n">sum_exp</span> <span class="o">+</span> <span class="n">sum_bc</span> <span class="o">-</span> <span class="n">absolute_error</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">normalize</span><span class="p">(</span><span class="n">sum_sign</span><span class="p">,</span> <span class="n">sum_man</span><span class="p">,</span> <span class="n">sum_exp</span><span class="p">,</span> <span class="n">sum_bc</span><span class="p">,</span> <span class="n">target_prec</span><span class="p">,</span>
-        <span class="n">rnd</span><span class="p">),</span> <span class="n">sum_accuracy</span>
-    <span class="c">#print &quot;returning&quot;, to_str(r[0],50), r[1]</span>
-    <span class="k">return</span> <span class="n">r</span>
-
-
-<span class="k">def</span> <span class="nf">evalf_add</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">):</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="n">pure_complex</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">res</span><span class="p">:</span>
-        <span class="n">h</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">res</span>
-        <span class="n">re</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-        <span class="n">im</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">im_acc</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_acc</span>
-
-    <span class="n">oldmaxprec</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;maxprec&#39;</span><span class="p">,</span> <span class="n">DEFAULT_MAXPREC</span><span class="p">)</span>
-
-    <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">target_prec</span> <span class="o">=</span> <span class="n">prec</span>
-    <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">options</span><span class="p">[</span><span class="s">&#39;maxprec&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">oldmaxprec</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">prec</span><span class="p">)</span>
-
-        <span class="n">terms</span> <span class="o">=</span> <span class="p">[</span><span class="n">evalf</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">prec</span> <span class="o">+</span> <span class="mi">10</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">v</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="n">re</span><span class="p">,</span> <span class="n">re_acc</span> <span class="o">=</span> <span class="n">add_terms</span><span class="p">(</span>
-            <span class="p">[</span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">::</span><span class="mi">2</span><span class="p">]</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">terms</span> <span class="k">if</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span> <span class="n">prec</span><span class="p">,</span> <span class="n">target_prec</span><span class="p">)</span>
-        <span class="n">im</span><span class="p">,</span> <span class="n">im_acc</span> <span class="o">=</span> <span class="n">add_terms</span><span class="p">(</span>
-            <span class="p">[</span><span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">::</span><span class="mi">2</span><span class="p">]</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">terms</span> <span class="k">if</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span> <span class="n">prec</span><span class="p">,</span> <span class="n">target_prec</span><span class="p">)</span>
-        <span class="n">acc</span> <span class="o">=</span> <span class="n">complex_accuracy</span><span class="p">((</span><span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_acc</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">acc</span> <span class="o">&gt;=</span> <span class="n">target_prec</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;verbose&#39;</span><span class="p">):</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&quot;ADD: wanted&quot;</span><span class="p">,</span> <span class="n">target_prec</span><span class="p">,</span> <span class="s">&quot;accurate bits, got&quot;</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_acc</span><span class="p">)</span>
-            <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">prec</span> <span class="o">-</span> <span class="n">target_prec</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">options</span><span class="p">[</span><span class="s">&#39;maxprec&#39;</span><span class="p">]:</span>
-                <span class="k">break</span>
-
-            <span class="n">prec</span> <span class="o">=</span> <span class="n">prec</span> <span class="o">+</span> <span class="nb">max</span><span class="p">(</span><span class="mi">10</span> <span class="o">+</span> <span class="mi">2</span><span class="o">**</span><span class="n">i</span><span class="p">,</span> <span class="n">target_prec</span> <span class="o">-</span> <span class="n">acc</span><span class="p">)</span>
-            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;verbose&#39;</span><span class="p">):</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&quot;ADD: restarting with prec&quot;</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-
-    <span class="n">options</span><span class="p">[</span><span class="s">&#39;maxprec&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">oldmaxprec</span>
-    <span class="k">if</span> <span class="n">iszero</span><span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="n">scaled</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="n">re</span> <span class="o">=</span> <span class="n">scaled_zero</span><span class="p">(</span><span class="n">re</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">iszero</span><span class="p">(</span><span class="n">im</span><span class="p">,</span> <span class="n">scaled</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="n">im</span> <span class="o">=</span> <span class="n">scaled_zero</span><span class="p">(</span><span class="n">im</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_acc</span>
-
-
-<span class="k">def</span> <span class="nf">evalf_mul</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">):</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="n">pure_complex</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">res</span><span class="p">:</span>
-        <span class="c"># the only pure complex that is a mul is h*I</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">res</span>
-        <span class="n">im</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">im_acc</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">None</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">im_acc</span>
-    <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">v</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="c"># With guard digits, multiplication in the real case does not destroy</span>
-    <span class="c"># accuracy. This is also true in the complex case when considering the</span>
-    <span class="c"># total accuracy; however accuracy for the real or imaginary parts</span>
-    <span class="c"># separately may be lower.</span>
-    <span class="n">acc</span> <span class="o">=</span> <span class="n">prec</span>
-
-    <span class="c"># XXX: big overestimate</span>
-    <span class="n">working_prec</span> <span class="o">=</span> <span class="n">prec</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">+</span> <span class="mi">5</span>
-
-    <span class="c"># Empty product is 1</span>
-    <span class="n">start</span> <span class="o">=</span> <span class="n">man</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">bc</span> <span class="o">=</span> <span class="n">MPZ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span>
-
-    <span class="c"># First, we multiply all pure real or pure imaginary numbers.</span>
-    <span class="c"># direction tells us that the result should be multiplied by</span>
-    <span class="c"># I**direction; all other numbers get put into complex_factors</span>
-    <span class="c"># to be multiplied out after the first phase.</span>
-    <span class="n">last</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-    <span class="n">direction</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span>
-    <span class="n">complex_factors</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">arg</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="n">last</span> <span class="ow">and</span> <span class="n">pure_complex</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">arg</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-            <span class="k">continue</span>
-        <span class="k">elif</span> <span class="n">i</span> <span class="o">==</span> <span class="n">last</span> <span class="ow">and</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-            <span class="k">continue</span>
-        <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_acc</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">working_prec</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">re</span> <span class="ow">and</span> <span class="n">im</span><span class="p">:</span>
-            <span class="n">complex_factors</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_acc</span><span class="p">))</span>
-            <span class="k">continue</span>
-        <span class="k">elif</span> <span class="n">re</span><span class="p">:</span>
-            <span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">w_acc</span> <span class="o">=</span> <span class="n">re</span><span class="p">,</span> <span class="n">re_acc</span>
-        <span class="k">elif</span> <span class="n">im</span><span class="p">:</span>
-            <span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">w_acc</span> <span class="o">=</span> <span class="n">im</span><span class="p">,</span> <span class="n">im_acc</span>
-            <span class="n">direction</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-        <span class="n">direction</span> <span class="o">+=</span> <span class="mi">2</span><span class="o">*</span><span class="n">s</span>
-        <span class="n">man</span> <span class="o">*=</span> <span class="n">m</span>
-        <span class="n">exp</span> <span class="o">+=</span> <span class="n">e</span>
-        <span class="n">bc</span> <span class="o">+=</span> <span class="n">b</span>
-        <span class="k">if</span> <span class="n">bc</span> <span class="o">&gt;</span> <span class="mi">3</span><span class="o">*</span><span class="n">working_prec</span><span class="p">:</span>
-            <span class="n">man</span> <span class="o">&gt;&gt;=</span> <span class="n">working_prec</span>
-            <span class="n">exp</span> <span class="o">+=</span> <span class="n">working_prec</span>
-        <span class="n">acc</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">acc</span><span class="p">,</span> <span class="n">w_acc</span><span class="p">)</span>
-    <span class="n">sign</span> <span class="o">=</span> <span class="p">(</span><span class="n">direction</span> <span class="o">&amp;</span> <span class="mi">2</span><span class="p">)</span> <span class="o">&gt;&gt;</span> <span class="mi">1</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">complex_factors</span><span class="p">:</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="n">normalize</span><span class="p">(</span><span class="n">sign</span><span class="p">,</span> <span class="n">man</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">bitcount</span><span class="p">(</span><span class="n">man</span><span class="p">),</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">)</span>
-        <span class="c"># multiply by i</span>
-        <span class="k">if</span> <span class="n">direction</span> <span class="o">&amp;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">acc</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">v</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">acc</span><span class="p">,</span> <span class="bp">None</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="c"># initialize with the first term</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">man</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">bc</span><span class="p">)</span> <span class="o">!=</span> <span class="n">start</span><span class="p">:</span>
-            <span class="c"># there was a real part; give it an imaginary part</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="p">(</span><span class="n">sign</span><span class="p">,</span> <span class="n">man</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">bitcount</span><span class="p">(</span><span class="n">man</span><span class="p">)),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">MPZ</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-            <span class="n">i0</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># there is no real part to start (other than the starting 1)</span>
-            <span class="n">wre</span><span class="p">,</span> <span class="n">wim</span><span class="p">,</span> <span class="n">wre_acc</span><span class="p">,</span> <span class="n">wim_acc</span> <span class="o">=</span> <span class="n">complex_factors</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">acc</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">acc</span><span class="p">,</span>
-                      <span class="n">complex_accuracy</span><span class="p">((</span><span class="n">wre</span><span class="p">,</span> <span class="n">wim</span><span class="p">,</span> <span class="n">wre_acc</span><span class="p">,</span> <span class="n">wim_acc</span><span class="p">)))</span>
-            <span class="n">re</span> <span class="o">=</span> <span class="n">wre</span>
-            <span class="n">im</span> <span class="o">=</span> <span class="n">wim</span>
-            <span class="n">i0</span> <span class="o">=</span> <span class="mi">1</span>
-
-        <span class="k">for</span> <span class="n">wre</span><span class="p">,</span> <span class="n">wim</span><span class="p">,</span> <span class="n">wre_acc</span><span class="p">,</span> <span class="n">wim_acc</span> <span class="ow">in</span> <span class="n">complex_factors</span><span class="p">[</span><span class="n">i0</span><span class="p">:]:</span>
-            <span class="c"># acc is the overall accuracy of the product; we aren&#39;t</span>
-            <span class="c"># computing exact accuracies of the product.</span>
-            <span class="n">acc</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">acc</span><span class="p">,</span>
-                      <span class="n">complex_accuracy</span><span class="p">((</span><span class="n">wre</span><span class="p">,</span> <span class="n">wim</span><span class="p">,</span> <span class="n">wre_acc</span><span class="p">,</span> <span class="n">wim_acc</span><span class="p">)))</span>
-
-            <span class="n">use_prec</span> <span class="o">=</span> <span class="n">working_prec</span>
-            <span class="n">A</span> <span class="o">=</span> <span class="n">mpf_mul</span><span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="n">wre</span><span class="p">,</span> <span class="n">use_prec</span><span class="p">)</span>
-            <span class="n">B</span> <span class="o">=</span> <span class="n">mpf_mul</span><span class="p">(</span><span class="n">mpf_neg</span><span class="p">(</span><span class="n">im</span><span class="p">),</span> <span class="n">wim</span><span class="p">,</span> <span class="n">use_prec</span><span class="p">)</span>
-            <span class="n">C</span> <span class="o">=</span> <span class="n">mpf_mul</span><span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="n">wim</span><span class="p">,</span> <span class="n">use_prec</span><span class="p">)</span>
-            <span class="n">D</span> <span class="o">=</span> <span class="n">mpf_mul</span><span class="p">(</span><span class="n">im</span><span class="p">,</span> <span class="n">wre</span><span class="p">,</span> <span class="n">use_prec</span><span class="p">)</span>
-            <span class="n">re</span> <span class="o">=</span> <span class="n">mpf_add</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">use_prec</span><span class="p">)</span>
-            <span class="n">im</span> <span class="o">=</span> <span class="n">mpf_add</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">use_prec</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;verbose&#39;</span><span class="p">):</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&quot;MUL: wanted&quot;</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="s">&quot;accurate bits, got&quot;</span><span class="p">,</span> <span class="n">acc</span><span class="p">)</span>
-        <span class="c"># multiply by I</span>
-        <span class="k">if</span> <span class="n">direction</span> <span class="o">&amp;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="n">mpf_neg</span><span class="p">(</span><span class="n">im</span><span class="p">),</span> <span class="n">re</span>
-        <span class="k">return</span> <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">acc</span><span class="p">,</span> <span class="n">acc</span>
-
-
-<span class="k">def</span> <span class="nf">evalf_pow</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">):</span>
-
-    <span class="n">target_prec</span> <span class="o">=</span> <span class="n">prec</span>
-    <span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="n">v</span><span class="o">.</span><span class="n">args</span>
-
-    <span class="c"># We handle x**n separately. This has two purposes: 1) it is much</span>
-    <span class="c"># faster, because we avoid calling evalf on the exponent, and 2) it</span>
-    <span class="c"># allows better handling of real/imaginary parts that are exactly zero</span>
-    <span class="k">if</span> <span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">exp</span><span class="o">.</span><span class="n">p</span>
-        <span class="c"># Exact</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">p</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">fone</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="bp">None</span>
-        <span class="c"># Exponentiation by p magnifies relative error by |p|, so the</span>
-        <span class="c"># base must be evaluated with increased precision if p is large</span>
-        <span class="n">prec</span> <span class="o">+=</span> <span class="nb">int</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">p</span><span class="p">),</span> <span class="mi">2</span><span class="p">))</span>
-        <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_acc</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">prec</span> <span class="o">+</span> <span class="mi">5</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-        <span class="c"># Real to integer power</span>
-        <span class="k">if</span> <span class="n">re</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">im</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">mpf_pow_int</span><span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">target_prec</span><span class="p">),</span> <span class="bp">None</span><span class="p">,</span> <span class="n">target_prec</span><span class="p">,</span> <span class="bp">None</span>
-        <span class="c"># (x*I)**n = I**n * x**n</span>
-        <span class="k">if</span> <span class="n">im</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">re</span><span class="p">:</span>
-            <span class="n">z</span> <span class="o">=</span> <span class="n">mpf_pow_int</span><span class="p">(</span><span class="n">im</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">target_prec</span><span class="p">)</span>
-            <span class="n">case</span> <span class="o">=</span> <span class="n">p</span> <span class="o">%</span> <span class="mi">4</span>
-            <span class="k">if</span> <span class="n">case</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">z</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">target_prec</span><span class="p">,</span> <span class="bp">None</span>
-            <span class="k">if</span> <span class="n">case</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">target_prec</span>
-            <span class="k">if</span> <span class="n">case</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">mpf_neg</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="bp">None</span><span class="p">,</span> <span class="n">target_prec</span><span class="p">,</span> <span class="bp">None</span>
-            <span class="k">if</span> <span class="n">case</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span><span class="p">,</span> <span class="n">mpf_neg</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="bp">None</span><span class="p">,</span> <span class="n">target_prec</span>
-        <span class="c"># Zero raised to an integer power</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">re</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-        <span class="c"># General complex number to arbitrary integer power</span>
-        <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="n">libmp</span><span class="o">.</span><span class="n">mpc_pow_int</span><span class="p">((</span><span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">),</span> <span class="n">p</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-        <span class="c"># Assumes full accuracy in input</span>
-        <span class="k">return</span> <span class="n">finalize_complex</span><span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">target_prec</span><span class="p">)</span>
-
-    <span class="c"># Pure square root</span>
-    <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">:</span>
-        <span class="n">xre</span><span class="p">,</span> <span class="n">xim</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">prec</span> <span class="o">+</span> <span class="mi">5</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-        <span class="c"># General complex square root</span>
-        <span class="k">if</span> <span class="n">xim</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="n">libmp</span><span class="o">.</span><span class="n">mpc_sqrt</span><span class="p">((</span><span class="n">xre</span> <span class="ow">or</span> <span class="n">fzero</span><span class="p">,</span> <span class="n">xim</span><span class="p">),</span> <span class="n">prec</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">finalize_complex</span><span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">xre</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-        <span class="c"># Square root of a negative real number</span>
-        <span class="k">if</span> <span class="n">mpf_lt</span><span class="p">(</span><span class="n">xre</span><span class="p">,</span> <span class="n">fzero</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">None</span><span class="p">,</span> <span class="n">mpf_sqrt</span><span class="p">(</span><span class="n">mpf_neg</span><span class="p">(</span><span class="n">xre</span><span class="p">),</span> <span class="n">prec</span><span class="p">),</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span>
-        <span class="c"># Positive square root</span>
-        <span class="k">return</span> <span class="n">mpf_sqrt</span><span class="p">(</span><span class="n">xre</span><span class="p">,</span> <span class="n">prec</span><span class="p">),</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="bp">None</span>
-
-    <span class="c"># We first evaluate the exponent to find its magnitude</span>
-    <span class="c"># This determines the working precision that must be used</span>
-    <span class="n">prec</span> <span class="o">+=</span> <span class="mi">10</span>
-    <span class="n">yre</span><span class="p">,</span> <span class="n">yim</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">exp</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-    <span class="c"># Special cases: x**0</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">yre</span> <span class="ow">or</span> <span class="n">yim</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">fone</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="bp">None</span>
-
-    <span class="n">ysize</span> <span class="o">=</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">yre</span><span class="p">)</span>
-    <span class="c"># Restart if too big</span>
-    <span class="c"># XXX: prec + ysize might exceed maxprec</span>
-    <span class="k">if</span> <span class="n">ysize</span> <span class="o">&gt;</span> <span class="mi">5</span><span class="p">:</span>
-        <span class="n">prec</span> <span class="o">+=</span> <span class="n">ysize</span>
-        <span class="n">yre</span><span class="p">,</span> <span class="n">yim</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">exp</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-
-    <span class="c"># Pure exponential function; no need to evalf the base</span>
-    <span class="k">if</span> <span class="n">base</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Exp1</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">yim</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="n">libmp</span><span class="o">.</span><span class="n">mpc_exp</span><span class="p">((</span><span class="n">yre</span> <span class="ow">or</span> <span class="n">fzero</span><span class="p">,</span> <span class="n">yim</span><span class="p">),</span> <span class="n">prec</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">finalize_complex</span><span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">target_prec</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">mpf_exp</span><span class="p">(</span><span class="n">yre</span><span class="p">,</span> <span class="n">target_prec</span><span class="p">),</span> <span class="bp">None</span><span class="p">,</span> <span class="n">target_prec</span><span class="p">,</span> <span class="bp">None</span>
-
-    <span class="n">xre</span><span class="p">,</span> <span class="n">xim</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">prec</span> <span class="o">+</span> <span class="mi">5</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-    <span class="c"># 0**y</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">xre</span> <span class="ow">or</span> <span class="n">xim</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-
-    <span class="c"># (real ** complex) or (complex ** complex)</span>
-    <span class="k">if</span> <span class="n">yim</span><span class="p">:</span>
-        <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="n">libmp</span><span class="o">.</span><span class="n">mpc_pow</span><span class="p">(</span>
-            <span class="p">(</span><span class="n">xre</span> <span class="ow">or</span> <span class="n">fzero</span><span class="p">,</span> <span class="n">xim</span> <span class="ow">or</span> <span class="n">fzero</span><span class="p">),</span> <span class="p">(</span><span class="n">yre</span> <span class="ow">or</span> <span class="n">fzero</span><span class="p">,</span> <span class="n">yim</span><span class="p">),</span>
-            <span class="n">target_prec</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">finalize_complex</span><span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">target_prec</span><span class="p">)</span>
-    <span class="c"># complex ** real</span>
-    <span class="k">if</span> <span class="n">xim</span><span class="p">:</span>
-        <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="n">libmp</span><span class="o">.</span><span class="n">mpc_pow_mpf</span><span class="p">((</span><span class="n">xre</span> <span class="ow">or</span> <span class="n">fzero</span><span class="p">,</span> <span class="n">xim</span><span class="p">),</span> <span class="n">yre</span><span class="p">,</span> <span class="n">target_prec</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">finalize_complex</span><span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">target_prec</span><span class="p">)</span>
-    <span class="c"># negative ** real</span>
-    <span class="k">elif</span> <span class="n">mpf_lt</span><span class="p">(</span><span class="n">xre</span><span class="p">,</span> <span class="n">fzero</span><span class="p">):</span>
-        <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="n">libmp</span><span class="o">.</span><span class="n">mpc_pow_mpf</span><span class="p">((</span><span class="n">xre</span><span class="p">,</span> <span class="n">fzero</span><span class="p">),</span> <span class="n">yre</span><span class="p">,</span> <span class="n">target_prec</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">finalize_complex</span><span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">target_prec</span><span class="p">)</span>
-    <span class="c"># positive ** real</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">mpf_pow</span><span class="p">(</span><span class="n">xre</span><span class="p">,</span> <span class="n">yre</span><span class="p">,</span> <span class="n">target_prec</span><span class="p">),</span> <span class="bp">None</span><span class="p">,</span> <span class="n">target_prec</span><span class="p">,</span> <span class="bp">None</span>
-
-
-<span class="c">#----------------------------------------------------------------------------#</span>
-<span class="c">#                                                                            #</span>
-<span class="c">#                            Special functions                               #</span>
-<span class="c">#                                                                            #</span>
-<span class="c">#----------------------------------------------------------------------------#</span>
-<span class="k">def</span> <span class="nf">evalf_trig</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    This function handles sin and cos of real arguments.</span>
-
-<span class="sd">    TODO: should also handle tan and complex arguments.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">v</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">C</span><span class="o">.</span><span class="n">cos</span><span class="p">:</span>
-        <span class="n">func</span> <span class="o">=</span> <span class="n">mpf_cos</span>
-    <span class="k">elif</span> <span class="n">v</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">C</span><span class="o">.</span><span class="n">sin</span><span class="p">:</span>
-        <span class="n">func</span> <span class="o">=</span> <span class="n">mpf_sin</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-    <span class="n">arg</span> <span class="o">=</span> <span class="n">v</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="c"># 20 extra bits is possibly overkill. It does make the need</span>
-    <span class="c"># to restart very unlikely</span>
-    <span class="n">xprec</span> <span class="o">=</span> <span class="n">prec</span> <span class="o">+</span> <span class="mi">20</span>
-    <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_acc</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">xprec</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">im</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">re</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">v</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">C</span><span class="o">.</span><span class="n">cos</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">fone</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="bp">None</span>
-        <span class="k">elif</span> <span class="n">v</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">C</span><span class="o">.</span><span class="n">sin</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-    <span class="c"># For trigonometric functions, we are interested in the</span>
-    <span class="c"># fixed-point (absolute) accuracy of the argument.</span>
-    <span class="n">xsize</span> <span class="o">=</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">re</span><span class="p">)</span>
-    <span class="c"># Magnitude &lt;= 1.0. OK to compute directly, because there is no</span>
-    <span class="c"># danger of hitting the first root of cos (with sin, magnitude</span>
-    <span class="c"># &lt;= 2.0 would actually be ok)</span>
-    <span class="k">if</span> <span class="n">xsize</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">func</span><span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">),</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="bp">None</span>
-    <span class="c"># Very large</span>
-    <span class="k">if</span> <span class="n">xsize</span> <span class="o">&gt;=</span> <span class="mi">10</span><span class="p">:</span>
-        <span class="n">xprec</span> <span class="o">=</span> <span class="n">prec</span> <span class="o">+</span> <span class="n">xsize</span>
-        <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_acc</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">xprec</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-    <span class="c"># Need to repeat in case the argument is very close to a</span>
-    <span class="c"># multiple of pi (or pi/2), hitting close to a root</span>
-    <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">y</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">)</span>
-        <span class="n">ysize</span> <span class="o">=</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-        <span class="n">gap</span> <span class="o">=</span> <span class="o">-</span><span class="n">ysize</span>
-        <span class="n">accuracy</span> <span class="o">=</span> <span class="p">(</span><span class="n">xprec</span> <span class="o">-</span> <span class="n">xsize</span><span class="p">)</span> <span class="o">-</span> <span class="n">gap</span>
-        <span class="k">if</span> <span class="n">accuracy</span> <span class="o">&lt;</span> <span class="n">prec</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;verbose&#39;</span><span class="p">):</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&quot;SIN/COS&quot;</span><span class="p">,</span> <span class="n">accuracy</span><span class="p">,</span> <span class="s">&quot;wanted&quot;</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="s">&quot;gap&quot;</span><span class="p">,</span> <span class="n">gap</span><span class="p">)</span>
-                <span class="k">print</span><span class="p">(</span><span class="n">to_str</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">xprec</span> <span class="o">&gt;</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;maxprec&#39;</span><span class="p">,</span> <span class="n">DEFAULT_MAXPREC</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">y</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">accuracy</span><span class="p">,</span> <span class="bp">None</span>
-            <span class="n">xprec</span> <span class="o">+=</span> <span class="n">gap</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_acc</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">xprec</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-            <span class="k">continue</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">y</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="bp">None</span>
-
-
-<span class="k">def</span> <span class="nf">evalf_log</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">):</span>
-    <span class="n">arg</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">workprec</span> <span class="o">=</span> <span class="n">prec</span> <span class="o">+</span> <span class="mi">10</span>
-    <span class="n">xre</span><span class="p">,</span> <span class="n">xim</span><span class="p">,</span> <span class="n">xacc</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">workprec</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">xim</span><span class="p">:</span>
-        <span class="c"># XXX: use get_abs etc instead</span>
-        <span class="n">re</span> <span class="o">=</span> <span class="n">evalf_log</span><span class="p">(</span>
-            <span class="n">C</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Abs</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">),</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">),</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-        <span class="n">im</span> <span class="o">=</span> <span class="n">mpf_atan2</span><span class="p">(</span><span class="n">xim</span><span class="p">,</span> <span class="n">xre</span> <span class="ow">or</span> <span class="n">fzero</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">re</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">im</span><span class="p">,</span> <span class="n">re</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">prec</span>
-
-    <span class="n">imaginary_term</span> <span class="o">=</span> <span class="p">(</span><span class="n">mpf_cmp</span><span class="p">(</span><span class="n">xre</span><span class="p">,</span> <span class="n">fzero</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">)</span>
-
-    <span class="n">re</span> <span class="o">=</span> <span class="n">mpf_log</span><span class="p">(</span><span class="n">mpf_abs</span><span class="p">(</span><span class="n">xre</span><span class="p">),</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">)</span>
-    <span class="n">size</span> <span class="o">=</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">re</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">prec</span> <span class="o">-</span> <span class="n">size</span> <span class="o">&gt;</span> <span class="n">workprec</span><span class="p">:</span>
-        <span class="c"># We actually need to compute 1+x accurately, not x</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Add</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">,</span> <span class="n">arg</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-        <span class="n">xre</span><span class="p">,</span> <span class="n">xim</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">evalf_add</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-        <span class="n">prec2</span> <span class="o">=</span> <span class="n">workprec</span> <span class="o">-</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">xre</span><span class="p">)</span>
-        <span class="n">re</span> <span class="o">=</span> <span class="n">mpf_log</span><span class="p">(</span><span class="n">mpf_add</span><span class="p">(</span><span class="n">xre</span><span class="p">,</span> <span class="n">fone</span><span class="p">,</span> <span class="n">prec2</span><span class="p">),</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">)</span>
-
-    <span class="n">re_acc</span> <span class="o">=</span> <span class="n">prec</span>
-
-    <span class="k">if</span> <span class="n">imaginary_term</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">re</span><span class="p">,</span> <span class="n">mpf_pi</span><span class="p">(</span><span class="n">prec</span><span class="p">),</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">prec</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">re</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="bp">None</span>
-
-
-<span class="k">def</span> <span class="nf">evalf_atan</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">):</span>
-    <span class="n">arg</span> <span class="o">=</span> <span class="n">v</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">xre</span><span class="p">,</span> <span class="n">xim</span><span class="p">,</span> <span class="n">reacc</span><span class="p">,</span> <span class="n">imacc</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">prec</span> <span class="o">+</span> <span class="mi">5</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">xre</span> <span class="ow">is</span> <span class="n">xim</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">None</span><span class="p">,)</span><span class="o">*</span><span class="mi">4</span>
-    <span class="k">if</span> <span class="n">xim</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-    <span class="k">return</span> <span class="n">mpf_atan</span><span class="p">(</span><span class="n">xre</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">),</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="bp">None</span>
-
-
-<span class="k">def</span> <span class="nf">evalf_subs</span><span class="p">(</span><span class="n">prec</span><span class="p">,</span> <span class="n">subs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Change all Float entries in `subs` to have precision prec. &quot;&quot;&quot;</span>
-    <span class="n">newsubs</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">subs</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-        <span class="n">newsubs</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="n">b</span>
-    <span class="k">return</span> <span class="n">newsubs</span>
-
-
-<span class="k">def</span> <span class="nf">evalf_piecewise</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">):</span>
-    <span class="k">if</span> <span class="s">&#39;subs&#39;</span> <span class="ow">in</span> <span class="n">options</span><span class="p">:</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">evalf_subs</span><span class="p">(</span><span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">[</span><span class="s">&#39;subs&#39;</span><span class="p">]))</span>
-        <span class="n">newopts</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="k">del</span> <span class="n">newopts</span><span class="p">[</span><span class="s">&#39;subs&#39;</span><span class="p">]</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&#39;func&#39;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">evalf</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">newopts</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span> <span class="o">==</span> <span class="nb">float</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">evalf</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Float</span><span class="p">(</span><span class="n">expr</span><span class="p">),</span> <span class="n">prec</span><span class="p">,</span> <span class="n">newopts</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span> <span class="o">==</span> <span class="nb">int</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">evalf</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Integer</span><span class="p">(</span><span class="n">expr</span><span class="p">),</span> <span class="n">prec</span><span class="p">,</span> <span class="n">newopts</span><span class="p">)</span>
-
-    <span class="c"># We still have undefined symbols</span>
-    <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-
-<span class="k">def</span> <span class="nf">evalf_bernoulli</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">):</span>
-    <span class="n">arg</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Bernoulli number index must be an integer&quot;</span><span class="p">)</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-    <span class="n">b</span> <span class="o">=</span> <span class="n">mpf_bernoulli</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">b</span> <span class="o">==</span> <span class="n">fzero</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-    <span class="k">return</span> <span class="n">b</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="bp">None</span>
-
-<span class="c">#----------------------------------------------------------------------------#</span>
-<span class="c">#                                                                            #</span>
-<span class="c">#                            High-level operations                           #</span>
-<span class="c">#                                                                            #</span>
-<span class="c">#----------------------------------------------------------------------------#</span>
-
-
-<span class="k">def</span> <span class="nf">as_mpmath</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">):</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Zero</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Infinity</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">mpf</span><span class="p">(</span><span class="s">&#39;inf&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">mpf</span><span class="p">(</span><span class="s">&#39;-inf&#39;</span><span class="p">)</span>
-    <span class="c"># XXX</span>
-    <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">im</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">mpc</span><span class="p">(</span><span class="n">re</span> <span class="ow">or</span> <span class="n">fzero</span><span class="p">,</span> <span class="n">im</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">mpf</span><span class="p">(</span><span class="n">re</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">do_integral</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">):</span>
-    <span class="n">func</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">x</span><span class="p">,</span> <span class="n">xlow</span><span class="p">,</span> <span class="n">xhigh</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-    <span class="n">orig</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">prec</span>
-
-    <span class="n">oldmaxprec</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;maxprec&#39;</span><span class="p">,</span> <span class="n">DEFAULT_MAXPREC</span><span class="p">)</span>
-    <span class="n">options</span><span class="p">[</span><span class="s">&#39;maxprec&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">oldmaxprec</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">prec</span><span class="p">)</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">mp</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="n">prec</span> <span class="o">+</span> <span class="mi">5</span>
-        <span class="n">xlow</span> <span class="o">=</span> <span class="n">as_mpmath</span><span class="p">(</span><span class="n">xlow</span><span class="p">,</span> <span class="n">prec</span> <span class="o">+</span> <span class="mi">15</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-        <span class="n">xhigh</span> <span class="o">=</span> <span class="n">as_mpmath</span><span class="p">(</span><span class="n">xhigh</span><span class="p">,</span> <span class="n">prec</span> <span class="o">+</span> <span class="mi">15</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-
-        <span class="c"># Integration is like summation, and we can phone home from</span>
-        <span class="c"># the integrand function to update accuracy summation style</span>
-        <span class="c"># Note that this accuracy is inaccurate, since it fails</span>
-        <span class="c"># to account for the variable quadrature weights,</span>
-        <span class="c"># but it is better than nothing</span>
-
-        <span class="n">have_part</span> <span class="o">=</span> <span class="p">[</span><span class="bp">False</span><span class="p">,</span> <span class="bp">False</span><span class="p">]</span>
-        <span class="n">max_real_term</span> <span class="o">=</span> <span class="p">[</span><span class="n">MINUS_INF</span><span class="p">]</span>
-        <span class="n">max_imag_term</span> <span class="o">=</span> <span class="p">[</span><span class="n">MINUS_INF</span><span class="p">]</span>
-
-        <span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_acc</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">mp</span><span class="o">.</span><span class="n">prec</span><span class="p">,</span> <span class="p">{</span><span class="s">&#39;subs&#39;</span><span class="p">:</span> <span class="p">{</span><span class="n">x</span><span class="p">:</span> <span class="n">t</span><span class="p">}})</span>
-
-            <span class="n">have_part</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">re</span> <span class="ow">or</span> <span class="n">have_part</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">have_part</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">im</span> <span class="ow">or</span> <span class="n">have_part</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-            <span class="n">max_real_term</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">max_real_term</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">re</span><span class="p">))</span>
-            <span class="n">max_imag_term</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">max_imag_term</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">im</span><span class="p">))</span>
-
-            <span class="k">if</span> <span class="n">im</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">mpc</span><span class="p">(</span><span class="n">re</span> <span class="ow">or</span> <span class="n">fzero</span><span class="p">,</span> <span class="n">im</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">mpf</span><span class="p">(</span><span class="n">re</span> <span class="ow">or</span> <span class="n">fzero</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;quad&#39;</span><span class="p">)</span> <span class="o">==</span> <span class="s">&#39;osc&#39;</span><span class="p">:</span>
-            <span class="n">A</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Wild</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">x</span><span class="p">])</span>
-            <span class="n">B</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Wild</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">x</span><span class="p">])</span>
-            <span class="n">D</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Wild</span><span class="p">(</span><span class="s">&#39;D&#39;</span><span class="p">)</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">A</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">B</span><span class="p">)</span><span class="o">*</span><span class="n">D</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">m</span><span class="p">:</span>
-                <span class="n">m</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">A</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">B</span><span class="p">)</span><span class="o">*</span><span class="n">D</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">m</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;An integrand of the form sin(A*x+B)*f(x) &quot;</span>
-                  <span class="s">&quot;or cos(A*x+B)*f(x) is required for oscillatory quadrature&quot;</span><span class="p">)</span>
-            <span class="n">period</span> <span class="o">=</span> <span class="n">as_mpmath</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="n">m</span><span class="p">[</span><span class="n">A</span><span class="p">],</span> <span class="n">prec</span> <span class="o">+</span> <span class="mi">15</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">quadosc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="n">xlow</span><span class="p">,</span> <span class="n">xhigh</span><span class="p">],</span> <span class="n">period</span><span class="o">=</span><span class="n">period</span><span class="p">)</span>
-            <span class="c"># XXX: quadosc does not do error detection yet</span>
-            <span class="n">quadrature_error</span> <span class="o">=</span> <span class="n">MINUS_INF</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">result</span><span class="p">,</span> <span class="n">quadrature_error</span> <span class="o">=</span> <span class="n">quadts</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="n">xlow</span><span class="p">,</span> <span class="n">xhigh</span><span class="p">],</span> <span class="n">error</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-            <span class="n">quadrature_error</span> <span class="o">=</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">quadrature_error</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">)</span>
-
-    <span class="k">finally</span><span class="p">:</span>
-        <span class="n">options</span><span class="p">[</span><span class="s">&#39;maxprec&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">oldmaxprec</span>
-        <span class="n">mp</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="n">orig</span>
-
-    <span class="k">if</span> <span class="n">have_part</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-        <span class="n">re</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">real</span><span class="o">.</span><span class="n">_mpf_</span>
-        <span class="k">if</span> <span class="n">re</span> <span class="o">==</span> <span class="n">fzero</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">re_acc</span> <span class="o">=</span> <span class="n">scaled_zero</span><span class="p">(</span>
-                <span class="nb">min</span><span class="p">(</span><span class="o">-</span><span class="n">prec</span><span class="p">,</span> <span class="o">-</span><span class="n">max_real_term</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="o">-</span><span class="n">quadrature_error</span><span class="p">))</span>
-            <span class="n">re</span> <span class="o">=</span> <span class="n">scaled_zero</span><span class="p">(</span><span class="n">re</span><span class="p">)</span>  <span class="c"># handled ok in evalf_integral</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">re_acc</span> <span class="o">=</span> <span class="o">-</span><span class="nb">max</span><span class="p">(</span><span class="n">max_real_term</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">re</span><span class="p">)</span> <span class="o">-</span>
-                          <span class="n">prec</span><span class="p">,</span> <span class="n">quadrature_error</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">re</span><span class="p">,</span> <span class="n">re_acc</span> <span class="o">=</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-
-    <span class="k">if</span> <span class="n">have_part</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span>
-        <span class="n">im</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">imag</span><span class="o">.</span><span class="n">_mpf_</span>
-        <span class="k">if</span> <span class="n">im</span> <span class="o">==</span> <span class="n">fzero</span><span class="p">:</span>
-            <span class="n">im</span><span class="p">,</span> <span class="n">im_acc</span> <span class="o">=</span> <span class="n">scaled_zero</span><span class="p">(</span>
-                <span class="nb">min</span><span class="p">(</span><span class="o">-</span><span class="n">prec</span><span class="p">,</span> <span class="o">-</span><span class="n">max_imag_term</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="o">-</span><span class="n">quadrature_error</span><span class="p">))</span>
-            <span class="n">im</span> <span class="o">=</span> <span class="n">scaled_zero</span><span class="p">(</span><span class="n">im</span><span class="p">)</span>  <span class="c"># handled ok in evalf_integral</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">im_acc</span> <span class="o">=</span> <span class="o">-</span><span class="nb">max</span><span class="p">(</span><span class="n">max_imag_term</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">im</span><span class="p">)</span> <span class="o">-</span>
-                          <span class="n">prec</span><span class="p">,</span> <span class="n">quadrature_error</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">im</span><span class="p">,</span> <span class="n">im_acc</span> <span class="o">=</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_acc</span>
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="k">def</span> <span class="nf">evalf_integral</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">):</span>
-    <span class="n">workprec</span> <span class="o">=</span> <span class="n">prec</span>
-    <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">maxprec</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;maxprec&#39;</span><span class="p">,</span> <span class="n">INF</span><span class="p">)</span>
-    <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">do_integral</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">workprec</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-        <span class="c"># if a scaled_zero comes back accuracy will compute to -1</span>
-        <span class="c"># which will cause workprec to increment by 1</span>
-        <span class="n">accuracy</span> <span class="o">=</span> <span class="n">complex_accuracy</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">accuracy</span> <span class="o">&gt;=</span> <span class="n">prec</span> <span class="ow">or</span> <span class="n">workprec</span> <span class="o">&gt;=</span> <span class="n">maxprec</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">result</span>
-        <span class="n">workprec</span> <span class="o">+=</span> <span class="n">prec</span> <span class="o">-</span> <span class="nb">max</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">**</span><span class="n">i</span><span class="p">,</span> <span class="n">accuracy</span><span class="p">)</span>
-        <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-
-
-<span class="k">def</span> <span class="nf">check_convergence</span><span class="p">(</span><span class="n">numer</span><span class="p">,</span> <span class="n">denom</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns (h, g, p) where</span>
-<span class="sd">    -- h is:</span>
-<span class="sd">        &gt; 0 for convergence of rate 1/factorial(n)**h</span>
-<span class="sd">        &lt; 0 for divergence of rate factorial(n)**(-h)</span>
-<span class="sd">        = 0 for geometric or polynomial convergence or divergence</span>
-
-<span class="sd">    -- abs(g) is:</span>
-<span class="sd">        &gt; 1 for geometric convergence of rate 1/h**n</span>
-<span class="sd">        &lt; 1 for geometric divergence of rate h**n</span>
-<span class="sd">        = 1 for polynomial convergence or divergence</span>
-
-<span class="sd">        (g &lt; 0 indicates an alternating series)</span>
-
-<span class="sd">    -- p is:</span>
-<span class="sd">        &gt; 1 for polynomial convergence of rate 1/n**h</span>
-<span class="sd">        &lt;= 1 for polynomial divergence of rate n**(-h)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">npol</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Poly</span><span class="p">(</span><span class="n">numer</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-    <span class="n">dpol</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Poly</span><span class="p">(</span><span class="n">denom</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="n">npol</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span>
-    <span class="n">q</span> <span class="o">=</span> <span class="n">dpol</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span>
-    <span class="n">rate</span> <span class="o">=</span> <span class="n">q</span> <span class="o">-</span> <span class="n">p</span>
-    <span class="k">if</span> <span class="n">rate</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">rate</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-    <span class="n">constant</span> <span class="o">=</span> <span class="n">dpol</span><span class="o">.</span><span class="n">LC</span><span class="p">()</span> <span class="o">/</span> <span class="n">npol</span><span class="o">.</span><span class="n">LC</span><span class="p">()</span>
-    <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="n">constant</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">rate</span><span class="p">,</span> <span class="n">constant</span><span class="p">,</span> <span class="bp">None</span>
-    <span class="k">if</span> <span class="n">npol</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span> <span class="o">==</span> <span class="n">dpol</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">rate</span><span class="p">,</span> <span class="n">constant</span><span class="p">,</span> <span class="mi">0</span>
-    <span class="n">pc</span> <span class="o">=</span> <span class="n">npol</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()[</span><span class="mi">1</span><span class="p">]</span>
-    <span class="n">qc</span> <span class="o">=</span> <span class="n">dpol</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()[</span><span class="mi">1</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">rate</span><span class="p">,</span> <span class="n">constant</span><span class="p">,</span> <span class="p">(</span><span class="n">qc</span> <span class="o">-</span> <span class="n">pc</span><span class="p">)</span><span class="o">/</span><span class="n">dpol</span><span class="o">.</span><span class="n">LC</span><span class="p">()</span>
-
-
-<span class="k">def</span> <span class="nf">hypsum</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Sum a rapidly convergent infinite hypergeometric series with</span>
-<span class="sd">    given general term, e.g. e = hypsum(1/factorial(n), n). The</span>
-<span class="sd">    quotient between successive terms must be a quotient of integer</span>
-<span class="sd">    polynomials.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">hypersimp</span><span class="p">,</span> <span class="n">lambdify</span>
-
-    <span class="k">if</span> <span class="n">start</span><span class="p">:</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="n">start</span><span class="p">)</span>
-    <span class="n">hs</span> <span class="o">=</span> <span class="n">hypersimp</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">hs</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;a hypergeometric series is required&quot;</span><span class="p">)</span>
-    <span class="n">num</span><span class="p">,</span> <span class="n">den</span> <span class="o">=</span> <span class="n">hs</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-
-    <span class="n">func1</span> <span class="o">=</span> <span class="n">lambdify</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">num</span><span class="p">)</span>
-    <span class="n">func2</span> <span class="o">=</span> <span class="n">lambdify</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">den</span><span class="p">)</span>
-
-    <span class="n">h</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="n">check_convergence</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">den</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">h</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Sum diverges like (n!)^</span><span class="si">%i</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="o">-</span><span class="n">h</span><span class="p">))</span>
-
-    <span class="c"># Direct summation if geometric or faster</span>
-    <span class="k">if</span> <span class="n">h</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="p">(</span><span class="n">h</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="nb">abs</span><span class="p">(</span><span class="n">g</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">term</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="n">term</span> <span class="o">=</span> <span class="p">(</span><span class="n">MPZ</span><span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">p</span><span class="p">)</span> <span class="o">&lt;&lt;</span> <span class="n">prec</span><span class="p">)</span> <span class="o">//</span> <span class="n">term</span><span class="o">.</span><span class="n">q</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">term</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">while</span> <span class="nb">abs</span><span class="p">(</span><span class="n">term</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">5</span><span class="p">:</span>
-            <span class="n">term</span> <span class="o">*=</span> <span class="n">MPZ</span><span class="p">(</span><span class="n">func1</span><span class="p">(</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-            <span class="n">term</span> <span class="o">//=</span> <span class="n">MPZ</span><span class="p">(</span><span class="n">func2</span><span class="p">(</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-            <span class="n">s</span> <span class="o">+=</span> <span class="n">term</span>
-            <span class="n">k</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="n">from_man_exp</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="o">-</span><span class="n">prec</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">alt</span> <span class="o">=</span> <span class="n">g</span> <span class="o">&lt;</span> <span class="mi">0</span>
-        <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="n">g</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Sum diverges like (</span><span class="si">%i</span><span class="s">)^n&quot;</span> <span class="o">%</span> <span class="nb">abs</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">g</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">p</span> <span class="o">&lt;</span> <span class="mi">1</span> <span class="ow">or</span> <span class="p">(</span><span class="n">p</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">alt</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Sum diverges like n^</span><span class="si">%i</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="o">-</span><span class="n">p</span><span class="p">))</span>
-        <span class="c"># We have polynomial convergence: use Richardson extrapolation</span>
-        <span class="c"># Need to use at least quad precision because a lot of cancellation</span>
-        <span class="c"># might occur in the extrapolation process</span>
-        <span class="n">prec2</span> <span class="o">=</span> <span class="mi">4</span><span class="o">*</span><span class="n">prec</span>
-        <span class="n">term</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="n">term</span> <span class="o">=</span> <span class="p">(</span><span class="n">MPZ</span><span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">p</span><span class="p">)</span> <span class="o">&lt;&lt;</span> <span class="n">prec2</span><span class="p">)</span> <span class="o">//</span> <span class="n">term</span><span class="o">.</span><span class="n">q</span>
-
-        <span class="k">def</span> <span class="nf">summand</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">_term</span><span class="o">=</span><span class="p">[</span><span class="n">term</span><span class="p">]):</span>
-            <span class="k">if</span> <span class="n">k</span><span class="p">:</span>
-                <span class="n">k</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-                <span class="n">_term</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*=</span> <span class="n">MPZ</span><span class="p">(</span><span class="n">func1</span><span class="p">(</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-                <span class="n">_term</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">//=</span> <span class="n">MPZ</span><span class="p">(</span><span class="n">func2</span><span class="p">(</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">make_mpf</span><span class="p">(</span><span class="n">from_man_exp</span><span class="p">(</span><span class="n">_term</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="o">-</span><span class="n">prec2</span><span class="p">))</span>
-
-        <span class="n">orig</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">prec</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">mp</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="n">prec</span>
-            <span class="n">v</span> <span class="o">=</span> <span class="n">nsum</span><span class="p">(</span><span class="n">summand</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">mpmath_inf</span><span class="p">],</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;richardson&#39;</span><span class="p">)</span>
-        <span class="k">finally</span><span class="p">:</span>
-            <span class="n">mp</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="n">orig</span>
-        <span class="k">return</span> <span class="n">v</span><span class="o">.</span><span class="n">_mpf_</span>
-
-
-<span class="k">def</span> <span class="nf">evalf_sum</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">):</span>
-    <span class="n">func</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">function</span>
-    <span class="n">limits</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">limits</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">limits</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span> <span class="ow">or</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">limits</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Tuple</span><span class="p">)</span> <span class="ow">or</span> \
-            <span class="nb">len</span><span class="p">(</span><span class="n">limits</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">!=</span> <span class="mi">3</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-    <span class="n">prec2</span> <span class="o">=</span> <span class="n">prec</span> <span class="o">+</span> <span class="mi">10</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">limits</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">b</span> <span class="o">!=</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span> <span class="ow">or</span> <span class="n">a</span> <span class="o">!=</span> <span class="nb">int</span><span class="p">(</span><span class="n">a</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-        <span class="c"># Use fast hypergeometric summation if possible</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="n">hypsum</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">prec2</span><span class="p">)</span>
-        <span class="n">delta</span> <span class="o">=</span> <span class="n">prec</span> <span class="o">-</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="o">&lt;</span> <span class="o">-</span><span class="mi">10</span><span class="p">:</span>
-            <span class="n">v</span> <span class="o">=</span> <span class="n">hypsum</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">delta</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">v</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="nb">min</span><span class="p">(</span><span class="n">prec</span><span class="p">,</span> <span class="n">delta</span><span class="p">),</span> <span class="bp">None</span>
-    <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-        <span class="c"># Euler-Maclaurin summation for general series</span>
-        <span class="n">eps</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Float</span><span class="p">(</span><span class="mf">2.0</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">prec</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">):</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">2</span><span class="o">**</span><span class="n">i</span> <span class="o">*</span> <span class="n">prec</span>
-            <span class="n">s</span><span class="p">,</span> <span class="n">err</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">euler_maclaurin</span><span class="p">(</span><span class="n">m</span><span class="o">=</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="n">eps</span><span class="p">,</span>
-                <span class="n">eval_integral</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-            <span class="n">err</span> <span class="o">=</span> <span class="n">err</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">err</span> <span class="o">&lt;=</span> <span class="n">eps</span><span class="p">:</span>
-                <span class="k">break</span>
-        <span class="n">err</span> <span class="o">=</span> <span class="n">fastlog</span><span class="p">(</span><span class="n">evalf</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">err</span><span class="p">),</span> <span class="mi">20</span><span class="p">,</span> <span class="n">options</span><span class="p">)[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_acc</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">prec2</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>  <span class="c"># issue 3174</span>
-            <span class="c"># when should it try subs if they are in options?</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-        <span class="k">if</span> <span class="n">re_acc</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">re_acc</span> <span class="o">=</span> <span class="o">-</span><span class="n">err</span>
-        <span class="k">if</span> <span class="n">im_acc</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">im_acc</span> <span class="o">=</span> <span class="o">-</span><span class="n">err</span>
-        <span class="k">return</span> <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_acc</span>
-
-
-<span class="c">#----------------------------------------------------------------------------#</span>
-<span class="c">#                                                                            #</span>
-<span class="c">#                            Symbolic interface                              #</span>
-<span class="c">#                                                                            #</span>
-<span class="c">#----------------------------------------------------------------------------#</span>
-
-<span class="k">def</span> <span class="nf">evalf_symbol</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">):</span>
-    <span class="n">val</span> <span class="o">=</span> <span class="n">options</span><span class="p">[</span><span class="s">&#39;subs&#39;</span><span class="p">][</span><span class="n">x</span><span class="p">]</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">val</span><span class="p">,</span> <span class="n">mpf</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">val</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-        <span class="k">return</span> <span class="n">val</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="bp">None</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="s">&#39;_cache&#39;</span> <span class="ow">in</span> <span class="n">options</span><span class="p">:</span>
-            <span class="n">options</span><span class="p">[</span><span class="s">&#39;_cache&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="n">cache</span> <span class="o">=</span> <span class="n">options</span><span class="p">[</span><span class="s">&#39;_cache&#39;</span><span class="p">]</span>
-        <span class="n">cached</span><span class="p">,</span> <span class="n">cached_prec</span> <span class="o">=</span> <span class="n">cache</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="n">MINUS_INF</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">cached_prec</span> <span class="o">&gt;=</span> <span class="n">prec</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cached</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">val</span><span class="p">),</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-        <span class="n">cache</span><span class="p">[</span><span class="n">x</span><span class="o">.</span><span class="n">name</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">v</span>
-
-<span class="n">evalf_table</span> <span class="o">=</span> <span class="bp">None</span>
-
-
-<span class="k">def</span> <span class="nf">_create_evalf_table</span><span class="p">():</span>
-    <span class="k">global</span> <span class="n">evalf_table</span>
-    <span class="n">evalf_table</span> <span class="o">=</span> <span class="p">{</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">:</span> <span class="n">evalf_symbol</span><span class="p">,</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">:</span> <span class="n">evalf_symbol</span><span class="p">,</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">Float</span><span class="p">:</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">:</span> <span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="bp">None</span><span class="p">),</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">Rational</span><span class="p">:</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">:</span> <span class="p">(</span><span class="n">from_rational</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="n">prec</span><span class="p">),</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="bp">None</span><span class="p">),</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">Integer</span><span class="p">:</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">:</span> <span class="p">(</span><span class="n">from_int</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="n">prec</span><span class="p">),</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="bp">None</span><span class="p">),</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">:</span> <span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="bp">None</span><span class="p">),</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">One</span><span class="p">:</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">:</span> <span class="p">(</span><span class="n">fone</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="bp">None</span><span class="p">),</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">Half</span><span class="p">:</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">:</span> <span class="p">(</span><span class="n">fhalf</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="bp">None</span><span class="p">),</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">Pi</span><span class="p">:</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">:</span> <span class="p">(</span><span class="n">mpf_pi</span><span class="p">(</span><span class="n">prec</span><span class="p">),</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="bp">None</span><span class="p">),</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">Exp1</span><span class="p">:</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">:</span> <span class="p">(</span><span class="n">mpf_e</span><span class="p">(</span><span class="n">prec</span><span class="p">),</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="bp">None</span><span class="p">),</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">:</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">:</span> <span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="n">fone</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span><span class="p">),</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">:</span> <span class="p">(</span><span class="n">fnone</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="bp">None</span><span class="p">),</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">NaN</span> <span class="p">:</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">:</span> <span class="p">(</span><span class="n">fnan</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="bp">None</span><span class="p">),</span>
-
-        <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">:</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">:</span> <span class="n">evalf_pow</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Pow</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Exp1</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span>
-        <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">),</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">),</span>
-
-        <span class="n">C</span><span class="o">.</span><span class="n">cos</span><span class="p">:</span> <span class="n">evalf_trig</span><span class="p">,</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">sin</span><span class="p">:</span> <span class="n">evalf_trig</span><span class="p">,</span>
-
-        <span class="n">C</span><span class="o">.</span><span class="n">Add</span><span class="p">:</span> <span class="n">evalf_add</span><span class="p">,</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">Mul</span><span class="p">:</span> <span class="n">evalf_mul</span><span class="p">,</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">Pow</span><span class="p">:</span> <span class="n">evalf_pow</span><span class="p">,</span>
-
-        <span class="n">C</span><span class="o">.</span><span class="n">log</span><span class="p">:</span> <span class="n">evalf_log</span><span class="p">,</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">atan</span><span class="p">:</span> <span class="n">evalf_atan</span><span class="p">,</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">Abs</span><span class="p">:</span> <span class="n">evalf_abs</span><span class="p">,</span>
-
-        <span class="n">C</span><span class="o">.</span><span class="n">re</span><span class="p">:</span> <span class="n">evalf_re</span><span class="p">,</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">im</span><span class="p">:</span> <span class="n">evalf_im</span><span class="p">,</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">floor</span><span class="p">:</span> <span class="n">evalf_floor</span><span class="p">,</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">ceiling</span><span class="p">:</span> <span class="n">evalf_ceiling</span><span class="p">,</span>
-
-        <span class="n">C</span><span class="o">.</span><span class="n">Integral</span><span class="p">:</span> <span class="n">evalf_integral</span><span class="p">,</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">Sum</span><span class="p">:</span> <span class="n">evalf_sum</span><span class="p">,</span>
-        <span class="n">C</span><span class="o">.</span><span class="n">Piecewise</span><span class="p">:</span> <span class="n">evalf_piecewise</span><span class="p">,</span>
-
-        <span class="n">C</span><span class="o">.</span><span class="n">bernoulli</span><span class="p">:</span> <span class="n">evalf_bernoulli</span><span class="p">,</span>
-    <span class="p">}</span>
-
-
-<span class="k">def</span> <span class="nf">evalf</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">):</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">re</span> <span class="k">as</span> <span class="n">re_</span><span class="p">,</span> <span class="n">im</span> <span class="k">as</span> <span class="n">im_</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">rf</span> <span class="o">=</span> <span class="n">evalf_table</span><span class="p">[</span><span class="n">x</span><span class="o">.</span><span class="n">func</span><span class="p">]</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">rf</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-    <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="c"># Fall back to ordinary evalf if possible</span>
-            <span class="k">if</span> <span class="s">&#39;subs&#39;</span> <span class="ow">in</span> <span class="n">options</span><span class="p">:</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">evalf_subs</span><span class="p">(</span><span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">[</span><span class="s">&#39;subs&#39;</span><span class="p">]))</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">re</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">re_</span><span class="p">)</span> <span class="ow">or</span> <span class="n">im</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">im_</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-            <span class="k">if</span> <span class="n">re</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">re</span> <span class="o">=</span> <span class="bp">None</span>
-                <span class="n">reprec</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">re</span> <span class="o">=</span> <span class="n">re</span><span class="o">.</span><span class="n">_to_mpmath</span><span class="p">(</span><span class="n">prec</span><span class="p">,</span> <span class="n">allow_ints</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span><span class="o">.</span><span class="n">_mpf_</span>
-                <span class="n">reprec</span> <span class="o">=</span> <span class="n">prec</span>
-            <span class="k">if</span> <span class="n">im</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">im</span> <span class="o">=</span> <span class="bp">None</span>
-                <span class="n">imprec</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">im</span> <span class="o">=</span> <span class="n">im</span><span class="o">.</span><span class="n">_to_mpmath</span><span class="p">(</span><span class="n">prec</span><span class="p">,</span> <span class="n">allow_ints</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span><span class="o">.</span><span class="n">_mpf_</span>
-                <span class="n">imprec</span> <span class="o">=</span> <span class="n">prec</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">reprec</span><span class="p">,</span> <span class="n">imprec</span>
-        <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-    <span class="k">if</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;verbose&quot;</span><span class="p">):</span>
-        <span class="k">print</span><span class="p">(</span><span class="s">&quot;### input&quot;</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">print</span><span class="p">(</span><span class="s">&quot;### output&quot;</span><span class="p">,</span> <span class="n">to_str</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">or</span> <span class="n">fzero</span><span class="p">,</span> <span class="mi">50</span><span class="p">))</span>
-        <span class="k">print</span><span class="p">(</span><span class="s">&quot;### raw&quot;</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span>  <span class="c"># r[0], r[2]</span>
-        <span class="k">print</span><span class="p">()</span>
-    <span class="n">chop</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;chop&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">chop</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">chop</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">chop_prec</span> <span class="o">=</span> <span class="n">prec</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># convert (approximately) from given tolerance;</span>
-            <span class="c"># the formula here will will make 1e-i rounds to 0 for</span>
-            <span class="c"># i in the range +/-27 while 2e-i will not be chopped</span>
-            <span class="n">chop_prec</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="nb">round</span><span class="p">(</span><span class="o">-</span><span class="mf">3.321</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">chop</span><span class="p">)</span> <span class="o">+</span> <span class="mf">2.5</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">chop_prec</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-                <span class="n">chop_prec</span> <span class="o">-=</span> <span class="mi">1</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">chop_parts</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">chop_prec</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;strict&quot;</span><span class="p">):</span>
-        <span class="n">check_target</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">r</span>
-
-
-<span class="k">class</span> <span class="nc">EvalfMixin</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Mixin class adding evalf capabililty.&quot;&quot;&quot;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">15</span><span class="p">,</span> <span class="n">subs</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">maxn</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">chop</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">quad</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Evaluate the given formula to an accuracy of n digits.</span>
-<span class="sd">        Optional keyword arguments:</span>
-
-<span class="sd">            subs=&lt;dict&gt;</span>
-<span class="sd">                Substitute numerical values for symbols, e.g.</span>
-<span class="sd">                subs={x:3, y:1+pi}.</span>
-
-<span class="sd">            maxn=&lt;integer&gt;</span>
-<span class="sd">                Allow a maximum temporary working precision of maxn digits</span>
-<span class="sd">                (default=100)</span>
-
-<span class="sd">            chop=&lt;bool&gt;</span>
-<span class="sd">                Replace tiny real or imaginary parts in subresults</span>
-<span class="sd">                by exact zeros (default=False)</span>
-
-<span class="sd">            strict=&lt;bool&gt;</span>
-<span class="sd">                Raise PrecisionExhausted if any subresult fails to evaluate</span>
-<span class="sd">                to full accuracy, given the available maxprec</span>
-<span class="sd">                (default=False)</span>
-
-<span class="sd">            quad=&lt;str&gt;</span>
-<span class="sd">                Choose algorithm for numerical quadrature. By default,</span>
-<span class="sd">                tanh-sinh quadrature is used. For oscillatory</span>
-<span class="sd">                integrals on an infinite interval, try quad=&#39;osc&#39;.</span>
-
-<span class="sd">            verbose=&lt;bool&gt;</span>
-<span class="sd">                Print debug information (default=False)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># for sake of sage that doesn&#39;t like evalf(1)</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Number</span><span class="p">):</span>
-            <span class="kn">from</span> <span class="nn">sympy.core.expr</span> <span class="kn">import</span> <span class="n">_mag</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">subs</span><span class="p">,</span> <span class="n">maxn</span><span class="p">,</span> <span class="n">chop</span><span class="p">,</span> <span class="n">strict</span><span class="p">,</span> <span class="n">quad</span><span class="p">,</span> <span class="n">verbose</span><span class="p">)</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="n">_mag</span><span class="p">(</span><span class="n">rv</span><span class="p">)</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="n">rv</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">m</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">rv</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">evalf_table</span><span class="p">:</span>
-            <span class="n">_create_evalf_table</span><span class="p">()</span>
-        <span class="n">prec</span> <span class="o">=</span> <span class="n">dps_to_prec</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="n">options</span> <span class="o">=</span> <span class="p">{</span><span class="s">&#39;maxprec&#39;</span><span class="p">:</span> <span class="nb">max</span><span class="p">(</span><span class="n">prec</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">maxn</span><span class="o">*</span><span class="n">LG10</span><span class="p">)),</span> <span class="s">&#39;chop&#39;</span><span class="p">:</span> <span class="n">chop</span><span class="p">,</span>
-               <span class="s">&#39;strict&#39;</span><span class="p">:</span> <span class="n">strict</span><span class="p">,</span> <span class="s">&#39;verbose&#39;</span><span class="p">:</span> <span class="n">verbose</span><span class="p">}</span>
-        <span class="k">if</span> <span class="n">subs</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">options</span><span class="p">[</span><span class="s">&#39;subs&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">subs</span>
-        <span class="k">if</span> <span class="n">quad</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">options</span><span class="p">[</span><span class="s">&#39;quad&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">quad</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span> <span class="o">+</span> <span class="mi">4</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-            <span class="c"># Fall back to the ordinary evalf</span>
-            <span class="n">v</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">v</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="c"># If the result is numerical, normalize it</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">options</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                <span class="c"># Probably contains symbols or unknown functions</span>
-                <span class="k">return</span> <span class="n">v</span>
-        <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">,</span> <span class="n">im_acc</span> <span class="o">=</span> <span class="n">result</span>
-        <span class="k">if</span> <span class="n">re</span><span class="p">:</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="nb">min</span><span class="p">(</span><span class="n">prec</span><span class="p">,</span> <span class="n">re_acc</span><span class="p">),</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="c">#re = mpf_pos(re, p, rnd)</span>
-            <span class="n">re</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">re</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">if</span> <span class="n">im</span><span class="p">:</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="nb">min</span><span class="p">(</span><span class="n">prec</span><span class="p">,</span> <span class="n">im_acc</span><span class="p">),</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="c">#im = mpf_pos(im, p, rnd)</span>
-            <span class="n">im</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">im</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">re</span> <span class="o">+</span> <span class="n">im</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">re</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="n">evalf</span>
-
-    <span class="k">def</span> <span class="nf">_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Helper for evalf. Does the same thing but takes binary precision&quot;&quot;&quot;</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">r</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="bp">self</span>
-        <span class="k">return</span> <span class="n">r</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">_to_mpmath</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">allow_ints</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="c"># mpmath functions accept ints as input</span>
-        <span class="n">errmsg</span> <span class="o">=</span> <span class="s">&quot;cannot convert to mpmath number&quot;</span>
-        <span class="k">if</span> <span class="n">allow_ints</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s">&#39;_as_mpf_val&#39;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">make_mpf</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_as_mpf_val</span><span class="p">(</span><span class="n">prec</span><span class="p">))</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="p">{})</span>
-            <span class="k">if</span> <span class="n">im</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">re</span><span class="p">:</span>
-                    <span class="n">re</span> <span class="o">=</span> <span class="n">fzero</span>
-                <span class="k">return</span> <span class="n">make_mpc</span><span class="p">((</span><span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">))</span>
-            <span class="k">elif</span> <span class="n">re</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">make_mpf</span><span class="p">(</span><span class="n">re</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">make_mpf</span><span class="p">(</span><span class="n">fzero</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-            <span class="n">v</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">v</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="n">errmsg</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">v</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">make_mpf</span><span class="p">(</span><span class="n">v</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">)</span>
-            <span class="c"># Number + Number*I is also fine</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="n">v</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">allow_ints</span> <span class="ow">and</span> <span class="n">re</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="n">re</span> <span class="o">=</span> <span class="n">from_int</span><span class="p">(</span><span class="n">re</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">re</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-                <span class="n">re</span> <span class="o">=</span> <span class="n">re</span><span class="o">.</span><span class="n">_mpf_</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="n">errmsg</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">allow_ints</span> <span class="ow">and</span> <span class="n">im</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="n">im</span> <span class="o">=</span> <span class="n">from_int</span><span class="p">(</span><span class="n">im</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">im</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-                <span class="n">im</span> <span class="o">=</span> <span class="n">im</span><span class="o">.</span><span class="n">_mpf_</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="n">errmsg</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">make_mpc</span><span class="p">((</span><span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">))</span>
-
-
-<div class="viewcode-block" id="N"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.evalf.N">[docs]</a><span class="k">def</span> <span class="nf">N</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">15</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Calls x.evalf(n, \*\*options).</span>
-
-<span class="sd">    Both .n() and N() are equivalent to .evalf(); use the one that you like better.</span>
-<span class="sd">    See also the docstring of .evalf() for information on the options.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Sum, Symbol, oo, N</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import k</span>
-<span class="sd">    &gt;&gt;&gt; Sum(1/k**k, (k, 1, oo))</span>
-<span class="sd">    Sum(k**(-k), (k, 1, oo))</span>
-<span class="sd">    &gt;&gt;&gt; N(_, 4)</span>
-<span class="sd">    1.291</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span></div>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
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-  <h3>Quick search</h3>
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diff --git a/dev-py3k/_modules/sympy/core/expr.html b/dev-py3k/_modules/sympy/core/expr.html
deleted file mode 100644
index 8c9f2267bfa..00000000000
--- a/dev-py3k/_modules/sympy/core/expr.html
+++ /dev/null
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-            
-  <h1>Source code for sympy.core.expr</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">.core</span> <span class="kn">import</span> <span class="n">C</span>
-<span class="kn">from</span> <span class="nn">.sympify</span> <span class="kn">import</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">.basic</span> <span class="kn">import</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">Atom</span>
-<span class="kn">from</span> <span class="nn">.singleton</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="kn">from</span> <span class="nn">.evalf</span> <span class="kn">import</span> <span class="n">EvalfMixin</span><span class="p">,</span> <span class="n">pure_complex</span>
-<span class="kn">from</span> <span class="nn">.decorators</span> <span class="kn">import</span> <span class="n">_sympifyit</span><span class="p">,</span> <span class="n">call_highest_priority</span>
-<span class="kn">from</span> <span class="nn">.cache</span> <span class="kn">import</span> <span class="n">cacheit</span>
-<span class="kn">from</span> <span class="nn">.compatibility</span> <span class="kn">import</span> <span class="nb">reduce</span><span class="p">,</span> <span class="n">as_int</span><span class="p">,</span> <span class="n">default_sort_key</span>
-<span class="kn">from</span> <span class="nn">sympy.mpmath.libmp</span> <span class="kn">import</span> <span class="n">mpf_log</span><span class="p">,</span> <span class="n">prec_to_dps</span>
-
-<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</span>
-<span class="kn">from</span> <span class="nn">inspect</span> <span class="kn">import</span> <span class="n">getmro</span>
-<span class="kn">import</span> <span class="nn">collections</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-
-<div class="viewcode-block" id="Expr"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr">[docs]</a><span class="k">class</span> <span class="nc">Expr</span><span class="p">(</span><span class="n">Basic</span><span class="p">,</span> <span class="n">EvalfMixin</span><span class="p">):</span>
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_diff_wrt</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Is it allowed to take derivative wrt to this instance.</span>
-
-<span class="sd">        This determines if it is allowed to take derivatives wrt this object.</span>
-<span class="sd">        Subclasses such as Symbol, Function and Derivative should return True</span>
-<span class="sd">        to enable derivatives wrt them. The implementation in Derivative</span>
-<span class="sd">        separates the Symbol and non-Symbol _diff_wrt=True variables and</span>
-<span class="sd">        temporarily converts the non-Symbol vars in Symbols when performing</span>
-<span class="sd">        the differentiation.</span>
-
-<span class="sd">        Note, see the docstring of Derivative for how this should work</span>
-<span class="sd">        mathematically.  In particular, note that expr.subs(yourclass, Symbol)</span>
-<span class="sd">        should be well-defined on a structural level, or this will lead to</span>
-<span class="sd">        inconsistent results.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Expr</span>
-<span class="sd">        &gt;&gt;&gt; e = Expr()</span>
-<span class="sd">        &gt;&gt;&gt; e._diff_wrt</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; class MyClass(Expr):</span>
-<span class="sd">        ...     _diff_wrt = True</span>
-<span class="sd">        ...</span>
-<span class="sd">        &gt;&gt;&gt; (2*MyClass()).diff(MyClass())</span>
-<span class="sd">        2</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">sort_key</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">expr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-            <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="n">expr</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">expr</span><span class="p">),)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="n">args</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_ordered_terms</span><span class="p">(</span><span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                <span class="n">args</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_ordered_factors</span><span class="p">(</span><span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">args</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span>
-
-            <span class="n">args</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span>
-                <span class="p">[</span> <span class="n">default_sort_key</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span> <span class="p">])</span>
-
-        <span class="n">args</span> <span class="o">=</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">),</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">args</span><span class="p">))</span>
-        <span class="n">exp</span> <span class="o">=</span> <span class="n">exp</span><span class="o">.</span><span class="n">sort_key</span><span class="p">(</span><span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">class_key</span><span class="p">(),</span> <span class="n">args</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">coeff</span>
-
-    <span class="k">def</span> <span class="nf">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="c"># (x+Lambda(y, 2*y))(z) -&gt; x+2*z</span>
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">_recursive_call</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">args</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_recursive_call</span><span class="p">(</span><span class="n">expr_to_call</span><span class="p">,</span> <span class="n">on_args</span><span class="p">):</span>
-        <span class="k">def</span> <span class="nf">the_call_method_is_overridden</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">cls</span> <span class="ow">in</span> <span class="n">getmro</span><span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">expr</span><span class="p">)):</span>
-                <span class="k">if</span> <span class="s">&#39;__call__&#39;</span> <span class="ow">in</span> <span class="n">cls</span><span class="o">.</span><span class="n">__dict__</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">cls</span> <span class="o">!=</span> <span class="n">Expr</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr_to_call</span><span class="p">,</span> <span class="n">collections</span><span class="o">.</span><span class="n">Callable</span><span class="p">)</span> <span class="ow">and</span> <span class="n">the_call_method_is_overridden</span><span class="p">(</span><span class="n">expr_to_call</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr_to_call</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">):</span>  <span class="c"># XXX When you call a Symbol it is</span>
-                <span class="k">return</span> <span class="n">expr_to_call</span>               <span class="c"># transformed into an UndefFunction</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">expr_to_call</span><span class="p">(</span><span class="o">*</span><span class="n">on_args</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">expr_to_call</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">Expr</span><span class="o">.</span><span class="n">_recursive_call</span><span class="p">(</span>
-                <span class="n">sub</span><span class="p">,</span> <span class="n">on_args</span><span class="p">)</span> <span class="k">for</span> <span class="n">sub</span> <span class="ow">in</span> <span class="n">expr_to_call</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-            <span class="k">return</span> <span class="nb">type</span><span class="p">(</span><span class="n">expr_to_call</span><span class="p">)(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr_to_call</span>
-
-    <span class="c"># ***************</span>
-    <span class="c"># * Arithmetics *</span>
-    <span class="c"># ***************</span>
-    <span class="c"># Expr and its sublcasses use _op_priority to determine which object</span>
-    <span class="c"># passed to a binary special method (__mul__, etc.) will handle the</span>
-    <span class="c"># operation. In general, the &#39;call_highest_priority&#39; decorator will choose</span>
-    <span class="c"># the object with the highest _op_priority to handle the call.</span>
-    <span class="c"># Custom subclasses that want to define their own binary special methods</span>
-    <span class="c"># should set an _op_priority value that is higher than the default.</span>
-    <span class="c">#</span>
-    <span class="c"># **NOTE**:</span>
-    <span class="c"># This is a temporary fix, and will eventually be replaced with</span>
-    <span class="c"># something better and more powerful.  See issue 2411.</span>
-    <span class="n">_op_priority</span> <span class="o">=</span> <span class="mf">10.0</span>
-
-    <span class="k">def</span> <span class="nf">__pos__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__abs__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Abs</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__radd__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__add__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__radd__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__rsub__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">-</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__sub__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__rsub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="o">-</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__rmul__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__mul__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__rpow__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__pow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Pow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__pow__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__rpow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Pow</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__rdiv__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Pow</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">))</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__div__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__rdiv__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Pow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">))</span>
-
-    <span class="n">__truediv__</span> <span class="o">=</span> <span class="n">__div__</span>
-    <span class="n">__rtruediv__</span> <span class="o">=</span> <span class="n">__rdiv__</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__rmod__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__mod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Mod</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__mod__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__rmod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Mod</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__int__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># Although we only need to round to the units position, we&#39;ll</span>
-        <span class="c"># get one more digit so the extra testing below can be avoided</span>
-        <span class="c"># unless the rounded value rounded to an integer, e.g. if an</span>
-        <span class="c"># expression were equal to 1.9 and we rounded to the unit position</span>
-        <span class="c"># we would get a 2 and would not know if this rounded up or not</span>
-        <span class="c"># without doing a test (as done below). But if we keep an extra</span>
-        <span class="c"># digit we know that 1.9 is not the same as 1 and there is no</span>
-        <span class="c"># need for further testing: our int value is correct. If the value</span>
-        <span class="c"># were 1.99, however, this would round to 2.0 and our int value is</span>
-        <span class="c"># off by one. So...if our round value is the same as the int value</span>
-        <span class="c"># (regardless of how much extra work we do to calculate extra decimal</span>
-        <span class="c"># places) we need to test whether we are off by one.</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;can&#39;t convert complex to int&quot;</span><span class="p">)</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">i</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">0</span>
-        <span class="c"># off-by-one check</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">r</span> <span class="ow">and</span> <span class="ow">not</span> <span class="p">(</span><span class="bp">self</span> <span class="o">-</span> <span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="mi">0</span><span class="p">):</span>
-            <span class="n">isign</span> <span class="o">=</span> <span class="mi">1</span> <span class="k">if</span> <span class="n">i</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="k">else</span> <span class="o">-</span><span class="mi">1</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">()</span>
-            <span class="c"># in the following (self - i).evalf(2) will not always work while</span>
-            <span class="c"># (self - r).evalf(2) and the use of subs does; if the test that</span>
-            <span class="c"># was added when this comment was added passes, it might be safe</span>
-            <span class="c"># to simply use sign to compute this rather than doing this by hand:</span>
-            <span class="n">diff_sign</span> <span class="o">=</span> <span class="mi">1</span> <span class="k">if</span> <span class="p">(</span><span class="bp">self</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">subs</span><span class="o">=</span><span class="p">{</span><span class="n">x</span><span class="p">:</span> <span class="n">i</span><span class="p">})</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="k">else</span> <span class="o">-</span><span class="mi">1</span>
-            <span class="k">if</span> <span class="n">diff_sign</span> <span class="o">!=</span> <span class="n">isign</span><span class="p">:</span>
-                <span class="n">i</span> <span class="o">-=</span> <span class="n">isign</span>
-        <span class="k">return</span> <span class="n">i</span>
-
-    <span class="k">def</span> <span class="nf">__float__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># Don&#39;t bother testing if it&#39;s a number; if it&#39;s not this is going</span>
-        <span class="c"># to fail, and if it is we still need to check that it evalf&#39;ed to</span>
-        <span class="c"># a number.</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">result</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">float</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">result</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="n">result</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()[</span><span class="mi">1</span><span class="p">]:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;can&#39;t convert complex to float&quot;</span><span class="p">)</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;can&#39;t convert expression to float&quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__complex__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-        <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-        <span class="k">return</span> <span class="nb">complex</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">re</span><span class="p">),</span> <span class="nb">float</span><span class="p">(</span><span class="n">im</span><span class="p">))</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>  <span class="c"># sympy &gt;  other</span>
-    <span class="k">def</span> <span class="nf">__ge__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="n">dif</span> <span class="o">=</span> <span class="bp">self</span> <span class="o">-</span> <span class="n">other</span>
-        <span class="k">if</span> <span class="n">dif</span><span class="o">.</span><span class="n">is_nonnegative</span> <span class="o">!=</span> <span class="n">dif</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">dif</span><span class="o">.</span><span class="n">is_nonnegative</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">GreaterThan</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>  <span class="c"># sympy &gt;  other</span>
-    <span class="k">def</span> <span class="nf">__le__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="n">dif</span> <span class="o">=</span> <span class="bp">self</span> <span class="o">-</span> <span class="n">other</span>
-        <span class="k">if</span> <span class="n">dif</span><span class="o">.</span><span class="n">is_nonpositive</span> <span class="o">!=</span> <span class="n">dif</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">dif</span><span class="o">.</span><span class="n">is_nonpositive</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">LessThan</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>  <span class="c"># sympy &gt;  other</span>
-    <span class="k">def</span> <span class="nf">__gt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="n">dif</span> <span class="o">=</span> <span class="bp">self</span> <span class="o">-</span> <span class="n">other</span>
-        <span class="k">if</span> <span class="n">dif</span><span class="o">.</span><span class="n">is_positive</span> <span class="o">!=</span> <span class="n">dif</span><span class="o">.</span><span class="n">is_nonpositive</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">dif</span><span class="o">.</span><span class="n">is_positive</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">StrictGreaterThan</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>  <span class="c"># sympy &gt;  other</span>
-    <span class="k">def</span> <span class="nf">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="n">dif</span> <span class="o">=</span> <span class="bp">self</span> <span class="o">-</span> <span class="n">other</span>
-        <span class="k">if</span> <span class="n">dif</span><span class="o">.</span><span class="n">is_negative</span> <span class="o">!=</span> <span class="n">dif</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">dif</span><span class="o">.</span><span class="n">is_negative</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">StrictLessThan</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_from_mpmath</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="s">&quot;_mpf_&quot;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="s">&quot;_mpc_&quot;</span><span class="p">):</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">_mpc_</span>
-            <span class="n">re</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-            <span class="n">im</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">im</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-            <span class="k">return</span> <span class="n">re</span> <span class="o">+</span> <span class="n">im</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;expected mpmath number (mpf or mpc)&quot;</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Expr.is_number"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.is_number">[docs]</a>    <span class="k">def</span> <span class="nf">is_number</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns True if &#39;self&#39; is a number.</span>
-
-<span class="sd">           &gt;&gt;&gt; from sympy import log, Integral</span>
-<span class="sd">           &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">           &gt;&gt;&gt; x.is_number</span>
-<span class="sd">           False</span>
-<span class="sd">           &gt;&gt;&gt; (2*x).is_number</span>
-<span class="sd">           False</span>
-<span class="sd">           &gt;&gt;&gt; (2 + log(2)).is_number</span>
-<span class="sd">           True</span>
-<span class="sd">           &gt;&gt;&gt; (2 + Integral(2, x)).is_number</span>
-<span class="sd">           False</span>
-<span class="sd">           &gt;&gt;&gt; (2 + Integral(2, (x, 1, 2))).is_number</span>
-<span class="sd">           True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="n">obj</span><span class="o">.</span><span class="n">is_number</span> <span class="k">for</span> <span class="n">obj</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">iter_basic_args</span><span class="p">())</span>
-</div>
-    <span class="k">def</span> <span class="nf">_random</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">re_min</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">im_min</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">re_max</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">im_max</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return self evaluated, if possible, replacing free symbols with</span>
-<span class="sd">        random complex values, if necessary.</span>
-
-<span class="sd">        The random complex value for each free symbol is generated</span>
-<span class="sd">        by the random_complex_number routine giving real and imaginary</span>
-<span class="sd">        parts in the range given by the re_min, re_max, im_min, and im_max</span>
-<span class="sd">        values. The returned value is evaluated to a precision of n</span>
-<span class="sd">        (if given) else the maximum of 15 and the precision needed</span>
-<span class="sd">        to get more than 1 digit of precision. If the expression</span>
-<span class="sd">        could not be evaluated to a number, or could not be evaluated</span>
-<span class="sd">        to more than 1 digit of precision, then None is returned.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sqrt</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; x._random()                         # doctest: +SKIP</span>
-<span class="sd">        0.0392918155679172 + 0.916050214307199*I</span>
-<span class="sd">        &gt;&gt;&gt; x._random(2)                        # doctest: +SKIP</span>
-<span class="sd">        -0.77 - 0.87*I</span>
-<span class="sd">        &gt;&gt;&gt; (x + y/2)._random(2)                # doctest: +SKIP</span>
-<span class="sd">        -0.57 + 0.16*I</span>
-<span class="sd">        &gt;&gt;&gt; sqrt(2)._random(2)</span>
-<span class="sd">        1.4</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.utilities.randtest.random_complex_number</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">free</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span>
-        <span class="n">prec</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">free</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">sympy.utilities.randtest</span> <span class="kn">import</span> <span class="n">random_complex_number</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">re_min</span><span class="p">,</span> <span class="n">re_max</span><span class="p">,</span> <span class="n">im_min</span><span class="p">,</span> <span class="n">im_max</span>
-            <span class="n">reps</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">free</span><span class="p">,</span> <span class="p">[</span><span class="n">random_complex_number</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">rational</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                           <span class="k">for</span> <span class="n">zi</span> <span class="ow">in</span> <span class="n">free</span><span class="p">])))</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">nmag</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">subs</span><span class="o">=</span><span class="n">reps</span><span class="p">))</span>
-            <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                <span class="c"># if an out of range value resulted in evalf problems</span>
-                <span class="c"># then return None -- XXX is there a way to know how to</span>
-                <span class="c"># select a good random number for a given expression?</span>
-                <span class="c"># e.g. when calculating n! negative values for n should not</span>
-                <span class="c"># be used</span>
-                <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">reps</span> <span class="o">=</span> <span class="p">{}</span>
-            <span class="n">nmag</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">nmag</span><span class="p">,</span> <span class="s">&#39;_prec&#39;</span><span class="p">):</span>
-            <span class="c"># e.g. exp_polar(2*I*pi) doesn&#39;t evaluate but is_number is True</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="k">if</span> <span class="n">nmag</span><span class="o">.</span><span class="n">_prec</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># increase the precision up to the default maximum</span>
-            <span class="c"># precision to see if we can get any significance</span>
-
-            <span class="c"># get the prec steps (patterned after giant_steps in</span>
-            <span class="c"># libintmath) which approximately doubles the prec</span>
-            <span class="c"># each step</span>
-            <span class="kn">from</span> <span class="nn">sympy.core.evalf</span> <span class="kn">import</span> <span class="n">DEFAULT_MAXPREC</span> <span class="k">as</span> <span class="n">target</span>
-            <span class="n">L</span> <span class="o">=</span> <span class="p">[</span><span class="n">target</span><span class="p">]</span>
-            <span class="n">start</span> <span class="o">=</span> <span class="mi">2</span>
-            <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">Li</span> <span class="o">=</span> <span class="n">L</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">//</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span>
-                <span class="k">if</span> <span class="n">Li</span> <span class="o">&gt;=</span> <span class="n">L</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="ow">or</span> <span class="n">Li</span> <span class="o">&lt;</span> <span class="n">start</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">L</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">!=</span> <span class="n">start</span><span class="p">:</span>
-                        <span class="n">L</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">start</span><span class="p">)</span>
-                    <span class="k">break</span>
-                <span class="n">L</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Li</span><span class="p">)</span>
-            <span class="n">L</span> <span class="o">=</span> <span class="n">L</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-
-            <span class="c"># evaluate</span>
-            <span class="k">for</span> <span class="n">prec</span> <span class="ow">in</span> <span class="n">L</span><span class="p">:</span>
-                <span class="n">nmag</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">,</span> <span class="n">subs</span><span class="o">=</span><span class="n">reps</span><span class="p">))</span>
-                <span class="k">if</span> <span class="n">nmag</span><span class="o">.</span><span class="n">_prec</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">break</span>
-
-        <span class="k">if</span> <span class="n">nmag</span><span class="o">.</span><span class="n">_prec</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">n</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">prec</span><span class="p">,</span> <span class="mi">15</span><span class="p">)</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">subs</span><span class="o">=</span><span class="n">reps</span><span class="p">)</span>
-
-        <span class="c"># never got any significance</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-<div class="viewcode-block" id="Expr.is_constant"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.is_constant">[docs]</a>    <span class="k">def</span> <span class="nf">is_constant</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">wrt</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return True if self is constant, False if not, or None if</span>
-<span class="sd">        the constancy could not be determined conclusively.</span>
-
-<span class="sd">        If an expression has no free symbols then it is a constant. If</span>
-<span class="sd">        there are free symbols it is possible that the expression is a</span>
-<span class="sd">        constant, perhaps (but not necessarily) zero. To test such</span>
-<span class="sd">        expressions, two strategies are tried:</span>
-
-<span class="sd">        1) numerical evaluation at two random points. If two such evaluations</span>
-<span class="sd">        give two different values and the values have a precision greater than</span>
-<span class="sd">        1 then self is not constant. If the evaluations agree or could not be</span>
-<span class="sd">        obtained with any precision, no decision is made. The numerical testing</span>
-<span class="sd">        is done only if ``wrt`` is different than the free symbols.</span>
-
-<span class="sd">        2) differentiation with respect to variables in &#39;wrt&#39; (or all free</span>
-<span class="sd">        symbols if omitted) to see if the expression is constant or not. This</span>
-<span class="sd">        will not always lead to an expression that is zero even though an</span>
-<span class="sd">        expression is constant (see added test in test_expr.py). If</span>
-<span class="sd">        all derivatives are zero then self is constant with respect to the</span>
-<span class="sd">        given symbols.</span>
-
-<span class="sd">        If neither evaluation nor differentiation can prove the expression is</span>
-<span class="sd">        constant, None is returned unless two numerical values happened to be</span>
-<span class="sd">        the same and the flag ``failing_number`` is True -- in that case the</span>
-<span class="sd">        numerical value will be returned.</span>
-
-<span class="sd">        If flag simplify=False is passed, self will not be simplified;</span>
-<span class="sd">        the default is True since self should be simplified before testing.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import cos, sin, Sum, S, pi</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import a, n, x, y</span>
-<span class="sd">        &gt;&gt;&gt; x.is_constant()</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; S(2).is_constant()</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Sum(x, (x, 1, 10)).is_constant()</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Sum(x, (x, 1, n)).is_constant()  # doctest: +SKIP</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Sum(x, (x, 1, n)).is_constant(y)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Sum(x, (x, 1, n)).is_constant(n) # doctest: +SKIP</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Sum(x, (x, 1, n)).is_constant(x)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; eq = a*cos(x)**2 + a*sin(x)**2 - a</span>
-<span class="sd">        &gt;&gt;&gt; eq.is_constant()</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; eq.subs({x:pi, a:2}) == eq.subs({x:pi, a:3}) == 0</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &gt;&gt;&gt; (0**x).is_constant()</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; x.is_constant()</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; (x**x).is_constant()</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; one = cos(x)**2 + sin(x)**2</span>
-<span class="sd">        &gt;&gt;&gt; one.is_constant()</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; ((one - 1)**(x + 1)).is_constant() # could be 0 or 1</span>
-<span class="sd">        False</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">simplify</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;simplify&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-
-        <span class="c"># Except for expressions that contain units, only one of these should</span>
-        <span class="c"># be necessary since if something is</span>
-        <span class="c"># known to be a number it should also know that there are no</span>
-        <span class="c"># free symbols. But is_number quits as soon as it hits a non-number</span>
-        <span class="c"># whereas free_symbols goes until all free symbols have been collected,</span>
-        <span class="c"># thus is_number should be faster. But a double check on free symbols</span>
-        <span class="c"># is made just in case there is a discrepancy between the two.</span>
-        <span class="n">free</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">free</span><span class="p">:</span>
-            <span class="c"># if the following assertion fails then that object&#39;s free_symbols</span>
-            <span class="c"># method needs attention: if an expression is a number it cannot</span>
-            <span class="c"># have free symbols</span>
-            <span class="k">assert</span> <span class="ow">not</span> <span class="n">free</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-        <span class="c"># if we are only interested in some symbols and they are not in the</span>
-        <span class="c"># free symbols then this expression is constant wrt those symbols</span>
-        <span class="n">wrt</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">wrt</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">wrt</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">wrt</span> <span class="o">&amp;</span> <span class="n">free</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="n">wrt</span> <span class="o">=</span> <span class="n">wrt</span> <span class="ow">or</span> <span class="n">free</span>
-
-        <span class="c"># simplify unless this has already been done</span>
-        <span class="k">if</span> <span class="n">simplify</span><span class="p">:</span>
-            <span class="bp">self</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-
-        <span class="c"># is_zero should be a quick assumptions check; it can be wrong for</span>
-        <span class="c"># numbers (see test_is_not_constant test), giving False when it</span>
-        <span class="c"># shouldn&#39;t, but hopefully it will never give True unless it is sure.</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-        <span class="c"># try numerical evaluation to see if we get two different values</span>
-        <span class="n">failing_number</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">wrt</span> <span class="o">==</span> <span class="n">free</span><span class="p">:</span>
-            <span class="c"># try 0 and 1</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">free</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="n">free</span><span class="p">))))</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_random</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">a</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">free</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="n">free</span><span class="p">))))</span>
-                <span class="k">if</span> <span class="n">b</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                    <span class="n">b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_random</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">b</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">b</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="bp">False</span>
-                <span class="c"># try random real</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_random</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">b</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">b</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span> <span class="ow">and</span> <span class="n">b</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-                <span class="c"># try random complex</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_random</span><span class="p">()</span>
-                <span class="k">if</span> <span class="n">b</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">b</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">a</span> <span class="o">!=</span> <span class="n">b</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="bp">False</span>
-                    <span class="n">failing_number</span> <span class="o">=</span> <span class="n">a</span> <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_number</span> <span class="k">else</span> <span class="n">b</span>
-
-        <span class="c"># now we will test each wrt symbol (or all free symbols) to see if the</span>
-        <span class="c"># expression depends on them or not using differentiation. This is</span>
-        <span class="c"># not sufficient for all expressions, however, so we don&#39;t return</span>
-        <span class="c"># False if we get a derivative other than 0 with free symbols.</span>
-        <span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">wrt</span><span class="p">:</span>
-            <span class="n">deriv</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">w</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">deriv</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">deriv</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">or</span> <span class="n">pure_complex</span><span class="p">(</span><span class="n">deriv</span><span class="p">)):</span>
-                    <span class="k">if</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;failing_number&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-                        <span class="k">return</span> <span class="n">failing_number</span>
-                    <span class="k">elif</span> <span class="n">deriv</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-                        <span class="c"># dead line provided _random returns None in such cases</span>
-                        <span class="k">return</span> <span class="bp">None</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="bp">True</span>
-</div>
-<div class="viewcode-block" id="Expr.equals"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.equals">[docs]</a>    <span class="k">def</span> <span class="nf">equals</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="n">failing_expression</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return True if self == other, False if it doesn&#39;t, or None. If</span>
-<span class="sd">        failing_expression is True then the expression which did not simplify</span>
-<span class="sd">        to a 0 will be returned instead of None.</span>
-
-<span class="sd">        If ``self`` is a Number (or complex number) that is not zero, then</span>
-<span class="sd">        the result is False.</span>
-
-<span class="sd">        If ``self`` is a number and has not evaluated to zero, evalf will be</span>
-<span class="sd">        used to test whether the expression evaluates to zero. If it does so</span>
-<span class="sd">        and the result has significance (i.e. the precision is either -1, for</span>
-<span class="sd">        a Rational result, or is greater than 1) then the evalf value will be</span>
-<span class="sd">        used to return True or False.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">other</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span> <span class="o">==</span> <span class="n">other</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-        <span class="c"># they aren&#39;t the same so see if we can make the difference 0;</span>
-        <span class="c"># don&#39;t worry about doing simplification steps one at a time</span>
-        <span class="c"># because if the expression ever goes to 0 then the subsequent</span>
-        <span class="c"># simplification steps that are done will be very fast.</span>
-        <span class="n">diff</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span> <span class="o">-</span> <span class="n">other</span><span class="p">)</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">diff</span> <span class="o">=</span> <span class="n">factor_terms</span><span class="p">(</span><span class="n">diff</span><span class="o">.</span><span class="n">simplify</span><span class="p">(),</span> <span class="n">radical</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">diff</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-        <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">is_Atom</span> <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">diff</span><span class="p">)</span>
-               <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">m</span><span class="p">)):</span>
-            <span class="c"># if there is no expanding to be done after simplifying</span>
-            <span class="c"># then this can&#39;t be a zero</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="n">constant</span> <span class="o">=</span> <span class="n">diff</span><span class="o">.</span><span class="n">is_constant</span><span class="p">(</span><span class="n">simplify</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">failing_number</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">constant</span> <span class="ow">is</span> <span class="bp">False</span> <span class="ow">or</span> \
-                <span class="ow">not</span> <span class="n">diff</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">diff</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="n">constant</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">ndiff</span> <span class="o">=</span> <span class="n">diff</span><span class="o">.</span><span class="n">_random</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">ndiff</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="c"># diff has not simplified to zero; constant is either None, True</span>
-        <span class="c"># or the number with significance (prec != 1) that was randomly</span>
-        <span class="c"># calculated twice as the same value.</span>
-        <span class="k">if</span> <span class="n">constant</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">(</span><span class="bp">True</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span> <span class="ow">and</span> <span class="n">constant</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="k">if</span> <span class="n">failing_expression</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">diff</span>
-        <span class="k">return</span> <span class="bp">None</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_is_positive</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="c"># check to see that we can get a value</span>
-                <span class="n">n2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-                <span class="n">n2</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">if</span> <span class="n">n2</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">i</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">if</span> <span class="n">i</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">_prec</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">elif</span> <span class="n">n</span><span class="o">.</span><span class="n">_prec</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-                <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_negative</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="c"># check to see that we can get a value</span>
-                <span class="n">n2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-                <span class="n">n2</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">if</span> <span class="n">n2</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">i</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">if</span> <span class="n">i</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">_prec</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">elif</span> <span class="n">n</span><span class="o">.</span><span class="n">_prec</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-                <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">_eval_interval</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns evaluation over an interval.  For most functions this is:</span>
-
-<span class="sd">        self.subs(x, b) - self.subs(x, a),</span>
-
-<span class="sd">        possibly using limit() if NaN is returned from subs.</span>
-
-<span class="sd">        If b or a is None, it only evaluates -self.subs(x, a) or self.subs(b, x),</span>
-<span class="sd">        respectively.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.series</span> <span class="kn">import</span> <span class="n">limit</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">a</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">b</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Both interval ends cannot be None.&#39;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">a</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">A</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">A</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">A</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">):</span>
-                <span class="n">A</span> <span class="o">=</span> <span class="n">limit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">A</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">A</span>
-
-        <span class="k">if</span> <span class="n">b</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">B</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">B</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">B</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">):</span>
-                <span class="n">B</span> <span class="o">=</span> <span class="n">limit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">B</span> <span class="o">-</span> <span class="n">A</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="c"># subclass to compute self**other for cases when</span>
-        <span class="c"># other is not NaN, 0, or 1</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="bp">self</span>
-
-    <span class="k">def</span> <span class="nf">conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.functions.elementary.complexes</span> <span class="kn">import</span> <span class="n">conjugate</span> <span class="k">as</span> <span class="n">c</span>
-        <span class="k">return</span> <span class="n">c</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_transpose</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.functions.elementary.complexes</span> <span class="kn">import</span> <span class="n">conjugate</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_complex</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_hermitian</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_antihermitian</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">transpose</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.functions.elementary.complexes</span> <span class="kn">import</span> <span class="n">transpose</span>
-        <span class="k">return</span> <span class="n">transpose</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_adjoint</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.functions.elementary.complexes</span> <span class="kn">import</span> <span class="n">conjugate</span><span class="p">,</span> <span class="n">transpose</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_hermitian</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_antihermitian</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="bp">self</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_conjugate</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">obj</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">transpose</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_transpose</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">obj</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">conjugate</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">adjoint</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.functions.elementary.complexes</span> <span class="kn">import</span> <span class="n">adjoint</span>
-        <span class="k">return</span> <span class="n">adjoint</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_parse_order</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">order</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Parse and configure the ordering of terms. &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.polys.monomialtools</span> <span class="kn">import</span> <span class="n">monomial_key</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">reverse</span> <span class="o">=</span> <span class="n">order</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39;rev-&#39;</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-            <span class="n">reverse</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">reverse</span><span class="p">:</span>
-                <span class="n">order</span> <span class="o">=</span> <span class="n">order</span><span class="p">[</span><span class="mi">4</span><span class="p">:]</span>
-
-        <span class="n">monom_key</span> <span class="o">=</span> <span class="n">monomial_key</span><span class="p">(</span><span class="n">order</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">neg</span><span class="p">(</span><span class="n">monom</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-
-            <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">monom</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span>
-                    <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">neg</span><span class="p">(</span><span class="n">m</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">-</span><span class="n">m</span><span class="p">)</span>
-
-            <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">key</span><span class="p">(</span><span class="n">term</span><span class="p">):</span>
-            <span class="n">_</span><span class="p">,</span> <span class="p">((</span><span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">),</span> <span class="n">monom</span><span class="p">,</span> <span class="n">ncpart</span><span class="p">)</span> <span class="o">=</span> <span class="n">term</span>
-
-            <span class="n">monom</span> <span class="o">=</span> <span class="n">neg</span><span class="p">(</span><span class="n">monom_key</span><span class="p">(</span><span class="n">monom</span><span class="p">))</span>
-            <span class="n">ncpart</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">([</span> <span class="n">e</span><span class="o">.</span><span class="n">sort_key</span><span class="p">(</span><span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">)</span> <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">ncpart</span> <span class="p">])</span>
-            <span class="n">coeff</span> <span class="o">=</span> <span class="p">((</span><span class="nb">bool</span><span class="p">(</span><span class="n">im</span><span class="p">),</span> <span class="n">im</span><span class="p">),</span> <span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">))</span>
-
-            <span class="k">return</span> <span class="n">monom</span><span class="p">,</span> <span class="n">ncpart</span><span class="p">,</span> <span class="n">coeff</span>
-
-        <span class="k">return</span> <span class="n">key</span><span class="p">,</span> <span class="n">reverse</span>
-
-<div class="viewcode-block" id="Expr.as_ordered_factors"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.as_ordered_factors">[docs]</a>    <span class="k">def</span> <span class="nf">as_ordered_factors</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return list of ordered factors (if Mul) else [self].&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">[</span><span class="bp">self</span><span class="p">]</span>
-</div>
-<div class="viewcode-block" id="Expr.as_ordered_terms"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.as_ordered_terms">[docs]</a>    <span class="k">def</span> <span class="nf">as_ordered_terms</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Transform an expression to an ordered list of terms.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sin, cos</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; (sin(x)**2*cos(x) + sin(x)**2 + 1).as_ordered_terms()</span>
-<span class="sd">        [sin(x)**2*cos(x), sin(x)**2, 1]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">key</span><span class="p">,</span> <span class="n">reverse</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_parse_order</span><span class="p">(</span><span class="n">order</span><span class="p">)</span>
-        <span class="n">terms</span><span class="p">,</span> <span class="n">gens</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_terms</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">any</span><span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">is_Order</span> <span class="k">for</span> <span class="n">term</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">):</span>
-            <span class="n">ordered</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">terms</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">key</span><span class="p">,</span> <span class="n">reverse</span><span class="o">=</span><span class="n">reverse</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">_terms</span><span class="p">,</span> <span class="n">_order</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
-
-            <span class="k">for</span> <span class="n">term</span><span class="p">,</span> <span class="nb">repr</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">term</span><span class="o">.</span><span class="n">is_Order</span><span class="p">:</span>
-                    <span class="n">_terms</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">term</span><span class="p">,</span> <span class="nb">repr</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">_order</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">term</span><span class="p">,</span> <span class="nb">repr</span><span class="p">))</span>
-
-            <span class="n">ordered</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">_terms</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">key</span><span class="p">,</span> <span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> \
-                <span class="o">+</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">_order</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">key</span><span class="p">,</span> <span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">data</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">ordered</span><span class="p">,</span> <span class="n">gens</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[</span> <span class="n">term</span> <span class="k">for</span> <span class="n">term</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">ordered</span> <span class="p">]</span>
-</div>
-<div class="viewcode-block" id="Expr.as_terms"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.as_terms">[docs]</a>    <span class="k">def</span> <span class="nf">as_terms</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Transform an expression to a list of terms. &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">S</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.exprtools</span> <span class="kn">import</span> <span class="n">decompose_power</span>
-
-        <span class="n">gens</span><span class="p">,</span> <span class="n">terms</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([]),</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-            <span class="n">coeff</span><span class="p">,</span> <span class="n">_term</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-
-            <span class="n">coeff</span> <span class="o">=</span> <span class="nb">complex</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-            <span class="n">cpart</span><span class="p">,</span> <span class="n">ncpart</span> <span class="o">=</span> <span class="p">{},</span> <span class="p">[]</span>
-
-            <span class="k">if</span> <span class="n">_term</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">factor</span> <span class="ow">in</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">_term</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">factor</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-                        <span class="k">try</span><span class="p">:</span>
-                            <span class="n">coeff</span> <span class="o">*=</span> <span class="nb">complex</span><span class="p">(</span><span class="n">factor</span><span class="p">)</span>
-                        <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                            <span class="k">pass</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="k">continue</span>
-
-                    <span class="k">if</span> <span class="n">factor</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-                        <span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="n">decompose_power</span><span class="p">(</span><span class="n">factor</span><span class="p">)</span>
-
-                        <span class="n">cpart</span><span class="p">[</span><span class="n">base</span><span class="p">]</span> <span class="o">=</span> <span class="n">exp</span>
-                        <span class="n">gens</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">ncpart</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">factor</span><span class="p">)</span>
-
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">coeff</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">coeff</span><span class="o">.</span><span class="n">imag</span>
-            <span class="n">ncpart</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">ncpart</span><span class="p">)</span>
-
-            <span class="n">terms</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">term</span><span class="p">,</span> <span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">cpart</span><span class="p">,</span> <span class="n">ncpart</span><span class="p">)))</span>
-
-        <span class="n">gens</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">gens</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-
-        <span class="n">k</span><span class="p">,</span> <span class="n">indices</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">),</span> <span class="p">{}</span>
-
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">g</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">gens</span><span class="p">):</span>
-            <span class="n">indices</span><span class="p">[</span><span class="n">g</span><span class="p">]</span> <span class="o">=</span> <span class="n">i</span>
-
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">term</span><span class="p">,</span> <span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">cpart</span><span class="p">,</span> <span class="n">ncpart</span><span class="p">)</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">:</span>
-            <span class="n">monom</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">k</span>
-
-            <span class="k">for</span> <span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="ow">in</span> <span class="n">cpart</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                <span class="n">monom</span><span class="p">[</span><span class="n">indices</span><span class="p">[</span><span class="n">base</span><span class="p">]]</span> <span class="o">=</span> <span class="n">exp</span>
-
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">term</span><span class="p">,</span> <span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">monom</span><span class="p">),</span> <span class="n">ncpart</span><span class="p">)))</span>
-
-        <span class="k">return</span> <span class="n">result</span><span class="p">,</span> <span class="n">gens</span>
-</div>
-<div class="viewcode-block" id="Expr.removeO"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.removeO">[docs]</a>    <span class="k">def</span> <span class="nf">removeO</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Removes the additive O(..) symbol if there is one&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span>
-</div>
-<div class="viewcode-block" id="Expr.getO"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.getO">[docs]</a>    <span class="k">def</span> <span class="nf">getO</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the additive O(..) symbol if there is one, else None.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">None</span>
-</div>
-<div class="viewcode-block" id="Expr.getn"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.getn">[docs]</a>    <span class="k">def</span> <span class="nf">getn</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the order of the expression.</span>
-
-<span class="sd">        The order is determined either from the O(...) term. If there</span>
-<span class="sd">        is no O(...) term, it returns None.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import O</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; (1 + x + O(x**2)).getn()</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; (1 + x).getn()</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">o</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">getO</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">o</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">elif</span> <span class="n">o</span><span class="o">.</span><span class="n">is_Order</span><span class="p">:</span>
-            <span class="n">o</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">expr</span>
-            <span class="k">if</span> <span class="n">o</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">if</span> <span class="n">o</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">if</span> <span class="n">o</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">o</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">o</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>  <span class="c"># x**n*log(x)**n or x**n/log(x)**n</span>
-                <span class="k">for</span> <span class="n">oi</span> <span class="ow">in</span> <span class="n">o</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">oi</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-                    <span class="k">if</span> <span class="n">oi</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-                        <span class="n">syms</span> <span class="o">=</span> <span class="n">oi</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">syms</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                            <span class="n">x</span> <span class="o">=</span> <span class="n">syms</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-                            <span class="n">oi</span> <span class="o">=</span> <span class="n">oi</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-                            <span class="k">if</span> <span class="n">oi</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_Symbol</span> <span class="ow">and</span> <span class="n">oi</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                                <span class="k">return</span> <span class="nb">abs</span><span class="p">(</span><span class="n">oi</span><span class="o">.</span><span class="n">exp</span><span class="p">)</span>
-
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;not sure of order of </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">o</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.count_ops"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.count_ops">[docs]</a>    <span class="k">def</span> <span class="nf">count_ops</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;wrapper for count_ops that returns the operation count.&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">.function</span> <span class="kn">import</span> <span class="n">count_ops</span>
-        <span class="k">return</span> <span class="n">count_ops</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">visual</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.args_cnc"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.args_cnc">[docs]</a>    <span class="k">def</span> <span class="nf">args_cnc</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">cset</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">warn</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">split_1</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return [commutative factors, non-commutative factors] of self.</span>
-
-<span class="sd">        self is treated as a Mul and the ordering of the factors is maintained.</span>
-<span class="sd">        If ``cset`` is True the commutative factors will be returned in a set.</span>
-<span class="sd">        If there were repeated factors (as may happen with an unevaluated Mul)</span>
-<span class="sd">        then an error will be raised unless it is explicitly supressed by</span>
-<span class="sd">        setting ``warn`` to False.</span>
-
-<span class="sd">        Note: -1 is always separated from a Number unless split_1 is False.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols, oo</span>
-<span class="sd">        &gt;&gt;&gt; A, B = symbols(&#39;A B&#39;, commutative=0)</span>
-<span class="sd">        &gt;&gt;&gt; x, y = symbols(&#39;x y&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; (-2*x*y).args_cnc()</span>
-<span class="sd">        [[-1, 2, x, y], []]</span>
-<span class="sd">        &gt;&gt;&gt; (-2.5*x).args_cnc()</span>
-<span class="sd">        [[-1, 2.5, x], []]</span>
-<span class="sd">        &gt;&gt;&gt; (-2*x*A*B*y).args_cnc()</span>
-<span class="sd">        [[-1, 2, x, y], [A, B]]</span>
-<span class="sd">        &gt;&gt;&gt; (-2*x*A*B*y).args_cnc(split_1=False)</span>
-<span class="sd">        [[-2, x, y], [A, B]]</span>
-<span class="sd">        &gt;&gt;&gt; (-2*x*y).args_cnc(cset=True)</span>
-<span class="sd">        [set([-1, 2, x, y]), []]</span>
-
-<span class="sd">        The arg is always treated as a Mul:</span>
-
-<span class="sd">        &gt;&gt;&gt; (-2 + x + A).args_cnc()</span>
-<span class="sd">        [[], [x - 2 + A]]</span>
-<span class="sd">        &gt;&gt;&gt; (-oo).args_cnc() # -oo is a singleton</span>
-<span class="sd">        [[-1, oo], []]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">mi</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">mi</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="n">args</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span>
-                <span class="n">nc</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">:]</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">args</span>
-            <span class="n">nc</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">if</span> <span class="n">c</span> <span class="ow">and</span> <span class="n">split_1</span> <span class="ow">and</span> <span class="p">(</span>
-            <span class="n">c</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">and</span>
-            <span class="n">c</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_negative</span> <span class="ow">and</span>
-                <span class="n">c</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">):</span>
-            <span class="n">c</span><span class="p">[:</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">,</span> <span class="o">-</span><span class="n">c</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span>
-
-        <span class="k">if</span> <span class="n">cset</span><span class="p">:</span>
-            <span class="n">clen</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">clen</span> <span class="ow">and</span> <span class="n">warn</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="o">!=</span> <span class="n">clen</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;repeated commutative arguments: </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span>
-                                 <span class="p">[</span><span class="n">ci</span> <span class="k">for</span> <span class="n">ci</span> <span class="ow">in</span> <span class="n">c</span> <span class="k">if</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="n">ci</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">])</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">c</span><span class="p">,</span> <span class="n">nc</span><span class="p">]</span>
-</div>
-<div class="viewcode-block" id="Expr.coeff"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.coeff">[docs]</a>    <span class="k">def</span> <span class="nf">coeff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">right</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the coefficient from the term containing &quot;x**n&quot; or None if</span>
-<span class="sd">        there is no such term. If ``n`` is zero then all terms independent of</span>
-<span class="sd">        x will be returned.</span>
-
-<span class="sd">        When x is noncommutative, the coeff to the left (default) or right of x</span>
-<span class="sd">        can be returned. The keyword &#39;right&#39; is ignored when x is commutative.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        as_coefficient</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-
-<span class="sd">        You can select terms that have an explicit negative in front of them:</span>
-
-<span class="sd">        &gt;&gt;&gt; (-x + 2*y).coeff(-1)</span>
-<span class="sd">        x</span>
-<span class="sd">        &gt;&gt;&gt; (x - 2*y).coeff(-1)</span>
-<span class="sd">        2*y</span>
-
-<span class="sd">        You can select terms with no Rational coefficient:</span>
-
-<span class="sd">        &gt;&gt;&gt; (x + 2*y).coeff(1)</span>
-<span class="sd">        x</span>
-<span class="sd">        &gt;&gt;&gt; (3 + 2*x + 4*x**2).coeff(1)</span>
-<span class="sd">        0</span>
-
-<span class="sd">        You can select terms independent of x by making n=0; in this case</span>
-<span class="sd">        expr.as_independent(x)[0] is returned (and 0 will be returned instead</span>
-<span class="sd">        of None):</span>
-
-<span class="sd">        &gt;&gt;&gt; (3 + 2*x + 4*x**2).coeff(x, 0)</span>
-<span class="sd">        3</span>
-<span class="sd">        &gt;&gt;&gt; eq = ((x + 1)**3).expand() + 1</span>
-<span class="sd">        &gt;&gt;&gt; eq</span>
-<span class="sd">        x**3 + 3*x**2 + 3*x + 2</span>
-<span class="sd">        &gt;&gt;&gt; [eq.coeff(x, i) for i in reversed(list(range(4)))]</span>
-<span class="sd">        [1, 3, 3, 2]</span>
-<span class="sd">        &gt;&gt;&gt; eq -= 2</span>
-<span class="sd">        &gt;&gt;&gt; [eq.coeff(x, i) for i in reversed(list(range(4)))]</span>
-<span class="sd">        [1, 3, 3, 0]</span>
-
-<span class="sd">        You can select terms that have a numerical term in front of them:</span>
-
-<span class="sd">        &gt;&gt;&gt; (-x - 2*y).coeff(2)</span>
-<span class="sd">        -y</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import sqrt</span>
-<span class="sd">        &gt;&gt;&gt; (x + sqrt(2)*x).coeff(sqrt(2))</span>
-<span class="sd">        x</span>
-
-<span class="sd">        The matching is exact:</span>
-
-<span class="sd">        &gt;&gt;&gt; (3 + 2*x + 4*x**2).coeff(x)</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; (3 + 2*x + 4*x**2).coeff(x**2)</span>
-<span class="sd">        4</span>
-<span class="sd">        &gt;&gt;&gt; (3 + 2*x + 4*x**2).coeff(x**3)</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; (z*(x + y)**2).coeff((x + y)**2)</span>
-<span class="sd">        z</span>
-<span class="sd">        &gt;&gt;&gt; (z*(x + y)**2).coeff(x + y)</span>
-<span class="sd">        0</span>
-
-<span class="sd">        In addition, no factoring is done, so 1 + z*(1 + y) is not obtained</span>
-<span class="sd">        from the following:</span>
-
-<span class="sd">        &gt;&gt;&gt; (x + z*(x + x*y)).coeff(x)</span>
-<span class="sd">        1</span>
-
-<span class="sd">        If such factoring is desired, factor_terms can be used first:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import factor_terms</span>
-<span class="sd">        &gt;&gt;&gt; factor_terms(x + z*(x + x*y)).coeff(x)</span>
-<span class="sd">        z*(y + 1) + 1</span>
-
-<span class="sd">        &gt;&gt;&gt; n, m, o = symbols(&#39;n m o&#39;, commutative=False)</span>
-<span class="sd">        &gt;&gt;&gt; n.coeff(n)</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; (3*n).coeff(n)</span>
-<span class="sd">        3</span>
-<span class="sd">        &gt;&gt;&gt; (n*m + m*n*m).coeff(n) # = (1 + m)*n*m</span>
-<span class="sd">        1 + m</span>
-<span class="sd">        &gt;&gt;&gt; (n*m + m*n*m).coeff(n, right=True) # = (1 + m)*n*m</span>
-<span class="sd">        m</span>
-
-<span class="sd">        If there is more than one possible coefficient 0 is returned:</span>
-
-<span class="sd">        &gt;&gt;&gt; (n*m + m*n).coeff(n)</span>
-<span class="sd">        0</span>
-
-<span class="sd">        If there is only one possible coefficient, it is returned:</span>
-
-<span class="sd">        &gt;&gt;&gt; (n*m + x*m*n).coeff(m*n)</span>
-<span class="sd">        x</span>
-<span class="sd">        &gt;&gt;&gt; (n*m + x*m*n).coeff(m*n, right=1)</span>
-<span class="sd">        1</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        as_coeff_Add: a method to separate the additive constant from an expression</span>
-<span class="sd">        as_coeff_Mul: a method to separate the multiplicative constant from an expression</span>
-<span class="sd">        as_independent: a method to separate x dependent terms/factors from others</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-        <span class="n">n</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">x</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="bp">self</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-            <span class="n">co</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-                  <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">]</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">co</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">co</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">right</span><span class="o">=</span><span class="n">right</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">c</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">right</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">self</span> <span class="o">-</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">a</span><span class="o">*</span><span class="n">x</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">c</span><span class="p">)])</span>
-                <span class="k">return</span> <span class="bp">self</span> <span class="o">-</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">x</span><span class="o">*</span><span class="n">a</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">c</span><span class="p">)])</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">as_Add</span><span class="o">=</span><span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="c"># continue with the full method, looking for this power of x:</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="n">n</span>
-
-        <span class="k">def</span> <span class="nf">incommon</span><span class="p">(</span><span class="n">l1</span><span class="p">,</span> <span class="n">l2</span><span class="p">):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">l1</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">l2</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[]</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">l1</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">l2</span><span class="p">))</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">l1</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">!=</span> <span class="n">l2</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                    <span class="k">return</span> <span class="n">l1</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">l1</span><span class="p">[:]</span>
-
-        <span class="k">def</span> <span class="nf">find</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">sub</span><span class="p">,</span> <span class="n">first</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot; Find where list sub appears in list l. When ``first`` is True</span>
-<span class="sd">            the first occurance from the left is returned, else the last</span>
-<span class="sd">            occurance is returned. Return None if sub is not in l.</span>
-
-<span class="sd">            &gt;&gt; l = range(5)*2</span>
-<span class="sd">            &gt;&gt; find(l, [2, 3])</span>
-<span class="sd">            2</span>
-<span class="sd">            &gt;&gt; find(l, [2, 3], first=0)</span>
-<span class="sd">            7</span>
-<span class="sd">            &gt;&gt; find(l, [2, 4])</span>
-<span class="sd">            None</span>
-
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">sub</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">l</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">sub</span><span class="p">)</span> <span class="o">&gt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">l</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">sub</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">first</span><span class="p">:</span>
-                <span class="n">l</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-                <span class="n">sub</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">l</span><span class="p">)</span> <span class="o">-</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">l</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="n">sub</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)):</span>
-                    <span class="k">break</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">i</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">first</span><span class="p">:</span>
-                <span class="n">l</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-                <span class="n">sub</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">first</span><span class="p">:</span>
-                <span class="n">i</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">l</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="n">n</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">i</span>
-
-        <span class="n">co</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="n">self_c</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_commutative</span>
-        <span class="n">x_c</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">is_commutative</span>
-        <span class="k">if</span> <span class="n">self_c</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">x_c</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-        <span class="k">if</span> <span class="n">self_c</span><span class="p">:</span>
-            <span class="n">xargs</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">(</span><span class="n">cset</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">warn</span><span class="o">=</span><span class="bp">False</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-                <span class="n">margs</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">(</span><span class="n">cset</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">warn</span><span class="o">=</span><span class="bp">False</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">xargs</span><span class="p">)</span> <span class="o">&gt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">margs</span><span class="p">):</span>
-                    <span class="k">continue</span>
-                <span class="n">resid</span> <span class="o">=</span> <span class="n">margs</span><span class="o">.</span><span class="n">difference</span><span class="p">(</span><span class="n">xargs</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">resid</span><span class="p">)</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">xargs</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">margs</span><span class="p">):</span>
-                    <span class="n">co</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">resid</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">co</span> <span class="o">==</span> <span class="p">[]:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">elif</span> <span class="n">co</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">co</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">x_c</span><span class="p">:</span>
-            <span class="n">xargs</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">(</span><span class="n">cset</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">warn</span><span class="o">=</span><span class="bp">False</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-                <span class="n">margs</span><span class="p">,</span> <span class="n">nc</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">(</span><span class="n">cset</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">xargs</span><span class="p">)</span> <span class="o">&gt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">margs</span><span class="p">):</span>
-                    <span class="k">continue</span>
-                <span class="n">resid</span> <span class="o">=</span> <span class="n">margs</span><span class="o">.</span><span class="n">difference</span><span class="p">(</span><span class="n">xargs</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">resid</span><span class="p">)</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">xargs</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">margs</span><span class="p">):</span>
-                    <span class="n">co</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">resid</span><span class="p">)</span> <span class="o">+</span> <span class="n">nc</span><span class="p">)))</span>
-            <span class="k">if</span> <span class="n">co</span> <span class="o">==</span> <span class="p">[]:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">elif</span> <span class="n">co</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">co</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># both nc</span>
-            <span class="n">xargs</span><span class="p">,</span> <span class="n">nx</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">(</span><span class="n">cset</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="c"># find the parts that pass the commutative terms</span>
-            <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-                <span class="n">margs</span><span class="p">,</span> <span class="n">nc</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">(</span><span class="n">cset</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">xargs</span><span class="p">)</span> <span class="o">&gt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">margs</span><span class="p">):</span>
-                    <span class="k">continue</span>
-                <span class="n">resid</span> <span class="o">=</span> <span class="n">margs</span><span class="o">.</span><span class="n">difference</span><span class="p">(</span><span class="n">xargs</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">resid</span><span class="p">)</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">xargs</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">margs</span><span class="p">):</span>
-                    <span class="n">co</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">resid</span><span class="p">,</span> <span class="n">nc</span><span class="p">))</span>
-            <span class="c"># now check the non-comm parts</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">co</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">n</span> <span class="o">==</span> <span class="n">co</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">r</span><span class="p">,</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">co</span><span class="p">):</span>
-                <span class="n">ii</span> <span class="o">=</span> <span class="n">find</span><span class="p">(</span><span class="n">co</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">],</span> <span class="n">nx</span><span class="p">,</span> <span class="n">right</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">ii</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">right</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">r</span><span class="p">)</span> <span class="k">for</span> <span class="n">r</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">co</span><span class="p">]),</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">co</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">][:</span><span class="n">ii</span><span class="p">]))</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">co</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">][</span><span class="n">ii</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">nx</span><span class="p">):])</span>
-            <span class="n">beg</span> <span class="o">=</span> <span class="nb">reduce</span><span class="p">(</span><span class="n">incommon</span><span class="p">,</span> <span class="p">(</span><span class="n">n</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">co</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">beg</span><span class="p">:</span>
-                <span class="n">ii</span> <span class="o">=</span> <span class="n">find</span><span class="p">(</span><span class="n">beg</span><span class="p">,</span> <span class="n">nx</span><span class="p">,</span> <span class="n">right</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">ii</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">right</span><span class="p">:</span>
-                        <span class="n">gcdc</span> <span class="o">=</span> <span class="n">co</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-                        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">co</span><span class="p">)):</span>
-                            <span class="n">gcdc</span> <span class="o">=</span> <span class="n">gcdc</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">co</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span>
-                            <span class="k">if</span> <span class="ow">not</span> <span class="n">gcdc</span><span class="p">:</span>
-                                <span class="k">break</span>
-                        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">gcdc</span><span class="p">)</span> <span class="o">+</span> <span class="n">beg</span><span class="p">[:</span><span class="n">ii</span><span class="p">]))</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">m</span> <span class="o">=</span> <span class="n">ii</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">nx</span><span class="p">)</span>
-                        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">r</span><span class="p">)</span> <span class="o">+</span> <span class="n">n</span><span class="p">[</span><span class="n">m</span><span class="p">:]))</span> <span class="k">for</span> <span class="n">r</span><span class="p">,</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">co</span><span class="p">])</span>
-            <span class="n">end</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span>
-                <span class="nb">reduce</span><span class="p">(</span><span class="n">incommon</span><span class="p">,</span> <span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">n</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">co</span><span class="p">))))</span>
-            <span class="k">if</span> <span class="n">end</span><span class="p">:</span>
-                <span class="n">ii</span> <span class="o">=</span> <span class="n">find</span><span class="p">(</span><span class="n">end</span><span class="p">,</span> <span class="n">nx</span><span class="p">,</span> <span class="n">right</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">ii</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">right</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">r</span><span class="p">)</span> <span class="o">+</span> <span class="n">n</span><span class="p">[:</span><span class="o">-</span><span class="nb">len</span><span class="p">(</span><span class="n">end</span><span class="p">)</span> <span class="o">+</span> <span class="n">ii</span><span class="p">]))</span> <span class="k">for</span> <span class="n">r</span><span class="p">,</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">co</span><span class="p">])</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">end</span><span class="p">[</span><span class="n">ii</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">nx</span><span class="p">):])</span>
-            <span class="c"># look for single match</span>
-            <span class="n">hit</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">co</span><span class="p">):</span>
-                <span class="n">ii</span> <span class="o">=</span> <span class="n">find</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">nx</span><span class="p">,</span> <span class="n">right</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">ii</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">hit</span><span class="p">:</span>
-                        <span class="n">hit</span> <span class="o">=</span> <span class="n">ii</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">n</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">break</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">hit</span><span class="p">:</span>
-                    <span class="n">ii</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">hit</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">right</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">r</span><span class="p">)</span> <span class="o">+</span> <span class="n">n</span><span class="p">[:</span><span class="n">ii</span><span class="p">]))</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">n</span><span class="p">[</span><span class="n">ii</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">nx</span><span class="p">):])</span>
-
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-</div>
-<div class="viewcode-block" id="Expr.as_expr"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.as_expr">[docs]</a>    <span class="k">def</span> <span class="nf">as_expr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Convert a polynomial to a SymPy expression.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sin</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; f = (x**2 + x*y).as_poly(x, y)</span>
-<span class="sd">        &gt;&gt;&gt; f.as_expr()</span>
-<span class="sd">        x**2 + x*y</span>
-
-<span class="sd">        &gt;&gt;&gt; sin(x).as_expr()</span>
-<span class="sd">        sin(x)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span>
-</div>
-<div class="viewcode-block" id="Expr.as_coefficient"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.as_coefficient">[docs]</a>    <span class="k">def</span> <span class="nf">as_coefficient</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Extracts symbolic coefficient at the given expression. In</span>
-<span class="sd">        other words, this functions separates &#39;self&#39; into the product</span>
-<span class="sd">        of &#39;expr&#39; and &#39;expr&#39;-free coefficient. If such separation</span>
-<span class="sd">        is not possible it will return None.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        coeff</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import E, pi, sin, I, symbols</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; E.as_coefficient(E)</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; (2*E).as_coefficient(E)</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; (2*sin(E)*E).as_coefficient(E)</span>
-
-<span class="sd">        &gt;&gt;&gt; (2*E + x*E).as_coefficient(E)</span>
-<span class="sd">        x + 2</span>
-<span class="sd">        &gt;&gt;&gt; (2*E*x + x).as_coefficient(E)</span>
-
-<span class="sd">        &gt;&gt;&gt; (E*(x + 1) + x).as_coefficient(E)</span>
-
-<span class="sd">        &gt;&gt;&gt; (2*pi*I).as_coefficient(pi*I)</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; (2*I).as_coefficient(pi*I)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">r</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">r</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">r</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">r</span>
-</div>
-<div class="viewcode-block" id="Expr.as_independent"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.as_independent">[docs]</a>    <span class="k">def</span> <span class="nf">as_independent</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">deps</span><span class="p">,</span> <span class="o">**</span><span class="n">hint</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        A mostly naive separation of a Mul or Add into arguments that are not</span>
-<span class="sd">        are dependent on deps. To obtain as complete a separation of variables</span>
-<span class="sd">        as possible, use a separation method first, e.g.:</span>
-
-<span class="sd">        * separatevars() to change Mul, Add and Pow (including exp) into Mul</span>
-<span class="sd">        * .expand(mul=True) to change Add or Mul into Add</span>
-<span class="sd">        * .expand(log=True) to change log expr into an Add</span>
-
-<span class="sd">        The only non-naive thing that is done here is to respect noncommutative</span>
-<span class="sd">        ordering of variables.</span>
-
-<span class="sd">        The returned tuple (i, d) has the following interpretation:</span>
-
-<span class="sd">        * i will has no variable that appears in deps</span>
-<span class="sd">        * d will be 1 or else have terms that contain variables that are in deps</span>
-<span class="sd">        * if self is an Add then self = i + d</span>
-<span class="sd">        * if self is a Mul then self = i*d</span>
-<span class="sd">        * if self is anything else, either tuple (self, S.One) or (S.One, self)</span>
-<span class="sd">          is returned.</span>
-
-<span class="sd">        To force the expression to be treated as an Add, use the hint as_Add=True</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        -- self is an Add</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sin, cos, exp</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-
-<span class="sd">        &gt;&gt;&gt; (x + x*y).as_independent(x)</span>
-<span class="sd">        (0, x*y + x)</span>
-<span class="sd">        &gt;&gt;&gt; (x + x*y).as_independent(y)</span>
-<span class="sd">        (x, x*y)</span>
-<span class="sd">        &gt;&gt;&gt; (2*x*sin(x) + y + x + z).as_independent(x)</span>
-<span class="sd">        (y + z, 2*x*sin(x) + x)</span>
-<span class="sd">        &gt;&gt;&gt; (2*x*sin(x) + y + x + z).as_independent(x, y)</span>
-<span class="sd">        (z, 2*x*sin(x) + x + y)</span>
-
-<span class="sd">        -- self is a Mul</span>
-
-<span class="sd">        &gt;&gt;&gt; (x*sin(x)*cos(y)).as_independent(x)</span>
-<span class="sd">        (cos(y), x*sin(x))</span>
-
-<span class="sd">        non-commutative terms cannot always be separated out when self is a Mul</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; n1, n2, n3 = symbols(&#39;n1 n2 n3&#39;, commutative=False)</span>
-<span class="sd">        &gt;&gt;&gt; (n1 + n1*n2).as_independent(n2)</span>
-<span class="sd">        (n1, n1*n2)</span>
-<span class="sd">        &gt;&gt;&gt; (n2*n1 + n1*n2).as_independent(n2)</span>
-<span class="sd">        (0, n1*n2 + n2*n1)</span>
-<span class="sd">        &gt;&gt;&gt; (n1*n2*n3).as_independent(n1)</span>
-<span class="sd">        (1, n1*n2*n3)</span>
-<span class="sd">        &gt;&gt;&gt; (n1*n2*n3).as_independent(n2)</span>
-<span class="sd">        (n1, n2*n3)</span>
-<span class="sd">        &gt;&gt;&gt; ((x-n1)*(x-y)).as_independent(x)</span>
-<span class="sd">        (1, (x - y)*(x - n1))</span>
-
-<span class="sd">        -- self is anything else:</span>
-
-<span class="sd">        &gt;&gt;&gt; (sin(x)).as_independent(x)</span>
-<span class="sd">        (1, sin(x))</span>
-<span class="sd">        &gt;&gt;&gt; (sin(x)).as_independent(y)</span>
-<span class="sd">        (sin(x), 1)</span>
-<span class="sd">        &gt;&gt;&gt; exp(x+y).as_independent(x)</span>
-<span class="sd">        (1, exp(x + y))</span>
-
-<span class="sd">        -- force self to be treated as an Add:</span>
-
-<span class="sd">        &gt;&gt;&gt; (3*x).as_independent(x, as_Add=True)</span>
-<span class="sd">        (0, 3*x)</span>
-
-<span class="sd">        -- force self to be treated as a Mul:</span>
-
-<span class="sd">        &gt;&gt;&gt; (3+x).as_independent(x, as_Add=False)</span>
-<span class="sd">        (1, x + 3)</span>
-<span class="sd">        &gt;&gt;&gt; (-3+x).as_independent(x, as_Add=False)</span>
-<span class="sd">        (1, x - 3)</span>
-
-<span class="sd">        Note how the below differs from the above in making the</span>
-<span class="sd">        constant on the dep term positive.</span>
-
-<span class="sd">        &gt;&gt;&gt; (y*(-3+x)).as_independent(x)</span>
-<span class="sd">        (y, x - 3)</span>
-
-<span class="sd">        -- use .as_independent() for true independence testing instead</span>
-<span class="sd">           of .has(). The former considers only symbols in the free</span>
-<span class="sd">           symbols while the latter considers all symbols</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Integral</span>
-<span class="sd">        &gt;&gt;&gt; I = Integral(x, (x, 1, 2))</span>
-<span class="sd">        &gt;&gt;&gt; I.has(x)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; x in I.free_symbols</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; I.as_independent(x) == (I, 1)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; (I + x).as_independent(x) == (I, x)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        Note: when trying to get independent terms, a separation method</span>
-<span class="sd">        might need to be used first. In this case, it is important to keep</span>
-<span class="sd">        track of what you send to this routine so you know how to interpret</span>
-<span class="sd">        the returned values</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import separatevars, log</span>
-<span class="sd">        &gt;&gt;&gt; separatevars(exp(x+y)).as_independent(x)</span>
-<span class="sd">        (exp(y), exp(x))</span>
-<span class="sd">        &gt;&gt;&gt; (x + x*y).as_independent(y)</span>
-<span class="sd">        (x, x*y)</span>
-<span class="sd">        &gt;&gt;&gt; separatevars(x + x*y).as_independent(y)</span>
-<span class="sd">        (x, y + 1)</span>
-<span class="sd">        &gt;&gt;&gt; (x*(1 + y)).as_independent(y)</span>
-<span class="sd">        (x, y + 1)</span>
-<span class="sd">        &gt;&gt;&gt; (x*(1 + y)).expand(mul=True).as_independent(y)</span>
-<span class="sd">        (x, x*y)</span>
-<span class="sd">        &gt;&gt;&gt; a, b=symbols(&#39;a b&#39;,positive=True)</span>
-<span class="sd">        &gt;&gt;&gt; (log(a*b).expand(log=True)).as_independent(b)</span>
-<span class="sd">        (log(a), log(b))</span>
-
-<span class="sd">        See also: .separatevars(), .expand(log=True),</span>
-<span class="sd">                  .as_two_terms(), .as_coeff_add(), .as_coeff_mul()</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">sift</span>
-
-        <span class="n">func</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span>
-        <span class="c"># sift out deps into symbolic and other and ignore</span>
-        <span class="c"># all symbols but those that are in the free symbols</span>
-        <span class="n">sym</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="n">other</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">deps</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">):</span>  <span class="c"># Symbol.is_Symbol is True</span>
-                <span class="n">sym</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">other</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">has</span><span class="p">(</span><span class="n">e</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;return the standard has() if there are no literal symbols, else</span>
-<span class="sd">            check to see that symbol-deps are in the free symbols.&quot;&quot;&quot;</span>
-            <span class="n">has_other</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">other</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">sym</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">has_other</span>
-            <span class="k">return</span> <span class="n">has_other</span> <span class="ow">or</span> <span class="n">e</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">free_symbols</span> <span class="o">&amp;</span> <span class="n">sym</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">hint</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;as_Add&#39;</span><span class="p">,</span> <span class="n">func</span> <span class="ow">is</span> <span class="n">Add</span><span class="p">):</span>
-            <span class="n">want</span> <span class="o">=</span> <span class="n">Add</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">want</span> <span class="o">=</span> <span class="n">Mul</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">want</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">func</span> <span class="ow">or</span>
-                <span class="n">func</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">Add</span> <span class="ow">and</span> <span class="n">func</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">Mul</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">has</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-                <span class="k">return</span> <span class="p">(</span><span class="n">want</span><span class="o">.</span><span class="n">identity</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">want</span><span class="o">.</span><span class="n">identity</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">func</span> <span class="ow">is</span> <span class="n">Add</span><span class="p">:</span>
-                <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">args</span><span class="p">,</span> <span class="n">nc</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-
-        <span class="n">d</span> <span class="o">=</span> <span class="n">sift</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-        <span class="n">depend</span> <span class="o">=</span> <span class="n">d</span><span class="p">[</span><span class="bp">True</span><span class="p">]</span>
-        <span class="n">indep</span> <span class="o">=</span> <span class="n">d</span><span class="p">[</span><span class="bp">False</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">func</span> <span class="ow">is</span> <span class="n">Add</span><span class="p">:</span>  <span class="c"># all terms were treated as commutative</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">indep</span><span class="p">),</span>
-                    <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">depend</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># handle noncommutative by stopping at first dependent term</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">nc</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">has</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-                    <span class="n">depend</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">nc</span><span class="p">[</span><span class="n">i</span><span class="p">:])</span>
-                    <span class="k">break</span>
-                <span class="n">indep</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">indep</span><span class="p">),</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">depend</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.as_real_imag"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.as_real_imag">[docs]</a>    <span class="k">def</span> <span class="nf">as_real_imag</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Performs complex expansion on &#39;self&#39; and returns a tuple</span>
-<span class="sd">           containing collected both real and imaginary parts. This</span>
-<span class="sd">           method can&#39;t be confused with re() and im() functions,</span>
-<span class="sd">           which does not perform complex expansion at evaluation.</span>
-
-<span class="sd">           However it is possible to expand both re() and im()</span>
-<span class="sd">           functions and get exactly the same results as with</span>
-<span class="sd">           a single call to this function.</span>
-
-<span class="sd">           &gt;&gt;&gt; from sympy import symbols, I</span>
-
-<span class="sd">           &gt;&gt;&gt; x, y = symbols(&#39;x,y&#39;, real=True)</span>
-
-<span class="sd">           &gt;&gt;&gt; (x + y*I).as_real_imag()</span>
-<span class="sd">           (x, y)</span>
-
-<span class="sd">           &gt;&gt;&gt; from sympy.abc import z, w</span>
-
-<span class="sd">           &gt;&gt;&gt; (z + w*I).as_real_imag()</span>
-<span class="sd">           (re(z) - im(w), re(w) + im(z))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;ignore&#39;</span><span class="p">)</span> <span class="o">==</span> <span class="bp">self</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">re</span><span class="p">(</span><span class="bp">self</span><span class="p">),</span> <span class="n">C</span><span class="o">.</span><span class="n">im</span><span class="p">(</span><span class="bp">self</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="Expr.as_powers_dict"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.as_powers_dict">[docs]</a>    <span class="k">def</span> <span class="nf">as_powers_dict</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return self as a dictionary of factors with each factor being</span>
-<span class="sd">        treated as a power. The keys are the bases of the factors and the</span>
-<span class="sd">        values, the corresponding exponents. The resulting dictionary should</span>
-<span class="sd">        be used with caution if the expression is a Mul and contains non-</span>
-<span class="sd">        commutative factors since the order that they appeared will be lost in</span>
-<span class="sd">        the dictionary.&quot;&quot;&quot;</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
-        <span class="n">d</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="nb">dict</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()]))</span>
-        <span class="k">return</span> <span class="n">d</span>
-</div>
-<div class="viewcode-block" id="Expr.as_coefficients_dict"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.as_coefficients_dict">[docs]</a>    <span class="k">def</span> <span class="nf">as_coefficients_dict</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return a dictionary mapping terms to their Rational coefficient.</span>
-<span class="sd">        Since the dictionary is a defaultdict, inquiries about terms which</span>
-<span class="sd">        were not present will return a coefficient of 0. If an expression is</span>
-<span class="sd">        not an Add it is considered to have a single term.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import a, x</span>
-<span class="sd">        &gt;&gt;&gt; (3*x + a*x + 4).as_coefficients_dict()</span>
-<span class="sd">        {1: 4, x: 3, a*x: 1}</span>
-<span class="sd">        &gt;&gt;&gt; _[a]</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; (3*a*x).as_coefficients_dict()</span>
-<span class="sd">        {a*x: 3}</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">c</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">c</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="bp">self</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
-        <span class="n">d</span><span class="o">.</span><span class="n">update</span><span class="p">({</span><span class="n">m</span><span class="p">:</span> <span class="n">c</span><span class="p">})</span>
-        <span class="k">return</span> <span class="n">d</span>
-</div>
-    <span class="k">def</span> <span class="nf">as_base_exp</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># a -&gt; b ** e</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-<div class="viewcode-block" id="Expr.as_coeff_terms"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.as_coeff_terms">[docs]</a>    <span class="k">def</span> <span class="nf">as_coeff_terms</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">deps</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        This method is deprecated. Use .as_coeff_mul() instead.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.utilities.exceptions</span> <span class="kn">import</span> <span class="n">SymPyDeprecationWarning</span>
-        <span class="n">SymPyDeprecationWarning</span><span class="p">(</span><span class="n">feature</span><span class="o">=</span><span class="s">&quot;as_coeff_terms()&quot;</span><span class="p">,</span>
-                                <span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;as_coeff_mul()&quot;</span><span class="p">,</span> <span class="n">issue</span><span class="o">=</span><span class="mi">3377</span><span class="p">,</span>
-                                <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.0&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">warn</span><span class="p">()</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">(</span><span class="o">*</span><span class="n">deps</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.as_coeff_factors"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.as_coeff_factors">[docs]</a>    <span class="k">def</span> <span class="nf">as_coeff_factors</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">deps</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        This method is deprecated.  Use .as_coeff_add() instead.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.utilities.exceptions</span> <span class="kn">import</span> <span class="n">SymPyDeprecationWarning</span>
-        <span class="n">SymPyDeprecationWarning</span><span class="p">(</span><span class="n">feature</span><span class="o">=</span><span class="s">&quot;as_coeff_factors()&quot;</span><span class="p">,</span>
-                                <span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;as_coeff_add()&quot;</span><span class="p">,</span> <span class="n">issue</span><span class="o">=</span><span class="mi">3377</span><span class="p">,</span>
-                                <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.0&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">warn</span><span class="p">()</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_coeff_add</span><span class="p">(</span><span class="o">*</span><span class="n">deps</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.as_coeff_mul"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.as_coeff_mul">[docs]</a>    <span class="k">def</span> <span class="nf">as_coeff_mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">deps</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the tuple (c, args) where self is written as a Mul, ``m``.</span>
-
-<span class="sd">        c should be a Rational multiplied by any terms of the Mul that are</span>
-<span class="sd">        independent of deps.</span>
-
-<span class="sd">        args should be a tuple of all other terms of m; args is empty</span>
-<span class="sd">        if self is a Number or if self is independent of deps (when given).</span>
-
-<span class="sd">        This should be used when you don&#39;t know if self is a Mul or not but</span>
-<span class="sd">        you want to treat self as a Mul or if you want to process the</span>
-<span class="sd">        individual arguments of the tail of self as a Mul.</span>
-
-<span class="sd">        - if you know self is a Mul and want only the head, use self.args[0];</span>
-<span class="sd">        - if you don&#39;t want to process the arguments of the tail but need the</span>
-<span class="sd">          tail then use self.as_two_terms() which gives the head and tail;</span>
-<span class="sd">        - if you want to split self into an independent and dependent parts</span>
-<span class="sd">          use ``self.as_independent(*deps)``</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import S</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; (S(3)).as_coeff_mul()</span>
-<span class="sd">        (3, ())</span>
-<span class="sd">        &gt;&gt;&gt; (3*x*y).as_coeff_mul()</span>
-<span class="sd">        (3, (x, y))</span>
-<span class="sd">        &gt;&gt;&gt; (3*x*y).as_coeff_mul(x)</span>
-<span class="sd">        (3*y, (x,))</span>
-<span class="sd">        &gt;&gt;&gt; (3*y).as_coeff_mul(x)</span>
-<span class="sd">        (3*y, ())</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">deps</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">deps</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="p">(</span><span class="bp">self</span><span class="p">,)</span>
-</div>
-<div class="viewcode-block" id="Expr.as_coeff_add"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.as_coeff_add">[docs]</a>    <span class="k">def</span> <span class="nf">as_coeff_add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">deps</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the tuple (c, args) where self is written as an Add, ``a``.</span>
-
-<span class="sd">        c should be a Rational added to any terms of the Add that are</span>
-<span class="sd">        independent of deps.</span>
-
-<span class="sd">        args should be a tuple of all other terms of ``a``; args is empty</span>
-<span class="sd">        if self is a Number or if self is independent of deps (when given).</span>
-
-<span class="sd">        This should be used when you don&#39;t know if self is an Add or not but</span>
-<span class="sd">        you want to treat self as an Add or if you want to process the</span>
-<span class="sd">        individual arguments of the tail of self as an Add.</span>
-
-<span class="sd">        - if you know self is an Add and want only the head, use self.args[0];</span>
-<span class="sd">        - if you don&#39;t want to process the arguments of the tail but need the</span>
-<span class="sd">          tail then use self.as_two_terms() which gives the head and tail.</span>
-<span class="sd">        - if you want to split self into an independent and dependent parts</span>
-<span class="sd">          use ``self.as_independent(*deps)``</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import S</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; (S(3)).as_coeff_add()</span>
-<span class="sd">        (3, ())</span>
-<span class="sd">        &gt;&gt;&gt; (3 + x).as_coeff_add()</span>
-<span class="sd">        (3, (x,))</span>
-<span class="sd">        &gt;&gt;&gt; (3 + x + y).as_coeff_add(x)</span>
-<span class="sd">        (y + 3, (x,))</span>
-<span class="sd">        &gt;&gt;&gt; (3 + y).as_coeff_add(x)</span>
-<span class="sd">        (y + 3, ())</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">deps</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">deps</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="p">(</span><span class="bp">self</span><span class="p">,)</span>
-</div>
-<div class="viewcode-block" id="Expr.primitive"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.primitive">[docs]</a>    <span class="k">def</span> <span class="nf">primitive</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the positive Rational that can be extracted non-recursively</span>
-<span class="sd">        from every term of self (i.e., self is treated like an Add). This is</span>
-<span class="sd">        like the as_coeff_Mul() method but primitive always extracts a positive</span>
-<span class="sd">        Rational (never a negative or a Float).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; (3*(x + 1)**2).primitive()</span>
-<span class="sd">        (3, (x + 1)**2)</span>
-<span class="sd">        &gt;&gt;&gt; a = (6*x + 2); a.primitive()</span>
-<span class="sd">        (2, 3*x + 1)</span>
-<span class="sd">        &gt;&gt;&gt; b = (x/2 + 3); b.primitive()</span>
-<span class="sd">        (1/2, x + 6)</span>
-<span class="sd">        &gt;&gt;&gt; (a*b).primitive() == (1, a*b)</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="n">c</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">(</span><span class="n">rational</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">c</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="n">c</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="o">-</span><span class="n">c</span><span class="p">,</span> <span class="o">-</span><span class="n">r</span>
-        <span class="k">return</span> <span class="n">c</span><span class="p">,</span> <span class="n">r</span>
-</div>
-<div class="viewcode-block" id="Expr.as_content_primitive"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.as_content_primitive">[docs]</a>    <span class="k">def</span> <span class="nf">as_content_primitive</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">radical</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;This method should recursively remove a Rational from all arguments</span>
-<span class="sd">        and return that (content) and the new self (primitive). The content</span>
-<span class="sd">        should always be positive and ``Mul(*foo.as_content_primitive()) == foo``.</span>
-<span class="sd">        The primitive need no be in canonical form and should try to preserve</span>
-<span class="sd">        the underlying structure if possible (i.e. expand_mul should not be</span>
-<span class="sd">        applied to self).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sqrt</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-
-<span class="sd">        &gt;&gt;&gt; eq = 2 + 2*x + 2*y*(3 + 3*y)</span>
-
-<span class="sd">        The as_content_primitive function is recursive and retains structure:</span>
-
-<span class="sd">        &gt;&gt;&gt; eq.as_content_primitive()</span>
-<span class="sd">        (2, x + 3*y*(y + 1) + 1)</span>
-
-<span class="sd">        Integer powers will have Rationals extracted from the base:</span>
-
-<span class="sd">        &gt;&gt;&gt; ((2 + 6*x)**2).as_content_primitive()</span>
-<span class="sd">        (4, (3*x + 1)**2)</span>
-<span class="sd">        &gt;&gt;&gt; ((2 + 6*x)**(2*y)).as_content_primitive()</span>
-<span class="sd">        (1, (2*(3*x + 1))**(2*y))</span>
-
-<span class="sd">        Terms may end up joining once their as_content_primitives are added:</span>
-
-<span class="sd">        &gt;&gt;&gt; ((5*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive()</span>
-<span class="sd">        (11, x*(y + 1))</span>
-<span class="sd">        &gt;&gt;&gt; ((3*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive()</span>
-<span class="sd">        (9, x*(y + 1))</span>
-<span class="sd">        &gt;&gt;&gt; ((3*(z*(1 + y)) + 2.0*x*(3 + 3*y))).as_content_primitive()</span>
-<span class="sd">        (1, 6.0*x*(y + 1) + 3*z*(y + 1))</span>
-<span class="sd">        &gt;&gt;&gt; ((5*(x*(1 + y)) + 2*x*(3 + 3*y))**2).as_content_primitive()</span>
-<span class="sd">        (121, x**2*(y + 1)**2)</span>
-<span class="sd">        &gt;&gt;&gt; ((5*(x*(1 + y)) + 2.0*x*(3 + 3*y))**2).as_content_primitive()</span>
-<span class="sd">        (1, 121.0*x**2*(y + 1)**2)</span>
-
-<span class="sd">        Radical content can also be factored out of the primitive:</span>
-
-<span class="sd">        &gt;&gt;&gt; (2*sqrt(2) + 4*sqrt(10)).as_content_primitive(radical=True)</span>
-<span class="sd">        (2, sqrt(2)*(1 + 2*sqrt(5)))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="bp">self</span>
-</div>
-<div class="viewcode-block" id="Expr.as_numer_denom"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.as_numer_denom">[docs]</a>    <span class="k">def</span> <span class="nf">as_numer_denom</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; expression -&gt; a/b -&gt; a, b</span>
-
-<span class="sd">        This is just a stub that should be defined by</span>
-<span class="sd">        an object&#39;s class methods to get anything else.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        normal: return a/b instead of a, b</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-</div>
-    <span class="k">def</span> <span class="nf">normal</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">d</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">n</span>
-        <span class="k">return</span> <span class="n">n</span><span class="o">/</span><span class="n">d</span>
-
-<div class="viewcode-block" id="Expr.extract_multiplicatively"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.extract_multiplicatively">[docs]</a>    <span class="k">def</span> <span class="nf">extract_multiplicatively</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return None if it&#39;s not possible to make self in the form</span>
-<span class="sd">           c * something in a nice way, i.e. preserving the properties</span>
-<span class="sd">           of arguments of self.</span>
-
-<span class="sd">           &gt;&gt;&gt; from sympy import symbols, Rational</span>
-
-<span class="sd">           &gt;&gt;&gt; x, y = symbols(&#39;x,y&#39;, real=True)</span>
-
-<span class="sd">           &gt;&gt;&gt; ((x*y)**3).extract_multiplicatively(x**2 * y)</span>
-<span class="sd">           x*y**2</span>
-
-<span class="sd">           &gt;&gt;&gt; ((x*y)**3).extract_multiplicatively(x**4 * y)</span>
-
-<span class="sd">           &gt;&gt;&gt; (2*x).extract_multiplicatively(2)</span>
-<span class="sd">           x</span>
-
-<span class="sd">           &gt;&gt;&gt; (2*x).extract_multiplicatively(3)</span>
-
-<span class="sd">           &gt;&gt;&gt; (Rational(1,2)*x).extract_multiplicatively(3)</span>
-<span class="sd">           x/6</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">elif</span> <span class="n">c</span> <span class="o">==</span> <span class="bp">self</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">if</span> <span class="n">c</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">cc</span><span class="p">,</span> <span class="n">pc</span> <span class="o">=</span> <span class="n">c</span><span class="o">.</span><span class="n">primitive</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">cc</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="n">cc</span><span class="p">,</span> <span class="n">pc</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">c</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">c</span><span class="o">.</span><span class="n">as_two_terms</span><span class="p">()</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">x</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="n">quotient</span> <span class="o">=</span> <span class="bp">self</span> <span class="o">/</span> <span class="n">c</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">c</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-            <span class="k">elif</span> <span class="bp">self</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">c</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-                <span class="k">elif</span> <span class="n">c</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-            <span class="k">elif</span> <span class="bp">self</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">c</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span>
-            <span class="k">elif</span> <span class="bp">self</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">quotient</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-                <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_positive</span> <span class="ow">and</span> <span class="n">quotient</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">quotient</span>
-            <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">quotient</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-                <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_positive</span> <span class="ow">and</span> <span class="n">quotient</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">quotient</span>
-            <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">quotient</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-                <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_positive</span> <span class="ow">and</span> <span class="n">quotient</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">quotient</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_NumberSymbol</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Symbol</span> <span class="ow">or</span> <span class="bp">self</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">quotient</span><span class="o">.</span><span class="n">is_Mul</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">quotient</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">quotient</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">quotient</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_positive</span> <span class="ow">and</span> <span class="n">quotient</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="bp">self</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">quotient</span>
-            <span class="k">elif</span> <span class="n">quotient</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">quotient</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">cs</span><span class="p">,</span> <span class="n">ps</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">primitive</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">cs</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="n">cs</span><span class="p">,</span> <span class="n">ps</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-            <span class="n">newargs</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="n">newarg</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">newarg</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">newargs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">newarg</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-            <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">newargs</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">arg</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-                <span class="n">newarg</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">newarg</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">newarg</span>
-                    <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">c</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">c</span><span class="o">.</span><span class="n">base</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="p">:</span>
-                <span class="n">new_exp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">extract_additively</span><span class="p">(</span><span class="n">c</span><span class="o">.</span><span class="n">exp</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">new_exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span> <span class="o">**</span> <span class="p">(</span><span class="n">new_exp</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">c</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="p">:</span>
-                <span class="n">new_exp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">extract_additively</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">new_exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span> <span class="o">**</span> <span class="p">(</span><span class="n">new_exp</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.extract_additively"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.extract_additively">[docs]</a>    <span class="k">def</span> <span class="nf">extract_additively</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return self - c if it&#39;s possible to subtract c from self and</span>
-<span class="sd">        make all matching coefficients move towards zero, else return None.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import S</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; e = 2*x + 3</span>
-<span class="sd">        &gt;&gt;&gt; e.extract_additively(x + 1)</span>
-<span class="sd">        x + 2</span>
-<span class="sd">        &gt;&gt;&gt; e.extract_additively(3*x)</span>
-<span class="sd">        &gt;&gt;&gt; e.extract_additively(4)</span>
-<span class="sd">        &gt;&gt;&gt; (y*(x + 1)).extract_additively(x + 1)</span>
-<span class="sd">        &gt;&gt;&gt; ((x + 1)*(x + 2*y + 1) + 3).extract_additively(x + 1)</span>
-<span class="sd">        (x + 1)*(x + 2*y) + 3</span>
-
-<span class="sd">        Sometimes auto-expansion will return a less simplified result</span>
-<span class="sd">        than desired; gcd_terms might be used in such cases:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import gcd_terms</span>
-<span class="sd">        &gt;&gt;&gt; (4*x*(y + 1) + y).extract_additively(x)</span>
-<span class="sd">        4*x*(y + 1) + x*(4*y + 3) - x*(4*y + 4) + y</span>
-<span class="sd">        &gt;&gt;&gt; gcd_terms(_)</span>
-<span class="sd">        x*(4*y + 3) + y</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        extract_multiplicatively</span>
-<span class="sd">        coeff</span>
-<span class="sd">        as_coefficient</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">c</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">elif</span> <span class="n">c</span> <span class="o">==</span> <span class="bp">self</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">elif</span> <span class="bp">self</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">c</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="n">co</span> <span class="o">=</span> <span class="bp">self</span>
-            <span class="n">diff</span> <span class="o">=</span> <span class="n">co</span> <span class="o">-</span> <span class="n">c</span>
-            <span class="c"># XXX should we match types? i.e should 3 - .1 succeed?</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">co</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">diff</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">diff</span> <span class="o">&lt;</span> <span class="n">co</span> <span class="ow">or</span>
-                    <span class="n">co</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">diff</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">diff</span> <span class="o">&gt;</span> <span class="n">co</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">diff</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="k">if</span> <span class="n">c</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="n">co</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_coeff_Add</span><span class="p">()</span>
-            <span class="n">xa</span> <span class="o">=</span> <span class="n">co</span><span class="o">.</span><span class="n">extract_additively</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">xa</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="k">return</span> <span class="n">xa</span> <span class="o">+</span> <span class="n">t</span>
-
-        <span class="c"># handle the args[0].is_Number case separately</span>
-        <span class="c"># since we will have trouble looking for the coeff of</span>
-        <span class="c"># a number.</span>
-        <span class="k">if</span> <span class="n">c</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">and</span> <span class="n">c</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="c"># whole term as a term factor</span>
-            <span class="n">co</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-            <span class="n">xa0</span> <span class="o">=</span> <span class="p">(</span><span class="n">co</span><span class="o">.</span><span class="n">extract_additively</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="ow">or</span> <span class="mi">0</span><span class="p">)</span><span class="o">*</span><span class="n">c</span>
-            <span class="k">if</span> <span class="n">xa0</span><span class="p">:</span>
-                <span class="n">diff</span> <span class="o">=</span> <span class="bp">self</span> <span class="o">-</span> <span class="n">co</span><span class="o">*</span><span class="n">c</span>
-                <span class="k">return</span> <span class="p">(</span><span class="n">xa0</span> <span class="o">+</span> <span class="p">(</span><span class="n">diff</span><span class="o">.</span><span class="n">extract_additively</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="ow">or</span> <span class="n">diff</span><span class="p">))</span> <span class="ow">or</span> <span class="bp">None</span>
-            <span class="c"># term-wise</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">c</span><span class="o">.</span><span class="n">as_coeff_Add</span><span class="p">()</span>
-            <span class="n">sh</span><span class="p">,</span> <span class="n">st</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_coeff_Add</span><span class="p">()</span>
-            <span class="n">xa</span> <span class="o">=</span> <span class="n">sh</span><span class="o">.</span><span class="n">extract_additively</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">xa</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="n">xa2</span> <span class="o">=</span> <span class="n">st</span><span class="o">.</span><span class="n">extract_additively</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">xa2</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="k">return</span> <span class="n">xa</span> <span class="o">+</span> <span class="n">xa2</span>
-
-        <span class="c"># whole term as a term factor</span>
-        <span class="n">co</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-        <span class="n">xa0</span> <span class="o">=</span> <span class="p">(</span><span class="n">co</span><span class="o">.</span><span class="n">extract_additively</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="ow">or</span> <span class="mi">0</span><span class="p">)</span><span class="o">*</span><span class="n">c</span>
-        <span class="k">if</span> <span class="n">xa0</span><span class="p">:</span>
-            <span class="n">diff</span> <span class="o">=</span> <span class="bp">self</span> <span class="o">-</span> <span class="n">co</span><span class="o">*</span><span class="n">c</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">xa0</span> <span class="o">+</span> <span class="p">(</span><span class="n">diff</span><span class="o">.</span><span class="n">extract_additively</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="ow">or</span> <span class="n">diff</span><span class="p">))</span> <span class="ow">or</span> <span class="bp">None</span>
-        <span class="c"># term-wise</span>
-        <span class="n">coeffs</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
-            <span class="n">ac</span><span class="p">,</span> <span class="n">at</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-            <span class="n">co</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">at</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">co</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="n">coc</span><span class="p">,</span> <span class="n">cot</span> <span class="o">=</span> <span class="n">co</span><span class="o">.</span><span class="n">as_coeff_Add</span><span class="p">()</span>
-            <span class="n">xa</span> <span class="o">=</span> <span class="n">coc</span><span class="o">.</span><span class="n">extract_additively</span><span class="p">(</span><span class="n">ac</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">xa</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="bp">self</span> <span class="o">-=</span> <span class="n">co</span><span class="o">*</span><span class="n">at</span>
-            <span class="n">coeffs</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">cot</span> <span class="o">+</span> <span class="n">xa</span><span class="p">)</span><span class="o">*</span><span class="n">at</span><span class="p">)</span>
-        <span class="n">coeffs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">coeffs</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.could_extract_minus_sign"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.could_extract_minus_sign">[docs]</a>    <span class="k">def</span> <span class="nf">could_extract_minus_sign</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Canonical way to choose an element in the set {e, -e} where</span>
-<span class="sd">           e is any expression. If the canonical element is e, we have</span>
-<span class="sd">           e.could_extract_minus_sign() == True, else</span>
-<span class="sd">           e.could_extract_minus_sign() == False.</span>
-
-<span class="sd">           For any expression, the set ``{e.could_extract_minus_sign(),</span>
-<span class="sd">           (-e).could_extract_minus_sign()}`` must be ``{True, False}``.</span>
-
-<span class="sd">           &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">           &gt;&gt;&gt; (x-y).could_extract_minus_sign() != (y-x).could_extract_minus_sign()</span>
-<span class="sd">           True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">negative_self</span> <span class="o">=</span> <span class="o">-</span><span class="bp">self</span>
-        <span class="n">self_has_minus</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">)</span>
-        <span class="n">negative_self_has_minus</span> <span class="o">=</span> <span class="p">(</span>
-            <span class="p">(</span><span class="n">negative_self</span><span class="p">)</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">self_has_minus</span> <span class="o">!=</span> <span class="n">negative_self_has_minus</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">self_has_minus</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="c"># We choose the one with less arguments with minus signs</span>
-                <span class="n">all_args</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-                <span class="n">negative_args</span> <span class="o">=</span> <span class="nb">len</span><span class="p">([</span><span class="bp">False</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">()])</span>
-                <span class="n">positive_args</span> <span class="o">=</span> <span class="n">all_args</span> <span class="o">-</span> <span class="n">negative_args</span>
-                <span class="k">if</span> <span class="n">positive_args</span> <span class="o">&gt;</span> <span class="n">negative_args</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-                <span class="k">elif</span> <span class="n">positive_args</span> <span class="o">&lt;</span> <span class="n">negative_args</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                <span class="c"># We choose the one with an odd number of minus signs</span>
-                <span class="n">num</span><span class="p">,</span> <span class="n">den</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-                <span class="n">args</span> <span class="o">=</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">num</span><span class="p">)</span> <span class="o">+</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">den</span><span class="p">)</span>
-                <span class="n">arg_signs</span> <span class="o">=</span> <span class="p">[</span><span class="n">arg</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">()</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span>
-                <span class="n">negative_args</span> <span class="o">=</span> <span class="p">[</span><span class="n">_f</span> <span class="k">for</span> <span class="n">_f</span> <span class="ow">in</span> <span class="n">arg_signs</span> <span class="k">if</span> <span class="n">_f</span><span class="p">]</span>
-                <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="n">negative_args</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">1</span>
-
-            <span class="c"># As a last resort, we choose the one with greater value of .sort_key()</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">sort_key</span><span class="p">()</span> <span class="o">&lt;</span> <span class="n">negative_self</span><span class="o">.</span><span class="n">sort_key</span><span class="p">()</span>
-</div>
-<div class="viewcode-block" id="Expr.extract_branch_factor"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.extract_branch_factor">[docs]</a>    <span class="k">def</span> <span class="nf">extract_branch_factor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">allow_half</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Try to write self as ``exp_polar(2*pi*I*n)*z`` in a nice way.</span>
-<span class="sd">        Return (z, n).</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import exp_polar, I, pi</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; exp_polar(I*pi).extract_branch_factor()</span>
-<span class="sd">        (exp_polar(I*pi), 0)</span>
-<span class="sd">        &gt;&gt;&gt; exp_polar(2*I*pi).extract_branch_factor()</span>
-<span class="sd">        (1, 1)</span>
-<span class="sd">        &gt;&gt;&gt; exp_polar(-pi*I).extract_branch_factor()</span>
-<span class="sd">        (exp_polar(I*pi), -1)</span>
-<span class="sd">        &gt;&gt;&gt; exp_polar(3*pi*I + x).extract_branch_factor()</span>
-<span class="sd">        (exp_polar(x + I*pi), 1)</span>
-<span class="sd">        &gt;&gt;&gt; (y*exp_polar(-5*pi*I)*exp_polar(3*pi*I + 2*pi*x)).extract_branch_factor()</span>
-<span class="sd">        (y*exp_polar(2*pi*x), -1)</span>
-<span class="sd">        &gt;&gt;&gt; exp_polar(-I*pi/2).extract_branch_factor()</span>
-<span class="sd">        (exp_polar(-I*pi/2), 0)</span>
-
-<span class="sd">        If allow_half is True, also extract exp_polar(I*pi):</span>
-
-<span class="sd">        &gt;&gt;&gt; exp_polar(I*pi).extract_branch_factor(allow_half=True)</span>
-<span class="sd">        (1, 1/2)</span>
-<span class="sd">        &gt;&gt;&gt; exp_polar(2*I*pi).extract_branch_factor(allow_half=True)</span>
-<span class="sd">        (1, 1)</span>
-<span class="sd">        &gt;&gt;&gt; exp_polar(3*I*pi).extract_branch_factor(allow_half=True)</span>
-<span class="sd">        (1, 3/2)</span>
-<span class="sd">        &gt;&gt;&gt; exp_polar(-I*pi).extract_branch_factor(allow_half=True)</span>
-<span class="sd">        (1, -1/2)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exp_polar</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">ceiling</span><span class="p">,</span> <span class="n">Add</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="n">exps</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">exp_polar</span><span class="p">:</span>
-                <span class="n">exps</span> <span class="o">+=</span> <span class="p">[</span><span class="n">arg</span><span class="o">.</span><span class="n">exp</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">res</span> <span class="o">*=</span> <span class="n">arg</span>
-        <span class="n">piimult</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="n">extras</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">while</span> <span class="n">exps</span><span class="p">:</span>
-            <span class="n">exp</span> <span class="o">=</span> <span class="n">exps</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">exp</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="n">exps</span> <span class="o">+=</span> <span class="n">exp</span><span class="o">.</span><span class="n">args</span>
-                <span class="k">continue</span>
-            <span class="k">if</span> <span class="n">exp</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                <span class="n">coeff</span> <span class="o">=</span> <span class="n">exp</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">piimult</span> <span class="o">+=</span> <span class="n">coeff</span>
-                    <span class="k">continue</span>
-            <span class="n">extras</span> <span class="o">+=</span> <span class="p">[</span><span class="n">exp</span><span class="p">]</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">piimult</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">piimult</span>
-            <span class="n">tail</span> <span class="o">=</span> <span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">coeff</span><span class="p">,</span> <span class="n">tail</span> <span class="o">=</span> <span class="n">piimult</span><span class="o">.</span><span class="n">as_coeff_add</span><span class="p">(</span><span class="o">*</span><span class="n">piimult</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">)</span>
-        <span class="c"># round down to nearest multiple of 2</span>
-        <span class="n">branchfact</span> <span class="o">=</span> <span class="n">ceiling</span><span class="p">(</span><span class="n">coeff</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="mi">2</span>
-        <span class="n">n</span> <span class="o">+=</span> <span class="n">branchfact</span><span class="o">/</span><span class="mi">2</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">coeff</span> <span class="o">-</span> <span class="n">branchfact</span>
-        <span class="k">if</span> <span class="n">allow_half</span><span class="p">:</span>
-            <span class="n">nc</span> <span class="o">=</span> <span class="n">c</span><span class="o">.</span><span class="n">extract_additively</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">nc</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">n</span> <span class="o">+=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="n">nc</span>
-        <span class="n">newexp</span> <span class="o">=</span> <span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">((</span><span class="n">c</span><span class="p">,</span> <span class="p">)</span> <span class="o">+</span> <span class="n">tail</span><span class="p">))</span> <span class="o">+</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">extras</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">newexp</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">res</span> <span class="o">*=</span> <span class="n">exp_polar</span><span class="p">(</span><span class="n">newexp</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">res</span><span class="p">,</span> <span class="n">n</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_is_polynomial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">syms</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">syms</span><span class="p">)</span> <span class="o">==</span> <span class="nb">set</span><span class="p">([]):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-<div class="viewcode-block" id="Expr.is_polynomial"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.is_polynomial">[docs]</a>    <span class="k">def</span> <span class="nf">is_polynomial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">syms</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if self is a polynomial in syms and False otherwise.</span>
-
-<span class="sd">        This checks if self is an exact polynomial in syms.  This function</span>
-<span class="sd">        returns False for expressions that are &quot;polynomials&quot; with symbolic</span>
-<span class="sd">        exponents.  Thus, you should be able to apply polynomial algorithms to</span>
-<span class="sd">        expressions for which this returns True, and Poly(expr, \*syms) should</span>
-<span class="sd">        work only if and only if expr.is_polynomial(\*syms) returns True. The</span>
-<span class="sd">        polynomial does not have to be in expanded form.  If no symbols are</span>
-<span class="sd">        given, all free symbols in the expression will be used.</span>
-
-<span class="sd">        This is not part of the assumptions system.  You cannot do</span>
-<span class="sd">        Symbol(&#39;z&#39;, polynomial=True).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; x = Symbol(&#39;x&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; ((x**2 + 1)**4).is_polynomial(x)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; ((x**2 + 1)**4).is_polynomial()</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; (2**x + 1).is_polynomial(x)</span>
-<span class="sd">        False</span>
-
-
-<span class="sd">        &gt;&gt;&gt; n = Symbol(&#39;n&#39;, nonnegative=True, integer=True)</span>
-<span class="sd">        &gt;&gt;&gt; (x**n + 1).is_polynomial(x)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        This function does not attempt any nontrivial simplifications that may</span>
-<span class="sd">        result in an expression that does not appear to be a polynomial to</span>
-<span class="sd">        become one.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sqrt, factor, cancel</span>
-<span class="sd">        &gt;&gt;&gt; y = Symbol(&#39;y&#39;, positive=True)</span>
-<span class="sd">        &gt;&gt;&gt; a = sqrt(y**2 + 2*y + 1)</span>
-<span class="sd">        &gt;&gt;&gt; a.is_polynomial(y)</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; factor(a)</span>
-<span class="sd">        y + 1</span>
-<span class="sd">        &gt;&gt;&gt; factor(a).is_polynomial(y)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &gt;&gt;&gt; b = (y**2 + 2*y + 1)/(y + 1)</span>
-<span class="sd">        &gt;&gt;&gt; b.is_polynomial(y)</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; cancel(b)</span>
-<span class="sd">        y + 1</span>
-<span class="sd">        &gt;&gt;&gt; cancel(b).is_polynomial(y)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See also .is_rational_function()</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">syms</span><span class="p">:</span>
-            <span class="n">syms</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="n">syms</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">syms</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span>
-
-        <span class="k">if</span> <span class="n">syms</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">)</span> <span class="o">==</span> <span class="nb">set</span><span class="p">([]):</span>
-            <span class="c"># constant polynomial</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_is_polynomial</span><span class="p">(</span><span class="n">syms</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_is_rational_function</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">syms</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">syms</span><span class="p">)</span> <span class="o">==</span> <span class="nb">set</span><span class="p">([]):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-<div class="viewcode-block" id="Expr.is_rational_function"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.is_rational_function">[docs]</a>    <span class="k">def</span> <span class="nf">is_rational_function</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">syms</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Test whether function is a ratio of two polynomials in the given</span>
-<span class="sd">        symbols, syms. When syms is not given, all free symbols will be used.</span>
-<span class="sd">        The rational function does not have to be in expanded or in any kind of</span>
-<span class="sd">        canonical form.</span>
-
-<span class="sd">        This function returns False for expressions that are &quot;rational</span>
-<span class="sd">        functions&quot; with symbolic exponents.  Thus, you should be able to call</span>
-<span class="sd">        .as_numer_denom() and apply polynomial algorithms to the result for</span>
-<span class="sd">        expressions for which this returns True.</span>
-
-<span class="sd">        This is not part of the assumptions system.  You cannot do</span>
-<span class="sd">        Symbol(&#39;z&#39;, rational_function=True).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol, sin</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; (x/y).is_rational_function()</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &gt;&gt;&gt; (x**2).is_rational_function()</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &gt;&gt;&gt; (x/sin(y)).is_rational_function(y)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &gt;&gt;&gt; n = Symbol(&#39;n&#39;, integer=True)</span>
-<span class="sd">        &gt;&gt;&gt; (x**n + 1).is_rational_function(x)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        This function does not attempt any nontrivial simplifications that may</span>
-<span class="sd">        result in an expression that does not appear to be a rational function</span>
-<span class="sd">        to become one.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sqrt, factor, cancel</span>
-<span class="sd">        &gt;&gt;&gt; y = Symbol(&#39;y&#39;, positive=True)</span>
-<span class="sd">        &gt;&gt;&gt; a = sqrt(y**2 + 2*y + 1)/y</span>
-<span class="sd">        &gt;&gt;&gt; a.is_rational_function(y)</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; factor(a)</span>
-<span class="sd">        (y + 1)/y</span>
-<span class="sd">        &gt;&gt;&gt; factor(a).is_rational_function(y)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See also is_rational_function().</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">syms</span><span class="p">:</span>
-            <span class="n">syms</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="n">syms</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">syms</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span>
-
-        <span class="k">if</span> <span class="n">syms</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">)</span> <span class="o">==</span> <span class="nb">set</span><span class="p">([]):</span>
-            <span class="c"># constant rational function</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_is_rational_function</span><span class="p">(</span><span class="n">syms</span><span class="p">)</span>
-
-    <span class="c">###################################################################################</span>
-    <span class="c">##################### SERIES, LEADING TERM, LIMIT, ORDER METHODS ##################</span>
-    <span class="c">###################################################################################</span>
-</div>
-<div class="viewcode-block" id="Expr.series"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.series">[docs]</a>    <span class="k">def</span> <span class="nf">series</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">x0</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">6</span><span class="p">,</span> <span class="nb">dir</span><span class="o">=</span><span class="s">&quot;+&quot;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Series expansion of &quot;self&quot; around ``x = x0`` yielding either terms of</span>
-<span class="sd">        the series one by one (the lazy series given when n=None), else</span>
-<span class="sd">        all the terms at once when n != None.</span>
-
-<span class="sd">        Note: when n != None, if an O() term is returned then the x in the</span>
-<span class="sd">        in it and the entire expression represents x - x0, the displacement</span>
-<span class="sd">        from x0. (If there is no O() term then the series was exact and x has</span>
-<span class="sd">        it&#39;s normal meaning.) This is currently necessary since sympy&#39;s O()</span>
-<span class="sd">        can only represent terms at x0=0. So instead of::</span>
-
-<span class="sd">          cos(x).series(x0=1, n=2) --&gt; (1 - x)*sin(1) + cos(1) + O((x - 1)**2)</span>
-
-<span class="sd">        which graphically looks like this::</span>
-
-<span class="sd">               |</span>
-<span class="sd">              .|.         . .</span>
-<span class="sd">             . | \      .     .</span>
-<span class="sd">            ---+----------------------</span>
-<span class="sd">               |   . .          . .</span>
-<span class="sd">               |    \</span>
-<span class="sd">              x=0</span>
-
-<span class="sd">        the following is returned instead::</span>
-
-<span class="sd">        -x*sin(1) + cos(1) + O(x**2)</span>
-
-<span class="sd">        whose graph is this::</span>
-
-<span class="sd">               \ |</span>
-<span class="sd">              . .|        . .</span>
-<span class="sd">             .   \      .     .</span>
-<span class="sd">            -----+\------------------.</span>
-<span class="sd">                 | . .          . .</span>
-<span class="sd">                 |  \</span>
-<span class="sd">                x=0</span>
-
-<span class="sd">        which is identical to ``cos(x + 1).series(n=2)``.</span>
-
-<span class="sd">        Usage:</span>
-<span class="sd">            Returns the series expansion of &quot;self&quot; around the point ``x = x0``</span>
-<span class="sd">            with respect to ``x`` up to O(x**n) (default n is 6).</span>
-
-<span class="sd">            If ``x=None`` and ``self`` is univariate, the univariate symbol will</span>
-<span class="sd">            be supplied, otherwise an error will be raised.</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy import cos, exp</span>
-<span class="sd">            &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">            &gt;&gt;&gt; cos(x).series()</span>
-<span class="sd">            1 - x**2/2 + x**4/24 + O(x**6)</span>
-<span class="sd">            &gt;&gt;&gt; cos(x).series(n=4)</span>
-<span class="sd">            1 - x**2/2 + O(x**4)</span>
-<span class="sd">            &gt;&gt;&gt; e = cos(x + exp(y))</span>
-<span class="sd">            &gt;&gt;&gt; e.series(y, n=2)</span>
-<span class="sd">            cos(x + 1) - y*sin(x + 1) + O(y**2)</span>
-<span class="sd">            &gt;&gt;&gt; e.series(x, n=2)</span>
-<span class="sd">            cos(exp(y)) - x*sin(exp(y)) + O(x**2)</span>
-
-<span class="sd">            If ``n=None`` then an iterator of the series terms will be returned.</span>
-
-<span class="sd">            &gt;&gt;&gt; term=cos(x).series(n=None)</span>
-<span class="sd">            &gt;&gt;&gt; [next(term) for i in range(2)]</span>
-<span class="sd">            [1, -x**2/2]</span>
-
-<span class="sd">            For ``dir=+`` (default) the series is calculated from the right and</span>
-<span class="sd">            for ``dir=-`` the series from the left. For smooth functions this</span>
-<span class="sd">            flag will not alter the results.</span>
-
-<span class="sd">            &gt;&gt;&gt; abs(x).series(dir=&quot;+&quot;)</span>
-<span class="sd">            x</span>
-<span class="sd">            &gt;&gt;&gt; abs(x).series(dir=&quot;-&quot;)</span>
-<span class="sd">            -x</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">collect</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">syms</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">syms</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;x must be given for multivariate functions.&#39;</span><span class="p">)</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">syms</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="n">s</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="p">[</span><span class="bp">self</span><span class="p">])</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span>
-
-        <span class="c">## it seems like the following should be doable, but several failures</span>
-        <span class="c">## then occur. Is this related to issue 1747 et al See also XPOS below.</span>
-        <span class="c">#if x.is_positive is x.is_negative is None:</span>
-        <span class="c">#    # replace x with an x that has a positive assumption</span>
-        <span class="c">#    xpos = C.Dummy(&#39;x&#39;, positive=True)</span>
-        <span class="c">#    rv = self.subs(x, xpos).series(xpos, x0, n, dir)</span>
-        <span class="c">#    if n is None:</span>
-        <span class="c">#        return (s.subs(xpos, x) for s in rv)</span>
-        <span class="c">#    else:</span>
-        <span class="c">#        return rv.subs(xpos, x)</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="nb">dir</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span> <span class="ow">or</span> <span class="nb">dir</span> <span class="ow">not</span> <span class="ow">in</span> <span class="s">&#39;+-&#39;</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Dir must be &#39;+&#39; or &#39;-&#39;&quot;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">x0</span> <span class="ow">in</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">]:</span>
-            <span class="nb">dir</span> <span class="o">=</span> <span class="p">{</span><span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span> <span class="s">&#39;+&#39;</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span> <span class="s">&#39;-&#39;</span><span class="p">}[</span><span class="n">x0</span><span class="p">]</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="nb">dir</span><span class="o">=</span><span class="nb">dir</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="n">si</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">si</span> <span class="ow">in</span> <span class="n">s</span><span class="p">)</span>
-            <span class="c"># don&#39;t include the order term since it will eat the larger terms</span>
-            <span class="k">return</span> <span class="n">s</span><span class="o">.</span><span class="n">removeO</span><span class="p">()</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="c"># use rep to shift origin to x0 and change sign (if dir is negative)</span>
-        <span class="c"># and undo the process with rep2</span>
-        <span class="k">if</span> <span class="n">x0</span> <span class="ow">or</span> <span class="nb">dir</span> <span class="o">==</span> <span class="s">&#39;-&#39;</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">dir</span> <span class="o">==</span> <span class="s">&#39;-&#39;</span><span class="p">:</span>
-                <span class="n">rep</span> <span class="o">=</span> <span class="o">-</span><span class="n">x</span> <span class="o">+</span> <span class="n">x0</span>
-                <span class="n">rep2</span> <span class="o">=</span> <span class="o">-</span><span class="n">x</span>
-                <span class="n">rep2b</span> <span class="o">=</span> <span class="n">x0</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">rep</span> <span class="o">=</span> <span class="n">x</span> <span class="o">+</span> <span class="n">x0</span>
-                <span class="n">rep2</span> <span class="o">=</span> <span class="n">x</span>
-                <span class="n">rep2b</span> <span class="o">=</span> <span class="o">-</span><span class="n">x0</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">rep</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x0</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="nb">dir</span><span class="o">=</span><span class="s">&#39;+&#39;</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>  <span class="c"># lseries...</span>
-                <span class="k">return</span> <span class="p">(</span><span class="n">si</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">rep2</span> <span class="o">+</span> <span class="n">rep2b</span><span class="p">)</span> <span class="k">for</span> <span class="n">si</span> <span class="ow">in</span> <span class="n">s</span><span class="p">)</span>
-            <span class="c"># nseries...</span>
-            <span class="n">o</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">getO</span><span class="p">()</span> <span class="ow">or</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">removeO</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">o</span> <span class="ow">and</span> <span class="n">x0</span><span class="p">:</span>
-                <span class="n">rep2b</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c"># when O() can handle x0 != 0 this can be removed</span>
-            <span class="k">return</span> <span class="n">s</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">rep2</span> <span class="o">+</span> <span class="n">rep2b</span><span class="p">)</span> <span class="o">+</span> <span class="n">o</span>
-
-        <span class="c"># from here on it&#39;s x0=0 and dir=&#39;+&#39; handling</span>
-
-        <span class="k">if</span> <span class="n">n</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>  <span class="c"># nseries handling</span>
-            <span class="n">s1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span>
-            <span class="n">o</span> <span class="o">=</span> <span class="n">s1</span><span class="o">.</span><span class="n">getO</span><span class="p">()</span> <span class="ow">or</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">if</span> <span class="n">o</span><span class="p">:</span>
-                <span class="c"># make sure the requested order is returned</span>
-                <span class="n">ngot</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">getn</span><span class="p">()</span>
-                <span class="k">if</span> <span class="n">ngot</span> <span class="o">&gt;</span> <span class="n">n</span><span class="p">:</span>
-                    <span class="c"># leave o in its current form (e.g. with x*log(x)) so</span>
-                    <span class="c"># it eats terms properly, then replace it below</span>
-                    <span class="n">s1</span> <span class="o">+=</span> <span class="n">o</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="n">C</span><span class="o">.</span><span class="n">Rational</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">ngot</span><span class="p">))</span>
-                <span class="k">elif</span> <span class="n">ngot</span> <span class="o">&lt;</span> <span class="n">n</span><span class="p">:</span>
-                    <span class="c"># increase the requested number of terms to get the desired</span>
-                    <span class="c"># number keep increasing (up to 9) until the received order</span>
-                    <span class="c"># is different than the original order and then predict how</span>
-                    <span class="c"># many additional terms are needed</span>
-                    <span class="k">for</span> <span class="n">more</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">9</span><span class="p">):</span>
-                        <span class="n">s1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span> <span class="o">+</span> <span class="n">more</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span>
-                        <span class="n">newn</span> <span class="o">=</span> <span class="n">s1</span><span class="o">.</span><span class="n">getn</span><span class="p">()</span>
-                        <span class="k">if</span> <span class="n">newn</span> <span class="o">!=</span> <span class="n">ngot</span><span class="p">:</span>
-                            <span class="n">ndo</span> <span class="o">=</span> <span class="n">n</span> <span class="o">+</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">ngot</span><span class="p">)</span><span class="o">*</span><span class="n">more</span><span class="o">/</span><span class="p">(</span><span class="n">newn</span> <span class="o">-</span> <span class="n">ngot</span><span class="p">)</span>
-                            <span class="n">s1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">ndo</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span>
-                            <span class="c"># if this assertion fails then our ndo calculation</span>
-                            <span class="c"># needs modification</span>
-                            <span class="k">assert</span> <span class="n">s1</span><span class="o">.</span><span class="n">getn</span><span class="p">()</span> <span class="o">==</span> <span class="n">n</span>
-                            <span class="k">break</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Could not calculate </span><span class="si">%s</span><span class="s"> terms for </span><span class="si">%s</span><span class="s">&#39;</span>
-                                         <span class="o">%</span> <span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="bp">self</span><span class="p">))</span>
-                <span class="n">o</span> <span class="o">=</span> <span class="n">s1</span><span class="o">.</span><span class="n">getO</span><span class="p">()</span>
-                <span class="n">s1</span> <span class="o">=</span> <span class="n">s1</span><span class="o">.</span><span class="n">removeO</span><span class="p">()</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">o</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="p">)</span>
-                <span class="k">if</span> <span class="p">(</span><span class="n">s1</span> <span class="o">+</span> <span class="n">o</span><span class="p">)</span><span class="o">.</span><span class="n">removeO</span><span class="p">()</span> <span class="o">==</span> <span class="n">s1</span><span class="p">:</span>
-                    <span class="n">o</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">collect</span><span class="p">(</span><span class="n">s1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">o</span>
-            <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">s1</span> <span class="o">+</span> <span class="n">o</span>
-
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># lseries handling</span>
-            <span class="k">def</span> <span class="nf">yield_lseries</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
-                <span class="sd">&quot;&quot;&quot;Return terms of lseries one at a time.&quot;&quot;&quot;</span>
-                <span class="k">for</span> <span class="n">si</span> <span class="ow">in</span> <span class="n">s</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">si</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                        <span class="k">yield</span> <span class="n">si</span>
-                        <span class="k">continue</span>
-                    <span class="c"># yield terms 1 at a time if possible</span>
-                    <span class="c"># by increasing order until all the</span>
-                    <span class="c"># terms have been returned</span>
-                    <span class="n">yielded</span> <span class="o">=</span> <span class="mi">0</span>
-                    <span class="n">o</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="n">si</span><span class="p">)</span><span class="o">*</span><span class="n">x</span>
-                    <span class="n">ndid</span> <span class="o">=</span> <span class="mi">0</span>
-                    <span class="n">ndo</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">si</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-                    <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="n">do</span> <span class="o">=</span> <span class="p">(</span><span class="n">si</span> <span class="o">-</span> <span class="n">yielded</span> <span class="o">+</span> <span class="n">o</span><span class="p">)</span><span class="o">.</span><span class="n">removeO</span><span class="p">()</span>
-                        <span class="n">o</span> <span class="o">*=</span> <span class="n">x</span>
-                        <span class="k">if</span> <span class="ow">not</span> <span class="n">do</span> <span class="ow">or</span> <span class="n">do</span><span class="o">.</span><span class="n">is_Order</span><span class="p">:</span>
-                            <span class="k">continue</span>
-                        <span class="k">if</span> <span class="n">do</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                            <span class="n">ndid</span> <span class="o">+=</span> <span class="nb">len</span><span class="p">(</span><span class="n">do</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="n">ndid</span> <span class="o">+=</span> <span class="mi">1</span>
-                        <span class="k">yield</span> <span class="n">do</span>
-                        <span class="k">if</span> <span class="n">ndid</span> <span class="o">==</span> <span class="n">ndo</span><span class="p">:</span>
-                            <span class="k">raise</span> <span class="ne">StopIteration</span>
-                        <span class="n">yielded</span> <span class="o">+=</span> <span class="n">do</span>
-
-            <span class="k">return</span> <span class="n">yield_lseries</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">removeO</span><span class="p">()</span><span class="o">.</span><span class="n">_eval_lseries</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="Expr.lseries"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.lseries">[docs]</a>    <span class="k">def</span> <span class="nf">lseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">x0</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="nb">dir</span><span class="o">=</span><span class="s">&#39;+&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Wrapper for series yielding an iterator of the terms of the series.</span>
-
-<span class="sd">        Note: an infinite series will yield an infinite iterator. The following,</span>
-<span class="sd">        for exaxmple, will never terminate. It will just keep printing terms</span>
-<span class="sd">        of the sin(x) series::</span>
-
-<span class="sd">          for term in sin(x).lseries(x):</span>
-<span class="sd">              print term</span>
-
-<span class="sd">        The advantage of lseries() over nseries() is that many times you are</span>
-<span class="sd">        just interested in the next term in the series (i.e. the first term for</span>
-<span class="sd">        example), but you don&#39;t know how many you should ask for in nseries()</span>
-<span class="sd">        using the &quot;n&quot; parameter.</span>
-
-<span class="sd">        See also nseries().</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="nb">dir</span><span class="o">=</span><span class="nb">dir</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_lseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="c"># default implementation of lseries is using nseries(), and adaptively</span>
-        <span class="c"># increasing the &quot;n&quot;. As you can see, it is not very efficient, because</span>
-        <span class="c"># we are calculating the series over and over again. Subclasses should</span>
-        <span class="c"># override this method and implement much more efficient yielding of</span>
-        <span class="c"># terms.</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">series</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">series</span><span class="o">.</span><span class="n">is_Order</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">series</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="k">yield</span> <span class="n">series</span><span class="o">.</span><span class="n">removeO</span><span class="p">()</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">yield</span> <span class="n">series</span>
-            <span class="k">raise</span> <span class="ne">StopIteration</span>
-
-        <span class="k">while</span> <span class="n">series</span><span class="o">.</span><span class="n">is_Order</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">series</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span>
-        <span class="n">e</span> <span class="o">=</span> <span class="n">series</span><span class="o">.</span><span class="n">removeO</span><span class="p">()</span>
-        <span class="k">yield</span> <span class="n">e</span>
-        <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">n</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="n">series</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span><span class="o">.</span><span class="n">removeO</span><span class="p">()</span>
-                <span class="k">if</span> <span class="n">e</span> <span class="o">!=</span> <span class="n">series</span><span class="p">:</span>
-                    <span class="k">break</span>
-            <span class="k">yield</span> <span class="n">series</span> <span class="o">-</span> <span class="n">e</span>
-            <span class="n">e</span> <span class="o">=</span> <span class="n">series</span>
-
-<div class="viewcode-block" id="Expr.nseries"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.nseries">[docs]</a>    <span class="k">def</span> <span class="nf">nseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">x0</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">6</span><span class="p">,</span> <span class="nb">dir</span><span class="o">=</span><span class="s">&#39;+&#39;</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Wrapper to _eval_nseries if assumptions allow, else to series.</span>
-
-<span class="sd">        If x is given, x0 is 0, dir=&#39;+&#39;, and self has x, then _eval_nseries is</span>
-<span class="sd">        called. This calculates &quot;n&quot; terms in the innermost expressions and</span>
-<span class="sd">        then builds up the final series just by &quot;cross-multiplying&quot; everything</span>
-<span class="sd">        out.</span>
-
-<span class="sd">        Advantage -- it&#39;s fast, because we don&#39;t have to determine how many</span>
-<span class="sd">        terms we need to calculate in advance.</span>
-
-<span class="sd">        Disadvantage -- you may end up with less terms than you may have</span>
-<span class="sd">        expected, but the O(x**n) term appended will always be correct and</span>
-<span class="sd">        so the result, though perhaps shorter, will also be correct.</span>
-
-<span class="sd">        If any of those assumptions is not met, this is treated like a</span>
-<span class="sd">        wrapper to series which will try harder to return the correct</span>
-<span class="sd">        number of terms.</span>
-
-<span class="sd">        See also lseries().</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">and</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">x0</span> <span class="ow">or</span> <span class="nb">dir</span> <span class="o">!=</span> <span class="s">&#39;+&#39;</span><span class="p">:</span>  <span class="c"># {see XPOS above} or (x.is_positive == x.is_negative == None):</span>
-            <span class="k">assert</span> <span class="n">logx</span> <span class="ow">is</span> <span class="bp">None</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="nb">dir</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_nseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return terms of series for self up to O(x**n) at x=0</span>
-<span class="sd">        from the positive direction.</span>
-
-<span class="sd">        This is a method that should be overridden in subclasses. Users should</span>
-<span class="sd">        never call this method directly (use .nseries() instead), so you don&#39;t</span>
-<span class="sd">        have to write docstrings for _eval_nseries().</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.utilities.misc</span> <span class="kn">import</span> <span class="n">filldedent</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&quot;&quot;&quot;</span>
-<span class="s">                     The _eval_nseries method should be added to</span>
-<span class="s">                     </span><span class="si">%s</span><span class="s"> to give terms up to O(x**n) at x=0</span>
-<span class="s">                     from the positive direction so it is available when</span>
-<span class="s">                     nseries calls it.&quot;&quot;&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">)</span>
-                     <span class="p">)</span>
-
-<div class="viewcode-block" id="Expr.limit"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.limit">[docs]</a>    <span class="k">def</span> <span class="nf">limit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">xlim</span><span class="p">,</span> <span class="nb">dir</span><span class="o">=</span><span class="s">&#39;+&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Compute limit x-&gt;xlim.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.series.limits</span> <span class="kn">import</span> <span class="n">limit</span>
-        <span class="k">return</span> <span class="n">limit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">xlim</span><span class="p">,</span> <span class="nb">dir</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.compute_leading_term"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.compute_leading_term">[docs]</a>    <span class="k">def</span> <span class="nf">compute_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">skip_abs</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        as_leading_term is only allowed for results of .series()</span>
-<span class="sd">        This is a wrapper to compute a series first.</span>
-<span class="sd">        If skip_abs is true, the absolute term is assumed to be zero.</span>
-<span class="sd">        (This is necessary because sometimes it cannot be simplified</span>
-<span class="sd">        to zero without a lot of work, but is still known to be zero.</span>
-<span class="sd">        See log._eval_nseries for an example.)</span>
-<span class="sd">        If skip_log is true, log(x) is treated as an independent symbol.</span>
-<span class="sd">        (This is needed for the gruntz algorithm.)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.series.gruntz</span> <span class="kn">import</span> <span class="n">calculate_series</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">cancel</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">removeO</span><span class="p">()</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">if</span> <span class="n">logx</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;logx&#39;</span><span class="p">)</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">calculate_series</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">skip_abs</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">calculate_series</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">skip_abs</span><span class="p">,</span> <span class="n">logx</span><span class="p">)</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">cancel</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">skip_abs</span><span class="p">:</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">s</span><span class="p">)</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">s</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-</div>
-    <span class="nd">@cacheit</span>
-<div class="viewcode-block" id="Expr.as_leading_term"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.as_leading_term">[docs]</a>    <span class="k">def</span> <span class="nf">as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the leading (nonzero) term of the series expansion of self.</span>
-
-<span class="sd">        The _eval_as_leading_term routines are used to do this, and they must</span>
-<span class="sd">        always return a non-zero value.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; (1 + x + x**2).as_leading_term(x)</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; (1/x**2 + x + x**2).as_leading_term(x)</span>
-<span class="sd">        x**(-2)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">powsimp</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="bp">self</span>
-            <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">symbols</span><span class="p">:</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="n">c</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">c</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="n">symbols</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">symbols</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">x</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;expecting a Symbol but got </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">not</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">obj</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">powsimp</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;exp&#39;</span><span class="p">)</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;as_leading_term(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-<div class="viewcode-block" id="Expr.as_coeff_exponent"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.as_coeff_exponent">[docs]</a>    <span class="k">def</span> <span class="nf">as_coeff_exponent</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; ``c*x**e -&gt; c,e`` where x can be any symbolic expression.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">collect</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">collect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="n">c</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">b</span> <span class="o">==</span> <span class="n">x</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">c</span><span class="p">,</span> <span class="n">e</span>
-        <span class="k">return</span> <span class="n">s</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-</div>
-<div class="viewcode-block" id="Expr.leadterm"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.leadterm">[docs]</a>    <span class="k">def</span> <span class="nf">leadterm</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the leading term a*x**b as a tuple (a, b).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; (1+x+x**2).leadterm(x)</span>
-<span class="sd">        (1, 0)</span>
-<span class="sd">        &gt;&gt;&gt; (1/x**2+x+x**2).leadterm(x)</span>
-<span class="sd">        (1, -2)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">c</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">as_coeff_exponent</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">c</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">sympy.utilities.misc</span> <span class="kn">import</span> <span class="n">filldedent</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&quot;&quot;&quot;</span>
-<span class="s">                cannot compute leadterm(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">). The coefficient</span>
-<span class="s">                should have been free of x but got </span><span class="si">%s</span><span class="s">&quot;&quot;&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">c</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">c</span><span class="p">,</span> <span class="n">e</span>
-</div>
-<div class="viewcode-block" id="Expr.as_coeff_Mul"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.as_coeff_Mul">[docs]</a>    <span class="k">def</span> <span class="nf">as_coeff_Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rational</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Efficiently extract the coefficient of a product. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="bp">self</span>
-</div>
-<div class="viewcode-block" id="Expr.as_coeff_Add"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.as_coeff_Add">[docs]</a>    <span class="k">def</span> <span class="nf">as_coeff_Add</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Efficiently extract the coefficient of a summation. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="bp">self</span>
-
-    <span class="c">###################################################################################</span>
-    <span class="c">##################### DERIVATIVE, INTEGRAL, FUNCTIONAL METHODS ####################</span>
-    <span class="c">###################################################################################</span>
-</div>
-    <span class="k">def</span> <span class="nf">diff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">):</span>
-        <span class="n">new_symbols</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="n">symbols</span><span class="p">))</span>  <span class="c"># e.g. x, 2, y, z</span>
-        <span class="n">assumptions</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="s">&quot;evaluate&quot;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">new_symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">)</span>
-
-    <span class="c">###########################################################################</span>
-    <span class="c">###################### EXPRESSION EXPANSION METHODS #######################</span>
-    <span class="c">###########################################################################</span>
-
-    <span class="c"># Relevant subclasses should override _eval_expand_hint() methods.  See</span>
-    <span class="c"># the docstring of expand() for more info.</span>
-
-    <span class="k">def</span> <span class="nf">_eval_expand_complex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="n">real</span><span class="p">,</span> <span class="n">imag</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">real</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">*</span><span class="n">imag</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_expand_hint</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">hint</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Helper for ``expand()``.  Recursively calls ``expr._eval_expand_hint()``.</span>
-
-<span class="sd">        Returns ``(expr, hit)``, where expr is the (possibly) expanded</span>
-<span class="sd">        ``expr`` and ``hit`` is ``True`` if ``expr`` was truly expanded and</span>
-<span class="sd">        ``False`` otherwise.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">hit</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="c"># XXX: Hack to support non-Basic args</span>
-        <span class="c">#              |</span>
-        <span class="c">#              V</span>
-        <span class="k">if</span> <span class="n">deep</span> <span class="ow">and</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&#39;args&#39;</span><span class="p">,</span> <span class="p">())</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-            <span class="n">sargs</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="n">arg</span><span class="p">,</span> <span class="n">arghit</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">_expand_hint</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">hint</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-                <span class="n">hit</span> <span class="o">|=</span> <span class="n">arghit</span>
-                <span class="n">sargs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">hit</span><span class="p">:</span>
-                <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">sargs</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&#39;_eval_expand_&#39;</span> <span class="o">+</span> <span class="n">hint</span><span class="p">):</span>
-            <span class="n">newexpr</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&#39;_eval_expand_&#39;</span> <span class="o">+</span> <span class="n">hint</span><span class="p">)(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">newexpr</span> <span class="o">!=</span> <span class="n">expr</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="n">newexpr</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">hit</span><span class="p">)</span>
-
-    <span class="nd">@cacheit</span>
-<div class="viewcode-block" id="Expr.expand"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.expand">[docs]</a>    <span class="k">def</span> <span class="nf">expand</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">modulus</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">power_base</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">power_exp</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span>
-            <span class="n">mul</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">log</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">multinomial</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">basic</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Expand an expression using hints.</span>
-
-<span class="sd">        See the docstring of the expand() function in sympy.core.function for</span>
-<span class="sd">        more information.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.simplify.simplify</span> <span class="kn">import</span> <span class="n">fraction</span>
-
-        <span class="n">hints</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">power_base</span><span class="o">=</span><span class="n">power_base</span><span class="p">,</span> <span class="n">power_exp</span><span class="o">=</span><span class="n">power_exp</span><span class="p">,</span> <span class="n">mul</span><span class="o">=</span><span class="n">mul</span><span class="p">,</span>
-           <span class="n">log</span><span class="o">=</span><span class="n">log</span><span class="p">,</span> <span class="n">multinomial</span><span class="o">=</span><span class="n">multinomial</span><span class="p">,</span> <span class="n">basic</span><span class="o">=</span><span class="n">basic</span><span class="p">)</span>
-
-        <span class="n">expr</span> <span class="o">=</span> <span class="bp">self</span>
-        <span class="k">if</span> <span class="n">hints</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;frac&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="o">=</span><span class="n">deep</span><span class="p">,</span> <span class="n">modulus</span><span class="o">=</span><span class="n">modulus</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-                    <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">fraction</span><span class="p">(</span><span class="bp">self</span><span class="p">)]</span>
-            <span class="k">return</span> <span class="n">n</span><span class="o">/</span><span class="n">d</span>
-        <span class="k">elif</span> <span class="n">hints</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;denom&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">fraction</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">n</span><span class="o">/</span><span class="n">d</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="o">=</span><span class="n">deep</span><span class="p">,</span> <span class="n">modulus</span><span class="o">=</span><span class="n">modulus</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">hints</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;numer&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">fraction</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">n</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="o">=</span><span class="n">deep</span><span class="p">,</span> <span class="n">modulus</span><span class="o">=</span><span class="n">modulus</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span><span class="o">/</span><span class="n">d</span>
-        <span class="c"># Although the hints are sorted here, an earlier hint may get applied</span>
-        <span class="c"># at a given node in the expression tree before another because of how</span>
-        <span class="c"># the hints are applied.  e.g. expand(log(x*(y + z))) -&gt; log(x*y +</span>
-        <span class="c"># x*z) because while applying log at the top level, log and mul are</span>
-        <span class="c"># applied at the deeper level in the tree so that when the log at the</span>
-        <span class="c"># upper level gets applied, the mul has already been applied at the</span>
-        <span class="c"># lower level.</span>
-
-        <span class="c"># Additionally, because hints are only applied once, the expression</span>
-        <span class="c"># may not be expanded all the way.   For example, if mul is applied</span>
-        <span class="c"># before multinomial, x*(x + 1)**2 won&#39;t be expanded all the way.  For</span>
-        <span class="c"># now, we just use a special case to make multinomial run before mul,</span>
-        <span class="c"># so that at least polynomials will be expanded all the way.  In the</span>
-        <span class="c"># future, smarter heuristics should be applied.</span>
-        <span class="c"># TODO: Smarter heuristics</span>
-
-        <span class="k">def</span> <span class="nf">_expand_hint_key</span><span class="p">(</span><span class="n">hint</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;Make multinomial come before mul&quot;&quot;&quot;</span>
-            <span class="k">if</span> <span class="n">hint</span> <span class="o">==</span> <span class="s">&#39;mul&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="s">&#39;mulz&#39;</span>
-            <span class="k">return</span> <span class="n">hint</span>
-
-        <span class="k">for</span> <span class="n">hint</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">hints</span><span class="o">.</span><span class="n">keys</span><span class="p">()),</span> <span class="n">key</span><span class="o">=</span><span class="n">_expand_hint_key</span><span class="p">):</span>
-            <span class="n">use_hint</span> <span class="o">=</span> <span class="n">hints</span><span class="p">[</span><span class="n">hint</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">use_hint</span><span class="p">:</span>
-                <span class="n">expr</span><span class="p">,</span> <span class="n">hit</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">_expand_hint</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">hint</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">modulus</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">modulus</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">modulus</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">modulus</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">or</span> <span class="n">modulus</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                    <span class="s">&quot;modulus must be a positive integer, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">modulus</span><span class="p">)</span>
-
-            <span class="n">terms</span> <span class="o">=</span> <span class="p">[]</span>
-
-            <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-                <span class="n">coeff</span><span class="p">,</span> <span class="n">tail</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">(</span><span class="n">rational</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-                <span class="n">coeff</span> <span class="o">%=</span> <span class="n">modulus</span>
-
-                <span class="k">if</span> <span class="n">coeff</span><span class="p">:</span>
-                    <span class="n">terms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">coeff</span><span class="o">*</span><span class="n">tail</span><span class="p">)</span>
-
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">terms</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="c">###########################################################################</span>
-    <span class="c">################### GLOBAL ACTION VERB WRAPPER METHODS ####################</span>
-    <span class="c">###########################################################################</span>
-</div>
-<div class="viewcode-block" id="Expr.integrate"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.integrate">[docs]</a>    <span class="k">def</span> <span class="nf">integrate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See the integrate function in sympy.integrals&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.integrals</span> <span class="kn">import</span> <span class="n">integrate</span>
-        <span class="k">return</span> <span class="n">integrate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.simplify"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.simplify">[docs]</a>    <span class="k">def</span> <span class="nf">simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ratio</span><span class="o">=</span><span class="mf">1.7</span><span class="p">,</span> <span class="n">measure</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See the simplify function in sympy.simplify&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">simplify</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">count_ops</span>
-        <span class="n">measure</span> <span class="o">=</span> <span class="n">measure</span> <span class="ow">or</span> <span class="n">count_ops</span>
-        <span class="k">return</span> <span class="n">simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ratio</span><span class="p">,</span> <span class="n">measure</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.nsimplify"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.nsimplify">[docs]</a>    <span class="k">def</span> <span class="nf">nsimplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">constants</span><span class="o">=</span><span class="p">[],</span> <span class="n">tolerance</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">full</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See the nsimplify function in sympy.simplify&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">nsimplify</span>
-        <span class="k">return</span> <span class="n">nsimplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">constants</span><span class="p">,</span> <span class="n">tolerance</span><span class="p">,</span> <span class="n">full</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.separate"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.separate">[docs]</a>    <span class="k">def</span> <span class="nf">separate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See the separate function in sympy.simplify&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">separate</span>
-        <span class="k">return</span> <span class="n">separate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.collect"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.collect">[docs]</a>    <span class="k">def</span> <span class="nf">collect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">syms</span><span class="p">,</span> <span class="n">func</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">distribute_order_term</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See the collect function in sympy.simplify&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">collect</span>
-        <span class="k">return</span> <span class="n">collect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">syms</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">evaluate</span><span class="p">,</span> <span class="n">exact</span><span class="p">,</span> <span class="n">distribute_order_term</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.together"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.together">[docs]</a>    <span class="k">def</span> <span class="nf">together</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See the together function in sympy.polys&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">together</span>
-        <span class="k">return</span> <span class="n">together</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.apart"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.apart">[docs]</a>    <span class="k">def</span> <span class="nf">apart</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See the apart function in sympy.polys&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">apart</span>
-        <span class="k">return</span> <span class="n">apart</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.ratsimp"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.ratsimp">[docs]</a>    <span class="k">def</span> <span class="nf">ratsimp</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See the ratsimp function in sympy.simplify&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">ratsimp</span>
-        <span class="k">return</span> <span class="n">ratsimp</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.trigsimp"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.trigsimp">[docs]</a>    <span class="k">def</span> <span class="nf">trigsimp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">recursive</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See the trigsimp function in sympy.simplify&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">trigsimp</span>
-        <span class="k">return</span> <span class="n">trigsimp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="p">,</span> <span class="n">recursive</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.radsimp"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.radsimp">[docs]</a>    <span class="k">def</span> <span class="nf">radsimp</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See the radsimp function in sympy.simplify&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">radsimp</span>
-        <span class="k">return</span> <span class="n">radsimp</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.powsimp"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.powsimp">[docs]</a>    <span class="k">def</span> <span class="nf">powsimp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;all&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See the powsimp function in sympy.simplify&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">powsimp</span>
-        <span class="k">return</span> <span class="n">powsimp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="p">,</span> <span class="n">combine</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.combsimp"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.combsimp">[docs]</a>    <span class="k">def</span> <span class="nf">combsimp</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See the combsimp function in sympy.simplify&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">combsimp</span>
-        <span class="k">return</span> <span class="n">combsimp</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.factor"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.factor">[docs]</a>    <span class="k">def</span> <span class="nf">factor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See the factor() function in sympy.polys.polytools&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">factor</span>
-        <span class="k">return</span> <span class="n">factor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.refine"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.refine">[docs]</a>    <span class="k">def</span> <span class="nf">refine</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">assumption</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See the refine function in sympy.assumptions&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.assumptions</span> <span class="kn">import</span> <span class="n">refine</span>
-        <span class="k">return</span> <span class="n">refine</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">assumption</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.cancel"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.cancel">[docs]</a>    <span class="k">def</span> <span class="nf">cancel</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See the cancel function in sympy.polys&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">cancel</span>
-        <span class="k">return</span> <span class="n">cancel</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.invert"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.invert">[docs]</a>    <span class="k">def</span> <span class="nf">invert</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See the invert function in sympy.polys&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">invert</span>
-        <span class="k">return</span> <span class="n">invert</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Expr.round"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.Expr.round">[docs]</a>    <span class="k">def</span> <span class="nf">round</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return x rounded to the given decimal place.</span>
-
-<span class="sd">        If a complex number would results, apply round to the real</span>
-<span class="sd">        and imaginary components of the number.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import pi, E, I, S, Add, Mul, Number</span>
-<span class="sd">        &gt;&gt;&gt; S(10.5).round()</span>
-<span class="sd">        11.</span>
-<span class="sd">        &gt;&gt;&gt; pi.round()</span>
-<span class="sd">        3.</span>
-<span class="sd">        &gt;&gt;&gt; pi.round(2)</span>
-<span class="sd">        3.14</span>
-<span class="sd">        &gt;&gt;&gt; (2*pi + E*I).round()              #doctest: +SKIP</span>
-<span class="sd">        6. + 3.*I</span>
-
-<span class="sd">        The round method has a chopping effect:</span>
-
-<span class="sd">        &gt;&gt;&gt; (2*pi + I/10).round()</span>
-<span class="sd">        6.</span>
-<span class="sd">        &gt;&gt;&gt; (pi/10 + 2*I).round()             #doctest: +SKIP</span>
-<span class="sd">        2.*I</span>
-<span class="sd">        &gt;&gt;&gt; (pi/10 + E*I).round(2)</span>
-<span class="sd">        0.31 + 2.72*I</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        Do not confuse the Python builtin function, round, with the</span>
-<span class="sd">        SymPy method of the same name. The former always returns a float</span>
-<span class="sd">        (or raises an error if applied to a complex value) while the</span>
-<span class="sd">        latter returns either a Number or a complex number:</span>
-
-<span class="sd">        &gt;&gt;&gt; isinstance(round(S(123), -2), Number)</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; isinstance(S(123).round(-2), Number)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; isinstance((3*I).round(), Mul)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; isinstance((1 + 3*I).round(), Add)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.functions.elementary.exponential</span> <span class="kn">import</span> <span class="n">log</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">x</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> is not a number&#39;</span> <span class="o">%</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">x</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="n">i</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-            <span class="k">return</span> <span class="n">i</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">*</span><span class="n">r</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">x</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">x</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-
-        <span class="n">precs</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="o">.</span><span class="n">_prec</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">x</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Float</span><span class="p">)]</span>
-        <span class="n">dps</span> <span class="o">=</span> <span class="n">prec_to_dps</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="n">precs</span><span class="p">))</span> <span class="k">if</span> <span class="n">precs</span> <span class="k">else</span> <span class="bp">None</span>
-
-        <span class="n">mag_first_dig</span> <span class="o">=</span> <span class="n">_mag</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="n">allow</span> <span class="o">=</span> <span class="n">digits_needed</span> <span class="o">=</span> <span class="n">mag_first_dig</span> <span class="o">+</span> <span class="n">p</span>
-        <span class="k">if</span> <span class="n">dps</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">allow</span> <span class="o">&gt;</span> <span class="n">dps</span><span class="p">:</span>
-            <span class="n">allow</span> <span class="o">=</span> <span class="n">dps</span>
-        <span class="n">mag</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>  <span class="c"># magnitude needed to bring digit p to units place</span>
-        <span class="n">x</span> <span class="o">+=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">mag</span><span class="p">)</span>  <span class="c"># add the half for rounding</span>
-        <span class="n">i10</span> <span class="o">=</span> <span class="mi">10</span><span class="o">*</span><span class="n">mag</span><span class="o">*</span><span class="n">x</span><span class="o">.</span><span class="n">n</span><span class="p">((</span><span class="n">dps</span> <span class="k">if</span> <span class="n">dps</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="k">else</span> <span class="n">digits_needed</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">Integer</span><span class="p">(</span><span class="n">i10</span><span class="p">)</span><span class="o">//</span><span class="mi">10</span>
-        <span class="n">q</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">p</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">q</span> <span class="o">=</span> <span class="n">mag</span>
-        <span class="k">elif</span> <span class="n">p</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">rv</span> <span class="o">/=</span> <span class="n">mag</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="n">rv</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">rv</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="c"># use str or else it won&#39;t be a float</span>
-            <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Float</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">rv</span><span class="p">),</span> <span class="n">digits_needed</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Float</span><span class="p">(</span><span class="n">rv</span><span class="p">,</span> <span class="n">allow</span><span class="p">)</span>
-
-</div></div>
-<div class="viewcode-block" id="AtomicExpr"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.expr.AtomicExpr">[docs]</a><span class="k">class</span> <span class="nc">AtomicExpr</span><span class="p">(</span><span class="n">Atom</span><span class="p">,</span> <span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A parent class for object which are both atoms and Exprs.</span>
-
-<span class="sd">    For example: Symbol, Number, Rational, Integer, ...</span>
-<span class="sd">    But not: Add, Mul, Pow, ...</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_Atom</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span> <span class="o">==</span> <span class="n">s</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_polynomial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">syms</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_rational_function</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">syms</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_eval_nseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_mag</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return integer ``i`` such that .1 &lt;= x/10**i &lt; 1</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.core.expr import _mag</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Float</span>
-<span class="sd">    &gt;&gt;&gt; _mag(Float(.1))</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; _mag(Float(.01))</span>
-<span class="sd">    -1</span>
-<span class="sd">    &gt;&gt;&gt; _mag(Float(1234))</span>
-<span class="sd">    4</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">log10</span><span class="p">,</span> <span class="n">ceil</span><span class="p">,</span> <span class="n">log</span>
-    <span class="n">xpos</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">n</span><span class="p">())</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">xpos</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">mag_first_dig</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">ceil</span><span class="p">(</span><span class="n">log10</span><span class="p">(</span><span class="n">xpos</span><span class="p">)))</span>
-    <span class="k">except</span> <span class="p">(</span><span class="ne">ValueError</span><span class="p">,</span> <span class="ne">OverflowError</span><span class="p">):</span>
-        <span class="n">mag_first_dig</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">ceil</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Float</span><span class="p">(</span><span class="n">mpf_log</span><span class="p">(</span><span class="n">xpos</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="mi">53</span><span class="p">))</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="mi">10</span><span class="p">)))</span>
-    <span class="c"># check that we aren&#39;t off by 1</span>
-    <span class="k">if</span> <span class="p">(</span><span class="n">xpos</span><span class="o">/</span><span class="mi">10</span><span class="o">**</span><span class="n">mag_first_dig</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">assert</span> <span class="mi">1</span> <span class="o">&lt;=</span> <span class="p">(</span><span class="n">xpos</span><span class="o">/</span><span class="mi">10</span><span class="o">**</span><span class="n">mag_first_dig</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">10</span>
-        <span class="n">mag_first_dig</span> <span class="o">+=</span> <span class="mi">1</span>
-    <span class="k">return</span> <span class="n">mag_first_dig</span>
-
-<span class="kn">from</span> <span class="nn">.mul</span> <span class="kn">import</span> <span class="n">Mul</span>
-<span class="kn">from</span> <span class="nn">.add</span> <span class="kn">import</span> <span class="n">Add</span>
-<span class="kn">from</span> <span class="nn">.power</span> <span class="kn">import</span> <span class="n">Pow</span>
-<span class="kn">from</span> <span class="nn">.function</span> <span class="kn">import</span> <span class="n">Derivative</span><span class="p">,</span> <span class="n">expand_mul</span>
-<span class="kn">from</span> <span class="nn">.mod</span> <span class="kn">import</span> <span class="n">Mod</span>
-<span class="kn">from</span> <span class="nn">.exprtools</span> <span class="kn">import</span> <span class="n">factor_terms</span>
-<span class="kn">from</span> <span class="nn">.numbers</span> <span class="kn">import</span> <span class="n">Integer</span><span class="p">,</span> <span class="n">Rational</span>
-</pre></div>
-
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-  <h1>Source code for sympy.core.exprtools</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Tools for manipulating of large commutative expressions. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.add</span> <span class="kn">import</span> <span class="n">Add</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">iterable</span><span class="p">,</span> <span class="n">is_sequence</span>
-<span class="kn">from</span> <span class="nn">sympy.core.mul</span> <span class="kn">import</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">_keep_coeff</span>
-<span class="kn">from</span> <span class="nn">sympy.core.power</span> <span class="kn">import</span> <span class="n">Pow</span>
-<span class="kn">from</span> <span class="nn">sympy.core.basic</span> <span class="kn">import</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">preorder_traversal</span>
-<span class="kn">from</span> <span class="nn">sympy.core.expr</span> <span class="kn">import</span> <span class="n">Expr</span>
-<span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">Rational</span><span class="p">,</span> <span class="n">Integer</span>
-<span class="kn">from</span> <span class="nn">sympy.core.singleton</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="kn">from</span> <span class="nn">sympy.core.symbol</span> <span class="kn">import</span> <span class="n">Dummy</span>
-<span class="kn">from</span> <span class="nn">sympy.core.coreerrors</span> <span class="kn">import</span> <span class="n">NonCommutativeExpression</span>
-<span class="kn">from</span> <span class="nn">sympy.core.containers</span> <span class="kn">import</span> <span class="n">Tuple</span><span class="p">,</span> <span class="n">Dict</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">default_sort_key</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="p">(</span><span class="n">common_prefix</span><span class="p">,</span> <span class="n">common_suffix</span><span class="p">,</span>
-        <span class="n">variations</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</span>
-
-
-<span class="k">def</span> <span class="nf">decompose_power</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Decompose power into symbolic base and integer exponent.</span>
-
-<span class="sd">    This is strictly only valid if the exponent from which</span>
-<span class="sd">    the integer is extracted is itself an integer or the</span>
-<span class="sd">    base is positive. These conditions are assumed and not</span>
-<span class="sd">    checked here.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.core.exprtools import decompose_power</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">    &gt;&gt;&gt; decompose_power(x)</span>
-<span class="sd">    (x, 1)</span>
-<span class="sd">    &gt;&gt;&gt; decompose_power(x**2)</span>
-<span class="sd">    (x, 2)</span>
-<span class="sd">    &gt;&gt;&gt; decompose_power(x**(2*y))</span>
-<span class="sd">    (x**y, 2)</span>
-<span class="sd">    &gt;&gt;&gt; decompose_power(x**(2*y/3))</span>
-<span class="sd">    (x**(y/3), 2)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="n">exp</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="n">base</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">exp</span><span class="o">.</span><span class="n">q</span><span class="p">))</span>
-
-            <span class="n">exp</span> <span class="o">=</span> <span class="n">exp</span><span class="o">.</span><span class="n">p</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="n">expr</span><span class="p">,</span> <span class="mi">1</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">exp</span><span class="p">,</span> <span class="n">tail</span> <span class="o">=</span> <span class="n">exp</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">(</span><span class="n">rational</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-            <span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">tail</span><span class="p">),</span> <span class="o">-</span><span class="mi">1</span>
-        <span class="k">elif</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-            <span class="n">tail</span> <span class="o">=</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">exp</span><span class="o">.</span><span class="n">q</span><span class="p">),</span> <span class="n">tail</span><span class="p">)</span>
-            <span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">tail</span><span class="p">),</span> <span class="n">exp</span><span class="o">.</span><span class="n">p</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="n">expr</span><span class="p">,</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="n">base</span><span class="p">,</span> <span class="n">exp</span>
-
-
-<span class="k">class</span> <span class="nc">Factors</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Efficient representation of ``f_1*f_2*...*f_n``. &quot;&quot;&quot;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;factors&#39;</span><span class="p">,</span> <span class="s">&#39;gens&#39;</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">factors</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>  <span class="c"># Factors</span>
-        <span class="k">if</span> <span class="n">factors</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">factors</span> <span class="o">=</span> <span class="p">{}</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">factors</span> <span class="o">=</span> <span class="n">factors</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">gens</span> <span class="o">=</span> <span class="nb">frozenset</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">factors</span><span class="o">.</span><span class="n">keys</span><span class="p">()))</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>  <span class="c"># Factors</span>
-        <span class="k">return</span> <span class="nb">hash</span><span class="p">((</span><span class="nb">tuple</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">factors</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>  <span class="c"># Factors</span>
-        <span class="k">return</span> <span class="s">&quot;Factors(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">factors</span>
-
-    <span class="k">def</span> <span class="nf">as_expr</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>  <span class="c"># Factors</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">factor</span><span class="p">,</span> <span class="n">exp</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">factors</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="k">if</span> <span class="n">exp</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">factor</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                <span class="n">e</span> <span class="o">=</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">Integer</span><span class="p">(</span><span class="n">exp</span><span class="p">),</span> <span class="n">e</span><span class="p">)</span>
-                <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">b</span><span class="o">**</span><span class="n">e</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">factor</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">normal</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Factors</span>
-        <span class="n">self_factors</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">factors</span><span class="p">)</span>
-        <span class="n">other_factors</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">factors</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">factor</span><span class="p">,</span> <span class="n">self_exp</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">factors</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">other_exp</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">factors</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span>
-            <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-                <span class="k">continue</span>
-
-            <span class="n">exp</span> <span class="o">=</span> <span class="n">self_exp</span> <span class="o">-</span> <span class="n">other_exp</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">exp</span><span class="p">:</span>
-                <span class="k">del</span> <span class="n">self_factors</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span>
-                <span class="k">del</span> <span class="n">other_factors</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">exp</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">self_factors</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span> <span class="o">=</span> <span class="n">exp</span>
-                    <span class="k">del</span> <span class="n">other_factors</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">del</span> <span class="n">self_factors</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span>
-                    <span class="n">other_factors</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">exp</span>
-
-        <span class="k">return</span> <span class="n">Factors</span><span class="p">(</span><span class="n">self_factors</span><span class="p">),</span> <span class="n">Factors</span><span class="p">(</span><span class="n">other_factors</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Factors</span>
-        <span class="n">factors</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">factors</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">factor</span><span class="p">,</span> <span class="n">exp</span> <span class="ow">in</span> <span class="n">other</span><span class="o">.</span><span class="n">factors</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="k">if</span> <span class="n">factor</span> <span class="ow">in</span> <span class="n">factors</span><span class="p">:</span>
-                <span class="n">exp</span> <span class="o">=</span> <span class="n">factors</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span> <span class="o">+</span> <span class="n">exp</span>
-
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">exp</span><span class="p">:</span>
-                    <span class="k">del</span> <span class="n">factors</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span>
-                    <span class="k">continue</span>
-
-            <span class="n">factors</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span> <span class="o">=</span> <span class="n">exp</span>
-
-        <span class="k">return</span> <span class="n">Factors</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">div</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Factors</span>
-        <span class="n">quo</span><span class="p">,</span> <span class="n">rem</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">factors</span><span class="p">),</span> <span class="p">{}</span>
-
-        <span class="k">for</span> <span class="n">factor</span><span class="p">,</span> <span class="n">exp</span> <span class="ow">in</span> <span class="n">other</span><span class="o">.</span><span class="n">factors</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="k">if</span> <span class="n">factor</span> <span class="ow">in</span> <span class="n">quo</span><span class="p">:</span>
-                <span class="n">exp</span> <span class="o">=</span> <span class="n">quo</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span> <span class="o">-</span> <span class="n">exp</span>
-
-                <span class="k">if</span> <span class="n">exp</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">del</span> <span class="n">quo</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span>
-
-                <span class="k">if</span> <span class="n">exp</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">exp</span><span class="p">:</span>
-                        <span class="n">quo</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span> <span class="o">=</span> <span class="n">exp</span>
-
-                    <span class="k">continue</span>
-
-                <span class="n">exp</span> <span class="o">=</span> <span class="o">-</span><span class="n">exp</span>
-
-            <span class="n">rem</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span> <span class="o">=</span> <span class="n">exp</span>
-
-        <span class="k">return</span> <span class="n">Factors</span><span class="p">(</span><span class="n">quo</span><span class="p">),</span> <span class="n">Factors</span><span class="p">(</span><span class="n">rem</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">quo</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Factors</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">div</span><span class="p">(</span><span class="n">other</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">rem</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Factors</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">div</span><span class="p">(</span><span class="n">other</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">pow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Factors</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">other</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">int</span> <span class="ow">and</span> <span class="n">other</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">factors</span> <span class="o">=</span> <span class="p">{}</span>
-
-            <span class="k">if</span> <span class="n">other</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">factor</span><span class="p">,</span> <span class="n">exp</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">factors</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                    <span class="n">factors</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span> <span class="o">=</span> <span class="n">exp</span><span class="o">*</span><span class="n">other</span>
-
-            <span class="k">return</span> <span class="n">Factors</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;expected non-negative integer, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">gcd</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Factors</span>
-        <span class="n">factors</span> <span class="o">=</span> <span class="p">{}</span>
-
-        <span class="k">for</span> <span class="n">factor</span><span class="p">,</span> <span class="n">exp</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">factors</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="k">if</span> <span class="n">factor</span> <span class="ow">in</span> <span class="n">other</span><span class="o">.</span><span class="n">factors</span><span class="p">:</span>
-                <span class="n">exp</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">exp</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">factors</span><span class="p">[</span><span class="n">factor</span><span class="p">])</span>
-                <span class="n">factors</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span> <span class="o">=</span> <span class="n">exp</span>
-
-        <span class="k">return</span> <span class="n">Factors</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">lcm</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Factors</span>
-        <span class="n">factors</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">factors</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">factor</span><span class="p">,</span> <span class="n">exp</span> <span class="ow">in</span> <span class="n">other</span><span class="o">.</span><span class="n">factors</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="k">if</span> <span class="n">factor</span> <span class="ow">in</span> <span class="n">factors</span><span class="p">:</span>
-                <span class="n">exp</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">exp</span><span class="p">,</span> <span class="n">factors</span><span class="p">[</span><span class="n">factor</span><span class="p">])</span>
-
-            <span class="n">factors</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span> <span class="o">=</span> <span class="n">exp</span>
-
-        <span class="k">return</span> <span class="n">Factors</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Factors</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Factors</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="nf">__divmod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Factors</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Factors</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">div</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Factors</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Factors</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="n">__truediv__</span> <span class="o">=</span> <span class="n">__div__</span>
-
-    <span class="k">def</span> <span class="nf">__mod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Factors</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Factors</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rem</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="nf">__pow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Factors</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">other</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">int</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Factors</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">factors</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">factors</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Factors</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">Term</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Efficient representation of ``coeff*(numer/denom)``. &quot;&quot;&quot;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;coeff&#39;</span><span class="p">,</span> <span class="s">&#39;numer&#39;</span><span class="p">,</span> <span class="s">&#39;denom&#39;</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">term</span><span class="p">,</span> <span class="n">numer</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">denom</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>  <span class="c"># Term</span>
-        <span class="k">if</span> <span class="n">numer</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">denom</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">term</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">NonCommutativeExpression</span><span class="p">(</span>
-                    <span class="s">&#39;commutative expression expected&#39;</span><span class="p">)</span>
-
-            <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">()</span>
-            <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">int</span><span class="p">),</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">factor</span> <span class="ow">in</span> <span class="n">factors</span><span class="p">:</span>
-                <span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="n">decompose_power</span><span class="p">(</span><span class="n">factor</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">base</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                    <span class="n">cont</span><span class="p">,</span> <span class="n">base</span> <span class="o">=</span> <span class="n">base</span><span class="o">.</span><span class="n">primitive</span><span class="p">()</span>
-                    <span class="n">coeff</span> <span class="o">*=</span> <span class="n">cont</span><span class="o">**</span><span class="n">exp</span>
-
-                <span class="k">if</span> <span class="n">exp</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">numer</span><span class="p">[</span><span class="n">base</span><span class="p">]</span> <span class="o">+=</span> <span class="n">exp</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">denom</span><span class="p">[</span><span class="n">base</span><span class="p">]</span> <span class="o">+=</span> <span class="o">-</span><span class="n">exp</span>
-
-            <span class="n">numer</span> <span class="o">=</span> <span class="n">Factors</span><span class="p">(</span><span class="n">numer</span><span class="p">)</span>
-            <span class="n">denom</span> <span class="o">=</span> <span class="n">Factors</span><span class="p">(</span><span class="n">denom</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">term</span>
-
-            <span class="k">if</span> <span class="n">numer</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">numer</span> <span class="o">=</span> <span class="n">Factors</span><span class="p">()</span>
-
-            <span class="k">if</span> <span class="n">denom</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">denom</span> <span class="o">=</span> <span class="n">Factors</span><span class="p">()</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">coeff</span> <span class="o">=</span> <span class="n">coeff</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">numer</span> <span class="o">=</span> <span class="n">numer</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">denom</span> <span class="o">=</span> <span class="n">denom</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>  <span class="c"># Term</span>
-        <span class="k">return</span> <span class="nb">hash</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">coeff</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">numer</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">denom</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>  <span class="c"># Term</span>
-        <span class="k">return</span> <span class="s">&quot;Term(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">coeff</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">numer</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">denom</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">as_expr</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>  <span class="c"># Term</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">coeff</span><span class="o">*</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">numer</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">denom</span><span class="o">.</span><span class="n">as_expr</span><span class="p">())</span>
-
-    <span class="k">def</span> <span class="nf">mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Term</span>
-        <span class="n">coeff</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">coeff</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">coeff</span>
-        <span class="n">numer</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">numer</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">numer</span><span class="p">)</span>
-        <span class="n">denom</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">denom</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">denom</span><span class="p">)</span>
-
-        <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="o">=</span> <span class="n">numer</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">denom</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">Term</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">inv</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>  <span class="c"># Term</span>
-        <span class="k">return</span> <span class="n">Term</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">coeff</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">denom</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">numer</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">quo</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Term</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">inv</span><span class="p">())</span>
-
-    <span class="k">def</span> <span class="nf">pow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Term</span>
-        <span class="k">if</span> <span class="n">other</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">inv</span><span class="p">()</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="o">-</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Term</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">coeff</span> <span class="o">**</span> <span class="n">other</span><span class="p">,</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">numer</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="n">other</span><span class="p">),</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">denom</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="n">other</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">gcd</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Term</span>
-        <span class="k">return</span> <span class="n">Term</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">coeff</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">coeff</span><span class="p">),</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">numer</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">numer</span><span class="p">),</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">denom</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">denom</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">lcm</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Term</span>
-        <span class="k">return</span> <span class="n">Term</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">coeff</span><span class="o">.</span><span class="n">lcm</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">coeff</span><span class="p">),</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">numer</span><span class="o">.</span><span class="n">lcm</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">numer</span><span class="p">),</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">denom</span><span class="o">.</span><span class="n">lcm</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">denom</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Term</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Term</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Term</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Term</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="n">__truediv__</span> <span class="o">=</span> <span class="n">__div__</span>
-
-    <span class="k">def</span> <span class="nf">__pow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Term</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">other</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">int</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Term</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">coeff</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">coeff</span> <span class="ow">and</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">numer</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">numer</span> <span class="ow">and</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">denom</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">denom</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>  <span class="c"># Term</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">_gcd_terms</span><span class="p">(</span><span class="n">terms</span><span class="p">,</span> <span class="n">isprimitive</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">fraction</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Helper function for :func:`gcd_terms`.</span>
-
-<span class="sd">    If ``isprimitive`` is True then the call to primitive</span>
-<span class="sd">    for an Add will be skipped. This is useful when the</span>
-<span class="sd">    content has already been extrated.</span>
-
-<span class="sd">    If ``fraction`` is True then the expression will appear over a common</span>
-<span class="sd">    denominator, the lcm of all term denominators.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">terms</span><span class="p">,</span> <span class="n">Basic</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">terms</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">):</span>
-        <span class="n">terms</span> <span class="o">=</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">terms</span><span class="p">)</span>
-
-    <span class="n">terms</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Term</span><span class="p">,</span> <span class="p">[</span><span class="n">t</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">terms</span> <span class="k">if</span> <span class="n">t</span><span class="p">]))</span>
-
-    <span class="c"># there is some simplification that may happen if we leave this</span>
-    <span class="c"># here rather than duplicate it before the mapping of Term onto</span>
-    <span class="c"># the terms</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">terms</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">terms</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">cont</span> <span class="o">=</span> <span class="n">terms</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">coeff</span>
-        <span class="n">numer</span> <span class="o">=</span> <span class="n">terms</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">numer</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-        <span class="n">denom</span> <span class="o">=</span> <span class="n">terms</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">denom</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">cont</span> <span class="o">=</span> <span class="n">terms</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="n">cont</span> <span class="o">=</span> <span class="n">cont</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">term</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">terms</span><span class="p">):</span>
-            <span class="n">terms</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">cont</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">fraction</span><span class="p">:</span>
-            <span class="n">denom</span> <span class="o">=</span> <span class="n">terms</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">denom</span>
-
-            <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-                <span class="n">denom</span> <span class="o">=</span> <span class="n">denom</span><span class="o">.</span><span class="n">lcm</span><span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">denom</span><span class="p">)</span>
-
-            <span class="n">numers</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">:</span>
-                <span class="n">numer</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">numer</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">denom</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">denom</span><span class="p">))</span>
-                <span class="n">numers</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">coeff</span><span class="o">*</span><span class="n">numer</span><span class="o">.</span><span class="n">as_expr</span><span class="p">())</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">numers</span> <span class="o">=</span> <span class="p">[</span><span class="n">t</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">]</span>
-            <span class="n">denom</span> <span class="o">=</span> <span class="n">Term</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">numer</span>
-
-        <span class="n">cont</span> <span class="o">=</span> <span class="n">cont</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-        <span class="n">numer</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">numers</span><span class="p">)</span>
-        <span class="n">denom</span> <span class="o">=</span> <span class="n">denom</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">isprimitive</span> <span class="ow">and</span> <span class="n">numer</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-        <span class="n">_cont</span><span class="p">,</span> <span class="n">numer</span> <span class="o">=</span> <span class="n">numer</span><span class="o">.</span><span class="n">primitive</span><span class="p">()</span>
-        <span class="n">cont</span> <span class="o">*=</span> <span class="n">_cont</span>
-
-    <span class="k">return</span> <span class="n">cont</span><span class="p">,</span> <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span>
-
-
-<div class="viewcode-block" id="gcd_terms"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.exprtools.gcd_terms">[docs]</a><span class="k">def</span> <span class="nf">gcd_terms</span><span class="p">(</span><span class="n">terms</span><span class="p">,</span> <span class="n">isprimitive</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">clear</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">fraction</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Compute the GCD of ``terms`` and put them together.</span>
-
-<span class="sd">    ``terms`` can be an expression or a non-Basic sequence of expressions</span>
-<span class="sd">    which will be handled as though they are terms from a sum.</span>
-
-<span class="sd">    If ``isprimitive`` is True the _gcd_terms will not run the primitive</span>
-<span class="sd">    method on the terms.</span>
-
-<span class="sd">    ``clear`` controls the removal of integers from the denominator of an Add</span>
-<span class="sd">    expression. When True (default), all numerical denominator will be cleared;</span>
-<span class="sd">    when False the denominators will be cleared only if all terms had numerical</span>
-<span class="sd">    denominators other than 1.</span>
-
-<span class="sd">    ``fraction``, when True (default), will put the expression over a common</span>
-<span class="sd">    denominator.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.core import gcd_terms</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">    &gt;&gt;&gt; gcd_terms((x + 1)**2*y + (x + 1)*y**2)</span>
-<span class="sd">    y*(x + 1)*(x + y + 1)</span>
-<span class="sd">    &gt;&gt;&gt; gcd_terms(x/2 + 1)</span>
-<span class="sd">    (x + 2)/2</span>
-<span class="sd">    &gt;&gt;&gt; gcd_terms(x/2 + 1, clear=False)</span>
-<span class="sd">    x/2 + 1</span>
-<span class="sd">    &gt;&gt;&gt; gcd_terms(x/2 + y/2, clear=False)</span>
-<span class="sd">    (x + y)/2</span>
-<span class="sd">    &gt;&gt;&gt; gcd_terms(x/2 + 1/x)</span>
-<span class="sd">    (x**2 + 2)/(2*x)</span>
-<span class="sd">    &gt;&gt;&gt; gcd_terms(x/2 + 1/x, fraction=False)</span>
-<span class="sd">    (x + 2/x)/2</span>
-<span class="sd">    &gt;&gt;&gt; gcd_terms(x/2 + 1/x, fraction=False, clear=False)</span>
-<span class="sd">    x/2 + 1/x</span>
-
-<span class="sd">    &gt;&gt;&gt; gcd_terms(x/2/y + 1/x/y)</span>
-<span class="sd">    (x**2 + 2)/(2*x*y)</span>
-<span class="sd">    &gt;&gt;&gt; gcd_terms(x/2/y + 1/x/y, fraction=False, clear=False)</span>
-<span class="sd">    (x + 2/x)/(2*y)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    factor_terms, sympy.polys.polytools.terms_gcd</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">mask</span><span class="p">(</span><span class="n">terms</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;replace nc portions of each term with a unique Dummy symbols</span>
-<span class="sd">        and return the replacements to restore them&quot;&quot;&quot;</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[(</span><span class="n">a</span><span class="p">,</span> <span class="p">[])</span> <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_commutative</span> <span class="k">else</span> <span class="n">a</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">]</span>
-        <span class="n">reps</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">nc</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">nc</span><span class="p">:</span>
-                <span class="n">nc</span> <span class="o">=</span> <span class="n">Mul</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">nc</span><span class="p">)</span>
-                <span class="n">d</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">()</span>
-                <span class="n">reps</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">d</span><span class="p">,</span> <span class="n">nc</span><span class="p">))</span>
-                <span class="n">c</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-                <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">Mul</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">c</span>
-        <span class="k">return</span> <span class="n">args</span><span class="p">,</span> <span class="nb">dict</span><span class="p">(</span><span class="n">reps</span><span class="p">)</span>
-
-    <span class="n">isadd</span> <span class="o">=</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">terms</span><span class="p">,</span> <span class="n">Add</span><span class="p">)</span>
-    <span class="n">addlike</span> <span class="o">=</span> <span class="n">isadd</span> <span class="ow">or</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">terms</span><span class="p">,</span> <span class="n">Basic</span><span class="p">)</span> <span class="ow">and</span> \
-        <span class="n">is_sequence</span><span class="p">(</span><span class="n">terms</span><span class="p">,</span> <span class="n">include</span><span class="o">=</span><span class="nb">set</span><span class="p">)</span> <span class="ow">and</span> \
-        <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">terms</span><span class="p">,</span> <span class="n">Dict</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">addlike</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">isadd</span><span class="p">:</span>  <span class="c"># i.e. an Add</span>
-            <span class="n">terms</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">terms</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">terms</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">terms</span><span class="p">)</span>
-        <span class="n">terms</span><span class="p">,</span> <span class="n">reps</span> <span class="o">=</span> <span class="n">mask</span><span class="p">(</span><span class="n">terms</span><span class="p">)</span>
-        <span class="n">cont</span><span class="p">,</span> <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="o">=</span> <span class="n">_gcd_terms</span><span class="p">(</span><span class="n">terms</span><span class="p">,</span> <span class="n">isprimitive</span><span class="p">,</span> <span class="n">fraction</span><span class="p">)</span>
-        <span class="n">numer</span> <span class="o">=</span> <span class="n">numer</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="n">reps</span><span class="p">)</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">cont</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span><span class="o">*</span><span class="n">numer</span><span class="o">/</span><span class="n">denom</span><span class="p">,</span> <span class="n">clear</span><span class="o">=</span><span class="n">clear</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">terms</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">terms</span>
-
-    <span class="k">if</span> <span class="n">terms</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">terms</span>
-
-    <span class="k">if</span> <span class="n">terms</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-        <span class="n">c</span><span class="p">,</span> <span class="n">args</span> <span class="o">=</span> <span class="n">terms</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">gcd_terms</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">isprimitive</span><span class="p">,</span> <span class="n">clear</span><span class="p">,</span> <span class="n">fraction</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]),</span> <span class="n">clear</span><span class="o">=</span><span class="n">clear</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">handle</span><span class="p">(</span><span class="n">a</span><span class="p">):</span>
-        <span class="c"># don&#39;t treat internal args like terms of an Add</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Expr</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">a</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">handle</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">a</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-            <span class="k">return</span> <span class="nb">type</span><span class="p">(</span><span class="n">a</span><span class="p">)([</span><span class="n">handle</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">a</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">gcd_terms</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">isprimitive</span><span class="p">,</span> <span class="n">clear</span><span class="p">,</span> <span class="n">fraction</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">terms</span><span class="p">,</span> <span class="n">Dict</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Dict</span><span class="p">(</span><span class="o">*</span><span class="p">[(</span><span class="n">k</span><span class="p">,</span> <span class="n">handle</span><span class="p">(</span><span class="n">v</span><span class="p">))</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">terms</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-    <span class="k">return</span> <span class="n">terms</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">handle</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">terms</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-</div>
-<div class="viewcode-block" id="factor_terms"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.exprtools.factor_terms">[docs]</a><span class="k">def</span> <span class="nf">factor_terms</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">radical</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">clear</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">fraction</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Remove common factors from terms in all arguments without</span>
-<span class="sd">    changing the underlying structure of the expr. No expansion or</span>
-<span class="sd">    simplification (and no processing of non-commutatives) is performed.</span>
-
-<span class="sd">    If radical=True then a radical common to all terms will be factored</span>
-<span class="sd">    out of any Add sub-expressions of the expr.</span>
-
-<span class="sd">    If clear=False (default) then coefficients will not be separated</span>
-<span class="sd">    from a single Add if they can be distributed to leave one or more</span>
-<span class="sd">    terms with integer coefficients.</span>
-
-<span class="sd">    If fraction=True (default is False) then a common denominator will be</span>
-<span class="sd">    constructed for the expression.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import factor_terms, Symbol, Mul, primitive</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; factor_terms(x + x*(2 + 4*y)**3)</span>
-<span class="sd">    x*(8*(2*y + 1)**3 + 1)</span>
-<span class="sd">    &gt;&gt;&gt; A = Symbol(&#39;A&#39;, commutative=False)</span>
-<span class="sd">    &gt;&gt;&gt; factor_terms(x*A + x*A + x*y*A)</span>
-<span class="sd">    x*(y*A + 2*A)</span>
-
-<span class="sd">    When ``clear`` is False, a rational will only be factored out of an</span>
-<span class="sd">    Add expression if all terms of the Add have coefficients that are</span>
-<span class="sd">    fractions:</span>
-
-<span class="sd">    &gt;&gt;&gt; factor_terms(x/2 + 1, clear=False)</span>
-<span class="sd">    x/2 + 1</span>
-<span class="sd">    &gt;&gt;&gt; factor_terms(x/2 + 1, clear=True)</span>
-<span class="sd">    (x + 2)/2</span>
-
-<span class="sd">    This only applies when there is a single Add that the coefficient</span>
-<span class="sd">    multiplies:</span>
-
-<span class="sd">    &gt;&gt;&gt; factor_terms(x*y/2 + y, clear=True)</span>
-<span class="sd">    y*(x + 2)/2</span>
-<span class="sd">    &gt;&gt;&gt; factor_terms(x*y/2 + y, clear=False) == _</span>
-<span class="sd">    True</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    gcd_terms, sympy.polys.polytools.terms_gcd</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="n">is_iterable</span> <span class="o">=</span> <span class="n">iterable</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Basic</span><span class="p">)</span> <span class="ow">or</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">is_iterable</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">type</span><span class="p">(</span><span class="n">expr</span><span class="p">)([</span><span class="n">factor_terms</span><span class="p">(</span><span class="n">i</span><span class="p">,</span>
-                <span class="n">radical</span><span class="o">=</span><span class="n">radical</span><span class="p">,</span>
-                <span class="n">clear</span><span class="o">=</span><span class="n">clear</span><span class="p">,</span>
-                <span class="n">fraction</span><span class="o">=</span><span class="n">fraction</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">or</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Function</span> <span class="ow">or</span> <span class="n">is_iterable</span> <span class="ow">or</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&#39;args_cnc&#39;</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span>
-        <span class="n">newargs</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">([</span><span class="n">factor_terms</span><span class="p">(</span><span class="n">i</span><span class="p">,</span>
-            <span class="n">radical</span><span class="o">=</span><span class="n">radical</span><span class="p">,</span>
-            <span class="n">clear</span><span class="o">=</span><span class="n">clear</span><span class="p">,</span>
-            <span class="n">fraction</span><span class="o">=</span><span class="n">fraction</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">args</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">newargs</span> <span class="o">==</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">newargs</span><span class="p">)</span>
-
-    <span class="n">cont</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">(</span><span class="n">radical</span><span class="o">=</span><span class="n">radical</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-        <span class="n">list_args</span> <span class="o">=</span> <span class="p">[</span><span class="n">gcd_terms</span><span class="p">(</span><span class="n">a</span><span class="p">,</span>
-        <span class="n">isprimitive</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span>
-        <span class="n">clear</span><span class="o">=</span><span class="n">clear</span><span class="p">,</span>
-        <span class="n">fraction</span><span class="o">=</span><span class="n">fraction</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">p</span><span class="p">)]</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">Add</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">list_args</span><span class="p">)</span>  <span class="c"># gcd_terms will fix up ordering</span>
-    <span class="k">elif</span> <span class="n">p</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">func</span><span class="p">(</span>
-            <span class="o">*</span><span class="p">[</span><span class="n">factor_terms</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">radical</span><span class="p">,</span> <span class="n">clear</span><span class="p">,</span> <span class="n">fraction</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">p</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="n">gcd_terms</span><span class="p">(</span><span class="n">p</span><span class="p">,</span>
-        <span class="n">isprimitive</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span>
-        <span class="n">clear</span><span class="o">=</span><span class="n">clear</span><span class="p">,</span>
-        <span class="n">fraction</span><span class="o">=</span><span class="n">fraction</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">cont</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">clear</span><span class="o">=</span><span class="n">clear</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_mask_nc</span><span class="p">(</span><span class="n">eq</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return ``eq`` with non-commutative objects replaced with dummy</span>
-<span class="sd">    symbols. A dictionary that can be used to restore the original</span>
-<span class="sd">    values is returned: if it is None, the expression is</span>
-<span class="sd">    noncommutative and cannot be made commutative. The third value</span>
-<span class="sd">    returned is a list of any non-commutative symbols that appear</span>
-<span class="sd">    in the returned equation.</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-<span class="sd">    All non-commutative objects other than Symbols are replaced with</span>
-<span class="sd">    a non-commutative Symbol. Identical objects will be identified</span>
-<span class="sd">    by identical symbols.</span>
-
-<span class="sd">    If there is only 1 non-commutative object in an expression it will</span>
-<span class="sd">    be replaced with a commutative symbol. Otherwise, the non-commutative</span>
-<span class="sd">    entities are retained and the calling routine should handle</span>
-<span class="sd">    replacements in this case since some care must be taken to keep</span>
-<span class="sd">    track of the ordering of symbols when they occur within Muls.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.secondquant import Commutator, NO, F, Fd</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Dummy, symbols, Sum, Mul, Basic</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.core.exprtools import _mask_nc</span>
-<span class="sd">    &gt;&gt;&gt; A, B, C = symbols(&#39;A,B,C&#39;, commutative=False)</span>
-<span class="sd">    &gt;&gt;&gt; Dummy._count = 0 # reset for doctest purposes</span>
-
-<span class="sd">    One nc-symbol:</span>
-
-<span class="sd">    &gt;&gt;&gt; _mask_nc(A**2 - x**2)</span>
-<span class="sd">    (_0**2 - x**2, {_0: A}, [])</span>
-
-<span class="sd">    Multiple nc-symbols:</span>
-
-<span class="sd">    &gt;&gt;&gt; _mask_nc(A**2 - B**2)</span>
-<span class="sd">    (A**2 - B**2, None, [A, B])</span>
-
-<span class="sd">    An nc-object with nc-symbols but no others outside of it:</span>
-
-<span class="sd">    &gt;&gt;&gt; _mask_nc(1 + x*Commutator(A, B))</span>
-<span class="sd">    (_1*x + 1, {_1: Commutator(A, B)}, [])</span>
-<span class="sd">    &gt;&gt;&gt; _mask_nc(NO(Fd(x)*F(y)))</span>
-<span class="sd">    (_2, {_2: NO(CreateFermion(x)*AnnihilateFermion(y))}, [])</span>
-
-<span class="sd">    Multiple nc-objects:</span>
-
-<span class="sd">    &gt;&gt;&gt; eq = x*Commutator(A, B) + x*Commutator(A, C)*Commutator(A, B)</span>
-<span class="sd">    &gt;&gt;&gt; _mask_nc(eq)</span>
-<span class="sd">    (x*_3 + x*_4*_3, {_3: Commutator(A, B), _4: Commutator(A, C)}, [_3, _4])</span>
-
-<span class="sd">    Multiple nc-objects and nc-symbols:</span>
-
-<span class="sd">    &gt;&gt;&gt; eq = A*Commutator(A, B) + B*Commutator(A, C)</span>
-<span class="sd">    &gt;&gt;&gt; _mask_nc(eq)</span>
-<span class="sd">    (A*_5 + B*_6, {_5: Commutator(A, B), _6: Commutator(A, C)}, [_5, _6, A, B])</span>
-
-<span class="sd">    If there is an object that:</span>
-
-<span class="sd">        - doesn&#39;t contain nc-symbols</span>
-<span class="sd">        - but has arguments which derive from Basic, not Expr</span>
-<span class="sd">        - and doesn&#39;t define an _eval_is_commutative routine</span>
-
-<span class="sd">    then it will give False (or None?) for the is_commutative test. Such</span>
-<span class="sd">    objects are also removed by this routine:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Basic, Mul</span>
-<span class="sd">    &gt;&gt;&gt; eq = (1 + Mul(Basic(), Basic(), evaluate=False))</span>
-<span class="sd">    &gt;&gt;&gt; eq.is_commutative</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; _mask_nc(eq)</span>
-<span class="sd">    (_7**2 + 1, {_7: Basic()}, [])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">eq</span>
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">eq</span><span class="p">,</span> <span class="p">{},</span> <span class="p">[]</span>
-
-    <span class="c"># identify nc-objects; symbols and other</span>
-    <span class="n">rep</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">nc_obj</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="n">nc_syms</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="n">pot</span> <span class="o">=</span> <span class="n">preorder_traversal</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">keys</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">pot</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">a</span> <span class="o">==</span> <span class="n">r</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">rep</span><span class="p">):</span>
-            <span class="n">pot</span><span class="o">.</span><span class="n">skip</span><span class="p">()</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="n">a</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">:</span>
-                <span class="n">nc_syms</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="ow">not</span> <span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">or</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Mul</span> <span class="ow">or</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">):</span>
-                <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">is_commutative</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">a</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">):</span>
-                    <span class="n">rep</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">a</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">()))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">nc_obj</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-                <span class="n">pot</span><span class="o">.</span><span class="n">skip</span><span class="p">()</span>
-
-    <span class="c"># If there is only one nc symbol or object, it can be factored regularly</span>
-    <span class="c"># but polys is going to complain, so replace it with a Dummy.</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">nc_obj</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">nc_syms</span><span class="p">:</span>
-        <span class="n">rep</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">nc_obj</span><span class="o">.</span><span class="n">pop</span><span class="p">(),</span> <span class="n">Dummy</span><span class="p">()))</span>
-    <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">nc_syms</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">nc_obj</span><span class="p">:</span>
-        <span class="n">rep</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">nc_syms</span><span class="o">.</span><span class="n">pop</span><span class="p">(),</span> <span class="n">Dummy</span><span class="p">()))</span>
-
-    <span class="c"># Any remaining nc-objects will be replaced with an nc-Dummy and</span>
-    <span class="c"># identified as an nc-Symbol to watch out for</span>
-    <span class="n">nc_obj</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">nc_obj</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">nc_obj</span><span class="p">:</span>
-        <span class="n">nc</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="n">commutative</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-        <span class="n">rep</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="n">nc</span><span class="p">))</span>
-        <span class="n">nc_syms</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">nc</span><span class="p">)</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">rep</span><span class="p">)</span>
-
-    <span class="n">nc_syms</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">nc_syms</span><span class="p">)</span>
-    <span class="n">nc_syms</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">expr</span><span class="p">,</span> <span class="nb">dict</span><span class="p">([(</span><span class="n">v</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">rep</span><span class="p">])</span> <span class="ow">or</span> <span class="bp">None</span><span class="p">,</span> <span class="n">nc_syms</span>
-
-
-<span class="k">def</span> <span class="nf">factor_nc</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return the factored form of ``expr`` while handling non-commutative</span>
-<span class="sd">    expressions.</span>
-
-<span class="sd">    **examples**</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.core.exprtools import factor_nc</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; A = Symbol(&#39;A&#39;, commutative=False)</span>
-<span class="sd">    &gt;&gt;&gt; B = Symbol(&#39;B&#39;, commutative=False)</span>
-<span class="sd">    &gt;&gt;&gt; factor_nc((x**2 + 2*A*x + A**2).expand())</span>
-<span class="sd">    (x + A)**2</span>
-<span class="sd">    &gt;&gt;&gt; factor_nc(((x + A)*(x + B)).expand())</span>
-<span class="sd">    (x + A)*(x + B)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.simplify.simplify</span> <span class="kn">import</span> <span class="n">_mexpand</span>
-    <span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">gcd</span><span class="p">,</span> <span class="n">factor</span>
-
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Expr</span><span class="p">)</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">factor_nc</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="n">expr</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="n">nc_symbols</span> <span class="o">=</span> <span class="n">_mask_nc</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">rep</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">factor</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">rep</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">expr</span><span class="p">)]</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">g</span> <span class="o">=</span> <span class="n">l</span> <span class="o">=</span> <span class="n">r</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="n">hit</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="c"># find any commutative gcd term</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="n">Mul</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="k">elif</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="n">gcd</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">Mul</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-            <span class="n">hit</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="n">c</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">c</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">g</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">cc</span><span class="p">,</span> <span class="n">_</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-                    <span class="n">cc</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">Mul</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">cc</span><span class="p">))</span><span class="o">/</span><span class="n">g</span><span class="p">))</span>
-                    <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">cc</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">cc</span><span class="p">,</span> <span class="n">_</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-                    <span class="n">cc</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">cc</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="n">c</span>
-                    <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">cc</span>
-        <span class="c"># find any noncommutative common prefix</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">n</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">][:]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">n</span> <span class="o">=</span> <span class="n">common_prefix</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-                <span class="c"># is there a power that can be extracted?</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]:</span>
-                    <span class="k">break</span>
-                <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                <span class="n">ok</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="ow">not</span> <span class="n">t</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span>
-                            <span class="k">break</span>
-                        <span class="n">bt</span><span class="p">,</span> <span class="n">et</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                        <span class="k">if</span> <span class="n">et</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">bt</span> <span class="o">==</span> <span class="n">b</span><span class="p">:</span>
-                            <span class="n">e</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">et</span><span class="p">)</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="k">break</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">ok</span> <span class="o">=</span> <span class="n">hit</span> <span class="o">=</span> <span class="bp">True</span>
-                        <span class="n">l</span> <span class="o">=</span> <span class="n">b</span><span class="o">**</span><span class="n">e</span>
-                        <span class="n">il</span> <span class="o">=</span> <span class="n">b</span><span class="o">**-</span><span class="n">e</span>
-                        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-                            <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">il</span><span class="o">*</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-                        <span class="k">break</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">ok</span><span class="p">:</span>
-                    <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">hit</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="n">lenn</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-            <span class="n">l</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">n</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-                <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">][</span><span class="n">lenn</span><span class="p">:]</span>
-        <span class="c"># find any noncommutative common suffix</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">n</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">][:]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">n</span> <span class="o">=</span> <span class="n">common_suffix</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-                <span class="c"># is there a power that can be extracted?</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]:</span>
-                    <span class="k">break</span>
-                <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">][</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                <span class="n">ok</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="ow">not</span> <span class="n">t</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span>
-                            <span class="k">break</span>
-                        <span class="n">bt</span><span class="p">,</span> <span class="n">et</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                        <span class="k">if</span> <span class="n">et</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">bt</span> <span class="o">==</span> <span class="n">b</span><span class="p">:</span>
-                            <span class="n">e</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">et</span><span class="p">)</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="k">break</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">ok</span> <span class="o">=</span> <span class="n">hit</span> <span class="o">=</span> <span class="bp">True</span>
-                        <span class="n">r</span> <span class="o">=</span> <span class="n">b</span><span class="o">**</span><span class="n">e</span>
-                        <span class="n">il</span> <span class="o">=</span> <span class="n">b</span><span class="o">**-</span><span class="n">e</span>
-                        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-                            <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">][</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">][</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">il</span>
-                        <span class="k">break</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">ok</span><span class="p">:</span>
-                    <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">hit</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="n">lenn</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">n</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-                <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">][:</span><span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="o">-</span> <span class="n">lenn</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">hit</span><span class="p">:</span>
-            <span class="n">mid</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">cc</span><span class="p">)</span><span class="o">*</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">nc</span><span class="p">)</span> <span class="k">for</span> <span class="n">cc</span><span class="p">,</span> <span class="n">nc</span> <span class="ow">in</span> <span class="n">args</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">mid</span> <span class="o">=</span> <span class="n">expr</span>
-
-        <span class="c"># sort the symbols so the Dummys would appear in the same</span>
-        <span class="c"># order as the original symbols, otherwise you may introduce</span>
-        <span class="c"># a factor of -1, e.g. A**2 - B**2) -- {A:y, B:x} --&gt; y**2 - x**2</span>
-        <span class="c"># and the former factors into two terms, (A - B)*(A + B) while the</span>
-        <span class="c"># latter factors into 3 terms, (-1)*(x - y)*(x + y)</span>
-        <span class="n">rep1</span> <span class="o">=</span> <span class="p">[(</span><span class="n">n</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">())</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">nc_symbols</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)]</span>
-        <span class="n">unrep1</span> <span class="o">=</span> <span class="p">[(</span><span class="n">v</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">rep1</span><span class="p">]</span>
-        <span class="n">unrep1</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-        <span class="n">new_mid</span><span class="p">,</span> <span class="n">r2</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">_mask_nc</span><span class="p">(</span><span class="n">mid</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">rep1</span><span class="p">))</span>
-        <span class="n">new_mid</span> <span class="o">=</span> <span class="n">factor</span><span class="p">(</span><span class="n">new_mid</span><span class="p">)</span>
-
-        <span class="n">new_mid</span> <span class="o">=</span> <span class="n">new_mid</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">r2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">unrep1</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">new_mid</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">g</span><span class="o">*</span><span class="n">l</span><span class="o">*</span><span class="n">new_mid</span><span class="o">*</span><span class="n">r</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">new_mid</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="c"># XXX TODO there should be a way to inspect what order the terms</span>
-            <span class="c"># must be in and just select the plausible ordering without</span>
-            <span class="c"># checking permutations</span>
-            <span class="n">cfac</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">ncfac</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">new_mid</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-                    <span class="n">cfac</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                    <span class="k">assert</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Integer</span>
-                    <span class="n">ncfac</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">b</span><span class="p">]</span><span class="o">*</span><span class="n">e</span><span class="p">)</span>
-            <span class="n">pre_mid</span> <span class="o">=</span> <span class="n">g</span><span class="o">*</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">cfac</span><span class="p">)</span><span class="o">*</span><span class="n">l</span>
-            <span class="n">target</span> <span class="o">=</span> <span class="n">_mexpand</span><span class="p">(</span><span class="n">expr</span><span class="o">/</span><span class="n">c</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">variations</span><span class="p">(</span><span class="n">ncfac</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">ncfac</span><span class="p">)):</span>
-                <span class="n">ok</span> <span class="o">=</span> <span class="n">pre_mid</span><span class="o">*</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="p">)</span><span class="o">*</span><span class="n">r</span>
-                <span class="k">if</span> <span class="n">_mexpand</span><span class="p">(</span><span class="n">ok</span><span class="p">)</span> <span class="o">==</span> <span class="n">target</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">ok</span><span class="p">)</span>
-
-        <span class="c"># mid was an Add that didn&#39;t factor successfully</span>
-        <span class="k">return</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">g</span><span class="o">*</span><span class="n">l</span><span class="o">*</span><span class="n">mid</span><span class="o">*</span><span class="n">r</span><span class="p">)</span>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/core/function.html b/dev-py3k/_modules/sympy/core/function.html
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-            
-  <h1>Source code for sympy.core.function</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">There are two types of functions:</span>
-<span class="sd">1) defined function like exp or sin that has a name and body</span>
-<span class="sd">   (in the sense that function can be evaluated).</span>
-<span class="sd">    e = exp</span>
-<span class="sd">2) undefined function with a name but no body. Undefined</span>
-<span class="sd">   functions can be defined using a Function class as follows:</span>
-<span class="sd">       f = Function(&#39;f&#39;)</span>
-<span class="sd">   (the result will be a Function instance)</span>
-<span class="sd">3) this isn&#39;t implemented yet: anonymous function or lambda function that has</span>
-<span class="sd">   no name but has body with dummy variables. Examples of anonymous function</span>
-<span class="sd">   creation:</span>
-<span class="sd">       f = Lambda(x, exp(x)*x)</span>
-<span class="sd">       f = Lambda(exp(x)*x) # free symbols of expr define the number of args</span>
-<span class="sd">       f = Lambda(exp(x)*x)  # free symbols in the expression define the number</span>
-<span class="sd">                             # of arguments</span>
-<span class="sd">       f = exp * Lambda(x,x)</span>
-<span class="sd">4) isn&#39;t implemented yet: composition of functions, like (sin+cos)(x), this</span>
-<span class="sd">   works in sympy core, but needs to be ported back to SymPy.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; import sympy</span>
-<span class="sd">    &gt;&gt;&gt; f = sympy.Function(&quot;f&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; f(x)</span>
-<span class="sd">    f(x)</span>
-<span class="sd">    &gt;&gt;&gt; print(sympy.srepr(f(x).func))</span>
-<span class="sd">    Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; f(x).args</span>
-<span class="sd">    (x,)</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">.add</span> <span class="kn">import</span> <span class="n">Add</span>
-<span class="kn">from</span> <span class="nn">.assumptions</span> <span class="kn">import</span> <span class="n">ManagedProperties</span>
-<span class="kn">from</span> <span class="nn">.basic</span> <span class="kn">import</span> <span class="n">Basic</span>
-<span class="kn">from</span> <span class="nn">.cache</span> <span class="kn">import</span> <span class="n">cacheit</span>
-<span class="kn">from</span> <span class="nn">.compatibility</span> <span class="kn">import</span> <span class="n">iterable</span><span class="p">,</span> <span class="n">is_sequence</span>
-<span class="kn">from</span> <span class="nn">.core</span> <span class="kn">import</span> <span class="n">BasicMeta</span><span class="p">,</span> <span class="n">C</span>
-<span class="kn">from</span> <span class="nn">.decorators</span> <span class="kn">import</span> <span class="n">_sympifyit</span>
-<span class="kn">from</span> <span class="nn">.expr</span> <span class="kn">import</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">AtomicExpr</span>
-<span class="kn">from</span> <span class="nn">.numbers</span> <span class="kn">import</span> <span class="n">Rational</span><span class="p">,</span> <span class="n">Float</span>
-<span class="kn">from</span> <span class="nn">.rules</span> <span class="kn">import</span> <span class="n">Transform</span>
-<span class="kn">from</span> <span class="nn">.singleton</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="kn">from</span> <span class="nn">.sympify</span> <span class="kn">import</span> <span class="n">sympify</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.containers</span> <span class="kn">import</span> <span class="n">Tuple</span><span class="p">,</span> <span class="n">Dict</span>
-<span class="kn">from</span> <span class="nn">sympy.core.logic</span> <span class="kn">import</span> <span class="n">fuzzy_and</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">default_sort_key</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">uniq</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">mpmath</span>
-<span class="kn">import</span> <span class="nn">sympy.mpmath.libmp</span> <span class="kn">as</span> <span class="nn">mlib</span>
-
-
-<span class="k">def</span> <span class="nf">_coeff_isneg</span><span class="p">(</span><span class="n">a</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return True if the leading Number is negative.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.core.function import _coeff_isneg</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import S, Symbol, oo, pi</span>
-<span class="sd">    &gt;&gt;&gt; _coeff_isneg(-3*pi)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; _coeff_isneg(S(3))</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; _coeff_isneg(-oo)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; _coeff_isneg(Symbol(&#39;n&#39;, negative=True)) # coeff is 1</span>
-<span class="sd">    False</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">and</span> <span class="n">a</span><span class="o">.</span><span class="n">is_negative</span>
-
-
-<div class="viewcode-block" id="PoleError"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.PoleError">[docs]</a><span class="k">class</span> <span class="nc">PoleError</span><span class="p">(</span><span class="ne">Exception</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<span class="k">class</span> <span class="nc">ArgumentIndexError</span><span class="p">(</span><span class="ne">ValueError</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="s">&quot;Invalid operation with argument number </span><span class="si">%s</span><span class="s"> for Function </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span>
-               <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-
-
-<div class="viewcode-block" id="FunctionClass"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.FunctionClass">[docs]</a><span class="k">class</span> <span class="nc">FunctionClass</span><span class="p">(</span><span class="n">ManagedProperties</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">BasicMeta</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Base class for function classes. FunctionClass is a subclass of type.</span>
-
-<span class="sd">    Use Function(&#39;&lt;function name&gt;&#39; [ , signature ]) to create</span>
-<span class="sd">    undefined function classes.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_new</span> <span class="o">=</span> <span class="nb">type</span><span class="o">.</span><span class="n">__new__</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="n">cls</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">__name__</span>
-
-</div>
-<span class="k">class</span> <span class="nc">Application</span><span class="p">(</span><span class="n">Basic</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">FunctionClass</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Base class for applied functions.</span>
-
-<span class="sd">    Instances of Application represent the result of applying an application of</span>
-<span class="sd">    any type to any object.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="n">is_Function</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="n">args</span><span class="p">))</span>
-        <span class="n">evaluate</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;evaluate&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">options</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Unknown options: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">options</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">evaluate</span><span class="p">:</span>
-            <span class="n">evaluated</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">evaluated</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">evaluated</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Application</span><span class="p">,</span> <span class="n">cls</span><span class="p">)</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns a canonical form of cls applied to arguments args.</span>
-
-<span class="sd">        The eval() method is called when the class cls is about to be</span>
-<span class="sd">        instantiated and it should return either some simplified instance</span>
-<span class="sd">        (possible of some other class), or if the class cls should be</span>
-<span class="sd">        unmodified, return None.</span>
-
-<span class="sd">        Examples of eval() for the function &quot;sign&quot;</span>
-<span class="sd">        ---------------------------------------------</span>
-
-<span class="sd">        @classmethod</span>
-<span class="sd">        def eval(cls, arg):</span>
-<span class="sd">            if arg is S.NaN:</span>
-<span class="sd">                return S.NaN</span>
-<span class="sd">            if arg is S.Zero: return S.Zero</span>
-<span class="sd">            if arg.is_positive: return S.One</span>
-<span class="sd">            if arg.is_negative: return S.NegativeOne</span>
-<span class="sd">            if isinstance(arg, C.Mul):</span>
-<span class="sd">                coeff, terms = arg.as_coeff_Mul(rational=True)</span>
-<span class="sd">                if coeff is not S.One:</span>
-<span class="sd">                    return cls(coeff) * cls(terms)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span>
-
-    <span class="k">def</span> <span class="nf">_eval_subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">):</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">old</span><span class="o">.</span><span class="n">is_Function</span> <span class="ow">and</span> <span class="n">new</span><span class="o">.</span><span class="n">is_Function</span> <span class="ow">and</span>
-            <span class="n">old</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span> <span class="ow">and</span>
-            <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">nargs</span> <span class="o">==</span> <span class="n">new</span><span class="o">.</span><span class="n">nargs</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">new</span><span class="o">.</span><span class="n">nargs</span> <span class="ow">or</span>
-             <span class="nb">isinstance</span><span class="p">(</span><span class="n">new</span><span class="o">.</span><span class="n">nargs</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">)</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">nargs</span> <span class="ow">in</span> <span class="n">new</span><span class="o">.</span><span class="n">nargs</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="n">new</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="Function"><a class="viewcode-back" href="../../../modules/functions/index.html#sympy.core.function.Function">[docs]</a><span class="k">class</span> <span class="nc">Function</span><span class="p">(</span><span class="n">Application</span><span class="p">,</span> <span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Base class for applied mathematical functions.</span>
-
-<span class="sd">    It also serves as a constructor for undefined function classes.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    First example shows how to use Function as a constructor for undefined</span>
-<span class="sd">    function classes:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, Symbol</span>
-<span class="sd">    &gt;&gt;&gt; x = Symbol(&#39;x&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; g = Function(&#39;g&#39;)(x)</span>
-<span class="sd">    &gt;&gt;&gt; f</span>
-<span class="sd">    f</span>
-<span class="sd">    &gt;&gt;&gt; f(x)</span>
-<span class="sd">    f(x)</span>
-<span class="sd">    &gt;&gt;&gt; g</span>
-<span class="sd">    g(x)</span>
-<span class="sd">    &gt;&gt;&gt; f(x).diff(x)</span>
-<span class="sd">    Derivative(f(x), x)</span>
-<span class="sd">    &gt;&gt;&gt; g.diff(x)</span>
-<span class="sd">    Derivative(g(x), x)</span>
-
-<span class="sd">    In the following example Function is used as a base class for</span>
-<span class="sd">    ``my_func`` that represents a mathematical function *my_func*. Suppose</span>
-<span class="sd">    that it is well known, that *my_func(0)* is *1* and *my_func* at infinity</span>
-<span class="sd">    goes to *0*, so we want those two simplifications to occur automatically.</span>
-<span class="sd">    Suppose also that *my_func(x)* is real exactly when *x* is real. Here is</span>
-<span class="sd">    an implementation that honours those requirements:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, S, oo, I, sin</span>
-<span class="sd">    &gt;&gt;&gt; class my_func(Function):</span>
-<span class="sd">    ...</span>
-<span class="sd">    ...     nargs = 1</span>
-<span class="sd">    ...</span>
-<span class="sd">    ...     @classmethod</span>
-<span class="sd">    ...     def eval(cls, x):</span>
-<span class="sd">    ...         if x.is_Number:</span>
-<span class="sd">    ...             if x is S.Zero:</span>
-<span class="sd">    ...                 return S.One</span>
-<span class="sd">    ...             elif x is S.Infinity:</span>
-<span class="sd">    ...                 return S.Zero</span>
-<span class="sd">    ...</span>
-<span class="sd">    ...     def _eval_is_real(self):</span>
-<span class="sd">    ...         return self.args[0].is_real</span>
-<span class="sd">    ...</span>
-<span class="sd">    &gt;&gt;&gt; x = S(&#39;x&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; my_func(0) + sin(0)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; my_func(oo)</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; my_func(3.54).n() # Not yet implemented for my_func.</span>
-<span class="sd">    my_func(3.54)</span>
-<span class="sd">    &gt;&gt;&gt; my_func(I).is_real</span>
-<span class="sd">    False</span>
-
-<span class="sd">    In order for ``my_func`` to become useful, several other methods would</span>
-<span class="sd">    need to be implemented. See source code of some of the already</span>
-<span class="sd">    implemented functions for more complete examples.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_diff_wrt</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Allow derivatives wrt functions.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Function, Symbol</span>
-<span class="sd">        &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; x = Symbol(&#39;x&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; f(x)._diff_wrt</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="c"># Handle calls like Function(&#39;f&#39;)</span>
-        <span class="k">if</span> <span class="n">cls</span> <span class="ow">is</span> <span class="n">Function</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">UndefinedFunction</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">cls</span><span class="o">.</span><span class="n">nargs</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">cls</span><span class="o">.</span><span class="n">nargs</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span>
-                <span class="n">nargs</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">nargs</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">nargs</span> <span class="o">=</span> <span class="p">(</span><span class="n">cls</span><span class="o">.</span><span class="n">nargs</span><span class="p">,)</span>
-
-            <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">n</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">nargs</span><span class="p">:</span>
-                <span class="c"># XXX: exception message must be in exactly this format to make</span>
-                <span class="c"># it work with NumPy&#39;s functions like vectorize(). The ideal</span>
-                <span class="c"># solution would be just to attach metadata to the exception</span>
-                <span class="c"># and change NumPy to take advantage of this.</span>
-                <span class="n">temp</span> <span class="o">=</span> <span class="p">(</span><span class="s">&#39;</span><span class="si">%(name)s</span><span class="s"> takes exactly </span><span class="si">%(args)s</span><span class="s"> &#39;</span>
-                       <span class="s">&#39;argument</span><span class="si">%(plural)s</span><span class="s"> (</span><span class="si">%(given)s</span><span class="s"> given)&#39;</span><span class="p">)</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="n">temp</span> <span class="o">%</span> <span class="p">{</span>
-                    <span class="s">&#39;name&#39;</span><span class="p">:</span> <span class="n">cls</span><span class="p">,</span>
-                    <span class="s">&#39;args&#39;</span><span class="p">:</span> <span class="n">cls</span><span class="o">.</span><span class="n">nargs</span><span class="p">,</span>
-                    <span class="s">&#39;plural&#39;</span><span class="p">:</span> <span class="s">&#39;s&#39;</span><span class="o">*</span><span class="p">(</span><span class="n">n</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">),</span>
-                    <span class="s">&#39;given&#39;</span><span class="p">:</span> <span class="n">n</span><span class="p">})</span>
-
-        <span class="n">evaluate</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;evaluate&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="nb">super</span><span class="p">(</span><span class="n">Function</span><span class="p">,</span> <span class="n">cls</span><span class="p">)</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">evaluate</span> <span class="ow">or</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">cls</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">result</span>
-
-        <span class="n">pr</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">cls</span><span class="o">.</span><span class="n">_should_evalf</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">result</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">pr2</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">cls</span><span class="o">.</span><span class="n">_should_evalf</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">result</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">pr2</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">result</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">libmpf</span><span class="o">.</span><span class="n">prec_to_dps</span><span class="p">(</span><span class="n">pr</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_should_evalf</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Decide if the function should automatically evalf().</span>
-
-<span class="sd">        By default (in this implementation), this happens if (and only if) the</span>
-<span class="sd">        ARG is a floating point number.</span>
-<span class="sd">        This function is used by __new__.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">arg</span><span class="o">.</span><span class="n">_prec</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-        <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-        <span class="n">l</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span><span class="o">.</span><span class="n">_prec</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="p">[</span><span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">]</span> <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Float</span><span class="p">]</span>
-        <span class="n">l</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-        <span class="k">return</span> <span class="nb">max</span><span class="p">(</span><span class="n">l</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">class_key</span><span class="p">(</span><span class="n">cls</span><span class="p">):</span>
-        <span class="n">funcs</span> <span class="o">=</span> <span class="p">{</span>
-            <span class="s">&#39;exp&#39;</span><span class="p">:</span> <span class="mi">10</span><span class="p">,</span>
-            <span class="s">&#39;log&#39;</span><span class="p">:</span> <span class="mi">11</span><span class="p">,</span>
-            <span class="s">&#39;sin&#39;</span><span class="p">:</span> <span class="mi">20</span><span class="p">,</span>
-            <span class="s">&#39;cos&#39;</span><span class="p">:</span> <span class="mi">21</span><span class="p">,</span>
-            <span class="s">&#39;tan&#39;</span><span class="p">:</span> <span class="mi">22</span><span class="p">,</span>
-            <span class="s">&#39;cot&#39;</span><span class="p">:</span> <span class="mi">23</span><span class="p">,</span>
-            <span class="s">&#39;sinh&#39;</span><span class="p">:</span> <span class="mi">30</span><span class="p">,</span>
-            <span class="s">&#39;cosh&#39;</span><span class="p">:</span> <span class="mi">31</span><span class="p">,</span>
-            <span class="s">&#39;tanh&#39;</span><span class="p">:</span> <span class="mi">32</span><span class="p">,</span>
-            <span class="s">&#39;coth&#39;</span><span class="p">:</span> <span class="mi">33</span><span class="p">,</span>
-            <span class="s">&#39;conjugate&#39;</span><span class="p">:</span> <span class="mi">40</span><span class="p">,</span>
-            <span class="s">&#39;re&#39;</span><span class="p">:</span> <span class="mi">41</span><span class="p">,</span>
-            <span class="s">&#39;im&#39;</span><span class="p">:</span> <span class="mi">42</span><span class="p">,</span>
-            <span class="s">&#39;arg&#39;</span><span class="p">:</span> <span class="mi">43</span><span class="p">,</span>
-        <span class="p">}</span>
-        <span class="n">name</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">__name__</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="n">funcs</span><span class="p">[</span><span class="n">name</span><span class="p">]</span>
-        <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-            <span class="n">nargs</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">nargs</span>
-
-            <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span> <span class="k">if</span> <span class="n">nargs</span> <span class="ow">is</span> <span class="bp">None</span> <span class="k">else</span> <span class="mi">10000</span>
-
-        <span class="k">return</span> <span class="mi">4</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">name</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Function.is_commutative"><a class="viewcode-back" href="../../../modules/functions/index.html#sympy.core.function.Function.is_commutative">[docs]</a>    <span class="k">def</span> <span class="nf">is_commutative</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns whether the functon is commutative.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="nb">getattr</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="s">&#39;is_commutative&#39;</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="c"># Lookup mpmath function based on name</span>
-        <span class="n">fname</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="o">.</span><span class="n">__name__</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">mpmath</span><span class="p">,</span> <span class="n">fname</span><span class="p">):</span>
-                <span class="kn">from</span> <span class="nn">sympy.utilities.lambdify</span> <span class="kn">import</span> <span class="n">MPMATH_TRANSLATIONS</span>
-                <span class="n">fname</span> <span class="o">=</span> <span class="n">MPMATH_TRANSLATIONS</span><span class="p">[</span><span class="n">fname</span><span class="p">]</span>
-            <span class="n">func</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">mpmath</span><span class="p">,</span> <span class="n">fname</span><span class="p">)</span>
-        <span class="k">except</span> <span class="p">(</span><span class="ne">AttributeError</span><span class="p">,</span> <span class="ne">KeyError</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Float</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_imp_</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">),</span> <span class="n">prec</span><span class="p">)</span>
-            <span class="k">except</span> <span class="p">(</span><span class="ne">AttributeError</span><span class="p">,</span> <span class="ne">TypeError</span><span class="p">):</span>
-                <span class="k">return</span>
-
-        <span class="c"># Convert all args to mpf or mpc</span>
-        <span class="c"># Convert the arguments to *higher* precision than requested for the</span>
-        <span class="c"># final result.</span>
-        <span class="c"># XXX + 5 is a guess, it is similar to what is used in evalf.py. Should</span>
-        <span class="c">#     we be more intelligent about it?</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">arg</span><span class="o">.</span><span class="n">_to_mpmath</span><span class="p">(</span><span class="n">prec</span> <span class="o">+</span> <span class="mi">5</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-            <span class="k">return</span>
-
-        <span class="c"># Set mpmath precision and apply. Make sure precision is restored</span>
-        <span class="c"># afterwards</span>
-        <span class="n">orig</span> <span class="o">=</span> <span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">prec</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="n">prec</span>
-            <span class="n">v</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">finally</span><span class="p">:</span>
-            <span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="n">orig</span>
-
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">_from_mpmath</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="c"># f(x).diff(s) -&gt; x.diff(s) * f.fdiff(1)(s)</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">l</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">da</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">da</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">df</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">fdiff</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">ArgumentIndexError</span><span class="p">:</span>
-                <span class="n">df</span> <span class="o">=</span> <span class="n">Function</span><span class="o">.</span><span class="n">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-            <span class="n">l</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">df</span> <span class="o">*</span> <span class="n">da</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">l</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_commutative</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">fuzzy_and</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">is_commutative</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Function.as_base_exp"><a class="viewcode-back" href="../../../modules/functions/index.html#sympy.core.function.Function.as_base_exp">[docs]</a>    <span class="k">def</span> <span class="nf">as_base_exp</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the method as the 2-tuple (base, exponent).</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_aseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">args0</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Compute an asymptotic expansion around args0, in terms of self.args.</span>
-<span class="sd">        This function is only used internally by _eval_nseries and should not</span>
-<span class="sd">        be called directly; derived classes can overwrite this to implement</span>
-<span class="sd">        asymptotic expansions.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.utilities.misc</span> <span class="kn">import</span> <span class="n">filldedent</span>
-        <span class="k">raise</span> <span class="n">PoleError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&#39;&#39;&#39;</span>
-<span class="s">            Asymptotic expansion of </span><span class="si">%s</span><span class="s"> around </span><span class="si">%s</span><span class="s"> is</span>
-<span class="s">            not implemented.&#39;&#39;&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">),</span> <span class="n">args0</span><span class="p">)))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_nseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        This function does compute series for multivariate functions,</span>
-<span class="sd">        but the expansion is always in terms of *one* variable.</span>
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import atan2, O</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; atan2(x, y).series(x, n=2)</span>
-<span class="sd">        atan2(0, y) + x/y + O(x**2)</span>
-<span class="sd">        &gt;&gt;&gt; atan2(x, y).series(y, n=2)</span>
-<span class="sd">        -y/x + atan2(x, 0) + O(y**2)</span>
-
-<span class="sd">        This function also computes asymptotic expansions, if necessary</span>
-<span class="sd">        and possible:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import loggamma</span>
-<span class="sd">        &gt;&gt;&gt; loggamma(1/x)._eval_nseries(x,0,None)</span>
-<span class="sd">        -1/x - log(x)/x + log(x)/2 + O(1)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="o">.</span><span class="n">nargs</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">sympy.utilities.misc</span> <span class="kn">import</span> <span class="n">filldedent</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&#39;&#39;&#39;</span>
-<span class="s">                series for user-defined functions are not</span>
-<span class="s">                supported.&#39;&#39;&#39;</span><span class="p">))</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="n">args0</span> <span class="o">=</span> <span class="p">[</span><span class="n">t</span><span class="o">.</span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">t</span><span class="o">.</span><span class="n">is_bounded</span> <span class="ow">is</span> <span class="bp">False</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">args0</span><span class="p">):</span>
-            <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">oo</span><span class="p">,</span> <span class="n">zoo</span><span class="p">,</span> <span class="n">nan</span>
-            <span class="c"># XXX could use t.as_leading_term(x) here but it&#39;s a little</span>
-            <span class="c"># slower</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="p">[</span><span class="n">t</span><span class="o">.</span><span class="n">compute_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span>
-            <span class="n">a0</span> <span class="o">=</span> <span class="p">[</span><span class="n">t</span><span class="o">.</span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">a</span><span class="p">]</span>
-            <span class="k">if</span> <span class="nb">any</span><span class="p">([</span><span class="n">t</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">oo</span><span class="p">,</span> <span class="o">-</span><span class="n">oo</span><span class="p">,</span> <span class="n">zoo</span><span class="p">,</span> <span class="n">nan</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">a0</span><span class="p">]):</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_aseries</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">args0</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">logx</span>
-                                          <span class="p">)</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">)</span>
-            <span class="c"># Careful: the argument goes to oo, but only logarithmically so. We</span>
-            <span class="c"># are supposed to do a power series expansion &quot;around the</span>
-            <span class="c"># logarithmic term&quot;. e.g.</span>
-            <span class="c">#      f(1+x+log(x))</span>
-            <span class="c">#     -&gt; f(1+logx) + x*f&#39;(1+logx) + O(x**2)</span>
-            <span class="c"># where &#39;logx&#39; is given in the argument</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="p">[</span><span class="n">t</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span>
-            <span class="n">z</span> <span class="o">=</span> <span class="p">[</span><span class="n">r</span> <span class="o">-</span> <span class="n">r0</span> <span class="k">for</span> <span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">r0</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">a0</span><span class="p">)]</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="p">[</span><span class="n">Dummy</span><span class="p">()</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">z</span><span class="p">]</span>
-            <span class="n">q</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">v</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">for</span> <span class="n">ai</span><span class="p">,</span> <span class="n">zi</span><span class="p">,</span> <span class="n">pi</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">a0</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">zi</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">v</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-                    <span class="n">q</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ai</span> <span class="o">+</span> <span class="n">pi</span><span class="p">)</span>
-                    <span class="n">v</span> <span class="o">=</span> <span class="n">pi</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">q</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ai</span><span class="p">)</span>
-            <span class="n">e1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">q</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">v</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">e1</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">e1</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">)</span>
-            <span class="n">o</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">getO</span><span class="p">()</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">removeO</span><span class="p">()</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">zi</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span> <span class="o">+</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="n">o</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">zi</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">s</span>
-        <span class="k">if</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="o">.</span><span class="n">nargs</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">args0</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="o">.</span><span class="n">nargs</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">e</span> <span class="o">=</span> <span class="bp">self</span>
-            <span class="n">e1</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">e</span> <span class="o">==</span> <span class="n">e1</span><span class="p">:</span>
-                <span class="c">#for example when e = sin(x+1) or e = sin(cos(x))</span>
-                <span class="c">#let&#39;s try the general algorithm</span>
-                <span class="n">term</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">term</span><span class="o">.</span><span class="n">is_bounded</span> <span class="ow">is</span> <span class="bp">False</span> <span class="ow">or</span> <span class="n">term</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="n">PoleError</span><span class="p">(</span><span class="s">&quot;Cannot expand </span><span class="si">%s</span><span class="s"> around 0&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="p">))</span>
-                <span class="n">series</span> <span class="o">=</span> <span class="n">term</span>
-                <span class="n">fact</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-                    <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-                    <span class="n">fact</span> <span class="o">*=</span> <span class="n">Rational</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                    <span class="n">e</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-                    <span class="n">subs</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">subs</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                        <span class="c"># try to evaluate a limit if we have to</span>
-                        <span class="n">subs</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">subs</span><span class="o">.</span><span class="n">is_bounded</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                        <span class="k">raise</span> <span class="n">PoleError</span><span class="p">(</span><span class="s">&quot;Cannot expand </span><span class="si">%s</span><span class="s"> around 0&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="p">))</span>
-                    <span class="n">term</span> <span class="o">=</span> <span class="n">subs</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">i</span><span class="p">)</span><span class="o">/</span><span class="n">fact</span>
-                    <span class="n">term</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-                    <span class="n">series</span> <span class="o">+=</span> <span class="n">term</span>
-                <span class="k">return</span> <span class="n">series</span> <span class="o">+</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">e1</span><span class="o">.</span><span class="n">nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">l</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">2</span><span class="p">):</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">taylor_term</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">arg</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span>
-            <span class="n">l</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">l</span><span class="p">)</span> <span class="o">+</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">pattern</span><span class="p">,</span> <span class="n">rule</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;deep&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span><span class="o">.</span><span class="n">_eval_rewrite</span><span class="p">(</span><span class="n">pattern</span><span class="p">,</span> <span class="n">rule</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-
-        <span class="k">if</span> <span class="n">pattern</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">,</span> <span class="n">pattern</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rule</span><span class="p">):</span>
-                <span class="n">rewritten</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rule</span><span class="p">)(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">rewritten</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">rewritten</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Function.fdiff"><a class="viewcode-back" href="../../../modules/functions/index.html#sympy.core.function.Function.fdiff">[docs]</a>    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the first derivative of the function.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">nargs</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">nargs</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span>
-                <span class="n">nargs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">nargs</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">nargs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">nargs</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="mi">1</span> <span class="o">&lt;=</span> <span class="n">argindex</span> <span class="o">&lt;=</span> <span class="n">nargs</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">argindex</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">:</span>
-            <span class="c"># See issue 1525 and issue 1620 and issue 2501</span>
-            <span class="n">arg_dummy</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;xi_</span><span class="si">%i</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">argindex</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Subs</span><span class="p">(</span><span class="n">Derivative</span><span class="p">(</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">argindex</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">arg_dummy</span><span class="p">),</span>
-                <span class="n">arg_dummy</span><span class="p">),</span> <span class="n">arg_dummy</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">argindex</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">Derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">argindex</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Stub that should be overridden by new Functions to return</span>
-<span class="sd">        the first non-zero term in a series if ever an x-dependent</span>
-<span class="sd">        argument whose leading term vanishes as x -&gt; 0 might be encountered.</span>
-<span class="sd">        See, for example, cos._eval_as_leading_term.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="n">o</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">x</span> <span class="ow">in</span> <span class="n">a</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="n">o</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">args</span><span class="p">):</span>
-            <span class="c"># Whereas x and any finite number are contained in O(1, x),</span>
-            <span class="c"># expressions like 1/x are not. If any arg simplified to a</span>
-            <span class="c"># vanishing expression as x -&gt; 0 (like x or x**2, but not</span>
-            <span class="c"># 3, 1/x, etc...) then the _eval_as_leading_term is needed</span>
-            <span class="c"># to supply the first non-zero term of the series,</span>
-            <span class="c">#</span>
-            <span class="c"># e.g. expression    leading term</span>
-            <span class="c">#      ----------    ------------</span>
-            <span class="c">#      cos(1/x)      cos(1/x)</span>
-            <span class="c">#      cos(cos(x))   cos(1)</span>
-            <span class="c">#      cos(x)        1        &lt;- _eval_as_leading_term needed</span>
-            <span class="c">#      sin(x)        x        &lt;- _eval_as_leading_term needed</span>
-            <span class="c">#</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                <span class="s">&#39;</span><span class="si">%s</span><span class="s"> has no _eval_as_leading_term routine&#39;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Function.taylor_term"><a class="viewcode-back" href="../../../modules/functions/index.html#sympy.core.function.Function.taylor_term">[docs]</a>    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;General method for the taylor term.</span>
-
-<span class="sd">        This method is slow, because it differentiates n-times. Subclasses can</span>
-<span class="sd">        redefine it to make it faster by using the &quot;previous_terms&quot;.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="n">n</span> <span class="o">/</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-
-</div></div>
-<span class="k">class</span> <span class="nc">AppliedUndef</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Base class for expressions resulting from the application of an undefined</span>
-<span class="sd">    function.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="n">args</span><span class="p">))</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="nb">super</span><span class="p">(</span><span class="n">AppliedUndef</span><span class="p">,</span> <span class="n">cls</span><span class="p">)</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="n">result</span><span class="o">.</span><span class="n">nargs</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="k">class</span> <span class="nc">UndefinedFunction</span><span class="p">(</span><span class="n">FunctionClass</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The (meta)class of undefined functions.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">mcl</span><span class="p">,</span> <span class="n">name</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">BasicMeta</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">mcl</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="p">(</span><span class="n">AppliedUndef</span><span class="p">,),</span> <span class="p">{})</span>
-
-
-<div class="viewcode-block" id="WildFunction"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.WildFunction">[docs]</a><span class="k">class</span> <span class="nc">WildFunction</span><span class="p">(</span><span class="n">Function</span><span class="p">,</span> <span class="n">AtomicExpr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A WildFunction function matches any function (with its arguments).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Wild, WildFunction, Function, cos</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">    &gt;&gt;&gt; F = WildFunction(&#39;F&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; x.match(F)</span>
-<span class="sd">    &gt;&gt;&gt; F.match(F)</span>
-<span class="sd">    {F_: F_}</span>
-<span class="sd">    &gt;&gt;&gt; f(x).match(F)</span>
-<span class="sd">    {F_: f(x)}</span>
-<span class="sd">    &gt;&gt;&gt; cos(x).match(F)</span>
-<span class="sd">    {F_: cos(x)}</span>
-<span class="sd">    &gt;&gt;&gt; f(x, y).match(F)</span>
-
-<span class="sd">    To match functions with more than 1 arguments, set ``nargs`` to the</span>
-<span class="sd">    desired value:</span>
-
-<span class="sd">    &gt;&gt;&gt; F.nargs = 2</span>
-<span class="sd">    &gt;&gt;&gt; f(x, y).match(F)</span>
-<span class="sd">    {F_: f(x, y)}</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">include</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">):</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Function</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="n">name</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">matches</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">repl_dict</span><span class="o">=</span><span class="p">{},</span> <span class="n">old</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">nargs</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&#39;nargs&#39;</span><span class="p">)</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">nargs</span> <span class="o">!=</span> <span class="n">expr</span><span class="o">.</span><span class="n">nargs</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="n">repl_dict</span> <span class="o">=</span> <span class="n">repl_dict</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="n">repl_dict</span><span class="p">[</span><span class="bp">self</span><span class="p">]</span> <span class="o">=</span> <span class="n">expr</span>
-        <span class="k">return</span> <span class="n">repl_dict</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_number</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-</div>
-<div class="viewcode-block" id="Derivative"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.Derivative">[docs]</a><span class="k">class</span> <span class="nc">Derivative</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Carries out differentiation of the given expression with respect to symbols.</span>
-
-<span class="sd">    expr must define ._eval_derivative(symbol) method that returns</span>
-<span class="sd">    the differentiation result. This function only needs to consider the</span>
-<span class="sd">    non-trivial case where expr contains symbol and it should call the diff()</span>
-<span class="sd">    method internally (not _eval_derivative); Derivative should be the only</span>
-<span class="sd">    one to call _eval_derivative.</span>
-
-<span class="sd">    Ordering of variables:</span>
-
-<span class="sd">    If evaluate is set to True and the expression can not be evaluated, the</span>
-<span class="sd">    list of differentiation symbols will be sorted, that is, the expression is</span>
-<span class="sd">    assumed to have continuous derivatives up to the order asked. This sorting</span>
-<span class="sd">    assumes that derivatives wrt Symbols commute, derivatives wrt non-Symbols</span>
-<span class="sd">    commute, but Symbol and non-Symbol derivatives don&#39;t commute with each</span>
-<span class="sd">    other.</span>
-
-<span class="sd">    Derivative wrt non-Symbols:</span>
-
-<span class="sd">    This class also allows derivatives wrt non-Symbols that have _diff_wrt</span>
-<span class="sd">    set to True, such as Function and Derivative. When a derivative wrt a non-</span>
-<span class="sd">    Symbol is attempted, the non-Symbol is temporarily converted to a Symbol</span>
-<span class="sd">    while the differentiation is performed.</span>
-
-<span class="sd">    Note that this may seem strange, that Derivative allows things like</span>
-<span class="sd">    f(g(x)).diff(g(x)), or even f(cos(x)).diff(cos(x)).  The motivation for</span>
-<span class="sd">    allowing this syntax is to make it easier to work with variational calculus</span>
-<span class="sd">    (i.e., the Euler-Lagrange method).  The best way to understand this is that</span>
-<span class="sd">    the action of derivative with respect to a non-Symbol is defined by the</span>
-<span class="sd">    above description:  the object is substituted for a Symbol and the</span>
-<span class="sd">    derivative is taken with respect to that.  This action is only allowed for</span>
-<span class="sd">    objects for which this can be done unambiguously, for example Function and</span>
-<span class="sd">    Derivative objects.  Note that this leads to what may appear to be</span>
-<span class="sd">    mathematically inconsistent results.  For example::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import cos, sin, sqrt</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; (2*cos(x)).diff(cos(x))</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; (2*sqrt(1 - sin(x)**2)).diff(cos(x))</span>
-<span class="sd">        0</span>
-
-<span class="sd">    This appears wrong because in fact 2*cos(x) and 2*sqrt(1 - sin(x)**2) are</span>
-<span class="sd">    identically equal.  However this is the wrong way to think of this.  Think</span>
-<span class="sd">    of it instead as if we have something like this::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import c, s</span>
-<span class="sd">        &gt;&gt;&gt; def F(u):</span>
-<span class="sd">        ...     return 2*u</span>
-<span class="sd">        ...</span>
-<span class="sd">        &gt;&gt;&gt; def G(u):</span>
-<span class="sd">        ...     return 2*sqrt(1 - u**2)</span>
-<span class="sd">        ...</span>
-<span class="sd">        &gt;&gt;&gt; F(cos(x))</span>
-<span class="sd">        2*cos(x)</span>
-<span class="sd">        &gt;&gt;&gt; G(sin(x))</span>
-<span class="sd">        2*sqrt(-sin(x)**2 + 1)</span>
-<span class="sd">        &gt;&gt;&gt; F(c).diff(c)</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; F(c).diff(c)</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; G(s).diff(c)</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; G(sin(x)).diff(cos(x))</span>
-<span class="sd">        0</span>
-
-<span class="sd">    Here, the Symbols c and s act just like the functions cos(x) and sin(x),</span>
-<span class="sd">    respectively. Think of 2*cos(x) as f(c).subs(c, cos(x)) (or f(c) *at*</span>
-<span class="sd">    c = cos(x)) and 2*sqrt(1 - sin(x)**2) as g(s).subs(s, sin(x)) (or g(s) *at*</span>
-<span class="sd">    s = sin(x)), where f(u) == 2*u and g(u) == 2*sqrt(1 - u**2).  Here, we</span>
-<span class="sd">    define the function first and evaluate it at the function, but we can</span>
-<span class="sd">    actually unambiguously do this in reverse in SymPy, because</span>
-<span class="sd">    expr.subs(Function, Symbol) is well-defined:  just structurally replace the</span>
-<span class="sd">    function everywhere it appears in the expression.</span>
-
-<span class="sd">    This is actually the same notational convenience used in the Euler-Lagrange</span>
-<span class="sd">    method when one says F(t, f(t), f&#39;(t)).diff(f(t)).  What is actually meant</span>
-<span class="sd">    is that the expression in question is represented by some F(t, u, v) at</span>
-<span class="sd">    u = f(t) and v = f&#39;(t), and F(t, f(t), f&#39;(t)).diff(f(t)) simply means</span>
-<span class="sd">    F(t, u, v).diff(u) at u = f(t).</span>
-
-<span class="sd">    We do not allow to take derivative with respect to expressions where this</span>
-<span class="sd">    is not so well defined.  For example, we do not allow expr.diff(x*y)</span>
-<span class="sd">    because there are multiple ways of structurally defining where x*y appears</span>
-<span class="sd">    in an expression, some of which may surprise the reader (for example, a</span>
-<span class="sd">    very strict definition would have that (x*y*z).diff(x*y) == 0).</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">        &gt;&gt;&gt; (x*y*z).diff(x*y)</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">        ...</span>
-<span class="sd">        ValueError: Can&#39;t differentiate wrt the variable: x*y, 1</span>
-
-<span class="sd">    Note that this definition also fits in nicely with the definition of the</span>
-<span class="sd">    chain rule.  Note how the chain rule in SymPy is defined using unevaluated</span>
-<span class="sd">    Subs objects::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols, Function</span>
-<span class="sd">        &gt;&gt;&gt; f, g = symbols(&#39;f g&#39;, cls=Function)</span>
-<span class="sd">        &gt;&gt;&gt; f(2*g(x)).diff(x)</span>
-<span class="sd">        2*Derivative(g(x), x)*Subs(Derivative(f(_xi_1), _xi_1),</span>
-<span class="sd">                                              (_xi_1,), (2*g(x),))</span>
-<span class="sd">        &gt;&gt;&gt; f(g(x)).diff(x)</span>
-<span class="sd">        Derivative(g(x), x)*Subs(Derivative(f(_xi_1), _xi_1),</span>
-<span class="sd">                                            (_xi_1,), (g(x),))</span>
-
-<span class="sd">    Finally, note that, to be consistent with variational calculus, and to</span>
-<span class="sd">    ensure that the definition of substituting a Function for a Symbol in an</span>
-<span class="sd">    expression is well-defined, derivatives of functions are assumed to not be</span>
-<span class="sd">    related to the function.  In other words, we have::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import diff</span>
-<span class="sd">        &gt;&gt;&gt; diff(f(x), x).diff(f(x))</span>
-<span class="sd">        0</span>
-
-<span class="sd">    The same is actually true for derivatives of different orders::</span>
-
-<span class="sd">        &gt;&gt;&gt; diff(f(x), x, 2).diff(diff(f(x), x, 1))</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; diff(f(x), x, 1).diff(diff(f(x), x, 2))</span>
-<span class="sd">        0</span>
-
-<span class="sd">    Note, any class can allow derivatives to be taken with respect to itself.</span>
-<span class="sd">    See the docstring of Expr._diff_wrt.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Some basic examples:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Derivative, Symbol, Function</span>
-<span class="sd">        &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; g = Function(&#39;g&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; x = Symbol(&#39;x&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; y = Symbol(&#39;y&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; Derivative(x**2, x, evaluate=True)</span>
-<span class="sd">        2*x</span>
-<span class="sd">        &gt;&gt;&gt; Derivative(Derivative(f(x,y), x), y)</span>
-<span class="sd">        Derivative(f(x, y), x, y)</span>
-<span class="sd">        &gt;&gt;&gt; Derivative(f(x), x, 3)</span>
-<span class="sd">        Derivative(f(x), x, x, x)</span>
-<span class="sd">        &gt;&gt;&gt; Derivative(f(x, y), y, x, evaluate=True)</span>
-<span class="sd">        Derivative(f(x, y), x, y)</span>
-
-<span class="sd">    Now some derivatives wrt functions:</span>
-
-<span class="sd">        &gt;&gt;&gt; Derivative(f(x)**2, f(x), evaluate=True)</span>
-<span class="sd">        2*f(x)</span>
-<span class="sd">        &gt;&gt;&gt; Derivative(f(g(x)), x, evaluate=True)</span>
-<span class="sd">        Derivative(g(x), x)*Subs(Derivative(f(_xi_1), _xi_1),</span>
-<span class="sd">                                            (_xi_1,), (g(x),))</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_Derivative</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_diff_wrt</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Allow derivatives wrt Derivatives if it contains a function.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy import Function, Symbol, Derivative</span>
-<span class="sd">            &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">            &gt;&gt;&gt; x = Symbol(&#39;x&#39;)</span>
-<span class="sd">            &gt;&gt;&gt; Derivative(f(x),x)._diff_wrt</span>
-<span class="sd">            True</span>
-<span class="sd">            &gt;&gt;&gt; Derivative(x**2,x)._diff_wrt</span>
-<span class="sd">            False</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">is_Function</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="n">variables</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">):</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-        <span class="c"># There are no variables, we differentiate wrt all of the free symbols</span>
-        <span class="c"># in expr.</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">variables</span><span class="p">:</span>
-            <span class="n">variables</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">free_symbols</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">variables</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="kn">from</span> <span class="nn">sympy.utilities.misc</span> <span class="kn">import</span> <span class="n">filldedent</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&#39;&#39;&#39;</span>
-<span class="s">                    Since there is more than one variable in the</span>
-<span class="s">                    expression, the variable(s) of differentiation</span>
-<span class="s">                    must be supplied to differentiate </span><span class="si">%s</span><span class="s">&#39;&#39;&#39;</span> <span class="o">%</span> <span class="n">expr</span><span class="p">))</span>
-
-        <span class="c"># Standardize the variables by sympifying them and making appending a</span>
-        <span class="c"># count of 1 if there is only one variable: diff(e,x)-&gt;diff(e,x,1).</span>
-        <span class="n">variables</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">variables</span><span class="p">))</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">variables</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">variables</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">variables</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span>
-
-        <span class="c"># Split the list of variables into a list of the variables we are diff</span>
-        <span class="c"># wrt, where each element of the list has the form (s, count) where</span>
-        <span class="c"># s is the entity to diff wrt and count is the order of the</span>
-        <span class="c"># derivative.</span>
-        <span class="n">variable_count</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">all_zero</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">variables</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>  <span class="c"># process up to final Integer</span>
-            <span class="n">v</span><span class="p">,</span> <span class="n">count</span> <span class="o">=</span> <span class="n">variables</span><span class="p">[</span><span class="n">i</span><span class="p">:</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">2</span><span class="p">]</span>
-            <span class="n">iwas</span> <span class="o">=</span> <span class="n">i</span>
-            <span class="k">if</span> <span class="n">v</span><span class="o">.</span><span class="n">_diff_wrt</span><span class="p">:</span>
-                <span class="c"># We need to test the more specific case of count being an</span>
-                <span class="c"># Integer first.</span>
-                <span class="k">if</span> <span class="n">count</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                    <span class="n">count</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">count</span><span class="p">)</span>
-                    <span class="n">i</span> <span class="o">+=</span> <span class="mi">2</span>
-                <span class="k">elif</span> <span class="n">count</span><span class="o">.</span><span class="n">_diff_wrt</span><span class="p">:</span>
-                    <span class="n">count</span> <span class="o">=</span> <span class="mi">1</span>
-                    <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-
-            <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">iwas</span><span class="p">:</span>  <span class="c"># didn&#39;t get an update because of bad input</span>
-                <span class="kn">from</span> <span class="nn">sympy.utilities.misc</span> <span class="kn">import</span> <span class="n">filldedent</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&#39;&#39;&#39;</span>
-<span class="s">                Can</span><span class="se">\&#39;</span><span class="s">t differentiate wrt the variable: </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">&#39;&#39;&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">count</span><span class="p">)))</span>
-
-            <span class="k">if</span> <span class="n">all_zero</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">count</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">all_zero</span> <span class="o">=</span> <span class="bp">False</span>
-
-            <span class="k">if</span> <span class="n">count</span><span class="p">:</span>
-                <span class="n">variable_count</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">v</span><span class="p">,</span> <span class="n">count</span><span class="p">))</span>
-
-        <span class="c"># We make a special case for 0th derivative, because there is no</span>
-        <span class="c"># good way to unambiguously print this.</span>
-        <span class="k">if</span> <span class="n">all_zero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span>
-
-        <span class="c"># Pop evaluate because it is not really an assumption and we will need</span>
-        <span class="c"># to track use it carefully below.</span>
-        <span class="n">evaluate</span> <span class="o">=</span> <span class="n">assumptions</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;evaluate&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-
-        <span class="c"># Look for a quick exit if there are symbols that don&#39;t appear in</span>
-        <span class="c"># expression at all. Note, this cannnot check non-symbols like</span>
-        <span class="c"># functions and Derivatives as those can be created by intermediate</span>
-        <span class="c"># derivatives.</span>
-        <span class="k">if</span> <span class="n">evaluate</span><span class="p">:</span>
-            <span class="n">symbol_set</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">sc</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">sc</span> <span class="ow">in</span> <span class="n">variable_count</span> <span class="k">if</span> <span class="n">sc</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">symbol_set</span><span class="o">.</span><span class="n">difference</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-        <span class="c"># We make a generator so as to only generate a variable when necessary.</span>
-        <span class="c"># If a high order of derivative is requested and the expr becomes 0</span>
-        <span class="c"># after a few differentiations, then we won&#39;t need the other variables.</span>
-        <span class="n">variablegen</span> <span class="o">=</span> <span class="p">(</span><span class="n">v</span> <span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">count</span> <span class="ow">in</span> <span class="n">variable_count</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">count</span><span class="p">))</span>
-
-        <span class="c"># If we can&#39;t compute the derivative of expr (but we wanted to) and</span>
-        <span class="c"># expr is itself not a Derivative, finish building an unevaluated</span>
-        <span class="c"># derivative class by calling Expr.__new__.</span>
-        <span class="k">if</span> <span class="p">(</span><span class="ow">not</span> <span class="p">(</span><span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&#39;_eval_derivative&#39;</span><span class="p">)</span> <span class="ow">and</span> <span class="n">evaluate</span><span class="p">)</span> <span class="ow">and</span>
-           <span class="p">(</span><span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">))):</span>
-            <span class="n">variables</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">variablegen</span><span class="p">)</span>
-            <span class="c"># If we wanted to evaluate, we sort the variables into standard</span>
-            <span class="c"># order for later comparisons. This is too agressive if evaluate</span>
-            <span class="c"># is False, so we don&#39;t do it in that case.</span>
-            <span class="k">if</span> <span class="n">evaluate</span><span class="p">:</span>
-                <span class="c">#TODO: check if assumption of discontinuous derivatives exist</span>
-                <span class="n">variables</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_sort_variables</span><span class="p">(</span><span class="n">variables</span><span class="p">)</span>
-            <span class="c"># Here we *don&#39;t* need to reinject evaluate into assumptions</span>
-            <span class="c"># because we are done with it and it is not an assumption that</span>
-            <span class="c"># Expr knows about.</span>
-            <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="n">variables</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">obj</span>
-
-        <span class="c"># Compute the derivative now by repeatedly calling the</span>
-        <span class="c"># _eval_derivative method of expr for each variable. When this method</span>
-        <span class="c"># returns None, the derivative couldn&#39;t be computed wrt that variable</span>
-        <span class="c"># and we save the variable for later.</span>
-        <span class="n">unhandled_variables</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="c"># Once we encouter a non_symbol that is unhandled, we stop taking</span>
-        <span class="c"># derivatives entirely. This is because derivatives wrt functions</span>
-        <span class="c"># don&#39;t commute with derivatives wrt symbols and we can&#39;t safely</span>
-        <span class="c"># continue.</span>
-        <span class="n">unhandled_non_symbol</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">variablegen</span><span class="p">:</span>
-            <span class="n">is_symbol</span> <span class="o">=</span> <span class="n">v</span><span class="o">.</span><span class="n">is_Symbol</span>
-
-            <span class="k">if</span> <span class="n">unhandled_non_symbol</span><span class="p">:</span>
-                <span class="n">obj</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">is_symbol</span><span class="p">:</span>
-                    <span class="n">new_v</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;xi_</span><span class="si">%i</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">i</span><span class="p">)</span>
-                    <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">new_v</span><span class="p">)</span>
-                    <span class="n">old_v</span> <span class="o">=</span> <span class="n">v</span>
-                    <span class="n">v</span> <span class="o">=</span> <span class="n">new_v</span>
-                <span class="n">obj</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">_eval_derivative</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">is_symbol</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">obj</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="n">obj</span> <span class="o">=</span> <span class="n">obj</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">old_v</span><span class="p">)</span>
-                    <span class="n">v</span> <span class="o">=</span> <span class="n">old_v</span>
-
-            <span class="k">if</span> <span class="n">obj</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">unhandled_variables</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">is_symbol</span><span class="p">:</span>
-                    <span class="n">unhandled_non_symbol</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">elif</span> <span class="n">obj</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">expr</span> <span class="o">=</span> <span class="n">obj</span>
-
-        <span class="k">if</span> <span class="n">unhandled_variables</span><span class="p">:</span>
-            <span class="n">unhandled_variables</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_sort_variables</span><span class="p">(</span><span class="n">unhandled_variables</span><span class="p">)</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="n">unhandled_variables</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># We got a Derivative at the end of it all, and we rebuild it by</span>
-            <span class="c"># sorting its variables.</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">):</span>
-                <span class="n">expr</span> <span class="o">=</span> <span class="n">Derivative</span><span class="p">(</span>
-                    <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="o">*</span><span class="n">cls</span><span class="o">.</span><span class="n">_sort_variables</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-                <span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_sort_variables</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="nb">vars</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Sort variables, but disallow sorting of non-symbols.</span>
-
-<span class="sd">        When taking derivatives, the following rules usually hold:</span>
-
-<span class="sd">        * Derivative wrt different symbols commute.</span>
-<span class="sd">        * Derivative wrt different non-symbols commute.</span>
-<span class="sd">        * Derivatives wrt symbols and non-symbols dont&#39; commute.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        --------</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Derivative, Function, symbols</span>
-<span class="sd">        &gt;&gt;&gt; vsort = Derivative._sort_variables</span>
-<span class="sd">        &gt;&gt;&gt; x, y, z = symbols(&#39;x y z&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; f, g, h = symbols(&#39;f g h&#39;, cls=Function)</span>
-
-<span class="sd">        &gt;&gt;&gt; vsort((x,y,z))</span>
-<span class="sd">        [x, y, z]</span>
-
-<span class="sd">        &gt;&gt;&gt; vsort((h(x),g(x),f(x)))</span>
-<span class="sd">        [f(x), g(x), h(x)]</span>
-
-<span class="sd">        &gt;&gt;&gt; vsort((z,y,x,h(x),g(x),f(x)))</span>
-<span class="sd">        [x, y, z, f(x), g(x), h(x)]</span>
-
-<span class="sd">        &gt;&gt;&gt; vsort((x,f(x),y,f(y)))</span>
-<span class="sd">        [x, f(x), y, f(y)]</span>
-
-<span class="sd">        &gt;&gt;&gt; vsort((y,x,g(x),f(x),z,h(x),y,x))</span>
-<span class="sd">        [x, y, f(x), g(x), z, h(x), x, y]</span>
-
-<span class="sd">        &gt;&gt;&gt; vsort((z,y,f(x),x,f(x),g(x)))</span>
-<span class="sd">        [y, z, f(x), x, f(x), g(x)]</span>
-
-<span class="sd">        &gt;&gt;&gt; vsort((z,y,f(x),x,f(x),g(x),z,z,y,x))</span>
-<span class="sd">        [y, z, f(x), x, f(x), g(x), x, y, z, z]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">sorted_vars</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">symbol_part</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">non_symbol_part</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">vars</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">v</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">symbol_part</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">sorted_vars</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="n">symbol_part</span><span class="p">,</span>
-                                              <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">))</span>
-                    <span class="n">symbol_part</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="n">non_symbol_part</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">non_symbol_part</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">sorted_vars</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="n">non_symbol_part</span><span class="p">,</span>
-                                              <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">))</span>
-                    <span class="n">non_symbol_part</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="n">symbol_part</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">non_symbol_part</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">sorted_vars</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="n">non_symbol_part</span><span class="p">,</span>
-                                      <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">))</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">symbol_part</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">sorted_vars</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="n">symbol_part</span><span class="p">,</span>
-                                      <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">sorted_vars</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_commutative</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">is_commutative</span>
-
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">v</span><span class="p">):</span>
-        <span class="c"># If the variable s we are diff wrt is not in self.variables, we</span>
-        <span class="c"># assume that we might be able to take the derivative.</span>
-        <span class="k">if</span> <span class="n">v</span> <span class="ow">not</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">:</span>
-            <span class="n">obj</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">obj</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">Derivative</span><span class="p">(</span><span class="n">obj</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span> <span class="o">+</span> <span class="n">obj</span><span class="o">.</span><span class="n">variables</span><span class="p">))</span>
-            <span class="c"># The derivative wrt s could have simplified things such that the</span>
-            <span class="c"># derivative wrt things in self.variables can now be done. Thus,</span>
-            <span class="c"># we set evaluate=True to see if there are any other derivatives</span>
-            <span class="c"># that can be done. The most common case is when obj is a simple</span>
-            <span class="c"># number so that the derivative wrt anything else will vanish.</span>
-            <span class="k">return</span> <span class="n">Derivative</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;evaluate&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">})</span>
-        <span class="c"># In this case s was in self.variables so the derivatve wrt s has</span>
-        <span class="c"># already been attempted and was not computed, either because it</span>
-        <span class="c"># couldn&#39;t be or evaluate=False originally.</span>
-        <span class="k">return</span> <span class="n">Derivative</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span> <span class="o">+</span> <span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="p">)),</span>
-                          <span class="o">**</span><span class="p">{</span><span class="s">&#39;evaluate&#39;</span><span class="p">:</span> <span class="bp">False</span><span class="p">})</span>
-
-    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span>
-        <span class="k">if</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;deep&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-        <span class="n">hints</span><span class="p">[</span><span class="s">&#39;evaluate&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="n">Derivative</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;z0&#39;</span><span class="p">,</span> <span class="ne">NotImplementedError</span><span class="p">)</span>
-<div class="viewcode-block" id="Derivative.doit_numerically"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.Derivative.doit_numerically">[docs]</a>    <span class="k">def</span> <span class="nf">doit_numerically</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z0</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Evaluate the derivative at z numerically.</span>
-
-<span class="sd">        When we can represent derivatives at a point, this should be folded</span>
-<span class="sd">        into the normal evalf. For now, we need a special method.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">mpmath</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.expr</span> <span class="kn">import</span> <span class="n">Expr</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;partials and higher order derivatives&#39;</span><span class="p">)</span>
-        <span class="n">z</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-            <span class="n">f0</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">Expr</span><span class="o">.</span><span class="n">_from_mpmath</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">prec</span><span class="p">))</span>
-            <span class="n">f0</span> <span class="o">=</span> <span class="n">f0</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">libmpf</span><span class="o">.</span><span class="n">prec_to_dps</span><span class="p">(</span><span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">prec</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">f0</span><span class="o">.</span><span class="n">_to_mpmath</span><span class="p">(</span><span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">prec</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">_from_mpmath</span><span class="p">(</span><span class="n">mpmath</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="nb">eval</span><span class="p">,</span>
-                                             <span class="n">z0</span><span class="o">.</span><span class="n">_to_mpmath</span><span class="p">(</span><span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">prec</span><span class="p">)),</span>
-                                 <span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">prec</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">expr</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">variables</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">free_symbols</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">free_symbols</span>
-
-    <span class="k">def</span> <span class="nf">_eval_subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">old</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">variables</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">new</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">:</span>
-            <span class="c"># Issue 1620</span>
-            <span class="k">return</span> <span class="n">Subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Derivative</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">x</span><span class="o">.</span><span class="n">_subs</span><span class="p">(</span><span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_lseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">dx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-        <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">lseries</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-            <span class="k">yield</span> <span class="n">Derivative</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="o">*</span><span class="n">dx</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_nseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span>
-        <span class="n">o</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">getO</span><span class="p">()</span>
-        <span class="n">dx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="p">[</span><span class="n">Derivative</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="o">*</span><span class="n">dx</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">removeO</span><span class="p">())]</span>
-        <span class="k">if</span> <span class="n">o</span><span class="p">:</span>
-            <span class="n">rv</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">o</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">rv</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="Lambda"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.Lambda">[docs]</a><span class="k">class</span> <span class="nc">Lambda</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Lambda(x, expr) represents a lambda function similar to Python&#39;s</span>
-<span class="sd">    &#39;lambda x: expr&#39;. A function of several variables is written as</span>
-<span class="sd">    Lambda((x, y, ...), expr).</span>
-
-<span class="sd">    A simple example:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Lambda</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; f = Lambda(x, x**2)</span>
-<span class="sd">    &gt;&gt;&gt; f(4)</span>
-<span class="sd">    16</span>
-
-<span class="sd">    For multivariate functions, use:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import y, z, t</span>
-<span class="sd">    &gt;&gt;&gt; f2 = Lambda((x, y, z, t), x + y**z + t**z)</span>
-<span class="sd">    &gt;&gt;&gt; f2(1, 2, 3, 4)</span>
-<span class="sd">    73</span>
-
-<span class="sd">    A handy shortcut for lots of arguments:</span>
-
-<span class="sd">    &gt;&gt;&gt; p = x, y, z</span>
-<span class="sd">    &gt;&gt;&gt; f = Lambda(p, x + y*z)</span>
-<span class="sd">    &gt;&gt;&gt; f(*p)</span>
-<span class="sd">    x + y*z</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_Function</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">variables</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">variables</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">variables</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-            <span class="n">variables</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="n">variables</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">variables</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">variables</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">expr</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">IdentityFunction</span>
-
-        <span class="c">#use dummy variables internally, just to be sure</span>
-        <span class="n">new_variables</span> <span class="o">=</span> <span class="p">[</span><span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">name</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">variables</span><span class="p">]</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">variables</span><span class="p">,</span> <span class="n">new_variables</span><span class="p">))))</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">new_variables</span><span class="p">),</span> <span class="n">expr</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Lambda.variables"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.Lambda.variables">[docs]</a>    <span class="k">def</span> <span class="nf">variables</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The variables used in the internal representation of the function&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Lambda.expr"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.Lambda.expr">[docs]</a>    <span class="k">def</span> <span class="nf">expr</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The return value of the function&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">free_symbols</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">free_symbols</span> <span class="o">-</span> <span class="nb">set</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Lambda.nargs"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.Lambda.nargs">[docs]</a>    <span class="k">def</span> <span class="nf">nargs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The number of arguments that this function takes&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-</div>
-    <span class="k">def</span> <span class="nf">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">nargs</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">sympy.utilities.misc</span> <span class="kn">import</span> <span class="n">filldedent</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&#39;&#39;&#39;</span>
-<span class="s">                </span><span class="si">%s</span><span class="s"> takes </span><span class="si">%d</span><span class="s"> arguments (</span><span class="si">%d</span><span class="s"> given)</span>
-<span class="s">                &#39;&#39;&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">nargs</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">))))</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">,</span> <span class="n">args</span><span class="p">))))</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Lambda</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">nargs</span> <span class="o">!=</span> <span class="n">other</span><span class="o">.</span><span class="n">nargs</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="n">selfexpr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">otherexpr</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">otherexpr</span> <span class="o">=</span> <span class="n">otherexpr</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]))))</span>
-        <span class="k">return</span> <span class="n">selfexpr</span> <span class="o">==</span> <span class="n">otherexpr</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span><span class="p">(</span><span class="bp">self</span> <span class="o">==</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Lambda</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">_hashable_content</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">nargs</span><span class="p">,</span> <span class="p">)</span> <span class="o">+</span> <span class="nb">tuple</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">))</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Lambda.is_identity"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.Lambda.is_identity">[docs]</a>    <span class="k">def</span> <span class="nf">is_identity</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return ``True`` if this ``Lambda`` is an identity function. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-</div></div>
-<div class="viewcode-block" id="Subs"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.Subs">[docs]</a><span class="k">class</span> <span class="nc">Subs</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Represents unevaluated substitutions of an expression.</span>
-
-<span class="sd">    ``Subs(expr, x, x0)`` receives 3 arguments: an expression, a variable or</span>
-<span class="sd">    list of distinct variables and a point or list of evaluation points</span>
-<span class="sd">    corresponding to those variables.</span>
-
-<span class="sd">    ``Subs`` objects are generally useful to represent unevaluated derivatives</span>
-<span class="sd">    calculated at a point.</span>
-
-<span class="sd">    The variables may be expressions, but they are subjected to the limitations</span>
-<span class="sd">    of subs(), so it is usually a good practice to use only symbols for</span>
-<span class="sd">    variables, since in that case there can be no ambiguity.</span>
-
-<span class="sd">    There&#39;s no automatic expansion - use the method .doit() to effect all</span>
-<span class="sd">    possible substitutions of the object and also of objects inside the</span>
-<span class="sd">    expression.</span>
-
-<span class="sd">    When evaluating derivatives at a point that is not a symbol, a Subs object</span>
-<span class="sd">    is returned. One is also able to calculate derivatives of Subs objects - in</span>
-<span class="sd">    this case the expression is always expanded (for the unevaluated form, use</span>
-<span class="sd">    Derivative()).</span>
-
-<span class="sd">    A simple example:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Subs, Function, sin</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; e = Subs(f(x).diff(x), x, y)</span>
-<span class="sd">    &gt;&gt;&gt; e.subs(y, 0)</span>
-<span class="sd">    Subs(Derivative(f(x), x), (x,), (0,))</span>
-<span class="sd">    &gt;&gt;&gt; e.subs(f, sin).doit()</span>
-<span class="sd">    cos(y)</span>
-
-<span class="sd">    An example with several variables:</span>
-
-<span class="sd">    &gt;&gt;&gt; Subs(f(x)*sin(y) + z, (x, y), (0, 1))</span>
-<span class="sd">    Subs(z + f(x)*sin(y), (x, y), (0, 1))</span>
-<span class="sd">    &gt;&gt;&gt; _.doit()</span>
-<span class="sd">    z + f(0)*sin(1)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">variables</span><span class="p">,</span> <span class="n">point</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">variables</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">):</span>
-            <span class="n">variables</span> <span class="o">=</span> <span class="p">[</span><span class="n">variables</span><span class="p">]</span>
-        <span class="n">variables</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">variables</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="nb">list</span><span class="p">(</span><span class="n">uniq</span><span class="p">(</span><span class="n">variables</span><span class="p">))</span> <span class="o">!=</span> <span class="n">variables</span><span class="p">:</span>
-            <span class="n">repeated</span> <span class="o">=</span> <span class="p">[</span> <span class="n">v</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">set</span><span class="p">(</span><span class="n">variables</span><span class="p">)</span>
-                         <span class="k">if</span> <span class="nb">list</span><span class="p">(</span><span class="n">variables</span><span class="p">)</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="p">]</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;cannot substitute expressions </span><span class="si">%s</span><span class="s"> more than &#39;</span>
-                             <span class="s">&#39;once.&#39;</span> <span class="o">%</span> <span class="n">repeated</span><span class="p">)</span>
-
-        <span class="n">point</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">point</span> <span class="k">if</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">point</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">)</span> <span class="k">else</span> <span class="p">[</span><span class="n">point</span><span class="p">]))</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">point</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">variables</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Number of point values must be the same as &#39;</span>
-                             <span class="s">&#39;the number of variables.&#39;</span><span class="p">)</span>
-
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-        <span class="c"># use symbols with names equal to the point value (with preppended _)</span>
-        <span class="c"># to give a variable-independent expression</span>
-        <span class="n">pre</span> <span class="o">=</span> <span class="s">&quot;_&quot;</span>
-        <span class="n">pts</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">point</span><span class="p">),</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">s_pts</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">([(</span><span class="n">p</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">pre</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">p</span><span class="p">)))</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">pts</span><span class="p">])</span>
-            <span class="n">reps</span> <span class="o">=</span> <span class="p">[(</span><span class="n">v</span><span class="p">,</span> <span class="n">s_pts</span><span class="p">[</span><span class="n">p</span><span class="p">])</span>
-                <span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">variables</span><span class="p">,</span> <span class="n">point</span><span class="p">)]</span>
-            <span class="c"># if any underscore-preppended symbol is already a free symbol</span>
-            <span class="c"># and is a variable with a different point value, then there</span>
-            <span class="c"># is a clash, e.g. _0 clashes in Subs(_0 + _1, (_0, _1), (1, 0))</span>
-            <span class="c"># because the new symbol that would be created is _1 but _1</span>
-            <span class="c"># is already mapped to 0 so __0 and __1 are used for the new</span>
-            <span class="c"># symbols</span>
-            <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">r</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span>
-                   <span class="n">r</span> <span class="ow">in</span> <span class="n">variables</span> <span class="ow">and</span>
-                   <span class="n">Symbol</span><span class="p">(</span><span class="n">pre</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">point</span><span class="p">[</span><span class="n">variables</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">r</span><span class="p">)]))</span> <span class="o">!=</span> <span class="n">r</span>
-                   <span class="k">for</span> <span class="n">_</span><span class="p">,</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">reps</span><span class="p">):</span>
-                <span class="n">pre</span> <span class="o">+=</span> <span class="s">&quot;_&quot;</span>
-                <span class="k">continue</span>
-            <span class="k">break</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">variables</span><span class="p">),</span> <span class="n">point</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">reps</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_commutative</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">is_commutative</span>
-
-    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">point</span><span class="p">)))</span>
-
-    <span class="k">def</span> <span class="nf">evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">prec</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="n">evalf</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Subs.variables"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.Subs.variables">[docs]</a>    <span class="k">def</span> <span class="nf">variables</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The variables to be evaluated&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Subs.expr"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.Subs.expr">[docs]</a>    <span class="k">def</span> <span class="nf">expr</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The expression on which the substitution operates&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Subs.point"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.Subs.point">[docs]</a>    <span class="k">def</span> <span class="nf">point</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The values for which the variables are to be substituted&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">free_symbols</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">free_symbols</span> <span class="o">-</span> <span class="nb">set</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span> <span class="o">|</span>
-            <span class="nb">set</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">point</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Subs</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_expr</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">_expr</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span><span class="p">(</span><span class="bp">self</span> <span class="o">==</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Subs</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">_hashable_content</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_expr</span><span class="p">,</span> <span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">old</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">:</span>
-            <span class="n">pts</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">point</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-            <span class="n">pts</span><span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">old</span><span class="p">)]</span> <span class="o">=</span> <span class="n">new</span>
-            <span class="k">return</span> <span class="n">Subs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">,</span> <span class="n">pts</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">s</span> <span class="ow">not</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">return</span> <span class="n">Subs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">point</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span> \
-            <span class="o">+</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">Subs</span><span class="p">(</span><span class="n">point</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">point</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span> <span class="k">for</span> <span class="n">arg</span><span class="p">,</span>
-                    <span class="n">point</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">point</span><span class="p">)</span> <span class="p">])</span>
-
-</div>
-<div class="viewcode-block" id="diff"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.diff">[docs]</a><span class="k">def</span> <span class="nf">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Differentiate f with respect to symbols.</span>
-
-<span class="sd">    This is just a wrapper to unify .diff() and the Derivative class; its</span>
-<span class="sd">    interface is similar to that of integrate().  You can use the same</span>
-<span class="sd">    shortcuts for multiple variables as with Derivative.  For example,</span>
-<span class="sd">    diff(f(x), x, x, x) and diff(f(x), x, 3) both return the third derivative</span>
-<span class="sd">    of f(x).</span>
-
-<span class="sd">    You can pass evaluate=False to get an unevaluated Derivative class.  Note</span>
-<span class="sd">    that if there are 0 symbols (such as diff(f(x), x, 0), then the result will</span>
-<span class="sd">    be the function (the zeroth derivative), even if evaluate=False.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import sin, cos, Function, diff</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-
-<span class="sd">    &gt;&gt;&gt; diff(sin(x), x)</span>
-<span class="sd">    cos(x)</span>
-<span class="sd">    &gt;&gt;&gt; diff(f(x), x, x, x)</span>
-<span class="sd">    Derivative(f(x), x, x, x)</span>
-<span class="sd">    &gt;&gt;&gt; diff(f(x), x, 3)</span>
-<span class="sd">    Derivative(f(x), x, x, x)</span>
-<span class="sd">    &gt;&gt;&gt; diff(sin(x)*cos(y), x, 2, y, 2)</span>
-<span class="sd">    sin(x)*cos(y)</span>
-
-<span class="sd">    &gt;&gt;&gt; type(diff(sin(x), x))</span>
-<span class="sd">    cos</span>
-<span class="sd">    &gt;&gt;&gt; type(diff(sin(x), x, evaluate=False))</span>
-<span class="sd">    &lt;class &#39;sympy.core.function.Derivative&#39;&gt;</span>
-<span class="sd">    &gt;&gt;&gt; type(diff(sin(x), x, 0))</span>
-<span class="sd">    sin</span>
-<span class="sd">    &gt;&gt;&gt; type(diff(sin(x), x, 0, evaluate=False))</span>
-<span class="sd">    sin</span>
-
-<span class="sd">    &gt;&gt;&gt; diff(sin(x))</span>
-<span class="sd">    cos(x)</span>
-<span class="sd">    &gt;&gt;&gt; diff(sin(x*y))</span>
-<span class="sd">    Traceback (most recent call last):</span>
-<span class="sd">    ...</span>
-<span class="sd">    ValueError: specify differentiation variables to differentiate sin(x*y)</span>
-
-<span class="sd">    Note that ``diff(sin(x))`` syntax is meant only for convenience</span>
-<span class="sd">    in interactive sessions and should be avoided in library code.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    http://reference.wolfram.com/legacy/v5_2/Built-inFunctions/AlgebraicComputation/Calculus/D.html</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Derivative</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">kwargs</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="s">&#39;evaluate&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="expand"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.expand">[docs]</a><span class="k">def</span> <span class="nf">expand</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">modulus</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">power_base</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">power_exp</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span>
-        <span class="n">mul</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">log</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">multinomial</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">basic</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Expand an expression using methods given as hints.</span>
-
-<span class="sd">    Hints evaluated unless explicitly set to False are:  ``basic``, ``log``,</span>
-<span class="sd">    ``multinomial``, ``mul``, ``power_base``, and ``power_exp`` The following</span>
-<span class="sd">    hints are supported but not applied unless set to True:  ``complex``,</span>
-<span class="sd">    ``func``, and ``trig``.  In addition, the following meta-hints are</span>
-<span class="sd">    supported by some or all of the other hints:  ``frac``, ``numer``,</span>
-<span class="sd">    ``denom``, ``modulus``, and ``force``.  ``deep`` is supported by all</span>
-<span class="sd">    hints.  Additionally, subclasses of Expr may define their own hints or</span>
-<span class="sd">    meta-hints.</span>
-
-<span class="sd">    The ``basic`` hint is used for any special rewriting of an object that</span>
-<span class="sd">    should be done automatically (along with the other hints like ``mul``)</span>
-<span class="sd">    when expand is called. This is a catch-all hint to handle any sort of</span>
-<span class="sd">    expansion that may not be described by the existing hint names. To use</span>
-<span class="sd">    this hint an object should override the ``_eval_expand_basic`` method.</span>
-<span class="sd">    Objects may also define their own expand methods, which are not run by</span>
-<span class="sd">    default.  See the API section below.</span>
-
-<span class="sd">    If ``deep`` is set to ``True`` (the default), things like arguments of</span>
-<span class="sd">    functions are recursively expanded.  Use ``deep=False`` to only expand on</span>
-<span class="sd">    the top level.</span>
-
-<span class="sd">    If the ``force`` hint is used, assumptions about variables will be ignored</span>
-<span class="sd">    in making the expansion.</span>
-
-<span class="sd">    Hints</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    These hints are run by default</span>
-
-<span class="sd">    mul</span>
-<span class="sd">    ---</span>
-
-<span class="sd">    Distributes multiplication over addition:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import cos, exp, sin</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">    &gt;&gt;&gt; (y*(x + z)).expand(mul=True)</span>
-<span class="sd">    x*y + y*z</span>
-
-<span class="sd">    multinomial</span>
-<span class="sd">    -----------</span>
-
-<span class="sd">    Expand (x + y + ...)**n where n is a positive integer.</span>
-
-<span class="sd">    &gt;&gt;&gt; ((x + y + z)**2).expand(multinomial=True)</span>
-<span class="sd">    x**2 + 2*x*y + 2*x*z + y**2 + 2*y*z + z**2</span>
-
-<span class="sd">    power_exp</span>
-<span class="sd">    ---------</span>
-
-<span class="sd">    Expand addition in exponents into multiplied bases.</span>
-
-<span class="sd">    &gt;&gt;&gt; exp(x + y).expand(power_exp=True)</span>
-<span class="sd">    exp(x)*exp(y)</span>
-<span class="sd">    &gt;&gt;&gt; (2**(x + y)).expand(power_exp=True)</span>
-<span class="sd">    2**x*2**y</span>
-
-<span class="sd">    power_base</span>
-<span class="sd">    ----------</span>
-
-<span class="sd">    Split powers of multiplied bases.</span>
-
-<span class="sd">    This only happens by default if assumptions allow, or if the</span>
-<span class="sd">    ``force`` meta-hint is used:</span>
-
-<span class="sd">    &gt;&gt;&gt; ((x*y)**z).expand(power_base=True)</span>
-<span class="sd">    (x*y)**z</span>
-<span class="sd">    &gt;&gt;&gt; ((x*y)**z).expand(power_base=True, force=True)</span>
-<span class="sd">    x**z*y**z</span>
-<span class="sd">    &gt;&gt;&gt; ((2*y)**z).expand(power_base=True)</span>
-<span class="sd">    2**z*y**z</span>
-
-<span class="sd">    Note that in some cases where this expansion always holds, SymPy performs</span>
-<span class="sd">    it automatically:</span>
-
-<span class="sd">    &gt;&gt;&gt; (x*y)**2</span>
-<span class="sd">    x**2*y**2</span>
-
-<span class="sd">    log</span>
-<span class="sd">    ---</span>
-
-<span class="sd">    Pull out power of an argument as a coefficient and split logs products</span>
-<span class="sd">    into sums of logs.</span>
-
-<span class="sd">    Note that these only work if the arguments of the log function have the</span>
-<span class="sd">    proper assumptions--the arguments must be positive and the exponents must</span>
-<span class="sd">    be real--or else the ``force`` hint must be True:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import log, symbols, oo</span>
-<span class="sd">    &gt;&gt;&gt; log(x**2*y).expand(log=True)</span>
-<span class="sd">    log(x**2*y)</span>
-<span class="sd">    &gt;&gt;&gt; log(x**2*y).expand(log=True, force=True)</span>
-<span class="sd">    2*log(x) + log(y)</span>
-<span class="sd">    &gt;&gt;&gt; x, y = symbols(&#39;x,y&#39;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; log(x**2*y).expand(log=True)</span>
-<span class="sd">    2*log(x) + log(y)</span>
-
-<span class="sd">    basic</span>
-<span class="sd">    -----</span>
-
-<span class="sd">    This hint is intended primarily as a way for custom subclasses to enable</span>
-<span class="sd">    expansion by default.</span>
-
-<span class="sd">    These hints are not run by default:</span>
-
-<span class="sd">    complex</span>
-<span class="sd">    -------</span>
-
-<span class="sd">    Split an expression into real and imaginary parts.</span>
-
-<span class="sd">    &gt;&gt;&gt; x, y = symbols(&#39;x,y&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; (x + y).expand(complex=True)</span>
-<span class="sd">    re(x) + re(y) + I*im(x) + I*im(y)</span>
-<span class="sd">    &gt;&gt;&gt; cos(x).expand(complex=True)</span>
-<span class="sd">    -I*sin(re(x))*sinh(im(x)) + cos(re(x))*cosh(im(x))</span>
-
-<span class="sd">    Note that this is just a wrapper around ``as_real_imag()``.  Most objects</span>
-<span class="sd">    that wish to redefine ``_eval_expand_complex()`` should consider</span>
-<span class="sd">    redefining ``as_real_imag()`` instead.</span>
-
-<span class="sd">    func</span>
-<span class="sd">    ----</span>
-
-<span class="sd">    Expand other functions.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import gamma</span>
-<span class="sd">    &gt;&gt;&gt; gamma(x + 1).expand(func=True)</span>
-<span class="sd">    x*gamma(x)</span>
-
-<span class="sd">    trig</span>
-<span class="sd">    ----</span>
-
-<span class="sd">    Do trigonometric expansions.</span>
-
-<span class="sd">    &gt;&gt;&gt; cos(x + y).expand(trig=True)</span>
-<span class="sd">    -sin(x)*sin(y) + cos(x)*cos(y)</span>
-<span class="sd">    &gt;&gt;&gt; sin(2*x).expand(trig=True)</span>
-<span class="sd">    2*sin(x)*cos(x)</span>
-
-<span class="sd">    Note that the forms of ``sin(n*x)`` and ``cos(n*x)`` in terms of ``sin(x)``</span>
-<span class="sd">    and ``cos(x)`` are not unique, due to the identity `\sin^2(x) + \cos^2(x)</span>
-<span class="sd">    = 1`.  The current implementation uses the form obtained from Chebyshev</span>
-<span class="sd">    polynomials, but this may change.  See `this MathWorld article</span>
-<span class="sd">    &lt;http://mathworld.wolfram.com/Multiple-AngleFormulas.html&gt;`_ for more</span>
-<span class="sd">    information.</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    - You can shut off unwanted methods::</span>
-
-<span class="sd">        &gt;&gt;&gt; (exp(x + y)*(x + y)).expand()</span>
-<span class="sd">        x*exp(x)*exp(y) + y*exp(x)*exp(y)</span>
-<span class="sd">        &gt;&gt;&gt; (exp(x + y)*(x + y)).expand(power_exp=False)</span>
-<span class="sd">        x*exp(x + y) + y*exp(x + y)</span>
-<span class="sd">        &gt;&gt;&gt; (exp(x + y)*(x + y)).expand(mul=False)</span>
-<span class="sd">        (x + y)*exp(x)*exp(y)</span>
-
-<span class="sd">    - Use deep=False to only expand on the top level::</span>
-
-<span class="sd">        &gt;&gt;&gt; exp(x + exp(x + y)).expand()</span>
-<span class="sd">        exp(x)*exp(exp(x)*exp(y))</span>
-<span class="sd">        &gt;&gt;&gt; exp(x + exp(x + y)).expand(deep=False)</span>
-<span class="sd">        exp(x)*exp(exp(x + y))</span>
-
-<span class="sd">    - Hints are applied in an arbitrary, but consistent order (in the current</span>
-<span class="sd">      implementation, they are applied in alphabetical order, except</span>
-<span class="sd">      multinomial comes before mul, but this may change).  Because of this,</span>
-<span class="sd">      some hints may prevent expansion by other hints if they are applied</span>
-<span class="sd">      first. For example, ``mul`` may distribute multiplications and prevent</span>
-<span class="sd">      ``log`` and ``power_base`` from expanding them. Also, if ``mul`` is</span>
-<span class="sd">      applied before ``multinomial`, the expression might not be fully</span>
-<span class="sd">      distributed. The solution is to use the various ``expand_hint`` helper</span>
-<span class="sd">      functions or to use ``hint=False`` to this function to finely control</span>
-<span class="sd">      which hints are applied. Here are some examples::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import expand_log, expand, expand_mul, expand_power_base</span>
-<span class="sd">        &gt;&gt;&gt; x, y, z = symbols(&#39;x,y,z&#39;, positive=True)</span>
-
-<span class="sd">        &gt;&gt;&gt; expand(log(x*(y + z)))</span>
-<span class="sd">        log(x) + log(y + z)</span>
-
-<span class="sd">      Here, we see that ``log`` was applied before ``mul``.  To get the log</span>
-<span class="sd">      expanded form, either of the following will work::</span>
-
-<span class="sd">        &gt;&gt;&gt; expand_log(log(x*(y + z)))</span>
-<span class="sd">        log(x) + log(y + z)</span>
-<span class="sd">        &gt;&gt;&gt; expand(log(x*(y + z)), mul=False)</span>
-<span class="sd">        log(x) + log(y + z)</span>
-
-<span class="sd">      A similar thing can happen with the ``power_base`` hint::</span>
-
-<span class="sd">        &gt;&gt;&gt; expand((x*(y + z))**x)</span>
-<span class="sd">        (x*y + x*z)**x</span>
-
-<span class="sd">      To get the ``power_base`` expanded form, either of the following will</span>
-<span class="sd">      work::</span>
-
-<span class="sd">        &gt;&gt;&gt; expand((x*(y + z))**x, mul=False)</span>
-<span class="sd">        x**x*(y + z)**x</span>
-<span class="sd">        &gt;&gt;&gt; expand_power_base((x*(y + z))**x)</span>
-<span class="sd">        x**x*(y + z)**x</span>
-
-<span class="sd">        &gt;&gt;&gt; expand((x + y)*y/x)</span>
-<span class="sd">        y + y**2/x</span>
-
-<span class="sd">      The parts of a rational expression can be targeted::</span>
-
-<span class="sd">        &gt;&gt;&gt; expand((x + y)*y/x/(x + 1), frac=True)</span>
-<span class="sd">        (x*y + y**2)/(x**2 + x)</span>
-<span class="sd">        &gt;&gt;&gt; expand((x + y)*y/x/(x + 1), numer=True)</span>
-<span class="sd">        (x*y + y**2)/(x*(x + 1))</span>
-<span class="sd">        &gt;&gt;&gt; expand((x + y)*y/x/(x + 1), denom=True)</span>
-<span class="sd">        y*(x + y)/(x**2 + x)</span>
-
-<span class="sd">    - The ``modulus`` meta-hint can be used to reduce the coefficients of an</span>
-<span class="sd">      expression post-expansion::</span>
-
-<span class="sd">        &gt;&gt;&gt; expand((3*x + 1)**2)</span>
-<span class="sd">        9*x**2 + 6*x + 1</span>
-<span class="sd">        &gt;&gt;&gt; expand((3*x + 1)**2, modulus=5)</span>
-<span class="sd">        4*x**2 + x + 1</span>
-
-<span class="sd">    - Either ``expand()`` the function or ``.expand()`` the method can be</span>
-<span class="sd">      used.  Both are equivalent::</span>
-
-<span class="sd">        &gt;&gt;&gt; expand((x + 1)**2)</span>
-<span class="sd">        x**2 + 2*x + 1</span>
-<span class="sd">        &gt;&gt;&gt; ((x + 1)**2).expand()</span>
-<span class="sd">        x**2 + 2*x + 1</span>
-
-<span class="sd">    API</span>
-<span class="sd">    ===</span>
-
-<span class="sd">    Objects can define their own expand hints by defining</span>
-<span class="sd">    ``_eval_expand_hint()``.  The function should take the form::</span>
-
-<span class="sd">        def _eval_expand_hint(self, **hints):</span>
-<span class="sd">            # Only apply the method to the top-level expression</span>
-<span class="sd">            ...</span>
-
-<span class="sd">    See also the example below.  Objects should define ``_eval_expand_hint()``</span>
-<span class="sd">    methods only if ``hint`` applies to that specific object.  The generic</span>
-<span class="sd">    ``_eval_expand_hint()`` method defined in Expr will handle the no-op case.</span>
-
-<span class="sd">    Each hint should be responsible for expanding that hint only.</span>
-<span class="sd">    Furthermore, the expansion should be applied to the top-level expression</span>
-<span class="sd">    only.  ``expand()`` takes care of the recursion that happens when</span>
-<span class="sd">    ``deep=True``.</span>
-
-<span class="sd">    You should only call ``_eval_expand_hint()`` methods directly if you are</span>
-<span class="sd">    100% sure that the object has the method, as otherwise you are liable to</span>
-<span class="sd">    get unexpected ``AttributeError``s.  Note, again, that you do not need to</span>
-<span class="sd">    recursively apply the hint to args of your object: this is handled</span>
-<span class="sd">    automatically by ``expand()``.  ``_eval_expand_hint()`` should</span>
-<span class="sd">    generally not be used at all outside of an ``_eval_expand_hint()`` method.</span>
-<span class="sd">    If you want to apply a specific expansion from within another method, use</span>
-<span class="sd">    the public ``expand()`` function, method, or ``expand_hint()`` functions.</span>
-
-<span class="sd">    In order for expand to work, objects must be rebuildable by their args,</span>
-<span class="sd">    i.e., ``obj.func(*obj.args) == obj`` must hold.</span>
-
-<span class="sd">    Expand methods are passed ``**hints`` so that expand hints may use</span>
-<span class="sd">    &#39;metahints&#39;--hints that control how different expand methods are applied.</span>
-<span class="sd">    For example, the ``force=True`` hint described above that causes</span>
-<span class="sd">    ``expand(log=True)`` to ignore assumptions is such a metahint.  The</span>
-<span class="sd">    ``deep`` meta-hint is handled exclusively by ``expand()`` and is not</span>
-<span class="sd">    passed to ``_eval_expand_hint()`` methods.</span>
-
-<span class="sd">    Note that expansion hints should generally be methods that perform some</span>
-<span class="sd">    kind of &#39;expansion&#39;.  For hints that simply rewrite an expression, use the</span>
-<span class="sd">    .rewrite() API.</span>
-
-<span class="sd">    Example</span>
-<span class="sd">    -------</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Expr, sympify</span>
-<span class="sd">    &gt;&gt;&gt; class MyClass(Expr):</span>
-<span class="sd">    ...     def __new__(cls, *args):</span>
-<span class="sd">    ...         args = sympify(args)</span>
-<span class="sd">    ...         return Expr.__new__(cls, *args)</span>
-<span class="sd">    ...</span>
-<span class="sd">    ...     def _eval_expand_double(self, **hints):</span>
-<span class="sd">    ...         &#39;&#39;&#39;</span>
-<span class="sd">    ...         Doubles the args of MyClass.</span>
-<span class="sd">    ...</span>
-<span class="sd">    ...         If there more than four args, doubling is not performed,</span>
-<span class="sd">    ...         unless force=True is also used (False by default).</span>
-<span class="sd">    ...         &#39;&#39;&#39;</span>
-<span class="sd">    ...         force = hints.pop(&#39;force&#39;, False)</span>
-<span class="sd">    ...         if not force and len(self.args) &gt; 4:</span>
-<span class="sd">    ...             return self</span>
-<span class="sd">    ...         return self.func(*(self.args + self.args))</span>
-<span class="sd">    ...</span>
-<span class="sd">    &gt;&gt;&gt; a = MyClass(1, 2, MyClass(3, 4))</span>
-<span class="sd">    &gt;&gt;&gt; a</span>
-<span class="sd">    MyClass(1, 2, MyClass(3, 4))</span>
-<span class="sd">    &gt;&gt;&gt; a.expand(double=True)</span>
-<span class="sd">    MyClass(1, 2, MyClass(3, 4, 3, 4), 1, 2, MyClass(3, 4, 3, 4))</span>
-<span class="sd">    &gt;&gt;&gt; a.expand(double=True, deep=False)</span>
-<span class="sd">    MyClass(1, 2, MyClass(3, 4), 1, 2, MyClass(3, 4))</span>
-
-<span class="sd">    &gt;&gt;&gt; b = MyClass(1, 2, 3, 4, 5)</span>
-<span class="sd">    &gt;&gt;&gt; b.expand(double=True)</span>
-<span class="sd">    MyClass(1, 2, 3, 4, 5)</span>
-<span class="sd">    &gt;&gt;&gt; b.expand(double=True, force=True)</span>
-<span class="sd">    MyClass(1, 2, 3, 4, 5, 1, 2, 3, 4, 5)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    expand_log, expand_mul, expand_multinomial, expand_complex, expand_trig,</span>
-<span class="sd">    expand_power_base, expand_power_exp, expand_func, hyperexpand</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># don&#39;t modify this; modify the Expr.expand method</span>
-    <span class="n">hints</span><span class="p">[</span><span class="s">&#39;power_base&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">power_base</span>
-    <span class="n">hints</span><span class="p">[</span><span class="s">&#39;power_exp&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">power_exp</span>
-    <span class="n">hints</span><span class="p">[</span><span class="s">&#39;mul&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">mul</span>
-    <span class="n">hints</span><span class="p">[</span><span class="s">&#39;log&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">log</span>
-    <span class="n">hints</span><span class="p">[</span><span class="s">&#39;multinomial&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">multinomial</span>
-    <span class="n">hints</span><span class="p">[</span><span class="s">&#39;basic&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">basic</span>
-    <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="n">e</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="o">=</span><span class="n">deep</span><span class="p">,</span> <span class="n">modulus</span><span class="o">=</span><span class="n">modulus</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-<span class="c"># These are simple wrappers around single hints.</span>
-
-</div>
-<div class="viewcode-block" id="expand_mul"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.expand_mul">[docs]</a><span class="k">def</span> <span class="nf">expand_mul</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Wrapper around expand that only uses the mul hint.  See the expand</span>
-<span class="sd">    docstring for more information.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, expand_mul, exp, log</span>
-<span class="sd">    &gt;&gt;&gt; x, y = symbols(&#39;x,y&#39;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; expand_mul(exp(x+y)*(x+y)*log(x*y**2))</span>
-<span class="sd">    x*exp(x + y)*log(x*y**2) + y*exp(x + y)*log(x*y**2)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="o">=</span><span class="n">deep</span><span class="p">,</span> <span class="n">mul</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">power_exp</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span>
-    <span class="n">power_base</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">basic</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">multinomial</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">log</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="expand_multinomial"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.expand_multinomial">[docs]</a><span class="k">def</span> <span class="nf">expand_multinomial</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Wrapper around expand that only uses the multinomial hint.  See the expand</span>
-<span class="sd">    docstring for more information.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, expand_multinomial, exp</span>
-<span class="sd">    &gt;&gt;&gt; x, y = symbols(&#39;x y&#39;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; expand_multinomial((x + exp(x + 1))**2)</span>
-<span class="sd">    x**2 + 2*x*exp(x + 1) + exp(2*x + 2)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="o">=</span><span class="n">deep</span><span class="p">,</span> <span class="n">mul</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">power_exp</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span>
-    <span class="n">power_base</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">basic</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">multinomial</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">log</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="expand_log"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.expand_log">[docs]</a><span class="k">def</span> <span class="nf">expand_log</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Wrapper around expand that only uses the log hint.  See the expand</span>
-<span class="sd">    docstring for more information.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, expand_log, exp, log</span>
-<span class="sd">    &gt;&gt;&gt; x, y = symbols(&#39;x,y&#39;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; expand_log(exp(x+y)*(x+y)*log(x*y**2))</span>
-<span class="sd">    (x + y)*(log(x) + 2*log(y))*exp(x + y)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="o">=</span><span class="n">deep</span><span class="p">,</span> <span class="n">log</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">mul</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span>
-        <span class="n">power_exp</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">power_base</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">multinomial</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span>
-        <span class="n">basic</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="n">force</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="expand_func"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.expand_func">[docs]</a><span class="k">def</span> <span class="nf">expand_func</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Wrapper around expand that only uses the func hint.  See the expand</span>
-<span class="sd">    docstring for more information.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import expand_func, gamma</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; expand_func(gamma(x + 2))</span>
-<span class="sd">    x*(x + 1)*gamma(x)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="o">=</span><span class="n">deep</span><span class="p">,</span> <span class="n">func</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">basic</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span>
-    <span class="n">log</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">mul</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">power_exp</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">power_base</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">multinomial</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="expand_trig"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.expand_trig">[docs]</a><span class="k">def</span> <span class="nf">expand_trig</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Wrapper around expand that only uses the trig hint.  See the expand</span>
-<span class="sd">    docstring for more information.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import expand_trig, sin, cos</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; expand_trig(sin(x+y)*(x+y))</span>
-<span class="sd">    (x + y)*(sin(x)*cos(y) + sin(y)*cos(x))</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="o">=</span><span class="n">deep</span><span class="p">,</span> <span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">basic</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span>
-    <span class="n">log</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">mul</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">power_exp</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">power_base</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">multinomial</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="expand_complex"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.expand_complex">[docs]</a><span class="k">def</span> <span class="nf">expand_complex</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Wrapper around expand that only uses the complex hint.  See the expand</span>
-<span class="sd">    docstring for more information.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import expand_complex, exp, sqrt, I</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import z</span>
-<span class="sd">    &gt;&gt;&gt; expand_complex(exp(z))</span>
-<span class="sd">    I*exp(re(z))*sin(im(z)) + exp(re(z))*cos(im(z))</span>
-<span class="sd">    &gt;&gt;&gt; expand_complex(sqrt(I))</span>
-<span class="sd">    sqrt(2)/2 + sqrt(2)*I/2</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    Expr.as_real_imag</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="o">=</span><span class="n">deep</span><span class="p">,</span> <span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">basic</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span>
-    <span class="n">log</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">mul</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">power_exp</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">power_base</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">multinomial</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="expand_power_base"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.expand_power_base">[docs]</a><span class="k">def</span> <span class="nf">expand_power_base</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Wrapper around expand that only uses the power_base hint.</span>
-
-<span class="sd">    See the expand docstring for more information.</span>
-
-<span class="sd">    A wrapper to expand(power_base=True) which separates a power with a base</span>
-<span class="sd">    that is a Mul into a product of powers, without performing any other</span>
-<span class="sd">    expansions, provided that assumptions about the power&#39;s base and exponent</span>
-<span class="sd">    allow.</span>
-
-<span class="sd">    deep=False (default is True) will only apply to the top-level expression.</span>
-
-<span class="sd">    force=True (default is False) will cause the expansion to ignore</span>
-<span class="sd">    assumptions about the base and exponent. When False, the expansion will</span>
-<span class="sd">    only happen if the base is non-negative or the exponent is an integer.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import expand_power_base, sin, cos, exp</span>
-
-<span class="sd">    &gt;&gt;&gt; (x*y)**2</span>
-<span class="sd">    x**2*y**2</span>
-
-<span class="sd">    &gt;&gt;&gt; (2*x)**y</span>
-<span class="sd">    (2*x)**y</span>
-<span class="sd">    &gt;&gt;&gt; expand_power_base(_)</span>
-<span class="sd">    2**y*x**y</span>
-
-<span class="sd">    &gt;&gt;&gt; expand_power_base((x*y)**z)</span>
-<span class="sd">    (x*y)**z</span>
-<span class="sd">    &gt;&gt;&gt; expand_power_base((x*y)**z, force=True)</span>
-<span class="sd">    x**z*y**z</span>
-<span class="sd">    &gt;&gt;&gt; expand_power_base(sin((x*y)**z), deep=False)</span>
-<span class="sd">    sin((x*y)**z)</span>
-<span class="sd">    &gt;&gt;&gt; expand_power_base(sin((x*y)**z), force=True)</span>
-<span class="sd">    sin(x**z*y**z)</span>
-
-<span class="sd">    &gt;&gt;&gt; expand_power_base((2*sin(x))**y + (2*cos(x))**y)</span>
-<span class="sd">    2**y*sin(x)**y + 2**y*cos(x)**y</span>
-
-<span class="sd">    &gt;&gt;&gt; expand_power_base((2*exp(y))**x)</span>
-<span class="sd">    2**x*exp(y)**x</span>
-
-<span class="sd">    &gt;&gt;&gt; expand_power_base((2*cos(x))**y)</span>
-<span class="sd">    2**y*cos(x)**y</span>
-
-<span class="sd">    Notice that sums are left untouched. If this is not the desired behavior,</span>
-<span class="sd">    apply full ``expand()`` to the expression:</span>
-
-<span class="sd">    &gt;&gt;&gt; expand_power_base(((x+y)*z)**2)</span>
-<span class="sd">    z**2*(x + y)**2</span>
-<span class="sd">    &gt;&gt;&gt; (((x+y)*z)**2).expand()</span>
-<span class="sd">    x**2*z**2 + 2*x*y*z**2 + y**2*z**2</span>
-
-<span class="sd">    &gt;&gt;&gt; expand_power_base((2*y)**(1+z))</span>
-<span class="sd">    2**(z + 1)*y**(z + 1)</span>
-<span class="sd">    &gt;&gt;&gt; ((2*y)**(1+z)).expand()</span>
-<span class="sd">    2*2**z*y*y**z</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="o">=</span><span class="n">deep</span><span class="p">,</span> <span class="n">log</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">mul</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span>
-        <span class="n">power_exp</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">power_base</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">multinomial</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span>
-        <span class="n">basic</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="n">force</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="expand_power_exp"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.expand_power_exp">[docs]</a><span class="k">def</span> <span class="nf">expand_power_exp</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Wrapper around expand that only uses the power_exp hint.</span>
-
-<span class="sd">    See the expand docstring for more information.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import expand_power_exp</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; expand_power_exp(x**(y + 2))</span>
-<span class="sd">    x**2*x**y</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="o">=</span><span class="n">deep</span><span class="p">,</span> <span class="nb">complex</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">basic</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span>
-    <span class="n">log</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">mul</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">power_exp</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">power_base</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">multinomial</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="count_ops"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.count_ops">[docs]</a><span class="k">def</span> <span class="nf">count_ops</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a representation (integer or expression) of the operations in expr.</span>
-
-<span class="sd">    If ``visual`` is ``False`` (default) then the sum of the coefficients of the</span>
-<span class="sd">    visual expression will be returned.</span>
-
-<span class="sd">    If ``visual`` is ``True`` then the number of each type of operation is shown</span>
-<span class="sd">    with the core class types (or their virtual equivalent) multiplied by the</span>
-<span class="sd">    number of times they occur.</span>
-
-<span class="sd">    If expr is an iterable, the sum of the op counts of the</span>
-<span class="sd">    items will be returned.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import a, b, x, y</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import sin, count_ops</span>
-
-<span class="sd">    Although there isn&#39;t a SUB object, minus signs are interpreted as</span>
-<span class="sd">    either negations or subtractions:</span>
-
-<span class="sd">    &gt;&gt;&gt; (x - y).count_ops(visual=True)</span>
-<span class="sd">    SUB</span>
-<span class="sd">    &gt;&gt;&gt; (-x).count_ops(visual=True)</span>
-<span class="sd">    NEG</span>
-
-<span class="sd">    Here, there are two Adds and a Pow:</span>
-
-<span class="sd">    &gt;&gt;&gt; (1 + a + b**2).count_ops(visual=True)</span>
-<span class="sd">    2*ADD + POW</span>
-
-<span class="sd">    In the following, an Add, Mul, Pow and two functions:</span>
-
-<span class="sd">    &gt;&gt;&gt; (sin(x)*x + sin(x)**2).count_ops(visual=True)</span>
-<span class="sd">    ADD + MUL + POW + 2*SIN</span>
-
-<span class="sd">    for a total of 5:</span>
-
-<span class="sd">    &gt;&gt;&gt; (sin(x)*x + sin(x)**2).count_ops(visual=False)</span>
-<span class="sd">    5</span>
-
-<span class="sd">    Note that &quot;what you type&quot; is not always what you get. The expression</span>
-<span class="sd">    1/x/y is translated by sympy into 1/(x*y) so it gives a DIV and MUL rather</span>
-<span class="sd">    than two DIVs:</span>
-
-<span class="sd">    &gt;&gt;&gt; (1/x/y).count_ops(visual=True)</span>
-<span class="sd">    DIV + MUL</span>
-
-<span class="sd">    The visual option can be used to demonstrate the difference in</span>
-<span class="sd">    operations for expressions in different forms. Here, the Horner</span>
-<span class="sd">    representation is compared with the expanded form of a polynomial:</span>
-
-<span class="sd">    &gt;&gt;&gt; eq=x*(1 + x*(2 + x*(3 + x)))</span>
-<span class="sd">    &gt;&gt;&gt; count_ops(eq.expand(), visual=True) - count_ops(eq, visual=True)</span>
-<span class="sd">    -MUL + 3*POW</span>
-
-<span class="sd">    The count_ops function also handles iterables:</span>
-
-<span class="sd">    &gt;&gt;&gt; count_ops([x, sin(x), None, True, x + 2], visual=False)</span>
-<span class="sd">    2</span>
-<span class="sd">    &gt;&gt;&gt; count_ops([x, sin(x), None, True, x + 2], visual=True)</span>
-<span class="sd">    ADD + SIN</span>
-<span class="sd">    &gt;&gt;&gt; count_ops({x: sin(x), x + 2: y + 1}, visual=True)</span>
-<span class="sd">    2*ADD + SIN</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.simplify.simplify</span> <span class="kn">import</span> <span class="n">fraction</span>
-
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Expr</span><span class="p">):</span>
-
-        <span class="n">ops</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">expr</span><span class="p">]</span>
-        <span class="n">NEG</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;NEG&#39;</span><span class="p">)</span>
-        <span class="n">DIV</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;DIV&#39;</span><span class="p">)</span>
-        <span class="n">SUB</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;SUB&#39;</span><span class="p">)</span>
-        <span class="n">ADD</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;ADD&#39;</span><span class="p">)</span>
-        <span class="k">while</span> <span class="n">args</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">args</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="c">#-1/3 = NEG + DIV</span>
-                <span class="k">if</span> <span class="n">a</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">p</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">ops</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">NEG</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">q</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="n">ops</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">DIV</span><span class="p">)</span>
-                    <span class="k">continue</span>
-            <span class="k">elif</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">a</span><span class="p">):</span>
-                    <span class="n">ops</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">NEG</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-                        <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">as_two_terms</span><span class="p">()[</span><span class="mi">1</span><span class="p">]</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">a</span> <span class="o">=</span> <span class="o">-</span><span class="n">a</span>
-                <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">fraction</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                    <span class="n">ops</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">DIV</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">ops</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">NEG</span><span class="p">)</span>
-                    <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-                    <span class="k">continue</span>  <span class="c"># won&#39;t be -Mul but could be Add</span>
-                <span class="k">elif</span> <span class="n">d</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">d</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                        <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-                    <span class="n">ops</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">DIV</span><span class="p">)</span>
-                    <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-                    <span class="k">continue</span>  <span class="c"># could be -Mul</span>
-            <span class="k">elif</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="n">aargs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-                <span class="n">negs</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">ai</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">aargs</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">ai</span><span class="p">):</span>
-                        <span class="n">negs</span> <span class="o">+=</span> <span class="mi">1</span>
-                        <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">-</span><span class="n">ai</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">i</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                            <span class="n">ops</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">SUB</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ai</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">i</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                            <span class="n">ops</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ADD</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">negs</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">aargs</span><span class="p">):</span>  <span class="c"># -x - y = NEG + SUB</span>
-                    <span class="n">ops</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">NEG</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">aargs</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>  <span class="c"># -x + y = SUB, but already recorded ADD</span>
-                    <span class="n">ops</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">SUB</span> <span class="o">-</span> <span class="n">ADD</span><span class="p">)</span>
-                <span class="k">continue</span>
-            <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">a</span><span class="o">.</span><span class="n">exp</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-                <span class="n">ops</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">DIV</span><span class="p">)</span>
-                <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">base</span><span class="p">)</span>  <span class="c"># won&#39;t be -Mul but could be Add</span>
-                <span class="k">continue</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">is_Mul</span> <span class="ow">or</span>
-                <span class="n">a</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">or</span>
-                <span class="n">a</span><span class="o">.</span><span class="n">is_Function</span> <span class="ow">or</span>
-                <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">)</span> <span class="ow">or</span>
-                    <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Integral</span><span class="p">)):</span>
-
-                <span class="n">o</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">func</span><span class="o">.</span><span class="n">__name__</span><span class="o">.</span><span class="n">upper</span><span class="p">())</span>
-                <span class="c"># count the args</span>
-                <span class="k">if</span> <span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">is_Mul</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">LatticeOp</span><span class="p">)):</span>
-                    <span class="n">ops</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">o</span><span class="o">*</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">ops</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">:</span>
-                <span class="n">args</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">elif</span> <span class="nb">type</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">dict</span><span class="p">:</span>
-        <span class="n">ops</span> <span class="o">=</span> <span class="p">[</span><span class="n">count_ops</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="n">visual</span><span class="p">)</span> <span class="o">+</span>
-               <span class="n">count_ops</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="n">visual</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">items</span><span class="p">()]</span>
-    <span class="k">elif</span> <span class="n">iterable</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="n">ops</span> <span class="o">=</span> <span class="p">[</span><span class="n">count_ops</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="n">visual</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span><span class="p">]</span>
-    <span class="k">elif</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-        <span class="n">ops</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">else</span><span class="p">:</span>  <span class="c"># it&#39;s Basic not isinstance(expr, Expr):</span>
-        <span class="k">assert</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Basic</span><span class="p">)</span>
-        <span class="n">ops</span> <span class="o">=</span> <span class="p">[</span><span class="n">count_ops</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="n">visual</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">ops</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">visual</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">return</span> <span class="mi">0</span>
-
-    <span class="n">ops</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">ops</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">visual</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">ops</span>
-
-    <span class="k">if</span> <span class="n">ops</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">ops</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">int</span><span class="p">((</span><span class="n">a</span><span class="o">.</span><span class="n">args</span> <span class="ow">or</span> <span class="p">[</span><span class="mi">1</span><span class="p">])[</span><span class="mi">0</span><span class="p">])</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">ops</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="nfloat"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.function.nfloat">[docs]</a><span class="k">def</span> <span class="nf">nfloat</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">15</span><span class="p">,</span> <span class="n">exponent</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Make all Rationals in expr Floats except those in exponents</span>
-<span class="sd">    (unless the exponents flag is set to True).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.core.function import nfloat</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import cos, pi, S, sqrt</span>
-<span class="sd">    &gt;&gt;&gt; nfloat(x**4 + x/2 + cos(pi/3) + 1 + sqrt(y))</span>
-<span class="sd">    x**4 + 0.5*x + sqrt(y) + 1.5</span>
-<span class="sd">    &gt;&gt;&gt; nfloat(x**4 + sqrt(y), exponent=True)</span>
-<span class="sd">    x**4.0 + y**0.5</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Pow</span>
-    <span class="kn">from</span> <span class="nn">sympy.core.basic</span> <span class="kn">import</span> <span class="n">_aresame</span>
-
-    <span class="k">if</span> <span class="n">iterable</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="nb">str</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="p">(</span><span class="nb">dict</span><span class="p">,</span> <span class="n">Dict</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="nb">type</span><span class="p">(</span><span class="n">expr</span><span class="p">)([(</span><span class="n">k</span><span class="p">,</span> <span class="n">nfloat</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">exponent</span><span class="p">))</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span>
-                               <span class="n">expr</span><span class="o">.</span><span class="n">items</span><span class="p">()])</span>
-        <span class="k">return</span> <span class="nb">type</span><span class="p">(</span><span class="n">expr</span><span class="p">)([</span><span class="n">nfloat</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">exponent</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">expr</span><span class="p">])</span>
-    <span class="n">rv</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">rv</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">Float</span><span class="p">(</span><span class="n">rv</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">rv</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-        <span class="c"># evalf doesn&#39;t always set the precision</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">rv</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">rv</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="n">Float</span><span class="p">(</span><span class="n">rv</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">n</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">pass</span>  <span class="c"># pure_complex(rv) is likely True</span>
-        <span class="k">return</span> <span class="n">rv</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">exponent</span><span class="p">:</span>
-        <span class="n">reps</span> <span class="o">=</span> <span class="p">[(</span><span class="n">p</span><span class="p">,</span> <span class="n">Pow</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">()))</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">rv</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Pow</span><span class="p">)]</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">rv</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">reps</span><span class="p">))</span>
-    <span class="n">rv</span> <span class="o">=</span> <span class="n">rv</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">exponent</span><span class="p">:</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">rv</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="nb">dict</span><span class="p">([(</span><span class="n">d</span><span class="o">.</span><span class="n">exp</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">exp</span><span class="p">)</span> <span class="k">for</span> <span class="n">p</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">reps</span><span class="p">]))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="c"># Pow._eval_evalf special cases Integer exponents so if</span>
-        <span class="c"># exponent is suppose to be handled we have to do so here</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">rv</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="n">Transform</span><span class="p">(</span>
-            <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">Pow</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">Float</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">exp</span><span class="p">,</span> <span class="n">n</span><span class="p">)),</span>
-            <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">x</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">rv</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="n">Transform</span><span class="p">(</span>
-        <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">nfloat</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">exponent</span><span class="p">)),</span>
-        <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">Function</span><span class="p">)))</span>
-
-</div>
-<span class="kn">from</span> <span class="nn">sympy.core.symbol</span> <span class="kn">import</span> <span class="n">Dummy</span>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.core.mod</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">.function</span> <span class="kn">import</span> <span class="n">Function</span>
-<span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">Float</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">expand_mul</span>
-
-
-<div class="viewcode-block" id="Mod"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.mod.Mod">[docs]</a><span class="k">class</span> <span class="nc">Mod</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Represents a modulo operation on symbolic expressions.</span>
-
-<span class="sd">    Receives two arguments, dividend p and divisor q.</span>
-
-<span class="sd">    The convention used is the same as python&#39;s: the remainder always has the</span>
-<span class="sd">    same sign as the divisor.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; x**2 % y</span>
-<span class="sd">    Mod(x**2, y)</span>
-<span class="sd">    &gt;&gt;&gt; _.subs({x: 5, y: 6})</span>
-<span class="sd">    1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.simplify.simplify</span> <span class="kn">import</span> <span class="n">nsimplify</span>
-        <span class="k">if</span> <span class="n">q</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="nb">float</span> <span class="o">=</span> <span class="ow">not</span> <span class="n">q</span><span class="o">.</span><span class="n">is_Rational</span>
-            <span class="n">pnew</span> <span class="o">=</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">pnew</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                <span class="nb">float</span> <span class="o">=</span> <span class="nb">float</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">pnew</span><span class="o">.</span><span class="n">is_Rational</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="nb">float</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">pnew</span> <span class="o">%</span> <span class="n">q</span>
-                <span class="k">return</span> <span class="n">Float</span><span class="p">(</span><span class="n">nsimplify</span><span class="p">(</span><span class="n">pnew</span><span class="p">)</span> <span class="o">%</span> <span class="n">nsimplify</span><span class="p">(</span><span class="n">q</span><span class="p">))</span>
-            <span class="k">elif</span> <span class="n">pnew</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">and</span> <span class="n">pnew</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                <span class="n">r</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="n">pnew</span><span class="o">.</span><span class="n">as_two_terms</span><span class="p">()</span>
-                <span class="n">p</span> <span class="o">+=</span> <span class="n">Mod</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Mod</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span></div>
-</pre></div>
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-            
-  <h1>Source code for sympy.core.mul</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</span>
-<span class="kn">import</span> <span class="nn">operator</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.core.basic</span> <span class="kn">import</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">C</span>
-<span class="kn">from</span> <span class="nn">sympy.core.singleton</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="kn">from</span> <span class="nn">sympy.core.operations</span> <span class="kn">import</span> <span class="n">AssocOp</span>
-<span class="kn">from</span> <span class="nn">sympy.core.cache</span> <span class="kn">import</span> <span class="n">cacheit</span>
-<span class="kn">from</span> <span class="nn">sympy.core.logic</span> <span class="kn">import</span> <span class="n">fuzzy_not</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">cmp_to_key</span>
-<span class="kn">from</span> <span class="nn">sympy.core.expr</span> <span class="kn">import</span> <span class="n">Expr</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-<span class="c"># internal marker to indicate:</span>
-<span class="c">#   &quot;there are still non-commutative objects -- don&#39;t forget to process them&quot;</span>
-
-
-<span class="k">class</span> <span class="nc">NC_Marker</span><span class="p">:</span>
-    <span class="n">is_Order</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Mul</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Number</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Poly</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">False</span>
-
-
-<div class="viewcode-block" id="Mul"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.mul.Mul">[docs]</a><span class="k">class</span> <span class="nc">Mul</span><span class="p">(</span><span class="n">Expr</span><span class="p">,</span> <span class="n">AssocOp</span><span class="p">):</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="n">is_Mul</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="c">#identity = S.One</span>
-    <span class="c"># cyclic import, so defined in numbers.py</span>
-
-    <span class="c"># Key for sorting commutative args in canonical order</span>
-    <span class="n">_args_sortkey</span> <span class="o">=</span> <span class="n">cmp_to_key</span><span class="p">(</span><span class="n">Basic</span><span class="o">.</span><span class="n">compare</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Mul.flatten"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.mul.Mul.flatten">[docs]</a>    <span class="k">def</span> <span class="nf">flatten</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">seq</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return commutative, noncommutative and order arguments by</span>
-<span class="sd">        combining related terms.</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-<span class="sd">            * In an expression like ``a*b*c``, python process this through sympy</span>
-<span class="sd">              as ``Mul(Mul(a, b), c)``. This can have undesirable consequences.</span>
-
-<span class="sd">              -  Sometimes terms are not combined as one would like:</span>
-<span class="sd">                 {c.f. http://code.google.com/p/sympy/issues/detail?id=1497}</span>
-
-<span class="sd">                &gt;&gt;&gt; from sympy import Mul, sqrt</span>
-<span class="sd">                &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">                &gt;&gt;&gt; 2*(x + 1) # this is the 2-arg Mul behavior</span>
-<span class="sd">                2*x + 2</span>
-<span class="sd">                &gt;&gt;&gt; y*(x + 1)*2</span>
-<span class="sd">                2*y*(x + 1)</span>
-<span class="sd">                &gt;&gt;&gt; 2*(x + 1)*y # 2-arg result will be obtained first</span>
-<span class="sd">                y*(2*x + 2)</span>
-<span class="sd">                &gt;&gt;&gt; Mul(2, x + 1, y) # all 3 args simultaneously processed</span>
-<span class="sd">                2*y*(x + 1)</span>
-<span class="sd">                &gt;&gt;&gt; 2*((x + 1)*y) # parentheses can control this behavior</span>
-<span class="sd">                2*y*(x + 1)</span>
-
-<span class="sd">                Powers with compound bases may not find a single base to</span>
-<span class="sd">                combine with unless all arguments are processed at once.</span>
-<span class="sd">                Post-processing may be necessary in such cases.</span>
-<span class="sd">                {c.f. http://code.google.com/p/sympy/issues/detail?id=2629}</span>
-
-<span class="sd">                &gt;&gt;&gt; a = sqrt(x*sqrt(y))</span>
-<span class="sd">                &gt;&gt;&gt; a**3</span>
-<span class="sd">                (x*sqrt(y))**(3/2)</span>
-<span class="sd">                &gt;&gt;&gt; Mul(a,a,a)</span>
-<span class="sd">                (x*sqrt(y))**(3/2)</span>
-<span class="sd">                &gt;&gt;&gt; a*a*a</span>
-<span class="sd">                x*sqrt(y)*sqrt(x*sqrt(y))</span>
-<span class="sd">                &gt;&gt;&gt; _.subs(a.base, z).subs(z, a.base)</span>
-<span class="sd">                (x*sqrt(y))**(3/2)</span>
-
-<span class="sd">              -  If more than two terms are being multiplied then all the</span>
-<span class="sd">                 previous terms will be re-processed for each new argument.</span>
-<span class="sd">                 So if each of ``a``, ``b`` and ``c`` were :class:`Mul`</span>
-<span class="sd">                 expression, then ``a*b*c`` (or building up the product</span>
-<span class="sd">                 with ``*=``) will process all the arguments of ``a`` and</span>
-<span class="sd">                 ``b`` twice: once when ``a*b`` is computed and again when</span>
-<span class="sd">                 ``c`` is multiplied.</span>
-
-<span class="sd">                 Using ``Mul(a, b, c)`` will process all arguments once.</span>
-
-<span class="sd">            * The results of Mul are cached according to arguments, so flatten</span>
-<span class="sd">              will only be called once for ``Mul(a, b, c)``. If you can</span>
-<span class="sd">              structure a calculation so the arguments are most likely to be</span>
-<span class="sd">              repeats then this can save time in computing the answer. For</span>
-<span class="sd">              example, say you had a Mul, M, that you wished to divide by ``d[i]``</span>
-<span class="sd">              and multiply by ``n[i]`` and you suspect there are many repeats</span>
-<span class="sd">              in ``n``. It would be better to compute ``M*n[i]/d[i]`` rather</span>
-<span class="sd">              than ``M/d[i]*n[i]`` since every time n[i] is a repeat, the</span>
-<span class="sd">              product, ``M*n[i]`` will be returned without flattening -- the</span>
-<span class="sd">              cached value will be returned. If you divide by the ``d[i]``</span>
-<span class="sd">              first (and those are more unique than the ``n[i]``) then that will</span>
-<span class="sd">              create a new Mul, ``M/d[i]`` the args of which will be traversed</span>
-<span class="sd">              again when it is multiplied by ``n[i]``.</span>
-
-<span class="sd">              {c.f. http://code.google.com/p/sympy/issues/detail?id=2607}</span>
-
-<span class="sd">              This consideration is moot if the cache is turned off.</span>
-
-<span class="sd">            NB</span>
-<span class="sd">            --</span>
-<span class="sd">              The validity of the above notes depends on the implementation</span>
-<span class="sd">              details of Mul and flatten which may change at any time. Therefore,</span>
-<span class="sd">              you should only consider them when your code is highly performance</span>
-<span class="sd">              sensitive.</span>
-
-<span class="sd">              Removal of 1 from the sequence is already handled by AssocOp.__new__.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">seq</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">seq</span>
-            <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="p">,</span> <span class="n">a</span>
-            <span class="k">assert</span> <span class="ow">not</span> <span class="n">a</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="ow">and</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="n">r</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-                <span class="n">a</span> <span class="o">*=</span> <span class="n">r</span>
-                <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                    <span class="n">bargs</span><span class="p">,</span> <span class="n">nc</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-                    <span class="n">rv</span> <span class="o">=</span> <span class="n">bargs</span><span class="p">,</span> <span class="n">nc</span><span class="p">,</span> <span class="bp">None</span>
-                    <span class="k">if</span> <span class="n">a</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                        <span class="n">bargs</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-
-                <span class="k">elif</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">and</span> <span class="n">b</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">a</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                        <span class="n">rv</span> <span class="o">=</span> <span class="p">[</span><span class="n">b</span><span class="p">],</span> <span class="p">[],</span> <span class="bp">None</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">r</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">as_coeff_Add</span><span class="p">()</span>
-                        <span class="n">bargs</span> <span class="o">=</span> <span class="p">[</span><span class="n">_keep_coeff</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">bi</span><span class="p">)</span> <span class="k">for</span> <span class="n">bi</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">b</span><span class="p">)]</span>
-                        <span class="n">bargs</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="nb">hash</span><span class="p">)</span>
-                        <span class="n">ar</span> <span class="o">=</span> <span class="n">a</span><span class="o">*</span><span class="n">r</span>
-                        <span class="k">if</span> <span class="n">ar</span><span class="p">:</span>
-                            <span class="n">bargs</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">ar</span><span class="p">)</span>
-                        <span class="n">bargs</span> <span class="o">=</span> <span class="p">[</span><span class="n">Add</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">bargs</span><span class="p">)]</span>
-                        <span class="n">rv</span> <span class="o">=</span> <span class="n">bargs</span><span class="p">,</span> <span class="p">[],</span> <span class="bp">None</span>
-            <span class="k">if</span> <span class="n">rv</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">rv</span>
-
-        <span class="c"># apply associativity, separate commutative part of seq</span>
-        <span class="n">c_part</span> <span class="o">=</span> <span class="p">[]</span>         <span class="c"># out: commutative factors</span>
-        <span class="n">nc_part</span> <span class="o">=</span> <span class="p">[]</span>        <span class="c"># out: non-commutative factors</span>
-
-        <span class="n">nc_seq</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="n">coeff</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>       <span class="c"># standalone term</span>
-                            <span class="c"># e.g. 3 * ...</span>
-
-        <span class="n">iu</span> <span class="o">=</span> <span class="p">[]</span>             <span class="c"># ImaginaryUnits, I</span>
-
-        <span class="n">c_powers</span> <span class="o">=</span> <span class="p">[]</span>       <span class="c"># (base,exp)      n</span>
-                            <span class="c"># e.g. (x,n) for x</span>
-
-        <span class="n">num_exp</span> <span class="o">=</span> <span class="p">[]</span>        <span class="c"># (num-base, exp)           y</span>
-                            <span class="c"># e.g.  (3, y)  for  ... * 3  * ...</span>
-
-        <span class="n">neg1e</span> <span class="o">=</span> <span class="mi">0</span>           <span class="c"># exponent on -1 extracted from Number-based Pow</span>
-
-        <span class="n">pnum_rat</span> <span class="o">=</span> <span class="p">{}</span>       <span class="c"># (num-base, Rat-exp)          1/2</span>
-                            <span class="c"># e.g.  (3, 1/2)  for  ... * 3     * ...</span>
-
-        <span class="n">order_symbols</span> <span class="o">=</span> <span class="bp">None</span>
-
-        <span class="c"># --- PART 1 ---</span>
-        <span class="c">#</span>
-        <span class="c"># &quot;collect powers and coeff&quot;:</span>
-        <span class="c">#</span>
-        <span class="c"># o coeff</span>
-        <span class="c"># o c_powers</span>
-        <span class="c"># o num_exp</span>
-        <span class="c"># o neg1e</span>
-        <span class="c"># o pnum_rat</span>
-        <span class="c">#</span>
-        <span class="c"># NOTE: this is optimized for all-objects-are-commutative case</span>
-        <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">seq</span><span class="p">:</span>
-            <span class="c"># O(x)</span>
-            <span class="k">if</span> <span class="n">o</span><span class="o">.</span><span class="n">is_Order</span><span class="p">:</span>
-                <span class="n">o</span><span class="p">,</span> <span class="n">order_symbols</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">as_expr_variables</span><span class="p">(</span><span class="n">order_symbols</span><span class="p">)</span>
-
-            <span class="c"># Mul([...])</span>
-            <span class="k">if</span> <span class="n">o</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">o</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-                    <span class="n">seq</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">o</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>    <span class="c"># XXX zerocopy?</span>
-
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="c"># NCMul can have commutative parts as well</span>
-                    <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">o</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">q</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-                            <span class="n">seq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="n">nc_seq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-
-                    <span class="c"># append non-commutative marker, so we don&#39;t forget to</span>
-                    <span class="c"># process scheduled non-commutative objects</span>
-                    <span class="n">seq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">NC_Marker</span><span class="p">)</span>
-
-                <span class="k">continue</span>
-
-            <span class="c"># 3</span>
-            <span class="k">elif</span> <span class="n">o</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">o</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span> <span class="ow">or</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span> <span class="ow">and</span> <span class="n">o</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                    <span class="c"># we know for sure the result will be nan</span>
-                    <span class="k">return</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">],</span> <span class="p">[],</span> <span class="bp">None</span>
-                <span class="k">elif</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>  <span class="c"># it could be zoo</span>
-                    <span class="n">coeff</span> <span class="o">*=</span> <span class="n">o</span>
-                    <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                        <span class="c"># we know for sure the result will be nan</span>
-                        <span class="k">return</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">],</span> <span class="p">[],</span> <span class="bp">None</span>
-                <span class="k">continue</span>
-
-            <span class="k">elif</span> <span class="n">o</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">coeff</span><span class="p">:</span>
-                    <span class="c"># 0 * zoo = NaN</span>
-                    <span class="k">return</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">],</span> <span class="p">[],</span> <span class="bp">None</span>
-                <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">:</span>
-                    <span class="c"># zoo * zoo = zoo</span>
-                    <span class="k">return</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">],</span> <span class="p">[],</span> <span class="bp">None</span>
-                <span class="n">coeff</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span>
-                <span class="k">continue</span>
-
-            <span class="k">elif</span> <span class="n">o</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">:</span>
-                <span class="n">iu</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-                <span class="k">continue</span>
-
-            <span class="k">elif</span> <span class="n">o</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-                <span class="c">#      e</span>
-                <span class="c"># o = b</span>
-                <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-
-                <span class="c">#  y</span>
-                <span class="c"># 3</span>
-                <span class="k">if</span> <span class="n">o</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-
-                    <span class="c"># get all the factors with numeric base so they can be</span>
-                    <span class="c"># combined below, but don&#39;t combine negatives unless</span>
-                    <span class="c"># the exponent is an integer</span>
-                    <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                            <span class="n">coeff</span> <span class="o">*=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span>  <span class="c"># it is an unevaluated power</span>
-                            <span class="k">continue</span>
-                        <span class="k">elif</span> <span class="n">e</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>    <span class="c"># also a sign of an unevaluated power</span>
-                            <span class="n">seq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">))</span>
-                            <span class="k">continue</span>
-                        <span class="k">elif</span> <span class="n">b</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                            <span class="n">neg1e</span> <span class="o">+=</span> <span class="n">e</span>
-                            <span class="n">b</span> <span class="o">=</span> <span class="o">-</span><span class="n">b</span>
-                        <span class="k">if</span> <span class="n">b</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                            <span class="n">pnum_rat</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="p">[])</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-                        <span class="k">continue</span>
-                    <span class="k">elif</span> <span class="n">b</span><span class="o">.</span><span class="n">is_positive</span> <span class="ow">or</span> <span class="n">e</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-                        <span class="n">num_exp</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">))</span>
-                        <span class="k">continue</span>
-                <span class="n">c_powers</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">))</span>
-
-            <span class="c"># NON-COMMUTATIVE</span>
-            <span class="c"># TODO: Make non-commutative exponents not combine automatically</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">o</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">NC_Marker</span><span class="p">:</span>
-                    <span class="n">nc_seq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-
-                <span class="c"># process nc_seq (if any)</span>
-                <span class="k">while</span> <span class="n">nc_seq</span><span class="p">:</span>
-                    <span class="n">o</span> <span class="o">=</span> <span class="n">nc_seq</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">nc_part</span><span class="p">:</span>
-                        <span class="n">nc_part</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-                        <span class="k">continue</span>
-
-                    <span class="c">#                             b    c       b+c</span>
-                    <span class="c"># try to combine last terms: a  * a   -&gt;  a</span>
-                    <span class="n">o1</span> <span class="o">=</span> <span class="n">nc_part</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-                    <span class="n">b1</span><span class="p">,</span> <span class="n">e1</span> <span class="o">=</span> <span class="n">o1</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                    <span class="n">b2</span><span class="p">,</span> <span class="n">e2</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                    <span class="n">new_exp</span> <span class="o">=</span> <span class="n">e1</span> <span class="o">+</span> <span class="n">e2</span>
-                    <span class="c"># Only allow powers to combine if the new exponent is</span>
-                    <span class="c"># not an Add. This allow things like a**2*b**3 == a**5</span>
-                    <span class="c"># if a.is_commutative == False, but prohibits</span>
-                    <span class="c"># a**x*a**y and x**a*x**b from combining (x,y commute).</span>
-                    <span class="k">if</span> <span class="n">b1</span> <span class="o">==</span> <span class="n">b2</span> <span class="ow">and</span> <span class="p">(</span><span class="ow">not</span> <span class="n">new_exp</span><span class="o">.</span><span class="n">is_Add</span><span class="p">):</span>
-                        <span class="n">o12</span> <span class="o">=</span> <span class="n">b1</span> <span class="o">**</span> <span class="n">new_exp</span>
-
-                        <span class="c"># now o12 could be a commutative object</span>
-                        <span class="k">if</span> <span class="n">o12</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-                            <span class="n">seq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">o12</span><span class="p">)</span>
-                            <span class="k">continue</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="n">nc_seq</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">o12</span><span class="p">)</span>
-
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">nc_part</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">o1</span><span class="p">)</span>
-                        <span class="n">nc_part</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-
-        <span class="c"># handle the ImaginaryUnits</span>
-        <span class="k">if</span> <span class="n">iu</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">iu</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">c_powers</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">iu</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># a product of I&#39;s has one of 4 values; select that value</span>
-                <span class="c"># based on the length of iu:</span>
-                <span class="c"># len(iu) % 4 of (0, 1, 2, 3) has a corresponding value of</span>
-                <span class="c">#                (1, I,-1,-I)</span>
-                <span class="n">niu</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">iu</span><span class="p">)</span> <span class="o">%</span> <span class="mi">4</span>
-                <span class="k">if</span> <span class="n">niu</span> <span class="o">%</span> <span class="mi">2</span><span class="p">:</span>
-                    <span class="n">c_powers</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">))</span>
-                <span class="k">if</span> <span class="n">niu</span> <span class="ow">in</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">):</span>
-                    <span class="n">coeff</span> <span class="o">=</span> <span class="o">-</span><span class="n">coeff</span>
-
-        <span class="c"># We do want a combined exponent if it would not be an Add, such as</span>
-        <span class="c">#  y    2y     3y</span>
-        <span class="c"># x  * x   -&gt; x</span>
-        <span class="c"># We determine if two exponents have the same term by using</span>
-        <span class="c"># as_coeff_Mul.</span>
-        <span class="c">#</span>
-        <span class="c"># Unfortunately, this isn&#39;t smart enough to consider combining into</span>
-        <span class="c"># exponents that might already be adds, so things like:</span>
-        <span class="c">#  z - y    y</span>
-        <span class="c"># x      * x  will be left alone.  This is because checking every possible</span>
-        <span class="c"># combination can slow things down.</span>
-
-        <span class="c"># gather exponents of common bases...</span>
-        <span class="k">def</span> <span class="nf">_gather</span><span class="p">(</span><span class="n">c_powers</span><span class="p">):</span>
-            <span class="n">new_c_powers</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">common_b</span> <span class="o">=</span> <span class="p">{}</span>  <span class="c"># b:e</span>
-            <span class="k">for</span> <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">c_powers</span><span class="p">:</span>
-                <span class="n">co</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-                <span class="n">common_b</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="p">{})</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="n">co</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[])</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">co</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="k">for</span> <span class="n">b</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">common_b</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-                <span class="k">for</span> <span class="n">di</span><span class="p">,</span> <span class="n">li</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-                    <span class="n">d</span><span class="p">[</span><span class="n">di</span><span class="p">]</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">li</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">common_b</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-                <span class="k">for</span> <span class="n">t</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-                    <span class="n">new_c_powers</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="o">*</span><span class="n">t</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">new_c_powers</span>
-
-        <span class="c"># in c_powers</span>
-        <span class="n">c_powers</span> <span class="o">=</span> <span class="n">_gather</span><span class="p">(</span><span class="n">c_powers</span><span class="p">)</span>
-
-        <span class="c"># and in num_exp</span>
-        <span class="n">num_exp</span> <span class="o">=</span> <span class="n">_gather</span><span class="p">(</span><span class="n">num_exp</span><span class="p">)</span>
-
-        <span class="c"># --- PART 2 ---</span>
-        <span class="c">#</span>
-        <span class="c"># o process collected powers  (x**0 -&gt; 1; x**1 -&gt; x; otherwise Pow)</span>
-        <span class="c"># o combine collected powers  (2**x * 3**x -&gt; 6**x)</span>
-        <span class="c">#   with numeric base</span>
-
-        <span class="c"># ................................</span>
-        <span class="c"># now we have:</span>
-        <span class="c"># - coeff:</span>
-        <span class="c"># - c_powers:    (b, e)</span>
-        <span class="c"># - num_exp:     (2, e)</span>
-        <span class="c"># - pnum_rat:    {(1/3, [1/3, 2/3, 1/4])}</span>
-
-        <span class="c">#  0             1</span>
-        <span class="c"># x  -&gt; 1       x  -&gt; x</span>
-        <span class="k">for</span> <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">c_powers</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">e</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                    <span class="n">coeff</span> <span class="o">*=</span> <span class="n">b</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">c_part</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">e</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="n">c_part</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">))</span>
-
-        <span class="c">#  x    x     x</span>
-        <span class="c"># 2  * 3  -&gt; 6</span>
-        <span class="n">inv_exp_dict</span> <span class="o">=</span> <span class="p">{}</span>   <span class="c"># exp:Mul(num-bases)     x    x</span>
-                            <span class="c"># e.g.  x:6  for  ... * 2  * 3  * ...</span>
-        <span class="k">for</span> <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">num_exp</span><span class="p">:</span>
-            <span class="n">inv_exp_dict</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="p">[])</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">inv_exp_dict</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-            <span class="n">inv_exp_dict</span><span class="p">[</span><span class="n">e</span><span class="p">]</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">b</span><span class="p">)</span>
-        <span class="n">c_part</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span> <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">inv_exp_dict</span><span class="o">.</span><span class="n">items</span><span class="p">()</span> <span class="k">if</span> <span class="n">e</span><span class="p">])</span>
-
-        <span class="c"># b, e -&gt; e&#39; = sum(e), b</span>
-        <span class="c"># {(1/5, [1/3]), (1/2, [1/12, 1/4]} -&gt; {(1/3, [1/5, 1/2])}</span>
-        <span class="n">comb_e</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">pnum_rat</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">comb_e</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">e</span><span class="p">),</span> <span class="p">[])</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="k">del</span> <span class="n">pnum_rat</span>
-        <span class="c"># process them, reducing exponents to values less than 1</span>
-        <span class="c"># and updating coeff if necessary else adding them to</span>
-        <span class="c"># num_rat for further processing</span>
-        <span class="n">num_rat</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">comb_e</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">b</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">q</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">coeff</span> <span class="o">*=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span>
-                <span class="k">continue</span>
-            <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">p</span> <span class="o">&gt;</span> <span class="n">e</span><span class="o">.</span><span class="n">q</span><span class="p">:</span>
-                <span class="n">e_i</span><span class="p">,</span> <span class="n">ep</span> <span class="o">=</span> <span class="nb">divmod</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="n">e</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-                <span class="n">coeff</span> <span class="o">*=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e_i</span><span class="p">)</span>
-                <span class="n">e</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="n">ep</span><span class="p">,</span> <span class="n">e</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-            <span class="n">num_rat</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">))</span>
-        <span class="k">del</span> <span class="n">comb_e</span>
-
-        <span class="c"># extract gcd of bases in num_rat</span>
-        <span class="c"># 2**(1/3)*6**(1/4) -&gt; 2**(1/3+1/4)*3**(1/4)</span>
-        <span class="n">pnew</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">list</span><span class="p">)</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c"># steps through num_rat which may grow</span>
-        <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">num_rat</span><span class="p">):</span>
-            <span class="n">bi</span><span class="p">,</span> <span class="n">ei</span> <span class="o">=</span> <span class="n">num_rat</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="n">grow</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">num_rat</span><span class="p">)):</span>
-                <span class="n">bj</span><span class="p">,</span> <span class="n">ej</span> <span class="o">=</span> <span class="n">num_rat</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-                <span class="n">g</span> <span class="o">=</span> <span class="n">_rgcd</span><span class="p">(</span><span class="n">bi</span><span class="p">,</span> <span class="n">bj</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">g</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                    <span class="c"># 4**r1*6**r2 -&gt; 2**(r1+r2)  *  2**r1 *  3**r2</span>
-                    <span class="c"># this might have a gcd with something else</span>
-                    <span class="n">e</span> <span class="o">=</span> <span class="n">ei</span> <span class="o">+</span> <span class="n">ej</span>
-                    <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">q</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="n">coeff</span> <span class="o">*=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">p</span> <span class="o">&gt;</span> <span class="n">e</span><span class="o">.</span><span class="n">q</span><span class="p">:</span>
-                            <span class="n">e_i</span><span class="p">,</span> <span class="n">ep</span> <span class="o">=</span> <span class="nb">divmod</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="n">e</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>  <span class="c"># change e in place</span>
-                            <span class="n">coeff</span> <span class="o">*=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">e_i</span><span class="p">)</span>
-                            <span class="n">e</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="n">ep</span><span class="p">,</span> <span class="n">e</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-                        <span class="n">grow</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">g</span><span class="p">,</span> <span class="n">e</span><span class="p">))</span>
-                    <span class="c"># update the jth item</span>
-                    <span class="n">num_rat</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">bj</span><span class="o">/</span><span class="n">g</span><span class="p">,</span> <span class="n">ej</span><span class="p">)</span>
-                    <span class="c"># update bi that we are checking with</span>
-                    <span class="n">bi</span> <span class="o">=</span> <span class="n">bi</span><span class="o">/</span><span class="n">g</span>
-                    <span class="k">if</span> <span class="n">bi</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                        <span class="k">break</span>
-            <span class="k">if</span> <span class="n">bi</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="n">obj</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">bi</span><span class="p">,</span> <span class="n">ei</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">obj</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                    <span class="n">coeff</span> <span class="o">*=</span> <span class="n">obj</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="c"># changes like sqrt(12) -&gt; 2*sqrt(3)</span>
-                    <span class="k">for</span> <span class="n">obj</span> <span class="ow">in</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">obj</span><span class="p">):</span>
-                        <span class="k">if</span> <span class="n">obj</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                            <span class="n">coeff</span> <span class="o">*=</span> <span class="n">obj</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="k">assert</span> <span class="n">obj</span><span class="o">.</span><span class="n">is_Pow</span>
-                            <span class="n">bi</span><span class="p">,</span> <span class="n">ei</span> <span class="o">=</span> <span class="n">obj</span><span class="o">.</span><span class="n">args</span>
-                            <span class="n">pnew</span><span class="p">[</span><span class="n">ei</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">bi</span><span class="p">)</span>
-
-            <span class="n">num_rat</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">grow</span><span class="p">)</span>
-            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-
-        <span class="c"># combine bases of the new powers</span>
-        <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">pnew</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">pnew</span><span class="p">[</span><span class="n">e</span><span class="p">]</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">b</span><span class="p">)</span>
-
-        <span class="c"># see if there is a base with matching coefficient</span>
-        <span class="c"># that the -1 can be joined with</span>
-        <span class="k">if</span> <span class="n">neg1e</span><span class="p">:</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">,</span> <span class="n">neg1e</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                <span class="n">coeff</span> <span class="o">*=</span> <span class="n">p</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">c</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-                <span class="n">coeff</span> <span class="o">*=</span> <span class="n">c</span>
-                <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">p</span><span class="o">.</span><span class="n">base</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-                    <span class="n">neg1e</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">exp</span>
-                <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">pnew</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                    <span class="k">if</span> <span class="n">e</span> <span class="o">==</span> <span class="n">neg1e</span> <span class="ow">and</span> <span class="n">b</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                        <span class="n">pnew</span><span class="p">[</span><span class="n">e</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">b</span>
-                        <span class="k">break</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">c_part</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-
-        <span class="c"># add all the pnew powers</span>
-        <span class="n">c_part</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span> <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">pnew</span><span class="o">.</span><span class="n">items</span><span class="p">()])</span>
-
-        <span class="c"># oo, -oo</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">coeff</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="n">coeff</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">):</span>
-            <span class="k">def</span> <span class="nf">_handle_for_oo</span><span class="p">(</span><span class="n">c_part</span><span class="p">,</span> <span class="n">coeff_sign</span><span class="p">):</span>
-                <span class="n">new_c_part</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">c_part</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                        <span class="k">continue</span>
-                    <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                        <span class="n">coeff_sign</span> <span class="o">*=</span> <span class="o">-</span><span class="mi">1</span>
-                        <span class="k">continue</span>
-                    <span class="n">new_c_part</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">new_c_part</span><span class="p">,</span> <span class="n">coeff_sign</span>
-            <span class="n">c_part</span><span class="p">,</span> <span class="n">coeff_sign</span> <span class="o">=</span> <span class="n">_handle_for_oo</span><span class="p">(</span><span class="n">c_part</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="n">nc_part</span><span class="p">,</span> <span class="n">coeff_sign</span> <span class="o">=</span> <span class="n">_handle_for_oo</span><span class="p">(</span><span class="n">nc_part</span><span class="p">,</span> <span class="n">coeff_sign</span><span class="p">)</span>
-            <span class="n">coeff</span> <span class="o">*=</span> <span class="n">coeff_sign</span>
-
-        <span class="c"># zoo</span>
-        <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">:</span>
-            <span class="c"># zoo might be</span>
-            <span class="c">#   unbounded_real + bounded_im</span>
-            <span class="c">#   bounded_real + unbounded_im</span>
-            <span class="c">#   unbounded_real + unbounded_im</span>
-            <span class="c"># and non-zero real or imaginary will not change that status.</span>
-            <span class="n">c_part</span> <span class="o">=</span> <span class="p">[</span><span class="n">c</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">c_part</span> <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">c</span><span class="o">.</span><span class="n">is_nonzero</span> <span class="ow">and</span>
-                                                <span class="n">c</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">)]</span>
-            <span class="n">nc_part</span> <span class="o">=</span> <span class="p">[</span><span class="n">c</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">nc_part</span> <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">c</span><span class="o">.</span><span class="n">is_nonzero</span> <span class="ow">and</span>
-                                                  <span class="n">c</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">)]</span>
-
-        <span class="c"># 0</span>
-        <span class="k">elif</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-            <span class="c"># we know for sure the result will be 0</span>
-            <span class="k">return</span> <span class="p">[</span><span class="n">coeff</span><span class="p">],</span> <span class="p">[],</span> <span class="n">order_symbols</span>
-
-        <span class="c"># order commutative part canonically</span>
-        <span class="n">c_part</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">cls</span><span class="o">.</span><span class="n">_args_sortkey</span><span class="p">)</span>
-
-        <span class="c"># current code expects coeff to be always in slot-0</span>
-        <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-            <span class="n">c_part</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">coeff</span><span class="p">)</span>
-
-        <span class="c"># we are done</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">c_part</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">c_part</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">and</span> <span class="n">c_part</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="c"># 2*(1+a) -&gt; 2 + 2 * a</span>
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">c_part</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">c_part</span> <span class="o">=</span> <span class="p">[</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">coeff</span><span class="o">*</span><span class="n">f</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">c_part</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">args</span><span class="p">])]</span>
-
-        <span class="k">return</span> <span class="n">c_part</span><span class="p">,</span> <span class="n">nc_part</span><span class="p">,</span> <span class="n">order_symbols</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-
-        <span class="c"># don&#39;t break up NC terms: (A*B)**3 != A**3*B**3, it is A*B*A*B*A*B</span>
-        <span class="n">cargs</span><span class="p">,</span> <span class="n">nc</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">(</span><span class="n">split_1</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">cargs</span><span class="p">])</span> <span class="o">*</span> \
-                <span class="n">Pow</span><span class="p">(</span><span class="n">Mul</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">nc</span><span class="p">),</span> <span class="n">e</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-        <span class="n">p</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">or</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">p</span><span class="o">.</span><span class="n">_eval_expand_power_base</span><span class="p">()</span>
-
-        <span class="k">return</span> <span class="n">p</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">class_key</span><span class="p">(</span><span class="n">cls</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">cls</span><span class="o">.</span><span class="n">__name__</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="n">c</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">m</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                <span class="n">rv</span> <span class="o">=</span> <span class="o">-</span><span class="n">AssocOp</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">mnew</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">mnew</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">m</span> <span class="o">=</span> <span class="n">mnew</span>
-                <span class="n">rv</span> <span class="o">=</span> <span class="o">-</span><span class="n">m</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="n">AssocOp</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">rv</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">rv</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">rv</span>
-
-    <span class="nd">@cacheit</span>
-<div class="viewcode-block" id="Mul.as_two_terms"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.mul.Mul.as_two_terms">[docs]</a>    <span class="k">def</span> <span class="nf">as_two_terms</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return head and tail of self.</span>
-
-<span class="sd">        This is the most efficient way to get the head and tail of an</span>
-<span class="sd">        expression.</span>
-
-<span class="sd">        - if you want only the head, use self.args[0];</span>
-<span class="sd">        - if you want to process the arguments of the tail then use</span>
-<span class="sd">          self.as_coef_mul() which gives the head and a tuple containing</span>
-<span class="sd">          the arguments of the tail when treated as a Mul.</span>
-<span class="sd">        - if you want the coefficient when self is treated as an Add</span>
-<span class="sd">          then use self.as_coeff_add()[0]</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; (3*x*y).as_two_terms()</span>
-<span class="sd">        (3, x*y)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="bp">self</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">args</span>
-
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new_rawargs</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-</div>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">as_coeff_mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">deps</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">deps</span><span class="p">:</span>
-            <span class="n">l1</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">l2</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">deps</span><span class="p">):</span>
-                    <span class="n">l2</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">l1</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new_rawargs</span><span class="p">(</span><span class="o">*</span><span class="n">l1</span><span class="p">),</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">l2</span><span class="p">)</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">if</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-        <span class="k">elif</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">,</span> <span class="p">(</span><span class="o">-</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],)</span> <span class="o">+</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">args</span>
-
-<div class="viewcode-block" id="Mul.as_coeff_Mul"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.mul.Mul.as_coeff_Mul">[docs]</a>    <span class="k">def</span> <span class="nf">as_coeff_Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rational</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Efficiently extract the coefficient of a product. &quot;&quot;&quot;</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-        <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">and</span> <span class="ow">not</span> <span class="p">(</span><span class="n">rational</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">coeff</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new_rawargs</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="bp">self</span>
-</div>
-    <span class="k">def</span> <span class="nf">as_real_imag</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="n">other</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">coeff</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-                <span class="n">coeff</span> <span class="o">*=</span> <span class="n">a</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">other</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;ignore&#39;</span><span class="p">)</span> <span class="o">==</span> <span class="n">m</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">coeff</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">re</span><span class="p">(</span><span class="n">m</span><span class="p">),</span> <span class="n">coeff</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">im</span><span class="p">(</span><span class="n">m</span><span class="p">))</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_expandsums</span><span class="p">(</span><span class="n">sums</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Helper function for _eval_expand_mul.</span>
-
-<span class="sd">        sums must be a list of instances of Basic.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">L</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">sums</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">L</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">sums</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">args</span>
-        <span class="n">terms</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">left</span> <span class="o">=</span> <span class="n">Mul</span><span class="o">.</span><span class="n">_expandsums</span><span class="p">(</span><span class="n">sums</span><span class="p">[:</span><span class="n">L</span><span class="o">//</span><span class="mi">2</span><span class="p">])</span>
-        <span class="n">right</span> <span class="o">=</span> <span class="n">Mul</span><span class="o">.</span><span class="n">_expandsums</span><span class="p">(</span><span class="n">sums</span><span class="p">[</span><span class="n">L</span><span class="o">//</span><span class="mi">2</span><span class="p">:])</span>
-
-        <span class="n">terms</span> <span class="o">=</span> <span class="p">[</span><span class="n">Mul</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">left</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">right</span><span class="p">]</span>
-        <span class="n">added</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">terms</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">added</span><span class="p">)</span>  <span class="c"># it may have collapsed down to one term</span>
-
-    <span class="k">def</span> <span class="nf">_eval_expand_mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">fraction</span><span class="p">,</span> <span class="n">expand_mul</span>
-
-        <span class="c"># Handle things like 1/(x*(x + 1)), which are automatically converted</span>
-        <span class="c"># to 1/x*1/(x + 1)</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="bp">self</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">fraction</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">d</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">n</span><span class="o">/</span><span class="n">d</span><span class="o">.</span><span class="n">_eval_expand_mul</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-        <span class="n">plain</span><span class="p">,</span> <span class="n">sums</span><span class="p">,</span> <span class="n">rewrite</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[],</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">factor</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">factor</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="n">sums</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">factor</span><span class="p">)</span>
-                <span class="n">rewrite</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">factor</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-                    <span class="n">plain</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">factor</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">sums</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Basic</span><span class="p">(</span><span class="n">factor</span><span class="p">))</span>  <span class="c"># Wrapper</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">rewrite</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">plain</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">plain</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">sums</span><span class="p">:</span>
-                <span class="n">terms</span> <span class="o">=</span> <span class="n">Mul</span><span class="o">.</span><span class="n">_expandsums</span><span class="p">(</span><span class="n">sums</span><span class="p">)</span>
-                <span class="n">args</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">:</span>
-                    <span class="n">t</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="n">plain</span><span class="p">,</span> <span class="n">term</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">is_Mul</span> <span class="ow">and</span> <span class="nb">any</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">is_Add</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">t</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-                        <span class="n">t</span> <span class="o">=</span> <span class="n">t</span><span class="o">.</span><span class="n">_eval_expand_mul</span><span class="p">()</span>
-                    <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">plain</span>
-
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="n">terms</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">factors</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">terms</span><span class="p">)):</span>
-            <span class="n">t</span> <span class="o">=</span> <span class="n">terms</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">t</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="n">factors</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">terms</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">t</span><span class="p">]</span> <span class="o">+</span> <span class="n">terms</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:])))</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">factors</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_matches_simple</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">repl_dict</span><span class="p">):</span>
-        <span class="c"># handle (w*3).matches(&#39;x*5&#39;) -&gt; {w: x*5/3}</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">terms</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-        <span class="n">terms</span> <span class="o">=</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">terms</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">terms</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">newexpr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">_combine_inverse</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">coeff</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">terms</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">matches</span><span class="p">(</span><span class="n">newexpr</span><span class="p">,</span> <span class="n">repl_dict</span><span class="p">)</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">matches</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">repl_dict</span><span class="o">=</span><span class="p">{},</span> <span class="n">old</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_commutative</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AssocOp</span><span class="o">.</span><span class="n">_matches_commutative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">repl_dict</span><span class="p">,</span> <span class="n">old</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_commutative</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="n">c1</span><span class="p">,</span> <span class="n">nc1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-        <span class="n">c2</span><span class="p">,</span> <span class="n">nc2</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-        <span class="n">repl_dict</span> <span class="o">=</span> <span class="n">repl_dict</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">c1</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">c2</span><span class="p">:</span>
-                <span class="n">c2</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">c1</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">AssocOp</span><span class="p">):</span>
-                <span class="n">repl_dict</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">_matches_commutative</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">c2</span><span class="p">),</span> <span class="n">repl_dict</span><span class="p">,</span> <span class="n">old</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">repl_dict</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">matches</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">c2</span><span class="p">),</span> <span class="n">repl_dict</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">repl_dict</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">nc1</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Mul</span><span class="p">):</span>
-                <span class="n">repl_dict</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">_matches</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">nc2</span><span class="p">),</span> <span class="n">repl_dict</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">repl_dict</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">matches</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">nc2</span><span class="p">),</span> <span class="n">repl_dict</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">repl_dict</span> <span class="ow">or</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">_matches</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">repl_dict</span><span class="o">=</span><span class="p">{}):</span>
-        <span class="c"># weed out negative one prefixes</span>
-        <span class="n">sign</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_two_terms</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">a</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                <span class="n">sign</span> <span class="o">=</span> <span class="o">-</span><span class="n">sign</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># the remainder, b, is not a Mul anymore</span>
-                <span class="k">return</span> <span class="n">b</span><span class="o">.</span><span class="n">matches</span><span class="p">(</span><span class="o">-</span><span class="n">expr</span><span class="p">,</span> <span class="n">repl_dict</span><span class="p">)</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="o">-</span><span class="n">expr</span>
-            <span class="n">sign</span> <span class="o">=</span> <span class="o">-</span><span class="n">sign</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="c"># expr can only match if it matches b and a matches +/- 1</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="c"># quickly test for equality</span>
-                <span class="k">if</span> <span class="n">b</span> <span class="o">==</span> <span class="n">expr</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">a</span><span class="o">.</span><span class="n">matches</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="n">sign</span><span class="p">),</span> <span class="n">repl_dict</span><span class="p">)</span>
-                <span class="c"># do more expensive match</span>
-                <span class="n">dd</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">matches</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">repl_dict</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">dd</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-                <span class="n">dd</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">matches</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="n">sign</span><span class="p">),</span> <span class="n">dd</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">dd</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="n">d</span> <span class="o">=</span> <span class="n">repl_dict</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-
-        <span class="c"># weed out identical terms</span>
-        <span class="n">pp</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">ee</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="n">ee</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-                <span class="n">pp</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-
-        <span class="c"># only one symbol left in pattern -&gt; match the remaining expression</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">pp</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">pp</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">C</span><span class="o">.</span><span class="n">Wild</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">ee</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">d</span><span class="p">[</span><span class="n">pp</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span> <span class="o">=</span> <span class="n">sign</span> <span class="o">*</span> <span class="n">ee</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">d</span><span class="p">[</span><span class="n">pp</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span> <span class="o">=</span> <span class="n">sign</span> <span class="o">*</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">ee</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">d</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">ee</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">pp</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="k">for</span> <span class="n">p</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">pp</span><span class="p">,</span> <span class="n">ee</span><span class="p">):</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="n">d</span><span class="p">)</span><span class="o">.</span><span class="n">matches</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">d</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">return</span> <span class="n">d</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_combine_inverse</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns lhs/rhs, but treats arguments like symbols, so things like</span>
-<span class="sd">        oo/oo return 1, instead of a nan.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">lhs</span> <span class="o">==</span> <span class="n">rhs</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-        <span class="k">def</span> <span class="nf">check</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">r</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">l</span><span class="o">.</span><span class="n">is_Float</span> <span class="ow">and</span> <span class="n">r</span><span class="o">.</span><span class="n">is_comparable</span><span class="p">:</span>
-                <span class="c"># if both objects are added to 0 they will share the same &quot;normalization&quot;</span>
-                <span class="c"># and are more likely to compare the same. Since Add(foo, 0) will not allow</span>
-                <span class="c"># the 0 to pass, we use __add__ directly.</span>
-                <span class="k">return</span> <span class="n">l</span><span class="o">.</span><span class="n">__add__</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="o">==</span> <span class="n">r</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span><span class="o">.</span><span class="n">__add__</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">check</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span> <span class="ow">or</span> <span class="n">check</span><span class="p">(</span><span class="n">rhs</span><span class="p">,</span> <span class="n">lhs</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">if</span> <span class="n">lhs</span><span class="o">.</span><span class="n">is_Mul</span> <span class="ow">and</span> <span class="n">rhs</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">lhs</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-            <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">rhs</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">a</span><span class="p">:</span>
-                    <span class="n">a</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="o">-</span><span class="n">x</span> <span class="ow">in</span> <span class="n">a</span><span class="p">:</span>
-                    <span class="n">a</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">)</span>
-                    <span class="n">b</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">b</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">b</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">lhs</span><span class="o">/</span><span class="n">rhs</span>
-
-    <span class="k">def</span> <span class="nf">as_powers_dict</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">list</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-            <span class="n">d</span><span class="p">[</span><span class="n">b</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">d</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">e</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">e</span> <span class="o">=</span> <span class="n">e</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">e</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">e</span><span class="p">)</span>
-            <span class="n">d</span><span class="p">[</span><span class="n">b</span><span class="p">]</span> <span class="o">=</span> <span class="n">e</span>
-        <span class="k">return</span> <span class="n">d</span>
-
-    <span class="k">def</span> <span class="nf">as_numer_denom</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># don&#39;t use _from_args to rebuild the numerators and denominators</span>
-        <span class="c"># as the order is not guaranteed to be the same once they have</span>
-        <span class="c"># been separated from each other</span>
-        <span class="n">numers</span><span class="p">,</span> <span class="n">denoms</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">f</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">]))</span>
-        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">numers</span><span class="p">),</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">denoms</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">as_base_exp</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">e1</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="n">bases</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">nc</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">b</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-                <span class="n">nc</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="n">e1</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">e1</span> <span class="o">=</span> <span class="n">e</span>
-            <span class="k">elif</span> <span class="n">e</span> <span class="o">!=</span> <span class="n">e1</span> <span class="ow">or</span> <span class="n">nc</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="n">bases</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">bases</span><span class="p">),</span> <span class="n">e1</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_polynomial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">syms</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">_eval_is_polynomial</span><span class="p">(</span><span class="n">syms</span><span class="p">)</span> <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_rational_function</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">syms</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">_eval_is_rational_function</span><span class="p">(</span><span class="n">syms</span><span class="p">)</span> <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="n">_eval_is_bounded</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span><span class="s">&#39;is_bounded&#39;</span><span class="p">)</span>
-    <span class="n">_eval_is_integer</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span>
-        <span class="s">&#39;is_integer&#39;</span><span class="p">,</span> <span class="n">when_multiple</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span>
-    <span class="n">_eval_is_commutative</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span>
-        <span class="s">&#39;is_commutative&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_polar</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">has_polar</span> <span class="o">=</span> <span class="nb">any</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">is_polar</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">has_polar</span> <span class="ow">and</span> \
-            <span class="nb">all</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">is_polar</span> <span class="ow">or</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_positive</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="c"># I*I -&gt; R,  I*I*I -&gt; -I</span>
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">im_count</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">is_neither</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">:</span>
-                <span class="n">im_count</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">continue</span>
-            <span class="n">t_real</span> <span class="o">=</span> <span class="n">t</span><span class="o">.</span><span class="n">is_real</span>
-            <span class="k">if</span> <span class="n">t_real</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="n">t_real</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">is_neither</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">is_neither</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">is_neither</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="k">return</span> <span class="p">(</span><span class="n">im_count</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_imaginary</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">im_count</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">is_neither</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">:</span>
-                <span class="n">im_count</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">continue</span>
-            <span class="n">t_real</span> <span class="o">=</span> <span class="n">t</span><span class="o">.</span><span class="n">is_real</span>
-            <span class="k">if</span> <span class="n">t_real</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="n">t_real</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">is_neither</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">is_neither</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">is_neither</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="k">return</span> <span class="p">(</span><span class="n">im_count</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_hermitian</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">nc_count</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">im_count</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">is_neither</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">t</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-                <span class="n">nc_count</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">if</span> <span class="n">nc_count</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-            <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">is_antihermitian</span><span class="p">:</span>
-                <span class="n">im_count</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">continue</span>
-            <span class="n">t_real</span> <span class="o">=</span> <span class="n">t</span><span class="o">.</span><span class="n">is_hermitian</span>
-            <span class="k">if</span> <span class="n">t_real</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="n">t_real</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">is_neither</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">is_neither</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">is_neither</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="k">return</span> <span class="p">(</span><span class="n">im_count</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_antihermitian</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">nc_count</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">im_count</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">is_neither</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">t</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-                <span class="n">nc_count</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">if</span> <span class="n">nc_count</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-            <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">is_antihermitian</span><span class="p">:</span>
-                <span class="n">im_count</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">continue</span>
-            <span class="n">t_real</span> <span class="o">=</span> <span class="n">t</span><span class="o">.</span><span class="n">is_hermitian</span>
-            <span class="k">if</span> <span class="n">t_real</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="n">t_real</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">is_neither</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">is_neither</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">is_neither</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="k">return</span> <span class="p">(</span><span class="n">im_count</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_irrational</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">t</span><span class="o">.</span><span class="n">is_irrational</span>
-            <span class="k">if</span> <span class="n">a</span><span class="p">:</span>
-                <span class="n">others</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-                <span class="n">others</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">is_rational</span> <span class="ow">is</span> <span class="bp">True</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">others</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_zero</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">zero</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-                <span class="n">zero</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">continue</span>
-            <span class="n">bound</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">is_bounded</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">bound</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">bound</span>
-        <span class="k">if</span> <span class="n">zero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_positive</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return True if self is positive, False if not, and None if it</span>
-<span class="sd">        cannot be determined.</span>
-
-<span class="sd">        This algorithm is non-recursive and works by keeping track of the</span>
-<span class="sd">        sign which changes when a negative or nonpositive is encountered.</span>
-<span class="sd">        Whether a nonpositive or nonnegative is seen is also tracked since</span>
-<span class="sd">        the presence of these makes it impossible to return True, but</span>
-<span class="sd">        possible to return False if the end result is nonpositive. e.g.</span>
-
-<span class="sd">            pos * neg * nonpositive -&gt; pos or zero -&gt; None is returned</span>
-<span class="sd">            pos * neg * nonnegative -&gt; neg or zero -&gt; False is returned</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">sign</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">saw_NON</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="n">t</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="n">sign</span> <span class="o">=</span> <span class="o">-</span><span class="n">sign</span>
-            <span class="k">elif</span> <span class="n">t</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">elif</span> <span class="n">t</span><span class="o">.</span><span class="n">is_nonpositive</span><span class="p">:</span>
-                <span class="n">sign</span> <span class="o">=</span> <span class="o">-</span><span class="n">sign</span>
-                <span class="n">saw_NON</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">elif</span> <span class="n">t</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">:</span>
-                <span class="n">saw_NON</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span>
-        <span class="k">if</span> <span class="n">sign</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">saw_NON</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="n">sign</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_negative</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return True if self is negative, False if not, and None if it</span>
-<span class="sd">        cannot be determined.</span>
-
-<span class="sd">        This algorithm is non-recursive and works by keeping track of the</span>
-<span class="sd">        sign which changes when a negative or nonpositive is encountered.</span>
-<span class="sd">        Whether a nonpositive or nonnegative is seen is also tracked since</span>
-<span class="sd">        the presence of these makes it impossible to return True, but</span>
-<span class="sd">        possible to return False if the end result is nonnegative. e.g.</span>
-
-<span class="sd">            pos * neg * nonpositive -&gt; pos or zero -&gt; False is returned</span>
-<span class="sd">            pos * neg * nonnegative -&gt; neg or zero -&gt; None is returned</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">sign</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">saw_NON</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="n">t</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="n">sign</span> <span class="o">=</span> <span class="o">-</span><span class="n">sign</span>
-            <span class="k">elif</span> <span class="n">t</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">elif</span> <span class="n">t</span><span class="o">.</span><span class="n">is_nonpositive</span><span class="p">:</span>
-                <span class="n">sign</span> <span class="o">=</span> <span class="o">-</span><span class="n">sign</span>
-                <span class="n">saw_NON</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">elif</span> <span class="n">t</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">:</span>
-                <span class="n">saw_NON</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span>
-        <span class="k">if</span> <span class="n">sign</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span> <span class="ow">and</span> <span class="n">saw_NON</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="n">sign</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_odd</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">is_integer</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_integer</span>
-
-        <span class="k">if</span> <span class="n">is_integer</span><span class="p">:</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-                <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">is_odd</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">r</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">return</span> <span class="n">r</span>
-
-        <span class="c"># !integer -&gt; !odd</span>
-        <span class="k">elif</span> <span class="n">is_integer</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_even</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">is_integer</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_integer</span>
-
-        <span class="k">if</span> <span class="n">is_integer</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">fuzzy_not</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_eval_is_odd</span><span class="p">())</span>
-
-        <span class="k">elif</span> <span class="n">is_integer</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">_eval_subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sign</span><span class="p">,</span> <span class="n">multiplicity</span>
-        <span class="kn">from</span> <span class="nn">sympy.simplify.simplify</span> <span class="kn">import</span> <span class="n">powdenest</span><span class="p">,</span> <span class="n">fraction</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">old</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="k">if</span> <span class="n">old</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_subs</span><span class="p">(</span><span class="o">-</span><span class="n">old</span><span class="p">,</span> <span class="o">-</span><span class="n">new</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">base_exp</span><span class="p">(</span><span class="n">a</span><span class="p">):</span>
-            <span class="c"># if I and -1 are in a Mul, they get both end up with</span>
-            <span class="c"># a -1 base (see issue 3322); all we want here are the</span>
-            <span class="c"># true Pow or exp separated into base and exponent</span>
-            <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">or</span> <span class="n">a</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">a</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-            <span class="k">return</span> <span class="n">a</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-        <span class="k">def</span> <span class="nf">breakup</span><span class="p">(</span><span class="n">eq</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;break up powers of eq when treated as a Mul:</span>
-<span class="sd">                   b**(Rational*e) -&gt; b**e, Rational</span>
-<span class="sd">                commutatives come back as a dictionary {b**e: Rational}</span>
-<span class="sd">                noncommutatives come back as a list [(b**e, Rational)]</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-
-            <span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">nc</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">defaultdict</span><span class="p">(</span><span class="nb">int</span><span class="p">),</span> <span class="nb">list</span><span class="p">())</span>
-            <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">eq</span><span class="p">):</span>
-                <span class="n">a</span> <span class="o">=</span> <span class="n">powdenest</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-                <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span> <span class="o">=</span> <span class="n">base_exp</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">e</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                    <span class="p">(</span><span class="n">co</span><span class="p">,</span> <span class="n">_</span><span class="p">)</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">()</span>
-                    <span class="n">b</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="o">/</span><span class="n">co</span><span class="p">)</span>
-                    <span class="n">e</span> <span class="o">=</span> <span class="n">co</span>
-                <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-                    <span class="n">c</span><span class="p">[</span><span class="n">b</span><span class="p">]</span> <span class="o">+=</span> <span class="n">e</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">nc</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">])</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">nc</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">rejoin</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">co</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            Put rational back with exponent; in general this is not ok, but</span>
-<span class="sd">            since we took it from the exponent for analysis, it&#39;s ok to put</span>
-<span class="sd">            it back.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-
-            <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span> <span class="o">=</span> <span class="n">base_exp</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="o">*</span><span class="n">co</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">ndiv</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;if b divides a in an extractive way (like 1/4 divides 1/2</span>
-<span class="sd">            but not vice versa, and 2/5 does not divide 1/3) then return</span>
-<span class="sd">            the integer number of times it divides, else return 0.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">b</span><span class="o">.</span><span class="n">q</span> <span class="o">%</span> <span class="n">a</span><span class="o">.</span><span class="n">q</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">a</span><span class="o">.</span><span class="n">q</span> <span class="o">%</span> <span class="n">b</span><span class="o">.</span><span class="n">q</span><span class="p">:</span>
-                <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">a</span><span class="o">/</span><span class="n">b</span><span class="p">)</span>
-            <span class="k">return</span> <span class="mi">0</span>
-
-        <span class="c"># give Muls in the denominator a chance to be changed (see issue 2552)</span>
-        <span class="c"># rv will be the default return value</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">fraction</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">d</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-            <span class="n">self2</span> <span class="o">=</span> <span class="n">n</span><span class="o">.</span><span class="n">_subs</span><span class="p">(</span><span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">)</span><span class="o">/</span><span class="n">d</span><span class="o">.</span><span class="n">_subs</span><span class="p">(</span><span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">self2</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">self2</span><span class="o">.</span><span class="n">_subs</span><span class="p">(</span><span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">self2</span> <span class="o">!=</span> <span class="bp">self</span><span class="p">:</span>
-                <span class="bp">self</span> <span class="o">=</span> <span class="n">rv</span> <span class="o">=</span> <span class="n">self2</span>
-
-        <span class="c"># Now continue with regular substitution.</span>
-
-        <span class="c"># handle the leading coefficient and use it to decide if anything</span>
-        <span class="c"># should even be started; we always know where to find the Rational</span>
-        <span class="c"># so it&#39;s a quick test</span>
-
-        <span class="n">co_self</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">co_old</span> <span class="o">=</span> <span class="n">old</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">co_xmul</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">co_old</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">co_self</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="c"># if coeffs are the same there will be no updating to do</span>
-            <span class="c"># below after breakup() step; so skip (and keep co_xmul=None)</span>
-            <span class="k">if</span> <span class="n">co_old</span> <span class="o">!=</span> <span class="n">co_self</span><span class="p">:</span>
-                <span class="n">co_xmul</span> <span class="o">=</span> <span class="n">co_self</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="n">co_old</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">co_old</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">rv</span>
-
-        <span class="c"># break self and old into factors</span>
-
-        <span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">nc</span><span class="p">)</span> <span class="o">=</span> <span class="n">breakup</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="p">(</span><span class="n">old_c</span><span class="p">,</span> <span class="n">old_nc</span><span class="p">)</span> <span class="o">=</span> <span class="n">breakup</span><span class="p">(</span><span class="n">old</span><span class="p">)</span>
-
-        <span class="c"># update the coefficients if we had an extraction</span>
-        <span class="c"># e.g. if co_self were 2*(3/35*x)**2 and co_old = 3/5</span>
-        <span class="c"># then co_self in c is replaced by (3/5)**2 and co_residual</span>
-        <span class="c"># is 2*(1/7)**2</span>
-
-        <span class="k">if</span> <span class="n">co_xmul</span> <span class="ow">and</span> <span class="n">co_xmul</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="n">n_old</span><span class="p">,</span> <span class="n">d_old</span> <span class="o">=</span> <span class="n">co_old</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-            <span class="n">n_self</span><span class="p">,</span> <span class="n">d_self</span> <span class="o">=</span> <span class="n">co_self</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-
-            <span class="k">def</span> <span class="nf">_multiplicity</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">p</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-                <span class="k">return</span> <span class="n">multiplicity</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="nb">abs</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-            <span class="n">mult</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="nb">min</span><span class="p">(</span><span class="n">_multiplicity</span><span class="p">(</span><span class="n">n_old</span><span class="p">,</span> <span class="n">n_self</span><span class="p">),</span>
-                         <span class="n">_multiplicity</span><span class="p">(</span><span class="n">d_old</span><span class="p">,</span> <span class="n">d_self</span><span class="p">)))</span>
-            <span class="n">c</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">co_self</span><span class="p">)</span>
-            <span class="n">c</span><span class="p">[</span><span class="n">co_old</span><span class="p">]</span> <span class="o">=</span> <span class="n">mult</span>
-            <span class="n">co_residual</span> <span class="o">=</span> <span class="n">co_self</span><span class="o">/</span><span class="n">co_old</span><span class="o">**</span><span class="n">mult</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">co_residual</span> <span class="o">=</span> <span class="mi">1</span>
-
-        <span class="c"># do quick tests to see if we can&#39;t succeed</span>
-
-        <span class="n">ok</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">old_nc</span><span class="p">)</span> <span class="o">&gt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">nc</span><span class="p">):</span>
-            <span class="c"># more non-commutative terms</span>
-            <span class="n">ok</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">old_c</span><span class="p">)</span> <span class="o">&gt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
-            <span class="c"># more commutative terms</span>
-            <span class="n">ok</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="nb">set</span><span class="p">(</span><span class="n">i</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">old_nc</span><span class="p">)</span><span class="o">.</span><span class="n">difference</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">i</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">nc</span><span class="p">)):</span>
-            <span class="c"># unmatched non-commutative bases</span>
-            <span class="n">ok</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="nb">set</span><span class="p">(</span><span class="n">old_c</span><span class="p">)</span><span class="o">.</span><span class="n">difference</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">c</span><span class="p">)):</span>
-            <span class="c"># unmatched commutative terms</span>
-            <span class="n">ok</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="nb">any</span><span class="p">(</span><span class="n">sign</span><span class="p">(</span><span class="n">c</span><span class="p">[</span><span class="n">b</span><span class="p">])</span> <span class="o">!=</span> <span class="n">sign</span><span class="p">(</span><span class="n">old_c</span><span class="p">[</span><span class="n">b</span><span class="p">])</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">old_c</span><span class="p">):</span>
-            <span class="c"># differences in sign</span>
-            <span class="n">ok</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">ok</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">rv</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">old_c</span><span class="p">:</span>
-            <span class="n">cdid</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">rat</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">old_e</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">old_c</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-                <span class="n">c_e</span> <span class="o">=</span> <span class="n">c</span><span class="p">[</span><span class="n">b</span><span class="p">]</span>
-                <span class="n">rat</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ndiv</span><span class="p">(</span><span class="n">c_e</span><span class="p">,</span> <span class="n">old_e</span><span class="p">))</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">rat</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-                    <span class="k">return</span> <span class="n">rv</span>
-            <span class="n">cdid</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">rat</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">old_nc</span><span class="p">:</span>
-            <span class="n">ncdid</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nc</span><span class="p">)):</span>
-                <span class="n">nc</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">rejoin</span><span class="p">(</span><span class="o">*</span><span class="n">nc</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">ncdid</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c"># number of nc replacements we did</span>
-            <span class="n">take</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">old_nc</span><span class="p">)</span>  <span class="c"># how much to look at each time</span>
-            <span class="n">limit</span> <span class="o">=</span> <span class="n">cdid</span> <span class="ow">or</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>  <span class="c"># max number that we can take</span>
-            <span class="n">failed</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c"># failed terms will need subs if other terms pass</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">while</span> <span class="n">limit</span> <span class="ow">and</span> <span class="n">i</span> <span class="o">+</span> <span class="n">take</span> <span class="o">&lt;=</span> <span class="nb">len</span><span class="p">(</span><span class="n">nc</span><span class="p">):</span>
-                <span class="n">hit</span> <span class="o">=</span> <span class="bp">False</span>
-
-                <span class="c"># the bases must be equivalent in succession, and</span>
-                <span class="c"># the powers must be extractively compatible on the</span>
-                <span class="c"># first and last factor but equal inbetween.</span>
-
-                <span class="n">rat</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">take</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">nc</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="n">old_nc</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="mi">0</span><span class="p">]:</span>
-                        <span class="k">break</span>
-                    <span class="k">elif</span> <span class="n">j</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">rat</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ndiv</span><span class="p">(</span><span class="n">nc</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">][</span><span class="mi">1</span><span class="p">],</span> <span class="n">old_nc</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="mi">1</span><span class="p">]))</span>
-                    <span class="k">elif</span> <span class="n">j</span> <span class="o">==</span> <span class="n">take</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="n">rat</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ndiv</span><span class="p">(</span><span class="n">nc</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">][</span><span class="mi">1</span><span class="p">],</span> <span class="n">old_nc</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="mi">1</span><span class="p">]))</span>
-                    <span class="k">elif</span> <span class="n">nc</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">!=</span> <span class="n">old_nc</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="mi">1</span><span class="p">]:</span>
-                        <span class="k">break</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">rat</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-                    <span class="n">j</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">ndo</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">rat</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">ndo</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">take</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                            <span class="k">if</span> <span class="n">cdid</span><span class="p">:</span>
-                                <span class="n">ndo</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">cdid</span><span class="p">,</span> <span class="n">ndo</span><span class="p">)</span>
-                            <span class="n">nc</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">new</span><span class="p">,</span> <span class="n">ndo</span><span class="p">)</span><span class="o">*</span><span class="n">rejoin</span><span class="p">(</span><span class="n">nc</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span>
-                                    <span class="n">nc</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">ndo</span><span class="o">*</span><span class="n">old_nc</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="n">ndo</span> <span class="o">=</span> <span class="mi">1</span>
-
-                            <span class="c"># the left residual</span>
-
-                            <span class="n">l</span> <span class="o">=</span> <span class="n">rejoin</span><span class="p">(</span><span class="n">nc</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">nc</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">ndo</span><span class="o">*</span>
-                                    <span class="n">old_nc</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span>
-
-                            <span class="c"># eliminate all middle terms</span>
-
-                            <span class="n">mid</span> <span class="o">=</span> <span class="n">new</span>
-
-                            <span class="c"># the right residual (which may be the same as the middle if take == 2)</span>
-
-                            <span class="n">ir</span> <span class="o">=</span> <span class="n">i</span> <span class="o">+</span> <span class="n">take</span> <span class="o">-</span> <span class="mi">1</span>
-                            <span class="n">r</span> <span class="o">=</span> <span class="p">(</span><span class="n">nc</span><span class="p">[</span><span class="n">ir</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">nc</span><span class="p">[</span><span class="n">ir</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">ndo</span><span class="o">*</span>
-                                 <span class="n">old_nc</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span>
-                            <span class="k">if</span> <span class="n">r</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span>
-                                <span class="k">if</span> <span class="n">i</span> <span class="o">+</span> <span class="n">take</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">nc</span><span class="p">):</span>
-                                    <span class="n">nc</span><span class="p">[</span><span class="n">i</span><span class="p">:</span><span class="n">i</span> <span class="o">+</span> <span class="n">take</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">l</span><span class="o">*</span><span class="n">mid</span><span class="p">,</span> <span class="n">r</span><span class="p">]</span>
-                                <span class="k">else</span><span class="p">:</span>
-                                    <span class="n">r</span> <span class="o">=</span> <span class="n">rejoin</span><span class="p">(</span><span class="o">*</span><span class="n">r</span><span class="p">)</span>
-                                    <span class="n">nc</span><span class="p">[</span><span class="n">i</span><span class="p">:</span><span class="n">i</span> <span class="o">+</span> <span class="n">take</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">l</span><span class="o">*</span><span class="n">mid</span><span class="o">*</span><span class="n">r</span><span class="p">]</span>
-                            <span class="k">else</span><span class="p">:</span>
-
-                                <span class="c"># there was nothing left on the right</span>
-
-                                <span class="n">nc</span><span class="p">[</span><span class="n">i</span><span class="p">:</span><span class="n">i</span> <span class="o">+</span> <span class="n">take</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">l</span><span class="o">*</span><span class="n">mid</span><span class="p">]</span>
-
-                        <span class="n">limit</span> <span class="o">-=</span> <span class="n">ndo</span>
-                        <span class="n">ncdid</span> <span class="o">+=</span> <span class="n">ndo</span>
-                        <span class="n">hit</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">hit</span><span class="p">:</span>
-
-                    <span class="c"># do the subs on this failing factor</span>
-
-                    <span class="n">failed</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">else</span><span class="p">:</span>
-
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">ncdid</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">rv</span>
-
-                <span class="c"># although we didn&#39;t fail, certain nc terms may have</span>
-                <span class="c"># failed so we rebuild them after attempting a partial</span>
-                <span class="c"># subs on them</span>
-
-                <span class="n">failed</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">nc</span><span class="p">))))</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">failed</span><span class="p">:</span>
-                    <span class="n">nc</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">rejoin</span><span class="p">(</span><span class="o">*</span><span class="n">nc</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">)</span>
-
-        <span class="c"># rebuild the expression</span>
-
-        <span class="k">if</span> <span class="n">cdid</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">do</span> <span class="o">=</span> <span class="n">ncdid</span>
-        <span class="k">elif</span> <span class="n">ncdid</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">do</span> <span class="o">=</span> <span class="n">cdid</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">do</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">ncdid</span><span class="p">,</span> <span class="n">cdid</span><span class="p">)</span>
-
-        <span class="n">margs</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">c</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">old_c</span><span class="p">:</span>
-
-                <span class="c"># calculate the new exponent</span>
-
-                <span class="n">e</span> <span class="o">=</span> <span class="n">c</span><span class="p">[</span><span class="n">b</span><span class="p">]</span> <span class="o">-</span> <span class="n">old_c</span><span class="p">[</span><span class="n">b</span><span class="p">]</span><span class="o">*</span><span class="n">do</span>
-                <span class="n">margs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">rejoin</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">margs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">rejoin</span><span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">),</span> <span class="n">c</span><span class="p">[</span><span class="n">b</span><span class="p">]))</span>
-        <span class="k">if</span> <span class="n">cdid</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">ncdid</span><span class="p">:</span>
-
-            <span class="c"># in case we are replacing commutative with non-commutative,</span>
-            <span class="c"># we want the new term to come at the front just like the</span>
-            <span class="c"># rest of this routine</span>
-
-            <span class="n">margs</span> <span class="o">=</span> <span class="p">[</span><span class="n">Pow</span><span class="p">(</span><span class="n">new</span><span class="p">,</span> <span class="n">cdid</span><span class="p">)]</span> <span class="o">+</span> <span class="n">margs</span>
-        <span class="k">return</span> <span class="n">co_residual</span><span class="o">*</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">margs</span><span class="p">)</span><span class="o">*</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">nc</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_nseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">powsimp</span>
-        <span class="n">terms</span> <span class="o">=</span> <span class="p">[</span><span class="n">t</span><span class="o">.</span><span class="n">nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">powsimp</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">terms</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(),</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;exp&#39;</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">t</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">t</span><span class="o">.</span><span class="n">conjugate</span><span class="p">()</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_transpose</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">t</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_adjoint</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">t</span><span class="o">.</span><span class="n">adjoint</span><span class="p">()</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]])</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">s</span> <span class="o">*=</span> <span class="n">x</span><span class="o">.</span><span class="n">_sage_</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">s</span>
-
-<div class="viewcode-block" id="Mul.as_content_primitive"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.mul.Mul.as_content_primitive">[docs]</a>    <span class="k">def</span> <span class="nf">as_content_primitive</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">radical</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the tuple (R, self/R) where R is the positive Rational</span>
-<span class="sd">        extracted from self.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sqrt</span>
-<span class="sd">        &gt;&gt;&gt; (-3*sqrt(2)*(2 - 2*sqrt(2))).as_content_primitive()</span>
-<span class="sd">        (6, -sqrt(2)*(-sqrt(2) + 1))</span>
-
-<span class="sd">        See docstring of Expr.as_content_primitive for more examples.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">coef</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="n">c</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">(</span><span class="n">radical</span><span class="o">=</span><span class="n">radical</span><span class="p">)</span>
-            <span class="n">coef</span> <span class="o">*=</span> <span class="n">c</span>
-            <span class="k">if</span> <span class="n">p</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-        <span class="c"># don&#39;t use self._from_args here to reconstruct args</span>
-        <span class="c"># since there may be identical args now that should be combined</span>
-        <span class="c"># e.g. (2+2*x)*(3+3*x) should be (6, (1 + x)**2) not (6, (1+x)*(1+x))</span>
-        <span class="k">return</span> <span class="n">coef</span><span class="p">,</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Mul.as_ordered_factors"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.mul.Mul.as_ordered_factors">[docs]</a>    <span class="k">def</span> <span class="nf">as_ordered_factors</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Transform an expression into an ordered list of factors.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sin, cos</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; (2*x*y*sin(x)*cos(x)).as_ordered_factors()</span>
-<span class="sd">        [2, x, y, sin(x), cos(x)]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">cpart</span><span class="p">,</span> <span class="n">ncpart</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-        <span class="n">cpart</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">expr</span><span class="p">:</span> <span class="n">expr</span><span class="o">.</span><span class="n">sort_key</span><span class="p">(</span><span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">cpart</span> <span class="o">+</span> <span class="n">ncpart</span>
-</div>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_sorted_args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_ordered_factors</span><span class="p">()</span>
-
-</div>
-<div class="viewcode-block" id="prod"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.mul.prod">[docs]</a><span class="k">def</span> <span class="nf">prod</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">start</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return product of elements of a. Start with int 1 so if only</span>
-<span class="sd">       ints are included then an int result is returned.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import prod, S</span>
-<span class="sd">    &gt;&gt;&gt; prod(list(range(3)))</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; type(_) is int</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; prod([S(2), 3])</span>
-<span class="sd">    6</span>
-<span class="sd">    &gt;&gt;&gt; _.is_Integer</span>
-<span class="sd">    True</span>
-
-<span class="sd">    You can start the product at something other than 1:</span>
-<span class="sd">    &gt;&gt;&gt; prod([1, 2], 3)</span>
-<span class="sd">    6</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="nb">reduce</span><span class="p">(</span><span class="n">operator</span><span class="o">.</span><span class="n">mul</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">start</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_keep_coeff</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span><span class="p">,</span> <span class="n">clear</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return ``coeff*factors`` unevaluated if necessary.</span>
-
-<span class="sd">    If clear is False, do not keep the coefficient as a factor</span>
-<span class="sd">    if it can be distributed on a single factor such that one or</span>
-<span class="sd">    more terms will still have integer coefficients.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.core.mul import _keep_coeff</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import S</span>
-
-<span class="sd">    &gt;&gt;&gt; _keep_coeff(S.Half, x + 2)</span>
-<span class="sd">    (x + 2)/2</span>
-<span class="sd">    &gt;&gt;&gt; _keep_coeff(S.Half, x + 2, clear=False)</span>
-<span class="sd">    x/2 + 1</span>
-<span class="sd">    &gt;&gt;&gt; _keep_coeff(S.Half, (x + 2)*y, clear=False)</span>
-<span class="sd">    y*(x + 2)/2</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">factors</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="n">factors</span><span class="p">,</span> <span class="n">coeff</span> <span class="o">=</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">coeff</span><span class="o">*</span><span class="n">factors</span>
-    <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">factors</span>
-    <span class="k">elif</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>  <span class="c"># don&#39;t keep sign?</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">factors</span>
-    <span class="k">elif</span> <span class="n">factors</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">clear</span> <span class="ow">and</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">coeff</span><span class="o">.</span><span class="n">q</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">q</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="n">coeff</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">factors</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="n">c</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">i</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-                <span class="n">r</span> <span class="o">=</span> <span class="n">c</span><span class="o">/</span><span class="n">q</span>
-                <span class="k">if</span> <span class="n">r</span> <span class="o">==</span> <span class="nb">int</span><span class="p">(</span><span class="n">r</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">coeff</span><span class="o">*</span><span class="n">factors</span>
-        <span class="k">return</span> <span class="n">Mul</span><span class="o">.</span><span class="n">_from_args</span><span class="p">((</span><span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span><span class="p">))</span>
-    <span class="k">elif</span> <span class="n">factors</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-        <span class="n">margs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">factors</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">margs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="n">margs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*=</span> <span class="n">coeff</span>
-            <span class="k">if</span> <span class="n">margs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">margs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">margs</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">coeff</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Mul</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">margs</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">coeff</span><span class="o">*</span><span class="n">factors</span>
-
-<span class="kn">from</span> <span class="nn">.numbers</span> <span class="kn">import</span> <span class="n">Rational</span><span class="p">,</span> <span class="n">igcd</span><span class="p">,</span> <span class="n">ilcm</span><span class="p">,</span> <span class="n">Integer</span>
-<span class="k">def</span> <span class="nf">_rgcd</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="n">Integer</span><span class="p">(</span><span class="n">igcd</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">p</span><span class="p">)),</span> <span class="n">Integer</span><span class="p">(</span><span class="n">ilcm</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">q</span><span class="p">)))</span>
-
-<span class="kn">from</span> <span class="nn">.power</span> <span class="kn">import</span> <span class="n">Pow</span>
-<span class="kn">from</span> <span class="nn">.add</span> <span class="kn">import</span> <span class="n">Add</span>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.core.multidimensional</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Provides functionality for multidimensional usage of scalar-functions.</span>
-
-<span class="sd">Read the vectorize docstring for more details.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">sympy.core.decorators</span> <span class="kn">import</span> <span class="n">wraps</span>
-
-
-<span class="k">def</span> <span class="nf">apply_on_element</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">args</span><span class="p">,</span> <span class="n">kwargs</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns a structure with the same dimension as the specified argument,</span>
-<span class="sd">    where each basic element is replaced by the function f applied on it. All</span>
-<span class="sd">    other arguments stay the same.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># Get the specified argument.</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-        <span class="n">structure</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-        <span class="n">is_arg</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-        <span class="n">structure</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-        <span class="n">is_arg</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="c"># Define reduced function that is only dependend of the specified argument.</span>
-    <span class="k">def</span> <span class="nf">f_reduced</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="s">&quot;__iter__&quot;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">f_reduced</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">is_arg</span><span class="p">:</span>
-                <span class="n">args</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">kwargs</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span>
-            <span class="k">return</span> <span class="n">f</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="c"># f_reduced will call itself recursively so that in the end f is applied to</span>
-    <span class="c"># all basic elements.</span>
-    <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">f_reduced</span><span class="p">,</span> <span class="n">structure</span><span class="p">))</span>
-
-
-<span class="k">def</span> <span class="nf">iter_copy</span><span class="p">(</span><span class="n">structure</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns a copy of an iterable object (also copying all embedded iterables).</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">l</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">structure</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="s">&quot;__iter__&quot;</span><span class="p">):</span>
-            <span class="n">l</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">iter_copy</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">l</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">l</span>
-
-
-<span class="k">def</span> <span class="nf">structure_copy</span><span class="p">(</span><span class="n">structure</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns a copy of the given structure (numpy-array, list, iterable, ..).</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">structure</span><span class="p">,</span> <span class="s">&quot;copy&quot;</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">structure</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-    <span class="k">return</span> <span class="n">iter_copy</span><span class="p">(</span><span class="n">structure</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="vectorize"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.multidimensional.vectorize">[docs]</a><span class="k">class</span> <span class="nc">vectorize</span><span class="p">:</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Generalizes a function taking scalars to accept multidimensional arguments.</span>
-
-<span class="sd">    For example::</span>
-
-<span class="sd">      (1) @vectorize(0)</span>
-<span class="sd">          def sin(x):</span>
-<span class="sd">              ....</span>
-
-<span class="sd">          sin([1, x, y])</span>
-<span class="sd">          --&gt; [sin(1), sin(x), sin(y)]</span>
-
-<span class="sd">      (2) @vectorize(0,1)</span>
-<span class="sd">          def diff(f(y), y)</span>
-<span class="sd">              ....</span>
-
-<span class="sd">          diff([f(x,y,z),g(x,y,z),h(x,y,z)], [x,y,z])</span>
-<span class="sd">          --&gt; [[d/dx f, d/dy f, d/dz f], [d/dx g, d/dy g, d/dz g], [d/dx h, d/dy h, d/dz h]]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">mdargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The given numbers and strings characterize the arguments that will be</span>
-<span class="sd">        treated as data structures, where the decorated function will be applied</span>
-<span class="sd">        to every single element.</span>
-<span class="sd">        If no argument is given, everything is treated multidimensional.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">mdargs</span><span class="p">:</span>
-            <span class="k">assert</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="nb">str</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">mdargs</span> <span class="o">=</span> <span class="n">mdargs</span>
-
-    <span class="k">def</span> <span class="nf">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns a wrapper for the one-dimensional function that can handle</span>
-<span class="sd">        multidimensional arguments.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="nd">@wraps</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-        <span class="k">def</span> <span class="nf">wrapper</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-            <span class="c"># Get arguments that should be treated multidimensional</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">mdargs</span><span class="p">:</span>
-                <span class="n">mdargs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">mdargs</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">mdargs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)))</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">kwargs</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-
-            <span class="n">arglength</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">mdargs</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;=</span> <span class="n">arglength</span><span class="p">:</span>
-                        <span class="k">continue</span>
-                    <span class="n">entry</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-                    <span class="n">is_arg</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-                    <span class="k">try</span><span class="p">:</span>
-                        <span class="n">entry</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-                    <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-                        <span class="k">continue</span>
-                    <span class="n">is_arg</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">entry</span><span class="p">,</span> <span class="s">&quot;__iter__&quot;</span><span class="p">):</span>
-                    <span class="c"># Create now a copy of the given array and manipulate then</span>
-                    <span class="c"># the entries directly.</span>
-                    <span class="k">if</span> <span class="n">is_arg</span><span class="p">:</span>
-                        <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-                        <span class="n">args</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">structure_copy</span><span class="p">(</span><span class="n">entry</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">kwargs</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">structure_copy</span><span class="p">(</span><span class="n">entry</span><span class="p">)</span>
-                    <span class="n">result</span> <span class="o">=</span> <span class="n">apply_on_element</span><span class="p">(</span><span class="n">wrapper</span><span class="p">,</span> <span class="n">args</span><span class="p">,</span> <span class="n">kwargs</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-                    <span class="k">return</span> <span class="n">result</span>
-            <span class="k">return</span> <span class="n">f</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">wrapper</span></div>
-</pre></div>
-
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-    <form class="search" action="../../../search.html" method="get">
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-            
-  <h1>Source code for sympy.core.numbers</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">.core</span> <span class="kn">import</span> <span class="n">C</span>
-<span class="kn">from</span> <span class="nn">.sympify</span> <span class="kn">import</span> <span class="n">converter</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">_sympify</span><span class="p">,</span> <span class="n">SympifyError</span>
-<span class="kn">from</span> <span class="nn">.basic</span> <span class="kn">import</span> <span class="n">Basic</span>
-<span class="kn">from</span> <span class="nn">.singleton</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">Singleton</span>
-<span class="kn">from</span> <span class="nn">.expr</span> <span class="kn">import</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">AtomicExpr</span>
-<span class="kn">from</span> <span class="nn">.decorators</span> <span class="kn">import</span> <span class="n">_sympifyit</span><span class="p">,</span> <span class="n">deprecated</span><span class="p">,</span> <span class="n">call_highest_priority</span>
-<span class="kn">from</span> <span class="nn">.cache</span> <span class="kn">import</span> <span class="n">cacheit</span><span class="p">,</span> <span class="n">clear_cache</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">as_int</span>
-<span class="kn">import</span> <span class="nn">sympy.mpmath</span> <span class="kn">as</span> <span class="nn">mpmath</span>
-<span class="kn">import</span> <span class="nn">sympy.mpmath.libmp</span> <span class="kn">as</span> <span class="nn">mlib</span>
-<span class="kn">from</span> <span class="nn">sympy.mpmath.libmp</span> <span class="kn">import</span> <span class="n">mpf_pow</span><span class="p">,</span> <span class="n">mpf_pi</span><span class="p">,</span> <span class="n">mpf_e</span><span class="p">,</span> <span class="n">phi_fixed</span>
-<span class="kn">from</span> <span class="nn">sympy.mpmath.ctx_mp</span> <span class="kn">import</span> <span class="n">mpnumeric</span>
-<span class="kn">from</span> <span class="nn">sympy.mpmath.libmp.libmpf</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">fnan</span> <span class="k">as</span> <span class="n">_mpf_nan</span><span class="p">,</span> <span class="n">fzero</span> <span class="k">as</span> <span class="n">_mpf_zero</span><span class="p">,</span> <span class="n">_normalize</span> <span class="k">as</span> <span class="n">mpf_normalize</span><span class="p">)</span>
-
-<span class="kn">import</span> <span class="nn">decimal</span>
-<span class="kn">import</span> <span class="nn">math</span>
-<span class="kn">import</span> <span class="nn">re</span> <span class="kn">as</span> <span class="nn">regex</span>
-<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</span>
-
-<span class="n">rnd</span> <span class="o">=</span> <span class="n">mlib</span><span class="o">.</span><span class="n">round_nearest</span>
-
-<span class="n">_LOG2</span> <span class="o">=</span> <span class="n">math</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">mpf_norm</span><span class="p">(</span><span class="n">mpf</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return the mpf tuple normalized appropriately for the indicated</span>
-<span class="sd">    precision.</span>
-
-<span class="sd">    This also contains a portion of code to not return zero if</span>
-<span class="sd">    the mantissa is 0; this is a necessary (but not sufficient) test of</span>
-<span class="sd">    zero since the mantissa for mpf&#39;s values &quot;+inf&quot;, &quot;-inf&quot; and &quot;nan&quot; have</span>
-<span class="sd">    a mantissa of zero, too.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">sign</span><span class="p">,</span> <span class="n">man</span><span class="p">,</span> <span class="n">expt</span><span class="p">,</span> <span class="n">bc</span> <span class="o">=</span> <span class="n">mpf</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">man</span><span class="p">:</span>
-        <span class="c"># hack for mpf_normalize which does not do this;</span>
-        <span class="c"># it assumes that if man is zero the result is 0</span>
-        <span class="c"># (see issue 3540)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">bc</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_mpf_zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># don&#39;t change anything; this should already</span>
-            <span class="c"># be a well formed mpf tuple</span>
-            <span class="k">return</span> <span class="n">mpf</span>
-    <span class="n">rv</span> <span class="o">=</span> <span class="n">mpf_normalize</span><span class="p">(</span><span class="n">sign</span><span class="p">,</span> <span class="n">man</span><span class="p">,</span> <span class="n">expt</span><span class="p">,</span> <span class="n">bc</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">rv</span>
-
-<span class="c"># TODO: we should use the warnings module</span>
-<span class="n">_errdict</span> <span class="o">=</span> <span class="p">{</span><span class="s">&quot;divide&quot;</span><span class="p">:</span> <span class="bp">False</span><span class="p">}</span>
-
-
-<div class="viewcode-block" id="seterr"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.numbers.seterr">[docs]</a><span class="k">def</span> <span class="nf">seterr</span><span class="p">(</span><span class="n">divide</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Should sympy raise an exception on 0/0 or return a nan?</span>
-
-<span class="sd">    divide == True .... raise an exception</span>
-<span class="sd">    divide == False ... return nan</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">_errdict</span><span class="p">[</span><span class="s">&quot;divide&quot;</span><span class="p">]</span> <span class="o">!=</span> <span class="n">divide</span><span class="p">:</span>
-        <span class="n">clear_cache</span><span class="p">()</span>
-        <span class="n">_errdict</span><span class="p">[</span><span class="s">&quot;divide&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">divide</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_as_integer_ratio</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;compatibility implementation for python &lt; 2.6&quot;&quot;&quot;</span>
-    <span class="n">neg_pow</span><span class="p">,</span> <span class="n">man</span><span class="p">,</span> <span class="n">expt</span><span class="p">,</span> <span class="n">bc</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="s">&#39;_mpf_&#39;</span><span class="p">,</span> <span class="n">mpmath</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">)</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="n">neg_pow</span> <span class="o">%</span> <span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">man</span>
-    <span class="k">if</span> <span class="n">expt</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">q</span> <span class="o">=</span> <span class="mi">2</span><span class="o">**-</span><span class="n">expt</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">q</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">p</span> <span class="o">*=</span> <span class="mi">2</span><span class="o">**</span><span class="n">expt</span>
-    <span class="k">return</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span>
-
-
-<span class="k">def</span> <span class="nf">_decimal_to_Rational_prec</span><span class="p">(</span><span class="n">dec</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Convert an ordinary decimal instance to a Rational.&quot;&quot;&quot;</span>
-    <span class="c"># _is_special is needed for Python 2.5 support; is_finite for Python 3.3</span>
-    <span class="c"># support</span>
-    <span class="n">nonfinite</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">dec</span><span class="p">,</span> <span class="s">&#39;_is_special&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">nonfinite</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">nonfinite</span> <span class="o">=</span> <span class="ow">not</span> <span class="n">dec</span><span class="o">.</span><span class="n">is_finite</span><span class="p">()</span>
-    <span class="k">if</span> <span class="n">nonfinite</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;dec must be finite, got </span><span class="si">%s</span><span class="s">.&quot;</span> <span class="o">%</span> <span class="n">dec</span><span class="p">)</span>
-    <span class="n">s</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">dec</span><span class="o">.</span><span class="n">as_tuple</span><span class="p">()</span>
-    <span class="n">prec</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">e</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>  <span class="c"># it&#39;s an integer</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">Integer</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">dec</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">s</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">([</span><span class="n">di</span><span class="o">*</span><span class="mi">10</span><span class="o">**</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">di</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">d</span><span class="p">))])</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="n">s</span><span class="o">*</span><span class="n">d</span><span class="p">,</span> <span class="mi">10</span><span class="o">**-</span><span class="n">e</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">rv</span><span class="p">,</span> <span class="n">prec</span>
-
-
-<span class="k">def</span> <span class="nf">_literal_float</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return True if n can be interpreted as a floating point number.&quot;&quot;&quot;</span>
-    <span class="n">pat</span> <span class="o">=</span> <span class="s">r&quot;[-+]?((\d*\.\d+)|(\d+\.?))(eE[-+]?\d+)?&quot;</span>
-    <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="n">regex</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">pat</span><span class="p">,</span> <span class="n">f</span><span class="p">))</span>
-
-<span class="c"># (a,b) -&gt; gcd(a,b)</span>
-<span class="n">_gcdcache</span> <span class="o">=</span> <span class="p">{}</span>
-
-<span class="c"># TODO caching with decorator, but not to degrade performance</span>
-
-
-<div class="viewcode-block" id="igcd"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.numbers.igcd">[docs]</a><span class="k">def</span> <span class="nf">igcd</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Computes positive, integer greatest common divisor of two numbers.</span>
-
-<span class="sd">       The algorithm is based on the well known Euclid&#39;s algorithm. To</span>
-<span class="sd">       improve speed, igcd() has its own caching mechanism implemented.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">_gcdcache</span><span class="p">[(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)]</span>
-    <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">as_int</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">a</span> <span class="ow">and</span> <span class="n">b</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">b</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="o">-</span><span class="n">b</span>
-
-            <span class="k">while</span> <span class="n">b</span><span class="p">:</span>
-                <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="p">,</span> <span class="n">a</span> <span class="o">%</span> <span class="n">b</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">a</span> <span class="ow">or</span> <span class="n">b</span><span class="p">)</span>
-
-        <span class="n">_gcdcache</span><span class="p">[(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)]</span> <span class="o">=</span> <span class="n">a</span>
-        <span class="k">return</span> <span class="n">a</span>
-
-</div>
-<div class="viewcode-block" id="ilcm"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.numbers.ilcm">[docs]</a><span class="k">def</span> <span class="nf">ilcm</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Computes integer least common multiple of two numbers. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">b</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">a</span><span class="o">*</span><span class="n">b</span> <span class="o">//</span> <span class="n">igcd</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">igcdex</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns x, y, g such that g = x*a + y*b = gcd(a, b).</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy.core.numbers import igcdex</span>
-<span class="sd">       &gt;&gt;&gt; igcdex(2, 3)</span>
-<span class="sd">       (-1, 1, 1)</span>
-<span class="sd">       &gt;&gt;&gt; igcdex(10, 12)</span>
-<span class="sd">       (-1, 1, 2)</span>
-
-<span class="sd">       &gt;&gt;&gt; x, y, g = igcdex(100, 2004)</span>
-<span class="sd">       &gt;&gt;&gt; x, y, g</span>
-<span class="sd">       (-20, 1, 4)</span>
-<span class="sd">       &gt;&gt;&gt; x*100 + y*2004</span>
-<span class="sd">       4</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="p">(</span><span class="ow">not</span> <span class="n">a</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="ow">not</span> <span class="n">b</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">a</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">b</span><span class="o">//</span><span class="nb">abs</span><span class="p">(</span><span class="n">b</span><span class="p">),</span> <span class="nb">abs</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">b</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">a</span><span class="o">//</span><span class="nb">abs</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">abs</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="n">a</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">x_sign</span> <span class="o">=</span> <span class="o">-</span><span class="n">a</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">x_sign</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="n">b</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">b</span><span class="p">,</span> <span class="n">y_sign</span> <span class="o">=</span> <span class="o">-</span><span class="n">b</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">y_sign</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">s</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span>
-
-    <span class="k">while</span> <span class="n">b</span><span class="p">:</span>
-        <span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">a</span> <span class="o">%</span> <span class="n">b</span><span class="p">,</span> <span class="n">a</span> <span class="o">//</span> <span class="n">b</span><span class="p">)</span>
-        <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">x</span> <span class="o">-</span> <span class="n">q</span><span class="o">*</span><span class="n">r</span><span class="p">,</span> <span class="n">y</span> <span class="o">-</span> <span class="n">q</span><span class="o">*</span><span class="n">s</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">x_sign</span><span class="p">,</span> <span class="n">y</span><span class="o">*</span><span class="n">y_sign</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="Number"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.numbers.Number">[docs]</a><span class="k">class</span> <span class="nc">Number</span><span class="p">(</span><span class="n">AtomicExpr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Represents any kind of number in sympy.</span>
-
-<span class="sd">    Floating point numbers are represented by the Float class.</span>
-<span class="sd">    Integer numbers (of any size), together with rational numbers (again,</span>
-<span class="sd">    there is no limit on their size) are represented by the Rational class.</span>
-
-<span class="sd">    If you want to represent, for example, ``1+sqrt(2)``, then you need to do::</span>
-
-<span class="sd">      Rational(1) + sqrt(Rational(2))</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_bounded</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_finite</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_number</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="c"># Used to make max(x._prec, y._prec) return x._prec when only x is a float</span>
-    <span class="n">_prec</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-
-    <span class="n">is_Number</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">obj</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">obj</span> <span class="o">=</span> <span class="n">obj</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">obj</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="o">*</span><span class="n">obj</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="p">(</span><span class="nb">float</span><span class="p">,</span> <span class="n">mpmath</span><span class="o">.</span><span class="n">mpf</span><span class="p">,</span> <span class="n">decimal</span><span class="o">.</span><span class="n">Decimal</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="n">Float</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="n">val</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">val</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">val</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;String &quot;</span><span class="si">%s</span><span class="s">&quot; does not denote a Number&#39;</span> <span class="o">%</span> <span class="n">obj</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">obj</span>
-        <span class="n">msg</span> <span class="o">=</span> <span class="s">&quot;expected str|int|long|float|Decimal|Number object but got </span><span class="si">%r</span><span class="s">&quot;</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="n">msg</span> <span class="o">%</span> <span class="nb">type</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span><span class="o">.</span><span class="n">__name__</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__divmod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">.containers</span> <span class="kn">import</span> <span class="n">Tuple</span>
-        <span class="kn">from</span> <span class="nn">sympy.functions.elementary.complexes</span> <span class="kn">import</span> <span class="n">sign</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">Number</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-            <span class="n">msg</span> <span class="o">=</span> <span class="s">&quot;unsupported operand type(s) for divmod(): &#39;</span><span class="si">%s</span><span class="s">&#39; and &#39;</span><span class="si">%s</span><span class="s">&#39;&quot;</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="n">msg</span> <span class="o">%</span> <span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="nb">type</span><span class="p">(</span><span class="n">other</span><span class="p">)</span><span class="o">.</span><span class="n">__name__</span><span class="p">))</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">other</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ZeroDivisionError</span><span class="p">(</span><span class="s">&#39;modulo by zero&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="nb">divmod</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">p</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">rat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">/</span><span class="n">other</span>
-        <span class="n">w</span> <span class="o">=</span> <span class="n">sign</span><span class="p">(</span><span class="n">rat</span><span class="p">)</span><span class="o">*</span><span class="nb">int</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">rat</span><span class="p">))</span>  <span class="c"># = rat.floor()</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="bp">self</span> <span class="o">-</span> <span class="n">other</span><span class="o">*</span><span class="n">w</span>
-        <span class="c">#w*other + r == self</span>
-        <span class="k">return</span> <span class="n">Tuple</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__rdivmod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">Number</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-            <span class="n">msg</span> <span class="o">=</span> <span class="s">&quot;unsupported operand type(s) for divmod(): &#39;</span><span class="si">%s</span><span class="s">&#39; and &#39;</span><span class="si">%s</span><span class="s">&#39;&quot;</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="n">msg</span> <span class="o">%</span> <span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">other</span><span class="p">)</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__name__</span><span class="p">))</span>
-        <span class="k">return</span> <span class="nb">divmod</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__round__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">round</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="bp">self</span><span class="p">),</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_as_mpf_val</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Evaluation of mpf tuple accurate to at least prec bits.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> needs ._as_mpf_val() method&#39;</span> <span class="o">%</span>
-            <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_as_mpf_val</span><span class="p">(</span><span class="n">prec</span><span class="p">),</span> <span class="n">prec</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_as_mpf_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="n">prec</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">prec</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_prec</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_as_mpf_val</span><span class="p">(</span><span class="n">prec</span><span class="p">),</span> <span class="n">prec</span>
-
-    <span class="k">def</span> <span class="nf">__float__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">mlib</span><span class="o">.</span><span class="n">to_float</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_as_mpf_val</span><span class="p">(</span><span class="mi">53</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="k">def</span> <span class="nf">_eval_order</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">):</span>
-        <span class="c"># Order(5, x, y) -&gt; Order(1,x,y)</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">old</span> <span class="o">==</span> <span class="o">-</span><span class="bp">self</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">new</span>
-        <span class="k">return</span> <span class="bp">self</span>  <span class="c"># there is no other possibility</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">class_key</span><span class="p">(</span><span class="n">cls</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&#39;Number&#39;</span>
-
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">sort_key</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">class_key</span><span class="p">(),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="p">()),</span> <span class="p">(),</span> <span class="bp">self</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-            <span class="k">elif</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-        <span class="k">return</span> <span class="n">AtomicExpr</span><span class="o">.</span><span class="n">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-            <span class="k">elif</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">return</span> <span class="n">AtomicExpr</span><span class="o">.</span><span class="n">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-                <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-            <span class="k">elif</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-                <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">return</span> <span class="n">AtomicExpr</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span> <span class="ow">or</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">return</span> <span class="n">AtomicExpr</span><span class="o">.</span><span class="n">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="n">__truediv__</span> <span class="o">=</span> <span class="n">__div__</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> needs .__eq__() method&#39;</span> <span class="o">%</span>
-            <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> needs .__ne__() method&#39;</span> <span class="o">%</span>
-            <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> needs .__lt__() method&#39;</span> <span class="o">%</span>
-            <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__le__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> needs .__le__() method&#39;</span> <span class="o">%</span>
-            <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__gt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span><span class="o">.</span><span class="n">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__ge__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span><span class="o">.</span><span class="n">__le__</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Number</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">is_constant</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">wrt</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_number</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">as_coeff_mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">deps</span><span class="p">):</span>
-        <span class="c"># a -&gt; c*t</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">()</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">,</span> <span class="p">(</span><span class="o">-</span><span class="bp">self</span><span class="p">,)</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="p">(</span><span class="bp">self</span><span class="p">,)</span>
-
-    <span class="k">def</span> <span class="nf">as_coeff_add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">deps</span><span class="p">):</span>
-        <span class="c"># a -&gt; c + t</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="p">(</span><span class="bp">self</span><span class="p">,)</span>
-
-<div class="viewcode-block" id="Number.as_coeff_Mul"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.numbers.Number.as_coeff_Mul">[docs]</a>    <span class="k">def</span> <span class="nf">as_coeff_Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rational</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Efficiently extract the coefficient of a product. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">rational</span> <span class="ow">and</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="bp">self</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-</div>
-<div class="viewcode-block" id="Number.as_coeff_Add"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.numbers.Number.as_coeff_Add">[docs]</a>    <span class="k">def</span> <span class="nf">as_coeff_Add</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Efficiently extract the coefficient of a summation. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-</div>
-<div class="viewcode-block" id="Number.gcd"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.numbers.Number.gcd">[docs]</a>    <span class="k">def</span> <span class="nf">gcd</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute GCD of `self` and `other`. &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">gcd</span>
-        <span class="k">return</span> <span class="n">gcd</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Number.lcm"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.numbers.Number.lcm">[docs]</a>    <span class="k">def</span> <span class="nf">lcm</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute LCM of `self` and `other`. &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">lcm</span>
-        <span class="k">return</span> <span class="n">lcm</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Number.cofactors"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.numbers.Number.cofactors">[docs]</a>    <span class="k">def</span> <span class="nf">cofactors</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute GCD and cofactors of `self` and `other`. &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">cofactors</span>
-        <span class="k">return</span> <span class="n">cofactors</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-</div></div>
-<div class="viewcode-block" id="Float"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.numbers.Float">[docs]</a><span class="k">class</span> <span class="nc">Float</span><span class="p">(</span><span class="n">Number</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Represents a floating point number. It is capable of representing</span>
-<span class="sd">    arbitrary-precision floating-point numbers.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Float, pi</span>
-<span class="sd">    &gt;&gt;&gt; Float(3.5)</span>
-<span class="sd">    3.50000000000000</span>
-<span class="sd">    &gt;&gt;&gt; Float(3)</span>
-<span class="sd">    3.00000000000000</span>
-
-<span class="sd">    Floats can be created from a string representations of Python floats</span>
-<span class="sd">    to force ints to Float or to enter high-precision (&gt; 15 significant</span>
-<span class="sd">    digits) values:</span>
-
-<span class="sd">    &gt;&gt;&gt; Float(&#39;.0010&#39;)</span>
-<span class="sd">    0.00100000000000000</span>
-<span class="sd">    &gt;&gt;&gt; Float(&#39;1e-3&#39;)</span>
-<span class="sd">    0.00100000000000000</span>
-<span class="sd">    &gt;&gt;&gt; Float(&#39;1e-3&#39;, 3)</span>
-<span class="sd">    0.00100</span>
-
-<span class="sd">    Float can automatically count significant figures if a null string</span>
-<span class="sd">    is sent for the precision; space are also allowed in the string. (Auto-</span>
-<span class="sd">    counting is only allowed for strings, ints and longs).</span>
-
-<span class="sd">    &gt;&gt;&gt; Float(&#39;123 456 789 . 123 456&#39;, &#39;&#39;)</span>
-<span class="sd">    123456789.123456</span>
-<span class="sd">    &gt;&gt;&gt; Float(&#39;12e-3&#39;, &#39;&#39;)</span>
-<span class="sd">    0.012</span>
-<span class="sd">    &gt;&gt;&gt; Float(3, &#39;&#39;)</span>
-<span class="sd">    3.</span>
-
-<span class="sd">    If a number is written in scientific notation, only the digits before the</span>
-<span class="sd">    exponent are considered significant if a decimal appears, otherwise the</span>
-<span class="sd">    &quot;e&quot; signifies only how to move the decimal:</span>
-
-<span class="sd">    &gt;&gt;&gt; Float(&#39;60.e2&#39;, &#39;&#39;)  # 2 digits significant</span>
-<span class="sd">    6.0e+3</span>
-<span class="sd">    &gt;&gt;&gt; Float(&#39;60e2&#39;, &#39;&#39;)  # 4 digits significant</span>
-<span class="sd">    6000.</span>
-<span class="sd">    &gt;&gt;&gt; Float(&#39;600e-2&#39;, &#39;&#39;)  # 3 digits significant</span>
-<span class="sd">    6.00</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    Floats are inexact by their nature unless their value is a binary-exact</span>
-<span class="sd">    value.</span>
-
-<span class="sd">    &gt;&gt;&gt; approx, exact = Float(.1, 1), Float(.125, 1)</span>
-
-<span class="sd">    For calculation purposes, evalf needs to be able to change the precision</span>
-<span class="sd">    but this will not increase the accuracy of the inexact value. The</span>
-<span class="sd">    following is the most accurate 5-digit approximation of a value of 0.1</span>
-<span class="sd">    that had only 1 digit of precision:</span>
-
-<span class="sd">    &gt;&gt;&gt; approx.evalf(5)</span>
-<span class="sd">    0.099609</span>
-
-<span class="sd">    By contrast, 0.125 is exact in binary (as it is in base 10) and so it</span>
-<span class="sd">    can be passed to Float or evalf to obtain an arbitrary precision with</span>
-<span class="sd">    matching accuracy:</span>
-
-<span class="sd">    &gt;&gt;&gt; Float(exact, 5)</span>
-<span class="sd">    0.12500</span>
-<span class="sd">    &gt;&gt;&gt; exact.evalf(20)</span>
-<span class="sd">    0.12500000000000000000</span>
-
-<span class="sd">    Trying to make a high-precision Float from a float is not disallowed,</span>
-<span class="sd">    but one must keep in mind that the *underlying float* (not the apparent</span>
-<span class="sd">    decimal value) is being obtained with high precision. For example, 0.3</span>
-<span class="sd">    does not have a finite binary representation. The closest rational is</span>
-<span class="sd">    the fraction 5404319552844595/2**54. So if you try to obtain a Float of</span>
-<span class="sd">    0.3 to 20 digits of precision you will not see the same thing as 0.3</span>
-<span class="sd">    followed by 19 zeros:</span>
-
-<span class="sd">    &gt;&gt;&gt; Float(0.3, 20)</span>
-<span class="sd">    0.29999999999999998890</span>
-
-<span class="sd">    If you want a 20-digit value of the decimal 0.3 (not the floating point</span>
-<span class="sd">    approximation of 0.3) you should send the 0.3 as a string. The underlying</span>
-<span class="sd">    representation is still binary but a higher precision than Python&#39;s float</span>
-<span class="sd">    is used:</span>
-
-<span class="sd">    &gt;&gt;&gt; Float(&#39;0.3&#39;, 20)</span>
-<span class="sd">    0.30000000000000000000</span>
-
-<span class="sd">    Although you can increase the precision of an existing Float using Float</span>
-<span class="sd">    it will not increase the accuracy -- the underlying value is not changed:</span>
-
-<span class="sd">    &gt;&gt;&gt; def show(f): # binary rep of Float</span>
-<span class="sd">    ...     from sympy import Mul, Pow</span>
-<span class="sd">    ...     s, m, e, b = f._mpf_</span>
-<span class="sd">    ...     v = Mul(m, Pow(2, e, evaluate=False), evaluate=False)</span>
-<span class="sd">    ...     print(&#39;%s at prec=%s&#39; % (v, f._prec))</span>
-<span class="sd">    ...</span>
-<span class="sd">    &gt;&gt;&gt; t = Float(&#39;0.3&#39;, 3)</span>
-<span class="sd">    &gt;&gt;&gt; show(t)</span>
-<span class="sd">    4915/2**14 at prec=13</span>
-<span class="sd">    &gt;&gt;&gt; show(Float(t, 20)) # higher prec, not higher accuracy</span>
-<span class="sd">    4915/2**14 at prec=70</span>
-<span class="sd">    &gt;&gt;&gt; show(Float(t, 2)) # lower prec</span>
-<span class="sd">    307/2**10 at prec=10</span>
-
-<span class="sd">    The same thing happens when evalf is used on a Float:</span>
-
-<span class="sd">    &gt;&gt;&gt; show(t.evalf(20))</span>
-<span class="sd">    4915/2**14 at prec=70</span>
-<span class="sd">    &gt;&gt;&gt; show(t.evalf(2))</span>
-<span class="sd">    307/2**10 at prec=10</span>
-
-<span class="sd">    Finally, Floats can be instantiated with an mpf tuple (n, c, p) to</span>
-<span class="sd">    produce the number (-1)**n*c*2**p:</span>
-
-<span class="sd">    &gt;&gt;&gt; n, c, p = 1, 5, 0</span>
-<span class="sd">    &gt;&gt;&gt; (-1)**n*c*2**p</span>
-<span class="sd">    -5</span>
-<span class="sd">    &gt;&gt;&gt; Float((1, 5, 0))</span>
-<span class="sd">    -5.00000000000000</span>
-
-<span class="sd">    An actual mpf tuple also contains the number of bits in c as the last</span>
-<span class="sd">    element of the tuple, but this is not needed for instantiation:</span>
-
-<span class="sd">    &gt;&gt;&gt; _._mpf_</span>
-<span class="sd">    (1, 5, 0, 3)</span>
-
-<span class="sd">    That last value can be used to create the corresponding Float, however,</span>
-<span class="sd">    by passing the tuple and prec=&#39;&#39; to Float to copy the precision from the</span>
-<span class="sd">    mpf tuple:</span>
-
-<span class="sd">    &gt;&gt;&gt; Float(pi.n(5)._mpf_, &#39;&#39;)</span>
-<span class="sd">    3.1416</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;_mpf_&#39;</span><span class="p">,</span> <span class="s">&#39;_prec&#39;</span><span class="p">]</span>
-
-    <span class="n">is_real</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_irrational</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_integer</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="n">is_Float</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">num</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="mi">15</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="n">num</span> <span class="o">=</span> <span class="n">num</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">num</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39;.&#39;</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">num</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">num</span> <span class="o">=</span> <span class="s">&#39;0&#39;</span> <span class="o">+</span> <span class="n">num</span>
-            <span class="k">elif</span> <span class="n">num</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39;-.&#39;</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">num</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">num</span> <span class="o">=</span> <span class="s">&#39;-0.&#39;</span> <span class="o">+</span> <span class="n">num</span><span class="p">[</span><span class="mi">2</span><span class="p">:]</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="nb">float</span><span class="p">)</span> <span class="ow">and</span> <span class="n">num</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">num</span> <span class="o">=</span> <span class="s">&#39;0&#39;</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="n">Integer</span><span class="p">)):</span>
-            <span class="n">num</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>  <span class="c"># faster than mlib.from_int</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">mpmath</span><span class="o">.</span><span class="n">mpf</span><span class="p">):</span>
-            <span class="n">num</span> <span class="o">=</span> <span class="n">num</span><span class="o">.</span><span class="n">_mpf_</span>
-
-        <span class="k">if</span> <span class="n">prec</span> <span class="o">==</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">num</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">tuple</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">num</span><span class="p">)</span> <span class="o">==</span> <span class="mi">4</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">num</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-            <span class="k">elif</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;The null string can only be used when &#39;</span>
-                <span class="s">&#39;the number to Float is passed as a string or is a tuple &#39;</span>
-                <span class="s">&#39;with length of 4.&#39;</span><span class="p">)</span>
-            <span class="n">ok</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">if</span> <span class="n">_literal_float</span><span class="p">(</span><span class="n">num</span><span class="p">):</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="n">Num</span> <span class="o">=</span> <span class="n">decimal</span><span class="o">.</span><span class="n">Decimal</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
-                <span class="k">except</span> <span class="n">decimal</span><span class="o">.</span><span class="n">InvalidOperation</span><span class="p">:</span>
-                    <span class="k">pass</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">isint</span> <span class="o">=</span> <span class="s">&#39;.&#39;</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">num</span>
-                    <span class="n">num</span><span class="p">,</span> <span class="n">dps</span> <span class="o">=</span> <span class="n">_decimal_to_Rational_prec</span><span class="p">(</span><span class="n">Num</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">num</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">isint</span><span class="p">:</span>
-                        <span class="n">dps</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">dps</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">num</span><span class="p">)</span><span class="o">.</span><span class="n">lstrip</span><span class="p">(</span><span class="s">&#39;-&#39;</span><span class="p">)))</span>
-                    <span class="n">ok</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">if</span> <span class="n">ok</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;string-float not recognized: </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">num</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">dps</span> <span class="o">=</span> <span class="n">prec</span>
-
-        <span class="n">prec</span> <span class="o">=</span> <span class="n">mlib</span><span class="o">.</span><span class="n">libmpf</span><span class="o">.</span><span class="n">dps_to_prec</span><span class="p">(</span><span class="n">dps</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="nb">float</span><span class="p">):</span>
-            <span class="n">_mpf_</span> <span class="o">=</span> <span class="n">mlib</span><span class="o">.</span><span class="n">from_float</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="n">_mpf_</span> <span class="o">=</span> <span class="n">mlib</span><span class="o">.</span><span class="n">from_str</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">decimal</span><span class="o">.</span><span class="n">Decimal</span><span class="p">):</span>
-            <span class="n">_mpf_</span> <span class="o">=</span> <span class="n">mlib</span><span class="o">.</span><span class="n">from_str</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">num</span><span class="p">),</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-            <span class="n">_mpf_</span> <span class="o">=</span> <span class="n">mlib</span><span class="o">.</span><span class="n">from_rational</span><span class="p">(</span><span class="n">num</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="n">num</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">num</span><span class="p">)</span> <span class="ow">in</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">num</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="ow">is</span> <span class="nb">str</span><span class="p">:</span>
-                <span class="c"># it&#39;s a hexadecimal (coming from a pickled object)</span>
-                <span class="c"># assume that it is in standard form</span>
-                <span class="n">num</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
-                <span class="n">num</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">num</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="mi">16</span><span class="p">)</span>
-                <span class="n">_mpf_</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">num</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">num</span><span class="p">)</span> <span class="o">==</span> <span class="mi">4</span><span class="p">:</span>
-                    <span class="c"># handle normalization hack</span>
-                    <span class="k">return</span> <span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">_mpf_</span> <span class="o">=</span> <span class="n">mpmath</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span>
-                        <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">**</span><span class="n">num</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">num</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="mi">2</span><span class="o">**</span><span class="n">num</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="o">.</span><span class="n">_mpf_</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">Float</span><span class="p">):</span>
-            <span class="n">_mpf_</span> <span class="o">=</span> <span class="n">num</span><span class="o">.</span><span class="n">_mpf_</span>
-            <span class="k">if</span> <span class="n">prec</span> <span class="o">&lt;</span> <span class="n">num</span><span class="o">.</span><span class="n">_prec</span><span class="p">:</span>
-                <span class="n">_mpf_</span> <span class="o">=</span> <span class="n">mpf_norm</span><span class="p">(</span><span class="n">_mpf_</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">_mpf_</span> <span class="o">=</span> <span class="n">mpmath</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span><span class="n">num</span><span class="p">)</span><span class="o">.</span><span class="n">_mpf_</span>
-
-        <span class="c"># special cases</span>
-        <span class="k">if</span> <span class="n">_mpf_</span> <span class="o">==</span> <span class="n">_mpf_zero</span><span class="p">:</span>
-            <span class="k">pass</span>  <span class="c"># we want a Float</span>
-        <span class="k">elif</span> <span class="n">_mpf_</span> <span class="o">==</span> <span class="n">_mpf_nan</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_mpf_</span> <span class="o">=</span> <span class="n">_mpf_</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_prec</span> <span class="o">=</span> <span class="n">prec</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_new</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">_mpf_</span><span class="p">,</span> <span class="n">_prec</span><span class="p">):</span>
-        <span class="c"># special cases</span>
-        <span class="k">if</span> <span class="n">_mpf_</span> <span class="o">==</span> <span class="n">_mpf_zero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>  <span class="c"># XXX this is different from Float which gives 0.0</span>
-        <span class="k">elif</span> <span class="n">_mpf_</span> <span class="o">==</span> <span class="n">_mpf_nan</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-
-        <span class="c"># the new Float should be normalized unless it is</span>
-        <span class="c"># an integer because Float doesn&#39;t return Floats</span>
-        <span class="c"># for Integers. If Integers can become Floats then</span>
-        <span class="c"># all the following (up to the first &#39;obj =&#39; line</span>
-        <span class="c"># can be replaced with ok = mpf_norm(_mpf_, _prec)</span>
-
-        <span class="n">sign</span><span class="p">,</span> <span class="n">man</span><span class="p">,</span> <span class="n">expt</span><span class="p">,</span> <span class="n">bc</span> <span class="o">=</span> <span class="n">_mpf_</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">man</span><span class="p">:</span>
-            <span class="c"># hack for mpf_normalize which does not do this</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">bc</span><span class="p">:</span>
-                <span class="n">ok</span> <span class="o">=</span> <span class="n">_mpf_zero</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">ok</span> <span class="o">=</span> <span class="p">(</span><span class="n">sign</span> <span class="o">%</span> <span class="mi">2</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">man</span><span class="p">),</span> <span class="n">expt</span><span class="p">,</span> <span class="n">bc</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">expt</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="c"># this is the non-hack normalization</span>
-            <span class="n">ok</span> <span class="o">=</span> <span class="n">mpf_normalize</span><span class="p">(</span><span class="n">sign</span><span class="p">,</span> <span class="n">man</span><span class="p">,</span> <span class="n">expt</span><span class="p">,</span> <span class="n">bc</span><span class="p">,</span> <span class="n">_prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">ok</span> <span class="o">=</span> <span class="n">_mpf_</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_mpf_</span> <span class="o">=</span> <span class="n">ok</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_prec</span> <span class="o">=</span> <span class="n">_prec</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="c"># mpz can&#39;t be pickled</span>
-    <span class="k">def</span> <span class="nf">__getnewargs__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">to_pickable</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">),)</span>
-
-    <span class="k">def</span> <span class="nf">__getstate__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">{</span><span class="s">&#39;_prec&#39;</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_prec</span><span class="p">}</span>
-
-    <span class="k">def</span> <span class="nf">_hashable_content</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_prec</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">floor</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Integer</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">to_int</span><span class="p">(</span>
-            <span class="n">mlib</span><span class="o">.</span><span class="n">mpf_floor</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_prec</span><span class="p">))))</span>
-
-    <span class="k">def</span> <span class="nf">ceiling</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Integer</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">to_int</span><span class="p">(</span>
-            <span class="n">mlib</span><span class="o">.</span><span class="n">mpf_ceil</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_prec</span><span class="p">))))</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">num</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">mpmath</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_as_mpf_val</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">mpf_norm</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-        <span class="c"># uncomment to see failures</span>
-        <span class="c">#if rv != was._mpf_ and self._prec == prec:</span>
-        <span class="c">#    print was._mpf_, rv</span>
-        <span class="k">return</span> <span class="n">rv</span>
-
-    <span class="k">def</span> <span class="nf">_as_mpf_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="nb">max</span><span class="p">(</span><span class="n">prec</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_prec</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_positive</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">num</span> <span class="o">&gt;</span> <span class="mi">0</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_negative</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">num</span> <span class="o">&lt;</span> <span class="mi">0</span>
-
-    <span class="k">def</span> <span class="nf">__bool__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># do not base answer on `man` alone:</span>
-        <span class="c"># &gt;&gt;&gt; mpf(&#39;0&#39;)._mpf_</span>
-        <span class="c"># (0, 0L, 0, 0)</span>
-        <span class="c"># &gt;&gt;&gt; mpf(&#39;nan&#39;)._mpf_</span>
-        <span class="c"># (0, 0L, -123, -1)</span>
-        <span class="c"># &gt;&gt;&gt; mpf(&#39;inf&#39;)._mpf_</span>
-        <span class="c"># (0, 0L, -456, -2)</span>
-        <span class="c"># &gt;&gt;&gt; mpf(&#39;-inf&#39;)._mpf_</span>
-        <span class="c"># (1, 0L, -789, -3)</span>
-        <span class="n">sign</span><span class="p">,</span> <span class="n">man</span><span class="p">,</span> <span class="n">expt</span><span class="p">,</span> <span class="n">bc</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="p">(</span><span class="n">man</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">bc</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">mpf_neg</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_prec</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="n">rhs</span><span class="p">,</span> <span class="n">prec</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">_as_mpf_op</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_prec</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">mpf_add</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="n">rhs</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">),</span> <span class="n">prec</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Number</span><span class="o">.</span><span class="n">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="n">rhs</span><span class="p">,</span> <span class="n">prec</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">_as_mpf_op</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_prec</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">mpf_sub</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="n">rhs</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">),</span> <span class="n">prec</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Number</span><span class="o">.</span><span class="n">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="n">rhs</span><span class="p">,</span> <span class="n">prec</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">_as_mpf_op</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_prec</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">mpf_mul</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="n">rhs</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">),</span> <span class="n">prec</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Number</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="n">rhs</span><span class="p">,</span> <span class="n">prec</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">_as_mpf_op</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_prec</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">mpf_div</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="n">rhs</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">),</span> <span class="n">prec</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Number</span><span class="o">.</span><span class="n">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="n">__truediv__</span> <span class="o">=</span> <span class="n">__div__</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__mod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="n">rhs</span><span class="p">,</span> <span class="n">prec</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">_as_mpf_op</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_prec</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">mpf_mod</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="n">rhs</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">),</span> <span class="n">prec</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Number</span><span class="o">.</span><span class="n">__mod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__rmod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="n">rhs</span><span class="p">,</span> <span class="n">prec</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">_as_mpf_op</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_prec</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">mpf_mod</span><span class="p">(</span><span class="n">rhs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">),</span> <span class="n">prec</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Number</span><span class="o">.</span><span class="n">__rmod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expt</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        expt is symbolic object but not equal to 0, 1</span>
-
-<span class="sd">        (-p)**r -&gt; exp(r*log(-p)) -&gt; exp(r*(log(p) + I*Pi)) -&gt;</span>
-<span class="sd">                  -&gt; p**r*(sin(Pi*r) + cos(Pi*r)*I)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expt</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expt</span><span class="p">,</span> <span class="n">Integer</span><span class="p">):</span>
-                <span class="n">prec</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_prec</span>
-                <span class="k">return</span> <span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span>
-                    <span class="n">mlib</span><span class="o">.</span><span class="n">mpf_pow_int</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="n">expt</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">),</span> <span class="n">prec</span><span class="p">)</span>
-            <span class="n">expt</span><span class="p">,</span> <span class="n">prec</span> <span class="o">=</span> <span class="n">expt</span><span class="o">.</span><span class="n">_as_mpf_op</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_prec</span><span class="p">)</span>
-            <span class="bp">self</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">y</span> <span class="o">=</span> <span class="n">mpf_pow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expt</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">mlib</span><span class="o">.</span><span class="n">ComplexResult</span><span class="p">:</span>
-                <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="n">mlib</span><span class="o">.</span><span class="n">mpc_pow</span><span class="p">(</span>
-                    <span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">_mpf_zero</span><span class="p">),</span> <span class="p">(</span><span class="n">expt</span><span class="p">,</span> <span class="n">_mpf_zero</span><span class="p">),</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span> <span class="o">+</span> \
-                    <span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">im</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-
-    <span class="k">def</span> <span class="nf">__abs__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">mpf_abs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_prec</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__int__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">to_int</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">))</span>  <span class="c"># uses round_fast = round_down</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="nb">float</span><span class="p">):</span>
-            <span class="c"># coerce to Float at same precision</span>
-            <span class="n">o</span> <span class="o">=</span> <span class="n">Float</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">ompf</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">_as_mpf_val</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_prec</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">mpf_eq</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="n">ompf</span><span class="p">))</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">SympifyError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>    <span class="c"># sympy != other  --&gt;  not ==</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">NumberSymbol</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">is_irrational</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">return</span> <span class="n">other</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Float</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">mpf_eq</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">))</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="c"># numbers should compare at the same precision;</span>
-            <span class="c"># all _as_mpf_val routines should be sure to abide</span>
-            <span class="c"># by the request to change the prec if necessary; if</span>
-            <span class="c"># they don&#39;t, the equality test will fail since it compares</span>
-            <span class="c"># the mpf tuples</span>
-            <span class="n">ompf</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">_as_mpf_val</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_prec</span><span class="p">)</span>
-            <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">mpf_eq</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="n">ompf</span><span class="p">))</span>
-        <span class="k">return</span> <span class="bp">False</span>    <span class="c"># Float != non-Number</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__gt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">SympifyError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>    <span class="c"># sympy &gt; other</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">NumberSymbol</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">other</span><span class="o">.</span><span class="n">__le__</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">is_comparable</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">mpf_gt</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">_as_mpf_val</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_prec</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__gt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__ge__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">SympifyError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>    <span class="c"># sympy &gt; other  --&gt;  ! &lt;=</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">NumberSymbol</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">other</span><span class="o">.</span><span class="n">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">is_comparable</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">mpf_ge</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">_as_mpf_val</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_prec</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__ge__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">SympifyError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>    <span class="c"># sympy &gt; other</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">NumberSymbol</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">other</span><span class="o">.</span><span class="n">__ge__</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">mpf_lt</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">_as_mpf_val</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_prec</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__le__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">SympifyError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>    <span class="c"># sympy &gt; other  --&gt;  ! &lt;=</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">NumberSymbol</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">other</span><span class="o">.</span><span class="n">__gt__</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">mpf_le</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">_as_mpf_val</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_prec</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__le__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Float</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">epsilon_eq</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="n">epsilon</span><span class="o">=</span><span class="s">&quot;10e-16&quot;</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">abs</span><span class="p">(</span><span class="bp">self</span> <span class="o">-</span> <span class="n">other</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">Float</span><span class="p">(</span><span class="n">epsilon</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">RealNumber</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="p">))</span>
-
-<span class="c"># Add sympify converters</span></div>
-<span class="n">converter</span><span class="p">[</span><span class="nb">float</span><span class="p">]</span> <span class="o">=</span> <span class="n">converter</span><span class="p">[</span><span class="n">decimal</span><span class="o">.</span><span class="n">Decimal</span><span class="p">]</span> <span class="o">=</span> <span class="n">Float</span>
-
-<span class="c"># this is here to work nicely in Sage</span>
-<span class="n">RealNumber</span> <span class="o">=</span> <span class="n">Float</span>
-
-
-<span class="nd">@deprecated</span><span class="p">(</span><span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;Float&quot;</span><span class="p">,</span> <span class="n">issue</span><span class="o">=</span><span class="mi">1721</span><span class="p">,</span> <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.0&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="Real"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.numbers.Real">[docs]</a><span class="k">def</span> <span class="nf">Real</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>  <span class="c"># pragma: no cover</span>
-    <span class="sd">&quot;&quot;&quot;Deprecated alias for the Float constructor.&quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">Float</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="Rational"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.numbers.Rational">[docs]</a><span class="k">class</span> <span class="nc">Rational</span><span class="p">(</span><span class="n">Number</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Represents integers and rational numbers (p/q) of any size.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Rational, nsimplify, S, pi</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; Rational(3)</span>
-<span class="sd">    3</span>
-<span class="sd">    &gt;&gt;&gt; Rational(1, 2)</span>
-<span class="sd">    1/2</span>
-
-<span class="sd">    Rational is unprejudiced in accepting input. If a float is passed, the</span>
-<span class="sd">    underlying value of the binary representation will be returned:</span>
-
-<span class="sd">    &gt;&gt;&gt; Rational(.5)</span>
-<span class="sd">    1/2</span>
-<span class="sd">    &gt;&gt;&gt; Rational(.2)</span>
-<span class="sd">    3602879701896397/18014398509481984</span>
-
-<span class="sd">    If the simpler representation of the float is desired then consider</span>
-<span class="sd">    limiting the denominator to the desired value or convert the float to</span>
-<span class="sd">    a string (which is roughly equivalent to limiting the denominator to</span>
-<span class="sd">    10**12):</span>
-
-<span class="sd">    &gt;&gt;&gt; Rational(str(.2))</span>
-<span class="sd">    1/5</span>
-<span class="sd">    &gt;&gt;&gt; Rational(.2).limit_denominator(10**12)</span>
-<span class="sd">    1/5</span>
-
-<span class="sd">    An arbitrarily precise Rational is obtained when a string literal is</span>
-<span class="sd">    passed:</span>
-
-<span class="sd">    &gt;&gt;&gt; Rational(&quot;1.23&quot;)</span>
-<span class="sd">    123/100</span>
-<span class="sd">    &gt;&gt;&gt; Rational(&#39;1e-2&#39;)</span>
-<span class="sd">    1/100</span>
-<span class="sd">    &gt;&gt;&gt; Rational(&quot;.1&quot;)</span>
-<span class="sd">    1/10</span>
-<span class="sd">    &gt;&gt;&gt; Rational(&#39;1e-2/3.2&#39;)</span>
-<span class="sd">    1/320</span>
-
-<span class="sd">    The conversion of other types of strings can be handled by</span>
-<span class="sd">    the sympify() function, and conversion of floats to expressions</span>
-<span class="sd">    or simple fractions can be handled with nsimplify:</span>
-
-<span class="sd">    &gt;&gt;&gt; S(&#39;.[3]&#39;)  # repeating digits in brackets</span>
-<span class="sd">    1/3</span>
-<span class="sd">    &gt;&gt;&gt; S(&#39;3**2/10&#39;)  # general expressions</span>
-<span class="sd">    9/10</span>
-<span class="sd">    &gt;&gt;&gt; nsimplify(.3)  # numbers that have a simple form</span>
-<span class="sd">    3/10</span>
-
-<span class="sd">    But if the input does not reduce to a literal Rational, an error will</span>
-<span class="sd">    be raised:</span>
-
-<span class="sd">    &gt;&gt;&gt; Rational(pi)</span>
-<span class="sd">    Traceback (most recent call last):</span>
-<span class="sd">    ...</span>
-<span class="sd">    TypeError: invalid input: pi</span>
-
-
-<span class="sd">    Low-level</span>
-<span class="sd">    ---------</span>
-
-<span class="sd">    Access numerator and denominator as .p and .q:</span>
-
-<span class="sd">    &gt;&gt;&gt; r = Rational(3, 4)</span>
-<span class="sd">    &gt;&gt;&gt; r</span>
-<span class="sd">    3/4</span>
-<span class="sd">    &gt;&gt;&gt; r.p</span>
-<span class="sd">    3</span>
-<span class="sd">    &gt;&gt;&gt; r.q</span>
-<span class="sd">    4</span>
-
-<span class="sd">    Note that p and q return integers (not sympy Integers) so some care</span>
-<span class="sd">    is needed when using them in expressions:</span>
-
-<span class="sd">    &gt;&gt;&gt; r.p//r.q</span>
-<span class="sd">    0</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    sympify, sympy.simplify.simplify.nsimplify</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_real</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_integer</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_rational</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;p&#39;</span><span class="p">,</span> <span class="s">&#39;q&#39;</span><span class="p">]</span>
-
-    <span class="n">is_Rational</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">q</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">p</span>
-
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="c"># we might have a Float</span>
-                    <span class="n">neg_pow</span><span class="p">,</span> <span class="n">digits</span><span class="p">,</span> <span class="n">expt</span> <span class="o">=</span> <span class="n">decimal</span><span class="o">.</span><span class="n">Decimal</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">as_tuple</span><span class="p">()</span>
-                    <span class="n">p</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="n">neg_pow</span><span class="p">]</span><span class="o">*</span><span class="nb">int</span><span class="p">(</span><span class="s">&quot;&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">digits</span><span class="p">))</span>
-                    <span class="k">if</span> <span class="n">expt</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="c"># TODO: this branch needs a test</span>
-                        <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">Pow</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="n">expt</span><span class="p">),</span> <span class="mi">1</span><span class="p">)</span>
-                    <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">Pow</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="o">-</span><span class="n">expt</span><span class="p">))</span>
-                <span class="k">except</span> <span class="n">decimal</span><span class="o">.</span><span class="n">InvalidOperation</span><span class="p">:</span>
-                    <span class="n">f</span> <span class="o">=</span> <span class="n">regex</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="s">&#39;^([-+]?[0-9]+)/([0-9]+)$&#39;</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">f</span><span class="p">:</span>
-                        <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">groups</span><span class="p">()</span>
-                        <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="nb">int</span><span class="p">(</span><span class="n">d</span><span class="p">))</span>
-                    <span class="k">elif</span> <span class="n">p</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="s">&#39;/&#39;</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;/&#39;</span><span class="p">)</span>
-                        <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="n">p</span><span class="p">),</span> <span class="n">Rational</span><span class="p">(</span><span class="n">q</span><span class="p">))</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">pass</span>  <span class="c"># error will raise below</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">decimal</span><span class="o">.</span><span class="n">Decimal</span><span class="p">):</span>
-                    <span class="n">rv</span> <span class="o">=</span>  <span class="n">_decimal_to_Rational_prec</span><span class="p">(</span><span class="n">p</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">fractions</span><span class="o">.</span><span class="n">Fraction</span><span class="p">):</span>
-                        <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">numerator</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">denominator</span><span class="p">)</span>
-                <span class="k">except</span> <span class="ne">NameError</span><span class="p">:</span>
-                    <span class="k">pass</span>  <span class="c"># error will raise below</span>
-
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="p">(</span><span class="nb">float</span><span class="p">,</span> <span class="n">Float</span><span class="p">)):</span>
-                    <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="o">*</span><span class="n">_as_integer_ratio</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="n">Rational</span><span class="p">)):</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;invalid input: </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">p</span><span class="p">)</span>
-            <span class="n">q</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-            <span class="n">q</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-            <span class="n">p</span> <span class="o">*=</span> <span class="n">q</span><span class="o">.</span><span class="n">q</span>
-            <span class="n">q</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">p</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-            <span class="n">q</span> <span class="o">*=</span> <span class="n">p</span><span class="o">.</span><span class="n">q</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">p</span>
-
-        <span class="c"># p and q are now integers</span>
-        <span class="k">if</span> <span class="n">q</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">p</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">_errdict</span><span class="p">[</span><span class="s">&quot;divide&quot;</span><span class="p">]:</span>
-                    <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Indeterminate 0/0&quot;</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">if</span> <span class="n">p</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">if</span> <span class="n">q</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">q</span> <span class="o">=</span> <span class="o">-</span><span class="n">q</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="o">-</span><span class="n">p</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">igcd</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">p</span><span class="p">),</span> <span class="n">q</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">p</span> <span class="o">//=</span> <span class="n">n</span>
-            <span class="n">q</span> <span class="o">//=</span> <span class="n">n</span>
-        <span class="k">if</span> <span class="n">q</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">p</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">q</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">p</span> <span class="o">=</span> <span class="n">p</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">q</span> <span class="o">=</span> <span class="n">q</span>
-        <span class="c">#obj._args = (p, q)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-<div class="viewcode-block" id="Rational.limit_denominator"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.numbers.Rational.limit_denominator">[docs]</a>    <span class="k">def</span> <span class="nf">limit_denominator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">max_denominator</span><span class="o">=</span><span class="mi">1000000</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Closest Rational to self with denominator at most max_denominator.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Rational</span>
-<span class="sd">        &gt;&gt;&gt; Rational(&#39;3.141592653589793&#39;).limit_denominator(10)</span>
-<span class="sd">        22/7</span>
-<span class="sd">        &gt;&gt;&gt; Rational(&#39;3.141592653589793&#39;).limit_denominator(100)</span>
-<span class="sd">        311/99</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># Algorithm notes: For any real number x, define a *best upper</span>
-        <span class="c"># approximation* to x to be a rational number p/q such that:</span>
-        <span class="c">#</span>
-        <span class="c">#   (1) p/q &gt;= x, and</span>
-        <span class="c">#   (2) if p/q &gt; r/s &gt;= x then s &gt; q, for any rational r/s.</span>
-        <span class="c">#</span>
-        <span class="c"># Define *best lower approximation* similarly.  Then it can be</span>
-        <span class="c"># proved that a rational number is a best upper or lower</span>
-        <span class="c"># approximation to x if, and only if, it is a convergent or</span>
-        <span class="c"># semiconvergent of the (unique shortest) continued fraction</span>
-        <span class="c"># associated to x.</span>
-        <span class="c">#</span>
-        <span class="c"># To find a best rational approximation with denominator &lt;= M,</span>
-        <span class="c"># we find the best upper and lower approximations with</span>
-        <span class="c"># denominator &lt;= M and take whichever of these is closer to x.</span>
-        <span class="c"># In the event of a tie, the bound with smaller denominator is</span>
-        <span class="c"># chosen.  If both denominators are equal (which can happen</span>
-        <span class="c"># only when max_denominator == 1 and self is midway between</span>
-        <span class="c"># two integers) the lower bound---i.e., the floor of self, is</span>
-        <span class="c"># taken.</span>
-
-        <span class="k">if</span> <span class="n">max_denominator</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;max_denominator should be at least 1&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span> <span class="o">&lt;=</span> <span class="n">max_denominator</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-
-        <span class="n">p0</span><span class="p">,</span> <span class="n">q0</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">q1</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span>
-        <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">n</span><span class="o">//</span><span class="n">d</span>
-            <span class="n">q2</span> <span class="o">=</span> <span class="n">q0</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="n">q1</span>
-            <span class="k">if</span> <span class="n">q2</span> <span class="o">&gt;</span> <span class="n">max_denominator</span><span class="p">:</span>
-                <span class="k">break</span>
-            <span class="n">p0</span><span class="p">,</span> <span class="n">q0</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">q1</span> <span class="o">=</span> <span class="n">p1</span><span class="p">,</span> <span class="n">q1</span><span class="p">,</span> <span class="n">p0</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="n">p1</span><span class="p">,</span> <span class="n">q2</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">d</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span>
-
-        <span class="n">k</span> <span class="o">=</span> <span class="p">(</span><span class="n">max_denominator</span> <span class="o">-</span> <span class="n">q0</span><span class="p">)</span><span class="o">//</span><span class="n">q1</span>
-        <span class="n">bound1</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="n">p0</span> <span class="o">+</span> <span class="n">k</span><span class="o">*</span><span class="n">p1</span><span class="p">,</span> <span class="n">q0</span> <span class="o">+</span> <span class="n">k</span><span class="o">*</span><span class="n">q1</span><span class="p">)</span>
-        <span class="n">bound2</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">q1</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="n">bound2</span> <span class="o">-</span> <span class="bp">self</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">bound1</span> <span class="o">-</span> <span class="bp">self</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">bound2</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">bound1</span>
-</div>
-    <span class="k">def</span> <span class="nf">__getnewargs__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_hashable_content</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_positive</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">&gt;</span> <span class="mi">0</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_zero</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">==</span> <span class="mi">0</span>
-
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">q</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Float</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">other</span> <span class="o">+</span> <span class="bp">self</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Number</span><span class="o">.</span><span class="n">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">q</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Float</span><span class="p">):</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">other</span> <span class="o">+</span> <span class="bp">self</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Number</span><span class="o">.</span><span class="n">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Float</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">other</span><span class="o">*</span><span class="bp">self</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Number</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Float</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Number</span><span class="o">.</span><span class="n">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="n">__truediv__</span> <span class="o">=</span> <span class="n">__div__</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__mod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">q</span><span class="p">)</span> <span class="o">//</span> <span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">p</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">q</span> <span class="o">-</span> <span class="n">n</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">p</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Float</span><span class="p">):</span>
-            <span class="n">evalf</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-            <span class="c"># In case self.evalf() does not return Float.</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">evalf</span><span class="p">,</span> <span class="n">Float</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">evalf</span> <span class="o">%</span> <span class="n">other</span>
-        <span class="k">return</span> <span class="n">Number</span><span class="o">.</span><span class="n">__mod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__rmod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Rational</span><span class="o">.</span><span class="n">__mod__</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Float</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">other</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">Number</span><span class="o">.</span><span class="n">__rmod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expt</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expt</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expt</span><span class="p">,</span> <span class="n">Float</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="n">expt</span><span class="o">.</span><span class="n">_prec</span><span class="p">)</span><span class="o">**</span><span class="n">expt</span>
-            <span class="k">if</span> <span class="n">expt</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="c"># (3/4)**-2 -&gt; (4/3)**2</span>
-                <span class="n">ne</span> <span class="o">=</span> <span class="o">-</span><span class="n">expt</span>
-                <span class="k">if</span> <span class="p">(</span><span class="n">ne</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">expt</span><span class="o">.</span><span class="n">q</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="o">-</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">)</span><span class="o">**</span><span class="p">((</span><span class="n">expt</span><span class="o">.</span><span class="n">p</span> <span class="o">%</span> <span class="n">expt</span><span class="o">.</span><span class="n">q</span><span class="p">)</span> <span class="o">/</span>
-                               <span class="n">S</span><span class="p">(</span><span class="n">expt</span><span class="o">.</span><span class="n">q</span><span class="p">))</span><span class="o">*</span><span class="n">Rational</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">)</span><span class="o">**</span><span class="n">ne</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">**</span><span class="n">ne</span><span class="o">*</span><span class="n">Rational</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">)</span><span class="o">**</span><span class="n">ne</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">)</span><span class="o">**</span><span class="n">ne</span>
-            <span class="k">if</span> <span class="n">expt</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>  <span class="c"># -oo already caught by test for negative</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">&gt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="p">:</span>
-                    <span class="c"># (3/2)**oo -&gt; oo</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">&lt;</span> <span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="p">:</span>
-                    <span class="c"># (-3/2)**oo -&gt; oo + I*oo</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expt</span><span class="p">,</span> <span class="n">Integer</span><span class="p">):</span>
-                <span class="c"># (4/3)**2 -&gt; 4**2 / 3**2</span>
-                <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">**</span><span class="n">expt</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="o">**</span><span class="n">expt</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expt</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="c"># (4/3)**(5/6) -&gt; 4**(5/6)*3**(-5/6)</span>
-                    <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">)</span><span class="o">**</span><span class="n">expt</span><span class="o">*</span><span class="n">Integer</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">expt</span><span class="p">)</span>
-                <span class="c"># as the above caught negative self.p, now self is positive</span>
-                <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="p">)</span><span class="o">**</span><span class="n">Rational</span><span class="p">(</span>
-                <span class="n">expt</span><span class="o">.</span><span class="n">p</span><span class="o">*</span><span class="p">(</span><span class="n">expt</span><span class="o">.</span><span class="n">q</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">expt</span><span class="o">.</span><span class="n">q</span><span class="p">)</span> <span class="o">/</span> \
-                    <span class="n">Integer</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="p">)</span><span class="o">**</span><span class="n">Integer</span><span class="p">(</span><span class="n">expt</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_negative</span> <span class="ow">and</span> <span class="n">expt</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="bp">self</span><span class="p">)</span><span class="o">**</span><span class="n">expt</span>
-
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">_as_mpf_val</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">mlib</span><span class="o">.</span><span class="n">from_rational</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_mpmath_</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">mpmath</span><span class="o">.</span><span class="n">make_mpf</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">from_rational</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__abs__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__int__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span>
-        <span class="k">if</span> <span class="n">p</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="p">(</span><span class="o">-</span><span class="n">p</span><span class="o">//</span><span class="n">q</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">p</span><span class="o">//</span><span class="n">q</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">SympifyError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>    <span class="c"># sympy != other  --&gt;  not ==</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">NumberSymbol</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">is_irrational</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">return</span> <span class="n">other</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Float</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">mlib</span><span class="o">.</span><span class="n">mpf_eq</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_as_mpf_val</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">_prec</span><span class="p">),</span> <span class="n">other</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">p</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">q</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__gt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">SympifyError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>    <span class="c"># sympy &gt; other  --&gt; not &lt;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">NumberSymbol</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">other</span><span class="o">.</span><span class="n">__le__</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Float</span><span class="p">):</span>
-                <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">mpf_gt</span><span class="p">(</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">_as_mpf_val</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">_prec</span><span class="p">),</span> <span class="n">other</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">))</span>
-            <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">q</span> <span class="o">&gt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__gt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__ge__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">SympifyError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>    <span class="c"># sympy &gt; other  --&gt;  not &lt;=</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">NumberSymbol</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">other</span><span class="o">.</span><span class="n">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Float</span><span class="p">):</span>
-                <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">mpf_ge</span><span class="p">(</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">_as_mpf_val</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">_prec</span><span class="p">),</span> <span class="n">other</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">))</span>
-            <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">q</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__ge__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">SympifyError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>    <span class="c"># sympy &gt; other  --&gt; not &lt;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">NumberSymbol</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">other</span><span class="o">.</span><span class="n">__ge__</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Float</span><span class="p">):</span>
-                <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">mpf_lt</span><span class="p">(</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">_as_mpf_val</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">_prec</span><span class="p">),</span> <span class="n">other</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">))</span>
-            <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">q</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__le__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">SympifyError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>    <span class="c"># sympy &gt; other  --&gt;  not &lt;=</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">NumberSymbol</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">other</span><span class="o">.</span><span class="n">__gt__</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Float</span><span class="p">):</span>
-                <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="n">mlib</span><span class="o">.</span><span class="n">mpf_le</span><span class="p">(</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">_as_mpf_val</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">_prec</span><span class="p">),</span> <span class="n">other</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">))</span>
-            <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">q</span> <span class="o">&lt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__le__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Rational</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-<div class="viewcode-block" id="Rational.factors"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.numbers.Rational.factors">[docs]</a>    <span class="k">def</span> <span class="nf">factors</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">limit</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">use_trial</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">use_rho</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span>
-                <span class="n">use_pm1</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A wrapper to factorint which return factors of self that are</span>
-<span class="sd">        smaller than limit (or cheap to compute). Special methods of</span>
-<span class="sd">        factoring are disabled by default so that only trial division is used.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">factorint</span>
-
-        <span class="n">f</span> <span class="o">=</span> <span class="n">factorint</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="n">limit</span><span class="o">=</span><span class="n">limit</span><span class="p">,</span> <span class="n">use_trial</span><span class="o">=</span><span class="n">use_trial</span><span class="p">,</span>
-                      <span class="n">use_rho</span><span class="o">=</span><span class="n">use_rho</span><span class="p">,</span> <span class="n">use_pm1</span><span class="o">=</span><span class="n">use_pm1</span><span class="p">,</span>
-                      <span class="n">verbose</span><span class="o">=</span><span class="n">verbose</span><span class="p">)</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">p</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">factorint</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="n">limit</span><span class="o">=</span><span class="n">limit</span><span class="p">,</span>
-                              <span class="n">use_trial</span><span class="o">=</span><span class="n">use_trial</span><span class="p">,</span>
-                              <span class="n">use_rho</span><span class="o">=</span><span class="n">use_rho</span><span class="p">,</span>
-                              <span class="n">use_pm1</span><span class="o">=</span><span class="n">use_pm1</span><span class="p">,</span>
-                              <span class="n">verbose</span><span class="o">=</span><span class="n">verbose</span><span class="p">)</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-            <span class="n">f</span><span class="p">[</span><span class="n">p</span><span class="p">]</span> <span class="o">+=</span> <span class="o">-</span><span class="n">e</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="ow">and</span> <span class="mi">1</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-            <span class="k">del</span> <span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="p">{</span><span class="mi">1</span><span class="p">:</span> <span class="mi">1</span><span class="p">}</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">visual</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">dict</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="o">-</span><span class="mi">1</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-                <span class="n">f</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-                <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">args</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">args</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">Pow</span><span class="p">(</span><span class="o">*</span><span class="n">i</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;evaluate&#39;</span><span class="p">:</span><span class="bp">False</span><span class="p">})</span>
-                         <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">items</span><span class="p">())])</span>
-            <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;evaluate&#39;</span><span class="p">:</span> <span class="bp">False</span><span class="p">})</span>
-</div>
-    <span class="k">def</span> <span class="nf">as_numer_denom</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">),</span> <span class="n">Integer</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">Integer</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">)</span><span class="o">/</span><span class="n">sage</span><span class="o">.</span><span class="n">Integer</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Rational.as_content_primitive"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.numbers.Rational.as_content_primitive">[docs]</a>    <span class="k">def</span> <span class="nf">as_content_primitive</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">radical</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the tuple (R, self/R) where R is the positive Rational</span>
-<span class="sd">        extracted from self.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import S</span>
-<span class="sd">        &gt;&gt;&gt; (S(-3)/2).as_content_primitive()</span>
-<span class="sd">        (3/2, -1)</span>
-
-<span class="sd">        See docstring of Expr.as_content_primitive for more examples.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">return</span> <span class="o">-</span><span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="bp">self</span>
-
-<span class="c"># int -&gt; Integer</span></div></div>
-<span class="n">_intcache</span> <span class="o">=</span> <span class="p">{}</span>
-
-
-<span class="c"># TODO move this tracing facility to sympy/core/trace.py  ?</span>
-<span class="k">def</span> <span class="nf">_intcache_printinfo</span><span class="p">():</span>
-    <span class="n">ints</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">_intcache</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-    <span class="n">nhit</span> <span class="o">=</span> <span class="n">_intcache_hits</span>
-    <span class="n">nmiss</span> <span class="o">=</span> <span class="n">_intcache_misses</span>
-
-    <span class="k">if</span> <span class="n">nhit</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">nmiss</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">print</span><span class="p">()</span>
-        <span class="k">print</span><span class="p">(</span><span class="s">&#39;Integer cache statistic was not collected&#39;</span><span class="p">)</span>
-        <span class="k">return</span>
-
-    <span class="n">miss_ratio</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">nmiss</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">nhit</span> <span class="o">+</span> <span class="n">nmiss</span><span class="p">)</span>
-
-    <span class="k">print</span><span class="p">()</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;Integer cache statistic&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;-----------------------&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">()</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;#items: </span><span class="si">%i</span><span class="s">&#39;</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">ints</span><span class="p">))</span>
-    <span class="k">print</span><span class="p">()</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39; #hit   #miss               #total&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">()</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%5i</span><span class="s">   </span><span class="si">%5i</span><span class="s"> (</span><span class="si">%7.5f</span><span class="s"> </span><span class="si">%%</span><span class="s">)   </span><span class="si">%5i</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span>
-        <span class="n">nhit</span><span class="p">,</span> <span class="n">nmiss</span><span class="p">,</span> <span class="n">miss_ratio</span><span class="o">*</span><span class="mi">100</span><span class="p">,</span> <span class="n">nhit</span> <span class="o">+</span> <span class="n">nmiss</span><span class="p">))</span>
-    <span class="k">print</span><span class="p">()</span>
-    <span class="k">print</span><span class="p">(</span><span class="n">ints</span><span class="p">)</span>
-
-<span class="n">_intcache_hits</span> <span class="o">=</span> <span class="mi">0</span>
-<span class="n">_intcache_misses</span> <span class="o">=</span> <span class="mi">0</span>
-
-
-<span class="k">def</span> <span class="nf">int_trace</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="kn">import</span> <span class="nn">os</span>
-    <span class="k">if</span> <span class="n">os</span><span class="o">.</span><span class="n">getenv</span><span class="p">(</span><span class="s">&#39;SYMPY_TRACE_INT&#39;</span><span class="p">,</span> <span class="s">&#39;no&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="o">!=</span> <span class="s">&#39;yes&#39;</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="k">def</span> <span class="nf">Integer_tracer</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="k">global</span> <span class="n">_intcache_hits</span><span class="p">,</span> <span class="n">_intcache_misses</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">_intcache_hits</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">return</span> <span class="n">_intcache</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-        <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-            <span class="n">_intcache_hits</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="n">_intcache_misses</span> <span class="o">+=</span> <span class="mi">1</span>
-
-            <span class="k">return</span> <span class="n">f</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-
-    <span class="c"># also we want to hook our _intcache_printinfo into sys.atexit</span>
-    <span class="kn">import</span> <span class="nn">atexit</span>
-    <span class="n">atexit</span><span class="o">.</span><span class="n">register</span><span class="p">(</span><span class="n">_intcache_printinfo</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">Integer_tracer</span>
-
-
-<div class="viewcode-block" id="Integer"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.numbers.Integer">[docs]</a><span class="k">class</span> <span class="nc">Integer</span><span class="p">(</span><span class="n">Rational</span><span class="p">):</span>
-
-    <span class="n">q</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">is_integer</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="n">is_Integer</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;p&#39;</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">_as_mpf_val</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">mlib</span><span class="o">.</span><span class="n">from_int</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_mpmath_</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">rnd</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">mpmath</span><span class="o">.</span><span class="n">make_mpf</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_as_mpf_val</span><span class="p">(</span><span class="n">prec</span><span class="p">))</span>
-
-    <span class="c"># TODO caching with decorator, but not to degrade performance</span>
-    <span class="nd">@int_trace</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="n">i</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-        <span class="c"># whereas we cannot, in general, make a Rational from an</span>
-        <span class="c"># arbitrary expression, we can make an Integer unambiguously</span>
-        <span class="c"># (except when a non-integer expression happens to round to</span>
-        <span class="c"># an integer). So we proceed by taking int() of the input and</span>
-        <span class="c"># let the int routines determine whether the expression can</span>
-        <span class="c"># be made into an int or whether an error should be raised.</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">ival</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span>
-                <span class="s">&#39;Integer can only work with integer expressions.&#39;</span><span class="p">)</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_intcache</span><span class="p">[</span><span class="n">ival</span><span class="p">]</span>
-        <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-            <span class="c"># We only work with well-behaved integer types. This converts, for</span>
-            <span class="c"># example, numpy.int32 instances.</span>
-            <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-            <span class="n">obj</span><span class="o">.</span><span class="n">p</span> <span class="o">=</span> <span class="n">ival</span>
-
-            <span class="n">_intcache</span><span class="p">[</span><span class="n">ival</span><span class="p">]</span> <span class="o">=</span> <span class="n">obj</span>
-            <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">__getnewargs__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">,)</span>
-
-    <span class="c"># Arithmetic operations are here for efficiency</span>
-    <span class="k">def</span> <span class="nf">__int__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span>
-
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__abs__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__divmod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">.containers</span> <span class="kn">import</span> <span class="n">Tuple</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Integer</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="nb">divmod</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">p</span><span class="p">)))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Number</span><span class="o">.</span><span class="n">__divmod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__rdivmod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">.containers</span> <span class="kn">import</span> <span class="n">Tuple</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="nb">divmod</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">)))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">other</span> <span class="o">=</span> <span class="n">Number</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                <span class="n">msg</span> <span class="o">=</span> <span class="s">&quot;unsupported operand type(s) for divmod(): &#39;</span><span class="si">%s</span><span class="s">&#39; and &#39;</span><span class="si">%s</span><span class="s">&#39;&quot;</span>
-                <span class="n">oname</span> <span class="o">=</span> <span class="nb">type</span><span class="p">(</span><span class="n">other</span><span class="p">)</span><span class="o">.</span><span class="n">__name__</span>
-                <span class="n">sname</span> <span class="o">=</span> <span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__name__</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="n">msg</span> <span class="o">%</span> <span class="p">(</span><span class="n">oname</span><span class="p">,</span> <span class="n">sname</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">Number</span><span class="o">.</span><span class="n">__divmod__</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span>
-
-    <span class="c"># TODO make it decorator + bytecodehacks?</span>
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">+</span> <span class="n">other</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Integer</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">+</span> <span class="n">other</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Rational</span><span class="o">.</span><span class="n">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__radd__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="n">other</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Rational</span><span class="o">.</span><span class="n">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">-</span> <span class="n">other</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Integer</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">-</span> <span class="n">other</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Rational</span><span class="o">.</span><span class="n">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__rsub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="n">other</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Rational</span><span class="o">.</span><span class="n">__rsub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">*</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Integer</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Rational</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="n">other</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Rational</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__mod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">%</span> <span class="n">other</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Integer</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">%</span> <span class="n">other</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Rational</span><span class="o">.</span><span class="n">__mod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__rmod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="n">other</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Integer</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">p</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Rational</span><span class="o">.</span><span class="n">__rmod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">==</span> <span class="n">other</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Integer</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Rational</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__gt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">&gt;</span> <span class="n">other</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Integer</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">&gt;</span> <span class="n">other</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Rational</span><span class="o">.</span><span class="n">__gt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">&lt;</span> <span class="n">other</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Integer</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">&lt;</span> <span class="n">other</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Rational</span><span class="o">.</span><span class="n">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__ge__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">&gt;=</span> <span class="n">other</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Integer</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">&gt;=</span> <span class="n">other</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Rational</span><span class="o">.</span><span class="n">__ge__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__le__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">&lt;=</span> <span class="n">other</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Integer</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">&lt;=</span> <span class="n">other</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Rational</span><span class="o">.</span><span class="n">__le__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Integer</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__index__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span>
-
-    <span class="c">########################################</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_odd</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">%</span> <span class="mi">2</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expt</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Tries to do some simplifications on self**expt</span>
-
-<span class="sd">        Returns None if no further simplifications can be done</span>
-
-<span class="sd">        When exponent is a fraction (so we have for example a square root),</span>
-<span class="sd">        we try to find a simpler representation by factoring the argument</span>
-<span class="sd">        up to factors of 2**15, e.g.</span>
-
-<span class="sd">          - sqrt(4) becomes 2</span>
-<span class="sd">          - sqrt(-4) becomes 2*I</span>
-<span class="sd">          - (2**(3+7)*3**(6+7))**Rational(1,7) becomes 6*18**(3/7)</span>
-
-<span class="sd">        Further simplification would require a special call to factorint on</span>
-<span class="sd">        the argument which is not done here for sake of speed.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">perfect_power</span>
-
-        <span class="k">if</span> <span class="n">expt</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">&gt;</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-            <span class="c"># cases -1, 0, 1 are done in their respective classes</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">if</span> <span class="n">expt</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">**</span><span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expt</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="c"># simplify when expt is even</span>
-            <span class="c"># (-2)**k --&gt; 2**k</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_negative</span> <span class="ow">and</span> <span class="n">expt</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="bp">self</span><span class="p">)</span><span class="o">**</span><span class="n">expt</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expt</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-            <span class="k">return</span>
-        <span class="k">if</span> <span class="n">expt</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="c"># we extract I for this special case since everyone is doing so</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">*</span><span class="n">Pow</span><span class="p">(</span><span class="o">-</span><span class="bp">self</span><span class="p">,</span> <span class="n">expt</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">expt</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="c"># invert base and change sign on exponent</span>
-            <span class="n">ne</span> <span class="o">=</span> <span class="o">-</span><span class="n">expt</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">expt</span><span class="o">.</span><span class="n">q</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="o">-</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">)</span><span class="o">**</span><span class="p">((</span><span class="n">expt</span><span class="o">.</span><span class="n">p</span> <span class="o">%</span> <span class="n">expt</span><span class="o">.</span><span class="n">q</span><span class="p">)</span> <span class="o">/</span>
-                            <span class="n">S</span><span class="p">(</span><span class="n">expt</span><span class="o">.</span><span class="n">q</span><span class="p">))</span><span class="o">*</span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="bp">self</span><span class="p">)</span><span class="o">**</span><span class="n">ne</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">)</span><span class="o">**</span><span class="n">ne</span><span class="o">*</span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="bp">self</span><span class="p">)</span><span class="o">**</span><span class="n">ne</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">)</span><span class="o">**</span><span class="n">ne</span>
-        <span class="c"># see if base is a perfect root, sqrt(4) --&gt; 2</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">xexact</span> <span class="o">=</span> <span class="n">integer_nthroot</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">),</span> <span class="n">expt</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">xexact</span><span class="p">:</span>
-            <span class="c"># if it&#39;s a perfect root we&#39;ve finished</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">Integer</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="nb">abs</span><span class="p">(</span><span class="n">expt</span><span class="o">.</span><span class="n">p</span><span class="p">))</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">*=</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">**</span><span class="n">expt</span>
-            <span class="k">return</span> <span class="n">result</span>
-
-        <span class="c"># The following is an algorithm where we collect perfect roots</span>
-        <span class="c"># from the factors of base.</span>
-
-        <span class="c"># if it&#39;s not an nth root, it still might be a perfect power</span>
-        <span class="n">b_pos</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">))</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">perfect_power</span><span class="p">(</span><span class="n">b_pos</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">p</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="nb">dict</span> <span class="o">=</span> <span class="p">{</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span> <span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]}</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="nb">dict</span> <span class="o">=</span> <span class="n">Integer</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">factors</span><span class="p">(</span><span class="n">limit</span><span class="o">=</span><span class="mi">2</span><span class="o">**</span><span class="mi">15</span><span class="p">)</span>
-
-        <span class="c"># now process the dict of factors</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="nb">dict</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">out_int</span> <span class="o">=</span> <span class="mi">1</span>  <span class="c"># integer part</span>
-        <span class="n">out_rad</span> <span class="o">=</span> <span class="mi">1</span>  <span class="c"># extracted radicals</span>
-        <span class="n">sqr_int</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">sqr_gcd</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">sqr_dict</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">prime</span><span class="p">,</span> <span class="n">exponent</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="nb">dict</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-            <span class="n">exponent</span> <span class="o">*=</span> <span class="n">expt</span><span class="o">.</span><span class="n">p</span>
-            <span class="c"># remove multiples of expt.q: (2**12)**(1/10) -&gt; 2*(2**2)**(1/10)</span>
-            <span class="n">div_e</span><span class="p">,</span> <span class="n">div_m</span> <span class="o">=</span> <span class="nb">divmod</span><span class="p">(</span><span class="n">exponent</span><span class="p">,</span> <span class="n">expt</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">div_e</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">out_int</span> <span class="o">*=</span> <span class="n">prime</span><span class="o">**</span><span class="n">div_e</span>
-            <span class="k">if</span> <span class="n">div_m</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="c"># see if the reduced exponent shares a gcd with e.q</span>
-                <span class="c"># (2**2)**(1/10) -&gt; 2**(1/5)</span>
-                <span class="n">g</span> <span class="o">=</span> <span class="n">igcd</span><span class="p">(</span><span class="n">div_m</span><span class="p">,</span> <span class="n">expt</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">g</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">out_rad</span> <span class="o">*=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">prime</span><span class="p">,</span> <span class="n">Rational</span><span class="p">(</span><span class="n">div_m</span><span class="o">//</span><span class="n">g</span><span class="p">,</span> <span class="n">expt</span><span class="o">.</span><span class="n">q</span><span class="o">//</span><span class="n">g</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">sqr_dict</span><span class="p">[</span><span class="n">prime</span><span class="p">]</span> <span class="o">=</span> <span class="n">div_m</span>
-        <span class="c"># identify gcd of remaining powers</span>
-        <span class="k">for</span> <span class="n">p</span><span class="p">,</span> <span class="n">ex</span> <span class="ow">in</span> <span class="n">sqr_dict</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="k">if</span> <span class="n">sqr_gcd</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">sqr_gcd</span> <span class="o">=</span> <span class="n">ex</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">sqr_gcd</span> <span class="o">=</span> <span class="n">igcd</span><span class="p">(</span><span class="n">sqr_gcd</span><span class="p">,</span> <span class="n">ex</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">sqr_gcd</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">break</span>
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">sqr_dict</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">sqr_int</span> <span class="o">*=</span> <span class="n">k</span><span class="o">**</span><span class="p">(</span><span class="n">v</span><span class="o">//</span><span class="n">sqr_gcd</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">sqr_int</span> <span class="o">==</span> <span class="bp">self</span> <span class="ow">and</span> <span class="n">out_int</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">out_rad</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">out_int</span><span class="o">*</span><span class="n">out_rad</span><span class="o">*</span><span class="n">Pow</span><span class="p">(</span><span class="n">sqr_int</span><span class="p">,</span> <span class="n">Rational</span><span class="p">(</span><span class="n">sqr_gcd</span><span class="p">,</span> <span class="n">expt</span><span class="o">.</span><span class="n">q</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_prime</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">isprime</span>
-
-        <span class="k">return</span> <span class="n">isprime</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">as_numer_denom</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-    <span class="k">def</span> <span class="nf">__floordiv__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p</span> <span class="o">//</span> <span class="n">Integer</span><span class="p">(</span><span class="n">other</span><span class="p">)</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__rfloordiv__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="n">Integer</span><span class="p">(</span><span class="n">other</span><span class="p">)</span><span class="o">.</span><span class="n">p</span> <span class="o">//</span> <span class="bp">self</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-
-<span class="c"># Add sympify converters</span></div>
-<span class="n">converter</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="n">converter</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="n">Integer</span>
-
-
-<span class="k">class</span> <span class="nc">RationalConstant</span><span class="p">(</span><span class="n">Rational</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Abstract base class for rationals with specific behaviors</span>
-
-<span class="sd">    Derived classes must define class attributes p and q and should probably all</span>
-<span class="sd">    be singletons.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">AtomicExpr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">IntegerConstant</span><span class="p">(</span><span class="n">Integer</span><span class="p">):</span>
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">AtomicExpr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">Zero</span><span class="p">(</span><span class="n">IntegerConstant</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">Singleton</span><span class="p">):</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">q</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">is_positive</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_negative</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_finite</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_zero</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_composite</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">__abs__</span><span class="p">():</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">():</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__rmul__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span> <span class="ow">or</span> \
-            <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span> <span class="ow">or</span> \
-            <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span> <span class="ow">or</span> \
-                <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expt</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expt</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">if</span> <span class="n">expt</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">if</span> <span class="n">expt</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-        <span class="c"># infinities are already handled with pos and neg</span>
-        <span class="c"># tests above; now throw away leading numbers on Mul</span>
-        <span class="c"># exponent</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">terms</span> <span class="o">=</span> <span class="n">expt</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="o">**</span><span class="n">terms</span>
-        <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>  <span class="c"># there is a Number to discard</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">**</span><span class="n">terms</span>
-
-    <span class="k">def</span> <span class="nf">_eval_order</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">):</span>
-        <span class="c"># Order(0,x) -&gt; 0</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="k">def</span> <span class="nf">__bool__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-
-<span class="k">class</span> <span class="nc">One</span><span class="p">(</span><span class="n">IntegerConstant</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">Singleton</span><span class="p">):</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">q</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">__abs__</span><span class="p">():</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">():</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expt</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="k">def</span> <span class="nf">_eval_order</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">):</span>
-        <span class="k">return</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">factors</span><span class="p">(</span><span class="n">limit</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">use_trial</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">use_rho</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">use_pm1</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span>
-                <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">visual</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">return</span> <span class="p">{</span><span class="mi">1</span><span class="p">:</span> <span class="mi">1</span><span class="p">}</span>
-
-
-<span class="k">class</span> <span class="nc">NegativeOne</span><span class="p">(</span><span class="n">IntegerConstant</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">Singleton</span><span class="p">):</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-    <span class="n">q</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">__abs__</span><span class="p">():</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">():</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expt</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expt</span><span class="o">.</span><span class="n">is_odd</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span>
-        <span class="k">if</span> <span class="n">expt</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expt</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expt</span><span class="p">,</span> <span class="n">Float</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">Float</span><span class="p">(</span><span class="o">-</span><span class="mf">1.0</span><span class="p">)</span><span class="o">**</span><span class="n">expt</span>
-            <span class="k">if</span> <span class="n">expt</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">if</span> <span class="n">expt</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span> <span class="ow">or</span> <span class="n">expt</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">if</span> <span class="n">expt</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expt</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">expt</span><span class="o">.</span><span class="n">q</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">**</span><span class="n">Integer</span><span class="p">(</span><span class="n">expt</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-                <span class="n">i</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="nb">divmod</span><span class="p">(</span><span class="n">expt</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="n">expt</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">i</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">self</span><span class="o">**</span><span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">**</span><span class="n">Rational</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">expt</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-        <span class="k">return</span>
-
-
-<span class="k">class</span> <span class="nc">Half</span><span class="p">(</span><span class="n">RationalConstant</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">Singleton</span><span class="p">):</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">q</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">__abs__</span><span class="p">():</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span>
-
-
-<span class="k">class</span> <span class="nc">Infinity</span><span class="p">(</span><span class="n">Number</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">Singleton</span><span class="p">):</span>
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_positive</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_bounded</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_finite</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_infinitesimal</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_integer</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">is_rational</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">is_odd</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">AtomicExpr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span> <span class="ow">or</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">other</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">other</span> <span class="o">==</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;-inf&#39;</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;inf&#39;</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">return</span> <span class="bp">NotImplemented</span>
-    <span class="n">__radd__</span> <span class="o">=</span> <span class="n">__add__</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span> <span class="ow">or</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">other</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">other</span> <span class="o">==</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;inf&#39;</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;inf&#39;</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span> <span class="ow">or</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">other</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">other</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-                <span class="k">if</span> <span class="n">other</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;inf&#39;</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;-inf&#39;</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">other</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-        <span class="k">return</span> <span class="bp">NotImplemented</span>
-    <span class="n">__rmul__</span> <span class="o">=</span> <span class="n">__mul__</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span> <span class="ow">or</span> \
-                <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span> <span class="ow">or</span> \
-                    <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">other</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">other</span> <span class="o">==</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;-inf&#39;</span><span class="p">)</span> <span class="ow">or</span> \
-                        <span class="n">other</span> <span class="o">==</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;inf&#39;</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-                <span class="k">elif</span> <span class="n">other</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;inf&#39;</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;-inf&#39;</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">other</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-        <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="n">__truediv__</span> <span class="o">=</span> <span class="n">__div__</span>
-
-    <span class="k">def</span> <span class="nf">__abs__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expt</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        ``expt`` is symbolic object but not equal to 0 or 1.</span>
-
-<span class="sd">        ================ ======= ==============================</span>
-<span class="sd">        Expression       Result  Notes</span>
-<span class="sd">        ================ ======= ==============================</span>
-<span class="sd">        ``oo ** nan``    ``nan``</span>
-<span class="sd">        ``oo ** -p``     ``0``   ``p`` is number, ``oo``</span>
-<span class="sd">        ================ ======= ==============================</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expt</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">if</span> <span class="n">expt</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">if</span> <span class="n">expt</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-
-        <span class="k">if</span> <span class="n">expt</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">**</span><span class="n">expt</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">_as_mpf_val</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">mlib</span><span class="o">.</span><span class="n">finf</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">oo</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Infinity</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">other</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-
-    <span class="k">def</span> <span class="nf">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__le__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-
-    <span class="k">def</span> <span class="nf">__gt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">other</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-
-    <span class="k">def</span> <span class="nf">__ge__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__mod__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-
-    <span class="n">__rmod__</span> <span class="o">=</span> <span class="n">__mod__</span>
-
-<span class="n">oo</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-
-
-<span class="k">class</span> <span class="nc">NegativeInfinity</span><span class="p">(</span><span class="n">Number</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">Singleton</span><span class="p">):</span>
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_real</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_positive</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_bounded</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_finite</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_infinitesimal</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_integer</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">is_rational</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">AtomicExpr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span> <span class="ow">or</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">other</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">other</span> <span class="o">==</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;inf&#39;</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;nan&#39;</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;-inf&#39;</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-        <span class="k">return</span> <span class="bp">NotImplemented</span>
-    <span class="n">__radd__</span> <span class="o">=</span> <span class="n">__add__</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span> <span class="ow">or</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">other</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">other</span> <span class="o">==</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;-inf&#39;</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;nan&#39;</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;-inf&#39;</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-        <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span> <span class="ow">or</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">other</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span> <span class="ow">or</span> <span class="n">other</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-                <span class="k">elif</span> <span class="n">other</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;-inf&#39;</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;inf&#39;</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">return</span> <span class="bp">NotImplemented</span>
-    <span class="n">__rmul__</span> <span class="o">=</span> <span class="n">__mul__</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span> <span class="ow">or</span> \
-                <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span> <span class="ow">or</span> \
-                    <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">other</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">other</span> <span class="o">==</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;-inf&#39;</span><span class="p">)</span> <span class="ow">or</span> \
-                    <span class="n">other</span> <span class="o">==</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;inf&#39;</span><span class="p">)</span> <span class="ow">or</span> \
-                        <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-                <span class="k">elif</span> <span class="n">other</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;-inf&#39;</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;inf&#39;</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">other</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="n">__truediv__</span> <span class="o">=</span> <span class="n">__div__</span>
-
-    <span class="k">def</span> <span class="nf">__abs__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expt</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        ``expt`` is symbolic object but not equal to 0 or 1.</span>
-
-<span class="sd">        ================ ======= ==============================</span>
-<span class="sd">        Expression       Result  Notes</span>
-<span class="sd">        ================ ======= ==============================</span>
-<span class="sd">        ``(-oo) ** nan`` ``nan``</span>
-<span class="sd">        ``(-oo) ** oo``  ``nan``</span>
-<span class="sd">        ``(-oo) ** -oo`` ``nan``</span>
-<span class="sd">        ``(-oo) ** e``   ``oo``  ``e`` is positive even integer</span>
-<span class="sd">        ``(-oo) ** o``   ``-oo`` ``o`` is positive odd integer</span>
-<span class="sd">        ================ ======= ==============================</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expt</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">expt</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span> <span class="ow">or</span> \
-                <span class="n">expt</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span> <span class="ow">or</span> \
-                    <span class="n">expt</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expt</span><span class="p">,</span> <span class="n">Integer</span><span class="p">)</span> <span class="ow">and</span> <span class="n">expt</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">expt</span><span class="o">.</span><span class="n">is_odd</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">**</span><span class="n">expt</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="o">**</span><span class="n">expt</span>
-
-    <span class="k">def</span> <span class="nf">_as_mpf_val</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">mlib</span><span class="o">.</span><span class="n">fninf</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="o">-</span><span class="p">(</span><span class="n">sage</span><span class="o">.</span><span class="n">oo</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">NegativeInfinity</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">other</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-
-    <span class="k">def</span> <span class="nf">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">other</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-
-    <span class="k">def</span> <span class="nf">__le__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__gt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__ge__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-
-
-<span class="k">class</span> <span class="nc">NaN</span><span class="p">(</span><span class="n">Number</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">Singleton</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Not a Number.</span>
-
-<span class="sd">    This represents the corresponding data type to floating point nan, which</span>
-<span class="sd">    is defined in the IEEE 754 floating point standard, and corresponds to the</span>
-<span class="sd">    Python ``float(&#39;nan&#39;)``.</span>
-
-<span class="sd">    NaN serves as a place holder for numeric values that are indeterminate,</span>
-<span class="sd">    but not infinite.  Most operations on nan, produce another nan.  Most</span>
-<span class="sd">    indeterminate forms, such as ``0/0`` or ``oo - oo` produce nan.  Three</span>
-<span class="sd">    exceptions are ``0**0``, ``1**oo``, and ``oo**0``, which all produce ``1``</span>
-<span class="sd">    (this is consistent with Python&#39;s float).</span>
-
-<span class="sd">    NaN is a singleton, and can be accessed by ``S.NaN``, or can be imported</span>
-<span class="sd">    as ``nan``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import nan, S, oo</span>
-<span class="sd">    &gt;&gt;&gt; nan is S.NaN</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; oo - oo</span>
-<span class="sd">    nan</span>
-<span class="sd">    &gt;&gt;&gt; nan + 1</span>
-<span class="sd">    nan</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - http://en.wikipedia.org/wiki/NaN</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_real</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">is_rational</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">is_integer</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">is_comparable</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_finite</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">is_bounded</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">is_zero</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">is_prime</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">is_positive</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">is_negative</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">AtomicExpr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;other&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="n">__truediv__</span> <span class="o">=</span> <span class="n">__div__</span>
-
-    <span class="k">def</span> <span class="nf">_as_mpf_val</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_mpf_nan</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">NaN</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">NaN</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">other</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">other</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-
-    <span class="k">def</span> <span class="nf">__gt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__ge__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__le__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-<span class="n">nan</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-
-
-<span class="k">class</span> <span class="nc">ComplexInfinity</span><span class="p">(</span><span class="n">AtomicExpr</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">Singleton</span><span class="p">):</span>
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_bounded</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_real</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">is_number</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">AtomicExpr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">__abs__</span><span class="p">():</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">():</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expt</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expt</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expt</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">expt</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">expt</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-<span class="n">zoo</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span>
-
-
-<div class="viewcode-block" id="NumberSymbol"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.numbers.NumberSymbol">[docs]</a><span class="k">class</span> <span class="nc">NumberSymbol</span><span class="p">(</span><span class="n">AtomicExpr</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">Singleton</span><span class="p">):</span>
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_bounded</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_finite</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_number</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="n">is_NumberSymbol</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">AtomicExpr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-
-<div class="viewcode-block" id="NumberSymbol.approximation"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.numbers.NumberSymbol.approximation">[docs]</a>    <span class="k">def</span> <span class="nf">approximation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">number_cls</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Return an interval with number_cls endpoints</span>
-<span class="sd">        that contains the value of NumberSymbol.</span>
-<span class="sd">        If not implemented, then return None.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_as_mpf_val</span><span class="p">(</span><span class="n">prec</span><span class="p">),</span> <span class="n">prec</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">SympifyError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>    <span class="c"># sympy != other  --&gt;  not ==</span>
-        <span class="k">if</span> <span class="bp">self</span> <span class="ow">is</span> <span class="n">other</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">)</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_irrational</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="k">return</span> <span class="bp">False</span>    <span class="c"># NumberSymbol != non-(Number|self)</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">SympifyError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>    <span class="c"># sympy &gt; other  --&gt; not &lt;</span>
-        <span class="k">if</span> <span class="bp">self</span> <span class="ow">is</span> <span class="n">other</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="n">approx</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">approximation_interval</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">__class__</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">approx</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">l</span><span class="p">,</span> <span class="n">u</span> <span class="o">=</span> <span class="n">approx</span>
-                <span class="k">if</span> <span class="n">other</span> <span class="o">&lt;</span> <span class="n">l</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-                <span class="k">if</span> <span class="n">other</span> <span class="o">&gt;</span> <span class="n">u</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span> <span class="o">&lt;</span> <span class="n">other</span>
-        <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span> <span class="o">&lt;</span> <span class="n">other</span>
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__le__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">SympifyError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>    <span class="c"># sympy &gt; other  --&gt; not &lt;=</span>
-        <span class="k">if</span> <span class="bp">self</span> <span class="ow">is</span> <span class="n">other</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="n">other</span>
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__le__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__gt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="bp">self</span><span class="p">)</span> <span class="o">&lt;</span> <span class="p">(</span><span class="o">-</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__ge__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="bp">self</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="p">(</span><span class="o">-</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__int__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># subclass with appropriate return value</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">NumberSymbol</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-</div>
-<span class="k">class</span> <span class="nc">Exp1</span><span class="p">(</span><span class="n">NumberSymbol</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">Singleton</span><span class="p">):</span>
-    <span class="n">is_real</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_positive</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_negative</span> <span class="o">=</span> <span class="bp">False</span>  <span class="c"># XXX Forces is_negative/is_nonnegative</span>
-    <span class="n">is_irrational</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">__abs__</span><span class="p">():</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Exp1</span>
-
-    <span class="k">def</span> <span class="nf">__int__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">2</span>
-
-    <span class="k">def</span> <span class="nf">_as_mpf_val</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">mpf_e</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">approximation_interval</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">number_cls</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">number_cls</span><span class="p">,</span> <span class="n">Integer</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">Integer</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">number_cls</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-            <span class="k">pass</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expt</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">expt</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">e</span>
-<span class="n">E</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Exp1</span>
-
-
-<span class="k">class</span> <span class="nc">Pi</span><span class="p">(</span><span class="n">NumberSymbol</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">Singleton</span><span class="p">):</span>
-    <span class="n">is_real</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_positive</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_negative</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_irrational</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">__abs__</span><span class="p">():</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-
-    <span class="k">def</span> <span class="nf">__int__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">3</span>
-
-    <span class="k">def</span> <span class="nf">_as_mpf_val</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">mpf_pi</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">approximation_interval</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">number_cls</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">number_cls</span><span class="p">,</span> <span class="n">Integer</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">Integer</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">4</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">number_cls</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="mi">223</span><span class="p">,</span> <span class="mi">71</span><span class="p">),</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">22</span><span class="p">,</span> <span class="mi">7</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">pi</span>
-<span class="n">pi</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-
-
-<span class="k">class</span> <span class="nc">GoldenRatio</span><span class="p">(</span><span class="n">NumberSymbol</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">Singleton</span><span class="p">):</span>
-    <span class="n">is_real</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_positive</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_negative</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_irrational</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">__int__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">_as_mpf_val</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-         <span class="c"># XXX track down why this has to be increased</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">mlib</span><span class="o">.</span><span class="n">from_man_exp</span><span class="p">(</span><span class="n">phi_fixed</span><span class="p">(</span><span class="n">prec</span> <span class="o">+</span> <span class="mi">10</span><span class="p">),</span> <span class="o">-</span><span class="n">prec</span> <span class="o">-</span> <span class="mi">10</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">mpf_norm</span><span class="p">(</span><span class="n">rv</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_expand_func</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">approximation_interval</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">number_cls</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">number_cls</span><span class="p">,</span> <span class="n">Integer</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">number_cls</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-            <span class="k">pass</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">golden_ratio</span>
-
-
-<span class="k">class</span> <span class="nc">EulerGamma</span><span class="p">(</span><span class="n">NumberSymbol</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">Singleton</span><span class="p">):</span>
-    <span class="n">is_real</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_positive</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_negative</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_irrational</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">__int__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">0</span>
-
-    <span class="k">def</span> <span class="nf">_as_mpf_val</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-         <span class="c"># XXX track down why this has to be increased</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="n">mlib</span><span class="o">.</span><span class="n">libhyper</span><span class="o">.</span><span class="n">euler_fixed</span><span class="p">(</span><span class="n">prec</span> <span class="o">+</span> <span class="mi">10</span><span class="p">)</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">mlib</span><span class="o">.</span><span class="n">from_man_exp</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="o">-</span><span class="n">prec</span> <span class="o">-</span> <span class="mi">10</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">mpf_norm</span><span class="p">(</span><span class="n">rv</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">approximation_interval</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">number_cls</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">number_cls</span><span class="p">,</span> <span class="n">Integer</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">number_cls</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">euler_gamma</span>
-
-
-<span class="k">class</span> <span class="nc">Catalan</span><span class="p">(</span><span class="n">NumberSymbol</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">Singleton</span><span class="p">):</span>
-    <span class="n">is_real</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_positive</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_negative</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_irrational</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">__int__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">0</span>
-
-    <span class="k">def</span> <span class="nf">_as_mpf_val</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="c"># XXX track down why this has to be increased</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="n">mlib</span><span class="o">.</span><span class="n">catalan_fixed</span><span class="p">(</span><span class="n">prec</span> <span class="o">+</span> <span class="mi">10</span><span class="p">)</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">mlib</span><span class="o">.</span><span class="n">from_man_exp</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="o">-</span><span class="n">prec</span> <span class="o">-</span> <span class="mi">10</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">mpf_norm</span><span class="p">(</span><span class="n">rv</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">approximation_interval</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">number_cls</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">number_cls</span><span class="p">,</span> <span class="n">Integer</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">number_cls</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mi">10</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">catalan</span>
-
-
-<span class="k">class</span> <span class="nc">ImaginaryUnit</span><span class="p">(</span><span class="n">AtomicExpr</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">Singleton</span><span class="p">):</span>
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_imaginary</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_bounded</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_finite</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_number</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">__abs__</span><span class="p">():</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expt</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        b is I = sqrt(-1)</span>
-<span class="sd">        e is symbolic object but not equal to 0, 1</span>
-
-<span class="sd">        I**r -&gt; (-1)**(r/2) -&gt; exp(r/2*Pi*I) -&gt; sin(Pi*r/2) + cos(Pi*r/2)*I, r is decimal</span>
-<span class="sd">        I**0 mod 4 -&gt; 1</span>
-<span class="sd">        I**1 mod 4 -&gt; I</span>
-<span class="sd">        I**2 mod 4 -&gt; -1</span>
-<span class="sd">        I**3 mod 4 -&gt; -I</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expt</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expt</span><span class="p">,</span> <span class="n">Integer</span><span class="p">):</span>
-                <span class="n">expt</span> <span class="o">=</span> <span class="n">expt</span><span class="o">.</span><span class="n">p</span> <span class="o">%</span> <span class="mi">4</span>
-                <span class="k">if</span> <span class="n">expt</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-                <span class="k">if</span> <span class="n">expt</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-                <span class="k">if</span> <span class="n">expt</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">One</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">expt</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">as_base_exp</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">I</span>
-
-<span class="n">I</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-
-<span class="k">try</span><span class="p">:</span>
-    <span class="c"># fractions is only available for python 2.6+</span>
-    <span class="kn">import</span> <span class="nn">fractions</span>
-
-    <span class="k">def</span> <span class="nf">sympify_fractions</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">numerator</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">denominator</span><span class="p">)</span>
-
-    <span class="n">converter</span><span class="p">[</span><span class="n">fractions</span><span class="o">.</span><span class="n">Fraction</span><span class="p">]</span> <span class="o">=</span> <span class="n">sympify_fractions</span>
-<span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>
-    <span class="k">pass</span>
-
-<span class="k">try</span><span class="p">:</span>
-    <span class="kn">import</span> <span class="nn">gmpy</span>
-
-    <span class="k">def</span> <span class="nf">sympify_mpz</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">sympify_mpq</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">numer</span><span class="p">()),</span> <span class="nb">int</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">denom</span><span class="p">()))</span>
-
-    <span class="n">converter</span><span class="p">[</span><span class="nb">type</span><span class="p">(</span><span class="n">gmpy</span><span class="o">.</span><span class="n">mpz</span><span class="p">(</span><span class="mi">1</span><span class="p">))]</span> <span class="o">=</span> <span class="n">sympify_mpz</span>
-    <span class="n">converter</span><span class="p">[</span><span class="nb">type</span><span class="p">(</span><span class="n">gmpy</span><span class="o">.</span><span class="n">mpq</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">))]</span> <span class="o">=</span> <span class="n">sympify_mpq</span>
-<span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>
-    <span class="k">pass</span>
-
-
-<span class="k">def</span> <span class="nf">sympify_mpmath</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">_from_mpmath</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">context</span><span class="o">.</span><span class="n">prec</span><span class="p">)</span>
-
-<span class="n">converter</span><span class="p">[</span><span class="n">mpnumeric</span><span class="p">]</span> <span class="o">=</span> <span class="n">sympify_mpmath</span>
-
-
-<span class="k">def</span> <span class="nf">sympify_complex</span><span class="p">(</span><span class="n">a</span><span class="p">):</span>
-    <span class="n">real</span><span class="p">,</span> <span class="n">imag</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">imag</span><span class="p">)))</span>
-    <span class="k">return</span> <span class="n">real</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">*</span><span class="n">imag</span>
-
-<span class="n">converter</span><span class="p">[</span><span class="nb">complex</span><span class="p">]</span> <span class="o">=</span> <span class="n">sympify_complex</span>
-
-<span class="n">_intcache</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-<span class="n">_intcache</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-<span class="n">_intcache</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span>
-
-<span class="kn">from</span> <span class="nn">.power</span> <span class="kn">import</span> <span class="n">Pow</span><span class="p">,</span> <span class="n">integer_nthroot</span>
-<span class="kn">from</span> <span class="nn">.mul</span> <span class="kn">import</span> <span class="n">Mul</span>
-<span class="n">Mul</span><span class="o">.</span><span class="n">identity</span> <span class="o">=</span> <span class="n">One</span><span class="p">()</span>
-<span class="kn">from</span> <span class="nn">.add</span> <span class="kn">import</span> <span class="n">Add</span>
-<span class="n">Add</span><span class="o">.</span><span class="n">identity</span> <span class="o">=</span> <span class="n">Zero</span><span class="p">()</span>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-<div id="searchbox" style="display: none">
-  <h3>Quick search</h3>
-    <form class="search" action="../../../search.html" method="get">
-      <input type="text" name="q" />
-      <input type="submit" value="Go" />
-      <input type="hidden" name="check_keywords" value="yes" />
-      <input type="hidden" name="area" value="default" />
-    </form>
-    <p class="searchtip" style="font-size: 90%">
-    Enter search terms or a module, class or function name.
-    </p>
-</div>
-<script type="text/javascript">$('#searchbox').show(0);</script>
-        </div>
-      </div>
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-    </div>
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-      <h3>Navigation</h3>
-      <ul>
-        <li class="right" style="margin-right: 10px">
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-             >index</a></li>
-        <li class="right" >
-          <a href="../../../py-modindex.html" title="Python Module Index"
-             >modules</a> |</li>
-        <li class="right" >
-          <a href="../../../np-modindex.html" title="Python Module Index"
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-          <li><a href="../../index.html" >Module code</a> &raquo;</li>
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-      Last updated on Jan 20, 2013.
-      Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.1.3.
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diff --git a/dev-py3k/_modules/sympy/core/power.html b/dev-py3k/_modules/sympy/core/power.html
deleted file mode 100644
index 72f1fc17719..00000000000
--- a/dev-py3k/_modules/sympy/core/power.html
+++ /dev/null
@@ -1,1204 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
-
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-  <head>
-    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
-    
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-    
-    <link rel="stylesheet" href="../../../_static/default.css" type="text/css" />
-    <link rel="stylesheet" href="../../../_static/pygments.css" type="text/css" />
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-    
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-      };
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-    </div>  
-
-    <div class="document">
-      <div class="documentwrapper">
-        <div class="bodywrapper">
-          <div class="body">
-            
-  <h1>Source code for sympy.core.power</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">log</span> <span class="k">as</span> <span class="n">_log</span>
-
-<span class="kn">from</span> <span class="nn">.sympify</span> <span class="kn">import</span> <span class="n">_sympify</span>
-<span class="kn">from</span> <span class="nn">.cache</span> <span class="kn">import</span> <span class="n">cacheit</span>
-<span class="kn">from</span> <span class="nn">.core</span> <span class="kn">import</span> <span class="n">C</span>
-<span class="kn">from</span> <span class="nn">.singleton</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="kn">from</span> <span class="nn">.expr</span> <span class="kn">import</span> <span class="n">Expr</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="p">(</span><span class="n">_coeff_isneg</span><span class="p">,</span> <span class="n">expand_complex</span><span class="p">,</span>
-    <span class="n">expand_multinomial</span><span class="p">,</span> <span class="n">expand_mul</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.core.logic</span> <span class="kn">import</span> <span class="n">fuzzy_bool</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">as_int</span>
-
-<span class="kn">from</span> <span class="nn">sympy.mpmath.libmp</span> <span class="kn">import</span> <span class="n">sqrtrem</span> <span class="k">as</span> <span class="n">mpmath_sqrtrem</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">sift</span>
-
-
-<div class="viewcode-block" id="integer_nthroot"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.power.integer_nthroot">[docs]</a><span class="k">def</span> <span class="nf">integer_nthroot</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a tuple containing x = floor(y**(1/n))</span>
-<span class="sd">    and a boolean indicating whether the result is exact (that is,</span>
-<span class="sd">    whether x**n == y).</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import integer_nthroot</span>
-<span class="sd">    &gt;&gt;&gt; integer_nthroot(16,2)</span>
-<span class="sd">    (4, True)</span>
-<span class="sd">    &gt;&gt;&gt; integer_nthroot(26,2)</span>
-<span class="sd">    (5, False)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">y</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">y</span><span class="p">),</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">y</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;y must be nonnegative&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;n must be positive&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">y</span> <span class="ow">in</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">y</span><span class="p">,</span> <span class="bp">True</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">y</span><span class="p">,</span> <span class="bp">True</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">rem</span> <span class="o">=</span> <span class="n">mpmath_sqrtrem</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-        <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="ow">not</span> <span class="n">rem</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="n">y</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">1</span><span class="p">,</span> <span class="bp">False</span>
-    <span class="c"># Get initial estimate for Newton&#39;s method. Care must be taken to</span>
-    <span class="c"># avoid overflow</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">guess</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">y</span><span class="o">**</span><span class="p">(</span><span class="mf">1.</span><span class="o">/</span><span class="n">n</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.5</span><span class="p">)</span>
-    <span class="k">except</span> <span class="ne">OverflowError</span><span class="p">:</span>
-        <span class="n">exp</span> <span class="o">=</span> <span class="n">_log</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">n</span>
-        <span class="k">if</span> <span class="n">exp</span> <span class="o">&gt;</span> <span class="mi">53</span><span class="p">:</span>
-            <span class="n">shift</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">exp</span> <span class="o">-</span> <span class="mi">53</span><span class="p">)</span>
-            <span class="n">guess</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="mf">2.0</span><span class="o">**</span><span class="p">(</span><span class="n">exp</span> <span class="o">-</span> <span class="n">shift</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">&lt;&lt;</span> <span class="n">shift</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">guess</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="mf">2.0</span><span class="o">**</span><span class="n">exp</span><span class="p">)</span>
-    <span class="c">#print n</span>
-    <span class="k">if</span> <span class="n">guess</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="o">**</span><span class="mi">50</span><span class="p">:</span>
-        <span class="c"># Newton iteration</span>
-        <span class="n">xprev</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">guess</span>
-        <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">t</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="c">#xprev, x = x, x - (t*x-y)//(n*t)</span>
-            <span class="n">xprev</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="p">,</span> <span class="p">((</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="o">//</span><span class="n">t</span><span class="p">)</span><span class="o">//</span><span class="n">n</span>
-            <span class="c">#print n, x-xprev, abs(x-xprev) &lt; 2</span>
-            <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">xprev</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="k">break</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">guess</span>
-    <span class="c"># Compensate</span>
-    <span class="n">t</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="n">n</span>
-    <span class="k">while</span> <span class="n">t</span> <span class="o">&lt;</span> <span class="n">y</span><span class="p">:</span>
-        <span class="n">x</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="n">n</span>
-    <span class="k">while</span> <span class="n">t</span> <span class="o">&gt;</span> <span class="n">y</span><span class="p">:</span>
-        <span class="n">x</span> <span class="o">-=</span> <span class="mi">1</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="n">n</span>
-    <span class="k">return</span> <span class="n">x</span><span class="p">,</span> <span class="n">t</span> <span class="o">==</span> <span class="n">y</span>
-
-</div>
-<div class="viewcode-block" id="Pow"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.power.Pow">[docs]</a><span class="k">class</span> <span class="nc">Pow</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-
-    <span class="n">is_Pow</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;is_commutative&#39;</span><span class="p">]</span>
-
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="c"># don&#39;t optimize &quot;if e==0; return 1&quot; here; it&#39;s better to handle that</span>
-        <span class="c"># in the calling routine so this doesn&#39;t get called</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="n">e</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">evaluate</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">e</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">elif</span> <span class="n">e</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">b</span>
-            <span class="k">elif</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span> <span class="ow">in</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">b</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>  <span class="c"># already handled e == 0 above</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">obj</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">_eval_power</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">obj</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">obj</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">is_commutative</span> <span class="o">=</span> <span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">is_commutative</span> <span class="ow">and</span> <span class="n">e</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">base</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">exp</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">class_key</span><span class="p">(</span><span class="n">cls</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">cls</span><span class="o">.</span><span class="n">__name__</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.functions.elementary.exponential</span> <span class="kn">import</span> <span class="n">log</span>
-
-        <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-        <span class="n">b_nneg</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">is_nonnegative</span>
-        <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">b_nneg</span> <span class="ow">and</span> <span class="n">e</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-            <span class="n">b_nneg</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="n">smallarg</span> <span class="o">=</span> <span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">e</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="n">b</span><span class="p">)))</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">q</span> <span class="o">==</span> <span class="mi">2</span> <span class="ow">and</span>
-                <span class="n">e</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">is</span> <span class="bp">False</span> <span class="ow">and</span> <span class="n">smallarg</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">):</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="o">*</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">or</span>
-            <span class="n">e</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span> <span class="p">(</span><span class="n">b_nneg</span> <span class="ow">or</span> <span class="nb">abs</span><span class="p">(</span><span class="n">e</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">)</span> <span class="ow">or</span>
-            <span class="n">e</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">is</span> <span class="bp">False</span> <span class="ow">and</span> <span class="n">smallarg</span> <span class="ow">is</span> <span class="bp">True</span> <span class="ow">or</span>
-                <span class="n">b</span><span class="o">.</span><span class="n">is_polar</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="o">*</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_even</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_even</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_positive</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_odd</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_nonpositive</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_odd</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_negative</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_odd</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_nonpositive</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_integer</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="n">c1</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">is_integer</span>
-        <span class="n">c2</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">is_integer</span>
-        <span class="k">if</span> <span class="n">c1</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">c2</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">c1</span> <span class="ow">and</span> <span class="n">e</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">:</span>  <span class="c"># rat**nonneg</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">c1</span> <span class="ow">and</span> <span class="n">c2</span><span class="p">:</span>  <span class="c"># int**int</span>
-            <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_nonnegative</span> <span class="ow">or</span> <span class="n">e</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">c1</span> <span class="ow">and</span> <span class="n">e</span><span class="o">.</span><span class="n">is_negative</span> <span class="ow">and</span> <span class="n">e</span><span class="o">.</span><span class="n">is_bounded</span><span class="p">:</span>  <span class="c"># int**neg</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">and</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="c"># int**nonneg or rat**?</span>
-            <span class="n">check</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">check</span><span class="o">.</span><span class="n">is_Integer</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">real_b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_real</span>
-        <span class="k">if</span> <span class="n">real_b</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="n">real_e</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_real</span>
-        <span class="k">if</span> <span class="n">real_e</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="k">if</span> <span class="n">real_b</span> <span class="ow">and</span> <span class="n">real_e</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>   <span class="c"># negative or zero (or positive)</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-                <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="bp">False</span>
-        <span class="n">im_b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_imaginary</span>
-        <span class="n">im_e</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_imaginary</span>
-        <span class="k">if</span> <span class="n">im_b</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-                <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_odd</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">elif</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">exp</span> <span class="ow">in</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">,</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">]</span> <span class="ow">and</span>
-                  <span class="bp">self</span><span class="o">.</span><span class="n">base</span> <span class="ow">in</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">,</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">]):</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="n">c</span><span class="p">,</span> <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">as_coeff_Add</span><span class="p">()</span>
-                <span class="k">if</span> <span class="n">c</span> <span class="ow">and</span> <span class="n">c</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Mul</span><span class="p">(</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">**</span><span class="n">a</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span><span class="o">.</span><span class="n">is_real</span>
-        <span class="k">if</span> <span class="n">real_b</span> <span class="ow">and</span> <span class="n">im_e</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">c</span><span class="p">:</span>
-                <span class="n">ok</span> <span class="o">=</span> <span class="p">(</span><span class="n">c</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="p">)</span><span class="o">/</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">)</span><span class="o">.</span><span class="n">is_Integer</span>
-                <span class="k">if</span> <span class="n">ok</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">ok</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_odd</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_odd</span>
-            <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_nonnegative</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_odd</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_bounded</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_infinitesimal</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_unbounded</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-        <span class="n">c1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_bounded</span>
-        <span class="k">if</span> <span class="n">c1</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="n">c2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_bounded</span>
-        <span class="k">if</span> <span class="n">c2</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="k">if</span> <span class="n">c1</span> <span class="ow">and</span> <span class="n">c2</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_nonnegative</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_nonzero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_polar</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_polar</span>
-
-    <span class="k">def</span> <span class="nf">_eval_subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">old</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span> <span class="o">==</span> <span class="n">old</span><span class="o">.</span><span class="n">base</span><span class="p">:</span>
-            <span class="n">coeff1</span><span class="p">,</span> <span class="n">terms1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">,</span> <span class="n">as_Add</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-            <span class="n">coeff2</span><span class="p">,</span> <span class="n">terms2</span> <span class="o">=</span> <span class="n">old</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">,</span> <span class="n">as_Add</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">terms1</span> <span class="o">==</span> <span class="n">terms2</span><span class="p">:</span>
-                <span class="nb">pow</span> <span class="o">=</span> <span class="n">coeff1</span><span class="o">/</span><span class="n">coeff2</span>
-                <span class="n">ok</span> <span class="o">=</span> <span class="bp">False</span>  <span class="c"># True if int(pow) == pow OR self.base.is_positive</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="nb">pow</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="nb">pow</span><span class="p">)</span>
-                    <span class="n">ok</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-                    <span class="n">ok</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_positive</span>
-                <span class="k">if</span> <span class="n">ok</span><span class="p">:</span>
-                    <span class="c"># issue 2081</span>
-                    <span class="k">return</span> <span class="n">Pow</span><span class="p">(</span><span class="n">new</span><span class="p">,</span> <span class="nb">pow</span><span class="p">)</span>  <span class="c"># (x**(6*y)).subs(x**(3*y),z)-&gt;z**2</span>
-        <span class="k">if</span> <span class="n">old</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-            <span class="n">coeff1</span><span class="p">,</span> <span class="n">terms1</span> <span class="o">=</span> <span class="n">old</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">,</span> <span class="n">as_Add</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-            <span class="c"># we can only do this when the base is positive AND the exponent</span>
-            <span class="c"># is real</span>
-            <span class="n">coeff2</span><span class="p">,</span> <span class="n">terms2</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="p">))</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span>
-                <span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">,</span> <span class="n">as_Add</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">terms1</span> <span class="o">==</span> <span class="n">terms2</span><span class="p">:</span>
-                <span class="nb">pow</span> <span class="o">=</span> <span class="n">coeff1</span><span class="o">/</span><span class="n">coeff2</span>
-                <span class="k">if</span> <span class="nb">pow</span> <span class="o">==</span> <span class="nb">int</span><span class="p">(</span><span class="nb">pow</span><span class="p">)</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">Pow</span><span class="p">(</span><span class="n">new</span><span class="p">,</span> <span class="nb">pow</span><span class="p">)</span>  <span class="c"># (2**x).subs(exp(x*log(2)), z) -&gt; z</span>
-
-<div class="viewcode-block" id="Pow.as_base_exp"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.power.Pow.as_base_exp">[docs]</a>    <span class="k">def</span> <span class="nf">as_base_exp</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return base and exp of self.</span>
-
-<span class="sd">        If base is 1/Integer, then return Integer, -exp. If this extra</span>
-<span class="sd">        processing is not needed, the base and exp properties will</span>
-<span class="sd">        give the raw arguments</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Pow, S</span>
-<span class="sd">        &gt;&gt;&gt; p = Pow(S.Half, 2, evaluate=False)</span>
-<span class="sd">        &gt;&gt;&gt; p.as_base_exp()</span>
-<span class="sd">        (2, -2)</span>
-<span class="sd">        &gt;&gt;&gt; p.args</span>
-<span class="sd">        (1/2, 2)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">b</span><span class="o">.</span><span class="n">p</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">q</span><span class="p">),</span> <span class="o">-</span><span class="n">e</span>
-        <span class="k">return</span> <span class="n">b</span><span class="p">,</span> <span class="n">e</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_adjoint</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.functions.elementary.complexes</span> <span class="kn">import</span> <span class="n">adjoint</span>
-        <span class="n">i</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_integer</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_positive</span>
-        <span class="k">if</span> <span class="n">i</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">adjoint</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="p">)</span><span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">exp</span>
-        <span class="k">if</span> <span class="n">p</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">**</span><span class="n">adjoint</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="ow">is</span> <span class="bp">False</span> <span class="ow">and</span> <span class="n">p</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="n">expanded</span> <span class="o">=</span> <span class="n">expand_complex</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">expanded</span> <span class="o">!=</span> <span class="bp">self</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">adjoint</span><span class="p">(</span><span class="n">expanded</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.functions.elementary.complexes</span> <span class="kn">import</span> <span class="n">conjugate</span> <span class="k">as</span> <span class="n">c</span>
-        <span class="n">i</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_integer</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_positive</span>
-        <span class="k">if</span> <span class="n">i</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">c</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="p">)</span><span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">exp</span>
-        <span class="k">if</span> <span class="n">p</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">**</span><span class="n">c</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="ow">is</span> <span class="bp">False</span> <span class="ow">and</span> <span class="n">p</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="n">expanded</span> <span class="o">=</span> <span class="n">expand_complex</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">expanded</span> <span class="o">!=</span> <span class="bp">self</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">c</span><span class="p">(</span><span class="n">expanded</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_transpose</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.functions.elementary.complexes</span> <span class="kn">import</span> <span class="n">transpose</span>
-        <span class="n">i</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_integer</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_complex</span>
-        <span class="k">if</span> <span class="n">p</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">exp</span>
-        <span class="k">if</span> <span class="n">i</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">transpose</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="p">)</span><span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">exp</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="ow">is</span> <span class="bp">False</span> <span class="ow">and</span> <span class="n">p</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="n">expanded</span> <span class="o">=</span> <span class="n">expand_complex</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">expanded</span> <span class="o">!=</span> <span class="bp">self</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">transpose</span><span class="p">(</span><span class="n">expanded</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_expand_power_exp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;a**(n+m) -&gt; a**n*a**m&quot;&quot;&quot;</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span>
-        <span class="n">e</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span>
-        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">and</span> <span class="n">e</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="n">expr</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Pow</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_expand_power_base</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;(a*b)**n -&gt; a**n * b**n&quot;&quot;&quot;</span>
-        <span class="n">force</span> <span class="o">=</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;force&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-
-        <span class="n">b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span>
-        <span class="n">e</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-
-        <span class="n">cargs</span><span class="p">,</span> <span class="n">nc</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">(</span><span class="n">split_1</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-        <span class="c"># expand each term - this is top-level-only</span>
-        <span class="c"># expansion but we have to watch out for things</span>
-        <span class="c"># that don&#39;t have an _eval_expand method</span>
-        <span class="k">if</span> <span class="n">nc</span><span class="p">:</span>
-            <span class="n">nc</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span><span class="o">.</span><span class="n">_eval_expand_power_base</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="s">&#39;_eval_expand_power_base&#39;</span><span class="p">)</span> <span class="k">else</span> <span class="n">i</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">nc</span><span class="p">]</span>
-
-            <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                    <span class="n">rv</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">nc</span><span class="o">*</span><span class="n">e</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">rv</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">nc</span><span class="o">*-</span><span class="n">e</span><span class="p">)</span>
-                <span class="k">assert</span> <span class="ow">not</span> <span class="n">cargs</span>
-                <span class="k">return</span> <span class="n">rv</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">cargs</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Pow</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">nc</span><span class="p">),</span> <span class="n">e</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-            <span class="n">nc</span> <span class="o">=</span> <span class="p">[</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">nc</span><span class="p">)]</span>
-
-        <span class="c"># sift the commutative bases</span>
-        <span class="k">def</span> <span class="nf">pred</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-            <span class="n">polar</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">is_polar</span>
-            <span class="k">if</span> <span class="n">polar</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">if</span> <span class="n">polar</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">fuzzy_bool</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">)</span>
-        <span class="n">sifted</span> <span class="o">=</span> <span class="n">sift</span><span class="p">(</span><span class="n">cargs</span><span class="p">,</span> <span class="n">pred</span><span class="p">)</span>
-        <span class="n">nonneg</span> <span class="o">=</span> <span class="n">sifted</span><span class="p">[</span><span class="bp">True</span><span class="p">]</span>
-        <span class="n">other</span> <span class="o">=</span> <span class="n">sifted</span><span class="p">[</span><span class="bp">None</span><span class="p">]</span>
-        <span class="n">neg</span> <span class="o">=</span> <span class="n">sifted</span><span class="p">[</span><span class="bp">False</span><span class="p">]</span>
-        <span class="n">imag</span> <span class="o">=</span> <span class="n">sifted</span><span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">imag</span><span class="p">:</span>
-            <span class="n">I</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">imag</span><span class="p">)</span> <span class="o">%</span> <span class="mi">4</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">pass</span>
-            <span class="k">elif</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">other</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">neg</span><span class="p">:</span>
-                    <span class="n">nonn</span> <span class="o">=</span> <span class="o">-</span><span class="n">neg</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-                    <span class="k">if</span> <span class="n">nonn</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                        <span class="n">nonneg</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">nonn</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">neg</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">neg</span><span class="p">:</span>
-                    <span class="n">nonn</span> <span class="o">=</span> <span class="o">-</span><span class="n">neg</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-                    <span class="k">if</span> <span class="n">nonn</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                        <span class="n">nonneg</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">nonn</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">neg</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">)</span>
-                <span class="n">other</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
-            <span class="k">del</span> <span class="n">imag</span>
-
-        <span class="c"># bring out the bases that can be separated from the base</span>
-
-        <span class="k">if</span> <span class="n">force</span> <span class="ow">or</span> <span class="n">e</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-            <span class="c"># treat all commutatives the same and put nc in other</span>
-            <span class="n">cargs</span> <span class="o">=</span> <span class="n">nonneg</span> <span class="o">+</span> <span class="n">neg</span> <span class="o">+</span> <span class="n">other</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">nc</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># this is just like what is happening automatically, except</span>
-            <span class="c"># that now we are doing it for an arbitrary exponent for which</span>
-            <span class="c"># no automatic expansion is done</span>
-
-            <span class="k">assert</span> <span class="ow">not</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Integer</span>
-
-            <span class="c"># handle negatives by making them all positive and putting</span>
-            <span class="c"># the residual -1 in other</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">neg</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">o</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">other</span> <span class="ow">and</span> <span class="n">neg</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                    <span class="n">o</span> <span class="o">*=</span> <span class="n">neg</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">neg</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span><span class="p">:</span>
-                    <span class="n">o</span> <span class="o">=</span> <span class="o">-</span><span class="n">o</span>
-                <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">neg</span><span class="p">:</span>
-                    <span class="n">nonneg</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">o</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                    <span class="n">other</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">neg</span> <span class="ow">and</span> <span class="n">other</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">neg</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">and</span> <span class="n">neg</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-                    <span class="n">other</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">)</span>
-                    <span class="n">nonneg</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">-</span><span class="n">neg</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">other</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">neg</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">other</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">neg</span><span class="p">)</span>
-            <span class="k">del</span> <span class="n">neg</span>
-
-            <span class="n">cargs</span> <span class="o">=</span> <span class="n">nonneg</span>
-            <span class="n">other</span> <span class="o">+=</span> <span class="n">nc</span>
-
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">if</span> <span class="n">cargs</span><span class="p">:</span>
-            <span class="n">rv</span> <span class="o">*=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">cargs</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">other</span><span class="p">:</span>
-            <span class="n">rv</span> <span class="o">*=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">other</span><span class="p">),</span> <span class="n">e</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">rv</span>
-
-    <span class="k">def</span> <span class="nf">_eval_expand_multinomial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;(a+b+..) ** n -&gt; a**n + n*a**(n-1)*b + .., n is nonzero integer&quot;&quot;&quot;</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span>
-        <span class="n">e</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span>
-
-        <span class="k">if</span> <span class="n">b</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">base</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">base</span> <span class="o">=</span> <span class="n">b</span>
-
-        <span class="k">if</span> <span class="n">e</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">exp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">exp</span> <span class="o">=</span> <span class="n">e</span>
-
-        <span class="k">if</span> <span class="n">e</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">b</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">result</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-                <span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">result</span><span class="o">.</span><span class="n">exp</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">result</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="bp">None</span>
-
-        <span class="k">if</span> <span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">exp</span><span class="o">.</span><span class="n">p</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">base</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="n">n</span> <span class="o">=</span> <span class="n">Integer</span><span class="p">(</span><span class="n">exp</span><span class="o">.</span><span class="n">p</span> <span class="o">//</span> <span class="n">exp</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">Pow</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">radical</span><span class="p">,</span> <span class="n">result</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">-</span> <span class="n">n</span><span class="p">),</span> <span class="p">[]</span>
-
-                    <span class="n">expanded_base_n</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">expanded_base_n</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-                        <span class="n">expanded_base_n</span> <span class="o">=</span> \
-                            <span class="n">expanded_base_n</span><span class="o">.</span><span class="n">_eval_expand_multinomial</span><span class="p">()</span>
-                    <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">expanded_base_n</span><span class="p">):</span>
-                        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">term</span><span class="o">*</span><span class="n">radical</span><span class="p">)</span>
-
-                    <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">result</span><span class="p">)</span>
-
-            <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">exp</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">base</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-                <span class="n">order_terms</span><span class="p">,</span> <span class="n">other_terms</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
-
-                <span class="k">for</span> <span class="n">order</span> <span class="ow">in</span> <span class="n">base</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">order</span><span class="o">.</span><span class="n">is_Order</span><span class="p">:</span>
-                        <span class="n">order_terms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">order</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">other_terms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">order</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">order_terms</span><span class="p">:</span>
-                    <span class="c"># (f(x) + O(x^n))^m -&gt; f(x)^m + m*f(x)^{m-1} *O(x^n)</span>
-                    <span class="n">f</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">other_terms</span><span class="p">)</span>
-
-                    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">expand_multinomial</span><span class="p">(</span><span class="n">f</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span> <span class="o">+</span> \
-                            <span class="n">n</span><span class="o">*</span><span class="n">f</span><span class="o">*</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">order_terms</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">g</span> <span class="o">=</span> <span class="n">expand_multinomial</span><span class="p">(</span><span class="n">f</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-                        <span class="k">return</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="n">g</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span> <span class="o">+</span> \
-                            <span class="n">n</span><span class="o">*</span><span class="n">g</span><span class="o">*</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">order_terms</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">base</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-                    <span class="c"># Efficiently expand expressions of the form (a + b*I)**n</span>
-                    <span class="c"># where &#39;a&#39; and &#39;b&#39; are real numbers and &#39;n&#39; is integer.</span>
-                    <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">base</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-
-                    <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="ow">not</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                            <span class="k">if</span> <span class="ow">not</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                                <span class="n">k</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">q</span> <span class="o">*</span> <span class="n">b</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-                                <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">p</span><span class="o">*</span><span class="n">b</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">q</span><span class="o">*</span><span class="n">b</span><span class="o">.</span><span class="n">p</span>
-                            <span class="k">else</span><span class="p">:</span>
-                                <span class="n">k</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-                                <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">q</span><span class="o">*</span><span class="n">b</span>
-                        <span class="k">elif</span> <span class="ow">not</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                            <span class="n">k</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-                            <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">a</span><span class="o">*</span><span class="n">b</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">p</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="n">k</span> <span class="o">=</span> <span class="mi">1</span>
-
-                        <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="nb">int</span><span class="p">(</span><span class="n">b</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span>
-
-                        <span class="k">while</span> <span class="n">n</span><span class="p">:</span>
-                            <span class="k">if</span> <span class="n">n</span> <span class="o">&amp;</span> <span class="mi">1</span><span class="p">:</span>
-                                <span class="n">c</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">a</span><span class="o">*</span><span class="n">c</span> <span class="o">-</span> <span class="n">b</span><span class="o">*</span><span class="n">d</span><span class="p">,</span> <span class="n">b</span><span class="o">*</span><span class="n">c</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span>
-                                <span class="n">n</span> <span class="o">-=</span> <span class="mi">1</span>
-                            <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">a</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="o">*</span><span class="n">b</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">b</span>
-                            <span class="n">n</span> <span class="o">//=</span> <span class="mi">2</span>
-
-                        <span class="n">I</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-
-                        <span class="k">if</span> <span class="n">k</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="n">c</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">d</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="n">c</span><span class="p">)</span><span class="o">/</span><span class="n">k</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">d</span><span class="o">/</span><span class="n">k</span>
-
-                <span class="n">p</span> <span class="o">=</span> <span class="n">other_terms</span>
-                <span class="c"># (x+y)**3 -&gt; x**3 + 3*x**2*y + 3*x*y**2 + y**3</span>
-                <span class="c"># in this particular example:</span>
-                <span class="c"># p = [x,y]; n = 3</span>
-                <span class="c"># so now it&#39;s easy to get the correct result -- we get the</span>
-                <span class="c"># coefficients first:</span>
-                <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">multinomial_coefficients</span>
-                <span class="n">expansion_dict</span> <span class="o">=</span> <span class="n">multinomial_coefficients</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="p">),</span> <span class="n">n</span><span class="p">)</span>
-                <span class="c"># in our example: {(3, 0): 1, (1, 2): 3, (0, 3): 1, (2, 1): 3}</span>
-                <span class="c"># and now construct the expression.</span>
-
-                <span class="c"># An elegant way would be to use Poly, but unfortunately it is</span>
-                <span class="c"># slower than the direct method below, so it is commented out:</span>
-                <span class="c">#b = {}</span>
-                <span class="c">#for k in expansion_dict:</span>
-                <span class="c">#    b[k] = Integer(expansion_dict[k])</span>
-                <span class="c">#return Poly(b, *p).as_expr()</span>
-
-                <span class="kn">from</span> <span class="nn">sympy.polys.polyutils</span> <span class="kn">import</span> <span class="n">basic_from_dict</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">basic_from_dict</span><span class="p">(</span><span class="n">expansion_dict</span><span class="p">,</span> <span class="o">*</span><span class="n">p</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">result</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">f</span><span class="o">*</span><span class="n">g</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">base</span><span class="o">.</span><span class="n">args</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">base</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">multi</span> <span class="o">=</span> <span class="p">(</span><span class="n">base</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">_eval_expand_multinomial</span><span class="p">()</span>
-                    <span class="k">if</span> <span class="n">multi</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">f</span><span class="o">*</span><span class="n">g</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">base</span><span class="o">.</span><span class="n">args</span>
-                            <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">multi</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">f</span><span class="o">*</span><span class="n">multi</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">base</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-        <span class="k">elif</span> <span class="p">(</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">exp</span><span class="o">.</span><span class="n">p</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">base</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">and</span>
-                <span class="nb">abs</span><span class="p">(</span><span class="n">exp</span><span class="o">.</span><span class="n">p</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">exp</span><span class="o">.</span><span class="n">q</span><span class="p">):</span>
-            <span class="k">return</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">Pow</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="o">-</span><span class="n">exp</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_expand_multinomial</span><span class="p">()</span>
-        <span class="k">elif</span> <span class="n">exp</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">and</span> <span class="n">base</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="c">#  a + b      a  b</span>
-            <span class="c"># n      --&gt; n  n  , where n, a, b are Numbers</span>
-
-            <span class="n">coeff</span><span class="p">,</span> <span class="n">tail</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-            <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">exp</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">term</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                    <span class="n">coeff</span> <span class="o">*=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">term</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">tail</span> <span class="o">+=</span> <span class="n">term</span>
-
-            <span class="k">return</span> <span class="n">coeff</span> <span class="o">*</span> <span class="n">Pow</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">tail</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">result</span>
-
-    <span class="k">def</span> <span class="nf">as_real_imag</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.polys.polytools</span> <span class="kn">import</span> <span class="n">poly</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="n">exp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">(</span><span class="n">deep</span><span class="o">=</span><span class="n">deep</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">re</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">C</span><span class="o">.</span><span class="n">re</span> <span class="ow">or</span> <span class="n">im</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">C</span><span class="o">.</span><span class="n">im</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">im</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a b&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Dummy</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">exp</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">re</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">and</span> <span class="n">im</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                    <span class="c"># We can be more efficient in this case</span>
-                    <span class="n">expr</span> <span class="o">=</span> <span class="n">expand_multinomial</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">**</span><span class="n">exp</span><span class="p">)</span>
-                    <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-
-                <span class="n">expr</span> <span class="o">=</span> <span class="n">poly</span><span class="p">(</span>
-                    <span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">exp</span><span class="p">)</span>  <span class="c"># a = re, b = im; expr = (a + b*I)**exp</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">mag</span> <span class="o">=</span> <span class="n">re</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">im</span><span class="o">**</span><span class="mi">2</span>
-                <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="n">re</span><span class="o">/</span><span class="n">mag</span><span class="p">,</span> <span class="o">-</span><span class="n">im</span><span class="o">/</span><span class="n">mag</span>
-                <span class="k">if</span> <span class="n">re</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">and</span> <span class="n">im</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                    <span class="c"># We can be more efficient in this case</span>
-                    <span class="n">expr</span> <span class="o">=</span> <span class="n">expand_multinomial</span><span class="p">((</span><span class="n">re</span> <span class="o">+</span> <span class="n">im</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span><span class="o">**-</span><span class="n">exp</span><span class="p">)</span>
-                    <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-
-                <span class="n">expr</span> <span class="o">=</span> <span class="n">poly</span><span class="p">((</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span><span class="o">**-</span><span class="n">exp</span><span class="p">)</span>
-
-            <span class="c"># Terms with even b powers will be real</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">terms</span><span class="p">()</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">i</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">%</span> <span class="mi">2</span><span class="p">]</span>
-            <span class="n">re_part</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">cc</span><span class="o">*</span><span class="n">a</span><span class="o">**</span><span class="n">aa</span><span class="o">*</span><span class="n">b</span><span class="o">**</span><span class="n">bb</span> <span class="k">for</span> <span class="p">(</span><span class="n">aa</span><span class="p">,</span> <span class="n">bb</span><span class="p">),</span> <span class="n">cc</span> <span class="ow">in</span> <span class="n">r</span><span class="p">])</span>
-            <span class="c"># Terms with odd b powers will be imaginary</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">terms</span><span class="p">()</span> <span class="k">if</span> <span class="n">i</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">%</span> <span class="mi">4</span> <span class="o">==</span> <span class="mi">1</span><span class="p">]</span>
-            <span class="n">im_part1</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">cc</span><span class="o">*</span><span class="n">a</span><span class="o">**</span><span class="n">aa</span><span class="o">*</span><span class="n">b</span><span class="o">**</span><span class="n">bb</span> <span class="k">for</span> <span class="p">(</span><span class="n">aa</span><span class="p">,</span> <span class="n">bb</span><span class="p">),</span> <span class="n">cc</span> <span class="ow">in</span> <span class="n">r</span><span class="p">])</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">terms</span><span class="p">()</span> <span class="k">if</span> <span class="n">i</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">%</span> <span class="mi">4</span> <span class="o">==</span> <span class="mi">3</span><span class="p">]</span>
-            <span class="n">im_part3</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">cc</span><span class="o">*</span><span class="n">a</span><span class="o">**</span><span class="n">aa</span><span class="o">*</span><span class="n">b</span><span class="o">**</span><span class="n">bb</span> <span class="k">for</span> <span class="p">(</span><span class="n">aa</span><span class="p">,</span> <span class="n">bb</span><span class="p">),</span> <span class="n">cc</span> <span class="ow">in</span> <span class="n">r</span><span class="p">])</span>
-
-            <span class="k">return</span> <span class="p">(</span><span class="n">re_part</span><span class="o">.</span><span class="n">subs</span><span class="p">({</span><span class="n">a</span><span class="p">:</span> <span class="n">re</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">*</span><span class="n">im</span><span class="p">}),</span>
-            <span class="n">im_part1</span><span class="o">.</span><span class="n">subs</span><span class="p">({</span><span class="n">a</span><span class="p">:</span> <span class="n">re</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">im</span><span class="p">})</span> <span class="o">+</span> <span class="n">im_part3</span><span class="o">.</span><span class="n">subs</span><span class="p">({</span><span class="n">a</span><span class="p">:</span> <span class="n">re</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="o">-</span><span class="n">im</span><span class="p">}))</span>
-
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="c"># NOTE: This is not totally correct since for x**(p/q) with</span>
-            <span class="c">#       x being imaginary there are actually q roots, but</span>
-            <span class="c">#       only a single one is returned from here.</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">(</span><span class="n">deep</span><span class="o">=</span><span class="n">deep</span><span class="p">)</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">Pow</span><span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">Pow</span><span class="p">(</span><span class="n">im</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span>
-            <span class="n">t</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">atan2</span><span class="p">(</span><span class="n">im</span><span class="p">,</span> <span class="n">re</span><span class="p">)</span>
-
-            <span class="n">rp</span><span class="p">,</span> <span class="n">tp</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">t</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">exp</span>
-
-            <span class="k">return</span> <span class="p">(</span><span class="n">rp</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">tp</span><span class="p">),</span> <span class="n">rp</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">tp</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-
-            <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-                <span class="n">hints</span><span class="p">[</span><span class="s">&#39;complex&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>
-
-                <span class="n">expanded</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;ignore&#39;</span><span class="p">)</span> <span class="o">==</span> <span class="n">expanded</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">re</span><span class="p">(</span><span class="n">expanded</span><span class="p">),</span> <span class="n">C</span><span class="o">.</span><span class="n">im</span><span class="p">(</span><span class="n">expanded</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">re</span><span class="p">(</span><span class="bp">self</span><span class="p">),</span> <span class="n">C</span><span class="o">.</span><span class="n">im</span><span class="p">(</span><span class="bp">self</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="n">dbase</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-        <span class="n">dexp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span> <span class="o">*</span> <span class="p">(</span><span class="n">dexp</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="p">)</span> <span class="o">+</span> <span class="n">dbase</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-        <span class="n">base</span> <span class="o">=</span> <span class="n">base</span><span class="o">.</span><span class="n">_evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="n">exp</span> <span class="o">=</span> <span class="n">exp</span><span class="o">.</span><span class="n">_evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">exp</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">base</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="n">base</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="n">base</span> <span class="o">=</span> <span class="n">base</span><span class="o">.</span><span class="n">conjugate</span><span class="p">()</span> <span class="o">/</span> <span class="p">(</span><span class="n">base</span> <span class="o">*</span> <span class="n">base</span><span class="o">.</span><span class="n">conjugate</span><span class="p">())</span><span class="o">.</span><span class="n">_evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-            <span class="n">exp</span> <span class="o">=</span> <span class="o">-</span><span class="n">exp</span>
-            <span class="k">return</span> <span class="n">Pow</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">Pow</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_polynomial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">syms</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">syms</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">syms</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">_eval_is_polynomial</span><span class="p">(</span><span class="n">syms</span><span class="p">)</span> <span class="ow">and</span> \
-                <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> \
-                <span class="bp">self</span><span class="o">.</span><span class="n">exp</span> <span class="o">&gt;=</span> <span class="mi">0</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_rational</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">())</span>  <span class="c"># in case it&#39;s unevaluated</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">p</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">p</span><span class="o">.</span><span class="n">is_rational</span>
-        <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="c"># we didn&#39;t check that e is not an Integer</span>
-            <span class="c"># because Rational**Integer autosimplifies</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">b</span><span class="o">.</span><span class="n">is_rational</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_rational_function</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">syms</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">syms</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">syms</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">_eval_is_rational_function</span><span class="p">(</span><span class="n">syms</span><span class="p">)</span> <span class="ow">and</span> \
-                <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">as_numer_denom</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">base</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-        <span class="c"># this should be the same as ExpBase.as_numer_denom wrt</span>
-        <span class="c"># exponent handling</span>
-        <span class="n">neg_exp</span> <span class="o">=</span> <span class="n">exp</span><span class="o">.</span><span class="n">is_negative</span>
-        <span class="n">int_exp</span> <span class="o">=</span> <span class="n">exp</span><span class="o">.</span><span class="n">is_integer</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">neg_exp</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">exp</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="n">neg_exp</span> <span class="o">=</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">exp</span><span class="p">)</span>
-        <span class="c"># the denominator cannot be separated from the numerator if</span>
-        <span class="c"># its sign is unknown unless the exponent is an integer, e.g.</span>
-        <span class="c"># sqrt(a/b) != sqrt(a)/sqrt(b) when a=1 and b=-1. But if the</span>
-        <span class="c"># denominator is negative the numerator and denominator can</span>
-        <span class="c"># be negated and the denominator (now positive) separated.</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">or</span> <span class="n">int_exp</span><span class="p">):</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">base</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="n">dnonpos</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">is_nonpositive</span>
-        <span class="k">if</span> <span class="n">dnonpos</span><span class="p">:</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="o">-</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="n">d</span>
-        <span class="k">elif</span> <span class="n">dnonpos</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">int_exp</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">base</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">if</span> <span class="n">neg_exp</span><span class="p">:</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">d</span><span class="p">,</span> <span class="n">n</span>
-            <span class="n">exp</span> <span class="o">=</span> <span class="o">-</span><span class="n">exp</span>
-        <span class="k">return</span> <span class="n">Pow</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">exp</span><span class="p">),</span> <span class="n">Pow</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">matches</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">repl_dict</span><span class="o">=</span><span class="p">{},</span> <span class="n">old</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-        <span class="c"># special case, pattern = 1 and expr.exp can match to 0</span>
-        <span class="k">if</span> <span class="n">expr</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="n">repl_dict</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">matches</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">d</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">d</span>
-
-        <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-
-        <span class="c"># special case number</span>
-        <span class="n">sb</span><span class="p">,</span> <span class="n">se</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">sb</span><span class="o">.</span><span class="n">is_Symbol</span> <span class="ow">and</span> <span class="n">se</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">expr</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_rational</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">sb</span><span class="o">.</span><span class="n">matches</span><span class="p">(</span><span class="n">b</span><span class="o">**</span><span class="p">(</span><span class="n">e</span><span class="o">/</span><span class="n">se</span><span class="p">),</span> <span class="n">repl_dict</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">sb</span><span class="o">.</span><span class="n">matches</span><span class="p">(</span><span class="n">expr</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">se</span><span class="p">),</span> <span class="n">repl_dict</span><span class="p">)</span>
-
-        <span class="n">d</span> <span class="o">=</span> <span class="n">repl_dict</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">matches</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">d</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="n">d</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="n">d</span><span class="p">)</span><span class="o">.</span><span class="n">matches</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">d</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">matches</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">repl_dict</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">d</span>
-
-    <span class="k">def</span> <span class="nf">_eval_nseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="c"># NOTE! This function is an important part of the gruntz algorithm</span>
-        <span class="c">#       for computing limits. It has to return a generalized power</span>
-        <span class="c">#       series with coefficients in C(log, log(x)). In more detail:</span>
-        <span class="c"># It has to return an expression</span>
-        <span class="c">#     c_0*x**e_0 + c_1*x**e_1 + ... (finitely many terms)</span>
-        <span class="c"># where e_i are numbers (not necessarily integers) and c_i are</span>
-        <span class="c"># expressions involving only numbers, the log function, and log(x).</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">powsimp</span><span class="p">,</span> <span class="n">collect</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">ceiling</span>
-        <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">e</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="c"># positive integer powers are easy to expand, e.g.:</span>
-                <span class="c"># sin(x)**4 = (x-x**3/3+...)**4 = ...</span>
-                <span class="k">return</span> <span class="n">expand_multinomial</span><span class="p">(</span><span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span>
-                    <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">),</span> <span class="n">e</span><span class="p">),</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">e</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-                <span class="c"># this is also easy to expand using the formula:</span>
-                <span class="c"># 1/(1 + x) = 1 - x + x**2 - x**3 ...</span>
-                <span class="c"># so we need to rewrite base to the form &quot;1+x&quot;</span>
-
-                <span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span>
-                <span class="n">prefactor</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-                <span class="c"># express &quot;rest&quot; as: rest = 1 + k*x**l + ... + O(x**n)</span>
-                <span class="n">rest</span> <span class="o">=</span> <span class="n">expand_mul</span><span class="p">((</span><span class="n">b</span> <span class="o">-</span> <span class="n">prefactor</span><span class="p">)</span><span class="o">/</span><span class="n">prefactor</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">rest</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="c"># if prefactor == w**4 + x**2*w**4 + 2*x*w**4, we need to</span>
-                    <span class="c"># factor the w**4 out using collect:</span>
-                    <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="n">collect</span><span class="p">(</span><span class="n">prefactor</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">rest</span><span class="o">.</span><span class="n">is_Order</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="n">prefactor</span> <span class="o">+</span> <span class="n">rest</span><span class="o">/</span><span class="n">prefactor</span>
-                <span class="n">n2</span> <span class="o">=</span> <span class="n">rest</span><span class="o">.</span><span class="n">getn</span><span class="p">()</span>
-                <span class="k">if</span> <span class="n">n2</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">n</span> <span class="o">=</span> <span class="n">n2</span>
-                    <span class="c"># remove the O - powering this is slow</span>
-                    <span class="k">if</span> <span class="n">logx</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="n">rest</span> <span class="o">=</span> <span class="n">rest</span><span class="o">.</span><span class="n">removeO</span><span class="p">()</span>
-
-                <span class="n">k</span><span class="p">,</span> <span class="n">l</span> <span class="o">=</span> <span class="n">rest</span><span class="o">.</span><span class="n">leadterm</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">l</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">l</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">pass</span>
-                <span class="k">elif</span> <span class="n">l</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="n">l</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">l</span> <span class="o">=</span> <span class="n">l</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-
-                <span class="n">terms</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="o">/</span><span class="n">prefactor</span><span class="p">]</span>
-                <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">ceiling</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="n">l</span><span class="p">)):</span>
-                    <span class="n">new_term</span> <span class="o">=</span> <span class="n">terms</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">rest</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">new_term</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-                        <span class="n">new_term</span> <span class="o">=</span> <span class="n">new_term</span><span class="o">.</span><span class="n">_eval_expand_multinomial</span><span class="p">(</span>
-                            <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">new_term</span> <span class="o">=</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">new_term</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-                    <span class="n">terms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">new_term</span><span class="p">)</span>
-
-                <span class="c"># Append O(...), we know the order.</span>
-                <span class="k">if</span> <span class="n">n2</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">logx</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">terms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">O</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="p">))</span>
-                <span class="k">return</span> <span class="n">powsimp</span><span class="p">(</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">terms</span><span class="p">),</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;exp&#39;</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># negative powers are rewritten to the cases above, for</span>
-                <span class="c"># example:</span>
-                <span class="c"># sin(x)**(-4) = 1/( sin(x)**4) = ...</span>
-                <span class="c"># and expand the denominator:</span>
-                <span class="n">denominator</span> <span class="o">=</span> <span class="p">(</span><span class="n">b</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">e</span><span class="p">))</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span>
-                <span class="k">if</span> <span class="mi">1</span><span class="o">/</span><span class="n">denominator</span> <span class="o">==</span> <span class="bp">self</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">self</span>
-                <span class="c"># now we have a type 1/f(x), that we know how to expand</span>
-                <span class="k">return</span> <span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">denominator</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Symbol</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">exp</span><span class="p">(</span><span class="n">e</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">b</span><span class="p">))</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span>
-
-        <span class="c"># see if the base is as simple as possible</span>
-        <span class="n">bx</span> <span class="o">=</span> <span class="n">b</span>
-        <span class="k">while</span> <span class="n">bx</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">bx</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="n">bx</span> <span class="o">=</span> <span class="n">bx</span><span class="o">.</span><span class="n">base</span>
-        <span class="k">if</span> <span class="n">bx</span> <span class="o">==</span> <span class="n">x</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-
-        <span class="c"># work for b(x)**e where e is not an Integer and does not contain x</span>
-        <span class="c"># and hopefully has no other symbols</span>
-
-        <span class="k">def</span> <span class="nf">e2int</span><span class="p">(</span><span class="n">e</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;return the integer value (if possible) of e and a</span>
-<span class="sd">            flag indicating whether it is bounded or not.&quot;&quot;&quot;</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-            <span class="n">unbounded</span> <span class="o">=</span> <span class="n">n</span><span class="o">.</span><span class="n">is_unbounded</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">unbounded</span><span class="p">:</span>
-                <span class="c"># XXX was int or floor intended? int used to behave like floor</span>
-                <span class="c"># so int(-Rational(1, 2)) returned -1 rather than int&#39;s 0</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-                <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                    <span class="c">#well, the n is something more complicated (like 1+log(2))</span>
-                    <span class="k">try</span><span class="p">:</span>
-                        <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="o">.</span><span class="n">evalf</span><span class="p">())</span> <span class="o">+</span> <span class="mi">1</span>  <span class="c"># XXX why is 1 being added?</span>
-                    <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                        <span class="k">pass</span>  <span class="c"># hope that base allows this to be resolved</span>
-                <span class="n">n</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">n</span><span class="p">,</span> <span class="n">unbounded</span>
-
-        <span class="n">order</span> <span class="o">=</span> <span class="n">O</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="n">ei</span><span class="p">,</span> <span class="n">unbounded</span> <span class="o">=</span> <span class="n">e2int</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-        <span class="n">b0</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">unbounded</span> <span class="ow">and</span> <span class="p">(</span><span class="n">b0</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="ow">or</span> <span class="n">b0</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Symbol</span><span class="p">)):</span>
-            <span class="c"># XXX what order</span>
-            <span class="k">if</span> <span class="n">b0</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="n">resid</span> <span class="o">=</span> <span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">resid</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-                <span class="k">elif</span> <span class="n">resid</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;cannot determine sign of </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">resid</span><span class="p">)</span>
-
-            <span class="k">return</span> <span class="n">b0</span><span class="o">**</span><span class="n">ei</span>
-
-        <span class="k">if</span> <span class="p">(</span><span class="n">b0</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span> <span class="ow">or</span> <span class="n">b0</span><span class="o">.</span><span class="n">is_unbounded</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">unbounded</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">b0</span><span class="o">**</span><span class="n">e</span>  <span class="c"># XXX what order</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">ei</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>  <span class="c"># if not, how will we proceed?</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                    <span class="s">&#39;expecting numerical exponent but got </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">ei</span><span class="p">)</span>
-
-            <span class="n">nuse</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="n">ei</span>
-            <span class="n">bs</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">nuse</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span>
-            <span class="n">terms</span> <span class="o">=</span> <span class="n">bs</span><span class="o">.</span><span class="n">removeO</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">terms</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="n">bs</span> <span class="o">=</span> <span class="n">terms</span>
-                <span class="n">lt</span> <span class="o">=</span> <span class="n">terms</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-                <span class="c"># bs -&gt; lt + rest -&gt; lt*(1 + (bs/lt - 1))</span>
-                <span class="k">return</span> <span class="p">((</span><span class="n">Pow</span><span class="p">(</span><span class="n">lt</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span> <span class="o">*</span> <span class="n">Pow</span><span class="p">((</span><span class="n">bs</span><span class="o">/</span><span class="n">lt</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(),</span> <span class="n">e</span><span class="p">)</span><span class="o">.</span><span class="n">nseries</span><span class="p">(</span>
-                    <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">nuse</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">))</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span> <span class="o">+</span> <span class="n">order</span><span class="p">)</span>
-
-            <span class="n">rv</span> <span class="o">=</span> <span class="n">bs</span><span class="o">**</span><span class="n">e</span>
-            <span class="k">if</span> <span class="n">terms</span> <span class="o">!=</span> <span class="n">bs</span><span class="p">:</span>
-                <span class="n">rv</span> <span class="o">+=</span> <span class="n">order</span>
-            <span class="k">return</span> <span class="n">rv</span>
-
-        <span class="c"># either b0 is bounded but neither 1 nor 0 or e is unbounded</span>
-        <span class="c"># b -&gt; b0 + (b-b0) -&gt; b0 * (1 + (b/b0-1))</span>
-        <span class="n">o2</span> <span class="o">=</span> <span class="n">order</span><span class="o">*</span><span class="p">(</span><span class="n">b0</span><span class="o">**-</span><span class="n">e</span><span class="p">)</span>
-        <span class="n">z</span> <span class="o">=</span> <span class="p">(</span><span class="n">b</span><span class="o">/</span><span class="n">b0</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">o</span> <span class="o">=</span> <span class="n">O</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="c">#r = self._compute_oseries3(z, o2, self.taylor_term)</span>
-        <span class="k">if</span> <span class="n">o</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span> <span class="ow">or</span> <span class="n">o2</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-            <span class="n">unbounded</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">o</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-                <span class="n">e2</span> <span class="o">=</span> <span class="n">log</span><span class="p">(</span><span class="n">o2</span><span class="o">.</span><span class="n">expr</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">e2</span> <span class="o">=</span> <span class="n">log</span><span class="p">(</span><span class="n">o2</span><span class="o">.</span><span class="n">expr</span><span class="p">)</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="n">o</span><span class="o">.</span><span class="n">expr</span><span class="p">)</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">unbounded</span> <span class="o">=</span> <span class="n">e2int</span><span class="p">(</span><span class="n">e2</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">unbounded</span><span class="p">:</span>
-            <span class="c"># requested accuracy gives infinite series,</span>
-            <span class="c"># order is probably non-polynomial e.g. O(exp(-1/x), x).</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">z</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">l</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">2</span><span class="p">):</span>
-                <span class="n">g</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">taylor_term</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-                <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span>
-                <span class="n">l</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">l</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">r</span><span class="o">*</span><span class="n">b0</span><span class="o">**</span><span class="n">e</span> <span class="o">+</span> <span class="n">order</span>
-
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Pow</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">exp</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="p">))</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>  <span class="c"># of (1+x)**e</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">binomial</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">*</span> <span class="n">Pow</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">()</span><span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">()</span>
-
-<div class="viewcode-block" id="Pow.as_content_primitive"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.power.Pow.as_content_primitive">[docs]</a>    <span class="k">def</span> <span class="nf">as_content_primitive</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">radical</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the tuple (R, self/R) where R is the positive Rational</span>
-<span class="sd">        extracted from self.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sqrt</span>
-<span class="sd">        &gt;&gt;&gt; sqrt(4 + 4*sqrt(2)).as_content_primitive()</span>
-<span class="sd">        (2, sqrt(1 + sqrt(2)))</span>
-<span class="sd">        &gt;&gt;&gt; sqrt(3 + 3*sqrt(2)).as_content_primitive()</span>
-<span class="sd">        (1, sqrt(3)*sqrt(1 + sqrt(2)))</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import expand_power_base, powsimp, Mul</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; ((2*x + 2)**2).as_content_primitive()</span>
-<span class="sd">        (4, (x + 1)**2)</span>
-<span class="sd">        &gt;&gt;&gt; (4**((1 + y)/2)).as_content_primitive()</span>
-<span class="sd">        (2, 4**(y/2))</span>
-<span class="sd">        &gt;&gt;&gt; (3**((1 + y)/2)).as_content_primitive()</span>
-<span class="sd">        (1, 3**((y + 1)/2))</span>
-<span class="sd">        &gt;&gt;&gt; (3**((5 + y)/2)).as_content_primitive()</span>
-<span class="sd">        (9, 3**((y + 1)/2))</span>
-<span class="sd">        &gt;&gt;&gt; eq = 3**(2 + 2*x)</span>
-<span class="sd">        &gt;&gt;&gt; powsimp(eq) == eq</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; eq.as_content_primitive()</span>
-<span class="sd">        (9, 3**(2*x))</span>
-<span class="sd">        &gt;&gt;&gt; powsimp(Mul(*_))</span>
-<span class="sd">        9**(x + 1)</span>
-
-<span class="sd">        &gt;&gt;&gt; eq = (2 + 2*x)**y</span>
-<span class="sd">        &gt;&gt;&gt; s = expand_power_base(eq); s.is_Mul, s</span>
-<span class="sd">        (False, (2*x + 2)**y)</span>
-<span class="sd">        &gt;&gt;&gt; eq.as_content_primitive()</span>
-<span class="sd">        (1, (2*(x + 1))**y)</span>
-<span class="sd">        &gt;&gt;&gt; s = expand_power_base(_[1]); s.is_Mul, s</span>
-<span class="sd">        (True, 2**y*(x + 1)**y)</span>
-
-<span class="sd">        See docstring of Expr.as_content_primitive for more examples.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="o">*</span><span class="n">b</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">(</span><span class="n">radical</span><span class="o">=</span><span class="n">radical</span><span class="p">))</span>
-        <span class="n">ce</span><span class="p">,</span> <span class="n">pe</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">(</span><span class="n">radical</span><span class="o">=</span><span class="n">radical</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="c">#e</span>
-            <span class="c">#= ce*pe</span>
-            <span class="c">#= ce*(h + t)</span>
-            <span class="c">#= ce*h + ce*t</span>
-            <span class="c">#=&gt; self</span>
-            <span class="c">#= b**(ce*h)*b**(ce*t)</span>
-            <span class="c">#= b**(cehp/cehq)*b**(ce*t)</span>
-            <span class="c">#= b**(iceh+r/cehq)*b**(ce*t)</span>
-            <span class="c">#= b**(iceh)*b**(r/cehq)*b**(ce*t)</span>
-            <span class="c">#= b**(iceh)*b**(ce*t + r/cehq)</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">pe</span><span class="o">.</span><span class="n">as_coeff_Add</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">h</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="n">ceh</span> <span class="o">=</span> <span class="n">ce</span><span class="o">*</span><span class="n">h</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">ceh</span><span class="p">)</span>
-                <span class="n">r</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">c</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                    <span class="n">iceh</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="nb">divmod</span><span class="p">(</span><span class="n">ceh</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="n">ceh</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-                    <span class="n">c</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">iceh</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">c</span><span class="p">,</span> <span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">ce</span><span class="p">,</span> <span class="n">t</span> <span class="o">+</span> <span class="n">r</span><span class="o">/</span><span class="n">ce</span><span class="o">/</span><span class="n">ceh</span><span class="o">.</span><span class="n">q</span><span class="p">))</span>
-        <span class="n">e</span> <span class="o">=</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">ce</span><span class="p">,</span> <span class="n">pe</span><span class="p">)</span>
-        <span class="c"># b**e = (h*t)**e = h**e*t**e = c*m*t**e</span>
-        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">(</span><span class="n">radical</span><span class="o">=</span><span class="n">radical</span><span class="p">)</span>  <span class="c"># h is positive</span>
-            <span class="n">c</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>  <span class="c"># so c is positive</span>
-            <span class="n">m</span><span class="p">,</span> <span class="n">me</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">m</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="ow">or</span> <span class="n">me</span> <span class="o">==</span> <span class="n">e</span><span class="p">:</span>  <span class="c"># probably always true</span>
-                <span class="c"># return the following, not return c, m*Pow(t, e)</span>
-                <span class="c"># which would change Pow into Mul; we let sympy</span>
-                <span class="c"># decide what to do by using the unevaluated Mul, e.g</span>
-                <span class="c"># should it stay as sqrt(2 + 2*sqrt(5)) or become</span>
-                <span class="c"># sqrt(2)*sqrt(1 + sqrt(5))</span>
-                <span class="k">return</span> <span class="n">c</span><span class="p">,</span> <span class="n">Pow</span><span class="p">(</span><span class="n">_keep_coeff</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">t</span><span class="p">),</span> <span class="n">e</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">is_constant</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">wrt</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;simplify&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-            <span class="bp">self</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-        <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-        <span class="n">bz</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">bz</span><span class="p">:</span>  <span class="c"># recalculate with assumptions in case it&#39;s unevaluated</span>
-            <span class="n">new</span> <span class="o">=</span> <span class="n">b</span><span class="o">**</span><span class="n">e</span>
-            <span class="k">if</span> <span class="n">new</span> <span class="o">!=</span> <span class="bp">self</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">new</span><span class="o">.</span><span class="n">is_constant</span><span class="p">()</span>
-        <span class="n">econ</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">is_constant</span><span class="p">(</span><span class="o">*</span><span class="n">wrt</span><span class="p">)</span>
-        <span class="n">bcon</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">is_constant</span><span class="p">(</span><span class="o">*</span><span class="n">wrt</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">bcon</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">econ</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="n">bz</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">bz</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="n">bcon</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="k">return</span> <span class="n">e</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-</div>
-<span class="kn">from</span> <span class="nn">.add</span> <span class="kn">import</span> <span class="n">Add</span>
-<span class="kn">from</span> <span class="nn">.numbers</span> <span class="kn">import</span> <span class="n">Integer</span>
-<span class="kn">from</span> <span class="nn">.mul</span> <span class="kn">import</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">_keep_coeff</span>
-<span class="kn">from</span> <span class="nn">.symbol</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">,</span> <span class="n">symbols</span>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/core/relational.html b/dev-py3k/_modules/sympy/core/relational.html
deleted file mode 100644
index 7fdcac4d99c..00000000000
--- a/dev-py3k/_modules/sympy/core/relational.html
+++ /dev/null
@@ -1,708 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
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-            
-  <h1>Source code for sympy.core.relational</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">.basic</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="kn">from</span> <span class="nn">.expr</span> <span class="kn">import</span> <span class="n">Expr</span>
-<span class="kn">from</span> <span class="nn">.evalf</span> <span class="kn">import</span> <span class="n">EvalfMixin</span>
-<span class="kn">from</span> <span class="nn">.sympify</span> <span class="kn">import</span> <span class="n">_sympify</span>
-
-<span class="kn">from</span> <span class="nn">sympy.logic.boolalg</span> <span class="kn">import</span> <span class="n">Boolean</span>
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">(</span>
-    <span class="s">&#39;Rel&#39;</span><span class="p">,</span> <span class="s">&#39;Eq&#39;</span><span class="p">,</span> <span class="s">&#39;Ne&#39;</span><span class="p">,</span> <span class="s">&#39;Lt&#39;</span><span class="p">,</span> <span class="s">&#39;Le&#39;</span><span class="p">,</span> <span class="s">&#39;Gt&#39;</span><span class="p">,</span> <span class="s">&#39;Ge&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Relational&#39;</span><span class="p">,</span> <span class="s">&#39;Equality&#39;</span><span class="p">,</span> <span class="s">&#39;Unequality&#39;</span><span class="p">,</span> <span class="s">&#39;StrictLessThan&#39;</span><span class="p">,</span> <span class="s">&#39;LessThan&#39;</span><span class="p">,</span>
-    <span class="s">&#39;StrictGreaterThan&#39;</span><span class="p">,</span> <span class="s">&#39;GreaterThan&#39;</span><span class="p">,</span>
-<span class="p">)</span>
-
-
-<div class="viewcode-block" id="Rel"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.relational.Rel">[docs]</a><span class="k">def</span> <span class="nf">Rel</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">op</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A handy wrapper around the Relational class.</span>
-<span class="sd">    Rel(a,b, op)</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Rel</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; Rel(y, x+x**2, &#39;==&#39;)</span>
-<span class="sd">    y == x**2 + x</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">Relational</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">op</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="Eq"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.relational.Eq">[docs]</a><span class="k">def</span> <span class="nf">Eq</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A handy wrapper around the Relational class.</span>
-<span class="sd">    Eq(a,b)</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Eq</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; Eq(y, x+x**2)</span>
-<span class="sd">    y == x**2 + x</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">Relational</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="s">&#39;==&#39;</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="Ne"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.relational.Ne">[docs]</a><span class="k">def</span> <span class="nf">Ne</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A handy wrapper around the Relational class.</span>
-<span class="sd">    Ne(a,b)</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Ne</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; Ne(y, x+x**2)</span>
-<span class="sd">    y != x**2 + x</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">Relational</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="s">&#39;!=&#39;</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="Lt"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.relational.Lt">[docs]</a><span class="k">def</span> <span class="nf">Lt</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A handy wrapper around the Relational class.</span>
-<span class="sd">    Lt(a,b)</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Lt</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; Lt(y, x+x**2)</span>
-<span class="sd">    y &lt; x**2 + x</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">Relational</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="s">&#39;&lt;&#39;</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="Le"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.relational.Le">[docs]</a><span class="k">def</span> <span class="nf">Le</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A handy wrapper around the Relational class.</span>
-<span class="sd">    Le(a,b)</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Le</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; Le(y, x+x**2)</span>
-<span class="sd">    y &lt;= x**2 + x</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">Relational</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="s">&#39;&lt;=&#39;</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="Gt"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.relational.Gt">[docs]</a><span class="k">def</span> <span class="nf">Gt</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A handy wrapper around the Relational class.</span>
-<span class="sd">    Gt(a,b)</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Gt</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; Gt(y, x + x**2)</span>
-<span class="sd">    y &gt; x**2 + x</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">Relational</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="s">&#39;&gt;&#39;</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="Ge"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.relational.Ge">[docs]</a><span class="k">def</span> <span class="nf">Ge</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A handy wrapper around the Relational class.</span>
-<span class="sd">    Ge(a,b)</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Ge</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; Ge(y, x + x**2)</span>
-<span class="sd">    y &gt;= x**2 + x</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">Relational</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="s">&#39;&gt;=&#39;</span><span class="p">)</span>
-
-<span class="c"># Note, see issue 1887.  Ideally, we wouldn&#39;t want to subclass both Boolean</span>
-<span class="c"># and Expr.</span>
-
-</div>
-<span class="k">class</span> <span class="nc">Relational</span><span class="p">(</span><span class="n">Boolean</span><span class="p">,</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">EvalfMixin</span><span class="p">):</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="n">is_Relational</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="c"># ValidRelationOperator - Defined below, because the necessary classes</span>
-    <span class="c">#   have not yet been defined</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">,</span> <span class="n">rop</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">):</span>
-        <span class="n">lhs</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">lhs</span><span class="p">)</span>
-        <span class="n">rhs</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">rhs</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">cls</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">Relational</span><span class="p">:</span>
-            <span class="n">rop_cls</span> <span class="o">=</span> <span class="n">cls</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">rop_cls</span> <span class="o">=</span> <span class="n">Relational</span><span class="o">.</span><span class="n">ValidRelationOperator</span><span class="p">[</span> <span class="n">rop</span> <span class="p">]</span>
-            <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-                <span class="n">msg</span> <span class="o">=</span> <span class="s">&quot;Invalid relational operator symbol: &#39;</span><span class="si">%r</span><span class="s">&#39;&quot;</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="n">msg</span> <span class="o">%</span> <span class="nb">repr</span><span class="p">(</span><span class="n">rop</span><span class="p">))</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">lhs</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="n">rhs</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span>
-           <span class="p">(</span><span class="n">rop_cls</span> <span class="ow">in</span> <span class="p">(</span><span class="n">Equality</span><span class="p">,</span> <span class="n">Unequality</span><span class="p">)</span> <span class="ow">or</span>
-                <span class="n">lhs</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span> <span class="n">rhs</span><span class="o">.</span><span class="n">is_real</span><span class="p">)):</span>
-            <span class="n">diff</span> <span class="o">=</span> <span class="n">lhs</span> <span class="o">-</span> <span class="n">rhs</span>
-            <span class="n">know</span> <span class="o">=</span> <span class="n">diff</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">failing_expression</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">know</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>  <span class="c"># exclude failing expression case</span>
-                <span class="n">Nlhs</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">elif</span> <span class="n">know</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sign</span>
-                <span class="n">Nlhs</span> <span class="o">=</span> <span class="n">sign</span><span class="p">(</span><span class="n">diff</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">Nlhs</span> <span class="o">=</span> <span class="bp">None</span>
-                <span class="n">lhs</span> <span class="o">=</span> <span class="n">know</span>
-                <span class="n">rhs</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">if</span> <span class="n">Nlhs</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">rop_cls</span><span class="o">.</span><span class="n">_eval_relation</span><span class="p">(</span><span class="n">Nlhs</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">rop_cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">lhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">s</span><span class="o">.</span><span class="n">_evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="n">lhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span>
-        <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span>
-        <span class="k">if</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;deep&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-            <span class="n">lhs</span> <span class="o">=</span> <span class="n">lhs</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-            <span class="n">rhs</span> <span class="o">=</span> <span class="n">rhs</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_relation_doit</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_relation_doit</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_eval_relation</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ratio</span><span class="p">,</span> <span class="n">measure</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span><span class="n">ratio</span><span class="o">=</span><span class="n">ratio</span><span class="p">),</span>
-                              <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span><span class="n">ratio</span><span class="o">=</span><span class="n">ratio</span><span class="p">))</span>
-
-
-<div class="viewcode-block" id="Equality"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.relational.Equality">[docs]</a><span class="k">class</span> <span class="nc">Equality</span><span class="p">(</span><span class="n">Relational</span><span class="p">):</span>
-
-    <span class="n">rel_op</span> <span class="o">=</span> <span class="s">&#39;==&#39;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="n">is_Equality</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_relation</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">lhs</span> <span class="o">==</span> <span class="n">rhs</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_relation_doit</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Eq</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__bool__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">compare</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span>
-
-</div>
-<div class="viewcode-block" id="Unequality"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.relational.Unequality">[docs]</a><span class="k">class</span> <span class="nc">Unequality</span><span class="p">(</span><span class="n">Relational</span><span class="p">):</span>
-
-    <span class="n">rel_op</span> <span class="o">=</span> <span class="s">&#39;!=&#39;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_relation</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">lhs</span> <span class="o">!=</span> <span class="n">rhs</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_relation_doit</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Ne</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__bool__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">compare</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span>
-
-</div>
-<span class="k">class</span> <span class="nc">_Greater</span><span class="p">(</span><span class="n">Relational</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Not intended for general use</span>
-
-<span class="sd">    _Greater is only used so that GreaterThan and StrictGreaterThan may subclass</span>
-<span class="sd">    it for the .gts and .lts properties.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">()</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">gts</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">lts</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-
-<span class="k">class</span> <span class="nc">_Less</span><span class="p">(</span><span class="n">Relational</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Not intended for general use.</span>
-
-<span class="sd">    _Less is only used so that LessThan and StrictLessThan may subclass it for</span>
-<span class="sd">    the .gts and .lts properties.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">()</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">gts</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">lts</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-
-<div class="viewcode-block" id="GreaterThan"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.relational.GreaterThan">[docs]</a><span class="k">class</span> <span class="nc">GreaterThan</span><span class="p">(</span><span class="n">_Greater</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Class representations of inequalities.</span>
-
-<span class="sd">    Extended Summary</span>
-<span class="sd">    ================</span>
-
-<span class="sd">    The ``*Than`` classes represent inequal relationships, where the left-hand</span>
-<span class="sd">    side is generally bigger or smaller than the right-hand side.  For example,</span>
-<span class="sd">    the GreaterThan class represents an inequal relationship where the</span>
-<span class="sd">    left-hand side is at least as big as the right side, if not bigger.  In</span>
-<span class="sd">    mathematical notation:</span>
-
-<span class="sd">    lhs &gt;= rhs</span>
-
-<span class="sd">    In total, there are four ``*Than`` classes, to represent the four</span>
-<span class="sd">    inequalities:</span>
-
-<span class="sd">    +-----------------+--------+</span>
-<span class="sd">    |Class Name       | Symbol |</span>
-<span class="sd">    +=================+========+</span>
-<span class="sd">    |GreaterThan      | (&gt;=)   |</span>
-<span class="sd">    +-----------------+--------+</span>
-<span class="sd">    |LessThan         | (&lt;=)   |</span>
-<span class="sd">    +-----------------+--------+</span>
-<span class="sd">    |StrictGreaterThan| (&gt;)    |</span>
-<span class="sd">    +-----------------+--------+</span>
-<span class="sd">    |StrictLessThan   | (&lt;)    |</span>
-<span class="sd">    +-----------------+--------+</span>
-
-<span class="sd">    All classes take two arguments, lhs and rhs.</span>
-
-<span class="sd">    +----------------------------+-----------------+</span>
-<span class="sd">    |Signature Example           | Math equivalent |</span>
-<span class="sd">    +============================+=================+</span>
-<span class="sd">    |GreaterThan(lhs, rhs)       |   lhs &gt;= rhs    |</span>
-<span class="sd">    +----------------------------+-----------------+</span>
-<span class="sd">    |LessThan(lhs, rhs)          |   lhs &lt;= rhs    |</span>
-<span class="sd">    +----------------------------+-----------------+</span>
-<span class="sd">    |StrictGreaterThan(lhs, rhs) |   lhs &gt;  rhs    |</span>
-<span class="sd">    +----------------------------+-----------------+</span>
-<span class="sd">    |StrictLessThan(lhs, rhs)    |   lhs &lt;  rhs    |</span>
-<span class="sd">    +----------------------------+-----------------+</span>
-
-<span class="sd">    In addition to the normal .lhs and .rhs of Relations, ``*Than`` inequality</span>
-<span class="sd">    objects also have the .lts and .gts properties, which represent the &quot;less</span>
-<span class="sd">    than side&quot; and &quot;greater than side&quot; of the operator.  Use of .lts and .gts</span>
-<span class="sd">    in an algorithm rather than .lhs and .rhs as an assumption of inequality</span>
-<span class="sd">    direction will make more explicit the intent of a certain section of code,</span>
-<span class="sd">    and will make it similarly more robust to client code changes:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import GreaterThan, StrictGreaterThan</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import LessThan,    StrictLessThan</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import And, Ge, Gt, Le, Lt, Rel, S</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.core.relational import Relational</span>
-
-<span class="sd">    &gt;&gt;&gt; e = GreaterThan(x, 1)</span>
-<span class="sd">    &gt;&gt;&gt; e</span>
-<span class="sd">    x &gt;= 1</span>
-<span class="sd">    &gt;&gt;&gt; &#39;%s &gt;= %s is the same as %s &lt;= %s&#39; % (e.gts, e.lts, e.lts, e.gts)</span>
-<span class="sd">    &#39;x &gt;= 1 is the same as 1 &lt;= x&#39;</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    One generally does not instantiate these classes directly, but uses various</span>
-<span class="sd">    convenience methods:</span>
-
-<span class="sd">    &gt;&gt;&gt; e1 = Ge( x, 2 )      # Ge is a convenience wrapper</span>
-<span class="sd">    &gt;&gt;&gt; print(e1)</span>
-<span class="sd">    x &gt;= 2</span>
-
-<span class="sd">    &gt;&gt;&gt; rels = Ge( x, 2 ), Gt( x, 2 ), Le( x, 2 ), Lt( x, 2 )</span>
-<span class="sd">    &gt;&gt;&gt; print(&#39;%s\\n%s\\n%s\\n%s&#39; % rels)</span>
-<span class="sd">    x &gt;= 2</span>
-<span class="sd">    x &gt; 2</span>
-<span class="sd">    x &lt;= 2</span>
-<span class="sd">    x &lt; 2</span>
-
-<span class="sd">    Another option is to use the Python inequality operators (&gt;=, &gt;, &lt;=, &lt;)</span>
-<span class="sd">    directly.  Their main advantage over the Ge, Gt, Le, and Lt counterparts, is</span>
-<span class="sd">    that one can write a more &quot;mathematical looking&quot; statement rather than</span>
-<span class="sd">    littering the math with oddball function calls.  However there are certain</span>
-<span class="sd">    (minor) caveats of which to be aware (search for &#39;gotcha&#39;, below).</span>
-
-<span class="sd">    &gt;&gt;&gt; e2 = x &gt;= 2</span>
-<span class="sd">    &gt;&gt;&gt; print(e2)</span>
-<span class="sd">    x &gt;= 2</span>
-<span class="sd">    &gt;&gt;&gt; print(&quot;e1: %s,    e2: %s&quot; % (e1, e2))</span>
-<span class="sd">    e1: x &gt;= 2,    e2: x &gt;= 2</span>
-<span class="sd">    &gt;&gt;&gt; e1 == e2</span>
-<span class="sd">    True</span>
-
-<span class="sd">    However, it is also perfectly valid to instantiate a ``*Than`` class less</span>
-<span class="sd">    succinctly and less conveniently:</span>
-
-<span class="sd">    &gt;&gt;&gt; rels = Rel(x, 1, &#39;&gt;=&#39;), Relational(x, 1, &#39;&gt;=&#39;), GreaterThan(x, 1)</span>
-<span class="sd">    &gt;&gt;&gt; print(&#39;%s\\n%s\\n%s&#39; % rels)</span>
-<span class="sd">    x &gt;= 1</span>
-<span class="sd">    x &gt;= 1</span>
-<span class="sd">    x &gt;= 1</span>
-
-<span class="sd">    &gt;&gt;&gt; rels = Rel(x, 1, &#39;&gt;&#39;), Relational(x, 1, &#39;&gt;&#39;), StrictGreaterThan(x, 1)</span>
-<span class="sd">    &gt;&gt;&gt; print(&#39;%s\\n%s\\n%s&#39; % rels)</span>
-<span class="sd">    x &gt; 1</span>
-<span class="sd">    x &gt; 1</span>
-<span class="sd">    x &gt; 1</span>
-
-<span class="sd">    &gt;&gt;&gt; rels = Rel(x, 1, &#39;&lt;=&#39;), Relational(x, 1, &#39;&lt;=&#39;), LessThan(x, 1)</span>
-<span class="sd">    &gt;&gt;&gt; print(&quot;%s\\n%s\\n%s&quot; % rels)</span>
-<span class="sd">    x &lt;= 1</span>
-<span class="sd">    x &lt;= 1</span>
-<span class="sd">    x &lt;= 1</span>
-
-<span class="sd">    &gt;&gt;&gt; rels = Rel(x, 1, &#39;&lt;&#39;), Relational(x, 1, &#39;&lt;&#39;), StrictLessThan(x, 1)</span>
-<span class="sd">    &gt;&gt;&gt; print(&#39;%s\\n%s\\n%s&#39; % rels)</span>
-<span class="sd">    x &lt; 1</span>
-<span class="sd">    x &lt; 1</span>
-<span class="sd">    x &lt; 1</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    There are a couple of &quot;gotchas&quot; when using Python&#39;s operators.</span>
-
-<span class="sd">    The first enters the mix when comparing against a literal number as the lhs</span>
-<span class="sd">    argument.  Due to the order that Python decides to parse a statement, it may</span>
-<span class="sd">    not immediately find two objects comparable.  For example, to evaluate the</span>
-<span class="sd">    statement (1 &lt; x), Python will first recognize the number 1 as a native</span>
-<span class="sd">    number, and then that x is *not* a native number.  At this point, because a</span>
-<span class="sd">    native Python number does not know how to compare itself with a SymPy object</span>
-<span class="sd">    Python will try the reflective operation, (x &gt; 1).  Unfortunately, there is</span>
-<span class="sd">    no way available to SymPy to recognize this has happened, so the statement</span>
-<span class="sd">    (1 &lt; x) will turn silently into (x &gt; 1).</span>
-
-<span class="sd">    &gt;&gt;&gt; e1 = x &gt;  1</span>
-<span class="sd">    &gt;&gt;&gt; e2 = x &gt;= 1</span>
-<span class="sd">    &gt;&gt;&gt; e3 = x &lt;  1</span>
-<span class="sd">    &gt;&gt;&gt; e4 = x &lt;= 1</span>
-<span class="sd">    &gt;&gt;&gt; e5 = 1 &gt;  x</span>
-<span class="sd">    &gt;&gt;&gt; e6 = 1 &gt;= x</span>
-<span class="sd">    &gt;&gt;&gt; e7 = 1 &lt;  x</span>
-<span class="sd">    &gt;&gt;&gt; e8 = 1 &lt;= x</span>
-<span class="sd">    &gt;&gt;&gt; print(&quot;%s     %s\\n&quot;*4 % (e1, e2, e3, e4, e5, e6, e7, e8))</span>
-<span class="sd">    x &gt; 1     x &gt;= 1</span>
-<span class="sd">    x &lt; 1     x &lt;= 1</span>
-<span class="sd">    x &lt; 1     x &lt;= 1</span>
-<span class="sd">    x &gt; 1     x &gt;= 1</span>
-
-<span class="sd">    If the order of the statement is important (for visual output to the</span>
-<span class="sd">    console, perhaps), one can work around this annoyance in a couple ways: (1)</span>
-<span class="sd">    &quot;sympify&quot; the literal before comparison, (2) use one of the wrappers, or (3)</span>
-<span class="sd">    use the less succinct methods described above:</span>
-
-<span class="sd">    &gt;&gt;&gt; e1 = S(1) &gt;  x</span>
-<span class="sd">    &gt;&gt;&gt; e2 = S(1) &gt;= x</span>
-<span class="sd">    &gt;&gt;&gt; e3 = S(1) &lt;  x</span>
-<span class="sd">    &gt;&gt;&gt; e4 = S(1) &lt;= x</span>
-<span class="sd">    &gt;&gt;&gt; e5 = Gt(1, x)</span>
-<span class="sd">    &gt;&gt;&gt; e6 = Ge(1, x)</span>
-<span class="sd">    &gt;&gt;&gt; e7 = Lt(1, x)</span>
-<span class="sd">    &gt;&gt;&gt; e8 = Le(1, x)</span>
-<span class="sd">    &gt;&gt;&gt; print(&quot;%s     %s\\n&quot;*4 % (e1, e2, e3, e4, e5, e6, e7, e8))</span>
-<span class="sd">    1 &gt; x     1 &gt;= x</span>
-<span class="sd">    1 &lt; x     1 &lt;= x</span>
-<span class="sd">    1 &gt; x     1 &gt;= x</span>
-<span class="sd">    1 &lt; x     1 &lt;= x</span>
-
-<span class="sd">    The other gotcha is with chained inequalities.  Occasionally, one may be</span>
-<span class="sd">    tempted to write statements like:</span>
-
-<span class="sd">    &gt;&gt;&gt; e = x &lt; y &lt; z  # silent error!  Where did ``x`` go?</span>
-<span class="sd">    &gt;&gt;&gt; e #doctest: +SKIP</span>
-<span class="sd">    y &lt; z</span>
-
-<span class="sd">    Due to an implementation detail or decision of Python [1]_, there is no way</span>
-<span class="sd">    for SymPy to reliably create that as a chained inequality.  To create a</span>
-<span class="sd">    chained inequality, the only method currently available is to make use of</span>
-<span class="sd">    And:</span>
-
-<span class="sd">    &gt;&gt;&gt; e = And(x &lt; y, y &lt; z)</span>
-<span class="sd">    &gt;&gt;&gt; type( e )</span>
-<span class="sd">    And</span>
-<span class="sd">    &gt;&gt;&gt; e</span>
-<span class="sd">    And(x &lt; y, y &lt; z)</span>
-
-<span class="sd">    Note that this is different than chaining an equality directly via use of</span>
-<span class="sd">    parenthesis (this is currently an open bug in SymPy [2]_):</span>
-
-<span class="sd">    &gt;&gt;&gt; e = (x &lt; y) &lt; z</span>
-<span class="sd">    &gt;&gt;&gt; type( e )</span>
-<span class="sd">    &lt;class &#39;sympy.core.relational.StrictLessThan&#39;&gt;</span>
-<span class="sd">    &gt;&gt;&gt; e</span>
-<span class="sd">    (x &lt; y) &lt; z</span>
-
-<span class="sd">    Any code that explicitly relies on this latter functionality will not be</span>
-<span class="sd">    robust as this behaviour is completely wrong and will be corrected at some</span>
-<span class="sd">    point.  For the time being (circa Jan 2012), use And to create chained</span>
-<span class="sd">    inequalities.</span>
-
-<span class="sd">    .. [1] This implementation detail is that Python provides no reliable</span>
-<span class="sd">       method to determine that a chained inequality is being built.  Chained</span>
-<span class="sd">       comparison operators are evaluated pairwise, using &quot;and&quot; logic (see</span>
-<span class="sd">       http://docs.python.org/reference/expressions.html#notin).  This is done</span>
-<span class="sd">       in an efficient way, so that each object being compared is only</span>
-<span class="sd">       evaluated once and the comparison can short-circuit.  For example, ``1</span>
-<span class="sd">       &gt; 2 &gt; 3`` is evaluated by Python as ``(1 &gt; 2) and (2 &gt; 3)``.  The</span>
-<span class="sd">       ``and`` operator coerces each side into a bool, returning the object</span>
-<span class="sd">       itself when it short-circuits.  Currently, the bool of the --Than</span>
-<span class="sd">       operators will give True or False arbitrarily.  Thus, if we were to</span>
-<span class="sd">       compute ``x &gt; y &gt; z``, with ``x``, ``y``, and ``z`` being Symbols,</span>
-<span class="sd">       Python converts the statement (roughly) into these steps:</span>
-
-<span class="sd">        (1) x &gt; y &gt; z</span>
-<span class="sd">        (2) (x &gt; y) and (y &gt; z)</span>
-<span class="sd">        (3) (GreaterThanObject) and (y &gt; z)</span>
-<span class="sd">        (4) (GreaterThanObject.__nonzero__()) and (y &gt; z)</span>
-<span class="sd">        (5) (True) and (y &gt; z)</span>
-<span class="sd">        (6) (y &gt; z)</span>
-<span class="sd">        (7) LessThanObject</span>
-
-<span class="sd">       Because of the &quot;and&quot; added at step 2, the statement gets turned into a</span>
-<span class="sd">       weak ternary statement.  If the first object evalutes __nonzero__ as</span>
-<span class="sd">       True, then the second object, (y &gt; z) is returned.  If the first object</span>
-<span class="sd">       evaluates __nonzero__ as False (step 5), then (x &gt; y) is returned.</span>
-
-<span class="sd">           In Python, there is no way to override the ``and`` operator, or to</span>
-<span class="sd">           control how it short circuits, so it is impossible to make something</span>
-<span class="sd">           like ``x &gt; y &gt; z`` work.  There is an open PEP to change this,</span>
-<span class="sd">           :pep:`335`, but until that is implemented, this cannot be made to work.</span>
-
-<span class="sd">    .. [2] For more information, see these two bug reports:</span>
-
-<span class="sd">       &quot;Separate boolean and symbolic relationals&quot;</span>
-<span class="sd">       `Issue 1887 &lt;http://code.google.com/p/sympy/issues/detail?id=1887&gt;`_</span>
-
-<span class="sd">       &quot;It right 0 &lt; x &lt; 1 ?&quot;</span>
-<span class="sd">       `Issue 2960 &lt;http://code.google.com/p/sympy/issues/detail?id=2960&gt;`_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">rel_op</span> <span class="o">=</span> <span class="s">&#39;&gt;=&#39;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">()</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_relation</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">lhs</span> <span class="o">&gt;=</span> <span class="n">rhs</span>
-
-    <span class="k">def</span> <span class="nf">__bool__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">compare</span><span class="p">(</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span> <span class="p">)</span> <span class="o">&gt;=</span> <span class="mi">0</span>
-
-</div>
-<div class="viewcode-block" id="LessThan"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.relational.LessThan">[docs]</a><span class="k">class</span> <span class="nc">LessThan</span><span class="p">(</span><span class="n">_Less</span><span class="p">):</span>
-    <span class="n">__doc__</span> <span class="o">=</span> <span class="n">GreaterThan</span><span class="o">.</span><span class="n">__doc__</span>
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">()</span>
-
-    <span class="n">rel_op</span> <span class="o">=</span> <span class="s">&#39;&lt;=&#39;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_relation</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">lhs</span> <span class="o">&lt;=</span> <span class="n">rhs</span>
-
-    <span class="k">def</span> <span class="nf">__bool__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">compare</span><span class="p">(</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span> <span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">0</span>
-
-</div>
-<div class="viewcode-block" id="StrictGreaterThan"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.relational.StrictGreaterThan">[docs]</a><span class="k">class</span> <span class="nc">StrictGreaterThan</span><span class="p">(</span><span class="n">_Greater</span><span class="p">):</span>
-    <span class="n">__doc__</span> <span class="o">=</span> <span class="n">GreaterThan</span><span class="o">.</span><span class="n">__doc__</span>
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">()</span>
-
-    <span class="n">rel_op</span> <span class="o">=</span> <span class="s">&#39;&gt;&#39;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_relation</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">lhs</span> <span class="o">&gt;</span> <span class="n">rhs</span>
-
-    <span class="k">def</span> <span class="nf">__bool__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">compare</span><span class="p">(</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span> <span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span>
-
-</div>
-<div class="viewcode-block" id="StrictLessThan"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.relational.StrictLessThan">[docs]</a><span class="k">class</span> <span class="nc">StrictLessThan</span><span class="p">(</span><span class="n">_Less</span><span class="p">):</span>
-    <span class="n">__doc__</span> <span class="o">=</span> <span class="n">GreaterThan</span><span class="o">.</span><span class="n">__doc__</span>
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">()</span>
-
-    <span class="n">rel_op</span> <span class="o">=</span> <span class="s">&#39;&lt;&#39;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_relation</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">lhs</span> <span class="o">&lt;</span> <span class="n">rhs</span>
-
-    <span class="k">def</span> <span class="nf">__bool__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">compare</span><span class="p">(</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span> <span class="p">)</span> <span class="o">&lt;</span> <span class="mi">0</span>
-
-<span class="c"># A class-specific (not object-specific) data item used for a minor speedup.  It</span>
-<span class="c"># is defined here, rather than directly in the class, because the classes that</span>
-<span class="c"># it references have not been defined until now (e.g. StrictLessThan).</span></div>
-<span class="n">Relational</span><span class="o">.</span><span class="n">ValidRelationOperator</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="bp">None</span><span class="p">:</span> <span class="n">Equality</span><span class="p">,</span>
-    <span class="s">&#39;==&#39;</span><span class="p">:</span> <span class="n">Equality</span><span class="p">,</span>
-    <span class="s">&#39;eq&#39;</span><span class="p">:</span> <span class="n">Equality</span><span class="p">,</span>
-    <span class="s">&#39;!=&#39;</span><span class="p">:</span> <span class="n">Unequality</span><span class="p">,</span>
-    <span class="s">&#39;&lt;&gt;&#39;</span><span class="p">:</span> <span class="n">Unequality</span><span class="p">,</span>
-    <span class="s">&#39;ne&#39;</span><span class="p">:</span> <span class="n">Unequality</span><span class="p">,</span>
-    <span class="s">&#39;&gt;=&#39;</span><span class="p">:</span> <span class="n">GreaterThan</span><span class="p">,</span>
-    <span class="s">&#39;ge&#39;</span><span class="p">:</span> <span class="n">GreaterThan</span><span class="p">,</span>
-    <span class="s">&#39;&lt;=&#39;</span><span class="p">:</span> <span class="n">LessThan</span><span class="p">,</span>
-    <span class="s">&#39;le&#39;</span><span class="p">:</span> <span class="n">LessThan</span><span class="p">,</span>
-    <span class="s">&#39;&gt;&#39;</span><span class="p">:</span> <span class="n">StrictGreaterThan</span><span class="p">,</span>
-    <span class="s">&#39;gt&#39;</span><span class="p">:</span> <span class="n">StrictGreaterThan</span><span class="p">,</span>
-    <span class="s">&#39;&lt;&#39;</span><span class="p">:</span> <span class="n">StrictLessThan</span><span class="p">,</span>
-    <span class="s">&#39;lt&#39;</span><span class="p">:</span> <span class="n">StrictLessThan</span><span class="p">,</span>
-<span class="p">}</span>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/core/sets.html b/dev-py3k/_modules/sympy/core/sets.html
deleted file mode 100644
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-            
-  <h1>Source code for sympy.core.sets</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="n">_sympify</span><span class="p">,</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.core.basic</span> <span class="kn">import</span> <span class="n">Basic</span>
-<span class="kn">from</span> <span class="nn">sympy.core.singleton</span> <span class="kn">import</span> <span class="n">Singleton</span><span class="p">,</span> <span class="n">S</span>
-<span class="kn">from</span> <span class="nn">sympy.core.evalf</span> <span class="kn">import</span> <span class="n">EvalfMixin</span>
-<span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">Float</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">iterable</span>
-<span class="kn">from</span> <span class="nn">sympy.core.decorators</span> <span class="kn">import</span> <span class="n">deprecated</span>
-
-<span class="kn">from</span> <span class="nn">sympy.mpmath</span> <span class="kn">import</span> <span class="n">mpi</span><span class="p">,</span> <span class="n">mpf</span>
-<span class="kn">from</span> <span class="nn">sympy.assumptions</span> <span class="kn">import</span> <span class="n">ask</span>
-<span class="kn">from</span> <span class="nn">sympy.logic.boolalg</span> <span class="kn">import</span> <span class="n">And</span><span class="p">,</span> <span class="n">Or</span>
-
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">default_sort_key</span>
-
-
-<div class="viewcode-block" id="Set"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Set">[docs]</a><span class="k">class</span> <span class="nc">Set</span><span class="p">(</span><span class="n">Basic</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The base class for any kind of set.</span>
-
-<span class="sd">    This is not meant to be used directly as a container of items.</span>
-<span class="sd">    It does not behave like the builtin set; see FiniteSet for that.</span>
-
-<span class="sd">    Real intervals are represented by the Interval class and unions of sets</span>
-<span class="sd">    by the Union class. The empty set is represented by the EmptySet class</span>
-<span class="sd">    and available as a singleton as S.EmptySet.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_number</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_iterable</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_interval</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="n">is_FiniteSet</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Interval</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_ProductSet</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Union</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Intersection</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">is_EmptySet</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">is_UniversalSet</span> <span class="o">=</span> <span class="bp">None</span>
-
-<div class="viewcode-block" id="Set.sort_key"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Set.sort_key">[docs]</a>    <span class="k">def</span> <span class="nf">sort_key</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Give sort_key of infimum (if possible) else sort_key of the set.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">infimum</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">inf</span>
-            <span class="k">if</span> <span class="n">infimum</span><span class="o">.</span><span class="n">is_comparable</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">default_sort_key</span><span class="p">(</span><span class="n">infimum</span><span class="p">,</span> <span class="n">order</span><span class="p">)</span>
-        <span class="k">except</span> <span class="p">(</span><span class="ne">NotImplementedError</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">):</span>
-            <span class="k">pass</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">([</span><span class="n">default_sort_key</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">order</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sorted_args</span><span class="p">])</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">class_key</span><span class="p">(),</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">),</span> <span class="n">args</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="o">.</span><span class="n">class_key</span><span class="p">(),</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-</div>
-<div class="viewcode-block" id="Set.union"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Set.union">[docs]</a>    <span class="k">def</span> <span class="nf">union</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the union of &#39;self&#39; and &#39;other&#39;.</span>
-
-<span class="sd">        As a shortcut it is possible to use the &#39;+&#39; operator:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Interval, FiniteSet</span>
-<span class="sd">        &gt;&gt;&gt; Interval(0, 1).union(Interval(2, 3))</span>
-<span class="sd">        [0, 1] U [2, 3]</span>
-<span class="sd">        &gt;&gt;&gt; Interval(0, 1) + Interval(2, 3)</span>
-<span class="sd">        [0, 1] U [2, 3]</span>
-<span class="sd">        &gt;&gt;&gt; Interval(1, 2, True, True) + FiniteSet(2, 3)</span>
-<span class="sd">        (1, 2] U {3}</span>
-
-<span class="sd">        Similarly it is possible to use the &#39;-&#39; operator for set differences:</span>
-
-<span class="sd">        &gt;&gt;&gt; Interval(0, 2) - Interval(0, 1)</span>
-<span class="sd">        (1, 2]</span>
-<span class="sd">        &gt;&gt;&gt; Interval(1, 3) - FiniteSet(2)</span>
-<span class="sd">        [1, 2) U (2, 3]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Union</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Set.intersect"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Set.intersect">[docs]</a>    <span class="k">def</span> <span class="nf">intersect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the intersection of &#39;self&#39; and &#39;other&#39;.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Interval</span>
-
-<span class="sd">        &gt;&gt;&gt; Interval(1, 3).intersect(Interval(1, 2))</span>
-<span class="sd">        [1, 2]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Intersection</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_intersect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        This function should only be used internally</span>
-
-<span class="sd">        self._intersect(other) returns a new, intersected set if self knows how</span>
-<span class="sd">        to intersect itself with other, otherwise it returns None</span>
-
-<span class="sd">        When making a new set class you can be assured that other will not</span>
-<span class="sd">        be a Union, FiniteSet, or EmptySet</span>
-
-<span class="sd">        Used within the Intersection class</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">_union</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        This function should only be used internally</span>
-
-<span class="sd">        self._union(other) returns a new, joined set if self knows how</span>
-<span class="sd">        to join itself with other, otherwise it returns None.</span>
-<span class="sd">        It may also return a python set of SymPy Sets if they are somehow</span>
-<span class="sd">        simpler. If it does this it must be idempotent i.e. the sets returned</span>
-<span class="sd">        must return None with _union&#39;ed with each other</span>
-
-<span class="sd">        Used within the Union class</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Set.complement"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Set.complement">[docs]</a>    <span class="k">def</span> <span class="nf">complement</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The complement of &#39;self&#39;.</span>
-
-<span class="sd">        As a shortcut it is possible to use the &#39;~&#39; or &#39;-&#39; operators:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Interval</span>
-
-<span class="sd">        &gt;&gt;&gt; Interval(0, 1).complement</span>
-<span class="sd">        (-oo, 0) U (1, oo)</span>
-<span class="sd">        &gt;&gt;&gt; ~Interval(0, 1)</span>
-<span class="sd">        (-oo, 0) U (1, oo)</span>
-<span class="sd">        &gt;&gt;&gt; -Interval(0, 1)</span>
-<span class="sd">        (-oo, 0) U (1, oo)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_complement</span>
-</div>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_complement</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;(</span><span class="si">%s</span><span class="s">)._complement&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Set.inf"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Set.inf">[docs]</a>    <span class="k">def</span> <span class="nf">inf</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The infimum of &#39;self&#39;</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Interval, Union</span>
-
-<span class="sd">        &gt;&gt;&gt; Interval(0, 1).inf</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; Union(Interval(0, 1), Interval(2, 3)).inf</span>
-<span class="sd">        0</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_inf</span>
-</div>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_inf</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;(</span><span class="si">%s</span><span class="s">)._inf&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Set.sup"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Set.sup">[docs]</a>    <span class="k">def</span> <span class="nf">sup</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The supremum of &#39;self&#39;</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Interval, Union</span>
-
-<span class="sd">        &gt;&gt;&gt; Interval(0, 1).sup</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; Union(Interval(0, 1), Interval(2, 3)).sup</span>
-<span class="sd">        3</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sup</span>
-</div>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_sup</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;(</span><span class="si">%s</span><span class="s">)._sup&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Set.contains"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Set.contains">[docs]</a>    <span class="k">def</span> <span class="nf">contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns True if &#39;other&#39; is contained in &#39;self&#39; as an element.</span>
-
-<span class="sd">        As a shortcut it is possible to use the &#39;in&#39; operator:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Interval</span>
-
-<span class="sd">        &gt;&gt;&gt; Interval(0, 1).contains(0.5)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; 0.5 in Interval(0, 1)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_contains</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-</div>
-    <span class="k">def</span> <span class="nf">_contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;(</span><span class="si">%s</span><span class="s">)._contains(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">))</span>
-
-<div class="viewcode-block" id="Set.subset"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Set.subset">[docs]</a>    <span class="k">def</span> <span class="nf">subset</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns True if &#39;other&#39; is a subset of &#39;self&#39;.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Interval</span>
-
-<span class="sd">        &gt;&gt;&gt; Interval(0, 1).contains(0)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Interval(0, 1, left_open=True).contains(0)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Set</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="n">other</span><span class="p">)</span> <span class="o">==</span> <span class="n">other</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Unknown argument &#39;</span><span class="si">%s</span><span class="s">&#39;&quot;</span> <span class="o">%</span> <span class="n">other</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Set.measure"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Set.measure">[docs]</a>    <span class="k">def</span> <span class="nf">measure</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The (Lebesgue) measure of &#39;self&#39;</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Interval, Union</span>
-
-<span class="sd">        &gt;&gt;&gt; Interval(0, 1).measure</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; Union(Interval(0, 1), Interval(2, 3)).measure</span>
-<span class="sd">        2</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_measure</span>
-</div>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_measure</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;(</span><span class="si">%s</span><span class="s">)._measure&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__or__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__and__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">ProductSet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__pow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">exp</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">sympify</span><span class="p">(</span><span class="n">exp</span><span class="p">)</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">exp</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s">: Exponent must be a positive Integer&quot;</span> <span class="o">%</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">ProductSet</span><span class="p">([</span><span class="bp">self</span><span class="p">]</span><span class="o">*</span><span class="n">exp</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">complement</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">complement</span>
-
-    <span class="k">def</span> <span class="nf">__invert__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">complement</span>
-
-    <span class="k">def</span> <span class="nf">__contains__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="n">symb</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">ask</span><span class="p">(</span><span class="n">symb</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;contains did not evaluate to a bool: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">symb</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-</div>
-<div class="viewcode-block" id="ProductSet"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.ProductSet">[docs]</a><span class="k">class</span> <span class="nc">ProductSet</span><span class="p">(</span><span class="n">Set</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Represents a Cartesian Product of Sets.</span>
-
-<span class="sd">    Returns a Cartesian product given several sets as either an iterable</span>
-<span class="sd">    or individual arguments.</span>
-
-<span class="sd">    Can use &#39;*&#39; operator on any sets for convenient shorthand.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Interval, FiniteSet, ProductSet</span>
-
-<span class="sd">        &gt;&gt;&gt; I = Interval(0, 5); S = FiniteSet(1, 2, 3)</span>
-<span class="sd">        &gt;&gt;&gt; ProductSet(I, S)</span>
-<span class="sd">        [0, 5] x {1, 2, 3}</span>
-
-<span class="sd">        &gt;&gt;&gt; (2, 2) in ProductSet(I, S)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &gt;&gt;&gt; Interval(0, 1) * Interval(0, 1) # The unit square</span>
-<span class="sd">        [0, 1] x [0, 1]</span>
-
-<span class="sd">        &gt;&gt;&gt; coin = FiniteSet(&#39;H&#39;, &#39;T&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; set(coin**2)</span>
-<span class="sd">        set([(H, H), (H, T), (T, H), (T, T)])</span>
-
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-<span class="sd">    - Passes most operations down to the argument sets</span>
-<span class="sd">    - Flattens Products of ProductSets</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-<span class="sd">    http://en.wikipedia.org/wiki/Cartesian_product</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_ProductSet</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">sets</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">def</span> <span class="nf">flatten</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Set</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_ProductSet</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">flatten</span><span class="p">,</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">)),</span> <span class="p">[])</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="p">[</span><span class="n">arg</span><span class="p">]</span>
-            <span class="k">elif</span> <span class="n">iterable</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-                <span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">flatten</span><span class="p">,</span> <span class="n">arg</span><span class="p">)),</span> <span class="p">[])</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Input must be Sets or iterables of Sets&quot;</span><span class="p">)</span>
-        <span class="n">sets</span> <span class="o">=</span> <span class="n">flatten</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">sets</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">EmptySet</span><span class="p">()</span> <span class="ow">in</span> <span class="n">sets</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">sets</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">EmptySet</span><span class="p">()</span>
-
-        <span class="k">return</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">sets</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">element</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        &#39;in&#39; operator for ProductSets</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Interval</span>
-
-<span class="sd">        &gt;&gt;&gt; (2, 3) in Interval(0, 5) * Interval(0, 5)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &gt;&gt;&gt; (10, 10) in Interval(0, 5) * Interval(0, 5)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        Passes operation on to constituent sets</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">element</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>  <span class="c"># maybe element isn&#39;t an iterable</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="n">And</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="nb">set</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">item</span><span class="p">)</span> <span class="k">for</span> <span class="nb">set</span><span class="p">,</span> <span class="n">item</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">sets</span><span class="p">,</span> <span class="n">element</span><span class="p">)])</span>
-
-    <span class="k">def</span> <span class="nf">_intersect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        This function should only be used internally</span>
-
-<span class="sd">        See Set._intersect for docstring</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">other</span><span class="o">.</span><span class="n">is_ProductSet</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">EmptySet</span>
-        <span class="k">return</span> <span class="n">ProductSet</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">sets</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">sets</span><span class="p">))</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">sets</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_complement</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># For each set consider it or it&#39;s complement</span>
-        <span class="c"># We need at least one of the sets to be complemented</span>
-        <span class="c"># Consider all 2^n combinations.</span>
-        <span class="c"># We can conveniently represent these options easily using a ProductSet</span>
-        <span class="n">switch_sets</span> <span class="o">=</span> <span class="n">ProductSet</span><span class="p">(</span><span class="n">FiniteSet</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">s</span><span class="o">.</span><span class="n">complement</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">sets</span><span class="p">)</span>
-        <span class="n">product_sets</span> <span class="o">=</span> <span class="p">(</span><span class="n">ProductSet</span><span class="p">(</span><span class="o">*</span><span class="nb">set</span><span class="p">)</span> <span class="k">for</span> <span class="nb">set</span> <span class="ow">in</span> <span class="n">switch_sets</span><span class="p">)</span>
-        <span class="c"># Union of all combinations but this one</span>
-        <span class="k">return</span> <span class="n">Union</span><span class="p">(</span><span class="n">p</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">product_sets</span> <span class="k">if</span> <span class="n">p</span> <span class="o">!=</span> <span class="bp">self</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="nb">set</span><span class="o">.</span><span class="n">is_real</span> <span class="k">for</span> <span class="nb">set</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">sets</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_iterable</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="nb">set</span><span class="o">.</span><span class="n">is_iterable</span> <span class="k">for</span> <span class="nb">set</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">sets</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_iterable</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">product</span>
-            <span class="k">return</span> <span class="n">product</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">sets</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Not all constituent sets are iterable&quot;</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_measure</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">measure</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">for</span> <span class="nb">set</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">sets</span><span class="p">:</span>
-            <span class="n">measure</span> <span class="o">*=</span> <span class="nb">set</span><span class="o">.</span><span class="n">measure</span>
-        <span class="k">return</span> <span class="n">measure</span>
-
-</div>
-<div class="viewcode-block" id="Interval"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Interval">[docs]</a><span class="k">class</span> <span class="nc">Interval</span><span class="p">(</span><span class="n">Set</span><span class="p">,</span> <span class="n">EvalfMixin</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Represents a real interval as a Set.</span>
-
-<span class="sd">    Usage:</span>
-<span class="sd">        Returns an interval with end points &quot;start&quot; and &quot;end&quot;.</span>
-
-<span class="sd">        For left_open=True (default left_open is False) the interval</span>
-<span class="sd">        will be open on the left. Similarly, for right_open=True the interval</span>
-<span class="sd">        will be open on the right.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, Interval, sets</span>
-
-<span class="sd">    &gt;&gt;&gt; Interval(0, 1)</span>
-<span class="sd">    [0, 1]</span>
-<span class="sd">    &gt;&gt;&gt; Interval(0, 1, False, True)</span>
-<span class="sd">    [0, 1)</span>
-
-<span class="sd">    &gt;&gt;&gt; a = Symbol(&#39;a&#39;, real=True)</span>
-<span class="sd">    &gt;&gt;&gt; Interval(0, a)</span>
-<span class="sd">    [0, a]</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-<span class="sd">    - Only real end points are supported</span>
-<span class="sd">    - Interval(a, b) with a &gt; b will return the empty set</span>
-<span class="sd">    - Use the evalf() method to turn an Interval into an mpmath</span>
-<span class="sd">      &#39;mpi&#39; interval instance</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    &lt;http://en.wikipedia.org/wiki/Interval_(mathematics)&gt;</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_Interval</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_real</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">end</span><span class="p">,</span> <span class="n">left_open</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">right_open</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-
-        <span class="n">start</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">start</span><span class="p">)</span>
-        <span class="n">end</span> <span class="o">=</span> <span class="n">_sympify</span><span class="p">(</span><span class="n">end</span><span class="p">)</span>
-
-        <span class="c"># Only allow real intervals (use symbols with &#39;is_real=True&#39;).</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">start</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">end</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Only real intervals are supported&quot;</span><span class="p">)</span>
-
-        <span class="c"># Make sure that the created interval will be valid.</span>
-        <span class="k">if</span> <span class="n">end</span><span class="o">.</span><span class="n">is_comparable</span> <span class="ow">and</span> <span class="n">start</span><span class="o">.</span><span class="n">is_comparable</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">end</span> <span class="o">&lt;</span> <span class="n">start</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">EmptySet</span>
-
-        <span class="k">if</span> <span class="n">end</span> <span class="o">==</span> <span class="n">start</span> <span class="ow">and</span> <span class="p">(</span><span class="n">left_open</span> <span class="ow">or</span> <span class="n">right_open</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">EmptySet</span>
-        <span class="k">if</span> <span class="n">end</span> <span class="o">==</span> <span class="n">start</span> <span class="ow">and</span> <span class="ow">not</span> <span class="p">(</span><span class="n">left_open</span> <span class="ow">or</span> <span class="n">right_open</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="n">end</span><span class="p">)</span>
-
-        <span class="c"># Make sure infinite interval end points are open.</span>
-        <span class="k">if</span> <span class="n">start</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-            <span class="n">left_open</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="n">end</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-            <span class="n">right_open</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="k">return</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">end</span><span class="p">,</span> <span class="n">left_open</span><span class="p">,</span> <span class="n">right_open</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Interval.start"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Interval.start">[docs]</a>    <span class="k">def</span> <span class="nf">start</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The left end point of &#39;self&#39;.</span>
-
-<span class="sd">        This property takes the same value as the &#39;inf&#39; property.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Interval</span>
-
-<span class="sd">        &gt;&gt;&gt; Interval(0, 1).start</span>
-<span class="sd">        0</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="n">_inf</span> <span class="o">=</span> <span class="n">left</span> <span class="o">=</span> <span class="n">start</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Interval.end"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Interval.end">[docs]</a>    <span class="k">def</span> <span class="nf">end</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The right end point of &#39;self&#39;.</span>
-
-<span class="sd">        This property takes the same value as the &#39;sup&#39; property.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Interval</span>
-
-<span class="sd">        &gt;&gt;&gt; Interval(0, 1).end</span>
-<span class="sd">        1</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="n">_sup</span> <span class="o">=</span> <span class="n">right</span> <span class="o">=</span> <span class="n">end</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Interval.left_open"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Interval.left_open">[docs]</a>    <span class="k">def</span> <span class="nf">left_open</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        True if &#39;self&#39; is left-open.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Interval</span>
-
-<span class="sd">        &gt;&gt;&gt; Interval(0, 1, left_open=True).left_open</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Interval(0, 1, left_open=False).left_open</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Interval.right_open"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Interval.right_open">[docs]</a>    <span class="k">def</span> <span class="nf">right_open</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        True if &#39;self&#39; is right-open.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Interval</span>
-
-<span class="sd">        &gt;&gt;&gt; Interval(0, 1, right_open=True).right_open</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Interval(0, 1, right_open=False).right_open</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span>
-</div>
-    <span class="k">def</span> <span class="nf">_intersect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        This function should only be used internally</span>
-
-<span class="sd">        See Set._intersect for docstring</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># We only know how to intersect with other intervals</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">other</span><span class="o">.</span><span class="n">is_Interval</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="c"># We can&#39;t intersect [0,3] with [x,6] -- we don&#39;t know if x&gt;0 or x&lt;0</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">_is_comparable</span><span class="p">(</span><span class="n">other</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="n">empty</span> <span class="o">=</span> <span class="bp">False</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">start</span> <span class="o">&lt;=</span> <span class="n">other</span><span class="o">.</span><span class="n">end</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">start</span> <span class="o">&lt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="p">:</span>
-            <span class="c"># Get topology right.</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">start</span> <span class="o">&lt;</span> <span class="n">other</span><span class="o">.</span><span class="n">start</span><span class="p">:</span>
-                <span class="n">start</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">start</span>
-                <span class="n">left_open</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">left_open</span>
-            <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">start</span> <span class="o">&gt;</span> <span class="n">other</span><span class="o">.</span><span class="n">start</span><span class="p">:</span>
-                <span class="n">start</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">start</span>
-                <span class="n">left_open</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">left_open</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">start</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">start</span>
-                <span class="n">left_open</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">left_open</span> <span class="ow">or</span> <span class="n">other</span><span class="o">.</span><span class="n">left_open</span>
-
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">end</span> <span class="o">&lt;</span> <span class="n">other</span><span class="o">.</span><span class="n">end</span><span class="p">:</span>
-                <span class="n">end</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">end</span>
-                <span class="n">right_open</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">right_open</span>
-            <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">end</span> <span class="o">&gt;</span> <span class="n">other</span><span class="o">.</span><span class="n">end</span><span class="p">:</span>
-                <span class="n">end</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">end</span>
-                <span class="n">right_open</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">right_open</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">end</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">end</span>
-                <span class="n">right_open</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">right_open</span> <span class="ow">or</span> <span class="n">other</span><span class="o">.</span><span class="n">right_open</span>
-
-            <span class="k">if</span> <span class="n">end</span> <span class="o">-</span> <span class="n">start</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="p">(</span><span class="n">left_open</span> <span class="ow">or</span> <span class="n">right_open</span><span class="p">):</span>
-                <span class="n">empty</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">empty</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="k">if</span> <span class="n">empty</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">EmptySet</span>
-
-        <span class="k">return</span> <span class="n">Interval</span><span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="n">end</span><span class="p">,</span> <span class="n">left_open</span><span class="p">,</span> <span class="n">right_open</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_union</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        This function should only be used internally</span>
-
-<span class="sd">        See Set._union for docstring</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">is_Interval</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">_is_comparable</span><span class="p">(</span><span class="n">other</span><span class="p">):</span>
-            <span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">Min</span><span class="p">,</span> <span class="n">Max</span>
-            <span class="c"># Non-overlapping intervals</span>
-            <span class="n">end</span> <span class="o">=</span> <span class="n">Min</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">end</span><span class="p">)</span>
-            <span class="n">start</span> <span class="o">=</span> <span class="n">Max</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">start</span><span class="p">)</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">end</span> <span class="o">&lt;</span> <span class="n">start</span> <span class="ow">or</span>
-               <span class="p">(</span><span class="n">end</span> <span class="o">==</span> <span class="n">start</span> <span class="ow">and</span> <span class="p">(</span><span class="n">end</span> <span class="ow">not</span> <span class="ow">in</span> <span class="bp">self</span> <span class="ow">and</span> <span class="n">end</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">other</span><span class="p">))):</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">start</span> <span class="o">=</span> <span class="n">Min</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">start</span><span class="p">)</span>
-                <span class="n">end</span> <span class="o">=</span> <span class="n">Max</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">end</span><span class="p">)</span>
-
-                <span class="n">left_open</span> <span class="o">=</span> <span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">start</span> <span class="o">!=</span> <span class="n">start</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">left_open</span><span class="p">)</span> <span class="ow">and</span>
-                             <span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">start</span> <span class="o">!=</span> <span class="n">start</span> <span class="ow">or</span> <span class="n">other</span><span class="o">.</span><span class="n">left_open</span><span class="p">))</span>
-                <span class="n">right_open</span> <span class="o">=</span> <span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">end</span> <span class="o">!=</span> <span class="n">end</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">right_open</span><span class="p">)</span> <span class="ow">and</span>
-                              <span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">end</span> <span class="o">!=</span> <span class="n">end</span> <span class="ow">or</span> <span class="n">other</span><span class="o">.</span><span class="n">right_open</span><span class="p">))</span>
-
-                <span class="k">return</span> <span class="n">Interval</span><span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="n">end</span><span class="p">,</span> <span class="n">left_open</span><span class="p">,</span> <span class="n">right_open</span><span class="p">)</span>
-
-        <span class="c"># If I have open end points and these endpoints are contained in other</span>
-        <span class="k">if</span> <span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">left_open</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">)</span> <span class="ow">or</span>
-                <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">right_open</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">)):</span>
-            <span class="c"># Fill in my end points and return</span>
-            <span class="n">open_left</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">left_open</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">start</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">other</span>
-            <span class="n">open_right</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">right_open</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">end</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">other</span>
-            <span class="n">new_self</span> <span class="o">=</span> <span class="n">Interval</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="p">,</span> <span class="n">open_left</span><span class="p">,</span> <span class="n">open_right</span><span class="p">)</span>
-            <span class="k">return</span> <span class="nb">set</span><span class="p">((</span><span class="n">new_self</span><span class="p">,</span> <span class="n">other</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_complement</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">Interval</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="p">,</span> <span class="bp">True</span><span class="p">,</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">left_open</span><span class="p">)</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">Interval</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">,</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">right_open</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Union</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">left_open</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">other</span> <span class="o">&gt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">start</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">other</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">start</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">right_open</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">And</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">other</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">And</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">other</span> <span class="o">&lt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_measure</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">end</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">start</span>
-
-    <span class="k">def</span> <span class="nf">to_mpi</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="mi">53</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">mpi</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)),</span> <span class="n">mpf</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Interval</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">left</span><span class="o">.</span><span class="n">evalf</span><span class="p">(),</span> <span class="bp">self</span><span class="o">.</span><span class="n">right</span><span class="o">.</span><span class="n">evalf</span><span class="p">(),</span>
-          <span class="n">left_open</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">left_open</span><span class="p">,</span> <span class="n">right_open</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">right_open</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_is_comparable</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="n">is_comparable</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="o">.</span><span class="n">is_comparable</span>
-        <span class="n">is_comparable</span> <span class="o">&amp;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="o">.</span><span class="n">is_comparable</span>
-        <span class="n">is_comparable</span> <span class="o">&amp;=</span> <span class="n">other</span><span class="o">.</span><span class="n">start</span><span class="o">.</span><span class="n">is_comparable</span>
-        <span class="n">is_comparable</span> <span class="o">&amp;=</span> <span class="n">other</span><span class="o">.</span><span class="n">end</span><span class="o">.</span><span class="n">is_comparable</span>
-
-        <span class="k">return</span> <span class="n">is_comparable</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Interval.is_left_unbounded"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Interval.is_left_unbounded">[docs]</a>    <span class="k">def</span> <span class="nf">is_left_unbounded</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return ``True`` if the left endpoint is negative infinity. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">left</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">left</span> <span class="o">==</span> <span class="n">Float</span><span class="p">(</span><span class="s">&quot;-inf&quot;</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Interval.is_right_unbounded"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Interval.is_right_unbounded">[docs]</a>    <span class="k">def</span> <span class="nf">is_right_unbounded</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return ``True`` if the right endpoint is positive infinity. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">right</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">right</span> <span class="o">==</span> <span class="n">Float</span><span class="p">(</span><span class="s">&quot;+inf&quot;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Interval.as_relational"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Interval.as_relational">[docs]</a>    <span class="k">def</span> <span class="nf">as_relational</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">symbol</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Rewrite an interval in terms of inequalities and logic operators. &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Lt</span><span class="p">,</span> <span class="n">Le</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_left_unbounded</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">left_open</span><span class="p">:</span>
-                <span class="n">left</span> <span class="o">=</span> <span class="n">Lt</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="p">,</span> <span class="n">symbol</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">left</span> <span class="o">=</span> <span class="n">Le</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="p">,</span> <span class="n">symbol</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_right_unbounded</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">right_open</span><span class="p">:</span>
-                <span class="n">right</span> <span class="o">=</span> <span class="n">Lt</span><span class="p">(</span><span class="n">symbol</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">right</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">right</span> <span class="o">=</span> <span class="n">Le</span><span class="p">(</span><span class="n">symbol</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">right</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_left_unbounded</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_right_unbounded</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>  <span class="c"># XXX: Contained(symbol, Floats)</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_left_unbounded</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">right</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_right_unbounded</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">left</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">And</span><span class="p">(</span><span class="n">left</span><span class="p">,</span> <span class="n">right</span><span class="p">)</span>
-
-</div></div>
-<div class="viewcode-block" id="Union"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Union">[docs]</a><span class="k">class</span> <span class="nc">Union</span><span class="p">(</span><span class="n">Set</span><span class="p">,</span> <span class="n">EvalfMixin</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Represents a union of sets as a Set.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Union, Interval</span>
-
-<span class="sd">        &gt;&gt;&gt; Union(Interval(1, 2), Interval(3, 4))</span>
-<span class="sd">        [1, 2] U [3, 4]</span>
-
-<span class="sd">        The Union constructor will always try to merge overlapping intervals,</span>
-<span class="sd">        if possible. For example:</span>
-
-<span class="sd">        &gt;&gt;&gt; Union(Interval(1, 2), Interval(2, 3))</span>
-<span class="sd">        [1, 3]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    Intersection</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-<span class="sd">    &lt;http://en.wikipedia.org/wiki/Union_(set_theory)&gt;</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_Union</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">evaluate</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;evaluate&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-
-        <span class="c"># flatten inputs to merge intersections and iterables</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">flatten</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Set</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Union</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">flatten</span><span class="p">,</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">)),</span> <span class="p">[])</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="p">[</span><span class="n">arg</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">iterable</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>  <span class="c"># and not isinstance(arg, Set) (implicit)</span>
-                <span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">flatten</span><span class="p">,</span> <span class="n">arg</span><span class="p">)),</span> <span class="p">[])</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Input must be Sets or iterables of Sets&quot;</span><span class="p">)</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="n">flatten</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="c"># Union of no sets is EmptySet</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">EmptySet</span>
-
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-
-        <span class="c"># Reduce sets using known rules</span>
-        <span class="k">if</span> <span class="n">evaluate</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Union</span><span class="o">.</span><span class="n">reduce</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="Union.reduce"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Union.reduce">[docs]</a>    <span class="k">def</span> <span class="nf">reduce</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Simplify a Union using known rules</span>
-
-<span class="sd">        We first start with global rules like</span>
-<span class="sd">        &#39;Merge all FiniteSets&#39;</span>
-
-<span class="sd">        Then we iterate through all pairs and ask the constituent sets if they</span>
-<span class="sd">        can simplify themselves with any other constituent</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="c"># ===== Global Rules =====</span>
-        <span class="c"># Merge all finite sets</span>
-        <span class="n">finite_sets</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">args</span> <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">is_FiniteSet</span><span class="p">]</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">finite_sets</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">finite_set</span> <span class="o">=</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="n">x</span> <span class="k">for</span> <span class="nb">set</span> <span class="ow">in</span> <span class="n">finite_sets</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">set</span><span class="p">)</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">finite_set</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">args</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">x</span><span class="o">.</span><span class="n">is_FiniteSet</span><span class="p">]</span>
-
-        <span class="c"># ===== Pair-wise Rules =====</span>
-        <span class="c"># Here we depend on rules built into the constituent sets</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">new_args</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">while</span><span class="p">(</span><span class="n">new_args</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-                <span class="n">new_args</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">args</span> <span class="o">-</span> <span class="nb">set</span><span class="p">((</span><span class="n">s</span><span class="p">,)):</span>
-                    <span class="n">new_set</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">_union</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-                    <span class="c"># This returns None if s does not know how to intersect</span>
-                    <span class="c"># with t. Returns the newly intersected set otherwise</span>
-                    <span class="k">if</span> <span class="n">new_set</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">new_set</span><span class="p">,</span> <span class="nb">set</span><span class="p">):</span>
-                            <span class="n">new_set</span> <span class="o">=</span> <span class="nb">set</span><span class="p">((</span><span class="n">new_set</span><span class="p">,</span> <span class="p">))</span>
-                        <span class="n">new_args</span> <span class="o">=</span> <span class="p">(</span><span class="n">args</span> <span class="o">-</span> <span class="nb">set</span><span class="p">((</span><span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">)))</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="n">new_set</span><span class="p">)</span>
-                        <span class="k">break</span>
-                <span class="k">if</span> <span class="n">new_args</span><span class="p">:</span>
-                    <span class="n">args</span> <span class="o">=</span> <span class="n">new_args</span>
-                    <span class="k">break</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">args</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Union</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_inf</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># We use Min so that sup is meaningful in combination with symbolic</span>
-        <span class="c"># interval end points.</span>
-        <span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">Min</span>
-        <span class="k">return</span> <span class="n">Min</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="nb">set</span><span class="o">.</span><span class="n">inf</span> <span class="k">for</span> <span class="nb">set</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_sup</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># We use Max so that sup is meaningful in combination with symbolic</span>
-        <span class="c"># end points.</span>
-        <span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">Max</span>
-        <span class="k">return</span> <span class="n">Max</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="nb">set</span><span class="o">.</span><span class="n">sup</span> <span class="k">for</span> <span class="nb">set</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_complement</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># De Morgan&#39;s formula.</span>
-        <span class="n">complement</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">complement</span>
-        <span class="k">for</span> <span class="nb">set</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="n">complement</span> <span class="o">=</span> <span class="n">complement</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="nb">set</span><span class="o">.</span><span class="n">complement</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">complement</span>
-
-    <span class="k">def</span> <span class="nf">_contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="n">or_args</span> <span class="o">=</span> <span class="p">[</span><span class="n">the_set</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">other</span><span class="p">)</span> <span class="k">for</span> <span class="n">the_set</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">Or</span><span class="p">(</span><span class="o">*</span><span class="n">or_args</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_measure</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># Measure of a union is the sum of the measures of the sets minus</span>
-        <span class="c"># the sum of their pairwise intersections plus the sum of their</span>
-        <span class="c"># triple-wise intersections minus ... etc...</span>
-
-        <span class="c"># Sets is a collection of intersections and a set of elementary</span>
-        <span class="c"># sets which made up those intersections (called &quot;sos&quot; for set of sets)</span>
-        <span class="c"># An example element might of this list might be:</span>
-        <span class="c">#    ( {A,B,C}, A.intersect(B).intersect(C) )</span>
-
-        <span class="c"># Start with just elementary sets (  ({A}, A), ({B}, B), ... )</span>
-        <span class="c"># Then get and subtract (  ({A,B}, (A int B), ... ) while non-zero</span>
-        <span class="n">sets</span> <span class="o">=</span> <span class="p">[(</span><span class="n">FiniteSet</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="n">measure</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">parity</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">while</span> <span class="n">sets</span><span class="p">:</span>
-            <span class="c"># Add up the measure of these sets and add or subtract it to total</span>
-            <span class="n">measure</span> <span class="o">+=</span> <span class="n">parity</span> <span class="o">*</span> <span class="nb">sum</span><span class="p">(</span><span class="n">inter</span><span class="o">.</span><span class="n">measure</span> <span class="k">for</span> <span class="n">sos</span><span class="p">,</span> <span class="n">inter</span> <span class="ow">in</span> <span class="n">sets</span><span class="p">)</span>
-
-            <span class="c"># For each intersection in sets, compute the intersection with every</span>
-            <span class="c"># other set not already part of the intersection.</span>
-            <span class="n">sets</span> <span class="o">=</span> <span class="p">((</span><span class="n">sos</span> <span class="o">+</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="n">newset</span><span class="p">),</span> <span class="n">newset</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="n">intersection</span><span class="p">))</span>
-                    <span class="k">for</span> <span class="n">sos</span><span class="p">,</span> <span class="n">intersection</span> <span class="ow">in</span> <span class="n">sets</span> <span class="k">for</span> <span class="n">newset</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-                    <span class="k">if</span> <span class="n">newset</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">sos</span><span class="p">)</span>
-
-            <span class="c"># Clear out sets with no measure</span>
-            <span class="n">sets</span> <span class="o">=</span> <span class="p">[(</span><span class="n">sos</span><span class="p">,</span> <span class="n">inter</span><span class="p">)</span> <span class="k">for</span> <span class="n">sos</span><span class="p">,</span> <span class="n">inter</span> <span class="ow">in</span> <span class="n">sets</span> <span class="k">if</span> <span class="n">inter</span><span class="o">.</span><span class="n">measure</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">]</span>
-
-            <span class="c"># Clear out duplicates</span>
-            <span class="n">sos_list</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">sets_list</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="nb">set</span> <span class="ow">in</span> <span class="n">sets</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">set</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">in</span> <span class="n">sos_list</span><span class="p">:</span>
-                    <span class="k">continue</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">sos_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">set</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-                    <span class="n">sets_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">set</span><span class="p">)</span>
-            <span class="n">sets</span> <span class="o">=</span> <span class="n">sets_list</span>
-
-            <span class="c"># Flip Parity - next time subtract/add if we added/subtracted here</span>
-            <span class="n">parity</span> <span class="o">*=</span> <span class="o">-</span><span class="mi">1</span>
-        <span class="k">return</span> <span class="n">measure</span>
-
-<div class="viewcode-block" id="Union.as_relational"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Union.as_relational">[docs]</a>    <span class="k">def</span> <span class="nf">as_relational</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">symbol</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Rewrite a Union in terms of equalities and logic operators. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Or</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="nb">set</span><span class="o">.</span><span class="n">as_relational</span><span class="p">(</span><span class="n">symbol</span><span class="p">)</span> <span class="k">for</span> <span class="nb">set</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-</div>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_iterable</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">is_iterable</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Union</span><span class="p">(</span><span class="nb">set</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span> <span class="k">for</span> <span class="nb">set</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">except</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Not all sets are evalf-able&quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">itertools</span>
-        <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="nb">set</span><span class="o">.</span><span class="n">is_iterable</span> <span class="k">for</span> <span class="nb">set</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">itertools</span><span class="o">.</span><span class="n">chain</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="nb">iter</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Not all constituent sets are iterable&quot;</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="nb">set</span><span class="o">.</span><span class="n">is_real</span> <span class="k">for</span> <span class="nb">set</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="Intersection"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Intersection">[docs]</a><span class="k">class</span> <span class="nc">Intersection</span><span class="p">(</span><span class="n">Set</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Represents an intersection of sets as a Set.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Intersection, Interval</span>
-
-<span class="sd">        &gt;&gt;&gt; Intersection(Interval(1, 3), Interval(2, 4))</span>
-<span class="sd">        [2, 3]</span>
-
-<span class="sd">        We often use the .intersect method</span>
-
-<span class="sd">        &gt;&gt;&gt; Interval(1,3).intersect(Interval(2,4))</span>
-<span class="sd">        [2, 3]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    Union</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-<span class="sd">    &lt;http://en.wikipedia.org/wiki/Intersection_(set_theory)&gt;</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_Intersection</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">evaluate</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;evaluate&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-
-        <span class="c"># flatten inputs to merge intersections and iterables</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">flatten</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Set</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Intersection</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">flatten</span><span class="p">,</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">)),</span> <span class="p">[])</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="p">[</span><span class="n">arg</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">iterable</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>  <span class="c"># and not isinstance(arg, Set) (implicit)</span>
-                <span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">flatten</span><span class="p">,</span> <span class="n">arg</span><span class="p">)),</span> <span class="p">[])</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Input must be Sets or iterables of Sets&quot;</span><span class="p">)</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="n">flatten</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="c"># Intersection of no sets is everything</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">UniversalSet</span>
-
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-
-        <span class="c"># Reduce sets using known rules</span>
-        <span class="k">if</span> <span class="n">evaluate</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Intersection</span><span class="o">.</span><span class="n">reduce</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_iterable</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">any</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">is_iterable</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_inf</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_sup</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_complement</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">_contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.logic.boolalg</span> <span class="kn">import</span> <span class="n">And</span>
-        <span class="k">return</span> <span class="n">And</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="nb">set</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">other</span><span class="p">)</span> <span class="k">for</span> <span class="nb">set</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">s</span><span class="o">.</span><span class="n">is_iterable</span><span class="p">:</span>
-                <span class="n">other_sets</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">-</span> <span class="nb">set</span><span class="p">((</span><span class="n">s</span><span class="p">,))</span>
-                <span class="n">other</span> <span class="o">=</span> <span class="n">Intersection</span><span class="p">(</span><span class="n">other_sets</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-                <span class="k">return</span> <span class="p">(</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">s</span> <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">other</span><span class="p">)</span>
-
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;None of the constituent sets are iterable&quot;</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="Intersection.reduce"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Intersection.reduce">[docs]</a>    <span class="k">def</span> <span class="nf">reduce</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Simplify an intersection using known rules</span>
-
-<span class="sd">        We first start with global rules like</span>
-<span class="sd">        &#39;if any empty sets return empty set&#39; and &#39;distribute any unions&#39;</span>
-
-<span class="sd">        Then we iterate through all pairs and ask the constituent sets if they</span>
-<span class="sd">        can simplify themselves with any other constituent</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="c"># ===== Global Rules =====</span>
-        <span class="c"># If any EmptySets return EmptySet</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">is_EmptySet</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">args</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">EmptySet</span>
-
-        <span class="c"># If any FiniteSets see which elements of that finite set occur within</span>
-        <span class="c"># all other sets in the intersection</span>
-        <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">s</span><span class="o">.</span><span class="n">is_FiniteSet</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">s</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">s</span>
-                        <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">x</span> <span class="ow">in</span> <span class="n">other</span> <span class="k">for</span> <span class="n">other</span> <span class="ow">in</span> <span class="n">args</span><span class="p">))</span>
-
-        <span class="c"># If any of the sets are unions, return a Union of Intersections</span>
-        <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">s</span><span class="o">.</span><span class="n">is_Union</span><span class="p">:</span>
-                <span class="n">other_sets</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">-</span> <span class="nb">set</span><span class="p">((</span><span class="n">s</span><span class="p">,))</span>
-                <span class="n">other</span> <span class="o">=</span> <span class="n">Intersection</span><span class="p">(</span><span class="n">other_sets</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">Union</span><span class="p">(</span><span class="n">Intersection</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">s</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="c"># At this stage we are guaranteed not to have any</span>
-        <span class="c"># EmptySets, FiniteSets, or Unions in the intersection</span>
-
-        <span class="c"># ===== Pair-wise Rules =====</span>
-        <span class="c"># Here we depend on rules built into the constituent sets</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">new_args</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">while</span><span class="p">(</span><span class="n">new_args</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-                <span class="n">new_args</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">args</span> <span class="o">-</span> <span class="nb">set</span><span class="p">((</span><span class="n">s</span><span class="p">,)):</span>
-                    <span class="n">new_set</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">_intersect</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-                    <span class="c"># This returns None if s does not know how to intersect</span>
-                    <span class="c"># with t. Returns the newly intersected set otherwise</span>
-                    <span class="k">if</span> <span class="n">new_set</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="n">new_args</span> <span class="o">=</span> <span class="p">(</span><span class="n">args</span> <span class="o">-</span> <span class="nb">set</span><span class="p">((</span><span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">)))</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="nb">set</span><span class="p">((</span><span class="n">new_set</span><span class="p">,</span> <span class="p">)))</span>
-                        <span class="k">break</span>
-                <span class="k">if</span> <span class="n">new_args</span><span class="p">:</span>
-                    <span class="n">args</span> <span class="o">=</span> <span class="n">new_args</span>
-                    <span class="k">break</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">args</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Intersection</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Intersection.as_relational"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.Intersection.as_relational">[docs]</a>    <span class="k">def</span> <span class="nf">as_relational</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">symbol</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Rewrite an Intersection in terms of equalities and logic operators&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">And</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="nb">set</span><span class="o">.</span><span class="n">as_relational</span><span class="p">(</span><span class="n">symbol</span><span class="p">)</span> <span class="k">for</span> <span class="nb">set</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-</div></div>
-<div class="viewcode-block" id="EmptySet"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.EmptySet">[docs]</a><span class="k">class</span> <span class="nc">EmptySet</span><span class="p">(</span><span class="n">Set</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">Singleton</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Represents the empty set. The empty set is available as a singleton</span>
-<span class="sd">    as S.EmptySet.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import S, Interval</span>
-
-<span class="sd">        &gt;&gt;&gt; S.EmptySet</span>
-<span class="sd">        EmptySet()</span>
-
-<span class="sd">        &gt;&gt;&gt; Interval(1, 2).intersect(S.EmptySet)</span>
-<span class="sd">        EmptySet()</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    UniversalSet</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-<span class="sd">    http://en.wikipedia.org/wiki/Empty_set</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_EmptySet</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_intersect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">EmptySet</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_complement</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">UniversalSet</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_measure</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">0</span>
-
-    <span class="k">def</span> <span class="nf">_contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">as_relational</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">symbol</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__len__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">0</span>
-
-    <span class="k">def</span> <span class="nf">_union</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">other</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">iter</span><span class="p">([])</span>
-
-</div>
-<div class="viewcode-block" id="UniversalSet"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.UniversalSet">[docs]</a><span class="k">class</span> <span class="nc">UniversalSet</span><span class="p">(</span><span class="n">Set</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">Singleton</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Represents the set of all things.</span>
-<span class="sd">    The universal set is available as a singleton as S.UniversalSet</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import S, Interval</span>
-
-<span class="sd">        &gt;&gt;&gt; S.UniversalSet</span>
-<span class="sd">        UniversalSet()</span>
-
-<span class="sd">        &gt;&gt;&gt; Interval(1, 2).intersect(S.UniversalSet)</span>
-<span class="sd">        [1, 2]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    EmptySet</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-<span class="sd">    http://en.wikipedia.org/wiki/Universal_set</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_UniversalSet</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_intersect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">other</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_complement</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">EmptySet</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_measure</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-
-    <span class="k">def</span> <span class="nf">_contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">as_relational</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">symbol</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_union</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-</div>
-<div class="viewcode-block" id="FiniteSet"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.FiniteSet">[docs]</a><span class="k">class</span> <span class="nc">FiniteSet</span><span class="p">(</span><span class="n">Set</span><span class="p">,</span> <span class="n">EvalfMixin</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Represents a finite set of discrete numbers</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol, FiniteSet, sets</span>
-
-<span class="sd">        &gt;&gt;&gt; FiniteSet(1, 2, 3, 4)</span>
-<span class="sd">        {1, 2, 3, 4}</span>
-<span class="sd">        &gt;&gt;&gt; 3 in FiniteSet(1, 2, 3, 4)</span>
-<span class="sd">        True</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-<span class="sd">    http://en.wikipedia.org/wiki/Finite_set</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_FiniteSet</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_iterable</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">evaluate</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;evaluate&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">evaluate</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">iterable</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
-                <span class="n">args</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-            <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="n">args</span><span class="p">))</span>
-
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">EmptySet</span><span class="p">()</span>
-
-
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">frozenset</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>  <span class="c"># remove duplicates</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_elements</span> <span class="o">=</span> <span class="n">args</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">iter</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_intersect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        This function should only be used internally</span>
-
-<span class="sd">        See Set._intersect for docstring</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_elements</span> <span class="o">&amp;</span> <span class="n">other</span><span class="o">.</span><span class="n">_elements</span><span class="p">))</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">el</span> <span class="k">for</span> <span class="n">el</span> <span class="ow">in</span> <span class="bp">self</span> <span class="k">if</span> <span class="n">el</span> <span class="ow">in</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_union</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        This function should only be used internally</span>
-
-<span class="sd">        See Set._union for docstring</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">is_FiniteSet</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_elements</span> <span class="o">|</span> <span class="n">other</span><span class="o">.</span><span class="n">_elements</span><span class="p">))</span>
-
-        <span class="c"># If other set contains one of my elements, remove it from myself</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">set</span><span class="p">((</span>
-                <span class="n">FiniteSet</span><span class="p">(</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span> <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">True</span><span class="p">),</span>
-                <span class="n">other</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">_contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Tests whether an element, other, is in the set.</span>
-
-<span class="sd">        Relies on Python&#39;s set class. This tests for object equality</span>
-<span class="sd">        All inputs are sympified</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import FiniteSet</span>
-
-<span class="sd">        &gt;&gt;&gt; 1 in FiniteSet(1, 2)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; 5 in FiniteSet(1, 2)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">other</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_elements</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_complement</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The complement of a real finite set is the Union of open Intervals</span>
-<span class="sd">        between the elements of the set.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import FiniteSet</span>
-<span class="sd">        &gt;&gt;&gt; FiniteSet(1, 2, 3).complement</span>
-<span class="sd">        (-oo, 1) U (1, 2) U (2, 3) U (3, oo)</span>
-
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">(</span><span class="n">elem</span><span class="o">.</span><span class="n">is_number</span> <span class="k">for</span> <span class="n">elem</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s">: Complement not defined for symbolic inputs&quot;</span>
-                    <span class="o">%</span> <span class="bp">self</span><span class="p">)</span>
-
-        <span class="c"># as there are only numbers involved, a straight sort is sufficient;</span>
-        <span class="c"># default_sort_key is not needed</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="n">intervals</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c"># Build up a list of intervals between the elements</span>
-        <span class="n">intervals</span> <span class="o">+=</span> <span class="p">[</span><span class="n">Interval</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">,</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">True</span><span class="p">,</span> <span class="bp">True</span><span class="p">)]</span>
-        <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">args</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]):</span>
-            <span class="n">intervals</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Interval</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="bp">True</span><span class="p">,</span> <span class="bp">True</span><span class="p">))</span>  <span class="c"># open intervals</span>
-        <span class="n">intervals</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Interval</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">,</span> <span class="bp">True</span><span class="p">,</span> <span class="bp">True</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">Union</span><span class="p">(</span><span class="n">intervals</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_inf</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">Min</span>
-        <span class="k">return</span> <span class="n">Min</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_sup</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">Max</span>
-        <span class="k">return</span> <span class="n">Max</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">measure</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">0</span>
-
-    <span class="k">def</span> <span class="nf">__len__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="n">el</span> <span class="k">for</span> <span class="n">el</span> <span class="ow">in</span> <span class="bp">self</span> <span class="k">if</span> <span class="n">el</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">other</span><span class="p">)</span>
-
-<div class="viewcode-block" id="FiniteSet.as_relational"><a class="viewcode-back" href="../../../modules/sets.html#sympy.core.sets.FiniteSet.as_relational">[docs]</a>    <span class="k">def</span> <span class="nf">as_relational</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">symbol</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Rewrite a FiniteSet in terms of equalities and logic operators. &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Eq</span>
-        <span class="k">return</span> <span class="n">Or</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">Eq</span><span class="p">(</span><span class="n">symbol</span><span class="p">,</span> <span class="n">elem</span><span class="p">)</span> <span class="k">for</span> <span class="n">elem</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">])</span>
-</div>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="n">el</span><span class="o">.</span><span class="n">is_real</span> <span class="k">for</span> <span class="n">el</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">compare</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="nb">hash</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-</span> <span class="nb">hash</span><span class="p">(</span><span class="n">other</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="n">elem</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span> <span class="k">for</span> <span class="n">elem</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_hashable_content</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_elements</span><span class="p">,)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_sorted_args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">default_sort_key</span>
-        <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span></div>
-</pre></div>
-
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-        </div>
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-            <p class="logo"><a href="../../../index.html">
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-  <h3>Quick search</h3>
-    <form class="search" action="../../../search.html" method="get">
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diff --git a/dev-py3k/_modules/sympy/core/singleton.html b/dev-py3k/_modules/sympy/core/singleton.html
deleted file mode 100644
index 961cc49d79b..00000000000
--- a/dev-py3k/_modules/sympy/core/singleton.html
+++ /dev/null
@@ -1,195 +0,0 @@
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-            
-  <h1>Source code for sympy.core.singleton</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Singleton mechanism&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">.core</span> <span class="kn">import</span> <span class="n">Registry</span>
-<span class="kn">from</span> <span class="nn">.assumptions</span> <span class="kn">import</span> <span class="n">ManagedProperties</span>
-<span class="kn">from</span> <span class="nn">.sympify</span> <span class="kn">import</span> <span class="n">sympify</span>
-
-
-<span class="k">class</span> <span class="nc">SingletonRegistry</span><span class="p">(</span><span class="n">Registry</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A map between singleton classes and the corresponding instances.</span>
-<span class="sd">    E.g. S.Exp == C.Exp()</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="n">__call__</span> <span class="o">=</span> <span class="nb">staticmethod</span><span class="p">(</span><span class="n">sympify</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;S&quot;</span>
-
-<span class="n">S</span> <span class="o">=</span> <span class="n">SingletonRegistry</span><span class="p">()</span>
-
-
-<span class="k">class</span> <span class="nc">Singleton</span><span class="p">(</span><span class="n">ManagedProperties</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Metaclass for singleton classes.</span>
-
-<span class="sd">    A singleton class has only one instance which is returned every time the</span>
-<span class="sd">    class is instantiated. Additionally, this instance can be accessed through</span>
-<span class="sd">    the global registry object S as S.&lt;class_name&gt;.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import S, Basic</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.core.singleton import Singleton</span>
-<span class="sd">        &gt;&gt;&gt; class MySingleton(Basic, metaclass=Singleton):</span>
-<span class="sd">        ...     pass</span>
-<span class="sd">        &gt;&gt;&gt; Basic() is Basic()</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; MySingleton() is MySingleton()</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; S.MySingleton is MySingleton()</span>
-<span class="sd">        True</span>
-
-<span class="sd">    ** Developer notes **</span>
-<span class="sd">        The class is instanciated immediately at the point where it is defined</span>
-<span class="sd">        by calling cls.__new__(cls). This instance is cached and cls.__new__ is</span>
-<span class="sd">        rebound to return it directly.</span>
-
-<span class="sd">        The original constructor is also cached to allow subclasses to access it</span>
-<span class="sd">        and have their own instance.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">bases</span><span class="p">,</span> <span class="n">dict_</span><span class="p">):</span>
-        <span class="nb">super</span><span class="p">(</span><span class="n">Singleton</span><span class="p">,</span> <span class="n">cls</span><span class="p">)</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">bases</span><span class="p">,</span> <span class="n">dict_</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">ancestor</span> <span class="ow">in</span> <span class="n">cls</span><span class="o">.</span><span class="n">mro</span><span class="p">():</span>
-            <span class="k">if</span> <span class="s">&#39;__new__&#39;</span> <span class="ow">in</span> <span class="n">ancestor</span><span class="o">.</span><span class="n">__dict__</span><span class="p">:</span>
-                <span class="k">break</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ancestor</span><span class="p">,</span> <span class="n">Singleton</span><span class="p">)</span> <span class="ow">and</span> <span class="n">ancestor</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">cls</span><span class="p">:</span>
-            <span class="n">ctor</span> <span class="o">=</span> <span class="n">ancestor</span><span class="o">.</span><span class="n">_new_instance</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">ctor</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">__new__</span>
-        <span class="n">cls</span><span class="o">.</span><span class="n">_new_instance</span> <span class="o">=</span> <span class="nb">staticmethod</span><span class="p">(</span><span class="n">ctor</span><span class="p">)</span>
-
-        <span class="n">the_instance</span> <span class="o">=</span> <span class="n">ctor</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">the_instance</span>
-        <span class="n">cls</span><span class="o">.</span><span class="n">__new__</span> <span class="o">=</span> <span class="nb">staticmethod</span><span class="p">(</span><span class="n">__new__</span><span class="p">)</span>
-
-        <span class="nb">setattr</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">the_instance</span><span class="p">)</span>
-
-        <span class="c"># Inject pickling support.</span>
-        <span class="k">def</span> <span class="nf">__getnewargs__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">()</span>
-        <span class="n">cls</span><span class="o">.</span><span class="n">__getnewargs__</span> <span class="o">=</span> <span class="n">__getnewargs__</span>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.core.symbol</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core.assumptions</span> <span class="kn">import</span> <span class="n">StdFactKB</span>
-<span class="kn">from</span> <span class="nn">.basic</span> <span class="kn">import</span> <span class="n">Basic</span>
-<span class="kn">from</span> <span class="nn">.core</span> <span class="kn">import</span> <span class="n">C</span>
-<span class="kn">from</span> <span class="nn">.sympify</span> <span class="kn">import</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">.singleton</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="kn">from</span> <span class="nn">.expr</span> <span class="kn">import</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">AtomicExpr</span>
-<span class="kn">from</span> <span class="nn">.cache</span> <span class="kn">import</span> <span class="n">cacheit</span>
-<span class="kn">from</span> <span class="nn">.function</span> <span class="kn">import</span> <span class="n">FunctionClass</span>
-<span class="kn">from</span> <span class="nn">sympy.core.logic</span> <span class="kn">import</span> <span class="n">fuzzy_bool</span>
-<span class="kn">from</span> <span class="nn">sympy.logic.boolalg</span> <span class="kn">import</span> <span class="n">Boolean</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.exceptions</span> <span class="kn">import</span> <span class="n">SymPyDeprecationWarning</span>
-
-<span class="kn">import</span> <span class="nn">re</span><span class="o">,</span> <span class="nn">string</span>
-
-
-<div class="viewcode-block" id="Symbol"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.symbol.Symbol">[docs]</a><span class="k">class</span> <span class="nc">Symbol</span><span class="p">(</span><span class="n">AtomicExpr</span><span class="p">,</span> <span class="n">Boolean</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Assumptions:</span>
-<span class="sd">       commutative = True</span>
-
-<span class="sd">    You can override the default assumptions in the constructor:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; A,B = symbols(&#39;A,B&#39;, commutative = False)</span>
-<span class="sd">    &gt;&gt;&gt; bool(A*B != B*A)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; bool(A*B*2 == 2*A*B) == True # multiplication by scalars is commutative</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_comparable</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;name&#39;</span><span class="p">]</span>
-
-    <span class="n">is_Symbol</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_diff_wrt</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Allow derivatives wrt Symbols.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">            &gt;&gt;&gt; x = Symbol(&#39;x&#39;)</span>
-<span class="sd">            &gt;&gt;&gt; x._diff_wrt</span>
-<span class="sd">            True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Symbols are identified by name and assumptions::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; Symbol(&quot;x&quot;) == Symbol(&quot;x&quot;)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Symbol(&quot;x&quot;, real=True) == Symbol(&quot;x&quot;, real=False)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="s">&#39;dummy&#39;</span> <span class="ow">in</span> <span class="n">assumptions</span><span class="p">:</span>
-            <span class="n">SymPyDeprecationWarning</span><span class="p">(</span>
-                <span class="n">feature</span><span class="o">=</span><span class="s">&quot;Symbol(&#39;x&#39;, dummy=True)&quot;</span><span class="p">,</span>
-                <span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;Dummy() or symbols(..., cls=Dummy)&quot;</span><span class="p">,</span>
-                <span class="n">issue</span><span class="o">=</span><span class="mi">3378</span><span class="p">,</span> <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.0&quot;</span><span class="p">,</span>
-            <span class="p">)</span><span class="o">.</span><span class="n">warn</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">assumptions</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;dummy&#39;</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">Dummy</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">assumptions</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;zero&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="n">is_commutative</span> <span class="o">=</span> <span class="n">fuzzy_bool</span><span class="p">(</span><span class="n">assumptions</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;commutative&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">is_commutative</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="sd">&#39;&#39;&#39;Symbol commutativity must be True or False.&#39;&#39;&#39;</span><span class="p">)</span>
-        <span class="n">assumptions</span><span class="p">[</span><span class="s">&#39;commutative&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">is_commutative</span>
-        <span class="k">return</span> <span class="n">Symbol</span><span class="o">.</span><span class="n">__xnew_cached_</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__new_stage2__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">assert</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="nb">str</span><span class="p">),</span> <span class="nb">repr</span><span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">name</span><span class="p">))</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="n">name</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_assumptions</span> <span class="o">=</span> <span class="n">StdFactKB</span><span class="p">(</span><span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="n">__xnew__</span> <span class="o">=</span> <span class="nb">staticmethod</span><span class="p">(</span>
-        <span class="n">__new_stage2__</span><span class="p">)</span>            <span class="c"># never cached (e.g. dummy)</span>
-    <span class="n">__xnew_cached_</span> <span class="o">=</span> <span class="nb">staticmethod</span><span class="p">(</span>
-        <span class="n">cacheit</span><span class="p">(</span><span class="n">__new_stage2__</span><span class="p">))</span>   <span class="c"># symbols are always cached</span>
-
-    <span class="k">def</span> <span class="nf">__getnewargs__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">,)</span>
-
-    <span class="k">def</span> <span class="nf">__getstate__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">{</span><span class="s">&#39;_assumptions&#39;</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_assumptions</span><span class="p">}</span>
-
-    <span class="k">def</span> <span class="nf">_hashable_content</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">,)</span> <span class="o">+</span> <span class="nb">tuple</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">items</span><span class="p">()))</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">assumptions0</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">dict</span><span class="p">((</span><span class="n">key</span><span class="p">,</span> <span class="n">value</span><span class="p">)</span> <span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="n">value</span>
-                <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_assumptions</span><span class="o">.</span><span class="n">items</span><span class="p">()</span> <span class="k">if</span> <span class="n">value</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">)</span>
-
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">sort_key</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">class_key</span><span class="p">(),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="p">),)),</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="o">.</span><span class="n">sort_key</span><span class="p">(),</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-    <span class="k">def</span> <span class="nf">as_dummy</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Dummy</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">assumptions0</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">.function</span> <span class="kn">import</span> <span class="n">Function</span>
-        <span class="k">return</span> <span class="n">Function</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">as_real_imag</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;ignore&#39;</span><span class="p">)</span> <span class="o">==</span> <span class="bp">self</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">re</span><span class="p">(</span><span class="bp">self</span><span class="p">),</span> <span class="n">C</span><span class="o">.</span><span class="n">im</span><span class="p">(</span><span class="bp">self</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">var</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">is_constant</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">wrt</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">wrt</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="bp">self</span> <span class="ow">in</span> <span class="n">wrt</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_number</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">free_symbols</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">set</span><span class="p">([</span><span class="bp">self</span><span class="p">])</span>
-
-</div>
-<div class="viewcode-block" id="Dummy"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.symbol.Dummy">[docs]</a><span class="k">class</span> <span class="nc">Dummy</span><span class="p">(</span><span class="n">Symbol</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Dummy symbols are each unique, identified by an internal count index:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Dummy</span>
-<span class="sd">    &gt;&gt;&gt; bool(Dummy(&quot;x&quot;) == Dummy(&quot;x&quot;)) == True</span>
-<span class="sd">    False</span>
-
-<span class="sd">    If a name is not supplied then a string value of the count index will be</span>
-<span class="sd">    used. This is useful when a temporary variable is needed and the name</span>
-<span class="sd">    of the variable used in the expression is not important.</span>
-
-<span class="sd">    &gt;&gt;&gt; Dummy._count = 0 # /!\ this should generally not be changed; it is being</span>
-<span class="sd">    &gt;&gt;&gt; Dummy()          # used here to make sure that the doctest passes.</span>
-<span class="sd">    _0</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_count</span> <span class="o">=</span> <span class="mi">0</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;dummy_index&#39;</span><span class="p">]</span>
-
-    <span class="n">is_Dummy</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">name</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">name</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">Dummy</span><span class="o">.</span><span class="n">_count</span><span class="p">)</span>
-
-        <span class="n">is_commutative</span> <span class="o">=</span> <span class="n">fuzzy_bool</span><span class="p">(</span><span class="n">assumptions</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;commutative&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">is_commutative</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="sd">&#39;&#39;&#39;Dummy&#39;s commutativity must be True or False.&#39;&#39;&#39;</span><span class="p">)</span>
-        <span class="n">assumptions</span><span class="p">[</span><span class="s">&#39;commutative&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">is_commutative</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Symbol</span><span class="o">.</span><span class="n">__xnew__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">)</span>
-
-        <span class="n">Dummy</span><span class="o">.</span><span class="n">_count</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">dummy_index</span> <span class="o">=</span> <span class="n">Dummy</span><span class="o">.</span><span class="n">_count</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">__getstate__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">{</span><span class="s">&#39;_assumptions&#39;</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_assumptions</span><span class="p">,</span> <span class="s">&#39;dummy_index&#39;</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">dummy_index</span><span class="p">}</span>
-
-    <span class="k">def</span> <span class="nf">_hashable_content</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Symbol</span><span class="o">.</span><span class="n">_hashable_content</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dummy_index</span><span class="p">,)</span>
-
-</div>
-<div class="viewcode-block" id="Wild"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.symbol.Wild">[docs]</a><span class="k">class</span> <span class="nc">Wild</span><span class="p">(</span><span class="n">Symbol</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A Wild symbol matches anything.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Wild, WildFunction, cos, pi</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; a = Wild(&#39;a&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; b = Wild(&#39;b&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; b.match(a)</span>
-<span class="sd">    {a_: b_}</span>
-<span class="sd">    &gt;&gt;&gt; x.match(a)</span>
-<span class="sd">    {a_: x}</span>
-<span class="sd">    &gt;&gt;&gt; pi.match(a)</span>
-<span class="sd">    {a_: pi}</span>
-<span class="sd">    &gt;&gt;&gt; (x**2).match(a)</span>
-<span class="sd">    {a_: x**2}</span>
-<span class="sd">    &gt;&gt;&gt; cos(x).match(a)</span>
-<span class="sd">    {a_: cos(x)}</span>
-<span class="sd">    &gt;&gt;&gt; A = WildFunction(&#39;A&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; A.match(a)</span>
-<span class="sd">    {a_: A_}</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;exclude&#39;</span><span class="p">,</span> <span class="s">&#39;properties&#39;</span><span class="p">]</span>
-    <span class="n">is_Wild</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">(),</span> <span class="n">properties</span><span class="o">=</span><span class="p">(),</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">):</span>
-        <span class="n">exclude</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">([</span><span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">exclude</span><span class="p">])</span>
-        <span class="n">properties</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">properties</span><span class="p">)</span>
-        <span class="n">is_commutative</span> <span class="o">=</span> <span class="n">fuzzy_bool</span><span class="p">(</span><span class="n">assumptions</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;commutative&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">is_commutative</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="sd">&#39;&#39;&#39;Wild&#39;s commutativity must be True or False.&#39;&#39;&#39;</span><span class="p">)</span>
-        <span class="n">assumptions</span><span class="p">[</span><span class="s">&#39;commutative&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">is_commutative</span>
-        <span class="k">return</span> <span class="n">Wild</span><span class="o">.</span><span class="n">__xnew__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">exclude</span><span class="p">,</span> <span class="n">properties</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__getnewargs__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">exclude</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">properties</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">__xnew__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">exclude</span><span class="p">,</span> <span class="n">properties</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">):</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Symbol</span><span class="o">.</span><span class="n">__xnew__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">exclude</span> <span class="o">=</span> <span class="n">exclude</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">properties</span> <span class="o">=</span> <span class="n">properties</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">_hashable_content</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Wild</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">_hashable_content</span><span class="p">()</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">exclude</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">properties</span><span class="p">)</span>
-
-    <span class="c"># TODO add check against another Wild</span>
-    <span class="k">def</span> <span class="nf">matches</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">repl_dict</span><span class="o">=</span><span class="p">{},</span> <span class="n">old</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">exclude</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="ow">not</span> <span class="n">f</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">properties</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="n">repl_dict</span> <span class="o">=</span> <span class="n">repl_dict</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="n">repl_dict</span><span class="p">[</span><span class="bp">self</span><span class="p">]</span> <span class="o">=</span> <span class="n">expr</span>
-        <span class="k">return</span> <span class="n">repl_dict</span>
-
-    <span class="k">def</span> <span class="nf">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;&#39;</span><span class="si">%s</span><span class="s">&#39; object is not callable&quot;</span> <span class="o">%</span> <span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__name__</span><span class="p">)</span>
-</div>
-<span class="n">_re_var_range</span> <span class="o">=</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="s">r&quot;^(.*?)(\d*):(\d+)$&quot;</span><span class="p">)</span>
-<span class="n">_re_var_scope</span> <span class="o">=</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="s">r&quot;^(.*?|)(.):(.)(.*?|)$&quot;</span><span class="p">)</span>
-<span class="n">_re_var_split</span> <span class="o">=</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="s">r&quot;\s*,\s*|\s+&quot;</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="symbols"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.symbol.symbols">[docs]</a><span class="k">def</span> <span class="nf">symbols</span><span class="p">(</span><span class="n">names</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Transform strings into instances of :class:`Symbol` class.</span>
-
-<span class="sd">    :func:`symbols` function returns a sequence of symbols with names taken</span>
-<span class="sd">    from ``names`` argument, which can be a comma or whitespace delimited</span>
-<span class="sd">    string, or a sequence of strings::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols, Function</span>
-
-<span class="sd">        &gt;&gt;&gt; x, y, z = symbols(&#39;x,y,z&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; a, b, c = symbols(&#39;a b c&#39;)</span>
-
-<span class="sd">    The type of output is dependent on the properties of input arguments::</span>
-
-<span class="sd">        &gt;&gt;&gt; symbols(&#39;x&#39;)</span>
-<span class="sd">        x</span>
-<span class="sd">        &gt;&gt;&gt; symbols(&#39;x,&#39;)</span>
-<span class="sd">        (x,)</span>
-<span class="sd">        &gt;&gt;&gt; symbols(&#39;x,y&#39;)</span>
-<span class="sd">        (x, y)</span>
-<span class="sd">        &gt;&gt;&gt; symbols((&#39;a&#39;, &#39;b&#39;, &#39;c&#39;))</span>
-<span class="sd">        (a, b, c)</span>
-<span class="sd">        &gt;&gt;&gt; symbols([&#39;a&#39;, &#39;b&#39;, &#39;c&#39;])</span>
-<span class="sd">        [a, b, c]</span>
-<span class="sd">        &gt;&gt;&gt; symbols(set([&#39;a&#39;, &#39;b&#39;, &#39;c&#39;]))</span>
-<span class="sd">        set([a, b, c])</span>
-
-<span class="sd">    If an iterable container is needed for a single symbol, set the ``seq``</span>
-<span class="sd">    argument to ``True`` or terminate the symbol name with a comma::</span>
-
-<span class="sd">        &gt;&gt;&gt; symbols(&#39;x&#39;, seq=True)</span>
-<span class="sd">        (x,)</span>
-
-<span class="sd">    To reduce typing, range syntax is supported to create indexed symbols::</span>
-
-<span class="sd">        &gt;&gt;&gt; symbols(&#39;x:10&#39;)</span>
-<span class="sd">        (x0, x1, x2, x3, x4, x5, x6, x7, x8, x9)</span>
-
-<span class="sd">        &gt;&gt;&gt; symbols(&#39;x5:10&#39;)</span>
-<span class="sd">        (x5, x6, x7, x8, x9)</span>
-
-<span class="sd">        &gt;&gt;&gt; symbols(&#39;x5:10,y:5&#39;)</span>
-<span class="sd">        (x5, x6, x7, x8, x9, y0, y1, y2, y3, y4)</span>
-
-<span class="sd">        &gt;&gt;&gt; symbols((&#39;x5:10&#39;, &#39;y:5&#39;))</span>
-<span class="sd">        ((x5, x6, x7, x8, x9), (y0, y1, y2, y3, y4))</span>
-
-<span class="sd">    To reduce typing even more, lexicographic range syntax is supported::</span>
-
-<span class="sd">        &gt;&gt;&gt; symbols(&#39;x:z&#39;)</span>
-<span class="sd">        (x, y, z)</span>
-
-<span class="sd">        &gt;&gt;&gt; symbols(&#39;a:d,x:z&#39;)</span>
-<span class="sd">        (a, b, c, d, x, y, z)</span>
-
-<span class="sd">        &gt;&gt;&gt; symbols((&#39;a:d&#39;, &#39;x:z&#39;))</span>
-<span class="sd">        ((a, b, c, d), (x, y, z))</span>
-
-<span class="sd">    All newly created symbols have assumptions set accordingly to ``args``::</span>
-
-<span class="sd">        &gt;&gt;&gt; a = symbols(&#39;a&#39;, integer=True)</span>
-<span class="sd">        &gt;&gt;&gt; a.is_integer</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &gt;&gt;&gt; x, y, z = symbols(&#39;x,y,z&#39;, real=True)</span>
-<span class="sd">        &gt;&gt;&gt; x.is_real and y.is_real and z.is_real</span>
-<span class="sd">        True</span>
-
-<span class="sd">    Despite its name, :func:`symbols` can create symbol--like objects of</span>
-<span class="sd">    other type, for example instances of Function or Wild classes. To</span>
-<span class="sd">    achieve this, set ``cls`` keyword argument to the desired type::</span>
-
-<span class="sd">        &gt;&gt;&gt; symbols(&#39;f,g,h&#39;, cls=Function)</span>
-<span class="sd">        (f, g, h)</span>
-
-<span class="sd">        &gt;&gt;&gt; type(_[0])</span>
-<span class="sd">        &lt;class &#39;sympy.core.function.UndefinedFunction&#39;&gt;</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">if</span> <span class="s">&#39;each_char&#39;</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">args</span><span class="p">[</span><span class="s">&#39;each_char&#39;</span><span class="p">]:</span>
-            <span class="n">value</span> <span class="o">=</span> <span class="s">&quot;Tip: &#39; &#39;.join(s) will transform a string s = &#39;xyz&#39; to &#39;x y z&#39;.&quot;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">value</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-        <span class="n">SymPyDeprecationWarning</span><span class="p">(</span>
-            <span class="n">feature</span><span class="o">=</span><span class="s">&quot;each_char in the options to symbols() and var()&quot;</span><span class="p">,</span>
-            <span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;spaces or commas between symbol names&quot;</span><span class="p">,</span>
-            <span class="n">issue</span><span class="o">=</span><span class="mi">1919</span><span class="p">,</span> <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.0&quot;</span><span class="p">,</span> <span class="n">value</span><span class="o">=</span><span class="n">value</span>
-        <span class="p">)</span><span class="o">.</span><span class="n">warn</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">names</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-        <span class="n">names</span> <span class="o">=</span> <span class="n">names</span><span class="o">.</span><span class="n">strip</span><span class="p">()</span>
-        <span class="n">as_seq</span> <span class="o">=</span> <span class="n">names</span><span class="o">.</span><span class="n">endswith</span><span class="p">(</span><span class="s">&#39;,&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">as_seq</span><span class="p">:</span>
-            <span class="n">names</span> <span class="o">=</span> <span class="n">names</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">rstrip</span><span class="p">()</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">names</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;no symbols given&#39;</span><span class="p">)</span>
-
-        <span class="n">names</span> <span class="o">=</span> <span class="n">_re_var_split</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="n">names</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">args</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;each_char&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">as_seq</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">names</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">symbols</span><span class="p">(</span><span class="nb">tuple</span><span class="p">(</span><span class="n">names</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="n">cls</span> <span class="o">=</span> <span class="n">args</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;cls&#39;</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">)</span>
-        <span class="n">seq</span> <span class="o">=</span> <span class="n">args</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;seq&#39;</span><span class="p">,</span> <span class="n">as_seq</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">name</span> <span class="ow">in</span> <span class="n">names</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">name</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;missing symbol&#39;</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="s">&#39;:&#39;</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">name</span><span class="p">:</span>
-                <span class="n">symbol</span> <span class="o">=</span> <span class="n">cls</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">symbol</span><span class="p">)</span>
-                <span class="k">continue</span>
-
-            <span class="n">match</span> <span class="o">=</span> <span class="n">_re_var_range</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">match</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">name</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">end</span> <span class="o">=</span> <span class="n">match</span><span class="o">.</span><span class="n">groups</span><span class="p">()</span>
-
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">start</span><span class="p">:</span>
-                    <span class="n">start</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">start</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">start</span><span class="p">)</span>
-
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">end</span><span class="p">)):</span>
-                    <span class="n">symbol</span> <span class="o">=</span> <span class="n">cls</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s%i</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">i</span><span class="p">),</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-                    <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">symbol</span><span class="p">)</span>
-
-                <span class="n">seq</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">continue</span>
-
-            <span class="n">match</span> <span class="o">=</span> <span class="n">_re_var_scope</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">match</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">name</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">end</span><span class="p">,</span> <span class="n">suffix</span> <span class="o">=</span> <span class="n">match</span><span class="o">.</span><span class="n">groups</span><span class="p">()</span>
-                <span class="n">letters</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">string</span><span class="o">.</span><span class="n">ascii_lowercase</span> <span class="o">+</span> <span class="n">string</span><span class="o">.</span><span class="n">ascii_uppercase</span>
-                        <span class="o">+</span> <span class="n">string</span><span class="o">.</span><span class="n">digits</span><span class="p">)</span>
-                <span class="n">start</span> <span class="o">=</span> <span class="n">letters</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">start</span><span class="p">)</span>
-                <span class="n">end</span> <span class="o">=</span> <span class="n">letters</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">end</span><span class="p">)</span>
-
-                <span class="k">for</span> <span class="n">subname</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="n">end</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                    <span class="n">symbol</span> <span class="o">=</span> <span class="n">cls</span><span class="p">(</span><span class="n">name</span> <span class="o">+</span> <span class="n">letters</span><span class="p">[</span><span class="n">subname</span><span class="p">]</span> <span class="o">+</span> <span class="n">suffix</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-                    <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">symbol</span><span class="p">)</span>
-
-                <span class="n">seq</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">continue</span>
-
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&quot;&#39;</span><span class="si">%s</span><span class="s">&#39; is not a valid symbol range specification&quot;</span> <span class="o">%</span> <span class="n">name</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">seq</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">result</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">result</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;missing symbol&#39;</span><span class="p">)</span>  <span class="c"># should never happen</span>
-            <span class="k">return</span> <span class="n">result</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">name</span> <span class="ow">in</span> <span class="n">names</span><span class="p">:</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">symbols</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="nb">type</span><span class="p">(</span><span class="n">names</span><span class="p">)(</span><span class="n">result</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="var"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.symbol.var">[docs]</a><span class="k">def</span> <span class="nf">var</span><span class="p">(</span><span class="n">names</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create symbols and inject them into the global namespace.</span>
-
-<span class="sd">    This calls :func:`symbols` with the same arguments and puts the results</span>
-<span class="sd">    into the *global* namespace. It&#39;s recommended not to use :func:`var` in</span>
-<span class="sd">    library code, where :func:`symbols` has to be used::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import var</span>
-
-<span class="sd">        &gt;&gt;&gt; var(&#39;x&#39;)</span>
-<span class="sd">        x</span>
-<span class="sd">        &gt;&gt;&gt; x</span>
-<span class="sd">        x</span>
-
-<span class="sd">        &gt;&gt;&gt; var(&#39;a,ab,abc&#39;)</span>
-<span class="sd">        (a, ab, abc)</span>
-<span class="sd">        &gt;&gt;&gt; abc</span>
-<span class="sd">        abc</span>
-
-<span class="sd">        &gt;&gt;&gt; var(&#39;x,y&#39;, real=True)</span>
-<span class="sd">        (x, y)</span>
-<span class="sd">        &gt;&gt;&gt; x.is_real and y.is_real</span>
-<span class="sd">        True</span>
-
-<span class="sd">    See :func:`symbol` documentation for more details on what kinds of</span>
-<span class="sd">    arguments can be passed to :func:`var`.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">traverse</span><span class="p">(</span><span class="n">symbols</span><span class="p">,</span> <span class="n">frame</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Recursively inject symbols to the global namespace. &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">symbol</span> <span class="ow">in</span> <span class="n">symbols</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">symbol</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-                <span class="n">frame</span><span class="o">.</span><span class="n">f_globals</span><span class="p">[</span><span class="n">symbol</span><span class="o">.</span><span class="n">name</span><span class="p">]</span> <span class="o">=</span> <span class="n">symbol</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">symbol</span><span class="p">,</span> <span class="n">FunctionClass</span><span class="p">):</span>
-                <span class="n">frame</span><span class="o">.</span><span class="n">f_globals</span><span class="p">[</span><span class="n">symbol</span><span class="o">.</span><span class="n">__name__</span><span class="p">]</span> <span class="o">=</span> <span class="n">symbol</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">traverse</span><span class="p">(</span><span class="n">symbol</span><span class="p">,</span> <span class="n">frame</span><span class="p">)</span>
-
-    <span class="kn">from</span> <span class="nn">inspect</span> <span class="kn">import</span> <span class="n">currentframe</span>
-    <span class="n">frame</span> <span class="o">=</span> <span class="n">currentframe</span><span class="p">()</span><span class="o">.</span><span class="n">f_back</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">syms</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="n">names</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">syms</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">syms</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-                <span class="n">frame</span><span class="o">.</span><span class="n">f_globals</span><span class="p">[</span><span class="n">syms</span><span class="o">.</span><span class="n">name</span><span class="p">]</span> <span class="o">=</span> <span class="n">syms</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">syms</span><span class="p">,</span> <span class="n">FunctionClass</span><span class="p">):</span>
-                <span class="n">frame</span><span class="o">.</span><span class="n">f_globals</span><span class="p">[</span><span class="n">syms</span><span class="o">.</span><span class="n">__name__</span><span class="p">]</span> <span class="o">=</span> <span class="n">syms</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">traverse</span><span class="p">(</span><span class="n">syms</span><span class="p">,</span> <span class="n">frame</span><span class="p">)</span>
-    <span class="k">finally</span><span class="p">:</span>
-        <span class="k">del</span> <span class="n">frame</span>  <span class="c"># break cyclic dependencies as stated in inspect docs</span>
-
-    <span class="k">return</span> <span class="n">syms</span></div>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
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-<div id="searchbox" style="display: none">
-  <h3>Quick search</h3>
-    <form class="search" action="../../../search.html" method="get">
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-        <li class="right" >
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deleted file mode 100644
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+++ /dev/null
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-  <h1>Source code for sympy.core.sympify</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;sympify -- convert objects SymPy internal format&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">inspect</span> <span class="kn">import</span> <span class="n">getmro</span>
-
-<span class="kn">from</span> <span class="nn">.core</span> <span class="kn">import</span> <span class="n">all_classes</span> <span class="k">as</span> <span class="n">sympy_classes</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">iterable</span>
-
-
-<span class="k">class</span> <span class="nc">SympifyError</span><span class="p">(</span><span class="ne">ValueError</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">base_exc</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">base_exc</span> <span class="o">=</span> <span class="n">base_exc</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">base_exc</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;SympifyError: </span><span class="si">%r</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">,)</span>
-
-        <span class="k">return</span> <span class="p">(</span><span class="s">&quot;Sympify of expression &#39;</span><span class="si">%s</span><span class="s">&#39; failed, because of exception being &quot;</span>
-            <span class="s">&quot;raised:</span><span class="se">\n</span><span class="si">%s</span><span class="s">: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">base_exc</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span>
-            <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">base_exc</span><span class="p">)))</span>
-
-<span class="n">converter</span> <span class="o">=</span> <span class="p">{}</span>  <span class="c"># See sympify docstring.</span>
-
-
-<div class="viewcode-block" id="sympify"><a class="viewcode-back" href="../../../modules/core.html#sympy.core.sympify.sympify">[docs]</a><span class="k">def</span> <span class="nf">sympify</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="nb">locals</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">convert_xor</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">rational</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Converts an arbitrary expression to a type that can be used inside sympy.</span>
-
-<span class="sd">    For example, it will convert python ints into instance of sympy.Rational,</span>
-<span class="sd">    floats into instances of sympy.Float, etc. It is also able to coerce symbolic</span>
-<span class="sd">    expressions which inherit from Basic. This can be useful in cooperation</span>
-<span class="sd">    with SAGE.</span>
-
-<span class="sd">    It currently accepts as arguments:</span>
-<span class="sd">       - any object defined in sympy (except matrices [TODO])</span>
-<span class="sd">       - standard numeric python types: int, long, float, Decimal</span>
-<span class="sd">       - strings (like &quot;0.09&quot; or &quot;2e-19&quot;)</span>
-<span class="sd">       - booleans, including ``None`` (will leave them unchanged)</span>
-<span class="sd">       - lists, sets or tuples containing any of the above</span>
-
-<span class="sd">    If the argument is already a type that sympy understands, it will do</span>
-<span class="sd">    nothing but return that value. This can be used at the beginning of a</span>
-<span class="sd">    function to ensure you are working with the correct type.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import sympify</span>
-
-<span class="sd">    &gt;&gt;&gt; sympify(2).is_integer</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; sympify(2).is_real</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &gt;&gt;&gt; sympify(2.0).is_real</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; sympify(&quot;2.0&quot;).is_real</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; sympify(&quot;2e-45&quot;).is_real</span>
-<span class="sd">    True</span>
-
-<span class="sd">    If the expression could not be converted, a SympifyError is raised.</span>
-
-<span class="sd">    &gt;&gt;&gt; sympify(&quot;x***2&quot;)</span>
-<span class="sd">    Traceback (most recent call last):</span>
-<span class="sd">    ...</span>
-<span class="sd">    SympifyError: SympifyError: &quot;could not parse u&#39;x***2&#39;&quot;</span>
-
-<span class="sd">    Locals</span>
-<span class="sd">    ------</span>
-
-<span class="sd">    The sympification happens with access to everything that is loaded</span>
-<span class="sd">    by ``from sympy import *``; anything used in a string that is not</span>
-<span class="sd">    defined by that import will be converted to a symbol. In the following,</span>
-<span class="sd">    the ``bitcout`` function is treated as a symbol and the ``O`` is</span>
-<span class="sd">    interpreted as the Order object (used with series) and it raises</span>
-<span class="sd">    an error when used improperly:</span>
-
-<span class="sd">    &gt;&gt;&gt; s = &#39;bitcount(42)&#39;</span>
-<span class="sd">    &gt;&gt;&gt; sympify(s)</span>
-<span class="sd">    bitcount(42)</span>
-<span class="sd">    &gt;&gt;&gt; sympify(&quot;O(x)&quot;)</span>
-<span class="sd">    O(x)</span>
-<span class="sd">    &gt;&gt;&gt; sympify(&quot;O + 1&quot;)</span>
-<span class="sd">    Traceback (most recent call last):</span>
-<span class="sd">    ...</span>
-<span class="sd">    TypeError: unbound method...</span>
-
-<span class="sd">    In order to have ``bitcount`` be recognized it can be imported into a</span>
-<span class="sd">    namespace dictionary and passed as locals:</span>
-
-<span class="sd">    &gt;&gt;&gt; ns = {}</span>
-<span class="sd">    &gt;&gt;&gt; exec(&#39;from sympy.core.evalf import bitcount&#39;, ns)</span>
-<span class="sd">    &gt;&gt;&gt; sympify(s, locals=ns)</span>
-<span class="sd">    6</span>
-
-<span class="sd">    In order to have the ``O`` interpreted as a Symbol, identify it as such</span>
-<span class="sd">    in the namespace dictionary. This can be done in a variety of ways; all</span>
-<span class="sd">    three of the following are possibilities:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">    &gt;&gt;&gt; ns[&quot;O&quot;] = Symbol(&quot;O&quot;)  # method 1</span>
-<span class="sd">    &gt;&gt;&gt; exec(&#39;from sympy.abc import O&#39;, ns)  # method 2</span>
-<span class="sd">    &gt;&gt;&gt; ns.update(dict(O=Symbol(&quot;O&quot;)))  # method 3</span>
-<span class="sd">    &gt;&gt;&gt; sympify(&quot;O + 1&quot;, locals=ns)</span>
-<span class="sd">    O + 1</span>
-
-<span class="sd">    If you want *all* single-letter and Greek-letter variables to be symbols</span>
-<span class="sd">    then you can use the clashing-symbols dictionaries that have been defined</span>
-<span class="sd">    there as private variables: _clash1 (single-letter variables), _clash2</span>
-<span class="sd">    (the multi-letter Greek names) or _clash (both single and multi-letter</span>
-<span class="sd">    names that are defined in abc).</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import _clash1</span>
-<span class="sd">    &gt;&gt;&gt; _clash1</span>
-<span class="sd">    {&#39;C&#39;: C, &#39;E&#39;: E, &#39;I&#39;: I, &#39;N&#39;: N, &#39;O&#39;: O, &#39;Q&#39;: Q, &#39;S&#39;: S}</span>
-<span class="sd">    &gt;&gt;&gt; sympify(&#39;C &amp; Q&#39;, _clash1)</span>
-<span class="sd">    And(C, Q)</span>
-
-<span class="sd">    Strict</span>
-<span class="sd">    ------</span>
-
-<span class="sd">    If the option ``strict`` is set to ``True``, only the types for which an</span>
-<span class="sd">    explicit conversion has been defined are converted. In the other</span>
-<span class="sd">    cases, a SympifyError is raised.</span>
-
-<span class="sd">    &gt;&gt;&gt; sympify(True)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; sympify(True, strict=True)</span>
-<span class="sd">    Traceback (most recent call last):</span>
-<span class="sd">    ...</span>
-<span class="sd">    SympifyError: SympifyError: True</span>
-
-<span class="sd">    Extending</span>
-<span class="sd">    ---------</span>
-
-<span class="sd">    To extend ``sympify`` to convert custom objects (not derived from ``Basic``),</span>
-<span class="sd">    just define a ``_sympy_`` method to your class. You can do that even to</span>
-<span class="sd">    classes that you do not own by subclassing or adding the method at runtime.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Matrix</span>
-<span class="sd">    &gt;&gt;&gt; class MyList1(object):</span>
-<span class="sd">    ...     def __iter__(self):</span>
-<span class="sd">    ...         yield 1</span>
-<span class="sd">    ...         yield 2</span>
-<span class="sd">    ...         raise StopIteration</span>
-<span class="sd">    ...     def __getitem__(self, i): return list(self)[i]</span>
-<span class="sd">    ...     def _sympy_(self): return Matrix(self)</span>
-<span class="sd">    &gt;&gt;&gt; sympify(MyList1())</span>
-<span class="sd">    [1]</span>
-<span class="sd">    [2]</span>
-
-<span class="sd">    If you do not have control over the class definition you could also use the</span>
-<span class="sd">    ``converter`` global dictionary. The key is the class and the value is a</span>
-<span class="sd">    function that takes a single argument and returns the desired SymPy</span>
-<span class="sd">    object, e.g. ``converter[MyList] = lambda x: Matrix(x)``.</span>
-
-<span class="sd">    &gt;&gt;&gt; class MyList2(object):   # XXX Do not do this if you control the class!</span>
-<span class="sd">    ...     def __iter__(self):  #     Use _sympy_!</span>
-<span class="sd">    ...         yield 1</span>
-<span class="sd">    ...         yield 2</span>
-<span class="sd">    ...         raise StopIteration</span>
-<span class="sd">    ...     def __getitem__(self, i): return list(self)[i]</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.core.sympify import converter</span>
-<span class="sd">    &gt;&gt;&gt; converter[MyList2] = lambda x: Matrix(x)</span>
-<span class="sd">    &gt;&gt;&gt; sympify(MyList2())</span>
-<span class="sd">    [1]</span>
-<span class="sd">    [2]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">cls</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">__class__</span>
-    <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>  <span class="c"># a is probably an old-style class object</span>
-        <span class="n">cls</span> <span class="o">=</span> <span class="nb">type</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">cls</span> <span class="ow">in</span> <span class="n">sympy_classes</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">a</span>
-    <span class="k">if</span> <span class="n">cls</span> <span class="ow">in</span> <span class="p">(</span><span class="nb">bool</span><span class="p">,</span> <span class="nb">type</span><span class="p">(</span><span class="bp">None</span><span class="p">)):</span>
-        <span class="k">if</span> <span class="n">strict</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">SympifyError</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">a</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">converter</span><span class="p">[</span><span class="n">cls</span><span class="p">](</span><span class="n">a</span><span class="p">)</span>
-    <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">superclass</span> <span class="ow">in</span> <span class="n">getmro</span><span class="p">(</span><span class="n">cls</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">converter</span><span class="p">[</span><span class="n">superclass</span><span class="p">](</span><span class="n">a</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-                <span class="k">continue</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">a</span><span class="o">.</span><span class="n">_sympy_</span><span class="p">()</span>
-    <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-        <span class="k">pass</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-        <span class="k">for</span> <span class="nb">coerce</span> <span class="ow">in</span> <span class="p">(</span><span class="nb">float</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="nb">coerce</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
-            <span class="k">except</span> <span class="p">(</span><span class="ne">TypeError</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">,</span> <span class="ne">AttributeError</span><span class="p">,</span> <span class="n">SympifyError</span><span class="p">):</span>
-                <span class="k">continue</span>
-
-    <span class="k">if</span> <span class="n">strict</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">SympifyError</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">iterable</span><span class="p">(</span><span class="n">a</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">type</span><span class="p">(</span><span class="n">a</span><span class="p">)([</span><span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="nb">locals</span><span class="o">=</span><span class="nb">locals</span><span class="p">,</span> <span class="n">convert_xor</span><span class="o">=</span><span class="n">convert_xor</span><span class="p">,</span>
-                <span class="n">rational</span><span class="o">=</span><span class="n">rational</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">a</span><span class="p">])</span>
-        <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-            <span class="c"># Not all iterables are rebuildable with their type.</span>
-            <span class="k">pass</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="nb">dict</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">type</span><span class="p">(</span><span class="n">a</span><span class="p">)([</span><span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="nb">locals</span><span class="o">=</span><span class="nb">locals</span><span class="p">,</span> <span class="n">convert_xor</span><span class="o">=</span><span class="n">convert_xor</span><span class="p">,</span>
-                <span class="n">rational</span><span class="o">=</span><span class="n">rational</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">a</span><span class="o">.</span><span class="n">items</span><span class="p">()])</span>
-        <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-            <span class="c"># Not all iterables are rebuildable with their type.</span>
-            <span class="k">pass</span>
-
-    <span class="c"># At this point we were given an arbitrary expression</span>
-    <span class="c"># which does not inherit from Basic and doesn&#39;t implement</span>
-    <span class="c"># _sympy_ (which is a canonical and robust way to convert</span>
-    <span class="c"># anything to SymPy expression).</span>
-    <span class="c">#</span>
-    <span class="c"># As a last chance, we try to take &quot;a&quot;&#39;s normal form via unicode()</span>
-    <span class="c"># and try to parse it. If it fails, then we have no luck and</span>
-    <span class="c"># return an exception</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-    <span class="k">except</span> <span class="ne">Exception</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">SympifyError</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="kn">from</span> <span class="nn">sympy.parsing.sympy_parser</span> <span class="kn">import</span> <span class="p">(</span><span class="n">parse_expr</span><span class="p">,</span> <span class="n">TokenError</span><span class="p">,</span>
-                                            <span class="n">standard_transformations</span><span class="p">)</span>
-    <span class="kn">from</span> <span class="nn">sympy.parsing.sympy_parser</span> <span class="kn">import</span> <span class="n">convert_xor</span> <span class="k">as</span> <span class="n">t_convert_xor</span>
-    <span class="kn">from</span> <span class="nn">sympy.parsing.sympy_parser</span> <span class="kn">import</span> <span class="n">rationalize</span> <span class="k">as</span> <span class="n">t_rationalize</span>
-
-    <span class="n">transformations</span> <span class="o">=</span> <span class="n">standard_transformations</span>
-
-    <span class="k">if</span> <span class="n">rational</span><span class="p">:</span>
-        <span class="n">transformations</span> <span class="o">+=</span> <span class="p">(</span><span class="n">t_rationalize</span><span class="p">,)</span>
-    <span class="k">if</span> <span class="n">convert_xor</span><span class="p">:</span>
-        <span class="n">transformations</span> <span class="o">+=</span> <span class="p">(</span><span class="n">t_convert_xor</span><span class="p">,)</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">parse_expr</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="nb">locals</span> <span class="ow">or</span> <span class="p">{},</span> <span class="n">transformations</span><span class="p">)</span>
-    <span class="k">except</span> <span class="p">(</span><span class="n">TokenError</span><span class="p">,</span> <span class="ne">SyntaxError</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="n">SympifyError</span><span class="p">(</span><span class="s">&#39;could not parse </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">a</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">expr</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_sympify</span><span class="p">(</span><span class="n">a</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Short version of sympify for internal usage for __add__ and __eq__</span>
-<span class="sd">       methods where it is ok to allow some things (like Python integers</span>
-<span class="sd">       and floats) in the expression. This excludes things (like strings)</span>
-<span class="sd">       that are unwise to allow into such an expression.</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy import Integer</span>
-<span class="sd">       &gt;&gt;&gt; Integer(1) == 1</span>
-<span class="sd">       True</span>
-
-<span class="sd">       &gt;&gt;&gt; Integer(1) == &#39;1&#39;</span>
-<span class="sd">       False</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">       &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">       &gt;&gt;&gt; x + 1</span>
-<span class="sd">       x + 1</span>
-
-<span class="sd">       &gt;&gt;&gt; x + &#39;1&#39;</span>
-<span class="sd">       Traceback (most recent call last):</span>
-<span class="sd">           ...</span>
-<span class="sd">       TypeError: unsupported operand type(s) for +: &#39;Symbol&#39; and &#39;str&#39;</span>
-
-<span class="sd">       see: sympify</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/diffgeom.html b/dev-py3k/_modules/sympy/diffgeom.html
deleted file mode 100644
index f6c41cecde9..00000000000
--- a/dev-py3k/_modules/sympy/diffgeom.html
+++ /dev/null
@@ -1,126 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
-
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-        <div class="bodywrapper">
-          <div class="body">
-            
-  <h1>Source code for sympy.diffgeom</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">.diffgeom</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">BaseCovarDerivativeOp</span><span class="p">,</span> <span class="n">BaseScalarField</span><span class="p">,</span> <span class="n">BaseVectorField</span><span class="p">,</span> <span class="n">Commutator</span><span class="p">,</span>
-    <span class="n">contravariant_order</span><span class="p">,</span> <span class="n">CoordSystem</span><span class="p">,</span> <span class="n">CovarDerivativeOp</span><span class="p">,</span> <span class="n">covariant_order</span><span class="p">,</span>
-    <span class="n">Differential</span><span class="p">,</span> <span class="n">intcurve_diffequ</span><span class="p">,</span> <span class="n">intcurve_series</span><span class="p">,</span> <span class="n">LieDerivative</span><span class="p">,</span>
-    <span class="n">Manifold</span><span class="p">,</span> <span class="n">metric_to_Christoffel_1st</span><span class="p">,</span> <span class="n">metric_to_Christoffel_2nd</span><span class="p">,</span>
-    <span class="n">metric_to_Ricci_components</span><span class="p">,</span> <span class="n">metric_to_Riemann_components</span><span class="p">,</span> <span class="n">Patch</span><span class="p">,</span>
-    <span class="n">Point</span><span class="p">,</span> <span class="n">TensorProduct</span><span class="p">,</span> <span class="n">twoform_to_matrix</span><span class="p">,</span> <span class="n">vectors_in_basis</span><span class="p">,</span>
-    <span class="n">WedgeProduct</span><span class="p">,</span>
-<span class="p">)</span>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../index.html">
-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
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-<div id="searchbox" style="display: none">
-  <h3>Quick search</h3>
-    <form class="search" action="../../search.html" method="get">
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-    </form>
-    <p class="searchtip" style="font-size: 90%">
-    Enter search terms or a module, class or function name.
-    </p>
-</div>
-<script type="text/javascript">$('#searchbox').show(0);</script>
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-          <li><a href="../index.html" >Module code</a> &raquo;</li>
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-      </ul>
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-    <div class="footer">
-        &copy; Copyright 2013 SymPy Development Team.
-      Last updated on Jan 20, 2013.
-      Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.1.3.
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-</html>
diff --git a/dev-py3k/_modules/sympy/functions/combinatorial/factorials.html b/dev-py3k/_modules/sympy/functions/combinatorial/factorials.html
deleted file mode 100644
index 1f8e6d8d49c..00000000000
--- a/dev-py3k/_modules/sympy/functions/combinatorial/factorials.html
+++ /dev/null
@@ -1,667 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
-
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-    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
-    
-    <title>sympy.functions.combinatorial.factorials &mdash; SymPy 0.7.2-git documentation</title>
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-        VERSION:     '0.7.2-git',
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-            
-  <h1>Source code for sympy.functions.combinatorial.factorials</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">ArgumentIndexError</span>
-<span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">sieve</span>
-<span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">sqrt</span> <span class="k">as</span> <span class="n">_sqrt</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="nb">reduce</span><span class="p">,</span> <span class="n">as_int</span>
-<span class="kn">from</span> <span class="nn">sympy.core.cache</span> <span class="kn">import</span> <span class="n">cacheit</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-
-
-<span class="k">class</span> <span class="nc">CombinatorialFunction</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Base class for combinatorial functions. &quot;&quot;&quot;</span>
-
-<span class="c">###############################################################################</span>
-<span class="c">######################## FACTORIAL and MULTI-FACTORIAL ########################</span>
-<span class="c">###############################################################################</span>
-
-
-<div class="viewcode-block" id="factorial"><a class="viewcode-back" href="../../../../modules/functions/combinatorial.html#sympy.functions.combinatorial.factorials.factorial">[docs]</a><span class="k">class</span> <span class="nc">factorial</span><span class="p">(</span><span class="n">CombinatorialFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Implementation of factorial function over nonnegative integers.</span>
-<span class="sd">       For the sake of convenience and simplicity of procedures using</span>
-<span class="sd">       this function it is defined for negative integers and returns</span>
-<span class="sd">       zero in this case.</span>
-
-<span class="sd">       The factorial is very important in combinatorics where it gives</span>
-<span class="sd">       the number of ways in which &#39;n&#39; objects can be permuted. It also</span>
-<span class="sd">       arises in calculus, probability, number theory etc.</span>
-
-<span class="sd">       There is strict relation of factorial with gamma function. In</span>
-<span class="sd">       fact n! = gamma(n+1) for nonnegative integers. Rewrite of this</span>
-<span class="sd">       kind is very useful in case of combinatorial simplification.</span>
-
-<span class="sd">       Computation of the factorial is done using two algorithms. For</span>
-<span class="sd">       small arguments naive product is evaluated. However for bigger</span>
-<span class="sd">       input algorithm Prime-Swing is used. It is the fastest algorithm</span>
-<span class="sd">       known and computes n! via prime factorization of special class</span>
-<span class="sd">       of numbers, called here the &#39;Swing Numbers&#39;.</span>
-
-<span class="sd">       Examples</span>
-<span class="sd">       ========</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy import Symbol, factorial</span>
-<span class="sd">       &gt;&gt;&gt; n = Symbol(&#39;n&#39;, integer=True)</span>
-
-<span class="sd">       &gt;&gt;&gt; factorial(-2)</span>
-<span class="sd">       0</span>
-
-<span class="sd">       &gt;&gt;&gt; factorial(0)</span>
-<span class="sd">       1</span>
-
-<span class="sd">       &gt;&gt;&gt; factorial(7)</span>
-<span class="sd">       5040</span>
-
-<span class="sd">       &gt;&gt;&gt; factorial(n)</span>
-<span class="sd">       n!</span>
-
-<span class="sd">       &gt;&gt;&gt; factorial(2*n)</span>
-<span class="sd">       (2*n)!</span>
-
-<span class="sd">       See Also</span>
-<span class="sd">       ========</span>
-
-<span class="sd">       factorial2, RisingFactorial, FallingFactorial</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">polygamma</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="n">_small_swing</span> <span class="o">=</span> <span class="p">[</span>
-        <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">35</span><span class="p">,</span> <span class="mi">35</span><span class="p">,</span> <span class="mi">315</span><span class="p">,</span> <span class="mi">63</span><span class="p">,</span> <span class="mi">693</span><span class="p">,</span> <span class="mi">231</span><span class="p">,</span> <span class="mi">3003</span><span class="p">,</span> <span class="mi">429</span><span class="p">,</span> <span class="mi">6435</span><span class="p">,</span> <span class="mi">6435</span><span class="p">,</span> <span class="mi">109395</span><span class="p">,</span>
-        <span class="mi">12155</span><span class="p">,</span> <span class="mi">230945</span><span class="p">,</span> <span class="mi">46189</span><span class="p">,</span> <span class="mi">969969</span><span class="p">,</span> <span class="mi">88179</span><span class="p">,</span> <span class="mi">2028117</span><span class="p">,</span> <span class="mi">676039</span><span class="p">,</span> <span class="mi">16900975</span><span class="p">,</span> <span class="mi">1300075</span><span class="p">,</span>
-        <span class="mi">35102025</span><span class="p">,</span> <span class="mi">5014575</span><span class="p">,</span> <span class="mi">145422675</span><span class="p">,</span> <span class="mi">9694845</span><span class="p">,</span> <span class="mi">300540195</span><span class="p">,</span> <span class="mi">300540195</span>
-    <span class="p">]</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_swing</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">33</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_small_swing</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">N</span><span class="p">,</span> <span class="n">primes</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">_sqrt</span><span class="p">(</span><span class="n">n</span><span class="p">)),</span> <span class="p">[]</span>
-
-            <span class="k">for</span> <span class="n">prime</span> <span class="ow">in</span> <span class="n">sieve</span><span class="o">.</span><span class="n">primerange</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">N</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span>
-
-                <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-                    <span class="n">q</span> <span class="o">//=</span> <span class="n">prime</span>
-
-                    <span class="k">if</span> <span class="n">q</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">q</span> <span class="o">&amp;</span> <span class="mi">1</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                            <span class="n">p</span> <span class="o">*=</span> <span class="n">prime</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">break</span>
-
-                <span class="k">if</span> <span class="n">p</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">primes</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">prime</span> <span class="ow">in</span> <span class="n">sieve</span><span class="o">.</span><span class="n">primerange</span><span class="p">(</span><span class="n">N</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">//</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="k">if</span> <span class="p">(</span><span class="n">n</span> <span class="o">//</span> <span class="n">prime</span><span class="p">)</span> <span class="o">&amp;</span> <span class="mi">1</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">primes</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">prime</span><span class="p">)</span>
-
-            <span class="n">L_product</span> <span class="o">=</span> <span class="n">R_product</span> <span class="o">=</span> <span class="mi">1</span>
-
-            <span class="k">for</span> <span class="n">prime</span> <span class="ow">in</span> <span class="n">sieve</span><span class="o">.</span><span class="n">primerange</span><span class="p">(</span><span class="n">n</span><span class="o">//</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="n">L_product</span> <span class="o">*=</span> <span class="n">prime</span>
-
-            <span class="k">for</span> <span class="n">prime</span> <span class="ow">in</span> <span class="n">primes</span><span class="p">:</span>
-                <span class="n">R_product</span> <span class="o">*=</span> <span class="n">prime</span>
-
-            <span class="k">return</span> <span class="n">L_product</span><span class="o">*</span><span class="n">R_product</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_recursive</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">cls</span><span class="o">.</span><span class="n">_recursive</span><span class="p">(</span><span class="n">n</span><span class="o">//</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">cls</span><span class="o">.</span><span class="n">_swing</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">elif</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">n</span><span class="p">,</span> <span class="n">result</span> <span class="o">=</span> <span class="n">n</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="mi">1</span>
-
-                    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">20</span><span class="p">:</span>
-                        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                            <span class="n">result</span> <span class="o">*=</span> <span class="n">i</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">N</span><span class="p">,</span> <span class="n">bits</span> <span class="o">=</span> <span class="n">n</span><span class="p">,</span> <span class="mi">0</span>
-
-                        <span class="k">while</span> <span class="n">N</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                            <span class="k">if</span> <span class="n">N</span> <span class="o">&amp;</span> <span class="mi">1</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                                <span class="n">bits</span> <span class="o">+=</span> <span class="mi">1</span>
-
-                            <span class="n">N</span> <span class="o">=</span> <span class="n">N</span> <span class="o">&gt;&gt;</span> <span class="mi">1</span>
-
-                        <span class="n">result</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_recursive</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">bits</span><span class="p">)</span>
-
-                    <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Integer</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_gamma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_integer</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_integer</span>
-
-</div>
-<div class="viewcode-block" id="MultiFactorial"><a class="viewcode-back" href="../../../../modules/functions/combinatorial.html#sympy.functions.combinatorial.factorials.MultiFactorial">[docs]</a><span class="k">class</span> <span class="nc">MultiFactorial</span><span class="p">(</span><span class="n">CombinatorialFunction</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<span class="k">class</span> <span class="nc">subfactorial</span><span class="p">(</span><span class="n">CombinatorialFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The subfactorial counts the derangements of n items and is</span>
-<span class="sd">    defined for non-negative integers as::</span>
-
-<span class="sd">              ,</span>
-<span class="sd">             |  1                             for n = 0</span>
-<span class="sd">        !n = {  0                             for n = 1</span>
-<span class="sd">             |  (n - 1)*(!(n - 1) + !(n - 2)) for n &gt; 1</span>
-<span class="sd">              `</span>
-
-<span class="sd">    It can also be written as int(round(n!/exp(1))) but the recursive</span>
-<span class="sd">    definition with caching is implemented for this function.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-<span class="sd">    * http://en.wikipedia.org/wiki/Subfactorial</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import subfactorial</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import n</span>
-<span class="sd">    &gt;&gt;&gt; subfactorial(n + 1)</span>
-<span class="sd">    !(n + 1)</span>
-<span class="sd">    &gt;&gt;&gt; subfactorial(5)</span>
-<span class="sd">    44</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    factorial, sympy.utilities.iterables.generate_derangements</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">_eval</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">1</span>
-        <span class="k">elif</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">0</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_eval</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="p">))</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">arg</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span>
-            <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Integer</span><span class="p">(</span><span class="n">cls</span><span class="o">.</span><span class="n">_eval</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span>
-        <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">sympify</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;argument must be a nonnegative integer&quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sympystr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;!</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">p</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;!(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="n">p</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-
-<div class="viewcode-block" id="factorial2"><a class="viewcode-back" href="../../../../modules/functions/combinatorial.html#sympy.functions.combinatorial.factorials.factorial2">[docs]</a><span class="k">class</span> <span class="nc">factorial2</span><span class="p">(</span><span class="n">CombinatorialFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The double factorial n!!, not to be confused with (n!)!</span>
-
-<span class="sd">    The double factorial is defined for integers &gt;= -1 as::</span>
-
-<span class="sd">               ,</span>
-<span class="sd">              |  n*(n - 2)*(n - 4)* ... * 1    for n odd</span>
-<span class="sd">        n!! = {  n*(n - 2)*(n - 4)* ... * 2    for n even</span>
-<span class="sd">              |  1                             for n = 0, -1</span>
-<span class="sd">               `</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import factorial2, var</span>
-<span class="sd">    &gt;&gt;&gt; var(&#39;n&#39;)</span>
-<span class="sd">    n</span>
-<span class="sd">    &gt;&gt;&gt; factorial2(n + 1)</span>
-<span class="sd">    (n + 1)!!</span>
-<span class="sd">    &gt;&gt;&gt; factorial2(5)</span>
-<span class="sd">    15</span>
-<span class="sd">    &gt;&gt;&gt; factorial2(-1)</span>
-<span class="sd">    1</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    factorial, RisingFactorial, FallingFactorial</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span> <span class="ow">or</span> <span class="n">arg</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">return</span> <span class="n">factorial2</span><span class="p">(</span><span class="n">arg</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">arg</span>
-
-    <span class="k">def</span> <span class="nf">_sympystr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">!!&quot;</span> <span class="o">%</span> <span class="n">p</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;(</span><span class="si">%s</span><span class="s">)!!&quot;</span> <span class="o">%</span> <span class="n">p</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-<span class="c">###############################################################################</span>
-<span class="c">######################## RISING and FALLING FACTORIALS ########################</span>
-<span class="c">###############################################################################</span>
-
-</div>
-<div class="viewcode-block" id="RisingFactorial"><a class="viewcode-back" href="../../../../modules/functions/combinatorial.html#sympy.functions.combinatorial.factorials.RisingFactorial">[docs]</a><span class="k">class</span> <span class="nc">RisingFactorial</span><span class="p">(</span><span class="n">CombinatorialFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Rising factorial (also called Pochhammer symbol) is a double valued</span>
-<span class="sd">       function arising in concrete mathematics, hypergeometric functions</span>
-<span class="sd">       and series expansions. It is defined by:</span>
-
-<span class="sd">                   rf(x, k) = x * (x+1) * ... * (x + k-1)</span>
-
-<span class="sd">       where &#39;x&#39; can be arbitrary expression and &#39;k&#39; is an integer. For</span>
-<span class="sd">       more information check &quot;Concrete mathematics&quot; by Graham, pp. 66</span>
-<span class="sd">       or visit http://mathworld.wolfram.com/RisingFactorial.html page.</span>
-
-<span class="sd">       Examples</span>
-<span class="sd">       ========</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy import rf</span>
-<span class="sd">       &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">       &gt;&gt;&gt; rf(x, 0)</span>
-<span class="sd">       1</span>
-
-<span class="sd">       &gt;&gt;&gt; rf(1, 5)</span>
-<span class="sd">       120</span>
-
-<span class="sd">       &gt;&gt;&gt; rf(x, 5) == x*(1 + x)*(2 + x)*(3 + x)*(4 + x)</span>
-<span class="sd">       True</span>
-
-<span class="sd">       See Also</span>
-<span class="sd">       ========</span>
-
-<span class="sd">       factorial, factorial2, FallingFactorial</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-        <span class="k">elif</span> <span class="n">x</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">factorial</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">k</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">k</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">k</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">k</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-                    <span class="k">elif</span> <span class="n">x</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">k</span><span class="o">.</span><span class="n">is_odd</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="nb">reduce</span><span class="p">(</span><span class="k">lambda</span> <span class="n">r</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="n">r</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">i</span><span class="p">),</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">k</span><span class="p">)),</span> <span class="mi">1</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-                    <span class="k">elif</span> <span class="n">x</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="nb">reduce</span><span class="p">(</span><span class="k">lambda</span> <span class="n">r</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="n">r</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">i</span><span class="p">),</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">abs</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">k</span><span class="p">))</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_gamma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span> <span class="o">/</span> <span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="FallingFactorial"><a class="viewcode-back" href="../../../../modules/functions/combinatorial.html#sympy.functions.combinatorial.factorials.FallingFactorial">[docs]</a><span class="k">class</span> <span class="nc">FallingFactorial</span><span class="p">(</span><span class="n">CombinatorialFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Falling factorial (related to rising factorial) is a double valued</span>
-<span class="sd">       function arising in concrete mathematics, hypergeometric functions</span>
-<span class="sd">       and series expansions. It is defined by</span>
-
-<span class="sd">                   ff(x, k) = x * (x-1) * ... * (x - k+1)</span>
-
-<span class="sd">       where &#39;x&#39; can be arbitrary expression and &#39;k&#39; is an integer. For</span>
-<span class="sd">       more information check &quot;Concrete mathematics&quot; by Graham, pp. 66</span>
-<span class="sd">       or visit http://mathworld.wolfram.com/FallingFactorial.html page.</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy import ff</span>
-<span class="sd">       &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">       &gt;&gt;&gt; ff(x, 0)</span>
-<span class="sd">       1</span>
-
-<span class="sd">       &gt;&gt;&gt; ff(5, 5)</span>
-<span class="sd">       120</span>
-
-<span class="sd">       &gt;&gt;&gt; ff(x, 5) == x*(x-1)*(x-2)*(x-3)*(x-4)</span>
-<span class="sd">       True</span>
-
-<span class="sd">       See Also</span>
-<span class="sd">       ========</span>
-
-<span class="sd">       factorial, factorial2, RisingFactorial</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-        <span class="k">elif</span> <span class="n">k</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">k</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">k</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">k</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-                    <span class="k">elif</span> <span class="n">x</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">k</span><span class="o">.</span><span class="n">is_odd</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="nb">reduce</span><span class="p">(</span><span class="k">lambda</span> <span class="n">r</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="n">r</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">i</span><span class="p">),</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">k</span><span class="p">)),</span> <span class="mi">1</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-                    <span class="k">elif</span> <span class="n">x</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="nb">reduce</span><span class="p">(</span><span class="k">lambda</span> <span class="n">r</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="n">r</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">i</span><span class="p">),</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">abs</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">k</span><span class="p">))</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_gamma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">k</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="o">-</span><span class="n">x</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span> <span class="o">/</span> <span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">)</span>
-</div>
-<span class="n">rf</span> <span class="o">=</span> <span class="n">RisingFactorial</span>
-<span class="n">ff</span> <span class="o">=</span> <span class="n">FallingFactorial</span>
-
-<span class="c">###############################################################################</span>
-<span class="c">########################### BINOMIAL COEFFICIENTS #############################</span>
-<span class="c">###############################################################################</span>
-
-
-<div class="viewcode-block" id="binomial"><a class="viewcode-back" href="../../../../modules/functions/combinatorial.html#sympy.functions.combinatorial.factorials.binomial">[docs]</a><span class="k">class</span> <span class="nc">binomial</span><span class="p">(</span><span class="n">CombinatorialFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Implementation of the binomial coefficient. It can be defined</span>
-<span class="sd">       in two ways depending on its desired interpretation:</span>
-
-<span class="sd">           C(n,k) = n!/(k!(n-k)!)   or   C(n, k) = ff(n, k)/k!</span>
-
-<span class="sd">       First, in a strict combinatorial sense it defines the</span>
-<span class="sd">       number of ways we can choose &#39;k&#39; elements from a set of</span>
-<span class="sd">       &#39;n&#39; elements. In this case both arguments are nonnegative</span>
-<span class="sd">       integers and binomial is computed using an efficient</span>
-<span class="sd">       algorithm based on prime factorization.</span>
-
-<span class="sd">       The other definition is generalization for arbitrary &#39;n&#39;,</span>
-<span class="sd">       however &#39;k&#39; must also be nonnegative. This case is very</span>
-<span class="sd">       useful when evaluating summations.</span>
-
-<span class="sd">       For the sake of convenience for negative &#39;k&#39; this function</span>
-<span class="sd">       will return zero no matter what valued is the other argument.</span>
-
-<span class="sd">       Examples</span>
-<span class="sd">       ========</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy import Symbol, Rational, binomial</span>
-<span class="sd">       &gt;&gt;&gt; n = Symbol(&#39;n&#39;, integer=True)</span>
-
-<span class="sd">       &gt;&gt;&gt; binomial(15, 8)</span>
-<span class="sd">       6435</span>
-
-<span class="sd">       &gt;&gt;&gt; binomial(n, -1)</span>
-<span class="sd">       0</span>
-
-<span class="sd">       &gt;&gt;&gt; [ binomial(0, i) for i in range(1)]</span>
-<span class="sd">       [1]</span>
-<span class="sd">       &gt;&gt;&gt; [ binomial(1, i) for i in range(2)]</span>
-<span class="sd">       [1, 1]</span>
-<span class="sd">       &gt;&gt;&gt; [ binomial(2, i) for i in range(3)]</span>
-<span class="sd">       [1, 2, 1]</span>
-<span class="sd">       &gt;&gt;&gt; [ binomial(3, i) for i in range(4)]</span>
-<span class="sd">       [1, 3, 3, 1]</span>
-<span class="sd">       &gt;&gt;&gt; [ binomial(4, i) for i in range(5)]</span>
-<span class="sd">       [1, 4, 6, 4, 1]</span>
-
-<span class="sd">       &gt;&gt;&gt; binomial(Rational(5,4), 3)</span>
-<span class="sd">       -5/128</span>
-
-<span class="sd">       &gt;&gt;&gt; binomial(n, 3)</span>
-<span class="sd">       n*(n - 2)*(n - 1)/6</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># http://functions.wolfram.com/GammaBetaErf/Binomial/20/01/01/</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">return</span> <span class="n">binomial</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">polygamma</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">C</span><span class="o">.</span><span class="n">polygamma</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="c"># http://functions.wolfram.com/GammaBetaErf/Binomial/20/01/02/</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">return</span> <span class="n">binomial</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">polygamma</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">C</span><span class="o">.</span><span class="n">polygamma</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">)))</span>
-
-        <span class="k">if</span> <span class="n">k</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">k</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">k</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-                <span class="k">elif</span> <span class="n">k</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">==</span> <span class="n">k</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-                <span class="k">elif</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">n</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">n</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="nb">int</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-
-                    <span class="k">if</span> <span class="n">k</span> <span class="o">&gt;</span> <span class="n">n</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-                    <span class="k">elif</span> <span class="n">k</span> <span class="o">&gt;</span> <span class="n">n</span> <span class="o">//</span> <span class="mi">2</span><span class="p">:</span>
-                        <span class="n">k</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="n">k</span>
-
-                    <span class="n">M</span><span class="p">,</span> <span class="n">result</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">_sqrt</span><span class="p">(</span><span class="n">n</span><span class="p">)),</span> <span class="mi">1</span>
-
-                    <span class="k">for</span> <span class="n">prime</span> <span class="ow">in</span> <span class="n">sieve</span><span class="o">.</span><span class="n">primerange</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                        <span class="k">if</span> <span class="n">prime</span> <span class="o">&gt;</span> <span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">:</span>
-                            <span class="n">result</span> <span class="o">*=</span> <span class="n">prime</span>
-                        <span class="k">elif</span> <span class="n">prime</span> <span class="o">&gt;</span> <span class="n">n</span> <span class="o">//</span> <span class="mi">2</span><span class="p">:</span>
-                            <span class="k">continue</span>
-                        <span class="k">elif</span> <span class="n">prime</span> <span class="o">&gt;</span> <span class="n">M</span><span class="p">:</span>
-                            <span class="k">if</span> <span class="n">n</span> <span class="o">%</span> <span class="n">prime</span> <span class="o">&lt;</span> <span class="n">k</span> <span class="o">%</span> <span class="n">prime</span><span class="p">:</span>
-                                <span class="n">result</span> <span class="o">*=</span> <span class="n">prime</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="n">N</span><span class="p">,</span> <span class="n">K</span> <span class="o">=</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span>
-                            <span class="n">exp</span> <span class="o">=</span> <span class="n">a</span> <span class="o">=</span> <span class="mi">0</span>
-
-                            <span class="k">while</span> <span class="n">N</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                                <span class="n">a</span> <span class="o">=</span> <span class="nb">int</span><span class="p">((</span><span class="n">N</span> <span class="o">%</span> <span class="n">prime</span><span class="p">)</span> <span class="o">&lt;</span> <span class="p">(</span><span class="n">K</span> <span class="o">%</span> <span class="n">prime</span> <span class="o">+</span> <span class="n">a</span><span class="p">))</span>
-                                <span class="n">N</span><span class="p">,</span> <span class="n">K</span> <span class="o">=</span> <span class="n">N</span> <span class="o">//</span> <span class="n">prime</span><span class="p">,</span> <span class="n">K</span> <span class="o">//</span> <span class="n">prime</span>
-                                <span class="n">exp</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="n">exp</span>
-
-                            <span class="k">if</span> <span class="n">exp</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                                <span class="n">result</span> <span class="o">*=</span> <span class="n">prime</span><span class="o">**</span><span class="n">exp</span>
-
-                    <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Integer</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">result</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span>
-
-                    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                        <span class="n">result</span> <span class="o">*=</span> <span class="n">n</span> <span class="o">-</span> <span class="n">k</span> <span class="o">+</span> <span class="n">i</span>
-                        <span class="n">result</span> <span class="o">/=</span> <span class="n">i</span>
-
-                    <span class="k">return</span> <span class="n">result</span>
-        <span class="k">elif</span> <span class="n">k</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">elif</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="n">k</span>
-
-            <span class="k">if</span> <span class="n">d</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_factorial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_gamma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_integer</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">is_integer</span></div>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.functions.combinatorial.numbers</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">This module implements some special functions that commonly appear in</span>
-<span class="sd">combinatorial contexts (e.g. in power series); in particular,</span>
-<span class="sd">sequences of rational numbers such as Bernoulli and Fibonacci numbers.</span>
-
-<span class="sd">Factorials, binomial coefficients and related functions are located in</span>
-<span class="sd">the separate &#39;factorials&#39; module.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Rational</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">Integer</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">Add</span><span class="p">,</span> <span class="n">expand_mul</span>
-
-<span class="kn">from</span> <span class="nn">sympy.mpmath</span> <span class="kn">import</span> <span class="n">bernfrac</span>
-<span class="kn">from</span> <span class="nn">sympy.mpmath.libmp</span> <span class="kn">import</span> <span class="n">ifib</span> <span class="k">as</span> <span class="n">_ifib</span>
-
-
-<span class="k">def</span> <span class="nf">_product</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">p</span> <span class="o">*=</span> <span class="n">k</span>
-    <span class="k">return</span> <span class="n">p</span>
-
-<span class="kn">from</span> <span class="nn">sympy.utilities.memoization</span> <span class="kn">import</span> <span class="n">recurrence_memo</span>
-
-
-<span class="c"># Dummy symbol used for computing polynomial sequences</span>
-<span class="n">_sym</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="n">_symbols</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-
-
-<span class="c">#----------------------------------------------------------------------------#</span>
-<span class="c">#                                                                            #</span>
-<span class="c">#                           Fibonacci numbers                                #</span>
-<span class="c">#                                                                            #</span>
-<span class="c">#----------------------------------------------------------------------------#</span>
-
-<div class="viewcode-block" id="fibonacci"><a class="viewcode-back" href="../../../../modules/functions/combinatorial.html#sympy.functions.combinatorial.numbers.fibonacci">[docs]</a><span class="k">class</span> <span class="nc">fibonacci</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Fibonacci numbers / Fibonacci polynomials</span>
-
-<span class="sd">    The Fibonacci numbers are the integer sequence defined by the</span>
-<span class="sd">    initial terms F_0 = 0, F_1 = 1 and the two-term recurrence</span>
-<span class="sd">    relation F_n = F_{n-1} + F_{n-2}.</span>
-
-<span class="sd">    The Fibonacci polynomials are defined by F_1(x) = 1,</span>
-<span class="sd">    F_2(x) = x, and F_n(x) = x*F_{n-1}(x) + F_{n-2}(x) for n &gt; 2.</span>
-<span class="sd">    For all positive integers n, F_n(1) = F_n.</span>
-
-<span class="sd">    * fibonacci(n) gives the nth Fibonacci number, F_n</span>
-<span class="sd">    * fibonacci(n, x) gives the nth Fibonacci polynomial in x, F_n(x)</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import fibonacci, Symbol</span>
-
-<span class="sd">    &gt;&gt;&gt; [fibonacci(x) for x in range(11)]</span>
-<span class="sd">    [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]</span>
-<span class="sd">    &gt;&gt;&gt; fibonacci(5, Symbol(&#39;t&#39;))</span>
-<span class="sd">    t**4 + 3*t**2 + 1</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    * http://en.wikipedia.org/wiki/Fibonacci_number</span>
-<span class="sd">    * http://mathworld.wolfram.com/FibonacciNumber.html</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    lucas</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_fib</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_ifib</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@recurrence_memo</span><span class="p">([</span><span class="bp">None</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">_sym</span><span class="p">])</span>
-    <span class="k">def</span> <span class="nf">_fibpoly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">prev</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">prev</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="n">_sym</span><span class="o">*</span><span class="n">prev</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">sym</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">fibonacci</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">sym</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="n">cls</span><span class="o">.</span><span class="n">_fib</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Fibonacci polynomials are defined &quot;</span>
-                       <span class="s">&quot;only for positive integer indices.&quot;</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_fibpoly</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">_sym</span><span class="p">,</span> <span class="n">sym</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="lucas"><a class="viewcode-back" href="../../../../modules/functions/combinatorial.html#sympy.functions.combinatorial.numbers.lucas">[docs]</a><span class="k">class</span> <span class="nc">lucas</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Lucas numbers</span>
-
-<span class="sd">    Lucas numbers satisfy a recurrence relation similar to that of</span>
-<span class="sd">    the Fibonacci sequence, in which each term is the sum of the</span>
-<span class="sd">    preceding two. They are generated by choosing the initial</span>
-<span class="sd">    values L_0 = 2 and L_1 = 1.</span>
-
-<span class="sd">    * lucas(n) gives the nth Lucas number</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import lucas</span>
-
-<span class="sd">    &gt;&gt;&gt; [lucas(x) for x in range(11)]</span>
-<span class="sd">    [2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123]</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    * http://en.wikipedia.org/wiki/Lucas_number</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    fibonacci</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">fibonacci</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">fibonacci</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-
-
-<span class="c">#----------------------------------------------------------------------------#</span>
-<span class="c">#                                                                            #</span>
-<span class="c">#                           Bernoulli numbers                                #</span>
-<span class="c">#                                                                            #</span>
-<span class="c">#----------------------------------------------------------------------------#</span>
-</div>
-<div class="viewcode-block" id="bernoulli"><a class="viewcode-back" href="../../../../modules/functions/combinatorial.html#sympy.functions.combinatorial.numbers.bernoulli">[docs]</a><span class="k">class</span> <span class="nc">bernoulli</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Bernoulli numbers / Bernoulli polynomials</span>
-
-<span class="sd">    The Bernoulli numbers are a sequence of rational numbers</span>
-<span class="sd">    defined by B_0 = 1 and the recursive relation (n &gt; 0)::</span>
-
-<span class="sd">                n</span>
-<span class="sd">               ___</span>
-<span class="sd">              \      / n + 1 \</span>
-<span class="sd">          0 =  )     |       | * B .</span>
-<span class="sd">              /___   \   k   /    k</span>
-<span class="sd">              k = 0</span>
-
-<span class="sd">    They are also commonly defined by their exponential generating</span>
-<span class="sd">    function, which is x/(exp(x) - 1). For odd indices &gt; 1, the</span>
-<span class="sd">    Bernoulli numbers are zero.</span>
-
-<span class="sd">    The Bernoulli polynomials satisfy the analogous formula::</span>
-
-<span class="sd">                    n</span>
-<span class="sd">                   ___</span>
-<span class="sd">                  \      / n \         n-k</span>
-<span class="sd">          B (x) =  )     |   | * B  * x   .</span>
-<span class="sd">           n      /___   \ k /    k</span>
-<span class="sd">                  k = 0</span>
-
-<span class="sd">    Bernoulli numbers and Bernoulli polynomials are related as</span>
-<span class="sd">    B_n(0) = B_n.</span>
-
-<span class="sd">    We compute Bernoulli numbers using Ramanujan&#39;s formula::</span>
-
-<span class="sd">                                   / n + 3 \</span>
-<span class="sd">          B   =  (A(n) - S(n))  /  |       |</span>
-<span class="sd">           n                       \   n   /</span>
-
-<span class="sd">    where A(n) = (n+3)/3 when n = 0 or 2 (mod 6), A(n) = -(n+3)/6</span>
-<span class="sd">    when n = 4 (mod 6), and::</span>
-
-<span class="sd">                 [n/6]</span>
-<span class="sd">                  ___</span>
-<span class="sd">                 \      /  n + 3  \</span>
-<span class="sd">          S(n) =  )     |         | * B</span>
-<span class="sd">                 /___   \ n - 6*k /    n-6*k</span>
-<span class="sd">                 k = 1</span>
-
-<span class="sd">    This formula is similar to the sum given in the definition, but</span>
-<span class="sd">    cuts 2/3 of the terms. For Bernoulli polynomials, we use the</span>
-<span class="sd">    formula in the definition.</span>
-
-<span class="sd">    * bernoulli(n) gives the nth Bernoulli number, B_n</span>
-<span class="sd">    * bernoulli(n, x) gives the nth Bernoulli polynomial in x, B_n(x)</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import bernoulli</span>
-
-<span class="sd">    &gt;&gt;&gt; [bernoulli(n) for n in range(11)]</span>
-<span class="sd">    [1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66]</span>
-<span class="sd">    &gt;&gt;&gt; bernoulli(1000001)</span>
-<span class="sd">    0</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    * http://en.wikipedia.org/wiki/Bernoulli_number</span>
-<span class="sd">    * http://en.wikipedia.org/wiki/Bernoulli_polynomial</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    euler, bell</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="c"># Calculates B_n for positive even n</span>
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_calc_bernoulli</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">binomial</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">3</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">6</span><span class="p">))</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">//</span><span class="mi">6</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">s</span> <span class="o">+=</span> <span class="n">a</span> <span class="o">*</span> <span class="n">bernoulli</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">6</span><span class="o">*</span><span class="n">j</span><span class="p">)</span>
-            <span class="c"># Avoid computing each binomial coefficient from scratch</span>
-            <span class="n">a</span> <span class="o">*=</span> <span class="n">_product</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">6</span> <span class="o">-</span> <span class="mi">6</span><span class="o">*</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">6</span><span class="o">*</span><span class="n">j</span><span class="p">)</span>
-            <span class="n">a</span> <span class="o">//=</span> <span class="n">_product</span><span class="p">(</span><span class="mi">6</span><span class="o">*</span><span class="n">j</span> <span class="o">+</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">6</span><span class="o">*</span><span class="n">j</span> <span class="o">+</span> <span class="mi">9</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">6</span> <span class="o">==</span> <span class="mi">4</span><span class="p">:</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="o">-</span><span class="n">Rational</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span> <span class="o">-</span> <span class="n">s</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="n">s</span>
-        <span class="k">return</span> <span class="n">s</span> <span class="o">/</span> <span class="n">C</span><span class="o">.</span><span class="n">binomial</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">3</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="c"># We implement a specialized memoization scheme to handle each</span>
-    <span class="c"># case modulo 6 separately</span>
-    <span class="n">_cache</span> <span class="o">=</span> <span class="p">{</span><span class="mi">0</span><span class="p">:</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="mi">2</span><span class="p">:</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="mi">4</span><span class="p">:</span> <span class="n">Rational</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">30</span><span class="p">)}</span>
-    <span class="n">_highest</span> <span class="o">=</span> <span class="p">{</span><span class="mi">0</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">:</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">:</span> <span class="mi">4</span><span class="p">}</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">sym</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">n</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">n</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-                <span class="k">elif</span> <span class="n">n</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">sym</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">sym</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span>
-                <span class="c"># Bernoulli numbers</span>
-                <span class="k">elif</span> <span class="n">sym</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_odd</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-                    <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-                    <span class="c"># Use mpmath for enormous Bernoulli numbers</span>
-                    <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">500</span><span class="p">:</span>
-                        <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">bernfrac</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-                        <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">p</span><span class="p">),</span> <span class="nb">int</span><span class="p">(</span><span class="n">q</span><span class="p">))</span>
-                    <span class="n">case</span> <span class="o">=</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">6</span>
-                    <span class="n">highest_cached</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_highest</span><span class="p">[</span><span class="n">case</span><span class="p">]</span>
-                    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;=</span> <span class="n">highest_cached</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_cache</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-                    <span class="c"># To avoid excessive recursion when, say, bernoulli(1000) is</span>
-                    <span class="c"># requested, calculate and cache the entire sequence ... B_988,</span>
-                    <span class="c"># B_994, B_1000 in increasing order</span>
-                    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">highest_cached</span> <span class="o">+</span> <span class="mi">6</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">6</span><span class="p">):</span>
-                        <span class="n">b</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_calc_bernoulli</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                        <span class="n">cls</span><span class="o">.</span><span class="n">_cache</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">b</span>
-                        <span class="n">cls</span><span class="o">.</span><span class="n">_highest</span><span class="p">[</span><span class="n">case</span><span class="p">]</span> <span class="o">=</span> <span class="n">i</span>
-                    <span class="k">return</span> <span class="n">b</span>
-                <span class="c"># Bernoulli polynomials</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">n</span><span class="p">,</span> <span class="n">result</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="p">[]</span>
-                    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">binomial</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="n">cls</span><span class="p">(</span><span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="n">sym</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">))</span>
-                    <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">result</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Bernoulli numbers are defined only&quot;</span>
-                                 <span class="s">&quot; for nonnegative integer indices.&quot;</span><span class="p">)</span>
-
-
-<span class="c">#----------------------------------------------------------------------------#</span>
-<span class="c">#                                                                            #</span>
-<span class="c">#                             Bell numbers                                   #</span>
-<span class="c">#                                                                            #</span>
-<span class="c">#----------------------------------------------------------------------------#</span>
-</div>
-<div class="viewcode-block" id="bell"><a class="viewcode-back" href="../../../../modules/functions/combinatorial.html#sympy.functions.combinatorial.numbers.bell">[docs]</a><span class="k">class</span> <span class="nc">bell</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Bell numbers / Bell polynomials</span>
-
-<span class="sd">    The Bell numbers satisfy `B_0 = 1` and</span>
-
-<span class="sd">    .. math:: B_n = \sum_{k=0}^{n-1} \binom{n-1}{k} B_k.</span>
-
-<span class="sd">    They are also given by:</span>
-
-<span class="sd">    .. math:: B_n = \frac{1}{e} \sum_{k=0}^{\infty} \frac{k^n}{k!}.</span>
-
-<span class="sd">    The Bell polynomials are given by `B_0(x) = 1` and</span>
-
-<span class="sd">    .. math:: B_n(x) = x \sum_{k=1}^{n-1} \binom{n-1}{k-1} B_{k-1}(x).</span>
-
-<span class="sd">    The second kind of Bell polynomials (are sometimes called &quot;partial&quot; Bell</span>
-<span class="sd">    polynomials or incomplete Bell polynomials) are defined as</span>
-
-<span class="sd">    .. math:: B_{n,k}(x_1, x_2,\dotsc x_{n-k+1}) =</span>
-<span class="sd">            \sum_{j_1+j_2+j_2+\dotsb=k \atop j_1+2j_2+3j_2+\dotsb=n}</span>
-<span class="sd">                \frac{n!}{j_1!j_2!\dotsb j_{n-k+1}!}</span>
-<span class="sd">                \left(\frac{x_1}{1!} \right)^{j_1}</span>
-<span class="sd">                \left(\frac{x_2}{2!} \right)^{j_2} \dotsb</span>
-<span class="sd">                \left(\frac{x_{n-k+1}}{(n-k+1)!} \right) ^{j_{n-k+1}}.</span>
-
-<span class="sd">    * bell(n) gives the `n^{th}` Bell number, `B_n`.</span>
-<span class="sd">    * bell(n, x) gives the `n^{th}` Bell polynomial, `B_n(x)`.</span>
-<span class="sd">    * bell(n, k, (x1, x2, ...)) gives Bell polynomials of the second kind,</span>
-<span class="sd">      `B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1})`.</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    Not to be confused with Bernoulli numbers and Bernoulli polynomials,</span>
-<span class="sd">    which use the same notation.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import bell, Symbol, symbols</span>
-
-<span class="sd">    &gt;&gt;&gt; [bell(n) for n in range(11)]</span>
-<span class="sd">    [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975]</span>
-<span class="sd">    &gt;&gt;&gt; bell(30)</span>
-<span class="sd">    846749014511809332450147</span>
-<span class="sd">    &gt;&gt;&gt; bell(4, Symbol(&#39;t&#39;))</span>
-<span class="sd">    t**4 + 6*t**3 + 7*t**2 + t</span>
-<span class="sd">    &gt;&gt;&gt; bell(6, 2, symbols(&#39;x:6&#39;)[1:])</span>
-<span class="sd">    6*x1*x5 + 15*x2*x4 + 10*x3**2</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    * http://en.wikipedia.org/wiki/Bell_number</span>
-<span class="sd">    * http://mathworld.wolfram.com/BellNumber.html</span>
-<span class="sd">    * http://mathworld.wolfram.com/BellPolynomial.html</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    euler, bernoulli</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@recurrence_memo</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-    <span class="k">def</span> <span class="nf">_bell</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">prev</span><span class="p">):</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">a</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span> <span class="o">//</span> <span class="n">k</span>
-            <span class="n">s</span> <span class="o">+=</span> <span class="n">a</span> <span class="o">*</span> <span class="n">prev</span><span class="p">[</span><span class="n">k</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">s</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@recurrence_memo</span><span class="p">([</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">_sym</span><span class="p">])</span>
-    <span class="k">def</span> <span class="nf">_bell_poly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">prev</span><span class="p">):</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">a</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">//</span> <span class="p">(</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="n">s</span> <span class="o">+=</span> <span class="n">a</span> <span class="o">*</span> <span class="n">prev</span><span class="p">[</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">_sym</span> <span class="o">*</span> <span class="n">s</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">_bell_incomplete_poly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">symbols</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">        The second kind of Bell polynomials (incomplete Bell polynomials).</span>
-
-<span class="sd">        Calculated by recurrence formula:</span>
-
-<span class="sd">        .. math:: B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1}) =</span>
-<span class="sd">                \sum_{m=1}^{n-k+1}</span>
-<span class="sd">                \x_m \binom{n-1}{m-1} B_{n-m,k-1}(x_1, x_2, \dotsc, x_{n-m-k})</span>
-
-<span class="sd">        where</span>
-<span class="sd">            B_{0,0} = 1;</span>
-<span class="sd">            B_{n,0} = 0; for n&gt;=1</span>
-<span class="sd">            B_{0,k} = 0; for k&gt;=1</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">k</span> <span class="o">==</span> <span class="mi">0</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">elif</span> <span class="p">(</span><span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="n">k</span> <span class="o">==</span> <span class="mi">0</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">2</span><span class="p">):</span>
-            <span class="n">s</span> <span class="o">+=</span> <span class="n">a</span><span class="o">*</span><span class="n">bell</span><span class="o">.</span><span class="n">_bell_incomplete_poly</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">m</span><span class="p">,</span> <span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">symbols</span><span class="p">)</span><span class="o">*</span><span class="n">symbols</span><span class="p">[</span><span class="n">m</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">m</span><span class="p">)</span><span class="o">/</span><span class="n">m</span>
-        <span class="k">return</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">k_sym</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">symbols</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">n</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">k_sym</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="n">cls</span><span class="o">.</span><span class="n">_bell</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)))</span>
-            <span class="k">elif</span> <span class="n">symbols</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_bell_poly</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">))</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">_sym</span><span class="p">,</span> <span class="n">k_sym</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">r</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_bell_incomplete_poly</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="nb">int</span><span class="p">(</span><span class="n">k_sym</span><span class="p">),</span> <span class="n">symbols</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">r</span>
-
-<span class="c">#----------------------------------------------------------------------------#</span>
-<span class="c">#                                                                            #</span>
-<span class="c">#                           Harmonic numbers                                 #</span>
-<span class="c">#                                                                            #</span>
-<span class="c">#----------------------------------------------------------------------------#</span>
-
-</div>
-<div class="viewcode-block" id="harmonic"><a class="viewcode-back" href="../../../../modules/functions/combinatorial.html#sympy.functions.combinatorial.numbers.harmonic">[docs]</a><span class="k">class</span> <span class="nc">harmonic</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Harmonic numbers</span>
-
-<span class="sd">    The nth harmonic number is given by 1 + 1/2 + 1/3 + ... + 1/n.</span>
-
-<span class="sd">    More generally::</span>
-
-<span class="sd">                   n</span>
-<span class="sd">                  ___</span>
-<span class="sd">                 \       -m</span>
-<span class="sd">          H    =  )     k   .</span>
-<span class="sd">           n,m   /___</span>
-<span class="sd">                 k = 1</span>
-
-<span class="sd">    As n -&gt; oo, H_{n,m} -&gt; zeta(m) (the Riemann zeta function)</span>
-
-<span class="sd">    * harmonic(n) gives the nth harmonic number, H_n</span>
-
-<span class="sd">    * harmonic(n, m) gives the nth generalized harmonic number</span>
-<span class="sd">      of order m, H_{n,m}, where harmonic(n) == harmonic(n, 1)</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import harmonic, oo</span>
-
-<span class="sd">    &gt;&gt;&gt; [harmonic(n) for n in range(6)]</span>
-<span class="sd">    [0, 1, 3/2, 11/6, 25/12, 137/60]</span>
-<span class="sd">    &gt;&gt;&gt; [harmonic(n, 2) for n in range(6)]</span>
-<span class="sd">    [0, 1, 5/4, 49/36, 205/144, 5269/3600]</span>
-<span class="sd">    &gt;&gt;&gt; harmonic(oo, 2)</span>
-<span class="sd">    pi**2/6</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    * http://en.wikipedia.org/wiki/Harmonic_number</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="c"># Generate one memoized Harmonic number-generating function for each</span>
-    <span class="c"># order and store it in a dictionary</span>
-    <span class="n">_functions</span> <span class="o">=</span> <span class="p">{}</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">m</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="n">oo</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">zeta</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">n</span><span class="o">.</span><span class="n">is_nonnegative</span> <span class="ow">and</span> <span class="n">m</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">cls</span><span class="o">.</span><span class="n">_functions</span><span class="p">:</span>
-                <span class="nd">@recurrence_memo</span><span class="p">([</span><span class="mi">0</span><span class="p">])</span>
-                <span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">prev</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">prev</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="o">/</span> <span class="n">n</span><span class="o">**</span><span class="n">m</span>
-                <span class="n">cls</span><span class="o">.</span><span class="n">_functions</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_functions</span><span class="p">[</span><span class="n">m</span><span class="p">](</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-
-<span class="c">#----------------------------------------------------------------------------#</span>
-<span class="c">#                                                                            #</span>
-<span class="c">#                           Euler numbers                                    #</span>
-<span class="c">#                                                                            #</span>
-<span class="c">#----------------------------------------------------------------------------#</span>
-
-</div>
-<div class="viewcode-block" id="euler"><a class="viewcode-back" href="../../../../modules/functions/combinatorial.html#sympy.functions.combinatorial.numbers.euler">[docs]</a><span class="k">class</span> <span class="nc">euler</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Euler numbers</span>
-
-<span class="sd">    The euler numbers are given by::</span>
-
-<span class="sd">                  2*n+1   k</span>
-<span class="sd">                   ___   ___            j          2*n+1</span>
-<span class="sd">                  \     \     / k \ (-1)  * (k-2*j)</span>
-<span class="sd">          E   = I  )     )    |   | --------------------</span>
-<span class="sd">           2n     /___  /___  \ j /      k    k</span>
-<span class="sd">                  k = 1 j = 0           2  * I  * k</span>
-
-<span class="sd">          E     = 0</span>
-<span class="sd">           2n+1</span>
-
-<span class="sd">    * euler(n) gives the n-th Euler number, E_n</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, euler</span>
-<span class="sd">    &gt;&gt;&gt; [euler(n) for n in range(10)]</span>
-<span class="sd">    [1, 0, -1, 0, 5, 0, -61, 0, 1385, 0]</span>
-<span class="sd">    &gt;&gt;&gt; n = Symbol(&quot;n&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; euler(n+2*n)</span>
-<span class="sd">    euler(3*n)</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    * http://en.wikipedia.org/wiki/Euler_numbers</span>
-<span class="sd">    * http://mathworld.wolfram.com/EulerNumber.html</span>
-<span class="sd">    * http://en.wikipedia.org/wiki/Alternating_permutation</span>
-<span class="sd">    * http://mathworld.wolfram.com/AlternatingPermutation.html</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    bernoulli, bell</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">evaluate</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="k">if</span> <span class="n">m</span><span class="o">.</span><span class="n">is_odd</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">if</span> <span class="n">m</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">m</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">sympy.mpmath</span> <span class="kn">import</span> <span class="n">mp</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">_to_mpmath</span><span class="p">(</span><span class="n">mp</span><span class="o">.</span><span class="n">prec</span><span class="p">)</span>
-            <span class="n">res</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">eulernum</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="n">res</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_Sum</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-            <span class="n">k</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="n">j</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;j&quot;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">/</span> <span class="mi">2</span>
-            <span class="n">Em</span> <span class="o">=</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">Sum</span><span class="p">(</span> <span class="n">C</span><span class="o">.</span><span class="n">Sum</span><span class="p">(</span> <span class="n">C</span><span class="o">.</span><span class="n">binomial</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="o">*</span> <span class="p">((</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">j</span> <span class="o">*</span> <span class="p">(</span><span class="n">k</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">j</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span> <span class="o">/</span>
-                  <span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">k</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">**</span><span class="n">k</span> <span class="o">*</span> <span class="n">k</span><span class="p">),</span> <span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">k</span><span class="p">)),</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)))</span>
-
-            <span class="k">return</span> <span class="n">Em</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">m</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">m</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">sympy.mpmath</span> <span class="kn">import</span> <span class="n">mp</span>
-            <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Expr</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">_to_mpmath</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-            <span class="n">oprec</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">prec</span>
-            <span class="n">mp</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="n">prec</span>
-            <span class="n">res</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">eulernum</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-            <span class="n">mp</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="n">oprec</span>
-            <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">_from_mpmath</span><span class="p">(</span><span class="n">res</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-
-<span class="c">#----------------------------------------------------------------------------#</span>
-<span class="c">#                                                                            #</span>
-<span class="c">#                           Catalan numbers                                  #</span>
-<span class="c">#                                                                            #</span>
-<span class="c">#----------------------------------------------------------------------------#</span>
-
-</div>
-<div class="viewcode-block" id="catalan"><a class="viewcode-back" href="../../../../modules/functions/combinatorial.html#sympy.functions.combinatorial.numbers.catalan">[docs]</a><span class="k">class</span> <span class="nc">catalan</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Catalan numbers</span>
-
-<span class="sd">    The n-th catalan number is given by::</span>
-
-<span class="sd">                 1   / 2*n \</span>
-<span class="sd">          C  = ----- |     |</span>
-<span class="sd">           n   n + 1 \  n  /</span>
-
-<span class="sd">    * catalan(n) gives the n-th Catalan number, C_n</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import (Symbol, binomial, gamma, hyper, polygamma,</span>
-<span class="sd">    ...             catalan, diff, combsimp, Rational, I)</span>
-
-<span class="sd">    &gt;&gt;&gt; [ catalan(i) for i in range(1,10) ]</span>
-<span class="sd">    [1, 2, 5, 14, 42, 132, 429, 1430, 4862]</span>
-
-<span class="sd">    &gt;&gt;&gt; n = Symbol(&quot;n&quot;, integer=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; catalan(n)</span>
-<span class="sd">    catalan(n)</span>
-
-<span class="sd">    Catalan numbers can be transformed into several other, identical</span>
-<span class="sd">    expressions involving other mathematical functions</span>
-
-<span class="sd">    &gt;&gt;&gt; catalan(n).rewrite(binomial)</span>
-<span class="sd">    binomial(2*n, n)/(n + 1)</span>
-
-<span class="sd">    &gt;&gt;&gt; catalan(n).rewrite(gamma)</span>
-<span class="sd">    4**n*gamma(n + 1/2)/(sqrt(pi)*gamma(n + 2))</span>
-
-<span class="sd">    &gt;&gt;&gt; catalan(n).rewrite(hyper)</span>
-<span class="sd">    hyper((-n + 1, -n), (2,), 1)</span>
-
-<span class="sd">    For some non-integer values of n we can get closed form</span>
-<span class="sd">    expressions by rewriting in terms of gamma functions:</span>
-
-<span class="sd">    &gt;&gt;&gt; catalan(Rational(1,2)).rewrite(gamma)</span>
-<span class="sd">    8/(3*pi)</span>
-
-<span class="sd">    We can differentiate the Catalan numbers C(n) interpreted as a</span>
-<span class="sd">    continuous real funtion in n:</span>
-
-<span class="sd">    &gt;&gt;&gt; diff(catalan(n), n)</span>
-<span class="sd">    (polygamma(0, n + 1/2) - polygamma(0, n + 2) + 2*log(2))*catalan(n)</span>
-
-<span class="sd">    As a more advanced example consider the following ratio</span>
-<span class="sd">    between consecutive numbers:</span>
-
-<span class="sd">    &gt;&gt;&gt; combsimp((catalan(n + 1)/catalan(n)).rewrite(binomial))</span>
-<span class="sd">    2*(2*n + 1)/(n + 2)</span>
-
-<span class="sd">    The Catalan numbers can be generalized to complex numbers:</span>
-
-<span class="sd">    &gt;&gt;&gt; catalan(I).rewrite(gamma)</span>
-<span class="sd">    4**I*gamma(1/2 + I)/(sqrt(pi)*gamma(2 + I))</span>
-
-<span class="sd">    and evaluated with arbitrary precision:</span>
-
-<span class="sd">    &gt;&gt;&gt; catalan(I).evalf(20)</span>
-<span class="sd">    0.39764993382373624267 - 0.020884341620842555705*I</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    * http://en.wikipedia.org/wiki/Catalan_number</span>
-<span class="sd">    * http://mathworld.wolfram.com/CatalanNumber.html</span>
-<span class="sd">    * http://geometer.org/mathcircles/catalan.pdf</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.functions.combinatorial.factorials.binomial</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">n</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">4</span><span class="o">**</span><span class="n">n</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">2</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">catalan</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">polygamma</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span> <span class="o">-</span> <span class="n">C</span><span class="o">.</span><span class="n">polygamma</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">C</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mi">4</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_binomial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">binomial</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_gamma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="c"># The gamma function allows to generalize Catalan numbers to complex n</span>
-        <span class="k">return</span> <span class="mi">4</span><span class="o">**</span><span class="n">n</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">2</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_hyper</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">hyper</span><span class="p">([</span><span class="mi">1</span> <span class="o">-</span> <span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="n">n</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span></div>
-</pre></div>
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-  <h1>Source code for sympy.functions.elementary.complexes</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">C</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">,</span> <span class="n">ArgumentIndexError</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.piecewise</span> <span class="kn">import</span> <span class="n">Piecewise</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Mul</span>
-<span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Eq</span>
-
-<span class="c">###############################################################################</span>
-<span class="c">######################### REAL and IMAGINARY PARTS ############################</span>
-<span class="c">###############################################################################</span>
-
-
-<div class="viewcode-block" id="re"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.complexes.re">[docs]</a><span class="k">class</span> <span class="nc">re</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns real part of expression. This function performs only</span>
-<span class="sd">       elementary analysis and so it will fail to decompose properly</span>
-<span class="sd">       more complicated expressions. If completely simplified result</span>
-<span class="sd">       is needed then use Basic.as_real_imag() or perform complex</span>
-<span class="sd">       expansion on instance of this function.</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy import re, im, I, E</span>
-<span class="sd">       &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">       &gt;&gt;&gt; re(2*E)</span>
-<span class="sd">       2*E</span>
-
-<span class="sd">       &gt;&gt;&gt; re(2*I + 17)</span>
-<span class="sd">       17</span>
-
-<span class="sd">       &gt;&gt;&gt; re(2*I)</span>
-<span class="sd">       0</span>
-
-<span class="sd">       &gt;&gt;&gt; re(im(x) + x*I + 2)</span>
-<span class="sd">       2</span>
-
-<span class="sd">       See Also</span>
-<span class="sd">       ========</span>
-
-<span class="sd">       im</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="n">is_real</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">unbranched</span> <span class="o">=</span> <span class="bp">True</span>  <span class="c"># implicitely works on the projection to C</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-        <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">arg</span>
-        <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Function</span> <span class="ow">and</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">conjugate</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">re</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-
-            <span class="n">included</span><span class="p">,</span> <span class="n">reverted</span><span class="p">,</span> <span class="n">excluded</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[]</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-                <span class="n">coeff</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-                        <span class="n">reverted</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="ow">not</span> <span class="n">term</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span> <span class="ow">and</span> <span class="n">term</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-                    <span class="n">excluded</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="c"># Try to do some advanced expansion.  If</span>
-                    <span class="c"># impossible, don&#39;t try to do re(arg) again</span>
-                    <span class="c"># (because this is what we are trying to do now).</span>
-                    <span class="n">real_imag</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">(</span><span class="n">ignore</span><span class="o">=</span><span class="n">arg</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">real_imag</span><span class="p">:</span>
-                        <span class="n">excluded</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">real_imag</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">included</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">included</span><span class="p">):</span>
-                <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="p">[</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">xs</span><span class="p">)</span> <span class="k">for</span> <span class="n">xs</span> <span class="ow">in</span> <span class="p">[</span><span class="n">included</span><span class="p">,</span> <span class="n">reverted</span><span class="p">,</span> <span class="n">excluded</span><span class="p">]]</span>
-
-                <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">-</span> <span class="n">im</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">+</span> <span class="n">c</span>
-
-<div class="viewcode-block" id="re.as_real_imag"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.complexes.re.as_real_imag">[docs]</a>    <span class="k">def</span> <span class="nf">as_real_imag</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the real number with a zero complex part.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">re</span><span class="p">(</span><span class="n">Derivative</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;evaluate&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">}))</span>
-        <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">is_imaginary</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> \
-                <span class="o">*</span> <span class="n">im</span><span class="p">(</span><span class="n">Derivative</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;evaluate&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">}))</span>
-
-</div>
-<span class="k">class</span> <span class="nc">im</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns imaginary part of expression. This function performs only</span>
-<span class="sd">    elementary analysis and so it will fail to decompose properly more</span>
-<span class="sd">    complicated expressions. If completely simplified result is needed then</span>
-<span class="sd">    use Basic.as_real_imag() or perform complex expansion on instance of</span>
-<span class="sd">    this function.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import re, im, E, I</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">    &gt;&gt;&gt; im(2*E)</span>
-<span class="sd">    0</span>
-
-<span class="sd">    &gt;&gt;&gt; re(2*I + 17)</span>
-<span class="sd">    17</span>
-
-<span class="sd">    &gt;&gt;&gt; im(x*I)</span>
-<span class="sd">    re(x)</span>
-
-<span class="sd">    &gt;&gt;&gt; im(re(x) + y)</span>
-<span class="sd">    im(y)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    re</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="n">is_real</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">unbranched</span> <span class="o">=</span> <span class="bp">True</span>  <span class="c"># implicitely works on the projection to C</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-        <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">arg</span>
-        <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Function</span> <span class="ow">and</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">conjugate</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">im</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">included</span><span class="p">,</span> <span class="n">reverted</span><span class="p">,</span> <span class="n">excluded</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[]</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-                <span class="n">coeff</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-                        <span class="n">reverted</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">excluded</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="n">term</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">term</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-                    <span class="c"># Try to do some advanced expansion.  If</span>
-                    <span class="c"># impossible, don&#39;t try to do im(arg) again</span>
-                    <span class="c"># (because this is what we are trying to do now).</span>
-                    <span class="n">real_imag</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">(</span><span class="n">ignore</span><span class="o">=</span><span class="n">arg</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">real_imag</span><span class="p">:</span>
-                        <span class="n">excluded</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">real_imag</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">included</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">included</span><span class="p">):</span>
-                <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="p">[</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">xs</span><span class="p">)</span> <span class="k">for</span> <span class="n">xs</span> <span class="ow">in</span> <span class="p">[</span><span class="n">included</span><span class="p">,</span> <span class="n">reverted</span><span class="p">,</span> <span class="n">excluded</span><span class="p">]]</span>
-
-                <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">+</span> <span class="n">re</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">+</span> <span class="n">c</span>
-
-    <span class="k">def</span> <span class="nf">as_real_imag</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the imaginary part with a zero real part.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.functions import im</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import I</span>
-<span class="sd">        &gt;&gt;&gt; im(2 + 3*I).as_real_imag()</span>
-<span class="sd">        (3, 0)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">im</span><span class="p">(</span><span class="n">Derivative</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;evaluate&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">}))</span>
-        <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">is_imaginary</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> \
-                <span class="o">*</span> <span class="n">re</span><span class="p">(</span><span class="n">Derivative</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;evaluate&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">}))</span>
-
-
-<span class="c">###############################################################################</span>
-<span class="c">############### SIGN, ABSOLUTE VALUE, ARGUMENT and CONJUGATION ################</span>
-<span class="c">###############################################################################</span>
-
-<div class="viewcode-block" id="sign"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.complexes.sign">[docs]</a><span class="k">class</span> <span class="nc">sign</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the sign of an expression, that is:</span>
-
-<span class="sd">    * 1 if expression is positive</span>
-<span class="sd">    * 0 if expression is equal to zero</span>
-<span class="sd">    * -1 if expression is negative</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.functions import sign</span>
-<span class="sd">    &gt;&gt;&gt; sign(-1)</span>
-<span class="sd">    -1</span>
-<span class="sd">    &gt;&gt;&gt; sign(0)</span>
-<span class="sd">    0</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Abs, conjugate</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="n">is_bounded</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_complex</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_nonzero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">/</span> <span class="n">Abs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-        <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Function</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">sign</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">arg</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">:</span>
-            <span class="n">arg2</span> <span class="o">=</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">arg</span>
-            <span class="k">if</span> <span class="n">arg2</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-            <span class="k">if</span> <span class="n">arg2</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="n">c</span><span class="p">,</span> <span class="n">args</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">()</span>
-            <span class="n">unk</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">is_imag</span> <span class="o">=</span> <span class="n">c</span><span class="o">.</span><span class="n">is_imaginary</span>
-            <span class="n">is_neg</span> <span class="o">=</span> <span class="n">c</span><span class="o">.</span><span class="n">is_negative</span>
-            <span class="k">for</span> <span class="n">ai</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-                <span class="n">ai2</span> <span class="o">=</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">ai</span>
-                <span class="k">if</span> <span class="n">ai</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                    <span class="n">is_neg</span> <span class="o">=</span> <span class="ow">not</span> <span class="n">is_neg</span>
-                <span class="k">elif</span> <span class="n">ai</span><span class="o">.</span><span class="n">is_imaginary</span> <span class="ow">and</span> <span class="n">ai2</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                    <span class="n">is_imag</span> <span class="o">=</span> <span class="ow">not</span> <span class="n">is_imag</span>
-                <span class="k">elif</span> <span class="n">ai</span><span class="o">.</span><span class="n">is_negative</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> \
-                        <span class="p">(</span><span class="n">ai</span><span class="o">.</span><span class="n">is_imaginary</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">ai2</span><span class="o">.</span><span class="n">is_positive</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">):</span>
-                    <span class="n">unk</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ai</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">unk</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span> <span class="k">if</span> <span class="n">is_neg</span> <span class="k">else</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span> \
-                <span class="o">*</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="k">if</span> <span class="n">is_imag</span> <span class="k">else</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span> \
-                <span class="o">*</span> <span class="n">cls</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">_new_rawargs</span><span class="p">(</span><span class="o">*</span><span class="n">unk</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_Abs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_nonzero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sign</span><span class="p">(</span><span class="n">conjugate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">sympy.functions.special.delta_functions</span> <span class="kn">import</span> <span class="n">DiracDelta</span>
-            <span class="k">return</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">Derivative</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;evaluate&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">})</span> \
-                <span class="o">*</span> <span class="n">DiracDelta</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">sympy.functions.special.delta_functions</span> <span class="kn">import</span> <span class="n">DiracDelta</span>
-            <span class="k">return</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">Derivative</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;evaluate&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">})</span> \
-                <span class="o">*</span> <span class="n">DiracDelta</span><span class="p">(</span><span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_imaginary</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_imaginary</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_integer</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_zero</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_zero</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="p">(</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_nonzero</span> <span class="ow">and</span>
-            <span class="n">other</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">and</span>
-            <span class="n">other</span><span class="o">.</span><span class="n">is_even</span>
-        <span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">sgn</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">())</span>
-
-</div>
-<div class="viewcode-block" id="Abs"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.complexes.Abs">[docs]</a><span class="k">class</span> <span class="nc">Abs</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the absolute value of the argument.</span>
-
-<span class="sd">    This is an extension of the built-in function abs() to accept symbolic</span>
-<span class="sd">    values.  If you pass a SymPy expression to the built-in abs(), it will</span>
-<span class="sd">    pass it automatically to Abs().</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Abs, Symbol, S</span>
-<span class="sd">    &gt;&gt;&gt; Abs(-1)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; x = Symbol(&#39;x&#39;, real=True)</span>
-<span class="sd">    &gt;&gt;&gt; Abs(-x)</span>
-<span class="sd">    Abs(x)</span>
-<span class="sd">    &gt;&gt;&gt; Abs(x**2)</span>
-<span class="sd">    x**2</span>
-<span class="sd">    &gt;&gt;&gt; abs(-x) # The Python built-in</span>
-<span class="sd">    Abs(x)</span>
-
-<span class="sd">    Note that the Python built-in will return either an Expr or int depending on</span>
-<span class="sd">    the argument::</span>
-
-<span class="sd">        &gt;&gt;&gt; type(abs(-1))</span>
-<span class="sd">        &lt;... &#39;int&#39;&gt;</span>
-<span class="sd">        &gt;&gt;&gt; type(abs(S.NegativeOne))</span>
-<span class="sd">        &lt;class &#39;sympy.core.numbers.One&#39;&gt;</span>
-
-<span class="sd">    Abs will always return a sympy object.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sign, conjugate</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="n">is_real</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_negative</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">unbranched</span> <span class="o">=</span> <span class="bp">True</span>
-
-<div class="viewcode-block" id="Abs.fdiff"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.complexes.Abs.fdiff">[docs]</a>    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Get the first derivative of the argument to Abs().</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.functions import Abs</span>
-<span class="sd">        &gt;&gt;&gt; Abs(-x).fdiff()</span>
-<span class="sd">        sign(x)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">sign</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="s">&#39;_eval_Abs&#39;</span><span class="p">):</span>
-            <span class="n">obj</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">_eval_Abs</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">obj</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">obj</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="n">known</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">unk</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="n">tnew</span> <span class="o">=</span> <span class="n">cls</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">tnew</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">cls</span><span class="p">:</span>
-                    <span class="n">unk</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tnew</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">known</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tnew</span><span class="p">)</span>
-            <span class="n">known</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">known</span><span class="p">)</span>
-            <span class="n">unk</span> <span class="o">=</span> <span class="n">cls</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">unk</span><span class="p">),</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span> <span class="k">if</span> <span class="n">unk</span> <span class="k">else</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">return</span> <span class="n">known</span><span class="o">*</span><span class="n">unk</span>
-        <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">arg</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_nonpositive</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">arg</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">:</span>
-            <span class="n">arg2</span> <span class="o">=</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">arg</span>
-            <span class="k">if</span> <span class="n">arg2</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">arg2</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">is</span> <span class="bp">False</span> <span class="ow">and</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_imaginary</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expand_mul</span>
-            <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">arg</span> <span class="o">*</span> <span class="n">arg</span><span class="o">.</span><span class="n">conjugate</span><span class="p">())</span> <span class="p">)</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-            <span class="n">base</span><span class="p">,</span> <span class="n">exponent</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">exponent</span><span class="o">.</span><span class="n">is_even</span> <span class="ow">and</span> <span class="n">base</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">arg</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_nonzero</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_nonzero</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_positive</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_nonzero</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="n">other</span>
-            <span class="k">elif</span> <span class="n">other</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="n">e</span> <span class="o">=</span> <span class="n">other</span> <span class="o">-</span> <span class="n">sign</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="n">e</span><span class="o">*</span><span class="bp">self</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">_eval_nseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="n">direction</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">leadterm</span><span class="p">(</span><span class="n">x</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span>
-        <span class="n">when</span> <span class="o">=</span> <span class="n">Eq</span><span class="p">(</span><span class="n">direction</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Piecewise</span><span class="p">(</span>
-            <span class="p">((</span><span class="n">s</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">direction</span><span class="p">,</span> <span class="mi">0</span><span class="p">)),</span> <span class="n">when</span><span class="p">),</span>
-            <span class="p">(</span><span class="n">sign</span><span class="p">(</span><span class="n">direction</span><span class="p">)</span><span class="o">*</span><span class="n">s</span><span class="p">,</span> <span class="bp">True</span><span class="p">),</span>
-        <span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">abs_symbolic</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">())</span>
-
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Derivative</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;evaluate&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">})</span> \
-                <span class="o">*</span> <span class="n">sign</span><span class="p">(</span><span class="n">conjugate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">re</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">*</span> <span class="n">Derivative</span><span class="p">(</span><span class="n">re</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">x</span><span class="p">,</span>
-            <span class="o">**</span><span class="p">{</span><span class="s">&#39;evaluate&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">})</span> <span class="o">+</span> <span class="n">im</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">*</span> <span class="n">Derivative</span><span class="p">(</span><span class="n">im</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span>
-                <span class="n">x</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;evaluate&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">}))</span> <span class="o">/</span> <span class="n">Abs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_Heaviside</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="c"># Note this only holds for real arg (since Heaviside is not defined</span>
-        <span class="c"># for complex arguments).</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">arg</span><span class="o">*</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Heaviside</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span> <span class="o">-</span> <span class="n">C</span><span class="o">.</span><span class="n">Heaviside</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-
-</div>
-<div class="viewcode-block" id="arg"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.complexes.arg">[docs]</a><span class="k">class</span> <span class="nc">arg</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns the argument (in radians) of a complex number&quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="n">is_real</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_bounded</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">re</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">im</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">atan2</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">arg</span>
-
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">re</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">im</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">x</span> <span class="o">*</span> <span class="n">Derivative</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;evaluate&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">})</span> <span class="o">-</span> <span class="n">y</span> <span class="o">*</span>
-                <span class="n">Derivative</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;evaluate&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">}))</span> <span class="o">/</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="conjugate"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.complexes.conjugate">[docs]</a><span class="k">class</span> <span class="nc">conjugate</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Changes the sign of the imaginary part of a complex number.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import conjugate, I</span>
-
-<span class="sd">    &gt;&gt;&gt; conjugate(1 + I)</span>
-<span class="sd">    1 - I</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sign, Abs</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">_eval_conjugate</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">obj</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">_eval_Abs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Abs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;evaluate&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">})</span>
-
-    <span class="k">def</span> <span class="nf">_eval_adjoint</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">transpose</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">conjugate</span><span class="p">(</span><span class="n">Derivative</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;evaluate&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">}))</span>
-        <span class="k">elif</span> <span class="n">x</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">conjugate</span><span class="p">(</span><span class="n">Derivative</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;evaluate&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">}))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_transpose</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">adjoint</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-</div>
-<span class="k">class</span> <span class="nc">transpose</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Linear map transposition.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">_eval_transpose</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">obj</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">_eval_adjoint</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">conjugate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">adjoint</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_transpose</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-
-<span class="k">class</span> <span class="nc">adjoint</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Conjugate transpose or Hermite conjugation.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">_eval_adjoint</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">obj</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">obj</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">_eval_transpose</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">obj</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">conjugate</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_adjoint</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">transpose</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_transpose</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">conjugate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&#39;</span><span class="si">%s</span><span class="s">^{\dag}&#39;</span> <span class="o">%</span> <span class="n">arg</span>
-        <span class="k">if</span> <span class="n">exp</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&#39;\left(</span><span class="si">%s</span><span class="s">\right)^{</span><span class="si">%s</span><span class="s">}&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">exp</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.printing.pretty.stringpict</span> <span class="kn">import</span> <span class="n">prettyForm</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">printer</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">pform</span><span class="o">**</span><span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\u2020</span><span class="s">&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">pform</span><span class="o">**</span><span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;+&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-<span class="c">###############################################################################</span>
-<span class="c">############### HANDLING OF POLAR NUMBERS #####################################</span>
-<span class="c">###############################################################################</span>
-
-
-<span class="k">class</span> <span class="nc">polar_lift</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Lift argument to the riemann surface of the logarithm, using the</span>
-<span class="sd">    standard branch.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, polar_lift, I</span>
-<span class="sd">    &gt;&gt;&gt; p = Symbol(&#39;p&#39;, polar=True)</span>
-<span class="sd">    &gt;&gt;&gt; x = Symbol(&#39;x&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; polar_lift(4)</span>
-<span class="sd">    4*exp_polar(0)</span>
-<span class="sd">    &gt;&gt;&gt; polar_lift(-4)</span>
-<span class="sd">    4*exp_polar(I*pi)</span>
-<span class="sd">    &gt;&gt;&gt; polar_lift(-I)</span>
-<span class="sd">    exp_polar(-I*pi/2)</span>
-<span class="sd">    &gt;&gt;&gt; polar_lift(I + 2)</span>
-<span class="sd">    polar_lift(2 + I)</span>
-
-<span class="sd">    &gt;&gt;&gt; polar_lift(4*x)</span>
-<span class="sd">    4*polar_lift(x)</span>
-<span class="sd">    &gt;&gt;&gt; polar_lift(4*p)</span>
-<span class="sd">    4*p</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.functions.elementary.exponential.exp_polar</span>
-<span class="sd">    periodic_argument</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="n">is_polar</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_comparable</span> <span class="o">=</span> <span class="bp">False</span>  <span class="c"># Cannot be evalf&#39;d.</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exp_polar</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">arg</span> <span class="k">as</span> <span class="n">argument</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="n">ar</span> <span class="o">=</span> <span class="n">argument</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="c">#if not ar.has(argument) and not ar.has(atan):</span>
-            <span class="k">if</span> <span class="n">ar</span> <span class="ow">in</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">pi</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">exp_polar</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">ar</span><span class="p">)</span><span class="o">*</span><span class="nb">abs</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">arg</span><span class="p">]</span>
-        <span class="n">included</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">excluded</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">positive</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_polar</span><span class="p">:</span>
-                <span class="n">included</span> <span class="o">+=</span> <span class="p">[</span><span class="n">arg</span><span class="p">]</span>
-            <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                <span class="n">positive</span> <span class="o">+=</span> <span class="p">[</span><span class="n">arg</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">excluded</span> <span class="o">+=</span> <span class="p">[</span><span class="n">arg</span><span class="p">]</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">excluded</span><span class="p">)</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">excluded</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">included</span> <span class="o">+</span> <span class="n">positive</span><span class="p">))</span><span class="o">*</span><span class="n">polar_lift</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">excluded</span><span class="p">))</span>
-            <span class="k">elif</span> <span class="n">included</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">included</span> <span class="o">+</span> <span class="n">positive</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">positive</span><span class="p">)</span><span class="o">*</span><span class="n">exp_polar</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Careful! any evalf of polar numbers is flaky &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">periodic_argument</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Represent the argument on a quotient of the riemann surface of the</span>
-<span class="sd">    logarithm. That is, given a period P, always return a value in</span>
-<span class="sd">    (-P/2, P/2], by using exp(P*I) == 1.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import exp, exp_polar, periodic_argument, unbranched_argument</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import I, pi</span>
-<span class="sd">    &gt;&gt;&gt; unbranched_argument(exp(5*I*pi))</span>
-<span class="sd">    pi</span>
-<span class="sd">    &gt;&gt;&gt; unbranched_argument(exp_polar(5*I*pi))</span>
-<span class="sd">    5*pi</span>
-<span class="sd">    &gt;&gt;&gt; periodic_argument(exp_polar(5*I*pi), 2*pi)</span>
-<span class="sd">    pi</span>
-<span class="sd">    &gt;&gt;&gt; periodic_argument(exp_polar(5*I*pi), 3*pi)</span>
-<span class="sd">    -pi</span>
-<span class="sd">    &gt;&gt;&gt; periodic_argument(exp_polar(5*I*pi), pi)</span>
-<span class="sd">    0</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.functions.elementary.exponential.exp_polar</span>
-<span class="sd">    polar_lift : Lift argument to the riemann surface of the logarithm</span>
-<span class="sd">    principal_branch</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_getunbranched</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">ar</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exp_polar</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">polar_lift</span>
-        <span class="k">if</span> <span class="n">ar</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="n">ar</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">ar</span><span class="p">]</span>
-        <span class="n">unbranched</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">a</span><span class="o">.</span><span class="n">is_polar</span><span class="p">:</span>
-                <span class="n">unbranched</span> <span class="o">+=</span> <span class="n">arg</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">a</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">exp_polar</span><span class="p">:</span>
-                <span class="n">unbranched</span> <span class="o">+=</span> <span class="n">a</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()[</span><span class="mi">1</span><span class="p">]</span>
-            <span class="k">elif</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-                <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-                <span class="n">unbranched</span> <span class="o">+=</span> <span class="n">re</span><span class="o">*</span><span class="n">unbranched_argument</span><span class="p">(</span>
-                    <span class="n">a</span><span class="o">.</span><span class="n">base</span><span class="p">)</span> <span class="o">+</span> <span class="n">im</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">base</span><span class="p">))</span>
-            <span class="k">elif</span> <span class="n">a</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">polar_lift</span><span class="p">:</span>
-                <span class="n">unbranched</span> <span class="o">+=</span> <span class="n">arg</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">return</span> <span class="n">unbranched</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">ar</span><span class="p">,</span> <span class="n">period</span><span class="p">):</span>
-        <span class="c"># Our strategy is to evaluate the argument on the riemann surface of the</span>
-        <span class="c"># logarithm, and then reduce.</span>
-        <span class="c"># NOTE evidently this means it is a rather bad idea to use this with</span>
-        <span class="c"># period != 2*pi and non-polar numbers.</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">ceiling</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">atan2</span><span class="p">,</span> <span class="n">atan</span><span class="p">,</span> <span class="n">polar_lift</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">Mul</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">period</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">period</span> <span class="o">==</span> <span class="n">oo</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ar</span><span class="p">,</span> <span class="n">principal_branch</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">periodic_argument</span><span class="p">(</span><span class="o">*</span><span class="n">ar</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">ar</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">polar_lift</span> <span class="ow">and</span> <span class="n">period</span> <span class="o">&gt;=</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">periodic_argument</span><span class="p">(</span><span class="n">ar</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">period</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">ar</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="n">newargs</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">ar</span><span class="o">.</span><span class="n">args</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">x</span><span class="o">.</span><span class="n">is_positive</span><span class="p">]</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">newargs</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">ar</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">periodic_argument</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">newargs</span><span class="p">),</span> <span class="n">period</span><span class="p">)</span>
-        <span class="n">unbranched</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_getunbranched</span><span class="p">(</span><span class="n">ar</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">unbranched</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">unbranched</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">periodic_argument</span><span class="p">,</span> <span class="n">atan2</span><span class="p">,</span> <span class="n">arg</span><span class="p">,</span> <span class="n">atan</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">period</span> <span class="o">==</span> <span class="n">oo</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">unbranched</span>
-        <span class="k">if</span> <span class="n">period</span> <span class="o">!=</span> <span class="n">oo</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">ceiling</span><span class="p">(</span><span class="n">unbranched</span><span class="o">/</span><span class="n">period</span> <span class="o">-</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">period</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">ceiling</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">unbranched</span> <span class="o">-</span> <span class="n">n</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">ceiling</span><span class="p">,</span> <span class="n">oo</span>
-        <span class="n">z</span><span class="p">,</span> <span class="n">period</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">if</span> <span class="n">period</span> <span class="o">==</span> <span class="n">oo</span><span class="p">:</span>
-            <span class="n">unbranched</span> <span class="o">=</span> <span class="n">periodic_argument</span><span class="o">.</span><span class="n">_getunbranched</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">unbranched</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span>
-            <span class="k">return</span> <span class="n">unbranched</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-        <span class="n">ub</span> <span class="o">=</span> <span class="n">periodic_argument</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">ub</span> <span class="o">-</span> <span class="n">ceiling</span><span class="p">(</span><span class="n">ub</span><span class="o">/</span><span class="n">period</span> <span class="o">-</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">period</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">unbranched_argument</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">oo</span>
-    <span class="k">return</span> <span class="n">periodic_argument</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">principal_branch</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Represent a polar number reduced to its principal branch on a quotient</span>
-<span class="sd">    of the riemann surface of the logarithm.</span>
-
-<span class="sd">    This is a function of two arguments. The first argument is a polar</span>
-<span class="sd">    number `z`, and the second one a positive real number of infinity, `p`.</span>
-<span class="sd">    The result is &quot;z mod exp_polar(I*p)&quot;.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import exp_polar, principal_branch, oo, I, pi</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import z</span>
-<span class="sd">    &gt;&gt;&gt; principal_branch(z, oo)</span>
-<span class="sd">    z</span>
-<span class="sd">    &gt;&gt;&gt; principal_branch(exp_polar(2*pi*I)*3, 2*pi)</span>
-<span class="sd">    3*exp_polar(0)</span>
-<span class="sd">    &gt;&gt;&gt; principal_branch(exp_polar(2*pi*I)*3*z, 2*pi)</span>
-<span class="sd">    3*principal_branch(z, 2*pi)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.functions.elementary.exponential.exp_polar</span>
-<span class="sd">    polar_lift : Lift argument to the riemann surface of the logarithm</span>
-<span class="sd">    periodic_argument</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-    <span class="n">is_polar</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_comparable</span> <span class="o">=</span> <span class="bp">False</span>  <span class="c"># cannot always be evalf&#39;d</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">period</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">oo</span><span class="p">,</span> <span class="n">exp_polar</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">polar_lift</span><span class="p">,</span> <span class="n">Symbol</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">polar_lift</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">principal_branch</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">period</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">period</span> <span class="o">==</span> <span class="n">oo</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">x</span>
-        <span class="n">ub</span> <span class="o">=</span> <span class="n">periodic_argument</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-        <span class="n">barg</span> <span class="o">=</span> <span class="n">periodic_argument</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">period</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">ub</span> <span class="o">!=</span> <span class="n">barg</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">ub</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">periodic_argument</span><span class="p">)</span> \
-                <span class="ow">and</span> <span class="ow">not</span> <span class="n">barg</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">periodic_argument</span><span class="p">):</span>
-            <span class="n">pl</span> <span class="o">=</span> <span class="n">polar_lift</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-            <span class="k">def</span> <span class="nf">mr</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">polar_lift</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">expr</span>
-            <span class="n">pl</span> <span class="o">=</span> <span class="n">pl</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">polar_lift</span><span class="p">,</span> <span class="n">mr</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">pl</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">polar_lift</span><span class="p">):</span>
-                <span class="n">res</span> <span class="o">=</span> <span class="n">exp_polar</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="p">(</span><span class="n">barg</span> <span class="o">-</span> <span class="n">ub</span><span class="p">))</span><span class="o">*</span><span class="n">pl</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">res</span><span class="o">.</span><span class="n">is_polar</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">res</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">exp_polar</span><span class="p">):</span>
-                    <span class="n">res</span> <span class="o">*=</span> <span class="n">exp_polar</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">res</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">x</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-            <span class="n">c</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">x</span><span class="p">,</span> <span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">c</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">(</span><span class="o">*</span><span class="n">x</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">)</span>
-        <span class="n">others</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">m</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">y</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                <span class="n">c</span> <span class="o">*=</span> <span class="n">y</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">others</span> <span class="o">+=</span> <span class="p">[</span><span class="n">y</span><span class="p">]</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">others</span><span class="p">)</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">periodic_argument</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">period</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">periodic_argument</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="p">(</span><span class="n">unbranched_argument</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="o">!=</span> <span class="n">arg</span> <span class="ow">or</span>
-                              <span class="p">(</span><span class="n">arg</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">m</span> <span class="o">!=</span> <span class="p">()</span> <span class="ow">and</span> <span class="n">c</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="nb">abs</span><span class="p">(</span><span class="n">c</span><span class="p">)</span><span class="o">*</span><span class="n">principal_branch</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">m</span><span class="p">),</span> <span class="n">period</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">principal_branch</span><span class="p">(</span><span class="n">exp_polar</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">arg</span><span class="p">)</span><span class="o">*</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">m</span><span class="p">),</span> <span class="n">period</span><span class="p">)</span><span class="o">*</span><span class="nb">abs</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="p">((</span><span class="nb">abs</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">period</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span> <span class="ow">or</span> <span class="n">arg</span> <span class="o">==</span> <span class="n">period</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> \
-                <span class="ow">and</span> <span class="n">m</span> <span class="o">==</span> <span class="p">():</span>
-            <span class="k">return</span> <span class="n">exp_polar</span><span class="p">(</span><span class="n">arg</span><span class="o">*</span><span class="n">I</span><span class="p">)</span><span class="o">*</span><span class="nb">abs</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exp</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">I</span>
-        <span class="n">z</span><span class="p">,</span> <span class="n">period</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">periodic_argument</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">period</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">pi</span> <span class="ow">or</span> <span class="n">p</span> <span class="o">==</span> <span class="o">-</span><span class="n">pi</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>  <span class="c"># Cannot evalf for this argument.</span>
-        <span class="k">return</span> <span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">p</span><span class="p">))</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-
-<span class="c"># /cyclic/</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">basic</span> <span class="k">as</span> <span class="n">_</span>
-<span class="n">_</span><span class="o">.</span><span class="n">abs_</span> <span class="o">=</span> <span class="n">Abs</span>
-<span class="k">del</span> <span class="n">_</span>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.functions.elementary.exponential</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">C</span><span class="p">,</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.core.add</span> <span class="kn">import</span> <span class="n">Add</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Lambda</span><span class="p">,</span> <span class="n">Function</span><span class="p">,</span> <span class="n">ArgumentIndexError</span>
-<span class="kn">from</span> <span class="nn">sympy.core.cache</span> <span class="kn">import</span> <span class="n">cacheit</span>
-<span class="kn">from</span> <span class="nn">sympy.core.singleton</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="kn">from</span> <span class="nn">sympy.core.symbol</span> <span class="kn">import</span> <span class="n">Wild</span><span class="p">,</span> <span class="n">Dummy</span>
-<span class="kn">from</span> <span class="nn">sympy.core.mul</span> <span class="kn">import</span> <span class="n">Mul</span>
-
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">multiplicity</span><span class="p">,</span> <span class="n">perfect_power</span>
-
-<span class="c"># NOTE IMPORTANT</span>
-<span class="c"># The series expansion code in this file is an important part of the gruntz</span>
-<span class="c"># algorithm for determining limits. _eval_nseries has to return a generalized</span>
-<span class="c"># power series with coefficients in C(log(x), log).</span>
-<span class="c"># In more detail, the result of _eval_nseries(self, x, n) must be</span>
-<span class="c">#   c_0*x**e_0 + ... (finitely many terms)</span>
-<span class="c"># where e_i are numbers (not necessarily integers) and c_i involve only</span>
-<span class="c"># numbers, the function log, and log(x). [This also means it must not contain</span>
-<span class="c"># log(x(1+p)), this *has* to be expanded to log(x)+log(1+p) if x.is_positive and</span>
-<span class="c"># p.is_positive.]</span>
-
-
-<span class="k">class</span> <span class="nc">ExpBase</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">unbranched</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the inverse function of ``exp(x)``.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">log</span>
-
-    <span class="k">def</span> <span class="nf">as_numer_denom</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns this with a positive exponent as a 2-tuple (a fraction).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.functions import exp</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; exp(-x).as_numer_denom()</span>
-<span class="sd">        (1, exp(x))</span>
-<span class="sd">        &gt;&gt;&gt; exp(x).as_numer_denom()</span>
-<span class="sd">        (exp(x), 1)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># this should be the same as Pow.as_numer_denom wrt</span>
-        <span class="c"># exponent handling</span>
-        <span class="n">exp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exp</span>
-        <span class="n">neg_exp</span> <span class="o">=</span> <span class="n">exp</span><span class="o">.</span><span class="n">is_negative</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">neg_exp</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">exp</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="n">neg_exp</span> <span class="o">=</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">exp</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">neg_exp</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">-</span><span class="n">exp</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">exp</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the exponent of the function.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">as_base_exp</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the 2-tuple (base, exponent).</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">conjugate</span><span class="p">())</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_bounded</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_unbounded</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_bounded</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_rational</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">s</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">s</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_rational</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">s</span><span class="o">.</span><span class="n">is_rational</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_zero</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;exp(arg)**e -&gt; exp(arg*e) if assumptions allow it.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">func</span>
-        <span class="n">be</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">exp</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">be</span><span class="o">*</span><span class="n">e</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">rv</span>
-        <span class="k">if</span> <span class="n">be</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">rv</span>
-        <span class="c"># &quot;is True&quot; needed below; exp.is_polar returns &lt;property object ...&gt;</span>
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_polar</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">rv</span>
-        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_polar</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">rv</span>
-        <span class="k">if</span> <span class="n">be</span><span class="o">.</span><span class="n">is_polar</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">rv</span>
-        <span class="n">besmall</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">be</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-        <span class="k">if</span> <span class="n">besmall</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">rv</span>
-        <span class="k">elif</span> <span class="n">besmall</span> <span class="ow">is</span> <span class="bp">False</span> <span class="ow">and</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">e</span><span class="o">.</span><span class="n">q</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">rv</span>
-
-    <span class="k">def</span> <span class="nf">_eval_expand_power_exp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">and</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="n">expr</span> <span class="o">*=</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">expr</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">exp_polar</span><span class="p">(</span><span class="n">ExpBase</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Represent a &#39;polar number&#39; (see g-function Sphinx documentation).</span>
-
-<span class="sd">    ``exp_polar`` represents the function</span>
-<span class="sd">    `Exp: \mathbb{C} \rightarrow \mathcal{S}`, sending the complex number</span>
-<span class="sd">    `z = a + bi` to the polar number `r = exp(a), \theta = b`. It is one of</span>
-<span class="sd">    the main functions to construct polar numbers.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import exp_polar, pi, I, exp</span>
-
-<span class="sd">    The main difference is that polar numbers don&#39;t &quot;wrap around&quot; at `2 \pi`:</span>
-
-<span class="sd">    &gt;&gt;&gt; exp(2*pi*I)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; exp_polar(2*pi*I)</span>
-<span class="sd">    exp_polar(2*I*pi)</span>
-
-<span class="sd">    apart from that they behave mostly like classical complex numbers:</span>
-
-<span class="sd">    &gt;&gt;&gt; exp_polar(2)*exp_polar(3)</span>
-<span class="sd">    exp_polar(5)</span>
-
-<span class="sd">    See also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.simplify.simplify.powsimp</span>
-<span class="sd">    sympy.functions.elementary.complexes.polar_lift</span>
-<span class="sd">    sympy.functions.elementary.complexes.periodic_argument</span>
-<span class="sd">    sympy.functions.elementary.complexes.principal_branch</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_polar</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_comparable</span> <span class="o">=</span> <span class="bp">False</span>  <span class="c"># cannot be evalf&#39;d</span>
-
-    <span class="k">def</span> <span class="nf">_eval_Abs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expand_mul</span>
-        <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span> <span class="n">expand_mul</span><span class="p">(</span><span class="bp">self</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">conjugate</span><span class="p">())</span> <span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Careful! any evalf of polar numbers is flaky &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">im</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">re</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="n">im</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="o">&lt;=</span> <span class="o">-</span><span class="n">pi</span> <span class="ow">or</span> <span class="n">i</span> <span class="o">&gt;</span> <span class="n">pi</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>  <span class="c"># cannot evalf for this argument</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">im</span><span class="p">(</span><span class="n">res</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="c"># i ~ pi, but exp(I*i) evaluated to argument slightly bigger than pi</span>
-            <span class="k">return</span> <span class="n">re</span><span class="p">(</span><span class="n">res</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">res</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">as_base_exp</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># XXX exp_polar(0) is special!</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">ExpBase</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="exp"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.exponential.exp">[docs]</a><span class="k">class</span> <span class="nc">exp</span><span class="p">(</span><span class="n">ExpBase</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The exponential function, :math:`e^x`.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    log</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-<div class="viewcode-block" id="exp.fdiff"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.exponential.exp.fdiff">[docs]</a>    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the first derivative of this function.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Exp1</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">log</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="n">Ioo</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">in</span> <span class="p">[</span><span class="n">Ioo</span><span class="p">,</span> <span class="o">-</span><span class="n">Ioo</span><span class="p">]:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">coeff</span><span class="p">:</span>
-                <span class="k">if</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">coeff</span><span class="p">)</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-                    <span class="k">elif</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_odd</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span>
-                    <span class="k">elif</span> <span class="p">(</span><span class="n">coeff</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-                    <span class="k">elif</span> <span class="p">(</span><span class="n">coeff</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span><span class="o">.</span><span class="n">is_odd</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-
-            <span class="c"># Warning: code in risch.py will be very sensitive to changes</span>
-            <span class="c"># in this (see DifferentialExtension).</span>
-
-            <span class="c"># look for a single log factor</span>
-
-            <span class="n">coeff</span><span class="p">,</span> <span class="n">terms</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-
-            <span class="c"># but it can&#39;t be multiplied by oo</span>
-            <span class="k">if</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">]:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-
-            <span class="n">coeffs</span><span class="p">,</span> <span class="n">log_term</span> <span class="o">=</span> <span class="p">[</span><span class="n">coeff</span><span class="p">],</span> <span class="bp">None</span>
-            <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">terms</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">term</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">log</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">log_term</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="n">log_term</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="bp">None</span>
-                <span class="k">elif</span> <span class="n">term</span><span class="o">.</span><span class="n">is_comparable</span><span class="p">:</span>
-                    <span class="n">coeffs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-
-            <span class="k">return</span> <span class="n">log_term</span><span class="o">**</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">coeffs</span><span class="p">)</span> <span class="k">if</span> <span class="n">log_term</span> <span class="k">else</span> <span class="bp">None</span>
-
-        <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">out</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">add</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">a</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                    <span class="n">add</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-                    <span class="k">continue</span>
-                <span class="n">newa</span> <span class="o">=</span> <span class="n">cls</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">newa</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">cls</span><span class="p">:</span>
-                    <span class="n">add</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">out</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">newa</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">out</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">out</span><span class="p">)</span><span class="o">*</span><span class="n">cls</span><span class="p">(</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">add</span><span class="p">),</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="exp.base"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.exponential.exp.base">[docs]</a>    <span class="k">def</span> <span class="nf">base</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the base of the exponential function.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Exp1</span>
-</div>
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@cacheit</span>
-<div class="viewcode-block" id="exp.taylor_term"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.exponential.exp.taylor_term">[docs]</a>    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Calculates the next term in the Taylor series expansion.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">previous_terms</span><span class="p">:</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">previous_terms</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">p</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">p</span> <span class="o">*</span> <span class="n">x</span> <span class="o">/</span> <span class="n">n</span>
-        <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">()(</span><span class="n">n</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="exp.as_real_imag"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.exponential.exp.as_real_imag">[docs]</a>    <span class="k">def</span> <span class="nf">as_real_imag</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns this function as a 2-tuple representing a complex number.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import I</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.functions import exp</span>
-<span class="sd">        &gt;&gt;&gt; exp(x).as_real_imag()</span>
-<span class="sd">        (exp(re(x))*cos(im(x)), exp(re(x))*sin(im(x)))</span>
-<span class="sd">        &gt;&gt;&gt; exp(1).as_real_imag()</span>
-<span class="sd">        (E, 0)</span>
-<span class="sd">        &gt;&gt;&gt; exp(I).as_real_imag()</span>
-<span class="sd">        (cos(1), sin(1))</span>
-<span class="sd">        &gt;&gt;&gt; exp(1+I).as_real_imag()</span>
-<span class="sd">        (E*cos(1), E*sin(1))</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.functions.elementary.complexes.re</span>
-<span class="sd">        sympy.functions.elementary.complexes.im</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-            <span class="n">re</span> <span class="o">=</span> <span class="n">re</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-            <span class="n">im</span> <span class="o">=</span> <span class="n">im</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-        <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">im</span><span class="p">),</span> <span class="n">C</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">im</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">re</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">,</span> <span class="n">exp</span><span class="p">(</span><span class="n">re</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">o</span> <span class="o">=</span> <span class="n">old</span>
-        <span class="k">if</span> <span class="n">old</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>  <span class="c"># handle (exp(3*log(x))).subs(x**2, z) -&gt; z**(3/2)</span>
-            <span class="n">o</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="n">o</span><span class="o">.</span><span class="n">exp</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">o</span><span class="o">.</span><span class="n">base</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">o</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">exp</span><span class="p">:</span>
-            <span class="c"># exp(a*expr) .subs( exp(b*expr), y )  -&gt;  y ** (a/b)</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">expr_terms</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span>
-                <span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">,</span> <span class="n">as_Add</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-            <span class="n">b</span><span class="p">,</span> <span class="n">expr_terms_</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span>
-                <span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">,</span> <span class="n">as_Add</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">expr_terms</span> <span class="o">==</span> <span class="n">expr_terms_</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">new</span><span class="o">**</span><span class="p">(</span><span class="n">a</span><span class="o">/</span><span class="n">b</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>  <span class="c"># exp(2*x+a).subs(exp(3*x),y) -&gt; y**(2/3) * exp(a)</span>
-                <span class="c"># exp(exp(x) + exp(x**2)).subs(exp(exp(x)), w) -&gt; w * exp(exp(x**2))</span>
-                <span class="n">oarg</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="n">new_l</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="n">o_al</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="n">coeff2</span><span class="p">,</span> <span class="n">terms2</span> <span class="o">=</span> <span class="n">oarg</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">()</span>
-                <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                    <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">_subs</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">new</span><span class="p">)</span>
-                    <span class="n">coeff1</span><span class="p">,</span> <span class="n">terms1</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">()</span>
-                    <span class="k">if</span> <span class="n">terms1</span> <span class="o">==</span> <span class="n">terms2</span><span class="p">:</span>
-                        <span class="n">new_l</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">new</span><span class="o">**</span><span class="p">(</span><span class="n">coeff1</span><span class="o">/</span><span class="n">coeff2</span><span class="p">))</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">o_al</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">_subs</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">new</span><span class="p">))</span>
-                <span class="k">if</span> <span class="n">new_l</span><span class="p">:</span>
-                    <span class="n">new_l</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">o_al</span><span class="p">)))</span>
-                    <span class="n">r</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">new_l</span><span class="p">)</span>
-                    <span class="k">return</span> <span class="n">r</span>
-        <span class="k">if</span> <span class="n">o</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Exp1</span><span class="p">:</span>
-            <span class="c"># treat this however Pow is being treated</span>
-            <span class="n">u</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;u&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">u</span><span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">xreplace</span><span class="p">({</span><span class="n">u</span><span class="p">:</span> <span class="n">new</span><span class="p">})</span>
-
-        <span class="k">return</span> <span class="n">Function</span><span class="o">.</span><span class="n">_eval_subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">,</span> <span class="n">new</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">:</span>
-            <span class="n">arg2</span> <span class="o">=</span> <span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">/</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-            <span class="k">return</span> <span class="n">arg2</span><span class="o">.</span><span class="n">is_even</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_positive</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">return</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">:</span>
-            <span class="n">arg2</span> <span class="o">=</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">/</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-            <span class="k">return</span> <span class="n">arg2</span><span class="o">.</span><span class="n">is_even</span>
-
-    <span class="k">def</span> <span class="nf">_eval_lseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">yield</span> <span class="n">exp</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">integrate</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;t&quot;</span><span class="p">)</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">f</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">t</span><span class="p">))</span><span class="o">.</span><span class="n">lseries</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
-            <span class="k">yield</span> <span class="n">integrate</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_nseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="c"># NOTE Please see the comment at the beginning of this file, labelled</span>
-        <span class="c">#      IMPORTANT.</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">limit</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">powsimp</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">arg_series</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">arg_series</span><span class="o">.</span><span class="n">is_Order</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">arg_series</span>
-        <span class="n">arg0</span> <span class="o">=</span> <span class="n">limit</span><span class="p">(</span><span class="n">arg_series</span><span class="o">.</span><span class="n">removeO</span><span class="p">(),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">arg0</span> <span class="ow">in</span> <span class="p">[</span><span class="o">-</span><span class="n">oo</span><span class="p">,</span> <span class="n">oo</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;t&quot;</span><span class="p">)</span>
-        <span class="n">exp_series</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">.</span><span class="n">_taylor</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-        <span class="n">o</span> <span class="o">=</span> <span class="n">exp_series</span><span class="o">.</span><span class="n">getO</span><span class="p">()</span>
-        <span class="n">exp_series</span> <span class="o">=</span> <span class="n">exp_series</span><span class="o">.</span><span class="n">removeO</span><span class="p">()</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="n">arg0</span><span class="p">)</span><span class="o">*</span><span class="n">exp_series</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">arg_series</span> <span class="o">-</span> <span class="n">arg0</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">+=</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="n">o</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="p">(</span><span class="n">arg_series</span> <span class="o">-</span> <span class="n">arg0</span><span class="p">)),</span> <span class="n">x</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">r</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">powsimp</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;exp&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_taylor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="n">l</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">taylor_term</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">g</span><span class="p">)</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">)</span>
-            <span class="n">l</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">l</span><span class="p">)</span> <span class="o">+</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">exp</span><span class="p">(</span><span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">return</span> <span class="n">exp</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_sin</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">I</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">arg</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="n">I</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_cos</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">I</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">arg</span><span class="p">)</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">arg</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">())</span>
-
-</div>
-<div class="viewcode-block" id="log"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.exponential.log">[docs]</a><span class="k">class</span> <span class="nc">log</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The logarithmic function :math:`ln(x)` or :math:`log(x)`.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    exp</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-
-<div class="viewcode-block" id="log.fdiff"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.exponential.log.fdiff">[docs]</a>    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the first derivative of the function.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Lambda</span><span class="p">(</span><span class="n">s</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">s</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="log.inverse"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.exponential.log.inverse">[docs]</a>    <span class="k">def</span> <span class="nf">inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the inverse function, log(x) (or ln(x)).</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">exp</span>
-</div>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">,</span> <span class="n">base</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">unpolarify</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">base</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">base</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">base</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">arg</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">base</span><span class="o">.</span><span class="n">is_positive</span> <span class="ow">and</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_positive</span><span class="p">):</span>
-                    <span class="k">raise</span> <span class="ne">ValueError</span>
-                <span class="n">n</span> <span class="o">=</span> <span class="n">multiplicity</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">arg</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">n</span> <span class="o">+</span> <span class="n">log</span><span class="p">(</span><span class="n">arg</span> <span class="o">//</span> <span class="n">base</span> <span class="o">**</span> <span class="n">n</span><span class="p">)</span> <span class="o">/</span> <span class="n">log</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-                <span class="k">pass</span>
-            <span class="k">if</span> <span class="n">base</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">Exp1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span><span class="o">/</span><span class="n">cls</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">*</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">+</span> <span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">q</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">p</span><span class="p">)</span> <span class="o">-</span> <span class="n">cls</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-                <span class="c"># remove perfect powers automatically</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="n">perfect_power</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span>
-                <span class="k">if</span> <span class="n">p</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">cls</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span>
-        <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Exp1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">exp</span> <span class="ow">and</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">exp_polar</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">unpolarify</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">exp</span><span class="p">)</span>
-        <span class="c">#don&#39;t autoexpand Pow or Mul (see the issue 252):</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-                <span class="k">elif</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-                <span class="k">elif</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">*</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="o">+</span> <span class="n">cls</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">*</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="o">+</span> <span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">coeff</span><span class="p">)</span>
-
-<div class="viewcode-block" id="log.as_base_exp"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.exponential.log.as_base_exp">[docs]</a>    <span class="k">def</span> <span class="nf">as_base_exp</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns this function in the form (base, exponent).</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-</div>
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@cacheit</span>
-<div class="viewcode-block" id="log.taylor_term"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.exponential.log.taylor_term">[docs]</a>    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>  <span class="c"># of log(1+x)</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the next term in the Taylor series expansion of log(1+x).</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">powsimp</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">x</span>
-        <span class="k">if</span> <span class="n">previous_terms</span><span class="p">:</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">previous_terms</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">p</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">powsimp</span><span class="p">((</span><span class="o">-</span><span class="n">n</span><span class="p">)</span> <span class="o">*</span> <span class="n">p</span> <span class="o">*</span> <span class="n">x</span> <span class="o">/</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;exp&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">n</span> <span class="o">%</span> <span class="mi">2</span><span class="p">))</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_expand_log</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">unpolarify</span>
-        <span class="n">force</span> <span class="o">=</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;force&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">nonpos</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">force</span> <span class="ow">or</span> <span class="n">x</span><span class="o">.</span><span class="n">is_positive</span> <span class="ow">or</span> <span class="n">x</span><span class="o">.</span><span class="n">is_polar</span><span class="p">:</span>
-                    <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">log</span><span class="p">):</span>
-                        <span class="n">expr</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_expand_log</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">))</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">expr</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">nonpos</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">expr</span><span class="p">)</span> <span class="o">+</span> <span class="n">log</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">nonpos</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">force</span> <span class="ow">or</span> <span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span> <span class="n">arg</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_positive</span><span class="p">)</span> <span class="ow">or</span> \
-                    <span class="n">arg</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_polar</span><span class="p">:</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">base</span>
-                <span class="n">e</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">exp</span>
-                <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">log</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">unpolarify</span><span class="p">(</span><span class="n">e</span><span class="p">)</span> <span class="o">*</span> <span class="n">a</span><span class="o">.</span><span class="n">_eval_expand_log</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">unpolarify</span><span class="p">(</span><span class="n">e</span><span class="p">)</span> <span class="o">*</span> <span class="n">a</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-<div class="viewcode-block" id="log.as_real_imag"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.exponential.log.as_real_imag">[docs]</a>    <span class="k">def</span> <span class="nf">as_real_imag</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns this function as a complex coordinate.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import I</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.functions import log</span>
-<span class="sd">        &gt;&gt;&gt; log(x).as_real_imag()</span>
-<span class="sd">        (log(Abs(x)), arg(x))</span>
-<span class="sd">        &gt;&gt;&gt; log(I).as_real_imag()</span>
-<span class="sd">        (0, pi/2)</span>
-<span class="sd">        &gt;&gt;&gt; log(1+I).as_real_imag()</span>
-<span class="sd">        (log(sqrt(2)), pi/4)</span>
-<span class="sd">        &gt;&gt;&gt; log(I*x).as_real_imag()</span>
-<span class="sd">        (log(Abs(x)), arg(I*x))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-            <span class="nb">abs</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Abs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">))</span>
-            <span class="n">arg</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">arg</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="nb">abs</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Abs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="n">arg</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">arg</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;log&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>  <span class="c"># Expand the log</span>
-            <span class="n">hints</span><span class="p">[</span><span class="s">&#39;complex&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="nb">abs</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">),</span> <span class="n">arg</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="nb">abs</span><span class="p">),</span> <span class="n">arg</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_is_rational</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">s</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">s</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_rational</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">s</span><span class="o">.</span><span class="n">is_rational</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_positive</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_bounded</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_infinitesimal</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_bounded</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_positive</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_unbounded</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_infinitesimal</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">arg</span> <span class="o">&gt;</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_zero</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># XXX This is not quite useless. Try evaluating log(0.5).is_negative</span>
-        <span class="c">#     without it. There&#39;s probably a nicer way though.</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">_eval_nseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="c"># NOTE Please see the comment at the beginning of this file, labelled</span>
-        <span class="c">#      IMPORTANT.</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">cancel</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">logx</span><span class="p">:</span>
-            <span class="n">logx</span> <span class="o">=</span> <span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">x</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">logx</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">k</span><span class="p">,</span> <span class="n">l</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">),</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&quot;l&quot;</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">k</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">l</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">r</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="c">#k = r.get(r, S.One)</span>
-            <span class="c">#l = r.get(l, S.Zero)</span>
-            <span class="n">k</span><span class="p">,</span> <span class="n">l</span> <span class="o">=</span> <span class="n">r</span><span class="p">[</span><span class="n">k</span><span class="p">],</span> <span class="n">r</span><span class="p">[</span><span class="n">l</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">l</span> <span class="o">!=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">l</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">k</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-                <span class="n">r</span> <span class="o">=</span> <span class="n">log</span><span class="p">(</span><span class="n">k</span><span class="p">)</span> <span class="o">+</span> <span class="n">l</span><span class="o">*</span><span class="n">logx</span>  <span class="c"># XXX true regardless of assumptions?</span>
-                <span class="k">return</span> <span class="n">r</span>
-
-        <span class="c"># TODO new and probably slow</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span>
-        <span class="k">while</span> <span class="n">s</span><span class="o">.</span><span class="n">is_Order</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">leadterm</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">cancel</span><span class="p">(</span><span class="n">s</span><span class="o">/</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">b</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="n">l</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">2</span><span class="p">):</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">log</span><span class="o">.</span><span class="n">taylor_term</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span>
-            <span class="n">l</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">log</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">logx</span> <span class="o">+</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">l</span><span class="p">)</span> <span class="o">+</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="n">p</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">())</span>
-
-</div>
-<div class="viewcode-block" id="LambertW"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.exponential.LambertW">[docs]</a><span class="k">class</span> <span class="nc">LambertW</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Lambert W function, defined as the inverse function of</span>
-<span class="sd">    x*exp(x). This function represents the principal branch</span>
-<span class="sd">    of this inverse function, which like the natural logarithm</span>
-<span class="sd">    is multivalued.</span>
-
-<span class="sd">    For more information, see:</span>
-<span class="sd">    http://en.wikipedia.org/wiki/Lambert_W_function</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Exp1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">S</span><span class="o">.</span><span class="n">Exp1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="o">-</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-
-<div class="viewcode-block" id="LambertW.fdiff"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.exponential.LambertW.fdiff">[docs]</a>    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the first derivative of this function.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">LambertW</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">LambertW</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-</div></div>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">_coeff_isneg</span>
-</pre></div>
-
-          </div>
-        </div>
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diff --git a/dev-py3k/_modules/sympy/functions/elementary/hyperbolic.html b/dev-py3k/_modules/sympy/functions/elementary/hyperbolic.html
deleted file mode 100644
index 70dc1fcec70..00000000000
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+++ /dev/null
@@ -1,992 +0,0 @@
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-            
-  <h1>Source code for sympy.functions.elementary.hyperbolic</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">cacheit</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">ArgumentIndexError</span><span class="p">,</span> <span class="n">_coeff_isneg</span>
-
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">sqrt</span>
-
-<span class="c">###############################################################################</span>
-<span class="c">########################### HYPERBOLIC FUNCTIONS ##############################</span>
-<span class="c">###############################################################################</span>
-
-
-<div class="viewcode-block" id="HyperbolicFunction"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.HyperbolicFunction">[docs]</a><span class="k">class</span> <span class="nc">HyperbolicFunction</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Base class for hyperbolic functions. &quot;&quot;&quot;</span>
-
-    <span class="n">unbranched</span> <span class="o">=</span> <span class="bp">True</span>
-
-</div>
-<div class="viewcode-block" id="sinh"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.sinh">[docs]</a><span class="k">class</span> <span class="nc">sinh</span><span class="p">(</span><span class="n">HyperbolicFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The hyperbolic sine function, :math:`\\frac{exp(x) - exp(-x)}{2}`.</span>
-
-<span class="sd">    * sinh(x) -&gt; Returns the hyperbolic sine of x</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    cosh, tanh, asinh</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-<div class="viewcode-block" id="sinh.fdiff"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.sinh.fdiff">[docs]</a>    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the first derivative of this function.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cosh</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="sinh.inverse"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.sinh.inverse">[docs]</a>    <span class="k">def</span> <span class="nf">inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the inverse of this function.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">asinh</span>
-</div>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-
-            <span class="n">i_coeff</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">i_coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">i_coeff</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">asinh</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">acosh</span><span class="p">:</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">atanh</span><span class="p">:</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="k">return</span> <span class="n">x</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">acoth</span><span class="p">:</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@cacheit</span>
-<div class="viewcode-block" id="sinh.taylor_term"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.sinh.taylor_term">[docs]</a>    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the next term in the Taylor series expansion.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">previous_terms</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="n">previous_terms</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span>
-                <span class="k">return</span> <span class="n">p</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span> <span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">/</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">conjugate</span><span class="p">())</span>
-
-<div class="viewcode-block" id="sinh.as_real_imag"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.sinh.as_real_imag">[docs]</a>    <span class="k">def</span> <span class="nf">as_real_imag</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns this function as a complex coordinate.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-                <span class="n">hints</span><span class="p">[</span><span class="s">&#39;complex&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">sinh</span><span class="p">(</span><span class="n">re</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">im</span><span class="p">),</span> <span class="n">cosh</span><span class="p">(</span><span class="n">re</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">im</span><span class="p">))</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_exp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span> <span class="o">-</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">))</span> <span class="o">/</span> <span class="mi">2</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_cosh</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="n">arg</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_tanh</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">tanh_half</span> <span class="o">=</span> <span class="n">tanh</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="o">*</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">tanh_half</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">tanh_half</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_coth</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">coth_half</span> <span class="o">=</span> <span class="n">coth</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="o">*</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">coth_half</span><span class="o">/</span><span class="p">(</span><span class="n">coth_half</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_bounded</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">sinh</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">())</span>
-
-</div>
-<div class="viewcode-block" id="cosh"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.cosh">[docs]</a><span class="k">class</span> <span class="nc">cosh</span><span class="p">(</span><span class="n">HyperbolicFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The hyperbolic cosine function, :math:`\\frac{exp(x) + exp(-x)}{2}`.</span>
-
-<span class="sd">    * cosh(x) -&gt; Returns the hyperbolic cosine of x</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sinh, tanh, acosh</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">sinh</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-<div class="viewcode-block" id="cosh.inverse"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.cosh.inverse">[docs]</a>    <span class="k">def</span> <span class="nf">inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the inverse of this function.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">acosh</span>
-</div>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-
-            <span class="n">i_coeff</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">i_coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">i_coeff</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">asinh</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">acosh</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">atanh</span><span class="p">:</span>
-                <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">acoth</span><span class="p">:</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="k">return</span> <span class="n">x</span><span class="o">/</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">previous_terms</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="n">previous_terms</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span>
-                <span class="k">return</span> <span class="n">p</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span> <span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">conjugate</span><span class="p">())</span>
-
-    <span class="k">def</span> <span class="nf">as_real_imag</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-                <span class="n">hints</span><span class="p">[</span><span class="s">&#39;complex&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-
-        <span class="k">return</span> <span class="p">(</span><span class="n">cosh</span><span class="p">(</span><span class="n">re</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">im</span><span class="p">),</span> <span class="n">sinh</span><span class="p">(</span><span class="n">re</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">im</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_exp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span> <span class="o">+</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">))</span> <span class="o">/</span> <span class="mi">2</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_sinh</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="n">arg</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_tanh</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">tanh_half</span> <span class="o">=</span> <span class="n">tanh</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="o">*</span><span class="n">arg</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-        <span class="k">return</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">tanh_half</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">tanh_half</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_coth</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">coth_half</span> <span class="o">=</span> <span class="n">coth</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="o">*</span><span class="n">arg</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">coth_half</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">coth_half</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_bounded</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">cosh</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">())</span>
-
-</div>
-<div class="viewcode-block" id="tanh"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.tanh">[docs]</a><span class="k">class</span> <span class="nc">tanh</span><span class="p">(</span><span class="n">HyperbolicFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The hyperbolic tangent function, :math:`\\frac{sinh(x)}{cosh(x)}`.</span>
-
-<span class="sd">    * tanh(x) -&gt; Returns the hyperbolic tangent of x</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sinh, cosh, atanh</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="o">-</span> <span class="n">tanh</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">**</span><span class="mi">2</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-<div class="viewcode-block" id="tanh.inverse"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.tanh.inverse">[docs]</a>    <span class="k">def</span> <span class="nf">inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the inverse of this function.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">atanh</span>
-</div>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-
-            <span class="n">i_coeff</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">i_coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">i_coeff</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">tan</span><span class="p">(</span><span class="o">-</span><span class="n">i_coeff</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">tan</span><span class="p">(</span><span class="n">i_coeff</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">asinh</span><span class="p">:</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="k">return</span> <span class="n">x</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">acosh</span><span class="p">:</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="n">x</span>
-
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">atanh</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">acoth</span><span class="p">:</span>
-                <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-            <span class="n">a</span> <span class="o">=</span> <span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-            <span class="n">B</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">bernoulli</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="n">F</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-            <span class="k">return</span> <span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">B</span><span class="o">/</span><span class="n">F</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="n">n</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">conjugate</span><span class="p">())</span>
-
-    <span class="k">def</span> <span class="nf">as_real_imag</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-                <span class="n">hints</span><span class="p">[</span><span class="s">&#39;complex&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-        <span class="n">denom</span> <span class="o">=</span> <span class="n">sinh</span><span class="p">(</span><span class="n">re</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">C</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">im</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">sinh</span><span class="p">(</span><span class="n">re</span><span class="p">)</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="n">re</span><span class="p">)</span><span class="o">/</span><span class="n">denom</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">im</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">im</span><span class="p">)</span><span class="o">/</span><span class="n">denom</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_exp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">neg_exp</span><span class="p">,</span> <span class="n">pos_exp</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">),</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">pos_exp</span> <span class="o">-</span> <span class="n">neg_exp</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">pos_exp</span> <span class="o">+</span> <span class="n">neg_exp</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_sinh</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span><span class="o">/</span><span class="n">sinh</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_cosh</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">arg</span><span class="p">)</span><span class="o">/</span><span class="n">cosh</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_coth</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="n">coth</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_bounded</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">tanh</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">())</span>
-
-</div>
-<div class="viewcode-block" id="coth"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.coth">[docs]</a><span class="k">class</span> <span class="nc">coth</span><span class="p">(</span><span class="n">HyperbolicFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The hyperbolic tangent function, :math:`\\frac{cosh(x)}{sinh(x)}`.</span>
-
-<span class="sd">    * coth(x) -&gt; Returns the hyperbolic cotangent of x</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">sinh</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">**</span><span class="mi">2</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-<div class="viewcode-block" id="coth.inverse"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.coth.inverse">[docs]</a>    <span class="k">def</span> <span class="nf">inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the inverse of this function.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">acoth</span>
-</div>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-
-            <span class="n">i_coeff</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">i_coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">i_coeff</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">cot</span><span class="p">(</span><span class="o">-</span><span class="n">i_coeff</span><span class="p">)</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">cot</span><span class="p">(</span><span class="n">i_coeff</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">asinh</span><span class="p">:</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">x</span>
-
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">acosh</span><span class="p">:</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="k">return</span> <span class="n">x</span><span class="o">/</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">atanh</span><span class="p">:</span>
-                <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">acoth</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-            <span class="n">B</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">bernoulli</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="n">F</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-            <span class="k">return</span> <span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">B</span><span class="o">/</span><span class="n">F</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="n">n</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">conjugate</span><span class="p">())</span>
-
-    <span class="k">def</span> <span class="nf">as_real_imag</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-                <span class="n">hints</span><span class="p">[</span><span class="s">&#39;complex&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-        <span class="n">denom</span> <span class="o">=</span> <span class="n">sinh</span><span class="p">(</span><span class="n">re</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">C</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">im</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">sinh</span><span class="p">(</span><span class="n">re</span><span class="p">)</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="n">re</span><span class="p">)</span><span class="o">/</span><span class="n">denom</span><span class="p">,</span> <span class="o">-</span><span class="n">C</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">im</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">im</span><span class="p">)</span><span class="o">/</span><span class="n">denom</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_exp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">neg_exp</span><span class="p">,</span> <span class="n">pos_exp</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">),</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">pos_exp</span> <span class="o">+</span> <span class="n">neg_exp</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">pos_exp</span> <span class="o">-</span> <span class="n">neg_exp</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_sinh</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">arg</span><span class="p">)</span><span class="o">/</span><span class="n">sinh</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_cosh</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span><span class="o">/</span><span class="n">cosh</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_tanh</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="n">tanh</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">coth</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">())</span>
-
-
-<span class="c">###############################################################################</span>
-<span class="c">############################# HYPERBOLIC INVERSES #############################</span>
-<span class="c">###############################################################################</span>
-</div>
-<div class="viewcode-block" id="asinh"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.asinh">[docs]</a><span class="k">class</span> <span class="nc">asinh</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The inverse hyperbolic sine function.</span>
-
-<span class="sd">    * asinh(x) -&gt; Returns the inverse hyperbolic sine of x</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    acosh, atanh, sinh</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span>
-
-            <span class="n">i_coeff</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">i_coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">asin</span><span class="p">(</span><span class="n">i_coeff</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">previous_terms</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="n">previous_terms</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">p</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">k</span> <span class="o">=</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>
-                <span class="n">R</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-                <span class="n">F</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-                <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">k</span> <span class="o">*</span> <span class="n">R</span> <span class="o">/</span> <span class="n">F</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="n">n</span> <span class="o">/</span> <span class="n">n</span>
-
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">arg</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">asinh</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">())</span>
-
-</div>
-<div class="viewcode-block" id="acosh"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.acosh">[docs]</a><span class="k">class</span> <span class="nc">acosh</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The inverse hyperbolic cosine function.</span>
-
-<span class="sd">    * acosh(x) -&gt; Returns the inverse hyperbolic cosine of x</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    asinh, atanh, cosh</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">/</span> <span class="mi">2</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="n">cst_table</span> <span class="o">=</span> <span class="p">{</span>
-                <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">:</span> <span class="n">C</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))),</span>
-                <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">:</span> <span class="n">C</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))),</span>
-                <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">:</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">3</span><span class="p">,</span>
-                <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">:</span> <span class="mi">2</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">3</span><span class="p">,</span>
-                <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span>
-                <span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span> <span class="mi">3</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span>
-                <span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">):</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span>
-                <span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">):</span> <span class="mi">3</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span>
-                <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">6</span><span class="p">,</span>
-                <span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span> <span class="mi">5</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">6</span><span class="p">,</span>
-                <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="mi">3</span><span class="p">):</span> <span class="mi">5</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">12</span><span class="p">,</span>
-                <span class="o">-</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="mi">3</span><span class="p">):</span> <span class="mi">7</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">12</span><span class="p">,</span>
-                <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">8</span><span class="p">,</span>
-                <span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span> <span class="mi">7</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">8</span><span class="p">,</span>
-                <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span> <span class="mi">3</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">8</span><span class="p">,</span>
-                <span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span> <span class="mi">5</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">8</span><span class="p">,</span>
-                <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)):</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">12</span><span class="p">,</span>
-                <span class="o">-</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)):</span> <span class="mi">11</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">12</span><span class="p">,</span>
-                <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">:</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">5</span><span class="p">,</span>
-                <span class="o">-</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">:</span> <span class="mi">4</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">5</span>
-            <span class="p">}</span>
-
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">cst_table</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">cst_table</span><span class="p">[</span><span class="n">arg</span><span class="p">]</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-                <span class="k">return</span> <span class="n">cst_table</span><span class="p">[</span><span class="n">arg</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-
-        <span class="n">i_coeff</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">i_coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">i_coeff</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">acos</span><span class="p">(</span><span class="n">i_coeff</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">acos</span><span class="p">(</span><span class="o">-</span><span class="n">i_coeff</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">/</span> <span class="mi">2</span>
-        <span class="k">elif</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">previous_terms</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="n">previous_terms</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span>
-                <span class="k">return</span> <span class="n">p</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">k</span> <span class="o">=</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>
-                <span class="n">R</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-                <span class="n">F</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">R</span> <span class="o">/</span> <span class="n">F</span> <span class="o">*</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="n">n</span> <span class="o">/</span> <span class="n">n</span>
-
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">arg</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">acosh</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">())</span>
-
-</div>
-<div class="viewcode-block" id="atanh"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.atanh">[docs]</a><span class="k">class</span> <span class="nc">atanh</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The inverse hyperbolic tangent function.</span>
-
-<span class="sd">    * atanh(x) -&gt; Returns the inverse hyperbolic tangent of x</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    asinh, acosh, tanh</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">atan</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">atan</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-
-            <span class="n">i_coeff</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">i_coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">atan</span><span class="p">(</span><span class="n">i_coeff</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="n">n</span> <span class="o">/</span> <span class="n">n</span>
-
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">arg</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">atanh</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">())</span>
-
-</div>
-<div class="viewcode-block" id="acoth"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.acoth">[docs]</a><span class="k">class</span> <span class="nc">acoth</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The inverse hyperbolic cotangent function.</span>
-
-<span class="sd">    * acoth(x) -&gt; Returns the inverse hyperbolic cotangent of x</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">/</span> <span class="mi">2</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-            <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="mi">0</span>
-
-            <span class="n">i_coeff</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">i_coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">acot</span><span class="p">(</span><span class="n">i_coeff</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">/</span> <span class="mi">2</span>
-        <span class="k">elif</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="n">n</span> <span class="o">/</span> <span class="n">n</span>
-
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">arg</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">acoth</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">())</span></div>
-</pre></div>
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diff --git a/dev-py3k/_modules/sympy/functions/elementary/integers.html b/dev-py3k/_modules/sympy/functions/elementary/integers.html
deleted file mode 100644
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-            
-  <h1>Source code for sympy.functions.elementary.integers</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core.basic</span> <span class="kn">import</span> <span class="n">C</span>
-<span class="kn">from</span> <span class="nn">sympy.core.singleton</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Function</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Add</span>
-<span class="kn">from</span> <span class="nn">sympy.core.evalf</span> <span class="kn">import</span> <span class="n">get_integer_part</span><span class="p">,</span> <span class="n">PrecisionExhausted</span>
-
-<span class="c">###############################################################################</span>
-<span class="c">######################### FLOOR and CEILING FUNCTIONS #########################</span>
-<span class="c">###############################################################################</span>
-
-
-<div class="viewcode-block" id="RoundFunction"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.integers.RoundFunction">[docs]</a><span class="k">class</span> <span class="nc">RoundFunction</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The base class for rounding functions.&quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">arg</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">im</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-
-        <span class="n">v</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_eval_number</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">v</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">v</span>
-
-        <span class="c"># Integral, numerical, symbolic part</span>
-        <span class="n">ipart</span> <span class="o">=</span> <span class="n">npart</span> <span class="o">=</span> <span class="n">spart</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-        <span class="c"># Extract integral (or complex integral) terms</span>
-        <span class="n">terms</span> <span class="o">=</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">or</span> <span class="p">(</span><span class="n">t</span><span class="o">.</span><span class="n">is_imaginary</span> <span class="ow">and</span> <span class="n">C</span><span class="o">.</span><span class="n">im</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">.</span><span class="n">is_integer</span><span class="p">):</span>
-                <span class="n">ipart</span> <span class="o">+=</span> <span class="n">t</span>
-            <span class="k">elif</span> <span class="n">t</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">):</span>
-                <span class="n">spart</span> <span class="o">+=</span> <span class="n">t</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">npart</span> <span class="o">+=</span> <span class="n">t</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">npart</span> <span class="ow">or</span> <span class="n">spart</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">ipart</span>
-
-        <span class="c"># Evaluate npart numerically if independent of spart</span>
-        <span class="k">if</span> <span class="n">npart</span> <span class="ow">and</span> <span class="p">(</span>
-            <span class="ow">not</span> <span class="n">spart</span> <span class="ow">or</span>
-            <span class="n">npart</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span> <span class="n">spart</span><span class="o">.</span><span class="n">is_imaginary</span> <span class="ow">or</span>
-                <span class="n">npart</span><span class="o">.</span><span class="n">is_imaginary</span> <span class="ow">and</span> <span class="n">spart</span><span class="o">.</span><span class="n">is_real</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="n">get_integer_part</span><span class="p">(</span>
-                    <span class="n">npart</span><span class="p">,</span> <span class="n">cls</span><span class="o">.</span><span class="n">_dir</span><span class="p">,</span> <span class="p">{},</span> <span class="n">return_ints</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                <span class="n">ipart</span> <span class="o">+=</span> <span class="n">C</span><span class="o">.</span><span class="n">Integer</span><span class="p">(</span><span class="n">re</span><span class="p">)</span> <span class="o">+</span> <span class="n">C</span><span class="o">.</span><span class="n">Integer</span><span class="p">(</span><span class="n">im</span><span class="p">)</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-                <span class="n">npart</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">except</span> <span class="p">(</span><span class="n">PrecisionExhausted</span><span class="p">,</span> <span class="ne">NotImplementedError</span><span class="p">):</span>
-                <span class="k">pass</span>
-
-        <span class="n">spart</span> <span class="o">=</span> <span class="n">npart</span> <span class="o">+</span> <span class="n">spart</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">spart</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">ipart</span>
-        <span class="k">elif</span> <span class="n">spart</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">ipart</span> <span class="o">+</span> <span class="n">cls</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">im</span><span class="p">(</span><span class="n">spart</span><span class="p">),</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">ipart</span> <span class="o">+</span> <span class="n">cls</span><span class="p">(</span><span class="n">spart</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_bounded</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_bounded</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_integer</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span>
-
-</div>
-<div class="viewcode-block" id="floor"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.integers.floor">[docs]</a><span class="k">class</span> <span class="nc">floor</span><span class="p">(</span><span class="n">RoundFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Floor is a univariate function which returns the largest integer</span>
-<span class="sd">    value not greater than its argument. However this implementation</span>
-<span class="sd">    generalizes floor to complex numbers.</span>
-
-<span class="sd">    More information can be found in &quot;Concrete mathematics&quot; by Graham,</span>
-<span class="sd">    pp. 87 or visit http://mathworld.wolfram.com/FloorFunction.html.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import floor, E, I, Float, Rational</span>
-<span class="sd">        &gt;&gt;&gt; floor(17)</span>
-<span class="sd">        17</span>
-<span class="sd">        &gt;&gt;&gt; floor(Rational(23, 10))</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; floor(2*E)</span>
-<span class="sd">        5</span>
-<span class="sd">        &gt;&gt;&gt; floor(-Float(0.567))</span>
-<span class="sd">        -1</span>
-<span class="sd">        &gt;&gt;&gt; floor(-I/2)</span>
-<span class="sd">        -I</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    ceiling</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">_dir</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_number</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Integer</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">p</span> <span class="o">//</span> <span class="n">arg</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Integer</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">floor</span><span class="p">()))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">arg</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_NumberSymbol</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">arg</span><span class="o">.</span><span class="n">approximation_interval</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Integer</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">_eval_nseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">args0</span> <span class="o">=</span> <span class="n">args</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">args0</span> <span class="o">==</span> <span class="n">r</span><span class="p">:</span>
-            <span class="n">direction</span> <span class="o">=</span> <span class="p">(</span><span class="n">args</span> <span class="o">-</span> <span class="n">args0</span><span class="p">)</span><span class="o">.</span><span class="n">leadterm</span><span class="p">(</span><span class="n">x</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">direction</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">r</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">r</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">r</span>
-
-</div>
-<div class="viewcode-block" id="ceiling"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.integers.ceiling">[docs]</a><span class="k">class</span> <span class="nc">ceiling</span><span class="p">(</span><span class="n">RoundFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Ceiling is a univariate function which returns the smallest integer</span>
-<span class="sd">    value not less than its argument. Ceiling function is generalized</span>
-<span class="sd">    in this implementation to complex numbers.</span>
-
-<span class="sd">    More information can be found in &quot;Concrete mathematics&quot; by Graham,</span>
-<span class="sd">    pp. 87 or visit http://mathworld.wolfram.com/CeilingFunction.html.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import ceiling, E, I, Float, Rational</span>
-<span class="sd">        &gt;&gt;&gt; ceiling(17)</span>
-<span class="sd">        17</span>
-<span class="sd">        &gt;&gt;&gt; ceiling(Rational(23, 10))</span>
-<span class="sd">        3</span>
-<span class="sd">        &gt;&gt;&gt; ceiling(2*E)</span>
-<span class="sd">        6</span>
-<span class="sd">        &gt;&gt;&gt; ceiling(-Float(0.567))</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; ceiling(I/2)</span>
-<span class="sd">        I</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    floor</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">_dir</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_number</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">C</span><span class="o">.</span><span class="n">Integer</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="o">.</span><span class="n">p</span> <span class="o">//</span> <span class="n">arg</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Integer</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">ceiling</span><span class="p">()))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">arg</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_NumberSymbol</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">arg</span><span class="o">.</span><span class="n">approximation_interval</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Integer</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">_eval_nseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">args0</span> <span class="o">=</span> <span class="n">args</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">args0</span> <span class="o">==</span> <span class="n">r</span><span class="p">:</span>
-            <span class="n">direction</span> <span class="o">=</span> <span class="p">(</span><span class="n">args</span> <span class="o">-</span> <span class="n">args0</span><span class="p">)</span><span class="o">.</span><span class="n">leadterm</span><span class="p">(</span><span class="n">x</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">direction</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">r</span> <span class="o">+</span> <span class="mi">1</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">r</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">r</span></div>
-</pre></div>
-
-          </div>
-        </div>
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diff --git a/dev-py3k/_modules/sympy/functions/elementary/miscellaneous.html b/dev-py3k/_modules/sympy/functions/elementary/miscellaneous.html
deleted file mode 100644
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+++ /dev/null
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-            
-  <h1>Source code for sympy.functions.elementary.miscellaneous</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.core.basic</span> <span class="kn">import</span> <span class="n">Basic</span>
-<span class="kn">from</span> <span class="nn">sympy.core.containers</span> <span class="kn">import</span> <span class="n">Tuple</span>
-<span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">Rational</span>
-<span class="kn">from</span> <span class="nn">sympy.core.operations</span> <span class="kn">import</span> <span class="n">LatticeOp</span><span class="p">,</span> <span class="n">ShortCircuit</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Application</span><span class="p">,</span> <span class="n">Lambda</span>
-<span class="kn">from</span> <span class="nn">sympy.core.expr</span> <span class="kn">import</span> <span class="n">Expr</span>
-<span class="kn">from</span> <span class="nn">sympy.core.singleton</span> <span class="kn">import</span> <span class="n">Singleton</span>
-<span class="kn">from</span> <span class="nn">sympy.core.rules</span> <span class="kn">import</span> <span class="n">Transform</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">as_int</span>
-
-
-<div class="viewcode-block" id="IdentityFunction"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.miscellaneous.IdentityFunction">[docs]</a><span class="k">class</span> <span class="nc">IdentityFunction</span><span class="p">(</span><span class="n">Lambda</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">Singleton</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The identity function</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Id, Symbol</span>
-<span class="sd">    &gt;&gt;&gt; x = Symbol(&#39;x&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; Id(x)</span>
-<span class="sd">    x</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-        <span class="c">#construct &quot;by hand&quot; to avoid infinite loop</span>
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span></div>
-<span class="n">Id</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">IdentityFunction</span>
-
-<span class="c">###############################################################################</span>
-<span class="c">############################# ROOT and SQUARE ROOT FUNCTION ###################</span>
-<span class="c">###############################################################################</span>
-
-
-<div class="viewcode-block" id="sqrt"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.miscellaneous.sqrt">[docs]</a><span class="k">def</span> <span class="nf">sqrt</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The square root function</span>
-
-<span class="sd">    sqrt(x) -&gt; Returns the principal square root of x.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import sqrt, Symbol</span>
-<span class="sd">    &gt;&gt;&gt; x = Symbol(&#39;x&#39;)</span>
-
-<span class="sd">    &gt;&gt;&gt; sqrt(x)</span>
-<span class="sd">    sqrt(x)</span>
-
-<span class="sd">    &gt;&gt;&gt; sqrt(x)**2</span>
-<span class="sd">    x</span>
-
-<span class="sd">    Note that sqrt(x**2) does not simplify to x.</span>
-
-<span class="sd">    &gt;&gt;&gt; sqrt(x**2)</span>
-<span class="sd">    sqrt(x**2)</span>
-
-<span class="sd">    This is because the two are not equal to each other in general.</span>
-<span class="sd">    For example, consider x == -1:</span>
-
-<span class="sd">    &gt;&gt;&gt; sqrt(x**2).subs(x, -1)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; x.subs(x, -1)</span>
-<span class="sd">    -1</span>
-
-<span class="sd">    This is because sqrt computes the principal square root, so the square may</span>
-<span class="sd">    put the argument in a different branch.  This identity does hold if x is</span>
-<span class="sd">    positive:</span>
-
-<span class="sd">    &gt;&gt;&gt; y = Symbol(&#39;y&#39;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; sqrt(y**2)</span>
-<span class="sd">    y</span>
-
-<span class="sd">    You can force this simplification by using the powdenest() function with</span>
-<span class="sd">    the force option set to True:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import powdenest</span>
-<span class="sd">    &gt;&gt;&gt; sqrt(x**2)</span>
-<span class="sd">    sqrt(x**2)</span>
-<span class="sd">    &gt;&gt;&gt; powdenest(sqrt(x**2), force=True)</span>
-<span class="sd">    x</span>
-
-<span class="sd">    To get both branches of the square root you can use the RootOf function:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import RootOf</span>
-
-<span class="sd">    &gt;&gt;&gt; [ RootOf(x**2-3,i) for i in (0,1) ]</span>
-<span class="sd">    [-sqrt(3), sqrt(3)]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.polys.rootoftools.RootOf, root</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    * http://en.wikipedia.org/wiki/Square_root</span>
-<span class="sd">    * http://en.wikipedia.org/wiki/Principal_value</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># arg = sympify(arg) is handled by Pow</span>
-    <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Pow</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="root"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.miscellaneous.root">[docs]</a><span class="k">def</span> <span class="nf">root</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The n-th root function (a shortcut for arg**(1/n))</span>
-
-<span class="sd">    root(x, n) -&gt; Returns the principal n-th root of x.</span>
-
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import root, Rational</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, n</span>
-
-<span class="sd">    &gt;&gt;&gt; root(x, 2)</span>
-<span class="sd">    sqrt(x)</span>
-
-<span class="sd">    &gt;&gt;&gt; root(x, 3)</span>
-<span class="sd">    x**(1/3)</span>
-
-<span class="sd">    &gt;&gt;&gt; root(x, n)</span>
-<span class="sd">    x**(1/n)</span>
-
-<span class="sd">    &gt;&gt;&gt; root(x, -Rational(2, 3))</span>
-<span class="sd">    x**(-3/2)</span>
-
-
-<span class="sd">    To get all n n-th roots you can use the RootOf function.</span>
-<span class="sd">    The following examples show the roots of unity for n</span>
-<span class="sd">    equal 2, 3 and 4:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import RootOf, I</span>
-
-<span class="sd">    &gt;&gt;&gt; [ RootOf(x**2-1,i) for i in (0,1) ]</span>
-<span class="sd">    [-1, 1]</span>
-
-<span class="sd">    &gt;&gt;&gt; [ RootOf(x**3-1,i) for i in (0,1,2) ]</span>
-<span class="sd">    [1, -1/2 - sqrt(3)*I/2, -1/2 + sqrt(3)*I/2]</span>
-
-<span class="sd">    &gt;&gt;&gt; [ RootOf(x**4-1,i) for i in (0,1,2,3) ]</span>
-<span class="sd">    [-1, 1, -I, I]</span>
-
-<span class="sd">    SymPy, like other symbolic algebra systems, returns the</span>
-<span class="sd">    complex root of negative numbers. This is the principal</span>
-<span class="sd">    root and differs from the text-book result that one might</span>
-<span class="sd">    be expecting. For example, the cube root of -8 does not</span>
-<span class="sd">    come back as -2:</span>
-
-<span class="sd">    &gt;&gt;&gt; root(-8, 3)</span>
-<span class="sd">    2*(-1)**(1/3)</span>
-
-<span class="sd">    The real_root function can be used to either make such a result</span>
-<span class="sd">    real or simply return the real root in the first place:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import real_root</span>
-<span class="sd">    &gt;&gt;&gt; real_root(_)</span>
-<span class="sd">    -2</span>
-<span class="sd">    &gt;&gt;&gt; real_root(-32, 5)</span>
-<span class="sd">    -2</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.polys.rootoftools.RootOf</span>
-<span class="sd">    sympy.core.power.integer_nthroot</span>
-<span class="sd">    sqrt, real_root</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    * http://en.wikipedia.org/wiki/Square_root</span>
-<span class="sd">    * http://en.wikipedia.org/wiki/real_root</span>
-<span class="sd">    * http://en.wikipedia.org/wiki/Root_of_unity</span>
-<span class="sd">    * http://en.wikipedia.org/wiki/Principal_value</span>
-<span class="sd">    * http://mathworld.wolfram.com/CubeRoot.html</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Pow</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">n</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">real_root</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return the real nth-root of arg if possible. If n is omitted then</span>
-<span class="sd">    all instances of -1**(1/odd) will be changed to -1.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import root, real_root, Rational</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, n</span>
-
-<span class="sd">    &gt;&gt;&gt; real_root(-8, 3)</span>
-<span class="sd">    -2</span>
-<span class="sd">    &gt;&gt;&gt; root(-8, 3)</span>
-<span class="sd">    2*(-1)**(1/3)</span>
-<span class="sd">    &gt;&gt;&gt; real_root(_)</span>
-<span class="sd">    -2</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.polys.rootoftools.RootOf</span>
-<span class="sd">    sympy.core.power.integer_nthroot</span>
-<span class="sd">    root, sqrt</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Pow</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">rv</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-    <span class="n">n1pow</span> <span class="o">=</span> <span class="n">Transform</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">,</span>
-                      <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span>
-                      <span class="n">x</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span>
-                      <span class="n">x</span><span class="o">.</span><span class="n">base</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span> <span class="ow">and</span>
-                      <span class="n">x</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span>
-                      <span class="n">x</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">p</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">x</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">q</span> <span class="o">%</span> <span class="mi">2</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">rv</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="n">n1pow</span><span class="p">)</span>
-
-<span class="c">###############################################################################</span>
-<span class="c">############################# MINIMUM and MAXIMUM #############################</span>
-<span class="c">###############################################################################</span>
-
-
-<span class="k">class</span> <span class="nc">MinMaxBase</span><span class="p">(</span><span class="n">Expr</span><span class="p">,</span> <span class="n">LatticeOp</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;The Max/Min functions must have arguments.&quot;</span><span class="p">)</span>
-
-        <span class="n">args</span> <span class="o">=</span> <span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">)</span>
-
-        <span class="c"># first standard filter, for cls.zero and cls.identity</span>
-        <span class="c"># also reshape Max(a, Max(b, c)) to Max(a, b, c)</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">_args</span> <span class="o">=</span> <span class="nb">frozenset</span><span class="p">(</span><span class="n">cls</span><span class="o">.</span><span class="n">_new_args_filter</span><span class="p">(</span><span class="n">args</span><span class="p">))</span>
-        <span class="k">except</span> <span class="n">ShortCircuit</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">zero</span>
-
-        <span class="c"># second filter</span>
-        <span class="c"># variant I: remove ones which can be removed</span>
-        <span class="c"># args = cls._collapse_arguments(set(_args), **assumptions)</span>
-
-        <span class="c"># variant II: find local zeros</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_find_localzeros</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">_args</span><span class="p">),</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">)</span>
-
-        <span class="n">_args</span> <span class="o">=</span> <span class="nb">frozenset</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">_args</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">identity</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">_args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">set</span><span class="p">(</span><span class="n">_args</span><span class="p">)</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># base creation</span>
-            <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">_args</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">)</span>
-            <span class="n">obj</span><span class="o">.</span><span class="n">_argset</span> <span class="o">=</span> <span class="n">_args</span>
-            <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_new_args_filter</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg_sequence</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Generator filtering args.</span>
-
-<span class="sd">        first standard filter, for cls.zero and cls.identity.</span>
-<span class="sd">        Also reshape Max(a, Max(b, c)) to Max(a, b, c),</span>
-<span class="sd">        and check arguments for comparability</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">arg_sequence</span><span class="p">:</span>
-
-            <span class="c"># pre-filter, checking comparability of arguments</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;The argument &#39;</span><span class="si">%s</span><span class="s">&#39; is not comparable.&quot;</span> <span class="o">%</span> <span class="n">arg</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">arg</span> <span class="o">==</span> <span class="n">cls</span><span class="o">.</span><span class="n">zero</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">ShortCircuit</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="o">==</span> <span class="n">cls</span><span class="o">.</span><span class="n">identity</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">cls</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">iter_basic_args</span><span class="p">():</span>
-                    <span class="k">yield</span> <span class="n">x</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">yield</span> <span class="n">arg</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_find_localzeros</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">values</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Sequentially allocate values to localzeros.</span>
-
-<span class="sd">        When a value is identified as being more extreme than another member it</span>
-<span class="sd">        replaces that member; if this is never true, then the value is simply</span>
-<span class="sd">        appended to the localzeros.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">localzeros</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">values</span><span class="p">:</span>
-            <span class="n">is_newzero</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="n">localzeros_</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">localzeros</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="n">localzeros_</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">id</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="o">==</span> <span class="nb">id</span><span class="p">(</span><span class="n">z</span><span class="p">):</span>
-                    <span class="n">is_newzero</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">elif</span> <span class="n">cls</span><span class="o">.</span><span class="n">_is_connected</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-                    <span class="n">is_newzero</span> <span class="o">=</span> <span class="bp">False</span>
-                    <span class="k">if</span> <span class="n">cls</span><span class="o">.</span><span class="n">_is_asneeded</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-                        <span class="n">localzeros</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-                        <span class="n">localzeros</span><span class="o">.</span><span class="n">update</span><span class="p">([</span><span class="n">v</span><span class="p">])</span>
-            <span class="k">if</span> <span class="n">is_newzero</span><span class="p">:</span>
-                <span class="n">localzeros</span><span class="o">.</span><span class="n">update</span><span class="p">([</span><span class="n">v</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">localzeros</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_is_connected</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Check if x and y are connected somehow.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">x</span> <span class="o">==</span> <span class="n">y</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span> <span class="o">&gt;</span> <span class="n">y</span><span class="p">,</span> <span class="nb">bool</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span> <span class="o">&lt;</span> <span class="n">y</span><span class="p">,</span> <span class="nb">bool</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">and</span> <span class="n">y</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_is_asneeded</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Check if x and y satisfy relation condition.</span>
-
-<span class="sd">        The relation condition for Max function is x &gt; y,</span>
-<span class="sd">        for Min function is x &lt; y. They are defined in children Max and Min</span>
-<span class="sd">        classes through the method _rel(cls, x, y)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">x</span> <span class="o">==</span> <span class="n">y</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">and</span> <span class="n">y</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">cls</span><span class="o">.</span><span class="n">_rel</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">True</span>
-        <span class="n">xy</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_rel</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">xy</span><span class="p">,</span> <span class="nb">bool</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">xy</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="n">yx</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_rel_inversed</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">yx</span><span class="p">,</span> <span class="nb">bool</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">yx</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>  <span class="c"># never occurs?</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-
-<div class="viewcode-block" id="Max"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.miscellaneous.Max">[docs]</a><span class="k">class</span> <span class="nc">Max</span><span class="p">(</span><span class="n">MinMaxBase</span><span class="p">,</span> <span class="n">Application</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return, if possible, the maximum value of the list.</span>
-
-<span class="sd">    When number of arguments is equal one, then</span>
-<span class="sd">    return this argument.</span>
-
-<span class="sd">    When number of arguments is equal two, then</span>
-<span class="sd">    return, if possible, the value from (a, b) that is &gt;= the other.</span>
-
-<span class="sd">    In common case, when the length of list greater than 2, the task</span>
-<span class="sd">    is more complicated. Return only the arguments, which are greater</span>
-<span class="sd">    than others, if it is possible to determine directional relation.</span>
-
-<span class="sd">    If is not possible to determine such a relation, return a partially</span>
-<span class="sd">    evaluated result.</span>
-
-<span class="sd">    Assumptions are used to make the decision too.</span>
-
-<span class="sd">    Also, only comparable arguments are permitted.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Max, Symbol, oo</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; p = Symbol(&#39;p&#39;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; n = Symbol(&#39;n&#39;, negative=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; Max(x, -2)                  #doctest: +SKIP</span>
-<span class="sd">    Max(x, -2)</span>
-
-<span class="sd">    &gt;&gt;&gt; Max(x, -2).subs(x, 3)</span>
-<span class="sd">    3</span>
-
-<span class="sd">    &gt;&gt;&gt; Max(p, -2)</span>
-<span class="sd">    p</span>
-
-<span class="sd">    &gt;&gt;&gt; Max(x, y)                   #doctest: +SKIP</span>
-<span class="sd">    Max(x, y)</span>
-
-<span class="sd">    &gt;&gt;&gt; Max(x, y) == Max(y, x)</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &gt;&gt;&gt; Max(x, Max(y, z))           #doctest: +SKIP</span>
-<span class="sd">    Max(x, y, z)</span>
-
-<span class="sd">    &gt;&gt;&gt; Max(n, 8, p, 7, -oo)        #doctest: +SKIP</span>
-<span class="sd">    Max(8, p)</span>
-
-<span class="sd">    &gt;&gt;&gt; Max (1, x, oo)</span>
-<span class="sd">    oo</span>
-
-<span class="sd">    Algorithm</span>
-
-<span class="sd">    The task can be considered as searching of supremums in the</span>
-<span class="sd">    directed complete partial orders [1]_.</span>
-
-<span class="sd">    The source values are sequentially allocated by the isolated subsets</span>
-<span class="sd">    in which supremums are searched and result as Max arguments.</span>
-
-<span class="sd">    If the resulted supremum is single, then it is returned.</span>
-
-<span class="sd">    The isolated subsets are the sets of values which are only the comparable</span>
-<span class="sd">    with each other in the current set. E.g. natural numbers are comparable with</span>
-<span class="sd">    each other, but not comparable with the `x` symbol. Another example: the</span>
-<span class="sd">    symbol `x` with negative assumption is comparable with a natural number.</span>
-
-<span class="sd">    Also there are &quot;least&quot; elements, which are comparable with all others,</span>
-<span class="sd">    and have a zero property (maximum or minimum for all elements). E.g. `oo`.</span>
-<span class="sd">    In case of it the allocation operation is terminated and only this value is</span>
-<span class="sd">    returned.</span>
-
-<span class="sd">    Assumption:</span>
-<span class="sd">       - if A &gt; B &gt; C then A &gt; C</span>
-<span class="sd">       - if A==B then B can be removed</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Directed_complete_partial_order</span>
-<span class="sd">    .. [2] http://en.wikipedia.org/wiki/Lattice_%28order%29</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Min : find minimum values</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">zero</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-    <span class="n">identity</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_rel</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Check if x &gt; y.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">x</span> <span class="o">&gt;</span> <span class="n">y</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_rel_inversed</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Check if x &lt; y.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">x</span> <span class="o">&lt;</span> <span class="n">y</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="Min"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.miscellaneous.Min">[docs]</a><span class="k">class</span> <span class="nc">Min</span><span class="p">(</span><span class="n">MinMaxBase</span><span class="p">,</span> <span class="n">Application</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return, if possible, the minimum value of the list.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Min, Symbol, oo</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; p = Symbol(&#39;p&#39;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; n = Symbol(&#39;n&#39;, negative=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; Min(x, -2)                  #doctest: +SKIP</span>
-<span class="sd">    Min(x, -2)</span>
-
-<span class="sd">    &gt;&gt;&gt; Min(x, -2).subs(x, 3)</span>
-<span class="sd">    -2</span>
-
-<span class="sd">    &gt;&gt;&gt; Min(p, -3)</span>
-<span class="sd">    -3</span>
-
-<span class="sd">    &gt;&gt;&gt; Min(x, y)                   #doctest: +SKIP</span>
-<span class="sd">    Min(x, y)</span>
-
-<span class="sd">    &gt;&gt;&gt; Min(n, 8, p, -7, p, oo)     #doctest: +SKIP</span>
-<span class="sd">    Min(n, -7)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Max : find maximum values</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">zero</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-    <span class="n">identity</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_rel</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Check if x &lt; y.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">x</span> <span class="o">&lt;</span> <span class="n">y</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_rel_inversed</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Check if x &gt; y.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">x</span> <span class="o">&gt;</span> <span class="n">y</span><span class="p">)</span></div>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/functions/elementary/piecewise.html b/dev-py3k/_modules/sympy/functions/elementary/piecewise.html
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-            
-  <h1>Source code for sympy.functions.elementary.piecewise</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">Function</span><span class="p">,</span> <span class="n">diff</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">,</span> <span class="n">Expr</span>
-<span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span><span class="p">,</span> <span class="n">Relational</span>
-<span class="kn">from</span> <span class="nn">sympy.core.symbol</span> <span class="kn">import</span> <span class="n">Dummy</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">Max</span><span class="p">,</span> <span class="n">Min</span>
-<span class="kn">from</span> <span class="nn">sympy.logic.boolalg</span> <span class="kn">import</span> <span class="n">And</span><span class="p">,</span> <span class="n">Boolean</span><span class="p">,</span> <span class="n">Or</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">default_sort_key</span>
-
-
-<div class="viewcode-block" id="ExprCondPair"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.piecewise.ExprCondPair">[docs]</a><span class="k">class</span> <span class="nc">ExprCondPair</span><span class="p">(</span><span class="n">Tuple</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Represents an expression, condition pair.&quot;&quot;&quot;</span>
-
-    <span class="n">true_sentinel</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;True&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">cond</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">cond</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">cond</span> <span class="o">=</span> <span class="n">ExprCondPair</span><span class="o">.</span><span class="n">true_sentinel</span>
-        <span class="k">return</span> <span class="n">Tuple</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">cond</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="ExprCondPair.expr"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.piecewise.ExprCondPair.expr">[docs]</a>    <span class="k">def</span> <span class="nf">expr</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the expression of this pair.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="ExprCondPair.cond"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.piecewise.ExprCondPair.cond">[docs]</a>    <span class="k">def</span> <span class="nf">cond</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the condition of this pair.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">ExprCondPair</span><span class="o">.</span><span class="n">true_sentinel</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="ExprCondPair.free_symbols"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.piecewise.ExprCondPair.free_symbols">[docs]</a>    <span class="k">def</span> <span class="nf">free_symbols</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the free symbols of this pair.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># Overload Basic.free_symbols because self.args[1] may contain non-Basic</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">free_symbols</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cond</span><span class="p">,</span> <span class="s">&#39;free_symbols&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">|=</span> <span class="bp">self</span><span class="o">.</span><span class="n">cond</span><span class="o">.</span><span class="n">free_symbols</span>
-        <span class="k">return</span> <span class="n">result</span>
-</div>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_commutative</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">is_commutative</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">yield</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span>
-        <span class="k">yield</span> <span class="bp">self</span><span class="o">.</span><span class="n">cond</span>
-
-</div>
-<div class="viewcode-block" id="Piecewise"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.piecewise.Piecewise">[docs]</a><span class="k">class</span> <span class="nc">Piecewise</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Represents a piecewise function.</span>
-
-<span class="sd">    Usage:</span>
-
-<span class="sd">      Piecewise( (expr,cond), (expr,cond), ... )</span>
-<span class="sd">        - Each argument is a 2-tuple defining a expression and condition</span>
-<span class="sd">        - The conds are evaluated in turn returning the first that is True.</span>
-<span class="sd">          If any of the evaluated conds are not determined explicitly False,</span>
-<span class="sd">          e.g. x &lt; 1, the function is returned in symbolic form.</span>
-<span class="sd">        - If the function is evaluated at a place where all conditions are False,</span>
-<span class="sd">          a ValueError exception will be raised.</span>
-<span class="sd">        - Pairs where the cond is explicitly False, will be removed.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">      &gt;&gt;&gt; from sympy import Piecewise, log</span>
-<span class="sd">      &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">      &gt;&gt;&gt; f = x**2</span>
-<span class="sd">      &gt;&gt;&gt; g = log(x)</span>
-<span class="sd">      &gt;&gt;&gt; p = Piecewise( (0, x&lt;-1), (f, x&lt;=1), (g, True))</span>
-<span class="sd">      &gt;&gt;&gt; p.subs(x,1)</span>
-<span class="sd">      1</span>
-<span class="sd">      &gt;&gt;&gt; p.subs(x,5)</span>
-<span class="sd">      log(5)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    piecewise_fold</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">is_Piecewise</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="c"># (Try to) sympify args first</span>
-        <span class="n">newargs</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">ec</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="n">pair</span> <span class="o">=</span> <span class="n">ExprCondPair</span><span class="p">(</span><span class="o">*</span><span class="n">ec</span><span class="p">)</span>
-            <span class="n">cond</span> <span class="o">=</span> <span class="n">pair</span><span class="o">.</span><span class="n">cond</span>
-            <span class="k">if</span> <span class="n">cond</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">cond</span><span class="p">,</span> <span class="p">(</span><span class="nb">bool</span><span class="p">,</span> <span class="n">Relational</span><span class="p">,</span> <span class="n">Boolean</span><span class="p">)):</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span>
-                    <span class="s">&quot;Cond </span><span class="si">%s</span><span class="s"> is of type </span><span class="si">%s</span><span class="s">, but must be a Relational,&quot;</span>
-                    <span class="s">&quot; Boolean, or a built-in bool.&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">cond</span><span class="p">,</span> <span class="nb">type</span><span class="p">(</span><span class="n">cond</span><span class="p">)))</span>
-            <span class="n">newargs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">pair</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">cond</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                <span class="k">break</span>
-
-        <span class="k">if</span> <span class="n">options</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;evaluate&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="o">*</span><span class="n">newargs</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="bp">None</span>
-
-        <span class="k">if</span> <span class="n">r</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">newargs</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">r</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="c"># Check for situations where we can evaluate the Piecewise object.</span>
-        <span class="c"># 1) Hit an unevaluable cond (e.g. x&lt;1) -&gt; keep object</span>
-        <span class="c"># 2) Hit a true condition -&gt; return that expr</span>
-        <span class="c"># 3) Remove false conditions, if no conditions left -&gt; raise ValueError</span>
-        <span class="n">all_conds_evaled</span> <span class="o">=</span> <span class="bp">True</span>    <span class="c"># Do all conds eval to a bool?</span>
-        <span class="n">piecewise_again</span> <span class="o">=</span> <span class="bp">False</span>    <span class="c"># Should we pass args to Piecewise again?</span>
-        <span class="n">non_false_ecpairs</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">or1</span> <span class="o">=</span> <span class="n">Or</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">cond</span> <span class="k">for</span> <span class="p">(</span><span class="n">_</span><span class="p">,</span> <span class="n">cond</span><span class="p">)</span> <span class="ow">in</span> <span class="n">args</span> <span class="k">if</span> <span class="n">cond</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">True</span><span class="p">])</span>
-        <span class="k">for</span> <span class="n">expr</span><span class="p">,</span> <span class="n">cond</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="c"># Check here if expr is a Piecewise and collapse if one of</span>
-            <span class="c"># the conds in expr matches cond. This allows the collapsing</span>
-            <span class="c"># of Piecewise((Piecewise(x,x&lt;0),x&lt;0)) to Piecewise((x,x&lt;0)).</span>
-            <span class="c"># This is important when using piecewise_fold to simplify</span>
-            <span class="c"># multiple Piecewise instances having the same conds.</span>
-            <span class="c"># Eventually, this code should be able to collapse Piecewise&#39;s</span>
-            <span class="c"># having different intervals, but this will probably require</span>
-            <span class="c"># using the new assumptions.</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Piecewise</span><span class="p">):</span>
-                <span class="n">or2</span> <span class="o">=</span> <span class="n">Or</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">c</span> <span class="k">for</span> <span class="p">(</span><span class="n">_</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span> <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">True</span><span class="p">])</span>
-                <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                    <span class="c"># Don&#39;t collapse if cond is &quot;True&quot; as this leads to</span>
-                    <span class="c"># incorrect simplifications with nested Piecewises.</span>
-                    <span class="k">if</span> <span class="n">c</span> <span class="o">==</span> <span class="n">cond</span> <span class="ow">and</span> <span class="p">(</span><span class="n">or1</span> <span class="o">==</span> <span class="n">or2</span> <span class="ow">or</span> <span class="n">cond</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">True</span><span class="p">):</span>
-                        <span class="n">expr</span> <span class="o">=</span> <span class="n">e</span>
-                        <span class="n">piecewise_again</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="n">cond_eval</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">__eval_cond</span><span class="p">(</span><span class="n">cond</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">cond_eval</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">all_conds_evaled</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="k">elif</span> <span class="n">cond_eval</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">all_conds_evaled</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">expr</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">non_false_ecpairs</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">non_false_ecpairs</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">cond</span> <span class="o">==</span> <span class="n">cond</span><span class="p">:</span>
-                    <span class="k">continue</span>
-                <span class="k">elif</span> <span class="n">non_false_ecpairs</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">expr</span> <span class="o">==</span> <span class="n">expr</span><span class="p">:</span>
-                    <span class="n">non_false_ecpairs</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">ExprCondPair</span><span class="p">(</span>
-                        <span class="n">expr</span><span class="p">,</span> <span class="n">Or</span><span class="p">(</span><span class="n">cond</span><span class="p">,</span> <span class="n">non_false_ecpairs</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">cond</span><span class="p">))</span>
-                    <span class="k">continue</span>
-            <span class="n">non_false_ecpairs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ExprCondPair</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">cond</span><span class="p">))</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">non_false_ecpairs</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="ow">or</span> <span class="n">piecewise_again</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Piecewise</span><span class="p">(</span><span class="o">*</span><span class="n">non_false_ecpairs</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="bp">None</span>
-
-<div class="viewcode-block" id="Piecewise.doit"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.piecewise.Piecewise.doit">[docs]</a>    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Evaluate this piecewise function.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">newargs</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;deep&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-                    <span class="n">e</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-                    <span class="n">c</span> <span class="o">=</span> <span class="n">c</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-            <span class="n">newargs</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">e</span><span class="p">,</span> <span class="n">c</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">Piecewise</span><span class="p">(</span><span class="o">*</span><span class="n">newargs</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="bp">True</span> <span class="ow">or</span> <span class="n">c</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">e</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_adjoint</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Piecewise</span><span class="p">(</span><span class="o">*</span><span class="p">[(</span><span class="n">e</span><span class="o">.</span><span class="n">adjoint</span><span class="p">(),</span> <span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Piecewise</span><span class="p">(</span><span class="o">*</span><span class="p">[(</span><span class="n">e</span><span class="o">.</span><span class="n">conjugate</span><span class="p">(),</span> <span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Piecewise</span><span class="p">(</span><span class="o">*</span><span class="p">[(</span><span class="n">diff</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Piecewise</span><span class="p">(</span><span class="o">*</span><span class="p">[(</span><span class="n">e</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">),</span> <span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_integral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.integrals</span> <span class="kn">import</span> <span class="n">integrate</span>
-        <span class="k">return</span> <span class="n">Piecewise</span><span class="p">(</span><span class="o">*</span><span class="p">[(</span><span class="n">integrate</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_interval</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sym</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Evaluates the function along the sym in a given interval ab&quot;&quot;&quot;</span>
-        <span class="c"># FIXME: Currently complex intervals are not supported.  A possible</span>
-        <span class="c"># replacement algorithm, discussed in issue 2128, can be found in the</span>
-        <span class="c"># following papers;</span>
-        <span class="c">#     http://portal.acm.org/citation.cfm?id=281649</span>
-        <span class="c">#     http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.70.4127&amp;rep=rep1&amp;type=pdf</span>
-
-        <span class="k">if</span> <span class="n">a</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">b</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="c"># In this case, it is just simple substitution</span>
-            <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Piecewise</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_interval</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-
-        <span class="n">mul</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">a</span> <span class="o">==</span> <span class="n">b</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">elif</span> <span class="p">(</span><span class="n">a</span> <span class="o">&gt;</span> <span class="n">b</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">mul</span> <span class="o">=</span> <span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span>
-        <span class="k">elif</span> <span class="p">(</span><span class="n">a</span> <span class="o">&lt;=</span> <span class="n">b</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">newargs</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="n">intervals</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sort_expr_cond</span><span class="p">(</span>
-                    <span class="n">sym</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
-                <span class="n">values</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">for</span> <span class="n">lower</span><span class="p">,</span> <span class="n">upper</span> <span class="ow">in</span> <span class="n">intervals</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="p">(</span><span class="n">a</span> <span class="o">&lt;</span> <span class="n">lower</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                        <span class="n">mid</span> <span class="o">=</span> <span class="n">lower</span>
-                        <span class="n">rep</span> <span class="o">=</span> <span class="n">b</span>
-                        <span class="n">val</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">-</span> <span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">mid</span><span class="p">)</span>
-                        <span class="n">val</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_interval</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">mid</span><span class="p">)</span>
-                    <span class="k">elif</span> <span class="p">(</span><span class="n">a</span> <span class="o">&gt;</span> <span class="n">upper</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                        <span class="n">mid</span> <span class="o">=</span> <span class="n">upper</span>
-                        <span class="n">rep</span> <span class="o">=</span> <span class="n">b</span>
-                        <span class="n">val</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">-</span> <span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">mid</span><span class="p">)</span>
-                        <span class="n">val</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_interval</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">mid</span><span class="p">)</span>
-                    <span class="k">elif</span> <span class="p">(</span><span class="n">a</span> <span class="o">&gt;=</span> <span class="n">lower</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span> <span class="ow">and</span> <span class="p">(</span><span class="n">a</span> <span class="o">&lt;=</span> <span class="n">upper</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                        <span class="n">rep</span> <span class="o">=</span> <span class="n">b</span>
-                        <span class="n">val</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">-</span> <span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-                    <span class="k">elif</span> <span class="p">(</span><span class="n">b</span> <span class="o">&lt;</span> <span class="n">lower</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                        <span class="n">mid</span> <span class="o">=</span> <span class="n">lower</span>
-                        <span class="n">rep</span> <span class="o">=</span> <span class="n">a</span>
-                        <span class="n">val</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">mid</span><span class="p">)</span> <span class="o">-</span> <span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-                        <span class="n">val</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_interval</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">mid</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-                    <span class="k">elif</span> <span class="p">(</span><span class="n">b</span> <span class="o">&gt;</span> <span class="n">upper</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                        <span class="n">mid</span> <span class="o">=</span> <span class="n">upper</span>
-                        <span class="n">rep</span> <span class="o">=</span> <span class="n">a</span>
-                        <span class="n">val</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">mid</span><span class="p">)</span> <span class="o">-</span> <span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-                        <span class="n">val</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_interval</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">mid</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-                    <span class="k">elif</span> <span class="p">((</span><span class="n">b</span> <span class="o">&gt;=</span> <span class="n">lower</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">)</span> <span class="ow">and</span> <span class="p">((</span><span class="n">b</span> <span class="o">&lt;=</span> <span class="n">upper</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">):</span>
-                        <span class="n">rep</span> <span class="o">=</span> <span class="n">a</span>
-                        <span class="n">val</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">-</span> <span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                            <span class="sd">&quot;&quot;&quot;The evaluation of a Piecewise interval when both the lower</span>
-<span class="sd">                            and the upper limit are symbolic is not yet implemented.&quot;&quot;&quot;</span><span class="p">)</span>
-                    <span class="n">values</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">values</span><span class="p">))</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">try</span><span class="p">:</span>
-                        <span class="n">c</span> <span class="o">=</span> <span class="n">c</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">rep</span><span class="p">)</span>
-                    <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-                        <span class="k">pass</span>
-                    <span class="n">e</span> <span class="o">=</span> <span class="n">values</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                    <span class="n">newargs</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">e</span><span class="p">,</span> <span class="n">c</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">values</span><span class="p">)):</span>
-                        <span class="n">newargs</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">values</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="p">(</span><span class="n">c</span> <span class="ow">is</span> <span class="bp">True</span> <span class="ow">and</span> <span class="n">i</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">values</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="ow">or</span>
-                            <span class="n">And</span><span class="p">(</span><span class="n">rep</span> <span class="o">&gt;=</span> <span class="n">intervals</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">rep</span> <span class="o">&lt;=</span> <span class="n">intervals</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">])))</span>
-            <span class="k">return</span> <span class="n">Piecewise</span><span class="p">(</span><span class="o">*</span><span class="n">newargs</span><span class="p">)</span>
-
-        <span class="c"># Determine what intervals the expr,cond pairs affect.</span>
-        <span class="n">int_expr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sort_expr_cond</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-
-        <span class="c"># Finally run through the intervals and sum the evaluation.</span>
-        <span class="n">ret_fun</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">int_a</span><span class="p">,</span> <span class="n">int_b</span><span class="p">,</span> <span class="n">expr</span> <span class="ow">in</span> <span class="n">int_expr</span><span class="p">:</span>
-            <span class="n">ret_fun</span> <span class="o">+=</span> <span class="n">expr</span><span class="o">.</span><span class="n">_eval_interval</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">Max</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">int_a</span><span class="p">),</span> <span class="n">Min</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">int_b</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">mul</span> <span class="o">*</span> <span class="n">ret_fun</span>
-
-    <span class="k">def</span> <span class="nf">_sort_expr_cond</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sym</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">targetcond</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Determine what intervals the expr, cond pairs affect.</span>
-
-<span class="sd">        1) If cond is True, then log it as default</span>
-<span class="sd">        1.1) Currently if cond can&#39;t be evaluated, throw NotImplementedError.</span>
-<span class="sd">        2) For each inequality, if previous cond defines part of the interval</span>
-<span class="sd">           update the new conds interval.</span>
-<span class="sd">           -  eg x &lt; 1, x &lt; 3 -&gt; [oo,1],[1,3] instead of [oo,1],[oo,3]</span>
-<span class="sd">        3) Sort the intervals to make it easier to find correct exprs</span>
-
-<span class="sd">        Under normal use, we return the expr,cond pairs in increasing order</span>
-<span class="sd">        along the real axis corresponding to the symbol sym.  If targetcond</span>
-<span class="sd">        is given, we return a list of (lowerbound, upperbound) pairs for</span>
-<span class="sd">        this condition.&quot;&quot;&quot;</span>
-        <span class="n">default</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="n">int_expr</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">expr_cond</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">or_cond</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="n">or_intervals</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">expr</span><span class="p">,</span> <span class="n">cond</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">cond</span><span class="p">,</span> <span class="n">Or</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">cond2</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">cond</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">):</span>
-                    <span class="n">expr_cond</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">expr</span><span class="p">,</span> <span class="n">cond2</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">expr_cond</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">expr</span><span class="p">,</span> <span class="n">cond</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">cond</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                <span class="k">break</span>
-        <span class="k">for</span> <span class="n">expr</span><span class="p">,</span> <span class="n">cond</span> <span class="ow">in</span> <span class="n">expr_cond</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">cond</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                <span class="n">default</span> <span class="o">=</span> <span class="n">expr</span>
-                <span class="k">break</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">cond</span><span class="p">,</span> <span class="n">Equality</span><span class="p">):</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">cond</span><span class="p">,</span> <span class="n">And</span><span class="p">):</span>
-                <span class="n">lower</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-                <span class="n">upper</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-                <span class="k">for</span> <span class="n">cond2</span> <span class="ow">in</span> <span class="n">cond</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">cond2</span><span class="o">.</span><span class="n">lts</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">sym</span><span class="p">):</span>
-                        <span class="n">upper</span> <span class="o">=</span> <span class="n">Min</span><span class="p">(</span><span class="n">cond2</span><span class="o">.</span><span class="n">gts</span><span class="p">,</span> <span class="n">upper</span><span class="p">)</span>
-                    <span class="k">elif</span> <span class="n">cond2</span><span class="o">.</span><span class="n">gts</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">sym</span><span class="p">):</span>
-                        <span class="n">lower</span> <span class="o">=</span> <span class="n">Max</span><span class="p">(</span><span class="n">cond2</span><span class="o">.</span><span class="n">lts</span><span class="p">,</span> <span class="n">lower</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">lower</span><span class="p">,</span> <span class="n">upper</span> <span class="o">=</span> <span class="n">cond</span><span class="o">.</span><span class="n">lts</span><span class="p">,</span> <span class="n">cond</span><span class="o">.</span><span class="n">gts</span>  <span class="c"># part 1: initialize with givens</span>
-                <span class="k">if</span> <span class="n">cond</span><span class="o">.</span><span class="n">lts</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">sym</span><span class="p">):</span>     <span class="c"># part 1a: expand the side ...</span>
-                    <span class="n">lower</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>   <span class="c"># e.g. x &lt;= 0 ---&gt; -oo &lt;= 0</span>
-                <span class="k">elif</span> <span class="n">cond</span><span class="o">.</span><span class="n">gts</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">sym</span><span class="p">):</span>   <span class="c"># part 1a: ... that can be expanded</span>
-                    <span class="n">upper</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>           <span class="c"># e.g. x &gt;= 0 ---&gt;  oo &gt;= 0</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                        <span class="s">&quot;Unable to handle interval evaluation of expression.&quot;</span><span class="p">)</span>
-
-            <span class="c"># part 1b: Reduce (-)infinity to what was passed in.</span>
-            <span class="n">lower</span><span class="p">,</span> <span class="n">upper</span> <span class="o">=</span> <span class="n">Max</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">lower</span><span class="p">),</span> <span class="n">Min</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">upper</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">int_expr</span><span class="p">)):</span>
-                <span class="c"># Part 2: remove any interval overlap.  For any conflicts, the</span>
-                <span class="c"># iterval already there wins, and the incoming interval updates</span>
-                <span class="c"># its bounds accordingly.</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">__eval_cond</span><span class="p">(</span><span class="n">lower</span> <span class="o">&lt;</span> <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span> <span class="ow">and</span> \
-                        <span class="bp">self</span><span class="o">.</span><span class="n">__eval_cond</span><span class="p">(</span><span class="n">lower</span> <span class="o">&gt;=</span> <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">0</span><span class="p">]):</span>
-                    <span class="n">lower</span> <span class="o">=</span> <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
-                <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">)</span> <span class="ow">and</span> \
-                        <span class="bp">self</span><span class="o">.</span><span class="n">__eval_cond</span><span class="p">(</span><span class="n">lower</span> <span class="o">&gt;=</span> <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">0</span><span class="p">]):</span>
-                    <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">__eval_cond</span><span class="p">(</span><span class="n">lower</span> <span class="o">==</span> <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">0</span><span class="p">]):</span>
-                        <span class="n">lower</span> <span class="o">=</span> <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">Min</span><span class="p">(</span><span class="n">lower</span><span class="p">,</span> <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span>
-                <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">)</span> <span class="ow">and</span> \
-                        <span class="n">lower</span> <span class="o">&lt;</span> <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">True</span><span class="p">:</span>
-                    <span class="n">upper</span> <span class="o">=</span> <span class="n">Min</span><span class="p">(</span><span class="n">upper</span><span class="p">,</span> <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span>
-                <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">__eval_cond</span><span class="p">(</span><span class="n">upper</span> <span class="o">&gt;</span> <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span> <span class="ow">and</span> \
-                        <span class="bp">self</span><span class="o">.</span><span class="n">__eval_cond</span><span class="p">(</span><span class="n">upper</span> <span class="o">&lt;=</span> <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">1</span><span class="p">]):</span>
-                    <span class="n">upper</span> <span class="o">=</span> <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-                <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">)</span> <span class="ow">and</span> \
-                        <span class="bp">self</span><span class="o">.</span><span class="n">__eval_cond</span><span class="p">(</span><span class="n">upper</span> <span class="o">&lt;</span> <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">1</span><span class="p">]):</span>
-                    <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">Max</span><span class="p">(</span><span class="n">upper</span><span class="p">,</span> <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span>
-
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">__eval_cond</span><span class="p">(</span><span class="n">lower</span> <span class="o">&gt;=</span> <span class="n">upper</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">True</span><span class="p">:</span>  <span class="c"># Is it still an interval?</span>
-                <span class="n">int_expr</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">lower</span><span class="p">,</span> <span class="n">upper</span><span class="p">,</span> <span class="n">expr</span><span class="p">])</span>
-            <span class="k">if</span> <span class="n">cond</span> <span class="ow">is</span> <span class="n">targetcond</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[(</span><span class="n">lower</span><span class="p">,</span> <span class="n">upper</span><span class="p">)]</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">targetcond</span><span class="p">,</span> <span class="n">Or</span><span class="p">)</span> <span class="ow">and</span> <span class="n">cond</span> <span class="ow">in</span> <span class="n">targetcond</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="n">or_cond</span> <span class="o">=</span> <span class="n">Or</span><span class="p">(</span><span class="n">or_cond</span><span class="p">,</span> <span class="n">cond</span><span class="p">)</span>
-                <span class="n">or_intervals</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">lower</span><span class="p">,</span> <span class="n">upper</span><span class="p">))</span>
-                <span class="k">if</span> <span class="n">or_cond</span> <span class="o">==</span> <span class="n">targetcond</span><span class="p">:</span>
-                    <span class="n">or_intervals</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-                    <span class="k">return</span> <span class="n">or_intervals</span>
-
-        <span class="n">int_expr</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">sort_key</span><span class="p">(</span>
-        <span class="p">)</span> <span class="k">if</span> <span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">is_number</span> <span class="k">else</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="o">.</span><span class="n">sort_key</span><span class="p">())</span>
-        <span class="n">int_expr</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">sort_key</span><span class="p">(</span>
-        <span class="p">)</span> <span class="k">if</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_number</span> <span class="k">else</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="o">.</span><span class="n">sort_key</span><span class="p">())</span>
-        <span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">MinMaxBase</span>
-        <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">int_expr</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">):</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">1</span><span class="p">],</span> <span class="n">Min</span><span class="p">)</span> <span class="ow">or</span> <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">b</span><span class="p">:</span>
-                    <span class="n">newval</span> <span class="o">=</span> <span class="n">Min</span><span class="p">(</span><span class="o">*</span><span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][:</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-                    <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">]:</span>
-                        <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">newval</span>
-                    <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">newval</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">newval</span> <span class="o">=</span> <span class="n">Max</span><span class="p">(</span><span class="o">*</span><span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][:</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-                    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">int_expr</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]:</span>
-                        <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">newval</span>
-                    <span class="n">int_expr</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">newval</span>
-
-        <span class="c"># Add holes to list of intervals if there is a default value,</span>
-        <span class="c"># otherwise raise a ValueError.</span>
-        <span class="n">holes</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">curr_low</span> <span class="o">=</span> <span class="n">a</span>
-        <span class="k">for</span> <span class="n">int_a</span><span class="p">,</span> <span class="n">int_b</span><span class="p">,</span> <span class="n">expr</span> <span class="ow">in</span> <span class="n">int_expr</span><span class="p">:</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">curr_low</span> <span class="o">&lt;</span> <span class="n">int_a</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                <span class="n">holes</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">curr_low</span><span class="p">,</span> <span class="n">Min</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">int_a</span><span class="p">),</span> <span class="n">default</span><span class="p">])</span>
-            <span class="k">elif</span> <span class="p">(</span><span class="n">curr_low</span> <span class="o">&gt;=</span> <span class="n">int_a</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">True</span><span class="p">:</span>
-                <span class="n">holes</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">curr_low</span><span class="p">,</span> <span class="n">Min</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">int_a</span><span class="p">),</span> <span class="n">default</span><span class="p">])</span>
-            <span class="n">curr_low</span> <span class="o">=</span> <span class="n">Min</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">int_b</span><span class="p">)</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">curr_low</span> <span class="o">&lt;</span> <span class="n">b</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">holes</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">Min</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">curr_low</span><span class="p">),</span> <span class="n">b</span><span class="p">,</span> <span class="n">default</span><span class="p">])</span>
-        <span class="k">elif</span> <span class="p">(</span><span class="n">curr_low</span> <span class="o">&gt;=</span> <span class="n">b</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">holes</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">Min</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">curr_low</span><span class="p">),</span> <span class="n">b</span><span class="p">,</span> <span class="n">default</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">holes</span> <span class="ow">and</span> <span class="n">default</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">int_expr</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">holes</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">targetcond</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[(</span><span class="n">h</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">h</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">holes</span><span class="p">]</span>
-        <span class="k">elif</span> <span class="n">holes</span> <span class="ow">and</span> <span class="n">default</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Called interval evaluation over piecewise &quot;</span>
-                             <span class="s">&quot;function on undefined intervals </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span>
-                             <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="nb">str</span><span class="p">((</span><span class="n">h</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">h</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span> <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">holes</span><span class="p">]))</span>
-
-        <span class="k">return</span> <span class="n">int_expr</span>
-
-    <span class="k">def</span> <span class="nf">_eval_nseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[(</span><span class="n">ec</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">),</span> <span class="n">ec</span><span class="o">.</span><span class="n">cond</span><span class="p">)</span> <span class="k">for</span> <span class="n">ec</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Piecewise</span><span class="p">(</span><span class="o">*</span><span class="p">[(</span><span class="n">e</span><span class="o">**</span><span class="n">s</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Piecewise conditions may contain bool which are not of Basic type.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">checksol</span><span class="p">,</span> <span class="n">solve</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-            <span class="n">e</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">_subs</span><span class="p">(</span><span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="nb">bool</span><span class="p">):</span>
-                <span class="k">pass</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="n">c</span><span class="o">.</span><span class="n">_subs</span><span class="p">(</span><span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">Equality</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">checksol</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="p">{},</span> <span class="n">minimal</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-                    <span class="c"># the equality is trivially solved</span>
-                    <span class="n">c</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="c"># try to solve the equality</span>
-                    <span class="k">try</span><span class="p">:</span>
-                        <span class="n">slns</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="nb">dict</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="ow">not</span> <span class="n">slns</span><span class="p">:</span>
-                            <span class="n">c</span> <span class="o">=</span> <span class="bp">False</span>
-                        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">slns</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                            <span class="n">c</span> <span class="o">=</span> <span class="n">And</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">Equality</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">value</span><span class="p">)</span>
-                                      <span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="n">value</span> <span class="ow">in</span> <span class="n">slns</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">items</span><span class="p">()])</span>
-                    <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                        <span class="k">pass</span>
-
-            <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">e</span><span class="p">,</span> <span class="n">c</span>
-
-        <span class="k">return</span> <span class="n">Piecewise</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_transpose</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Piecewise</span><span class="p">(</span><span class="o">*</span><span class="p">[(</span><span class="n">e</span><span class="o">.</span><span class="n">transpose</span><span class="p">(),</span> <span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_template_is_attr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">is_attr</span><span class="p">,</span> <span class="n">when_multiple</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">for</span> <span class="n">expr</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">is_attr</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="k">if</span> <span class="n">b</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="n">a</span>
-            <span class="k">elif</span> <span class="n">b</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">a</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">when_multiple</span>
-        <span class="k">return</span> <span class="n">b</span>
-
-    <span class="n">_eval_is_bounded</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span>
-        <span class="s">&#39;is_bounded&#39;</span><span class="p">,</span> <span class="n">when_multiple</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-    <span class="n">_eval_is_complex</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span><span class="s">&#39;is_complex&#39;</span><span class="p">)</span>
-    <span class="n">_eval_is_even</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span><span class="s">&#39;is_even&#39;</span><span class="p">)</span>
-    <span class="n">_eval_is_imaginary</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span>
-        <span class="s">&#39;is_imaginary&#39;</span><span class="p">)</span>
-    <span class="n">_eval_is_integer</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span><span class="s">&#39;is_integer&#39;</span><span class="p">)</span>
-    <span class="n">_eval_is_irrational</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span>
-        <span class="s">&#39;is_irrational&#39;</span><span class="p">)</span>
-    <span class="n">_eval_is_negative</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span><span class="s">&#39;is_negative&#39;</span><span class="p">)</span>
-    <span class="n">_eval_is_nonnegative</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span>
-        <span class="s">&#39;is_nonnegative&#39;</span><span class="p">)</span>
-    <span class="n">_eval_is_nonpositive</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span>
-        <span class="s">&#39;is_nonpositive&#39;</span><span class="p">)</span>
-    <span class="n">_eval_is_nonzero</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span>
-        <span class="s">&#39;is_nonzero&#39;</span><span class="p">,</span> <span class="n">when_multiple</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">_eval_is_odd</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span><span class="s">&#39;is_odd&#39;</span><span class="p">)</span>
-    <span class="n">_eval_is_polar</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span><span class="s">&#39;is_polar&#39;</span><span class="p">)</span>
-    <span class="n">_eval_is_positive</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span><span class="s">&#39;is_positive&#39;</span><span class="p">)</span>
-    <span class="n">_eval_is_real</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span><span class="s">&#39;is_real&#39;</span><span class="p">)</span>
-    <span class="n">_eval_is_zero</span> <span class="o">=</span> <span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_template_is_attr</span><span class="p">(</span>
-        <span class="s">&#39;is_zero&#39;</span><span class="p">,</span> <span class="n">when_multiple</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">__eval_cond</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">cond</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the truth value of the condition.&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">cond</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-</div>
-<div class="viewcode-block" id="piecewise_fold"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.piecewise.piecewise_fold">[docs]</a><span class="k">def</span> <span class="nf">piecewise_fold</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Takes an expression containing a piecewise function and returns the</span>
-<span class="sd">    expression in piecewise form.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Piecewise, piecewise_fold, sympify as S</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; p = Piecewise((x, x &lt; 1), (1, S(1) &lt;= x))</span>
-<span class="sd">    &gt;&gt;&gt; piecewise_fold(x*p)</span>
-<span class="sd">    Piecewise((x**2, x &lt; 1), (x, 1 &lt;= x))</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Piecewise</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Basic</span><span class="p">)</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Piecewise</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span>
-    <span class="n">new_args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">piecewise_fold</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">ExprCondPair</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">ExprCondPair</span><span class="p">(</span><span class="o">*</span><span class="n">new_args</span><span class="p">)</span>
-    <span class="n">piecewise_args</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">arg</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">new_args</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Piecewise</span><span class="p">:</span>
-            <span class="n">piecewise_args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">piecewise_args</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">piecewise_args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">new_args</span> <span class="o">=</span> <span class="p">[(</span><span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">new_args</span><span class="p">[:</span><span class="n">n</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">e</span><span class="p">]</span> <span class="o">+</span> <span class="n">new_args</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:])),</span> <span class="n">c</span><span class="p">)</span>
-                    <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">new_args</span><span class="p">[</span><span class="n">n</span><span class="p">]</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">piecewise_args</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">piecewise_fold</span><span class="p">(</span><span class="n">Piecewise</span><span class="p">(</span><span class="o">*</span><span class="n">new_args</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">Piecewise</span><span class="p">(</span><span class="o">*</span><span class="n">new_args</span><span class="p">)</span></div>
-</pre></div>
-
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-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../../index.html">
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-<div id="searchbox" style="display: none">
-  <h3>Quick search</h3>
-    <form class="search" action="../../../../search.html" method="get">
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-            
-  <h1>Source code for sympy.functions.elementary.trigonometric</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core.add</span> <span class="kn">import</span> <span class="n">Add</span>
-<span class="kn">from</span> <span class="nn">sympy.core.basic</span> <span class="kn">import</span> <span class="n">C</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">cacheit</span>
-<span class="kn">from</span> <span class="nn">sympy.core.singleton</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">igcdex</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">ArgumentIndexError</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.exponential</span> <span class="kn">import</span> <span class="n">log</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.hyperbolic</span> <span class="kn">import</span> <span class="n">HyperbolicFunction</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">numbered_symbols</span>
-
-<span class="c">###############################################################################</span>
-<span class="c">########################## TRIGONOMETRIC FUNCTIONS ############################</span>
-<span class="c">###############################################################################</span>
-
-
-<span class="k">class</span> <span class="nc">TrigonometricFunction</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Base class for trigonometric functions. &quot;&quot;&quot;</span>
-
-    <span class="n">unbranched</span> <span class="o">=</span> <span class="bp">True</span>
-
-
-<span class="k">def</span> <span class="nf">_peeloff_pi</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Split ARG into two parts, a &quot;rest&quot; and a multiple of pi/2.</span>
-<span class="sd">    This assumes ARG to be an Add.</span>
-<span class="sd">    The multiple of pi returned in the second position is always a Rational.</span>
-
-<span class="sd">    Examples:</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.functions.elementary.trigonometric import _peeloff_pi as peel</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import pi</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; peel(x + pi/2)</span>
-<span class="sd">    (x, pi/2)</span>
-<span class="sd">    &gt;&gt;&gt; peel(x + 2*pi/3 + pi*y)</span>
-<span class="sd">    (x + pi*y + pi/6, pi/2)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">a</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">:</span>
-            <span class="n">K</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">break</span>
-        <span class="k">elif</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="n">K</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">as_two_terms</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">p</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="ow">and</span> <span class="n">K</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="k">break</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">arg</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="n">m1</span> <span class="o">=</span> <span class="p">(</span><span class="n">K</span> <span class="o">%</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span> <span class="o">*</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-    <span class="n">m2</span> <span class="o">=</span> <span class="n">K</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">-</span> <span class="n">m1</span>
-    <span class="k">return</span> <span class="n">arg</span> <span class="o">-</span> <span class="n">m2</span><span class="p">,</span> <span class="n">m2</span>
-
-
-<span class="k">def</span> <span class="nf">_pi_coeff</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">cycles</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    When arg is a Number times pi (e.g. 3*pi/2) then return the Number</span>
-<span class="sd">    normalized to be in the range [0, 2], else None.</span>
-
-<span class="sd">    When an even multiple of pi is encountered, if it is multiplying</span>
-<span class="sd">    something with known parity then the multiple is returned as 0 otherwise</span>
-<span class="sd">    as 2.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.functions.elementary.trigonometric import _pi_coeff as coeff</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import pi</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; coeff(3*x*pi)</span>
-<span class="sd">    3*x</span>
-<span class="sd">    &gt;&gt;&gt; coeff(11*pi/7)</span>
-<span class="sd">    11/7</span>
-<span class="sd">    &gt;&gt;&gt; coeff(-11*pi/7)</span>
-<span class="sd">    3/7</span>
-<span class="sd">    &gt;&gt;&gt; coeff(4*pi)</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; coeff(5*pi)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; coeff(5.0*pi)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; coeff(5.5*pi)</span>
-<span class="sd">    3/2</span>
-<span class="sd">    &gt;&gt;&gt; coeff(2 + pi)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">arg</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-    <span class="k">elif</span> <span class="ow">not</span> <span class="n">arg</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-    <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-        <span class="n">cx</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">cx</span><span class="p">:</span>
-            <span class="n">c</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="n">cx</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>  <span class="c"># pi is not included as coeff</span>
-            <span class="k">if</span> <span class="n">c</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-                <span class="c"># recast exact binary fractions to Rationals</span>
-                <span class="n">f</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="o">%</span> <span class="mi">1</span>
-                <span class="k">if</span> <span class="n">f</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">p</span> <span class="o">=</span> <span class="o">-</span><span class="nb">int</span><span class="p">(</span><span class="nb">round</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">()))</span>
-                    <span class="n">m</span> <span class="o">=</span> <span class="mi">2</span><span class="o">**</span><span class="n">p</span>
-                    <span class="n">cm</span> <span class="o">=</span> <span class="n">c</span><span class="o">*</span><span class="n">m</span>
-                    <span class="n">i</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">cm</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">cm</span><span class="p">:</span>
-                        <span class="n">c</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Rational</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-                        <span class="n">cx</span> <span class="o">=</span> <span class="n">c</span><span class="o">*</span><span class="n">x</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">c</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Rational</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">c</span><span class="p">))</span>
-                    <span class="n">cx</span> <span class="o">=</span> <span class="n">c</span><span class="o">*</span><span class="n">x</span>
-            <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-                <span class="n">c2</span> <span class="o">=</span> <span class="n">c</span> <span class="o">%</span> <span class="mi">2</span>
-                <span class="k">if</span> <span class="n">c2</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">x</span>
-                <span class="k">elif</span> <span class="ow">not</span> <span class="n">c2</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">is_even</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>  <span class="c"># known parity</span>
-                        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-                    <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">c2</span><span class="o">*</span><span class="n">x</span>
-            <span class="k">return</span> <span class="n">cx</span>
-
-
-<div class="viewcode-block" id="sin"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.trigonometric.sin">[docs]</a><span class="k">class</span> <span class="nc">sin</span><span class="p">(</span><span class="n">TrigonometricFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The sine function.</span>
-
-<span class="sd">    * sin(x) -&gt; Returns the sine of x (measured in radians)</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    * sin(x) will evaluate automatically in the case x</span>
-<span class="sd">      is a multiple of pi, pi/2, pi/3, pi/4 and pi/6.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import sin, pi</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; sin(x**2).diff(x)</span>
-<span class="sd">    2*x*cos(x**2)</span>
-<span class="sd">    &gt;&gt;&gt; sin(1).diff(x)</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; sin(pi)</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; sin(pi/2)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; sin(pi/6)</span>
-<span class="sd">    1/2</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    cos, tan, asin</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    U{Definitions in trigonometry&lt;http://planetmath.org/encyclopedia/DefinitionsInTrigonometry.html&gt;}</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cos</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-<div class="viewcode-block" id="sin.inverse"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.trigonometric.sin.inverse">[docs]</a>    <span class="k">def</span> <span class="nf">inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the inverse of this function.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">asin</span>
-</div>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span> <span class="ow">or</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">return</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">():</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="n">i_coeff</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">i_coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">sinh</span><span class="p">(</span><span class="n">i_coeff</span><span class="p">)</span>
-
-        <span class="n">pi_coeff</span> <span class="o">=</span> <span class="n">_pi_coeff</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">pi_coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">pi_coeff</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">pi_coeff</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="n">narg</span> <span class="o">=</span> <span class="n">pi_coeff</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-                <span class="k">if</span> <span class="n">narg</span> <span class="o">!=</span> <span class="n">arg</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">narg</span><span class="p">)</span>
-                <span class="k">return</span> <span class="bp">None</span>
-
-            <span class="c"># http://code.google.com/p/sympy/issues/detail?id=2949</span>
-            <span class="c"># transform a sine to a cosine, to avoid redundant code</span>
-            <span class="k">if</span> <span class="n">pi_coeff</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="n">pi_coeff</span> <span class="o">%</span> <span class="mi">2</span>
-                <span class="k">if</span> <span class="n">x</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">((</span><span class="n">x</span> <span class="o">%</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">)</span>
-                <span class="k">if</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">cls</span><span class="p">((</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">)</span>
-                <span class="n">narg</span> <span class="o">=</span> <span class="p">((</span><span class="n">pi_coeff</span> <span class="o">+</span> <span class="n">C</span><span class="o">.</span><span class="n">Rational</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span> <span class="o">%</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">cos</span><span class="p">(</span><span class="n">narg</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">cos</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">result</span>
-                <span class="k">if</span> <span class="n">pi_coeff</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">!=</span> <span class="n">arg</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">pi_coeff</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">)</span>
-                <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">x</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">_peeloff_pi</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">m</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">sin</span><span class="p">(</span><span class="n">m</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">cos</span><span class="p">(</span><span class="n">m</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">asin</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">atan</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">x</span> <span class="o">/</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">atan2</span><span class="p">:</span>
-            <span class="n">y</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">return</span> <span class="n">y</span> <span class="o">/</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">acos</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">acot</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="mi">1</span> <span class="o">/</span> <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">previous_terms</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="n">previous_terms</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">p</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span> <span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="o">//</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_exp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">exp</span><span class="p">,</span> <span class="n">I</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">TrigonometricFunction</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">HyperbolicFunction</span><span class="p">):</span>
-            <span class="n">arg</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">exp</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">arg</span><span class="o">*</span><span class="n">I</span><span class="p">)</span> <span class="o">-</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="o">*</span><span class="n">I</span><span class="p">))</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">I</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_Pow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">log</span><span class="p">:</span>
-            <span class="n">I</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="o">**-</span><span class="n">I</span> <span class="o">/</span> <span class="mi">2</span> <span class="o">-</span> <span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">I</span> <span class="o">/</span><span class="mi">2</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_cos</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="n">arg</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_tan</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">tan_half</span> <span class="o">=</span> <span class="n">tan</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="o">*</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">tan_half</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">tan_half</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_sincos</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sin</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_cot</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">cot_half</span> <span class="o">=</span> <span class="n">cot</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="o">*</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">cot_half</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">cot_half</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_pow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">cos</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="nb">pow</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_sqrt</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">cos</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">sqrt</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">conjugate</span><span class="p">())</span>
-
-    <span class="k">def</span> <span class="nf">as_real_imag</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-                <span class="n">hints</span><span class="p">[</span><span class="s">&#39;complex&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">re</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">cosh</span><span class="p">(</span><span class="n">im</span><span class="p">),</span> <span class="n">cos</span><span class="p">(</span><span class="n">re</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">sinh</span><span class="p">(</span><span class="n">im</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_expand_trig</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expand_mul</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>  <span class="c"># TODO, implement more if deep stuff here</span>
-            <span class="c"># TODO: Do this more efficiently for more than two terms</span>
-            <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_two_terms</span><span class="p">()</span>
-            <span class="n">sx</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_expand_trig</span><span class="p">()</span>
-            <span class="n">sy</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_expand_trig</span><span class="p">()</span>
-            <span class="n">cx</span> <span class="o">=</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_expand_trig</span><span class="p">()</span>
-            <span class="n">cy</span> <span class="o">=</span> <span class="n">cos</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_expand_trig</span><span class="p">()</span>
-            <span class="k">return</span> <span class="n">sx</span><span class="o">*</span><span class="n">cy</span> <span class="o">+</span> <span class="n">sy</span><span class="o">*</span><span class="n">cx</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">(</span><span class="n">rational</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>  <span class="c"># n will be positive because of .eval</span>
-                <span class="c"># canonicalization</span>
-
-                <span class="c"># See http://mathworld.wolfram.com/Multiple-AngleFormulas.html</span>
-                <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_odd</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">((</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">chebyshevt</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">expand_mul</span><span class="p">((</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">chebyshevu</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span>
-                        <span class="mi">1</span><span class="p">,</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-            <span class="n">pi_coeff</span> <span class="o">=</span> <span class="n">_pi_coeff</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">pi_coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">pi_coeff</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">sqrt</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">sin</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">arg</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_bounded</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">())</span>
-
-</div>
-<div class="viewcode-block" id="cos"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.trigonometric.cos">[docs]</a><span class="k">class</span> <span class="nc">cos</span><span class="p">(</span><span class="n">TrigonometricFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The cosine function.</span>
-
-<span class="sd">    * cos(x) -&gt; Returns the cosine of x (measured in radians)</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    * cos(x) will evaluate automatically in the case x</span>
-<span class="sd">      is a multiple of pi, pi/2, pi/3, pi/4 and pi/6.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import cos, pi</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; cos(x**2).diff(x)</span>
-<span class="sd">    -2*x*sin(x**2)</span>
-<span class="sd">    &gt;&gt;&gt; cos(1).diff(x)</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; cos(pi)</span>
-<span class="sd">    -1</span>
-<span class="sd">    &gt;&gt;&gt; cos(pi/2)</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; cos(2*pi/3)</span>
-<span class="sd">    -1/2</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sin, tan, acos</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    U{Definitions in trigonometry&lt;http://planetmath.org/encyclopedia/DefinitionsInTrigonometry.html&gt;}</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">acos</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span> <span class="ow">or</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="c"># In this cases, it is unclear if we should</span>
-                <span class="c"># return S.NaN or leave un-evaluated.  One</span>
-                <span class="c"># useful test case is how &quot;limit(sin(x)/x,x,oo)&quot;</span>
-                <span class="c"># is handled.</span>
-                <span class="c"># See test_sin_cos_with_infinity() an</span>
-                <span class="c"># Test for issue 209</span>
-                <span class="c"># http://code.google.com/p/sympy/issues/detail?id=2097</span>
-                <span class="c"># For now, we return un-evaluated.</span>
-                <span class="k">return</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">():</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="n">i_coeff</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">i_coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">cosh</span><span class="p">(</span><span class="n">i_coeff</span><span class="p">)</span>
-
-        <span class="n">pi_coeff</span> <span class="o">=</span> <span class="n">_pi_coeff</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">pi_coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">pi_coeff</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">)</span><span class="o">**</span><span class="n">pi_coeff</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">pi_coeff</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="n">narg</span> <span class="o">=</span> <span class="n">pi_coeff</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-                <span class="k">if</span> <span class="n">narg</span> <span class="o">!=</span> <span class="n">arg</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">narg</span><span class="p">)</span>
-                <span class="k">return</span> <span class="bp">None</span>
-
-            <span class="c"># cosine formula #####################</span>
-            <span class="c"># http://code.google.com/p/sympy/issues/detail?id=2949</span>
-            <span class="c"># explicit calculations are preformed for</span>
-            <span class="c"># cos(k pi / 8), cos(k pi /10), and cos(k pi / 12)</span>
-            <span class="c"># Some other exact values like cos(k pi/15) can be</span>
-            <span class="c"># calculated using a partial-fraction decomposition</span>
-            <span class="c"># by calling cos( X ).rewrite(sqrt)</span>
-            <span class="n">cst_table_some</span> <span class="o">=</span> <span class="p">{</span>
-                <span class="mi">3</span><span class="p">:</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span>
-                <span class="mi">5</span><span class="p">:</span> <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span>
-            <span class="p">}</span>
-            <span class="k">if</span> <span class="n">pi_coeff</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="n">q</span> <span class="o">=</span> <span class="n">pi_coeff</span><span class="o">.</span><span class="n">q</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="n">pi_coeff</span><span class="o">.</span><span class="n">p</span> <span class="o">%</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">q</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">p</span> <span class="o">&gt;</span> <span class="n">q</span><span class="p">:</span>
-                    <span class="n">narg</span> <span class="o">=</span> <span class="p">(</span><span class="n">pi_coeff</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-                    <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">(</span><span class="n">narg</span><span class="p">)</span>
-                <span class="k">if</span> <span class="mi">2</span><span class="o">*</span><span class="n">p</span> <span class="o">&gt;</span> <span class="n">q</span><span class="p">:</span>
-                    <span class="n">narg</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">pi_coeff</span><span class="p">)</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-                    <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">(</span><span class="n">narg</span><span class="p">)</span>
-
-                <span class="c"># If nested sqrt&#39;s are worse than un-evaluation</span>
-                <span class="c"># you can require q in (1, 2, 3, 4, 6)</span>
-                <span class="c"># q &lt;= 12 returns expressions with 2 or fewer nestings.</span>
-                <span class="k">if</span> <span class="n">q</span> <span class="o">&gt;</span> <span class="mi">12</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-
-                <span class="k">if</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">cst_table_some</span><span class="p">:</span>
-                    <span class="n">cts</span> <span class="o">=</span> <span class="n">cst_table_some</span><span class="p">[</span><span class="n">pi_coeff</span><span class="o">.</span><span class="n">q</span><span class="p">]</span>
-                    <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">chebyshevt</span><span class="p">(</span><span class="n">pi_coeff</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="n">cts</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-
-                <span class="k">if</span> <span class="mi">0</span> <span class="o">==</span> <span class="n">q</span> <span class="o">%</span> <span class="mi">2</span><span class="p">:</span>
-                    <span class="n">narg</span> <span class="o">=</span> <span class="p">(</span><span class="n">pi_coeff</span><span class="o">*</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-                    <span class="n">nval</span> <span class="o">=</span> <span class="n">cls</span><span class="p">(</span><span class="n">narg</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="bp">None</span> <span class="o">==</span> <span class="n">nval</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="bp">None</span>
-                    <span class="n">x</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi_coeff</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-                    <span class="n">sign_cos</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">((</span><span class="o">-</span><span class="mi">1</span> <span class="k">if</span> <span class="n">x</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="k">else</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="nb">int</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
-                    <span class="k">return</span> <span class="n">sign_cos</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">nval</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="p">)</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">x</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">_peeloff_pi</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">m</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">cos</span><span class="p">(</span><span class="n">m</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">sin</span><span class="p">(</span><span class="n">m</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">acos</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">atan</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">atan2</span><span class="p">:</span>
-            <span class="n">y</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">return</span> <span class="n">x</span> <span class="o">/</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">asin</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">acot</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">previous_terms</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="n">previous_terms</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">p</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span> <span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="o">//</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_exp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">exp</span><span class="p">,</span> <span class="n">I</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">TrigonometricFunction</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">HyperbolicFunction</span><span class="p">):</span>
-            <span class="n">arg</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">exp</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">arg</span><span class="o">*</span><span class="n">I</span><span class="p">)</span> <span class="o">+</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="o">*</span><span class="n">I</span><span class="p">))</span> <span class="o">/</span> <span class="mi">2</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_Pow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">log</span><span class="p">:</span>
-            <span class="n">I</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="n">I</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">**-</span><span class="n">I</span><span class="o">/</span><span class="mi">2</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_sin</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sin</span><span class="p">(</span><span class="n">arg</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_tan</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">tan_half</span> <span class="o">=</span> <span class="n">tan</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="o">*</span><span class="n">arg</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-        <span class="k">return</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">tan_half</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">tan_half</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_sincos</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sin</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span><span class="o">/</span><span class="n">sin</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_cot</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">cot_half</span> <span class="o">=</span> <span class="n">cot</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="o">*</span><span class="n">arg</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">cot_half</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">cot_half</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_pow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_rewrite_as_sqrt</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_sqrt</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">_EXPAND_INTS</span> <span class="o">=</span> <span class="bp">False</span>
-
-        <span class="k">def</span> <span class="nf">migcdex</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-            <span class="c"># recursive calcuation of gcd and linear combination</span>
-            <span class="c"># for a sequence of integers.</span>
-            <span class="c"># Given  (x1, x2, x3)</span>
-            <span class="c"># Returns (y1, y1, y3, g)</span>
-            <span class="c"># such that g is the gcd and x1*y1+x2*y2+x3*y3 - g = 0</span>
-            <span class="c"># Note, that this is only one such linear combination.</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">igcdex</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">migcdex</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-            <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">igcdex</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">g</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-            <span class="k">return</span> <span class="nb">tuple</span><span class="p">([</span><span class="n">u</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">v</span><span class="o">*</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">g</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">h</span><span class="p">])</span>
-
-        <span class="k">def</span> <span class="nf">ipartfrac</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">factors</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">r</span>
-            <span class="k">assert</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Rational</span><span class="p">)</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">r</span><span class="o">.</span><span class="n">q</span>
-            <span class="k">if</span> <span class="mi">2</span> <span class="o">&gt;</span> <span class="n">r</span><span class="o">.</span><span class="n">q</span><span class="o">*</span><span class="n">r</span><span class="o">.</span><span class="n">q</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">r</span><span class="o">.</span><span class="n">q</span>
-
-            <span class="k">if</span> <span class="bp">None</span> <span class="o">==</span> <span class="n">factors</span><span class="p">:</span>
-                <span class="n">a</span> <span class="o">=</span> <span class="p">[</span><span class="n">n</span><span class="o">//</span><span class="n">x</span><span class="o">**</span><span class="n">y</span> <span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">factorint</span><span class="p">(</span><span class="n">r</span><span class="o">.</span><span class="n">q</span><span class="p">)</span><span class="o">.</span><span class="n">items</span><span class="p">()]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">a</span> <span class="o">=</span> <span class="p">[</span><span class="n">n</span><span class="o">//</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">factors</span><span class="p">]</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span> <span class="n">r</span> <span class="p">]</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">migcdex</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-            <span class="n">ans</span> <span class="o">=</span> <span class="p">[</span> <span class="n">r</span><span class="o">.</span><span class="n">p</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">Rational</span><span class="p">(</span><span class="n">i</span><span class="o">*</span><span class="n">j</span><span class="p">,</span> <span class="n">r</span><span class="o">.</span><span class="n">q</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">h</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">a</span><span class="p">)</span> <span class="p">]</span>
-            <span class="k">assert</span> <span class="n">r</span> <span class="o">==</span> <span class="nb">sum</span><span class="p">(</span><span class="n">ans</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">ans</span>
-        <span class="n">pi_coeff</span> <span class="o">=</span> <span class="n">_pi_coeff</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">pi_coeff</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="k">assert</span> <span class="ow">not</span> <span class="n">pi_coeff</span><span class="o">.</span><span class="n">is_integer</span><span class="p">,</span> <span class="s">&quot;should have been simplified already&quot;</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">pi_coeff</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="n">cst_table_some</span> <span class="o">=</span> <span class="p">{</span>
-            <span class="mi">3</span><span class="p">:</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span>
-            <span class="mi">5</span><span class="p">:</span> <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span>
-            <span class="mi">17</span><span class="p">:</span> <span class="n">sqrt</span><span class="p">((</span><span class="mi">15</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">17</span><span class="p">))</span><span class="o">/</span><span class="mi">32</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">17</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">17</span><span class="p">))</span> <span class="o">+</span>
-                <span class="n">sqrt</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">17</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">17</span><span class="p">))</span> <span class="o">-</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">17</span><span class="p">))</span>
-                <span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">17</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">17</span><span class="p">)))</span> <span class="o">+</span> <span class="mi">6</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">17</span><span class="p">)</span> <span class="o">+</span> <span class="mi">34</span><span class="p">))</span><span class="o">/</span><span class="mi">32</span><span class="p">)</span>
-            <span class="c"># 65537 and 257 are the only other known Fermat primes</span>
-            <span class="c"># Please add if you would like them</span>
-        <span class="p">}</span>
-
-        <span class="k">def</span> <span class="nf">fermatCoords</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="k">assert</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="nb">int</span><span class="p">)</span>
-            <span class="k">assert</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">0</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="mi">0</span> <span class="o">==</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="n">primes</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span> <span class="p">[(</span><span class="n">p</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">cst_table_some</span> <span class="p">]</span> <span class="p">)</span>
-            <span class="k">assert</span> <span class="mi">1</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">primes</span>
-            <span class="k">for</span> <span class="n">p_i</span> <span class="ow">in</span> <span class="n">primes</span><span class="p">:</span>
-                <span class="k">while</span> <span class="mi">0</span> <span class="o">==</span> <span class="n">n</span> <span class="o">%</span> <span class="n">p_i</span><span class="p">:</span>
-                    <span class="n">n</span> <span class="o">=</span> <span class="n">n</span><span class="o">/</span><span class="n">p_i</span>
-                    <span class="n">primes</span><span class="p">[</span><span class="n">p_i</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="mi">1</span> <span class="o">!=</span> <span class="n">n</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">if</span> <span class="nb">max</span><span class="p">(</span><span class="n">primes</span><span class="o">.</span><span class="n">values</span><span class="p">())</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">return</span> <span class="nb">tuple</span><span class="p">([</span> <span class="n">p</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">primes</span> <span class="k">if</span> <span class="n">primes</span><span class="p">[</span><span class="n">p</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">pi_coeff</span><span class="o">.</span><span class="n">q</span> <span class="ow">in</span> <span class="n">cst_table_some</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">chebyshevt</span><span class="p">(</span><span class="n">pi_coeff</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="n">cst_table_some</span><span class="p">[</span><span class="n">pi_coeff</span><span class="o">.</span><span class="n">q</span><span class="p">])</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="mi">0</span> <span class="o">==</span> <span class="n">pi_coeff</span><span class="o">.</span><span class="n">q</span> <span class="o">%</span> <span class="mi">2</span><span class="p">:</span>  <span class="c"># recursively remove powers of 2</span>
-            <span class="n">narg</span> <span class="o">=</span> <span class="p">(</span><span class="n">pi_coeff</span><span class="o">*</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-            <span class="n">nval</span> <span class="o">=</span> <span class="n">cos</span><span class="p">(</span><span class="n">narg</span><span class="p">)</span>
-            <span class="k">if</span> <span class="bp">None</span> <span class="o">==</span> <span class="n">nval</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="n">nval</span> <span class="o">=</span> <span class="n">nval</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">sqrt</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">_EXPAND_INTS</span><span class="p">:</span>
-                <span class="k">if</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">nval</span><span class="p">,</span> <span class="n">cos</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="o">-</span><span class="n">nval</span><span class="p">,</span> <span class="n">cos</span><span class="p">)):</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi_coeff</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-            <span class="n">sign_cos</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">((</span><span class="o">-</span><span class="mi">1</span> <span class="k">if</span> <span class="n">x</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="k">else</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="nb">int</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
-            <span class="k">return</span> <span class="n">sign_cos</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">nval</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="p">)</span>
-
-        <span class="n">FC</span> <span class="o">=</span> <span class="n">fermatCoords</span><span class="p">(</span><span class="n">pi_coeff</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">FC</span><span class="p">:</span>
-            <span class="n">decomp</span> <span class="o">=</span> <span class="n">ipartfrac</span><span class="p">(</span><span class="n">pi_coeff</span><span class="p">,</span> <span class="n">FC</span><span class="p">)</span>
-            <span class="n">X</span> <span class="o">=</span> <span class="p">[(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">decomp</span><span class="p">,</span> <span class="n">numbered_symbols</span><span class="p">(</span><span class="s">&#39;z&#39;</span><span class="p">))]</span>
-            <span class="n">pcls</span> <span class="o">=</span> <span class="n">cos</span><span class="p">(</span><span class="nb">sum</span><span class="p">([</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">X</span><span class="p">]))</span><span class="o">.</span><span class="n">_eval_expand_trig</span><span class="p">()</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">pcls</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">sqrt</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">_EXPAND_INTS</span><span class="p">:</span>
-            <span class="n">decomp</span> <span class="o">=</span> <span class="n">ipartfrac</span><span class="p">(</span><span class="n">pi_coeff</span><span class="p">)</span>
-            <span class="n">X</span> <span class="o">=</span> <span class="p">[(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">decomp</span><span class="p">,</span> <span class="n">numbered_symbols</span><span class="p">(</span><span class="s">&#39;z&#39;</span><span class="p">))]</span>
-            <span class="n">pcls</span> <span class="o">=</span> <span class="n">cos</span><span class="p">(</span><span class="nb">sum</span><span class="p">([</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">X</span><span class="p">]))</span><span class="o">.</span><span class="n">_eval_expand_trig</span><span class="p">()</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">pcls</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">conjugate</span><span class="p">())</span>
-
-    <span class="k">def</span> <span class="nf">as_real_imag</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-                <span class="n">hints</span><span class="p">[</span><span class="s">&#39;complex&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">re</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">cosh</span><span class="p">(</span><span class="n">im</span><span class="p">),</span> <span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">re</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">sinh</span><span class="p">(</span><span class="n">im</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_expand_trig</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>  <span class="c"># TODO: Do this more efficiently for more than two terms</span>
-            <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_two_terms</span><span class="p">()</span>
-            <span class="n">sx</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_expand_trig</span><span class="p">()</span>
-            <span class="n">sy</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_expand_trig</span><span class="p">()</span>
-            <span class="n">cx</span> <span class="o">=</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_expand_trig</span><span class="p">()</span>
-            <span class="n">cy</span> <span class="o">=</span> <span class="n">cos</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_expand_trig</span><span class="p">()</span>
-            <span class="k">return</span> <span class="n">cx</span><span class="o">*</span><span class="n">cy</span> <span class="o">-</span> <span class="n">sx</span><span class="o">*</span><span class="n">sy</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">coeff</span><span class="p">,</span> <span class="n">terms</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">(</span><span class="n">rational</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">chebyshevt</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">cos</span><span class="p">(</span><span class="n">terms</span><span class="p">))</span>
-            <span class="n">pi_coeff</span> <span class="o">=</span> <span class="n">_pi_coeff</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">pi_coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">pi_coeff</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">sqrt</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">cos</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_bounded</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">())</span>
-
-</div>
-<span class="k">class</span> <span class="nc">sec</span><span class="p">(</span><span class="n">TrigonometricFunction</span><span class="p">):</span>  <span class="c"># TODO implement rest all functions for sec. see cos, sin, tan.</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_cos</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_sincos</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sin</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span>
-
-
-<span class="k">class</span> <span class="nc">csc</span><span class="p">(</span><span class="n">TrigonometricFunction</span><span class="p">):</span>  <span class="c"># TODO implement rest all functions for csc. see cos, sin, tan.</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_sin</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">sin</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_sincos</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">cos</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span>
-
-
-<div class="viewcode-block" id="tan"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.trigonometric.tan">[docs]</a><span class="k">class</span> <span class="nc">tan</span><span class="p">(</span><span class="n">TrigonometricFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    tan(x) -&gt; Returns the tangent of x (measured in radians)</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    * tan(x) will evaluate automatically in the case x is a</span>
-<span class="sd">      multiple of pi.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import tan</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; tan(x**2).diff(x)</span>
-<span class="sd">    2*x*(tan(x**2)**2 + 1)</span>
-<span class="sd">    &gt;&gt;&gt; tan(1).diff(x)</span>
-<span class="sd">    0</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sin, cos, atan</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    U{Definitions in trigonometry&lt;http://planetmath.org/encyclopedia/DefinitionsInTrigonometry.html&gt;}</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="o">+</span> <span class="bp">self</span><span class="o">**</span><span class="mi">2</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-<div class="viewcode-block" id="tan.inverse"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.trigonometric.tan.inverse">[docs]</a>    <span class="k">def</span> <span class="nf">inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the inverse of this function.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">atan</span>
-</div>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">():</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="n">i_coeff</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">i_coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">tanh</span><span class="p">(</span><span class="n">i_coeff</span><span class="p">)</span>
-
-        <span class="n">pi_coeff</span> <span class="o">=</span> <span class="n">_pi_coeff</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">pi_coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">pi_coeff</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">pi_coeff</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="n">narg</span> <span class="o">=</span> <span class="n">pi_coeff</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-                <span class="k">if</span> <span class="n">narg</span> <span class="o">!=</span> <span class="n">arg</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">narg</span><span class="p">)</span>
-                <span class="k">return</span> <span class="bp">None</span>
-
-            <span class="k">if</span> <span class="n">pi_coeff</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="n">narg</span> <span class="o">=</span> <span class="p">((</span><span class="n">pi_coeff</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span> <span class="o">%</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-                <span class="c"># see cos() to specify which expressions should  be</span>
-                <span class="c"># expanded automatically in terms of radicals</span>
-                <span class="n">cresult</span><span class="p">,</span> <span class="n">sresult</span> <span class="o">=</span> <span class="n">cos</span><span class="p">(</span><span class="n">narg</span><span class="p">),</span> <span class="n">cos</span><span class="p">(</span><span class="n">narg</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">cresult</span><span class="p">,</span> <span class="n">cos</span><span class="p">)</span> \
-                        <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">sresult</span><span class="p">,</span> <span class="n">cos</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">cresult</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span>
-                    <span class="k">return</span> <span class="p">(</span><span class="n">sresult</span><span class="o">/</span><span class="n">cresult</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">narg</span> <span class="o">!=</span> <span class="n">arg</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">narg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">x</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">_peeloff_pi</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">m</span><span class="p">:</span>
-                <span class="n">tanm</span> <span class="o">=</span> <span class="n">tan</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-                <span class="n">tanx</span> <span class="o">=</span> <span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">tanm</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="o">-</span><span class="n">cot</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-                <span class="k">return</span> <span class="p">(</span><span class="n">tanm</span> <span class="o">+</span> <span class="n">tanx</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">tanm</span><span class="o">*</span><span class="n">tanx</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">atan</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">atan2</span><span class="p">:</span>
-            <span class="n">y</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">return</span> <span class="n">y</span><span class="o">/</span><span class="n">x</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">asin</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">x</span> <span class="o">/</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">acos</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="n">x</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">acot</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">x</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-            <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="p">((</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span><span class="p">),</span> <span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-            <span class="n">B</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">bernoulli</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="n">F</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-            <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">a</span> <span class="o">*</span> <span class="n">b</span><span class="o">*</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">B</span><span class="o">/</span><span class="n">F</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="n">n</span>
-
-    <span class="k">def</span> <span class="nf">_eval_nseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span><span class="o">*</span><span class="mi">2</span><span class="o">/</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="ow">and</span> <span class="n">i</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">cos</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Function</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_Pow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">log</span><span class="p">:</span>
-            <span class="n">I</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">I</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">**-</span><span class="n">I</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="n">I</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">**-</span><span class="n">I</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="n">I</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">conjugate</span><span class="p">())</span>
-
-    <span class="k">def</span> <span class="nf">as_real_imag</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-                <span class="n">hints</span><span class="p">[</span><span class="s">&#39;complex&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-        <span class="n">denom</span> <span class="o">=</span> <span class="n">cos</span><span class="p">(</span><span class="n">re</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">C</span><span class="o">.</span><span class="n">sinh</span><span class="p">(</span><span class="n">im</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">re</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">re</span><span class="p">)</span><span class="o">/</span><span class="n">denom</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">sinh</span><span class="p">(</span><span class="n">im</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">cosh</span><span class="p">(</span><span class="n">im</span><span class="p">)</span><span class="o">/</span><span class="n">denom</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_expand_trig</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symmetric_poly</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-            <span class="n">TX</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="n">tx</span> <span class="o">=</span> <span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_expand_trig</span><span class="p">()</span>
-                <span class="n">TX</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tx</span><span class="p">)</span>
-
-            <span class="n">Yg</span> <span class="o">=</span> <span class="n">numbered_symbols</span><span class="p">(</span><span class="s">&#39;Y&#39;</span><span class="p">)</span>
-            <span class="n">Y</span> <span class="o">=</span> <span class="p">[</span> <span class="nb">next</span><span class="p">(</span><span class="n">Yg</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="p">]</span>
-
-            <span class="n">p</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="n">p</span><span class="p">[</span><span class="mi">1</span> <span class="o">-</span> <span class="n">i</span> <span class="o">%</span> <span class="mi">2</span><span class="p">]</span> <span class="o">+=</span> <span class="n">symmetric_poly</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">((</span><span class="n">i</span> <span class="o">%</span> <span class="mi">4</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span><span class="p">)</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="n">TX</span><span class="p">)))</span>
-
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">coeff</span><span class="p">,</span> <span class="n">terms</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">(</span><span class="n">rational</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">coeff</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">I</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-                <span class="n">z</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;dummy&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                <span class="n">P</span> <span class="o">=</span> <span class="p">((</span><span class="mi">1</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="n">coeff</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-                <span class="k">return</span> <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">im</span><span class="p">(</span><span class="n">P</span><span class="p">)</span><span class="o">/</span><span class="n">C</span><span class="o">.</span><span class="n">re</span><span class="p">(</span><span class="n">P</span><span class="p">))</span><span class="o">.</span><span class="n">subs</span><span class="p">([(</span><span class="n">z</span><span class="p">,</span> <span class="n">tan</span><span class="p">(</span><span class="n">terms</span><span class="p">))])</span>
-        <span class="k">return</span> <span class="n">tan</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_exp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">exp</span><span class="p">,</span> <span class="n">I</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">TrigonometricFunction</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">HyperbolicFunction</span><span class="p">):</span>
-            <span class="n">arg</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">exp</span><span class="p">)</span>
-        <span class="n">neg_exp</span><span class="p">,</span> <span class="n">pos_exp</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="o">*</span><span class="n">I</span><span class="p">),</span> <span class="n">exp</span><span class="p">(</span><span class="n">arg</span><span class="o">*</span><span class="n">I</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">I</span><span class="o">*</span><span class="p">(</span><span class="n">neg_exp</span> <span class="o">-</span> <span class="n">pos_exp</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">neg_exp</span> <span class="o">+</span> <span class="n">pos_exp</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_sin</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_cos</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_sincos</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sin</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_cot</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="n">cot</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_pow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">y</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">cos</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="nb">pow</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">y</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">cos</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">return</span> <span class="n">y</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_sqrt</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">y</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">cos</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">sqrt</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">y</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">cos</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">return</span> <span class="n">y</span>
-
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">arg</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_bounded</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">tan</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">())</span>
-
-</div>
-<div class="viewcode-block" id="cot"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.trigonometric.cot">[docs]</a><span class="k">class</span> <span class="nc">cot</span><span class="p">(</span><span class="n">TrigonometricFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    cot(x) -&gt; Returns the cotangent of x (measured in radians)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span> <span class="o">-</span> <span class="bp">self</span><span class="o">**</span><span class="mi">2</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-<div class="viewcode-block" id="cot.inverse"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.trigonometric.cot.inverse">[docs]</a>    <span class="k">def</span> <span class="nf">inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the inverse of this function.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">acot</span>
-</div>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">():</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="n">i_coeff</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">i_coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">coth</span><span class="p">(</span><span class="n">i_coeff</span><span class="p">)</span>
-
-        <span class="n">pi_coeff</span> <span class="o">=</span> <span class="n">_pi_coeff</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">pi_coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">pi_coeff</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">pi_coeff</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="n">narg</span> <span class="o">=</span> <span class="n">pi_coeff</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-                <span class="k">if</span> <span class="n">narg</span> <span class="o">!=</span> <span class="n">arg</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">narg</span><span class="p">)</span>
-                <span class="k">return</span> <span class="bp">None</span>
-
-            <span class="k">if</span> <span class="n">pi_coeff</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="n">narg</span> <span class="o">=</span> <span class="p">(((</span><span class="n">pi_coeff</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span> <span class="o">%</span> <span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-                <span class="c"># see cos() to specify which expressions should be</span>
-                <span class="c"># expanded automatically in terms of radicals</span>
-                <span class="n">cresult</span><span class="p">,</span> <span class="n">sresult</span> <span class="o">=</span> <span class="n">cos</span><span class="p">(</span><span class="n">narg</span><span class="p">),</span> <span class="n">cos</span><span class="p">(</span><span class="n">narg</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">cresult</span><span class="p">,</span> <span class="n">cos</span><span class="p">)</span> \
-                        <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">sresult</span><span class="p">,</span> <span class="n">cos</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">sresult</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span>
-                    <span class="k">return</span> <span class="n">cresult</span> <span class="o">/</span> <span class="n">sresult</span>
-                <span class="k">if</span> <span class="n">narg</span> <span class="o">!=</span> <span class="n">arg</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">narg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">x</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">_peeloff_pi</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">m</span><span class="p">:</span>
-                <span class="n">cotm</span> <span class="o">=</span> <span class="n">cot</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">cotm</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="o">-</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-                <span class="n">cotx</span> <span class="o">=</span> <span class="n">cot</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">cotm</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">cotx</span>
-                <span class="k">if</span> <span class="n">cotm</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="p">(</span><span class="n">cotm</span><span class="o">*</span><span class="n">cotx</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">cotm</span> <span class="o">+</span> <span class="n">cotx</span><span class="p">)</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">acot</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">atan</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">x</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">atan2</span><span class="p">:</span>
-            <span class="n">y</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">return</span> <span class="n">x</span><span class="o">/</span><span class="n">y</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">asin</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="n">x</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">acos</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">x</span> <span class="o">/</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-            <span class="n">B</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">bernoulli</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="n">F</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-            <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">((</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">B</span><span class="o">/</span><span class="n">F</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="n">n</span>
-
-    <span class="k">def</span> <span class="nf">_eval_nseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span><span class="o">/</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="ow">and</span> <span class="n">i</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">cos</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Function</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">conjugate</span><span class="p">())</span>
-
-    <span class="k">def</span> <span class="nf">as_real_imag</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-                <span class="n">hints</span><span class="p">[</span><span class="s">&#39;complex&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-        <span class="n">denom</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">re</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">C</span><span class="o">.</span><span class="n">sinh</span><span class="p">(</span><span class="n">im</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">re</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">re</span><span class="p">)</span><span class="o">/</span><span class="n">denom</span><span class="p">,</span> <span class="o">-</span><span class="n">C</span><span class="o">.</span><span class="n">sinh</span><span class="p">(</span><span class="n">im</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">cosh</span><span class="p">(</span><span class="n">im</span><span class="p">)</span><span class="o">/</span><span class="n">denom</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_exp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">exp</span><span class="p">,</span> <span class="n">I</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">TrigonometricFunction</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">HyperbolicFunction</span><span class="p">):</span>
-            <span class="n">arg</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">exp</span><span class="p">)</span>
-        <span class="n">neg_exp</span><span class="p">,</span> <span class="n">pos_exp</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="o">*</span><span class="n">I</span><span class="p">),</span> <span class="n">exp</span><span class="p">(</span><span class="n">arg</span><span class="o">*</span><span class="n">I</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">I</span><span class="o">*</span><span class="p">(</span><span class="n">pos_exp</span> <span class="o">+</span> <span class="n">neg_exp</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">pos_exp</span> <span class="o">-</span> <span class="n">neg_exp</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_Pow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">log</span><span class="p">:</span>
-            <span class="n">I</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">**-</span><span class="n">I</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="n">I</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">**-</span><span class="n">I</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="n">I</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_sin</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_cos</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_sincos</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">cos</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span><span class="o">/</span><span class="n">sin</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_tan</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="n">tan</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_pow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">y</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">cos</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="nb">pow</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">y</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">cos</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">return</span> <span class="n">y</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_sqrt</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">y</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">cos</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">sqrt</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">y</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">cos</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">return</span> <span class="n">y</span>
-
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="n">arg</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span>
-
-    <span class="k">def</span> <span class="nf">_eval_expand_trig</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symmetric_poly</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-            <span class="n">CX</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="n">cx</span> <span class="o">=</span> <span class="n">cot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_expand_trig</span><span class="p">()</span>
-                <span class="n">CX</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">cx</span><span class="p">)</span>
-
-            <span class="n">Yg</span> <span class="o">=</span> <span class="n">numbered_symbols</span><span class="p">(</span><span class="s">&#39;Y&#39;</span><span class="p">)</span>
-            <span class="n">Y</span> <span class="o">=</span> <span class="p">[</span> <span class="nb">next</span><span class="p">(</span><span class="n">Yg</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="p">]</span>
-
-            <span class="n">p</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-                <span class="n">p</span><span class="p">[(</span><span class="n">n</span> <span class="o">-</span> <span class="n">i</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span><span class="p">]</span> <span class="o">+=</span> <span class="n">symmetric_poly</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(((</span><span class="n">n</span> <span class="o">-</span> <span class="n">i</span><span class="p">)</span> <span class="o">%</span> <span class="mi">4</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span><span class="p">)</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="n">CX</span><span class="p">)))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">coeff</span><span class="p">,</span> <span class="n">terms</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">(</span><span class="n">rational</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">coeff</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">I</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-                <span class="n">z</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;dummy&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                <span class="n">P</span> <span class="o">=</span> <span class="p">((</span><span class="n">z</span> <span class="o">+</span> <span class="n">I</span><span class="p">)</span><span class="o">**</span><span class="n">coeff</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-                <span class="k">return</span> <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">re</span><span class="p">(</span><span class="n">P</span><span class="p">)</span><span class="o">/</span><span class="n">C</span><span class="o">.</span><span class="n">im</span><span class="p">(</span><span class="n">P</span><span class="p">))</span><span class="o">.</span><span class="n">subs</span><span class="p">([(</span><span class="n">z</span><span class="p">,</span> <span class="n">cot</span><span class="p">(</span><span class="n">terms</span><span class="p">))])</span>
-        <span class="k">return</span> <span class="n">cot</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">cot</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">())</span>
-
-<span class="c">###############################################################################</span>
-<span class="c">########################### TRIGONOMETRIC INVERSES ############################</span>
-<span class="c">###############################################################################</span>
-
-</div>
-<div class="viewcode-block" id="asin"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.trigonometric.asin">[docs]</a><span class="k">class</span> <span class="nc">asin</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    asin(x) -&gt; Returns the arc sine of x (measured in radians)</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    * asin(x) will evaluate automatically in the cases</span>
-<span class="sd">      oo, -oo, 0, 1, -1</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import asin, oo, pi</span>
-<span class="sd">    &gt;&gt;&gt; asin(1)</span>
-<span class="sd">    pi/2</span>
-<span class="sd">    &gt;&gt;&gt; asin(-1)</span>
-<span class="sd">    -pi/2</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    acos, atan, sin</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span> <span class="o">*</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span> <span class="o">*</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">/</span> <span class="mi">2</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">/</span> <span class="mi">2</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">():</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="n">cst_table</span> <span class="o">=</span> <span class="p">{</span>
-                <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span> <span class="mi">3</span><span class="p">,</span>
-                <span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span>
-                <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span> <span class="mi">4</span><span class="p">,</span>
-                <span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span> <span class="o">-</span><span class="mi">4</span><span class="p">,</span>
-                <span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">):</span> <span class="mi">4</span><span class="p">,</span>
-                <span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">):</span> <span class="o">-</span><span class="mi">4</span><span class="p">,</span>
-                <span class="n">sqrt</span><span class="p">((</span><span class="mi">5</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span><span class="o">/</span><span class="mi">8</span><span class="p">):</span> <span class="mi">5</span><span class="p">,</span>
-                <span class="o">-</span><span class="n">sqrt</span><span class="p">((</span><span class="mi">5</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span><span class="o">/</span><span class="mi">8</span><span class="p">):</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span>
-                <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">:</span> <span class="mi">6</span><span class="p">,</span>
-                <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">:</span> <span class="o">-</span><span class="mi">6</span><span class="p">,</span>
-                <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span> <span class="mi">8</span><span class="p">,</span>
-                <span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span> <span class="o">-</span><span class="mi">8</span><span class="p">,</span>
-                <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">:</span> <span class="mi">10</span><span class="p">,</span>
-                <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span><span class="o">/</span><span class="mi">4</span><span class="p">:</span> <span class="o">-</span><span class="mi">10</span><span class="p">,</span>
-                <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="mi">3</span><span class="p">):</span> <span class="mi">12</span><span class="p">,</span>
-                <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="mi">3</span><span class="p">):</span> <span class="o">-</span><span class="mi">12</span><span class="p">,</span>
-                <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">:</span> <span class="n">S</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span><span class="p">,</span>
-                <span class="o">-</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">:</span> <span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span>
-            <span class="p">}</span>
-
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">cst_table</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">/</span> <span class="n">cst_table</span><span class="p">[</span><span class="n">arg</span><span class="p">]</span>
-
-        <span class="n">i_coeff</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">i_coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">asinh</span><span class="p">(</span><span class="n">i_coeff</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">previous_terms</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="n">previous_terms</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span>
-                <span class="k">return</span> <span class="n">p</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">k</span> <span class="o">=</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>
-                <span class="n">R</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-                <span class="n">F</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">R</span> <span class="o">/</span> <span class="n">F</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="n">n</span> <span class="o">/</span> <span class="n">n</span>
-
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">arg</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_acos</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">acos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_atan</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">atan</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_log</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&gt;=</span> <span class="o">-</span><span class="mi">1</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">asin</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">())</span>
-
-</div>
-<div class="viewcode-block" id="acos"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.trigonometric.acos">[docs]</a><span class="k">class</span> <span class="nc">acos</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    acos(x) -&gt; Returns the arc cosine of x (measured in radians)</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    * acos(x) will evaluate automatically in the cases</span>
-<span class="sd">      oo, -oo, 0, 1, -1</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import acos, oo, pi</span>
-<span class="sd">    &gt;&gt;&gt; acos(1)</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; acos(0)</span>
-<span class="sd">    pi/2</span>
-<span class="sd">    &gt;&gt;&gt; acos(oo)</span>
-<span class="sd">    oo*I</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    asin, atan, cos</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span> <span class="o">*</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span> <span class="o">*</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">/</span> <span class="mi">2</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="n">cst_table</span> <span class="o">=</span> <span class="p">{</span>
-                <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">:</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">3</span><span class="p">,</span>
-                <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">:</span> <span class="mi">2</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">3</span><span class="p">,</span>
-                <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span>
-                <span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span> <span class="mi">3</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span>
-                <span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">):</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span>
-                <span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">):</span> <span class="mi">3</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span>
-                <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">6</span><span class="p">,</span>
-                <span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span> <span class="mi">5</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">6</span><span class="p">,</span>
-            <span class="p">}</span>
-
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">cst_table</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">cst_table</span><span class="p">[</span><span class="n">arg</span><span class="p">]</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">/</span> <span class="mi">2</span>
-        <span class="k">elif</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">previous_terms</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="n">previous_terms</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span>
-                <span class="k">return</span> <span class="n">p</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">k</span> <span class="o">=</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>
-                <span class="n">R</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-                <span class="n">F</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">R</span> <span class="o">/</span> <span class="n">F</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="n">n</span> <span class="o">/</span> <span class="n">n</span>
-
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">arg</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&gt;=</span> <span class="o">-</span><span class="mi">1</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_log</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">x</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_asin</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">asin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_atan</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="o">&gt;</span> <span class="o">-</span><span class="mi">1</span> <span class="ow">and</span> <span class="n">x</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">atan</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&quot;The argument must be bounded in the interval (-1,1]&quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">acos</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">())</span>
-
-</div>
-<div class="viewcode-block" id="atan"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.trigonometric.atan">[docs]</a><span class="k">class</span> <span class="nc">atan</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    atan(x) -&gt; Returns the arc tangent of x (measured in radians)</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    * atan(x) will evaluate automatically in the cases</span>
-<span class="sd">      oo, -oo, 0, 1, -1</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import atan, oo, pi</span>
-<span class="sd">    &gt;&gt;&gt; atan(0)</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; atan(1)</span>
-<span class="sd">    pi/4</span>
-<span class="sd">    &gt;&gt;&gt; atan(oo)</span>
-<span class="sd">    pi/2</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    acos, asin, tan</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">/</span> <span class="mi">2</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">/</span> <span class="mi">2</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">/</span> <span class="mi">4</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">/</span> <span class="mi">4</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">():</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="n">cst_table</span> <span class="o">=</span> <span class="p">{</span>
-                <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span><span class="p">:</span> <span class="mi">6</span><span class="p">,</span>
-                <span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span><span class="p">:</span> <span class="o">-</span><span class="mi">6</span><span class="p">,</span>
-                <span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">):</span> <span class="mi">6</span><span class="p">,</span>
-                <span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">):</span> <span class="o">-</span><span class="mi">6</span><span class="p">,</span>
-                <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">):</span> <span class="mi">3</span><span class="p">,</span>
-                <span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">):</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span>
-                <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)):</span> <span class="n">S</span><span class="p">(</span><span class="mi">8</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span><span class="p">,</span>
-                <span class="o">-</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)):</span> <span class="n">S</span><span class="p">(</span><span class="mi">8</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span><span class="p">,</span>
-                <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span> <span class="mi">8</span><span class="p">,</span>
-                <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)):</span> <span class="o">-</span><span class="mi">8</span><span class="p">,</span>
-                <span class="n">sqrt</span><span class="p">((</span><span class="mi">5</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">))):</span> <span class="n">S</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span>
-                <span class="o">-</span><span class="n">sqrt</span><span class="p">((</span><span class="mi">5</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">))):</span> <span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span>
-                <span class="p">(</span><span class="mi">2</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)):</span> <span class="mi">12</span><span class="p">,</span>
-                <span class="o">-</span><span class="p">(</span><span class="mi">2</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)):</span> <span class="o">-</span><span class="mi">12</span>
-            <span class="p">}</span>
-
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">cst_table</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">/</span> <span class="n">cst_table</span><span class="p">[</span><span class="n">arg</span><span class="p">]</span>
-
-        <span class="n">i_coeff</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">i_coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">atanh</span><span class="p">(</span><span class="n">i_coeff</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">((</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="n">n</span> <span class="o">/</span> <span class="n">n</span>
-
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">arg</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_log</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">/</span><span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">log</span><span class="p">(</span>
-            <span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">x</span><span class="p">)))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_aseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">args0</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">args0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">atan</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">elif</span> <span class="n">args0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">atan</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">atan</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_aseries</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">args0</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">logx</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">atan</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">())</span>
-
-</div>
-<div class="viewcode-block" id="acot"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.trigonometric.acot">[docs]</a><span class="k">class</span> <span class="nc">acot</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    acot(x) -&gt; Returns the arc cotangent of x (measured in radians)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="mi">1</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span> <span class="mi">2</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">/</span> <span class="mi">4</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">/</span> <span class="mi">4</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">():</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="n">cst_table</span> <span class="o">=</span> <span class="p">{</span>
-                <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span><span class="p">:</span> <span class="mi">3</span><span class="p">,</span>
-                <span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span><span class="p">:</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span>
-                <span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">):</span> <span class="mi">3</span><span class="p">,</span>
-                <span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">):</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span>
-                <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">):</span> <span class="mi">6</span><span class="p">,</span>
-                <span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">):</span> <span class="o">-</span><span class="mi">6</span><span class="p">,</span>
-                <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)):</span> <span class="mi">8</span><span class="p">,</span>
-                <span class="o">-</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)):</span> <span class="o">-</span><span class="mi">8</span><span class="p">,</span>
-                <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)):</span> <span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">8</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span><span class="p">,</span>
-                <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span> <span class="n">S</span><span class="p">(</span><span class="mi">8</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span><span class="p">,</span>
-                <span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">)):</span> <span class="mi">10</span><span class="p">,</span>
-                <span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">)):</span> <span class="o">-</span><span class="mi">10</span><span class="p">,</span>
-                <span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)):</span> <span class="mi">12</span><span class="p">,</span>
-                <span class="o">-</span><span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)):</span> <span class="o">-</span><span class="mi">12</span><span class="p">,</span>
-                <span class="p">(</span><span class="mi">2</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)):</span> <span class="n">S</span><span class="p">(</span><span class="mi">12</span><span class="p">)</span><span class="o">/</span><span class="mi">5</span><span class="p">,</span>
-                <span class="o">-</span><span class="p">(</span><span class="mi">2</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)):</span> <span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">12</span><span class="p">)</span><span class="o">/</span><span class="mi">5</span><span class="p">,</span>
-            <span class="p">}</span>
-
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">cst_table</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">/</span> <span class="n">cst_table</span><span class="p">[</span><span class="n">arg</span><span class="p">]</span>
-
-        <span class="n">i_coeff</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">i_coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">acoth</span><span class="p">(</span><span class="n">i_coeff</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">/</span> <span class="mi">2</span>  <span class="c"># FIX THIS</span>
-        <span class="k">elif</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">((</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="n">n</span> <span class="o">/</span> <span class="n">n</span>
-
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">arg</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span>
-
-    <span class="k">def</span> <span class="nf">_eval_aseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">args0</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">args0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">acot</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">elif</span> <span class="n">args0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">3</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">acot</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">atan</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_aseries</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">args0</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">logx</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">acot</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">())</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_log</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">/</span><span class="mi">2</span> <span class="o">*</span> \
-            <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">log</span><span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)))</span>
-
-</div>
-<div class="viewcode-block" id="atan2"><a class="viewcode-back" href="../../../../modules/functions/elementary.html#sympy.functions.elementary.trigonometric.atan2">[docs]</a><span class="k">class</span> <span class="nc">atan2</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    atan2(y,x) -&gt; Returns the atan(y/x) taking two arguments y and x.</span>
-<span class="sd">    Signs of both y and x are considered to determine the appropriate</span>
-<span class="sd">    quadrant of atan(y/x). The range is (-pi, pi].</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">sign_y</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">y</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">elif</span> <span class="n">x</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">x</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-        <span class="k">elif</span> <span class="n">x</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">sign_y</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">sign_y</span> <span class="o">*</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">2</span>
-        <span class="k">elif</span> <span class="n">x</span><span class="o">.</span><span class="n">is_zero</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="n">abs_yx</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Abs</span><span class="p">(</span><span class="n">y</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">sign_y</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">and</span> <span class="n">abs_yx</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-                <span class="n">phi</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">atan</span><span class="p">(</span><span class="n">abs_yx</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">sign_y</span> <span class="o">*</span> <span class="n">phi</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">sign_y</span> <span class="o">*</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">-</span> <span class="n">phi</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">):</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">y</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">x</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">import</span> <span class="nn">sage.all</span> <span class="kn">as</span> <span class="nn">sage</span>
-        <span class="k">return</span> <span class="n">sage</span><span class="o">.</span><span class="n">atan2</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">(),</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">_sage_</span><span class="p">())</span></div>
-</pre></div>
-
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-        <div class="sphinxsidebarwrapper">
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deleted file mode 100644
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-            
-  <h1>Source code for sympy.functions.special.bessel</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Bessel type functions&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">I</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">ArgumentIndexError</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.trigonometric</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">sqrt</span>
-
-<span class="c"># TODO</span>
-<span class="c"># o Airy Ai and Bi functions</span>
-<span class="c"># o Scorer functions G1 and G2</span>
-<span class="c"># o Asymptotic expansions</span>
-<span class="c">#   These are possible, e.g. for fixed order, but since the bessel type</span>
-<span class="c">#   functions are oscillatory they are not actually tractable at</span>
-<span class="c">#   infinity, so this is not particularly useful right now.</span>
-<span class="c"># o Series Expansions for functions of the second kind about zero</span>
-<span class="c"># o Nicer series expansions.</span>
-<span class="c"># o More rewriting.</span>
-<span class="c"># o Add solvers to ode.py (or rather add solvers for the hypergeometric equation).</span>
-
-
-<div class="viewcode-block" id="BesselBase"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.bessel.BesselBase">[docs]</a><span class="k">class</span> <span class="nc">BesselBase</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Abstract base class for bessel-type functions.</span>
-
-<span class="sd">    This class is meant to reduce code duplication.</span>
-<span class="sd">    All bessel type functions can 1) be differentiated, and the derivatives</span>
-<span class="sd">    expressed in terms of similar functions and 2) be rewritten in terms</span>
-<span class="sd">    of other bessel-type functions.</span>
-
-<span class="sd">    Here &quot;bessel-type functions&quot; are assumed to have one complex parameter.</span>
-
-<span class="sd">    To use this base class, define class attributes ``_a`` and ``_b`` such that</span>
-<span class="sd">    ``2*F_n&#39; = -_a*F_{n+1} b*F_{n-1}``.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="BesselBase.order"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.bessel.BesselBase.order">[docs]</a>    <span class="k">def</span> <span class="nf">order</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; The order of the bessel-type function. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="BesselBase.argument"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.bessel.BesselBase.argument">[docs]</a>    <span class="k">def</span> <span class="nf">argument</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; The argument of the bessel-type function. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">2</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">!=</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_b</span><span class="o">/</span><span class="mi">2</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">order</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">argument</span><span class="p">)</span> \
-            <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_a</span><span class="o">/</span><span class="mi">2</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">order</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">argument</span><span class="p">)</span> \
-
-
-</div>
-<div class="viewcode-block" id="besselj"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.bessel.besselj">[docs]</a><span class="k">class</span> <span class="nc">besselj</span><span class="p">(</span><span class="n">BesselBase</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Bessel function of the first kind.</span>
-
-<span class="sd">    The Bessel J function of order :math:`\nu` is defined to be the function</span>
-<span class="sd">    satisfying Bessel&#39;s differential equation</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2}</span>
-<span class="sd">        + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu^2) w = 0,</span>
-
-<span class="sd">    with Laurent expansion</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        J_\nu(z) = z^\nu \left(\frac{1}{\Gamma(\nu + 1) 2^\nu} + O(z^2) \right),</span>
-
-<span class="sd">    if :math:`\nu` is not a negative integer. If :math:`\nu=-n \in \mathbb{Z}_{&lt;0}`</span>
-<span class="sd">    *is* a negative integer, then the definition is</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        J_{-n}(z) = (-1)^n J_n(z).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Create a bessel function object:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import besselj, jn</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import z, n</span>
-<span class="sd">    &gt;&gt;&gt; b = besselj(n, z)</span>
-
-<span class="sd">    Differentiate it:</span>
-
-<span class="sd">    &gt;&gt;&gt; b.diff(z)</span>
-<span class="sd">    besselj(n - 1, z)/2 - besselj(n + 1, z)/2</span>
-
-<span class="sd">    Rewrite in terms of spherical bessel functions:</span>
-
-<span class="sd">    &gt;&gt;&gt; b.rewrite(jn)</span>
-<span class="sd">    sqrt(2)*sqrt(z)*jn(n - 1/2, z)/sqrt(pi)</span>
-
-<span class="sd">    Access the parameter and argument:</span>
-
-<span class="sd">    &gt;&gt;&gt; b.order</span>
-<span class="sd">    n</span>
-<span class="sd">    &gt;&gt;&gt; b.argument</span>
-<span class="sd">    z</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    bessely, besseli, besselk</span>
-
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - Abramowitz, Milton; Stegun, Irene A., eds. (1965), &quot;Chapter 9&quot;,</span>
-<span class="sd">      Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical</span>
-<span class="sd">      Tables</span>
-<span class="sd">    - Luke, Y. L. (1969), The Special Functions and Their Approximations,</span>
-<span class="sd">      Volume 1</span>
-<span class="sd">    - http://en.wikipedia.org/wiki/Bessel_function</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_a</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-    <span class="n">_b</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_jn</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">expand</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="n">jn_part</span> <span class="o">=</span> <span class="n">jn</span><span class="p">(</span><span class="n">nu</span> <span class="o">-</span> <span class="n">S</span><span class="p">(</span><span class="s">&#39;1/2&#39;</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">argument</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">expand</span><span class="p">:</span>
-            <span class="n">jn_part</span> <span class="o">=</span> <span class="n">jn_part</span><span class="o">.</span><span class="n">_eval_expand_func</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="o">/</span><span class="n">pi</span><span class="p">)</span> <span class="o">*</span> <span class="n">jn_part</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">nu</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">nu</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">nu</span><span class="o">*</span><span class="n">besselj</span><span class="p">(</span><span class="o">-</span><span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">z</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">():</span>
-                <span class="k">return</span> <span class="n">S</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">nu</span><span class="o">*</span><span class="n">besselj</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="o">-</span><span class="n">z</span><span class="p">)</span>
-            <span class="n">newz</span> <span class="o">=</span> <span class="n">z</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">newz</span><span class="p">:</span>  <span class="c"># NOTE we don&#39;t want to change the function if z==0</span>
-                <span class="k">return</span> <span class="n">I</span><span class="o">**</span><span class="p">(</span><span class="n">nu</span><span class="p">)</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">newz</span><span class="p">)</span>
-
-        <span class="c"># branch handling:</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">unpolarify</span><span class="p">,</span> <span class="n">exp</span>
-        <span class="k">if</span> <span class="n">nu</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-            <span class="n">newz</span> <span class="o">=</span> <span class="n">unpolarify</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">newz</span> <span class="o">!=</span> <span class="n">z</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">besselj</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">newz</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">newz</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">z</span><span class="o">.</span><span class="n">extract_branch_factor</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">nu</span><span class="o">*</span><span class="n">I</span><span class="p">)</span><span class="o">*</span><span class="n">besselj</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">newz</span><span class="p">)</span>
-        <span class="n">nnu</span> <span class="o">=</span> <span class="n">unpolarify</span><span class="p">(</span><span class="n">nu</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">nu</span> <span class="o">!=</span> <span class="n">nnu</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">besselj</span><span class="p">(</span><span class="n">nnu</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_expand_func</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span><span class="o">.</span><span class="n">q</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_rewrite_as_jn</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;expand&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">})</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_besseli</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">polar_lift</span><span class="p">,</span> <span class="n">exp</span>
-        <span class="k">return</span> <span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">nu</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">polar_lift</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="bessely"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.bessel.bessely">[docs]</a><span class="k">class</span> <span class="nc">bessely</span><span class="p">(</span><span class="n">BesselBase</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Bessel function of the second kind.</span>
-
-<span class="sd">    The Bessel Y function of order :math:`\nu` is defined as</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        Y_\nu(z) = \lim_{\mu \to \nu} \frac{J_\mu(z) \cos(\pi \mu)</span>
-<span class="sd">                                            - J_{-\mu}(z)}{\sin(\pi \mu)},</span>
-
-<span class="sd">    where :math:`J_\mu(z)` is the Bessel function of the first kind.</span>
-
-<span class="sd">    It is a solution to Bessel&#39;s equation, and linearly independent from</span>
-<span class="sd">    :math:`J_\nu`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import bessely, yn</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import z, n</span>
-<span class="sd">    &gt;&gt;&gt; b = bessely(n, z)</span>
-<span class="sd">    &gt;&gt;&gt; b.diff(z)</span>
-<span class="sd">    bessely(n - 1, z)/2 - bessely(n + 1, z)/2</span>
-<span class="sd">    &gt;&gt;&gt; b.rewrite(yn)</span>
-<span class="sd">    sqrt(2)*sqrt(z)*yn(n - 1/2, z)/sqrt(pi)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    besselj, besseli, besselk</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_a</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-    <span class="n">_b</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_yn</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">expand</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="n">yn_part</span> <span class="o">=</span> <span class="n">yn</span><span class="p">(</span><span class="n">nu</span> <span class="o">-</span> <span class="n">S</span><span class="p">(</span><span class="s">&#39;1/2&#39;</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">argument</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">expand</span><span class="p">:</span>
-            <span class="n">yn_part</span> <span class="o">=</span> <span class="n">yn_part</span><span class="o">.</span><span class="n">_eval_expand_func</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="o">/</span><span class="n">pi</span><span class="p">)</span> <span class="o">*</span> <span class="n">yn_part</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">nu</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">nu</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">nu</span><span class="o">*</span><span class="n">bessely</span><span class="p">(</span><span class="o">-</span><span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_expand_func</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span><span class="o">.</span><span class="n">q</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_rewrite_as_yn</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;expand&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">})</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-</div>
-<div class="viewcode-block" id="besseli"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.bessel.besseli">[docs]</a><span class="k">class</span> <span class="nc">besseli</span><span class="p">(</span><span class="n">BesselBase</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Modified Bessel function of the first kind.</span>
-
-<span class="sd">    The Bessel I function is a solution to the modified Bessel equation</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2}</span>
-<span class="sd">        + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 + \nu^2)^2 w = 0.</span>
-
-<span class="sd">    It can be defined as</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        I_\nu(z) = i^{-\nu} J_\nu(iz),</span>
-
-<span class="sd">    where :math:`J_\mu(z)` is the Bessel function of the first kind.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import besseli</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import z, n</span>
-<span class="sd">    &gt;&gt;&gt; besseli(n, z).diff(z)</span>
-<span class="sd">    besseli(n - 1, z)/2 + besseli(n + 1, z)/2</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    besselj, bessely, besselk</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_a</span> <span class="o">=</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">One</span>
-    <span class="n">_b</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">nu</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="n">newz</span> <span class="o">=</span> <span class="n">z</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">newz</span><span class="p">:</span>  <span class="c"># NOTE we don&#39;t want to change the function if z==0</span>
-                <span class="k">return</span> <span class="n">I</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">nu</span><span class="p">)</span><span class="o">*</span><span class="n">besselj</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="o">-</span><span class="n">newz</span><span class="p">)</span>
-
-        <span class="c"># branch handling:</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">unpolarify</span><span class="p">,</span> <span class="n">exp</span>
-        <span class="k">if</span> <span class="n">nu</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-            <span class="n">newz</span> <span class="o">=</span> <span class="n">unpolarify</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">newz</span> <span class="o">!=</span> <span class="n">z</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">besseli</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">newz</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">newz</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">z</span><span class="o">.</span><span class="n">extract_branch_factor</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">nu</span><span class="o">*</span><span class="n">I</span><span class="p">)</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">newz</span><span class="p">)</span>
-        <span class="n">nnu</span> <span class="o">=</span> <span class="n">unpolarify</span><span class="p">(</span><span class="n">nu</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">nu</span> <span class="o">!=</span> <span class="n">nnu</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">besseli</span><span class="p">(</span><span class="n">nnu</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_besselj</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">polar_lift</span><span class="p">,</span> <span class="n">exp</span>
-        <span class="k">return</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">nu</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">besselj</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">polar_lift</span><span class="p">(</span><span class="n">I</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="besselk"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.bessel.besselk">[docs]</a><span class="k">class</span> <span class="nc">besselk</span><span class="p">(</span><span class="n">BesselBase</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Modified Bessel function of the second kind.</span>
-
-<span class="sd">    The Bessel K function of order :math:`\nu` is defined as</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        K_\nu(z) = \lim_{\mu \to \nu} \frac{\pi}{2}</span>
-<span class="sd">                   \frac{I_{-\mu}(z) -I_\mu(z)}{\sin(\pi \mu)},</span>
-
-<span class="sd">    where :math:`I_\mu(z)` is the modified Bessel function of the first kind.</span>
-
-<span class="sd">    It is a solution of the modified Bessel equation, and linearly independent</span>
-<span class="sd">    from :math:`Y_\nu`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import besselk</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import z, n</span>
-<span class="sd">    &gt;&gt;&gt; besselk(n, z).diff(z)</span>
-<span class="sd">    -besselk(n - 1, z)/2 - besselk(n + 1, z)/2</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    besselj, besseli, bessely</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_a</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-    <span class="n">_b</span> <span class="o">=</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-</div>
-<div class="viewcode-block" id="hankel1"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.bessel.hankel1">[docs]</a><span class="k">class</span> <span class="nc">hankel1</span><span class="p">(</span><span class="n">BesselBase</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Hankel function of the first kind.</span>
-
-<span class="sd">    This function is defined as</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        H_\nu^{(1)} = J_\nu(z) + iY_\nu(z),</span>
-
-<span class="sd">    where :math:`J_\nu(z)` is the Bessel function of the first kind, and</span>
-<span class="sd">    :math:`Y_\nu(z)` is the Bessel function of the second kind.</span>
-
-<span class="sd">    It is a solution to Bessel&#39;s equation.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import hankel1</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import z, n</span>
-<span class="sd">    &gt;&gt;&gt; hankel1(n, z).diff(z)</span>
-<span class="sd">    hankel1(n - 1, z)/2 - hankel1(n + 1, z)/2</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    hankel2, besselj, bessely</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_a</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-    <span class="n">_b</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-</div>
-<div class="viewcode-block" id="hankel2"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.bessel.hankel2">[docs]</a><span class="k">class</span> <span class="nc">hankel2</span><span class="p">(</span><span class="n">BesselBase</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Hankel function of the second kind.</span>
-
-<span class="sd">    This function is defined as</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        H_\nu^{(2)} = J_\nu(z) - iY_\nu(z),</span>
-
-<span class="sd">    where :math:`J_\nu(z)` is the Bessel function of the first kind, and</span>
-<span class="sd">    :math:`Y_\nu(z)` is the Bessel function of the second kind.</span>
-
-<span class="sd">    It is a solution to Bessel&#39;s equation, and linearly independent from</span>
-<span class="sd">    :math:`H_\nu^{(1)}`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import hankel2</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import z, n</span>
-<span class="sd">    &gt;&gt;&gt; hankel2(n, z).diff(z)</span>
-<span class="sd">    hankel2(n - 1, z)/2 - hankel2(n + 1, z)/2</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    hankel1, besselj, bessely</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_a</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-    <span class="n">_b</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-</div>
-<span class="kn">from</span> <span class="nn">sympy.polys.orthopolys</span> <span class="kn">import</span> <span class="n">spherical_bessel_fn</span> <span class="k">as</span> <span class="n">fn</span>
-
-
-<span class="k">class</span> <span class="nc">SphericalBesselBase</span><span class="p">(</span><span class="n">BesselBase</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Base class for spherical bessel functions.</span>
-
-<span class="sd">    These are thin wrappers around ordinary bessel functions,</span>
-<span class="sd">    since spherical bessel functions differ from the ordinary</span>
-<span class="sd">    ones just by a slight change in order.</span>
-
-<span class="sd">    To use this class, define the _rewrite and _expand methods.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_expand</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Expand self into a polynomial. Nu is guaranteed to be Integer. &quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;expansion&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_rewrite</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Rewrite self in terms of ordinary bessel functions. &quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;rewriting&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_expand_func</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_expand</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rewrite</span><span class="p">()</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">2</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">!=</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">order</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">argument</span><span class="p">)</span> <span class="o">-</span> \
-            <span class="bp">self</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">order</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">argument</span>
-
-
-<div class="viewcode-block" id="jn"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.bessel.jn">[docs]</a><span class="k">class</span> <span class="nc">jn</span><span class="p">(</span><span class="n">SphericalBesselBase</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Spherical Bessel function of the first kind.</span>
-
-<span class="sd">    This function is a solution to the spherical bessel equation</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2}</span>
-<span class="sd">          + 2z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu(\nu + 1)) w = 0.</span>
-
-<span class="sd">    It can be defined as</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        j_\nu(z) = \sqrt{\frac{\pi}{2z}} J_{\nu + \frac{1}{2}}(z),</span>
-
-<span class="sd">    where :math:`J_\nu(z)` is the Bessel function of the first kind.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, jn, sin, cos, expand_func</span>
-<span class="sd">    &gt;&gt;&gt; z = Symbol(&quot;z&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; print(jn(0, z).expand(func=True))</span>
-<span class="sd">    sin(z)/z</span>
-<span class="sd">    &gt;&gt;&gt; jn(1, z).expand(func=True) == sin(z)/z**2 - cos(z)/z</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; expand_func(jn(3, z))</span>
-<span class="sd">    (-6/z**2 + 15/z**4)*sin(z) + (1/z - 15/z**3)*cos(z)</span>
-
-<span class="sd">    The spherical Bessel functions of integral order</span>
-<span class="sd">    are calculated using the formula:</span>
-
-<span class="sd">    .. math:: j_n(z) = f_n(z) \sin{z} + (-1)^{n+1} f_{-n-1}(z) \cos{z},</span>
-
-<span class="sd">    where the coefficients :math:`f_n(z)` are available as</span>
-<span class="sd">    :func:`polys.orthopolys.spherical_bessel_fn`.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    besselj, bessely, besselk, yn</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_rewrite</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_rewrite_as_besselj</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">order</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">argument</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_besselj</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="p">))</span> <span class="o">*</span> <span class="n">besselj</span><span class="p">(</span><span class="n">nu</span> <span class="o">+</span> <span class="n">S</span><span class="p">(</span><span class="s">&#39;1/2&#39;</span><span class="p">),</span> <span class="n">z</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_expand</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span>
-        <span class="n">z</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">argument</span>
-        <span class="k">return</span> <span class="n">fn</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span> <span class="o">*</span> <span class="n">sin</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">fn</span><span class="p">(</span><span class="o">-</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span> <span class="o">*</span> <span class="n">cos</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="yn"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.bessel.yn">[docs]</a><span class="k">class</span> <span class="nc">yn</span><span class="p">(</span><span class="n">SphericalBesselBase</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Spherical Bessel function of the second kind.</span>
-
-<span class="sd">    This function is another solution to the spherical bessel equation, and</span>
-<span class="sd">    linearly independent from :math:`j_n`. It can be defined as</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        j_\nu(z) = \sqrt{\frac{\pi}{2z}} Y_{\nu + \frac{1}{2}}(z),</span>
-
-<span class="sd">    where :math:`Y_\nu(z)` is the Bessel function of the second kind.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, yn, sin, cos, expand_func</span>
-<span class="sd">    &gt;&gt;&gt; z = Symbol(&quot;z&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; print(expand_func(yn(0, z)))</span>
-<span class="sd">    -cos(z)/z</span>
-<span class="sd">    &gt;&gt;&gt; expand_func(yn(1, z)) == -cos(z)/z**2-sin(z)/z</span>
-<span class="sd">    True</span>
-
-<span class="sd">    For integral orders :math:`n`, :math:`y_n` is calculated using the formula:</span>
-
-<span class="sd">    .. math:: y_n(z) = (-1)^{n+1} j_{-n-1}(z)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    besselj, bessely, besselk, jn</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_rewrite</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_rewrite_as_bessely</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">order</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">argument</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_bessely</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="p">))</span> <span class="o">*</span> <span class="n">bessely</span><span class="p">(</span><span class="n">nu</span> <span class="o">+</span> <span class="n">S</span><span class="p">(</span><span class="s">&#39;1/2&#39;</span><span class="p">),</span> <span class="n">z</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_expand</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span>
-        <span class="n">z</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">argument</span>
-        <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> \
-               <span class="p">(</span><span class="n">fn</span><span class="p">(</span><span class="o">-</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span> <span class="o">*</span> <span class="n">sin</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">)</span> <span class="o">*</span> <span class="n">fn</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span> <span class="o">*</span> <span class="n">cos</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="jn_zeros"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.bessel.jn_zeros">[docs]</a><span class="k">def</span> <span class="nf">jn_zeros</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">&quot;sympy&quot;</span><span class="p">,</span> <span class="n">dps</span><span class="o">=</span><span class="mi">15</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Zeros of the spherical Bessel function of the first kind.</span>
-
-<span class="sd">    This returns an array of zeros of jn up to the k-th zero.</span>
-
-<span class="sd">    * method = &quot;sympy&quot;: uses mpmath besseljzero</span>
-<span class="sd">    * method = &quot;scipy&quot;: uses the SciPy&#39;s sph_jn and newton to find all</span>
-<span class="sd">      roots, which is faster than computing the zeros using a general</span>
-<span class="sd">      numerical solver, but it requires SciPy and only works with low</span>
-<span class="sd">      precision floating point numbers.  [the function used with</span>
-<span class="sd">      method=&quot;sympy&quot; is a recent addition to mpmath, before that a general</span>
-<span class="sd">      solver was used]</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import jn_zeros</span>
-<span class="sd">    &gt;&gt;&gt; jn_zeros(2, 4, dps=5)</span>
-<span class="sd">    [5.7635, 9.095, 12.323, 15.515]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    jn, yn, besselj, besselk, bessely</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">pi</span>
-
-    <span class="k">if</span> <span class="n">method</span> <span class="o">==</span> <span class="s">&quot;sympy&quot;</span><span class="p">:</span>
-        <span class="kn">from</span> <span class="nn">sympy.mpmath</span> <span class="kn">import</span> <span class="n">besseljzero</span>
-        <span class="kn">from</span> <span class="nn">sympy.mpmath.libmp.libmpf</span> <span class="kn">import</span> <span class="n">dps_to_prec</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Expr</span>
-        <span class="n">prec</span> <span class="o">=</span> <span class="n">dps_to_prec</span><span class="p">(</span><span class="n">dps</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">Expr</span><span class="o">.</span><span class="n">_from_mpmath</span><span class="p">(</span><span class="n">besseljzero</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mf">0.5</span><span class="p">)</span><span class="o">.</span><span class="n">_to_mpmath</span><span class="p">(</span><span class="n">prec</span><span class="p">),</span>
-                                              <span class="nb">int</span><span class="p">(</span><span class="n">k</span><span class="p">)),</span> <span class="n">prec</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
-    <span class="k">elif</span> <span class="n">method</span> <span class="o">==</span> <span class="s">&quot;scipy&quot;</span><span class="p">:</span>
-        <span class="kn">from</span> <span class="nn">scipy.special</span> <span class="kn">import</span> <span class="n">sph_jn</span>
-        <span class="kn">from</span> <span class="nn">scipy.optimize</span> <span class="kn">import</span> <span class="n">newton</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">sph_jn</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)[</span><span class="mi">0</span><span class="p">][</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Unknown method.&quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">solver</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">method</span> <span class="o">==</span> <span class="s">&quot;scipy&quot;</span><span class="p">:</span>
-            <span class="n">root</span> <span class="o">=</span> <span class="n">newton</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Unknown method.&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">root</span>
-
-    <span class="c"># we need to approximate the position of the first root:</span>
-    <span class="n">root</span> <span class="o">=</span> <span class="n">n</span> <span class="o">+</span> <span class="n">pi</span>
-    <span class="c"># determine the first root exactly:</span>
-    <span class="n">root</span> <span class="o">=</span> <span class="n">solver</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">root</span><span class="p">)</span>
-    <span class="n">roots</span> <span class="o">=</span> <span class="p">[</span><span class="n">root</span><span class="p">]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="c"># estimate the position of the next root using the last root + pi:</span>
-        <span class="n">root</span> <span class="o">=</span> <span class="n">solver</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">root</span> <span class="o">+</span> <span class="n">pi</span><span class="p">)</span>
-        <span class="n">roots</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">roots</span></div>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.functions.special.bsplines</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">expand</span>
-<span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">Piecewise</span><span class="p">,</span> <span class="n">piecewise_fold</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.piecewise</span> <span class="kn">import</span> <span class="n">ExprCondPair</span>
-<span class="kn">from</span> <span class="nn">sympy.core.sets</span> <span class="kn">import</span> <span class="n">Interval</span>
-
-
-<span class="k">def</span> <span class="nf">_add_splines</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">b1</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">b2</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Construct c*b1 + d*b2.&quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">b1</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span> <span class="ow">or</span> <span class="n">c</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expand</span><span class="p">(</span><span class="n">piecewise_fold</span><span class="p">(</span><span class="n">d</span><span class="o">*</span><span class="n">b2</span><span class="p">))</span>
-    <span class="k">if</span> <span class="n">b2</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span> <span class="ow">or</span> <span class="n">d</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expand</span><span class="p">(</span><span class="n">piecewise_fold</span><span class="p">(</span><span class="n">c</span><span class="o">*</span><span class="n">b1</span><span class="p">))</span>
-    <span class="n">new_args</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">n_intervals</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">b1</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">assert</span><span class="p">(</span><span class="n">n_intervals</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">b2</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-    <span class="n">new_args</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">expand</span><span class="p">(</span><span class="n">c</span><span class="o">*</span><span class="n">b1</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">expr</span><span class="p">),</span> <span class="n">b1</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">cond</span><span class="p">))</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_intervals</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">new_args</span><span class="o">.</span><span class="n">append</span><span class="p">((</span>
-            <span class="n">expand</span><span class="p">(</span><span class="n">c</span><span class="o">*</span><span class="n">b1</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">expr</span> <span class="o">+</span> <span class="n">d</span><span class="o">*</span><span class="n">b2</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">expr</span><span class="p">),</span>
-            <span class="n">b1</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">cond</span>
-        <span class="p">))</span>
-    <span class="n">new_args</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">expand</span><span class="p">(</span><span class="n">d</span><span class="o">*</span><span class="n">b2</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">expr</span><span class="p">),</span> <span class="n">b2</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">cond</span><span class="p">))</span>
-    <span class="n">new_args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">b2</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-    <span class="k">return</span> <span class="n">Piecewise</span><span class="p">(</span><span class="o">*</span><span class="n">new_args</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="bspline_basis"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.bsplines.bspline_basis">[docs]</a><span class="k">def</span> <span class="nf">bspline_basis</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">knots</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">close</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The n-th B-spline at x of degree d with knots.</span>
-
-<span class="sd">    B-Splines are piecewise polynomials of degree d [1].  They are defined on</span>
-<span class="sd">    a set of knots, which is a sequence of integers or floats.</span>
-
-<span class="sd">    The 0th degree splines have a value of one on a single interval:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import bspline_basis</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; d = 0</span>
-<span class="sd">        &gt;&gt;&gt; knots = list(range(5))</span>
-<span class="sd">        &gt;&gt;&gt; bspline_basis(d, knots, 0, x)</span>
-<span class="sd">        Piecewise((1, And(x &lt;= 1, x &gt;= 0)), (0, True))</span>
-
-<span class="sd">    For a given (d, knots) there are len(knots)-d-1 B-splines defined, that</span>
-<span class="sd">    are indexed by n (starting at 0).</span>
-
-<span class="sd">    Here is an example of a cubic B-spline:</span>
-
-<span class="sd">        &gt;&gt;&gt; bspline_basis(3, list(range(5)), 0, x)</span>
-<span class="sd">        Piecewise((x**3/6, And(x &lt; 1, x &gt;= 0)),</span>
-<span class="sd">                  (-x**3/2 + 2*x**2 - 2*x + 2/3, And(x &lt; 2, x &gt;= 1)),</span>
-<span class="sd">                  (x**3/2 - 4*x**2 + 10*x - 22/3, And(x &lt; 3, x &gt;= 2)),</span>
-<span class="sd">                  (-x**3/6 + 2*x**2 - 8*x + 32/3, And(x &lt;= 4, x &gt;= 3)),</span>
-<span class="sd">                  (0, True))</span>
-
-<span class="sd">    By repeating knot points, you can introduce discontinuities in the</span>
-<span class="sd">    B-splines and their derivatives:</span>
-
-<span class="sd">        &gt;&gt;&gt; d = 1</span>
-<span class="sd">        &gt;&gt;&gt; knots = [0,0,2,3,4]</span>
-<span class="sd">        &gt;&gt;&gt; bspline_basis(d, knots, 0, x)</span>
-<span class="sd">        Piecewise((-x/2 + 1, And(x &lt;= 2, x &gt;= 0)), (0, True))</span>
-
-<span class="sd">    It is quite time consuming to construct and evaluate B-splines. If you</span>
-<span class="sd">    need to evaluate a B-splines many times, it is best to lambdify them</span>
-<span class="sd">    first:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import lambdify</span>
-<span class="sd">        &gt;&gt;&gt; d = 3</span>
-<span class="sd">        &gt;&gt;&gt; knots = list(range(10))</span>
-<span class="sd">        &gt;&gt;&gt; b0 = bspline_basis(d, knots, 0, x)</span>
-<span class="sd">        &gt;&gt;&gt; f = lambdify(x, b0)</span>
-<span class="sd">        &gt;&gt;&gt; y = f(0.5)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    bsplines_basis_set</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/B-spline</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">knots</span> <span class="o">=</span> <span class="p">[</span><span class="n">sympify</span><span class="p">(</span><span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">knots</span><span class="p">]</span>
-    <span class="n">d</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="n">n_knots</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span>
-    <span class="n">n_intervals</span> <span class="o">=</span> <span class="n">n_knots</span> <span class="o">-</span> <span class="mi">1</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">+</span> <span class="n">d</span> <span class="o">+</span> <span class="mi">1</span> <span class="o">&gt;</span> <span class="n">n_intervals</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;n+d+1 must not exceed len(knots)-1&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">d</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">Piecewise</span><span class="p">(</span>
-            <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">Interval</span><span class="p">(</span><span class="n">knots</span><span class="p">[</span><span class="n">n</span><span class="p">],</span> <span class="n">knots</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="bp">False</span><span class="p">,</span>
-             <span class="ow">not</span> <span class="n">close</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span>
-            <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-        <span class="p">)</span>
-    <span class="k">elif</span> <span class="n">d</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">denom</span> <span class="o">=</span> <span class="n">knots</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="n">d</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">knots</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">denom</span> <span class="o">!=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-            <span class="n">B</span> <span class="o">=</span> <span class="p">(</span><span class="n">knots</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="n">d</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">denom</span>
-            <span class="n">b2</span> <span class="o">=</span> <span class="n">bspline_basis</span><span class="p">(</span><span class="n">d</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">knots</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">close</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">b2</span> <span class="o">=</span> <span class="n">B</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-        <span class="n">denom</span> <span class="o">=</span> <span class="n">knots</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="n">d</span><span class="p">]</span> <span class="o">-</span> <span class="n">knots</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">denom</span> <span class="o">!=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-            <span class="n">A</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">knots</span><span class="p">[</span><span class="n">n</span><span class="p">])</span><span class="o">/</span><span class="n">denom</span>
-            <span class="n">b1</span> <span class="o">=</span> <span class="n">bspline_basis</span><span class="p">(</span>
-                <span class="n">d</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">knots</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">close</span> <span class="ow">and</span> <span class="p">(</span><span class="n">B</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span> <span class="ow">or</span> <span class="n">b2</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">b1</span> <span class="o">=</span> <span class="n">A</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-        <span class="n">result</span> <span class="o">=</span> <span class="n">_add_splines</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">b1</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">b2</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;degree must be non-negative: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="bspline_basis_set"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.bsplines.bspline_basis_set">[docs]</a><span class="k">def</span> <span class="nf">bspline_basis_set</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">knots</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return the len(knots)-d-1 B-splines at x of degree d with knots.</span>
-
-<span class="sd">    This function returns a list of Piecewise polynomials that are the</span>
-<span class="sd">    len(knots)-d-1 B-splines of degree d for the given knots. This function</span>
-<span class="sd">    calls bspline_basis(d, knots, n, x) for different values of n.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import bspline_basis_set</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; d = 2</span>
-<span class="sd">    &gt;&gt;&gt; knots = list(range(5))</span>
-<span class="sd">    &gt;&gt;&gt; splines = bspline_basis_set(d, knots, x)</span>
-<span class="sd">    &gt;&gt;&gt; splines</span>
-<span class="sd">    [Piecewise((x**2/2, And(x &lt; 1, x &gt;= 0)),</span>
-<span class="sd">               (-x**2 + 3*x - 3/2, And(x &lt; 2, x &gt;= 1)),</span>
-<span class="sd">               (x**2/2 - 3*x + 9/2, And(x &lt;= 3, x &gt;= 2)),</span>
-<span class="sd">               (0, True)),</span>
-<span class="sd">     Piecewise((x**2/2 - x + 1/2, And(x &lt; 2, x &gt;= 1)),</span>
-<span class="sd">               (-x**2 + 5*x - 11/2, And(x &lt; 3, x &gt;= 2)),</span>
-<span class="sd">               (x**2/2 - 4*x + 8, And(x &lt;= 4, x &gt;= 3)),</span>
-<span class="sd">               (0, True))]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    bsplines_basis</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n_splines</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">d</span> <span class="o">-</span> <span class="mi">1</span>
-    <span class="k">return</span> <span class="p">[</span><span class="n">bspline_basis</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">knots</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_splines</span><span class="p">)]</span></div>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/functions/special/delta_functions.html b/dev-py3k/_modules/sympy/functions/special/delta_functions.html
deleted file mode 100644
index f0905db5eae..00000000000
--- a/dev-py3k/_modules/sympy/functions/special/delta_functions.html
+++ /dev/null
@@ -1,347 +0,0 @@
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-            
-  <h1>Source code for sympy.functions.special.delta_functions</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">diff</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">ArgumentIndexError</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="n">PolynomialError</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.complexes</span> <span class="kn">import</span> <span class="n">im</span>
-
-<span class="c">###############################################################################</span>
-<span class="c">################################ DELTA FUNCTION ###############################</span>
-<span class="c">###############################################################################</span>
-
-
-<div class="viewcode-block" id="DiracDelta"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.delta_functions.DiracDelta">[docs]</a><span class="k">class</span> <span class="nc">DiracDelta</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The DiracDelta function and its derivatives.</span>
-
-<span class="sd">    DiracDelta function has the following properties:</span>
-
-<span class="sd">    1) ``diff(Heaviside(x),x) = DiracDelta(x)``</span>
-<span class="sd">    2) ``integrate(DiracDelta(x-a)*f(x),(x,-oo,oo)) = f(a)`` and</span>
-<span class="sd">       ``integrate(DiracDelta(x-a)*f(x),(x,a-e,a+e)) = f(a)``</span>
-<span class="sd">    3) ``DiracDelta(x) = 0`` for all ``x != 0``</span>
-<span class="sd">    4) ``DiracDelta(g(x)) = Sum_i(DiracDelta(x-x_i)/abs(g&#39;(x_i)))``</span>
-<span class="sd">       Where ``x_i``-s are the roots of ``g``</span>
-
-<span class="sd">    Derivatives of ``k``-th order of DiracDelta have the following property:</span>
-
-<span class="sd">    5) ``DiracDelta(x,k) = 0``, for all ``x != 0``</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Heaviside</span>
-<span class="sd">    simplify, is_simple</span>
-<span class="sd">    sympy.functions.special.tensor_functions.KroneckerDelta</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    http://mathworld.wolfram.com/DeltaFunction.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-
-    <span class="n">is_real</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c">#I didn&#39;t know if there is a better way to handle default arguments</span>
-            <span class="n">k</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">k</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">DiracDelta</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">k</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">or</span> <span class="n">k</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Error: the second argument of DiracDelta must be </span><span class="se">\</span>
-<span class="s">            a non-negative integer, </span><span class="si">%s</span><span class="s"> given instead.&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">k</span><span class="p">,))</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_positive</span> <span class="ow">or</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-<div class="viewcode-block" id="DiracDelta.simplify"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.delta_functions.DiracDelta.simplify">[docs]</a>    <span class="k">def</span> <span class="nf">simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;simplify(self, x)</span>
-
-<span class="sd">           Compute a simplified representation of the function using</span>
-<span class="sd">           property number 4.</span>
-
-<span class="sd">           x can be:</span>
-
-<span class="sd">           - a symbol</span>
-
-<span class="sd">           Examples</span>
-<span class="sd">           ========</span>
-
-<span class="sd">           &gt;&gt;&gt; from sympy import DiracDelta</span>
-<span class="sd">           &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">           &gt;&gt;&gt; DiracDelta(x*y).simplify(x)</span>
-<span class="sd">           DiracDelta(x)/Abs(y)</span>
-<span class="sd">           &gt;&gt;&gt; DiracDelta(x*y).simplify(y)</span>
-<span class="sd">           DiracDelta(y)/Abs(x)</span>
-
-<span class="sd">           &gt;&gt;&gt; DiracDelta(x**2 + x - 2).simplify(x)</span>
-<span class="sd">           DiracDelta(x - 1)/3 + DiracDelta(x + 2)/3</span>
-
-<span class="sd">           See Also</span>
-<span class="sd">           ========</span>
-
-<span class="sd">           is_simple, Directdelta</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.polys.polyroots</span> <span class="kn">import</span> <span class="n">roots</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">0</span> <span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">argroots</span> <span class="o">=</span> <span class="n">roots</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">,</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="n">valid</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="n">darg</span> <span class="o">=</span> <span class="n">diff</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">argroots</span><span class="p">:</span>
-                <span class="c">#should I care about multiplicities of roots?</span>
-                <span class="k">if</span> <span class="n">r</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">darg</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-                    <span class="n">result</span> <span class="o">+=</span> <span class="n">DiracDelta</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">r</span><span class="p">)</span><span class="o">/</span><span class="nb">abs</span><span class="p">(</span><span class="n">darg</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">r</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">valid</span> <span class="o">=</span> <span class="bp">False</span>
-                    <span class="k">break</span>
-            <span class="k">if</span> <span class="n">valid</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">result</span>
-        <span class="k">except</span> <span class="n">PolynomialError</span><span class="p">:</span>
-            <span class="k">pass</span>
-        <span class="k">return</span> <span class="bp">self</span>
-</div>
-<div class="viewcode-block" id="DiracDelta.is_simple"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.delta_functions.DiracDelta.is_simple">[docs]</a>    <span class="k">def</span> <span class="nf">is_simple</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;is_simple(self, x)</span>
-
-<span class="sd">           Tells whether the argument(args[0]) of DiracDelta is a linear</span>
-<span class="sd">           expression in x.</span>
-
-<span class="sd">           x can be:</span>
-
-<span class="sd">           - a symbol</span>
-
-<span class="sd">           Examples</span>
-<span class="sd">           ========</span>
-
-<span class="sd">           &gt;&gt;&gt; from sympy import DiracDelta, cos</span>
-<span class="sd">           &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">           &gt;&gt;&gt; DiracDelta(x*y).is_simple(x)</span>
-<span class="sd">           True</span>
-<span class="sd">           &gt;&gt;&gt; DiracDelta(x*y).is_simple(y)</span>
-<span class="sd">           True</span>
-
-<span class="sd">           &gt;&gt;&gt; DiracDelta(x**2+x-2).is_simple(x)</span>
-<span class="sd">           False</span>
-
-<span class="sd">           &gt;&gt;&gt; DiracDelta(cos(x)).is_simple(x)</span>
-<span class="sd">           False</span>
-
-<span class="sd">           See Also</span>
-<span class="sd">           ========</span>
-
-<span class="sd">           simplify, Directdelta</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_poly</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">p</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">p</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span> <span class="o">==</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-<span class="c">###############################################################################</span>
-<span class="c">############################## HEAVISIDE FUNCTION #############################</span>
-<span class="c">###############################################################################</span>
-
-</div></div>
-<div class="viewcode-block" id="Heaviside"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.delta_functions.Heaviside">[docs]</a><span class="k">class</span> <span class="nc">Heaviside</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Heaviside Piecewise function</span>
-
-<span class="sd">    Heaviside function has the following properties [*]_:</span>
-
-<span class="sd">    1) ``diff(Heaviside(x),x) = DiracDelta(x)``</span>
-<span class="sd">                        ``( 0, if x &lt; 0``</span>
-<span class="sd">    2) ``Heaviside(x) = &lt; ( 1/2 if x==0 [*]``</span>
-<span class="sd">                        ``( 1, if x &gt; 0``</span>
-
-<span class="sd">    .. [*] Regarding to the value at 0, Mathematica defines ``H(0) = 1``,</span>
-<span class="sd">           but Maple uses ``H(0) = undefined``</span>
-
-<span class="sd">    I think is better to have H(0) = 1/2, due to the following::</span>
-
-<span class="sd">        integrate(DiracDelta(x), x) = Heaviside(x)</span>
-<span class="sd">        integrate(DiracDelta(x), (x, -oo, oo)) = 1</span>
-
-<span class="sd">    and since DiracDelta is a symmetric function,</span>
-<span class="sd">    ``integrate(DiracDelta(x), (x, 0, oo))`` should be 1/2 (which is what</span>
-<span class="sd">    Maple returns).</span>
-
-<span class="sd">    If we take ``Heaviside(0) = 1/2``, we would have</span>
-<span class="sd">    ``integrate(DiracDelta(x), (x, 0, oo)) = ``</span>
-<span class="sd">    ``Heaviside(oo) - Heaviside(0) = 1 - 1/2 = 1/2``</span>
-<span class="sd">    and</span>
-<span class="sd">    ``integrate(DiracDelta(x), (x, -oo, 0)) = ``</span>
-<span class="sd">    ``Heaviside(0) - Heaviside(-oo) = 1/2 - 0 = 1/2``</span>
-
-<span class="sd">    If we consider, instead ``Heaviside(0) = 1``, we would have</span>
-<span class="sd">    ``integrate(DiracDelta(x), (x, 0, oo)) = Heaviside(oo) - Heaviside(0) = 0``</span>
-<span class="sd">    and</span>
-<span class="sd">    ``integrate(DiracDelta(x), (x, -oo, 0)) = Heaviside(0) - Heaviside(-oo) = 1``</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    DiracDelta</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    http://mathworld.wolfram.com/HeavisideStepFunction.html</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="n">is_real</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># property number 1</span>
-            <span class="k">return</span> <span class="n">DiracDelta</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-        <span class="k">elif</span> <span class="n">im</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span><span class="o">.</span><span class="n">is_nonzero</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Function defined only for Real Values. Complex part: </span><span class="si">%s</span><span class="s">  found in </span><span class="si">%s</span><span class="s"> .&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="nb">repr</span><span class="p">(</span><span class="n">im</span><span class="p">(</span><span class="n">arg</span><span class="p">)),</span> <span class="nb">repr</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span> <span class="p">)</span>
-        <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span>
-        <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span></div>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
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-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../../index.html">
-              <img class="logo" src="../../../../_static/sympylogo.png" alt="Logo"/>
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-<div id="searchbox" style="display: none">
-  <h3>Quick search</h3>
-    <form class="search" action="../../../../search.html" method="get">
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-    Enter search terms or a module, class or function name.
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diff --git a/dev-py3k/_modules/sympy/functions/special/error_functions.html b/dev-py3k/_modules/sympy/functions/special/error_functions.html
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-            
-  <h1>Source code for sympy.functions.special.error_functions</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot; This module contains various functions that are special cases</span>
-<span class="sd">    of incomplete gamma functions. It should probably be renamed. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Add</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">cacheit</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">I</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">ArgumentIndexError</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">root</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.complexes</span> <span class="kn">import</span> <span class="n">polar_lift</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.special.hyper</span> <span class="kn">import</span> <span class="n">hyper</span><span class="p">,</span> <span class="n">meijerg</span>
-
-<span class="c"># TODO series expansions</span>
-<span class="c"># TODO see the &quot;Note:&quot; in Ei</span>
-
-<span class="c">###############################################################################</span>
-<span class="c">################################ ERROR FUNCTION ###############################</span>
-<span class="c">###############################################################################</span>
-
-
-<div class="viewcode-block" id="erf"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.error_functions.erf">[docs]</a><span class="k">class</span> <span class="nc">erf</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The Gauss error function.</span>
-
-<span class="sd">    This function is defined as:</span>
-
-<span class="sd">    :math:`\\mathrm{erf}(x)=\\frac{2}{\\sqrt{\\pi}} \\int_0^x e^{-t^2} \\, \\mathrm{d}x`</span>
-
-<span class="sd">    Or, in ASCII::</span>
-
-<span class="sd">                x</span>
-<span class="sd">            /</span>
-<span class="sd">           |</span>
-<span class="sd">           |     2</span>
-<span class="sd">           |   -t</span>
-<span class="sd">        2* |  e    dt</span>
-<span class="sd">           |</span>
-<span class="sd">          /</span>
-<span class="sd">          0</span>
-<span class="sd">        -------------</span>
-<span class="sd">              ____</span>
-<span class="sd">            \/ pi</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import I, oo, erf</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import z</span>
-
-<span class="sd">    Several special values are known:</span>
-
-<span class="sd">    &gt;&gt;&gt; erf(0)</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; erf(oo)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; erf(-oo)</span>
-<span class="sd">    -1</span>
-<span class="sd">    &gt;&gt;&gt; erf(I*oo)</span>
-<span class="sd">    oo*I</span>
-<span class="sd">    &gt;&gt;&gt; erf(-I*oo)</span>
-<span class="sd">    -oo*I</span>
-
-<span class="sd">    In general one can pull out factors of -1 and I from the argument:</span>
-
-<span class="sd">    &gt;&gt;&gt; erf(-z)</span>
-<span class="sd">    -erf(z)</span>
-
-<span class="sd">    The error function obeys the mirror symmetry:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import conjugate</span>
-<span class="sd">    &gt;&gt;&gt; conjugate(erf(z))</span>
-<span class="sd">    erf(conjugate(z))</span>
-
-<span class="sd">    Differentiation with respect to z is supported:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import diff</span>
-<span class="sd">    &gt;&gt;&gt; diff(erf(z), z)</span>
-<span class="sd">    2*exp(-z**2)/sqrt(pi)</span>
-
-<span class="sd">    We can numerically evaluate the error function to arbitrary precision</span>
-<span class="sd">    on the whole complex plane:</span>
-
-<span class="sd">    &gt;&gt;&gt; erf(4).evalf(30)</span>
-<span class="sd">    0.999999984582742099719981147840</span>
-
-<span class="sd">    &gt;&gt;&gt; erf(-4*I).evalf(30)</span>
-<span class="sd">    -1296959.73071763923152794095062*I</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Error_function</span>
-<span class="sd">    .. [2] http://dlmf.nist.gov/7</span>
-<span class="sd">    .. [3] http://mathworld.wolfram.com/Erf.html</span>
-<span class="sd">    .. [4] http://functions.wolfram.com/GammaBetaErf/Erf</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">unbranched</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-        <span class="n">t</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">t</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span> <span class="ow">or</span> <span class="n">t</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">arg</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">():</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="n">k</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">floor</span><span class="p">((</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">S</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">previous_terms</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="k">return</span> <span class="o">-</span><span class="n">previous_terms</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">k</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">k</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">conjugate</span><span class="p">())</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_uppergamma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">z</span><span class="o">*</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="o">-</span> <span class="n">C</span><span class="o">.</span><span class="n">uppergamma</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_tractable</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="o">-</span> <span class="n">_erfs</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_as_leading_term</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-
-<span class="c">###############################################################################</span>
-<span class="c">#################### EXPONENTIAL INTEGRALS ####################################</span>
-<span class="c">###############################################################################</span>
-</div>
-<div class="viewcode-block" id="Ei"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.error_functions.Ei">[docs]</a><span class="k">class</span> <span class="nc">Ei</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    The classical exponential integral.</span>
-
-<span class="sd">    For the use in SymPy, this function is defined as</span>
-
-<span class="sd">    .. math:: \operatorname{Ei}(x) = \sum_{n=1}^\infty \frac{x^n}{n\, n!}</span>
-<span class="sd">                                     + \log(x) + \gamma,</span>
-
-<span class="sd">    where :math:`\gamma` is the Euler-Mascheroni constant.</span>
-
-<span class="sd">    If :math:`x` is a polar number, this defines an analytic function on the</span>
-<span class="sd">    riemann surface of the logarithm. Otherwise this defines an analytic</span>
-<span class="sd">    function in the cut plane :math:`\mathbb{C} \setminus (-\infty, 0]`.</span>
-
-<span class="sd">    **Background**</span>
-
-<span class="sd">    The name &#39;exponential integral&#39; comes from the following statement:</span>
-
-<span class="sd">    .. math:: \operatorname{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} \mathrm{d}t</span>
-
-<span class="sd">    If the integral is interpreted as a Cauchy principal value, this statement</span>
-<span class="sd">    holds for :math:`x &gt; 0` and :math:`\operatorname{Ei}(x)` as defined above.</span>
-
-<span class="sd">    Note that we carefully avoided defining :math:`\operatorname{Ei}(x)` for</span>
-<span class="sd">    negative real x. This is because above integral formula does not hold for</span>
-<span class="sd">    any polar lift of such :math:`x`, indeed all branches of</span>
-<span class="sd">    :math:`\operatorname{Ei}(x)` above the negative reals are imaginary.</span>
-
-<span class="sd">    However, the following statement holds for all :math:`x \in \mathbb{R}^*`:</span>
-
-<span class="sd">    .. math:: \int_{-\infty}^x \frac{e^t}{t} \mathrm{d}t =</span>
-<span class="sd">              \frac{\operatorname{Ei}\left(|x|e^{i \arg(x)}\right) +</span>
-<span class="sd">                    \operatorname{Ei}\left(|x|e^{- i \arg(x)}\right)}{2},</span>
-
-<span class="sd">    where the integral is again understood to be a principal value if</span>
-<span class="sd">    :math:`x &gt; 0`, and :math:`|x|e^{i \arg(x)}`,</span>
-<span class="sd">    :math:`|x|e^{- i \arg(x)}` denote two conjugate polar lifts of :math:`x`.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    expint, sympy.functions.special.gamma_functions.uppergamma</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - Abramowitz &amp; Stegun, section 5: http://www.math.sfu.ca/~cbm/aands/page_228.htm</span>
-<span class="sd">    - http://en.wikipedia.org/wiki/Exponential_integral</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Ei, polar_lift, exp_polar, I, pi</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    The exponential integral in SymPy is strictly undefined for negative values</span>
-<span class="sd">    of the argument. For convenience, exponential integrals with negative</span>
-<span class="sd">    arguments are immediately converted into an expression that agrees with</span>
-<span class="sd">    the classical integral definition:</span>
-
-<span class="sd">    &gt;&gt;&gt; Ei(-1)</span>
-<span class="sd">    -I*pi + Ei(exp_polar(I*pi))</span>
-
-<span class="sd">    This yields a real value:</span>
-
-<span class="sd">    &gt;&gt;&gt; Ei(-1).n(chop=True)</span>
-<span class="sd">    -0.219383934395520</span>
-
-<span class="sd">    On the other hand the analytic continuation is not real:</span>
-
-<span class="sd">    &gt;&gt;&gt; Ei(polar_lift(-1)).n(chop=True)</span>
-<span class="sd">    -0.21938393439552 + 3.14159265358979*I</span>
-
-<span class="sd">    The exponential integral has a logarithmic branch point at the origin:</span>
-
-<span class="sd">    &gt;&gt;&gt; Ei(x*exp_polar(2*I*pi))</span>
-<span class="sd">    Ei(x) + 2*I*pi</span>
-
-<span class="sd">    Differentiation is supported:</span>
-
-<span class="sd">    &gt;&gt;&gt; Ei(x).diff(x)</span>
-<span class="sd">    exp(x)/x</span>
-
-<span class="sd">    The exponential integral is related to many other special functions.</span>
-<span class="sd">    For example:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import uppergamma, expint, Shi</span>
-<span class="sd">    &gt;&gt;&gt; Ei(x).rewrite(expint)</span>
-<span class="sd">    -expint(1, x*exp_polar(I*pi)) - I*pi</span>
-<span class="sd">    &gt;&gt;&gt; Ei(x).rewrite(Shi)</span>
-<span class="sd">    Chi(x) + Shi(x)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">z</span><span class="o">.</span><span class="n">is_polar</span> <span class="ow">and</span> <span class="n">z</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="c"># Note: is this a good idea?</span>
-            <span class="k">return</span> <span class="n">Ei</span><span class="p">(</span><span class="n">polar_lift</span><span class="p">(</span><span class="n">z</span><span class="p">))</span> <span class="o">-</span> <span class="n">pi</span><span class="o">*</span><span class="n">I</span>
-        <span class="n">nz</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">z</span><span class="o">.</span><span class="n">extract_branch_factor</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Ei</span><span class="p">(</span><span class="n">nz</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">n</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">unpolarify</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">unpolarify</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span><span class="o">/</span><span class="n">arg</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">if</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="n">polar_lift</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Function</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Function</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_uppergamma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">uppergamma</span>
-        <span class="c"># XXX this does not currently work usefully because uppergamma</span>
-        <span class="c">#     immediately turns into expint</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">uppergamma</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">polar_lift</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">)</span> <span class="o">-</span> <span class="n">I</span><span class="o">*</span><span class="n">pi</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_expint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">expint</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">polar_lift</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">)</span> <span class="o">-</span> <span class="n">I</span><span class="o">*</span><span class="n">pi</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_Si</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Shi</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="n">Chi</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-    <span class="n">_eval_rewrite_as_Ci</span> <span class="o">=</span> <span class="n">_eval_rewrite_as_Si</span>
-    <span class="n">_eval_rewrite_as_Chi</span> <span class="o">=</span> <span class="n">_eval_rewrite_as_Si</span>
-    <span class="n">_eval_rewrite_as_Shi</span> <span class="o">=</span> <span class="n">_eval_rewrite_as_Si</span>
-
-</div>
-<div class="viewcode-block" id="expint"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.error_functions.expint">[docs]</a><span class="k">class</span> <span class="nc">expint</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Generalized exponential integral.</span>
-
-<span class="sd">    This function is defined as</span>
-
-<span class="sd">    .. math:: \operatorname{E}_\nu(z) = z^{\nu - 1} \Gamma(1 - \nu, z),</span>
-
-<span class="sd">    where `\Gamma(1 - \nu, z)` is the upper incomplete gamma function</span>
-<span class="sd">    (``uppergamma``).</span>
-
-<span class="sd">    Hence for :math:`z` with positive real part we have</span>
-
-<span class="sd">    .. math:: \operatorname{E}_\nu(z)</span>
-<span class="sd">              =   \int_1^\infty \frac{e^{-zt}}{z^\nu} \mathrm{d}t,</span>
-
-<span class="sd">    which explains the name.</span>
-
-<span class="sd">    The representation as an incomplete gamma function provides an analytic</span>
-<span class="sd">    continuation for :math:`\operatorname{E}_\nu(z)`. If :math:`\nu` is a</span>
-<span class="sd">    non-positive integer the exponential integral is thus an unbranched</span>
-<span class="sd">    function of :math:`z`, otherwise there is a branch point at the origin.</span>
-<span class="sd">    Refer to the incomplete gamma function documentation for details of the</span>
-<span class="sd">    branching behavior.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    E1: The classical case, returns expint(1, z).</span>
-<span class="sd">    Ei: Another related function called exponential integral.</span>
-<span class="sd">    sympy.functions.special.gamma_functions.uppergamma</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - http://dlmf.nist.gov/8.19</span>
-<span class="sd">    - http://functions.wolfram.com/GammaBetaErf/ExpIntegralE/</span>
-<span class="sd">    - http://en.wikipedia.org/wiki/Exponential_integral</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import expint, S</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import nu, z</span>
-
-<span class="sd">    Differentiation is supported. Differentiation with respect to z explains</span>
-<span class="sd">    further the name: for integral orders, the exponential integral is an</span>
-<span class="sd">    iterated integral of the exponential function.</span>
-
-<span class="sd">    &gt;&gt;&gt; expint(nu, z).diff(z)</span>
-<span class="sd">    -expint(nu - 1, z)</span>
-
-<span class="sd">    Differentiation with respect to nu has no classical expression:</span>
-
-<span class="sd">    &gt;&gt;&gt; expint(nu, z).diff(nu)</span>
-<span class="sd">    -z**(nu - 1)*meijerg(((), (1, 1)), ((0, 0, -nu + 1), ()), z)</span>
-
-<span class="sd">    At non-postive integer orders, the exponential integral reduces to the</span>
-<span class="sd">    exponential function:</span>
-
-<span class="sd">    &gt;&gt;&gt; expint(0, z)</span>
-<span class="sd">    exp(-z)/z</span>
-<span class="sd">    &gt;&gt;&gt; expint(-1, z)</span>
-<span class="sd">    exp(-z)/z + exp(-z)/z**2</span>
-
-<span class="sd">    At half-integers it reduces to error functions:</span>
-
-<span class="sd">    &gt;&gt;&gt; expint(S(1)/2, z)</span>
-<span class="sd">    -sqrt(pi)*erf(sqrt(z))/sqrt(z) + sqrt(pi)/sqrt(z)</span>
-
-<span class="sd">    At positive integer orders it can be rewritten in terms of exponentials</span>
-<span class="sd">    and expint(1, z). Use expand_func() to do this:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import expand_func</span>
-<span class="sd">    &gt;&gt;&gt; expand_func(expint(5, z))</span>
-<span class="sd">    z**4*expint(1, z)/24 + (-z**3 + z**2 - 2*z + 6)*exp(-z)/24</span>
-
-<span class="sd">    The generalised exponential integral is essentially equivalent to the</span>
-<span class="sd">    incomplete gamma function:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import uppergamma</span>
-<span class="sd">    &gt;&gt;&gt; expint(nu, z).rewrite(uppergamma)</span>
-<span class="sd">    z**(nu - 1)*uppergamma(-nu + 1, z)</span>
-
-<span class="sd">    As such it is branched at the origin:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import exp_polar, pi, I</span>
-<span class="sd">    &gt;&gt;&gt; expint(4, z*exp_polar(2*pi*I))</span>
-<span class="sd">    I*pi*z**3/3 + expint(4, z)</span>
-<span class="sd">    &gt;&gt;&gt; expint(nu, z*exp_polar(2*pi*I))</span>
-<span class="sd">    z**(nu - 1)*(exp(2*I*pi*nu) - 1)*gamma(-nu + 1) + expint(nu, z)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="p">(</span><span class="n">unpolarify</span><span class="p">,</span> <span class="n">expand_mul</span><span class="p">,</span> <span class="n">uppergamma</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span>
-                           <span class="n">factorial</span><span class="p">)</span>
-        <span class="n">nu2</span> <span class="o">=</span> <span class="n">unpolarify</span><span class="p">(</span><span class="n">nu</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">nu</span> <span class="o">!=</span> <span class="n">nu2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expint</span><span class="p">(</span><span class="n">nu2</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">nu</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">nu</span> <span class="o">&lt;=</span> <span class="mi">0</span> <span class="ow">or</span> <span class="p">(</span><span class="ow">not</span> <span class="n">nu</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">nu</span><span class="p">)</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">unpolarify</span><span class="p">(</span><span class="n">expand_mul</span><span class="p">(</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">nu</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">uppergamma</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">)))</span>
-
-        <span class="c"># Extract branching information. This can be deduced from what is</span>
-        <span class="c"># explained in lowergamma.eval().</span>
-        <span class="n">z</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">z</span><span class="o">.</span><span class="n">extract_branch_factor</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="k">if</span> <span class="n">nu</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">nu</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">True</span><span class="p">:</span>
-                <span class="k">return</span>
-            <span class="k">return</span> <span class="n">expint</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span> \
-                <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">nu</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span><span class="n">nu</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">unpolarify</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">nu</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">nu</span><span class="o">*</span><span class="n">n</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">nu</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">nu</span><span class="p">)</span> <span class="o">+</span> <span class="n">expint</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">meijerg</span>
-        <span class="n">nu</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">nu</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">meijerg</span><span class="p">([],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">nu</span><span class="p">],</span> <span class="p">[],</span> <span class="n">z</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">expint</span><span class="p">(</span><span class="n">nu</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_uppergamma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">uppergamma</span>
-        <span class="k">return</span> <span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">nu</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">uppergamma</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_Ei</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exp_polar</span><span class="p">,</span> <span class="n">unpolarify</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">factorial</span>
-        <span class="k">if</span> <span class="n">nu</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">Ei</span><span class="p">(</span><span class="n">z</span><span class="o">*</span><span class="n">exp_polar</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span> <span class="o">-</span> <span class="n">I</span><span class="o">*</span><span class="n">pi</span>
-        <span class="k">elif</span> <span class="n">nu</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">nu</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># DLMF, 8.19.7</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="o">-</span><span class="n">unpolarify</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">nu</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span><span class="n">nu</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">E1</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">Ei</span><span class="p">)</span> <span class="o">+</span> \
-                <span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span><span class="n">nu</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> \
-                <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">factorial</span><span class="p">(</span><span class="n">nu</span> <span class="o">-</span> <span class="n">k</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">k</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nu</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="k">def</span> <span class="nf">_eval_expand_func</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">Ei</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">expint</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_Si</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">nu</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">return</span> <span class="n">Shi</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">-</span> <span class="n">Chi</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-    <span class="n">_eval_rewrite_as_Ci</span> <span class="o">=</span> <span class="n">_eval_rewrite_as_Si</span>
-    <span class="n">_eval_rewrite_as_Chi</span> <span class="o">=</span> <span class="n">_eval_rewrite_as_Si</span>
-    <span class="n">_eval_rewrite_as_Shi</span> <span class="o">=</span> <span class="n">_eval_rewrite_as_Si</span>
-
-</div>
-<div class="viewcode-block" id="E1"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.error_functions.E1">[docs]</a><span class="k">def</span> <span class="nf">E1</span><span class="p">(</span><span class="n">z</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Classical case of the generalized exponential integral.</span>
-
-<span class="sd">    This is equivalent to ``expint(1, z)``.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">expint</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-
-
-<span class="c">###############################################################################</span>
-<span class="c">#################### TRIGONOMETRIC INTEGRALS ##################################</span>
-<span class="c">###############################################################################</span>
-</div>
-<span class="k">class</span> <span class="nc">TrigonometricIntegral</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Base class for trigonometric integrals. &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">z</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_atzero</span>
-        <span class="k">elif</span> <span class="n">z</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_atinf</span>
-        <span class="k">elif</span> <span class="n">z</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_atneginf</span>
-
-        <span class="n">nz</span> <span class="o">=</span> <span class="n">z</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="n">polar_lift</span><span class="p">(</span><span class="n">I</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">nz</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">cls</span><span class="o">.</span><span class="n">_trigfunc</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">nz</span> <span class="o">=</span> <span class="n">z</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">nz</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_Ifactor</span><span class="p">(</span><span class="n">nz</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">nz</span> <span class="o">=</span> <span class="n">z</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="n">polar_lift</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">nz</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_Ifactor</span><span class="p">(</span><span class="n">nz</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-
-        <span class="n">nz</span> <span class="o">=</span> <span class="n">z</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="n">polar_lift</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">nz</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">cls</span><span class="o">.</span><span class="n">_trigfunc</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">nz</span> <span class="o">=</span> <span class="n">z</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">nz</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_minusfactor</span><span class="p">(</span><span class="n">nz</span><span class="p">)</span>
-
-        <span class="n">nz</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">z</span><span class="o">.</span><span class="n">extract_branch_factor</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">nz</span> <span class="o">==</span> <span class="n">z</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">cls</span><span class="o">.</span><span class="n">_trigfunc</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="o">+</span> <span class="n">cls</span><span class="p">(</span><span class="n">nz</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">unpolarify</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">unpolarify</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_trigfunc</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span><span class="o">/</span><span class="n">arg</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_Ei</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_rewrite_as_expint</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">Ei</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_uppergamma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">uppergamma</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_rewrite_as_expint</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">uppergamma</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_nseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="c"># NOTE this is fairly inefficient</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">log</span><span class="p">,</span> <span class="n">EulerGamma</span><span class="p">,</span> <span class="n">Pow</span>
-        <span class="n">n</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">TrigonometricIntegral</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">)</span>
-        <span class="n">baseseries</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_trigfunc</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_trigfunc</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">baseseries</span> <span class="o">-=</span> <span class="mi">1</span>
-        <span class="n">baseseries</span> <span class="o">=</span> <span class="n">baseseries</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">Pow</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">,</span> <span class="n">n</span><span class="p">:</span> <span class="n">t</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="n">n</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_trigfunc</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">baseseries</span> <span class="o">+=</span> <span class="n">EulerGamma</span> <span class="o">+</span> <span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">baseseries</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="Si"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.error_functions.Si">[docs]</a><span class="k">class</span> <span class="nc">Si</span><span class="p">(</span><span class="n">TrigonometricIntegral</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Sine integral.</span>
-
-<span class="sd">    This function is defined by</span>
-
-<span class="sd">    .. math:: \operatorname{Si}(z) = \int_0^z \frac{\sin{t}}{t} \mathrm{d}t.</span>
-
-<span class="sd">    It is an entire function.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Ci: Cosine integral.</span>
-<span class="sd">    Shi: Sinh integral.</span>
-<span class="sd">    Chi: Cosh integral.</span>
-<span class="sd">    expint: The generalised exponential integral.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - http://en.wikipedia.org/wiki/Trigonometric_integral</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Si</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import z</span>
-
-<span class="sd">    The sine integral is an antiderivative of sin(z)/z:</span>
-
-<span class="sd">    &gt;&gt;&gt; Si(z).diff(z)</span>
-<span class="sd">    sin(z)/z</span>
-
-<span class="sd">    It is unbranched:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import exp_polar, I, pi</span>
-<span class="sd">    &gt;&gt;&gt; Si(z*exp_polar(2*I*pi))</span>
-<span class="sd">    Si(z)</span>
-
-<span class="sd">    Sine integral behaves much like ordinary sine under multiplication by I:</span>
-
-<span class="sd">    &gt;&gt;&gt; Si(I*z)</span>
-<span class="sd">    I*Shi(z)</span>
-<span class="sd">    &gt;&gt;&gt; Si(-z)</span>
-<span class="sd">    -Si(z)</span>
-
-<span class="sd">    It can also be expressed in terms of exponential integrals, but beware</span>
-<span class="sd">    that the latter is branched:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import expint</span>
-<span class="sd">    &gt;&gt;&gt; Si(z).rewrite(expint)</span>
-<span class="sd">    -I*(-expint(1, z*exp_polar(-I*pi/2))/2 +</span>
-<span class="sd">         expint(1, z*exp_polar(I*pi/2))/2) + pi/2</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_trigfunc</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">sin</span>
-    <span class="n">_atzero</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-    <span class="n">_atinf</span> <span class="o">=</span> <span class="n">pi</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span>
-    <span class="n">_atneginf</span> <span class="o">=</span> <span class="o">-</span><span class="n">pi</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_minusfactor</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">Si</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_Ifactor</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">sign</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">I</span><span class="o">*</span><span class="n">Shi</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">sign</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_expint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="c"># XXX should we polarify z?</span>
-        <span class="k">return</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="p">(</span><span class="n">E1</span><span class="p">(</span><span class="n">polar_lift</span><span class="p">(</span><span class="n">I</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">)</span> <span class="o">-</span> <span class="n">E1</span><span class="p">(</span><span class="n">polar_lift</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span><span class="o">/</span><span class="n">I</span>
-
-</div>
-<div class="viewcode-block" id="Ci"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.error_functions.Ci">[docs]</a><span class="k">class</span> <span class="nc">Ci</span><span class="p">(</span><span class="n">TrigonometricIntegral</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Cosine integral.</span>
-
-<span class="sd">    This function is defined for positive :math:`x` by</span>
-
-<span class="sd">    .. math:: \operatorname{Ci}(x) = \gamma + \log{x}</span>
-<span class="sd">                         + \int_0^x \frac{\cos{t} - 1}{t} \mathrm{d}t</span>
-<span class="sd">           = -\int_x^\infty \frac{\cos{t}}{t} \mathrm{d}t,</span>
-
-<span class="sd">    where :math:`\gamma` is the Euler-Mascheroni constant.</span>
-
-<span class="sd">    We have</span>
-
-<span class="sd">    .. math:: \operatorname{Ci}(z) =</span>
-<span class="sd">        -\frac{\operatorname{E}_1\left(e^{i\pi/2} z\right)</span>
-<span class="sd">               + \operatorname{E}_1\left(e^{-i \pi/2} z\right)}{2}</span>
-
-<span class="sd">    which holds for all polar :math:`z` and thus provides an analytic</span>
-<span class="sd">    continuation to the Riemann surface of the logarithm.</span>
-
-<span class="sd">    The formula also holds as stated</span>
-<span class="sd">    for :math:`z \in \mathbb{C}` with :math:`Re(z) &gt; 0`.</span>
-<span class="sd">    By lifting to the principal branch we obtain an analytic function on the</span>
-<span class="sd">    cut complex plane.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Si: Sine integral.</span>
-<span class="sd">    Shi: Sinh integral.</span>
-<span class="sd">    Chi: Cosh integral.</span>
-<span class="sd">    expint: The generalised exponential integral.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - http://en.wikipedia.org/wiki/Trigonometric_integral</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Ci</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import z</span>
-
-<span class="sd">    The cosine integral is a primitive of cos(z)/z:</span>
-
-<span class="sd">    &gt;&gt;&gt; Ci(z).diff(z)</span>
-<span class="sd">    cos(z)/z</span>
-
-<span class="sd">    It has a logarithmic branch point at the origin:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import exp_polar, I, pi</span>
-<span class="sd">    &gt;&gt;&gt; Ci(z*exp_polar(2*I*pi))</span>
-<span class="sd">    Ci(z) + 2*I*pi</span>
-
-<span class="sd">    Cosine integral behaves somewhat like ordinary cos under multiplication by I:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import polar_lift</span>
-<span class="sd">    &gt;&gt;&gt; Ci(polar_lift(I)*z)</span>
-<span class="sd">    Chi(z) + I*pi/2</span>
-<span class="sd">    &gt;&gt;&gt; Ci(polar_lift(-1)*z)</span>
-<span class="sd">    Ci(z) + I*pi</span>
-
-<span class="sd">    It can also be expressed in terms of exponential integrals:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import expint</span>
-<span class="sd">    &gt;&gt;&gt; Ci(z).rewrite(expint)</span>
-<span class="sd">    -expint(1, z*exp_polar(-I*pi/2))/2 - expint(1, z*exp_polar(I*pi/2))/2</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_trigfunc</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">cos</span>
-    <span class="n">_atzero</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span>
-    <span class="n">_atinf</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-    <span class="n">_atneginf</span> <span class="o">=</span> <span class="n">I</span><span class="o">*</span><span class="n">pi</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_minusfactor</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Ci</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">pi</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_Ifactor</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">sign</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Chi</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="o">*</span><span class="n">sign</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_expint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="p">(</span><span class="n">E1</span><span class="p">(</span><span class="n">polar_lift</span><span class="p">(</span><span class="n">I</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="n">E1</span><span class="p">(</span><span class="n">polar_lift</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span>
-
-</div>
-<div class="viewcode-block" id="Shi"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.error_functions.Shi">[docs]</a><span class="k">class</span> <span class="nc">Shi</span><span class="p">(</span><span class="n">TrigonometricIntegral</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Sinh integral.</span>
-
-<span class="sd">    This function is defined by</span>
-
-<span class="sd">    .. math:: \operatorname{Shi}(z) = \int_0^z \frac{\sinh{t}}{t} \mathrm{d}t.</span>
-
-<span class="sd">    It is an entire function.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Si: Sine integral.</span>
-<span class="sd">    Ci: Cosine integral.</span>
-<span class="sd">    Chi: Cosh integral.</span>
-<span class="sd">    expint: The generalised exponential integral.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - http://en.wikipedia.org/wiki/Trigonometric_integral</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Shi</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import z</span>
-
-<span class="sd">    The Sinh integral is a primitive of sinh(z)/z:</span>
-
-<span class="sd">    &gt;&gt;&gt; Shi(z).diff(z)</span>
-<span class="sd">    sinh(z)/z</span>
-
-<span class="sd">    It is unbranched:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import exp_polar, I, pi</span>
-<span class="sd">    &gt;&gt;&gt; Shi(z*exp_polar(2*I*pi))</span>
-<span class="sd">    Shi(z)</span>
-
-<span class="sd">    Sinh integral behaves much like ordinary sinh under multiplication by I:</span>
-
-<span class="sd">    &gt;&gt;&gt; Shi(I*z)</span>
-<span class="sd">    I*Si(z)</span>
-<span class="sd">    &gt;&gt;&gt; Shi(-z)</span>
-<span class="sd">    -Shi(z)</span>
-
-<span class="sd">    It can also be expressed in terms of exponential integrals, but beware</span>
-<span class="sd">    that the latter is branched:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import expint</span>
-<span class="sd">    &gt;&gt;&gt; Shi(z).rewrite(expint)</span>
-<span class="sd">    expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_trigfunc</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">sinh</span>
-    <span class="n">_atzero</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-    <span class="n">_atinf</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-    <span class="n">_atneginf</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_minusfactor</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">Shi</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_Ifactor</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">sign</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">I</span><span class="o">*</span><span class="n">Si</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">sign</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_expint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exp_polar</span>
-        <span class="c"># XXX should we polarify z?</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">E1</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">-</span> <span class="n">E1</span><span class="p">(</span><span class="n">exp_polar</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span>
-
-</div>
-<div class="viewcode-block" id="Chi"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.error_functions.Chi">[docs]</a><span class="k">class</span> <span class="nc">Chi</span><span class="p">(</span><span class="n">TrigonometricIntegral</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Cosh integral.</span>
-
-<span class="sd">    This function is defined for positive :math:`x` by</span>
-
-<span class="sd">    .. math:: \operatorname{Chi}(x) = \gamma + \log{x}</span>
-<span class="sd">                         + \int_0^x \frac{\cosh{t} - 1}{t} \mathrm{d}t,</span>
-
-<span class="sd">    where :math:`\gamma` is the Euler-Mascheroni constant.</span>
-
-<span class="sd">    We have</span>
-
-<span class="sd">    .. math:: \operatorname{Chi}(z) = \operatorname{Ci}\left(e^{i \pi/2}z\right)</span>
-<span class="sd">                         - i\frac{\pi}{2},</span>
-
-<span class="sd">    which holds for all polar :math:`z` and thus provides an analytic</span>
-<span class="sd">    continuation to the Riemann surface of the logarithm.</span>
-<span class="sd">    By lifting to the principal branch we obtain an analytic function on the</span>
-<span class="sd">    cut complex plane.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Si: Sine integral.</span>
-<span class="sd">    Ci: Cosine integral.</span>
-<span class="sd">    Shi: Sinh integral.</span>
-<span class="sd">    expint: The generalised exponential integral.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - http://en.wikipedia.org/wiki/Trigonometric_integral</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Chi</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import z</span>
-
-<span class="sd">    The cosh integral is a primitive of cosh(z)/z:</span>
-
-<span class="sd">    &gt;&gt;&gt; Chi(z).diff(z)</span>
-<span class="sd">    cosh(z)/z</span>
-
-<span class="sd">    It has a logarithmic branch point at the origin:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import exp_polar, I, pi</span>
-<span class="sd">    &gt;&gt;&gt; Chi(z*exp_polar(2*I*pi))</span>
-<span class="sd">    Chi(z) + 2*I*pi</span>
-
-<span class="sd">    Cosh integral behaves somewhat like ordinary cosh under multiplication by I:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import polar_lift</span>
-<span class="sd">    &gt;&gt;&gt; Chi(polar_lift(I)*z)</span>
-<span class="sd">    Ci(z) + I*pi/2</span>
-<span class="sd">    &gt;&gt;&gt; Chi(polar_lift(-1)*z)</span>
-<span class="sd">    Chi(z) + I*pi</span>
-
-<span class="sd">    It can also be expressed in terms of exponential integrals:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import expint</span>
-<span class="sd">    &gt;&gt;&gt; Chi(z).rewrite(expint)</span>
-<span class="sd">    -expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_trigfunc</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">cosh</span>
-    <span class="n">_atzero</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span>
-    <span class="n">_atinf</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-    <span class="n">_atneginf</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_minusfactor</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Chi</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">pi</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_Ifactor</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">sign</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Ci</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="o">*</span><span class="n">sign</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_expint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exp_polar</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="p">(</span><span class="n">E1</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="n">E1</span><span class="p">(</span><span class="n">exp_polar</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span>
-
-
-<span class="c">###############################################################################</span>
-<span class="c">#################### FRESNEL INTEGRALS ########################################</span>
-<span class="c">###############################################################################</span>
-</div>
-<div class="viewcode-block" id="FresnelIntegral"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.error_functions.FresnelIntegral">[docs]</a><span class="k">class</span> <span class="nc">FresnelIntegral</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Base class for the Fresnel integrals.&quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">unbranched</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="c"># Value at zero</span>
-        <span class="k">if</span> <span class="n">z</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-
-        <span class="c"># Try to pull out factors of -1 and I</span>
-        <span class="n">prefact</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="n">newarg</span> <span class="o">=</span> <span class="n">z</span>
-        <span class="n">changed</span> <span class="o">=</span> <span class="bp">False</span>
-
-        <span class="n">nz</span> <span class="o">=</span> <span class="n">newarg</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">nz</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">prefact</span> <span class="o">=</span> <span class="o">-</span><span class="n">prefact</span>
-            <span class="n">newarg</span> <span class="o">=</span> <span class="n">nz</span>
-            <span class="n">changed</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="n">nz</span> <span class="o">=</span> <span class="n">newarg</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">nz</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">prefact</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_sign</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">prefact</span>
-            <span class="n">newarg</span> <span class="o">=</span> <span class="n">nz</span>
-            <span class="n">changed</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="k">if</span> <span class="n">changed</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">prefact</span><span class="o">*</span><span class="n">cls</span><span class="p">(</span><span class="n">newarg</span><span class="p">)</span>
-
-        <span class="c"># Values at positive infinities signs</span>
-        <span class="c"># if any were extracted automatically</span>
-        <span class="k">if</span> <span class="n">z</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span>
-        <span class="k">elif</span> <span class="n">z</span> <span class="ow">is</span> <span class="n">I</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_sign</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_trigfunc</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">conjugate</span><span class="p">())</span>
-
-    <span class="k">def</span> <span class="nf">_as_real_imag</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-                <span class="n">hints</span><span class="p">[</span><span class="s">&#39;complex&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">as_real_imag</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="c"># Fresnel S</span>
-        <span class="c"># http://functions.wolfram.com/06.32.19.0003.01</span>
-        <span class="c"># http://functions.wolfram.com/06.32.19.0006.01</span>
-        <span class="c"># Fresnel C</span>
-        <span class="c"># http://functions.wolfram.com/06.33.19.0003.01</span>
-        <span class="c"># http://functions.wolfram.com/06.33.19.0006.01</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_as_real_imag</span><span class="p">(</span><span class="n">deep</span><span class="o">=</span><span class="n">deep</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-        <span class="n">sq</span> <span class="o">=</span> <span class="o">-</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span>
-        <span class="n">re</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="o">*</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">sq</span><span class="p">))</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">x</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">sq</span><span class="p">)))</span>
-        <span class="n">im</span> <span class="o">=</span> <span class="n">x</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="p">)</span> <span class="o">*</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">sq</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">x</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">sq</span><span class="p">))</span> <span class="o">-</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">sq</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="fresnels"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.error_functions.fresnels">[docs]</a><span class="k">class</span> <span class="nc">fresnels</span><span class="p">(</span><span class="n">FresnelIntegral</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Fresnel integral S.</span>
-
-<span class="sd">    This function is defined by</span>
-
-<span class="sd">    .. math:: \operatorname{S}(z) = \int_0^z \sin{\frac{\pi}{2} t^2} \mathrm{d}t.</span>
-
-<span class="sd">    It is an entire function.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import I, oo, fresnels</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import z</span>
-
-<span class="sd">    Several special values are known:</span>
-
-<span class="sd">    &gt;&gt;&gt; fresnels(0)</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; fresnels(oo)</span>
-<span class="sd">    1/2</span>
-<span class="sd">    &gt;&gt;&gt; fresnels(-oo)</span>
-<span class="sd">    -1/2</span>
-<span class="sd">    &gt;&gt;&gt; fresnels(I*oo)</span>
-<span class="sd">    -I/2</span>
-<span class="sd">    &gt;&gt;&gt; fresnels(-I*oo)</span>
-<span class="sd">    I/2</span>
-
-<span class="sd">    In general one can pull out factors of -1 and I from the argument:</span>
-<span class="sd">    &gt;&gt;&gt; fresnels(-z)</span>
-<span class="sd">    -fresnels(z)</span>
-
-<span class="sd">    &gt;&gt;&gt; fresnels(I*z)</span>
-<span class="sd">    -I*fresnels(z)</span>
-
-<span class="sd">    The Fresnel S integral obeys the mirror symmetry:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import conjugate</span>
-<span class="sd">    &gt;&gt;&gt; conjugate(fresnels(z))</span>
-<span class="sd">    fresnels(conjugate(z))</span>
-
-<span class="sd">    Differentiation with respect to z is supported:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import diff</span>
-<span class="sd">    &gt;&gt;&gt; diff(fresnels(z), z)</span>
-<span class="sd">    sin(pi*z**2/2)</span>
-
-<span class="sd">    Defining the Fresnel functions via an integral</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import integrate, pi, sin, gamma, expand_func</span>
-<span class="sd">    &gt;&gt;&gt; integrate(sin(pi*z**2/2), z)</span>
-<span class="sd">    3*fresnels(z)*gamma(3/4)/(4*gamma(7/4))</span>
-<span class="sd">    &gt;&gt;&gt; expand_func(integrate(sin(pi*z**2/2), z))</span>
-<span class="sd">    fresnels(z)</span>
-
-<span class="sd">    We can numerically evaluate the Fresnel integral to arbitrary precision</span>
-<span class="sd">    on the whole complex plane:</span>
-
-<span class="sd">    &gt;&gt;&gt; fresnels(2).evalf(30)</span>
-<span class="sd">    0.343415678363698242195300815958</span>
-
-<span class="sd">    &gt;&gt;&gt; fresnels(-2*I).evalf(30)</span>
-<span class="sd">    0.343415678363698242195300815958*I</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    fresnelc</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Fresnel_integral</span>
-<span class="sd">    .. [2] http://dlmf.nist.gov/7</span>
-<span class="sd">    .. [3] http://mathworld.wolfram.com/FresnelIntegrals.html</span>
-<span class="sd">    .. [4] http://functions.wolfram.com/GammaBetaErf/FresnelS</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_trigfunc</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">sin</span>
-    <span class="n">_sign</span> <span class="o">=</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">previous_terms</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="n">previous_terms</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-                <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span><span class="o">*</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">8</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">n</span> <span class="o">+</span> <span class="mi">3</span><span class="p">)))</span> <span class="o">*</span> <span class="n">p</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">*</span> <span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span><span class="p">)</span><span class="o">**</span><span class="n">n</span> <span class="o">*</span> <span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">pi</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span> <span class="o">/</span> <span class="p">((</span><span class="mi">4</span><span class="o">*</span><span class="n">n</span> <span class="o">+</span> <span class="mi">3</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_erf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="o">+</span> <span class="n">I</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span> <span class="o">*</span> <span class="p">(</span><span class="n">erf</span><span class="p">((</span><span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="o">+</span> <span class="n">I</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">)</span> <span class="o">-</span> <span class="n">I</span><span class="o">*</span><span class="n">erf</span><span class="p">((</span><span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="o">-</span> <span class="n">I</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_hyper</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">3</span><span class="o">/</span><span class="mi">6</span> <span class="o">*</span> <span class="n">hyper</span><span class="p">([</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">7</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">],</span> <span class="o">-</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">4</span><span class="o">/</span><span class="mi">16</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_meijerg</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">9</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">))</span>
-                <span class="o">*</span> <span class="n">meijerg</span><span class="p">([],</span> <span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="o">-</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">4</span><span class="o">/</span><span class="mi">16</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="fresnelc"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.error_functions.fresnelc">[docs]</a><span class="k">class</span> <span class="nc">fresnelc</span><span class="p">(</span><span class="n">FresnelIntegral</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Fresnel integral C.</span>
-
-<span class="sd">    This function is defined by</span>
-
-<span class="sd">    .. math:: \operatorname{C}(z) = \int_0^z \cos{\frac{\pi}{2} t^2} \mathrm{d}t.</span>
-
-<span class="sd">    It is an entire function.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import I, oo, fresnelc</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import z</span>
-
-<span class="sd">    Several special values are known:</span>
-
-<span class="sd">    &gt;&gt;&gt; fresnelc(0)</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; fresnelc(oo)</span>
-<span class="sd">    1/2</span>
-<span class="sd">    &gt;&gt;&gt; fresnelc(-oo)</span>
-<span class="sd">    -1/2</span>
-<span class="sd">    &gt;&gt;&gt; fresnelc(I*oo)</span>
-<span class="sd">    I/2</span>
-<span class="sd">    &gt;&gt;&gt; fresnelc(-I*oo)</span>
-<span class="sd">    -I/2</span>
-
-<span class="sd">    In general one can pull out factors of -1 and I from the argument:</span>
-<span class="sd">    &gt;&gt;&gt; fresnelc(-z)</span>
-<span class="sd">    -fresnelc(z)</span>
-
-<span class="sd">    &gt;&gt;&gt; fresnelc(I*z)</span>
-<span class="sd">    I*fresnelc(z)</span>
-
-<span class="sd">    The Fresnel C integral obeys the mirror symmetry:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import conjugate</span>
-<span class="sd">    &gt;&gt;&gt; conjugate(fresnelc(z))</span>
-<span class="sd">    fresnelc(conjugate(z))</span>
-
-<span class="sd">    Differentiation with respect to z is supported:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import diff</span>
-<span class="sd">    &gt;&gt;&gt; diff(fresnelc(z), z)</span>
-<span class="sd">    cos(pi*z**2/2)</span>
-
-<span class="sd">    Defining the Fresnel functions via an integral</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import integrate, pi, cos, gamma, expand_func</span>
-<span class="sd">    &gt;&gt;&gt; integrate(cos(pi*z**2/2), z)</span>
-<span class="sd">    fresnelc(z)*gamma(1/4)/(4*gamma(5/4))</span>
-<span class="sd">    &gt;&gt;&gt; expand_func(integrate(cos(pi*z**2/2), z))</span>
-<span class="sd">    fresnelc(z)</span>
-
-<span class="sd">    We can numerically evaluate the Fresnel integral to arbitrary precision</span>
-<span class="sd">    on the whole complex plane:</span>
-
-<span class="sd">    &gt;&gt;&gt; fresnelc(2).evalf(30)</span>
-<span class="sd">    0.488253406075340754500223503357</span>
-
-<span class="sd">    &gt;&gt;&gt; fresnelc(-2*I).evalf(30)</span>
-<span class="sd">    -0.488253406075340754500223503357*I</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    fresnels</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Fresnel_integral</span>
-<span class="sd">    .. [2] http://dlmf.nist.gov/7</span>
-<span class="sd">    .. [3] http://mathworld.wolfram.com/FresnelIntegrals.html</span>
-<span class="sd">    .. [4] http://functions.wolfram.com/GammaBetaErf/FresnelC</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_trigfunc</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">cos</span>
-    <span class="n">_sign</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">taylor_term</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">*</span><span class="n">previous_terms</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">previous_terms</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="n">previous_terms</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-                <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span><span class="o">*</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">n</span> <span class="o">-</span> <span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">8</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)))</span> <span class="o">*</span> <span class="n">p</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">x</span> <span class="o">*</span> <span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span><span class="p">)</span><span class="o">**</span><span class="n">n</span> <span class="o">*</span> <span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">pi</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="p">))</span> <span class="o">/</span> <span class="p">((</span><span class="mi">4</span><span class="o">*</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_erf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="o">-</span> <span class="n">I</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span> <span class="o">*</span> <span class="p">(</span><span class="n">erf</span><span class="p">((</span><span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="o">+</span> <span class="n">I</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">erf</span><span class="p">((</span><span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="o">-</span> <span class="n">I</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_hyper</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">z</span> <span class="o">*</span> <span class="n">hyper</span><span class="p">([</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="o">/</span><span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">],</span> <span class="o">-</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">4</span><span class="o">/</span><span class="mi">16</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_meijerg</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">root</span><span class="p">(</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="n">root</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
-                <span class="o">*</span> <span class="n">meijerg</span><span class="p">([],</span> <span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="o">-</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">4</span><span class="o">/</span><span class="mi">16</span><span class="p">))</span>
-
-<span class="c">###############################################################################</span>
-<span class="c">#################### HELPER FUNCTIONS #########################################</span>
-<span class="c">###############################################################################</span>
-
-</div>
-<span class="k">class</span> <span class="nc">_erfs</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Helper function to make the :math:`erf(z)` function</span>
-<span class="sd">    tractable for the Gruntz algorithm.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">_eval_aseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">args0</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">args0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">_erfs</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_aseries</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">args0</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">logx</span><span class="p">)</span>
-
-        <span class="n">z</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">l</span> <span class="o">=</span> <span class="p">[</span> <span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">S</span><span class="p">(</span>
-            <span class="mi">4</span><span class="p">))</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">k</span><span class="p">)</span><span class="o">/</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">k</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="p">]</span>
-        <span class="n">o</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-        <span class="c"># It is very inefficient to first add the order and then do the nseries</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">l</span><span class="p">))</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">)</span> <span class="o">+</span> <span class="n">o</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">z</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="o">-</span><span class="mi">2</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="o">*</span><span class="n">_erfs</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_intractable</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="o">-</span> <span class="n">erf</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-</pre></div>
-
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-  <h3>Quick search</h3>
-    <form class="search" action="../../../../search.html" method="get">
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-  <h1>Source code for sympy.functions.special.gamma_functions</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Add</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">pi</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">ArgumentIndexError</span>
-<span class="kn">from</span> <span class="nn">.zeta_functions</span> <span class="kn">import</span> <span class="n">zeta</span>
-<span class="kn">from</span> <span class="nn">.error_functions</span> <span class="kn">import</span> <span class="n">erf</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Dummy</span><span class="p">,</span> <span class="n">Rational</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.exponential</span> <span class="kn">import</span> <span class="n">log</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.integers</span> <span class="kn">import</span> <span class="n">floor</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.combinatorial.numbers</span> <span class="kn">import</span> <span class="n">bernoulli</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.combinatorial.factorials</span> <span class="kn">import</span> <span class="n">rf</span>
-
-<span class="c">###############################################################################</span>
-<span class="c">############################ COMPLETE GAMMA FUNCTION ##########################</span>
-<span class="c">###############################################################################</span>
-
-
-<div class="viewcode-block" id="gamma"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.gamma_functions.gamma">[docs]</a><span class="k">class</span> <span class="nc">gamma</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The gamma function returns a function which passes through the integral</span>
-<span class="sd">    values of the factorial function, i.e. though defined in the complex plane,</span>
-<span class="sd">    when n is an integer, `gamma(n) = (n - 1)!`</span>
-
-<span class="sd">    Reference:</span>
-<span class="sd">        http://en.wikipedia.org/wiki/Gamma_function</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">unbranched</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">gamma</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">*</span><span class="n">polygamma</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">arg</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-            <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">arg</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span>
-            <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">q</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                    <span class="n">n</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">p</span><span class="p">)</span> <span class="o">//</span> <span class="n">arg</span><span class="o">.</span><span class="n">q</span>
-
-                    <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                        <span class="n">k</span><span class="p">,</span> <span class="n">coeff</span> <span class="o">=</span> <span class="n">n</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">n</span> <span class="o">=</span> <span class="n">k</span> <span class="o">=</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span>
-
-                        <span class="k">if</span> <span class="n">n</span> <span class="o">&amp;</span> <span class="mi">1</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                            <span class="n">coeff</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="n">coeff</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span>
-
-                    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">,</span> <span class="mi">2</span><span class="p">):</span>
-                        <span class="n">coeff</span> <span class="o">*=</span> <span class="n">i</span>
-
-                    <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">coeff</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span><span class="o">**</span><span class="n">n</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="mi">2</span><span class="o">**</span><span class="n">n</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">)</span> <span class="o">/</span> <span class="n">coeff</span>
-
-    <span class="k">def</span> <span class="nf">_eval_expand_func</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">p</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">arg</span><span class="o">.</span><span class="n">q</span><span class="p">:</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-                <span class="n">n</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">p</span> <span class="o">//</span> <span class="n">arg</span><span class="o">.</span><span class="n">q</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">p</span> <span class="o">-</span> <span class="n">n</span><span class="o">*</span><span class="n">arg</span><span class="o">.</span><span class="n">q</span>
-                <span class="k">return</span> <span class="n">gamma</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_expand_func</span><span class="p">()</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">Rational</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">arg</span><span class="o">.</span><span class="n">q</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">coeff</span><span class="p">,</span> <span class="n">tail</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coeff_add</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">coeff</span> <span class="ow">and</span> <span class="n">coeff</span><span class="o">.</span><span class="n">q</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">intpart</span> <span class="o">=</span> <span class="n">floor</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-                <span class="n">tail</span> <span class="o">=</span> <span class="p">(</span><span class="n">coeff</span> <span class="o">-</span> <span class="n">intpart</span><span class="p">,)</span> <span class="o">+</span> <span class="n">tail</span>
-                <span class="n">coeff</span> <span class="o">=</span> <span class="n">intpart</span>
-            <span class="n">tail</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">_new_rawargs</span><span class="p">(</span><span class="o">*</span><span class="n">tail</span><span class="p">,</span> <span class="o">**</span><span class="nb">dict</span><span class="p">(</span><span class="n">reeval</span><span class="o">=</span><span class="bp">False</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">gamma</span><span class="p">(</span><span class="n">tail</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span><span class="n">tail</span><span class="p">,</span> <span class="n">coeff</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_tractable</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">loggamma</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_nseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="n">x0</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">x0</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">x0</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">gamma</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">)</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">x0</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">gamma</span><span class="p">(</span><span class="n">t</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">rf</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="o">-</span><span class="n">x0</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">)</span>
-
-
-<span class="c">###############################################################################</span>
-<span class="c">################## LOWER and UPPER INCOMPLETE GAMMA FUNCTIONS #################</span>
-<span class="c">###############################################################################</span>
-</div>
-<div class="viewcode-block" id="lowergamma"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.gamma_functions.lowergamma">[docs]</a><span class="k">class</span> <span class="nc">lowergamma</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    The lower incomplete gamma function.</span>
-
-<span class="sd">    It can be defined as the meromorphic continuation of</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        \gamma(s, x) = \int_0^x t^{s-1} e^{-t} \mathrm{d}t.</span>
-
-<span class="sd">    This can be shown to be the same as</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        \gamma(s, x) = \frac{x^s}{s} {}_1F_1\left({s \atop s+1} \middle| -x\right),</span>
-
-<span class="sd">    where :math:`{}_1F_1` is the (confluent) hypergeometric function.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    gamma, uppergamma</span>
-<span class="sd">    sympy.functions.special.hyper.hyper</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import lowergamma, S</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import s, x</span>
-<span class="sd">    &gt;&gt;&gt; lowergamma(s, x)</span>
-<span class="sd">    lowergamma(s, x)</span>
-<span class="sd">    &gt;&gt;&gt; lowergamma(3, x)</span>
-<span class="sd">    -x**2*exp(-x) - 2*x*exp(-x) + 2 - 2*exp(-x)</span>
-<span class="sd">    &gt;&gt;&gt; lowergamma(-S(1)/2, x)</span>
-<span class="sd">    -2*sqrt(pi)*erf(sqrt(x)) - 2*exp(-x)/sqrt(x)</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6, Section 5,</span>
-<span class="sd">      Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical</span>
-<span class="sd">      Tables</span>
-<span class="sd">    - http://en.wikipedia.org/wiki/Incomplete_gamma_function</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">2</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">meijerg</span><span class="p">,</span> <span class="n">unpolarify</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">unpolarify</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">return</span> <span class="n">gamma</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">digamma</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">-</span> <span class="n">log</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">uppergamma</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span> \
-                <span class="o">+</span> <span class="n">meijerg</span><span class="p">([],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">a</span><span class="p">],</span> <span class="p">[],</span> <span class="n">z</span><span class="p">)</span>
-
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="c"># For lack of a better place, we use this one to extract branching</span>
-        <span class="c"># information. The following can be</span>
-        <span class="c"># found in the literature (c/f references given above), albeit scattered:</span>
-        <span class="c"># 1) For fixed x != 0, lowergamma(s, x) is an entire function of s</span>
-        <span class="c"># 2) For fixed positive integers s, lowergamma(s, x) is an entire</span>
-        <span class="c">#    function of x.</span>
-        <span class="c"># 3) For fixed non-positive integers s,</span>
-        <span class="c">#    lowergamma(s, exp(I*2*pi*n)*x) =</span>
-        <span class="c">#              2*pi*I*n*(-1)**(-s)/factorial(-s) + lowergamma(s, x)</span>
-        <span class="c">#    (this follows from lowergamma(s, x).diff(x) = x**(s-1)*exp(-x)).</span>
-        <span class="c"># 4) For fixed non-integral s,</span>
-        <span class="c">#    lowergamma(s, x) = x**s*gamma(s)*lowergamma_unbranched(s, x),</span>
-        <span class="c">#    where lowergamma_unbranched(s, x) is an entire function (in fact</span>
-        <span class="c">#    of both s and x), i.e.</span>
-        <span class="c">#    lowergamma(s, exp(2*I*pi*n)*x) = exp(2*pi*I*n*a)*lowergamma(a, x)</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">unpolarify</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">exp</span>
-        <span class="n">nx</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">extract_branch_factor</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">and</span> <span class="n">a</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">nx</span> <span class="o">=</span> <span class="n">unpolarify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">nx</span> <span class="o">!=</span> <span class="n">x</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">lowergamma</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">nx</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">a</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">and</span> <span class="n">a</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="p">)</span> <span class="o">+</span> <span class="n">lowergamma</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">nx</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">n</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">lowergamma</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">nx</span><span class="p">)</span>
-
-        <span class="c"># Special values.</span>
-        <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="c"># TODO this should be non-recursive</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="o">-</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">a</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">erf</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-            <span class="k">elif</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">or</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="n">a</span> <span class="o">-</span> <span class="mi">1</span>
-                <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">b</span><span class="o">*</span><span class="n">cls</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="n">b</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="p">(</span><span class="n">cls</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="n">a</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">))</span><span class="o">/</span><span class="n">a</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.mpmath</span> <span class="kn">import</span> <span class="n">mp</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Expr</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_to_mpmath</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-        <span class="n">z</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">_to_mpmath</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-        <span class="n">oprec</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">prec</span>
-        <span class="n">mp</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="n">prec</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">gammainc</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-        <span class="n">mp</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="n">oprec</span>
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">_from_mpmath</span><span class="p">(</span><span class="n">res</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_uppergamma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">gamma</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="o">-</span> <span class="n">uppergamma</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_expint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expint</span>
-        <span class="k">if</span> <span class="n">s</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">and</span> <span class="n">s</span><span class="o">.</span><span class="n">is_nonpositive</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">uppergamma</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">expint</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="uppergamma"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.gamma_functions.uppergamma">[docs]</a><span class="k">class</span> <span class="nc">uppergamma</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Upper incomplete gamma function</span>
-
-<span class="sd">    It can be defined as the meromorphic continuation of</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        \Gamma(s, x) = \int_x^\infty t^{s-1} e^{-t} \mathrm{d}t</span>
-<span class="sd">                     = \Gamma(s) - \gamma(s, x).</span>
-
-<span class="sd">    This can be shown to be the same as</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        \Gamma(s, x) = \Gamma(s)</span>
-<span class="sd">                - \frac{x^s}{s} {}_1F_1\left({s \atop s+1} \middle| -x\right),</span>
-
-<span class="sd">    where :math:`{}_1F_1` is the (confluent) hypergeometric function.</span>
-
-<span class="sd">    The upper incomplete gamma function is also essentially equivalent to the</span>
-<span class="sd">    generalized exponential integral.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import uppergamma, S</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import s, x</span>
-<span class="sd">    &gt;&gt;&gt; uppergamma(s, x)</span>
-<span class="sd">    uppergamma(s, x)</span>
-<span class="sd">    &gt;&gt;&gt; uppergamma(3, x)</span>
-<span class="sd">    x**2*exp(-x) + 2*x*exp(-x) + 2*exp(-x)</span>
-<span class="sd">    &gt;&gt;&gt; uppergamma(-S(1)/2, x)</span>
-<span class="sd">    -2*sqrt(pi)*(-erf(sqrt(x)) + 1) + 2*exp(-x)/sqrt(x)</span>
-<span class="sd">    &gt;&gt;&gt; uppergamma(-2, x)</span>
-<span class="sd">    expint(3, x)/x**2</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    gamma, lowergamma</span>
-<span class="sd">    sympy.functions.special.hyper.hyper</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6, Section 5,</span>
-<span class="sd">      Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical</span>
-<span class="sd">      Tables</span>
-<span class="sd">    - http://en.wikipedia.org/wiki/Incomplete_gamma_function</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">2</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">meijerg</span><span class="p">,</span> <span class="n">unpolarify</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">unpolarify</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">return</span> <span class="n">uppergamma</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="n">meijerg</span><span class="p">([],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">a</span><span class="p">],</span> <span class="p">[],</span> <span class="n">z</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.mpmath</span> <span class="kn">import</span> <span class="n">mp</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Expr</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_to_mpmath</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-        <span class="n">z</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">_to_mpmath</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-        <span class="n">oprec</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">prec</span>
-        <span class="n">mp</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="n">prec</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">gammainc</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">mp</span><span class="o">.</span><span class="n">inf</span><span class="p">)</span>
-        <span class="n">mp</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="n">oprec</span>
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">_from_mpmath</span><span class="p">(</span><span class="n">res</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">unpolarify</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">expint</span>
-        <span class="k">if</span> <span class="n">z</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">z</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">z</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">elif</span> <span class="n">z</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">gamma</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-
-        <span class="c"># We extract branching information here. C/f lowergamma.</span>
-        <span class="n">nx</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">z</span><span class="o">.</span><span class="n">extract_branch_factor</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">and</span> <span class="n">a</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">nx</span> <span class="o">=</span> <span class="n">unpolarify</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">z</span> <span class="o">!=</span> <span class="n">nx</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">uppergamma</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">nx</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">a</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">and</span> <span class="n">a</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="p">)</span> <span class="o">+</span> <span class="n">uppergamma</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">nx</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">n</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">gamma</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">a</span><span class="p">))</span> <span class="o">+</span> <span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">uppergamma</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">nx</span><span class="p">)</span>
-
-        <span class="c"># Special values.</span>
-        <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="c"># TODO this should be non-recursive</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">a</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">erf</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)))</span>  <span class="c"># TODO could use erfc...</span>
-            <span class="k">elif</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">or</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="n">a</span> <span class="o">-</span> <span class="mi">1</span>
-                <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">b</span><span class="o">*</span><span class="n">cls</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="n">z</span><span class="o">**</span><span class="n">b</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">expint</span><span class="p">(</span><span class="o">-</span><span class="n">b</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">unpolarify</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="p">(</span><span class="n">cls</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span> <span class="o">-</span> <span class="n">z</span><span class="o">**</span><span class="n">a</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">))</span><span class="o">/</span><span class="n">a</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_lowergamma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">gamma</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="o">-</span> <span class="n">lowergamma</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_expint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expint</span>
-        <span class="k">return</span> <span class="n">expint</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">s</span>
-
-
-<span class="c">###############################################################################</span>
-<span class="c">########################### GAMMA RELATED FUNCTIONS ###########################</span>
-<span class="c">###############################################################################</span></div>
-<div class="viewcode-block" id="polygamma"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.gamma_functions.polygamma">[docs]</a><span class="k">class</span> <span class="nc">polygamma</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The function `polygamma(n, z)` returns `log(gamma(z)).diff(n + 1)`</span>
-
-<span class="sd">       See Also</span>
-<span class="sd">       ========</span>
-
-<span class="sd">       gamma, digamma, trigamma</span>
-
-<span class="sd">    Reference:</span>
-<span class="sd">        http://en.wikipedia.org/wiki/Polygamma_function</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">2</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[:</span><span class="mi">2</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">polygamma</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_positive</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">is_positive</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_odd</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_negative</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">is_positive</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_even</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span>
-
-    <span class="k">def</span> <span class="nf">_eval_aseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">args0</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">args0</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">!=</span> <span class="n">oo</span> <span class="ow">or</span> <span class="ow">not</span> \
-                <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">polygamma</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_aseries</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">args0</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">logx</span><span class="p">)</span>
-        <span class="n">z</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">N</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">N</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="c"># digamma function series</span>
-            <span class="c"># Abramowitz &amp; Stegun, p. 259, 6.3.18</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">log</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
-            <span class="n">o</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">o</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">m</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">ceiling</span><span class="p">((</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span><span class="p">)</span>
-                <span class="n">l</span> <span class="o">=</span> <span class="p">[</span><span class="n">bernoulli</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">))</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span><span class="p">)]</span>
-                <span class="n">r</span> <span class="o">-=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">l</span><span class="p">)</span>
-                <span class="n">o</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">m</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">r</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">)</span> <span class="o">+</span> <span class="n">o</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># proper polygamma function</span>
-            <span class="c"># Abramowitz &amp; Stegun, p. 260, 6.4.10</span>
-            <span class="c"># We return terms to order higher than O(x**n) on purpose</span>
-            <span class="c"># -- otherwise we would not be able to return any terms for</span>
-            <span class="c">#    quite a long time!</span>
-            <span class="n">fac</span> <span class="o">=</span> <span class="n">gamma</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-            <span class="n">e0</span> <span class="o">=</span> <span class="n">fac</span> <span class="o">+</span> <span class="n">N</span><span class="o">*</span><span class="n">fac</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">ceiling</span><span class="p">((</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-                <span class="n">fac</span> <span class="o">=</span> <span class="n">fac</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span> <span class="o">+</span> <span class="n">N</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span> <span class="o">+</span> <span class="n">N</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-                <span class="n">e0</span> <span class="o">+=</span> <span class="n">bernoulli</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="n">fac</span><span class="o">/</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">)</span>
-            <span class="n">o</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">m</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">o</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">o</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">e0</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">)</span> <span class="o">+</span> <span class="n">o</span>
-            <span class="k">return</span> <span class="o">-</span><span class="mi">1</span> <span class="o">*</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="n">N</span> <span class="o">*</span> <span class="n">r</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">z</span><span class="p">)))</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">unpolarify</span>
-
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">:</span>
-                <span class="n">nz</span> <span class="o">=</span> <span class="n">unpolarify</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">z</span> <span class="o">!=</span> <span class="n">nz</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">polygamma</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">nz</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">loggamma</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">z</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">z</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-                    <span class="k">elif</span> <span class="n">z</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                            <span class="k">if</span> <span class="n">n</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-                            <span class="k">else</span><span class="p">:</span>
-                                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-                    <span class="k">elif</span> <span class="n">z</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">z</span><span class="o">.</span><span class="n">is_nonpositive</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="k">if</span> <span class="n">n</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                                <span class="k">return</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">EulerGamma</span> <span class="o">+</span> <span class="n">C</span><span class="o">.</span><span class="n">harmonic</span><span class="p">(</span><span class="n">z</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-                            <span class="k">elif</span> <span class="n">n</span><span class="o">.</span><span class="n">is_odd</span><span class="p">:</span>
-                                <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">zeta</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">z</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="c"># TODO actually *any* n/m can be done, but that is messy</span>
-            <span class="n">lookup</span> <span class="o">=</span> <span class="p">{</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span> <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">EulerGamma</span><span class="p">,</span>
-                      <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span><span class="p">:</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">2</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">EulerGamma</span><span class="p">,</span>
-                      <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">:</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">EulerGamma</span><span class="p">,</span>
-                      <span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">:</span> <span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">EulerGamma</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span>
-                      <span class="n">S</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span><span class="p">:</span> <span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">2</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">EulerGamma</span><span class="p">}</span>
-            <span class="k">if</span> <span class="n">z</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">n</span> <span class="o">=</span> <span class="n">floor</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-                <span class="n">z0</span> <span class="o">=</span> <span class="n">z</span> <span class="o">-</span> <span class="n">n</span>
-                <span class="k">if</span> <span class="n">z0</span> <span class="ow">in</span> <span class="n">lookup</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">lookup</span><span class="p">[</span><span class="n">z0</span><span class="p">]</span> <span class="o">+</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">z0</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)])</span>
-            <span class="k">elif</span> <span class="n">z</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">n</span> <span class="o">=</span> <span class="n">floor</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">z</span><span class="p">)</span>
-                <span class="n">z0</span> <span class="o">=</span> <span class="n">z</span> <span class="o">+</span> <span class="n">n</span>
-                <span class="k">if</span> <span class="n">z0</span> <span class="ow">in</span> <span class="n">lookup</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">lookup</span><span class="p">[</span><span class="n">z0</span><span class="p">]</span> <span class="o">-</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">z0</span> <span class="o">-</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)])</span>
-
-        <span class="c"># TODO n == 1 also can do some rational z</span>
-
-    <span class="k">def</span> <span class="nf">_eval_expand_func</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">n</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">z</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="n">coeff</span> <span class="o">=</span> <span class="n">z</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                    <span class="n">e</span> <span class="o">=</span> <span class="o">-</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">coeff</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">tail</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">C</span><span class="o">.</span><span class="n">Pow</span><span class="p">(</span>
-                            <span class="n">z</span> <span class="o">-</span> <span class="n">i</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)])</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">tail</span> <span class="o">=</span> <span class="o">-</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">C</span><span class="o">.</span><span class="n">Pow</span><span class="p">(</span>
-                            <span class="n">z</span> <span class="o">+</span> <span class="n">i</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="o">-</span><span class="n">coeff</span><span class="p">))])</span>
-                    <span class="k">return</span> <span class="n">polygamma</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">z</span> <span class="o">-</span> <span class="n">coeff</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">tail</span>
-
-            <span class="k">elif</span> <span class="n">z</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                <span class="n">coeff</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="n">z</span><span class="o">.</span><span class="n">as_two_terms</span><span class="p">()</span>
-                <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                    <span class="n">tail</span> <span class="o">=</span> <span class="p">[</span> <span class="n">polygamma</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">z</span> <span class="o">+</span> <span class="n">C</span><span class="o">.</span><span class="n">Rational</span><span class="p">(</span>
-                        <span class="n">i</span><span class="p">,</span> <span class="n">coeff</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">coeff</span><span class="p">))</span> <span class="p">]</span>
-                    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">tail</span><span class="p">)</span><span class="o">/</span><span class="n">coeff</span> <span class="o">+</span> <span class="n">log</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">tail</span><span class="p">)</span><span class="o">/</span><span class="n">coeff</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-                <span class="n">z</span> <span class="o">*=</span> <span class="n">coeff</span>
-
-        <span class="k">return</span> <span class="n">polygamma</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_zeta</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">zeta</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">z</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="loggamma"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.gamma_functions.loggamma">[docs]</a><span class="k">class</span> <span class="nc">loggamma</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The loggamma function is `ln(gamma(x))`.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    http://mathworld.wolfram.com/LogGammaFunction.html</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">_eval_aseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">args0</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">args0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="n">oo</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">loggamma</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_aseries</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">args0</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">logx</span><span class="p">)</span>
-        <span class="n">z</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">ceiling</span><span class="p">((</span><span class="n">n</span> <span class="o">+</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">log</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">z</span> <span class="o">-</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="n">z</span> <span class="o">+</span> <span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-        <span class="n">l</span> <span class="o">=</span> <span class="p">[</span><span class="n">bernoulli</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span><span class="p">)]</span>
-        <span class="n">o</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">o</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">o</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">m</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-        <span class="c"># It is very inefficient to first add the order and then do the nseries</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">r</span> <span class="o">+</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">l</span><span class="p">))</span><span class="o">.</span><span class="n">_eval_nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">)</span> <span class="o">+</span> <span class="n">o</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_intractable</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">log</span><span class="p">(</span><span class="n">gamma</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">polygamma</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="digamma"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.gamma_functions.digamma">[docs]</a><span class="k">def</span> <span class="nf">digamma</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The digamma function is the logarithmic derivative of the gamma function.</span>
-
-
-<span class="sd">    In this case, `digamma(x) = polygamma(0, x)`.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    gamma, trigamma, polygamma</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">polygamma</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="trigamma"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.gamma_functions.trigamma">[docs]</a><span class="k">def</span> <span class="nf">trigamma</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The trigamma function is the second of the polygamma functions.</span>
-
-<span class="sd">    In this case, `trigamma(x) = polygamma(1, x)`.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    gamma, digamma, polygamma</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">polygamma</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="beta"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.gamma_functions.beta">[docs]</a><span class="k">def</span> <span class="nf">beta</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Euler Beta function</span>
-
-<span class="sd">    ``beta(x, y) == gamma(x)*gamma(y) / gamma(x+y)``</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">gamma</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="n">y</span><span class="p">)</span> <span class="o">/</span> <span class="n">gamma</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span></div>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/functions/special/hyper.html b/dev-py3k/_modules/sympy/functions/special/hyper.html
deleted file mode 100644
index c83bb4ab437..00000000000
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+++ /dev/null
@@ -1,1132 +0,0 @@
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-            
-  <h1>Source code for sympy.functions.special.hyper</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Hypergeometric and Meijer G-functions&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">ilcm</span><span class="p">,</span> <span class="n">Mod</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">,</span> <span class="n">ArgumentIndexError</span>
-<span class="kn">from</span> <span class="nn">sympy.core.containers</span> <span class="kn">import</span> <span class="n">Tuple</span>
-<span class="kn">from</span> <span class="nn">sympy.core.mul</span> <span class="kn">import</span> <span class="n">Mul</span>
-
-<span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="p">(</span><span class="n">sqrt</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">asin</span><span class="p">,</span> <span class="n">atan</span><span class="p">,</span>
-        <span class="n">sinh</span><span class="p">,</span> <span class="n">cosh</span><span class="p">,</span> <span class="n">asinh</span><span class="p">,</span> <span class="n">acosh</span><span class="p">,</span> <span class="n">atanh</span><span class="p">,</span> <span class="n">acoth</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-<span class="c"># TODO should __new__ accept **options?</span>
-<span class="c"># TODO should constructors should check if parameters are sensible?</span>
-
-
-<span class="k">def</span> <span class="nf">_prep_tuple</span><span class="p">(</span><span class="n">v</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Turn an iterable argument V into a Tuple and unpolarify, since both</span>
-<span class="sd">    hypergeometric and meijer g-functions are unbranched in their parameters.</span>
-
-<span class="sd">    Examples:</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.functions.special.hyper import _prep_tuple</span>
-<span class="sd">    &gt;&gt;&gt; _prep_tuple([1, 2, 3])</span>
-<span class="sd">    (1, 2, 3)</span>
-<span class="sd">    &gt;&gt;&gt; _prep_tuple((4, 5))</span>
-<span class="sd">    (4, 5)</span>
-<span class="sd">    &gt;&gt;&gt; _prep_tuple((7, 8, 9))</span>
-<span class="sd">    (7, 8, 9)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.simplify.simplify</span> <span class="kn">import</span> <span class="n">unpolarify</span>
-    <span class="k">return</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">unpolarify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">v</span><span class="p">])</span>
-
-
-<span class="k">class</span> <span class="nc">TupleParametersBase</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Base class that takes care of differentiation, when some of</span>
-<span class="sd">        the arguments are actually tuples. &quot;&quot;&quot;</span>
-    <span class="c"># This is not deduced automatically since there are Tuples as arguments.</span>
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">res</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_diffargs</span><span class="p">):</span>
-                    <span class="n">m</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_diffargs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">m</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">res</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">fdiff</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="n">i</span><span class="p">))</span><span class="o">*</span><span class="n">m</span>
-            <span class="k">return</span> <span class="n">res</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">fdiff</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-        <span class="k">except</span> <span class="p">(</span><span class="n">ArgumentIndexError</span><span class="p">,</span> <span class="ne">NotImplementedError</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="hyper"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.hyper.hyper">[docs]</a><span class="k">class</span> <span class="nc">hyper</span><span class="p">(</span><span class="n">TupleParametersBase</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    The (generalized) hypergeometric function is defined by a series where</span>
-<span class="sd">    the ratios of successive terms are a rational function of the summation</span>
-<span class="sd">    index. When convergent, it is continued analytically to the largest</span>
-<span class="sd">    possible domain.</span>
-
-<span class="sd">    The hypergeometric function depends on two vectors of parameters, called</span>
-<span class="sd">    the numerator parameters :math:`a_p`, and the denominator parameters</span>
-<span class="sd">    :math:`b_q`. It also has an argument :math:`z`. The series definition is</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        {}_pF_q\left(\begin{matrix} a_1, \dots, a_p \\ b_1, \dots, b_q \end{matrix}</span>
-<span class="sd">                     \middle| z \right)</span>
-<span class="sd">        = \sum_{n=0}^\infty \frac{(a_1)_n \dots (a_p)_n}{(b_1)_n \dots (b_q)_n}</span>
-<span class="sd">                            \frac{z^n}{n!},</span>
-
-<span class="sd">    where :math:`(a)_n = (a)(a+1)\dots(a+n-1)` denotes the rising factorial.</span>
-
-<span class="sd">    If one of the :math:`b_q` is a non-positive integer then the series is</span>
-<span class="sd">    undefined unless one of the `a_p` is a larger (i.e. smaller in</span>
-<span class="sd">    magnitude) non-positive integer. If none of the :math:`b_q` is a</span>
-<span class="sd">    non-positive integer and one of the :math:`a_p` is a non-positive</span>
-<span class="sd">    integer, then the series reduces to a polynomial. To simplify the</span>
-<span class="sd">    following discussion, we assume that none of the :math:`a_p` or</span>
-<span class="sd">    :math:`b_q` is a non-positive integer. For more details, see the</span>
-<span class="sd">    references.</span>
-
-<span class="sd">    The series converges for all :math:`z` if :math:`p \le q`, and thus</span>
-<span class="sd">    defines an entire single-valued function in this case. If :math:`p =</span>
-<span class="sd">    q+1` the series converges for :math:`|z| &lt; 1`, and can be continued</span>
-<span class="sd">    analytically into a half-plane. If :math:`p &gt; q+1` the series is</span>
-<span class="sd">    divergent for all :math:`z`.</span>
-
-<span class="sd">    Note: The hypergeometric function constructor currently does *not* check</span>
-<span class="sd">    if the parameters actually yield a well-defined function.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    The parameters :math:`a_p` and :math:`b_q` can be passed as arbitrary</span>
-<span class="sd">    iterables, for example:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.functions import hyper</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, n, a</span>
-<span class="sd">    &gt;&gt;&gt; hyper((1, 2, 3), [3, 4], x)</span>
-<span class="sd">    hyper((1, 2, 3), (3, 4), x)</span>
-
-<span class="sd">    There is also pretty printing (it looks better using unicode):</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import pprint</span>
-<span class="sd">    &gt;&gt;&gt; pprint(hyper((1, 2, 3), [3, 4], x), use_unicode=False)</span>
-<span class="sd">      _</span>
-<span class="sd">     |_  /1, 2, 3 |  \</span>
-<span class="sd">     |   |        | x|</span>
-<span class="sd">    3  2 \  3, 4  |  /</span>
-
-<span class="sd">    The parameters must always be iterables, even if they are vectors of</span>
-<span class="sd">    length one or zero:</span>
-
-<span class="sd">    &gt;&gt;&gt; hyper((1, ), [], x)</span>
-<span class="sd">    hyper((1,), (), x)</span>
-
-<span class="sd">    But of course they may be variables (but if they depend on x then you</span>
-<span class="sd">    should not expect much implemented functionality):</span>
-
-<span class="sd">    &gt;&gt;&gt; hyper((n, a), (n**2,), x)</span>
-<span class="sd">    hyper((n, a), (n**2,), x)</span>
-
-<span class="sd">    The hypergeometric function generalizes many named special functions.</span>
-<span class="sd">    The function hyperexpand() tries to express a hypergeometric function</span>
-<span class="sd">    using named special functions.</span>
-<span class="sd">    For example:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import hyperexpand</span>
-<span class="sd">    &gt;&gt;&gt; hyperexpand(hyper([], [], x))</span>
-<span class="sd">    exp(x)</span>
-
-<span class="sd">    You can also use expand_func:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import expand_func</span>
-<span class="sd">    &gt;&gt;&gt; expand_func(x*hyper([1, 1], [2], -x))</span>
-<span class="sd">    log(x + 1)</span>
-
-<span class="sd">    More examples:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import S</span>
-<span class="sd">    &gt;&gt;&gt; hyperexpand(hyper([], [S(1)/2], -x**2/4))</span>
-<span class="sd">    cos(x)</span>
-<span class="sd">    &gt;&gt;&gt; hyperexpand(x*hyper([S(1)/2, S(1)/2], [S(3)/2], x**2))</span>
-<span class="sd">    asin(x)</span>
-
-<span class="sd">    We can also sometimes hyperexpand parametric functions:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import a</span>
-<span class="sd">    &gt;&gt;&gt; hyperexpand(hyper([-a], [], x))</span>
-<span class="sd">    (-x + 1)**a</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.simplify.hyperexpand</span>
-<span class="sd">    sympy.functions.special.gamma_functions.gamma</span>
-<span class="sd">    meijerg</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - Luke, Y. L. (1969), The Special Functions and Their Approximations,</span>
-<span class="sd">      Volume 1</span>
-<span class="sd">    - http://en.wikipedia.org/wiki/Generalized_hypergeometric_function</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">3</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="c"># TODO should we check convergence conditions?</span>
-        <span class="k">return</span> <span class="n">Function</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">_prep_tuple</span><span class="p">(</span><span class="n">ap</span><span class="p">),</span> <span class="n">_prep_tuple</span><span class="p">(</span><span class="n">bq</span><span class="p">),</span> <span class="n">z</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">unpolarify</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">ap</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="nb">len</span><span class="p">(</span><span class="n">bq</span><span class="p">):</span>
-            <span class="n">nz</span> <span class="o">=</span> <span class="n">unpolarify</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">z</span> <span class="o">!=</span> <span class="n">nz</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">hyper</span><span class="p">(</span><span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">nz</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">3</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">!=</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-        <span class="n">nap</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">ap</span><span class="p">])</span>
-        <span class="n">nbq</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">b</span> <span class="o">+</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">bq</span><span class="p">])</span>
-        <span class="n">fac</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">ap</span><span class="p">)</span><span class="o">/</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">bq</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">fac</span><span class="o">*</span><span class="n">hyper</span><span class="p">(</span><span class="n">nap</span><span class="p">,</span> <span class="n">nbq</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">argument</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_expand_func</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">hyperexpand</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ap</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bq</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">argument</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ap</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">bq</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">gamma</span><span class="p">(</span><span class="n">c</span><span class="p">)</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="n">c</span> <span class="o">-</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span><span class="o">/</span><span class="n">gamma</span><span class="p">(</span><span class="n">c</span> <span class="o">-</span> <span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="n">gamma</span><span class="p">(</span><span class="n">c</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">hyperexpand</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="hyper.argument"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.hyper.hyper.argument">[docs]</a>    <span class="k">def</span> <span class="nf">argument</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Argument of the hypergeometric function. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="hyper.ap"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.hyper.hyper.ap">[docs]</a>    <span class="k">def</span> <span class="nf">ap</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Numerator parameters of the hypergeometric function. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="hyper.bq"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.hyper.hyper.bq">[docs]</a>    <span class="k">def</span> <span class="nf">bq</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Denominator parameters of the hypergeometric function. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_diffargs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ap</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">bq</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="hyper.eta"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.hyper.hyper.eta">[docs]</a>    <span class="k">def</span> <span class="nf">eta</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; A quantity related to the convergence of the series. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ap</span><span class="p">)</span> <span class="o">-</span> <span class="nb">sum</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bq</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="hyper.radius_of_convergence"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.hyper.hyper.radius_of_convergence">[docs]</a>    <span class="k">def</span> <span class="nf">radius_of_convergence</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Compute the radius of convergence of the defining series.</span>
-
-<span class="sd">        Note that even if this is not oo, the function may still be evaluated</span>
-<span class="sd">        outside of the radius of convergence by analytic continuation. But if</span>
-<span class="sd">        this is zero, then the function is not actually defined anywhere else.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.functions import hyper</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import z</span>
-<span class="sd">        &gt;&gt;&gt; hyper((1, 2), [3], z).radius_of_convergence</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; hyper((1, 2, 3), [4], z).radius_of_convergence</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; hyper((1, 2), (3, 4), z).radius_of_convergence</span>
-<span class="sd">        oo</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">and</span> <span class="n">a</span> <span class="o">&lt;=</span> <span class="mi">0</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">ap</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">bq</span><span class="p">):</span>
-            <span class="n">aints</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">ap</span> <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">a</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">]</span>
-            <span class="n">bints</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">bq</span> <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">a</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">]</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">aints</span><span class="p">)</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">bints</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-            <span class="n">popped</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">bints</span><span class="p">:</span>
-                <span class="n">cancelled</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">while</span> <span class="n">aints</span><span class="p">:</span>
-                    <span class="n">a</span> <span class="o">=</span> <span class="n">aints</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-                    <span class="k">if</span> <span class="n">a</span> <span class="o">&gt;=</span> <span class="n">b</span><span class="p">:</span>
-                        <span class="n">cancelled</span> <span class="o">=</span> <span class="bp">True</span>
-                        <span class="k">break</span>
-                    <span class="n">popped</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">cancelled</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">aints</span> <span class="ow">or</span> <span class="n">popped</span><span class="p">:</span>
-                <span class="c"># There are still non-positive numerator parameters.</span>
-                <span class="c"># This is a polynomial.</span>
-                <span class="k">return</span> <span class="n">oo</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ap</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bq</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ap</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bq</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">oo</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="hyper.convergence_statement"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.hyper.hyper.convergence_statement">[docs]</a>    <span class="k">def</span> <span class="nf">convergence_statement</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Return a condition on z under which the series converges. &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">And</span><span class="p">,</span> <span class="n">Or</span><span class="p">,</span> <span class="n">re</span><span class="p">,</span> <span class="n">Ne</span><span class="p">,</span> <span class="n">oo</span>
-        <span class="n">R</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">radius_of_convergence</span>
-        <span class="k">if</span> <span class="n">R</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">R</span> <span class="o">==</span> <span class="n">oo</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="c"># The special functions and their approximations, page 44</span>
-        <span class="n">e</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">eta</span>
-        <span class="n">z</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">argument</span>
-        <span class="n">c1</span> <span class="o">=</span> <span class="n">And</span><span class="p">(</span><span class="n">re</span><span class="p">(</span><span class="n">e</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">abs</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">c2</span> <span class="o">=</span> <span class="n">And</span><span class="p">(</span><span class="mi">0</span> <span class="o">&lt;=</span> <span class="n">re</span><span class="p">(</span><span class="n">e</span><span class="p">),</span> <span class="n">re</span><span class="p">(</span><span class="n">e</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">abs</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">,</span> <span class="n">Ne</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-        <span class="n">c3</span> <span class="o">=</span> <span class="n">And</span><span class="p">(</span><span class="n">re</span><span class="p">(</span><span class="n">e</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">abs</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Or</span><span class="p">(</span><span class="n">c1</span><span class="p">,</span> <span class="n">c2</span><span class="p">,</span> <span class="n">c3</span><span class="p">)</span>
-
-</div></div>
-<div class="viewcode-block" id="meijerg"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.hyper.meijerg">[docs]</a><span class="k">class</span> <span class="nc">meijerg</span><span class="p">(</span><span class="n">TupleParametersBase</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    The Meijer G-function is defined by a Mellin-Barnes type integral that</span>
-<span class="sd">    resembles an inverse Mellin transform. It generalizes the hypergeometric</span>
-<span class="sd">    functions.</span>
-
-<span class="sd">    The Meijer G-function depends on four sets of parameters. There are</span>
-<span class="sd">    &quot;*numerator parameters*&quot;</span>
-<span class="sd">    :math:`a_1, \dots, a_n` and :math:`a_{n+1}, \dots, a_p`, and there are</span>
-<span class="sd">    &quot;*denominator parameters*&quot;</span>
-<span class="sd">    :math:`b_1, \dots, b_m` and :math:`b_{m+1}, \dots, b_q`.</span>
-<span class="sd">    Confusingly, it is traditionally denoted as follows (note the position</span>
-<span class="sd">    of `m`, `n`, `p`, `q`, and how they relate to the lengths of the four</span>
-<span class="sd">    parameter vectors):</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        G_{p,q}^{m,n} \left(\begin{matrix}a_1, \dots, a_n &amp; a_{n+1}, \dots, a_p \\</span>
-<span class="sd">                                        b_1, \dots, b_m &amp; b_{m+1}, \dots, b_q</span>
-<span class="sd">                          \end{matrix} \middle| z \right).</span>
-
-<span class="sd">    However, in sympy the four parameter vectors are always available</span>
-<span class="sd">    separately (see examples), so that there is no need to keep track of the</span>
-<span class="sd">    decorating sub- and super-scripts on the G symbol.</span>
-
-<span class="sd">    The G function is defined as the following integral:</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">         \frac{1}{2 \pi i} \int_L \frac{\prod_{j=1}^m \Gamma(b_j - s)</span>
-<span class="sd">         \prod_{j=1}^n \Gamma(1 - a_j + s)}{\prod_{j=m+1}^q \Gamma(1- b_j +s)</span>
-<span class="sd">         \prod_{j=n+1}^p \Gamma(a_j - s)} z^s \mathrm{d}s,</span>
-
-<span class="sd">    where :math:`\Gamma(z)` is the gamma function. There are three possible</span>
-<span class="sd">    contours which we will not describe in detail here (see the references).</span>
-<span class="sd">    If the integral converges along more than one of them the definitions</span>
-<span class="sd">    agree. The contours all separate the poles of :math:`\Gamma(1-a_j+s)`</span>
-<span class="sd">    from the poles of :math:`\Gamma(b_k-s)`, so in particular the G function</span>
-<span class="sd">    is undefined if :math:`a_j - b_k \in \mathbb{Z}_{&gt;0}` for some</span>
-<span class="sd">    :math:`j \le n` and :math:`k \le m`.</span>
-
-<span class="sd">    The conditions under which one of the contours yields a convergent integral</span>
-<span class="sd">    are complicated and we do not state them here, see the references.</span>
-
-<span class="sd">    Note: Currently the Meijer G-function constructor does *not* check any</span>
-<span class="sd">    convergence conditions.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    You can pass the parameters either as four separate vectors:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.functions import meijerg</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, a</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.core.containers import Tuple</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import pprint</span>
-<span class="sd">    &gt;&gt;&gt; pprint(meijerg((1, 2), (a, 4), (5,), [], x), use_unicode=False)</span>
-<span class="sd">     __1, 2 /1, 2  a, 4 |  \</span>
-<span class="sd">    /__     |           | x|</span>
-<span class="sd">    \_|4, 1 \ 5         |  /</span>
-
-<span class="sd">    or as two nested vectors:</span>
-
-<span class="sd">    &gt;&gt;&gt; pprint(meijerg([(1, 2), (3, 4)], ([5], Tuple()), x), use_unicode=False)</span>
-<span class="sd">     __1, 2 /1, 2  3, 4 |  \</span>
-<span class="sd">    /__     |           | x|</span>
-<span class="sd">    \_|4, 1 \ 5         |  /</span>
-
-<span class="sd">    As with the hypergeometric function, the parameters may be passed as</span>
-<span class="sd">    arbitrary iterables. Vectors of length zero and one also have to be</span>
-<span class="sd">    passed as iterables. The parameters need not be constants, but if they</span>
-<span class="sd">    depend on the argument then not much implemented functionality should be</span>
-<span class="sd">    expected.</span>
-
-<span class="sd">    All the subvectors of parameters are available:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import pprint</span>
-<span class="sd">    &gt;&gt;&gt; g = meijerg([1], [2], [3], [4], x)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(g, use_unicode=False)</span>
-<span class="sd">     __1, 1 /1  2 |  \</span>
-<span class="sd">    /__     |     | x|</span>
-<span class="sd">    \_|2, 2 \3  4 |  /</span>
-<span class="sd">    &gt;&gt;&gt; g.an</span>
-<span class="sd">    (1,)</span>
-<span class="sd">    &gt;&gt;&gt; g.ap</span>
-<span class="sd">    (1, 2)</span>
-<span class="sd">    &gt;&gt;&gt; g.aother</span>
-<span class="sd">    (2,)</span>
-<span class="sd">    &gt;&gt;&gt; g.bm</span>
-<span class="sd">    (3,)</span>
-<span class="sd">    &gt;&gt;&gt; g.bq</span>
-<span class="sd">    (3, 4)</span>
-<span class="sd">    &gt;&gt;&gt; g.bother</span>
-<span class="sd">    (4,)</span>
-
-<span class="sd">    The Meijer G-function generalizes the hypergeometric functions.</span>
-<span class="sd">    In some cases it can be expressed in terms of hypergeometric functions,</span>
-<span class="sd">    using Slater&#39;s theorem. For example:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import hyperexpand</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import a, b, c</span>
-<span class="sd">    &gt;&gt;&gt; hyperexpand(meijerg([a], [], [c], [b], x), allow_hyper=True)</span>
-<span class="sd">    x**c*gamma(-a + c + 1)*hyper((-a + c + 1,),</span>
-<span class="sd">                                 (-b + c + 1,), -x)/gamma(-b + c + 1)</span>
-
-<span class="sd">    Thus the Meijer G-function also subsumes many named functions as special</span>
-<span class="sd">    cases. You can use expand_func or hyperexpand to (try to) rewrite a</span>
-<span class="sd">    Meijer G-function in terms of named special functions. For example:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import expand_func, S</span>
-<span class="sd">    &gt;&gt;&gt; expand_func(meijerg([[],[]], [[0],[]], -x))</span>
-<span class="sd">    exp(x)</span>
-<span class="sd">    &gt;&gt;&gt; hyperexpand(meijerg([[],[]], [[S(1)/2],[0]], (x/2)**2))</span>
-<span class="sd">    sin(x)/sqrt(pi)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    hyper</span>
-<span class="sd">    sympy.simplify.hyperexpand</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - Luke, Y. L. (1969), The Special Functions and Their Approximations,</span>
-<span class="sd">      Volume 1</span>
-<span class="sd">    - http://en.wikipedia.org/wiki/Meijer_G-function</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">3</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">5</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="p">[(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">args</span><span class="p">[</span><span class="mi">3</span><span class="p">]),</span> <span class="n">args</span><span class="p">[</span><span class="mi">4</span><span class="p">]]</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;args must eiter be as, as&#39;, bs, bs&#39;, z or &quot;</span>
-                            <span class="s">&quot;as, bs, z&quot;</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">tr</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;wrong argument&quot;</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Tuple</span><span class="p">(</span><span class="n">_prep_tuple</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">_prep_tuple</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-
-        <span class="c"># TODO should we check convergence conditions?</span>
-        <span class="k">return</span> <span class="n">Function</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">tr</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">tr</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">3</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">!=</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_diff_wrt_parameter</span><span class="p">(</span><span class="n">argindex</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">an</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">an</span><span class="p">)</span>
-            <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="n">G</span> <span class="o">=</span> <span class="n">meijerg</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">aother</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">bm</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">bother</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">argument</span><span class="p">)</span>
-            <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">argument</span> <span class="o">*</span> <span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">an</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span> <span class="o">+</span> <span class="n">G</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bm</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bm</span><span class="p">)</span>
-            <span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">G</span> <span class="o">=</span> <span class="n">meijerg</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">an</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">aother</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">bother</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">argument</span><span class="p">)</span>
-            <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">argument</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bm</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="bp">self</span> <span class="o">-</span> <span class="n">G</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="k">def</span> <span class="nf">_diff_wrt_parameter</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">idx</span><span class="p">):</span>
-        <span class="c"># Differentiation wrt a parameter can only be done in very special</span>
-        <span class="c"># cases. In particular, if we want to differentiate with respect to</span>
-        <span class="c"># `a`, all other gamma factors have to reduce to rational functions.</span>
-        <span class="c">#</span>
-        <span class="c"># Let MT denote mellin transform. Suppose T(-s) is the gamma factor</span>
-        <span class="c"># appearing in the definition of G. Then</span>
-        <span class="c">#</span>
-        <span class="c">#   MT(log(z)G(z)) = d/ds T(s) = d/da T(s) + ...</span>
-        <span class="c">#</span>
-        <span class="c"># Thus d/da G(z) = log(z)G(z) - ...</span>
-        <span class="c"># The ... can be evaluated as a G function under the above conditions,</span>
-        <span class="c"># the formula being most easily derived by using</span>
-        <span class="c">#</span>
-        <span class="c"># d  Gamma(s + n)    Gamma(s + n) / 1    1                1     \</span>
-        <span class="c"># -- ------------ =  ------------ | - + ----  + ... + --------- |</span>
-        <span class="c"># ds Gamma(s)        Gamma(s)     \ s   s + 1         s + n - 1 /</span>
-        <span class="c">#</span>
-        <span class="c"># which follows from the difference equation of the digamma function.</span>
-        <span class="c"># (There is a similar equation for -n instead of +n).</span>
-
-        <span class="c"># We first figure out how to pair the parameters.</span>
-        <span class="n">an</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">an</span><span class="p">)</span>
-        <span class="n">ap</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">aother</span><span class="p">)</span>
-        <span class="n">bm</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bm</span><span class="p">)</span>
-        <span class="n">bq</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bother</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">idx</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">an</span><span class="p">):</span>
-            <span class="n">an</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">idx</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">idx</span> <span class="o">-=</span> <span class="nb">len</span><span class="p">(</span><span class="n">an</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">idx</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">ap</span><span class="p">):</span>
-                <span class="n">ap</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">idx</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">idx</span> <span class="o">-=</span> <span class="nb">len</span><span class="p">(</span><span class="n">ap</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">idx</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">bm</span><span class="p">):</span>
-                    <span class="n">bm</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">idx</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">bq</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">idx</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">bm</span><span class="p">))</span>
-        <span class="n">pairs1</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">pairs2</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">l1</span><span class="p">,</span> <span class="n">l2</span><span class="p">,</span> <span class="n">pairs</span> <span class="ow">in</span> <span class="p">[(</span><span class="n">an</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">pairs1</span><span class="p">),</span> <span class="p">(</span><span class="n">ap</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">pairs2</span><span class="p">)]:</span>
-            <span class="k">while</span> <span class="n">l1</span><span class="p">:</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="n">l1</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-                <span class="n">found</span> <span class="o">=</span> <span class="bp">None</span>
-                <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">y</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">l2</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">Mod</span><span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">(),</span> <span class="mi">1</span><span class="p">):</span>
-                        <span class="n">found</span> <span class="o">=</span> <span class="n">i</span>
-                        <span class="k">break</span>
-                <span class="k">if</span> <span class="n">found</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;Derivative not expressible &#39;</span>
-                                              <span class="s">&#39;as G-function?&#39;</span><span class="p">)</span>
-                <span class="n">y</span> <span class="o">=</span> <span class="n">l2</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-                <span class="n">l2</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                <span class="n">pairs</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span>
-
-        <span class="c"># Now build the result.</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="n">log</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">argument</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span>
-
-        <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">pairs1</span><span class="p">:</span>
-            <span class="n">sign</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span>
-            <span class="n">base</span> <span class="o">=</span> <span class="n">b</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">sign</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-                <span class="n">n</span> <span class="o">=</span> <span class="n">b</span> <span class="o">-</span> <span class="n">a</span>
-                <span class="n">base</span> <span class="o">=</span> <span class="n">a</span>
-            <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-                <span class="n">res</span> <span class="o">-=</span> <span class="n">sign</span><span class="o">*</span><span class="n">meijerg</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">an</span> <span class="o">+</span> <span class="p">(</span><span class="n">base</span> <span class="o">+</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,),</span> <span class="bp">self</span><span class="o">.</span><span class="n">aother</span><span class="p">,</span>
-                                    <span class="bp">self</span><span class="o">.</span><span class="n">bm</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">bother</span> <span class="o">+</span> <span class="p">(</span><span class="n">base</span> <span class="o">+</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">0</span><span class="p">,),</span>
-                                    <span class="bp">self</span><span class="o">.</span><span class="n">argument</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">pairs2</span><span class="p">:</span>
-            <span class="n">sign</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">b</span> <span class="o">-</span> <span class="n">a</span>
-            <span class="n">base</span> <span class="o">=</span> <span class="n">a</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">sign</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-                <span class="n">n</span> <span class="o">=</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span>
-                <span class="n">base</span> <span class="o">=</span> <span class="n">b</span>
-            <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-                <span class="n">res</span> <span class="o">-=</span> <span class="n">sign</span><span class="o">*</span><span class="n">meijerg</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">an</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">aother</span> <span class="o">+</span> <span class="p">(</span><span class="n">base</span> <span class="o">+</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,),</span>
-                                    <span class="bp">self</span><span class="o">.</span><span class="n">bm</span> <span class="o">+</span> <span class="p">(</span><span class="n">base</span> <span class="o">+</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">0</span><span class="p">,),</span> <span class="bp">self</span><span class="o">.</span><span class="n">bother</span><span class="p">,</span>
-                                    <span class="bp">self</span><span class="o">.</span><span class="n">argument</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">res</span>
-
-<div class="viewcode-block" id="meijerg.get_period"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.hyper.meijerg.get_period">[docs]</a>    <span class="k">def</span> <span class="nf">get_period</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return a number P such that G(x*exp(I*P)) == G(x).</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.functions.special.hyper import meijerg</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import z</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import pi, S</span>
-
-<span class="sd">        &gt;&gt;&gt; meijerg([1], [], [], [], z).get_period()</span>
-<span class="sd">        2*pi</span>
-<span class="sd">        &gt;&gt;&gt; meijerg([pi], [], [], [], z).get_period()</span>
-<span class="sd">        oo</span>
-<span class="sd">        &gt;&gt;&gt; meijerg([1, 2], [], [], [], z).get_period()</span>
-<span class="sd">        oo</span>
-<span class="sd">        &gt;&gt;&gt; meijerg([1,1], [2], [1, S(1)/2, S(1)/3], [1], z).get_period()</span>
-<span class="sd">        12*pi</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># This follows from slater&#39;s theorem.</span>
-        <span class="k">def</span> <span class="nf">compute</span><span class="p">(</span><span class="n">l</span><span class="p">):</span>
-            <span class="c"># first check that no two differ by an integer</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">l</span><span class="p">):</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">oo</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">l</span><span class="p">)):</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">Mod</span><span class="p">((</span><span class="n">b</span> <span class="o">-</span> <span class="n">l</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="o">.</span><span class="n">simplify</span><span class="p">(),</span> <span class="mi">1</span><span class="p">):</span>
-                        <span class="k">return</span> <span class="n">oo</span>
-            <span class="k">return</span> <span class="nb">reduce</span><span class="p">(</span><span class="n">ilcm</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">q</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">l</span><span class="p">),</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">beta</span> <span class="o">=</span> <span class="n">compute</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bm</span><span class="p">)</span>
-        <span class="n">alpha</span> <span class="o">=</span> <span class="n">compute</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">an</span><span class="p">)</span>
-        <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ap</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bq</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">p</span> <span class="o">==</span> <span class="n">q</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">beta</span> <span class="o">==</span> <span class="n">oo</span> <span class="ow">or</span> <span class="n">alpha</span> <span class="o">==</span> <span class="n">oo</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">oo</span>
-            <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">ilcm</span><span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">p</span> <span class="o">&lt;</span> <span class="n">q</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">beta</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">alpha</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_expand_func</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">hyperexpand</span>
-        <span class="k">return</span> <span class="n">hyperexpand</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="c"># The default code is insufficient for polar arguments.</span>
-        <span class="c"># mpmath provides an optional argument &quot;r&quot;, which evaluates</span>
-        <span class="c"># G(z**(1/r)). I am not sure what its intended use is, but we hijack it</span>
-        <span class="c"># here in the following way: to evaluate at a number z of |argument|</span>
-        <span class="c"># less than (say) n*pi, we put r=1/n, compute z&#39; = root(z, n)</span>
-        <span class="c"># (carefully so as not to loose the branch information), and evaluate</span>
-        <span class="c"># G(z&#39;**(1/r)) = G(z&#39;**n) = G(z).</span>
-        <span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">exp_polar</span><span class="p">,</span> <span class="n">ceiling</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">mpmath</span><span class="p">,</span> <span class="n">Expr</span>
-        <span class="n">z</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">argument</span>
-        <span class="n">znum</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">argument</span><span class="o">.</span><span class="n">_eval_evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">znum</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">exp_polar</span><span class="p">):</span>
-            <span class="n">znum</span><span class="p">,</span> <span class="n">branch</span> <span class="o">=</span> <span class="n">znum</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">(</span><span class="n">exp_polar</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">branch</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span>
-            <span class="n">branch</span> <span class="o">=</span> <span class="n">branch</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="n">I</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">branch</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">ceiling</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">branch</span><span class="o">/</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">))</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="n">znum</span> <span class="o">=</span> <span class="n">znum</span><span class="o">**</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">branch</span> <span class="o">/</span> <span class="n">n</span><span class="p">)</span>
-        <span class="c">#print znum, branch, n</span>
-
-        <span class="c"># Convert all args to mpf or mpc</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="p">[</span><span class="n">z</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">arg</span><span class="o">.</span><span class="n">_to_mpmath</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-                    <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="p">[</span><span class="n">znum</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">n</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]]]</span>
-        <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-            <span class="k">return</span>
-
-        <span class="c"># Set mpmath precision and apply. Make sure precision is restored</span>
-        <span class="c"># afterwards</span>
-        <span class="n">orig</span> <span class="o">=</span> <span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">prec</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="n">prec</span>
-            <span class="n">v</span> <span class="o">=</span> <span class="n">mpmath</span><span class="o">.</span><span class="n">meijerg</span><span class="p">(</span><span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span>
-            <span class="c">#print ap, bq, z, r, v</span>
-        <span class="k">finally</span><span class="p">:</span>
-            <span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="n">orig</span>
-
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">_from_mpmath</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-
-<div class="viewcode-block" id="meijerg.integrand"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.hyper.meijerg.integrand">[docs]</a>    <span class="k">def</span> <span class="nf">integrand</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Get the defining integrand D(s). &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">gamma</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">argument</span><span class="o">**</span><span class="n">s</span> \
-            <span class="o">*</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">gamma</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">bm</span><span class="p">))</span> \
-            <span class="o">*</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">gamma</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">a</span> <span class="o">+</span> <span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">an</span><span class="p">))</span> \
-            <span class="o">/</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">gamma</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">b</span> <span class="o">+</span> <span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">bother</span><span class="p">))</span> \
-            <span class="o">/</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">gamma</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">aother</span><span class="p">))</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="meijerg.argument"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.hyper.meijerg.argument">[docs]</a>    <span class="k">def</span> <span class="nf">argument</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Argument of the Meijer G-function. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="meijerg.an"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.hyper.meijerg.an">[docs]</a>    <span class="k">def</span> <span class="nf">an</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; First set of numerator parameters. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="meijerg.ap"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.hyper.meijerg.ap">[docs]</a>    <span class="k">def</span> <span class="nf">ap</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Combined numerator parameters. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="meijerg.aother"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.hyper.meijerg.aother">[docs]</a>    <span class="k">def</span> <span class="nf">aother</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Second set of numerator parameters. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="meijerg.bm"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.hyper.meijerg.bm">[docs]</a>    <span class="k">def</span> <span class="nf">bm</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; First set of denominator parameters. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="meijerg.bq"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.hyper.meijerg.bq">[docs]</a>    <span class="k">def</span> <span class="nf">bq</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Combined denominator parameters. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="meijerg.bother"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.hyper.meijerg.bother">[docs]</a>    <span class="k">def</span> <span class="nf">bother</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Second set of denominator parameters. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_diffargs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ap</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">bq</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="meijerg.nu"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.hyper.meijerg.nu">[docs]</a>    <span class="k">def</span> <span class="nf">nu</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; A quantity related to the convergence region of the integral,</span>
-<span class="sd">            c.f. references. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bq</span><span class="p">)</span> <span class="o">-</span> <span class="nb">sum</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ap</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="meijerg.delta"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.hyper.meijerg.delta">[docs]</a>    <span class="k">def</span> <span class="nf">delta</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; A quantity related to the convergence region of the integral,</span>
-<span class="sd">            c.f. references. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bm</span><span class="p">)</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">an</span><span class="p">)</span> <span class="o">-</span> <span class="n">S</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ap</span><span class="p">)</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bq</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span>
-
-</div></div>
-<span class="k">class</span> <span class="nc">HyperRep</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A base class for &quot;hyper representation functions&quot;.</span>
-
-<span class="sd">    This is used exclusively in hyperexpand(), but fits more logically here.</span>
-
-<span class="sd">    pFq is branched at 1 if p == q+1. For use with slater-expansion, we want</span>
-<span class="sd">    define an &quot;analytic continuation&quot; to all polar numbers, which is</span>
-<span class="sd">    continuous on circles and on the ray t*exp_polar(I*pi). Moreover, we want</span>
-<span class="sd">    a &quot;nice&quot; expression for the various cases.</span>
-
-<span class="sd">    This base class contains the core logic, concrete derived classes only</span>
-<span class="sd">    supply the actual functions.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">unpolarify</span>
-        <span class="n">nargs</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">unpolarify</span><span class="p">,</span> <span class="n">args</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]))</span> <span class="o">+</span> <span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">:]</span>
-        <span class="k">if</span> <span class="n">args</span> <span class="o">!=</span> <span class="n">nargs</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="o">*</span><span class="n">nargs</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; An expression for F(x) which holds for |x| &lt; 1. &quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; An expression for F(-x) which holds for |x| &lt; 1. &quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_big</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; An expression for F(exp_polar(2*I*pi*n)*x), |x| &gt; 1. &quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_big_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; An expression for F(exp_polar(2*I*pi*n + pi*I)*x), |x| &gt; 1. &quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_nonrep</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Piecewise</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">extract_branch_factor</span><span class="p">(</span><span class="n">allow_half</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="n">minus</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="n">nargs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="p">(</span><span class="n">x</span><span class="p">,)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="n">minus</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="n">n</span> <span class="o">-=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-        <span class="n">nnargs</span> <span class="o">=</span> <span class="n">nargs</span> <span class="o">+</span> <span class="p">(</span><span class="n">n</span><span class="p">,)</span>
-        <span class="k">if</span> <span class="n">minus</span><span class="p">:</span>
-            <span class="n">small</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_expr_small_minus</span><span class="p">(</span><span class="o">*</span><span class="n">nargs</span><span class="p">)</span>
-            <span class="n">big</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_expr_big_minus</span><span class="p">(</span><span class="o">*</span><span class="n">nnargs</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">small</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_expr_small</span><span class="p">(</span><span class="o">*</span><span class="n">nargs</span><span class="p">)</span>
-            <span class="n">big</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_expr_big</span><span class="p">(</span><span class="o">*</span><span class="n">nnargs</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">big</span> <span class="o">==</span> <span class="n">small</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">small</span>
-        <span class="k">return</span> <span class="n">Piecewise</span><span class="p">((</span><span class="n">big</span><span class="p">,</span> <span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="n">small</span><span class="p">,</span> <span class="bp">True</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_nonrepsmall</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">extract_branch_factor</span><span class="p">(</span><span class="n">allow_half</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="p">(</span><span class="n">x</span><span class="p">,)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_expr_small_minus</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_expr_small</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">HyperRep_power1</span><span class="p">(</span><span class="n">HyperRep</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Return a representative for hyper([-a], [], z) == (1 - z)**a. &quot;&quot;&quot;</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="n">a</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="n">a</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_big</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_expr_small</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">a</span><span class="o">*</span><span class="n">exp</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">a</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_big_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_expr_small_minus</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="n">a</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">a</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">HyperRep_power2</span><span class="p">(</span><span class="n">HyperRep</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Return a representative for hyper([a, a - 1/2], [2*a], z). &quot;&quot;&quot;</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="p">))</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="p">))</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_big</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="n">sgn</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_odd</span><span class="p">:</span>
-            <span class="n">sgn</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="n">n</span> <span class="o">-=</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">sgn</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span> \
-            <span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">a</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_big_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="n">sgn</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_odd</span><span class="p">:</span>
-            <span class="n">sgn</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-        <span class="k">return</span> <span class="n">sgn</span><span class="o">*</span><span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">sgn</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">n</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">HyperRep_log1</span><span class="p">(</span><span class="n">HyperRep</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Represent -z*hyper([1, 1], [2], z) == log(1 - z). &quot;&quot;&quot;</span>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">log</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">log</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_big</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">log</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_big_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">log</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span>
-
-
-<span class="k">class</span> <span class="nc">HyperRep_atanh</span><span class="p">(</span><span class="n">HyperRep</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Represent hyper([1/2, 1], [3/2], z) == atanh(sqrt(z))/sqrt(z). &quot;&quot;&quot;</span>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">atanh</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_expr_small_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">atan</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_expr_big</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">acoth</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">acoth</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="o">-</span> <span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_expr_big_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">atan</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">atan</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="o">-</span> <span class="n">pi</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">HyperRep_asin1</span><span class="p">(</span><span class="n">HyperRep</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Represent hyper([1/2, 1/2], [3/2], z) == asin(sqrt(z))/sqrt(z). &quot;&quot;&quot;</span>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">asin</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">asinh</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_big</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="o">*</span><span class="p">((</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">acosh</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_big_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">asinh</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="n">n</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-
-
-<span class="k">class</span> <span class="nc">HyperRep_asin2</span><span class="p">(</span><span class="n">HyperRep</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Represent hyper([1, 1], [3/2], z) == asin(sqrt(z))/sqrt(z)/sqrt(1-z). &quot;&quot;&quot;</span>
-    <span class="c"># TODO this can be nicer</span>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">HyperRep_asin1</span><span class="o">.</span><span class="n">_expr_small</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> \
-            <span class="o">/</span><span class="n">HyperRep_power1</span><span class="o">.</span><span class="n">_expr_small</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">HyperRep_asin1</span><span class="o">.</span><span class="n">_expr_small_minus</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> \
-            <span class="o">/</span><span class="n">HyperRep_power1</span><span class="o">.</span><span class="n">_expr_small_minus</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_big</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">HyperRep_asin1</span><span class="o">.</span><span class="n">_expr_big</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> \
-            <span class="o">/</span><span class="n">HyperRep_power1</span><span class="o">.</span><span class="n">_expr_big</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_big_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">HyperRep_asin1</span><span class="o">.</span><span class="n">_expr_big_minus</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> \
-            <span class="o">/</span><span class="n">HyperRep_power1</span><span class="o">.</span><span class="n">_expr_big_minus</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">HyperRep_sqrts1</span><span class="p">(</span><span class="n">HyperRep</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Return a representative for hyper([-a, 1/2 - a], [1/2], z). &quot;&quot;&quot;</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">((</span><span class="mi">1</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="n">a</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">atan</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)))</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_big</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">((</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">a</span><span class="p">)</span> <span class="o">+</span>
-                    <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">a</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="k">return</span> <span class="p">((</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span> <span class="o">+</span>
-                    <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">n</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_big_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="n">a</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">atan</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="n">a</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">atan</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">a</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">HyperRep_sqrts2</span><span class="p">(</span><span class="n">HyperRep</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Return a representative for</span>
-<span class="sd">          sqrt(z)/2*[(1-sqrt(z))**2a - (1 + sqrt(z))**2a]</span>
-<span class="sd">          == -2*z/(2*a+1) d/dz hyper([-a - 1/2, -a], [1/2], z)&quot;&quot;&quot;</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="p">((</span><span class="mi">1</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="n">a</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">atan</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)))</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_big</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="o">*</span><span class="p">((</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span> <span class="o">-</span>
-                              <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">n</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="o">*</span><span class="p">((</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span> <span class="o">-</span>
-                              <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">n</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_expr_big_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="n">a</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">atan</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="n">a</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> \
-                <span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">atan</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">a</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">HyperRep_log2</span><span class="p">(</span><span class="n">HyperRep</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Represent log(1/2 + sqrt(1 - z)/2) == -z/4*hyper([3/2, 1, 1], [2, 2], z) &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">log</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">log</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_big</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span> <span class="o">+</span> <span class="n">log</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">asin</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span> <span class="o">+</span> <span class="n">log</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="n">I</span><span class="o">*</span><span class="n">asin</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_expr_big_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">n</span> <span class="o">+</span> <span class="n">log</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">n</span> <span class="o">+</span> <span class="n">log</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">HyperRep_cosasin</span><span class="p">(</span><span class="n">HyperRep</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Represent hyper([a, -a], [1/2], z) == cos(2*a*asin(sqrt(z))). &quot;&quot;&quot;</span>
-    <span class="c"># Note there are many alternative expressions, e.g. as powers of a sum of</span>
-    <span class="c"># square roots.</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">asin</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)))</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">cosh</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">asinh</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)))</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_big</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">cosh</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">acosh</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_big_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">cosh</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">asinh</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">n</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">HyperRep_sinasin</span><span class="p">(</span><span class="n">HyperRep</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Represent 2*a*z*hyper([1 - a, 1 + a], [3/2], z)</span>
-<span class="sd">        == sqrt(z)/sqrt(1-z)*sin(2*a*asin(sqrt(z))) &quot;&quot;&quot;</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">asin</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)))</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_small_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">asinh</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)))</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_big</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">acosh</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_expr_big_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">asinh</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">n</span><span class="p">)</span>
-</pre></div>
-
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-        </div>
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diff --git a/dev-py3k/_modules/sympy/functions/special/polynomials.html b/dev-py3k/_modules/sympy/functions/special/polynomials.html
deleted file mode 100644
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--- a/dev-py3k/_modules/sympy/functions/special/polynomials.html
+++ /dev/null
@@ -1,1277 +0,0 @@
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-            
-  <h1>Source code for sympy.functions.special.polynomials</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">This module mainly implements special orthogonal polynomials.</span>
-
-<span class="sd">See also functions.combinatorial.numbers which contains some</span>
-<span class="sd">combinatorial polynomials.</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.basic</span> <span class="kn">import</span> <span class="n">C</span>
-<span class="kn">from</span> <span class="nn">sympy.core.singleton</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Rational</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">ArgumentIndexError</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">sqrt</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.orthopolys</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">jacobi_poly</span><span class="p">,</span>
-    <span class="n">gegenbauer_poly</span><span class="p">,</span>
-    <span class="n">chebyshevt_poly</span><span class="p">,</span>
-    <span class="n">chebyshevu_poly</span><span class="p">,</span>
-    <span class="n">laguerre_poly</span><span class="p">,</span>
-    <span class="n">hermite_poly</span><span class="p">,</span>
-    <span class="n">legendre_poly</span>
-<span class="p">)</span>
-
-<span class="n">_x</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">OrthogonalPolynomial</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Base class for orthogonal polynomials.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_at_order</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">and</span> <span class="n">n</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_ortho_poly</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">_x</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">_x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">conjugate</span><span class="p">())</span>
-
-<span class="c">#----------------------------------------------------------------------------</span>
-<span class="c"># Jacobi polynomials</span>
-<span class="c">#</span>
-
-
-<div class="viewcode-block" id="jacobi"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.polynomials.jacobi">[docs]</a><span class="k">class</span> <span class="nc">jacobi</span><span class="p">(</span><span class="n">OrthogonalPolynomial</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Jacobi polynomial :math:`P_n^{\left(\alpha, \beta\right)}(x)`</span>
-
-<span class="sd">    jacobi(n, alpha, beta, x) gives the nth Jacobi polynomial</span>
-<span class="sd">    in x, :math:`P_n^{\left(\alpha, \beta\right)}(x)`.</span>
-
-<span class="sd">    The Jacobi polynomials are orthogonal on :math:`[-1, 1]` with respect</span>
-<span class="sd">    to the weight :math:`\left(1-x\right)^\alpha \left(1+x\right)^\beta`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import jacobi, S, conjugate, diff</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import n,a,b,x</span>
-
-<span class="sd">    &gt;&gt;&gt; jacobi(0, a, b, x)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; jacobi(1, a, b, x)</span>
-<span class="sd">    a/2 - b/2 + x*(a/2 + b/2 + 1)</span>
-<span class="sd">    &gt;&gt;&gt; jacobi(2, a, b, x)   # doctest:+SKIP</span>
-<span class="sd">    (a**2/8 - a*b/4 - a/8 + b**2/8 - b/8 + x**2*(a**2/8 + a*b/4 + 7*a/8 +</span>
-<span class="sd">    b**2/8 + 7*b/8 + 3/2) + x*(a**2/4 + 3*a/4 - b**2/4 - 3*b/4) - 1/2)</span>
-
-<span class="sd">    &gt;&gt;&gt; jacobi(n, a, b, x)</span>
-<span class="sd">    jacobi(n, a, b, x)</span>
-
-<span class="sd">    &gt;&gt;&gt; jacobi(n, a, a, x)</span>
-<span class="sd">    RisingFactorial(a + 1, n)*gegenbauer(n,</span>
-<span class="sd">        a + 1/2, x)/RisingFactorial(2*a + 1, n)</span>
-
-<span class="sd">    &gt;&gt;&gt; jacobi(n, 0, 0, x)</span>
-<span class="sd">    legendre(n, x)</span>
-
-<span class="sd">    &gt;&gt;&gt; jacobi(n, S(1)/2, S(1)/2, x)</span>
-<span class="sd">    RisingFactorial(3/2, n)*chebyshevu(n, x)/(n + 1)!</span>
-
-<span class="sd">    &gt;&gt;&gt; jacobi(n, -S(1)/2, -S(1)/2, x)</span>
-<span class="sd">    RisingFactorial(1/2, n)*chebyshevt(n, x)/n!</span>
-
-<span class="sd">    &gt;&gt;&gt; jacobi(n, a, b, -x)</span>
-<span class="sd">    (-1)**n*jacobi(n, b, a, x)</span>
-
-<span class="sd">    &gt;&gt;&gt; jacobi(n, a, b, 0)</span>
-<span class="sd">    2**(-n)*gamma(a + n + 1)*hyper((-b - n, -n), (a + 1,), -1)/(n!*gamma(a + 1))</span>
-<span class="sd">    &gt;&gt;&gt; jacobi(n, a, b, 1)</span>
-<span class="sd">    RisingFactorial(a + 1, n)/n!</span>
-
-<span class="sd">    &gt;&gt;&gt; conjugate(jacobi(n, a, b, x))</span>
-<span class="sd">    jacobi(n, conjugate(a), conjugate(b), conjugate(x))</span>
-
-<span class="sd">    &gt;&gt;&gt; diff(jacobi(n,a,b,x), x)</span>
-<span class="sd">    (a/2 + b/2 + n/2 + 1/2)*jacobi(n - 1, a + 1, b + 1, x)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    gegenbauer,</span>
-<span class="sd">    chebyshevt_root, chebyshevu, chebyshevu_root,</span>
-<span class="sd">    legendre, assoc_legendre,</span>
-<span class="sd">    hermite,</span>
-<span class="sd">    laguerre, assoc_laguerre,</span>
-<span class="sd">    sympy.polys.orthopolys.jacobi_poly,</span>
-<span class="sd">    sympy.polys.orthopolys.gegenbauer_poly</span>
-<span class="sd">    sympy.polys.orthopolys.chebyshevt_poly</span>
-<span class="sd">    sympy.polys.orthopolys.chebyshevu_poly</span>
-<span class="sd">    sympy.polys.orthopolys.hermite_poly</span>
-<span class="sd">    sympy.polys.orthopolys.legendre_poly</span>
-<span class="sd">    sympy.polys.orthopolys.laguerre_poly</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Jacobi_polynomials</span>
-<span class="sd">    .. [2] http://mathworld.wolfram.com/JacobiPolynomial.html</span>
-<span class="sd">    .. [3] http://functions.wolfram.com/Polynomials/JacobiP/</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">4</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="c"># Simplify to other polynomials</span>
-        <span class="c"># P^{a, a}_n(x)</span>
-        <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="n">b</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">/</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">*</span> <span class="n">chebyshevt</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">a</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">a</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">/</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">chebyshevu</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">/</span> <span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">*</span> <span class="n">gegenbauer</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">b</span> <span class="o">==</span> <span class="o">-</span><span class="n">a</span><span class="p">:</span>
-            <span class="c"># P^{a, -a}_n(x)</span>
-            <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">a</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">a</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">assoc_legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">a</span> <span class="o">==</span> <span class="o">-</span><span class="n">b</span><span class="p">:</span>
-            <span class="c"># P^{-b, b}_n(x)</span>
-            <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">b</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">b</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">b</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">assoc_legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="c"># Symbolic result P^{a,b}_n(x)</span>
-            <span class="c"># P^{a,b}_n(-x)  ---&gt;  (-1)**n * P^{b,a}_n(-x)</span>
-            <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">():</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">**</span><span class="n">n</span> <span class="o">*</span> <span class="n">jacobi</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="o">-</span><span class="n">x</span><span class="p">)</span>
-            <span class="c"># We can evaluate for some special values of x</span>
-            <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">))</span> <span class="o">*</span>
-                        <span class="n">C</span><span class="o">.</span><span class="n">hyper</span><span class="p">([</span><span class="o">-</span><span class="n">b</span> <span class="o">-</span> <span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="n">n</span><span class="p">],</span> <span class="p">[</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="o">-</span><span class="mi">1</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">/</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                    <span class="c"># TODO: Make sure a+b+2*n \notin Z</span>
-                    <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">*</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># n is a given fixed integer, evaluate into polynomial</span>
-            <span class="k">return</span> <span class="n">jacobi_poly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">4</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># Diff wrt n</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="c"># Diff wrt a</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-            <span class="n">k</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">)</span>
-            <span class="n">f1</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">/</span> <span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="n">n</span> <span class="o">+</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="n">f2</span> <span class="o">=</span> <span class="p">((</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span> <span class="o">/</span>
-                  <span class="p">((</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">)))</span>
-            <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Sum</span><span class="p">(</span><span class="n">f1</span> <span class="o">*</span> <span class="p">(</span><span class="n">jacobi</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f2</span><span class="o">*</span><span class="n">jacobi</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">)),</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="c"># Diff wrt b</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-            <span class="n">k</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">)</span>
-            <span class="n">f1</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">/</span> <span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="n">n</span> <span class="o">+</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="n">f2</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span> <span class="o">*</span> <span class="p">((</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span> <span class="o">/</span>
-                  <span class="p">((</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">)))</span>
-            <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Sum</span><span class="p">(</span><span class="n">f1</span> <span class="o">*</span> <span class="p">(</span><span class="n">jacobi</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f2</span><span class="o">*</span><span class="n">jacobi</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">)),</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">4</span><span class="p">:</span>
-            <span class="c"># Diff wrt x</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="o">*</span> <span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">jacobi</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">b</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_polynomial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="c"># TODO: Make sure n \in N</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">)</span>
-        <span class="n">kern</span> <span class="o">=</span> <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span> <span class="o">/</span>
-                <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">k</span><span class="p">)</span> <span class="o">*</span> <span class="p">((</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="n">k</span><span class="p">)</span>
-        <span class="k">return</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">Sum</span><span class="p">(</span><span class="n">kern</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">conjugate</span><span class="p">(),</span> <span class="n">b</span><span class="o">.</span><span class="n">conjugate</span><span class="p">(),</span> <span class="n">x</span><span class="o">.</span><span class="n">conjugate</span><span class="p">())</span>
-
-<span class="c">#----------------------------------------------------------------------------</span>
-<span class="c"># Gegenbauer polynomials</span>
-<span class="c">#</span>
-
-</div>
-<div class="viewcode-block" id="gegenbauer"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.polynomials.gegenbauer">[docs]</a><span class="k">class</span> <span class="nc">gegenbauer</span><span class="p">(</span><span class="n">OrthogonalPolynomial</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Gegenbauer polynomial :math:`C_n^{\left(\alpha\right)}(x)`</span>
-
-<span class="sd">    gegenbauer(n, alpha, x) gives the nth Gegenbauer polynomial</span>
-<span class="sd">    in x, :math:`C_n^{\left(\alpha\right)}(x)`.</span>
-
-<span class="sd">    The Gegenbauer polynomials are orthogonal on :math:`[-1, 1]` with</span>
-<span class="sd">    respect to the weight :math:`\left(1-x^2\right)^{\alpha-\frac{1}{2}}`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import gegenbauer, conjugate, diff</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import n,a,x</span>
-<span class="sd">    &gt;&gt;&gt; gegenbauer(0, a, x)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; gegenbauer(1, a, x)</span>
-<span class="sd">    2*a*x</span>
-<span class="sd">    &gt;&gt;&gt; gegenbauer(2, a, x)</span>
-<span class="sd">    -a + x**2*(2*a**2 + 2*a)</span>
-<span class="sd">    &gt;&gt;&gt; gegenbauer(3, a, x)</span>
-<span class="sd">    x**3*(4*a**3/3 + 4*a**2 + 8*a/3) + x*(-2*a**2 - 2*a)</span>
-
-<span class="sd">    &gt;&gt;&gt; gegenbauer(n, a, x)</span>
-<span class="sd">    gegenbauer(n, a, x)</span>
-<span class="sd">    &gt;&gt;&gt; gegenbauer(n, a, -x)</span>
-<span class="sd">    (-1)**n*gegenbauer(n, a, x)</span>
-
-<span class="sd">    &gt;&gt;&gt; gegenbauer(n, a, 0)</span>
-<span class="sd">    2**n*sqrt(pi)*gamma(a + n/2)/(gamma(a)*gamma(-n/2 + 1/2)*gamma(n + 1))</span>
-<span class="sd">    &gt;&gt;&gt; gegenbauer(n, a, 1)</span>
-<span class="sd">    gamma(2*a + n)/(gamma(2*a)*gamma(n + 1))</span>
-
-<span class="sd">    &gt;&gt;&gt; conjugate(gegenbauer(n, a, x))</span>
-<span class="sd">    gegenbauer(n, conjugate(a), conjugate(x))</span>
-
-<span class="sd">    &gt;&gt;&gt; diff(gegenbauer(n, a, x), x)</span>
-<span class="sd">    2*a*gegenbauer(n - 1, a + 1, x)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    jacobi,</span>
-<span class="sd">    chebyshevt_root, chebyshevu, chebyshevu_root,</span>
-<span class="sd">    legendre, assoc_legendre,</span>
-<span class="sd">    hermite,</span>
-<span class="sd">    laguerre, assoc_laguerre,</span>
-<span class="sd">    sympy.polys.orthopolys.jacobi_poly</span>
-<span class="sd">    sympy.polys.orthopolys.gegenbauer_poly</span>
-<span class="sd">    sympy.polys.orthopolys.chebyshevt_poly</span>
-<span class="sd">    sympy.polys.orthopolys.chebyshevu_poly</span>
-<span class="sd">    sympy.polys.orthopolys.hermite_poly</span>
-<span class="sd">    sympy.polys.orthopolys.legendre_poly</span>
-<span class="sd">    sympy.polys.orthopolys.laguerre_poly</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Gegenbauer_polynomials</span>
-<span class="sd">    .. [2] http://mathworld.wolfram.com/GegenbauerPolynomial.html</span>
-<span class="sd">    .. [3] http://functions.wolfram.com/Polynomials/GegenbauerC3/</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">3</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="c"># For negative n the polynomials vanish</span>
-        <span class="c"># See http://functions.wolfram.com/Polynomials/GegenbauerC3/03/01/03/0012/</span>
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-        <span class="c"># Some special values for fixed a</span>
-        <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">a</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">chebyshevu</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">a</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="c"># Handle this before the general sign extraction rule</span>
-            <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">C</span><span class="o">.</span><span class="n">re</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="c"># No sec function available yet</span>
-                    <span class="c">#return (C.cos(S.Pi*(a+n)) * C.sec(S.Pi*a) * C.gamma(2*a+n) /</span>
-                    <span class="c">#            (C.gamma(2*a) * C.gamma(n+1)))</span>
-                    <span class="k">return</span> <span class="n">cls</span>
-
-            <span class="c"># Symbolic result C^a_n(x)</span>
-            <span class="c"># C^a_n(-x)  ---&gt;  (-1)**n * C^a_n(x)</span>
-            <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">():</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">**</span><span class="n">n</span> <span class="o">*</span> <span class="n">gegenbauer</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="o">-</span><span class="n">x</span><span class="p">)</span>
-            <span class="c"># We can evaluate for some special values of x</span>
-            <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">n</span> <span class="o">*</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="o">*</span><span class="n">n</span><span class="p">)</span> <span class="o">/</span>
-                        <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">((</span><span class="mi">1</span> <span class="o">-</span> <span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="p">)</span>
-            <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">+</span> <span class="n">n</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-            <span class="k">elif</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">*</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># n is a given fixed integer, evaluate into polynomial</span>
-            <span class="k">return</span> <span class="n">gegenbauer_poly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">3</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># Diff wrt n</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="c"># Diff wrt a</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-            <span class="n">k</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">)</span>
-            <span class="n">factor1</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">))</span> <span class="o">*</span> <span class="p">(</span><span class="n">k</span> <span class="o">+</span> <span class="n">a</span><span class="p">)</span> <span class="o">/</span> <span class="p">((</span><span class="n">k</span> <span class="o">+</span>
-                           <span class="n">n</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">))</span>
-            <span class="n">factor2</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="p">((</span><span class="n">k</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span> <span class="o">+</span> \
-                <span class="mi">2</span> <span class="o">/</span> <span class="p">(</span><span class="n">k</span> <span class="o">+</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span>
-            <span class="n">kern</span> <span class="o">=</span> <span class="n">factor1</span><span class="o">*</span><span class="n">gegenbauer</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">factor2</span><span class="o">*</span><span class="n">gegenbauer</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Sum</span><span class="p">(</span><span class="n">kern</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="c"># Diff wrt x</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">gegenbauer</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_polynomial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">)</span>
-        <span class="n">kern</span> <span class="o">=</span> <span class="p">((</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">k</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">)</span> <span class="o">/</span>
-                <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">k</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Sum</span><span class="p">(</span><span class="n">kern</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">floor</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="mi">2</span><span class="p">)))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">conjugate</span><span class="p">(),</span> <span class="n">x</span><span class="o">.</span><span class="n">conjugate</span><span class="p">())</span>
-
-<span class="c">#----------------------------------------------------------------------------</span>
-<span class="c"># Chebyshev polynomials of first and second kind</span>
-<span class="c">#</span>
-
-</div>
-<div class="viewcode-block" id="chebyshevt"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.polynomials.chebyshevt">[docs]</a><span class="k">class</span> <span class="nc">chebyshevt</span><span class="p">(</span><span class="n">OrthogonalPolynomial</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Chebyshev polynomial of the first kind, :math:`T_n(x)`</span>
-
-<span class="sd">    chebyshevt(n, x) gives the nth Chebyshev polynomial (of the first</span>
-<span class="sd">    kind) in x, :math:`T_n(x)`.</span>
-
-<span class="sd">    The Chebyshev polynomials of the first kind are orthogonal on</span>
-<span class="sd">    :math:`[-1, 1]` with respect to the weight :math:`\frac{1}{\sqrt{1-x^2}}`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import chebyshevt, chebyshevu, diff</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import n,x</span>
-<span class="sd">    &gt;&gt;&gt; chebyshevt(0, x)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; chebyshevt(1, x)</span>
-<span class="sd">    x</span>
-<span class="sd">    &gt;&gt;&gt; chebyshevt(2, x)</span>
-<span class="sd">    2*x**2 - 1</span>
-
-<span class="sd">    &gt;&gt;&gt; chebyshevt(n, x)</span>
-<span class="sd">    chebyshevt(n, x)</span>
-<span class="sd">    &gt;&gt;&gt; chebyshevt(n, -x)</span>
-<span class="sd">    (-1)**n*chebyshevt(n, x)</span>
-<span class="sd">    &gt;&gt;&gt; chebyshevt(-n, x)</span>
-<span class="sd">    chebyshevt(n, x)</span>
-
-<span class="sd">    &gt;&gt;&gt; chebyshevt(n, 0)</span>
-<span class="sd">    cos(pi*n/2)</span>
-<span class="sd">    &gt;&gt;&gt; chebyshevt(n, -1)</span>
-<span class="sd">    (-1)**n</span>
-
-<span class="sd">    &gt;&gt;&gt; diff(chebyshevt(n, x), x)</span>
-<span class="sd">    n*chebyshevu(n - 1, x)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    jacobi, gegenbauer,</span>
-<span class="sd">    chebyshevt_root, chebyshevu, chebyshevu_root,</span>
-<span class="sd">    legendre, assoc_legendre,</span>
-<span class="sd">    hermite,</span>
-<span class="sd">    laguerre, assoc_laguerre,</span>
-<span class="sd">    sympy.polys.orthopolys.jacobi_poly</span>
-<span class="sd">    sympy.polys.orthopolys.gegenbauer_poly</span>
-<span class="sd">    sympy.polys.orthopolys.chebyshevt_poly</span>
-<span class="sd">    sympy.polys.orthopolys.chebyshevu_poly</span>
-<span class="sd">    sympy.polys.orthopolys.hermite_poly</span>
-<span class="sd">    sympy.polys.orthopolys.legendre_poly</span>
-<span class="sd">    sympy.polys.orthopolys.laguerre_poly</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Chebyshev_polynomial</span>
-<span class="sd">    .. [2] http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html</span>
-<span class="sd">    .. [3] http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html</span>
-<span class="sd">    .. [4] http://functions.wolfram.com/Polynomials/ChebyshevT/</span>
-<span class="sd">    .. [5] http://functions.wolfram.com/Polynomials/ChebyshevU/</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_ortho_poly</span> <span class="o">=</span> <span class="nb">staticmethod</span><span class="p">(</span><span class="n">chebyshevt_poly</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="c"># Symbolic result T_n(x)</span>
-            <span class="c"># T_n(-x)  ---&gt;  (-1)**n * T_n(x)</span>
-            <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">():</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">**</span><span class="n">n</span> <span class="o">*</span> <span class="n">chebyshevt</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="n">x</span><span class="p">)</span>
-            <span class="c"># T_{-n}(x)  ---&gt;  T_n(x)</span>
-            <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">():</span>
-                <span class="k">return</span> <span class="n">chebyshevt</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-            <span class="c"># We can evaluate for some special values of x</span>
-            <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="o">*</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">*</span> <span class="n">n</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">elif</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># n is a given fixed integer, evaluate into polynomial</span>
-            <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="c"># T_{-n}(x) == T_n(x)</span>
-                <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_eval_at_order</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_eval_at_order</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">2</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># Diff wrt n</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="c"># Diff wrt x</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">return</span> <span class="n">n</span> <span class="o">*</span> <span class="n">chebyshevu</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_polynomial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">)</span>
-        <span class="n">kern</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">binomial</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">k</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Sum</span><span class="p">(</span><span class="n">kern</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">floor</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="mi">2</span><span class="p">)))</span>
-
-</div>
-<div class="viewcode-block" id="chebyshevu"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.polynomials.chebyshevu">[docs]</a><span class="k">class</span> <span class="nc">chebyshevu</span><span class="p">(</span><span class="n">OrthogonalPolynomial</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Chebyshev polynomial of the second kind, :math:`U_n(x)`</span>
-
-<span class="sd">    chebyshevu(n, x) gives the nth Chebyshev polynomial of the second</span>
-<span class="sd">    kind in x, :math:`U_n(x)`.</span>
-
-<span class="sd">    The Chebyshev polynomials of the second kind are orthogonal on</span>
-<span class="sd">    :math:`[-1, 1]` with respect to the weight :math:`\sqrt{1-x^2}`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import chebyshevt, chebyshevu, diff</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import n,x</span>
-<span class="sd">    &gt;&gt;&gt; chebyshevu(0, x)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; chebyshevu(1, x)</span>
-<span class="sd">    2*x</span>
-<span class="sd">    &gt;&gt;&gt; chebyshevu(2, x)</span>
-<span class="sd">    4*x**2 - 1</span>
-
-<span class="sd">    &gt;&gt;&gt; chebyshevu(n, x)</span>
-<span class="sd">    chebyshevu(n, x)</span>
-<span class="sd">    &gt;&gt;&gt; chebyshevu(n, -x)</span>
-<span class="sd">    (-1)**n*chebyshevu(n, x)</span>
-<span class="sd">    &gt;&gt;&gt; chebyshevu(-n, x)</span>
-<span class="sd">    -chebyshevu(n - 2, x)</span>
-
-<span class="sd">    &gt;&gt;&gt; chebyshevu(n, 0)</span>
-<span class="sd">    cos(pi*n/2)</span>
-<span class="sd">    &gt;&gt;&gt; chebyshevu(n, 1)</span>
-<span class="sd">    n + 1</span>
-
-<span class="sd">    &gt;&gt;&gt; diff(chebyshevu(n, x), x)</span>
-<span class="sd">    (-x*chebyshevu(n, x) + (n + 1)*chebyshevt(n + 1, x))/(x**2 - 1)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    jacobi, gegenbauer,</span>
-<span class="sd">    chebyshevt, chebyshevt_root, chebyshevu_root,</span>
-<span class="sd">    legendre, assoc_legendre,</span>
-<span class="sd">    hermite,</span>
-<span class="sd">    laguerre, assoc_laguerre,</span>
-<span class="sd">    sympy.polys.orthopolys.jacobi_poly</span>
-<span class="sd">    sympy.polys.orthopolys.gegenbauer_poly</span>
-<span class="sd">    sympy.polys.orthopolys.chebyshevt_poly</span>
-<span class="sd">    sympy.polys.orthopolys.chebyshevu_poly</span>
-<span class="sd">    sympy.polys.orthopolys.hermite_poly</span>
-<span class="sd">    sympy.polys.orthopolys.legendre_poly</span>
-<span class="sd">    sympy.polys.orthopolys.laguerre_poly</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Chebyshev_polynomial</span>
-<span class="sd">    .. [2] http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html</span>
-<span class="sd">    .. [3] http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html</span>
-<span class="sd">    .. [4] http://functions.wolfram.com/Polynomials/ChebyshevT/</span>
-<span class="sd">    .. [5] http://functions.wolfram.com/Polynomials/ChebyshevU/</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_ortho_poly</span> <span class="o">=</span> <span class="nb">staticmethod</span><span class="p">(</span><span class="n">chebyshevu_poly</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="c"># Symbolic result U_n(x)</span>
-            <span class="c"># U_n(-x)  ---&gt;  (-1)**n * U_n(x)</span>
-            <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">():</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">**</span><span class="n">n</span> <span class="o">*</span> <span class="n">chebyshevu</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="n">x</span><span class="p">)</span>
-            <span class="c"># U_{-n}(x)  ---&gt;  -U_{n-2}(x)</span>
-            <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">():</span>
-                <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="o">-</span><span class="n">chebyshevu</span><span class="p">(</span><span class="o">-</span><span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-            <span class="c"># We can evaluate for some special values of x</span>
-            <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="o">*</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">*</span> <span class="n">n</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="o">+</span> <span class="n">n</span>
-            <span class="k">elif</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># n is a given fixed integer, evaluate into polynomial</span>
-            <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="c"># U_{-n}(x)  ---&gt;  -U_{n-2}(x)</span>
-                <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="o">-</span><span class="n">cls</span><span class="o">.</span><span class="n">_eval_at_order</span><span class="p">(</span><span class="o">-</span><span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_eval_at_order</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">2</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># Diff wrt n</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="c"># Diff wrt x</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">return</span> <span class="p">((</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">chebyshevt</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">x</span> <span class="o">*</span> <span class="n">chebyshevu</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span> <span class="o">/</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_polynomial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">)</span>
-        <span class="n">kern</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">**</span><span class="n">k</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span>
-            <span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">k</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Sum</span><span class="p">(</span><span class="n">kern</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">floor</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="mi">2</span><span class="p">)))</span>
-
-</div>
-<div class="viewcode-block" id="chebyshevt_root"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.polynomials.chebyshevt_root">[docs]</a><span class="k">class</span> <span class="nc">chebyshevt_root</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    chebyshev_root(n, k) returns the kth root (indexed from zero) of</span>
-<span class="sd">    the nth Chebyshev polynomial of the first kind; that is, if</span>
-<span class="sd">    0 &lt;= k &lt; n, chebyshevt(n, chebyshevt_root(n, k)) == 0.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import chebyshevt, chebyshevt_root</span>
-<span class="sd">    &gt;&gt;&gt; chebyshevt_root(3, 2)</span>
-<span class="sd">    -sqrt(3)/2</span>
-<span class="sd">    &gt;&gt;&gt; chebyshevt(3, chebyshevt_root(3, 2))</span>
-<span class="sd">    0</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    jacobi, gegenbauer,</span>
-<span class="sd">    chebyshevt, chebyshevu, chebyshevu_root,</span>
-<span class="sd">    legendre, assoc_legendre,</span>
-<span class="sd">    hermite,</span>
-<span class="sd">    laguerre, assoc_laguerre,</span>
-<span class="sd">    sympy.polys.orthopolys.jacobi_poly</span>
-<span class="sd">    sympy.polys.orthopolys.gegenbauer_poly</span>
-<span class="sd">    sympy.polys.orthopolys.chebyshevt_poly</span>
-<span class="sd">    sympy.polys.orthopolys.chebyshevu_poly</span>
-<span class="sd">    sympy.polys.orthopolys.hermite_poly</span>
-<span class="sd">    sympy.polys.orthopolys.legendre_poly</span>
-<span class="sd">    sympy.polys.orthopolys.laguerre_poly</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="mi">0</span> <span class="o">&lt;=</span> <span class="n">k</span> <span class="o">&lt;</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;must have 0 &lt;= k &lt; n&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="chebyshevu_root"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.polynomials.chebyshevu_root">[docs]</a><span class="k">class</span> <span class="nc">chebyshevu_root</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    chebyshevu_root(n, k) returns the kth root (indexed from zero) of the</span>
-<span class="sd">    nth Chebyshev polynomial of the second kind; that is, if 0 &lt;= k &lt; n,</span>
-<span class="sd">    chebyshevu(n, chebyshevu_root(n, k)) == 0.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import chebyshevu, chebyshevu_root</span>
-<span class="sd">    &gt;&gt;&gt; chebyshevu_root(3, 2)</span>
-<span class="sd">    -sqrt(2)/2</span>
-<span class="sd">    &gt;&gt;&gt; chebyshevu(3, chebyshevu_root(3, 2))</span>
-<span class="sd">    0</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    chebyshevt, chebyshevt_root, chebyshevu,</span>
-<span class="sd">    legendre, assoc_legendre,</span>
-<span class="sd">    hermite,</span>
-<span class="sd">    laguerre, assoc_laguerre,</span>
-<span class="sd">    sympy.polys.orthopolys.jacobi_poly</span>
-<span class="sd">    sympy.polys.orthopolys.gegenbauer_poly</span>
-<span class="sd">    sympy.polys.orthopolys.chebyshevt_poly</span>
-<span class="sd">    sympy.polys.orthopolys.chebyshevu_poly</span>
-<span class="sd">    sympy.polys.orthopolys.hermite_poly</span>
-<span class="sd">    sympy.polys.orthopolys.legendre_poly</span>
-<span class="sd">    sympy.polys.orthopolys.laguerre_poly</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="mi">0</span> <span class="o">&lt;=</span> <span class="n">k</span> <span class="o">&lt;</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;must have 0 &lt;= k &lt; n&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">*</span><span class="p">(</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-
-<span class="c">#----------------------------------------------------------------------------</span>
-<span class="c"># Legendre polynomials and Associated Legendre polynomials</span>
-<span class="c">#</span>
-
-</div>
-<div class="viewcode-block" id="legendre"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.polynomials.legendre">[docs]</a><span class="k">class</span> <span class="nc">legendre</span><span class="p">(</span><span class="n">OrthogonalPolynomial</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    legendre(n, x) gives the nth Legendre polynomial of x, :math:`P_n(x)`</span>
-
-<span class="sd">    The Legendre polynomials are orthogonal on [-1, 1] with respect to</span>
-<span class="sd">    the constant weight 1. They satisfy :math:`P_n(1) = 1` for all n; further,</span>
-<span class="sd">    :math:`P_n` is odd for odd n and even for even n.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import legendre, diff</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, n</span>
-<span class="sd">    &gt;&gt;&gt; legendre(0, x)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; legendre(1, x)</span>
-<span class="sd">    x</span>
-<span class="sd">    &gt;&gt;&gt; legendre(2, x)</span>
-<span class="sd">    3*x**2/2 - 1/2</span>
-<span class="sd">    &gt;&gt;&gt; legendre(n, x)</span>
-<span class="sd">    legendre(n, x)</span>
-<span class="sd">    &gt;&gt;&gt; diff(legendre(n,x), x)</span>
-<span class="sd">    n*(x*legendre(n, x) - legendre(n - 1, x))/(x**2 - 1)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    jacobi, gegenbauer,</span>
-<span class="sd">    chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,</span>
-<span class="sd">    assoc_legendre,</span>
-<span class="sd">    hermite,</span>
-<span class="sd">    laguerre, assoc_laguerre,</span>
-<span class="sd">    sympy.polys.orthopolys.jacobi_poly</span>
-<span class="sd">    sympy.polys.orthopolys.gegenbauer_poly</span>
-<span class="sd">    sympy.polys.orthopolys.chebyshevt_poly</span>
-<span class="sd">    sympy.polys.orthopolys.chebyshevu_poly</span>
-<span class="sd">    sympy.polys.orthopolys.hermite_poly</span>
-<span class="sd">    sympy.polys.orthopolys.legendre_poly</span>
-<span class="sd">    sympy.polys.orthopolys.laguerre_poly</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Legendre_polynomial</span>
-<span class="sd">    .. [2] http://mathworld.wolfram.com/LegendrePolynomial.html</span>
-<span class="sd">    .. [3] http://functions.wolfram.com/Polynomials/LegendreP/</span>
-<span class="sd">    .. [4] http://functions.wolfram.com/Polynomials/LegendreP2/</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_ortho_poly</span> <span class="o">=</span> <span class="nb">staticmethod</span><span class="p">(</span><span class="n">legendre_poly</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="c"># Symbolic result L_n(x)</span>
-            <span class="c"># L_n(-x)  ---&gt;  (-1)**n * L_n(x)</span>
-            <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">():</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">**</span><span class="n">n</span> <span class="o">*</span> <span class="n">legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="n">x</span><span class="p">)</span>
-            <span class="c"># L_{-n}(x)  ---&gt;  L_{n-1}(x)</span>
-            <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">():</span>
-                <span class="k">return</span> <span class="n">legendre</span><span class="p">(</span><span class="o">-</span><span class="n">n</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-            <span class="c"># We can evaluate for some special values of x</span>
-            <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="o">-</span> <span class="n">n</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="o">+</span> <span class="n">n</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-            <span class="k">elif</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">elif</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># n is a given fixed integer, evaluate into polynomial</span>
-            <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                    <span class="s">&quot;The index n must be nonnegative integer (got </span><span class="si">%r</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_eval_at_order</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">2</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># Diff wrt n</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="c"># Diff wrt x</span>
-            <span class="c"># Find better formula, this is unsuitable for x = 1</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">return</span> <span class="n">n</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">legendre</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_polynomial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">)</span>
-        <span class="n">kern</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">k</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">binomial</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">((</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="p">((</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="n">k</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Sum</span><span class="p">(</span><span class="n">kern</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="assoc_legendre"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.polynomials.assoc_legendre">[docs]</a><span class="k">class</span> <span class="nc">assoc_legendre</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    assoc_legendre(n,m, x) gives :math:`P_n^m(x)`, where n and m are</span>
-<span class="sd">    the degree and order or an expression which is related to the nth</span>
-<span class="sd">    order Legendre polynomial, :math:`P_n(x)` in the following manner:</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        P_n^m(x) = (-1)^m (1 - x^2)^{\frac{m}{2}}</span>
-<span class="sd">                   \frac{\mathrm{d}^m P_n(x)}{\mathrm{d} x^m}</span>
-
-<span class="sd">    Associated Legendre polynomial are orthogonal on [-1, 1] with:</span>
-
-<span class="sd">    - weight = 1            for the same m, and different n.</span>
-<span class="sd">    - weight = 1/(1-x**2)   for the same n, and different m.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import assoc_legendre</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, m, n</span>
-<span class="sd">    &gt;&gt;&gt; assoc_legendre(0,0, x)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; assoc_legendre(1,0, x)</span>
-<span class="sd">    x</span>
-<span class="sd">    &gt;&gt;&gt; assoc_legendre(1,1, x)</span>
-<span class="sd">    -sqrt(-x**2 + 1)</span>
-<span class="sd">    &gt;&gt;&gt; assoc_legendre(n,m,x)</span>
-<span class="sd">    assoc_legendre(n, m, x)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    jacobi, gegenbauer,</span>
-<span class="sd">    chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,</span>
-<span class="sd">    legendre,</span>
-<span class="sd">    hermite,</span>
-<span class="sd">    laguerre, assoc_laguerre,</span>
-<span class="sd">    sympy.polys.orthopolys.jacobi_poly</span>
-<span class="sd">    sympy.polys.orthopolys.gegenbauer_poly</span>
-<span class="sd">    sympy.polys.orthopolys.chebyshevt_poly</span>
-<span class="sd">    sympy.polys.orthopolys.chebyshevu_poly</span>
-<span class="sd">    sympy.polys.orthopolys.hermite_poly</span>
-<span class="sd">    sympy.polys.orthopolys.legendre_poly</span>
-<span class="sd">    sympy.polys.orthopolys.laguerre_poly</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Associated_Legendre_polynomials</span>
-<span class="sd">    .. [2] http://mathworld.wolfram.com/LegendrePolynomial.html</span>
-<span class="sd">    .. [3] http://functions.wolfram.com/Polynomials/LegendreP/</span>
-<span class="sd">    .. [4] http://functions.wolfram.com/Polynomials/LegendreP2/</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">3</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_at_order</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-        <span class="n">P</span> <span class="o">=</span> <span class="n">legendre_poly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">_x</span><span class="p">,</span> <span class="n">polys</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">((</span><span class="n">_x</span><span class="p">,</span> <span class="n">m</span><span class="p">))</span>
-        <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">m</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">_x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="n">Rational</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">P</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">m</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">():</span>
-            <span class="c"># P^{-m}_n  ---&gt;  F * P^m_n</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">m</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">m</span> <span class="o">+</span> <span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">m</span><span class="p">))</span> <span class="o">*</span> <span class="n">assoc_legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="n">m</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="c"># P^0_n  ---&gt;  L_n</span>
-            <span class="k">return</span> <span class="n">legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">2</span><span class="o">**</span><span class="n">m</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">((</span><span class="mi">1</span> <span class="o">-</span> <span class="n">m</span> <span class="o">-</span> <span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="p">(</span><span class="n">m</span> <span class="o">-</span> <span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">and</span> <span class="n">m</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">and</span> <span class="n">n</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">and</span> <span class="n">m</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> : 1st index must be nonnegative integer (got </span><span class="si">%r</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-            <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">n</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> : abs(&#39;2nd index&#39;) must be &lt;= &#39;1st index&#39; (got </span><span class="si">%r</span><span class="s">, </span><span class="si">%r</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_eval_at_order</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="nb">abs</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">m</span><span class="p">)))</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">_x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">3</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># Diff wrt n</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="c"># Diff wrt m</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="c"># Diff wrt x</span>
-            <span class="c"># Find better formula, this is unsuitable for x = 1</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">assoc_legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="n">m</span> <span class="o">+</span> <span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">assoc_legendre</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_polynomial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">)</span>
-        <span class="n">kern</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">n</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span>
-            <span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">k</span> <span class="o">-</span> <span class="n">m</span><span class="p">))</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">k</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">m</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">m</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">Sum</span><span class="p">(</span><span class="n">kern</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">floor</span><span class="p">((</span><span class="n">n</span> <span class="o">-</span> <span class="n">m</span><span class="p">)</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)))</span>
-
-<span class="c">#----------------------------------------------------------------------------</span>
-<span class="c"># Hermite polynomials</span>
-<span class="c">#</span>
-
-</div>
-<div class="viewcode-block" id="hermite"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.polynomials.hermite">[docs]</a><span class="k">class</span> <span class="nc">hermite</span><span class="p">(</span><span class="n">OrthogonalPolynomial</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    hermite(n, x) gives the nth Hermite polynomial in x, :math:`H_n(x)`</span>
-
-<span class="sd">    The Hermite polynomials are orthogonal on :math:`(-\infty, \infty)`</span>
-<span class="sd">    with respect to the weight :math:`\exp\left(-\frac{x^2}{2}\right)`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import hermite, diff</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, n</span>
-<span class="sd">    &gt;&gt;&gt; hermite(0, x)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; hermite(1, x)</span>
-<span class="sd">    2*x</span>
-<span class="sd">    &gt;&gt;&gt; hermite(2, x)</span>
-<span class="sd">    4*x**2 - 2</span>
-<span class="sd">    &gt;&gt;&gt; hermite(n, x)</span>
-<span class="sd">    hermite(n, x)</span>
-<span class="sd">    &gt;&gt;&gt; diff(hermite(n,x), x)</span>
-<span class="sd">    2*n*hermite(n - 1, x)</span>
-<span class="sd">    &gt;&gt;&gt; diff(hermite(n,x), x)</span>
-<span class="sd">    2*n*hermite(n - 1, x)</span>
-<span class="sd">    &gt;&gt;&gt; hermite(n, -x)</span>
-<span class="sd">    (-1)**n*hermite(n, x)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    jacobi, gegenbauer,</span>
-<span class="sd">    chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,</span>
-<span class="sd">    legendre, assoc_legendre,</span>
-<span class="sd">    laguerre, assoc_laguerre,</span>
-<span class="sd">    sympy.polys.orthopolys.jacobi_poly</span>
-<span class="sd">    sympy.polys.orthopolys.gegenbauer_poly</span>
-<span class="sd">    sympy.polys.orthopolys.chebyshevt_poly</span>
-<span class="sd">    sympy.polys.orthopolys.chebyshevu_poly</span>
-<span class="sd">    sympy.polys.orthopolys.hermite_poly</span>
-<span class="sd">    sympy.polys.orthopolys.legendre_poly</span>
-<span class="sd">    sympy.polys.orthopolys.laguerre_poly</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Hermite_polynomial</span>
-<span class="sd">    .. [2] http://mathworld.wolfram.com/HermitePolynomial.html</span>
-<span class="sd">    .. [3] http://functions.wolfram.com/Polynomials/HermiteH/</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_ortho_poly</span> <span class="o">=</span> <span class="nb">staticmethod</span><span class="p">(</span><span class="n">hermite_poly</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="c"># Symbolic result H_n(x)</span>
-            <span class="c"># H_n(-x)  ---&gt;  (-1)**n * H_n(x)</span>
-            <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">():</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">**</span><span class="n">n</span> <span class="o">*</span> <span class="n">hermite</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="n">x</span><span class="p">)</span>
-            <span class="c"># We can evaluate for some special values of x</span>
-            <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="mi">2</span><span class="o">**</span><span class="n">n</span> <span class="o">*</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">)</span> <span class="o">/</span> <span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">((</span><span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="o">-</span> <span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># n is a given fixed integer, evaluate into polynomial</span>
-            <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                    <span class="s">&quot;The index n must be nonnegative integer (got </span><span class="si">%r</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_eval_at_order</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">2</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># Diff wrt n</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="c"># Diff wrt x</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">hermite</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_polynomial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">)</span>
-        <span class="n">kern</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">k</span> <span class="o">/</span> <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">))</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">Sum</span><span class="p">(</span><span class="n">kern</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">floor</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="mi">2</span><span class="p">)))</span>
-
-<span class="c">#----------------------------------------------------------------------------</span>
-<span class="c"># Laguerre polynomials</span>
-<span class="c">#</span>
-
-</div>
-<div class="viewcode-block" id="laguerre"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.polynomials.laguerre">[docs]</a><span class="k">class</span> <span class="nc">laguerre</span><span class="p">(</span><span class="n">OrthogonalPolynomial</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Returns the nth Laguerre polynomial in x, :math:`L_n(x)`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    n : int</span>
-<span class="sd">        Degree of Laguerre polynomial. Must be ``n &gt;= 0``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import laguerre, diff</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, n</span>
-<span class="sd">    &gt;&gt;&gt; laguerre(0, x)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; laguerre(1, x)</span>
-<span class="sd">    -x + 1</span>
-<span class="sd">    &gt;&gt;&gt; laguerre(2, x)</span>
-<span class="sd">    x**2/2 - 2*x + 1</span>
-<span class="sd">    &gt;&gt;&gt; laguerre(3, x)</span>
-<span class="sd">    -x**3/6 + 3*x**2/2 - 3*x + 1</span>
-
-<span class="sd">    &gt;&gt;&gt; laguerre(n, x)</span>
-<span class="sd">    laguerre(n, x)</span>
-
-<span class="sd">    &gt;&gt;&gt; diff(laguerre(n, x), x)</span>
-<span class="sd">    -assoc_laguerre(n - 1, 1, x)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    jacobi, gegenbauer,</span>
-<span class="sd">    chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,</span>
-<span class="sd">    legendre, assoc_legendre,</span>
-<span class="sd">    hermite,</span>
-<span class="sd">    assoc_laguerre,</span>
-<span class="sd">    sympy.polys.orthopolys.jacobi_poly</span>
-<span class="sd">    sympy.polys.orthopolys.gegenbauer_poly</span>
-<span class="sd">    sympy.polys.orthopolys.chebyshevt_poly</span>
-<span class="sd">    sympy.polys.orthopolys.chebyshevu_poly</span>
-<span class="sd">    sympy.polys.orthopolys.hermite_poly</span>
-<span class="sd">    sympy.polys.orthopolys.legendre_poly</span>
-<span class="sd">    sympy.polys.orthopolys.laguerre_poly</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Laguerre_polynomial</span>
-<span class="sd">    .. [2] http://mathworld.wolfram.com/LaguerrePolynomial.html</span>
-<span class="sd">    .. [3] http://functions.wolfram.com/Polynomials/LaguerreL/</span>
-<span class="sd">    .. [4] http://functions.wolfram.com/Polynomials/LaguerreL3/</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="c"># Symbolic result L_n(x)</span>
-            <span class="c"># L_{n}(-x)  ---&gt;  exp(-x) * L_{-n-1}(x)</span>
-            <span class="c"># L_{-n}(x)  ---&gt;  exp(x) * L_{n-1}(-x)</span>
-            <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">():</span>
-                <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="n">laguerre</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="n">x</span><span class="p">)</span>
-            <span class="c"># We can evaluate for some special values of x</span>
-            <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">elif</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-            <span class="k">elif</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">**</span><span class="n">n</span> <span class="o">*</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># n is a given fixed integer, evaluate into polynomial</span>
-            <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                    <span class="s">&quot;The index n must be nonnegative integer (got </span><span class="si">%r</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">laguerre_poly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">2</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># Diff wrt n</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="c"># Diff wrt x</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">assoc_laguerre</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_polynomial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="c"># TODO: Should make sure n is in N_0</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">)</span>
-        <span class="n">kern</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="o">/</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">k</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="n">k</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Sum</span><span class="p">(</span><span class="n">kern</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="assoc_laguerre"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.polynomials.assoc_laguerre">[docs]</a><span class="k">class</span> <span class="nc">assoc_laguerre</span><span class="p">(</span><span class="n">OrthogonalPolynomial</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Returns the nth generalized Laguerre polynomial in x, :math:`L_n(x)`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    n : int</span>
-<span class="sd">        Degree of Laguerre polynomial. Must be ``n &gt;= 0``.</span>
-
-<span class="sd">    alpha : Expr</span>
-<span class="sd">        Arbitrary expression. For ``alpha=0`` regular Laguerre</span>
-<span class="sd">        polynomials will be generated.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import laguerre, assoc_laguerre, diff</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, n, a</span>
-<span class="sd">    &gt;&gt;&gt; assoc_laguerre(0, a, x)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; assoc_laguerre(1, a, x)</span>
-<span class="sd">    a - x + 1</span>
-<span class="sd">    &gt;&gt;&gt; assoc_laguerre(2, a, x)</span>
-<span class="sd">    a**2/2 + 3*a/2 + x**2/2 + x*(-a - 2) + 1</span>
-<span class="sd">    &gt;&gt;&gt; assoc_laguerre(3, a, x)</span>
-<span class="sd">    a**3/6 + a**2 + 11*a/6 - x**3/6 + x**2*(a/2 + 3/2) +</span>
-<span class="sd">        x*(-a**2/2 - 5*a/2 - 3) + 1</span>
-
-<span class="sd">    &gt;&gt;&gt; assoc_laguerre(n, a, 0)</span>
-<span class="sd">    binomial(a + n, a)</span>
-
-<span class="sd">    &gt;&gt;&gt; assoc_laguerre(n, a, x)</span>
-<span class="sd">    assoc_laguerre(n, a, x)</span>
-
-<span class="sd">    &gt;&gt;&gt; assoc_laguerre(n, 0, x)</span>
-<span class="sd">    laguerre(n, x)</span>
-
-<span class="sd">    &gt;&gt;&gt; diff(assoc_laguerre(n, a, x), x)</span>
-<span class="sd">    -assoc_laguerre(n - 1, a + 1, x)</span>
-
-<span class="sd">    &gt;&gt;&gt; diff(assoc_laguerre(n, a, x), a)</span>
-<span class="sd">    Sum(assoc_laguerre(_k, a, x)/(-a + n), (_k, 0, n - 1))</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    jacobi, gegenbauer,</span>
-<span class="sd">    chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,</span>
-<span class="sd">    legendre, assoc_legendre,</span>
-<span class="sd">    hermite,</span>
-<span class="sd">    laguerre,</span>
-<span class="sd">    sympy.polys.orthopolys.jacobi_poly</span>
-<span class="sd">    sympy.polys.orthopolys.gegenbauer_poly</span>
-<span class="sd">    sympy.polys.orthopolys.chebyshevt_poly</span>
-<span class="sd">    sympy.polys.orthopolys.chebyshevu_poly</span>
-<span class="sd">    sympy.polys.orthopolys.hermite_poly</span>
-<span class="sd">    sympy.polys.orthopolys.legendre_poly</span>
-<span class="sd">    sympy.polys.orthopolys.laguerre_poly</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Laguerre_polynomial#Assoc_laguerre_polynomials</span>
-<span class="sd">    .. [2] http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html</span>
-<span class="sd">    .. [3] http://functions.wolfram.com/Polynomials/LaguerreL/</span>
-<span class="sd">    .. [4] http://functions.wolfram.com/Polynomials/LaguerreL3/</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">3</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="c"># L_{n}^{0}(x)  ---&gt;  L_{n}(x)</span>
-        <span class="k">if</span> <span class="n">alpha</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">laguerre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="c"># We can evaluate for some special values of x</span>
-            <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">binomial</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">alpha</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span> <span class="ow">and</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">**</span><span class="n">n</span> <span class="o">*</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-            <span class="k">elif</span> <span class="n">x</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span> <span class="ow">and</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># n is a given fixed integer, evaluate into polynomial</span>
-            <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                    <span class="s">&quot;The index n must be nonnegative integer (got </span><span class="si">%r</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">laguerre_poly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">alpha</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">3</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># Diff wrt n</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="c"># Diff wrt alpha</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-            <span class="n">k</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Sum</span><span class="p">(</span><span class="n">assoc_laguerre</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">alpha</span><span class="p">),</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="c"># Diff wrt x</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">assoc_laguerre</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">alpha</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_polynomial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="c"># TODO: Should make sure n is in N_0</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">)</span>
-        <span class="n">kern</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">RisingFactorial</span><span class="p">(</span>
-            <span class="o">-</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">k</span> <span class="o">+</span> <span class="n">alpha</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">k</span><span class="p">))</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="n">k</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="n">alpha</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">Sum</span><span class="p">(</span><span class="n">kern</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span></div>
-</pre></div>
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-  <h1>Source code for sympy.functions.special.spherical_harmonics</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">C</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">I</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Dummy</span><span class="p">,</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">legendre</span><span class="p">,</span> <span class="n">assoc_legendre</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">sqrt</span>
-
-<span class="n">Pl</span> <span class="o">=</span> <span class="n">legendre</span>
-<span class="n">Plm</span> <span class="o">=</span> <span class="n">assoc_legendre</span>
-
-<span class="n">_x</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="Plmcos"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.spherical_harmonics.Plmcos">[docs]</a><span class="k">def</span> <span class="nf">Plmcos</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">th</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Plm(cos(th)).</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">l</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">l</span><span class="p">)</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-    <span class="n">sin</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">sin</span>
-    <span class="n">cos</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">cos</span>
-    <span class="n">P</span> <span class="o">=</span> <span class="n">Plm</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">_x</span><span class="p">,</span> <span class="n">cos</span><span class="p">(</span><span class="n">th</span><span class="p">))</span>
-    <span class="c"># assume th in (0,pi) =&gt; sin(th) is nonnegative</span>
-    <span class="n">_sinth</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;_sinth&quot;</span><span class="p">,</span> <span class="n">nonnegative</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">P</span> <span class="o">=</span> <span class="n">P</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">cos</span><span class="p">(</span><span class="n">th</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">_sinth</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">_sinth</span><span class="p">,</span> <span class="n">sin</span><span class="p">(</span><span class="n">th</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">P</span>
-
-</div>
-<div class="viewcode-block" id="Ylm"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.spherical_harmonics.Ylm">[docs]</a><span class="k">def</span> <span class="nf">Ylm</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Spherical harmonics Ylm.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, Ylm</span>
-<span class="sd">    &gt;&gt;&gt; theta, phi = symbols(&quot;theta phi&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; Ylm(0, 0, theta, phi)</span>
-<span class="sd">    1/(2*sqrt(pi))</span>
-<span class="sd">    &gt;&gt;&gt; Ylm(1, -1, theta, phi)</span>
-<span class="sd">    sqrt(6)*exp(-I*phi)*sin(theta)/(4*sqrt(pi))</span>
-<span class="sd">    &gt;&gt;&gt; Ylm(1, 0, theta, phi)</span>
-<span class="sd">    sqrt(3)*cos(theta)/(2*sqrt(pi))</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">l</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span> <span class="o">=</span> <span class="p">[</span><span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)]</span>
-    <span class="n">factorial</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span>
-    <span class="k">return</span> <span class="n">sqrt</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">l</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span> <span class="o">*</span> <span class="n">factorial</span><span class="p">(</span><span class="n">l</span> <span class="o">-</span> <span class="n">m</span><span class="p">)</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span><span class="n">l</span> <span class="o">+</span> <span class="n">m</span><span class="p">))</span> <span class="o">*</span> \
-        <span class="n">Plmcos</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">m</span><span class="o">*</span><span class="n">phi</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="Ylm_c"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.spherical_harmonics.Ylm_c">[docs]</a><span class="k">def</span> <span class="nf">Ylm_c</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Conjugate spherical harmonics.&quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">m</span> <span class="o">*</span> <span class="n">Ylm</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="o">-</span><span class="n">m</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="Zlm"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.spherical_harmonics.Zlm">[docs]</a><span class="k">def</span> <span class="nf">Zlm</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">th</span><span class="p">,</span> <span class="n">ph</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Real spherical harmonics.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">simplify</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">zz</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">()</span><span class="o">**</span><span class="n">m</span> <span class="o">*</span> \
-            <span class="p">(</span><span class="n">Ylm</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">th</span><span class="p">,</span> <span class="n">ph</span><span class="p">)</span> <span class="o">+</span> <span class="n">Ylm_c</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">th</span><span class="p">,</span> <span class="n">ph</span><span class="p">))</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">Ylm</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">th</span><span class="p">,</span> <span class="n">ph</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">zz</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">()</span><span class="o">**</span><span class="n">m</span> <span class="o">*</span> \
-            <span class="p">(</span><span class="n">Ylm</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="o">-</span><span class="n">m</span><span class="p">,</span> <span class="n">th</span><span class="p">,</span> <span class="n">ph</span><span class="p">)</span> <span class="o">-</span> <span class="n">Ylm_c</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="o">-</span><span class="n">m</span><span class="p">,</span> <span class="n">th</span><span class="p">,</span> <span class="n">ph</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-
-    <span class="n">zz</span> <span class="o">=</span> <span class="n">zz</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">zz</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">zz</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">zz</span></div>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.functions.special.tensor_functions</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">C</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">Integer</span>
-<span class="kn">from</span> <span class="nn">sympy.core.mul</span> <span class="kn">import</span> <span class="n">prod</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">has_dups</span>
-
-<span class="c">###############################################################################</span>
-<span class="c">###################### Kronecker Delta, Levi-Civita etc. ######################</span>
-<span class="c">###############################################################################</span>
-
-
-<div class="viewcode-block" id="Eijk"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.tensor_functions.Eijk">[docs]</a><span class="k">def</span> <span class="nf">Eijk</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Represent the Levi-Civita symbol.</span>
-
-<span class="sd">    This is just compatibility wrapper to LeviCivita().</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    LeviCivita</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">LeviCivita</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="eval_levicivita"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.tensor_functions.eval_levicivita">[docs]</a><span class="k">def</span> <span class="nf">eval_levicivita</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Evaluate Levi-Civita symbol.&quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">prod</span><span class="p">(</span>
-        <span class="n">prod</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-        <span class="o">/</span> <span class="n">factorial</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-    <span class="c"># converting factorial(i) to int is slightly faster</span>
-
-</div>
-<div class="viewcode-block" id="LeviCivita"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.tensor_functions.LeviCivita">[docs]</a><span class="k">class</span> <span class="nc">LeviCivita</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Represent the Levi-Civita symbol.</span>
-
-<span class="sd">    For even permutations of indices it returns 1, for odd permutations -1, and</span>
-<span class="sd">    for everything else (a repeated index) it returns 0.</span>
-
-<span class="sd">    Thus it represents an alternating pseudotensor.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import LeviCivita</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import i, j, k</span>
-<span class="sd">    &gt;&gt;&gt; LeviCivita(1, 2, 3)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; LeviCivita(1, 3, 2)</span>
-<span class="sd">    -1</span>
-<span class="sd">    &gt;&gt;&gt; LeviCivita(1, 2, 2)</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; LeviCivita(i, j, k)</span>
-<span class="sd">    LeviCivita(i, j, k)</span>
-<span class="sd">    &gt;&gt;&gt; LeviCivita(i, j, i)</span>
-<span class="sd">    0</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Eijk</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_integer</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="n">Integer</span><span class="p">))</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">args</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">eval_levicivita</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">has_dups</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">eval_levicivita</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="KroneckerDelta"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.tensor_functions.KroneckerDelta">[docs]</a><span class="k">class</span> <span class="nc">KroneckerDelta</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The discrete, or Kronecker, delta function.</span>
-
-<span class="sd">    A function that takes in two integers i and j. It returns 0 if i and j are</span>
-<span class="sd">    not equal or it returns 1 if i and j are equal.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    i : Number, Symbol</span>
-<span class="sd">        The first index of the delta function.</span>
-<span class="sd">    j : Number, Symbol</span>
-<span class="sd">        The second index of the delta function.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    A simple example with integer indices::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.functions.special.tensor_functions import KroneckerDelta</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(1, 2)</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(3, 3)</span>
-<span class="sd">        1</span>
-
-<span class="sd">    Symbolic indices::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import i, j, k</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(i, j)</span>
-<span class="sd">        KroneckerDelta(i, j)</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(i, i)</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(i, i + 1)</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(i, i + 1 + k)</span>
-<span class="sd">        KroneckerDelta(i, i + k + 1)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    eval</span>
-<span class="sd">    sympy.functions.special.delta_functions.DiracDelta</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    http://en.wikipedia.org/wiki/Kronecker_delta</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="n">is_integer</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="KroneckerDelta.eval"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.tensor_functions.KroneckerDelta.eval">[docs]</a>    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Evaluates the discrete delta function.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.functions.special.tensor_functions import KroneckerDelta</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import i, j, k</span>
-
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(i, j)</span>
-<span class="sd">        KroneckerDelta(i, j)</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(i, i)</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(i, i + 1)</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(i, i + 1 + k)</span>
-<span class="sd">        KroneckerDelta(i, i + k + 1)</span>
-
-<span class="sd">        # indirect doctest</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">i</span> <span class="o">&gt;</span> <span class="n">j</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-
-        <span class="n">diff</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Abs</span><span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="n">j</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">diff</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">elif</span> <span class="n">diff</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">elif</span> <span class="n">i</span> <span class="o">!=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">diff</span><span class="o">.</span><span class="n">is_nonzero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">diff</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;below_fermi&quot;</span><span class="p">)</span> <span class="ow">and</span> \
-                <span class="n">j</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;above_fermi&quot;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">if</span> <span class="n">j</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;below_fermi&quot;</span><span class="p">)</span> <span class="ow">and</span> \
-                <span class="n">i</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;above_fermi&quot;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="KroneckerDelta.is_above_fermi"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.tensor_functions.KroneckerDelta.is_above_fermi">[docs]</a>    <span class="k">def</span> <span class="nf">is_above_fermi</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        True if Delta can be non-zero above fermi</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.functions.special.tensor_functions import KroneckerDelta</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = Symbol(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; q = Symbol(&#39;q&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(p, a).is_above_fermi</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(p, i).is_above_fermi</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(p, q).is_above_fermi</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        is_below_fermi, is_only_below_fermi, is_only_above_fermi</span>
-
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;below_fermi&quot;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;below_fermi&quot;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="bp">True</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="KroneckerDelta.is_below_fermi"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.tensor_functions.KroneckerDelta.is_below_fermi">[docs]</a>    <span class="k">def</span> <span class="nf">is_below_fermi</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        True if Delta can be non-zero below fermi</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.functions.special.tensor_functions import KroneckerDelta</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = Symbol(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; q = Symbol(&#39;q&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(p, a).is_below_fermi</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(p, i).is_below_fermi</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(p, q).is_below_fermi</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        is_above_fermi, is_only_above_fermi, is_only_below_fermi</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;above_fermi&quot;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;above_fermi&quot;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="bp">True</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="KroneckerDelta.is_only_above_fermi"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.tensor_functions.KroneckerDelta.is_only_above_fermi">[docs]</a>    <span class="k">def</span> <span class="nf">is_only_above_fermi</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        True if Delta is restricted to above fermi</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.functions.special.tensor_functions import KroneckerDelta</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = Symbol(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; q = Symbol(&#39;q&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(p, a).is_only_above_fermi</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(p, q).is_only_above_fermi</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(p, i).is_only_above_fermi</span>
-<span class="sd">        False</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        is_above_fermi, is_below_fermi, is_only_below_fermi</span>
-
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">(</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;above_fermi&quot;</span><span class="p">)</span>
-                <span class="ow">or</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;above_fermi&quot;</span><span class="p">)</span>
-                <span class="p">)</span> <span class="ow">or</span> <span class="bp">False</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="KroneckerDelta.is_only_below_fermi"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.tensor_functions.KroneckerDelta.is_only_below_fermi">[docs]</a>    <span class="k">def</span> <span class="nf">is_only_below_fermi</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        True if Delta is restricted to below fermi</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.functions.special.tensor_functions import KroneckerDelta</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = Symbol(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; q = Symbol(&#39;q&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(p, i).is_only_below_fermi</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(p, q).is_only_below_fermi</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(p, a).is_only_below_fermi</span>
-<span class="sd">        False</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        is_above_fermi, is_below_fermi, is_only_above_fermi</span>
-
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">(</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;below_fermi&quot;</span><span class="p">)</span>
-                <span class="ow">or</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;below_fermi&quot;</span><span class="p">)</span>
-                <span class="p">)</span> <span class="ow">or</span> <span class="bp">False</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="KroneckerDelta.indices_contain_equal_information"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.tensor_functions.KroneckerDelta.indices_contain_equal_information">[docs]</a>    <span class="k">def</span> <span class="nf">indices_contain_equal_information</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns True if indices are either both above or below fermi.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.functions.special.tensor_functions import KroneckerDelta</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = Symbol(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; q = Symbol(&#39;q&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(p, q).indices_contain_equal_information</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(p, q+1).indices_contain_equal_information</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(i, p).indices_contain_equal_information</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;below_fermi&quot;</span><span class="p">)</span> <span class="ow">and</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;below_fermi&quot;</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;above_fermi&quot;</span><span class="p">)</span>
-                <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;above_fermi&quot;</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-        <span class="c"># if both indices are general we are True, else false</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_below_fermi</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_above_fermi</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="KroneckerDelta.preferred_index"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.tensor_functions.KroneckerDelta.preferred_index">[docs]</a>    <span class="k">def</span> <span class="nf">preferred_index</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the index which is preferred to keep in the final expression.</span>
-
-<span class="sd">        The preferred index is the index with more information regarding fermi</span>
-<span class="sd">        level.  If indices contain same information, &#39;a&#39; is preferred before</span>
-<span class="sd">        &#39;b&#39;.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.functions.special.tensor_functions import KroneckerDelta</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = Symbol(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; j = Symbol(&#39;j&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(p, i).preferred_index</span>
-<span class="sd">        i</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(p, a).preferred_index</span>
-<span class="sd">        a</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(i, j).preferred_index</span>
-<span class="sd">        i</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        killable_index</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_preferred_index</span><span class="p">():</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="KroneckerDelta.killable_index"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.tensor_functions.KroneckerDelta.killable_index">[docs]</a>    <span class="k">def</span> <span class="nf">killable_index</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the index which is preferred to substitute in the final</span>
-<span class="sd">        expression.</span>
-
-<span class="sd">        The index to substitute is the index with less information regarding</span>
-<span class="sd">        fermi level.  If indices contain same information, &#39;a&#39; is preferred</span>
-<span class="sd">        before &#39;b&#39;.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.functions.special.tensor_functions import KroneckerDelta</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = Symbol(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; j = Symbol(&#39;j&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(p, i).killable_index</span>
-<span class="sd">        p</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(p, a).killable_index</span>
-<span class="sd">        p</span>
-<span class="sd">        &gt;&gt;&gt; KroneckerDelta(i, j).killable_index</span>
-<span class="sd">        j</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        preferred_index</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_preferred_index</span><span class="p">():</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="k">def</span> <span class="nf">_get_preferred_index</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the index which is preferred to keep in the final expression.</span>
-
-<span class="sd">        The preferred index is the index with more information regarding fermi</span>
-<span class="sd">        level.  If indices contain same information, index 0 is returned.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_above_fermi</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;below_fermi&quot;</span><span class="p">):</span>
-                <span class="k">return</span> <span class="mi">0</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="mi">1</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_below_fermi</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;above_fermi&quot;</span><span class="p">):</span>
-                <span class="k">return</span> <span class="mi">0</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">0</span></div>
-</pre></div>
-
-          </div>
-        </div>
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-        <div class="sphinxsidebarwrapper">
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-  <h3>Quick search</h3>
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-            
-  <h1>Source code for sympy.functions.special.zeta_functions</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot; Riemann zeta and related function. &quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">pi</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">ArgumentIndexError</span>
-
-<span class="c">###############################################################################</span>
-<span class="c">###################### LERCH TRANSCENDENT #####################################</span>
-<span class="c">###############################################################################</span>
-
-
-<div class="viewcode-block" id="lerchphi"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.zeta_functions.lerchphi">[docs]</a><span class="k">class</span> <span class="nc">lerchphi</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Lerch transcendent (Lerch phi function).</span>
-
-<span class="sd">    For :math:`Re(a) &gt; 0`, :math:`|z| &lt; 1` and :math:`s \in \mathbb{C}`, the</span>
-<span class="sd">    Lerch transcendent is defined as</span>
-
-<span class="sd">    .. math :: \Phi(z, s, a) = \sum_{n=0}^\infty \frac{z^n}{(n + a)^s},</span>
-
-<span class="sd">    where the standard branch of the argument is used for :math:`n + a`,</span>
-<span class="sd">    and by analytic continuation for other values of the parameters.</span>
-
-<span class="sd">    A commonly used related function is the Lerch zeta function, defined by</span>
-
-<span class="sd">    .. math:: L(q, s, a) = \Phi(e^{2\pi i q}, s, a).</span>
-
-<span class="sd">    **Analytic Continuation and Branching Behavior**</span>
-
-<span class="sd">    It can be shown that</span>
-
-<span class="sd">    .. math:: \Phi(z, s, a) = z\Phi(z, s, a+1) + a^{-s}.</span>
-
-<span class="sd">    This provides the analytic continuation to :math:`Re(a) \le 0`.</span>
-
-<span class="sd">    Assume now :math:`Re(a) &gt; 0`. The integral representation</span>
-
-<span class="sd">    .. math:: \Phi_0(z, s, a) = \int_0^\infty \frac{t^{s-1} e^{-at}}{1 - ze^{-t}}</span>
-<span class="sd">                                \frac{\mathrm{d}t}{\Gamma(s)}</span>
-
-<span class="sd">    provides an analytic continuation to :math:`\mathbb{C} - [1, \infty)`.</span>
-<span class="sd">    Finally, for :math:`x \in (1, \infty)` we find</span>
-
-<span class="sd">    .. math:: \lim_{\epsilon \to 0^+} \Phi_0(x + i\epsilon, s, a)</span>
-<span class="sd">             -\lim_{\epsilon \to 0^+} \Phi_0(x - i\epsilon, s, a)</span>
-<span class="sd">             = \frac{2\pi i \log^{s-1}{x}}{x^a \Gamma(s)},</span>
-
-<span class="sd">    using the standard branch for both :math:`\log{x}` and</span>
-<span class="sd">    :math:`\log{\log{x}}` (a branch of :math:`\log{\log{x}}` is needed to</span>
-<span class="sd">    evaluate :math:`\log{x}^{s-1}`).</span>
-<span class="sd">    This concludes the analytic continuation. The Lerch transcendent is thus</span>
-<span class="sd">    branched at :math:`z \in \{0, 1, \infty\}` and</span>
-<span class="sd">    :math:`a \in \mathbb{Z}_{\le 0}`. For fixed :math:`z, a` outside these</span>
-<span class="sd">    branch points, it is an entire function of :math:`s`.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    polylog, zeta</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - Bateman, H.; Erdelyi, A. (1953), Higher Transcendental Functions,</span>
-<span class="sd">      Vol. I, New York: McGraw-Hill. Section 1.11.</span>
-<span class="sd">    - http://dlmf.nist.gov/25.14</span>
-<span class="sd">    - http://en.wikipedia.org/wiki/Lerch_transcendent</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    The Lerch transcendent is a fairly general function, for this reason it does</span>
-<span class="sd">    not automatically evaluate to simpler functions. Use expand_func() to</span>
-<span class="sd">    achieve this.</span>
-
-<span class="sd">    If :math:`z=1`, the Lerch transcendent reduces to the Hurwitz zeta function:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import lerchphi, expand_func</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import z, s, a</span>
-<span class="sd">    &gt;&gt;&gt; expand_func(lerchphi(1, s, a))</span>
-<span class="sd">    zeta(s, a)</span>
-
-<span class="sd">    More generally, if :math:`z` is a root of unity, the Lerch transcendent</span>
-<span class="sd">    reduces to a sum of Hurwitz zeta functions:</span>
-
-<span class="sd">    &gt;&gt;&gt; expand_func(lerchphi(-1, s, a))</span>
-<span class="sd">    2**(-s)*zeta(s, a/2) - 2**(-s)*zeta(s, a/2 + 1/2)</span>
-
-<span class="sd">    If :math:`a=1`, the Lerch transcendent reduces to the polylogarithm:</span>
-
-<span class="sd">    &gt;&gt;&gt; expand_func(lerchphi(z, s, 1))</span>
-<span class="sd">    polylog(s, z)/z</span>
-
-<span class="sd">    More generally, if :math:`a` is rational, the Lerch transcendent reduces</span>
-<span class="sd">    to a sum of polylogarithms:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import S</span>
-<span class="sd">    &gt;&gt;&gt; expand_func(lerchphi(z, s, S(1)/2))</span>
-<span class="sd">    2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) -</span>
-<span class="sd">                polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z))</span>
-<span class="sd">    &gt;&gt;&gt; expand_func(lerchphi(z, s, S(3)/2))</span>
-<span class="sd">    -2**s/z + 2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) -</span>
-<span class="sd">                          polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z))/z</span>
-
-<span class="sd">    The derivatives with respect to :math:`z` and :math:`a` can be computed in</span>
-<span class="sd">    closed form:</span>
-
-<span class="sd">    &gt;&gt;&gt; lerchphi(z, s, a).diff(z)</span>
-<span class="sd">    (-a*lerchphi(z, s, a) + lerchphi(z, s - 1, a))/z</span>
-<span class="sd">    &gt;&gt;&gt; lerchphi(z, s, a).diff(a)</span>
-<span class="sd">    -s*lerchphi(z, s + 1, a)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">3</span>
-
-    <span class="k">def</span> <span class="nf">_eval_expand_func</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exp</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">floor</span><span class="p">,</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">,</span> <span class="n">exp_polar</span><span class="p">,</span> <span class="n">unpolarify</span>
-        <span class="n">z</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">if</span> <span class="n">z</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">zeta</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">s</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">s</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">t</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">)</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">((</span><span class="n">t</span> <span class="o">+</span> <span class="n">a</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">s</span><span class="p">),</span> <span class="n">t</span><span class="p">)</span>
-            <span class="n">start</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">t</span><span class="p">)</span>
-            <span class="n">res</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()):</span>
-                <span class="n">res</span> <span class="o">+=</span> <span class="n">c</span><span class="o">*</span><span class="n">start</span>
-                <span class="n">start</span> <span class="o">=</span> <span class="n">t</span><span class="o">*</span><span class="n">start</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">res</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="c"># See section 18 of</span>
-            <span class="c">#   Kelly B. Roach.  Hypergeometric Function Representations.</span>
-            <span class="c">#   In: Proceedings of the 1997 International Symposium on Symbolic and</span>
-            <span class="c">#   Algebraic Computation, pages 205-211, New York, 1997. ACM.</span>
-            <span class="c"># TODO should something be polarified here?</span>
-            <span class="n">add</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-            <span class="n">mul</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-            <span class="c"># First reduce a to the interaval (0, 1]</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">n</span> <span class="o">=</span> <span class="n">floor</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="n">a</span><span class="p">:</span>
-                    <span class="n">n</span> <span class="o">-=</span> <span class="mi">1</span>
-                <span class="n">a</span> <span class="o">-=</span> <span class="n">n</span>
-                <span class="n">mul</span> <span class="o">=</span> <span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">)</span>
-                <span class="n">add</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="o">-</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">k</span> <span class="o">-</span> <span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span><span class="o">**</span><span class="n">s</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)])</span>
-            <span class="k">elif</span> <span class="n">a</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">n</span> <span class="o">=</span> <span class="n">floor</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
-                <span class="n">a</span> <span class="o">+=</span> <span class="n">n</span>
-                <span class="n">mul</span> <span class="o">=</span> <span class="n">z</span><span class="o">**</span><span class="n">n</span>
-                <span class="n">add</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">s</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)])</span>
-
-            <span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">S</span><span class="p">([</span><span class="n">a</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">q</span><span class="p">])</span>
-            <span class="n">zet</span> <span class="o">=</span> <span class="n">exp_polar</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">/</span><span class="n">n</span><span class="p">)</span>
-            <span class="n">root</span> <span class="o">=</span> <span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">n</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">add</span> <span class="o">+</span> <span class="n">mul</span><span class="o">*</span><span class="n">n</span><span class="o">**</span><span class="p">(</span><span class="n">s</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">Add</span><span class="p">(</span>
-                <span class="o">*</span><span class="p">[</span><span class="n">polylog</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">zet</span><span class="o">**</span><span class="n">k</span><span class="o">*</span><span class="n">root</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_expand_func</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-                  <span class="o">/</span> <span class="p">(</span><span class="n">unpolarify</span><span class="p">(</span><span class="n">zet</span><span class="p">)</span><span class="o">**</span><span class="n">k</span><span class="o">*</span><span class="n">root</span><span class="p">)</span><span class="o">**</span><span class="n">m</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)])</span>
-
-        <span class="c"># TODO use minpoly instead of ad-hoc methods when issue 2789 is fixed</span>
-        <span class="k">if</span> <span class="n">z</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">exp</span> <span class="ow">and</span> <span class="p">(</span><span class="n">z</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="p">))</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">or</span> <span class="n">z</span> <span class="ow">in</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="o">-</span><span class="n">I</span><span class="p">]:</span>
-            <span class="c"># TODO reference?</span>
-            <span class="k">if</span> <span class="n">z</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">S</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-            <span class="k">elif</span> <span class="n">z</span> <span class="o">==</span> <span class="n">I</span><span class="p">:</span>
-                <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">S</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
-            <span class="k">elif</span> <span class="n">z</span> <span class="o">==</span> <span class="o">-</span><span class="n">I</span><span class="p">:</span>
-                <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">S</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">arg</span> <span class="o">=</span> <span class="n">z</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="p">)</span>
-                <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">S</span><span class="p">([</span><span class="n">arg</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="n">arg</span><span class="o">.</span><span class="n">q</span><span class="p">])</span>
-            <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">k</span><span class="o">*</span><span class="n">p</span><span class="o">/</span><span class="n">q</span><span class="p">)</span><span class="o">/</span><span class="n">q</span><span class="o">**</span><span class="n">s</span><span class="o">*</span><span class="n">zeta</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span> <span class="o">+</span> <span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="n">q</span><span class="p">)</span>
-                         <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">q</span><span class="p">)])</span>
-
-        <span class="k">return</span> <span class="n">lerchphi</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="n">z</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">s</span><span class="o">*</span><span class="n">lerchphi</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">s</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">lerchphi</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">s</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="n">lerchphi</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">))</span><span class="o">/</span><span class="n">z</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_helper</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">target</span><span class="p">):</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_expand_func</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">res</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">target</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">res</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_zeta</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_rewrite_helper</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">zeta</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_polylog</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_rewrite_helper</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">polylog</span><span class="p">)</span>
-
-<span class="c">###############################################################################</span>
-<span class="c">###################### POLYLOGARITHM ##########################################</span>
-<span class="c">###############################################################################</span>
-
-</div>
-<div class="viewcode-block" id="polylog"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.zeta_functions.polylog">[docs]</a><span class="k">class</span> <span class="nc">polylog</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Polylogarithm function.</span>
-
-<span class="sd">    For :math:`|z| &lt; 1` and :math:`s \in \mathbb{C}`, the polylogarithm is</span>
-<span class="sd">    defined by</span>
-
-<span class="sd">    .. math:: \operatorname{Li}_s(z) = \sum_{n=1}^\infty \frac{z^n}{n^s},</span>
-
-<span class="sd">    where the standard branch of the argument is used for :math:`n`. It admits</span>
-<span class="sd">    an analytic continuation which is branched at :math:`z=1` (notably not on the</span>
-<span class="sd">    sheet of initial definition), :math:`z=0` and :math:`z=\infty`.</span>
-
-<span class="sd">    The name polylogarithm comes from the fact that for :math:`s=1`, the</span>
-<span class="sd">    polylogarithm is related to the ordinary logarithm (see examples), and that</span>
-
-<span class="sd">    .. math:: \operatorname{Li}_{s+1}(z) =</span>
-<span class="sd">                    \int_0^z \frac{\operatorname{Li}_s(t)}{t} \mathrm{d}t.</span>
-
-<span class="sd">    The polylogarithm is a special case of the Lerch transcendent:</span>
-
-<span class="sd">    .. math:: \operatorname{Li}_{s}(z) = z \Phi(z, s, 1)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    zeta, lerchphi</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    For :math:`z \in \{0, 1, -1\}`, the polylogarithm is automatically expressed</span>
-<span class="sd">    using other functions:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import polylog</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import s</span>
-<span class="sd">    &gt;&gt;&gt; polylog(s, 0)</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; polylog(s, 1)</span>
-<span class="sd">    zeta(s)</span>
-<span class="sd">    &gt;&gt;&gt; polylog(s, -1)</span>
-<span class="sd">    dirichlet_eta(s)</span>
-
-<span class="sd">    If :math:`s` is a negative integer, :math:`0` or :math:`1`, the</span>
-<span class="sd">    polylogarithm can be expressed using elementary functions. This can be</span>
-<span class="sd">    done using expand_func():</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import expand_func</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import z</span>
-<span class="sd">    &gt;&gt;&gt; expand_func(polylog(1, z))</span>
-<span class="sd">    -log(z*exp_polar(-I*pi) + 1)</span>
-<span class="sd">    &gt;&gt;&gt; expand_func(polylog(0, z))</span>
-<span class="sd">    z/(-z + 1)</span>
-
-<span class="sd">    The derivative with respect to :math:`z` can be computed in closed form:</span>
-
-<span class="sd">    &gt;&gt;&gt; polylog(s, z).diff(z)</span>
-<span class="sd">    polylog(s - 1, z)/z</span>
-
-<span class="sd">    The polylogarithm can be expressed in terms of the lerch transcendent:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import lerchphi</span>
-<span class="sd">    &gt;&gt;&gt; polylog(s, z).rewrite(lerchphi)</span>
-<span class="sd">    z*lerchphi(z, s, 1)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">z</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">zeta</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">z</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">dirichlet_eta</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">z</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">0</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="n">s</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">polylog</span><span class="p">(</span><span class="n">s</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="n">z</span>
-        <span class="k">raise</span> <span class="n">ArgumentIndexError</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_lerchphi</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">z</span><span class="o">*</span><span class="n">lerchphi</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_expand_func</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">log</span><span class="p">,</span> <span class="n">expand_mul</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">,</span> <span class="n">exp_polar</span><span class="p">,</span> <span class="n">I</span>
-        <span class="n">s</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">if</span> <span class="n">s</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">log</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">exp_polar</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">s</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">s</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">u</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;u&#39;</span><span class="p">)</span>
-            <span class="n">start</span> <span class="o">=</span> <span class="n">u</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">u</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="n">s</span><span class="p">):</span>
-                <span class="n">start</span> <span class="o">=</span> <span class="n">u</span><span class="o">*</span><span class="n">start</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">start</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">polylog</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-
-<span class="c">###############################################################################</span>
-<span class="c">###################### HURWITZ GENERALIZED ZETA FUNCTION ######################</span>
-<span class="c">###############################################################################</span>
-
-</div>
-<div class="viewcode-block" id="zeta"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.zeta_functions.zeta">[docs]</a><span class="k">class</span> <span class="nc">zeta</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Hurwitz zeta function (or Riemann zeta function).</span>
-
-<span class="sd">    For :math:`Re(a) &gt; 0` and :math:`Re(s) &gt; 1`, this function is defined as</span>
-
-<span class="sd">    .. math:: \zeta(s, a) = \sum_{n=0}^\infty \frac{1}{(n + a)^s},</span>
-
-<span class="sd">    where the standard choice of argument for :math:`n + a` is used. For fixed</span>
-<span class="sd">    :math:`a` with :math:`Re(a) &gt; 0` the Hurwitz zeta function admits a</span>
-<span class="sd">    meromorphic continuation to all of :math:`\mathbb{C}`, it is an unbranched</span>
-<span class="sd">    function with a simple pole at :math:`s = 1`.</span>
-
-<span class="sd">    Analytic continuation to other :math:`a` is possible under some circumstances,</span>
-<span class="sd">    but this is not typically done.</span>
-
-<span class="sd">    The Hurwitz zeta function is a special case of the Lerch transcendent:</span>
-
-<span class="sd">    .. math:: \zeta(s, a) = \Phi(1, s, a).</span>
-
-<span class="sd">    This formula defines an analytic continuation for all possible values of</span>
-<span class="sd">    :math:`s` and :math:`a` (also :math:`Re(a) &lt; 0`), see the documentation of</span>
-<span class="sd">    :class:`lerchphi` for a description of the branching behavior.</span>
-
-<span class="sd">    If no value is passed for :math:`a`, by this function assumes a default value</span>
-<span class="sd">    of :math:`a = 1`, yielding the Riemann zeta function.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    dirichlet_eta, lerchphi, polylog</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - http://dlmf.nist.gov/25.11</span>
-<span class="sd">    - http://en.wikipedia.org/wiki/Hurwitz_zeta_function</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    For :math:`a = 1` the Hurwitz zeta function reduces to the famous Riemann</span>
-<span class="sd">    zeta function:</span>
-
-<span class="sd">    .. math:: \zeta(s, 1) = \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import zeta</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import s</span>
-<span class="sd">    &gt;&gt;&gt; zeta(s, 1)</span>
-<span class="sd">    zeta(s)</span>
-
-<span class="sd">    The Riemann zeta function can also be expressed using the Dirichlet eta</span>
-<span class="sd">    function:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import dirichlet_eta</span>
-<span class="sd">    &gt;&gt;&gt; zeta(s).rewrite(dirichlet_eta)</span>
-<span class="sd">    dirichlet_eta(s)/(-2**(-s + 1) + 1)</span>
-
-<span class="sd">    The Riemann zeta function at positive even integer and negative odd integer</span>
-<span class="sd">    values is related to the Bernoulli numbers:</span>
-
-<span class="sd">    &gt;&gt;&gt; zeta(2)</span>
-<span class="sd">    pi**2/6</span>
-<span class="sd">    &gt;&gt;&gt; zeta(4)</span>
-<span class="sd">    pi**4/90</span>
-<span class="sd">    &gt;&gt;&gt; zeta(-1)</span>
-<span class="sd">    -1/12</span>
-
-<span class="sd">    The specific formulae are:</span>
-
-<span class="sd">    .. math:: \zeta(2n) = (-1)^{n+1} \frac{B_{2n} (2\pi)^{2n}}{2(2n)!}</span>
-<span class="sd">    .. math:: \zeta(-n) = -\frac{B_{n+1}}{n+1}</span>
-
-<span class="sd">    At negative even integers the Riemann zeta function is zero:</span>
-
-<span class="sd">    &gt;&gt;&gt; zeta(-4)</span>
-<span class="sd">    0</span>
-
-<span class="sd">    No closed-form expressions are known at positive odd integers, but</span>
-<span class="sd">    numerical evaluation is possible:</span>
-
-<span class="sd">    &gt;&gt;&gt; zeta(3).n()</span>
-<span class="sd">    1.20205690315959</span>
-
-<span class="sd">    The derivative of :math:`\zeta(s, a)` with respect to :math:`a` is easily</span>
-<span class="sd">    computed:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import a</span>
-<span class="sd">    &gt;&gt;&gt; zeta(s, a).diff(a)</span>
-<span class="sd">    -s*zeta(s + 1, a)</span>
-
-<span class="sd">    However the derivative with respect to :math:`s` has no useful closed form</span>
-<span class="sd">    expression:</span>
-
-<span class="sd">    &gt;&gt;&gt; zeta(s, a).diff(s)</span>
-<span class="sd">    Derivative(zeta(s, a), s)</span>
-
-<span class="sd">    The Hurwitz zeta function can be expressed in terms of the Lerch transcendent,</span>
-<span class="sd">    :class:`sympy.functions.special.lerchphi`:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import lerchphi</span>
-<span class="sd">    &gt;&gt;&gt; zeta(s, a).rewrite(lerchphi)</span>
-<span class="sd">    lerchphi(1, s, a)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">a_</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">a_</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">z</span><span class="p">,</span> <span class="n">a</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="mi">1</span><span class="p">)))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">z</span><span class="p">,</span> <span class="n">a</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">a_</span><span class="p">)))</span>
-
-        <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">a</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="ow">and</span> <span class="n">a_</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-            <span class="c"># TODO Should a == 0 return S.NaN as well?</span>
-
-        <span class="k">if</span> <span class="n">z</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">z</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-            <span class="k">elif</span> <span class="n">z</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">elif</span> <span class="n">z</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="o">-</span> <span class="n">a</span> <span class="o">-</span> <span class="mi">1</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="o">-</span> <span class="n">a</span>
-            <span class="k">elif</span> <span class="n">z</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span>
-            <span class="k">elif</span> <span class="n">z</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">z</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                        <span class="n">zeta</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">z</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">bernoulli</span><span class="p">(</span><span class="o">-</span><span class="n">z</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="o">-</span><span class="n">z</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-                    <span class="k">elif</span> <span class="n">z</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-                        <span class="n">B</span><span class="p">,</span> <span class="n">F</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">bernoulli</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-                        <span class="n">zeta</span> <span class="o">=</span> <span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="n">z</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="nb">abs</span><span class="p">(</span><span class="n">B</span><span class="p">)</span> <span class="o">*</span> <span class="n">pi</span><span class="o">**</span><span class="n">z</span> <span class="o">/</span> <span class="n">F</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">return</span>
-
-                    <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">zeta</span> <span class="o">+</span> <span class="n">C</span><span class="o">.</span><span class="n">harmonic</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">z</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">zeta</span> <span class="o">-</span> <span class="n">C</span><span class="o">.</span><span class="n">harmonic</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_dirichlet_eta</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">a</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">dirichlet_eta</span><span class="p">(</span><span class="n">s</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">s</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_lerchphi</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">lerchphi</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">s</span><span class="p">,</span> <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">s</span><span class="p">,</span> <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span><span class="p">,)</span>
-        <span class="k">if</span> <span class="n">argindex</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">s</span><span class="o">*</span><span class="n">zeta</span><span class="p">(</span><span class="n">s</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ArgumentIndexError</span>
-
-</div>
-<div class="viewcode-block" id="dirichlet_eta"><a class="viewcode-back" href="../../../../modules/functions/special.html#sympy.functions.special.zeta_functions.dirichlet_eta">[docs]</a><span class="k">class</span> <span class="nc">dirichlet_eta</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Dirichlet eta function.</span>
-
-<span class="sd">    For :math:`Re(s) &gt; 0`, this function is defined as</span>
-
-<span class="sd">    .. math:: \eta(s) = \sum_{n=1}^\infty \frac{(-1)^n}{n^s}.</span>
-
-<span class="sd">    It admits a unique analytic continuation to all of :math:`\mathbb{C}`.</span>
-<span class="sd">    It is an entire, unbranched function.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    zeta</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - http://en.wikipedia.org/wiki/Dirichlet_eta_function</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    The Dirichlet eta function is closely related to the Riemann zeta function:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import dirichlet_eta, zeta</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import s</span>
-<span class="sd">    &gt;&gt;&gt; dirichlet_eta(s).rewrite(zeta)</span>
-<span class="sd">    (-2**(-s + 1) + 1)*zeta(s)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">s</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-        <span class="n">z</span> <span class="o">=</span> <span class="n">zeta</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">z</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">zeta</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">s</span><span class="p">))</span><span class="o">*</span><span class="n">z</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_zeta</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">s</span><span class="p">))</span> <span class="o">*</span> <span class="n">zeta</span><span class="p">(</span><span class="n">s</span><span class="p">)</span></div>
-</pre></div>
-
-          </div>
-        </div>
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diff --git a/dev-py3k/_modules/sympy/galgebra/GA.html b/dev-py3k/_modules/sympy/galgebra/GA.html
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+++ /dev/null
@@ -1,2695 +0,0 @@
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-            
-  <h1>Source code for sympy.galgebra.GA</h1><div class="highlight"><pre>
-<span class="c">#!/usr/bin/python</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">The module symbolicGA implements symbolic Geometric Algebra in python.</span>
-<span class="sd">The relevant references for this module are:</span>
-
-<span class="sd">    1. &quot;Geometric Algebra for Physicists&quot; by C. Doran and A. Lazenby,</span>
-<span class="sd">       Cambridge University Press, 2003.</span>
-
-<span class="sd">    2. &quot;Geometric Algebra for Computer Science&quot; by Leo Dorst,</span>
-<span class="sd">       Daniel Fontijne, and Stephen Mann, Morgan Kaufmann Publishers, 2007.</span>
-
-<span class="sd">    3. SymPy Tutorial, http://docs.sympy.org/</span>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">import</span> <span class="nn">sys</span>
-<span class="kn">import</span> <span class="nn">numpy</span>
-<span class="kn">import</span> <span class="nn">sympy</span>
-<span class="kn">import</span> <span class="nn">re</span> <span class="kn">as</span> <span class="nn">regrep</span>
-<span class="kn">import</span> <span class="nn">sympy.galgebra.latex_ex</span>
-<span class="kn">from</span> <span class="nn">sympy.core.decorators</span> <span class="kn">import</span> <span class="n">deprecated</span>
-
-<span class="n">NUMPAT</span> <span class="o">=</span> <span class="n">regrep</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span> <span class="s">&#39;([\-0-9])|([\-0-9]/[0-9])&#39;</span><span class="p">)</span>
-<span class="sd">&quot;&quot;&quot;Re pattern for rational number&quot;&quot;&quot;</span>
-
-<span class="n">ZERO</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Rational</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="n">ONE</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="n">TWO</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Rational</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="n">HALF</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Pow</span> <span class="k">as</span> <span class="n">pow_type</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Abs</span> <span class="k">as</span> <span class="n">abs_type</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Mul</span> <span class="k">as</span> <span class="n">mul_type</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Add</span> <span class="k">as</span> <span class="n">add_type</span>
-
-
-<span class="nd">@sympy.vectorize</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">substitute_array</span><span class="p">(</span><span class="n">array</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-    <span class="k">return</span><span class="p">(</span><span class="n">array</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">))</span>
-
-
-<div class="viewcode-block" id="is_quasi_unit_numpy_array"><a class="viewcode-back" href="../../../modules/galgebra/GA.html#sympy.galgebra.GA.is_quasi_unit_numpy_array">[docs]</a><span class="k">def</span> <span class="nf">is_quasi_unit_numpy_array</span><span class="p">(</span><span class="n">array</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Determine if a array is square and diagonal with</span>
-<span class="sd">    entries of +1 or -1.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">shape</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">shape</span><span class="p">(</span><span class="n">array</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span> <span class="ow">and</span> <span class="p">(</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]):</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">ix</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">while</span> <span class="n">ix</span> <span class="o">&lt;</span> <span class="n">n</span><span class="p">:</span>
-            <span class="n">iy</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">while</span> <span class="n">iy</span> <span class="o">&lt;</span> <span class="n">ix</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">array</span><span class="p">[</span><span class="n">ix</span><span class="p">][</span><span class="n">iy</span><span class="p">]</span> <span class="o">!=</span> <span class="n">ZERO</span><span class="p">:</span>
-                    <span class="k">return</span><span class="p">(</span><span class="bp">False</span><span class="p">)</span>
-                <span class="n">iy</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Abs</span><span class="p">(</span><span class="n">array</span><span class="p">[</span><span class="n">ix</span><span class="p">][</span><span class="n">ix</span><span class="p">])</span> <span class="o">!=</span> <span class="n">ONE</span><span class="p">:</span>
-                <span class="k">return</span><span class="p">(</span><span class="bp">False</span><span class="p">)</span>
-            <span class="n">ix</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">return</span><span class="p">(</span><span class="bp">True</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span><span class="p">(</span><span class="bp">False</span><span class="p">)</span>
-
-</div>
-<span class="nd">@deprecated</span><span class="p">(</span><span class="n">value</span><span class="o">=</span><span class="s">&quot;This function is no longer needed.&quot;</span><span class="p">,</span> <span class="n">issue</span><span class="o">=</span><span class="mi">3379</span><span class="p">,</span>
-            <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.2&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">set_main</span><span class="p">(</span><span class="n">main_program</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-
-<span class="k">def</span> <span class="nf">plist</span><span class="p">(</span><span class="n">lst</span><span class="p">):</span>
-    <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">lst</span><span class="p">)</span> <span class="o">==</span> <span class="nb">list</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">lst</span><span class="p">:</span>
-            <span class="n">plist</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">sys</span><span class="o">.</span><span class="n">stderr</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="n">lst</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">)</span>
-    <span class="k">return</span>
-
-
-<div class="viewcode-block" id="numeric"><a class="viewcode-back" href="../../../modules/galgebra/GA.html#sympy.galgebra.GA.numeric">[docs]</a><span class="k">def</span> <span class="nf">numeric</span><span class="p">(</span><span class="n">num_str</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns rational numbers compatible with symbols.</span>
-<span class="sd">    Input is a string representing a fraction or integer</span>
-<span class="sd">    or a simple integer.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">num_str</span><span class="p">)</span> <span class="o">==</span> <span class="nb">int</span><span class="p">:</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">num_str</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">tmp</span> <span class="o">=</span> <span class="n">num_str</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;/&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">tmp</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">tmp</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">tmp</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">tmp</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-    <span class="k">return</span><span class="p">(</span><span class="n">sympy</span><span class="o">.</span><span class="n">Rational</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="collect"><a class="viewcode-back" href="../../../modules/galgebra/GA.html#sympy.galgebra.GA.collect">[docs]</a><span class="k">def</span> <span class="nf">collect</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">lst</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Wrapper for sympy.collect.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">lst</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">scalar_to_symbol</span><span class="p">(</span><span class="n">lst</span><span class="p">)</span>
-    <span class="k">return</span><span class="p">(</span><span class="n">sympy</span><span class="o">.</span><span class="n">collect</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">lst</span><span class="p">))</span>
-
-</div>
-<span class="k">def</span> <span class="nf">sqrt</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="k">return</span><span class="p">(</span><span class="n">sympy</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-
-
-<div class="viewcode-block" id="isint"><a class="viewcode-back" href="../../../modules/galgebra/GA.html#sympy.galgebra.GA.isint">[docs]</a><span class="k">def</span> <span class="nf">isint</span><span class="p">(</span><span class="n">a</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Test for integer.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span><span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">==</span> <span class="nb">int</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="make_null_array"><a class="viewcode-back" href="../../../modules/galgebra/GA.html#sympy.galgebra.GA.make_null_array">[docs]</a><span class="k">def</span> <span class="nf">make_null_array</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return list of n empty lists.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-        <span class="n">a</span><span class="o">.</span><span class="n">append</span><span class="p">([])</span>
-    <span class="k">return</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="test_int_flgs"><a class="viewcode-back" href="../../../modules/galgebra/GA.html#sympy.galgebra.GA.test_int_flgs">[docs]</a><span class="k">def</span> <span class="nf">test_int_flgs</span><span class="p">(</span><span class="n">lst</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Test if all elements in list are 0.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">lst</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">i</span><span class="p">:</span>
-            <span class="k">return</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-    <span class="k">return</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="comb"><a class="viewcode-back" href="../../../modules/galgebra/GA.html#sympy.galgebra.GA.comb">[docs]</a><span class="k">def</span> <span class="nf">comb</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">P</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Calculates the combinations of the integers [0,N-1] taken P at a time.</span>
-<span class="sd">    The output is a list of lists of integers where the inner lists are</span>
-<span class="sd">    the different combinations. Each combination is sorted in ascending</span>
-<span class="sd">    order.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">rloop</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">combs</span><span class="p">,</span> <span class="n">comb</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">p</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-                <span class="n">newcomb</span> <span class="o">=</span> <span class="n">comb</span> <span class="o">+</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span>
-                <span class="n">np</span> <span class="o">=</span> <span class="n">p</span> <span class="o">-</span> <span class="mi">1</span>
-                <span class="n">rloop</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">np</span><span class="p">,</span> <span class="n">combs</span><span class="p">,</span> <span class="n">newcomb</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">combs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">comb</span><span class="p">)</span>
-    <span class="n">combs</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">rloop</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">combs</span><span class="p">,</span> <span class="p">[])</span>
-    <span class="k">for</span> <span class="n">comb</span> <span class="ow">in</span> <span class="n">combs</span><span class="p">:</span>
-        <span class="n">comb</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-    <span class="k">return</span><span class="p">(</span><span class="n">combs</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="diagpq"><a class="viewcode-back" href="../../../modules/galgebra/GA.html#sympy.galgebra.GA.diagpq">[docs]</a><span class="k">def</span> <span class="nf">diagpq</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return string equivalent metric tensor for signature (p, q).</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">p</span> <span class="o">+</span> <span class="n">q</span>
-    <span class="n">D</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">rn</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">rn</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">rn</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">p</span><span class="p">:</span>
-                    <span class="n">D</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;1 &#39;</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">D</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;-1 &#39;</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">D</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;0 &#39;</span><span class="p">)</span>
-    <span class="k">return</span> <span class="s">&#39;,&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">D</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="make_scalars"><a class="viewcode-back" href="../../../modules/galgebra/GA.html#sympy.galgebra.GA.make_scalars">[docs]</a><span class="k">def</span> <span class="nf">make_scalars</span><span class="p">(</span><span class="n">symnamelst</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    make_scalars takes a string of symbol names separated by</span>
-<span class="sd">    blanks and converts them to MV scalars and returns a list</span>
-<span class="sd">    of the symbols.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">symlst</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">symbols</span><span class="p">(</span><span class="n">symnamelst</span><span class="p">)</span>
-    <span class="n">scalar_lst</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">symlst</span><span class="p">:</span>
-        <span class="n">tmp</span> <span class="o">=</span> <span class="n">MV</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="s">&#39;scalar&#39;</span><span class="p">)</span>
-        <span class="n">scalar_lst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tmp</span><span class="p">)</span>
-    <span class="k">return</span><span class="p">(</span><span class="n">scalar_lst</span><span class="p">)</span>
-
-</div>
-<span class="nd">@deprecated</span><span class="p">(</span><span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;sympy.symbols()&quot;</span><span class="p">,</span> <span class="n">issue</span><span class="o">=</span><span class="mi">3379</span><span class="p">,</span>
-            <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.2&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">make_symbols</span><span class="p">(</span><span class="n">symnamelst</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">sympy</span><span class="o">.</span><span class="n">symbols</span><span class="p">(</span><span class="n">symnamelst</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="israt"><a class="viewcode-back" href="../../../modules/galgebra/GA.html#sympy.galgebra.GA.israt">[docs]</a><span class="k">def</span> <span class="nf">israt</span><span class="p">(</span><span class="n">numstr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Test if string represents a rational number.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">global</span> <span class="n">NUMPAT</span>
-    <span class="k">if</span> <span class="n">NUMPAT</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">numstr</span><span class="p">):</span>
-        <span class="k">return</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-    <span class="k">return</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="dualsort"><a class="viewcode-back" href="../../../modules/galgebra/GA.html#sympy.galgebra.GA.dualsort">[docs]</a><span class="k">def</span> <span class="nf">dualsort</span><span class="p">(</span><span class="n">lst1</span><span class="p">,</span> <span class="n">lst2</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Inplace dual sort of lst1 and lst2 keyed on sorted lst1.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">_indices</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">lst1</span><span class="p">)))</span>
-    <span class="n">_indices</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">lst2</span><span class="o">.</span><span class="n">__getitem__</span><span class="p">)</span>
-    <span class="n">lst1</span><span class="p">[:]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">lst1</span><span class="o">.</span><span class="n">__getitem__</span><span class="p">,</span> <span class="n">_indices</span><span class="p">))</span>
-    <span class="n">lst2</span><span class="p">[:]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">lst2</span><span class="o">.</span><span class="n">__getitem__</span><span class="p">,</span> <span class="n">_indices</span><span class="p">))</span>
-    <span class="k">return</span>
-
-</div>
-<div class="viewcode-block" id="cp"><a class="viewcode-back" href="../../../modules/galgebra/GA.html#sympy.galgebra.GA.cp">[docs]</a><span class="k">def</span> <span class="nf">cp</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Calculates the commutator product (A*B-B*A)/2 for</span>
-<span class="sd">    the objects A and B.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span><span class="p">(</span><span class="n">HALF</span><span class="o">*</span><span class="p">(</span><span class="n">A</span><span class="o">*</span><span class="n">B</span> <span class="o">-</span> <span class="n">B</span><span class="o">*</span><span class="n">A</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="reduce_base"><a class="viewcode-back" href="../../../modules/galgebra/GA.html#sympy.galgebra.GA.reduce_base">[docs]</a><span class="k">def</span> <span class="nf">reduce_base</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">base</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    If base is a list of sorted integers [i_1,...,i_R] then reduce_base</span>
-<span class="sd">    sorts the list [k,i_1,...,i_R] and calculates whether an odd or even</span>
-<span class="sd">    number of permutations is required to sort the list. The sorted list</span>
-<span class="sd">    is returned and +1 for even permutations or -1 for odd permutations.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">base</span><span class="p">:</span>
-        <span class="k">return</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">base</span><span class="p">)</span>
-    <span class="n">grade</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">grade</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">k</span> <span class="o">&lt;</span> <span class="n">base</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-            <span class="k">return</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">base</span><span class="p">[</span><span class="mi">0</span><span class="p">]])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">[</span><span class="n">base</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">k</span><span class="p">])</span>
-    <span class="n">ilo</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">ihi</span> <span class="o">=</span> <span class="n">grade</span> <span class="o">-</span> <span class="mi">1</span>
-    <span class="k">if</span> <span class="n">k</span> <span class="o">&lt;</span> <span class="n">base</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-        <span class="k">return</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">+</span> <span class="n">base</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">k</span> <span class="o">&gt;</span> <span class="n">base</span><span class="p">[</span><span class="n">ihi</span><span class="p">]:</span>
-        <span class="k">if</span> <span class="n">grade</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">base</span> <span class="o">+</span> <span class="p">[</span><span class="n">k</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">base</span> <span class="o">+</span> <span class="p">[</span><span class="n">k</span><span class="p">])</span>
-    <span class="n">imid</span> <span class="o">=</span> <span class="n">ihi</span> <span class="o">+</span> <span class="n">ilo</span>
-    <span class="k">if</span> <span class="n">grade</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">return</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">[</span><span class="n">base</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">k</span><span class="p">,</span> <span class="n">base</span><span class="p">[</span><span class="mi">1</span><span class="p">]])</span>
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">ihi</span> <span class="o">-</span> <span class="n">ilo</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">break</span>
-        <span class="k">if</span> <span class="n">base</span><span class="p">[</span><span class="n">imid</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">k</span><span class="p">:</span>
-            <span class="n">ihi</span> <span class="o">=</span> <span class="n">imid</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">ilo</span> <span class="o">=</span> <span class="n">imid</span>
-        <span class="n">imid</span> <span class="o">=</span> <span class="p">(</span><span class="n">ilo</span> <span class="o">+</span> <span class="n">ihi</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-    <span class="k">if</span> <span class="n">ilo</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">base</span><span class="p">[:</span><span class="n">ihi</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">+</span> <span class="n">base</span><span class="p">[</span><span class="n">ihi</span><span class="p">:])</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">base</span><span class="p">[:</span><span class="n">ihi</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">+</span> <span class="n">base</span><span class="p">[</span><span class="n">ihi</span><span class="p">:])</span>
-
-</div>
-<div class="viewcode-block" id="sub_base"><a class="viewcode-back" href="../../../modules/galgebra/GA.html#sympy.galgebra.GA.sub_base">[docs]</a><span class="k">def</span> <span class="nf">sub_base</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">base</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    If base is a list of sorted integers [i_1,...,i_R] then sub_base returns</span>
-<span class="sd">    a list with the k^th element removed. Note that k=0 removes the first</span>
-<span class="sd">    element.  There is no test to see if k is in the range of the list.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span><span class="p">([])</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">k</span> <span class="o">==</span> <span class="n">base</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-            <span class="k">return</span><span class="p">([</span><span class="n">base</span><span class="p">[</span><span class="mi">1</span><span class="p">]])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span><span class="p">([</span><span class="n">base</span><span class="p">[</span><span class="mi">0</span><span class="p">]])</span>
-    <span class="k">return</span><span class="p">(</span><span class="n">base</span><span class="p">[:</span><span class="n">k</span><span class="p">]</span> <span class="o">+</span> <span class="n">base</span><span class="p">[</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:])</span>
-
-</div>
-<div class="viewcode-block" id="magnitude"><a class="viewcode-back" href="../../../modules/galgebra/GA.html#sympy.galgebra.GA.magnitude">[docs]</a><span class="k">def</span> <span class="nf">magnitude</span><span class="p">(</span><span class="n">vector</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Calculate magnitude of vector containing trig expressions</span>
-<span class="sd">    and simplify.  This is a hack because of way he sign of</span>
-<span class="sd">    magsq is determined and because of the way that absolute</span>
-<span class="sd">    values are removed.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">magsq</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">expand</span><span class="p">((</span><span class="n">vector</span> <span class="o">|</span> <span class="n">vector</span><span class="p">)())</span>
-    <span class="n">magsq</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">trigsimp</span><span class="p">(</span><span class="n">magsq</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">recursive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="c">#print magsq</span>
-    <span class="n">magsq_str</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">latex_ex</span><span class="o">.</span><span class="n">LatexPrinter</span><span class="p">()</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">magsq</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">magsq_str</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;-&#39;</span><span class="p">:</span>
-        <span class="n">magsq</span> <span class="o">=</span> <span class="o">-</span><span class="n">magsq</span>
-    <span class="n">mag</span> <span class="o">=</span> <span class="n">unabs</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">magsq</span><span class="p">))</span>
-    <span class="c">#print mag</span>
-    <span class="k">return</span><span class="p">(</span><span class="n">mag</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="LaTeX_lst"><a class="viewcode-back" href="../../../modules/galgebra/GA.html#sympy.galgebra.GA.LaTeX_lst">[docs]</a><span class="k">def</span> <span class="nf">LaTeX_lst</span><span class="p">(</span><span class="n">lst</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s">&#39;&#39;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Output a list in LaTeX format.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">title</span> <span class="o">!=</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-        <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">latex_ex</span><span class="o">.</span><span class="n">LaTeX</span><span class="p">(</span><span class="n">title</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">lst</span><span class="p">:</span>
-        <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">latex_ex</span><span class="o">.</span><span class="n">LaTeX</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="k">return</span>
-
-</div>
-<div class="viewcode-block" id="unabs"><a class="viewcode-back" href="../../../modules/galgebra/GA.html#sympy.galgebra.GA.unabs">[docs]</a><span class="k">def</span> <span class="nf">unabs</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Remove absolute values from expressions so a = sqrt(a**2).</span>
-<span class="sd">    This is a hack.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">==</span> <span class="n">mul_type</span><span class="p">:</span>
-        <span class="n">y</span> <span class="o">=</span> <span class="n">unabs</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">for</span> <span class="n">yi</span> <span class="ow">in</span> <span class="n">x</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="n">y</span> <span class="o">*=</span> <span class="n">unabs</span><span class="p">(</span><span class="n">yi</span><span class="p">)</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">==</span> <span class="n">pow_type</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">HALF</span> <span class="ow">and</span> <span class="nb">type</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">==</span> <span class="n">add_type</span><span class="p">:</span>
-            <span class="k">return</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="n">y</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">unabs</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">==</span> <span class="n">abs_type</span><span class="p">:</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-    <span class="k">return</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">function_lst</span><span class="p">(</span><span class="n">fstr</span><span class="p">,</span> <span class="n">xtuple</span><span class="p">):</span>
-    <span class="n">sys</span><span class="o">.</span><span class="n">stderr</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="n">fstr</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">)</span>
-    <span class="n">fct_lst</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">xstr</span> <span class="ow">in</span> <span class="n">fstr</span><span class="o">.</span><span class="n">split</span><span class="p">():</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Function</span><span class="p">(</span><span class="n">xstr</span><span class="p">)(</span><span class="o">*</span><span class="n">xtuple</span><span class="p">)</span>
-        <span class="n">fct_lst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="k">return</span><span class="p">(</span><span class="n">fct_lst</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="vector_fct"><a class="viewcode-back" href="../../../modules/galgebra/GA.html#sympy.galgebra.GA.vector_fct">[docs]</a><span class="k">def</span> <span class="nf">vector_fct</span><span class="p">(</span><span class="n">Fstr</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create a list of functions of arguments x.  One function is</span>
-<span class="sd">    created for each variable in x.  Fstr is a string that is</span>
-<span class="sd">    the base name of each function while each function in the</span>
-<span class="sd">    list is given the name Fstr+&#39;__&#39;+str(x[ix]) so that if</span>
-<span class="sd">    Fstr = &#39;f&#39; and str(x[1]) = &#39;theta&#39; then the LaTeX output</span>
-<span class="sd">    of the second element in the output list would be &#39;f^{\\theta}&#39;.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">nx</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="n">Fvec</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">ix</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nx</span><span class="p">):</span>
-        <span class="n">ftmp</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Function</span><span class="p">(</span><span class="n">Fstr</span> <span class="o">+</span> <span class="s">&#39;__&#39;</span> <span class="o">+</span> <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">latex_ex</span><span class="o">.</span><span class="n">LatexPrinter</span><span class="o">.</span><span class="n">str_basic</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">ix</span><span class="p">]))(</span><span class="o">*</span><span class="nb">tuple</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-        <span class="n">Fvec</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ftmp</span><span class="p">)</span>
-    <span class="k">return</span><span class="p">(</span><span class="n">Fvec</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">print_lst</span><span class="p">(</span><span class="n">lst</span><span class="p">):</span>
-    <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">lst</span><span class="p">:</span>
-        <span class="k">print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="k">return</span>
-
-
-<div class="viewcode-block" id="normalize"><a class="viewcode-back" href="../../../modules/galgebra/GA.html#sympy.galgebra.GA.normalize">[docs]</a><span class="k">def</span> <span class="nf">normalize</span><span class="p">(</span><span class="n">elst</span><span class="p">,</span> <span class="n">nname_lst</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Normalize a list of vectors and rename the normalized vectors. &#39;elist&#39; is the list</span>
-<span class="sd">    (or array) of vectors to be normalized and nname_lst is a list of the names for the</span>
-<span class="sd">    normalized vectors.  The function returns the numpy arrays enlst and mags containing</span>
-<span class="sd">    the normalized vectors (enlst) and the magnitudes of the original vectors (mags).</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">mags</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="o">*</span><span class="p">[</span><span class="n">ZERO</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-    <span class="n">enlst</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="o">*</span><span class="p">[</span><span class="n">ZERO</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-    <span class="k">for</span> <span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">nname</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">elst</span><span class="p">,</span> <span class="n">nname_lst</span><span class="p">):</span>
-        <span class="n">emag</span> <span class="o">=</span> <span class="n">magnitude</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-        <span class="n">emaginv</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">emag</span>
-        <span class="n">mags</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">emag</span>
-        <span class="n">enorm</span> <span class="o">=</span> <span class="n">emaginv</span><span class="o">*</span><span class="n">e</span>
-        <span class="n">enorm</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="n">nname</span>
-        <span class="n">enlst</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">enorm</span>
-        <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-    <span class="k">return</span><span class="p">(</span><span class="n">enlst</span><span class="p">,</span> <span class="n">mags</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">build_base</span><span class="p">(</span><span class="n">base_index</span><span class="p">,</span> <span class="n">base_vectors</span><span class="p">,</span> <span class="n">reverse</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="n">base</span> <span class="o">=</span> <span class="n">base_vectors</span><span class="p">[</span><span class="n">base_index</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">base_index</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">base_index</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="n">base</span> <span class="o">=</span> <span class="n">base</span> <span class="o">^</span> <span class="n">base_vectors</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">reverse</span><span class="p">:</span>
-        <span class="n">base</span> <span class="o">=</span> <span class="n">base</span><span class="o">.</span><span class="n">rev</span><span class="p">()</span>
-    <span class="k">return</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">MV</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-
-    <span class="n">is_setup</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">basislabel_lst</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">curvilinear_flg</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">coords</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">pad_zeros</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Pad list with zeros to length n. If length is &gt; n</span>
-<span class="sd">        truncate list.  Return padded list.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">nvalue</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">value</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">nvalue</span> <span class="o">&lt;</span> <span class="n">n</span><span class="p">:</span>
-            <span class="n">value</span> <span class="o">=</span> <span class="n">value</span> <span class="o">+</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">nvalue</span><span class="p">)</span><span class="o">*</span><span class="p">[</span><span class="n">ZERO</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">nvalue</span> <span class="o">&gt;</span> <span class="n">n</span><span class="p">:</span>
-            <span class="n">value</span> <span class="o">=</span> <span class="n">value</span><span class="p">[:</span><span class="n">n</span><span class="p">]</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">value</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">define_basis</span><span class="p">(</span><span class="n">basis</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Calculates all the MV static variables needed for</span>
-<span class="sd">        basis operations.  See reference 5 section 2.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">vbasis</span> <span class="o">=</span> <span class="n">basis</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">vsyms</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">symbols</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">vbasis</span><span class="p">)</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">vbasis</span><span class="p">)</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">nrg</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">))</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">n1</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n1</span><span class="p">))</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">npow</span> <span class="o">=</span> <span class="mi">2</span><span class="o">**</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">index</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">))</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">gabasis</span> <span class="o">=</span> <span class="p">[[]]</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">basis</span> <span class="o">=</span> <span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">basislabel</span> <span class="o">=</span> <span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">basis</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">basislabel</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="s">&#39;1&#39;</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">basislabel_lst</span> <span class="o">=</span> <span class="p">[[</span><span class="s">&#39;1&#39;</span><span class="p">]]</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">((</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">igrade</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">MV</span><span class="o">.</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">tmp</span> <span class="o">=</span> <span class="n">comb</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">,</span> <span class="n">igrade</span><span class="p">)</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">gabasis</span> <span class="o">+=</span> <span class="p">[</span><span class="n">tmp</span><span class="p">]</span>
-            <span class="n">ntmp</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">tmp</span><span class="p">)</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span> <span class="o">=</span> <span class="n">ntmp</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">basis</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span> <span class="o">=</span> <span class="n">tmp</span>
-            <span class="n">gradelabels</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">gradelabel_lst</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">ntmp</span><span class="p">):</span>
-                <span class="n">tmp_lst</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="n">bstr</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">tmp</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                    <span class="n">bstr</span> <span class="o">+=</span> <span class="n">MV</span><span class="o">.</span><span class="n">vbasis</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-                    <span class="n">tmp_lst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">vbasis</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-                <span class="n">gradelabel_lst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tmp_lst</span><span class="p">)</span>
-                <span class="n">gradelabels</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">bstr</span><span class="p">)</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">basislabel_lst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">gradelabel_lst</span><span class="p">)</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">basislabel</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span> <span class="o">=</span> <span class="n">gradelabels</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">basis_map</span> <span class="o">=</span> <span class="p">[{</span><span class="s">&#39;&#39;</span><span class="p">:</span><span class="mi">0</span><span class="p">}]</span>
-        <span class="n">igrade</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">while</span> <span class="n">igrade</span> <span class="o">&lt;=</span> <span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">:</span>
-            <span class="n">tmpdict</span> <span class="o">=</span> <span class="p">{}</span>
-            <span class="n">bases</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">gabasis</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span>
-            <span class="n">ibases</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="n">bases</span><span class="p">:</span>
-                <span class="n">tmpdict</span><span class="p">[</span><span class="nb">str</span><span class="p">(</span><span class="n">base</span><span class="p">)]</span> <span class="o">=</span> <span class="n">ibases</span>
-                <span class="n">ibases</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">basis_map</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tmpdict</span><span class="p">)</span>
-            <span class="n">igrade</span> <span class="o">+=</span> <span class="mi">1</span>
-
-        <span class="k">if</span> <span class="n">MV</span><span class="o">.</span><span class="n">debug</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;basis strings =&#39;</span><span class="p">,</span> <span class="n">MV</span><span class="o">.</span><span class="n">vbasis</span><span class="p">)</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;basis symbols =&#39;</span><span class="p">,</span> <span class="n">MV</span><span class="o">.</span><span class="n">vsyms</span><span class="p">)</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;basis labels  =&#39;</span><span class="p">,</span> <span class="n">MV</span><span class="o">.</span><span class="n">basislabel</span><span class="p">)</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;basis         =&#39;</span><span class="p">,</span> <span class="n">MV</span><span class="o">.</span><span class="n">basis</span><span class="p">)</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;grades        =&#39;</span><span class="p">,</span> <span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">)</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;index         =&#39;</span><span class="p">,</span> <span class="n">MV</span><span class="o">.</span><span class="n">index</span><span class="p">)</span>
-        <span class="k">return</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">define_metric</span><span class="p">(</span><span class="n">metric</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Calculates all the MV static variables needed for</span>
-<span class="sd">        metric operations.  See reference 5 section 2.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">MV</span><span class="o">.</span><span class="n">metric_str</span><span class="p">:</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">g</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">metric</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="o">*</span><span class="p">[</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="o">*</span><span class="p">[</span><span class="n">ZERO</span><span class="p">]],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">metric</span> <span class="o">==</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-                <span class="n">metric</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="o">*</span><span class="p">[</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="o">*</span><span class="p">[</span><span class="s">&#39;#&#39;</span><span class="p">]],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">index</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">index</span><span class="p">:</span>
-                    <span class="n">gij</span> <span class="o">=</span> <span class="n">metric</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
-                    <span class="k">if</span> <span class="n">israt</span><span class="p">(</span><span class="n">gij</span><span class="p">):</span>
-                        <span class="n">MV</span><span class="o">.</span><span class="n">metric</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">numeric</span><span class="p">(</span><span class="n">gij</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">gij</span> <span class="o">==</span> <span class="s">&#39;#&#39;</span><span class="p">:</span>
-                            <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
-                                <span class="n">gij</span> <span class="o">=</span> <span class="s">&#39;(&#39;</span> <span class="o">+</span> <span class="n">MV</span><span class="o">.</span><span class="n">vbasis</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+</span> <span class="s">&#39;**2)&#39;</span>
-                            <span class="k">else</span><span class="p">:</span>
-                                <span class="n">gij</span> <span class="o">=</span> <span class="s">&#39;(&#39;</span> <span class="o">+</span> <span class="n">MV</span><span class="o">.</span><span class="n">vbasis</span><span class="p">[</span><span class="nb">min</span><span class="p">(</span>
-                                    <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)]</span> <span class="o">+</span> <span class="s">&#39;.&#39;</span> <span class="o">+</span> <span class="n">MV</span><span class="o">.</span><span class="n">vbasis</span><span class="p">[</span><span class="nb">max</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)]</span> <span class="o">+</span> <span class="s">&#39;)&#39;</span>
-                        <span class="n">tmp</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="n">gij</span><span class="p">)</span>
-                        <span class="n">MV</span><span class="o">.</span><span class="n">metric</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">tmp</span>
-                        <span class="k">if</span> <span class="n">i</span> <span class="o">&lt;=</span> <span class="n">j</span><span class="p">:</span>
-                            <span class="n">MV</span><span class="o">.</span><span class="n">g</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tmp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">metric</span> <span class="o">=</span> <span class="n">metric</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">g</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="n">metric</span><span class="p">:</span>
-                <span class="n">g_row</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">for</span> <span class="n">col</span> <span class="ow">in</span> <span class="n">metric</span><span class="p">:</span>
-                    <span class="n">g_row</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">col</span><span class="p">)</span>
-                    <span class="n">MV</span><span class="o">.</span><span class="n">g</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">g_row</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">MV</span><span class="o">.</span><span class="n">debug</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;metric =&#39;</span><span class="p">,</span> <span class="n">MV</span><span class="o">.</span><span class="n">metric</span><span class="p">)</span>
-        <span class="k">return</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">define_reciprocal_frame</span><span class="p">():</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Calculates unscaled reciprocal vectors (MV.brecp) and scale</span>
-<span class="sd">        factor (MV.Esq). The ith scaled reciprocal vector is</span>
-<span class="sd">        (1/MV.Esq)*MV.brecp[i].  The pseudoscalar for the set of</span>
-<span class="sd">        basis vectors is MV.E.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">MV</span><span class="o">.</span><span class="n">tables_flg</span><span class="p">:</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">E</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">bvec</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">brecp</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">):</span>
-                <span class="n">MV</span><span class="o">.</span><span class="n">E</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">E</span> <span class="o">^</span> <span class="n">MV</span><span class="o">.</span><span class="n">bvec</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">):</span>
-                <span class="n">tmp</span> <span class="o">=</span> <span class="n">ONE</span>
-                <span class="k">if</span> <span class="n">i</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">tmp</span> <span class="o">=</span> <span class="o">-</span><span class="n">ONE</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="n">j</span><span class="p">:</span>
-                        <span class="n">tmp</span> <span class="o">=</span> <span class="n">tmp</span> <span class="o">^</span> <span class="n">MV</span><span class="o">.</span><span class="n">bvec</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-                <span class="n">tmp</span> <span class="o">=</span> <span class="n">tmp</span><span class="o">*</span><span class="n">MV</span><span class="o">.</span><span class="n">E</span>
-                <span class="n">MV</span><span class="o">.</span><span class="n">brecp</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tmp</span><span class="p">)</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">Esq</span> <span class="o">=</span> <span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">E</span><span class="o">*</span><span class="n">MV</span><span class="o">.</span><span class="n">E</span><span class="p">)()</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">Esq_inv</span> <span class="o">=</span> <span class="n">ONE</span><span class="o">/</span><span class="n">MV</span><span class="o">.</span><span class="n">Esq</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">):</span>
-                <span class="n">MV</span><span class="o">.</span><span class="n">brecp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">brecp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">MV</span><span class="o">.</span><span class="n">Esq_inv</span>
-        <span class="k">return</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">reduce_basis_loop</span><span class="p">(</span><span class="n">blst</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Makes one pass through basis product representation for</span>
-<span class="sd">        reduction of representation to normal form.</span>
-<span class="sd">        See reference 5 section 3.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">nblst</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">blst</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">nblst</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-        <span class="n">jstep</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">while</span> <span class="n">jstep</span> <span class="o">&lt;</span> <span class="n">nblst</span><span class="p">:</span>
-            <span class="n">istep</span> <span class="o">=</span> <span class="n">jstep</span> <span class="o">-</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="n">blst</span><span class="p">[</span><span class="n">istep</span><span class="p">]</span> <span class="o">==</span> <span class="n">blst</span><span class="p">[</span><span class="n">jstep</span><span class="p">]:</span>
-                <span class="n">i</span> <span class="o">=</span> <span class="n">blst</span><span class="p">[</span><span class="n">istep</span><span class="p">]</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">blst</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-                    <span class="n">blst</span> <span class="o">=</span> <span class="n">blst</span><span class="p">[:</span><span class="n">istep</span><span class="p">]</span> <span class="o">+</span> <span class="n">blst</span><span class="p">[</span><span class="n">jstep</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">blst</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">blst</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">jstep</span> <span class="o">==</span> <span class="n">nblst</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">blst_flg</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">blst_flg</span> <span class="o">=</span> <span class="mi">1</span>
-                <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">metric</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">i</span><span class="p">],</span> <span class="n">blst</span><span class="p">,</span> <span class="n">blst_flg</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">blst</span><span class="p">[</span><span class="n">istep</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">blst</span><span class="p">[</span><span class="n">jstep</span><span class="p">]:</span>
-                <span class="n">blst1</span> <span class="o">=</span> <span class="n">blst</span><span class="p">[:</span><span class="n">istep</span><span class="p">]</span> <span class="o">+</span> <span class="n">blst</span><span class="p">[</span><span class="n">jstep</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-                <span class="n">a1</span> <span class="o">=</span> <span class="n">TWO</span><span class="o">*</span><span class="n">MV</span><span class="o">.</span><span class="n">metric</span><span class="p">[</span><span class="n">blst</span><span class="p">[</span><span class="n">jstep</span><span class="p">]][</span><span class="n">blst</span><span class="p">[</span><span class="n">istep</span><span class="p">]]</span>
-                <span class="n">blst</span> <span class="o">=</span> <span class="n">blst</span><span class="p">[:</span><span class="n">istep</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">blst</span><span class="p">[</span><span class="n">jstep</span><span class="p">]]</span> <span class="o">+</span> <span class="p">[</span><span class="n">blst</span><span class="p">[</span><span class="n">istep</span><span class="p">]]</span> <span class="o">+</span> <span class="n">blst</span><span class="p">[</span><span class="n">jstep</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">blst1</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">blst1_flg</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">blst1_flg</span> <span class="o">=</span> <span class="mi">1</span>
-                <span class="k">return</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span> <span class="n">blst1</span><span class="p">,</span> <span class="n">blst1_flg</span><span class="p">,</span> <span class="n">blst</span><span class="p">)</span>
-            <span class="n">jstep</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">return</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">reduce_basis</span><span class="p">(</span><span class="n">blst</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Repetitively applies reduce_basis_loop to basis</span>
-<span class="sd">        product representation until normal form is</span>
-<span class="sd">        realized.  See reference 5 section 3.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">blst</span> <span class="o">==</span> <span class="p">[]:</span>
-            <span class="n">blst_coef</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">blst_expand</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-                <span class="n">blst_coef</span><span class="o">.</span><span class="n">append</span><span class="p">([])</span>
-                <span class="n">blst_expand</span><span class="o">.</span><span class="n">append</span><span class="p">([])</span>
-            <span class="n">blst_expand</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">([])</span>
-            <span class="n">blst_coef</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ONE</span><span class="p">)</span>
-            <span class="k">return</span><span class="p">(</span><span class="n">blst_coef</span><span class="p">,</span> <span class="n">blst_expand</span><span class="p">)</span>
-        <span class="n">blst_expand</span> <span class="o">=</span> <span class="p">[</span><span class="n">blst</span><span class="p">]</span>
-        <span class="n">blst_coef</span> <span class="o">=</span> <span class="p">[</span><span class="n">ONE</span><span class="p">]</span>
-        <span class="n">blst_flg</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">while</span> <span class="n">test_int_flgs</span><span class="p">(</span><span class="n">blst_flg</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">blst_flg</span><span class="p">)):</span>
-                <span class="k">if</span> <span class="n">blst_flg</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                    <span class="n">tmp</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">reduce_basis_loop</span><span class="p">(</span><span class="n">blst_expand</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-                    <span class="k">if</span> <span class="n">tmp</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="n">blst_flg</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">tmp</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-                            <span class="n">blst_coef</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">tmp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">blst_coef</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-                            <span class="n">blst_expand</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">tmp</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-                            <span class="n">blst_flg</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">tmp</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="n">blst_coef</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">blst_coef</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-                            <span class="n">blst_flg</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-                            <span class="n">blst_expand</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">tmp</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span>
-                            <span class="n">blst_coef</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">-</span><span class="n">blst_coef</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">tmp</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-                            <span class="n">blst_expand</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tmp</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-                            <span class="n">blst_flg</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tmp</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-        <span class="p">(</span><span class="n">blst_coef</span><span class="p">,</span> <span class="n">blst_expand</span><span class="p">)</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">combine_common_factors</span><span class="p">(</span>
-            <span class="n">blst_coef</span><span class="p">,</span> <span class="n">blst_expand</span><span class="p">)</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">blst_coef</span><span class="p">,</span> <span class="n">blst_expand</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">combine_common_factors</span><span class="p">(</span><span class="n">blst_coef</span><span class="p">,</span> <span class="n">blst_expand</span><span class="p">):</span>
-        <span class="n">new_blst_coef</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">new_blst_expand</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n1</span><span class="p">):</span>
-            <span class="n">new_blst_coef</span><span class="o">.</span><span class="n">append</span><span class="p">([])</span>
-            <span class="n">new_blst_expand</span><span class="o">.</span><span class="n">append</span><span class="p">([])</span>
-        <span class="n">nfac</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">blst_coef</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">ifac</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nfac</span><span class="p">):</span>
-            <span class="n">blen</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">blst_expand</span><span class="p">[</span><span class="n">ifac</span><span class="p">])</span>
-            <span class="n">new_blst_coef</span><span class="p">[</span><span class="n">blen</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">blst_coef</span><span class="p">[</span><span class="n">ifac</span><span class="p">])</span>
-            <span class="n">new_blst_expand</span><span class="p">[</span><span class="n">blen</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">blst_expand</span><span class="p">[</span><span class="n">ifac</span><span class="p">])</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n1</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">new_blst_coef</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">MV</span><span class="o">.</span><span class="n">contract</span><span class="p">(</span><span class="n">new_blst_coef</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">new_blst_expand</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">new_blst_coef</span><span class="p">,</span> <span class="n">new_blst_expand</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">contract</span><span class="p">(</span><span class="n">coefs</span><span class="p">,</span> <span class="n">bases</span><span class="p">):</span>
-        <span class="n">dualsort</span><span class="p">(</span><span class="n">coefs</span><span class="p">,</span> <span class="n">bases</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">bases</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">n</span><span class="p">:</span>
-            <span class="n">j</span> <span class="o">=</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="n">bases</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="n">bases</span><span class="p">[</span><span class="n">j</span><span class="p">]:</span>
-                <span class="n">coefs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+=</span> <span class="n">coefs</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-                <span class="n">bases</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-                <span class="n">coefs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-                <span class="n">n</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">coefs</span><span class="p">)</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">coefs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="n">ZERO</span><span class="p">:</span>
-                <span class="n">coefs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                <span class="n">bases</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                <span class="n">n</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">return</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">convert</span><span class="p">(</span><span class="n">coefs</span><span class="p">,</span> <span class="n">bases</span><span class="p">):</span>
-        <span class="n">mv</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-        <span class="n">mv</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">igrade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="n">coef</span> <span class="o">=</span> <span class="n">coefs</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span>
-            <span class="n">base</span> <span class="o">=</span> <span class="n">bases</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">coef</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">nbases</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span>
-                <span class="n">mv</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">nbases</span><span class="o">*</span><span class="p">[</span><span class="n">ZERO</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-                <span class="n">nbaserg</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">)))</span>
-                <span class="k">for</span> <span class="n">ibase</span> <span class="ow">in</span> <span class="n">nbaserg</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">igrade</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">k</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">basis</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">base</span><span class="p">[</span><span class="n">ibase</span><span class="p">])</span>
-                        <span class="n">mv</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">coef</span><span class="p">[</span><span class="n">ibase</span><span class="p">]</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">mv</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
-                            <span class="p">[</span><span class="n">coef</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">mv</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">set_str_format</span><span class="p">(</span><span class="n">str_mode</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">str_mode</span> <span class="o">=</span> <span class="n">str_mode</span>
-        <span class="k">return</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">str_rep</span><span class="p">(</span><span class="n">mv</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">         Converts internal representation of a multivector to a string</span>
-<span class="sd">         for outputting.  If lst_mode = 1, str_rep outputs a list of</span>
-<span class="sd">         strings where each string contains one multivector coefficient</span>
-<span class="sd">         concatenated with the corresponding base or blade symbol.</span>
-
-<span class="sd">            MV.str_mode     Effect</span>
-<span class="sd">                0           Print entire multivector on one line (default)</span>
-<span class="sd">                1           Print each grade on a single line</span>
-<span class="sd">                2           Print each base on a single line</span>
-<span class="sd">         &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="n">MV</span><span class="o">.</span><span class="n">bladeprint</span><span class="p">:</span>
-            <span class="n">mv</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-            <span class="n">labels</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">bladelabel</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">mv</span><span class="o">.</span><span class="n">bladeflg</span><span class="p">:</span>
-                <span class="n">labels</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">basislabel</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">labels</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">bladelabel</span>
-        <span class="n">mv</span><span class="o">.</span><span class="n">compact</span><span class="p">()</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="n">value</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">value</span> <span class="o">=</span> <span class="p">(</span><span class="n">mv</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">__str__</span><span class="p">()</span>
-            <span class="n">value</span> <span class="o">=</span> <span class="n">value</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-        <span class="n">dummy</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;dummy&#39;</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">igrade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">mv</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]:</span>
-                    <span class="k">if</span> <span class="n">x</span> <span class="o">!=</span> <span class="n">ZERO</span><span class="p">:</span>
-                        <span class="n">xstr</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">dummy</span><span class="p">)</span><span class="o">.</span><span class="n">__str__</span><span class="p">()</span>
-                        <span class="n">xstr</span> <span class="o">=</span> <span class="n">xstr</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">xstr</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="s">&#39;-&#39;</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">value</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                            <span class="n">xstr</span> <span class="o">=</span> <span class="s">&#39;+&#39;</span> <span class="o">+</span> <span class="n">xstr</span>
-                        <span class="k">if</span> <span class="n">xstr</span><span class="o">.</span><span class="n">find</span><span class="p">(</span><span class="s">&#39;_dummy&#39;</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">xstr</span><span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">:]</span> <span class="o">!=</span> <span class="s">&#39;_dummy&#39;</span><span class="p">:</span>
-                            <span class="n">xstr</span> <span class="o">=</span> <span class="n">xstr</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span>
-                                <span class="s">&#39;_dummy*&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39;*&#39;</span> <span class="o">+</span> <span class="n">labels</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="n">xstr</span> <span class="o">=</span> <span class="n">xstr</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;_dummy&#39;</span><span class="p">,</span> <span class="n">labels</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">j</span><span class="p">])</span>
-                        <span class="k">if</span> <span class="n">MV</span><span class="o">.</span><span class="n">str_mode</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                            <span class="n">xstr</span> <span class="o">+=</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span>
-                        <span class="n">value</span> <span class="o">+=</span> <span class="n">xstr</span>
-                    <span class="n">j</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">if</span> <span class="n">MV</span><span class="o">.</span><span class="n">str_mode</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">value</span> <span class="o">+=</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span>
-        <span class="k">if</span> <span class="n">value</span> <span class="o">==</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-            <span class="n">value</span> <span class="o">=</span> <span class="s">&#39;0&#39;</span>
-        <span class="c">#value = value.replace(&#39; &#39;,&#39;&#39;)</span>
-        <span class="n">value</span> <span class="o">=</span> <span class="n">value</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;dummy&#39;</span><span class="p">,</span> <span class="s">&#39;1&#39;</span><span class="p">)</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">value</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">xstr_rep</span><span class="p">(</span><span class="n">mv</span><span class="p">,</span> <span class="n">lst_mode</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">         Converts internal representation of a multivector to a string</span>
-<span class="sd">         for outputing.  If lst_mode = 1, str_rep outputs a list of</span>
-<span class="sd">         strings where each string contains one multivector coefficient</span>
-<span class="sd">         concatenated with the corresponding base or blade symbol.</span>
-<span class="sd">         &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">lst_mode</span><span class="p">:</span>
-            <span class="n">outlst</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">if</span> <span class="n">MV</span><span class="o">.</span><span class="n">bladeprint</span><span class="p">:</span>
-            <span class="n">mv</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-            <span class="n">labels</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">bladelabel</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">mv</span><span class="o">.</span><span class="n">bladeflg</span><span class="p">:</span>
-                <span class="n">labels</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">basislabel</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">labels</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">bladelabel</span>
-        <span class="n">value</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-        <span class="k">for</span> <span class="n">igrade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="n">tmp</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">mv</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]:</span>
-                    <span class="k">if</span> <span class="n">x</span> <span class="o">!=</span> <span class="n">ZERO</span><span class="p">:</span>
-                        <span class="n">xstr</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">__str__</span><span class="p">()</span>
-                        <span class="k">if</span> <span class="n">xstr</span> <span class="o">==</span> <span class="s">&#39;+1&#39;</span> <span class="ow">or</span> <span class="n">xstr</span> <span class="o">==</span> <span class="s">&#39;1&#39;</span> <span class="ow">or</span> <span class="n">xstr</span> <span class="o">==</span> <span class="s">&#39;-1&#39;</span><span class="p">:</span>
-                            <span class="k">if</span> <span class="n">xstr</span> <span class="o">==</span> <span class="s">&#39;+1&#39;</span> <span class="ow">or</span> <span class="n">xstr</span> <span class="o">==</span> <span class="s">&#39;1&#39;</span><span class="p">:</span>
-                                <span class="n">xstr</span> <span class="o">=</span> <span class="s">&#39;+&#39;</span>
-                            <span class="k">else</span><span class="p">:</span>
-                                <span class="n">xstr</span> <span class="o">=</span> <span class="s">&#39;-&#39;</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="k">if</span> <span class="n">xstr</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="s">&#39;+&#39;</span><span class="p">:</span>
-                                <span class="n">xstr</span> <span class="o">=</span> <span class="s">&#39;+(&#39;</span> <span class="o">+</span> <span class="n">xstr</span> <span class="o">+</span> <span class="s">&#39;)&#39;</span>
-                            <span class="k">else</span><span class="p">:</span>
-                                <span class="n">xstr</span> <span class="o">=</span> <span class="s">&#39;+(&#39;</span> <span class="o">+</span> <span class="n">xstr</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span> <span class="o">+</span> <span class="s">&#39;)&#39;</span>
-                        <span class="n">value</span> <span class="o">+=</span> <span class="n">xstr</span> <span class="o">+</span> <span class="n">labels</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
-                        <span class="k">if</span> <span class="n">MV</span><span class="o">.</span><span class="n">str_mode</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">lst_mode</span><span class="p">:</span>
-                            <span class="n">value</span> <span class="o">+=</span> <span class="n">value</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span>
-                        <span class="k">if</span> <span class="n">lst_mode</span><span class="p">:</span>
-                            <span class="n">tmp</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">value</span><span class="p">)</span>
-                    <span class="n">j</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="n">lst_mode</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">tmp</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">outlst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tmp</span><span class="p">)</span>
-                <span class="n">value</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">lst_mode</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">value</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">value</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;+&#39;</span><span class="p">:</span>
-                <span class="n">value</span> <span class="o">=</span> <span class="n">value</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">value</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">value</span> <span class="o">=</span> <span class="s">&#39;0&#39;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">value</span> <span class="o">=</span> <span class="n">outlst</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">value</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">setup</span><span class="p">(</span><span class="n">basis</span><span class="p">,</span> <span class="n">metric</span><span class="o">=</span><span class="s">&#39;&#39;</span><span class="p">,</span> <span class="n">rframe</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">coords</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">debug</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">offset</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        MV.setup initializes the MV class by calculating the static</span>
-<span class="sd">        multivector tables required for geometric algebra operations</span>
-<span class="sd">        on multivectors.  See reference 5 section 2 for details on</span>
-<span class="sd">        basis and metric arguments.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">is_setup</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">metric_str</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">debug</span> <span class="o">=</span> <span class="n">debug</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">bladeprint</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">tables_flg</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">str_mode</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">basisroot</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">index_offset</span> <span class="o">=</span> <span class="n">offset</span>
-        <span class="k">if</span> <span class="n">coords</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">coords</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">coords</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">coords</span><span class="p">)</span>
-            <span class="n">rframe</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">basis</span><span class="p">)</span> <span class="o">==</span> <span class="nb">str</span><span class="p">:</span>
-            <span class="n">basislst</span> <span class="o">=</span> <span class="n">basis</span><span class="o">.</span><span class="n">split</span><span class="p">()</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">basislst</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">MV</span><span class="o">.</span><span class="n">basisroot</span> <span class="o">=</span> <span class="n">basislst</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="n">basislst</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">for</span> <span class="n">coord</span> <span class="ow">in</span> <span class="n">coords</span><span class="p">:</span>
-                    <span class="n">basislst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">basisroot</span> <span class="o">+</span> <span class="s">&#39;_&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">coord</span><span class="p">))</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">define_basis</span><span class="p">(</span><span class="n">basislst</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">metric</span><span class="p">)</span> <span class="o">==</span> <span class="nb">str</span><span class="p">:</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">metric_str</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">metric</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">metric</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;[&#39;</span> <span class="ow">and</span> <span class="n">metric</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;]&#39;</span><span class="p">:</span>
-                    <span class="n">tmps</span> <span class="o">=</span> <span class="n">metric</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;,&#39;</span><span class="p">)</span>
-                    <span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">tmps</span><span class="p">)</span>
-                    <span class="n">metric</span> <span class="o">=</span> <span class="p">[]</span>
-                    <span class="n">itmp</span> <span class="o">=</span> <span class="mi">0</span>
-                    <span class="k">for</span> <span class="n">tmp</span> <span class="ow">in</span> <span class="n">tmps</span><span class="p">:</span>
-                        <span class="n">xtmp</span> <span class="o">=</span> <span class="n">N</span><span class="o">*</span><span class="p">[</span><span class="s">&#39;0&#39;</span><span class="p">]</span>
-                        <span class="n">xtmp</span><span class="p">[</span><span class="n">itmp</span><span class="p">]</span> <span class="o">=</span> <span class="n">tmp</span>
-                        <span class="n">itmp</span> <span class="o">+=</span> <span class="mi">1</span>
-                        <span class="n">metric</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">xtmp</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">tmps</span> <span class="o">=</span> <span class="n">metric</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;,&#39;</span><span class="p">)</span>
-                    <span class="n">metric</span> <span class="o">=</span> <span class="p">[]</span>
-                    <span class="k">for</span> <span class="n">tmp</span> <span class="ow">in</span> <span class="n">tmps</span><span class="p">:</span>
-                        <span class="n">xlst</span> <span class="o">=</span> <span class="n">tmp</span><span class="o">.</span><span class="n">split</span><span class="p">()</span>
-                        <span class="n">xtmp</span> <span class="o">=</span> <span class="p">[]</span>
-                        <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xlst</span><span class="p">:</span>
-                            <span class="n">xtmp</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-                        <span class="n">metric</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">xtmp</span><span class="p">)</span>
-
-        <span class="n">MV</span><span class="o">.</span><span class="n">define_metric</span><span class="p">(</span><span class="n">metric</span><span class="p">)</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">multiplication_table</span><span class="p">()</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">blade_table</span><span class="p">()</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">inverse_blade_table</span><span class="p">()</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">tables_flg</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">isym</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">bvec</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">name</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">vbasis</span><span class="p">:</span>
-            <span class="n">bvar</span> <span class="o">=</span> <span class="n">MV</span><span class="p">(</span><span class="n">value</span><span class="o">=</span><span class="n">isym</span><span class="p">,</span> <span class="n">mvtype</span><span class="o">=</span><span class="s">&#39;basisvector&#39;</span><span class="p">,</span> <span class="n">mvname</span><span class="o">=</span><span class="n">name</span><span class="p">)</span>
-            <span class="n">bvar</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">bvec</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">bvar</span><span class="p">)</span>
-            <span class="n">isym</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">rframe</span><span class="p">:</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">define_reciprocal_frame</span><span class="p">()</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">I</span> <span class="o">=</span> <span class="n">MV</span><span class="p">(</span><span class="n">ONE</span><span class="p">,</span> <span class="s">&#39;pseudo&#39;</span><span class="p">,</span> <span class="s">&#39;I&#39;</span><span class="p">)</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">ZERO</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-        <span class="n">Isq</span> <span class="o">=</span> <span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">I</span><span class="o">*</span><span class="n">MV</span><span class="o">.</span><span class="n">I</span><span class="p">)()</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">Iinv</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">Isq</span><span class="p">)</span><span class="o">*</span><span class="n">MV</span><span class="o">.</span><span class="n">I</span>
-        <span class="k">return</span> <span class="n">MV</span><span class="o">.</span><span class="n">bvec</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">set_coords</span><span class="p">(</span><span class="n">coords</span><span class="p">):</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">coords</span> <span class="o">=</span> <span class="n">coords</span>
-        <span class="k">return</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">scalar_fct</span><span class="p">(</span><span class="n">fct_name</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Create multivector scalar function with name fct_name (string) and</span>
-<span class="sd">        independent variables coords (list of variable).  Default variables are</span>
-<span class="sd">        those associated with each dimension of vector space.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">phi</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Function</span><span class="p">(</span><span class="n">fct_name</span><span class="p">)(</span><span class="o">*</span><span class="n">MV</span><span class="o">.</span><span class="n">coords</span><span class="p">)</span>
-        <span class="n">Phi</span> <span class="o">=</span> <span class="n">MV</span><span class="p">(</span><span class="n">phi</span><span class="p">,</span> <span class="s">&#39;scalar&#39;</span><span class="p">)</span>
-        <span class="n">Phi</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="n">fct_name</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">Phi</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">vector_fct</span><span class="p">(</span><span class="n">fct_name</span><span class="p">,</span> <span class="nb">vars</span><span class="o">=</span><span class="s">&#39;&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Create multivector vector function with name fct_name (string) and</span>
-<span class="sd">        independent variables coords (list of variable).  Default variables are</span>
-<span class="sd">        those associated with each dimension of vector space.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="nb">vars</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="n">Acoefs</span> <span class="o">=</span> <span class="n">vector_fct</span><span class="p">(</span><span class="n">fct_name</span><span class="p">,</span> <span class="n">MV</span><span class="o">.</span><span class="n">coords</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">Acoefs</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="o">*</span><span class="p">[</span><span class="n">ZERO</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">coords</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="nb">vars</span><span class="p">,</span> <span class="n">sympy</span><span class="o">.</span><span class="n">core</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">Symbol</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">icoef</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">nrg</span><span class="p">:</span>
-                    <span class="n">Acoefs</span><span class="p">[</span><span class="n">icoef</span><span class="p">]</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Function</span><span class="p">(</span><span class="n">fct_name</span> <span class="o">+</span> <span class="s">&#39;__&#39;</span> <span class="o">+</span>
-                                   <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">latex_ex</span><span class="o">.</span><span class="n">LatexPrinter</span><span class="o">.</span><span class="n">str_basic</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">icoef</span><span class="p">]))(</span><span class="nb">vars</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">icoef</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">nrg</span><span class="p">:</span>
-                    <span class="n">Acoefs</span><span class="p">[</span><span class="n">icoef</span><span class="p">]</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Function</span><span class="p">(</span><span class="n">fct_name</span> <span class="o">+</span> <span class="s">&#39;__&#39;</span> <span class="o">+</span>
-                                   <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">latex_ex</span><span class="o">.</span><span class="n">LatexPrinter</span><span class="o">.</span><span class="n">str_basic</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">icoef</span><span class="p">]))(</span><span class="o">*</span><span class="nb">tuple</span><span class="p">(</span><span class="nb">vars</span><span class="p">))</span>
-        <span class="n">A</span> <span class="o">=</span> <span class="n">MV</span><span class="p">(</span><span class="n">Acoefs</span><span class="p">,</span> <span class="s">&#39;vector&#39;</span><span class="p">,</span> <span class="n">fct_name</span><span class="p">)</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">rebase</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">coords</span><span class="p">,</span> <span class="n">base_name</span><span class="o">=</span><span class="s">&#39;&#39;</span><span class="p">,</span> <span class="n">debug</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">debug_level</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Define curvilinear coordinates for previously defined vector (multivector) space (MV.setup has been run)</span>
-<span class="sd">        with position vector, x, that is a vector function of the independent coordinates, coords (list of</span>
-<span class="sd">        sympy variables equal in length to dimension of vector space), and calculate:</span>
-
-<span class="sd">            1. Frame (basis) vectors</span>
-<span class="sd">            2. Normalized frame (basis) vectors.</span>
-<span class="sd">            3. Metric tensor</span>
-<span class="sd">            4. Reciprocal frame vectors</span>
-<span class="sd">            5. Reciprocal metric tensor</span>
-<span class="sd">            6. Connection multivectors</span>
-
-<span class="sd">        The basis vectors are named with the base_name (string) and a subscript derived from the name of each</span>
-<span class="sd">        coordinate.  So that if the base name is &#39;e&#39; and the coordinated are [r,theta,z] the variable names</span>
-<span class="sd">        of the frame vectors would be e_r, e_theta, and e_z.  For LaTeX output the names of the frame vectors</span>
-<span class="sd">        would be e_{r}, e_{\theta}, and e_{z}.  Everything needed to compute the geometric, outer, and inner</span>
-<span class="sd">        derivatives of multivector functions in curvilinear coordinates is calculated.</span>
-
-<span class="sd">        If debug is True all the quantities in the above list are output in LaTeX format.</span>
-
-<span class="sd">        Currently rebase works with cylindrical and spherical coordinates in any dimension.  The limitation is the</span>
-<span class="sd">        ability to automatically simplify complex sympy expressions generated while calculating the quantities in</span>
-<span class="sd">        the above list.  This is why the debug option is included.  The debug_level can equal 0,1,2, or 3 and</span>
-<span class="sd">        determines how far in the list to calculate (input 0 to do the entire list) while debugging.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c">#Form root names for basis, reciprocal basis, normalized basis, and normalized reciprocal basis</span>
-
-        <span class="k">if</span> <span class="n">base_name</span> <span class="o">==</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-            <span class="n">base_name</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">basisroot</span> <span class="o">+</span> <span class="s">&#39;prm&#39;</span>
-
-        <span class="n">LaTeX_base</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">latex_ex</span><span class="o">.</span><span class="n">LatexPrinter</span><span class="o">.</span><span class="n">extended_symbol</span><span class="p">(</span>
-            <span class="n">base_name</span><span class="p">)</span>
-        <span class="n">bm</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">bm{&#39;</span> <span class="o">+</span> <span class="n">LaTeX_base</span> <span class="o">+</span> <span class="s">&#39;}&#39;</span>
-        <span class="n">bmhat</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">hat{&#39;</span> <span class="o">+</span> <span class="n">bm</span> <span class="o">+</span> <span class="s">&#39;}&#39;</span>
-        <span class="n">bstr</span> <span class="o">=</span> <span class="n">bmhat</span> <span class="o">+</span> <span class="s">&#39;_{[i_{1},\dots, i_{R}]}&#39;</span>
-        <span class="n">base_name</span> <span class="o">+=</span> <span class="s">&#39;bm&#39;</span>
-        <span class="n">base_name_hat</span> <span class="o">=</span> <span class="n">base_name</span> <span class="o">+</span> <span class="s">&#39;hat&#39;</span>
-
-        <span class="n">base_name_lst</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">nbase_name_lst</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">rbase_name_lst</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">rnbase_name_lst</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">coords_lst</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">coord</span> <span class="ow">in</span> <span class="n">coords</span><span class="p">:</span>
-            <span class="n">coord_str</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">latex_ex</span><span class="o">.</span><span class="n">LatexPrinter</span><span class="o">.</span><span class="n">str_basic</span><span class="p">(</span><span class="n">coord</span><span class="p">)</span>
-            <span class="n">coords_lst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">coord_str</span><span class="p">)</span>
-            <span class="n">base_name_lst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">base_name</span> <span class="o">+</span> <span class="s">&#39;_&#39;</span> <span class="o">+</span> <span class="n">coord_str</span><span class="p">)</span>
-            <span class="n">rbase_name_lst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">base_name</span> <span class="o">+</span> <span class="s">&#39;__&#39;</span> <span class="o">+</span> <span class="n">coord_str</span><span class="p">)</span>
-            <span class="n">nbase_name_lst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">base_name_hat</span> <span class="o">+</span> <span class="s">&#39;_&#39;</span> <span class="o">+</span> <span class="n">coord_str</span><span class="p">)</span>
-            <span class="n">rnbase_name_lst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">base_name_hat</span> <span class="o">+</span> <span class="s">&#39;__&#39;</span> <span class="o">+</span> <span class="n">coord_str</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">coords</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">base_name_lst</span><span class="p">)):</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;rebaseMV inputs not congruent:&#39;</span><span class="p">)</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;MV.n =&#39;</span><span class="p">,</span> <span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">)</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;coords =&#39;</span><span class="p">,</span> <span class="n">coords</span><span class="p">)</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;bases =&#39;</span><span class="p">,</span> <span class="n">base_name</span><span class="p">)</span>
-            <span class="n">sys</span><span class="o">.</span><span class="n">exit</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">MV</span><span class="p">):</span>
-
-            <span class="c">#Calculate basis vectors from derivatives of position vector x</span>
-
-            <span class="n">bases</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="o">*</span><span class="p">[</span><span class="n">ZERO</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">for</span> <span class="n">coord</span> <span class="ow">in</span> <span class="n">coords</span><span class="p">:</span>
-                <span class="n">ei</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">coords</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-                <span class="n">ei</span><span class="o">.</span><span class="n">set_name</span><span class="p">(</span><span class="n">base_name_lst</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-                <span class="n">bases</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">ei</span>
-                <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-
-            <span class="c">#Calculate normalizee basis vectors and basis vector magnitudes</span>
-
-            <span class="k">if</span> <span class="n">debug</span><span class="p">:</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&#39;Coordinate Generating Vector&#39;</span><span class="p">)</span>
-                <span class="k">print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&#39;Basis Vectors&#39;</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="n">bases</span><span class="p">:</span>
-                    <span class="k">print</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-
-        <span class="k">else</span><span class="p">:</span>
-
-            <span class="c">#Input basis vectors as N vector fields</span>
-
-            <span class="n">bases</span> <span class="o">=</span> <span class="n">x</span>
-
-            <span class="k">for</span> <span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">name</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">bases</span><span class="p">,</span> <span class="n">base_name_lst</span><span class="p">):</span>
-                <span class="n">base</span><span class="o">.</span><span class="n">set_name</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">debug</span><span class="p">:</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&#39;Basis Vectors&#39;</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="n">bases</span><span class="p">:</span>
-                    <span class="k">print</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">debug_level</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span>
-
-        <span class="k">if</span> <span class="n">debug_level</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span>
-
-        <span class="c">#Calculate normalized basis vectors and magnitudes of</span>
-        <span class="c">#unormalized basis vectors</span>
-
-        <span class="p">(</span><span class="n">nbases</span><span class="p">,</span> <span class="n">mags</span><span class="p">)</span> <span class="o">=</span> <span class="n">normalize</span><span class="p">(</span><span class="n">bases</span><span class="p">,</span> <span class="n">nbase_name_lst</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">debug</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;Magnitudes&#39;</span><span class="p">)</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\\</span><span class="s">abs{&#39;</span> <span class="o">+</span> <span class="n">LaTeX_base</span> <span class="o">+</span> <span class="s">&#39;_{i}} = &#39;</span><span class="p">,</span> <span class="n">mags</span><span class="p">)</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;Normalized Basis Vectors&#39;</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">nbase</span> <span class="ow">in</span> <span class="n">nbases</span><span class="p">:</span>
-                <span class="k">print</span><span class="p">(</span><span class="n">nbase</span><span class="p">)</span>
-
-        <span class="n">g</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="o">*</span><span class="p">[</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="o">*</span><span class="p">[</span><span class="n">ZERO</span><span class="p">]],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">irow</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">nrg</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">icol</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">nrg</span><span class="p">:</span>
-                <span class="n">magsq</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">expand</span><span class="p">((</span><span class="n">nbases</span><span class="p">[</span><span class="n">irow</span><span class="p">]</span> <span class="o">|</span> <span class="n">nbases</span><span class="p">[</span><span class="n">icol</span><span class="p">])())</span>
-                <span class="n">g</span><span class="p">[</span><span class="n">irow</span><span class="p">][</span><span class="n">icol</span><span class="p">]</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span>
-                    <span class="n">sympy</span><span class="o">.</span><span class="n">trigsimp</span><span class="p">(</span><span class="n">magsq</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">recursive</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">debug</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;Metric $</span><span class="se">\\</span><span class="s">hat{g}_{ij} = </span><span class="se">\\</span><span class="s">hat{&#39;</span> <span class="o">+</span> <span class="n">LaTeX_base</span> <span class="o">+</span> \
-                <span class="s">&#39;}_{i}</span><span class="se">\\</span><span class="s">cdot </span><span class="se">\\</span><span class="s">hat{&#39;</span> <span class="o">+</span> <span class="n">LaTeX_base</span> <span class="o">+</span> <span class="s">&#39;}_{j}$&#39;</span><span class="p">)</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">r&#39;\hat{g}_{ij} =&#39;</span><span class="p">,</span> <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">latex_ex</span><span class="o">.</span><span class="n">LaTeX</span><span class="p">(</span><span class="n">g</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">debug_level</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span>
-
-        <span class="c">#Calculate reciprocal normalized basis vectors</span>
-
-        <span class="n">rnbases</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">if</span> <span class="n">is_quasi_unit_numpy_array</span><span class="p">(</span><span class="n">g</span><span class="p">):</span>
-            <span class="n">ibasis</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">while</span> <span class="n">ibasis</span> <span class="o">&lt;</span> <span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">:</span>
-                <span class="n">base</span> <span class="o">=</span> <span class="n">g</span><span class="p">[</span><span class="n">ibasis</span><span class="p">][</span><span class="n">ibasis</span><span class="p">]</span><span class="o">*</span><span class="n">nbases</span><span class="p">[</span><span class="n">ibasis</span><span class="p">]</span>
-                <span class="n">base</span><span class="o">.</span><span class="n">set_name</span><span class="p">(</span><span class="n">rnbase_name_lst</span><span class="p">[</span><span class="n">ibasis</span><span class="p">])</span>
-                <span class="n">base</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-                <span class="n">base</span><span class="o">.</span><span class="n">trigsimp</span><span class="p">()</span>
-                <span class="n">rnbases</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-                <span class="n">ibasis</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">rnbases</span> <span class="o">=</span> <span class="n">reciprocal_frame</span><span class="p">(</span><span class="n">nbases</span><span class="p">,</span> <span class="n">rnbase_name_lst</span><span class="p">)</span>
-            <span class="n">ibase</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="n">rnbases</span><span class="p">:</span>
-                <span class="n">base</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-                <span class="n">base</span><span class="o">.</span><span class="n">trigsimp</span><span class="p">()</span>
-                <span class="n">rnbases</span><span class="p">[</span><span class="n">ibase</span><span class="p">]</span> <span class="o">=</span> <span class="n">base</span>
-                <span class="n">ibase</span> <span class="o">+=</span> <span class="mi">1</span>
-
-        <span class="k">if</span> <span class="n">debug</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">debug_level</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">latex_ex</span><span class="o">.</span><span class="n">MV_format</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;Reciprocal Normalized Basis Vectors&#39;</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">rnbase</span> <span class="ow">in</span> <span class="n">rnbases</span><span class="p">:</span>
-                <span class="k">print</span><span class="p">(</span><span class="n">rnbase</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">debug_level</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="k">return</span>
-
-        <span class="c">#Calculate components of inverse vectors</span>
-
-        <span class="n">Acoef</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">ibasis</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">nrg</span><span class="p">:</span>
-            <span class="n">evec</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="o">*</span><span class="p">[</span><span class="n">ZERO</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">jbasis</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">nrg</span><span class="p">:</span>
-                <span class="n">evec</span><span class="p">[</span><span class="n">jbasis</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">bvec</span><span class="p">[</span><span class="n">ibasis</span><span class="p">]</span> <span class="o">|</span> <span class="n">rnbases</span><span class="p">[</span><span class="n">jbasis</span><span class="p">])()</span>
-            <span class="n">Acoef</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">evec</span><span class="p">)</span>
-
-        <span class="c">#Calculat metric tensors</span>
-
-        <span class="n">gr</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="o">*</span><span class="p">[</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="o">*</span><span class="p">[</span><span class="n">ZERO</span><span class="p">]],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">irow</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">nrg</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">icol</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">nrg</span><span class="p">:</span>
-                <span class="n">magsq</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">expand</span><span class="p">((</span><span class="n">rnbases</span><span class="p">[</span><span class="n">irow</span><span class="p">]</span> <span class="o">|</span> <span class="n">rnbases</span><span class="p">[</span><span class="n">icol</span><span class="p">])())</span>
-                <span class="n">gr</span><span class="p">[</span><span class="n">irow</span><span class="p">][</span><span class="n">icol</span><span class="p">]</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span>
-                    <span class="n">sympy</span><span class="o">.</span><span class="n">trigsimp</span><span class="p">(</span><span class="n">magsq</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">recursive</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">debug</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;Metric $</span><span class="se">\\</span><span class="s">hat{g}^{ij} = </span><span class="se">\\</span><span class="s">hat{&#39;</span> <span class="o">+</span> <span class="n">LaTeX_base</span> <span class="o">+</span> \
-                <span class="s">&#39;}^{i}</span><span class="se">\\</span><span class="s">cdot </span><span class="se">\\</span><span class="s">hat{&#39;</span> <span class="o">+</span> <span class="n">LaTeX_base</span> <span class="o">+</span> <span class="s">&#39;}^{j}$&#39;</span><span class="p">)</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">r&#39;\hat{g}^{ij} =&#39;</span><span class="p">,</span> <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">latex_ex</span><span class="o">.</span><span class="n">LaTeX</span><span class="p">(</span><span class="n">gr</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">debug_level</span> <span class="o">==</span> <span class="mi">4</span><span class="p">:</span>
-            <span class="k">return</span>
-
-        <span class="c">#Calculate bases and reciprocal bases for curvilinear mulitvectors</span>
-
-        <span class="n">MV_bases</span> <span class="o">=</span> <span class="p">[[</span><span class="n">ONE</span><span class="p">]]</span>
-        <span class="n">MV_rbases</span> <span class="o">=</span> <span class="p">[[</span><span class="n">ONE</span><span class="p">]]</span>
-        <span class="n">igrade</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">while</span> <span class="n">igrade</span> <span class="o">&lt;=</span> <span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">:</span>
-            <span class="n">base_index</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">gabasis</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span>
-            <span class="n">grade_bases</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">rgrade_bases</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">index</span> <span class="ow">in</span> <span class="n">base_index</span><span class="p">:</span>
-                <span class="n">base</span> <span class="o">=</span> <span class="n">build_base</span><span class="p">(</span><span class="n">index</span><span class="p">,</span> <span class="n">nbases</span><span class="p">)</span>
-                <span class="n">base</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-                <span class="n">base</span><span class="o">.</span><span class="n">trigsimp</span><span class="p">()</span>
-                <span class="n">rbase</span> <span class="o">=</span> <span class="n">build_base</span><span class="p">(</span><span class="n">index</span><span class="p">,</span> <span class="n">rnbases</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-                <span class="n">rbase</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-                <span class="n">rbase</span><span class="o">.</span><span class="n">trigsimp</span><span class="p">()</span>
-                <span class="n">grade_bases</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-                <span class="n">rgrade_bases</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">rbase</span><span class="p">)</span>
-            <span class="n">igrade</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">MV_bases</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">grade_bases</span><span class="p">)</span>
-            <span class="n">MV_rbases</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">rgrade_bases</span><span class="p">)</span>
-
-        <span class="c">#Calculate connection multivectors for geometric derivative</span>
-
-        <span class="n">MV_connect</span> <span class="o">=</span> <span class="p">[[</span><span class="n">ZERO</span><span class="p">]]</span>
-        <span class="n">igrade</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">while</span> <span class="n">igrade</span> <span class="o">&lt;=</span> <span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">:</span>
-            <span class="n">grade_connect</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">ibase</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="n">MV_bases</span><span class="p">[</span><span class="n">igrade</span><span class="p">]:</span>
-                <span class="nb">sum</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-                <span class="n">itheta</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="k">for</span> <span class="p">(</span><span class="n">theta</span><span class="p">,</span> <span class="n">etheta</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">coords</span><span class="p">,</span> <span class="n">rnbases</span><span class="p">):</span>
-                    <span class="n">psum</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">mags</span><span class="p">[</span><span class="n">itheta</span><span class="p">])</span><span class="o">*</span><span class="n">etheta</span><span class="o">*</span><span class="n">base</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
-                    <span class="n">psum</span><span class="o">.</span><span class="n">trigsimp</span><span class="p">()</span>
-                    <span class="nb">sum</span> <span class="o">+=</span> <span class="n">psum</span>
-                    <span class="n">itheta</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="nb">sum</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-                <span class="nb">sum</span><span class="o">.</span><span class="n">trigsimp</span><span class="p">()</span>
-                <span class="n">grade_connect</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">sum</span><span class="p">)</span>
-                <span class="n">ibase</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">MV_connect</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">grade_connect</span><span class="p">)</span>
-            <span class="n">igrade</span> <span class="o">+=</span> <span class="mi">1</span>
-
-        <span class="k">if</span> <span class="n">debug</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;Curvilinear Bases: $&#39;</span> <span class="o">+</span> <span class="n">bstr</span> <span class="o">+</span> <span class="s">&#39; = &#39;</span> <span class="o">+</span> <span class="n">bmhat</span> <span class="o">+</span> \
-                <span class="s">&#39;_{i_{1}}</span><span class="se">\\</span><span class="s">W</span><span class="se">\\</span><span class="s">dots</span><span class="se">\\</span><span class="s">W&#39;</span> <span class="o">+</span> <span class="n">bmhat</span> <span class="o">+</span> <span class="s">&#39;_{i_{R}}$&#39;</span><span class="p">)</span>
-            <span class="n">igrade</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="k">for</span> <span class="n">grade</span> <span class="ow">in</span> <span class="n">MV_bases</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-                <span class="n">ibase</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="n">grade</span><span class="p">:</span>
-                    <span class="n">index</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">gabasis</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span>
-                    <span class="n">sub_str</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-                    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">index</span><span class="p">:</span>
-                        <span class="n">sub_str</span> <span class="o">+=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">latex_ex</span><span class="o">.</span><span class="n">LatexPrinter</span><span class="o">.</span><span class="n">extended_symbol</span><span class="p">(</span><span class="n">coords_lst</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-                    <span class="n">base_str</span> <span class="o">=</span> <span class="n">bmhat</span> <span class="o">+</span> <span class="s">&#39;_{[&#39;</span> <span class="o">+</span> <span class="n">sub_str</span> <span class="o">+</span> <span class="s">&#39;]} = &#39;</span>
-                    <span class="k">print</span><span class="p">(</span><span class="n">base_str</span><span class="p">,</span> <span class="n">base</span><span class="p">)</span>
-                    <span class="n">ibase</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="n">igrade</span> <span class="o">+=</span> <span class="mi">1</span>
-
-        <span class="k">if</span> <span class="n">debug_level</span> <span class="o">==</span> <span class="mi">5</span><span class="p">:</span>
-            <span class="k">return</span>
-
-        <span class="c">#Calculate representation of connection multivectors in curvilinear system</span>
-
-        <span class="n">MV_Connect</span> <span class="o">=</span> <span class="p">[[</span><span class="n">ZERO</span><span class="p">]]</span>
-        <span class="n">igrade</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">while</span> <span class="n">igrade</span> <span class="o">&lt;=</span> <span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">:</span>
-            <span class="n">grade_connect</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">ibase</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="n">nbase</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">MV_bases</span><span class="p">[</span><span class="n">igrade</span><span class="p">])</span>
-
-            <span class="k">if</span> <span class="n">igrade</span> <span class="o">&lt;</span> <span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">:</span>
-                <span class="n">ibase</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="n">p1base</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">MV_bases</span><span class="p">[</span><span class="n">igrade</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span>
-                <span class="n">m1base</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">MV_bases</span><span class="p">[</span><span class="n">igrade</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span>
-                <span class="k">while</span> <span class="n">ibase</span> <span class="o">&lt;</span> <span class="n">nbase</span><span class="p">:</span>
-                    <span class="n">Cm1</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">m1base</span><span class="o">*</span><span class="p">[</span><span class="n">ZERO</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-                    <span class="n">Cp1</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">p1base</span><span class="o">*</span><span class="p">[</span><span class="n">ZERO</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-                    <span class="n">C</span> <span class="o">=</span> <span class="n">MV_connect</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span>
-                    <span class="k">if</span> <span class="n">igrade</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="n">X</span> <span class="o">=</span> <span class="n">C</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">X</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-                    <span class="n">jbase</span> <span class="o">=</span> <span class="mi">0</span>
-                    <span class="k">while</span> <span class="n">jbase</span> <span class="o">&lt;</span> <span class="n">m1base</span><span class="p">:</span>
-                        <span class="n">Cm1</span><span class="p">[</span><span class="n">jbase</span><span class="p">]</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">trigsimp</span><span class="p">((</span><span class="n">MV</span><span class="o">.</span><span class="n">inner_product</span><span class="p">(</span><span class="n">MV_rbases</span><span class="p">[</span><span class="n">igrade</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">jbase</span><span class="p">],</span> <span class="n">C</span><span class="p">))(),</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">recursive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                        <span class="n">jbase</span> <span class="o">+=</span> <span class="mi">1</span>
-                    <span class="n">jbase</span> <span class="o">=</span> <span class="mi">0</span>
-                    <span class="k">while</span> <span class="n">jbase</span> <span class="o">&lt;</span> <span class="n">p1base</span><span class="p">:</span>
-                        <span class="n">Cp1</span><span class="p">[</span><span class="n">jbase</span><span class="p">]</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">trigsimp</span><span class="p">((</span><span class="n">MV</span><span class="o">.</span><span class="n">inner_product</span><span class="p">(</span><span class="n">MV_rbases</span><span class="p">[</span><span class="n">igrade</span> <span class="o">+</span> <span class="mi">1</span><span class="p">][</span><span class="n">jbase</span><span class="p">],</span> <span class="n">C</span><span class="p">))(),</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">recursive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                        <span class="n">jbase</span> <span class="o">+=</span> <span class="mi">1</span>
-                    <span class="n">X</span> <span class="o">+=</span> <span class="n">MV</span><span class="p">(</span>
-                        <span class="p">(</span><span class="n">igrade</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">Cm1</span><span class="p">),</span> <span class="s">&#39;grade&#39;</span><span class="p">)</span> <span class="o">+</span> <span class="n">MV</span><span class="p">((</span><span class="n">igrade</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">Cp1</span><span class="p">),</span> <span class="s">&#39;grade&#39;</span><span class="p">)</span>
-                    <span class="n">X</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-                    <span class="n">X</span><span class="o">.</span><span class="n">trigsimp</span><span class="p">()</span>
-                    <span class="n">grade_connect</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-                    <span class="n">ibase</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">ibase</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="n">m1base</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">MV_bases</span><span class="p">[</span><span class="n">igrade</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span>
-                <span class="k">while</span> <span class="n">ibase</span> <span class="o">&lt;</span> <span class="n">nbase</span><span class="p">:</span>
-                    <span class="n">Cm1</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">m1base</span><span class="o">*</span><span class="p">[</span><span class="n">ZERO</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-                    <span class="n">C</span> <span class="o">=</span> <span class="n">MV_connect</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span>
-                    <span class="n">jbase</span> <span class="o">=</span> <span class="mi">0</span>
-                    <span class="k">while</span> <span class="n">jbase</span> <span class="o">&lt;</span> <span class="n">m1base</span><span class="p">:</span>
-                        <span class="n">Cm1</span><span class="p">[</span><span class="n">jbase</span><span class="p">]</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">trigsimp</span><span class="p">((</span><span class="n">MV</span><span class="o">.</span><span class="n">inner_product</span><span class="p">(</span><span class="n">MV_rbases</span><span class="p">[</span><span class="n">igrade</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">jbase</span><span class="p">],</span> <span class="n">C</span><span class="p">))(),</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">recursive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                        <span class="n">jbase</span> <span class="o">+=</span> <span class="mi">1</span>
-                    <span class="n">X</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-                    <span class="n">X</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">Cm1</span>
-                    <span class="n">X</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-                    <span class="n">X</span><span class="o">.</span><span class="n">trigsimp</span><span class="p">()</span>
-                    <span class="n">grade_connect</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-                    <span class="n">ibase</span> <span class="o">+=</span> <span class="mi">1</span>
-
-            <span class="n">MV_Connect</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">grade_connect</span><span class="p">)</span>
-            <span class="n">igrade</span> <span class="o">+=</span> <span class="mi">1</span>
-
-        <span class="n">base_str</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-        <span class="k">for</span> <span class="n">coord</span> <span class="ow">in</span> <span class="n">coords</span><span class="p">:</span>
-            <span class="n">base_str</span> <span class="o">+=</span> <span class="n">base_name</span> <span class="o">+</span> <span class="s">&#39;_&#39;</span> <span class="o">+</span> \
-                <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">latex_ex</span><span class="o">.</span><span class="n">LatexPrinter</span><span class="o">.</span><span class="n">str_basic</span><span class="p">(</span><span class="n">coord</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39; &#39;</span>
-        <span class="n">base_str</span> <span class="o">=</span> <span class="n">base_str</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-
-        <span class="n">old_names</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">vbasis</span>
-
-        <span class="n">MV</span><span class="o">.</span><span class="n">setup</span><span class="p">(</span><span class="n">base_str</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="bp">True</span><span class="p">,</span> <span class="n">coords</span><span class="p">)</span>
-
-        <span class="n">MV</span><span class="o">.</span><span class="n">curvilinear_flg</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">Connect</span> <span class="o">=</span> <span class="n">MV_Connect</span>
-        <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">latex_ex</span><span class="o">.</span><span class="n">LatexPrinter</span><span class="o">.</span><span class="n">latex_bases</span><span class="p">()</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">Rframe</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="o">*</span><span class="p">[</span><span class="n">ZERO</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-        <span class="n">ibasis</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">while</span> <span class="n">ibasis</span> <span class="o">&lt;</span> <span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">:</span>
-            <span class="n">base</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-            <span class="n">jbasis</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">while</span> <span class="n">jbasis</span> <span class="o">&lt;</span> <span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">:</span>
-                <span class="n">base</span><span class="o">.</span><span class="n">add_in_place</span><span class="p">(</span><span class="n">gr</span><span class="p">[</span><span class="n">ibasis</span><span class="p">][</span><span class="n">jbasis</span><span class="p">]</span><span class="o">*</span><span class="n">MV</span><span class="o">.</span><span class="n">bvec</span><span class="p">[</span><span class="n">jbasis</span><span class="p">])</span>
-                <span class="n">jbasis</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">base</span><span class="o">.</span><span class="n">scalar_mul_inplace</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">mags</span><span class="p">[</span><span class="n">ibasis</span><span class="p">])</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">Rframe</span><span class="p">[</span><span class="n">ibasis</span><span class="p">]</span> <span class="o">=</span> <span class="n">base</span>
-            <span class="n">ibasis</span> <span class="o">+=</span> <span class="mi">1</span>
-
-        <span class="n">MV</span><span class="o">.</span><span class="n">org_basis</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">ibasis</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">nrg</span><span class="p">:</span>
-            <span class="n">evec</span> <span class="o">=</span> <span class="n">MV</span><span class="p">(</span><span class="n">Acoef</span><span class="p">[</span><span class="n">ibasis</span><span class="p">],</span> <span class="s">&#39;vector&#39;</span><span class="p">,</span> <span class="n">old_names</span><span class="p">[</span><span class="n">ibasis</span><span class="p">])</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">org_basis</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">evec</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">MV</span><span class="o">.</span><span class="n">coords</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">):</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">dedt</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="c"># I can&#39;t fix this no name error because I have no clue what the</span>
-            <span class="c"># original writer was trying to do.</span>
-            <span class="k">for</span> <span class="n">coef</span> <span class="ow">in</span> <span class="n">dedt_coef</span><span class="p">:</span>
-                <span class="n">MV</span><span class="o">.</span><span class="n">dedt</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">MV</span><span class="p">(</span><span class="n">coef</span><span class="p">,</span> <span class="s">&#39;vector&#39;</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">dedt</span> <span class="o">=</span> <span class="bp">None</span>
-
-        <span class="k">if</span> <span class="n">debug</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;Representation of Original Basis Vectors&#39;</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">evec</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">org_basis</span><span class="p">:</span>
-                <span class="k">print</span><span class="p">(</span><span class="n">evec</span><span class="p">)</span>
-
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;Renormalized Reciprocal Vectors &#39;</span> <span class="o">+</span> <span class="s">&#39;$</span><span class="se">\\</span><span class="s">bfrac{&#39;</span> <span class="o">+</span> <span class="n">bmhat</span> <span class="o">+</span> \
-                <span class="s">&#39;^{k}}{</span><span class="se">\\</span><span class="s">abs{</span><span class="se">\\</span><span class="s">bm{&#39;</span> <span class="o">+</span> <span class="n">LaTeX_base</span> <span class="o">+</span> <span class="s">&#39;}_{k}}}$&#39;</span><span class="p">)</span>
-
-            <span class="n">ibasis</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">while</span> <span class="n">ibasis</span> <span class="o">&lt;</span> <span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">:</span>
-                <span class="n">c_str</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">latex_ex</span><span class="o">.</span><span class="n">LatexPrinter</span><span class="o">.</span><span class="n">extended_symbol</span><span class="p">(</span>
-                    <span class="n">coords_lst</span><span class="p">[</span><span class="n">ibasis</span><span class="p">])</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\\</span><span class="s">bfrac{</span><span class="se">\\</span><span class="s">bm{</span><span class="se">\\</span><span class="s">hat{&#39;</span> <span class="o">+</span> <span class="n">LaTeX_base</span> <span class="o">+</span> <span class="s">&#39;}}^{&#39;</span> <span class="o">+</span> <span class="n">c_str</span> <span class="o">+</span> \
-                    <span class="s">&#39;}}{</span><span class="se">\\</span><span class="s">abs{</span><span class="se">\\</span><span class="s">bm{&#39;</span> <span class="o">+</span> <span class="n">LaTeX_base</span> <span class="o">+</span> <span class="s">&#39;}_{&#39;</span> <span class="o">+</span> <span class="n">c_str</span> <span class="o">+</span> <span class="s">&#39;}}} =&#39;</span><span class="p">,</span> \
-                    <span class="n">MV</span><span class="o">.</span><span class="n">Rframe</span><span class="p">[</span><span class="n">ibasis</span><span class="p">])</span>
-                <span class="n">ibasis</span> <span class="o">+=</span> <span class="mi">1</span>
-
-            <span class="n">title_str</span> <span class="o">=</span> <span class="s">&#39;Connection Multivectors: $C</span><span class="se">\\</span><span class="s">lbrc&#39;</span> <span class="o">+</span> <span class="n">bstr</span> <span class="o">+</span> \
-                <span class="s">&#39;</span><span class="se">\\</span><span class="s">rbrc = &#39;</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">bfrac{&#39;</span> <span class="o">+</span> <span class="n">bmhat</span> <span class="o">+</span> <span class="s">&#39;^{k}}{</span><span class="se">\\</span><span class="s">abs{&#39;</span> <span class="o">+</span> <span class="n">bmhat</span> <span class="o">+</span> \
-                <span class="s">&#39;_{k}}}</span><span class="se">\\</span><span class="s">pdiff{&#39;</span> <span class="o">+</span> <span class="n">bstr</span> <span class="o">+</span> <span class="s">&#39;}{</span><span class="se">\\</span><span class="s">theta^{k}}$&#39;</span>
-
-            <span class="k">print</span><span class="p">(</span><span class="n">title_str</span><span class="p">)</span>
-            <span class="n">igrade</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="k">for</span> <span class="n">grade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">Connect</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-                <span class="n">ibase</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="n">grade</span><span class="p">:</span>
-                    <span class="n">index</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">gabasis</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span>
-                    <span class="n">sub_str</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-                    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">index</span><span class="p">:</span>
-                        <span class="n">sub_str</span> <span class="o">+=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">latex_ex</span><span class="o">.</span><span class="n">LatexPrinter</span><span class="o">.</span><span class="n">extended_symbol</span><span class="p">(</span><span class="n">coords_lst</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-
-                    <span class="n">base_str</span> <span class="o">=</span> <span class="s">&#39;C</span><span class="se">\\</span><span class="s">lbrc</span><span class="se">\\</span><span class="s">hat{&#39;</span> <span class="o">+</span> \
-                        <span class="n">LaTeX_base</span> <span class="o">+</span> <span class="s">&#39;}_{[&#39;</span> <span class="o">+</span> <span class="n">sub_str</span> <span class="o">+</span> <span class="s">&#39;]}</span><span class="se">\\</span><span class="s">rbrc = &#39;</span>
-                    <span class="k">print</span><span class="p">(</span><span class="n">base_str</span><span class="p">,</span> <span class="n">base</span><span class="p">)</span>
-                    <span class="n">ibase</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="n">igrade</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">return</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">print_blades</span><span class="p">():</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Set multivector output to blade representation.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">bladeprint</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">return</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">print_bases</span><span class="p">():</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Set multivector output to base representation.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">bladeprint</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">return</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">multiplication_table</span><span class="p">():</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Calculate geometric product base multiplication table.</span>
-<span class="sd">        See reference 5 section 3 for details.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">mtable</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">igrade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">mtable</span><span class="o">.</span><span class="n">append</span><span class="p">([])</span>
-            <span class="k">for</span> <span class="n">ibase</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">igrade</span><span class="p">]):</span>
-                <span class="n">MV</span><span class="o">.</span><span class="n">mtable</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">([])</span>
-                <span class="k">if</span> <span class="n">igrade</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">base1</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">base1</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">basis</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span>
-                <span class="k">for</span> <span class="n">jgrade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-                    <span class="n">MV</span><span class="o">.</span><span class="n">mtable</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">([])</span>
-                    <span class="k">for</span> <span class="n">jbase</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">jgrade</span><span class="p">]):</span>
-                        <span class="k">if</span> <span class="n">jgrade</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                            <span class="n">base2</span> <span class="o">=</span> <span class="p">[]</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="n">base2</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">basis</span><span class="p">[</span><span class="n">jgrade</span><span class="p">][</span><span class="n">jbase</span><span class="p">]</span>
-                        <span class="n">base</span> <span class="o">=</span> <span class="n">base1</span> <span class="o">+</span> <span class="n">base2</span>
-                        <span class="p">(</span><span class="n">coefs</span><span class="p">,</span> <span class="n">bases</span><span class="p">)</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">reduce_basis</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-                        <span class="n">product</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">coefs</span><span class="p">,</span> <span class="n">bases</span><span class="p">)</span>
-                        <span class="n">product</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="s">&#39;(&#39;</span> <span class="o">+</span> <span class="n">MV</span><span class="o">.</span><span class="n">basislabel</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span> <span class="o">+</span> \
-                            <span class="s">&#39;)(&#39;</span> <span class="o">+</span> <span class="n">MV</span><span class="o">.</span><span class="n">basislabel</span><span class="p">[</span><span class="n">jgrade</span><span class="p">][</span><span class="n">jbase</span><span class="p">]</span> <span class="o">+</span> <span class="s">&#39;)&#39;</span>
-                        <span class="n">MV</span><span class="o">.</span><span class="n">mtable</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">][</span><span class="n">jgrade</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">product</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">MV</span><span class="o">.</span><span class="n">debug</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;Multiplication Table:&#39;</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">level1</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">mtable</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">level2</span> <span class="ow">in</span> <span class="n">level1</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="n">level3</span> <span class="ow">in</span> <span class="n">level2</span><span class="p">:</span>
-                        <span class="k">for</span> <span class="n">mv</span> <span class="ow">in</span> <span class="n">level3</span><span class="p">:</span>
-                            <span class="n">mv</span><span class="o">.</span><span class="n">printmv</span><span class="p">()</span>
-        <span class="k">return</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">geometric_product</span><span class="p">(</span><span class="n">mv1</span><span class="p">,</span> <span class="n">mv2</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        MV.geometric_product(mv1,mv2) calculates the geometric</span>
-<span class="sd">        product the multivectors mv1 and mv2 (mv1*mv2). See</span>
-<span class="sd">        reference 5 section 3.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">product</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv1</span><span class="p">,</span> <span class="n">MV</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv2</span><span class="p">,</span> <span class="n">MV</span><span class="p">):</span>
-            <span class="n">bladeflg1</span> <span class="o">=</span> <span class="n">mv1</span><span class="o">.</span><span class="n">bladeflg</span>
-            <span class="n">bladeflg2</span> <span class="o">=</span> <span class="n">mv2</span><span class="o">.</span><span class="n">bladeflg</span>
-            <span class="k">if</span> <span class="n">bladeflg1</span><span class="p">:</span>
-                <span class="n">mv1</span><span class="o">.</span><span class="n">convert_from_blades</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">bladeflg2</span><span class="p">:</span>
-                <span class="n">mv2</span><span class="o">.</span><span class="n">convert_from_blades</span><span class="p">()</span>
-            <span class="k">for</span> <span class="n">igrade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-                <span class="n">gradei</span> <span class="o">=</span> <span class="n">mv1</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">gradei</span><span class="p">,</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                    <span class="k">for</span> <span class="n">ibase</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">igrade</span><span class="p">]):</span>
-                        <span class="n">xi</span> <span class="o">=</span> <span class="n">gradei</span><span class="p">[</span><span class="n">ibase</span><span class="p">]</span>
-                        <span class="k">if</span> <span class="n">xi</span> <span class="o">!=</span> <span class="n">ZERO</span><span class="p">:</span>
-                            <span class="k">for</span> <span class="n">jgrade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-                                <span class="n">gradej</span> <span class="o">=</span> <span class="n">mv2</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">jgrade</span><span class="p">]</span>
-                                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">gradej</span><span class="p">,</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                                    <span class="k">for</span> <span class="n">jbase</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">jgrade</span><span class="p">]):</span>
-                                        <span class="n">xj</span> <span class="o">=</span> <span class="n">gradej</span><span class="p">[</span><span class="n">jbase</span><span class="p">]</span>
-                                        <span class="k">if</span> <span class="n">xj</span> <span class="o">!=</span> <span class="n">ZERO</span><span class="p">:</span>
-                                            <span class="n">xixj</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">mtable</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">][</span><span class="n">jgrade</span><span class="p">][</span><span class="n">jbase</span><span class="p">]</span><span class="o">.</span><span class="n">scalar_mul</span><span class="p">(</span><span class="n">xi</span><span class="o">*</span><span class="n">xj</span><span class="p">)</span>
-                                            <span class="n">product</span><span class="o">.</span><span class="n">add_in_place</span><span class="p">(</span><span class="n">xixj</span><span class="p">)</span>
-            <span class="n">product</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">if</span> <span class="n">bladeflg1</span><span class="p">:</span>
-                <span class="n">mv1</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">bladeflg2</span><span class="p">:</span>
-                <span class="n">mv1</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">bladeflg1</span> <span class="ow">and</span> <span class="n">bladeflg2</span><span class="p">:</span>
-                <span class="n">product</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv1</span><span class="p">,</span> <span class="n">MV</span><span class="p">):</span>
-                <span class="n">product</span> <span class="o">=</span> <span class="n">mv1</span><span class="o">.</span><span class="n">scalar_mul</span><span class="p">(</span><span class="n">mv2</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">product</span> <span class="o">=</span> <span class="n">mv2</span><span class="o">.</span><span class="n">scalar_mul</span><span class="p">(</span><span class="n">mv1</span><span class="p">)</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">product</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">wedge</span><span class="p">(</span><span class="n">igrade1</span><span class="p">,</span> <span class="n">blade1</span><span class="p">,</span> <span class="n">vector2</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s">&#39;&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Calculate the outer product of a multivector blade</span>
-<span class="sd">        and a vector.  See reference 5 section 5 for details.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">w12</span> <span class="o">=</span> <span class="n">blade1</span><span class="o">*</span><span class="n">vector2</span>
-        <span class="n">w21</span> <span class="o">=</span> <span class="n">vector2</span><span class="o">*</span><span class="n">blade1</span>
-        <span class="k">if</span> <span class="n">igrade1</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">w</span> <span class="o">=</span> <span class="n">w12</span> <span class="o">-</span> <span class="n">w21</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">w</span> <span class="o">=</span> <span class="n">w12</span> <span class="o">+</span> <span class="n">w21</span>
-        <span class="n">w</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="n">name</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">w</span><span class="o">*</span><span class="n">HALF</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">blade_table</span><span class="p">():</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Calculate basis blades in terms of bases. See reference 5</span>
-<span class="sd">        section 5 for details. Used to convert from blade to base</span>
-<span class="sd">        representation.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">btable</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">bladelabel</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">basis_str</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">basislabel</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">igrade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">bladelabel</span><span class="o">.</span><span class="n">append</span><span class="p">([])</span>
-            <span class="k">if</span> <span class="n">igrade</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">MV</span><span class="o">.</span><span class="n">bladelabel</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;1&#39;</span><span class="p">)</span>
-                <span class="n">tmp</span> <span class="o">=</span> <span class="p">[</span><span class="n">MV</span><span class="p">(</span><span class="n">value</span><span class="o">=</span><span class="n">ONE</span><span class="p">,</span> <span class="n">mvtype</span><span class="o">=</span><span class="s">&#39;scalar&#39;</span><span class="p">,</span> <span class="n">mvname</span><span class="o">=</span><span class="s">&#39;1&#39;</span><span class="p">)]</span>
-            <span class="k">if</span> <span class="n">igrade</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">tmp</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">for</span> <span class="n">ibase</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="mi">1</span><span class="p">]):</span>
-                    <span class="n">MV</span><span class="o">.</span><span class="n">bladelabel</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">basis_str</span><span class="p">[</span><span class="n">ibase</span><span class="p">])</span>
-                    <span class="n">tmp</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">MV</span><span class="p">(</span><span class="n">value</span><span class="o">=</span><span class="n">ibase</span><span class="p">,</span>
-                               <span class="n">mvtype</span><span class="o">=</span><span class="s">&#39;basisvector&#39;</span><span class="p">,</span> <span class="n">mvname</span><span class="o">=</span><span class="n">basis_str</span><span class="p">[</span><span class="n">ibase</span><span class="p">]))</span>
-            <span class="k">if</span> <span class="n">igrade</span> <span class="o">&gt;=</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">tmp</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="n">basis</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">basis</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span>
-                <span class="k">for</span> <span class="n">blade</span> <span class="ow">in</span> <span class="n">basis</span><span class="p">:</span>
-                    <span class="n">name</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-                    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">blade</span><span class="p">:</span>
-                        <span class="n">name</span> <span class="o">+=</span> <span class="n">basis_str</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="s">&#39;^&#39;</span>
-                    <span class="n">name</span> <span class="o">=</span> <span class="n">name</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-                    <span class="n">MV</span><span class="o">.</span><span class="n">bladelabel</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-                    <span class="n">lblade</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">basis</span><span class="p">[</span><span class="n">igrade</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">blade</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-                    <span class="n">rblade</span> <span class="o">=</span> <span class="n">blade</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-                    <span class="n">igrade1</span> <span class="o">=</span> <span class="n">igrade</span> <span class="o">-</span> <span class="mi">1</span>
-                    <span class="n">blade1</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">btable</span><span class="p">[</span><span class="n">igrade1</span><span class="p">][</span><span class="n">lblade</span><span class="p">]</span>
-                    <span class="n">vector2</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">btable</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="n">rblade</span><span class="p">]</span>
-                    <span class="n">b1Wv2</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">wedge</span><span class="p">(</span><span class="n">igrade1</span><span class="p">,</span> <span class="n">blade1</span><span class="p">,</span> <span class="n">vector2</span><span class="p">,</span> <span class="n">name</span><span class="p">)</span>
-                    <span class="n">tmp</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">b1Wv2</span><span class="p">)</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">btable</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tmp</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">MV</span><span class="o">.</span><span class="n">debug</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;Blade Tabel:&#39;</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">grade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">btable</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">mv</span> <span class="ow">in</span> <span class="n">grade</span><span class="p">:</span>
-                    <span class="k">print</span><span class="p">(</span><span class="n">mv</span><span class="p">)</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;Blade Labels:&#39;</span><span class="p">)</span>
-            <span class="k">print</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">bladelabel</span><span class="p">)</span>
-        <span class="k">return</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">inverse_blade_table</span><span class="p">():</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Calculate bases in terms of basis blades. See reference 5</span>
-<span class="sd">        section 5 for details. Used to convert from base to blade</span>
-<span class="sd">        representation.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">MV</span><span class="o">.</span><span class="n">ibtable</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">igrade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">igrade</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">tmp</span> <span class="o">=</span> <span class="p">[</span><span class="n">MV</span><span class="p">(</span><span class="n">value</span><span class="o">=</span><span class="n">ONE</span><span class="p">,</span> <span class="n">mvtype</span><span class="o">=</span><span class="s">&#39;scalar&#39;</span><span class="p">,</span> <span class="n">mvname</span><span class="o">=</span><span class="s">&#39;1&#39;</span><span class="p">)]</span>
-            <span class="k">if</span> <span class="n">igrade</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">tmp</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">for</span> <span class="n">ibase</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="mi">1</span><span class="p">]):</span>
-                    <span class="n">tmp</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">MV</span><span class="p">(</span><span class="n">value</span><span class="o">=</span><span class="n">ibase</span><span class="p">,</span> <span class="n">mvtype</span><span class="o">=</span><span class="s">&#39;basisvector&#39;</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">igrade</span> <span class="o">&gt;=</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">tmp</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="n">iblade</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="k">for</span> <span class="n">blade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">btable</span><span class="p">[</span><span class="n">igrade</span><span class="p">]:</span>
-                    <span class="n">invblade</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-                    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">igrade</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-                        <span class="n">invblade</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">blade</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-                        <span class="n">invblade</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span> <span class="o">=</span> <span class="o">+</span><span class="n">blade</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span>
-                        <span class="n">invblade</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="mi">1</span>
-                    <span class="k">if</span> <span class="n">igrade</span> <span class="o">&gt;=</span> <span class="mi">4</span><span class="p">:</span>
-                        <span class="n">jgrade</span> <span class="o">=</span> <span class="n">igrade</span> <span class="o">-</span> <span class="mi">2</span>
-                        <span class="k">while</span> <span class="n">jgrade</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                            <span class="k">for</span> <span class="n">ibase</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">jgrade</span><span class="p">]):</span>
-                                <span class="n">invblade</span><span class="o">.</span><span class="n">substitute_base</span><span class="p">(</span>
-                                    <span class="n">jgrade</span><span class="p">,</span> <span class="n">ibase</span><span class="p">,</span> <span class="n">MV</span><span class="o">.</span><span class="n">ibtable</span><span class="p">[</span><span class="n">jgrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">])</span>
-                            <span class="n">jgrade</span> <span class="o">-=</span> <span class="mi">2</span>
-                    <span class="n">invblade</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">basislabel</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">iblade</span><span class="p">]</span>
-                    <span class="n">iblade</span> <span class="o">+=</span> <span class="mi">1</span>
-                    <span class="n">tmp</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">invblade</span><span class="p">)</span>
-            <span class="n">MV</span><span class="o">.</span><span class="n">ibtable</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tmp</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">MV</span><span class="o">.</span><span class="n">debug</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;Inverse Blade Tabel:&#39;</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">grade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">ibtable</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">mv</span> <span class="ow">in</span> <span class="n">grade</span><span class="p">:</span>
-                    <span class="n">mv</span><span class="o">.</span><span class="n">printmv</span><span class="p">()</span>
-        <span class="k">return</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">outer_product</span><span class="p">(</span><span class="n">mv1</span><span class="p">,</span> <span class="n">mv2</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        MV.outer_product(mv1,mv2) calculates the outer (exterior,wedge)</span>
-<span class="sd">        product of the multivectors mv1 and mv2 (mv1^mv2). See</span>
-<span class="sd">        reference 5 section 6.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">mv1</span><span class="p">)</span> <span class="o">==</span> <span class="nb">type</span><span class="p">(</span><span class="n">MV</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">type</span><span class="p">(</span><span class="n">mv2</span><span class="p">)</span> <span class="o">==</span> <span class="nb">type</span><span class="p">(</span><span class="n">MV</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">mv1</span><span class="o">.</span><span class="n">is_scalar</span><span class="p">()</span> <span class="ow">and</span> <span class="n">mv2</span><span class="o">.</span><span class="n">is_scalar</span><span class="p">():</span>
-                <span class="k">return</span><span class="p">(</span><span class="n">mv1</span><span class="o">*</span><span class="n">mv2</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv1</span><span class="p">,</span> <span class="n">MV</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv2</span><span class="p">,</span> <span class="n">MV</span><span class="p">):</span>
-            <span class="n">product</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-            <span class="n">product</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="n">mv1</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-            <span class="n">mv2</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-            <span class="k">for</span> <span class="n">igrade1</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv1</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade1</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                    <span class="n">pg1</span> <span class="o">=</span> <span class="n">mv1</span><span class="o">.</span><span class="n">project</span><span class="p">(</span><span class="n">igrade1</span><span class="p">)</span>
-                    <span class="k">for</span> <span class="n">igrade2</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-                        <span class="n">igrade</span> <span class="o">=</span> <span class="n">igrade1</span> <span class="o">+</span> <span class="n">igrade2</span>
-                        <span class="k">if</span> <span class="n">igrade</span> <span class="o">&lt;=</span> <span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">:</span>
-                            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv2</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade2</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                                <span class="n">pg2</span> <span class="o">=</span> <span class="n">mv2</span><span class="o">.</span><span class="n">project</span><span class="p">(</span><span class="n">igrade2</span><span class="p">)</span>
-                                <span class="n">pg1pg2</span> <span class="o">=</span> <span class="n">pg1</span><span class="o">*</span><span class="n">pg2</span>
-                                <span class="n">product</span><span class="o">.</span><span class="n">add_in_place</span><span class="p">(</span><span class="n">pg1pg2</span><span class="o">.</span><span class="n">project</span><span class="p">(</span><span class="n">igrade</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv1</span><span class="p">,</span> <span class="n">MV</span><span class="p">):</span>
-                <span class="n">product</span> <span class="o">=</span> <span class="n">mv1</span><span class="o">.</span><span class="n">scalar_mul</span><span class="p">(</span><span class="n">mv2</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv2</span><span class="p">,</span> <span class="n">MV</span><span class="p">):</span>
-                <span class="n">product</span> <span class="o">=</span> <span class="n">mv2</span><span class="o">.</span><span class="n">scalar_mul</span><span class="p">(</span><span class="n">mv1</span><span class="p">)</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">product</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">inner_product</span><span class="p">(</span><span class="n">mv1</span><span class="p">,</span> <span class="n">mv2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;s&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        MV.inner_product(mv1,mv2) calculates the inner</span>
-
-<span class="sd">        mode = &#39;s&#39; - symmetric (Doran &amp; Lasenby)</span>
-<span class="sd">        mode = &#39;l&#39; - left contraction (Dorst)</span>
-<span class="sd">        mode = &#39;r&#39; - right contraction (Dorst)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">mv1</span><span class="p">)</span> <span class="o">==</span> <span class="nb">type</span><span class="p">(</span><span class="n">MV</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">type</span><span class="p">(</span><span class="n">mv2</span><span class="p">)</span> <span class="o">==</span> <span class="nb">type</span><span class="p">(</span><span class="n">MV</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">mv1</span><span class="o">.</span><span class="n">is_scalar</span><span class="p">()</span> <span class="ow">and</span> <span class="n">mv2</span><span class="o">.</span><span class="n">is_scalar</span><span class="p">():</span>
-                <span class="k">return</span><span class="p">(</span><span class="n">mv1</span><span class="o">*</span><span class="n">mv2</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv1</span><span class="p">,</span> <span class="n">MV</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv2</span><span class="p">,</span> <span class="n">MV</span><span class="p">):</span>
-            <span class="n">product</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-            <span class="n">product</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="n">mv1</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-            <span class="n">mv2</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-            <span class="k">for</span> <span class="n">igrade1</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n1</span><span class="p">):</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv1</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade1</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                    <span class="n">pg1</span> <span class="o">=</span> <span class="n">mv1</span><span class="o">.</span><span class="n">project</span><span class="p">(</span><span class="n">igrade1</span><span class="p">)</span>
-                    <span class="k">for</span> <span class="n">igrade2</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n1</span><span class="p">):</span>
-                        <span class="n">igrade</span> <span class="o">=</span> <span class="n">igrade1</span> <span class="o">-</span> <span class="n">igrade2</span>
-                        <span class="k">if</span> <span class="n">mode</span> <span class="o">==</span> <span class="s">&#39;s&#39;</span><span class="p">:</span>
-                            <span class="n">igrade</span> <span class="o">=</span> <span class="n">igrade</span><span class="o">.</span><span class="n">__abs__</span><span class="p">()</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="k">if</span> <span class="n">mode</span> <span class="o">==</span> <span class="s">&#39;l&#39;</span><span class="p">:</span>
-                                <span class="n">igrade</span> <span class="o">=</span> <span class="o">-</span><span class="n">igrade</span>
-                        <span class="k">if</span> <span class="n">igrade</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-                            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv2</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade2</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                                <span class="n">pg2</span> <span class="o">=</span> <span class="n">mv2</span><span class="o">.</span><span class="n">project</span><span class="p">(</span><span class="n">igrade2</span><span class="p">)</span>
-                                <span class="n">pg1pg2</span> <span class="o">=</span> <span class="n">pg1</span><span class="o">*</span><span class="n">pg2</span>
-                                <span class="n">product</span><span class="o">.</span><span class="n">add_in_place</span><span class="p">(</span><span class="n">pg1pg2</span><span class="o">.</span><span class="n">project</span><span class="p">(</span><span class="n">igrade</span><span class="p">))</span>
-            <span class="k">return</span><span class="p">(</span><span class="n">product</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">mode</span> <span class="o">==</span> <span class="s">&#39;s&#39;</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv1</span><span class="p">,</span> <span class="n">MV</span><span class="p">):</span>
-                    <span class="n">product</span> <span class="o">=</span> <span class="n">mv1</span><span class="o">.</span><span class="n">scalar_mul</span><span class="p">(</span><span class="n">mv2</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv2</span><span class="p">,</span> <span class="n">MV</span><span class="p">):</span>
-                    <span class="n">product</span> <span class="o">=</span> <span class="n">mv2</span><span class="o">.</span><span class="n">scalar_mul</span><span class="p">(</span><span class="n">mv1</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">product</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">product</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">addition</span><span class="p">(</span><span class="n">mv1</span><span class="p">,</span> <span class="n">mv2</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        MV.addition(mv1,mv2) calculates the sum</span>
-<span class="sd">        of the multivectors mv1 and mv2 (mv1+mv2).</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="nb">sum</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv1</span><span class="p">,</span> <span class="n">MV</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv2</span><span class="p">,</span> <span class="n">MV</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">mv1</span><span class="o">.</span><span class="n">bladeflg</span> <span class="ow">or</span> <span class="n">mv2</span><span class="o">.</span><span class="n">bladeflg</span><span class="p">:</span>
-                <span class="n">mv1</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-                <span class="n">mv2</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-                <span class="nb">sum</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv1</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv2</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                    <span class="nb">sum</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">mv1</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">mv2</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv1</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv2</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                        <span class="nb">sum</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="o">+</span><span class="n">mv1</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv2</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv1</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                            <span class="nb">sum</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="o">+</span><span class="n">mv2</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="k">return</span><span class="p">(</span><span class="nb">sum</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv1</span><span class="p">,</span> <span class="n">MV</span><span class="p">):</span>
-                <span class="k">return</span><span class="p">(</span><span class="n">mv1</span> <span class="o">+</span> <span class="n">MV</span><span class="p">(</span><span class="n">mv2</span><span class="p">,</span> <span class="s">&#39;scalar&#39;</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="p">(</span><span class="n">mv1</span><span class="p">,</span> <span class="s">&#39;scalar&#39;</span><span class="p">)</span> <span class="o">+</span> <span class="n">mv2</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">subtraction</span><span class="p">(</span><span class="n">mv1</span><span class="p">,</span> <span class="n">mv2</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        MV.subtraction(mv1,mv2) calculates the difference</span>
-<span class="sd">        of the multivectors mv1 and mv2 (mv1-mv2).</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">diff</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv1</span><span class="p">,</span> <span class="n">MV</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv2</span><span class="p">,</span> <span class="n">MV</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">mv1</span><span class="o">.</span><span class="n">bladeflg</span> <span class="ow">or</span> <span class="n">mv2</span><span class="o">.</span><span class="n">bladeflg</span><span class="p">:</span>
-                <span class="n">mv1</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-                <span class="n">mv2</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-                <span class="n">diff</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv1</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv2</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                    <span class="n">diff</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">mv1</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">mv2</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv1</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv2</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                        <span class="n">diff</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="o">+</span><span class="n">mv1</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv1</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv2</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                            <span class="n">diff</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">mv2</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="k">return</span><span class="p">(</span><span class="n">diff</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv1</span><span class="p">,</span> <span class="n">MV</span><span class="p">):</span>
-                <span class="k">return</span><span class="p">(</span><span class="n">mv1</span> <span class="o">-</span> <span class="n">MV</span><span class="p">(</span><span class="n">mv2</span><span class="p">,</span> <span class="s">&#39;scalar&#39;</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="p">(</span><span class="n">mv1</span><span class="p">,</span> <span class="s">&#39;scalar&#39;</span><span class="p">)</span> <span class="o">-</span> <span class="n">mv2</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">vdiff</span><span class="p">(</span><span class="n">vec</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">dvec</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">vec</span><span class="p">)</span><span class="o">*</span><span class="p">[</span><span class="n">ZERO</span><span class="p">])</span>
-        <span class="n">ivec</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">veci</span> <span class="ow">in</span> <span class="n">vec</span><span class="p">:</span>
-            <span class="n">dvec</span><span class="p">[</span><span class="n">ivec</span><span class="p">]</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">veci</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-            <span class="n">ivec</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">dvec</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">scalar_to_symbol</span><span class="p">(</span><span class="n">scalar</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">scalar</span><span class="p">,</span> <span class="n">MV</span><span class="p">):</span>
-            <span class="k">return</span><span class="p">(</span><span class="n">scalar</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">scalar</span><span class="p">)</span> <span class="o">==</span> <span class="nb">list</span><span class="p">:</span>
-            <span class="n">sym</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">scalar</span><span class="p">:</span>
-                <span class="n">sym</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">scalar_to_symbol</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-            <span class="k">return</span><span class="p">(</span><span class="n">sym</span><span class="p">)</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">scalar</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">value</span><span class="o">=</span><span class="s">&#39;&#39;</span><span class="p">,</span> <span class="n">mvtype</span><span class="o">=</span><span class="s">&#39;&#39;</span><span class="p">,</span> <span class="n">mvname</span><span class="o">=</span><span class="s">&#39;&#39;</span><span class="p">,</span> <span class="n">fct</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="nb">vars</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Initialization of multivector X. Inputs are as follows</span>
-
-<span class="sd">        mvtype           value                       result</span>
-
-<span class="sd">        default         default                  Zero multivector</span>
-<span class="sd">        &#39;basisvector&#39;   int i                    ith basis vector</span>
-<span class="sd">        &#39;basisbivector&#39; int i                    ith basis bivector</span>
-<span class="sd">        &#39;scalar&#39;        symbol x                 scalar of value x</span>
-<span class="sd">                        string s</span>
-<span class="sd">        &#39;grade&#39;         [int i, symbol array A]  X.grade(i) = A</span>
-<span class="sd">                        [int i, string s]</span>
-<span class="sd">        &#39;vector&#39;        symbol array A           X.grade(1) = A</span>
-<span class="sd">                        string s</span>
-<span class="sd">        &#39;grade2&#39;        symbol array A           X.grade(2) = A</span>
-<span class="sd">                        string s</span>
-<span class="sd">        &#39;pseudo&#39;        symbol x                 X.grade(n) = x</span>
-<span class="sd">                        string s</span>
-<span class="sd">        &#39;spinor&#39;        string s                 spinor with coefficients</span>
-<span class="sd">                                                 s__indices and name sbm</span>
-
-<span class="sd">        mvname is name of multivector.</span>
-<span class="sd">        If fct is &#39;True&#39; and MV.coords is defined in MV.setup then a</span>
-<span class="sd">        multivector field of MV.coords is instantiated.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="n">mvname</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">mv</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1</span><span class="o">*</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c"># 1 for blade expansion</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">puregrade</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">mvtype</span> <span class="o">==</span> <span class="s">&#39;basisvector&#39;</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="p">[</span><span class="n">ZERO</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="n">value</span><span class="p">]</span> <span class="o">=</span> <span class="n">ONE</span>
-        <span class="k">if</span> <span class="n">mvtype</span> <span class="o">==</span> <span class="s">&#39;basisbivector&#39;</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="p">[</span><span class="n">ZERO</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="n">value</span><span class="p">]</span> <span class="o">=</span> <span class="n">ONE</span>
-        <span class="k">if</span> <span class="n">mvtype</span> <span class="o">==</span> <span class="s">&#39;scalar&#39;</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-                <span class="n">value</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="n">value</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-                <span class="n">value</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Rational</span><span class="p">(</span><span class="n">value</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">value</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">mvtype</span> <span class="o">==</span> <span class="s">&#39;pseudo&#39;</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-                <span class="n">value</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="n">value</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">value</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">mvtype</span> <span class="o">==</span> <span class="s">&#39;vector&#39;</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>  <span class="c"># Most general vector</span>
-                <span class="n">symbol_str</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-                <span class="k">for</span> <span class="n">ibase</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">nrg</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">MV</span><span class="o">.</span><span class="n">coords</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="n">symbol</span> <span class="o">=</span> <span class="n">value</span> <span class="o">+</span> <span class="s">&#39;__&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">ibase</span> <span class="o">+</span> <span class="n">MV</span><span class="o">.</span><span class="n">index_offset</span><span class="p">)</span>
-                        <span class="n">symbol_str</span> <span class="o">+=</span> <span class="n">symbol</span> <span class="o">+</span> <span class="s">&#39; &#39;</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">symbol</span> <span class="o">=</span> <span class="n">value</span> <span class="o">+</span> <span class="s">&#39;__&#39;</span> <span class="o">+</span> <span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">coords</span><span class="p">[</span><span class="n">ibase</span><span class="p">])</span><span class="o">.</span><span class="n">name</span>
-                        <span class="n">symbol_str</span> <span class="o">+=</span> <span class="n">symbol</span> <span class="o">+</span> <span class="s">&#39; &#39;</span>
-                <span class="n">symbol_lst</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">symbols</span><span class="p">(</span><span class="n">symbol_str</span><span class="p">)</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">symbol_lst</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="n">value</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">value</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">pad_zeros</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">mvtype</span> <span class="o">==</span> <span class="s">&#39;grade2&#39;</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>  <span class="c"># Most general grade-2 multivector</span>
-                <span class="k">if</span> <span class="n">value</span> <span class="o">!=</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-                    <span class="n">symbol_str</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-                    <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">basis</span><span class="p">[</span><span class="mi">2</span><span class="p">]:</span>
-                        <span class="n">symbol</span> <span class="o">=</span> <span class="n">value</span> <span class="o">+</span> <span class="n">MV</span><span class="o">.</span><span class="n">construct_index</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-                        <span class="n">symbol_str</span> <span class="o">+=</span> <span class="n">symbol</span> <span class="o">+</span> <span class="s">&#39; &#39;</span>
-                    <span class="n">symbol_lst</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">symbols</span><span class="p">(</span><span class="n">symbol_str</span><span class="p">)</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">symbol_lst</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">value</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">pad_zeros</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">mvtype</span> <span class="o">==</span> <span class="s">&#39;grade&#39;</span><span class="p">:</span>
-            <span class="n">igrade</span> <span class="o">=</span> <span class="n">value</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">coefs</span> <span class="o">=</span> <span class="n">value</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">coefs</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>  <span class="c"># Most general pure grade multivector</span>
-                <span class="n">base_symbol</span> <span class="o">=</span> <span class="n">coefs</span>
-                <span class="n">coefs</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="n">bases</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">basis</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span>
-                <span class="k">if</span> <span class="n">igrade</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
-                        <span class="p">[</span><span class="n">sympy</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="n">base_symbol</span><span class="p">)],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="n">bases</span><span class="p">:</span>
-                        <span class="n">coef</span> <span class="o">=</span> <span class="n">base_symbol</span> <span class="o">+</span> <span class="n">MV</span><span class="o">.</span><span class="n">construct_index</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-                        <span class="n">coef</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="n">coef</span><span class="p">)</span>
-                        <span class="n">coefs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">coef</span><span class="p">)</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">coefs</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span> <span class="o">=</span> <span class="n">coefs</span>
-        <span class="k">if</span> <span class="n">mvtype</span> <span class="o">==</span> <span class="s">&#39;base&#39;</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">value</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
-                <span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">value</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span><span class="o">*</span><span class="p">[</span><span class="n">ZERO</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">value</span><span class="p">[</span><span class="mi">0</span><span class="p">]][</span><span class="n">value</span><span class="p">[</span><span class="mi">1</span><span class="p">]]</span> <span class="o">=</span> <span class="n">ONE</span>
-        <span class="k">if</span> <span class="n">mvtype</span> <span class="o">==</span> <span class="s">&#39;spinor&#39;</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>  <span class="c"># Most general spinor</span>
-                <span class="k">for</span> <span class="n">grade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">grade</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">symbol_str</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-                        <span class="k">if</span> <span class="n">grade</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                            <span class="n">symbol_lst</span> <span class="o">=</span> <span class="p">[]</span>
-                            <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">basis</span><span class="p">[</span><span class="n">grade</span><span class="p">]:</span>
-                                <span class="n">symbol</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span>
-                                    <span class="n">value</span> <span class="o">+</span> <span class="n">MV</span><span class="o">.</span><span class="n">construct_index</span><span class="p">(</span><span class="n">base</span><span class="p">))</span>
-                                <span class="n">symbol_lst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">symbol</span><span class="p">)</span>
-                            <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">grade</span><span class="p">]</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
-                                <span class="n">symbol_lst</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
-                                <span class="p">[</span><span class="n">sympy</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="n">value</span><span class="p">)],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="n">value</span> <span class="o">+</span> <span class="s">&#39;bm&#39;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="nb">str</span><span class="p">)</span> <span class="ow">and</span> <span class="n">mvtype</span> <span class="o">==</span> <span class="s">&#39;&#39;</span><span class="p">:</span>  <span class="c"># Most general multivector</span>
-            <span class="k">if</span> <span class="n">value</span> <span class="o">!=</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">grade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-                    <span class="n">symbol_str</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-                    <span class="k">if</span> <span class="n">grade</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">symbol_lst</span> <span class="o">=</span> <span class="p">[]</span>
-                        <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">basis</span><span class="p">[</span><span class="n">grade</span><span class="p">]:</span>
-                            <span class="n">symbol</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span>
-                                <span class="n">value</span> <span class="o">+</span> <span class="n">MV</span><span class="o">.</span><span class="n">construct_index</span><span class="p">(</span><span class="n">base</span><span class="p">))</span>
-                            <span class="n">symbol_lst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">symbol</span><span class="p">)</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">grade</span><span class="p">]</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
-                            <span class="n">symbol_lst</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
-                            <span class="p">[</span><span class="n">sympy</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="n">value</span><span class="p">)],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="n">value</span> <span class="o">+</span> <span class="s">&#39;bm&#39;</span>
-        <span class="k">if</span> <span class="n">fct</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">vars</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="nb">vars</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="nb">vars</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">grade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">grade</span><span class="p">],</span> <span class="nb">int</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">grade</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">coef</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">latex_ex</span><span class="o">.</span><span class="n">LatexPrinter</span><span class="o">.</span><span class="n">str_basic</span><span class="p">(</span>
-                            <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span>
-                        <span class="k">if</span> <span class="nb">vars</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">MV</span><span class="o">.</span><span class="n">coords</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                            <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">sympy</span><span class="o">.</span><span class="n">Function</span><span class="p">(</span>
-                                <span class="n">coef</span><span class="p">)(</span><span class="o">*</span><span class="n">MV</span><span class="o">.</span><span class="n">coords</span><span class="p">)],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">sympy</span><span class="o">.</span><span class="n">Function</span><span class="p">(</span>
-                                <span class="n">coef</span><span class="p">)(</span><span class="o">*</span><span class="nb">vars</span><span class="p">)],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">grade</span><span class="p">]):</span>
-                            <span class="n">coef</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">latex_ex</span><span class="o">.</span><span class="n">LatexPrinter</span><span class="o">.</span><span class="n">str_basic</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">grade</span><span class="p">][</span><span class="n">base</span><span class="p">])</span>
-                            <span class="k">if</span> <span class="nb">vars</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">MV</span><span class="o">.</span><span class="n">coords</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                                <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">grade</span><span class="p">][</span>
-                                    <span class="n">base</span><span class="p">]</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Function</span><span class="p">(</span><span class="n">coef</span><span class="p">)(</span><span class="o">*</span><span class="n">MV</span><span class="o">.</span><span class="n">coords</span><span class="p">)</span>
-                            <span class="k">else</span><span class="p">:</span>
-                                <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span>
-                                    <span class="n">grade</span><span class="p">][</span><span class="n">base</span><span class="p">]</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Function</span><span class="p">(</span><span class="n">coef</span><span class="p">)(</span><span class="o">*</span><span class="nb">vars</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">construct_index</span><span class="p">(</span><span class="n">base</span><span class="p">):</span>
-        <span class="n">index_str</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">base</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span><span class="p">(</span><span class="s">&#39;&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">MV</span><span class="o">.</span><span class="n">coords</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">ix</span> <span class="ow">in</span> <span class="n">base</span><span class="p">:</span>
-                <span class="n">index_str</span> <span class="o">+=</span> <span class="nb">str</span><span class="p">(</span><span class="n">ix</span> <span class="o">+</span> <span class="n">MV</span><span class="o">.</span><span class="n">index_offset</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">ix</span> <span class="ow">in</span> <span class="n">base</span><span class="p">:</span>
-                <span class="n">index_str</span> <span class="o">+=</span> <span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">coords</span><span class="p">[</span><span class="n">ix</span><span class="p">])</span><span class="o">.</span><span class="n">name</span>
-        <span class="k">return</span><span class="p">(</span><span class="s">&#39;__&#39;</span> <span class="o">+</span> <span class="n">index_str</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">set_name</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">namestr</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="n">namestr</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">max_grade</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        X.max_grade() is maximum grade of non-zero grades of X.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="k">return</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-        <span class="k">return</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">coord</span><span class="p">(</span><span class="n">xname</span><span class="p">,</span> <span class="n">offset</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="n">xi_str</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">nrg</span><span class="p">:</span>
-            <span class="n">xi_str</span> <span class="o">+=</span> <span class="n">xname</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="n">offset</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39; &#39;</span>
-        <span class="n">xi</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">symbols</span><span class="p">(</span><span class="n">xi_str</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">MV</span><span class="p">(</span><span class="n">xi</span><span class="p">,</span> <span class="s">&#39;vector&#39;</span><span class="p">)</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">x</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">isint</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">1</span><span class="p">]):</span>
-            <span class="k">return</span><span class="p">(</span><span class="n">ZERO</span><span class="p">)</span>
-        <span class="k">return</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="n">i</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">set_coef</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">grade</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">grade</span><span class="p">],</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">grade</span><span class="p">]</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
-                <span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">grade</span><span class="p">]</span><span class="o">*</span><span class="p">[</span><span class="n">ZERO</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">grade</span><span class="p">][</span><span class="n">base</span><span class="p">]</span> <span class="o">=</span> <span class="n">value</span>
-        <span class="k">return</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">named</span><span class="p">(</span><span class="n">mvname</span><span class="p">,</span> <span class="n">value</span><span class="o">=</span><span class="s">&#39;&#39;</span><span class="p">,</span> <span class="n">mvtype</span><span class="o">=</span><span class="s">&#39;&#39;</span><span class="p">):</span>
-        <span class="n">name</span> <span class="o">=</span> <span class="n">mvname</span>
-        <span class="n">tmp</span> <span class="o">=</span> <span class="n">MV</span><span class="p">(</span><span class="n">value</span><span class="o">=</span><span class="n">value</span><span class="p">,</span> <span class="n">mvtype</span><span class="o">=</span><span class="n">mvtype</span><span class="p">,</span> <span class="n">mvname</span><span class="o">=</span><span class="n">name</span><span class="p">)</span>
-        <span class="nb">setattr</span><span class="p">(</span><span class="n">sys</span><span class="o">.</span><span class="n">modules</span><span class="p">[</span><span class="n">__name__</span><span class="p">],</span> <span class="n">name</span><span class="p">,</span> <span class="n">tmp</span><span class="p">)</span>
-        <span class="k">return</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">printnm</span><span class="p">(</span><span class="n">tpl</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">tpl</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="s">&#39; =&#39;</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">mv</span><span class="p">)</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">str_rep</span><span class="p">(</span><span class="bp">self</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">printmv</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s">&#39;&#39;</span><span class="p">):</span>
-        <span class="n">title</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-        <span class="k">if</span> <span class="n">name</span><span class="p">:</span>
-            <span class="n">title</span> <span class="o">+=</span> <span class="n">name</span> <span class="o">+</span> <span class="s">&#39; = &#39;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">:</span>
-                <span class="n">title</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">name</span> <span class="o">+</span> <span class="s">&#39; = &#39;</span>
-        <span class="k">print</span><span class="p">(</span><span class="n">title</span> <span class="o">+</span> <span class="n">MV</span><span class="o">.</span><span class="n">str_rep</span><span class="p">(</span><span class="bp">self</span><span class="p">))</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">set_value</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">igrade</span><span class="p">,</span> <span class="n">ibase</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span> <span class="o">=</span> <span class="n">value</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
-                <span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span><span class="o">*</span><span class="p">[</span><span class="n">ZERO</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">object</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span> <span class="o">=</span> <span class="n">value</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">add_in_place</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        X.add_in_place(mv) increments multivector X by multivector</span>
-<span class="sd">        mv.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">bladeflg</span> <span class="ow">or</span> <span class="n">mv</span><span class="o">.</span><span class="n">bladeflg</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-            <span class="n">mv</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+=</span> <span class="n">mv</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="o">+</span><span class="n">mv</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">sub_in_place</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        X.sub_in_place(mv) decrements multivector X by multivector</span>
-<span class="sd">        mv.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">bladeflg</span> <span class="ow">or</span> <span class="n">mv</span><span class="o">.</span><span class="n">bladeflg</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-            <span class="n">mv</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-=</span> <span class="n">mv</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="o">+</span><span class="n">mv</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">__pos__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-        <span class="n">p</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">bladeflg</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="n">p</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="o">+</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-        <span class="n">n</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">bladeflg</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="n">n</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__add_ab__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">add_in_place</span><span class="p">(</span><span class="n">mv</span><span class="p">)</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">__sub_ab__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">sub_in_place</span><span class="p">(</span><span class="n">mv</span><span class="p">)</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See MV.addition(self,mv)&quot;&quot;&quot;</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">addition</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__radd__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See MV.addition(mv,self)&quot;&quot;&quot;</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">addition</span><span class="p">(</span><span class="n">mv</span><span class="p">,</span> <span class="bp">self</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See MV.subtraction(self,mv)&quot;&quot;&quot;</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">subtraction</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__rsub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See MV.subtraction(mv,self)&quot;&quot;&quot;</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">subtraction</span><span class="p">(</span><span class="n">mv</span><span class="p">,</span> <span class="bp">self</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__xor__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See MV.outer_product(self,mv)&quot;&quot;&quot;</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">outer_product</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__pow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See MV.outer_product(self,mv)&quot;&quot;&quot;</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">outer_product</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__rxor__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See MV.outer_product(mv,self)&quot;&quot;&quot;</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">outer_product</span><span class="p">(</span><span class="n">mv</span><span class="p">,</span> <span class="bp">self</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__or__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See MV.inner_product(self,mv)&quot;&quot;&quot;</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">inner_product</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__ror__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See MV.inner_product(mv,self)&quot;&quot;&quot;</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">inner_product</span><span class="p">(</span><span class="n">mv</span><span class="p">,</span> <span class="bp">self</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See MV.inner_product(self,mv)&quot;&quot;&quot;</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">inner_product</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">,</span> <span class="s">&#39;l&#39;</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__lshift__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See MV.inner_product(self,mv)&quot;&quot;&quot;</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">inner_product</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">,</span> <span class="s">&#39;l&#39;</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__rlshift__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See MV.inner_product(self,mv)&quot;&quot;&quot;</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">inner_product</span><span class="p">(</span><span class="n">mv</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="s">&#39;l&#39;</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">lc</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">inner_product</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">,</span> <span class="s">&#39;l&#39;</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__gt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See MV.inner_product(self,mv)&quot;&quot;&quot;</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">inner_product</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">,</span> <span class="s">&#39;r&#39;</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__rshift__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See MV.inner_product(self,mv)&quot;&quot;&quot;</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">inner_product</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">,</span> <span class="s">&#39;r&#39;</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__rrshift__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See MV.inner_product(self,mv)&quot;&quot;&quot;</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">inner_product</span><span class="p">(</span><span class="n">mv</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="s">&#39;r&#39;</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">rc</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">inner_product</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">,</span> <span class="s">&#39;r&#39;</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">scalar_mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Y = X.scalar_mul(c), multiply multivector X by scalar c and return</span>
-<span class="sd">        result.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">mv</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-        <span class="n">mv</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">bladeflg</span>
-        <span class="n">mv</span><span class="o">.</span><span class="n">puregrade</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">puregrade</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="c">#print self.mv[i]</span>
-                <span class="c">#print c,type(c)</span>
-                <span class="n">mv</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">c</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">mv</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">scalar_mul_inplace</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        X.scalar_mul_inplace(c), multiply multivector X by scalar c and save</span>
-<span class="sd">        result in X.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">c</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See MV.geometric_product(self,mv)&quot;&quot;&quot;</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">geometric_product</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;See MV.geometric_product(mv,self)&quot;&quot;&quot;</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">geometric_product</span><span class="p">(</span><span class="n">mv</span><span class="p">,</span> <span class="bp">self</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">scalar</span><span class="p">):</span>
-        <span class="n">div</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-        <span class="n">div</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">bladeflg</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="n">div</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">/</span><span class="n">scalar</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">div</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__div_ab__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">scalar</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">/=</span> <span class="n">scalar</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">igrade</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">ibase</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        X(i,j) returns symbol in ith grade and jth base or blade of</span>
-<span class="sd">        multivector X.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-            <span class="k">return</span><span class="p">(</span><span class="n">ZERO</span><span class="p">)</span>
-        <span class="k">return</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">])</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">equal</span><span class="p">(</span><span class="n">mv1</span><span class="p">,</span> <span class="n">mv2</span><span class="p">):</span>
-        <span class="n">mv1</span><span class="o">.</span><span class="n">compact</span><span class="p">()</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv2</span><span class="p">,</span> <span class="n">MV</span><span class="p">):</span>
-            <span class="n">mv2</span><span class="o">.</span><span class="n">compact</span><span class="p">()</span>
-        <span class="n">pure_grade</span> <span class="o">=</span> <span class="n">mv1</span><span class="o">.</span><span class="n">is_pure</span><span class="p">()</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv2</span><span class="p">,</span> <span class="n">MV</span><span class="p">)</span> <span class="ow">and</span> <span class="n">pure_grade</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span><span class="p">(</span><span class="bp">False</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv2</span><span class="p">,</span> <span class="n">MV</span><span class="p">)</span> <span class="ow">and</span> <span class="n">pure_grade</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv1</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="nb">int</span><span class="p">):</span>
-                <span class="k">return</span><span class="p">(</span><span class="n">mv2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span><span class="p">(</span><span class="n">mv1</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">mv2</span><span class="p">)</span>
-        <span class="k">for</span> <span class="p">(</span><span class="n">mvi</span><span class="p">,</span> <span class="n">mvj</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">mv1</span><span class="o">.</span><span class="n">mv</span><span class="p">,</span> <span class="n">mv2</span><span class="o">.</span><span class="n">mv</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">isint</span><span class="p">(</span><span class="n">mvi</span><span class="p">)</span> <span class="o">^</span> <span class="n">isint</span><span class="p">(</span><span class="n">mvj</span><span class="p">):</span>
-                <span class="k">return</span><span class="p">(</span><span class="bp">False</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mvi</span><span class="p">,</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mvj</span><span class="p">,</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="k">for</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">mvi</span><span class="p">,</span> <span class="n">mvj</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">x</span> <span class="o">!=</span> <span class="n">y</span><span class="p">:</span>
-                        <span class="k">return</span><span class="p">(</span><span class="bp">False</span><span class="p">)</span>
-        <span class="k">return</span><span class="p">(</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">equal</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">copy</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sub</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Y = X.copy(), make a deep copy of multivector X in multivector</span>
-<span class="sd">        Y so that Y can be modified without affecting X.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">cpy</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-        <span class="n">cpy</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">name</span>
-        <span class="n">cpy</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">bladeflg</span>
-        <span class="n">cpy</span><span class="o">.</span><span class="n">puregrade</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">puregrade</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">sub</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                    <span class="n">cpy</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                    <span class="n">cpy</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="o">+</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">cpy</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">substitute_base</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">igrade</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="n">mv</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-            <span class="k">return</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-            <span class="n">ibase</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">basis</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">ibase</span> <span class="o">=</span> <span class="n">base</span>
-        <span class="n">coef</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">coef</span> <span class="o">==</span> <span class="n">ZERO</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span> <span class="o">=</span> <span class="n">ZERO</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">add_in_place</span><span class="p">(</span><span class="n">mv</span><span class="o">*</span><span class="n">coef</span><span class="p">)</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">convert_to_blades</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        X.convert_to_blades(), inplace convert base representation</span>
-<span class="sd">        to blade representation.  See reference 5 section 5.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">bladeflg</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">for</span> <span class="n">igrade</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">ibase</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">igrade</span><span class="p">]):</span>
-                    <span class="n">coef</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span>
-                    <span class="k">if</span> <span class="p">(</span><span class="ow">not</span> <span class="n">coef</span> <span class="o">==</span> <span class="n">ZERO</span><span class="p">):</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span> <span class="o">=</span> <span class="n">ZERO</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">add_in_place</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">ibtable</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span><span class="o">*</span><span class="n">coef</span><span class="p">)</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">convert_from_blades</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        X.convert_from_blades(), inplace convert blade representation</span>
-<span class="sd">        to base representation.  See reference 5 section 5.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">bladeflg</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">igrade</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">ibase</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">igrade</span><span class="p">]):</span>
-                    <span class="n">coef</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span>
-                    <span class="k">if</span> <span class="p">(</span><span class="ow">not</span> <span class="n">coef</span> <span class="o">==</span> <span class="n">ZERO</span><span class="p">):</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span> <span class="o">=</span> <span class="n">ZERO</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">add_in_place</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">btable</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span><span class="o">*</span><span class="n">coef</span><span class="p">)</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">project</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">r</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Grade projection operator. For multivector X, X.project(r)</span>
-<span class="sd">        returns multivector of grade r components of X if r is an</span>
-<span class="sd">        integer. If r is a multivector X.project(r) returns a</span>
-<span class="sd">        multivector consisting of the grade of X for which r has non-</span>
-<span class="sd">        zero grades. For example if X is a general multivector and</span>
-<span class="sd">        r is a general spinor then X.project(r) will return the even</span>
-<span class="sd">        grades of X.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="n">grade_r</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">r</span> <span class="o">&gt;</span> <span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">:</span>
-                <span class="k">return</span><span class="p">(</span><span class="n">grade_r</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">r</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="k">return</span><span class="p">(</span><span class="n">grade_r</span><span class="p">)</span>
-            <span class="n">grade_r</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="n">grade_r</span><span class="o">.</span><span class="n">puregrade</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="n">grade_r</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">r</span><span class="p">]</span> <span class="o">=</span> <span class="o">+</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">r</span><span class="p">]</span>
-            <span class="k">return</span><span class="p">(</span><span class="n">grade_r</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">MV</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-            <span class="n">r</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-            <span class="n">proj</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">r</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="nb">int</span><span class="p">):</span>
-                    <span class="n">proj</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="n">proj</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">bladeflg</span>
-            <span class="k">return</span><span class="p">(</span><span class="n">proj</span><span class="p">)</span>
-        <span class="k">return</span><span class="p">(</span><span class="bp">None</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">even</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Even grade projection operator. For multivector X, X.even()</span>
-<span class="sd">        returns multivector of even grade components of X.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">egrades</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-        <span class="n">egrades</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">bladeflg</span>
-        <span class="n">egrades</span><span class="o">.</span><span class="n">puregrade</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">puregrade</span>
-        <span class="k">for</span> <span class="n">igrade</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1</span><span class="p">,</span> <span class="mi">2</span><span class="p">):</span>
-            <span class="n">egrades</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span> <span class="o">=</span> <span class="o">+</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">egrades</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">odd</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Odd grade projection operator. For multivector X, X.odd()</span>
-<span class="sd">        returns multivector of odd grade components of X.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">ogrades</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-        <span class="n">ogrades</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">bladeflg</span>
-        <span class="n">ogrades</span><span class="o">.</span><span class="n">puregrade</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">puregrade</span>
-        <span class="k">for</span> <span class="n">igrade</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1</span><span class="p">,</span> <span class="mi">2</span><span class="p">):</span>
-            <span class="n">ogrades</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span> <span class="o">=</span> <span class="o">+</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">ogrades</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">rev</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Revision operator. For multivector X, X.rev()</span>
-<span class="sd">        returns reversed multivector of X.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">revmv</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-        <span class="n">revmv</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">bladeflg</span>
-        <span class="n">revmv</span><span class="o">.</span><span class="n">puregrade</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">puregrade</span>
-        <span class="k">for</span> <span class="n">igrade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">igrade</span> <span class="o">&lt;</span> <span class="mi">2</span> <span class="ow">or</span> <span class="p">(</span><span class="ow">not</span> <span class="p">(((</span><span class="n">igrade</span><span class="o">*</span><span class="p">(</span><span class="n">igrade</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span><span class="p">)):</span>
-                    <span class="n">revmv</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span> <span class="o">=</span> <span class="o">+</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">revmv</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">revmv</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">cse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">grade</span><span class="p">):</span>
-        <span class="n">cse_lst</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">grade</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">ibase</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">grade</span><span class="p">]):</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">grade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span> <span class="o">!=</span> <span class="n">ZERO</span><span class="p">:</span>
-                    <span class="n">cse_lst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sympy</span><span class="o">.</span><span class="n">cse</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">grade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]))</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">cse_lst</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">div</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">grade</span><span class="p">,</span> <span class="n">divisor</span><span class="p">):</span>
-        <span class="n">div_lst</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">grade</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">ibase</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">grade</span><span class="p">]):</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">grade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span> <span class="o">!=</span> <span class="n">ZERO</span><span class="p">:</span>
-                    <span class="n">div_lst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">grade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">divisor</span><span class="p">))</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">div_lst</span><span class="p">)</span>
-
-    <span class="c"># don&#39;t know which one is correctly named</span>
-
-    <span class="k">def</span> <span class="nf">div</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">grad_int</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">collect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">lst</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Applies sympy collect function to each component</span>
-<span class="sd">        of multivector.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">igrade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">ibase</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">igrade</span><span class="p">]):</span>
-                    <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span> <span class="o">!=</span> <span class="n">ZERO</span><span class="p">:</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span> <span class="o">=</span> \
-                            <span class="n">sympy</span><span class="o">.</span><span class="n">collect</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">],</span> <span class="n">lst</span><span class="p">)</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">sqrfree</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">lst</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Applies sympy sqrfree function to each component</span>
-<span class="sd">        of multivector.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">igrade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">ibase</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">igrade</span><span class="p">]):</span>
-                    <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span> <span class="o">!=</span> <span class="n">ZERO</span><span class="p">:</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span> <span class="o">=</span> \
-                            <span class="n">sympy</span><span class="o">.</span><span class="n">sqrfree</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">],</span> <span class="n">lst</span><span class="p">)</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">flatten</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">flst</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">igrade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">],</span> <span class="nb">int</span><span class="p">):</span>
-                <span class="n">flst</span> <span class="o">+=</span> <span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span><span class="o">*</span><span class="p">[</span><span class="n">ZERO</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">coef</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]:</span>
-                    <span class="n">flst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">coef</span><span class="p">)</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">flst</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">X</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-        <span class="n">X</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">bladeflg</span>
-        <span class="n">X</span><span class="o">.</span><span class="n">puregrade</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">puregrade</span>
-        <span class="k">for</span> <span class="n">igrade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="n">X</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
-                    <span class="n">substitute_array</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">],</span> <span class="o">*</span><span class="n">args</span><span class="p">))</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">sub_mv</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mv1</span><span class="p">,</span> <span class="n">mv2</span><span class="p">):</span>
-        <span class="n">mv1_flat</span> <span class="o">=</span> <span class="n">mv1</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span>
-        <span class="n">mv2_flat</span> <span class="o">=</span> <span class="n">mv2</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">sub_scalar</span><span class="p">(</span><span class="n">mv1_flat</span><span class="p">,</span> <span class="n">mv2_flat</span><span class="p">)</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">sub_scalar</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr1</span><span class="p">,</span> <span class="n">expr2</span><span class="p">):</span>
-        <span class="k">if</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">expr1</span><span class="p">,</span> <span class="nb">list</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr2</span><span class="p">,</span> <span class="nb">list</span><span class="p">))</span> <span class="ow">or</span> \
-           <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">expr1</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr2</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">)):</span>
-            <span class="k">for</span> <span class="p">(</span><span class="n">var1</span><span class="p">,</span> <span class="n">var2</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">expr1</span><span class="p">,</span> <span class="n">expr2</span><span class="p">):</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">sub_scalar</span><span class="p">(</span><span class="n">var1</span><span class="p">,</span> <span class="n">var2</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">igrade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">],</span> <span class="nb">int</span><span class="p">):</span>
-                    <span class="k">for</span> <span class="n">ibase</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">igrade</span><span class="p">]):</span>
-                        <span class="k">if</span> <span class="n">expr1</span> <span class="o">!=</span> <span class="n">ZERO</span><span class="p">:</span>
-                            <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span>
-                                <span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">expr1</span><span class="p">,</span> <span class="n">expr2</span><span class="p">)</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Applies sympy simplify function</span>
-<span class="sd">        to each component of multivector.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">igrade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">ibase</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">igrade</span><span class="p">]):</span>
-                    <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span> <span class="o">!=</span> <span class="n">ZERO</span><span class="p">:</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span> <span class="o">=</span> \
-                            <span class="n">sympy</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">])</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">trigsimp</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Applies sympy trigsimp function</span>
-<span class="sd">        to each component of multivector</span>
-<span class="sd">        using the recursive option.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">igrade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">ibase</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">igrade</span><span class="p">]):</span>
-                    <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span> <span class="o">!=</span> <span class="n">ZERO</span><span class="p">:</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span> <span class="o">=</span> \
-                            <span class="n">sympy</span><span class="o">.</span><span class="n">trigsimp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span>
-                                           <span class="p">[</span><span class="n">ibase</span><span class="p">],</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">recursive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">cancel</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Applies sympy cancel function</span>
-<span class="sd">        to each component of multivector.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">igrade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">ibase</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">igrade</span><span class="p">]):</span>
-                    <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span> <span class="o">!=</span> <span class="n">ZERO</span><span class="p">:</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span> <span class="o">=</span> \
-                            <span class="n">sympy</span><span class="o">.</span><span class="n">cancel</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">])</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">expand</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Applies sympy expand function to each component</span>
-<span class="sd">        of multivector.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">igrade</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">ibase</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">nbasis</span><span class="p">[</span><span class="n">igrade</span><span class="p">]):</span>
-                    <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span> <span class="o">!=</span> <span class="n">ZERO</span><span class="p">:</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span> <span class="o">=</span> \
-                            <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">is_pure</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">igrade</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-        <span class="n">ngrade</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">compact</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                    <span class="k">if</span> <span class="n">base</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">igrade</span> <span class="o">=</span> <span class="n">i</span>
-                        <span class="n">ngrade</span> <span class="o">+=</span> <span class="mi">1</span>
-                        <span class="k">break</span>
-                <span class="k">if</span> <span class="n">ngrade</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">igrade</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">igrade</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">is_scalar</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_pure</span><span class="p">()</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span><span class="p">(</span><span class="bp">True</span><span class="p">)</span>
-        <span class="k">return</span><span class="p">(</span><span class="bp">False</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">compact</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Convert zero numpy arrays to single integer zero place holder</span>
-<span class="sd">        in grade list for instantiated multivector. For example if</span>
-<span class="sd">        numpy array of grade one components is a zero array then replace</span>
-<span class="sd">        with single integer equal to zero.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">isint</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]):</span>
-                <span class="n">zero_flg</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                    <span class="k">if</span> <span class="n">x</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">zero_flg</span> <span class="o">=</span> <span class="bp">False</span>
-                        <span class="k">break</span>
-                <span class="k">if</span> <span class="n">zero_flg</span><span class="p">:</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">diff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Calculate partial derivative of multivector with respect to</span>
-<span class="sd">        argument x.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-        <span class="n">igrade</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">grade</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">grade</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-                <span class="n">D</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">vdiff</span><span class="p">(</span><span class="n">grade</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-            <span class="n">igrade</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">D</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">ddt</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">MV</span><span class="o">.</span><span class="n">coords</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">):</span>
-            <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="p">())</span>
-        <span class="n">dxdt</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">coords</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">for</span> <span class="n">ibase</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">nrg</span><span class="p">:</span>
-            <span class="n">dxdt</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="n">ibase</span><span class="p">]</span><span class="o">*</span><span class="n">MV</span><span class="o">.</span><span class="n">dedt</span><span class="p">[</span><span class="n">ibase</span><span class="p">]</span>
-        <span class="c">#dxdt.simplify()</span>
-        <span class="c">#dxdt.trigsimp()</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">dxdt</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">grad</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Calculate geometric (grad) derivative of multivector function</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">dD</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">theta</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">coords</span><span class="p">:</span>
-            <span class="n">D</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">theta</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">MV</span><span class="o">.</span><span class="n">curvilinear_flg</span><span class="p">:</span>
-            <span class="n">recp</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">Rframe</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">recp</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">brecp</span>
-        <span class="k">for</span> <span class="p">(</span><span class="n">rbase</span><span class="p">,</span> <span class="n">iD</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">recp</span><span class="p">,</span> <span class="n">D</span><span class="p">):</span>
-            <span class="n">dD</span><span class="o">.</span><span class="n">add_in_place</span><span class="p">(</span><span class="n">rbase</span><span class="o">*</span><span class="n">iD</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">MV</span><span class="o">.</span><span class="n">curvilinear_flg</span><span class="p">:</span>  <span class="c"># Add Connection</span>
-            <span class="n">igrade</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="k">while</span> <span class="n">igrade</span> <span class="o">&lt;=</span> <span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">:</span>
-                <span class="n">coefs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span>
-                <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">coefs</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">int</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="p">(</span><span class="n">coef</span><span class="p">,</span> <span class="n">connect</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">coefs</span><span class="p">,</span> <span class="n">MV</span><span class="o">.</span><span class="n">Connect</span><span class="p">[</span><span class="n">igrade</span><span class="p">]):</span>
-                        <span class="n">dD</span><span class="o">.</span><span class="n">add_in_place</span><span class="p">(</span><span class="n">coef</span><span class="o">*</span><span class="n">connect</span><span class="p">)</span>
-                <span class="n">igrade</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">dD</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">grad_ext</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Calculate outer (exterior,curl) derivative of multivector function.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">dD</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">ix</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">coords</span><span class="p">:</span>
-            <span class="n">D</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">ix</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">MV</span><span class="o">.</span><span class="n">curvilinear_flg</span><span class="p">:</span>
-            <span class="n">recp</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">Rframe</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">recp</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">brecp</span>
-        <span class="k">for</span> <span class="p">(</span><span class="n">irbase</span><span class="p">,</span> <span class="n">iD</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">recp</span><span class="p">,</span> <span class="n">D</span><span class="p">):</span>
-            <span class="n">dD</span><span class="o">.</span><span class="n">add_in_place</span><span class="p">(</span><span class="n">irbase</span> <span class="o">^</span> <span class="n">iD</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">MV</span><span class="o">.</span><span class="n">curvilinear_flg</span><span class="p">:</span>  <span class="c"># Add Connection</span>
-            <span class="n">igrade</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="k">while</span> <span class="n">igrade</span> <span class="o">&lt;=</span> <span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">:</span>
-                <span class="n">coefs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span>
-                <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">coefs</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">int</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="p">(</span><span class="n">coef</span><span class="p">,</span> <span class="n">connect</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">coefs</span><span class="p">,</span> <span class="n">MV</span><span class="o">.</span><span class="n">Connect</span><span class="p">[</span><span class="n">igrade</span><span class="p">]):</span>
-                        <span class="k">if</span> <span class="n">igrade</span> <span class="o">&lt;</span> <span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">:</span>
-                            <span class="n">dD</span><span class="o">.</span><span class="n">add_in_place</span><span class="p">(</span><span class="n">coef</span><span class="o">*</span><span class="n">connect</span><span class="o">.</span><span class="n">project</span><span class="p">(</span><span class="n">igrade</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-                <span class="n">igrade</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">dD</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">curl</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">grad_ext</span><span class="p">())</span>
-
-    <span class="k">def</span> <span class="nf">grad_int</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Calculate inner (interior,div) derivative of multivector function.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">dD</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">ix</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">coords</span><span class="p">:</span>
-            <span class="n">D</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">ix</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">MV</span><span class="o">.</span><span class="n">curvilinear_flg</span><span class="p">:</span>
-            <span class="n">recp</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">Rframe</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">recp</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">brecp</span>
-        <span class="k">for</span> <span class="p">(</span><span class="n">irbase</span><span class="p">,</span> <span class="n">iD</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">recp</span><span class="p">,</span> <span class="n">D</span><span class="p">):</span>
-            <span class="n">dD</span><span class="o">.</span><span class="n">add_in_place</span><span class="p">(</span><span class="n">irbase</span> <span class="o">|</span> <span class="n">iD</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">MV</span><span class="o">.</span><span class="n">curvilinear_flg</span><span class="p">:</span>  <span class="c"># Add Connection</span>
-            <span class="n">igrade</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="k">while</span> <span class="n">igrade</span> <span class="o">&lt;=</span> <span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">:</span>
-                <span class="n">coefs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade</span><span class="p">]</span>
-                <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">coefs</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">int</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="p">(</span><span class="n">coef</span><span class="p">,</span> <span class="n">connect</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">coefs</span><span class="p">,</span> <span class="n">MV</span><span class="o">.</span><span class="n">Connect</span><span class="p">[</span><span class="n">igrade</span><span class="p">]):</span>
-                        <span class="n">dD</span><span class="o">.</span><span class="n">add_in_place</span><span class="p">(</span><span class="n">coef</span><span class="o">*</span><span class="n">connect</span><span class="o">.</span><span class="n">project</span><span class="p">(</span><span class="n">igrade</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-                <span class="n">igrade</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">dD</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">mag2</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Calculate scalar component of square of multivector.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span><span class="p">((</span><span class="bp">self</span> <span class="o">|</span> <span class="bp">self</span><span class="p">)())</span>
-
-    <span class="k">def</span> <span class="nf">Name</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Get LaTeX name of multivector.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">latex_ex</span><span class="o">.</span><span class="n">LatexPrinter</span><span class="o">.</span><span class="n">extended_symbol</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">))</span>
-
-
-<div class="viewcode-block" id="set_names"><a class="viewcode-back" href="../../../modules/galgebra/GA.html#sympy.galgebra.GA.set_names">[docs]</a><span class="k">def</span> <span class="nf">set_names</span><span class="p">(</span><span class="n">var_lst</span><span class="p">,</span> <span class="n">var_str</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Set the names of a list of multivectors (var_lst) for a space delimited</span>
-<span class="sd">    string (var_str) containing the names.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">var_str_lst</span> <span class="o">=</span> <span class="n">var_str</span><span class="o">.</span><span class="n">split</span><span class="p">()</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">var_lst</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">var_str_lst</span><span class="p">):</span>
-        <span class="k">for</span> <span class="p">(</span><span class="n">var</span><span class="p">,</span> <span class="n">var_name</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">var_lst</span><span class="p">,</span> <span class="n">var_str_lst</span><span class="p">):</span>
-            <span class="n">var</span><span class="o">.</span><span class="n">set_name</span><span class="p">(</span><span class="n">var_name</span><span class="p">)</span>
-        <span class="k">return</span>
-    <span class="n">sys</span><span class="o">.</span><span class="n">stderr</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&#39;Error in set_names. Lists incongruent!</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">)</span>
-    <span class="n">sys</span><span class="o">.</span><span class="n">exit</span><span class="p">()</span>
-    <span class="k">return</span>
-
-</div>
-<div class="viewcode-block" id="reciprocal_frame"><a class="viewcode-back" href="../../../modules/galgebra/GA.html#sympy.galgebra.GA.reciprocal_frame">[docs]</a><span class="k">def</span> <span class="nf">reciprocal_frame</span><span class="p">(</span><span class="n">vlst</span><span class="p">,</span> <span class="n">names</span><span class="o">=</span><span class="s">&#39;&#39;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Calculate reciprocal frame of list (vlst) of vectors.  If desired name each</span>
-<span class="sd">    vector in list of reciprocal vectors with names in space delimited string</span>
-<span class="sd">    (names).</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">E</span> <span class="o">=</span> <span class="n">vlst</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">recp</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">names</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">str</span><span class="p">:</span>
-        <span class="n">name_lst</span> <span class="o">=</span> <span class="n">names</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">names</span> <span class="o">!=</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-            <span class="n">name_lst</span> <span class="o">=</span> <span class="n">names</span><span class="o">.</span><span class="n">split</span><span class="p">()</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">):</span>
-        <span class="n">E</span> <span class="o">=</span> <span class="n">E</span> <span class="o">^</span> <span class="n">vlst</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">):</span>
-        <span class="n">tmp</span> <span class="o">=</span> <span class="n">ONE</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">tmp</span> <span class="o">=</span> <span class="o">-</span><span class="n">ONE</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="n">j</span><span class="p">:</span>
-                <span class="n">tmp</span> <span class="o">=</span> <span class="n">tmp</span> <span class="o">^</span> <span class="n">vlst</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-        <span class="n">tmp</span> <span class="o">=</span> <span class="n">tmp</span><span class="o">*</span><span class="n">E</span>
-        <span class="n">recp</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tmp</span><span class="p">)</span>
-    <span class="n">Esq</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">trigsimp</span><span class="p">(</span><span class="n">E</span><span class="o">.</span><span class="n">mag2</span><span class="p">(),</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">recursive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="n">Esq</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="n">sympy</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span><span class="n">Esq</span><span class="p">))</span>
-    <span class="n">Esq_inv</span> <span class="o">=</span> <span class="n">ONE</span><span class="o">/</span><span class="n">Esq</span>
-    <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">):</span>
-        <span class="n">recp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">trigsimp</span><span class="p">()</span>
-        <span class="n">recp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">recp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">Esq_inv</span>
-        <span class="k">if</span> <span class="n">names</span> <span class="o">!=</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-            <span class="n">recp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set_name</span><span class="p">(</span><span class="n">name_lst</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-    <span class="k">return</span><span class="p">(</span><span class="n">recp</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">S</span><span class="p">(</span><span class="n">value</span><span class="p">):</span>
-    <span class="k">return</span><span class="p">(</span><span class="n">MV</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="s">&#39;scalar&#39;</span><span class="p">))</span>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
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-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
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-  <h3>Quick search</h3>
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diff --git a/dev-py3k/_modules/sympy/galgebra/latex_ex.html b/dev-py3k/_modules/sympy/galgebra/latex_ex.html
deleted file mode 100644
index 3101fecb352..00000000000
--- a/dev-py3k/_modules/sympy/galgebra/latex_ex.html
+++ /dev/null
@@ -1,1410 +0,0 @@
-
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-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
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-    
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-            
-  <h1>Source code for sympy.galgebra.latex_ex</h1><div class="highlight"><pre>
-<span class="c">#latex_ex.py</span>
-
-
-
-<span class="kn">import</span> <span class="nn">sys</span>
-<span class="c">#if sys.version.find(&#39;Stackless&#39;) &gt;= 0:</span>
-<span class="c">#    sys.path.append(&#39;/usr/lib/python2.5/site-packages&#39;)</span>
-
-<span class="kn">import</span> <span class="nn">os</span>
-<span class="kn">import</span> <span class="nn">types</span>
-<span class="kn">import</span> <span class="nn">io</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">_coeff_isneg</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.printer</span> <span class="kn">import</span> <span class="n">Printer</span>
-<span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">fraction</span>
-<span class="kn">import</span> <span class="nn">re</span> <span class="kn">as</span> <span class="nn">regrep</span>
-
-<span class="kn">import</span> <span class="nn">sympy.galgebra.GA</span>
-<span class="c">#import sympy.galgebra.OGA</span>
-<span class="kn">import</span> <span class="nn">numpy</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">cmp_to_key</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">default_sort_key</span>
-
-<span class="kn">from</span> <span class="nn">sympy.printing.latex</span> <span class="kn">import</span> <span class="n">accepted_latex_functions</span>
-
-
-<span class="k">def</span> <span class="nf">debug</span><span class="p">(</span><span class="n">txt</span><span class="p">):</span>
-    <span class="n">sys</span><span class="o">.</span><span class="n">stderr</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="n">txt</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">)</span>
-    <span class="k">return</span>
-
-
-<div class="viewcode-block" id="find_executable"><a class="viewcode-back" href="../../../modules/galgebra/latex_ex.html#sympy.galgebra.latex_ex.find_executable">[docs]</a><span class="k">def</span> <span class="nf">find_executable</span><span class="p">(</span><span class="n">executable</span><span class="p">,</span> <span class="n">path</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Try to find &#39;executable&#39; in the directories listed in &#39;path&#39; (a</span>
-<span class="sd">    string listing directories separated by &#39;os.pathsep&#39;; defaults to</span>
-<span class="sd">    os.environ[&#39;PATH&#39;]).  Returns the complete filename or None if not</span>
-<span class="sd">    found</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">path</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">path</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">environ</span><span class="p">[</span><span class="s">&#39;PATH&#39;</span><span class="p">]</span>
-    <span class="n">paths</span> <span class="o">=</span> <span class="n">path</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">pathsep</span><span class="p">)</span>
-    <span class="n">extlist</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;&#39;</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">os</span><span class="o">.</span><span class="n">name</span> <span class="o">==</span> <span class="s">&#39;os2&#39;</span><span class="p">:</span>
-        <span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">ext</span><span class="p">)</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">splitext</span><span class="p">(</span><span class="n">executable</span><span class="p">)</span>
-        <span class="c"># executable files on OS/2 can have an arbitrary extension, but</span>
-        <span class="c"># .exe is automatically appended if no dot is present in the name</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">ext</span><span class="p">:</span>
-            <span class="n">executable</span> <span class="o">=</span> <span class="n">executable</span> <span class="o">+</span> <span class="s">&quot;.exe&quot;</span>
-    <span class="k">elif</span> <span class="n">sys</span><span class="o">.</span><span class="n">platform</span> <span class="o">==</span> <span class="s">&#39;win32&#39;</span><span class="p">:</span>
-        <span class="n">pathext</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">environ</span><span class="p">[</span><span class="s">&#39;PATHEXT&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">pathsep</span><span class="p">)</span>
-        <span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">ext</span><span class="p">)</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">splitext</span><span class="p">(</span><span class="n">executable</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">ext</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">pathext</span><span class="p">:</span>
-            <span class="n">extlist</span> <span class="o">=</span> <span class="n">pathext</span>
-    <span class="k">for</span> <span class="n">ext</span> <span class="ow">in</span> <span class="n">extlist</span><span class="p">:</span>
-        <span class="n">execname</span> <span class="o">=</span> <span class="n">executable</span> <span class="o">+</span> <span class="n">ext</span>
-        <span class="k">if</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">isfile</span><span class="p">(</span><span class="n">execname</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">execname</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">paths</span><span class="p">:</span>
-                <span class="n">f</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">execname</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">isfile</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">f</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-</div>
-<span class="k">def</span> <span class="nf">len_cmp</span><span class="p">(</span><span class="n">str1</span><span class="p">,</span> <span class="n">str2</span><span class="p">):</span>
-    <span class="k">return</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">str2</span><span class="p">)</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">str1</span><span class="p">))</span>
-
-
-<span class="k">def</span> <span class="nf">process_equals</span><span class="p">(</span><span class="n">xstr</span><span class="p">):</span>
-    <span class="n">eq1</span> <span class="o">=</span> <span class="n">xstr</span><span class="o">.</span><span class="n">find</span><span class="p">(</span><span class="s">&#39;=&#39;</span><span class="p">)</span>
-    <span class="n">eq2</span> <span class="o">=</span> <span class="n">xstr</span><span class="o">.</span><span class="n">rfind</span><span class="p">(</span><span class="s">&#39;=&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">eq1</span> <span class="o">==</span> <span class="n">eq2</span><span class="p">:</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">xstr</span><span class="p">)</span>
-    <span class="n">xstr</span> <span class="o">=</span> <span class="n">xstr</span><span class="p">[:</span><span class="n">eq1</span><span class="p">]</span> <span class="o">+</span> <span class="n">xstr</span><span class="p">[</span><span class="n">eq2</span><span class="p">:]</span>
-    <span class="k">return</span><span class="p">(</span><span class="n">xstr</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="LatexPrinter"><a class="viewcode-back" href="../../../modules/galgebra/latex_ex.html#sympy.galgebra.latex_ex.LatexPrinter">[docs]</a><span class="k">class</span> <span class="nc">LatexPrinter</span><span class="p">(</span><span class="n">Printer</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    A printer class which converts an expression into its LaTeX equivalent.</span>
-<span class="sd">    This class extends the LatexPrinter class currently in sympy in the</span>
-<span class="sd">    following ways:</span>
-
-<span class="sd">        1.  Variable and function names can now encode multiple Greek symbols,</span>
-<span class="sd">            number, Greek, and roman super and subscripts and accents plus bold</span>
-<span class="sd">            math in an alphanumeric ASCII string consisting of ``[A-Za-z0-9_]``</span>
-<span class="sd">            symbols</span>
-
-<span class="sd">            a)  Accents and bold math are implemented in reverse notation. For</span>
-<span class="sd">                example if you wished the LaTeX output to be ``\bm{\hat{\sigma}}``</span>
-<span class="sd">                you would give the variable the name sigmahatbm.</span>
-
-<span class="sd">            b)  Subscripts are denoted by a single underscore and superscripts</span>
-<span class="sd">                by a double underscore so that ``A_{\rho\beta}^{25}`` would be</span>
-<span class="sd">                input as A_rhobeta__25.</span>
-
-<span class="sd">        2.  Some standard function names have been improved such as asin is now</span>
-<span class="sd">            denoted by sin^{-1} and log by ln.</span>
-
-<span class="sd">        3.  Several LaTeX formats for multivectors are available:</span>
-
-<span class="sd">            a)  Print multivector on one line</span>
-
-<span class="sd">            b)  Print each grade of multivector on one line</span>
-
-<span class="sd">            c)  Print each base of multivector on one line</span>
-
-<span class="sd">        4.  A LaTeX output for numpy arrays containing sympy expressions is</span>
-<span class="sd">            implemented for up to a three dimensional array.</span>
-
-<span class="sd">        5.  LaTeX formatting for raw LaTeX, eqnarray, and array is available</span>
-<span class="sd">            in simple output strings.</span>
-
-<span class="sd">            a)  The delimiter for raw LaTeX input is &#39;%&#39;.  The raw input starts</span>
-<span class="sd">                on the line where &#39;%&#39; is first encountered and continues until</span>
-<span class="sd">                the next line where &#39;%&#39; is encountered. It does not matter where</span>
-<span class="sd">                &#39;%&#39; is in the line.</span>
-
-<span class="sd">            b)  The delimiter for eqnarray input is &#39;@&#39;. The rules are the same</span>
-<span class="sd">                as for raw input except that &#39;=&#39; in the first line is replaced</span>
-<span class="sd">                be &#39;&amp;=&amp;&#39; and &#39;\begin{eqnarray*}&#39; is added before the first line</span>
-<span class="sd">                and &#39;\end{eqnarray*}&#39; to after the last line in the group of</span>
-<span class="sd">                lines.</span>
-
-<span class="sd">            c)  The delimiter for array input is &#39;#&#39;. The rules are the same</span>
-<span class="sd">                as for raw input except that &#39;\begin{equation*}&#39; is added before</span>
-<span class="sd">                the first line and &#39;\end{equation*}&#39; to after the last line in</span>
-<span class="sd">                the group of lines.</span>
-
-<span class="sd">        6.  Additional formats for partial derivatives:</span>
-
-<span class="sd">            a)  Same as sympy latex module</span>
-
-<span class="sd">            b)  Use subscript notation with partial symbol to indicate which</span>
-<span class="sd">                variable the differentiation is with respect to.  Symbol is of</span>
-<span class="sd">                form \partial_{differentiation variable}</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="c">#printmethod =&#39;_latex_ex&#39;</span>
-    <span class="n">sym_fmt</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">fct_fmt</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">pdiff_fmt</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">mv_fmt</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">str_fmt</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">LaTeX_flg</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="n">mode</span> <span class="o">=</span> <span class="p">(</span><span class="s">&#39;_&#39;</span><span class="p">,</span> <span class="s">&#39;^&#39;</span><span class="p">)</span>
-
-    <span class="n">fmt_dict</span> <span class="o">=</span> <span class="p">{</span><span class="s">&#39;sym&#39;</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&#39;fct&#39;</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&#39;pdiff&#39;</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&#39;mv&#39;</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&#39;str&#39;</span><span class="p">:</span> <span class="mi">1</span><span class="p">}</span>
-
-    <span class="n">fct_dict</span> <span class="o">=</span> <span class="p">{</span><span class="s">&#39;sin&#39;</span><span class="p">:</span> <span class="s">&#39;sin&#39;</span><span class="p">,</span> <span class="s">&#39;cos&#39;</span><span class="p">:</span> <span class="s">&#39;cos&#39;</span><span class="p">,</span> <span class="s">&#39;tan&#39;</span><span class="p">:</span> <span class="s">&#39;tan&#39;</span><span class="p">,</span> <span class="s">&#39;cot&#39;</span><span class="p">:</span> <span class="s">&#39;cot&#39;</span><span class="p">,</span>
-                <span class="s">&#39;asin&#39;</span><span class="p">:</span> <span class="s">&#39;Sin^{-1}&#39;</span><span class="p">,</span> <span class="s">&#39;acos&#39;</span><span class="p">:</span> <span class="s">&#39;Cos^{-1}&#39;</span><span class="p">,</span>
-                <span class="s">&#39;atan&#39;</span><span class="p">:</span> <span class="s">&#39;Tan^{-1}&#39;</span><span class="p">,</span> <span class="s">&#39;acot&#39;</span><span class="p">:</span> <span class="s">&#39;Cot^{-1}&#39;</span><span class="p">,</span>
-                <span class="s">&#39;sinh&#39;</span><span class="p">:</span> <span class="s">&#39;sinh&#39;</span><span class="p">,</span> <span class="s">&#39;cosh&#39;</span><span class="p">:</span> <span class="s">&#39;cosh&#39;</span><span class="p">,</span> <span class="s">&#39;tanh&#39;</span><span class="p">:</span> <span class="s">&#39;tanh&#39;</span><span class="p">,</span> <span class="s">&#39;coth&#39;</span><span class="p">:</span> <span class="s">&#39;coth&#39;</span><span class="p">,</span>
-                <span class="s">&#39;asinh&#39;</span><span class="p">:</span> <span class="s">&#39;Sinh^{-1}&#39;</span><span class="p">,</span> <span class="s">&#39;acosh&#39;</span><span class="p">:</span> <span class="s">&#39;Cosh^{-1}&#39;</span><span class="p">,</span>
-                <span class="s">&#39;atanh&#39;</span><span class="p">:</span> <span class="s">&#39;Tanh^{-1}&#39;</span><span class="p">,</span> <span class="s">&#39;acoth&#39;</span><span class="p">:</span> <span class="s">&#39;Coth^{-1}&#39;</span><span class="p">,</span>
-                <span class="s">&#39;sqrt&#39;</span><span class="p">:</span> <span class="s">&#39;sqrt&#39;</span><span class="p">,</span> <span class="s">&#39;exp&#39;</span><span class="p">:</span> <span class="s">&#39;exp&#39;</span><span class="p">,</span> <span class="s">&#39;log&#39;</span><span class="p">:</span> <span class="s">&#39;ln&#39;</span><span class="p">}</span>
-
-    <span class="n">fct_dict_keys</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">fct_dict</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-
-    <span class="n">greek_keys</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span>
-        <span class="p">(</span><span class="s">&#39;alpha&#39;</span><span class="p">,</span> <span class="s">&#39;beta&#39;</span><span class="p">,</span> <span class="s">&#39;gamma&#39;</span><span class="p">,</span> <span class="s">&#39;delta&#39;</span><span class="p">,</span> <span class="s">&#39;varepsilon&#39;</span><span class="p">,</span> <span class="s">&#39;epsilon&#39;</span><span class="p">,</span> <span class="s">&#39;zeta&#39;</span><span class="p">,</span>
-         <span class="s">&#39;vartheta&#39;</span><span class="p">,</span> <span class="s">&#39;theta&#39;</span><span class="p">,</span> <span class="s">&#39;iota&#39;</span><span class="p">,</span> <span class="s">&#39;kappa&#39;</span><span class="p">,</span> <span class="s">&#39;lambda&#39;</span><span class="p">,</span> <span class="s">&#39;mu&#39;</span><span class="p">,</span> <span class="s">&#39;nu&#39;</span><span class="p">,</span> <span class="s">&#39;xi&#39;</span><span class="p">,</span>
-         <span class="s">&#39;varpi&#39;</span><span class="p">,</span> <span class="s">&#39;pi&#39;</span><span class="p">,</span> <span class="s">&#39;rho&#39;</span><span class="p">,</span> <span class="s">&#39;varrho&#39;</span><span class="p">,</span> <span class="s">&#39;varsigma&#39;</span><span class="p">,</span> <span class="s">&#39;sigma&#39;</span><span class="p">,</span> <span class="s">&#39;tau&#39;</span><span class="p">,</span> <span class="s">&#39;upsilon&#39;</span><span class="p">,</span>
-         <span class="s">&#39;varphi&#39;</span><span class="p">,</span> <span class="s">&#39;phi&#39;</span><span class="p">,</span> <span class="s">&#39;chi&#39;</span><span class="p">,</span> <span class="s">&#39;psi&#39;</span><span class="p">,</span> <span class="s">&#39;omega&#39;</span><span class="p">,</span> <span class="s">&#39;Gamma&#39;</span><span class="p">,</span> <span class="s">&#39;Delta&#39;</span><span class="p">,</span> <span class="s">&#39;Theta&#39;</span><span class="p">,</span>
-         <span class="s">&#39;Lambda&#39;</span><span class="p">,</span> <span class="s">&#39;Xi&#39;</span><span class="p">,</span> <span class="s">&#39;Pi&#39;</span><span class="p">,</span> <span class="s">&#39;Sigma&#39;</span><span class="p">,</span> <span class="s">&#39;Upsilon&#39;</span><span class="p">,</span> <span class="s">&#39;Phi&#39;</span><span class="p">,</span> <span class="s">&#39;Psi&#39;</span><span class="p">,</span> <span class="s">&#39;Omega&#39;</span><span class="p">,</span> <span class="s">&#39;partial&#39;</span><span class="p">,</span>
-                         <span class="s">&#39;nabla&#39;</span><span class="p">,</span> <span class="s">&#39;eta&#39;</span><span class="p">),</span> <span class="n">key</span><span class="o">=</span><span class="n">cmp_to_key</span><span class="p">(</span><span class="n">len_cmp</span><span class="p">))</span>
-
-    <span class="n">accent_keys</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span>
-        <span class="p">(</span><span class="s">&#39;hat&#39;</span><span class="p">,</span> <span class="s">&#39;check&#39;</span><span class="p">,</span> <span class="s">&#39;dot&#39;</span><span class="p">,</span> <span class="s">&#39;breve&#39;</span><span class="p">,</span> <span class="s">&#39;acute&#39;</span><span class="p">,</span> <span class="s">&#39;ddot&#39;</span><span class="p">,</span> <span class="s">&#39;grave&#39;</span><span class="p">,</span> <span class="s">&#39;tilde&#39;</span><span class="p">,</span>
-         <span class="s">&#39;mathring&#39;</span><span class="p">,</span> <span class="s">&#39;bar&#39;</span><span class="p">,</span> <span class="s">&#39;vec&#39;</span><span class="p">,</span> <span class="s">&#39;bm&#39;</span><span class="p">,</span> <span class="s">&#39;prm&#39;</span><span class="p">,</span> <span class="s">&#39;abs&#39;</span><span class="p">),</span> <span class="n">key</span><span class="o">=</span><span class="n">cmp_to_key</span><span class="p">(</span><span class="n">len_cmp</span><span class="p">))</span>
-
-    <span class="n">greek_cnt</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">greek_dict</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="n">accent_cnt</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">accent_dict</span> <span class="o">=</span> <span class="p">{}</span>
-
-    <span class="n">preamble</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">documentclass[10pt,letter,fleqn]{report}</span><span class="se">\n</span><span class="s">&#39;</span> \
-               <span class="s">&#39;</span><span class="se">\\</span><span class="s">pagestyle{empty}</span><span class="se">\n</span><span class="s">&#39;</span> \
-               <span class="s">&#39;</span><span class="se">\\</span><span class="s">usepackage[latin1]{inputenc}</span><span class="se">\n</span><span class="s">&#39;</span> \
-               <span class="s">&#39;</span><span class="se">\\</span><span class="s">usepackage[dvips,landscape,top=1cm,nohead,nofoot]{geometry}</span><span class="se">\n</span><span class="s">&#39;</span> \
-               <span class="s">&#39;</span><span class="se">\\</span><span class="s">usepackage{amsmath}</span><span class="se">\n</span><span class="s">&#39;</span> \
-               <span class="s">&#39;</span><span class="se">\\</span><span class="s">usepackage{bm}</span><span class="se">\n</span><span class="s">&#39;</span> \
-               <span class="s">&#39;</span><span class="se">\\</span><span class="s">usepackage{amsfonts}</span><span class="se">\n</span><span class="s">&#39;</span> \
-               <span class="s">&#39;</span><span class="se">\\</span><span class="s">usepackage{amssymb}</span><span class="se">\n</span><span class="s">&#39;</span> \
-               <span class="s">&#39;</span><span class="se">\\</span><span class="s">setlength{</span><span class="se">\\</span><span class="s">parindent}{0pt}</span><span class="se">\n</span><span class="s">&#39;</span> \
-               <span class="s">&#39;</span><span class="se">\\</span><span class="s">newcommand{</span><span class="se">\\</span><span class="s">bfrac}[2]{</span><span class="se">\\</span><span class="s">displaystyle</span><span class="se">\\</span><span class="s">frac{#1}{#2}}</span><span class="se">\n</span><span class="s">&#39;</span> \
-               <span class="s">&#39;</span><span class="se">\\</span><span class="s">newcommand{</span><span class="se">\\</span><span class="s">lp}{</span><span class="se">\\</span><span class="s">left (}</span><span class="se">\n</span><span class="s">&#39;</span> \
-               <span class="s">&#39;</span><span class="se">\\</span><span class="s">newcommand{</span><span class="se">\\</span><span class="s">rp}{</span><span class="se">\\</span><span class="s">right )}</span><span class="se">\n</span><span class="s">&#39;</span> \
-               <span class="s">&#39;</span><span class="se">\\</span><span class="s">newcommand{</span><span class="se">\\</span><span class="s">half}{</span><span class="se">\\</span><span class="s">frac{1}{2}}</span><span class="se">\n</span><span class="s">&#39;</span> \
-               <span class="s">&#39;</span><span class="se">\\</span><span class="s">newcommand{</span><span class="se">\\</span><span class="s">llt}{</span><span class="se">\\</span><span class="s">left &lt;}</span><span class="se">\n</span><span class="s">&#39;</span> \
-               <span class="s">&#39;</span><span class="se">\\</span><span class="s">newcommand{</span><span class="se">\\</span><span class="s">rgt}{</span><span class="se">\\</span><span class="s">right &gt;}</span><span class="se">\n</span><span class="s">&#39;</span> \
-               <span class="s">&#39;</span><span class="se">\\</span><span class="s">newcommand{</span><span class="se">\\</span><span class="s">abs}[1]{</span><span class="se">\\</span><span class="s">left |{#1}</span><span class="se">\\</span><span class="s">right | }</span><span class="se">\n</span><span class="s">&#39;</span> \
-               <span class="s">&#39;</span><span class="se">\\</span><span class="s">newcommand{</span><span class="se">\\</span><span class="s">pdiff}[2]{</span><span class="se">\\</span><span class="s">bfrac{</span><span class="se">\\</span><span class="s">partial {#1}}{</span><span class="se">\\</span><span class="s">partial {#2}}}</span><span class="se">\n</span><span class="s">&#39;</span> \
-               <span class="s">&#39;</span><span class="se">\\</span><span class="s">newcommand{</span><span class="se">\\</span><span class="s">lbrc}{</span><span class="se">\\</span><span class="s">left </span><span class="se">\\</span><span class="s">{}</span><span class="se">\n</span><span class="s">&#39;</span> \
-               <span class="s">&#39;</span><span class="se">\\</span><span class="s">newcommand{</span><span class="se">\\</span><span class="s">rbrc}{</span><span class="se">\\</span><span class="s">right </span><span class="se">\\</span><span class="s">}}</span><span class="se">\n</span><span class="s">&#39;</span> \
-               <span class="s">&#39;</span><span class="se">\\</span><span class="s">newcommand{</span><span class="se">\\</span><span class="s">W}{</span><span class="se">\\</span><span class="s">wedge}</span><span class="se">\n</span><span class="s">&#39;</span> \
-               <span class="s">&quot;</span><span class="se">\\</span><span class="s">newcommand{</span><span class="se">\\</span><span class="s">prm}[1]{{#1}&#39;}</span><span class="se">\n</span><span class="s">&quot;</span> \
-               <span class="s">&#39;</span><span class="se">\\</span><span class="s">newcommand{</span><span class="se">\\</span><span class="s">ddt}[1]{</span><span class="se">\\</span><span class="s">bfrac{d{#1}}{dt}}</span><span class="se">\n</span><span class="s">&#39;</span> \
-               <span class="s">&#39;</span><span class="se">\\</span><span class="s">newcommand{</span><span class="se">\\</span><span class="s">R}{</span><span class="se">\\</span><span class="s">dagger}</span><span class="se">\n</span><span class="s">&#39;</span> \
-               <span class="s">&#39;</span><span class="se">\\</span><span class="s">begin{document}</span><span class="se">\n</span><span class="s">&#39;</span>
-    <span class="n">postscript</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">end{document}</span><span class="se">\n</span><span class="s">&#39;</span>
-
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="LatexPrinter.latex_bases"><a class="viewcode-back" href="../../../modules/galgebra/latex_ex.html#sympy.galgebra.latex_ex.LatexPrinter.latex_bases">[docs]</a>    <span class="k">def</span> <span class="nf">latex_bases</span><span class="p">():</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Generate LaTeX strings for multivector bases</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">GA</span><span class="o">.</span><span class="n">MV</span><span class="o">.</span><span class="n">basislabel_lst</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="n">sys</span><span class="o">.</span><span class="n">stderr</span><span class="o">.</span><span class="n">write</span><span class="p">(</span>
-                <span class="s">&#39;MV.setup() must be executed before LatexPrinter.format()!</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">)</span>
-            <span class="n">sys</span><span class="o">.</span><span class="n">exit</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">latexbasis_lst</span> <span class="o">=</span> <span class="p">[[</span><span class="s">&#39;&#39;</span><span class="p">]]</span>
-        <span class="k">for</span> <span class="n">grades</span> <span class="ow">in</span> <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">GA</span><span class="o">.</span><span class="n">MV</span><span class="o">.</span><span class="n">basislabel_lst</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="n">grades_lst</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">grade</span> <span class="ow">in</span> <span class="n">grades</span><span class="p">:</span>
-                <span class="n">grade_lst</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="n">grade</span><span class="p">:</span>
-                    <span class="n">latex_base</span> <span class="o">=</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">extended_symbol</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-                    <span class="n">grade_lst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">latex_base</span><span class="p">)</span>
-                <span class="n">grades_lst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">grade_lst</span><span class="p">)</span>
-            <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">latexbasis_lst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">grades_lst</span><span class="p">)</span>
-        <span class="k">return</span>
-</div>
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">build_base</span><span class="p">(</span><span class="n">igrade</span><span class="p">,</span> <span class="n">iblade</span><span class="p">,</span> <span class="n">bld_flg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">igrade</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span><span class="p">(</span><span class="s">&#39;&#39;</span><span class="p">)</span>
-        <span class="n">base_lst</span> <span class="o">=</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">latexbasis_lst</span><span class="p">[</span><span class="n">igrade</span><span class="p">][</span><span class="n">iblade</span><span class="p">]</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">base_lst</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span><span class="p">(</span><span class="n">base_lst</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="n">base_str</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-        <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="n">base_lst</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-            <span class="k">if</span> <span class="n">bld_flg</span><span class="p">:</span>
-                <span class="n">base_str</span> <span class="o">+=</span> <span class="n">base</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">W &#39;</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">base_str</span> <span class="o">+=</span> <span class="n">base</span>
-        <span class="n">base_str</span> <span class="o">+=</span> <span class="n">base_lst</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">base_str</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">format</span><span class="p">(</span><span class="n">sym</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">fct</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">pdiff</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">mv</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">LaTeX_flg</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;sym&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">sym</span>
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;fct&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">fct</span>
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;pdiff&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">pdiff</span>
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;mv&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">mv</span>
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;str&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">GA</span><span class="o">.</span><span class="n">MV</span><span class="o">.</span><span class="n">is_setup</span><span class="p">:</span>
-            <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">latex_bases</span><span class="p">()</span>
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">redirect</span><span class="p">()</span>
-        <span class="k">return</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">str_basic</span><span class="p">(</span><span class="n">in_str</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">LaTeX_flg</span><span class="p">:</span>
-            <span class="k">return</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">in_str</span><span class="p">))</span>
-        <span class="n">Basic</span><span class="o">.</span><span class="n">__str__</span> <span class="o">=</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">Basic__str__</span>
-        <span class="n">out_str</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">in_str</span><span class="p">)</span>
-        <span class="n">Basic</span><span class="o">.</span><span class="n">__str__</span> <span class="o">=</span> <span class="n">LaTeX</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">out_str</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">redirect</span><span class="p">():</span>
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">Basic__str__</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__str__</span>
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">MV__str__</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">GA</span><span class="o">.</span><span class="n">MV</span><span class="o">.</span><span class="n">__str__</span>
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">stdout</span> <span class="o">=</span> <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span>
-        <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span> <span class="o">=</span> <span class="n">io</span><span class="o">.</span><span class="n">StringIO</span><span class="p">()</span>
-        <span class="n">Basic</span><span class="o">.</span><span class="n">__str__</span> <span class="o">=</span> <span class="n">LaTeX</span>
-        <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">GA</span><span class="o">.</span><span class="n">MV</span><span class="o">.</span><span class="n">__str__</span> <span class="o">=</span> <span class="n">LaTeX</span>
-        <span class="k">return</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">restore</span><span class="p">():</span>
-        <span class="n">LatexPrinter_stdout</span> <span class="o">=</span> <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span>
-        <span class="n">LatexPrinter_Basic__str__</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__str__</span>
-        <span class="n">LatexPrinter_MV__str__</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">GA</span><span class="o">.</span><span class="n">MV</span><span class="o">.</span><span class="n">__str__</span>
-
-        <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span> <span class="o">=</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">stdout</span>
-        <span class="n">Basic</span><span class="o">.</span><span class="n">__str__</span> <span class="o">=</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">Basic__str__</span>
-        <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">GA</span><span class="o">.</span><span class="n">MV</span><span class="o">.</span><span class="n">__str__</span> <span class="o">=</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">MV__str__</span>
-
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">stdout</span> <span class="o">=</span> <span class="n">LatexPrinter_stdout</span>
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">Basic__str__</span> <span class="o">=</span> <span class="n">LatexPrinter_Basic__str__</span>
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">MV__str__</span> <span class="o">=</span> <span class="n">LatexPrinter_MV__str__</span>
-        <span class="k">return</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">format_str</span><span class="p">(</span><span class="n">fmt</span><span class="o">=</span><span class="s">&#39;0 0 0 0&#39;</span><span class="p">):</span>
-        <span class="n">fmt_lst</span> <span class="o">=</span> <span class="n">fmt</span><span class="o">.</span><span class="n">split</span><span class="p">()</span>
-        <span class="k">if</span> <span class="s">&#39;=&#39;</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">fmt</span><span class="p">:</span>
-            <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;sym&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">fmt_lst</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;fct&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">fmt_lst</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-            <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;pdiff&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">fmt_lst</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-            <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;mv&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">fmt_lst</span><span class="p">[</span><span class="mi">3</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">fmt</span> <span class="ow">in</span> <span class="n">fmt_lst</span><span class="p">:</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="n">fmt</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;=&#39;</span><span class="p">)</span>
-                <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">LaTeX_flg</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">sympy</span><span class="o">.</span><span class="n">galgebra</span><span class="o">.</span><span class="n">GA</span><span class="o">.</span><span class="n">MV</span><span class="o">.</span><span class="n">is_setup</span><span class="p">:</span>
-                <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">latex_bases</span><span class="p">()</span>
-            <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">redirect</span><span class="p">()</span>
-            <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">LaTeX_flg</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">return</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">append_body</span><span class="p">(</span><span class="n">xstr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">body_flg</span><span class="p">:</span>
-            <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">body</span> <span class="o">+=</span> <span class="n">xstr</span>
-            <span class="k">return</span><span class="p">(</span><span class="s">&#39;&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span><span class="p">(</span><span class="n">xstr</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">tokenize_greek</span><span class="p">(</span><span class="n">name_str</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">sym</span> <span class="ow">in</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">greek_keys</span><span class="p">:</span>
-            <span class="n">isym</span> <span class="o">=</span> <span class="n">name_str</span><span class="o">.</span><span class="n">find</span><span class="p">(</span><span class="n">sym</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">isym</span> <span class="o">&gt;</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                <span class="n">keystr</span> <span class="o">=</span> <span class="s">&#39;@&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">LatexPrinter</span><span class="o">.</span><span class="n">greek_cnt</span><span class="p">)</span>
-                <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">greek_cnt</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">greek_dict</span><span class="p">[</span><span class="n">keystr</span><span class="p">]</span> <span class="o">=</span> <span class="n">sym</span>
-                <span class="n">name_str</span> <span class="o">=</span> <span class="n">name_str</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">keystr</span><span class="p">)</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">name_str</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">tokenize_accents</span><span class="p">(</span><span class="n">name_str</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">sym</span> <span class="ow">in</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">accent_keys</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">name_str</span><span class="o">.</span><span class="n">find</span><span class="p">(</span><span class="n">sym</span><span class="p">)</span> <span class="o">&gt;</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                <span class="n">keystr</span> <span class="o">=</span> <span class="s">&#39;#&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">LatexPrinter</span><span class="o">.</span><span class="n">accent_cnt</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39;#&#39;</span>
-                <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">accent_cnt</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">accent_dict</span><span class="p">[</span><span class="n">keystr</span><span class="p">]</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">&#39;</span> <span class="o">+</span> <span class="n">sym</span>
-                <span class="n">name_str</span> <span class="o">=</span> <span class="n">name_str</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">keystr</span><span class="p">)</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">name_str</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">replace_greek_tokens</span><span class="p">(</span><span class="n">name_str</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">name_str</span><span class="o">.</span><span class="n">find</span><span class="p">(</span><span class="s">&#39;@&#39;</span><span class="p">)</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span><span class="p">(</span><span class="n">name_str</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">token</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">LatexPrinter</span><span class="o">.</span><span class="n">greek_dict</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
-            <span class="n">name_str</span> <span class="o">=</span> <span class="n">name_str</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span>
-                <span class="n">token</span><span class="p">,</span> <span class="s">&#39;{</span><span class="se">\\</span><span class="s">&#39;</span> <span class="o">+</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">greek_dict</span><span class="p">[</span><span class="n">token</span><span class="p">]</span> <span class="o">+</span> <span class="s">&#39;}&#39;</span><span class="p">)</span>
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">greek_cnt</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">greek_dict</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">name_str</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">replace_accent_tokens</span><span class="p">(</span><span class="n">name_str</span><span class="p">):</span>
-        <span class="n">tmp_lst</span> <span class="o">=</span> <span class="n">name_str</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;#&#39;</span><span class="p">)</span>
-        <span class="n">name_str</span> <span class="o">=</span> <span class="n">tmp_lst</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">tmp_lst</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span><span class="p">(</span><span class="n">name_str</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">tmp_lst</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="k">if</span> <span class="n">x</span> <span class="o">!=</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-                <span class="n">name_str</span> <span class="o">=</span> <span class="s">&#39;{}&#39;</span> <span class="o">+</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">accent_dict</span><span class="p">[</span>
-                    <span class="s">&#39;#&#39;</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="s">&#39;#&#39;</span><span class="p">]</span> <span class="o">+</span> <span class="s">&#39;{&#39;</span> <span class="o">+</span> <span class="n">name_str</span> <span class="o">+</span> <span class="s">&#39;}&#39;</span>
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">accent_cnt</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">accent_dict</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">name_str</span><span class="p">)</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">extended_symbol</span><span class="p">(</span><span class="n">name_str</span><span class="p">):</span>
-        <span class="n">name_str</span> <span class="o">=</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">tokenize_greek</span><span class="p">(</span><span class="n">name_str</span><span class="p">)</span>
-        <span class="n">tmp_lst</span> <span class="o">=</span> <span class="n">name_str</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;_&#39;</span><span class="p">)</span>
-        <span class="n">subsup_str</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-        <span class="n">sym_str</span> <span class="o">=</span> <span class="n">tmp_lst</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">sym_str</span> <span class="o">=</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">tokenize_accents</span><span class="p">(</span><span class="n">sym_str</span><span class="p">)</span>
-        <span class="n">sym_str</span> <span class="o">=</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">replace_accent_tokens</span><span class="p">(</span><span class="n">sym_str</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">tmp_lst</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">imode</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">tmp_lst</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-                <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-                    <span class="n">imode</span> <span class="o">=</span> <span class="p">(</span><span class="n">imode</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">subsup_str</span> <span class="o">+=</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">mode</span><span class="p">[</span><span class="n">imode</span><span class="p">]</span> <span class="o">+</span> <span class="s">&#39;{&#39;</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="s">&#39;}&#39;</span>
-                    <span class="c">#subsup_str += LatexPrinter.mode[imode]+x+&#39; &#39;</span>
-                    <span class="n">imode</span> <span class="o">=</span> <span class="p">(</span><span class="n">imode</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span>
-        <span class="n">name_str</span> <span class="o">=</span> <span class="n">sym_str</span> <span class="o">+</span> <span class="n">subsup_str</span>
-        <span class="n">name_str</span> <span class="o">=</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">replace_greek_tokens</span><span class="p">(</span><span class="n">name_str</span><span class="p">)</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">name_str</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">coefficient</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">coef</span><span class="p">,</span> <span class="n">first_flg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">coef</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">AssocOp</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="o">-</span><span class="n">coef</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">AssocOp</span><span class="p">):</span>
-            <span class="n">coef_str</span> <span class="o">=</span> <span class="s">r&quot;\lp </span><span class="si">%s</span><span class="s">\rp &quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">coef</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">coef_str</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">coef</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">first_flg</span><span class="p">:</span>
-            <span class="n">first_flg</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="k">if</span> <span class="n">coef_str</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;+&#39;</span><span class="p">:</span>
-                <span class="n">coef_str</span> <span class="o">=</span> <span class="n">coef_str</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">coef_str</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="s">&#39;-&#39;</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">coef_str</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="s">&#39;+&#39;</span><span class="p">:</span>
-                    <span class="n">coef_str</span> <span class="o">=</span> <span class="s">&#39;+&#39;</span> <span class="o">+</span> <span class="n">coef_str</span>
-        <span class="k">if</span> <span class="n">coef_str</span> <span class="ow">in</span> <span class="p">(</span><span class="s">&#39;1&#39;</span><span class="p">,</span> <span class="s">&#39;+1&#39;</span><span class="p">,</span> <span class="s">&#39;-1&#39;</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">coef_str</span> <span class="o">==</span> <span class="s">&#39;1&#39;</span><span class="p">:</span>
-                <span class="n">coef_str</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">coef_str</span> <span class="o">=</span> <span class="n">coef_str</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">coef_str</span><span class="p">,</span> <span class="n">first_flg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">inline</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="n">Printer</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_inline</span> <span class="o">=</span> <span class="n">inline</span>
-
-    <span class="k">def</span> <span class="nf">doprint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="n">Printer</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-        <span class="n">xstr</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_inline</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;fct&#39;</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">xstr</span> <span class="o">=</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">tex</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">xstr</span> <span class="o">=</span> <span class="s">r&quot;$</span><span class="si">%s</span><span class="s">$&quot;</span> <span class="o">%</span> <span class="n">tex</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">xstr</span> <span class="o">=</span> <span class="s">r&quot;\begin{equation*}</span><span class="si">%s</span><span class="s">\end{equation*}&quot;</span> <span class="o">%</span> <span class="n">tex</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">xstr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_needs_brackets</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="p">((</span><span class="n">expr</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">)</span> <span class="ow">or</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_do_exponent</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="k">def</span> <span class="nf">_print_Add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-
-        <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="k">if</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">term</span><span class="p">):</span>
-                <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot; </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot; + </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">tail</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="n">coeff</span> <span class="o">=</span> <span class="o">-</span><span class="n">coeff</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">&quot;- &quot;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-
-        <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="o">=</span> <span class="n">fraction</span><span class="p">(</span><span class="n">tail</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">convert</span><span class="p">(</span><span class="n">terms</span><span class="p">):</span>
-            <span class="n">product</span> <span class="o">=</span> <span class="p">[]</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">terms</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">terms</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">terms</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                    <span class="n">pretty</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-
-                    <span class="k">if</span> <span class="n">term</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                        <span class="n">product</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="n">pretty</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">product</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">pretty</span><span class="p">))</span>
-
-                <span class="k">return</span> <span class="s">r&quot; &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">product</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">denom</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="n">tex</span> <span class="o">+=</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">coeff</span><span class="p">))</span> <span class="o">+</span> <span class="s">&quot; &quot;</span>
-
-            <span class="k">if</span> <span class="n">numer</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="n">convert</span><span class="p">(</span><span class="n">numer</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">convert</span><span class="p">(</span><span class="n">numer</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">numer</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                    <span class="n">numer</span> <span class="o">*=</span> <span class="n">coeff</span><span class="o">.</span><span class="n">p</span>
-                <span class="k">elif</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">p</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="n">numer</span> <span class="o">*=</span> <span class="n">coeff</span><span class="o">.</span><span class="n">p</span>
-
-                    <span class="n">denom</span> <span class="o">*=</span> <span class="n">coeff</span><span class="o">.</span><span class="n">q</span>
-                <span class="k">elif</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                    <span class="n">tex</span> <span class="o">+=</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">coeff</span><span class="p">))</span> <span class="o">+</span> <span class="s">&quot; &quot;</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">coeff</span><span class="o">.</span><span class="n">p</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">denom</span> <span class="o">*=</span> <span class="n">coeff</span><span class="o">.</span><span class="n">q</span>
-                <span class="k">elif</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                    <span class="n">tex</span> <span class="o">+=</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">coeff</span><span class="p">))</span> <span class="o">+</span> <span class="s">&quot; &quot;</span>
-
-            <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot;\frac{</span><span class="si">%s</span><span class="s">}{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> \
-                <span class="p">(</span><span class="n">convert</span><span class="p">(</span><span class="n">numer</span><span class="p">),</span> <span class="n">convert</span><span class="p">(</span><span class="n">denom</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_Pow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">q</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="nb">abs</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">exp</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\sqrt{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="n">base</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\sqrt[</span><span class="si">%s</span><span class="s">]{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">exp</span><span class="p">,</span> <span class="n">base</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="k">return</span> <span class="s">r&quot;\frac{1}{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="n">tex</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">tex</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_Function</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-                    <span class="c">#solves issue 1030</span>
-                    <span class="c">#As Mul always simplify 1/x to x**-1</span>
-                    <span class="c">#The objective is achieved with this hack</span>
-                    <span class="c">#first we get the latex for -1 * expr,</span>
-                    <span class="c">#which is a Mul expression</span>
-                    <span class="n">tex</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span> <span class="o">*</span> <span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">strip</span><span class="p">()</span>
-                    <span class="c">#the result comes with a minus and a space, so we remove</span>
-                    <span class="k">if</span> <span class="n">tex</span><span class="p">[:</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="s">&quot;-&quot;</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">tex</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span><span class="o">.</span><span class="n">strip</span><span class="p">()</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_needs_brackets</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">):</span>
-                    <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)^{</span><span class="si">%s</span><span class="s">}&quot;</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;{</span><span class="si">%s</span><span class="s">}^{</span><span class="si">%s</span><span class="s">}&quot;</span>
-
-                <span class="k">return</span> <span class="n">tex</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span>
-                              <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">dim</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">dim</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;pdiff&#39;</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">tex</span> <span class="o">=</span> <span class="s">r&#39;\partial_{</span><span class="si">%s</span><span class="s">}&#39;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">variables</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\frac{\partial}{\partial </span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span>
-                    <span class="n">expr</span><span class="o">.</span><span class="n">variables</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">multiplicity</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">tex</span> <span class="o">=</span> <span class="p">[],</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&quot;&quot;</span>
-            <span class="n">current</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">variables</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">for</span> <span class="n">symbol</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">variables</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-                <span class="k">if</span> <span class="n">symbol</span> <span class="o">==</span> <span class="n">current</span><span class="p">:</span>
-                    <span class="n">i</span> <span class="o">=</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">multiplicity</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">current</span><span class="p">,</span> <span class="n">i</span><span class="p">))</span>
-                    <span class="n">current</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="n">symbol</span><span class="p">,</span> <span class="mi">1</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">multiplicity</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">current</span><span class="p">,</span> <span class="n">i</span><span class="p">))</span>
-
-            <span class="k">if</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;pdiff&#39;</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">multiplicity</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot;\partial_{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot;\partial^{</span><span class="si">%s</span><span class="s">}_{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">multiplicity</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot;\partial </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot;\partial^{</span><span class="si">%s</span><span class="s">} </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-
-                <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\frac{\partial^{</span><span class="si">%s</span><span class="s">}}{</span><span class="si">%s</span><span class="s">} &quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">dim</span><span class="p">,</span> <span class="n">tex</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">AssocOp</span><span class="p">):</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">expr</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">expr</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Integral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">tex</span><span class="p">,</span> <span class="n">symbols</span> <span class="o">=</span> <span class="s">&quot;&quot;</span><span class="p">,</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">symbol</span><span class="p">,</span> <span class="n">limits</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">limits</span><span class="p">):</span>
-            <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot;\int&quot;</span>
-
-            <span class="k">if</span> <span class="n">limits</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">_inline</span><span class="p">:</span>
-                    <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot;\limits&quot;</span>
-
-                <span class="n">tex</span> <span class="o">+=</span> <span class="s">&quot;_{</span><span class="si">%s</span><span class="s">}^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">limits</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span>
-                                       <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">limits</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-
-            <span class="n">symbols</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="s">&quot;d</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">symbol</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">\,</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span>
-            <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">function</span><span class="p">)),</span> <span class="s">&quot; &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">symbols</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Limit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\lim_{</span><span class="si">%s</span><span class="s"> \to </span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">var</span><span class="p">),</span>
-                                     <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">varlim</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">AssocOp</span><span class="p">):</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">expr</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">expr</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Function</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">func</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="o">.</span><span class="n">__name__</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s">&#39;_print_&#39;</span> <span class="o">+</span> <span class="n">func</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s">&#39;_print_&#39;</span> <span class="o">+</span> <span class="n">func</span><span class="p">)(</span><span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="p">[</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span> <span class="p">]</span>
-
-            <span class="k">if</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;fct&#39;</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">func</span> <span class="ow">in</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fct_dict_keys</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">func</span> <span class="ow">in</span> <span class="n">accepted_latex_functions</span><span class="p">:</span>
-                            <span class="n">name</span> <span class="o">=</span> <span class="s">r&quot;\</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span>
-                                <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fct_dict</span><span class="p">[</span><span class="n">func</span><span class="p">],</span> <span class="n">exp</span><span class="p">)</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="n">name</span> <span class="o">=</span> <span class="s">r&quot;\operatorname{</span><span class="si">%s</span><span class="s">}^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span>
-                                <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fct_dict</span><span class="p">[</span><span class="n">func</span><span class="p">],</span> <span class="n">exp</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fct_dict</span><span class="p">[</span><span class="n">func</span><span class="p">]</span> <span class="ow">in</span> <span class="n">accepted_latex_functions</span><span class="p">:</span>
-                            <span class="n">name</span> <span class="o">=</span> <span class="s">r&quot;\</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fct_dict</span><span class="p">[</span><span class="n">func</span><span class="p">]</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="n">name</span> <span class="o">=</span> <span class="s">r&quot;\operatorname{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fct_dict</span><span class="p">[</span><span class="n">func</span><span class="p">]</span>
-                    <span class="n">name</span> <span class="o">+=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="s">&quot;,&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-                    <span class="k">return</span> <span class="n">name</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">func</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">print_Symbol_name</span><span class="p">(</span><span class="n">func</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="n">name</span> <span class="o">=</span> <span class="s">r&quot;{</span><span class="si">%s</span><span class="s">}^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">name</span> <span class="o">=</span> <span class="s">r&quot;{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="n">func</span>
-                    <span class="k">return</span> <span class="n">name</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">func</span> <span class="ow">in</span> <span class="n">accepted_latex_functions</span><span class="p">:</span>
-                        <span class="n">name</span> <span class="o">=</span> <span class="s">r&quot;\</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">name</span> <span class="o">=</span> <span class="s">r&quot;\operatorname{</span><span class="si">%s</span><span class="s">}^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">func</span> <span class="ow">in</span> <span class="n">accepted_latex_functions</span><span class="p">:</span>
-                        <span class="n">name</span> <span class="o">=</span> <span class="s">r&quot;\</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">func</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">name</span> <span class="o">=</span> <span class="s">r&quot;\operatorname{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="n">func</span>
-                <span class="k">return</span> <span class="n">name</span> <span class="o">+</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="s">&quot;,&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_floor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\lfloor{</span><span class="si">%s</span><span class="s">}\rfloor&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_ceiling</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\lceil{</span><span class="si">%s</span><span class="s">}\rceil&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_abs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\lvert{</span><span class="si">%s</span><span class="s">}\rvert&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_re</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_needs_brackets</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\Re\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\Re{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_do_exponent</span><span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_im</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_needs_brackets</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\Im\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\Im{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_do_exponent</span><span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\overline{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_exp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;{e}^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_do_exponent</span><span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_gamma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\Gamma^{</span><span class="si">%s</span><span class="s">}</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">exp</span><span class="p">,</span> <span class="n">tex</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\Gamma</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_factorial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_needs_brackets</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)!&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot;!&quot;</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_factorial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_needs_brackets</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;!\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">&quot;!&quot;</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_binomial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;{{</span><span class="si">%s</span><span class="s">}\choose{</span><span class="si">%s</span><span class="s">}}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span>
-                                      <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_RisingFactorial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;{\left(</span><span class="si">%s</span><span class="s">\right)}^{\left(</span><span class="si">%s</span><span class="s">\right)}&quot;</span> <span class="o">%</span> \
-            <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_do_exponent</span><span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_FallingFactorial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;{\left(</span><span class="si">%s</span><span class="s">\right)}_{\left(</span><span class="si">%s</span><span class="s">\right)}&quot;</span> <span class="o">%</span> \
-            <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_do_exponent</span><span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Rational</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">q</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">sign</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">p</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">p</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">sign</span> <span class="o">=</span> <span class="s">&quot;- &quot;</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="o">-</span><span class="n">p</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">\frac{</span><span class="si">%d</span><span class="s">}{</span><span class="si">%d</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">sign</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Infinity</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\infty&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_NegativeInfinity</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;-\infty&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_ComplexInfinity</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\tilde{\infty}&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_ImaginaryUnit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\mathbf{\imath}&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_NaN</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\bot&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Pi</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\pi&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Exp1</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;e&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_EulerGamma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\gamma&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Order</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;</span><span class="se">\\</span><span class="s">mathcal{O}\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> \
-            <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">print_Symbol_name</span><span class="p">(</span><span class="n">name_str</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">name_str</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">name_str</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;sym&#39;</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">extended_symbol</span><span class="p">(</span><span class="n">name_str</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span><span class="p">(</span><span class="n">name_str</span><span class="p">)</span>
-
-        <span class="c">#convert trailing digits to subscript</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="n">regrep</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="s">&#39;(^[a-zA-Z]+)([0-9]+)$&#39;</span><span class="p">,</span> <span class="n">name_str</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">m</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">name</span><span class="p">,</span> <span class="n">sub</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">groups</span><span class="p">()</span>
-                <span class="n">tex</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Symbol</span><span class="p">(</span><span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">))</span>
-                <span class="n">tex</span> <span class="o">=</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">_{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">sub</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">tex</span>
-
-            <span class="c"># insert braces to expresions containing &#39;_&#39; or &#39;^&#39;</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="n">regrep</span><span class="o">.</span><span class="n">match</span><span class="p">(</span>
-                <span class="s">&#39;(^[a-zA-Z0-9]+)([_\^]{1})([a-zA-Z0-9]+)$&#39;</span><span class="p">,</span> <span class="n">name_str</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">m</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">name</span><span class="p">,</span> <span class="n">sep</span><span class="p">,</span> <span class="n">rest</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">groups</span><span class="p">()</span>
-                <span class="n">tex</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Symbol</span><span class="p">(</span><span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">))</span>
-                <span class="n">tex</span> <span class="o">=</span> <span class="s">&quot;</span><span class="si">%s%s</span><span class="s">{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">sep</span><span class="p">,</span> <span class="n">rest</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">tex</span>
-
-            <span class="n">greek</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span> <span class="s">&#39;alpha&#39;</span><span class="p">,</span> <span class="s">&#39;beta&#39;</span><span class="p">,</span> <span class="s">&#39;gamma&#39;</span><span class="p">,</span> <span class="s">&#39;delta&#39;</span><span class="p">,</span> <span class="s">&#39;epsilon&#39;</span><span class="p">,</span> <span class="s">&#39;zeta&#39;</span><span class="p">,</span>
-                          <span class="s">&#39;eta&#39;</span><span class="p">,</span> <span class="s">&#39;theta&#39;</span><span class="p">,</span> <span class="s">&#39;iota&#39;</span><span class="p">,</span> <span class="s">&#39;kappa&#39;</span><span class="p">,</span> <span class="s">&#39;lambda&#39;</span><span class="p">,</span> <span class="s">&#39;mu&#39;</span><span class="p">,</span> <span class="s">&#39;nu&#39;</span><span class="p">,</span>
-                          <span class="s">&#39;xi&#39;</span><span class="p">,</span> <span class="s">&#39;omicron&#39;</span><span class="p">,</span> <span class="s">&#39;pi&#39;</span><span class="p">,</span> <span class="s">&#39;rho&#39;</span><span class="p">,</span> <span class="s">&#39;sigma&#39;</span><span class="p">,</span> <span class="s">&#39;tau&#39;</span><span class="p">,</span> <span class="s">&#39;upsilon&#39;</span><span class="p">,</span>
-                          <span class="s">&#39;phi&#39;</span><span class="p">,</span> <span class="s">&#39;chi&#39;</span><span class="p">,</span> <span class="s">&#39;psi&#39;</span><span class="p">,</span> <span class="s">&#39;omega&#39;</span> <span class="p">])</span>
-
-            <span class="n">other</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span> <span class="p">[</span><span class="s">&#39;aleph&#39;</span><span class="p">,</span> <span class="s">&#39;beth&#39;</span><span class="p">,</span> <span class="s">&#39;daleth&#39;</span><span class="p">,</span> <span class="s">&#39;gimel&#39;</span><span class="p">,</span> <span class="s">&#39;ell&#39;</span><span class="p">,</span> <span class="s">&#39;eth&#39;</span><span class="p">,</span>
-                          <span class="s">&#39;hbar&#39;</span><span class="p">,</span> <span class="s">&#39;hslash&#39;</span><span class="p">,</span> <span class="s">&#39;mho&#39;</span> <span class="p">])</span>
-
-            <span class="k">if</span> <span class="n">name_str</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="ow">in</span> <span class="n">greek</span><span class="p">:</span>
-                <span class="k">return</span> <span class="s">&quot;</span><span class="se">\\</span><span class="s">&quot;</span> <span class="o">+</span> <span class="n">name_str</span>
-            <span class="k">elif</span> <span class="n">name_str</span> <span class="ow">in</span> <span class="n">other</span><span class="p">:</span>
-                <span class="k">return</span> <span class="s">&quot;</span><span class="se">\\</span><span class="s">&quot;</span> <span class="o">+</span> <span class="n">name_str</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">name_str</span>
-
-    <span class="k">def</span> <span class="nf">_print_Symbol</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">print_Symbol_name</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_str</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;str&#39;</span><span class="p">]</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;^&#39;</span><span class="p">,</span> <span class="s">&#39;{</span><span class="se">\\</span><span class="s">wedge}&#39;</span><span class="p">)</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;|&#39;</span><span class="p">,</span> <span class="s">&#39;{</span><span class="se">\\</span><span class="s">cdot}&#39;</span><span class="p">)</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;__&#39;</span><span class="p">,</span> <span class="s">&#39;^&#39;</span><span class="p">)</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_ndarray</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">shape</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">shape</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="n">ndim</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">shape</span><span class="p">)</span>
-        <span class="n">expr_str</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-
-        <span class="k">if</span> <span class="n">ndim</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">expr_str</span> <span class="o">+=</span> <span class="s">&#39;#</span><span class="se">\\</span><span class="s">left [ </span><span class="se">\\</span><span class="s">begin{array}{&#39;</span> <span class="o">+</span> <span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="s">&#39;c&#39;</span> <span class="o">+</span> <span class="s">&#39;}  </span><span class="se">\n</span><span class="s">&#39;</span>
-            <span class="k">for</span> <span class="n">col</span> <span class="ow">in</span> <span class="n">expr</span><span class="p">:</span>
-                <span class="n">expr_str</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">col</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39; &amp; &#39;</span>
-            <span class="n">expr_str</span> <span class="o">=</span> <span class="n">expr_str</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\n\\</span><span class="s">end{array}</span><span class="se">\\</span><span class="s">right ]#</span><span class="se">\n</span><span class="s">&#39;</span>
-            <span class="k">return</span><span class="p">(</span><span class="n">expr_str</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">ndim</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">expr_str</span> <span class="o">+=</span> <span class="s">&#39;#</span><span class="se">\\</span><span class="s">left [ </span><span class="se">\\</span><span class="s">begin{array}{&#39;</span> <span class="o">+</span> <span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="s">&#39;c&#39;</span> <span class="o">+</span> <span class="s">&#39;}  </span><span class="se">\n</span><span class="s">&#39;</span>
-            <span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="n">expr</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-                <span class="k">for</span> <span class="n">xij</span> <span class="ow">in</span> <span class="n">row</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-                    <span class="n">expr_str</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">xij</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39; &amp; &#39;</span>
-                <span class="n">expr_str</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">row</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span> <span class="o">+</span> <span class="s">&#39; </span><span class="se">\\\\</span><span class="s"> </span><span class="se">\n</span><span class="s">&#39;</span>
-            <span class="k">for</span> <span class="n">xij</span> <span class="ow">in</span> <span class="n">expr</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">][:</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-                <span class="n">expr_str</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">xij</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39; &amp; &#39;</span>
-            <span class="n">expr_str</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span>
-                <span class="n">expr</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s"> </span><span class="se">\\</span><span class="s">end{array} </span><span class="se">\\</span><span class="s">right ] #</span><span class="se">\n</span><span class="s">&#39;</span>
-            <span class="k">return</span><span class="p">(</span><span class="n">expr_str</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">ndim</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="n">expr_str</span> <span class="o">=</span> <span class="s">&#39;#</span><span class="se">\\</span><span class="s">left </span><span class="se">\\</span><span class="s">{ </span><span class="se">\\</span><span class="s">begin{array}{&#39;</span> <span class="o">+</span> <span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="s">&#39;c&#39;</span> <span class="o">+</span> <span class="s">&#39;} </span><span class="se">\n</span><span class="s">&#39;</span>
-            <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">expr</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-                <span class="n">xstr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;#&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-                <span class="n">expr_str</span> <span class="o">+=</span> <span class="n">xstr</span> <span class="o">+</span> <span class="s">&#39; , &amp; &#39;</span>
-            <span class="n">xstr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;#&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-            <span class="n">expr_str</span> <span class="o">+=</span> <span class="n">xstr</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\n\\</span><span class="s">end{array} </span><span class="se">\\</span><span class="s">right </span><span class="se">\\</span><span class="s">}#</span><span class="se">\n</span><span class="s">&#39;</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">expr_str</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_MV</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">igrade</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">MV_str</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-        <span class="n">line_lst</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">grade</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">mv</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">grade</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-                <span class="n">ibase</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="n">grade</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">base</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">tmp</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;XYZW&#39;</span><span class="p">)</span>
-                        <span class="n">base_str</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">base</span><span class="o">*</span><span class="n">tmp</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">base_str</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="s">&#39;-&#39;</span><span class="p">:</span>
-                            <span class="n">base_str</span> <span class="o">=</span> <span class="s">&#39;+&#39;</span> <span class="o">+</span> <span class="n">base_str</span>
-                        <span class="n">base_str</span> <span class="o">=</span> <span class="n">base_str</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;- &#39;</span><span class="p">,</span> <span class="s">&#39;-&#39;</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">base_str</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="mi">5</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;XYZW&#39;</span><span class="p">:</span>
-                            <span class="n">base_str</span> <span class="o">=</span> <span class="n">base_str</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;XYZW&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="n">base_str</span> <span class="o">=</span> <span class="n">base_str</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;XYZW&#39;</span><span class="p">,</span> <span class="s">&#39;1&#39;</span><span class="p">)</span>
-                        <span class="n">MV_str</span> <span class="o">+=</span> <span class="n">base_str</span> <span class="o">+</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">build_base</span><span class="p">(</span>
-                            <span class="n">igrade</span><span class="p">,</span> <span class="n">ibase</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">bladeflg</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;mv&#39;</span><span class="p">]</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-                            <span class="n">line_lst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">MV_str</span><span class="p">)</span>
-                            <span class="n">MV_str</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-                    <span class="n">ibase</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">if</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;mv&#39;</span><span class="p">]</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">MV_str</span> <span class="o">!=</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-                        <span class="n">line_lst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">MV_str</span><span class="p">)</span>
-                        <span class="n">MV_str</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-            <span class="n">igrade</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="n">n_lines</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">line_lst</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">MV_str</span> <span class="o">==</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">n_lines</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">line_lst</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;+&#39;</span><span class="p">:</span>
-                <span class="n">line_lst</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">line_lst</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">:]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">MV_str</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;+&#39;</span><span class="p">:</span>
-                <span class="n">MV_str</span> <span class="o">=</span> <span class="n">MV_str</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-        <span class="k">if</span> <span class="n">n_lines</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">MV_str</span> <span class="o">=</span> <span class="n">line_lst</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">n_lines</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">if</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;mv&#39;</span><span class="p">]</span> <span class="o">&gt;=</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">MV_str</span> <span class="o">=</span> <span class="s">&#39;@&#39;</span> <span class="o">+</span> <span class="n">line_lst</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="s">&#39; </span><span class="se">\\\\</span><span class="s"> </span><span class="se">\n</span><span class="s">&#39;</span>
-            <span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="n">line_lst</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-                <span class="n">MV_str</span> <span class="o">+=</span> <span class="s">&#39;&amp; &#39;</span> <span class="o">+</span> <span class="n">line</span> <span class="o">+</span> <span class="s">&#39; </span><span class="se">\\\\</span><span class="s"> </span><span class="se">\n</span><span class="s">&#39;</span>
-            <span class="n">MV_str</span> <span class="o">+=</span> <span class="s">&#39;&amp; &#39;</span> <span class="o">+</span> <span class="n">line_lst</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="s">&#39;@</span><span class="se">\n</span><span class="s">&#39;</span>
-        <span class="k">if</span> <span class="n">MV_str</span> <span class="o">==</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-            <span class="n">MV_str</span> <span class="o">=</span> <span class="s">&#39;0&#39;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">name</span> <span class="o">!=</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-            <span class="n">MV_str</span> <span class="o">=</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">extended_symbol</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">name</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39; = &#39;</span> <span class="o">+</span> <span class="n">MV_str</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV_str</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_OMV</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">igrade</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">MV_str</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-        <span class="n">line_lst</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">grade</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">mv</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">grade</span><span class="p">,</span> <span class="bp">None</span><span class="p">):</span>
-                <span class="n">ibase</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="n">grade</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">base</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">tmp</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;XYZW&#39;</span><span class="p">)</span>
-                        <span class="n">base_str</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">base</span><span class="o">*</span><span class="n">tmp</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">base_str</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="s">&#39;-&#39;</span><span class="p">:</span>
-                            <span class="n">base_str</span> <span class="o">=</span> <span class="s">&#39;+&#39;</span> <span class="o">+</span> <span class="n">base_str</span>
-                        <span class="n">base_str</span> <span class="o">=</span> <span class="n">base_str</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;- &#39;</span><span class="p">,</span> <span class="s">&#39;-&#39;</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">base_str</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="mi">5</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;XYZW&#39;</span><span class="p">:</span>
-                            <span class="n">base_str</span> <span class="o">=</span> <span class="n">base_str</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;XYZW&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="n">base_str</span> <span class="o">=</span> <span class="n">base_str</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;XYZW&#39;</span><span class="p">,</span> <span class="s">&#39;1&#39;</span><span class="p">)</span>
-                        <span class="n">MV_str</span> <span class="o">+=</span> <span class="n">base_str</span> <span class="o">+</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">build_base</span><span class="p">(</span>
-                            <span class="n">igrade</span><span class="p">,</span> <span class="n">ibase</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">bladeflg</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;mv&#39;</span><span class="p">]</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-                            <span class="n">line_lst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">MV_str</span><span class="p">)</span>
-                            <span class="n">MV_str</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-                    <span class="n">ibase</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">if</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;mv&#39;</span><span class="p">]</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">MV_str</span> <span class="o">!=</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-                        <span class="n">line_lst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">MV_str</span><span class="p">)</span>
-                        <span class="n">MV_str</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-            <span class="n">igrade</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="n">n_lines</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">line_lst</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">MV_str</span> <span class="o">==</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">n_lines</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">line_lst</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;+&#39;</span><span class="p">:</span>
-                <span class="n">line_lst</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">line_lst</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">:]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">MV_str</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;+&#39;</span><span class="p">:</span>
-                <span class="n">MV_str</span> <span class="o">=</span> <span class="n">MV_str</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-        <span class="k">if</span> <span class="n">n_lines</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">MV_str</span> <span class="o">=</span> <span class="n">line_lst</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">n_lines</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">if</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;mv&#39;</span><span class="p">]</span> <span class="o">&gt;=</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">MV_str</span> <span class="o">=</span> <span class="s">&#39;@&#39;</span> <span class="o">+</span> <span class="n">line_lst</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="s">&#39; </span><span class="se">\\\\</span><span class="s"> </span><span class="se">\n</span><span class="s">&#39;</span>
-            <span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="n">line_lst</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-                <span class="n">MV_str</span> <span class="o">+=</span> <span class="s">&#39;&amp; &#39;</span> <span class="o">+</span> <span class="n">line</span> <span class="o">+</span> <span class="s">&#39; </span><span class="se">\\\\</span><span class="s"> </span><span class="se">\n</span><span class="s">&#39;</span>
-            <span class="n">MV_str</span> <span class="o">+=</span> <span class="s">&#39;&amp; &#39;</span> <span class="o">+</span> <span class="n">line_lst</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="s">&#39;@</span><span class="se">\n</span><span class="s">&#39;</span>
-        <span class="k">if</span> <span class="n">MV_str</span> <span class="o">==</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-            <span class="n">MV_str</span> <span class="o">=</span> <span class="s">&#39;0&#39;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">name</span> <span class="o">!=</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-            <span class="n">MV_str</span> <span class="o">=</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">extended_symbol</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">name</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39; = &#39;</span> <span class="o">+</span> <span class="n">MV_str</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">MV_str</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Relational</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">charmap</span> <span class="o">=</span> <span class="p">{</span>
-            <span class="s">&quot;==&quot;</span><span class="p">:</span> <span class="s">&quot;=&quot;</span><span class="p">,</span>
-            <span class="s">&quot;&lt;&quot;</span><span class="p">:</span> <span class="s">&quot;&lt;&quot;</span><span class="p">,</span>
-            <span class="s">&quot;&lt;=&quot;</span><span class="p">:</span> <span class="s">r&quot;\leq&quot;</span><span class="p">,</span>
-            <span class="s">&quot;!=&quot;</span><span class="p">:</span> <span class="s">r&quot;\neq&quot;</span><span class="p">,</span>
-        <span class="p">}</span>
-
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">lhs</span><span class="p">),</span>
-            <span class="n">charmap</span><span class="p">[</span><span class="n">expr</span><span class="o">.</span><span class="n">rel_op</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">rhs</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">lines</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">lines</span><span class="p">):</span>  <span class="c"># horrible, should be &#39;rows&#39;</span>
-            <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot; &amp; &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span><span class="p">[</span><span class="n">line</span><span class="p">,</span> <span class="p">:]</span> <span class="p">]))</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_inline</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(\begin{smallmatrix}</span><span class="si">%s</span><span class="s">\end{smallmatrix}\right)&quot;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\begin{pmatrix}</span><span class="si">%s</span><span class="s">\end{pmatrix}&quot;</span>
-
-        <span class="k">return</span> <span class="n">tex</span> <span class="o">%</span> <span class="s">r&quot;</span><span class="se">\\</span><span class="s">&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">lines</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_tuple</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\begin{pmatrix}</span><span class="si">%s</span><span class="s">\end{pmatrix}&quot;</span> <span class="o">%</span> \
-            <span class="s">r&quot;, &amp; &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span> <span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_print_list</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\begin{bmatrix}</span><span class="si">%s</span><span class="s">\end{bmatrix}&quot;</span> <span class="o">%</span> \
-            <span class="s">r&quot;, &amp; &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span> <span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_print_dict</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">items</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="n">keys</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-        <span class="n">keys</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">key</span> <span class="ow">in</span> <span class="n">keys</span><span class="p">:</span>
-            <span class="n">val</span> <span class="o">=</span> <span class="n">expr</span><span class="p">[</span><span class="n">key</span><span class="p">]</span>
-            <span class="n">items</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> : </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">key</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">val</span><span class="p">)))</span>
-
-        <span class="k">return</span> <span class="s">r&quot;\begin{Bmatrix}</span><span class="si">%s</span><span class="s">\end{Bmatrix}&quot;</span> <span class="o">%</span> <span class="s">r&quot;, &amp; &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">items</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_DiracDelta</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\delta\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\delta^{\left( </span><span class="si">%s</span><span class="s"> \right)}\left( </span><span class="si">%s</span><span class="s"> \right)&quot;</span> <span class="o">%</span> <span class="p">(</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-        <span class="k">return</span> <span class="n">tex</span>
-
-</div>
-<div class="viewcode-block" id="LaTeX"><a class="viewcode-back" href="../../../modules/galgebra/latex_ex.html#sympy.galgebra.latex_ex.LaTeX">[docs]</a><span class="k">def</span> <span class="nf">LaTeX</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">inline</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Convert the given expression to LaTeX representation.</span>
-
-<span class="sd">    You can specify how the generated code will be delimited.</span>
-<span class="sd">    If the &#39;inline&#39; keyword is set then inline LaTeX $ $ will</span>
-<span class="sd">    be used. Otherwise the resulting code will be enclosed in</span>
-<span class="sd">    &#39;equation*&#39; environment (remember to import &#39;amsmath&#39;).</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Rational</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import tau, mu</span>
-
-<span class="sd">    &gt;&gt;&gt; latex((2*tau)**Rational(7,2))</span>
-<span class="sd">    &#39;$8 \\\\sqrt{2} \\\\sqrt[7]{\\\\tau}$&#39;</span>
-
-<span class="sd">    &gt;&gt;&gt; latex((2*mu)**Rational(7,2), inline=False)</span>
-<span class="sd">    &#39;\\\\begin{equation*}8 \\\\sqrt{2} \\\\sqrt[7]{\\\\mu}\\\\end{equation*}&#39;</span>
-
-<span class="sd">    Besides all Basic based expressions, you can recursively</span>
-<span class="sd">    convert Python containers (lists, tuples and dicts) and</span>
-<span class="sd">    also SymPy matrices:</span>
-
-<span class="sd">    &gt;&gt;&gt; latex([2/x, y])</span>
-<span class="sd">    &#39;$\\\\begin{bmatrix}\\\\frac{2}{x}, &amp; y\\\\end{bmatrix}$&#39;</span>
-
-<span class="sd">    The extended latex printer will also append the output to a</span>
-<span class="sd">    string (LatexPrinter.body) that will be processed by xdvi()</span>
-<span class="sd">    for immediate display one xdvi() is called.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">xstr</span> <span class="o">=</span> <span class="n">LatexPrinter</span><span class="p">(</span><span class="n">inline</span><span class="p">)</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">return</span> <span class="p">(</span><span class="n">xstr</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="print_LaTeX"><a class="viewcode-back" href="../../../modules/galgebra/latex_ex.html#sympy.galgebra.latex_ex.print_LaTeX">[docs]</a><span class="k">def</span> <span class="nf">print_LaTeX</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Prints LaTeX representation of the given expression.&quot;&quot;&quot;</span>
-    <span class="k">print</span><span class="p">(</span><span class="n">LaTeX</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-
-</div>
-<span class="k">def</span> <span class="nf">Format</span><span class="p">(</span><span class="n">fmt</span><span class="o">=</span><span class="s">&#39;1 1 1 1&#39;</span><span class="p">):</span>
-    <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">format_str</span><span class="p">(</span><span class="n">fmt</span><span class="p">)</span>
-    <span class="k">return</span>
-
-
-<div class="viewcode-block" id="xdvi"><a class="viewcode-back" href="../../../modules/galgebra/latex_ex.html#sympy.galgebra.latex_ex.xdvi">[docs]</a><span class="k">def</span> <span class="nf">xdvi</span><span class="p">(</span><span class="n">filename</span><span class="o">=</span><span class="s">&#39;tmplatex.tex&#39;</span><span class="p">,</span> <span class="n">debug</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Post processes LaTeX output (see comments below), adds preamble and</span>
-<span class="sd">    postscript, generates tex file, inputs file to latex, displays resulting</span>
-<span class="sd">    dvi file with xdvi or yap.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">LaTeX_flg</span><span class="p">:</span>
-        <span class="k">return</span>
-    <span class="n">body</span> <span class="o">=</span> <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span><span class="o">.</span><span class="n">getvalue</span><span class="p">()</span>
-    <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">restore</span><span class="p">()</span>
-
-    <span class="n">body_lst</span> <span class="o">=</span> <span class="n">body</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">)</span>
-    <span class="n">body</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-    <span class="n">array_flg</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">eqnarray_flg</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">raw_flg</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">i</span> <span class="o">=</span> <span class="nb">iter</span><span class="p">(</span><span class="n">body_lst</span><span class="p">)</span>
-    <span class="n">line</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="k">if</span> <span class="s">&#39;$&#39;</span> <span class="ow">in</span> <span class="n">line</span><span class="p">:</span>  <span class="c"># Inline math expression(s)</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">line</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">line</span> <span class="o">+=</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">newline </span><span class="se">\n</span><span class="s">&#39;</span>
-            <span class="n">body</span> <span class="o">+=</span> <span class="n">line</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">line</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">StopIteration</span><span class="p">:</span>
-                <span class="k">break</span>
-
-        <span class="k">elif</span> <span class="s">&#39;%&#39;</span> <span class="ow">in</span> <span class="n">line</span><span class="p">:</span>  <span class="c"># Raw LaTeX input</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            If % in line assume line is beginning of raw LaTeX input and stop</span>
-<span class="sd">            post processing</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="n">line</span> <span class="o">=</span> <span class="n">line</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;%&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-            <span class="n">raw_flg</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">while</span> <span class="n">raw_flg</span><span class="p">:</span>
-                <span class="k">if</span> <span class="s">&#39;%&#39;</span> <span class="ow">in</span> <span class="n">line</span><span class="p">:</span>
-                    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">                    If % in line assume line is end of LaTeX input and begin</span>
-<span class="sd">                    post processing</span>
-<span class="sd">                    &quot;&quot;&quot;</span>
-                    <span class="n">raw_flg</span> <span class="o">=</span> <span class="bp">False</span>
-                    <span class="n">line</span> <span class="o">=</span> <span class="n">line</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;%&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">line</span> <span class="o">+=</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span>
-                <span class="n">line</span> <span class="o">=</span> <span class="n">process_equals</span><span class="p">(</span><span class="n">line</span><span class="p">)</span>
-                <span class="n">body</span> <span class="o">+=</span> <span class="n">line</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="n">line</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                <span class="k">except</span> <span class="ne">StopIteration</span><span class="p">:</span>
-                    <span class="k">break</span>
-
-        <span class="k">elif</span> <span class="s">&#39;#&#39;</span> <span class="ow">in</span> <span class="n">line</span><span class="p">:</span>  <span class="c"># Array input</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            If # in line assume line is beginning of array input and contains</span>
-<span class="sd">            \begin{array} statement</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="n">line</span> <span class="o">=</span> <span class="n">line</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;#&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-            <span class="n">array_flg</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="n">line</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">begin{equation*}</span><span class="se">\n</span><span class="s">&#39;</span> <span class="o">+</span> <span class="n">line</span>
-            <span class="k">while</span> <span class="n">array_flg</span><span class="p">:</span>
-                <span class="k">if</span> <span class="s">&#39;#&#39;</span> <span class="ow">in</span> <span class="n">line</span><span class="p">:</span>
-                    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">                    If # in line assume line is end of array input and contains</span>
-<span class="sd">                    \end{array} statement</span>
-<span class="sd">                    &quot;&quot;&quot;</span>
-                    <span class="n">array_flg</span> <span class="o">=</span> <span class="bp">False</span>
-                    <span class="n">line</span> <span class="o">=</span> <span class="n">line</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;#&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-                    <span class="n">line</span> <span class="o">+=</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">end{equation*}</span><span class="se">\n</span><span class="s">&#39;</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">line</span> <span class="o">+=</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span>
-                <span class="n">line</span> <span class="o">=</span> <span class="n">process_equals</span><span class="p">(</span><span class="n">line</span><span class="p">)</span>
-                <span class="n">body</span> <span class="o">+=</span> <span class="n">line</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="n">line</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                <span class="k">except</span> <span class="ne">StopIteration</span><span class="p">:</span>
-                    <span class="k">break</span>
-
-        <span class="k">elif</span> <span class="s">&#39;@&#39;</span> <span class="ow">in</span> <span class="n">line</span><span class="p">:</span>  <span class="c"># Align input</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            If @ in line assume line is beginning of align input</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="n">line</span> <span class="o">=</span> <span class="n">line</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;@&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-            <span class="n">line</span> <span class="o">=</span> <span class="n">line</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;=&#39;</span><span class="p">,</span> <span class="s">&#39;&amp; = &#39;</span><span class="p">)</span>
-            <span class="n">eqnarray_flg</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="n">line</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">begin{align*}</span><span class="se">\n</span><span class="s">&#39;</span> <span class="o">+</span> <span class="n">line</span>
-            <span class="n">line</span> <span class="o">=</span> <span class="n">process_equals</span><span class="p">(</span><span class="n">line</span><span class="p">)</span>
-            <span class="n">body</span> <span class="o">+=</span> <span class="n">line</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">line</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">StopIteration</span><span class="p">:</span>
-                <span class="k">break</span>
-            <span class="k">while</span> <span class="n">eqnarray_flg</span><span class="p">:</span>
-                <span class="k">if</span> <span class="s">&#39;@&#39;</span> <span class="ow">in</span> <span class="n">line</span><span class="p">:</span>
-                    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">                    If @ in line assume line is end of align input</span>
-<span class="sd">                    &quot;&quot;&quot;</span>
-                    <span class="n">eqnarray_flg</span> <span class="o">=</span> <span class="bp">False</span>
-                    <span class="n">line</span> <span class="o">=</span> <span class="n">line</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;@&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-                    <span class="n">line</span> <span class="o">+=</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">end{align*}</span><span class="se">\n</span><span class="s">&#39;</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">line</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span>
-                <span class="n">line</span> <span class="o">=</span> <span class="n">process_equals</span><span class="p">(</span><span class="n">line</span><span class="p">)</span>
-                <span class="n">body</span> <span class="o">+=</span> <span class="n">line</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="n">line</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                <span class="k">except</span> <span class="ne">StopIteration</span><span class="p">:</span>
-                    <span class="k">break</span>
-
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="s">&#39;=&#39;</span> <span class="ow">in</span> <span class="n">line</span><span class="p">:</span>  <span class="c"># Single line equation</span>
-                <span class="n">line</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">begin{equation*}</span><span class="se">\n</span><span class="s">&#39;</span> <span class="o">+</span> <span class="n">line</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\n\\</span><span class="s">end{equation*}&#39;</span>
-            <span class="k">else</span><span class="p">:</span>  <span class="c"># Text with no math expression(s)unless \ or _ in line</span>
-                <span class="k">if</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">&#39;</span> <span class="ow">in</span> <span class="n">line</span> <span class="ow">or</span> <span class="s">&#39;_&#39;</span> <span class="ow">in</span> <span class="n">line</span> <span class="ow">or</span> <span class="s">&#39;^&#39;</span> <span class="ow">in</span> <span class="n">line</span><span class="p">:</span>
-                    <span class="n">line</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">begin{equation*}</span><span class="se">\n</span><span class="s">&#39;</span> <span class="o">+</span> <span class="n">line</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\n\\</span><span class="s">end{equation*}&#39;</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">line</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">line</span> <span class="o">+=</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">newline </span><span class="se">\n</span><span class="s">&#39;</span>
-            <span class="n">line</span> <span class="o">=</span> <span class="n">process_equals</span><span class="p">(</span><span class="n">line</span><span class="p">)</span>
-            <span class="n">body</span> <span class="o">+=</span> <span class="n">line</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">line</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">StopIteration</span><span class="p">:</span>
-                <span class="k">break</span>
-    <span class="n">body</span> <span class="o">=</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">preamble</span> <span class="o">+</span> <span class="n">body</span> <span class="o">+</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">postscript</span>
-
-    <span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="n">filename</span><span class="p">,</span> <span class="s">&#39;w&#39;</span><span class="p">)</span> <span class="k">as</span> <span class="n">latex_file</span><span class="p">:</span>
-        <span class="n">latex_file</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="n">body</span><span class="p">)</span>
-
-    <span class="n">latex_str</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">xdvi_str</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="k">if</span> <span class="n">find_executable</span><span class="p">(</span><span class="s">&#39;latex&#39;</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">latex_str</span> <span class="o">=</span> <span class="s">&#39;latex&#39;</span>
-
-    <span class="k">if</span> <span class="n">find_executable</span><span class="p">(</span><span class="s">&#39;xdvi&#39;</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">xdvi_str</span> <span class="o">=</span> <span class="s">&#39;xdvi&#39;</span>
-
-    <span class="k">if</span> <span class="n">find_executable</span><span class="p">(</span><span class="s">&#39;yap&#39;</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">xdvi_str</span> <span class="o">=</span> <span class="s">&#39;yap&#39;</span>
-
-    <span class="k">if</span> <span class="n">latex_str</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">xdvi_str</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">debug</span><span class="p">:</span>  <span class="c"># Display latex excution output for debugging purposes</span>
-            <span class="n">os</span><span class="o">.</span><span class="n">system</span><span class="p">(</span><span class="n">latex_str</span> <span class="o">+</span> <span class="s">&#39; &#39;</span> <span class="o">+</span> <span class="n">filename</span><span class="p">[:</span><span class="o">-</span><span class="mi">4</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># Works for Linux don&#39;t know about Windows</span>
-            <span class="k">if</span> <span class="n">sys</span><span class="o">.</span><span class="n">platform</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39;linux&#39;</span><span class="p">):</span>
-                <span class="n">os</span><span class="o">.</span><span class="n">system</span><span class="p">(</span><span class="n">latex_str</span> <span class="o">+</span> <span class="s">&#39; &#39;</span> <span class="o">+</span> <span class="n">filename</span><span class="p">[:</span><span class="o">-</span><span class="mi">4</span><span class="p">]</span> <span class="o">+</span> <span class="s">&#39; &gt; /dev/null&#39;</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">os</span><span class="o">.</span><span class="n">system</span><span class="p">(</span><span class="n">latex_str</span> <span class="o">+</span> <span class="s">&#39; &#39;</span> <span class="o">+</span> <span class="n">filename</span><span class="p">[:</span><span class="o">-</span><span class="mi">4</span><span class="p">]</span> <span class="o">+</span> <span class="s">&#39; &gt; NUL&#39;</span><span class="p">)</span>
-        <span class="n">os</span><span class="o">.</span><span class="n">system</span><span class="p">(</span><span class="n">xdvi_str</span> <span class="o">+</span> <span class="s">&#39; &#39;</span> <span class="o">+</span> <span class="n">filename</span><span class="p">[:</span><span class="o">-</span><span class="mi">4</span><span class="p">]</span> <span class="o">+</span> <span class="s">&#39; &amp;&#39;</span><span class="p">)</span>
-    <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">LaTeX_flg</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="k">return</span>
-
-</div>
-<div class="viewcode-block" id="MV_format"><a class="viewcode-back" href="../../../modules/galgebra/latex_ex.html#sympy.galgebra.latex_ex.MV_format">[docs]</a><span class="k">def</span> <span class="nf">MV_format</span><span class="p">(</span><span class="n">mv_fmt</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    0 or 1 - Print multivector on one line</span>
-
-<span class="sd">    2      - Print each multivector grade on one line</span>
-
-<span class="sd">    3      - Print each multivector base on one line</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">LaTeX_flg</span><span class="p">:</span>
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;mv&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">mv_fmt</span>
-    <span class="k">return</span>
-
-</div>
-<div class="viewcode-block" id="fct_format"><a class="viewcode-back" href="../../../modules/galgebra/latex_ex.html#sympy.galgebra.latex_ex.fct_format">[docs]</a><span class="k">def</span> <span class="nf">fct_format</span><span class="p">(</span><span class="n">fct_fmt</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    0 - Default sympy latex format</span>
-
-<span class="sd">    1 - Do not print arguments of arbitrary functions.</span>
-<span class="sd">        Use symbol font for arbitrary functions.</span>
-<span class="sd">        Use enhanced symbol naming for arbitrary functions.</span>
-<span class="sd">        Use new names for standard functions (acos -&gt; Cos^{-1})</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">LaTeX_flg</span><span class="p">:</span>
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fct</span> <span class="o">=</span> <span class="n">fct_fmt</span>
-    <span class="k">return</span>
-
-</div>
-<div class="viewcode-block" id="pdiff_format"><a class="viewcode-back" href="../../../modules/galgebra/latex_ex.html#sympy.galgebra.latex_ex.pdiff_format">[docs]</a><span class="k">def</span> <span class="nf">pdiff_format</span><span class="p">(</span><span class="n">pdiff_fmt</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    0 - Use default sympy partial derivative format</span>
-
-<span class="sd">    1 - Contracted derivative format (no fraction symbols)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">LaTeX_flg</span><span class="p">:</span>
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;pdiff&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">pdiff_fmt</span>
-    <span class="k">return</span>
-
-</div>
-<div class="viewcode-block" id="sym_format"><a class="viewcode-back" href="../../../modules/galgebra/latex_ex.html#sympy.galgebra.latex_ex.sym_format">[docs]</a><span class="k">def</span> <span class="nf">sym_format</span><span class="p">(</span><span class="n">sym_fmt</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    0 - Use default sympy format</span>
-
-<span class="sd">    1 - Use extended symbol format including multiple Greek letters in</span>
-<span class="sd">        basic symbol (symbol preceding sub and superscripts)and in</span>
-<span class="sd">        sub and superscripts of basic symbol and accents in basic symbol</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">LaTeX_flg</span><span class="p">:</span>
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;sym&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">sym_fmt</span>
-    <span class="k">return</span>
-
-</div>
-<div class="viewcode-block" id="str_format"><a class="viewcode-back" href="../../../modules/galgebra/latex_ex.html#sympy.galgebra.latex_ex.str_format">[docs]</a><span class="k">def</span> <span class="nf">str_format</span><span class="p">(</span><span class="n">str_fmt</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    0 - Use default sympy format</span>
-
-<span class="sd">    1 - Use extended symbol format including multiple Greek letters in</span>
-<span class="sd">        basic symbol (symbol preceding sub and superscripts)and in</span>
-<span class="sd">        sub and superscripts of basic symbol and accents in basic symbol</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">LaTeX_flg</span><span class="p">:</span>
-        <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">fmt_dict</span><span class="p">[</span><span class="s">&#39;str&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">str_fmt</span>
-    <span class="k">return</span>
-
-</div>
-<span class="k">def</span> <span class="nf">ext_str</span><span class="p">(</span><span class="n">xstr</span><span class="p">):</span>
-    <span class="k">return</span><span class="p">(</span><span class="n">LatexPrinter</span><span class="o">.</span><span class="n">extended_symbol</span><span class="p">(</span><span class="n">xstr</span><span class="p">))</span>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-<div id="searchbox" style="display: none">
-  <h3>Quick search</h3>
-    <form class="search" action="../../../search.html" method="get">
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-  <h1>Source code for sympy.geometry.curve</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Curves in 2-dimensional Euclidean space.</span>
-
-<span class="sd">Contains</span>
-<span class="sd">========</span>
-<span class="sd">Curve</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">is_sequence</span>
-<span class="kn">from</span> <span class="nn">sympy.core.containers</span> <span class="kn">import</span> <span class="n">Tuple</span>
-<span class="kn">from</span> <span class="nn">sympy.geometry.entity</span> <span class="kn">import</span> <span class="n">GeometryEntity</span>
-<span class="kn">from</span> <span class="nn">sympy.geometry.point</span> <span class="kn">import</span> <span class="n">Point</span>
-<span class="kn">from</span> <span class="nn">.util</span> <span class="kn">import</span> <span class="n">_symbol</span>
-
-
-<div class="viewcode-block" id="Curve"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.curve.Curve">[docs]</a><span class="k">class</span> <span class="nc">Curve</span><span class="p">(</span><span class="n">GeometryEntity</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A curve in space.</span>
-
-<span class="sd">    A curve is defined by parametric functions for the coordinates, a</span>
-<span class="sd">    parameter and the lower and upper bounds for the parameter value.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    function : list of functions</span>
-<span class="sd">    limits : 3-tuple</span>
-<span class="sd">        Function parameter and lower and upper bounds.</span>
-
-<span class="sd">    Attributes</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    functions</span>
-<span class="sd">    parameter</span>
-<span class="sd">    limits</span>
-
-<span class="sd">    Raises</span>
-<span class="sd">    ======</span>
-
-<span class="sd">    ValueError</span>
-<span class="sd">        When `functions` are specified incorrectly.</span>
-<span class="sd">        When `limits` are specified incorrectly.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.core.function.Function</span>
-<span class="sd">    sympy.polys.polyfuncs.interpolate</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import sin, cos, Symbol, interpolate</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import t, a</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.geometry import Curve</span>
-<span class="sd">    &gt;&gt;&gt; C = Curve((sin(t), cos(t)), (t, 0, 2))</span>
-<span class="sd">    &gt;&gt;&gt; C.functions</span>
-<span class="sd">    (sin(t), cos(t))</span>
-<span class="sd">    &gt;&gt;&gt; C.limits</span>
-<span class="sd">    (t, 0, 2)</span>
-<span class="sd">    &gt;&gt;&gt; C.parameter</span>
-<span class="sd">    t</span>
-<span class="sd">    &gt;&gt;&gt; C = Curve((t, interpolate([1, 4, 9, 16], t)), (t, 0, 1)); C</span>
-<span class="sd">    Curve((t, t**2), (t, 0, 1))</span>
-<span class="sd">    &gt;&gt;&gt; C.subs(t, 4)</span>
-<span class="sd">    Point(4, 16)</span>
-<span class="sd">    &gt;&gt;&gt; C.arbitrary_point(a)</span>
-<span class="sd">    Point(a, a**2)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">function</span><span class="p">,</span> <span class="n">limits</span><span class="p">):</span>
-        <span class="n">fun</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">function</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">fun</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">fun</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Function argument should be (x(t), y(t)) &quot;</span>
-                <span class="s">&quot;but got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">str</span><span class="p">(</span><span class="n">function</span><span class="p">))</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">limits</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">limits</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Limit argument should be (t, tmin, tmax) &quot;</span>
-                <span class="s">&quot;but got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">str</span><span class="p">(</span><span class="n">limits</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">GeometryEntity</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">fun</span><span class="p">),</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">limits</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">old</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">parameter</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">f</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">)</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">functions</span><span class="p">])</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Curve.free_symbols"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.curve.Curve.free_symbols">[docs]</a>    <span class="k">def</span> <span class="nf">free_symbols</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return a set of symbols other than the bound symbols used to</span>
-<span class="sd">        parametrically define the Curve.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import t, a</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Curve</span>
-<span class="sd">        &gt;&gt;&gt; Curve((t, t**2), (t, 0, 2)).free_symbols</span>
-<span class="sd">        set()</span>
-<span class="sd">        &gt;&gt;&gt; Curve((t, t**2), (t, a, 2)).free_symbols</span>
-<span class="sd">        set([a])</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">free</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">functions</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">limits</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="n">free</span> <span class="o">|=</span> <span class="n">a</span><span class="o">.</span><span class="n">free_symbols</span>
-        <span class="n">free</span> <span class="o">=</span> <span class="n">free</span><span class="o">.</span><span class="n">difference</span><span class="p">(</span><span class="nb">set</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">parameter</span><span class="p">]))</span>
-        <span class="k">return</span> <span class="n">free</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Curve.functions"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.curve.Curve.functions">[docs]</a>    <span class="k">def</span> <span class="nf">functions</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The functions specifying the curve.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        functions : list of parameterized coordinate functions.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        parameter</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import t</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Curve</span>
-<span class="sd">        &gt;&gt;&gt; C = Curve((t, t**2), (t, 0, 2))</span>
-<span class="sd">        &gt;&gt;&gt; C.functions</span>
-<span class="sd">        (t, t**2)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Curve.parameter"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.curve.Curve.parameter">[docs]</a>    <span class="k">def</span> <span class="nf">parameter</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The curve function variable.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        parameter : SymPy symbol</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        functions</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import t</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Curve</span>
-<span class="sd">        &gt;&gt;&gt; C = Curve([t, t**2], (t, 0, 2))</span>
-<span class="sd">        &gt;&gt;&gt; C.parameter</span>
-<span class="sd">        t</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Curve.limits"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.curve.Curve.limits">[docs]</a>    <span class="k">def</span> <span class="nf">limits</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The limits for the curve.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        limits : tuple</span>
-<span class="sd">            Contains parameter and lower and upper limits.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        plot_interval</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import t</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Curve</span>
-<span class="sd">        &gt;&gt;&gt; C = Curve([t, t**3], (t, -2, 2))</span>
-<span class="sd">        &gt;&gt;&gt; C.limits</span>
-<span class="sd">        (t, -2, 2)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-</div>
-<div class="viewcode-block" id="Curve.rotate"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.curve.Curve.rotate">[docs]</a>    <span class="k">def</span> <span class="nf">rotate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">angle</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">pt</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Rotate ``angle`` radians counterclockwise about Point ``pt``.</span>
-
-<span class="sd">        The default pt is the origin, Point(0, 0).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry.curve import Curve</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import pi</span>
-<span class="sd">        &gt;&gt;&gt; Curve((x, x), (x, 0, 1)).rotate(pi/2)</span>
-<span class="sd">        Curve((-x, x), (x, 0, 1))</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">rot_axis3</span>
-        <span class="n">pt</span> <span class="o">=</span> <span class="o">-</span><span class="n">Point</span><span class="p">(</span><span class="n">pt</span> <span class="ow">or</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="o">*</span><span class="n">pt</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">rv</span><span class="o">.</span><span class="n">functions</span><span class="p">)</span>
-        <span class="n">f</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
-        <span class="n">f</span> <span class="o">*=</span> <span class="n">rot_axis3</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">tolist</span><span class="p">()[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">limits</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">pt</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">pt</span> <span class="o">=</span> <span class="o">-</span><span class="n">pt</span>
-            <span class="k">return</span> <span class="n">rv</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="o">*</span><span class="n">pt</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">rv</span>
-</div>
-<div class="viewcode-block" id="Curve.scale"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.curve.Curve.scale">[docs]</a>    <span class="k">def</span> <span class="nf">scale</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">pt</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Override GeometryEntity.scale since Curve is not made up of Points.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry.curve import Curve</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import pi</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; Curve((x, x), (x, 0, 1)).scale(2)</span>
-<span class="sd">        Curve((2*x, x), (x, 0, 1))</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">pt</span><span class="p">:</span>
-            <span class="n">pt</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">pt</span><span class="p">)</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">pt</span><span class="p">)</span><span class="o">.</span><span class="n">args</span><span class="p">)</span><span class="o">.</span><span class="n">scale</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="o">*</span><span class="n">pt</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">fx</span><span class="p">,</span> <span class="n">fy</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">functions</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">((</span><span class="n">fx</span><span class="o">*</span><span class="n">x</span><span class="p">,</span> <span class="n">fy</span><span class="o">*</span><span class="n">y</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">limits</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Curve.translate"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.curve.Curve.translate">[docs]</a>    <span class="k">def</span> <span class="nf">translate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Translate the Curve by (x, y).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry.curve import Curve</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import pi</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; Curve((x, x), (x, 0, 1)).translate(1, 2)</span>
-<span class="sd">        Curve((x + 1, x + 2), (x, 0, 1))</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">fx</span><span class="p">,</span> <span class="n">fy</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">functions</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">((</span><span class="n">fx</span> <span class="o">+</span> <span class="n">x</span><span class="p">,</span> <span class="n">fy</span> <span class="o">+</span> <span class="n">y</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">limits</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Curve.arbitrary_point"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.curve.Curve.arbitrary_point">[docs]</a>    <span class="k">def</span> <span class="nf">arbitrary_point</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">parameter</span><span class="o">=</span><span class="s">&#39;t&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        A parameterized point on the curve.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        parameter : str or Symbol, optional</span>
-<span class="sd">            Default value is &#39;t&#39;;</span>
-<span class="sd">            the Curve&#39;s parameter is selected with None or self.parameter</span>
-<span class="sd">            otherwise the provided symbol is used.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        arbitrary_point : Point</span>
-
-<span class="sd">        Raises</span>
-<span class="sd">        ======</span>
-
-<span class="sd">        ValueError</span>
-<span class="sd">            When `parameter` already appears in the functions.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import s</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Curve</span>
-<span class="sd">        &gt;&gt;&gt; C = Curve([2*s, s**2], (s, 0, 2))</span>
-<span class="sd">        &gt;&gt;&gt; C.arbitrary_point()</span>
-<span class="sd">        Point(2*t, t**2)</span>
-<span class="sd">        &gt;&gt;&gt; C.arbitrary_point(C.parameter)</span>
-<span class="sd">        Point(2*s, s**2)</span>
-<span class="sd">        &gt;&gt;&gt; C.arbitrary_point(None)</span>
-<span class="sd">        Point(2*s, s**2)</span>
-<span class="sd">        &gt;&gt;&gt; C.arbitrary_point(Symbol(&#39;a&#39;))</span>
-<span class="sd">        Point(2*a, a**2)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">parameter</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">functions</span><span class="p">)</span>
-
-        <span class="n">tnew</span> <span class="o">=</span> <span class="n">_symbol</span><span class="p">(</span><span class="n">parameter</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">parameter</span><span class="p">)</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">parameter</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">tnew</span><span class="o">.</span><span class="n">name</span> <span class="o">!=</span> <span class="n">t</span><span class="o">.</span><span class="n">name</span> <span class="ow">and</span>
-                <span class="n">tnew</span><span class="o">.</span><span class="n">name</span> <span class="ow">in</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">name</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">)):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Symbol </span><span class="si">%s</span><span class="s"> already appears in object &#39;</span>
-                <span class="s">&#39;and cannot be used as a parameter.&#39;</span> <span class="o">%</span> <span class="n">tnew</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">w</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">tnew</span><span class="p">)</span> <span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">functions</span><span class="p">])</span>
-</div>
-<div class="viewcode-block" id="Curve.plot_interval"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.curve.Curve.plot_interval">[docs]</a>    <span class="k">def</span> <span class="nf">plot_interval</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">parameter</span><span class="o">=</span><span class="s">&#39;t&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The plot interval for the default geometric plot of the curve.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        parameter : str or Symbol, optional</span>
-<span class="sd">            Default value is &#39;t&#39;;</span>
-<span class="sd">            otherwise the provided symbol is used.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        plot_interval : list (plot interval)</span>
-<span class="sd">            [parameter, lower_bound, upper_bound]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        limits : Returns limits of the parameter interval</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Curve, sin</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, t, s</span>
-<span class="sd">        &gt;&gt;&gt; Curve((x, sin(x)), (x, 1, 2)).plot_interval()</span>
-<span class="sd">        [t, 1, 2]</span>
-<span class="sd">        &gt;&gt;&gt; Curve((x, sin(x)), (x, 1, 2)).plot_interval(s)</span>
-<span class="sd">        [s, 1, 2]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">_symbol</span><span class="p">(</span><span class="n">parameter</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">parameter</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">t</span><span class="p">]</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">limits</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span></div></div>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.geometry.ellipse</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Elliptical geometrical entities.</span>
-
-<span class="sd">Contains</span>
-<span class="sd">* Ellipse</span>
-<span class="sd">* Circle</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">Dummy</span>
-<span class="kn">from</span> <span class="nn">sympy.core.logic</span> <span class="kn">import</span> <span class="n">fuzzy_bool</span>
-<span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">trigsimp</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">Max</span><span class="p">,</span> <span class="n">Min</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.complexes</span> <span class="kn">import</span> <span class="n">im</span>
-<span class="kn">from</span> <span class="nn">sympy.geometry.exceptions</span> <span class="kn">import</span> <span class="n">GeometryError</span>
-<span class="kn">from</span> <span class="nn">sympy.solvers</span> <span class="kn">import</span> <span class="n">solve</span>
-<span class="kn">from</span> <span class="nn">.entity</span> <span class="kn">import</span> <span class="n">GeometryEntity</span>
-<span class="kn">from</span> <span class="nn">.point</span> <span class="kn">import</span> <span class="n">Point</span>
-<span class="kn">from</span> <span class="nn">.line</span> <span class="kn">import</span> <span class="n">LinearEntity</span><span class="p">,</span> <span class="n">Line</span>
-<span class="kn">from</span> <span class="nn">.util</span> <span class="kn">import</span> <span class="n">_symbol</span><span class="p">,</span> <span class="n">idiff</span>
-
-<span class="kn">import</span> <span class="nn">random</span>
-
-
-<div class="viewcode-block" id="Ellipse"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse">[docs]</a><span class="k">class</span> <span class="nc">Ellipse</span><span class="p">(</span><span class="n">GeometryEntity</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;An elliptical GeometryEntity.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    center : Point, optional</span>
-<span class="sd">        Default value is Point(0, 0)</span>
-<span class="sd">    hradius : number or SymPy expression, optional</span>
-<span class="sd">    vradius : number or SymPy expression, optional</span>
-<span class="sd">    eccentricity : number or SymPy expression, optional</span>
-<span class="sd">        Two of `hradius`, `vradius` and `eccentricity` must be supplied to</span>
-<span class="sd">        create an Ellipse. The third is derived from the two supplied.</span>
-
-<span class="sd">    Attributes</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    center</span>
-<span class="sd">    hradius</span>
-<span class="sd">    vradius</span>
-<span class="sd">    area</span>
-<span class="sd">    circumference</span>
-<span class="sd">    eccentricity</span>
-<span class="sd">    periapsis</span>
-<span class="sd">    apoapsis</span>
-<span class="sd">    focus_distance</span>
-<span class="sd">    foci</span>
-
-<span class="sd">    Raises</span>
-<span class="sd">    ======</span>
-
-<span class="sd">    GeometryError</span>
-<span class="sd">        When `hradius`, `vradius` and `eccentricity` are incorrectly supplied</span>
-<span class="sd">        as parameters.</span>
-<span class="sd">    TypeError</span>
-<span class="sd">        When `center` is not a Point.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Circle</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    -----</span>
-<span class="sd">    Constructed from a center and two radii, the first being the horizontal</span>
-<span class="sd">    radius (along the x-axis) and the second being the vertical radius (along</span>
-<span class="sd">    the y-axis).</span>
-
-<span class="sd">    When symbolic value for hradius and vradius are used, any calculation that</span>
-<span class="sd">    refers to the foci or the major or minor axis will assume that the ellipse</span>
-<span class="sd">    has its major radius on the x-axis. If this is not true then a manual</span>
-<span class="sd">    rotation is necessary.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Ellipse, Point, Rational</span>
-<span class="sd">    &gt;&gt;&gt; e1 = Ellipse(Point(0, 0), 5, 1)</span>
-<span class="sd">    &gt;&gt;&gt; e1.hradius, e1.vradius</span>
-<span class="sd">    (5, 1)</span>
-<span class="sd">    &gt;&gt;&gt; e2 = Ellipse(Point(3, 1), hradius=3, eccentricity=Rational(4, 5))</span>
-<span class="sd">    &gt;&gt;&gt; e2</span>
-<span class="sd">    Ellipse(Point(3, 1), 3, 9/5)</span>
-
-<span class="sd">    Plotting:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Circle, Plot, Segment</span>
-<span class="sd">    &gt;&gt;&gt; c1 = Circle(Point(0,0), 1)</span>
-<span class="sd">    &gt;&gt;&gt; Plot(c1)                                # doctest: +SKIP</span>
-<span class="sd">    [0]: cos(t), sin(t), &#39;mode=parametric&#39;</span>
-<span class="sd">    &gt;&gt;&gt; p = Plot()                              # doctest: +SKIP</span>
-<span class="sd">    &gt;&gt;&gt; p[0] = c1                               # doctest: +SKIP</span>
-<span class="sd">    &gt;&gt;&gt; radius = Segment(c1.center, c1.random_point())  # doctest: +SKIP</span>
-<span class="sd">    &gt;&gt;&gt; p[1] = radius                           # doctest: +SKIP</span>
-<span class="sd">    &gt;&gt;&gt; p                                       # doctest: +SKIP</span>
-<span class="sd">    [0]: cos(t), sin(t), &#39;mode=parametric&#39;</span>
-<span class="sd">    [1]: t*cos(1.546086215036205357975518382),</span>
-<span class="sd">    t*sin(1.546086215036205357975518382), &#39;mode=parametric&#39;</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span>
-        <span class="n">cls</span><span class="p">,</span> <span class="n">center</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">hradius</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">vradius</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">eccentricity</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span>
-            <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">hradius</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">hradius</span><span class="p">)</span>
-        <span class="n">vradius</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">vradius</span><span class="p">)</span>
-
-        <span class="n">eccentricity</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">eccentricity</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">center</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">center</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">center</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">center</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">([</span><span class="n">_f</span> <span class="k">for</span> <span class="n">_f</span> <span class="ow">in</span> <span class="p">(</span><span class="n">hradius</span><span class="p">,</span> <span class="n">vradius</span><span class="p">,</span> <span class="n">eccentricity</span><span class="p">)</span> <span class="k">if</span> <span class="n">_f</span><span class="p">])</span> <span class="o">!=</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Exactly two arguments of &quot;hradius&quot;, &#39;</span>
-                <span class="s">&#39;&quot;vradius&quot;, and &quot;eccentricity&quot; must not be None.&quot;&#39;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">eccentricity</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">hradius</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">hradius</span> <span class="o">=</span> <span class="n">vradius</span> <span class="o">/</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">eccentricity</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">vradius</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">vradius</span> <span class="o">=</span> <span class="n">hradius</span> <span class="o">*</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">eccentricity</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">hradius</span> <span class="o">==</span> <span class="n">vradius</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Circle</span><span class="p">(</span><span class="n">center</span><span class="p">,</span> <span class="n">hradius</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">GeometryEntity</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">center</span><span class="p">,</span> <span class="n">hradius</span><span class="p">,</span> <span class="n">vradius</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Ellipse.center"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse.center">[docs]</a>    <span class="k">def</span> <span class="nf">center</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The center of the ellipse.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        center : number</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ellipse</span>
-<span class="sd">        &gt;&gt;&gt; p1 = Point(0, 0)</span>
-<span class="sd">        &gt;&gt;&gt; e1 = Ellipse(p1, 3, 1)</span>
-<span class="sd">        &gt;&gt;&gt; e1.center</span>
-<span class="sd">        Point(0, 0)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Ellipse.hradius"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse.hradius">[docs]</a>    <span class="k">def</span> <span class="nf">hradius</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The horizontal radius of the ellipse.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        hradius : number</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        vradius, major, minor</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ellipse</span>
-<span class="sd">        &gt;&gt;&gt; p1 = Point(0, 0)</span>
-<span class="sd">        &gt;&gt;&gt; e1 = Ellipse(p1, 3, 1)</span>
-<span class="sd">        &gt;&gt;&gt; e1.hradius</span>
-<span class="sd">        3</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Ellipse.vradius"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse.vradius">[docs]</a>    <span class="k">def</span> <span class="nf">vradius</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The vertical radius of the ellipse.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        vradius : number</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        hradius, major, minor</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ellipse</span>
-<span class="sd">        &gt;&gt;&gt; p1 = Point(0, 0)</span>
-<span class="sd">        &gt;&gt;&gt; e1 = Ellipse(p1, 3, 1)</span>
-<span class="sd">        &gt;&gt;&gt; e1.vradius</span>
-<span class="sd">        1</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Ellipse.minor"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse.minor">[docs]</a>    <span class="k">def</span> <span class="nf">minor</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Shorter axis of the ellipse (if it can be determined) else vradius.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        minor : number or expression</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        hradius, vradius, major</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ellipse, Symbol</span>
-<span class="sd">        &gt;&gt;&gt; p1 = Point(0, 0)</span>
-<span class="sd">        &gt;&gt;&gt; e1 = Ellipse(p1, 3, 1)</span>
-<span class="sd">        &gt;&gt;&gt; e1.minor</span>
-<span class="sd">        1</span>
-
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; b = Symbol(&#39;b&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; Ellipse(p1, a, b).minor</span>
-<span class="sd">        b</span>
-<span class="sd">        &gt;&gt;&gt; Ellipse(p1, b, a).minor</span>
-<span class="sd">        a</span>
-
-<span class="sd">        &gt;&gt;&gt; m = Symbol(&#39;m&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; M = m + 1</span>
-<span class="sd">        &gt;&gt;&gt; Ellipse(p1, m, M).minor</span>
-<span class="sd">        m</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">Min</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="mi">3</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">rv</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Min</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">vradius</span>
-        <span class="k">return</span> <span class="n">rv</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Ellipse.major"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse.major">[docs]</a>    <span class="k">def</span> <span class="nf">major</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Longer axis of the ellipse (if it can be determined) else hradius.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        major : number or expression</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        hradius, vradius, minor</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ellipse, Symbol</span>
-<span class="sd">        &gt;&gt;&gt; p1 = Point(0, 0)</span>
-<span class="sd">        &gt;&gt;&gt; e1 = Ellipse(p1, 3, 1)</span>
-<span class="sd">        &gt;&gt;&gt; e1.major</span>
-<span class="sd">        3</span>
-
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; b = Symbol(&#39;b&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; Ellipse(p1, a, b).major</span>
-<span class="sd">        a</span>
-<span class="sd">        &gt;&gt;&gt; Ellipse(p1, b, a).major</span>
-<span class="sd">        b</span>
-
-<span class="sd">        &gt;&gt;&gt; m = Symbol(&#39;m&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; M = m + 1</span>
-<span class="sd">        &gt;&gt;&gt; Ellipse(p1, m, M).major</span>
-<span class="sd">        m + 1</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">Max</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="mi">3</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">rv</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Max</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">hradius</span>
-        <span class="k">return</span> <span class="n">rv</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Ellipse.area"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse.area">[docs]</a>    <span class="k">def</span> <span class="nf">area</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The area of the ellipse.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        area : number</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ellipse</span>
-<span class="sd">        &gt;&gt;&gt; p1 = Point(0, 0)</span>
-<span class="sd">        &gt;&gt;&gt; e1 = Ellipse(p1, 3, 1)</span>
-<span class="sd">        &gt;&gt;&gt; e1.area</span>
-<span class="sd">        3*pi</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">simplify</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">hradius</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">vradius</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Ellipse.circumference"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse.circumference">[docs]</a>    <span class="k">def</span> <span class="nf">circumference</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The circumference of the ellipse.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ellipse</span>
-<span class="sd">        &gt;&gt;&gt; p1 = Point(0, 0)</span>
-<span class="sd">        &gt;&gt;&gt; e1 = Ellipse(p1, 3, 1)</span>
-<span class="sd">        &gt;&gt;&gt; e1.circumference</span>
-<span class="sd">        12*Integral(sqrt((-8*_x**2/9 + 1)/(-_x**2 + 1)), (_x, 0, 1))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">eccentricity</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">hradius</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="k">return</span> <span class="mi">4</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">major</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">Integral</span><span class="p">(</span>
-                <span class="n">sqrt</span><span class="p">((</span><span class="mi">1</span> <span class="o">-</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">eccentricity</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Ellipse.eccentricity"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse.eccentricity">[docs]</a>    <span class="k">def</span> <span class="nf">eccentricity</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The eccentricity of the ellipse.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        eccentricity : number</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ellipse, sqrt</span>
-<span class="sd">        &gt;&gt;&gt; p1 = Point(0, 0)</span>
-<span class="sd">        &gt;&gt;&gt; e1 = Ellipse(p1, 3, sqrt(2))</span>
-<span class="sd">        &gt;&gt;&gt; e1.eccentricity</span>
-<span class="sd">        sqrt(7)/3</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">focus_distance</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">major</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Ellipse.periapsis"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse.periapsis">[docs]</a>    <span class="k">def</span> <span class="nf">periapsis</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The periapsis of the ellipse.</span>
-
-<span class="sd">        The shortest distance between the focus and the contour.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        periapsis : number</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        apoapsis : Returns greatest distance between focus and contour</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ellipse</span>
-<span class="sd">        &gt;&gt;&gt; p1 = Point(0, 0)</span>
-<span class="sd">        &gt;&gt;&gt; e1 = Ellipse(p1, 3, 1)</span>
-<span class="sd">        &gt;&gt;&gt; e1.periapsis</span>
-<span class="sd">        -2*sqrt(2) + 3</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">major</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">eccentricity</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Ellipse.apoapsis"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse.apoapsis">[docs]</a>    <span class="k">def</span> <span class="nf">apoapsis</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The apoapsis of the ellipse.</span>
-
-<span class="sd">        The greatest distance between the focus and the contour.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        apoapsis : number</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        periapsis : Returns shortest distance between foci and contour</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ellipse</span>
-<span class="sd">        &gt;&gt;&gt; p1 = Point(0, 0)</span>
-<span class="sd">        &gt;&gt;&gt; e1 = Ellipse(p1, 3, 1)</span>
-<span class="sd">        &gt;&gt;&gt; e1.apoapsis</span>
-<span class="sd">        2*sqrt(2) + 3</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">major</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">eccentricity</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Ellipse.focus_distance"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse.focus_distance">[docs]</a>    <span class="k">def</span> <span class="nf">focus_distance</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The focale distance of the ellipse.</span>
-
-<span class="sd">        The distance between the center and one focus.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        focus_distance : number</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        foci</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ellipse</span>
-<span class="sd">        &gt;&gt;&gt; p1 = Point(0, 0)</span>
-<span class="sd">        &gt;&gt;&gt; e1 = Ellipse(p1, 3, 1)</span>
-<span class="sd">        &gt;&gt;&gt; e1.focus_distance</span>
-<span class="sd">        2*sqrt(2)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Point</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">center</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">foci</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Ellipse.foci"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse.foci">[docs]</a>    <span class="k">def</span> <span class="nf">foci</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The foci of the ellipse.</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        -----</span>
-<span class="sd">        The foci can only be calculated if the major/minor axes are known.</span>
-
-<span class="sd">        Raises</span>
-<span class="sd">        ======</span>
-
-<span class="sd">        ValueError</span>
-<span class="sd">            When the major and minor axis cannot be determined.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point</span>
-<span class="sd">        focus_distance : Returns the distance between focus and center</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ellipse</span>
-<span class="sd">        &gt;&gt;&gt; p1 = Point(0, 0)</span>
-<span class="sd">        &gt;&gt;&gt; e1 = Ellipse(p1, 3, 1)</span>
-<span class="sd">        &gt;&gt;&gt; e1.foci</span>
-<span class="sd">        (Point(-2*sqrt(2), 0), Point(2*sqrt(2), 0))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">center</span>
-        <span class="n">hr</span><span class="p">,</span> <span class="n">vr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">hradius</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">vradius</span>
-        <span class="k">if</span> <span class="n">hr</span> <span class="o">==</span> <span class="n">vr</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
-
-        <span class="c"># calculate focus distance manually, since focus_distance calls this routine</span>
-        <span class="n">fd</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">major</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">minor</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">hr</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">minor</span><span class="p">:</span>
-            <span class="c"># foci on the y-axis</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">c</span> <span class="o">+</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="n">fd</span><span class="p">),</span> <span class="n">c</span> <span class="o">+</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">fd</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="n">hr</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">major</span><span class="p">:</span>
-            <span class="c"># foci on the x-axis</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">c</span> <span class="o">+</span> <span class="n">Point</span><span class="p">(</span><span class="o">-</span><span class="n">fd</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">c</span> <span class="o">+</span> <span class="n">Point</span><span class="p">(</span><span class="n">fd</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="Ellipse.rotate"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse.rotate">[docs]</a>    <span class="k">def</span> <span class="nf">rotate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">angle</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">pt</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Rotate ``angle`` radians counterclockwise about Point ``pt``.</span>
-
-<span class="sd">        Note: since the general ellipse is not supported, the axes of</span>
-<span class="sd">        the ellipse will not be rotated. Only the center is rotated to</span>
-<span class="sd">        a new position.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Ellipse, pi</span>
-<span class="sd">        &gt;&gt;&gt; Ellipse((1, 0), 2, 1).rotate(pi/2)</span>
-<span class="sd">        Ellipse(Point(0, 1), 2, 1)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Ellipse</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">rotate</span><span class="p">(</span><span class="n">angle</span><span class="p">,</span> <span class="n">pt</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Ellipse.scale"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse.scale">[docs]</a>    <span class="k">def</span> <span class="nf">scale</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">pt</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Override GeometryEntity.scale since it is the major and minor</span>
-<span class="sd">        axes which must be scaled and they are not GeometryEntities.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Ellipse</span>
-<span class="sd">        &gt;&gt;&gt; Ellipse((0, 0), 2, 1).scale(2, 4)</span>
-<span class="sd">        Circle(Point(0, 0), 4)</span>
-<span class="sd">        &gt;&gt;&gt; Ellipse((0, 0), 2, 1).scale(2)</span>
-<span class="sd">        Ellipse(Point(0, 0), 4, 1)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">center</span>
-        <span class="k">if</span> <span class="n">pt</span><span class="p">:</span>
-            <span class="n">pt</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">pt</span><span class="p">)</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">pt</span><span class="p">)</span><span class="o">.</span><span class="n">args</span><span class="p">)</span><span class="o">.</span><span class="n">scale</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="o">*</span><span class="n">pt</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">hradius</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">vradius</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">c</span><span class="o">.</span><span class="n">scale</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="n">hradius</span><span class="o">=</span><span class="n">h</span><span class="o">*</span><span class="n">x</span><span class="p">,</span> <span class="n">vradius</span><span class="o">=</span><span class="n">v</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Ellipse.encloses_point"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse.encloses_point">[docs]</a>    <span class="k">def</span> <span class="nf">encloses_point</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if p is enclosed by (is inside of) self.</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        -----</span>
-<span class="sd">        Being on the border of self is considered False.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        p : Point</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        encloses_point : True, False or None</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Ellipse, S</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import t</span>
-<span class="sd">        &gt;&gt;&gt; e = Ellipse((0, 0), 3, 2)</span>
-<span class="sd">        &gt;&gt;&gt; e.encloses_point((0, 0))</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; e.encloses_point(e.arbitrary_point(t).subs(t, S.Half))</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; e.encloses_point((4, 0))</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">p</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">foci</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="c"># if the combined distance from the foci to p (h1 + h2) is less</span>
-            <span class="c"># than the combined distance from the foci to the minor axis</span>
-            <span class="c"># (which is the same as the major axis length) then p is inside</span>
-            <span class="c"># the ellipse</span>
-            <span class="n">h1</span><span class="p">,</span> <span class="n">h2</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">foci</span><span class="p">]</span>
-            <span class="n">test</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">major</span> <span class="o">-</span> <span class="p">(</span><span class="n">h1</span> <span class="o">+</span> <span class="n">h2</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">test</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">radius</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">center</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">fuzzy_bool</span><span class="p">(</span><span class="n">test</span><span class="o">.</span><span class="n">is_positive</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Ellipse.tangent_lines"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse.tangent_lines">[docs]</a>    <span class="k">def</span> <span class="nf">tangent_lines</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Tangent lines between `p` and the ellipse.</span>
-
-<span class="sd">        If `p` is on the ellipse, returns the tangent line through point `p`.</span>
-<span class="sd">        Otherwise, returns the tangent line(s) from `p` to the ellipse, or</span>
-<span class="sd">        None if no tangent line is possible (e.g., `p` inside ellipse).</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        p : Point</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        tangent_lines : list with 1 or 2 Lines</span>
-
-<span class="sd">        Raises</span>
-<span class="sd">        ======</span>
-
-<span class="sd">        NotImplementedError</span>
-<span class="sd">            Can only find tangent lines for a point, `p`, on the ellipse.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point, sympy.geometry.line.Line</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ellipse</span>
-<span class="sd">        &gt;&gt;&gt; e1 = Ellipse(Point(0, 0), 3, 2)</span>
-<span class="sd">        &gt;&gt;&gt; e1.tangent_lines(Point(3, 0))</span>
-<span class="sd">        [Line(Point(3, 0), Point(3, -12))]</span>
-
-<span class="sd">        &gt;&gt;&gt; # This will plot an ellipse together with a tangent line.</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ellipse, Plot</span>
-<span class="sd">        &gt;&gt;&gt; e = Ellipse(Point(0,0), 3, 2)</span>
-<span class="sd">        &gt;&gt;&gt; t = e.tangent_lines(e.random_point()) # doctest: +SKIP</span>
-<span class="sd">        &gt;&gt;&gt; p = Plot() # doctest: +SKIP</span>
-<span class="sd">        &gt;&gt;&gt; p[0] = e # doctest: +SKIP</span>
-<span class="sd">        &gt;&gt;&gt; p[1] = t # doctest: +SKIP</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">encloses_point</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">[]</span>
-
-        <span class="k">if</span> <span class="n">p</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">:</span>
-            <span class="n">delta</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">center</span> <span class="o">-</span> <span class="n">p</span>
-            <span class="n">rise</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">vradius</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">delta</span><span class="o">.</span><span class="n">x</span>
-            <span class="n">run</span> <span class="o">=</span> <span class="o">-</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">hradius</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">delta</span><span class="o">.</span><span class="n">y</span>
-            <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">simplify</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">run</span><span class="p">),</span>
-                       <span class="n">simplify</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">y</span> <span class="o">+</span> <span class="n">rise</span><span class="p">))</span>
-            <span class="k">return</span> <span class="p">[</span><span class="n">Line</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">p2</span><span class="p">)]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">foci</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">f1</span><span class="p">,</span> <span class="n">f2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">foci</span>
-                <span class="n">maj</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">hradius</span>
-                <span class="n">test</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">maj</span> <span class="o">-</span>
-                        <span class="n">Point</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">f1</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span> <span class="o">-</span>
-                        <span class="n">Point</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">f2</span><span class="p">,</span> <span class="n">p</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">test</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">radius</span> <span class="o">-</span> <span class="n">Point</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">center</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">test</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="n">test</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[]</span>
-            <span class="c"># else p is outside the ellipse or we can&#39;t tell. In case of the</span>
-            <span class="c"># latter, the solutions returned will only be valid if</span>
-            <span class="c"># the point is not inside the ellipse; if it is, nan will result.</span>
-            <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">),</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-            <span class="n">eq</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">equation</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-            <span class="n">dydx</span> <span class="o">=</span> <span class="n">idiff</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-            <span class="n">slope</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">Point</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">slope</span>
-            <span class="n">tangent_points</span> <span class="o">=</span> <span class="n">solve</span><span class="p">([</span><span class="n">slope</span> <span class="o">-</span> <span class="n">dydx</span><span class="p">,</span> <span class="n">eq</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">])</span>
-
-            <span class="c"># handle horizontal and vertical tangent lines</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">tangent_points</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">assert</span> <span class="n">tangent_points</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span>
-                    <span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">p</span><span class="o">.</span><span class="n">x</span> <span class="ow">or</span> <span class="n">tangent_points</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">p</span><span class="o">.</span><span class="n">y</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">Line</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">p</span> <span class="o">+</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)),</span> <span class="n">Line</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">p</span> <span class="o">+</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))]</span>
-
-            <span class="c"># others</span>
-            <span class="k">return</span> <span class="p">[</span><span class="n">Line</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">tangent_points</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">Line</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">tangent_points</span><span class="p">[</span><span class="mi">1</span><span class="p">])]</span>
-</div>
-<div class="viewcode-block" id="Ellipse.is_tangent"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse.is_tangent">[docs]</a>    <span class="k">def</span> <span class="nf">is_tangent</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Is `o` tangent to the ellipse?</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        o : GeometryEntity</span>
-<span class="sd">            An Ellipse, LinearEntity or Polygon</span>
-
-<span class="sd">        Raises</span>
-<span class="sd">        ======</span>
-
-<span class="sd">        NotImplementedError</span>
-<span class="sd">            When the wrong type of argument is supplied.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        is_tangent: boolean</span>
-<span class="sd">            True if o is tangent to the ellipse, False otherwise.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        tangent_lines</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ellipse, Line</span>
-<span class="sd">        &gt;&gt;&gt; p0, p1, p2 = Point(0, 0), Point(3, 0), Point(3, 3)</span>
-<span class="sd">        &gt;&gt;&gt; e1 = Ellipse(p0, 3, 2)</span>
-<span class="sd">        &gt;&gt;&gt; l1 = Line(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; e1.is_tangent(l1)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">inter</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Ellipse</span><span class="p">):</span>
-            <span class="n">inter</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">inter</span><span class="p">,</span> <span class="n">Ellipse</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">inter</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">inter</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Point</span><span class="p">)</span>
-                    <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">inter</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">LinearEntity</span><span class="p">):</span>
-            <span class="n">inter</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_do_line_intersection</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">inter</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">inter</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">inter</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">in</span> <span class="n">o</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Polygon</span><span class="p">):</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">for</span> <span class="n">seg</span> <span class="ow">in</span> <span class="n">o</span><span class="o">.</span><span class="n">sides</span><span class="p">:</span>
-                <span class="n">inter</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_do_line_intersection</span><span class="p">(</span><span class="n">seg</span><span class="p">)</span>
-                <span class="n">c</span> <span class="o">+=</span> <span class="nb">len</span><span class="p">([</span><span class="bp">True</span> <span class="k">for</span> <span class="n">point</span> <span class="ow">in</span> <span class="n">inter</span> <span class="k">if</span> <span class="n">point</span> <span class="ow">in</span> <span class="n">seg</span><span class="p">])</span>
-            <span class="k">return</span> <span class="n">c</span> <span class="o">==</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Unknown argument type&quot;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Ellipse.arbitrary_point"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse.arbitrary_point">[docs]</a>    <span class="k">def</span> <span class="nf">arbitrary_point</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">parameter</span><span class="o">=</span><span class="s">&#39;t&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A parameterized point on the ellipse.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        parameter : str, optional</span>
-<span class="sd">            Default value is &#39;t&#39;.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        arbitrary_point : Point</span>
-
-<span class="sd">        Raises</span>
-<span class="sd">        ======</span>
-
-<span class="sd">        ValueError</span>
-<span class="sd">            When `parameter` already appears in the functions.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ellipse</span>
-<span class="sd">        &gt;&gt;&gt; e1 = Ellipse(Point(0, 0), 3, 2)</span>
-<span class="sd">        &gt;&gt;&gt; e1.arbitrary_point()</span>
-<span class="sd">        Point(3*cos(t), 2*sin(t))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">_symbol</span><span class="p">(</span><span class="n">parameter</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">name</span> <span class="ow">in</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">name</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Symbol </span><span class="si">%s</span><span class="s"> already appears in object and cannot be used as a parameter.&#39;</span> <span class="o">%</span> <span class="n">t</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">center</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">hradius</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">t</span><span class="p">),</span>
-                     <span class="bp">self</span><span class="o">.</span><span class="n">center</span><span class="o">.</span><span class="n">y</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">vradius</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="Ellipse.plot_interval"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse.plot_interval">[docs]</a>    <span class="k">def</span> <span class="nf">plot_interval</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">parameter</span><span class="o">=</span><span class="s">&#39;t&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The plot interval for the default geometric plot of the Ellipse.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        parameter : str, optional</span>
-<span class="sd">            Default value is &#39;t&#39;.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        plot_interval : list</span>
-<span class="sd">            [parameter, lower_bound, upper_bound]</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ellipse</span>
-<span class="sd">        &gt;&gt;&gt; e1 = Ellipse(Point(0, 0), 3, 2)</span>
-<span class="sd">        &gt;&gt;&gt; e1.plot_interval()</span>
-<span class="sd">        [t, -pi, pi]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">_symbol</span><span class="p">(</span><span class="n">parameter</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">t</span><span class="p">,</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">]</span>
-</div>
-<div class="viewcode-block" id="Ellipse.random_point"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse.random_point">[docs]</a>    <span class="k">def</span> <span class="nf">random_point</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A random point on the ellipse.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        point : Point</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point</span>
-<span class="sd">        arbitrary_point : Returns parameterized point on ellipse</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        -----</span>
-
-<span class="sd">        A random point may not appear to be on the ellipse, ie, `p in e` may</span>
-<span class="sd">        return False. This is because the coordinates of the point will be</span>
-<span class="sd">        floating point values, and when these values are substituted into the</span>
-<span class="sd">        equation for the ellipse the result may not be zero because of floating</span>
-<span class="sd">        point rounding error.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ellipse, Segment</span>
-<span class="sd">        &gt;&gt;&gt; e1 = Ellipse(Point(0, 0), 3, 2)</span>
-<span class="sd">        &gt;&gt;&gt; e1.random_point() # gives some random point</span>
-<span class="sd">        Point(...)</span>
-<span class="sd">        &gt;&gt;&gt; p1 = e1.random_point(seed=0); p1.n(2)</span>
-<span class="sd">        Point(2.1, 1.4)</span>
-
-<span class="sd">        The random_point method assures that the point will test as being</span>
-<span class="sd">        in the ellipse:</span>
-
-<span class="sd">        &gt;&gt;&gt; p1 in e1</span>
-<span class="sd">        True</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        An arbitrary_point with a random value of t substituted into it may</span>
-<span class="sd">        not test as being on the ellipse because the expression tested that</span>
-<span class="sd">        a point is on the ellipse doesn&#39;t simplify to zero and doesn&#39;t evaluate</span>
-<span class="sd">        exactly to zero:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import t</span>
-<span class="sd">        &gt;&gt;&gt; e1.arbitrary_point(t)</span>
-<span class="sd">        Point(3*cos(t), 2*sin(t))</span>
-<span class="sd">        &gt;&gt;&gt; p2 = _.subs(t, 0.1)</span>
-<span class="sd">        &gt;&gt;&gt; p2 in e1</span>
-<span class="sd">        False</span>
-
-<span class="sd">        Note that arbitrary_point routine does not take this approach. A value for</span>
-<span class="sd">        cos(t) and sin(t) (not t) is substituted into the arbitrary point. There is</span>
-<span class="sd">        a small chance that this will give a point that will not test as being</span>
-<span class="sd">        in the ellipse, so the process is repeated (up to 10 times) until a</span>
-<span class="sd">        valid point is obtained.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">nsimplify</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">_symbol</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">)</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">.</span><span class="n">args</span>
-        <span class="c"># get a random value in [-1, 1) corresponding to cos(t)</span>
-        <span class="c"># and confirm that it will test as being in the ellipse</span>
-        <span class="k">if</span> <span class="n">seed</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">rng</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">Random</span><span class="p">(</span><span class="n">seed</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">rng</span> <span class="o">=</span> <span class="n">random</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">):</span>  <span class="c"># should be enough?</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">nsimplify</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">rng</span><span class="o">.</span><span class="n">random</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">c</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-            <span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="n">c</span><span class="p">),</span> <span class="n">y</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="n">s</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">p1</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">p1</span>
-        <span class="k">raise</span> <span class="n">GeometryError</span><span class="p">(</span>
-            <span class="s">&#39;Having problems generating a point in the ellipse.&#39;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Ellipse.equation"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse.equation">[docs]</a>    <span class="k">def</span> <span class="nf">equation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s">&#39;y&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The equation of the ellipse.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        x : str, optional</span>
-<span class="sd">            Label for the x-axis. Default value is &#39;x&#39;.</span>
-<span class="sd">        y : str, optional</span>
-<span class="sd">            Label for the y-axis. Default value is &#39;y&#39;.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        equation : sympy expression</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        arbitrary_point : Returns parameterized point on ellipse</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ellipse</span>
-<span class="sd">        &gt;&gt;&gt; e1 = Ellipse(Point(1, 0), 3, 2)</span>
-<span class="sd">        &gt;&gt;&gt; e1.equation()</span>
-<span class="sd">        y**2/4 + (x/3 - 1/3)**2 - 1</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">_symbol</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="n">y</span> <span class="o">=</span> <span class="n">_symbol</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-        <span class="n">t1</span> <span class="o">=</span> <span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">center</span><span class="o">.</span><span class="n">x</span><span class="p">)</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">hradius</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-        <span class="n">t2</span> <span class="o">=</span> <span class="p">((</span><span class="n">y</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">center</span><span class="o">.</span><span class="n">y</span><span class="p">)</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">vradius</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-        <span class="k">return</span> <span class="n">t1</span> <span class="o">+</span> <span class="n">t2</span> <span class="o">-</span> <span class="mi">1</span>
-</div>
-    <span class="k">def</span> <span class="nf">_do_line_intersection</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Find the intersection of a LinearEntity and the ellipse.</span>
-
-<span class="sd">        All LinearEntities are treated as a line and filtered at</span>
-<span class="sd">        the end to see that they lie in o.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">hr_sq</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">hradius</span> <span class="o">**</span> <span class="mi">2</span>
-        <span class="n">vr_sq</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">vradius</span> <span class="o">**</span> <span class="mi">2</span>
-        <span class="n">lp</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">points</span>
-
-        <span class="n">ldir</span> <span class="o">=</span> <span class="n">lp</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">lp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">diff</span> <span class="o">=</span> <span class="n">lp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">center</span>
-        <span class="n">mdir</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">ldir</span><span class="o">.</span><span class="n">x</span><span class="o">/</span><span class="n">hr_sq</span><span class="p">,</span> <span class="n">ldir</span><span class="o">.</span><span class="n">y</span><span class="o">/</span><span class="n">vr_sq</span><span class="p">)</span>
-        <span class="n">mdiff</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">diff</span><span class="o">.</span><span class="n">x</span><span class="o">/</span><span class="n">hr_sq</span><span class="p">,</span> <span class="n">diff</span><span class="o">.</span><span class="n">y</span><span class="o">/</span><span class="n">vr_sq</span><span class="p">)</span>
-
-        <span class="n">a</span> <span class="o">=</span> <span class="n">ldir</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">mdir</span><span class="p">)</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">ldir</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">mdiff</span><span class="p">)</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">diff</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">mdiff</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="n">det</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">b</span><span class="o">*</span><span class="n">b</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="n">c</span><span class="p">)</span>
-
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">if</span> <span class="n">det</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">t</span> <span class="o">=</span> <span class="o">-</span><span class="n">b</span> <span class="o">/</span> <span class="n">a</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">lp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="p">(</span><span class="n">lp</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">lp</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">*</span> <span class="n">t</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">is_good</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">is_good</span> <span class="o">=</span> <span class="p">(</span><span class="n">det</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>  <span class="c"># symbolic, allow</span>
-                <span class="n">is_good</span> <span class="o">=</span> <span class="bp">True</span>
-
-            <span class="k">if</span> <span class="n">is_good</span><span class="p">:</span>
-                <span class="n">root</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">det</span><span class="p">)</span>
-                <span class="n">t_a</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="n">b</span> <span class="o">-</span> <span class="n">root</span><span class="p">)</span> <span class="o">/</span> <span class="n">a</span>
-                <span class="n">t_b</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="n">b</span> <span class="o">+</span> <span class="n">root</span><span class="p">)</span> <span class="o">/</span> <span class="n">a</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">lp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="p">(</span><span class="n">lp</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">lp</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">*</span> <span class="n">t_a</span> <span class="p">)</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">lp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="p">(</span><span class="n">lp</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">lp</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">*</span> <span class="n">t_b</span> <span class="p">)</span>
-
-        <span class="k">return</span> <span class="p">[</span><span class="n">r</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">result</span> <span class="k">if</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">o</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">_do_circle_intersection</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The intersection of an Ellipse and a Circle.</span>
-
-<span class="sd">        Private helper method for `intersection`.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">variables</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">equation</span><span class="p">()</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">variables</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">center</span> <span class="o">==</span> <span class="n">o</span><span class="o">.</span><span class="n">center</span><span class="p">:</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">major</span><span class="p">,</span> <span class="n">o</span><span class="o">.</span><span class="n">minor</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">radius</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">a</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">simplify</span><span class="p">((</span><span class="n">r</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">b</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">a</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">b</span><span class="o">**</span><span class="mi">2</span><span class="p">)))</span>
-            <span class="n">y</span> <span class="o">=</span> <span class="n">b</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">simplify</span><span class="p">((</span><span class="n">a</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">a</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">b</span><span class="o">**</span><span class="mi">2</span><span class="p">)))</span>
-            <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">set</span><span class="p">([</span><span class="n">Point</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">y</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span>
-                             <span class="n">Point</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">y</span><span class="p">)]))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">variables</span>
-            <span class="n">xx</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">equation</span><span class="p">(),</span> <span class="n">x</span><span class="p">)</span>
-            <span class="n">intersect</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">xi</span> <span class="ow">in</span> <span class="n">xx</span><span class="p">:</span>
-                <span class="n">yy</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">o</span><span class="o">.</span><span class="n">equation</span><span class="p">()</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">xi</span><span class="p">),</span> <span class="n">y</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">yi</span> <span class="ow">in</span> <span class="n">yy</span><span class="p">:</span>
-                    <span class="n">intersect</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="n">xi</span><span class="p">,</span> <span class="n">yi</span><span class="p">))</span>
-            <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">intersect</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_do_ellipse_intersection</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The intersection of two ellipses.</span>
-
-<span class="sd">        Private helper method for `intersection`.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-        <span class="n">y</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-        <span class="n">seq</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">equation</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-        <span class="n">oeq</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">equation</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">solve</span><span class="p">([</span><span class="n">seq</span><span class="p">,</span> <span class="n">oeq</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">])</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">Point</span><span class="p">(</span><span class="o">*</span><span class="n">r</span><span class="p">)</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">result</span> <span class="k">if</span> <span class="n">im</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">is_zero</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span> <span class="ow">and</span> <span class="n">im</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">is_zero</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">]</span>
-
-<div class="viewcode-block" id="Ellipse.intersection"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Ellipse.intersection">[docs]</a>    <span class="k">def</span> <span class="nf">intersection</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The intersection of this ellipse and another geometrical entity</span>
-<span class="sd">        `o`.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        o : GeometryEntity</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        intersection : list of GeometryEntity objects</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        -----</span>
-<span class="sd">        Currently supports intersections with Point, Line, Segment, Ray,</span>
-<span class="sd">        Circle and Ellipse types.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.entity.GeometryEntity</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Ellipse, Point, Line, sqrt</span>
-<span class="sd">        &gt;&gt;&gt; e = Ellipse(Point(0, 0), 5, 7)</span>
-<span class="sd">        &gt;&gt;&gt; e.intersection(Point(0, 0))</span>
-<span class="sd">        []</span>
-<span class="sd">        &gt;&gt;&gt; e.intersection(Point(5, 0))</span>
-<span class="sd">        [Point(5, 0)]</span>
-<span class="sd">        &gt;&gt;&gt; e.intersection(Line(Point(0,0), Point(0, 1)))</span>
-<span class="sd">        [Point(0, -7), Point(0, 7)]</span>
-<span class="sd">        &gt;&gt;&gt; e.intersection(Line(Point(5,0), Point(5, 1)))</span>
-<span class="sd">        [Point(5, 0)]</span>
-<span class="sd">        &gt;&gt;&gt; e.intersection(Line(Point(6,0), Point(6, 1)))</span>
-<span class="sd">        []</span>
-<span class="sd">        &gt;&gt;&gt; e = Ellipse(Point(-1, 0), 4, 3)</span>
-<span class="sd">        &gt;&gt;&gt; e.intersection(Ellipse(Point(1, 0), 4, 3))</span>
-<span class="sd">        [Point(0, -3*sqrt(15)/4), Point(0, 3*sqrt(15)/4)]</span>
-<span class="sd">        &gt;&gt;&gt; e.intersection(Ellipse(Point(5, 0), 4, 3))</span>
-<span class="sd">        [Point(2, -3*sqrt(7)/4), Point(2, 3*sqrt(7)/4)]</span>
-<span class="sd">        &gt;&gt;&gt; e.intersection(Ellipse(Point(100500, 0), 4, 3))</span>
-<span class="sd">        []</span>
-<span class="sd">        &gt;&gt;&gt; e.intersection(Ellipse(Point(0, 0), 3, 4))</span>
-<span class="sd">        [Point(-363/175, -48*sqrt(111)/175), Point(-363/175, 48*sqrt(111)/175),</span>
-<span class="sd">        Point(3, 0)]</span>
-<span class="sd">        &gt;&gt;&gt; e.intersection(Ellipse(Point(-1, 0), 3, 4))</span>
-<span class="sd">        [Point(-17/5, -12/5), Point(-17/5, 12/5), Point(7/5, -12/5),</span>
-<span class="sd">        Point(7/5, 12/5)]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Point</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">o</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">o</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[]</span>
-
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">LinearEntity</span><span class="p">):</span>
-            <span class="c"># LinearEntity may be a ray/segment, so check the points</span>
-            <span class="c"># of intersection for coincidence first</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_do_line_intersection</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Circle</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_do_circle_intersection</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Ellipse</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">o</span> <span class="o">==</span> <span class="bp">self</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_do_ellipse_intersection</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">o</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Is the other GeometryEntity the same as this ellipse?&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">GeometryEntity</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">center</span> <span class="o">==</span> <span class="n">o</span><span class="o">.</span><span class="n">center</span> <span class="ow">and</span>
-                                                  <span class="bp">self</span><span class="o">.</span><span class="n">hradius</span> <span class="o">==</span> <span class="n">o</span><span class="o">.</span><span class="n">hradius</span> <span class="ow">and</span>
-                                                  <span class="bp">self</span><span class="o">.</span><span class="n">vradius</span> <span class="o">==</span> <span class="n">o</span><span class="o">.</span><span class="n">vradius</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Ellipse</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__contains__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Point</span><span class="p">):</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="n">y</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-            <span class="n">res</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">equation</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">({</span><span class="n">x</span><span class="p">:</span> <span class="n">o</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">o</span><span class="o">.</span><span class="n">y</span><span class="p">})</span>
-            <span class="k">return</span> <span class="n">trigsimp</span><span class="p">(</span><span class="n">simplify</span><span class="p">(</span><span class="n">res</span><span class="p">))</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Ellipse</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span> <span class="o">==</span> <span class="n">o</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-</div>
-<div class="viewcode-block" id="Circle"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Circle">[docs]</a><span class="k">class</span> <span class="nc">Circle</span><span class="p">(</span><span class="n">Ellipse</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A circle in space.</span>
-
-<span class="sd">    Constructed simply from a center and a radius, or from three</span>
-<span class="sd">    non-collinear points.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    center : Point</span>
-<span class="sd">    radius : number or sympy expression</span>
-<span class="sd">    points : sequence of three Points</span>
-
-<span class="sd">    Attributes</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    radius (synonymous with hradius, vradius, major and minor)</span>
-<span class="sd">    circumference</span>
-<span class="sd">    equation</span>
-
-<span class="sd">    Raises</span>
-<span class="sd">    ======</span>
-
-<span class="sd">    GeometryError</span>
-<span class="sd">        When trying to construct circle from three collinear points.</span>
-<span class="sd">        When trying to construct circle from incorrect parameters.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Ellipse, sympy.geometry.point.Point</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.geometry import Point, Circle</span>
-<span class="sd">    &gt;&gt;&gt; # a circle constructed from a center and radius</span>
-<span class="sd">    &gt;&gt;&gt; c1 = Circle(Point(0, 0), 5)</span>
-<span class="sd">    &gt;&gt;&gt; c1.hradius, c1.vradius, c1.radius</span>
-<span class="sd">    (5, 5, 5)</span>
-
-<span class="sd">    &gt;&gt;&gt; # a circle costructed from three points</span>
-<span class="sd">    &gt;&gt;&gt; c2 = Circle(Point(0, 0), Point(1, 1), Point(1, 0))</span>
-<span class="sd">    &gt;&gt;&gt; c2.hradius, c2.vradius, c2.radius, c2.center</span>
-<span class="sd">    (sqrt(2)/2, sqrt(2)/2, sqrt(2)/2, Point(1/2, 1/2))</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">c</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">Point</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">Point</span><span class="o">.</span><span class="n">is_collinear</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="n">GeometryError</span><span class="p">(</span>
-                    <span class="s">&quot;Cannot construct a circle from three collinear points&quot;</span><span class="p">)</span>
-            <span class="kn">from</span> <span class="nn">.polygon</span> <span class="kn">import</span> <span class="n">Triangle</span>
-            <span class="n">t</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">t</span><span class="o">.</span><span class="n">circumcenter</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">t</span><span class="o">.</span><span class="n">circumradius</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="c"># Assume (center, radius) pair</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">c</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">r</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">GeometryEntity</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-        <span class="k">raise</span> <span class="n">GeometryError</span><span class="p">(</span><span class="s">&quot;Circle.__new__ received unknown arguments&quot;</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Circle.radius"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Circle.radius">[docs]</a>    <span class="k">def</span> <span class="nf">radius</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The radius of the circle.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        radius : number or sympy expression</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        Ellipse.major, Ellipse.minor, Ellipse.hradius, Ellipse.vradius</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Circle</span>
-<span class="sd">        &gt;&gt;&gt; c1 = Circle(Point(3, 4), 6)</span>
-<span class="sd">        &gt;&gt;&gt; c1.radius</span>
-<span class="sd">        6</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Circle.vradius"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Circle.vradius">[docs]</a>    <span class="k">def</span> <span class="nf">vradius</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        This Ellipse property is an alias for the Circle&#39;s radius.</span>
-
-<span class="sd">        Whereas hradius, major and minor can use Ellipse&#39;s conventions,</span>
-<span class="sd">        the vradius does not exist for a circle.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Circle</span>
-<span class="sd">        &gt;&gt;&gt; c1 = Circle(Point(3, 4), 6)</span>
-<span class="sd">        &gt;&gt;&gt; c1.vradius</span>
-<span class="sd">        6</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">radius</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Circle.circumference"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Circle.circumference">[docs]</a>    <span class="k">def</span> <span class="nf">circumference</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The circumference of the circle.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        circumference : number or SymPy expression</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Circle</span>
-<span class="sd">        &gt;&gt;&gt; c1 = Circle(Point(3, 4), 6)</span>
-<span class="sd">        &gt;&gt;&gt; c1.circumference</span>
-<span class="sd">        12*pi</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">radius</span>
-</div>
-<div class="viewcode-block" id="Circle.equation"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Circle.equation">[docs]</a>    <span class="k">def</span> <span class="nf">equation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s">&#39;y&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The equation of the circle.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        x : str or Symbol, optional</span>
-<span class="sd">            Default value is &#39;x&#39;.</span>
-<span class="sd">        y : str or Symbol, optional</span>
-<span class="sd">            Default value is &#39;y&#39;.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        equation : SymPy expression</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Circle</span>
-<span class="sd">        &gt;&gt;&gt; c1 = Circle(Point(0, 0), 5)</span>
-<span class="sd">        &gt;&gt;&gt; c1.equation()</span>
-<span class="sd">        x**2 + y**2 - 25</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">_symbol</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="n">y</span> <span class="o">=</span> <span class="n">_symbol</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-        <span class="n">t1</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">center</span><span class="o">.</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-        <span class="n">t2</span> <span class="o">=</span> <span class="p">(</span><span class="n">y</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">center</span><span class="o">.</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-        <span class="k">return</span> <span class="n">t1</span> <span class="o">+</span> <span class="n">t2</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">major</span><span class="o">**</span><span class="mi">2</span>
-</div>
-<div class="viewcode-block" id="Circle.intersection"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Circle.intersection">[docs]</a>    <span class="k">def</span> <span class="nf">intersection</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The intersection of this circle with another geometrical entity.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        o : GeometryEntity</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        intersection : list of GeometryEntities</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Circle, Line, Ray</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(5, 5), Point(6, 0)</span>
-<span class="sd">        &gt;&gt;&gt; p4 = Point(5, 0)</span>
-<span class="sd">        &gt;&gt;&gt; c1 = Circle(p1, 5)</span>
-<span class="sd">        &gt;&gt;&gt; c1.intersection(p2)</span>
-<span class="sd">        []</span>
-<span class="sd">        &gt;&gt;&gt; c1.intersection(p4)</span>
-<span class="sd">        [Point(5, 0)]</span>
-<span class="sd">        &gt;&gt;&gt; c1.intersection(Ray(p1, p2))</span>
-<span class="sd">        [Point(5*sqrt(2)/2, 5*sqrt(2)/2)]</span>
-<span class="sd">        &gt;&gt;&gt; c1.intersection(Line(p2, p3))</span>
-<span class="sd">        []</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Circle</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">o</span><span class="o">.</span><span class="n">center</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">center</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">o</span><span class="o">.</span><span class="n">radius</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">radius</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">o</span>
-                <span class="k">return</span> <span class="p">[]</span>
-            <span class="n">dx</span><span class="p">,</span> <span class="n">dy</span> <span class="o">=</span> <span class="p">(</span><span class="n">o</span><span class="o">.</span><span class="n">center</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">center</span><span class="p">)</span><span class="o">.</span><span class="n">args</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">simplify</span><span class="p">(</span><span class="n">dy</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">dx</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-            <span class="n">R</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">radius</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">radius</span>
-            <span class="k">if</span> <span class="n">d</span> <span class="o">&gt;</span> <span class="n">R</span> <span class="ow">or</span> <span class="n">d</span> <span class="o">&lt;</span> <span class="nb">abs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">radius</span> <span class="o">-</span> <span class="n">o</span><span class="o">.</span><span class="n">radius</span><span class="p">):</span>
-                <span class="k">return</span> <span class="p">[]</span>
-
-            <span class="n">a</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">radius</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">o</span><span class="o">.</span><span class="n">radius</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">d</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">d</span><span class="p">))</span>
-
-            <span class="n">x2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">center</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="p">(</span><span class="n">dx</span> <span class="o">*</span> <span class="n">a</span><span class="o">/</span><span class="n">d</span><span class="p">)</span>
-            <span class="n">y2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">center</span><span class="o">.</span><span class="n">y</span> <span class="o">+</span> <span class="p">(</span><span class="n">dy</span> <span class="o">*</span> <span class="n">a</span><span class="o">/</span><span class="n">d</span><span class="p">)</span>
-
-            <span class="n">h</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">simplify</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">radius</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">a</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-            <span class="n">rx</span> <span class="o">=</span> <span class="o">-</span><span class="n">dy</span> <span class="o">*</span> <span class="p">(</span><span class="n">h</span><span class="o">/</span><span class="n">d</span><span class="p">)</span>
-            <span class="n">ry</span> <span class="o">=</span> <span class="n">dx</span> <span class="o">*</span> <span class="p">(</span><span class="n">h</span><span class="o">/</span><span class="n">d</span><span class="p">)</span>
-
-            <span class="n">xi_1</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">x2</span> <span class="o">+</span> <span class="n">rx</span><span class="p">)</span>
-            <span class="n">xi_2</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">x2</span> <span class="o">-</span> <span class="n">rx</span><span class="p">)</span>
-            <span class="n">yi_1</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">y2</span> <span class="o">+</span> <span class="n">ry</span><span class="p">)</span>
-            <span class="n">yi_2</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">y2</span> <span class="o">-</span> <span class="n">ry</span><span class="p">)</span>
-
-            <span class="n">ret</span> <span class="o">=</span> <span class="p">[</span><span class="n">Point</span><span class="p">(</span><span class="n">xi_1</span><span class="p">,</span> <span class="n">yi_1</span><span class="p">)]</span>
-            <span class="k">if</span> <span class="n">xi_1</span> <span class="o">!=</span> <span class="n">xi_2</span> <span class="ow">or</span> <span class="n">yi_1</span> <span class="o">!=</span> <span class="n">yi_2</span><span class="p">:</span>
-                <span class="n">ret</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="n">xi_2</span><span class="p">,</span> <span class="n">yi_2</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">ret</span>
-
-        <span class="k">return</span> <span class="n">Ellipse</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Circle.scale"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.ellipse.Circle.scale">[docs]</a>    <span class="k">def</span> <span class="nf">scale</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">pt</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Override GeometryEntity.scale since the radius</span>
-<span class="sd">        is not a GeometryEntity.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Circle</span>
-<span class="sd">        &gt;&gt;&gt; Circle((0, 0), 1).scale(2, 2)</span>
-<span class="sd">        Circle(Point(0, 0), 2)</span>
-<span class="sd">        &gt;&gt;&gt; Circle((0, 0), 1).scale(2, 4)</span>
-<span class="sd">        Ellipse(Point(0, 0), 2, 4)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">center</span>
-        <span class="k">if</span> <span class="n">pt</span><span class="p">:</span>
-            <span class="n">pt</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">pt</span><span class="p">)</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">pt</span><span class="p">)</span><span class="o">.</span><span class="n">args</span><span class="p">)</span><span class="o">.</span><span class="n">scale</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="o">*</span><span class="n">pt</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">c</span><span class="o">.</span><span class="n">scale</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="n">y</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">x</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">radius</span><span class="p">)</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">v</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">radius</span>
-        <span class="k">return</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">hradius</span><span class="o">=</span><span class="n">h</span><span class="o">*</span><span class="n">x</span><span class="p">,</span> <span class="n">vradius</span><span class="o">=</span><span class="n">v</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
-
-</div></div>
-<span class="kn">from</span> <span class="nn">.polygon</span> <span class="kn">import</span> <span class="n">Polygon</span>
-</pre></div>
-
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-        </div>
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diff --git a/dev-py3k/_modules/sympy/geometry/entity.html b/dev-py3k/_modules/sympy/geometry/entity.html
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-            
-  <h1>Source code for sympy.geometry.entity</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;The definition of the base geometrical entity with attributes common to</span>
-<span class="sd">all derived geometrical entities.</span>
-
-<span class="sd">Contains</span>
-<span class="sd">========</span>
-<span class="sd">GeometryEntity</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="nb">cmp</span><span class="p">,</span> <span class="n">is_sequence</span>
-<span class="kn">from</span> <span class="nn">sympy.core.basic</span> <span class="kn">import</span> <span class="n">Basic</span>
-<span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span>
-<span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">eye</span>
-
-<span class="c"># How entities are ordered; used by __cmp__ in GeometryEntity</span>
-<span class="n">ordering_of_classes</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="s">&quot;Point&quot;</span><span class="p">,</span>
-    <span class="s">&quot;Segment&quot;</span><span class="p">,</span>
-    <span class="s">&quot;Ray&quot;</span><span class="p">,</span>
-    <span class="s">&quot;Line&quot;</span><span class="p">,</span>
-    <span class="s">&quot;Triangle&quot;</span><span class="p">,</span>
-    <span class="s">&quot;RegularPolygon&quot;</span><span class="p">,</span>
-    <span class="s">&quot;Polygon&quot;</span><span class="p">,</span>
-    <span class="s">&quot;Circle&quot;</span><span class="p">,</span>
-    <span class="s">&quot;Ellipse&quot;</span><span class="p">,</span>
-    <span class="s">&quot;Curve&quot;</span>
-<span class="p">]</span>
-
-
-<div class="viewcode-block" id="GeometryEntity"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.entity.GeometryEntity">[docs]</a><span class="k">class</span> <span class="nc">GeometryEntity</span><span class="p">(</span><span class="n">Basic</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The base class for all geometrical entities.</span>
-
-<span class="sd">    This class doesn&#39;t represent any particular geometric entity, it only</span>
-<span class="sd">    provides the implementation of some methods common to all subclasses.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="n">args</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sympy_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="k">def</span> <span class="nf">__getnewargs__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-<div class="viewcode-block" id="GeometryEntity.intersection"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.entity.GeometryEntity.intersection">[docs]</a>    <span class="k">def</span> <span class="nf">intersection</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns a list of all of the intersections of self with o.</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        An entity is not required to implement this method.</span>
-
-<span class="sd">        If two different types of entities can intersect, the item with</span>
-<span class="sd">        higher index in ordering_of_classes should implement</span>
-<span class="sd">        intersections with anything having a lower index.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.util.intersection</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-</div>
-<div class="viewcode-block" id="GeometryEntity.rotate"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.entity.GeometryEntity.rotate">[docs]</a>    <span class="k">def</span> <span class="nf">rotate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">angle</span><span class="p">,</span> <span class="n">pt</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Rotate ``angle`` radians counterclockwise about Point ``pt``.</span>
-
-<span class="sd">        The default pt is the origin, Point(0, 0)</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        scale, translate</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, RegularPolygon, Polygon, pi</span>
-<span class="sd">        &gt;&gt;&gt; t = Polygon(*RegularPolygon(Point(0, 0), 1, 3).vertices)</span>
-<span class="sd">        &gt;&gt;&gt; t # vertex on x axis</span>
-<span class="sd">        Triangle(Point(1, 0), Point(-1/2, sqrt(3)/2), Point(-1/2, -sqrt(3)/2))</span>
-<span class="sd">        &gt;&gt;&gt; t.rotate(pi/2) # vertex on y axis now</span>
-<span class="sd">        Triangle(Point(0, 1), Point(-sqrt(3)/2, -1/2), Point(sqrt(3)/2, -1/2))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">newargs</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">GeometryEntity</span><span class="p">):</span>
-                <span class="n">newargs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">rotate</span><span class="p">(</span><span class="n">angle</span><span class="p">,</span> <span class="n">pt</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">newargs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="k">return</span> <span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">)(</span><span class="o">*</span><span class="n">newargs</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="GeometryEntity.scale"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.entity.GeometryEntity.scale">[docs]</a>    <span class="k">def</span> <span class="nf">scale</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">pt</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Scale the object by multiplying the x,y-coordinates by x and y.</span>
-
-<span class="sd">        If pt is given, the scaling is done relative to that point; the</span>
-<span class="sd">        object is shifted by -pt, scaled, and shifted by pt.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        rotate, translate</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import RegularPolygon, Point, Polygon</span>
-<span class="sd">        &gt;&gt;&gt; t = Polygon(*RegularPolygon(Point(0, 0), 1, 3).vertices)</span>
-<span class="sd">        &gt;&gt;&gt; t</span>
-<span class="sd">        Triangle(Point(1, 0), Point(-1/2, sqrt(3)/2), Point(-1/2, -sqrt(3)/2))</span>
-<span class="sd">        &gt;&gt;&gt; t.scale(2)</span>
-<span class="sd">        Triangle(Point(2, 0), Point(-1, sqrt(3)/2), Point(-1, -sqrt(3)/2))</span>
-<span class="sd">        &gt;&gt;&gt; t.scale(2,2)</span>
-<span class="sd">        Triangle(Point(2, 0), Point(-1, sqrt(3)), Point(-1, -sqrt(3)))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.geometry.point</span> <span class="kn">import</span> <span class="n">Point</span>
-        <span class="k">if</span> <span class="n">pt</span><span class="p">:</span>
-            <span class="n">pt</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">pt</span><span class="p">)</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">pt</span><span class="p">)</span><span class="o">.</span><span class="n">args</span><span class="p">)</span><span class="o">.</span><span class="n">scale</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="o">*</span><span class="n">pt</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">)(</span><span class="o">*</span><span class="p">[</span><span class="n">a</span><span class="o">.</span><span class="n">scale</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>  <span class="c"># if this fails, override this class</span>
-</div>
-<div class="viewcode-block" id="GeometryEntity.translate"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.entity.GeometryEntity.translate">[docs]</a>    <span class="k">def</span> <span class="nf">translate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Shift the object by adding to the x,y-coordinates the values x and y.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        rotate, scale</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import RegularPolygon, Point, Polygon</span>
-<span class="sd">        &gt;&gt;&gt; t = Polygon(*RegularPolygon(Point(0, 0), 1, 3).vertices)</span>
-<span class="sd">        &gt;&gt;&gt; t</span>
-<span class="sd">        Triangle(Point(1, 0), Point(-1/2, sqrt(3)/2), Point(-1/2, -sqrt(3)/2))</span>
-<span class="sd">        &gt;&gt;&gt; t.translate(2)</span>
-<span class="sd">        Triangle(Point(3, 0), Point(3/2, sqrt(3)/2), Point(3/2, -sqrt(3)/2))</span>
-<span class="sd">        &gt;&gt;&gt; t.translate(2, 2)</span>
-<span class="sd">        Triangle(Point(3, 2), Point(3/2, sqrt(3)/2 + 2),</span>
-<span class="sd">            Point(3/2, -sqrt(3)/2 + 2))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">newargs</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">GeometryEntity</span><span class="p">):</span>
-                <span class="n">newargs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">newargs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="k">return</span> <span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">)(</span><span class="o">*</span><span class="n">newargs</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="GeometryEntity.encloses"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.entity.GeometryEntity.encloses">[docs]</a>    <span class="k">def</span> <span class="nf">encloses</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if o is inside (not on or outside) the boundaries of self.</span>
-
-<span class="sd">        The object will be decomposed into Points and individual Entities need</span>
-<span class="sd">        only define an encloses_point method for their class.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.ellipse.Ellipse.encloses_point</span>
-<span class="sd">        sympy.geometry.polygon.Polygon.encloses_point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import RegularPolygon, Point, Polygon</span>
-<span class="sd">        &gt;&gt;&gt; t  = Polygon(*RegularPolygon(Point(0, 0), 1, 3).vertices)</span>
-<span class="sd">        &gt;&gt;&gt; t2 = Polygon(*RegularPolygon(Point(0, 0), 2, 3).vertices)</span>
-<span class="sd">        &gt;&gt;&gt; t2.encloses(t)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; t.encloses(t2)</span>
-<span class="sd">        False</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.geometry.point</span> <span class="kn">import</span> <span class="n">Point</span>
-        <span class="kn">from</span> <span class="nn">sympy.geometry.line</span> <span class="kn">import</span> <span class="n">Segment</span><span class="p">,</span> <span class="n">Ray</span><span class="p">,</span> <span class="n">Line</span>
-        <span class="kn">from</span> <span class="nn">sympy.geometry.ellipse</span> <span class="kn">import</span> <span class="n">Ellipse</span>
-        <span class="kn">from</span> <span class="nn">sympy.geometry.polygon</span> <span class="kn">import</span> <span class="n">Polygon</span><span class="p">,</span> <span class="n">RegularPolygon</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Point</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">encloses_point</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Segment</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">encloses_point</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">o</span><span class="o">.</span><span class="n">points</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Ray</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Line</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Ellipse</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">encloses_point</span><span class="p">(</span><span class="n">o</span><span class="o">.</span><span class="n">center</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Polygon</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">RegularPolygon</span><span class="p">):</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">encloses_point</span><span class="p">(</span><span class="n">o</span><span class="o">.</span><span class="n">center</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">encloses_point</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">o</span><span class="o">.</span><span class="n">vertices</span><span class="p">)</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-</div>
-<div class="viewcode-block" id="GeometryEntity.is_similar"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.entity.GeometryEntity.is_similar">[docs]</a>    <span class="k">def</span> <span class="nf">is_similar</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Is this geometrical entity similar to another geometrical entity?</span>
-
-<span class="sd">        Two entities are similar if a uniform scaling (enlarging or</span>
-<span class="sd">        shrinking) of one of the entities will allow one to obtain the other.</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        This method is not intended to be used directly but rather</span>
-<span class="sd">        through the `are_similar` function found in util.py.</span>
-<span class="sd">        An entity is not required to implement this method.</span>
-<span class="sd">        If two different types of entities can be similar, it is only</span>
-<span class="sd">        required that one of them be able to determine this.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        scale</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-</div>
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Test inequality of two geometrical entities.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__radd__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">a</span><span class="o">.</span><span class="n">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__rsub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">a</span><span class="o">.</span><span class="n">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">a</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__rdiv__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">a</span><span class="o">.</span><span class="n">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;String representation of a GeometryEntity.&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.printing</span> <span class="kn">import</span> <span class="n">sstr</span>
-        <span class="k">return</span> <span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__name__</span> <span class="o">+</span> <span class="n">sstr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;String representation of a GeometryEntity that can be evaluated</span>
-<span class="sd">        by sympy.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__name__</span> <span class="o">+</span> <span class="nb">repr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__cmp__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Comparison of two GeometryEntities.&quot;&quot;&quot;</span>
-        <span class="n">n1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span>
-        <span class="n">n2</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="nb">cmp</span><span class="p">(</span><span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">c</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">0</span>
-
-        <span class="n">i1</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-        <span class="k">for</span> <span class="n">cls</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__mro__</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">i1</span> <span class="o">=</span> <span class="n">ordering_of_classes</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">cls</span><span class="o">.</span><span class="n">__name__</span><span class="p">)</span>
-                <span class="k">break</span>
-            <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-                <span class="n">i1</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-        <span class="k">if</span> <span class="n">i1</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">c</span>
-
-        <span class="n">i2</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-        <span class="k">for</span> <span class="n">cls</span> <span class="ow">in</span> <span class="n">other</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__mro__</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">i2</span> <span class="o">=</span> <span class="n">ordering_of_classes</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">cls</span><span class="o">.</span><span class="n">__name__</span><span class="p">)</span>
-                <span class="k">break</span>
-            <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-                <span class="n">i2</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-        <span class="k">if</span> <span class="n">i2</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">c</span>
-
-        <span class="k">return</span> <span class="nb">cmp</span><span class="p">(</span><span class="n">i1</span><span class="p">,</span> <span class="n">i2</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__contains__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Subclasses should implement this method for anything more complex than equality.&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">==</span> <span class="nb">type</span><span class="p">(</span><span class="n">other</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span> <span class="o">==</span> <span class="n">other</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">_eval_subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.geometry.point</span> <span class="kn">import</span> <span class="n">Point</span>
-        <span class="k">if</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">old</span><span class="p">)</span> <span class="ow">or</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">new</span><span class="p">):</span>
-            <span class="n">old</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">old</span><span class="p">)</span>
-            <span class="n">new</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">new</span><span class="p">)</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_subs</span><span class="p">(</span><span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">translate</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return the matrix to translate a 2-D point by x and y.&quot;&quot;&quot;</span>
-    <span class="n">rv</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-    <span class="n">rv</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span>
-    <span class="n">rv</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">y</span>
-    <span class="k">return</span> <span class="n">rv</span>
-
-
-<span class="k">def</span> <span class="nf">scale</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">pt</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return the matrix to multiply a 2-D point&#39;s coordinates by x and y.</span>
-
-<span class="sd">    If pt is given, the scaling is done relative to that point.&quot;&quot;&quot;</span>
-    <span class="n">rv</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-    <span class="n">rv</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span>
-    <span class="n">rv</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">y</span>
-    <span class="k">if</span> <span class="n">pt</span><span class="p">:</span>
-        <span class="kn">from</span> <span class="nn">sympy.geometry.point</span> <span class="kn">import</span> <span class="n">Point</span>
-        <span class="n">pt</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">pt</span><span class="p">)</span>
-        <span class="n">tr1</span> <span class="o">=</span> <span class="n">translate</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">pt</span><span class="p">)</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">tr2</span> <span class="o">=</span> <span class="n">translate</span><span class="p">(</span><span class="o">*</span><span class="n">pt</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">tr1</span><span class="o">*</span><span class="n">rv</span><span class="o">*</span><span class="n">tr2</span>
-    <span class="k">return</span> <span class="n">rv</span>
-
-
-<span class="k">def</span> <span class="nf">rotate</span><span class="p">(</span><span class="n">th</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return the matrix to rotate a 2-D point about the origin by ``angle``.</span>
-
-<span class="sd">    The angle is measured in radians. To Point a point about a point other</span>
-<span class="sd">    then the origin, translate the Point, do the rotation, and</span>
-<span class="sd">    translate it back:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.geometry.entity import rotate, translate</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Point, pi</span>
-<span class="sd">    &gt;&gt;&gt; rot_about_11 = translate(-1, -1)*rotate(pi/2)*translate(1, 1)</span>
-<span class="sd">    &gt;&gt;&gt; Point(1, 1).transform(rot_about_11)</span>
-<span class="sd">    Point(1, 1)</span>
-<span class="sd">    &gt;&gt;&gt; Point(0, 0).transform(rot_about_11)</span>
-<span class="sd">    Point(2, 0)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">th</span><span class="p">)</span>
-    <span class="n">rv</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">th</span><span class="p">)</span>
-    <span class="n">rv</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">s</span>
-    <span class="n">rv</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">s</span>
-    <span class="n">rv</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="k">return</span> <span class="n">rv</span>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
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diff --git a/dev-py3k/_modules/sympy/geometry/line.html b/dev-py3k/_modules/sympy/geometry/line.html
deleted file mode 100644
index 60a37ba5136..00000000000
--- a/dev-py3k/_modules/sympy/geometry/line.html
+++ /dev/null
@@ -1,1861 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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-    <div class="document">
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-          <div class="body">
-            
-  <h1>Source code for sympy.geometry.line</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Line-like geometrical entities.</span>
-
-<span class="sd">Contains</span>
-<span class="sd">========</span>
-<span class="sd">LinearEntity</span>
-<span class="sd">Line</span>
-<span class="sd">Ray</span>
-<span class="sd">Segment</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">Dummy</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.trigonometric</span> <span class="kn">import</span> <span class="n">_pi_coeff</span> <span class="k">as</span> <span class="n">pi_coeff</span>
-<span class="kn">from</span> <span class="nn">sympy.core.logic</span> <span class="kn">import</span> <span class="n">fuzzy_and</span>
-<span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">simplify</span>
-<span class="kn">from</span> <span class="nn">sympy.solvers</span> <span class="kn">import</span> <span class="n">solve</span>
-<span class="kn">from</span> <span class="nn">sympy.geometry.exceptions</span> <span class="kn">import</span> <span class="n">GeometryError</span>
-<span class="kn">from</span> <span class="nn">.entity</span> <span class="kn">import</span> <span class="n">GeometryEntity</span>
-<span class="kn">from</span> <span class="nn">.point</span> <span class="kn">import</span> <span class="n">Point</span>
-<span class="kn">from</span> <span class="nn">.util</span> <span class="kn">import</span> <span class="n">_symbol</span>
-
-<span class="c"># TODO: this should be placed elsewhere and reused in other modules</span>
-
-
-<span class="k">class</span> <span class="nc">Undecidable</span><span class="p">(</span><span class="ne">ValueError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-
-<div class="viewcode-block" id="LinearEntity"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.LinearEntity">[docs]</a><span class="k">class</span> <span class="nc">LinearEntity</span><span class="p">(</span><span class="n">GeometryEntity</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;An abstract base class for all linear entities (line, ray and segment)</span>
-<span class="sd">    in a 2-dimensional Euclidean space.</span>
-
-<span class="sd">    Attributes</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    p1</span>
-<span class="sd">    p2</span>
-<span class="sd">    coefficients</span>
-<span class="sd">    slope</span>
-<span class="sd">    points</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    This is an abstract class and is not meant to be instantiated.</span>
-<span class="sd">    Subclasses should implement the following methods:</span>
-
-<span class="sd">        * __eq__</span>
-<span class="sd">        * contains</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.geometry.entity.GeometryEntity</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">p1</span><span class="p">)</span>
-        <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">p2</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">p1</span> <span class="o">==</span> <span class="n">p2</span><span class="p">:</span>
-            <span class="c"># Rolygon returns lower priority classes...should LinearEntity, too?</span>
-            <span class="k">return</span> <span class="n">p1</span>  <span class="c"># raise ValueError(&quot;%s.__new__ requires two unique Points.&quot; % cls.__name__)</span>
-
-        <span class="k">return</span> <span class="n">GeometryEntity</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="LinearEntity.p1"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.LinearEntity.p1">[docs]</a>    <span class="k">def</span> <span class="nf">p1</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The first defining point of a linear entity.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Line</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(0, 0), Point(5, 3)</span>
-<span class="sd">        &gt;&gt;&gt; l = Line(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; l.p1</span>
-<span class="sd">        Point(0, 0)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="LinearEntity.p2"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.LinearEntity.p2">[docs]</a>    <span class="k">def</span> <span class="nf">p2</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The second defining point of a linear entity.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Line</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(0, 0), Point(5, 3)</span>
-<span class="sd">        &gt;&gt;&gt; l = Line(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; l.p2</span>
-<span class="sd">        Point(5, 3)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="LinearEntity.coefficients"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.LinearEntity.coefficients">[docs]</a>    <span class="k">def</span> <span class="nf">coefficients</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The coefficients (`a`, `b`, `c`) for the linear equation `ax + by + c = 0`.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.line.Line.equation</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Line</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(0, 0), Point(5, 3)</span>
-<span class="sd">        &gt;&gt;&gt; l = Line(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; l.coefficients</span>
-<span class="sd">        (-3, 5, 0)</span>
-
-<span class="sd">        &gt;&gt;&gt; p3 = Point(x, y)</span>
-<span class="sd">        &gt;&gt;&gt; l2 = Line(p1, p3)</span>
-<span class="sd">        &gt;&gt;&gt; l2.coefficients</span>
-<span class="sd">        (-y, x, 0)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">points</span>
-        <span class="k">if</span> <span class="n">p1</span><span class="o">.</span><span class="n">x</span> <span class="o">==</span> <span class="n">p2</span><span class="o">.</span><span class="n">x</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="o">-</span><span class="n">p1</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">p1</span><span class="o">.</span><span class="n">y</span> <span class="o">==</span> <span class="n">p2</span><span class="o">.</span><span class="n">y</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="o">-</span><span class="n">p1</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">([</span><span class="n">simplify</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span>
-               <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">y</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="o">.</span><span class="n">y</span><span class="p">,</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="o">.</span><span class="n">x</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">x</span><span class="p">,</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">x</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="o">.</span><span class="n">y</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">y</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="o">.</span><span class="n">x</span><span class="p">)])</span>
-</div>
-<div class="viewcode-block" id="LinearEntity.is_concurrent"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.LinearEntity.is_concurrent">[docs]</a>    <span class="k">def</span> <span class="nf">is_concurrent</span><span class="p">(</span><span class="o">*</span><span class="n">lines</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Is a sequence of linear entities concurrent?</span>
-
-<span class="sd">        Two or more linear entities are concurrent if they all</span>
-<span class="sd">        intersect at a single point.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        lines : a sequence of linear entities.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        True : if the set of linear entities are concurrent,</span>
-<span class="sd">        False : otherwise.</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        Simply take the first two lines and find their intersection.</span>
-<span class="sd">        If there is no intersection, then the first two lines were</span>
-<span class="sd">        parallel and had no intersection so concurrency is impossible</span>
-<span class="sd">        amongst the whole set. Otherwise, check to see if the</span>
-<span class="sd">        intersection point of the first two lines is a member on</span>
-<span class="sd">        the rest of the lines. If so, the lines are concurrent.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.util.intersection</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Line</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(0, 0), Point(3, 5)</span>
-<span class="sd">        &gt;&gt;&gt; p3, p4 = Point(-2, -2), Point(0, 2)</span>
-<span class="sd">        &gt;&gt;&gt; l1, l2, l3 = Line(p1, p2), Line(p1, p3), Line(p1, p4)</span>
-<span class="sd">        &gt;&gt;&gt; l1.is_concurrent(l2, l3)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &gt;&gt;&gt; l4 = Line(p2, p3)</span>
-<span class="sd">        &gt;&gt;&gt; l4.is_concurrent(l2, l3)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="c"># Concurrency requires intersection at a single point; One linear</span>
-        <span class="c"># entity cannot be concurrent.</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">lines</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="c"># Get the intersection (if parallel)</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">lines</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">lines</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-
-            <span class="c"># Make sure the intersection is on every linear entity</span>
-            <span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="n">lines</span><span class="p">[</span><span class="mi">2</span><span class="p">:]:</span>
-                <span class="k">if</span> <span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">line</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-</div>
-<div class="viewcode-block" id="LinearEntity.is_parallel"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.LinearEntity.is_parallel">[docs]</a>    <span class="k">def</span> <span class="nf">is_parallel</span><span class="p">(</span><span class="n">l1</span><span class="p">,</span> <span class="n">l2</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Are two linear entities parallel?</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        l1 : LinearEntity</span>
-<span class="sd">        l2 : LinearEntity</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        True : if l1 and l2 are parallel,</span>
-<span class="sd">        False : otherwise.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        coefficients</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Line</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(0, 0), Point(1, 1)</span>
-<span class="sd">        &gt;&gt;&gt; p3, p4 = Point(3, 4), Point(6, 7)</span>
-<span class="sd">        &gt;&gt;&gt; l1, l2 = Line(p1, p2), Line(p3, p4)</span>
-<span class="sd">        &gt;&gt;&gt; Line.is_parallel(l1, l2)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &gt;&gt;&gt; p5 = Point(6, 6)</span>
-<span class="sd">        &gt;&gt;&gt; l3 = Line(p3, p5)</span>
-<span class="sd">        &gt;&gt;&gt; Line.is_parallel(l1, l3)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">a1</span><span class="p">,</span> <span class="n">b1</span><span class="p">,</span> <span class="n">c1</span> <span class="o">=</span> <span class="n">l1</span><span class="o">.</span><span class="n">coefficients</span>
-            <span class="n">a2</span><span class="p">,</span> <span class="n">b2</span><span class="p">,</span> <span class="n">c2</span> <span class="o">=</span> <span class="n">l2</span><span class="o">.</span><span class="n">coefficients</span>
-            <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="n">simplify</span><span class="p">(</span><span class="n">a1</span><span class="o">*</span><span class="n">b2</span> <span class="o">-</span> <span class="n">b1</span><span class="o">*</span><span class="n">a2</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-</div>
-<div class="viewcode-block" id="LinearEntity.is_perpendicular"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.LinearEntity.is_perpendicular">[docs]</a>    <span class="k">def</span> <span class="nf">is_perpendicular</span><span class="p">(</span><span class="n">l1</span><span class="p">,</span> <span class="n">l2</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Are two linear entities perpendicular?</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        l1 : LinearEntity</span>
-<span class="sd">        l2 : LinearEntity</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        True : if l1 and l2 are perpendicular,</span>
-<span class="sd">        False : otherwise.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        coefficients</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Line</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(1, 1), Point(-1, 1)</span>
-<span class="sd">        &gt;&gt;&gt; l1, l2 = Line(p1, p2), Line(p1, p3)</span>
-<span class="sd">        &gt;&gt;&gt; l1.is_perpendicular(l2)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &gt;&gt;&gt; p4 = Point(5, 3)</span>
-<span class="sd">        &gt;&gt;&gt; l3 = Line(p1, p4)</span>
-<span class="sd">        &gt;&gt;&gt; l1.is_perpendicular(l3)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">a1</span><span class="p">,</span> <span class="n">b1</span><span class="p">,</span> <span class="n">c1</span> <span class="o">=</span> <span class="n">l1</span><span class="o">.</span><span class="n">coefficients</span>
-            <span class="n">a2</span><span class="p">,</span> <span class="n">b2</span><span class="p">,</span> <span class="n">c2</span> <span class="o">=</span> <span class="n">l2</span><span class="o">.</span><span class="n">coefficients</span>
-            <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="n">simplify</span><span class="p">(</span><span class="n">a1</span><span class="o">*</span><span class="n">a2</span> <span class="o">+</span> <span class="n">b1</span><span class="o">*</span><span class="n">b2</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-</div>
-<div class="viewcode-block" id="LinearEntity.angle_between"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.LinearEntity.angle_between">[docs]</a>    <span class="k">def</span> <span class="nf">angle_between</span><span class="p">(</span><span class="n">l1</span><span class="p">,</span> <span class="n">l2</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The angle formed between the two linear entities.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        l1 : LinearEntity</span>
-<span class="sd">        l2 : LinearEntity</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        angle : angle in radians</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        From the dot product of vectors v1 and v2 it is known that:</span>
-
-<span class="sd">            ``dot(v1, v2) = |v1|*|v2|*cos(A)``</span>
-
-<span class="sd">        where A is the angle formed between the two vectors. We can</span>
-<span class="sd">        get the directional vectors of the two lines and readily</span>
-<span class="sd">        find the angle between the two using the above formula.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        is_perpendicular</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Line</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(0, 4), Point(2, 0)</span>
-<span class="sd">        &gt;&gt;&gt; l1, l2 = Line(p1, p2), Line(p1, p3)</span>
-<span class="sd">        &gt;&gt;&gt; l1.angle_between(l2)</span>
-<span class="sd">        pi/2</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">v1</span> <span class="o">=</span> <span class="n">l1</span><span class="o">.</span><span class="n">p2</span> <span class="o">-</span> <span class="n">l1</span><span class="o">.</span><span class="n">p1</span>
-        <span class="n">v2</span> <span class="o">=</span> <span class="n">l2</span><span class="o">.</span><span class="n">p2</span> <span class="o">-</span> <span class="n">l2</span><span class="o">.</span><span class="n">p1</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">acos</span><span class="p">(</span><span class="n">v1</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">v2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">v1</span><span class="p">)</span><span class="o">*</span><span class="nb">abs</span><span class="p">(</span><span class="n">v2</span><span class="p">)))</span>
-</div>
-<div class="viewcode-block" id="LinearEntity.parallel_line"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.LinearEntity.parallel_line">[docs]</a>    <span class="k">def</span> <span class="nf">parallel_line</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Create a new Line parallel to this linear entity which passes</span>
-<span class="sd">        through the point `p`.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        p : Point</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        line : Line</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        is_parallel</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Line</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(2, 3), Point(-2, 2)</span>
-<span class="sd">        &gt;&gt;&gt; l1 = Line(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; l2 = l1.parallel_line(p3)</span>
-<span class="sd">        &gt;&gt;&gt; p3 in l2</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; l1.is_parallel(l2)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span>
-        <span class="k">return</span> <span class="n">Line</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">p</span> <span class="o">+</span> <span class="n">d</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="LinearEntity.perpendicular_line"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.LinearEntity.perpendicular_line">[docs]</a>    <span class="k">def</span> <span class="nf">perpendicular_line</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Create a new Line perpendicular to this linear entity which passes</span>
-<span class="sd">        through the point `p`.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        p : Point</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        line : Line</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        is_perpendicular, perpendicular_segment</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Line</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(2, 3), Point(-2, 2)</span>
-<span class="sd">        &gt;&gt;&gt; l1 = Line(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; l2 = l1.perpendicular_line(p3)</span>
-<span class="sd">        &gt;&gt;&gt; p3 in l2</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; l1.is_perpendicular(l2)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">d1</span><span class="p">,</span> <span class="n">d2</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="p">)</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">if</span> <span class="n">d2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>  <span class="c"># If a horizontal line</span>
-            <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">y</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">y</span><span class="p">:</span>  <span class="c"># if p is on this linear entity</span>
-                <span class="k">return</span> <span class="n">Line</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">p</span> <span class="o">+</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">Line</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">x</span> <span class="o">-</span> <span class="n">d2</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">y</span> <span class="o">+</span> <span class="n">d1</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Line</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="LinearEntity.perpendicular_segment"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.LinearEntity.perpendicular_segment">[docs]</a>    <span class="k">def</span> <span class="nf">perpendicular_segment</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Create a perpendicular line segment from `p` to this line.</span>
-
-<span class="sd">        The enpoints of the segment are ``p`` and the closest point in</span>
-<span class="sd">        the line containing self. (If self is not a line, the point might</span>
-<span class="sd">        not be in self.)</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        p : Point</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        segment : Segment</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        Returns `p` itself if `p` is on this linear entity.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        perpendicular_line</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Line</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, 2)</span>
-<span class="sd">        &gt;&gt;&gt; l1 = Line(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; s1 = l1.perpendicular_segment(p3)</span>
-<span class="sd">        &gt;&gt;&gt; l1.is_perpendicular(s1)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; p3 in s1</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; l1.perpendicular_segment(Point(4, 0))</span>
-<span class="sd">        Segment(Point(2, 2), Point(4, 0))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">p</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">p</span>
-        <span class="n">pl</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">perpendicular_line</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-        <span class="n">p2</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">pl</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">Segment</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="LinearEntity.length"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.LinearEntity.length">[docs]</a>    <span class="k">def</span> <span class="nf">length</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The length of the line.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Line</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(0, 0), Point(3, 5)</span>
-<span class="sd">        &gt;&gt;&gt; l1 = Line(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; l1.length</span>
-<span class="sd">        oo</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="LinearEntity.slope"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.LinearEntity.slope">[docs]</a>    <span class="k">def</span> <span class="nf">slope</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The slope of this linear entity, or infinity if vertical.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        slope : number or sympy expression</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        coefficients</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Line</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(0, 0), Point(3, 5)</span>
-<span class="sd">        &gt;&gt;&gt; l1 = Line(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; l1.slope</span>
-<span class="sd">        5/3</span>
-
-<span class="sd">        &gt;&gt;&gt; p3 = Point(0, 4)</span>
-<span class="sd">        &gt;&gt;&gt; l2 = Line(p1, p3)</span>
-<span class="sd">        &gt;&gt;&gt; l2.slope</span>
-<span class="sd">        oo</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">d1</span><span class="p">,</span> <span class="n">d2</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="p">)</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">if</span> <span class="n">d1</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">return</span> <span class="n">simplify</span><span class="p">(</span><span class="n">d2</span><span class="o">/</span><span class="n">d1</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="LinearEntity.points"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.LinearEntity.points">[docs]</a>    <span class="k">def</span> <span class="nf">points</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The two points used to define this linear entity.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        points : tuple of Points</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Line</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(0, 0), Point(5, 11)</span>
-<span class="sd">        &gt;&gt;&gt; l1 = Line(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; l1.points</span>
-<span class="sd">        (Point(0, 0), Point(5, 11))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="LinearEntity.projection"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.LinearEntity.projection">[docs]</a>    <span class="k">def</span> <span class="nf">projection</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Project a point, line, ray, or segment onto this linear entity.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        other : Point or LinearEntity (Line, Ray, Segment)</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        projection : Point or LinearEntity (Line, Ray, Segment)</span>
-<span class="sd">            The return type matches the type of the parameter ``other``.</span>
-
-<span class="sd">        Raises</span>
-<span class="sd">        ======</span>
-
-<span class="sd">        GeometryError</span>
-<span class="sd">            When method is unable to perform projection.</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        A projection involves taking the two points that define</span>
-<span class="sd">        the linear entity and projecting those points onto a</span>
-<span class="sd">        Line and then reforming the linear entity using these</span>
-<span class="sd">        projections.</span>
-<span class="sd">        A point P is projected onto a line L by finding the point</span>
-<span class="sd">        on L that is closest to P. This is done by creating a</span>
-<span class="sd">        perpendicular line through P and L and finding its</span>
-<span class="sd">        intersection with L.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point, perpendicular_line</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Line, Segment, Rational</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(1, 1), Point(Rational(1, 2), 0)</span>
-<span class="sd">        &gt;&gt;&gt; l1 = Line(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; l1.projection(p3)</span>
-<span class="sd">        Point(1/4, 1/4)</span>
-
-<span class="sd">        &gt;&gt;&gt; p4, p5 = Point(10, 0), Point(12, 1)</span>
-<span class="sd">        &gt;&gt;&gt; s1 = Segment(p4, p5)</span>
-<span class="sd">        &gt;&gt;&gt; l1.projection(s1)</span>
-<span class="sd">        Segment(Point(5, 5), Point(13/2, 13/2))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">tline</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">_project</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;Project a point onto the line representing self.&quot;&quot;&quot;</span>
-            <span class="k">if</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">tline</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">p</span>
-            <span class="n">l1</span> <span class="o">=</span> <span class="n">tline</span><span class="o">.</span><span class="n">perpendicular_line</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">tline</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">l1</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="n">projected</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Point</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">_project</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">LinearEntity</span><span class="p">):</span>
-            <span class="n">n_p1</span> <span class="o">=</span> <span class="n">_project</span><span class="p">(</span><span class="n">o</span><span class="o">.</span><span class="n">p1</span><span class="p">)</span>
-            <span class="n">n_p2</span> <span class="o">=</span> <span class="n">_project</span><span class="p">(</span><span class="n">o</span><span class="o">.</span><span class="n">p2</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">n_p1</span> <span class="o">==</span> <span class="n">n_p2</span><span class="p">:</span>
-                <span class="n">projected</span> <span class="o">=</span> <span class="n">n_p1</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">projected</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">n_p1</span><span class="p">,</span> <span class="n">n_p2</span><span class="p">)</span>
-
-        <span class="c"># Didn&#39;t know how to project so raise an error</span>
-        <span class="k">if</span> <span class="n">projected</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">n1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span>
-            <span class="n">n2</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span>
-            <span class="k">raise</span> <span class="n">GeometryError</span><span class="p">(</span>
-                <span class="s">&quot;Do not know how to project </span><span class="si">%s</span><span class="s"> onto </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n2</span><span class="p">,</span> <span class="n">n1</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">projected</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-<div class="viewcode-block" id="LinearEntity.intersection"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.LinearEntity.intersection">[docs]</a>    <span class="k">def</span> <span class="nf">intersection</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The intersection with another geometrical entity.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        o : Point or LinearEntity</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        intersection : list of geometrical entities</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Line, Segment</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(1, 1), Point(7, 7)</span>
-<span class="sd">        &gt;&gt;&gt; l1 = Line(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; l1.intersection(p3)</span>
-<span class="sd">        [Point(7, 7)]</span>
-
-<span class="sd">        &gt;&gt;&gt; p4, p5 = Point(5, 0), Point(0, 3)</span>
-<span class="sd">        &gt;&gt;&gt; l2 = Line(p4, p5)</span>
-<span class="sd">        &gt;&gt;&gt; l1.intersection(l2)</span>
-<span class="sd">        [Point(15/8, 15/8)]</span>
-
-<span class="sd">        &gt;&gt;&gt; p6, p7 = Point(0, 5), Point(2, 6)</span>
-<span class="sd">        &gt;&gt;&gt; s1 = Segment(p6, p7)</span>
-<span class="sd">        &gt;&gt;&gt; l1.intersection(s1)</span>
-<span class="sd">        []</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Point</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">o</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">o</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[]</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">LinearEntity</span><span class="p">):</span>
-            <span class="n">a1</span><span class="p">,</span> <span class="n">b1</span><span class="p">,</span> <span class="n">c1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">coefficients</span>
-            <span class="n">a2</span><span class="p">,</span> <span class="n">b2</span><span class="p">,</span> <span class="n">c2</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">coefficients</span>
-            <span class="n">t</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">a1</span><span class="o">*</span><span class="n">b2</span> <span class="o">-</span> <span class="n">a2</span><span class="o">*</span><span class="n">b1</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">:</span>  <span class="c"># assume they are parallel</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Line</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">o</span><span class="o">.</span><span class="n">p1</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="p">[</span><span class="n">o</span><span class="p">]</span>
-                    <span class="k">return</span> <span class="p">[]</span>
-                <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Line</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span> <span class="ow">in</span> <span class="n">o</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="p">[</span><span class="bp">self</span><span class="p">]</span>
-                    <span class="k">return</span> <span class="p">[]</span>
-                <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Ray</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Ray</span><span class="p">):</span>
-                        <span class="c"># case 1, rays in the same direction</span>
-                        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">xdirection</span> <span class="o">==</span> <span class="n">o</span><span class="o">.</span><span class="n">xdirection</span><span class="p">:</span>
-                            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">source</span><span class="o">.</span><span class="n">x</span> <span class="o">&lt;</span> <span class="n">o</span><span class="o">.</span><span class="n">source</span><span class="o">.</span><span class="n">x</span><span class="p">:</span>
-                                <span class="k">return</span> <span class="p">[</span><span class="n">o</span><span class="p">]</span>
-                            <span class="k">return</span> <span class="p">[</span><span class="bp">self</span><span class="p">]</span>
-                        <span class="c"># case 2, rays in the opposite directions</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="k">if</span> <span class="n">o</span><span class="o">.</span><span class="n">source</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">:</span>
-                                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">source</span> <span class="o">==</span> <span class="n">o</span><span class="o">.</span><span class="n">source</span><span class="p">:</span>
-                                    <span class="k">return</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">source</span><span class="p">]</span>
-                                <span class="k">return</span> <span class="p">[</span><span class="n">Segment</span><span class="p">(</span><span class="n">o</span><span class="o">.</span><span class="n">source</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">source</span><span class="p">)]</span>
-                            <span class="k">return</span> <span class="p">[]</span>
-                    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Segment</span><span class="p">):</span>
-                        <span class="k">if</span> <span class="n">o</span><span class="o">.</span><span class="n">p1</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">:</span>
-                            <span class="k">if</span> <span class="n">o</span><span class="o">.</span><span class="n">p2</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">:</span>
-                                <span class="k">return</span> <span class="p">[</span><span class="n">o</span><span class="p">]</span>
-                            <span class="k">return</span> <span class="p">[</span><span class="n">Segment</span><span class="p">(</span><span class="n">o</span><span class="o">.</span><span class="n">p1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">source</span><span class="p">)]</span>
-                        <span class="k">elif</span> <span class="n">o</span><span class="o">.</span><span class="n">p2</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="p">[</span><span class="n">Segment</span><span class="p">(</span><span class="n">o</span><span class="o">.</span><span class="n">p2</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">source</span><span class="p">)]</span>
-                        <span class="k">return</span> <span class="p">[]</span>
-                <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Segment</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Ray</span><span class="p">):</span>
-                        <span class="k">return</span> <span class="n">o</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-                    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Segment</span><span class="p">):</span>
-                        <span class="c"># A reminder that the points of Segments are ordered</span>
-                        <span class="c"># in such a way that the following works. See</span>
-                        <span class="c"># Segment.__new__ for details on the ordering.</span>
-                        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">o</span><span class="p">:</span>
-                            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">o</span><span class="p">:</span>
-                                <span class="c"># Neither of the endpoints are in o so either</span>
-                                <span class="c"># o is contained in this segment or it isn&#39;t</span>
-                                <span class="k">if</span> <span class="n">o</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">:</span>
-                                    <span class="k">return</span> <span class="p">[</span><span class="bp">self</span><span class="p">]</span>
-                                <span class="k">return</span> <span class="p">[]</span>
-                            <span class="k">else</span><span class="p">:</span>
-                                <span class="c"># p1 not in o but p2 is. Either there is a</span>
-                                <span class="c"># segment as an intersection, or they only</span>
-                                <span class="c"># intersect at an endpoint</span>
-                                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span> <span class="o">==</span> <span class="n">o</span><span class="o">.</span><span class="n">p1</span><span class="p">:</span>
-                                    <span class="k">return</span> <span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">p1</span><span class="p">]</span>
-                                <span class="k">return</span> <span class="p">[</span><span class="n">Segment</span><span class="p">(</span><span class="n">o</span><span class="o">.</span><span class="n">p1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="p">)]</span>
-                        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">o</span><span class="p">:</span>
-                            <span class="c"># p2 not in o but p1 is. Either there is a</span>
-                            <span class="c"># segment as an intersection, or they only</span>
-                            <span class="c"># intersect at an endpoint</span>
-                            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span> <span class="o">==</span> <span class="n">o</span><span class="o">.</span><span class="n">p2</span><span class="p">:</span>
-                                <span class="k">return</span> <span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">p2</span><span class="p">]</span>
-                            <span class="k">return</span> <span class="p">[</span><span class="n">Segment</span><span class="p">(</span><span class="n">o</span><span class="o">.</span><span class="n">p2</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="p">)]</span>
-
-                        <span class="c"># Both points of self in o so the whole segment</span>
-                        <span class="c"># is in o</span>
-                        <span class="k">return</span> <span class="p">[</span><span class="bp">self</span><span class="p">]</span>
-
-                <span class="c"># Unknown linear entity</span>
-                <span class="k">return</span> <span class="p">[]</span>
-
-            <span class="c"># Not parallel, so find the point of intersection</span>
-            <span class="n">px</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">((</span><span class="n">b1</span><span class="o">*</span><span class="n">c2</span> <span class="o">-</span> <span class="n">c1</span><span class="o">*</span><span class="n">b2</span><span class="p">)</span> <span class="o">/</span> <span class="n">t</span><span class="p">)</span>
-            <span class="n">py</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">((</span><span class="n">a2</span><span class="o">*</span><span class="n">c1</span> <span class="o">-</span> <span class="n">a1</span><span class="o">*</span><span class="n">c2</span><span class="p">)</span> <span class="o">/</span> <span class="n">t</span><span class="p">)</span>
-            <span class="n">inter</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">px</span><span class="p">,</span> <span class="n">py</span><span class="p">)</span>
-            <span class="c"># we do not use a simplistic &#39;inter in self and inter in o&#39;</span>
-            <span class="c"># because that requires an equality test that is fragile;</span>
-            <span class="c"># instead we employ some diagnostics to see if the intersection</span>
-            <span class="c"># is valid</span>
-
-            <span class="k">def</span> <span class="nf">inseg</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-                <span class="k">def</span> <span class="nf">_between</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">c</span> <span class="o">&gt;=</span> <span class="n">a</span> <span class="ow">and</span> <span class="n">c</span> <span class="o">&lt;=</span> <span class="n">b</span> <span class="ow">or</span> <span class="n">c</span> <span class="o">&lt;=</span> <span class="n">a</span> <span class="ow">and</span> <span class="n">c</span> <span class="o">&gt;=</span> <span class="n">b</span>
-                <span class="k">if</span> <span class="n">_between</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">inter</span><span class="o">.</span><span class="n">x</span><span class="p">)</span> <span class="ow">and</span> \
-                        <span class="n">_between</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">y</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="o">.</span><span class="n">y</span><span class="p">,</span> <span class="n">inter</span><span class="o">.</span><span class="n">y</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-
-            <span class="k">def</span> <span class="nf">inray</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-                <span class="n">sray</span> <span class="o">=</span> <span class="n">Ray</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="p">,</span> <span class="n">inter</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">sray</span><span class="o">.</span><span class="n">xdirection</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">xdirection</span> <span class="ow">and</span> \
-                        <span class="n">sray</span><span class="o">.</span><span class="n">ydirection</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">ydirection</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">):</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Line</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Line</span><span class="p">):</span>
-                        <span class="k">return</span> <span class="p">[</span><span class="n">inter</span><span class="p">]</span>
-                    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Ray</span><span class="p">)</span> <span class="ow">and</span> <span class="n">inray</span><span class="p">(</span><span class="n">o</span><span class="p">):</span>
-                        <span class="k">return</span> <span class="p">[</span><span class="n">inter</span><span class="p">]</span>
-                    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Segment</span><span class="p">)</span> <span class="ow">and</span> <span class="n">inseg</span><span class="p">(</span><span class="n">o</span><span class="p">):</span>
-                        <span class="k">return</span> <span class="p">[</span><span class="n">inter</span><span class="p">]</span>
-                <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Ray</span><span class="p">)</span> <span class="ow">and</span> <span class="n">inray</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Ray</span><span class="p">)</span> <span class="ow">and</span> <span class="n">inray</span><span class="p">(</span><span class="n">o</span><span class="p">):</span>
-                        <span class="k">return</span> <span class="p">[</span><span class="n">inter</span><span class="p">]</span>
-                    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Segment</span><span class="p">)</span> <span class="ow">and</span> <span class="n">inseg</span><span class="p">(</span><span class="n">o</span><span class="p">):</span>
-                        <span class="k">return</span> <span class="p">[</span><span class="n">inter</span><span class="p">]</span>
-                <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Segment</span><span class="p">)</span> <span class="ow">and</span> <span class="n">inseg</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Segment</span><span class="p">)</span> <span class="ow">and</span> <span class="n">inseg</span><span class="p">(</span><span class="n">o</span><span class="p">):</span>
-                        <span class="k">return</span> <span class="p">[</span><span class="n">inter</span><span class="p">]</span>
-                <span class="bp">self</span><span class="p">,</span> <span class="n">o</span> <span class="o">=</span> <span class="n">o</span><span class="p">,</span> <span class="bp">self</span>
-            <span class="k">return</span> <span class="p">[]</span>
-
-        <span class="k">return</span> <span class="n">o</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="LinearEntity.random_point"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.LinearEntity.random_point">[docs]</a>    <span class="k">def</span> <span class="nf">random_point</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A random point on a LinearEntity.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        point : Point</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Line</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(0, 0), Point(5, 3)</span>
-<span class="sd">        &gt;&gt;&gt; l1 = Line(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; p3 = l1.random_point()</span>
-<span class="sd">        &gt;&gt;&gt; # random point - don&#39;t know its coords in advance</span>
-<span class="sd">        &gt;&gt;&gt; p3 # doctest: +ELLIPSIS</span>
-<span class="sd">        Point(...)</span>
-<span class="sd">        &gt;&gt;&gt; # point should belong to the line</span>
-<span class="sd">        &gt;&gt;&gt; p3 in l1</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">randint</span>
-
-        <span class="c"># The lower and upper</span>
-        <span class="n">lower</span><span class="p">,</span> <span class="n">upper</span> <span class="o">=</span> <span class="o">-</span><span class="mi">2</span><span class="o">**</span><span class="mi">32</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">**</span><span class="mi">32</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">slope</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Ray</span><span class="p">):</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">ydirection</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                    <span class="n">lower</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">y</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">upper</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">y</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Segment</span><span class="p">):</span>
-                <span class="n">lower</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">y</span>
-                <span class="n">upper</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="o">.</span><span class="n">y</span>
-
-            <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">x</span>
-            <span class="n">y</span> <span class="o">=</span> <span class="n">randint</span><span class="p">(</span><span class="n">lower</span><span class="p">,</span> <span class="n">upper</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Ray</span><span class="p">):</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">xdirection</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                    <span class="n">lower</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">x</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">upper</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">x</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Segment</span><span class="p">):</span>
-                <span class="n">lower</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">x</span>
-                <span class="n">upper</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="o">.</span><span class="n">x</span>
-
-            <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">coefficients</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">randint</span><span class="p">(</span><span class="n">lower</span><span class="p">,</span> <span class="n">upper</span><span class="p">)</span>
-            <span class="n">y</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">((</span><span class="o">-</span><span class="n">c</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="n">x</span><span class="p">)</span> <span class="o">/</span> <span class="n">b</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="LinearEntity.is_similar"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.LinearEntity.is_similar">[docs]</a>    <span class="k">def</span> <span class="nf">is_similar</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if self and other are contained in the same line.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Line</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 1), Point(3, 4), Point(2, 3)</span>
-<span class="sd">        &gt;&gt;&gt; l1 = Line(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; l2 = Line(p1, p3)</span>
-<span class="sd">        &gt;&gt;&gt; l1.is_similar(l2)</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">def</span> <span class="nf">_norm</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="mi">1</span><span class="p">,</span> <span class="n">b</span><span class="o">/</span><span class="n">a</span><span class="p">,</span> <span class="n">c</span><span class="o">/</span><span class="n">a</span>
-            <span class="k">elif</span> <span class="n">b</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">a</span><span class="o">/</span><span class="n">b</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">c</span><span class="o">/</span><span class="n">b</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">c</span>
-        <span class="k">return</span> <span class="n">_norm</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">coefficients</span><span class="p">)</span> <span class="o">==</span> <span class="n">_norm</span><span class="p">(</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">coefficients</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">__contains__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return a definitive answer or else raise an error if it cannot</span>
-<span class="sd">        be determined that other is on the boundaries of self.&quot;&quot;&quot;</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">result</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">Undecidable</span><span class="p">(</span>
-                <span class="s">&quot;can&#39;t decide whether &#39;</span><span class="si">%s</span><span class="s">&#39; contains &#39;</span><span class="si">%s</span><span class="s">&#39;&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">))</span>
-
-<div class="viewcode-block" id="LinearEntity.contains"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.LinearEntity.contains">[docs]</a>    <span class="k">def</span> <span class="nf">contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Subclasses should implement this method and should return</span>
-<span class="sd">            True if other is on the boundaries of self;</span>
-<span class="sd">            False if not on the boundaries of self;</span>
-<span class="sd">            None if a determination cannot be made.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-</div>
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Subclasses should implement this method.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">LinearEntity</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-</div>
-<div class="viewcode-block" id="Line"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.Line">[docs]</a><span class="k">class</span> <span class="nc">Line</span><span class="p">(</span><span class="n">LinearEntity</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;An infinite line in space.</span>
-
-<span class="sd">    A line is declared with two distinct points or a point and slope</span>
-<span class="sd">    as defined using keyword `slope`.</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    At the moment only lines in a 2D space can be declared, because</span>
-<span class="sd">    Points can be defined only for 2D spaces.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    p1 : Point</span>
-<span class="sd">    pt : Point</span>
-<span class="sd">    slope : sympy expression</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.geometry.point.Point</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; import sympy</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Point</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import L</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.geometry import Line, Segment</span>
-<span class="sd">    &gt;&gt;&gt; L = Line(Point(2,3), Point(3,5))</span>
-<span class="sd">    &gt;&gt;&gt; L</span>
-<span class="sd">    Line(Point(2, 3), Point(3, 5))</span>
-<span class="sd">    &gt;&gt;&gt; L.points</span>
-<span class="sd">    (Point(2, 3), Point(3, 5))</span>
-<span class="sd">    &gt;&gt;&gt; L.equation()</span>
-<span class="sd">    -2*x + y + 1</span>
-<span class="sd">    &gt;&gt;&gt; L.coefficients</span>
-<span class="sd">    (-2, 1, 1)</span>
-
-<span class="sd">    Instantiate with keyword ``slope``:</span>
-
-<span class="sd">    &gt;&gt;&gt; Line(Point(0, 0), slope=0)</span>
-<span class="sd">    Line(Point(0, 0), Point(1, 0))</span>
-
-<span class="sd">    Instantiate with another linear object</span>
-
-<span class="sd">    &gt;&gt;&gt; s = Segment((0, 0), (0, 1))</span>
-<span class="sd">    &gt;&gt;&gt; Line(s).equation()</span>
-<span class="sd">    x</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">pt</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">slope</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">LinearEntity</span><span class="p">):</span>
-            <span class="n">p1</span><span class="p">,</span> <span class="n">pt</span> <span class="o">=</span> <span class="n">p1</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">p1</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">pt</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">slope</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">pt</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;The 2nd argument was not a valid Point; if it was meant to be a slope it should be given with keyword &quot;slope&quot;.&#39;</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">p1</span> <span class="o">==</span> <span class="n">p2</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;A line requires two distinct points.&#39;</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">slope</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">pt</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">slope</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">slope</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">slope</span><span class="o">.</span><span class="n">is_bounded</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                <span class="c"># when unbounded slope, don&#39;t change x</span>
-                <span class="n">p2</span> <span class="o">=</span> <span class="n">p1</span> <span class="o">+</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># go over 1 up slope</span>
-                <span class="n">p2</span> <span class="o">=</span> <span class="n">p1</span> <span class="o">+</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">slope</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;A 2nd Point or keyword &quot;slope&quot; must be used.&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">LinearEntity</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Line.arbitrary_point"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.Line.arbitrary_point">[docs]</a>    <span class="k">def</span> <span class="nf">arbitrary_point</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">parameter</span><span class="o">=</span><span class="s">&#39;t&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A parameterized point on the Line.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        parameter : str, optional</span>
-<span class="sd">            The name of the parameter which will be used for the parametric</span>
-<span class="sd">            point. The default value is &#39;t&#39;.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        point : Point</span>
-
-<span class="sd">        Raises</span>
-<span class="sd">        ======</span>
-
-<span class="sd">        ValueError</span>
-<span class="sd">            When ``parameter`` already appears in the Line&#39;s definition.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Line</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(1, 0), Point(5, 3)</span>
-<span class="sd">        &gt;&gt;&gt; l1 = Line(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; l1.arbitrary_point()</span>
-<span class="sd">        Point(4*t + 1, 3*t)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">_symbol</span><span class="p">(</span><span class="n">parameter</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">name</span> <span class="ow">in</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">name</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Symbol </span><span class="si">%s</span><span class="s"> already appears in object and cannot be used as a parameter.&#39;</span> <span class="o">%</span> <span class="n">t</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">t</span><span class="o">*</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="o">.</span><span class="n">x</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">x</span><span class="p">))</span>
-        <span class="n">y</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">y</span> <span class="o">+</span> <span class="n">t</span><span class="o">*</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="o">.</span><span class="n">y</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">y</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Line.plot_interval"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.Line.plot_interval">[docs]</a>    <span class="k">def</span> <span class="nf">plot_interval</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">parameter</span><span class="o">=</span><span class="s">&#39;t&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The plot interval for the default geometric plot of line.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        parameter : str, optional</span>
-<span class="sd">            Default value is &#39;t&#39;.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        plot_interval : list (plot interval)</span>
-<span class="sd">            [parameter, lower_bound, upper_bound]</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Line</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(0, 0), Point(5, 3)</span>
-<span class="sd">        &gt;&gt;&gt; l1 = Line(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; l1.plot_interval()</span>
-<span class="sd">        [t, -5, 5]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">_symbol</span><span class="p">(</span><span class="n">parameter</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">t</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">]</span>
-</div>
-<div class="viewcode-block" id="Line.equation"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.Line.equation">[docs]</a>    <span class="k">def</span> <span class="nf">equation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s">&#39;y&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The equation of the line: ax + by + c.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        x : str, optional</span>
-<span class="sd">            The name to use for the x-axis, default value is &#39;x&#39;.</span>
-<span class="sd">        y : str, optional</span>
-<span class="sd">            The name to use for the y-axis, default value is &#39;y&#39;.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        equation : sympy expression</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        LinearEntity.coefficients</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Line</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(1, 0), Point(5, 3)</span>
-<span class="sd">        &gt;&gt;&gt; l1 = Line(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; l1.equation()</span>
-<span class="sd">        -3*x + 4*y + 3</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">_symbol</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">_symbol</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-        <span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">points</span>
-        <span class="k">if</span> <span class="n">p1</span><span class="o">.</span><span class="n">x</span> <span class="o">==</span> <span class="n">p2</span><span class="o">.</span><span class="n">x</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">x</span> <span class="o">-</span> <span class="n">p1</span><span class="o">.</span><span class="n">x</span>
-        <span class="k">elif</span> <span class="n">p1</span><span class="o">.</span><span class="n">y</span> <span class="o">==</span> <span class="n">p2</span><span class="o">.</span><span class="n">y</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">y</span> <span class="o">-</span> <span class="n">p1</span><span class="o">.</span><span class="n">y</span>
-
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">coefficients</span>
-        <span class="k">return</span> <span class="n">simplify</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="n">c</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Line.contains"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.Line.contains">[docs]</a>    <span class="k">def</span> <span class="nf">contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return True if o is on this Line, or False otherwise.&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Point</span><span class="p">):</span>
-            <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(),</span> <span class="n">Dummy</span><span class="p">()</span>
-            <span class="n">eq</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">equation</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">eq</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">y</span><span class="p">):</span>
-                <span class="k">return</span> <span class="p">(</span><span class="n">solve</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">x</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">o</span><span class="o">.</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">eq</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-                <span class="k">return</span> <span class="p">(</span><span class="n">solve</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">y</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">o</span><span class="o">.</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">solve</span><span class="p">(</span><span class="n">eq</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">o</span><span class="o">.</span><span class="n">x</span><span class="p">),</span> <span class="n">y</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">o</span><span class="o">.</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">LinearEntity</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Line</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_similar</span><span class="p">(</span><span class="n">o</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">o</span><span class="o">.</span><span class="n">p1</span> <span class="ow">in</span> <span class="bp">self</span> <span class="ow">and</span> <span class="n">o</span><span class="o">.</span><span class="n">p2</span> <span class="ow">in</span> <span class="bp">self</span>
-</div>
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return True if other is equal to this Line, or False otherwise.&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Line</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="n">Point</span><span class="o">.</span><span class="n">is_collinear</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">p1</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">p2</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Line</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-</div>
-<div class="viewcode-block" id="Ray"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.Ray">[docs]</a><span class="k">class</span> <span class="nc">Ray</span><span class="p">(</span><span class="n">LinearEntity</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A Ray is a semi-line in the space with a source point and a direction.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    p1 : Point</span>
-<span class="sd">        The source of the Ray</span>
-<span class="sd">    p2 : Point or radian value</span>
-<span class="sd">        This point determines the direction in which the Ray propagates.</span>
-<span class="sd">        If given as an angle it is interpreted in radians with the positive</span>
-<span class="sd">        direction being ccw.</span>
-
-<span class="sd">    Attributes</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    source</span>
-<span class="sd">    xdirection</span>
-<span class="sd">    ydirection</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.geometry.point.Point, Line</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    At the moment only rays in a 2D space can be declared, because</span>
-<span class="sd">    Points can be defined only for 2D spaces.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; import sympy</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Point, pi</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import r</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.geometry import Ray</span>
-<span class="sd">    &gt;&gt;&gt; r = Ray(Point(2, 3), Point(3, 5))</span>
-<span class="sd">    &gt;&gt;&gt; r = Ray(Point(2, 3), Point(3, 5))</span>
-<span class="sd">    &gt;&gt;&gt; r</span>
-<span class="sd">    Ray(Point(2, 3), Point(3, 5))</span>
-<span class="sd">    &gt;&gt;&gt; r.points</span>
-<span class="sd">    (Point(2, 3), Point(3, 5))</span>
-<span class="sd">    &gt;&gt;&gt; r.source</span>
-<span class="sd">    Point(2, 3)</span>
-<span class="sd">    &gt;&gt;&gt; r.xdirection</span>
-<span class="sd">    oo</span>
-<span class="sd">    &gt;&gt;&gt; r.ydirection</span>
-<span class="sd">    oo</span>
-<span class="sd">    &gt;&gt;&gt; r.slope</span>
-<span class="sd">    2</span>
-<span class="sd">    &gt;&gt;&gt; Ray(Point(0, 0), angle=pi/4).slope</span>
-<span class="sd">    1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">pt</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">angle</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">p1</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">pt</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">angle</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">pt</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;The 2nd argument was not a valid Point;</span><span class="se">\n</span><span class="s">if it was meant to be an angle it should be given with keyword &quot;angle&quot;.&#39;</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">p1</span> <span class="o">==</span> <span class="n">p2</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;A Ray requires two distinct points.&#39;</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">angle</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">pt</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="c"># we need to know if the angle is an odd multiple of pi/2</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">pi_coeff</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">angle</span><span class="p">))</span>
-            <span class="n">p2</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">c</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">c</span><span class="o">.</span><span class="n">q</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">c</span><span class="o">.</span><span class="n">p</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                            <span class="n">p2</span> <span class="o">=</span> <span class="n">p1</span> <span class="o">+</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-                        <span class="k">elif</span> <span class="n">c</span><span class="o">.</span><span class="n">p</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-                            <span class="n">p2</span> <span class="o">=</span> <span class="n">p1</span> <span class="o">+</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-                    <span class="k">elif</span> <span class="n">c</span><span class="o">.</span><span class="n">q</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">c</span><span class="o">.</span><span class="n">p</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                            <span class="n">p2</span> <span class="o">=</span> <span class="n">p1</span> <span class="o">+</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-                        <span class="k">elif</span> <span class="n">c</span><span class="o">.</span><span class="n">p</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                            <span class="n">p2</span> <span class="o">=</span> <span class="n">p1</span> <span class="o">+</span> <span class="n">Point</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">p2</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">c</span> <span class="o">*=</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="n">angle</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">p2</span><span class="p">:</span>
-                <span class="n">p2</span> <span class="o">=</span> <span class="n">p1</span> <span class="o">+</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">tan</span><span class="p">(</span><span class="n">c</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;A 2nd point or keyword &quot;angle&quot; must be used.&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">LinearEntity</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Ray.source"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.Ray.source">[docs]</a>    <span class="k">def</span> <span class="nf">source</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The point from which the ray emanates.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ray</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(0, 0), Point(4, 1)</span>
-<span class="sd">        &gt;&gt;&gt; r1 = Ray(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; r1.source</span>
-<span class="sd">        Point(0, 0)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Ray.xdirection"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.Ray.xdirection">[docs]</a>    <span class="k">def</span> <span class="nf">xdirection</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The x direction of the ray.</span>
-
-<span class="sd">        Positive infinity if the ray points in the positive x direction,</span>
-<span class="sd">        negative infinity if the ray points in the negative x direction,</span>
-<span class="sd">        or 0 if the ray is vertical.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        ydirection</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ray</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, -1)</span>
-<span class="sd">        &gt;&gt;&gt; r1, r2 = Ray(p1, p2), Ray(p1, p3)</span>
-<span class="sd">        &gt;&gt;&gt; r1.xdirection</span>
-<span class="sd">        oo</span>
-<span class="sd">        &gt;&gt;&gt; r2.xdirection</span>
-<span class="sd">        0</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">x</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="o">.</span><span class="n">x</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">x</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="o">.</span><span class="n">x</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Ray.ydirection"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.Ray.ydirection">[docs]</a>    <span class="k">def</span> <span class="nf">ydirection</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The y direction of the ray.</span>
-
-<span class="sd">        Positive infinity if the ray points in the positive y direction,</span>
-<span class="sd">        negative infinity if the ray points in the negative y direction,</span>
-<span class="sd">        or 0 if the ray is horizontal.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        xdirection</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ray</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(-1, -1), Point(-1, 0)</span>
-<span class="sd">        &gt;&gt;&gt; r1, r2 = Ray(p1, p2), Ray(p1, p3)</span>
-<span class="sd">        &gt;&gt;&gt; r1.ydirection</span>
-<span class="sd">        -oo</span>
-<span class="sd">        &gt;&gt;&gt; r2.ydirection</span>
-<span class="sd">        0</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">y</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="o">.</span><span class="n">y</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">y</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="o">.</span><span class="n">y</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-</div>
-<div class="viewcode-block" id="Ray.arbitrary_point"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.Ray.arbitrary_point">[docs]</a>    <span class="k">def</span> <span class="nf">arbitrary_point</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">parameter</span><span class="o">=</span><span class="s">&#39;t&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A parameterized point on the Ray.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        parameter : str, optional</span>
-<span class="sd">            The name of the parameter which will be used for the parametric</span>
-<span class="sd">            point. The default value is &#39;t&#39;.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        point : Point</span>
-
-<span class="sd">        Raises</span>
-<span class="sd">        ======</span>
-
-<span class="sd">        ValueError</span>
-<span class="sd">            When ``parameter`` already appears in the Ray&#39;s definition.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Ray, Point, Segment, S, simplify, solve</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import t</span>
-<span class="sd">        &gt;&gt;&gt; r = Ray(Point(0, 0), Point(2, 3))</span>
-
-<span class="sd">        &gt;&gt;&gt; p = r.arbitrary_point(t)</span>
-
-<span class="sd">        The parameter `t` used in the arbitrary point maps 0 to the</span>
-<span class="sd">        origin of the ray and 1 to the end of the ray at infinity</span>
-<span class="sd">        (which will show up as NaN).</span>
-
-<span class="sd">        &gt;&gt;&gt; p.subs(t, 0), p.subs(t, 1)</span>
-<span class="sd">        (Point(0, 0), Point(oo, oo))</span>
-
-<span class="sd">        The unit that `t` moves you is based on the spacing of the</span>
-<span class="sd">        points used to define the ray.</span>
-
-<span class="sd">        &gt;&gt;&gt; p.subs(t, 1/(S(1) + 1)) # one unit</span>
-<span class="sd">        Point(2, 3)</span>
-<span class="sd">        &gt;&gt;&gt; p.subs(t, 2/(S(1) + 2)) # two units out</span>
-<span class="sd">        Point(4, 6)</span>
-<span class="sd">        &gt;&gt;&gt; p.subs(t, S.Half/(S(1) + S.Half)) # half a unit out</span>
-<span class="sd">        Point(1, 3/2)</span>
-
-<span class="sd">        If you want to be located a distance of 1 from the origin of the</span>
-<span class="sd">        ray, what value of `t` is needed?</span>
-
-<span class="sd">        a)  Find the unit length and pick `t` accordingly.</span>
-
-<span class="sd">            &gt;&gt;&gt; u = Segment(r.p1, p.subs(t, S.Half)).length # S.Half = 1/(1 + 1)</span>
-<span class="sd">            &gt;&gt;&gt; want = 1</span>
-<span class="sd">            &gt;&gt;&gt; t_need = want/u</span>
-<span class="sd">            &gt;&gt;&gt; p_want = p.subs(t, t_need/(1 + t_need))</span>
-<span class="sd">            &gt;&gt;&gt; simplify(Segment(r.p1, p_want).length)</span>
-<span class="sd">            1</span>
-
-<span class="sd">        b)  Find the `t` that makes the length from origin to `p` equal to 1.</span>
-
-<span class="sd">            &gt;&gt;&gt; l = Segment(r.p1, p).length</span>
-<span class="sd">            &gt;&gt;&gt; t_need = solve(l**2 - want**2, t) # use the square to remove abs() if it is there</span>
-<span class="sd">            &gt;&gt;&gt; t_need = [w for w in t_need if w.n() &gt; 0][0] # take positive t</span>
-<span class="sd">            &gt;&gt;&gt; p_want = p.subs(t, t_need)</span>
-<span class="sd">            &gt;&gt;&gt; simplify(Segment(r.p1, p_want).length)</span>
-<span class="sd">            1</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">_symbol</span><span class="p">(</span><span class="n">parameter</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">name</span> <span class="ow">in</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">name</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Symbol </span><span class="si">%s</span><span class="s"> already appears in object and cannot be used as a parameter.&#39;</span> <span class="o">%</span> <span class="n">t</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">t</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">t</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="o">.</span><span class="n">x</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">x</span><span class="p">))</span>
-        <span class="n">y</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">y</span> <span class="o">+</span> <span class="n">t</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">t</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="o">.</span><span class="n">y</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">y</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Ray.plot_interval"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.Ray.plot_interval">[docs]</a>    <span class="k">def</span> <span class="nf">plot_interval</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">parameter</span><span class="o">=</span><span class="s">&#39;t&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The plot interval for the default geometric plot of the Ray.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        parameter : str, optional</span>
-<span class="sd">            Default value is &#39;t&#39;.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        plot_interval : list</span>
-<span class="sd">            [parameter, lower_bound, upper_bound]</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Ray, pi</span>
-<span class="sd">        &gt;&gt;&gt; r = Ray((0, 0), angle=pi/4)</span>
-<span class="sd">        &gt;&gt;&gt; r.plot_interval()</span>
-<span class="sd">        [t, 0, 5*sqrt(2)/(1 + 5*sqrt(2))]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">_symbol</span><span class="p">(</span><span class="n">parameter</span><span class="p">)</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-        <span class="c"># get a t corresponding to length of 10</span>
-        <span class="n">want</span> <span class="o">=</span> <span class="mi">10</span>
-        <span class="n">u</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">))</span><span class="o">.</span><span class="n">length</span>  <span class="c"># gives unit length</span>
-        <span class="n">t_need</span> <span class="o">=</span> <span class="n">want</span><span class="o">/</span><span class="n">u</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">t</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">t_need</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">t_need</span><span class="p">)]</span>
-</div>
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Is the other GeometryEntity equal to this Ray?&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Ray</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">source</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">source</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">p2</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Ray</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-<div class="viewcode-block" id="Ray.contains"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.Ray.contains">[docs]</a>    <span class="k">def</span> <span class="nf">contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Is other GeometryEntity contained in this Ray?&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Ray</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">Point</span><span class="o">.</span><span class="n">is_collinear</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="p">,</span> <span class="n">o</span><span class="o">.</span><span class="n">p1</span><span class="p">,</span> <span class="n">o</span><span class="o">.</span><span class="n">p2</span><span class="p">)</span> <span class="ow">and</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">xdirection</span> <span class="o">==</span> <span class="n">o</span><span class="o">.</span><span class="n">xdirection</span> <span class="ow">and</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">ydirection</span> <span class="o">==</span> <span class="n">o</span><span class="o">.</span><span class="n">ydirection</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Segment</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">o</span><span class="o">.</span><span class="n">p1</span> <span class="ow">in</span> <span class="bp">self</span> <span class="ow">and</span> <span class="n">o</span><span class="o">.</span><span class="n">p2</span> <span class="ow">in</span> <span class="bp">self</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Point</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">Point</span><span class="o">.</span><span class="n">is_collinear</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">xdirection</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                    <span class="n">rv</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">x</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">source</span><span class="o">.</span><span class="n">x</span>
-                <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">xdirection</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                    <span class="n">rv</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">x</span> <span class="o">&lt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">source</span><span class="o">.</span><span class="n">x</span>
-                <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">ydirection</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                    <span class="n">rv</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">y</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">source</span><span class="o">.</span><span class="n">y</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">rv</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">y</span> <span class="o">&lt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">source</span><span class="o">.</span><span class="n">y</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">rv</span><span class="p">,</span> <span class="nb">bool</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">rv</span>
-                <span class="k">raise</span> <span class="n">Undecidable</span><span class="p">(</span>
-                    <span class="s">&#39;Cannot determine if </span><span class="si">%s</span><span class="s"> is in </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="bp">self</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># Points are not collinear, so the rays are not parallel</span>
-                <span class="c"># and hence it is impossible for self to contain o</span>
-                <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="c"># No other known entity can be contained in a Ray</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-</div></div>
-<div class="viewcode-block" id="Segment"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.Segment">[docs]</a><span class="k">class</span> <span class="nc">Segment</span><span class="p">(</span><span class="n">LinearEntity</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;An undirected line segment in space.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    p1 : Point</span>
-<span class="sd">    p2 : Point</span>
-
-<span class="sd">    Attributes</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    length : number or sympy expression</span>
-<span class="sd">    midpoint : Point</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.geometry.point.Point, Line</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    At the moment only segments in a 2D space can be declared, because</span>
-<span class="sd">    Points can be defined only for 2D spaces.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; import sympy</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Point</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import s</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.geometry import Segment</span>
-<span class="sd">    &gt;&gt;&gt; Segment((1, 0), (1, 1)) # tuples are interpreted as pts</span>
-<span class="sd">    Segment(Point(1, 0), Point(1, 1))</span>
-<span class="sd">    &gt;&gt;&gt; s = Segment(Point(4, 3), Point(1, 1))</span>
-<span class="sd">    &gt;&gt;&gt; s</span>
-<span class="sd">    Segment(Point(1, 1), Point(4, 3))</span>
-<span class="sd">    &gt;&gt;&gt; s.points</span>
-<span class="sd">    (Point(1, 1), Point(4, 3))</span>
-<span class="sd">    &gt;&gt;&gt; s.slope</span>
-<span class="sd">    2/3</span>
-<span class="sd">    &gt;&gt;&gt; s.length</span>
-<span class="sd">    sqrt(13)</span>
-<span class="sd">    &gt;&gt;&gt; s.midpoint</span>
-<span class="sd">    Point(5/2, 2)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="c"># Reorder the two points under the following ordering:</span>
-        <span class="c">#   if p1.x != p2.x then p1.x &lt; p2.x</span>
-        <span class="c">#   if p1.x == p2.x then p1.y &lt; p2.y</span>
-        <span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">p1</span><span class="p">)</span>
-        <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">p2</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">p1</span> <span class="o">==</span> <span class="n">p2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="n">p1</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">p1</span><span class="o">.</span><span class="n">x</span> <span class="o">&gt;</span> <span class="n">p2</span><span class="o">.</span><span class="n">x</span><span class="p">:</span>
-            <span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p1</span>
-        <span class="k">elif</span> <span class="n">p1</span><span class="o">.</span><span class="n">x</span> <span class="o">==</span> <span class="n">p2</span><span class="o">.</span><span class="n">x</span> <span class="ow">and</span> <span class="n">p1</span><span class="o">.</span><span class="n">y</span> <span class="o">&gt;</span> <span class="n">p2</span><span class="o">.</span><span class="n">y</span><span class="p">:</span>
-            <span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p1</span>
-        <span class="k">return</span> <span class="n">LinearEntity</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Segment.arbitrary_point"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.Segment.arbitrary_point">[docs]</a>    <span class="k">def</span> <span class="nf">arbitrary_point</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">parameter</span><span class="o">=</span><span class="s">&#39;t&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A parameterized point on the Segment.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        parameter : str, optional</span>
-<span class="sd">            The name of the parameter which will be used for the parametric</span>
-<span class="sd">            point. The default value is &#39;t&#39;.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        point : Point</span>
-
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        parameter : str, optional</span>
-<span class="sd">            The name of the parameter which will be used for the parametric</span>
-<span class="sd">            point. The default value is &#39;t&#39;.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        point : Point</span>
-
-<span class="sd">        Raises</span>
-<span class="sd">        ======</span>
-
-<span class="sd">        ValueError</span>
-<span class="sd">            When ``parameter`` already appears in the Segment&#39;s definition.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Segment</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(1, 0), Point(5, 3)</span>
-<span class="sd">        &gt;&gt;&gt; s1 = Segment(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; s1.arbitrary_point()</span>
-<span class="sd">        Point(4*t + 1, 3*t)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">_symbol</span><span class="p">(</span><span class="n">parameter</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">name</span> <span class="ow">in</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">name</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Symbol </span><span class="si">%s</span><span class="s"> already appears in object and cannot be used as a parameter.&#39;</span> <span class="o">%</span> <span class="n">t</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">t</span><span class="o">*</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="o">.</span><span class="n">x</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">x</span><span class="p">))</span>
-        <span class="n">y</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">y</span> <span class="o">+</span> <span class="n">t</span><span class="o">*</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="o">.</span><span class="n">y</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">y</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Segment.plot_interval"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.Segment.plot_interval">[docs]</a>    <span class="k">def</span> <span class="nf">plot_interval</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">parameter</span><span class="o">=</span><span class="s">&#39;t&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The plot interval for the default geometric plot of the Segment.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        parameter : str, optional</span>
-<span class="sd">            Default value is &#39;t&#39;.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        plot_interval : list</span>
-<span class="sd">            [parameter, lower_bound, upper_bound]</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Segment</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(0, 0), Point(5, 3)</span>
-<span class="sd">        &gt;&gt;&gt; s1 = Segment(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; s1.plot_interval()</span>
-<span class="sd">        [t, 0, 1]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">_symbol</span><span class="p">(</span><span class="n">parameter</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">t</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
-</div>
-<div class="viewcode-block" id="Segment.perpendicular_bisector"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.Segment.perpendicular_bisector">[docs]</a>    <span class="k">def</span> <span class="nf">perpendicular_bisector</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The perpendicular bisector of this segment.</span>
-
-<span class="sd">        If no point is specified or the point specified is not on the</span>
-<span class="sd">        bisector then the bisector is returned as a Line. Otherwise a</span>
-<span class="sd">        Segment is returned that joins the point specified and the</span>
-<span class="sd">        intersection of the bisector and the segment.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        p : Point</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        bisector : Line or Segment</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        LinearEntity.perpendicular_segment</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Segment</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(6, 6), Point(5, 1)</span>
-<span class="sd">        &gt;&gt;&gt; s1 = Segment(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; s1.perpendicular_bisector()</span>
-<span class="sd">        Line(Point(3, 3), Point(9, -3))</span>
-
-<span class="sd">        &gt;&gt;&gt; s1.perpendicular_bisector(p3)</span>
-<span class="sd">        Segment(Point(3, 3), Point(5, 1))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">l</span> <span class="o">=</span> <span class="n">LinearEntity</span><span class="o">.</span><span class="n">perpendicular_line</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">midpoint</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">p</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">p</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">l</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">l</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Segment</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">midpoint</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Segment.length"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.Segment.length">[docs]</a>    <span class="k">def</span> <span class="nf">length</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The length of the line segment.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point.distance</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Segment</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(0, 0), Point(4, 3)</span>
-<span class="sd">        &gt;&gt;&gt; s1 = Segment(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; s1.length</span>
-<span class="sd">        5</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Point</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Segment.midpoint"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.Segment.midpoint">[docs]</a>    <span class="k">def</span> <span class="nf">midpoint</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The midpoint of the line segment.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point.midpoint</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Segment</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(0, 0), Point(4, 3)</span>
-<span class="sd">        &gt;&gt;&gt; s1 = Segment(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; s1.midpoint</span>
-<span class="sd">        Point(2, 3/2)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Point</span><span class="o">.</span><span class="n">midpoint</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Segment.distance"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.Segment.distance">[docs]</a>    <span class="k">def</span> <span class="nf">distance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Finds the shortest distance between a line segment and a point.</span>
-
-<span class="sd">        Raises</span>
-<span class="sd">        ======</span>
-
-<span class="sd">        NotImplementedError is raised if o is not a Point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Segment</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(0, 1), Point(3, 4)</span>
-<span class="sd">        &gt;&gt;&gt; s = Segment(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; s.distance(Point(10, 15))</span>
-<span class="sd">        sqrt(170)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Point</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_do_point_distance</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-</div>
-    <span class="k">def</span> <span class="nf">_do_point_distance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">pt</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Calculates the distance between a point and a line segment.&quot;&quot;&quot;</span>
-
-        <span class="n">seg_vector</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span>
-        <span class="n">pt_vector</span> <span class="o">=</span> <span class="n">pt</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">seg_vector</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">pt_vector</span><span class="p">)</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">length</span><span class="o">**</span><span class="mi">2</span>
-        <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">distance</span> <span class="o">=</span> <span class="n">Point</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="p">,</span> <span class="n">pt</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">t</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">distance</span> <span class="o">=</span> <span class="n">Point</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="p">,</span> <span class="n">pt</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">distance</span> <span class="o">=</span> <span class="n">Point</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">p1</span> <span class="o">+</span> <span class="n">Point</span><span class="p">(</span><span class="n">t</span><span class="o">*</span><span class="n">seg_vector</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="o">*</span><span class="n">seg_vector</span><span class="o">.</span><span class="n">y</span><span class="p">),</span> <span class="n">pt</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">distance</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Is the other GeometryEntity equal to this Ray?&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Segment</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">p1</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p2</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">p2</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Segment</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-<div class="viewcode-block" id="Segment.contains"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.line.Segment.contains">[docs]</a>    <span class="k">def</span> <span class="nf">contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Is the other GeometryEntity contained within this Segment?</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Segment</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(0, 1), Point(3, 4)</span>
-<span class="sd">        &gt;&gt;&gt; s = Segment(p1, p2)</span>
-<span class="sd">        &gt;&gt;&gt; s2 = Segment(p2, p1)</span>
-<span class="sd">        &gt;&gt;&gt; s.contains(s2)</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Segment</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">other</span><span class="o">.</span><span class="n">p1</span> <span class="ow">in</span> <span class="bp">self</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">p2</span> <span class="ow">in</span> <span class="bp">self</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Point</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">Point</span><span class="o">.</span><span class="n">is_collinear</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-                <span class="n">t</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">)</span>
-                <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">.</span><span class="n">args</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">p1</span><span class="o">.</span><span class="n">x</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">p2</span><span class="o">.</span><span class="n">x</span><span class="p">:</span>
-                    <span class="n">ti</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">other</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">ti</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">y</span> <span class="o">-</span> <span class="n">other</span><span class="o">.</span><span class="n">y</span><span class="p">,</span> <span class="n">t</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="k">if</span> <span class="n">ti</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="mi">0</span> <span class="o">&lt;=</span> <span class="n">ti</span> <span class="o">&lt;=</span> <span class="mi">1</span>
-                <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="c"># No other known entity can be contained in a Ray</span>
-        <span class="k">return</span> <span class="bp">False</span></div></div>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.geometry.point</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Geometrical Points.</span>
-
-<span class="sd">Contains</span>
-<span class="sd">========</span>
-<span class="sd">Point</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">iterable</span>
-<span class="kn">from</span> <span class="nn">sympy.core.containers</span> <span class="kn">import</span> <span class="n">Tuple</span>
-<span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">simplify</span>
-<span class="kn">from</span> <span class="nn">sympy.geometry.exceptions</span> <span class="kn">import</span> <span class="n">GeometryError</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="kn">from</span> <span class="nn">.entity</span> <span class="kn">import</span> <span class="n">GeometryEntity</span>
-<span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">Float</span>
-<span class="kn">from</span> <span class="nn">sympy.simplify.simplify</span> <span class="kn">import</span> <span class="n">nsimplify</span>
-
-
-<div class="viewcode-block" id="Point"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.point.Point">[docs]</a><span class="k">class</span> <span class="nc">Point</span><span class="p">(</span><span class="n">GeometryEntity</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A point in a 2-dimensional Euclidean space.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    coords : sequence of 2 coordinate values.</span>
-
-<span class="sd">    Attributes</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    x</span>
-<span class="sd">    y</span>
-<span class="sd">    length</span>
-
-<span class="sd">    Raises</span>
-<span class="sd">    ======</span>
-
-<span class="sd">    NotImplementedError</span>
-<span class="sd">        When trying to create a point with more than two dimensions.</span>
-<span class="sd">        When `intersection` is called with object other than a Point.</span>
-<span class="sd">    TypeError</span>
-<span class="sd">        When trying to add or subtract points with different dimensions.</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    Currently only 2-dimensional points are supported.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.geometry.line.Segment : Connects two Points</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.geometry import Point</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; Point(1, 2)</span>
-<span class="sd">    Point(1, 2)</span>
-<span class="sd">    &gt;&gt;&gt; Point([1, 2])</span>
-<span class="sd">    Point(1, 2)</span>
-<span class="sd">    &gt;&gt;&gt; Point(0, x)</span>
-<span class="sd">    Point(0, x)</span>
-
-<span class="sd">    Floats are automatically converted to Rational unless the</span>
-<span class="sd">    evaluate flag is False:</span>
-
-<span class="sd">    &gt;&gt;&gt; Point(0.5, 0.25)</span>
-<span class="sd">    Point(1/2, 1/4)</span>
-<span class="sd">    &gt;&gt;&gt; Point(0.5, 0.25, evaluate=False)</span>
-<span class="sd">    Point(0.5, 0.25)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">iterable</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
-            <span class="n">coords</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Point</span><span class="p">):</span>
-            <span class="n">coords</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">coords</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">coords</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                <span class="s">&quot;Only two dimensional points currently supported&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;evaluate&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-            <span class="n">coords</span> <span class="o">=</span> <span class="p">[</span><span class="n">nsimplify</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">coords</span><span class="p">]</span>
-
-        <span class="k">return</span> <span class="n">GeometryEntity</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">coords</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Point</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="n">ts</span><span class="p">,</span> <span class="n">to</span> <span class="o">=</span> <span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">),</span> <span class="nb">type</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">ts</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">to</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">args</span>
-
-    <span class="k">def</span> <span class="nf">__lt__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">&lt;</span> <span class="n">other</span><span class="o">.</span><span class="n">args</span>
-
-    <span class="k">def</span> <span class="nf">__contains__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">item</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">item</span> <span class="o">==</span> <span class="bp">self</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Point.x"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.point.Point.x">[docs]</a>    <span class="k">def</span> <span class="nf">x</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the X coordinate of the Point.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point</span>
-<span class="sd">        &gt;&gt;&gt; p = Point(0, 1)</span>
-<span class="sd">        &gt;&gt;&gt; p.x</span>
-<span class="sd">        0</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Point.y"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.point.Point.y">[docs]</a>    <span class="k">def</span> <span class="nf">y</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the Y coordinate of the Point.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point</span>
-<span class="sd">        &gt;&gt;&gt; p = Point(0, 1)</span>
-<span class="sd">        &gt;&gt;&gt; p.y</span>
-<span class="sd">        1</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Point.length"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.point.Point.length">[docs]</a>    <span class="k">def</span> <span class="nf">length</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Treating a Point as a Line, this returns 0 for the length of a Point.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point</span>
-<span class="sd">        &gt;&gt;&gt; p = Point(0, 1)</span>
-<span class="sd">        &gt;&gt;&gt; p.length</span>
-<span class="sd">        0</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-</div>
-<div class="viewcode-block" id="Point.is_collinear"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.point.Point.is_collinear">[docs]</a>    <span class="k">def</span> <span class="nf">is_collinear</span><span class="p">(</span><span class="o">*</span><span class="n">points</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Is a sequence of points collinear?</span>
-
-<span class="sd">        Test whether or not a set of points are collinear. Returns True if</span>
-<span class="sd">        the set of points are collinear, or False otherwise.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        points : sequence of Point</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        is_collinear : boolean</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        Slope is preserved everywhere on a line, so the slope between</span>
-<span class="sd">        any two points on the line should be the same. Take the first</span>
-<span class="sd">        two points, p1 and p2, and create a translated point v1</span>
-<span class="sd">        with p1 as the origin. Now for every other point we create</span>
-<span class="sd">        a translated point, vi with p1 also as the origin. Note that</span>
-<span class="sd">        these translations preserve slope since everything is</span>
-<span class="sd">        consistently translated to a new origin of p1. Since slope</span>
-<span class="sd">        is preserved then we have the following equality:</span>
-
-<span class="sd">              * v1_slope = vi_slope</span>
-<span class="sd">              * v1.y/v1.x = vi.y/vi.x (due to translation)</span>
-<span class="sd">              * v1.y*vi.x = vi.y*v1.x</span>
-<span class="sd">              * v1.y*vi.x - vi.y*v1.x = 0           (*)</span>
-
-<span class="sd">        Hence, if we have a vi such that the equality in (*) is False</span>
-<span class="sd">        then the points are not collinear. We do this test for every</span>
-<span class="sd">        point in the list, and if all pass then they are collinear.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.line.Line</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(0, 0), Point(1, 1)</span>
-<span class="sd">        &gt;&gt;&gt; p3, p4, p5 = Point(2, 2), Point(x, x), Point(1, 2)</span>
-<span class="sd">        &gt;&gt;&gt; Point.is_collinear(p1, p2, p3, p4)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Point.is_collinear(p1, p2, p3, p5)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">points</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">points</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>  <span class="c"># two points always form a line</span>
-        <span class="n">points</span> <span class="o">=</span> <span class="p">[</span><span class="n">Point</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">points</span><span class="p">]</span>
-
-        <span class="c"># XXX Cross product is used now, but that only extends to three</span>
-        <span class="c">#     dimensions. If the concept needs to extend to greater</span>
-        <span class="c">#     dimensions then another method would have to be used</span>
-        <span class="n">p1</span> <span class="o">=</span> <span class="n">points</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">p2</span> <span class="o">=</span> <span class="n">points</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">v1</span> <span class="o">=</span> <span class="n">p2</span> <span class="o">-</span> <span class="n">p1</span>
-        <span class="n">x1</span><span class="p">,</span> <span class="n">y1</span> <span class="o">=</span> <span class="n">v1</span><span class="o">.</span><span class="n">args</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">for</span> <span class="n">p3</span> <span class="ow">in</span> <span class="n">points</span><span class="p">[</span><span class="mi">2</span><span class="p">:]:</span>
-            <span class="n">x2</span><span class="p">,</span> <span class="n">y2</span> <span class="o">=</span> <span class="p">(</span><span class="n">p3</span> <span class="o">-</span> <span class="n">p1</span><span class="p">)</span><span class="o">.</span><span class="n">args</span>
-            <span class="n">test</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">x1</span><span class="o">*</span><span class="n">y2</span> <span class="o">-</span> <span class="n">y1</span><span class="o">*</span><span class="n">x2</span><span class="p">)</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">test</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">if</span> <span class="n">rv</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">test</span><span class="p">:</span>
-                <span class="n">rv</span> <span class="o">=</span> <span class="n">test</span>
-        <span class="k">return</span> <span class="n">rv</span>
-</div>
-<div class="viewcode-block" id="Point.is_concyclic"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.point.Point.is_concyclic">[docs]</a>    <span class="k">def</span> <span class="nf">is_concyclic</span><span class="p">(</span><span class="o">*</span><span class="n">points</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Is a sequence of points concyclic?</span>
-
-<span class="sd">        Test whether or not a sequence of points are concyclic (i.e., they lie</span>
-<span class="sd">        on a circle).</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        points : sequence of Points</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        is_concyclic : boolean</span>
-<span class="sd">            True if points are concyclic, False otherwise.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.ellipse.Circle</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        No points are not considered to be concyclic. One or two points</span>
-<span class="sd">        are definitely concyclic and three points are conyclic iff they</span>
-<span class="sd">        are not collinear.</span>
-
-<span class="sd">        For more than three points, create a circle from the first three</span>
-<span class="sd">        points. If the circle cannot be created (i.e., they are collinear)</span>
-<span class="sd">        then all of the points cannot be concyclic. If the circle is created</span>
-<span class="sd">        successfully then simply check the remaining points for containment</span>
-<span class="sd">        in the circle.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Point</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(-1, 0), Point(1, 0)</span>
-<span class="sd">        &gt;&gt;&gt; p3, p4 = Point(0, 1), Point(-1, 2)</span>
-<span class="sd">        &gt;&gt;&gt; Point.is_concyclic(p1, p2, p3)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Point.is_concyclic(p1, p2, p3, p4)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">points</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">points</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="n">points</span> <span class="o">=</span> <span class="p">[</span><span class="n">Point</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">points</span><span class="p">]</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">points</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="ow">not</span> <span class="n">Point</span><span class="o">.</span><span class="n">is_collinear</span><span class="p">(</span><span class="o">*</span><span class="n">points</span><span class="p">))</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">.ellipse</span> <span class="kn">import</span> <span class="n">Circle</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">Circle</span><span class="p">(</span><span class="n">points</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">points</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">points</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-            <span class="k">for</span> <span class="n">point</span> <span class="ow">in</span> <span class="n">points</span><span class="p">[</span><span class="mi">3</span><span class="p">:]:</span>
-                <span class="k">if</span> <span class="n">point</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">c</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">except</span> <span class="n">GeometryError</span><span class="p">:</span>
-            <span class="c"># Circle could not be created, because of collinearity of the</span>
-            <span class="c"># three points passed in, hence they are not concyclic.</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-<span class="c">#       &quot;&quot;&quot;</span>
-<span class="c">#       # This code is from Maple</span>
-<span class="c">#       def f(u):</span>
-<span class="c">#           dd = u[0]**2 + u[1]**2 + 1</span>
-<span class="c">#           u1 = 2*u[0] / dd</span>
-<span class="c">#           u2 = 2*u[1] / dd</span>
-<span class="c">#           u3 = (dd - 2) / dd</span>
-<span class="c">#           return u1,u2,u3</span>
-
-<span class="c">#       u1,u2,u3 = f(points[0])</span>
-<span class="c">#       v1,v2,v3 = f(points[1])</span>
-<span class="c">#       w1,w2,w3 = f(points[2])</span>
-<span class="c">#       p = [v1 - u1, v2 - u2, v3 - u3]</span>
-<span class="c">#       q = [w1 - u1, w2 - u2, w3 - u3]</span>
-<span class="c">#       r = [p[1]*q[2] - p[2]*q[1], p[2]*q[0] - p[0]*q[2], p[0]*q[1] - p[1]*q[0]]</span>
-<span class="c">#       for ind in xrange(3, len(points)):</span>
-<span class="c">#           s1,s2,s3 = f(points[ind])</span>
-<span class="c">#           test = simplify(r[0]*(s1-u1) + r[1]*(s2-u2) + r[2]*(s3-u3))</span>
-<span class="c">#           if test != 0:</span>
-<span class="c">#               return False</span>
-<span class="c">#       return True</span>
-<span class="c">#       &quot;&quot;&quot;</span>
-</div>
-<div class="viewcode-block" id="Point.distance"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.point.Point.distance">[docs]</a>    <span class="k">def</span> <span class="nf">distance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The Euclidean distance from self to point p.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        p : Point</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        distance : number or symbolic expression.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.line.Segment.length</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Point</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(1, 1), Point(4, 5)</span>
-<span class="sd">        &gt;&gt;&gt; p1.distance(p2)</span>
-<span class="sd">        5</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; p3 = Point(x, y)</span>
-<span class="sd">        &gt;&gt;&gt; p3.distance(Point(0, 0))</span>
-<span class="sd">        sqrt(x**2 + y**2)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="nb">sum</span><span class="p">([(</span><span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">args</span><span class="p">)]))</span>
-</div>
-<div class="viewcode-block" id="Point.midpoint"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.point.Point.midpoint">[docs]</a>    <span class="k">def</span> <span class="nf">midpoint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The midpoint between self and point p.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        p : Point</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        midpoint : Point</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.line.Segment.midpoint</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Point</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = Point(1, 1), Point(13, 5)</span>
-<span class="sd">        &gt;&gt;&gt; p1.midpoint(p2)</span>
-<span class="sd">        Point(7, 3)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Point</span><span class="p">([</span><span class="n">simplify</span><span class="p">((</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">args</span><span class="p">)])</span>
-</div>
-<div class="viewcode-block" id="Point.evalf"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.point.Point.evalf">[docs]</a>    <span class="k">def</span> <span class="nf">evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Evaluate the coordinates of the point.</span>
-
-<span class="sd">        This method will, where possible, create and return a new Point</span>
-<span class="sd">        where the coordinates are evaluated as floating point numbers to</span>
-<span class="sd">        the precision indicated (default=15).</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        point : Point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Rational</span>
-<span class="sd">        &gt;&gt;&gt; p1 = Point(Rational(1, 2), Rational(3, 2))</span>
-<span class="sd">        &gt;&gt;&gt; p1</span>
-<span class="sd">        Point(1/2, 3/2)</span>
-<span class="sd">        &gt;&gt;&gt; p1.evalf()</span>
-<span class="sd">        Point(0.5, 1.5)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">prec</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">coords</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">**</span><span class="n">options</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">coords</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="o">*</span><span class="n">coords</span><span class="p">,</span> <span class="o">**</span><span class="nb">dict</span><span class="p">(</span><span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">))</span>
-</div>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">evalf</span>
-
-<div class="viewcode-block" id="Point.intersection"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.point.Point.intersection">[docs]</a>    <span class="k">def</span> <span class="nf">intersection</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The intersection between this point and another point.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        other : Point</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        intersection : list of Points</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        The return value will either be an empty list if there is no</span>
-<span class="sd">        intersection, otherwise it will contain this point.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(1, 1), Point(0, 0)</span>
-<span class="sd">        &gt;&gt;&gt; p1.intersection(p2)</span>
-<span class="sd">        []</span>
-<span class="sd">        &gt;&gt;&gt; p1.intersection(p3)</span>
-<span class="sd">        [Point(0, 0)]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Point</span><span class="p">):</span>
-            <span class="k">if</span> <span class="bp">self</span> <span class="o">==</span> <span class="n">o</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="bp">self</span><span class="p">]</span>
-            <span class="k">return</span> <span class="p">[]</span>
-
-        <span class="k">return</span> <span class="n">o</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Point.rotate"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.point.Point.rotate">[docs]</a>    <span class="k">def</span> <span class="nf">rotate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">angle</span><span class="p">,</span> <span class="n">pt</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Rotate ``angle`` radians counterclockwise about Point ``pt``.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        rotate, scale</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, pi</span>
-<span class="sd">        &gt;&gt;&gt; t = Point(1, 0)</span>
-<span class="sd">        &gt;&gt;&gt; t.rotate(pi/2)</span>
-<span class="sd">        Point(0, 1)</span>
-<span class="sd">        &gt;&gt;&gt; t.rotate(pi/2, (2, 0))</span>
-<span class="sd">        Point(2, -1)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">Point</span>
-
-        <span class="n">c</span> <span class="o">=</span> <span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)</span>
-
-        <span class="n">rv</span> <span class="o">=</span> <span class="bp">self</span>
-        <span class="k">if</span> <span class="n">pt</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">pt</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">pt</span><span class="p">)</span>
-            <span class="n">rv</span> <span class="o">-=</span> <span class="n">pt</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">rv</span><span class="o">.</span><span class="n">args</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">c</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="n">s</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="n">s</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">c</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">pt</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">rv</span> <span class="o">+=</span> <span class="n">pt</span>
-        <span class="k">return</span> <span class="n">rv</span>
-</div>
-<div class="viewcode-block" id="Point.scale"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.point.Point.scale">[docs]</a>    <span class="k">def</span> <span class="nf">scale</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">pt</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Scale the coordinates of the Point by multiplying by</span>
-<span class="sd">        ``x`` and ``y`` after subtracting ``pt`` -- default is (0, 0) --</span>
-<span class="sd">        and then adding ``pt`` back again (i.e. ``pt`` is the point of</span>
-<span class="sd">        reference for the scaling).</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        rotate, translate</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point</span>
-<span class="sd">        &gt;&gt;&gt; t = Point(1, 1)</span>
-<span class="sd">        &gt;&gt;&gt; t.scale(2)</span>
-<span class="sd">        Point(2, 1)</span>
-<span class="sd">        &gt;&gt;&gt; t.scale(2, 2)</span>
-<span class="sd">        Point(2, 2)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">pt</span><span class="p">:</span>
-            <span class="n">pt</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">pt</span><span class="p">)</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">pt</span><span class="p">)</span><span class="o">.</span><span class="n">args</span><span class="p">)</span><span class="o">.</span><span class="n">scale</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="o">*</span><span class="n">pt</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="o">*</span><span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">y</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Point.translate"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.point.Point.translate">[docs]</a>    <span class="k">def</span> <span class="nf">translate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Shift the Point by adding x and y to the coordinates of the Point.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        rotate, scale</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point</span>
-<span class="sd">        &gt;&gt;&gt; t = Point(0, 1)</span>
-<span class="sd">        &gt;&gt;&gt; t.translate(2)</span>
-<span class="sd">        Point(2, 1)</span>
-<span class="sd">        &gt;&gt;&gt; t.translate(2, 2)</span>
-<span class="sd">        Point(2, 3)</span>
-<span class="sd">        &gt;&gt;&gt; t + Point(2, 2)</span>
-<span class="sd">        Point(2, 3)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">y</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Point.transform"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.point.Point.transform">[docs]</a>    <span class="k">def</span> <span class="nf">transform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">matrix</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the point after applying the transformation described</span>
-<span class="sd">        by the 3x3 Matrix, ``matrix``.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        geometry.entity.rotate</span>
-<span class="sd">        geometry.entity.scale</span>
-<span class="sd">        geometry.entity.translate</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">Matrix</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span><span class="o">*</span><span class="n">matrix</span><span class="p">)</span><span class="o">.</span><span class="n">tolist</span><span class="p">()[</span><span class="mi">0</span><span class="p">][:</span><span class="mi">2</span><span class="p">])</span>
-</div>
-<div class="viewcode-block" id="Point.dot"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.point.Point.dot">[docs]</a>    <span class="k">def</span> <span class="nf">dot</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p2</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return dot product of self with another Point.&quot;&quot;&quot;</span>
-        <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">p2</span><span class="p">)</span>
-        <span class="n">x1</span><span class="p">,</span> <span class="n">y1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="n">x2</span><span class="p">,</span> <span class="n">y2</span> <span class="o">=</span> <span class="n">p2</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">return</span> <span class="n">x1</span><span class="o">*</span><span class="n">x2</span> <span class="o">+</span> <span class="n">y1</span><span class="o">*</span><span class="n">y2</span>
-</div>
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Add other to self by incrementing self&#39;s coordinates by those of other.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.entity.translate</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Point</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">simplify</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span>
-                               <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">args</span><span class="p">)])</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span>
-                    <span class="s">&quot;Points must have the same number of dimensions&quot;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Cannot add non-Point, </span><span class="si">%s</span><span class="s">, to a Point&#39;</span> <span class="o">%</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Subtract two points, or subtract a factor from this point&#39;s</span>
-<span class="sd">        coordinates.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span> <span class="o">+</span> <span class="p">(</span><span class="o">-</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">factor</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Multiply point&#39;s coordinates by a factor.&quot;&quot;&quot;</span>
-        <span class="n">factor</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">factor</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Point</span><span class="p">([</span><span class="n">x</span><span class="o">*</span><span class="n">factor</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">divisor</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Divide point&#39;s coordinates by a factor.&quot;&quot;&quot;</span>
-        <span class="n">divisor</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">divisor</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Point</span><span class="p">([</span><span class="n">x</span><span class="o">/</span><span class="n">divisor</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="n">__truediv__</span> <span class="o">=</span> <span class="n">__div__</span>
-
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Negate the point.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Point</span><span class="p">([</span><span class="o">-</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">__abs__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the distance between this point and the origin.&quot;&quot;&quot;</span>
-        <span class="n">origin</span> <span class="o">=</span> <span class="n">Point</span><span class="p">([</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">Point</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">origin</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span></div>
-</pre></div>
-
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-  <h1>Source code for sympy.geometry.polygon</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">zoo</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">as_int</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.piecewise</span> <span class="kn">import</span> <span class="n">Piecewise</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.trigonometric</span> <span class="kn">import</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">tan</span><span class="p">,</span> <span class="n">sqrt</span>
-<span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">nsimplify</span>
-<span class="kn">from</span> <span class="nn">sympy.geometry.exceptions</span> <span class="kn">import</span> <span class="n">GeometryError</span>
-<span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="kn">from</span> <span class="nn">sympy.solvers</span> <span class="kn">import</span> <span class="n">solve</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">has_variety</span><span class="p">,</span> <span class="n">has_dups</span>
-
-<span class="kn">from</span> <span class="nn">.entity</span> <span class="kn">import</span> <span class="n">GeometryEntity</span>
-<span class="kn">from</span> <span class="nn">.point</span> <span class="kn">import</span> <span class="n">Point</span>
-<span class="kn">from</span> <span class="nn">.ellipse</span> <span class="kn">import</span> <span class="n">Circle</span>
-<span class="kn">from</span> <span class="nn">.line</span> <span class="kn">import</span> <span class="n">Line</span><span class="p">,</span> <span class="n">Segment</span>
-<span class="kn">from</span> <span class="nn">.util</span> <span class="kn">import</span> <span class="n">_symbol</span>
-
-<span class="kn">import</span> <span class="nn">warnings</span>
-
-
-<div class="viewcode-block" id="Polygon"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Polygon">[docs]</a><span class="k">class</span> <span class="nc">Polygon</span><span class="p">(</span><span class="n">GeometryEntity</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A two-dimensional polygon.</span>
-
-<span class="sd">    A simple polygon in space. Can be constructed from a sequence of points</span>
-<span class="sd">    or from a center, radius, number of sides and rotation angle.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    vertices : sequence of Points</span>
-
-<span class="sd">    Attributes</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    area</span>
-<span class="sd">    angles</span>
-<span class="sd">    perimeter</span>
-<span class="sd">    vertices</span>
-<span class="sd">    centroid</span>
-<span class="sd">    sides</span>
-
-<span class="sd">    Raises</span>
-<span class="sd">    ======</span>
-
-<span class="sd">    GeometryError</span>
-<span class="sd">        If all parameters are not Points.</span>
-
-<span class="sd">        If the Polygon has intersecting sides.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.geometry.point.Point, sympy.geometry.line.Segment, Triangle</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    Polygons are treated as closed paths rather than 2D areas so</span>
-<span class="sd">    some calculations can be be negative or positive (e.g., area)</span>
-<span class="sd">    based on the orientation of the points.</span>
-
-<span class="sd">    Any consecutive identical points are reduced to a single point</span>
-<span class="sd">    and any points collinear and between two points will be removed</span>
-<span class="sd">    unless they are needed to define an explicit intersection (see examples).</span>
-
-<span class="sd">    A Triangle, Segment or Point will be returned when there are 3 or</span>
-<span class="sd">    fewer points provided.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Point, Polygon, pi</span>
-<span class="sd">    &gt;&gt;&gt; p1, p2, p3, p4, p5 = [(0, 0), (1, 0), (5, 1), (0, 1), (3, 0)]</span>
-<span class="sd">    &gt;&gt;&gt; Polygon(p1, p2, p3, p4)</span>
-<span class="sd">    Polygon(Point(0, 0), Point(1, 0), Point(5, 1), Point(0, 1))</span>
-<span class="sd">    &gt;&gt;&gt; Polygon(p1, p2)</span>
-<span class="sd">    Segment(Point(0, 0), Point(1, 0))</span>
-<span class="sd">    &gt;&gt;&gt; Polygon(p1, p2, p5)</span>
-<span class="sd">    Segment(Point(0, 0), Point(3, 0))</span>
-
-<span class="sd">    While the sides of a polygon are not allowed to cross implicitly, they</span>
-<span class="sd">    can do so explicitly. For example, a polygon shaped like a Z with the top</span>
-<span class="sd">    left connecting to the bottom right of the Z must have the point in the</span>
-<span class="sd">    middle of the Z explicitly given:</span>
-
-<span class="sd">    &gt;&gt;&gt; mid = Point(1, 1)</span>
-<span class="sd">    &gt;&gt;&gt; Polygon((0, 2), (2, 2), mid, (0, 0), (2, 0), mid).area</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; Polygon((0, 2), (2, 2), mid, (2, 0), (0, 0), mid).area</span>
-<span class="sd">    -2</span>
-
-<span class="sd">    When the the keyword `n` is used to define the number of sides of the</span>
-<span class="sd">    Polygon then a RegularPolygon is created and the other arguments are</span>
-<span class="sd">    interpreted as center, radius and rotation. The unrotated RegularPolygon</span>
-<span class="sd">    will always have a vertex at Point(r, 0) where `r` is the radius of the</span>
-<span class="sd">    circle that circumscribes the RegularPolygon. Its method `spin` can be</span>
-<span class="sd">    used to increment that angle.</span>
-
-<span class="sd">    &gt;&gt;&gt; p = Polygon((0,0), 1, n=3)</span>
-<span class="sd">    &gt;&gt;&gt; p</span>
-<span class="sd">    RegularPolygon(Point(0, 0), 1, 3, 0)</span>
-<span class="sd">    &gt;&gt;&gt; p.vertices[0]</span>
-<span class="sd">    Point(1, 0)</span>
-<span class="sd">    &gt;&gt;&gt; p.args[0]</span>
-<span class="sd">    Point(0, 0)</span>
-<span class="sd">    &gt;&gt;&gt; p.spin(pi/2)</span>
-<span class="sd">    &gt;&gt;&gt; p.vertices[0]</span>
-<span class="sd">    Point(0, 1)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">):</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">)</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-            <span class="c"># return a virtual polygon with n sides</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>  <span class="c"># center, radius</span>
-                <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>  <span class="c"># center, radius, rotation</span>
-                <span class="n">args</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">RegularPolygon</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-        <span class="n">vertices</span> <span class="o">=</span> <span class="p">[</span><span class="n">Point</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span>
-
-        <span class="c"># remove consecutive duplicates</span>
-        <span class="n">nodup</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">vertices</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">nodup</span> <span class="ow">and</span> <span class="n">p</span> <span class="o">==</span> <span class="n">nodup</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-                <span class="k">continue</span>
-            <span class="n">nodup</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">nodup</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">nodup</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">nodup</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-            <span class="n">nodup</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>  <span class="c"># last point was same as first</span>
-
-        <span class="c"># remove collinear points unless they are shared points</span>
-        <span class="n">got</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="n">shared</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">nodup</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">got</span><span class="p">:</span>
-                <span class="n">shared</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">got</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="o">-</span><span class="mi">3</span>
-        <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">nodup</span><span class="p">)</span> <span class="o">-</span> <span class="mi">3</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">nodup</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">([</span><span class="n">nodup</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">nodup</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">nodup</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">2</span><span class="p">]])</span>
-            <span class="k">if</span> <span class="n">b</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">shared</span> <span class="ow">and</span> <span class="n">Point</span><span class="o">.</span><span class="n">is_collinear</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
-                <span class="n">nodup</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">a</span>
-                <span class="n">nodup</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="bp">None</span>
-                <span class="n">nodup</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-
-        <span class="n">vertices</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">nodup</span> <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">vertices</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="n">GeometryEntity</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">vertices</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">vertices</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Triangle</span><span class="p">(</span><span class="o">*</span><span class="n">vertices</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">vertices</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Segment</span><span class="p">(</span><span class="o">*</span><span class="n">vertices</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="o">*</span><span class="n">vertices</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-        <span class="c"># reject polygons that have intersecting sides unless the</span>
-        <span class="c"># intersection is a shared point or a generalized intersection.</span>
-        <span class="c"># A self-intersecting polygon is easier to detect than a</span>
-        <span class="c"># random set of segments since only those sides that are not</span>
-        <span class="c"># part of the convex hull can possibly intersect with other</span>
-        <span class="c"># sides of the polygon...but for now we use the n**2 algorithm</span>
-        <span class="c"># and check all sides with intersection with any preceding sides</span>
-        <span class="n">hit</span> <span class="o">=</span> <span class="n">_symbol</span><span class="p">(</span><span class="s">&#39;hit&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">rv</span><span class="o">.</span><span class="n">is_convex</span><span class="p">:</span>
-            <span class="n">sides</span> <span class="o">=</span> <span class="n">rv</span><span class="o">.</span><span class="n">sides</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">si</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">sides</span><span class="p">):</span>
-                <span class="n">pts</span> <span class="o">=</span> <span class="n">si</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">si</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-                <span class="n">ai</span> <span class="o">=</span> <span class="n">si</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">(</span><span class="n">hit</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
-                    <span class="n">sj</span> <span class="o">=</span> <span class="n">sides</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-                    <span class="k">if</span> <span class="n">sj</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">pts</span> <span class="ow">and</span> <span class="n">sj</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">pts</span><span class="p">:</span>
-                        <span class="n">aj</span> <span class="o">=</span> <span class="n">si</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">(</span><span class="n">hit</span><span class="p">)</span>
-                        <span class="n">tx</span> <span class="o">=</span> <span class="p">(</span><span class="n">solve</span><span class="p">(</span><span class="n">ai</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">aj</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="ow">or</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">])[</span><span class="mi">0</span><span class="p">]</span>
-                        <span class="k">if</span> <span class="n">tx</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="mi">0</span> <span class="o">&lt;=</span> <span class="n">tx</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">:</span>
-                            <span class="n">ty</span> <span class="o">=</span> <span class="p">(</span><span class="n">solve</span><span class="p">(</span><span class="n">ai</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">aj</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="ow">or</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">])[</span><span class="mi">0</span><span class="p">]</span>
-                            <span class="k">if</span> <span class="p">(</span><span class="n">tx</span> <span class="ow">or</span> <span class="n">ty</span><span class="p">)</span> <span class="ow">and</span> <span class="n">ty</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="mi">0</span> <span class="o">&lt;=</span> <span class="n">ty</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">:</span>
-                                <span class="k">print</span><span class="p">(</span><span class="n">ai</span><span class="p">,</span> <span class="n">aj</span><span class="p">)</span>
-                                <span class="k">raise</span> <span class="n">GeometryError</span><span class="p">(</span>
-                                    <span class="s">&quot;Polygon has intersecting sides.&quot;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">rv</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Polygon.area"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Polygon.area">[docs]</a>    <span class="k">def</span> <span class="nf">area</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The area of the polygon.</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        The area calculation can be positive or negative based on the</span>
-<span class="sd">        orientation of the points.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.ellipse.Ellipse.area</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Polygon</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3, p4 = list(map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]))</span>
-<span class="sd">        &gt;&gt;&gt; poly = Polygon(p1, p2, p3, p4)</span>
-<span class="sd">        &gt;&gt;&gt; poly.area</span>
-<span class="sd">        3</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">area</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)):</span>
-            <span class="n">x1</span><span class="p">,</span> <span class="n">y1</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">args</span>
-            <span class="n">x2</span><span class="p">,</span> <span class="n">y2</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">args</span>
-            <span class="n">area</span> <span class="o">+=</span> <span class="n">x1</span><span class="o">*</span><span class="n">y2</span> <span class="o">-</span> <span class="n">x2</span><span class="o">*</span><span class="n">y1</span>
-        <span class="k">return</span> <span class="n">simplify</span><span class="p">(</span><span class="n">area</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Polygon.angles"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Polygon.angles">[docs]</a>    <span class="k">def</span> <span class="nf">angles</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The internal angle at each vertex.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        angles : dict</span>
-<span class="sd">            A dictionary where each key is a vertex and each value is the</span>
-<span class="sd">            internal angle at that vertex. The vertices are represented as</span>
-<span class="sd">            Points.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point, sympy.geometry.line.LinearEntity.angle_between</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Polygon</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3, p4 = list(map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]))</span>
-<span class="sd">        &gt;&gt;&gt; poly = Polygon(p1, p2, p3, p4)</span>
-<span class="sd">        &gt;&gt;&gt; poly.angles[p1]</span>
-<span class="sd">        pi/2</span>
-<span class="sd">        &gt;&gt;&gt; poly.angles[p2]</span>
-<span class="sd">        acos(-4*sqrt(17)/17)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">def</span> <span class="nf">_isright</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
-            <span class="n">ba</span> <span class="o">=</span> <span class="n">b</span> <span class="o">-</span> <span class="n">a</span>
-            <span class="n">ca</span> <span class="o">=</span> <span class="n">c</span> <span class="o">-</span> <span class="n">a</span>
-            <span class="n">t_area</span> <span class="o">=</span> <span class="n">ba</span><span class="o">.</span><span class="n">x</span><span class="o">*</span><span class="n">ca</span><span class="o">.</span><span class="n">y</span> <span class="o">-</span> <span class="n">ca</span><span class="o">.</span><span class="n">x</span><span class="o">*</span><span class="n">ba</span><span class="o">.</span><span class="n">y</span>
-            <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="n">t_area</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">)</span>
-
-        <span class="c"># Determine orientation of points</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">vertices</span>
-        <span class="n">cw</span> <span class="o">=</span> <span class="n">_isright</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-
-        <span class="n">ret</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)):</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">2</span><span class="p">],</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="n">ang</span> <span class="o">=</span> <span class="n">Line</span><span class="o">.</span><span class="n">angle_between</span><span class="p">(</span><span class="n">Line</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">),</span> <span class="n">Line</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">cw</span> <span class="o">^</span> <span class="n">_isright</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
-                <span class="n">ret</span><span class="p">[</span><span class="n">b</span><span class="p">]</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span> <span class="o">-</span> <span class="n">ang</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">ret</span><span class="p">[</span><span class="n">b</span><span class="p">]</span> <span class="o">=</span> <span class="n">ang</span>
-        <span class="k">return</span> <span class="n">ret</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Polygon.perimeter"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Polygon.perimeter">[docs]</a>    <span class="k">def</span> <span class="nf">perimeter</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The perimeter of the polygon.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        perimeter : number or Basic instance</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.line.Segment.length</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Polygon</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3, p4 = list(map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]))</span>
-<span class="sd">        &gt;&gt;&gt; poly = Polygon(p1, p2, p3, p4)</span>
-<span class="sd">        &gt;&gt;&gt; poly.perimeter</span>
-<span class="sd">        sqrt(17) + 7</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">vertices</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)):</span>
-            <span class="n">p</span> <span class="o">+=</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">simplify</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Polygon.vertices"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Polygon.vertices">[docs]</a>    <span class="k">def</span> <span class="nf">vertices</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The vertices of the polygon.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        vertices : tuple of Points</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        When iterating over the vertices, it is more efficient to index self</span>
-<span class="sd">        rather than to request the vertices and index them. Only use the</span>
-<span class="sd">        vertices when you want to process all of them at once. This is even</span>
-<span class="sd">        more important with RegularPolygons that calculate each vertex.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Polygon</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3, p4 = list(map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]))</span>
-<span class="sd">        &gt;&gt;&gt; poly = Polygon(p1, p2, p3, p4)</span>
-<span class="sd">        &gt;&gt;&gt; poly.vertices</span>
-<span class="sd">        (Point(0, 0), Point(1, 0), Point(5, 1), Point(0, 1))</span>
-<span class="sd">        &gt;&gt;&gt; poly.args[0]</span>
-<span class="sd">        Point(0, 0)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Polygon.centroid"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Polygon.centroid">[docs]</a>    <span class="k">def</span> <span class="nf">centroid</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The centroid of the polygon.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        centroid : Point</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point, sympy.geometry.util.centroid</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Polygon</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3, p4 = list(map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]))</span>
-<span class="sd">        &gt;&gt;&gt; poly = Polygon(p1, p2, p3, p4)</span>
-<span class="sd">        &gt;&gt;&gt; poly.centroid</span>
-<span class="sd">        Point(31/18, 11/18)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">A</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">6</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">area</span><span class="p">)</span>
-        <span class="n">cx</span><span class="p">,</span> <span class="n">cy</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)):</span>
-            <span class="n">x1</span><span class="p">,</span> <span class="n">y1</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">args</span>
-            <span class="n">x2</span><span class="p">,</span> <span class="n">y2</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">args</span>
-            <span class="n">v</span> <span class="o">=</span> <span class="n">x1</span><span class="o">*</span><span class="n">y2</span> <span class="o">-</span> <span class="n">x2</span><span class="o">*</span><span class="n">y1</span>
-            <span class="n">cx</span> <span class="o">+=</span> <span class="n">v</span><span class="o">*</span><span class="p">(</span><span class="n">x1</span> <span class="o">+</span> <span class="n">x2</span><span class="p">)</span>
-            <span class="n">cy</span> <span class="o">+=</span> <span class="n">v</span><span class="o">*</span><span class="p">(</span><span class="n">y1</span> <span class="o">+</span> <span class="n">y2</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="n">simplify</span><span class="p">(</span><span class="n">A</span><span class="o">*</span><span class="n">cx</span><span class="p">),</span> <span class="n">simplify</span><span class="p">(</span><span class="n">A</span><span class="o">*</span><span class="n">cy</span><span class="p">))</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Polygon.sides"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Polygon.sides">[docs]</a>    <span class="k">def</span> <span class="nf">sides</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The line segments that form the sides of the polygon.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        sides : list of sides</span>
-<span class="sd">            Each side is a Segment.</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        The Segments that represent the sides are an undirected</span>
-<span class="sd">        line segment so cannot be used to tell the orientation of</span>
-<span class="sd">        the polygon.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point, sympy.geometry.line.Segment</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Polygon</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3, p4 = list(map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]))</span>
-<span class="sd">        &gt;&gt;&gt; poly = Polygon(p1, p2, p3, p4)</span>
-<span class="sd">        &gt;&gt;&gt; poly.sides</span>
-<span class="sd">        [Segment(Point(0, 0), Point(1, 0)),</span>
-<span class="sd">        Segment(Point(1, 0), Point(5, 1)),</span>
-<span class="sd">        Segment(Point(0, 1), Point(5, 1)), Segment(Point(0, 0), Point(0, 1))]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">vertices</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">),</span> <span class="mi">0</span><span class="p">):</span>
-            <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Segment</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]))</span>
-        <span class="k">return</span> <span class="n">res</span>
-</div>
-<div class="viewcode-block" id="Polygon.is_convex"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Polygon.is_convex">[docs]</a>    <span class="k">def</span> <span class="nf">is_convex</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Is the polygon convex?</span>
-
-<span class="sd">        A polygon is convex if all its interior angles are less than 180</span>
-<span class="sd">        degrees.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        is_convex : boolean</span>
-<span class="sd">            True if this polygon is convex, False otherwise.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.util.convex_hull</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Polygon</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3, p4 = list(map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]))</span>
-<span class="sd">        &gt;&gt;&gt; poly = Polygon(p1, p2, p3, p4)</span>
-<span class="sd">        &gt;&gt;&gt; poly.is_convex()</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">def</span> <span class="nf">_isright</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
-            <span class="n">ba</span> <span class="o">=</span> <span class="n">b</span> <span class="o">-</span> <span class="n">a</span>
-            <span class="n">ca</span> <span class="o">=</span> <span class="n">c</span> <span class="o">-</span> <span class="n">a</span>
-            <span class="n">t_area</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">ba</span><span class="o">.</span><span class="n">x</span><span class="o">*</span><span class="n">ca</span><span class="o">.</span><span class="n">y</span> <span class="o">-</span> <span class="n">ca</span><span class="o">.</span><span class="n">x</span><span class="o">*</span><span class="n">ba</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-            <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="n">t_area</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">)</span>
-
-        <span class="c"># Determine orientation of points</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">vertices</span>
-        <span class="n">cw</span> <span class="o">=</span> <span class="n">_isright</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="n">cw</span> <span class="o">^</span> <span class="n">_isright</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">2</span><span class="p">],</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]):</span>
-                <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="k">return</span> <span class="bp">True</span>
-</div>
-<div class="viewcode-block" id="Polygon.encloses_point"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Polygon.encloses_point">[docs]</a>    <span class="k">def</span> <span class="nf">encloses_point</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if p is enclosed by (is inside of) self.</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        Being on the border of self is considered False.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        p : Point</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        encloses_point : True, False or None</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point, sympy.geometry.ellipse.Ellipse.encloses_point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Polygon, Point</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import t</span>
-<span class="sd">        &gt;&gt;&gt; p = Polygon((0, 0), (4, 0), (4, 4))</span>
-<span class="sd">        &gt;&gt;&gt; p.encloses_point(Point(2, 1))</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; p.encloses_point(Point(2, 2))</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; p.encloses_point(Point(5, 5))</span>
-<span class="sd">        False</span>
-
-<span class="sd">        References</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        [1] http://www.ariel.com.au/a/python-point-int-poly.html</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">p</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">vertices</span> <span class="ow">or</span> <span class="nb">any</span><span class="p">(</span><span class="n">p</span> <span class="ow">in</span> <span class="n">s</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">sides</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="c"># move to p, checking that the result is numeric</span>
-        <span class="n">lit</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">vertices</span><span class="p">:</span>
-            <span class="n">lit</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">v</span> <span class="o">-</span> <span class="n">p</span><span class="p">)</span>  <span class="c"># the difference is simplified</span>
-            <span class="k">if</span> <span class="n">lit</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-        <span class="bp">self</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">(</span><span class="o">*</span><span class="n">lit</span><span class="p">)</span>
-
-        <span class="c"># polygon closure is assumed in the following test but Polygon removes duplicate pts so</span>
-        <span class="c"># the last point has to be added so all sides are computed. Using Polygon.sides is</span>
-        <span class="c"># not good since Segments are unordered.</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="n">indices</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">),</span> <span class="mi">1</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_convex</span><span class="p">():</span>
-            <span class="n">orientation</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">indices</span><span class="p">:</span>
-                <span class="n">a</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
-                <span class="n">test</span> <span class="o">=</span> <span class="p">((</span><span class="o">-</span><span class="n">a</span><span class="o">.</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">x</span> <span class="o">-</span> <span class="n">a</span><span class="o">.</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="o">.</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">y</span> <span class="o">-</span> <span class="n">a</span><span class="o">.</span><span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">is_negative</span>
-                <span class="k">if</span> <span class="n">orientation</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">orientation</span> <span class="o">=</span> <span class="n">test</span>
-                <span class="k">elif</span> <span class="n">test</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">orientation</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-        <span class="n">hit_odd</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="n">p1x</span><span class="p">,</span> <span class="n">p1y</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">indices</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="n">p2x</span><span class="p">,</span> <span class="n">p2y</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">if</span> <span class="mi">0</span> <span class="o">&gt;</span> <span class="nb">min</span><span class="p">(</span><span class="n">p1y</span><span class="p">,</span> <span class="n">p2y</span><span class="p">):</span>
-                <span class="k">if</span> <span class="mi">0</span> <span class="o">&lt;=</span> <span class="nb">max</span><span class="p">(</span><span class="n">p1y</span><span class="p">,</span> <span class="n">p2y</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="mi">0</span> <span class="o">&lt;=</span> <span class="nb">max</span><span class="p">(</span><span class="n">p1x</span><span class="p">,</span> <span class="n">p2x</span><span class="p">):</span>
-                        <span class="k">if</span> <span class="n">p1y</span> <span class="o">!=</span> <span class="n">p2y</span><span class="p">:</span>
-                            <span class="n">xinters</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="n">p1y</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">p2x</span> <span class="o">-</span> <span class="n">p1x</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">p2y</span> <span class="o">-</span> <span class="n">p1y</span><span class="p">)</span> <span class="o">+</span> <span class="n">p1x</span>
-                        <span class="k">if</span> <span class="n">p1x</span> <span class="o">==</span> <span class="n">p2x</span> <span class="ow">or</span> <span class="mi">0</span> <span class="o">&lt;=</span> <span class="n">xinters</span><span class="p">:</span>
-                            <span class="n">hit_odd</span> <span class="o">=</span> <span class="ow">not</span> <span class="n">hit_odd</span>
-            <span class="n">p1x</span><span class="p">,</span> <span class="n">p1y</span> <span class="o">=</span> <span class="n">p2x</span><span class="p">,</span> <span class="n">p2y</span>
-        <span class="k">return</span> <span class="n">hit_odd</span>
-</div>
-<div class="viewcode-block" id="Polygon.arbitrary_point"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Polygon.arbitrary_point">[docs]</a>    <span class="k">def</span> <span class="nf">arbitrary_point</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">parameter</span><span class="o">=</span><span class="s">&#39;t&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A parameterized point on the polygon.</span>
-
-<span class="sd">        The parameter, varying from 0 to 1, assigns points to the position on</span>
-<span class="sd">        the perimeter that is that fraction of the total perimeter. So the</span>
-<span class="sd">        point evaluated at t=1/2 would return the point from the first vertex</span>
-<span class="sd">        that is 1/2 way around the polygon.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        parameter : str, optional</span>
-<span class="sd">            Default value is &#39;t&#39;.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        arbitrary_point : Point</span>
-
-<span class="sd">        Raises</span>
-<span class="sd">        ======</span>
-
-<span class="sd">        ValueError</span>
-<span class="sd">            When `parameter` already appears in the Polygon&#39;s definition.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Polygon, S, Symbol</span>
-<span class="sd">        &gt;&gt;&gt; t = Symbol(&#39;t&#39;, real=True)</span>
-<span class="sd">        &gt;&gt;&gt; tri = Polygon((0, 0), (1, 0), (1, 1))</span>
-<span class="sd">        &gt;&gt;&gt; p = tri.arbitrary_point(&#39;t&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; perimeter = tri.perimeter</span>
-<span class="sd">        &gt;&gt;&gt; s1, s2 = [s.length for s in tri.sides[:2]]</span>
-<span class="sd">        &gt;&gt;&gt; p.subs(t, (s1 + s2/2)/perimeter)</span>
-<span class="sd">        Point(1, 1/2)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">_symbol</span><span class="p">(</span><span class="n">parameter</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">name</span> <span class="ow">in</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">name</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Symbol </span><span class="si">%s</span><span class="s"> already appears in object and cannot be used as a parameter.&#39;</span> <span class="o">%</span> <span class="n">t</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">sides</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">perimeter</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">perimeter</span>
-        <span class="n">perim_fraction_start</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">sides</span><span class="p">:</span>
-            <span class="n">side_perim_fraction</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">length</span><span class="o">/</span><span class="n">perimeter</span>
-            <span class="n">perim_fraction_end</span> <span class="o">=</span> <span class="n">perim_fraction_start</span> <span class="o">+</span> <span class="n">side_perim_fraction</span>
-            <span class="n">pt</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">(</span><span class="n">parameter</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span>
-                <span class="n">t</span><span class="p">,</span> <span class="p">(</span><span class="n">t</span> <span class="o">-</span> <span class="n">perim_fraction_start</span><span class="p">)</span><span class="o">/</span><span class="n">side_perim_fraction</span><span class="p">)</span>
-            <span class="n">sides</span><span class="o">.</span><span class="n">append</span><span class="p">(</span>
-                <span class="p">(</span><span class="n">pt</span><span class="p">,</span> <span class="p">(</span><span class="n">perim_fraction_start</span> <span class="o">&lt;=</span> <span class="n">t</span> <span class="o">&lt;</span> <span class="n">perim_fraction_end</span><span class="p">)))</span>
-            <span class="n">perim_fraction_start</span> <span class="o">=</span> <span class="n">perim_fraction_end</span>
-        <span class="k">return</span> <span class="n">Piecewise</span><span class="p">(</span><span class="o">*</span><span class="n">sides</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Polygon.plot_interval"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Polygon.plot_interval">[docs]</a>    <span class="k">def</span> <span class="nf">plot_interval</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">parameter</span><span class="o">=</span><span class="s">&#39;t&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The plot interval for the default geometric plot of the polygon.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        parameter : str, optional</span>
-<span class="sd">            Default value is &#39;t&#39;.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        plot_interval : list (plot interval)</span>
-<span class="sd">            [parameter, lower_bound, upper_bound]</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Polygon</span>
-<span class="sd">        &gt;&gt;&gt; p = Polygon((0, 0), (1, 0), (1, 1))</span>
-<span class="sd">        &gt;&gt;&gt; p.plot_interval()</span>
-<span class="sd">        [t, 0, 1]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">parameter</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">t</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
-</div>
-<div class="viewcode-block" id="Polygon.intersection"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Polygon.intersection">[docs]</a>    <span class="k">def</span> <span class="nf">intersection</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The intersection of two polygons.</span>
-
-<span class="sd">        The intersection may be empty and can contain individual Points and</span>
-<span class="sd">        complete Line Segments.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        other: Polygon</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        intersection : list</span>
-<span class="sd">            The list of Segments and Points</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point, sympy.geometry.line.Segment</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Polygon</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3, p4 = list(map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)]))</span>
-<span class="sd">        &gt;&gt;&gt; poly1 = Polygon(p1, p2, p3, p4)</span>
-<span class="sd">        &gt;&gt;&gt; p5, p6, p7 = list(map(Point, [(3, 2), (1, -1), (0, 2)]))</span>
-<span class="sd">        &gt;&gt;&gt; poly2 = Polygon(p5, p6, p7)</span>
-<span class="sd">        &gt;&gt;&gt; poly1.intersection(poly2)</span>
-<span class="sd">        [Point(2/3, 0), Point(9/5, 1/5), Point(7/3, 1), Point(1/3, 1)]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">side</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">sides</span><span class="p">:</span>
-            <span class="n">inter</span> <span class="o">=</span> <span class="n">side</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">inter</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">res</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">inter</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">res</span>
-</div>
-<div class="viewcode-block" id="Polygon.distance"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Polygon.distance">[docs]</a>    <span class="k">def</span> <span class="nf">distance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the shortest distance between self and o.</span>
-
-<span class="sd">        If o is a point, then self does not need to be convex.</span>
-<span class="sd">        If o is another polygon self and o must be complex.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, Polygon, RegularPolygon</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2 = list(map(Point, [(0, 0), (7, 5)]))</span>
-<span class="sd">        &gt;&gt;&gt; poly = Polygon(*RegularPolygon(p1, 1, 3).vertices)</span>
-<span class="sd">        &gt;&gt;&gt; poly.distance(p2)</span>
-<span class="sd">        sqrt(61)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Point</span><span class="p">):</span>
-            <span class="n">dist</span> <span class="o">=</span> <span class="n">oo</span>
-            <span class="k">for</span> <span class="n">side</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">sides</span><span class="p">:</span>
-                <span class="n">current</span> <span class="o">=</span> <span class="n">side</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">current</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-                <span class="k">elif</span> <span class="n">current</span> <span class="o">&lt;</span> <span class="n">dist</span><span class="p">:</span>
-                    <span class="n">dist</span> <span class="o">=</span> <span class="n">current</span>
-            <span class="k">return</span> <span class="n">dist</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Polygon</span><span class="p">)</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_convex</span><span class="p">()</span> <span class="ow">and</span> <span class="n">o</span><span class="o">.</span><span class="n">is_convex</span><span class="p">():</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_do_poly_distance</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-</div>
-    <span class="k">def</span> <span class="nf">_do_poly_distance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e2</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Calculates the least distance between the exteriors of two</span>
-<span class="sd">        convex polygons e1 and e2. Does not check for the convexity</span>
-<span class="sd">        of the polygons as this is checked by Polygon.distance.</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">            - Prints a warning if the two polygons possibly intersect as the return</span>
-<span class="sd">              value will not be valid in such a case. For a more through test of</span>
-<span class="sd">              intersection use intersection().</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point.distance</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Point, Polygon</span>
-<span class="sd">        &gt;&gt;&gt; square = Polygon(Point(0, 0), Point(0, 1), Point(1, 1), Point(1, 0))</span>
-<span class="sd">        &gt;&gt;&gt; triangle = Polygon(Point(1, 2), Point(2, 2), Point(2, 1))</span>
-<span class="sd">        &gt;&gt;&gt; square._do_poly_distance(triangle)</span>
-<span class="sd">        sqrt(2)/2</span>
-
-<span class="sd">        Description of method used</span>
-<span class="sd">        ==========================</span>
-
-<span class="sd">        Method:</span>
-<span class="sd">        [1] http://cgm.cs.mcgill.ca/~orm/mind2p.html</span>
-<span class="sd">        Uses rotating calipers:</span>
-<span class="sd">        [2] http://en.wikipedia.org/wiki/Rotating_calipers</span>
-<span class="sd">        and antipodal points:</span>
-<span class="sd">        [3] http://en.wikipedia.org/wiki/Antipodal_point</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">e1</span> <span class="o">=</span> <span class="bp">self</span>
-
-        <span class="sd">&#39;&#39;&#39;Tests for a possible intersection between the polygons and outputs a warning&#39;&#39;&#39;</span>
-        <span class="n">e1_center</span> <span class="o">=</span> <span class="n">e1</span><span class="o">.</span><span class="n">centroid</span>
-        <span class="n">e2_center</span> <span class="o">=</span> <span class="n">e2</span><span class="o">.</span><span class="n">centroid</span>
-        <span class="n">e1_max_radius</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="n">e2_max_radius</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">for</span> <span class="n">vertex</span> <span class="ow">in</span> <span class="n">e1</span><span class="o">.</span><span class="n">vertices</span><span class="p">:</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">Point</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">e1_center</span><span class="p">,</span> <span class="n">vertex</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">e1_max_radius</span> <span class="o">&lt;</span> <span class="n">r</span><span class="p">:</span>
-                <span class="n">e1_max_radius</span> <span class="o">=</span> <span class="n">r</span>
-        <span class="k">for</span> <span class="n">vertex</span> <span class="ow">in</span> <span class="n">e2</span><span class="o">.</span><span class="n">vertices</span><span class="p">:</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">Point</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">e2_center</span><span class="p">,</span> <span class="n">vertex</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">e2_max_radius</span> <span class="o">&lt;</span> <span class="n">r</span><span class="p">:</span>
-                <span class="n">e2_max_radius</span> <span class="o">=</span> <span class="n">r</span>
-        <span class="n">center_dist</span> <span class="o">=</span> <span class="n">Point</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">e1_center</span><span class="p">,</span> <span class="n">e2_center</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">center_dist</span> <span class="o">&lt;=</span> <span class="n">e1_max_radius</span> <span class="o">+</span> <span class="n">e2_max_radius</span><span class="p">:</span>
-            <span class="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="s">&quot;Polygons may intersect producing erroneous output&quot;</span><span class="p">)</span>
-
-        <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">        Find the upper rightmost vertex of e1 and the lowest leftmost vertex of e2</span>
-<span class="sd">        &#39;&#39;&#39;</span>
-        <span class="n">e1_ymax</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="n">oo</span><span class="p">)</span>
-        <span class="n">e2_ymin</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">vertex</span> <span class="ow">in</span> <span class="n">e1</span><span class="o">.</span><span class="n">vertices</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">vertex</span><span class="o">.</span><span class="n">y</span> <span class="o">&gt;</span> <span class="n">e1_ymax</span><span class="o">.</span><span class="n">y</span> <span class="ow">or</span> <span class="p">(</span><span class="n">vertex</span><span class="o">.</span><span class="n">y</span> <span class="o">==</span> <span class="n">e1_ymax</span><span class="o">.</span><span class="n">y</span> <span class="ow">and</span> <span class="n">vertex</span><span class="o">.</span><span class="n">x</span> <span class="o">&gt;</span> <span class="n">e1_ymax</span><span class="o">.</span><span class="n">x</span><span class="p">):</span>
-                <span class="n">e1_ymax</span> <span class="o">=</span> <span class="n">vertex</span>
-        <span class="k">for</span> <span class="n">vertex</span> <span class="ow">in</span> <span class="n">e2</span><span class="o">.</span><span class="n">vertices</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">vertex</span><span class="o">.</span><span class="n">y</span> <span class="o">&lt;</span> <span class="n">e2_ymin</span><span class="o">.</span><span class="n">y</span> <span class="ow">or</span> <span class="p">(</span><span class="n">vertex</span><span class="o">.</span><span class="n">y</span> <span class="o">==</span> <span class="n">e2_ymin</span><span class="o">.</span><span class="n">y</span> <span class="ow">and</span> <span class="n">vertex</span><span class="o">.</span><span class="n">x</span> <span class="o">&lt;</span> <span class="n">e2_ymin</span><span class="o">.</span><span class="n">x</span><span class="p">):</span>
-                <span class="n">e2_ymin</span> <span class="o">=</span> <span class="n">vertex</span>
-        <span class="n">min_dist</span> <span class="o">=</span> <span class="n">Point</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">e1_ymax</span><span class="p">,</span> <span class="n">e2_ymin</span><span class="p">)</span>
-
-        <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">        Produce a dictionary with vertices of e1 as the keys and, for each vertex, the points</span>
-<span class="sd">        to which the vertex is connected as its value. The same is then done for e2.</span>
-<span class="sd">        &#39;&#39;&#39;</span>
-        <span class="n">e1_connections</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="n">e2_connections</span> <span class="o">=</span> <span class="p">{}</span>
-
-        <span class="k">for</span> <span class="n">side</span> <span class="ow">in</span> <span class="n">e1</span><span class="o">.</span><span class="n">sides</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">side</span><span class="o">.</span><span class="n">p1</span> <span class="ow">in</span> <span class="n">e1_connections</span><span class="p">:</span>
-                <span class="n">e1_connections</span><span class="p">[</span><span class="n">side</span><span class="o">.</span><span class="n">p1</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">side</span><span class="o">.</span><span class="n">p2</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">e1_connections</span><span class="p">[</span><span class="n">side</span><span class="o">.</span><span class="n">p1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">side</span><span class="o">.</span><span class="n">p2</span><span class="p">]</span>
-
-            <span class="k">if</span> <span class="n">side</span><span class="o">.</span><span class="n">p2</span> <span class="ow">in</span> <span class="n">e1_connections</span><span class="p">:</span>
-                <span class="n">e1_connections</span><span class="p">[</span><span class="n">side</span><span class="o">.</span><span class="n">p2</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">side</span><span class="o">.</span><span class="n">p1</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">e1_connections</span><span class="p">[</span><span class="n">side</span><span class="o">.</span><span class="n">p2</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">side</span><span class="o">.</span><span class="n">p1</span><span class="p">]</span>
-
-        <span class="k">for</span> <span class="n">side</span> <span class="ow">in</span> <span class="n">e2</span><span class="o">.</span><span class="n">sides</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">side</span><span class="o">.</span><span class="n">p1</span> <span class="ow">in</span> <span class="n">e2_connections</span><span class="p">:</span>
-                <span class="n">e2_connections</span><span class="p">[</span><span class="n">side</span><span class="o">.</span><span class="n">p1</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">side</span><span class="o">.</span><span class="n">p2</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">e2_connections</span><span class="p">[</span><span class="n">side</span><span class="o">.</span><span class="n">p1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">side</span><span class="o">.</span><span class="n">p2</span><span class="p">]</span>
-
-            <span class="k">if</span> <span class="n">side</span><span class="o">.</span><span class="n">p2</span> <span class="ow">in</span> <span class="n">e2_connections</span><span class="p">:</span>
-                <span class="n">e2_connections</span><span class="p">[</span><span class="n">side</span><span class="o">.</span><span class="n">p2</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">side</span><span class="o">.</span><span class="n">p1</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">e2_connections</span><span class="p">[</span><span class="n">side</span><span class="o">.</span><span class="n">p2</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">side</span><span class="o">.</span><span class="n">p1</span><span class="p">]</span>
-
-        <span class="n">e1_current</span> <span class="o">=</span> <span class="n">e1_ymax</span>
-        <span class="n">e2_current</span> <span class="o">=</span> <span class="n">e2_ymin</span>
-        <span class="n">support_line</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">))</span>
-
-        <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">        Determine which point in e1 and e2 will be selected after e2_ymin and e1_ymax,</span>
-<span class="sd">        this information combined with the above produced dictionaries determines the</span>
-<span class="sd">        path that will be taken around the polygons</span>
-<span class="sd">        &#39;&#39;&#39;</span>
-        <span class="n">point1</span> <span class="o">=</span> <span class="n">e1_connections</span><span class="p">[</span><span class="n">e1_ymax</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">point2</span> <span class="o">=</span> <span class="n">e1_connections</span><span class="p">[</span><span class="n">e1_ymax</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">angle1</span> <span class="o">=</span> <span class="n">support_line</span><span class="o">.</span><span class="n">angle_between</span><span class="p">(</span><span class="n">Line</span><span class="p">(</span><span class="n">e1_ymax</span><span class="p">,</span> <span class="n">point1</span><span class="p">))</span>
-        <span class="n">angle2</span> <span class="o">=</span> <span class="n">support_line</span><span class="o">.</span><span class="n">angle_between</span><span class="p">(</span><span class="n">Line</span><span class="p">(</span><span class="n">e1_ymax</span><span class="p">,</span> <span class="n">point2</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">angle1</span> <span class="o">&lt;</span> <span class="n">angle2</span><span class="p">:</span>
-            <span class="n">e1_next</span> <span class="o">=</span> <span class="n">point1</span>
-        <span class="k">elif</span> <span class="n">angle2</span> <span class="o">&lt;</span> <span class="n">angle1</span><span class="p">:</span>
-            <span class="n">e1_next</span> <span class="o">=</span> <span class="n">point2</span>
-        <span class="k">elif</span> <span class="n">Point</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">e1_ymax</span><span class="p">,</span> <span class="n">point1</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">Point</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">e1_ymax</span><span class="p">,</span> <span class="n">point2</span><span class="p">):</span>
-            <span class="n">e1_next</span> <span class="o">=</span> <span class="n">point2</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">e1_next</span> <span class="o">=</span> <span class="n">point1</span>
-
-        <span class="n">point1</span> <span class="o">=</span> <span class="n">e2_connections</span><span class="p">[</span><span class="n">e2_ymin</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">point2</span> <span class="o">=</span> <span class="n">e2_connections</span><span class="p">[</span><span class="n">e2_ymin</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">angle1</span> <span class="o">=</span> <span class="n">support_line</span><span class="o">.</span><span class="n">angle_between</span><span class="p">(</span><span class="n">Line</span><span class="p">(</span><span class="n">e2_ymin</span><span class="p">,</span> <span class="n">point1</span><span class="p">))</span>
-        <span class="n">angle2</span> <span class="o">=</span> <span class="n">support_line</span><span class="o">.</span><span class="n">angle_between</span><span class="p">(</span><span class="n">Line</span><span class="p">(</span><span class="n">e2_ymin</span><span class="p">,</span> <span class="n">point2</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">angle1</span> <span class="o">&gt;</span> <span class="n">angle2</span><span class="p">:</span>
-            <span class="n">e2_next</span> <span class="o">=</span> <span class="n">point1</span>
-        <span class="k">elif</span> <span class="n">angle2</span> <span class="o">&gt;</span> <span class="n">angle1</span><span class="p">:</span>
-            <span class="n">e2_next</span> <span class="o">=</span> <span class="n">point2</span>
-        <span class="k">elif</span> <span class="n">Point</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">e2_ymin</span><span class="p">,</span> <span class="n">point1</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">Point</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">e2_ymin</span><span class="p">,</span> <span class="n">point2</span><span class="p">):</span>
-            <span class="n">e2_next</span> <span class="o">=</span> <span class="n">point2</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">e2_next</span> <span class="o">=</span> <span class="n">point1</span>
-
-        <span class="sd">&#39;&#39;&#39;</span>
-<span class="sd">        Loop which determins the distance between anti-podal pairs and updates the</span>
-<span class="sd">        minimum distance accordingly. It repeats until it reaches the starting position.</span>
-<span class="sd">        &#39;&#39;&#39;</span>
-        <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">e1_angle</span> <span class="o">=</span> <span class="n">support_line</span><span class="o">.</span><span class="n">angle_between</span><span class="p">(</span><span class="n">Line</span><span class="p">(</span><span class="n">e1_current</span><span class="p">,</span> <span class="n">e1_next</span><span class="p">))</span>
-            <span class="n">e2_angle</span> <span class="o">=</span> <span class="n">pi</span> <span class="o">-</span> <span class="n">support_line</span><span class="o">.</span><span class="n">angle_between</span><span class="p">(</span><span class="n">Line</span><span class="p">(</span>
-                <span class="n">e2_current</span><span class="p">,</span> <span class="n">e2_next</span><span class="p">))</span>
-
-            <span class="k">if</span> <span class="n">e1_angle</span> <span class="o">&lt;</span> <span class="n">e2_angle</span><span class="p">:</span>
-                <span class="n">support_line</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">e1_current</span><span class="p">,</span> <span class="n">e1_next</span><span class="p">)</span>
-                <span class="n">e1_segment</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="n">e1_current</span><span class="p">,</span> <span class="n">e1_next</span><span class="p">)</span>
-                <span class="n">min_dist_current</span> <span class="o">=</span> <span class="n">e1_segment</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">e2_current</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">min_dist_current</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span> <span class="o">&lt;</span> <span class="n">min_dist</span><span class="o">.</span><span class="n">evalf</span><span class="p">():</span>
-                    <span class="n">min_dist</span> <span class="o">=</span> <span class="n">min_dist_current</span>
-
-                <span class="k">if</span> <span class="n">e1_connections</span><span class="p">[</span><span class="n">e1_next</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="n">e1_current</span><span class="p">:</span>
-                    <span class="n">e1_current</span> <span class="o">=</span> <span class="n">e1_next</span>
-                    <span class="n">e1_next</span> <span class="o">=</span> <span class="n">e1_connections</span><span class="p">[</span><span class="n">e1_next</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">e1_current</span> <span class="o">=</span> <span class="n">e1_next</span>
-                    <span class="n">e1_next</span> <span class="o">=</span> <span class="n">e1_connections</span><span class="p">[</span><span class="n">e1_next</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
-            <span class="k">elif</span> <span class="n">e1_angle</span> <span class="o">&gt;</span> <span class="n">e2_angle</span><span class="p">:</span>
-                <span class="n">support_line</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">e2_next</span><span class="p">,</span> <span class="n">e2_current</span><span class="p">)</span>
-                <span class="n">e2_segment</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="n">e2_current</span><span class="p">,</span> <span class="n">e2_next</span><span class="p">)</span>
-                <span class="n">min_dist_current</span> <span class="o">=</span> <span class="n">e2_segment</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">e1_current</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">min_dist_current</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span> <span class="o">&lt;</span> <span class="n">min_dist</span><span class="o">.</span><span class="n">evalf</span><span class="p">():</span>
-                    <span class="n">min_dist</span> <span class="o">=</span> <span class="n">min_dist_current</span>
-
-                <span class="k">if</span> <span class="n">e2_connections</span><span class="p">[</span><span class="n">e2_next</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="n">e2_current</span><span class="p">:</span>
-                    <span class="n">e2_current</span> <span class="o">=</span> <span class="n">e2_next</span>
-                    <span class="n">e2_next</span> <span class="o">=</span> <span class="n">e2_connections</span><span class="p">[</span><span class="n">e2_next</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">e2_current</span> <span class="o">=</span> <span class="n">e2_next</span>
-                    <span class="n">e2_next</span> <span class="o">=</span> <span class="n">e2_connections</span><span class="p">[</span><span class="n">e2_next</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">support_line</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">e1_current</span><span class="p">,</span> <span class="n">e1_next</span><span class="p">)</span>
-                <span class="n">e1_segment</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="n">e1_current</span><span class="p">,</span> <span class="n">e1_next</span><span class="p">)</span>
-                <span class="n">e2_segment</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="n">e2_current</span><span class="p">,</span> <span class="n">e2_next</span><span class="p">)</span>
-                <span class="n">min1</span> <span class="o">=</span> <span class="n">e1_segment</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">e2_next</span><span class="p">)</span>
-                <span class="n">min2</span> <span class="o">=</span> <span class="n">e2_segment</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">e1_next</span><span class="p">)</span>
-
-                <span class="n">min_dist_current</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">min1</span><span class="p">,</span> <span class="n">min2</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">min_dist_current</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span> <span class="o">&lt;</span> <span class="n">min_dist</span><span class="o">.</span><span class="n">evalf</span><span class="p">():</span>
-                    <span class="n">min_dist</span> <span class="o">=</span> <span class="n">min_dist_current</span>
-
-                <span class="k">if</span> <span class="n">e1_connections</span><span class="p">[</span><span class="n">e1_next</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="n">e1_current</span><span class="p">:</span>
-                    <span class="n">e1_current</span> <span class="o">=</span> <span class="n">e1_next</span>
-                    <span class="n">e1_next</span> <span class="o">=</span> <span class="n">e1_connections</span><span class="p">[</span><span class="n">e1_next</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">e1_current</span> <span class="o">=</span> <span class="n">e1_next</span>
-                    <span class="n">e1_next</span> <span class="o">=</span> <span class="n">e1_connections</span><span class="p">[</span><span class="n">e1_next</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
-
-                <span class="k">if</span> <span class="n">e2_connections</span><span class="p">[</span><span class="n">e2_next</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="n">e2_current</span><span class="p">:</span>
-                    <span class="n">e2_current</span> <span class="o">=</span> <span class="n">e2_next</span>
-                    <span class="n">e2_next</span> <span class="o">=</span> <span class="n">e2_connections</span><span class="p">[</span><span class="n">e2_next</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">e2_current</span> <span class="o">=</span> <span class="n">e2_next</span>
-                    <span class="n">e2_next</span> <span class="o">=</span> <span class="n">e2_connections</span><span class="p">[</span><span class="n">e2_next</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">e1_current</span> <span class="o">==</span> <span class="n">e1_ymax</span> <span class="ow">and</span> <span class="n">e2_current</span> <span class="o">==</span> <span class="n">e2_ymin</span><span class="p">:</span>
-                <span class="k">break</span>
-        <span class="k">return</span> <span class="n">min_dist</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Polygon</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">o</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="c"># See if self can ever be traversed (cw or ccw) from any of its</span>
-        <span class="c"># vertices to match all points of o</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="n">oargs</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">args</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">o0</span> <span class="o">=</span> <span class="n">oargs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">i0</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">args</span><span class="p">[</span><span class="n">i0</span><span class="p">]</span> <span class="o">==</span> <span class="n">o0</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">args</span><span class="p">[(</span><span class="n">i0</span> <span class="o">+</span> <span class="n">i</span><span class="p">)</span> <span class="o">%</span> <span class="n">n</span><span class="p">]</span> <span class="o">==</span> <span class="n">oargs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">)):</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-                <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">args</span><span class="p">[(</span><span class="n">i0</span> <span class="o">-</span> <span class="n">i</span><span class="p">)</span> <span class="o">%</span> <span class="n">n</span><span class="p">]</span> <span class="o">==</span> <span class="n">oargs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">)):</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Polygon</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__contains__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if o is contained within the boundary lines of self.altitudes</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        other : GeometryEntity</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        contained in : bool</span>
-<span class="sd">            The points (and sides, if applicable) are contained in self.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.entity.GeometryEntity.encloses</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Line, Segment, Point</span>
-<span class="sd">        &gt;&gt;&gt; p = Point(0, 0)</span>
-<span class="sd">        &gt;&gt;&gt; q = Point(1, 1)</span>
-<span class="sd">        &gt;&gt;&gt; s = Segment(p, q*2)</span>
-<span class="sd">        &gt;&gt;&gt; l = Line(p, q)</span>
-<span class="sd">        &gt;&gt;&gt; p in q</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; p in s</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; q*3 in s</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; s in l</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Polygon</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span> <span class="o">==</span> <span class="n">o</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Segment</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">any</span><span class="p">(</span><span class="n">o</span> <span class="ow">in</span> <span class="n">s</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">sides</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Point</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">o</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">vertices</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">for</span> <span class="n">side</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">sides</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">side</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-
-        <span class="k">return</span> <span class="bp">False</span>
-
-</div>
-<div class="viewcode-block" id="RegularPolygon"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.RegularPolygon">[docs]</a><span class="k">class</span> <span class="nc">RegularPolygon</span><span class="p">(</span><span class="n">Polygon</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A regular polygon.</span>
-
-<span class="sd">    Such a polygon has all internal angles equal and all sides the same length.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    center : Point</span>
-<span class="sd">    radius : number or Basic instance</span>
-<span class="sd">        The distance from the center to a vertex</span>
-<span class="sd">    n : int</span>
-<span class="sd">        The number of sides</span>
-
-<span class="sd">    Attributes</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    vertices</span>
-<span class="sd">    center</span>
-<span class="sd">    radius</span>
-<span class="sd">    rotation</span>
-<span class="sd">    apothem</span>
-<span class="sd">    interior_angle</span>
-<span class="sd">    exterior_angle</span>
-<span class="sd">    circumcircle</span>
-<span class="sd">    incircle</span>
-<span class="sd">    angles</span>
-
-<span class="sd">    Raises</span>
-<span class="sd">    ======</span>
-
-<span class="sd">    GeometryError</span>
-<span class="sd">        If the `center` is not a Point, or the `radius` is not a number or Basic</span>
-<span class="sd">        instance, or the number of sides, `n`, is less than three.</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    A RegularPolygon can be instantiated with Polygon with the kwarg n.</span>
-
-<span class="sd">    Regular polygons are instantiated with a center, radius, number of sides</span>
-<span class="sd">    and a rotation angle. Whereas the arguments of a Polygon are vertices, the</span>
-<span class="sd">    vertices of the RegularPolygon must be obtained with the vertices method.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.geometry.point.Point, Polygon</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.geometry import RegularPolygon, Point</span>
-<span class="sd">    &gt;&gt;&gt; r = RegularPolygon(Point(0, 0), 5, 3)</span>
-<span class="sd">    &gt;&gt;&gt; r</span>
-<span class="sd">    RegularPolygon(Point(0, 0), 5, 3, 0)</span>
-<span class="sd">    &gt;&gt;&gt; r.vertices[0]</span>
-<span class="sd">    Point(5, 0)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;_n&#39;</span><span class="p">,</span> <span class="s">&#39;_center&#39;</span><span class="p">,</span> <span class="s">&#39;_radius&#39;</span><span class="p">,</span> <span class="s">&#39;_rot&#39;</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">rot</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">r</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">rot</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">rot</span><span class="p">)))</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">Expr</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="n">GeometryError</span><span class="p">(</span><span class="s">&quot;r must be an Expr object, not </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">r</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="n">as_int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>  <span class="c"># let an error raise if necessary</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">3</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">GeometryError</span><span class="p">(</span><span class="s">&quot;n must be a &gt;= 3, not </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">GeometryEntity</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_n</span> <span class="o">=</span> <span class="n">n</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_center</span> <span class="o">=</span> <span class="n">c</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_radius</span> <span class="o">=</span> <span class="n">r</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_rot</span> <span class="o">=</span> <span class="n">rot</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="RegularPolygon.args"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.RegularPolygon.args">[docs]</a>    <span class="k">def</span> <span class="nf">args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the center point, the radius,</span>
-<span class="sd">        the number of sides, and the orientation angle.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import RegularPolygon, Point</span>
-<span class="sd">        &gt;&gt;&gt; r = RegularPolygon(Point(0, 0), 5, 3)</span>
-<span class="sd">        &gt;&gt;&gt; r.args</span>
-<span class="sd">        (Point(0, 0), 5, 3, 0)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_center</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_radius</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_n</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rot</span>
-</div>
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;RegularPolygon(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="nb">tuple</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;RegularPolygon(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="nb">tuple</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="RegularPolygon.area"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.RegularPolygon.area">[docs]</a>    <span class="k">def</span> <span class="nf">area</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the area.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import RegularPolygon</span>
-<span class="sd">        &gt;&gt;&gt; square = RegularPolygon((0, 0), 1, 4)</span>
-<span class="sd">        &gt;&gt;&gt; square.area</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; _ == square.length**2</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">c</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">rot</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">return</span> <span class="n">n</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">length</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="n">n</span><span class="p">))</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="RegularPolygon.length"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.RegularPolygon.length">[docs]</a>    <span class="k">def</span> <span class="nf">length</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the length of the sides.</span>
-
-<span class="sd">        The half-length of the side and the apothem form two legs</span>
-<span class="sd">        of a right triangle whose hypotenuse is the radius of the</span>
-<span class="sd">        regular polygon.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import RegularPolygon</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import sqrt</span>
-<span class="sd">        &gt;&gt;&gt; s = square_in_unit_circle = RegularPolygon((0, 0), 1, 4)</span>
-<span class="sd">        &gt;&gt;&gt; s.length</span>
-<span class="sd">        sqrt(2)</span>
-<span class="sd">        &gt;&gt;&gt; sqrt((_/2)**2 + s.apothem**2) == s.radius</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">radius</span><span class="o">*</span><span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">_n</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="RegularPolygon.center"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.RegularPolygon.center">[docs]</a>    <span class="k">def</span> <span class="nf">center</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The center of the RegularPolygon</span>
-
-<span class="sd">        This is also the center of the circumscribing circle.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        center : Point</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point, sympy.geometry.ellipse.Ellipse.center</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import RegularPolygon, Point</span>
-<span class="sd">        &gt;&gt;&gt; rp = RegularPolygon(Point(0, 0), 5, 4)</span>
-<span class="sd">        &gt;&gt;&gt; rp.center</span>
-<span class="sd">        Point(0, 0)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_center</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="RegularPolygon.circumcenter"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.RegularPolygon.circumcenter">[docs]</a>    <span class="k">def</span> <span class="nf">circumcenter</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Alias for center.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import RegularPolygon, Point</span>
-<span class="sd">        &gt;&gt;&gt; rp = RegularPolygon(Point(0, 0), 5, 4)</span>
-<span class="sd">        &gt;&gt;&gt; rp.circumcenter</span>
-<span class="sd">        Point(0, 0)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">center</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="RegularPolygon.radius"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.RegularPolygon.radius">[docs]</a>    <span class="k">def</span> <span class="nf">radius</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Radius of the RegularPolygon</span>
-
-<span class="sd">        This is also the radius of the circumscribing circle.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        radius : number or instance of Basic</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.line.Segment.length, sympy.geometry.ellipse.Circle.radius</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import RegularPolygon, Point</span>
-<span class="sd">        &gt;&gt;&gt; radius = Symbol(&#39;r&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; rp = RegularPolygon(Point(0, 0), radius, 4)</span>
-<span class="sd">        &gt;&gt;&gt; rp.radius</span>
-<span class="sd">        r</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_radius</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="RegularPolygon.circumradius"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.RegularPolygon.circumradius">[docs]</a>    <span class="k">def</span> <span class="nf">circumradius</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Alias for radius.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import RegularPolygon, Point</span>
-<span class="sd">        &gt;&gt;&gt; radius = Symbol(&#39;r&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; rp = RegularPolygon(Point(0, 0), radius, 4)</span>
-<span class="sd">        &gt;&gt;&gt; rp.circumradius</span>
-<span class="sd">        r</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">radius</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="RegularPolygon.rotation"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.RegularPolygon.rotation">[docs]</a>    <span class="k">def</span> <span class="nf">rotation</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;CCW angle by which the RegularPolygon is rotated</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        rotation : number or instance of Basic</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import pi</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import RegularPolygon, Point</span>
-<span class="sd">        &gt;&gt;&gt; RegularPolygon(Point(0, 0), 3, 4, pi).rotation</span>
-<span class="sd">        pi</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rot</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="RegularPolygon.apothem"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.RegularPolygon.apothem">[docs]</a>    <span class="k">def</span> <span class="nf">apothem</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The inradius of the RegularPolygon.</span>
-
-<span class="sd">        The apothem/inradius is the radius of the inscribed circle.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        apothem : number or instance of Basic</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.line.Segment.length, sympy.geometry.ellipse.Circle.radius</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import RegularPolygon, Point</span>
-<span class="sd">        &gt;&gt;&gt; radius = Symbol(&#39;r&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; rp = RegularPolygon(Point(0, 0), radius, 4)</span>
-<span class="sd">        &gt;&gt;&gt; rp.apothem</span>
-<span class="sd">        sqrt(2)*r/2</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">radius</span> <span class="o">*</span> <span class="n">cos</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">_n</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="RegularPolygon.inradius"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.RegularPolygon.inradius">[docs]</a>    <span class="k">def</span> <span class="nf">inradius</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Alias for apothem.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import RegularPolygon, Point</span>
-<span class="sd">        &gt;&gt;&gt; radius = Symbol(&#39;r&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; rp = RegularPolygon(Point(0, 0), radius, 4)</span>
-<span class="sd">        &gt;&gt;&gt; rp.inradius</span>
-<span class="sd">        sqrt(2)*r/2</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">apothem</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="RegularPolygon.interior_angle"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.RegularPolygon.interior_angle">[docs]</a>    <span class="k">def</span> <span class="nf">interior_angle</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Measure of the interior angles.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        interior_angle : number</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.line.LinearEntity.angle_between</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import RegularPolygon, Point</span>
-<span class="sd">        &gt;&gt;&gt; rp = RegularPolygon(Point(0, 0), 4, 8)</span>
-<span class="sd">        &gt;&gt;&gt; rp.interior_angle</span>
-<span class="sd">        3*pi/4</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_n</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">_n</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="RegularPolygon.exterior_angle"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.RegularPolygon.exterior_angle">[docs]</a>    <span class="k">def</span> <span class="nf">exterior_angle</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Measure of the exterior angles.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        exterior_angle : number</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.line.LinearEntity.angle_between</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import RegularPolygon, Point</span>
-<span class="sd">        &gt;&gt;&gt; rp = RegularPolygon(Point(0, 0), 4, 8)</span>
-<span class="sd">        &gt;&gt;&gt; rp.exterior_angle</span>
-<span class="sd">        pi/4</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">_n</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="RegularPolygon.circumcircle"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.RegularPolygon.circumcircle">[docs]</a>    <span class="k">def</span> <span class="nf">circumcircle</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The circumcircle of the RegularPolygon.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        circumcircle : Circle</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        circumcenter, sympy.geometry.ellipse.Circle</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import RegularPolygon, Point</span>
-<span class="sd">        &gt;&gt;&gt; rp = RegularPolygon(Point(0, 0), 4, 8)</span>
-<span class="sd">        &gt;&gt;&gt; rp.circumcircle</span>
-<span class="sd">        Circle(Point(0, 0), 4)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Circle</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">center</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">radius</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="RegularPolygon.incircle"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.RegularPolygon.incircle">[docs]</a>    <span class="k">def</span> <span class="nf">incircle</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The incircle of the RegularPolygon.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        incircle : Circle</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        inradius, sympy.geometry.ellipse.Circle</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import RegularPolygon, Point</span>
-<span class="sd">        &gt;&gt;&gt; rp = RegularPolygon(Point(0, 0), 4, 7)</span>
-<span class="sd">        &gt;&gt;&gt; rp.incircle</span>
-<span class="sd">        Circle(Point(0, 0), 4*cos(pi/7))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Circle</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">center</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">apothem</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="RegularPolygon.angles"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.RegularPolygon.angles">[docs]</a>    <span class="k">def</span> <span class="nf">angles</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns a dictionary with keys, the vertices of the Polygon,</span>
-<span class="sd">        and values, the interior angle at each vertex.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import RegularPolygon, Point</span>
-<span class="sd">        &gt;&gt;&gt; r = RegularPolygon(Point(0, 0), 5, 3)</span>
-<span class="sd">        &gt;&gt;&gt; r.angles</span>
-<span class="sd">        {Point(-5/2, -5*sqrt(3)/2): pi/3,</span>
-<span class="sd">         Point(-5/2, 5*sqrt(3)/2): pi/3,</span>
-<span class="sd">         Point(5, 0): pi/3}</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">ret</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="n">ang</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">interior_angle</span>
-        <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">vertices</span><span class="p">:</span>
-            <span class="n">ret</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="n">ang</span>
-        <span class="k">return</span> <span class="n">ret</span>
-</div>
-<div class="viewcode-block" id="RegularPolygon.encloses_point"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.RegularPolygon.encloses_point">[docs]</a>    <span class="k">def</span> <span class="nf">encloses_point</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if p is enclosed by (is inside of) self.</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        Being on the border of self is considered False.</span>
-
-<span class="sd">        The general Polygon.encloses_point method is called only if</span>
-<span class="sd">        a point is not within or beyond the incircle or circumcircle,</span>
-<span class="sd">        respectively.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        p : Point</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        encloses_point : True, False or None</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.ellipse.Ellipse.encloses_point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import RegularPolygon, S, Point, Symbol</span>
-<span class="sd">        &gt;&gt;&gt; p = RegularPolygon((0, 0), 3, 4)</span>
-<span class="sd">        &gt;&gt;&gt; p.encloses_point(Point(0, 0))</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; r, R = p.inradius, p.circumradius</span>
-<span class="sd">        &gt;&gt;&gt; p.encloses_point(Point((r + R)/2, 0))</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; p.encloses_point(Point(R/2, R/2 + (R - r)/10))</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; t = Symbol(&#39;t&#39;, real=True)</span>
-<span class="sd">        &gt;&gt;&gt; p.encloses_point(p.arbitrary_point().subs(t, S.Half))</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; p.encloses_point(Point(5, 5))</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">c</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">center</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">length</span>
-        <span class="k">if</span> <span class="n">d</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">radius</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="n">d</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">inradius</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># now enumerate the RegularPolygon like a general polygon.</span>
-            <span class="k">return</span> <span class="n">Polygon</span><span class="o">.</span><span class="n">encloses_point</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="RegularPolygon.spin"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.RegularPolygon.spin">[docs]</a>    <span class="k">def</span> <span class="nf">spin</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">angle</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Increment *in place* the virtual Polygon&#39;s rotation by ccw angle.</span>
-
-<span class="sd">        See also: rotate method which moves the center.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Polygon, Point, pi</span>
-<span class="sd">        &gt;&gt;&gt; r = Polygon(Point(0,0), 1, n=3)</span>
-<span class="sd">        &gt;&gt;&gt; r.vertices[0]</span>
-<span class="sd">        Point(1, 0)</span>
-<span class="sd">        &gt;&gt;&gt; r.spin(pi/6)</span>
-<span class="sd">        &gt;&gt;&gt; r.vertices[0]</span>
-<span class="sd">        Point(sqrt(3)/2, 1/2)</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        rotation</span>
-<span class="sd">        rotate : Creates a copy of the RegularPolygon rotated about a Point</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_rot</span> <span class="o">+=</span> <span class="n">angle</span>
-</div>
-<div class="viewcode-block" id="RegularPolygon.rotate"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.RegularPolygon.rotate">[docs]</a>    <span class="k">def</span> <span class="nf">rotate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">angle</span><span class="p">,</span> <span class="n">pt</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Override GeometryEntity.rotate to first rotate the RegularPolygon</span>
-<span class="sd">        about its center.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Point, RegularPolygon, Polygon, pi</span>
-<span class="sd">        &gt;&gt;&gt; t = RegularPolygon(Point(1, 0), 1, 3)</span>
-<span class="sd">        &gt;&gt;&gt; t.vertices[0] # vertex on x-axis</span>
-<span class="sd">        Point(2, 0)</span>
-<span class="sd">        &gt;&gt;&gt; t.rotate(pi/2).vertices[0] # vertex on y axis now</span>
-<span class="sd">        Point(0, 2)</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        rotation</span>
-<span class="sd">        spin : Rotates a RegularPolygon in place</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">r</span> <span class="o">=</span> <span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">)(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>  <span class="c"># need a copy or else changes are in-place</span>
-        <span class="n">r</span><span class="o">.</span><span class="n">_rot</span> <span class="o">+=</span> <span class="n">angle</span>
-        <span class="k">return</span> <span class="n">GeometryEntity</span><span class="o">.</span><span class="n">rotate</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">angle</span><span class="p">,</span> <span class="n">pt</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="RegularPolygon.scale"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.RegularPolygon.scale">[docs]</a>    <span class="k">def</span> <span class="nf">scale</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">pt</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Override GeometryEntity.scale since it is the radius that must be</span>
-<span class="sd">        scaled (if x == y) or else a new Polygon must be returned.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import RegularPolygon</span>
-
-<span class="sd">        Symmetric scaling returns a RegularPolygon:</span>
-
-<span class="sd">        &gt;&gt;&gt; RegularPolygon((0, 0), 1, 4).scale(2, 2)</span>
-<span class="sd">        RegularPolygon(Point(0, 0), 2, 4, 0)</span>
-
-<span class="sd">        Asymmetric scaling returns a kite as a Polygon:</span>
-
-<span class="sd">        &gt;&gt;&gt; RegularPolygon((0, 0), 1, 4).scale(2, 1)</span>
-<span class="sd">        Polygon(Point(2, 0), Point(0, 1), Point(-2, 0), Point(0, -1))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">pt</span><span class="p">:</span>
-            <span class="n">pt</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">pt</span><span class="p">)</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">pt</span><span class="p">)</span><span class="o">.</span><span class="n">args</span><span class="p">)</span><span class="o">.</span><span class="n">scale</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="o">*</span><span class="n">pt</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="o">!=</span> <span class="n">y</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Polygon</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">vertices</span><span class="p">)</span><span class="o">.</span><span class="n">scale</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-        <span class="n">c</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">rot</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="n">r</span> <span class="o">*=</span> <span class="n">x</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">rot</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="RegularPolygon.vertices"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.RegularPolygon.vertices">[docs]</a>    <span class="k">def</span> <span class="nf">vertices</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The vertices of the RegularPolygon.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        vertices : list</span>
-<span class="sd">            Each vertex is a Point.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import RegularPolygon, Point</span>
-<span class="sd">        &gt;&gt;&gt; rp = RegularPolygon(Point(0, 0), 5, 4)</span>
-<span class="sd">        &gt;&gt;&gt; rp.vertices</span>
-<span class="sd">        [Point(5, 0), Point(0, 5), Point(-5, 0), Point(0, -5)]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_center</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_radius</span>
-        <span class="n">rot</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rot</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">_n</span>
-
-        <span class="k">return</span> <span class="p">[</span><span class="n">Point</span><span class="p">(</span><span class="n">c</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">r</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">k</span><span class="o">*</span><span class="n">v</span> <span class="o">+</span> <span class="n">rot</span><span class="p">),</span> <span class="n">c</span><span class="o">.</span><span class="n">y</span> <span class="o">+</span> <span class="n">r</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">k</span><span class="o">*</span><span class="n">v</span> <span class="o">+</span> <span class="n">rot</span><span class="p">))</span>
-                <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_n</span><span class="p">)]</span>
-</div>
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">Polygon</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">RegularPolygon</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Polygon</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">==</span> <span class="n">o</span><span class="o">.</span><span class="n">args</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">RegularPolygon</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-</div>
-<div class="viewcode-block" id="Triangle"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Triangle">[docs]</a><span class="k">class</span> <span class="nc">Triangle</span><span class="p">(</span><span class="n">Polygon</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A polygon with three vertices and three sides.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    points : sequence of Points</span>
-<span class="sd">    keyword: asa, sas, or sss to specify sides/angles of the triangle</span>
-
-<span class="sd">    Attributes</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    vertices</span>
-<span class="sd">    altitudes</span>
-<span class="sd">    orthocenter</span>
-<span class="sd">    circumcenter</span>
-<span class="sd">    circumradius</span>
-<span class="sd">    circumcircle</span>
-<span class="sd">    inradius</span>
-<span class="sd">    incircle</span>
-<span class="sd">    medians</span>
-<span class="sd">    medial</span>
-
-<span class="sd">    Raises</span>
-<span class="sd">    ======</span>
-
-<span class="sd">    GeometryError</span>
-<span class="sd">        If the number of vertices is not equal to three, or one of the vertices</span>
-<span class="sd">        is not a Point, or a valid keyword is not given.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.geometry.point.Point, Polygon</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.geometry import Triangle, Point</span>
-<span class="sd">    &gt;&gt;&gt; Triangle(Point(0, 0), Point(4, 0), Point(4, 3))</span>
-<span class="sd">    Triangle(Point(0, 0), Point(4, 0), Point(4, 3))</span>
-
-<span class="sd">    Keywords sss, sas, or asa can be used to give the desired</span>
-<span class="sd">    side lengths (in order) and interior angles (in degrees) that</span>
-<span class="sd">    define the triangle:</span>
-
-<span class="sd">    &gt;&gt;&gt; Triangle(sss=(3, 4, 5))</span>
-<span class="sd">    Triangle(Point(0, 0), Point(3, 0), Point(3, 4))</span>
-<span class="sd">    &gt;&gt;&gt; Triangle(asa=(30, 1, 30))</span>
-<span class="sd">    Triangle(Point(0, 0), Point(1, 0), Point(1/2, sqrt(3)/6))</span>
-<span class="sd">    &gt;&gt;&gt; Triangle(sas=(1, 45, 2))</span>
-<span class="sd">    Triangle(Point(0, 0), Point(2, 0), Point(sqrt(2)/2, sqrt(2)/2))</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="k">if</span> <span class="s">&#39;sss&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">_sss</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">nsimplify</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;sss&#39;</span><span class="p">]])</span>
-            <span class="k">if</span> <span class="s">&#39;asa&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">_asa</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">nsimplify</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;asa&#39;</span><span class="p">]])</span>
-            <span class="k">if</span> <span class="s">&#39;sas&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">_sas</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">nsimplify</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;sas&#39;</span><span class="p">]])</span>
-            <span class="n">msg</span> <span class="o">=</span> <span class="s">&quot;Triangle instantiates with three points or a valid keyword.&quot;</span>
-            <span class="k">raise</span> <span class="n">GeometryError</span><span class="p">(</span><span class="n">msg</span><span class="p">)</span>
-
-        <span class="n">vertices</span> <span class="o">=</span> <span class="p">[</span><span class="n">Point</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span>
-
-        <span class="c"># remove consecutive duplicates</span>
-        <span class="n">nodup</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">vertices</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">nodup</span> <span class="ow">and</span> <span class="n">p</span> <span class="o">==</span> <span class="n">nodup</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-                <span class="k">continue</span>
-            <span class="n">nodup</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">nodup</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">nodup</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">nodup</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-            <span class="n">nodup</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>  <span class="c"># last point was same as first</span>
-
-        <span class="c"># remove collinear points</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="o">-</span><span class="mi">3</span>
-        <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">nodup</span><span class="p">)</span> <span class="o">-</span> <span class="mi">3</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">nodup</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">([</span><span class="n">nodup</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">nodup</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">nodup</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">2</span><span class="p">]])</span>
-            <span class="k">if</span> <span class="n">Point</span><span class="o">.</span><span class="n">is_collinear</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
-                <span class="n">nodup</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">a</span>
-                <span class="n">nodup</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="bp">None</span>
-                <span class="n">nodup</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-
-        <span class="n">vertices</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">nodup</span> <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">vertices</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">GeometryEntity</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">vertices</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">vertices</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Segment</span><span class="p">(</span><span class="o">*</span><span class="n">vertices</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="o">*</span><span class="n">vertices</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Triangle.vertices"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Triangle.vertices">[docs]</a>    <span class="k">def</span> <span class="nf">vertices</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The triangle&#39;s vertices</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        vertices : tuple</span>
-<span class="sd">            Each element in the tuple is a Point</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Triangle, Point</span>
-<span class="sd">        &gt;&gt;&gt; t = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))</span>
-<span class="sd">        &gt;&gt;&gt; t.vertices</span>
-<span class="sd">        (Point(0, 0), Point(4, 0), Point(4, 3))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-</div>
-<div class="viewcode-block" id="Triangle.is_similar"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Triangle.is_similar">[docs]</a>    <span class="k">def</span> <span class="nf">is_similar</span><span class="p">(</span><span class="n">t1</span><span class="p">,</span> <span class="n">t2</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Is another triangle similar to this one.</span>
-
-<span class="sd">        Two triangles are similar if one can be uniformly scaled to the other.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        other: Triangle</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        is_similar : boolean</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.entity.GeometryEntity.is_similar</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Triangle, Point</span>
-<span class="sd">        &gt;&gt;&gt; t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))</span>
-<span class="sd">        &gt;&gt;&gt; t2 = Triangle(Point(0, 0), Point(-4, 0), Point(-4, -3))</span>
-<span class="sd">        &gt;&gt;&gt; t1.is_similar(t2)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &gt;&gt;&gt; t2 = Triangle(Point(0, 0), Point(-4, 0), Point(-4, -4))</span>
-<span class="sd">        &gt;&gt;&gt; t1.is_similar(t2)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">t2</span><span class="p">,</span> <span class="n">Polygon</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="n">s1_1</span><span class="p">,</span> <span class="n">s1_2</span><span class="p">,</span> <span class="n">s1_3</span> <span class="o">=</span> <span class="p">[</span><span class="n">side</span><span class="o">.</span><span class="n">length</span> <span class="k">for</span> <span class="n">side</span> <span class="ow">in</span> <span class="n">t1</span><span class="o">.</span><span class="n">sides</span><span class="p">]</span>
-        <span class="n">s2</span> <span class="o">=</span> <span class="p">[</span><span class="n">side</span><span class="o">.</span><span class="n">length</span> <span class="k">for</span> <span class="n">side</span> <span class="ow">in</span> <span class="n">t2</span><span class="o">.</span><span class="n">sides</span><span class="p">]</span>
-
-        <span class="k">def</span> <span class="nf">_are_similar</span><span class="p">(</span><span class="n">u1</span><span class="p">,</span> <span class="n">u2</span><span class="p">,</span> <span class="n">u3</span><span class="p">,</span> <span class="n">v1</span><span class="p">,</span> <span class="n">v2</span><span class="p">,</span> <span class="n">v3</span><span class="p">):</span>
-            <span class="n">e1</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">u1</span><span class="o">/</span><span class="n">v1</span><span class="p">)</span>
-            <span class="n">e2</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">u2</span><span class="o">/</span><span class="n">v2</span><span class="p">)</span>
-            <span class="n">e3</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">u3</span><span class="o">/</span><span class="n">v3</span><span class="p">)</span>
-            <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="n">e1</span> <span class="o">==</span> <span class="n">e2</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">bool</span><span class="p">(</span><span class="n">e2</span> <span class="o">==</span> <span class="n">e3</span><span class="p">)</span>
-
-        <span class="c"># There&#39;s only 6 permutations, so write them out</span>
-        <span class="k">return</span> <span class="n">_are_similar</span><span class="p">(</span><span class="n">s1_1</span><span class="p">,</span> <span class="n">s1_2</span><span class="p">,</span> <span class="n">s1_3</span><span class="p">,</span> <span class="o">*</span><span class="n">s2</span><span class="p">)</span> <span class="ow">or</span> \
-            <span class="n">_are_similar</span><span class="p">(</span><span class="n">s1_1</span><span class="p">,</span> <span class="n">s1_3</span><span class="p">,</span> <span class="n">s1_2</span><span class="p">,</span> <span class="o">*</span><span class="n">s2</span><span class="p">)</span> <span class="ow">or</span> \
-            <span class="n">_are_similar</span><span class="p">(</span><span class="n">s1_2</span><span class="p">,</span> <span class="n">s1_1</span><span class="p">,</span> <span class="n">s1_3</span><span class="p">,</span> <span class="o">*</span><span class="n">s2</span><span class="p">)</span> <span class="ow">or</span> \
-            <span class="n">_are_similar</span><span class="p">(</span><span class="n">s1_2</span><span class="p">,</span> <span class="n">s1_3</span><span class="p">,</span> <span class="n">s1_1</span><span class="p">,</span> <span class="o">*</span><span class="n">s2</span><span class="p">)</span> <span class="ow">or</span> \
-            <span class="n">_are_similar</span><span class="p">(</span><span class="n">s1_3</span><span class="p">,</span> <span class="n">s1_1</span><span class="p">,</span> <span class="n">s1_2</span><span class="p">,</span> <span class="o">*</span><span class="n">s2</span><span class="p">)</span> <span class="ow">or</span> \
-            <span class="n">_are_similar</span><span class="p">(</span><span class="n">s1_3</span><span class="p">,</span> <span class="n">s1_2</span><span class="p">,</span> <span class="n">s1_1</span><span class="p">,</span> <span class="o">*</span><span class="n">s2</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Triangle.is_equilateral"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Triangle.is_equilateral">[docs]</a>    <span class="k">def</span> <span class="nf">is_equilateral</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Are all the sides the same length?</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        is_equilateral : boolean</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.entity.GeometryEntity.is_similar, RegularPolygon</span>
-<span class="sd">        is_isosceles, is_right, is_scalene</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Triangle, Point</span>
-<span class="sd">        &gt;&gt;&gt; t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))</span>
-<span class="sd">        &gt;&gt;&gt; t1.is_equilateral()</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sqrt</span>
-<span class="sd">        &gt;&gt;&gt; t2 = Triangle(Point(0, 0), Point(10, 0), Point(5, 5*sqrt(3)))</span>
-<span class="sd">        &gt;&gt;&gt; t2.is_equilateral()</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="n">has_variety</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">length</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">sides</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Triangle.is_isosceles"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Triangle.is_isosceles">[docs]</a>    <span class="k">def</span> <span class="nf">is_isosceles</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Are two or more of the sides the same length?</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        is_isosceles : boolean</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        is_equilateral, is_right, is_scalene</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Triangle, Point</span>
-<span class="sd">        &gt;&gt;&gt; t1 = Triangle(Point(0, 0), Point(4, 0), Point(2, 4))</span>
-<span class="sd">        &gt;&gt;&gt; t1.is_isosceles()</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">has_dups</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">length</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">sides</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Triangle.is_scalene"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Triangle.is_scalene">[docs]</a>    <span class="k">def</span> <span class="nf">is_scalene</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Are all the sides of the triangle of different lengths?</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        is_scalene : boolean</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        is_equilateral, is_isosceles, is_right</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Triangle, Point</span>
-<span class="sd">        &gt;&gt;&gt; t1 = Triangle(Point(0, 0), Point(4, 0), Point(1, 4))</span>
-<span class="sd">        &gt;&gt;&gt; t1.is_scalene()</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="n">has_dups</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">length</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">sides</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Triangle.is_right"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Triangle.is_right">[docs]</a>    <span class="k">def</span> <span class="nf">is_right</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Is the triangle right-angled.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        is_right : boolean</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.line.LinearEntity.is_perpendicular</span>
-<span class="sd">        is_equilateral, is_isosceles, is_scalene</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Triangle, Point</span>
-<span class="sd">        &gt;&gt;&gt; t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))</span>
-<span class="sd">        &gt;&gt;&gt; t1.is_right()</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sides</span>
-        <span class="k">return</span> <span class="n">Segment</span><span class="o">.</span><span class="n">is_perpendicular</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="ow">or</span> \
-            <span class="n">Segment</span><span class="o">.</span><span class="n">is_perpendicular</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">s</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span> <span class="ow">or</span> \
-            <span class="n">Segment</span><span class="o">.</span><span class="n">is_perpendicular</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">s</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Triangle.altitudes"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Triangle.altitudes">[docs]</a>    <span class="k">def</span> <span class="nf">altitudes</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The altitudes of the triangle.</span>
-
-<span class="sd">        An altitude of a triangle is a segment through a vertex,</span>
-<span class="sd">        perpendicular to the opposite side, with length being the</span>
-<span class="sd">        height of the vertex measured from the line containing the side.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        altitudes : dict</span>
-<span class="sd">            The dictionary consists of keys which are vertices and values</span>
-<span class="sd">            which are Segments.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point, sympy.geometry.line.Segment.length</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Point, Triangle</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)</span>
-<span class="sd">        &gt;&gt;&gt; t = Triangle(p1, p2, p3)</span>
-<span class="sd">        &gt;&gt;&gt; t.altitudes[p1]</span>
-<span class="sd">        Segment(Point(0, 0), Point(1/2, 1/2))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sides</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">vertices</span>
-        <span class="k">return</span> <span class="p">{</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span> <span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">perpendicular_segment</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span>
-                <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span> <span class="n">s</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">perpendicular_segment</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span>
-                <span class="n">v</span><span class="p">[</span><span class="mi">2</span><span class="p">]:</span> <span class="n">s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">perpendicular_segment</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">2</span><span class="p">])}</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Triangle.orthocenter"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Triangle.orthocenter">[docs]</a>    <span class="k">def</span> <span class="nf">orthocenter</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The orthocenter of the triangle.</span>
-
-<span class="sd">        The orthocenter is the intersection of the altitudes of a triangle.</span>
-<span class="sd">        It may lie inside, outside or on the triangle.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        orthocenter : Point</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Point, Triangle</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)</span>
-<span class="sd">        &gt;&gt;&gt; t = Triangle(p1, p2, p3)</span>
-<span class="sd">        &gt;&gt;&gt; t.orthocenter</span>
-<span class="sd">        Point(0, 0)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">altitudes</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">vertices</span>
-        <span class="k">return</span> <span class="n">Line</span><span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]])</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">Line</span><span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]]))[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Triangle.circumcenter"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Triangle.circumcenter">[docs]</a>    <span class="k">def</span> <span class="nf">circumcenter</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The circumcenter of the triangle</span>
-
-<span class="sd">        The circumcenter is the center of the circumcircle.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        circumcenter : Point</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Point, Triangle</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)</span>
-<span class="sd">        &gt;&gt;&gt; t = Triangle(p1, p2, p3)</span>
-<span class="sd">        &gt;&gt;&gt; t.circumcenter</span>
-<span class="sd">        Point(1/2, 1/2)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="o">.</span><span class="n">perpendicular_bisector</span><span class="p">()</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">sides</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">a</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">b</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Triangle.circumradius"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Triangle.circumradius">[docs]</a>    <span class="k">def</span> <span class="nf">circumradius</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The radius of the circumcircle of the triangle.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        circumradius : number of Basic instance</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.ellipse.Circle.radius</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Point, Triangle</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, a)</span>
-<span class="sd">        &gt;&gt;&gt; t = Triangle(p1, p2, p3)</span>
-<span class="sd">        &gt;&gt;&gt; t.circumradius</span>
-<span class="sd">        sqrt(a**2/4 + 1/4)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Point</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">circumcenter</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">vertices</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Triangle.circumcircle"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Triangle.circumcircle">[docs]</a>    <span class="k">def</span> <span class="nf">circumcircle</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The circle which passes through the three vertices of the triangle.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        circumcircle : Circle</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.ellipse.Circle</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Point, Triangle</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)</span>
-<span class="sd">        &gt;&gt;&gt; t = Triangle(p1, p2, p3)</span>
-<span class="sd">        &gt;&gt;&gt; t.circumcircle</span>
-<span class="sd">        Circle(Point(1/2, 1/2), sqrt(2)/2)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Circle</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">circumcenter</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">circumradius</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Triangle.bisectors"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Triangle.bisectors">[docs]</a>    <span class="k">def</span> <span class="nf">bisectors</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The angle bisectors of the triangle.</span>
-
-<span class="sd">        An angle bisector of a triangle is a straight line through a vertex</span>
-<span class="sd">        which cuts the corresponding angle in half.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        bisectors : dict</span>
-<span class="sd">            Each key is a vertex (Point) and each value is the corresponding</span>
-<span class="sd">            bisector (Segment).</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point, sympy.geometry.line.Segment</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Point, Triangle, Segment</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)</span>
-<span class="sd">        &gt;&gt;&gt; t = Triangle(p1, p2, p3)</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import sqrt</span>
-<span class="sd">        &gt;&gt;&gt; t.bisectors()[p2] == Segment(Point(0, sqrt(2) - 1), Point(1, 0))</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sides</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">vertices</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">incenter</span>
-        <span class="n">l1</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Line</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">c</span><span class="p">)</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="p">])[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="n">l2</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">Line</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="p">)</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="mi">2</span><span class="p">])[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="n">l3</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">Line</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">c</span><span class="p">)</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="mi">0</span><span class="p">])[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">return</span> <span class="p">{</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span> <span class="n">l1</span><span class="p">,</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span> <span class="n">l2</span><span class="p">,</span> <span class="n">v</span><span class="p">[</span><span class="mi">2</span><span class="p">]:</span> <span class="n">l3</span><span class="p">}</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Triangle.incenter"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Triangle.incenter">[docs]</a>    <span class="k">def</span> <span class="nf">incenter</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The center of the incircle.</span>
-
-<span class="sd">        The incircle is the circle which lies inside the triangle and touches</span>
-<span class="sd">        all three sides.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        incenter : Point</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        incircle, sympy.geometry.point.Point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Point, Triangle</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)</span>
-<span class="sd">        &gt;&gt;&gt; t = Triangle(p1, p2, p3)</span>
-<span class="sd">        &gt;&gt;&gt; t.incenter</span>
-<span class="sd">        Point(-sqrt(2)/2 + 1, -sqrt(2)/2 + 1)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sides</span>
-        <span class="n">l</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">length</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">l</span><span class="p">)</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">vertices</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">l</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">Matrix</span><span class="p">([</span><span class="n">vi</span><span class="o">.</span><span class="n">x</span> <span class="k">for</span> <span class="n">vi</span> <span class="ow">in</span> <span class="n">v</span><span class="p">]))</span><span class="o">/</span><span class="n">p</span><span class="p">)</span>
-        <span class="n">y</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">l</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">Matrix</span><span class="p">([</span><span class="n">vi</span><span class="o">.</span><span class="n">y</span> <span class="k">for</span> <span class="n">vi</span> <span class="ow">in</span> <span class="n">v</span><span class="p">]))</span><span class="o">/</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Triangle.inradius"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Triangle.inradius">[docs]</a>    <span class="k">def</span> <span class="nf">inradius</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The radius of the incircle.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        inradius : number of Basic instance</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        incircle, sympy.geometry.ellipse.Circle.radius</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Point, Triangle</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(4, 0), Point(0, 3)</span>
-<span class="sd">        &gt;&gt;&gt; t = Triangle(p1, p2, p3)</span>
-<span class="sd">        &gt;&gt;&gt; t.inradius</span>
-<span class="sd">        1</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">simplify</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">area</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">perimeter</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Triangle.incircle"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Triangle.incircle">[docs]</a>    <span class="k">def</span> <span class="nf">incircle</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The incircle of the triangle.</span>
-
-<span class="sd">        The incircle is the circle which lies inside the triangle and touches</span>
-<span class="sd">        all three sides.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        incircle : Circle</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.ellipse.Circle</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Point, Triangle</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(2, 0), Point(0, 2)</span>
-<span class="sd">        &gt;&gt;&gt; t = Triangle(p1, p2, p3)</span>
-<span class="sd">        &gt;&gt;&gt; t.incircle</span>
-<span class="sd">        Circle(Point(-sqrt(2) + 2, -sqrt(2) + 2), -sqrt(2) + 2)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Circle</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">incenter</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">inradius</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Triangle.medians"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Triangle.medians">[docs]</a>    <span class="k">def</span> <span class="nf">medians</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The medians of the triangle.</span>
-
-<span class="sd">        A median of a triangle is a straight line through a vertex and the</span>
-<span class="sd">        midpoint of the opposite side, and divides the triangle into two</span>
-<span class="sd">        equal areas.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        medians : dict</span>
-<span class="sd">            Each key is a vertex (Point) and each value is the median (Segment)</span>
-<span class="sd">            at that point.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.point.Point.midpoint, sympy.geometry.line.Segment.midpoint</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Point, Triangle</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)</span>
-<span class="sd">        &gt;&gt;&gt; t = Triangle(p1, p2, p3)</span>
-<span class="sd">        &gt;&gt;&gt; t.medians[p1]</span>
-<span class="sd">        Segment(Point(0, 0), Point(1/2, 1/2))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sides</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">vertices</span>
-        <span class="k">return</span> <span class="p">{</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span> <span class="n">Segment</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">midpoint</span><span class="p">,</span> <span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span>
-                <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span> <span class="n">Segment</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">midpoint</span><span class="p">,</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span>
-                <span class="n">v</span><span class="p">[</span><span class="mi">2</span><span class="p">]:</span> <span class="n">Segment</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">midpoint</span><span class="p">,</span> <span class="n">v</span><span class="p">[</span><span class="mi">2</span><span class="p">])}</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Triangle.medial"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.polygon.Triangle.medial">[docs]</a>    <span class="k">def</span> <span class="nf">medial</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The medial triangle of the triangle.</span>
-
-<span class="sd">        The triangle which is formed from the midpoints of the three sides.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        medial : Triangle</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.geometry.line.Segment.midpoint</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.geometry import Point, Triangle</span>
-<span class="sd">        &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)</span>
-<span class="sd">        &gt;&gt;&gt; t = Triangle(p1, p2, p3)</span>
-<span class="sd">        &gt;&gt;&gt; t.medial</span>
-<span class="sd">        Triangle(Point(1/2, 0), Point(1/2, 1/2), Point(0, 1/2))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sides</span>
-        <span class="k">return</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">midpoint</span><span class="p">,</span> <span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">midpoint</span><span class="p">,</span> <span class="n">s</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">midpoint</span><span class="p">)</span>
-
-    <span class="c">#@property</span>
-    <span class="c">#def excircles(self):</span>
-    <span class="c">#    &quot;&quot;&quot;Returns a list of the three excircles for this triangle.&quot;&quot;&quot;</span>
-    <span class="c">#    pass</span>
-
-</div></div>
-<span class="k">def</span> <span class="nf">rad</span><span class="p">(</span><span class="n">d</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return the radian value for the given degrees (pi = 180 degrees).&quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">d</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">180</span>
-
-
-<span class="k">def</span> <span class="nf">deg</span><span class="p">(</span><span class="n">r</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return the degree value for the given radians (pi = 180 degrees).&quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">r</span><span class="o">/</span><span class="n">pi</span><span class="o">*</span><span class="mi">180</span>
-
-
-<span class="k">def</span> <span class="nf">_slope</span><span class="p">(</span><span class="n">d</span><span class="p">):</span>
-    <span class="n">rv</span> <span class="o">=</span> <span class="n">tan</span><span class="p">(</span><span class="n">rad</span><span class="p">(</span><span class="n">d</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">rv</span>
-
-
-<span class="k">def</span> <span class="nf">_asa</span><span class="p">(</span><span class="n">d1</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="n">d2</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return triangle having side with length l on the x-axis.&quot;&quot;&quot;</span>
-    <span class="n">xy</span> <span class="o">=</span> <span class="n">Line</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">slope</span><span class="o">=</span><span class="n">_slope</span><span class="p">(</span><span class="n">d1</span><span class="p">))</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span>
-        <span class="n">Line</span><span class="p">((</span><span class="n">l</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">slope</span><span class="o">=</span><span class="n">_slope</span><span class="p">(</span><span class="mi">180</span> <span class="o">-</span> <span class="n">d2</span><span class="p">)))[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">Triangle</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">xy</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">_sss</span><span class="p">(</span><span class="n">l1</span><span class="p">,</span> <span class="n">l2</span><span class="p">,</span> <span class="n">l3</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return triangle having side of length l1 on the x-axis.&quot;&quot;&quot;</span>
-    <span class="n">c1</span> <span class="o">=</span> <span class="n">Circle</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">l3</span><span class="p">)</span>
-    <span class="n">c2</span> <span class="o">=</span> <span class="n">Circle</span><span class="p">((</span><span class="n">l1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">l2</span><span class="p">)</span>
-    <span class="n">inter</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">c1</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">c2</span><span class="p">)</span> <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">y</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">]</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">inter</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-    <span class="n">pt</span> <span class="o">=</span> <span class="n">inter</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">Triangle</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="n">l1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">pt</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">_sas</span><span class="p">(</span><span class="n">l1</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">l2</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return triangle having side with length l2 on the x-axis.&quot;&quot;&quot;</span>
-    <span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-    <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">l2</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-    <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">rad</span><span class="p">(</span><span class="n">d</span><span class="p">))</span><span class="o">*</span><span class="n">l1</span><span class="p">,</span> <span class="n">sin</span><span class="p">(</span><span class="n">rad</span><span class="p">(</span><span class="n">d</span><span class="p">))</span><span class="o">*</span><span class="n">l1</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">)</span>
-</pre></div>
-
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-  <h1>Source code for sympy.geometry.util</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Utility functions for geometrical entities.</span>
-
-<span class="sd">Contains</span>
-<span class="sd">========</span>
-<span class="sd">intersection</span>
-<span class="sd">convex_hull</span>
-<span class="sd">are_similar</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Function</span><span class="p">,</span> <span class="n">solve</span>
-
-
-<span class="k">def</span> <span class="nf">idiff</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">dep</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return dy/dx assuming that y and any other variables given in dep</span>
-<span class="sd">    depend on x.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, a</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.geometry.util import idiff</span>
-
-<span class="sd">    &gt;&gt;&gt; idiff(x**2 + y**2 - 4, y, x)</span>
-<span class="sd">    -x/y</span>
-<span class="sd">    &gt;&gt;&gt; idiff(x + a + y, y, x)</span>
-<span class="sd">    -1</span>
-<span class="sd">    &gt;&gt;&gt; idiff(x + a + y, y, x, [a])</span>
-<span class="sd">    -Derivative(a, x) - 1</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.core.function.Derivative</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">dep</span><span class="p">:</span>
-        <span class="n">dep</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">dep</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">dep</span><span class="p">)</span>
-    <span class="n">dep</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-
-    <span class="n">f</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">([(</span><span class="n">s</span><span class="p">,</span> <span class="n">Function</span><span class="p">(</span>
-        <span class="n">s</span><span class="o">.</span><span class="n">name</span><span class="p">)(</span><span class="n">x</span><span class="p">))</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">eq</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Symbol</span><span class="p">)</span> <span class="k">if</span> <span class="n">s</span> <span class="o">!=</span> <span class="n">x</span> <span class="ow">and</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">dep</span><span class="p">])</span>
-    <span class="n">dydx</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="n">y</span><span class="o">.</span><span class="n">name</span><span class="p">)(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">solve</span><span class="p">(</span><span class="n">eq</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">dydx</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span>
-        <span class="p">[(</span><span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">items</span><span class="p">()])</span>
-
-
-<span class="k">def</span> <span class="nf">_symbol</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">matching_symbol</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return s if s is a Symbol, else return either a new Symbol (real=True)</span>
-<span class="sd">    with the same name s or the matching_symbol if s is a string and it matches</span>
-<span class="sd">    the name of the matching_symbol.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.geometry.util import _symbol</span>
-<span class="sd">    &gt;&gt;&gt; x = Symbol(&#39;x&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; _symbol(&#39;y&#39;)</span>
-<span class="sd">    y</span>
-<span class="sd">    &gt;&gt;&gt; _.is_real</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; _symbol(x)</span>
-<span class="sd">    x</span>
-<span class="sd">    &gt;&gt;&gt; _.is_real is None</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; arb = Symbol(&#39;foo&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; _symbol(&#39;arb&#39;, arb) # arb&#39;s name is foo so foo will not be returned</span>
-<span class="sd">    arb</span>
-<span class="sd">    &gt;&gt;&gt; _symbol(&#39;foo&#39;, arb) # now it will</span>
-<span class="sd">    foo</span>
-
-<span class="sd">    NB: the symbol here may not be the same as a symbol with the same</span>
-<span class="sd">    name defined elsewhere as a result of different assumptions.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.core.symbol.Symbol</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">matching_symbol</span> <span class="ow">and</span> <span class="n">matching_symbol</span><span class="o">.</span><span class="n">name</span> <span class="o">==</span> <span class="n">s</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">matching_symbol</span>
-        <span class="k">return</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">s</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;symbol must be string for symbol name or Symbol&#39;</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="intersection"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.util.intersection">[docs]</a><span class="k">def</span> <span class="nf">intersection</span><span class="p">(</span><span class="o">*</span><span class="n">entities</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The intersection of a collection of GeometryEntity instances.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    entities : sequence of GeometryEntity</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    intersection : list of GeometryEntity</span>
-
-<span class="sd">    Raises</span>
-<span class="sd">    ======</span>
-
-<span class="sd">    NotImplementedError</span>
-<span class="sd">        When unable to calculate intersection.</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    The intersection of any geometrical entity with itself should return</span>
-<span class="sd">    a list with one item: the entity in question.</span>
-<span class="sd">    An intersection requires two or more entities. If only a single</span>
-<span class="sd">    entity is given then the function will return an empty list.</span>
-<span class="sd">    It is possible for `intersection` to miss intersections that one</span>
-<span class="sd">    knows exists because the required quantities were not fully</span>
-<span class="sd">    simplified internally.</span>
-<span class="sd">    Reals should be converted to Rationals, e.g. Rational(str(real_num))</span>
-<span class="sd">    or else failures due to floating point issues may result.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.geometry.entity.GeometryEntity.intersection</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.geometry import Point, Line, Circle, intersection</span>
-<span class="sd">    &gt;&gt;&gt; p1, p2, p3 = Point(0, 0), Point(1, 1), Point(-1, 5)</span>
-<span class="sd">    &gt;&gt;&gt; l1, l2 = Line(p1, p2), Line(p3, p2)</span>
-<span class="sd">    &gt;&gt;&gt; c = Circle(p2, 1)</span>
-<span class="sd">    &gt;&gt;&gt; intersection(l1, p2)</span>
-<span class="sd">    [Point(1, 1)]</span>
-<span class="sd">    &gt;&gt;&gt; intersection(l1, l2)</span>
-<span class="sd">    [Point(1, 1)]</span>
-<span class="sd">    &gt;&gt;&gt; intersection(c, p2)</span>
-<span class="sd">    []</span>
-<span class="sd">    &gt;&gt;&gt; intersection(c, Point(1, 0))</span>
-<span class="sd">    [Point(1, 0)]</span>
-<span class="sd">    &gt;&gt;&gt; intersection(c, l2)</span>
-<span class="sd">    [Point(-sqrt(5)/5 + 1, 2*sqrt(5)/5 + 1),</span>
-<span class="sd">     Point(sqrt(5)/5 + 1, -2*sqrt(5)/5 + 1)]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">.entity</span> <span class="kn">import</span> <span class="n">GeometryEntity</span>
-    <span class="kn">from</span> <span class="nn">.point</span> <span class="kn">import</span> <span class="n">Point</span>
-
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">entities</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">entities</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">GeometryEntity</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">entities</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> is not a GeometryEntity and cannot be made into Point&#39;</span> <span class="o">%</span> <span class="nb">str</span><span class="p">(</span><span class="n">e</span><span class="p">))</span>
-
-    <span class="n">res</span> <span class="o">=</span> <span class="n">entities</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">entities</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-    <span class="k">for</span> <span class="n">entity</span> <span class="ow">in</span> <span class="n">entities</span><span class="p">[</span><span class="mi">2</span><span class="p">:]:</span>
-        <span class="n">newres</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">res</span><span class="p">:</span>
-            <span class="n">newres</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">entity</span><span class="p">))</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="n">newres</span>
-    <span class="k">return</span> <span class="n">res</span>
-
-</div>
-<div class="viewcode-block" id="convex_hull"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.util.convex_hull">[docs]</a><span class="k">def</span> <span class="nf">convex_hull</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The convex hull surrounding the Points contained in the list of entities.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    args : a collection of Points, Segments and/or Polygons</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    convex_hull : Polygon</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    This can only be performed on a set of non-symbolic points.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Graham_scan</span>
-
-<span class="sd">    [2] Andrew&#39;s Monotone Chain Algorithm</span>
-<span class="sd">    (A.M. Andrew,</span>
-<span class="sd">    &quot;Another Efficient Algorithm for Convex Hulls in Two Dimensions&quot;, 1979)</span>
-<span class="sd">    http://softsurfer.com/Archive/algorithm_0109/algorithm_0109.htm</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.geometry.point.Point, sympy.geometry.polygon.Polygon</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.geometry import Point, convex_hull</span>
-<span class="sd">    &gt;&gt;&gt; points = [(1,1), (1,2), (3,1), (-5,2), (15,4)]</span>
-<span class="sd">    &gt;&gt;&gt; convex_hull(*points)</span>
-<span class="sd">    Polygon(Point(-5, 2), Point(1, 1), Point(3, 1), Point(15, 4))</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">.entity</span> <span class="kn">import</span> <span class="n">GeometryEntity</span>
-    <span class="kn">from</span> <span class="nn">.point</span> <span class="kn">import</span> <span class="n">Point</span>
-    <span class="kn">from</span> <span class="nn">.line</span> <span class="kn">import</span> <span class="n">Segment</span>
-    <span class="kn">from</span> <span class="nn">.polygon</span> <span class="kn">import</span> <span class="n">Polygon</span>
-
-    <span class="n">p</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">GeometryEntity</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">e</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> is not a GeometryEntity and cannot be made into Point&#39;</span> <span class="o">%</span> <span class="nb">str</span><span class="p">(</span><span class="n">e</span><span class="p">))</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Point</span><span class="p">):</span>
-            <span class="n">p</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Segment</span><span class="p">):</span>
-            <span class="n">p</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">points</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Polygon</span><span class="p">):</span>
-            <span class="n">p</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">vertices</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                <span class="s">&#39;Convex hull for </span><span class="si">%s</span><span class="s"> not implemented.&#39;</span> <span class="o">%</span> <span class="nb">type</span><span class="p">(</span><span class="n">e</span><span class="p">))</span>
-
-    <span class="n">p</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">Segment</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_orientation</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">r</span><span class="p">):</span>
-        <span class="sd">&#39;&#39;&#39;Return positive if p-q-r are clockwise, neg if ccw, zero if</span>
-<span class="sd">        collinear.&#39;&#39;&#39;</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">y</span> <span class="o">-</span> <span class="n">p</span><span class="o">.</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">r</span><span class="o">.</span><span class="n">x</span> <span class="o">-</span> <span class="n">p</span><span class="o">.</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">x</span> <span class="o">-</span> <span class="n">p</span><span class="o">.</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">r</span><span class="o">.</span><span class="n">y</span> <span class="o">-</span> <span class="n">p</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-
-    <span class="c"># scan to find upper and lower convex hulls of a set of 2d points.</span>
-    <span class="n">U</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">L</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">p</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">p_i</span> <span class="ow">in</span> <span class="n">p</span><span class="p">:</span>
-        <span class="k">while</span> <span class="nb">len</span><span class="p">(</span><span class="n">U</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">_orientation</span><span class="p">(</span><span class="n">U</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">U</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">p_i</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">U</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-        <span class="k">while</span> <span class="nb">len</span><span class="p">(</span><span class="n">L</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">_orientation</span><span class="p">(</span><span class="n">L</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">L</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">p_i</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">L</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-        <span class="n">U</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">p_i</span><span class="p">)</span>
-        <span class="n">L</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">p_i</span><span class="p">)</span>
-    <span class="n">U</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-    <span class="n">convexHull</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">L</span> <span class="o">+</span> <span class="n">U</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">convexHull</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">Segment</span><span class="p">(</span><span class="n">convexHull</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">convexHull</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-    <span class="k">return</span> <span class="n">Polygon</span><span class="p">(</span><span class="o">*</span><span class="n">convexHull</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="are_similar"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.util.are_similar">[docs]</a><span class="k">def</span> <span class="nf">are_similar</span><span class="p">(</span><span class="n">e1</span><span class="p">,</span> <span class="n">e2</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Are two geometrical entities similar.</span>
-
-<span class="sd">    Can one geometrical entity be uniformly scaled to the other?</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    e1 : GeometryEntity</span>
-<span class="sd">    e2 : GeometryEntity</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    are_similar : boolean</span>
-
-<span class="sd">    Raises</span>
-<span class="sd">    ======</span>
-
-<span class="sd">    GeometryError</span>
-<span class="sd">        When `e1` and `e2` cannot be compared.</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    If the two objects are equal then they are similar.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.geometry.entity.GeometryEntity.is_similar</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Point, Circle, Triangle, are_similar</span>
-<span class="sd">    &gt;&gt;&gt; c1, c2 = Circle(Point(0, 0), 4), Circle(Point(1, 4), 3)</span>
-<span class="sd">    &gt;&gt;&gt; t1 = Triangle(Point(0, 0), Point(1, 0), Point(0, 1))</span>
-<span class="sd">    &gt;&gt;&gt; t2 = Triangle(Point(0, 0), Point(2, 0), Point(0, 2))</span>
-<span class="sd">    &gt;&gt;&gt; t3 = Triangle(Point(0, 0), Point(3, 0), Point(0, 1))</span>
-<span class="sd">    &gt;&gt;&gt; are_similar(t1, t2)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; are_similar(t1, t3)</span>
-<span class="sd">    False</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">.exceptions</span> <span class="kn">import</span> <span class="n">GeometryError</span>
-
-    <span class="k">if</span> <span class="n">e1</span> <span class="o">==</span> <span class="n">e2</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">True</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">e1</span><span class="o">.</span><span class="n">is_similar</span><span class="p">(</span><span class="n">e2</span><span class="p">)</span>
-    <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">e2</span><span class="o">.</span><span class="n">is_similar</span><span class="p">(</span><span class="n">e1</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-            <span class="n">n1</span> <span class="o">=</span> <span class="n">e1</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span>
-            <span class="n">n2</span> <span class="o">=</span> <span class="n">e2</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span>
-            <span class="k">raise</span> <span class="n">GeometryError</span><span class="p">(</span>
-                <span class="s">&quot;Cannot test similarity between </span><span class="si">%s</span><span class="s"> and </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="centroid"><a class="viewcode-back" href="../../../modules/geometry.html#sympy.geometry.util.centroid">[docs]</a><span class="k">def</span> <span class="nf">centroid</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Find the centroid (center of mass) of the collection containing only Points,</span>
-<span class="sd">    Segments or Polygons. The centroid is the weighted average of the individual centroid</span>
-<span class="sd">    where the weights are the lengths (of segments) or areas (of polygons).</span>
-<span class="sd">    Overlapping regions will add to the weight of that region.</span>
-
-<span class="sd">    If there are no objects (or a mixture of objects) then None is returned.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.geometry.point.Point, sympy.geometry.line.Segment,</span>
-<span class="sd">    sympy.geometry.polygon.Polygon</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Point, Segment, Polygon</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.geometry.util import centroid</span>
-<span class="sd">    &gt;&gt;&gt; p = Polygon((0, 0), (10, 0), (10, 10))</span>
-<span class="sd">    &gt;&gt;&gt; q = p.translate(0, 20)</span>
-<span class="sd">    &gt;&gt;&gt; p.centroid, q.centroid</span>
-<span class="sd">    (Point(20/3, 10/3), Point(20/3, 70/3))</span>
-<span class="sd">    &gt;&gt;&gt; centroid(p, q)</span>
-<span class="sd">    Point(20/3, 40/3)</span>
-<span class="sd">    &gt;&gt;&gt; p, q = Segment((0, 0), (2, 0)), Segment((0, 0), (2, 2))</span>
-<span class="sd">    &gt;&gt;&gt; centroid(p, q)</span>
-<span class="sd">    Point(1, -sqrt(2) + 2)</span>
-<span class="sd">    &gt;&gt;&gt; centroid(Point(0, 0), Point(2, 0))</span>
-<span class="sd">    Point(1, 0)</span>
-
-<span class="sd">    Stacking 3 polygons on top of each other effectively triples the</span>
-<span class="sd">    weight of that polygon:</span>
-
-<span class="sd">        &gt;&gt;&gt; p = Polygon((0, 0), (1, 0), (1, 1), (0, 1))</span>
-<span class="sd">        &gt;&gt;&gt; q = Polygon((1, 0), (3, 0), (3, 1), (1, 1))</span>
-<span class="sd">        &gt;&gt;&gt; centroid(p, q)</span>
-<span class="sd">        Point(3/2, 1/2)</span>
-<span class="sd">        &gt;&gt;&gt; centroid(p, p, p, q) # centroid x-coord shifts left</span>
-<span class="sd">        Point(11/10, 1/2)</span>
-
-<span class="sd">    Stacking the squares vertically above and below p has the same</span>
-<span class="sd">    effect:</span>
-
-<span class="sd">        &gt;&gt;&gt; centroid(p, p.translate(0, 1), p.translate(0, -1), q)</span>
-<span class="sd">        Point(11/10, 1/2)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Polygon</span><span class="p">,</span> <span class="n">Segment</span><span class="p">,</span> <span class="n">Point</span>
-    <span class="k">if</span> <span class="n">args</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">Point</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">args</span><span class="p">):</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-                <span class="n">c</span> <span class="o">+=</span> <span class="n">g</span>
-            <span class="k">return</span> <span class="n">c</span><span class="o">/</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">all</span><span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">Segment</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">args</span><span class="p">):</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-            <span class="n">L</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-                <span class="n">l</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">length</span>
-                <span class="n">c</span> <span class="o">+=</span> <span class="n">g</span><span class="o">.</span><span class="n">midpoint</span><span class="o">*</span><span class="n">l</span>
-                <span class="n">L</span> <span class="o">+=</span> <span class="n">l</span>
-            <span class="k">return</span> <span class="n">c</span><span class="o">/</span><span class="n">L</span>
-        <span class="k">elif</span> <span class="nb">all</span><span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">Polygon</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">args</span><span class="p">):</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-            <span class="n">A</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-                <span class="n">a</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">area</span>
-                <span class="n">c</span> <span class="o">+=</span> <span class="n">g</span><span class="o">.</span><span class="n">centroid</span><span class="o">*</span><span class="n">a</span>
-                <span class="n">A</span> <span class="o">+=</span> <span class="n">a</span>
-            <span class="k">return</span> <span class="n">c</span><span class="o">/</span><span class="n">A</span></div>
-</pre></div>
-
-          </div>
-        </div>
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-            
-  <h1>Source code for sympy.integrals</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Integration functions that integrates a sympy expression.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    --------</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import integrate, sin</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; integrate(1/x,x)</span>
-<span class="sd">    log(x)</span>
-<span class="sd">    &gt;&gt;&gt; integrate(sin(x),x)</span>
-<span class="sd">    -cos(x)</span>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">.integrals</span> <span class="kn">import</span> <span class="n">integrate</span><span class="p">,</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">line_integrate</span>
-<span class="kn">from</span> <span class="nn">.transforms</span> <span class="kn">import</span> <span class="p">(</span><span class="n">mellin_transform</span><span class="p">,</span> <span class="n">inverse_mellin_transform</span><span class="p">,</span>
-                        <span class="n">MellinTransform</span><span class="p">,</span> <span class="n">InverseMellinTransform</span><span class="p">,</span>
-                        <span class="n">laplace_transform</span><span class="p">,</span> <span class="n">inverse_laplace_transform</span><span class="p">,</span>
-                        <span class="n">LaplaceTransform</span><span class="p">,</span> <span class="n">InverseLaplaceTransform</span><span class="p">,</span>
-                        <span class="n">fourier_transform</span><span class="p">,</span> <span class="n">inverse_fourier_transform</span><span class="p">,</span>
-                        <span class="n">FourierTransform</span><span class="p">,</span> <span class="n">InverseFourierTransform</span><span class="p">,</span>
-                        <span class="n">sine_transform</span><span class="p">,</span> <span class="n">inverse_sine_transform</span><span class="p">,</span>
-                        <span class="n">SineTransform</span><span class="p">,</span> <span class="n">InverseSineTransform</span><span class="p">,</span>
-                        <span class="n">cosine_transform</span><span class="p">,</span> <span class="n">inverse_cosine_transform</span><span class="p">,</span>
-                        <span class="n">CosineTransform</span><span class="p">,</span> <span class="n">InverseCosineTransform</span><span class="p">,</span>
-                        <span class="n">hankel_transform</span><span class="p">,</span> <span class="n">inverse_hankel_transform</span><span class="p">,</span>
-                        <span class="n">HankelTransform</span><span class="p">,</span> <span class="n">InverseHankelTransform</span><span class="p">)</span>
-</pre></div>
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-  <h1>Source code for sympy.integrals.transforms</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot; Integral Transforms &quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">sympy.integrals</span> <span class="kn">import</span> <span class="n">integrate</span><span class="p">,</span> <span class="n">Integral</span>
-<span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">oo</span>
-<span class="kn">from</span> <span class="nn">sympy.core.symbol</span> <span class="kn">import</span> <span class="n">Dummy</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Function</span>
-<span class="kn">from</span> <span class="nn">sympy.logic.boolalg</span> <span class="kn">import</span> <span class="n">to_cnf</span><span class="p">,</span> <span class="n">conjuncts</span><span class="p">,</span> <span class="n">disjuncts</span><span class="p">,</span> <span class="n">Or</span><span class="p">,</span> <span class="n">And</span>
-<span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">simplify</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span>
-
-<span class="kn">from</span> <span class="nn">sympy.integrals.meijerint</span> <span class="kn">import</span> <span class="n">_dummy</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-<span class="c">##########################################################################</span>
-<span class="c"># Helpers / Utilities</span>
-<span class="c">##########################################################################</span>
-
-
-<span class="k">class</span> <span class="nc">IntegralTransformError</span><span class="p">(</span><span class="ne">NotImplementedError</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Exception raised in relation to problems computing transforms.</span>
-
-<span class="sd">    This class is mostly used internally; if integrals cannot be computed</span>
-<span class="sd">    objects representing unevaluated transforms are usually returned.</span>
-
-<span class="sd">    The hint ``needeval=True`` can be used to disable returning transform</span>
-<span class="sd">    objects, and instead raise this exception if an integral cannot be</span>
-<span class="sd">    computed.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">transform</span><span class="p">,</span> <span class="n">function</span><span class="p">,</span> <span class="n">msg</span><span class="p">):</span>
-        <span class="nb">super</span><span class="p">(</span><span class="n">IntegralTransformError</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span>
-            <span class="s">&quot;</span><span class="si">%s</span><span class="s"> Transform could not be computed: </span><span class="si">%s</span><span class="s">.&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">transform</span><span class="p">,</span> <span class="n">msg</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">function</span> <span class="o">=</span> <span class="n">function</span>
-
-
-<span class="k">class</span> <span class="nc">IntegralTransform</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Base class for integral transforms.</span>
-
-<span class="sd">    This class represents unevaluated transforms.</span>
-
-<span class="sd">    To implement a concrete transform, derive from this class and implement</span>
-<span class="sd">    the _compute_transform(f, x, s, **hints) and _as_integral(f, x, s)</span>
-<span class="sd">    functions. If the transform cannot be computed, raise IntegralTransformError.</span>
-
-<span class="sd">    Also set cls._name.</span>
-
-<span class="sd">    Implement self._collapse_extra if your function returns more than just a</span>
-<span class="sd">    number and possibly a convergence condition.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">3</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">function</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; The function to be transformed. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">function_variable</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; The dependent variable of the function to be transformed. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">transform_variable</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; The independent transform variable. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">free_symbols</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        This method returns the symbols that will exist when the transform</span>
-<span class="sd">        is evaluated.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="o">.</span><span class="n">free_symbols</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="nb">set</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">transform_variable</span><span class="p">]))</span> \
-            <span class="o">-</span> <span class="nb">set</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">function_variable</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_compute_transform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">def</span> <span class="nf">_as_integral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">def</span> <span class="nf">_collapse_extra</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">extra</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">And</span>
-        <span class="n">cond</span> <span class="o">=</span> <span class="n">And</span><span class="p">(</span><span class="o">*</span><span class="n">extra</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">cond</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">IntegralTransformError</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Try to evaluate the transform in closed form.</span>
-
-<span class="sd">        This general function handles linearity, but apart from that leaves</span>
-<span class="sd">        pretty much everything to _compute_transform.</span>
-
-<span class="sd">        Standard hints are the following:</span>
-
-<span class="sd">        - ``simplify``: whether or not to simplify the result</span>
-<span class="sd">        - ``noconds``: if True, don&#39;t return convergence conditions</span>
-<span class="sd">        - ``needeval``: if True, raise IntegralTransformError instead of</span>
-<span class="sd">                        returning IntegralTransform objects</span>
-
-<span class="sd">        The default values of these hints depend on the concrete transform,</span>
-<span class="sd">        usually the default is</span>
-<span class="sd">        ``(simplify, noconds, needeval) = (True, False, False)``.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Add</span><span class="p">,</span> <span class="n">expand_mul</span><span class="p">,</span> <span class="n">Mul</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">AppliedUndef</span>
-        <span class="n">needeval</span> <span class="o">=</span> <span class="n">hints</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;needeval&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-        <span class="n">try_directly</span> <span class="o">=</span> <span class="ow">not</span> <span class="nb">any</span><span class="p">(</span><span class="n">func</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">function_variable</span><span class="p">)</span>
-                               <span class="k">for</span> <span class="n">func</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">AppliedUndef</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">try_directly</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_compute_transform</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="p">,</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">function_variable</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">transform_variable</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">IntegralTransformError</span><span class="p">:</span>
-                <span class="k">pass</span>
-
-        <span class="n">fn</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">function</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">fn</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">fn</span> <span class="o">=</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">fn</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">fn</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">hints</span><span class="p">[</span><span class="s">&#39;needeval&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">needeval</span>
-            <span class="n">res</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="o">*</span><span class="p">([</span><span class="n">x</span><span class="p">]</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:])))</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-                   <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">fn</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-            <span class="n">extra</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">ress</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">res</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span>
-                    <span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="p">]</span>
-                <span class="n">ress</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">extra</span> <span class="o">+=</span> <span class="p">[</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">:]]</span>
-            <span class="n">res</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">ress</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">extra</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">res</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">extra</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_collapse_extra</span><span class="p">(</span><span class="n">extra</span><span class="p">)</span>
-                <span class="k">return</span> <span class="nb">tuple</span><span class="p">([</span><span class="n">res</span><span class="p">])</span> <span class="o">+</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">extra</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">IntegralTransformError</span><span class="p">:</span>
-                <span class="k">pass</span>
-
-        <span class="k">if</span> <span class="n">needeval</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">IntegralTransformError</span><span class="p">(</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">_name</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="p">,</span> <span class="s">&#39;needeval&#39;</span><span class="p">)</span>
-
-        <span class="c"># TODO handle derivatives etc</span>
-
-        <span class="c"># pull out constant coefficients</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">rest</span> <span class="o">=</span> <span class="n">fn</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">function_variable</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">coeff</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="o">*</span><span class="p">([</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">rest</span><span class="p">)]</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:])))</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">as_integral</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_as_integral</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">function_variable</span><span class="p">,</span>
-                                 <span class="bp">self</span><span class="o">.</span><span class="n">transform_variable</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_Integral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_integral</span>
-
-<span class="kn">from</span> <span class="nn">sympy.solvers.inequalities</span> <span class="kn">import</span> <span class="n">_solve_inequality</span>
-
-
-<span class="k">def</span> <span class="nf">_simplify</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">doit</span><span class="p">):</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">powdenest</span><span class="p">,</span> <span class="n">piecewise_fold</span>
-    <span class="k">if</span> <span class="n">doit</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">simplify</span><span class="p">(</span><span class="n">powdenest</span><span class="p">(</span><span class="n">piecewise_fold</span><span class="p">(</span><span class="n">expr</span><span class="p">),</span> <span class="n">polar</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">expr</span>
-
-
-<span class="k">def</span> <span class="nf">_noconds_</span><span class="p">(</span><span class="n">default</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    This is a decorator generator for dropping convergence conditions.</span>
-
-<span class="sd">    Suppose you define a function ``transform(*args)`` which returns a tuple of</span>
-<span class="sd">    the form ``(result, cond1, cond2, ...)``.</span>
-
-<span class="sd">    Decorating it ``@_noconds_(default)`` will add a new keyword argument</span>
-<span class="sd">    ``noconds`` to it. If ``noconds=True``, the return value will be altered to</span>
-<span class="sd">    be only ``result``, whereas if ``noconds=False`` the return value will not</span>
-<span class="sd">    be altered.</span>
-
-<span class="sd">    The default value of the ``noconds`` keyword will be ``default`` (i.e. the</span>
-<span class="sd">    argument of this function).</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">make_wrapper</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.decorators</span> <span class="kn">import</span> <span class="n">wraps</span>
-
-        <span class="nd">@wraps</span><span class="p">(</span><span class="n">func</span><span class="p">)</span>
-        <span class="k">def</span> <span class="nf">wrapper</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-            <span class="n">noconds</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;noconds&#39;</span><span class="p">,</span> <span class="n">default</span><span class="p">)</span>
-            <span class="n">res</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">noconds</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">res</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">res</span>
-        <span class="k">return</span> <span class="n">wrapper</span>
-    <span class="k">return</span> <span class="n">make_wrapper</span>
-<span class="n">_noconds</span> <span class="o">=</span> <span class="n">_noconds_</span><span class="p">(</span><span class="bp">False</span><span class="p">)</span>
-
-
-<span class="c">##########################################################################</span>
-<span class="c"># Mellin Transform</span>
-<span class="c">##########################################################################</span>
-
-<span class="k">def</span> <span class="nf">_default_integrator</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">integrate</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-
-
-<span class="nd">@_noconds</span>
-<span class="k">def</span> <span class="nf">_mellin_transform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">s_</span><span class="p">,</span> <span class="n">integrator</span><span class="o">=</span><span class="n">_default_integrator</span><span class="p">,</span> <span class="n">simplify</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Backend function to compute Mellin transforms. &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">re</span><span class="p">,</span> <span class="n">Max</span><span class="p">,</span> <span class="n">Min</span><span class="p">,</span> <span class="n">count_ops</span>
-    <span class="c"># We use a fresh dummy, because assumptions on s might drop conditions on</span>
-    <span class="c"># convergence of the integral.</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="n">_dummy</span><span class="p">(</span><span class="s">&#39;s&#39;</span><span class="p">,</span> <span class="s">&#39;mellin-transform&#39;</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
-    <span class="n">F</span> <span class="o">=</span> <span class="n">integrator</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">s</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">F</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Integral</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_simplify</span><span class="p">(</span><span class="n">F</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">s_</span><span class="p">),</span> <span class="n">simplify</span><span class="p">),</span> <span class="p">(</span><span class="o">-</span><span class="n">oo</span><span class="p">,</span> <span class="n">oo</span><span class="p">),</span> <span class="bp">True</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">F</span><span class="o">.</span><span class="n">is_Piecewise</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">IntegralTransformError</span><span class="p">(</span><span class="s">&#39;Mellin&#39;</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="s">&#39;could not compute integral&#39;</span><span class="p">)</span>
-
-    <span class="n">F</span><span class="p">,</span> <span class="n">cond</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">F</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Integral</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="n">IntegralTransformError</span><span class="p">(</span>
-            <span class="s">&#39;Mellin&#39;</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="s">&#39;integral in unexpected form&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">process_conds</span><span class="p">(</span><span class="n">cond</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Turn ``cond`` into a strip (a, b), and auxiliary conditions.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="o">-</span><span class="n">oo</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">oo</span>
-        <span class="n">aux</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="n">conds</span> <span class="o">=</span> <span class="n">conjuncts</span><span class="p">(</span><span class="n">to_cnf</span><span class="p">(</span><span class="n">cond</span><span class="p">))</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">conds</span><span class="p">:</span>
-            <span class="n">a_</span> <span class="o">=</span> <span class="n">oo</span>
-            <span class="n">b_</span> <span class="o">=</span> <span class="o">-</span><span class="n">oo</span>
-            <span class="n">aux_</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">disjuncts</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
-                <span class="n">d_</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span>
-                    <span class="n">re</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()[</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">re</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="n">t</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">d</span><span class="o">.</span><span class="n">is_Relational</span> <span class="ow">or</span> \
-                    <span class="n">d</span><span class="o">.</span><span class="n">rel_op</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">(</span><span class="s">&#39;&gt;&#39;</span><span class="p">,</span> <span class="s">&#39;&gt;=&#39;</span><span class="p">,</span> <span class="s">&#39;&lt;&#39;</span><span class="p">,</span> <span class="s">&#39;&lt;=&#39;</span><span class="p">)</span> \
-                        <span class="ow">or</span> <span class="n">d_</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">d_</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
-                    <span class="n">aux_</span> <span class="o">+=</span> <span class="p">[</span><span class="n">d</span><span class="p">]</span>
-                    <span class="k">continue</span>
-                <span class="n">soln</span> <span class="o">=</span> <span class="n">_solve_inequality</span><span class="p">(</span><span class="n">d_</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">soln</span><span class="o">.</span><span class="n">is_Relational</span> <span class="ow">or</span> \
-                        <span class="n">soln</span><span class="o">.</span><span class="n">rel_op</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">(</span><span class="s">&#39;&gt;&#39;</span><span class="p">,</span> <span class="s">&#39;&gt;=&#39;</span><span class="p">,</span> <span class="s">&#39;&lt;&#39;</span><span class="p">,</span> <span class="s">&#39;&lt;=&#39;</span><span class="p">):</span>
-                    <span class="n">aux_</span> <span class="o">+=</span> <span class="p">[</span><span class="n">d</span><span class="p">]</span>
-                    <span class="k">continue</span>
-                <span class="k">if</span> <span class="n">soln</span><span class="o">.</span><span class="n">lts</span> <span class="o">==</span> <span class="n">t</span><span class="p">:</span>
-                    <span class="n">b_</span> <span class="o">=</span> <span class="n">Max</span><span class="p">(</span><span class="n">soln</span><span class="o">.</span><span class="n">gts</span><span class="p">,</span> <span class="n">b_</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">a_</span> <span class="o">=</span> <span class="n">Min</span><span class="p">(</span><span class="n">soln</span><span class="o">.</span><span class="n">lts</span><span class="p">,</span> <span class="n">a_</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">a_</span> <span class="o">!=</span> <span class="n">oo</span> <span class="ow">and</span> <span class="n">a_</span> <span class="o">!=</span> <span class="n">b</span><span class="p">:</span>
-                <span class="n">a</span> <span class="o">=</span> <span class="n">Max</span><span class="p">(</span><span class="n">a_</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">b_</span> <span class="o">!=</span> <span class="o">-</span><span class="n">oo</span> <span class="ow">and</span> <span class="n">b_</span> <span class="o">!=</span> <span class="n">a</span><span class="p">:</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="n">Min</span><span class="p">(</span><span class="n">b_</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">aux</span> <span class="o">=</span> <span class="n">And</span><span class="p">(</span><span class="n">aux</span><span class="p">,</span> <span class="n">Or</span><span class="p">(</span><span class="o">*</span><span class="n">aux_</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">aux</span>
-
-    <span class="n">conds</span> <span class="o">=</span> <span class="p">[</span><span class="n">process_conds</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">disjuncts</span><span class="p">(</span><span class="n">cond</span><span class="p">)]</span>
-    <span class="n">conds</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">conds</span> <span class="k">if</span> <span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">]</span>
-    <span class="n">conds</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">count_ops</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">])))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">conds</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">IntegralTransformError</span><span class="p">(</span><span class="s">&#39;Mellin&#39;</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="s">&#39;no convergence found&#39;</span><span class="p">)</span>
-
-    <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">aux</span> <span class="o">=</span> <span class="n">conds</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">_simplify</span><span class="p">(</span><span class="n">F</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">s_</span><span class="p">),</span> <span class="n">simplify</span><span class="p">),</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">aux</span>
-
-
-<span class="k">class</span> <span class="nc">MellinTransform</span><span class="p">(</span><span class="n">IntegralTransform</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Class representing unevaluated Mellin transforms.</span>
-
-<span class="sd">    For usage of this class, see the :class:`IntegralTransform` docstring.</span>
-
-<span class="sd">    For how to compute Mellin transforms, see the :func:`mellin_transform`</span>
-<span class="sd">    docstring.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_name</span> <span class="o">=</span> <span class="s">&#39;Mellin&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_compute_transform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_mellin_transform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_as_integral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-        <span class="k">return</span> <span class="n">Integral</span><span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">s</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_collapse_extra</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">extra</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">And</span><span class="p">,</span> <span class="n">Max</span><span class="p">,</span> <span class="n">Min</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">cond</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="p">(</span><span class="n">sa</span><span class="p">,</span> <span class="n">sb</span><span class="p">),</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">extra</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">+=</span> <span class="p">[</span><span class="n">sa</span><span class="p">]</span>
-            <span class="n">b</span> <span class="o">+=</span> <span class="p">[</span><span class="n">sb</span><span class="p">]</span>
-            <span class="n">cond</span> <span class="o">+=</span> <span class="p">[</span><span class="n">c</span><span class="p">]</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="p">(</span><span class="n">Max</span><span class="p">(</span><span class="o">*</span><span class="n">a</span><span class="p">),</span> <span class="n">Min</span><span class="p">(</span><span class="o">*</span><span class="n">b</span><span class="p">)),</span> <span class="n">And</span><span class="p">(</span><span class="o">*</span><span class="n">cond</span><span class="p">)</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">res</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">&gt;=</span> <span class="n">res</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span> <span class="ow">is</span> <span class="bp">True</span> <span class="ow">or</span> <span class="n">res</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">IntegralTransformError</span><span class="p">(</span>
-                <span class="s">&#39;Mellin&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="s">&#39;no combined convergence.&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">res</span>
-
-
-<div class="viewcode-block" id="mellin_transform"><a class="viewcode-back" href="../../../modules/integrals/integrals.html#sympy.integrals.transforms.mellin_transform">[docs]</a><span class="k">def</span> <span class="nf">mellin_transform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Compute the Mellin transform `F(s)` of `f(x)`,</span>
-
-<span class="sd">    .. math :: F(s) = \int_0^\infty x^{s-1} f(x) \mathrm{d}x.</span>
-
-<span class="sd">    For all &quot;sensible&quot; functions, this converges absolutely in a strip</span>
-<span class="sd">      `a &lt; Re(s) &lt; b`.</span>
-
-<span class="sd">    The Mellin transform is related via change of variables to the Fourier</span>
-<span class="sd">    transform, and also to the (bilateral) Laplace transform.</span>
-
-<span class="sd">    This function returns (F, (a, b), cond)</span>
-<span class="sd">    where `F` is the Mellin transform of `f`, `(a, b)` is the fundamental strip</span>
-<span class="sd">    (as above), and cond are auxiliary convergence conditions.</span>
-
-<span class="sd">    If the integral cannot be computed in closed form, this function returns</span>
-<span class="sd">    an unevaluated MellinTransform object.</span>
-
-<span class="sd">    For a description of possible hints, refer to the docstring of</span>
-<span class="sd">    :func:`sympy.integrals.transforms.IntegralTransform.doit`. If ``noconds=False``,</span>
-<span class="sd">    then only `F` will be returned (i.e. not ``cond``, and also not the strip</span>
-<span class="sd">    ``(a, b)``).</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.integrals.transforms import mellin_transform</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import exp</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, s</span>
-<span class="sd">    &gt;&gt;&gt; mellin_transform(exp(-x), x, s)</span>
-<span class="sd">    (gamma(s), (0, oo), True)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    inverse_mellin_transform, laplace_transform, fourier_transform</span>
-<span class="sd">    hankel_transform, inverse_hankel_transform</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">MellinTransform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_rewrite_sin</span><span class="p">(</span><span class="n">xxx_todo_changeme</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Re-write the sine function sin(m*s + n) as gamma functions, compatible</span>
-<span class="sd">    with the strip (a, b).</span>
-
-<span class="sd">    Return (gamma1, gamma2, fac) so that f == fac/(gamma1 * gamma2).</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.integrals.transforms import _rewrite_sin</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import pi, S</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import s</span>
-<span class="sd">    &gt;&gt;&gt; _rewrite_sin((pi, 0), s, 0, 1)</span>
-<span class="sd">    (gamma(s), gamma(-s + 1), pi)</span>
-<span class="sd">    &gt;&gt;&gt; _rewrite_sin((pi, 0), s, 1, 0)</span>
-<span class="sd">    (gamma(s - 1), gamma(-s + 2), -pi)</span>
-<span class="sd">    &gt;&gt;&gt; _rewrite_sin((pi, 0), s, -1, 0)</span>
-<span class="sd">    (gamma(s + 1), gamma(-s), -pi)</span>
-<span class="sd">    &gt;&gt;&gt; _rewrite_sin((pi, pi/2), s, S(1)/2, S(3)/2)</span>
-<span class="sd">    (gamma(s - 1/2), gamma(-s + 3/2), -pi)</span>
-<span class="sd">    &gt;&gt;&gt; _rewrite_sin((pi, pi), s, 0, 1)</span>
-<span class="sd">    (gamma(s), gamma(-s + 1), -pi)</span>
-<span class="sd">    &gt;&gt;&gt; _rewrite_sin((2*pi, 0), s, 0, S(1)/2)</span>
-<span class="sd">    (gamma(2*s), gamma(-2*s + 1), pi)</span>
-<span class="sd">    &gt;&gt;&gt; _rewrite_sin((2*pi, 0), s, S(1)/2, 1)</span>
-<span class="sd">    (gamma(2*s - 1), gamma(-2*s + 2), -pi)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">=</span> <span class="n">xxx_todo_changeme</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expand_mul</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">ceiling</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">re</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">m</span><span class="o">/</span><span class="n">pi</span><span class="p">)</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="n">pi</span><span class="p">)</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">ceiling</span><span class="p">(</span><span class="o">-</span><span class="n">m</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="n">n</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()[</span><span class="mi">0</span><span class="p">])</span>  <span class="c"># Don&#39;t use re(n), does not expand</span>
-    <span class="k">return</span> <span class="n">gamma</span><span class="p">(</span><span class="n">m</span><span class="o">*</span><span class="n">s</span> <span class="o">+</span> <span class="n">n</span> <span class="o">+</span> <span class="n">r</span><span class="p">),</span> <span class="n">gamma</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">n</span> <span class="o">-</span> <span class="n">r</span> <span class="o">-</span> <span class="n">m</span><span class="o">*</span><span class="n">s</span><span class="p">),</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">r</span><span class="o">*</span><span class="n">pi</span>
-
-
-<span class="k">class</span> <span class="nc">MellinTransformStripError</span><span class="p">(</span><span class="ne">ValueError</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Exception raised by _rewrite_gamma. Mainly for internal use.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">pass</span>
-
-
-<span class="k">def</span> <span class="nf">_rewrite_gamma</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Try to rewrite the product f(s) as a product of gamma functions,</span>
-<span class="sd">    so that the inverse Mellin transform of f can be expressed as a meijer</span>
-<span class="sd">    G function.</span>
-
-<span class="sd">    Return (an, ap), (bm, bq), arg, exp, fac such that</span>
-<span class="sd">    G((an, ap), (bm, bq), arg/z**exp)*fac is the inverse Mellin transform of f(s).</span>
-
-<span class="sd">    Raises IntegralTransformError or MellinTransformStripError on failure.</span>
-
-<span class="sd">    It is asserted that f has no poles in the fundamental strip designated by</span>
-<span class="sd">    (a, b). One of a and b is allowed to be None. The fundamental strip is</span>
-<span class="sd">    important, because it determines the inversion contour.</span>
-
-<span class="sd">    This function can handle exponentials, linear factors, trigonometric</span>
-<span class="sd">    functions.</span>
-
-<span class="sd">    This is a helper function for inverse_mellin_transform that will not</span>
-<span class="sd">    attempt any transformations on f.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.integrals.transforms import _rewrite_gamma</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import s</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import oo</span>
-<span class="sd">    &gt;&gt;&gt; _rewrite_gamma(s*(s+3)*(s-1), s, -oo, oo)</span>
-<span class="sd">    (([], [-3, 0, 1]), ([-2, 1, 2], []), 1, 1, -1)</span>
-<span class="sd">    &gt;&gt;&gt; _rewrite_gamma((s-1)**2, s, -oo, oo)</span>
-<span class="sd">    (([], [1, 1]), ([2, 2], []), 1, 1, 1)</span>
-
-<span class="sd">    Importance of the fundamental strip:</span>
-
-<span class="sd">    &gt;&gt;&gt; _rewrite_gamma(1/s, s, 0, oo)</span>
-<span class="sd">    (([1], []), ([], [0]), 1, 1, 1)</span>
-<span class="sd">    &gt;&gt;&gt; _rewrite_gamma(1/s, s, None, oo)</span>
-<span class="sd">    (([1], []), ([], [0]), 1, 1, 1)</span>
-<span class="sd">    &gt;&gt;&gt; _rewrite_gamma(1/s, s, 0, None)</span>
-<span class="sd">    (([1], []), ([], [0]), 1, 1, 1)</span>
-<span class="sd">    &gt;&gt;&gt; _rewrite_gamma(1/s, s, -oo, 0)</span>
-<span class="sd">    (([], [1]), ([0], []), 1, 1, -1)</span>
-<span class="sd">    &gt;&gt;&gt; _rewrite_gamma(1/s, s, None, 0)</span>
-<span class="sd">    (([], [1]), ([0], []), 1, 1, -1)</span>
-<span class="sd">    &gt;&gt;&gt; _rewrite_gamma(1/s, s, -oo, None)</span>
-<span class="sd">    (([], [1]), ([0], []), 1, 1, -1)</span>
-
-<span class="sd">    &gt;&gt;&gt; _rewrite_gamma(2**(-s+3), s, -oo, oo)</span>
-<span class="sd">    (([], []), ([], []), 1/2, 1, 8)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">itertools</span> <span class="kn">import</span> <span class="n">repeat</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="p">(</span><span class="n">Poly</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">re</span><span class="p">,</span> <span class="n">RootOf</span><span class="p">,</span> <span class="n">exp</span> <span class="k">as</span> <span class="n">exp_</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">expand</span><span class="p">,</span>
-                       <span class="n">roots</span><span class="p">,</span> <span class="n">ilcm</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">tan</span><span class="p">,</span> <span class="n">cot</span><span class="p">,</span> <span class="n">igcd</span><span class="p">,</span> <span class="n">exp_polar</span><span class="p">)</span>
-    <span class="c"># Our strategy will be as follows:</span>
-    <span class="c"># 1) Guess a constant c such that the inversion integral should be</span>
-    <span class="c">#    performed wrt s&#39;=c*s (instead of plain s). Write s for s&#39;.</span>
-    <span class="c"># 2) Process all factors, rewrite them independently as gamma functions in</span>
-    <span class="c">#    argument s, or exponentials of s.</span>
-    <span class="c"># 3) Try to transform all gamma functions s.t. they have argument</span>
-    <span class="c">#    a+s or a-s.</span>
-    <span class="c"># 4) Check that the resulting G function parameters are valid.</span>
-    <span class="c"># 5) Combine all the exponentials.</span>
-
-    <span class="n">a_</span><span class="p">,</span> <span class="n">b_</span> <span class="o">=</span> <span class="n">S</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">left</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">is_numer</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Decide whether pole at c lies to the left of the fundamental strip.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># heuristically, this is the best chance for us to solve the inequalities</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">expand</span><span class="p">(</span><span class="n">re</span><span class="p">(</span><span class="n">c</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">a_</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">c</span> <span class="o">&lt;</span> <span class="n">b_</span>
-        <span class="k">if</span> <span class="n">b_</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">c</span> <span class="o">&lt;=</span> <span class="n">a_</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">c</span> <span class="o">&gt;=</span> <span class="n">b_</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">c</span> <span class="o">&lt;=</span> <span class="n">a_</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="n">is_numer</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">a_</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">or</span> <span class="n">b_</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">or</span> <span class="n">c</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>  <span class="c"># XXX</span>
-            <span class="c">#raise IntegralTransformError(&#39;Inverse Mellin&#39;, f,</span>
-            <span class="c">#                     &#39;Could not determine position of singularity %s&#39;</span>
-            <span class="c">#                     &#39; relative to fundamental strip&#39; % c)</span>
-        <span class="k">raise</span> <span class="n">MellinTransformStripError</span><span class="p">(</span><span class="s">&#39;Pole inside critical strip?&#39;</span><span class="p">)</span>
-
-    <span class="c"># 1)</span>
-    <span class="n">s_multipliers</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">gamma</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
-            <span class="k">continue</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">arg</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">s</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-        <span class="n">s_multipliers</span> <span class="o">+=</span> <span class="p">[</span><span class="n">coeff</span><span class="p">]</span>
-    <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">tan</span><span class="p">,</span> <span class="n">cot</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
-            <span class="k">continue</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">arg</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">s</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-        <span class="n">s_multipliers</span> <span class="o">+=</span> <span class="p">[</span><span class="n">coeff</span><span class="o">/</span><span class="n">pi</span><span class="p">]</span>
-    <span class="n">s_multipliers</span> <span class="o">=</span> <span class="p">[</span><span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">s_multipliers</span> <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">is_real</span><span class="p">]</span>
-    <span class="n">common_coefficient</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">s_multipliers</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">x</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="n">common_coefficient</span> <span class="o">=</span> <span class="n">x</span>
-            <span class="k">break</span>
-    <span class="n">s_multipliers</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="o">/</span><span class="n">common_coefficient</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">s_multipliers</span><span class="p">]</span>
-    <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="ow">not</span> <span class="n">x</span><span class="o">.</span><span class="n">is_Rational</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">s_multipliers</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-    <span class="n">s_multiplier</span> <span class="o">=</span> <span class="n">common_coefficient</span><span class="o">/</span><span class="nb">reduce</span><span class="p">(</span><span class="n">ilcm</span><span class="p">,</span> <span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-                                             <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">s_multipliers</span><span class="p">],</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span>
-    <span class="k">if</span> <span class="n">s_multiplier</span> <span class="o">==</span> <span class="n">common_coefficient</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">s_multipliers</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">s_multiplier</span> <span class="o">=</span> <span class="n">common_coefficient</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">s_multiplier</span> <span class="o">=</span> <span class="n">common_coefficient</span> \
-                <span class="o">*</span><span class="nb">reduce</span><span class="p">(</span><span class="n">igcd</span><span class="p">,</span> <span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">p</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">s_multipliers</span><span class="p">])</span>
-
-    <span class="n">exponent</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-    <span class="n">fac</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">s</span><span class="o">/</span><span class="n">s_multiplier</span><span class="p">)</span>
-    <span class="n">fac</span> <span class="o">/=</span> <span class="n">s_multiplier</span>
-    <span class="n">exponent</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">s_multiplier</span>
-    <span class="k">if</span> <span class="n">a_</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">a_</span> <span class="o">*=</span> <span class="n">s_multiplier</span>
-    <span class="k">if</span> <span class="n">b_</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">b_</span> <span class="o">*=</span> <span class="n">s_multiplier</span>
-
-    <span class="c"># 2)</span>
-    <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-    <span class="n">numer</span> <span class="o">=</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">numer</span><span class="p">)</span>
-    <span class="n">denom</span> <span class="o">=</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">denom</span><span class="p">)</span>
-    <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">numer</span><span class="p">,</span> <span class="n">repeat</span><span class="p">(</span><span class="bp">True</span><span class="p">)))</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">denom</span><span class="p">,</span> <span class="n">repeat</span><span class="p">(</span><span class="bp">False</span><span class="p">)))</span>
-
-    <span class="n">facs</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">dfacs</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="c"># *_gammas will contain pairs (a, c) representing Gamma(a*s + c)</span>
-    <span class="n">numer_gammas</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">denom_gammas</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="c"># exponentials will contain bases for exponentials of s</span>
-    <span class="n">exponentials</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">exception</span><span class="p">(</span><span class="n">fact</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">IntegralTransformError</span><span class="p">(</span><span class="s">&quot;Inverse Mellin&quot;</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="s">&quot;Unrecognised form &#39;</span><span class="si">%s</span><span class="s">&#39;.&quot;</span> <span class="o">%</span> <span class="n">fact</span><span class="p">)</span>
-    <span class="k">while</span> <span class="n">args</span><span class="p">:</span>
-        <span class="n">fact</span><span class="p">,</span> <span class="n">is_numer</span> <span class="o">=</span> <span class="n">args</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">is_numer</span><span class="p">:</span>
-            <span class="n">ugammas</span><span class="p">,</span> <span class="n">lgammas</span> <span class="o">=</span> <span class="n">numer_gammas</span><span class="p">,</span> <span class="n">denom_gammas</span>
-            <span class="n">ufacs</span><span class="p">,</span> <span class="n">lfacs</span> <span class="o">=</span> <span class="n">facs</span><span class="p">,</span> <span class="n">dfacs</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">ugammas</span><span class="p">,</span> <span class="n">lgammas</span> <span class="o">=</span> <span class="n">denom_gammas</span><span class="p">,</span> <span class="n">numer_gammas</span>
-            <span class="n">ufacs</span><span class="p">,</span> <span class="n">lfacs</span> <span class="o">=</span> <span class="n">dfacs</span><span class="p">,</span> <span class="n">facs</span>
-
-        <span class="k">def</span> <span class="nf">linear_arg</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot; Test if arg is of form a*s+b, raise exception if not. &quot;&quot;&quot;</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_polynomial</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="n">exception</span><span class="p">(</span><span class="n">fact</span><span class="p">)</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">exception</span><span class="p">(</span><span class="n">fact</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">p</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()</span>
-
-        <span class="c"># constants</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">fact</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
-            <span class="n">ufacs</span> <span class="o">+=</span> <span class="p">[</span><span class="n">fact</span><span class="p">]</span>
-        <span class="c"># exponentials</span>
-        <span class="k">elif</span> <span class="n">fact</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">fact</span><span class="p">,</span> <span class="n">exp_</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">fact</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-                <span class="n">base</span> <span class="o">=</span> <span class="n">fact</span><span class="o">.</span><span class="n">base</span>
-                <span class="n">exp</span> <span class="o">=</span> <span class="n">fact</span><span class="o">.</span><span class="n">exp</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">base</span> <span class="o">=</span> <span class="n">exp_polar</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-                <span class="n">exp</span> <span class="o">=</span> <span class="n">fact</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="n">cond</span> <span class="o">=</span> <span class="n">is_numer</span>
-                <span class="k">if</span> <span class="n">exp</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">cond</span> <span class="o">=</span> <span class="ow">not</span> <span class="n">cond</span>
-                <span class="n">args</span> <span class="o">+=</span> <span class="p">[(</span><span class="n">base</span><span class="p">,</span> <span class="n">cond</span><span class="p">)]</span><span class="o">*</span><span class="nb">abs</span><span class="p">(</span><span class="n">exp</span><span class="p">)</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="ow">not</span> <span class="n">base</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
-                <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">linear_arg</span><span class="p">(</span><span class="n">exp</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">is_numer</span><span class="p">:</span>
-                    <span class="n">base</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">base</span>
-                <span class="n">exponentials</span> <span class="o">+=</span> <span class="p">[</span><span class="n">base</span><span class="o">**</span><span class="n">a</span><span class="p">]</span>
-                <span class="n">facs</span> <span class="o">+=</span> <span class="p">[</span><span class="n">base</span><span class="o">**</span><span class="n">b</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">exception</span><span class="p">(</span><span class="n">fact</span><span class="p">)</span>
-        <span class="c"># linear factors</span>
-        <span class="k">elif</span> <span class="n">fact</span><span class="o">.</span><span class="n">is_polynomial</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">fact</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="c"># We completely factor the poly. For this we need the roots.</span>
-                <span class="c"># Now roots() only works in some cases (low degree), and RootOf</span>
-                <span class="c"># only works without parameters. So try both...</span>
-                <span class="n">coeff</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">LT</span><span class="p">()[</span><span class="mi">1</span><span class="p">]</span>
-                <span class="n">rs</span> <span class="o">=</span> <span class="n">roots</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">rs</span><span class="p">)</span> <span class="o">!=</span> <span class="n">p</span><span class="o">.</span><span class="n">degree</span><span class="p">():</span>
-                    <span class="n">rs</span> <span class="o">=</span> <span class="n">RootOf</span><span class="o">.</span><span class="n">all_roots</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-                <span class="n">ufacs</span> <span class="o">+=</span> <span class="p">[</span><span class="n">coeff</span><span class="p">]</span>
-                <span class="n">args</span> <span class="o">+=</span> <span class="p">[(</span><span class="n">s</span> <span class="o">-</span> <span class="n">c</span><span class="p">,</span> <span class="n">is_numer</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">rs</span><span class="p">]</span>
-                <span class="k">continue</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()</span>
-            <span class="n">ufacs</span> <span class="o">+=</span> <span class="p">[</span><span class="n">a</span><span class="p">]</span>
-            <span class="n">c</span> <span class="o">/=</span> <span class="o">-</span><span class="n">a</span>
-            <span class="c"># Now need to convert s - c</span>
-            <span class="k">if</span> <span class="n">left</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">is_numer</span><span class="p">):</span>
-                <span class="n">ugammas</span> <span class="o">+=</span> <span class="p">[(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="o">-</span><span class="n">c</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
-                <span class="n">lgammas</span> <span class="o">+=</span> <span class="p">[(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="o">-</span><span class="n">c</span><span class="p">)]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">ufacs</span> <span class="o">+=</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-                <span class="n">ugammas</span> <span class="o">+=</span> <span class="p">[(</span><span class="n">S</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">c</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
-                <span class="n">lgammas</span> <span class="o">+=</span> <span class="p">[(</span><span class="n">S</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">c</span><span class="p">)]</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">fact</span><span class="p">,</span> <span class="n">gamma</span><span class="p">):</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">linear_arg</span><span class="p">(</span><span class="n">fact</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="k">if</span> <span class="n">is_numer</span><span class="p">:</span>
-                <span class="k">if</span> <span class="p">(</span><span class="n">a</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="ow">and</span> <span class="p">(</span><span class="n">left</span><span class="p">(</span><span class="o">-</span><span class="n">b</span><span class="o">/</span><span class="n">a</span><span class="p">,</span> <span class="n">is_numer</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">))</span> <span class="ow">or</span> \
-                   <span class="p">(</span><span class="n">a</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">and</span> <span class="p">(</span><span class="n">left</span><span class="p">(</span><span class="o">-</span><span class="n">b</span><span class="o">/</span><span class="n">a</span><span class="p">,</span> <span class="n">is_numer</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">)):</span>
-                    <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                        <span class="s">&#39;Gammas partially over the strip.&#39;</span><span class="p">)</span>
-            <span class="n">ugammas</span> <span class="o">+=</span> <span class="p">[(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)]</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">fact</span><span class="p">,</span> <span class="n">sin</span><span class="p">):</span>
-            <span class="c"># We try to re-write all trigs as gammas. This is not in</span>
-            <span class="c"># general the best strategy, since sometimes this is impossible,</span>
-            <span class="c"># but rewriting as exponentials would work. However trig functions</span>
-            <span class="c"># in inverse mellin transforms usually all come from simplifying</span>
-            <span class="c"># gamma terms, so this should work.</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">fact</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">is_numer</span><span class="p">:</span>
-                <span class="c"># No problem with the poles.</span>
-                <span class="n">gamma1</span><span class="p">,</span> <span class="n">gamma2</span><span class="p">,</span> <span class="n">fac_</span> <span class="o">=</span> <span class="n">gamma</span><span class="p">(</span><span class="n">a</span><span class="o">/</span><span class="n">pi</span><span class="p">),</span> <span class="n">gamma</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">a</span><span class="o">/</span><span class="n">pi</span><span class="p">),</span> <span class="n">pi</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">gamma1</span><span class="p">,</span> <span class="n">gamma2</span><span class="p">,</span> <span class="n">fac_</span> <span class="o">=</span> <span class="n">_rewrite_sin</span><span class="p">(</span><span class="n">linear_arg</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">s</span><span class="p">,</span> <span class="n">a_</span><span class="p">,</span> <span class="n">b_</span><span class="p">)</span>
-            <span class="n">args</span> <span class="o">+=</span> <span class="p">[(</span><span class="n">gamma1</span><span class="p">,</span> <span class="ow">not</span> <span class="n">is_numer</span><span class="p">),</span> <span class="p">(</span><span class="n">gamma2</span><span class="p">,</span> <span class="ow">not</span> <span class="n">is_numer</span><span class="p">)]</span>
-            <span class="n">ufacs</span> <span class="o">+=</span> <span class="p">[</span><span class="n">fac_</span><span class="p">]</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">fact</span><span class="p">,</span> <span class="n">tan</span><span class="p">):</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">fact</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">args</span> <span class="o">+=</span> <span class="p">[(</span><span class="n">sin</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">),</span> <span class="n">is_numer</span><span class="p">),</span>
-                     <span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">a</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">),</span> <span class="ow">not</span> <span class="n">is_numer</span><span class="p">)]</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">fact</span><span class="p">,</span> <span class="n">cos</span><span class="p">):</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">fact</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">args</span> <span class="o">+=</span> <span class="p">[(</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">a</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">),</span> <span class="n">is_numer</span><span class="p">)]</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">fact</span><span class="p">,</span> <span class="n">cot</span><span class="p">):</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">fact</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">args</span> <span class="o">+=</span> <span class="p">[(</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">a</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">),</span> <span class="n">is_numer</span><span class="p">),</span>
-                     <span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">),</span> <span class="ow">not</span> <span class="n">is_numer</span><span class="p">)]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">exception</span><span class="p">(</span><span class="n">fact</span><span class="p">)</span>
-
-    <span class="n">fac</span> <span class="o">*=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">facs</span><span class="p">)</span><span class="o">/</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">dfacs</span><span class="p">)</span>
-
-    <span class="c"># 3)</span>
-    <span class="n">an</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">bq</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">gammas</span><span class="p">,</span> <span class="n">plus</span><span class="p">,</span> <span class="n">minus</span><span class="p">,</span> <span class="n">is_numer</span> <span class="ow">in</span> <span class="p">[(</span><span class="n">numer_gammas</span><span class="p">,</span> <span class="n">an</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="bp">True</span><span class="p">),</span>
-                                          <span class="p">(</span><span class="n">denom_gammas</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="bp">False</span><span class="p">)]:</span>
-        <span class="k">while</span> <span class="n">gammas</span><span class="p">:</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">gammas</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="o">!=</span> <span class="o">-</span><span class="mi">1</span> <span class="ow">and</span> <span class="n">a</span> <span class="o">!=</span> <span class="o">+</span><span class="mi">1</span><span class="p">:</span>
-                <span class="c"># We use the gamma function multiplication theorem.</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
-                <span class="n">newa</span> <span class="o">=</span> <span class="n">a</span><span class="o">/</span><span class="n">p</span>
-                <span class="n">newc</span> <span class="o">=</span> <span class="n">c</span><span class="o">/</span><span class="n">p</span>
-                <span class="k">assert</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Integer</span>
-                <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
-                    <span class="n">gammas</span> <span class="o">+=</span> <span class="p">[(</span><span class="n">newa</span><span class="p">,</span> <span class="n">newc</span> <span class="o">+</span> <span class="n">k</span><span class="o">/</span><span class="n">p</span><span class="p">)]</span>
-                <span class="k">if</span> <span class="n">is_numer</span><span class="p">:</span>
-                    <span class="n">fac</span> <span class="o">*=</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">**</span><span class="p">((</span><span class="mi">1</span> <span class="o">-</span> <span class="n">p</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">p</span><span class="o">**</span><span class="p">(</span><span class="n">c</span> <span class="o">-</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-                    <span class="n">exponentials</span> <span class="o">+=</span> <span class="p">[</span><span class="n">p</span><span class="o">**</span><span class="n">a</span><span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">fac</span> <span class="o">/=</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">**</span><span class="p">((</span><span class="mi">1</span> <span class="o">-</span> <span class="n">p</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">p</span><span class="o">**</span><span class="p">(</span><span class="n">c</span> <span class="o">-</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-                    <span class="n">exponentials</span> <span class="o">+=</span> <span class="p">[</span><span class="n">p</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="p">)]</span>
-                <span class="k">continue</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="o">+</span><span class="mi">1</span><span class="p">:</span>
-                <span class="n">plus</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">c</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">minus</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-
-    <span class="c"># 4)</span>
-    <span class="c"># TODO</span>
-
-    <span class="c"># 5)</span>
-    <span class="n">arg</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">exponentials</span><span class="p">)</span>
-
-    <span class="c"># for testability, sort the arguments</span>
-    <span class="n">an</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-    <span class="n">ap</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-    <span class="n">bm</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-    <span class="n">bq</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-
-    <span class="k">return</span> <span class="p">(</span><span class="n">an</span><span class="p">,</span> <span class="n">ap</span><span class="p">),</span> <span class="p">(</span><span class="n">bm</span><span class="p">,</span> <span class="n">bq</span><span class="p">),</span> <span class="n">arg</span><span class="p">,</span> <span class="n">exponent</span><span class="p">,</span> <span class="n">fac</span>
-
-
-<span class="nd">@_noconds_</span><span class="p">(</span><span class="bp">True</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_inverse_mellin_transform</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">x_</span><span class="p">,</span> <span class="n">strip</span><span class="p">,</span> <span class="n">as_meijerg</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; A helper for the real inverse_mellin_transform function, this one here</span>
-<span class="sd">        assumes x to be real and positive. &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="p">(</span><span class="n">expand</span><span class="p">,</span> <span class="n">expand_mul</span><span class="p">,</span> <span class="n">hyperexpand</span><span class="p">,</span> <span class="n">meijerg</span><span class="p">,</span> <span class="n">And</span><span class="p">,</span> <span class="n">Or</span><span class="p">,</span>
-                       <span class="n">arg</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">re</span><span class="p">,</span> <span class="n">factor</span><span class="p">,</span> <span class="n">Heaviside</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">Add</span><span class="p">)</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">_dummy</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">,</span> <span class="s">&#39;inverse-mellin-transform&#39;</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="c"># Actually, we won&#39;t try integration at all. Instead we use the definition</span>
-    <span class="c"># of the Meijer G function as a fairly general inverse mellin transform.</span>
-    <span class="n">F</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">gamma</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="p">[</span><span class="n">factor</span><span class="p">(</span><span class="n">F</span><span class="p">),</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">F</span><span class="p">),</span> <span class="n">expand</span><span class="p">(</span><span class="n">F</span><span class="p">)]:</span>
-        <span class="k">if</span> <span class="n">g</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="c"># do all terms separately</span>
-            <span class="n">ress</span> <span class="o">=</span> <span class="p">[</span><span class="n">_inverse_mellin_transform</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">strip</span><span class="p">,</span> <span class="n">as_meijerg</span><span class="p">,</span>
-                                              <span class="n">noconds</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-                    <span class="k">for</span> <span class="n">G</span> <span class="ow">in</span> <span class="n">g</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-            <span class="n">conds</span> <span class="o">=</span> <span class="p">[</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">ress</span><span class="p">]</span>
-            <span class="n">ress</span> <span class="o">=</span> <span class="p">[</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">ress</span><span class="p">]</span>
-            <span class="n">res</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">ress</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">as_meijerg</span><span class="p">:</span>
-                <span class="n">res</span> <span class="o">=</span> <span class="n">factor</span><span class="p">(</span><span class="n">res</span><span class="p">,</span> <span class="n">gens</span><span class="o">=</span><span class="n">res</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Heaviside</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">res</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x_</span><span class="p">),</span> <span class="n">And</span><span class="p">(</span><span class="o">*</span><span class="n">conds</span><span class="p">)</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">fac</span> <span class="o">=</span> <span class="n">_rewrite_gamma</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">strip</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">strip</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="k">except</span> <span class="n">IntegralTransformError</span><span class="p">:</span>
-            <span class="k">continue</span>
-        <span class="n">G</span> <span class="o">=</span> <span class="n">meijerg</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">C</span><span class="o">/</span><span class="n">x</span><span class="o">**</span><span class="n">e</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">as_meijerg</span><span class="p">:</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">G</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">hyperexpand</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">h</span><span class="o">.</span><span class="n">is_Piecewise</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">h</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-                <span class="c"># XXX we break modularity here!</span>
-                <span class="n">h</span> <span class="o">=</span> <span class="n">Heaviside</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="nb">abs</span><span class="p">(</span><span class="n">C</span><span class="p">))</span><span class="o">*</span><span class="n">h</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> \
-                    <span class="o">+</span> <span class="n">Heaviside</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">C</span><span class="p">)</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">h</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="c"># We must ensure that the intgral along the line we want converges,</span>
-        <span class="c"># and return that value.</span>
-        <span class="c"># See [L], 5.2</span>
-        <span class="n">cond</span> <span class="o">=</span> <span class="p">[</span><span class="nb">abs</span><span class="p">(</span><span class="n">arg</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">argument</span><span class="p">))</span> <span class="o">&lt;</span> <span class="n">G</span><span class="o">.</span><span class="n">delta</span><span class="o">*</span><span class="n">pi</span><span class="p">]</span>
-        <span class="c"># Note: we allow &quot;&gt;=&quot; here, this corresponds to convergence if we let</span>
-        <span class="c"># limits go to oo symetrically. &quot;&gt;&quot; corresponds to absolute convergence.</span>
-        <span class="n">cond</span> <span class="o">+=</span> <span class="p">[</span><span class="n">And</span><span class="p">(</span><span class="n">Or</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">ap</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">bq</span><span class="p">),</span> <span class="mi">0</span> <span class="o">&gt;=</span> <span class="n">re</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">nu</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span>
-                     <span class="nb">abs</span><span class="p">(</span><span class="n">arg</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">argument</span><span class="p">))</span> <span class="o">==</span> <span class="n">G</span><span class="o">.</span><span class="n">delta</span><span class="o">*</span><span class="n">pi</span><span class="p">)]</span>
-        <span class="n">cond</span> <span class="o">=</span> <span class="n">Or</span><span class="p">(</span><span class="o">*</span><span class="n">cond</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">cond</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">IntegralTransformError</span><span class="p">(</span>
-                <span class="s">&#39;Inverse Mellin&#39;</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="s">&#39;does not converge&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">h</span><span class="o">*</span><span class="n">fac</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x_</span><span class="p">),</span> <span class="n">cond</span>
-
-    <span class="k">raise</span> <span class="n">IntegralTransformError</span><span class="p">(</span><span class="s">&#39;Inverse Mellin&#39;</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-
-<span class="n">_allowed</span> <span class="o">=</span> <span class="bp">None</span>
-
-
-<span class="k">class</span> <span class="nc">InverseMellinTransform</span><span class="p">(</span><span class="n">IntegralTransform</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Class representing unevaluated inverse Mellin transforms.</span>
-
-<span class="sd">    For usage of this class, see the :class:`IntegralTransform` docstring.</span>
-
-<span class="sd">    For how to compute inverse Mellin transforms, see the</span>
-<span class="sd">    :func:`inverse_mellin_transform` docstring.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">5</span>
-
-    <span class="n">_name</span> <span class="o">=</span> <span class="s">&#39;Inverse Mellin&#39;</span>
-    <span class="n">_none_sentinel</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;None&#39;</span><span class="p">)</span>
-    <span class="n">_c</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;c&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="o">**</span><span class="n">opts</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">a</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">InverseMellinTransform</span><span class="o">.</span><span class="n">_none_sentinel</span>
-        <span class="k">if</span> <span class="n">b</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="n">InverseMellinTransform</span><span class="o">.</span><span class="n">_none_sentinel</span>
-        <span class="k">return</span> <span class="n">IntegralTransform</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="o">**</span><span class="n">opts</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">fundamental_strip</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">4</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">a</span> <span class="ow">is</span> <span class="n">InverseMellinTransform</span><span class="o">.</span><span class="n">_none_sentinel</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">b</span> <span class="ow">is</span> <span class="n">InverseMellinTransform</span><span class="o">.</span><span class="n">_none_sentinel</span><span class="p">:</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">return</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span>
-
-    <span class="k">def</span> <span class="nf">_compute_transform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">postorder_traversal</span>
-        <span class="k">global</span> <span class="n">_allowed</span>
-        <span class="k">if</span> <span class="n">_allowed</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="p">(</span>
-                <span class="n">exp</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">tan</span><span class="p">,</span> <span class="n">cot</span><span class="p">,</span> <span class="n">cosh</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">tanh</span><span class="p">,</span>
-                <span class="n">coth</span><span class="p">,</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">rf</span><span class="p">)</span>
-            <span class="n">_allowed</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span>
-                <span class="p">[</span><span class="n">exp</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">tan</span><span class="p">,</span> <span class="n">cot</span><span class="p">,</span> <span class="n">cosh</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">tanh</span><span class="p">,</span> <span class="n">coth</span><span class="p">,</span>
-                 <span class="n">factorial</span><span class="p">,</span> <span class="n">rf</span><span class="p">])</span>
-        <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">postorder_traversal</span><span class="p">(</span><span class="n">F</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_Function</span> <span class="ow">and</span> <span class="n">f</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="ow">and</span> <span class="n">f</span><span class="o">.</span><span class="n">func</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">_allowed</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">IntegralTransformError</span><span class="p">(</span><span class="s">&#39;Inverse Mellin&#39;</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span>
-                                     <span class="s">&#39;Component </span><span class="si">%s</span><span class="s"> not recognised.&#39;</span> <span class="o">%</span> <span class="n">f</span><span class="p">)</span>
-        <span class="n">strip</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">fundamental_strip</span>
-        <span class="k">return</span> <span class="n">_inverse_mellin_transform</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">strip</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_as_integral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">oo</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">_c</span>
-        <span class="k">return</span> <span class="n">Integral</span><span class="p">(</span><span class="n">F</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">s</span><span class="p">),</span> <span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">c</span> <span class="o">-</span> <span class="n">I</span><span class="o">*</span><span class="n">oo</span><span class="p">,</span> <span class="n">c</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">oo</span><span class="p">))</span>
-
-
-<div class="viewcode-block" id="inverse_mellin_transform"><a class="viewcode-back" href="../../../modules/integrals/integrals.html#sympy.integrals.transforms.inverse_mellin_transform">[docs]</a><span class="k">def</span> <span class="nf">inverse_mellin_transform</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">strip</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Compute the inverse Mellin transform of `F(s)` over the fundamental</span>
-<span class="sd">    strip given by ``strip=(a, b)``.</span>
-
-<span class="sd">    This can be defined as</span>
-
-<span class="sd">    .. math:: f(x) = \int_{c - i\infty}^{c + i\infty} x^{-s} F(s) \mathrm{d}s,</span>
-
-<span class="sd">    for any `c` in the fundamental strip. Under certain regularity</span>
-<span class="sd">    conditions on `F` and/or `f`,</span>
-<span class="sd">    this recovers `f` from its Mellin transform `F`</span>
-<span class="sd">    (and vice versa), for positive real `x`.</span>
-
-<span class="sd">    One of `a` or `b` may be passed as None; a suitable `c` will be</span>
-<span class="sd">    inferred.</span>
-
-<span class="sd">    If the integral cannot be computed in closed form, this function returns</span>
-<span class="sd">    an unevaluated InverseMellinTransform object.</span>
-
-<span class="sd">    Note that this function will assume x to be positive and real, regardless</span>
-<span class="sd">    of the sympy assumptions!</span>
-
-<span class="sd">    For a description of possible hints, refer to the docstring of</span>
-<span class="sd">    :func:`sympy.integrals.transforms.IntegralTransform.doit`.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.integrals.transforms import inverse_mellin_transform</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import oo, gamma</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, s</span>
-<span class="sd">    &gt;&gt;&gt; inverse_mellin_transform(gamma(s), s, x, (0, oo))</span>
-<span class="sd">    exp(-x)</span>
-
-<span class="sd">    The fundamental strip matters:</span>
-
-<span class="sd">    &gt;&gt;&gt; f = 1/(s**2 - 1)</span>
-<span class="sd">    &gt;&gt;&gt; inverse_mellin_transform(f, s, x, (-oo, -1))</span>
-<span class="sd">    x*(1 - 1/x**2)*Heaviside(x - 1)/2</span>
-<span class="sd">    &gt;&gt;&gt; inverse_mellin_transform(f, s, x, (-1, 1))</span>
-<span class="sd">    -x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x)</span>
-<span class="sd">    &gt;&gt;&gt; inverse_mellin_transform(f, s, x, (1, oo))</span>
-<span class="sd">    (-x**2/2 + 1/2)*Heaviside(-x + 1)/x</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    mellin_transform</span>
-<span class="sd">    hankel_transform, inverse_hankel_transform</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">InverseMellinTransform</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">strip</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">strip</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-
-<span class="c">##########################################################################</span>
-<span class="c"># Laplace Transform</span>
-<span class="c">##########################################################################</span>
-</div>
-<span class="k">def</span> <span class="nf">_simplifyconds</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Naively simplify some conditions occuring in ``expr``, given that Re(s) &gt; a.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.integrals.transforms import _simplifyconds as simp</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import sympify as S</span>
-<span class="sd">    &gt;&gt;&gt; simp(abs(x**2) &lt; 1, x, 1)</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; simp(abs(x**2) &lt; 1, x, 2)</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; simp(abs(x**2) &lt; 1, x, 0)</span>
-<span class="sd">    Abs(x**2) &lt; 1</span>
-<span class="sd">    &gt;&gt;&gt; simp(abs(1/x**2) &lt; 1, x, 1)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; simp(S(1) &lt; abs(x), x, 1)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; simp(S(1) &lt; abs(1/x), x, 1)</span>
-<span class="sd">    False</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Ne</span>
-<span class="sd">    &gt;&gt;&gt; simp(Ne(1, x**3), x, 1)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; simp(Ne(1, x**3), x, 2)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; simp(Ne(1, x**3), x, 0)</span>
-<span class="sd">    1 != x**3</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="p">(</span> <span class="n">StrictGreaterThan</span><span class="p">,</span> <span class="n">StrictLessThan</span><span class="p">,</span>
-        <span class="n">Unequality</span> <span class="p">)</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Abs</span>
-
-    <span class="k">def</span> <span class="nf">power</span><span class="p">(</span><span class="n">ex</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">ex</span> <span class="o">==</span> <span class="n">s</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">ex</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">ex</span><span class="o">.</span><span class="n">base</span> <span class="o">==</span> <span class="n">s</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">ex</span><span class="o">.</span><span class="n">exp</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">bigger</span><span class="p">(</span><span class="n">ex1</span><span class="p">,</span> <span class="n">ex2</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Return True only if |ex1| &gt; |ex2|, False only if |ex1| &lt; |ex2|.</span>
-<span class="sd">            Else return None. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">ex1</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="ow">and</span> <span class="n">ex2</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">ex1</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Abs</span><span class="p">:</span>
-            <span class="n">ex1</span> <span class="o">=</span> <span class="n">ex1</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">ex2</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Abs</span><span class="p">:</span>
-            <span class="n">ex2</span> <span class="o">=</span> <span class="n">ex2</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">ex1</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">bigger</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">ex2</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">ex1</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">power</span><span class="p">(</span><span class="n">ex2</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="ow">and</span> <span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">ex1</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">and</span> <span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">ex1</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">replie</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; simplify x &lt; y &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">is_positive</span> <span class="ow">or</span> <span class="n">x</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Abs</span><span class="p">)</span> \
-                <span class="ow">or</span> <span class="ow">not</span> <span class="p">(</span><span class="n">y</span><span class="o">.</span><span class="n">is_positive</span> <span class="ow">or</span> <span class="n">y</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Abs</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">x</span> <span class="o">&lt;</span> <span class="n">y</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">bigger</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">r</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="ow">not</span> <span class="n">r</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">x</span> <span class="o">&lt;</span> <span class="n">y</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">replue</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">bigger</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span> <span class="ow">in</span> <span class="p">(</span><span class="bp">True</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="n">Unequality</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">repl</span><span class="p">(</span><span class="n">ex</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ex</span><span class="p">,</span> <span class="nb">bool</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">ex</span>
-        <span class="k">return</span> <span class="n">ex</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">repl</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">StrictLessThan</span><span class="p">,</span> <span class="n">replie</span><span class="p">)</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">repl</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">StrictGreaterThan</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">replie</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">repl</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Unequality</span><span class="p">,</span> <span class="n">replue</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">expr</span>
-
-
-<span class="nd">@_noconds</span>
-<span class="k">def</span> <span class="nf">_laplace_transform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">s_</span><span class="p">,</span> <span class="n">simplify</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; The backend function for Laplace transforms. &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="n">Max</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">Abs</span><span class="p">,</span> <span class="n">Min</span><span class="p">,</span> <span class="n">periodic_argument</span> <span class="k">as</span> <span class="n">arg</span><span class="p">,</span>
-                       <span class="n">cos</span><span class="p">,</span> <span class="n">Wild</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">polar_lift</span><span class="p">)</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;s&#39;</span><span class="p">)</span>
-    <span class="n">F</span> <span class="o">=</span> <span class="n">integrate</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">s</span><span class="o">*</span><span class="n">t</span><span class="p">)</span> <span class="o">*</span> <span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">F</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Integral</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_simplify</span><span class="p">(</span><span class="n">F</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">s_</span><span class="p">),</span> <span class="n">simplify</span><span class="p">),</span> <span class="o">-</span><span class="n">oo</span><span class="p">,</span> <span class="bp">True</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">F</span><span class="o">.</span><span class="n">is_Piecewise</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">IntegralTransformError</span><span class="p">(</span>
-            <span class="s">&#39;Laplace&#39;</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="s">&#39;could not compute integral&#39;</span><span class="p">)</span>
-
-    <span class="n">F</span><span class="p">,</span> <span class="n">cond</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">F</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Integral</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="n">IntegralTransformError</span><span class="p">(</span>
-            <span class="s">&#39;Laplace&#39;</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="s">&#39;integral in unexpected form&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">process_conds</span><span class="p">(</span><span class="n">conds</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Turn ``conds`` into a strip and auxiliary conditions. &quot;&quot;&quot;</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="o">-</span><span class="n">oo</span>
-        <span class="n">aux</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="n">conds</span> <span class="o">=</span> <span class="n">conjuncts</span><span class="p">(</span><span class="n">to_cnf</span><span class="p">(</span><span class="n">conds</span><span class="p">))</span>
-        <span class="n">u</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;u&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">w1</span><span class="p">,</span> <span class="n">w2</span><span class="p">,</span> <span class="n">w3</span><span class="p">,</span> <span class="n">w4</span><span class="p">,</span> <span class="n">w5</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span>
-            <span class="s">&#39;p q w1 w2 w3 w4 w5&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Wild</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">s</span><span class="p">])</span>
-        <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">conds</span><span class="p">:</span>
-            <span class="n">a_</span> <span class="o">=</span> <span class="n">oo</span>
-            <span class="n">aux_</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">disjuncts</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
-                <span class="n">m</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">arg</span><span class="p">((</span><span class="n">s</span> <span class="o">+</span> <span class="n">w3</span><span class="p">)</span><span class="o">**</span><span class="n">p</span><span class="o">*</span><span class="n">q</span><span class="p">,</span> <span class="n">w1</span><span class="p">))</span> <span class="o">&lt;</span> <span class="n">w2</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">m</span><span class="p">:</span>
-                    <span class="n">m</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">arg</span><span class="p">((</span><span class="n">s</span> <span class="o">+</span> <span class="n">w3</span><span class="p">)</span><span class="o">**</span><span class="n">p</span><span class="o">*</span><span class="n">q</span><span class="p">,</span> <span class="n">w1</span><span class="p">))</span> <span class="o">&lt;=</span> <span class="n">w2</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">m</span><span class="p">:</span>
-                    <span class="n">m</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">arg</span><span class="p">((</span><span class="n">polar_lift</span><span class="p">(</span><span class="n">s</span> <span class="o">+</span> <span class="n">w3</span><span class="p">))</span><span class="o">**</span><span class="n">p</span><span class="o">*</span><span class="n">q</span><span class="p">,</span> <span class="n">w1</span><span class="p">))</span> <span class="o">&lt;</span> <span class="n">w2</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">m</span><span class="p">:</span>
-                    <span class="n">m</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">arg</span><span class="p">((</span><span class="n">polar_lift</span><span class="p">(</span><span class="n">s</span> <span class="o">+</span> <span class="n">w3</span><span class="p">))</span><span class="o">**</span><span class="n">p</span><span class="o">*</span><span class="n">q</span><span class="p">,</span> <span class="n">w1</span><span class="p">))</span> <span class="o">&lt;=</span> <span class="n">w2</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">m</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">m</span><span class="p">[</span><span class="n">q</span><span class="p">]</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">m</span><span class="p">[</span><span class="n">w2</span><span class="p">]</span><span class="o">/</span><span class="n">m</span><span class="p">[</span><span class="n">p</span><span class="p">]</span> <span class="o">==</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span>
-                        <span class="n">d</span> <span class="o">=</span> <span class="n">re</span><span class="p">(</span><span class="n">s</span> <span class="o">+</span> <span class="n">m</span><span class="p">[</span><span class="n">w3</span><span class="p">])</span> <span class="o">&gt;</span> <span class="mi">0</span>
-                <span class="n">m</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">match</span><span class="p">(</span>
-                    <span class="mi">0</span> <span class="o">&lt;</span> <span class="n">cos</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">arg</span><span class="p">(</span><span class="n">s</span><span class="o">**</span><span class="n">w1</span><span class="o">*</span><span class="n">w5</span><span class="p">,</span> <span class="n">q</span><span class="p">))</span><span class="o">*</span><span class="n">w2</span><span class="p">)</span><span class="o">*</span><span class="nb">abs</span><span class="p">(</span><span class="n">s</span><span class="o">**</span><span class="n">w3</span><span class="p">)</span><span class="o">**</span><span class="n">w4</span> <span class="o">-</span> <span class="n">p</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">m</span><span class="p">:</span>
-                    <span class="n">m</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="mi">0</span> <span class="o">&lt;</span> <span class="n">cos</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span>
-                        <span class="n">arg</span><span class="p">(</span><span class="n">polar_lift</span><span class="p">(</span><span class="n">s</span><span class="p">)</span><span class="o">**</span><span class="n">w1</span><span class="o">*</span><span class="n">w5</span><span class="p">,</span> <span class="n">q</span><span class="p">))</span><span class="o">*</span><span class="n">w2</span><span class="p">)</span><span class="o">*</span><span class="nb">abs</span><span class="p">(</span><span class="n">s</span><span class="o">**</span><span class="n">w3</span><span class="p">)</span><span class="o">**</span><span class="n">w4</span> <span class="o">-</span> <span class="n">p</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">m</span> <span class="ow">and</span> <span class="nb">all</span><span class="p">(</span><span class="n">m</span><span class="p">[</span><span class="n">wild</span><span class="p">]</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="k">for</span> <span class="n">wild</span> <span class="ow">in</span> <span class="p">[</span><span class="n">w1</span><span class="p">,</span> <span class="n">w2</span><span class="p">,</span> <span class="n">w3</span><span class="p">,</span> <span class="n">w4</span><span class="p">,</span> <span class="n">w5</span><span class="p">]):</span>
-                    <span class="n">d</span> <span class="o">=</span> <span class="n">re</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">m</span><span class="p">[</span><span class="n">p</span><span class="p">]</span>
-                <span class="n">d_</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span>
-                    <span class="n">re</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()[</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">re</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="n">t</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">d</span><span class="o">.</span><span class="n">is_Relational</span> <span class="ow">or</span> \
-                    <span class="n">d</span><span class="o">.</span><span class="n">rel_op</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">(</span><span class="s">&#39;&gt;&#39;</span><span class="p">,</span> <span class="s">&#39;&gt;=&#39;</span><span class="p">,</span> <span class="s">&#39;&lt;&#39;</span><span class="p">,</span> <span class="s">&#39;&lt;=&#39;</span><span class="p">)</span> \
-                        <span class="ow">or</span> <span class="n">d_</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">d_</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
-                    <span class="n">aux_</span> <span class="o">+=</span> <span class="p">[</span><span class="n">d</span><span class="p">]</span>
-                    <span class="k">continue</span>
-                <span class="n">soln</span> <span class="o">=</span> <span class="n">_solve_inequality</span><span class="p">(</span><span class="n">d_</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">soln</span><span class="o">.</span><span class="n">is_Relational</span> <span class="ow">or</span> \
-                        <span class="n">soln</span><span class="o">.</span><span class="n">rel_op</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">(</span><span class="s">&#39;&gt;&#39;</span><span class="p">,</span> <span class="s">&#39;&gt;=&#39;</span><span class="p">,</span> <span class="s">&#39;&lt;&#39;</span><span class="p">,</span> <span class="s">&#39;&lt;=&#39;</span><span class="p">):</span>
-                    <span class="n">aux_</span> <span class="o">+=</span> <span class="p">[</span><span class="n">d</span><span class="p">]</span>
-                    <span class="k">continue</span>
-                <span class="k">if</span> <span class="n">soln</span><span class="o">.</span><span class="n">lts</span> <span class="o">==</span> <span class="n">t</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="n">IntegralTransformError</span><span class="p">(</span><span class="s">&#39;Laplace&#39;</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span>
-                                         <span class="s">&#39;convergence not in half-plane?&#39;</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">a_</span> <span class="o">=</span> <span class="n">Min</span><span class="p">(</span><span class="n">soln</span><span class="o">.</span><span class="n">lts</span><span class="p">,</span> <span class="n">a_</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">a_</span> <span class="o">!=</span> <span class="n">oo</span><span class="p">:</span>
-                <span class="n">a</span> <span class="o">=</span> <span class="n">Max</span><span class="p">(</span><span class="n">a_</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">aux</span> <span class="o">=</span> <span class="n">And</span><span class="p">(</span><span class="n">aux</span><span class="p">,</span> <span class="n">Or</span><span class="p">(</span><span class="o">*</span><span class="n">aux_</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">a</span><span class="p">,</span> <span class="n">aux</span>
-
-    <span class="n">conds</span> <span class="o">=</span> <span class="p">[</span><span class="n">process_conds</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">disjuncts</span><span class="p">(</span><span class="n">cond</span><span class="p">)]</span>
-    <span class="n">conds2</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">conds</span> <span class="k">if</span> <span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span> <span class="ow">and</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="o">-</span><span class="n">oo</span><span class="p">]</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">conds2</span><span class="p">:</span>
-        <span class="n">conds2</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">conds</span> <span class="k">if</span> <span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">]</span>
-    <span class="n">conds</span> <span class="o">=</span> <span class="n">conds2</span>
-
-    <span class="k">def</span> <span class="nf">cnt</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="nb">bool</span><span class="p">):</span>
-            <span class="k">return</span> <span class="mi">0</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">count_ops</span><span class="p">()</span>
-    <span class="n">conds</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">cnt</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">])))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">conds</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">IntegralTransformError</span><span class="p">(</span><span class="s">&#39;Laplace&#39;</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="s">&#39;no convergence found&#39;</span><span class="p">)</span>
-    <span class="n">a</span><span class="p">,</span> <span class="n">aux</span> <span class="o">=</span> <span class="n">conds</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">sbs</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="nb">bool</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">expr</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">s_</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">simplify</span><span class="p">:</span>
-        <span class="n">F</span> <span class="o">=</span> <span class="n">_simplifyconds</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-        <span class="n">aux</span> <span class="o">=</span> <span class="n">_simplifyconds</span><span class="p">(</span><span class="n">aux</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">_simplify</span><span class="p">(</span><span class="n">F</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">s_</span><span class="p">),</span> <span class="n">simplify</span><span class="p">),</span> <span class="n">sbs</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">sbs</span><span class="p">(</span><span class="n">aux</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">LaplaceTransform</span><span class="p">(</span><span class="n">IntegralTransform</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Class representing unevaluated Laplace transforms.</span>
-
-<span class="sd">    For usage of this class, see the :class:`IntegralTransform` docstring.</span>
-
-<span class="sd">    For how to compute Laplace transforms, see the :func:`laplace_transform`</span>
-<span class="sd">    docstring.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_name</span> <span class="o">=</span> <span class="s">&#39;Laplace&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_compute_transform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_laplace_transform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_as_integral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">exp</span>
-        <span class="k">return</span> <span class="n">Integral</span><span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">s</span><span class="o">*</span><span class="n">t</span><span class="p">),</span> <span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Class representing unevaluated Laplace transforms.</span>
-
-<span class="sd">    For usage of this class, see the :class:`IntegralTransform` docstring.</span>
-<span class="sd">    For how to compute Laplace transforms, see the :func:`laplace_transform`</span>
-<span class="sd">    docstring.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">_collapse_extra</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">extra</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">And</span><span class="p">,</span> <span class="n">Max</span>
-        <span class="n">conds</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">planes</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">plane</span><span class="p">,</span> <span class="n">cond</span> <span class="ow">in</span> <span class="n">extra</span><span class="p">:</span>
-            <span class="n">conds</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">cond</span><span class="p">)</span>
-            <span class="n">planes</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">plane</span><span class="p">)</span>
-        <span class="n">cond</span> <span class="o">=</span> <span class="n">And</span><span class="p">(</span><span class="o">*</span><span class="n">conds</span><span class="p">)</span>
-        <span class="n">plane</span> <span class="o">=</span> <span class="n">Max</span><span class="p">(</span><span class="o">*</span><span class="n">planes</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">cond</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">IntegralTransformError</span><span class="p">(</span>
-                <span class="s">&#39;Laplace&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="s">&#39;No combined convergence.&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">plane</span><span class="p">,</span> <span class="n">cond</span>
-
-
-<div class="viewcode-block" id="laplace_transform"><a class="viewcode-back" href="../../../modules/integrals/integrals.html#sympy.integrals.transforms.laplace_transform">[docs]</a><span class="k">def</span> <span class="nf">laplace_transform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Compute the Laplace Transform `F(s)` of `f(t)`,</span>
-
-<span class="sd">    .. math :: F(s) = \int_0^\infty e^{-st} f(t) \mathrm{d}t.</span>
-
-<span class="sd">    For all &quot;sensible&quot; functions, this converges absolutely in a</span>
-<span class="sd">    half plane  `a &lt; Re(s)`.</span>
-
-<span class="sd">    This function returns (F, a, cond)</span>
-<span class="sd">    where `F` is the Laplace transform of `f`, `Re(s) &gt; a` is the half-plane</span>
-<span class="sd">    of convergence, and cond are auxiliary convergence conditions.</span>
-
-<span class="sd">    If the integral cannot be computed in closed form, this function returns</span>
-<span class="sd">    an unevaluated LaplaceTransform object.</span>
-
-<span class="sd">    For a description of possible hints, refer to the docstring of</span>
-<span class="sd">    :func:`sympy.integrals.transforms.IntegralTransform.doit`. If ``noconds=True``,</span>
-<span class="sd">    only `F` will be returned (i.e. not ``cond``, and also not the plane ``a``).</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.integrals import laplace_transform</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import t, s, a</span>
-<span class="sd">    &gt;&gt;&gt; laplace_transform(t**a, t, s)</span>
-<span class="sd">    (s**(-a - 1)*gamma(a + 1), 0, -re(a) &lt; 1)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    inverse_laplace_transform, mellin_transform, fourier_transform</span>
-<span class="sd">    hankel_transform, inverse_hankel_transform</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">LaplaceTransform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-</div>
-<span class="nd">@_noconds_</span><span class="p">(</span><span class="bp">True</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_inverse_laplace_transform</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t_</span><span class="p">,</span> <span class="n">plane</span><span class="p">,</span> <span class="n">simplify</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; The backend function for inverse Laplace transforms. &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exp</span><span class="p">,</span> <span class="n">Heaviside</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">expand_complex</span><span class="p">,</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">Piecewise</span>
-    <span class="kn">from</span> <span class="nn">sympy.integrals.meijerint</span> <span class="kn">import</span> <span class="n">meijerint_inversion</span><span class="p">,</span> <span class="n">_get_coeff_exp</span>
-    <span class="c"># There are two strategies we can try:</span>
-    <span class="c"># 1) Use inverse mellin transforms - related by a simple change of variables.</span>
-    <span class="c"># 2) Use the inversion integral.</span>
-
-    <span class="n">t</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">pw_simp</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Simplify a piecewise expression from hyperexpand. &quot;&quot;&quot;</span>
-        <span class="c"># XXX we break modularity here!</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Piecewise</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">argument</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">exponent</span> <span class="o">=</span> <span class="n">_get_coeff_exp</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
-        <span class="n">e1</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">e2</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">Heaviside</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="nb">abs</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span> <span class="o">-</span> <span class="n">t</span><span class="o">**</span><span class="n">exponent</span><span class="p">)</span><span class="o">*</span><span class="n">e1</span> \
-            <span class="o">+</span> <span class="n">Heaviside</span><span class="p">(</span><span class="n">t</span><span class="o">**</span><span class="n">exponent</span> <span class="o">-</span> <span class="mi">1</span><span class="o">/</span><span class="nb">abs</span><span class="p">(</span><span class="n">coeff</span><span class="p">))</span><span class="o">*</span><span class="n">e2</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">f</span><span class="p">,</span> <span class="n">cond</span> <span class="o">=</span> <span class="n">inverse_mellin_transform</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">t</span><span class="p">),</span> <span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="n">oo</span><span class="p">),</span>
-                                           <span class="n">needeval</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">noconds</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">IntegralTransformError</span><span class="p">:</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="k">if</span> <span class="n">f</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">meijerint_inversion</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">f</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">IntegralTransformError</span><span class="p">(</span><span class="s">&#39;Inverse Laplace&#39;</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_Piecewise</span><span class="p">:</span>
-            <span class="n">f</span><span class="p">,</span> <span class="n">cond</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Integral</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="n">IntegralTransformError</span><span class="p">(</span><span class="s">&#39;Inverse Laplace&#39;</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span>
-                                     <span class="s">&#39;inversion integral of unrecognised form.&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">cond</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">Piecewise</span><span class="p">,</span> <span class="n">pw_simp</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_Piecewise</span><span class="p">:</span>
-        <span class="c"># many of the functions called below can&#39;t work with piecewise</span>
-        <span class="c"># (b/c it has a bool in args)</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">t_</span><span class="p">),</span> <span class="n">cond</span>
-
-    <span class="n">u</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;u&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">simp_heaviside</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">t</span><span class="p">),</span> <span class="n">u</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Heaviside</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-        <span class="n">rel</span> <span class="o">=</span> <span class="n">_solve_inequality</span><span class="p">(</span><span class="n">a</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">rel</span><span class="o">.</span><span class="n">lts</span> <span class="o">==</span> <span class="n">u</span><span class="p">:</span>
-            <span class="n">k</span> <span class="o">=</span> <span class="n">log</span><span class="p">(</span><span class="n">rel</span><span class="o">.</span><span class="n">gts</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Heaviside</span><span class="p">(</span><span class="n">t</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">k</span> <span class="o">=</span> <span class="n">log</span><span class="p">(</span><span class="n">rel</span><span class="o">.</span><span class="n">lts</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Heaviside</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">t</span> <span class="o">+</span> <span class="n">k</span><span class="p">))</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">Heaviside</span><span class="p">,</span> <span class="n">simp_heaviside</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">simp_exp</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expand_complex</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">exp</span><span class="p">,</span> <span class="n">simp_exp</span><span class="p">)</span>
-
-    <span class="c"># TODO it would be nice to fix cosh and sinh ... simplify messes these</span>
-    <span class="c">#      exponentials up</span>
-
-    <span class="k">return</span> <span class="n">_simplify</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">t_</span><span class="p">),</span> <span class="n">simplify</span><span class="p">),</span> <span class="n">cond</span>
-
-
-<span class="k">class</span> <span class="nc">InverseLaplaceTransform</span><span class="p">(</span><span class="n">IntegralTransform</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Class representing unevaluated inverse Laplace transforms.</span>
-
-<span class="sd">    For usage of this class, see the :class:`IntegralTransform` docstring.</span>
-
-<span class="sd">    For how to compute inverse Laplace transforms, see the</span>
-<span class="sd">    :func:`inverse_laplace_transform` docstring.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">4</span>
-
-    <span class="n">_name</span> <span class="o">=</span> <span class="s">&#39;Inverse Laplace&#39;</span>
-    <span class="n">_none_sentinel</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;None&#39;</span><span class="p">)</span>
-    <span class="n">_c</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;c&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">plane</span><span class="p">,</span> <span class="o">**</span><span class="n">opts</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">plane</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">plane</span> <span class="o">=</span> <span class="n">InverseLaplaceTransform</span><span class="o">.</span><span class="n">_none_sentinel</span>
-        <span class="k">return</span> <span class="n">IntegralTransform</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">plane</span><span class="p">,</span> <span class="o">**</span><span class="n">opts</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">fundamental_plane</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">plane</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">plane</span> <span class="ow">is</span> <span class="n">InverseLaplaceTransform</span><span class="o">.</span><span class="n">_none_sentinel</span><span class="p">:</span>
-            <span class="n">plane</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">return</span> <span class="n">plane</span>
-
-    <span class="k">def</span> <span class="nf">_compute_transform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_inverse_laplace_transform</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">fundamental_plane</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_as_integral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">I</span><span class="p">,</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">exp</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">_c</span>
-        <span class="k">return</span> <span class="n">Integral</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">s</span><span class="o">*</span><span class="n">t</span><span class="p">)</span><span class="o">*</span><span class="n">F</span><span class="p">,</span> <span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">c</span> <span class="o">-</span> <span class="n">I</span><span class="o">*</span><span class="n">oo</span><span class="p">,</span> <span class="n">c</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">oo</span><span class="p">))</span>
-
-
-<div class="viewcode-block" id="inverse_laplace_transform"><a class="viewcode-back" href="../../../modules/integrals/integrals.html#sympy.integrals.transforms.inverse_laplace_transform">[docs]</a><span class="k">def</span> <span class="nf">inverse_laplace_transform</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">plane</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Compute the inverse Laplace transform of `F(s)`, defined as</span>
-
-<span class="sd">    .. math :: f(t) = \int_{c-i\infty}^{c+i\infty} e^{st} F(s) \mathrm{d}s,</span>
-
-<span class="sd">    for `c` so large that `F(s)` has no singularites in the</span>
-<span class="sd">    half-plane `Re(s) &gt; c-\epsilon`.</span>
-
-<span class="sd">    The plane can be specified by</span>
-<span class="sd">    argument ``plane``, but will be inferred if passed as None.</span>
-
-<span class="sd">    Under certain regularity conditions, this recovers `f(t)` from its</span>
-<span class="sd">    Laplace Transform `F(s)`, for non-negative `t`, and vice</span>
-<span class="sd">    versa.</span>
-
-<span class="sd">    If the integral cannot be computed in closed form, this function returns</span>
-<span class="sd">    an unevaluated InverseLaplaceTransform object.</span>
-
-<span class="sd">    Note that this function will always assume `t` to be real,</span>
-<span class="sd">    regardless of the sympy assumption on `t`.</span>
-
-<span class="sd">    For a description of possible hints, refer to the docstring of</span>
-<span class="sd">    :func:`sympy.integrals.transforms.IntegralTransform.doit`.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.integrals.transforms import inverse_laplace_transform</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import exp, Symbol</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import s, t</span>
-<span class="sd">    &gt;&gt;&gt; a = Symbol(&#39;a&#39;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; inverse_laplace_transform(exp(-a*s)/s, s, t)</span>
-<span class="sd">    Heaviside(-a + t)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    laplace_transform</span>
-<span class="sd">    hankel_transform, inverse_hankel_transform</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">InverseLaplaceTransform</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">plane</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-
-<span class="c">##########################################################################</span>
-<span class="c"># Fourier Transform</span>
-<span class="c">##########################################################################</span>
-</div>
-<span class="nd">@_noconds_</span><span class="p">(</span><span class="bp">True</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_fourier_transform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">simplify</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute a general Fourier-type transform</span>
-<span class="sd">        F(k) = a int_-oo^oo exp(b*I*x*k) f(x) dx.</span>
-
-<span class="sd">    For suitable choice of a and b, this reduces to the standard Fourier</span>
-<span class="sd">    and inverse Fourier transforms.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exp</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">oo</span>
-    <span class="n">F</span> <span class="o">=</span> <span class="n">integrate</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">f</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">b</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">k</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">oo</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">F</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Integral</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_simplify</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">simplify</span><span class="p">),</span> <span class="bp">True</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">F</span><span class="o">.</span><span class="n">is_Piecewise</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">IntegralTransformError</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="s">&#39;could not compute integral&#39;</span><span class="p">)</span>
-
-    <span class="n">F</span><span class="p">,</span> <span class="n">cond</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">F</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Integral</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="n">IntegralTransformError</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="s">&#39;integral in unexpected form&#39;</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">_simplify</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">simplify</span><span class="p">),</span> <span class="n">cond</span>
-
-
-<span class="k">class</span> <span class="nc">FourierTypeTransform</span><span class="p">(</span><span class="n">IntegralTransform</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Base class for Fourier transforms.</span>
-<span class="sd">        Specify cls._a and cls._b.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_compute_transform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_fourier_transform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span>
-                                  <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">_a</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">_b</span><span class="p">,</span>
-                                  <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">_name</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_as_integral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">I</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">_a</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">_b</span>
-        <span class="k">return</span> <span class="n">Integral</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">f</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">b</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">k</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">oo</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-
-
-<span class="k">class</span> <span class="nc">FourierTransform</span><span class="p">(</span><span class="n">FourierTypeTransform</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Class representing unevaluated Fourier transforms.</span>
-
-<span class="sd">    For usage of this class, see the :class:`IntegralTransform` docstring.</span>
-
-<span class="sd">    For how to compute Fourier transforms, see the :func:`fourier_transform`</span>
-<span class="sd">    docstring.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_name</span> <span class="o">=</span> <span class="s">&#39;Fourier&#39;</span>
-    <span class="n">_a</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">_b</span> <span class="o">=</span> <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-
-
-<div class="viewcode-block" id="fourier_transform"><a class="viewcode-back" href="../../../modules/integrals/integrals.html#sympy.integrals.transforms.fourier_transform">[docs]</a><span class="k">def</span> <span class="nf">fourier_transform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Compute the unitary, ordinary-frequency Fourier transform of `f`, defined</span>
-<span class="sd">    as</span>
-
-<span class="sd">    .. math:: F(k) = \int_{-\infty}^\infty f(x) e^{-2\pi i x k} \mathrm{d} x.</span>
-
-<span class="sd">    If the transform cannot be computed in closed form, this</span>
-<span class="sd">    function returns an unevaluated FourierTransform object.</span>
-
-<span class="sd">    For other Fourier transform conventions, see the function</span>
-<span class="sd">    :func:`sympy.integrals.transforms._fourier_transform`.</span>
-
-<span class="sd">    For a description of possible hints, refer to the docstring of</span>
-<span class="sd">    :func:`sympy.integrals.transforms.IntegralTransform.doit`.</span>
-<span class="sd">    Note that for this transform, by default ``noconds=True``.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import fourier_transform, exp</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, k</span>
-<span class="sd">    &gt;&gt;&gt; fourier_transform(exp(-x**2), x, k)</span>
-<span class="sd">    sqrt(pi)*exp(-pi**2*k**2)</span>
-<span class="sd">    &gt;&gt;&gt; fourier_transform(exp(-x**2), x, k, noconds=False)</span>
-<span class="sd">    (sqrt(pi)*exp(-pi**2*k**2), True)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    inverse_fourier_transform</span>
-<span class="sd">    sine_transform, inverse_sine_transform</span>
-<span class="sd">    cosine_transform, inverse_cosine_transform</span>
-<span class="sd">    hankel_transform, inverse_hankel_transform</span>
-<span class="sd">    mellin_transform, laplace_transform</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">FourierTransform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-</div>
-<span class="k">class</span> <span class="nc">InverseFourierTransform</span><span class="p">(</span><span class="n">FourierTypeTransform</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Class representing unevaluated inverse Fourier transforms.</span>
-
-<span class="sd">    For usage of this class, see the :class:`IntegralTransform` docstring.</span>
-
-<span class="sd">    For how to compute inverse Fourier transforms, see the</span>
-<span class="sd">    :func:`inverse_fourier_transform` docstring.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_name</span> <span class="o">=</span> <span class="s">&#39;Inverse Fourier&#39;</span>
-    <span class="n">_a</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">_b</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-
-
-<div class="viewcode-block" id="inverse_fourier_transform"><a class="viewcode-back" href="../../../modules/integrals/integrals.html#sympy.integrals.transforms.inverse_fourier_transform">[docs]</a><span class="k">def</span> <span class="nf">inverse_fourier_transform</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Compute the unitary, ordinary-frequency inverse Fourier transform of `F`,</span>
-<span class="sd">    defined as</span>
-
-<span class="sd">    .. math:: f(x) = \int_{-\infty}^\infty F(k) e^{2\pi i x k} \mathrm{d} k.</span>
-
-<span class="sd">    If the transform cannot be computed in closed form, this</span>
-<span class="sd">    function returns an unevaluated InverseFourierTransform object.</span>
-
-<span class="sd">    For other Fourier transform conventions, see the function</span>
-<span class="sd">    :func:`sympy.integrals.transforms._fourier_transform`.</span>
-
-<span class="sd">    For a description of possible hints, refer to the docstring of</span>
-<span class="sd">    :func:`sympy.integrals.transforms.IntegralTransform.doit`.</span>
-<span class="sd">    Note that for this transform, by default ``noconds=True``.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import inverse_fourier_transform, exp, sqrt, pi</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, k</span>
-<span class="sd">    &gt;&gt;&gt; inverse_fourier_transform(sqrt(pi)*exp(-(pi*k)**2), k, x)</span>
-<span class="sd">    exp(-x**2)</span>
-<span class="sd">    &gt;&gt;&gt; inverse_fourier_transform(sqrt(pi)*exp(-(pi*k)**2), k, x, noconds=False)</span>
-<span class="sd">    (exp(-x**2), True)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    fourier_transform</span>
-<span class="sd">    sine_transform, inverse_sine_transform</span>
-<span class="sd">    cosine_transform, inverse_cosine_transform</span>
-<span class="sd">    hankel_transform, inverse_hankel_transform</span>
-<span class="sd">    mellin_transform, laplace_transform</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">InverseFourierTransform</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-
-<span class="c">##########################################################################</span>
-<span class="c"># Fourier Sine and Cosine Transform</span>
-<span class="c">##########################################################################</span>
-</div>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">oo</span>
-
-
-<span class="nd">@_noconds_</span><span class="p">(</span><span class="bp">True</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_sine_cosine_transform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">simplify</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute a general sine or cosine-type transform</span>
-<span class="sd">        F(k) = a int_0^oo b*sin(x*k) f(x) dx.</span>
-<span class="sd">        F(k) = a int_0^oo b*cos(x*k) f(x) dx.</span>
-
-<span class="sd">    For suitable choice of a and b, this reduces to the standard sine/cosine</span>
-<span class="sd">    and inverse sine/cosine transforms.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">F</span> <span class="o">=</span> <span class="n">integrate</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">f</span><span class="o">*</span><span class="n">K</span><span class="p">(</span><span class="n">b</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">k</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">F</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Integral</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_simplify</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">simplify</span><span class="p">),</span> <span class="bp">True</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">F</span><span class="o">.</span><span class="n">is_Piecewise</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">IntegralTransformError</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="s">&#39;could not compute integral&#39;</span><span class="p">)</span>
-
-    <span class="n">F</span><span class="p">,</span> <span class="n">cond</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">F</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Integral</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="n">IntegralTransformError</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="s">&#39;integral in unexpected form&#39;</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">_simplify</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">simplify</span><span class="p">),</span> <span class="n">cond</span>
-
-
-<span class="k">class</span> <span class="nc">SineCosineTypeTransform</span><span class="p">(</span><span class="n">IntegralTransform</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Base class for sine and cosine transforms.</span>
-<span class="sd">    Specify cls._a and cls._b and cls._kern.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_compute_transform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_sine_cosine_transform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span>
-                                      <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">_a</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">_b</span><span class="p">,</span>
-                                      <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">_kern</span><span class="p">,</span>
-                                      <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">_name</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_as_integral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">I</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">_a</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">_b</span>
-        <span class="n">K</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">_kern</span>
-        <span class="k">return</span> <span class="n">Integral</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">f</span><span class="o">*</span><span class="n">K</span><span class="p">(</span><span class="n">b</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">k</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-
-
-<span class="k">class</span> <span class="nc">SineTransform</span><span class="p">(</span><span class="n">SineCosineTypeTransform</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Class representing unevaluated sine transforms.</span>
-
-<span class="sd">    For usage of this class, see the :class:`IntegralTransform` docstring.</span>
-
-<span class="sd">    For how to compute sine transforms, see the :func:`sine_transform`</span>
-<span class="sd">    docstring.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_name</span> <span class="o">=</span> <span class="s">&#39;Sine&#39;</span>
-    <span class="n">_kern</span> <span class="o">=</span> <span class="n">sin</span>
-    <span class="n">_a</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-    <span class="n">_b</span> <span class="o">=</span> <span class="mi">1</span>
-
-
-<div class="viewcode-block" id="sine_transform"><a class="viewcode-back" href="../../../modules/integrals/integrals.html#sympy.integrals.transforms.sine_transform">[docs]</a><span class="k">def</span> <span class="nf">sine_transform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Compute the unitary, ordinary-frequency sine transform of `f`, defined</span>
-<span class="sd">    as</span>
-
-<span class="sd">    .. math:: F(k) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty f(x) \sin(2\pi x k) \mathrm{d} x.</span>
-
-<span class="sd">    If the transform cannot be computed in closed form, this</span>
-<span class="sd">    function returns an unevaluated SineTransform object.</span>
-
-<span class="sd">    For a description of possible hints, refer to the docstring of</span>
-<span class="sd">    :func:`sympy.integrals.transforms.IntegralTransform.doit`.</span>
-<span class="sd">    Note that for this transform, by default ``noconds=True``.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import sine_transform, exp</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, k, a</span>
-<span class="sd">    &gt;&gt;&gt; sine_transform(x*exp(-a*x**2), x, k)</span>
-<span class="sd">    sqrt(2)*k*exp(-k**2/(4*a))/(4*a**(3/2))</span>
-<span class="sd">    &gt;&gt;&gt; sine_transform(x**(-a), x, k)</span>
-<span class="sd">    2**(-a + 1/2)*k**(a - 1)*gamma(-a/2 + 1)/gamma(a/2 + 1/2)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    fourier_transform, inverse_fourier_transform</span>
-<span class="sd">    inverse_sine_transform</span>
-<span class="sd">    cosine_transform, inverse_cosine_transform</span>
-<span class="sd">    hankel_transform, inverse_hankel_transform</span>
-<span class="sd">    mellin_transform, laplace_transform</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">SineTransform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-</div>
-<span class="k">class</span> <span class="nc">InverseSineTransform</span><span class="p">(</span><span class="n">SineCosineTypeTransform</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Class representing unevaluated inverse sine transforms.</span>
-
-<span class="sd">    For usage of this class, see the :class:`IntegralTransform` docstring.</span>
-
-<span class="sd">    For how to compute inverse sine transforms, see the</span>
-<span class="sd">    :func:`inverse_sine_transform` docstring.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_name</span> <span class="o">=</span> <span class="s">&#39;Inverse Sine&#39;</span>
-    <span class="n">_kern</span> <span class="o">=</span> <span class="n">sin</span>
-    <span class="n">_a</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-    <span class="n">_b</span> <span class="o">=</span> <span class="mi">1</span>
-
-
-<div class="viewcode-block" id="inverse_sine_transform"><a class="viewcode-back" href="../../../modules/integrals/integrals.html#sympy.integrals.transforms.inverse_sine_transform">[docs]</a><span class="k">def</span> <span class="nf">inverse_sine_transform</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Compute the unitary, ordinary-frequency inverse sine transform of `F`,</span>
-<span class="sd">    defined as</span>
-
-<span class="sd">    .. math:: f(x) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty F(k) \sin(2\pi x k) \mathrm{d} k.</span>
-
-<span class="sd">    If the transform cannot be computed in closed form, this</span>
-<span class="sd">    function returns an unevaluated InverseSineTransform object.</span>
-
-<span class="sd">    For a description of possible hints, refer to the docstring of</span>
-<span class="sd">    :func:`sympy.integrals.transforms.IntegralTransform.doit`.</span>
-<span class="sd">    Note that for this transform, by default ``noconds=True``.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import inverse_sine_transform, exp, sqrt, gamma, pi</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, k, a</span>
-<span class="sd">    &gt;&gt;&gt; inverse_sine_transform(2**((1-2*a)/2)*k**(a - 1)*</span>
-<span class="sd">    ...     gamma(-a/2 + 1)/gamma((a+1)/2), k, x)</span>
-<span class="sd">    x**(-a)</span>
-<span class="sd">    &gt;&gt;&gt; inverse_sine_transform(sqrt(2)*k*exp(-k**2/(4*a))/(4*sqrt(a)**3), k, x)</span>
-<span class="sd">    x*exp(-a*x**2)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    fourier_transform, inverse_fourier_transform</span>
-<span class="sd">    sine_transform</span>
-<span class="sd">    cosine_transform, inverse_cosine_transform</span>
-<span class="sd">    hankel_transform, inverse_hankel_transform</span>
-<span class="sd">    mellin_transform, laplace_transform</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">InverseSineTransform</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-</div>
-<span class="k">class</span> <span class="nc">CosineTransform</span><span class="p">(</span><span class="n">SineCosineTypeTransform</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Class representing unevaluated cosine transforms.</span>
-
-<span class="sd">    For usage of this class, see the :class:`IntegralTransform` docstring.</span>
-
-<span class="sd">    For how to compute cosine transforms, see the :func:`cosine_transform`</span>
-<span class="sd">    docstring.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_name</span> <span class="o">=</span> <span class="s">&#39;Cosine&#39;</span>
-    <span class="n">_kern</span> <span class="o">=</span> <span class="n">cos</span>
-    <span class="n">_a</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-    <span class="n">_b</span> <span class="o">=</span> <span class="mi">1</span>
-
-
-<div class="viewcode-block" id="cosine_transform"><a class="viewcode-back" href="../../../modules/integrals/integrals.html#sympy.integrals.transforms.cosine_transform">[docs]</a><span class="k">def</span> <span class="nf">cosine_transform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Compute the unitary, ordinary-frequency cosine transform of `f`, defined</span>
-<span class="sd">    as</span>
-
-<span class="sd">    .. math:: F(k) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty f(x) \cos(2\pi x k) \mathrm{d} x.</span>
-
-<span class="sd">    If the transform cannot be computed in closed form, this</span>
-<span class="sd">    function returns an unevaluated CosineTransform object.</span>
-
-<span class="sd">    For a description of possible hints, refer to the docstring of</span>
-<span class="sd">    :func:`sympy.integrals.transforms.IntegralTransform.doit`.</span>
-<span class="sd">    Note that for this transform, by default ``noconds=True``.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import cosine_transform, exp, sqrt, cos</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, k, a</span>
-<span class="sd">    &gt;&gt;&gt; cosine_transform(exp(-a*x), x, k)</span>
-<span class="sd">    sqrt(2)*a/(sqrt(pi)*(a**2 + k**2))</span>
-<span class="sd">    &gt;&gt;&gt; cosine_transform(exp(-a*sqrt(x))*cos(a*sqrt(x)), x, k)</span>
-<span class="sd">    a*(-sinh(a**2/(2*k)) + cosh(a**2/(2*k)))/(2*k**(3/2))</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    fourier_transform, inverse_fourier_transform,</span>
-<span class="sd">    sine_transform, inverse_sine_transform</span>
-<span class="sd">    inverse_cosine_transform</span>
-<span class="sd">    hankel_transform, inverse_hankel_transform</span>
-<span class="sd">    mellin_transform, laplace_transform</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">CosineTransform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-</div>
-<span class="k">class</span> <span class="nc">InverseCosineTransform</span><span class="p">(</span><span class="n">SineCosineTypeTransform</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Class representing unevaluated inverse cosine transforms.</span>
-
-<span class="sd">    For usage of this class, see the :class:`IntegralTransform` docstring.</span>
-
-<span class="sd">    For how to compute inverse cosine transforms, see the</span>
-<span class="sd">    :func:`inverse_cosine_transform` docstring.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_name</span> <span class="o">=</span> <span class="s">&#39;Inverse Cosine&#39;</span>
-    <span class="n">_kern</span> <span class="o">=</span> <span class="n">cos</span>
-    <span class="n">_a</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-    <span class="n">_b</span> <span class="o">=</span> <span class="mi">1</span>
-
-
-<div class="viewcode-block" id="inverse_cosine_transform"><a class="viewcode-back" href="../../../modules/integrals/integrals.html#sympy.integrals.transforms.inverse_cosine_transform">[docs]</a><span class="k">def</span> <span class="nf">inverse_cosine_transform</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Compute the unitary, ordinary-frequency inverse cosine transform of `F`,</span>
-<span class="sd">    defined as</span>
-
-<span class="sd">    .. math:: f(x) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty F(k) \cos(2\pi x k) \mathrm{d} k.</span>
-
-<span class="sd">    If the transform cannot be computed in closed form, this</span>
-<span class="sd">    function returns an unevaluated InverseCosineTransform object.</span>
-
-<span class="sd">    For a description of possible hints, refer to the docstring of</span>
-<span class="sd">    :func:`sympy.integrals.transforms.IntegralTransform.doit`.</span>
-<span class="sd">    Note that for this transform, by default ``noconds=True``.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import inverse_cosine_transform, exp, sqrt, pi</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, k, a</span>
-<span class="sd">    &gt;&gt;&gt; inverse_cosine_transform(sqrt(2)*a/(sqrt(pi)*(a**2 + k**2)), k, x)</span>
-<span class="sd">    -sinh(a*x) + cosh(a*x)</span>
-<span class="sd">    &gt;&gt;&gt; inverse_cosine_transform(1/sqrt(k), k, x)</span>
-<span class="sd">    1/sqrt(x)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    fourier_transform, inverse_fourier_transform,</span>
-<span class="sd">    sine_transform, inverse_sine_transform</span>
-<span class="sd">    cosine_transform</span>
-<span class="sd">    hankel_transform, inverse_hankel_transform</span>
-<span class="sd">    mellin_transform, laplace_transform</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">InverseCosineTransform</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-
-<span class="c">##########################################################################</span>
-<span class="c"># Hankel Transform</span>
-<span class="c">##########################################################################</span>
-</div>
-<span class="nd">@_noconds_</span><span class="p">(</span><span class="bp">True</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_hankel_transform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">simplify</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute a general Hankel transform</span>
-
-<span class="sd">    .. math:: F_\nu(k) = \int_{0}^\infty f(r) J_\nu(k r) r \mathrm{d} r.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">besselj</span><span class="p">,</span> <span class="n">oo</span>
-    <span class="n">F</span> <span class="o">=</span> <span class="n">integrate</span><span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="n">besselj</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">k</span><span class="o">*</span><span class="n">r</span><span class="p">)</span><span class="o">*</span><span class="n">r</span><span class="p">,</span> <span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">F</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Integral</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_simplify</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">simplify</span><span class="p">),</span> <span class="bp">True</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">F</span><span class="o">.</span><span class="n">is_Piecewise</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">IntegralTransformError</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="s">&#39;could not compute integral&#39;</span><span class="p">)</span>
-
-    <span class="n">F</span><span class="p">,</span> <span class="n">cond</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">F</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Integral</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="n">IntegralTransformError</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="s">&#39;integral in unexpected form&#39;</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">_simplify</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">simplify</span><span class="p">),</span> <span class="n">cond</span>
-
-
-<span class="k">class</span> <span class="nc">HankelTypeTransform</span><span class="p">(</span><span class="n">IntegralTransform</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Base class for Hankel transforms.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">4</span>
-
-    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_compute_transform</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="p">,</span>
-                                       <span class="bp">self</span><span class="o">.</span><span class="n">function_variable</span><span class="p">,</span>
-                                       <span class="bp">self</span><span class="o">.</span><span class="n">transform_variable</span><span class="p">,</span>
-                                       <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">3</span><span class="p">],</span>
-                                       <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_compute_transform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_hankel_transform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_name</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_as_integral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">nu</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">besselj</span><span class="p">,</span> <span class="n">oo</span>
-        <span class="k">return</span> <span class="n">Integral</span><span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="n">besselj</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">k</span><span class="o">*</span><span class="n">r</span><span class="p">)</span><span class="o">*</span><span class="n">r</span><span class="p">,</span> <span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">as_integral</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_as_integral</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="p">,</span>
-                                 <span class="bp">self</span><span class="o">.</span><span class="n">function_variable</span><span class="p">,</span>
-                                 <span class="bp">self</span><span class="o">.</span><span class="n">transform_variable</span><span class="p">,</span>
-                                 <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">3</span><span class="p">])</span>
-
-
-<span class="k">class</span> <span class="nc">HankelTransform</span><span class="p">(</span><span class="n">HankelTypeTransform</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Class representing unevaluated Hankel transforms.</span>
-
-<span class="sd">    For usage of this class, see the :class:`IntegralTransform` docstring.</span>
-
-<span class="sd">    For how to compute Hankel transforms, see the :func:`hankel_transform`</span>
-<span class="sd">    docstring.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_name</span> <span class="o">=</span> <span class="s">&#39;Hankel&#39;</span>
-
-
-<div class="viewcode-block" id="hankel_transform"><a class="viewcode-back" href="../../../modules/integrals/integrals.html#sympy.integrals.transforms.hankel_transform">[docs]</a><span class="k">def</span> <span class="nf">hankel_transform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Compute the Hankel transform of `f`, defined as</span>
-
-<span class="sd">    .. math:: F_\nu(k) = \int_{0}^\infty f(r) J_\nu(k r) r \mathrm{d} r.</span>
-
-<span class="sd">    If the transform cannot be computed in closed form, this</span>
-<span class="sd">    function returns an unevaluated :class:`HankelTransform` object.</span>
-
-<span class="sd">    For a description of possible hints, refer to the docstring of</span>
-<span class="sd">    :func:`sympy.integrals.transforms.IntegralTransform.doit`.</span>
-<span class="sd">    Note that for this transform, by default ``noconds=True``.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import hankel_transform, inverse_hankel_transform</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import gamma, exp, sinh, cosh</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import r, k, m, nu, a</span>
-
-<span class="sd">    &gt;&gt;&gt; ht = hankel_transform(1/r**m, r, k, nu)</span>
-<span class="sd">    &gt;&gt;&gt; ht</span>
-<span class="sd">    2**(-m + 1)*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2)</span>
-
-<span class="sd">    &gt;&gt;&gt; inverse_hankel_transform(ht, k, r, nu)</span>
-<span class="sd">    r**(-m)</span>
-
-<span class="sd">    &gt;&gt;&gt; ht = hankel_transform(exp(-a*r), r, k, 0)</span>
-<span class="sd">    &gt;&gt;&gt; ht</span>
-<span class="sd">    a/(k**3*(a**2/k**2 + 1)**(3/2))</span>
-
-<span class="sd">    &gt;&gt;&gt; inverse_hankel_transform(ht, k, r, 0)</span>
-<span class="sd">    -sinh(a*r) + cosh(a*r)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    fourier_transform, inverse_fourier_transform</span>
-<span class="sd">    sine_transform, inverse_sine_transform</span>
-<span class="sd">    cosine_transform, inverse_cosine_transform</span>
-<span class="sd">    inverse_hankel_transform</span>
-<span class="sd">    mellin_transform, laplace_transform</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">HankelTransform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">nu</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-</div>
-<span class="k">class</span> <span class="nc">InverseHankelTransform</span><span class="p">(</span><span class="n">HankelTypeTransform</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Class representing unevaluated inverse Hankel transforms.</span>
-
-<span class="sd">    For usage of this class, see the :class:`IntegralTransform` docstring.</span>
-
-<span class="sd">    For how to compute inverse Hankel transforms, see the</span>
-<span class="sd">    :func:`inverse_hankel_transform` docstring.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_name</span> <span class="o">=</span> <span class="s">&#39;Inverse Hankel&#39;</span>
-
-
-<div class="viewcode-block" id="inverse_hankel_transform"><a class="viewcode-back" href="../../../modules/integrals/integrals.html#sympy.integrals.transforms.inverse_hankel_transform">[docs]</a><span class="k">def</span> <span class="nf">inverse_hankel_transform</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Compute the inverse Hankel transform of `F` defined as</span>
-
-<span class="sd">    .. math:: f(r) = \int_{0}^\infty F_\nu(k) J_\nu(k r) k \mathrm{d} k.</span>
-
-<span class="sd">    If the transform cannot be computed in closed form, this</span>
-<span class="sd">    function returns an unevaluated :class:`InverseHankelTransform` object.</span>
-
-<span class="sd">    For a description of possible hints, refer to the docstring of</span>
-<span class="sd">    :func:`sympy.integrals.transforms.IntegralTransform.doit`.</span>
-<span class="sd">    Note that for this transform, by default ``noconds=True``.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import hankel_transform, inverse_hankel_transform, gamma</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import gamma, exp, sinh, cosh</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import r, k, m, nu, a</span>
-
-<span class="sd">    &gt;&gt;&gt; ht = hankel_transform(1/r**m, r, k, nu)</span>
-<span class="sd">    &gt;&gt;&gt; ht</span>
-<span class="sd">    2**(-m + 1)*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2)</span>
-
-<span class="sd">    &gt;&gt;&gt; inverse_hankel_transform(ht, k, r, nu)</span>
-<span class="sd">    r**(-m)</span>
-
-<span class="sd">    &gt;&gt;&gt; ht = hankel_transform(exp(-a*r), r, k, 0)</span>
-<span class="sd">    &gt;&gt;&gt; ht</span>
-<span class="sd">    a/(k**3*(a**2/k**2 + 1)**(3/2))</span>
-
-<span class="sd">    &gt;&gt;&gt; inverse_hankel_transform(ht, k, r, 0)</span>
-<span class="sd">    -sinh(a*r) + cosh(a*r)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    fourier_transform, inverse_fourier_transform</span>
-<span class="sd">    sine_transform, inverse_sine_transform</span>
-<span class="sd">    cosine_transform, inverse_cosine_transform</span>
-<span class="sd">    hankel_transform</span>
-<span class="sd">    mellin_transform, laplace_transform</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">InverseHankelTransform</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">nu</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span></div>
-</pre></div>
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-  <h1>Source code for sympy.logic.boolalg</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Boolean algebra module for SymPy&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.basic</span> <span class="kn">import</span> <span class="n">Basic</span>
-<span class="kn">from</span> <span class="nn">sympy.core.decorators</span> <span class="kn">import</span> <span class="n">deprecated</span>
-<span class="kn">from</span> <span class="nn">sympy.core.operations</span> <span class="kn">import</span> <span class="n">LatticeOp</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Application</span><span class="p">,</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">ordered</span><span class="p">,</span> <span class="n">product</span>
-
-
-<span class="k">class</span> <span class="nc">Boolean</span><span class="p">(</span><span class="n">Basic</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A boolean object is an object for which logic operations make sense.&quot;&quot;&quot;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">__and__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Overloading for &amp; operator&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">And</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__or__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Overloading for |&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Or</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__invert__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Overloading for ~&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Not</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__rshift__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Overloading for &gt;&gt;&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Implies</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__lshift__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Overloading for &lt;&lt;&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Implies</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__xor__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Xor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">BooleanFunction</span><span class="p">(</span><span class="n">Application</span><span class="p">,</span> <span class="n">Boolean</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Boolean function is a function that lives in a boolean space</span>
-<span class="sd">    It is used as base class for And, Or, Not, etc.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_Boolean</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">arg</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ratio</span><span class="p">,</span> <span class="n">measure</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">simplify_logic</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="And"><a class="viewcode-back" href="../../../modules/logic.html#sympy.logic.boolalg.And">[docs]</a><span class="k">class</span> <span class="nc">And</span><span class="p">(</span><span class="n">LatticeOp</span><span class="p">,</span> <span class="n">BooleanFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Logical AND function.</span>
-
-<span class="sd">    It evaluates its arguments in order, giving False immediately</span>
-<span class="sd">    if any of them are False, and True if they are all True.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.core import symbols</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; x &amp; y</span>
-<span class="sd">    And(x, y)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">zero</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">identity</span> <span class="o">=</span> <span class="bp">True</span>
-
-</div>
-<div class="viewcode-block" id="Or"><a class="viewcode-back" href="../../../modules/logic.html#sympy.logic.boolalg.Or">[docs]</a><span class="k">class</span> <span class="nc">Or</span><span class="p">(</span><span class="n">LatticeOp</span><span class="p">,</span> <span class="n">BooleanFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Logical OR function</span>
-
-<span class="sd">    It evaluates its arguments in order, giving True immediately</span>
-<span class="sd">    if any of them are True, and False if they are all False.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">zero</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">identity</span> <span class="o">=</span> <span class="bp">False</span>
-
-</div>
-<div class="viewcode-block" id="Xor"><a class="viewcode-back" href="../../../modules/logic.html#sympy.logic.boolalg.Xor">[docs]</a><span class="k">class</span> <span class="nc">Xor</span><span class="p">(</span><span class="n">BooleanFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Logical XOR (exclusive OR) function.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Logical XOR (exclusive OR) function.</span>
-
-<span class="sd">        Returns True if an odd number of the arguments are True</span>
-<span class="sd">            and the rest are False.</span>
-<span class="sd">        Returns False if an even number of the arguments are True</span>
-<span class="sd">            and the rest are False.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.logic.boolalg import Xor</span>
-<span class="sd">        &gt;&gt;&gt; Xor(True, False)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Xor(True, True)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &gt;&gt;&gt; Xor(True, False, True, True, False)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Xor(True, False, True, False)</span>
-<span class="sd">        False</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">A</span> <span class="o">=</span> <span class="n">args</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-        <span class="k">while</span> <span class="n">args</span><span class="p">:</span>
-            <span class="n">B</span> <span class="o">=</span> <span class="n">args</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-            <span class="n">A</span> <span class="o">=</span> <span class="n">Or</span><span class="p">(</span><span class="n">And</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">Not</span><span class="p">(</span><span class="n">B</span><span class="p">)),</span> <span class="n">And</span><span class="p">(</span><span class="n">Not</span><span class="p">(</span><span class="n">A</span><span class="p">),</span> <span class="n">B</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">A</span>
-
-</div>
-<div class="viewcode-block" id="Not"><a class="viewcode-back" href="../../../modules/logic.html#sympy.logic.boolalg.Not">[docs]</a><span class="k">class</span> <span class="nc">Not</span><span class="p">(</span><span class="n">BooleanFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Logical Not function (negation)</span>
-
-<span class="sd">    Note: De Morgan rules applied automatically</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_Not</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Logical Not function (negation)</span>
-
-<span class="sd">        Returns True if the statement is False</span>
-<span class="sd">        Returns False if the statement is True</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.logic.boolalg import Not, And, Or</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; Not(True)</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Not(False)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Not(And(True, False))</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Not(Or(True, False))</span>
-<span class="sd">        False</span>
-
-<span class="sd">        If multiple statements are given, returns an array of each result</span>
-
-<span class="sd">        &gt;&gt;&gt; Not(True, False)</span>
-<span class="sd">        [False, True]</span>
-<span class="sd">        &gt;&gt;&gt; Not(True and False, True or False, True)</span>
-<span class="sd">        [True, False, False]</span>
-
-<span class="sd">        &gt;&gt;&gt; Not(And(And(True, x), Or(x, False)))</span>
-<span class="sd">        Not(x)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">))</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">arg</span> <span class="ow">in</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">):</span>  <span class="c"># includes True and False, too</span>
-            <span class="k">return</span> <span class="ow">not</span> <span class="nb">bool</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-        <span class="c"># apply De Morgan Rules</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">And</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Or</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">Not</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Or</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">And</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">Not</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Not</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="Nand"><a class="viewcode-back" href="../../../modules/logic.html#sympy.logic.boolalg.Nand">[docs]</a><span class="k">class</span> <span class="nc">Nand</span><span class="p">(</span><span class="n">BooleanFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Logical NAND function.</span>
-
-<span class="sd">    It evaluates its arguments in order, giving True immediately if any</span>
-<span class="sd">    of them are False, and False if they are all True.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Logical NAND function.</span>
-
-<span class="sd">        Returns True if any of the arguments are False</span>
-<span class="sd">        Returns False if all arguments are True</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.logic.boolalg import Nand</span>
-<span class="sd">        &gt;&gt;&gt; Nand(False, True)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Nand(True, True)</span>
-<span class="sd">        False</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Not</span><span class="p">(</span><span class="n">And</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="Nor"><a class="viewcode-back" href="../../../modules/logic.html#sympy.logic.boolalg.Nor">[docs]</a><span class="k">class</span> <span class="nc">Nor</span><span class="p">(</span><span class="n">BooleanFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Logical NOR function.</span>
-
-<span class="sd">    It evaluates its arguments in order, giving False immediately if any</span>
-<span class="sd">    of them are True, and True if they are all False.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Logical NOR function.</span>
-
-<span class="sd">        Returns False if any argument is True</span>
-<span class="sd">        Returns True if all arguments are False</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.logic.boolalg import Nor</span>
-<span class="sd">        &gt;&gt;&gt; Nor(True, False)</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Nor(True, True)</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Nor(False, True)</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Nor(False, False)</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Not</span><span class="p">(</span><span class="n">Or</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="Implies"><a class="viewcode-back" href="../../../modules/logic.html#sympy.logic.boolalg.Implies">[docs]</a><span class="k">class</span> <span class="nc">Implies</span><span class="p">(</span><span class="n">BooleanFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Logical implication.</span>
-
-<span class="sd">    A implies B is equivalent to !A v B</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Logical implication.</span>
-
-<span class="sd">        Accepts two Boolean arguments; A and B.</span>
-<span class="sd">        Returns False if A is True and B is False</span>
-<span class="sd">        Returns True otherwise.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.logic.boolalg import Implies</span>
-<span class="sd">        &gt;&gt;&gt; Implies(True, False)</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Implies(False, False)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Implies(True, True)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Implies(False, True)</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">A</span><span class="p">,</span> <span class="n">B</span> <span class="o">=</span> <span class="n">args</span>
-        <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&quot;</span><span class="si">%d</span><span class="s"> operand(s) used for an Implies &quot;</span>
-                <span class="s">&quot;(pairs are required): </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">),</span> <span class="nb">str</span><span class="p">(</span><span class="n">args</span><span class="p">)))</span>
-        <span class="k">if</span> <span class="n">A</span> <span class="ow">is</span> <span class="bp">True</span> <span class="ow">or</span> <span class="n">A</span> <span class="ow">is</span> <span class="bp">False</span> <span class="ow">or</span> <span class="n">B</span> <span class="ow">is</span> <span class="bp">True</span> <span class="ow">or</span> <span class="n">B</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Or</span><span class="p">(</span><span class="n">Not</span><span class="p">(</span><span class="n">A</span><span class="p">),</span> <span class="n">B</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="Equivalent"><a class="viewcode-back" href="../../../modules/logic.html#sympy.logic.boolalg.Equivalent">[docs]</a><span class="k">class</span> <span class="nc">Equivalent</span><span class="p">(</span><span class="n">BooleanFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Equivalence relation.</span>
-
-<span class="sd">    Equivalent(A, B) is True iff A and B are both True or both False</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Equivalence relation.</span>
-
-<span class="sd">        Returns True if all of the arguments are logically equivalent.</span>
-<span class="sd">        Returns False otherwise.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.logic.boolalg import Equivalent, And</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; Equivalent(False, False, False)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Equivalent(True, False, False)</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Equivalent(x, And(x, True))</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">argset</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">argset</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="bp">True</span> <span class="ow">in</span> <span class="n">argset</span><span class="p">:</span>
-            <span class="n">argset</span><span class="o">.</span><span class="n">discard</span><span class="p">(</span><span class="bp">True</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">And</span><span class="p">(</span><span class="o">*</span><span class="n">argset</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">False</span> <span class="ow">in</span> <span class="n">argset</span><span class="p">:</span>
-            <span class="n">argset</span><span class="o">.</span><span class="n">discard</span><span class="p">(</span><span class="bp">False</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Nor</span><span class="p">(</span><span class="o">*</span><span class="n">argset</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="nb">set</span><span class="p">(</span><span class="n">args</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="ITE"><a class="viewcode-back" href="../../../modules/logic.html#sympy.logic.boolalg.ITE">[docs]</a><span class="k">class</span> <span class="nc">ITE</span><span class="p">(</span><span class="n">BooleanFunction</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    If then else clause.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        If then else clause</span>
-
-<span class="sd">        ITE(A, B, C) evaluates and returns the result of B if A is true</span>
-<span class="sd">        else it returns the result of C</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.logic.boolalg import ITE, And, Xor, Or</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">        &gt;&gt;&gt; x = True</span>
-<span class="sd">        &gt;&gt;&gt; y = False</span>
-<span class="sd">        &gt;&gt;&gt; z = True</span>
-<span class="sd">        &gt;&gt;&gt; ITE(x, y, z)</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; ITE(Or(x, y), And(x, z), Xor(z, x))</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Or</span><span class="p">(</span><span class="n">And</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="n">And</span><span class="p">(</span><span class="n">Not</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;ITE expects 3 arguments, but got </span><span class="si">%d</span><span class="s">: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span>
-                         <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">),</span> <span class="nb">str</span><span class="p">(</span><span class="n">args</span><span class="p">)))</span>
-
-<span class="c">### end class definitions. Some useful methods</span>
-
-</div>
-<span class="k">def</span> <span class="nf">conjuncts</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return a list of the conjuncts in the expr s.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.logic.boolalg import conjuncts</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import A, B</span>
-<span class="sd">    &gt;&gt;&gt; conjuncts(A &amp; B)</span>
-<span class="sd">    frozenset([A, B])</span>
-<span class="sd">    &gt;&gt;&gt; conjuncts(A | B)</span>
-<span class="sd">    frozenset([Or(A, B)])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">And</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">disjuncts</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return a list of the disjuncts in the sentence s.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.logic.boolalg import disjuncts</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import A, B</span>
-<span class="sd">    &gt;&gt;&gt; disjuncts(A | B)</span>
-<span class="sd">    frozenset([A, B])</span>
-<span class="sd">    &gt;&gt;&gt; disjuncts(A &amp; B)</span>
-<span class="sd">    frozenset([And(A, B)])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">Or</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">distribute_and_over_or</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Given a sentence s consisting of conjunctions and disjunctions</span>
-<span class="sd">    of literals, return an equivalent sentence in CNF.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.logic.boolalg import distribute_and_over_or, And, Or, Not</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import A, B, C</span>
-<span class="sd">    &gt;&gt;&gt; distribute_and_over_or(Or(A, And(Not(B), Not(C))))</span>
-<span class="sd">    And(Or(A, Not(B)), Or(A, Not(C)))</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Or</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">And</span><span class="p">:</span>
-                <span class="n">conj</span> <span class="o">=</span> <span class="n">arg</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span>
-        <span class="n">rest</span> <span class="o">=</span> <span class="n">Or</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">a</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span> <span class="k">if</span> <span class="n">a</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">conj</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">And</span><span class="p">(</span><span class="o">*</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">distribute_and_over_or</span><span class="p">,</span>
-                   <span class="p">[</span><span class="n">Or</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">rest</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">conj</span><span class="o">.</span><span class="n">args</span><span class="p">])))</span>
-    <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">And</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">And</span><span class="p">(</span><span class="o">*</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">distribute_and_over_or</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-
-<div class="viewcode-block" id="to_cnf"><a class="viewcode-back" href="../../../modules/logic.html#sympy.logic.boolalg.to_cnf">[docs]</a><span class="k">def</span> <span class="nf">to_cnf</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Convert a propositional logical sentence s to conjunctive normal form.</span>
-<span class="sd">    That is, of the form ((A | ~B | ...) &amp; (B | C | ...) &amp; ...)</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.logic.boolalg import to_cnf</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import A, B, D</span>
-<span class="sd">    &gt;&gt;&gt; to_cnf(~(A | B) | D)</span>
-<span class="sd">    And(Or(D, Not(A)), Or(D, Not(B)))</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># Don&#39;t convert unless we have to</span>
-    <span class="k">if</span> <span class="n">is_cnf</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">BooleanFunction</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">eliminate_implications</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">distribute_and_over_or</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">is_cnf</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Test whether or not an expression is in conjunctive normal form.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.logic.boolalg import is_cnf</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import A, B, C</span>
-<span class="sd">    &gt;&gt;&gt; is_cnf(A | B | C)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; is_cnf(A &amp; B &amp; C)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; is_cnf((A &amp; B) | C)</span>
-<span class="sd">    False</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="c"># Special case of a single disjunction</span>
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Or</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">lit</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">lit</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Not</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">lit</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">lit</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="c"># Special case of a single negation</span>
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Not</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">And</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">for</span> <span class="n">cls</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">cls</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-            <span class="k">continue</span>
-        <span class="k">if</span> <span class="n">cls</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Not</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">cls</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="n">cls</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">Or</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">lit</span> <span class="ow">in</span> <span class="n">cls</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">lit</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Not</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">lit</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">lit</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">return</span> <span class="bp">True</span>
-
-
-<span class="k">def</span> <span class="nf">eliminate_implications</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Change &gt;&gt;, &lt;&lt;, and Equivalent into &amp;, |, and ~. That is, return an</span>
-<span class="sd">    expression that is equivalent to s, but has only &amp;, |, and ~ as logical</span>
-<span class="sd">    operators.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.logic.boolalg import Implies, Equivalent, \</span>
-<span class="sd">         eliminate_implications</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import A, B, C</span>
-<span class="sd">    &gt;&gt;&gt; eliminate_implications(Implies(A, B))</span>
-<span class="sd">    Or(B, Not(A))</span>
-<span class="sd">    &gt;&gt;&gt; eliminate_implications(Equivalent(A, B))</span>
-<span class="sd">    And(Or(A, Not(B)), Or(B, Not(A)))</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span>  <span class="c"># (Atoms are unchanged.)</span>
-    <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">eliminate_implications</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Implies</span><span class="p">:</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">return</span> <span class="p">(</span><span class="o">~</span><span class="n">a</span><span class="p">)</span> <span class="o">|</span> <span class="n">b</span>
-    <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Equivalent</span><span class="p">:</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">a</span> <span class="o">|</span> <span class="n">Not</span><span class="p">(</span><span class="n">b</span><span class="p">))</span> <span class="o">&amp;</span> <span class="p">(</span><span class="n">b</span> <span class="o">|</span> <span class="n">Not</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-
-<span class="nd">@deprecated</span><span class="p">(</span>
-    <span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;sympify&quot;</span><span class="p">,</span> <span class="n">issue</span><span class="o">=</span><span class="mi">2947</span><span class="p">,</span> <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.3&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">compile_rule</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Transforms a rule into a SymPy expression</span>
-<span class="sd">    A rule is a string of the form &quot;symbol1 &amp; symbol2 | ...&quot;</span>
-
-<span class="sd">    Note: this is nearly the same as sympifying the expression, but</span>
-<span class="sd">    this function converts all variables to Symbols -- there are no</span>
-<span class="sd">    special function names recognized.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.logic.boolalg import compile_rule</span>
-<span class="sd">    &gt;&gt;&gt; compile_rule(&#39;A &amp; B&#39;)</span>
-<span class="sd">    And(A, B)</span>
-<span class="sd">    &gt;&gt;&gt; compile_rule(&#39;(~B &amp; ~C)|A&#39;)</span>
-<span class="sd">    Or(A, And(Not(B), Not(C)))</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">import</span> <span class="nn">re</span>
-    <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="n">re</span><span class="o">.</span><span class="n">sub</span><span class="p">(</span><span class="s">r&#39;([a-zA-Z_][a-zA-Z0-9_]*)&#39;</span><span class="p">,</span> <span class="s">r&#39;Symbol(&quot;\1&quot;)&#39;</span><span class="p">,</span> <span class="n">s</span><span class="p">))</span>
-
-
-<span class="k">def</span> <span class="nf">to_int_repr</span><span class="p">(</span><span class="n">clauses</span><span class="p">,</span> <span class="n">symbols</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Takes clauses in CNF format and puts them into an integer representation.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.logic.boolalg import to_int_repr</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; to_int_repr([x | y, y], [x, y]) == [set([1, 2]), set([2])]</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="c"># Convert the symbol list into a dict</span>
-    <span class="n">symbols</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">symbols</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)))))</span>
-
-    <span class="k">def</span> <span class="nf">append_symbol</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">symbols</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Not</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">symbols</span><span class="p">[</span><span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">symbols</span><span class="p">[</span><span class="n">arg</span><span class="p">]</span>
-
-    <span class="k">return</span> <span class="p">[</span><span class="nb">set</span><span class="p">(</span><span class="n">append_symbol</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">symbols</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">Or</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">c</span><span class="p">))</span>
-            <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">clauses</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">_check_pair</span><span class="p">(</span><span class="n">minterm1</span><span class="p">,</span> <span class="n">minterm2</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Checks if a pair of minterms differs by only one bit. If yes, returns</span>
-<span class="sd">    index, else returns -1.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">index</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-    <span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">minterm1</span><span class="p">,</span> <span class="n">minterm2</span><span class="p">))):</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="n">j</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">index</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                <span class="n">index</span> <span class="o">=</span> <span class="n">x</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-    <span class="k">return</span> <span class="n">index</span>
-
-
-<span class="k">def</span> <span class="nf">_convert_to_varsSOP</span><span class="p">(</span><span class="n">minterm</span><span class="p">,</span> <span class="n">variables</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Converts a term in the expansion of a function from binary to it&#39;s</span>
-<span class="sd">    variable form (for SOP).</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">temp</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">m</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">minterm</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">temp</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Not</span><span class="p">(</span><span class="n">variables</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span>
-        <span class="k">elif</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">temp</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">variables</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">pass</span>  <span class="c"># ignore the 3s</span>
-    <span class="k">return</span> <span class="n">And</span><span class="p">(</span><span class="o">*</span><span class="n">temp</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">_convert_to_varsPOS</span><span class="p">(</span><span class="n">maxterm</span><span class="p">,</span> <span class="n">variables</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Converts a term in the expansion of a function from binary to it&#39;s</span>
-<span class="sd">    variable form (for POS).</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">temp</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">m</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">maxterm</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">temp</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Not</span><span class="p">(</span><span class="n">variables</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span>
-        <span class="k">elif</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">temp</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">variables</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">pass</span>  <span class="c"># ignore the 3s</span>
-    <span class="k">return</span> <span class="n">Or</span><span class="p">(</span><span class="o">*</span><span class="n">temp</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">_simplified_pairs</span><span class="p">(</span><span class="n">terms</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Reduces a set of minterms, if possible, to a simplified set of minterms</span>
-<span class="sd">    with one less variable in the terms using QM method.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">simplified_terms</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">todo</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">terms</span><span class="p">)))</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">ti</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">terms</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]):</span>
-        <span class="k">for</span> <span class="n">j_i</span><span class="p">,</span> <span class="n">tj</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">terms</span><span class="p">[(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):]):</span>
-            <span class="n">index</span> <span class="o">=</span> <span class="n">_check_pair</span><span class="p">(</span><span class="n">ti</span><span class="p">,</span> <span class="n">tj</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">index</span> <span class="o">!=</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                <span class="n">todo</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">todo</span><span class="p">[</span><span class="n">j_i</span> <span class="o">+</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="bp">None</span>
-                <span class="n">newterm</span> <span class="o">=</span> <span class="n">ti</span><span class="p">[:]</span>
-                <span class="n">newterm</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="o">=</span> <span class="mi">3</span>
-                <span class="k">if</span> <span class="n">newterm</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">simplified_terms</span><span class="p">:</span>
-                    <span class="n">simplified_terms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">newterm</span><span class="p">)</span>
-    <span class="n">simplified_terms</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span>
-        <span class="p">[</span><span class="n">terms</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="n">_</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">todo</span> <span class="k">if</span> <span class="n">_</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">]])</span>
-    <span class="k">return</span> <span class="n">simplified_terms</span>
-
-
-<span class="k">def</span> <span class="nf">_compare_term</span><span class="p">(</span><span class="n">minterm</span><span class="p">,</span> <span class="n">term</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return True if a binary term is satisfied by the given term. Used</span>
-<span class="sd">    for recognizing prime implicants.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">term</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="o">!=</span> <span class="mi">3</span> <span class="ow">and</span> <span class="n">x</span> <span class="o">!=</span> <span class="n">minterm</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-    <span class="k">return</span> <span class="bp">True</span>
-
-
-<span class="k">def</span> <span class="nf">_rem_redundancy</span><span class="p">(</span><span class="n">l1</span><span class="p">,</span> <span class="n">terms</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    After the truth table has been sufficiently simplified, use the prime</span>
-<span class="sd">    implicant table method to recognize and eliminate redundant pairs,</span>
-<span class="sd">    and return the essential arguments.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">essential</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">:</span>
-        <span class="n">temporary</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">l1</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">_compare_term</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
-                <span class="n">temporary</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">temporary</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">temporary</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">essential</span><span class="p">:</span>
-                <span class="n">essential</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">temporary</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-    <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">essential</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">_compare_term</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="n">l1</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">_compare_term</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">z</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">essential</span><span class="p">:</span>
-                        <span class="n">essential</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-                    <span class="k">break</span>
-
-    <span class="k">return</span> <span class="n">essential</span>
-
-
-<span class="k">def</span> <span class="nf">SOPform</span><span class="p">(</span><span class="n">variables</span><span class="p">,</span> <span class="n">minterms</span><span class="p">,</span> <span class="n">dontcares</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The SOPform function uses simplified_pairs and a redundant group-</span>
-<span class="sd">    eliminating algorithm to convert the list of all input combos that</span>
-<span class="sd">    generate &#39;1&#39; (the minterms) into the smallest Sum of Products form.</span>
-
-<span class="sd">    The variables must be given as the first argument.</span>
-
-<span class="sd">    Return a logical Or function (i.e., the &quot;sum of products&quot; or &quot;SOP&quot;</span>
-<span class="sd">    form) that gives the desired outcome. If there are inputs that can</span>
-<span class="sd">    be ignored, pass them as a list, too.</span>
-
-<span class="sd">    The result will be one of the (perhaps many) functions that satisfy</span>
-<span class="sd">    the conditions.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.logic import SOPform</span>
-<span class="sd">    &gt;&gt;&gt; minterms = [[0, 0, 0, 1], [0, 0, 1, 1],</span>
-<span class="sd">    ...             [0, 1, 1, 1], [1, 0, 1, 1], [1, 1, 1, 1]]</span>
-<span class="sd">    &gt;&gt;&gt; dontcares = [[0, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 1]]</span>
-<span class="sd">    &gt;&gt;&gt; SOPform([&#39;w&#39;,&#39;x&#39;,&#39;y&#39;,&#39;z&#39;], minterms, dontcares)</span>
-<span class="sd">    Or(And(Not(w), z), And(y, z))</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] en.wikipedia.org/wiki/Quine-McCluskey_algorithm</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.core.symbol</span> <span class="kn">import</span> <span class="n">Symbol</span>
-
-    <span class="n">variables</span> <span class="o">=</span> <span class="p">[</span><span class="n">Symbol</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">)</span> <span class="k">else</span> <span class="n">v</span>
-                 <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">variables</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">minterms</span> <span class="o">==</span> <span class="p">[]:</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="n">minterms</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">minterms</span><span class="p">]</span>
-    <span class="n">dontcares</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">(</span><span class="n">dontcares</span> <span class="ow">or</span> <span class="p">[])]</span>
-    <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">dontcares</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">minterms</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> in minterms is also in dontcares&#39;</span> <span class="o">%</span> <span class="n">d</span><span class="p">)</span>
-
-    <span class="n">old</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">new</span> <span class="o">=</span> <span class="n">minterms</span> <span class="o">+</span> <span class="n">dontcares</span>
-    <span class="k">while</span> <span class="n">new</span> <span class="o">!=</span> <span class="n">old</span><span class="p">:</span>
-        <span class="n">old</span> <span class="o">=</span> <span class="n">new</span>
-        <span class="n">new</span> <span class="o">=</span> <span class="n">_simplified_pairs</span><span class="p">(</span><span class="n">old</span><span class="p">)</span>
-    <span class="n">essential</span> <span class="o">=</span> <span class="n">_rem_redundancy</span><span class="p">(</span><span class="n">new</span><span class="p">,</span> <span class="n">minterms</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">Or</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">_convert_to_varsSOP</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">variables</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">essential</span><span class="p">])</span>
-
-
-<span class="k">def</span> <span class="nf">POSform</span><span class="p">(</span><span class="n">variables</span><span class="p">,</span> <span class="n">minterms</span><span class="p">,</span> <span class="n">dontcares</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The POSform function uses simplified_pairs and a redundant-group</span>
-<span class="sd">    eliminating algorithm to convert the list of all input combinations</span>
-<span class="sd">    that generate &#39;1&#39; (the minterms) into the smallest Product of Sums form.</span>
-
-<span class="sd">    The variables must be given as the first argument.</span>
-
-<span class="sd">    Return a logical And function (i.e., the &quot;product of sums&quot; or &quot;POS&quot;</span>
-<span class="sd">    form) that gives the desired outcome. If there are inputs that can</span>
-<span class="sd">    be ignored, pass them as a list, too.</span>
-
-<span class="sd">    The result will be one of the (perhaps many) functions that satisfy</span>
-<span class="sd">    the conditions.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.logic import POSform</span>
-<span class="sd">    &gt;&gt;&gt; minterms = [[0, 0, 0, 1], [0, 0, 1, 1], [0, 1, 1, 1],</span>
-<span class="sd">    ...             [1, 0, 1, 1], [1, 1, 1, 1]]</span>
-<span class="sd">    &gt;&gt;&gt; dontcares = [[0, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 1]]</span>
-<span class="sd">    &gt;&gt;&gt; POSform([&#39;w&#39;,&#39;x&#39;,&#39;y&#39;,&#39;z&#39;], minterms, dontcares)</span>
-<span class="sd">    And(Or(Not(w), y), z)</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] en.wikipedia.org/wiki/Quine-McCluskey_algorithm</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.core.symbol</span> <span class="kn">import</span> <span class="n">Symbol</span>
-
-    <span class="n">variables</span> <span class="o">=</span> <span class="p">[</span><span class="n">Symbol</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">)</span> <span class="k">else</span> <span class="n">v</span>
-                 <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">variables</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">minterms</span> <span class="o">==</span> <span class="p">[]:</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="n">minterms</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">minterms</span><span class="p">]</span>
-    <span class="n">dontcares</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">(</span><span class="n">dontcares</span> <span class="ow">or</span> <span class="p">[])]</span>
-    <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">dontcares</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">minterms</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> in minterms is also in dontcares&#39;</span> <span class="o">%</span> <span class="n">d</span><span class="p">)</span>
-
-    <span class="n">maxterms</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">product</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">repeat</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">variables</span><span class="p">)):</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">t</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">minterms</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">t</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">dontcares</span><span class="p">):</span>
-            <span class="n">maxterms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-    <span class="n">old</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">new</span> <span class="o">=</span> <span class="n">maxterms</span> <span class="o">+</span> <span class="n">dontcares</span>
-    <span class="k">while</span> <span class="n">new</span> <span class="o">!=</span> <span class="n">old</span><span class="p">:</span>
-        <span class="n">old</span> <span class="o">=</span> <span class="n">new</span>
-        <span class="n">new</span> <span class="o">=</span> <span class="n">_simplified_pairs</span><span class="p">(</span><span class="n">old</span><span class="p">)</span>
-    <span class="n">essential</span> <span class="o">=</span> <span class="n">_rem_redundancy</span><span class="p">(</span><span class="n">new</span><span class="p">,</span> <span class="n">maxterms</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">And</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">_convert_to_varsPOS</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">variables</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">essential</span><span class="p">])</span>
-
-
-<span class="k">def</span> <span class="nf">simplify_logic</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    This function simplifies a boolean function to its</span>
-<span class="sd">    simplified version in SOP or POS form. The return type is an</span>
-<span class="sd">    Or or And object in SymPy. The input can be a string or a boolean</span>
-<span class="sd">    expression.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.logic import simplify_logic</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import S</span>
-
-<span class="sd">    &gt;&gt;&gt; b = &#39;(~x &amp; ~y &amp; ~z) | ( ~x &amp; ~y &amp; z)&#39;</span>
-<span class="sd">    &gt;&gt;&gt; simplify_logic(b)</span>
-<span class="sd">    And(Not(x), Not(y))</span>
-
-<span class="sd">    &gt;&gt;&gt; S(b)</span>
-<span class="sd">    Or(And(Not(x), Not(y), Not(z)), And(Not(x), Not(y), z))</span>
-<span class="sd">    &gt;&gt;&gt; simplify_logic(_)</span>
-<span class="sd">    And(Not(x), Not(y))</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">BooleanFunction</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span>
-    <span class="n">variables</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">)</span>
-    <span class="n">truthtable</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">product</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">repeat</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">variables</span><span class="p">)):</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">variables</span><span class="p">,</span> <span class="n">t</span><span class="p">)))</span> <span class="o">==</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">truthtable</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-    <span class="k">if</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">truthtable</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="p">(</span><span class="mi">2</span> <span class="o">**</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">variables</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))):</span>
-        <span class="k">return</span> <span class="n">SOPform</span><span class="p">(</span><span class="n">variables</span><span class="p">,</span> <span class="n">truthtable</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">POSform</span><span class="p">(</span><span class="n">variables</span><span class="p">,</span> <span class="n">truthtable</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">_finger</span><span class="p">(</span><span class="n">eq</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Assign a 5-item fingerprint to each symbol in the equation:</span>
-<span class="sd">    [</span>
-<span class="sd">    # of times it appeared as a Symbol,</span>
-<span class="sd">    # of times it appeared as a Not(symbol),</span>
-<span class="sd">    # of times it appeared as a Symbol in an And or Or,</span>
-<span class="sd">    # of times it appeared as a Not(Symbol) in an And or Or,</span>
-<span class="sd">    sum of the number of arguments with which it appeared,</span>
-<span class="sd">    counting Symbol as 1 and Not(Symbol) as 2</span>
-<span class="sd">    ]</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.logic.boolalg import _finger as finger</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import And, Or, Not</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import a, b, x, y</span>
-<span class="sd">    &gt;&gt;&gt; eq = Or(And(Not(y), a), And(Not(y), b), And(x, y))</span>
-<span class="sd">    &gt;&gt;&gt; dict(finger(eq))</span>
-<span class="sd">    {(0, 0, 1, 0, 2): [x], (0, 0, 1, 0, 3): [a, b], (0, 0, 1, 2, 8): [y]}</span>
-
-<span class="sd">    So y and x have unique fingerprints, but a and b do not.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">eq</span><span class="o">.</span><span class="n">free_symbols</span>
-    <span class="n">d</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="mi">5</span> <span class="k">for</span> <span class="n">fi</span> <span class="ow">in</span> <span class="n">f</span><span class="p">])))</span>
-    <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">eq</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">:</span>
-            <span class="n">d</span><span class="p">[</span><span class="n">a</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">elif</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Not</span><span class="p">:</span>
-            <span class="n">d</span><span class="p">[</span><span class="n">a</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]][</span><span class="mi">1</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">o</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">+</span> <span class="nb">sum</span><span class="p">(</span><span class="n">ai</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Not</span> <span class="k">for</span> <span class="n">ai</span> <span class="ow">in</span> <span class="n">a</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">ai</span> <span class="ow">in</span> <span class="n">a</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">ai</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">:</span>
-                    <span class="n">d</span><span class="p">[</span><span class="n">ai</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-                    <span class="n">d</span><span class="p">[</span><span class="n">ai</span><span class="p">][</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+=</span> <span class="n">o</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">d</span><span class="p">[</span><span class="n">ai</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]][</span><span class="mi">3</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-                    <span class="n">d</span><span class="p">[</span><span class="n">ai</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]][</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+=</span> <span class="n">o</span>
-    <span class="n">inv</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">list</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">ordered</span><span class="p">(</span><span class="nb">iter</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">items</span><span class="p">())):</span>
-        <span class="n">inv</span><span class="p">[</span><span class="nb">tuple</span><span class="p">(</span><span class="n">v</span><span class="p">)]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">inv</span>
-
-
-<span class="k">def</span> <span class="nf">bool_equal</span><span class="p">(</span><span class="n">bool1</span><span class="p">,</span> <span class="n">bool2</span><span class="p">,</span> <span class="n">info</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return True if the two expressions represent the same logical</span>
-<span class="sd">    behavior for some correspondence between the variables of each</span>
-<span class="sd">    (which may be different). For example, And(x, y) is logically</span>
-<span class="sd">    equivalent to And(a, b) for {x: a, y: b} (or vice versa). If the</span>
-<span class="sd">    mapping is desired, then set ``info`` to True and the simplified</span>
-<span class="sd">    form of the functions and the mapping of variables will be</span>
-<span class="sd">    returned.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import SOPform, bool_equal, Or, And, Not, Xor</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import w, x, y, z, a, b, c, d</span>
-<span class="sd">    &gt;&gt;&gt; function1 = SOPform([&#39;x&#39;,&#39;z&#39;,&#39;y&#39;],[[1, 0, 1], [0, 0, 1]])</span>
-<span class="sd">    &gt;&gt;&gt; function2 = SOPform([&#39;a&#39;,&#39;b&#39;,&#39;c&#39;],[[1, 0, 1], [1, 0, 0]])</span>
-<span class="sd">    &gt;&gt;&gt; bool_equal(function1, function2, info=True)</span>
-<span class="sd">    (And(Not(z), y), {y: a, z: b})</span>
-
-<span class="sd">    The results are not necessarily unique, but they are canonical. Here,</span>
-<span class="sd">    ``(w, z)`` could be ``(a, d)`` or ``(d, a)``:</span>
-
-<span class="sd">    &gt;&gt;&gt; eq =  Or(And(Not(y), w), And(Not(y), z), And(x, y))</span>
-<span class="sd">    &gt;&gt;&gt; eq2 = Or(And(Not(c), a), And(Not(c), d), And(b, c))</span>
-<span class="sd">    &gt;&gt;&gt; bool_equal(eq, eq2)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; bool_equal(eq, eq2, info=True)</span>
-<span class="sd">    (Or(And(Not(y), w), And(Not(y), z), And(x, y)), {w: a, x: b, y: c, z: d})</span>
-<span class="sd">    &gt;&gt;&gt; eq = And(Xor(a, b), c, And(c,d))</span>
-<span class="sd">    &gt;&gt;&gt; bool_equal(eq, eq.subs(c, x), info=True)</span>
-<span class="sd">    (And(Or(Not(a), Not(b)), Or(a, b), c, d), {a: a, b: b, c: d, d: x})</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">match</span><span class="p">(</span><span class="n">function1</span><span class="p">,</span> <span class="n">function2</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the mapping that equates variables between two</span>
-<span class="sd">        simplified boolean expressions if possible.</span>
-
-<span class="sd">        By &quot;simplified&quot; we mean that a function has been denested</span>
-<span class="sd">        and is either an And (or an Or) whose arguments are either</span>
-<span class="sd">        symbols (x), negated symbols (Not(x)), or Or (or an And) whose</span>
-<span class="sd">        arguments are only symbols or negated symbols. For example,</span>
-<span class="sd">        And(x, Not(y), Or(w, Not(z))).</span>
-
-<span class="sd">        Basic.match is not robust enough (see issue 1736) so this is</span>
-<span class="sd">        a workaround that is valid for simplified boolean expressions</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="c"># do some quick checks</span>
-        <span class="k">if</span> <span class="n">function1</span><span class="o">.</span><span class="n">__class__</span> <span class="o">!=</span> <span class="n">function2</span><span class="o">.</span><span class="n">__class__</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">function1</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">function2</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">function1</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">{</span><span class="n">function1</span><span class="p">:</span> <span class="n">function2</span><span class="p">}</span>
-
-        <span class="c"># get the fingerprint dictionaries</span>
-        <span class="n">f1</span> <span class="o">=</span> <span class="n">_finger</span><span class="p">(</span><span class="n">function1</span><span class="p">)</span>
-        <span class="n">f2</span> <span class="o">=</span> <span class="n">_finger</span><span class="p">(</span><span class="n">function2</span><span class="p">)</span>
-
-        <span class="c"># more quick checks</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">f1</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">f2</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="c"># assemble the match dictionary if possible</span>
-        <span class="n">matchdict</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">f1</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
-            <span class="k">if</span> <span class="n">k</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">f2</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">f1</span><span class="p">[</span><span class="n">k</span><span class="p">])</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">f2</span><span class="p">[</span><span class="n">k</span><span class="p">]):</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">f1</span><span class="p">[</span><span class="n">k</span><span class="p">]):</span>
-                <span class="n">matchdict</span><span class="p">[</span><span class="n">x</span><span class="p">]</span> <span class="o">=</span> <span class="n">f2</span><span class="p">[</span><span class="n">k</span><span class="p">][</span><span class="n">i</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">matchdict</span>
-
-    <span class="n">a</span> <span class="o">=</span> <span class="n">simplify_logic</span><span class="p">(</span><span class="n">bool1</span><span class="p">)</span>
-    <span class="n">b</span> <span class="o">=</span> <span class="n">simplify_logic</span><span class="p">(</span><span class="n">bool2</span><span class="p">)</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">match</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="ow">and</span> <span class="n">info</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">a</span><span class="p">,</span> <span class="n">m</span>
-    <span class="k">return</span> <span class="n">m</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span>
-</pre></div>
-
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-  <h1>Source code for sympy.logic.inference</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Inference in propositional logic&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">sympy.logic.boolalg</span> <span class="kn">import</span> <span class="n">And</span><span class="p">,</span> <span class="n">Or</span><span class="p">,</span> <span class="n">Not</span><span class="p">,</span> <span class="n">Implies</span><span class="p">,</span> <span class="n">Equivalent</span><span class="p">,</span> \
-    <span class="n">conjuncts</span><span class="p">,</span> <span class="n">to_cnf</span>
-<span class="kn">from</span> <span class="nn">sympy.core.basic</span> <span class="kn">import</span> <span class="n">C</span>
-<span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="n">sympify</span>
-
-
-<span class="k">def</span> <span class="nf">literal_symbol</span><span class="p">(</span><span class="n">literal</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The symbol in this literal (without the negation).</span>
-
-<span class="sd">    Examples:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import A</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.logic.inference import literal_symbol</span>
-<span class="sd">    &gt;&gt;&gt; literal_symbol(A)</span>
-<span class="sd">    A</span>
-<span class="sd">    &gt;&gt;&gt; literal_symbol(~A)</span>
-<span class="sd">    A</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="n">literal</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Not</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">literal</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">literal</span>
-
-
-<div class="viewcode-block" id="satisfiable"><a class="viewcode-back" href="../../../modules/logic.html#sympy.logic.inference.satisfiable">[docs]</a><span class="k">def</span> <span class="nf">satisfiable</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">algorithm</span><span class="o">=</span><span class="s">&quot;dpll2&quot;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Check satisfiability of a propositional sentence.</span>
-<span class="sd">    Returns a model when it succeeds</span>
-
-<span class="sd">    Examples:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import A, B</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.logic.inference import satisfiable</span>
-<span class="sd">    &gt;&gt;&gt; satisfiable(A &amp; ~B)</span>
-<span class="sd">    {A: True, B: False}</span>
-<span class="sd">    &gt;&gt;&gt; satisfiable(A &amp; ~A)</span>
-<span class="sd">    False</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">to_cnf</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">algorithm</span> <span class="o">==</span> <span class="s">&quot;dpll&quot;</span><span class="p">:</span>
-        <span class="kn">from</span> <span class="nn">sympy.logic.algorithms.dpll</span> <span class="kn">import</span> <span class="n">dpll_satisfiable</span>
-        <span class="k">return</span> <span class="n">dpll_satisfiable</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">algorithm</span> <span class="o">==</span> <span class="s">&quot;dpll2&quot;</span><span class="p">:</span>
-        <span class="kn">from</span> <span class="nn">sympy.logic.algorithms.dpll2</span> <span class="kn">import</span> <span class="n">dpll_satisfiable</span>
-        <span class="k">return</span> <span class="n">dpll_satisfiable</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-</div>
-<span class="k">def</span> <span class="nf">pl_true</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">model</span><span class="o">=</span><span class="p">{}):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return True if the propositional logic expression is true in the model,</span>
-<span class="sd">    and False if it is false. If the model does not specify the value for</span>
-<span class="sd">    every proposition, this may return None to indicate &#39;not obvious&#39;;</span>
-<span class="sd">    this may happen even when the expression is tautological.</span>
-
-<span class="sd">    The model is implemented as a dict containing the pair symbol, boolean value.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import A, B</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.logic.inference import pl_true</span>
-<span class="sd">    &gt;&gt;&gt; pl_true( A &amp; B, {A: True, B : True})</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="nb">bool</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">model</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="n">args</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span>
-    <span class="n">func</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span>
-
-    <span class="k">if</span> <span class="n">func</span> <span class="ow">is</span> <span class="n">Not</span><span class="p">:</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">pl_true</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">model</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">p</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="ow">not</span> <span class="n">p</span>
-    <span class="k">elif</span> <span class="n">func</span> <span class="ow">is</span> <span class="n">Or</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">pl_true</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">model</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">p</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">if</span> <span class="n">p</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">return</span> <span class="n">result</span>
-    <span class="k">elif</span> <span class="n">func</span> <span class="ow">is</span> <span class="n">And</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">pl_true</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">model</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">p</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">if</span> <span class="n">p</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="k">elif</span> <span class="n">func</span> <span class="ow">is</span> <span class="n">Implies</span><span class="p">:</span>
-        <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">args</span>
-        <span class="k">return</span> <span class="n">pl_true</span><span class="p">(</span><span class="n">Or</span><span class="p">(</span><span class="n">Not</span><span class="p">(</span><span class="n">p</span><span class="p">),</span> <span class="n">q</span><span class="p">),</span> <span class="n">model</span><span class="p">)</span>
-
-    <span class="k">elif</span> <span class="n">func</span> <span class="ow">is</span> <span class="n">Equivalent</span><span class="p">:</span>
-        <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">args</span>
-        <span class="n">pt</span> <span class="o">=</span> <span class="n">pl_true</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">model</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">pt</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="n">qt</span> <span class="o">=</span> <span class="n">pl_true</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">model</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">qt</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">return</span> <span class="n">pt</span> <span class="o">==</span> <span class="n">qt</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Illegal operator in logic expression&quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-
-
-<span class="k">class</span> <span class="nc">KB</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Base class for all knowledge bases&quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sentence</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">clauses</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">if</span> <span class="n">sentence</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">tell</span><span class="p">(</span><span class="n">sentence</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">tell</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sentence</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">def</span> <span class="nf">ask</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">query</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">def</span> <span class="nf">retract</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sentence</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-
-<span class="k">class</span> <span class="nc">PropKB</span><span class="p">(</span><span class="n">KB</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A KB for Propositional Logic.  Inefficient, with no indexing.&quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">tell</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sentence</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Add the sentence&#39;s clauses to the KB</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.logic.inference import PropKB</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; l = PropKB()</span>
-<span class="sd">        &gt;&gt;&gt; l.clauses</span>
-<span class="sd">        []</span>
-
-<span class="sd">        &gt;&gt;&gt; l.tell(x | y)</span>
-<span class="sd">        &gt;&gt;&gt; l.clauses</span>
-<span class="sd">        [Or(x, y)]</span>
-
-<span class="sd">        &gt;&gt;&gt; l.tell(y)</span>
-<span class="sd">        &gt;&gt;&gt; l.clauses</span>
-<span class="sd">        [Or(x, y), y]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">conjuncts</span><span class="p">(</span><span class="n">to_cnf</span><span class="p">(</span><span class="n">sentence</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">c</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">clauses</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">clauses</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">ask</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">query</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Checks if the query is true given the set of clauses.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.logic.inference import PropKB</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; l = PropKB()</span>
-<span class="sd">        &gt;&gt;&gt; l.tell(x &amp; ~y)</span>
-<span class="sd">        &gt;&gt;&gt; l.ask(x)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; l.ask(y)</span>
-<span class="sd">        False</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">clauses</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="kn">from</span> <span class="nn">sympy.logic.algorithms.dpll</span> <span class="kn">import</span> <span class="n">dpll</span>
-        <span class="n">query_conjuncts</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">clauses</span><span class="p">[:]</span>
-        <span class="n">query_conjuncts</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">conjuncts</span><span class="p">(</span><span class="n">to_cnf</span><span class="p">(</span><span class="n">query</span><span class="p">)))</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">query_conjuncts</span><span class="p">:</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">))</span>
-        <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="n">dpll</span><span class="p">(</span><span class="n">query_conjuncts</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="p">{}))</span>
-
-    <span class="k">def</span> <span class="nf">retract</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sentence</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Remove the sentence&#39;s clauses from the KB</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.logic.inference import PropKB</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; l = PropKB()</span>
-<span class="sd">        &gt;&gt;&gt; l.clauses</span>
-<span class="sd">        []</span>
-
-<span class="sd">        &gt;&gt;&gt; l.tell(x | y)</span>
-<span class="sd">        &gt;&gt;&gt; l.clauses</span>
-<span class="sd">        [Or(x, y)]</span>
-
-<span class="sd">        &gt;&gt;&gt; l.retract(x | y)</span>
-<span class="sd">        &gt;&gt;&gt; l.clauses</span>
-<span class="sd">        []</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">conjuncts</span><span class="p">(</span><span class="n">to_cnf</span><span class="p">(</span><span class="n">sentence</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="n">c</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">clauses</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">clauses</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-</pre></div>
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diff --git a/dev-py3k/_modules/sympy/matrices/dense.html b/dev-py3k/_modules/sympy/matrices/dense.html
deleted file mode 100644
index 4bee54d73e4..00000000000
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+++ /dev/null
@@ -1,1712 +0,0 @@
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-            
-  <h1>Source code for sympy.matrices.dense</h1><div class="highlight"><pre>
-<span class="kn">import</span> <span class="nn">random</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.basic</span> <span class="kn">import</span> <span class="n">Basic</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">is_sequence</span><span class="p">,</span> <span class="n">as_int</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">count_ops</span>
-<span class="kn">from</span> <span class="nn">sympy.core.decorators</span> <span class="kn">import</span> <span class="n">call_highest_priority</span>
-<span class="kn">from</span> <span class="nn">sympy.core.singleton</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="kn">from</span> <span class="nn">sympy.core.symbol</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.trigonometric</span> <span class="kn">import</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">simplify</span> <span class="k">as</span> <span class="n">_simplify</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.exceptions</span> <span class="kn">import</span> <span class="n">SymPyDeprecationWarning</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.misc</span> <span class="kn">import</span> <span class="n">filldedent</span>
-
-<span class="kn">from</span> <span class="nn">sympy.matrices.matrices</span> <span class="kn">import</span> <span class="p">(</span><span class="n">MatrixBase</span><span class="p">,</span>
-    <span class="n">ShapeError</span><span class="p">,</span> <span class="n">a2idx</span><span class="p">,</span> <span class="n">classof</span><span class="p">)</span>
-<span class="kn">import</span> <span class="nn">collections</span>
-
-
-<span class="k">def</span> <span class="nf">_iszero</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns True if x is zero.&quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">x</span><span class="o">.</span><span class="n">is_zero</span>
-
-
-<span class="k">class</span> <span class="nc">DenseMatrix</span><span class="p">(</span><span class="n">MatrixBase</span><span class="p">):</span>
-
-    <span class="n">is_MatrixExpr</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="n">_op_priority</span> <span class="o">=</span> <span class="mf">10.01</span>
-    <span class="n">_class_priority</span> <span class="o">=</span> <span class="mi">4</span>
-
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return portion of self defined by key. If the key involves a slice</span>
-<span class="sd">        then a list will be returned (if key is a single slice) or a matrix</span>
-<span class="sd">        (if key was a tuple involving a slice).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, I</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix([</span>
-<span class="sd">        ... [1, 2 + I],</span>
-<span class="sd">        ... [3, 4    ]])</span>
-
-<span class="sd">        If the key is a tuple that doesn&#39;t involve a slice then that element</span>
-<span class="sd">        is returned:</span>
-
-<span class="sd">        &gt;&gt;&gt; m[1, 0]</span>
-<span class="sd">        3</span>
-
-<span class="sd">        When a tuple key involves a slice, a matrix is returned. Here, the</span>
-<span class="sd">        first column is selected (all rows, column 0):</span>
-
-<span class="sd">        &gt;&gt;&gt; m[:, 0]</span>
-<span class="sd">        [1]</span>
-<span class="sd">        [3]</span>
-
-<span class="sd">        If the slice is not a tuple then it selects from the underlying</span>
-<span class="sd">        list of elements that are arranged in row order and a list is</span>
-<span class="sd">        returned if a slice is involved:</span>
-
-<span class="sd">        &gt;&gt;&gt; m[0]</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; m[::2]</span>
-<span class="sd">        [1, 3]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">key</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">tuple</span><span class="p">:</span>
-            <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">key</span>
-            <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">slice</span> <span class="ow">or</span> <span class="nb">type</span><span class="p">(</span><span class="n">j</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">slice</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">submatrix</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">key2ij</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">+</span> <span class="n">j</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># row-wise decomposition of matrix</span>
-            <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">key</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">slice</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">[</span><span class="n">key</span><span class="p">]</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">[</span><span class="n">a2idx</span><span class="p">(</span><span class="n">key</span><span class="p">)]</span>
-
-    <span class="k">def</span> <span class="nf">__setitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># issue 880 suggests that there should be no hash for a mutable</span>
-        <span class="c"># object...but at least we aren&#39;t caching the result</span>
-        <span class="k">return</span> <span class="nb">hash</span><span class="p">((</span><span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__name__</span><span class="p">,)</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">)))</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_Identity</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">(</span><span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="ow">or</span> <span class="bp">self</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">]:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">tolist</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the Matrix as a nested Python list.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, ones</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix(3, 3, list(range(9)))</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [0, 1, 2]</span>
-<span class="sd">        [3, 4, 5]</span>
-<span class="sd">        [6, 7, 8]</span>
-<span class="sd">        &gt;&gt;&gt; m.tolist()</span>
-<span class="sd">        [[0, 1, 2], [3, 4, 5], [6, 7, 8]]</span>
-<span class="sd">        &gt;&gt;&gt; ones(3, 0).tolist()</span>
-<span class="sd">        [[], [], []]</span>
-
-<span class="sd">        When there are no rows then it will not be possible to tell how</span>
-<span class="sd">        many columns were in the original matrix:</span>
-
-<span class="sd">        &gt;&gt;&gt; ones(0, 3).tolist()</span>
-<span class="sd">        []</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[]</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[[]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)]</span>
-        <span class="k">return</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">[</span><span class="n">i</span><span class="p">:</span> <span class="n">i</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">]</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">)]</span>
-
-    <span class="k">def</span> <span class="nf">row</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">f</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Elementary row selector.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import eye</span>
-<span class="sd">        &gt;&gt;&gt; eye(2).row(0)</span>
-<span class="sd">        [1, 0]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        col</span>
-<span class="sd">        row_op</span>
-<span class="sd">        row_swap</span>
-<span class="sd">        row_del</span>
-<span class="sd">        row_join</span>
-<span class="sd">        row_insert</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">f</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span>
-        <span class="n">SymPyDeprecationWarning</span><span class="p">(</span>
-            <span class="n">feature</span><span class="o">=</span><span class="s">&quot;calling .row(i, f)&quot;</span><span class="p">,</span>
-            <span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;.row_op(i, f)&quot;</span><span class="p">,</span>
-            <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.2&quot;</span><span class="p">,</span>
-        <span class="p">)</span><span class="o">.</span><span class="n">warn</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">row_op</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">col</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">f</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Elementary column selector.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import eye</span>
-<span class="sd">        &gt;&gt;&gt; eye(2).col(0)</span>
-<span class="sd">        [1]</span>
-<span class="sd">        [0]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        row</span>
-<span class="sd">        col_op</span>
-<span class="sd">        col_swap</span>
-<span class="sd">        col_del</span>
-<span class="sd">        col_join</span>
-<span class="sd">        col_insert</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">f</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="p">[:,</span> <span class="n">j</span><span class="p">]</span>
-        <span class="n">SymPyDeprecationWarning</span><span class="p">(</span>
-            <span class="n">feature</span><span class="o">=</span><span class="s">&quot;calling .col(j, f)&quot;</span><span class="p">,</span>
-            <span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;.col_op(j, f)&quot;</span><span class="p">,</span>
-            <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.2&quot;</span><span class="p">,</span>
-        <span class="p">)</span><span class="o">.</span><span class="n">warn</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">col_op</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_trace</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Calculate the trace of a square matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import eye</span>
-<span class="sd">        &gt;&gt;&gt; eye(3).trace()</span>
-<span class="sd">        3</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">trace</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-            <span class="n">trace</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">+</span> <span class="n">i</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">trace</span>
-
-    <span class="k">def</span> <span class="nf">_eval_transpose</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Matrix transposition.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, I</span>
-<span class="sd">        &gt;&gt;&gt; m=Matrix(((1, 2+I), (3, 4)))</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [1, 2 + I]</span>
-<span class="sd">        [3,     4]</span>
-<span class="sd">        &gt;&gt;&gt; m.transpose()</span>
-<span class="sd">        [    1, 3]</span>
-<span class="sd">        [2 + I, 4]</span>
-<span class="sd">        &gt;&gt;&gt; m.T == m.transpose()</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        conjugate: By-element conjugation</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-            <span class="n">a</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">[</span><span class="n">i</span><span class="p">::</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">])</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;By-element conjugation.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        transpose: Matrix transposition</span>
-<span class="sd">        H: Hermite conjugation</span>
-<span class="sd">        D: Dirac conjugation</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">out</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span>
-                <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">conjugate</span><span class="p">())</span>
-        <span class="k">return</span> <span class="n">out</span>
-
-    <span class="k">def</span> <span class="nf">_eval_adjoint</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="o">.</span><span class="n">C</span>
-
-    <span class="k">def</span> <span class="nf">_eval_inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the matrix inverse using the method indicated (default</span>
-<span class="sd">        is Gauss elimination).</span>
-
-<span class="sd">        kwargs</span>
-<span class="sd">        ======</span>
-
-<span class="sd">        method : (&#39;GE&#39;, &#39;LU&#39;, or &#39;ADJ&#39;)</span>
-<span class="sd">        iszerofunc</span>
-<span class="sd">        try_block_diag</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        According to the ``method`` keyword, it calls the appropriate method:</span>
-
-<span class="sd">          GE .... inverse_GE(); default</span>
-<span class="sd">          LU .... inverse_LU()</span>
-<span class="sd">          ADJ ... inverse_ADJ()</span>
-
-<span class="sd">        According to the ``try_block_diag`` keyword, it will try to form block</span>
-<span class="sd">        diagonal matrices using the method get_diag_blocks(), invert these</span>
-<span class="sd">        individually, and then reconstruct the full inverse matrix.</span>
-
-<span class="sd">        Note, the GE and LU methods may require the matrix to be simplified</span>
-<span class="sd">        before it is inverted in order to properly detect zeros during</span>
-<span class="sd">        pivoting. In difficult cases a custom zero detection function can</span>
-<span class="sd">        be provided by setting the ``iszerosfunc`` argument to a function that</span>
-<span class="sd">        should return True if its argument is zero. The ADJ routine computes</span>
-<span class="sd">        the determinant and uses that to detect singular matrices in addition</span>
-<span class="sd">        to testing for zeros on the diagonal.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        inverse_LU</span>
-<span class="sd">        inverse_GE</span>
-<span class="sd">        inverse_ADJ</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">diag</span>
-
-        <span class="n">method</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;method&#39;</span><span class="p">,</span> <span class="s">&#39;GE&#39;</span><span class="p">)</span>
-        <span class="n">iszerofunc</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;iszerofunc&#39;</span><span class="p">,</span> <span class="n">_iszero</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;try_block_diag&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-            <span class="n">blocks</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_diag_blocks</span><span class="p">()</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">block</span> <span class="ow">in</span> <span class="n">blocks</span><span class="p">:</span>
-                <span class="n">r</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">block</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">method</span><span class="o">=</span><span class="n">method</span><span class="p">,</span> <span class="n">iszerofunc</span><span class="o">=</span><span class="n">iszerofunc</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">diag</span><span class="p">(</span><span class="o">*</span><span class="n">r</span><span class="p">)</span>
-
-        <span class="n">M</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_mutable</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">method</span> <span class="o">==</span> <span class="s">&quot;GE&quot;</span><span class="p">:</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">inverse_GE</span><span class="p">(</span><span class="n">iszerofunc</span><span class="o">=</span><span class="n">iszerofunc</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">method</span> <span class="o">==</span> <span class="s">&quot;LU&quot;</span><span class="p">:</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">inverse_LU</span><span class="p">(</span><span class="n">iszerofunc</span><span class="o">=</span><span class="n">iszerofunc</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">method</span> <span class="o">==</span> <span class="s">&quot;ADJ&quot;</span><span class="p">:</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">inverse_ADJ</span><span class="p">(</span><span class="n">iszerofunc</span><span class="o">=</span><span class="n">iszerofunc</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># make sure to add an invertibility check (as in inverse_LU)</span>
-            <span class="c"># if a new method is added.</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Inversion method unrecognized&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">rv</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">equals</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="n">failing_expression</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Applies ``equals`` to corresponding elements of the matrices,</span>
-<span class="sd">        trying to prove that the elements are equivalent, returning True</span>
-<span class="sd">        if they are, False if any pair is not, and None (or the first</span>
-<span class="sd">        failing expression if failing_expression is True) if it cannot</span>
-<span class="sd">        be decided if the expressions are equivalent or not. This is, in</span>
-<span class="sd">        general, an expensive operation.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import cos</span>
-<span class="sd">        &gt;&gt;&gt; A = Matrix([x*(x - 1), 0])</span>
-<span class="sd">        &gt;&gt;&gt; B = Matrix([x**2 - x, 0])</span>
-<span class="sd">        &gt;&gt;&gt; A == B</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; A.simplify() == B.simplify()</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; A.equals(B)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; A.equals(2)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        sympy.core.expr.equals</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">shape</span> <span class="o">!=</span> <span class="n">other</span><span class="o">.</span><span class="n">shape</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-                    <span class="n">ans</span> <span class="o">=</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="n">other</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">],</span> <span class="n">failing_expression</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">ans</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="bp">False</span>
-                    <span class="k">elif</span> <span class="n">ans</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">True</span> <span class="ow">and</span> <span class="n">rv</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                        <span class="n">rv</span> <span class="o">=</span> <span class="n">ans</span>
-            <span class="k">return</span> <span class="n">rv</span>
-        <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">shape</span> <span class="o">!=</span> <span class="n">other</span><span class="o">.</span><span class="n">shape</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">_mat</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">MatrixBase</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span> <span class="o">==</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">other</span><span class="p">)</span><span class="o">.</span><span class="n">_mat</span>
-        <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="bp">self</span> <span class="o">==</span> <span class="n">other</span>
-
-    <span class="k">def</span> <span class="nf">_cholesky</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Helper function of cholesky.</span>
-<span class="sd">        Without the error checks.</span>
-<span class="sd">        To be used privately. &quot;&quot;&quot;</span>
-        <span class="n">L</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
-                <span class="n">L</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="n">L</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">])</span><span class="o">*</span><span class="p">(</span><span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">-</span>
-                    <span class="nb">sum</span><span class="p">(</span><span class="n">L</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">L</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">j</span><span class="p">)))</span>
-            <span class="n">L</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">-</span>
-                    <span class="nb">sum</span><span class="p">(</span><span class="n">L</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">L</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_LDLdecomposition</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Helper function of LDLdecomposition.</span>
-<span class="sd">        Without the error checks.</span>
-<span class="sd">        To be used privately.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)</span>
-        <span class="n">L</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
-                <span class="n">L</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="n">D</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">])</span><span class="o">*</span><span class="p">(</span><span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="nb">sum</span><span class="p">(</span>
-                    <span class="n">L</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">L</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">D</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">j</span><span class="p">)))</span>
-            <span class="n">D</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="nb">sum</span><span class="p">(</span><span class="n">L</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">D</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span>
-                <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">L</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">D</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_lower_triangular_solve</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Helper function of function lower_triangular_solve.</span>
-<span class="sd">        Without the error checks.</span>
-<span class="sd">        To be used privately.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">X</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="n">rhs</span><span class="o">.</span><span class="n">cols</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">rhs</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Matrix must be non-singular.&quot;</span><span class="p">)</span>
-                <span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">rhs</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="nb">sum</span><span class="p">(</span><span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">X</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-                    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">)))</span> <span class="o">/</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_upper_triangular_solve</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Helper function of function upper_triangular_solve.</span>
-<span class="sd">        Without the error checks, to be used privately. &quot;&quot;&quot;</span>
-        <span class="n">X</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="n">rhs</span><span class="o">.</span><span class="n">cols</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">rhs</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">))):</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Matrix must be non-singular.&quot;</span><span class="p">)</span>
-                <span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">rhs</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="nb">sum</span><span class="p">(</span><span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">X</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-                    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)))</span> <span class="o">/</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_diagonal_solve</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Helper function of function diagonal_solve,</span>
-<span class="sd">        without the error checks, to be used privately.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">rhs</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="n">rhs</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="n">rhs</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">/</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">applyfunc</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Apply a function to each element of the matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix(2, 2, lambda i, j: i*2+j)</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [0, 1]</span>
-<span class="sd">        [2, 3]</span>
-<span class="sd">        &gt;&gt;&gt; m.applyfunc(lambda i: 2*i)</span>
-<span class="sd">        [0, 2]</span>
-<span class="sd">        [4, 6]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">collections</span><span class="o">.</span><span class="n">Callable</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;`f` must be callable.&quot;</span><span class="p">)</span>
-
-        <span class="n">out</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">out</span>
-
-    <span class="k">def</span> <span class="nf">reshape</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rows</span><span class="p">,</span> <span class="n">cols</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Reshape the matrix. Total number of elements must remain the same.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix(2, 3, lambda i, j: 1)</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [1, 1, 1]</span>
-<span class="sd">        [1, 1, 1]</span>
-<span class="sd">        &gt;&gt;&gt; m.reshape(1, 6)</span>
-<span class="sd">        [1, 1, 1, 1, 1, 1]</span>
-<span class="sd">        &gt;&gt;&gt; m.reshape(3, 2)</span>
-<span class="sd">        [1, 1]</span>
-<span class="sd">        [1, 1]</span>
-<span class="sd">        [1, 1]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">!=</span> <span class="n">rows</span><span class="o">*</span><span class="n">cols</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Invalid reshape parameters </span><span class="si">%d</span><span class="s"> </span><span class="si">%d</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">rows</span><span class="p">,</span> <span class="n">cols</span><span class="p">))</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">rows</span><span class="p">,</span> <span class="n">cols</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="n">cols</span> <span class="o">+</span> <span class="n">j</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">as_mutable</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a mutable version of this matrix</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import ImmutableMatrix</span>
-<span class="sd">        &gt;&gt;&gt; X = ImmutableMatrix([[1, 2], [3, 4]])</span>
-<span class="sd">        &gt;&gt;&gt; Y = X.as_mutable()</span>
-<span class="sd">        &gt;&gt;&gt; Y[1, 1] = 5 # Can set values in Y</span>
-<span class="sd">        &gt;&gt;&gt; Y</span>
-<span class="sd">        [1, 2]</span>
-<span class="sd">        [3, 5]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">as_immutable</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns an Immutable version of this Matrix</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">.immutable</span> <span class="kn">import</span> <span class="n">ImmutableMatrix</span> <span class="k">as</span> <span class="n">cls</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">tolist</span><span class="p">())</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span> <span class="p">[])</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">zeros</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return an r x c matrix of zeros, square if c is omitted.&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">r</span><span class="p">):</span>
-            <span class="n">SymPyDeprecationWarning</span><span class="p">(</span>
-                <span class="n">feature</span><span class="o">=</span><span class="s">&quot;The syntax zeros([</span><span class="si">%i</span><span class="s">, </span><span class="si">%i</span><span class="s">])&quot;</span> <span class="o">%</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">r</span><span class="p">),</span>
-                <span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;zeros(</span><span class="si">%i</span><span class="s">, </span><span class="si">%i</span><span class="s">).&quot;</span> <span class="o">%</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">r</span><span class="p">),</span>
-                <span class="n">issue</span><span class="o">=</span><span class="mi">3381</span><span class="p">,</span> <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.2&quot;</span><span class="p">,</span>
-            <span class="p">)</span><span class="o">.</span><span class="n">warn</span><span class="p">()</span>
-            <span class="n">r</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">r</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">r</span> <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="bp">None</span> <span class="k">else</span> <span class="n">c</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">]</span><span class="o">*</span><span class="n">r</span><span class="o">*</span><span class="n">c</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eye</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return an n x n identity matrix.&quot;&quot;&quot;</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="n">mat</span> <span class="o">=</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">]</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">n</span>
-        <span class="n">mat</span><span class="p">[::</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">]</span><span class="o">*</span><span class="n">n</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">mat</span><span class="p">)</span>
-
-    <span class="c">############################</span>
-    <span class="c"># Mutable matrix operators #</span>
-    <span class="c">############################</span>
-
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__radd__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">_force_mutable</span><span class="p">(</span><span class="n">other</span><span class="p">))</span>
-
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__add__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__radd__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">__radd__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">_force_mutable</span><span class="p">(</span><span class="n">other</span><span class="p">))</span>
-
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__rsub__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">_force_mutable</span><span class="p">(</span><span class="n">other</span><span class="p">))</span>
-
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__sub__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__rsub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">__rsub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">_force_mutable</span><span class="p">(</span><span class="n">other</span><span class="p">))</span>
-
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__rmul__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">_force_mutable</span><span class="p">(</span><span class="n">other</span><span class="p">))</span>
-
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__mul__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">_force_mutable</span><span class="p">(</span><span class="n">other</span><span class="p">))</span>
-
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__div__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">_force_mutable</span><span class="p">(</span><span class="n">other</span><span class="p">))</span>
-
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__truediv__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__truediv__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">__truediv__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">_force_mutable</span><span class="p">(</span><span class="n">other</span><span class="p">))</span>
-
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__rpow__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__pow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">__pow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@call_highest_priority</span><span class="p">(</span><span class="s">&#39;__pow__&#39;</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__rpow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Matrix Power not defined&quot;</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">_force_mutable</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return a matrix as a Matrix, otherwise return x.&quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="s">&#39;is_Matrix&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">x</span><span class="o">.</span><span class="n">as_mutable</span><span class="p">()</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">x</span>
-    <span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="s">&#39;__array__&#39;</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">__array__</span><span class="p">()</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">x</span>
-
-
-<div class="viewcode-block" id="MutableDenseMatrix"><a class="viewcode-back" href="../../../modules/matrices/dense.html#sympy.matrices.dense.MutableDenseMatrix">[docs]</a><span class="k">class</span> <span class="nc">MutableDenseMatrix</span><span class="p">(</span><span class="n">DenseMatrix</span><span class="p">,</span> <span class="n">MatrixBase</span><span class="p">):</span>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_new</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">rows</span><span class="p">,</span> <span class="n">cols</span><span class="p">,</span> <span class="n">flat_list</span> <span class="o">=</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">_handle_creation_inputs</span><span class="p">(</span>
-            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="bp">self</span> <span class="o">=</span> <span class="nb">object</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">=</span> <span class="n">rows</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">=</span> <span class="n">cols</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">flat_list</span><span class="p">)</span>  <span class="c"># create a shallow copy</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__setitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, I, zeros, ones</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix(((1, 2+I), (3, 4)))</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [1, 2 + I]</span>
-<span class="sd">        [3,     4]</span>
-<span class="sd">        &gt;&gt;&gt; m[1, 0] = 9</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [1, 2 + I]</span>
-<span class="sd">        [9,     4]</span>
-<span class="sd">        &gt;&gt;&gt; m[1, 0] = [[0, 1]]</span>
-
-<span class="sd">        To replace row r you assign to position r*m where m</span>
-<span class="sd">        is the number of columns:</span>
-
-<span class="sd">        &gt;&gt;&gt; M = zeros(4)</span>
-<span class="sd">        &gt;&gt;&gt; m = M.cols</span>
-<span class="sd">        &gt;&gt;&gt; M[3*m] = ones(1, m)*2; M</span>
-<span class="sd">        [0, 0, 0, 0]</span>
-<span class="sd">        [0, 0, 0, 0]</span>
-<span class="sd">        [0, 0, 0, 0]</span>
-<span class="sd">        [2, 2, 2, 2]</span>
-
-<span class="sd">        And to replace column c you can assign to position c:</span>
-
-<span class="sd">        &gt;&gt;&gt; M[2] = ones(m, 1)*4; M</span>
-<span class="sd">        [0, 0, 4, 0]</span>
-<span class="sd">        [0, 0, 4, 0]</span>
-<span class="sd">        [0, 0, 4, 0]</span>
-<span class="sd">        [2, 2, 4, 2]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_setitem</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">value</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">rv</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">value</span> <span class="o">=</span> <span class="n">rv</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">+</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">value</span>
-
-<div class="viewcode-block" id="MutableDenseMatrix.copyin_matrix"><a class="viewcode-back" href="../../../modules/matrices/dense.html#sympy.matrices.dense.MutableDenseMatrix.copyin_matrix">[docs]</a>    <span class="k">def</span> <span class="nf">copyin_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Copy in values from a matrix into the given bounds.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        key : slice</span>
-<span class="sd">            The section of this matrix to replace.</span>
-<span class="sd">        value : Matrix</span>
-<span class="sd">            The matrix to copy values from.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import Matrix, eye</span>
-<span class="sd">        &gt;&gt;&gt; M = Matrix([[0, 1], [2, 3], [4, 5]])</span>
-<span class="sd">        &gt;&gt;&gt; I = eye(3)</span>
-<span class="sd">        &gt;&gt;&gt; I[:3, :2] = M</span>
-<span class="sd">        &gt;&gt;&gt; I</span>
-<span class="sd">        [0, 1, 0]</span>
-<span class="sd">        [2, 3, 0]</span>
-<span class="sd">        [4, 5, 1]</span>
-<span class="sd">        &gt;&gt;&gt; I[0, 1] = M</span>
-<span class="sd">        &gt;&gt;&gt; I</span>
-<span class="sd">        [0, 0, 1]</span>
-<span class="sd">        [2, 2, 3]</span>
-<span class="sd">        [4, 4, 5]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        copyin_list</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">rlo</span><span class="p">,</span> <span class="n">rhi</span><span class="p">,</span> <span class="n">clo</span><span class="p">,</span> <span class="n">chi</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">key2bounds</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
-        <span class="n">shape</span> <span class="o">=</span> <span class="n">value</span><span class="o">.</span><span class="n">shape</span>
-        <span class="n">dr</span><span class="p">,</span> <span class="n">dc</span> <span class="o">=</span> <span class="n">rhi</span> <span class="o">-</span> <span class="n">rlo</span><span class="p">,</span> <span class="n">chi</span> <span class="o">-</span> <span class="n">clo</span>
-        <span class="k">if</span> <span class="n">shape</span> <span class="o">!=</span> <span class="p">(</span><span class="n">dr</span><span class="p">,</span> <span class="n">dc</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="n">ShapeError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&quot;The Matrix `value` doesn&#39;t have the &quot;</span>
-                <span class="s">&quot;same dimensions &quot;</span>
-                <span class="s">&quot;as the in sub-Matrix given by `key`.&quot;</span><span class="p">))</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">value</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">value</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-                <span class="bp">self</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="n">rlo</span><span class="p">,</span> <span class="n">j</span> <span class="o">+</span> <span class="n">clo</span><span class="p">]</span> <span class="o">=</span> <span class="n">value</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-</div>
-<div class="viewcode-block" id="MutableDenseMatrix.copyin_list"><a class="viewcode-back" href="../../../modules/matrices/dense.html#sympy.matrices.dense.MutableDenseMatrix.copyin_list">[docs]</a>    <span class="k">def</span> <span class="nf">copyin_list</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Copy in elements from a list.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        key : slice</span>
-<span class="sd">            The section of this matrix to replace.</span>
-<span class="sd">        value : iterable</span>
-<span class="sd">            The iterable to copy values from.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import eye</span>
-<span class="sd">        &gt;&gt;&gt; I = eye(3)</span>
-<span class="sd">        &gt;&gt;&gt; I[:2, 0] = [1, 2] # col</span>
-<span class="sd">        &gt;&gt;&gt; I</span>
-<span class="sd">        [1, 0, 0]</span>
-<span class="sd">        [2, 1, 0]</span>
-<span class="sd">        [0, 0, 1]</span>
-<span class="sd">        &gt;&gt;&gt; I[1, :2] = [[3, 4]]</span>
-<span class="sd">        &gt;&gt;&gt; I</span>
-<span class="sd">        [1, 0, 0]</span>
-<span class="sd">        [3, 4, 0]</span>
-<span class="sd">        [0, 0, 1]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        copyin_matrix</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">value</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span>
-                <span class="s">&quot;`value` must be an ordered iterable, not </span><span class="si">%s</span><span class="s">.&quot;</span> <span class="o">%</span> <span class="nb">type</span><span class="p">(</span><span class="n">value</span><span class="p">))</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">copyin_matrix</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">value</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="MutableDenseMatrix.row_op"><a class="viewcode-back" href="../../../modules/matrices/dense.html#sympy.matrices.dense.MutableDenseMatrix.row_op">[docs]</a>    <span class="k">def</span> <span class="nf">row_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;In-place operation on row i using two-arg functor whose args are</span>
-<span class="sd">        interpreted as (self[i, j], j).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import eye</span>
-<span class="sd">        &gt;&gt;&gt; M = eye(3)</span>
-<span class="sd">        &gt;&gt;&gt; M.row_op(1, lambda v, j: v + 2*M[0, j]); M</span>
-<span class="sd">        [1, 0, 0]</span>
-<span class="sd">        [2, 1, 0]</span>
-<span class="sd">        [0, 0, 1]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        row</span>
-<span class="sd">        col_op</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">i0</span> <span class="o">=</span> <span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">[</span><span class="n">i0</span><span class="p">:</span> <span class="n">i0</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="p">(</span><span class="o">*</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">[</span><span class="n">i0</span><span class="p">:</span> <span class="n">i0</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">],</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">)))]</span>
-</div>
-<div class="viewcode-block" id="MutableDenseMatrix.col_op"><a class="viewcode-back" href="../../../modules/matrices/dense.html#sympy.matrices.dense.MutableDenseMatrix.col_op">[docs]</a>    <span class="k">def</span> <span class="nf">col_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;In-place operation on col j using two-arg functor whose args are</span>
-<span class="sd">        interpreted as (self[i, j], i).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import eye</span>
-<span class="sd">        &gt;&gt;&gt; M = eye(3)</span>
-<span class="sd">        &gt;&gt;&gt; M.col_op(1, lambda v, i: v + 2*M[i, 0]); M</span>
-<span class="sd">        [1, 2, 0]</span>
-<span class="sd">        [0, 1, 0]</span>
-<span class="sd">        [0, 0, 1]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        col</span>
-<span class="sd">        row_op</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">[</span><span class="n">j</span><span class="p">::</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="p">(</span><span class="o">*</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">[</span><span class="n">j</span><span class="p">::</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">],</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)))]</span>
-</div>
-<div class="viewcode-block" id="MutableDenseMatrix.row_swap"><a class="viewcode-back" href="../../../modules/matrices/dense.html#sympy.matrices.dense.MutableDenseMatrix.row_swap">[docs]</a>    <span class="k">def</span> <span class="nf">row_swap</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Swap the two given rows of the matrix in-place.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; M = Matrix([[0, 1], [1, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; M</span>
-<span class="sd">        [0, 1]</span>
-<span class="sd">        [1, 0]</span>
-<span class="sd">        &gt;&gt;&gt; M.row_swap(0, 1)</span>
-<span class="sd">        &gt;&gt;&gt; M</span>
-<span class="sd">        [1, 0]</span>
-<span class="sd">        [0, 1]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        row</span>
-<span class="sd">        col_swap</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-            <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">],</span> <span class="bp">self</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">],</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span>
-</div>
-<div class="viewcode-block" id="MutableDenseMatrix.col_swap"><a class="viewcode-back" href="../../../modules/matrices/dense.html#sympy.matrices.dense.MutableDenseMatrix.col_swap">[docs]</a>    <span class="k">def</span> <span class="nf">col_swap</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Swap the two given columns of the matrix in-place.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; M = Matrix([[1, 0], [1, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; M</span>
-<span class="sd">        [1, 0]</span>
-<span class="sd">        [1, 0]</span>
-<span class="sd">        &gt;&gt;&gt; M.col_swap(0, 1)</span>
-<span class="sd">        &gt;&gt;&gt; M</span>
-<span class="sd">        [0, 1]</span>
-<span class="sd">        [0, 1]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        col</span>
-<span class="sd">        row_swap</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="bp">self</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">i</span><span class="p">],</span> <span class="bp">self</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">j</span><span class="p">],</span> <span class="bp">self</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span>
-</div>
-<div class="viewcode-block" id="MutableDenseMatrix.row_del"><a class="viewcode-back" href="../../../modules/matrices/dense.html#sympy.matrices.dense.MutableDenseMatrix.row_del">[docs]</a>    <span class="k">def</span> <span class="nf">row_del</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Delete the given row.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import eye</span>
-<span class="sd">        &gt;&gt;&gt; M = eye(3)</span>
-<span class="sd">        &gt;&gt;&gt; M.row_del(1)</span>
-<span class="sd">        &gt;&gt;&gt; M</span>
-<span class="sd">        [1, 0, 0]</span>
-<span class="sd">        [0, 0, 1]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        row</span>
-<span class="sd">        col_del</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">[:</span><span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">]</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">[(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">:]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">-=</span> <span class="mi">1</span>
-</div>
-<div class="viewcode-block" id="MutableDenseMatrix.col_del"><a class="viewcode-back" href="../../../modules/matrices/dense.html#sympy.matrices.dense.MutableDenseMatrix.col_del">[docs]</a>    <span class="k">def</span> <span class="nf">col_del</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Delete the given column.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import eye</span>
-<span class="sd">        &gt;&gt;&gt; M = eye(3)</span>
-<span class="sd">        &gt;&gt;&gt; M.col_del(1)</span>
-<span class="sd">        &gt;&gt;&gt; M</span>
-<span class="sd">        [1, 0]</span>
-<span class="sd">        [0, 0]</span>
-<span class="sd">        [0, 1]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        col</span>
-<span class="sd">        row_del</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-            <span class="k">del</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">-=</span> <span class="mi">1</span>
-
-    <span class="c"># Utility functions</span></div>
-<div class="viewcode-block" id="MutableDenseMatrix.simplify"><a class="viewcode-back" href="../../../modules/matrices/dense.html#sympy.matrices.dense.MutableDenseMatrix.simplify">[docs]</a>    <span class="k">def</span> <span class="nf">simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ratio</span><span class="o">=</span><span class="mf">1.7</span><span class="p">,</span> <span class="n">measure</span><span class="o">=</span><span class="n">count_ops</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Applies simplify to the elements of a matrix in place.</span>
-
-<span class="sd">        This is a shortcut for M.applyfunc(lambda x: simplify(x, ratio, measure))</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.simplify.simplify.simplify</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">)):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">_simplify</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">ratio</span><span class="o">=</span><span class="n">ratio</span><span class="p">,</span>
-                                     <span class="n">measure</span><span class="o">=</span><span class="n">measure</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MutableDenseMatrix.fill"><a class="viewcode-back" href="../../../modules/matrices/dense.html#sympy.matrices.dense.MutableDenseMatrix.fill">[docs]</a>    <span class="k">def</span> <span class="nf">fill</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Fill the matrix with the scalar value.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        zeros</span>
-<span class="sd">        ones</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span> <span class="o">=</span> <span class="p">[</span><span class="n">value</span><span class="p">]</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-</div></div>
-<span class="n">MutableMatrix</span> <span class="o">=</span> <span class="n">Matrix</span> <span class="o">=</span> <span class="n">MutableDenseMatrix</span>
-
-<span class="c">###########</span>
-<span class="c"># Numpy Utility Functions:</span>
-<span class="c"># list2numpy, matrix2numpy, symmarray, rot_axis[123]</span>
-<span class="c">###########</span>
-
-
-<div class="viewcode-block" id="list2numpy"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.dense.list2numpy">[docs]</a><span class="k">def</span> <span class="nf">list2numpy</span><span class="p">(</span><span class="n">l</span><span class="p">):</span>  <span class="c"># pragma: no cover</span>
-    <span class="sd">&quot;&quot;&quot;Converts python list of SymPy expressions to a NumPy array.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    matrix2numpy</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">empty</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">empty</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">l</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">object</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">s</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">l</span><span class="p">):</span>
-        <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">s</span>
-    <span class="k">return</span> <span class="n">a</span>
-
-</div>
-<div class="viewcode-block" id="matrix2numpy"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.dense.matrix2numpy">[docs]</a><span class="k">def</span> <span class="nf">matrix2numpy</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>  <span class="c"># pragma: no cover</span>
-    <span class="sd">&quot;&quot;&quot;Converts SymPy&#39;s matrix to a NumPy array.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    list2numpy</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">empty</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">empty</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">object</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-            <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">m</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">a</span>
-
-</div>
-<div class="viewcode-block" id="symarray"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.dense.symarray">[docs]</a><span class="k">def</span> <span class="nf">symarray</span><span class="p">(</span><span class="n">prefix</span><span class="p">,</span> <span class="n">shape</span><span class="p">):</span>  <span class="c"># pragma: no cover</span>
-    <span class="sd">&quot;&quot;&quot;Create a numpy ndarray of symbols (as an object array).</span>
-
-<span class="sd">    The created symbols are named ``prefix_i1_i2_``...  You should thus provide a</span>
-<span class="sd">    non-empty prefix if you want your symbols to be unique for different output</span>
-<span class="sd">    arrays, as SymPy symbols with identical names are the same object.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-
-<span class="sd">    prefix : string</span>
-<span class="sd">      A prefix prepended to the name of every symbol.</span>
-
-<span class="sd">    shape : int or tuple</span>
-<span class="sd">      Shape of the created array.  If an int, the array is one-dimensional; for</span>
-<span class="sd">      more than one dimension the shape must be a tuple.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    --------</span>
-<span class="sd">    These doctests require numpy.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symarray</span>
-<span class="sd">    &gt;&gt;&gt; symarray(&#39;&#39;, 3) #doctest: +SKIP</span>
-<span class="sd">    [_0, _1, _2]</span>
-
-<span class="sd">    If you want multiple symarrays to contain distinct symbols, you *must*</span>
-<span class="sd">    provide unique prefixes:</span>
-
-<span class="sd">    &gt;&gt;&gt; a = symarray(&#39;&#39;, 3) #doctest: +SKIP</span>
-<span class="sd">    &gt;&gt;&gt; b = symarray(&#39;&#39;, 3) #doctest: +SKIP</span>
-<span class="sd">    &gt;&gt;&gt; a[0] is b[0] #doctest: +SKIP</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; a = symarray(&#39;a&#39;, 3) #doctest: +SKIP</span>
-<span class="sd">    &gt;&gt;&gt; b = symarray(&#39;b&#39;, 3) #doctest: +SKIP</span>
-<span class="sd">    &gt;&gt;&gt; a[0] is b[0] #doctest: +SKIP</span>
-<span class="sd">    False</span>
-
-<span class="sd">    Creating symarrays with a prefix:</span>
-
-<span class="sd">    &gt;&gt;&gt; symarray(&#39;a&#39;, 3) #doctest: +SKIP</span>
-<span class="sd">    [a_0, a_1, a_2]</span>
-
-<span class="sd">    For more than one dimension, the shape must be given as a tuple:</span>
-
-<span class="sd">    &gt;&gt;&gt; symarray(&#39;a&#39;, (2, 3)) #doctest: +SKIP</span>
-<span class="sd">    [[a_0_0, a_0_1, a_0_2],</span>
-<span class="sd">     [a_1_0, a_1_1, a_1_2]]</span>
-<span class="sd">    &gt;&gt;&gt; symarray(&#39;a&#39;, (2, 3, 2)) #doctest: +SKIP</span>
-<span class="sd">    [[[a_0_0_0, a_0_0_1],</span>
-<span class="sd">      [a_0_1_0, a_0_1_1],</span>
-<span class="sd">      [a_0_2_0, a_0_2_1]],</span>
-<span class="sd">    &lt;BLANKLINE&gt;</span>
-<span class="sd">     [[a_1_0_0, a_1_0_1],</span>
-<span class="sd">      [a_1_1_0, a_1_1_1],</span>
-<span class="sd">      [a_1_2_0, a_1_2_1]]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">empty</span><span class="p">,</span> <span class="n">ndindex</span>
-    <span class="n">arr</span> <span class="o">=</span> <span class="n">empty</span><span class="p">(</span><span class="n">shape</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">object</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">index</span> <span class="ow">in</span> <span class="n">ndindex</span><span class="p">(</span><span class="n">shape</span><span class="p">):</span>
-        <span class="n">arr</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s">_</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">prefix</span><span class="p">,</span> <span class="s">&#39;_&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">str</span><span class="p">,</span> <span class="n">index</span><span class="p">))))</span>
-    <span class="k">return</span> <span class="n">arr</span>
-
-</div>
-<div class="viewcode-block" id="rot_axis3"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.dense.rot_axis3">[docs]</a><span class="k">def</span> <span class="nf">rot_axis3</span><span class="p">(</span><span class="n">theta</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns a rotation matrix for a rotation of theta (in radians) about</span>
-<span class="sd">    the 3-axis.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import pi</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.matrices import rot_axis3</span>
-
-<span class="sd">    A rotation of pi/3 (60 degrees):</span>
-
-<span class="sd">    &gt;&gt;&gt; theta = pi/3</span>
-<span class="sd">    &gt;&gt;&gt; rot_axis3(theta)</span>
-<span class="sd">    [       1/2, sqrt(3)/2, 0]</span>
-<span class="sd">    [-sqrt(3)/2,       1/2, 0]</span>
-<span class="sd">    [         0,         0, 1]</span>
-
-<span class="sd">    If we rotate by pi/2 (90 degrees):</span>
-
-<span class="sd">    &gt;&gt;&gt; rot_axis3(pi/2)</span>
-<span class="sd">    [ 0, 1, 0]</span>
-<span class="sd">    [-1, 0, 0]</span>
-<span class="sd">    [ 0, 0, 1]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    rot_axis1: Returns a rotation matrix for a rotation of theta (in radians)</span>
-<span class="sd">        about the 1-axis</span>
-<span class="sd">    rot_axis2: Returns a rotation matrix for a rotation of theta (in radians)</span>
-<span class="sd">        about the 2-axis</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">ct</span> <span class="o">=</span> <span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
-    <span class="n">st</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
-    <span class="n">lil</span> <span class="o">=</span> <span class="p">((</span><span class="n">ct</span><span class="p">,</span> <span class="n">st</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
-           <span class="p">(</span><span class="o">-</span><span class="n">st</span><span class="p">,</span> <span class="n">ct</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
-           <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">lil</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="rot_axis2"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.dense.rot_axis2">[docs]</a><span class="k">def</span> <span class="nf">rot_axis2</span><span class="p">(</span><span class="n">theta</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns a rotation matrix for a rotation of theta (in radians) about</span>
-<span class="sd">    the 2-axis.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import pi</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.matrices import rot_axis2</span>
-
-<span class="sd">    A rotation of pi/3 (60 degrees):</span>
-
-<span class="sd">    &gt;&gt;&gt; theta = pi/3</span>
-<span class="sd">    &gt;&gt;&gt; rot_axis2(theta)</span>
-<span class="sd">    [      1/2, 0, -sqrt(3)/2]</span>
-<span class="sd">    [        0, 1,          0]</span>
-<span class="sd">    [sqrt(3)/2, 0,        1/2]</span>
-
-<span class="sd">    If we rotate by pi/2 (90 degrees):</span>
-
-<span class="sd">    &gt;&gt;&gt; rot_axis2(pi/2)</span>
-<span class="sd">    [0, 0, -1]</span>
-<span class="sd">    [0, 1,  0]</span>
-<span class="sd">    [1, 0,  0]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    rot_axis1: Returns a rotation matrix for a rotation of theta (in radians)</span>
-<span class="sd">        about the 1-axis</span>
-<span class="sd">    rot_axis3: Returns a rotation matrix for a rotation of theta (in radians)</span>
-<span class="sd">        about the 3-axis</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">ct</span> <span class="o">=</span> <span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
-    <span class="n">st</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
-    <span class="n">lil</span> <span class="o">=</span> <span class="p">((</span><span class="n">ct</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="n">st</span><span class="p">),</span>
-           <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
-           <span class="p">(</span><span class="n">st</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">ct</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">lil</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="rot_axis1"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.dense.rot_axis1">[docs]</a><span class="k">def</span> <span class="nf">rot_axis1</span><span class="p">(</span><span class="n">theta</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns a rotation matrix for a rotation of theta (in radians) about</span>
-<span class="sd">    the 1-axis.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import pi</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.matrices import rot_axis1</span>
-
-<span class="sd">    A rotation of pi/3 (60 degrees):</span>
-
-<span class="sd">    &gt;&gt;&gt; theta = pi/3</span>
-<span class="sd">    &gt;&gt;&gt; rot_axis1(theta)</span>
-<span class="sd">    [1,          0,         0]</span>
-<span class="sd">    [0,        1/2, sqrt(3)/2]</span>
-<span class="sd">    [0, -sqrt(3)/2,       1/2]</span>
-
-<span class="sd">    If we rotate by pi/2 (90 degrees):</span>
-
-<span class="sd">    &gt;&gt;&gt; rot_axis1(pi/2)</span>
-<span class="sd">    [1,  0, 0]</span>
-<span class="sd">    [0,  0, 1]</span>
-<span class="sd">    [0, -1, 0]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    rot_axis2: Returns a rotation matrix for a rotation of theta (in radians)</span>
-<span class="sd">        about the 2-axis</span>
-<span class="sd">    rot_axis3: Returns a rotation matrix for a rotation of theta (in radians)</span>
-<span class="sd">        about the 3-axis</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">ct</span> <span class="o">=</span> <span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
-    <span class="n">st</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
-    <span class="n">lil</span> <span class="o">=</span> <span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
-           <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">ct</span><span class="p">,</span> <span class="n">st</span><span class="p">),</span>
-           <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="n">st</span><span class="p">,</span> <span class="n">ct</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">lil</span><span class="p">)</span>
-
-<span class="c">###############</span>
-<span class="c"># Functions</span>
-<span class="c">###############</span>
-
-</div>
-<span class="k">def</span> <span class="nf">matrix_add</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">):</span>
-    <span class="n">SymPyDeprecationWarning</span><span class="p">(</span>
-        <span class="n">feature</span><span class="o">=</span><span class="s">&quot;matrix_add(A, B)&quot;</span><span class="p">,</span>
-        <span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;A + B&quot;</span><span class="p">,</span>
-        <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.2&quot;</span><span class="p">,</span>
-    <span class="p">)</span><span class="o">.</span><span class="n">warn</span><span class="p">()</span>
-    <span class="k">return</span> <span class="n">A</span> <span class="o">+</span> <span class="n">B</span>
-
-
-<span class="k">def</span> <span class="nf">matrix_multiply</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">):</span>
-    <span class="n">SymPyDeprecationWarning</span><span class="p">(</span>
-        <span class="n">feature</span><span class="o">=</span><span class="s">&quot;matrix_multiply(A, B)&quot;</span><span class="p">,</span>
-        <span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;A*B&quot;</span><span class="p">,</span>
-        <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.2&quot;</span><span class="p">,</span>
-    <span class="p">)</span><span class="o">.</span><span class="n">warn</span><span class="p">()</span>
-    <span class="k">return</span> <span class="n">A</span><span class="o">*</span><span class="n">B</span>
-
-
-<div class="viewcode-block" id="matrix_multiply_elementwise"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.dense.matrix_multiply_elementwise">[docs]</a><span class="k">def</span> <span class="nf">matrix_multiply_elementwise</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return the Hadamard product (elementwise product) of A and B</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.matrices import matrix_multiply_elementwise</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.matrices import Matrix</span>
-<span class="sd">    &gt;&gt;&gt; A = Matrix([[0, 1, 2], [3, 4, 5]])</span>
-<span class="sd">    &gt;&gt;&gt; B = Matrix([[1, 10, 100], [100, 10, 1]])</span>
-<span class="sd">    &gt;&gt;&gt; matrix_multiply_elementwise(A, B)</span>
-<span class="sd">    [  0, 10, 200]</span>
-<span class="sd">    [300, 40,   5]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    __mul__</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">A</span><span class="o">.</span><span class="n">shape</span> <span class="o">!=</span> <span class="n">B</span><span class="o">.</span><span class="n">shape</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ShapeError</span><span class="p">()</span>
-    <span class="n">shape</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">shape</span>
-    <span class="k">return</span> <span class="n">classof</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span>
-        <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">*</span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">])</span>
-
-</div>
-<div class="viewcode-block" id="ones"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.dense.ones">[docs]</a><span class="k">def</span> <span class="nf">ones</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns a matrix of ones with ``r`` rows and ``c`` columns;</span>
-<span class="sd">    if ``c`` is omitted a square matrix will be returned.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    zeros</span>
-<span class="sd">    eye</span>
-<span class="sd">    diag</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">.dense</span> <span class="kn">import</span> <span class="n">Matrix</span>
-
-    <span class="k">if</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">r</span><span class="p">):</span>
-        <span class="n">SymPyDeprecationWarning</span><span class="p">(</span>
-            <span class="n">feature</span><span class="o">=</span><span class="s">&quot;The syntax ones([</span><span class="si">%i</span><span class="s">, </span><span class="si">%i</span><span class="s">])&quot;</span> <span class="o">%</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">r</span><span class="p">),</span>
-            <span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;ones(</span><span class="si">%i</span><span class="s">, </span><span class="si">%i</span><span class="s">).&quot;</span> <span class="o">%</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">r</span><span class="p">),</span>
-            <span class="n">issue</span><span class="o">=</span><span class="mi">3381</span><span class="p">,</span> <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.2&quot;</span><span class="p">,</span>
-        <span class="p">)</span><span class="o">.</span><span class="n">warn</span><span class="p">()</span>
-        <span class="n">r</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">r</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">r</span> <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="bp">None</span> <span class="k">else</span> <span class="n">c</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-    <span class="n">c</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">]</span><span class="o">*</span><span class="n">r</span><span class="o">*</span><span class="n">c</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="zeros"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.dense.zeros">[docs]</a><span class="k">def</span> <span class="nf">zeros</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns a matrix of zeros with ``r`` rows and ``c`` columns;</span>
-<span class="sd">    if ``c`` is omitted a square matrix will be returned.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    ones</span>
-<span class="sd">    eye</span>
-<span class="sd">    diag</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">cls</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="kn">from</span> <span class="nn">.dense</span> <span class="kn">import</span> <span class="n">Matrix</span> <span class="k">as</span> <span class="n">cls</span>
-    <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="eye"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.dense.eye">[docs]</a><span class="k">def</span> <span class="nf">eye</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Create square identity matrix n x n</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    diag</span>
-<span class="sd">    zeros</span>
-<span class="sd">    ones</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">cls</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span> <span class="k">as</span> <span class="n">cls</span>
-    <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="diag"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.dense.diag">[docs]</a><span class="k">def</span> <span class="nf">diag</span><span class="p">(</span><span class="o">*</span><span class="n">values</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Create a sparse, diagonal matrix from a list of diagonal values.</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    When arguments are matrices they are fitted in resultant matrix.</span>
-
-<span class="sd">    The returned matrix is a mutable, dense matrix. To make it a different</span>
-<span class="sd">    type, send the desired class for keyword ``cls``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.matrices import diag, Matrix, ones</span>
-<span class="sd">    &gt;&gt;&gt; diag(1, 2, 3)</span>
-<span class="sd">    [1, 0, 0]</span>
-<span class="sd">    [0, 2, 0]</span>
-<span class="sd">    [0, 0, 3]</span>
-<span class="sd">    &gt;&gt;&gt; diag(*[1, 2, 3])</span>
-<span class="sd">    [1, 0, 0]</span>
-<span class="sd">    [0, 2, 0]</span>
-<span class="sd">    [0, 0, 3]</span>
-
-<span class="sd">    The diagonal elements can be matrices; diagonal filling will</span>
-<span class="sd">    continue on the diagonal from the last element of the matrix:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">    &gt;&gt;&gt; a = Matrix([x, y, z])</span>
-<span class="sd">    &gt;&gt;&gt; b = Matrix([[1, 2], [3, 4]])</span>
-<span class="sd">    &gt;&gt;&gt; c = Matrix([[5, 6]])</span>
-<span class="sd">    &gt;&gt;&gt; diag(a, 7, b, c)</span>
-<span class="sd">    [x, 0, 0, 0, 0, 0]</span>
-<span class="sd">    [y, 0, 0, 0, 0, 0]</span>
-<span class="sd">    [z, 0, 0, 0, 0, 0]</span>
-<span class="sd">    [0, 7, 0, 0, 0, 0]</span>
-<span class="sd">    [0, 0, 1, 2, 0, 0]</span>
-<span class="sd">    [0, 0, 3, 4, 0, 0]</span>
-<span class="sd">    [0, 0, 0, 0, 5, 6]</span>
-
-<span class="sd">    When diagonal elements are lists, they will be treated as arguments</span>
-<span class="sd">    to Matrix:</span>
-
-<span class="sd">    &gt;&gt;&gt; diag([1, 2, 3], 4)</span>
-<span class="sd">    [1, 0]</span>
-<span class="sd">    [2, 0]</span>
-<span class="sd">    [3, 0]</span>
-<span class="sd">    [0, 4]</span>
-<span class="sd">    &gt;&gt;&gt; diag([[1, 2, 3]], 4)</span>
-<span class="sd">    [1, 2, 3, 0]</span>
-<span class="sd">    [0, 0, 0, 4]</span>
-
-<span class="sd">    A given band off the diagonal can be made by padding with a</span>
-<span class="sd">    vertical or horizontal &quot;kerning&quot; vector:</span>
-
-<span class="sd">    &gt;&gt;&gt; hpad = ones(0, 2)</span>
-<span class="sd">    &gt;&gt;&gt; vpad = ones(2, 0)</span>
-<span class="sd">    &gt;&gt;&gt; diag(vpad, 1, 2, 3, hpad) + diag(hpad, 4, 5, 6, vpad)</span>
-<span class="sd">    [0, 0, 4, 0, 0]</span>
-<span class="sd">    [0, 0, 0, 5, 0]</span>
-<span class="sd">    [1, 0, 0, 0, 6]</span>
-<span class="sd">    [0, 2, 0, 0, 0]</span>
-<span class="sd">    [0, 0, 3, 0, 0]</span>
-
-
-<span class="sd">    The type is mutable by default but can be made immutable by setting</span>
-<span class="sd">    the ``mutable`` flag to False:</span>
-
-<span class="sd">    &gt;&gt;&gt; type(diag(1))</span>
-<span class="sd">    &lt;class &#39;sympy.matrices.dense.MutableDenseMatrix&#39;&gt;</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.matrices import ImmutableMatrix</span>
-<span class="sd">    &gt;&gt;&gt; type(diag(1, cls=ImmutableMatrix))</span>
-<span class="sd">    &lt;class &#39;sympy.matrices.immutable.ImmutableMatrix&#39;&gt;</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    eye</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">.sparse</span> <span class="kn">import</span> <span class="n">MutableSparseMatrix</span>
-
-    <span class="n">cls</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;cls&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">cls</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="kn">from</span> <span class="nn">.dense</span> <span class="kn">import</span> <span class="n">Matrix</span> <span class="k">as</span> <span class="n">cls</span>
-
-    <span class="k">if</span> <span class="n">kwargs</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;unrecognized keyword</span><span class="si">%s</span><span class="s">: </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span>
-            <span class="s">&#39;s&#39;</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">kwargs</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="k">else</span> <span class="s">&#39;&#39;</span><span class="p">,</span>
-            <span class="s">&#39;, &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">kwargs</span><span class="o">.</span><span class="n">keys</span><span class="p">()))))</span>
-    <span class="n">rows</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">cols</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">values</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">values</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">values</span><span class="p">)):</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">values</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">MatrixBase</span><span class="p">):</span>
-            <span class="n">rows</span> <span class="o">+=</span> <span class="n">m</span><span class="o">.</span><span class="n">rows</span>
-            <span class="n">cols</span> <span class="o">+=</span> <span class="n">m</span><span class="o">.</span><span class="n">cols</span>
-        <span class="k">elif</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="n">values</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-            <span class="n">rows</span> <span class="o">+=</span> <span class="n">m</span><span class="o">.</span><span class="n">rows</span>
-            <span class="n">cols</span> <span class="o">+=</span> <span class="n">m</span><span class="o">.</span><span class="n">cols</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">rows</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">cols</span> <span class="o">+=</span> <span class="mi">1</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="n">MutableSparseMatrix</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">rows</span><span class="p">,</span> <span class="n">cols</span><span class="p">)</span>
-    <span class="n">i_row</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">i_col</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">values</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">MatrixBase</span><span class="p">):</span>
-            <span class="n">res</span><span class="p">[</span><span class="n">i_row</span><span class="p">:</span><span class="n">i_row</span> <span class="o">+</span> <span class="n">m</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="n">i_col</span><span class="p">:</span><span class="n">i_col</span> <span class="o">+</span> <span class="n">m</span><span class="o">.</span><span class="n">cols</span><span class="p">]</span> <span class="o">=</span> <span class="n">m</span>
-            <span class="n">i_row</span> <span class="o">+=</span> <span class="n">m</span><span class="o">.</span><span class="n">rows</span>
-            <span class="n">i_col</span> <span class="o">+=</span> <span class="n">m</span><span class="o">.</span><span class="n">cols</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">res</span><span class="p">[</span><span class="n">i_row</span><span class="p">,</span> <span class="n">i_col</span><span class="p">]</span> <span class="o">=</span> <span class="n">m</span>
-            <span class="n">i_row</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">i_col</span> <span class="o">+=</span> <span class="mi">1</span>
-    <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">res</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="jordan_cell"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.dense.jordan_cell">[docs]</a><span class="k">def</span> <span class="nf">jordan_cell</span><span class="p">(</span><span class="n">eigenval</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create matrix of Jordan cell kind:</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.matrices import jordan_cell</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; jordan_cell(x, 4)</span>
-<span class="sd">    [x, 1, 0, 0]</span>
-<span class="sd">    [0, x, 1, 0]</span>
-<span class="sd">    [0, 0, x, 1]</span>
-<span class="sd">    [0, 0, 0, x]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="n">out</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">out</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">eigenval</span>
-        <span class="n">out</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-    <span class="n">out</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">eigenval</span>
-    <span class="k">return</span> <span class="n">out</span>
-
-</div>
-<div class="viewcode-block" id="hessian"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.dense.hessian">[docs]</a><span class="k">def</span> <span class="nf">hessian</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">varlist</span><span class="p">,</span> <span class="n">constraints</span><span class="o">=</span><span class="p">[]):</span>
-    <span class="sd">&quot;&quot;&quot;Compute Hessian matrix for a function f wrt parameters in varlist</span>
-<span class="sd">    which may be given as a sequence or a row/column vector. A list of</span>
-<span class="sd">    constraints may optionally be given.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, hessian, pprint</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)(x, y)</span>
-<span class="sd">    &gt;&gt;&gt; g1 = Function(&#39;g&#39;)(x, y)</span>
-<span class="sd">    &gt;&gt;&gt; g2 = x**2 + 3*y</span>
-<span class="sd">    &gt;&gt;&gt; pprint(hessian(f, (x, y), [g1, g2]))</span>
-<span class="sd">    [                   d               d            ]</span>
-<span class="sd">    [     0        0    --(g(x, y))     --(g(x, y))  ]</span>
-<span class="sd">    [                   dx              dy           ]</span>
-<span class="sd">    [                                                ]</span>
-<span class="sd">    [     0        0        2*x              3       ]</span>
-<span class="sd">    [                                                ]</span>
-<span class="sd">    [                     2               2          ]</span>
-<span class="sd">    [d                   d               d           ]</span>
-<span class="sd">    [--(g(x, y))  2*x   ---(f(x, y))   -----(f(x, y))]</span>
-<span class="sd">    [dx                   2            dy dx         ]</span>
-<span class="sd">    [                   dx                           ]</span>
-<span class="sd">    [                                                ]</span>
-<span class="sd">    [                     2               2          ]</span>
-<span class="sd">    [d                   d               d           ]</span>
-<span class="sd">    [--(g(x, y))   3   -----(f(x, y))   ---(f(x, y)) ]</span>
-<span class="sd">    [dy                dy dx              2          ]</span>
-<span class="sd">    [                                   dy           ]</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    http://en.wikipedia.org/wiki/Hessian_matrix</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.matrices.mutable.Matrix.jacobian</span>
-<span class="sd">    wronskian</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># f is the expression representing a function f, return regular matrix</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">varlist</span><span class="p">,</span> <span class="n">MatrixBase</span><span class="p">):</span>
-        <span class="k">if</span> <span class="mi">1</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">varlist</span><span class="o">.</span><span class="n">shape</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ShapeError</span><span class="p">(</span><span class="s">&quot;`varlist` must be a column or row vector.&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">varlist</span><span class="o">.</span><span class="n">cols</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">varlist</span> <span class="o">=</span> <span class="n">varlist</span><span class="o">.</span><span class="n">T</span>
-        <span class="n">varlist</span> <span class="o">=</span> <span class="n">varlist</span><span class="o">.</span><span class="n">tolist</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">varlist</span><span class="p">):</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">varlist</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ShapeError</span><span class="p">(</span><span class="s">&quot;`len(varlist)` must not be zero.&quot;</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Improper variable list in hessian function&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;diff&#39;</span><span class="p">):</span>
-        <span class="c"># check differentiability</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Function `f` (</span><span class="si">%s</span><span class="s">) is not differentiable&quot;</span> <span class="o">%</span> <span class="n">f</span><span class="p">)</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">constraints</span><span class="p">)</span>
-    <span class="n">N</span> <span class="o">=</span> <span class="n">m</span> <span class="o">+</span> <span class="n">n</span>
-    <span class="n">out</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">g</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">constraints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="s">&#39;diff&#39;</span><span class="p">):</span>
-            <span class="c"># check differentiability</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Function `f` (</span><span class="si">%s</span><span class="s">) is not differentiable&quot;</span> <span class="o">%</span> <span class="n">f</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="n">out</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="n">m</span><span class="p">]</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">varlist</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-            <span class="n">out</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="n">m</span><span class="p">,</span> <span class="n">j</span> <span class="o">+</span> <span class="n">m</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">varlist</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">varlist</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="p">):</span>
-            <span class="n">out</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">out</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">out</span>
-
-</div>
-<div class="viewcode-block" id="GramSchmidt"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.dense.GramSchmidt">[docs]</a><span class="k">def</span> <span class="nf">GramSchmidt</span><span class="p">(</span><span class="n">vlist</span><span class="p">,</span> <span class="n">orthog</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Apply the Gram-Schmidt process to a set of vectors.</span>
-
-<span class="sd">    see: http://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">out</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">vlist</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
-        <span class="n">tmp</span> <span class="o">=</span> <span class="n">vlist</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
-            <span class="n">tmp</span> <span class="o">-=</span> <span class="n">vlist</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">project</span><span class="p">(</span><span class="n">out</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">list</span><span class="p">(</span><span class="n">tmp</span><span class="o">.</span><span class="n">values</span><span class="p">()):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&quot;GramSchmidt: vector set not linearly independent&quot;</span><span class="p">)</span>
-        <span class="n">out</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tmp</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">orthog</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">out</span><span class="p">)):</span>
-            <span class="n">out</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">out</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">normalized</span><span class="p">()</span>
-    <span class="k">return</span> <span class="n">out</span>
-
-</div>
-<div class="viewcode-block" id="wronskian"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.dense.wronskian">[docs]</a><span class="k">def</span> <span class="nf">wronskian</span><span class="p">(</span><span class="n">functions</span><span class="p">,</span> <span class="n">var</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;bareis&#39;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute Wronskian for [] of functions</span>
-
-<span class="sd">    ::</span>
-
-<span class="sd">                         | f1       f2        ...   fn      |</span>
-<span class="sd">                         | f1&#39;      f2&#39;       ...   fn&#39;     |</span>
-<span class="sd">                         |  .        .        .      .      |</span>
-<span class="sd">        W(f1, ..., fn) = |  .        .         .     .      |</span>
-<span class="sd">                         |  .        .          .    .      |</span>
-<span class="sd">                         |  (n)      (n)            (n)     |</span>
-<span class="sd">                         | D   (f1) D   (f2)  ...  D   (fn) |</span>
-
-<span class="sd">    see: http://en.wikipedia.org/wiki/Wronskian</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.matrices.mutable.Matrix.jacobian</span>
-<span class="sd">    hessian</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">.dense</span> <span class="kn">import</span> <span class="n">Matrix</span>
-
-    <span class="k">for</span> <span class="n">index</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">functions</span><span class="p">)):</span>
-        <span class="n">functions</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">functions</span><span class="p">[</span><span class="n">index</span><span class="p">])</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">functions</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">1</span>
-    <span class="n">W</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="n">functions</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">var</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">W</span><span class="o">.</span><span class="n">det</span><span class="p">(</span><span class="n">method</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="casoratian"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.dense.casoratian">[docs]</a><span class="k">def</span> <span class="nf">casoratian</span><span class="p">(</span><span class="n">seqs</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">zero</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Given linear difference operator L of order &#39;k&#39; and homogeneous</span>
-<span class="sd">       equation Ly = 0 we want to compute kernel of L, which is a set</span>
-<span class="sd">       of &#39;k&#39; sequences: a(n), b(n), ... z(n).</span>
-
-<span class="sd">       Solutions of L are linearly independent iff their Casoratian,</span>
-<span class="sd">       denoted as C(a, b, ..., z), do not vanish for n = 0.</span>
-
-<span class="sd">       Casoratian is defined by k x k determinant::</span>
-
-<span class="sd">                  +  a(n)     b(n)     . . . z(n)     +</span>
-<span class="sd">                  |  a(n+1)   b(n+1)   . . . z(n+1)   |</span>
-<span class="sd">                  |    .         .     .        .     |</span>
-<span class="sd">                  |    .         .       .      .     |</span>
-<span class="sd">                  |    .         .         .    .     |</span>
-<span class="sd">                  +  a(n+k-1) b(n+k-1) . . . z(n+k-1) +</span>
-
-<span class="sd">       It proves very useful in rsolve_hyper() where it is applied</span>
-<span class="sd">       to a generating set of a recurrence to factor out linearly</span>
-<span class="sd">       dependent solutions and return a basis:</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy import Symbol, casoratian, factorial</span>
-<span class="sd">       &gt;&gt;&gt; n = Symbol(&#39;n&#39;, integer=True)</span>
-
-<span class="sd">       Exponential and factorial are linearly independent:</span>
-
-<span class="sd">       &gt;&gt;&gt; casoratian([2**n, factorial(n)], n) != 0</span>
-<span class="sd">       True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">.dense</span> <span class="kn">import</span> <span class="n">Matrix</span>
-
-    <span class="n">seqs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="n">seqs</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">zero</span><span class="p">:</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="n">seqs</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="n">i</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="n">seqs</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-
-    <span class="n">k</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">seqs</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">det</span><span class="p">()</span>
-
-</div>
-<div class="viewcode-block" id="randMatrix"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.dense.randMatrix">[docs]</a><span class="k">def</span> <span class="nf">randMatrix</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="nb">min</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="nb">max</span><span class="o">=</span><span class="mi">99</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">symmetric</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">percent</span><span class="o">=</span><span class="mi">100</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Create random matrix with dimensions ``r`` x ``c``. If ``c`` is omitted</span>
-<span class="sd">    the matrix will be square. If ``symmetric`` is True the matrix must be</span>
-<span class="sd">    square. If ``percent`` is less than 100 then only approximately the given</span>
-<span class="sd">    percentage of elements will be non-zero.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.matrices import randMatrix</span>
-<span class="sd">    &gt;&gt;&gt; randMatrix(3) # doctest:+SKIP</span>
-<span class="sd">    [25, 45, 27]</span>
-<span class="sd">    [44, 54,  9]</span>
-<span class="sd">    [23, 96, 46]</span>
-<span class="sd">    &gt;&gt;&gt; randMatrix(3, 2) # doctest:+SKIP</span>
-<span class="sd">    [87, 29]</span>
-<span class="sd">    [23, 37]</span>
-<span class="sd">    [90, 26]</span>
-<span class="sd">    &gt;&gt;&gt; randMatrix(3, 3, 0, 2) # doctest:+SKIP</span>
-<span class="sd">    [0, 2, 0]</span>
-<span class="sd">    [2, 0, 1]</span>
-<span class="sd">    [0, 0, 1]</span>
-<span class="sd">    &gt;&gt;&gt; randMatrix(3, symmetric=True) # doctest:+SKIP</span>
-<span class="sd">    [85, 26, 29]</span>
-<span class="sd">    [26, 71, 43]</span>
-<span class="sd">    [29, 43, 57]</span>
-<span class="sd">    &gt;&gt;&gt; A = randMatrix(3, seed=1)</span>
-<span class="sd">    &gt;&gt;&gt; B = randMatrix(3, seed=2)</span>
-<span class="sd">    &gt;&gt;&gt; A == B # doctest:+SKIP</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; A == randMatrix(3, seed=1)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; randMatrix(3, symmetric=True, percent=50) # doctest:+SKIP</span>
-<span class="sd">    [0, 68, 43]</span>
-<span class="sd">    [0, 68,  0]</span>
-<span class="sd">    [0, 91, 34]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">r</span>
-    <span class="k">if</span> <span class="n">seed</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">prng</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">Random</span><span class="p">()</span>  <span class="c"># use system time</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">prng</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">Random</span><span class="p">(</span><span class="n">seed</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">symmetric</span> <span class="ow">and</span> <span class="n">r</span> <span class="o">!=</span> <span class="n">c</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-            <span class="s">&#39;For symmetric matrices, r must equal c, but </span><span class="si">%i</span><span class="s"> != </span><span class="si">%i</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">c</span><span class="p">))</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">symmetric</span><span class="p">:</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="n">prng</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="nb">min</span><span class="p">,</span> <span class="nb">max</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">r</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">r</span><span class="p">):</span>
-                <span class="n">m</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">prng</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="nb">min</span><span class="p">,</span> <span class="nb">max</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">r</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
-                <span class="n">m</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">m</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">percent</span> <span class="o">==</span> <span class="mi">100</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">m</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">z</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">r</span><span class="o">*</span><span class="n">c</span><span class="o">*</span><span class="n">percent</span> <span class="o">//</span> <span class="mi">100</span><span class="p">)</span>
-        <span class="n">m</span><span class="o">.</span><span class="n">_mat</span><span class="p">[:</span><span class="n">z</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">]</span><span class="o">*</span><span class="n">z</span>
-        <span class="n">random</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">_mat</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">m</span></div>
-</pre></div>
-
-          </div>
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diff --git a/dev-py3k/_modules/sympy/matrices/expressions.html b/dev-py3k/_modules/sympy/matrices/expressions.html
deleted file mode 100644
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+++ /dev/null
@@ -1,131 +0,0 @@
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-  <h1>Source code for sympy.matrices.expressions</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot; A module which handles Matrix Expressions &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">.blockmatrix</span> <span class="kn">import</span> <span class="n">BlockMatrix</span><span class="p">,</span> <span class="n">BlockDiagMatrix</span><span class="p">,</span> <span class="n">block_collapse</span>
-<span class="kn">from</span> <span class="nn">.funcmatrix</span> <span class="kn">import</span> <span class="n">FunctionMatrix</span>
-<span class="kn">from</span> <span class="nn">.inverse</span> <span class="kn">import</span> <span class="n">Inverse</span>
-<span class="kn">from</span> <span class="nn">.matadd</span> <span class="kn">import</span> <span class="n">MatAdd</span>
-<span class="kn">from</span> <span class="nn">.matexpr</span> <span class="kn">import</span> <span class="p">(</span><span class="n">Identity</span><span class="p">,</span> <span class="n">MatrixExpr</span><span class="p">,</span> <span class="n">MatrixSymbol</span><span class="p">,</span> <span class="n">ZeroMatrix</span><span class="p">,</span>
-     <span class="n">matrix_symbols</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">.matmul</span> <span class="kn">import</span> <span class="n">MatMul</span>
-<span class="kn">from</span> <span class="nn">.matpow</span> <span class="kn">import</span> <span class="n">MatPow</span>
-<span class="kn">from</span> <span class="nn">.trace</span> <span class="kn">import</span> <span class="n">Trace</span>
-<span class="kn">from</span> <span class="nn">.transpose</span> <span class="kn">import</span> <span class="n">Transpose</span>
-<span class="kn">from</span> <span class="nn">.adjoint</span> <span class="kn">import</span> <span class="n">Adjoint</span>
-<span class="kn">from</span> <span class="nn">.hadamard</span> <span class="kn">import</span> <span class="n">hadamard_product</span><span class="p">,</span> <span class="n">HadamardProduct</span>
-</pre></div>
-
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-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
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diff --git a/dev-py3k/_modules/sympy/matrices/expressions/blockmatrix.html b/dev-py3k/_modules/sympy/matrices/expressions/blockmatrix.html
deleted file mode 100644
index 4c9055e6c41..00000000000
--- a/dev-py3k/_modules/sympy/matrices/expressions/blockmatrix.html
+++ /dev/null
@@ -1,481 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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-            
-  <h1>Source code for sympy.matrices.expressions.blockmatrix</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Tuple</span><span class="p">,</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">Add</span>
-<span class="kn">from</span> <span class="nn">sympy.rules</span> <span class="kn">import</span> <span class="n">typed</span><span class="p">,</span> <span class="n">canon</span><span class="p">,</span> <span class="n">debug</span><span class="p">,</span> <span class="n">do_one</span><span class="p">,</span> <span class="n">unpack</span>
-<span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">transpose</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">sift</span>
-
-<span class="kn">from</span> <span class="nn">sympy.matrices.expressions.matexpr</span> <span class="kn">import</span> <span class="n">MatrixExpr</span><span class="p">,</span> <span class="n">ZeroMatrix</span><span class="p">,</span> <span class="n">Identity</span>
-<span class="kn">from</span> <span class="nn">sympy.matrices.expressions.matmul</span> <span class="kn">import</span> <span class="n">MatMul</span>
-<span class="kn">from</span> <span class="nn">sympy.matrices.expressions.matadd</span> <span class="kn">import</span> <span class="n">MatAdd</span>
-<span class="kn">from</span> <span class="nn">sympy.matrices.expressions.matpow</span> <span class="kn">import</span> <span class="n">MatPow</span>
-<span class="kn">from</span> <span class="nn">sympy.matrices.expressions.transpose</span> <span class="kn">import</span> <span class="n">Transpose</span>
-<span class="kn">from</span> <span class="nn">sympy.matrices.expressions.trace</span> <span class="kn">import</span> <span class="n">Trace</span>
-<span class="kn">from</span> <span class="nn">sympy.matrices.expressions.inverse</span> <span class="kn">import</span> <span class="n">Inverse</span>
-<span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">eye</span>
-
-
-<div class="viewcode-block" id="BlockMatrix"><a class="viewcode-back" href="../../../../modules/matrices/expressions.html#sympy.matrices.expressions.blockmatrix.BlockMatrix">[docs]</a><span class="k">class</span> <span class="nc">BlockMatrix</span><span class="p">(</span><span class="n">MatrixExpr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A BlockMatrix is a Matrix composed of other smaller, submatrices</span>
-
-<span class="sd">    The submatrices are stored in a SymPy Matrix object but accessed as part of</span>
-<span class="sd">    a Matrix Expression</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import (MatrixSymbol, BlockMatrix, symbols,</span>
-<span class="sd">    ...     Identity, ZeroMatrix, block_collapse)</span>
-<span class="sd">    &gt;&gt;&gt; n,m,l = symbols(&#39;n m l&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; X = MatrixSymbol(&#39;X&#39;, n, n)</span>
-<span class="sd">    &gt;&gt;&gt; Y = MatrixSymbol(&#39;Y&#39;, m ,m)</span>
-<span class="sd">    &gt;&gt;&gt; Z = MatrixSymbol(&#39;Z&#39;, n, m)</span>
-<span class="sd">    &gt;&gt;&gt; B = BlockMatrix([[X, Z], [ZeroMatrix(m,n), Y]])</span>
-<span class="sd">    &gt;&gt;&gt; print(B)</span>
-<span class="sd">    [X, Z]</span>
-<span class="sd">    [0, Y]</span>
-
-<span class="sd">    &gt;&gt;&gt; C = BlockMatrix([[Identity(n), Z]])</span>
-<span class="sd">    &gt;&gt;&gt; print(C)</span>
-<span class="sd">    [I, Z]</span>
-
-<span class="sd">    &gt;&gt;&gt; print(block_collapse(C*B))</span>
-<span class="sd">    [X, Z + Z*Y]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices.immutable</span> <span class="kn">import</span> <span class="n">ImmutableMatrix</span>
-        <span class="n">mat</span> <span class="o">=</span> <span class="n">ImmutableMatrix</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">mat</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">shape</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">numrows</span> <span class="o">=</span> <span class="n">numcols</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">M</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">blocks</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
-            <span class="n">numrows</span> <span class="o">+=</span> <span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]):</span>
-            <span class="n">numcols</span> <span class="o">+=</span> <span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">numrows</span><span class="p">,</span> <span class="n">numcols</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">blockshape</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">blocks</span><span class="o">.</span><span class="n">shape</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">blocks</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">rowblocksizes</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">blocks</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rows</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">blockshape</span><span class="p">[</span><span class="mi">0</span><span class="p">])]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">colblocksizes</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">blocks</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">cols</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">blockshape</span><span class="p">[</span><span class="mi">1</span><span class="p">])]</span>
-
-    <span class="k">def</span> <span class="nf">structurally_equal</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">BlockMatrix</span><span class="p">)</span>
-            <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">shape</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">shape</span>
-            <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">blockshape</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">blockshape</span>
-            <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">rowblocksizes</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">rowblocksizes</span>
-            <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">colblocksizes</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">colblocksizes</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_blockmul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">BlockMatrix</span><span class="p">)</span> <span class="ow">and</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">colblocksizes</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">rowblocksizes</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">BlockMatrix</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">blocks</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">blocks</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="bp">self</span> <span class="o">*</span> <span class="n">other</span>
-
-    <span class="k">def</span> <span class="nf">_blockadd</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">BlockMatrix</span><span class="p">)</span>
-                <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">structurally_equal</span><span class="p">(</span><span class="n">other</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="n">BlockMatrix</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">blocks</span> <span class="o">+</span> <span class="n">other</span><span class="o">.</span><span class="n">blocks</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="bp">self</span> <span class="o">+</span> <span class="n">other</span>
-
-    <span class="k">def</span> <span class="nf">_eval_transpose</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># Flip all the individual matrices</span>
-        <span class="n">matrices</span> <span class="o">=</span> <span class="p">[</span><span class="n">transpose</span><span class="p">(</span><span class="n">matrix</span><span class="p">)</span> <span class="k">for</span> <span class="n">matrix</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">blocks</span><span class="p">]</span>
-        <span class="c"># Make a copy</span>
-        <span class="n">M</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">blockshape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">blockshape</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">matrices</span><span class="p">)</span>
-        <span class="c"># Transpose the block structure</span>
-        <span class="n">M</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">BlockMatrix</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_trace</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">rowblocksizes</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">colblocksizes</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">Trace</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">blocks</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">])</span>
-                        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">blockshape</span><span class="p">[</span><span class="mi">0</span><span class="p">])])</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-            <span class="s">&quot;Can&#39;t perform trace of irregular blockshape&quot;</span><span class="p">)</span>
-
-<div class="viewcode-block" id="BlockMatrix.transpose"><a class="viewcode-back" href="../../../../modules/matrices/expressions.html#sympy.matrices.expressions.blockmatrix.BlockMatrix.transpose">[docs]</a>    <span class="k">def</span> <span class="nf">transpose</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return transpose of matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import MatrixSymbol, BlockMatrix, ZeroMatrix</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import l, m, n</span>
-<span class="sd">        &gt;&gt;&gt; X = MatrixSymbol(&#39;X&#39;, n, n)</span>
-<span class="sd">        &gt;&gt;&gt; Y = MatrixSymbol(&#39;Y&#39;, m ,m)</span>
-<span class="sd">        &gt;&gt;&gt; Z = MatrixSymbol(&#39;Z&#39;, n, m)</span>
-<span class="sd">        &gt;&gt;&gt; B = BlockMatrix([[X, Z], [ZeroMatrix(m,n), Y]])</span>
-<span class="sd">        &gt;&gt;&gt; B.transpose()</span>
-<span class="sd">        [X&#39;,  0]</span>
-<span class="sd">        [Z&#39;, Y&#39;]</span>
-<span class="sd">        &gt;&gt;&gt; _.transpose()</span>
-<span class="sd">        [X, Z]</span>
-<span class="sd">        [0, Y]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_transpose</span><span class="p">()</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expand</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="c"># Inverse of one by one block matrix is easy</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">blockshape</span> <span class="o">==</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">mat</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">blocks</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">inverse</span><span class="p">(),))</span>
-            <span class="k">return</span> <span class="n">BlockMatrix</span><span class="p">(</span><span class="n">mat</span><span class="p">)</span>
-        <span class="c"># Inverse of a two by two block matrix is known</span>
-        <span class="k">elif</span> <span class="n">expand</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">blockshape</span> <span class="o">==</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">):</span>
-            <span class="c"># Cite: The Matrix Cookbook Section 9.1.3</span>
-            <span class="n">A11</span><span class="p">,</span> <span class="n">A12</span><span class="p">,</span> <span class="n">A21</span><span class="p">,</span> <span class="n">A22</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">blocks</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">blocks</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">blocks</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">blocks</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-            <span class="n">C1</span> <span class="o">=</span> <span class="n">A11</span> <span class="o">-</span> <span class="n">A12</span><span class="o">*</span><span class="n">A22</span><span class="o">.</span><span class="n">I</span><span class="o">*</span><span class="n">A21</span>
-            <span class="n">C2</span> <span class="o">=</span> <span class="n">A22</span> <span class="o">-</span> <span class="n">A21</span><span class="o">*</span><span class="n">A11</span><span class="o">.</span><span class="n">I</span><span class="o">*</span><span class="n">A12</span>
-            <span class="n">mat</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="n">C1</span><span class="o">.</span><span class="n">I</span><span class="p">,</span> <span class="p">(</span><span class="o">-</span><span class="n">A11</span><span class="p">)</span><span class="o">.</span><span class="n">I</span><span class="o">*</span><span class="n">A12</span><span class="o">*</span><span class="n">C2</span><span class="o">.</span><span class="n">I</span><span class="p">],</span>
-                <span class="p">[</span><span class="o">-</span><span class="n">C2</span><span class="o">.</span><span class="n">I</span><span class="o">*</span><span class="n">A21</span><span class="o">*</span><span class="n">A11</span><span class="o">.</span><span class="n">I</span><span class="p">,</span> <span class="n">C2</span><span class="o">.</span><span class="n">I</span><span class="p">]])</span>
-            <span class="k">return</span> <span class="n">BlockMatrix</span><span class="p">(</span><span class="n">mat</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-<div class="viewcode-block" id="BlockMatrix.inverse"><a class="viewcode-back" href="../../../../modules/matrices/expressions.html#sympy.matrices.expressions.blockmatrix.BlockMatrix.inverse">[docs]</a>    <span class="k">def</span> <span class="nf">inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expand</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return inverse of matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import MatrixSymbol, BlockMatrix, ZeroMatrix</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import l, m, n</span>
-<span class="sd">        &gt;&gt;&gt; X = MatrixSymbol(&#39;X&#39;, n, n)</span>
-<span class="sd">        &gt;&gt;&gt; BlockMatrix([[X]]).inverse()</span>
-<span class="sd">        [X^-1]</span>
-
-<span class="sd">        &gt;&gt;&gt; Y = MatrixSymbol(&#39;Y&#39;, m ,m)</span>
-<span class="sd">        &gt;&gt;&gt; Z = MatrixSymbol(&#39;Z&#39;, n, m)</span>
-<span class="sd">        &gt;&gt;&gt; B = BlockMatrix([[X, Z], [ZeroMatrix(m,n), Y]])</span>
-<span class="sd">        &gt;&gt;&gt; B</span>
-<span class="sd">        [X, Z]</span>
-<span class="sd">        [0, Y]</span>
-<span class="sd">        &gt;&gt;&gt; B.inverse(expand=True)</span>
-<span class="sd">        [X^-1, (-1)*X^-1*Z*Y^-1]</span>
-<span class="sd">        [   0,             Y^-1]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_inverse</span><span class="p">(</span><span class="n">expand</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_entry</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">):</span>
-        <span class="c"># Find row entry</span>
-        <span class="k">for</span> <span class="n">row_block</span><span class="p">,</span> <span class="n">numrows</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rowblocksizes</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">numrows</span><span class="p">:</span>
-                <span class="k">break</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">i</span> <span class="o">-=</span> <span class="n">numrows</span>
-        <span class="k">for</span> <span class="n">col_block</span><span class="p">,</span> <span class="n">numcols</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">colblocksizes</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="n">numcols</span><span class="p">:</span>
-                <span class="k">break</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">j</span> <span class="o">-=</span> <span class="n">numcols</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">blocks</span><span class="p">[</span><span class="n">row_block</span><span class="p">,</span> <span class="n">col_block</span><span class="p">][</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_Identity</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">blockshape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">blockshape</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">blockshape</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">blockshape</span><span class="p">[</span><span class="mi">1</span><span class="p">]):</span>
-                <span class="k">if</span> <span class="n">i</span><span class="o">==</span><span class="n">j</span> <span class="ow">and</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">blocks</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">is_Identity</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-                <span class="k">if</span> <span class="n">i</span><span class="o">!=</span><span class="n">j</span> <span class="ow">and</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">blocks</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">is_ZeroMatrix</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_structurally_symmetric</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rowblocksizes</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">colblocksizes</span>
-
-    <span class="k">def</span> <span class="nf">equals</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span> <span class="o">==</span> <span class="n">other</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">BlockMatrix</span><span class="p">)</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">blocks</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">blocks</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">BlockMatrix</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="BlockDiagMatrix"><a class="viewcode-back" href="../../../../modules/matrices/expressions.html#sympy.matrices.expressions.blockmatrix.BlockDiagMatrix">[docs]</a><span class="k">class</span> <span class="nc">BlockDiagMatrix</span><span class="p">(</span><span class="n">BlockMatrix</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A BlockDiagMatrix is a BlockMatrix with matrices only along the diagonal</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import MatrixSymbol, BlockDiagMatrix, symbols, Identity</span>
-<span class="sd">    &gt;&gt;&gt; n,m,l = symbols(&#39;n m l&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; X = MatrixSymbol(&#39;X&#39;, n, n)</span>
-<span class="sd">    &gt;&gt;&gt; Y = MatrixSymbol(&#39;Y&#39;, m ,m)</span>
-<span class="sd">    &gt;&gt;&gt; BlockDiagMatrix(X, Y)</span>
-<span class="sd">    [X, 0]</span>
-<span class="sd">    [0, Y]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">mats</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">BlockDiagMatrix</span><span class="p">,</span> <span class="o">*</span><span class="n">mats</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">diag</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">blocks</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices.immutable</span> <span class="kn">import</span> <span class="n">ImmutableMatrix</span>
-        <span class="n">mats</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="n">data</span> <span class="o">=</span> <span class="p">[[</span><span class="n">mats</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span> <span class="k">else</span> <span class="n">ZeroMatrix</span><span class="p">(</span><span class="n">mats</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="n">mats</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">cols</span><span class="p">)</span>
-                        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">mats</span><span class="p">))]</span>
-                        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">mats</span><span class="p">))]</span>
-        <span class="k">return</span> <span class="n">ImmutableMatrix</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">shape</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="nb">sum</span><span class="p">(</span><span class="n">block</span><span class="o">.</span><span class="n">rows</span> <span class="k">for</span> <span class="n">block</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">),</span>
-                <span class="nb">sum</span><span class="p">(</span><span class="n">block</span><span class="o">.</span><span class="n">cols</span> <span class="k">for</span> <span class="n">block</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">blockshape</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">rowblocksizes</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">block</span><span class="o">.</span><span class="n">rows</span> <span class="k">for</span> <span class="n">block</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">colblocksizes</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">block</span><span class="o">.</span><span class="n">cols</span> <span class="k">for</span> <span class="n">block</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">_eval_inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expand</span><span class="o">=</span><span class="s">&#39;ignored&#39;</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">BlockDiagMatrix</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">mat</span><span class="o">.</span><span class="n">inverse</span><span class="p">()</span> <span class="k">for</span> <span class="n">mat</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_blockmul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">BlockDiagMatrix</span><span class="p">)</span> <span class="ow">and</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">colblocksizes</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">rowblocksizes</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">BlockDiagMatrix</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">a</span><span class="o">*</span><span class="n">b</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">args</span><span class="p">)])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">BlockMatrix</span><span class="o">.</span><span class="n">_blockmul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_blockadd</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">BlockDiagMatrix</span><span class="p">)</span> <span class="ow">and</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">blockshape</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">blockshape</span> <span class="ow">and</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">rowblocksizes</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">rowblocksizes</span> <span class="ow">and</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">colblocksizes</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">colblocksizes</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">BlockDiagMatrix</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">args</span><span class="p">)])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">BlockMatrix</span><span class="o">.</span><span class="n">_blockadd</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="block_collapse"><a class="viewcode-back" href="../../../../modules/matrices/expressions.html#sympy.matrices.expressions.blockmatrix.block_collapse">[docs]</a><span class="k">def</span> <span class="nf">block_collapse</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Evaluates a block matrix expression</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import MatrixSymbol, BlockMatrix, symbols, \</span>
-<span class="sd">                          Identity, Matrix, ZeroMatrix, block_collapse</span>
-<span class="sd">    &gt;&gt;&gt; n,m,l = symbols(&#39;n m l&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; X = MatrixSymbol(&#39;X&#39;, n, n)</span>
-<span class="sd">    &gt;&gt;&gt; Y = MatrixSymbol(&#39;Y&#39;, m ,m)</span>
-<span class="sd">    &gt;&gt;&gt; Z = MatrixSymbol(&#39;Z&#39;, n, m)</span>
-<span class="sd">    &gt;&gt;&gt; B = BlockMatrix([[X, Z], [ZeroMatrix(m, n), Y]])</span>
-<span class="sd">    &gt;&gt;&gt; print(B)</span>
-<span class="sd">    [X, Z]</span>
-<span class="sd">    [0, Y]</span>
-
-<span class="sd">    &gt;&gt;&gt; C = BlockMatrix([[Identity(n), Z]])</span>
-<span class="sd">    &gt;&gt;&gt; print(C)</span>
-<span class="sd">    [I, Z]</span>
-
-<span class="sd">    &gt;&gt;&gt; print(block_collapse(C*B))</span>
-<span class="sd">    [X, Z + Z*Y]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">rule</span> <span class="o">=</span> <span class="n">canon</span><span class="p">(</span><span class="n">typed</span><span class="p">({</span><span class="n">MatAdd</span><span class="p">:</span> <span class="n">do_one</span><span class="p">(</span><span class="n">bc_matadd</span><span class="p">,</span> <span class="n">bc_block_plus_ident</span><span class="p">),</span>
-                        <span class="n">MatMul</span><span class="p">:</span> <span class="n">do_one</span><span class="p">(</span><span class="n">bc_matmul</span><span class="p">,</span> <span class="n">bc_dist</span><span class="p">),</span>
-                        <span class="n">BlockMatrix</span><span class="p">:</span> <span class="n">bc_unpack</span><span class="p">}))</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="n">rule</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-    <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-</div>
-<span class="k">def</span> <span class="nf">bc_unpack</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">blockshape</span> <span class="o">==</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">blocks</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">expr</span>
-
-<span class="k">def</span> <span class="nf">bc_matadd</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="n">args</span> <span class="o">=</span> <span class="n">sift</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">M</span><span class="p">:</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">BlockMatrix</span><span class="p">))</span>
-    <span class="n">blocks</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="bp">True</span><span class="p">]</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">blocks</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="n">nonblocks</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="bp">False</span><span class="p">]</span>
-    <span class="n">block</span> <span class="o">=</span> <span class="n">blocks</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">blocks</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-        <span class="n">block</span> <span class="o">=</span> <span class="n">block</span><span class="o">.</span><span class="n">_blockadd</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">nonblocks</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">MatAdd</span><span class="p">(</span><span class="o">*</span><span class="n">nonblocks</span><span class="p">)</span> <span class="o">+</span> <span class="n">block</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">block</span>
-
-<span class="k">def</span> <span class="nf">bc_block_plus_ident</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="n">idents</span> <span class="o">=</span> <span class="p">[</span><span class="n">arg</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span> <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Identity</span><span class="p">]</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">idents</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="n">blocks</span> <span class="o">=</span> <span class="p">[</span><span class="n">arg</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">BlockMatrix</span><span class="p">)]</span>
-    <span class="k">if</span> <span class="p">(</span><span class="n">blocks</span> <span class="ow">and</span> <span class="nb">all</span><span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">structurally_equal</span><span class="p">(</span><span class="n">blocks</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">blocks</span><span class="p">)</span>
-               <span class="ow">and</span> <span class="n">blocks</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_structurally_symmetric</span><span class="p">):</span>
-        <span class="n">block_id</span> <span class="o">=</span> <span class="n">BlockDiagMatrix</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">Identity</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-                                        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">blocks</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rowblocksizes</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">MatAdd</span><span class="p">(</span><span class="n">block_id</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">idents</span><span class="p">),</span> <span class="o">*</span><span class="n">blocks</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-
-    <span class="k">return</span> <span class="n">expr</span>
-
-<span class="k">def</span> <span class="nf">bc_dist</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Turn  a*[X, Y] into [a*X, a*Y] &quot;&quot;&quot;</span>
-    <span class="n">factor</span><span class="p">,</span> <span class="n">mat</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_coeff_mmul</span><span class="p">()</span>
-    <span class="k">if</span> <span class="n">factor</span> <span class="o">!=</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">unpack</span><span class="p">(</span><span class="n">mat</span><span class="p">),</span> <span class="n">BlockMatrix</span><span class="p">):</span>
-        <span class="n">B</span> <span class="o">=</span> <span class="n">unpack</span><span class="p">(</span><span class="n">mat</span><span class="p">)</span><span class="o">.</span><span class="n">blocks</span>
-        <span class="k">return</span> <span class="n">BlockMatrix</span><span class="p">([[</span><span class="n">factor</span> <span class="o">*</span> <span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">B</span><span class="o">.</span><span class="n">cols</span><span class="p">)]</span>
-                                              <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">B</span><span class="o">.</span><span class="n">rows</span><span class="p">)])</span>
-    <span class="k">return</span> <span class="n">expr</span>
-
-<span class="k">def</span> <span class="nf">bc_matmul</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="n">factor</span><span class="p">,</span> <span class="n">matrices</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_coeff_matrices</span><span class="p">()</span>
-    <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">while</span> <span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">matrices</span><span class="p">)):</span>
-        <span class="n">A</span><span class="p">,</span> <span class="n">B</span> <span class="o">=</span> <span class="n">matrices</span><span class="p">[</span><span class="n">i</span><span class="p">:</span><span class="n">i</span><span class="o">+</span><span class="mi">2</span><span class="p">]</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">BlockMatrix</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">BlockMatrix</span><span class="p">):</span>
-            <span class="n">matrices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">_blockmul</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
-            <span class="n">matrices</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">i</span><span class="o">+=</span><span class="mi">1</span>
-    <span class="k">return</span> <span class="n">MatMul</span><span class="p">(</span><span class="n">factor</span><span class="p">,</span> <span class="o">*</span><span class="n">matrices</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-</pre></div>
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-  <h1>Source code for sympy.matrices.immutable</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">Integer</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">,</span> <span class="n">Dict</span>
-<span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="n">converter</span> <span class="k">as</span> <span class="n">sympify_converter</span>
-
-<span class="kn">from</span> <span class="nn">sympy.matrices.matrices</span> <span class="kn">import</span> <span class="n">MatrixBase</span>
-<span class="kn">from</span> <span class="nn">sympy.matrices.dense</span> <span class="kn">import</span> <span class="n">DenseMatrix</span>
-<span class="kn">from</span> <span class="nn">sympy.matrices.sparse</span> <span class="kn">import</span> <span class="n">SparseMatrix</span><span class="p">,</span> <span class="n">MutableSparseMatrix</span>
-<span class="kn">from</span> <span class="nn">sympy.matrices.expressions</span> <span class="kn">import</span> <span class="n">MatrixExpr</span>
-
-
-<span class="k">def</span> <span class="nf">sympify_matrix</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">ImmutableMatrix</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-<span class="n">sympify_converter</span><span class="p">[</span><span class="n">MatrixBase</span><span class="p">]</span> <span class="o">=</span> <span class="n">sympify_matrix</span>
-
-<div class="viewcode-block" id="ImmutableMatrix"><a class="viewcode-back" href="../../../modules/matrices/immutablematrices.html#sympy.matrices.immutable.ImmutableMatrix">[docs]</a><span class="k">class</span> <span class="nc">ImmutableMatrix</span><span class="p">(</span><span class="n">MatrixExpr</span><span class="p">,</span> <span class="n">DenseMatrix</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Create an immutable version of a matrix.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import eye</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.matrices import ImmutableMatrix</span>
-<span class="sd">    &gt;&gt;&gt; ImmutableMatrix(eye(3))</span>
-<span class="sd">    [1, 0, 0]</span>
-<span class="sd">    [0, 1, 0]</span>
-<span class="sd">    [0, 0, 1]</span>
-<span class="sd">    &gt;&gt;&gt; _[0, 0] = 42</span>
-<span class="sd">    Traceback (most recent call last):</span>
-<span class="sd">    ...</span>
-<span class="sd">    TypeError: Cannot set values of ImmutableDenseMatrix</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_class_priority</span> <span class="o">=</span> <span class="mi">8</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_new</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">ImmutableMatrix</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">rows</span><span class="p">,</span> <span class="n">cols</span><span class="p">,</span> <span class="n">flat_list</span> <span class="o">=</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">_handle_creation_inputs</span><span class="p">(</span>
-            <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="n">rows</span> <span class="o">=</span> <span class="n">Integer</span><span class="p">(</span><span class="n">rows</span><span class="p">)</span>
-        <span class="n">cols</span> <span class="o">=</span> <span class="n">Integer</span><span class="p">(</span><span class="n">cols</span><span class="p">)</span>
-        <span class="n">mat</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">flat_list</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">rows</span><span class="p">,</span> <span class="n">cols</span><span class="p">,</span> <span class="n">mat</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">shape</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">([</span><span class="nb">int</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[:</span><span class="mi">2</span><span class="p">]])</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_mat</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_entry</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">DenseMatrix</span><span class="o">.</span><span class="n">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span>
-
-    <span class="n">__getitem__</span> <span class="o">=</span> <span class="n">DenseMatrix</span><span class="o">.</span><span class="n">__getitem__</span>
-
-    <span class="k">def</span> <span class="nf">__setitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Cannot set values of ImmutableMatrix&quot;</span><span class="p">)</span>
-
-    <span class="n">adjoint</span> <span class="o">=</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">adjoint</span>
-    <span class="n">conjugate</span> <span class="o">=</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">conjugate</span>
-    <span class="c"># C and T are defined in MatrixExpr...I don&#39;t know why C alone</span>
-    <span class="c"># needs to be defined here</span>
-    <span class="n">C</span> <span class="o">=</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">C</span>
-
-    <span class="n">as_mutable</span> <span class="o">=</span> <span class="n">DenseMatrix</span><span class="o">.</span><span class="n">as_mutable</span>
-    <span class="n">_eval_trace</span> <span class="o">=</span> <span class="n">DenseMatrix</span><span class="o">.</span><span class="n">_eval_trace</span>
-    <span class="n">_eval_transpose</span> <span class="o">=</span> <span class="n">DenseMatrix</span><span class="o">.</span><span class="n">_eval_transpose</span>
-    <span class="n">_eval_conjugate</span> <span class="o">=</span> <span class="n">DenseMatrix</span><span class="o">.</span><span class="n">_eval_conjugate</span>
-    <span class="n">_eval_adjoint</span> <span class="o">=</span> <span class="n">DenseMatrix</span><span class="o">.</span><span class="n">_eval_adjoint</span>
-    <span class="n">_eval_inverse</span> <span class="o">=</span> <span class="n">DenseMatrix</span><span class="o">.</span><span class="n">_eval_inverse</span>
-    <span class="n">_eval_simplify</span> <span class="o">=</span> <span class="n">DenseMatrix</span><span class="o">.</span><span class="n">_eval_simplify</span>
-
-    <span class="n">equals</span> <span class="o">=</span> <span class="n">DenseMatrix</span><span class="o">.</span><span class="n">equals</span>
-    <span class="n">is_Identity</span> <span class="o">=</span> <span class="n">DenseMatrix</span><span class="o">.</span><span class="n">is_Identity</span>
-
-    <span class="n">__add__</span> <span class="o">=</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">__add__</span>
-    <span class="n">__radd__</span> <span class="o">=</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">__radd__</span>
-    <span class="n">__mul__</span> <span class="o">=</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">__mul__</span>
-    <span class="n">__rmul__</span> <span class="o">=</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">__rmul__</span>
-    <span class="n">__pow__</span> <span class="o">=</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">__pow__</span>
-    <span class="n">__sub__</span> <span class="o">=</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">__sub__</span>
-    <span class="n">__rsub__</span> <span class="o">=</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">__rsub__</span>
-    <span class="n">__neg__</span> <span class="o">=</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">__neg__</span>
-    <span class="n">__div__</span> <span class="o">=</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">__div__</span>
-    <span class="n">__truediv__</span> <span class="o">=</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">__truediv__</span>
-
-</div>
-<div class="viewcode-block" id="ImmutableSparseMatrix"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.immutable.ImmutableSparseMatrix">[docs]</a><span class="k">class</span> <span class="nc">ImmutableSparseMatrix</span><span class="p">(</span><span class="n">Basic</span><span class="p">,</span> <span class="n">SparseMatrix</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Create an immutable version of a sparse matrix.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import eye</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.matrices.immutable import ImmutableSparseMatrix</span>
-<span class="sd">    &gt;&gt;&gt; ImmutableSparseMatrix(1, 1, {})</span>
-<span class="sd">    [0]</span>
-<span class="sd">    &gt;&gt;&gt; ImmutableSparseMatrix(eye(3))</span>
-<span class="sd">    [1, 0, 0]</span>
-<span class="sd">    [0, 1, 0]</span>
-<span class="sd">    [0, 0, 1]</span>
-<span class="sd">    &gt;&gt;&gt; _[0, 0] = 42</span>
-<span class="sd">    Traceback (most recent call last):</span>
-<span class="sd">    ...</span>
-<span class="sd">    TypeError: Cannot set values of ImmutableSparseMatrix</span>
-<span class="sd">    &gt;&gt;&gt; _.shape</span>
-<span class="sd">    (3, 3)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_class_priority</span> <span class="o">=</span> <span class="mi">9</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_new</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">MutableSparseMatrix</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">rows</span> <span class="o">=</span> <span class="n">Integer</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">rows</span><span class="p">)</span>
-        <span class="n">cols</span> <span class="o">=</span> <span class="n">Integer</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">cols</span><span class="p">)</span>
-        <span class="n">mat</span> <span class="o">=</span> <span class="n">Dict</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">_smat</span><span class="p">)</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">rows</span><span class="p">,</span> <span class="n">cols</span><span class="p">,</span> <span class="n">mat</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">rows</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">rows</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">cols</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">cols</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_smat</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">_smat</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__setitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Cannot set values of ImmutableSparseMatrix&quot;</span><span class="p">)</span>
-
-    <span class="n">subs</span> <span class="o">=</span> <span class="n">MatrixBase</span><span class="o">.</span><span class="n">subs</span></div>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/matrices/matrices.html b/dev-py3k/_modules/sympy/matrices/matrices.html
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-  <h1>Source code for sympy.matrices.matrices</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core.add</span> <span class="kn">import</span> <span class="n">Add</span>
-<span class="kn">from</span> <span class="nn">sympy.core.basic</span> <span class="kn">import</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">C</span>
-<span class="kn">from</span> <span class="nn">sympy.core.expr</span> <span class="kn">import</span> <span class="n">Expr</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">count_ops</span>
-<span class="kn">from</span> <span class="nn">sympy.core.power</span> <span class="kn">import</span> <span class="n">Pow</span>
-<span class="kn">from</span> <span class="nn">sympy.core.symbol</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Dummy</span>
-<span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">Integer</span><span class="p">,</span> <span class="n">ilcm</span><span class="p">,</span> <span class="n">Rational</span><span class="p">,</span> <span class="n">Float</span>
-<span class="kn">from</span> <span class="nn">sympy.core.singleton</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">SympifyError</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">is_sequence</span><span class="p">,</span> <span class="n">default_sort_key</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">PurePoly</span><span class="p">,</span> <span class="n">roots</span><span class="p">,</span> <span class="n">cancel</span>
-<span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">simplify</span> <span class="k">as</span> <span class="n">_simplify</span><span class="p">,</span> <span class="n">signsimp</span><span class="p">,</span> <span class="n">nsimplify</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">flatten</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">Max</span><span class="p">,</span> <span class="n">Min</span>
-<span class="kn">from</span> <span class="nn">sympy.printing</span> <span class="kn">import</span> <span class="n">sstr</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="nb">callable</span><span class="p">,</span> <span class="nb">reduce</span><span class="p">,</span> <span class="n">as_int</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.exceptions</span> <span class="kn">import</span> <span class="n">SymPyDeprecationWarning</span>
-
-<span class="kn">from</span> <span class="nn">types</span> <span class="kn">import</span> <span class="n">FunctionType</span>
-<span class="kn">import</span> <span class="nn">collections</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-
-<span class="k">def</span> <span class="nf">_iszero</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns True if x is zero.&quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">x</span><span class="o">.</span><span class="n">is_zero</span>
-
-
-<div class="viewcode-block" id="MatrixError"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixError">[docs]</a><span class="k">class</span> <span class="nc">MatrixError</span><span class="p">(</span><span class="ne">Exception</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="ShapeError"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.ShapeError">[docs]</a><span class="k">class</span> <span class="nc">ShapeError</span><span class="p">(</span><span class="ne">ValueError</span><span class="p">,</span> <span class="n">MatrixError</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Wrong matrix shape&quot;&quot;&quot;</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="NonSquareMatrixError"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.NonSquareMatrixError">[docs]</a><span class="k">class</span> <span class="nc">NonSquareMatrixError</span><span class="p">(</span><span class="n">ShapeError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<span class="k">class</span> <span class="nc">DeferredVector</span><span class="p">(</span><span class="n">Symbol</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A vector whose components are deferred (e.g. for use with lambdify)</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import DeferredVector, lambdify</span>
-<span class="sd">    &gt;&gt;&gt; X = DeferredVector( &#39;X&#39; )</span>
-<span class="sd">    &gt;&gt;&gt; X</span>
-<span class="sd">    X</span>
-<span class="sd">    &gt;&gt;&gt; expr = (X[0] + 2, X[2] + 3)</span>
-<span class="sd">    &gt;&gt;&gt; func = lambdify( X, expr )</span>
-<span class="sd">    &gt;&gt;&gt; func( [1, 2, 3] )</span>
-<span class="sd">    (3, 6)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="o">-</span><span class="mi">0</span><span class="p">:</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s">&#39;DeferredVector index out of range&#39;</span><span class="p">)</span>
-        <span class="n">component_name</span> <span class="o">=</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">[</span><span class="si">%d</span><span class="s">]&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">component_name</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sstr</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;DeferredVector(&#39;</span><span class="si">%s</span><span class="s">&#39;)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="MatrixBase"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase">[docs]</a><span class="k">class</span> <span class="nc">MatrixBase</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-
-    <span class="c"># Added just for numpy compatibility</span>
-    <span class="c"># TODO: investigate about __array_priority__</span>
-    <span class="n">__array_priority__</span> <span class="o">=</span> <span class="mf">10.0</span>
-
-    <span class="n">is_Matrix</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_Identity</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">_class_priority</span> <span class="o">=</span> <span class="mi">3</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_handle_creation_inputs</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the number of rows, cols and flat matrix elements.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, I</span>
-
-<span class="sd">        Matrix can be constructed as follows:</span>
-
-<span class="sd">        * from a nested list of iterables</span>
-
-<span class="sd">        &gt;&gt;&gt; Matrix( ((1, 2+I), (3, 4)) )</span>
-<span class="sd">        [1, 2 + I]</span>
-<span class="sd">        [3,     4]</span>
-
-<span class="sd">        * from un-nested iterable (interpreted as a column)</span>
-
-<span class="sd">        &gt;&gt;&gt; Matrix( [1, 2] )</span>
-<span class="sd">        [1]</span>
-<span class="sd">        [2]</span>
-
-<span class="sd">        * from un-nested iterable with dimensions</span>
-
-<span class="sd">        &gt;&gt;&gt; Matrix(1, 2, [1, 2] )</span>
-<span class="sd">        [1, 2]</span>
-
-<span class="sd">        * from no arguments (a 0 x 0 matrix)</span>
-
-<span class="sd">        &gt;&gt;&gt; Matrix()</span>
-<span class="sd">        []</span>
-
-<span class="sd">        * from a rule</span>
-
-<span class="sd">        &gt;&gt;&gt; Matrix(2, 2, lambda i, j: i/(j + 1) )</span>
-<span class="sd">        [0,   0]</span>
-<span class="sd">        [1, 1/2]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices.sparse</span> <span class="kn">import</span> <span class="n">SparseMatrix</span>
-
-        <span class="c"># Matrix(SparseMatrix(...))</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">SparseMatrix</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span> <span class="n">flatten</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">tolist</span><span class="p">())</span>
-
-        <span class="c"># Matrix(Matrix(...))</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">MatrixBase</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_mat</span>
-
-        <span class="c"># Matrix(MatrixSymbol(&#39;X&#39;, 2, 2))</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Basic</span><span class="p">)</span> <span class="ow">and</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Matrix</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">as_explicit</span><span class="p">()</span><span class="o">.</span><span class="n">_mat</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="n">rows</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="n">cols</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-
-        <span class="c"># Matrix(2, 2, lambda i, j: i+j)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">collections</span><span class="o">.</span><span class="n">Callable</span><span class="p">):</span>
-            <span class="n">operation</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-            <span class="n">flat_list</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">rows</span><span class="p">):</span>
-                <span class="n">flat_list</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">sympify</span><span class="p">(</span><span class="n">operation</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">i</span><span class="p">),</span> <span class="n">j</span><span class="p">))</span>
-                    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">cols</span><span class="p">)])</span>
-
-        <span class="c"># Matrix(2, 2, [1, 2, 3, 4])</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span> <span class="ow">and</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]):</span>
-            <span class="n">flat_list</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">flat_list</span><span class="p">)</span> <span class="o">!=</span> <span class="n">rows</span><span class="o">*</span><span class="n">cols</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;List length should be equal to rows*columns&#39;</span><span class="p">)</span>
-            <span class="n">flat_list</span> <span class="o">=</span> <span class="p">[</span><span class="n">sympify</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">flat_list</span><span class="p">]</span>
-
-        <span class="c"># Matrix(numpy.ones((2, 2)))</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="s">&quot;__array__&quot;</span><span class="p">):</span>  <span class="c"># pragma: no cover</span>
-            <span class="c"># NumPy array or matrix or some other object that implements</span>
-            <span class="c"># __array__. So let&#39;s first use this method to get a</span>
-            <span class="c"># numpy.array() and then make a python list out of it.</span>
-            <span class="n">arr</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">__array__</span><span class="p">()</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">arr</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">rows</span><span class="p">,</span> <span class="n">cols</span> <span class="o">=</span> <span class="n">arr</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">arr</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-                <span class="n">flat_list</span> <span class="o">=</span> <span class="p">[</span><span class="n">sympify</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">arr</span><span class="o">.</span><span class="n">ravel</span><span class="p">()]</span>
-                <span class="k">return</span> <span class="n">rows</span><span class="p">,</span> <span class="n">cols</span><span class="p">,</span> <span class="n">flat_list</span>
-            <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">arr</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">rows</span><span class="p">,</span> <span class="n">cols</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="n">arr</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="n">flat_list</span> <span class="o">=</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">]</span><span class="o">*</span><span class="n">cols</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">arr</span><span class="p">)):</span>
-                    <span class="n">flat_list</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">arr</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-                <span class="k">return</span> <span class="n">rows</span><span class="p">,</span> <span class="n">cols</span><span class="p">,</span> <span class="n">flat_list</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                    <span class="s">&quot;SymPy supports just 1D and 2D matrices&quot;</span><span class="p">)</span>
-
-        <span class="c"># Matrix([1, 2, 3]) or Matrix([[1, 2], [3, 4]])</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
-            <span class="n">in_mat</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">ncol</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-            <span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">row</span><span class="p">,</span> <span class="n">MatrixBase</span><span class="p">):</span>
-                    <span class="n">in_mat</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">row</span><span class="o">.</span><span class="n">tolist</span><span class="p">())</span>
-                    <span class="k">if</span> <span class="n">row</span><span class="o">.</span><span class="n">cols</span> <span class="ow">or</span> <span class="n">row</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>  <span class="c"># only pay attention if it&#39;s not 0x0</span>
-                        <span class="n">ncol</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">row</span><span class="o">.</span><span class="n">cols</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">in_mat</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">row</span><span class="p">)</span>
-                    <span class="k">try</span><span class="p">:</span>
-                        <span class="n">ncol</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">row</span><span class="p">))</span>
-                    <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                        <span class="n">ncol</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">ncol</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Got rows of variable lengths: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span>
-                    <span class="nb">sorted</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">ncol</span><span class="p">)))</span>
-            <span class="n">rows</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">in_mat</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">rows</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">in_mat</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
-                    <span class="n">cols</span> <span class="o">=</span> <span class="mi">1</span>
-                    <span class="n">flat_list</span> <span class="o">=</span> <span class="p">[</span><span class="n">sympify</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">in_mat</span><span class="p">]</span>
-                    <span class="k">return</span> <span class="n">rows</span><span class="p">,</span> <span class="n">cols</span><span class="p">,</span> <span class="n">flat_list</span>
-                <span class="n">cols</span> <span class="o">=</span> <span class="n">ncol</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">cols</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="n">flat_list</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">rows</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">cols</span><span class="p">):</span>
-                    <span class="n">flat_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">in_mat</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]))</span>
-
-        <span class="c"># Matrix()</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="c"># Empty Matrix</span>
-            <span class="n">rows</span> <span class="o">=</span> <span class="n">cols</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="n">flat_list</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Data type not understood&quot;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">rows</span><span class="p">,</span> <span class="n">cols</span><span class="p">,</span> <span class="n">flat_list</span>
-
-    <span class="k">def</span> <span class="nf">_setitem</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Helper to set value at location given by key.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, I, zeros, ones</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix(((1, 2+I), (3, 4)))</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [1, 2 + I]</span>
-<span class="sd">        [3,     4]</span>
-<span class="sd">        &gt;&gt;&gt; m[1, 0] = 9</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [1, 2 + I]</span>
-<span class="sd">        [9,     4]</span>
-<span class="sd">        &gt;&gt;&gt; m[1, 0] = [[0, 1]]</span>
-
-<span class="sd">        To replace row r you assign to position r*m where m</span>
-<span class="sd">        is the number of columns:</span>
-
-<span class="sd">        &gt;&gt;&gt; M = zeros(4)</span>
-<span class="sd">        &gt;&gt;&gt; m = M.cols</span>
-<span class="sd">        &gt;&gt;&gt; M[3*m] = ones(1, m)*2; M</span>
-<span class="sd">        [0, 0, 0, 0]</span>
-<span class="sd">        [0, 0, 0, 0]</span>
-<span class="sd">        [0, 0, 0, 0]</span>
-<span class="sd">        [2, 2, 2, 2]</span>
-
-<span class="sd">        And to replace column c you can assign to position c:</span>
-
-<span class="sd">        &gt;&gt;&gt; M[2] = ones(m, 1)*4; M</span>
-<span class="sd">        [0, 0, 4, 0]</span>
-<span class="sd">        [0, 0, 4, 0]</span>
-<span class="sd">        [0, 0, 4, 0]</span>
-<span class="sd">        [2, 2, 4, 2]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">.dense</span> <span class="kn">import</span> <span class="n">Matrix</span>
-
-        <span class="n">is_slice</span> <span class="o">=</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="nb">slice</span><span class="p">)</span>
-        <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">key</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">key2ij</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
-        <span class="n">is_mat</span> <span class="o">=</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">MatrixBase</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">slice</span> <span class="ow">or</span> <span class="nb">type</span><span class="p">(</span><span class="n">j</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">slice</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">is_mat</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">copyin_matrix</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">value</span><span class="p">)</span>
-                <span class="k">return</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">Expr</span><span class="p">)</span> <span class="ow">and</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">value</span><span class="p">):</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">copyin_list</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">value</span><span class="p">)</span>
-                <span class="k">return</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;unexpected value: </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">value</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">is_mat</span> <span class="ow">and</span> \
-                    <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">Expr</span><span class="p">)</span> <span class="ow">and</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">value</span><span class="p">):</span>
-                <span class="n">value</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">value</span><span class="p">)</span>
-                <span class="n">is_mat</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">if</span> <span class="n">is_mat</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">is_slice</span><span class="p">:</span>
-                    <span class="n">key</span> <span class="o">=</span> <span class="p">(</span><span class="nb">slice</span><span class="p">(</span><span class="o">*</span><span class="nb">divmod</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">)),</span>
-                           <span class="nb">slice</span><span class="p">(</span><span class="o">*</span><span class="nb">divmod</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">)))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">key</span> <span class="o">=</span> <span class="p">(</span><span class="nb">slice</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="n">value</span><span class="o">.</span><span class="n">rows</span><span class="p">),</span>
-                           <span class="nb">slice</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span> <span class="o">+</span> <span class="n">value</span><span class="o">.</span><span class="n">cols</span><span class="p">))</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">copyin_matrix</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">value</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">sympify</span><span class="p">(</span><span class="n">value</span><span class="p">)</span>
-            <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">copy</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">trace</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">NonSquareMatrixError</span><span class="p">()</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_trace</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">inv</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">NonSquareMatrixError</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">method</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;method&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">method</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_inverse</span><span class="p">(</span><span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">transpose</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_transpose</span><span class="p">()</span>
-
-    <span class="n">T</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="n">transpose</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="s">&quot;Matrix transposition.&quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_conjugate</span><span class="p">()</span>
-
-    <span class="n">C</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="n">conjugate</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="s">&quot;By-element conjugation.&quot;</span><span class="p">)</span>
-
-<div class="viewcode-block" id="MatrixBase.adjoint"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.adjoint">[docs]</a>    <span class="k">def</span> <span class="nf">adjoint</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Conjugate transpose or Hermitian conjugation.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="o">.</span><span class="n">C</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="MatrixBase.H"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.H">[docs]</a>    <span class="k">def</span> <span class="nf">H</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return Hermite conjugate.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, I, eye</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix((0, 1 + I, 2, 3))</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [    0]</span>
-<span class="sd">        [1 + I]</span>
-<span class="sd">        [    2]</span>
-<span class="sd">        [    3]</span>
-<span class="sd">        &gt;&gt;&gt; m.H</span>
-<span class="sd">        [0, 1 - I, 2, 3]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        conjugate: By-element conjugation</span>
-<span class="sd">        D: Dirac conjugation</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="o">.</span><span class="n">C</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="MatrixBase.D"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.D">[docs]</a>    <span class="k">def</span> <span class="nf">D</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return Dirac conjugate (if self.rows == 4).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, I, eye</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix((0, 1 + I, 2, 3))</span>
-<span class="sd">        &gt;&gt;&gt; m.D</span>
-<span class="sd">        [0, 1 - I, -2, -3]</span>
-<span class="sd">        &gt;&gt;&gt; m = (eye(4) + I*eye(4))</span>
-<span class="sd">        &gt;&gt;&gt; m[0, 3] = 2</span>
-<span class="sd">        &gt;&gt;&gt; m.D</span>
-<span class="sd">        [1 - I,     0,      0,      0]</span>
-<span class="sd">        [    0, 1 - I,      0,      0]</span>
-<span class="sd">        [    0,     0, -1 + I,      0]</span>
-<span class="sd">        [    2,     0,      0, -1 + I]</span>
-
-<span class="sd">        If the matrix does not have 4 rows an AttributeError will be raised</span>
-<span class="sd">        because this property is only defined for matrices with 4 rows.</span>
-
-<span class="sd">        &gt;&gt;&gt; Matrix(eye(2)).D</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">        ...</span>
-<span class="sd">        AttributeError: Matrix has no attribute D.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        conjugate: By-element conjugation</span>
-<span class="sd">        H: Hermite conjugation</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.physics.matrices</span> <span class="kn">import</span> <span class="n">mgamma</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">!=</span> <span class="mi">4</span><span class="p">:</span>
-            <span class="c"># In Python 3.2, properties can only return an AttributeError</span>
-            <span class="c"># so we can&#39;t raise a ShapeError -- see commit which added the</span>
-            <span class="c"># first line of this inline comment. Also, there is no need</span>
-            <span class="c"># for a message since MatrixBase will raise the AttributeError</span>
-            <span class="k">raise</span> <span class="ne">AttributeError</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">H</span><span class="o">*</span><span class="n">mgamma</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">__array__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">.dense</span> <span class="kn">import</span> <span class="n">matrix2numpy</span>
-        <span class="k">return</span> <span class="n">matrix2numpy</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__len__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the number of elements of self.</span>
-
-<span class="sd">        Implemented mainly so bool(Matrix()) == False.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="MatrixBase.shape"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.shape">[docs]</a>    <span class="k">def</span> <span class="nf">shape</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The shape (dimensions) of the matrix as the 2-tuple (rows, cols).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import zeros</span>
-<span class="sd">        &gt;&gt;&gt; M = zeros(2, 3)</span>
-<span class="sd">        &gt;&gt;&gt; M.shape</span>
-<span class="sd">        (2, 3)</span>
-<span class="sd">        &gt;&gt;&gt; M.rows</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; M.cols</span>
-<span class="sd">        3</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span> <span class="o">+</span> <span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__rsub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="bp">self</span><span class="p">)</span> <span class="o">+</span> <span class="n">a</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return self*other where other is either a scalar or a matrix</span>
-<span class="sd">        of compatible dimensions.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; A = Matrix([[1, 2, 3], [4, 5, 6]])</span>
-<span class="sd">        &gt;&gt;&gt; 2*A == A*2 == Matrix([[2, 4, 6], [8, 10, 12]])</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; B = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])</span>
-<span class="sd">        &gt;&gt;&gt; A*B</span>
-<span class="sd">        [30, 36, 42]</span>
-<span class="sd">        [66, 81, 96]</span>
-<span class="sd">        &gt;&gt;&gt; B*A</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">        ...</span>
-<span class="sd">        ShapeError: Matrices size mismatch.</span>
-<span class="sd">        &gt;&gt;&gt;</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        matrix_multiply_elementwise</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="s">&#39;is_Matrix&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-            <span class="c"># The following implmentation is equivalent, but about 5% slower</span>
-            <span class="c">#ma, na = A.shape</span>
-            <span class="c">#mb, nb = B.shape</span>
-            <span class="c">#</span>
-            <span class="c">#if na != mb:</span>
-            <span class="c">#    raise ShapeError()</span>
-            <span class="c">#product = Matrix(ma, nb, lambda i, j: 0)</span>
-            <span class="c">#for i in range(ma):</span>
-            <span class="c">#    for j in range(nb):</span>
-            <span class="c">#        s = 0</span>
-            <span class="c">#        for k in range(na):</span>
-            <span class="c">#            s += A[i, k]*B[k, j]</span>
-            <span class="c">#        product[i, j] = s</span>
-            <span class="c">#return product</span>
-            <span class="n">A</span> <span class="o">=</span> <span class="bp">self</span>
-            <span class="n">B</span> <span class="o">=</span> <span class="n">other</span>
-            <span class="k">if</span> <span class="n">A</span><span class="o">.</span><span class="n">cols</span> <span class="o">!=</span> <span class="n">B</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">ShapeError</span><span class="p">(</span><span class="s">&quot;Matrices size mismatch.&quot;</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">A</span><span class="o">.</span><span class="n">cols</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">classof</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="n">B</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="mi">0</span><span class="p">)</span>
-            <span class="n">blst</span> <span class="o">=</span> <span class="n">B</span><span class="o">.</span><span class="n">T</span><span class="o">.</span><span class="n">tolist</span><span class="p">()</span>
-            <span class="n">alst</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">tolist</span><span class="p">()</span>
-            <span class="k">return</span> <span class="n">classof</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="n">B</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span>
-                <span class="nb">reduce</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">,</span> <span class="n">l</span><span class="p">:</span> <span class="n">k</span> <span class="o">+</span> <span class="n">l</span><span class="p">,</span>
-                    <span class="p">[</span><span class="n">a_ik</span> <span class="o">*</span> <span class="n">b_kj</span> <span class="k">for</span> <span class="n">a_ik</span><span class="p">,</span> <span class="n">b_kj</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">alst</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">blst</span><span class="p">[</span><span class="n">j</span><span class="p">])]))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span>
-                <span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="n">other</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="s">&#39;is_Matrix&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">*</span><span class="n">a</span>
-
-    <span class="k">def</span> <span class="nf">__pow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">num</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">eye</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">NonSquareMatrixError</span><span class="p">()</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="nb">int</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">Integer</span><span class="p">):</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">inv</span><span class="p">()</span><span class="o">**-</span><span class="n">n</span>   <span class="c"># A**-2 = (A**-1)**2</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">)</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="bp">self</span>
-            <span class="k">while</span> <span class="n">n</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span><span class="p">:</span>
-                    <span class="n">a</span> <span class="o">*=</span> <span class="n">s</span>
-                    <span class="n">n</span> <span class="o">-=</span> <span class="mi">1</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-                    <span class="k">break</span>
-                <span class="n">s</span> <span class="o">*=</span> <span class="n">s</span>
-                <span class="n">n</span> <span class="o">//=</span> <span class="mi">2</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">P</span><span class="p">,</span> <span class="n">D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">diagonalize</span><span class="p">()</span>
-            <span class="k">except</span> <span class="n">MatrixError</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                    <span class="s">&quot;Implemented only for diagonalizable matrices&quot;</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">D</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-                <span class="n">D</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">D</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span><span class="o">**</span><span class="n">num</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">P</span><span class="o">*</span><span class="n">D</span><span class="o">*</span><span class="n">P</span><span class="o">.</span><span class="n">inv</span><span class="p">())</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                <span class="s">&quot;Only integer and rational values are supported&quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return self + other, raising ShapeError if shapes don&#39;t match.&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="s">&#39;is_Matrix&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-            <span class="n">A</span> <span class="o">=</span> <span class="bp">self</span>
-            <span class="n">B</span> <span class="o">=</span> <span class="n">other</span>
-            <span class="k">if</span> <span class="n">A</span><span class="o">.</span><span class="n">shape</span> <span class="o">!=</span> <span class="n">B</span><span class="o">.</span><span class="n">shape</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">ShapeError</span><span class="p">(</span><span class="s">&quot;Matrices size mismatch.&quot;</span><span class="p">)</span>
-            <span class="n">alst</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">tolist</span><span class="p">()</span>
-            <span class="n">blst</span> <span class="o">=</span> <span class="n">B</span><span class="o">.</span><span class="n">tolist</span><span class="p">()</span>
-            <span class="n">ret</span> <span class="o">=</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">]</span><span class="o">*</span><span class="n">A</span><span class="o">.</span><span class="n">rows</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
-                <span class="n">ret</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="k">lambda</span> <span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">:</span> <span class="n">j</span> <span class="o">+</span> <span class="n">k</span><span class="p">,</span> <span class="n">alst</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">blst</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span>
-            <span class="k">return</span> <span class="n">classof</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">ret</span><span class="p">)</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;cannot add matrix and </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="nb">type</span><span class="p">(</span><span class="n">other</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__radd__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span> <span class="o">+</span> <span class="n">other</span>
-
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">*</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="o">/</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__truediv__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__div__</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="mi">1</span><span class="o">*</span><span class="bp">self</span>
-
-<div class="viewcode-block" id="MatrixBase.multiply"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.multiply">[docs]</a>    <span class="k">def</span> <span class="nf">multiply</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns self*b</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        dot</span>
-<span class="sd">        cross</span>
-<span class="sd">        multiply_elementwise</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">*</span><span class="n">b</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.add"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.add">[docs]</a>    <span class="k">def</span> <span class="nf">add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return self + b &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span> <span class="o">+</span> <span class="n">b</span>
-</div>
-    <span class="k">def</span> <span class="nf">_format_str</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">strfunc</span><span class="p">,</span> <span class="n">rowsep</span><span class="o">=</span><span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">):</span>
-        <span class="c"># Handle zero dimensions:</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&#39;[]&#39;</span>
-        <span class="c"># Build table of string representations of the elements</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="c"># Track per-column max lengths for pretty alignment</span>
-        <span class="n">maxlen</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">([])</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-                <span class="n">string</span> <span class="o">=</span> <span class="n">strfunc</span><span class="p">(</span><span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">])</span>
-                <span class="n">res</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">string</span><span class="p">)</span>
-                <span class="n">maxlen</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">string</span><span class="p">),</span> <span class="n">maxlen</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-        <span class="c"># Patch strings together</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">row</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">res</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">elem</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">row</span><span class="p">):</span>
-                <span class="c"># Pad each element up to maxlen so the columns line up</span>
-                <span class="n">row</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">elem</span><span class="o">.</span><span class="n">rjust</span><span class="p">(</span><span class="n">maxlen</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-            <span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="s">&quot;[&quot;</span> <span class="o">+</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">row</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot;]&quot;</span>
-        <span class="k">return</span> <span class="n">rowsep</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">res</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sstr</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sstr</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-<div class="viewcode-block" id="MatrixBase.cholesky"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.cholesky">[docs]</a>    <span class="k">def</span> <span class="nf">cholesky</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the Cholesky decomposition L of a matrix A</span>
-<span class="sd">        such that L * L.T = A</span>
-
-<span class="sd">        A must be a square, symmetric, positive-definite</span>
-<span class="sd">        and non-singular matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; A = Matrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))</span>
-<span class="sd">        &gt;&gt;&gt; A.cholesky()</span>
-<span class="sd">        [ 5, 0, 0]</span>
-<span class="sd">        [ 3, 3, 0]</span>
-<span class="sd">        [-1, 1, 3]</span>
-<span class="sd">        &gt;&gt;&gt; A.cholesky() * A.cholesky().T</span>
-<span class="sd">        [25, 15, -5]</span>
-<span class="sd">        [15, 18,  0]</span>
-<span class="sd">        [-5,  0, 11]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        LDLdecomposition</span>
-<span class="sd">        LUdecomposition</span>
-<span class="sd">        QRdecomposition</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">NonSquareMatrixError</span><span class="p">(</span><span class="s">&quot;Matrix must be square.&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_symmetric</span><span class="p">():</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Matrix must be symmetric.&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_cholesky</span><span class="p">()</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.LDLdecomposition"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.LDLdecomposition">[docs]</a>    <span class="k">def</span> <span class="nf">LDLdecomposition</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the LDL Decomposition (L, D) of matrix A,</span>
-<span class="sd">        such that L * D * L.T == A</span>
-<span class="sd">        This method eliminates the use of square root.</span>
-<span class="sd">        Further this ensures that all the diagonal entries of L are 1.</span>
-<span class="sd">        A must be a square, symmetric, positive-definite</span>
-<span class="sd">        and non-singular matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import Matrix, eye</span>
-<span class="sd">        &gt;&gt;&gt; A = Matrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))</span>
-<span class="sd">        &gt;&gt;&gt; L, D = A.LDLdecomposition()</span>
-<span class="sd">        &gt;&gt;&gt; L</span>
-<span class="sd">        [   1,   0, 0]</span>
-<span class="sd">        [ 3/5,   1, 0]</span>
-<span class="sd">        [-1/5, 1/3, 1]</span>
-<span class="sd">        &gt;&gt;&gt; D</span>
-<span class="sd">        [25, 0, 0]</span>
-<span class="sd">        [ 0, 9, 0]</span>
-<span class="sd">        [ 0, 0, 9]</span>
-<span class="sd">        &gt;&gt;&gt; L * D * L.T * A.inv() == eye(A.rows)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        cholesky</span>
-<span class="sd">        LUdecomposition</span>
-<span class="sd">        QRdecomposition</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">NonSquareMatrixError</span><span class="p">(</span><span class="s">&quot;Matrix must be square.&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_symmetric</span><span class="p">():</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Matrix must be symmetric.&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_LDLdecomposition</span><span class="p">()</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.lower_triangular_solve"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.lower_triangular_solve">[docs]</a>    <span class="k">def</span> <span class="nf">lower_triangular_solve</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Solves Ax = B, where A is a lower triangular matrix.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        upper_triangular_solve</span>
-<span class="sd">        cholesky_solve</span>
-<span class="sd">        diagonal_solve</span>
-<span class="sd">        LDLsolve</span>
-<span class="sd">        LUsolve</span>
-<span class="sd">        QRsolve</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">NonSquareMatrixError</span><span class="p">(</span><span class="s">&quot;Matrix must be square.&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">rhs</span><span class="o">.</span><span class="n">rows</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ShapeError</span><span class="p">(</span><span class="s">&quot;Matrices size mismatch.&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_lower</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Matrix must be lower triangular.&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_lower_triangular_solve</span><span class="p">(</span><span class="n">rhs</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.upper_triangular_solve"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.upper_triangular_solve">[docs]</a>    <span class="k">def</span> <span class="nf">upper_triangular_solve</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Solves Ax = B, where A is an upper triangular matrix.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        lower_triangular_solve</span>
-<span class="sd">        cholesky_solve</span>
-<span class="sd">        diagonal_solve</span>
-<span class="sd">        LDLsolve</span>
-<span class="sd">        LUsolve</span>
-<span class="sd">        QRsolve</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">NonSquareMatrixError</span><span class="p">(</span><span class="s">&quot;Matrix must be square.&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">rhs</span><span class="o">.</span><span class="n">rows</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Matrix size mismatch.&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_upper</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Matrix is not upper triangular.&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_upper_triangular_solve</span><span class="p">(</span><span class="n">rhs</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.cholesky_solve"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.cholesky_solve">[docs]</a>    <span class="k">def</span> <span class="nf">cholesky_solve</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Solves Ax = B using Cholesky decomposition,</span>
-<span class="sd">        for a general square non-singular matrix.</span>
-<span class="sd">        For a non-square matrix with rows &gt; cols,</span>
-<span class="sd">        the least squares solution is returned.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        lower_triangular_solve</span>
-<span class="sd">        upper_triangular_solve</span>
-<span class="sd">        diagonal_solve</span>
-<span class="sd">        LDLsolve</span>
-<span class="sd">        LUsolve</span>
-<span class="sd">        QRsolve</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_symmetric</span><span class="p">():</span>
-            <span class="n">L</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_cholesky</span><span class="p">()</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">:</span>
-            <span class="n">L</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="o">*</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">_cholesky</span><span class="p">()</span>
-            <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="o">*</span><span class="n">rhs</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Under-determined System.&quot;</span><span class="p">)</span>
-        <span class="n">Y</span> <span class="o">=</span> <span class="n">L</span><span class="o">.</span><span class="n">_lower_triangular_solve</span><span class="p">(</span><span class="n">rhs</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">L</span><span class="o">.</span><span class="n">T</span><span class="p">)</span><span class="o">.</span><span class="n">_upper_triangular_solve</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.diagonal_solve"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.diagonal_solve">[docs]</a>    <span class="k">def</span> <span class="nf">diagonal_solve</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Solves Ax = B efficiently, where A is a diagonal Matrix,</span>
-<span class="sd">        with non-zero diagonal entries.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import Matrix, eye</span>
-<span class="sd">        &gt;&gt;&gt; A = eye(2)*2</span>
-<span class="sd">        &gt;&gt;&gt; B = Matrix([[1, 2], [3, 4]])</span>
-<span class="sd">        &gt;&gt;&gt; A.diagonal_solve(B) == B/2</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        lower_triangular_solve</span>
-<span class="sd">        upper_triangular_solve</span>
-<span class="sd">        cholesky_solve</span>
-<span class="sd">        LDLsolve</span>
-<span class="sd">        LUsolve</span>
-<span class="sd">        QRsolve</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_diagonal</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Matrix should be diagonal&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">rhs</span><span class="o">.</span><span class="n">rows</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Size mis-match&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_diagonal_solve</span><span class="p">(</span><span class="n">rhs</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.LDLsolve"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.LDLsolve">[docs]</a>    <span class="k">def</span> <span class="nf">LDLsolve</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Solves Ax = B using LDL decomposition,</span>
-<span class="sd">        for a general square and non-singular matrix.</span>
-
-<span class="sd">        For a non-square matrix with rows &gt; cols,</span>
-<span class="sd">        the least squares solution is returned.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import Matrix, eye</span>
-<span class="sd">        &gt;&gt;&gt; A = eye(2)*2</span>
-<span class="sd">        &gt;&gt;&gt; B = Matrix([[1, 2], [3, 4]])</span>
-<span class="sd">        &gt;&gt;&gt; A.LDLsolve(B) == B/2</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        LDLdecomposition</span>
-<span class="sd">        lower_triangular_solve</span>
-<span class="sd">        upper_triangular_solve</span>
-<span class="sd">        cholesky_solve</span>
-<span class="sd">        diagonal_solve</span>
-<span class="sd">        LUsolve</span>
-<span class="sd">        QRsolve</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_symmetric</span><span class="p">():</span>
-            <span class="n">L</span><span class="p">,</span> <span class="n">D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">LDLdecomposition</span><span class="p">()</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">:</span>
-            <span class="n">L</span><span class="p">,</span> <span class="n">D</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="o">*</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">LDLdecomposition</span><span class="p">()</span>
-            <span class="n">rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="o">*</span><span class="n">rhs</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Under-determined System.&quot;</span><span class="p">)</span>
-        <span class="n">Y</span> <span class="o">=</span> <span class="n">L</span><span class="o">.</span><span class="n">_lower_triangular_solve</span><span class="p">(</span><span class="n">rhs</span><span class="p">)</span>
-        <span class="n">Z</span> <span class="o">=</span> <span class="n">D</span><span class="o">.</span><span class="n">_diagonal_solve</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">L</span><span class="o">.</span><span class="n">T</span><span class="p">)</span><span class="o">.</span><span class="n">_upper_triangular_solve</span><span class="p">(</span><span class="n">Z</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.solve_least_squares"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.solve_least_squares">[docs]</a>    <span class="k">def</span> <span class="nf">solve_least_squares</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rhs</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;CH&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the least-square fit to the data.</span>
-
-<span class="sd">        By default the cholesky_solve routine is used (method=&#39;CH&#39;); other</span>
-<span class="sd">        methods of matrix inversion can be used. To find out which are</span>
-<span class="sd">        available, see the docstring of the .inv() method.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import Matrix, Matrix, ones</span>
-<span class="sd">        &gt;&gt;&gt; A = Matrix([1, 2, 3])</span>
-<span class="sd">        &gt;&gt;&gt; B = Matrix([2, 3, 4])</span>
-<span class="sd">        &gt;&gt;&gt; S = Matrix(A.row_join(B))</span>
-<span class="sd">        &gt;&gt;&gt; S</span>
-<span class="sd">        [1, 2]</span>
-<span class="sd">        [2, 3]</span>
-<span class="sd">        [3, 4]</span>
-
-<span class="sd">        If each line of S represent coefficients of Ax + By</span>
-<span class="sd">        and x and y are [2, 3] then S*xy is:</span>
-
-<span class="sd">        &gt;&gt;&gt; r = S*Matrix([2, 3]); r</span>
-<span class="sd">        [ 8]</span>
-<span class="sd">        [13]</span>
-<span class="sd">        [18]</span>
-
-<span class="sd">        But let&#39;s add 1 to the middle value and then solve for the</span>
-<span class="sd">        least-squares value of xy:</span>
-
-<span class="sd">        &gt;&gt;&gt; xy = S.solve_least_squares(Matrix([8, 14, 18])); xy</span>
-<span class="sd">        [ 5/3]</span>
-<span class="sd">        [10/3]</span>
-
-<span class="sd">        The error is given by S*xy - r:</span>
-
-<span class="sd">        &gt;&gt;&gt; S*xy - r</span>
-<span class="sd">        [1/3]</span>
-<span class="sd">        [1/3]</span>
-<span class="sd">        [1/3]</span>
-<span class="sd">        &gt;&gt;&gt; _.norm().n(2)</span>
-<span class="sd">        0.58</span>
-
-<span class="sd">        If a different xy is used, the norm will be higher:</span>
-
-<span class="sd">        &gt;&gt;&gt; xy += ones(2, 1)/10</span>
-<span class="sd">        &gt;&gt;&gt; (S*xy - r).norm().n(2)</span>
-<span class="sd">        1.5</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">method</span> <span class="o">==</span> <span class="s">&#39;CH&#39;</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">cholesky_solve</span><span class="p">(</span><span class="n">rhs</span><span class="p">)</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">T</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">t</span><span class="o">*</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">method</span><span class="o">=</span><span class="n">method</span><span class="p">)</span><span class="o">*</span><span class="n">t</span><span class="o">*</span><span class="n">rhs</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.solve"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.solve">[docs]</a>    <span class="k">def</span> <span class="nf">solve</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rhs</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;GE&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return solution to self*soln = rhs using given inversion method.</span>
-
-<span class="sd">        For a list of possible inversion methods, see the .inv() docstring.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Under-determined system.&#39;</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">&gt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;For over-determined system, M, having &#39;</span>
-                    <span class="s">&#39;more rows than columns, try M.solve_least_squares(rhs).&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">method</span><span class="o">=</span><span class="n">method</span><span class="p">)</span><span class="o">*</span><span class="n">rhs</span>
-</div>
-    <span class="k">def</span> <span class="nf">__mathml__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">mml</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="n">mml</span> <span class="o">+=</span> <span class="s">&quot;&lt;matrixrow&gt;&quot;</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-                <span class="n">mml</span> <span class="o">+=</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">__mathml__</span><span class="p">()</span>
-            <span class="n">mml</span> <span class="o">+=</span> <span class="s">&quot;&lt;/matrixrow&gt;&quot;</span>
-        <span class="k">return</span> <span class="s">&quot;&lt;matrix&gt;&quot;</span> <span class="o">+</span> <span class="n">mml</span> <span class="o">+</span> <span class="s">&quot;&lt;/matrix&gt;&quot;</span>
-
-<div class="viewcode-block" id="MatrixBase.submatrix"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.submatrix">[docs]</a>    <span class="k">def</span> <span class="nf">submatrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">keys</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Get a slice/submatrix of the matrix using the given slice.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix(4, 4, lambda i, j: i+j)</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [0, 1, 2, 3]</span>
-<span class="sd">        [1, 2, 3, 4]</span>
-<span class="sd">        [2, 3, 4, 5]</span>
-<span class="sd">        [3, 4, 5, 6]</span>
-<span class="sd">        &gt;&gt;&gt; m[:1, 1]</span>
-<span class="sd">        [1]</span>
-<span class="sd">        &gt;&gt;&gt; m[:2, :1]</span>
-<span class="sd">        [0]</span>
-<span class="sd">        [1]</span>
-<span class="sd">        &gt;&gt;&gt; m[2:4, 2:4]</span>
-<span class="sd">        [4, 5]</span>
-<span class="sd">        [5, 6]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        extract</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">rlo</span><span class="p">,</span> <span class="n">rhi</span><span class="p">,</span> <span class="n">clo</span><span class="p">,</span> <span class="n">chi</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">key2bounds</span><span class="p">(</span><span class="n">keys</span><span class="p">)</span>
-        <span class="n">rows</span><span class="p">,</span> <span class="n">cols</span> <span class="o">=</span> <span class="n">rhi</span> <span class="o">-</span> <span class="n">rlo</span><span class="p">,</span> <span class="n">chi</span> <span class="o">-</span> <span class="n">clo</span>
-        <span class="n">mat</span> <span class="o">=</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">]</span><span class="o">*</span><span class="n">rows</span><span class="o">*</span><span class="n">cols</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">rows</span><span class="p">):</span>
-            <span class="n">mat</span><span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="n">cols</span><span class="p">:(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">cols</span><span class="p">]</span> <span class="o">=</span> \
-                <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">[(</span><span class="n">i</span> <span class="o">+</span> <span class="n">rlo</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">+</span> <span class="n">clo</span><span class="p">:(</span><span class="n">i</span> <span class="o">+</span> <span class="n">rlo</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">+</span> <span class="n">chi</span><span class="p">]</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">rows</span><span class="p">,</span> <span class="n">cols</span><span class="p">,</span> <span class="n">mat</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.extract"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.extract">[docs]</a>    <span class="k">def</span> <span class="nf">extract</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rowsList</span><span class="p">,</span> <span class="n">colsList</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return a submatrix by specifying a list of rows and columns.</span>
-<span class="sd">        Negative indices can be given. All indices must be in the range</span>
-<span class="sd">        -n &lt;= i &lt; n where n is the number of rows or columns.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix(4, 3, list(range(12)))</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [0,  1,  2]</span>
-<span class="sd">        [3,  4,  5]</span>
-<span class="sd">        [6,  7,  8]</span>
-<span class="sd">        [9, 10, 11]</span>
-<span class="sd">        &gt;&gt;&gt; m.extract([0, 1, 3], [0, 1])</span>
-<span class="sd">        [0,  1]</span>
-<span class="sd">        [3,  4]</span>
-<span class="sd">        [9, 10]</span>
-
-<span class="sd">        Rows or columns can be repeated:</span>
-
-<span class="sd">        &gt;&gt;&gt; m.extract([0, 0, 1], [-1])</span>
-<span class="sd">        [2]</span>
-<span class="sd">        [2]</span>
-<span class="sd">        [5]</span>
-
-<span class="sd">        Every other row can be taken by using range to provide the indices:</span>
-
-<span class="sd">        &gt;&gt;&gt; m.extract(list(range(0, m.rows, 2)), [-1])</span>
-<span class="sd">        [2]</span>
-<span class="sd">        [8]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        submatrix</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">cols</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span>
-        <span class="n">flat_list</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span>
-        <span class="n">rowsList</span> <span class="o">=</span> <span class="p">[</span><span class="n">a2idx</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">rowsList</span><span class="p">]</span>
-        <span class="n">colsList</span> <span class="o">=</span> <span class="p">[</span><span class="n">a2idx</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">colsList</span><span class="p">]</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">rowsList</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">colsList</span><span class="p">),</span>
-                <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="n">flat_list</span><span class="p">[</span><span class="n">rowsList</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">cols</span> <span class="o">+</span> <span class="n">colsList</span><span class="p">[</span><span class="n">j</span><span class="p">]])</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.key2bounds"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.key2bounds">[docs]</a>    <span class="k">def</span> <span class="nf">key2bounds</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">keys</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Converts a key with potentially mixed types of keys (integer and slice)</span>
-<span class="sd">        into a tuple of ranges and raises an error if any index is out of self&#39;s</span>
-<span class="sd">        range.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        key2ij</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">islice</span><span class="p">,</span> <span class="n">jslice</span> <span class="o">=</span> <span class="p">[</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="nb">slice</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">keys</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">islice</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-                <span class="n">rlo</span> <span class="o">=</span> <span class="n">rhi</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">rlo</span><span class="p">,</span> <span class="n">rhi</span> <span class="o">=</span> <span class="n">keys</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">indices</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)[:</span><span class="mi">2</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">rlo</span> <span class="o">=</span> <span class="n">a2idx</span><span class="p">(</span><span class="n">keys</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)</span>
-            <span class="n">rhi</span> <span class="o">=</span> <span class="n">rlo</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">jslice</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">:</span>
-                <span class="n">clo</span> <span class="o">=</span> <span class="n">chi</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">clo</span><span class="p">,</span> <span class="n">chi</span> <span class="o">=</span> <span class="n">keys</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">indices</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">)[:</span><span class="mi">2</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">clo</span> <span class="o">=</span> <span class="n">a2idx</span><span class="p">(</span><span class="n">keys</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">)</span>
-            <span class="n">chi</span> <span class="o">=</span> <span class="n">clo</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="n">rlo</span><span class="p">,</span> <span class="n">rhi</span><span class="p">,</span> <span class="n">clo</span><span class="p">,</span> <span class="n">chi</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.key2ij"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.key2ij">[docs]</a>    <span class="k">def</span> <span class="nf">key2ij</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Converts key into canonical form, converting integers or indexable</span>
-<span class="sd">        items into valid integers for self&#39;s range or returning slices</span>
-<span class="sd">        unchanged.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        key2bounds</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">key</span><span class="p">):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">len</span><span class="p">(</span><span class="n">key</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;key must be a sequence of length 2&#39;</span><span class="p">)</span>
-            <span class="k">return</span> <span class="p">[</span><span class="n">a2idx</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="nb">slice</span><span class="p">)</span> <span class="k">else</span> <span class="n">i</span>
-                <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">shape</span><span class="p">)]</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="nb">slice</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">key</span><span class="o">.</span><span class="n">indices</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="p">))[:</span><span class="mi">2</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">divmod</span><span class="p">(</span><span class="n">a2idx</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="p">)),</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.evalf"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.evalf">[docs]</a>    <span class="k">def</span> <span class="nf">evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Apply evalf() to each element of self.&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">prec</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">applyfunc</span><span class="p">(</span><span class="k">lambda</span> <span class="n">i</span><span class="p">:</span> <span class="n">i</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="o">**</span><span class="n">options</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">applyfunc</span><span class="p">(</span><span class="k">lambda</span> <span class="n">i</span><span class="p">:</span> <span class="n">i</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">))</span>
-</div>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">evalf</span>
-
-<div class="viewcode-block" id="MatrixBase.subs"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.subs">[docs]</a>    <span class="k">def</span> <span class="nf">subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>  <span class="c"># should mirror core.basic.subs</span>
-        <span class="sd">&quot;&quot;&quot;Return a new matrix with subs applied to each entry.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix, Matrix</span>
-<span class="sd">        &gt;&gt;&gt; SparseMatrix(1, 1, [x])</span>
-<span class="sd">        [x]</span>
-<span class="sd">        &gt;&gt;&gt; _.subs(x, y)</span>
-<span class="sd">        [y]</span>
-<span class="sd">        &gt;&gt;&gt; Matrix(_).subs(y, x)</span>
-<span class="sd">        [x]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">applyfunc</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.expand"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.expand">[docs]</a>    <span class="k">def</span> <span class="nf">expand</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">modulus</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">power_base</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">power_exp</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span>
-            <span class="n">mul</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">log</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">multinomial</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">basic</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Apply core.function.expand to each entry of the matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; Matrix(1, 1, [x*(x+1)])</span>
-<span class="sd">        [x*(x + 1)]</span>
-<span class="sd">        &gt;&gt;&gt; _.expand()</span>
-<span class="sd">        [x**2 + x]</span>
-<span class="sd">        &gt;&gt;&gt;</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">applyfunc</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span>
-                              <span class="n">deep</span><span class="p">,</span> <span class="n">modulus</span><span class="p">,</span> <span class="n">power_base</span><span class="p">,</span> <span class="n">power_exp</span><span class="p">,</span> <span class="n">mul</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">multinomial</span><span class="p">,</span> <span class="n">basic</span><span class="p">,</span>
-        <span class="o">**</span><span class="n">hints</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.simplify"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.simplify">[docs]</a>    <span class="k">def</span> <span class="nf">simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ratio</span><span class="o">=</span><span class="mf">1.7</span><span class="p">,</span> <span class="n">measure</span><span class="o">=</span><span class="n">count_ops</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Apply simplify to each element of the matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import sin, cos</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix</span>
-<span class="sd">        &gt;&gt;&gt; SparseMatrix(1, 1, [x*sin(y)**2 + x*cos(y)**2])</span>
-<span class="sd">        [x*sin(y)**2 + x*cos(y)**2]</span>
-<span class="sd">        &gt;&gt;&gt; _.simplify()</span>
-<span class="sd">        [x]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">applyfunc</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span><span class="n">ratio</span><span class="p">,</span> <span class="n">measure</span><span class="p">))</span></div>
-    <span class="n">_eval_simplify</span> <span class="o">=</span> <span class="n">simplify</span>
-
-<div class="viewcode-block" id="MatrixBase.print_nonzero"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.print_nonzero">[docs]</a>    <span class="k">def</span> <span class="nf">print_nonzero</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">symb</span><span class="o">=</span><span class="s">&quot;X&quot;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Shows location of non-zero entries for fast shape lookup.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import Matrix, eye</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix(2, 3, lambda i, j: i*3+j)</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [0, 1, 2]</span>
-<span class="sd">        [3, 4, 5]</span>
-<span class="sd">        &gt;&gt;&gt; m.print_nonzero()</span>
-<span class="sd">        [ XX]</span>
-<span class="sd">        [XXX]</span>
-<span class="sd">        &gt;&gt;&gt; m = eye(4)</span>
-<span class="sd">        &gt;&gt;&gt; m.print_nonzero(&quot;x&quot;)</span>
-<span class="sd">        [x   ]</span>
-<span class="sd">        [ x  ]</span>
-<span class="sd">        [  x ]</span>
-<span class="sd">        [   x]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="n">line</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">line</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot; &quot;</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">line</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">symb</span><span class="p">))</span>
-            <span class="n">s</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;[</span><span class="si">%s</span><span class="s">]&quot;</span> <span class="o">%</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">line</span><span class="p">))</span>
-        <span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">s</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.LUsolve"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.LUsolve">[docs]</a>    <span class="k">def</span> <span class="nf">LUsolve</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rhs</span><span class="p">,</span> <span class="n">iszerofunc</span><span class="o">=</span><span class="n">_iszero</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Solve the linear system Ax = rhs for x where A = self.</span>
-
-<span class="sd">        This is for symbolic matrices, for real or complex ones use</span>
-<span class="sd">        sympy.mpmath.lu_solve or sympy.mpmath.qr_solve.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        lower_triangular_solve</span>
-<span class="sd">        upper_triangular_solve</span>
-<span class="sd">        cholesky_solve</span>
-<span class="sd">        diagonal_solve</span>
-<span class="sd">        LDLsolve</span>
-<span class="sd">        QRsolve</span>
-<span class="sd">        LUdecomposition</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">rhs</span><span class="o">.</span><span class="n">rows</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ShapeError</span><span class="p">(</span>
-                <span class="s">&quot;`self` and `rhs` must have the same number of rows.&quot;</span><span class="p">)</span>
-
-        <span class="n">A</span><span class="p">,</span> <span class="n">perm</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">LUdecomposition_Simple</span><span class="p">(</span><span class="n">iszerofunc</span><span class="o">=</span><span class="n">_iszero</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">rhs</span><span class="o">.</span><span class="n">permuteFwd</span><span class="p">(</span><span class="n">perm</span><span class="p">)</span><span class="o">.</span><span class="n">as_mutable</span><span class="p">()</span>
-        <span class="c"># forward substitution, all diag entries are scaled to 1</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
-                <span class="n">b</span><span class="o">.</span><span class="n">row_op</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">:</span> <span class="n">x</span> <span class="o">-</span> <span class="n">b</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">])</span>
-        <span class="c"># backward substitution</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-                <span class="n">b</span><span class="o">.</span><span class="n">row_op</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">:</span> <span class="n">x</span> <span class="o">-</span> <span class="n">b</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">])</span>
-            <span class="n">b</span><span class="o">.</span><span class="n">row_op</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">:</span> <span class="n">x</span> <span class="o">/</span> <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">rhs</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.LUdecomposition"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.LUdecomposition">[docs]</a>    <span class="k">def</span> <span class="nf">LUdecomposition</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">iszerofunc</span><span class="o">=</span><span class="n">_iszero</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the decomposition LU and the row swaps p.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; a = Matrix([[4, 3], [6, 3]])</span>
-<span class="sd">        &gt;&gt;&gt; L, U, _ = a.LUdecomposition()</span>
-<span class="sd">        &gt;&gt;&gt; L</span>
-<span class="sd">        [  1, 0]</span>
-<span class="sd">        [3/2, 1]</span>
-<span class="sd">        &gt;&gt;&gt; U</span>
-<span class="sd">        [4,    3]</span>
-<span class="sd">        [0, -3/2]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        cholesky</span>
-<span class="sd">        LDLdecomposition</span>
-<span class="sd">        QRdecomposition</span>
-<span class="sd">        LUdecomposition_Simple</span>
-<span class="sd">        LUdecompositionFF</span>
-<span class="sd">        LUsolve</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">combined</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">LUdecomposition_Simple</span><span class="p">(</span><span class="n">iszerofunc</span><span class="o">=</span><span class="n">_iszero</span><span class="p">)</span>
-        <span class="n">L</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)</span>
-        <span class="n">U</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">i</span> <span class="o">&gt;</span> <span class="n">j</span><span class="p">:</span>
-                    <span class="n">L</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">combined</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
-                        <span class="n">L</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-                    <span class="n">U</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">combined</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">L</span><span class="p">,</span> <span class="n">U</span><span class="p">,</span> <span class="n">p</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.LUdecomposition_Simple"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.LUdecomposition_Simple">[docs]</a>    <span class="k">def</span> <span class="nf">LUdecomposition_Simple</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">iszerofunc</span><span class="o">=</span><span class="n">_iszero</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns A comprised of L, U (L&#39;s diag entries are 1) and</span>
-<span class="sd">        p which is the list of the row swaps (in order).</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        LUdecomposition</span>
-<span class="sd">        LUdecompositionFF</span>
-<span class="sd">        LUsolve</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">NonSquareMatrixError</span><span class="p">(</span><span class="s">&quot;A Matrix must be square to apply LUdecomposition_Simple().&quot;</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span>
-        <span class="n">A</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_mutable</span><span class="p">()</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="c"># factorization</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">j</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
-                    <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">A</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-            <span class="n">pivot</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">j</span><span class="p">):</span>
-                    <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">A</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-                <span class="c"># find the first non-zero pivot, includes any expression</span>
-                <span class="k">if</span> <span class="n">pivot</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">iszerofunc</span><span class="p">(</span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]):</span>
-                    <span class="n">pivot</span> <span class="o">=</span> <span class="n">i</span>
-            <span class="k">if</span> <span class="n">pivot</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="c"># this result is based on iszerofunc&#39;s analysis of the possible pivots, so even though</span>
-                <span class="c"># the element may not be strictly zero, the supplied iszerofunc&#39;s evaluation gave True</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;No nonzero pivot found; inversion failed.&quot;</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">pivot</span> <span class="o">!=</span> <span class="n">j</span><span class="p">:</span>  <span class="c"># row must be swapped</span>
-                <span class="n">A</span><span class="o">.</span><span class="n">row_swap</span><span class="p">(</span><span class="n">pivot</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
-                <span class="n">p</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">pivot</span><span class="p">,</span> <span class="n">j</span><span class="p">])</span>
-            <span class="n">scale</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">A</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-                <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">*</span><span class="n">scale</span>
-        <span class="k">return</span> <span class="n">A</span><span class="p">,</span> <span class="n">p</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.LUdecompositionFF"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.LUdecompositionFF">[docs]</a>    <span class="k">def</span> <span class="nf">LUdecompositionFF</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute a fraction-free LU decomposition.</span>
-
-<span class="sd">        Returns 4 matrices P, L, D, U such that PA = L D**-1 U.</span>
-<span class="sd">        If the elements of the matrix belong to some integral domain I, then all</span>
-<span class="sd">        elements of L, D and U are guaranteed to belong to I.</span>
-
-<span class="sd">        **Reference**</span>
-<span class="sd">            - W. Zhou &amp; D.J. Jeffrey, &quot;Fraction-free matrix factors: new forms</span>
-<span class="sd">              for LU and QR factors&quot;. Frontiers in Computer Science in China,</span>
-<span class="sd">              Vol 2, no. 1, pp. 67-80, 2008.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        LUdecomposition</span>
-<span class="sd">        LUdecomposition_Simple</span>
-<span class="sd">        LUsolve</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span>
-        <span class="n">zeros</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="o">.</span><span class="n">zeros</span>
-        <span class="n">eye</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="o">.</span><span class="n">eye</span>
-
-        <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span>
-        <span class="n">U</span><span class="p">,</span> <span class="n">L</span><span class="p">,</span> <span class="n">P</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_mutable</span><span class="p">(),</span> <span class="n">eye</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">eye</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="n">DD</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-        <span class="n">oldpivot</span> <span class="o">=</span> <span class="mi">1</span>
-
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">U</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">kpivot</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">U</span><span class="p">[</span><span class="n">kpivot</span><span class="p">,</span> <span class="n">k</span><span class="p">]:</span>
-                        <span class="k">break</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Matrix is not full rank&quot;</span><span class="p">)</span>
-                <span class="n">U</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">k</span><span class="p">:],</span> <span class="n">U</span><span class="p">[</span><span class="n">kpivot</span><span class="p">,</span> <span class="n">k</span><span class="p">:]</span> <span class="o">=</span> <span class="n">U</span><span class="p">[</span><span class="n">kpivot</span><span class="p">,</span> <span class="n">k</span><span class="p">:],</span> <span class="n">U</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">k</span><span class="p">:]</span>
-                <span class="n">L</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="p">:</span><span class="n">k</span><span class="p">],</span> <span class="n">L</span><span class="p">[</span><span class="n">kpivot</span><span class="p">,</span> <span class="p">:</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">L</span><span class="p">[</span><span class="n">kpivot</span><span class="p">,</span> <span class="p">:</span><span class="n">k</span><span class="p">],</span> <span class="n">L</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="p">:</span><span class="n">k</span><span class="p">]</span>
-                <span class="n">P</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="p">:],</span> <span class="n">P</span><span class="p">[</span><span class="n">kpivot</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">kpivot</span><span class="p">,</span> <span class="p">:],</span> <span class="n">P</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="p">:]</span>
-            <span class="n">L</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">Ukk</span> <span class="o">=</span> <span class="n">U</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span>
-            <span class="n">DD</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">oldpivot</span><span class="o">*</span><span class="n">Ukk</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-                <span class="n">L</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">Uik</span> <span class="o">=</span> <span class="n">U</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-                    <span class="n">U</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">Ukk</span><span class="o">*</span><span class="n">U</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">U</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">*</span><span class="n">Uik</span><span class="p">)</span> <span class="o">/</span> <span class="n">oldpivot</span>
-                <span class="n">U</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="n">oldpivot</span> <span class="o">=</span> <span class="n">Ukk</span>
-        <span class="n">DD</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">oldpivot</span>
-        <span class="k">return</span> <span class="n">P</span><span class="p">,</span> <span class="n">L</span><span class="p">,</span> <span class="n">DD</span><span class="p">,</span> <span class="n">U</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.cofactorMatrix"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.cofactorMatrix">[docs]</a>    <span class="k">def</span> <span class="nf">cofactorMatrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">&quot;berkowitz&quot;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return a matrix containing the cofactor of each element.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        cofactor</span>
-<span class="sd">        minorEntry</span>
-<span class="sd">        minorMatrix</span>
-<span class="sd">        adjugate</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">out</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">cofactor</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">method</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">out</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.minorEntry"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.minorEntry">[docs]</a>    <span class="k">def</span> <span class="nf">minorEntry</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">&quot;berkowitz&quot;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Calculate the minor of an element.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        minorMatrix</span>
-<span class="sd">        cofactor</span>
-<span class="sd">        cofactorMatrix</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="mi">0</span> <span class="o">&lt;=</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="ow">or</span> <span class="ow">not</span> <span class="mi">0</span> <span class="o">&lt;=</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;`i` and `j` must satisfy 0 &lt;= i &lt; `self.rows` &quot;</span> <span class="o">+</span>
-                <span class="s">&quot;(</span><span class="si">%d</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">+</span> <span class="s">&quot;and 0 &lt;= j &lt; `self.cols` (</span><span class="si">%d</span><span class="s">).&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">minorMatrix</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span><span class="o">.</span><span class="n">det</span><span class="p">(</span><span class="n">method</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.minorMatrix"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.minorMatrix">[docs]</a>    <span class="k">def</span> <span class="nf">minorMatrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Creates the minor matrix of a given element.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        minorEntry</span>
-<span class="sd">        cofactor</span>
-<span class="sd">        cofactorMatrix</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="mi">0</span> <span class="o">&lt;=</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="ow">or</span> <span class="ow">not</span> <span class="mi">0</span> <span class="o">&lt;=</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;`i` and `j` must satisfy 0 &lt;= i &lt; `self.rows` &quot;</span> <span class="o">+</span>
-                <span class="s">&quot;(</span><span class="si">%d</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">+</span> <span class="s">&quot;and 0 &lt;= j &lt; `self.cols` (</span><span class="si">%d</span><span class="s">).&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">)</span>
-        <span class="n">M</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_mutable</span><span class="p">()</span>
-        <span class="n">M</span><span class="o">.</span><span class="n">row_del</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-        <span class="n">M</span><span class="o">.</span><span class="n">col_del</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.cofactor"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.cofactor">[docs]</a>    <span class="k">def</span> <span class="nf">cofactor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">&quot;berkowitz&quot;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Calculate the cofactor of an element.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        cofactorMatrix</span>
-<span class="sd">        minorEntry</span>
-<span class="sd">        minorMatrix</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">minorEntry</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">method</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="mi">1</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">minorEntry</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">method</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.jacobian"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.jacobian">[docs]</a>    <span class="k">def</span> <span class="nf">jacobian</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Calculates the Jacobian matrix (derivative of a vectorial function).</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        self : vector of expressions representing functions f_i(x_1, ..., x_n).</span>
-<span class="sd">        X : set of x_i&#39;s in order, it can be a list or a Matrix</span>
-
-<span class="sd">        Both self and X can be a row or a column matrix in any order</span>
-<span class="sd">        (i.e., jacobian() should always work).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sin, cos, Matrix</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import rho, phi</span>
-<span class="sd">        &gt;&gt;&gt; X = Matrix([rho*cos(phi), rho*sin(phi), rho**2])</span>
-<span class="sd">        &gt;&gt;&gt; Y = Matrix([rho, phi])</span>
-<span class="sd">        &gt;&gt;&gt; X.jacobian(Y)</span>
-<span class="sd">        [cos(phi), -rho*sin(phi)]</span>
-<span class="sd">        [sin(phi),  rho*cos(phi)]</span>
-<span class="sd">        [   2*rho,             0]</span>
-<span class="sd">        &gt;&gt;&gt; X = Matrix([rho*cos(phi), rho*sin(phi)])</span>
-<span class="sd">        &gt;&gt;&gt; X.jacobian(Y)</span>
-<span class="sd">        [cos(phi), -rho*sin(phi)]</span>
-<span class="sd">        [sin(phi),  rho*cos(phi)]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        hessian</span>
-<span class="sd">        wronskian</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">MatrixBase</span><span class="p">):</span>
-            <span class="n">X</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-        <span class="c"># Both X and self can be a row or a column matrix, so we need to make</span>
-        <span class="c"># sure all valid combinations work, but everything else fails:</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;self must be a row or a column matrix&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">elif</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;X must be a row or a column matrix&quot;</span><span class="p">)</span>
-
-        <span class="c"># m is the number of functions and n is the number of variables</span>
-        <span class="c"># computing the Jacobian is now easy:</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="bp">self</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.QRdecomposition"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.QRdecomposition">[docs]</a>    <span class="k">def</span> <span class="nf">QRdecomposition</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return Q, R where A = Q*R, Q is orthogonal and R is upper triangular.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        This is the example from wikipedia:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, eye</span>
-<span class="sd">        &gt;&gt;&gt; A = Matrix([[12, -51, 4], [6, 167, -68], [-4, 24, -41]])</span>
-<span class="sd">        &gt;&gt;&gt; Q, R = A.QRdecomposition()</span>
-<span class="sd">        &gt;&gt;&gt; Q</span>
-<span class="sd">        [ 6/7, -69/175, -58/175]</span>
-<span class="sd">        [ 3/7, 158/175,   6/175]</span>
-<span class="sd">        [-2/7,    6/35,  -33/35]</span>
-<span class="sd">        &gt;&gt;&gt; R</span>
-<span class="sd">        [14,  21, -14]</span>
-<span class="sd">        [ 0, 175, -70]</span>
-<span class="sd">        [ 0,   0,  35]</span>
-<span class="sd">        &gt;&gt;&gt; A == Q*R</span>
-<span class="sd">        True</span>
-
-<span class="sd">        QR factorization of an identity matrix:</span>
-
-<span class="sd">        &gt;&gt;&gt; A = Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])</span>
-<span class="sd">        &gt;&gt;&gt; Q, R = A.QRdecomposition()</span>
-<span class="sd">        &gt;&gt;&gt; Q</span>
-<span class="sd">        [1, 0, 0]</span>
-<span class="sd">        [0, 1, 0]</span>
-<span class="sd">        [0, 0, 1]</span>
-<span class="sd">        &gt;&gt;&gt; R</span>
-<span class="sd">        [1, 0, 0]</span>
-<span class="sd">        [0, 1, 0]</span>
-<span class="sd">        [0, 0, 1]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        cholesky</span>
-<span class="sd">        LDLdecomposition</span>
-<span class="sd">        LUdecomposition</span>
-<span class="sd">        QRsolve</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">cls</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span>
-        <span class="bp">self</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_mutable</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">MatrixError</span><span class="p">(</span>
-                <span class="s">&quot;The number of rows must be greater than columns&quot;</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span>
-        <span class="n">rank</span> <span class="o">=</span> <span class="n">n</span>
-        <span class="n">row_reduced</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rref</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">row_reduced</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">row_reduced</span><span class="o">.</span><span class="n">row</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">norm</span><span class="p">()</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">rank</span> <span class="o">-=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">rank</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">MatrixError</span><span class="p">(</span><span class="s">&quot;The rank of the matrix must match the columns&quot;</span><span class="p">)</span>
-        <span class="n">Q</span><span class="p">,</span> <span class="n">R</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>      <span class="c"># for each column vector</span>
-            <span class="n">tmp</span> <span class="o">=</span> <span class="bp">self</span><span class="p">[:,</span> <span class="n">j</span><span class="p">]</span>     <span class="c"># take original v</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">j</span><span class="p">):</span>
-                <span class="c"># subtract the project of self on new vector</span>
-                <span class="n">tmp</span> <span class="o">-=</span> <span class="n">Q</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="bp">self</span><span class="p">[:,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span> <span class="n">i</span><span class="p">])</span>
-                <span class="n">tmp</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-            <span class="c"># normalize it</span>
-            <span class="n">R</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">tmp</span><span class="o">.</span><span class="n">norm</span><span class="p">()</span>
-            <span class="n">Q</span><span class="p">[:,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">tmp</span> <span class="o">/</span> <span class="n">R</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">Q</span><span class="p">[:,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">norm</span><span class="p">()</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                    <span class="s">&quot;Could not normalize the vector </span><span class="si">%d</span><span class="s">.&quot;</span> <span class="o">%</span> <span class="n">j</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">j</span><span class="p">):</span>
-                <span class="n">R</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">Q</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="bp">self</span><span class="p">[:,</span> <span class="n">j</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">Q</span><span class="p">),</span> <span class="n">cls</span><span class="p">(</span><span class="n">R</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.QRsolve"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.QRsolve">[docs]</a>    <span class="k">def</span> <span class="nf">QRsolve</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Solve the linear system &#39;Ax = b&#39;.</span>
-
-<span class="sd">        &#39;self&#39; is the matrix &#39;A&#39;, the method argument is the vector</span>
-<span class="sd">        &#39;b&#39;.  The method returns the solution vector &#39;x&#39;.  If &#39;b&#39; is a</span>
-<span class="sd">        matrix, the system is solved for each column of &#39;b&#39; and the</span>
-<span class="sd">        return value is a matrix of the same shape as &#39;b&#39;.</span>
-
-<span class="sd">        This method is slower (approximately by a factor of 2) but</span>
-<span class="sd">        more stable for floating-point arithmetic than the LUsolve method.</span>
-<span class="sd">        However, LUsolve usually uses an exact arithmetic, so you don&#39;t need</span>
-<span class="sd">        to use QRsolve.</span>
-
-<span class="sd">        This is mainly for educational purposes and symbolic matrices, for real</span>
-<span class="sd">        (or complex) matrices use sympy.mpmath.qr_solve.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        lower_triangular_solve</span>
-<span class="sd">        upper_triangular_solve</span>
-<span class="sd">        cholesky_solve</span>
-<span class="sd">        diagonal_solve</span>
-<span class="sd">        LDLsolve</span>
-<span class="sd">        LUsolve</span>
-<span class="sd">        QRdecomposition</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">Q</span><span class="p">,</span> <span class="n">R</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_mutable</span><span class="p">()</span><span class="o">.</span><span class="n">QRdecomposition</span><span class="p">()</span>
-        <span class="n">y</span> <span class="o">=</span> <span class="n">Q</span><span class="o">.</span><span class="n">T</span><span class="o">*</span><span class="n">b</span>
-
-        <span class="c"># back substitution to solve R*x = y:</span>
-        <span class="c"># We build up the result &quot;backwards&quot; in the vector &#39;x&#39; and reverse it</span>
-        <span class="c"># only in the end.</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">R</span><span class="o">.</span><span class="n">rows</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-            <span class="n">tmp</span> <span class="o">=</span> <span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="p">:]</span>
-            <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-                <span class="n">tmp</span> <span class="o">-=</span> <span class="n">R</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">k</span><span class="p">]</span>
-            <span class="n">x</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tmp</span> <span class="o">/</span> <span class="n">R</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">])</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">([</span><span class="n">row</span><span class="o">.</span><span class="n">_mat</span> <span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">x</span><span class="p">)])</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.cross"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.cross">[docs]</a>    <span class="k">def</span> <span class="nf">cross</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Calculate the cross product of ``self`` and ``b``.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        dot</span>
-<span class="sd">        multiply</span>
-<span class="sd">        multiply_elementwise</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">b</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;`b` must be an ordered iterable or Matrix, not </span><span class="si">%s</span><span class="s">.&quot;</span> <span class="o">%</span>
-                <span class="nb">type</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">==</span> <span class="mi">3</span> <span class="ow">or</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">==</span> <span class="mi">3</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)</span> <span class="ow">and</span> \
-                <span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">rows</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">b</span><span class="o">.</span><span class="n">cols</span> <span class="o">==</span> <span class="mi">3</span> <span class="ow">or</span>
-                <span class="n">b</span><span class="o">.</span><span class="n">rows</span> <span class="o">==</span> <span class="mi">3</span> <span class="ow">and</span> <span class="n">b</span><span class="o">.</span><span class="n">cols</span> <span class="o">==</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="n">ShapeError</span><span class="p">(</span><span class="s">&quot;Dimensions incorrect for cross product.&quot;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="p">((</span><span class="bp">self</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">b</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">-</span> <span class="bp">self</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">b</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span>
-                               <span class="p">(</span><span class="bp">self</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="bp">self</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">b</span><span class="p">[</span><span class="mi">2</span><span class="p">]),</span>
-                               <span class="p">(</span><span class="bp">self</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">b</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="bp">self</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">])))</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.dot"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.dot">[docs]</a>    <span class="k">def</span> <span class="nf">dot</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the dot product of Matrix self and b relaxing the condition</span>
-<span class="sd">        of compatible dimensions: if either the number of rows or columns are</span>
-<span class="sd">        the same as the length of b then the dot product is returned. If self</span>
-<span class="sd">        is a row or column vector, a scalar is returned. Otherwise, a list</span>
-<span class="sd">        of results is returned (and in that case the number of columns in self</span>
-<span class="sd">        must match the length of b).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])</span>
-<span class="sd">        &gt;&gt;&gt; v = [1, 1, 1]</span>
-<span class="sd">        &gt;&gt;&gt; M.row(0).dot(v)</span>
-<span class="sd">        6</span>
-<span class="sd">        &gt;&gt;&gt; M.col(0).dot(v)</span>
-<span class="sd">        12</span>
-<span class="sd">        &gt;&gt;&gt; M.dot(v)</span>
-<span class="sd">        [6, 15, 24]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        cross</span>
-<span class="sd">        multiply</span>
-<span class="sd">        multiply_elementwise</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">.dense</span> <span class="kn">import</span> <span class="n">Matrix</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">MatrixBase</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">b</span><span class="p">):</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="n">ShapeError</span><span class="p">(</span><span class="s">&quot;Dimensions incorrect for dot product.&quot;</span><span class="p">)</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">Matrix</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;`b` must be an ordered iterable or Matrix, not </span><span class="si">%s</span><span class="s">.&quot;</span> <span class="o">%</span>
-                <span class="nb">type</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">==</span> <span class="n">b</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">cols</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="bp">self</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">T</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">T</span>
-            <span class="n">prod</span> <span class="o">=</span> <span class="n">flatten</span><span class="p">((</span><span class="bp">self</span><span class="o">*</span><span class="n">b</span><span class="p">)</span><span class="o">.</span><span class="n">tolist</span><span class="p">())</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">prod</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">prod</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">prod</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">==</span> <span class="n">b</span><span class="o">.</span><span class="n">cols</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">==</span> <span class="n">b</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ShapeError</span><span class="p">(</span><span class="s">&quot;Dimensions incorrect for dot product.&quot;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.multiply_elementwise"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.multiply_elementwise">[docs]</a>    <span class="k">def</span> <span class="nf">multiply_elementwise</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the Hadamard product (elementwise product) of A and B</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; A = Matrix([[0, 1, 2], [3, 4, 5]])</span>
-<span class="sd">        &gt;&gt;&gt; B = Matrix([[1, 10, 100], [100, 10, 1]])</span>
-<span class="sd">        &gt;&gt;&gt; A.multiply_elementwise(B)</span>
-<span class="sd">        [  0, 10, 200]</span>
-<span class="sd">        [300, 40,   5]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        cross</span>
-<span class="sd">        dot</span>
-<span class="sd">        multiply</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">matrix_multiply_elementwise</span>
-
-        <span class="k">return</span> <span class="n">matrix_multiply_elementwise</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.values"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.values">[docs]</a>    <span class="k">def</span> <span class="nf">values</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return non-zero values of self.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">flatten</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">tolist</span><span class="p">())</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">i</span><span class="o">.</span><span class="n">is_zero</span><span class="p">]</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.norm"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.norm">[docs]</a>    <span class="k">def</span> <span class="nf">norm</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">ord</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the Norm of a Matrix or Vector.</span>
-<span class="sd">        In the simplest case this is the geometric size of the vector</span>
-<span class="sd">        Other norms can be specified by the ord parameter</span>
-
-
-<span class="sd">        =====  ============================  ==========================</span>
-<span class="sd">        ord    norm for matrices             norm for vectors</span>
-<span class="sd">        =====  ============================  ==========================</span>
-<span class="sd">        None   Frobenius norm                2-norm</span>
-<span class="sd">        &#39;fro&#39;  Frobenius norm                - does not exist</span>
-<span class="sd">        inf    --                            max(abs(x))</span>
-<span class="sd">        -inf   --                            min(abs(x))</span>
-<span class="sd">        1      --                            as below</span>
-<span class="sd">        -1     --                            as below</span>
-<span class="sd">        2      2-norm (largest sing. value)  as below</span>
-<span class="sd">        -2     smallest singular value       as below</span>
-<span class="sd">        other  - does not exist              sum(abs(x)**ord)**(1./ord)</span>
-<span class="sd">        =====  ============================  ==========================</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, Symbol, trigsimp, cos, sin, oo</span>
-<span class="sd">        &gt;&gt;&gt; x = Symbol(&#39;x&#39;, real=True)</span>
-<span class="sd">        &gt;&gt;&gt; v = Matrix([cos(x), sin(x)])</span>
-<span class="sd">        &gt;&gt;&gt; trigsimp( v.norm() )</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; v.norm(10)</span>
-<span class="sd">        (sin(x)**10 + cos(x)**10)**(1/10)</span>
-<span class="sd">        &gt;&gt;&gt; A = Matrix([[1, 1], [1, 1]])</span>
-<span class="sd">        &gt;&gt;&gt; A.norm(2)# Spectral norm (max of |Ax|/|x| under 2-vector-norm)</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; A.norm(-2) # Inverse spectral norm (smallest singular value)</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; A.norm() # Frobenius Norm</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; Matrix([1, -2]).norm(oo)</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; Matrix([-1, 2]).norm(-oo)</span>
-<span class="sd">        1</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        normalized</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># Row or Column Vector Norms</span>
-        <span class="n">vals</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">values</span><span class="p">())</span> <span class="ow">or</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">ord</span> <span class="o">==</span> <span class="mi">2</span> <span class="ow">or</span> <span class="nb">ord</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>  <span class="c"># Common case sqrt(&lt;x, x&gt;)</span>
-                <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">vals</span><span class="p">)))</span>
-
-            <span class="k">elif</span> <span class="nb">ord</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>  <span class="c"># sum(abs(x))</span>
-                <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">vals</span><span class="p">))</span>
-
-            <span class="k">elif</span> <span class="nb">ord</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>  <span class="c"># max(abs(x))</span>
-                <span class="k">return</span> <span class="n">Max</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="nb">abs</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">vals</span><span class="p">])</span>
-
-            <span class="k">elif</span> <span class="nb">ord</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>  <span class="c"># min(abs(x))</span>
-                <span class="k">return</span> <span class="n">Min</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="nb">abs</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">vals</span><span class="p">])</span>
-
-            <span class="c"># Otherwise generalize the 2-norm, Sum(x_i**ord)**(1/ord)</span>
-            <span class="c"># Note that while useful this is not mathematically a norm</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Pow</span><span class="p">(</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">**</span><span class="nb">ord</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">vals</span><span class="p">)),</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="nb">ord</span><span class="p">)</span>
-            <span class="k">except</span> <span class="p">(</span><span class="ne">NotImplementedError</span><span class="p">,</span> <span class="ne">TypeError</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Expected order to be Number, Symbol, oo&quot;</span><span class="p">)</span>
-
-        <span class="c"># Matrix Norms</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">ord</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>  <span class="c"># Spectral Norm</span>
-                <span class="c"># Maximum singular value</span>
-                <span class="k">return</span> <span class="n">Max</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">singular_values</span><span class="p">())</span>
-
-            <span class="k">elif</span> <span class="nb">ord</span> <span class="o">==</span> <span class="o">-</span><span class="mi">2</span><span class="p">:</span>
-                <span class="c"># Minimum singular value</span>
-                <span class="k">return</span> <span class="n">Min</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">singular_values</span><span class="p">())</span>
-
-            <span class="k">elif</span> <span class="p">(</span><span class="nb">ord</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="nb">ord</span><span class="p">,</span> <span class="nb">str</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">ord</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="ow">in</span>
-                    <span class="p">[</span><span class="s">&#39;f&#39;</span><span class="p">,</span> <span class="s">&#39;fro&#39;</span><span class="p">,</span> <span class="s">&#39;frobenius&#39;</span><span class="p">,</span> <span class="s">&#39;vector&#39;</span><span class="p">]):</span>
-                <span class="c"># Reshape as vector and send back to norm function</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">vec</span><span class="p">()</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="nb">ord</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
-
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Matrix Norms under development&quot;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.normalized"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.normalized">[docs]</a>    <span class="k">def</span> <span class="nf">normalized</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the normalized version of ``self``.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        norm</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">!=</span> <span class="mi">1</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ShapeError</span><span class="p">(</span><span class="s">&quot;A Matrix must be a vector to normalize.&quot;</span><span class="p">)</span>
-        <span class="n">norm</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">norm</span><span class="p">()</span>
-        <span class="n">out</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">applyfunc</span><span class="p">(</span><span class="k">lambda</span> <span class="n">i</span><span class="p">:</span> <span class="n">i</span> <span class="o">/</span> <span class="n">norm</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">out</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.project"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.project">[docs]</a>    <span class="k">def</span> <span class="nf">project</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">v</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the projection of ``self`` onto the line containing ``v``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, S, sqrt</span>
-<span class="sd">        &gt;&gt;&gt; V = Matrix([sqrt(3)/2, S.Half])</span>
-<span class="sd">        &gt;&gt;&gt; x = Matrix([[1, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; V.project(x)</span>
-<span class="sd">        [sqrt(3)/2, 0]</span>
-<span class="sd">        &gt;&gt;&gt; V.project(-x)</span>
-<span class="sd">        [sqrt(3)/2, 0]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">v</span><span class="o">*</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="o">/</span> <span class="n">v</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">v</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.permuteBkwd"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.permuteBkwd">[docs]</a>    <span class="k">def</span> <span class="nf">permuteBkwd</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">perm</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Permute the rows of the matrix with the given permutation in reverse.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import eye</span>
-<span class="sd">        &gt;&gt;&gt; M = eye(3)</span>
-<span class="sd">        &gt;&gt;&gt; M.permuteBkwd([[0, 1], [0, 2]])</span>
-<span class="sd">        [0, 1, 0]</span>
-<span class="sd">        [0, 0, 1]</span>
-<span class="sd">        [1, 0, 0]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        permuteFwd</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">copy</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">perm</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-            <span class="n">copy</span><span class="o">.</span><span class="n">row_swap</span><span class="p">(</span><span class="n">perm</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">perm</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">copy</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.permuteFwd"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.permuteFwd">[docs]</a>    <span class="k">def</span> <span class="nf">permuteFwd</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">perm</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Permute the rows of the matrix with the given permutation.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import eye</span>
-<span class="sd">        &gt;&gt;&gt; M = eye(3)</span>
-<span class="sd">        &gt;&gt;&gt; M.permuteFwd([[0, 1], [0, 2]])</span>
-<span class="sd">        [0, 0, 1]</span>
-<span class="sd">        [1, 0, 0]</span>
-<span class="sd">        [0, 1, 0]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        permuteBkwd</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">copy</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">perm</span><span class="p">)):</span>
-            <span class="n">copy</span><span class="o">.</span><span class="n">row_swap</span><span class="p">(</span><span class="n">perm</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">perm</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">copy</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.exp"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.exp">[docs]</a>    <span class="k">def</span> <span class="nf">exp</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the exponentiation of a square matrix.&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">NonSquareMatrixError</span><span class="p">(</span>
-                <span class="s">&quot;Exponentiation is valid only for square matrices&quot;</span><span class="p">)</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">U</span><span class="p">,</span> <span class="n">D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">diagonalize</span><span class="p">()</span>
-        <span class="k">except</span> <span class="n">MatrixError</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Exponentiation is implemented only for diagonalizable matrices&quot;</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">D</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="n">D</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">D</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">U</span><span class="o">*</span><span class="n">D</span><span class="o">*</span><span class="n">U</span><span class="o">.</span><span class="n">inv</span><span class="p">()</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="MatrixBase.is_square"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_square">[docs]</a>    <span class="k">def</span> <span class="nf">is_square</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Checks if a matrix is square.</span>
-
-<span class="sd">        A matrix is square if the number of rows equals the number of columns.</span>
-<span class="sd">        The empty matrix is square by definition, since the number of rows and</span>
-<span class="sd">        the number of columns are both zero.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; a = Matrix([[1, 2, 3], [4, 5, 6]])</span>
-<span class="sd">        &gt;&gt;&gt; b = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])</span>
-<span class="sd">        &gt;&gt;&gt; c = Matrix([])</span>
-<span class="sd">        &gt;&gt;&gt; a.is_square</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; b.is_square</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; c.is_square</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="MatrixBase.is_zero"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_zero">[docs]</a>    <span class="k">def</span> <span class="nf">is_zero</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Checks if a matrix is a zero matrix.</span>
-
-<span class="sd">        A matrix is zero if every element is zero.  A matrix need not be square</span>
-<span class="sd">        to be considered zero.  The empty matrix is zero by the principle of</span>
-<span class="sd">        vacuous truth.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, zeros</span>
-<span class="sd">        &gt;&gt;&gt; a = Matrix([[0, 0], [0, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; b = zeros(3, 4)</span>
-<span class="sd">        &gt;&gt;&gt; c = Matrix([[0, 1], [0, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; d = Matrix([])</span>
-<span class="sd">        &gt;&gt;&gt; a.is_zero</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; b.is_zero</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; c.is_zero</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; d.is_zero</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">values</span><span class="p">())</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.is_nilpotent"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_nilpotent">[docs]</a>    <span class="k">def</span> <span class="nf">is_nilpotent</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Checks if a matrix is nilpotent.</span>
-
-<span class="sd">        A matrix B is nilpotent if for some integer k, B**k is</span>
-<span class="sd">        a zero matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; a = Matrix([[0, 0, 0], [1, 0, 0], [1, 1, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; a.is_nilpotent()</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &gt;&gt;&gt; a = Matrix([[1, 0, 1], [1, 0, 0], [1, 1, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; a.is_nilpotent()</span>
-<span class="sd">        False</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">NonSquareMatrixError</span><span class="p">(</span>
-                <span class="s">&quot;Nilpotency is valid only for square matrices&quot;</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">charpoly</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">x</span><span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="bp">False</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="MatrixBase.is_upper"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_upper">[docs]</a>    <span class="k">def</span> <span class="nf">is_upper</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Check if matrix is an upper triangular matrix. True can be returned</span>
-<span class="sd">        even if the matrix is not square.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix(2, 2, [1, 0, 0, 1])</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [1, 0]</span>
-<span class="sd">        [0, 1]</span>
-<span class="sd">        &gt;&gt;&gt; m.is_upper</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &gt;&gt;&gt; m = Matrix(4, 3, [5, 1, 9, 0, 4 , 6, 0, 0, 5, 0, 0, 0])</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [5, 1, 9]</span>
-<span class="sd">        [0, 4, 6]</span>
-<span class="sd">        [0, 0, 5]</span>
-<span class="sd">        [0, 0, 0]</span>
-<span class="sd">        &gt;&gt;&gt; m.is_upper</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &gt;&gt;&gt; m = Matrix(2, 3, [4, 2, 5, 6, 1, 1])</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [4, 2, 5]</span>
-<span class="sd">        [6, 1, 1]</span>
-<span class="sd">        &gt;&gt;&gt; m.is_upper</span>
-<span class="sd">        False</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        is_lower</span>
-<span class="sd">        is_diagonal</span>
-<span class="sd">        is_upper_hessenberg</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">is_zero</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="MatrixBase.is_lower"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_lower">[docs]</a>    <span class="k">def</span> <span class="nf">is_lower</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Check if matrix is a lower triangular matrix. True can be returned</span>
-<span class="sd">        even if the matrix is not square.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix(2, 2, [1, 0, 0, 1])</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [1, 0]</span>
-<span class="sd">        [0, 1]</span>
-<span class="sd">        &gt;&gt;&gt; m.is_lower</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &gt;&gt;&gt; m = Matrix(4, 3, [0, 0, 0, 2, 0, 0, 1, 4 , 0, 6, 6, 5])</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [0, 0, 0]</span>
-<span class="sd">        [2, 0, 0]</span>
-<span class="sd">        [1, 4, 0]</span>
-<span class="sd">        [6, 6, 5]</span>
-<span class="sd">        &gt;&gt;&gt; m.is_lower</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix(2, 2, [x**2 + y, y**2 + x, 0, x + y])</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [x**2 + y, x + y**2]</span>
-<span class="sd">        [       0,    x + y]</span>
-<span class="sd">        &gt;&gt;&gt; m.is_lower</span>
-<span class="sd">        False</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        is_upper</span>
-<span class="sd">        is_diagonal</span>
-<span class="sd">        is_lower_hessenberg</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">is_zero</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">))</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="MatrixBase.is_upper_hessenberg"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_upper_hessenberg">[docs]</a>    <span class="k">def</span> <span class="nf">is_upper_hessenberg</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Checks if the matrix is the upper-Hessenberg form.</span>
-
-<span class="sd">        The upper hessenberg matrix has zero entries</span>
-<span class="sd">        below the first subdiagonal.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; a = Matrix([[1, 4, 2, 3], [3, 4, 1, 7], [0, 2, 3, 4], [0, 0, 1, 3]])</span>
-<span class="sd">        &gt;&gt;&gt; a</span>
-<span class="sd">        [1, 4, 2, 3]</span>
-<span class="sd">        [3, 4, 1, 7]</span>
-<span class="sd">        [0, 2, 3, 4]</span>
-<span class="sd">        [0, 0, 1, 3]</span>
-<span class="sd">        &gt;&gt;&gt; a.is_upper_hessenberg</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        is_lower_hessenberg</span>
-<span class="sd">        is_upper</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">is_zero</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="MatrixBase.is_lower_hessenberg"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_lower_hessenberg">[docs]</a>    <span class="k">def</span> <span class="nf">is_lower_hessenberg</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;Checks if the matrix is in the lower-Hessenberg form.</span>
-
-<span class="sd">        The lower hessenberg matrix has zero entries</span>
-<span class="sd">        above the first superdiagonal.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; a = Matrix([[1, 2, 0, 0], [5, 2, 3, 0], [3, 4, 3, 7], [5, 6, 1, 1]])</span>
-<span class="sd">        &gt;&gt;&gt; a</span>
-<span class="sd">        [1, 2, 0, 0]</span>
-<span class="sd">        [5, 2, 3, 0]</span>
-<span class="sd">        [3, 4, 3, 7]</span>
-<span class="sd">        [5, 6, 1, 1]</span>
-<span class="sd">        &gt;&gt;&gt; a.is_lower_hessenberg</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        is_upper_hessenberg</span>
-<span class="sd">        is_lower</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">is_zero</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">2</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.is_symbolic"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_symbolic">[docs]</a>    <span class="k">def</span> <span class="nf">is_symbolic</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Checks if any elements contain Symbols.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; M = Matrix([[x, y], [1, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; M.is_symbolic()</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">any</span><span class="p">(</span><span class="n">element</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Symbol</span><span class="p">)</span> <span class="k">for</span> <span class="n">element</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">values</span><span class="p">()))</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.is_symmetric"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_symmetric">[docs]</a>    <span class="k">def</span> <span class="nf">is_symmetric</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">simplify</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Check if matrix is symmetric matrix,</span>
-<span class="sd">        that is square matrix and is equal to its transpose.</span>
-
-<span class="sd">        By default, simplifications occur before testing symmetry.</span>
-<span class="sd">        They can be skipped using &#39;simplify=False&#39;; while speeding things a bit,</span>
-<span class="sd">        this may however induce false negatives.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix(2, 2, [0, 1, 1, 2])</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [0, 1]</span>
-<span class="sd">        [1, 2]</span>
-<span class="sd">        &gt;&gt;&gt; m.is_symmetric()</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &gt;&gt;&gt; m = Matrix(2, 2, [0, 1, 2, 0])</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [0, 1]</span>
-<span class="sd">        [2, 0]</span>
-<span class="sd">        &gt;&gt;&gt; m.is_symmetric()</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &gt;&gt;&gt; m = Matrix(2, 3, [0, 0, 0, 0, 0, 0])</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [0, 0, 0]</span>
-<span class="sd">        [0, 0, 0]</span>
-<span class="sd">        &gt;&gt;&gt; m.is_symmetric()</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix(3, 3, [1, x**2 + 2*x + 1, y, (x + 1)**2 , 2, 0, y, 0, 3])</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [         1, x**2 + 2*x + 1, y]</span>
-<span class="sd">        [(x + 1)**2,              2, 0]</span>
-<span class="sd">        [         y,              0, 3]</span>
-<span class="sd">        &gt;&gt;&gt; m.is_symmetric()</span>
-<span class="sd">        True</span>
-
-<span class="sd">        If the matrix is already simplified, you may speed-up is_symmetric()</span>
-<span class="sd">        test by using &#39;simplify=False&#39;.</span>
-
-<span class="sd">        &gt;&gt;&gt; m.is_symmetric(simplify=False)</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; m1 = m.expand()</span>
-<span class="sd">        &gt;&gt;&gt; m1.is_symmetric(simplify=False)</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">simplify</span><span class="p">:</span>
-            <span class="n">delta</span> <span class="o">=</span> <span class="bp">self</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span>
-            <span class="n">delta</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-            <span class="k">return</span> <span class="n">delta</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.is_anti_symmetric"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_anti_symmetric">[docs]</a>    <span class="k">def</span> <span class="nf">is_anti_symmetric</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">simplify</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Check if matrix M is an antisymmetric matrix,</span>
-<span class="sd">        that is, M is a square matrix with all M[i, j] == -M[j, i].</span>
-
-<span class="sd">        When ``simplify=True`` (default), the sum M[i, j] + M[j, i] is</span>
-<span class="sd">        simplified before testing to see if it is zero. By default,</span>
-<span class="sd">        the SymPy simplify function is used. To use a custom function</span>
-<span class="sd">        set simplify to a function that accepts a single argument which</span>
-<span class="sd">        returns a simplified expression. To skip simplification, set</span>
-<span class="sd">        simplify to False but note that although this will be faster,</span>
-<span class="sd">        it may induce false negatives.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, symbols</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix(2, 2, [0, 1, -1, 0])</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [ 0, 1]</span>
-<span class="sd">        [-1, 0]</span>
-<span class="sd">        &gt;&gt;&gt; m.is_anti_symmetric()</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; x, y = symbols(&#39;x y&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix(2, 3, [0, 0, x, -y, 0, 0])</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [ 0, 0, x]</span>
-<span class="sd">        [-y, 0, 0]</span>
-<span class="sd">        &gt;&gt;&gt; m.is_anti_symmetric()</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix(3, 3, [0, x**2 + 2*x + 1, y,</span>
-<span class="sd">        ...                   -(x + 1)**2 , 0, x*y,</span>
-<span class="sd">        ...                   -y, -x*y, 0])</span>
-
-<span class="sd">        Simplification of matrix elements is done by default so even</span>
-<span class="sd">        though two elements which should be equal and opposite wouldn&#39;t</span>
-<span class="sd">        pass an equality test, the matrix is still reported as</span>
-<span class="sd">        anti-symmetric:</span>
-
-<span class="sd">        &gt;&gt;&gt; m[0, 1] == -m[1, 0]</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; m.is_anti_symmetric()</span>
-<span class="sd">        True</span>
-
-<span class="sd">        If &#39;simplify=False&#39; is used for the case when a Matrix is already</span>
-<span class="sd">        simplified, this will speed things up. Here, we see that without</span>
-<span class="sd">        simplification the matrix does not appear anti-symmetric:</span>
-
-<span class="sd">        &gt;&gt;&gt; m.is_anti_symmetric(simplify=False)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        But if the matrix were already expanded, then it would appear</span>
-<span class="sd">        anti-symmetric and simplification in the is_anti_symmetric routine</span>
-<span class="sd">        is not needed:</span>
-
-<span class="sd">        &gt;&gt;&gt; m = m.expand()</span>
-<span class="sd">        &gt;&gt;&gt; m.is_anti_symmetric(simplify=False)</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># accept custom simplification</span>
-        <span class="n">simpfunc</span> <span class="o">=</span> <span class="n">simplify</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">simplify</span><span class="p">,</span> <span class="n">FunctionType</span><span class="p">)</span> <span class="k">else</span> \
-            <span class="n">_simplify</span> <span class="k">if</span> <span class="n">simplify</span> <span class="k">else</span> <span class="bp">False</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span>
-        <span class="k">if</span> <span class="n">simplify</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-                <span class="c"># diagonal</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">simpfunc</span><span class="p">(</span><span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">])</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-                <span class="c"># others</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-                    <span class="n">diff</span> <span class="o">=</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">+</span> <span class="bp">self</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">simpfunc</span><span class="p">(</span><span class="n">diff</span><span class="p">)</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">!=</span> <span class="o">-</span><span class="bp">self</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">]:</span>
-                        <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">return</span> <span class="bp">True</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.is_diagonal"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_diagonal">[docs]</a>    <span class="k">def</span> <span class="nf">is_diagonal</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Check if matrix is diagonal,</span>
-<span class="sd">        that is matrix in which the entries outside the main diagonal are all zero.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, diag</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix(2, 2, [1, 0, 0, 2])</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [1, 0]</span>
-<span class="sd">        [0, 2]</span>
-<span class="sd">        &gt;&gt;&gt; m.is_diagonal()</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &gt;&gt;&gt; m = Matrix(2, 2, [1, 1, 0, 2])</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [1, 1]</span>
-<span class="sd">        [0, 2]</span>
-<span class="sd">        &gt;&gt;&gt; m.is_diagonal()</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &gt;&gt;&gt; m = diag(1, 2, 3)</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [1, 0, 0]</span>
-<span class="sd">        [0, 2, 0]</span>
-<span class="sd">        [0, 0, 3]</span>
-<span class="sd">        &gt;&gt;&gt; m.is_diagonal()</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        is_lower</span>
-<span class="sd">        is_upper</span>
-<span class="sd">        is_diagonalizable</span>
-<span class="sd">        diagonalize</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="n">j</span> <span class="ow">and</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="bp">True</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.det"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.det">[docs]</a>    <span class="k">def</span> <span class="nf">det</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">&quot;bareis&quot;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes the matrix determinant using the method &quot;method&quot;.</span>
-
-<span class="sd">        Possible values for &quot;method&quot;:</span>
-<span class="sd">          bareis ... det_bareis</span>
-<span class="sd">          berkowitz ... berkowitz_det</span>
-<span class="sd">          lu_decomposition ... det_LU</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        det_bareis</span>
-<span class="sd">        berkowitz_det</span>
-<span class="sd">        det_LU</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="c"># if methods were made internal and all determinant calculations</span>
-        <span class="c"># passed through here, then these lines could be factored out of</span>
-        <span class="c"># the method routines</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">NonSquareMatrixError</span><span class="p">()</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">if</span> <span class="n">method</span> <span class="o">==</span> <span class="s">&quot;bareis&quot;</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">det_bareis</span><span class="p">()</span>
-        <span class="k">elif</span> <span class="n">method</span> <span class="o">==</span> <span class="s">&quot;berkowitz&quot;</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">berkowitz_det</span><span class="p">()</span>
-        <span class="k">elif</span> <span class="n">method</span> <span class="o">==</span> <span class="s">&quot;det_LU&quot;</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">det_LU_decomposition</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Determinant method &#39;</span><span class="si">%s</span><span class="s">&#39; unrecognized&quot;</span> <span class="o">%</span> <span class="n">method</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.det_bareis"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.det_bareis">[docs]</a>    <span class="k">def</span> <span class="nf">det_bareis</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute matrix determinant using Bareis&#39; fraction-free</span>
-<span class="sd">        algorithm which is an extension of the well known Gaussian</span>
-<span class="sd">        elimination method. This approach is best suited for dense</span>
-<span class="sd">        symbolic matrices and will result in a determinant with</span>
-<span class="sd">        minimal number of fractions. It means that less term</span>
-<span class="sd">        rewriting is needed on resulting formulae.</span>
-
-<span class="sd">        TODO: Implement algorithm for sparse matrices (SFF),</span>
-<span class="sd">        http://www.eecis.udel.edu/~saunders/papers/sffge/it5.ps.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        det</span>
-<span class="sd">        berkowitz_det</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">NonSquareMatrixError</span><span class="p">()</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-        <span class="n">M</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">copy</span><span class="p">(),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span>
-
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">det</span> <span class="o">=</span> <span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-        <span class="k">elif</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">det</span> <span class="o">=</span> <span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">M</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">M</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">sign</span> <span class="o">=</span> <span class="mi">1</span>  <span class="c"># track current sign in case of column swap</span>
-
-            <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="c"># look for a pivot in the current column</span>
-                <span class="c"># and assume det == 0 if none is found</span>
-                <span class="k">if</span> <span class="n">M</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-                        <span class="k">if</span> <span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">]:</span>
-                            <span class="n">M</span><span class="o">.</span><span class="n">row_swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-                            <span class="n">sign</span> <span class="o">*=</span> <span class="o">-</span><span class="mi">1</span>
-                            <span class="k">break</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-                <span class="c"># proceed with Bareis&#39; fraction-free (FF)</span>
-                <span class="c"># form of Gaussian elimination algorithm</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-                    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-                        <span class="n">D</span> <span class="o">=</span> <span class="n">M</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">M</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-
-                        <span class="k">if</span> <span class="n">k</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                            <span class="n">D</span> <span class="o">/=</span> <span class="n">M</span><span class="p">[</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-
-                        <span class="k">if</span> <span class="n">D</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-                            <span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">D</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">cancel</span><span class="p">(</span><span class="n">D</span><span class="p">)</span>
-
-            <span class="n">det</span> <span class="o">=</span> <span class="n">sign</span><span class="o">*</span><span class="n">M</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-
-        <span class="k">return</span> <span class="n">det</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.det_LU_decomposition"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.det_LU_decomposition">[docs]</a>    <span class="k">def</span> <span class="nf">det_LU_decomposition</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute matrix determinant using LU decomposition</span>
-
-<span class="sd">        Note that this method fails if the LU decomposition itself</span>
-<span class="sd">        fails. In particular, if the matrix has no inverse this method</span>
-<span class="sd">        will fail.</span>
-
-<span class="sd">        TODO: Implement algorithm for sparse matrices (SFF),</span>
-<span class="sd">        http://www.eecis.udel.edu/~saunders/papers/sffge/it5.ps.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        det</span>
-<span class="sd">        det_bareis</span>
-<span class="sd">        berkowitz_det</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">NonSquareMatrixError</span><span class="p">()</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-        <span class="n">M</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">copy</span><span class="p">(),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span>
-        <span class="n">p</span><span class="p">,</span> <span class="n">prod</span> <span class="o">=</span> <span class="p">[],</span> <span class="mi">1</span>
-        <span class="n">l</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">LUdecomposition</span><span class="p">()</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">prod</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="n">prod</span> <span class="o">=</span> <span class="n">prod</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">l</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span>
-
-        <span class="k">return</span> <span class="n">prod</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.adjugate"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.adjugate">[docs]</a>    <span class="k">def</span> <span class="nf">adjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">&quot;berkowitz&quot;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the adjugate matrix.</span>
-
-<span class="sd">        Adjugate matrix is the transpose of the cofactor matrix.</span>
-
-<span class="sd">        http://en.wikipedia.org/wiki/Adjugate</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        cofactorMatrix</span>
-<span class="sd">        transpose</span>
-<span class="sd">        berkowitz</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">cofactorMatrix</span><span class="p">(</span><span class="n">method</span><span class="p">)</span><span class="o">.</span><span class="n">T</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.inverse_LU"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.inverse_LU">[docs]</a>    <span class="k">def</span> <span class="nf">inverse_LU</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">iszerofunc</span><span class="o">=</span><span class="n">_iszero</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Calculates the inverse using LU decomposition.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        inv</span>
-<span class="sd">        inverse_GE</span>
-<span class="sd">        inverse_ADJ</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">NonSquareMatrixError</span><span class="p">()</span>
-
-        <span class="n">ok</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rref</span><span class="p">(</span><span class="n">simplify</span><span class="o">=</span><span class="bp">True</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">iszerofunc</span><span class="p">(</span><span class="n">ok</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">])</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">ok</span><span class="o">.</span><span class="n">rows</span><span class="p">)):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Matrix det == 0; not invertible.&quot;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">LUsolve</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">),</span> <span class="n">iszerofunc</span><span class="o">=</span><span class="n">_iszero</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.inverse_GE"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.inverse_GE">[docs]</a>    <span class="k">def</span> <span class="nf">inverse_GE</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">iszerofunc</span><span class="o">=</span><span class="n">_iszero</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Calculates the inverse using Gaussian elimination.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        inv</span>
-<span class="sd">        inverse_LU</span>
-<span class="sd">        inverse_ADJ</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">.dense</span> <span class="kn">import</span> <span class="n">Matrix</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">NonSquareMatrixError</span><span class="p">(</span><span class="s">&quot;A Matrix must be square to invert.&quot;</span><span class="p">)</span>
-
-        <span class="n">big</span> <span class="o">=</span> <span class="n">Matrix</span><span class="o">.</span><span class="n">hstack</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">as_mutable</span><span class="p">(),</span> <span class="n">Matrix</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">))</span>
-        <span class="n">red</span> <span class="o">=</span> <span class="n">big</span><span class="o">.</span><span class="n">rref</span><span class="p">(</span><span class="n">iszerofunc</span><span class="o">=</span><span class="n">iszerofunc</span><span class="p">,</span> <span class="n">simplify</span><span class="o">=</span><span class="bp">True</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">iszerofunc</span><span class="p">(</span><span class="n">red</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">])</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">red</span><span class="o">.</span><span class="n">rows</span><span class="p">)):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Matrix det == 0; not invertible.&quot;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">red</span><span class="p">[:,</span> <span class="n">big</span><span class="o">.</span><span class="n">rows</span><span class="p">:])</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.inverse_ADJ"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.inverse_ADJ">[docs]</a>    <span class="k">def</span> <span class="nf">inverse_ADJ</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">iszerofunc</span><span class="o">=</span><span class="n">_iszero</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Calculates the inverse using the adjugate matrix and a determinant.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        inv</span>
-<span class="sd">        inverse_LU</span>
-<span class="sd">        inverse_GE</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">NonSquareMatrixError</span><span class="p">(</span><span class="s">&quot;A Matrix must be square to invert.&quot;</span><span class="p">)</span>
-
-        <span class="n">d</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">berkowitz_det</span><span class="p">()</span>
-        <span class="n">zero</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">zero</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="c"># if equals() can&#39;t decide, will rref be able to?</span>
-            <span class="n">ok</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rref</span><span class="p">(</span><span class="n">simplify</span><span class="o">=</span><span class="bp">True</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">zero</span> <span class="o">=</span> <span class="nb">any</span><span class="p">(</span><span class="n">iszerofunc</span><span class="p">(</span><span class="n">ok</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">])</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">ok</span><span class="o">.</span><span class="n">rows</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">zero</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Matrix det == 0; not invertible.&quot;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">adjugate</span><span class="p">()</span> <span class="o">/</span> <span class="n">d</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.rref"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.rref">[docs]</a>    <span class="k">def</span> <span class="nf">rref</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">simplified</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">iszerofunc</span><span class="o">=</span><span class="n">_iszero</span><span class="p">,</span>
-            <span class="n">simplify</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return reduced row-echelon form of matrix and indices of pivot vars.</span>
-
-<span class="sd">        To simplify elements before finding nonzero pivots set simplify=True</span>
-<span class="sd">        (to use the default SymPy simplify function) or pass a custom</span>
-<span class="sd">        simplify function.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix([[1, 2], [x, 1 - 1/x]])</span>
-<span class="sd">        &gt;&gt;&gt; m.rref()</span>
-<span class="sd">        ([1, 0]</span>
-<span class="sd">        [0, 1], [0, 1])</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">simplified</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="n">SymPyDeprecationWarning</span><span class="p">(</span>
-                <span class="n">feature</span><span class="o">=</span><span class="s">&quot;&#39;simplified&#39; as a keyword to rref&quot;</span><span class="p">,</span>
-                <span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;simplify=True, or set simplify equal to your &quot;</span>
-                <span class="s">&quot;own custom simplification function&quot;</span><span class="p">,</span>
-                <span class="n">issue</span><span class="o">=</span><span class="mi">3382</span><span class="p">,</span> <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.2&quot;</span><span class="p">,</span>
-            <span class="p">)</span><span class="o">.</span><span class="n">warn</span><span class="p">()</span>
-            <span class="n">simplify</span> <span class="o">=</span> <span class="n">simplify</span> <span class="ow">or</span> <span class="bp">True</span>
-        <span class="n">simpfunc</span> <span class="o">=</span> <span class="n">simplify</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span>
-            <span class="n">simplify</span><span class="p">,</span> <span class="n">FunctionType</span><span class="p">)</span> <span class="k">else</span> <span class="n">_simplify</span>
-        <span class="c"># pivot: index of next row to contain a pivot</span>
-        <span class="n">pivot</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_mutable</span><span class="p">()</span>
-        <span class="c"># pivotlist: indices of pivot variables (non-free)</span>
-        <span class="n">pivotlist</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">r</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">pivot</span> <span class="o">==</span> <span class="n">r</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-                <span class="k">break</span>
-            <span class="k">if</span> <span class="n">simplify</span><span class="p">:</span>
-                <span class="n">r</span><span class="p">[</span><span class="n">pivot</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">simpfunc</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">pivot</span><span class="p">,</span> <span class="n">i</span><span class="p">])</span>
-            <span class="k">if</span> <span class="n">iszerofunc</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">pivot</span><span class="p">,</span> <span class="n">i</span><span class="p">]):</span>
-                <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">pivot</span><span class="p">,</span> <span class="n">r</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">simplify</span> <span class="ow">and</span> <span class="n">k</span> <span class="o">&gt;</span> <span class="n">pivot</span><span class="p">:</span>
-                        <span class="n">r</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">simpfunc</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">i</span><span class="p">])</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">iszerofunc</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">i</span><span class="p">]):</span>
-                        <span class="k">break</span>
-                <span class="k">if</span> <span class="n">k</span> <span class="o">==</span> <span class="n">r</span><span class="o">.</span><span class="n">rows</span> <span class="o">-</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">iszerofunc</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">i</span><span class="p">]):</span>
-                    <span class="k">continue</span>
-                <span class="n">r</span><span class="o">.</span><span class="n">row_swap</span><span class="p">(</span><span class="n">pivot</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-            <span class="n">scale</span> <span class="o">=</span> <span class="n">r</span><span class="p">[</span><span class="n">pivot</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span>
-            <span class="n">r</span><span class="o">.</span><span class="n">row_op</span><span class="p">(</span><span class="n">pivot</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">_</span><span class="p">:</span> <span class="n">x</span> <span class="o">/</span> <span class="n">scale</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">r</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">j</span> <span class="o">==</span> <span class="n">pivot</span><span class="p">:</span>
-                    <span class="k">continue</span>
-                <span class="n">scale</span> <span class="o">=</span> <span class="n">r</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span>
-                <span class="n">r</span><span class="o">.</span><span class="n">row_op</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">:</span> <span class="n">x</span> <span class="o">-</span> <span class="n">scale</span><span class="o">*</span><span class="n">r</span><span class="p">[</span><span class="n">pivot</span><span class="p">,</span> <span class="n">k</span><span class="p">])</span>
-            <span class="n">pivotlist</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="n">pivot</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">r</span><span class="p">),</span> <span class="n">pivotlist</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.nullspace"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.nullspace">[docs]</a>    <span class="k">def</span> <span class="nf">nullspace</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">simplified</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">simplify</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns list of vectors (Matrix objects) that span nullspace of self</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">zeros</span>
-
-        <span class="k">if</span> <span class="n">simplified</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="n">SymPyDeprecationWarning</span><span class="p">(</span>
-                <span class="n">feature</span><span class="o">=</span><span class="s">&quot;&#39;simplified&#39; as a keyword to nullspace&quot;</span><span class="p">,</span>
-                <span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;simplify=True, or set simplify equal to your &quot;</span>
-                <span class="s">&quot;own custom simplification function&quot;</span><span class="p">,</span>
-                <span class="n">issue</span><span class="o">=</span><span class="mi">3382</span><span class="p">,</span> <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.2&quot;</span><span class="p">,</span>
-            <span class="p">)</span><span class="o">.</span><span class="n">warn</span><span class="p">()</span>
-            <span class="n">simplify</span> <span class="o">=</span> <span class="n">simplify</span> <span class="ow">or</span> <span class="bp">True</span>
-        <span class="n">simpfunc</span> <span class="o">=</span> <span class="n">simplify</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span>
-            <span class="n">simplify</span><span class="p">,</span> <span class="n">FunctionType</span><span class="p">)</span> <span class="k">else</span> <span class="n">_simplify</span>
-        <span class="n">reduced</span><span class="p">,</span> <span class="n">pivots</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rref</span><span class="p">(</span><span class="n">simplify</span><span class="o">=</span><span class="n">simpfunc</span><span class="p">)</span>
-
-        <span class="n">basis</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="c"># create a set of vectors for the basis</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">pivots</span><span class="p">)):</span>
-            <span class="n">basis</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-        <span class="c"># contains the variable index to which the vector corresponds</span>
-        <span class="n">basiskey</span><span class="p">,</span> <span class="n">cur</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="n">basis</span><span class="p">),</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">pivots</span><span class="p">:</span>
-                <span class="n">basiskey</span><span class="p">[</span><span class="n">cur</span><span class="p">]</span> <span class="o">=</span> <span class="n">i</span>
-                <span class="n">cur</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">pivots</span><span class="p">:</span>  <span class="c"># free var, just set vector&#39;s ith place to 1</span>
-                <span class="n">basis</span><span class="p">[</span><span class="n">basiskey</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">i</span><span class="p">)][</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="k">else</span><span class="p">:</span>               <span class="c"># add negative of nonpivot entry to corr vector</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-                    <span class="n">line</span> <span class="o">=</span> <span class="n">pivots</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                    <span class="n">v</span> <span class="o">=</span> <span class="n">reduced</span><span class="p">[</span><span class="n">line</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-                    <span class="k">if</span> <span class="n">simplify</span><span class="p">:</span>
-                        <span class="n">v</span> <span class="o">=</span> <span class="n">simpfunc</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">v</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">pivots</span><span class="p">:</span>
-                            <span class="c"># XXX: Is this the correct error?</span>
-                            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                                <span class="s">&quot;Could not compute the nullspace of `self`.&quot;</span><span class="p">)</span>
-                        <span class="n">basis</span><span class="p">[</span><span class="n">basiskey</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">j</span><span class="p">)][</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">v</span>
-        <span class="k">return</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">basis</span><span class="p">]</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.berkowitz"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.berkowitz">[docs]</a>    <span class="k">def</span> <span class="nf">berkowitz</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The Berkowitz algorithm.</span>
-
-<span class="sd">           Given N x N matrix with symbolic content, compute efficiently</span>
-<span class="sd">           coefficients of characteristic polynomials of &#39;self&#39; and all</span>
-<span class="sd">           its square sub-matrices composed by removing both i-th row</span>
-<span class="sd">           and column, without division in the ground domain.</span>
-
-<span class="sd">           This method is particularly useful for computing determinant,</span>
-<span class="sd">           principal minors and characteristic polynomial, when &#39;self&#39;</span>
-<span class="sd">           has complicated coefficients e.g. polynomials. Semi-direct</span>
-<span class="sd">           usage of this algorithm is also important in computing</span>
-<span class="sd">           efficiently sub-resultant PRS.</span>
-
-<span class="sd">           Assuming that M is a square matrix of dimension N x N and</span>
-<span class="sd">           I is N x N identity matrix,  then the following following</span>
-<span class="sd">           definition of characteristic polynomial is begin used:</span>
-
-<span class="sd">                          charpoly(M) = det(t*I - M)</span>
-
-<span class="sd">           As a consequence, all polynomials generated by Berkowitz</span>
-<span class="sd">           algorithm are monic.</span>
-
-<span class="sd">           &gt;&gt;&gt; from sympy import Matrix</span>
-<span class="sd">           &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-
-<span class="sd">           &gt;&gt;&gt; M = Matrix([[x, y, z], [1, 0, 0], [y, z, x]])</span>
-
-<span class="sd">           &gt;&gt;&gt; p, q, r = M.berkowitz()</span>
-
-<span class="sd">           &gt;&gt;&gt; p # 1 x 1 M&#39;s sub-matrix</span>
-<span class="sd">           (1, -x)</span>
-
-<span class="sd">           &gt;&gt;&gt; q # 2 x 2 M&#39;s sub-matrix</span>
-<span class="sd">           (1, -x, -y)</span>
-
-<span class="sd">           &gt;&gt;&gt; r # 3 x 3 M&#39;s sub-matrix</span>
-<span class="sd">           (1, -2*x, x**2 - y*z - y, x*y - z**2)</span>
-
-<span class="sd">           For more information on the implemented algorithm refer to:</span>
-
-<span class="sd">           [1] S.J. Berkowitz, On computing the determinant in small</span>
-<span class="sd">               parallel time using a small number of processors, ACM,</span>
-<span class="sd">               Information Processing Letters 18, 1984, pp. 147-150</span>
-
-<span class="sd">           [2] M. Keber, Division-Free computation of sub-resultants</span>
-<span class="sd">               using Bezout matrices, Tech. Report MPI-I-2006-1-006,</span>
-<span class="sd">               Saarbrucken, 2006</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        berkowitz_det</span>
-<span class="sd">        berkowitz_minors</span>
-<span class="sd">        berkowitz_charpoly</span>
-<span class="sd">        berkowitz_eigenvals</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">zeros</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">NonSquareMatrixError</span><span class="p">()</span>
-
-        <span class="n">A</span><span class="p">,</span> <span class="n">N</span> <span class="o">=</span> <span class="bp">self</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span>
-        <span class="n">transforms</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">N</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-            <span class="n">T</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">),</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span>
-
-            <span class="n">R</span><span class="p">,</span> <span class="n">C</span> <span class="o">=</span> <span class="o">-</span><span class="n">A</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="p">:</span><span class="n">k</span><span class="p">],</span> <span class="n">A</span><span class="p">[:</span><span class="n">k</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span>
-            <span class="n">A</span><span class="p">,</span> <span class="n">a</span> <span class="o">=</span> <span class="n">A</span><span class="p">[:</span><span class="n">k</span><span class="p">,</span> <span class="p">:</span><span class="n">k</span><span class="p">],</span> <span class="o">-</span><span class="n">A</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span>
-
-            <span class="n">items</span> <span class="o">=</span> <span class="p">[</span><span class="n">C</span><span class="p">]</span>
-
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="p">):</span>
-                <span class="n">items</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">A</span><span class="o">*</span><span class="n">items</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">B</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">items</span><span class="p">):</span>
-                <span class="n">items</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">R</span><span class="o">*</span><span class="n">B</span><span class="p">)[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-
-            <span class="n">items</span> <span class="o">=</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">a</span><span class="p">]</span> <span class="o">+</span> <span class="n">items</span>
-
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-                <span class="n">T</span><span class="p">[</span><span class="n">i</span><span class="p">:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">items</span><span class="p">[:</span><span class="n">n</span> <span class="o">-</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
-
-            <span class="n">transforms</span><span class="p">[</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">T</span>
-
-        <span class="n">polys</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">([</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="o">-</span><span class="n">A</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])]</span>
-
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">T</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">transforms</span><span class="p">):</span>
-            <span class="n">polys</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">T</span><span class="o">*</span><span class="n">polys</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="n">polys</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.berkowitz_det"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.berkowitz_det">[docs]</a>    <span class="k">def</span> <span class="nf">berkowitz_det</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes determinant using Berkowitz method.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        det</span>
-<span class="sd">        berkowitz</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">NonSquareMatrixError</span><span class="p">()</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">berkowitz</span><span class="p">()[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">sign</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">poly</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">sign</span><span class="o">*</span><span class="n">poly</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.berkowitz_minors"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.berkowitz_minors">[docs]</a>    <span class="k">def</span> <span class="nf">berkowitz_minors</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes principal minors using Berkowitz method.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        berkowitz</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">sign</span><span class="p">,</span> <span class="n">minors</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">,</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">poly</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">berkowitz</span><span class="p">():</span>
-            <span class="n">minors</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sign</span><span class="o">*</span><span class="n">poly</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-            <span class="n">sign</span> <span class="o">=</span> <span class="o">-</span><span class="n">sign</span>
-
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">minors</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.berkowitz_charpoly"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.berkowitz_charpoly">[docs]</a>    <span class="k">def</span> <span class="nf">berkowitz_charpoly</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;lambda&#39;</span><span class="p">),</span> <span class="n">simplify</span><span class="o">=</span><span class="n">_simplify</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes characteristic polynomial minors using Berkowitz method.</span>
-
-<span class="sd">        A PurePoly is returned so using different variables for ``x`` does</span>
-<span class="sd">        not affect the comparison or the polynomials:</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; A = Matrix([[1, 3], [2, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; A.berkowitz_charpoly(x) == A.berkowitz_charpoly(y)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        Specifying ``x`` is optional; a Dummy with name ``lambda`` is used by</span>
-<span class="sd">        default (which looks good when pretty-printed in unicode):</span>
-
-<span class="sd">        &gt;&gt;&gt; A.berkowitz_charpoly().as_expr()</span>
-<span class="sd">        _lambda**2 - _lambda - 6</span>
-
-<span class="sd">        No test is done to see that ``x`` doesn&#39;t clash with an existing</span>
-<span class="sd">        symbol, so using the default (``lambda``) or your own Dummy symbol is</span>
-<span class="sd">        the safest option:</span>
-
-<span class="sd">        &gt;&gt;&gt; A = Matrix([[1, 2], [x, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; A.charpoly().as_expr()</span>
-<span class="sd">        _lambda**2 - _lambda - 2*x</span>
-<span class="sd">        &gt;&gt;&gt; A.charpoly(x).as_expr()</span>
-<span class="sd">        x**2 - 3*x</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        berkowitz</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">PurePoly</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">simplify</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">berkowitz</span><span class="p">()[</span><span class="o">-</span><span class="mi">1</span><span class="p">])),</span> <span class="n">x</span><span class="p">)</span>
-</div>
-    <span class="n">charpoly</span> <span class="o">=</span> <span class="n">berkowitz_charpoly</span>
-
-<div class="viewcode-block" id="MatrixBase.berkowitz_eigenvals"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.berkowitz_eigenvals">[docs]</a>    <span class="k">def</span> <span class="nf">berkowitz_eigenvals</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes eigenvalues of a Matrix using Berkowitz method.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        berkowitz</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">roots</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">berkowitz_charpoly</span><span class="p">(</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)),</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.eigenvals"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.eigenvals">[docs]</a>    <span class="k">def</span> <span class="nf">eigenvals</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return eigen values using the berkowitz_eigenvals routine.</span>
-
-<span class="sd">        Since the roots routine doesn&#39;t always work well with Floats,</span>
-<span class="sd">        they will be replaced with Rationals before calling that</span>
-<span class="sd">        routine. If this is not desired, set flag ``rational`` to False.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># roots doesn&#39;t like Floats, so replace them with Rationals</span>
-        <span class="c"># unless the nsimplify flag indicates that this has already</span>
-        <span class="c"># been done, e.g. in eigenvects</span>
-        <span class="k">if</span> <span class="n">flags</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;rational&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">v</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Float</span><span class="p">)</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">):</span>
-                <span class="bp">self</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span>
-                    <span class="p">[</span><span class="n">nsimplify</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">rational</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">])</span>
-
-        <span class="n">flags</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;simplify&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>  <span class="c"># pop unsupported flag</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">berkowitz_eigenvals</span><span class="p">(</span><span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.eigenvects"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.eigenvects">[docs]</a>    <span class="k">def</span> <span class="nf">eigenvects</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return list of triples (eigenval, multiplicity, basis).</span>
-
-<span class="sd">        The flag ``simplify`` has two effects:</span>
-<span class="sd">            1) if bool(simplify) is True, as_content_primitive()</span>
-<span class="sd">            will be used to tidy up normalization artifacts;</span>
-<span class="sd">            2) if nullspace needs simplification to compute the</span>
-<span class="sd">            basis, the simplify flag will be passed on to the</span>
-<span class="sd">            nullspace routine which will interpret it there.</span>
-
-<span class="sd">        If the matrix contains any Floats, they will be changed to Rationals</span>
-<span class="sd">        for computation purposes, but the answers will be returned after being</span>
-<span class="sd">        evaluated with evalf. If it is desired to removed small imaginary</span>
-<span class="sd">        portions during the evalf step, pass a value for the ``chop`` flag.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">eye</span>
-
-        <span class="n">simplify</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;simplify&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-        <span class="n">primitive</span> <span class="o">=</span> <span class="nb">bool</span><span class="p">(</span><span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;simplify&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">))</span>
-        <span class="n">chop</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;chop&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-
-        <span class="n">flags</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;multiple&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>  <span class="c"># remove this if it&#39;s there</span>
-
-        <span class="c"># roots doesn&#39;t like Floats, so replace them with Rationals</span>
-        <span class="nb">float</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">v</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Float</span><span class="p">)</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">):</span>
-            <span class="nb">float</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="bp">self</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span> <span class="p">[</span><span class="n">nsimplify</span><span class="p">(</span>
-                <span class="n">v</span><span class="p">,</span> <span class="n">rational</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">])</span>
-            <span class="n">flags</span><span class="p">[</span><span class="s">&#39;rational&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>  <span class="c"># to tell eigenvals not to do this</span>
-
-        <span class="n">out</span><span class="p">,</span> <span class="n">vlist</span> <span class="o">=</span> <span class="p">[],</span> <span class="bp">self</span><span class="o">.</span><span class="n">eigenvals</span><span class="p">(</span><span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-        <span class="n">vlist</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">vlist</span><span class="o">.</span><span class="n">items</span><span class="p">())</span>
-        <span class="n">vlist</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="n">flags</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;rational&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">r</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">vlist</span><span class="p">:</span>
-            <span class="n">tmp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_mutable</span><span class="p">()</span> <span class="o">-</span> <span class="n">eye</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)</span><span class="o">*</span><span class="n">r</span>
-            <span class="n">basis</span> <span class="o">=</span> <span class="n">tmp</span><span class="o">.</span><span class="n">nullspace</span><span class="p">()</span>
-            <span class="c"># whether tmp.is_symbolic() is True or False, it is possible that</span>
-            <span class="c"># the basis will come back as [] in which case simplification is</span>
-            <span class="c"># necessary.</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">basis</span><span class="p">:</span>
-                <span class="c"># The nullspace routine failed, try it again with simplification</span>
-                <span class="n">basis</span> <span class="o">=</span> <span class="n">tmp</span><span class="o">.</span><span class="n">nullspace</span><span class="p">(</span><span class="n">simplify</span><span class="o">=</span><span class="n">simplify</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">basis</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                        <span class="s">&quot;Can&#39;t evaluate eigenvector for eigenvalue </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">r</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">primitive</span><span class="p">:</span>
-                <span class="c"># the relationship A*e = lambda*e will still hold if we change the</span>
-                <span class="c"># eigenvector; so if simplify is True we tidy up any normalization</span>
-                <span class="c"># artifacts with as_content_primtive (default) and remove any pure Integer</span>
-                <span class="c"># denominators.</span>
-                <span class="n">l</span> <span class="o">=</span> <span class="mi">1</span>
-                <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">basis</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
-                    <span class="n">c</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="n">signsimp</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-                    <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                        <span class="n">b</span> <span class="o">=</span> <span class="n">c</span><span class="o">*</span><span class="n">p</span>
-                        <span class="n">l</span> <span class="o">=</span> <span class="n">ilcm</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">c</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-                    <span class="n">basis</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">b</span>
-                <span class="k">if</span> <span class="n">l</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">basis</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*=</span> <span class="n">l</span>
-            <span class="k">if</span> <span class="nb">float</span><span class="p">:</span>
-                <span class="n">out</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">r</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">chop</span><span class="o">=</span><span class="n">chop</span><span class="p">),</span> <span class="n">k</span><span class="p">,</span> <span class="p">[</span>
-                           <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">chop</span><span class="o">=</span><span class="n">chop</span><span class="p">)</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">basis</span><span class="p">]))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">out</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">r</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">basis</span><span class="p">]))</span>
-        <span class="k">return</span> <span class="n">out</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.singular_values"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.singular_values">[docs]</a>    <span class="k">def</span> <span class="nf">singular_values</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute the singular values of a Matrix</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, Symbol, eye</span>
-<span class="sd">        &gt;&gt;&gt; x = Symbol(&#39;x&#39;, real=True)</span>
-<span class="sd">        &gt;&gt;&gt; A = Matrix([[0, 1, 0], [0, x, 0], [-1, 0, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; A.singular_values()</span>
-<span class="sd">        [sqrt(x**2 + 1), 1, 0]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        condition_number</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_mutable</span><span class="p">()</span>
-        <span class="c"># Compute eigenvalues of A.H A</span>
-        <span class="n">valmultpairs</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">H</span><span class="o">*</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">eigenvals</span><span class="p">()</span>
-
-        <span class="c"># Expands result from eigenvals into a simple list</span>
-        <span class="n">vals</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">valmultpairs</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-            <span class="n">vals</span> <span class="o">+=</span> <span class="p">[</span><span class="n">sqrt</span><span class="p">(</span><span class="n">k</span><span class="p">)]</span><span class="o">*</span><span class="n">v</span>  <span class="c"># dangerous! same k in several spots!</span>
-        <span class="c"># sort them in descending order</span>
-        <span class="n">vals</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">vals</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.condition_number"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.condition_number">[docs]</a>    <span class="k">def</span> <span class="nf">condition_number</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the condition number of a matrix.</span>
-
-<span class="sd">        This is the maximum singular value divided by the minimum singular value</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, S</span>
-<span class="sd">        &gt;&gt;&gt; A = Matrix([[1, 0, 0], [0, 10, 0], [0, 0, S.One/10]])</span>
-<span class="sd">        &gt;&gt;&gt; A.condition_number()</span>
-<span class="sd">        100</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        singular_values</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">singularvalues</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">singular_values</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">Max</span><span class="p">(</span><span class="o">*</span><span class="n">singularvalues</span><span class="p">)</span> <span class="o">/</span> <span class="n">Min</span><span class="p">(</span><span class="o">*</span><span class="n">singularvalues</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">__getattr__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">attr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">attr</span> <span class="ow">in</span> <span class="p">(</span><span class="s">&#39;diff&#39;</span><span class="p">,</span> <span class="s">&#39;integrate&#39;</span><span class="p">,</span> <span class="s">&#39;limit&#39;</span><span class="p">):</span>
-            <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">):</span>
-                <span class="n">item_doit</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">item</span><span class="p">:</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">item</span><span class="p">,</span> <span class="n">attr</span><span class="p">)(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">applyfunc</span><span class="p">(</span><span class="n">item_doit</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">doit</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span>
-                <span class="s">&quot;</span><span class="si">%s</span><span class="s"> has no attribute </span><span class="si">%s</span><span class="s">.&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="n">attr</span><span class="p">))</span>
-
-<div class="viewcode-block" id="MatrixBase.integrate"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.integrate">[docs]</a>    <span class="k">def</span> <span class="nf">integrate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Integrate each element of the matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; M = Matrix([[x, y], [1, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; M.integrate((x, ))</span>
-<span class="sd">        [x**2/2, x*y]</span>
-<span class="sd">        [     x,   0]</span>
-<span class="sd">        &gt;&gt;&gt; M.integrate((x, 0, 2))</span>
-<span class="sd">        [2, 2*y]</span>
-<span class="sd">        [2,   0]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        limit</span>
-<span class="sd">        diff</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span>
-                <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">integrate</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.limit"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.limit">[docs]</a>    <span class="k">def</span> <span class="nf">limit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Calculate the limit of each element in the matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; M = Matrix([[x, y], [1, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; M.limit(x, 2)</span>
-<span class="sd">        [2, y]</span>
-<span class="sd">        [1, 0]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        integrate</span>
-<span class="sd">        diff</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span>
-                <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">limit</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.diff"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.diff">[docs]</a>    <span class="k">def</span> <span class="nf">diff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Calculate the derivative of each element in the matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; M = Matrix([[x, y], [1, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; M.diff(x)</span>
-<span class="sd">        [1, 0]</span>
-<span class="sd">        [0, 0]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        integrate</span>
-<span class="sd">        limit</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span>
-                <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.vec"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.vec">[docs]</a>    <span class="k">def</span> <span class="nf">vec</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the Matrix converted into a one column matrix by stacking columns</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; m=Matrix([[1, 3], [2, 4]])</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [1, 3]</span>
-<span class="sd">        [2, 4]</span>
-<span class="sd">        &gt;&gt;&gt; m.vec()</span>
-<span class="sd">        [1]</span>
-<span class="sd">        [2]</span>
-<span class="sd">        [3]</span>
-<span class="sd">        [4]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        vech</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">T</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="p">),</span> <span class="mi">1</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.vech"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.vech">[docs]</a>    <span class="k">def</span> <span class="nf">vech</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">diagonal</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">check_symmetry</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the unique elements of a symmetric Matrix as a one column matrix</span>
-<span class="sd">        by stacking the elements in the lower triangle.</span>
-
-<span class="sd">        Arguments:</span>
-<span class="sd">        diagonal -- include the diagonal cells of self or not</span>
-<span class="sd">        check_symmetry -- checks symmetry of self but not completely reliably</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; m=Matrix([[1, 2], [2, 3]])</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [1, 2]</span>
-<span class="sd">        [2, 3]</span>
-<span class="sd">        &gt;&gt;&gt; m.vech()</span>
-<span class="sd">        [1]</span>
-<span class="sd">        [2]</span>
-<span class="sd">        [3]</span>
-<span class="sd">        &gt;&gt;&gt; m.vech(diagonal=False)</span>
-<span class="sd">        [2]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        vec</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">zeros</span>
-
-        <span class="n">c</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span>
-        <span class="k">if</span> <span class="n">c</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ShapeError</span><span class="p">(</span><span class="s">&quot;Matrix must be square&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">check_symmetry</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-            <span class="k">if</span> <span class="bp">self</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">transpose</span><span class="p">():</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Matrix appears to be asymmetric; consider check_symmetry=False&quot;</span><span class="p">)</span>
-        <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">if</span> <span class="n">diagonal</span><span class="p">:</span>
-            <span class="n">v</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">c</span><span class="o">*</span><span class="p">(</span><span class="n">c</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
-                    <span class="n">v</span><span class="p">[</span><span class="n">count</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-                    <span class="n">count</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">v</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">c</span><span class="o">*</span><span class="p">(</span><span class="n">c</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
-                    <span class="n">v</span><span class="p">[</span><span class="n">count</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-                    <span class="n">count</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="n">v</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.get_diag_blocks"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.get_diag_blocks">[docs]</a>    <span class="k">def</span> <span class="nf">get_diag_blocks</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Obtains the square sub-matrices on the main diagonal of a square matrix.</span>
-
-<span class="sd">        Useful for inverting symbolic matrices or solving systems of</span>
-<span class="sd">        linear equations which may be decoupled by having a block diagonal</span>
-<span class="sd">        structure.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, symbols</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">        &gt;&gt;&gt; A = Matrix([[1, 3, 0, 0], [y, z*z, 0, 0], [0, 0, x, 0], [0, 0, 0, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; a1, a2, a3 = A.get_diag_blocks()</span>
-<span class="sd">        &gt;&gt;&gt; a1</span>
-<span class="sd">        [1,    3]</span>
-<span class="sd">        [y, z**2]</span>
-<span class="sd">        &gt;&gt;&gt; a2</span>
-<span class="sd">        [x]</span>
-<span class="sd">        &gt;&gt;&gt; a3</span>
-<span class="sd">        [0]</span>
-<span class="sd">        &gt;&gt;&gt;</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">sub_blocks</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">def</span> <span class="nf">recurse_sub_blocks</span><span class="p">(</span><span class="n">M</span><span class="p">):</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;=</span> <span class="n">M</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-                <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">to_the_right</span> <span class="o">=</span> <span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">i</span><span class="p">:]</span>
-                    <span class="n">to_the_bottom</span> <span class="o">=</span> <span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">:,</span> <span class="mi">0</span><span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">to_the_right</span> <span class="o">=</span> <span class="n">M</span><span class="p">[:</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">:]</span>
-                    <span class="n">to_the_bottom</span> <span class="o">=</span> <span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">:,</span> <span class="p">:</span><span class="n">i</span><span class="p">]</span>
-                <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">to_the_right</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">any</span><span class="p">(</span><span class="n">to_the_bottom</span><span class="p">):</span>
-                    <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-                    <span class="k">continue</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">sub_blocks</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">M</span><span class="p">[:</span><span class="n">i</span><span class="p">,</span> <span class="p">:</span><span class="n">i</span><span class="p">])</span>
-                    <span class="k">if</span> <span class="n">M</span><span class="o">.</span><span class="n">shape</span> <span class="o">==</span> <span class="n">M</span><span class="p">[:</span><span class="n">i</span><span class="p">,</span> <span class="p">:</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span><span class="p">:</span>
-                        <span class="k">return</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">recurse_sub_blocks</span><span class="p">(</span><span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">:,</span> <span class="n">i</span><span class="p">:])</span>
-                        <span class="k">return</span>
-        <span class="n">recurse_sub_blocks</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">sub_blocks</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.diagonalize"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.diagonalize">[docs]</a>    <span class="k">def</span> <span class="nf">diagonalize</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">reals_only</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">sort</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">normalize</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return diagonalized matrix D and transformation P such as</span>
-
-<span class="sd">            D = P^-1 * M * P</span>
-
-<span class="sd">        where M is current matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix(3, 3, [1, 2, 0, 0, 3, 0, 2, -4, 2])</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [1,  2, 0]</span>
-<span class="sd">        [0,  3, 0]</span>
-<span class="sd">        [2, -4, 2]</span>
-<span class="sd">        &gt;&gt;&gt; (P, D) = m.diagonalize()</span>
-<span class="sd">        &gt;&gt;&gt; D</span>
-<span class="sd">        [1, 0, 0]</span>
-<span class="sd">        [0, 2, 0]</span>
-<span class="sd">        [0, 0, 3]</span>
-<span class="sd">        &gt;&gt;&gt; P</span>
-<span class="sd">        [-1, 0, -1]</span>
-<span class="sd">        [ 0, 0, -1]</span>
-<span class="sd">        [ 2, 1,  2]</span>
-<span class="sd">        &gt;&gt;&gt; P.inv() * m * P</span>
-<span class="sd">        [1, 0, 0]</span>
-<span class="sd">        [0, 2, 0]</span>
-<span class="sd">        [0, 0, 3]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        is_diagonal</span>
-<span class="sd">        is_diagonalizable</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">diag</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">NonSquareMatrixError</span><span class="p">()</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_diagonalizable</span><span class="p">(</span><span class="n">reals_only</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_diagonalize_clear_subproducts</span><span class="p">()</span>
-            <span class="k">raise</span> <span class="n">MatrixError</span><span class="p">(</span><span class="s">&quot;Matrix is not diagonalizable&quot;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eigenvects</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_eigenvects</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">eigenvects</span><span class="p">(</span><span class="n">simplify</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">sort</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_eigenvects</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_eigenvects</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-            <span class="n">diagvals</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">P</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="p">[])</span>
-            <span class="k">for</span> <span class="n">eigenval</span><span class="p">,</span> <span class="n">multiplicity</span><span class="p">,</span> <span class="n">vects</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eigenvects</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">multiplicity</span><span class="p">):</span>
-                    <span class="n">diagvals</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">eigenval</span><span class="p">)</span>
-                    <span class="n">vec</span> <span class="o">=</span> <span class="n">vects</span><span class="p">[</span><span class="n">k</span><span class="p">]</span>
-                    <span class="k">if</span> <span class="n">normalize</span><span class="p">:</span>
-                        <span class="n">vec</span> <span class="o">=</span> <span class="n">vec</span> <span class="o">/</span> <span class="n">vec</span><span class="o">.</span><span class="n">norm</span><span class="p">()</span>
-                    <span class="n">P</span> <span class="o">=</span> <span class="n">P</span><span class="o">.</span><span class="n">col_insert</span><span class="p">(</span><span class="n">P</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span> <span class="n">vec</span><span class="p">)</span>
-            <span class="n">D</span> <span class="o">=</span> <span class="n">diag</span><span class="p">(</span><span class="o">*</span><span class="n">diagvals</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_diagonalize_clear_subproducts</span><span class="p">()</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">D</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.is_diagonalizable"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_diagonalizable">[docs]</a>    <span class="k">def</span> <span class="nf">is_diagonalizable</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">reals_only</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">clear_subproducts</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Check if matrix is diagonalizable.</span>
-
-<span class="sd">        If reals_only==True then check that diagonalized matrix consists of the only not complex values.</span>
-
-<span class="sd">        Some subproducts could be used further in other methods to avoid double calculations,</span>
-<span class="sd">        By default (if clear_subproducts==True) they will be deleted.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix(3, 3, [1, 2, 0, 0, 3, 0, 2, -4, 2])</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [1,  2, 0]</span>
-<span class="sd">        [0,  3, 0]</span>
-<span class="sd">        [2, -4, 2]</span>
-<span class="sd">        &gt;&gt;&gt; m.is_diagonalizable()</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix(2, 2, [0, 1, 0, 0])</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [0, 1]</span>
-<span class="sd">        [0, 0]</span>
-<span class="sd">        &gt;&gt;&gt; m.is_diagonalizable()</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix(2, 2, [0, 1, -1, 0])</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [ 0, 1]</span>
-<span class="sd">        [-1, 0]</span>
-<span class="sd">        &gt;&gt;&gt; m.is_diagonalizable()</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; m.is_diagonalizable(True)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        is_diagonal</span>
-<span class="sd">        diagonalize</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_is_symbolic</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_symbolic</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_is_symmetric</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_symmetric</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_eigenvects</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="c">#if self._is_symbolic:</span>
-        <span class="c">#    self._diagonalize_clear_subproducts()</span>
-        <span class="c">#    raise NotImplementedError(&quot;Symbolic matrices are not implemented for diagonalization yet&quot;)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_eigenvects</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">eigenvects</span><span class="p">(</span><span class="n">simplify</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="n">all_iscorrect</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">for</span> <span class="n">eigenval</span><span class="p">,</span> <span class="n">multiplicity</span><span class="p">,</span> <span class="n">vects</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eigenvects</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">vects</span><span class="p">)</span> <span class="o">!=</span> <span class="n">multiplicity</span><span class="p">:</span>
-                <span class="n">all_iscorrect</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">break</span>
-            <span class="k">elif</span> <span class="n">reals_only</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">eigenval</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-                <span class="n">all_iscorrect</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">break</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="n">all_iscorrect</span>
-        <span class="k">if</span> <span class="n">clear_subproducts</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_diagonalize_clear_subproducts</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">res</span>
-</div>
-    <span class="k">def</span> <span class="nf">_diagonalize_clear_subproducts</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">del</span> <span class="bp">self</span><span class="o">.</span><span class="n">_is_symbolic</span>
-        <span class="k">del</span> <span class="bp">self</span><span class="o">.</span><span class="n">_is_symmetric</span>
-        <span class="k">del</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eigenvects</span>
-
-<div class="viewcode-block" id="MatrixBase.jordan_form"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.jordan_form">[docs]</a>    <span class="k">def</span> <span class="nf">jordan_form</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">calc_transformation</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return Jordan form J of current matrix.</span>
-
-<span class="sd">        If calc_transformation is specified as False, then transformation P such that</span>
-
-<span class="sd">              J = P^-1 * M * P</span>
-
-<span class="sd">        will not be calculated.</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        Calculation of transformation P is not implemented yet.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix([</span>
-<span class="sd">        ...        [ 6,  5, -2, -3],</span>
-<span class="sd">        ...        [-3, -1,  3,  3],</span>
-<span class="sd">        ...        [ 2,  1, -2, -3],</span>
-<span class="sd">        ...        [-1,  1,  5,  5]])</span>
-<span class="sd">        &gt;&gt;&gt; (P, J) = m.jordan_form()</span>
-<span class="sd">        &gt;&gt;&gt; J</span>
-<span class="sd">        [2, 1, 0, 0]</span>
-<span class="sd">        [0, 2, 0, 0]</span>
-<span class="sd">        [0, 0, 2, 1]</span>
-<span class="sd">        [0, 0, 0, 2]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        jordan_cells</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">diag</span>
-
-        <span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">Jcells</span><span class="p">)</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">jordan_cells</span><span class="p">(</span><span class="n">calc_transformation</span><span class="p">)</span>
-        <span class="n">J</span> <span class="o">=</span> <span class="n">diag</span><span class="p">(</span><span class="o">*</span><span class="n">Jcells</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">J</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.jordan_cells"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.jordan_cells">[docs]</a>    <span class="k">def</span> <span class="nf">jordan_cells</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">calc_transformation</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return a list of Jordan cells of current matrix.</span>
-<span class="sd">        This list shape Jordan matrix J.</span>
-
-<span class="sd">        If calc_transformation is specified as False, then transformation P such that</span>
-
-<span class="sd">              J = P^-1 * M * P</span>
-
-<span class="sd">        will not be calculated.</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        Calculation of transformation P is not implemented yet.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix(4, 4, [</span>
-<span class="sd">        ...  6,  5, -2, -3,</span>
-<span class="sd">        ... -3, -1,  3,  3,</span>
-<span class="sd">        ...  2,  1, -2, -3,</span>
-<span class="sd">        ... -1,  1,  5,  5])</span>
-
-<span class="sd">        &gt;&gt;&gt; (P, Jcells) = m.jordan_cells()</span>
-<span class="sd">        &gt;&gt;&gt; Jcells[0]</span>
-<span class="sd">        [2, 1]</span>
-<span class="sd">        [0, 2]</span>
-<span class="sd">        &gt;&gt;&gt; Jcells[1]</span>
-<span class="sd">        [2, 1]</span>
-<span class="sd">        [0, 2]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        jordan_form</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">jordan_cell</span><span class="p">,</span> <span class="n">diag</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">NonSquareMatrixError</span><span class="p">()</span>
-        <span class="n">_eigenvects</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">eigenvects</span><span class="p">()</span>
-        <span class="n">Jcells</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">eigenval</span><span class="p">,</span> <span class="n">multiplicity</span><span class="p">,</span> <span class="n">vects</span> <span class="ow">in</span> <span class="n">_eigenvects</span><span class="p">:</span>
-            <span class="n">geometrical</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">vects</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">geometrical</span> <span class="o">==</span> <span class="n">multiplicity</span><span class="p">:</span>
-                <span class="n">Jcell</span> <span class="o">=</span> <span class="n">diag</span><span class="p">(</span><span class="o">*</span><span class="p">([</span><span class="n">eigenval</span><span class="p">]</span><span class="o">*</span><span class="n">multiplicity</span><span class="p">))</span>
-                <span class="n">Jcells</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Jcell</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">sizes</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_jordan_split</span><span class="p">(</span><span class="n">multiplicity</span><span class="p">,</span> <span class="n">geometrical</span><span class="p">)</span>
-                <span class="n">cells</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">for</span> <span class="n">size</span> <span class="ow">in</span> <span class="n">sizes</span><span class="p">:</span>
-                    <span class="n">cell</span> <span class="o">=</span> <span class="n">jordan_cell</span><span class="p">(</span><span class="n">eigenval</span><span class="p">,</span> <span class="n">size</span><span class="p">)</span>
-                    <span class="n">cells</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">cell</span><span class="p">)</span>
-                <span class="n">Jcells</span> <span class="o">+=</span> <span class="n">cells</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="n">Jcells</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_jordan_split</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">algebraical</span><span class="p">,</span> <span class="n">geometrical</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return a list of integers with sum equal to &#39;algebraical&#39;</span>
-<span class="sd">        and length equal to &#39;geometrical&#39;&quot;&quot;&quot;</span>
-        <span class="n">n1</span> <span class="o">=</span> <span class="n">algebraical</span> <span class="o">//</span> <span class="n">geometrical</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="p">[</span><span class="n">n1</span><span class="p">]</span><span class="o">*</span><span class="n">geometrical</span>
-        <span class="n">res</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="n">res</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+=</span> <span class="n">algebraical</span> <span class="o">%</span> <span class="n">geometrical</span>
-        <span class="k">assert</span> <span class="nb">sum</span><span class="p">(</span><span class="n">res</span><span class="p">)</span> <span class="o">==</span> <span class="n">algebraical</span>
-        <span class="k">return</span> <span class="n">res</span>
-
-<div class="viewcode-block" id="MatrixBase.has"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.has">[docs]</a>    <span class="k">def</span> <span class="nf">has</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">patterns</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Test whether any subexpression matches any of the patterns.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, Float</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; A = Matrix(((1, x), (0.2, 3)))</span>
-<span class="sd">        &gt;&gt;&gt; A.has(x)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; A.has(y)</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; A.has(Float)</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">any</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">patterns</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.dual"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.dual">[docs]</a>    <span class="k">def</span> <span class="nf">dual</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the dual of a matrix, which is:</span>
-
-<span class="sd">        `(1/2)*levicivita(i, j, k, l)*M(k, l)` summed over indices `k` and `l`</span>
-
-<span class="sd">        Since the levicivita method is anti_symmetric for any pairwise</span>
-<span class="sd">        exchange of indices, the dual of a symmetric matrix is the zero</span>
-<span class="sd">        matrix. Strictly speaking the dual defined here assumes that the</span>
-<span class="sd">        &#39;matrix&#39; `M` is a contravariant anti_symmetric second rank tensor,</span>
-<span class="sd">        so that the dual is a covariant second rank tensor.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">LeviCivita</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">zeros</span>
-
-        <span class="n">M</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="p">[:,</span> <span class="p">:],</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span>
-        <span class="n">work</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_symmetric</span><span class="p">():</span>
-            <span class="k">return</span> <span class="n">work</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-                <span class="n">acum</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-                    <span class="n">acum</span> <span class="o">+=</span> <span class="n">LeviCivita</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span>
-                <span class="n">work</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">acum</span>
-                <span class="n">work</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">acum</span>
-
-        <span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-            <span class="n">acum</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-                    <span class="n">acum</span> <span class="o">+=</span> <span class="n">LeviCivita</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">M</span><span class="p">[</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">]</span>
-            <span class="n">acum</span> <span class="o">/=</span> <span class="mi">2</span>
-            <span class="n">work</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">acum</span>
-            <span class="n">work</span><span class="p">[</span><span class="n">l</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">acum</span>
-
-        <span class="k">return</span> <span class="n">work</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="MatrixBase.hstack"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.hstack">[docs]</a>    <span class="k">def</span> <span class="nf">hstack</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return a matrix formed by joining args horizontally (i.e.</span>
-<span class="sd">        by repeated application of row_join).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import Matrix, eye</span>
-<span class="sd">        &gt;&gt;&gt; Matrix.hstack(eye(2), 2*eye(2))</span>
-<span class="sd">        [1, 0, 2, 0]</span>
-<span class="sd">        [0, 1, 0, 2]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">reduce</span><span class="p">(</span><span class="n">cls</span><span class="o">.</span><span class="n">row_join</span><span class="p">,</span> <span class="n">args</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="MatrixBase.vstack"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.vstack">[docs]</a>    <span class="k">def</span> <span class="nf">vstack</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return a matrix formed by joining args vertically (i.e.</span>
-<span class="sd">        by repeated application of col_join).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import Matrix, eye</span>
-<span class="sd">        &gt;&gt;&gt; Matrix.vstack(eye(2), 2*eye(2))</span>
-<span class="sd">        [1, 0]</span>
-<span class="sd">        [0, 1]</span>
-<span class="sd">        [2, 0]</span>
-<span class="sd">        [0, 2]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">reduce</span><span class="p">(</span><span class="n">cls</span><span class="o">.</span><span class="n">col_join</span><span class="p">,</span> <span class="n">args</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.row_join"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.row_join">[docs]</a>    <span class="k">def</span> <span class="nf">row_join</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Concatenates two matrices along self&#39;s last and rhs&#39;s first column</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, zeros, ones</span>
-<span class="sd">        &gt;&gt;&gt; M = zeros(3)</span>
-<span class="sd">        &gt;&gt;&gt; V = ones(3, 1)</span>
-<span class="sd">        &gt;&gt;&gt; M.row_join(V)</span>
-<span class="sd">        [0, 0, 0, 1]</span>
-<span class="sd">        [0, 0, 0, 1]</span>
-<span class="sd">        [0, 0, 0, 1]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        row</span>
-<span class="sd">        col_join</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">!=</span> <span class="n">rhs</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ShapeError</span><span class="p">(</span>
-                <span class="s">&quot;`self` and `rhs` must have the same number of rows.&quot;</span><span class="p">)</span>
-
-        <span class="n">newmat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">+</span> <span class="n">rhs</span><span class="o">.</span><span class="n">cols</span><span class="p">)</span>
-        <span class="n">newmat</span><span class="p">[:,</span> <span class="p">:</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span>
-        <span class="n">newmat</span><span class="p">[:,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">:]</span> <span class="o">=</span> <span class="n">rhs</span>
-        <span class="k">return</span> <span class="n">newmat</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.col_join"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.col_join">[docs]</a>    <span class="k">def</span> <span class="nf">col_join</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">bott</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Concatenates two matrices along self&#39;s last and bott&#39;s first row</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, zeros, ones</span>
-<span class="sd">        &gt;&gt;&gt; M = zeros(3)</span>
-<span class="sd">        &gt;&gt;&gt; V = ones(1, 3)</span>
-<span class="sd">        &gt;&gt;&gt; M.col_join(V)</span>
-<span class="sd">        [0, 0, 0]</span>
-<span class="sd">        [0, 0, 0]</span>
-<span class="sd">        [0, 0, 0]</span>
-<span class="sd">        [1, 1, 1]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        col</span>
-<span class="sd">        row_join</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">!=</span> <span class="n">bott</span><span class="o">.</span><span class="n">cols</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ShapeError</span><span class="p">(</span>
-                <span class="s">&quot;`self` and `bott` must have the same number of columns.&quot;</span><span class="p">)</span>
-
-        <span class="n">newmat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">+</span> <span class="n">bott</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">)</span>
-        <span class="n">newmat</span><span class="p">[:</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="bp">self</span>
-        <span class="n">newmat</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">:,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">bott</span>
-        <span class="k">return</span> <span class="n">newmat</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.row_insert"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.row_insert">[docs]</a>    <span class="k">def</span> <span class="nf">row_insert</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">pos</span><span class="p">,</span> <span class="n">mti</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Insert one or more rows at the given row position.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, zeros, ones</span>
-<span class="sd">        &gt;&gt;&gt; M = zeros(3)</span>
-<span class="sd">        &gt;&gt;&gt; V = ones(1, 3)</span>
-<span class="sd">        &gt;&gt;&gt; M.row_insert(1, V)</span>
-<span class="sd">        [0, 0, 0]</span>
-<span class="sd">        [1, 1, 1]</span>
-<span class="sd">        [0, 0, 0]</span>
-<span class="sd">        [0, 0, 0]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        row</span>
-<span class="sd">        col_insert</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">pos</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">mti</span><span class="o">.</span><span class="n">col_join</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">pos</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">pos</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">+</span> <span class="n">pos</span>
-        <span class="k">if</span> <span class="n">pos</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">pos</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">elif</span> <span class="n">pos</span> <span class="o">&gt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-            <span class="n">pos</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">!=</span> <span class="n">mti</span><span class="o">.</span><span class="n">cols</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ShapeError</span><span class="p">(</span>
-                <span class="s">&quot;`self` and `mti` must have the same number of columns.&quot;</span><span class="p">)</span>
-
-        <span class="n">newmat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">+</span> <span class="n">mti</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">)</span>
-        <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">pos</span><span class="p">,</span> <span class="n">pos</span> <span class="o">+</span> <span class="n">mti</span><span class="o">.</span><span class="n">rows</span>
-        <span class="n">newmat</span><span class="p">[:</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="bp">self</span><span class="p">[:</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span>
-        <span class="n">newmat</span><span class="p">[</span><span class="n">i</span><span class="p">:</span> <span class="n">j</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">mti</span>
-        <span class="n">newmat</span><span class="p">[</span><span class="n">j</span><span class="p">:,</span> <span class="p">:]</span> <span class="o">=</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">:,</span> <span class="p">:]</span>
-        <span class="k">return</span> <span class="n">newmat</span>
-</div>
-<div class="viewcode-block" id="MatrixBase.col_insert"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.col_insert">[docs]</a>    <span class="k">def</span> <span class="nf">col_insert</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">pos</span><span class="p">,</span> <span class="n">mti</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Insert one or more columns at the given column position.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, zeros, ones</span>
-<span class="sd">        &gt;&gt;&gt; M = zeros(3)</span>
-<span class="sd">        &gt;&gt;&gt; V = ones(3, 1)</span>
-<span class="sd">        &gt;&gt;&gt; M.col_insert(1, V)</span>
-<span class="sd">        [0, 1, 0, 0]</span>
-<span class="sd">        [0, 1, 0, 0]</span>
-<span class="sd">        [0, 1, 0, 0]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        col</span>
-<span class="sd">        row_insert</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">pos</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">mti</span><span class="o">.</span><span class="n">row_join</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">pos</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">pos</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">+</span> <span class="n">pos</span>
-        <span class="k">if</span> <span class="n">pos</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">pos</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">elif</span> <span class="n">pos</span> <span class="o">&gt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">:</span>
-            <span class="n">pos</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">!=</span> <span class="n">mti</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ShapeError</span><span class="p">(</span><span class="s">&quot;self and mti must have the same number of rows.&quot;</span><span class="p">)</span>
-
-        <span class="n">newmat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">+</span> <span class="n">mti</span><span class="o">.</span><span class="n">cols</span><span class="p">)</span>
-        <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">pos</span><span class="p">,</span> <span class="n">pos</span> <span class="o">+</span> <span class="n">mti</span><span class="o">.</span><span class="n">cols</span>
-        <span class="n">newmat</span><span class="p">[:,</span> <span class="p">:</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="p">[:,</span> <span class="p">:</span><span class="n">i</span><span class="p">]</span>
-        <span class="n">newmat</span><span class="p">[:,</span> <span class="n">i</span><span class="p">:</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">mti</span>
-        <span class="n">newmat</span><span class="p">[:,</span> <span class="n">j</span><span class="p">:]</span> <span class="o">=</span> <span class="bp">self</span><span class="p">[:,</span> <span class="n">i</span><span class="p">:]</span>
-        <span class="k">return</span> <span class="n">newmat</span>
-
-</div></div>
-<div class="viewcode-block" id="classof"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.classof">[docs]</a><span class="k">def</span> <span class="nf">classof</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Get the type of the result when combining matrices of different types.</span>
-
-<span class="sd">    Currently the strategy is that immutability is contagious.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Matrix, ImmutableMatrix</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.matrices.matrices import classof</span>
-<span class="sd">    &gt;&gt;&gt; M = Matrix([[1, 2], [3, 4]]) # a Mutable Matrix</span>
-<span class="sd">    &gt;&gt;&gt; IM = ImmutableMatrix([[1, 2], [3, 4]])</span>
-<span class="sd">    &gt;&gt;&gt; classof(M, IM)</span>
-<span class="sd">    &lt;class &#39;sympy.matrices.immutable.ImmutableMatrix&#39;&gt;</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">A</span><span class="o">.</span><span class="n">_class_priority</span> <span class="o">&gt;</span> <span class="n">B</span><span class="o">.</span><span class="n">_class_priority</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">A</span><span class="o">.</span><span class="n">__class__</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">B</span><span class="o">.</span><span class="n">__class__</span>
-    <span class="k">except</span><span class="p">:</span>
-        <span class="k">pass</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="kn">import</span> <span class="nn">numpy</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">B</span><span class="o">.</span><span class="n">__class__</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">A</span><span class="o">.</span><span class="n">__class__</span>
-    <span class="k">except</span><span class="p">:</span>
-        <span class="k">pass</span>
-    <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Incompatible classes </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">__class__</span><span class="p">,</span> <span class="n">B</span><span class="o">.</span><span class="n">__class__</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="a2idx"><a class="viewcode-back" href="../../../modules/matrices/matrices.html#sympy.matrices.matrices.a2idx">[docs]</a><span class="k">def</span> <span class="nf">a2idx</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return integer after making positive and validating against n.&quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="nb">slice</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">j</span>
-    <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">j</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="nb">int</span><span class="p">:</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">j</span> <span class="o">=</span> <span class="n">j</span><span class="o">.</span><span class="n">__index__</span><span class="p">()</span>
-        <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s">&quot;Invalid index a[</span><span class="si">%r</span><span class="s">]&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="p">))</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">j</span> <span class="o">+=</span> <span class="n">n</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">j</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="n">n</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s">&quot;Index out of range: a[</span><span class="si">%s</span><span class="s">]&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="p">))</span>
-    <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">j</span><span class="p">)</span></div>
-</pre></div>
-
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-          <div class="body">
-            
-  <h1>Source code for sympy.matrices.sparse</h1><div class="highlight"><pre>
-<span class="kn">import</span> <span class="nn">copy</span>
-<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.containers</span> <span class="kn">import</span> <span class="n">Dict</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">is_sequence</span><span class="p">,</span> <span class="n">as_int</span>
-<span class="kn">from</span> <span class="nn">sympy.core.singleton</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.exceptions</span> <span class="kn">import</span> <span class="n">SymPyDeprecationWarning</span>
-
-<span class="kn">from</span> <span class="nn">.matrices</span> <span class="kn">import</span> <span class="n">MatrixBase</span><span class="p">,</span> <span class="n">ShapeError</span><span class="p">,</span> <span class="n">a2idx</span>
-<span class="kn">from</span> <span class="nn">.dense</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="kn">import</span> <span class="nn">collections</span>
-
-
-<div class="viewcode-block" id="SparseMatrix"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix">[docs]</a><span class="k">class</span> <span class="nc">SparseMatrix</span><span class="p">(</span><span class="n">MatrixBase</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A sparse matrix (a matrix with a large number of zero elements).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.matrices import SparseMatrix</span>
-<span class="sd">    &gt;&gt;&gt; SparseMatrix(2, 2, list(range(4)))</span>
-<span class="sd">    [0, 1]</span>
-<span class="sd">    [2, 3]</span>
-<span class="sd">    &gt;&gt;&gt; SparseMatrix(2, 2, {(1, 1): 2})</span>
-<span class="sd">    [0, 0]</span>
-<span class="sd">    [0, 2]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    sympy.matrices.dense.Matrix</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">SparseMatrix</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rows</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">cols</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">_smat</span><span class="p">)</span>
-            <span class="k">return</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span> <span class="o">=</span> <span class="p">{}</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">collections</span><span class="o">.</span><span class="n">Callable</span><span class="p">):</span>
-                <span class="n">op</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-                    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-                        <span class="n">value</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">op</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span>
-                        <span class="k">if</span> <span class="n">value</span><span class="p">:</span>
-                            <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)]</span> <span class="o">=</span> <span class="n">value</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="p">(</span><span class="nb">dict</span><span class="p">,</span> <span class="n">Dict</span><span class="p">)):</span>
-                <span class="c"># manual copy, copy.deepcopy() doesn&#39;t work</span>
-                <span class="k">for</span> <span class="n">key</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
-                    <span class="n">v</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="n">key</span><span class="p">]</span>
-                    <span class="k">if</span> <span class="n">v</span><span class="p">:</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">=</span> <span class="n">v</span>
-            <span class="k">elif</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]):</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                        <span class="s">&#39;List length (</span><span class="si">%s</span><span class="s">) != rows*columns (</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span>
-                        <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">))</span>
-                <span class="n">flat_list</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-                    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-                        <span class="n">value</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">flat_list</span><span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">+</span> <span class="n">j</span><span class="p">])</span>
-                        <span class="k">if</span> <span class="n">value</span><span class="p">:</span>
-                            <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)]</span> <span class="o">=</span> <span class="n">value</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># handle full matrix forms with _handle_creation_inputs</span>
-            <span class="n">r</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">_list</span> <span class="o">=</span> <span class="n">Matrix</span><span class="o">.</span><span class="n">_handle_creation_inputs</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">=</span> <span class="n">r</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">=</span> <span class="n">c</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-                    <span class="n">value</span> <span class="o">=</span> <span class="n">_list</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="o">*</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">]</span>
-                    <span class="k">if</span> <span class="n">value</span><span class="p">:</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)]</span> <span class="o">=</span> <span class="n">value</span>
-
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">):</span>
-
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">key</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">tuple</span><span class="p">:</span>
-            <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">key</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="nb">int</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-                <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">key2ij</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
-                <span class="n">rv</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">get</span><span class="p">((</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">rv</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="nb">slice</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="nb">slice</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">submatrix</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
-
-        <span class="c"># check for single arg, like M[:] or M[3]</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="nb">slice</span><span class="p">):</span>
-            <span class="n">lo</span><span class="p">,</span> <span class="n">hi</span> <span class="o">=</span> <span class="n">key</span><span class="o">.</span><span class="n">indices</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="p">))[:</span><span class="mi">2</span><span class="p">]</span>
-            <span class="n">L</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">lo</span><span class="p">,</span> <span class="n">hi</span><span class="p">):</span>
-                <span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="nb">divmod</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">)</span>
-                <span class="n">L</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">get</span><span class="p">((</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">L</span>
-
-        <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="nb">divmod</span><span class="p">(</span><span class="n">a2idx</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="p">)),</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">get</span><span class="p">((</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__setitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">copy</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_Identity</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">(</span><span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span>
-
-<div class="viewcode-block" id="SparseMatrix.tolist"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.tolist">[docs]</a>    <span class="k">def</span> <span class="nf">tolist</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Convert this sparse matrix into a list of nested Python lists.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix, ones</span>
-<span class="sd">        &gt;&gt;&gt; a = SparseMatrix(((1, 2), (3, 4)))</span>
-<span class="sd">        &gt;&gt;&gt; a.tolist()</span>
-<span class="sd">        [[1, 2], [3, 4]]</span>
-
-<span class="sd">        When there are no rows then it will not be possible to tell how</span>
-<span class="sd">        many columns were in the original matrix:</span>
-
-<span class="sd">        &gt;&gt;&gt; SparseMatrix(ones(0, 3)).tolist()</span>
-<span class="sd">        []</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[]</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[[]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)]</span>
-        <span class="n">I</span><span class="p">,</span> <span class="n">J</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">shape</span>
-        <span class="k">return</span> <span class="p">[[</span><span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">J</span><span class="p">)]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">I</span><span class="p">)]</span>
-</div>
-<div class="viewcode-block" id="SparseMatrix.row"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.row">[docs]</a>    <span class="k">def</span> <span class="nf">row</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns column i from self as a row vector.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix</span>
-<span class="sd">        &gt;&gt;&gt; a = SparseMatrix(((1, 2), (3, 4)))</span>
-<span class="sd">        &gt;&gt;&gt; a.row(0)</span>
-<span class="sd">        [1, 2]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        col</span>
-<span class="sd">        row_list</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">smat</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">:</span>
-                <span class="n">smat</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span> <span class="n">smat</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="SparseMatrix.col"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.col">[docs]</a>    <span class="k">def</span> <span class="nf">col</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">j</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns column j from self as a column vector.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix</span>
-<span class="sd">        &gt;&gt;&gt; a = SparseMatrix(((1, 2), (3, 4)))</span>
-<span class="sd">        &gt;&gt;&gt; a.col(0)</span>
-<span class="sd">        [1]</span>
-<span class="sd">        [3]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        row</span>
-<span class="sd">        col_list</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">smat</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">:</span>
-                <span class="n">smat</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">smat</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="SparseMatrix.row_list"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.row_list">[docs]</a>    <span class="k">def</span> <span class="nf">row_list</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a row-sorted list of non-zero elements of the matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix</span>
-<span class="sd">        &gt;&gt;&gt; a = SparseMatrix(((1, 2), (3, 4)))</span>
-<span class="sd">        &gt;&gt;&gt; a</span>
-<span class="sd">        [1, 2]</span>
-<span class="sd">        [3, 4]</span>
-<span class="sd">        &gt;&gt;&gt; a.RL</span>
-<span class="sd">        [(0, 0, 1), (0, 1, 2), (1, 0, 3), (1, 1, 4)]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        row_op</span>
-<span class="sd">        col_list</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">new</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-                <span class="n">value</span> <span class="o">=</span> <span class="bp">self</span><span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)]</span>
-                <span class="k">if</span> <span class="n">value</span><span class="p">:</span>
-                    <span class="n">new</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">value</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">new</span>
-</div>
-    <span class="n">RL</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="n">row_list</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="s">&quot;Alternate faster representation&quot;</span><span class="p">)</span>
-
-<div class="viewcode-block" id="SparseMatrix.col_list"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.col_list">[docs]</a>    <span class="k">def</span> <span class="nf">col_list</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a column-sorted list of non-zero elements of the matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix</span>
-<span class="sd">        &gt;&gt;&gt; a=SparseMatrix(((1, 2), (3, 4)))</span>
-<span class="sd">        &gt;&gt;&gt; a</span>
-<span class="sd">        [1, 2]</span>
-<span class="sd">        [3, 4]</span>
-<span class="sd">        &gt;&gt;&gt; a.CL</span>
-<span class="sd">        [(0, 0, 1), (1, 0, 3), (0, 1, 2), (1, 1, 4)]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        col_op</span>
-<span class="sd">        row_list</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">new</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-                <span class="n">value</span> <span class="o">=</span> <span class="bp">self</span><span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)]</span>
-                <span class="k">if</span> <span class="n">value</span><span class="p">:</span>
-                    <span class="n">new</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">value</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">new</span>
-</div>
-    <span class="n">CL</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="n">col_list</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="s">&quot;Alternate faster representation&quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_trace</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Calculate the trace of a square matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import eye</span>
-<span class="sd">        &gt;&gt;&gt; eye(3).trace()</span>
-<span class="sd">        3</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">trace</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-            <span class="n">trace</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">get</span><span class="p">((</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">),</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">trace</span>
-
-    <span class="k">def</span> <span class="nf">_eval_transpose</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the transposed SparseMatrix of this SparseMatrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix</span>
-<span class="sd">        &gt;&gt;&gt; a = SparseMatrix(((1, 2), (3, 4)))</span>
-<span class="sd">        &gt;&gt;&gt; a</span>
-<span class="sd">        [1, 2]</span>
-<span class="sd">        [3, 4]</span>
-<span class="sd">        &gt;&gt;&gt; a.T</span>
-<span class="sd">        [1, 3]</span>
-<span class="sd">        [2, 4]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">tran</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="n">value</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">key</span> <span class="o">=</span> <span class="n">key</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">key</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>  <span class="c"># reverse</span>
-            <span class="n">tran</span><span class="o">.</span><span class="n">_smat</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">=</span> <span class="n">value</span>
-        <span class="k">return</span> <span class="n">tran</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the by-element conjugation.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import I</span>
-<span class="sd">        &gt;&gt;&gt; a = SparseMatrix(((1, 2 + I), (3, 4), (I, -I)))</span>
-<span class="sd">        &gt;&gt;&gt; a</span>
-<span class="sd">        [1, 2 + I]</span>
-<span class="sd">        [3,     4]</span>
-<span class="sd">        [I,    -I]</span>
-<span class="sd">        &gt;&gt;&gt; a.C</span>
-<span class="sd">        [ 1, 2 - I]</span>
-<span class="sd">        [ 3,     4]</span>
-<span class="sd">        [-I,     I]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        transpose: Matrix transposition</span>
-<span class="sd">        H: Hermite conjugation</span>
-<span class="sd">        D: Dirac conjugation</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">conj</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="n">value</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">conj</span><span class="o">.</span><span class="n">_smat</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">=</span> <span class="n">value</span><span class="o">.</span><span class="n">conjugate</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">conj</span>
-
-<div class="viewcode-block" id="SparseMatrix.multiply"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.multiply">[docs]</a>    <span class="k">def</span> <span class="nf">multiply</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Fast multiplication exploiting the sparsity of the matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix, ones</span>
-<span class="sd">        &gt;&gt;&gt; A, B = SparseMatrix(ones(4, 3)), SparseMatrix(ones(3, 4))</span>
-<span class="sd">        &gt;&gt;&gt; A.multiply(B) == 3*ones(4)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        add</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">A</span> <span class="o">=</span> <span class="bp">self</span>
-        <span class="n">B</span> <span class="o">=</span> <span class="n">other</span>
-        <span class="c"># sort B&#39;s row_list into list of rows</span>
-        <span class="n">Blist</span> <span class="o">=</span> <span class="p">[[]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">B</span><span class="o">.</span><span class="n">rows</span><span class="p">)]</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">B</span><span class="o">.</span><span class="n">row_list</span><span class="p">():</span>
-            <span class="n">Blist</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">j</span><span class="p">,</span> <span class="n">v</span><span class="p">))</span>
-        <span class="n">Cdict</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">Akj</span> <span class="ow">in</span> <span class="n">A</span><span class="o">.</span><span class="n">row_list</span><span class="p">():</span>
-            <span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">Bjn</span> <span class="ow">in</span> <span class="n">Blist</span><span class="p">[</span><span class="n">j</span><span class="p">]:</span>
-                <span class="n">temp</span> <span class="o">=</span> <span class="n">Akj</span><span class="o">*</span><span class="n">Bjn</span>
-                <span class="n">Cdict</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">n</span><span class="p">]</span> <span class="o">+=</span> <span class="n">temp</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="n">B</span><span class="o">.</span><span class="n">cols</span><span class="p">)</span>
-        <span class="n">rv</span><span class="o">.</span><span class="n">_smat</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">([(</span><span class="n">k</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">Cdict</span><span class="o">.</span><span class="n">items</span><span class="p">()</span> <span class="k">if</span> <span class="n">v</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">rv</span>
-</div>
-<div class="viewcode-block" id="SparseMatrix.scalar_multiply"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.scalar_multiply">[docs]</a>    <span class="k">def</span> <span class="nf">scalar_multiply</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">scalar</span><span class="p">):</span>
-        <span class="s">&quot;Scalar element-wise multiplication&quot;</span>
-        <span class="n">M</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">scalar</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">:</span>
-                <span class="n">v</span> <span class="o">=</span> <span class="n">scalar</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-                <span class="k">if</span> <span class="n">v</span><span class="p">:</span>
-                    <span class="n">M</span><span class="o">.</span><span class="n">_smat</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">v</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">M</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">M</span>
-</div>
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Multiply self and other, watching for non-matrix entities.</span>
-
-<span class="sd">        When multiplying be a non-sparse matrix, the result is no longer</span>
-<span class="sd">        sparse.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix, eye, zeros</span>
-<span class="sd">        &gt;&gt;&gt; I = SparseMatrix(eye(3))</span>
-<span class="sd">        &gt;&gt;&gt; I*I == I</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Z = zeros(3)</span>
-<span class="sd">        &gt;&gt;&gt; I*Z</span>
-<span class="sd">        [0, 0, 0]</span>
-<span class="sd">        [0, 0, 0]</span>
-<span class="sd">        [0, 0, 0]</span>
-<span class="sd">        &gt;&gt;&gt; I*2 == 2*I</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">SparseMatrix</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">multiply</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">MatrixBase</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">other</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="bp">self</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">other</span><span class="p">))</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">scalar_multiply</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return product the same type as other (if a Matrix).</span>
-
-<span class="sd">        When multiplying be a non-sparse matrix, the result is no longer</span>
-<span class="sd">        sparse.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import Matrix, SparseMatrix</span>
-<span class="sd">        &gt;&gt;&gt; A = Matrix(2, 2, list(range(1, 5)))</span>
-<span class="sd">        &gt;&gt;&gt; S = SparseMatrix(2, 2, list(range(2, 6)))</span>
-<span class="sd">        &gt;&gt;&gt; A*S == S*A</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; (isinstance(A*S, SparseMatrix) ==</span>
-<span class="sd">        ...  isinstance(S*A, SparseMatrix) == False)</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">MatrixBase</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">other</span><span class="o">*</span><span class="n">other</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">scalar_multiply</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Add other to self, efficiently if possible.</span>
-
-<span class="sd">        When adding a non-sparse matrix, the result is no longer</span>
-<span class="sd">        sparse.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix, eye</span>
-<span class="sd">        &gt;&gt;&gt; A = SparseMatrix(eye(3)) + SparseMatrix(eye(3))</span>
-<span class="sd">        &gt;&gt;&gt; B = SparseMatrix(eye(3)) + eye(3)</span>
-<span class="sd">        &gt;&gt;&gt; A</span>
-<span class="sd">        [2, 0, 0]</span>
-<span class="sd">        [0, 2, 0]</span>
-<span class="sd">        [0, 0, 2]</span>
-<span class="sd">        &gt;&gt;&gt; A == B</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; isinstance(A, SparseMatrix) and isinstance(B, SparseMatrix)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">SparseMatrix</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">MatrixBase</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">other</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">other</span> <span class="o">+</span> <span class="bp">self</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                <span class="s">&quot;Cannot add </span><span class="si">%s</span><span class="s"> to </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span>
-                <span class="nb">tuple</span><span class="p">([</span><span class="n">c</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">)]))</span>
-
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Negate all elements of self.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix, eye</span>
-<span class="sd">        &gt;&gt;&gt; -SparseMatrix(eye(3))</span>
-<span class="sd">        [-1,  0,  0]</span>
-<span class="sd">        [ 0, -1,  0]</span>
-<span class="sd">        [ 0,  0, -1]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">rv</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">rv</span><span class="o">.</span><span class="n">_smat</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">v</span>
-        <span class="k">return</span> <span class="n">rv</span>
-
-<div class="viewcode-block" id="SparseMatrix.add"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.add">[docs]</a>    <span class="k">def</span> <span class="nf">add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Add two sparse matrices with dictionary representation.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix, eye, ones</span>
-<span class="sd">        &gt;&gt;&gt; SparseMatrix(eye(3)).add(SparseMatrix(ones(3)))</span>
-<span class="sd">        [2, 1, 1]</span>
-<span class="sd">        [1, 2, 1]</span>
-<span class="sd">        [1, 1, 2]</span>
-<span class="sd">        &gt;&gt;&gt; SparseMatrix(eye(3)).add(-SparseMatrix(eye(3)))</span>
-<span class="sd">        [0, 0, 0]</span>
-<span class="sd">        [0, 0, 0]</span>
-<span class="sd">        [0, 0, 0]</span>
-
-<span class="sd">        Only the non-zero elements are stored, so the resulting dictionary</span>
-<span class="sd">        that is used to represent the sparse matrix is empty:</span>
-<span class="sd">        &gt;&gt;&gt; _._smat</span>
-<span class="sd">        {}</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        multiply</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">SparseMatrix</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;only use add with </span><span class="si">%s</span><span class="s">, not </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span>
-                <span class="nb">tuple</span><span class="p">([</span><span class="n">c</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)]))</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">shape</span> <span class="o">!=</span> <span class="n">other</span><span class="o">.</span><span class="n">shape</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ShapeError</span><span class="p">()</span>
-        <span class="n">M</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">other</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">v</span> <span class="o">=</span> <span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">v</span>
-            <span class="k">if</span> <span class="n">v</span><span class="p">:</span>
-                <span class="n">M</span><span class="o">.</span><span class="n">_smat</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">v</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">M</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">M</span>
-</div>
-    <span class="k">def</span> <span class="nf">submatrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">keys</span><span class="p">):</span>
-        <span class="n">rlo</span><span class="p">,</span> <span class="n">rhi</span><span class="p">,</span> <span class="n">clo</span><span class="p">,</span> <span class="n">chi</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">key2bounds</span><span class="p">(</span><span class="n">keys</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">rhi</span> <span class="o">-</span> <span class="n">rlo</span><span class="p">,</span> <span class="n">chi</span> <span class="o">-</span> <span class="n">clo</span><span class="p">,</span>
-            <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="n">rlo</span><span class="p">,</span> <span class="n">j</span> <span class="o">+</span> <span class="n">clo</span><span class="p">])</span>
-
-<div class="viewcode-block" id="SparseMatrix.is_symmetric"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.is_symmetric">[docs]</a>    <span class="k">def</span> <span class="nf">is_symmetric</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">simplify</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return True if self is symmetric.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix, eye</span>
-<span class="sd">        &gt;&gt;&gt; M = SparseMatrix(eye(3))</span>
-<span class="sd">        &gt;&gt;&gt; M.is_symmetric()</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; M[0, 2] = 1</span>
-<span class="sd">        &gt;&gt;&gt; M.is_symmetric()</span>
-<span class="sd">        False</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">simplify</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">all</span><span class="p">((</span><span class="n">k</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">k</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span> <span class="ow">and</span>
-                <span class="ow">not</span> <span class="p">(</span><span class="bp">self</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">-</span> <span class="bp">self</span><span class="p">[(</span><span class="n">k</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">k</span><span class="p">[</span><span class="mi">0</span><span class="p">])])</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-                <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">all</span><span class="p">((</span><span class="n">k</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">k</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span> <span class="ow">and</span>
-                <span class="bp">self</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">==</span> <span class="bp">self</span><span class="p">[(</span><span class="n">k</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">k</span><span class="p">[</span><span class="mi">0</span><span class="p">])]</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="SparseMatrix.has"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.has">[docs]</a>    <span class="k">def</span> <span class="nf">has</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">patterns</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Test whether any subexpression matches any of the patterns.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import SparseMatrix, Float</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; A = SparseMatrix(((1, x), (0.2, 3)))</span>
-<span class="sd">        &gt;&gt;&gt; A.has(x)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; A.has(y)</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; A.has(Float)</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">any</span><span class="p">(</span><span class="bp">self</span><span class="p">[</span><span class="n">key</span><span class="p">]</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">patterns</span><span class="p">)</span> <span class="k">for</span> <span class="n">key</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="SparseMatrix.applyfunc"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.applyfunc">[docs]</a>    <span class="k">def</span> <span class="nf">applyfunc</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Apply a function to each element of the matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix</span>
-<span class="sd">        &gt;&gt;&gt; m = SparseMatrix(2, 2, lambda i, j: i*2+j)</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [0, 1]</span>
-<span class="sd">        [2, 3]</span>
-<span class="sd">        &gt;&gt;&gt; m.applyfunc(lambda i: 2*i)</span>
-<span class="sd">        [0, 2]</span>
-<span class="sd">        [4, 6]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">collections</span><span class="o">.</span><span class="n">Callable</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;`f` must be callable.&quot;</span><span class="p">)</span>
-
-        <span class="n">out</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">fv</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">fv</span><span class="p">:</span>
-                <span class="n">out</span><span class="o">.</span><span class="n">_smat</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">fv</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">out</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">out</span>
-</div>
-<div class="viewcode-block" id="SparseMatrix.reshape"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.reshape">[docs]</a>    <span class="k">def</span> <span class="nf">reshape</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rows</span><span class="p">,</span> <span class="n">cols</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Reshape matrix while retaining original size.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix</span>
-<span class="sd">        &gt;&gt;&gt; S = SparseMatrix(4, 2, list(range(8)))</span>
-<span class="sd">        &gt;&gt;&gt; S.reshape(2, 4)</span>
-<span class="sd">        [0, 1, 2, 3]</span>
-<span class="sd">        [4, 5, 6, 7]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">!=</span> <span class="n">rows</span><span class="o">*</span><span class="n">cols</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Invalid reshape parameters </span><span class="si">%d</span><span class="s"> </span><span class="si">%d</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">rows</span><span class="p">,</span> <span class="n">cols</span><span class="p">))</span>
-        <span class="n">smat</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">k</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">+</span> <span class="n">j</span>
-            <span class="n">ii</span><span class="p">,</span> <span class="n">jj</span> <span class="o">=</span> <span class="nb">divmod</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">cols</span><span class="p">)</span>
-            <span class="n">smat</span><span class="p">[(</span><span class="n">ii</span><span class="p">,</span> <span class="n">jj</span><span class="p">)]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)]</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">rows</span><span class="p">,</span> <span class="n">cols</span><span class="p">,</span> <span class="n">smat</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="SparseMatrix.liupc"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.liupc">[docs]</a>    <span class="k">def</span> <span class="nf">liupc</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Liu&#39;s algorithm, for pre-determination of the Elimination Tree of</span>
-<span class="sd">        the given matrix, used in row-based symbolic Cholesky factorization.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix</span>
-<span class="sd">        &gt;&gt;&gt; S = SparseMatrix([</span>
-<span class="sd">        ... [1, 0, 3, 2],</span>
-<span class="sd">        ... [0, 0, 1, 0],</span>
-<span class="sd">        ... [4, 0, 0, 5],</span>
-<span class="sd">        ... [0, 6, 7, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; S.liupc()</span>
-<span class="sd">        ([[0], [], [0], [1, 2]], [4, 3, 4, 4])</span>
-
-<span class="sd">        References</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        Symbolic Sparse Cholesky Factorization using Elimination Trees,</span>
-<span class="sd">        Jeroen Van Grondelle (1999)</span>
-<span class="sd">        http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.39.7582,</span>
-<span class="sd">        downloaded from http://tinyurl.com/9o2jsxj</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># Algorithm 2.4, p 17 of reference</span>
-
-        <span class="c"># get the indices of the elements that are non-zero on or below diag</span>
-        <span class="n">R</span> <span class="o">=</span> <span class="p">[[]</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)]</span>
-        <span class="k">for</span> <span class="n">r</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">row_list</span><span class="p">():</span>
-            <span class="k">if</span> <span class="n">c</span> <span class="o">&lt;=</span> <span class="n">r</span><span class="p">:</span>
-                <span class="n">R</span><span class="p">[</span><span class="n">r</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-
-        <span class="n">inf</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">R</span><span class="p">)</span>  <span class="c"># nothing will be this large</span>
-        <span class="n">parent</span> <span class="o">=</span> <span class="p">[</span><span class="n">inf</span><span class="p">]</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span>
-        <span class="n">virtual</span> <span class="o">=</span> <span class="p">[</span><span class="n">inf</span><span class="p">]</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span>
-        <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">R</span><span class="p">[</span><span class="n">r</span><span class="p">][:</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-                <span class="k">while</span> <span class="n">virtual</span><span class="p">[</span><span class="n">c</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">r</span><span class="p">:</span>
-                    <span class="n">t</span> <span class="o">=</span> <span class="n">virtual</span><span class="p">[</span><span class="n">c</span><span class="p">]</span>
-                    <span class="n">virtual</span><span class="p">[</span><span class="n">c</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span>
-                    <span class="n">c</span> <span class="o">=</span> <span class="n">t</span>
-                <span class="k">if</span> <span class="n">virtual</span><span class="p">[</span><span class="n">c</span><span class="p">]</span> <span class="o">==</span> <span class="n">inf</span><span class="p">:</span>
-                    <span class="n">parent</span><span class="p">[</span><span class="n">c</span><span class="p">]</span> <span class="o">=</span> <span class="n">virtual</span><span class="p">[</span><span class="n">c</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span>
-        <span class="k">return</span> <span class="n">R</span><span class="p">,</span> <span class="n">parent</span>
-</div>
-<div class="viewcode-block" id="SparseMatrix.row_structure_symbolic_cholesky"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.row_structure_symbolic_cholesky">[docs]</a>    <span class="k">def</span> <span class="nf">row_structure_symbolic_cholesky</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Symbolic cholesky factorization, for pre-determination of the</span>
-<span class="sd">        non-zero structure of the Cholesky factororization.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix</span>
-<span class="sd">        &gt;&gt;&gt; S = SparseMatrix([</span>
-<span class="sd">        ... [1, 0, 3, 2],</span>
-<span class="sd">        ... [0, 0, 1, 0],</span>
-<span class="sd">        ... [4, 0, 0, 5],</span>
-<span class="sd">        ... [0, 6, 7, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; S.row_structure_symbolic_cholesky()</span>
-<span class="sd">        [[0], [], [0], [1, 2]]</span>
-
-<span class="sd">        References</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        Symbolic Sparse Cholesky Factorization using Elimination Trees,</span>
-<span class="sd">        Jeroen Van Grondelle (1999)</span>
-<span class="sd">        http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.39.7582,</span>
-<span class="sd">        downloaded from http://tinyurl.com/9o2jsxj</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">R</span><span class="p">,</span> <span class="n">parent</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">liupc</span><span class="p">()</span>
-        <span class="n">inf</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">R</span><span class="p">)</span>  <span class="c"># this acts as infinity</span>
-        <span class="n">Lrow</span> <span class="o">=</span> <span class="n">copy</span><span class="o">.</span><span class="n">deepcopy</span><span class="p">(</span><span class="n">R</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">R</span><span class="p">[</span><span class="n">k</span><span class="p">]:</span>
-                <span class="k">while</span> <span class="n">j</span> <span class="o">!=</span> <span class="n">inf</span> <span class="ow">and</span> <span class="n">j</span> <span class="o">!=</span> <span class="n">k</span><span class="p">:</span>
-                    <span class="n">Lrow</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-                    <span class="n">j</span> <span class="o">=</span> <span class="n">parent</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-            <span class="n">Lrow</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">Lrow</span><span class="p">[</span><span class="n">k</span><span class="p">])))</span>
-        <span class="k">return</span> <span class="n">Lrow</span>
-</div>
-    <span class="k">def</span> <span class="nf">_cholesky_sparse</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Algorithm for numeric Cholesky factorization of a sparse matrix.&quot;&quot;&quot;</span>
-        <span class="n">Crowstruc</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">row_structure_symbolic_cholesky</span><span class="p">()</span>
-        <span class="n">C</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">Crowstruc</span><span class="p">)):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">Crowstruc</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                <span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="n">j</span><span class="p">:</span>
-                    <span class="n">C</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-                    <span class="n">summ</span> <span class="o">=</span> <span class="mi">0</span>
-                    <span class="k">for</span> <span class="n">p1</span> <span class="ow">in</span> <span class="n">Crowstruc</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                        <span class="k">if</span> <span class="n">p1</span> <span class="o">&lt;</span> <span class="n">j</span><span class="p">:</span>
-                            <span class="k">for</span> <span class="n">p2</span> <span class="ow">in</span> <span class="n">Crowstruc</span><span class="p">[</span><span class="n">j</span><span class="p">]:</span>
-                                <span class="k">if</span> <span class="n">p2</span> <span class="o">&lt;</span> <span class="n">j</span><span class="p">:</span>
-                                    <span class="k">if</span> <span class="n">p1</span> <span class="o">==</span> <span class="n">p2</span><span class="p">:</span>
-                                        <span class="n">summ</span> <span class="o">+=</span> <span class="n">C</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">p1</span><span class="p">]</span><span class="o">*</span><span class="n">C</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">p1</span><span class="p">]</span>
-                                <span class="k">else</span><span class="p">:</span>
-                                    <span class="k">break</span>
-                            <span class="k">else</span><span class="p">:</span>
-                                <span class="k">break</span>
-                    <span class="n">C</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">-=</span> <span class="n">summ</span>
-                    <span class="n">C</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">/=</span> <span class="n">C</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">C</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-                    <span class="n">summ</span> <span class="o">=</span> <span class="mi">0</span>
-                    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">Crowstruc</span><span class="p">[</span><span class="n">j</span><span class="p">]:</span>
-                        <span class="k">if</span> <span class="n">k</span> <span class="o">&lt;</span> <span class="n">j</span><span class="p">:</span>
-                            <span class="n">summ</span> <span class="o">+=</span> <span class="n">C</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="k">break</span>
-                    <span class="n">C</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">-=</span> <span class="n">summ</span>
-                    <span class="n">C</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">C</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">])</span>
-
-        <span class="k">return</span> <span class="n">C</span>
-
-    <span class="k">def</span> <span class="nf">_LDL_sparse</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Algorithm for numeric LDL factization, exploiting sparse structure.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">Lrowstruc</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">row_structure_symbolic_cholesky</span><span class="p">()</span>
-        <span class="n">L</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">Lrowstruc</span><span class="p">)):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">Lrowstruc</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                <span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="n">j</span><span class="p">:</span>
-                    <span class="n">L</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-                    <span class="n">summ</span> <span class="o">=</span> <span class="mi">0</span>
-                    <span class="k">for</span> <span class="n">p1</span> <span class="ow">in</span> <span class="n">Lrowstruc</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                        <span class="k">if</span> <span class="n">p1</span> <span class="o">&lt;</span> <span class="n">j</span><span class="p">:</span>
-                            <span class="k">for</span> <span class="n">p2</span> <span class="ow">in</span> <span class="n">Lrowstruc</span><span class="p">[</span><span class="n">j</span><span class="p">]:</span>
-                                <span class="k">if</span> <span class="n">p2</span> <span class="o">&lt;</span> <span class="n">j</span><span class="p">:</span>
-                                    <span class="k">if</span> <span class="n">p1</span> <span class="o">==</span> <span class="n">p2</span><span class="p">:</span>
-                                        <span class="n">summ</span> <span class="o">+=</span> <span class="n">L</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">p1</span><span class="p">]</span><span class="o">*</span><span class="n">L</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">p1</span><span class="p">]</span><span class="o">*</span><span class="n">D</span><span class="p">[</span><span class="n">p1</span><span class="p">,</span> <span class="n">p1</span><span class="p">]</span>
-                                <span class="k">else</span><span class="p">:</span>
-                                    <span class="k">break</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="k">break</span>
-                    <span class="n">L</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">-=</span> <span class="n">summ</span>
-                    <span class="n">L</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">/=</span> <span class="n">D</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-                <span class="k">elif</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
-                    <span class="n">D</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span>
-                    <span class="n">summ</span> <span class="o">=</span> <span class="mi">0</span>
-                    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">Lrowstruc</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                        <span class="k">if</span> <span class="n">k</span> <span class="o">&lt;</span> <span class="n">i</span><span class="p">:</span>
-                            <span class="n">summ</span> <span class="o">+=</span> <span class="n">L</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">D</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="k">break</span>
-                    <span class="n">D</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">-=</span> <span class="n">summ</span>
-
-        <span class="k">return</span> <span class="n">L</span><span class="p">,</span> <span class="n">D</span>
-
-    <span class="k">def</span> <span class="nf">_lower_triangular_solve</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Fast algorithm for solving a lower-triangular system,</span>
-<span class="sd">        exploiting the sparsity of the given matrix.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">rows</span> <span class="o">=</span> <span class="p">[[]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)]</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">row_list</span><span class="p">():</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">&gt;</span> <span class="n">j</span><span class="p">:</span>
-                <span class="n">rows</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">j</span><span class="p">,</span> <span class="n">v</span><span class="p">))</span>
-        <span class="n">X</span> <span class="o">=</span> <span class="n">rhs</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">rows</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                <span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-=</span> <span class="n">v</span><span class="o">*</span><span class="n">X</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-            <span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">/=</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_upper_triangular_solve</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Fast algorithm for solving an upper-triangular system,</span>
-<span class="sd">        exploiting the sparsity of the given matrix.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">rows</span> <span class="o">=</span> <span class="p">[[]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)]</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">row_list</span><span class="p">():</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">j</span><span class="p">:</span>
-                <span class="n">rows</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">j</span><span class="p">,</span> <span class="n">v</span><span class="p">))</span>
-        <span class="n">X</span> <span class="o">=</span> <span class="n">rhs</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-            <span class="n">rows</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-            <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">rows</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                <span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">-=</span> <span class="n">v</span><span class="o">*</span><span class="n">X</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-            <span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">/=</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_diagonal_solve</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="s">&quot;Diagonal solve.&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="n">rhs</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">/</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_cholesky_solve</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="c"># for speed reasons, this is not uncommented, but if you are</span>
-        <span class="c"># having difficulties, try uncommenting to make sure that the</span>
-        <span class="c"># input matrix is symmetric</span>
-
-        <span class="c">#assert self.is_symmetric()</span>
-        <span class="n">L</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_cholesky_sparse</span><span class="p">()</span>
-        <span class="n">Y</span> <span class="o">=</span> <span class="n">L</span><span class="o">.</span><span class="n">_lower_triangular_solve</span><span class="p">(</span><span class="n">rhs</span><span class="p">)</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">L</span><span class="o">.</span><span class="n">T</span><span class="o">.</span><span class="n">_upper_triangular_solve</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">rv</span>
-
-    <span class="k">def</span> <span class="nf">_LDL_solve</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
-        <span class="c"># for speed reasons, this is not uncommented, but if you are</span>
-        <span class="c"># having difficulties, try uncommenting to make sure that the</span>
-        <span class="c"># input matrix is symmetric</span>
-
-        <span class="c">#assert self.is_symmetric()</span>
-        <span class="n">L</span><span class="p">,</span> <span class="n">D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_LDL_sparse</span><span class="p">()</span>
-        <span class="n">Z</span> <span class="o">=</span> <span class="n">L</span><span class="o">.</span><span class="n">_lower_triangular_solve</span><span class="p">(</span><span class="n">rhs</span><span class="p">)</span>
-        <span class="n">Y</span> <span class="o">=</span> <span class="n">D</span><span class="o">.</span><span class="n">_diagonal_solve</span><span class="p">(</span><span class="n">Z</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">L</span><span class="o">.</span><span class="n">T</span><span class="o">.</span><span class="n">_upper_triangular_solve</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span>
-
-<div class="viewcode-block" id="SparseMatrix.cholesky"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.cholesky">[docs]</a>    <span class="k">def</span> <span class="nf">cholesky</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the Cholesky decomposition L of a matrix A</span>
-<span class="sd">        such that L * L.T = A</span>
-
-<span class="sd">        A must be a square, symmetric, positive-definite</span>
-<span class="sd">        and non-singular matrix</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix</span>
-<span class="sd">        &gt;&gt;&gt; A = SparseMatrix(((25,15,-5),(15,18,0),(-5,0,11)))</span>
-<span class="sd">        &gt;&gt;&gt; A.cholesky()</span>
-<span class="sd">        [ 5, 0, 0]</span>
-<span class="sd">        [ 3, 3, 0]</span>
-<span class="sd">        [-1, 1, 3]</span>
-<span class="sd">        &gt;&gt;&gt; A.cholesky() * A.cholesky().T == A</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">nan</span><span class="p">,</span> <span class="n">oo</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_symmetric</span><span class="p">():</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Cholesky decomposition applies only to &#39;</span>
-                <span class="s">&#39;symmetric matrices.&#39;</span><span class="p">)</span>
-        <span class="n">M</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_mutable</span><span class="p">()</span><span class="o">.</span><span class="n">_cholesky_sparse</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">M</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">nan</span><span class="p">)</span> <span class="ow">or</span> <span class="n">M</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">oo</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Cholesky decomposition applies only to &#39;</span>
-                <span class="s">&#39;positive-definite matrices&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="SparseMatrix.LDLdecomposition"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.LDLdecomposition">[docs]</a>    <span class="k">def</span> <span class="nf">LDLdecomposition</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the LDL Decomposition (matrices ``L`` and ``D``) of matrix</span>
-<span class="sd">        ``A``, such that ``L * D * L.T == A``. ``A`` must be a square,</span>
-<span class="sd">        symmetric, positive-definite and non-singular.</span>
-
-<span class="sd">        This method eliminates the use of square root and ensures that all</span>
-<span class="sd">        the diagonal entries of L are 1.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix</span>
-<span class="sd">        &gt;&gt;&gt; A = SparseMatrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))</span>
-<span class="sd">        &gt;&gt;&gt; L, D = A.LDLdecomposition()</span>
-<span class="sd">        &gt;&gt;&gt; L</span>
-<span class="sd">        [   1,   0, 0]</span>
-<span class="sd">        [ 3/5,   1, 0]</span>
-<span class="sd">        [-1/5, 1/3, 1]</span>
-<span class="sd">        &gt;&gt;&gt; D</span>
-<span class="sd">        [25, 0, 0]</span>
-<span class="sd">        [ 0, 9, 0]</span>
-<span class="sd">        [ 0, 0, 9]</span>
-<span class="sd">        &gt;&gt;&gt; L * D * L.T == A</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">nan</span><span class="p">,</span> <span class="n">oo</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_symmetric</span><span class="p">():</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;LDL decomposition applies only to &#39;</span>
-                <span class="s">&#39;symmetric matrices.&#39;</span><span class="p">)</span>
-        <span class="n">L</span><span class="p">,</span> <span class="n">D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_mutable</span><span class="p">()</span><span class="o">.</span><span class="n">_LDL_sparse</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">L</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">nan</span><span class="p">)</span> <span class="ow">or</span> <span class="n">L</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">oo</span><span class="p">)</span> <span class="ow">or</span> <span class="n">D</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">nan</span><span class="p">)</span> <span class="ow">or</span> <span class="n">D</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">oo</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;LDL decomposition applies only to &#39;</span>
-                <span class="s">&#39;positive-definite matrices&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">L</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">D</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="SparseMatrix.solve_least_squares"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.solve_least_squares">[docs]</a>    <span class="k">def</span> <span class="nf">solve_least_squares</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rhs</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;LDL&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the least-square fit to the data.</span>
-
-<span class="sd">        By default the cholesky_solve routine is used (method=&#39;CH&#39;); other</span>
-<span class="sd">        methods of matrix inversion can be used. To find out which are</span>
-<span class="sd">        available, see the docstring of the .inv() method.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix, Matrix, ones</span>
-<span class="sd">        &gt;&gt;&gt; A = Matrix([1, 2, 3])</span>
-<span class="sd">        &gt;&gt;&gt; B = Matrix([2, 3, 4])</span>
-<span class="sd">        &gt;&gt;&gt; S = SparseMatrix(A.row_join(B))</span>
-<span class="sd">        &gt;&gt;&gt; S</span>
-<span class="sd">        [1, 2]</span>
-<span class="sd">        [2, 3]</span>
-<span class="sd">        [3, 4]</span>
-
-<span class="sd">        If each line of S represent coefficients of Ax + By</span>
-<span class="sd">        and x and y are [2, 3] then S*xy is:</span>
-
-<span class="sd">        &gt;&gt;&gt; r = S*Matrix([2, 3]); r</span>
-<span class="sd">        [ 8]</span>
-<span class="sd">        [13]</span>
-<span class="sd">        [18]</span>
-
-<span class="sd">        But let&#39;s add 1 to the middle value and then solve for the</span>
-<span class="sd">        least-squares value of xy:</span>
-
-<span class="sd">        &gt;&gt;&gt; xy = S.solve_least_squares(Matrix([8, 14, 18])); xy</span>
-<span class="sd">        [ 5/3]</span>
-<span class="sd">        [10/3]</span>
-
-<span class="sd">        The error is given by S*xy - r:</span>
-
-<span class="sd">        &gt;&gt;&gt; S*xy - r</span>
-<span class="sd">        [1/3]</span>
-<span class="sd">        [1/3]</span>
-<span class="sd">        [1/3]</span>
-<span class="sd">        &gt;&gt;&gt; _.norm().n(2)</span>
-<span class="sd">        0.58</span>
-
-<span class="sd">        If a different xy is used, the norm will be higher:</span>
-
-<span class="sd">        &gt;&gt;&gt; xy += ones(2, 1)/10</span>
-<span class="sd">        &gt;&gt;&gt; (S*xy - r).norm().n(2)</span>
-<span class="sd">        1.5</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">T</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">t</span><span class="o">*</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">method</span><span class="o">=</span><span class="n">method</span><span class="p">)</span><span class="o">*</span><span class="n">t</span><span class="o">*</span><span class="n">rhs</span>
-</div>
-<div class="viewcode-block" id="SparseMatrix.solve"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.solve">[docs]</a>    <span class="k">def</span> <span class="nf">solve</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rhs</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;LDL&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return solution to self*soln = rhs using given inversion method.</span>
-
-<span class="sd">        For a list of possible inversion methods, see the .inv() docstring.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_square</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Under-determined system.&#39;</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">&gt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;For over-determined system, M, having &#39;</span>
-                    <span class="s">&#39;more rows than columns, try M.solve_least_squares(rhs).&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">method</span><span class="o">=</span><span class="n">method</span><span class="p">)</span><span class="o">*</span><span class="n">rhs</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the matrix inverse using Cholesky or LDL (default)</span>
-<span class="sd">        decomposition as selected with the ``method`` keyword: &#39;CH&#39; or &#39;LDL&#39;,</span>
-<span class="sd">        respectively.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import SparseMatrix, Matrix</span>
-<span class="sd">        &gt;&gt;&gt; A = SparseMatrix([</span>
-<span class="sd">        ... [ 2, -1,  0],</span>
-<span class="sd">        ... [-1,  2, -1],</span>
-<span class="sd">        ... [ 0,  0,  2]])</span>
-<span class="sd">        &gt;&gt;&gt; A.inv(&#39;CH&#39;)</span>
-<span class="sd">        [2/3, 1/3, 1/6]</span>
-<span class="sd">        [1/3, 2/3, 1/3]</span>
-<span class="sd">        [  0,   0, 1/2]</span>
-<span class="sd">        &gt;&gt;&gt; A.inv(method=&#39;LDL&#39;) # use of &#39;method=&#39; is optional</span>
-<span class="sd">        [2/3, 1/3, 1/6]</span>
-<span class="sd">        [1/3, 2/3, 1/3]</span>
-<span class="sd">        [  0,   0, 1/2]</span>
-<span class="sd">        &gt;&gt;&gt; A * _</span>
-<span class="sd">        [1, 0, 0]</span>
-<span class="sd">        [0, 1, 0]</span>
-<span class="sd">        [0, 0, 1]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">sym</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_symmetric</span><span class="p">()</span>
-        <span class="n">M</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_mutable</span><span class="p">()</span>
-        <span class="n">I</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">rows</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">sym</span><span class="p">:</span>
-            <span class="n">t</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">T</span>
-            <span class="n">r1</span> <span class="o">=</span> <span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:]</span>
-            <span class="n">M</span> <span class="o">=</span> <span class="n">t</span><span class="o">*</span><span class="n">M</span>
-            <span class="n">I</span> <span class="o">=</span> <span class="n">t</span><span class="o">*</span><span class="n">I</span>
-        <span class="n">method</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;method&#39;</span><span class="p">,</span> <span class="s">&#39;LDL&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">method</span> <span class="ow">in</span> <span class="s">&quot;LDL&quot;</span><span class="p">:</span>
-            <span class="n">solve</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">_LDL_solve</span>
-        <span class="k">elif</span> <span class="n">method</span> <span class="o">==</span> <span class="s">&quot;CH&quot;</span><span class="p">:</span>
-            <span class="n">solve</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">_cholesky_solve</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                <span class="s">&#39;Method may be &quot;CH&quot; or &quot;LDL&quot;, not </span><span class="si">%s</span><span class="s">.&#39;</span> <span class="o">%</span> <span class="n">method</span><span class="p">)</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">hstack</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">solve</span><span class="p">(</span><span class="n">I</span><span class="p">[:,</span> <span class="n">i</span><span class="p">])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">I</span><span class="o">.</span><span class="n">cols</span><span class="p">)])</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">sym</span><span class="p">:</span>
-            <span class="n">scale</span> <span class="o">=</span> <span class="p">(</span><span class="n">r1</span><span class="o">*</span><span class="n">rv</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-            <span class="n">rv</span> <span class="o">/=</span> <span class="n">scale</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">rv</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">shape</span> <span class="o">!=</span> <span class="n">other</span><span class="o">.</span><span class="n">shape</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">SparseMatrix</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">_smat</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">MatrixBase</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span> <span class="o">==</span> <span class="n">MutableSparseMatrix</span><span class="p">(</span><span class="n">other</span><span class="p">)</span><span class="o">.</span><span class="n">_smat</span>
-        <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="bp">self</span> <span class="o">==</span> <span class="n">other</span>
-
-<div class="viewcode-block" id="SparseMatrix.as_mutable"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.as_mutable">[docs]</a>    <span class="k">def</span> <span class="nf">as_mutable</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a mutable version of this matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import ImmutableMatrix</span>
-<span class="sd">        &gt;&gt;&gt; X = ImmutableMatrix([[1, 2], [3, 4]])</span>
-<span class="sd">        &gt;&gt;&gt; Y = X.as_mutable()</span>
-<span class="sd">        &gt;&gt;&gt; Y[1, 1] = 5 # Can set values in Y</span>
-<span class="sd">        &gt;&gt;&gt; Y</span>
-<span class="sd">        [1, 2]</span>
-<span class="sd">        [3, 5]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">MutableSparseMatrix</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="SparseMatrix.as_immutable"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.as_immutable">[docs]</a>    <span class="k">def</span> <span class="nf">as_immutable</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns an Immutable version of this Matrix.&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">.immutable</span> <span class="kn">import</span> <span class="n">ImmutableSparseMatrix</span>
-        <span class="k">return</span> <span class="n">ImmutableSparseMatrix</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="SparseMatrix.zeros"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.zeros">[docs]</a>    <span class="k">def</span> <span class="nf">zeros</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return an r x c matrix of zeros, square if c is omitted.&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">r</span><span class="p">):</span>
-            <span class="n">SymPyDeprecationWarning</span><span class="p">(</span>
-                <span class="n">feature</span><span class="o">=</span><span class="s">&quot;The syntax zeros([</span><span class="si">%i</span><span class="s">, </span><span class="si">%i</span><span class="s">])&quot;</span> <span class="o">%</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">r</span><span class="p">),</span>
-                <span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;zeros(</span><span class="si">%i</span><span class="s">, </span><span class="si">%i</span><span class="s">).&quot;</span> <span class="o">%</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">r</span><span class="p">),</span>
-                <span class="n">issue</span><span class="o">=</span><span class="mi">3381</span><span class="p">,</span> <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.2&quot;</span><span class="p">,</span>
-            <span class="p">)</span><span class="o">.</span><span class="n">warn</span><span class="p">()</span>
-            <span class="n">r</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">r</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">r</span> <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="bp">None</span> <span class="k">else</span> <span class="n">c</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="p">{})</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="SparseMatrix.eye"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.eye">[docs]</a>    <span class="k">def</span> <span class="nf">eye</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return an n x n identity matrix.&quot;&quot;&quot;</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="nb">dict</span><span class="p">([((</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)]))</span>
-</div>
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">hash</span><span class="p">((</span><span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__name__</span><span class="p">,)</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">)))</span>
-
-</div>
-<div class="viewcode-block" id="MutableSparseMatrix"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.MutableSparseMatrix">[docs]</a><span class="k">class</span> <span class="nc">MutableSparseMatrix</span><span class="p">(</span><span class="n">SparseMatrix</span><span class="p">,</span> <span class="n">MatrixBase</span><span class="p">):</span>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_new</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__setitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Assign value to position designated by key.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix, ones</span>
-<span class="sd">        &gt;&gt;&gt; M = SparseMatrix(2, 2, {})</span>
-<span class="sd">        &gt;&gt;&gt; M[1] = 1; M</span>
-<span class="sd">        [0, 1]</span>
-<span class="sd">        [0, 0]</span>
-<span class="sd">        &gt;&gt;&gt; M[1, 1] = 2; M</span>
-<span class="sd">        [0, 1]</span>
-<span class="sd">        [0, 2]</span>
-<span class="sd">        &gt;&gt;&gt; M = SparseMatrix(2, 2, {})</span>
-<span class="sd">        &gt;&gt;&gt; M[:, 1] = [1, 1]; M</span>
-<span class="sd">        [0, 1]</span>
-<span class="sd">        [0, 1]</span>
-<span class="sd">        &gt;&gt;&gt; M = SparseMatrix(2, 2, {})</span>
-<span class="sd">        &gt;&gt;&gt; M[1, :] = [[1, 1]]; M</span>
-<span class="sd">        [0, 0]</span>
-<span class="sd">        [1, 1]</span>
-
-
-<span class="sd">        To replace row r you assign to position r*m where m</span>
-<span class="sd">        is the number of columns:</span>
-
-<span class="sd">        &gt;&gt;&gt; M = SparseMatrix(4, 4, {})</span>
-<span class="sd">        &gt;&gt;&gt; m = M.cols</span>
-<span class="sd">        &gt;&gt;&gt; M[3*m] = ones(1, m)*2; M</span>
-<span class="sd">        [0, 0, 0, 0]</span>
-<span class="sd">        [0, 0, 0, 0]</span>
-<span class="sd">        [0, 0, 0, 0]</span>
-<span class="sd">        [2, 2, 2, 2]</span>
-
-<span class="sd">        And to replace column c you can assign to position c:</span>
-
-<span class="sd">        &gt;&gt;&gt; M[2] = ones(m, 1)*4; M</span>
-<span class="sd">        [0, 0, 4, 0]</span>
-<span class="sd">        [0, 0, 4, 0]</span>
-<span class="sd">        [0, 0, 4, 0]</span>
-<span class="sd">        [2, 2, 4, 2]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_setitem</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">value</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">rv</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">value</span> <span class="o">=</span> <span class="n">rv</span>
-            <span class="k">if</span> <span class="n">value</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)]</span> <span class="o">=</span> <span class="n">value</span>
-            <span class="k">elif</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">:</span>
-                <span class="k">del</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)]</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">hash</span><span class="p">((</span><span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__name__</span><span class="p">,)</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_mat</span><span class="p">)))</span>
-
-<div class="viewcode-block" id="MutableSparseMatrix.row_del"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.MutableSparseMatrix.row_del">[docs]</a>    <span class="k">def</span> <span class="nf">row_del</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Delete the given row of the matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix</span>
-<span class="sd">        &gt;&gt;&gt; M = SparseMatrix([[0, 0], [0, 1]])</span>
-<span class="sd">        &gt;&gt;&gt; M</span>
-<span class="sd">        [0, 0]</span>
-<span class="sd">        [0, 1]</span>
-<span class="sd">        &gt;&gt;&gt; M.row_del(0)</span>
-<span class="sd">        &gt;&gt;&gt; M</span>
-<span class="sd">        [0, 1]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        col_del</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">newD</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">a2idx</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)</span>
-        <span class="k">for</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">k</span><span class="p">:</span>
-                <span class="k">pass</span>
-            <span class="k">elif</span> <span class="n">i</span> <span class="o">&gt;</span> <span class="n">k</span><span class="p">:</span>
-                <span class="n">newD</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">newD</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span> <span class="o">=</span> <span class="n">newD</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">rows</span> <span class="o">-=</span> <span class="mi">1</span>
-</div>
-<div class="viewcode-block" id="MutableSparseMatrix.col_del"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.MutableSparseMatrix.col_del">[docs]</a>    <span class="k">def</span> <span class="nf">col_del</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Delete the given column of the matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix</span>
-<span class="sd">        &gt;&gt;&gt; M = SparseMatrix([[0, 0], [0, 1]])</span>
-<span class="sd">        &gt;&gt;&gt; M</span>
-<span class="sd">        [0, 0]</span>
-<span class="sd">        [0, 1]</span>
-<span class="sd">        &gt;&gt;&gt; M.col_del(0)</span>
-<span class="sd">        &gt;&gt;&gt; M</span>
-<span class="sd">        [0]</span>
-<span class="sd">        [1]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        row_del</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">newD</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">a2idx</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">)</span>
-        <span class="k">for</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">j</span> <span class="o">==</span> <span class="n">k</span><span class="p">:</span>
-                <span class="k">pass</span>
-            <span class="k">elif</span> <span class="n">j</span> <span class="o">&gt;</span> <span class="n">k</span><span class="p">:</span>
-                <span class="n">newD</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">newD</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span> <span class="o">=</span> <span class="n">newD</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">cols</span> <span class="o">-=</span> <span class="mi">1</span>
-</div>
-<div class="viewcode-block" id="MutableSparseMatrix.row_swap"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.MutableSparseMatrix.row_swap">[docs]</a>    <span class="k">def</span> <span class="nf">row_swap</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Swap, in place, columns i and j.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix</span>
-<span class="sd">        &gt;&gt;&gt; S = SparseMatrix.eye(3); S[2, 1] = 2</span>
-<span class="sd">        &gt;&gt;&gt; S.row_swap(1, 0); S</span>
-<span class="sd">        [0, 1, 0]</span>
-<span class="sd">        [1, 0, 0]</span>
-<span class="sd">        [0, 2, 1]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="o">&gt;</span> <span class="n">j</span><span class="p">:</span>
-            <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">j</span><span class="p">,</span> <span class="n">i</span>
-        <span class="n">rows</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">row_list</span><span class="p">()</span>
-        <span class="n">temp</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">ii</span><span class="p">,</span> <span class="n">jj</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">rows</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">ii</span> <span class="o">==</span> <span class="n">i</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">pop</span><span class="p">((</span><span class="n">ii</span><span class="p">,</span> <span class="n">jj</span><span class="p">))</span>
-                <span class="n">temp</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">jj</span><span class="p">,</span> <span class="n">v</span><span class="p">))</span>
-            <span class="k">elif</span> <span class="n">ii</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">pop</span><span class="p">((</span><span class="n">ii</span><span class="p">,</span> <span class="n">jj</span><span class="p">))</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">jj</span><span class="p">]</span> <span class="o">=</span> <span class="n">v</span>
-            <span class="k">elif</span> <span class="n">ii</span> <span class="o">&gt;</span> <span class="n">j</span><span class="p">:</span>
-                <span class="k">break</span>
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">temp</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">v</span>
-</div>
-<div class="viewcode-block" id="MutableSparseMatrix.col_swap"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.MutableSparseMatrix.col_swap">[docs]</a>    <span class="k">def</span> <span class="nf">col_swap</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Swap, in place, columns i and j.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix</span>
-<span class="sd">        &gt;&gt;&gt; S = SparseMatrix.eye(3); S[2, 1] = 2</span>
-<span class="sd">        &gt;&gt;&gt; S.col_swap(1, 0); S</span>
-<span class="sd">        [0, 1, 0]</span>
-<span class="sd">        [1, 0, 0]</span>
-<span class="sd">        [2, 0, 1]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="o">&gt;</span> <span class="n">j</span><span class="p">:</span>
-            <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">j</span><span class="p">,</span> <span class="n">i</span>
-        <span class="n">rows</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">col_list</span><span class="p">()</span>
-        <span class="n">temp</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">ii</span><span class="p">,</span> <span class="n">jj</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">rows</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">jj</span> <span class="o">==</span> <span class="n">i</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">pop</span><span class="p">((</span><span class="n">ii</span><span class="p">,</span> <span class="n">jj</span><span class="p">))</span>
-                <span class="n">temp</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">ii</span><span class="p">,</span> <span class="n">v</span><span class="p">))</span>
-            <span class="k">elif</span> <span class="n">jj</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">pop</span><span class="p">((</span><span class="n">ii</span><span class="p">,</span> <span class="n">jj</span><span class="p">))</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">[</span><span class="n">ii</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">v</span>
-            <span class="k">elif</span> <span class="n">jj</span> <span class="o">&gt;</span> <span class="n">j</span><span class="p">:</span>
-                <span class="k">break</span>
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">temp</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">v</span>
-</div>
-<div class="viewcode-block" id="MutableSparseMatrix.row_join"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.MutableSparseMatrix.row_join">[docs]</a>    <span class="k">def</span> <span class="nf">row_join</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns B appended after A (column-wise augmenting)::</span>
-
-<span class="sd">            [A B]</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import SparseMatrix, Matrix</span>
-<span class="sd">        &gt;&gt;&gt; A = SparseMatrix(((1, 0, 1), (0, 1, 0), (1, 1, 0)))</span>
-<span class="sd">        &gt;&gt;&gt; A</span>
-<span class="sd">        [1, 0, 1]</span>
-<span class="sd">        [0, 1, 0]</span>
-<span class="sd">        [1, 1, 0]</span>
-<span class="sd">        &gt;&gt;&gt; B = SparseMatrix(((1, 0, 0), (0, 1, 0), (0, 0, 1)))</span>
-<span class="sd">        &gt;&gt;&gt; B</span>
-<span class="sd">        [1, 0, 0]</span>
-<span class="sd">        [0, 1, 0]</span>
-<span class="sd">        [0, 0, 1]</span>
-<span class="sd">        &gt;&gt;&gt; C = A.row_join(B); C</span>
-<span class="sd">        [1, 0, 1, 1, 0, 0]</span>
-<span class="sd">        [0, 1, 0, 0, 1, 0]</span>
-<span class="sd">        [1, 1, 0, 0, 0, 1]</span>
-<span class="sd">        &gt;&gt;&gt; C == A.row_join(Matrix(B))</span>
-<span class="sd">        True</span>
-
-<span class="sd">        Joining at row ends is the same as appending columns at the end</span>
-<span class="sd">        of the matrix:</span>
-
-<span class="sd">        &gt;&gt;&gt; C == A.col_insert(A.cols, B)</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">A</span><span class="p">,</span> <span class="n">B</span> <span class="o">=</span> <span class="bp">self</span><span class="p">,</span> <span class="n">other</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">A</span><span class="o">.</span><span class="n">rows</span> <span class="o">==</span> <span class="n">B</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ShapeError</span><span class="p">()</span>
-        <span class="n">A</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">SparseMatrix</span><span class="p">):</span>
-            <span class="n">k</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="n">B</span><span class="o">.</span><span class="n">_mat</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">B</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">B</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-                    <span class="n">v</span> <span class="o">=</span> <span class="n">b</span><span class="p">[</span><span class="n">k</span><span class="p">]</span>
-                    <span class="k">if</span> <span class="n">v</span><span class="p">:</span>
-                        <span class="n">A</span><span class="o">.</span><span class="n">_smat</span><span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">+</span> <span class="n">A</span><span class="o">.</span><span class="n">cols</span><span class="p">)]</span> <span class="o">=</span> <span class="n">v</span>
-                    <span class="n">k</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">),</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">B</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                <span class="n">A</span><span class="o">.</span><span class="n">_smat</span><span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">+</span> <span class="n">A</span><span class="o">.</span><span class="n">cols</span><span class="p">)]</span> <span class="o">=</span> <span class="n">v</span>
-        <span class="n">A</span><span class="o">.</span><span class="n">cols</span> <span class="o">+=</span> <span class="n">B</span><span class="o">.</span><span class="n">cols</span>
-        <span class="k">return</span> <span class="n">A</span>
-</div>
-<div class="viewcode-block" id="MutableSparseMatrix.col_join"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.MutableSparseMatrix.col_join">[docs]</a>    <span class="k">def</span> <span class="nf">col_join</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns B augmented beneath A (row-wise joining)::</span>
-
-<span class="sd">            [A]</span>
-<span class="sd">            [B]</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import SparseMatrix, Matrix, ones</span>
-<span class="sd">        &gt;&gt;&gt; A = SparseMatrix(ones(3))</span>
-<span class="sd">        &gt;&gt;&gt; A</span>
-<span class="sd">        [1, 1, 1]</span>
-<span class="sd">        [1, 1, 1]</span>
-<span class="sd">        [1, 1, 1]</span>
-<span class="sd">        &gt;&gt;&gt; B = SparseMatrix.eye(3)</span>
-<span class="sd">        &gt;&gt;&gt; B</span>
-<span class="sd">        [1, 0, 0]</span>
-<span class="sd">        [0, 1, 0]</span>
-<span class="sd">        [0, 0, 1]</span>
-<span class="sd">        &gt;&gt;&gt; C = A.col_join(B); C</span>
-<span class="sd">        [1, 1, 1]</span>
-<span class="sd">        [1, 1, 1]</span>
-<span class="sd">        [1, 1, 1]</span>
-<span class="sd">        [1, 0, 0]</span>
-<span class="sd">        [0, 1, 0]</span>
-<span class="sd">        [0, 0, 1]</span>
-<span class="sd">        &gt;&gt;&gt; C == A.col_join(Matrix(B))</span>
-<span class="sd">        True</span>
-
-<span class="sd">        Joining along columns is the same as appending rows at the end</span>
-<span class="sd">        of the matrix:</span>
-
-<span class="sd">        &gt;&gt;&gt; C == A.row_insert(A.rows, Matrix(B))</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">A</span><span class="p">,</span> <span class="n">B</span> <span class="o">=</span> <span class="bp">self</span><span class="p">,</span> <span class="n">other</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">A</span><span class="o">.</span><span class="n">cols</span> <span class="o">==</span> <span class="n">B</span><span class="o">.</span><span class="n">cols</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ShapeError</span><span class="p">()</span>
-        <span class="n">A</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">SparseMatrix</span><span class="p">):</span>
-            <span class="n">k</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="n">B</span><span class="o">.</span><span class="n">_mat</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">B</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">B</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-                    <span class="n">v</span> <span class="o">=</span> <span class="n">b</span><span class="p">[</span><span class="n">k</span><span class="p">]</span>
-                    <span class="k">if</span> <span class="n">v</span><span class="p">:</span>
-                        <span class="n">A</span><span class="o">.</span><span class="n">_smat</span><span class="p">[(</span><span class="n">i</span> <span class="o">+</span> <span class="n">A</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="n">j</span><span class="p">)]</span> <span class="o">=</span> <span class="n">v</span>
-                    <span class="n">k</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">),</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">B</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                <span class="n">A</span><span class="o">.</span><span class="n">_smat</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="n">A</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">v</span>
-        <span class="n">A</span><span class="o">.</span><span class="n">rows</span> <span class="o">+=</span> <span class="n">B</span><span class="o">.</span><span class="n">rows</span>
-        <span class="k">return</span> <span class="n">A</span>
-</div>
-    <span class="k">def</span> <span class="nf">copyin_list</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">value</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;`value` must be of type list or tuple.&quot;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">copyin_matrix</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">value</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">copyin_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
-        <span class="c"># include this here because it&#39;s not part of BaseMatrix</span>
-        <span class="n">rlo</span><span class="p">,</span> <span class="n">rhi</span><span class="p">,</span> <span class="n">clo</span><span class="p">,</span> <span class="n">chi</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">key2bounds</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
-        <span class="n">shape</span> <span class="o">=</span> <span class="n">value</span><span class="o">.</span><span class="n">shape</span>
-        <span class="n">dr</span><span class="p">,</span> <span class="n">dc</span> <span class="o">=</span> <span class="n">rhi</span> <span class="o">-</span> <span class="n">rlo</span><span class="p">,</span> <span class="n">chi</span> <span class="o">-</span> <span class="n">clo</span>
-        <span class="k">if</span> <span class="n">shape</span> <span class="o">!=</span> <span class="p">(</span><span class="n">dr</span><span class="p">,</span> <span class="n">dc</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="n">ShapeError</span><span class="p">(</span>
-                <span class="s">&quot;The Matrix `value` doesn&#39;t have the same dimensions &quot;</span>
-                <span class="s">&quot;as the in sub-Matrix given by `key`.&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">value</span><span class="p">,</span> <span class="n">SparseMatrix</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">value</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">value</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-                    <span class="bp">self</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="n">rlo</span><span class="p">,</span> <span class="n">j</span> <span class="o">+</span> <span class="n">clo</span><span class="p">]</span> <span class="o">=</span> <span class="n">value</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">rhi</span> <span class="o">-</span> <span class="n">rlo</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">chi</span> <span class="o">-</span> <span class="n">clo</span><span class="p">)</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">rlo</span><span class="p">,</span> <span class="n">rhi</span><span class="p">):</span>
-                    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">clo</span><span class="p">,</span> <span class="n">chi</span><span class="p">):</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">pop</span><span class="p">((</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">),</span> <span class="bp">None</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">row_list</span><span class="p">():</span>
-                    <span class="k">if</span> <span class="n">rlo</span> <span class="o">&lt;=</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">rhi</span> <span class="ow">and</span> <span class="n">clo</span> <span class="o">&lt;=</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="n">chi</span><span class="p">:</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">pop</span><span class="p">((</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">),</span> <span class="bp">None</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">value</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">k</span>
-                <span class="bp">self</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="n">rlo</span><span class="p">,</span> <span class="n">j</span> <span class="o">+</span> <span class="n">clo</span><span class="p">]</span> <span class="o">=</span> <span class="n">value</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-
-<div class="viewcode-block" id="MutableSparseMatrix.row_op"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.MutableSparseMatrix.row_op">[docs]</a>    <span class="k">def</span> <span class="nf">row_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;In-place operation on row i using two-arg functor whose args are</span>
-<span class="sd">        interpreted as (self[i, j], j) for j in range(self.cols).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix</span>
-<span class="sd">        &gt;&gt;&gt; M = SparseMatrix.eye(3)*2</span>
-<span class="sd">        &gt;&gt;&gt; M[0, 1] = -1</span>
-<span class="sd">        &gt;&gt;&gt; M.row_op(1, lambda v, j: v + 2*M[0, j]); M</span>
-<span class="sd">        [2, -1, 0]</span>
-<span class="sd">        [4,  0, 0]</span>
-<span class="sd">        [0,  0, 2]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-            <span class="n">v</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">get</span><span class="p">((</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-            <span class="n">fv</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">fv</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)]</span> <span class="o">=</span> <span class="n">fv</span>
-            <span class="k">elif</span> <span class="n">v</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">pop</span><span class="p">((</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="MutableSparseMatrix.col_op"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.MutableSparseMatrix.col_op">[docs]</a>    <span class="k">def</span> <span class="nf">col_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;In-place operation on col j using two-arg functor whose args are</span>
-<span class="sd">        interpreted as (self[i, j], i) for i in range(self.rows).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix</span>
-<span class="sd">        &gt;&gt;&gt; M = SparseMatrix.eye(3)*2</span>
-<span class="sd">        &gt;&gt;&gt; M[1, 0] = -1</span>
-<span class="sd">        &gt;&gt;&gt; M.col_op(1, lambda v, i: v + 2*M[i, 0]); M</span>
-<span class="sd">        [ 2, 4, 0]</span>
-<span class="sd">        [-1, 0, 0]</span>
-<span class="sd">        [ 0, 0, 2]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="n">v</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">get</span><span class="p">((</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-            <span class="n">fv</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">fv</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)]</span> <span class="o">=</span> <span class="n">fv</span>
-            <span class="k">elif</span> <span class="n">v</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span><span class="o">.</span><span class="n">pop</span><span class="p">((</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="MutableSparseMatrix.fill"><a class="viewcode-back" href="../../../modules/matrices/sparse.html#sympy.matrices.sparse.MutableSparseMatrix.fill">[docs]</a>    <span class="k">def</span> <span class="nf">fill</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Fill self with the given value.</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        Unless many values are going to be deleted (i.e. set to zero)</span>
-<span class="sd">        this will create a matrix that is slower than a dense matrix in</span>
-<span class="sd">        operations.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.matrices import SparseMatrix</span>
-<span class="sd">        &gt;&gt;&gt; M = SparseMatrix.zeros(3); M</span>
-<span class="sd">        [0, 0, 0]</span>
-<span class="sd">        [0, 0, 0]</span>
-<span class="sd">        [0, 0, 0]</span>
-<span class="sd">        &gt;&gt;&gt; M.fill(1); M</span>
-<span class="sd">        [1, 1, 1]</span>
-<span class="sd">        [1, 1, 1]</span>
-<span class="sd">        [1, 1, 1]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">value</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">v</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">value</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_smat</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">([((</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">),</span> <span class="n">v</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rows</span><span class="p">)</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">cols</span><span class="p">)])</span></div></div>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/ntheory/factor_.html b/dev-py3k/_modules/sympy/ntheory/factor_.html
deleted file mode 100644
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+++ /dev/null
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-            
-  <h1>Source code for sympy.ntheory.factor_</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Integer factorization</span>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">import</span> <span class="nn">random</span>
-<span class="kn">import</span> <span class="nn">math</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.evalf</span> <span class="kn">import</span> <span class="n">bitcount</span>
-<span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">igcd</span>
-<span class="kn">from</span> <span class="nn">sympy.core.power</span> <span class="kn">import</span> <span class="n">integer_nthroot</span><span class="p">,</span> <span class="n">Pow</span>
-<span class="kn">from</span> <span class="nn">sympy.core.mul</span> <span class="kn">import</span> <span class="n">Mul</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">as_int</span>
-<span class="kn">from</span> <span class="nn">.primetest</span> <span class="kn">import</span> <span class="n">isprime</span>
-<span class="kn">from</span> <span class="nn">.generate</span> <span class="kn">import</span> <span class="n">sieve</span><span class="p">,</span> <span class="n">primerange</span><span class="p">,</span> <span class="n">nextprime</span>
-<span class="kn">from</span> <span class="nn">sympy.core.singleton</span> <span class="kn">import</span> <span class="n">S</span>
-
-<span class="n">small_trailing</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="ow">and</span> <span class="nb">max</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="ow">not</span> <span class="n">i</span> <span class="o">%</span> <span class="mi">2</span><span class="o">**</span><span class="n">j</span><span class="p">)</span> <span class="ow">and</span> <span class="n">j</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">256</span><span class="p">)]</span>
-
-
-<div class="viewcode-block" id="smoothness"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.factor_.smoothness">[docs]</a><span class="k">def</span> <span class="nf">smoothness</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the B-smooth and B-power smooth values of n.</span>
-
-<span class="sd">    The smoothness of n is the largest prime factor of n; the power-</span>
-<span class="sd">    smoothness is the largest divisor raised to its multiplicity.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory.factor_ import smoothness</span>
-<span class="sd">    &gt;&gt;&gt; smoothness(2**7*3**2)</span>
-<span class="sd">    (3, 128)</span>
-<span class="sd">    &gt;&gt;&gt; smoothness(2**4*13)</span>
-<span class="sd">    (13, 16)</span>
-<span class="sd">    &gt;&gt;&gt; smoothness(2)</span>
-<span class="sd">    (2, 2)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    factorint, smoothness_p</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>  <span class="c"># not prime, but otherwise this causes headaches</span>
-    <span class="n">facs</span> <span class="o">=</span> <span class="n">factorint</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">return</span> <span class="nb">max</span><span class="p">(</span><span class="n">facs</span><span class="p">),</span> <span class="nb">max</span><span class="p">([</span><span class="n">m</span><span class="o">**</span><span class="n">facs</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">facs</span><span class="p">])</span>
-
-</div>
-<div class="viewcode-block" id="smoothness_p"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.factor_.smoothness_p">[docs]</a><span class="k">def</span> <span class="nf">smoothness_p</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">power</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a list of [m, (p, (M, sm(p + m), psm(p + m)))...]</span>
-<span class="sd">    where:</span>
-
-<span class="sd">    1. p**M is the base-p divisor of n</span>
-<span class="sd">    2. sm(p + m) is the smoothness of p + m (m = -1 by default)</span>
-<span class="sd">    3. psm(p + n) is the power smoothness of p + m</span>
-
-<span class="sd">    The list is sorted according to smoothness (default) or by power smoothness</span>
-<span class="sd">    if power=1.</span>
-
-<span class="sd">    The smoothness of the numbers to the left (m = -1) or right (m = 1) of a</span>
-<span class="sd">    factor govern the results that are obtained from the p +/- 1 type factoring</span>
-<span class="sd">    methods.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.ntheory.factor_ import smoothness_p, factorint</span>
-<span class="sd">        &gt;&gt;&gt; smoothness_p(10431, m=1)</span>
-<span class="sd">        (1, [(3, (2, 2, 4)), (19, (1, 5, 5)), (61, (1, 31, 31))])</span>
-<span class="sd">        &gt;&gt;&gt; smoothness_p(10431)</span>
-<span class="sd">        (-1, [(3, (2, 2, 2)), (19, (1, 3, 9)), (61, (1, 5, 5))])</span>
-<span class="sd">        &gt;&gt;&gt; smoothness_p(10431, power=1)</span>
-<span class="sd">        (-1, [(3, (2, 2, 2)), (61, (1, 5, 5)), (19, (1, 3, 9))])</span>
-
-<span class="sd">    If visual=True then an annotated string will be returned:</span>
-
-<span class="sd">        &gt;&gt;&gt; print(smoothness_p(21477639576571, visual=1))</span>
-<span class="sd">        p**i=4410317**1 has p-1 B=1787, B-pow=1787</span>
-<span class="sd">        p**i=4869863**1 has p-1 B=2434931, B-pow=2434931</span>
-
-<span class="sd">    This string can also be generated directly from a factorization dictionary</span>
-<span class="sd">    and vice versa:</span>
-
-<span class="sd">        &gt;&gt;&gt; factorint(17*9)</span>
-<span class="sd">        {3: 2, 17: 1}</span>
-<span class="sd">        &gt;&gt;&gt; smoothness_p(_)</span>
-<span class="sd">        &#39;p**i=3**2 has p-1 B=2, B-pow=2\\np**i=17**1 has p-1 B=2, B-pow=16&#39;</span>
-<span class="sd">        &gt;&gt;&gt; smoothness_p(_)</span>
-<span class="sd">        {3: 2, 17: 1}</span>
-
-<span class="sd">    The table of the output logic is:</span>
-
-<span class="sd">        ====== ====== ======= =======</span>
-<span class="sd">        |              Visual</span>
-<span class="sd">        ------ ----------------------</span>
-<span class="sd">        Input  True   False   other</span>
-<span class="sd">        ====== ====== ======= =======</span>
-<span class="sd">        dict    str    tuple   str</span>
-<span class="sd">        str     str    tuple   dict</span>
-<span class="sd">        tuple   str    tuple   str</span>
-<span class="sd">        n       str    tuple   tuple</span>
-<span class="sd">        mul     str    tuple   tuple</span>
-<span class="sd">        ====== ====== ======= =======</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    factorint, smoothness</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">flatten</span>
-
-    <span class="c"># visual must be True, False or other (stored as None)</span>
-    <span class="k">if</span> <span class="n">visual</span> <span class="ow">in</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">):</span>
-        <span class="n">visual</span> <span class="o">=</span> <span class="nb">bool</span><span class="p">(</span><span class="n">visual</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">visual</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">(</span><span class="bp">True</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="n">visual</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">str</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">visual</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">n</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">li</span> <span class="ow">in</span> <span class="n">n</span><span class="o">.</span><span class="n">splitlines</span><span class="p">():</span>
-            <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span>
-                    <span class="n">li</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;has&#39;</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;=&#39;</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;**&#39;</span><span class="p">)]</span>
-            <span class="n">d</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">v</span>
-        <span class="k">if</span> <span class="n">visual</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">True</span> <span class="ow">and</span> <span class="n">visual</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">d</span>
-        <span class="k">return</span> <span class="n">smoothness_p</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="nb">type</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="nb">tuple</span><span class="p">:</span>
-        <span class="n">facs</span> <span class="o">=</span> <span class="n">factorint</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">power</span><span class="p">:</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="nb">tuple</span><span class="p">:</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="nb">sorted</span><span class="p">([(</span><span class="n">f</span><span class="p">,</span>
-                          <span class="nb">tuple</span><span class="p">([</span><span class="n">M</span><span class="p">]</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">smoothness</span><span class="p">(</span><span class="n">f</span> <span class="o">+</span> <span class="n">m</span><span class="p">))))</span>
-                         <span class="k">for</span> <span class="n">f</span><span class="p">,</span> <span class="n">M</span> <span class="ow">in</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">facs</span><span class="o">.</span><span class="n">items</span><span class="p">())]],</span>
-                        <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="n">k</span><span class="p">],</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">])))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">n</span>
-
-    <span class="k">if</span> <span class="n">visual</span> <span class="ow">is</span> <span class="bp">False</span> <span class="ow">or</span> <span class="p">(</span><span class="n">visual</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">True</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="ow">in</span> <span class="p">[</span><span class="nb">int</span><span class="p">,</span> <span class="n">Mul</span><span class="p">]):</span>
-        <span class="k">return</span> <span class="n">rv</span>
-    <span class="n">lines</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">dat</span> <span class="ow">in</span> <span class="n">rv</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span>
-        <span class="n">dat</span> <span class="o">=</span> <span class="n">flatten</span><span class="p">(</span><span class="n">dat</span><span class="p">)</span>
-        <span class="n">dat</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-        <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;p**i=</span><span class="si">%i</span><span class="s">**</span><span class="si">%i</span><span class="s"> has p</span><span class="si">%+i</span><span class="s"> B=</span><span class="si">%i</span><span class="s">, B-pow=</span><span class="si">%i</span><span class="s">&#39;</span> <span class="o">%</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">dat</span><span class="p">))</span>
-    <span class="k">return</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">lines</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="trailing"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.factor_.trailing">[docs]</a><span class="k">def</span> <span class="nf">trailing</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Count the number of trailing zero digits in the binary</span>
-<span class="sd">    representation of n, i.e. determine the largest power of 2</span>
-<span class="sd">    that divides n.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import trailing</span>
-<span class="sd">    &gt;&gt;&gt; trailing(128)</span>
-<span class="sd">    7</span>
-<span class="sd">    &gt;&gt;&gt; trailing(63)</span>
-<span class="sd">    0</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="n">low_byte</span> <span class="o">=</span> <span class="n">n</span> <span class="o">&amp;</span> <span class="mh">0xff</span>
-    <span class="k">if</span> <span class="n">low_byte</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">small_trailing</span><span class="p">[</span><span class="n">low_byte</span><span class="p">]</span>
-
-    <span class="c"># 2**m is quick for z up through 2**30</span>
-    <span class="n">z</span> <span class="o">=</span> <span class="n">bitcount</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-    <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">int</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span> <span class="o">&lt;&lt;</span> <span class="n">z</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">z</span>
-
-    <span class="n">t</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="mi">8</span>
-    <span class="k">while</span> <span class="ow">not</span> <span class="n">n</span> <span class="o">&amp;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">while</span> <span class="ow">not</span> <span class="n">n</span> <span class="o">&amp;</span> <span class="p">((</span><span class="mi">1</span> <span class="o">&lt;&lt;</span> <span class="n">p</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">n</span> <span class="o">&gt;&gt;=</span> <span class="n">p</span>
-            <span class="n">t</span> <span class="o">+=</span> <span class="n">p</span>
-            <span class="n">p</span> <span class="o">*=</span> <span class="mi">2</span>
-        <span class="n">p</span> <span class="o">//=</span> <span class="mi">2</span>
-    <span class="k">return</span> <span class="n">t</span>
-
-</div>
-<div class="viewcode-block" id="multiplicity"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.factor_.multiplicity">[docs]</a><span class="k">def</span> <span class="nf">multiplicity</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Find the greatest integer m such that p**m divides n.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory import multiplicity</span>
-<span class="sd">    &gt;&gt;&gt; [multiplicity(5, n) for n in [8, 5, 25, 125, 250]]</span>
-<span class="sd">    [0, 1, 2, 3, 3]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">p</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">p</span><span class="p">),</span> <span class="n">as_int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">p</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">trailing</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">p</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;p must be an integer, 2 or larger, but got </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">p</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">p</span> <span class="o">==</span> <span class="n">n</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">1</span>
-
-    <span class="n">m</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">n</span><span class="p">,</span> <span class="n">rem</span> <span class="o">=</span> <span class="nb">divmod</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-    <span class="k">while</span> <span class="ow">not</span> <span class="n">rem</span><span class="p">:</span>
-        <span class="n">m</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">m</span> <span class="o">&gt;</span> <span class="mi">5</span><span class="p">:</span>
-            <span class="c"># The multiplicity could be very large. Better</span>
-            <span class="c"># to increment in powers of two</span>
-            <span class="n">e</span> <span class="o">=</span> <span class="mi">2</span>
-            <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">ppow</span> <span class="o">=</span> <span class="n">p</span><span class="o">**</span><span class="n">e</span>
-                <span class="k">if</span> <span class="n">ppow</span> <span class="o">&lt;</span> <span class="n">n</span><span class="p">:</span>
-                    <span class="n">nnew</span><span class="p">,</span> <span class="n">rem</span> <span class="o">=</span> <span class="nb">divmod</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">ppow</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">rem</span><span class="p">:</span>
-                        <span class="n">m</span> <span class="o">+=</span> <span class="n">e</span>
-                        <span class="n">e</span> <span class="o">*=</span> <span class="mi">2</span>
-                        <span class="n">n</span> <span class="o">=</span> <span class="n">nnew</span>
-                        <span class="k">continue</span>
-                <span class="k">return</span> <span class="n">m</span> <span class="o">+</span> <span class="n">multiplicity</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">rem</span> <span class="o">=</span> <span class="nb">divmod</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">m</span>
-
-</div>
-<div class="viewcode-block" id="perfect_power"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.factor_.perfect_power">[docs]</a><span class="k">def</span> <span class="nf">perfect_power</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">candidates</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">big</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">factor</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return ``(b, e)`` such that ``n`` == ``b**e`` if ``n`` is a</span>
-<span class="sd">    perfect power; otherwise return ``False``.</span>
-
-<span class="sd">    By default, the base is recursively decomposed and the exponents</span>
-<span class="sd">    collected so the largest possible ``e`` is sought. If ``big=False``</span>
-<span class="sd">    then the smallest possible ``e`` (thus prime) will be chosen.</span>
-
-<span class="sd">    If ``candidates`` for exponents are given, they are assumed to be sorted</span>
-<span class="sd">    and the first one that is larger than the computed maximum will signal</span>
-<span class="sd">    failure for the routine.</span>
-
-<span class="sd">    If ``factor=True`` then simultaneous factorization of n is attempted</span>
-<span class="sd">    since finding a factor indicates the only possible root for n. This</span>
-<span class="sd">    is True by default since only a few small factors will be tested in</span>
-<span class="sd">    the course of searching for the perfect power.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import perfect_power</span>
-<span class="sd">    &gt;&gt;&gt; perfect_power(16)</span>
-<span class="sd">    (2, 4)</span>
-<span class="sd">    &gt;&gt;&gt; perfect_power(16, big = False)</span>
-<span class="sd">    (4, 2)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">3</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">False</span>
-    <span class="n">logn</span> <span class="o">=</span> <span class="n">math</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-    <span class="n">max_possible</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">logn</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span>  <span class="c"># only check values less than this</span>
-    <span class="n">not_square</span> <span class="o">=</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">10</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">]</span>  <span class="c"># squares cannot end in 2, 3, 7, 8</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">candidates</span><span class="p">:</span>
-        <span class="n">candidates</span> <span class="o">=</span> <span class="n">primerange</span><span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="n">not_square</span><span class="p">,</span> <span class="n">max_possible</span><span class="p">)</span>
-
-    <span class="n">afactor</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span>
-    <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">candidates</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">e</span> <span class="o">&lt;</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">e</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">e</span> <span class="o">==</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">not_square</span><span class="p">:</span>
-                <span class="k">continue</span>
-        <span class="k">if</span> <span class="n">e</span> <span class="o">&gt;</span> <span class="n">max_possible</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="c"># see if there is a factor present</span>
-        <span class="k">if</span> <span class="n">factor</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">%</span> <span class="n">afactor</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="c"># find what the potential power is</span>
-                <span class="k">if</span> <span class="n">afactor</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                    <span class="n">e</span> <span class="o">=</span> <span class="n">trailing</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">e</span> <span class="o">=</span> <span class="n">multiplicity</span><span class="p">(</span><span class="n">afactor</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-                <span class="c"># if it&#39;s a trivial power we are done</span>
-                <span class="k">if</span> <span class="n">e</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-
-                <span class="c"># maybe the bth root of n is exact</span>
-                <span class="n">r</span><span class="p">,</span> <span class="n">exact</span> <span class="o">=</span> <span class="n">integer_nthroot</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">exact</span><span class="p">:</span>
-                    <span class="c"># then remove this factor and check to see if</span>
-                    <span class="c"># any of e&#39;s factors are a common exponent; if</span>
-                    <span class="c"># not then it&#39;s not a perfect power</span>
-                    <span class="n">n</span> <span class="o">//=</span> <span class="n">afactor</span><span class="o">**</span><span class="n">e</span>
-                    <span class="n">m</span> <span class="o">=</span> <span class="n">perfect_power</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">candidates</span><span class="o">=</span><span class="n">primefactors</span><span class="p">(</span><span class="n">e</span><span class="p">),</span> <span class="n">big</span><span class="o">=</span><span class="n">big</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">m</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="bp">False</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">r</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">m</span>
-                        <span class="c"># adjust the two exponents so the bases can</span>
-                        <span class="c"># be combined</span>
-                        <span class="n">g</span> <span class="o">=</span> <span class="n">igcd</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">g</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="bp">False</span>
-                        <span class="n">m</span> <span class="o">//=</span> <span class="n">g</span>
-                        <span class="n">e</span> <span class="o">//=</span> <span class="n">g</span>
-                        <span class="n">r</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">r</span><span class="o">**</span><span class="n">m</span><span class="o">*</span><span class="n">afactor</span><span class="o">**</span><span class="n">e</span><span class="p">,</span> <span class="n">g</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">big</span><span class="p">:</span>
-                    <span class="n">e0</span> <span class="o">=</span> <span class="n">primefactors</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">e0</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">e0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="n">e</span><span class="p">:</span>
-                        <span class="n">e0</span> <span class="o">=</span> <span class="n">e0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                        <span class="n">r</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">r</span><span class="o">**</span><span class="p">(</span><span class="n">e</span><span class="o">//</span><span class="n">e0</span><span class="p">),</span> <span class="n">e0</span>
-                <span class="k">return</span> <span class="n">r</span><span class="p">,</span> <span class="n">e</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># get the next factor ready for the next pass through the loop</span>
-                <span class="n">afactor</span> <span class="o">=</span> <span class="n">nextprime</span><span class="p">(</span><span class="n">afactor</span><span class="p">)</span>
-
-        <span class="c"># Weed out downright impossible candidates</span>
-        <span class="k">if</span> <span class="n">logn</span><span class="o">/</span><span class="n">e</span> <span class="o">&lt;</span> <span class="mi">40</span><span class="p">:</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="mf">2.0</span><span class="o">**</span><span class="p">(</span><span class="n">logn</span><span class="o">/</span><span class="n">e</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="mf">0.5</span><span class="p">)</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mf">0.01</span><span class="p">:</span>
-                <span class="k">continue</span>
-
-        <span class="c"># now see if the plausible e makes a perfect power</span>
-        <span class="n">r</span><span class="p">,</span> <span class="n">exact</span> <span class="o">=</span> <span class="n">integer_nthroot</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">exact</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">big</span><span class="p">:</span>
-                <span class="n">m</span> <span class="o">=</span> <span class="n">perfect_power</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">big</span><span class="o">=</span><span class="n">big</span><span class="p">,</span> <span class="n">factor</span><span class="o">=</span><span class="n">factor</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">m</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">:</span>
-                    <span class="n">r</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">m</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">e</span><span class="o">*</span><span class="n">m</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-            <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">r</span><span class="p">),</span> <span class="n">e</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-</div>
-<div class="viewcode-block" id="pollard_rho"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.factor_.pollard_rho">[docs]</a><span class="k">def</span> <span class="nf">pollard_rho</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">retries</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">1234</span><span class="p">,</span> <span class="n">max_steps</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Use Pollard&#39;s rho method to try to extract a nontrivial factor</span>
-<span class="sd">    of ``n``. The returned factor may be a composite number. If no</span>
-<span class="sd">    factor is found, ``None`` is returned.</span>
-
-<span class="sd">    The algorithm generates pseudo-random values of x with a generator</span>
-<span class="sd">    function, replacing x with F(x). If F is not supplied then the</span>
-<span class="sd">    function x**2 + ``a`` is used. The first value supplied to F(x) is ``s``.</span>
-<span class="sd">    Upon failure (if ``retries`` is &gt; 0) a new ``a`` and ``s`` will be</span>
-<span class="sd">    supplied; the ``a`` will be ignored if F was supplied.</span>
-
-<span class="sd">    The sequence of numbers generated by such functions generally have a</span>
-<span class="sd">    a lead-up to some number and then loop around back to that number and</span>
-<span class="sd">    begin to repeat the sequence, e.g. 1, 2, 3, 4, 5, 3, 4, 5 -- this leader</span>
-<span class="sd">    and loop look a bit like the Greek letter rho, and thus the name, &#39;rho&#39;.</span>
-
-<span class="sd">    For a given function, very different leader-loop values can be obtained</span>
-<span class="sd">    so it is a good idea to allow for retries:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory.generate import cycle_length</span>
-<span class="sd">    &gt;&gt;&gt; n = 16843009</span>
-<span class="sd">    &gt;&gt;&gt; F = lambda x:(2048*pow(x, 2, n) + 32767) % n</span>
-<span class="sd">    &gt;&gt;&gt; for s in range(5):</span>
-<span class="sd">    ...     print(&#39;loop length = %4i; leader length = %3i&#39; % next(cycle_length(F, s)))</span>
-<span class="sd">    ...</span>
-<span class="sd">    loop length = 2489; leader length =  42</span>
-<span class="sd">    loop length =   78; leader length = 120</span>
-<span class="sd">    loop length = 1482; leader length =  99</span>
-<span class="sd">    loop length = 1482; leader length = 285</span>
-<span class="sd">    loop length = 1482; leader length = 100</span>
-
-<span class="sd">    Here is an explicit example where there is a two element leadup to</span>
-<span class="sd">    a sequence of 3 numbers (11, 14, 4) that then repeat:</span>
-
-<span class="sd">    &gt;&gt;&gt; x=2</span>
-<span class="sd">    &gt;&gt;&gt; for i in range(9):</span>
-<span class="sd">    ...     x=(x**2+12)%17</span>
-<span class="sd">    ...     print(x, end=&#39; &#39;)</span>
-<span class="sd">    ...</span>
-<span class="sd">    16 13 11 14 4 11 14 4 11</span>
-<span class="sd">    &gt;&gt;&gt; next(cycle_length(lambda x: (x**2+12)%17, 2))</span>
-<span class="sd">    (3, 2)</span>
-<span class="sd">    &gt;&gt;&gt; list(cycle_length(lambda x: (x**2+12)%17, 2, values=True))</span>
-<span class="sd">    [16, 13, 11, 14, 4]</span>
-
-<span class="sd">    Instead of checking the differences of all generated values for a gcd</span>
-<span class="sd">    with n, only the kth and 2*kth numbers are checked, e.g. 1st and 2nd,</span>
-<span class="sd">    2nd and 4th, 3rd and 6th until it has been detected that the loop has been</span>
-<span class="sd">    traversed. Loops may be many thousands of steps long before rho finds a</span>
-<span class="sd">    factor or reports failure. If ``max_steps`` is specified, the iteration</span>
-<span class="sd">    is cancelled with a failure after the specified number of steps.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import pollard_rho</span>
-<span class="sd">    &gt;&gt;&gt; n=16843009</span>
-<span class="sd">    &gt;&gt;&gt; F=lambda x:(2048*pow(x,2,n) + 32767) % n</span>
-<span class="sd">    &gt;&gt;&gt; pollard_rho(n, F=F)</span>
-<span class="sd">    257</span>
-
-<span class="sd">    Use the default setting with a bad value of ``a`` and no retries:</span>
-
-<span class="sd">    &gt;&gt;&gt; pollard_rho(n, a=n-2, retries=0)</span>
-
-<span class="sd">    If retries is &gt; 0 then perhaps the problem will correct itself when</span>
-<span class="sd">    new values are generated for a:</span>
-
-<span class="sd">    &gt;&gt;&gt; pollard_rho(n, a=n-2, retries=1)</span>
-<span class="sd">    257</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - Richard Crandall &amp; Carl Pomerance (2005), &quot;Prime Numbers:</span>
-<span class="sd">      A Computational Perspective&quot;, Springer, 2nd edition, 229-231</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">5</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;pollard_rho should receive n &gt; 4&#39;</span><span class="p">)</span>
-    <span class="n">prng</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">Random</span><span class="p">(</span><span class="n">seed</span> <span class="o">+</span> <span class="n">retries</span><span class="p">)</span>
-    <span class="n">V</span> <span class="o">=</span> <span class="n">s</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">retries</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">U</span> <span class="o">=</span> <span class="n">V</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">F</span><span class="p">:</span>
-            <span class="n">F</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="p">(</span><span class="nb">pow</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">+</span> <span class="n">a</span><span class="p">)</span> <span class="o">%</span> <span class="n">n</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">max_steps</span> <span class="ow">and</span> <span class="p">(</span><span class="n">j</span> <span class="o">&gt;</span> <span class="n">max_steps</span><span class="p">):</span>
-                <span class="k">break</span>
-            <span class="n">j</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">U</span> <span class="o">=</span> <span class="n">F</span><span class="p">(</span><span class="n">U</span><span class="p">)</span>
-            <span class="n">V</span> <span class="o">=</span> <span class="n">F</span><span class="p">(</span><span class="n">F</span><span class="p">(</span><span class="n">V</span><span class="p">))</span>  <span class="c"># V is 2x further along than U</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">igcd</span><span class="p">(</span><span class="n">U</span> <span class="o">-</span> <span class="n">V</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">g</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">if</span> <span class="n">g</span> <span class="o">==</span> <span class="n">n</span><span class="p">:</span>
-                <span class="k">break</span>
-            <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="n">V</span> <span class="o">=</span> <span class="n">prng</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">prng</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">3</span><span class="p">)</span>  <span class="c"># for x**2 + a, a%n should not be 0 or -2</span>
-        <span class="n">F</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="k">return</span> <span class="bp">None</span>
-
-</div>
-<div class="viewcode-block" id="pollard_pm1"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.factor_.pollard_pm1">[docs]</a><span class="k">def</span> <span class="nf">pollard_pm1</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">B</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">retries</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">1234</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Use Pollard&#39;s p-1 method to try to extract a nontrivial factor</span>
-<span class="sd">    of ``n``. Either a divisor (perhaps composite) or ``None`` is returned.</span>
-
-<span class="sd">    The value of ``a`` is the base that is used in the test gcd(a**M - 1, n).</span>
-<span class="sd">    The default is 2.  If ``retries`` &gt; 0 then if no factor is found after the</span>
-<span class="sd">    first attempt, a new ``a`` will be generated randomly (using the ``seed``)</span>
-<span class="sd">    and the process repeated.</span>
-
-<span class="sd">    Note: the value of M is lcm(1..B) = reduce(ilcm, range(2, B + 1)).</span>
-
-<span class="sd">    A search is made for factors next to even numbers having a power smoothness</span>
-<span class="sd">    less than ``B``. Choosing a larger B increases the likelihood of finding a</span>
-<span class="sd">    larger factor but takes longer. Whether a factor of n is found or not</span>
-<span class="sd">    depends on ``a`` and the power smoothness of the even mumber just less than</span>
-<span class="sd">    the factor p (hence the name p - 1).</span>
-
-<span class="sd">    Although some discussion of what constitutes a good ``a`` some</span>
-<span class="sd">    descriptions are hard to interpret. At the modular.math site referenced</span>
-<span class="sd">    below it is stated that if gcd(a**M - 1, n) = N then a**M % q**r is 1</span>
-<span class="sd">    for every prime power divisor of N. But consider the following:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.ntheory.factor_ import smoothness_p, pollard_pm1</span>
-<span class="sd">        &gt;&gt;&gt; n=257*1009</span>
-<span class="sd">        &gt;&gt;&gt; smoothness_p(n)</span>
-<span class="sd">        (-1, [(257, (1, 2, 256)), (1009, (1, 7, 16))])</span>
-
-<span class="sd">    So we should (and can) find a root with B=16:</span>
-
-<span class="sd">        &gt;&gt;&gt; pollard_pm1(n, B=16, a=3)</span>
-<span class="sd">        1009</span>
-
-<span class="sd">    If we attempt to increase B to 256 we find that it doesn&#39;t work:</span>
-
-<span class="sd">        &gt;&gt;&gt; pollard_pm1(n, B=256)</span>
-<span class="sd">        &gt;&gt;&gt;</span>
-
-<span class="sd">    But if the value of ``a`` is changed we find that only multiples of</span>
-<span class="sd">    257 work, e.g.:</span>
-
-<span class="sd">        &gt;&gt;&gt; pollard_pm1(n, B=256, a=257)</span>
-<span class="sd">        1009</span>
-
-<span class="sd">    Checking different ``a`` values shows that all the ones that didn&#39;t</span>
-<span class="sd">    work had a gcd value not equal to ``n`` but equal to one of the</span>
-<span class="sd">    factors:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.core.numbers import ilcm, igcd</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import factorint, Pow</span>
-<span class="sd">        &gt;&gt;&gt; M = 1</span>
-<span class="sd">        &gt;&gt;&gt; for i in range(2, 256):</span>
-<span class="sd">        ...     M = ilcm(M, i)</span>
-<span class="sd">        ...</span>
-<span class="sd">        &gt;&gt;&gt; set([igcd(pow(a, M, n) - 1, n) for a in range(2, 256) if</span>
-<span class="sd">        ...      igcd(pow(a, M, n) - 1, n) != n])</span>
-<span class="sd">        set([1009])</span>
-
-<span class="sd">    But does aM % d for every divisor of n give 1?</span>
-
-<span class="sd">        &gt;&gt;&gt; aM = pow(255, M, n)</span>
-<span class="sd">        &gt;&gt;&gt; [(d, aM%Pow(*d.args)) for d in factorint(n, visual=True).args]</span>
-<span class="sd">        [(257**1, 1), (1009**1, 1)]</span>
-
-<span class="sd">    No, only one of them. So perhaps the principle is that a root will</span>
-<span class="sd">    be found for a given value of B provided that:</span>
-
-<span class="sd">    1) the power smoothness of the p - 1 value next to the root</span>
-<span class="sd">       does not exceed B</span>
-<span class="sd">    2) a**M % p != 1 for any of the divisors of n.</span>
-
-<span class="sd">    By trying more than one ``a`` it is possible that one of them</span>
-<span class="sd">    will yield a factor.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    With the default smoothness bound, this number can&#39;t be cracked:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.ntheory import pollard_pm1, primefactors</span>
-<span class="sd">        &gt;&gt;&gt; pollard_pm1(21477639576571)</span>
-
-<span class="sd">    Increasing the smoothness bound helps:</span>
-
-<span class="sd">        &gt;&gt;&gt; pollard_pm1(21477639576571, B=2000)</span>
-<span class="sd">        4410317</span>
-
-<span class="sd">    Looking at the smoothness of the factors of this number we find:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.utilities import flatten</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.ntheory.factor_ import smoothness_p, factorint</span>
-<span class="sd">        &gt;&gt;&gt; print(smoothness_p(21477639576571, visual=1))</span>
-<span class="sd">        p**i=4410317**1 has p-1 B=1787, B-pow=1787</span>
-<span class="sd">        p**i=4869863**1 has p-1 B=2434931, B-pow=2434931</span>
-
-<span class="sd">    The B and B-pow are the same for the p - 1 factorizations of the divisors</span>
-<span class="sd">    because those factorizations had a very large prime factor:</span>
-
-<span class="sd">        &gt;&gt;&gt; factorint(4410317 - 1)</span>
-<span class="sd">        {2: 2, 617: 1, 1787: 1}</span>
-<span class="sd">        &gt;&gt;&gt; factorint(4869863-1)</span>
-<span class="sd">        {2: 1, 2434931: 1}</span>
-
-<span class="sd">    Note that until B reaches the B-pow value of 1787, the number is not cracked;</span>
-
-<span class="sd">        &gt;&gt;&gt; pollard_pm1(21477639576571, B=1786)</span>
-<span class="sd">        &gt;&gt;&gt; pollard_pm1(21477639576571, B=1787)</span>
-<span class="sd">        4410317</span>
-
-<span class="sd">    The B value has to do with the factors of the number next to the divisor,</span>
-<span class="sd">    not the divisors themselves. A worst case scenario is that the number next</span>
-<span class="sd">    to the factor p has a large prime divisisor or is a perfect power. If these</span>
-<span class="sd">    conditions apply then the power-smoothness will be about p/2 or p. The more</span>
-<span class="sd">    realistic is that there will be a large prime factor next to p requiring</span>
-<span class="sd">    a B value on the order of p/2. Although primes may have been searched for</span>
-<span class="sd">    up to this level, the p/2 is a factor of p - 1, something that we don&#39;t</span>
-<span class="sd">    know. The modular.math reference below states that 15% of numbers in the</span>
-<span class="sd">    range of 10**15 to 15**15 + 10**4 are 10**6 power smooth so a B of 10**6</span>
-<span class="sd">    will fail 85% of the time in that range. From 10**8 to 10**8 + 10**3 the</span>
-<span class="sd">    percentages are nearly reversed...but in that range the simple trial</span>
-<span class="sd">    division is quite fast.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - Richard Crandall &amp; Carl Pomerance (2005), &quot;Prime Numbers:</span>
-<span class="sd">      A Computational Perspective&quot;, Springer, 2nd edition, 236-238</span>
-<span class="sd">    - http://modular.math.washington.edu/edu/2007/spring/ent/ent-html/</span>
-<span class="sd">            node81.html</span>
-<span class="sd">    - http://www.cs.toronto.edu/~yuvalf/Factorization.pdf</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">4</span> <span class="ow">or</span> <span class="n">B</span> <span class="o">&lt;</span> <span class="mi">3</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;pollard_pm1 should receive n &gt; 3 and B &gt; 2&#39;</span><span class="p">)</span>
-    <span class="n">prng</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">Random</span><span class="p">(</span><span class="n">seed</span> <span class="o">+</span> <span class="n">B</span><span class="p">)</span>
-
-    <span class="c"># computing a**lcm(1,2,3,..B) % n for B &gt; 2</span>
-    <span class="c"># it looks weird, but it&#39;s right: primes run [2, B]</span>
-    <span class="c"># and the answer&#39;s not right until the loop is done.</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">retries</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">aM</span> <span class="o">=</span> <span class="n">a</span>
-        <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">sieve</span><span class="o">.</span><span class="n">primerange</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">B</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">e</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">p</span><span class="p">))</span>
-            <span class="n">aM</span> <span class="o">=</span> <span class="nb">pow</span><span class="p">(</span><span class="n">aM</span><span class="p">,</span> <span class="nb">pow</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">e</span><span class="p">),</span> <span class="n">n</span><span class="p">)</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">igcd</span><span class="p">(</span><span class="n">aM</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-        <span class="k">if</span> <span class="mi">1</span> <span class="o">&lt;</span> <span class="n">g</span> <span class="o">&lt;</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="c"># get a new a:</span>
-        <span class="c"># since the exponent, lcm(1..B), is even, if we allow &#39;a&#39; to be &#39;n-1&#39;</span>
-        <span class="c"># then (n - 1)**even % n will be 1 which will give a g of 0 and 1 will</span>
-        <span class="c"># give a zero, too, so we set the range as [2, n-2]. Some references</span>
-        <span class="c"># say &#39;a&#39; should be coprime to n, but either will detect factors.</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">prng</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_trial</span><span class="p">(</span><span class="n">factors</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">candidates</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Helper function for integer factorization. Trial factors ``n`</span>
-<span class="sd">    against all integers given in the sequence ``candidates``</span>
-<span class="sd">    and updates the dict ``factors`` in-place. Returns the reduced</span>
-<span class="sd">    value of ``n`` and a flag indicating whether any factors were found.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-        <span class="n">factors0</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">factors</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-    <span class="n">nfactors</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">candidates</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">%</span> <span class="n">d</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="n">multiplicity</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-            <span class="n">n</span> <span class="o">//=</span> <span class="n">d</span><span class="o">**</span><span class="n">m</span>
-            <span class="n">factors</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">=</span> <span class="n">m</span>
-    <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span><span class="o">.</span><span class="n">difference</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">factors0</span><span class="p">))):</span>
-            <span class="k">print</span><span class="p">(</span><span class="n">factor_msg</span> <span class="o">%</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">factors</span><span class="p">[</span><span class="n">k</span><span class="p">]))</span>
-    <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span> <span class="o">!=</span> <span class="n">nfactors</span>
-
-
-<span class="k">def</span> <span class="nf">_check_termination</span><span class="p">(</span><span class="n">factors</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">limitp1</span><span class="p">,</span> <span class="n">use_trial</span><span class="p">,</span> <span class="n">use_rho</span><span class="p">,</span> <span class="n">use_pm1</span><span class="p">,</span>
-                       <span class="n">verbose</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Helper function for integer factorization. Checks if ``n``</span>
-<span class="sd">    is a prime or a perfect power, and in those cases updates</span>
-<span class="sd">    the factorization and raises ``StopIteration``.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-        <span class="k">print</span><span class="p">(</span><span class="s">&#39;Check for termination&#39;</span><span class="p">)</span>
-
-    <span class="c"># since we&#39;ve already been factoring there is no need to do</span>
-    <span class="c"># simultaneous factoring with the power check</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="n">perfect_power</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">factor</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">p</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">:</span>
-        <span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="n">p</span>
-        <span class="k">if</span> <span class="n">limitp1</span><span class="p">:</span>
-            <span class="n">limit</span> <span class="o">=</span> <span class="n">limitp1</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">limit</span> <span class="o">=</span> <span class="n">limitp1</span>
-        <span class="n">facs</span> <span class="o">=</span> <span class="n">factorint</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">limit</span><span class="p">,</span> <span class="n">use_trial</span><span class="p">,</span> <span class="n">use_rho</span><span class="p">,</span> <span class="n">use_pm1</span><span class="p">,</span>
-                         <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">facs</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-            <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-                <span class="k">print</span><span class="p">(</span><span class="n">factor_msg</span> <span class="o">%</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">))</span>
-            <span class="n">factors</span><span class="p">[</span><span class="n">b</span><span class="p">]</span> <span class="o">=</span> <span class="n">exp</span><span class="o">*</span><span class="n">e</span>
-        <span class="k">raise</span> <span class="ne">StopIteration</span>
-
-    <span class="k">if</span> <span class="n">isprime</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-        <span class="n">factors</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)]</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">raise</span> <span class="ne">StopIteration</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">StopIteration</span>
-
-<span class="n">trial_int_msg</span> <span class="o">=</span> <span class="s">&quot;Trial division with ints [</span><span class="si">%i</span><span class="s"> ... </span><span class="si">%i</span><span class="s">] and fail_max=</span><span class="si">%i</span><span class="s">&quot;</span>
-<span class="n">trial_msg</span> <span class="o">=</span> <span class="s">&quot;Trial division with primes [</span><span class="si">%i</span><span class="s"> ... </span><span class="si">%i</span><span class="s">]&quot;</span>
-<span class="n">rho_msg</span> <span class="o">=</span> <span class="s">&quot;Pollard&#39;s rho with retries </span><span class="si">%i</span><span class="s">, max_steps </span><span class="si">%i</span><span class="s"> and seed </span><span class="si">%i</span><span class="s">&quot;</span>
-<span class="n">pm1_msg</span> <span class="o">=</span> <span class="s">&quot;Pollard&#39;s p-1 with smoothness bound </span><span class="si">%i</span><span class="s"> and seed </span><span class="si">%i</span><span class="s">&quot;</span>
-<span class="n">factor_msg</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\t</span><span class="si">%i</span><span class="s"> ** </span><span class="si">%i</span><span class="s">&#39;</span>
-<span class="n">fermat_msg</span> <span class="o">=</span> <span class="s">&#39;Close factors satisying Fermat condition found.&#39;</span>
-<span class="n">complete_msg</span> <span class="o">=</span> <span class="s">&#39;Factorization is complete.&#39;</span>
-
-
-<span class="k">def</span> <span class="nf">_factorint_small</span><span class="p">(</span><span class="n">factors</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">limit</span><span class="p">,</span> <span class="n">fail_max</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the value of n and either a 0 (indicating that factorization up</span>
-<span class="sd">    to the limit was complete) or else the next near-prime that would have</span>
-<span class="sd">    been tested.</span>
-
-<span class="sd">    Factoring stops if there are fail_max unsuccessful tests in a row.</span>
-
-<span class="sd">    If factors of n were found they will be in the factors dictionary as</span>
-<span class="sd">    {factor: multiplicity} and the returned value of n will have had those</span>
-<span class="sd">    factors removed. The factors dictionary is modified in-place.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">done</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">d</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;return n, d if the sqrt(n) wasn&#39;t reached yet, else</span>
-<span class="sd">           n, 0 indicating that factoring is done.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">d</span><span class="o">*</span><span class="n">d</span> <span class="o">&lt;=</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">n</span><span class="p">,</span> <span class="n">d</span>
-        <span class="k">return</span> <span class="n">n</span><span class="p">,</span> <span class="mi">0</span>
-
-    <span class="n">d</span> <span class="o">=</span> <span class="mi">2</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">trailing</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">m</span><span class="p">:</span>
-        <span class="n">factors</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">=</span> <span class="n">m</span>
-        <span class="n">n</span> <span class="o">&gt;&gt;=</span> <span class="n">m</span>
-    <span class="n">d</span> <span class="o">=</span> <span class="mi">3</span>
-    <span class="k">if</span> <span class="n">limit</span> <span class="o">&lt;</span> <span class="n">d</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">factors</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="n">done</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
-    <span class="c"># reduce</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">while</span> <span class="n">n</span> <span class="o">%</span> <span class="n">d</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">n</span> <span class="o">//=</span> <span class="n">d</span>
-        <span class="n">m</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">20</span><span class="p">:</span>
-            <span class="n">mm</span> <span class="o">=</span> <span class="n">multiplicity</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-            <span class="n">m</span> <span class="o">+=</span> <span class="n">mm</span>
-            <span class="n">n</span> <span class="o">//=</span> <span class="n">d</span><span class="o">**</span><span class="n">mm</span>
-            <span class="k">break</span>
-    <span class="k">if</span> <span class="n">m</span><span class="p">:</span>
-        <span class="n">factors</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">=</span> <span class="n">m</span>
-
-    <span class="c"># when d*d exceeds maxx or n we are done; if limit**2 is greater</span>
-    <span class="c"># than n then maxx is set to zero so the value of n will flag the finish</span>
-    <span class="k">if</span> <span class="n">limit</span><span class="o">*</span><span class="n">limit</span> <span class="o">&gt;</span> <span class="n">n</span><span class="p">:</span>
-        <span class="n">maxx</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">maxx</span> <span class="o">=</span> <span class="n">limit</span><span class="o">*</span><span class="n">limit</span>
-
-    <span class="n">dd</span> <span class="o">=</span> <span class="n">maxx</span> <span class="ow">or</span> <span class="n">n</span>
-    <span class="n">d</span> <span class="o">=</span> <span class="mi">5</span>
-    <span class="n">fails</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">while</span> <span class="n">fails</span> <span class="o">&lt;</span> <span class="n">fail_max</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">d</span><span class="o">*</span><span class="n">d</span> <span class="o">&gt;</span> <span class="n">dd</span><span class="p">:</span>
-            <span class="k">break</span>
-        <span class="c"># d = 6*i - 1</span>
-        <span class="c"># reduce</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">while</span> <span class="n">n</span> <span class="o">%</span> <span class="n">d</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">//=</span> <span class="n">d</span>
-            <span class="n">m</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">20</span><span class="p">:</span>
-                <span class="n">mm</span> <span class="o">=</span> <span class="n">multiplicity</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-                <span class="n">m</span> <span class="o">+=</span> <span class="n">mm</span>
-                <span class="n">n</span> <span class="o">//=</span> <span class="n">d</span><span class="o">**</span><span class="n">mm</span>
-                <span class="k">break</span>
-        <span class="k">if</span> <span class="n">m</span><span class="p">:</span>
-            <span class="n">factors</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">=</span> <span class="n">m</span>
-            <span class="n">dd</span> <span class="o">=</span> <span class="n">maxx</span> <span class="ow">or</span> <span class="n">n</span>
-            <span class="n">fails</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">fails</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="n">d</span> <span class="o">+=</span> <span class="mi">2</span>
-        <span class="k">if</span> <span class="n">d</span><span class="o">*</span><span class="n">d</span> <span class="o">&gt;</span> <span class="n">dd</span><span class="p">:</span>
-            <span class="k">break</span>
-        <span class="c"># d = 6*i - 1</span>
-        <span class="c"># reduce</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">while</span> <span class="n">n</span> <span class="o">%</span> <span class="n">d</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">//=</span> <span class="n">d</span>
-            <span class="n">m</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">20</span><span class="p">:</span>
-                <span class="n">mm</span> <span class="o">=</span> <span class="n">multiplicity</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-                <span class="n">m</span> <span class="o">+=</span> <span class="n">mm</span>
-                <span class="n">n</span> <span class="o">//=</span> <span class="n">d</span><span class="o">**</span><span class="n">mm</span>
-                <span class="k">break</span>
-        <span class="k">if</span> <span class="n">m</span><span class="p">:</span>
-            <span class="n">factors</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">=</span> <span class="n">m</span>
-            <span class="n">dd</span> <span class="o">=</span> <span class="n">maxx</span> <span class="ow">or</span> <span class="n">n</span>
-            <span class="n">fails</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">fails</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="c"># d = 6*(i+1) - 1</span>
-        <span class="n">d</span> <span class="o">+=</span> <span class="mi">4</span>
-
-    <span class="k">return</span> <span class="n">done</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="factorint"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.factor_.factorint">[docs]</a><span class="k">def</span> <span class="nf">factorint</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">limit</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">use_trial</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">use_rho</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">use_pm1</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span>
-              <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Given a positive integer ``n``, ``factorint(n)`` returns a dict containing</span>
-<span class="sd">    the prime factors of ``n`` as keys and their respective multiplicities</span>
-<span class="sd">    as values. For example:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory import factorint</span>
-<span class="sd">    &gt;&gt;&gt; factorint(2000)    # 2000 = (2**4) * (5**3)</span>
-<span class="sd">    {2: 4, 5: 3}</span>
-<span class="sd">    &gt;&gt;&gt; factorint(65537)   # This number is prime</span>
-<span class="sd">    {65537: 1}</span>
-
-<span class="sd">    For input less than 2, factorint behaves as follows:</span>
-
-<span class="sd">        - ``factorint(1)`` returns the empty factorization, ``{}``</span>
-<span class="sd">        - ``factorint(0)`` returns ``{0:1}``</span>
-<span class="sd">        - ``factorint(-n)`` adds ``-1:1`` to the factors and then factors ``n``</span>
-
-<span class="sd">    Partial Factorization:</span>
-
-<span class="sd">    If ``limit`` (&gt; 3) is specified, the search is stopped after performing</span>
-<span class="sd">    trial division up to (and including) the limit (or taking a</span>
-<span class="sd">    corresponding number of rho/p-1 steps). This is useful if one has</span>
-<span class="sd">    a large number and only is interested in finding small factors (if</span>
-<span class="sd">    any). Note that setting a limit does not prevent larger factors</span>
-<span class="sd">    from being found early; it simply means that the largest factor may</span>
-<span class="sd">    be composite. Since checking for perfect power is relatively cheap, it is</span>
-<span class="sd">    done regardless of the limit setting.</span>
-
-<span class="sd">    This number, for example, has two small factors and a huge</span>
-<span class="sd">    semi-prime factor that cannot be reduced easily:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory import isprime</span>
-<span class="sd">    &gt;&gt;&gt; a = 1407633717262338957430697921446883</span>
-<span class="sd">    &gt;&gt;&gt; f = factorint(a, limit=10000)</span>
-<span class="sd">    &gt;&gt;&gt; f == {991: 1, 202916782076162456022877024859: 1, 7: 1}</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; isprime(max(f))</span>
-<span class="sd">    False</span>
-
-<span class="sd">    This number has a small factor and a residual perfect power whose</span>
-<span class="sd">    base is greater than the limit:</span>
-
-<span class="sd">    &gt;&gt;&gt; factorint(3*101**7, limit=5)</span>
-<span class="sd">    {3: 1, 101: 7}</span>
-
-<span class="sd">    Visual Factorization:</span>
-
-<span class="sd">    If ``visual`` is set to ``True``, then it will return a visual</span>
-<span class="sd">    factorization of the integer.  For example:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import pprint</span>
-<span class="sd">    &gt;&gt;&gt; pprint(factorint(4200, visual=True))</span>
-<span class="sd">     3  1  2  1</span>
-<span class="sd">    2 *3 *5 *7</span>
-
-<span class="sd">    Note that this is achieved by using the evaluate=False flag in Mul</span>
-<span class="sd">    and Pow. If you do other manipulations with an expression where</span>
-<span class="sd">    evaluate=False, it may evaluate.  Therefore, you should use the</span>
-<span class="sd">    visual option only for visualization, and use the normal dictionary</span>
-<span class="sd">    returned by visual=False if you want to perform operations on the</span>
-<span class="sd">    factors.</span>
-
-<span class="sd">    You can easily switch between the two forms by sending them back to</span>
-<span class="sd">    factorint:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Mul, Pow</span>
-<span class="sd">    &gt;&gt;&gt; regular = factorint(1764); regular</span>
-<span class="sd">    {2: 2, 3: 2, 7: 2}</span>
-<span class="sd">    &gt;&gt;&gt; pprint(factorint(regular))</span>
-<span class="sd">     2  2  2</span>
-<span class="sd">    2 *3 *7</span>
-
-<span class="sd">    &gt;&gt;&gt; visual = factorint(1764, visual=True); pprint(visual)</span>
-<span class="sd">     2  2  2</span>
-<span class="sd">    2 *3 *7</span>
-<span class="sd">    &gt;&gt;&gt; print(factorint(visual))</span>
-<span class="sd">    {2: 2, 3: 2, 7: 2}</span>
-
-<span class="sd">    If you want to send a number to be factored in a partially factored form</span>
-<span class="sd">    you can do so with a dictionary or unevaluated expression:</span>
-
-<span class="sd">    &gt;&gt;&gt; factorint(factorint({4: 2, 12: 3})) # twice to toggle to dict form</span>
-<span class="sd">    {2: 10, 3: 3}</span>
-<span class="sd">    &gt;&gt;&gt; factorint(Mul(4, 12, **dict(evaluate=False)))</span>
-<span class="sd">    {2: 4, 3: 1}</span>
-
-<span class="sd">    The table of the output logic is:</span>
-
-<span class="sd">        ====== ====== ======= =======</span>
-<span class="sd">                       Visual</span>
-<span class="sd">        ------ ----------------------</span>
-<span class="sd">        Input  True   False   other</span>
-<span class="sd">        ====== ====== ======= =======</span>
-<span class="sd">        dict    mul    dict    mul</span>
-<span class="sd">        n       mul    dict    dict</span>
-<span class="sd">        mul     mul    dict    dict</span>
-<span class="sd">        ====== ====== ======= =======</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    Algorithm:</span>
-
-<span class="sd">    The function switches between multiple algorithms. Trial division</span>
-<span class="sd">    quickly finds small factors (of the order 1-5 digits), and finds</span>
-<span class="sd">    all large factors if given enough time. The Pollard rho and p-1</span>
-<span class="sd">    algorithms are used to find large factors ahead of time; they</span>
-<span class="sd">    will often find factors of the order of 10 digits within a few</span>
-<span class="sd">    seconds:</span>
-
-<span class="sd">    &gt;&gt;&gt; factors = factorint(12345678910111213141516)</span>
-<span class="sd">    &gt;&gt;&gt; for base, exp in sorted(factors.items()):</span>
-<span class="sd">    ...     print(base, exp)</span>
-<span class="sd">    ...</span>
-<span class="sd">    2 2</span>
-<span class="sd">    2507191691 1</span>
-<span class="sd">    1231026625769 1</span>
-
-<span class="sd">    Any of these methods can optionally be disabled with the following</span>
-<span class="sd">    boolean parameters:</span>
-
-<span class="sd">        - ``use_trial``: Toggle use of trial division</span>
-<span class="sd">        - ``use_rho``: Toggle use of Pollard&#39;s rho method</span>
-<span class="sd">        - ``use_pm1``: Toggle use of Pollard&#39;s p-1 method</span>
-
-<span class="sd">    ``factorint`` also periodically checks if the remaining part is</span>
-<span class="sd">    a prime number or a perfect power, and in those cases stops.</span>
-
-
-<span class="sd">    If ``verbose`` is set to ``True``, detailed progress is printed.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    smoothness, smoothness_p, divisors</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">factordict</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">if</span> <span class="n">visual</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">Mul</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="nb">dict</span><span class="p">):</span>
-        <span class="n">factordict</span> <span class="o">=</span> <span class="n">factorint</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">limit</span><span class="o">=</span><span class="n">limit</span><span class="p">,</span> <span class="n">use_trial</span><span class="o">=</span><span class="n">use_trial</span><span class="p">,</span>
-                               <span class="n">use_rho</span><span class="o">=</span><span class="n">use_rho</span><span class="p">,</span> <span class="n">use_pm1</span><span class="o">=</span><span class="n">use_pm1</span><span class="p">,</span>
-                               <span class="n">verbose</span><span class="o">=</span><span class="n">verbose</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">Mul</span><span class="p">):</span>
-        <span class="n">factordict</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">([(</span><span class="nb">int</span><span class="p">(</span><span class="n">k</span><span class="p">),</span> <span class="nb">int</span><span class="p">(</span><span class="n">v</span><span class="p">))</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span>
-                           <span class="nb">list</span><span class="p">(</span><span class="n">n</span><span class="o">.</span><span class="n">as_powers_dict</span><span class="p">()</span><span class="o">.</span><span class="n">items</span><span class="p">())])</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="nb">dict</span><span class="p">):</span>
-        <span class="n">factordict</span> <span class="o">=</span> <span class="n">n</span>
-    <span class="k">if</span> <span class="n">factordict</span> <span class="ow">and</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">Mul</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="nb">dict</span><span class="p">)):</span>
-        <span class="c"># check it</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">factordict</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
-            <span class="k">if</span> <span class="n">isprime</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
-                <span class="k">continue</span>
-            <span class="n">e</span> <span class="o">=</span> <span class="n">factordict</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="n">factorint</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">limit</span><span class="o">=</span><span class="n">limit</span><span class="p">,</span> <span class="n">use_trial</span><span class="o">=</span><span class="n">use_trial</span><span class="p">,</span> <span class="n">use_rho</span><span class="o">=</span><span class="n">use_rho</span><span class="p">,</span>
-                          <span class="n">use_pm1</span><span class="o">=</span><span class="n">use_pm1</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="n">verbose</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-                <span class="k">if</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">factordict</span><span class="p">:</span>
-                    <span class="n">factordict</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">+=</span> <span class="n">v</span><span class="o">*</span><span class="n">e</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">factordict</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">v</span><span class="o">*</span><span class="n">e</span>
-    <span class="k">if</span> <span class="n">visual</span> <span class="ow">or</span> <span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">dict</span> <span class="ow">and</span>
-                  <span class="n">visual</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">True</span> <span class="ow">and</span>
-                  <span class="n">visual</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">factordict</span> <span class="o">==</span> <span class="p">{}:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">if</span> <span class="o">-</span><span class="mi">1</span> <span class="ow">in</span> <span class="n">factordict</span><span class="p">:</span>
-            <span class="n">factordict</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">args</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">Pow</span><span class="p">(</span><span class="o">*</span><span class="n">i</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;evaluate&#39;</span><span class="p">:</span><span class="bp">False</span><span class="p">})</span>
-                     <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">factordict</span><span class="o">.</span><span class="n">items</span><span class="p">())])</span>
-        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;evaluate&#39;</span><span class="p">:</span> <span class="bp">False</span><span class="p">})</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="nb">dict</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">Mul</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">factordict</span>
-
-    <span class="k">assert</span> <span class="n">use_trial</span> <span class="ow">or</span> <span class="n">use_rho</span> <span class="ow">or</span> <span class="n">use_pm1</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">limit</span><span class="p">:</span>
-        <span class="n">limit</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">limit</span><span class="p">)</span>
-
-    <span class="c"># special cases</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">factors</span> <span class="o">=</span> <span class="n">factorint</span><span class="p">(</span>
-            <span class="o">-</span><span class="n">n</span><span class="p">,</span> <span class="n">limit</span><span class="o">=</span><span class="n">limit</span><span class="p">,</span> <span class="n">use_trial</span><span class="o">=</span><span class="n">use_trial</span><span class="p">,</span> <span class="n">use_rho</span><span class="o">=</span><span class="n">use_rho</span><span class="p">,</span>
-            <span class="n">use_pm1</span><span class="o">=</span><span class="n">use_pm1</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="n">verbose</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-        <span class="n">factors</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="n">factors</span>
-
-    <span class="k">if</span> <span class="n">limit</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">limit</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">{}</span>
-            <span class="k">return</span> <span class="p">{</span><span class="n">n</span><span class="p">:</span> <span class="mi">1</span><span class="p">}</span>
-    <span class="k">elif</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">10</span><span class="p">:</span>
-        <span class="c"># doing this we are assured of getting a limit &gt; 2</span>
-        <span class="c"># when we have to compute it later</span>
-        <span class="k">return</span> <span class="p">[{</span><span class="mi">0</span><span class="p">:</span> <span class="mi">1</span><span class="p">},</span> <span class="p">{},</span> <span class="p">{</span><span class="mi">2</span><span class="p">:</span> <span class="mi">1</span><span class="p">},</span> <span class="p">{</span><span class="mi">3</span><span class="p">:</span> <span class="mi">1</span><span class="p">},</span> <span class="p">{</span><span class="mi">2</span><span class="p">:</span> <span class="mi">2</span><span class="p">},</span> <span class="p">{</span><span class="mi">5</span><span class="p">:</span> <span class="mi">1</span><span class="p">},</span>
-                <span class="p">{</span><span class="mi">2</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">:</span> <span class="mi">1</span><span class="p">},</span> <span class="p">{</span><span class="mi">7</span><span class="p">:</span> <span class="mi">1</span><span class="p">},</span> <span class="p">{</span><span class="mi">2</span><span class="p">:</span> <span class="mi">3</span><span class="p">},</span> <span class="p">{</span><span class="mi">3</span><span class="p">:</span> <span class="mi">2</span><span class="p">}][</span><span class="n">n</span><span class="p">]</span>
-
-    <span class="n">factors</span> <span class="o">=</span> <span class="p">{}</span>
-
-    <span class="c"># do simplistic factorization</span>
-    <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-        <span class="n">sn</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">sn</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">50</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;Factoring </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">sn</span><span class="p">[:</span><span class="mi">5</span><span class="p">]</span> <span class="o">+</span> \
-                  <span class="s">&#39;..(</span><span class="si">%i</span><span class="s"> other digits)..&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">sn</span><span class="p">)</span> <span class="o">-</span> <span class="mi">10</span><span class="p">)</span> <span class="o">+</span> <span class="n">sn</span><span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">:])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;Factoring&#39;</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">use_trial</span><span class="p">:</span>
-        <span class="c"># this is the preliminary factorization for small factors</span>
-        <span class="n">small</span> <span class="o">=</span> <span class="mi">2</span><span class="o">**</span><span class="mi">15</span>
-        <span class="n">fail_max</span> <span class="o">=</span> <span class="mi">600</span>
-        <span class="n">small</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">small</span><span class="p">,</span> <span class="n">limit</span> <span class="ow">or</span> <span class="n">small</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="n">trial_int_msg</span> <span class="o">%</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">small</span><span class="p">,</span> <span class="n">fail_max</span><span class="p">))</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">next_p</span> <span class="o">=</span> <span class="n">_factorint_small</span><span class="p">(</span><span class="n">factors</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">small</span><span class="p">,</span> <span class="n">fail_max</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">next_p</span> <span class="o">=</span> <span class="mi">2</span>
-    <span class="k">if</span> <span class="n">factors</span> <span class="ow">and</span> <span class="n">verbose</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">factors</span><span class="p">):</span>
-            <span class="k">print</span><span class="p">(</span><span class="n">factor_msg</span> <span class="o">%</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">factors</span><span class="p">[</span><span class="n">k</span><span class="p">]))</span>
-    <span class="k">if</span> <span class="n">next_p</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">factors</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)]</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="n">complete_msg</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">factors</span>
-
-    <span class="c"># continue with more advanced factorization methods</span>
-
-    <span class="c"># first check if the simplistic run didn&#39;t finish</span>
-    <span class="c"># because of the limit and check for a perfect</span>
-    <span class="c"># power before exiting</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">limit</span> <span class="ow">and</span> <span class="n">next_p</span> <span class="o">&gt;</span> <span class="n">limit</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&#39;Exceeded limit:&#39;</span><span class="p">,</span> <span class="n">limit</span><span class="p">)</span>
-
-            <span class="n">_check_termination</span><span class="p">(</span><span class="n">factors</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">limit</span><span class="p">,</span> <span class="n">use_trial</span><span class="p">,</span> <span class="n">use_rho</span><span class="p">,</span> <span class="n">use_pm1</span><span class="p">,</span>
-                               <span class="n">verbose</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">factors</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)]</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="k">return</span> <span class="n">factors</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># Before quitting (or continuing on)...</span>
-
-            <span class="c"># ...do a Fermat test since it&#39;s so easy and we need the</span>
-            <span class="c"># square root anyway. Finding 2 factors is easy if they are</span>
-            <span class="c"># &quot;close enough.&quot; This is the big root equivalent of dividing by</span>
-            <span class="c"># 2, 3, 5.</span>
-            <span class="n">sqrt_n</span> <span class="o">=</span> <span class="n">integer_nthroot</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">sqrt_n</span> <span class="o">+</span> <span class="mi">1</span>
-            <span class="n">a2</span> <span class="o">=</span> <span class="n">a</span><span class="o">**</span><span class="mi">2</span>
-            <span class="n">b2</span> <span class="o">=</span> <span class="n">a2</span> <span class="o">-</span> <span class="n">n</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">):</span>
-                <span class="n">b</span><span class="p">,</span> <span class="n">fermat</span> <span class="o">=</span> <span class="n">integer_nthroot</span><span class="p">(</span><span class="n">b2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">fermat</span><span class="p">:</span>
-                    <span class="k">break</span>
-                <span class="n">b2</span> <span class="o">+=</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span>  <span class="c"># equiv to (a+1)**2 - n</span>
-                <span class="n">a</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="n">fermat</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-                    <span class="k">print</span><span class="p">(</span><span class="n">fermat_msg</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">limit</span><span class="p">:</span>
-                    <span class="n">limit</span> <span class="o">-=</span> <span class="mi">1</span>
-                <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="p">[</span><span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">]:</span>
-                    <span class="n">facs</span> <span class="o">=</span> <span class="n">factorint</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">limit</span><span class="o">=</span><span class="n">limit</span><span class="p">,</span> <span class="n">use_trial</span><span class="o">=</span><span class="n">use_trial</span><span class="p">,</span>
-                                     <span class="n">use_rho</span><span class="o">=</span><span class="n">use_rho</span><span class="p">,</span> <span class="n">use_pm1</span><span class="o">=</span><span class="n">use_pm1</span><span class="p">,</span>
-                                     <span class="n">verbose</span><span class="o">=</span><span class="n">verbose</span><span class="p">)</span>
-                    <span class="n">factors</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">facs</span><span class="p">)</span>
-                <span class="k">raise</span> <span class="ne">StopIteration</span>
-
-            <span class="c"># ...see if factorization can be terminated</span>
-            <span class="n">_check_termination</span><span class="p">(</span><span class="n">factors</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">limit</span><span class="p">,</span> <span class="n">use_trial</span><span class="p">,</span> <span class="n">use_rho</span><span class="p">,</span> <span class="n">use_pm1</span><span class="p">,</span>
-                               <span class="n">verbose</span><span class="p">)</span>
-
-    <span class="k">except</span> <span class="ne">StopIteration</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="n">complete_msg</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">factors</span>
-
-    <span class="c"># these are the limits for trial division which will</span>
-    <span class="c"># be attempted in parallel with pollard methods</span>
-    <span class="n">low</span><span class="p">,</span> <span class="n">high</span> <span class="o">=</span> <span class="n">next_p</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">next_p</span>
-
-    <span class="n">limit</span> <span class="o">=</span> <span class="n">limit</span> <span class="ow">or</span> <span class="n">sqrt_n</span>
-    <span class="c"># add 1 to make sure limit is reached in primerange calls</span>
-    <span class="n">limit</span> <span class="o">+=</span> <span class="mi">1</span>
-
-    <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">high_</span> <span class="o">=</span> <span class="n">high</span>
-            <span class="k">if</span> <span class="n">limit</span> <span class="o">&lt;</span> <span class="n">high_</span><span class="p">:</span>
-                <span class="n">high_</span> <span class="o">=</span> <span class="n">limit</span>
-
-            <span class="c"># Trial division</span>
-            <span class="k">if</span> <span class="n">use_trial</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-                    <span class="k">print</span><span class="p">(</span><span class="n">trial_msg</span> <span class="o">%</span> <span class="p">(</span><span class="n">low</span><span class="p">,</span> <span class="n">high_</span><span class="p">))</span>
-                <span class="n">ps</span> <span class="o">=</span> <span class="n">sieve</span><span class="o">.</span><span class="n">primerange</span><span class="p">(</span><span class="n">low</span><span class="p">,</span> <span class="n">high_</span><span class="p">)</span>
-                <span class="n">n</span><span class="p">,</span> <span class="n">found_trial</span> <span class="o">=</span> <span class="n">_trial</span><span class="p">(</span><span class="n">factors</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">ps</span><span class="p">,</span> <span class="n">verbose</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">found_trial</span><span class="p">:</span>
-                    <span class="n">_check_termination</span><span class="p">(</span><span class="n">factors</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">limit</span><span class="p">,</span> <span class="n">use_trial</span><span class="p">,</span> <span class="n">use_rho</span><span class="p">,</span>
-                                       <span class="n">use_pm1</span><span class="p">,</span> <span class="n">verbose</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">found_trial</span> <span class="o">=</span> <span class="bp">False</span>
-
-            <span class="k">if</span> <span class="n">high</span> <span class="o">&gt;</span> <span class="n">limit</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-                    <span class="k">print</span><span class="p">(</span><span class="s">&#39;Exceeded limit:&#39;</span><span class="p">,</span> <span class="n">limit</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">factors</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)]</span> <span class="o">=</span> <span class="mi">1</span>
-                <span class="k">raise</span> <span class="ne">StopIteration</span>
-
-            <span class="c"># Only used advanced methods when no small factors were found</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">found_trial</span><span class="p">:</span>
-                <span class="k">if</span> <span class="p">(</span><span class="n">use_pm1</span> <span class="ow">or</span> <span class="n">use_rho</span><span class="p">):</span>
-                    <span class="n">high_root</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">high_</span><span class="o">**</span><span class="mf">0.7</span><span class="p">)),</span> <span class="n">low</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-
-                    <span class="c"># Pollard p-1</span>
-                    <span class="k">if</span> <span class="n">use_pm1</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-                            <span class="k">print</span><span class="p">((</span><span class="n">pm1_msg</span> <span class="o">%</span> <span class="p">(</span><span class="n">high_root</span><span class="p">,</span> <span class="n">high_</span><span class="p">)))</span>
-                        <span class="n">c</span> <span class="o">=</span> <span class="n">pollard_pm1</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">B</span><span class="o">=</span><span class="n">high_root</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="n">high_</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">c</span><span class="p">:</span>
-                            <span class="c"># factor it and let _trial do the update</span>
-                            <span class="n">ps</span> <span class="o">=</span> <span class="n">factorint</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">limit</span><span class="o">=</span><span class="n">limit</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span>
-                                           <span class="n">use_trial</span><span class="o">=</span><span class="n">use_trial</span><span class="p">,</span>
-                                           <span class="n">use_rho</span><span class="o">=</span><span class="n">use_rho</span><span class="p">,</span>
-                                           <span class="n">use_pm1</span><span class="o">=</span><span class="n">use_pm1</span><span class="p">,</span>
-                                           <span class="n">verbose</span><span class="o">=</span><span class="n">verbose</span><span class="p">)</span>
-                            <span class="n">n</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">_trial</span><span class="p">(</span><span class="n">factors</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">ps</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-                            <span class="n">_check_termination</span><span class="p">(</span><span class="n">factors</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">limit</span><span class="p">,</span> <span class="n">use_trial</span><span class="p">,</span>
-                                               <span class="n">use_rho</span><span class="p">,</span> <span class="n">use_pm1</span><span class="p">,</span> <span class="n">verbose</span><span class="p">)</span>
-
-                    <span class="c"># Pollard rho</span>
-                    <span class="k">if</span> <span class="n">use_rho</span><span class="p">:</span>
-                        <span class="n">max_steps</span> <span class="o">=</span> <span class="n">high_root</span>
-                        <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-                            <span class="k">print</span><span class="p">((</span><span class="n">rho_msg</span> <span class="o">%</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">max_steps</span><span class="p">,</span> <span class="n">high_</span><span class="p">)))</span>
-                        <span class="n">c</span> <span class="o">=</span> <span class="n">pollard_rho</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">retries</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">max_steps</span><span class="o">=</span><span class="n">max_steps</span><span class="p">,</span>
-                                        <span class="n">seed</span><span class="o">=</span><span class="n">high_</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">c</span><span class="p">:</span>
-                            <span class="c"># factor it and let _trial do the update</span>
-                            <span class="n">ps</span> <span class="o">=</span> <span class="n">factorint</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">limit</span><span class="o">=</span><span class="n">limit</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span>
-                                           <span class="n">use_trial</span><span class="o">=</span><span class="n">use_trial</span><span class="p">,</span>
-                                           <span class="n">use_rho</span><span class="o">=</span><span class="n">use_rho</span><span class="p">,</span>
-                                           <span class="n">use_pm1</span><span class="o">=</span><span class="n">use_pm1</span><span class="p">,</span>
-                                           <span class="n">verbose</span><span class="o">=</span><span class="n">verbose</span><span class="p">)</span>
-                            <span class="n">n</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">_trial</span><span class="p">(</span><span class="n">factors</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">ps</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-                            <span class="n">_check_termination</span><span class="p">(</span><span class="n">factors</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">limit</span><span class="p">,</span> <span class="n">use_trial</span><span class="p">,</span>
-                                               <span class="n">use_rho</span><span class="p">,</span> <span class="n">use_pm1</span><span class="p">,</span> <span class="n">verbose</span><span class="p">)</span>
-
-        <span class="k">except</span> <span class="ne">StopIteration</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-                <span class="k">print</span><span class="p">(</span><span class="n">complete_msg</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">factors</span>
-
-        <span class="n">low</span><span class="p">,</span> <span class="n">high</span> <span class="o">=</span> <span class="n">high</span><span class="p">,</span> <span class="n">high</span><span class="o">*</span><span class="mi">2</span>
-
-</div>
-<div class="viewcode-block" id="primefactors"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.factor_.primefactors">[docs]</a><span class="k">def</span> <span class="nf">primefactors</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">limit</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return a sorted list of n&#39;s prime factors, ignoring multiplicity</span>
-<span class="sd">    and any composite factor that remains if the limit was set too low</span>
-<span class="sd">    for complete factorization. Unlike factorint(), primefactors() does</span>
-<span class="sd">    not return -1 or 0.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory import primefactors, factorint, isprime</span>
-<span class="sd">    &gt;&gt;&gt; primefactors(6)</span>
-<span class="sd">    [2, 3]</span>
-<span class="sd">    &gt;&gt;&gt; primefactors(-5)</span>
-<span class="sd">    [5]</span>
-
-<span class="sd">    &gt;&gt;&gt; sorted(factorint(123456).items())</span>
-<span class="sd">    [(2, 6), (3, 1), (643, 1)]</span>
-<span class="sd">    &gt;&gt;&gt; primefactors(123456)</span>
-<span class="sd">    [2, 3, 643]</span>
-
-<span class="sd">    &gt;&gt;&gt; sorted(factorint(10000000001, limit=200).items())</span>
-<span class="sd">    [(101, 1), (99009901, 1)]</span>
-<span class="sd">    &gt;&gt;&gt; isprime(99009901)</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; primefactors(10000000001, limit=300)</span>
-<span class="sd">    [101]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    divisors</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">factors</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">factorint</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">limit</span><span class="o">=</span><span class="n">limit</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="n">verbose</span><span class="p">)</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">factors</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">:]</span> <span class="k">if</span> <span class="n">f</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]]</span>
-    <span class="k">if</span> <span class="n">factors</span> <span class="ow">and</span> <span class="n">isprime</span><span class="p">(</span><span class="n">factors</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]):</span>
-        <span class="n">s</span> <span class="o">+=</span> <span class="p">[</span><span class="n">factors</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span>
-    <span class="k">return</span> <span class="n">s</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_divisors</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Helper function for divisors which generates the divisors.&quot;&quot;&quot;</span>
-
-    <span class="n">factordict</span> <span class="o">=</span> <span class="n">factorint</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="n">ps</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">factordict</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-
-    <span class="k">def</span> <span class="nf">rec_gen</span><span class="p">(</span><span class="n">n</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">ps</span><span class="p">):</span>
-            <span class="k">yield</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">pows</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">factordict</span><span class="p">[</span><span class="n">ps</span><span class="p">[</span><span class="n">n</span><span class="p">]]):</span>
-                <span class="n">pows</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">pows</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">ps</span><span class="p">[</span><span class="n">n</span><span class="p">])</span>
-            <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">rec_gen</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">pows</span><span class="p">:</span>
-                    <span class="k">yield</span> <span class="n">p</span> <span class="o">*</span> <span class="n">q</span>
-
-    <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">rec_gen</span><span class="p">():</span>
-        <span class="k">yield</span> <span class="n">p</span>
-
-
-<div class="viewcode-block" id="divisors"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.factor_.divisors">[docs]</a><span class="k">def</span> <span class="nf">divisors</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">generator</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Return all divisors of n sorted from 1..n by default.</span>
-<span class="sd">    If generator is True an unordered generator is returned.</span>
-
-<span class="sd">    The number of divisors of n can be quite large if there are many</span>
-<span class="sd">    prime factors (counting repeated factors). If only the number of</span>
-<span class="sd">    factors is desired use divisor_count(n).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import divisors, divisor_count</span>
-<span class="sd">    &gt;&gt;&gt; divisors(24)</span>
-<span class="sd">    [1, 2, 3, 4, 6, 8, 12, 24]</span>
-<span class="sd">    &gt;&gt;&gt; divisor_count(24)</span>
-<span class="sd">    8</span>
-
-<span class="sd">    &gt;&gt;&gt; list(divisors(120, generator=True))</span>
-<span class="sd">    [1, 2, 4, 8, 3, 6, 12, 24, 5, 10, 20, 40, 15, 30, 60, 120]</span>
-
-<span class="sd">    This is a slightly modified version of Tim Peters referenced at:</span>
-<span class="sd">    http://stackoverflow.com/questions/1010381/python-factorization</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    primefactors, factorint, divisor_count</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">isprime</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">]</span>
-    <span class="k">elif</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-    <span class="k">elif</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">_divisors</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">generator</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">rv</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">rv</span>
-
-</div>
-<div class="viewcode-block" id="divisor_count"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.factor_.divisor_count">[docs]</a><span class="k">def</span> <span class="nf">divisor_count</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">modulus</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the number of divisors of ``n``. If ``modulus`` is not 1 then only</span>
-<span class="sd">    those that are divisible by ``modulus`` are counted.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - http://www.mayer.dial.pipex.com/maths/formulae.htm</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import divisor_count</span>
-<span class="sd">    &gt;&gt;&gt; divisor_count(6)</span>
-<span class="sd">    4</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    factorint, divisors, totient</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">modulus</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="k">elif</span> <span class="n">modulus</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="nb">divmod</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">modulus</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">r</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">0</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">v</span> <span class="o">+</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">factorint</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">items</span><span class="p">())</span> <span class="k">if</span> <span class="n">k</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">])</span>
-
-</div>
-<div class="viewcode-block" id="totient"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.factor_.totient">[docs]</a><span class="k">def</span> <span class="nf">totient</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Calculate the Euler totient function phi(n)</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory import totient</span>
-<span class="sd">    &gt;&gt;&gt; totient(1)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; totient(25)</span>
-<span class="sd">    20</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    divisor_count</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;n must be a positive integer&quot;</span><span class="p">)</span>
-    <span class="n">factors</span> <span class="o">=</span> <span class="n">factorint</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="n">t</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="k">for</span> <span class="n">p</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">factors</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="n">t</span> <span class="o">*=</span> <span class="p">(</span><span class="n">p</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">p</span><span class="o">**</span><span class="p">(</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">t</span></div>
-</pre></div>
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-  <h1>Source code for sympy.ntheory.generate</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Generating and counting primes.</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">import</span> <span class="nn">random</span>
-<span class="kn">from</span> <span class="nn">bisect</span> <span class="kn">import</span> <span class="n">bisect</span>
-<span class="c"># Using arrays for sieving instead of lists greatly reduces</span>
-<span class="c"># memory consumption</span>
-<span class="kn">from</span> <span class="nn">array</span> <span class="kn">import</span> <span class="n">array</span> <span class="k">as</span> <span class="n">_array</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">as_int</span>
-<span class="kn">from</span> <span class="nn">.primetest</span> <span class="kn">import</span> <span class="n">isprime</span>
-
-
-<span class="k">def</span> <span class="nf">_arange</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="n">ar</span> <span class="o">=</span> <span class="n">_array</span><span class="p">(</span><span class="s">&#39;l&#39;</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="n">a</span><span class="p">))</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)):</span>
-        <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">e</span>
-    <span class="k">return</span> <span class="n">ar</span>
-
-
-<div class="viewcode-block" id="Sieve"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.generate.Sieve">[docs]</a><span class="k">class</span> <span class="nc">Sieve</span><span class="p">:</span>
-    <span class="sd">&quot;&quot;&quot;An infinite list of prime numbers, implemented as a dynamically</span>
-<span class="sd">    growing sieve of Eratosthenes. When a lookup is requested involving</span>
-<span class="sd">    an odd number that has not been sieved, the sieve is automatically</span>
-<span class="sd">    extended up to that number.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import sieve</span>
-<span class="sd">    &gt;&gt;&gt; from array import array # this line and next for doctest only</span>
-<span class="sd">    &gt;&gt;&gt; sieve._list = array(&#39;l&#39;, [2, 3, 5, 7, 11, 13])</span>
-
-<span class="sd">    &gt;&gt;&gt; 25 in sieve</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; sieve._list</span>
-<span class="sd">    array(&#39;l&#39;, [2, 3, 5, 7, 11, 13, 17, 19, 23])</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="c"># data shared (and updated) by all Sieve instances</span>
-    <span class="n">_list</span> <span class="o">=</span> <span class="n">_array</span><span class="p">(</span><span class="s">&#39;l&#39;</span><span class="p">,</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">13</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;&lt;Sieve with </span><span class="si">%i</span><span class="s"> primes sieved: 2, 3, 5, ... </span><span class="si">%i</span><span class="s">, </span><span class="si">%i</span><span class="s">&gt;&quot;</span> <span class="o">%</span> \
-            <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_list</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_list</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">_list</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-
-<div class="viewcode-block" id="Sieve.extend"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.generate.Sieve.extend">[docs]</a>    <span class="k">def</span> <span class="nf">extend</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Grow the sieve to cover all primes &lt;= n (a real number).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sieve</span>
-<span class="sd">        &gt;&gt;&gt; from array import array # this line and next for doctest only</span>
-<span class="sd">        &gt;&gt;&gt; sieve._list = array(&#39;l&#39;, [2, 3, 5, 7, 11, 13])</span>
-
-<span class="sd">        &gt;&gt;&gt; sieve.extend(30)</span>
-<span class="sd">        &gt;&gt;&gt; sieve[10] == 29</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_list</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-            <span class="k">return</span>
-
-        <span class="c"># We need to sieve against all bases up to sqrt(n).</span>
-        <span class="c"># This is a recursive call that will do nothing if there are enough</span>
-        <span class="c"># known bases already.</span>
-        <span class="n">maxbase</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="o">**</span><span class="mf">0.5</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">maxbase</span><span class="p">)</span>
-
-        <span class="c"># Create a new sieve starting from sqrt(n)</span>
-        <span class="n">begin</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_list</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="n">newsieve</span> <span class="o">=</span> <span class="n">_arange</span><span class="p">(</span><span class="n">begin</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-        <span class="c"># Now eliminate all multiples of primes in [2, sqrt(n)]</span>
-        <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">primerange</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">maxbase</span><span class="p">):</span>
-            <span class="c"># Start counting at a multiple of p, offsetting</span>
-            <span class="c"># the index to account for the new sieve&#39;s base index</span>
-            <span class="n">startindex</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="n">begin</span><span class="p">)</span> <span class="o">%</span> <span class="n">p</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">startindex</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">newsieve</span><span class="p">),</span> <span class="n">p</span><span class="p">):</span>
-                <span class="n">newsieve</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-
-        <span class="c"># Merge the sieves</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_list</span> <span class="o">+=</span> <span class="n">_array</span><span class="p">(</span><span class="s">&#39;l&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">newsieve</span> <span class="k">if</span> <span class="n">x</span><span class="p">])</span>
-</div>
-<div class="viewcode-block" id="Sieve.extend_to_no"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.generate.Sieve.extend_to_no">[docs]</a>    <span class="k">def</span> <span class="nf">extend_to_no</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Extend to include the ith prime number.</span>
-
-<span class="sd">        i must be an integer.</span>
-
-<span class="sd">        The list is extended by 50% if it is too short, so it is</span>
-<span class="sd">        likely that it will be longer than requested.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sieve</span>
-<span class="sd">        &gt;&gt;&gt; from array import array # this line and next for doctest only</span>
-<span class="sd">        &gt;&gt;&gt; sieve._list = array(&#39;l&#39;, [2, 3, 5, 7, 11, 13])</span>
-
-<span class="sd">        &gt;&gt;&gt; sieve.extend_to_no(9)</span>
-<span class="sd">        &gt;&gt;&gt; sieve._list</span>
-<span class="sd">        array(&#39;l&#39;, [2, 3, 5, 7, 11, 13, 17, 19, 23])</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-        <span class="k">while</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_list</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">i</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_list</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="mf">1.5</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="Sieve.primerange"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.generate.Sieve.primerange">[docs]</a>    <span class="k">def</span> <span class="nf">primerange</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Generate all prime numbers in the range [a, b).</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sieve</span>
-<span class="sd">        &gt;&gt;&gt; print([i for i in sieve.primerange(7, 18)])</span>
-<span class="sd">        [7, 11, 13, 17]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.functions.elementary.integers</span> <span class="kn">import</span> <span class="n">ceiling</span>
-
-        <span class="c"># wrapping ceiling in int will raise an error if there was a problem</span>
-        <span class="c"># determining whether the expression was exactly an integer or not</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">ceiling</span><span class="p">(</span><span class="n">a</span><span class="p">)))</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">ceiling</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">a</span> <span class="o">&gt;=</span> <span class="n">b</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">search</span><span class="p">(</span><span class="n">a</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">maxi</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_list</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">maxi</span><span class="p">:</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_list</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">p</span> <span class="o">&lt;</span> <span class="n">b</span><span class="p">:</span>
-                <span class="k">yield</span> <span class="n">p</span>
-                <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span>
-</div>
-<div class="viewcode-block" id="Sieve.search"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.generate.Sieve.search">[docs]</a>    <span class="k">def</span> <span class="nf">search</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the indices i, j of the primes that bound n.</span>
-
-<span class="sd">        If n is prime then i == j.</span>
-
-<span class="sd">        Although n can be an expression, if ceiling cannot convert</span>
-<span class="sd">        it to an integer then an n error will be raised.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sieve</span>
-<span class="sd">        &gt;&gt;&gt; sieve.search(25)</span>
-<span class="sd">        (9, 10)</span>
-<span class="sd">        &gt;&gt;&gt; sieve.search(23)</span>
-<span class="sd">        (9, 9)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.functions.elementary.integers</span> <span class="kn">import</span> <span class="n">ceiling</span>
-
-        <span class="c"># wrapping ceiling in int will raise an error if there was a problem</span>
-        <span class="c"># determining whether the expression was exactly an integer or not</span>
-        <span class="n">test</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">ceiling</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;n should be &gt;= 2 but got: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">_list</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">bisect</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_list</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_list</span><span class="p">[</span><span class="n">b</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">test</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">b</span><span class="p">,</span> <span class="n">b</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">b</span><span class="p">,</span> <span class="n">b</span> <span class="o">+</span> <span class="mi">1</span>
-</div>
-    <span class="k">def</span> <span class="nf">__contains__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-            <span class="k">assert</span> <span class="n">n</span> <span class="o">&gt;=</span> <span class="mi">2</span>
-        <span class="k">except</span> <span class="p">(</span><span class="ne">ValueError</span><span class="p">,</span> <span class="ne">AssertionError</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">2</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">search</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">a</span> <span class="o">==</span> <span class="n">b</span>
-
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the nth prime number&quot;&quot;&quot;</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">extend_to_no</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_list</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-
-<span class="c"># Generate a global object for repeated use in trial division etc</span></div>
-<span class="n">sieve</span> <span class="o">=</span> <span class="n">Sieve</span><span class="p">()</span>
-
-
-<div class="viewcode-block" id="prime"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.generate.prime">[docs]</a><span class="k">def</span> <span class="nf">prime</span><span class="p">(</span><span class="n">nth</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Return the nth prime, with the primes indexed as prime(1) = 2,</span>
-<span class="sd">        prime(2) = 3, etc.... The nth prime is approximately n*log(n) and</span>
-<span class="sd">        can never be larger than 2**n.</span>
-
-<span class="sd">        References</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        - http://primes.utm.edu/glossary/xpage/BertrandsPostulate.html</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import prime</span>
-<span class="sd">        &gt;&gt;&gt; prime(10)</span>
-<span class="sd">        29</span>
-<span class="sd">        &gt;&gt;&gt; prime(1)</span>
-<span class="sd">        2</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.ntheory.primetest.isprime : Test if n is prime</span>
-<span class="sd">        primerange : Generate all primes in a given range</span>
-<span class="sd">        primepi : Return the number of primes less than or equal to n</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">nth</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;nth must be a positive integer; prime(1) == 2&quot;</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">sieve</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="primepi"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.generate.primepi">[docs]</a><span class="k">def</span> <span class="nf">primepi</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Return the value of the prime counting function pi(n) = the number</span>
-<span class="sd">        of prime numbers less than or equal to n.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import primepi</span>
-<span class="sd">        &gt;&gt;&gt; primepi(25)</span>
-<span class="sd">        9</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        sympy.ntheory.primetest.isprime : Test if n is prime</span>
-<span class="sd">        primerange : Generate all primes in a given range</span>
-<span class="sd">        prime : Return the nth prime</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">sieve</span><span class="o">.</span><span class="n">search</span><span class="p">(</span><span class="n">n</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="nextprime"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.generate.nextprime">[docs]</a><span class="k">def</span> <span class="nf">nextprime</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">ith</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Return the ith prime greater than n.</span>
-
-<span class="sd">        i must be an integer.</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        Potential primes are located at 6*j +/- 1. This</span>
-<span class="sd">        property is used during searching.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import nextprime</span>
-<span class="sd">        &gt;&gt;&gt; [(i, nextprime(i)) for i in range(10, 15)]</span>
-<span class="sd">        [(10, 11), (11, 13), (12, 13), (13, 17), (14, 17)]</span>
-<span class="sd">        &gt;&gt;&gt; nextprime(2, ith=2) # the 2nd prime after 2</span>
-<span class="sd">        5</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        prevprime : Return the largest prime smaller than n</span>
-<span class="sd">        primerange : Generate all primes in a given range</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="n">i</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">ith</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">i</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">pr</span> <span class="o">=</span> <span class="n">n</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">pr</span> <span class="o">=</span> <span class="n">nextprime</span><span class="p">(</span><span class="n">pr</span><span class="p">)</span>
-            <span class="n">j</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="n">j</span> <span class="o">&gt;</span> <span class="n">i</span><span class="p">:</span>
-                <span class="k">break</span>
-        <span class="k">return</span> <span class="n">pr</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">2</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">7</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">{</span><span class="mi">2</span><span class="p">:</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">:</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">:</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">:</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">6</span><span class="p">:</span> <span class="mi">7</span><span class="p">}[</span><span class="n">n</span><span class="p">]</span>
-    <span class="n">nn</span> <span class="o">=</span> <span class="mi">6</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">//</span><span class="mi">6</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">nn</span> <span class="o">==</span> <span class="n">n</span><span class="p">:</span>
-        <span class="n">n</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">isprime</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">n</span>
-        <span class="n">n</span> <span class="o">+=</span> <span class="mi">4</span>
-    <span class="k">elif</span> <span class="n">n</span> <span class="o">-</span> <span class="n">nn</span> <span class="o">==</span> <span class="mi">5</span><span class="p">:</span>
-        <span class="n">n</span> <span class="o">+=</span> <span class="mi">2</span>
-        <span class="k">if</span> <span class="n">isprime</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">n</span>
-        <span class="n">n</span> <span class="o">+=</span> <span class="mi">4</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">nn</span> <span class="o">+</span> <span class="mi">5</span>
-    <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">isprime</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">n</span>
-        <span class="n">n</span> <span class="o">+=</span> <span class="mi">2</span>
-        <span class="k">if</span> <span class="n">isprime</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">n</span>
-        <span class="n">n</span> <span class="o">+=</span> <span class="mi">4</span>
-
-</div>
-<div class="viewcode-block" id="prevprime"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.generate.prevprime">[docs]</a><span class="k">def</span> <span class="nf">prevprime</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Return the largest prime smaller than n.</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        Potential primes are located at 6*j +/- 1. This</span>
-<span class="sd">        property is used during searching.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import prevprime</span>
-<span class="sd">        &gt;&gt;&gt; [(i, prevprime(i)) for i in range(10, 15)]</span>
-<span class="sd">        [(10, 7), (11, 7), (12, 11), (13, 11), (14, 13)]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        nextprime : Return the ith prime greater than n</span>
-<span class="sd">        primerange : Generates all primes in a given range</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.functions.elementary.integers</span> <span class="kn">import</span> <span class="n">ceiling</span>
-
-    <span class="c"># wrapping ceiling in int will raise an error if there was a problem</span>
-    <span class="c"># determining whether the expression was exactly an integer or not</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">ceiling</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">3</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;no preceding primes&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">8</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">{</span><span class="mi">3</span><span class="p">:</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">:</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">:</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">6</span><span class="p">:</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">:</span> <span class="mi">5</span><span class="p">}[</span><span class="n">n</span><span class="p">]</span>
-    <span class="n">nn</span> <span class="o">=</span> <span class="mi">6</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">//</span><span class="mi">6</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">-</span> <span class="n">nn</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">nn</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">isprime</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">n</span>
-        <span class="n">n</span> <span class="o">-=</span> <span class="mi">4</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">nn</span> <span class="o">+</span> <span class="mi">1</span>
-    <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">isprime</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">n</span>
-        <span class="n">n</span> <span class="o">-=</span> <span class="mi">2</span>
-        <span class="k">if</span> <span class="n">isprime</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">n</span>
-        <span class="n">n</span> <span class="o">-=</span> <span class="mi">4</span>
-
-</div>
-<div class="viewcode-block" id="primerange"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.generate.primerange">[docs]</a><span class="k">def</span> <span class="nf">primerange</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Generate a list of all prime numbers in the range [a, b).</span>
-
-<span class="sd">        If the range exists in the default sieve, the values will</span>
-<span class="sd">        be returned from there; otherwise values will be returned</span>
-<span class="sd">        but will not modifiy the sieve.</span>
-
-<span class="sd">        Notes</span>
-<span class="sd">        =====</span>
-
-<span class="sd">        Some famous conjectures about the occurence of primes in a given</span>
-<span class="sd">        range are [1]:</span>
-
-<span class="sd">        - Twin primes: though often not, the following will give 2 primes</span>
-<span class="sd">                    an infinite number of times:</span>
-<span class="sd">                        primerange(6*n - 1, 6*n + 2)</span>
-<span class="sd">        - Legendre&#39;s: the following always yields at least one prime</span>
-<span class="sd">                        primerange(n**2, (n+1)**2+1)</span>
-<span class="sd">        - Bertrand&#39;s (proven): there is always a prime in the range</span>
-<span class="sd">                        primerange(n, 2*n)</span>
-<span class="sd">        - Brocard&#39;s: there are at least four primes in the range</span>
-<span class="sd">                        primerange(prime(n)**2, prime(n+1)**2)</span>
-
-<span class="sd">        The average gap between primes is log(n) [2]; the gap between</span>
-<span class="sd">        primes can be arbitrarily large since sequences of composite</span>
-<span class="sd">        numbers are arbitrarily large, e.g. the numbers in the sequence</span>
-<span class="sd">        n! + 2, n! + 3 ... n! + n are all composite.</span>
-
-<span class="sd">        References</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        1. http://en.wikipedia.org/wiki/Prime_number</span>
-<span class="sd">        2. http://primes.utm.edu/notes/gaps.html</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import primerange, sieve</span>
-<span class="sd">        &gt;&gt;&gt; print([i for i in primerange(1, 30)])</span>
-<span class="sd">        [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]</span>
-
-<span class="sd">        The Sieve method, primerange, is generally faster but it will</span>
-<span class="sd">        occupy more memory as the sieve stores values. The default</span>
-<span class="sd">        instance of Sieve, named sieve, can be used:</span>
-
-<span class="sd">        &gt;&gt;&gt; list(sieve.primerange(1, 30))</span>
-<span class="sd">        [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        nextprime : Return the ith prime greater than n</span>
-<span class="sd">        prevprime : Return the largest prime smaller than n</span>
-<span class="sd">        randprime : Returns a random prime in a given range</span>
-<span class="sd">        primorial : Returns the product of primes based on condition</span>
-<span class="sd">        Sieve.primerange : return range from already computed primes</span>
-<span class="sd">                           or extend the sieve to contain the requested</span>
-<span class="sd">                           range.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.functions.elementary.integers</span> <span class="kn">import</span> <span class="n">ceiling</span>
-
-    <span class="c"># if we already have the range, return it</span>
-    <span class="k">if</span> <span class="n">b</span> <span class="o">&lt;=</span> <span class="n">sieve</span><span class="o">.</span><span class="n">_list</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">sieve</span><span class="o">.</span><span class="n">primerange</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-            <span class="k">yield</span> <span class="n">i</span>
-        <span class="k">return</span>
-    <span class="c"># otherwise compute, without storing, the desired range</span>
-    <span class="k">if</span> <span class="n">a</span> <span class="o">&gt;=</span> <span class="n">b</span><span class="p">:</span>
-        <span class="k">return</span>
-    <span class="c"># wrapping ceiling in int will raise an error if there was a problem</span>
-    <span class="c"># determining whether the expression was exactly an integer or not</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">ceiling</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="o">-</span> <span class="mi">1</span>
-    <span class="n">b</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">ceiling</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
-    <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">nextprime</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">a</span> <span class="o">&lt;</span> <span class="n">b</span><span class="p">:</span>
-            <span class="k">yield</span> <span class="n">a</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span>
-
-</div>
-<div class="viewcode-block" id="randprime"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.generate.randprime">[docs]</a><span class="k">def</span> <span class="nf">randprime</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Return a random prime number in the range [a, b).</span>
-
-<span class="sd">        Bertrand&#39;s postulate assures that</span>
-<span class="sd">        randprime(a, 2*a) will always succeed for a &gt; 1.</span>
-
-<span class="sd">        References</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        - http://en.wikipedia.org/wiki/Bertrand&#39;s_postulate</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import randprime, isprime</span>
-<span class="sd">        &gt;&gt;&gt; randprime(1, 30) #doctest: +SKIP</span>
-<span class="sd">        13</span>
-<span class="sd">        &gt;&gt;&gt; isprime(randprime(1, 30))</span>
-<span class="sd">        True</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        primerange : Generate all primes in a given range</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">a</span> <span class="o">&gt;=</span> <span class="n">b</span><span class="p">:</span>
-        <span class="k">return</span>
-    <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)))</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="n">nextprime</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">p</span> <span class="o">&gt;=</span> <span class="n">b</span><span class="p">:</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">prevprime</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">p</span> <span class="o">&lt;</span> <span class="n">a</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;no primes exist in the specified range&quot;</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">p</span>
-
-</div>
-<div class="viewcode-block" id="primorial"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.generate.primorial">[docs]</a><span class="k">def</span> <span class="nf">primorial</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">nth</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the product of the first n primes (default) or</span>
-<span class="sd">    the primes less than or equal to n (when ``nth=False``).</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory.generate import primorial, randprime, primerange</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import factorint, Mul, primefactors, sqrt</span>
-<span class="sd">    &gt;&gt;&gt; primorial(4) # the first 4 primes are 2, 3, 5, 7</span>
-<span class="sd">    210</span>
-<span class="sd">    &gt;&gt;&gt; primorial(4, nth=False) # primes &lt;= 4 are 2 and 3</span>
-<span class="sd">    6</span>
-<span class="sd">    &gt;&gt;&gt; primorial(1)</span>
-<span class="sd">    2</span>
-<span class="sd">    &gt;&gt;&gt; primorial(1, nth=False)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; primorial(sqrt(101), nth=False)</span>
-<span class="sd">    210</span>
-
-<span class="sd">    One can argue that the primes are infinite since if you take</span>
-<span class="sd">    a set of primes and multiply them together (e.g. the primorial) and</span>
-<span class="sd">    then add or subtract 1, the result cannot be divided by any of the</span>
-<span class="sd">    original factors, hence either 1 or more new primes must divide this</span>
-<span class="sd">    product of primes.</span>
-
-<span class="sd">    In this case, the number itself is a new prime:</span>
-
-<span class="sd">    &gt;&gt;&gt; factorint(primorial(4) + 1)</span>
-<span class="sd">    {211: 1}</span>
-
-<span class="sd">    In this case two new primes are the factors:</span>
-
-<span class="sd">    &gt;&gt;&gt; factorint(primorial(4) - 1)</span>
-<span class="sd">    {11: 1, 19: 1}</span>
-
-<span class="sd">    Here, some primes smaller and larger than the primes multiplied together</span>
-<span class="sd">    are obtained:</span>
-
-<span class="sd">    &gt;&gt;&gt; p = list(primerange(10, 20))</span>
-<span class="sd">    &gt;&gt;&gt; sorted(set(primefactors(Mul(*p) + 1)).difference(set(p)))</span>
-<span class="sd">    [2, 5, 31, 149]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    primerange : Generate all primes in a given range</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">nth</span><span class="p">:</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;primorial argument must be &gt;= 1&quot;</span><span class="p">)</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="k">if</span> <span class="n">nth</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">p</span> <span class="o">*=</span> <span class="n">prime</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">primerange</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">p</span> <span class="o">*=</span> <span class="n">i</span>
-    <span class="k">return</span> <span class="n">p</span>
-
-</div>
-<div class="viewcode-block" id="cycle_length"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.generate.cycle_length">[docs]</a><span class="k">def</span> <span class="nf">cycle_length</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">nmax</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">values</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;For a given iterated sequence, return a generator that gives</span>
-<span class="sd">    the length of the iterated cycle (lambda) and the length of terms</span>
-<span class="sd">    before the cycle begins (mu); if ``values`` is True then the</span>
-<span class="sd">    terms of the sequence will be returned instead. The sequence is</span>
-<span class="sd">    started with value ``x0``.</span>
-
-<span class="sd">    Note: more than the first lambda + mu terms may be returned and this</span>
-<span class="sd">    is the cost of cycle detection with Brent&#39;s method; there are, however,</span>
-<span class="sd">    generally less terms calculated than would have been calculated if the</span>
-<span class="sd">    proper ending point were determined, e.g. by using Floyd&#39;s method.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory.generate import cycle_length</span>
-
-<span class="sd">    This will yield successive values of i &lt;-- func(i):</span>
-
-<span class="sd">        &gt;&gt;&gt; def iter(func, i):</span>
-<span class="sd">        ...     while 1:</span>
-<span class="sd">        ...         ii = func(i)</span>
-<span class="sd">        ...         yield ii</span>
-<span class="sd">        ...         i = ii</span>
-<span class="sd">        ...</span>
-
-<span class="sd">    A function is defined:</span>
-
-<span class="sd">        &gt;&gt;&gt; func = lambda i: (i**2 + 1) % 51</span>
-
-<span class="sd">    and given a seed of 4 and the mu and lambda terms calculated:</span>
-
-<span class="sd">        &gt;&gt;&gt; next(cycle_length(func, 4))</span>
-<span class="sd">        (6, 2)</span>
-
-<span class="sd">    We can see what is meant by looking at the output:</span>
-
-<span class="sd">        &gt;&gt;&gt; n = cycle_length(func, 4, values=True)</span>
-<span class="sd">        &gt;&gt;&gt; list(ni for ni in n)</span>
-<span class="sd">        [17, 35, 2, 5, 26, 14, 44, 50, 2, 5, 26, 14]</span>
-
-<span class="sd">    There are 6 repeating values after the first 2.</span>
-
-<span class="sd">    If a sequence is suspected of being longer than you might wish, ``nmax``</span>
-<span class="sd">    can be used to exit early (and mu will be returned as None):</span>
-
-<span class="sd">        &gt;&gt;&gt; next(cycle_length(func, 4, nmax = 4))</span>
-<span class="sd">        (4, None)</span>
-<span class="sd">        &gt;&gt;&gt; [ni for ni in cycle_length(func, 4, nmax = 4, values=True)]</span>
-<span class="sd">        [17, 35, 2, 5]</span>
-
-<span class="sd">    Code modified from:</span>
-<span class="sd">        http://en.wikipedia.org/wiki/Cycle_detection.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nmax</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">nmax</span> <span class="ow">or</span> <span class="mi">0</span><span class="p">)</span>
-
-    <span class="c"># main phase: search successive powers of two</span>
-    <span class="n">power</span> <span class="o">=</span> <span class="n">lam</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">tortoise</span><span class="p">,</span> <span class="n">hare</span> <span class="o">=</span> <span class="n">x0</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span>  <span class="c"># f(x0) is the element/node next to x0.</span>
-    <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">while</span> <span class="n">tortoise</span> <span class="o">!=</span> <span class="n">hare</span> <span class="ow">and</span> <span class="p">(</span><span class="ow">not</span> <span class="n">nmax</span> <span class="ow">or</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">nmax</span><span class="p">):</span>
-        <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">power</span> <span class="o">==</span> <span class="n">lam</span><span class="p">:</span>   <span class="c"># time to start a new power of two?</span>
-            <span class="n">tortoise</span> <span class="o">=</span> <span class="n">hare</span>
-            <span class="n">power</span> <span class="o">*=</span> <span class="mi">2</span>
-            <span class="n">lam</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">if</span> <span class="n">values</span><span class="p">:</span>
-            <span class="k">yield</span> <span class="n">hare</span>
-        <span class="n">hare</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">hare</span><span class="p">)</span>
-        <span class="n">lam</span> <span class="o">+=</span> <span class="mi">1</span>
-    <span class="k">if</span> <span class="n">nmax</span> <span class="ow">and</span> <span class="n">i</span> <span class="o">==</span> <span class="n">nmax</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">values</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">yield</span> <span class="n">nmax</span><span class="p">,</span> <span class="bp">None</span>
-            <span class="k">return</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">values</span><span class="p">:</span>
-        <span class="c"># Find the position of the first repetition of length lambda</span>
-        <span class="n">mu</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">tortoise</span> <span class="o">=</span> <span class="n">hare</span> <span class="o">=</span> <span class="n">x0</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">lam</span><span class="p">):</span>
-            <span class="n">hare</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">hare</span><span class="p">)</span>
-        <span class="k">while</span> <span class="n">tortoise</span> <span class="o">!=</span> <span class="n">hare</span><span class="p">:</span>
-            <span class="n">tortoise</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">tortoise</span><span class="p">)</span>
-            <span class="n">hare</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">hare</span><span class="p">)</span>
-            <span class="n">mu</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">mu</span><span class="p">:</span>
-            <span class="n">mu</span> <span class="o">-=</span> <span class="mi">1</span>
-        <span class="k">yield</span> <span class="n">lam</span><span class="p">,</span> <span class="n">mu</span></div>
-</pre></div>
-
-          </div>
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-  <h1>Source code for sympy.ntheory.modular</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">igcdex</span><span class="p">,</span> <span class="n">igcd</span>
-<span class="kn">from</span> <span class="nn">sympy.core.mul</span> <span class="kn">import</span> <span class="n">prod</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">as_int</span>
-<span class="kn">from</span> <span class="nn">sympy.ntheory.primetest</span> <span class="kn">import</span> <span class="n">isprime</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_crt</span><span class="p">,</span> <span class="n">gf_crt1</span><span class="p">,</span> <span class="n">gf_crt2</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-
-<div class="viewcode-block" id="symmetric_residue"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.modular.symmetric_residue">[docs]</a><span class="k">def</span> <span class="nf">symmetric_residue</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return the residual mod m such that it is within half of the modulus.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory.modular import symmetric_residue</span>
-<span class="sd">    &gt;&gt;&gt; symmetric_residue(1, 6)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; symmetric_residue(4, 6)</span>
-<span class="sd">    -2</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">a</span> <span class="o">&lt;=</span> <span class="n">m</span> <span class="o">//</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">a</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">a</span> <span class="o">-</span> <span class="n">m</span>
-
-</div>
-<div class="viewcode-block" id="crt"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.modular.crt">[docs]</a><span class="k">def</span> <span class="nf">crt</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">symmetric</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">check</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;Chinese Remainder Theorem.</span>
-
-<span class="sd">    The moduli in m are assumed to be pairwise coprime.  The output</span>
-<span class="sd">    is then an integer f, such that f = v_i mod m_i for each pair out</span>
-<span class="sd">    of v and m. If ``symmetric`` is False a positive integer will be</span>
-<span class="sd">    returned, else \|f\| will be less than or equal to the LCM of the</span>
-<span class="sd">    moduli, and thus f may be negative.</span>
-
-<span class="sd">    If the moduli are not co-prime the correct result will be returned</span>
-<span class="sd">    if/when the test of the result is found to be incorrect. This result</span>
-<span class="sd">    will be None if there is no solution.</span>
-
-<span class="sd">    The keyword ``check`` can be set to False if it is known that the moduli</span>
-<span class="sd">    are coprime.</span>
-
-<span class="sd">    As an example consider a set of residues ``U = [49, 76, 65]``</span>
-<span class="sd">    and a set of moduli ``M = [99, 97, 95]``. Then we have::</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy.ntheory.modular import crt, solve_congruence</span>
-
-<span class="sd">       &gt;&gt;&gt; crt([99, 97, 95], [49, 76, 65])</span>
-<span class="sd">       (639985, 912285)</span>
-
-<span class="sd">    This is the correct result because::</span>
-
-<span class="sd">       &gt;&gt;&gt; [639985 % m for m in [99, 97, 95]]</span>
-<span class="sd">       [49, 76, 65]</span>
-
-<span class="sd">    If the moduli are not co-prime, you may receive an incorrect result</span>
-<span class="sd">    if you use ``check=False``:</span>
-
-<span class="sd">       &gt;&gt;&gt; crt([12, 6, 17], [3, 4, 2], check=False)</span>
-<span class="sd">       (954, 1224)</span>
-<span class="sd">       &gt;&gt;&gt; [954 % m for m in [12, 6, 17]]</span>
-<span class="sd">       [6, 0, 2]</span>
-<span class="sd">       &gt;&gt;&gt; crt([12, 6, 17], [3, 4, 2]) is None</span>
-<span class="sd">       True</span>
-<span class="sd">       &gt;&gt;&gt; crt([3, 6], [2, 5])</span>
-<span class="sd">       (5, 6)</span>
-
-<span class="sd">    Note: the order of gf_crt&#39;s arguments is reversed relative to crt,</span>
-<span class="sd">    and that solve_congruence takes residue, modulus pairs.</span>
-
-<span class="sd">    Programmer&#39;s note: rather than checking that all pairs of moduli share</span>
-<span class="sd">    no GCD (an O(n**2) test) and rather than factoring all moduli and seeing</span>
-<span class="sd">    that there is no factor in common, a check that the result gives the</span>
-<span class="sd">    indicated residuals is performed -- an O(n) operation.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    solve_congruence</span>
-<span class="sd">    sympy.polys.galoistools.gf_crt : low level crt routine used by this routine</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">check</span><span class="p">:</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">as_int</span><span class="p">,</span> <span class="n">m</span><span class="p">))</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">as_int</span><span class="p">,</span> <span class="n">v</span><span class="p">))</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">gf_crt</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-    <span class="n">mm</span> <span class="o">=</span> <span class="n">prod</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">check</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">(</span><span class="n">v</span> <span class="o">%</span> <span class="n">m</span> <span class="o">==</span> <span class="n">result</span> <span class="o">%</span> <span class="n">m</span> <span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">m</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">m</span><span class="p">)):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">solve_congruence</span><span class="p">(</span><span class="o">*</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">m</span><span class="p">)),</span>
-                                      <span class="o">**</span><span class="nb">dict</span><span class="p">(</span><span class="n">check</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span>
-                                             <span class="n">symmetric</span><span class="o">=</span><span class="n">symmetric</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">result</span>
-            <span class="n">result</span><span class="p">,</span> <span class="n">mm</span> <span class="o">=</span> <span class="n">result</span>
-
-    <span class="k">if</span> <span class="n">symmetric</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">symmetric_residue</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">mm</span><span class="p">),</span> <span class="n">mm</span>
-    <span class="k">return</span> <span class="n">result</span><span class="p">,</span> <span class="n">mm</span>
-
-</div>
-<div class="viewcode-block" id="crt1"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.modular.crt1">[docs]</a><span class="k">def</span> <span class="nf">crt1</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;First part of Chinese Remainder Theorem, for multiple application.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory.modular import crt1</span>
-<span class="sd">    &gt;&gt;&gt; crt1([18, 42, 6])</span>
-<span class="sd">    (4536, [252, 108, 756], [0, 2, 0])</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">gf_crt1</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="crt2"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.modular.crt2">[docs]</a><span class="k">def</span> <span class="nf">crt2</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">mm</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">symmetric</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Second part of Chinese Remainder Theorem, for multiple application.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory.modular import crt1, crt2</span>
-<span class="sd">    &gt;&gt;&gt; mm, e, s = crt1([18, 42, 6])</span>
-<span class="sd">    &gt;&gt;&gt; crt2([18, 42, 6], [0, 0, 0], mm, e, s)</span>
-<span class="sd">    (0, 4536)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">gf_crt2</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">mm</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">symmetric</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">symmetric_residue</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">mm</span><span class="p">),</span> <span class="n">mm</span>
-    <span class="k">return</span> <span class="n">result</span><span class="p">,</span> <span class="n">mm</span>
-
-</div>
-<div class="viewcode-block" id="solve_congruence"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.modular.solve_congruence">[docs]</a><span class="k">def</span> <span class="nf">solve_congruence</span><span class="p">(</span><span class="o">*</span><span class="n">remainder_modulus_pairs</span><span class="p">,</span> <span class="o">**</span><span class="n">hint</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Compute the integer ``n`` that has the residual ``ai`` when it is</span>
-<span class="sd">    divided by ``mi`` where the ``ai`` and ``mi`` are given as pairs to</span>
-<span class="sd">    this function: ((a1, m1), (a2, m2), ...). If there is no solution,</span>
-<span class="sd">    return None. Otherwise return ``n`` and its modulus.</span>
-
-<span class="sd">    The ``mi`` values need not be co-prime. If it is known that the moduli are</span>
-<span class="sd">    not co-prime then the hint ``check`` can be set to False (default=True) and</span>
-<span class="sd">    the check for a quicker solution via crt() (valid when the moduli are</span>
-<span class="sd">    co-prime) will be skipped.</span>
-
-<span class="sd">    If the hint ``symmetric`` is True (default is False), the value of ``n``</span>
-<span class="sd">    will be within 1/2 of the modulus, possibly negative.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory.modular import solve_congruence</span>
-
-<span class="sd">    What number is 2 mod 3, 3 mod 5 and 2 mod 7?</span>
-
-<span class="sd">    &gt;&gt;&gt; solve_congruence((2, 3), (3, 5), (2, 7))</span>
-<span class="sd">    (23, 105)</span>
-<span class="sd">    &gt;&gt;&gt; [23 % m for m in [3, 5, 7]]</span>
-<span class="sd">    [2, 3, 2]</span>
-
-<span class="sd">    If you prefer to work with all remainder in one list and</span>
-<span class="sd">    all moduli in another, send the arguments like this:</span>
-
-<span class="sd">    &gt;&gt;&gt; solve_congruence(*list(zip((2, 3, 2), (3, 5, 7))))</span>
-<span class="sd">    (23, 105)</span>
-
-<span class="sd">    The moduli need not be co-prime; in this case there may or</span>
-<span class="sd">    may not be a solution:</span>
-
-<span class="sd">    &gt;&gt;&gt; solve_congruence((2, 3), (4, 6)) is None</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &gt;&gt;&gt; solve_congruence((2, 3), (5, 6))</span>
-<span class="sd">    (5, 6)</span>
-
-<span class="sd">    The symmetric flag will make the result be within 1/2 of the modulus:</span>
-
-<span class="sd">    &gt;&gt;&gt; solve_congruence((2, 3), (5, 6), symmetric=True)</span>
-<span class="sd">    (-1, 6)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    crt : high level routine implementing the Chinese Remainder Theorem</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">combine</span><span class="p">(</span><span class="n">c1</span><span class="p">,</span> <span class="n">c2</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the tuple (a, m) which satisfies the requirement</span>
-<span class="sd">        that n = a + i*m satisfy n = a1 + j*m1 and n = a2 = k*m2.</span>
-
-<span class="sd">        References</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        - http://en.wikipedia.org/wiki/Method_of_successive_substitution</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">a1</span><span class="p">,</span> <span class="n">m1</span> <span class="o">=</span> <span class="n">c1</span>
-        <span class="n">a2</span><span class="p">,</span> <span class="n">m2</span> <span class="o">=</span> <span class="n">c2</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">m1</span><span class="p">,</span> <span class="n">a2</span> <span class="o">-</span> <span class="n">a1</span><span class="p">,</span> <span class="n">m2</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="nb">reduce</span><span class="p">(</span><span class="n">igcd</span><span class="p">,</span> <span class="p">[</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">])</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span><span class="o">//</span><span class="n">g</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">]]</span>
-        <span class="k">if</span> <span class="n">a</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">inv_a</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">igcdex</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">g</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="n">b</span> <span class="o">*=</span> <span class="n">inv_a</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">a1</span> <span class="o">+</span> <span class="n">m1</span><span class="o">*</span><span class="n">b</span><span class="p">,</span> <span class="n">m1</span><span class="o">*</span><span class="n">c</span>
-        <span class="k">return</span> <span class="n">a</span><span class="p">,</span> <span class="n">m</span>
-
-    <span class="n">rm</span> <span class="o">=</span> <span class="n">remainder_modulus_pairs</span>
-    <span class="n">symmetric</span> <span class="o">=</span> <span class="n">hint</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;symmetric&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">hint</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;check&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-        <span class="n">rm</span> <span class="o">=</span> <span class="p">[(</span><span class="n">as_int</span><span class="p">(</span><span class="n">r</span><span class="p">),</span> <span class="n">as_int</span><span class="p">(</span><span class="n">m</span><span class="p">))</span> <span class="k">for</span> <span class="n">r</span><span class="p">,</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">rm</span><span class="p">]</span>
-
-        <span class="c"># ignore redundant pairs but raise an error otherwise; also</span>
-        <span class="c"># make sure that a unique set of bases is sent to gf_crt if</span>
-        <span class="c"># they are all prime.</span>
-        <span class="c">#</span>
-        <span class="c"># The routine will work out less-trivial violations and</span>
-        <span class="c"># return None, e.g. for the pairs (1,3) and (14,42) there</span>
-        <span class="c"># is no answer because 14 mod 42 (having a gcd of 14) implies</span>
-        <span class="c"># (14/2) mod (42/2), (14/7) mod (42/7) and (14/14) mod (42/14)</span>
-        <span class="c"># which, being 0 mod 3, is inconsistent with 1 mod 3. But to</span>
-        <span class="c"># preprocess the input beyond checking of another pair with 42</span>
-        <span class="c"># or 3 as the modulus (for this example) is not necessary.</span>
-        <span class="n">uniq</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">r</span><span class="p">,</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">rm</span><span class="p">:</span>
-            <span class="n">r</span> <span class="o">%=</span> <span class="n">m</span>
-            <span class="k">if</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">uniq</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">r</span> <span class="o">!=</span> <span class="n">uniq</span><span class="p">[</span><span class="n">m</span><span class="p">]:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-                <span class="k">continue</span>
-            <span class="n">uniq</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span>
-        <span class="n">rm</span> <span class="o">=</span> <span class="p">[(</span><span class="n">r</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span> <span class="k">for</span> <span class="n">m</span><span class="p">,</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">uniq</span><span class="o">.</span><span class="n">items</span><span class="p">()]</span>
-        <span class="k">del</span> <span class="n">uniq</span>
-
-        <span class="c"># if the moduli are co-prime, the crt will be significantly faster;</span>
-        <span class="c"># checking all pairs for being co-prime gets to be slow but a prime</span>
-        <span class="c"># test is a good trade-off</span>
-        <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">isprime</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="k">for</span> <span class="n">r</span><span class="p">,</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">rm</span><span class="p">):</span>
-            <span class="n">r</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">rm</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">crt</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">symmetric</span><span class="o">=</span><span class="n">symmetric</span><span class="p">,</span> <span class="n">check</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-    <span class="n">rv</span> <span class="o">=</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">rmi</span> <span class="ow">in</span> <span class="n">rm</span><span class="p">:</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">combine</span><span class="p">(</span><span class="n">rv</span><span class="p">,</span> <span class="n">rmi</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">rv</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">break</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">rv</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">n</span> <span class="o">%</span> <span class="n">m</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">symmetric</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">symmetric_residue</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="p">),</span> <span class="n">m</span>
-        <span class="k">return</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span></div>
-</pre></div>
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-            
-  <h1>Source code for sympy.ntheory.multinomial</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</span>
-
-
-<span class="k">def</span> <span class="nf">binomial_coefficients</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-<div class="viewcode-block" id="binomial_coefficients"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.multinomial.binomial_coefficients">[docs]</a>    <span class="sd">&quot;&quot;&quot;Return a dictionary containing pairs :math:`{(k1,k2) : C_kn}` where</span>
-<span class="sd">    :math:`C_kn` are binomial coefficients and :math:`n=k1+k2`.</span>
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory import binomial_coefficients</span>
-<span class="sd">    &gt;&gt;&gt; binomial_coefficients(9)</span>
-<span class="sd">    {(0, 9): 1, (1, 8): 9, (2, 7): 36, (3, 6): 84,</span>
-<span class="sd">     (4, 5): 126, (5, 4): 126, (6, 3): 84, (7, 2): 36, (8, 1): 9, (9, 0): 1}</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    binomial_coefficients_list, multinomial_coefficients</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">d</span> <span class="o">=</span> <span class="p">{(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">):</span> <span class="mi">1</span><span class="p">}</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">//</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="p">(</span><span class="n">a</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span><span class="o">//</span><span class="n">k</span>
-        <span class="n">d</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">d</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">a</span>
-    <span class="k">return</span> <span class="n">d</span>
-
-
-<span class="k">def</span> <span class="nf">binomial_coefficients_list</span><span class="p">(</span><span class="n">n</span><span class="p">):</span></div>
-<div class="viewcode-block" id="binomial_coefficients_list"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.multinomial.binomial_coefficients_list">[docs]</a>    <span class="sd">&quot;&quot;&quot; Return a list of binomial coefficients as rows of the Pascal&#39;s</span>
-<span class="sd">    triangle.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory import binomial_coefficients_list</span>
-<span class="sd">    &gt;&gt;&gt; binomial_coefficients_list(9)</span>
-<span class="sd">    [1, 9, 36, 84, 126, 126, 84, 36, 9, 1]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    binomial_coefficients, multinomial_coefficients</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">d</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">//</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="p">(</span><span class="n">a</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span><span class="o">//</span><span class="n">k</span>
-        <span class="n">d</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">d</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">a</span>
-    <span class="k">return</span> <span class="n">d</span>
-
-
-<span class="k">def</span> <span class="nf">multinomial_coefficients0</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">_tuple</span><span class="o">=</span><span class="nb">tuple</span><span class="p">,</span> <span class="n">_zip</span><span class="o">=</span><span class="nb">zip</span><span class="p">):</span></div>
-    <span class="sd">&quot;&quot;&quot;Return a dictionary containing pairs ``{(k1,k2,..,km) : C_kn}``</span>
-<span class="sd">    where ``C_kn`` are multinomial coefficients such that</span>
-<span class="sd">    ``n=k1+k2+..+km``.</span>
-
-<span class="sd">    For example:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import multinomial_coefficients</span>
-<span class="sd">    &gt;&gt;&gt; multinomial_coefficients(2, 5) # indirect doctest</span>
-<span class="sd">    {(0, 5): 1, (1, 4): 5, (2, 3): 10, (3, 2): 10, (4, 1): 5, (5, 0): 1}</span>
-
-<span class="sd">    The algorithm is based on the following result:</span>
-
-<span class="sd">       Consider a polynomial and its ``n``-th exponent::</span>
-
-<span class="sd">         P(x) = sum_{i=0}^m p_i x^i</span>
-<span class="sd">         P(x)^n = sum_{k=0}^{m n} a(n,k) x^k</span>
-
-<span class="sd">       The coefficients ``a(n,k)`` can be computed using the</span>
-<span class="sd">       J.C.P. Miller Pure Recurrence [see D.E.Knuth, Seminumerical</span>
-<span class="sd">       Algorithms, The art of Computer Programming v.2, Addison</span>
-<span class="sd">       Wesley, Reading, 1981;]::</span>
-
-<span class="sd">         a(n,k) = 1/(k p_0) sum_{i=1}^m p_i ((n+1)i-k) a(n,k-i),</span>
-
-<span class="sd">       where ``a(n,0) = p_0^n``.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">m</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">{}</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">{():</span> <span class="mi">1</span><span class="p">}</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">binomial_coefficients</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="n">symbols</span> <span class="o">=</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,)</span><span class="o">*</span><span class="n">i</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span><span class="p">,)</span> <span class="o">+</span> <span class="p">(</span><span class="mi">0</span><span class="p">,)</span><span class="o">*</span><span class="p">(</span><span class="n">m</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="p">)]</span>
-    <span class="n">s0</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">p0</span> <span class="o">=</span> <span class="p">[</span><span class="n">_tuple</span><span class="p">(</span><span class="n">aa</span> <span class="o">-</span> <span class="n">bb</span> <span class="k">for</span> <span class="n">aa</span><span class="p">,</span> <span class="n">bb</span> <span class="ow">in</span> <span class="n">_zip</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">s0</span><span class="p">))</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">symbols</span><span class="p">]</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="p">{</span><span class="n">_tuple</span><span class="p">(</span><span class="n">aa</span><span class="o">*</span><span class="n">n</span> <span class="k">for</span> <span class="n">aa</span> <span class="ow">in</span> <span class="n">s0</span><span class="p">):</span> <span class="mi">1</span><span class="p">}</span>
-    <span class="n">l</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">m</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="n">l</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">r</span><span class="o">.</span><span class="n">items</span><span class="p">())</span>
-    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">m</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">min</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)):</span>
-            <span class="n">nn</span> <span class="o">=</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">i</span> <span class="o">-</span> <span class="n">k</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">nn</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="n">t</span> <span class="o">=</span> <span class="n">p0</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="k">for</span> <span class="n">t2</span><span class="p">,</span> <span class="n">c2</span> <span class="ow">in</span> <span class="n">l</span><span class="p">[</span><span class="n">k</span> <span class="o">-</span> <span class="n">i</span><span class="p">]:</span>
-                <span class="n">tt</span> <span class="o">=</span> <span class="n">_tuple</span><span class="p">([</span><span class="n">aa</span> <span class="o">+</span> <span class="n">bb</span> <span class="k">for</span> <span class="n">aa</span><span class="p">,</span> <span class="n">bb</span> <span class="ow">in</span> <span class="n">_zip</span><span class="p">(</span><span class="n">t2</span><span class="p">,</span> <span class="n">t</span><span class="p">)])</span>
-                <span class="n">d</span><span class="p">[</span><span class="n">tt</span><span class="p">]</span> <span class="o">+=</span> <span class="n">nn</span><span class="o">*</span><span class="n">c2</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">d</span><span class="p">[</span><span class="n">tt</span><span class="p">]:</span>
-                    <span class="k">del</span> <span class="n">d</span><span class="p">[</span><span class="n">tt</span><span class="p">]</span>
-        <span class="n">r1</span> <span class="o">=</span> <span class="p">[(</span><span class="n">t</span><span class="p">,</span> <span class="n">c</span><span class="o">//</span><span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span> <span class="ow">in</span> <span class="n">d</span><span class="o">.</span><span class="n">items</span><span class="p">()]</span>
-        <span class="n">l</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">r1</span>
-        <span class="n">r</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">r1</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">r</span>
-
-
-<span class="k">def</span> <span class="nf">multinomial_coefficients</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-<div class="viewcode-block" id="multinomial_coefficients"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.multinomial.multinomial_coefficients">[docs]</a>    <span class="sd">r&quot;&quot;&quot;Return a dictionary containing pairs ``{(k1,k2,..,km) : C_kn}``</span>
-<span class="sd">    where ``C_kn`` are multinomial coefficients such that</span>
-<span class="sd">    ``n=k1+k2+..+km``.</span>
-
-<span class="sd">    For example:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory import multinomial_coefficients</span>
-<span class="sd">    &gt;&gt;&gt; multinomial_coefficients(2, 5) # indirect doctest</span>
-<span class="sd">    {(0, 5): 1, (1, 4): 5, (2, 3): 10, (3, 2): 10, (4, 1): 5, (5, 0): 1}</span>
-
-<span class="sd">    The algorithm is based on the following result:</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        \binom{n}{k_1, \ldots, k_m} =</span>
-<span class="sd">        \frac{k_1 + 1}{n - k_1} \sum_{i=2}^m \binom{n}{k_1 + 1, \ldots, k_i - 1, \ldots}</span>
-
-<span class="sd">    Code contributed to Sage by Yann Laigle-Chapuy, copied with permission</span>
-<span class="sd">    of the author.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    binomial_coefficients_list, binomial_coefficients</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">m</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">{}</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">{():</span> <span class="mi">1</span><span class="p">}</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">binomial_coefficients</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">&gt;=</span> <span class="mi">2</span><span class="o">*</span><span class="n">n</span> <span class="ow">and</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">dict</span><span class="p">(</span><span class="n">multinomial_coefficients_iterator</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-    <span class="n">t</span> <span class="o">=</span> <span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">m</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="p">{</span><span class="nb">tuple</span><span class="p">(</span><span class="n">t</span><span class="p">):</span> <span class="mi">1</span><span class="p">}</span>
-    <span class="k">if</span> <span class="n">n</span><span class="p">:</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c"># j will be the leftmost nonzero position</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="n">m</span>
-    <span class="c"># enumerate tuples in co-lex order</span>
-    <span class="k">while</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="n">m</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="c"># compute next tuple</span>
-        <span class="n">tj</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">j</span><span class="p">:</span>
-            <span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="n">t</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">tj</span>
-        <span class="k">if</span> <span class="n">tj</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">t</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="n">start</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="n">v</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">j</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">start</span> <span class="o">=</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">1</span>
-            <span class="n">v</span> <span class="o">=</span> <span class="n">r</span><span class="p">[</span><span class="nb">tuple</span><span class="p">(</span><span class="n">t</span><span class="p">)]</span>
-            <span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="c"># compute the value</span>
-        <span class="c"># NB: the initialization of v was done above</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">t</span><span class="p">[</span><span class="n">k</span><span class="p">]:</span>
-                <span class="n">t</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">1</span>
-                <span class="n">v</span> <span class="o">+=</span> <span class="n">r</span><span class="p">[</span><span class="nb">tuple</span><span class="p">(</span><span class="n">t</span><span class="p">)]</span>
-                <span class="n">t</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="n">t</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">1</span>
-        <span class="n">r</span><span class="p">[</span><span class="nb">tuple</span><span class="p">(</span><span class="n">t</span><span class="p">)]</span> <span class="o">=</span> <span class="p">(</span><span class="n">v</span> <span class="o">*</span> <span class="n">tj</span><span class="p">)</span> <span class="o">//</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">t</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-    <span class="k">return</span> <span class="n">r</span>
-
-
-<span class="k">def</span> <span class="nf">multinomial_coefficients_iterator</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">_tuple</span><span class="o">=</span><span class="nb">tuple</span><span class="p">):</span></div>
-<div class="viewcode-block" id="multinomial_coefficients_iterator"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.multinomial.multinomial_coefficients_iterator">[docs]</a>    <span class="sd">&quot;&quot;&quot;multinomial coefficient iterator</span>
-
-<span class="sd">    This routine has been optimized for `m` large with respect to `n` by taking</span>
-<span class="sd">    advantage of the fact that when the monomial tuples `t` are stripped of</span>
-<span class="sd">    zeros, their coefficient is the same as that of the monomial tuples from</span>
-<span class="sd">    ``multinomial_coefficients(n, n)``. Therefore, the latter coefficients are</span>
-<span class="sd">    precomputed to save memory and time.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory.multinomial import multinomial_coefficients</span>
-<span class="sd">    &gt;&gt;&gt; m53, m33 = multinomial_coefficients(5,3), multinomial_coefficients(3,3)</span>
-<span class="sd">    &gt;&gt;&gt; m53[(0,0,0,1,2)] == m53[(0,0,1,0,2)] == m53[(1,0,2,0,0)] == m33[(0,1,2)]</span>
-<span class="sd">    True</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory.multinomial import multinomial_coefficients_iterator</span>
-<span class="sd">    &gt;&gt;&gt; it = multinomial_coefficients_iterator(20,3)</span>
-<span class="sd">    &gt;&gt;&gt; next(it)</span>
-<span class="sd">    ((3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), 1)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="o">*</span><span class="n">n</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">mc</span> <span class="o">=</span> <span class="n">multinomial_coefficients</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">mc</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="k">yield</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">mc</span> <span class="o">=</span> <span class="n">multinomial_coefficients</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-        <span class="n">mc1</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">mc</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">mc1</span><span class="p">[</span><span class="n">_tuple</span><span class="p">(</span><span class="nb">filter</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="n">k</span><span class="p">))]</span> <span class="o">=</span> <span class="n">v</span>
-        <span class="n">mc</span> <span class="o">=</span> <span class="n">mc1</span>
-
-        <span class="n">t</span> <span class="o">=</span> <span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">m</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">t1</span> <span class="o">=</span> <span class="n">_tuple</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">_tuple</span><span class="p">(</span><span class="nb">filter</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="n">t1</span><span class="p">))</span>
-        <span class="k">yield</span> <span class="p">(</span><span class="n">t1</span><span class="p">,</span> <span class="n">mc</span><span class="p">[</span><span class="n">b</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">n</span><span class="p">:</span>
-            <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c"># j will be the leftmost nonzero position</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">j</span> <span class="o">=</span> <span class="n">m</span>
-        <span class="c"># enumerate tuples in co-lex order</span>
-        <span class="k">while</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="n">m</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># compute next tuple</span>
-            <span class="n">tj</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">j</span><span class="p">:</span>
-                <span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="n">t</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">tj</span>
-            <span class="k">if</span> <span class="n">tj</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">t</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">j</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-
-            <span class="n">t</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="n">t1</span> <span class="o">=</span> <span class="n">_tuple</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="n">_tuple</span><span class="p">(</span><span class="nb">filter</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="n">t1</span><span class="p">))</span>
-            <span class="k">yield</span> <span class="p">(</span><span class="n">t1</span><span class="p">,</span> <span class="n">mc</span><span class="p">[</span><span class="n">b</span><span class="p">])</span>
-</pre></div></div>
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-  <h1>Source code for sympy.ntheory.partitions_</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.mpmath.libmp</span> <span class="kn">import</span> <span class="p">(</span><span class="n">fzero</span><span class="p">,</span>
-    <span class="n">from_man_exp</span><span class="p">,</span> <span class="n">from_int</span><span class="p">,</span> <span class="n">from_rational</span><span class="p">,</span>
-    <span class="n">fone</span><span class="p">,</span> <span class="n">fhalf</span><span class="p">,</span> <span class="n">bitcount</span><span class="p">,</span> <span class="n">to_int</span><span class="p">,</span> <span class="n">to_str</span><span class="p">,</span> <span class="n">mpf_mul</span><span class="p">,</span> <span class="n">mpf_div</span><span class="p">,</span> <span class="n">mpf_sub</span><span class="p">,</span>
-    <span class="n">mpf_add</span><span class="p">,</span> <span class="n">mpf_sqrt</span><span class="p">,</span> <span class="n">mpf_pi</span><span class="p">,</span> <span class="n">mpf_cosh_sinh</span><span class="p">,</span> <span class="n">pi_fixed</span><span class="p">,</span> <span class="n">mpf_cos</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">igcd</span>
-<span class="kn">import</span> <span class="nn">math</span>
-
-
-<span class="k">def</span> <span class="nf">_a</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Compute the inner sum in the HRR formula.&quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">j</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">fone</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="n">fzero</span>
-    <span class="n">pi</span> <span class="o">=</span> <span class="n">pi_fixed</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">j</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">igcd</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">continue</span>
-        <span class="c"># &amp; with mask to compute fractional part of fixed-point number</span>
-        <span class="n">one</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">&lt;&lt;</span> <span class="n">prec</span>
-        <span class="n">onemask</span> <span class="o">=</span> <span class="n">one</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="n">half</span> <span class="o">=</span> <span class="n">one</span> <span class="o">&gt;&gt;</span> <span class="mi">1</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">if</span> <span class="n">j</span> <span class="o">&gt;=</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">j</span><span class="p">):</span>
-                <span class="n">t</span> <span class="o">=</span> <span class="n">h</span><span class="o">*</span><span class="n">k</span><span class="o">*</span><span class="n">one</span><span class="o">//</span><span class="n">j</span>
-                <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">frac</span> <span class="o">=</span> <span class="n">t</span> <span class="o">&amp;</span> <span class="n">onemask</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">frac</span> <span class="o">=</span> <span class="o">-</span><span class="p">((</span><span class="o">-</span><span class="n">t</span><span class="p">)</span> <span class="o">&amp;</span> <span class="n">onemask</span><span class="p">)</span>
-                <span class="n">g</span> <span class="o">+=</span> <span class="n">k</span><span class="o">*</span><span class="p">(</span><span class="n">frac</span> <span class="o">-</span> <span class="n">half</span><span class="p">)</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="p">((</span><span class="n">g</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">h</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">one</span><span class="p">)</span><span class="o">*</span><span class="n">pi</span><span class="o">//</span><span class="n">j</span><span class="p">)</span> <span class="o">&gt;&gt;</span> <span class="n">prec</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">mpf_add</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">mpf_cos</span><span class="p">(</span><span class="n">from_man_exp</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="o">-</span><span class="n">prec</span><span class="p">),</span> <span class="n">prec</span><span class="p">),</span> <span class="n">prec</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">s</span>
-
-
-<span class="k">def</span> <span class="nf">_d</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">prec</span><span class="p">,</span> <span class="n">sq23pi</span><span class="p">,</span> <span class="n">sqrt8</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute the sinh term in the outer sum of the HRR formula.</span>
-<span class="sd">    The constants sqrt(2/3*pi) and sqrt(8) must be precomputed.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">j</span> <span class="o">=</span> <span class="n">from_int</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-    <span class="n">pi</span> <span class="o">=</span> <span class="n">mpf_pi</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">mpf_div</span><span class="p">(</span><span class="n">sq23pi</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-    <span class="n">b</span> <span class="o">=</span> <span class="n">mpf_sub</span><span class="p">(</span><span class="n">from_int</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">from_rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">24</span><span class="p">,</span> <span class="n">prec</span><span class="p">),</span> <span class="n">prec</span><span class="p">)</span>
-    <span class="n">c</span> <span class="o">=</span> <span class="n">mpf_sqrt</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-    <span class="n">ch</span><span class="p">,</span> <span class="n">sh</span> <span class="o">=</span> <span class="n">mpf_cosh_sinh</span><span class="p">(</span><span class="n">mpf_mul</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">c</span><span class="p">),</span> <span class="n">prec</span><span class="p">)</span>
-    <span class="n">D</span> <span class="o">=</span> <span class="n">mpf_div</span><span class="p">(</span><span class="n">mpf_sqrt</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">prec</span><span class="p">),</span> <span class="n">mpf_mul</span><span class="p">(</span><span class="n">mpf_mul</span><span class="p">(</span><span class="n">sqrt8</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">pi</span><span class="p">),</span> <span class="n">prec</span><span class="p">)</span>
-    <span class="n">E</span> <span class="o">=</span> <span class="n">mpf_sub</span><span class="p">(</span><span class="n">mpf_mul</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">ch</span><span class="p">),</span> <span class="n">mpf_div</span><span class="p">(</span><span class="n">sh</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">prec</span><span class="p">),</span> <span class="n">prec</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">mpf_mul</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">E</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="npartitions"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.partitions_.npartitions">[docs]</a><span class="k">def</span> <span class="nf">npartitions</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Calculate the partition function P(n), i.e. the number of ways that</span>
-<span class="sd">    n can be written as a sum of positive integers.</span>
-
-<span class="sd">    P(n) is computed using the Hardy-Ramanujan-Rademacher formula,</span>
-<span class="sd">    described e.g. at http://mathworld.wolfram.com/PartitionFunctionP.html</span>
-
-<span class="sd">    The correctness of this implementation has been tested for 10**n</span>
-<span class="sd">    up to n = 8.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory import npartitions</span>
-<span class="sd">    &gt;&gt;&gt; npartitions(25)</span>
-<span class="sd">    1958</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;=</span> <span class="mi">5</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">][</span><span class="n">n</span><span class="p">]</span>
-    <span class="c"># Estimate number of bits in p(n). This formula could be tidied</span>
-    <span class="n">pbits</span> <span class="o">=</span> <span class="nb">int</span><span class="p">((</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">/</span><span class="mf">3.</span><span class="p">)</span><span class="o">**</span><span class="mf">0.5</span> <span class="o">-</span> <span class="n">math</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">n</span><span class="p">))</span><span class="o">/</span><span class="n">math</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> \
-        <span class="n">math</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-    <span class="n">prec</span> <span class="o">=</span> <span class="n">p</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">pbits</span><span class="o">*</span><span class="mf">1.1</span> <span class="o">+</span> <span class="mi">100</span><span class="p">)</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="n">fzero</span>
-    <span class="n">M</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="mf">0.24</span><span class="o">*</span><span class="n">n</span><span class="o">**</span><span class="mf">0.5</span> <span class="o">+</span> <span class="mi">4</span><span class="p">))</span>
-    <span class="n">sq23pi</span> <span class="o">=</span> <span class="n">mpf_mul</span><span class="p">(</span><span class="n">mpf_sqrt</span><span class="p">(</span><span class="n">from_rational</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">p</span><span class="p">),</span> <span class="n">p</span><span class="p">),</span> <span class="n">mpf_pi</span><span class="p">(</span><span class="n">p</span><span class="p">),</span> <span class="n">p</span><span class="p">)</span>
-    <span class="n">sqrt8</span> <span class="o">=</span> <span class="n">mpf_sqrt</span><span class="p">(</span><span class="n">from_int</span><span class="p">(</span><span class="mi">8</span><span class="p">),</span> <span class="n">p</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">M</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">_a</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">_d</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">sq23pi</span><span class="p">,</span> <span class="n">sqrt8</span><span class="p">)</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">mpf_add</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">mpf_mul</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">d</span><span class="p">),</span> <span class="n">prec</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&quot;step&quot;</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="s">&quot;of&quot;</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">to_str</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">10</span><span class="p">),</span> <span class="n">to_str</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
-        <span class="c"># On average, the terms decrease rapidly in magnitude. Dynamically</span>
-        <span class="c"># reducing the precision greatly improves performance.</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">bitcount</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">to_int</span><span class="p">(</span><span class="n">d</span><span class="p">)))</span> <span class="o">+</span> <span class="mi">50</span>
-    <span class="n">np</span> <span class="o">=</span> <span class="n">to_int</span><span class="p">(</span><span class="n">mpf_add</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">fhalf</span><span class="p">,</span> <span class="n">prec</span><span class="p">))</span>
-    <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">np</span><span class="p">)</span>
-</div>
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;npartitions&#39;</span><span class="p">]</span>
-</pre></div>
-
-          </div>
-        </div>
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-            
-  <h1>Source code for sympy.ntheory.primetest</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Primality testing</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="c"># prime list to use when number must be tested as a probable prime.</span>
-<span class="c">#&gt;&gt;&gt; list(primerange(2, 200))</span>
-<span class="n">_isprime_fallback_primes</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">17</span><span class="p">,</span> <span class="mi">19</span><span class="p">,</span> <span class="mi">23</span><span class="p">,</span> <span class="mi">29</span><span class="p">,</span> <span class="mi">31</span><span class="p">,</span> <span class="mi">37</span><span class="p">,</span> <span class="mi">41</span><span class="p">,</span> <span class="mi">43</span><span class="p">,</span> <span class="mi">47</span><span class="p">,</span>
-    <span class="mi">53</span><span class="p">,</span> <span class="mi">59</span><span class="p">,</span> <span class="mi">61</span><span class="p">,</span> <span class="mi">67</span><span class="p">,</span> <span class="mi">71</span><span class="p">,</span> <span class="mi">73</span><span class="p">,</span> <span class="mi">79</span><span class="p">,</span> <span class="mi">83</span><span class="p">,</span> <span class="mi">89</span><span class="p">,</span> <span class="mi">97</span><span class="p">,</span> <span class="mi">101</span><span class="p">,</span> <span class="mi">103</span><span class="p">,</span> <span class="mi">107</span><span class="p">,</span>
-    <span class="mi">109</span><span class="p">,</span> <span class="mi">113</span><span class="p">,</span> <span class="mi">127</span><span class="p">,</span> <span class="mi">131</span><span class="p">,</span> <span class="mi">137</span><span class="p">,</span> <span class="mi">139</span><span class="p">,</span> <span class="mi">149</span><span class="p">,</span> <span class="mi">151</span><span class="p">,</span> <span class="mi">157</span><span class="p">,</span> <span class="mi">163</span><span class="p">,</span> <span class="mi">167</span><span class="p">,</span>
-    <span class="mi">173</span><span class="p">,</span> <span class="mi">179</span><span class="p">,</span> <span class="mi">181</span><span class="p">,</span> <span class="mi">191</span><span class="p">,</span> <span class="mi">193</span><span class="p">,</span> <span class="mi">197</span><span class="p">,</span> <span class="mi">199</span><span class="p">]</span>
-<span class="c">#&gt;&gt;&gt; len(_)</span>
-<span class="c">#46</span>
-<span class="c"># pseudoprimes that will pass through last mr_safe test</span>
-<span class="n">_pseudos</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span>
-            <span class="mi">669094855201</span><span class="p">,</span>
-           <span class="mi">1052516956501</span><span class="p">,</span> <span class="mi">2007193456621</span><span class="p">,</span> <span class="mi">2744715551581</span><span class="p">,</span> <span class="mi">9542968210729</span><span class="p">,</span>
-          <span class="mi">17699592963781</span><span class="p">,</span> <span class="mi">19671510288601</span><span class="p">,</span>
-          <span class="mi">24983920772821</span><span class="p">,</span> <span class="mi">24984938689453</span><span class="p">,</span> <span class="mi">29661584268781</span><span class="p">,</span> <span class="mi">37473222618541</span><span class="p">,</span>
-          <span class="mi">46856248255981</span><span class="p">,</span> <span class="mi">47922612926653</span><span class="p">,</span> <span class="mi">48103703944453</span><span class="p">,</span> <span class="mi">49110566041153</span><span class="p">,</span>
-          <span class="mi">49752242681221</span><span class="p">,</span> <span class="mi">91206655032481</span><span class="p">,</span> <span class="mi">91481980096033</span><span class="p">,</span> <span class="mi">119034193492321</span><span class="p">,</span>
-         <span class="mi">123645258399601</span><span class="p">,</span> <span class="mi">128928036060253</span><span class="p">,</span> <span class="mi">137364148720147</span><span class="p">,</span> <span class="mi">150753857310253</span><span class="p">,</span>
-         <span class="mi">153131886327421</span><span class="p">,</span> <span class="mi">155216912613121</span><span class="p">,</span> <span class="mi">185610214763821</span><span class="p">,</span> <span class="mi">224334357392701</span><span class="p">,</span>
-         <span class="mi">227752294950181</span><span class="p">,</span> <span class="mi">230058334559041</span><span class="p">,</span> <span class="mi">304562854940401</span><span class="p">,</span> <span class="mi">306001576998253</span><span class="p">,</span>
-         <span class="mi">335788261073821</span><span class="p">,</span> <span class="mi">377133492079081</span><span class="p">,</span> <span class="mi">379242177424951</span><span class="p">,</span> <span class="mi">389970770948461</span><span class="p">,</span>
-         <span class="mi">397319638319521</span><span class="p">,</span> <span class="mi">448114903362253</span><span class="p">,</span> <span class="mi">523235160050221</span><span class="p">,</span> <span class="mi">628999496281621</span><span class="p">,</span>
-         <span class="mi">699349238838253</span><span class="p">,</span> <span class="mi">746667678235753</span><span class="p">,</span> <span class="mi">790198268451301</span><span class="p">,</span> <span class="mi">794036495175661</span><span class="p">,</span>
-         <span class="mi">823820871230281</span><span class="p">,</span> <span class="mi">867739535711821</span><span class="p">,</span> <span class="mi">1039918661294761</span><span class="p">,</span> <span class="mi">1099127938585141</span><span class="p">,</span>
-        <span class="mi">1104388025338153</span><span class="p">,</span> <span class="mi">1173374598605653</span><span class="p">,</span> <span class="mi">1262797719066157</span><span class="p">,</span> <span class="mi">1265872947674653</span><span class="p">,</span>
-        <span class="mi">1325898212229667</span><span class="p">,</span> <span class="mi">1327034517143653</span><span class="p">,</span> <span class="mi">1418575746675583</span><span class="p">,</span> <span class="mi">1666122072463621</span><span class="p">,</span>
-        <span class="mi">1837400535259453</span><span class="p">,</span> <span class="mi">1857422490084961</span><span class="p">,</span> <span class="mi">1870756820971741</span><span class="p">,</span> <span class="mi">1914550540480717</span><span class="p">,</span>
-        <span class="mi">2018963273468221</span><span class="p">,</span> <span class="mi">2163829000939453</span><span class="p">,</span> <span class="mi">2206020317369221</span><span class="p">,</span> <span class="mi">2301037384029121</span><span class="p">,</span>
-        <span class="mi">2416062055125421</span><span class="p">,</span> <span class="mi">2435076500074921</span><span class="p">,</span> <span class="mi">2545656135020833</span><span class="p">,</span> <span class="mi">2594428516569781</span><span class="p">,</span>
-        <span class="mi">2669983768115821</span><span class="p">,</span> <span class="mi">2690937050990653</span><span class="p">,</span> <span class="mi">2758640869506607</span><span class="p">,</span> <span class="mi">2833525461416653</span><span class="p">,</span>
-        <span class="mi">2876662942007221</span><span class="p">,</span> <span class="mi">2932155806957821</span><span class="p">,</span> <span class="mi">2957010595723801</span><span class="p">,</span> <span class="mi">3183606449929153</span><span class="p">,</span>
-        <span class="mi">3220133449185901</span><span class="p">,</span> <span class="mi">3424103775720253</span><span class="p">,</span> <span class="mi">3625360152399541</span><span class="p">,</span> <span class="mi">3939300299037421</span><span class="p">,</span>
-        <span class="mi">3947917710714841</span><span class="p">,</span> <span class="mi">3980273496750253</span><span class="p">,</span> <span class="mi">4182256679324041</span><span class="p">,</span> <span class="mi">4450605887818261</span><span class="p">,</span>
-        <span class="mi">4727893739521501</span><span class="p">,</span> <span class="mi">4750350311306953</span><span class="p">,</span> <span class="mi">4755334362931153</span><span class="p">,</span> <span class="mi">5756440863559753</span><span class="p">,</span>
-        <span class="mi">5760976603475341</span><span class="p">,</span> <span class="mi">5794399356078761</span><span class="p">,</span> <span class="mi">5954850603819253</span><span class="p">,</span> <span class="mi">6125544931991761</span><span class="p">,</span>
-        <span class="mi">6320931714094861</span><span class="p">,</span> <span class="mi">6347593619672581</span><span class="p">,</span> <span class="mi">6406268028524101</span><span class="p">,</span> <span class="mi">6510632945054941</span><span class="p">,</span>
-        <span class="mi">6620082224794741</span><span class="p">,</span> <span class="mi">6627325072566061</span><span class="p">,</span> <span class="mi">6844056606431101</span><span class="p">,</span> <span class="mi">6989404981060153</span><span class="p">,</span>
-        <span class="mi">7144293947609521</span><span class="p">,</span> <span class="mi">7288348593229021</span><span class="p">,</span> <span class="mi">7288539837129253</span><span class="p">,</span> <span class="mi">7406102904971689</span><span class="p">,</span>
-        <span class="mi">7430233301822341</span><span class="p">,</span> <span class="mi">7576425305871193</span><span class="p">,</span> <span class="mi">7601696719033861</span><span class="p">,</span> <span class="mi">7803926845356487</span><span class="p">,</span>
-        <span class="mi">7892007967006633</span><span class="p">,</span> <span class="mi">7947797946559453</span><span class="p">,</span> <span class="mi">8207000460596953</span><span class="p">,</span> <span class="mi">8295064717807513</span><span class="p">,</span>
-        <span class="mi">8337196000698841</span><span class="p">,</span> <span class="mi">8352714234009421</span><span class="p">,</span> <span class="mi">8389755717406381</span><span class="p">,</span> <span class="mi">8509654470665701</span><span class="p">,</span>
-        <span class="mi">8757647355282841</span><span class="p">,</span> <span class="mi">8903933671696381</span><span class="p">,</span> <span class="mi">8996133652295653</span><span class="p">,</span> <span class="mi">9074421465661261</span><span class="p">,</span>
-        <span class="mi">9157536631454221</span><span class="p">,</span> <span class="mi">9188353522314541</span><span class="p">])</span>
-
-
-<span class="k">def</span> <span class="nf">_test</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">base</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Miller-Rabin strong pseudoprime test for one base.</span>
-<span class="sd">    Return False if n is definitely composite, True if n is</span>
-<span class="sd">    probably prime, with a probability greater than 3/4.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.ntheory.factor_</span> <span class="kn">import</span> <span class="n">trailing</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">False</span>
-    <span class="c"># remove powers of 2 from n (= t * 2**s)</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="n">trailing</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="n">t</span> <span class="o">=</span> <span class="n">n</span> <span class="o">&gt;&gt;</span> <span class="n">s</span>
-    <span class="c"># do the Fermat test</span>
-    <span class="n">b</span> <span class="o">=</span> <span class="nb">pow</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">b</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">b</span> <span class="o">==</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">True</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="p">(</span><span class="n">b</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">%</span> <span class="n">n</span>
-            <span class="k">if</span> <span class="n">b</span> <span class="o">==</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-    <span class="k">return</span> <span class="bp">False</span>
-
-
-<div class="viewcode-block" id="mr"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.primetest.mr">[docs]</a><span class="k">def</span> <span class="nf">mr</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">bases</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Perform a Miller-Rabin strong pseudoprime test on n using a</span>
-<span class="sd">    given list of bases/witnesses.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - Richard Crandall &amp; Carl Pomerance (2005), &quot;Prime Numbers:</span>
-<span class="sd">      A Computational Perspective&quot;, Springer, 2nd edition, 135-138</span>
-
-<span class="sd">    A list of thresholds and the bases they require are here:</span>
-<span class="sd">    http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test#Deterministic_variants_of_the_test</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory.primetest import mr</span>
-<span class="sd">    &gt;&gt;&gt; mr(1373651, [2, 3])</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; mr(479001599, [31, 73])</span>
-<span class="sd">    True</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="n">bases</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">_test</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">base</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-    <span class="k">return</span> <span class="bp">True</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_mr_safe</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;For n &lt; 10**16, use the Miller-Rabin test to determine with</span>
-<span class="sd">    certainty (unless the code is buggy!) whether n is prime.</span>
-
-<span class="sd">    Although the primes 2 through 17 are sufficient to confirm that a number</span>
-<span class="sd">    less than 341550071728322 (that is not prime 2 through 17) is prime, this</span>
-<span class="sd">    range is broken up into smaller ranges with earlier ranges requiring less</span>
-<span class="sd">    work. For example, for n &lt; 1373653 only the bases 2 and 3 need be tested.</span>
-
-<span class="sd">    What makes this a &quot;safe&quot; Miller-Rabin routine is that for n less than</span>
-<span class="sd">    the indicated limit, the given bases have been confirmed to detect all</span>
-<span class="sd">    composite numbers. What can potentially make this routine &quot;unsafe&quot; is</span>
-<span class="sd">    including ranges for which previous tests do not removes prime factors of</span>
-<span class="sd">    the bases being used. For example, this routine assumes that 2 and 3 have</span>
-<span class="sd">    already been removed as prime; but if the first test were the one for</span>
-<span class="sd">    n &lt; 170584961 (that uses bases 350 and 3958281543) the routine would have</span>
-<span class="sd">    to ensure that the primes 5, 7, 29, 67, 679067 are already removed or else</span>
-<span class="sd">    they will be reported as being composite. For this reason it is helpful to</span>
-<span class="sd">    list the prime factors of the bases being tested as is done below. The</span>
-<span class="sd">    _mr_safe_helper can be used to generate this info-tag.</span>
-
-<span class="sd">    References for the bounds:</span>
-<span class="sd">    ==========================</span>
-
-<span class="sd">    1. http://primes.utm.edu/prove/prove2_3.html</span>
-<span class="sd">    2. http://www.trnicely.net/misc/mpzspsp.html</span>
-<span class="sd">    3. http://en.wikipedia.org/wiki/Miller-Rabin_primality_test#</span>
-<span class="sd">        Accuracy_of_the_test</span>
-<span class="sd">    4. http://zdu.spaces.live.com/?_c11_BlogPart_pagedir=</span>
-<span class="sd">        Next&amp;_c11_BlogPart_handle=cns!C95152CB25EF2037!</span>
-<span class="sd">        138&amp;_c11_BlogPart_BlogPart=blogview&amp;_c=BlogPart</span>
-<span class="sd">    5. http://primes.utm.edu/glossary/xpage/Pseudoprime.html</span>
-<span class="sd">    6. http://uucode.com/obf/dalbec/alg.html#sprp</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">1373653</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">mr</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-        <span class="c">#[2, 3] stot = 1 clear == bases</span>
-        <span class="c"># these two (and similar below) are commented out since they are</span>
-        <span class="c"># more expensive in terms of stot than a later test.</span>
-        <span class="c">#if n &lt; 9080191: return mr(n, [31, 73]) # ref [3]</span>
-        <span class="c"># [31, 73] stot = 4 clear == bases</span>
-        <span class="c">#if n &lt; 25326001: return mr(n, [2, 3, 5])</span>
-        <span class="c"># [2, 3, 5] stot = 3 clear == bases</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">170584961</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">mr</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="p">[</span><span class="mi">350</span><span class="p">,</span> <span class="mi">3958281543</span><span class="p">])</span>
-        <span class="c"># [350, 3958281543] stot = 1 clear [2, 3, 5, 7, 29, 67, 679067]</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">4759123141</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">mr</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">61</span><span class="p">])</span>  <span class="c"># ref [3]</span>
-        <span class="c"># [2, 7, 61] stot = 3 clear == bases</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">75792980677</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">mr</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">379215</span><span class="p">,</span> <span class="mi">457083754</span><span class="p">])</span>
-        <span class="c"># [2, 379215, 457083754] stot = 1 clear [2, 3, 5, 53, 228541877]</span>
-        <span class="c">#if n &lt; 118670087467: return n is not 3215031751 and mr(n, [2, 3, 5, 7]) # ref [3]</span>
-        <span class="c"># [2, 3, 5, 7] stot = 4 clear == bases</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">1000000000000</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">mr</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">23</span><span class="p">,</span> <span class="mi">1662803</span><span class="p">])</span>
-        <span class="c"># [2, 13, 23, 1662803] stot = 4 clear == bases</span>
-        <span class="c">#if n &lt; 2152302898747: return mr(n, [2, 3, 5, 7, 11])</span>
-        <span class="c"># [2, 3, 5, 7, 11] stot = 5 clear == bases</span>
-        <span class="c">#if n &lt; 3474749660383: return mr(n, [2, 3, 5, 7, 11, 13])</span>
-        <span class="c"># [2, 3, 5, 7, 11, 13] stot = 7 clear == bases</span>
-        <span class="c">#if n &lt; 21652684502221: return mr(n, [2, 1215, 34862, 574237825])</span>
-        <span class="c"># [2, 1215, 34862, 574237825] stot = 8 clear [2, 3, 5, 7, 17431, 3281359]</span>
-        <span class="c">#if n &lt; 341550071728321: return mr(n, [2, 3, 5, 7, 11, 13, 17])</span>
-        <span class="c"># [2, 3, 5, 7, 11, 13, 17] stot = 11 clear == bases</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">10000000000000000</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">mr</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">61</span><span class="p">,</span> <span class="mi">24251</span><span class="p">])</span> <span class="ow">and</span> <span class="n">n</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">_pseudos</span>
-        <span class="c"># [2, 3, 7, 61, 24251] stot = 5 clear == bases</span>
-    <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;n too large&quot;</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="isprime"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.primetest.isprime">[docs]</a><span class="k">def</span> <span class="nf">isprime</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Test if n is a prime number (True) or not (False). For n &lt; 10**16 the</span>
-<span class="sd">    answer is accurate; greater n values have a small probability of actually</span>
-<span class="sd">    being pseudoprimes.</span>
-
-<span class="sd">    Negative primes (e.g. -2) are not considered prime.</span>
-
-<span class="sd">    The function first looks for trivial factors, and if none is found,</span>
-<span class="sd">    performs a safe Miller-Rabin strong pseudoprime test with bases</span>
-<span class="sd">    that are known to prove a number prime. Finally, a general Miller-Rabin</span>
-<span class="sd">    test is done with the first k bases which, which will report a</span>
-<span class="sd">    pseudoprime as a prime with an error of about 4**-k. The current value</span>
-<span class="sd">    of k is 46 so the error is about 2 x 10**-28.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory import isprime</span>
-<span class="sd">    &gt;&gt;&gt; isprime(13)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; isprime(15)</span>
-<span class="sd">    False</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.ntheory.generate.primerange : Generates all primes in a given range</span>
-<span class="sd">    sympy.ntheory.generate.primepi : Return the number of primes less than or equal to n</span>
-<span class="sd">    sympy.ntheory.generate.prime : Return the nth prime</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">False</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&amp;</span> <span class="mi">1</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">2</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;=</span> <span class="mi">23001</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">pow</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">n</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">341</span><span class="p">,</span> <span class="mi">561</span><span class="p">,</span> <span class="mi">645</span><span class="p">,</span> <span class="mi">1105</span><span class="p">,</span> <span class="mi">1387</span><span class="p">,</span> <span class="mi">1729</span><span class="p">,</span>
-                                               <span class="mi">1905</span><span class="p">,</span> <span class="mi">2047</span><span class="p">,</span> <span class="mi">2465</span><span class="p">,</span> <span class="mi">2701</span><span class="p">,</span> <span class="mi">2821</span><span class="p">,</span>
-                                               <span class="mi">3277</span><span class="p">,</span> <span class="mi">4033</span><span class="p">,</span> <span class="mi">4369</span><span class="p">,</span> <span class="mi">4371</span><span class="p">,</span> <span class="mi">4681</span><span class="p">,</span>
-                                               <span class="mi">5461</span><span class="p">,</span> <span class="mi">6601</span><span class="p">,</span> <span class="mi">7957</span><span class="p">,</span> <span class="mi">8321</span><span class="p">,</span> <span class="mi">8481</span><span class="p">,</span>
-                                               <span class="mi">8911</span><span class="p">,</span> <span class="mi">10261</span><span class="p">,</span> <span class="mi">10585</span><span class="p">,</span> <span class="mi">11305</span><span class="p">,</span>
-                                               <span class="mi">12801</span><span class="p">,</span> <span class="mi">13741</span><span class="p">,</span> <span class="mi">13747</span><span class="p">,</span> <span class="mi">13981</span><span class="p">,</span>
-                                               <span class="mi">14491</span><span class="p">,</span> <span class="mi">15709</span><span class="p">,</span> <span class="mi">15841</span><span class="p">,</span> <span class="mi">16705</span><span class="p">,</span>
-                                               <span class="mi">18705</span><span class="p">,</span> <span class="mi">18721</span><span class="p">,</span> <span class="mi">19951</span><span class="p">,</span> <span class="mi">23001</span><span class="p">]</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">_mr_safe</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">mr</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">_isprime_fallback_primes</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_mr_safe_helper</span><span class="p">(</span><span class="n">_s</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Analyze a (new) mr_safe line for for total number of s&#39;s to</span>
-<span class="sd">    be tested in _test along with the primes that must be cleared</span>
-<span class="sd">    by a previous test.</span>
-
-<span class="sd">    e.g.</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory.primetest import _mr_safe_helper</span>
-<span class="sd">    &gt;&gt;&gt; print(_mr_safe_helper(&quot;if n &lt; 170584961: return mr(n, [350, 3958281543])&quot;))</span>
-<span class="sd">     # [350, 3958281543] stot = 1 clear [2, 3, 5, 7, 29, 67, 679067]</span>
-<span class="sd">    &gt;&gt;&gt; print(_mr_safe_helper(&#39;return mr(n, [2, 379215, 457083754])&#39;))</span>
-<span class="sd">     # [2, 379215, 457083754] stot = 1 clear [2, 3, 5, 53, 228541877]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_info</span><span class="p">(</span><span class="n">bases</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Analyze the list of bases, reporting the number of &#39;j-loops&#39; that</span>
-<span class="sd">        will be required if this list is passed to _test (stot) and the primes</span>
-<span class="sd">        that must be cleared by a previous test.</span>
-
-<span class="sd">        This info tag should then be appended to any new mr_safe line</span>
-<span class="sd">        that is added so someone can easily see whether that line satisfies</span>
-<span class="sd">        the requirements of mr_safe (see docstring there for details).</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.ntheory.factor_</span> <span class="kn">import</span> <span class="n">factorint</span><span class="p">,</span> <span class="n">trailing</span>
-
-        <span class="n">factors</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">tot</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">bases</span><span class="p">:</span>
-            <span class="n">tot</span> <span class="o">+=</span> <span class="n">trailing</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">factorint</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-            <span class="n">factors</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-        <span class="n">factors</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">factors</span><span class="p">))</span>
-        <span class="n">bases</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">bases</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">bases</span> <span class="o">==</span> <span class="n">factors</span><span class="p">:</span>
-            <span class="n">factors</span> <span class="o">=</span> <span class="s">&#39;== bases&#39;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">factors</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&#39; # </span><span class="si">%s</span><span class="s"> stot = </span><span class="si">%s</span><span class="s"> clear </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="nb">tuple</span><span class="p">(</span>
-            <span class="p">[</span><span class="nb">str</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;L&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">bases</span><span class="p">),</span> <span class="n">tot</span><span class="p">,</span> <span class="n">factors</span><span class="p">)])</span>
-
-    <span class="n">_r</span> <span class="o">=</span> <span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">_x</span><span class="p">)</span> <span class="k">for</span> <span class="n">_x</span> <span class="ow">in</span> <span class="n">_s</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;[&#39;</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;]&#39;</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;,&#39;</span><span class="p">)]</span>
-    <span class="k">return</span> <span class="n">_info</span><span class="p">(</span><span class="n">_r</span><span class="p">)</span>
-</pre></div>
-
-          </div>
-        </div>
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-            <p class="logo"><a href="../../../index.html">
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-  <h3>Quick search</h3>
-    <form class="search" action="../../../search.html" method="get">
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-      <input type="hidden" name="check_keywords" value="yes" />
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diff --git a/dev-py3k/_modules/sympy/ntheory/residue_ntheory.html b/dev-py3k/_modules/sympy/ntheory/residue_ntheory.html
deleted file mode 100644
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--- a/dev-py3k/_modules/sympy/ntheory/residue_ntheory.html
+++ /dev/null
@@ -1,334 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
-
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-  <head>
-    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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-  <h1>Source code for sympy.ntheory.residue_ntheory</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">igcd</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">as_int</span>
-<span class="kn">from</span> <span class="nn">.primetest</span> <span class="kn">import</span> <span class="n">isprime</span>
-<span class="kn">from</span> <span class="nn">.factor_</span> <span class="kn">import</span> <span class="n">factorint</span><span class="p">,</span> <span class="n">trailing</span><span class="p">,</span> <span class="n">totient</span>
-
-
-<div class="viewcode-block" id="n_order"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.residue_ntheory.n_order">[docs]</a><span class="k">def</span> <span class="nf">n_order</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns the order of ``a`` modulo ``n``.</span>
-
-<span class="sd">    The order of ``a`` modulo ``n`` is the smallest integer</span>
-<span class="sd">    ``k`` such that ``a**k`` leaves a remainder of 1 with ``n``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory import n_order</span>
-<span class="sd">    &gt;&gt;&gt; n_order(3, 7)</span>
-<span class="sd">    6</span>
-<span class="sd">    &gt;&gt;&gt; n_order(4, 7)</span>
-<span class="sd">    3</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">a</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">as_int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">igcd</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;The two numbers should be relatively prime&quot;</span><span class="p">)</span>
-    <span class="n">group_order</span> <span class="o">=</span> <span class="n">totient</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="n">factors</span> <span class="o">=</span> <span class="n">factorint</span><span class="p">(</span><span class="n">group_order</span><span class="p">)</span>
-    <span class="n">order</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="k">if</span> <span class="n">a</span> <span class="o">&gt;</span> <span class="n">n</span><span class="p">:</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">a</span> <span class="o">%</span> <span class="n">n</span>
-    <span class="k">for</span> <span class="n">p</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">factors</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="n">exponent</span> <span class="o">=</span> <span class="n">group_order</span>
-        <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">e</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">a</span> <span class="o">**</span> <span class="p">(</span><span class="n">exponent</span><span class="p">))</span> <span class="o">%</span> <span class="n">n</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">order</span> <span class="o">*=</span> <span class="n">p</span> <span class="o">**</span> <span class="p">(</span><span class="n">e</span> <span class="o">-</span> <span class="n">f</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-                <span class="k">break</span>
-            <span class="n">exponent</span> <span class="o">=</span> <span class="n">exponent</span> <span class="o">//</span> <span class="n">p</span>
-    <span class="k">return</span> <span class="n">order</span>
-
-</div>
-<div class="viewcode-block" id="is_primitive_root"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.residue_ntheory.is_primitive_root">[docs]</a><span class="k">def</span> <span class="nf">is_primitive_root</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns True if ``a`` is a primitive root of ``p``</span>
-
-<span class="sd">    ``a`` is said to be the primitive root of ``p`` if gcd(a, p) == 1 and</span>
-<span class="sd">    totient(p) is the smallest positive number s.t.</span>
-
-<span class="sd">        a**totient(p) cong 1 mod(p)</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory import is_primitive_root, n_order, totient</span>
-<span class="sd">    &gt;&gt;&gt; is_primitive_root(3, 10)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; is_primitive_root(9, 10)</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; n_order(3, 10) == totient(10)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; n_order(9, 10) == totient(10)</span>
-<span class="sd">    False</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">a</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">as_int</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">igcd</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;The two numbers should be relatively prime&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">a</span> <span class="o">&gt;</span> <span class="n">p</span><span class="p">:</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">a</span> <span class="o">%</span> <span class="n">p</span>
-    <span class="k">if</span> <span class="n">n_order</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span> <span class="o">==</span> <span class="n">totient</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-</div>
-<div class="viewcode-block" id="is_quad_residue"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.residue_ntheory.is_quad_residue">[docs]</a><span class="k">def</span> <span class="nf">is_quad_residue</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns True if ``a`` (mod ``p``) is in the set of squares mod ``p``,</span>
-<span class="sd">    i.e a % p in set([i**2 % p for i in range(p)]). If ``p`` is an odd</span>
-<span class="sd">    prime, an iterative method is used to make the determination:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory import is_quad_residue</span>
-<span class="sd">    &gt;&gt;&gt; list(set([i**2 % 7 for i in range(7)]))</span>
-<span class="sd">    [0, 1, 2, 4]</span>
-<span class="sd">    &gt;&gt;&gt; [j for j in range(7) if is_quad_residue(j, 7)]</span>
-<span class="sd">    [0, 1, 2, 4]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    legendre_symbol, jacobi_symbol</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">a</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">as_int</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">p</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;p must be &gt; 0&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">a</span> <span class="o">&gt;=</span> <span class="n">p</span> <span class="ow">or</span> <span class="n">a</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">a</span> <span class="o">%</span> <span class="n">p</span>
-    <span class="k">if</span> <span class="n">a</span> <span class="o">&lt;</span> <span class="mi">2</span> <span class="ow">or</span> <span class="n">p</span> <span class="o">&lt;</span> <span class="mi">3</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">True</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">isprime</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">p</span> <span class="o">%</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">jacobi_symbol</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">p</span><span class="o">//</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">i</span><span class="o">**</span><span class="mi">2</span> <span class="o">%</span> <span class="n">p</span> <span class="o">==</span> <span class="n">a</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">square_and_multiply</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">a</span>
-        <span class="k">elif</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">((</span><span class="n">square_and_multiply</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span> <span class="o">//</span> <span class="mi">2</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">a</span><span class="p">)</span> <span class="o">%</span> <span class="n">p</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">square_and_multiply</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span> <span class="o">//</span> <span class="mi">2</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">%</span> <span class="n">p</span>
-
-    <span class="k">return</span> <span class="p">(</span><span class="n">square_and_multiply</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="p">(</span><span class="n">p</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span> <span class="o">%</span> <span class="n">p</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span>
-
-</div>
-<div class="viewcode-block" id="legendre_symbol"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.residue_ntheory.legendre_symbol">[docs]</a><span class="k">def</span> <span class="nf">legendre_symbol</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    1. 0 if a is multiple of p</span>
-<span class="sd">    2. 1 if a is a quadratic residue of p</span>
-<span class="sd">    3. -1 otherwise</span>
-
-<span class="sd">    p should be an odd prime by definition</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory import legendre_symbol</span>
-<span class="sd">    &gt;&gt;&gt; [legendre_symbol(i, 7) for i in range(7)]</span>
-<span class="sd">    [0, 1, 1, -1, 1, -1, -1]</span>
-<span class="sd">    &gt;&gt;&gt; list(set([i**2 % 7 for i in range(7)]))</span>
-<span class="sd">    [0, 1, 2, 4]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    is_quad_residue, jacobi_symbol</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">a</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">as_int</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">isprime</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="ow">or</span> <span class="n">p</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;p should be an odd prime&quot;</span><span class="p">)</span>
-    <span class="n">_</span><span class="p">,</span> <span class="n">a</span> <span class="o">=</span> <span class="nb">divmod</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">a</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="k">if</span> <span class="n">is_quad_residue</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">1</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-
-</div>
-<div class="viewcode-block" id="jacobi_symbol"><a class="viewcode-back" href="../../../modules/ntheory.html#sympy.ntheory.residue_ntheory.jacobi_symbol">[docs]</a><span class="k">def</span> <span class="nf">jacobi_symbol</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the product of the legendre_symbol(m, p)</span>
-<span class="sd">    for all the prime factors, p, of n.</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    1. 0 if m cong 0 mod(n)</span>
-<span class="sd">    2. 1 if x**2 cong m mod(n) has a solution</span>
-<span class="sd">    3. -1 otherwise</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.ntheory import jacobi_symbol, legendre_symbol</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Mul, S</span>
-<span class="sd">    &gt;&gt;&gt; jacobi_symbol(45, 77)</span>
-<span class="sd">    -1</span>
-<span class="sd">    &gt;&gt;&gt; jacobi_symbol(60, 121)</span>
-<span class="sd">    1</span>
-
-<span class="sd">    The relationship between the jacobi_symbol and legendre_symbol can</span>
-<span class="sd">    be demonstrated as follows:</span>
-
-<span class="sd">    &gt;&gt;&gt; L = legendre_symbol</span>
-<span class="sd">    &gt;&gt;&gt; S(45).factors()</span>
-<span class="sd">    {3: 2, 5: 1}</span>
-<span class="sd">    &gt;&gt;&gt; jacobi_symbol(7, 45) == L(7, 3)**2 * L(7, 5)**1</span>
-<span class="sd">    True</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    is_quad_residue, legendre_symbol</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">m</span><span class="p">),</span> <span class="n">as_int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;n should be an odd integer&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">m</span> <span class="o">&gt;</span> <span class="n">n</span><span class="p">:</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">m</span> <span class="o">%</span> <span class="n">n</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">m</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">1</span>
-    <span class="k">if</span> <span class="n">igcd</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-
-    <span class="n">j</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="n">trailing</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">m</span> <span class="o">&gt;&gt;</span> <span class="n">s</span>
-    <span class="k">if</span> <span class="n">s</span> <span class="o">%</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">8</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">]:</span>
-        <span class="n">j</span> <span class="o">*=</span> <span class="o">-</span><span class="mi">1</span>
-
-    <span class="k">while</span> <span class="n">m</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">m</span> <span class="o">%</span> <span class="mi">4</span> <span class="o">==</span> <span class="mi">3</span> <span class="ow">and</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">4</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="n">j</span> <span class="o">*=</span> <span class="o">-</span><span class="mi">1</span>
-        <span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">n</span> <span class="o">%</span> <span class="n">m</span><span class="p">,</span> <span class="n">m</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">trailing</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">m</span> <span class="o">&gt;&gt;</span> <span class="n">s</span>
-        <span class="k">if</span> <span class="n">s</span> <span class="o">%</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">8</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">]:</span>
-            <span class="n">j</span> <span class="o">*=</span> <span class="o">-</span><span class="mi">1</span>
-    <span class="k">return</span> <span class="n">j</span></div>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/parsing/mathematica.html b/dev-py3k/_modules/sympy/parsing/mathematica.html
deleted file mode 100644
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@@ -1,172 +0,0 @@
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-            
-  <h1>Source code for sympy.parsing.mathematica</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">re</span> <span class="kn">import</span> <span class="n">match</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sympify</span>
-
-
-<div class="viewcode-block" id="mathematica"><a class="viewcode-back" href="../../../modules/parsing.html#sympy.parsing.mathematica.mathematica">[docs]</a><span class="k">def</span> <span class="nf">mathematica</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="n">parse</span><span class="p">(</span><span class="n">s</span><span class="p">))</span>
-
-</div>
-<span class="k">def</span> <span class="nf">parse</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">strip</span><span class="p">()</span>
-
-    <span class="c">#Begin rules</span>
-    <span class="n">rules</span> <span class="o">=</span> <span class="p">(</span>
-        <span class="p">(</span><span class="s">r&quot;\A(\w+)\[([^\]]+[^\[]*)\]\Z&quot;</span><span class="p">,</span>  <span class="c"># Function call</span>
-        <span class="k">lambda</span> <span class="n">m</span><span class="p">:</span> <span class="n">translateFunction</span><span class="p">(</span>
-            <span class="n">m</span><span class="o">.</span><span class="n">group</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span> <span class="o">+</span> <span class="s">&quot;(&quot;</span> <span class="o">+</span> <span class="n">parse</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">group</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span> <span class="o">+</span> <span class="s">&quot;)&quot;</span> <span class="p">),</span>
-
-        <span class="p">(</span><span class="s">r&quot;\((.+)\)\((.+)\)&quot;</span><span class="p">,</span>  <span class="c"># Parenthesized implied multiplication</span>
-        <span class="k">lambda</span> <span class="n">m</span><span class="p">:</span> <span class="s">&quot;(&quot;</span> <span class="o">+</span> <span class="n">parse</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">group</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span> <span class="o">+</span> <span class="s">&quot;)*(&quot;</span> <span class="o">+</span> <span class="n">parse</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">group</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span> <span class="o">+</span> <span class="s">&quot;)&quot;</span> <span class="p">),</span>
-
-        <span class="p">(</span><span class="s">r&quot;\A\((.+)\)\Z&quot;</span><span class="p">,</span>  <span class="c"># Parenthesized expression</span>
-        <span class="k">lambda</span> <span class="n">m</span><span class="p">:</span> <span class="s">&quot;(&quot;</span> <span class="o">+</span> <span class="n">parse</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">group</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span> <span class="o">+</span> <span class="s">&quot;)&quot;</span> <span class="p">),</span>
-
-        <span class="p">(</span><span class="s">r&quot;\A(.*[\w\.])\((.+)\)\Z&quot;</span><span class="p">,</span>  <span class="c"># Implied multiplication - a(b)</span>
-        <span class="k">lambda</span> <span class="n">m</span><span class="p">:</span> <span class="n">parse</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">group</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span> <span class="o">+</span> <span class="s">&quot;*(&quot;</span> <span class="o">+</span> <span class="n">parse</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">group</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span> <span class="o">+</span> <span class="s">&quot;)&quot;</span> <span class="p">),</span>
-
-        <span class="p">(</span><span class="s">r&quot;\A\((.+)\)([\w\.].*)\Z&quot;</span><span class="p">,</span>  <span class="c"># Implied multiplication - (a)b</span>
-        <span class="k">lambda</span> <span class="n">m</span><span class="p">:</span> <span class="s">&quot;(&quot;</span> <span class="o">+</span> <span class="n">parse</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">group</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span> <span class="o">+</span> <span class="s">&quot;)*&quot;</span> <span class="o">+</span> <span class="n">parse</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">group</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span> <span class="p">),</span>
-
-        <span class="p">(</span><span class="s">r&quot;\A([\d\.]+)([a-zA-Z].*)\Z&quot;</span><span class="p">,</span>  <span class="c"># Implied multiplicatin - 2a</span>
-        <span class="k">lambda</span> <span class="n">m</span><span class="p">:</span> <span class="n">parse</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">group</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span> <span class="o">+</span> <span class="s">&quot;*&quot;</span> <span class="o">+</span> <span class="n">parse</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">group</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span> <span class="p">),</span>
-
-        <span class="p">(</span><span class="s">r&quot;\A([^=]+)([\^\-\*/\+=]=?)(.+)\Z&quot;</span><span class="p">,</span>  <span class="c"># Infix operator</span>
-        <span class="k">lambda</span> <span class="n">m</span><span class="p">:</span> <span class="n">parse</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">group</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span> <span class="o">+</span> <span class="n">translateOperator</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">group</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span> <span class="o">+</span> <span class="n">parse</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">group</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span> <span class="p">))</span>
-    <span class="c">#End rules</span>
-
-    <span class="k">for</span> <span class="n">rule</span><span class="p">,</span> <span class="n">action</span> <span class="ow">in</span> <span class="n">rules</span><span class="p">:</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">match</span><span class="p">(</span><span class="n">rule</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">m</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">action</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">s</span>
-
-
-<span class="k">def</span> <span class="nf">translateFunction</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
-    <span class="k">if</span> <span class="n">s</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&quot;Arc&quot;</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;a&quot;</span> <span class="o">+</span> <span class="n">s</span><span class="p">[</span><span class="mi">3</span><span class="p">:]</span>
-    <span class="k">return</span> <span class="n">s</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span>
-
-
-<span class="k">def</span> <span class="nf">translateOperator</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
-    <span class="n">dictionary</span> <span class="o">=</span> <span class="p">{</span><span class="s">&#39;^&#39;</span><span class="p">:</span> <span class="s">&#39;**&#39;</span><span class="p">}</span>
-    <span class="k">if</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">dictionary</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dictionary</span><span class="p">[</span><span class="n">s</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">s</span>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/parsing/maxima.html b/dev-py3k/_modules/sympy/parsing/maxima.html
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--- a/dev-py3k/_modules/sympy/parsing/maxima.html
+++ /dev/null
@@ -1,185 +0,0 @@
-
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-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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-  <head>
-    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
-    
-    <title>sympy.parsing.maxima &mdash; SymPy 0.7.2-git documentation</title>
-    
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-  <h1>Source code for sympy.parsing.maxima</h1><div class="highlight"><pre>
-<span class="kn">import</span> <span class="nn">re</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">Sum</span><span class="p">,</span> <span class="n">product</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span>
-
-
-<span class="k">class</span> <span class="nc">MaximaHelpers</span><span class="p">:</span>
-    <span class="k">def</span> <span class="nf">maxima_expand</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">maxima_float</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">maxima_trigexpand</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">maxima_sum</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span> <span class="n">a2</span><span class="p">,</span> <span class="n">a3</span><span class="p">,</span> <span class="n">a4</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Sum</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span> <span class="p">(</span><span class="n">a2</span><span class="p">,</span> <span class="n">a3</span><span class="p">,</span> <span class="n">a4</span><span class="p">))</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">maxima_product</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span> <span class="n">a2</span><span class="p">,</span> <span class="n">a3</span><span class="p">,</span> <span class="n">a4</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">product</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span> <span class="p">(</span><span class="n">a2</span><span class="p">,</span> <span class="n">a3</span><span class="p">,</span> <span class="n">a4</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">maxima_csc</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="n">sin</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">maxima_sec</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-<span class="n">sub_dict</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="s">&#39;pi&#39;</span><span class="p">:</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="s">&#39;%pi&#39;</span><span class="p">),</span>
-    <span class="s">&#39;E&#39;</span><span class="p">:</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%e</span><span class="s">&#39;</span><span class="p">),</span>
-    <span class="s">&#39;I&#39;</span><span class="p">:</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%i</span><span class="s">&#39;</span><span class="p">),</span>
-    <span class="s">&#39;**&#39;</span><span class="p">:</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="s">&#39;\^&#39;</span><span class="p">),</span>
-    <span class="s">&#39;oo&#39;</span><span class="p">:</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="s">r&#39;\binf\b&#39;</span><span class="p">),</span>
-    <span class="s">&#39;-oo&#39;</span><span class="p">:</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="s">r&#39;\bminf\b&#39;</span><span class="p">),</span>
-    <span class="s">&quot;&#39;-&#39;&quot;</span><span class="p">:</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="s">r&#39;\bminus\b&#39;</span><span class="p">),</span>
-    <span class="s">&#39;maxima_expand&#39;</span><span class="p">:</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="s">r&#39;\bexpand\b&#39;</span><span class="p">),</span>
-    <span class="s">&#39;maxima_float&#39;</span><span class="p">:</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="s">r&#39;\bfloat\b&#39;</span><span class="p">),</span>
-    <span class="s">&#39;maxima_trigexpand&#39;</span><span class="p">:</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="s">r&#39;\btrigexpand&#39;</span><span class="p">),</span>
-    <span class="s">&#39;maxima_sum&#39;</span><span class="p">:</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="s">r&#39;\bsum\b&#39;</span><span class="p">),</span>
-    <span class="s">&#39;maxima_product&#39;</span><span class="p">:</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="s">r&#39;\bproduct\b&#39;</span><span class="p">),</span>
-    <span class="s">&#39;cancel&#39;</span><span class="p">:</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="s">r&#39;\bratsimp\b&#39;</span><span class="p">),</span>
-    <span class="s">&#39;maxima_csc&#39;</span><span class="p">:</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="s">r&#39;\bcsc\b&#39;</span><span class="p">),</span>
-    <span class="s">&#39;maxima_sec&#39;</span><span class="p">:</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="s">r&#39;\bsec\b&#39;</span><span class="p">)</span>
-<span class="p">}</span>
-
-<span class="n">var_name</span> <span class="o">=</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="s">&#39;^\s*(\w+)\s*:&#39;</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="parse_maxima"><a class="viewcode-back" href="../../../modules/parsing.html#sympy.parsing.maxima.parse_maxima">[docs]</a><span class="k">def</span> <span class="nf">parse_maxima</span><span class="p">(</span><span class="nb">str</span><span class="p">,</span> <span class="nb">globals</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">name_dict</span><span class="o">=</span><span class="p">{}):</span>
-    <span class="nb">str</span> <span class="o">=</span> <span class="nb">str</span><span class="o">.</span><span class="n">strip</span><span class="p">()</span>
-    <span class="nb">str</span> <span class="o">=</span> <span class="nb">str</span><span class="o">.</span><span class="n">rstrip</span><span class="p">(</span><span class="s">&#39;; &#39;</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">sub_dict</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-        <span class="nb">str</span> <span class="o">=</span> <span class="n">v</span><span class="o">.</span><span class="n">sub</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="nb">str</span><span class="p">)</span>
-
-    <span class="n">assign_var</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">var_match</span> <span class="o">=</span> <span class="n">var_name</span><span class="o">.</span><span class="n">search</span><span class="p">(</span><span class="nb">str</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">var_match</span><span class="p">:</span>
-        <span class="n">assign_var</span> <span class="o">=</span> <span class="n">var_match</span><span class="o">.</span><span class="n">group</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-        <span class="nb">str</span> <span class="o">=</span> <span class="nb">str</span><span class="p">[</span><span class="n">var_match</span><span class="o">.</span><span class="n">end</span><span class="p">():]</span><span class="o">.</span><span class="n">strip</span><span class="p">()</span>
-
-    <span class="n">dct</span> <span class="o">=</span> <span class="n">MaximaHelpers</span><span class="o">.</span><span class="n">__dict__</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-    <span class="n">dct</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">name_dict</span><span class="p">)</span>
-    <span class="n">obj</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="nb">str</span><span class="p">,</span> <span class="nb">locals</span><span class="o">=</span><span class="n">dct</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">assign_var</span> <span class="ow">and</span> <span class="nb">globals</span><span class="p">:</span>
-        <span class="nb">globals</span><span class="p">[</span><span class="n">assign_var</span><span class="p">]</span> <span class="o">=</span> <span class="n">obj</span>
-
-    <span class="k">return</span> <span class="n">obj</span></div>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/parsing/sympy_parser.html b/dev-py3k/_modules/sympy/parsing/sympy_parser.html
deleted file mode 100644
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+++ /dev/null
@@ -1,675 +0,0 @@
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-  <h1>Source code for sympy.parsing.sympy_parser</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Transform a string with Python-like source code into SymPy expression. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">.sympy_tokenize</span> <span class="kn">import</span> \
-    <span class="n">generate_tokens</span><span class="p">,</span> <span class="n">untokenize</span><span class="p">,</span> <span class="n">TokenError</span><span class="p">,</span> \
-    <span class="n">NUMBER</span><span class="p">,</span> <span class="n">STRING</span><span class="p">,</span> <span class="n">NAME</span><span class="p">,</span> <span class="n">OP</span><span class="p">,</span> <span class="n">ENDMARKER</span>
-
-<span class="kn">from</span> <span class="nn">keyword</span> <span class="kn">import</span> <span class="n">iskeyword</span>
-<span class="kn">from</span> <span class="nn">io</span> <span class="kn">import</span> <span class="n">StringIO</span>
-<span class="kn">import</span> <span class="nn">re</span>
-<span class="kn">import</span> <span class="nn">unicodedata</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.basic</span> <span class="kn">import</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">C</span>
-<span class="kn">import</span> <span class="nn">collections</span>
-
-<span class="n">_re_repeated</span> <span class="o">=</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="s">r&quot;^(\d*)\.(\d*)\[(\d+)\]$&quot;</span><span class="p">)</span>
-<span class="n">UNSPLITTABLE_TOKEN_NAMES</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;_kern&#39;</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">_token_splittable</span><span class="p">(</span><span class="n">token</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Predicate for whether a token name can be split into multiple tokens.</span>
-
-<span class="sd">    A token is splittable if it does not contain an underscore charater and</span>
-<span class="sd">    it is not the name of a Greek letter. This is used to implicitly convert</span>
-<span class="sd">    expressions like &#39;xyz&#39; into &#39;x*y*z&#39;.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="s">&#39;_&#39;</span> <span class="ow">in</span> <span class="n">token</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">False</span>
-    <span class="k">elif</span> <span class="n">token</span> <span class="ow">in</span> <span class="n">UNSPLITTABLE_TOKEN_NAMES</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">False</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="ow">not</span> <span class="n">unicodedata</span><span class="o">.</span><span class="n">lookup</span><span class="p">(</span><span class="s">&#39;GREEK SMALL LETTER &#39;</span> <span class="o">+</span> <span class="n">token</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-            <span class="k">pass</span>
-    <span class="k">return</span> <span class="bp">True</span>
-
-
-<span class="k">def</span> <span class="nf">_add_factorial_tokens</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">result</span><span class="p">):</span>
-    <span class="k">if</span> <span class="n">result</span> <span class="o">==</span> <span class="p">[]</span> <span class="ow">or</span> <span class="n">result</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;(&#39;</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">TokenError</span><span class="p">()</span>
-
-    <span class="n">beginning</span> <span class="o">=</span> <span class="p">[(</span><span class="n">NAME</span><span class="p">,</span> <span class="n">name</span><span class="p">),</span> <span class="p">(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;(&#39;</span><span class="p">)]</span>
-    <span class="n">end</span> <span class="o">=</span> <span class="p">[(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;)&#39;</span><span class="p">)]</span>
-
-    <span class="n">diff</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">length</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">index</span><span class="p">,</span> <span class="n">token</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">result</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]):</span>
-        <span class="n">toknum</span><span class="p">,</span> <span class="n">tokval</span> <span class="o">=</span> <span class="n">token</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="n">length</span> <span class="o">-</span> <span class="n">index</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="k">if</span> <span class="n">tokval</span> <span class="o">==</span> <span class="s">&#39;)&#39;</span><span class="p">:</span>
-            <span class="n">diff</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">elif</span> <span class="n">tokval</span> <span class="o">==</span> <span class="s">&#39;(&#39;</span><span class="p">:</span>
-            <span class="n">diff</span> <span class="o">-=</span> <span class="mi">1</span>
-
-        <span class="k">if</span> <span class="n">diff</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">result</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">NAME</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">result</span><span class="p">[:</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">beginning</span> <span class="o">+</span> <span class="n">result</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:]</span> <span class="o">+</span> <span class="n">end</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">result</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">beginning</span> <span class="o">+</span> <span class="n">result</span><span class="p">[</span><span class="n">i</span><span class="p">:]</span> <span class="o">+</span> <span class="n">end</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="k">class</span> <span class="nc">AppliedFunction</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A group of tokens representing a function and its arguments.</span>
-
-<span class="sd">    `exponent` is for handling the shorthand sin^2, ln^2, etc.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">function</span><span class="p">,</span> <span class="n">args</span><span class="p">,</span> <span class="n">exponent</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">exponent</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">exponent</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">function</span> <span class="o">=</span> <span class="n">function</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">=</span> <span class="n">args</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">exponent</span> <span class="o">=</span> <span class="n">exponent</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">items</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;function&#39;</span><span class="p">,</span> <span class="s">&#39;args&#39;</span><span class="p">,</span> <span class="s">&#39;exponent&#39;</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">expand</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return a list of tokens representing the function&quot;&quot;&quot;</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="p">)</span>
-        <span class="n">result</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">index</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">items</span><span class="p">[</span><span class="n">index</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;AppliedFunction(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">function</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">,</span>
-                                                <span class="bp">self</span><span class="o">.</span><span class="n">exponent</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">ParenthesisGroup</span><span class="p">(</span><span class="nb">list</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;List of tokens representing an expression in parentheses.&quot;&quot;&quot;</span>
-    <span class="k">pass</span>
-
-
-<span class="k">def</span> <span class="nf">_flatten</span><span class="p">(</span><span class="n">result</span><span class="p">):</span>
-    <span class="n">result2</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">tok</span> <span class="ow">in</span> <span class="n">result</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">tok</span><span class="p">,</span> <span class="n">AppliedFunction</span><span class="p">):</span>
-            <span class="n">result2</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">tok</span><span class="o">.</span><span class="n">expand</span><span class="p">())</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">result2</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tok</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">result2</span>
-
-
-<span class="k">def</span> <span class="nf">_group_parentheses</span><span class="p">(</span><span class="n">tokens</span><span class="p">,</span> <span class="n">local_dict</span><span class="p">,</span> <span class="n">global_dict</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Group tokens between parentheses with ParenthesisGroup.</span>
-
-<span class="sd">    Also processes those tokens recursively.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">stacks</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">stacklevel</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">token</span> <span class="ow">in</span> <span class="n">tokens</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">token</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">OP</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">token</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;(&#39;</span><span class="p">:</span>
-                <span class="n">stacks</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ParenthesisGroup</span><span class="p">([]))</span>
-                <span class="n">stacklevel</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">elif</span> <span class="n">token</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;)&#39;</span><span class="p">:</span>
-                <span class="n">stacks</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">token</span><span class="p">)</span>
-                <span class="n">stack</span> <span class="o">=</span> <span class="n">stacks</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">stacks</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="c"># We don&#39;t recurse here since the upper-level stack</span>
-                    <span class="c"># would reprocess these tokens</span>
-                    <span class="n">stacks</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">stack</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="c"># Recurse here to handle nested parentheses</span>
-                    <span class="c"># Strip off the outer parentheses to avoid an infinite loop</span>
-                    <span class="n">inner</span> <span class="o">=</span> <span class="n">stack</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-                    <span class="n">inner</span> <span class="o">=</span> <span class="n">implicit_multiplication_application</span><span class="p">(</span><span class="n">inner</span><span class="p">,</span>
-                                                                <span class="n">local_dict</span><span class="p">,</span>
-                                                                <span class="n">global_dict</span><span class="p">)</span>
-                    <span class="n">parenGroup</span> <span class="o">=</span> <span class="p">[</span><span class="n">stack</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span> <span class="o">+</span> <span class="n">inner</span> <span class="o">+</span> <span class="p">[</span><span class="n">stack</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span>
-                    <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ParenthesisGroup</span><span class="p">(</span><span class="n">parenGroup</span><span class="p">))</span>
-                <span class="n">stacklevel</span> <span class="o">-=</span> <span class="mi">1</span>
-                <span class="k">continue</span>
-        <span class="k">if</span> <span class="n">stacklevel</span><span class="p">:</span>
-            <span class="n">stacks</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">token</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">token</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="k">def</span> <span class="nf">_apply_functions</span><span class="p">(</span><span class="n">tokens</span><span class="p">,</span> <span class="n">local_dict</span><span class="p">,</span> <span class="n">global_dict</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Convert a NAME token + ParenthesisGroup into an AppliedFunction.</span>
-
-<span class="sd">    Note that ParenthesisGroups, if not applied to any function, are</span>
-<span class="sd">    converted back into lists of tokens.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">symbol</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="k">for</span> <span class="n">tok</span> <span class="ow">in</span> <span class="n">tokens</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">tok</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">NAME</span><span class="p">:</span>
-            <span class="n">symbol</span> <span class="o">=</span> <span class="n">tok</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tok</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">tok</span><span class="p">,</span> <span class="n">ParenthesisGroup</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">symbol</span><span class="p">:</span>
-                <span class="n">result</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">AppliedFunction</span><span class="p">(</span><span class="n">symbol</span><span class="p">,</span> <span class="n">tok</span><span class="p">)</span>
-                <span class="n">symbol</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">tok</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">symbol</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tok</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="k">def</span> <span class="nf">_split_symbols</span><span class="p">(</span><span class="n">tokens</span><span class="p">,</span> <span class="n">local_dict</span><span class="p">,</span> <span class="n">global_dict</span><span class="p">):</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">tok</span> <span class="ow">in</span> <span class="n">tokens</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">tok</span><span class="p">,</span> <span class="n">AppliedFunction</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">tok</span><span class="o">.</span><span class="n">function</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;Symbol&#39;</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">tok</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-                <span class="c"># Middle token, get value, strip quotes</span>
-                <span class="n">symbol</span> <span class="o">=</span> <span class="n">tok</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-                <span class="k">if</span> <span class="n">_token_splittable</span><span class="p">(</span><span class="n">symbol</span><span class="p">):</span>
-                    <span class="k">for</span> <span class="n">char</span> <span class="ow">in</span> <span class="n">symbol</span><span class="p">:</span>
-                        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">AppliedFunction</span><span class="p">(</span>
-                            <span class="n">tok</span><span class="o">.</span><span class="n">function</span><span class="p">,</span>
-                            <span class="p">[(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;(&#39;</span><span class="p">),</span> <span class="p">(</span><span class="n">NAME</span><span class="p">,</span> <span class="nb">str</span><span class="p">(</span><span class="nb">repr</span><span class="p">(</span><span class="n">char</span><span class="p">))),</span> <span class="p">(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;)&#39;</span><span class="p">)]</span>
-                        <span class="p">))</span>
-                    <span class="k">continue</span>
-        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tok</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="k">def</span> <span class="nf">_implicit_multiplication</span><span class="p">(</span><span class="n">tokens</span><span class="p">,</span> <span class="n">local_dict</span><span class="p">,</span> <span class="n">global_dict</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Implicitly adds &#39;*&#39; tokens.</span>
-
-<span class="sd">    Cases:</span>
-
-<span class="sd">    - Two AppliedFunctions next to each other (&quot;sin(x)cos(x)&quot;)</span>
-
-<span class="sd">    - AppliedFunction next to an open parenthesis (&quot;sin x (cos x + 1)&quot;)</span>
-
-<span class="sd">    - A close parenthesis next to an AppliedFunction (&quot;(x+2)sin x&quot;)\</span>
-
-<span class="sd">    - A closeparenthesis next to an open parenthesis (&quot;(x+2)(x+3)&quot;)</span>
-
-<span class="sd">    - An AppliedFunction next to an implicitly applied function (&quot;sin(x)cos</span>
-<span class="sd">      x&quot;)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">tok</span><span class="p">,</span> <span class="n">nextTok</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">tokens</span><span class="p">,</span> <span class="n">tokens</span><span class="p">[</span><span class="mi">1</span><span class="p">:]):</span>
-        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tok</span><span class="p">)</span>
-        <span class="k">if</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">tok</span><span class="p">,</span> <span class="n">AppliedFunction</span><span class="p">)</span> <span class="ow">and</span>
-                <span class="nb">isinstance</span><span class="p">(</span><span class="n">nextTok</span><span class="p">,</span> <span class="n">AppliedFunction</span><span class="p">)):</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;*&#39;</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">tok</span><span class="p">,</span> <span class="n">AppliedFunction</span><span class="p">)</span> <span class="ow">and</span>
-              <span class="n">nextTok</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">OP</span> <span class="ow">and</span> <span class="n">nextTok</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;(&#39;</span><span class="p">):</span>
-            <span class="c"># Applied function followed by an open parenthesis</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;*&#39;</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="p">(</span><span class="n">tok</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">OP</span> <span class="ow">and</span> <span class="n">tok</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;)&#39;</span> <span class="ow">and</span>
-              <span class="nb">isinstance</span><span class="p">(</span><span class="n">nextTok</span><span class="p">,</span> <span class="n">AppliedFunction</span><span class="p">)):</span>
-            <span class="c"># Close parenthesis followed by an applied function</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;*&#39;</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="p">(</span><span class="n">tok</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">OP</span> <span class="ow">and</span> <span class="n">tok</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;)&#39;</span> <span class="ow">and</span>
-              <span class="n">nextTok</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">NAME</span><span class="p">):</span>
-            <span class="c"># Close parenthesis followed by an implicitly applied function</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;*&#39;</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="p">(</span><span class="n">tok</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">nextTok</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">OP</span>
-              <span class="ow">and</span> <span class="n">tok</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;)&#39;</span> <span class="ow">and</span> <span class="n">nextTok</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;(&#39;</span><span class="p">):</span>
-            <span class="c"># Close parenthesis followed by an open parenthesis</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;*&#39;</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">tok</span><span class="p">,</span> <span class="n">AppliedFunction</span><span class="p">)</span> <span class="ow">and</span> <span class="n">nextTok</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">NAME</span><span class="p">):</span>
-            <span class="c"># Applied function followed by implicitly applied function</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;*&#39;</span><span class="p">))</span>
-    <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tokens</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="k">def</span> <span class="nf">_implicit_application</span><span class="p">(</span><span class="n">tokens</span><span class="p">,</span> <span class="n">local_dict</span><span class="p">,</span> <span class="n">global_dict</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Adds parentheses as needed after functions.</span>
-
-<span class="sd">    Also processes functions raised to powers.&quot;&quot;&quot;</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">appendParen</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c"># number of closing parentheses to add</span>
-    <span class="n">skip</span> <span class="o">=</span> <span class="bp">False</span>  <span class="c"># number of tokens to delay before adding a &#39;)&#39; (to</span>
-                  <span class="c"># capture **, ^, etc.)</span>
-    <span class="n">exponent</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="k">for</span> <span class="n">tok</span><span class="p">,</span> <span class="n">nextTok</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">tokens</span><span class="p">,</span> <span class="n">tokens</span><span class="p">[</span><span class="mi">1</span><span class="p">:]):</span>
-        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tok</span><span class="p">)</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">tok</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">NAME</span> <span class="ow">and</span>
-            <span class="n">nextTok</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="n">OP</span> <span class="ow">and</span>
-                <span class="n">nextTok</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="n">ENDMARKER</span><span class="p">):</span>
-            <span class="n">func</span> <span class="o">=</span> <span class="n">global_dict</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">tok</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-            <span class="n">is_Function</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="s">&#39;is_Function&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">is_Function</span> <span class="ow">or</span>
-                <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">collections</span><span class="o">.</span><span class="n">Callable</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="s">&#39;is_Function&#39;</span><span class="p">))</span> <span class="ow">or</span>
-                    <span class="nb">isinstance</span><span class="p">(</span><span class="n">nextTok</span><span class="p">,</span> <span class="n">AppliedFunction</span><span class="p">)):</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;(&#39;</span><span class="p">))</span>
-                <span class="n">appendParen</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">tok</span><span class="p">,</span> <span class="n">AppliedFunction</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">tok</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="c"># This occurs when we had a function raised to a power - it has</span>
-            <span class="c"># no arguments</span>
-            <span class="n">result</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">tok</span><span class="o">.</span><span class="n">function</span>
-            <span class="n">exponent</span> <span class="o">=</span> <span class="n">tok</span><span class="o">.</span><span class="n">exponent</span>
-            <span class="k">if</span> <span class="n">nextTok</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="n">OP</span> <span class="ow">and</span> <span class="n">nextTok</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">!=</span> <span class="s">&#39;(&#39;</span><span class="p">:</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;(&#39;</span><span class="p">))</span>
-                <span class="n">appendParen</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">elif</span> <span class="n">appendParen</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">nextTok</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">OP</span> <span class="ow">and</span> <span class="n">nextTok</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">in</span> <span class="p">(</span><span class="s">&#39;^&#39;</span><span class="p">,</span> <span class="s">&#39;**&#39;</span><span class="p">):</span>
-                <span class="n">skip</span> <span class="o">=</span> <span class="mi">1</span>
-                <span class="k">continue</span>
-            <span class="k">if</span> <span class="n">skip</span><span class="p">:</span>
-                <span class="n">skip</span> <span class="o">-=</span> <span class="mi">1</span>
-                <span class="k">continue</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;)&#39;</span><span class="p">))</span>
-            <span class="n">appendParen</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="n">exponent</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">appendParen</span><span class="p">:</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">exponent</span><span class="p">)</span>
-                <span class="n">exponent</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tokens</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-
-    <span class="k">if</span> <span class="n">appendParen</span><span class="p">:</span>
-        <span class="n">result</span><span class="o">.</span><span class="n">extend</span><span class="p">([(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;)&#39;</span><span class="p">)]</span> <span class="o">*</span> <span class="n">appendParen</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">exponent</span><span class="p">:</span>
-        <span class="n">result</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">exponent</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="k">def</span> <span class="nf">_function_exponents</span><span class="p">(</span><span class="n">tokens</span><span class="p">,</span> <span class="n">local_dict</span><span class="p">,</span> <span class="n">global_dict</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Preprocess functions raised to powers.&quot;&quot;&quot;</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">need_exponent</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="k">for</span> <span class="n">tok</span><span class="p">,</span> <span class="n">nextTok</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">tokens</span><span class="p">,</span> <span class="n">tokens</span><span class="p">[</span><span class="mi">1</span><span class="p">:]):</span>
-        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tok</span><span class="p">)</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">tok</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">NAME</span> <span class="ow">and</span> <span class="n">nextTok</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">OP</span>
-                <span class="ow">and</span> <span class="n">nextTok</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">in</span> <span class="p">(</span><span class="s">&#39;**&#39;</span><span class="p">,</span> <span class="s">&#39;^&#39;</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">global_dict</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">tok</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="s">&#39;is_Function&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-                <span class="n">result</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">AppliedFunction</span><span class="p">(</span><span class="n">tok</span><span class="p">,</span> <span class="p">[])</span>
-                <span class="n">need_exponent</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">elif</span> <span class="n">need_exponent</span><span class="p">:</span>
-            <span class="k">del</span> <span class="n">result</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-            <span class="n">result</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">exponent</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tok</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">tok</span><span class="p">,</span> <span class="n">AppliedFunction</span><span class="p">):</span>
-                <span class="n">need_exponent</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tokens</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="k">def</span> <span class="nf">implicit_multiplication_application</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">local_dict</span><span class="p">,</span> <span class="n">global_dict</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Allows a slightly relaxed syntax.</span>
-
-<span class="sd">    - Parentheses for single-argument method calls are optional.</span>
-
-<span class="sd">    - Multiplication is implicit.</span>
-
-<span class="sd">    - Symbol names can be split (i.e. spaces are not needed between</span>
-<span class="sd">      symbols).</span>
-
-<span class="sd">    - Functions can be exponentiated.</span>
-
-<span class="sd">    Example:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.parsing.sympy_parser import (parse_expr,</span>
-<span class="sd">    ... standard_transformations, implicit_multiplication_application)</span>
-<span class="sd">    &gt;&gt;&gt; parse_expr(&quot;10sin**2 x**2 + 3xyz + tan theta&quot;,</span>
-<span class="sd">    ... transformations=standard_transformations +</span>
-<span class="sd">    ... (implicit_multiplication_application,))</span>
-<span class="sd">    3*x*y*z + 10*sin(x**2)**2 + tan(theta)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># These are interdependent steps, so we don&#39;t expose them separately</span>
-    <span class="k">for</span> <span class="n">step</span> <span class="ow">in</span> <span class="p">(</span><span class="n">_group_parentheses</span><span class="p">,</span>
-                 <span class="n">_apply_functions</span><span class="p">,</span>
-                 <span class="n">_split_symbols</span><span class="p">,</span>
-                 <span class="n">_function_exponents</span><span class="p">,</span>
-                 <span class="n">_implicit_application</span><span class="p">,</span>
-                 <span class="n">_implicit_multiplication</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">step</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">local_dict</span><span class="p">,</span> <span class="n">global_dict</span><span class="p">)</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">_flatten</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="k">def</span> <span class="nf">auto_symbol</span><span class="p">(</span><span class="n">tokens</span><span class="p">,</span> <span class="n">local_dict</span><span class="p">,</span> <span class="n">global_dict</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Inserts calls to ``Symbol`` for undefined variables.&quot;&quot;&quot;</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">toknum</span><span class="p">,</span> <span class="n">tokval</span> <span class="ow">in</span> <span class="n">tokens</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">toknum</span> <span class="o">==</span> <span class="n">NAME</span><span class="p">:</span>
-            <span class="n">name</span> <span class="o">=</span> <span class="n">tokval</span>
-
-            <span class="k">if</span> <span class="p">(</span><span class="n">name</span> <span class="ow">in</span> <span class="p">[</span><span class="s">&#39;True&#39;</span><span class="p">,</span> <span class="s">&#39;False&#39;</span><span class="p">,</span> <span class="s">&#39;None&#39;</span><span class="p">]</span>
-                <span class="ow">or</span> <span class="n">iskeyword</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-                    <span class="ow">or</span> <span class="n">name</span> <span class="ow">in</span> <span class="n">local_dict</span><span class="p">):</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">NAME</span><span class="p">,</span> <span class="n">name</span><span class="p">))</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="n">name</span> <span class="ow">in</span> <span class="n">global_dict</span><span class="p">:</span>
-                <span class="n">obj</span> <span class="o">=</span> <span class="n">global_dict</span><span class="p">[</span><span class="n">name</span><span class="p">]</span>
-
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="p">(</span><span class="n">Basic</span><span class="p">,</span> <span class="nb">type</span><span class="p">))</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="n">collections</span><span class="o">.</span><span class="n">Callable</span><span class="p">):</span>
-                    <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">NAME</span><span class="p">,</span> <span class="n">name</span><span class="p">))</span>
-                    <span class="k">continue</span>
-
-            <span class="n">result</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span>
-                <span class="p">(</span><span class="n">NAME</span><span class="p">,</span> <span class="s">&#39;Symbol&#39;</span><span class="p">),</span>
-                <span class="p">(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;(&#39;</span><span class="p">),</span>
-                <span class="p">(</span><span class="n">NAME</span><span class="p">,</span> <span class="nb">repr</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">name</span><span class="p">))),</span>
-                <span class="p">(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;)&#39;</span><span class="p">),</span>
-            <span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">toknum</span><span class="p">,</span> <span class="n">tokval</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="k">def</span> <span class="nf">factorial_notation</span><span class="p">(</span><span class="n">tokens</span><span class="p">,</span> <span class="n">local_dict</span><span class="p">,</span> <span class="n">global_dict</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Allows standard notation for factorial.&quot;&quot;&quot;</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">prevtoken</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-    <span class="k">for</span> <span class="n">toknum</span><span class="p">,</span> <span class="n">tokval</span> <span class="ow">in</span> <span class="n">tokens</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">toknum</span> <span class="o">==</span> <span class="n">OP</span><span class="p">:</span>
-            <span class="n">op</span> <span class="o">=</span> <span class="n">tokval</span>
-
-            <span class="k">if</span> <span class="n">op</span> <span class="o">==</span> <span class="s">&#39;!!&#39;</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">prevtoken</span> <span class="o">==</span> <span class="s">&#39;!&#39;</span> <span class="ow">or</span> <span class="n">prevtoken</span> <span class="o">==</span> <span class="s">&#39;!!&#39;</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="n">TokenError</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">_add_factorial_tokens</span><span class="p">(</span><span class="s">&#39;factorial2&#39;</span><span class="p">,</span> <span class="n">result</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">op</span> <span class="o">==</span> <span class="s">&#39;!&#39;</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">prevtoken</span> <span class="o">==</span> <span class="s">&#39;!&#39;</span> <span class="ow">or</span> <span class="n">prevtoken</span> <span class="o">==</span> <span class="s">&#39;!!&#39;</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="n">TokenError</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">_add_factorial_tokens</span><span class="p">(</span><span class="s">&#39;factorial&#39;</span><span class="p">,</span> <span class="n">result</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">OP</span><span class="p">,</span> <span class="n">op</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">toknum</span><span class="p">,</span> <span class="n">tokval</span><span class="p">))</span>
-
-        <span class="n">prevtoken</span> <span class="o">=</span> <span class="n">tokval</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="k">def</span> <span class="nf">convert_xor</span><span class="p">(</span><span class="n">tokens</span><span class="p">,</span> <span class="n">local_dict</span><span class="p">,</span> <span class="n">global_dict</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Treats XOR, &#39;^&#39;, as exponentiation, &#39;**&#39;.&quot;&quot;&quot;</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">toknum</span><span class="p">,</span> <span class="n">tokval</span> <span class="ow">in</span> <span class="n">tokens</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">toknum</span> <span class="o">==</span> <span class="n">OP</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">tokval</span> <span class="o">==</span> <span class="s">&#39;^&#39;</span><span class="p">:</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;**&#39;</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">toknum</span><span class="p">,</span> <span class="n">tokval</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">toknum</span><span class="p">,</span> <span class="n">tokval</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="k">def</span> <span class="nf">auto_number</span><span class="p">(</span><span class="n">tokens</span><span class="p">,</span> <span class="n">local_dict</span><span class="p">,</span> <span class="n">global_dict</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Converts numeric literals to use SymPy equivalents.</span>
-
-<span class="sd">    Complex numbers use ``I``; integer literals use ``Integer``, float</span>
-<span class="sd">    literals use ``Float``, and repeating decimals use ``Rational``.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">prevtoken</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-
-    <span class="k">for</span> <span class="n">toknum</span><span class="p">,</span> <span class="n">tokval</span> <span class="ow">in</span> <span class="n">tokens</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">toknum</span> <span class="o">==</span> <span class="n">NUMBER</span><span class="p">:</span>
-            <span class="n">number</span> <span class="o">=</span> <span class="n">tokval</span>
-            <span class="n">postfix</span> <span class="o">=</span> <span class="p">[]</span>
-
-            <span class="k">if</span> <span class="n">number</span><span class="o">.</span><span class="n">endswith</span><span class="p">(</span><span class="s">&#39;j&#39;</span><span class="p">)</span> <span class="ow">or</span> <span class="n">number</span><span class="o">.</span><span class="n">endswith</span><span class="p">(</span><span class="s">&#39;J&#39;</span><span class="p">):</span>
-                <span class="n">number</span> <span class="o">=</span> <span class="n">number</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-                <span class="n">postfix</span> <span class="o">=</span> <span class="p">[(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;*&#39;</span><span class="p">),</span> <span class="p">(</span><span class="n">NAME</span><span class="p">,</span> <span class="s">&#39;I&#39;</span><span class="p">)]</span>
-
-            <span class="k">if</span> <span class="s">&#39;.&#39;</span> <span class="ow">in</span> <span class="n">number</span> <span class="ow">or</span> <span class="p">((</span><span class="s">&#39;e&#39;</span> <span class="ow">in</span> <span class="n">number</span> <span class="ow">or</span> <span class="s">&#39;E&#39;</span> <span class="ow">in</span> <span class="n">number</span><span class="p">)</span> <span class="ow">and</span>
-                    <span class="ow">not</span> <span class="p">(</span><span class="n">number</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39;0x&#39;</span><span class="p">)</span> <span class="ow">or</span> <span class="n">number</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39;0X&#39;</span><span class="p">))):</span>
-                <span class="n">match</span> <span class="o">=</span> <span class="n">_re_repeated</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">number</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">match</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="c"># Clear repeating decimals, e.g. 3.4[31] -&gt; (3 + 4/10 + 31/990)</span>
-                    <span class="n">pre</span><span class="p">,</span> <span class="n">post</span><span class="p">,</span> <span class="n">repetend</span> <span class="o">=</span> <span class="n">match</span><span class="o">.</span><span class="n">groups</span><span class="p">()</span>
-
-                    <span class="n">zeros</span> <span class="o">=</span> <span class="s">&#39;0&#39;</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="n">post</span><span class="p">)</span>
-                    <span class="n">post</span><span class="p">,</span> <span class="n">repetends</span> <span class="o">=</span> <span class="p">[</span><span class="n">w</span><span class="o">.</span><span class="n">lstrip</span><span class="p">(</span><span class="s">&#39;0&#39;</span><span class="p">)</span> <span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="p">[</span><span class="n">post</span><span class="p">,</span> <span class="n">repetend</span><span class="p">]]</span>
-                                                <span class="c"># or else interpreted as octal</span>
-
-                    <span class="n">a</span> <span class="o">=</span> <span class="n">pre</span> <span class="ow">or</span> <span class="s">&#39;0&#39;</span>
-                    <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">post</span> <span class="ow">or</span> <span class="s">&#39;0&#39;</span><span class="p">,</span> <span class="s">&#39;1&#39;</span> <span class="o">+</span> <span class="n">zeros</span>
-                    <span class="n">d</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">repetends</span><span class="p">,</span> <span class="p">(</span><span class="s">&#39;9&#39;</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="n">repetend</span><span class="p">))</span> <span class="o">+</span> <span class="n">zeros</span>
-
-                    <span class="n">seq</span> <span class="o">=</span> <span class="p">[</span>
-                        <span class="p">(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;(&#39;</span><span class="p">),</span>
-                        <span class="p">(</span><span class="n">NAME</span><span class="p">,</span>
-                         <span class="s">&#39;Integer&#39;</span><span class="p">),</span> <span class="p">(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;(&#39;</span><span class="p">),</span> <span class="p">(</span><span class="n">NUMBER</span><span class="p">,</span> <span class="n">a</span><span class="p">),</span> <span class="p">(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;)&#39;</span><span class="p">),</span>
-                        <span class="p">(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;+&#39;</span><span class="p">),</span>
-                        <span class="p">(</span><span class="n">NAME</span><span class="p">,</span> <span class="s">&#39;Rational&#39;</span><span class="p">),</span> <span class="p">(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;(&#39;</span><span class="p">),</span> <span class="p">(</span>
-                            <span class="n">NUMBER</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="p">(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;,&#39;</span><span class="p">),</span> <span class="p">(</span><span class="n">NUMBER</span><span class="p">,</span> <span class="n">c</span><span class="p">),</span> <span class="p">(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;)&#39;</span><span class="p">),</span>
-                        <span class="p">(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;+&#39;</span><span class="p">),</span>
-                        <span class="p">(</span><span class="n">NAME</span><span class="p">,</span> <span class="s">&#39;Rational&#39;</span><span class="p">),</span> <span class="p">(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;(&#39;</span><span class="p">),</span> <span class="p">(</span>
-                            <span class="n">NUMBER</span><span class="p">,</span> <span class="n">d</span><span class="p">),</span> <span class="p">(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;,&#39;</span><span class="p">),</span> <span class="p">(</span><span class="n">NUMBER</span><span class="p">,</span> <span class="n">e</span><span class="p">),</span> <span class="p">(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;)&#39;</span><span class="p">),</span>
-                        <span class="p">(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;)&#39;</span><span class="p">),</span>
-                    <span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">seq</span> <span class="o">=</span> <span class="p">[(</span><span class="n">NAME</span><span class="p">,</span> <span class="s">&#39;Float&#39;</span><span class="p">),</span> <span class="p">(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;(&#39;</span><span class="p">),</span>
-                           <span class="p">(</span><span class="n">NUMBER</span><span class="p">,</span> <span class="nb">repr</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">number</span><span class="p">))),</span> <span class="p">(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;)&#39;</span><span class="p">)]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">seq</span> <span class="o">=</span> <span class="p">[(</span><span class="n">NAME</span><span class="p">,</span> <span class="s">&#39;Integer&#39;</span><span class="p">),</span> <span class="p">(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;(&#39;</span><span class="p">),</span> <span class="p">(</span>
-                    <span class="n">NUMBER</span><span class="p">,</span> <span class="n">number</span><span class="p">),</span> <span class="p">(</span><span class="n">OP</span><span class="p">,</span> <span class="s">&#39;)&#39;</span><span class="p">)]</span>
-
-            <span class="n">result</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">seq</span> <span class="o">+</span> <span class="n">postfix</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">toknum</span><span class="p">,</span> <span class="n">tokval</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="k">def</span> <span class="nf">rationalize</span><span class="p">(</span><span class="n">tokens</span><span class="p">,</span> <span class="n">local_dict</span><span class="p">,</span> <span class="n">global_dict</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Converts floats into ``Rational``s. Run AFTER ``auto_number``.&quot;&quot;&quot;</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">passed_float</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="k">for</span> <span class="n">toknum</span><span class="p">,</span> <span class="n">tokval</span> <span class="ow">in</span> <span class="n">tokens</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">toknum</span> <span class="o">==</span> <span class="n">NAME</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">tokval</span> <span class="o">==</span> <span class="s">&#39;Float&#39;</span><span class="p">:</span>
-                <span class="n">passed_float</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="n">tokval</span> <span class="o">=</span> <span class="s">&#39;Rational&#39;</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">toknum</span><span class="p">,</span> <span class="n">tokval</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="n">passed_float</span> <span class="o">==</span> <span class="bp">True</span> <span class="ow">and</span> <span class="n">toknum</span> <span class="o">==</span> <span class="n">NUMBER</span><span class="p">:</span>
-            <span class="n">passed_float</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">STRING</span><span class="p">,</span> <span class="n">tokval</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">toknum</span><span class="p">,</span> <span class="n">tokval</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-<span class="n">standard_transformations</span> <span class="o">=</span> <span class="p">(</span><span class="n">auto_symbol</span><span class="p">,</span> <span class="n">auto_number</span><span class="p">,</span> <span class="n">factorial_notation</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="parse_expr"><a class="viewcode-back" href="../../../modules/parsing.html#sympy.parsing.sympy_parser.parse_expr">[docs]</a><span class="k">def</span> <span class="nf">parse_expr</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">local_dict</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">transformations</span><span class="o">=</span><span class="n">standard_transformations</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Converts the string ``s`` to a SymPy expression, in ``local_dict``</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.parsing.sympy_parser import parse_expr</span>
-
-<span class="sd">    &gt;&gt;&gt; parse_expr(&quot;1/2&quot;)</span>
-<span class="sd">    1/2</span>
-<span class="sd">    &gt;&gt;&gt; type(_)</span>
-<span class="sd">    &lt;class &#39;sympy.core.numbers.Half&#39;&gt;</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="n">local_dict</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">local_dict</span> <span class="o">=</span> <span class="p">{}</span>
-
-    <span class="n">global_dict</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">exec</span><span class="p">(</span><span class="s">&#39;from sympy import *&#39;</span><span class="p">,</span> <span class="n">global_dict</span><span class="p">)</span>
-
-    <span class="c"># keep autosimplification from joining Integer or</span>
-    <span class="c"># minus sign into a Mul; this modification doesn&#39;t</span>
-    <span class="c"># prevent the 2-arg Mul from becoming an Add, however.</span>
-    <span class="n">hit</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="k">if</span> <span class="s">&#39;(&#39;</span> <span class="ow">in</span> <span class="n">s</span><span class="p">:</span>
-        <span class="n">kern</span> <span class="o">=</span> <span class="s">&#39;_kern&#39;</span>
-        <span class="k">while</span> <span class="n">kern</span> <span class="ow">in</span> <span class="n">s</span><span class="p">:</span>
-            <span class="n">kern</span> <span class="o">+=</span> <span class="s">&quot;_&quot;</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">re</span><span class="o">.</span><span class="n">sub</span><span class="p">(</span><span class="s">r&#39;(\d *\*|-) *\(&#39;</span><span class="p">,</span> <span class="s">r&#39;\1</span><span class="si">%s</span><span class="s">*(&#39;</span> <span class="o">%</span> <span class="n">kern</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span>
-        <span class="n">hit</span> <span class="o">=</span> <span class="n">kern</span> <span class="ow">in</span> <span class="n">s</span>
-
-    <span class="n">tokens</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">input_code</span> <span class="o">=</span> <span class="n">StringIO</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">strip</span><span class="p">())</span>
-    <span class="k">for</span> <span class="n">toknum</span><span class="p">,</span> <span class="n">tokval</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">generate_tokens</span><span class="p">(</span><span class="n">input_code</span><span class="o">.</span><span class="n">readline</span><span class="p">):</span>
-        <span class="n">tokens</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">toknum</span><span class="p">,</span> <span class="n">tokval</span><span class="p">))</span>
-
-    <span class="k">for</span> <span class="n">transform</span> <span class="ow">in</span> <span class="n">transformations</span><span class="p">:</span>
-        <span class="n">tokens</span> <span class="o">=</span> <span class="n">transform</span><span class="p">(</span><span class="n">tokens</span><span class="p">,</span> <span class="n">local_dict</span><span class="p">,</span> <span class="n">global_dict</span><span class="p">)</span>
-
-    <span class="n">code</span> <span class="o">=</span> <span class="n">untokenize</span><span class="p">(</span><span class="n">tokens</span><span class="p">)</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="nb">eval</span><span class="p">(</span>
-        <span class="n">code</span><span class="p">,</span> <span class="n">global_dict</span><span class="p">,</span> <span class="n">local_dict</span><span class="p">)</span>  <span class="c"># take local objects in preference</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">hit</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span>
-    <span class="n">rep</span> <span class="o">=</span> <span class="p">{</span><span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="n">kern</span><span class="p">):</span> <span class="mi">1</span><span class="p">}</span>
-
-    <span class="k">def</span> <span class="nf">_clear</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&#39;xreplace&#39;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="n">rep</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="p">(</span><span class="nb">list</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">,</span> <span class="nb">set</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="nb">type</span><span class="p">(</span><span class="n">expr</span><span class="p">)([</span><span class="n">_clear</span><span class="p">(</span><span class="n">e</span><span class="p">)</span> <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">expr</span><span class="p">])</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&#39;subs&#39;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">rep</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">expr</span>
-    <span class="k">return</span> <span class="n">_clear</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span></div>
-</pre></div>
-
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@@ -1,561 +0,0 @@
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-            
-  <h1>Source code for sympy.parsing.sympy_tokenize</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Tokenization help for Python programs.</span>
-
-<span class="sd">generate_tokens(readline) is a generator that breaks a stream of</span>
-<span class="sd">text into Python tokens.  It accepts a readline-like method which is called</span>
-<span class="sd">repeatedly to get the next line of input (or &quot;&quot; for EOF).  It generates</span>
-<span class="sd">5-tuples with these members:</span>
-
-<span class="sd">    the token type (see token.py)</span>
-<span class="sd">    the token (a string)</span>
-<span class="sd">    the starting (row, column) indices of the token (a 2-tuple of ints)</span>
-<span class="sd">    the ending (row, column) indices of the token (a 2-tuple of ints)</span>
-<span class="sd">    the original line (string)</span>
-
-<span class="sd">It is designed to match the working of the Python tokenizer exactly, except</span>
-<span class="sd">that it produces COMMENT tokens for comments and gives type OP for all</span>
-<span class="sd">operators</span>
-
-<span class="sd">Older entry points</span>
-<span class="sd">    tokenize_loop(readline, tokeneater)</span>
-<span class="sd">    tokenize(readline, tokeneater=printtoken)</span>
-<span class="sd">are the same, except instead of generating tokens, tokeneater is a callback</span>
-<span class="sd">function to which the 5 fields described above are passed as 5 arguments,</span>
-<span class="sd">each time a new token is found.&quot;&quot;&quot;</span>
-
-<span class="n">__author__</span> <span class="o">=</span> <span class="s">&#39;Ka-Ping Yee &lt;ping@lfw.org&gt;&#39;</span>
-<span class="n">__credits__</span> <span class="o">=</span> \
-    <span class="s">&#39;GvR, ESR, Tim Peters, Thomas Wouters, Fred Drake, Skip Montanaro, Raymond Hettinger&#39;</span>
-
-<span class="kn">import</span> <span class="nn">string</span>
-<span class="kn">import</span> <span class="nn">re</span>
-<span class="kn">from</span> <span class="nn">token</span> <span class="kn">import</span> <span class="o">*</span>
-
-<span class="kn">import</span> <span class="nn">token</span>
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">dir</span><span class="p">(</span><span class="n">token</span><span class="p">)</span> <span class="k">if</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="s">&#39;_&#39;</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="s">&quot;COMMENT&quot;</span><span class="p">,</span> <span class="s">&quot;tokenize&quot;</span><span class="p">,</span>
-           <span class="s">&quot;generate_tokens&quot;</span><span class="p">,</span> <span class="s">&quot;NL&quot;</span><span class="p">,</span> <span class="s">&quot;untokenize&quot;</span><span class="p">]</span>
-<span class="k">del</span> <span class="n">token</span>
-
-<span class="n">COMMENT</span> <span class="o">=</span> <span class="n">N_TOKENS</span>
-<span class="n">tok_name</span><span class="p">[</span><span class="n">COMMENT</span><span class="p">]</span> <span class="o">=</span> <span class="s">&#39;COMMENT&#39;</span>
-<span class="n">NL</span> <span class="o">=</span> <span class="n">N_TOKENS</span> <span class="o">+</span> <span class="mi">1</span>
-<span class="n">tok_name</span><span class="p">[</span><span class="n">NL</span><span class="p">]</span> <span class="o">=</span> <span class="s">&#39;NL&#39;</span>
-<span class="n">N_TOKENS</span> <span class="o">+=</span> <span class="mi">2</span>
-
-
-<div class="viewcode-block" id="group"><a class="viewcode-back" href="../../../modules/parsing.html#sympy.parsing.sympy_tokenize.group">[docs]</a><span class="k">def</span> <span class="nf">group</span><span class="p">(</span><span class="o">*</span><span class="n">choices</span><span class="p">):</span>
-    <span class="k">return</span> <span class="s">&#39;(&#39;</span> <span class="o">+</span> <span class="s">&#39;|&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">choices</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39;)&#39;</span>
-
-</div>
-<div class="viewcode-block" id="any"><a class="viewcode-back" href="../../../modules/parsing.html#sympy.parsing.sympy_tokenize.any">[docs]</a><span class="k">def</span> <span class="nf">any</span><span class="p">(</span><span class="o">*</span><span class="n">choices</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">group</span><span class="p">(</span><span class="o">*</span><span class="n">choices</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39;*&#39;</span>
-
-</div>
-<div class="viewcode-block" id="maybe"><a class="viewcode-back" href="../../../modules/parsing.html#sympy.parsing.sympy_tokenize.maybe">[docs]</a><span class="k">def</span> <span class="nf">maybe</span><span class="p">(</span><span class="o">*</span><span class="n">choices</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">group</span><span class="p">(</span><span class="o">*</span><span class="n">choices</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39;?&#39;</span>
-</div>
-<span class="n">Whitespace</span> <span class="o">=</span> <span class="s">r&#39;[ \f\t]*&#39;</span>
-<span class="n">Comment</span> <span class="o">=</span> <span class="s">r&#39;#[^\r\n]*&#39;</span>
-<span class="n">Ignore</span> <span class="o">=</span> <span class="n">Whitespace</span> <span class="o">+</span> <span class="nb">any</span><span class="p">(</span><span class="s">r&#39;</span><span class="se">\\</span><span class="s">\r?\n&#39;</span> <span class="o">+</span> <span class="n">Whitespace</span><span class="p">)</span> <span class="o">+</span> <span class="n">maybe</span><span class="p">(</span><span class="n">Comment</span><span class="p">)</span>
-<span class="n">Name</span> <span class="o">=</span> <span class="s">r&#39;[a-zA-Z_]\w*&#39;</span>
-
-<span class="n">Hexnumber</span> <span class="o">=</span> <span class="s">r&#39;0[xX][\da-fA-F]+[lL]?&#39;</span>
-<span class="n">Octnumber</span> <span class="o">=</span> <span class="s">r&#39;(0[oO][0-7]+)|(0[0-7]*)[lL]?&#39;</span>
-<span class="n">Binnumber</span> <span class="o">=</span> <span class="s">r&#39;0[bB][01]+[lL]?&#39;</span>
-<span class="n">Decnumber</span> <span class="o">=</span> <span class="s">r&#39;[1-9]\d*[lL]?&#39;</span>
-<span class="n">Intnumber</span> <span class="o">=</span> <span class="n">group</span><span class="p">(</span><span class="n">Hexnumber</span><span class="p">,</span> <span class="n">Binnumber</span><span class="p">,</span> <span class="n">Octnumber</span><span class="p">,</span> <span class="n">Decnumber</span><span class="p">)</span>
-<span class="n">Exponent</span> <span class="o">=</span> <span class="s">r&#39;[eE][-+]?\d+&#39;</span>
-<span class="n">Pointfloat</span> <span class="o">=</span> <span class="n">group</span><span class="p">(</span><span class="s">r&#39;\d+\.\d*&#39;</span><span class="p">,</span> <span class="s">r&#39;\.\d+&#39;</span><span class="p">)</span> <span class="o">+</span> <span class="n">maybe</span><span class="p">(</span><span class="n">Exponent</span><span class="p">)</span>
-<span class="n">Repeatedfloat</span> <span class="o">=</span> <span class="s">r&#39;\d*\.\d*\[\d+\]&#39;</span>
-<span class="n">Expfloat</span> <span class="o">=</span> <span class="s">r&#39;\d+&#39;</span> <span class="o">+</span> <span class="n">Exponent</span>
-<span class="n">Floatnumber</span> <span class="o">=</span> <span class="n">group</span><span class="p">(</span><span class="n">Repeatedfloat</span><span class="p">,</span> <span class="n">Pointfloat</span><span class="p">,</span> <span class="n">Expfloat</span><span class="p">)</span>
-<span class="n">Imagnumber</span> <span class="o">=</span> <span class="n">group</span><span class="p">(</span><span class="s">r&#39;\d+[jJ]&#39;</span><span class="p">,</span> <span class="n">Floatnumber</span> <span class="o">+</span> <span class="s">r&#39;[jJ]&#39;</span><span class="p">)</span>
-<span class="n">Number</span> <span class="o">=</span> <span class="n">group</span><span class="p">(</span><span class="n">Imagnumber</span><span class="p">,</span> <span class="n">Floatnumber</span><span class="p">,</span> <span class="n">Intnumber</span><span class="p">)</span>
-
-<span class="c"># Tail end of &#39; string.</span>
-<span class="n">Single</span> <span class="o">=</span> <span class="s">r&quot;[^&#39;</span><span class="se">\\</span><span class="s">]*(?:</span><span class="se">\\</span><span class="s">.[^&#39;</span><span class="se">\\</span><span class="s">]*)*&#39;&quot;</span>
-<span class="c"># Tail end of &quot; string.</span>
-<span class="n">Double</span> <span class="o">=</span> <span class="s">r&#39;[^&quot;</span><span class="se">\\</span><span class="s">]*(?:</span><span class="se">\\</span><span class="s">.[^&quot;</span><span class="se">\\</span><span class="s">]*)*&quot;&#39;</span>
-<span class="c"># Tail end of &#39;&#39;&#39; string.</span>
-<span class="n">Single3</span> <span class="o">=</span> <span class="s">r&quot;[^&#39;</span><span class="se">\\</span><span class="s">]*(?:(?:</span><span class="se">\\</span><span class="s">.|&#39;(?!&#39;&#39;))[^&#39;</span><span class="se">\\</span><span class="s">]*)*&#39;&#39;&#39;&quot;</span>
-<span class="c"># Tail end of &quot;&quot;&quot; string.</span>
-<span class="n">Double3</span> <span class="o">=</span> <span class="s">r&#39;[^&quot;</span><span class="se">\\</span><span class="s">]*(?:(?:</span><span class="se">\\</span><span class="s">.|&quot;(?!&quot;&quot;))[^&quot;</span><span class="se">\\</span><span class="s">]*)*&quot;&quot;&quot;&#39;</span>
-<span class="n">Triple</span> <span class="o">=</span> <span class="n">group</span><span class="p">(</span><span class="s">&quot;[uU]?[rR]?&#39;&#39;&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;[uU]?[rR]?&quot;&quot;&quot;&#39;</span><span class="p">)</span>
-<span class="c"># Single-line &#39; or &quot; string.</span>
-<span class="n">String</span> <span class="o">=</span> <span class="n">group</span><span class="p">(</span><span class="s">r&quot;[uU]?[rR]?&#39;[^\n&#39;</span><span class="se">\\</span><span class="s">]*(?:</span><span class="se">\\</span><span class="s">.[^\n&#39;</span><span class="se">\\</span><span class="s">]*)*&#39;&quot;</span><span class="p">,</span>
-               <span class="s">r&#39;[uU]?[rR]?&quot;[^\n&quot;</span><span class="se">\\</span><span class="s">]*(?:</span><span class="se">\\</span><span class="s">.[^\n&quot;</span><span class="se">\\</span><span class="s">]*)*&quot;&#39;</span><span class="p">)</span>
-
-<span class="c"># Because of leftmost-then-longest match semantics, be sure to put the</span>
-<span class="c"># longest operators first (e.g., if = came before ==, == would get</span>
-<span class="c"># recognized as two instances of =).</span>
-<span class="n">Operator</span> <span class="o">=</span> <span class="n">group</span><span class="p">(</span><span class="s">r&quot;\*\*=?&quot;</span><span class="p">,</span> <span class="s">r&quot;&gt;&gt;=?&quot;</span><span class="p">,</span> <span class="s">r&quot;&lt;&lt;=?&quot;</span><span class="p">,</span> <span class="s">r&quot;&lt;&gt;&quot;</span><span class="p">,</span> <span class="s">r&quot;!=&quot;</span><span class="p">,</span>
-                 <span class="s">r&quot;//=?&quot;</span><span class="p">,</span>
-                 <span class="s">r&quot;[+\-*/%&amp;|^=&lt;&gt;]=?&quot;</span><span class="p">,</span>
-                 <span class="s">r&quot;~&quot;</span><span class="p">)</span>
-
-<span class="n">Bracket</span> <span class="o">=</span> <span class="s">&#39;[][(){}]&#39;</span>
-<span class="n">Special</span> <span class="o">=</span> <span class="n">group</span><span class="p">(</span><span class="s">r&#39;\r?\n&#39;</span><span class="p">,</span> <span class="s">r&#39;[:;.,`@]&#39;</span><span class="p">,</span> <span class="s">r&#39;\!\!&#39;</span><span class="p">,</span> <span class="s">r&#39;\!&#39;</span><span class="p">)</span>
-<span class="n">Funny</span> <span class="o">=</span> <span class="n">group</span><span class="p">(</span><span class="n">Operator</span><span class="p">,</span> <span class="n">Bracket</span><span class="p">,</span> <span class="n">Special</span><span class="p">)</span>
-
-<span class="n">PlainToken</span> <span class="o">=</span> <span class="n">group</span><span class="p">(</span><span class="n">Number</span><span class="p">,</span> <span class="n">Funny</span><span class="p">,</span> <span class="n">String</span><span class="p">,</span> <span class="n">Name</span><span class="p">)</span>
-<span class="n">Token</span> <span class="o">=</span> <span class="n">Ignore</span> <span class="o">+</span> <span class="n">PlainToken</span>
-
-<span class="c"># First (or only) line of &#39; or &quot; string.</span>
-<span class="n">ContStr</span> <span class="o">=</span> <span class="n">group</span><span class="p">(</span><span class="s">r&quot;[uU]?[rR]?&#39;[^\n&#39;</span><span class="se">\\</span><span class="s">]*(?:</span><span class="se">\\</span><span class="s">.[^\n&#39;</span><span class="se">\\</span><span class="s">]*)*&quot;</span> <span class="o">+</span>
-                <span class="n">group</span><span class="p">(</span><span class="s">&quot;&#39;&quot;</span><span class="p">,</span> <span class="s">r&#39;</span><span class="se">\\</span><span class="s">\r?\n&#39;</span><span class="p">),</span>
-                <span class="s">r&#39;[uU]?[rR]?&quot;[^\n&quot;</span><span class="se">\\</span><span class="s">]*(?:</span><span class="se">\\</span><span class="s">.[^\n&quot;</span><span class="se">\\</span><span class="s">]*)*&#39;</span> <span class="o">+</span>
-                <span class="n">group</span><span class="p">(</span><span class="s">&#39;&quot;&#39;</span><span class="p">,</span> <span class="s">r&#39;</span><span class="se">\\</span><span class="s">\r?\n&#39;</span><span class="p">))</span>
-<span class="n">PseudoExtras</span> <span class="o">=</span> <span class="n">group</span><span class="p">(</span><span class="s">r&#39;</span><span class="se">\\</span><span class="s">\r?\n&#39;</span><span class="p">,</span> <span class="n">Comment</span><span class="p">,</span> <span class="n">Triple</span><span class="p">)</span>
-<span class="n">PseudoToken</span> <span class="o">=</span> <span class="n">Whitespace</span> <span class="o">+</span> <span class="n">group</span><span class="p">(</span><span class="n">PseudoExtras</span><span class="p">,</span> <span class="n">Number</span><span class="p">,</span> <span class="n">Funny</span><span class="p">,</span> <span class="n">ContStr</span><span class="p">,</span> <span class="n">Name</span><span class="p">)</span>
-
-<span class="n">tokenprog</span><span class="p">,</span> <span class="n">pseudoprog</span><span class="p">,</span> <span class="n">single3prog</span><span class="p">,</span> <span class="n">double3prog</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span>
-    <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">,</span> <span class="p">(</span><span class="n">Token</span><span class="p">,</span> <span class="n">PseudoToken</span><span class="p">,</span> <span class="n">Single3</span><span class="p">,</span> <span class="n">Double3</span><span class="p">)))</span>
-<span class="n">endprogs</span> <span class="o">=</span> <span class="p">{</span><span class="s">&quot;&#39;&quot;</span><span class="p">:</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="n">Single</span><span class="p">),</span> <span class="s">&#39;&quot;&#39;</span><span class="p">:</span> <span class="n">re</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="n">Double</span><span class="p">),</span>
-            <span class="s">&quot;&#39;&#39;&#39;&quot;</span><span class="p">:</span> <span class="n">single3prog</span><span class="p">,</span> <span class="s">&#39;&quot;&quot;&quot;&#39;</span><span class="p">:</span> <span class="n">double3prog</span><span class="p">,</span>
-            <span class="s">&quot;r&#39;&#39;&#39;&quot;</span><span class="p">:</span> <span class="n">single3prog</span><span class="p">,</span> <span class="s">&#39;r&quot;&quot;&quot;&#39;</span><span class="p">:</span> <span class="n">double3prog</span><span class="p">,</span>
-            <span class="s">&quot;u&#39;&#39;&#39;&quot;</span><span class="p">:</span> <span class="n">single3prog</span><span class="p">,</span> <span class="s">&#39;u&quot;&quot;&quot;&#39;</span><span class="p">:</span> <span class="n">double3prog</span><span class="p">,</span>
-            <span class="s">&quot;ur&#39;&#39;&#39;&quot;</span><span class="p">:</span> <span class="n">single3prog</span><span class="p">,</span> <span class="s">&#39;ur&quot;&quot;&quot;&#39;</span><span class="p">:</span> <span class="n">double3prog</span><span class="p">,</span>
-            <span class="s">&quot;R&#39;&#39;&#39;&quot;</span><span class="p">:</span> <span class="n">single3prog</span><span class="p">,</span> <span class="s">&#39;R&quot;&quot;&quot;&#39;</span><span class="p">:</span> <span class="n">double3prog</span><span class="p">,</span>
-            <span class="s">&quot;U&#39;&#39;&#39;&quot;</span><span class="p">:</span> <span class="n">single3prog</span><span class="p">,</span> <span class="s">&#39;U&quot;&quot;&quot;&#39;</span><span class="p">:</span> <span class="n">double3prog</span><span class="p">,</span>
-            <span class="s">&quot;uR&#39;&#39;&#39;&quot;</span><span class="p">:</span> <span class="n">single3prog</span><span class="p">,</span> <span class="s">&#39;uR&quot;&quot;&quot;&#39;</span><span class="p">:</span> <span class="n">double3prog</span><span class="p">,</span>
-            <span class="s">&quot;Ur&#39;&#39;&#39;&quot;</span><span class="p">:</span> <span class="n">single3prog</span><span class="p">,</span> <span class="s">&#39;Ur&quot;&quot;&quot;&#39;</span><span class="p">:</span> <span class="n">double3prog</span><span class="p">,</span>
-            <span class="s">&quot;UR&#39;&#39;&#39;&quot;</span><span class="p">:</span> <span class="n">single3prog</span><span class="p">,</span> <span class="s">&#39;UR&quot;&quot;&quot;&#39;</span><span class="p">:</span> <span class="n">double3prog</span><span class="p">,</span>
-            <span class="s">&quot;b&#39;&#39;&#39;&quot;</span><span class="p">:</span> <span class="n">single3prog</span><span class="p">,</span> <span class="s">&#39;b&quot;&quot;&quot;&#39;</span><span class="p">:</span> <span class="n">double3prog</span><span class="p">,</span>
-            <span class="s">&quot;br&#39;&#39;&#39;&quot;</span><span class="p">:</span> <span class="n">single3prog</span><span class="p">,</span> <span class="s">&#39;br&quot;&quot;&quot;&#39;</span><span class="p">:</span> <span class="n">double3prog</span><span class="p">,</span>
-            <span class="s">&quot;B&#39;&#39;&#39;&quot;</span><span class="p">:</span> <span class="n">single3prog</span><span class="p">,</span> <span class="s">&#39;B&quot;&quot;&quot;&#39;</span><span class="p">:</span> <span class="n">double3prog</span><span class="p">,</span>
-            <span class="s">&quot;bR&#39;&#39;&#39;&quot;</span><span class="p">:</span> <span class="n">single3prog</span><span class="p">,</span> <span class="s">&#39;bR&quot;&quot;&quot;&#39;</span><span class="p">:</span> <span class="n">double3prog</span><span class="p">,</span>
-            <span class="s">&quot;Br&#39;&#39;&#39;&quot;</span><span class="p">:</span> <span class="n">single3prog</span><span class="p">,</span> <span class="s">&#39;Br&quot;&quot;&quot;&#39;</span><span class="p">:</span> <span class="n">double3prog</span><span class="p">,</span>
-            <span class="s">&quot;BR&#39;&#39;&#39;&quot;</span><span class="p">:</span> <span class="n">single3prog</span><span class="p">,</span> <span class="s">&#39;BR&quot;&quot;&quot;&#39;</span><span class="p">:</span> <span class="n">double3prog</span><span class="p">,</span>
-            <span class="s">&#39;r&#39;</span><span class="p">:</span> <span class="bp">None</span><span class="p">,</span> <span class="s">&#39;R&#39;</span><span class="p">:</span> <span class="bp">None</span><span class="p">,</span> <span class="s">&#39;u&#39;</span><span class="p">:</span> <span class="bp">None</span><span class="p">,</span> <span class="s">&#39;U&#39;</span><span class="p">:</span> <span class="bp">None</span><span class="p">,</span>
-            <span class="s">&#39;b&#39;</span><span class="p">:</span> <span class="bp">None</span><span class="p">,</span> <span class="s">&#39;B&#39;</span><span class="p">:</span> <span class="bp">None</span><span class="p">}</span>
-
-<span class="n">triple_quoted</span> <span class="o">=</span> <span class="p">{}</span>
-<span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="p">(</span><span class="s">&quot;&#39;&#39;&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;&quot;&quot;&quot;&#39;</span><span class="p">,</span>
-          <span class="s">&quot;r&#39;&#39;&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;r&quot;&quot;&quot;&#39;</span><span class="p">,</span> <span class="s">&quot;R&#39;&#39;&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;R&quot;&quot;&quot;&#39;</span><span class="p">,</span>
-          <span class="s">&quot;u&#39;&#39;&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;u&quot;&quot;&quot;&#39;</span><span class="p">,</span> <span class="s">&quot;U&#39;&#39;&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;U&quot;&quot;&quot;&#39;</span><span class="p">,</span>
-          <span class="s">&quot;ur&#39;&#39;&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;ur&quot;&quot;&quot;&#39;</span><span class="p">,</span> <span class="s">&quot;Ur&#39;&#39;&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;Ur&quot;&quot;&quot;&#39;</span><span class="p">,</span>
-          <span class="s">&quot;uR&#39;&#39;&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;uR&quot;&quot;&quot;&#39;</span><span class="p">,</span> <span class="s">&quot;UR&#39;&#39;&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;UR&quot;&quot;&quot;&#39;</span><span class="p">,</span>
-          <span class="s">&quot;b&#39;&#39;&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;b&quot;&quot;&quot;&#39;</span><span class="p">,</span> <span class="s">&quot;B&#39;&#39;&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;B&quot;&quot;&quot;&#39;</span><span class="p">,</span>
-          <span class="s">&quot;br&#39;&#39;&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;br&quot;&quot;&quot;&#39;</span><span class="p">,</span> <span class="s">&quot;Br&#39;&#39;&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;Br&quot;&quot;&quot;&#39;</span><span class="p">,</span>
-          <span class="s">&quot;bR&#39;&#39;&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;bR&quot;&quot;&quot;&#39;</span><span class="p">,</span> <span class="s">&quot;BR&#39;&#39;&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;BR&quot;&quot;&quot;&#39;</span><span class="p">):</span>
-    <span class="n">triple_quoted</span><span class="p">[</span><span class="n">t</span><span class="p">]</span> <span class="o">=</span> <span class="n">t</span>
-<span class="n">single_quoted</span> <span class="o">=</span> <span class="p">{}</span>
-<span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="p">(</span><span class="s">&quot;&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;&quot;&#39;</span><span class="p">,</span>
-          <span class="s">&quot;r&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;r&quot;&#39;</span><span class="p">,</span> <span class="s">&quot;R&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;R&quot;&#39;</span><span class="p">,</span>
-          <span class="s">&quot;u&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;u&quot;&#39;</span><span class="p">,</span> <span class="s">&quot;U&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;U&quot;&#39;</span><span class="p">,</span>
-          <span class="s">&quot;ur&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;ur&quot;&#39;</span><span class="p">,</span> <span class="s">&quot;Ur&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;Ur&quot;&#39;</span><span class="p">,</span>
-          <span class="s">&quot;uR&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;uR&quot;&#39;</span><span class="p">,</span> <span class="s">&quot;UR&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;UR&quot;&#39;</span><span class="p">,</span>
-          <span class="s">&quot;b&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;b&quot;&#39;</span><span class="p">,</span> <span class="s">&quot;B&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;B&quot;&#39;</span><span class="p">,</span>
-          <span class="s">&quot;br&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;br&quot;&#39;</span><span class="p">,</span> <span class="s">&quot;Br&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;Br&quot;&#39;</span><span class="p">,</span>
-          <span class="s">&quot;bR&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;bR&quot;&#39;</span><span class="p">,</span> <span class="s">&quot;BR&#39;&quot;</span><span class="p">,</span> <span class="s">&#39;BR&quot;&#39;</span> <span class="p">):</span>
-    <span class="n">single_quoted</span><span class="p">[</span><span class="n">t</span><span class="p">]</span> <span class="o">=</span> <span class="n">t</span>
-
-<span class="n">tabsize</span> <span class="o">=</span> <span class="mi">8</span>
-
-
-<div class="viewcode-block" id="TokenError"><a class="viewcode-back" href="../../../modules/parsing.html#sympy.parsing.sympy_tokenize.TokenError">[docs]</a><span class="k">class</span> <span class="nc">TokenError</span><span class="p">(</span><span class="ne">Exception</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="StopTokenizing"><a class="viewcode-back" href="../../../modules/parsing.html#sympy.parsing.sympy_tokenize.StopTokenizing">[docs]</a><span class="k">class</span> <span class="nc">StopTokenizing</span><span class="p">(</span><span class="ne">Exception</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="printtoken"><a class="viewcode-back" href="../../../modules/parsing.html#sympy.parsing.sympy_tokenize.printtoken">[docs]</a><span class="k">def</span> <span class="nf">printtoken</span><span class="p">(</span><span class="nb">type</span><span class="p">,</span> <span class="n">token</span><span class="p">,</span> <span class="n">srow_scol</span><span class="p">,</span> <span class="n">erow_ecol</span><span class="p">,</span> <span class="n">line</span><span class="p">):</span>  <span class="c"># for testing</span>
-    <span class="n">srow</span><span class="p">,</span> <span class="n">scol</span> <span class="o">=</span> <span class="n">srow_scol</span>
-    <span class="n">erow</span><span class="p">,</span> <span class="n">ecol</span> <span class="o">=</span> <span class="n">erow_ecol</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%d</span><span class="s">,</span><span class="si">%d</span><span class="s">-</span><span class="si">%d</span><span class="s">,</span><span class="si">%d</span><span class="s">:</span><span class="se">\t</span><span class="si">%s</span><span class="se">\t</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> \
-        <span class="p">(</span><span class="n">srow</span><span class="p">,</span> <span class="n">scol</span><span class="p">,</span> <span class="n">erow</span><span class="p">,</span> <span class="n">ecol</span><span class="p">,</span> <span class="n">tok_name</span><span class="p">[</span><span class="nb">type</span><span class="p">],</span> <span class="nb">repr</span><span class="p">(</span><span class="n">token</span><span class="p">)))</span>
-
-</div>
-<div class="viewcode-block" id="tokenize"><a class="viewcode-back" href="../../../modules/parsing.html#sympy.parsing.sympy_tokenize.tokenize">[docs]</a><span class="k">def</span> <span class="nf">tokenize</span><span class="p">(</span><span class="n">readline</span><span class="p">,</span> <span class="n">tokeneater</span><span class="o">=</span><span class="n">printtoken</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The tokenize() function accepts two parameters: one representing the</span>
-<span class="sd">    input stream, and one providing an output mechanism for tokenize().</span>
-
-<span class="sd">    The first parameter, readline, must be a callable object which provides</span>
-<span class="sd">    the same interface as the readline() method of built-in file objects.</span>
-<span class="sd">    Each call to the function should return one line of input as a string.</span>
-
-<span class="sd">    The second parameter, tokeneater, must also be a callable object. It is</span>
-<span class="sd">    called once for each token, with five arguments, corresponding to the</span>
-<span class="sd">    tuples generated by generate_tokens().</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">tokenize_loop</span><span class="p">(</span><span class="n">readline</span><span class="p">,</span> <span class="n">tokeneater</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">StopTokenizing</span><span class="p">:</span>
-        <span class="k">pass</span>
-
-<span class="c"># backwards compatible interface</span>
-
-</div>
-<span class="k">def</span> <span class="nf">tokenize_loop</span><span class="p">(</span><span class="n">readline</span><span class="p">,</span> <span class="n">tokeneater</span><span class="p">):</span>
-    <span class="k">for</span> <span class="n">token_info</span> <span class="ow">in</span> <span class="n">generate_tokens</span><span class="p">(</span><span class="n">readline</span><span class="p">):</span>
-        <span class="n">tokeneater</span><span class="p">(</span><span class="o">*</span><span class="n">token_info</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">Untokenizer</span><span class="p">:</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">tokens</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">prev_row</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">prev_col</span> <span class="o">=</span> <span class="mi">0</span>
-
-    <span class="k">def</span> <span class="nf">add_whitespace</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">start</span><span class="p">):</span>
-        <span class="n">row</span><span class="p">,</span> <span class="n">col</span> <span class="o">=</span> <span class="n">start</span>
-        <span class="k">assert</span> <span class="n">row</span> <span class="o">&lt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">prev_row</span>
-        <span class="n">col_offset</span> <span class="o">=</span> <span class="n">col</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">prev_col</span>
-        <span class="k">if</span> <span class="n">col_offset</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">tokens</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot; &quot;</span> <span class="o">*</span> <span class="n">col_offset</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">untokenize</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">iterable</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">iterable</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">compat</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">iterable</span><span class="p">)</span>
-                <span class="k">break</span>
-            <span class="n">tok_type</span><span class="p">,</span> <span class="n">token</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">end</span><span class="p">,</span> <span class="n">line</span> <span class="o">=</span> <span class="n">t</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">add_whitespace</span><span class="p">(</span><span class="n">start</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">tokens</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">token</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">prev_row</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">prev_col</span> <span class="o">=</span> <span class="n">end</span>
-            <span class="k">if</span> <span class="n">tok_type</span> <span class="ow">in</span> <span class="p">(</span><span class="n">NEWLINE</span><span class="p">,</span> <span class="n">NL</span><span class="p">):</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">prev_row</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">prev_col</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">return</span> <span class="s">&quot;&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">tokens</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">compat</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">token</span><span class="p">,</span> <span class="n">iterable</span><span class="p">):</span>
-        <span class="n">startline</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="n">indents</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">toks_append</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">tokens</span><span class="o">.</span><span class="n">append</span>
-        <span class="n">toknum</span><span class="p">,</span> <span class="n">tokval</span> <span class="o">=</span> <span class="n">token</span>
-        <span class="k">if</span> <span class="n">toknum</span> <span class="ow">in</span> <span class="p">(</span><span class="n">NAME</span><span class="p">,</span> <span class="n">NUMBER</span><span class="p">):</span>
-            <span class="n">tokval</span> <span class="o">+=</span> <span class="s">&#39; &#39;</span>
-        <span class="k">if</span> <span class="n">toknum</span> <span class="ow">in</span> <span class="p">(</span><span class="n">NEWLINE</span><span class="p">,</span> <span class="n">NL</span><span class="p">):</span>
-            <span class="n">startline</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="n">prevstring</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">tok</span> <span class="ow">in</span> <span class="n">iterable</span><span class="p">:</span>
-            <span class="n">toknum</span><span class="p">,</span> <span class="n">tokval</span> <span class="o">=</span> <span class="n">tok</span><span class="p">[:</span><span class="mi">2</span><span class="p">]</span>
-
-            <span class="k">if</span> <span class="n">toknum</span> <span class="ow">in</span> <span class="p">(</span><span class="n">NAME</span><span class="p">,</span> <span class="n">NUMBER</span><span class="p">):</span>
-                <span class="n">tokval</span> <span class="o">+=</span> <span class="s">&#39; &#39;</span>
-
-            <span class="c"># Insert a space between two consecutive strings</span>
-            <span class="k">if</span> <span class="n">toknum</span> <span class="o">==</span> <span class="n">STRING</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">prevstring</span><span class="p">:</span>
-                    <span class="n">tokval</span> <span class="o">=</span> <span class="s">&#39; &#39;</span> <span class="o">+</span> <span class="n">tokval</span>
-                <span class="n">prevstring</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">prevstring</span> <span class="o">=</span> <span class="bp">False</span>
-
-            <span class="k">if</span> <span class="n">toknum</span> <span class="o">==</span> <span class="n">INDENT</span><span class="p">:</span>
-                <span class="n">indents</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tokval</span><span class="p">)</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="n">toknum</span> <span class="o">==</span> <span class="n">DEDENT</span><span class="p">:</span>
-                <span class="n">indents</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="n">toknum</span> <span class="ow">in</span> <span class="p">(</span><span class="n">NEWLINE</span><span class="p">,</span> <span class="n">NL</span><span class="p">):</span>
-                <span class="n">startline</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">elif</span> <span class="n">startline</span> <span class="ow">and</span> <span class="n">indents</span><span class="p">:</span>
-                <span class="n">toks_append</span><span class="p">(</span><span class="n">indents</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-                <span class="n">startline</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="n">toks_append</span><span class="p">(</span><span class="n">tokval</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="untokenize"><a class="viewcode-back" href="../../../modules/parsing.html#sympy.parsing.sympy_tokenize.untokenize">[docs]</a><span class="k">def</span> <span class="nf">untokenize</span><span class="p">(</span><span class="n">iterable</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Transform tokens back into Python source code.</span>
-
-<span class="sd">    Each element returned by the iterable must be a token sequence</span>
-<span class="sd">    with at least two elements, a token number and token value.  If</span>
-<span class="sd">    only two tokens are passed, the resulting output is poor.</span>
-
-<span class="sd">    Round-trip invariant for full input:</span>
-<span class="sd">        Untokenized source will match input source exactly</span>
-
-<span class="sd">    Round-trip invariant for limited intput::</span>
-
-<span class="sd">        # Output text will tokenize the back to the input</span>
-<span class="sd">        t1 = [tok[:2] for tok in generate_tokens(f.readline)]</span>
-<span class="sd">        newcode = untokenize(t1)</span>
-<span class="sd">        readline = iter(newcode.splitlines(1)).next</span>
-<span class="sd">        t2 = [tok[:2] for tok in generate_tokens(readline)]</span>
-<span class="sd">        assert t1 == t2</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">ut</span> <span class="o">=</span> <span class="n">Untokenizer</span><span class="p">()</span>
-    <span class="k">return</span> <span class="n">ut</span><span class="o">.</span><span class="n">untokenize</span><span class="p">(</span><span class="n">iterable</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="generate_tokens"><a class="viewcode-back" href="../../../modules/parsing.html#sympy.parsing.sympy_tokenize.generate_tokens">[docs]</a><span class="k">def</span> <span class="nf">generate_tokens</span><span class="p">(</span><span class="n">readline</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The generate_tokens() generator requires one argment, readline, which</span>
-<span class="sd">    must be a callable object which provides the same interface as the</span>
-<span class="sd">    readline() method of built-in file objects. Each call to the function</span>
-<span class="sd">    should return one line of input as a string.  Alternately, readline</span>
-<span class="sd">    can be a callable function terminating with StopIteration::</span>
-
-<span class="sd">        readline = open(myfile).next    # Example of alternate readline</span>
-
-<span class="sd">    The generator produces 5-tuples with these members: the token type; the</span>
-<span class="sd">    token string; a 2-tuple (srow, scol) of ints specifying the row and</span>
-<span class="sd">    column where the token begins in the source; a 2-tuple (erow, ecol) of</span>
-<span class="sd">    ints specifying the row and column where the token ends in the source;</span>
-<span class="sd">    and the line on which the token was found. The line passed is the</span>
-<span class="sd">    logical line; continuation lines are included.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">lnum</span> <span class="o">=</span> <span class="n">parenlev</span> <span class="o">=</span> <span class="n">continued</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">namechars</span><span class="p">,</span> <span class="n">numchars</span> <span class="o">=</span> <span class="n">string</span><span class="o">.</span><span class="n">ascii_letters</span> <span class="o">+</span> <span class="s">&#39;_&#39;</span><span class="p">,</span> <span class="s">&#39;0123456789&#39;</span>
-    <span class="n">contstr</span><span class="p">,</span> <span class="n">needcont</span> <span class="o">=</span> <span class="s">&#39;&#39;</span><span class="p">,</span> <span class="mi">0</span>
-    <span class="n">contline</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">indents</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>                                   <span class="c"># loop over lines in stream</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">line</span> <span class="o">=</span> <span class="n">readline</span><span class="p">()</span>
-        <span class="k">except</span> <span class="ne">StopIteration</span><span class="p">:</span>
-            <span class="n">line</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-        <span class="n">lnum</span> <span class="o">=</span> <span class="n">lnum</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="n">pos</span><span class="p">,</span> <span class="nb">max</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">line</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">contstr</span><span class="p">:</span>                            <span class="c"># continued string</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">line</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">TokenError</span><span class="p">(</span><span class="s">&quot;EOF in multi-line string&quot;</span><span class="p">,</span> <span class="n">strstart</span><span class="p">)</span>
-            <span class="n">endmatch</span> <span class="o">=</span> <span class="n">endprog</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">line</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">endmatch</span><span class="p">:</span>
-                <span class="n">pos</span> <span class="o">=</span> <span class="n">end</span> <span class="o">=</span> <span class="n">endmatch</span><span class="o">.</span><span class="n">end</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-                <span class="k">yield</span> <span class="p">(</span><span class="n">STRING</span><span class="p">,</span> <span class="n">contstr</span> <span class="o">+</span> <span class="n">line</span><span class="p">[:</span><span class="n">end</span><span class="p">],</span>
-                       <span class="n">strstart</span><span class="p">,</span> <span class="p">(</span><span class="n">lnum</span><span class="p">,</span> <span class="n">end</span><span class="p">),</span> <span class="n">contline</span> <span class="o">+</span> <span class="n">line</span><span class="p">)</span>
-                <span class="n">contstr</span><span class="p">,</span> <span class="n">needcont</span> <span class="o">=</span> <span class="s">&#39;&#39;</span><span class="p">,</span> <span class="mi">0</span>
-                <span class="n">contline</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">elif</span> <span class="n">needcont</span> <span class="ow">and</span> <span class="n">line</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span> <span class="o">!=</span> <span class="s">&#39;</span><span class="se">\\\n</span><span class="s">&#39;</span> <span class="ow">and</span> <span class="n">line</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">:]</span> <span class="o">!=</span> <span class="s">&#39;</span><span class="se">\\\r\n</span><span class="s">&#39;</span><span class="p">:</span>
-                <span class="k">yield</span> <span class="p">(</span><span class="n">ERRORTOKEN</span><span class="p">,</span> <span class="n">contstr</span> <span class="o">+</span> <span class="n">line</span><span class="p">,</span>
-                       <span class="n">strstart</span><span class="p">,</span> <span class="p">(</span><span class="n">lnum</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">line</span><span class="p">)),</span> <span class="n">contline</span><span class="p">)</span>
-                <span class="n">contstr</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-                <span class="n">contline</span> <span class="o">=</span> <span class="bp">None</span>
-                <span class="k">continue</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">contstr</span> <span class="o">=</span> <span class="n">contstr</span> <span class="o">+</span> <span class="n">line</span>
-                <span class="n">contline</span> <span class="o">=</span> <span class="n">contline</span> <span class="o">+</span> <span class="n">line</span>
-                <span class="k">continue</span>
-
-        <span class="k">elif</span> <span class="n">parenlev</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">continued</span><span class="p">:</span>  <span class="c"># new statement</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">line</span><span class="p">:</span>
-                <span class="k">break</span>
-            <span class="n">column</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">while</span> <span class="n">pos</span> <span class="o">&lt;</span> <span class="nb">max</span><span class="p">:</span>                   <span class="c"># measure leading whitespace</span>
-                <span class="k">if</span> <span class="n">line</span><span class="p">[</span><span class="n">pos</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39; &#39;</span><span class="p">:</span>
-                    <span class="n">column</span> <span class="o">=</span> <span class="n">column</span> <span class="o">+</span> <span class="mi">1</span>
-                <span class="k">elif</span> <span class="n">line</span><span class="p">[</span><span class="n">pos</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;</span><span class="se">\t</span><span class="s">&#39;</span><span class="p">:</span>
-                    <span class="n">column</span> <span class="o">=</span> <span class="p">(</span><span class="n">column</span><span class="o">/</span><span class="n">tabsize</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">tabsize</span>
-                <span class="k">elif</span> <span class="n">line</span><span class="p">[</span><span class="n">pos</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;</span><span class="se">\f</span><span class="s">&#39;</span><span class="p">:</span>
-                    <span class="n">column</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">break</span>
-                <span class="n">pos</span> <span class="o">=</span> <span class="n">pos</span> <span class="o">+</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="n">pos</span> <span class="o">==</span> <span class="nb">max</span><span class="p">:</span>
-                <span class="k">break</span>
-
-            <span class="k">if</span> <span class="n">line</span><span class="p">[</span><span class="n">pos</span><span class="p">]</span> <span class="ow">in</span> <span class="s">&#39;#</span><span class="se">\r\n</span><span class="s">&#39;</span><span class="p">:</span>           <span class="c"># skip comments or blank lines</span>
-                <span class="k">if</span> <span class="n">line</span><span class="p">[</span><span class="n">pos</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;#&#39;</span><span class="p">:</span>
-                    <span class="n">comment_token</span> <span class="o">=</span> <span class="n">line</span><span class="p">[</span><span class="n">pos</span><span class="p">:]</span><span class="o">.</span><span class="n">rstrip</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\r\n</span><span class="s">&#39;</span><span class="p">)</span>
-                    <span class="n">nl_pos</span> <span class="o">=</span> <span class="n">pos</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">comment_token</span><span class="p">)</span>
-                    <span class="k">yield</span> <span class="p">(</span><span class="n">COMMENT</span><span class="p">,</span> <span class="n">comment_token</span><span class="p">,</span>
-                           <span class="p">(</span><span class="n">lnum</span><span class="p">,</span> <span class="n">pos</span><span class="p">),</span> <span class="p">(</span><span class="n">lnum</span><span class="p">,</span> <span class="n">pos</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">comment_token</span><span class="p">)),</span> <span class="n">line</span><span class="p">)</span>
-                    <span class="k">yield</span> <span class="p">(</span><span class="n">NL</span><span class="p">,</span> <span class="n">line</span><span class="p">[</span><span class="n">nl_pos</span><span class="p">:],</span>
-                           <span class="p">(</span><span class="n">lnum</span><span class="p">,</span> <span class="n">nl_pos</span><span class="p">),</span> <span class="p">(</span><span class="n">lnum</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">line</span><span class="p">)),</span> <span class="n">line</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">yield</span> <span class="p">((</span><span class="n">NL</span><span class="p">,</span> <span class="n">COMMENT</span><span class="p">)[</span><span class="n">line</span><span class="p">[</span><span class="n">pos</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;#&#39;</span><span class="p">],</span> <span class="n">line</span><span class="p">[</span><span class="n">pos</span><span class="p">:],</span>
-                           <span class="p">(</span><span class="n">lnum</span><span class="p">,</span> <span class="n">pos</span><span class="p">),</span> <span class="p">(</span><span class="n">lnum</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">line</span><span class="p">)),</span> <span class="n">line</span><span class="p">)</span>
-                <span class="k">continue</span>
-
-            <span class="k">if</span> <span class="n">column</span> <span class="o">&gt;</span> <span class="n">indents</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>           <span class="c"># count indents or dedents</span>
-                <span class="n">indents</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">column</span><span class="p">)</span>
-                <span class="k">yield</span> <span class="p">(</span><span class="n">INDENT</span><span class="p">,</span> <span class="n">line</span><span class="p">[:</span><span class="n">pos</span><span class="p">],</span> <span class="p">(</span><span class="n">lnum</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="n">lnum</span><span class="p">,</span> <span class="n">pos</span><span class="p">),</span> <span class="n">line</span><span class="p">)</span>
-            <span class="k">while</span> <span class="n">column</span> <span class="o">&lt;</span> <span class="n">indents</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-                <span class="k">if</span> <span class="n">column</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">indents</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">IndentationError</span><span class="p">(</span>
-                        <span class="s">&quot;unindent does not match any outer indentation level&quot;</span><span class="p">,</span>
-                        <span class="p">(</span><span class="s">&quot;&lt;tokenize&gt;&quot;</span><span class="p">,</span> <span class="n">lnum</span><span class="p">,</span> <span class="n">pos</span><span class="p">,</span> <span class="n">line</span><span class="p">))</span>
-                <span class="n">indents</span> <span class="o">=</span> <span class="n">indents</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-                <span class="k">yield</span> <span class="p">(</span><span class="n">DEDENT</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">,</span> <span class="p">(</span><span class="n">lnum</span><span class="p">,</span> <span class="n">pos</span><span class="p">),</span> <span class="p">(</span><span class="n">lnum</span><span class="p">,</span> <span class="n">pos</span><span class="p">),</span> <span class="n">line</span><span class="p">)</span>
-
-        <span class="k">else</span><span class="p">:</span>                                  <span class="c"># continued statement</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">line</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">TokenError</span><span class="p">(</span><span class="s">&quot;EOF in multi-line statement&quot;</span><span class="p">,</span> <span class="p">(</span><span class="n">lnum</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-            <span class="n">continued</span> <span class="o">=</span> <span class="mi">0</span>
-
-        <span class="k">while</span> <span class="n">pos</span> <span class="o">&lt;</span> <span class="nb">max</span><span class="p">:</span>
-            <span class="n">pseudomatch</span> <span class="o">=</span> <span class="n">pseudoprog</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">line</span><span class="p">,</span> <span class="n">pos</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">pseudomatch</span><span class="p">:</span>                                <span class="c"># scan for tokens</span>
-                <span class="n">start</span><span class="p">,</span> <span class="n">end</span> <span class="o">=</span> <span class="n">pseudomatch</span><span class="o">.</span><span class="n">span</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-                <span class="n">spos</span><span class="p">,</span> <span class="n">epos</span><span class="p">,</span> <span class="n">pos</span> <span class="o">=</span> <span class="p">(</span><span class="n">lnum</span><span class="p">,</span> <span class="n">start</span><span class="p">),</span> <span class="p">(</span><span class="n">lnum</span><span class="p">,</span> <span class="n">end</span><span class="p">),</span> <span class="n">end</span>
-                <span class="n">token</span><span class="p">,</span> <span class="n">initial</span> <span class="o">=</span> <span class="n">line</span><span class="p">[</span><span class="n">start</span><span class="p">:</span><span class="n">end</span><span class="p">],</span> <span class="n">line</span><span class="p">[</span><span class="n">start</span><span class="p">]</span>
-
-                <span class="k">if</span> <span class="n">initial</span> <span class="ow">in</span> <span class="n">numchars</span> <span class="ow">or</span> \
-                        <span class="p">(</span><span class="n">initial</span> <span class="o">==</span> <span class="s">&#39;.&#39;</span> <span class="ow">and</span> <span class="n">token</span> <span class="o">!=</span> <span class="s">&#39;.&#39;</span><span class="p">):</span>      <span class="c"># ordinary number</span>
-                    <span class="k">yield</span> <span class="p">(</span><span class="n">NUMBER</span><span class="p">,</span> <span class="n">token</span><span class="p">,</span> <span class="n">spos</span><span class="p">,</span> <span class="n">epos</span><span class="p">,</span> <span class="n">line</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="n">initial</span> <span class="ow">in</span> <span class="s">&#39;</span><span class="se">\r\n</span><span class="s">&#39;</span><span class="p">:</span>
-                    <span class="k">yield</span> <span class="p">(</span><span class="n">NL</span> <span class="k">if</span> <span class="n">parenlev</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="k">else</span> <span class="n">NEWLINE</span><span class="p">,</span> <span class="n">token</span><span class="p">,</span> <span class="n">spos</span><span class="p">,</span> <span class="n">epos</span><span class="p">,</span> <span class="n">line</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="n">initial</span> <span class="o">==</span> <span class="s">&#39;#&#39;</span><span class="p">:</span>
-                    <span class="k">assert</span> <span class="ow">not</span> <span class="n">token</span><span class="o">.</span><span class="n">endswith</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-                    <span class="k">yield</span> <span class="p">(</span><span class="n">COMMENT</span><span class="p">,</span> <span class="n">token</span><span class="p">,</span> <span class="n">spos</span><span class="p">,</span> <span class="n">epos</span><span class="p">,</span> <span class="n">line</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="n">token</span> <span class="ow">in</span> <span class="n">triple_quoted</span><span class="p">:</span>
-                    <span class="n">endprog</span> <span class="o">=</span> <span class="n">endprogs</span><span class="p">[</span><span class="n">token</span><span class="p">]</span>
-                    <span class="n">endmatch</span> <span class="o">=</span> <span class="n">endprog</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">line</span><span class="p">,</span> <span class="n">pos</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">endmatch</span><span class="p">:</span>                           <span class="c"># all on one line</span>
-                        <span class="n">pos</span> <span class="o">=</span> <span class="n">endmatch</span><span class="o">.</span><span class="n">end</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-                        <span class="n">token</span> <span class="o">=</span> <span class="n">line</span><span class="p">[</span><span class="n">start</span><span class="p">:</span><span class="n">pos</span><span class="p">]</span>
-                        <span class="k">yield</span> <span class="p">(</span><span class="n">STRING</span><span class="p">,</span> <span class="n">token</span><span class="p">,</span> <span class="n">spos</span><span class="p">,</span> <span class="p">(</span><span class="n">lnum</span><span class="p">,</span> <span class="n">pos</span><span class="p">),</span> <span class="n">line</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">contstr</span> <span class="o">=</span> <span class="n">line</span><span class="p">[</span><span class="n">start</span><span class="p">:]</span>
-                        <span class="n">contline</span> <span class="o">=</span> <span class="n">line</span>
-                        <span class="k">break</span>
-                <span class="k">elif</span> <span class="n">initial</span> <span class="ow">in</span> <span class="n">single_quoted</span> <span class="ow">or</span> \
-                    <span class="n">token</span><span class="p">[:</span><span class="mi">2</span><span class="p">]</span> <span class="ow">in</span> <span class="n">single_quoted</span> <span class="ow">or</span> \
-                        <span class="n">token</span><span class="p">[:</span><span class="mi">3</span><span class="p">]</span> <span class="ow">in</span> <span class="n">single_quoted</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">token</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">:</span>                  <span class="c"># continued string</span>
-                        <span class="n">endprog</span> <span class="o">=</span> <span class="p">(</span><span class="n">endprogs</span><span class="p">[</span><span class="n">initial</span><span class="p">]</span> <span class="ow">or</span> <span class="n">endprogs</span><span class="p">[</span><span class="n">token</span><span class="p">[</span><span class="mi">1</span><span class="p">]]</span> <span class="ow">or</span>
-                                   <span class="n">endprogs</span><span class="p">[</span><span class="n">token</span><span class="p">[</span><span class="mi">2</span><span class="p">]])</span>
-                        <span class="n">contstr</span><span class="p">,</span> <span class="n">needcont</span> <span class="o">=</span> <span class="n">line</span><span class="p">[</span><span class="n">start</span><span class="p">:],</span> <span class="mi">1</span>
-                        <span class="n">contline</span> <span class="o">=</span> <span class="n">line</span>
-                        <span class="k">break</span>
-                    <span class="k">else</span><span class="p">:</span>                                  <span class="c"># ordinary string</span>
-                        <span class="k">yield</span> <span class="p">(</span><span class="n">STRING</span><span class="p">,</span> <span class="n">token</span><span class="p">,</span> <span class="n">spos</span><span class="p">,</span> <span class="n">epos</span><span class="p">,</span> <span class="n">line</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="n">initial</span> <span class="ow">in</span> <span class="n">namechars</span><span class="p">:</span>                 <span class="c"># ordinary name</span>
-                    <span class="k">yield</span> <span class="p">(</span><span class="n">NAME</span><span class="p">,</span> <span class="n">token</span><span class="p">,</span> <span class="n">spos</span><span class="p">,</span> <span class="n">epos</span><span class="p">,</span> <span class="n">line</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="n">initial</span> <span class="o">==</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">&#39;</span><span class="p">:</span>                      <span class="c"># continued stmt</span>
-                    <span class="n">continued</span> <span class="o">=</span> <span class="mi">1</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">initial</span> <span class="ow">in</span> <span class="s">&#39;([{&#39;</span><span class="p">:</span>
-                        <span class="n">parenlev</span> <span class="o">=</span> <span class="n">parenlev</span> <span class="o">+</span> <span class="mi">1</span>
-                    <span class="k">elif</span> <span class="n">initial</span> <span class="ow">in</span> <span class="s">&#39;)]}&#39;</span><span class="p">:</span>
-                        <span class="n">parenlev</span> <span class="o">=</span> <span class="n">parenlev</span> <span class="o">-</span> <span class="mi">1</span>
-                    <span class="k">yield</span> <span class="p">(</span><span class="n">OP</span><span class="p">,</span> <span class="n">token</span><span class="p">,</span> <span class="n">spos</span><span class="p">,</span> <span class="n">epos</span><span class="p">,</span> <span class="n">line</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">yield</span> <span class="p">(</span><span class="n">ERRORTOKEN</span><span class="p">,</span> <span class="n">line</span><span class="p">[</span><span class="n">pos</span><span class="p">],</span>
-                       <span class="p">(</span><span class="n">lnum</span><span class="p">,</span> <span class="n">pos</span><span class="p">),</span> <span class="p">(</span><span class="n">lnum</span><span class="p">,</span> <span class="n">pos</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">line</span><span class="p">)</span>
-                <span class="n">pos</span> <span class="o">=</span> <span class="n">pos</span> <span class="o">+</span> <span class="mi">1</span>
-
-    <span class="k">for</span> <span class="n">indent</span> <span class="ow">in</span> <span class="n">indents</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>                 <span class="c"># pop remaining indent levels</span>
-        <span class="k">yield</span> <span class="p">(</span><span class="n">DEDENT</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">,</span> <span class="p">(</span><span class="n">lnum</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="n">lnum</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-    <span class="k">yield</span> <span class="p">(</span><span class="n">ENDMARKER</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">,</span> <span class="p">(</span><span class="n">lnum</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="n">lnum</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-</div>
-<span class="k">if</span> <span class="n">__name__</span> <span class="o">==</span> <span class="s">&#39;__main__&#39;</span><span class="p">:</span>                     <span class="c"># testing</span>
-    <span class="kn">import</span> <span class="nn">sys</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">sys</span><span class="o">.</span><span class="n">argv</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">tokenize</span><span class="p">(</span><span class="nb">open</span><span class="p">(</span><span class="n">sys</span><span class="o">.</span><span class="n">argv</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">readline</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">tokenize</span><span class="p">(</span><span class="n">sys</span><span class="o">.</span><span class="n">stdin</span><span class="o">.</span><span class="n">readline</span><span class="p">)</span>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.physics.gaussopt</h1><div class="highlight"><pre>
-<span class="c"># -*- encoding: utf-8 -*-</span>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Gaussian optics.</span>
-
-<span class="sd">The module implements:</span>
-
-<span class="sd">- Ray transfer matrices for geometrical and gaussian optics.</span>
-
-<span class="sd">  See RayTransferMatrix, GeometricRay and BeamParameter</span>
-
-<span class="sd">- Conjugation relations for geometrical and gaussian optics.</span>
-
-<span class="sd">  See geometric_conj*, gauss_conj and conjugate_gauss_beams</span>
-
-<span class="sd">The conventions for the distances are as follows:</span>
-
-<span class="sd">focal distance</span>
-<span class="sd">    positive for convergent lenses</span>
-<span class="sd">object distance</span>
-<span class="sd">    positive for real objects</span>
-<span class="sd">image distance</span>
-<span class="sd">    positive for real images</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="p">(</span><span class="n">atan2</span><span class="p">,</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">re</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span>
-    <span class="n">together</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.misc</span> <span class="kn">import</span> <span class="n">filldedent</span>
-
-<span class="c">###</span>
-<span class="c"># A, B, C, D matrices</span>
-<span class="c">###</span>
-
-
-<div class="viewcode-block" id="RayTransferMatrix"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.RayTransferMatrix">[docs]</a><span class="k">class</span> <span class="nc">RayTransferMatrix</span><span class="p">(</span><span class="n">Matrix</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Base class for a Ray Transfer Matrix.</span>
-
-<span class="sd">    It should be used if there isn&#39;t already a more specific subclass mentioned</span>
-<span class="sd">    in See Also.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    parameters : A, B, C and D or 2x2 matrix (Matrix(2, 2, [A, B, C, D]))</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.gaussopt import RayTransferMatrix, ThinLens</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, Matrix</span>
-
-<span class="sd">    &gt;&gt;&gt; mat = RayTransferMatrix(1, 2, 3, 4)</span>
-<span class="sd">    &gt;&gt;&gt; mat</span>
-<span class="sd">    [1,  2]</span>
-<span class="sd">    [3,  4]</span>
-
-<span class="sd">    &gt;&gt;&gt; RayTransferMatrix(Matrix([[1, 2], [3, 4]]))</span>
-<span class="sd">    [1,  2]</span>
-<span class="sd">    [3,  4]</span>
-
-<span class="sd">    &gt;&gt;&gt; mat.A</span>
-<span class="sd">    1</span>
-
-<span class="sd">    &gt;&gt;&gt; f = Symbol(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; lens = ThinLens(f)</span>
-<span class="sd">    &gt;&gt;&gt; lens</span>
-<span class="sd">    [   1, 0]</span>
-<span class="sd">    [-1/f, 1]</span>
-
-<span class="sd">    &gt;&gt;&gt; lens.C</span>
-<span class="sd">    -1/f</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    GeometricRay, BeamParameter,</span>
-<span class="sd">    FreeSpace, FlatRefraction, CurvedRefraction,</span>
-<span class="sd">    FlatMirror, CurvedMirror, ThinLens</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Ray_transfer_matrix_analysis</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">4</span><span class="p">:</span>
-            <span class="n">temp</span> <span class="o">=</span> <span class="p">((</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">args</span><span class="p">[</span><span class="mi">3</span><span class="p">]))</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> \
-            <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Matrix</span><span class="p">)</span> \
-                <span class="ow">and</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span> <span class="o">==</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">):</span>
-            <span class="n">temp</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&#39;&#39;&#39;</span>
-<span class="s">                Expecting 2x2 Matrix or the 4 elements of</span>
-<span class="s">                the Matrix but got </span><span class="si">%s</span><span class="s">&#39;&#39;&#39;</span> <span class="o">%</span> <span class="nb">str</span><span class="p">(</span><span class="n">args</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">Matrix</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">temp</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">RayTransferMatrix</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">RayTransferMatrix</span><span class="p">(</span><span class="n">Matrix</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">GeometricRay</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">GeometricRay</span><span class="p">(</span><span class="n">Matrix</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">BeamParameter</span><span class="p">):</span>
-            <span class="n">temp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">*</span><span class="n">Matrix</span><span class="p">(((</span><span class="n">other</span><span class="o">.</span><span class="n">q</span><span class="p">,),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,)))</span>
-            <span class="n">q</span> <span class="o">=</span> <span class="p">(</span><span class="n">temp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="n">temp</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">BeamParameter</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">wavelen</span><span class="p">,</span>
-                                 <span class="n">together</span><span class="p">(</span><span class="n">re</span><span class="p">(</span><span class="n">q</span><span class="p">)),</span>
-                                 <span class="n">z_r</span><span class="o">=</span><span class="n">together</span><span class="p">(</span><span class="n">im</span><span class="p">(</span><span class="n">q</span><span class="p">)))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Matrix</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="RayTransferMatrix.A"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.RayTransferMatrix.A">[docs]</a>    <span class="k">def</span> <span class="nf">A</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The A parameter of the Matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.gaussopt import RayTransferMatrix</span>
-<span class="sd">        &gt;&gt;&gt; mat = RayTransferMatrix(1, 2, 3, 4)</span>
-<span class="sd">        &gt;&gt;&gt; mat.A</span>
-<span class="sd">        1</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="RayTransferMatrix.B"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.RayTransferMatrix.B">[docs]</a>    <span class="k">def</span> <span class="nf">B</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The B parameter of the Matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.gaussopt import RayTransferMatrix</span>
-<span class="sd">        &gt;&gt;&gt; mat = RayTransferMatrix(1, 2, 3, 4)</span>
-<span class="sd">        &gt;&gt;&gt; mat.B</span>
-<span class="sd">        2</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="RayTransferMatrix.C"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.RayTransferMatrix.C">[docs]</a>    <span class="k">def</span> <span class="nf">C</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The C parameter of the Matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.gaussopt import RayTransferMatrix</span>
-<span class="sd">        &gt;&gt;&gt; mat = RayTransferMatrix(1, 2, 3, 4)</span>
-<span class="sd">        &gt;&gt;&gt; mat.C</span>
-<span class="sd">        3</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="RayTransferMatrix.D"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.RayTransferMatrix.D">[docs]</a>    <span class="k">def</span> <span class="nf">D</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The D parameter of the Matrix.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.gaussopt import RayTransferMatrix</span>
-<span class="sd">        &gt;&gt;&gt; mat = RayTransferMatrix(1, 2, 3, 4)</span>
-<span class="sd">        &gt;&gt;&gt; mat.D</span>
-<span class="sd">        4</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
-
-</div></div>
-<div class="viewcode-block" id="FreeSpace"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.FreeSpace">[docs]</a><span class="k">class</span> <span class="nc">FreeSpace</span><span class="p">(</span><span class="n">RayTransferMatrix</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Ray Transfer Matrix for free space.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    distance</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    RayTransferMatrix</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.gaussopt import FreeSpace</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; d = symbols(&#39;d&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; FreeSpace(d)</span>
-<span class="sd">    [1, d]</span>
-<span class="sd">    [0, 1]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">d</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">RayTransferMatrix</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="FlatRefraction"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.FlatRefraction">[docs]</a><span class="k">class</span> <span class="nc">FlatRefraction</span><span class="p">(</span><span class="n">RayTransferMatrix</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Ray Transfer Matrix for refraction.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    n1 : refractive index of one medium</span>
-<span class="sd">    n2 : refractive index of other medium</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    RayTransferMatrix</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.gaussopt import FlatRefraction</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; n1, n2 = symbols(&#39;n1 n2&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; FlatRefraction(n1, n2)</span>
-<span class="sd">    [1,     0]</span>
-<span class="sd">    [0, n1/n2]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">):</span>
-        <span class="n">n1</span><span class="p">,</span> <span class="n">n2</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">RayTransferMatrix</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n1</span><span class="o">/</span><span class="n">n2</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="CurvedRefraction"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.CurvedRefraction">[docs]</a><span class="k">class</span> <span class="nc">CurvedRefraction</span><span class="p">(</span><span class="n">RayTransferMatrix</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Ray Transfer Matrix for refraction on curved interface.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    R : radius of curvature (positive for concave)</span>
-<span class="sd">    n1 : refractive index of one medium</span>
-<span class="sd">    n2 : refractive index of other medium</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    RayTransferMatrix</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.gaussopt import CurvedRefraction</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; R, n1, n2 = symbols(&#39;R n1 n2&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; CurvedRefraction(R, n1, n2)</span>
-<span class="sd">    [               1,     0]</span>
-<span class="sd">    [(n1 - n2)/(R*n2), n1/n2]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">):</span>
-        <span class="n">R</span><span class="p">,</span> <span class="n">n1</span><span class="p">,</span> <span class="n">n2</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">RayTransferMatrix</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="p">(</span><span class="n">n1</span> <span class="o">-</span> <span class="n">n2</span><span class="p">)</span><span class="o">/</span><span class="n">R</span><span class="o">/</span><span class="n">n2</span><span class="p">,</span> <span class="n">n1</span><span class="o">/</span><span class="n">n2</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="FlatMirror"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.FlatMirror">[docs]</a><span class="k">class</span> <span class="nc">FlatMirror</span><span class="p">(</span><span class="n">RayTransferMatrix</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Ray Transfer Matrix for reflection.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    RayTransferMatrix</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.gaussopt import FlatMirror</span>
-<span class="sd">    &gt;&gt;&gt; FlatMirror()</span>
-<span class="sd">    [1, 0]</span>
-<span class="sd">    [0, 1]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">RayTransferMatrix</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="CurvedMirror"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.CurvedMirror">[docs]</a><span class="k">class</span> <span class="nc">CurvedMirror</span><span class="p">(</span><span class="n">RayTransferMatrix</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Ray Transfer Matrix for reflection from curved surface.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    R : radius of curvature (positive for concave)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    RayTransferMatrix</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.gaussopt import CurvedMirror</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; R = symbols(&#39;R&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; CurvedMirror(R)</span>
-<span class="sd">    [   1, 0]</span>
-<span class="sd">    [-2/R, 1]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">R</span><span class="p">):</span>
-        <span class="n">R</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">R</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">RayTransferMatrix</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="o">/</span><span class="n">R</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="ThinLens"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.ThinLens">[docs]</a><span class="k">class</span> <span class="nc">ThinLens</span><span class="p">(</span><span class="n">RayTransferMatrix</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Ray Transfer Matrix for a thin lens.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    f : the focal distance</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    RayTransferMatrix</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.gaussopt import ThinLens</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; f = symbols(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; ThinLens(f)</span>
-<span class="sd">    [   1, 0]</span>
-<span class="sd">    [-1/f, 1]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">f</span><span class="p">):</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">RayTransferMatrix</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-
-
-<span class="c">###</span>
-<span class="c"># Representation for geometric ray</span>
-<span class="c">###</span>
-</div>
-<div class="viewcode-block" id="GeometricRay"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.GeometricRay">[docs]</a><span class="k">class</span> <span class="nc">GeometricRay</span><span class="p">(</span><span class="n">Matrix</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Representation for a geometric ray in the Ray Transfer Matrix formalism.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    h : height, and</span>
-<span class="sd">    angle : angle, or</span>
-<span class="sd">    matrix : a 2x1 matrix (Matrix(2, 1, [height, angle]))</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.gaussopt import GeometricRay, FreeSpace</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, Matrix</span>
-<span class="sd">    &gt;&gt;&gt; d, h, angle = symbols(&#39;d, h, angle&#39;)</span>
-
-<span class="sd">    &gt;&gt;&gt; GeometricRay(h, angle)</span>
-<span class="sd">    [    h]</span>
-<span class="sd">    [angle]</span>
-
-<span class="sd">    &gt;&gt;&gt; FreeSpace(d)*GeometricRay(h, angle)</span>
-<span class="sd">    [angle*d + h]</span>
-<span class="sd">    [      angle]</span>
-
-<span class="sd">    &gt;&gt;&gt; GeometricRay( Matrix( ((h,), (angle,)) ) )</span>
-<span class="sd">    [    h]</span>
-<span class="sd">    [angle]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    RayTransferMatrix</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Matrix</span><span class="p">)</span> \
-                <span class="ow">and</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span> <span class="o">==</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">temp</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">temp</span> <span class="o">=</span> <span class="p">((</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],),</span> <span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">],))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&#39;&#39;&#39;</span>
-<span class="s">                Expecting 2x1 Matrix or the 2 elements of</span>
-<span class="s">                the Matrix but got </span><span class="si">%s</span><span class="s">&#39;&#39;&#39;</span> <span class="o">%</span> <span class="nb">str</span><span class="p">(</span><span class="n">args</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">Matrix</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">temp</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="GeometricRay.height"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.GeometricRay.height">[docs]</a>    <span class="k">def</span> <span class="nf">height</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The distance from the optical axis.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.gaussopt import GeometricRay</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; h, angle = symbols(&#39;h, angle&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; gRay = GeometricRay(h, angle)</span>
-<span class="sd">        &gt;&gt;&gt; gRay.height</span>
-<span class="sd">        h</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="GeometricRay.angle"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.GeometricRay.angle">[docs]</a>    <span class="k">def</span> <span class="nf">angle</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The angle with the optical axis.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.gaussopt import GeometricRay</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; h, angle = symbols(&#39;h, angle&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; gRay = GeometricRay(h, angle)</span>
-<span class="sd">        &gt;&gt;&gt; gRay.angle</span>
-<span class="sd">        angle</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-
-<span class="c">###</span>
-<span class="c"># Representation for gauss beam</span>
-<span class="c">###</span>
-</div></div>
-<div class="viewcode-block" id="BeamParameter"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.BeamParameter">[docs]</a><span class="k">class</span> <span class="nc">BeamParameter</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Representation for a gaussian ray in the Ray Transfer Matrix formalism.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    wavelen : the wavelength,</span>
-<span class="sd">    z : the distance to waist, and</span>
-<span class="sd">    w : the waist, or</span>
-<span class="sd">    z_r : the rayleigh range</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.gaussopt import BeamParameter</span>
-<span class="sd">    &gt;&gt;&gt; p = BeamParameter(530e-9, 1, w=1e-3)</span>
-<span class="sd">    &gt;&gt;&gt; p.q</span>
-<span class="sd">    1 + 1.88679245283019*I*pi</span>
-
-<span class="sd">    &gt;&gt;&gt; p.q.n()</span>
-<span class="sd">    1.0 + 5.92753330865999*I</span>
-<span class="sd">    &gt;&gt;&gt; p.w_0.n()</span>
-<span class="sd">    0.00100000000000000</span>
-<span class="sd">    &gt;&gt;&gt; p.z_r.n()</span>
-<span class="sd">    5.92753330865999</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.gaussopt import FreeSpace</span>
-<span class="sd">    &gt;&gt;&gt; fs = FreeSpace(10)</span>
-<span class="sd">    &gt;&gt;&gt; p1 = fs*p</span>
-<span class="sd">    &gt;&gt;&gt; p.w.n()</span>
-<span class="sd">    0.00101413072159615</span>
-<span class="sd">    &gt;&gt;&gt; p1.w.n()</span>
-<span class="sd">    0.00210803120913829</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    RayTransferMatrix</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Complex_beam_parameter</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c">#TODO A class Complex may be implemented. The BeamParameter may</span>
-    <span class="c"># subclass it. See:</span>
-    <span class="c"># https://groups.google.com/d/topic/sympy/7XkU07NRBEs/discussion</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;z&#39;</span><span class="p">,</span> <span class="s">&#39;z_r&#39;</span><span class="p">,</span> <span class="s">&#39;wavelen&#39;</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">wavelen</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">wavelen</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">wavelen</span><span class="p">,</span> <span class="n">z</span><span class="p">)))</span>
-        <span class="n">inst</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">wavelen</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-        <span class="n">inst</span><span class="o">.</span><span class="n">wavelen</span> <span class="o">=</span> <span class="n">wavelen</span>
-        <span class="n">inst</span><span class="o">.</span><span class="n">z</span> <span class="o">=</span> <span class="n">z</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">kwargs</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Constructor expects exactly one named argument.&#39;</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="s">&#39;z_r&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
-            <span class="n">inst</span><span class="o">.</span><span class="n">z_r</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;z_r&#39;</span><span class="p">])</span>
-        <span class="k">elif</span> <span class="s">&#39;w&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
-            <span class="n">inst</span><span class="o">.</span><span class="n">z_r</span> <span class="o">=</span> <span class="n">waist2rayleigh</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;w&#39;</span><span class="p">]),</span> <span class="n">wavelen</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;The constructor needs named argument w or z_r&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">inst</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="BeamParameter.q"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.BeamParameter.q">[docs]</a>    <span class="k">def</span> <span class="nf">q</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The complex parameter representing the beam.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.gaussopt import BeamParameter</span>
-<span class="sd">        &gt;&gt;&gt; p = BeamParameter(530e-9, 1, w=1e-3)</span>
-<span class="sd">        &gt;&gt;&gt; p.q</span>
-<span class="sd">        1 + 1.88679245283019*I*pi</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">z</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">z_r</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="BeamParameter.radius"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.BeamParameter.radius">[docs]</a>    <span class="k">def</span> <span class="nf">radius</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The radius of curvature of the phase front.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.gaussopt import BeamParameter</span>
-<span class="sd">        &gt;&gt;&gt; p = BeamParameter(530e-9, 1, w=1e-3)</span>
-<span class="sd">        &gt;&gt;&gt; p.radius</span>
-<span class="sd">        0.2809/pi**2 + 1</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">z</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">z</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">z_r</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="BeamParameter.w"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.BeamParameter.w">[docs]</a>    <span class="k">def</span> <span class="nf">w</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The beam radius at `1/e^2` intensity.</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        w_0 : the minimal radius of beam</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.gaussopt import BeamParameter</span>
-<span class="sd">        &gt;&gt;&gt; p = BeamParameter(530e-9, 1, w=1e-3)</span>
-<span class="sd">        &gt;&gt;&gt; p.w</span>
-<span class="sd">        0.001*sqrt(0.2809/pi**2 + 1)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">w_0</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">z</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">z_r</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="BeamParameter.w_0"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.BeamParameter.w_0">[docs]</a>    <span class="k">def</span> <span class="nf">w_0</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The beam waist (minimal radius).</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        w : the beam radius at `1/e^2` intensity</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.gaussopt import BeamParameter</span>
-<span class="sd">        &gt;&gt;&gt; p = BeamParameter(530e-9, 1, w=1e-3)</span>
-<span class="sd">        &gt;&gt;&gt; p.w_0</span>
-<span class="sd">        0.00100000000000000</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">z_r</span><span class="o">/</span><span class="n">pi</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">wavelen</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="BeamParameter.divergence"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.BeamParameter.divergence">[docs]</a>    <span class="k">def</span> <span class="nf">divergence</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Half of the total angular spread.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.gaussopt import BeamParameter</span>
-<span class="sd">        &gt;&gt;&gt; p = BeamParameter(530e-9, 1, w=1e-3)</span>
-<span class="sd">        &gt;&gt;&gt; p.divergence</span>
-<span class="sd">        0.00053/pi</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">wavelen</span><span class="o">/</span><span class="n">pi</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">w_0</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="BeamParameter.gouy"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.BeamParameter.gouy">[docs]</a>    <span class="k">def</span> <span class="nf">gouy</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The Gouy phase.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.gaussopt import BeamParameter</span>
-<span class="sd">        &gt;&gt;&gt; p = BeamParameter(530e-9, 1, w=1e-3)</span>
-<span class="sd">        &gt;&gt;&gt; p.gouy</span>
-<span class="sd">        atan(0.53/pi)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">atan2</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">z</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">z_r</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="BeamParameter.waist_approximation_limit"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.BeamParameter.waist_approximation_limit">[docs]</a>    <span class="k">def</span> <span class="nf">waist_approximation_limit</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The minimal waist for which the gauss beam approximation is valid.</span>
-
-<span class="sd">        The gauss beam is a solution to the paraxial equation. For curvatures</span>
-<span class="sd">        that are too great it is not a valid approximation.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.gaussopt import BeamParameter</span>
-<span class="sd">        &gt;&gt;&gt; p = BeamParameter(530e-9, 1, w=1e-3)</span>
-<span class="sd">        &gt;&gt;&gt; p.waist_approximation_limit</span>
-<span class="sd">        1.06e-6/pi</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">wavelen</span><span class="o">/</span><span class="n">pi</span>
-
-
-<span class="c">###</span>
-<span class="c"># Utilities</span>
-<span class="c">###</span>
-</div></div>
-<div class="viewcode-block" id="waist2rayleigh"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.waist2rayleigh">[docs]</a><span class="k">def</span> <span class="nf">waist2rayleigh</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">wavelen</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Calculate the rayleigh range from the waist of a gaussian beam.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    rayleigh2waist, BeamParameter</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.gaussopt import waist2rayleigh</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; w, wavelen = symbols(&#39;w wavelen&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; waist2rayleigh(w, wavelen)</span>
-<span class="sd">    pi*w**2/wavelen</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">w</span><span class="p">,</span> <span class="n">wavelen</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">wavelen</span><span class="p">)))</span>
-    <span class="k">return</span> <span class="n">w</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="n">wavelen</span>
-
-</div>
-<div class="viewcode-block" id="rayleigh2waist"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.rayleigh2waist">[docs]</a><span class="k">def</span> <span class="nf">rayleigh2waist</span><span class="p">(</span><span class="n">z_r</span><span class="p">,</span> <span class="n">wavelen</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Calculate the waist from the rayleigh range of a gaussian beam.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    waist2rayleigh, BeamParameter</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.gaussopt import rayleigh2waist</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; z_r, wavelen = symbols(&#39;z_r wavelen&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; rayleigh2waist(z_r, wavelen)</span>
-<span class="sd">    sqrt(wavelen*z_r)/sqrt(pi)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">z_r</span><span class="p">,</span> <span class="n">wavelen</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">z_r</span><span class="p">,</span> <span class="n">wavelen</span><span class="p">)))</span>
-    <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z_r</span><span class="o">/</span><span class="n">pi</span><span class="o">*</span><span class="n">wavelen</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="geometric_conj_ab"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.geometric_conj_ab">[docs]</a><span class="k">def</span> <span class="nf">geometric_conj_ab</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Conjugation relation for geometrical beams under paraxial conditions.</span>
-
-<span class="sd">    Takes the distances to the optical element and returns the needed</span>
-<span class="sd">    focal distance.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    geometric_conj_af, geometric_conj_bf</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.gaussopt import geometric_conj_ab</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; a, b = symbols(&#39;a b&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; geometric_conj_ab(a, b)</span>
-<span class="sd">    a*b/(a + b)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)))</span>
-    <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">==</span> <span class="n">oo</span> <span class="ow">or</span> <span class="nb">abs</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">==</span> <span class="n">oo</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">a</span> <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">==</span> <span class="n">oo</span> <span class="k">else</span> <span class="n">b</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">a</span><span class="o">*</span><span class="n">b</span><span class="o">/</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="geometric_conj_af"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.geometric_conj_af">[docs]</a><span class="k">def</span> <span class="nf">geometric_conj_af</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Conjugation relation for geometrical beams under paraxial conditions.</span>
-
-<span class="sd">    Takes the object distance (for geometric_conj_af) or the image distance</span>
-<span class="sd">    (for geometric_conj_bf) to the optical element and the focal distance.</span>
-<span class="sd">    Then it returns the other distance needed for conjugation.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    geometric_conj_ab</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.gaussopt import geometric_conj_af, geometric_conj_bf</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; a, b, f = symbols(&#39;a b f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; geometric_conj_af(a, f)</span>
-<span class="sd">    a*f/(a - f)</span>
-<span class="sd">    &gt;&gt;&gt; geometric_conj_bf(b, f)</span>
-<span class="sd">    b*f/(b - f)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">a</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">f</span><span class="p">)))</span>
-    <span class="k">return</span> <span class="o">-</span><span class="n">geometric_conj_ab</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="o">-</span><span class="n">f</span><span class="p">)</span>
-</div>
-<span class="n">geometric_conj_bf</span> <span class="o">=</span> <span class="n">geometric_conj_af</span>
-
-
-<div class="viewcode-block" id="gaussian_conj"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.gaussian_conj">[docs]</a><span class="k">def</span> <span class="nf">gaussian_conj</span><span class="p">(</span><span class="n">s_in</span><span class="p">,</span> <span class="n">z_r_in</span><span class="p">,</span> <span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Conjugation relation for gaussian beams.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    s_in : the distance to optical element from the waist</span>
-<span class="sd">    z_r_in : the rayleigh range of the incident beam</span>
-<span class="sd">    f : the focal length of the optical element</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    a tuple containing (s_out, z_r_out, m)</span>
-<span class="sd">    s_out : the distance between the new waist and the optical element</span>
-<span class="sd">    z_r_out : the rayleigh range of the emergent beam</span>
-<span class="sd">    m : the ration between the new and the old waists</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.gaussopt import gaussian_conj</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; s_in, z_r_in, f = symbols(&#39;s_in z_r_in f&#39;)</span>
-
-<span class="sd">    &gt;&gt;&gt; gaussian_conj(s_in, z_r_in, f)[0]</span>
-<span class="sd">    1/(-1/(s_in + z_r_in**2/(-f + s_in)) + 1/f)</span>
-
-<span class="sd">    &gt;&gt;&gt; gaussian_conj(s_in, z_r_in, f)[1]</span>
-<span class="sd">    z_r_in/(1 - s_in**2/f**2 + z_r_in**2/f**2)</span>
-
-<span class="sd">    &gt;&gt;&gt; gaussian_conj(s_in, z_r_in, f)[2]</span>
-<span class="sd">    1/sqrt(1 - s_in**2/f**2 + z_r_in**2/f**2)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">s_in</span><span class="p">,</span> <span class="n">z_r_in</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">s_in</span><span class="p">,</span> <span class="n">z_r_in</span><span class="p">,</span> <span class="n">f</span><span class="p">)))</span>
-    <span class="n">s_out</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">/</span> <span class="p">(</span> <span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">s_in</span> <span class="o">+</span> <span class="n">z_r_in</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="n">s_in</span> <span class="o">-</span> <span class="n">f</span><span class="p">))</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">f</span> <span class="p">)</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">((</span><span class="mi">1</span> <span class="o">-</span> <span class="p">(</span><span class="n">s_in</span><span class="o">/</span><span class="n">f</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">z_r_in</span><span class="o">/</span><span class="n">f</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-    <span class="n">z_r_out</span> <span class="o">=</span> <span class="n">z_r_in</span> <span class="o">/</span> <span class="p">((</span><span class="mi">1</span> <span class="o">-</span> <span class="p">(</span><span class="n">s_in</span><span class="o">/</span><span class="n">f</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">z_r_in</span><span class="o">/</span><span class="n">f</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-    <span class="k">return</span> <span class="p">(</span><span class="n">s_out</span><span class="p">,</span> <span class="n">z_r_out</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="conjugate_gauss_beams"><a class="viewcode-back" href="../../../modules/physics/gaussopt.html#sympy.physics.gaussopt.conjugate_gauss_beams">[docs]</a><span class="k">def</span> <span class="nf">conjugate_gauss_beams</span><span class="p">(</span><span class="n">wavelen</span><span class="p">,</span> <span class="n">waist_in</span><span class="p">,</span> <span class="n">waist_out</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Find the optical setup conjugating the object/image waists.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    wavelen : the wavelength of the beam</span>
-<span class="sd">    waist_in and waist_out : the waists to be conjugated</span>
-<span class="sd">    f : the focal distance of the element used in the conjugation</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    a tuple containing (s_in, s_out, f)</span>
-<span class="sd">    s_in : the distance before the optical element</span>
-<span class="sd">    s_out : the distance after the optical element</span>
-<span class="sd">    f : the focal distance of the optical element</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.gaussopt import conjugate_gauss_beams</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, factor</span>
-<span class="sd">    &gt;&gt;&gt; l, w_i, w_o, f = symbols(&#39;l w_i w_o f&#39;)</span>
-
-<span class="sd">    &gt;&gt;&gt; conjugate_gauss_beams(l, w_i, w_o, f=f)[0]</span>
-<span class="sd">    f*(-sqrt(w_i**2/w_o**2 - pi**2*w_i**4/(f**2*l**2)) + 1)</span>
-
-<span class="sd">    &gt;&gt;&gt; factor(conjugate_gauss_beams(l, w_i, w_o, f=f)[1])</span>
-<span class="sd">    f*w_o**2*(w_i**2/w_o**2 - sqrt(w_i**2/w_o**2 -</span>
-<span class="sd">              pi**2*w_i**4/(f**2*l**2)))/w_i**2</span>
-
-<span class="sd">    &gt;&gt;&gt; conjugate_gauss_beams(l, w_i, w_o, f=f)[2]</span>
-<span class="sd">    f</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c">#TODO add the other possible arguments</span>
-    <span class="n">wavelen</span><span class="p">,</span> <span class="n">waist_in</span><span class="p">,</span> <span class="n">waist_out</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">wavelen</span><span class="p">,</span> <span class="n">waist_in</span><span class="p">,</span> <span class="n">waist_out</span><span class="p">)))</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">waist_out</span> <span class="o">/</span> <span class="n">waist_in</span>
-    <span class="n">z</span> <span class="o">=</span> <span class="n">waist2rayleigh</span><span class="p">(</span><span class="n">waist_in</span><span class="p">,</span> <span class="n">wavelen</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">kwargs</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;The function expects only one named argument&quot;</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="s">&#39;dist&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&#39;&#39;&#39;</span>
-<span class="s">            Currently only focal length is supported as a parameter&#39;&#39;&#39;</span><span class="p">))</span>
-    <span class="k">elif</span> <span class="s">&#39;f&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;f&#39;</span><span class="p">])</span>
-        <span class="n">s_in</span> <span class="o">=</span> <span class="n">f</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">m</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="n">f</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-        <span class="n">s_out</span> <span class="o">=</span> <span class="n">gaussian_conj</span><span class="p">(</span><span class="n">s_in</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">f</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">elif</span> <span class="s">&#39;s_in&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&#39;&#39;&#39;</span>
-<span class="s">            Currently only focal length is supported as a parameter&#39;&#39;&#39;</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&#39;&#39;&#39;</span>
-<span class="s">            The functions expects the focal length as a named argument&#39;&#39;&#39;</span><span class="p">))</span>
-    <span class="k">return</span> <span class="p">(</span><span class="n">s_in</span><span class="p">,</span> <span class="n">s_out</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
-
-<span class="c">#TODO</span>
-<span class="c">#def plot_beam():</span>
-<span class="c">#    &quot;&quot;&quot;Plot the beam radius as it propagates in space.&quot;&quot;&quot;</span>
-<span class="c">#    pass</span>
-
-<span class="c">#TODO</span>
-<span class="c">#def plot_beam_conjugation():</span>
-<span class="c">#    &quot;&quot;&quot;</span>
-<span class="c">#    Plot the intersection of two beams.</span>
-<span class="c">#</span>
-<span class="c">#    Represents the conjugation relation.</span>
-<span class="c">#</span>
-<span class="c">#    See Also</span>
-<span class="c">#    ========</span>
-<span class="c">#</span>
-<span class="c">#    conjugate_gauss_beams</span>
-<span class="c">#    &quot;&quot;&quot;</span>
-<span class="c">#    pass</span></div>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.physics.hydrogen</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">assoc_laguerre</span><span class="p">,</span> <span class="n">Float</span>
-
-
-<div class="viewcode-block" id="R_nl"><a class="viewcode-back" href="../../../modules/physics/hydrogen.html#sympy.physics.hydrogen.R_nl">[docs]</a><span class="k">def</span> <span class="nf">R_nl</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">Z</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the Hydrogen radial wavefunction R_{nl}.</span>
-
-<span class="sd">    n, l</span>
-<span class="sd">        quantum numbers &#39;n&#39; and &#39;l&#39;</span>
-<span class="sd">    r</span>
-<span class="sd">        radial coordinate</span>
-<span class="sd">    Z</span>
-<span class="sd">        atomic number (1 for Hydrogen, 2 for Helium, ...)</span>
-
-<span class="sd">    Everything is in Hartree atomic units.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.hydrogen import R_nl</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import var</span>
-<span class="sd">    &gt;&gt;&gt; var(&quot;r Z&quot;)</span>
-<span class="sd">    (r, Z)</span>
-<span class="sd">    &gt;&gt;&gt; R_nl(1, 0, r, Z)</span>
-<span class="sd">    2*sqrt(Z**3)*exp(-Z*r)</span>
-<span class="sd">    &gt;&gt;&gt; R_nl(2, 0, r, Z)</span>
-<span class="sd">    sqrt(2)*(-Z*r + 2)*sqrt(Z**3)*exp(-Z*r/2)/4</span>
-<span class="sd">    &gt;&gt;&gt; R_nl(2, 1, r, Z)</span>
-<span class="sd">    sqrt(6)*Z*r*sqrt(Z**3)*exp(-Z*r/2)/12</span>
-
-<span class="sd">    For Hydrogen atom, you can just use the default value of Z=1:</span>
-
-<span class="sd">    &gt;&gt;&gt; R_nl(1, 0, r)</span>
-<span class="sd">    2*exp(-r)</span>
-<span class="sd">    &gt;&gt;&gt; R_nl(2, 0, r)</span>
-<span class="sd">    sqrt(2)*(-r + 2)*exp(-r/2)/4</span>
-<span class="sd">    &gt;&gt;&gt; R_nl(3, 0, r)</span>
-<span class="sd">    2*sqrt(3)*(2*r**2/9 - 2*r + 3)*exp(-r/3)/27</span>
-
-<span class="sd">    For Silver atom, you would use Z=47:</span>
-
-<span class="sd">    &gt;&gt;&gt; R_nl(1, 0, r, Z=47)</span>
-<span class="sd">    94*sqrt(47)*exp(-47*r)</span>
-<span class="sd">    &gt;&gt;&gt; R_nl(2, 0, r, Z=47)</span>
-<span class="sd">    47*sqrt(94)*(-47*r + 2)*exp(-47*r/2)/4</span>
-<span class="sd">    &gt;&gt;&gt; R_nl(3, 0, r, Z=47)</span>
-<span class="sd">    94*sqrt(141)*(4418*r**2/9 - 94*r + 3)*exp(-47*r/3)/27</span>
-
-<span class="sd">    The normalization of the radial wavefunction is:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import integrate, oo</span>
-<span class="sd">    &gt;&gt;&gt; integrate(R_nl(1, 0, r)**2 * r**2, (r, 0, oo))</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; integrate(R_nl(2, 0, r)**2 * r**2, (r, 0, oo))</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; integrate(R_nl(2, 1, r)**2 * r**2, (r, 0, oo))</span>
-<span class="sd">    1</span>
-
-<span class="sd">    It holds for any atomic number:</span>
-
-<span class="sd">    &gt;&gt;&gt; integrate(R_nl(1, 0, r, Z=2)**2 * r**2, (r, 0, oo))</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; integrate(R_nl(2, 0, r, Z=3)**2 * r**2, (r, 0, oo))</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; integrate(R_nl(2, 1, r, Z=4)**2 * r**2, (r, 0, oo))</span>
-<span class="sd">    1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># sympify arguments</span>
-    <span class="n">n</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">Z</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">S</span><span class="p">(</span><span class="n">l</span><span class="p">),</span> <span class="n">S</span><span class="p">(</span><span class="n">r</span><span class="p">),</span> <span class="n">S</span><span class="p">(</span><span class="n">Z</span><span class="p">)</span>
-    <span class="c"># radial quantum number</span>
-    <span class="n">n_r</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="n">l</span> <span class="o">-</span> <span class="mi">1</span>
-    <span class="c"># rescaled &quot;r&quot;</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">Z</span>  <span class="c"># Bohr radius</span>
-    <span class="n">r0</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">r</span> <span class="o">/</span> <span class="p">(</span><span class="n">n</span> <span class="o">*</span> <span class="n">a</span><span class="p">)</span>
-    <span class="c"># normalization coefficient</span>
-    <span class="n">C</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">((</span><span class="n">S</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">a</span><span class="p">))</span><span class="o">**</span><span class="mi">3</span> <span class="o">*</span> <span class="n">factorial</span><span class="p">(</span><span class="n">n_r</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="n">l</span><span class="p">)))</span>
-    <span class="c"># This is an equivalent normalization coefficient, that can be found in</span>
-    <span class="c"># some books. Both coefficients seem to be the same fast:</span>
-    <span class="c"># C =  S(2)/n**2 * sqrt(1/a**3 * factorial(n_r) / (factorial(n+l)))</span>
-    <span class="k">return</span> <span class="n">C</span> <span class="o">*</span> <span class="n">r0</span><span class="o">**</span><span class="n">l</span> <span class="o">*</span> <span class="n">assoc_laguerre</span><span class="p">(</span><span class="n">n_r</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">l</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">r0</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span> <span class="o">*</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">r0</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="E_nl"><a class="viewcode-back" href="../../../modules/physics/hydrogen.html#sympy.physics.hydrogen.E_nl">[docs]</a><span class="k">def</span> <span class="nf">E_nl</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">Z</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the energy of the state (n, l) in Hartree atomic units.</span>
-
-<span class="sd">    The energy doesn&#39;t depend on &quot;l&quot;.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import var</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.hydrogen import E_nl</span>
-<span class="sd">    &gt;&gt;&gt; var(&quot;n Z&quot;)</span>
-<span class="sd">    (n, Z)</span>
-<span class="sd">    &gt;&gt;&gt; E_nl(n, Z)</span>
-<span class="sd">    -Z**2/(2*n**2)</span>
-<span class="sd">    &gt;&gt;&gt; E_nl(1)</span>
-<span class="sd">    -1/2</span>
-<span class="sd">    &gt;&gt;&gt; E_nl(2)</span>
-<span class="sd">    -1/8</span>
-<span class="sd">    &gt;&gt;&gt; E_nl(3)</span>
-<span class="sd">    -1/18</span>
-<span class="sd">    &gt;&gt;&gt; E_nl(3, 47)</span>
-<span class="sd">    -2209/18</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span><span class="p">,</span> <span class="n">Z</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">S</span><span class="p">(</span><span class="n">Z</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">and</span> <span class="p">(</span><span class="n">n</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;&#39;n&#39; must be positive integer&quot;</span><span class="p">)</span>
-    <span class="k">return</span> <span class="o">-</span><span class="n">Z</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="E_nl_dirac"><a class="viewcode-back" href="../../../modules/physics/hydrogen.html#sympy.physics.hydrogen.E_nl_dirac">[docs]</a><span class="k">def</span> <span class="nf">E_nl_dirac</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="n">spin_up</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">Z</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">Float</span><span class="p">(</span><span class="s">&quot;137.035999037&quot;</span><span class="p">)):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the relativistic energy of the state (n, l, spin) in Hartree atomic</span>
-<span class="sd">    units.</span>
-
-<span class="sd">    The energy is calculated from the Dirac equation. The rest mass energy is</span>
-<span class="sd">    *not* included.</span>
-
-<span class="sd">    n, l</span>
-<span class="sd">        quantum numbers &#39;n&#39; and &#39;l&#39;</span>
-<span class="sd">    spin_up</span>
-<span class="sd">        True if the electron spin is up (default), otherwise down</span>
-<span class="sd">    Z</span>
-<span class="sd">        atomic number (1 for Hydrogen, 2 for Helium, ...)</span>
-<span class="sd">    c</span>
-<span class="sd">        speed of light in atomic units. Default value is 137.035999037,</span>
-<span class="sd">        taken from: http://arxiv.org/abs/1012.3627</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.hydrogen import E_nl_dirac</span>
-<span class="sd">    &gt;&gt;&gt; E_nl_dirac(1, 0)</span>
-<span class="sd">    -0.500006656595360</span>
-
-<span class="sd">    &gt;&gt;&gt; E_nl_dirac(2, 0)</span>
-<span class="sd">    -0.125002080189006</span>
-<span class="sd">    &gt;&gt;&gt; E_nl_dirac(2, 1)</span>
-<span class="sd">    -0.125000416028342</span>
-<span class="sd">    &gt;&gt;&gt; E_nl_dirac(2, 1, False)</span>
-<span class="sd">    -0.125002080189006</span>
-
-<span class="sd">    &gt;&gt;&gt; E_nl_dirac(3, 0)</span>
-<span class="sd">    -0.0555562951740285</span>
-<span class="sd">    &gt;&gt;&gt; E_nl_dirac(3, 1)</span>
-<span class="sd">    -0.0555558020932949</span>
-<span class="sd">    &gt;&gt;&gt; E_nl_dirac(3, 1, False)</span>
-<span class="sd">    -0.0555562951740285</span>
-<span class="sd">    &gt;&gt;&gt; E_nl_dirac(3, 2)</span>
-<span class="sd">    -0.0555556377366884</span>
-<span class="sd">    &gt;&gt;&gt; E_nl_dirac(3, 2, False)</span>
-<span class="sd">    -0.0555558020932949</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">l</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;&#39;l&#39; must be positive or zero&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">n</span> <span class="o">&gt;</span> <span class="n">l</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;&#39;n&#39; must be greater than &#39;l&#39;&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="p">(</span><span class="n">l</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">spin_up</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Spin must be up for l==0.&quot;</span><span class="p">)</span>
-    <span class="c"># skappa is sign*kappa, where sign contains the correct sign</span>
-    <span class="k">if</span> <span class="n">spin_up</span><span class="p">:</span>
-        <span class="n">skappa</span> <span class="o">=</span> <span class="o">-</span><span class="n">l</span> <span class="o">-</span> <span class="mi">1</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">skappa</span> <span class="o">=</span> <span class="o">-</span><span class="n">l</span>
-    <span class="n">c</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-    <span class="n">beta</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">skappa</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">Z</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="n">c</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">c</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">Z</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="n">skappa</span> <span class="o">+</span> <span class="n">beta</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="n">c</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="n">c</span><span class="o">**</span><span class="mi">2</span></div>
-</pre></div>
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-            
-  <h1>Source code for sympy.physics.matrices</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Known matrices related to physics&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">I</span>
-
-
-<div class="viewcode-block" id="msigma"><a class="viewcode-back" href="../../../modules/physics/matrices.html#sympy.physics.matrices.msigma">[docs]</a><span class="k">def</span> <span class="nf">msigma</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns a Pauli matrix sigma_i. i=1,2,3</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    http://en.wikipedia.org/wiki/Pauli_matrices</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.matrices import msigma</span>
-<span class="sd">    &gt;&gt;&gt; msigma(1)</span>
-<span class="sd">    [0, 1]</span>
-<span class="sd">    [1, 0]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">mat</span> <span class="o">=</span> <span class="p">(</span> <span class="p">(</span>
-            <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span>
-            <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="p">)</span> <span class="p">)</span>
-    <span class="k">elif</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="n">mat</span> <span class="o">=</span> <span class="p">(</span> <span class="p">(</span>
-            <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="n">I</span><span class="p">),</span>
-            <span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="p">)</span> <span class="p">)</span>
-    <span class="k">elif</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-        <span class="n">mat</span> <span class="o">=</span> <span class="p">(</span> <span class="p">(</span>
-            <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
-            <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-        <span class="p">)</span> <span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s">&quot;Invalid Pauli index&quot;</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">mat</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="pat_matrix"><a class="viewcode-back" href="../../../modules/physics/matrices.html#sympy.physics.matrices.pat_matrix">[docs]</a><span class="k">def</span> <span class="nf">pat_matrix</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">dx</span><span class="p">,</span> <span class="n">dy</span><span class="p">,</span> <span class="n">dz</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns the Parallel Axis Theorem matrix to translate the inertia</span>
-<span class="sd">    matrix a distance of (dx, dy, dz) for a body of mass m.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    --------</span>
-<span class="sd">    If the point we want the inertia about is a distance of 2 units of</span>
-<span class="sd">    length and 1 unit along the x-axis we get:</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.matrices import pat_matrix</span>
-<span class="sd">    &gt;&gt;&gt; pat_matrix(2,1,0,0)</span>
-<span class="sd">    [0, 0, 0]</span>
-<span class="sd">    [0, 2, 0]</span>
-<span class="sd">    [0, 0, 2]</span>
-
-<span class="sd">    In case we want to find the inertia along a vector of (1,1,1):</span>
-<span class="sd">    &gt;&gt;&gt; pat_matrix(2,1,1,1)</span>
-<span class="sd">    [ 4, -2, -2]</span>
-<span class="sd">    [-2,  4, -2]</span>
-<span class="sd">    [-2, -2,  4]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">dxdy</span> <span class="o">=</span> <span class="o">-</span><span class="n">dx</span><span class="o">*</span><span class="n">dy</span>
-    <span class="n">dydz</span> <span class="o">=</span> <span class="o">-</span><span class="n">dy</span><span class="o">*</span><span class="n">dz</span>
-    <span class="n">dzdx</span> <span class="o">=</span> <span class="o">-</span><span class="n">dz</span><span class="o">*</span><span class="n">dx</span>
-    <span class="n">dxdx</span> <span class="o">=</span> <span class="n">dx</span><span class="o">**</span><span class="mi">2</span>
-    <span class="n">dydy</span> <span class="o">=</span> <span class="n">dy</span><span class="o">**</span><span class="mi">2</span>
-    <span class="n">dzdz</span> <span class="o">=</span> <span class="n">dz</span><span class="o">**</span><span class="mi">2</span>
-    <span class="n">mat</span> <span class="o">=</span> <span class="p">((</span><span class="n">dydy</span> <span class="o">+</span> <span class="n">dzdz</span><span class="p">,</span> <span class="n">dxdy</span><span class="p">,</span> <span class="n">dzdx</span><span class="p">),</span>
-           <span class="p">(</span><span class="n">dxdy</span><span class="p">,</span> <span class="n">dxdx</span> <span class="o">+</span> <span class="n">dzdz</span><span class="p">,</span> <span class="n">dydz</span><span class="p">),</span>
-           <span class="p">(</span><span class="n">dzdx</span><span class="p">,</span> <span class="n">dydz</span><span class="p">,</span> <span class="n">dydy</span> <span class="o">+</span> <span class="n">dxdx</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">m</span><span class="o">*</span><span class="n">Matrix</span><span class="p">(</span><span class="n">mat</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="mgamma"><a class="viewcode-back" href="../../../modules/physics/matrices.html#sympy.physics.matrices.mgamma">[docs]</a><span class="k">def</span> <span class="nf">mgamma</span><span class="p">(</span><span class="n">mu</span><span class="p">,</span> <span class="n">lower</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns a Dirac gamma matrix gamma^mu in the standard</span>
-<span class="sd">    (Dirac) representation.</span>
-
-<span class="sd">    If you want gamma_mu, use gamma(mu, True).</span>
-
-<span class="sd">    We use a convention:</span>
-
-<span class="sd">    gamma^5 = I * gamma^0 * gamma^1 * gamma^2 * gamma^3</span>
-<span class="sd">    gamma_5 = I * gamma_0 * gamma_1 * gamma_2 * gamma_3 = - gamma^5</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    http://en.wikipedia.org/wiki/Gamma_matrices</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.matrices import mgamma</span>
-<span class="sd">    &gt;&gt;&gt; mgamma(1)</span>
-<span class="sd">    [ 0,  0, 0, 1]</span>
-<span class="sd">    [ 0,  0, 1, 0]</span>
-<span class="sd">    [ 0, -1, 0, 0]</span>
-<span class="sd">    [-1,  0, 0, 0]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">mu</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">]:</span>
-        <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s">&quot;Invalid Dirac index&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">mu</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">mat</span> <span class="o">=</span> <span class="p">(</span>
-            <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
-            <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
-            <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
-            <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-        <span class="p">)</span>
-    <span class="k">elif</span> <span class="n">mu</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">mat</span> <span class="o">=</span> <span class="p">(</span>
-            <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span>
-            <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
-            <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
-            <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="p">)</span>
-    <span class="k">elif</span> <span class="n">mu</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="n">mat</span> <span class="o">=</span> <span class="p">(</span>
-            <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="n">I</span><span class="p">),</span>
-            <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
-            <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
-            <span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="p">)</span>
-    <span class="k">elif</span> <span class="n">mu</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-        <span class="n">mat</span> <span class="o">=</span> <span class="p">(</span>
-            <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
-            <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">),</span>
-            <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
-            <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="p">)</span>
-    <span class="k">elif</span> <span class="n">mu</span> <span class="o">==</span> <span class="mi">5</span><span class="p">:</span>
-        <span class="n">mat</span> <span class="o">=</span> <span class="p">(</span>
-            <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
-            <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span>
-            <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
-            <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="p">)</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">mat</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">lower</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">mu</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">]:</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="o">-</span><span class="n">m</span>
-    <span class="k">return</span> <span class="n">m</span>
-
-<span class="c">#Minkowski tensor using the convention (+,-,-,-) used in the Quantum Field</span>
-<span class="c">#Theory</span></div>
-<span class="n">minkowski_tensor</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span> <span class="p">(</span>
-    <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
-    <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
-    <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
-    <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="p">))</span>
-</pre></div>
-
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-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
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@@ -1,2050 +0,0 @@
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-            
-  <h1>Source code for sympy.physics.mechanics.essential</h1><div class="highlight"><pre>
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;ReferenceFrame&#39;</span><span class="p">,</span> <span class="s">&#39;Vector&#39;</span><span class="p">,</span> <span class="s">&#39;Dyadic&#39;</span><span class="p">,</span> <span class="s">&#39;dynamicsymbols&#39;</span><span class="p">,</span>
-           <span class="s">&#39;MechanicsStrPrinter&#39;</span><span class="p">,</span> <span class="s">&#39;MechanicsPrettyPrinter&#39;</span><span class="p">,</span>
-           <span class="s">&#39;MechanicsLatexPrinter&#39;</span><span class="p">]</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">Matrix</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">eye</span><span class="p">,</span> <span class="n">trigsimp</span><span class="p">,</span> <span class="n">diff</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span>
-    <span class="n">expand</span><span class="p">,</span> <span class="n">zeros</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">,</span> <span class="n">Function</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">Add</span><span class="p">,</span>
-    <span class="n">solve</span><span class="p">,</span> <span class="n">S</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">C</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">UndefinedFunction</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.conventions</span> <span class="kn">import</span> <span class="n">split_super_sub</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.latex</span> <span class="kn">import</span> <span class="n">LatexPrinter</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.pretty.pretty</span> <span class="kn">import</span> <span class="n">PrettyPrinter</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.pretty.stringpict</span> <span class="kn">import</span> <span class="n">prettyForm</span><span class="p">,</span> <span class="n">stringPict</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.str</span> <span class="kn">import</span> <span class="n">StrPrinter</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">group</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="nb">reduce</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-
-<div class="viewcode-block" id="Dyadic"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Dyadic">[docs]</a><span class="k">class</span> <span class="nc">Dyadic</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A Dyadic object.</span>
-
-<span class="sd">    See:</span>
-<span class="sd">    http://en.wikipedia.org/wiki/Dyadic_tensor</span>
-<span class="sd">    Kane, T., Levinson, D. Dynamics Theory and Applications. 1985 McGraw-Hill</span>
-
-<span class="sd">    A more powerful way to represent a rigid body&#39;s inertia. While it is more</span>
-<span class="sd">    complex, by choosing Dyadic components to be in body fixed basis vectors,</span>
-<span class="sd">    the resulting matrix is equivalent to the inertia tensor.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">inlist</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Just like Vector&#39;s init, you shouldn&#39;t call this.</span>
-
-<span class="sd">        Stores a Dyadic as a list of lists; the inner list has the measure number</span>
-<span class="sd">        and the two unit vectors; the outerlist holds each unique unit vector</span>
-<span class="sd">        pair.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">while</span> <span class="nb">len</span><span class="p">(</span><span class="n">inlist</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">added</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-                <span class="k">if</span> <span class="p">((</span><span class="nb">str</span><span class="p">(</span><span class="n">inlist</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span> <span class="o">==</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]))</span> <span class="ow">and</span>
-                        <span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">inlist</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">2</span><span class="p">])</span> <span class="o">==</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">2</span><span class="p">]))):</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span>
-                        <span class="n">inlist</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">inlist</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">],</span> <span class="n">inlist</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">2</span><span class="p">])</span>
-                    <span class="n">inlist</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">inlist</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-                    <span class="n">added</span> <span class="o">=</span> <span class="mi">1</span>
-                    <span class="k">break</span>
-            <span class="k">if</span> <span class="n">added</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">inlist</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-                <span class="n">inlist</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">inlist</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="c"># This code is to remove empty parts from the list</span>
-        <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">if</span> <span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span> <span class="o">|</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span> <span class="o">|</span>
-                    <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)):</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-                <span class="n">i</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The add operator for Dyadic. &quot;&quot;&quot;</span>
-        <span class="n">other</span> <span class="o">=</span> <span class="n">_check_dyadic</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Dyadic</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">+</span> <span class="n">other</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__and__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The inner product operator for a Dyadic and a Dyadic or Vector.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        other : Dyadic or Vector</span>
-<span class="sd">            The other Dyadic or Vector to take the inner product with</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, outer</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; D1 = outer(N.x, N.y)</span>
-<span class="sd">        &gt;&gt;&gt; D2 = outer(N.y, N.y)</span>
-<span class="sd">        &gt;&gt;&gt; D1.dot(D2)</span>
-<span class="sd">        (N.x|N.y)</span>
-<span class="sd">        &gt;&gt;&gt; D1.dot(N.y)</span>
-<span class="sd">        N.x</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Dyadic</span><span class="p">):</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">_check_dyadic</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-            <span class="n">ol</span> <span class="o">=</span> <span class="n">Dyadic</span><span class="p">([])</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">i2</span><span class="p">,</span> <span class="n">v2</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-                    <span class="n">ol</span> <span class="o">+=</span> <span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">v2</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">&amp;</span> <span class="n">v2</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="o">*</span> <span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">|</span> <span class="n">v2</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">_check_vector</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-            <span class="n">ol</span> <span class="o">=</span> <span class="n">Vector</span><span class="p">([])</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-                <span class="n">ol</span> <span class="o">+=</span> <span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">&amp;</span> <span class="n">other</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">ol</span>
-
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Divides the Dyadic by a sympifyable expression. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="n">__truediv__</span> <span class="o">=</span> <span class="n">__div__</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Tests for equality.</span>
-
-<span class="sd">        Is currently weak; needs stronger comparison testing</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">other</span> <span class="o">=</span> <span class="n">_check_dyadic</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">if</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">==</span> <span class="p">[])</span> <span class="ow">and</span> <span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">args</span> <span class="o">==</span> <span class="p">[]):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">elif</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">==</span> <span class="p">[])</span> <span class="ow">or</span> <span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">args</span> <span class="o">==</span> <span class="p">[]):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="nb">set</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="nb">set</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Multiplies the Dyadic by a sympifyable expression.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        other : Sympafiable</span>
-<span class="sd">            The scalar to multiply this Dyadic with</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, outer</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; d = outer(N.x, N.x)</span>
-<span class="sd">        &gt;&gt;&gt; 5 * d</span>
-<span class="sd">        5*(N.x|N.x)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">newlist</span> <span class="o">=</span> <span class="p">[</span><span class="n">v</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">newlist</span><span class="p">):</span>
-            <span class="n">newlist</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span> <span class="o">*</span> <span class="n">newlist</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">newlist</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">],</span>
-                    <span class="n">newlist</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">2</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">Dyadic</span><span class="p">(</span><span class="n">newlist</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span> <span class="o">*</span> <span class="o">-</span><span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">ar</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>  <span class="c"># just to shorten things</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">ar</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="n">ol</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c"># output list, to be concatenated to a string</span>
-        <span class="n">mlp</span> <span class="o">=</span> <span class="n">MechanicsLatexPrinter</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">ar</span><span class="p">):</span>
-            <span class="c"># if the coef of the dyadic is 1, we skip the 1</span>
-            <span class="k">if</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">ol</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39; + &#39;</span> <span class="o">+</span> <span class="n">mlp</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span> <span class="o">+</span> <span class="s">r&quot;\otimes &quot;</span> <span class="o">+</span>
-                        <span class="n">mlp</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">2</span><span class="p">]))</span>
-            <span class="c"># if the coef of the dyadic is -1, we skip the 1</span>
-            <span class="k">elif</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                <span class="n">ol</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39; - &#39;</span> <span class="o">+</span>
-                          <span class="n">mlp</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span> <span class="o">+</span>
-                          <span class="s">r&quot;\otimes &quot;</span> <span class="o">+</span>
-                          <span class="n">mlp</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">2</span><span class="p">]))</span>
-            <span class="c"># If the coefficient of the dyadic is not 1 or -1,</span>
-            <span class="c"># we might wrap it in parentheses, for readability.</span>
-            <span class="k">elif</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">arg_str</span> <span class="o">=</span> <span class="n">mlp</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">Add</span><span class="p">):</span>
-                    <span class="n">arg_str</span> <span class="o">=</span> <span class="s">&#39;(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="n">arg_str</span>
-                <span class="k">if</span> <span class="n">arg_str</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39;-&#39;</span><span class="p">):</span>
-                    <span class="n">arg_str</span> <span class="o">=</span> <span class="n">arg_str</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-                    <span class="n">str_start</span> <span class="o">=</span> <span class="s">&#39; - &#39;</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">str_start</span> <span class="o">=</span> <span class="s">&#39; + &#39;</span>
-                <span class="n">ol</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">str_start</span> <span class="o">+</span> <span class="n">arg_str</span> <span class="o">+</span> <span class="s">r&quot; &quot;</span> <span class="o">+</span>
-                          <span class="n">mlp</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span> <span class="o">+</span>
-                          <span class="s">r&quot;\otimes &quot;</span> <span class="o">+</span>
-                          <span class="n">mlp</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">2</span><span class="p">]))</span>
-        <span class="n">outstr</span> <span class="o">=</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">ol</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">outstr</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39; + &#39;</span><span class="p">):</span>
-            <span class="n">outstr</span> <span class="o">=</span> <span class="n">outstr</span><span class="p">[</span><span class="mi">3</span><span class="p">:]</span>
-        <span class="k">elif</span> <span class="n">outstr</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">):</span>
-            <span class="n">outstr</span> <span class="o">=</span> <span class="n">outstr</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-        <span class="k">return</span> <span class="n">outstr</span>
-
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">e</span> <span class="o">=</span> <span class="bp">self</span>
-
-        <span class="k">class</span> <span class="nc">Fake</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-            <span class="n">baseline</span> <span class="o">=</span> <span class="mi">0</span>
-
-            <span class="k">def</span> <span class="nf">render</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-                <span class="bp">self</span> <span class="o">=</span> <span class="n">e</span>
-                <span class="n">ar</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>  <span class="c"># just to shorten things</span>
-                <span class="n">mpp</span> <span class="o">=</span> <span class="n">MechanicsPrettyPrinter</span><span class="p">()</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">ar</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-                <span class="n">ol</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c"># output list, to be concatenated to a string</span>
-                <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">ar</span><span class="p">):</span>
-                    <span class="c"># if the coef of the dyadic is 1, we skip the 1</span>
-                    <span class="k">if</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="n">ol</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot; + &quot;</span> <span class="o">+</span>
-                                  <span class="n">mpp</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span> <span class="o">+</span>
-                                  <span class="s">&quot;</span><span class="se">\u2a02</span><span class="s"> &quot;</span> <span class="o">+</span>
-                                  <span class="n">mpp</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">2</span><span class="p">]))</span>
-                    <span class="c"># if the coef of the dyadic is -1, we skip the 1</span>
-                    <span class="k">elif</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                        <span class="n">ol</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot; - &quot;</span> <span class="o">+</span>
-                                  <span class="n">mpp</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span> <span class="o">+</span>
-                                  <span class="s">&quot;</span><span class="se">\u2a02</span><span class="s"> &quot;</span> <span class="o">+</span>
-                                  <span class="n">mpp</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">2</span><span class="p">]))</span>
-                    <span class="c"># If the coefficient of the dyadic is not 1 or -1,</span>
-                    <span class="c"># we might wrap it in parentheses, for readability.</span>
-                    <span class="k">elif</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">arg_str</span> <span class="o">=</span> <span class="n">mpp</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span>
-                        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">Add</span><span class="p">):</span>
-                            <span class="n">arg_str</span> <span class="o">=</span> <span class="s">&quot;(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="n">arg_str</span>
-                        <span class="k">if</span> <span class="n">arg_str</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&quot;-&quot;</span><span class="p">):</span>
-                            <span class="n">arg_str</span> <span class="o">=</span> <span class="n">arg_str</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-                            <span class="n">str_start</span> <span class="o">=</span> <span class="s">&quot; - &quot;</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="n">str_start</span> <span class="o">=</span> <span class="s">&quot; + &quot;</span>
-                        <span class="n">ol</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">str_start</span> <span class="o">+</span> <span class="n">arg_str</span> <span class="o">+</span> <span class="s">&quot; &quot;</span> <span class="o">+</span>
-                                  <span class="n">mpp</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span> <span class="o">+</span>
-                                  <span class="s">&quot;</span><span class="se">\u2a02</span><span class="s"> &quot;</span> <span class="o">+</span>
-                                  <span class="n">mpp</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">2</span><span class="p">]))</span>
-                <span class="n">outstr</span> <span class="o">=</span> <span class="s">&quot;&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">ol</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">outstr</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&quot; + &quot;</span><span class="p">):</span>
-                    <span class="n">outstr</span> <span class="o">=</span> <span class="n">outstr</span><span class="p">[</span><span class="mi">3</span><span class="p">:]</span>
-                <span class="k">elif</span> <span class="n">outstr</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&quot; &quot;</span><span class="p">):</span>
-                    <span class="n">outstr</span> <span class="o">=</span> <span class="n">outstr</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-                <span class="k">return</span> <span class="n">outstr</span>
-        <span class="k">return</span> <span class="n">Fake</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__rand__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The inner product operator for a Vector or Dyadic, and a Dyadic</span>
-
-<span class="sd">        This is for: Vector dot Dyadic</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        other : Vector</span>
-<span class="sd">            The vector we are dotting with</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, dot, outer</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; d = outer(N.x, N.x)</span>
-<span class="sd">        &gt;&gt;&gt; dot(N.x, d)</span>
-<span class="sd">        N.x</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">other</span> <span class="o">=</span> <span class="n">_check_vector</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="n">ol</span> <span class="o">=</span> <span class="n">Vector</span><span class="p">([])</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="n">ol</span> <span class="o">+=</span> <span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">v</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">&amp;</span> <span class="n">other</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">ol</span>
-
-    <span class="k">def</span> <span class="nf">__rsub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span> <span class="o">*</span> <span class="bp">self</span><span class="p">)</span> <span class="o">+</span> <span class="n">other</span>
-
-    <span class="k">def</span> <span class="nf">__rxor__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;For a cross product in the form: Vector x Dyadic</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        other : Vector</span>
-<span class="sd">            The Vector that we are crossing this Dyadic with</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, outer, cross</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; d = outer(N.x, N.x)</span>
-<span class="sd">        &gt;&gt;&gt; cross(N.y, d)</span>
-<span class="sd">        - (N.z|N.x)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">other</span> <span class="o">=</span> <span class="n">_check_vector</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="n">ol</span> <span class="o">=</span> <span class="n">Dyadic</span><span class="p">([])</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="n">ol</span> <span class="o">+=</span> <span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">((</span><span class="n">other</span> <span class="o">^</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="o">|</span> <span class="n">v</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">ol</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Printing method. &quot;&quot;&quot;</span>
-        <span class="n">ar</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>  <span class="c"># just to shorten things</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">ar</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="n">ol</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c"># output list, to be concatenated to a string</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">ar</span><span class="p">):</span>
-            <span class="c"># if the coef of the dyadic is 1, we skip the 1</span>
-            <span class="k">if</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">ol</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39; + (&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span> <span class="o">+</span> <span class="s">&#39;|&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">2</span><span class="p">])</span> <span class="o">+</span> <span class="s">&#39;)&#39;</span><span class="p">)</span>
-            <span class="c"># if the coef of the dyadic is -1, we skip the 1</span>
-            <span class="k">elif</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                <span class="n">ol</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39; - (&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span> <span class="o">+</span> <span class="s">&#39;|&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">2</span><span class="p">])</span> <span class="o">+</span> <span class="s">&#39;)&#39;</span><span class="p">)</span>
-            <span class="c"># If the coefficient of the dyadic is not 1 or -1,</span>
-            <span class="c"># we might wrap it in parentheses, for readability.</span>
-            <span class="k">elif</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">arg_str</span> <span class="o">=</span> <span class="n">MechanicsStrPrinter</span><span class="p">()</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">Add</span><span class="p">):</span>
-                    <span class="n">arg_str</span> <span class="o">=</span> <span class="s">&quot;(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="n">arg_str</span>
-                <span class="k">if</span> <span class="n">arg_str</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;-&#39;</span><span class="p">:</span>
-                    <span class="n">arg_str</span> <span class="o">=</span> <span class="n">arg_str</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-                    <span class="n">str_start</span> <span class="o">=</span> <span class="s">&#39; - &#39;</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">str_start</span> <span class="o">=</span> <span class="s">&#39; + &#39;</span>
-                <span class="n">ol</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">str_start</span> <span class="o">+</span> <span class="n">arg_str</span> <span class="o">+</span> <span class="s">&#39;*(&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span> <span class="o">+</span>
-                          <span class="s">&#39;|&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">2</span><span class="p">])</span> <span class="o">+</span> <span class="s">&#39;)&#39;</span><span class="p">)</span>
-        <span class="n">outstr</span> <span class="o">=</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">ol</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">outstr</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39; + &#39;</span><span class="p">):</span>
-            <span class="n">outstr</span> <span class="o">=</span> <span class="n">outstr</span><span class="p">[</span><span class="mi">3</span><span class="p">:]</span>
-        <span class="k">elif</span> <span class="n">outstr</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">):</span>
-            <span class="n">outstr</span> <span class="o">=</span> <span class="n">outstr</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-        <span class="k">return</span> <span class="n">outstr</span>
-
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The subtraction operator. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__add__</span><span class="p">(</span><span class="n">other</span> <span class="o">*</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__xor__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;For a cross product in the form: Dyadic x Vector.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        other : Vector</span>
-<span class="sd">            The Vector that we are crossing this Dyadic with</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, outer, cross</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; d = outer(N.x, N.x)</span>
-<span class="sd">        &gt;&gt;&gt; cross(d, N.y)</span>
-<span class="sd">        (N.x|N.z)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">other</span> <span class="o">=</span> <span class="n">_check_vector</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="n">ol</span> <span class="o">=</span> <span class="n">Dyadic</span><span class="p">([])</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="n">ol</span> <span class="o">+=</span> <span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">|</span> <span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">^</span> <span class="n">other</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">ol</span>
-
-    <span class="n">_sympystr</span> <span class="o">=</span> <span class="n">__str__</span>
-    <span class="n">_sympyrepr</span> <span class="o">=</span> <span class="n">_sympystr</span>
-    <span class="n">__repr__</span> <span class="o">=</span> <span class="n">__str__</span>
-    <span class="n">__radd__</span> <span class="o">=</span> <span class="n">__add__</span>
-    <span class="n">__rmul__</span> <span class="o">=</span> <span class="n">__mul__</span>
-
-<div class="viewcode-block" id="Dyadic.express"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Dyadic.express">[docs]</a>    <span class="k">def</span> <span class="nf">express</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">frame1</span><span class="p">,</span> <span class="n">frame2</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Expresses this Dyadic in alternate frame(s)</span>
-
-<span class="sd">        The first frame is the list side expression, the second frame is the</span>
-<span class="sd">        right side; if Dyadic is in form A.x|B.y, you can express it in two</span>
-<span class="sd">        different frames. If no second frame is given, the Dyadic is expressed in</span>
-<span class="sd">        only one frame.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        frame1 : ReferenceFrame</span>
-<span class="sd">            The frame to express the left side of the Dyadic in</span>
-<span class="sd">        frame2 : ReferenceFrame</span>
-<span class="sd">            If provided, the frame to express the right side of the Dyadic in</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, outer, dynamicsymbols</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; q = dynamicsymbols(&#39;q&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; B = N.orientnew(&#39;B&#39;, &#39;Axis&#39;, [q, N.z])</span>
-<span class="sd">        &gt;&gt;&gt; d = outer(N.x, N.x)</span>
-<span class="sd">        &gt;&gt;&gt; d.express(B, N)</span>
-<span class="sd">        cos(q)*(B.x|N.x) - sin(q)*(B.y|N.x)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="n">frame2</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">frame2</span> <span class="o">=</span> <span class="n">frame1</span>
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">frame1</span><span class="p">)</span>
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">frame2</span><span class="p">)</span>
-        <span class="n">ol</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="n">ol</span> <span class="o">+=</span> <span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">express</span><span class="p">(</span><span class="n">frame1</span><span class="p">)</span> <span class="o">|</span> <span class="n">v</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">express</span><span class="p">(</span><span class="n">frame2</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">ol</span>
-</div>
-<div class="viewcode-block" id="Dyadic.doit"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Dyadic.doit">[docs]</a>    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Calls .doit() on each term in the Dyadic&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">sum</span><span class="p">([</span><span class="n">Dyadic</span><span class="p">(</span> <span class="p">[</span> <span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">),</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span> <span class="p">])</span> <span class="k">for</span>
-                    <span class="n">v</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-</div>
-<div class="viewcode-block" id="Dyadic.dt"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Dyadic.dt">[docs]</a>    <span class="k">def</span> <span class="nf">dt</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">frame</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Take the time derivative of this Dyadic in a frame.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        frame : ReferenceFrame</span>
-<span class="sd">            The frame to take the time derivative in</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, outer, dynamicsymbols</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; q = dynamicsymbols(&#39;q&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; B = N.orientnew(&#39;B&#39;, &#39;Axis&#39;, [q, N.z])</span>
-<span class="sd">        &gt;&gt;&gt; d = outer(N.x, N.x)</span>
-<span class="sd">        &gt;&gt;&gt; d.dt(B)</span>
-<span class="sd">        - q&#39;*(N.y|N.x) - q&#39;*(N.x|N.y)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">frame</span><span class="p">)</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span>
-        <span class="n">ol</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="n">ol</span> <span class="o">+=</span> <span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">|</span> <span class="n">v</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span>
-            <span class="n">ol</span> <span class="o">+=</span> <span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">dt</span><span class="p">(</span><span class="n">frame</span><span class="p">)</span> <span class="o">|</span> <span class="n">v</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span>
-            <span class="n">ol</span> <span class="o">+=</span> <span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">|</span> <span class="n">v</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">dt</span><span class="p">(</span><span class="n">frame</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">ol</span>
-</div>
-<div class="viewcode-block" id="Dyadic.simplify"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Dyadic.simplify">[docs]</a>    <span class="k">def</span> <span class="nf">simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Simplify the elements in the Dyadic in-place.&quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">simplify</span><span class="p">(),</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-</div>
-<div class="viewcode-block" id="Dyadic.subs"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Dyadic.subs">[docs]</a>    <span class="k">def</span> <span class="nf">subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Substituion on the Dyadic.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; s = Symbol(&#39;s&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; a = s * (N.x|N.x)</span>
-<span class="sd">        &gt;&gt;&gt; a.subs({s: 2})</span>
-<span class="sd">        2*(N.x|N.x)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">return</span> <span class="nb">sum</span><span class="p">([</span> <span class="n">Dyadic</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">2</span><span class="p">])])</span>
-                     <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-</div>
-    <span class="n">dot</span> <span class="o">=</span> <span class="n">__and__</span>
-    <span class="n">cross</span> <span class="o">=</span> <span class="n">__xor__</span>
-
-</div>
-<div class="viewcode-block" id="ReferenceFrame"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.ReferenceFrame">[docs]</a><span class="k">class</span> <span class="nc">ReferenceFrame</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A reference frame in classical mechanics.</span>
-
-<span class="sd">    ReferenceFrame is a class used to represent a reference frame in classical</span>
-<span class="sd">    mechanics. It has a standard basis of three unit vectors in the frame&#39;s</span>
-<span class="sd">    x, y, and z directions.</span>
-
-<span class="sd">    It also can have a rotation relative to a parent frame; this rotation is</span>
-<span class="sd">    defined by a direction cosine matrix relating this frame&#39;s basis vectors to</span>
-<span class="sd">    the parent frame&#39;s basis vectors.  It can also have an angular velocity</span>
-<span class="sd">    vector, defined in another frame.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">indices</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">latexs</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;ReferenceFrame initialization method.</span>
-
-<span class="sd">        A ReferenceFrame has a set of orthonormal basis vectors, along with</span>
-<span class="sd">        orientations relative to other ReferenceFrames and angular velocities</span>
-<span class="sd">        relative to other ReferenceFrames.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        indices : list (of strings)</span>
-<span class="sd">            If custom indices are desired for console, pretty, and LaTeX</span>
-<span class="sd">            printing, supply three as a list. The basis vectors can then be</span>
-<span class="sd">            accessed with the get_item method.</span>
-<span class="sd">        latexs : list (of strings)</span>
-<span class="sd">            If custom names are desired for LaTeX printing of each basis</span>
-<span class="sd">            vector, supply the names here in a list.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, mlatex</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; N.x</span>
-<span class="sd">        N.x</span>
-<span class="sd">        &gt;&gt;&gt; O = ReferenceFrame(&#39;O&#39;, (&#39;1&#39;, &#39;2&#39;, &#39;3&#39;))</span>
-<span class="sd">        &gt;&gt;&gt; O.x</span>
-<span class="sd">        O[&#39;1&#39;]</span>
-<span class="sd">        &gt;&gt;&gt; O[&#39;1&#39;]</span>
-<span class="sd">        O[&#39;1&#39;]</span>
-<span class="sd">        &gt;&gt;&gt; P = ReferenceFrame(&#39;P&#39;, latexs=(&#39;A1&#39;, &#39;A2&#39;, &#39;A3&#39;))</span>
-<span class="sd">        &gt;&gt;&gt; mlatex(P.x)</span>
-<span class="sd">        &#39;A1&#39;</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Need to supply a valid name&#39;</span><span class="p">)</span>
-        <span class="c"># The if statements below are for custom printing of basis-vectors for</span>
-        <span class="c"># each frame.</span>
-        <span class="c"># First case, when custom indices are supplied</span>
-        <span class="k">if</span> <span class="n">indices</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">indices</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="nb">list</span><span class="p">)):</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Supply the indices as a list&#39;</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">indices</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">3</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Supply 3 indices&#39;</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">indices</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-                    <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Indices must be strings&#39;</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">str_vecs</span> <span class="o">=</span> <span class="p">[(</span><span class="n">name</span> <span class="o">+</span> <span class="s">&#39;[</span><span class="se">\&#39;</span><span class="s">&#39;</span> <span class="o">+</span> <span class="n">indices</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\&#39;</span><span class="s">]&#39;</span><span class="p">),</span>
-                             <span class="p">(</span><span class="n">name</span> <span class="o">+</span> <span class="s">&#39;[</span><span class="se">\&#39;</span><span class="s">&#39;</span> <span class="o">+</span> <span class="n">indices</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\&#39;</span><span class="s">]&#39;</span><span class="p">),</span>
-                             <span class="p">(</span><span class="n">name</span> <span class="o">+</span> <span class="s">&#39;[</span><span class="se">\&#39;</span><span class="s">&#39;</span> <span class="o">+</span> <span class="n">indices</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\&#39;</span><span class="s">]&#39;</span><span class="p">)]</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">pretty_vecs</span> <span class="o">=</span> <span class="p">[(</span><span class="s">&quot;</span><span class="se">\033</span><span class="s">[94m</span><span class="se">\033</span><span class="s">[1m&quot;</span> <span class="o">+</span> <span class="n">name</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="o">+</span> <span class="s">&quot;_&quot;</span> <span class="o">+</span>
-                                <span class="n">indices</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="s">&quot;</span><span class="se">\033</span><span class="s">[0;0m</span><span class="se">\x1b</span><span class="s">[0;0m&quot;</span><span class="p">),</span>
-                                <span class="p">(</span><span class="s">&quot;</span><span class="se">\033</span><span class="s">[94m</span><span class="se">\033</span><span class="s">[1m&quot;</span> <span class="o">+</span> <span class="n">name</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="o">+</span> <span class="s">&quot;_&quot;</span> <span class="o">+</span>
-                                <span class="n">indices</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="s">&quot;</span><span class="se">\033</span><span class="s">[0;0m</span><span class="se">\x1b</span><span class="s">[0;0m&quot;</span><span class="p">),</span>
-                                <span class="p">(</span><span class="s">&quot;</span><span class="se">\033</span><span class="s">[94m</span><span class="se">\033</span><span class="s">[1m&quot;</span> <span class="o">+</span> <span class="n">name</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="o">+</span> <span class="s">&quot;_&quot;</span> <span class="o">+</span>
-                                <span class="n">indices</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="s">&quot;</span><span class="se">\033</span><span class="s">[0;0m</span><span class="se">\x1b</span><span class="s">[0;0m&quot;</span><span class="p">)]</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">latex_vecs</span> <span class="o">=</span> <span class="p">[(</span><span class="s">r&quot;\mathbf{\hat{</span><span class="si">%s</span><span class="s">}_{</span><span class="si">%s</span><span class="s">}}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">name</span><span class="o">.</span><span class="n">lower</span><span class="p">(),</span>
-                               <span class="n">indices</span><span class="p">[</span><span class="mi">0</span><span class="p">])),</span> <span class="p">(</span><span class="s">r&quot;\mathbf{\hat{</span><span class="si">%s</span><span class="s">}_{</span><span class="si">%s</span><span class="s">}}&quot;</span> <span class="o">%</span>
-                               <span class="p">(</span><span class="n">name</span><span class="o">.</span><span class="n">lower</span><span class="p">(),</span> <span class="n">indices</span><span class="p">[</span><span class="mi">1</span><span class="p">])),</span>
-                               <span class="p">(</span><span class="s">r&quot;\mathbf{\hat{</span><span class="si">%s</span><span class="s">}_{</span><span class="si">%s</span><span class="s">}}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">name</span><span class="o">.</span><span class="n">lower</span><span class="p">(),</span>
-                               <span class="n">indices</span><span class="p">[</span><span class="mi">2</span><span class="p">]))]</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">indices</span> <span class="o">=</span> <span class="n">indices</span>
-        <span class="c"># Second case, when no custom indices are supplied</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">str_vecs</span> <span class="o">=</span> <span class="p">[(</span><span class="n">name</span> <span class="o">+</span> <span class="s">&#39;.x&#39;</span><span class="p">),</span> <span class="p">(</span><span class="n">name</span> <span class="o">+</span> <span class="s">&#39;.y&#39;</span><span class="p">),</span> <span class="p">(</span><span class="n">name</span> <span class="o">+</span> <span class="s">&#39;.z&#39;</span><span class="p">)]</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">pretty_vecs</span> <span class="o">=</span> <span class="p">[(</span><span class="s">&quot;</span><span class="se">\033</span><span class="s">[94m</span><span class="se">\033</span><span class="s">[1m&quot;</span> <span class="o">+</span> <span class="n">name</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="o">+</span>
-                                <span class="s">&quot;_x</span><span class="se">\033</span><span class="s">[0;0m</span><span class="se">\x1b</span><span class="s">[0;0m&quot;</span><span class="p">),</span>
-                                <span class="p">(</span><span class="s">&quot;</span><span class="se">\033</span><span class="s">[94m</span><span class="se">\033</span><span class="s">[1m&quot;</span> <span class="o">+</span> <span class="n">name</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="o">+</span>
-                                <span class="s">&quot;_y</span><span class="se">\033</span><span class="s">[0;0m</span><span class="se">\x1b</span><span class="s">[0;0m&quot;</span><span class="p">),</span>
-                                <span class="p">(</span><span class="s">&quot;</span><span class="se">\033</span><span class="s">[94m</span><span class="se">\033</span><span class="s">[1m&quot;</span> <span class="o">+</span> <span class="n">name</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="o">+</span>
-                                <span class="s">&quot;_z</span><span class="se">\033</span><span class="s">[0;0m</span><span class="se">\x1b</span><span class="s">[0;0m&quot;</span><span class="p">)]</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">latex_vecs</span> <span class="o">=</span> <span class="p">[(</span><span class="s">r&quot;\mathbf{\hat{</span><span class="si">%s</span><span class="s">}_x}&quot;</span> <span class="o">%</span> <span class="n">name</span><span class="o">.</span><span class="n">lower</span><span class="p">()),</span>
-                               <span class="p">(</span><span class="s">r&quot;\mathbf{\hat{</span><span class="si">%s</span><span class="s">}_y}&quot;</span> <span class="o">%</span> <span class="n">name</span><span class="o">.</span><span class="n">lower</span><span class="p">()),</span>
-                               <span class="p">(</span><span class="s">r&quot;\mathbf{\hat{</span><span class="si">%s</span><span class="s">}_z}&quot;</span> <span class="o">%</span> <span class="n">name</span><span class="o">.</span><span class="n">lower</span><span class="p">())]</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">indices</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="s">&#39;y&#39;</span><span class="p">,</span> <span class="s">&#39;z&#39;</span><span class="p">]</span>
-        <span class="c"># Different step, for custom latex basis vectors</span>
-        <span class="k">if</span> <span class="n">latexs</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">latexs</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="nb">list</span><span class="p">)):</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Supply the indices as a list&#39;</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">latexs</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">3</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Supply 3 indices&#39;</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">latexs</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-                    <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Latex entries must be strings&#39;</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">latex_vecs</span> <span class="o">=</span> <span class="n">latexs</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="n">name</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_dcm_dict</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_ang_vel_dict</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_ang_acc_dict</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_dlist</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_dcm_dict</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ang_vel_dict</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ang_acc_dict</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_cur</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_x</span> <span class="o">=</span> <span class="n">Vector</span><span class="p">([(</span><span class="n">Matrix</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]),</span> <span class="bp">self</span><span class="p">)])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_y</span> <span class="o">=</span> <span class="n">Vector</span><span class="p">([(</span><span class="n">Matrix</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]),</span> <span class="bp">self</span><span class="p">)])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_z</span> <span class="o">=</span> <span class="n">Vector</span><span class="p">([(</span><span class="n">Matrix</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]),</span> <span class="bp">self</span><span class="p">)])</span>
-
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ind</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns basis vector for the provided index (index being an str)&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ind</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Supply a valid str for the index&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">indices</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">ind</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">x</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">indices</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">ind</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">y</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">indices</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="n">ind</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">z</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Not a defined index&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">iter</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">y</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">z</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the name of the frame. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">name</span>
-
-    <span class="n">__repr__</span> <span class="o">=</span> <span class="n">__str__</span>
-
-    <span class="k">def</span> <span class="nf">_dict_list</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="n">num</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Creates a list from self to other using _dcm_dict. &quot;&quot;&quot;</span>
-        <span class="n">outlist</span> <span class="o">=</span> <span class="p">[[</span><span class="bp">self</span><span class="p">]]</span>
-        <span class="n">oldlist</span> <span class="o">=</span> <span class="p">[[]]</span>
-        <span class="k">while</span> <span class="n">outlist</span> <span class="o">!=</span> <span class="n">oldlist</span><span class="p">:</span>
-            <span class="n">oldlist</span> <span class="o">=</span> <span class="n">outlist</span><span class="p">[:]</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">outlist</span><span class="p">):</span>
-                <span class="n">templist</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">_dlist</span><span class="p">[</span><span class="n">num</span><span class="p">]</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-                <span class="k">for</span> <span class="n">i2</span><span class="p">,</span> <span class="n">v2</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">templist</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">v</span><span class="o">.</span><span class="n">__contains__</span><span class="p">(</span><span class="n">v2</span><span class="p">):</span>
-                        <span class="n">littletemplist</span> <span class="o">=</span> <span class="n">v</span> <span class="o">+</span> <span class="p">[</span><span class="n">v2</span><span class="p">]</span>
-                        <span class="k">if</span> <span class="ow">not</span> <span class="n">outlist</span><span class="o">.</span><span class="n">__contains__</span><span class="p">(</span><span class="n">littletemplist</span><span class="p">):</span>
-                            <span class="n">outlist</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">littletemplist</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">oldlist</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">v</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">!=</span> <span class="n">other</span><span class="p">:</span>
-                <span class="n">outlist</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-        <span class="n">outlist</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="nb">len</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">outlist</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">outlist</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;No Connecting Path found between &#39;</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">name</span> <span class="o">+</span>
-                         <span class="s">&#39; and &#39;</span> <span class="o">+</span> <span class="n">other</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_w_diff_dcm</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">otherframe</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Angular velocity from time differentiating the DCM. &quot;&quot;&quot;</span>
-        <span class="n">dcm2diff</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dcm</span><span class="p">(</span><span class="n">otherframe</span><span class="p">)</span>
-        <span class="n">diffed</span> <span class="o">=</span> <span class="n">dcm2diff</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span><span class="p">)</span>
-        <span class="n">angvelmat</span> <span class="o">=</span> <span class="n">diffed</span> <span class="o">*</span> <span class="n">dcm2diff</span><span class="o">.</span><span class="n">T</span>
-        <span class="n">w1</span> <span class="o">=</span> <span class="n">trigsimp</span><span class="p">(</span><span class="n">expand</span><span class="p">(</span><span class="n">angvelmat</span><span class="p">[</span><span class="mi">7</span><span class="p">]),</span> <span class="n">recursive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="n">w2</span> <span class="o">=</span> <span class="n">trigsimp</span><span class="p">(</span><span class="n">expand</span><span class="p">(</span><span class="n">angvelmat</span><span class="p">[</span><span class="mi">2</span><span class="p">]),</span> <span class="n">recursive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="n">w3</span> <span class="o">=</span> <span class="n">trigsimp</span><span class="p">(</span><span class="n">expand</span><span class="p">(</span><span class="n">angvelmat</span><span class="p">[</span><span class="mi">3</span><span class="p">]),</span> <span class="n">recursive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">Vector</span><span class="p">([(</span><span class="n">Matrix</span><span class="p">([</span><span class="n">w1</span><span class="p">,</span> <span class="n">w2</span><span class="p">,</span> <span class="n">w3</span><span class="p">]),</span> <span class="bp">self</span><span class="p">)])</span>
-
-<div class="viewcode-block" id="ReferenceFrame.ang_acc_in"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.ReferenceFrame.ang_acc_in">[docs]</a>    <span class="k">def</span> <span class="nf">ang_acc_in</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">otherframe</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the angular acceleration Vector of the ReferenceFrame.</span>
-
-<span class="sd">        Effectively returns the Vector:</span>
-<span class="sd">        ^N alpha ^B</span>
-<span class="sd">        which represent the angular acceleration of B in N, where B is self, and</span>
-<span class="sd">        N is otherframe.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        otherframe : ReferenceFrame</span>
-<span class="sd">            The ReferenceFrame which the angular acceleration is returned in.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, Vector</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; A = ReferenceFrame(&#39;A&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; V = 10 * N.x</span>
-<span class="sd">        &gt;&gt;&gt; A.set_ang_acc(N, V)</span>
-<span class="sd">        &gt;&gt;&gt; A.ang_acc_in(N)</span>
-<span class="sd">        10*N.x</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">otherframe</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">otherframe</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ang_acc_dict</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ang_acc_dict</span><span class="p">[</span><span class="n">otherframe</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">otherframe</span><span class="p">)</span><span class="o">.</span><span class="n">dt</span><span class="p">(</span><span class="n">otherframe</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="ReferenceFrame.ang_vel_in"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.ReferenceFrame.ang_vel_in">[docs]</a>    <span class="k">def</span> <span class="nf">ang_vel_in</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">otherframe</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the angular velocity Vector of the ReferenceFrame.</span>
-
-<span class="sd">        Effectively returns the Vector:</span>
-<span class="sd">        ^N omega ^B</span>
-<span class="sd">        which represent the angular velocity of B in N, where B is self, and</span>
-<span class="sd">        N is otherframe.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        otherframe : ReferenceFrame</span>
-<span class="sd">            The ReferenceFrame which the angular velocity is returned in.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, Vector</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; A = ReferenceFrame(&#39;A&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; V = 10 * N.x</span>
-<span class="sd">        &gt;&gt;&gt; A.set_ang_vel(N, V)</span>
-<span class="sd">        &gt;&gt;&gt; A.ang_vel_in(N)</span>
-<span class="sd">        10*N.x</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">otherframe</span><span class="p">)</span>
-        <span class="n">flist</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_dict_list</span><span class="p">(</span><span class="n">otherframe</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">outvec</span> <span class="o">=</span> <span class="n">Vector</span><span class="p">([])</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">flist</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">outvec</span> <span class="o">+=</span> <span class="n">flist</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">_ang_vel_dict</span><span class="p">[</span><span class="n">flist</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]]</span>
-        <span class="k">return</span> <span class="n">outvec</span>
-</div>
-<div class="viewcode-block" id="ReferenceFrame.dcm"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.ReferenceFrame.dcm">[docs]</a>    <span class="k">def</span> <span class="nf">dcm</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">otherframe</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The direction cosine matrix between frames.</span>
-
-<span class="sd">        This gives the DCM between this frame and the otherframe.</span>
-<span class="sd">        The format is N.xyz = N.dcm(B) * B.xyz</span>
-<span class="sd">        A SymPy Matrix is returned.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        otherframe : ReferenceFrame</span>
-<span class="sd">            The otherframe which the DCM is generated to.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, Vector</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; q1 = symbols(&#39;q1&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; A = N.orientnew(&#39;A&#39;, &#39;Axis&#39;, [q1, N.x])</span>
-<span class="sd">        &gt;&gt;&gt; N.dcm(A)</span>
-<span class="sd">        [1,       0,        0]</span>
-<span class="sd">        [0, cos(q1), -sin(q1)]</span>
-<span class="sd">        [0, sin(q1),  cos(q1)]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">otherframe</span><span class="p">)</span>
-        <span class="n">flist</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_dict_list</span><span class="p">(</span><span class="n">otherframe</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="n">outdcm</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">flist</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">outdcm</span> <span class="o">=</span> <span class="n">outdcm</span> <span class="o">*</span> <span class="n">flist</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">_dcm_dict</span><span class="p">[</span><span class="n">flist</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span>
-        <span class="k">return</span> <span class="n">outdcm</span>
-</div>
-<div class="viewcode-block" id="ReferenceFrame.orient"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.ReferenceFrame.orient">[docs]</a>    <span class="k">def</span> <span class="nf">orient</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">parent</span><span class="p">,</span> <span class="n">rot_type</span><span class="p">,</span> <span class="n">amounts</span><span class="p">,</span> <span class="n">rot_order</span><span class="o">=</span><span class="s">&#39;&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Defines the orientation of this frame relative to a parent frame.</span>
-
-<span class="sd">        Supported orientation types are Body, Space, Quaternion, Axis.</span>
-<span class="sd">        Examples show correct usage.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        parent : ReferenceFrame</span>
-<span class="sd">            The frame that this ReferenceFrame will have its orientation matrix</span>
-<span class="sd">            defined in relation to.</span>
-<span class="sd">        rot_type : str</span>
-<span class="sd">            The type of orientation matrix that is being created.</span>
-<span class="sd">        amounts : list OR value</span>
-<span class="sd">            The quantities that the orientation matrix will be defined by.</span>
-<span class="sd">        rot_order : str</span>
-<span class="sd">            If applicable, the order of a series of rotations.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, Vector</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; q0, q1, q2, q3, q4 = symbols(&#39;q0 q1 q2 q3 q4&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; B = ReferenceFrame(&#39;B&#39;)</span>
-
-<span class="sd">        Now we have a choice of how to implement the orientation. First is</span>
-<span class="sd">        Body. Body orientation takes this reference frame through three</span>
-<span class="sd">        successive simple rotations. Acceptable rotation orders are of length</span>
-<span class="sd">        3, expressed in XYZ or 123, and cannot have a rotation about about an</span>
-<span class="sd">        axis twice in a row.</span>
-
-<span class="sd">        &gt;&gt;&gt; B.orient(N, &#39;Body&#39;, [q1, q2, q3], &#39;123&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; B.orient(N, &#39;Body&#39;, [q1, q2, 0], &#39;ZXZ&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; B.orient(N, &#39;Body&#39;, [0, 0, 0], &#39;XYX&#39;)</span>
-
-<span class="sd">        Next is Space. Space is like Body, but the rotations are applied in the</span>
-<span class="sd">        opposite order.</span>
-
-<span class="sd">        &gt;&gt;&gt; B.orient(N, &#39;Space&#39;, [q1, q2, q3], &#39;312&#39;)</span>
-
-<span class="sd">        Next is Quaternion. This orients the new ReferenceFrame with</span>
-<span class="sd">        Quaternions, defined as a finite rotation about lambda, a unit vector,</span>
-<span class="sd">        by some amount theta.</span>
-<span class="sd">        This orientation is described by four parameters:</span>
-<span class="sd">        q0 = cos(theta/2)</span>
-<span class="sd">        q1 = lambda_x sin(theta/2)</span>
-<span class="sd">        q2 = lambda_y sin(theta/2)</span>
-<span class="sd">        q3 = lambda_z sin(theta/2)</span>
-<span class="sd">        Quaternion does not take in a rotation order.</span>
-
-<span class="sd">        &gt;&gt;&gt; B.orient(N, &#39;Quaternion&#39;, [q0, q1, q2, q3])</span>
-
-<span class="sd">        Last is Axis. This is a rotation about an arbitrary, non-time-varying</span>
-<span class="sd">        axis by some angle. The axis is supplied as a Vector. This is how</span>
-<span class="sd">        simple rotations are defined.</span>
-
-<span class="sd">        &gt;&gt;&gt; B.orient(N, &#39;Axis&#39;, [q1, N.x + 2 * N.y])</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">parent</span><span class="p">)</span>
-        <span class="n">amounts</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">amounts</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">amounts</span><span class="p">):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">Vector</span><span class="p">):</span>
-                <span class="n">amounts</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">_rot</span><span class="p">(</span><span class="n">axis</span><span class="p">,</span> <span class="n">angle</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;DCM for simple axis 1,2,or 3 rotations. &quot;&quot;&quot;</span>
-            <span class="k">if</span> <span class="n">axis</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
-                    <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">),</span> <span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)],</span>
-                    <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">),</span> <span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)]])</span>
-            <span class="k">elif</span> <span class="n">axis</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Matrix</span><span class="p">([[</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)],</span>
-                    <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
-                    <span class="p">[</span><span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)]])</span>
-            <span class="k">elif</span> <span class="n">axis</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Matrix</span><span class="p">([[</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">),</span> <span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">),</span> <span class="mi">0</span><span class="p">],</span>
-                    <span class="p">[</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">),</span> <span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">),</span> <span class="mi">0</span><span class="p">],</span>
-                    <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-
-        <span class="n">approved_orders</span> <span class="o">=</span> <span class="p">(</span><span class="s">&#39;123&#39;</span><span class="p">,</span> <span class="s">&#39;231&#39;</span><span class="p">,</span> <span class="s">&#39;312&#39;</span><span class="p">,</span> <span class="s">&#39;132&#39;</span><span class="p">,</span> <span class="s">&#39;213&#39;</span><span class="p">,</span> <span class="s">&#39;321&#39;</span><span class="p">,</span> <span class="s">&#39;121&#39;</span><span class="p">,</span>
-                           <span class="s">&#39;131&#39;</span><span class="p">,</span> <span class="s">&#39;212&#39;</span><span class="p">,</span> <span class="s">&#39;232&#39;</span><span class="p">,</span> <span class="s">&#39;313&#39;</span><span class="p">,</span> <span class="s">&#39;323&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-        <span class="n">rot_order</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span>
-            <span class="n">rot_order</span><span class="p">)</span><span class="o">.</span><span class="n">upper</span><span class="p">()</span>  <span class="c"># Now we need to make sure XYZ = 123</span>
-        <span class="n">rot_type</span> <span class="o">=</span> <span class="n">rot_type</span><span class="o">.</span><span class="n">upper</span><span class="p">()</span>
-        <span class="n">rot_order</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="s">&#39;1&#39;</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">rot_order</span><span class="p">]</span>
-        <span class="n">rot_order</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;Y&#39;</span><span class="p">,</span> <span class="s">&#39;2&#39;</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">rot_order</span><span class="p">]</span>
-        <span class="n">rot_order</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;Z&#39;</span><span class="p">,</span> <span class="s">&#39;3&#39;</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">rot_order</span><span class="p">]</span>
-        <span class="n">rot_order</span> <span class="o">=</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">rot_order</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">rot_order</span> <span class="ow">in</span> <span class="n">approved_orders</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;The supplied order is not an approved type&#39;</span><span class="p">)</span>
-        <span class="n">parent_orient</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">if</span> <span class="n">rot_type</span> <span class="o">==</span> <span class="s">&#39;AXIS&#39;</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Axis orientation takes no rotation order&#39;</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">amounts</span><span class="p">,</span> <span class="p">(</span><span class="nb">list</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">))</span> <span class="o">&amp;</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">amounts</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">)):</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Amounts are a list or tuple of length 2&#39;</span><span class="p">)</span>
-            <span class="n">theta</span> <span class="o">=</span> <span class="n">amounts</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">axis</span> <span class="o">=</span> <span class="n">amounts</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-            <span class="n">axis</span> <span class="o">=</span> <span class="n">_check_vector</span><span class="p">(</span><span class="n">axis</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">axis</span><span class="o">.</span><span class="n">dt</span><span class="p">(</span><span class="n">parent</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Axis cannot be time-varying&#39;</span><span class="p">)</span>
-            <span class="n">axis</span> <span class="o">=</span> <span class="n">axis</span><span class="o">.</span><span class="n">express</span><span class="p">(</span><span class="n">parent</span><span class="p">)</span><span class="o">.</span><span class="n">normalize</span><span class="p">()</span>
-            <span class="n">axis</span> <span class="o">=</span> <span class="n">axis</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">parent_orient</span> <span class="o">=</span> <span class="p">((</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="n">axis</span> <span class="o">*</span> <span class="n">axis</span><span class="o">.</span><span class="n">T</span><span class="p">)</span> <span class="o">*</span> <span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span> <span class="o">+</span>
-                    <span class="n">Matrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="n">axis</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">axis</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span> <span class="p">[</span><span class="n">axis</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="n">axis</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span>
-                        <span class="p">[</span><span class="o">-</span><span class="n">axis</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">axis</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">0</span><span class="p">]])</span> <span class="o">*</span> <span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span> <span class="o">+</span> <span class="n">axis</span> <span class="o">*</span> <span class="n">axis</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">rot_type</span> <span class="o">==</span> <span class="s">&#39;QUATERNION&#39;</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span>
-                    <span class="s">&#39;Quaternion orientation takes no rotation order&#39;</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">amounts</span><span class="p">,</span> <span class="p">(</span><span class="nb">list</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">))</span> <span class="o">&amp;</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">amounts</span><span class="p">)</span> <span class="o">==</span> <span class="mi">4</span><span class="p">)):</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Amounts are a list or tuple of length 4&#39;</span><span class="p">)</span>
-            <span class="n">q0</span><span class="p">,</span> <span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span> <span class="o">=</span> <span class="n">amounts</span>
-            <span class="n">parent_orient</span> <span class="o">=</span> <span class="p">(</span><span class="n">Matrix</span><span class="p">([[</span><span class="n">q0</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">q1</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">-</span> <span class="n">q2</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">-</span> <span class="n">q3</span> <span class="o">**</span>
-                <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="n">q1</span> <span class="o">*</span> <span class="n">q2</span> <span class="o">-</span> <span class="n">q0</span> <span class="o">*</span> <span class="n">q3</span><span class="p">),</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="n">q0</span> <span class="o">*</span> <span class="n">q2</span> <span class="o">+</span> <span class="n">q1</span> <span class="o">*</span> <span class="n">q3</span><span class="p">)],</span>
-                <span class="p">[</span><span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="n">q1</span> <span class="o">*</span> <span class="n">q2</span> <span class="o">+</span> <span class="n">q0</span> <span class="o">*</span> <span class="n">q3</span><span class="p">),</span> <span class="n">q0</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">-</span> <span class="n">q1</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">q2</span> <span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">q3</span> <span class="o">**</span> <span class="mi">2</span><span class="p">,</span>
-                <span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="n">q2</span> <span class="o">*</span> <span class="n">q3</span> <span class="o">-</span> <span class="n">q0</span> <span class="o">*</span> <span class="n">q1</span><span class="p">)],</span> <span class="p">[</span><span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="n">q1</span> <span class="o">*</span> <span class="n">q3</span> <span class="o">-</span> <span class="n">q0</span> <span class="o">*</span> <span class="n">q2</span><span class="p">),</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="n">q0</span> <span class="o">*</span>
-                <span class="n">q1</span> <span class="o">+</span> <span class="n">q2</span> <span class="o">*</span> <span class="n">q3</span><span class="p">),</span> <span class="n">q0</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">-</span> <span class="n">q1</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">-</span> <span class="n">q2</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">q3</span> <span class="o">**</span> <span class="mi">2</span><span class="p">]]))</span>
-        <span class="k">elif</span> <span class="n">rot_type</span> <span class="o">==</span> <span class="s">&#39;BODY&#39;</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">amounts</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span> <span class="o">&amp;</span> <span class="nb">len</span><span class="p">(</span><span class="n">rot_order</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Body orientation takes 3 values &amp; 3 orders&#39;</span><span class="p">)</span>
-            <span class="n">a1</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">rot_order</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="n">a2</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">rot_order</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-            <span class="n">a3</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">rot_order</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-            <span class="n">parent_orient</span> <span class="o">=</span> <span class="p">(</span><span class="n">_rot</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span> <span class="n">amounts</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">*</span> <span class="n">_rot</span><span class="p">(</span><span class="n">a2</span><span class="p">,</span> <span class="n">amounts</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-                    <span class="o">*</span> <span class="n">_rot</span><span class="p">(</span><span class="n">a3</span><span class="p">,</span> <span class="n">amounts</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span>
-        <span class="k">elif</span> <span class="n">rot_type</span> <span class="o">==</span> <span class="s">&#39;SPACE&#39;</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">amounts</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span> <span class="o">&amp;</span> <span class="nb">len</span><span class="p">(</span><span class="n">rot_order</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Space orientation takes 3 values &amp; 3 orders&#39;</span><span class="p">)</span>
-            <span class="n">a1</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">rot_order</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="n">a2</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">rot_order</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-            <span class="n">a3</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">rot_order</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-            <span class="n">parent_orient</span> <span class="o">=</span> <span class="p">(</span><span class="n">_rot</span><span class="p">(</span><span class="n">a3</span><span class="p">,</span> <span class="n">amounts</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span> <span class="o">*</span> <span class="n">_rot</span><span class="p">(</span><span class="n">a2</span><span class="p">,</span> <span class="n">amounts</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-                    <span class="o">*</span> <span class="n">_rot</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span> <span class="n">amounts</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;That is not an implemented rotation&#39;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_dcm_dict</span><span class="o">.</span><span class="n">update</span><span class="p">({</span><span class="n">parent</span><span class="p">:</span> <span class="n">parent_orient</span><span class="p">})</span>
-        <span class="n">parent</span><span class="o">.</span><span class="n">_dcm_dict</span><span class="o">.</span><span class="n">update</span><span class="p">({</span><span class="bp">self</span><span class="p">:</span> <span class="n">parent_orient</span><span class="o">.</span><span class="n">T</span><span class="p">})</span>
-        <span class="k">if</span> <span class="n">rot_type</span> <span class="o">==</span> <span class="s">&#39;QUATERNION&#39;</span><span class="p">:</span>
-            <span class="n">t</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span>
-            <span class="n">q0</span><span class="p">,</span> <span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span> <span class="o">=</span> <span class="n">amounts</span>
-            <span class="n">q0d</span> <span class="o">=</span> <span class="n">diff</span><span class="p">(</span><span class="n">q0</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
-            <span class="n">q1d</span> <span class="o">=</span> <span class="n">diff</span><span class="p">(</span><span class="n">q1</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
-            <span class="n">q2d</span> <span class="o">=</span> <span class="n">diff</span><span class="p">(</span><span class="n">q2</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
-            <span class="n">q3d</span> <span class="o">=</span> <span class="n">diff</span><span class="p">(</span><span class="n">q3</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
-            <span class="n">w1</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="n">q1d</span> <span class="o">*</span> <span class="n">q0</span> <span class="o">+</span> <span class="n">q2d</span> <span class="o">*</span> <span class="n">q3</span> <span class="o">-</span> <span class="n">q3d</span> <span class="o">*</span> <span class="n">q2</span> <span class="o">-</span> <span class="n">q0d</span> <span class="o">*</span> <span class="n">q1</span><span class="p">)</span>
-            <span class="n">w2</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="n">q2d</span> <span class="o">*</span> <span class="n">q0</span> <span class="o">+</span> <span class="n">q3d</span> <span class="o">*</span> <span class="n">q1</span> <span class="o">-</span> <span class="n">q1d</span> <span class="o">*</span> <span class="n">q3</span> <span class="o">-</span> <span class="n">q0d</span> <span class="o">*</span> <span class="n">q2</span><span class="p">)</span>
-            <span class="n">w3</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="n">q3d</span> <span class="o">*</span> <span class="n">q0</span> <span class="o">+</span> <span class="n">q1d</span> <span class="o">*</span> <span class="n">q2</span> <span class="o">-</span> <span class="n">q2d</span> <span class="o">*</span> <span class="n">q1</span> <span class="o">-</span> <span class="n">q0d</span> <span class="o">*</span> <span class="n">q3</span><span class="p">)</span>
-            <span class="n">wvec</span> <span class="o">=</span> <span class="n">Vector</span><span class="p">([(</span><span class="n">Matrix</span><span class="p">([</span><span class="n">w1</span><span class="p">,</span> <span class="n">w2</span><span class="p">,</span> <span class="n">w3</span><span class="p">]),</span> <span class="bp">self</span><span class="p">)])</span>
-        <span class="k">elif</span> <span class="n">rot_type</span> <span class="o">==</span> <span class="s">&#39;AXIS&#39;</span><span class="p">:</span>
-            <span class="n">thetad</span> <span class="o">=</span> <span class="p">(</span><span class="n">amounts</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span><span class="p">)</span>
-            <span class="n">wvec</span> <span class="o">=</span> <span class="n">thetad</span> <span class="o">*</span> <span class="n">amounts</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">express</span><span class="p">(</span><span class="n">parent</span><span class="p">)</span><span class="o">.</span><span class="n">normalize</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="n">CoercionFailed</span>
-                <span class="kn">from</span> <span class="nn">sympy.physics.mechanics.functions</span> <span class="kn">import</span> <span class="n">kinematic_equations</span>
-                <span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span> <span class="o">=</span> <span class="n">amounts</span>
-                <span class="n">u1</span><span class="p">,</span> <span class="n">u2</span><span class="p">,</span> <span class="n">u3</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;u1, u2, u3&#39;</span><span class="p">)</span>
-                <span class="n">templist</span> <span class="o">=</span> <span class="n">kinematic_equations</span><span class="p">([</span><span class="n">u1</span><span class="p">,</span> <span class="n">u2</span><span class="p">,</span> <span class="n">u3</span><span class="p">],</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span><span class="p">],</span>
-                                               <span class="n">rot_type</span><span class="p">,</span> <span class="n">rot_order</span><span class="p">)</span>
-                <span class="n">templist</span> <span class="o">=</span> <span class="p">[</span><span class="n">expand</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">templist</span><span class="p">]</span>
-                <span class="n">td</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">templist</span><span class="p">,</span> <span class="p">[</span><span class="n">u1</span><span class="p">,</span> <span class="n">u2</span><span class="p">,</span> <span class="n">u3</span><span class="p">])</span>
-                <span class="n">u1</span> <span class="o">=</span> <span class="n">expand</span><span class="p">(</span><span class="n">td</span><span class="p">[</span><span class="n">u1</span><span class="p">])</span>
-                <span class="n">u2</span> <span class="o">=</span> <span class="n">expand</span><span class="p">(</span><span class="n">td</span><span class="p">[</span><span class="n">u2</span><span class="p">])</span>
-                <span class="n">u3</span> <span class="o">=</span> <span class="n">expand</span><span class="p">(</span><span class="n">td</span><span class="p">[</span><span class="n">u3</span><span class="p">])</span>
-                <span class="n">wvec</span> <span class="o">=</span> <span class="n">u1</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">u2</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">y</span> <span class="o">+</span> <span class="n">u3</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">z</span>
-            <span class="k">except</span> <span class="p">(</span><span class="n">CoercionFailed</span><span class="p">,</span> <span class="ne">AssertionError</span><span class="p">):</span>
-                <span class="n">wvec</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_w_diff_dcm</span><span class="p">(</span><span class="n">parent</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_ang_vel_dict</span><span class="o">.</span><span class="n">update</span><span class="p">({</span><span class="n">parent</span><span class="p">:</span> <span class="n">wvec</span><span class="p">})</span>
-        <span class="n">parent</span><span class="o">.</span><span class="n">_ang_vel_dict</span><span class="o">.</span><span class="n">update</span><span class="p">({</span><span class="bp">self</span><span class="p">:</span> <span class="o">-</span><span class="n">wvec</span><span class="p">})</span>
-</div>
-<div class="viewcode-block" id="ReferenceFrame.orientnew"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.ReferenceFrame.orientnew">[docs]</a>    <span class="k">def</span> <span class="nf">orientnew</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">newname</span><span class="p">,</span> <span class="n">rot_type</span><span class="p">,</span> <span class="n">amounts</span><span class="p">,</span> <span class="n">rot_order</span><span class="o">=</span><span class="s">&#39;&#39;</span><span class="p">,</span> <span class="n">indices</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span>
-            <span class="n">latexs</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Creates a new ReferenceFrame oriented with respect to this Frame.</span>
-
-<span class="sd">        See ReferenceFrame.orient() for acceptable rotation types, amounts,</span>
-<span class="sd">        and orders. Parent is going to be self.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        newname : str</span>
-<span class="sd">            The name for the new ReferenceFrame</span>
-<span class="sd">        rot_type : str</span>
-<span class="sd">            The type of orientation matrix that is being created.</span>
-<span class="sd">        amounts : list OR value</span>
-<span class="sd">            The quantities that the orientation matrix will be defined by.</span>
-<span class="sd">        rot_order : str</span>
-<span class="sd">            If applicable, the order of a series of rotations.</span>
-
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, Vector</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; q1 = symbols(&#39;q1&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; A = N.orientnew(&#39;A&#39;, &#39;Axis&#39;, [q1, N.x])</span>
-
-
-<span class="sd">        .orient() documentation:\n</span>
-<span class="sd">        ========================</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">newframe</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="n">newname</span><span class="p">,</span> <span class="n">indices</span><span class="p">,</span> <span class="n">latexs</span><span class="p">)</span>
-        <span class="n">newframe</span><span class="o">.</span><span class="n">orient</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rot_type</span><span class="p">,</span> <span class="n">amounts</span><span class="p">,</span> <span class="n">rot_order</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">newframe</span>
-</div>
-    <span class="n">orientnew</span><span class="o">.</span><span class="n">__doc__</span> <span class="o">+=</span> <span class="n">orient</span><span class="o">.</span><span class="n">__doc__</span>
-
-<div class="viewcode-block" id="ReferenceFrame.set_ang_acc"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.ReferenceFrame.set_ang_acc">[docs]</a>    <span class="k">def</span> <span class="nf">set_ang_acc</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">otherframe</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Define the angular acceleration Vector in a ReferenceFrame.</span>
-
-<span class="sd">        Defines the angular acceleration of this ReferenceFrame, in another.</span>
-<span class="sd">        Angular acceleration can be defined with respect to multiple different</span>
-<span class="sd">        ReferenceFrames. Care must be taken to not create loops which are</span>
-<span class="sd">        inconsistent.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        otherframe : ReferenceFrame</span>
-<span class="sd">            A ReferenceFrame to define the angular acceleration in</span>
-<span class="sd">        value : Vector</span>
-<span class="sd">            The Vector representing angular acceleration</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, Vector</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; A = ReferenceFrame(&#39;A&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; V = 10 * N.x</span>
-<span class="sd">        &gt;&gt;&gt; A.set_ang_acc(N, V)</span>
-<span class="sd">        &gt;&gt;&gt; A.ang_acc_in(N)</span>
-<span class="sd">        10*N.x</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">value</span> <span class="o">=</span> <span class="n">_check_vector</span><span class="p">(</span><span class="n">value</span><span class="p">)</span>
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">otherframe</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_ang_acc_dict</span><span class="o">.</span><span class="n">update</span><span class="p">({</span><span class="n">otherframe</span><span class="p">:</span> <span class="n">value</span><span class="p">})</span>
-        <span class="n">otherframe</span><span class="o">.</span><span class="n">_ang_acc_dict</span><span class="o">.</span><span class="n">update</span><span class="p">({</span><span class="bp">self</span><span class="p">:</span> <span class="o">-</span><span class="n">value</span><span class="p">})</span>
-</div>
-<div class="viewcode-block" id="ReferenceFrame.set_ang_vel"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.ReferenceFrame.set_ang_vel">[docs]</a>    <span class="k">def</span> <span class="nf">set_ang_vel</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">otherframe</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Define the angular velocity vector in a ReferenceFrame.</span>
-
-<span class="sd">        Defines the angular velocity of this ReferenceFrame, in another.</span>
-<span class="sd">        Angular velocity can be defined with respect to multiple different</span>
-<span class="sd">        ReferenceFrames. Care must be taken to not create loops which are</span>
-<span class="sd">        inconsistent.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        otherframe : ReferenceFrame</span>
-<span class="sd">            A ReferenceFrame to define the angular velocity in</span>
-<span class="sd">        value : Vector</span>
-<span class="sd">            The Vector representing angular velocity</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, Vector</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; A = ReferenceFrame(&#39;A&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; V = 10 * N.x</span>
-<span class="sd">        &gt;&gt;&gt; A.set_ang_vel(N, V)</span>
-<span class="sd">        &gt;&gt;&gt; A.ang_vel_in(N)</span>
-<span class="sd">        10*N.x</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">value</span> <span class="o">=</span> <span class="n">_check_vector</span><span class="p">(</span><span class="n">value</span><span class="p">)</span>
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">otherframe</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_ang_vel_dict</span><span class="o">.</span><span class="n">update</span><span class="p">({</span><span class="n">otherframe</span><span class="p">:</span> <span class="n">value</span><span class="p">})</span>
-        <span class="n">otherframe</span><span class="o">.</span><span class="n">_ang_vel_dict</span><span class="o">.</span><span class="n">update</span><span class="p">({</span><span class="bp">self</span><span class="p">:</span> <span class="o">-</span><span class="n">value</span><span class="p">})</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="ReferenceFrame.x"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.ReferenceFrame.x">[docs]</a>    <span class="k">def</span> <span class="nf">x</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The basis Vector for the ReferenceFrame, in the x direction. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_x</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="ReferenceFrame.y"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.ReferenceFrame.y">[docs]</a>    <span class="k">def</span> <span class="nf">y</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The basis Vector for the ReferenceFrame, in the y direction. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_y</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="ReferenceFrame.z"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.ReferenceFrame.z">[docs]</a>    <span class="k">def</span> <span class="nf">z</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The basis Vector for the ReferenceFrame, in the z direction. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_z</span>
-
-</div></div>
-<div class="viewcode-block" id="Vector"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector">[docs]</a><span class="k">class</span> <span class="nc">Vector</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The class used to define vectors.</span>
-
-<span class="sd">    It along with ReferenceFrame are the building blocks of describing a</span>
-<span class="sd">    classical mechanics system in PyDy.</span>
-
-<span class="sd">    Attributes</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    simp : Boolean</span>
-<span class="sd">        Let certain methods use trigsimp on their outputs</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">simp</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">inlist</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;This is the constructor for the Vector class.  You shouldn&#39;t be</span>
-<span class="sd">        calling this, it should only be used by other functions. You should be</span>
-<span class="sd">        treating Vectors like you would with if you were doing the math by</span>
-<span class="sd">        hand, and getting the first 3 from the standard basis vectors from a</span>
-<span class="sd">        ReferenceFrame.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">while</span> <span class="nb">len</span><span class="p">(</span><span class="n">inlist</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">added</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">inlist</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]:</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span>
-                            <span class="n">inlist</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">inlist</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span>
-                    <span class="n">inlist</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">inlist</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-                    <span class="n">added</span> <span class="o">=</span> <span class="mi">1</span>
-                    <span class="k">break</span>
-            <span class="k">if</span> <span class="n">added</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">inlist</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-                <span class="n">inlist</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">inlist</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="c"># This code is to remove empty frames from the list</span>
-        <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">Matrix</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]):</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-                <span class="n">i</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">hash</span><span class="p">(</span><span class="nb">tuple</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The add operator for Vector. &quot;&quot;&quot;</span>
-        <span class="n">other</span> <span class="o">=</span> <span class="n">_check_vector</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Vector</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">+</span> <span class="n">other</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__and__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Dot product of two vectors.</span>
-
-<span class="sd">        Returns a scalar, the dot product of the two Vectors</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        other : Vector</span>
-<span class="sd">            The Vector which we are dotting with</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, Vector, dot</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; q1 = symbols(&#39;q1&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; dot(N.x, N.x)</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; dot(N.x, N.y)</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; A = N.orientnew(&#39;A&#39;, &#39;Axis&#39;, [q1, N.x])</span>
-<span class="sd">        &gt;&gt;&gt; dot(N.y, A.y)</span>
-<span class="sd">        cos(q1)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Dyadic</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-        <span class="n">other</span> <span class="o">=</span> <span class="n">_check_vector</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="n">out</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v1</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">v2</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-                <span class="n">out</span> <span class="o">+=</span> <span class="p">((</span><span class="n">v2</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
-                        <span class="o">*</span> <span class="p">(</span><span class="n">v2</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">dcm</span><span class="p">(</span><span class="n">v1</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-                        <span class="o">*</span> <span class="p">(</span><span class="n">v1</span><span class="p">[</span><span class="mi">0</span><span class="p">]))[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">Vector</span><span class="o">.</span><span class="n">simp</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">trigsimp</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">out</span><span class="p">),</span> <span class="n">recursive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="n">out</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;This uses mul and inputs self and 1 divided by other. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="n">__truediv__</span> <span class="o">=</span> <span class="n">__div__</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Tests for equality.</span>
-
-<span class="sd">        It is very import to note that this is only as good as the SymPy</span>
-<span class="sd">        equality test; False does not always mean they are not equivalent</span>
-<span class="sd">        Vectors.</span>
-<span class="sd">        If other is 0, and self is empty, returns True.</span>
-<span class="sd">        If other is 0 and self is not empty, returns False.</span>
-<span class="sd">        If none of the above, only accepts other as a Vector.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">other</span> <span class="o">=</span> <span class="n">_check_vector</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">if</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">==</span> <span class="p">[])</span> <span class="ow">and</span> <span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">args</span> <span class="o">==</span> <span class="p">[]):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">elif</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">==</span> <span class="p">[])</span> <span class="ow">or</span> <span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">args</span> <span class="o">==</span> <span class="p">[]):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="n">frame</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">frame</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">expand</span><span class="p">((</span><span class="bp">self</span> <span class="o">-</span> <span class="n">other</span><span class="p">)</span> <span class="o">&amp;</span> <span class="n">v</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Multiplies the Vector by a sympifyable expression.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        other : Sympifyable</span>
-<span class="sd">            The scalar to multiply this Vector with</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, Vector</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; b = Symbol(&#39;b&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; V = 10 * b * N.x</span>
-<span class="sd">        &gt;&gt;&gt; print(V)</span>
-<span class="sd">        10*b*N.x</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">newlist</span> <span class="o">=</span> <span class="p">[</span><span class="n">v</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">newlist</span><span class="p">):</span>
-            <span class="n">newlist</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span> <span class="o">*</span> <span class="n">newlist</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">newlist</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">Vector</span><span class="p">(</span><span class="n">newlist</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span> <span class="o">*</span> <span class="o">-</span><span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">__or__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Outer product between two Vectors.</span>
-
-<span class="sd">        A rank increasing operation, which returns a Dyadic from two Vectors</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        other : Vector</span>
-<span class="sd">            The Vector to take the outer product with</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, outer</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; outer(N.x, N.x)</span>
-<span class="sd">        (N.x|N.x)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">other</span> <span class="o">=</span> <span class="n">_check_vector</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="n">ol</span> <span class="o">=</span> <span class="n">Dyadic</span><span class="p">([])</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">i2</span><span class="p">,</span> <span class="n">v2</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-                <span class="c"># it looks this way because if we are in the same frame and</span>
-                <span class="c"># use the enumerate function on the same frame in a nested</span>
-                <span class="c"># fashion, then bad things happen</span>
-                <span class="n">ol</span> <span class="o">+=</span> <span class="n">Dyadic</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">v2</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">v2</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="p">)])</span>
-                <span class="n">ol</span> <span class="o">+=</span> <span class="n">Dyadic</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">v2</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">v2</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">y</span><span class="p">)])</span>
-                <span class="n">ol</span> <span class="o">+=</span> <span class="n">Dyadic</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">v2</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">2</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">v2</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">z</span><span class="p">)])</span>
-                <span class="n">ol</span> <span class="o">+=</span> <span class="n">Dyadic</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">v2</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">y</span><span class="p">,</span> <span class="n">v2</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="p">)])</span>
-                <span class="n">ol</span> <span class="o">+=</span> <span class="n">Dyadic</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">v2</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">y</span><span class="p">,</span> <span class="n">v2</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">y</span><span class="p">)])</span>
-                <span class="n">ol</span> <span class="o">+=</span> <span class="n">Dyadic</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">v2</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">2</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">y</span><span class="p">,</span> <span class="n">v2</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">z</span><span class="p">)])</span>
-                <span class="n">ol</span> <span class="o">+=</span> <span class="n">Dyadic</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span> <span class="o">*</span> <span class="n">v2</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">z</span><span class="p">,</span> <span class="n">v2</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="p">)])</span>
-                <span class="n">ol</span> <span class="o">+=</span> <span class="n">Dyadic</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span> <span class="o">*</span> <span class="n">v2</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">z</span><span class="p">,</span> <span class="n">v2</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">y</span><span class="p">)])</span>
-                <span class="n">ol</span> <span class="o">+=</span> <span class="n">Dyadic</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span> <span class="o">*</span> <span class="n">v2</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">2</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">z</span><span class="p">,</span> <span class="n">v2</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">z</span><span class="p">)])</span>
-        <span class="k">return</span> <span class="n">ol</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Latex Printing method. &quot;&quot;&quot;</span>
-        <span class="n">ar</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>  <span class="c"># just to shorten things</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">ar</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="n">ol</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c"># output list, to be concatenated to a string</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">ar</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="c"># if the coef of the basis vector is 1, we skip the 1</span>
-                <span class="k">if</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">ol</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39; + &#39;</span> <span class="o">+</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">latex_vecs</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-                <span class="c"># if the coef of the basis vector is -1, we skip the 1</span>
-                <span class="k">elif</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                    <span class="n">ol</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39; - &#39;</span> <span class="o">+</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">latex_vecs</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-                <span class="k">elif</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="c"># If the coefficient of the basis vector is not 1 or -1;</span>
-                    <span class="c"># also, we might wrap it in parentheses, for readability.</span>
-                    <span class="n">arg_str</span> <span class="o">=</span> <span class="n">MechanicsStrPrinter</span><span class="p">()</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">])</span>
-                    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">],</span> <span class="n">Add</span><span class="p">):</span>
-                        <span class="n">arg_str</span> <span class="o">=</span> <span class="s">&quot;(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="n">arg_str</span>
-                    <span class="k">if</span> <span class="n">arg_str</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;-&#39;</span><span class="p">:</span>
-                        <span class="n">arg_str</span> <span class="o">=</span> <span class="n">arg_str</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-                        <span class="n">str_start</span> <span class="o">=</span> <span class="s">&#39; - &#39;</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">str_start</span> <span class="o">=</span> <span class="s">&#39; + &#39;</span>
-                    <span class="n">ol</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">str_start</span> <span class="o">+</span> <span class="n">arg_str</span> <span class="o">+</span> <span class="s">&#39;*&#39;</span> <span class="o">+</span>
-                              <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">latex_vecs</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-        <span class="n">outstr</span> <span class="o">=</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">ol</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">outstr</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39; + &#39;</span><span class="p">):</span>
-            <span class="n">outstr</span> <span class="o">=</span> <span class="n">outstr</span><span class="p">[</span><span class="mi">3</span><span class="p">:]</span>
-        <span class="k">elif</span> <span class="n">outstr</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">):</span>
-            <span class="n">outstr</span> <span class="o">=</span> <span class="n">outstr</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-        <span class="k">return</span> <span class="n">outstr</span>
-
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Pretty Printing method. &quot;&quot;&quot;</span>
-        <span class="n">e</span> <span class="o">=</span> <span class="bp">self</span>
-
-        <span class="k">class</span> <span class="nc">Fake</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-            <span class="n">baseline</span> <span class="o">=</span> <span class="mi">0</span>
-
-            <span class="k">def</span> <span class="nf">render</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-                <span class="bp">self</span> <span class="o">=</span> <span class="n">e</span>
-                <span class="n">ar</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>  <span class="c"># just to shorten things</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">ar</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-                <span class="n">ol</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c"># output list, to be concatenated to a string</span>
-                <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">ar</span><span class="p">):</span>
-                    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">:</span>
-                        <span class="c"># if the coef of the basis vector is 1, we skip the 1</span>
-                        <span class="k">if</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                            <span class="n">ol</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot; + &quot;</span> <span class="o">+</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">pretty_vecs</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-                        <span class="c"># if the coef of the basis vector is -1, we skip the 1</span>
-                        <span class="k">elif</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                            <span class="n">ol</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot; - &quot;</span> <span class="o">+</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">pretty_vecs</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-                        <span class="k">elif</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                            <span class="c"># If the basis vector coeff is not 1 or -1,</span>
-                            <span class="c"># we might wrap it in parentheses, for readability.</span>
-                            <span class="n">arg_str</span> <span class="o">=</span> <span class="p">(</span><span class="n">MechanicsPrettyPrinter</span><span class="p">()</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span>
-                                <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]))</span>
-                            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">],</span> <span class="n">Add</span><span class="p">):</span>
-                                <span class="n">arg_str</span> <span class="o">=</span> <span class="s">&quot;(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="n">arg_str</span>
-                            <span class="k">if</span> <span class="n">arg_str</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s">&quot;-&quot;</span><span class="p">:</span>
-                                <span class="n">arg_str</span> <span class="o">=</span> <span class="n">arg_str</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-                                <span class="n">str_start</span> <span class="o">=</span> <span class="s">&quot; - &quot;</span>
-                            <span class="k">else</span><span class="p">:</span>
-                                <span class="n">str_start</span> <span class="o">=</span> <span class="s">&quot; + &quot;</span>
-                            <span class="n">ol</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">str_start</span> <span class="o">+</span> <span class="n">arg_str</span> <span class="o">+</span> <span class="s">&#39;*&#39;</span> <span class="o">+</span>
-                                      <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">pretty_vecs</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-                <span class="n">outstr</span> <span class="o">=</span> <span class="s">&quot;&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">ol</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">outstr</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&quot; + &quot;</span><span class="p">):</span>
-                    <span class="n">outstr</span> <span class="o">=</span> <span class="n">outstr</span><span class="p">[</span><span class="mi">3</span><span class="p">:]</span>
-                <span class="k">elif</span> <span class="n">outstr</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&quot; &quot;</span><span class="p">):</span>
-                    <span class="n">outstr</span> <span class="o">=</span> <span class="n">outstr</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-                <span class="k">return</span> <span class="n">outstr</span>
-        <span class="k">return</span> <span class="n">Fake</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__ror__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Outer product between two Vectors.</span>
-
-<span class="sd">        A rank increasing operation, which returns a Dyadic from two Vectors</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        other : Vector</span>
-<span class="sd">            The Vector to take the outer product with</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, outer</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; outer(N.x, N.x)</span>
-<span class="sd">        (N.x|N.x)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">other</span> <span class="o">=</span> <span class="n">_check_vector</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="n">ol</span> <span class="o">=</span> <span class="n">Dyadic</span><span class="p">([])</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">i2</span><span class="p">,</span> <span class="n">v2</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-                <span class="c"># it looks this way because if we are in the same frame and</span>
-                <span class="c"># use the enumerate function on the same frame in a nested</span>
-                <span class="c"># fashion, then bad things happen</span>
-                <span class="n">ol</span> <span class="o">+=</span> <span class="n">Dyadic</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">v2</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">v2</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="p">)])</span>
-                <span class="n">ol</span> <span class="o">+=</span> <span class="n">Dyadic</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">v2</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">v2</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">y</span><span class="p">)])</span>
-                <span class="n">ol</span> <span class="o">+=</span> <span class="n">Dyadic</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">v2</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">2</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">v2</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">z</span><span class="p">)])</span>
-                <span class="n">ol</span> <span class="o">+=</span> <span class="n">Dyadic</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">v2</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">y</span><span class="p">,</span> <span class="n">v2</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="p">)])</span>
-                <span class="n">ol</span> <span class="o">+=</span> <span class="n">Dyadic</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">v2</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">y</span><span class="p">,</span> <span class="n">v2</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">y</span><span class="p">)])</span>
-                <span class="n">ol</span> <span class="o">+=</span> <span class="n">Dyadic</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">v2</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">2</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">y</span><span class="p">,</span> <span class="n">v2</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">z</span><span class="p">)])</span>
-                <span class="n">ol</span> <span class="o">+=</span> <span class="n">Dyadic</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span> <span class="o">*</span> <span class="n">v2</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">z</span><span class="p">,</span> <span class="n">v2</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">x</span><span class="p">)])</span>
-                <span class="n">ol</span> <span class="o">+=</span> <span class="n">Dyadic</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span> <span class="o">*</span> <span class="n">v2</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">z</span><span class="p">,</span> <span class="n">v2</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">y</span><span class="p">)])</span>
-                <span class="n">ol</span> <span class="o">+=</span> <span class="n">Dyadic</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span> <span class="o">*</span> <span class="n">v2</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">2</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">z</span><span class="p">,</span> <span class="n">v2</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">z</span><span class="p">)])</span>
-        <span class="k">return</span> <span class="n">ol</span>
-
-    <span class="k">def</span> <span class="nf">__rsub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span> <span class="o">*</span> <span class="bp">self</span><span class="p">)</span> <span class="o">+</span> <span class="n">other</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Printing method. &quot;&quot;&quot;</span>
-        <span class="n">ar</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>  <span class="c"># just to shorten things</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">ar</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="n">ol</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c"># output list, to be concatenated to a string</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">ar</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="c"># if the coef of the basis vector is 1, we skip the 1</span>
-                <span class="k">if</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">ol</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39; + &#39;</span> <span class="o">+</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">str_vecs</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-                <span class="c"># if the coef of the basis vector is -1, we skip the 1</span>
-                <span class="k">elif</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                    <span class="n">ol</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39; - &#39;</span> <span class="o">+</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">str_vecs</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-                <span class="k">elif</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="c"># If the coefficient of the basis vector is not 1 or -1;</span>
-                    <span class="c"># also, we might wrap it in parentheses, for readability.</span>
-                    <span class="n">arg_str</span> <span class="o">=</span> <span class="n">MechanicsStrPrinter</span><span class="p">()</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">])</span>
-                    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">],</span> <span class="n">Add</span><span class="p">):</span>
-                        <span class="n">arg_str</span> <span class="o">=</span> <span class="s">&quot;(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="n">arg_str</span>
-                    <span class="k">if</span> <span class="n">arg_str</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;-&#39;</span><span class="p">:</span>
-                        <span class="n">arg_str</span> <span class="o">=</span> <span class="n">arg_str</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-                        <span class="n">str_start</span> <span class="o">=</span> <span class="s">&#39; - &#39;</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">str_start</span> <span class="o">=</span> <span class="s">&#39; + &#39;</span>
-                    <span class="n">ol</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">str_start</span> <span class="o">+</span> <span class="n">arg_str</span> <span class="o">+</span> <span class="s">&#39;*&#39;</span> <span class="o">+</span> <span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">str_vecs</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-        <span class="n">outstr</span> <span class="o">=</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">ol</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">outstr</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39; + &#39;</span><span class="p">):</span>
-            <span class="n">outstr</span> <span class="o">=</span> <span class="n">outstr</span><span class="p">[</span><span class="mi">3</span><span class="p">:]</span>
-        <span class="k">elif</span> <span class="n">outstr</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">):</span>
-            <span class="n">outstr</span> <span class="o">=</span> <span class="n">outstr</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-        <span class="k">return</span> <span class="n">outstr</span>
-
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The subraction operator. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__add__</span><span class="p">(</span><span class="n">other</span> <span class="o">*</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__xor__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The cross product operator for two Vectors.</span>
-
-<span class="sd">        Returns a Vector, expressed in the same ReferenceFrames as self.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        other : Vector</span>
-<span class="sd">            The Vector which we are crossing with</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, Vector</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; q1 = symbols(&#39;q1&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; N.x ^ N.y</span>
-<span class="sd">        N.z</span>
-<span class="sd">        &gt;&gt;&gt; A = N.orientnew(&#39;A&#39;, &#39;Axis&#39;, [q1, N.x])</span>
-<span class="sd">        &gt;&gt;&gt; A.x ^ N.y</span>
-<span class="sd">        N.z</span>
-<span class="sd">        &gt;&gt;&gt; N.y ^ A.x</span>
-<span class="sd">        - sin(q1)*A.y - cos(q1)*A.z</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Dyadic</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-        <span class="n">other</span> <span class="o">=</span> <span class="n">_check_vector</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">args</span> <span class="o">==</span> <span class="p">[]:</span>
-            <span class="k">return</span> <span class="bp">self</span> <span class="o">*</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">_det</span><span class="p">(</span><span class="n">mat</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;This is needed as a little method for to find the determinant</span>
-<span class="sd">            of a list in python; needs to work for a 3x3 list.</span>
-<span class="sd">            SymPy&#39;s Matrix won&#39;t take in Vector, so need a custom function.</span>
-<span class="sd">            You shouldn&#39;t be calling this.</span>
-
-<span class="sd">            &quot;&quot;&quot;</span>
-
-            <span class="k">return</span> <span class="p">(</span><span class="n">mat</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">mat</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">mat</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span> <span class="o">-</span> <span class="n">mat</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span> <span class="o">*</span> <span class="n">mat</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span>
-                    <span class="o">+</span> <span class="n">mat</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">mat</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span> <span class="o">*</span> <span class="n">mat</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">mat</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span>
-                    <span class="n">mat</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="mi">2</span><span class="p">])</span> <span class="o">+</span> <span class="n">mat</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">mat</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">mat</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span>
-                    <span class="n">mat</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">mat</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="mi">0</span><span class="p">]))</span>
-
-        <span class="n">outvec</span> <span class="o">=</span> <span class="n">Vector</span><span class="p">([])</span>
-        <span class="n">ar</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">args</span>  <span class="c"># For brevity</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">ar</span><span class="p">):</span>
-            <span class="n">tempx</span> <span class="o">=</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">x</span>
-            <span class="n">tempy</span> <span class="o">=</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">y</span>
-            <span class="n">tempz</span> <span class="o">=</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">z</span>
-            <span class="n">tempm</span> <span class="o">=</span> <span class="p">([[</span><span class="n">tempx</span><span class="p">,</span> <span class="n">tempy</span><span class="p">,</span> <span class="n">tempz</span><span class="p">],</span> <span class="p">[</span><span class="bp">self</span> <span class="o">&amp;</span> <span class="n">tempx</span><span class="p">,</span> <span class="bp">self</span> <span class="o">&amp;</span> <span class="n">tempy</span><span class="p">,</span>
-                <span class="bp">self</span> <span class="o">&amp;</span> <span class="n">tempz</span><span class="p">],</span> <span class="p">[</span><span class="n">Vector</span><span class="p">([</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span> <span class="o">&amp;</span> <span class="n">tempx</span><span class="p">,</span>
-                <span class="n">Vector</span><span class="p">([</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span> <span class="o">&amp;</span> <span class="n">tempy</span><span class="p">,</span> <span class="n">Vector</span><span class="p">([</span><span class="n">ar</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span> <span class="o">&amp;</span> <span class="n">tempz</span><span class="p">]])</span>
-            <span class="n">outvec</span> <span class="o">+=</span> <span class="n">_det</span><span class="p">(</span><span class="n">tempm</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">outvec</span>
-
-    <span class="n">_sympystr</span> <span class="o">=</span> <span class="n">__str__</span>
-    <span class="n">_sympyrepr</span> <span class="o">=</span> <span class="n">_sympystr</span>
-    <span class="n">__repr__</span> <span class="o">=</span> <span class="n">__str__</span>
-    <span class="n">__radd__</span> <span class="o">=</span> <span class="n">__add__</span>
-    <span class="n">__rand__</span> <span class="o">=</span> <span class="n">__and__</span>
-    <span class="n">__rmul__</span> <span class="o">=</span> <span class="n">__mul__</span>
-
-<div class="viewcode-block" id="Vector.dot"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector.dot">[docs]</a>    <span class="k">def</span> <span class="nf">dot</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span> <span class="o">&amp;</span> <span class="n">other</span></div>
-    <span class="n">dot</span><span class="o">.</span><span class="n">__doc__</span> <span class="o">=</span> <span class="n">__and__</span><span class="o">.</span><span class="n">__doc__</span>
-
-<div class="viewcode-block" id="Vector.cross"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector.cross">[docs]</a>    <span class="k">def</span> <span class="nf">cross</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span> <span class="o">^</span> <span class="n">other</span></div>
-    <span class="n">cross</span><span class="o">.</span><span class="n">__doc__</span> <span class="o">=</span> <span class="n">__xor__</span><span class="o">.</span><span class="n">__doc__</span>
-
-<div class="viewcode-block" id="Vector.outer"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector.outer">[docs]</a>    <span class="k">def</span> <span class="nf">outer</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span> <span class="o">|</span> <span class="n">other</span></div>
-    <span class="n">outer</span><span class="o">.</span><span class="n">__doc__</span> <span class="o">=</span> <span class="n">__or__</span><span class="o">.</span><span class="n">__doc__</span>
-
-<div class="viewcode-block" id="Vector.diff"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector.diff">[docs]</a>    <span class="k">def</span> <span class="nf">diff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">wrt</span><span class="p">,</span> <span class="n">otherframe</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Takes the partial derivative, with respect to a value, in a frame.</span>
-
-<span class="sd">        Returns a Vector.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        wrt : Symbol</span>
-<span class="sd">            What the partial derivative is taken with respect to.</span>
-<span class="sd">        otherframe : ReferenceFrame</span>
-<span class="sd">            The ReferenceFrame that the partial derivative is taken in.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, Vector, dynamicsymbols</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; Vector.simp = True</span>
-<span class="sd">        &gt;&gt;&gt; t = Symbol(&#39;t&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; q1 = dynamicsymbols(&#39;q1&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; A = N.orientnew(&#39;A&#39;, &#39;Axis&#39;, [q1, N.y])</span>
-<span class="sd">        &gt;&gt;&gt; A.x.diff(t, N)</span>
-<span class="sd">        - q1&#39;*A.z</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">wrt</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">wrt</span><span class="p">)</span>
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">otherframe</span><span class="p">)</span>
-        <span class="n">outvec</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">otherframe</span><span class="p">:</span>
-                <span class="n">outvec</span> <span class="o">+=</span> <span class="n">Vector</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">wrt</span><span class="p">),</span> <span class="n">otherframe</span><span class="p">)])</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">otherframe</span><span class="o">.</span><span class="n">dcm</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">wrt</span><span class="p">)</span> <span class="o">==</span> <span class="n">zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">):</span>
-                    <span class="n">d</span> <span class="o">=</span> <span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">wrt</span><span class="p">)</span>
-                    <span class="n">outvec</span> <span class="o">+=</span> <span class="n">Vector</span><span class="p">([(</span><span class="n">d</span><span class="p">,</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">])])</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">d</span> <span class="o">=</span> <span class="p">(</span><span class="n">Vector</span><span class="p">([</span><span class="n">v</span><span class="p">])</span><span class="o">.</span><span class="n">express</span><span class="p">(</span><span class="n">otherframe</span><span class="p">))</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">wrt</span><span class="p">)</span>
-                    <span class="n">outvec</span> <span class="o">+=</span> <span class="n">Vector</span><span class="p">([(</span><span class="n">d</span><span class="p">,</span> <span class="n">otherframe</span><span class="p">)])</span><span class="o">.</span><span class="n">express</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">outvec</span>
-</div>
-<div class="viewcode-block" id="Vector.doit"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector.doit">[docs]</a>    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Calls .doit() on each term in the Vector&quot;&quot;&quot;</span>
-        <span class="n">ov</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="n">ov</span> <span class="o">+=</span> <span class="n">Vector</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">applyfunc</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)),</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">])])</span>
-        <span class="k">return</span> <span class="n">ov</span>
-</div>
-<div class="viewcode-block" id="Vector.dt"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector.dt">[docs]</a>    <span class="k">def</span> <span class="nf">dt</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">otherframe</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the time derivative of the Vector in a ReferenceFrame.</span>
-
-<span class="sd">        Returns a Vector which is the time derivative of the self Vector, taken</span>
-<span class="sd">        in frame otherframe.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        otherframe : ReferenceFrame</span>
-<span class="sd">            The ReferenceFrame that the partial derivative is taken in.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, Vector, dynamicsymbols</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; q1 = Symbol(&#39;q1&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; u1 = dynamicsymbols(&#39;u1&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; A = N.orientnew(&#39;A&#39;, &#39;Axis&#39;, [q1, N.x])</span>
-<span class="sd">        &gt;&gt;&gt; v = u1 * N.x</span>
-<span class="sd">        &gt;&gt;&gt; A.set_ang_vel(N, 10*A.x)</span>
-<span class="sd">        &gt;&gt;&gt; A.x.dt(N) == 0</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; v.dt(N)</span>
-<span class="sd">        u1&#39;*N.x</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">outvec</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">otherframe</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">otherframe</span><span class="p">:</span>
-                <span class="n">outvec</span> <span class="o">+=</span> <span class="n">Vector</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span><span class="p">),</span> <span class="n">otherframe</span><span class="p">)])</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">outvec</span> <span class="o">+=</span> <span class="p">(</span><span class="n">Vector</span><span class="p">([</span><span class="n">v</span><span class="p">])</span><span class="o">.</span><span class="n">dt</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="o">+</span>
-                    <span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">otherframe</span><span class="p">)</span> <span class="o">^</span> <span class="n">Vector</span><span class="p">([</span><span class="n">v</span><span class="p">])))</span>
-        <span class="k">return</span> <span class="n">outvec</span>
-</div>
-<div class="viewcode-block" id="Vector.express"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector.express">[docs]</a>    <span class="k">def</span> <span class="nf">express</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">otherframe</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a vector, expressed in the other frame.</span>
-
-<span class="sd">        A new Vector is returned, equalivalent to this Vector, but its</span>
-<span class="sd">        components are all defined in only the otherframe.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        otherframe : ReferenceFrame</span>
-<span class="sd">            The frame for this Vector to be described in</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, Vector, dynamicsymbols</span>
-<span class="sd">        &gt;&gt;&gt; q1 = dynamicsymbols(&#39;q1&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; A = N.orientnew(&#39;A&#39;, &#39;Axis&#39;, [q1, N.y])</span>
-<span class="sd">        &gt;&gt;&gt; A.x.express(N)</span>
-<span class="sd">        cos(q1)*N.x - sin(q1)*N.z</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">otherframe</span><span class="p">)</span>
-        <span class="n">outvec</span> <span class="o">=</span> <span class="n">Vector</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">+</span> <span class="p">[])</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">!=</span> <span class="n">otherframe</span><span class="p">:</span>
-                <span class="n">temp</span> <span class="o">=</span> <span class="n">otherframe</span><span class="o">.</span><span class="n">dcm</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="o">*</span> <span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="k">for</span> <span class="n">i2</span><span class="p">,</span> <span class="n">v2</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">temp</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">Vector</span><span class="o">.</span><span class="n">simp</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                        <span class="n">temp</span><span class="p">[</span><span class="n">i2</span><span class="p">]</span> <span class="o">=</span> <span class="n">trigsimp</span><span class="p">(</span><span class="n">v2</span><span class="p">,</span> <span class="n">recursive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">temp</span><span class="p">[</span><span class="n">i2</span><span class="p">]</span> <span class="o">=</span> <span class="n">v2</span>
-                <span class="n">outvec</span> <span class="o">+=</span> <span class="n">Vector</span><span class="p">([(</span><span class="n">temp</span><span class="p">,</span> <span class="n">otherframe</span><span class="p">)])</span>
-                <span class="n">outvec</span> <span class="o">-=</span> <span class="n">Vector</span><span class="p">([</span><span class="n">v</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">outvec</span>
-</div>
-<div class="viewcode-block" id="Vector.simplify"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector.simplify">[docs]</a>    <span class="k">def</span> <span class="nf">simplify</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Simplify the elements in the Vector in place. &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">i</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-</div>
-<div class="viewcode-block" id="Vector.subs"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector.subs">[docs]</a>    <span class="k">def</span> <span class="nf">subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Substituion on the Vector.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; s = Symbol(&#39;s&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; a = N.x * s</span>
-<span class="sd">        &gt;&gt;&gt; a.subs({s: 2})</span>
-<span class="sd">        2*N.x</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">ov</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="n">ov</span> <span class="o">+=</span> <span class="n">Vector</span><span class="p">([(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">])])</span>
-        <span class="k">return</span> <span class="n">ov</span>
-</div>
-<div class="viewcode-block" id="Vector.magnitude"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector.magnitude">[docs]</a>    <span class="k">def</span> <span class="nf">magnitude</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the magnitude (Euclidean norm) of self.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="bp">self</span> <span class="o">&amp;</span> <span class="bp">self</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Vector.normalize"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector.normalize">[docs]</a>    <span class="k">def</span> <span class="nf">normalize</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a Vector of magnitude 1, codirectional with self.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Vector</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">+</span> <span class="p">[])</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">magnitude</span><span class="p">()</span>
-
-</div></div>
-<span class="k">class</span> <span class="nc">MechanicsStrPrinter</span><span class="p">(</span><span class="n">StrPrinter</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;String Printer for mechanics. &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span>
-        <span class="k">if</span> <span class="p">(</span><span class="nb">bool</span><span class="p">(</span><span class="nb">sum</span><span class="p">([</span><span class="n">i</span> <span class="o">==</span> <span class="n">t</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">variables</span><span class="p">]))</span> <span class="o">&amp;</span>
-                <span class="nb">isinstance</span><span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">UndefinedFunction</span><span class="p">)):</span>
-            <span class="n">ol</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">func</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">variables</span><span class="p">):</span>
-                <span class="n">ol</span> <span class="o">+=</span> <span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_str</span>
-            <span class="k">return</span> <span class="n">ol</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">StrPrinter</span><span class="p">()</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Function</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">e</span><span class="p">),</span> <span class="n">UndefinedFunction</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">StrPrinter</span><span class="p">()</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">e</span><span class="p">)</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&quot;(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="n">t</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">e</span><span class="o">.</span><span class="n">func</span><span class="o">.</span><span class="n">__name__</span> <span class="o">+</span> <span class="s">&quot;(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">stringify</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="s">&quot;, &quot;</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">MechanicsLatexPrinter</span><span class="p">(</span><span class="n">LatexPrinter</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Latex Printer for mechanics. &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Function</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">func</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="o">.</span><span class="n">__name__</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s">&#39;_print_&#39;</span> <span class="o">+</span> <span class="n">func</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s">&#39;_print_&#39;</span> <span class="o">+</span> <span class="n">func</span><span class="p">)(</span><span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">expr</span><span class="p">),</span> <span class="n">UndefinedFunction</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span> <span class="o">==</span> <span class="p">(</span><span class="n">t</span><span class="p">,)):</span>
-            <span class="n">name</span><span class="p">,</span> <span class="n">sup</span><span class="p">,</span> <span class="n">sub</span> <span class="o">=</span> <span class="n">split_super_sub</span><span class="p">(</span><span class="n">func</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">sup</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">sup</span> <span class="o">=</span> <span class="s">r&quot;^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="s">&quot;&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">sup</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">sup</span> <span class="o">=</span> <span class="s">r&quot;&quot;</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">sub</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">sub</span> <span class="o">=</span> <span class="s">r&quot;_{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="s">&quot;&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">sub</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">sub</span> <span class="o">=</span> <span class="s">r&quot;&quot;</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">name</span> <span class="o">+</span> <span class="n">sup</span> <span class="o">+</span> <span class="n">sub</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="p">[</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span> <span class="p">]</span>
-            <span class="c"># How inverse trig functions should be displayed, formats are:</span>
-            <span class="c"># abbreviated: asin, full: arcsin, power: sin^-1</span>
-            <span class="n">inv_trig_style</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;inv_trig_style&#39;</span><span class="p">]</span>
-            <span class="c"># If we are dealing with a power-style inverse trig function</span>
-            <span class="n">inv_trig_power_case</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="c"># If it is applicable to fold the argument brackets</span>
-            <span class="n">can_fold_brackets</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;fold_func_brackets&#39;</span><span class="p">]</span> <span class="ow">and</span> \
-                <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> \
-                <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">_needs_function_brackets</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-            <span class="n">inv_trig_table</span> <span class="o">=</span> <span class="p">[</span><span class="s">&quot;asin&quot;</span><span class="p">,</span> <span class="s">&quot;acos&quot;</span><span class="p">,</span> <span class="s">&quot;atan&quot;</span><span class="p">,</span> <span class="s">&quot;acot&quot;</span><span class="p">]</span>
-
-            <span class="c"># If the function is an inverse trig function, handle the style</span>
-            <span class="k">if</span> <span class="n">func</span> <span class="ow">in</span> <span class="n">inv_trig_table</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">inv_trig_style</span> <span class="o">==</span> <span class="s">&quot;abbreviated&quot;</span><span class="p">:</span>
-                    <span class="n">func</span> <span class="o">=</span> <span class="n">func</span>
-                <span class="k">elif</span> <span class="n">inv_trig_style</span> <span class="o">==</span> <span class="s">&quot;full&quot;</span><span class="p">:</span>
-                    <span class="n">func</span> <span class="o">=</span> <span class="s">&quot;arc&quot;</span> <span class="o">+</span> <span class="n">func</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-                <span class="k">elif</span> <span class="n">inv_trig_style</span> <span class="o">==</span> <span class="s">&quot;power&quot;</span><span class="p">:</span>
-                    <span class="n">func</span> <span class="o">=</span> <span class="n">func</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-                    <span class="n">inv_trig_power_case</span> <span class="o">=</span> <span class="bp">True</span>
-
-                    <span class="c"># Can never fold brackets if we&#39;re raised to a power</span>
-                    <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="n">can_fold_brackets</span> <span class="o">=</span> <span class="bp">False</span>
-
-            <span class="k">if</span> <span class="n">inv_trig_power_case</span><span class="p">:</span>
-                <span class="n">name</span> <span class="o">=</span> <span class="s">r&quot;\operatorname{</span><span class="si">%s</span><span class="s">}^{-1}&quot;</span> <span class="o">%</span> <span class="n">func</span>
-            <span class="k">elif</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">name</span> <span class="o">=</span> <span class="s">r&quot;\operatorname{</span><span class="si">%s</span><span class="s">}^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">name</span> <span class="o">=</span> <span class="s">r&quot;\operatorname{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="n">func</span>
-
-            <span class="k">if</span> <span class="n">can_fold_brackets</span><span class="p">:</span>
-                <span class="n">name</span> <span class="o">+=</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">&quot;</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">name</span> <span class="o">+=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span>
-
-            <span class="k">if</span> <span class="n">inv_trig_power_case</span> <span class="ow">and</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">name</span> <span class="o">+=</span> <span class="s">r&quot;^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="n">exp</span>
-
-            <span class="k">return</span> <span class="n">name</span> <span class="o">%</span> <span class="s">&quot;,&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">der_expr</span><span class="p">):</span>
-        <span class="c"># make sure it is an the right form</span>
-        <span class="n">der_expr</span> <span class="o">=</span> <span class="n">der_expr</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">der_expr</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">der_expr</span><span class="p">)</span>
-
-        <span class="c"># check if expr is a dynamicsymbol</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">AppliedUndef</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">der_expr</span><span class="o">.</span><span class="n">expr</span>
-        <span class="n">red</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">AppliedUndef</span><span class="p">)</span>
-        <span class="n">syms</span> <span class="o">=</span> <span class="n">der_expr</span><span class="o">.</span><span class="n">variables</span>
-        <span class="n">test1</span> <span class="o">=</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">([</span><span class="bp">True</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">red</span> <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">atoms</span><span class="p">()</span> <span class="o">==</span> <span class="nb">set</span><span class="p">([</span><span class="n">t</span><span class="p">])])</span>
-        <span class="n">test2</span> <span class="o">=</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">([(</span><span class="n">t</span> <span class="o">==</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">syms</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">test1</span> <span class="ow">or</span> <span class="n">test2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">LatexPrinter</span><span class="p">()</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">der_expr</span><span class="p">)</span>
-
-        <span class="c"># done checking</span>
-        <span class="n">dots</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">syms</span><span class="p">)</span>
-        <span class="n">base</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Function</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="n">base_split</span> <span class="o">=</span> <span class="n">base</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;_&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">base</span> <span class="o">=</span> <span class="n">base_split</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">dots</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">base</span> <span class="o">=</span> <span class="s">r&quot;\dot{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="n">base</span>
-        <span class="k">elif</span> <span class="n">dots</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">base</span> <span class="o">=</span> <span class="s">r&quot;\ddot{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="n">base</span>
-        <span class="k">elif</span> <span class="n">dots</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="n">base</span> <span class="o">=</span> <span class="s">r&quot;\dddot{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="n">base</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">base_split</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">base</span> <span class="o">+=</span> <span class="s">&#39;_&#39;</span> <span class="o">+</span> <span class="n">base_split</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">base</span>
-
-
-<span class="k">class</span> <span class="nc">MechanicsPrettyPrinter</span><span class="p">(</span><span class="n">PrettyPrinter</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Pretty Printer for mechanics. &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deriv</span><span class="p">):</span>
-        <span class="c"># XXX use U(&#39;PARTIAL DIFFERENTIAL&#39;) here ?</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span>
-        <span class="n">dots</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">can_break</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="n">syms</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">deriv</span><span class="o">.</span><span class="n">variables</span><span class="p">))</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">None</span>
-
-        <span class="k">while</span> <span class="nb">len</span><span class="p">(</span><span class="n">syms</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">syms</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">t</span><span class="p">:</span>
-                <span class="n">syms</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-                <span class="n">dots</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">break</span>
-
-        <span class="n">f</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">binding</span><span class="o">=</span><span class="n">prettyForm</span><span class="o">.</span><span class="n">FUNC</span><span class="p">,</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">deriv</span><span class="o">.</span><span class="n">expr</span><span class="p">))</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">deriv</span><span class="o">.</span><span class="n">expr</span><span class="p">),</span> <span class="n">UndefinedFunction</span><span class="p">)</span>
-                <span class="ow">and</span> <span class="p">(</span><span class="n">deriv</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span> <span class="o">==</span> <span class="p">(</span><span class="n">t</span><span class="p">,))):</span>
-            <span class="n">dots</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="n">can_break</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">binding</span><span class="o">=</span><span class="n">prettyForm</span><span class="o">.</span><span class="n">FUNC</span><span class="p">,</span>
-                    <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">deriv</span><span class="o">.</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-
-        <span class="k">if</span> <span class="n">dots</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">dots</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-        <span class="k">elif</span> <span class="n">dots</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">dots</span> <span class="o">=</span> <span class="s">&quot;</span><span class="se">\u0307</span><span class="s">&quot;</span>
-        <span class="k">elif</span> <span class="n">dots</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">dots</span> <span class="o">=</span> <span class="s">&quot;</span><span class="se">\u0308</span><span class="s">&quot;</span>
-        <span class="k">elif</span> <span class="n">dots</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="n">dots</span> <span class="o">=</span> <span class="s">&quot;</span><span class="se">\u20db</span><span class="s">&quot;</span>
-        <span class="k">elif</span> <span class="n">dots</span> <span class="o">==</span> <span class="mi">4</span><span class="p">:</span>
-            <span class="n">dots</span> <span class="o">=</span> <span class="s">&quot;</span><span class="se">\u20dc</span><span class="s">&quot;</span>
-
-        <span class="n">uni_subs</span> <span class="o">=</span> <span class="p">[</span><span class="s">&quot;</span><span class="se">\u2080</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u2081</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u2082</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u2083</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u2084</span><span class="s">&quot;</span><span class="p">,</span>
-                    <span class="s">&quot;</span><span class="se">\u2085</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u2086</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u2087</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u2088</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u2089</span><span class="s">&quot;</span><span class="p">,</span>
-                    <span class="s">&quot;</span><span class="se">\u208a</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u208b</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u208c</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u208d</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u208e</span><span class="s">&quot;</span><span class="p">,</span>
-                    <span class="s">&quot;</span><span class="se">\u208f</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u2090</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u2091</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u2092</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u2093</span><span class="s">&quot;</span><span class="p">,</span>
-                    <span class="s">&quot;</span><span class="se">\u2094</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u2095</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u2096</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u2097</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u2098</span><span class="s">&quot;</span><span class="p">,</span>
-                    <span class="s">&quot;</span><span class="se">\u2099</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u209a</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u209b</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u209c</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u209d</span><span class="s">&quot;</span><span class="p">,</span>
-                    <span class="s">&quot;</span><span class="se">\u209e</span><span class="s">&quot;</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u209f</span><span class="s">&quot;</span><span class="p">]</span>
-
-        <span class="n">fpic</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">__dict__</span><span class="p">[</span><span class="s">&#39;picture&#39;</span><span class="p">]</span>
-        <span class="n">funi</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">__dict__</span><span class="p">[</span><span class="s">&#39;unicode&#39;</span><span class="p">]</span>
-        <span class="n">ind</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">funi</span><span class="p">)</span>
-        <span class="n">val</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">uni_subs</span><span class="p">:</span>
-            <span class="n">cur_ind</span> <span class="o">=</span> <span class="n">funi</span><span class="o">.</span><span class="n">find</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">cur_ind</span> <span class="o">!=</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">cur_ind</span> <span class="o">&lt;</span> <span class="n">ind</span><span class="p">):</span>
-                <span class="n">ind</span> <span class="o">=</span> <span class="n">cur_ind</span>
-                <span class="n">val</span> <span class="o">=</span> <span class="n">i</span>
-        <span class="k">if</span> <span class="n">ind</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">funi</span><span class="p">):</span>
-            <span class="n">funi</span> <span class="o">+=</span> <span class="n">dots</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">funi</span> <span class="o">=</span> <span class="n">funi</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">val</span><span class="p">,</span> <span class="n">dots</span> <span class="o">+</span> <span class="n">val</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">__dict__</span><span class="p">[</span><span class="s">&#39;picture&#39;</span><span class="p">]</span> <span class="o">==</span> <span class="p">[</span><span class="n">f</span><span class="o">.</span><span class="n">__dict__</span><span class="p">[</span><span class="s">&#39;unicode&#39;</span><span class="p">]]:</span>
-            <span class="n">fpic</span> <span class="o">=</span> <span class="p">[</span><span class="n">funi</span><span class="p">]</span>
-        <span class="n">f</span><span class="o">.</span><span class="n">__dict__</span><span class="p">[</span><span class="s">&#39;picture&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">fpic</span>
-        <span class="n">f</span><span class="o">.</span><span class="n">__dict__</span><span class="p">[</span><span class="s">&#39;unicode&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">funi</span>
-
-        <span class="k">if</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">syms</span><span class="p">))</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">can_break</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span>
-
-        <span class="k">for</span> <span class="n">sym</span><span class="p">,</span> <span class="n">num</span> <span class="ow">in</span> <span class="n">group</span><span class="p">(</span><span class="n">syms</span><span class="p">,</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">sym</span><span class="p">)</span>
-            <span class="n">ds</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39;d&#39;</span><span class="p">))</span>
-
-            <span class="k">if</span> <span class="n">num</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">ds</span> <span class="o">=</span> <span class="n">ds</span><span class="o">**</span><span class="n">prettyForm</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">num</span><span class="p">))</span>
-
-            <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="n">ds</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">x</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">))</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">x</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">ds</span><span class="p">))</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;d&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">syms</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">pform</span><span class="o">**</span><span class="n">prettyForm</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">syms</span><span class="p">)))</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="n">stringPict</span><span class="o">.</span><span class="n">LINE</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-        <span class="n">pform</span><span class="o">.</span><span class="n">baseline</span> <span class="o">=</span> <span class="n">pform</span><span class="o">.</span><span class="n">baseline</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">stringPict</span><span class="o">.</span><span class="n">next</span><span class="p">(</span><span class="n">pform</span><span class="p">,</span> <span class="n">f</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_Function</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span>
-        <span class="c"># XXX works only for applied functions</span>
-        <span class="n">func</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">func</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span>
-        <span class="n">func_name</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">__name__</span>
-        <span class="n">prettyFunc</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="n">func_name</span><span class="p">))</span>
-        <span class="n">prettyArgs</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">args</span><span class="p">)</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-        <span class="c"># If this function is an Undefined function of t, it is probably a</span>
-        <span class="c"># dynamic symbol, so we&#39;ll skip the (t). The rest of the code is</span>
-        <span class="c"># identical to the normal PrettyPrinter code</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">UndefinedFunction</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">args</span> <span class="o">==</span> <span class="p">(</span><span class="n">t</span><span class="p">,)):</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">binding</span><span class="o">=</span><span class="n">prettyForm</span><span class="o">.</span><span class="n">FUNC</span><span class="p">,</span>
-                       <span class="o">*</span><span class="n">stringPict</span><span class="o">.</span><span class="n">next</span><span class="p">(</span><span class="n">prettyFunc</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">binding</span><span class="o">=</span><span class="n">prettyForm</span><span class="o">.</span><span class="n">FUNC</span><span class="p">,</span>
-                       <span class="o">*</span><span class="n">stringPict</span><span class="o">.</span><span class="n">next</span><span class="p">(</span><span class="n">prettyFunc</span><span class="p">,</span> <span class="n">prettyArgs</span><span class="p">))</span>
-        <span class="c"># store pform parts so it can be reassembled e.g. when powered</span>
-        <span class="n">pform</span><span class="o">.</span><span class="n">prettyFunc</span> <span class="o">=</span> <span class="n">prettyFunc</span>
-        <span class="n">pform</span><span class="o">.</span><span class="n">prettyArgs</span> <span class="o">=</span> <span class="n">prettyArgs</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-
-<span class="k">def</span> <span class="nf">_check_dyadic</span><span class="p">(</span><span class="n">other</span><span class="p">):</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Dyadic</span><span class="p">):</span>
-        <span class="n">other</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">other</span> <span class="o">!=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;A Dyadic must be supplied&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">Dyadic</span><span class="p">([])</span>
-    <span class="k">return</span> <span class="n">other</span>
-
-
-<span class="k">def</span> <span class="nf">_check_frame</span><span class="p">(</span><span class="n">other</span><span class="p">):</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">ReferenceFrame</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;A ReferenceFrame must be supplied&#39;</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">_check_vector</span><span class="p">(</span><span class="n">other</span><span class="p">):</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Vector</span><span class="p">):</span>
-        <span class="n">other</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">other</span> <span class="o">!=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;A Vector must be supplied&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">other</span> <span class="o">=</span> <span class="n">Vector</span><span class="p">([])</span>
-    <span class="k">return</span> <span class="n">other</span>
-
-
-<div class="viewcode-block" id="dynamicsymbols"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/functions.html#sympy.physics.mechanics.essential.dynamicsymbols">[docs]</a><span class="k">def</span> <span class="nf">dynamicsymbols</span><span class="p">(</span><span class="n">names</span><span class="p">,</span> <span class="n">level</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Uses symbols and Function for functions of time.</span>
-
-<span class="sd">    Creates a SymPy UndefinedFunction, which is then initialized as a function</span>
-<span class="sd">    of a variable, the default being Symbol(&#39;t&#39;).</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    names : str</span>
-<span class="sd">        Names of the dynamic symbols you want to create; works the same way as</span>
-<span class="sd">        inputs to symbols</span>
-<span class="sd">    level : int</span>
-<span class="sd">        Level of differentiation of the returned function; d/dt once of t,</span>
-<span class="sd">        twice of t, etc.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import dynamicsymbols</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import diff, Symbol</span>
-<span class="sd">    &gt;&gt;&gt; q1 = dynamicsymbols(&#39;q1&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; q1</span>
-<span class="sd">    q1(t)</span>
-<span class="sd">    &gt;&gt;&gt; diff(q1, Symbol(&#39;t&#39;))</span>
-<span class="sd">    Derivative(q1(t), t)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">esses</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="n">names</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Function</span><span class="p">)</span>
-    <span class="n">t</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span>
-    <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">esses</span><span class="p">,</span> <span class="s">&#39;__iter__&#39;</span><span class="p">):</span>
-        <span class="n">esses</span> <span class="o">=</span> <span class="p">[</span><span class="nb">reduce</span><span class="p">(</span><span class="n">diff</span><span class="p">,</span> <span class="p">[</span><span class="n">t</span><span class="p">]</span><span class="o">*</span><span class="n">level</span><span class="p">,</span> <span class="n">e</span><span class="p">(</span><span class="n">t</span><span class="p">))</span> <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">esses</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">esses</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">reduce</span><span class="p">(</span><span class="n">diff</span><span class="p">,</span> <span class="p">[</span><span class="n">t</span><span class="p">]</span><span class="o">*</span><span class="n">level</span><span class="p">,</span> <span class="n">esses</span><span class="p">(</span><span class="n">t</span><span class="p">))</span>
-</div>
-<span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">)</span>
-<span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_str</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\&#39;</span><span class="s">&#39;</span>
-</pre></div>
-
-          </div>
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-  <h1>Source code for sympy.physics.mechanics.functions</h1><div class="highlight"><pre>
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;cross&#39;</span><span class="p">,</span>
-           <span class="s">&#39;dot&#39;</span><span class="p">,</span>
-           <span class="s">&#39;express&#39;</span><span class="p">,</span>
-           <span class="s">&#39;outer&#39;</span><span class="p">,</span>
-           <span class="s">&#39;inertia&#39;</span><span class="p">,</span>
-           <span class="s">&#39;mechanics_printing&#39;</span><span class="p">,</span>
-           <span class="s">&#39;mprint&#39;</span><span class="p">,</span>
-           <span class="s">&#39;mpprint&#39;</span><span class="p">,</span>
-           <span class="s">&#39;mlatex&#39;</span><span class="p">,</span>
-           <span class="s">&#39;kinematic_equations&#39;</span><span class="p">,</span>
-           <span class="s">&#39;inertia_of_point_mass&#39;</span><span class="p">,</span>
-           <span class="s">&#39;partial_velocity&#39;</span><span class="p">,</span>
-           <span class="s">&#39;linear_momentum&#39;</span><span class="p">,</span>
-           <span class="s">&#39;angular_momentum&#39;</span><span class="p">,</span>
-           <span class="s">&#39;kinetic_energy&#39;</span><span class="p">,</span>
-           <span class="s">&#39;potential_energy&#39;</span><span class="p">,</span>
-           <span class="s">&#39;Lagrangian&#39;</span><span class="p">]</span>
-
-<span class="kn">from</span> <span class="nn">sympy.physics.mechanics.essential</span> <span class="kn">import</span> <span class="p">(</span><span class="n">Vector</span><span class="p">,</span> <span class="n">Dyadic</span><span class="p">,</span> <span class="n">ReferenceFrame</span><span class="p">,</span>
-                                               <span class="n">MechanicsStrPrinter</span><span class="p">,</span>
-                                               <span class="n">MechanicsPrettyPrinter</span><span class="p">,</span>
-                                               <span class="n">MechanicsLatexPrinter</span><span class="p">,</span>
-                                               <span class="n">dynamicsymbols</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.mechanics.particle</span> <span class="kn">import</span> <span class="n">Particle</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.mechanics.rigidbody</span> <span class="kn">import</span> <span class="n">RigidBody</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.mechanics.point</span> <span class="kn">import</span> <span class="n">Point</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">diff</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">Matrix</span>
-<span class="kn">from</span> <span class="nn">sympy.core.basic</span> <span class="kn">import</span> <span class="n">S</span>
-
-
-<div class="viewcode-block" id="cross"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/functions.html#sympy.physics.mechanics.functions.cross">[docs]</a><span class="k">def</span> <span class="nf">cross</span><span class="p">(</span><span class="n">vec1</span><span class="p">,</span> <span class="n">vec2</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Cross product convenience wrapper for Vector.cross(): \n&quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">vec1</span><span class="p">,</span> <span class="p">(</span><span class="n">Vector</span><span class="p">,</span> <span class="n">Dyadic</span><span class="p">)):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Cross product is between two vectors&#39;</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">vec1</span> <span class="o">^</span> <span class="n">vec2</span></div>
-<span class="n">cross</span><span class="o">.</span><span class="n">__doc__</span> <span class="o">+=</span> <span class="n">Vector</span><span class="o">.</span><span class="n">cross</span><span class="o">.</span><span class="n">__doc__</span>
-
-
-<div class="viewcode-block" id="dot"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/functions.html#sympy.physics.mechanics.functions.dot">[docs]</a><span class="k">def</span> <span class="nf">dot</span><span class="p">(</span><span class="n">vec1</span><span class="p">,</span> <span class="n">vec2</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Dot product convenience wrapper for Vector.dot(): \n&quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">vec1</span><span class="p">,</span> <span class="p">(</span><span class="n">Vector</span><span class="p">,</span> <span class="n">Dyadic</span><span class="p">)):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Dot product is between two vectors&#39;</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">vec1</span> <span class="o">&amp;</span> <span class="n">vec2</span></div>
-<span class="n">dot</span><span class="o">.</span><span class="n">__doc__</span> <span class="o">+=</span> <span class="n">Vector</span><span class="o">.</span><span class="n">dot</span><span class="o">.</span><span class="n">__doc__</span>
-
-
-<div class="viewcode-block" id="express"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/functions.html#sympy.physics.mechanics.functions.express">[docs]</a><span class="k">def</span> <span class="nf">express</span><span class="p">(</span><span class="n">vec</span><span class="p">,</span> <span class="n">frame</span><span class="p">,</span> <span class="n">frame2</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Express convenience wrapper for Vector.express(): \n&quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">vec</span><span class="p">,</span> <span class="p">(</span><span class="n">Vector</span><span class="p">,</span> <span class="n">Dyadic</span><span class="p">)):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Can only express Vectors&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">vec</span><span class="p">,</span> <span class="n">Vector</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">vec</span><span class="o">.</span><span class="n">express</span><span class="p">(</span><span class="n">frame</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">vec</span><span class="o">.</span><span class="n">express</span><span class="p">(</span><span class="n">frame</span><span class="p">,</span> <span class="n">frame2</span><span class="p">)</span>
-</div>
-<span class="n">express</span><span class="o">.</span><span class="n">__doc__</span> <span class="o">+=</span> <span class="n">Vector</span><span class="o">.</span><span class="n">express</span><span class="o">.</span><span class="n">__doc__</span>
-
-
-<div class="viewcode-block" id="outer"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/functions.html#sympy.physics.mechanics.functions.outer">[docs]</a><span class="k">def</span> <span class="nf">outer</span><span class="p">(</span><span class="n">vec1</span><span class="p">,</span> <span class="n">vec2</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Outer prodcut convenience wrapper for Vector.outer():\n&quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">vec1</span><span class="p">,</span> <span class="n">Vector</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Outer product is between two Vectors&#39;</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">vec1</span> <span class="o">|</span> <span class="n">vec2</span></div>
-<span class="n">outer</span><span class="o">.</span><span class="n">__doc__</span> <span class="o">+=</span> <span class="n">Vector</span><span class="o">.</span><span class="n">express</span><span class="o">.</span><span class="n">__doc__</span>
-
-
-<div class="viewcode-block" id="inertia"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.functions.inertia">[docs]</a><span class="k">def</span> <span class="nf">inertia</span><span class="p">(</span><span class="n">frame</span><span class="p">,</span> <span class="n">ixx</span><span class="p">,</span> <span class="n">iyy</span><span class="p">,</span> <span class="n">izz</span><span class="p">,</span> <span class="n">ixy</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">iyz</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">izx</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Simple way to create inertia Dyadic object.</span>
-
-<span class="sd">    If you don&#39;t know what a Dyadic is, just treat this like the inertia</span>
-<span class="sd">    tensor.  Then, do the easy thing and define it in a body-fixed frame.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    frame : ReferenceFrame</span>
-<span class="sd">        The frame the inertia is defined in</span>
-<span class="sd">    ixx : Sympifyable</span>
-<span class="sd">        the xx element in the inertia dyadic</span>
-<span class="sd">    iyy : Sympifyable</span>
-<span class="sd">        the yy element in the inertia dyadic</span>
-<span class="sd">    izz : Sympifyable</span>
-<span class="sd">        the zz element in the inertia dyadic</span>
-<span class="sd">    ixy : Sympifyable</span>
-<span class="sd">        the xy element in the inertia dyadic</span>
-<span class="sd">    iyz : Sympifyable</span>
-<span class="sd">        the yz element in the inertia dyadic</span>
-<span class="sd">    izx : Sympifyable</span>
-<span class="sd">        the zx element in the inertia dyadic</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, inertia</span>
-<span class="sd">    &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; inertia(N, 1, 2, 3)</span>
-<span class="sd">    (N.x|N.x) + 2*(N.y|N.y) + 3*(N.z|N.z)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">frame</span><span class="p">,</span> <span class="n">ReferenceFrame</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Need to define the inertia in a frame&#39;</span><span class="p">)</span>
-    <span class="n">ol</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">ixx</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">frame</span><span class="o">.</span><span class="n">x</span> <span class="o">|</span> <span class="n">frame</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-    <span class="n">ol</span> <span class="o">+=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">ixy</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">frame</span><span class="o">.</span><span class="n">x</span> <span class="o">|</span> <span class="n">frame</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-    <span class="n">ol</span> <span class="o">+=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">izx</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">frame</span><span class="o">.</span><span class="n">x</span> <span class="o">|</span> <span class="n">frame</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-    <span class="n">ol</span> <span class="o">+=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">ixy</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">frame</span><span class="o">.</span><span class="n">y</span> <span class="o">|</span> <span class="n">frame</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-    <span class="n">ol</span> <span class="o">+=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">iyy</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">frame</span><span class="o">.</span><span class="n">y</span> <span class="o">|</span> <span class="n">frame</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-    <span class="n">ol</span> <span class="o">+=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">iyz</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">frame</span><span class="o">.</span><span class="n">y</span> <span class="o">|</span> <span class="n">frame</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-    <span class="n">ol</span> <span class="o">+=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">izx</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">frame</span><span class="o">.</span><span class="n">z</span> <span class="o">|</span> <span class="n">frame</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-    <span class="n">ol</span> <span class="o">+=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">iyz</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">frame</span><span class="o">.</span><span class="n">z</span> <span class="o">|</span> <span class="n">frame</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-    <span class="n">ol</span> <span class="o">+=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">izz</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">frame</span><span class="o">.</span><span class="n">z</span> <span class="o">|</span> <span class="n">frame</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">ol</span>
-
-</div>
-<div class="viewcode-block" id="inertia_of_point_mass"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.functions.inertia_of_point_mass">[docs]</a><span class="k">def</span> <span class="nf">inertia_of_point_mass</span><span class="p">(</span><span class="n">mass</span><span class="p">,</span> <span class="n">pos_vec</span><span class="p">,</span> <span class="n">frame</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Inertia dyadic of a point mass realtive to point O.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    mass : Sympifyable</span>
-<span class="sd">        Mass of the point mass</span>
-<span class="sd">    pos_vec : Vector</span>
-<span class="sd">        Position from point O to point mass</span>
-<span class="sd">    frame : ReferenceFrame</span>
-<span class="sd">        Reference frame to express the dyadic in</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, inertia_of_point_mass</span>
-<span class="sd">    &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; r, m = symbols(&#39;r m&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; px = r * N.x</span>
-<span class="sd">    &gt;&gt;&gt; inertia_of_point_mass(m, px, N)</span>
-<span class="sd">    m*r**2*(N.y|N.y) + m*r**2*(N.z|N.z)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">mass</span> <span class="o">*</span> <span class="p">(((</span><span class="n">frame</span><span class="o">.</span><span class="n">x</span> <span class="o">|</span> <span class="n">frame</span><span class="o">.</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">frame</span><span class="o">.</span><span class="n">y</span> <span class="o">|</span> <span class="n">frame</span><span class="o">.</span><span class="n">y</span><span class="p">)</span> <span class="o">+</span>
-                   <span class="p">(</span><span class="n">frame</span><span class="o">.</span><span class="n">z</span> <span class="o">|</span> <span class="n">frame</span><span class="o">.</span><span class="n">z</span><span class="p">))</span> <span class="o">*</span> <span class="p">(</span><span class="n">pos_vec</span> <span class="o">&amp;</span> <span class="n">pos_vec</span><span class="p">)</span> <span class="o">-</span>
-                   <span class="p">(</span><span class="n">pos_vec</span> <span class="o">|</span> <span class="n">pos_vec</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="mechanics_printing"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/printing.html#sympy.physics.mechanics.functions.mechanics_printing">[docs]</a><span class="k">def</span> <span class="nf">mechanics_printing</span><span class="p">():</span>
-    <span class="sd">&quot;&quot;&quot;Sets up interactive printing for mechanics&#39; derivatives.</span>
-
-<span class="sd">    The main benefit of this is for printing of time derivatives;</span>
-<span class="sd">    instead of displaying as Derivative(f(t),t), it will display f&#39;</span>
-<span class="sd">    This is only actually needed for when derivatives are present and are not</span>
-<span class="sd">    in a physics.mechanics object.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; # 2 lines below are for tests to function properly</span>
-<span class="sd">    &gt;&gt;&gt; import sys</span>
-<span class="sd">    &gt;&gt;&gt; sys.displayhook = sys.__displayhook__</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, Symbol, diff</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import mechanics_printing</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; t = Symbol(&#39;t&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; x = Symbol(&#39;x&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; diff(f(t), t)</span>
-<span class="sd">    Derivative(f(t), t)</span>
-<span class="sd">    &gt;&gt;&gt; mechanics_printing()</span>
-<span class="sd">    &gt;&gt;&gt; diff(f(t), t)</span>
-<span class="sd">    f&#39;</span>
-<span class="sd">    &gt;&gt;&gt; diff(f(x), x)</span>
-<span class="sd">    Derivative(f(x), x)</span>
-<span class="sd">    &gt;&gt;&gt; # 2 lines below are for tests to function properly</span>
-<span class="sd">    &gt;&gt;&gt; import sys</span>
-<span class="sd">    &gt;&gt;&gt; sys.displayhook = sys.__displayhook__</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="kn">import</span> <span class="nn">sys</span>
-    <span class="n">sys</span><span class="o">.</span><span class="n">displayhook</span> <span class="o">=</span> <span class="n">mprint</span>
-
-</div>
-<div class="viewcode-block" id="mprint"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/printing.html#sympy.physics.mechanics.functions.mprint">[docs]</a><span class="k">def</span> <span class="nf">mprint</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;Function for printing of expressions generated in mechanics.</span>
-
-<span class="sd">    Extends SymPy&#39;s StrPrinter; mprint is equivalent to:</span>
-<span class="sd">    print sstr()</span>
-<span class="sd">    mprint takes the same options as sstr.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    expr : valid sympy object</span>
-<span class="sd">        SymPy expression to print</span>
-<span class="sd">    settings : args</span>
-<span class="sd">        Same as print for SymPy</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import mprint, dynamicsymbols</span>
-<span class="sd">    &gt;&gt;&gt; u1 = dynamicsymbols(&#39;u1&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; print(u1)</span>
-<span class="sd">    u1(t)</span>
-<span class="sd">    &gt;&gt;&gt; mprint(u1)</span>
-<span class="sd">    u1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">pr</span> <span class="o">=</span> <span class="n">MechanicsStrPrinter</span><span class="p">(</span><span class="n">settings</span><span class="p">)</span>
-    <span class="n">outstr</span> <span class="o">=</span> <span class="n">pr</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="kn">import</span> <span class="nn">builtins</span>
-    <span class="k">if</span> <span class="p">(</span><span class="n">outstr</span> <span class="o">!=</span> <span class="s">&#39;None&#39;</span><span class="p">):</span>
-        <span class="n">builtins</span><span class="o">.</span><span class="n">_</span> <span class="o">=</span> <span class="n">outstr</span>
-        <span class="k">print</span><span class="p">(</span><span class="n">outstr</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="mpprint"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/printing.html#sympy.physics.mechanics.functions.mpprint">[docs]</a><span class="k">def</span> <span class="nf">mpprint</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;Function for pretty printing of expressions generated in mechanics.</span>
-
-<span class="sd">    Mainly used for expressions not inside a vector; the output of running</span>
-<span class="sd">    scripts and generating equations of motion. Takes the same options as</span>
-<span class="sd">    SymPy&#39;s pretty_print(); see that function for more information.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    expr : valid sympy object</span>
-<span class="sd">        SymPy expression to pretty print</span>
-<span class="sd">    settings : args</span>
-<span class="sd">        Same as pretty print</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Use in the same way as pprint</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">mp</span> <span class="o">=</span> <span class="n">MechanicsPrettyPrinter</span><span class="p">(</span><span class="n">settings</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">((</span><span class="n">mp</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">expr</span><span class="p">)))</span>
-
-</div>
-<div class="viewcode-block" id="mlatex"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/printing.html#sympy.physics.mechanics.functions.mlatex">[docs]</a><span class="k">def</span> <span class="nf">mlatex</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;Function for printing latex representation of mechanics objects.</span>
-
-<span class="sd">    For latex representation of Vectors, Dyadics, and dynamicsymbols. Takes the</span>
-<span class="sd">    same options as SymPy&#39;s latex(); see that function for more information;</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    expr : valid sympy object</span>
-<span class="sd">        SymPy expression to represent in LaTeX form</span>
-<span class="sd">    settings : args</span>
-<span class="sd">        Same as latex()</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import mlatex, ReferenceFrame, dynamicsymbols</span>
-<span class="sd">    &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; q1, q2 = dynamicsymbols(&#39;q1 q2&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; q1d, q2d = dynamicsymbols(&#39;q1 q2&#39;, 1)</span>
-<span class="sd">    &gt;&gt;&gt; q1dd, q2dd = dynamicsymbols(&#39;q1 q2&#39;, 2)</span>
-<span class="sd">    &gt;&gt;&gt; mlatex(N.x + N.y)</span>
-<span class="sd">    &#39;\\mathbf{\\hat{n}_x} + \\mathbf{\\hat{n}_y}&#39;</span>
-<span class="sd">    &gt;&gt;&gt; mlatex(q1 + q2)</span>
-<span class="sd">    &#39;q_{1} + q_{2}&#39;</span>
-<span class="sd">    &gt;&gt;&gt; mlatex(q1d)</span>
-<span class="sd">    &#39;\\dot{q}_{1}&#39;</span>
-<span class="sd">    &gt;&gt;&gt; mlatex(q1 * q2d)</span>
-<span class="sd">    &#39;q_{1} \\dot{q}_{2}&#39;</span>
-<span class="sd">    &gt;&gt;&gt; mlatex(q1dd * q1 / q1d)</span>
-<span class="sd">    &#39;\\frac{q_{1} \\ddot{q}_{1}}{\\dot{q}_{1}}&#39;</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">MechanicsLatexPrinter</span><span class="p">(</span><span class="n">settings</span><span class="p">)</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="kinematic_equations"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kinematics.html#sympy.physics.mechanics.functions.kinematic_equations">[docs]</a><span class="k">def</span> <span class="nf">kinematic_equations</span><span class="p">(</span><span class="n">speeds</span><span class="p">,</span> <span class="n">coords</span><span class="p">,</span> <span class="n">rot_type</span><span class="p">,</span> <span class="n">rot_order</span><span class="o">=</span><span class="s">&#39;&#39;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Gives equations relating the qdot&#39;s to u&#39;s for a rotation type.</span>
-
-<span class="sd">    Supply rotation type and order as in orient. Speeds are assumed to be</span>
-<span class="sd">    body-fixed; if we are defining the orientation of B in A using by rot_type,</span>
-<span class="sd">    the angular velocity of B in A is assumed to be in the form: speed[0]*B.x +</span>
-<span class="sd">    speed[1]*B.y + speed[2]*B.z</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    speeds : list of length 3</span>
-<span class="sd">        The body fixed angular velocity measure numbers.</span>
-<span class="sd">    coords : list of length 3 or 4</span>
-<span class="sd">        The coordinates used to define the orientation of the two frames.</span>
-<span class="sd">    rot_type : str</span>
-<span class="sd">        The type of rotation used to create the equations. Body, Space, or</span>
-<span class="sd">        Quaternion only</span>
-<span class="sd">    rot_order : str</span>
-<span class="sd">        If applicable, the order of a series of rotations.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import dynamicsymbols</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import kinematic_equations, mprint</span>
-<span class="sd">    &gt;&gt;&gt; u1, u2, u3 = dynamicsymbols(&#39;u1 u2 u3&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; q1, q2, q3 = dynamicsymbols(&#39;q1 q2 q3&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; mprint(kinematic_equations([u1,u2,u3], [q1,q2,q3], &#39;body&#39;, &#39;313&#39;),</span>
-<span class="sd">    ...     order=None)</span>
-<span class="sd">    [-(u1*sin(q3) + u2*cos(q3))/sin(q2) + q1&#39;, -u1*cos(q3) + u2*sin(q3) + q2&#39;, (u1*sin(q3) + u2*cos(q3))*cos(q2)/sin(q2) - u3 + q3&#39;]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="c"># Code below is checking and sanitizing input</span>
-    <span class="n">approved_orders</span> <span class="o">=</span> <span class="p">(</span><span class="s">&#39;123&#39;</span><span class="p">,</span> <span class="s">&#39;231&#39;</span><span class="p">,</span> <span class="s">&#39;312&#39;</span><span class="p">,</span> <span class="s">&#39;132&#39;</span><span class="p">,</span> <span class="s">&#39;213&#39;</span><span class="p">,</span> <span class="s">&#39;321&#39;</span><span class="p">,</span> <span class="s">&#39;121&#39;</span><span class="p">,</span> <span class="s">&#39;131&#39;</span><span class="p">,</span>
-                       <span class="s">&#39;212&#39;</span><span class="p">,</span> <span class="s">&#39;232&#39;</span><span class="p">,</span> <span class="s">&#39;313&#39;</span><span class="p">,</span> <span class="s">&#39;323&#39;</span><span class="p">,</span> <span class="s">&#39;1&#39;</span><span class="p">,</span> <span class="s">&#39;2&#39;</span><span class="p">,</span> <span class="s">&#39;3&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-    <span class="n">rot_order</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">rot_order</span><span class="p">)</span><span class="o">.</span><span class="n">upper</span><span class="p">()</span>  <span class="c"># Now we need to make sure XYZ = 123</span>
-    <span class="n">rot_type</span> <span class="o">=</span> <span class="n">rot_type</span><span class="o">.</span><span class="n">upper</span><span class="p">()</span>
-    <span class="n">rot_order</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="s">&#39;1&#39;</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">rot_order</span><span class="p">]</span>
-    <span class="n">rot_order</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;Y&#39;</span><span class="p">,</span> <span class="s">&#39;2&#39;</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">rot_order</span><span class="p">]</span>
-    <span class="n">rot_order</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;Z&#39;</span><span class="p">,</span> <span class="s">&#39;3&#39;</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">rot_order</span><span class="p">]</span>
-    <span class="n">rot_order</span> <span class="o">=</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">rot_order</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">speeds</span><span class="p">,</span> <span class="p">(</span><span class="nb">list</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">)):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Need to supply speeds in a list&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">speeds</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">3</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Need to supply 3 body-fixed speeds&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">coords</span><span class="p">,</span> <span class="p">(</span><span class="nb">list</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">)):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Need to supply coordinates in a list&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">rot_type</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="ow">in</span> <span class="p">[</span><span class="s">&#39;body&#39;</span><span class="p">,</span> <span class="s">&#39;space&#39;</span><span class="p">]:</span>
-        <span class="k">if</span> <span class="n">rot_order</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">approved_orders</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Not an acceptable rotation order&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">coords</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Need 3 coordinates for body or space&#39;</span><span class="p">)</span>
-        <span class="c"># Actual hard-coded kinematic differential equations</span>
-        <span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span> <span class="o">=</span> <span class="n">coords</span>
-        <span class="n">q1d</span><span class="p">,</span> <span class="n">q2d</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">=</span> <span class="p">[</span><span class="n">diff</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">coords</span><span class="p">]</span>
-        <span class="n">w1</span><span class="p">,</span> <span class="n">w2</span><span class="p">,</span> <span class="n">w3</span> <span class="o">=</span> <span class="n">speeds</span>
-        <span class="n">s1</span><span class="p">,</span> <span class="n">s2</span><span class="p">,</span> <span class="n">s3</span> <span class="o">=</span> <span class="p">[</span><span class="n">sin</span><span class="p">(</span><span class="n">q1</span><span class="p">),</span> <span class="n">sin</span><span class="p">(</span><span class="n">q2</span><span class="p">),</span> <span class="n">sin</span><span class="p">(</span><span class="n">q3</span><span class="p">)]</span>
-        <span class="n">c1</span><span class="p">,</span> <span class="n">c2</span><span class="p">,</span> <span class="n">c3</span> <span class="o">=</span> <span class="p">[</span><span class="n">cos</span><span class="p">(</span><span class="n">q1</span><span class="p">),</span> <span class="n">cos</span><span class="p">(</span><span class="n">q2</span><span class="p">),</span> <span class="n">cos</span><span class="p">(</span><span class="n">q3</span><span class="p">)]</span>
-        <span class="k">if</span> <span class="n">rot_type</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="o">==</span> <span class="s">&#39;body&#39;</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;123&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">c3</span> <span class="o">-</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">s3</span><span class="p">)</span> <span class="o">/</span> <span class="n">c2</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">w1</span> <span class="o">*</span> <span class="n">s3</span> <span class="o">-</span> <span class="n">w2</span> <span class="o">*</span>
-                        <span class="n">c3</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="p">(</span><span class="o">-</span><span class="n">w1</span> <span class="o">*</span> <span class="n">c3</span> <span class="o">+</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">s3</span><span class="p">)</span> <span class="o">*</span> <span class="n">s2</span> <span class="o">/</span> <span class="n">c2</span> <span class="o">-</span> <span class="n">w3</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;231&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w2</span> <span class="o">*</span> <span class="n">c3</span> <span class="o">-</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">s3</span><span class="p">)</span> <span class="o">/</span> <span class="n">c2</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">s3</span> <span class="o">-</span> <span class="n">w3</span> <span class="o">*</span>
-                        <span class="n">c3</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="n">w1</span> <span class="o">-</span> <span class="p">(</span><span class="o">-</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">c3</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">s3</span><span class="p">)</span> <span class="o">*</span> <span class="n">s2</span> <span class="o">/</span> <span class="n">c2</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;312&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="p">(</span><span class="o">-</span><span class="n">w1</span> <span class="o">*</span> <span class="n">s3</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">c3</span><span class="p">)</span> <span class="o">/</span> <span class="n">c2</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">w1</span> <span class="o">*</span> <span class="n">c3</span> <span class="o">-</span> <span class="n">w3</span> <span class="o">*</span>
-                        <span class="n">s3</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">s3</span> <span class="o">-</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">c3</span><span class="p">)</span> <span class="o">*</span> <span class="n">s2</span> <span class="o">/</span> <span class="n">c2</span> <span class="o">-</span> <span class="n">w2</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;132&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">c3</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">s3</span><span class="p">)</span> <span class="o">/</span> <span class="n">c2</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">+</span> <span class="n">w1</span> <span class="o">*</span> <span class="n">s3</span> <span class="o">-</span> <span class="n">w3</span> <span class="o">*</span>
-                        <span class="n">c3</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">c3</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">s3</span><span class="p">)</span> <span class="o">*</span> <span class="n">s2</span> <span class="o">/</span> <span class="n">c2</span> <span class="o">-</span> <span class="n">w2</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;213&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">s3</span> <span class="o">+</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">c3</span><span class="p">)</span> <span class="o">/</span> <span class="n">c2</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">w1</span> <span class="o">*</span> <span class="n">c3</span> <span class="o">+</span> <span class="n">w2</span> <span class="o">*</span>
-                        <span class="n">s3</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">s3</span> <span class="o">+</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">c3</span><span class="p">)</span> <span class="o">*</span> <span class="n">s2</span> <span class="o">/</span> <span class="n">c2</span> <span class="o">-</span> <span class="n">w3</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;321&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w2</span> <span class="o">*</span> <span class="n">s3</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">c3</span><span class="p">)</span> <span class="o">/</span> <span class="n">c2</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">c3</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span>
-                        <span class="n">s3</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="n">w1</span> <span class="o">-</span> <span class="p">(</span><span class="n">w2</span> <span class="o">*</span> <span class="n">s3</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">c3</span><span class="p">)</span> <span class="o">*</span> <span class="n">s2</span> <span class="o">/</span> <span class="n">c2</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;121&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w2</span> <span class="o">*</span> <span class="n">s3</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">c3</span><span class="p">)</span> <span class="o">/</span> <span class="n">s2</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">c3</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span>
-                        <span class="n">s3</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="n">w1</span> <span class="o">+</span> <span class="p">(</span><span class="n">w2</span> <span class="o">*</span> <span class="n">s3</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">c3</span><span class="p">)</span> <span class="o">*</span> <span class="n">c2</span> <span class="o">/</span> <span class="n">s2</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;131&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="p">(</span><span class="o">-</span><span class="n">w2</span> <span class="o">*</span> <span class="n">c3</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">s3</span><span class="p">)</span> <span class="o">/</span> <span class="n">s2</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">s3</span> <span class="o">-</span> <span class="n">w3</span> <span class="o">*</span>
-                        <span class="n">c3</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="n">w1</span> <span class="o">-</span> <span class="p">(</span><span class="n">w2</span> <span class="o">*</span> <span class="n">c3</span> <span class="o">-</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">s3</span><span class="p">)</span> <span class="o">*</span> <span class="n">c2</span> <span class="o">/</span> <span class="n">s2</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;212&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">s3</span> <span class="o">-</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">c3</span><span class="p">)</span> <span class="o">/</span> <span class="n">s2</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">w1</span> <span class="o">*</span> <span class="n">c3</span> <span class="o">-</span> <span class="n">w3</span> <span class="o">*</span>
-                        <span class="n">s3</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="p">(</span><span class="o">-</span><span class="n">w1</span> <span class="o">*</span> <span class="n">s3</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">c3</span><span class="p">)</span> <span class="o">*</span> <span class="n">c2</span> <span class="o">/</span> <span class="n">s2</span> <span class="o">-</span> <span class="n">w2</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;232&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">c3</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">s3</span><span class="p">)</span> <span class="o">/</span> <span class="n">s2</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">+</span> <span class="n">w1</span> <span class="o">*</span> <span class="n">s3</span> <span class="o">-</span> <span class="n">w3</span> <span class="o">*</span>
-                        <span class="n">c3</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">+</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">c3</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">s3</span><span class="p">)</span> <span class="o">*</span> <span class="n">c2</span> <span class="o">/</span> <span class="n">s2</span> <span class="o">-</span> <span class="n">w2</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;313&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">s3</span> <span class="o">+</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">c3</span><span class="p">)</span> <span class="o">/</span> <span class="n">s2</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">w1</span> <span class="o">*</span> <span class="n">c3</span> <span class="o">+</span> <span class="n">w2</span> <span class="o">*</span>
-                        <span class="n">s3</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">+</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">s3</span> <span class="o">+</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">c3</span><span class="p">)</span> <span class="o">*</span> <span class="n">c2</span> <span class="o">/</span> <span class="n">s2</span> <span class="o">-</span> <span class="n">w3</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;323&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="p">(</span><span class="o">-</span><span class="n">w1</span> <span class="o">*</span> <span class="n">c3</span> <span class="o">+</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">s3</span><span class="p">)</span> <span class="o">/</span> <span class="n">s2</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">w1</span> <span class="o">*</span> <span class="n">s3</span> <span class="o">-</span> <span class="n">w2</span> <span class="o">*</span>
-                        <span class="n">c3</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">c3</span> <span class="o">-</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">s3</span><span class="p">)</span> <span class="o">*</span> <span class="n">c2</span> <span class="o">/</span> <span class="n">s2</span> <span class="o">-</span> <span class="n">w3</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">rot_type</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="o">==</span> <span class="s">&#39;space&#39;</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;123&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="n">w1</span> <span class="o">-</span> <span class="p">(</span><span class="n">w2</span> <span class="o">*</span> <span class="n">s1</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">c1</span><span class="p">)</span> <span class="o">*</span> <span class="n">s2</span> <span class="o">/</span> <span class="n">c2</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">w2</span> <span class="o">*</span>
-                        <span class="n">c1</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">s1</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w2</span> <span class="o">*</span> <span class="n">s1</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">c1</span><span class="p">)</span> <span class="o">/</span> <span class="n">c2</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;231&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">c1</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">s1</span><span class="p">)</span> <span class="o">*</span> <span class="n">s2</span> <span class="o">/</span> <span class="n">c2</span> <span class="o">-</span> <span class="n">w2</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">+</span> <span class="n">w1</span> <span class="o">*</span>
-                        <span class="n">s1</span> <span class="o">-</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">c1</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">c1</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">s1</span><span class="p">)</span> <span class="o">/</span> <span class="n">c2</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;312&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">s1</span> <span class="o">+</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">c1</span><span class="p">)</span> <span class="o">*</span> <span class="n">s2</span> <span class="o">/</span> <span class="n">c2</span> <span class="o">-</span> <span class="n">w3</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">w1</span> <span class="o">*</span>
-                        <span class="n">c1</span> <span class="o">+</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">s1</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">s1</span> <span class="o">+</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">c1</span><span class="p">)</span> <span class="o">/</span> <span class="n">c2</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;132&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="n">w1</span> <span class="o">-</span> <span class="p">(</span><span class="o">-</span><span class="n">w2</span> <span class="o">*</span> <span class="n">c1</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">s1</span><span class="p">)</span> <span class="o">*</span> <span class="n">s2</span> <span class="o">/</span> <span class="n">c2</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">w2</span> <span class="o">*</span>
-                        <span class="n">s1</span> <span class="o">-</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">c1</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w2</span> <span class="o">*</span> <span class="n">c1</span> <span class="o">-</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">s1</span><span class="p">)</span> <span class="o">/</span> <span class="n">c2</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;213&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">s1</span> <span class="o">-</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">c1</span><span class="p">)</span> <span class="o">*</span> <span class="n">s2</span> <span class="o">/</span> <span class="n">c2</span> <span class="o">-</span> <span class="n">w2</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">w1</span> <span class="o">*</span>
-                        <span class="n">c1</span> <span class="o">-</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">s1</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="p">(</span><span class="o">-</span><span class="n">w1</span> <span class="o">*</span> <span class="n">s1</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">c1</span><span class="p">)</span> <span class="o">/</span> <span class="n">c2</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;321&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="p">(</span><span class="o">-</span><span class="n">w1</span> <span class="o">*</span> <span class="n">c1</span> <span class="o">+</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">s1</span><span class="p">)</span> <span class="o">*</span> <span class="n">s2</span> <span class="o">/</span> <span class="n">c2</span> <span class="o">-</span> <span class="n">w3</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">w1</span> <span class="o">*</span>
-                        <span class="n">s1</span> <span class="o">-</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">c1</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">c1</span> <span class="o">-</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">s1</span><span class="p">)</span> <span class="o">/</span> <span class="n">c2</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;121&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="n">w1</span> <span class="o">+</span> <span class="p">(</span><span class="n">w2</span> <span class="o">*</span> <span class="n">s1</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">c1</span><span class="p">)</span> <span class="o">*</span> <span class="n">c2</span> <span class="o">/</span> <span class="n">s2</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">w2</span> <span class="o">*</span>
-                        <span class="n">c1</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">s1</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w2</span> <span class="o">*</span> <span class="n">s1</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">c1</span><span class="p">)</span> <span class="o">/</span> <span class="n">s2</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;131&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="n">w1</span> <span class="o">-</span> <span class="p">(</span><span class="n">w2</span> <span class="o">*</span> <span class="n">c1</span> <span class="o">-</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">s1</span><span class="p">)</span> <span class="o">*</span> <span class="n">c2</span> <span class="o">/</span> <span class="n">s2</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">w2</span> <span class="o">*</span>
-                        <span class="n">s1</span> <span class="o">-</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">c1</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="p">(</span><span class="o">-</span><span class="n">w2</span> <span class="o">*</span> <span class="n">c1</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">s1</span><span class="p">)</span> <span class="o">/</span> <span class="n">s2</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;212&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="p">(</span><span class="o">-</span><span class="n">w1</span> <span class="o">*</span> <span class="n">s1</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">c1</span><span class="p">)</span> <span class="o">*</span> <span class="n">c2</span> <span class="o">/</span> <span class="n">s2</span> <span class="o">-</span> <span class="n">w2</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">w1</span> <span class="o">*</span>
-                        <span class="n">c1</span> <span class="o">-</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">s1</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">s1</span> <span class="o">-</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">c1</span><span class="p">)</span> <span class="o">/</span> <span class="n">s2</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;232&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">+</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">c1</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">s1</span><span class="p">)</span> <span class="o">*</span> <span class="n">c2</span> <span class="o">/</span> <span class="n">s2</span> <span class="o">-</span> <span class="n">w2</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">+</span> <span class="n">w1</span> <span class="o">*</span>
-                        <span class="n">s1</span> <span class="o">-</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">c1</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">c1</span> <span class="o">+</span> <span class="n">w3</span> <span class="o">*</span> <span class="n">s1</span><span class="p">)</span> <span class="o">/</span> <span class="n">s2</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;313&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">+</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">s1</span> <span class="o">+</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">c1</span><span class="p">)</span> <span class="o">*</span> <span class="n">c2</span> <span class="o">/</span> <span class="n">s2</span> <span class="o">-</span> <span class="n">w3</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">w1</span> <span class="o">*</span>
-                        <span class="n">c1</span> <span class="o">+</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">s1</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">s1</span> <span class="o">+</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">c1</span><span class="p">)</span> <span class="o">/</span> <span class="n">s2</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">rot_order</span> <span class="o">==</span> <span class="s">&#39;323&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="p">(</span><span class="n">w1</span> <span class="o">*</span> <span class="n">c1</span> <span class="o">-</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">s1</span><span class="p">)</span> <span class="o">*</span> <span class="n">c2</span> <span class="o">/</span> <span class="n">s2</span> <span class="o">-</span> <span class="n">w3</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">w1</span> <span class="o">*</span>
-                        <span class="n">s1</span> <span class="o">-</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">c1</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="p">(</span><span class="o">-</span><span class="n">w1</span> <span class="o">*</span> <span class="n">c1</span> <span class="o">+</span> <span class="n">w2</span> <span class="o">*</span> <span class="n">s1</span><span class="p">)</span> <span class="o">/</span> <span class="n">s2</span><span class="p">]</span>
-    <span class="k">elif</span> <span class="n">rot_type</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="o">==</span> <span class="s">&#39;quaternion&#39;</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">rot_order</span> <span class="o">!=</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Cannot have rotation order for quaternion&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">coords</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">4</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Need 4 coordinates for quaternion&#39;</span><span class="p">)</span>
-        <span class="c"># Actual hard-coded kinematic differential equations</span>
-        <span class="n">e0</span><span class="p">,</span> <span class="n">e1</span><span class="p">,</span> <span class="n">e2</span><span class="p">,</span> <span class="n">e3</span> <span class="o">=</span> <span class="n">coords</span>
-        <span class="n">w</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">speeds</span> <span class="o">+</span> <span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="n">E</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="n">e0</span><span class="p">,</span> <span class="o">-</span><span class="n">e3</span><span class="p">,</span> <span class="n">e2</span><span class="p">,</span> <span class="n">e1</span><span class="p">],</span> <span class="p">[</span><span class="n">e3</span><span class="p">,</span> <span class="n">e0</span><span class="p">,</span> <span class="o">-</span><span class="n">e1</span><span class="p">,</span> <span class="n">e2</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="n">e2</span><span class="p">,</span> <span class="n">e1</span><span class="p">,</span> <span class="n">e0</span><span class="p">,</span> <span class="n">e3</span><span class="p">],</span>
-            <span class="p">[</span><span class="o">-</span><span class="n">e1</span><span class="p">,</span> <span class="o">-</span><span class="n">e2</span><span class="p">,</span> <span class="o">-</span><span class="n">e3</span><span class="p">,</span> <span class="n">e0</span><span class="p">]])</span>
-        <span class="n">edots</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span><span class="n">diff</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="n">e1</span><span class="p">,</span> <span class="n">e2</span><span class="p">,</span> <span class="n">e3</span><span class="p">,</span> <span class="n">e0</span><span class="p">]])</span>
-        <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="n">edots</span><span class="o">.</span><span class="n">T</span> <span class="o">-</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">w</span><span class="o">.</span><span class="n">T</span> <span class="o">*</span> <span class="n">E</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Not an approved rotation type for this function&#39;</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="partial_velocity"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.functions.partial_velocity">[docs]</a><span class="k">def</span> <span class="nf">partial_velocity</span><span class="p">(</span><span class="n">vel_list</span><span class="p">,</span> <span class="n">u_list</span><span class="p">,</span> <span class="n">frame</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns a list of partial velocities.</span>
-
-<span class="sd">    For a list of velocity or angular velocity vectors the partial derivatives</span>
-<span class="sd">    with respect to the supplied generalized speeds are computed, in the</span>
-<span class="sd">    specified ReferenceFrame.</span>
-
-<span class="sd">    The output is a list of lists. The outer list has a number of elements</span>
-<span class="sd">    equal to the number of supplied velocity vectors. The inner lists are, for</span>
-<span class="sd">    each velocity vector, the partial derivatives of that velocity vector with</span>
-<span class="sd">    respect to the generalized speeds supplied.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    vel_list : list</span>
-<span class="sd">        List of velocities of Point&#39;s and angular velocities of ReferenceFrame&#39;s</span>
-<span class="sd">    u_list : list</span>
-<span class="sd">        List of independent generalized speeds.</span>
-<span class="sd">    frame : ReferenceFrame</span>
-<span class="sd">        The ReferenceFrame the partial derivatives are going to be taken in.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import Point, ReferenceFrame</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import dynamicsymbols</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import partial_velocity</span>
-<span class="sd">    &gt;&gt;&gt; u = dynamicsymbols(&#39;u&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; P = Point(&#39;P&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; P.set_vel(N, u * N.x)</span>
-<span class="sd">    &gt;&gt;&gt; vel_list = [P.vel(N)]</span>
-<span class="sd">    &gt;&gt;&gt; u_list = [u]</span>
-<span class="sd">    &gt;&gt;&gt; partial_velocity(vel_list, u_list, N)</span>
-<span class="sd">    [[N.x]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">vel_list</span><span class="p">,</span> <span class="s">&#39;__iter__&#39;</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Provide velocities in an iterable&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">u_list</span><span class="p">,</span> <span class="s">&#39;__iter__&#39;</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Provide speeds in an iterable&#39;</span><span class="p">)</span>
-    <span class="n">list_of_pvlists</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">vel_list</span><span class="p">:</span>
-        <span class="n">pvlist</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">u_list</span><span class="p">:</span>
-            <span class="n">vel</span> <span class="o">=</span> <span class="n">i</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">frame</span><span class="p">)</span>
-            <span class="n">pvlist</span> <span class="o">+=</span> <span class="p">[</span><span class="n">vel</span><span class="p">]</span>
-        <span class="n">list_of_pvlists</span> <span class="o">+=</span> <span class="p">[</span><span class="n">pvlist</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">list_of_pvlists</span>
-
-</div>
-<div class="viewcode-block" id="linear_momentum"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.functions.linear_momentum">[docs]</a><span class="k">def</span> <span class="nf">linear_momentum</span><span class="p">(</span><span class="n">frame</span><span class="p">,</span> <span class="o">*</span><span class="n">body</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Linear momentum of the system.</span>
-
-<span class="sd">    This function returns the linear momentum of a system of Particle&#39;s and/or</span>
-<span class="sd">    RigidBody&#39;s. The linear momentum of a system is equal to the vector sum of</span>
-<span class="sd">    the linear momentum of its constituents. Consider a system, S, comprised of</span>
-<span class="sd">    a rigid body, A, and a particle, P. The linear momentum of the system, L,</span>
-<span class="sd">    is equal to the vector sum of the linear momentum of the particle, L1, and</span>
-<span class="sd">    the linear momentum of the rigid body, L2, i.e-</span>
-
-<span class="sd">    L = L1 + L2</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    frame : ReferenceFrame</span>
-<span class="sd">        The frame in which linear momentum is desired.</span>
-<span class="sd">    body1, body2, body3... : Particle and/or RigidBody</span>
-<span class="sd">        The body (or bodies) whose kinetic energy is required.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import Point, Particle, ReferenceFrame</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import RigidBody, outer, linear_momentum</span>
-<span class="sd">    &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; P = Point(&#39;P&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; P.set_vel(N, 10 * N.x)</span>
-<span class="sd">    &gt;&gt;&gt; Pa = Particle(&#39;Pa&#39;, P, 1)</span>
-<span class="sd">    &gt;&gt;&gt; Ac = Point(&#39;Ac&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; Ac.set_vel(N, 25 * N.y)</span>
-<span class="sd">    &gt;&gt;&gt; I = outer(N.x, N.x)</span>
-<span class="sd">    &gt;&gt;&gt; A = RigidBody(&#39;A&#39;, Ac, N, 20, (I, Ac))</span>
-<span class="sd">    &gt;&gt;&gt; linear_momentum(N, A, Pa)</span>
-<span class="sd">    10*N.x + 500*N.y</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">frame</span><span class="p">,</span> <span class="n">ReferenceFrame</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Please specify a valid ReferenceFrame&#39;</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">linear_momentum_sys</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">body</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="p">(</span><span class="n">RigidBody</span><span class="p">,</span> <span class="n">Particle</span><span class="p">)):</span>
-                <span class="n">linear_momentum_sys</span> <span class="o">+=</span> <span class="n">e</span><span class="o">.</span><span class="n">linear_momentum</span><span class="p">(</span><span class="n">frame</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;*body must have only Particle or RigidBody&#39;</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">linear_momentum_sys</span>
-
-</div>
-<div class="viewcode-block" id="angular_momentum"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.functions.angular_momentum">[docs]</a><span class="k">def</span> <span class="nf">angular_momentum</span><span class="p">(</span><span class="n">point</span><span class="p">,</span> <span class="n">frame</span><span class="p">,</span> <span class="o">*</span><span class="n">body</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Angular momentum of a system</span>
-
-<span class="sd">    This function returns the angular momentum of a system of Particle&#39;s and/or</span>
-<span class="sd">    RigidBody&#39;s. The angular momentum of such a system is equal to the vector</span>
-<span class="sd">    sum of the angular momentum of its constituents. Consider a system, S,</span>
-<span class="sd">    comprised of a rigid body, A, and a particle, P. The angular momentum of</span>
-<span class="sd">    the system, H, is equal to the vector sum of the linear momentum of the</span>
-<span class="sd">    particle, H1, and the linear momentum of the rigid body, H2, i.e-</span>
-
-<span class="sd">    H = H1 + H2</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    point : Point</span>
-<span class="sd">        The point about which angular momentum of the system is desired.</span>
-<span class="sd">    frame : ReferenceFrame</span>
-<span class="sd">        The frame in which angular momentum is desired.</span>
-<span class="sd">    body1, body2, body3... : Particle and/or RigidBody</span>
-<span class="sd">        The body (or bodies) whose kinetic energy is required.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import Point, Particle, ReferenceFrame</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import RigidBody, outer, angular_momentum</span>
-<span class="sd">    &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; O = Point(&#39;O&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; O.set_vel(N, 0 * N.x)</span>
-<span class="sd">    &gt;&gt;&gt; P = O.locatenew(&#39;P&#39;, 1 * N.x)</span>
-<span class="sd">    &gt;&gt;&gt; P.set_vel(N, 10 * N.x)</span>
-<span class="sd">    &gt;&gt;&gt; Pa = Particle(&#39;Pa&#39;, P, 1)</span>
-<span class="sd">    &gt;&gt;&gt; Ac = O.locatenew(&#39;Ac&#39;, 2 * N.y)</span>
-<span class="sd">    &gt;&gt;&gt; Ac.set_vel(N, 5 * N.y)</span>
-<span class="sd">    &gt;&gt;&gt; a = ReferenceFrame(&#39;a&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; a.set_ang_vel(N, 10 * N.z)</span>
-<span class="sd">    &gt;&gt;&gt; I = outer(N.z, N.z)</span>
-<span class="sd">    &gt;&gt;&gt; A = RigidBody(&#39;A&#39;, Ac, a, 20, (I, Ac))</span>
-<span class="sd">    &gt;&gt;&gt; angular_momentum(O, N, Pa, A)</span>
-<span class="sd">    10*N.z</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">frame</span><span class="p">,</span> <span class="n">ReferenceFrame</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Please enter a valid ReferenceFrame&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">point</span><span class="p">,</span> <span class="n">Point</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Please specify a valid Point&#39;</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">angular_momentum_sys</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">body</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="p">(</span><span class="n">RigidBody</span><span class="p">,</span> <span class="n">Particle</span><span class="p">)):</span>
-                <span class="n">angular_momentum_sys</span> <span class="o">+=</span> <span class="n">e</span><span class="o">.</span><span class="n">angular_momentum</span><span class="p">(</span><span class="n">point</span><span class="p">,</span> <span class="n">frame</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;*body must have only Particle or RigidBody&#39;</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">angular_momentum_sys</span>
-
-</div>
-<div class="viewcode-block" id="kinetic_energy"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.functions.kinetic_energy">[docs]</a><span class="k">def</span> <span class="nf">kinetic_energy</span><span class="p">(</span><span class="n">frame</span><span class="p">,</span> <span class="o">*</span><span class="n">body</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Kinetic energy of a multibody system.</span>
-
-<span class="sd">    This function returns the kinetic energy of a system of Particle&#39;s and/or</span>
-<span class="sd">    RigidBody&#39;s. The kinetic energy of such a system is equal to the sum of</span>
-<span class="sd">    the kinetic energies of its constituents. Consider a system, S, comprising</span>
-<span class="sd">    a rigid body, A, and a particle, P. The kinetic energy of the system, T,</span>
-<span class="sd">    is equal to the vector sum of the kinetic energy of the particle, T1, and</span>
-<span class="sd">    the kinetic energy of the rigid body, T2, i.e.</span>
-
-<span class="sd">    T = T1 + T2</span>
-
-<span class="sd">    Kinetic energy is a scalar.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    frame : ReferenceFrame</span>
-<span class="sd">        The frame in which the velocity or angular velocity of the body is</span>
-<span class="sd">        defined.</span>
-<span class="sd">    body1, body2, body3... : Particle and/or RigidBody</span>
-<span class="sd">        The body (or bodies) whose kinetic energy is required.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import Point, Particle, ReferenceFrame</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import RigidBody, outer, kinetic_energy</span>
-<span class="sd">    &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; O = Point(&#39;O&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; O.set_vel(N, 0 * N.x)</span>
-<span class="sd">    &gt;&gt;&gt; P = O.locatenew(&#39;P&#39;, 1 * N.x)</span>
-<span class="sd">    &gt;&gt;&gt; P.set_vel(N, 10 * N.x)</span>
-<span class="sd">    &gt;&gt;&gt; Pa = Particle(&#39;Pa&#39;, P, 1)</span>
-<span class="sd">    &gt;&gt;&gt; Ac = O.locatenew(&#39;Ac&#39;, 2 * N.y)</span>
-<span class="sd">    &gt;&gt;&gt; Ac.set_vel(N, 5 * N.y)</span>
-<span class="sd">    &gt;&gt;&gt; a = ReferenceFrame(&#39;a&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; a.set_ang_vel(N, 10 * N.z)</span>
-<span class="sd">    &gt;&gt;&gt; I = outer(N.z, N.z)</span>
-<span class="sd">    &gt;&gt;&gt; A = RigidBody(&#39;A&#39;, Ac, a, 20, (I, Ac))</span>
-<span class="sd">    &gt;&gt;&gt; kinetic_energy(N, Pa, A)</span>
-<span class="sd">    350</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">frame</span><span class="p">,</span> <span class="n">ReferenceFrame</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Please enter a valid ReferenceFrame&#39;</span><span class="p">)</span>
-    <span class="n">ke_sys</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">body</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="p">(</span><span class="n">RigidBody</span><span class="p">,</span> <span class="n">Particle</span><span class="p">)):</span>
-            <span class="n">ke_sys</span> <span class="o">+=</span> <span class="n">e</span><span class="o">.</span><span class="n">kinetic_energy</span><span class="p">(</span><span class="n">frame</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;*body must have only Particle or RigidBody&#39;</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">ke_sys</span>
-
-</div>
-<div class="viewcode-block" id="potential_energy"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.functions.potential_energy">[docs]</a><span class="k">def</span> <span class="nf">potential_energy</span><span class="p">(</span><span class="o">*</span><span class="n">body</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Potential energy of a multibody system.</span>
-
-<span class="sd">    This function returns the potential energy of a system of Particle&#39;s and/or</span>
-<span class="sd">    RigidBody&#39;s. The potential energy of such a system is equal to the sum of</span>
-<span class="sd">    the potential energy of its constituents. Consider a system, S, comprising</span>
-<span class="sd">    a rigid body, A, and a particle, P. The potential energy of the system, V,</span>
-<span class="sd">    is equal to the vector sum of the potential energy of the particle, V1, and</span>
-<span class="sd">    the potential energy of the rigid body, V2, i.e.</span>
-
-<span class="sd">    V = V1 + V2</span>
-
-<span class="sd">    Potential energy is a scalar.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    body1, body2, body3... : Particle and/or RigidBody</span>
-<span class="sd">        The body (or bodies) whose potential energy is required.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import Point, Particle, ReferenceFrame</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import RigidBody, outer, potential_energy</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; M, m, g, h = symbols(&#39;M m g h&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; O = Point(&#39;O&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; O.set_vel(N, 0 * N.x)</span>
-<span class="sd">    &gt;&gt;&gt; P = O.locatenew(&#39;P&#39;, 1 * N.x)</span>
-<span class="sd">    &gt;&gt;&gt; Pa = Particle(&#39;Pa&#39;, P, m)</span>
-<span class="sd">    &gt;&gt;&gt; Ac = O.locatenew(&#39;Ac&#39;, 2 * N.y)</span>
-<span class="sd">    &gt;&gt;&gt; a = ReferenceFrame(&#39;a&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; I = outer(N.z, N.z)</span>
-<span class="sd">    &gt;&gt;&gt; A = RigidBody(&#39;A&#39;, Ac, a, M, (I, Ac))</span>
-<span class="sd">    &gt;&gt;&gt; Pa.set_potential_energy(m * g * h)</span>
-<span class="sd">    &gt;&gt;&gt; A.set_potential_energy(M * g * h)</span>
-<span class="sd">    &gt;&gt;&gt; potential_energy(Pa, A)</span>
-<span class="sd">    M*g*h + g*h*m</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">pe_sys</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">body</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="p">(</span><span class="n">RigidBody</span><span class="p">,</span> <span class="n">Particle</span><span class="p">)):</span>
-            <span class="n">pe_sys</span> <span class="o">+=</span> <span class="n">e</span><span class="o">.</span><span class="n">potential_energy</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;*body must have only Particle or RigidBody&#39;</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">pe_sys</span>
-
-</div>
-<div class="viewcode-block" id="Lagrangian"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.functions.Lagrangian">[docs]</a><span class="k">def</span> <span class="nf">Lagrangian</span><span class="p">(</span><span class="n">frame</span><span class="p">,</span> <span class="o">*</span><span class="n">body</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Lagrangian of a multibody system.</span>
-
-<span class="sd">    This function returns the Lagrangian of a system of Particle&#39;s and/or</span>
-<span class="sd">    RigidBody&#39;s. The Lagrangian of such a system is equal to the difference</span>
-<span class="sd">    between the kinetic energies and potential energies of its constituents. If</span>
-<span class="sd">    T and V are the kinetic and potential energies of a system then it&#39;s</span>
-<span class="sd">    Lagrangian, L, is defined as</span>
-
-<span class="sd">    L = T - V</span>
-
-<span class="sd">    The Lagrangian is a scalar.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    frame : ReferenceFrame</span>
-<span class="sd">        The frame in which the velocity or angular velocity of the body is</span>
-<span class="sd">        defined to determine the kinetic energy.</span>
-
-<span class="sd">    body1, body2, body3... : Particle and/or RigidBody</span>
-<span class="sd">        The body (or bodies) whose kinetic energy is required.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import Point, Particle, ReferenceFrame</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import RigidBody, outer, Lagrangian</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; M, m, g, h = symbols(&#39;M m g h&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; O = Point(&#39;O&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; O.set_vel(N, 0 * N.x)</span>
-<span class="sd">    &gt;&gt;&gt; P = O.locatenew(&#39;P&#39;, 1 * N.x)</span>
-<span class="sd">    &gt;&gt;&gt; P.set_vel(N, 10 * N.x)</span>
-<span class="sd">    &gt;&gt;&gt; Pa = Particle(&#39;Pa&#39;, P, 1)</span>
-<span class="sd">    &gt;&gt;&gt; Ac = O.locatenew(&#39;Ac&#39;, 2 * N.y)</span>
-<span class="sd">    &gt;&gt;&gt; Ac.set_vel(N, 5 * N.y)</span>
-<span class="sd">    &gt;&gt;&gt; a = ReferenceFrame(&#39;a&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; a.set_ang_vel(N, 10 * N.z)</span>
-<span class="sd">    &gt;&gt;&gt; I = outer(N.z, N.z)</span>
-<span class="sd">    &gt;&gt;&gt; A = RigidBody(&#39;A&#39;, Ac, a, 20, (I, Ac))</span>
-<span class="sd">    &gt;&gt;&gt; Pa.set_potential_energy(m * g * h)</span>
-<span class="sd">    &gt;&gt;&gt; A.set_potential_energy(M * g * h)</span>
-<span class="sd">    &gt;&gt;&gt; Lagrangian(N, Pa, A)</span>
-<span class="sd">    -M*g*h - g*h*m + 350</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">frame</span><span class="p">,</span> <span class="n">ReferenceFrame</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Please supply a valid ReferenceFrame&#39;</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">body</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="p">(</span><span class="n">RigidBody</span><span class="p">,</span> <span class="n">Particle</span><span class="p">)):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;*body must have only Particle or RigidBody&#39;</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">kinetic_energy</span><span class="p">(</span><span class="n">frame</span><span class="p">,</span> <span class="o">*</span><span class="n">body</span><span class="p">)</span> <span class="o">-</span> <span class="n">potential_energy</span><span class="p">(</span><span class="o">*</span><span class="n">body</span><span class="p">)</span></div>
-</pre></div>
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-  <h1>Source code for sympy.physics.mechanics.kane</h1><div class="highlight"><pre>
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;KanesMethod&#39;</span><span class="p">]</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">zeros</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">diff</span><span class="p">,</span> <span class="n">solve_linear_system_LU</span><span class="p">,</span> <span class="n">eye</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">default_sort_key</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.mechanics.essential</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">dynamicsymbols</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.mechanics.particle</span> <span class="kn">import</span> <span class="n">Particle</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.mechanics.point</span> <span class="kn">import</span> <span class="n">Point</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.mechanics.rigidbody</span> <span class="kn">import</span> <span class="n">RigidBody</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.mechanics.functions</span> <span class="kn">import</span> <span class="p">(</span><span class="n">inertia_of_point_mass</span><span class="p">,</span>
-                                               <span class="n">partial_velocity</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-
-<div class="viewcode-block" id="KanesMethod"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.kane.KanesMethod">[docs]</a><span class="k">class</span> <span class="nc">KanesMethod</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Kane&#39;s method object.</span>
-
-<span class="sd">    This object is used to do the &quot;book-keeping&quot; as you go through and form</span>
-<span class="sd">    equations of motion in the way Kane presents in:</span>
-<span class="sd">    Kane, T., Levinson, D. Dynamics Theory and Applications. 1985 McGraw-Hill</span>
-
-<span class="sd">    The attributes are for equations in the form [M] udot = forcing.</span>
-
-<span class="sd">    Attributes</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    auxiliary : Matrix</span>
-<span class="sd">        If applicable, the set of auxiliary Kane&#39;s</span>
-<span class="sd">        equations used to solve for non-contributing</span>
-<span class="sd">        forces.</span>
-<span class="sd">    mass_matrix : Matrix</span>
-<span class="sd">        The system&#39;s mass matrix</span>
-<span class="sd">    forcing : Matrix</span>
-<span class="sd">        The system&#39;s forcing vector</span>
-<span class="sd">    mass_matrix_full : Matrix</span>
-<span class="sd">        The &quot;mass matrix&quot; for the u&#39;s and q&#39;s</span>
-<span class="sd">    forcing_full : Matrix</span>
-<span class="sd">        The &quot;forcing vector&quot; for the u&#39;s and q&#39;s</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    This is a simple example for a one degree of freedom translational</span>
-<span class="sd">    spring-mass-damper.</span>
-
-<span class="sd">    In this example, we first need to do the kinematics.</span>
-<span class="sd">    This involves creating generalized speeds and coordinates and their</span>
-<span class="sd">    derivatives.</span>
-<span class="sd">    Then we create a point and set its velocity in a frame::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import dynamicsymbols, ReferenceFrame</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import Point, Particle, KanesMethod</span>
-<span class="sd">        &gt;&gt;&gt; q, u = dynamicsymbols(&#39;q u&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; qd, ud = dynamicsymbols(&#39;q u&#39;, 1)</span>
-<span class="sd">        &gt;&gt;&gt; m, c, k = symbols(&#39;m c k&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; P = Point(&#39;P&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; P.set_vel(N, u * N.x)</span>
-
-<span class="sd">    Next we need to arrange/store information in the way that KanesMethod</span>
-<span class="sd">    requires.  The kinematic differential equations need to be stored in a</span>
-<span class="sd">    dict.  A list of forces/torques must be constructed, where each entry in</span>
-<span class="sd">    the list is a (Point, Vector) or (ReferenceFrame, Vector) tuple, where the</span>
-<span class="sd">    Vectors represent the Force or Torque.</span>
-<span class="sd">    Next a particle needs to be created, and it needs to have a point and mass</span>
-<span class="sd">    assigned to it.</span>
-<span class="sd">    Finally, a list of all bodies and particles needs to be created::</span>
-
-<span class="sd">    &gt;&gt;&gt; kd = [qd - u]</span>
-<span class="sd">    &gt;&gt;&gt; FL = [(P, (-k * q - c * u) * N.x)]</span>
-<span class="sd">    &gt;&gt;&gt; pa = Particle(&#39;pa&#39;, P, m)</span>
-<span class="sd">    &gt;&gt;&gt; BL = [pa]</span>
-
-<span class="sd">    Finally we can generate the equations of motion.</span>
-<span class="sd">    First we create the KanesMethod object and supply an inertial frame,</span>
-<span class="sd">    coordinates, generalized speeds, and the kinematic differential equations.</span>
-<span class="sd">    Additional quantities such as configuration and motion constraints,</span>
-<span class="sd">    dependent coordinates and speeds, and auxiliary speeds are also supplied</span>
-<span class="sd">    here (see the online documentation).</span>
-<span class="sd">    Next we form FR* and FR to complete: Fr + Fr* = 0.</span>
-<span class="sd">    We have the equations of motion at this point.</span>
-<span class="sd">    It makes sense to rearrnge them though, so we calculate the mass matrix and</span>
-<span class="sd">    the forcing terms, for E.o.M. in the form: [MM] udot = forcing, where MM is</span>
-<span class="sd">    the mass matrix, udot is a vector of the time derivatives of the</span>
-<span class="sd">    generalized speeds, and forcing is a vector representing &quot;forcing&quot; terms::</span>
-
-<span class="sd">        &gt;&gt;&gt; KM = KanesMethod(N, q_ind=[q], u_ind=[u], kd_eqs=kd)</span>
-<span class="sd">        &gt;&gt;&gt; (fr, frstar) = KM.kanes_equations(FL, BL)</span>
-<span class="sd">        &gt;&gt;&gt; MM = KM.mass_matrix</span>
-<span class="sd">        &gt;&gt;&gt; forcing = KM.forcing</span>
-<span class="sd">        &gt;&gt;&gt; rhs = MM.inv() * forcing</span>
-<span class="sd">        &gt;&gt;&gt; rhs</span>
-<span class="sd">        [(-c*u(t) - k*q(t))/m]</span>
-<span class="sd">        &gt;&gt;&gt; KM.linearize()[0]</span>
-<span class="sd">        [ 0,  1]</span>
-<span class="sd">        [-k, -c]</span>
-
-<span class="sd">    Please look at the documentation pages for more information on how to</span>
-<span class="sd">    perform linearization and how to deal with dependent coordinates &amp; speeds,</span>
-<span class="sd">    and how do deal with bringing non-contributing forces into evidence.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">simp</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">frame</span><span class="p">,</span> <span class="n">q_ind</span><span class="p">,</span> <span class="n">u_ind</span><span class="p">,</span> <span class="n">kd_eqs</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">q_dependent</span><span class="o">=</span><span class="p">[],</span>
-            <span class="n">configuration_constraints</span><span class="o">=</span><span class="p">[],</span> <span class="n">u_dependent</span><span class="o">=</span><span class="p">[],</span>
-            <span class="n">velocity_constraints</span><span class="o">=</span><span class="p">[],</span> <span class="n">acceleration_constraints</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span>
-            <span class="n">u_auxiliary</span><span class="o">=</span><span class="p">[]):</span>
-
-        <span class="sd">&quot;&quot;&quot;Please read the online documentation. &quot;&quot;&quot;</span>
-        <span class="c"># Big storage things</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">frame</span><span class="p">,</span> <span class="n">ReferenceFrame</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;An intertial ReferenceFrame must be supplied&#39;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_inertial</span> <span class="o">=</span> <span class="n">frame</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_forcelist</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_bodylist</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_fr</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_frstar</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_rhs</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_aux_eq</span> <span class="o">=</span> <span class="bp">None</span>
-
-        <span class="c"># States</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_q</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_qdep</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_qdot</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_u</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_udep</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_udot</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_uaux</span> <span class="o">=</span> <span class="bp">None</span>
-
-        <span class="c"># Differential Equations Matrices</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_k_d</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_f_d</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_k_kqdot</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_k_ku</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_f_k</span> <span class="o">=</span> <span class="bp">None</span>
-
-        <span class="c"># Constraint Matrices</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_f_h</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_k_nh</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_f_nh</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_k_dnh</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_f_dnh</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([])</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_coords</span><span class="p">(</span><span class="n">q_ind</span><span class="p">,</span> <span class="n">q_dependent</span><span class="p">,</span> <span class="n">configuration_constraints</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_speeds</span><span class="p">(</span><span class="n">u_ind</span><span class="p">,</span> <span class="n">u_dependent</span><span class="p">,</span> <span class="n">velocity_constraints</span><span class="p">,</span>
-                <span class="n">acceleration_constraints</span><span class="p">,</span> <span class="n">u_auxiliary</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">kd_eqs</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_kindiffeq</span><span class="p">(</span><span class="n">kd_eqs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_find_dynamicsymbols</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">inlist</span><span class="p">,</span> <span class="n">insyms</span><span class="o">=</span><span class="p">[]):</span>
-        <span class="sd">&quot;&quot;&quot;Finds all non-supplied dynamicsymbols in the expressions.&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">AppliedUndef</span><span class="p">,</span> <span class="n">Derivative</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span>
-        <span class="k">return</span> <span class="nb">reduce</span><span class="p">(</span><span class="nb">set</span><span class="o">.</span><span class="n">union</span><span class="p">,</span> <span class="p">[</span><span class="nb">set</span><span class="p">([</span><span class="n">i</span><span class="p">])</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">inlist</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">j</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">AppliedUndef</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">atoms</span><span class="p">()</span> <span class="o">==</span> <span class="nb">set</span><span class="p">([</span><span class="n">t</span><span class="p">])],</span> <span class="nb">set</span><span class="p">())</span> <span class="o">-</span> <span class="n">insyms</span>
-
-        <span class="n">temp_f</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">i</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">AppliedUndef</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">inlist</span><span class="p">])</span>
-        <span class="n">temp_d</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">i</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Derivative</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">inlist</span><span class="p">])</span>
-        <span class="n">set_f</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span><span class="n">a</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">temp_f</span> <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">args</span> <span class="o">==</span> <span class="p">(</span><span class="n">t</span><span class="p">,)])</span>
-        <span class="n">set_d</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span><span class="n">a</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">temp_d</span> <span class="k">if</span> <span class="p">((</span><span class="n">a</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">in</span> <span class="n">set_f</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">all</span><span class="p">([</span><span class="n">i</span> <span class="o">==</span> <span class="n">t</span>
-                     <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">a</span><span class="o">.</span><span class="n">variables</span><span class="p">]))])</span>
-        <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">set</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="n">set_f</span><span class="p">,</span> <span class="n">set_d</span><span class="p">)</span> <span class="o">-</span> <span class="nb">set</span><span class="p">(</span><span class="n">insyms</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_find_othersymbols</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">inlist</span><span class="p">,</span> <span class="n">insyms</span><span class="o">=</span><span class="p">[]):</span>
-        <span class="sd">&quot;&quot;&quot;Finds all non-dynamic symbols in the expressions.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">reduce</span><span class="p">(</span><span class="nb">set</span><span class="o">.</span><span class="n">union</span><span class="p">,</span> <span class="p">[</span><span class="n">i</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Symbol</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">inlist</span><span class="p">])</span> <span class="o">-</span>
-                    <span class="nb">set</span><span class="p">(</span><span class="n">insyms</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_mat_inv_mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Internal Function</span>
-
-<span class="sd">        Computes A^-1 * B symbolically w/ substitution, where B is not</span>
-<span class="sd">        necessarily a vector, but can be a matrix.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">r1</span><span class="p">,</span> <span class="n">c1</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">shape</span>
-        <span class="n">r2</span><span class="p">,</span> <span class="n">c2</span> <span class="o">=</span> <span class="n">B</span><span class="o">.</span><span class="n">shape</span>
-        <span class="n">temp1</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">r1</span><span class="p">,</span> <span class="n">c1</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="n">r1</span> <span class="o">*</span> <span class="n">i</span><span class="p">)))</span>
-        <span class="n">temp2</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">r2</span><span class="p">,</span> <span class="n">c2</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="n">r2</span> <span class="o">*</span> <span class="n">i</span><span class="p">)))</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">temp1</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">temp1</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">temp2</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">temp2</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">temp3</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">c2</span><span class="p">):</span>
-            <span class="n">temp3</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">temp1</span><span class="o">.</span><span class="n">LDLsolve</span><span class="p">(</span><span class="n">temp2</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]))</span>
-        <span class="n">temp3</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span><span class="n">i</span><span class="o">.</span><span class="n">T</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">temp3</span><span class="p">])</span><span class="o">.</span><span class="n">T</span>
-        <span class="k">return</span> <span class="n">temp3</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">temp1</span><span class="p">,</span> <span class="n">A</span><span class="p">))))</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">temp2</span><span class="p">,</span> <span class="n">B</span><span class="p">))))</span>
-
-    <span class="k">def</span> <span class="nf">_coords</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">qind</span><span class="p">,</span> <span class="n">qdep</span><span class="o">=</span><span class="p">[],</span> <span class="n">coneqs</span><span class="o">=</span><span class="p">[]):</span>
-        <span class="sd">&quot;&quot;&quot;Supply all the generalized coordiantes in a list.</span>
-
-<span class="sd">        If some coordinates are dependent, supply them as part of qdep. Their</span>
-<span class="sd">        dependent nature will only show up in the linearization process though.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        qind : list</span>
-<span class="sd">            A list of independent generalized coords</span>
-<span class="sd">        qdep : list</span>
-<span class="sd">            List of dependent coordinates</span>
-<span class="sd">        coneq : list</span>
-<span class="sd">            List of expressions which are equal to zero; these are the</span>
-<span class="sd">            configuration constraint equations</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">qind</span><span class="p">,</span> <span class="p">(</span><span class="nb">list</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">)):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Generalized coords. must be supplied in a list.&#39;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_q</span> <span class="o">=</span> <span class="n">qind</span> <span class="o">+</span> <span class="n">qdep</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_qdot</span> <span class="o">=</span> <span class="p">[</span><span class="n">diff</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_q</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">qdep</span><span class="p">,</span> <span class="p">(</span><span class="nb">list</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">)):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Dependent coordinates and constraints must each be &#39;</span>
-                            <span class="s">&#39;provided in their own list.&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">qdep</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">coneqs</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;There must be an equal number of dependent &#39;</span>
-                             <span class="s">&#39;coordinates and constraints.&#39;</span><span class="p">)</span>
-        <span class="n">coneqs</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">coneqs</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_qdep</span> <span class="o">=</span> <span class="n">qdep</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_f_h</span> <span class="o">=</span> <span class="n">coneqs</span>
-
-    <span class="k">def</span> <span class="nf">_speeds</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">uind</span><span class="p">,</span> <span class="n">udep</span><span class="o">=</span><span class="p">[],</span> <span class="n">coneqs</span><span class="o">=</span><span class="p">[],</span> <span class="n">diffconeqs</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">u_auxiliary</span><span class="o">=</span><span class="p">[]):</span>
-        <span class="sd">&quot;&quot;&quot;Supply all the generalized speeds in a list.</span>
-
-<span class="sd">        If there are motion constraints or auxiliary speeds, they are provided</span>
-<span class="sd">        here as well (as well as motion constraints).</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        uind : list</span>
-<span class="sd">            A list of independent generalized speeds</span>
-<span class="sd">        udep : list</span>
-<span class="sd">            Optional list of dependent speeds</span>
-<span class="sd">        coneqs : list</span>
-<span class="sd">            Optional List of constraint expressions; these are expressions</span>
-<span class="sd">            which are equal to zero which define a speed (motion) constraint.</span>
-<span class="sd">        diffconeqs : list</span>
-<span class="sd">            Optional, calculated automatically otherwise; list of constraint</span>
-<span class="sd">            equations; again equal to zero, but define an acceleration</span>
-<span class="sd">            constraint.</span>
-<span class="sd">        u_auxiliary : list</span>
-<span class="sd">            An optional list of auxiliary speeds used for brining</span>
-<span class="sd">            non-contributing forces into evidence</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">uind</span><span class="p">,</span> <span class="s">&#39;__iter__&#39;</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Supply generalized speeds in an iterable.&#39;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_u</span> <span class="o">=</span> <span class="n">uind</span> <span class="o">+</span> <span class="n">udep</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_udot</span> <span class="o">=</span> <span class="p">[</span><span class="n">diff</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_u</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_uaux</span> <span class="o">=</span> <span class="n">u_auxiliary</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">udep</span><span class="p">,</span> <span class="s">&#39;__iter__&#39;</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Supply dependent speeds in an iterable.&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">udep</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">coneqs</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;There must be an equal number of dependent &#39;</span>
-                             <span class="s">&#39;speeds and constraints.&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">diffconeqs</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">udep</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">diffconeqs</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;There must be an equal number of dependent &#39;</span>
-                                 <span class="s">&#39;speeds and constraints.&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">udep</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">u</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_u</span>
-            <span class="n">uzero</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">u</span><span class="p">))))</span>
-            <span class="n">coneqs</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">coneqs</span><span class="p">)</span>
-            <span class="n">udot</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_udot</span>
-            <span class="n">udotzero</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">udot</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">udot</span><span class="p">))))</span>
-
-            <span class="bp">self</span><span class="o">.</span><span class="n">_udep</span> <span class="o">=</span> <span class="n">udep</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_f_nh</span> <span class="o">=</span> <span class="n">coneqs</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">uzero</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_k_nh</span> <span class="o">=</span> <span class="p">(</span><span class="n">coneqs</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_f_nh</span><span class="p">)</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
-            <span class="c"># if no differentiated non holonomic constraints were given, calculate</span>
-            <span class="k">if</span> <span class="n">diffconeqs</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_k_dnh</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_k_nh</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_f_dnh</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_k_nh</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span><span class="p">)</span> <span class="o">*</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">u</span><span class="p">)</span> <span class="o">+</span>
-                               <span class="bp">self</span><span class="o">.</span><span class="n">_f_nh</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_f_dnh</span> <span class="o">=</span> <span class="n">diffconeqs</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">udotzero</span><span class="p">)</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_k_dnh</span> <span class="o">=</span> <span class="p">(</span><span class="n">diffconeqs</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_f_dnh</span><span class="p">)</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">udot</span><span class="p">)</span>
-
-            <span class="n">o</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>  <span class="c"># number of generalized speeds</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">udep</span><span class="p">)</span>  <span class="c"># number of motion constraints</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">o</span> <span class="o">-</span> <span class="n">m</span>  <span class="c"># number of independent speeds</span>
-            <span class="c"># For a reminder, form of non-holonomic constraints is:</span>
-            <span class="c"># B u + C = 0</span>
-            <span class="n">B</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_k_nh</span><span class="p">[:</span><span class="n">m</span><span class="p">,</span> <span class="p">:]</span>
-            <span class="n">C</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_f_nh</span><span class="p">[:</span><span class="n">m</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-
-            <span class="c"># We partition B into indenpendent and dependent columns</span>
-            <span class="c"># Ars is then -Bdep.inv() * Bind, and it relates depedent speeds to</span>
-            <span class="c"># independent speeds as: udep = Ars uind, neglecting the C term here.</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_depB</span> <span class="o">=</span> <span class="n">B</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_depC</span> <span class="o">=</span> <span class="n">C</span>
-            <span class="n">mr1</span> <span class="o">=</span> <span class="n">B</span><span class="p">[:</span><span class="n">m</span><span class="p">,</span> <span class="p">:</span><span class="n">p</span><span class="p">]</span>
-            <span class="n">ml1</span> <span class="o">=</span> <span class="n">B</span><span class="p">[:</span><span class="n">m</span><span class="p">,</span> <span class="n">p</span><span class="p">:</span><span class="n">o</span><span class="p">]</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_Ars</span> <span class="o">=</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat_inv_mul</span><span class="p">(</span><span class="n">ml1</span><span class="p">,</span> <span class="n">mr1</span><span class="p">)</span>
-
-<div class="viewcode-block" id="KanesMethod.kindiffdict"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.kane.KanesMethod.kindiffdict">[docs]</a>    <span class="k">def</span> <span class="nf">kindiffdict</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the qdot&#39;s in a dictionary. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_k_kqdot</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Kin. diff. eqs need to be supplied first.&#39;</span><span class="p">)</span>
-        <span class="n">sub_dict</span> <span class="o">=</span> <span class="n">solve_linear_system_LU</span><span class="p">(</span><span class="n">Matrix</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">_k_kqdot</span><span class="o">.</span><span class="n">T</span><span class="p">,</span>
-            <span class="o">-</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_k_ku</span> <span class="o">*</span> <span class="n">Matrix</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_u</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_f_k</span><span class="p">)</span><span class="o">.</span><span class="n">T</span><span class="p">])</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_qdot</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">sub_dict</span>
-</div>
-    <span class="k">def</span> <span class="nf">_kindiffeq</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">kdeqs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Supply all the kinematic differential equations in a list.</span>
-
-<span class="sd">        They should be in the form [Expr1, Expr2, ...] where Expri is equal to</span>
-<span class="sd">        zero</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        kdeqs : list (of Expr)</span>
-<span class="sd">            The listof kinematic differential equations</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_q</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">kdeqs</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;There must be an equal number of kinematic &#39;</span>
-                             <span class="s">&#39;differential equations and coordinates.&#39;</span><span class="p">)</span>
-
-        <span class="n">uaux</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_uaux</span>
-        <span class="c"># dictionary of auxiliary speeds which are equal to zero</span>
-        <span class="n">uaz</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">uaux</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">uaux</span><span class="p">))))</span>
-
-        <span class="n">kdeqs</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">kdeqs</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">uaz</span><span class="p">)</span>
-
-        <span class="n">qdot</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_qdot</span>
-        <span class="n">qdotzero</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">qdot</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">qdot</span><span class="p">))))</span>
-        <span class="n">u</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_u</span>
-        <span class="n">uzero</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">u</span><span class="p">))))</span>
-
-        <span class="n">f_k</span> <span class="o">=</span> <span class="n">kdeqs</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">uzero</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">qdotzero</span><span class="p">)</span>
-        <span class="n">k_kqdot</span> <span class="o">=</span> <span class="p">(</span><span class="n">kdeqs</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">uzero</span><span class="p">)</span> <span class="o">-</span> <span class="n">f_k</span><span class="p">)</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">Matrix</span><span class="p">(</span><span class="n">qdot</span><span class="p">))</span>
-        <span class="n">k_ku</span> <span class="o">=</span> <span class="p">(</span><span class="n">kdeqs</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">qdotzero</span><span class="p">)</span> <span class="o">-</span> <span class="n">f_k</span><span class="p">)</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">Matrix</span><span class="p">(</span><span class="n">u</span><span class="p">))</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_k_ku</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat_inv_mul</span><span class="p">(</span><span class="n">k_kqdot</span><span class="p">,</span> <span class="n">k_ku</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_f_k</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat_inv_mul</span><span class="p">(</span><span class="n">k_kqdot</span><span class="p">,</span> <span class="n">f_k</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_k_kqdot</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">qdot</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_form_fr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">fl</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Form the generalized active force.</span>
-
-<span class="sd">        Computes the vector of the generalized active force vector.</span>
-<span class="sd">        Used to compute E.o.M. in the form Fr + Fr* = 0.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        fl : list</span>
-<span class="sd">            Takes in a list of (Point, Vector) or (ReferenceFrame, Vector)</span>
-<span class="sd">            tuples which represent the force at a point or torque on a frame.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">fl</span><span class="p">,</span> <span class="s">&#39;__iter__&#39;</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Force pairs must be supplied in an iterable.&#39;</span><span class="p">)</span>
-
-        <span class="n">N</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_inertial</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_forcelist</span> <span class="o">=</span> <span class="n">fl</span><span class="p">[:]</span>
-        <span class="n">u</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_u</span>
-        <span class="n">o</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>  <span class="c"># number of gen. speeds</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">fl</span><span class="p">)</span>  <span class="c"># number of forces</span>
-
-        <span class="n">FR</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-
-        <span class="c"># pull out relevant velocities for constructing partial velocities</span>
-        <span class="n">vel_list</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">f_list</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">fl</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">i</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">ReferenceFrame</span><span class="p">):</span>
-                <span class="n">vel_list</span> <span class="o">+=</span> <span class="p">[</span><span class="n">i</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">N</span><span class="p">)]</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">i</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Point</span><span class="p">):</span>
-                <span class="n">vel_list</span> <span class="o">+=</span> <span class="p">[</span><span class="n">i</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">N</span><span class="p">)]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;First entry in pair must be point or frame.&#39;</span><span class="p">)</span>
-            <span class="n">f_list</span> <span class="o">+=</span> <span class="p">[</span><span class="n">i</span><span class="p">[</span><span class="mi">1</span><span class="p">]]</span>
-        <span class="n">partials</span> <span class="o">=</span> <span class="n">partial_velocity</span><span class="p">(</span><span class="n">vel_list</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
-
-        <span class="c"># Fill Fr with dot product of partial velocities and forces</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">o</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">b</span><span class="p">):</span>
-                <span class="n">FR</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-=</span> <span class="n">partials</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="o">&amp;</span> <span class="n">f_list</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-
-        <span class="c"># In case there are dependent speeds</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_udep</span><span class="p">)</span>  <span class="c"># number of dependent speeds</span>
-        <span class="k">if</span> <span class="n">m</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">o</span> <span class="o">-</span> <span class="n">m</span>
-            <span class="n">FRtilde</span> <span class="o">=</span> <span class="n">FR</span><span class="p">[:</span><span class="n">p</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-            <span class="n">FRold</span> <span class="o">=</span> <span class="n">FR</span><span class="p">[</span><span class="n">p</span><span class="p">:</span><span class="n">o</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-            <span class="n">FRtilde</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_Ars</span><span class="o">.</span><span class="n">T</span> <span class="o">*</span> <span class="n">FRold</span>
-            <span class="n">FR</span> <span class="o">=</span> <span class="n">FRtilde</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_fr</span> <span class="o">=</span> <span class="n">FR</span>
-        <span class="k">return</span> <span class="n">FR</span>
-
-    <span class="k">def</span> <span class="nf">_form_frstar</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">bl</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Form the generalized inertia force.</span>
-
-<span class="sd">        Computes the vector of the generalized inertia force vector.</span>
-<span class="sd">        Used to compute E.o.M. in the form Fr + Fr* = 0.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        bl : list</span>
-<span class="sd">            A list of all RigidBody&#39;s and Particle&#39;s in the system.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">bl</span><span class="p">,</span> <span class="s">&#39;__iter__&#39;</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Bodies must be supplied in an iterable.&#39;</span><span class="p">)</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span>
-        <span class="n">N</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_inertial</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_bodylist</span> <span class="o">=</span> <span class="n">bl</span>
-        <span class="n">u</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_u</span>  <span class="c"># all speeds</span>
-        <span class="n">udep</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_udep</span>  <span class="c"># dependent speeds</span>
-        <span class="n">o</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">udep</span><span class="p">)</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">o</span> <span class="o">-</span> <span class="n">m</span>
-        <span class="n">udot</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_udot</span>
-        <span class="n">udotzero</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">udot</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">o</span><span class="p">)))</span>
-        <span class="c"># auxiliary speeds</span>
-        <span class="n">uaux</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_uaux</span>
-        <span class="n">uauxdot</span> <span class="o">=</span> <span class="p">[</span><span class="n">diff</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">uaux</span><span class="p">]</span>
-        <span class="c"># dictionary of auxiliary speeds which are equal to zero</span>
-        <span class="n">uaz</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">uaux</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">uaux</span><span class="p">))))</span>
-        <span class="n">uadz</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">uauxdot</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">uauxdot</span><span class="p">))))</span>
-
-        <span class="n">MM</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">o</span><span class="p">)</span>
-        <span class="n">nonMM</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">partials</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="c"># Fill up the list of partials: format is a list with no. elements</span>
-        <span class="c"># equal to number of entries in body list. Each of these elements is a</span>
-        <span class="c"># list - either of length 1 for the translational components of</span>
-        <span class="c"># particles or of length 2 for the translational and rotational</span>
-        <span class="c"># components of rigid bodies. The inner most list is the list of</span>
-        <span class="c"># partial velocities.</span>
-        <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">bl</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">RigidBody</span><span class="p">):</span>
-                <span class="n">partials</span> <span class="o">+=</span> <span class="p">[</span><span class="n">partial_velocity</span><span class="p">([</span><span class="n">v</span><span class="o">.</span><span class="n">masscenter</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">N</span><span class="p">),</span>
-                                               <span class="n">v</span><span class="o">.</span><span class="n">frame</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">N</span><span class="p">)],</span> <span class="n">u</span><span class="p">,</span> <span class="n">N</span><span class="p">)]</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">Particle</span><span class="p">):</span>
-                <span class="n">partials</span> <span class="o">+=</span> <span class="p">[</span><span class="n">partial_velocity</span><span class="p">([</span><span class="n">v</span><span class="o">.</span><span class="n">point</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">N</span><span class="p">)],</span> <span class="n">u</span><span class="p">,</span> <span class="n">N</span><span class="p">)]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;The body list needs RigidBody or &#39;</span>
-                                <span class="s">&#39;Particle as list elements.&#39;</span><span class="p">)</span>
-
-        <span class="c"># This section does 2 things - computes the parts of Fr* that are</span>
-        <span class="c"># associated with the udots, and the parts that are not associated with</span>
-        <span class="c"># the udots. This happens for RigidBody and Particle a little</span>
-        <span class="c"># differently, but similar process overall.</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">bl</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">RigidBody</span><span class="p">):</span>
-                <span class="n">M</span> <span class="o">=</span> <span class="n">v</span><span class="o">.</span><span class="n">mass</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">uaz</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-                <span class="n">I</span><span class="p">,</span> <span class="n">P</span> <span class="o">=</span> <span class="n">v</span><span class="o">.</span><span class="n">inertia</span>
-                <span class="k">if</span> <span class="n">P</span> <span class="o">!=</span> <span class="n">v</span><span class="o">.</span><span class="n">masscenter</span><span class="p">:</span>
-                    <span class="c"># redefine I about the center of mass</span>
-                    <span class="c"># have I S/O, want I S/S*</span>
-                    <span class="c"># I S/O = I S/S* + I S*/O; I S/S* = I S/O - I S*/O</span>
-                    <span class="n">f</span> <span class="o">=</span> <span class="n">v</span><span class="o">.</span><span class="n">frame</span>
-                    <span class="n">d</span> <span class="o">=</span> <span class="n">v</span><span class="o">.</span><span class="n">masscenter</span><span class="o">.</span><span class="n">pos_from</span><span class="p">(</span><span class="n">P</span><span class="p">)</span>
-                    <span class="n">I</span> <span class="o">-=</span> <span class="n">inertia_of_point_mass</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
-                <span class="n">I</span> <span class="o">=</span> <span class="n">I</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">uaz</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">o</span><span class="p">):</span>
-                    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">o</span><span class="p">):</span>
-                        <span class="c"># translational</span>
-                        <span class="n">MM</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">+=</span> <span class="n">M</span> <span class="o">*</span> <span class="p">(</span><span class="n">partials</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">uaz</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span> <span class="o">&amp;</span>
-                                         <span class="n">partials</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">k</span><span class="p">])</span>
-                        <span class="c"># rotational</span>
-                        <span class="n">temp</span> <span class="o">=</span> <span class="p">(</span><span class="n">I</span> <span class="o">&amp;</span> <span class="n">partials</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">uaz</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">())</span>
-                        <span class="n">MM</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">+=</span> <span class="p">(</span><span class="n">temp</span> <span class="o">&amp;</span>
-                                     <span class="n">partials</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">][</span><span class="n">k</span><span class="p">])</span>
-                    <span class="c"># translational components</span>
-                    <span class="n">nonMM</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="p">(</span> <span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="o">*</span>
-                                   <span class="n">v</span><span class="o">.</span><span class="n">masscenter</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">N</span><span class="p">))</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">uaz</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span> <span class="o">&amp;</span>
-                                 <span class="n">partials</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">])</span>
-                    <span class="n">nonMM</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="p">(</span><span class="n">M</span> <span class="o">*</span>
-                            <span class="n">v</span><span class="o">.</span><span class="n">masscenter</span><span class="o">.</span><span class="n">acc</span><span class="p">(</span><span class="n">N</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">udotzero</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">uaz</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-                            <span class="o">&amp;</span> <span class="n">partials</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">])</span>
-                    <span class="c"># rotational components</span>
-                    <span class="n">omega</span> <span class="o">=</span> <span class="n">v</span><span class="o">.</span><span class="n">frame</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">N</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">uaz</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-                    <span class="n">nonMM</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="p">((</span><span class="n">I</span><span class="o">.</span><span class="n">dt</span><span class="p">(</span><span class="n">v</span><span class="o">.</span><span class="n">frame</span><span class="p">)</span> <span class="o">&amp;</span> <span class="n">omega</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">uaz</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span> <span class="o">&amp;</span>
-                                 <span class="n">partials</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">])</span>
-                    <span class="n">nonMM</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="p">((</span><span class="n">I</span> <span class="o">&amp;</span>
-                        <span class="n">v</span><span class="o">.</span><span class="n">frame</span><span class="o">.</span><span class="n">ang_acc_in</span><span class="p">(</span><span class="n">N</span><span class="p">))</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">udotzero</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">uaz</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-                        <span class="o">&amp;</span> <span class="n">partials</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">])</span>
-                    <span class="n">nonMM</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="p">((</span><span class="n">omega</span> <span class="o">^</span> <span class="p">(</span><span class="n">I</span> <span class="o">&amp;</span> <span class="n">omega</span><span class="p">))</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">uaz</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span> <span class="o">&amp;</span>
-                                 <span class="n">partials</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">])</span>
-
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">Particle</span><span class="p">):</span>
-                <span class="n">M</span> <span class="o">=</span> <span class="n">v</span><span class="o">.</span><span class="n">mass</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">uaz</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">o</span><span class="p">):</span>
-                    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">o</span><span class="p">):</span>
-                        <span class="n">MM</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">+=</span> <span class="n">M</span> <span class="o">*</span> <span class="p">(</span><span class="n">partials</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">uaz</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span> <span class="o">&amp;</span>
-                                         <span class="n">partials</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">k</span><span class="p">])</span>
-                    <span class="n">nonMM</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="n">M</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">v</span><span class="o">.</span><span class="n">point</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">N</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">uaz</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span> <span class="o">&amp;</span>
-                                             <span class="n">partials</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">])</span>
-                    <span class="n">nonMM</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="p">(</span><span class="n">M</span> <span class="o">*</span>
-                            <span class="n">v</span><span class="o">.</span><span class="n">point</span><span class="o">.</span><span class="n">acc</span><span class="p">(</span><span class="n">N</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">udotzero</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">uaz</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span> <span class="o">&amp;</span>
-                            <span class="n">partials</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">])</span>
-        <span class="n">FRSTAR</span> <span class="o">=</span> <span class="n">MM</span> <span class="o">*</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">udot</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">uadz</span><span class="p">)</span> <span class="o">+</span> <span class="n">nonMM</span>
-
-        <span class="c"># For motion constraints, m is the number of constraints</span>
-        <span class="c"># Really, one should just look at Kane&#39;s book for descriptions of this</span>
-        <span class="c"># process</span>
-        <span class="k">if</span> <span class="n">m</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">FRSTARtilde</span> <span class="o">=</span> <span class="n">FRSTAR</span><span class="p">[:</span><span class="n">p</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-            <span class="n">FRSTARold</span> <span class="o">=</span> <span class="n">FRSTAR</span><span class="p">[</span><span class="n">p</span><span class="p">:</span><span class="n">o</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-            <span class="n">FRSTARtilde</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_Ars</span><span class="o">.</span><span class="n">T</span> <span class="o">*</span> <span class="n">FRSTARold</span>
-            <span class="n">FRSTAR</span> <span class="o">=</span> <span class="n">FRSTARtilde</span>
-
-            <span class="n">MMi</span> <span class="o">=</span> <span class="n">MM</span><span class="p">[:</span><span class="n">p</span><span class="p">,</span> <span class="p">:]</span>
-            <span class="n">MMd</span> <span class="o">=</span> <span class="n">MM</span><span class="p">[</span><span class="n">p</span><span class="p">:</span><span class="n">o</span><span class="p">,</span> <span class="p">:]</span>
-            <span class="n">MM</span> <span class="o">=</span> <span class="n">MMi</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_Ars</span><span class="o">.</span><span class="n">T</span> <span class="o">*</span> <span class="n">MMd</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_frstar</span> <span class="o">=</span> <span class="n">FRSTAR</span>
-
-        <span class="n">zeroeq</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_fr</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_frstar</span>
-        <span class="n">zeroeq</span> <span class="o">=</span> <span class="n">zeroeq</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">udotzero</span><span class="p">)</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_k_d</span> <span class="o">=</span> <span class="n">MM</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_f_d</span> <span class="o">=</span> <span class="n">zeroeq</span>
-        <span class="k">return</span> <span class="n">FRSTAR</span>
-
-<div class="viewcode-block" id="KanesMethod.kanes_equations"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.kane.KanesMethod.kanes_equations">[docs]</a>    <span class="k">def</span> <span class="nf">kanes_equations</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">FL</span><span class="p">,</span> <span class="n">BL</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Method to form Kane&#39;s equations, Fr + Fr* = 0.</span>
-
-<span class="sd">        Returns (Fr, Fr*). In the case where auxiliary generalized speeds are</span>
-<span class="sd">        present (say, s auxiliary speeds, o generalized speeds, and m motion</span>
-<span class="sd">        constraints) the length of the returned vectors will be o - m + s in</span>
-<span class="sd">        length. The first o - m equations will be the constrained Kane&#39;s</span>
-<span class="sd">        equations, then the s auxiliary Kane&#39;s equations. These auxiliary</span>
-<span class="sd">        equations can be accessed with the auxiliary_eqs().</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        FL : list</span>
-<span class="sd">            Takes in a list of (Point, Vector) or (ReferenceFrame, Vector)</span>
-<span class="sd">            tuples which represent the force at a point or torque on a frame.</span>
-<span class="sd">        BL : list</span>
-<span class="sd">            A list of all RigidBody&#39;s and Particle&#39;s in the system.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_q</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_u</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Speeds and coordinates must be supplied first.&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_k_kqdot</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&#39;Supply kinematic differential equations, please.&#39;</span><span class="p">)</span>
-
-        <span class="n">fr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_form_fr</span><span class="p">(</span><span class="n">FL</span><span class="p">)</span>
-        <span class="n">frstar</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_form_frstar</span><span class="p">(</span><span class="n">BL</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_uaux</span> <span class="o">!=</span> <span class="p">[]:</span>
-            <span class="n">km</span> <span class="o">=</span> <span class="n">KanesMethod</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_inertial</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_q</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_uaux</span><span class="p">,</span>
-                             <span class="n">u_auxiliary</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">_uaux</span><span class="p">)</span>
-            <span class="n">fraux</span> <span class="o">=</span> <span class="n">km</span><span class="o">.</span><span class="n">_form_fr</span><span class="p">(</span><span class="n">FL</span><span class="p">)</span>
-            <span class="n">frstaraux</span> <span class="o">=</span> <span class="n">km</span><span class="o">.</span><span class="n">_form_frstar</span><span class="p">(</span><span class="n">BL</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_aux_eq</span> <span class="o">=</span> <span class="n">fraux</span> <span class="o">+</span> <span class="n">frstaraux</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_fr</span> <span class="o">=</span> <span class="n">fr</span><span class="o">.</span><span class="n">col_join</span><span class="p">(</span><span class="n">fraux</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_frstar</span> <span class="o">=</span> <span class="n">frstar</span><span class="o">.</span><span class="n">col_join</span><span class="p">(</span><span class="n">frstaraux</span><span class="p">)</span>
-            <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_fr</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_frstar</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">fr</span><span class="p">,</span> <span class="n">frstar</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="KanesMethod.linearize"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.kane.KanesMethod.linearize">[docs]</a>    <span class="k">def</span> <span class="nf">linearize</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Method used to generate linearized equations.</span>
-
-<span class="sd">        Note that for linearization, it is assumed that time is not perturbed,</span>
-<span class="sd">        but only coordinates and positions. The &quot;forcing&quot; vector&#39;s jacobian is</span>
-<span class="sd">        computed with respect to the state vector in the form [Qi, Qd, Ui, Ud].</span>
-<span class="sd">        This is the &quot;f_lin_A&quot; matrix.</span>
-
-<span class="sd">        It also finds any non-state dynamicsymbols and computes the jacobian of</span>
-<span class="sd">        the &quot;forcing&quot; vector with respect to them. This is the &quot;f_lin_B&quot;</span>
-<span class="sd">        matrix; if this is empty, an empty matrix is created.</span>
-
-<span class="sd">        Consider the following:</span>
-<span class="sd">        If our equations are: [M]qudot = f, where [M] is the full mass matrix,</span>
-<span class="sd">        qudot is a vector of the deriatives of the coordinates and speeds, and</span>
-<span class="sd">        f in the full forcing vector, the linearization process is as follows:</span>
-<span class="sd">        [M]qudot = [f_lin_A]qu + [f_lin_B]y, where qu is the state vector,</span>
-<span class="sd">        f_lin_A is the jacobian of the full forcing vector with respect to the</span>
-<span class="sd">        state vector, f_lin_B is the jacobian of the full forcing vector with</span>
-<span class="sd">        respect to any non-speed/coordinate dynamicsymbols which show up in the</span>
-<span class="sd">        full forcing vector, and y is a vector of those dynamic symbols (each</span>
-<span class="sd">        column in f_lin_B corresponds to a row of the y vector, each of which</span>
-<span class="sd">        is a non-speed/coordinate dynamicsymbol).</span>
-
-<span class="sd">        To get the traditional state-space A and B matrix, you need to multiply</span>
-<span class="sd">        the f_lin_A and f_lin_B matrices by the inverse of the mass matrix.</span>
-<span class="sd">        Caution needs to be taken when inverting large symbolic matrices;</span>
-<span class="sd">        substituting in numerical values before inverting will work better.</span>
-
-<span class="sd">        A tuple of (f_lin_A, f_lin_B, other_dynamicsymbols) is returned.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_fr</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_frstar</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Need to compute Fr, Fr* first.&#39;</span><span class="p">)</span>
-
-        <span class="c"># Note that this is now unneccessary, and it should never be</span>
-        <span class="c"># encountered; I still think it should be in here in case the user</span>
-        <span class="c"># manually sets these matrices incorrectly.</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_q</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_k_kqdot</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">_k_kqdot</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Matrix K_kqdot must not depend on any q.&#39;</span><span class="p">)</span>
-
-        <span class="n">t</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span>
-        <span class="n">uaux</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_uaux</span>
-        <span class="n">uauxdot</span> <span class="o">=</span> <span class="p">[</span><span class="n">diff</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">uaux</span><span class="p">]</span>
-        <span class="c"># dictionary of auxiliary speeds &amp; derivatives which are equal to zero</span>
-        <span class="n">subdict</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">uaux</span> <span class="o">+</span> <span class="n">uauxdot</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">uaux</span><span class="p">)</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">uauxdot</span><span class="p">)))))</span>
-
-        <span class="c"># Checking for dynamic symbols outside the dynamic differential</span>
-        <span class="c"># equations; throws error if there is.</span>
-        <span class="n">insyms</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_q</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_qdot</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_u</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_udot</span> <span class="o">+</span> <span class="n">uaux</span> <span class="o">+</span> <span class="n">uauxdot</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_find_dynamicsymbols</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">insyms</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_k_kqdot</span><span class="p">,</span>
-                                                              <span class="bp">self</span><span class="o">.</span><span class="n">_k_ku</span><span class="p">,</span>
-                                                              <span class="bp">self</span><span class="o">.</span><span class="n">_f_k</span><span class="p">,</span>
-                                                              <span class="bp">self</span><span class="o">.</span><span class="n">_k_dnh</span><span class="p">,</span>
-                                                              <span class="bp">self</span><span class="o">.</span><span class="n">_f_dnh</span><span class="p">,</span>
-                                                              <span class="bp">self</span><span class="o">.</span><span class="n">_k_d</span><span class="p">]):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Cannot have dynamicsymbols outside dynamic &#39;</span>
-                             <span class="s">&#39;forcing vector.&#39;</span><span class="p">)</span>
-        <span class="n">other_dyns</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_find_dynamicsymbols</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_f_d</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">subdict</span><span class="p">),</span>
-                                             <span class="n">insyms</span><span class="p">))</span>
-
-        <span class="c"># make it canonically ordered so the jacobian is canonical</span>
-        <span class="n">other_dyns</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">other_dyns</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">diff</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span><span class="p">)</span> <span class="ow">in</span> <span class="n">other_dyns</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Cannot have derivatives of specified &#39;</span>
-                                 <span class="s">&#39;quantities when linearizing forcing terms.&#39;</span><span class="p">)</span>
-
-        <span class="n">o</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_u</span><span class="p">)</span>  <span class="c"># number of speeds</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_q</span><span class="p">)</span>  <span class="c"># number of coordinates</span>
-        <span class="n">l</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_qdep</span><span class="p">)</span>  <span class="c"># number of configuration constraints</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_udep</span><span class="p">)</span>  <span class="c"># number of motion constraints</span>
-        <span class="n">qi</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_q</span><span class="p">[:</span> <span class="n">n</span> <span class="o">-</span> <span class="n">l</span><span class="p">])</span>  <span class="c"># independent coords</span>
-        <span class="n">qd</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_q</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="n">l</span><span class="p">:</span> <span class="n">n</span><span class="p">])</span>  <span class="c"># dependent coords; could be empty</span>
-        <span class="n">ui</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_u</span><span class="p">[:</span> <span class="n">o</span> <span class="o">-</span> <span class="n">m</span><span class="p">])</span>  <span class="c"># independent speeds</span>
-        <span class="n">ud</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_u</span><span class="p">[</span><span class="n">o</span> <span class="o">-</span> <span class="n">m</span><span class="p">:</span> <span class="n">o</span><span class="p">])</span>  <span class="c"># dependent speeds; could be empty</span>
-        <span class="n">qdot</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_qdot</span><span class="p">)</span>  <span class="c"># time derivatives of coordinates</span>
-
-        <span class="c"># with equations in the form MM udot = forcing, expand that to:</span>
-        <span class="c"># MM_full [q,u].T = forcing_full. This combines coordinates and</span>
-        <span class="c"># speeds together for the linearization, which is necessary for the</span>
-        <span class="c"># linearization process, due to dependent coordinates. f1 is the rows</span>
-        <span class="c"># from the kinematic differential equations, f2 is the rows from the</span>
-        <span class="c"># dynamic differential equations (and differentiated non-holonomic</span>
-        <span class="c"># constraints).</span>
-        <span class="n">f1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_k_ku</span> <span class="o">*</span> <span class="n">Matrix</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_u</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_f_k</span>
-        <span class="n">f2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_f_d</span>
-        <span class="c"># Only want to do this if these matrices have been filled in, which</span>
-        <span class="c"># occurs when there are dependent speeds</span>
-        <span class="k">if</span> <span class="n">m</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">f2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_f_d</span><span class="o">.</span><span class="n">col_join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_f_dnh</span><span class="p">)</span>
-            <span class="n">fnh</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_f_nh</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_k_nh</span> <span class="o">*</span> <span class="n">Matrix</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_u</span><span class="p">)</span>
-        <span class="n">f1</span> <span class="o">=</span> <span class="n">f1</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">subdict</span><span class="p">)</span>
-        <span class="n">f2</span> <span class="o">=</span> <span class="n">f2</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">subdict</span><span class="p">)</span>
-        <span class="n">fh</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_f_h</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">subdict</span><span class="p">)</span>
-        <span class="n">fku</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_k_ku</span> <span class="o">*</span> <span class="n">Matrix</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_u</span><span class="p">))</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">subdict</span><span class="p">)</span>
-        <span class="n">fkf</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_f_k</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">subdict</span><span class="p">)</span>
-
-        <span class="c"># In the code below, we are applying the chain rule by hand on these</span>
-        <span class="c"># things. All the matrices have been changed into vectors (by</span>
-        <span class="c"># multiplying the dynamic symbols which it is paired with), so we can</span>
-        <span class="c"># take the jacobian of them. The basic operation is take the jacobian</span>
-        <span class="c"># of the f1, f2 vectors wrt all of the q&#39;s and u&#39;s. f1 is a function of</span>
-        <span class="c"># q, u, and t; f2 is a function of q, qdot, u, and t. In the code</span>
-        <span class="c"># below, we are not considering perturbations in t. So if f1 is a</span>
-        <span class="c"># function of the q&#39;s, u&#39;s but some of the q&#39;s or u&#39;s could be</span>
-        <span class="c"># dependent on other q&#39;s or u&#39;s (qd&#39;s might be dependent on qi&#39;s, ud&#39;s</span>
-        <span class="c"># might be dependent on ui&#39;s or qi&#39;s), so what we do is take the</span>
-        <span class="c"># jacobian of the f1 term wrt qi&#39;s and qd&#39;s, the jacobian wrt the qd&#39;s</span>
-        <span class="c"># gets multiplied by the jacobian of qd wrt qi, this is extended for</span>
-        <span class="c"># the ud&#39;s as well. dqd_dqi is computed by taking a taylor expansion of</span>
-        <span class="c"># the holonomic constraint equations about q*, treating q* - q as dq,</span>
-        <span class="c"># seperating into dqd (depedent q&#39;s) and dqi (independent q&#39;s) and the</span>
-        <span class="c"># rearranging for dqd/dqi. This is again extended for the speeds.</span>
-
-        <span class="c"># First case: configuration and motion constraints</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">l</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">m</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">):</span>
-            <span class="n">fh_jac_qi</span> <span class="o">=</span> <span class="n">fh</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qi</span><span class="p">)</span>
-            <span class="n">fh_jac_qd</span> <span class="o">=</span> <span class="n">fh</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qd</span><span class="p">)</span>
-            <span class="n">fnh_jac_qi</span> <span class="o">=</span> <span class="n">fnh</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qi</span><span class="p">)</span>
-            <span class="n">fnh_jac_qd</span> <span class="o">=</span> <span class="n">fnh</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qd</span><span class="p">)</span>
-            <span class="n">fnh_jac_ui</span> <span class="o">=</span> <span class="n">fnh</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ui</span><span class="p">)</span>
-            <span class="n">fnh_jac_ud</span> <span class="o">=</span> <span class="n">fnh</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ud</span><span class="p">)</span>
-            <span class="n">fku_jac_qi</span> <span class="o">=</span> <span class="n">fku</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qi</span><span class="p">)</span>
-            <span class="n">fku_jac_qd</span> <span class="o">=</span> <span class="n">fku</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qd</span><span class="p">)</span>
-            <span class="n">fku_jac_ui</span> <span class="o">=</span> <span class="n">fku</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ui</span><span class="p">)</span>
-            <span class="n">fku_jac_ud</span> <span class="o">=</span> <span class="n">fku</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ud</span><span class="p">)</span>
-            <span class="n">fkf_jac_qi</span> <span class="o">=</span> <span class="n">fkf</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qi</span><span class="p">)</span>
-            <span class="n">fkf_jac_qd</span> <span class="o">=</span> <span class="n">fkf</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qd</span><span class="p">)</span>
-            <span class="n">f1_jac_qi</span> <span class="o">=</span> <span class="n">f1</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qi</span><span class="p">)</span>
-            <span class="n">f1_jac_qd</span> <span class="o">=</span> <span class="n">f1</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qd</span><span class="p">)</span>
-            <span class="n">f1_jac_ui</span> <span class="o">=</span> <span class="n">f1</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ui</span><span class="p">)</span>
-            <span class="n">f1_jac_ud</span> <span class="o">=</span> <span class="n">f1</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ud</span><span class="p">)</span>
-            <span class="n">f2_jac_qi</span> <span class="o">=</span> <span class="n">f2</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qi</span><span class="p">)</span>
-            <span class="n">f2_jac_qd</span> <span class="o">=</span> <span class="n">f2</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qd</span><span class="p">)</span>
-            <span class="n">f2_jac_ui</span> <span class="o">=</span> <span class="n">f2</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ui</span><span class="p">)</span>
-            <span class="n">f2_jac_ud</span> <span class="o">=</span> <span class="n">f2</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ud</span><span class="p">)</span>
-            <span class="n">f2_jac_qdot</span> <span class="o">=</span> <span class="n">f2</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qdot</span><span class="p">)</span>
-
-            <span class="n">dqd_dqi</span> <span class="o">=</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat_inv_mul</span><span class="p">(</span><span class="n">fh_jac_qd</span><span class="p">,</span> <span class="n">fh_jac_qi</span><span class="p">)</span>
-            <span class="n">dud_dqi</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat_inv_mul</span><span class="p">(</span><span class="n">fnh_jac_ud</span><span class="p">,</span> <span class="p">(</span><span class="n">fnh_jac_qd</span> <span class="o">*</span>
-                                        <span class="n">dqd_dqi</span> <span class="o">-</span> <span class="n">fnh_jac_qi</span><span class="p">))</span>
-            <span class="n">dud_dui</span> <span class="o">=</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat_inv_mul</span><span class="p">(</span><span class="n">fnh_jac_ud</span><span class="p">,</span> <span class="n">fnh_jac_ui</span><span class="p">)</span>
-            <span class="n">dqdot_dui</span> <span class="o">=</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_k_kqdot</span><span class="o">.</span><span class="n">inv</span><span class="p">()</span> <span class="o">*</span> <span class="p">(</span><span class="n">fku_jac_ui</span> <span class="o">+</span>
-                                                <span class="n">fku_jac_ud</span> <span class="o">*</span> <span class="n">dud_dui</span><span class="p">)</span>
-            <span class="n">dqdot_dqi</span> <span class="o">=</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_k_kqdot</span><span class="o">.</span><span class="n">inv</span><span class="p">()</span> <span class="o">*</span> <span class="p">(</span><span class="n">fku_jac_qi</span> <span class="o">+</span> <span class="n">fkf_jac_qi</span> <span class="o">+</span>
-                    <span class="p">(</span><span class="n">fku_jac_qd</span> <span class="o">+</span> <span class="n">fkf_jac_qd</span><span class="p">)</span> <span class="o">*</span> <span class="n">dqd_dqi</span> <span class="o">+</span> <span class="n">fku_jac_ud</span> <span class="o">*</span> <span class="n">dud_dqi</span><span class="p">)</span>
-            <span class="n">f1_q</span> <span class="o">=</span> <span class="n">f1_jac_qi</span> <span class="o">+</span> <span class="n">f1_jac_qd</span> <span class="o">*</span> <span class="n">dqd_dqi</span> <span class="o">+</span> <span class="n">f1_jac_ud</span> <span class="o">*</span> <span class="n">dud_dqi</span>
-            <span class="n">f1_u</span> <span class="o">=</span> <span class="n">f1_jac_ui</span> <span class="o">+</span> <span class="n">f1_jac_ud</span> <span class="o">*</span> <span class="n">dud_dui</span>
-            <span class="n">f2_q</span> <span class="o">=</span> <span class="p">(</span><span class="n">f2_jac_qi</span> <span class="o">+</span> <span class="n">f2_jac_qd</span> <span class="o">*</span> <span class="n">dqd_dqi</span> <span class="o">+</span> <span class="n">f2_jac_qdot</span> <span class="o">*</span> <span class="n">dqdot_dqi</span> <span class="o">+</span>
-                    <span class="n">f2_jac_ud</span> <span class="o">*</span> <span class="n">dud_dqi</span><span class="p">)</span>
-            <span class="n">f2_u</span> <span class="o">=</span> <span class="n">f2_jac_ui</span> <span class="o">+</span> <span class="n">f2_jac_ud</span> <span class="o">*</span> <span class="n">dud_dui</span> <span class="o">+</span> <span class="n">f2_jac_qdot</span> <span class="o">*</span> <span class="n">dqdot_dui</span>
-        <span class="c"># Second case: configuration constraints only</span>
-        <span class="k">elif</span> <span class="n">l</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">dqd_dqi</span> <span class="o">=</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat_inv_mul</span><span class="p">(</span><span class="n">fh</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qd</span><span class="p">),</span> <span class="n">fh</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qi</span><span class="p">))</span>
-            <span class="n">dqdot_dui</span> <span class="o">=</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_k_kqdot</span><span class="o">.</span><span class="n">inv</span><span class="p">()</span> <span class="o">*</span> <span class="n">fku</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ui</span><span class="p">)</span>
-            <span class="n">dqdot_dqi</span> <span class="o">=</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_k_kqdot</span><span class="o">.</span><span class="n">inv</span><span class="p">()</span> <span class="o">*</span> <span class="p">(</span><span class="n">fku</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qi</span><span class="p">)</span> <span class="o">+</span>
-                <span class="n">fkf</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qi</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">fku</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qd</span><span class="p">)</span> <span class="o">+</span> <span class="n">fkf</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qd</span><span class="p">))</span> <span class="o">*</span>
-                <span class="n">dqd_dqi</span><span class="p">)</span>
-            <span class="n">f1_q</span> <span class="o">=</span> <span class="p">(</span><span class="n">f1</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qi</span><span class="p">)</span> <span class="o">+</span> <span class="n">f1</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qd</span><span class="p">)</span> <span class="o">*</span> <span class="n">dqd_dqi</span><span class="p">)</span>
-            <span class="n">f1_u</span> <span class="o">=</span> <span class="n">f1</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ui</span><span class="p">)</span>
-            <span class="n">f2_jac_qdot</span> <span class="o">=</span> <span class="n">f2</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qdot</span><span class="p">)</span>
-            <span class="n">f2_q</span> <span class="o">=</span> <span class="p">(</span><span class="n">f2</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qi</span><span class="p">)</span> <span class="o">+</span> <span class="n">f2</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qd</span><span class="p">)</span> <span class="o">*</span> <span class="n">dqd_dqi</span> <span class="o">+</span>
-                    <span class="n">f2</span><span class="o">.</span><span class="n">jac_qdot</span> <span class="o">*</span> <span class="n">dqdot_dqi</span><span class="p">)</span>
-            <span class="n">f2_u</span> <span class="o">=</span> <span class="n">f2</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ui</span><span class="p">)</span> <span class="o">+</span> <span class="n">f2_jac_qdot</span> <span class="o">*</span> <span class="n">dqdot_dui</span>
-        <span class="c"># Third case: motion constraints only</span>
-        <span class="k">elif</span> <span class="n">m</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">dud_dqi</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat_inv_mul</span><span class="p">(</span><span class="n">fnh</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ud</span><span class="p">),</span> <span class="o">-</span> <span class="n">fnh</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qi</span><span class="p">))</span>
-            <span class="n">dud_dui</span> <span class="o">=</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat_inv_mul</span><span class="p">(</span><span class="n">fnh</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ud</span><span class="p">),</span> <span class="n">fnh</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ui</span><span class="p">))</span>
-            <span class="n">dqdot_dui</span> <span class="o">=</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_k_kqdot</span><span class="o">.</span><span class="n">inv</span><span class="p">()</span> <span class="o">*</span> <span class="p">(</span><span class="n">fku</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ui</span><span class="p">)</span> <span class="o">+</span>
-                                                <span class="n">fku</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ud</span><span class="p">)</span> <span class="o">*</span> <span class="n">dud_dui</span><span class="p">)</span>
-            <span class="n">dqdot_dqi</span> <span class="o">=</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_k_kqdot</span><span class="o">.</span><span class="n">inv</span><span class="p">()</span> <span class="o">*</span> <span class="p">(</span><span class="n">fku</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qi</span><span class="p">)</span> <span class="o">+</span>
-                    <span class="n">fkf</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qi</span><span class="p">)</span> <span class="o">+</span> <span class="n">fku</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ud</span><span class="p">)</span> <span class="o">*</span> <span class="n">dud_dqi</span><span class="p">)</span>
-            <span class="n">f1_jac_ud</span> <span class="o">=</span> <span class="n">f1</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ud</span><span class="p">)</span>
-            <span class="n">f2_jac_qdot</span> <span class="o">=</span> <span class="n">f2</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qdot</span><span class="p">)</span>
-            <span class="n">f2_jac_ud</span> <span class="o">=</span> <span class="n">f2</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ud</span><span class="p">)</span>
-            <span class="n">f1_q</span> <span class="o">=</span> <span class="n">f1</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qi</span><span class="p">)</span> <span class="o">+</span> <span class="n">f1_jac_ud</span> <span class="o">*</span> <span class="n">dud_dqi</span>
-            <span class="n">f1_u</span> <span class="o">=</span> <span class="n">f1</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ui</span><span class="p">)</span> <span class="o">+</span> <span class="n">f1_jac_ud</span> <span class="o">*</span> <span class="n">dud_dui</span>
-            <span class="n">f2_q</span> <span class="o">=</span> <span class="p">(</span><span class="n">f2</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qi</span><span class="p">)</span> <span class="o">+</span> <span class="n">f2_jac_qdot</span> <span class="o">*</span> <span class="n">dqdot_dqi</span> <span class="o">+</span> <span class="n">f2_jac_ud</span>
-                    <span class="o">*</span> <span class="n">dud_dqi</span><span class="p">)</span>
-            <span class="n">f2_u</span> <span class="o">=</span> <span class="p">(</span><span class="n">f2</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ui</span><span class="p">)</span> <span class="o">+</span> <span class="n">f2_jac_ud</span> <span class="o">*</span> <span class="n">dud_dui</span> <span class="o">+</span> <span class="n">f2_jac_qdot</span> <span class="o">*</span>
-                    <span class="n">dqdot_dui</span><span class="p">)</span>
-        <span class="c"># Fourth case: No constraints</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">dqdot_dui</span> <span class="o">=</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_k_kqdot</span><span class="o">.</span><span class="n">inv</span><span class="p">()</span> <span class="o">*</span> <span class="n">fku</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ui</span><span class="p">)</span>
-            <span class="n">dqdot_dqi</span> <span class="o">=</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_k_kqdot</span><span class="o">.</span><span class="n">inv</span><span class="p">()</span> <span class="o">*</span> <span class="p">(</span><span class="n">fku</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qi</span><span class="p">)</span> <span class="o">+</span>
-                    <span class="n">fkf</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qi</span><span class="p">))</span>
-            <span class="n">f1_q</span> <span class="o">=</span> <span class="n">f1</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qi</span><span class="p">)</span>
-            <span class="n">f1_u</span> <span class="o">=</span> <span class="n">f1</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ui</span><span class="p">)</span>
-            <span class="n">f2_jac_qdot</span> <span class="o">=</span> <span class="n">f2</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qdot</span><span class="p">)</span>
-            <span class="n">f2_q</span> <span class="o">=</span> <span class="n">f2</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qi</span><span class="p">)</span> <span class="o">+</span> <span class="n">f2_jac_qdot</span> <span class="o">*</span> <span class="n">dqdot_dqi</span>
-            <span class="n">f2_u</span> <span class="o">=</span> <span class="n">f2</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ui</span><span class="p">)</span> <span class="o">+</span> <span class="n">f2_jac_qdot</span> <span class="o">*</span> <span class="n">dqdot_dui</span>
-        <span class="n">f_lin_A</span> <span class="o">=</span> <span class="o">-</span><span class="p">(</span><span class="n">f1_q</span><span class="o">.</span><span class="n">row_join</span><span class="p">(</span><span class="n">f1_u</span><span class="p">))</span><span class="o">.</span><span class="n">col_join</span><span class="p">(</span><span class="n">f2_q</span><span class="o">.</span><span class="n">row_join</span><span class="p">(</span><span class="n">f2_u</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">other_dyns</span><span class="p">:</span>
-            <span class="n">f1_oths</span> <span class="o">=</span> <span class="n">f1</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">other_dyns</span><span class="p">)</span>
-            <span class="n">f2_oths</span> <span class="o">=</span> <span class="n">f2</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">other_dyns</span><span class="p">)</span>
-            <span class="n">f_lin_B</span> <span class="o">=</span> <span class="o">-</span><span class="n">f1_oths</span><span class="o">.</span><span class="n">col_join</span><span class="p">(</span><span class="n">f2_oths</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">f_lin_B</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([])</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">f_lin_A</span><span class="p">,</span> <span class="n">f_lin_B</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">other_dyns</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="KanesMethod.rhs"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.kane.KanesMethod.rhs">[docs]</a>    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">inv_method</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Returns the system&#39;s equations of motion in first order form.</span>
-
-<span class="sd">        The output of this will be the right hand side of:</span>
-
-<span class="sd">        [qdot, udot].T = f(q, u, t)</span>
-
-<span class="sd">        Or, the equations of motion in first order form.  The right hand side</span>
-<span class="sd">        is what is needed by most numerical ODE integrators.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        inv_method : str</span>
-<span class="sd">            The specific sympy inverse matrix calculation method to use.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="n">inv_method</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_rhs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mat_inv_mul</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mass_matrix_full</span><span class="p">,</span>
-                                          <span class="bp">self</span><span class="o">.</span><span class="n">forcing_full</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_rhs</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mass_matrix_full</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">inv_method</span><span class="p">,</span>
-                         <span class="n">try_block_diag</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">forcing_full</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rhs</span>
-</div>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">auxiliary_eqs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_fr</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_frstar</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Need to compute Fr, Fr* first.&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_uaux</span> <span class="o">==</span> <span class="p">[]:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;No auxiliary speeds have been declared.&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_aux_eq</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">mass_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># Returns the mass matrix, which is augmented by the differentiated non</span>
-        <span class="c"># holonomic equations if necessary</span>
-        <span class="k">if</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_frstar</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">)</span> <span class="o">&amp;</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_fr</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Need to compute Fr, Fr* first.&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Matrix</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">_k_d</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_k_dnh</span><span class="p">])</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">mass_matrix_full</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># Returns the mass matrix from above, augmented by kin diff&#39;s k_kqdot</span>
-        <span class="k">if</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_frstar</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">)</span> <span class="o">&amp;</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_fr</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Need to compute Fr, Fr* first.&#39;</span><span class="p">)</span>
-        <span class="n">o</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_u</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_q</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">_k_kqdot</span><span class="p">)</span><span class="o">.</span><span class="n">row_join</span><span class="p">(</span><span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">o</span><span class="p">)))</span><span class="o">.</span><span class="n">col_join</span><span class="p">((</span><span class="n">zeros</span><span class="p">(</span><span class="n">o</span><span class="p">,</span>
-                <span class="n">n</span><span class="p">))</span><span class="o">.</span><span class="n">row_join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mass_matrix</span><span class="p">))</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">forcing</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># Returns the forcing vector, which is augmented by the differentiated</span>
-        <span class="c"># non holonomic equations if necessary</span>
-        <span class="k">if</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_frstar</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">)</span> <span class="o">&amp;</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_fr</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Need to compute Fr, Fr* first.&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">Matrix</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">_f_d</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_f_dnh</span><span class="p">])</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">forcing_full</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># Returns the forcing vector, which is augmented by the differentiated</span>
-        <span class="c"># non holonomic equations if necessary</span>
-        <span class="k">if</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_frstar</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">)</span> <span class="o">&amp;</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_fr</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Need to compute Fr, Fr* first.&#39;</span><span class="p">)</span>
-        <span class="n">f1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_k_ku</span> <span class="o">*</span> <span class="n">Matrix</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_u</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_f_k</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">Matrix</span><span class="p">([</span><span class="n">f1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_f_d</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_f_dnh</span><span class="p">])</span></div>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.physics.mechanics.lagrange</h1><div class="highlight"><pre>
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;LagrangesMethod&#39;</span><span class="p">]</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">diff</span><span class="p">,</span> <span class="n">zeros</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">eye</span><span class="p">,</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="p">(</span><span class="n">dynamicsymbols</span><span class="p">,</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">Point</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="LagrangesMethod"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.lagrange.LagrangesMethod">[docs]</a><span class="k">class</span> <span class="nc">LagrangesMethod</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Lagrange&#39;s method object.</span>
-
-<span class="sd">    This object generates the equations of motion in a two step procedure. The</span>
-<span class="sd">    first step involves the initialization of LagrangesMethod by supplying the</span>
-<span class="sd">    Lagrangian and a list of the generalized coordinates, at the bare minimum.</span>
-<span class="sd">    If there are any constraint equations, they can be supplied as keyword</span>
-<span class="sd">    arguments. The Lagrangian multipliers are automatically generated and are</span>
-<span class="sd">    equal in number to the constraint equations.Similarly any non-conservative</span>
-<span class="sd">    forces can be supplied in a list (as described below and also shown in the</span>
-<span class="sd">    example) along with a ReferenceFrame. This is also discussed further in the</span>
-<span class="sd">    __init__ method.</span>
-
-<span class="sd">    Attributes</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    mass_matrix : Matrix</span>
-<span class="sd">        The system&#39;s mass matrix</span>
-
-<span class="sd">    forcing : Matrix</span>
-<span class="sd">        The system&#39;s forcing vector</span>
-
-<span class="sd">    mass_matrix_full : Matrix</span>
-<span class="sd">        The &quot;mass matrix&quot; for the qdot&#39;s, qdoubledot&#39;s, and the</span>
-<span class="sd">        lagrange multipliers (lam)</span>
-
-<span class="sd">    forcing_full : Matrix</span>
-<span class="sd">        The forcing vector for the qdot&#39;s, qdoubledot&#39;s and</span>
-<span class="sd">        lagrange multipliers (lam)</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    This is a simple example for a one degree of freedom translational</span>
-<span class="sd">    spring-mass-damper.</span>
-
-<span class="sd">    In this example, we first need to do the kinematics.$</span>
-<span class="sd">    This involves creating generalized coordinates and its derivative.</span>
-<span class="sd">    Then we create a point and set its velocity in a frame::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import LagrangesMethod, Lagrangian</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, Particle, Point</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import dynamicsymbols, kinetic_energy</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; q = dynamicsymbols(&#39;q&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; qd = dynamicsymbols(&#39;q&#39;, 1)</span>
-<span class="sd">        &gt;&gt;&gt; m, k, b = symbols(&#39;m k b&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; P = Point(&#39;P&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; P.set_vel(N, qd * N.x)</span>
-
-<span class="sd">    We need to then prepare the information as required by LagrangesMethod to</span>
-<span class="sd">    generate equations of motion.</span>
-<span class="sd">    First we create the Particle, which has a point attached to it.</span>
-<span class="sd">    Following this the lagrangian is created from the kinetic and potential</span>
-<span class="sd">    energies.</span>
-<span class="sd">    Then, a list of nonconservative forces/torques must be constructed, where</span>
-<span class="sd">    each entry in is a (Point, Vector) or (ReferenceFrame, Vector) tuple, where</span>
-<span class="sd">    the Vectors represent the nonconservative force or torque.</span>
-
-<span class="sd">        &gt;&gt;&gt; Pa = Particle(&#39;Pa&#39;, P, m)</span>
-<span class="sd">        &gt;&gt;&gt; Pa.set_potential_energy(k * q**2 / 2.0)</span>
-<span class="sd">        &gt;&gt;&gt; L = Lagrangian(N, Pa)</span>
-<span class="sd">        &gt;&gt;&gt; fl = [(P, -b * qd * N.x)]</span>
-
-<span class="sd">     Finally we can generate the equations of motion.</span>
-<span class="sd">     First we create the LagrangesMethod object.To do this one must supply an</span>
-<span class="sd">     the Lagrangian, the list of generalized coordinates. Also supplied are the</span>
-<span class="sd">     constraint equations, the forcelist and the inertial frame, if relevant.</span>
-<span class="sd">     Next we generate Lagrange&#39;s equations of motion, such that:</span>
-<span class="sd">     Lagrange&#39;s equations of motion = 0.</span>
-<span class="sd">     We have the equations of motion at this point.</span>
-
-<span class="sd">        &gt;&gt;&gt; l = LagrangesMethod(L, [q], forcelist = fl, frame = N)</span>
-<span class="sd">        &gt;&gt;&gt; print(l.form_lagranges_equations())</span>
-<span class="sd">        [b*Derivative(q(t), t) + 1.0*k*q(t) + m*Derivative(q(t), t, t)]</span>
-
-<span class="sd">    We can also solve for the states using the &#39;rhs&#39; method.</span>
-
-<span class="sd">        &gt;&gt;&gt; print(l.rhs())</span>
-<span class="sd">        [                    Derivative(q(t), t)]</span>
-<span class="sd">        [(-b*Derivative(q(t), t) - 1.0*k*q(t))/m]</span>
-
-<span class="sd">    Please refer to the docstrings on each method for more details.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Lagrangian</span><span class="p">,</span> <span class="n">q_list</span><span class="p">,</span> <span class="n">coneqs</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">forcelist</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">frame</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Supply the following for the initialization of LagrangesMethod</span>
-
-<span class="sd">        Lagrangian : Sympifyable</span>
-
-<span class="sd">        q_list : list</span>
-<span class="sd">            A list of the generalized coordinates</span>
-
-<span class="sd">        coneqs : list</span>
-<span class="sd">            A list of the holonomic and non-holonomic constraint equations.</span>
-<span class="sd">            VERY IMPORTANT NOTE- The holonomic constraints must be</span>
-<span class="sd">            differentiated with respect to time and then included in coneqs.</span>
-
-<span class="sd">        forcelist : list</span>
-<span class="sd">            Takes a list of (Point, Vector) or (ReferenceFrame, Vector) tuples</span>
-<span class="sd">            which represent the force at a point or torque on a frame. This</span>
-<span class="sd">            feature is primarily to account for the nonconservative forces</span>
-<span class="sd">            amd/or moments.</span>
-
-<span class="sd">        frame : ReferenceFrame</span>
-<span class="sd">            Supply the inertial frame. This is used to determine the</span>
-<span class="sd">            generalized forces due to non-sonservative forces.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_L</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">Lagrangian</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">eom</span> <span class="o">=</span> <span class="bp">None</span>  <span class="c"># initializing the eom Matrix</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_m_cd</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([])</span>  <span class="c"># Mass Matrix of differentiated coneqs</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_m_d</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([])</span>  <span class="c"># Mass Matrix of dynamic equations</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_f_cd</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([])</span>  <span class="c"># Forcing part of the diff coneqs</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_f_d</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([])</span>  <span class="c"># Forcing part of the dynamic equations</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">lam_coeffs</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([])</span>  <span class="c"># Initializing the coeffecients of lams</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">forcelist</span> <span class="o">=</span> <span class="n">forcelist</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">inertial</span> <span class="o">=</span> <span class="n">frame</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">lam_vec</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([])</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_term1</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_term2</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_term3</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_term4</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([])</span>
-
-        <span class="c"># Creating the qs, qdots and qdoubledots</span>
-
-        <span class="n">q_list</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">q_list</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">q_list</span><span class="p">,</span> <span class="nb">list</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Generalized coords. must be supplied in a list&#39;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_q</span> <span class="o">=</span> <span class="n">q_list</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_qdots</span> <span class="o">=</span> <span class="p">[</span><span class="n">diff</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_q</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_qdoubledots</span> <span class="o">=</span> <span class="p">[</span><span class="n">diff</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_qdots</span><span class="p">]</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">coneqs</span> <span class="o">=</span> <span class="n">coneqs</span>
-
-<div class="viewcode-block" id="LagrangesMethod.form_lagranges_equations"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.lagrange.LagrangesMethod.form_lagranges_equations">[docs]</a>    <span class="k">def</span> <span class="nf">form_lagranges_equations</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Method to form Lagrange&#39;s equations of motion.</span>
-
-<span class="sd">        Returns a vector of equations of motion using Lagrange&#39;s equations of</span>
-<span class="sd">        the second kind.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">q</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_q</span>
-        <span class="n">qd</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_qdots</span>
-        <span class="n">qdd</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_qdoubledots</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-
-        <span class="c">#Putting the Lagrangian in a Matrix</span>
-        <span class="n">L</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">_L</span><span class="p">])</span>
-
-        <span class="c">#Determining the first term in Lagrange&#39;s EOM</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_term1</span> <span class="o">=</span> <span class="n">L</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qd</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_term1</span> <span class="o">=</span> <span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">_term1</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span><span class="p">))</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span>
-
-        <span class="c">#Determining the second term in Lagrange&#39;s EOM</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_term2</span> <span class="o">=</span> <span class="p">(</span><span class="n">L</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">q</span><span class="p">))</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span>
-
-        <span class="c">#term1 and term2 will be there no matter what so leave them as they are</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">coneqs</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">coneqs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">coneqs</span>
-            <span class="c">#If there are coneqs supplied then the following will be created</span>
-            <span class="n">coneqs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">coneqs</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">coneqs</span><span class="p">,</span> <span class="nb">list</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Enter the constraint equations in a list&#39;</span><span class="p">)</span>
-
-            <span class="n">o</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">coneqs</span><span class="p">)</span>
-
-            <span class="c">#Creating the multipliers</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">lam_vec</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;lam1:&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">o</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)))</span>
-
-            <span class="c">#Extracting the coeffecients of the multipliers</span>
-            <span class="n">coneqs_mat</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">coneqs</span><span class="p">)</span>
-            <span class="n">qd</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_qdots</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">lam_coeffs</span> <span class="o">=</span> <span class="o">-</span><span class="n">coneqs_mat</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qd</span><span class="p">)</span>
-
-            <span class="c">#Determining the third term in Lagrange&#39;s EOM</span>
-            <span class="c">#term3 = ((self.lam_vec).transpose() * self.lam_coeffs).transpose()</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_term3</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lam_coeffs</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">lam_vec</span>
-
-            <span class="c">#Taking the time derivative of the constraint equations</span>
-            <span class="n">diffconeqs</span> <span class="o">=</span> <span class="p">[</span><span class="n">diff</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">coneqs</span><span class="p">]</span>
-
-            <span class="c">#Extracting the coeffecients of the qdds from the diff coneqs</span>
-            <span class="n">diffconeqs_mat</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">diffconeqs</span><span class="p">)</span>
-            <span class="n">qdd</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_qdoubledots</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_m_cd</span> <span class="o">=</span> <span class="n">diffconeqs_mat</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">qdd</span><span class="p">)</span>
-
-            <span class="c">#The remaining terms i.e. the &#39;forcing&#39; terms in diff coneqs</span>
-            <span class="n">qddzero</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">qdd</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">n</span><span class="p">)))</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_f_cd</span> <span class="o">=</span> <span class="o">-</span><span class="n">diffconeqs_mat</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">qddzero</span><span class="p">)</span>
-
-        <span class="k">else</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_term3</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">forcelist</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">forcelist</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">forcelist</span>
-            <span class="n">N</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">inertial</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">ReferenceFrame</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Enter a valid ReferenceFrame&#39;</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">forcelist</span><span class="p">,</span> <span class="p">(</span><span class="nb">list</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">)):</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Forces must be supplied in a list of: lists&#39;</span>
-                        <span class="s">&#39; or tuples&#39;</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_term4</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">qd</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">w</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">forcelist</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">w</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">ReferenceFrame</span><span class="p">):</span>
-                        <span class="n">speed</span> <span class="o">=</span> <span class="n">w</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">_term4</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+=</span> <span class="n">speed</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span> <span class="o">&amp;</span> <span class="n">w</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-                    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">w</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Point</span><span class="p">):</span>
-                        <span class="n">speed</span> <span class="o">=</span> <span class="n">w</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">_term4</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+=</span> <span class="n">speed</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span> <span class="o">&amp;</span> <span class="n">w</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;First entry in force pair is a point&#39;</span>
-                                        <span class="s">&#39; or frame&#39;</span><span class="p">)</span>
-
-        <span class="k">else</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_term4</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">eom</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_term1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_term2</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_term3</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_term4</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">eom</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="LagrangesMethod.mass_matrix"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.lagrange.LagrangesMethod.mass_matrix">[docs]</a>    <span class="k">def</span> <span class="nf">mass_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Returns the mass matrix, which is augmented by the Lagrange</span>
-<span class="sd">        multipliers, if necessary.</span>
-
-<span class="sd">        If the system is described by &#39;n&#39; generalized coordinates and there are</span>
-<span class="sd">        no constraint equations then an n X n matrix is returned.</span>
-
-<span class="sd">        If there are &#39;n&#39; generalized coordinates and &#39;m&#39; constraint equations</span>
-<span class="sd">        have been supplied during initialization then an n X (n+m) matrix is</span>
-<span class="sd">        returned. The (n + m - 1)th and (n + m)th columns contain the</span>
-<span class="sd">        coefficients of the Lagrange multipliers.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">eom</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Need to compute the equations of motion first&#39;</span><span class="p">)</span>
-
-        <span class="c">#The &#39;dynamic&#39; mass matrix is generated by the following</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_m_d</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">eom</span><span class="p">)</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_qdoubledots</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lam_coeffs</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_m_d</span><span class="p">)</span><span class="o">.</span><span class="n">row_join</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">lam_coeffs</span><span class="p">)</span><span class="o">.</span><span class="n">transpose</span><span class="p">())</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_m_d</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="LagrangesMethod.mass_matrix_full"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.lagrange.LagrangesMethod.mass_matrix_full">[docs]</a>    <span class="k">def</span> <span class="nf">mass_matrix_full</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Augments the coefficients of qdots to the mass_matrix. &quot;&quot;&quot;</span>
-
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_q</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">eom</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Need to compute the equations of motion first&#39;</span><span class="p">)</span>
-        <span class="c">#THE FIRST TWO ROWS OF THE MATRIX</span>
-        <span class="n">row1</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">row_join</span><span class="p">(</span><span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-        <span class="n">row2</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">row_join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mass_matrix</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">coneqs</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">coneqs</span><span class="p">)</span>
-            <span class="n">I</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">row_join</span><span class="p">(</span><span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="n">m</span><span class="p">))</span>
-            <span class="n">below_eye</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-            <span class="n">A</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mass_matrix</span><span class="p">)</span><span class="o">.</span><span class="n">col_join</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">_m_cd</span><span class="p">)</span><span class="o">.</span><span class="n">row_join</span><span class="p">(</span><span class="n">zeros</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">m</span><span class="p">)))</span>
-            <span class="n">below_I</span> <span class="o">=</span> <span class="n">below_eye</span><span class="o">.</span><span class="n">row_join</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">I</span><span class="o">.</span><span class="n">col_join</span><span class="p">(</span><span class="n">below_I</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">A</span> <span class="o">=</span> <span class="n">row1</span><span class="o">.</span><span class="n">col_join</span><span class="p">(</span><span class="n">row2</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">A</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="LagrangesMethod.forcing"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.lagrange.LagrangesMethod.forcing">[docs]</a>    <span class="k">def</span> <span class="nf">forcing</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Returns the forcing vector from &#39;lagranges_equations&#39; method. &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">eom</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Need to compute the equations of motion first&#39;</span><span class="p">)</span>
-
-        <span class="n">qdd</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_qdoubledots</span>
-        <span class="n">qddzero</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">qdd</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">qdd</span><span class="p">))))</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">coneqs</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">lam</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lam_vec</span>
-            <span class="n">lamzero</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">lam</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">lam</span><span class="p">))))</span>
-
-            <span class="c">#The forcing terms from the eoms</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_f_d</span> <span class="o">=</span> <span class="o">-</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">eom</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">qddzero</span><span class="p">))</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">lamzero</span><span class="p">)</span>
-
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c">#The forcing terms from the eoms</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_f_d</span> <span class="o">=</span> <span class="o">-</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">eom</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">qddzero</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_f_d</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="LagrangesMethod.forcing_full"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.lagrange.LagrangesMethod.forcing_full">[docs]</a>    <span class="k">def</span> <span class="nf">forcing_full</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Augments qdots to the forcing vector above. &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">eom</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Need to compute the equations of motion first&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">coneqs</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">Matrix</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_qdots</span><span class="p">))</span><span class="o">.</span><span class="n">col_join</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">forcing</span><span class="p">)</span><span class="o">.</span><span class="n">col_join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_f_cd</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">Matrix</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_qdots</span><span class="p">))</span><span class="o">.</span><span class="n">col_join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">forcing</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="LagrangesMethod.rhs"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.lagrange.LagrangesMethod.rhs">[docs]</a>    <span class="k">def</span> <span class="nf">rhs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">&quot;GE&quot;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Returns equations that can be solved numerically</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        method : string</span>
-<span class="sd">            The method by which matrix inversion of mass_matrix_full must be</span>
-<span class="sd">            performed such as Gauss Elimination or LU decomposition.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="c"># TODO- should probably use the matinvmul method from Kane</span>
-
-        <span class="k">return</span> <span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">mass_matrix_full</span><span class="p">)</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">method</span><span class="p">,</span> <span class="n">try_block_diag</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="o">*</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">forcing_full</span><span class="p">)</span></div></div>
-</pre></div>
-
-          </div>
-        </div>
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-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../../index.html">
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diff --git a/dev-py3k/_modules/sympy/physics/mechanics/particle.html b/dev-py3k/_modules/sympy/physics/mechanics/particle.html
deleted file mode 100644
index b50dc581071..00000000000
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+++ /dev/null
@@ -1,339 +0,0 @@
-
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-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
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-            
-  <h1>Source code for sympy.physics.mechanics.particle</h1><div class="highlight"><pre>
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;Particle&#39;</span><span class="p">]</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.mechanics.point</span> <span class="kn">import</span> <span class="n">Point</span>
-
-
-<div class="viewcode-block" id="Particle"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.particle.Particle">[docs]</a><span class="k">class</span> <span class="nc">Particle</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A particle.</span>
-
-<span class="sd">    Particles have a non-zero mass and lack spatial extension; they take up no</span>
-<span class="sd">    space.</span>
-
-<span class="sd">    Values need to be supplied on initialization, but can be changed later.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-<span class="sd">    name : str</span>
-<span class="sd">        Name of particle</span>
-<span class="sd">    point : Point</span>
-<span class="sd">        A physics/mechanics Point which represents the position, velocity, and</span>
-<span class="sd">        acceleration of this Particle</span>
-<span class="sd">    mass : sympifyable</span>
-<span class="sd">        A SymPy expression representing the Particle&#39;s mass</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import Particle, Point</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">    &gt;&gt;&gt; po = Point(&#39;po&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; m = Symbol(&#39;m&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; pa = Particle(&#39;pa&#39;, po, m)</span>
-<span class="sd">    &gt;&gt;&gt; # Or you could change these later</span>
-<span class="sd">    &gt;&gt;&gt; pa.mass = m</span>
-<span class="sd">    &gt;&gt;&gt; pa.point = po</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">point</span><span class="p">,</span> <span class="n">mass</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Supply a valid name.&#39;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_name</span> <span class="o">=</span> <span class="n">name</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">set_mass</span><span class="p">(</span><span class="n">mass</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">set_point</span><span class="p">(</span><span class="n">point</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_pe</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_name</span>
-
-    <span class="n">__repr__</span> <span class="o">=</span> <span class="n">__str__</span>
-
-<div class="viewcode-block" id="Particle.get_mass"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.particle.Particle.get_mass">[docs]</a>    <span class="k">def</span> <span class="nf">get_mass</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Mass of the particle.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mass</span>
-</div>
-    <span class="k">def</span> <span class="nf">set_mass</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mass</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_mass</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">mass</span><span class="p">)</span>
-
-    <span class="n">mass</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="n">get_mass</span><span class="p">,</span> <span class="n">set_mass</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Particle.get_point"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.particle.Particle.get_point">[docs]</a>    <span class="k">def</span> <span class="nf">get_point</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Point of the particle.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_point</span>
-</div>
-    <span class="k">def</span> <span class="nf">set_point</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">Point</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Particle point attribute must be a Point object.&quot;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_point</span> <span class="o">=</span> <span class="n">p</span>
-
-    <span class="n">point</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="n">get_point</span><span class="p">,</span> <span class="n">set_point</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Particle.linear_momentum"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.particle.Particle.linear_momentum">[docs]</a>    <span class="k">def</span> <span class="nf">linear_momentum</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">frame</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Linear momentum of the particle.</span>
-
-<span class="sd">        The linear momentum L, of a particle P, with respect to frame N is</span>
-<span class="sd">        given by</span>
-
-<span class="sd">        L = m * v</span>
-
-<span class="sd">        where m is the mass of the particle, and v is the velocity of the</span>
-<span class="sd">        particle in the frame N.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        frame : ReferenceFrame</span>
-<span class="sd">            The frame in which linear momentum is desired.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import Particle, Point, ReferenceFrame</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import dynamicsymbols</span>
-<span class="sd">        &gt;&gt;&gt; m, v = dynamicsymbols(&#39;m v&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; P = Point(&#39;P&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; A = Particle(&#39;A&#39;, P, m)</span>
-<span class="sd">        &gt;&gt;&gt; P.set_vel(N, v * N.x)</span>
-<span class="sd">        &gt;&gt;&gt; A.linear_momentum(N)</span>
-<span class="sd">        m*v*N.x</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">mass</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">point</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">frame</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Particle.angular_momentum"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.particle.Particle.angular_momentum">[docs]</a>    <span class="k">def</span> <span class="nf">angular_momentum</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">point</span><span class="p">,</span> <span class="n">frame</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Angular momentum of the particle about the point.</span>
-
-<span class="sd">        The angular momentum H, about some point O of a particle, P, is given</span>
-<span class="sd">        by:</span>
-
-<span class="sd">        H = r x m * v</span>
-
-<span class="sd">        where r is the position vector from point O to the particle P, m is</span>
-<span class="sd">        the mass of the particle, and v is the velocity of the particle in</span>
-<span class="sd">        the inertial frame, N.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        point : Point</span>
-<span class="sd">            The point about which angular momentum of the particle is desired.</span>
-
-<span class="sd">        frame : ReferenceFrame</span>
-<span class="sd">            The frame in which angular momentum is desired.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import Particle, Point, ReferenceFrame</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import dynamicsymbols</span>
-<span class="sd">        &gt;&gt;&gt; m, v, r = dynamicsymbols(&#39;m v r&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; O = Point(&#39;O&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; A = O.locatenew(&#39;A&#39;, r * N.x)</span>
-<span class="sd">        &gt;&gt;&gt; P = Particle(&#39;P&#39;, A, m)</span>
-<span class="sd">        &gt;&gt;&gt; P.point.set_vel(N, v * N.y)</span>
-<span class="sd">        &gt;&gt;&gt; P.angular_momentum(O, N)</span>
-<span class="sd">        m*r*v*N.z</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">point</span><span class="o">.</span><span class="n">pos_from</span><span class="p">(</span><span class="n">point</span><span class="p">)</span> <span class="o">^</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mass</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">point</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">frame</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="Particle.kinetic_energy"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.particle.Particle.kinetic_energy">[docs]</a>    <span class="k">def</span> <span class="nf">kinetic_energy</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">frame</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Kinetic energy of the particle</span>
-
-<span class="sd">        The kinetic energy, T, of a particle, P, is given by</span>
-
-<span class="sd">        &#39;T = 1/2 m v^2&#39;</span>
-
-<span class="sd">        where m is the mass of particle P, and v is the velocity of the</span>
-<span class="sd">        particle in the supplied ReferenceFrame.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        frame : ReferenceFrame</span>
-<span class="sd">            The Particle&#39;s velocity is typically defined with respect to</span>
-<span class="sd">            an inertial frame but any relevant frame in which the velocity is</span>
-<span class="sd">            known can be supplied.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import Particle, Point, ReferenceFrame</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; m, v, r = symbols(&#39;m v r&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; O = Point(&#39;O&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; P = Particle(&#39;P&#39;, O, m)</span>
-<span class="sd">        &gt;&gt;&gt; P.point.set_vel(N, v * N.y)</span>
-<span class="sd">        &gt;&gt;&gt; P.kinetic_energy(N)</span>
-<span class="sd">        m*v**2/2</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mass</span> <span class="o">/</span> <span class="n">sympify</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">point</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">frame</span><span class="p">)</span> <span class="o">&amp;</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">point</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">frame</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="Particle.set_potential_energy"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.particle.Particle.set_potential_energy">[docs]</a>    <span class="k">def</span> <span class="nf">set_potential_energy</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">scalar</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Used to set the potential energy of the Particle.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        scalar : Sympifyable</span>
-<span class="sd">            The potential energy (a scalar) of the Particle.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import Particle, Point</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; m, g, h = symbols(&#39;m g h&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; O = Point(&#39;O&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; P = Particle(&#39;P&#39;, O, m)</span>
-<span class="sd">        &gt;&gt;&gt; P.set_potential_energy(m * g * h)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_pe</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">scalar</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Particle.potential_energy"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.particle.Particle.potential_energy">[docs]</a>    <span class="k">def</span> <span class="nf">potential_energy</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The potential energy of the Particle.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import Particle, Point</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; m, g, h = symbols(&#39;m g h&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; O = Point(&#39;O&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; P = Particle(&#39;P&#39;, O, m)</span>
-<span class="sd">        &gt;&gt;&gt; P.set_potential_energy(m * g * h)</span>
-<span class="sd">        &gt;&gt;&gt; P.potential_energy</span>
-<span class="sd">        g*h*m</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_pe</span></div></div>
-</pre></div>
-
-          </div>
-        </div>
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-            
-  <h1>Source code for sympy.physics.mechanics.point</h1><div class="highlight"><pre>
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;Point&#39;</span><span class="p">]</span>
-
-<span class="kn">from</span> <span class="nn">sympy.physics.mechanics.essential</span> <span class="kn">import</span> <span class="n">_check_frame</span><span class="p">,</span> <span class="n">_check_vector</span>
-
-
-<div class="viewcode-block" id="Point"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kinematics.html#sympy.physics.mechanics.point.Point">[docs]</a><span class="k">class</span> <span class="nc">Point</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;This object represents a point in a dynamic system.</span>
-
-<span class="sd">    It stores the: position, velocity, and acceleration of a point.</span>
-<span class="sd">    The position is a vector defined as the vector distance from a parent</span>
-<span class="sd">    point to this point.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Initialization of a Point object. &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="n">name</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_pos_dict</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_vel_dict</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_acc_dict</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_pdlist</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_pos_dict</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_vel_dict</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_acc_dict</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">name</span>
-
-    <span class="n">__repr__</span> <span class="o">=</span> <span class="n">__str__</span>
-
-    <span class="k">def</span> <span class="nf">_check_point</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Point</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;A Point must be supplied&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_pdict_list</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="n">num</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Creates a list from self to other using _dcm_dict. &quot;&quot;&quot;</span>
-        <span class="n">outlist</span> <span class="o">=</span> <span class="p">[[</span><span class="bp">self</span><span class="p">]]</span>
-        <span class="n">oldlist</span> <span class="o">=</span> <span class="p">[[]]</span>
-        <span class="k">while</span> <span class="n">outlist</span> <span class="o">!=</span> <span class="n">oldlist</span><span class="p">:</span>
-            <span class="n">oldlist</span> <span class="o">=</span> <span class="n">outlist</span><span class="p">[:]</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">outlist</span><span class="p">):</span>
-                <span class="n">templist</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">_pdlist</span><span class="p">[</span><span class="n">num</span><span class="p">]</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-                <span class="k">for</span> <span class="n">i2</span><span class="p">,</span> <span class="n">v2</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">templist</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">v</span><span class="o">.</span><span class="n">__contains__</span><span class="p">(</span><span class="n">v2</span><span class="p">):</span>
-                        <span class="n">littletemplist</span> <span class="o">=</span> <span class="n">v</span> <span class="o">+</span> <span class="p">[</span><span class="n">v2</span><span class="p">]</span>
-                        <span class="k">if</span> <span class="ow">not</span> <span class="n">outlist</span><span class="o">.</span><span class="n">__contains__</span><span class="p">(</span><span class="n">littletemplist</span><span class="p">):</span>
-                            <span class="n">outlist</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">littletemplist</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">oldlist</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">v</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">!=</span> <span class="n">other</span><span class="p">:</span>
-                <span class="n">outlist</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-        <span class="n">outlist</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="nb">len</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">outlist</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">outlist</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;No Connecting Path found between &#39;</span> <span class="o">+</span> <span class="n">other</span><span class="o">.</span><span class="n">name</span> <span class="o">+</span>
-                         <span class="s">&#39; and &#39;</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Point.a1pt_theory"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kinematics.html#sympy.physics.mechanics.point.Point.a1pt_theory">[docs]</a>    <span class="k">def</span> <span class="nf">a1pt_theory</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">otherpoint</span><span class="p">,</span> <span class="n">outframe</span><span class="p">,</span> <span class="n">interframe</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Sets the acceleration of this point with the 1-point theory.</span>
-
-<span class="sd">        The 1-point theory for point acceleration looks like this:</span>
-
-<span class="sd">        ^N a^P = ^B a^P + ^N a^O + ^N alpha^B x r^OP + ^N omega^B x (^N omega^B</span>
-<span class="sd">        x r^OP) + 2 ^N omega^B x ^B v^P</span>
-
-<span class="sd">        where O is a point fixed in B, P is a point moving in B, and B is</span>
-<span class="sd">        rotating in frame N.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        otherpoint : Point</span>
-<span class="sd">            The first point of the 1-point theory (O)</span>
-<span class="sd">        outframe : ReferenceFrame</span>
-<span class="sd">            The frame we want this point&#39;s acceleration defined in (N)</span>
-<span class="sd">        fixedframe : ReferenceFrame</span>
-<span class="sd">            The intermediate frame in this calculation (B)</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import Point, ReferenceFrame, dynamicsymbols</span>
-<span class="sd">        &gt;&gt;&gt; q = dynamicsymbols(&#39;q&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; q2 = dynamicsymbols(&#39;q2&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; qd = dynamicsymbols(&#39;q&#39;, 1)</span>
-<span class="sd">        &gt;&gt;&gt; q2d = dynamicsymbols(&#39;q2&#39;, 1)</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; B = ReferenceFrame(&#39;B&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; B.set_ang_vel(N, 5 * B.y)</span>
-<span class="sd">        &gt;&gt;&gt; O = Point(&#39;O&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; P = O.locatenew(&#39;P&#39;, q * B.x)</span>
-<span class="sd">        &gt;&gt;&gt; P.set_vel(B, qd * B.x + q2d * B.y)</span>
-<span class="sd">        &gt;&gt;&gt; O.set_vel(N, 0)</span>
-<span class="sd">        &gt;&gt;&gt; P.a1pt_theory(O, N, B)</span>
-<span class="sd">        (-25*q + q&#39;&#39;)*B.x + q2&#39;&#39;*B.y - 10*q&#39;*B.z</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">outframe</span><span class="p">)</span>
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">interframe</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_check_point</span><span class="p">(</span><span class="n">otherpoint</span><span class="p">)</span>
-        <span class="n">dist</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">pos_from</span><span class="p">(</span><span class="n">otherpoint</span><span class="p">)</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">interframe</span><span class="p">)</span>
-        <span class="n">a1</span> <span class="o">=</span> <span class="n">otherpoint</span><span class="o">.</span><span class="n">acc</span><span class="p">(</span><span class="n">outframe</span><span class="p">)</span>
-        <span class="n">a2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">acc</span><span class="p">(</span><span class="n">interframe</span><span class="p">)</span>
-        <span class="n">omega</span> <span class="o">=</span> <span class="n">interframe</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">outframe</span><span class="p">)</span>
-        <span class="n">alpha</span> <span class="o">=</span> <span class="n">interframe</span><span class="o">.</span><span class="n">ang_acc_in</span><span class="p">(</span><span class="n">outframe</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">set_acc</span><span class="p">(</span><span class="n">outframe</span><span class="p">,</span> <span class="n">a2</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="n">omega</span> <span class="o">^</span> <span class="n">v</span><span class="p">)</span> <span class="o">+</span> <span class="n">a1</span> <span class="o">+</span> <span class="p">(</span><span class="n">alpha</span> <span class="o">^</span> <span class="n">dist</span><span class="p">)</span> <span class="o">+</span>
-                <span class="p">(</span><span class="n">omega</span> <span class="o">^</span> <span class="p">(</span><span class="n">omega</span> <span class="o">^</span> <span class="n">dist</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">acc</span><span class="p">(</span><span class="n">outframe</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Point.a2pt_theory"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kinematics.html#sympy.physics.mechanics.point.Point.a2pt_theory">[docs]</a>    <span class="k">def</span> <span class="nf">a2pt_theory</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">otherpoint</span><span class="p">,</span> <span class="n">outframe</span><span class="p">,</span> <span class="n">fixedframe</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Sets the acceleration of this point with the 2-point theory.</span>
-
-<span class="sd">        The 2-point theory for point acceleration looks like this:</span>
-
-<span class="sd">        ^N a^P = ^N a^O + ^N alpha^B x r^OP + ^N omega^B x (^N omega^B x r^OP)</span>
-
-<span class="sd">        where O and P are both points fixed in frame B, which is rotating in</span>
-<span class="sd">        frame N.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        otherpoint : Point</span>
-<span class="sd">            The first point of the 2-point theory (O)</span>
-<span class="sd">        outframe : ReferenceFrame</span>
-<span class="sd">            The frame we want this point&#39;s acceleration defined in (N)</span>
-<span class="sd">        fixedframe : ReferenceFrame</span>
-<span class="sd">            The frame in which both points are fixed (B)</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import Point, ReferenceFrame, dynamicsymbols</span>
-<span class="sd">        &gt;&gt;&gt; q = dynamicsymbols(&#39;q&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; qd = dynamicsymbols(&#39;q&#39;, 1)</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; B = N.orientnew(&#39;B&#39;, &#39;Axis&#39;, [q, N.z])</span>
-<span class="sd">        &gt;&gt;&gt; O = Point(&#39;O&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; P = O.locatenew(&#39;P&#39;, 10 * B.x)</span>
-<span class="sd">        &gt;&gt;&gt; O.set_vel(N, 5 * N.x)</span>
-<span class="sd">        &gt;&gt;&gt; P.a2pt_theory(O, N, B)</span>
-<span class="sd">        - 10*q&#39;**2*B.x + 10*q&#39;&#39;*B.y</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">outframe</span><span class="p">)</span>
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">fixedframe</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_check_point</span><span class="p">(</span><span class="n">otherpoint</span><span class="p">)</span>
-        <span class="n">dist</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">pos_from</span><span class="p">(</span><span class="n">otherpoint</span><span class="p">)</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">otherpoint</span><span class="o">.</span><span class="n">acc</span><span class="p">(</span><span class="n">outframe</span><span class="p">)</span>
-        <span class="n">omega</span> <span class="o">=</span> <span class="n">fixedframe</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">outframe</span><span class="p">)</span>
-        <span class="n">alpha</span> <span class="o">=</span> <span class="n">fixedframe</span><span class="o">.</span><span class="n">ang_acc_in</span><span class="p">(</span><span class="n">outframe</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">set_acc</span><span class="p">(</span><span class="n">outframe</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="p">(</span><span class="n">alpha</span> <span class="o">^</span> <span class="n">dist</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">omega</span> <span class="o">^</span> <span class="p">(</span><span class="n">omega</span> <span class="o">^</span> <span class="n">dist</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">acc</span><span class="p">(</span><span class="n">outframe</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Point.acc"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kinematics.html#sympy.physics.mechanics.point.Point.acc">[docs]</a>    <span class="k">def</span> <span class="nf">acc</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">frame</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The acceleration Vector of this Point in a ReferenceFrame.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        frame : ReferenceFrame</span>
-<span class="sd">            The frame in which the returned acceleration vector will be defined in</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import Point, ReferenceFrame</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; p1 = Point(&#39;p1&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; p1.set_acc(N, 10 * N.x)</span>
-<span class="sd">        &gt;&gt;&gt; p1.acc(N)</span>
-<span class="sd">        10*N.x</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">frame</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">frame</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_acc_dict</span><span class="p">):</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_vel_dict</span><span class="p">[</span><span class="n">frame</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_vel_dict</span><span class="p">[</span><span class="n">frame</span><span class="p">])</span><span class="o">.</span><span class="n">dt</span><span class="p">(</span><span class="n">frame</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="mi">0</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_acc_dict</span><span class="p">[</span><span class="n">frame</span><span class="p">]</span>
-</div>
-<div class="viewcode-block" id="Point.locatenew"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kinematics.html#sympy.physics.mechanics.point.Point.locatenew">[docs]</a>    <span class="k">def</span> <span class="nf">locatenew</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Creates a new point with a position defined from this point.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        name : str</span>
-<span class="sd">            The name for the new point</span>
-<span class="sd">        value : Vector</span>
-<span class="sd">            The position of the new point relative to this point</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, Point</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; P1 = Point(&#39;P1&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; P2 = P1.locatenew(&#39;P2&#39;, 10 * N.x)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Must supply a valid name&#39;</span><span class="p">)</span>
-        <span class="n">value</span> <span class="o">=</span> <span class="n">_check_vector</span><span class="p">(</span><span class="n">value</span><span class="p">)</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">p</span><span class="o">.</span><span class="n">set_pos</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">value</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">set_pos</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="o">-</span><span class="n">value</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">p</span>
-</div>
-<div class="viewcode-block" id="Point.pos_from"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kinematics.html#sympy.physics.mechanics.point.Point.pos_from">[docs]</a>    <span class="k">def</span> <span class="nf">pos_from</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">otherpoint</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a Vector distance between this Point and the other Point.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        otherpoint : Point</span>
-<span class="sd">            The otherpoint we are locating this one relative to</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import Point, ReferenceFrame</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; p1 = Point(&#39;p1&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; p2 = Point(&#39;p2&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; p1.set_pos(p2, 10 * N.x)</span>
-<span class="sd">        &gt;&gt;&gt; p1.pos_from(p2)</span>
-<span class="sd">        10*N.x</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">outvec</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">plist</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_pdict_list</span><span class="p">(</span><span class="n">otherpoint</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">plist</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">outvec</span> <span class="o">+=</span> <span class="n">plist</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">_pos_dict</span><span class="p">[</span><span class="n">plist</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]]</span>
-        <span class="k">return</span> <span class="n">outvec</span>
-</div>
-<div class="viewcode-block" id="Point.set_acc"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kinematics.html#sympy.physics.mechanics.point.Point.set_acc">[docs]</a>    <span class="k">def</span> <span class="nf">set_acc</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">frame</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Used to set the acceleration of this Point in a ReferenceFrame.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        value : Vector</span>
-<span class="sd">            The vector value of this point&#39;s acceleration in the frame</span>
-<span class="sd">        frame : ReferenceFrame</span>
-<span class="sd">            The frame in which this point&#39;s acceleration is defined</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import Point, ReferenceFrame</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; p1 = Point(&#39;p1&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; p1.set_acc(N, 10 * N.x)</span>
-<span class="sd">        &gt;&gt;&gt; p1.acc(N)</span>
-<span class="sd">        10*N.x</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">value</span> <span class="o">=</span> <span class="n">_check_vector</span><span class="p">(</span><span class="n">value</span><span class="p">)</span>
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">frame</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_acc_dict</span><span class="o">.</span><span class="n">update</span><span class="p">({</span><span class="n">frame</span><span class="p">:</span> <span class="n">value</span><span class="p">})</span>
-</div>
-<div class="viewcode-block" id="Point.set_pos"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kinematics.html#sympy.physics.mechanics.point.Point.set_pos">[docs]</a>    <span class="k">def</span> <span class="nf">set_pos</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">otherpoint</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Used to set the position of this point w.r.t. another point.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        value : Vector</span>
-<span class="sd">            The vector which defines the location of this point</span>
-<span class="sd">        point : Point</span>
-<span class="sd">            The other point which this point&#39;s location is defined relative to</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import Point, ReferenceFrame</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; p1 = Point(&#39;p1&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; p2 = Point(&#39;p2&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; p1.set_pos(p2, 10 * N.x)</span>
-<span class="sd">        &gt;&gt;&gt; p1.pos_from(p2)</span>
-<span class="sd">        10*N.x</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">value</span> <span class="o">=</span> <span class="n">_check_vector</span><span class="p">(</span><span class="n">value</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_check_point</span><span class="p">(</span><span class="n">otherpoint</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_pos_dict</span><span class="o">.</span><span class="n">update</span><span class="p">({</span><span class="n">otherpoint</span><span class="p">:</span> <span class="n">value</span><span class="p">})</span>
-        <span class="n">otherpoint</span><span class="o">.</span><span class="n">_pos_dict</span><span class="o">.</span><span class="n">update</span><span class="p">({</span><span class="bp">self</span><span class="p">:</span> <span class="o">-</span><span class="n">value</span><span class="p">})</span>
-</div>
-<div class="viewcode-block" id="Point.set_vel"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kinematics.html#sympy.physics.mechanics.point.Point.set_vel">[docs]</a>    <span class="k">def</span> <span class="nf">set_vel</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">frame</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Sets the velocity Vector of this Point in a ReferenceFrame.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        value : Vector</span>
-<span class="sd">            The vector value of this point&#39;s velocity in the frame</span>
-<span class="sd">        frame : ReferenceFrame</span>
-<span class="sd">            The frame in which this point&#39;s velocity is defined</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import Point, ReferenceFrame</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; p1 = Point(&#39;p1&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; p1.set_vel(N, 10 * N.x)</span>
-<span class="sd">        &gt;&gt;&gt; p1.vel(N)</span>
-<span class="sd">        10*N.x</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">value</span> <span class="o">=</span> <span class="n">_check_vector</span><span class="p">(</span><span class="n">value</span><span class="p">)</span>
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">frame</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_vel_dict</span><span class="o">.</span><span class="n">update</span><span class="p">({</span><span class="n">frame</span><span class="p">:</span> <span class="n">value</span><span class="p">})</span>
-</div>
-<div class="viewcode-block" id="Point.v1pt_theory"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kinematics.html#sympy.physics.mechanics.point.Point.v1pt_theory">[docs]</a>    <span class="k">def</span> <span class="nf">v1pt_theory</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">otherpoint</span><span class="p">,</span> <span class="n">outframe</span><span class="p">,</span> <span class="n">interframe</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Sets the velocity of this point with the 1-point theory.</span>
-
-<span class="sd">        The 1-point theory for point velocity looks like this:</span>
-
-<span class="sd">        ^N v^P = ^B v^P + ^N v^O + ^N omega^B x r^OP</span>
-
-<span class="sd">        where O is a point fixed in B, P is a point moving in B, and B is</span>
-<span class="sd">        rotating in frame N.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        otherpoint : Point</span>
-<span class="sd">            The first point of the 2-point theory (O)</span>
-<span class="sd">        outframe : ReferenceFrame</span>
-<span class="sd">            The frame we want this point&#39;s velocity defined in (N)</span>
-<span class="sd">        interframe : ReferenceFrame</span>
-<span class="sd">            The intermediate frame in this calculation (B)</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import Point, ReferenceFrame, dynamicsymbols</span>
-<span class="sd">        &gt;&gt;&gt; q = dynamicsymbols(&#39;q&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; q2 = dynamicsymbols(&#39;q2&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; qd = dynamicsymbols(&#39;q&#39;, 1)</span>
-<span class="sd">        &gt;&gt;&gt; q2d = dynamicsymbols(&#39;q2&#39;, 1)</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; B = ReferenceFrame(&#39;B&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; B.set_ang_vel(N, 5 * B.y)</span>
-<span class="sd">        &gt;&gt;&gt; O = Point(&#39;O&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; P = O.locatenew(&#39;P&#39;, q * B.x)</span>
-<span class="sd">        &gt;&gt;&gt; P.set_vel(B, qd * B.x + q2d * B.y)</span>
-<span class="sd">        &gt;&gt;&gt; O.set_vel(N, 0)</span>
-<span class="sd">        &gt;&gt;&gt; P.v1pt_theory(O, N, B)</span>
-<span class="sd">        q&#39;*B.x + q2&#39;*B.y - 5*q*B.z</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">outframe</span><span class="p">)</span>
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">interframe</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_check_point</span><span class="p">(</span><span class="n">otherpoint</span><span class="p">)</span>
-        <span class="n">dist</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">pos_from</span><span class="p">(</span><span class="n">otherpoint</span><span class="p">)</span>
-        <span class="n">v1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">interframe</span><span class="p">)</span>
-        <span class="n">v2</span> <span class="o">=</span> <span class="n">otherpoint</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">outframe</span><span class="p">)</span>
-        <span class="n">omega</span> <span class="o">=</span> <span class="n">interframe</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">outframe</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">outframe</span><span class="p">,</span> <span class="n">v1</span> <span class="o">+</span> <span class="n">v2</span> <span class="o">+</span> <span class="p">(</span><span class="n">omega</span> <span class="o">^</span> <span class="n">dist</span><span class="p">))</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">outframe</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Point.v2pt_theory"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kinematics.html#sympy.physics.mechanics.point.Point.v2pt_theory">[docs]</a>    <span class="k">def</span> <span class="nf">v2pt_theory</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">otherpoint</span><span class="p">,</span> <span class="n">outframe</span><span class="p">,</span> <span class="n">fixedframe</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Sets the velocity of this point with the 2-point theory.</span>
-
-<span class="sd">        The 2-point theory for point velocity looks like this:</span>
-
-<span class="sd">        ^N v^P = ^N v^O + ^N omega^B x r^OP</span>
-
-<span class="sd">        where O and P are both points fixed in frame B, which is rotating in</span>
-<span class="sd">        frame N.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        otherpoint : Point</span>
-<span class="sd">            The first point of the 2-point theory (O)</span>
-<span class="sd">        outframe : ReferenceFrame</span>
-<span class="sd">            The frame we want this point&#39;s velocity defined in (N)</span>
-<span class="sd">        fixedframe : ReferenceFrame</span>
-<span class="sd">            The frame in which both points are fixed (B)</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import Point, ReferenceFrame, dynamicsymbols</span>
-<span class="sd">        &gt;&gt;&gt; q = dynamicsymbols(&#39;q&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; qd = dynamicsymbols(&#39;q&#39;, 1)</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; B = N.orientnew(&#39;B&#39;, &#39;Axis&#39;, [q, N.z])</span>
-<span class="sd">        &gt;&gt;&gt; O = Point(&#39;O&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; P = O.locatenew(&#39;P&#39;, 10 * B.x)</span>
-<span class="sd">        &gt;&gt;&gt; O.set_vel(N, 5 * N.x)</span>
-<span class="sd">        &gt;&gt;&gt; P.v2pt_theory(O, N, B)</span>
-<span class="sd">        5*N.x + 10*q&#39;*B.y</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">outframe</span><span class="p">)</span>
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">fixedframe</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_check_point</span><span class="p">(</span><span class="n">otherpoint</span><span class="p">)</span>
-        <span class="n">dist</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">pos_from</span><span class="p">(</span><span class="n">otherpoint</span><span class="p">)</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="n">otherpoint</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">outframe</span><span class="p">)</span>
-        <span class="n">omega</span> <span class="o">=</span> <span class="n">fixedframe</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">outframe</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">outframe</span><span class="p">,</span> <span class="n">v</span> <span class="o">+</span> <span class="p">(</span><span class="n">omega</span> <span class="o">^</span> <span class="n">dist</span><span class="p">))</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">outframe</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Point.vel"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/kinematics.html#sympy.physics.mechanics.point.Point.vel">[docs]</a>    <span class="k">def</span> <span class="nf">vel</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">frame</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The velocity Vector of this Point in the ReferenceFrame.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        frame : ReferenceFrame</span>
-<span class="sd">            The frame in which the returned velocity vector will be defined in</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import Point, ReferenceFrame</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; p1 = Point(&#39;p1&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; p1.set_vel(N, 10 * N.x)</span>
-<span class="sd">        &gt;&gt;&gt; p1.vel(N)</span>
-<span class="sd">        10*N.x</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">_check_frame</span><span class="p">(</span><span class="n">frame</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">frame</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_vel_dict</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Velocity of point &#39;</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">name</span> <span class="o">+</span> <span class="s">&#39; has not been&#39;</span>
-                             <span class="s">&#39; defined in ReferenceFrame &#39;</span> <span class="o">+</span> <span class="n">frame</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_vel_dict</span><span class="p">[</span><span class="n">frame</span><span class="p">]</span></div></div>
-</pre></div>
-
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-  <h1>Source code for sympy.physics.mechanics.rigidbody</h1><div class="highlight"><pre>
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;RigidBody&#39;</span><span class="p">]</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.mechanics.point</span> <span class="kn">import</span> <span class="n">Point</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.mechanics.essential</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">Dyadic</span>
-
-
-<div class="viewcode-block" id="RigidBody"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.rigidbody.RigidBody">[docs]</a><span class="k">class</span> <span class="nc">RigidBody</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;An idealized rigid body.</span>
-
-<span class="sd">    This is essentially a container which holds the various components which</span>
-<span class="sd">    describe a rigid body: a name, mass, center of mass, reference frame, and</span>
-<span class="sd">    inertia.</span>
-
-<span class="sd">    All of these need to be supplied on creation, but can be changed</span>
-<span class="sd">    afterwards.</span>
-
-<span class="sd">    Attributes</span>
-<span class="sd">    ==========</span>
-<span class="sd">    name : string</span>
-<span class="sd">        The body&#39;s name.</span>
-<span class="sd">    masscenter : Point</span>
-<span class="sd">        The point which represents the center of mass of the rigid body.</span>
-<span class="sd">    frame : ReferenceFrame</span>
-<span class="sd">        The ReferenceFrame which the rigid body is fixed in.</span>
-<span class="sd">    mass : Sympifyable</span>
-<span class="sd">        The body&#39;s mass.</span>
-<span class="sd">    inertia : (Dyadic, Point)</span>
-<span class="sd">        The body&#39;s inertia about a point; stored in a tuple as shown above.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import ReferenceFrame, Point, RigidBody</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.mechanics import outer</span>
-<span class="sd">    &gt;&gt;&gt; m = Symbol(&#39;m&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; A = ReferenceFrame(&#39;A&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; P = Point(&#39;P&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; I = outer (A.x, A.x)</span>
-<span class="sd">    &gt;&gt;&gt; inertia_tuple = (I, P)</span>
-<span class="sd">    &gt;&gt;&gt; B = RigidBody(&#39;B&#39;, P, A, m, inertia_tuple)</span>
-<span class="sd">    &gt;&gt;&gt; # Or you could change them afterwards</span>
-<span class="sd">    &gt;&gt;&gt; m2 = Symbol(&#39;m2&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; B.mass = m2</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">masscenter</span><span class="p">,</span> <span class="n">frame</span><span class="p">,</span> <span class="n">mass</span><span class="p">,</span> <span class="n">inertia</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Supply a valid name.&#39;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_name</span> <span class="o">=</span> <span class="n">name</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">set_masscenter</span><span class="p">(</span><span class="n">masscenter</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">set_mass</span><span class="p">(</span><span class="n">mass</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">set_frame</span><span class="p">(</span><span class="n">frame</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">set_inertia</span><span class="p">(</span><span class="n">inertia</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_pe</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_name</span>
-
-    <span class="n">__repr__</span> <span class="o">=</span> <span class="n">__str__</span>
-
-    <span class="k">def</span> <span class="nf">get_frame</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_frame</span>
-
-    <span class="k">def</span> <span class="nf">set_frame</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">F</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">ReferenceFrame</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;RigdBody frame must be a ReferenceFrame object.&quot;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_frame</span> <span class="o">=</span> <span class="n">F</span>
-
-    <span class="n">frame</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="n">get_frame</span><span class="p">,</span> <span class="n">set_frame</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">get_masscenter</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_masscenter</span>
-
-    <span class="k">def</span> <span class="nf">set_masscenter</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">Point</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;RigidBody center of mass must be a Point object.&quot;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_masscenter</span> <span class="o">=</span> <span class="n">p</span>
-
-    <span class="n">masscenter</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="n">get_masscenter</span><span class="p">,</span> <span class="n">set_masscenter</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">get_mass</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mass</span>
-
-    <span class="k">def</span> <span class="nf">set_mass</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_mass</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-
-    <span class="n">mass</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="n">get_mass</span><span class="p">,</span> <span class="n">set_mass</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">get_inertia</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_inertia</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_inertia_point</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">set_inertia</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">I</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">I</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Dyadic</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;RigidBody inertia must be a Dyadic object.&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">I</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">Point</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;RigidBody inertia must be about a Point.&quot;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_inertia</span> <span class="o">=</span> <span class="n">I</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_inertia_point</span> <span class="o">=</span> <span class="n">I</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="n">inertia</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="n">get_inertia</span><span class="p">,</span> <span class="n">set_inertia</span><span class="p">)</span>
-
-<div class="viewcode-block" id="RigidBody.linear_momentum"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.rigidbody.RigidBody.linear_momentum">[docs]</a>    <span class="k">def</span> <span class="nf">linear_momentum</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">frame</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Linear momentum of the rigid body.</span>
-
-<span class="sd">        The linear momentum L, of a rigid body B, with respect to frame N is</span>
-<span class="sd">        given by</span>
-
-<span class="sd">        L = M * v*</span>
-
-<span class="sd">        where M is the mass of the rigid body and v* is the velocity of</span>
-<span class="sd">        the mass center of B in the frame, N.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        frame : ReferenceFrame</span>
-<span class="sd">            The frame in which linear momentum is desired.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import Point, ReferenceFrame, outer</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import RigidBody, dynamicsymbols</span>
-<span class="sd">        &gt;&gt;&gt; M, v = dynamicsymbols(&#39;M v&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; P = Point(&#39;P&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; P.set_vel(N, v * N.x)</span>
-<span class="sd">        &gt;&gt;&gt; I = outer (N.x, N.x)</span>
-<span class="sd">        &gt;&gt;&gt; Inertia_tuple = (I, P)</span>
-<span class="sd">        &gt;&gt;&gt; B = RigidBody(&#39;B&#39;, P, N, M, Inertia_tuple)</span>
-<span class="sd">        &gt;&gt;&gt; B.linear_momentum(N)</span>
-<span class="sd">        M*v*N.x</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">mass</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">masscenter</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">frame</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="RigidBody.angular_momentum"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.rigidbody.RigidBody.angular_momentum">[docs]</a>    <span class="k">def</span> <span class="nf">angular_momentum</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">point</span><span class="p">,</span> <span class="n">frame</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Angular momentum of the rigid body.</span>
-
-<span class="sd">        The angular momentum H, about some point O, of a rigid body B, in a</span>
-<span class="sd">        frame N is given by</span>
-
-<span class="sd">        H = I* . omega + r* x (M * v)</span>
-
-<span class="sd">        where I* is the central inertia dyadic of B, omega is the angular</span>
-<span class="sd">        velocity of body B in the frame, N, r* is the position vector from</span>
-<span class="sd">        point O to the mass center of B, and v is the velocity of point O in</span>
-<span class="sd">        the frame, N.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        point : Point</span>
-<span class="sd">            The point about which angular momentum is desired.</span>
-
-<span class="sd">        frame : ReferenceFrame</span>
-<span class="sd">            The frame in which angular momentum is desired.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import Point, ReferenceFrame, outer</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import RigidBody, dynamicsymbols</span>
-<span class="sd">        &gt;&gt;&gt; M, v, r, omega = dynamicsymbols(&#39;M v r omega&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; b = ReferenceFrame(&#39;b&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; b.set_ang_vel(N, omega * b.x)</span>
-<span class="sd">        &gt;&gt;&gt; P = Point(&#39;P&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; P.set_vel(N, 1 * N.x)</span>
-<span class="sd">        &gt;&gt;&gt; I = outer (b.x, b.x)</span>
-<span class="sd">        &gt;&gt;&gt; Inertia_tuple = (I, P)</span>
-<span class="sd">        &gt;&gt;&gt; B = RigidBody(&#39;B&#39;, P, b, M, Inertia_tuple)</span>
-<span class="sd">        &gt;&gt;&gt; B.angular_momentum(P, N)</span>
-<span class="sd">        omega*b.x</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">return</span> <span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">inertia</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&amp;</span> <span class="bp">self</span><span class="o">.</span><span class="n">frame</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">frame</span><span class="p">))</span> <span class="o">+</span>
-                <span class="p">(</span><span class="n">point</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">frame</span><span class="p">)</span> <span class="o">^</span> <span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">masscenter</span><span class="o">.</span><span class="n">pos_from</span><span class="p">(</span><span class="n">point</span><span class="p">))</span> <span class="o">*</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">mass</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="RigidBody.kinetic_energy"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.rigidbody.RigidBody.kinetic_energy">[docs]</a>    <span class="k">def</span> <span class="nf">kinetic_energy</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">frame</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Kinetic energy of the rigid body</span>
-
-<span class="sd">        The kinetic energy, T, of a rigid body, B, is given by</span>
-
-<span class="sd">        &#39;T = 1/2 (I omega^2 + m v^2)&#39;</span>
-
-<span class="sd">        where I and m are the central inertia dyadic and mass of rigid body B,</span>
-<span class="sd">        respectively, omega is the body&#39;s angular velocity and v is the</span>
-<span class="sd">        velocity of the body&#39;s mass center in the supplied ReferenceFrame.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        frame : ReferenceFrame</span>
-<span class="sd">            The RigidBody&#39;s angular velocity and the velocity of it&#39;s mass</span>
-<span class="sd">            center is typically defined with respect to an inertial frame but</span>
-<span class="sd">            any relevant frame in which the velocity is known can be supplied.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import Point, ReferenceFrame, outer</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import RigidBody</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; M, v, r, omega = symbols(&#39;M v r omega&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; N = ReferenceFrame(&#39;N&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; b = ReferenceFrame(&#39;b&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; b.set_ang_vel(N, omega * b.x)</span>
-<span class="sd">        &gt;&gt;&gt; P = Point(&#39;P&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; P.set_vel(N, v * N.x)</span>
-<span class="sd">        &gt;&gt;&gt; I = outer (b.x, b.x)</span>
-<span class="sd">        &gt;&gt;&gt; inertia_tuple = (I, P)</span>
-<span class="sd">        &gt;&gt;&gt; B = RigidBody(&#39;B&#39;, P, b, M, inertia_tuple)</span>
-<span class="sd">        &gt;&gt;&gt; B.kinetic_energy(N)</span>
-<span class="sd">        M*v**2/2 + omega**2/2</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">rotational_KE</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">frame</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">frame</span><span class="p">)</span> <span class="o">&amp;</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">inertia</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&amp;</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">frame</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">frame</span><span class="p">))</span> <span class="o">/</span> <span class="n">sympify</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-
-        <span class="n">translational_KE</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mass</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">masscenter</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">frame</span><span class="p">)</span> <span class="o">&amp;</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">masscenter</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">frame</span><span class="p">))</span> <span class="o">/</span> <span class="n">sympify</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">rotational_KE</span> <span class="o">+</span> <span class="n">translational_KE</span>
-</div>
-<div class="viewcode-block" id="RigidBody.set_potential_energy"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.rigidbody.RigidBody.set_potential_energy">[docs]</a>    <span class="k">def</span> <span class="nf">set_potential_energy</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">scalar</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Used to set the potential energy of this RigidBody.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        scalar: Sympifyable</span>
-<span class="sd">            The potential energy (a scalar) of the RigidBody.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import Particle, Point, outer</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import RigidBody, ReferenceFrame</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; b = ReferenceFrame(&#39;b&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; M, g, h = symbols(&#39;M g h&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; P = Point(&#39;P&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; I = outer (b.x, b.x)</span>
-<span class="sd">        &gt;&gt;&gt; Inertia_tuple = (I, P)</span>
-<span class="sd">        &gt;&gt;&gt; B = RigidBody(&#39;B&#39;, P, b, M, Inertia_tuple)</span>
-<span class="sd">        &gt;&gt;&gt; B.set_potential_energy(M * g * h)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_pe</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">scalar</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="RigidBody.potential_energy"><a class="viewcode-back" href="../../../../modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.rigidbody.RigidBody.potential_energy">[docs]</a>    <span class="k">def</span> <span class="nf">potential_energy</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The potential energy of the RigidBody.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.mechanics import RigidBody, Point, outer, ReferenceFrame</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; M, g, h = symbols(&#39;M g h&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; b = ReferenceFrame(&#39;b&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; P = Point(&#39;P&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; I = outer (b.x, b.x)</span>
-<span class="sd">        &gt;&gt;&gt; Inertia_tuple = (I, P)</span>
-<span class="sd">        &gt;&gt;&gt; B = RigidBody(&#39;B&#39;, P, b, M, Inertia_tuple)</span>
-<span class="sd">        &gt;&gt;&gt; B.set_potential_energy(M * g * h)</span>
-<span class="sd">        &gt;&gt;&gt; B.potential_energy</span>
-<span class="sd">        M*g*h</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_pe</span></div></div>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.physics.paulialgebra</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">This module implements Pauli algebra by subclassing Symbol. Only algebraic</span>
-<span class="sd">properties of Pauli matrices are used (we don&#39;t use the Matrix class).</span>
-
-<span class="sd">See the documentation to the class Pauli for examples.</span>
-
-<span class="sd">See also:</span>
-<span class="sd">    http://en.wikipedia.org/wiki/Pauli_matrices</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">I</span>
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;evaluate_pauli_product&#39;</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">delta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns 1 if i == j, else 0.</span>
-
-<span class="sd">    This is used in the multiplication of Pauli matrices.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.paulialgebra import delta</span>
-<span class="sd">    &gt;&gt;&gt; delta(1, 1)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; delta(2, 3)</span>
-<span class="sd">    0</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">1</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-
-
-<span class="k">def</span> <span class="nf">epsilon</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return 1 if i,j,k is equal to (1,2,3), (2,3,1), or (3,1,2);</span>
-<span class="sd">    -1 if i,j,k is equal to (1,3,2), (3,2,1), or (2,1,3);</span>
-<span class="sd">    else return 0.</span>
-
-<span class="sd">    This is used in the multiplication of Pauli matrices.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.paulialgebra import epsilon</span>
-<span class="sd">    &gt;&gt;&gt; epsilon(1, 2, 3)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; epsilon(1, 3, 2)</span>
-<span class="sd">    -1</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="ow">in</span> <span class="p">[(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)]:</span>
-        <span class="k">return</span> <span class="mi">1</span>
-    <span class="k">elif</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="ow">in</span> <span class="p">[(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)]:</span>
-        <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-
-
-<span class="k">class</span> <span class="nc">Pauli</span><span class="p">(</span><span class="n">Symbol</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The class representing algebraic properties of Pauli matrices</span>
-
-<span class="sd">    If the left multiplication of symbol or number with Pauli matrix is needed,</span>
-<span class="sd">    please use parentheses  to separate Pauli and symbolic multiplication</span>
-<span class="sd">    (for example: 2*I*(Pauli(3)*Pauli(2)))</span>
-
-<span class="sd">    Another variant is to use evaluate_pauli_product function to evaluate</span>
-<span class="sd">    the product of Pauli matrices and other symbols (with commutative</span>
-<span class="sd">    multiply rules)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    =======</span>
-<span class="sd">    evaluate_pauli_product</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.paulialgebra import Pauli</span>
-<span class="sd">    &gt;&gt;&gt; Pauli(1)</span>
-<span class="sd">    sigma1</span>
-<span class="sd">    &gt;&gt;&gt; Pauli(1)*Pauli(2)</span>
-<span class="sd">    I*sigma3</span>
-<span class="sd">    &gt;&gt;&gt; Pauli(1)*Pauli(1)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; Pauli(3)**4</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; Pauli(1)*Pauli(2)*Pauli(3)</span>
-<span class="sd">    I</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import I</span>
-<span class="sd">    &gt;&gt;&gt; I*(Pauli(2)*Pauli(3))</span>
-<span class="sd">    -sigma1</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.paulialgebra import evaluate_pauli_product</span>
-<span class="sd">    &gt;&gt;&gt; f = I*Pauli(2)*Pauli(3)</span>
-<span class="sd">    &gt;&gt;&gt; f</span>
-<span class="sd">    I*sigma2*sigma3</span>
-<span class="sd">    &gt;&gt;&gt; evaluate_pauli_product(f)</span>
-<span class="sd">    -sigma1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&quot;i&quot;</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]:</span>
-            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s">&quot;Invalid Pauli index&quot;</span><span class="p">)</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Symbol</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="s">&quot;sigma</span><span class="si">%d</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">i</span><span class="p">,</span> <span class="n">commutative</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">i</span> <span class="o">=</span> <span class="n">i</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">__getnewargs__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">i</span><span class="p">,)</span>
-
-    <span class="c"># FIXME don&#39;t work for -I*Pauli(2)*Pauli(3)</span>
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Pauli</span><span class="p">):</span>
-            <span class="n">j</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">i</span>
-            <span class="n">k</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">i</span>
-            <span class="k">return</span> <span class="n">delta</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> \
-                <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">epsilon</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">Pauli</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> \
-                <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">epsilon</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">Pauli</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> \
-                <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">epsilon</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span><span class="o">*</span><span class="n">Pauli</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Pauli</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">e</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Pauli</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span><span class="o">.</span><span class="n">__pow__</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">e</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="evaluate_pauli_product"><a class="viewcode-back" href="../../../modules/physics/paulialgebra.html#sympy.physics.paulialgebra.evaluate_pauli_product">[docs]</a><span class="k">def</span> <span class="nf">evaluate_pauli_product</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-    <span class="sd">&#39;&#39;&#39;Help function to evaluate Pauli matrices product</span>
-<span class="sd">    with symbolic objects</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    arg: symbolic expression that contains Paulimatrices</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.paulialgebra import Pauli, evaluate_pauli_product</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import I</span>
-<span class="sd">    &gt;&gt;&gt; evaluate_pauli_product(I*Pauli(1)*Pauli(2))</span>
-<span class="sd">    -sigma3</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x,y</span>
-<span class="sd">    &gt;&gt;&gt; evaluate_pauli_product(x**2*Pauli(2)*Pauli(1))</span>
-<span class="sd">    -I*x**2*sigma3</span>
-<span class="sd">    &#39;&#39;&#39;</span>
-    <span class="n">tmp</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">()</span>
-    <span class="n">sigma_product</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">com_product</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="k">for</span> <span class="n">el</span> <span class="ow">in</span> <span class="n">tmp</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">el</span><span class="p">,</span> <span class="n">Pauli</span><span class="p">):</span>
-            <span class="n">sigma_product</span> <span class="o">*=</span> <span class="n">el</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">com_product</span> <span class="o">*=</span> <span class="n">el</span>
-    <span class="k">return</span> <span class="p">(</span><span class="n">tmp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">sigma_product</span><span class="o">*</span><span class="n">com_product</span><span class="p">)</span></div>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/physics/qho_1d.html b/dev-py3k/_modules/sympy/physics/qho_1d.html
deleted file mode 100644
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-            
-  <h1>Source code for sympy.physics.qho_1d</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">Rational</span>
-<span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">hermite</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">factorial</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.constants</span> <span class="kn">import</span> <span class="n">hbar</span>
-
-
-<div class="viewcode-block" id="psi_n"><a class="viewcode-back" href="../../../modules/physics/qho_1d.html#sympy.physics.qho_1d.psi_n">[docs]</a><span class="k">def</span> <span class="nf">psi_n</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">omega</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the wavefunction psi_{n} for the One-dimensional harmonic oscillator.</span>
-
-<span class="sd">    ``n``</span>
-<span class="sd">        the &quot;nodal&quot; quantum number.  Corresponds to the number of nodes in the</span>
-<span class="sd">        wavefunction.  n &gt;= 0</span>
-<span class="sd">    ``x``</span>
-<span class="sd">        x coordinate</span>
-<span class="sd">    ``m``</span>
-<span class="sd">        mass of the particle</span>
-<span class="sd">    ``omega``</span>
-<span class="sd">        angular frequency of the oscillator</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.qho_1d import psi_n</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import var</span>
-<span class="sd">    &gt;&gt;&gt; var(&quot;x m omega&quot;)</span>
-<span class="sd">    (x, m, omega)</span>
-<span class="sd">    &gt;&gt;&gt; psi_n(0, x, m, omega)</span>
-<span class="sd">    (m*omega)**(1/4)*exp(-m*omega*x**2/(2*hbar))/(hbar**(1/4)*pi**(1/4))</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="c"># sympify arguments</span>
-    <span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">omega</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">omega</span><span class="p">]))</span>
-    <span class="n">nu</span> <span class="o">=</span> <span class="n">m</span> <span class="o">*</span> <span class="n">omega</span> <span class="o">/</span> <span class="n">hbar</span>
-    <span class="c"># normalization coefficient</span>
-    <span class="n">C</span> <span class="o">=</span> <span class="p">(</span><span class="n">nu</span><span class="o">/</span><span class="n">pi</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span> <span class="o">*</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">n</span><span class="o">*</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)))</span>
-
-    <span class="k">return</span> <span class="n">C</span> <span class="o">*</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">nu</span><span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">hermite</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">nu</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="E_n"><a class="viewcode-back" href="../../../modules/physics/qho_1d.html#sympy.physics.qho_1d.E_n">[docs]</a><span class="k">def</span> <span class="nf">E_n</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">omega</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the Energy of the One-dimensional harmonic oscillator</span>
-
-<span class="sd">    ``n``</span>
-<span class="sd">        the &quot;nodal&quot; quantum number</span>
-<span class="sd">    ``omega``</span>
-<span class="sd">        the harmonic oscillator angular frequency</span>
-
-<span class="sd">    The unit of the returned value matches the unit of hw, since the energy is</span>
-<span class="sd">    calculated as:</span>
-
-<span class="sd">        E_n = hbar * omega*(n + 1/2)</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.qho_1d import E_n</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import var</span>
-<span class="sd">    &gt;&gt;&gt; var(&quot;x omega&quot;)</span>
-<span class="sd">    (x, omega)</span>
-<span class="sd">    &gt;&gt;&gt; E_n(x, omega)</span>
-<span class="sd">    hbar*omega*(x + 1/2)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">hbar</span> <span class="o">*</span> <span class="n">omega</span><span class="o">*</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span></div>
-</pre></div>
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diff --git a/dev-py3k/_modules/sympy/physics/quantum/anticommutator.html b/dev-py3k/_modules/sympy/physics/quantum/anticommutator.html
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-  <h1>Source code for sympy.physics.quantum.anticommutator</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;The anti-commutator: ``{A,B} = A*B + B*A``.&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Integer</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.pretty.stringpict</span> <span class="kn">import</span> <span class="n">prettyForm</span>
-
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.operator</span> <span class="kn">import</span> <span class="n">Operator</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.dagger</span> <span class="kn">import</span> <span class="n">Dagger</span>
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="s">&#39;AntiCommutator&#39;</span>
-<span class="p">]</span>
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># Anti-commutator</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-
-<div class="viewcode-block" id="AntiCommutator"><a class="viewcode-back" href="../../../../modules/physics/quantum/anticommutator.html#sympy.physics.quantum.anticommutator.AntiCommutator">[docs]</a><span class="k">class</span> <span class="nc">AntiCommutator</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The standard anticommutator, in an unevaluated state.</span>
-
-<span class="sd">    Evaluating an anticommutator is defined [1]_ as: ``{A, B} = A*B + B*A``.</span>
-<span class="sd">    This class returns the anticommutator in an unevaluated form.  To evaluate</span>
-<span class="sd">    the anticommutator, use the ``.doit()`` method.</span>
-
-<span class="sd">    Cannonical ordering of an anticommutator is ``{A, B}`` for ``A &lt; B``. The</span>
-<span class="sd">    arguments of the anticommutator are put into canonical order using</span>
-<span class="sd">    ``__cmp__``. If ``B &lt; A``, then ``{A, B}`` is returned as ``{B, A}``.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    A : Expr</span>
-<span class="sd">        The first argument of the anticommutator {A,B}.</span>
-<span class="sd">    B : Expr</span>
-<span class="sd">        The second argument of the anticommutator {A,B}.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum import AntiCommutator</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum import Operator, Dagger</span>
-<span class="sd">    &gt;&gt;&gt; x, y = symbols(&#39;x,y&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; A = Operator(&#39;A&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; B = Operator(&#39;B&#39;)</span>
-
-<span class="sd">    Create an anticommutator and use ``doit()`` to multiply them out.</span>
-
-<span class="sd">    &gt;&gt;&gt; ac = AntiCommutator(A,B); ac</span>
-<span class="sd">    {A,B}</span>
-<span class="sd">    &gt;&gt;&gt; ac.doit()</span>
-<span class="sd">    A*B + B*A</span>
-
-<span class="sd">    The commutator orders it arguments in canonical order:</span>
-
-<span class="sd">    &gt;&gt;&gt; ac = AntiCommutator(B,A); ac</span>
-<span class="sd">    {A,B}</span>
-
-<span class="sd">    Commutative constants are factored out:</span>
-
-<span class="sd">    &gt;&gt;&gt; AntiCommutator(3*x*A,x*y*B)</span>
-<span class="sd">    3*x**2*y*{A,B}</span>
-
-<span class="sd">    Adjoint operations applied to the anticommutator are properly applied to</span>
-<span class="sd">    the arguments:</span>
-
-<span class="sd">    &gt;&gt;&gt; Dagger(AntiCommutator(A,B))</span>
-<span class="sd">    {Dagger(A),Dagger(B)}</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Commutator</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">):</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">r</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">r</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">a</span> <span class="ow">and</span> <span class="n">b</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="n">b</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">a</span><span class="o">**</span><span class="mi">2</span>
-        <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_commutative</span> <span class="ow">or</span> <span class="n">b</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">b</span>
-
-        <span class="c"># [xA,yB]  -&gt;  xy*[A,B]</span>
-        <span class="c"># from sympy.physics.qmul import QMul</span>
-        <span class="n">ca</span><span class="p">,</span> <span class="n">nca</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-        <span class="n">cb</span><span class="p">,</span> <span class="n">ncb</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-        <span class="n">c_part</span> <span class="o">=</span> <span class="n">ca</span> <span class="o">+</span> <span class="n">cb</span>
-        <span class="k">if</span> <span class="n">c_part</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">c_part</span><span class="p">),</span> <span class="n">cls</span><span class="p">(</span><span class="n">Mul</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">nca</span><span class="p">),</span> <span class="n">Mul</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">ncb</span><span class="p">)))</span>
-
-        <span class="c"># Canonical ordering of arguments</span>
-        <span class="c">#The Commutator [A,B] is on canonical form if A &lt; B.</span>
-        <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">compare</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-
-<div class="viewcode-block" id="AntiCommutator.doit"><a class="viewcode-back" href="../../../../modules/physics/quantum/anticommutator.html#sympy.physics.quantum.anticommutator.AntiCommutator.doit">[docs]</a>    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Evaluate anticommutator &quot;&quot;&quot;</span>
-        <span class="n">A</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">B</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">Operator</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">Operator</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">comm</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">_eval_anticommutator</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="n">comm</span> <span class="o">=</span> <span class="n">B</span><span class="o">.</span><span class="n">_eval_anticommutator</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-                <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                    <span class="n">comm</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">if</span> <span class="n">comm</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">comm</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">A</span><span class="o">*</span><span class="n">B</span> <span class="o">+</span> <span class="n">B</span><span class="o">*</span><span class="n">A</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_adjoint</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">AntiCommutator</span><span class="p">(</span><span class="n">Dagger</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">Dagger</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-
-    <span class="k">def</span> <span class="nf">_sympyrepr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sympystr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;{</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">((</span><span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;,&#39;</span><span class="p">))))</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">((</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="o">*</span><span class="n">args</span><span class="p">))))</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">parens</span><span class="p">(</span><span class="n">left</span><span class="o">=</span><span class="s">&#39;{&#39;</span><span class="p">,</span> <span class="n">right</span><span class="o">=</span><span class="s">&#39;}&#39;</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="se">\\</span><span class="s">left</span><span class="se">\\</span><span class="s">{</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="se">\\</span><span class="s">right</span><span class="se">\\</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="nb">tuple</span><span class="p">([</span>
-            <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span></div>
-</pre></div>
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-  <h1>Source code for sympy.physics.quantum.cartesian</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Operators and states for 1D cartesian position and momentum.</span>
-
-<span class="sd">TODO:</span>
-
-<span class="sd">* Add 3D classes to mappings in operatorset.py</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">DiracDelta</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">Interval</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">sqrt</span>
-
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.constants</span> <span class="kn">import</span> <span class="n">hbar</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.hilbert</span> <span class="kn">import</span> <span class="n">L2</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.operator</span> <span class="kn">import</span> <span class="n">DifferentialOperator</span><span class="p">,</span> <span class="n">HermitianOperator</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.state</span> <span class="kn">import</span> <span class="n">Ket</span><span class="p">,</span> <span class="n">Bra</span><span class="p">,</span> <span class="n">State</span>
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="s">&#39;XOp&#39;</span><span class="p">,</span>
-    <span class="s">&#39;YOp&#39;</span><span class="p">,</span>
-    <span class="s">&#39;ZOp&#39;</span><span class="p">,</span>
-    <span class="s">&#39;PxOp&#39;</span><span class="p">,</span>
-    <span class="s">&#39;X&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Y&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Z&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Px&#39;</span><span class="p">,</span>
-    <span class="s">&#39;XKet&#39;</span><span class="p">,</span>
-    <span class="s">&#39;XBra&#39;</span><span class="p">,</span>
-    <span class="s">&#39;PxKet&#39;</span><span class="p">,</span>
-    <span class="s">&#39;PxBra&#39;</span><span class="p">,</span>
-    <span class="s">&#39;PositionState3D&#39;</span><span class="p">,</span>
-    <span class="s">&#39;PositionKet3D&#39;</span><span class="p">,</span>
-    <span class="s">&#39;PositionBra3D&#39;</span>
-<span class="p">]</span>
-
-<span class="c">#-------------------------------------------------------------------------</span>
-<span class="c"># Position operators</span>
-<span class="c">#-------------------------------------------------------------------------</span>
-
-
-<div class="viewcode-block" id="XOp"><a class="viewcode-back" href="../../../../modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.XOp">[docs]</a><span class="k">class</span> <span class="nc">XOp</span><span class="p">(</span><span class="n">HermitianOperator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;1D cartesian position operator.&quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">default_args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="s">&quot;X&quot;</span><span class="p">,)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_hilbert_space</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">L2</span><span class="p">(</span><span class="n">Interval</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_PxOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">I</span><span class="o">*</span><span class="n">hbar</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_XKet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">ket</span><span class="o">.</span><span class="n">position</span><span class="o">*</span><span class="n">ket</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_PositionKet3D</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">ket</span><span class="o">.</span><span class="n">position_x</span><span class="o">*</span><span class="n">ket</span>
-
-    <span class="k">def</span> <span class="nf">_represent_PxKet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">index</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&quot;index&quot;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-
-        <span class="n">states</span> <span class="o">=</span> <span class="n">basis</span><span class="o">.</span><span class="n">_enumerate_state</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">start_index</span><span class="o">=</span><span class="n">index</span><span class="p">)</span>
-        <span class="n">coord1</span> <span class="o">=</span> <span class="n">states</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">momentum</span>
-        <span class="n">coord2</span> <span class="o">=</span> <span class="n">states</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">momentum</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">DifferentialOperator</span><span class="p">(</span><span class="n">coord1</span><span class="p">)</span>
-        <span class="n">delta</span> <span class="o">=</span> <span class="n">DiracDelta</span><span class="p">(</span><span class="n">coord1</span> <span class="o">-</span> <span class="n">coord2</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">I</span><span class="o">*</span><span class="n">hbar</span><span class="o">*</span><span class="p">(</span><span class="n">d</span><span class="o">*</span><span class="n">delta</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="YOp"><a class="viewcode-back" href="../../../../modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.YOp">[docs]</a><span class="k">class</span> <span class="nc">YOp</span><span class="p">(</span><span class="n">HermitianOperator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Y cartesian coordinate operator (for 2D or 3D systems) &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">default_args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="s">&quot;Y&quot;</span><span class="p">,)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_hilbert_space</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">L2</span><span class="p">(</span><span class="n">Interval</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_PositionKet3D</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">ket</span><span class="o">.</span><span class="n">position_y</span><span class="o">*</span><span class="n">ket</span>
-
-</div>
-<div class="viewcode-block" id="ZOp"><a class="viewcode-back" href="../../../../modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.ZOp">[docs]</a><span class="k">class</span> <span class="nc">ZOp</span><span class="p">(</span><span class="n">HermitianOperator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Z cartesian coordinate operator (for 3D systems) &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">default_args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="s">&quot;Z&quot;</span><span class="p">,)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_hilbert_space</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">L2</span><span class="p">(</span><span class="n">Interval</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_PositionKet3D</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">ket</span><span class="o">.</span><span class="n">position_z</span><span class="o">*</span><span class="n">ket</span>
-
-<span class="c">#-------------------------------------------------------------------------</span>
-<span class="c"># Momentum operators</span>
-<span class="c">#-------------------------------------------------------------------------</span>
-
-</div>
-<div class="viewcode-block" id="PxOp"><a class="viewcode-back" href="../../../../modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.PxOp">[docs]</a><span class="k">class</span> <span class="nc">PxOp</span><span class="p">(</span><span class="n">HermitianOperator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;1D cartesian momentum operator.&quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">default_args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="s">&quot;Px&quot;</span><span class="p">,)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_hilbert_space</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">L2</span><span class="p">(</span><span class="n">Interval</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_PxKet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">ket</span><span class="o">.</span><span class="n">momentum</span><span class="o">*</span><span class="n">ket</span>
-
-    <span class="k">def</span> <span class="nf">_represent_XKet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">index</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&quot;index&quot;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-
-        <span class="n">states</span> <span class="o">=</span> <span class="n">basis</span><span class="o">.</span><span class="n">_enumerate_state</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">start_index</span><span class="o">=</span><span class="n">index</span><span class="p">)</span>
-        <span class="n">coord1</span> <span class="o">=</span> <span class="n">states</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">position</span>
-        <span class="n">coord2</span> <span class="o">=</span> <span class="n">states</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">position</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">DifferentialOperator</span><span class="p">(</span><span class="n">coord1</span><span class="p">)</span>
-        <span class="n">delta</span> <span class="o">=</span> <span class="n">DiracDelta</span><span class="p">(</span><span class="n">coord1</span> <span class="o">-</span> <span class="n">coord2</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">hbar</span><span class="o">*</span><span class="p">(</span><span class="n">d</span><span class="o">*</span><span class="n">delta</span><span class="p">)</span>
-</div>
-<span class="n">X</span> <span class="o">=</span> <span class="n">XOp</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">)</span>
-<span class="n">Y</span> <span class="o">=</span> <span class="n">YOp</span><span class="p">(</span><span class="s">&#39;Y&#39;</span><span class="p">)</span>
-<span class="n">Z</span> <span class="o">=</span> <span class="n">ZOp</span><span class="p">(</span><span class="s">&#39;Z&#39;</span><span class="p">)</span>
-<span class="n">Px</span> <span class="o">=</span> <span class="n">PxOp</span><span class="p">(</span><span class="s">&#39;Px&#39;</span><span class="p">)</span>
-
-<span class="c">#-------------------------------------------------------------------------</span>
-<span class="c"># Position eigenstates</span>
-<span class="c">#-------------------------------------------------------------------------</span>
-
-
-<div class="viewcode-block" id="XKet"><a class="viewcode-back" href="../../../../modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.XKet">[docs]</a><span class="k">class</span> <span class="nc">XKet</span><span class="p">(</span><span class="n">Ket</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;1D cartesian position eigenket.&quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_operators_to_state</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">op</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">_lowercase_labels</span><span class="p">(</span><span class="n">op</span><span class="p">),</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_state_to_operators</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">op_class</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">op_class</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">op_class</span><span class="p">,</span>
-                                <span class="o">*</span><span class="n">_uppercase_labels</span><span class="p">(</span><span class="bp">self</span><span class="p">),</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">default_args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">XBra</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="XKet.position"><a class="viewcode-back" href="../../../../modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.XKet.position">[docs]</a>    <span class="k">def</span> <span class="nf">position</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The position of the state.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="k">def</span> <span class="nf">_enumerate_state</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">num_states</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_enumerate_continuous_1D</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">num_states</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_innerproduct_XBra</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">bra</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">DiracDelta</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">position</span> <span class="o">-</span> <span class="n">bra</span><span class="o">.</span><span class="n">position</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_innerproduct_PxBra</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">bra</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">position</span><span class="o">*</span><span class="n">bra</span><span class="o">.</span><span class="n">momentum</span><span class="o">/</span><span class="n">hbar</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">hbar</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="XBra"><a class="viewcode-back" href="../../../../modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.XBra">[docs]</a><span class="k">class</span> <span class="nc">XBra</span><span class="p">(</span><span class="n">Bra</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;1D cartesian position eigenbra.&quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">default_args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">XKet</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="XBra.position"><a class="viewcode-back" href="../../../../modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.XBra.position">[docs]</a>    <span class="k">def</span> <span class="nf">position</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The position of the state.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-</div></div>
-<div class="viewcode-block" id="PositionState3D"><a class="viewcode-back" href="../../../../modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.PositionState3D">[docs]</a><span class="k">class</span> <span class="nc">PositionState3D</span><span class="p">(</span><span class="n">State</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Base class for 3D cartesian position eigenstates &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_operators_to_state</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">op</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">_lowercase_labels</span><span class="p">(</span><span class="n">op</span><span class="p">),</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_state_to_operators</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">op_class</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">op_class</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">op_class</span><span class="p">,</span>
-                                <span class="o">*</span><span class="n">_uppercase_labels</span><span class="p">(</span><span class="bp">self</span><span class="p">),</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">default_args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="s">&quot;y&quot;</span><span class="p">,</span> <span class="s">&quot;z&quot;</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="PositionState3D.position_x"><a class="viewcode-back" href="../../../../modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.PositionState3D.position_x">[docs]</a>    <span class="k">def</span> <span class="nf">position_x</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; The x coordinate of the state &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="PositionState3D.position_y"><a class="viewcode-back" href="../../../../modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.PositionState3D.position_y">[docs]</a>    <span class="k">def</span> <span class="nf">position_y</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; The y coordinate of the state &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="PositionState3D.position_z"><a class="viewcode-back" href="../../../../modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.PositionState3D.position_z">[docs]</a>    <span class="k">def</span> <span class="nf">position_z</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; The z coordinate of the state &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-
-</div></div>
-<div class="viewcode-block" id="PositionKet3D"><a class="viewcode-back" href="../../../../modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.PositionKet3D">[docs]</a><span class="k">class</span> <span class="nc">PositionKet3D</span><span class="p">(</span><span class="n">Ket</span><span class="p">,</span> <span class="n">PositionState3D</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; 3D cartesian position eigenket &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_eval_innerproduct_PositionBra3D</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">bra</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">x_diff</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">position_x</span> <span class="o">-</span> <span class="n">bra</span><span class="o">.</span><span class="n">position_x</span>
-        <span class="n">y_diff</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">position_y</span> <span class="o">-</span> <span class="n">bra</span><span class="o">.</span><span class="n">position_y</span>
-        <span class="n">z_diff</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">position_z</span> <span class="o">-</span> <span class="n">bra</span><span class="o">.</span><span class="n">position_z</span>
-
-        <span class="k">return</span> <span class="n">DiracDelta</span><span class="p">(</span><span class="n">x_diff</span><span class="p">)</span><span class="o">*</span><span class="n">DiracDelta</span><span class="p">(</span><span class="n">y_diff</span><span class="p">)</span><span class="o">*</span><span class="n">DiracDelta</span><span class="p">(</span><span class="n">z_diff</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">PositionBra3D</span>
-
-</div>
-<div class="viewcode-block" id="PositionBra3D"><a class="viewcode-back" href="../../../../modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.PositionBra3D">[docs]</a><span class="k">class</span> <span class="nc">PositionBra3D</span><span class="p">(</span><span class="n">Bra</span><span class="p">,</span> <span class="n">PositionState3D</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; 3D cartesian position eigenbra &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">PositionKet3D</span>
-
-<span class="c">#-------------------------------------------------------------------------</span>
-<span class="c"># Momentum eigenstates</span>
-<span class="c">#-------------------------------------------------------------------------</span>
-
-</div>
-<div class="viewcode-block" id="PxKet"><a class="viewcode-back" href="../../../../modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.PxKet">[docs]</a><span class="k">class</span> <span class="nc">PxKet</span><span class="p">(</span><span class="n">Ket</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;1D cartesian momentum eigenket.&quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_operators_to_state</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">op</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">_lowercase_labels</span><span class="p">(</span><span class="n">op</span><span class="p">),</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_state_to_operators</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">op_class</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">op_class</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">op_class</span><span class="p">,</span>
-                                <span class="o">*</span><span class="n">_uppercase_labels</span><span class="p">(</span><span class="bp">self</span><span class="p">),</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">default_args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="s">&quot;px&quot;</span><span class="p">,)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">PxBra</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="PxKet.momentum"><a class="viewcode-back" href="../../../../modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.PxKet.momentum">[docs]</a>    <span class="k">def</span> <span class="nf">momentum</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The momentum of the state.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="k">def</span> <span class="nf">_enumerate_state</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_enumerate_continuous_1D</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_innerproduct_XBra</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">bra</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">momentum</span><span class="o">*</span><span class="n">bra</span><span class="o">.</span><span class="n">position</span><span class="o">/</span><span class="n">hbar</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">hbar</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_innerproduct_PxBra</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">bra</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">DiracDelta</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">momentum</span> <span class="o">-</span> <span class="n">bra</span><span class="o">.</span><span class="n">momentum</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="PxBra"><a class="viewcode-back" href="../../../../modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.PxBra">[docs]</a><span class="k">class</span> <span class="nc">PxBra</span><span class="p">(</span><span class="n">Bra</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;1D cartesian momentum eigenbra.&quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">default_args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="s">&quot;px&quot;</span><span class="p">,)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">PxKet</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="PxBra.momentum"><a class="viewcode-back" href="../../../../modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.PxBra.momentum">[docs]</a>    <span class="k">def</span> <span class="nf">momentum</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The momentum of the state.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-<span class="c">#-------------------------------------------------------------------------</span>
-<span class="c"># Global helper functions</span>
-<span class="c">#-------------------------------------------------------------------------</span>
-
-</div></div>
-<span class="k">def</span> <span class="nf">_enumerate_continuous_1D</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-    <span class="n">state</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">num_states</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-    <span class="n">state_class</span> <span class="o">=</span> <span class="n">state</span><span class="o">.</span><span class="n">__class__</span>
-    <span class="n">index_list</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;index_list&#39;</span><span class="p">,</span> <span class="p">[])</span>
-
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">index_list</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">start_index</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;start_index&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">index_list</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">start_index</span><span class="p">,</span> <span class="n">start_index</span> <span class="o">+</span> <span class="n">num_states</span><span class="p">))</span>
-
-    <span class="n">enum_states</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">index_list</span><span class="p">))]</span>
-
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">ind</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">index_list</span><span class="p">):</span>
-        <span class="n">label</span> <span class="o">=</span> <span class="n">state</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">enum_states</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">state_class</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">label</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot;_&quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">ind</span><span class="p">),</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">enum_states</span>
-
-
-<span class="k">def</span> <span class="nf">_lowercase_labels</span><span class="p">(</span><span class="n">ops</span><span class="p">):</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ops</span><span class="p">,</span> <span class="nb">set</span><span class="p">):</span>
-        <span class="n">ops</span> <span class="o">=</span> <span class="p">[</span><span class="n">ops</span><span class="p">]</span>
-
-    <span class="k">return</span> <span class="p">[</span><span class="nb">str</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">ops</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">_uppercase_labels</span><span class="p">(</span><span class="n">ops</span><span class="p">):</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ops</span><span class="p">,</span> <span class="nb">set</span><span class="p">):</span>
-        <span class="n">ops</span> <span class="o">=</span> <span class="p">[</span><span class="n">ops</span><span class="p">]</span>
-
-    <span class="n">new_args</span> <span class="o">=</span> <span class="p">[</span><span class="nb">str</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">])[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">upper</span><span class="p">()</span> <span class="o">+</span>
-                <span class="nb">str</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">])[</span><span class="mi">1</span><span class="p">:]</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">ops</span><span class="p">]</span>
-
-    <span class="k">return</span> <span class="n">new_args</span>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.physics.quantum.cg</h1><div class="highlight"><pre>
-<span class="c">#TODO:</span>
-<span class="c"># -Implement Clebsch-Gordan symmetries</span>
-<span class="c"># -Improve simplification method</span>
-<span class="c"># -Implement new simpifications</span>
-<span class="sd">&quot;&quot;&quot;Clebsch-Gordon Coefficients.&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="p">(</span><span class="n">Add</span><span class="p">,</span> <span class="n">expand</span><span class="p">,</span> <span class="n">Eq</span><span class="p">,</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Piecewise</span><span class="p">,</span> <span class="n">Pow</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">Sum</span><span class="p">,</span>
-                   <span class="n">symbols</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">Wild</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.pretty.stringpict</span> <span class="kn">import</span> <span class="n">prettyForm</span><span class="p">,</span> <span class="n">stringPict</span>
-
-<span class="kn">from</span> <span class="nn">sympy.functions.special.tensor_functions</span> <span class="kn">import</span> <span class="n">KroneckerDelta</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.wigner</span> <span class="kn">import</span> <span class="n">clebsch_gordan</span><span class="p">,</span> <span class="n">wigner_3j</span><span class="p">,</span> <span class="n">wigner_6j</span><span class="p">,</span> <span class="n">wigner_9j</span>
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="s">&#39;CG&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Wigner3j&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Wigner6j&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Wigner9j&#39;</span><span class="p">,</span>
-    <span class="s">&#39;cg_simp&#39;</span>
-<span class="p">]</span>
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># CG Coefficients</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-
-<div class="viewcode-block" id="Wigner3j"><a class="viewcode-back" href="../../../../modules/physics/quantum/cg.html#sympy.physics.quantum.cg.Wigner3j">[docs]</a><span class="k">class</span> <span class="nc">Wigner3j</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Class for the Wigner-3j symbols</span>
-
-<span class="sd">    Wigner 3j-symbols are coefficients determined by the coupling of</span>
-<span class="sd">    two angular momenta. When created, they are expressed as symbolic</span>
-<span class="sd">    quantities that, for numerical parameters, can be evaluated using the</span>
-<span class="sd">    ``.doit()`` method [1]_.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    j1, m1, j2, m2, j3, m3 : Number, Symbol</span>
-<span class="sd">        Terms determining the angular momentum of coupled angular momentum</span>
-<span class="sd">        systems.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Declare a Wigner-3j coefficient and calcualte its value</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.cg import Wigner3j</span>
-<span class="sd">        &gt;&gt;&gt; w3j = Wigner3j(6,0,4,0,2,0)</span>
-<span class="sd">        &gt;&gt;&gt; w3j</span>
-<span class="sd">        Wigner3j(6, 0, 4, 0, 2, 0)</span>
-<span class="sd">        &gt;&gt;&gt; w3j.doit()</span>
-<span class="sd">        sqrt(715)/143</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    CG: Clebsch-Gordan coefficients</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">j1</span><span class="p">,</span> <span class="n">m1</span><span class="p">,</span> <span class="n">j2</span><span class="p">,</span> <span class="n">m2</span><span class="p">,</span> <span class="n">j3</span><span class="p">,</span> <span class="n">m3</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">j1</span><span class="p">,</span> <span class="n">m1</span><span class="p">,</span> <span class="n">j2</span><span class="p">,</span> <span class="n">m2</span><span class="p">,</span> <span class="n">j3</span><span class="p">,</span> <span class="n">m3</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">j1</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">m1</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">j2</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">m2</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">j3</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">4</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">m3</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_symbolic</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">([</span><span class="n">arg</span><span class="o">.</span><span class="n">is_number</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="c"># This is modified from the _print_Matrix method</span>
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="p">((</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j1</span><span class="p">),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">m1</span><span class="p">)),</span>
-            <span class="p">(</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j2</span><span class="p">),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">m2</span><span class="p">)),</span>
-            <span class="p">(</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j3</span><span class="p">),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">m3</span><span class="p">)))</span>
-        <span class="n">hsep</span> <span class="o">=</span> <span class="mi">2</span>
-        <span class="n">vsep</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">maxw</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="mi">3</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">):</span>
-            <span class="n">maxw</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">([</span> <span class="n">m</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">width</span><span class="p">()</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="p">])</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">):</span>
-            <span class="n">D_row</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">):</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">m</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span>
-                <span class="n">wdelta</span> <span class="o">=</span> <span class="n">maxw</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">s</span><span class="o">.</span><span class="n">width</span><span class="p">()</span>
-                <span class="n">wleft</span> <span class="o">=</span> <span class="n">wdelta</span> <span class="o">//</span><span class="mi">2</span>
-                <span class="n">wright</span> <span class="o">=</span> <span class="n">wdelta</span> <span class="o">-</span> <span class="n">wleft</span>
-
-                <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">wright</span><span class="p">))</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">wleft</span><span class="p">))</span>
-
-                <span class="k">if</span> <span class="n">D_row</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">D_row</span> <span class="o">=</span> <span class="n">s</span>
-                    <span class="k">continue</span>
-                <span class="n">D_row</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D_row</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">hsep</span><span class="p">))</span>
-                <span class="n">D_row</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D_row</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">s</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">D</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">D</span> <span class="o">=</span> <span class="n">D_row</span>
-                <span class="k">continue</span>
-            <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">vsep</span><span class="p">):</span>
-                <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">))</span>
-            <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="n">D_row</span><span class="p">))</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-        <span class="k">return</span> <span class="n">D</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">label</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j2</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j3</span><span class="p">,</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">m1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">m2</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">m3</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="s">r&#39;\left(\begin{array}{ccc} </span><span class="si">%s</span><span class="s"> &amp; </span><span class="si">%s</span><span class="s"> &amp; </span><span class="si">%s</span><span class="s"> </span><span class="se">\\</span><span class="s"> </span><span class="si">%s</span><span class="s"> &amp; </span><span class="si">%s</span><span class="s"> &amp; </span><span class="si">%s</span><span class="s"> \end{array}\right)&#39;</span> <span class="o">%</span> \
-            <span class="nb">tuple</span><span class="p">(</span><span class="n">label</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_symbolic</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Coefficients must be numerical&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">wigner_3j</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j2</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j3</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">m1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">m2</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">m3</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="CG"><a class="viewcode-back" href="../../../../modules/physics/quantum/cg.html#sympy.physics.quantum.cg.CG">[docs]</a><span class="k">class</span> <span class="nc">CG</span><span class="p">(</span><span class="n">Wigner3j</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Class for Clebsch-Gordan coefficient</span>
-
-<span class="sd">    Clebsch-Gordan coefficients describe the angular momentum coupling between</span>
-<span class="sd">    two systems. The coefficients give the expansion of a coupled total angular</span>
-<span class="sd">    momentum state and an uncoupled tensor product state. The Clebsch-Gordan</span>
-<span class="sd">    coefficients are defined as [1]_:</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        C^{j_1,m_1}_{j_2,m_2,j_3,m_3} = \langle j_1,m_1;j_2,m_2 | j_3,m_3\\rangle</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    j1, m1, j2, m2, j3, m3 : Number, Symbol</span>
-<span class="sd">        Terms determining the angular momentum of coupled angular momentum</span>
-<span class="sd">        systems.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Define a Clebsch-Gordan coefficient and evaluate its value</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.cg import CG</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import S</span>
-<span class="sd">        &gt;&gt;&gt; cg = CG(S(3)/2, S(3)/2, S(1)/2, -S(1)/2, 1, 1)</span>
-<span class="sd">        &gt;&gt;&gt; cg</span>
-<span class="sd">        CG(3/2, 3/2, 1/2, -1/2, 1, 1)</span>
-<span class="sd">        &gt;&gt;&gt; cg.doit()</span>
-<span class="sd">        sqrt(3)/2</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Wigner3j: Wigner-3j symbols</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_symbolic</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Coefficients must be numerical&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">clebsch_gordan</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j2</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j3</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">m1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">m2</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">m3</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">bot</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span>
-            <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">m1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j2</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">m2</span><span class="p">),</span> <span class="n">delimiter</span><span class="o">=</span><span class="s">&#39;,&#39;</span><span class="p">)</span>
-        <span class="n">top</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">j3</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">m3</span><span class="p">),</span> <span class="n">delimiter</span><span class="o">=</span><span class="s">&#39;,&#39;</span><span class="p">)</span>
-
-        <span class="n">pad</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">top</span><span class="o">.</span><span class="n">width</span><span class="p">(),</span> <span class="n">bot</span><span class="o">.</span><span class="n">width</span><span class="p">())</span>
-        <span class="n">bot</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">bot</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">))</span>
-        <span class="n">top</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">top</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">pad</span> <span class="o">==</span> <span class="n">bot</span><span class="o">.</span><span class="n">width</span><span class="p">():</span>
-            <span class="n">bot</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">bot</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span> <span class="o">*</span> <span class="p">(</span><span class="n">pad</span> <span class="o">-</span> <span class="n">bot</span><span class="o">.</span><span class="n">width</span><span class="p">())))</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">pad</span> <span class="o">==</span> <span class="n">top</span><span class="o">.</span><span class="n">width</span><span class="p">():</span>
-            <span class="n">top</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">top</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span> <span class="o">*</span> <span class="p">(</span><span class="n">pad</span> <span class="o">-</span> <span class="n">top</span><span class="o">.</span><span class="n">width</span><span class="p">())))</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="s">&#39;C&#39;</span> <span class="o">+</span> <span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">pad</span><span class="p">)</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="n">bot</span><span class="p">))</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">above</span><span class="p">(</span><span class="n">top</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">s</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">label</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j3</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">m3</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j1</span><span class="p">,</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">m1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j2</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">m2</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="s">r&#39;C^{</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">}_{</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">}&#39;</span> <span class="o">%</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">label</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="Wigner6j"><a class="viewcode-back" href="../../../../modules/physics/quantum/cg.html#sympy.physics.quantum.cg.Wigner6j">[docs]</a><span class="k">class</span> <span class="nc">Wigner6j</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Class for the Wigner-6j symbols</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Wigner3j: Wigner-3j symbols</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">j1</span><span class="p">,</span> <span class="n">j2</span><span class="p">,</span> <span class="n">j12</span><span class="p">,</span> <span class="n">j3</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">j23</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">j1</span><span class="p">,</span> <span class="n">j2</span><span class="p">,</span> <span class="n">j12</span><span class="p">,</span> <span class="n">j3</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">j23</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">j1</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">j2</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">j12</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">j3</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">j</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">4</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">j23</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_symbolic</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">([</span><span class="n">arg</span><span class="o">.</span><span class="n">is_number</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="c"># This is modified from the _print_Matrix method</span>
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="p">((</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j1</span><span class="p">),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j3</span><span class="p">)),</span>
-            <span class="p">(</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j2</span><span class="p">),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">)),</span>
-            <span class="p">(</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j12</span><span class="p">),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j23</span><span class="p">)))</span>
-        <span class="n">hsep</span> <span class="o">=</span> <span class="mi">2</span>
-        <span class="n">vsep</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">maxw</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="mi">3</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">):</span>
-            <span class="n">maxw</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">([</span> <span class="n">m</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">width</span><span class="p">()</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="p">])</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">):</span>
-            <span class="n">D_row</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">):</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">m</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span>
-                <span class="n">wdelta</span> <span class="o">=</span> <span class="n">maxw</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">s</span><span class="o">.</span><span class="n">width</span><span class="p">()</span>
-                <span class="n">wleft</span> <span class="o">=</span> <span class="n">wdelta</span> <span class="o">//</span><span class="mi">2</span>
-                <span class="n">wright</span> <span class="o">=</span> <span class="n">wdelta</span> <span class="o">-</span> <span class="n">wleft</span>
-
-                <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">wright</span><span class="p">))</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">wleft</span><span class="p">))</span>
-
-                <span class="k">if</span> <span class="n">D_row</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">D_row</span> <span class="o">=</span> <span class="n">s</span>
-                    <span class="k">continue</span>
-                <span class="n">D_row</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D_row</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">hsep</span><span class="p">))</span>
-                <span class="n">D_row</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D_row</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">s</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">D</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">D</span> <span class="o">=</span> <span class="n">D_row</span>
-                <span class="k">continue</span>
-            <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">vsep</span><span class="p">):</span>
-                <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">))</span>
-            <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="n">D_row</span><span class="p">))</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">parens</span><span class="p">(</span><span class="n">left</span><span class="o">=</span><span class="s">&#39;{&#39;</span><span class="p">,</span> <span class="n">right</span><span class="o">=</span><span class="s">&#39;}&#39;</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">D</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">label</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j2</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j12</span><span class="p">,</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">j3</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j23</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="s">r&#39;\left\{\begin{array}{ccc} </span><span class="si">%s</span><span class="s"> &amp; </span><span class="si">%s</span><span class="s"> &amp; </span><span class="si">%s</span><span class="s"> </span><span class="se">\\</span><span class="s"> </span><span class="si">%s</span><span class="s"> &amp; </span><span class="si">%s</span><span class="s"> &amp; </span><span class="si">%s</span><span class="s"> \end{array}\right\}&#39;</span> <span class="o">%</span> \
-            <span class="nb">tuple</span><span class="p">(</span><span class="n">label</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_symbolic</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Coefficients must be numerical&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">wigner_6j</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j2</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j12</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j3</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j3</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="Wigner9j"><a class="viewcode-back" href="../../../../modules/physics/quantum/cg.html#sympy.physics.quantum.cg.Wigner9j">[docs]</a><span class="k">class</span> <span class="nc">Wigner9j</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Class for the Wigner-9j symbols</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Wigner3j: Wigner-3j symbols</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">j1</span><span class="p">,</span> <span class="n">j2</span><span class="p">,</span> <span class="n">j12</span><span class="p">,</span> <span class="n">j3</span><span class="p">,</span> <span class="n">j4</span><span class="p">,</span> <span class="n">j34</span><span class="p">,</span> <span class="n">j13</span><span class="p">,</span> <span class="n">j24</span><span class="p">,</span> <span class="n">j</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">j1</span><span class="p">,</span> <span class="n">j2</span><span class="p">,</span> <span class="n">j12</span><span class="p">,</span> <span class="n">j3</span><span class="p">,</span> <span class="n">j4</span><span class="p">,</span> <span class="n">j34</span><span class="p">,</span> <span class="n">j13</span><span class="p">,</span> <span class="n">j24</span><span class="p">,</span> <span class="n">j</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">j1</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">j2</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">j12</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">j3</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">j4</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">4</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">j34</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">j13</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">6</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">j24</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">7</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">j</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">8</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_symbolic</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">([</span><span class="n">arg</span><span class="o">.</span><span class="n">is_number</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="c"># This is modified from the _print_Matrix method</span>
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="p">(</span>
-            <span class="p">(</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">j1</span><span class="p">),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j3</span><span class="p">),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j13</span><span class="p">)),</span>
-            <span class="p">(</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">j2</span><span class="p">),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j4</span><span class="p">),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j24</span><span class="p">)),</span>
-            <span class="p">(</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j12</span><span class="p">),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j34</span><span class="p">),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">)))</span>
-        <span class="n">hsep</span> <span class="o">=</span> <span class="mi">2</span>
-        <span class="n">vsep</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">maxw</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="mi">3</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">):</span>
-            <span class="n">maxw</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">([</span> <span class="n">m</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">width</span><span class="p">()</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="p">])</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">):</span>
-            <span class="n">D_row</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">):</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">m</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span>
-                <span class="n">wdelta</span> <span class="o">=</span> <span class="n">maxw</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">s</span><span class="o">.</span><span class="n">width</span><span class="p">()</span>
-                <span class="n">wleft</span> <span class="o">=</span> <span class="n">wdelta</span> <span class="o">//</span><span class="mi">2</span>
-                <span class="n">wright</span> <span class="o">=</span> <span class="n">wdelta</span> <span class="o">-</span> <span class="n">wleft</span>
-
-                <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">wright</span><span class="p">))</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">wleft</span><span class="p">))</span>
-
-                <span class="k">if</span> <span class="n">D_row</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">D_row</span> <span class="o">=</span> <span class="n">s</span>
-                    <span class="k">continue</span>
-                <span class="n">D_row</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D_row</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">hsep</span><span class="p">))</span>
-                <span class="n">D_row</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D_row</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">s</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">D</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">D</span> <span class="o">=</span> <span class="n">D_row</span>
-                <span class="k">continue</span>
-            <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">vsep</span><span class="p">):</span>
-                <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">))</span>
-            <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="n">D_row</span><span class="p">))</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">parens</span><span class="p">(</span><span class="n">left</span><span class="o">=</span><span class="s">&#39;{&#39;</span><span class="p">,</span> <span class="n">right</span><span class="o">=</span><span class="s">&#39;}&#39;</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">D</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">label</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j2</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j12</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j3</span><span class="p">,</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">j4</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j34</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j13</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j24</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="s">r&#39;\left\{\begin{array}{ccc} </span><span class="si">%s</span><span class="s"> &amp; </span><span class="si">%s</span><span class="s"> &amp; </span><span class="si">%s</span><span class="s"> </span><span class="se">\\</span><span class="s"> </span><span class="si">%s</span><span class="s"> &amp; </span><span class="si">%s</span><span class="s"> &amp; </span><span class="si">%s</span><span class="s"> </span><span class="se">\\</span><span class="s"> </span><span class="si">%s</span><span class="s"> &amp; </span><span class="si">%s</span><span class="s"> &amp; </span><span class="si">%s</span><span class="s"> \end{array}\right\}&#39;</span> <span class="o">%</span> \
-            <span class="nb">tuple</span><span class="p">(</span><span class="n">label</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_symbolic</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Coefficients must be numerical&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">wigner_9j</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j2</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j12</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j3</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j4</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j34</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j13</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j24</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="cg_simp"><a class="viewcode-back" href="../../../../modules/physics/quantum/cg.html#sympy.physics.quantum.cg.cg_simp">[docs]</a><span class="k">def</span> <span class="nf">cg_simp</span><span class="p">(</span><span class="n">e</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Simplify and combine CG coefficients</span>
-
-<span class="sd">    This function uses various symmetry and properties of sums and</span>
-<span class="sd">    products of Clebsch-Gordan coefficients to simplify statements</span>
-<span class="sd">    involving these terms [1]_.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Simplify the sum over CG(a,alpha,0,0,a,alpha) for all alpha to</span>
-<span class="sd">    2*a+1</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.cg import CG, cg_simp</span>
-<span class="sd">        &gt;&gt;&gt; a = CG(1,1,0,0,1,1)</span>
-<span class="sd">        &gt;&gt;&gt; b = CG(1,0,0,0,1,0)</span>
-<span class="sd">        &gt;&gt;&gt; c = CG(1,-1,0,0,1,-1)</span>
-<span class="sd">        &gt;&gt;&gt; cg_simp(a+b+c)</span>
-<span class="sd">        3</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    CG: Clebsh-Gordan coefficients</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Add</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_cg_simp_add</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Sum</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_cg_simp_sum</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Mul</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">cg_simp</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Pow</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Pow</span><span class="p">(</span><span class="n">cg_simp</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">e</span><span class="o">.</span><span class="n">exp</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">e</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_cg_simp_add</span><span class="p">(</span><span class="n">e</span><span class="p">):</span>
-    <span class="c">#TODO: Improve simplification method</span>
-    <span class="sd">&quot;&quot;&quot;Takes a sum of terms involving Clebsch-Gordan coefficients and</span>
-<span class="sd">    simplifies the terms.</span>
-
-<span class="sd">    First, we create two lists, cg_part, which is all the terms involving CG</span>
-<span class="sd">    coefficients, and other_part, which is all other terms. The cg_part list</span>
-<span class="sd">    is then passed to the simplification methods, which return the new cg_part</span>
-<span class="sd">    and any additional terms that are added to other_part</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">cg_part</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">other_part</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="n">e</span> <span class="o">=</span> <span class="n">expand</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">CG</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Sum</span><span class="p">):</span>
-                <span class="n">other_part</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">_cg_simp_sum</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Mul</span><span class="p">):</span>
-                <span class="n">terms</span> <span class="o">=</span> <span class="mi">1</span>
-                <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">Sum</span><span class="p">):</span>
-                        <span class="n">terms</span> <span class="o">*=</span> <span class="n">_cg_simp_sum</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">terms</span> <span class="o">*=</span> <span class="n">term</span>
-                <span class="k">if</span> <span class="n">terms</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">CG</span><span class="p">):</span>
-                    <span class="n">cg_part</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">terms</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">other_part</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">terms</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">cg_part</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">other_part</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="n">cg_part</span><span class="p">,</span> <span class="n">other</span> <span class="o">=</span> <span class="n">_check_varsh_871_1</span><span class="p">(</span><span class="n">cg_part</span><span class="p">)</span>
-    <span class="n">other_part</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-    <span class="n">cg_part</span><span class="p">,</span> <span class="n">other</span> <span class="o">=</span> <span class="n">_check_varsh_871_2</span><span class="p">(</span><span class="n">cg_part</span><span class="p">)</span>
-    <span class="n">other_part</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-    <span class="n">cg_part</span><span class="p">,</span> <span class="n">other</span> <span class="o">=</span> <span class="n">_check_varsh_872_9</span><span class="p">(</span><span class="n">cg_part</span><span class="p">)</span>
-    <span class="n">other_part</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">cg_part</span><span class="p">)</span> <span class="o">+</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">other_part</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">_check_varsh_871_1</span><span class="p">(</span><span class="n">term_list</span><span class="p">):</span>
-    <span class="c"># Sum( CG(a,alpha,b,0,a,alpha), (alpha, -a, a)) == KroneckerDelta(b,0)</span>
-    <span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">lt</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Wild</span><span class="p">,</span> <span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="s">&#39;alpha&#39;</span><span class="p">,</span> <span class="s">&#39;b&#39;</span><span class="p">,</span> <span class="s">&#39;lt&#39;</span><span class="p">)))</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">lt</span><span class="o">*</span><span class="n">CG</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">)</span>
-    <span class="n">simp</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-    <span class="n">sign</span> <span class="o">=</span> <span class="n">lt</span><span class="o">/</span><span class="nb">abs</span><span class="p">(</span><span class="n">lt</span><span class="p">)</span>
-    <span class="n">build_expr</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span>
-    <span class="n">index_expr</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="n">alpha</span>
-    <span class="k">return</span> <span class="n">_check_cg_simp</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">simp</span><span class="p">,</span> <span class="n">sign</span><span class="p">,</span> <span class="n">lt</span><span class="p">,</span> <span class="n">term_list</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">lt</span><span class="p">),</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">),</span> <span class="n">build_expr</span><span class="p">,</span> <span class="n">index_expr</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">_check_varsh_871_2</span><span class="p">(</span><span class="n">term_list</span><span class="p">):</span>
-    <span class="c"># Sum((-1)**(a-alpha)*CG(a,alpha,a,-alpha,c,0),(alpha,-a,a))</span>
-    <span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">lt</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Wild</span><span class="p">,</span> <span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="s">&#39;alpha&#39;</span><span class="p">,</span> <span class="s">&#39;c&#39;</span><span class="p">,</span> <span class="s">&#39;lt&#39;</span><span class="p">)))</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">lt</span><span class="o">*</span><span class="n">CG</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="o">-</span><span class="n">alpha</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-    <span class="n">simp</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-    <span class="n">sign</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">alpha</span><span class="p">)</span><span class="o">*</span><span class="n">lt</span><span class="o">/</span><span class="nb">abs</span><span class="p">(</span><span class="n">lt</span><span class="p">)</span>
-    <span class="n">build_expr</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span>
-    <span class="n">index_expr</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="n">alpha</span>
-    <span class="k">return</span> <span class="n">_check_cg_simp</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">simp</span><span class="p">,</span> <span class="n">sign</span><span class="p">,</span> <span class="n">lt</span><span class="p">,</span> <span class="n">term_list</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">lt</span><span class="p">),</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">c</span><span class="p">),</span> <span class="n">build_expr</span><span class="p">,</span> <span class="n">index_expr</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">_check_varsh_872_9</span><span class="p">(</span><span class="n">term_list</span><span class="p">):</span>
-    <span class="c"># Sum( CG(a,alpha,b,beta,c,gamma)*CG(a,alpha&#39;,b,beta&#39;,c,gamma), (gamma, -c, c), (c, abs(a-b), a+b))</span>
-    <span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">alphap</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">betap</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">lt</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Wild</span><span class="p">,</span> <span class="p">(</span>
-        <span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="s">&#39;alpha&#39;</span><span class="p">,</span> <span class="s">&#39;alphap&#39;</span><span class="p">,</span> <span class="s">&#39;b&#39;</span><span class="p">,</span> <span class="s">&#39;beta&#39;</span><span class="p">,</span> <span class="s">&#39;betap&#39;</span><span class="p">,</span> <span class="s">&#39;c&#39;</span><span class="p">,</span> <span class="s">&#39;gamma&#39;</span><span class="p">,</span> <span class="s">&#39;lt&#39;</span><span class="p">)))</span>
-    <span class="c"># Case alpha==alphap, beta==betap</span>
-
-    <span class="c"># For numerical alpha,beta</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">lt</span><span class="o">*</span><span class="n">CG</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">gamma</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-    <span class="n">simp</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">sign</span> <span class="o">=</span> <span class="n">lt</span><span class="o">/</span><span class="nb">abs</span><span class="p">(</span><span class="n">lt</span><span class="p">)</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span>
-    <span class="n">y</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">alpha</span> <span class="o">+</span> <span class="n">beta</span><span class="p">)</span>
-    <span class="n">build_expr</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">Piecewise</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span> <span class="o">&gt;</span> <span class="n">y</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">Eq</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)),</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">y</span> <span class="o">&gt;</span> <span class="n">x</span><span class="p">))</span>
-    <span class="n">index_expr</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">-</span> <span class="n">c</span>
-    <span class="n">term_list</span><span class="p">,</span> <span class="n">other1</span> <span class="o">=</span> <span class="n">_check_cg_simp</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">simp</span><span class="p">,</span> <span class="n">sign</span><span class="p">,</span> <span class="n">lt</span><span class="p">,</span> <span class="n">term_list</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">lt</span><span class="p">),</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">beta</span><span class="p">),</span> <span class="n">build_expr</span><span class="p">,</span> <span class="n">index_expr</span><span class="p">)</span>
-
-    <span class="c"># For symbolic alpha,beta</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span>
-    <span class="n">y</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span>
-    <span class="n">build_expr</span> <span class="o">=</span> <span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="n">index_expr</span> <span class="o">=</span> <span class="p">(</span><span class="n">c</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">c</span><span class="p">)</span> <span class="o">+</span> <span class="n">c</span> <span class="o">+</span> <span class="n">gamma</span>
-    <span class="n">term_list</span><span class="p">,</span> <span class="n">other2</span> <span class="o">=</span> <span class="n">_check_cg_simp</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">simp</span><span class="p">,</span> <span class="n">sign</span><span class="p">,</span> <span class="n">lt</span><span class="p">,</span> <span class="n">term_list</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">lt</span><span class="p">),</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">beta</span><span class="p">),</span> <span class="n">build_expr</span><span class="p">,</span> <span class="n">index_expr</span><span class="p">)</span>
-
-    <span class="c"># Case alpha!=alphap or beta!=betap</span>
-    <span class="c"># Note: this only works with leading term of 1, pattern matching is unable to match when there is a Wild leading term</span>
-    <span class="c"># For numerical alpha,alphap,beta,betap</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">CG</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">gamma</span><span class="p">)</span><span class="o">*</span><span class="n">CG</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">alphap</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">betap</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">gamma</span><span class="p">)</span>
-    <span class="n">simp</span> <span class="o">=</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="n">alphap</span><span class="p">)</span><span class="o">*</span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">beta</span><span class="p">,</span> <span class="n">betap</span><span class="p">)</span>
-    <span class="n">sign</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span>
-    <span class="n">y</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">alpha</span> <span class="o">+</span> <span class="n">beta</span><span class="p">)</span>
-    <span class="n">build_expr</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">Piecewise</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span> <span class="o">&gt;</span> <span class="n">y</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">Eq</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)),</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">y</span> <span class="o">&gt;</span> <span class="n">x</span><span class="p">))</span>
-    <span class="n">index_expr</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">-</span> <span class="n">c</span>
-    <span class="n">term_list</span><span class="p">,</span> <span class="n">other3</span> <span class="o">=</span> <span class="n">_check_cg_simp</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">simp</span><span class="p">,</span> <span class="n">sign</span><span class="p">,</span> <span class="n">sympify</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">term_list</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">alphap</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">betap</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">gamma</span><span class="p">),</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">alphap</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">betap</span><span class="p">),</span> <span class="n">build_expr</span><span class="p">,</span> <span class="n">index_expr</span><span class="p">)</span>
-
-    <span class="c"># For symbolic alpha,alphap,beta,betap</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span>
-    <span class="n">y</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span>
-    <span class="n">build_expr</span> <span class="o">=</span> <span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="n">index_expr</span> <span class="o">=</span> <span class="p">(</span><span class="n">c</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">c</span><span class="p">)</span> <span class="o">+</span> <span class="n">c</span> <span class="o">+</span> <span class="n">gamma</span>
-    <span class="n">term_list</span><span class="p">,</span> <span class="n">other4</span> <span class="o">=</span> <span class="n">_check_cg_simp</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">simp</span><span class="p">,</span> <span class="n">sign</span><span class="p">,</span> <span class="n">sympify</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">term_list</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">alphap</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">betap</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">gamma</span><span class="p">),</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">alphap</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">betap</span><span class="p">),</span> <span class="n">build_expr</span><span class="p">,</span> <span class="n">index_expr</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">term_list</span><span class="p">,</span> <span class="n">other1</span> <span class="o">+</span> <span class="n">other2</span> <span class="o">+</span> <span class="n">other4</span>
-
-
-<span class="k">def</span> <span class="nf">_check_cg_simp</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">simp</span><span class="p">,</span> <span class="n">sign</span><span class="p">,</span> <span class="n">lt</span><span class="p">,</span> <span class="n">term_list</span><span class="p">,</span> <span class="n">variables</span><span class="p">,</span> <span class="n">dep_variables</span><span class="p">,</span> <span class="n">build_index_expr</span><span class="p">,</span> <span class="n">index_expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Checks for simplifications that can be made, returning a tuple of the</span>
-<span class="sd">    simplified list of terms and any terms generated by simplification.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    expr: expression</span>
-<span class="sd">        The expression with Wild terms that will be matched to the terms in</span>
-<span class="sd">        the sum</span>
-
-<span class="sd">    simp: expression</span>
-<span class="sd">        The expression with Wild terms that is substituted in place of the CG</span>
-<span class="sd">        terms in the case of simplification</span>
-
-<span class="sd">    sign: expression</span>
-<span class="sd">        The expression with Wild terms denoting the sign that is on expr that</span>
-<span class="sd">        must match</span>
-
-<span class="sd">    lt: expression</span>
-<span class="sd">        The expression with Wild terms that gives the leading term of the</span>
-<span class="sd">        matched expr</span>
-
-<span class="sd">    term_list: list</span>
-<span class="sd">        A list of all of the terms is the sum to be simplified</span>
-
-<span class="sd">    variables: list</span>
-<span class="sd">        A list of all the variables that appears in expr</span>
-
-<span class="sd">    dep_variables: list</span>
-<span class="sd">        A list of the variables that must match for all the terms in the sum,</span>
-<span class="sd">        i.e. the dependant variables</span>
-
-<span class="sd">    build_index_expr: expression</span>
-<span class="sd">        Expression with Wild terms giving the number of elements in cg_index</span>
-
-<span class="sd">    index_expr: expression</span>
-<span class="sd">        Expression with Wild terms giving the index terms have when storing</span>
-<span class="sd">        them to cg_index</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">other_part</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">term_list</span><span class="p">):</span>
-        <span class="n">sub_1</span> <span class="o">=</span> <span class="n">_check_cg</span><span class="p">(</span><span class="n">term_list</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">expr</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">variables</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">sub_1</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">continue</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">sympify</span><span class="p">(</span><span class="n">build_index_expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sub_1</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">continue</span>
-        <span class="n">sub_dep</span> <span class="o">=</span> <span class="p">[(</span><span class="n">x</span><span class="p">,</span> <span class="n">sub_1</span><span class="p">[</span><span class="n">x</span><span class="p">])</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">dep_variables</span><span class="p">]</span>
-        <span class="n">cg_index</span> <span class="o">=</span> <span class="p">[</span><span class="bp">None</span><span class="p">]</span> <span class="o">*</span> <span class="n">build_index_expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sub_1</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">term_list</span><span class="p">)):</span>
-            <span class="n">sub_2</span> <span class="o">=</span> <span class="n">_check_cg</span><span class="p">(</span><span class="n">term_list</span><span class="p">[</span><span class="n">j</span><span class="p">],</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sub_dep</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">variables</span><span class="p">)</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">dep_variables</span><span class="p">),</span> <span class="n">sign</span><span class="o">=</span><span class="p">(</span><span class="n">sign</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sub_1</span><span class="p">),</span> <span class="n">sign</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sub_dep</span><span class="p">)))</span>
-            <span class="k">if</span> <span class="n">sub_2</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">sympify</span><span class="p">(</span><span class="n">index_expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sub_dep</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sub_2</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="n">cg_index</span><span class="p">[</span><span class="n">index_expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sub_dep</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sub_2</span><span class="p">)]</span> <span class="o">=</span> <span class="n">j</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">lt</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sub_dep</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sub_2</span><span class="p">),</span> <span class="n">lt</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sub_2</span><span class="p">),</span> <span class="n">sign</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sub_dep</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sub_2</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">i</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">cg_index</span><span class="p">):</span>
-            <span class="n">min_lt</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="nb">abs</span><span class="p">(</span><span class="n">term</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span> <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">cg_index</span> <span class="p">])</span>
-            <span class="n">indicies</span> <span class="o">=</span> <span class="p">[</span> <span class="n">term</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">cg_index</span><span class="p">]</span>
-            <span class="n">indicies</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-            <span class="n">indicies</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-            <span class="p">[</span> <span class="n">term_list</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">indicies</span> <span class="p">]</span>
-            <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">cg_index</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="n">term</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span> <span class="o">&gt;</span> <span class="n">min_lt</span><span class="p">:</span>
-                    <span class="n">term_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="p">(</span><span class="n">term</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">-</span> <span class="n">min_lt</span><span class="o">*</span><span class="n">term</span><span class="p">[</span><span class="mi">3</span><span class="p">])</span> <span class="o">*</span> <span class="n">term</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="p">)</span>
-            <span class="n">other_part</span> <span class="o">+=</span> <span class="n">min_lt</span> <span class="o">*</span> <span class="p">(</span><span class="n">sign</span><span class="o">*</span><span class="n">simp</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sub_1</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-    <span class="k">return</span> <span class="n">term_list</span><span class="p">,</span> <span class="n">other_part</span>
-
-
-<span class="k">def</span> <span class="nf">_check_cg</span><span class="p">(</span><span class="n">cg_term</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">length</span><span class="p">,</span> <span class="n">sign</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Checks whether a term matches the given expression&quot;&quot;&quot;</span>
-    <span class="c"># TODO: Check for symmetries</span>
-    <span class="n">matches</span> <span class="o">=</span> <span class="n">cg_term</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">matches</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span>
-    <span class="k">if</span> <span class="n">sign</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">sign</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;sign must be a tuple&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">sign</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="p">(</span><span class="n">sign</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">matches</span><span class="p">):</span>
-            <span class="k">return</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">matches</span><span class="p">)</span> <span class="o">==</span> <span class="n">length</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">matches</span>
-
-
-<span class="k">def</span> <span class="nf">_cg_simp_sum</span><span class="p">(</span><span class="n">e</span><span class="p">):</span>
-    <span class="n">e</span> <span class="o">=</span> <span class="n">_check_varsh_sum_871_1</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-    <span class="n">e</span> <span class="o">=</span> <span class="n">_check_varsh_sum_871_2</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-    <span class="n">e</span> <span class="o">=</span> <span class="n">_check_varsh_sum_872_4</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">e</span>
-
-
-<span class="k">def</span> <span class="nf">_check_varsh_sum_871_1</span><span class="p">(</span><span class="n">e</span><span class="p">):</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">)</span>
-    <span class="n">alpha</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;alpha&#39;</span><span class="p">)</span>
-    <span class="n">b</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;b&#39;</span><span class="p">)</span>
-    <span class="n">match</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">Sum</span><span class="p">(</span><span class="n">CG</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">),</span> <span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="o">-</span><span class="n">a</span><span class="p">,</span> <span class="n">a</span><span class="p">)))</span>
-    <span class="k">if</span> <span class="n">match</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">match</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">match</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">e</span>
-
-
-<span class="k">def</span> <span class="nf">_check_varsh_sum_871_2</span><span class="p">(</span><span class="n">e</span><span class="p">):</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">)</span>
-    <span class="n">alpha</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;alpha&#39;</span><span class="p">)</span>
-    <span class="n">c</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;c&#39;</span><span class="p">)</span>
-    <span class="n">match</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">match</span><span class="p">(</span>
-        <span class="n">Sum</span><span class="p">((</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">alpha</span><span class="p">)</span><span class="o">*</span><span class="n">CG</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="o">-</span><span class="n">alpha</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="o">-</span><span class="n">a</span><span class="p">,</span> <span class="n">a</span><span class="p">)))</span>
-    <span class="k">if</span> <span class="n">match</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">match</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">match</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">e</span>
-
-
-<span class="k">def</span> <span class="nf">_check_varsh_sum_872_4</span><span class="p">(</span><span class="n">e</span><span class="p">):</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">)</span>
-    <span class="n">alpha</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;alpha&#39;</span><span class="p">)</span>
-    <span class="n">b</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;b&#39;</span><span class="p">)</span>
-    <span class="n">beta</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;beta&#39;</span><span class="p">)</span>
-    <span class="n">c</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;c&#39;</span><span class="p">)</span>
-    <span class="n">cp</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;cp&#39;</span><span class="p">)</span>
-    <span class="n">gamma</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;gamma&#39;</span><span class="p">)</span>
-    <span class="n">gammap</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;gammap&#39;</span><span class="p">)</span>
-    <span class="n">match1</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">Sum</span><span class="p">(</span><span class="n">CG</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">gamma</span><span class="p">)</span><span class="o">*</span><span class="n">CG</span><span class="p">(</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">cp</span><span class="p">,</span> <span class="n">gammap</span><span class="p">),</span> <span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="o">-</span><span class="n">a</span><span class="p">,</span> <span class="n">a</span><span class="p">),</span> <span class="p">(</span><span class="n">beta</span><span class="p">,</span> <span class="o">-</span><span class="n">b</span><span class="p">,</span> <span class="n">b</span><span class="p">)))</span>
-    <span class="k">if</span> <span class="n">match1</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">match1</span><span class="p">)</span> <span class="o">==</span> <span class="mi">8</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">cp</span><span class="p">)</span><span class="o">*</span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">gamma</span><span class="p">,</span> <span class="n">gammap</span><span class="p">))</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">match1</span><span class="p">)</span>
-    <span class="n">match2</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">Sum</span><span class="p">(</span>
-        <span class="n">CG</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">gamma</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="o">-</span><span class="n">a</span><span class="p">,</span> <span class="n">a</span><span class="p">),</span> <span class="p">(</span><span class="n">beta</span><span class="p">,</span> <span class="o">-</span><span class="n">b</span><span class="p">,</span> <span class="n">b</span><span class="p">)))</span>
-    <span class="k">if</span> <span class="n">match2</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">match2</span><span class="p">)</span> <span class="o">==</span> <span class="mi">6</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">1</span>
-    <span class="k">return</span> <span class="n">e</span>
-
-
-<span class="k">def</span> <span class="nf">_cg_list</span><span class="p">(</span><span class="n">term</span><span class="p">):</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">CG</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">term</span><span class="p">,),</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span>
-    <span class="n">cg</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">coeff</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">Mul</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">Pow</span><span class="p">)):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;term must be CG, Add, Mul or Pow&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">Pow</span><span class="p">)</span> <span class="ow">and</span> <span class="n">sympify</span><span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">exp</span><span class="p">)</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">sympify</span><span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">exp</span><span class="p">)</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="p">[</span> <span class="n">cg</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">base</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">exp</span><span class="p">)</span> <span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">term</span><span class="p">,),</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">Mul</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">term</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">CG</span><span class="p">):</span>
-                <span class="n">cg</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">coeff</span> <span class="o">*=</span> <span class="n">arg</span>
-        <span class="k">return</span> <span class="n">cg</span><span class="p">,</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">coeff</span><span class="o">/</span><span class="nb">abs</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-</pre></div>
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-  <h1>Source code for sympy.physics.quantum.commutator</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;The commutator: [A,B] = A*B - B*A.&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Add</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.pretty.stringpict</span> <span class="kn">import</span> <span class="n">prettyForm</span>
-
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.dagger</span> <span class="kn">import</span> <span class="n">Dagger</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.operator</span> <span class="kn">import</span> <span class="n">Operator</span>
-
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="s">&#39;Commutator&#39;</span>
-<span class="p">]</span>
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># Commutator</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-
-<div class="viewcode-block" id="Commutator"><a class="viewcode-back" href="../../../../modules/physics/quantum/commutator.html#sympy.physics.quantum.commutator.Commutator">[docs]</a><span class="k">class</span> <span class="nc">Commutator</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The standard commutator, in an unevaluated state.</span>
-
-<span class="sd">    Evaluating a commutator is defined [1]_ as: ``[A, B] = A*B - B*A``. This</span>
-<span class="sd">    class returns the commutator in an unevaluated form. To evaluate the</span>
-<span class="sd">    commutator, use the ``.doit()`` method.</span>
-
-<span class="sd">    Cannonical ordering of a commutator is ``[A, B]`` for ``A &lt; B``. The</span>
-<span class="sd">    arguments of the commutator are put into canonical order using ``__cmp__``.</span>
-<span class="sd">    If ``B &lt; A``, then ``[B, A]`` is returned as ``-[A, B]``.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    A : Expr</span>
-<span class="sd">        The first argument of the commutator [A,B].</span>
-<span class="sd">    B : Expr</span>
-<span class="sd">        The second argument of the commutator [A,B].</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum import Commutator, Dagger, Operator</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; A = Operator(&#39;A&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; B = Operator(&#39;B&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; C = Operator(&#39;C&#39;)</span>
-
-<span class="sd">    Create a commutator and use ``.doit()`` to evaluate it:</span>
-
-<span class="sd">    &gt;&gt;&gt; comm = Commutator(A, B)</span>
-<span class="sd">    &gt;&gt;&gt; comm</span>
-<span class="sd">    [A,B]</span>
-<span class="sd">    &gt;&gt;&gt; comm.doit()</span>
-<span class="sd">    A*B - B*A</span>
-
-<span class="sd">    The commutator orders it arguments in canonical order:</span>
-
-<span class="sd">    &gt;&gt;&gt; comm = Commutator(B, A); comm</span>
-<span class="sd">    -[A,B]</span>
-
-<span class="sd">    Commutative constants are factored out:</span>
-
-<span class="sd">    &gt;&gt;&gt; Commutator(3*x*A, x*y*B)</span>
-<span class="sd">    3*x**2*y*[A,B]</span>
-
-<span class="sd">    Using ``.expand(commutator=True)``, the standard commutator expansion rules</span>
-<span class="sd">    can be applied:</span>
-
-<span class="sd">    &gt;&gt;&gt; Commutator(A+B, C).expand(commutator=True)</span>
-<span class="sd">    [A,C] + [B,C]</span>
-<span class="sd">    &gt;&gt;&gt; Commutator(A, B+C).expand(commutator=True)</span>
-<span class="sd">    [A,B] + [A,C]</span>
-<span class="sd">    &gt;&gt;&gt; Commutator(A*B, C).expand(commutator=True)</span>
-<span class="sd">    [A,C]*B + A*[B,C]</span>
-<span class="sd">    &gt;&gt;&gt; Commutator(A, B*C).expand(commutator=True)</span>
-<span class="sd">    [A,B]*C + B*[A,C]</span>
-
-<span class="sd">    Adjoint operations applied to the commutator are properly applied to the</span>
-<span class="sd">    arguments:</span>
-
-<span class="sd">    &gt;&gt;&gt; Dagger(Commutator(A, B))</span>
-<span class="sd">    -[Dagger(A),Dagger(B)]</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Commutator</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">):</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">r</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">r</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">a</span> <span class="ow">and</span> <span class="n">b</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="n">b</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_commutative</span> <span class="ow">or</span> <span class="n">b</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-        <span class="c"># [xA,yB]  -&gt;  xy*[A,B]</span>
-        <span class="c"># from sympy.physics.qmul import QMul</span>
-        <span class="n">ca</span><span class="p">,</span> <span class="n">nca</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-        <span class="n">cb</span><span class="p">,</span> <span class="n">ncb</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-        <span class="n">c_part</span> <span class="o">=</span> <span class="n">ca</span> <span class="o">+</span> <span class="n">cb</span>
-        <span class="k">if</span> <span class="n">c_part</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">c_part</span><span class="p">),</span> <span class="n">cls</span><span class="p">(</span><span class="n">Mul</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">nca</span><span class="p">),</span> <span class="n">Mul</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">ncb</span><span class="p">)))</span>
-
-        <span class="c"># Canonical ordering of arguments</span>
-        <span class="c"># The Commutator [A, B] is in canonical form if A &lt; B.</span>
-        <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">compare</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">*</span><span class="n">cls</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_expand_commutator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="n">A</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">B</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">Add</span><span class="p">):</span>
-            <span class="c"># [A + B, C]  -&gt;  [A, C] + [B, C]</span>
-            <span class="n">sargs</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">A</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="n">comm</span> <span class="o">=</span> <span class="n">Commutator</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">comm</span><span class="p">,</span> <span class="n">Commutator</span><span class="p">):</span>
-                    <span class="n">comm</span> <span class="o">=</span> <span class="n">comm</span><span class="o">.</span><span class="n">_eval_expand_commutator</span><span class="p">()</span>
-                <span class="n">sargs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">comm</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">sargs</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">Add</span><span class="p">):</span>
-            <span class="c"># [A, B + C]  -&gt;  [A, B] + [A, C]</span>
-            <span class="n">sargs</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">B</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="n">comm</span> <span class="o">=</span> <span class="n">Commutator</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">term</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">comm</span><span class="p">,</span> <span class="n">Commutator</span><span class="p">):</span>
-                    <span class="n">comm</span> <span class="o">=</span> <span class="n">comm</span><span class="o">.</span><span class="n">_eval_expand_commutator</span><span class="p">()</span>
-                <span class="n">sargs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">comm</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">sargs</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">Mul</span><span class="p">):</span>
-            <span class="c"># [A*B, C] -&gt; A*[B, C] + [A, C]*B</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">A</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">B</span>
-            <span class="n">comm1</span> <span class="o">=</span> <span class="n">Commutator</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
-            <span class="n">comm2</span> <span class="o">=</span> <span class="n">Commutator</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">comm1</span><span class="p">,</span> <span class="n">Commutator</span><span class="p">):</span>
-                <span class="n">comm1</span> <span class="o">=</span> <span class="n">comm1</span><span class="o">.</span><span class="n">_eval_expand_commutator</span><span class="p">()</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">comm2</span><span class="p">,</span> <span class="n">Commutator</span><span class="p">):</span>
-                <span class="n">comm2</span> <span class="o">=</span> <span class="n">comm2</span><span class="o">.</span><span class="n">_eval_expand_commutator</span><span class="p">()</span>
-            <span class="n">first</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">comm1</span><span class="p">)</span>
-            <span class="n">second</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="n">comm2</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="n">first</span><span class="p">,</span> <span class="n">second</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">Mul</span><span class="p">):</span>
-            <span class="c"># [A, B*C] -&gt; [A, B]*C + B*[A, C]</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">A</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="n">B</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">B</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-            <span class="n">comm1</span> <span class="o">=</span> <span class="n">Commutator</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-            <span class="n">comm2</span> <span class="o">=</span> <span class="n">Commutator</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">comm1</span><span class="p">,</span> <span class="n">Commutator</span><span class="p">):</span>
-                <span class="n">comm1</span> <span class="o">=</span> <span class="n">comm1</span><span class="o">.</span><span class="n">_eval_expand_commutator</span><span class="p">()</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">comm2</span><span class="p">,</span> <span class="n">Commutator</span><span class="p">):</span>
-                <span class="n">comm2</span> <span class="o">=</span> <span class="n">comm2</span><span class="o">.</span><span class="n">_eval_expand_commutator</span><span class="p">()</span>
-            <span class="n">first</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="n">comm1</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
-            <span class="n">second</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">comm2</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="n">first</span><span class="p">,</span> <span class="n">second</span><span class="p">)</span>
-
-        <span class="c"># No changes, so return self</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-<div class="viewcode-block" id="Commutator.doit"><a class="viewcode-back" href="../../../../modules/physics/quantum/commutator.html#sympy.physics.quantum.commutator.Commutator.doit">[docs]</a>    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Evaluate commutator &quot;&quot;&quot;</span>
-        <span class="n">A</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">B</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">Operator</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">Operator</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">comm</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">_eval_commutator</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="n">comm</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span><span class="o">*</span><span class="n">B</span><span class="o">.</span><span class="n">_eval_commutator</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-                <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                    <span class="n">comm</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">if</span> <span class="n">comm</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">comm</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">A</span><span class="o">*</span><span class="n">B</span> <span class="o">-</span> <span class="n">B</span><span class="o">*</span><span class="n">A</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_adjoint</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Commutator</span><span class="p">(</span><span class="n">Dagger</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="n">Dagger</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-
-    <span class="k">def</span> <span class="nf">_sympyrepr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sympystr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;[</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">]&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">((</span><span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;,&#39;</span><span class="p">))))</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">((</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="o">*</span><span class="n">args</span><span class="p">))))</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">parens</span><span class="p">(</span><span class="n">left</span><span class="o">=</span><span class="s">&#39;[&#39;</span><span class="p">,</span> <span class="n">right</span><span class="o">=</span><span class="s">&#39;]&#39;</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="se">\\</span><span class="s">left[</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="se">\\</span><span class="s">right]&quot;</span> <span class="o">%</span> <span class="nb">tuple</span><span class="p">([</span>
-            <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span></div>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.physics.quantum.dagger</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Hermitian conjugation.&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Expr</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.complexes</span> <span class="kn">import</span> <span class="n">adjoint</span>
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="s">&#39;Dagger&#39;</span>
-<span class="p">]</span>
-
-
-<div class="viewcode-block" id="Dagger"><a class="viewcode-back" href="../../../../modules/physics/quantum/dagger.html#sympy.physics.quantum.dagger.Dagger">[docs]</a><span class="k">class</span> <span class="nc">Dagger</span><span class="p">(</span><span class="n">adjoint</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;General Hermitian conjugate operation.</span>
-
-<span class="sd">    Take the Hermetian conjugate of an argument [1]_. For matrices this</span>
-<span class="sd">    operation is equivalent to transpose and complex conjugate [2]_.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    arg : Expr</span>
-<span class="sd">        The sympy expression that we want to take the dagger of.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Daggering various quantum objects:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.dagger import Dagger</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.state import Ket, Bra</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.operator import Operator</span>
-<span class="sd">        &gt;&gt;&gt; Dagger(Ket(&#39;psi&#39;))</span>
-<span class="sd">        &lt;psi|</span>
-<span class="sd">        &gt;&gt;&gt; Dagger(Bra(&#39;phi&#39;))</span>
-<span class="sd">        |phi&gt;</span>
-<span class="sd">        &gt;&gt;&gt; Dagger(Operator(&#39;A&#39;))</span>
-<span class="sd">        Dagger(A)</span>
-
-<span class="sd">    Inner and outer products::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum import InnerProduct, OuterProduct</span>
-<span class="sd">        &gt;&gt;&gt; Dagger(InnerProduct(Bra(&#39;a&#39;), Ket(&#39;b&#39;)))</span>
-<span class="sd">        &lt;b|a&gt;</span>
-<span class="sd">        &gt;&gt;&gt; Dagger(OuterProduct(Ket(&#39;a&#39;), Bra(&#39;b&#39;)))</span>
-<span class="sd">        |b&gt;&lt;a|</span>
-
-<span class="sd">    Powers, sums and products::</span>
-
-<span class="sd">        &gt;&gt;&gt; A = Operator(&#39;A&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; B = Operator(&#39;B&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; Dagger(A*B)</span>
-<span class="sd">        Dagger(B)*Dagger(A)</span>
-<span class="sd">        &gt;&gt;&gt; Dagger(A+B)</span>
-<span class="sd">        Dagger(A) + Dagger(B)</span>
-<span class="sd">        &gt;&gt;&gt; Dagger(A**2)</span>
-<span class="sd">        Dagger(A)**2</span>
-
-<span class="sd">    Dagger also seamlessly handles complex numbers and matrices::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Matrix, I</span>
-<span class="sd">        &gt;&gt;&gt; m = Matrix([[1,I],[2,I]])</span>
-<span class="sd">        &gt;&gt;&gt; m</span>
-<span class="sd">        [1, I]</span>
-<span class="sd">        [2, I]</span>
-<span class="sd">        &gt;&gt;&gt; Dagger(m)</span>
-<span class="sd">        [ 1,  2]</span>
-<span class="sd">        [-I, -I]</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Hermitian_adjoint</span>
-<span class="sd">    .. [2] http://en.wikipedia.org/wiki/Hermitian_transpose</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="s">&#39;adjoint&#39;</span><span class="p">):</span>
-            <span class="n">obj</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">adjoint</span><span class="p">()</span>
-        <span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="s">&#39;conjugate&#39;</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="s">&#39;transpose&#39;</span><span class="p">):</span>
-            <span class="n">obj</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">conjugate</span><span class="p">()</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">obj</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">obj</span>
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">)</span>
-</div>
-<span class="n">adjoint</span><span class="o">.</span><span class="n">__name__</span> <span class="o">=</span> <span class="s">&quot;Dagger&quot;</span>
-<span class="n">adjoint</span><span class="o">.</span><span class="n">_sympyrepr</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="s">&quot;Dagger(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="n">b</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-</pre></div>
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-            
-  <h1>Source code for sympy.physics.quantum.gate</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;An implementation of gates that act on qubits.</span>
-
-<span class="sd">Gates are unitary operators that act on the space of qubits.</span>
-
-<span class="sd">Medium Term Todo:</span>
-
-<span class="sd">* Optimize Gate._apply_operators_Qubit to remove the creation of many</span>
-<span class="sd">  intermediate Qubit objects.</span>
-<span class="sd">* Add commutation relationships to all operators and use this in gate_sort.</span>
-<span class="sd">* Fix gate_sort and gate_simp.</span>
-<span class="sd">* Get multi-target UGates plotting properly.</span>
-<span class="sd">* Get UGate to work with either sympy/numpy matrices and output either</span>
-<span class="sd">  format. This should also use the matrix slots.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">itertools</span> <span class="kn">import</span> <span class="n">chain</span>
-<span class="kn">import</span> <span class="nn">random</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Add</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">Integer</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Pow</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">Tuple</span>
-<span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">Number</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">is_sequence</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.pretty.stringpict</span> <span class="kn">import</span> <span class="n">prettyForm</span><span class="p">,</span> <span class="n">stringPict</span>
-
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.anticommutator</span> <span class="kn">import</span> <span class="n">AntiCommutator</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.commutator</span> <span class="kn">import</span> <span class="n">Commutator</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.qexpr</span> <span class="kn">import</span> <span class="n">QuantumError</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.hilbert</span> <span class="kn">import</span> <span class="n">ComplexSpace</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.operator</span> <span class="kn">import</span> <span class="p">(</span><span class="n">UnitaryOperator</span><span class="p">,</span> <span class="n">Operator</span><span class="p">,</span>
-                                            <span class="n">HermitianOperator</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.matrixutils</span> <span class="kn">import</span> <span class="n">matrix_tensor_product</span><span class="p">,</span> <span class="n">matrix_eye</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.matrixcache</span> <span class="kn">import</span> <span class="n">matrix_cache</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.dagger</span> <span class="kn">import</span> <span class="n">Dagger</span>
-
-<span class="kn">from</span> <span class="nn">sympy.matrices.matrices</span> <span class="kn">import</span> <span class="n">MatrixBase</span>
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="s">&#39;Gate&#39;</span><span class="p">,</span>
-    <span class="s">&#39;CGate&#39;</span><span class="p">,</span>
-    <span class="s">&#39;UGate&#39;</span><span class="p">,</span>
-    <span class="s">&#39;OneQubitGate&#39;</span><span class="p">,</span>
-    <span class="s">&#39;TwoQubitGate&#39;</span><span class="p">,</span>
-    <span class="s">&#39;IdentityGate&#39;</span><span class="p">,</span>
-    <span class="s">&#39;HadamardGate&#39;</span><span class="p">,</span>
-    <span class="s">&#39;XGate&#39;</span><span class="p">,</span>
-    <span class="s">&#39;YGate&#39;</span><span class="p">,</span>
-    <span class="s">&#39;ZGate&#39;</span><span class="p">,</span>
-    <span class="s">&#39;TGate&#39;</span><span class="p">,</span>
-    <span class="s">&#39;PhaseGate&#39;</span><span class="p">,</span>
-    <span class="s">&#39;SwapGate&#39;</span><span class="p">,</span>
-    <span class="s">&#39;CNotGate&#39;</span><span class="p">,</span>
-    <span class="c"># Aliased gate names</span>
-    <span class="s">&#39;CNOT&#39;</span><span class="p">,</span>
-    <span class="s">&#39;SWAP&#39;</span><span class="p">,</span>
-    <span class="s">&#39;H&#39;</span><span class="p">,</span>
-    <span class="s">&#39;X&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Y&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Z&#39;</span><span class="p">,</span>
-    <span class="s">&#39;T&#39;</span><span class="p">,</span>
-    <span class="s">&#39;S&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Phase&#39;</span><span class="p">,</span>
-    <span class="s">&#39;normalized&#39;</span><span class="p">,</span>
-    <span class="s">&#39;gate_sort&#39;</span><span class="p">,</span>
-    <span class="s">&#39;gate_simp&#39;</span><span class="p">,</span>
-    <span class="s">&#39;random_circuit&#39;</span><span class="p">,</span>
-<span class="p">]</span>
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># Gate Super-Classes</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-<span class="n">_normalized</span> <span class="o">=</span> <span class="bp">True</span>
-
-
-<div class="viewcode-block" id="normalized"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.normalized">[docs]</a><span class="k">def</span> <span class="nf">normalized</span><span class="p">(</span><span class="n">normalize</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Should Hadamard gates be normalized by a 1/sqrt(2).</span>
-
-<span class="sd">    This is a global setting that can be used to simplify the look of various</span>
-<span class="sd">    expressions, by leaving of the leading 1/sqrt(2) of the Hadamard gate.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    normalize : bool</span>
-<span class="sd">        Should the Hadamard gate include the 1/sqrt(2) normalization factor?</span>
-<span class="sd">        When True, the Hadamard gate will have the 1/sqrt(2). When False, the</span>
-<span class="sd">        Hadamard gate will not have this factor.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">global</span> <span class="n">_normalized</span>
-    <span class="n">_normalized</span> <span class="o">=</span> <span class="n">normalize</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_validate_targets_controls</span><span class="p">(</span><span class="n">tandc</span><span class="p">):</span>
-    <span class="n">tandc</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">tandc</span><span class="p">)</span>
-    <span class="c"># Check for integers</span>
-    <span class="k">for</span> <span class="n">bit</span> <span class="ow">in</span> <span class="n">tandc</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">bit</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">bit</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Integer expected, got: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">tandc</span><span class="p">[</span><span class="n">bit</span><span class="p">])</span>
-    <span class="c"># Detect duplicates</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">tandc</span><span class="p">)))</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">tandc</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="n">QuantumError</span><span class="p">(</span>
-            <span class="s">&#39;Target/control qubits in a gate cannot be duplicated&#39;</span>
-        <span class="p">)</span>
-
-
-<div class="viewcode-block" id="Gate"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.Gate">[docs]</a><span class="k">class</span> <span class="nc">Gate</span><span class="p">(</span><span class="n">UnitaryOperator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Non-controlled unitary gate operator that acts on qubits.</span>
-
-<span class="sd">    This is a general abstract gate that needs to be subclassed to do anything</span>
-<span class="sd">    useful.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    label : tuple, int</span>
-<span class="sd">        A list of the target qubits (as ints) that the gate will apply to.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    --------</span>
-
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_label_separator</span> <span class="o">=</span> <span class="s">&#39;,&#39;</span>
-
-    <span class="n">gate_name</span> <span class="o">=</span> <span class="s">&#39;G&#39;</span>
-    <span class="n">gate_name_latex</span> <span class="o">=</span> <span class="s">&#39;G&#39;</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Initialization/creation</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_args</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">UnitaryOperator</span><span class="o">.</span><span class="n">_eval_args</span><span class="p">(</span><span class="n">args</span><span class="p">))</span>
-        <span class="n">_validate_targets_controls</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">args</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_hilbert_space</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;This returns the smallest possible Hilbert space.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">ComplexSpace</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Properties</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Gate.nqubits"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.Gate.nqubits">[docs]</a>    <span class="k">def</span> <span class="nf">nqubits</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The total number of qubits this gate acts on.</span>
-
-<span class="sd">        For controlled gate subclasses this includes both target and control</span>
-<span class="sd">        qubits, so that, for examples the CNOT gate acts on 2 qubits.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Gate.min_qubits"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.Gate.min_qubits">[docs]</a>    <span class="k">def</span> <span class="nf">min_qubits</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The minimum number of qubits this gate needs to act on.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">max</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Gate.targets"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.Gate.targets">[docs]</a>    <span class="k">def</span> <span class="nf">targets</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A tuple of target qubits.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span>
-</div>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">gate_name_plot</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&#39;$</span><span class="si">%s</span><span class="s">$&#39;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">gate_name_latex</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Gate methods</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-<div class="viewcode-block" id="Gate.get_target_matrix"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.Gate.get_target_matrix">[docs]</a>    <span class="k">def</span> <span class="nf">get_target_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="s">&#39;sympy&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The matrix rep. of the target part of the gate.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ----------</span>
-<span class="sd">        format : str</span>
-<span class="sd">            The format string (&#39;sympy&#39;,&#39;numpy&#39;, etc.)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-            <span class="s">&#39;get_target_matrix is not implemented in Gate.&#39;</span><span class="p">)</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Apply</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-</div>
-    <span class="k">def</span> <span class="nf">_apply_operator_IntQubit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">qubits</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Redirect an apply from IntQubit to Qubit&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_apply_operator_Qubit</span><span class="p">(</span><span class="n">qubits</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_Qubit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">qubits</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Apply this gate to a Qubit.&quot;&quot;&quot;</span>
-
-        <span class="c"># Check number of qubits this gate acts on.</span>
-        <span class="k">if</span> <span class="n">qubits</span><span class="o">.</span><span class="n">nqubits</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">min_qubits</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">QuantumError</span><span class="p">(</span>
-                <span class="s">&#39;Gate needs a minimum of </span><span class="si">%r</span><span class="s"> qubits to act on, got: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span>
-                <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">min_qubits</span><span class="p">,</span> <span class="n">qubits</span><span class="o">.</span><span class="n">nqubits</span><span class="p">)</span>
-            <span class="p">)</span>
-
-        <span class="c"># If the controls are not met, just return</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">CGate</span><span class="p">):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">eval_controls</span><span class="p">(</span><span class="n">qubits</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">qubits</span>
-
-        <span class="n">targets</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">targets</span>
-        <span class="n">target_matrix</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_target_matrix</span><span class="p">(</span><span class="n">format</span><span class="o">=</span><span class="s">&#39;sympy&#39;</span><span class="p">)</span>
-
-        <span class="c"># Find which column of the target matrix this applies to.</span>
-        <span class="n">column_index</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">for</span> <span class="n">target</span> <span class="ow">in</span> <span class="n">targets</span><span class="p">:</span>
-            <span class="n">column_index</span> <span class="o">+=</span> <span class="n">n</span><span class="o">*</span><span class="n">qubits</span><span class="p">[</span><span class="n">target</span><span class="p">]</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">n</span> <span class="o">&lt;&lt;</span> <span class="mi">1</span>
-        <span class="n">column</span> <span class="o">=</span> <span class="n">target_matrix</span><span class="p">[:,</span> <span class="nb">int</span><span class="p">(</span><span class="n">column_index</span><span class="p">)]</span>
-
-        <span class="c"># Now apply each column element to the qubit.</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">index</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">column</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="c"># TODO: This can be optimized to reduce the number of Qubit</span>
-            <span class="c"># creations. We should simply manipulate the raw list of qubit</span>
-            <span class="c"># values and then build the new Qubit object once.</span>
-            <span class="c"># Make a copy of the incoming qubits.</span>
-            <span class="n">new_qubit</span> <span class="o">=</span> <span class="n">qubits</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="o">*</span><span class="n">qubits</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-            <span class="c"># Flip the bits that need to be flipped.</span>
-            <span class="k">for</span> <span class="n">bit</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">targets</span><span class="p">)):</span>
-                <span class="k">if</span> <span class="n">new_qubit</span><span class="p">[</span><span class="n">targets</span><span class="p">[</span><span class="n">bit</span><span class="p">]]</span> <span class="o">!=</span> <span class="p">(</span><span class="n">index</span> <span class="o">&gt;&gt;</span> <span class="n">bit</span><span class="p">)</span> <span class="o">&amp;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">new_qubit</span> <span class="o">=</span> <span class="n">new_qubit</span><span class="o">.</span><span class="n">flip</span><span class="p">(</span><span class="n">targets</span><span class="p">[</span><span class="n">bit</span><span class="p">])</span>
-            <span class="c"># The value in that row and column times the flipped-bit qubit</span>
-            <span class="c"># is the result for that part.</span>
-            <span class="n">result</span> <span class="o">+=</span> <span class="n">column</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="o">*</span><span class="n">new_qubit</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Represent</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="k">def</span> <span class="nf">_represent_default_basis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_ZGate</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_ZGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">format</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;format&#39;</span><span class="p">,</span> <span class="s">&#39;sympy&#39;</span><span class="p">)</span>
-        <span class="n">nqubits</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;nqubits&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">nqubits</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">QuantumError</span><span class="p">(</span>
-                <span class="s">&#39;The number of qubits must be given as nqubits.&#39;</span><span class="p">)</span>
-
-        <span class="c"># Make sure we have enough qubits for the gate.</span>
-        <span class="k">if</span> <span class="n">nqubits</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">min_qubits</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">QuantumError</span><span class="p">(</span>
-                <span class="s">&#39;The number of qubits </span><span class="si">%r</span><span class="s"> is too small for the gate.&#39;</span> <span class="o">%</span> <span class="n">nqubits</span>
-            <span class="p">)</span>
-
-        <span class="n">target_matrix</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_target_matrix</span><span class="p">(</span><span class="n">format</span><span class="p">)</span>
-        <span class="n">targets</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">targets</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">CGate</span><span class="p">):</span>
-            <span class="n">controls</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">controls</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">controls</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">represent_zbasis</span><span class="p">(</span>
-            <span class="n">controls</span><span class="p">,</span> <span class="n">targets</span><span class="p">,</span> <span class="n">target_matrix</span><span class="p">,</span> <span class="n">nqubits</span><span class="p">,</span> <span class="n">format</span>
-        <span class="p">)</span>
-        <span class="k">return</span> <span class="n">m</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Print methods</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="k">def</span> <span class="nf">_sympystr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">label</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_label</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gate_name</span><span class="p">,</span> <span class="n">label</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gate_name</span><span class="p">))</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_label_pretty</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_subscript_pretty</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">label</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_label</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">_{</span><span class="si">%s</span><span class="s">}&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gate_name_latex</span><span class="p">,</span> <span class="n">label</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">plot_gate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">axes</span><span class="p">,</span> <span class="n">gate_idx</span><span class="p">,</span> <span class="n">gate_grid</span><span class="p">,</span> <span class="n">wire_grid</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;plot_gate is not implemented.&#39;</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="CGate"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CGate">[docs]</a><span class="k">class</span> <span class="nc">CGate</span><span class="p">(</span><span class="n">Gate</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A general unitary gate with control qubits.</span>
-
-<span class="sd">    A general control gate applies a target gate to a set of targets if all</span>
-<span class="sd">    of the control qubits have a particular values (set by</span>
-<span class="sd">    ``CGate.control_value``).</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    label : tuple</span>
-<span class="sd">        The label in this case has the form (controls, gate), where controls</span>
-<span class="sd">        is a tuple/list of control qubits (as ints) and gate is a ``Gate``</span>
-<span class="sd">        instance that is the target operator.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    --------</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">gate_name</span> <span class="o">=</span> <span class="s">&#39;C&#39;</span>
-    <span class="n">gate_name_latex</span> <span class="o">=</span> <span class="s">&#39;C&#39;</span>
-
-    <span class="c"># The values this class controls for.</span>
-    <span class="n">control_value</span> <span class="o">=</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Initialization</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_args</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="c"># _eval_args has the right logic for the controls argument.</span>
-        <span class="n">controls</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">gate</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">controls</span><span class="p">):</span>
-            <span class="n">controls</span> <span class="o">=</span> <span class="p">(</span><span class="n">controls</span><span class="p">,)</span>
-        <span class="n">controls</span> <span class="o">=</span> <span class="n">UnitaryOperator</span><span class="o">.</span><span class="n">_eval_args</span><span class="p">(</span><span class="n">controls</span><span class="p">)</span>
-        <span class="n">_validate_targets_controls</span><span class="p">(</span><span class="n">chain</span><span class="p">(</span><span class="n">controls</span><span class="p">,</span> <span class="n">gate</span><span class="o">.</span><span class="n">targets</span><span class="p">))</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">controls</span><span class="p">),</span> <span class="n">gate</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_hilbert_space</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;This returns the smallest possible Hilbert space.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">ComplexSpace</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="nb">max</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">min_qubits</span><span class="p">)</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Properties</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="CGate.nqubits"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CGate.nqubits">[docs]</a>    <span class="k">def</span> <span class="nf">nqubits</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The total number of qubits this gate acts on.</span>
-
-<span class="sd">        For controlled gate subclasses this includes both target and control</span>
-<span class="sd">        qubits, so that, for examples the CNOT gate acts on 2 qubits.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">)</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">controls</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="CGate.min_qubits"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CGate.min_qubits">[docs]</a>    <span class="k">def</span> <span class="nf">min_qubits</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The minimum number of qubits this gate needs to act on.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">max</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">controls</span><span class="p">),</span> <span class="nb">max</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">))</span> <span class="o">+</span> <span class="mi">1</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="CGate.targets"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CGate.targets">[docs]</a>    <span class="k">def</span> <span class="nf">targets</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A tuple of target qubits.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">gate</span><span class="o">.</span><span class="n">targets</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="CGate.controls"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CGate.controls">[docs]</a>    <span class="k">def</span> <span class="nf">controls</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A tuple of control qubits.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="CGate.gate"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CGate.gate">[docs]</a>    <span class="k">def</span> <span class="nf">gate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The non-controlled gate that will be applied to the targets.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Gate methods</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-</div>
-    <span class="k">def</span> <span class="nf">get_target_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="s">&#39;sympy&#39;</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">gate</span><span class="o">.</span><span class="n">get_target_matrix</span><span class="p">(</span><span class="n">format</span><span class="p">)</span>
-
-<div class="viewcode-block" id="CGate.eval_controls"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CGate.eval_controls">[docs]</a>    <span class="k">def</span> <span class="nf">eval_controls</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">qubit</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return True/False to indicate if the controls are satisfied.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="n">qubit</span><span class="p">[</span><span class="n">bit</span><span class="p">]</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">control_value</span> <span class="k">for</span> <span class="n">bit</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">controls</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="CGate.decompose"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CGate.decompose">[docs]</a>    <span class="k">def</span> <span class="nf">decompose</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Decompose the controlled gate into CNOT and single qubits gates.&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">controls</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">controls</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">t</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">gate</span><span class="o">.</span><span class="n">targets</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gate</span><span class="p">,</span> <span class="n">YGate</span><span class="p">):</span>
-                <span class="n">g1</span> <span class="o">=</span> <span class="n">PhaseGate</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-                <span class="n">g2</span> <span class="o">=</span> <span class="n">CNotGate</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
-                <span class="n">g3</span> <span class="o">=</span> <span class="n">PhaseGate</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-                <span class="n">g4</span> <span class="o">=</span> <span class="n">ZGate</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">g1</span><span class="o">*</span><span class="n">g2</span><span class="o">*</span><span class="n">g3</span><span class="o">*</span><span class="n">g4</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gate</span><span class="p">,</span> <span class="n">ZGate</span><span class="p">):</span>
-                <span class="n">g1</span> <span class="o">=</span> <span class="n">HadamardGate</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-                <span class="n">g2</span> <span class="o">=</span> <span class="n">CNotGate</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
-                <span class="n">g3</span> <span class="o">=</span> <span class="n">HadamardGate</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">g1</span><span class="o">*</span><span class="n">g2</span><span class="o">*</span><span class="n">g3</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Print methods</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-</div>
-    <span class="k">def</span> <span class="nf">_print_label</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">controls</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_sequence</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">controls</span><span class="p">,</span> <span class="s">&#39;,&#39;</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">gate</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gate</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&#39;(</span><span class="si">%s</span><span class="s">),</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">controls</span><span class="p">,</span> <span class="n">gate</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">controls</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_sequence_pretty</span><span class="p">(</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">controls</span><span class="p">,</span> <span class="s">&#39;,&#39;</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">gate</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gate</span><span class="p">)</span>
-        <span class="n">gate_name</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gate_name</span><span class="p">))</span>
-        <span class="n">first</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_subscript_pretty</span><span class="p">(</span><span class="n">gate_name</span><span class="p">,</span> <span class="n">controls</span><span class="p">)</span>
-        <span class="n">gate</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_parens_pretty</span><span class="p">(</span><span class="n">gate</span><span class="p">)</span>
-        <span class="n">final</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">first</span><span class="o">.</span><span class="n">right</span><span class="p">((</span><span class="n">gate</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">final</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">controls</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_sequence</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">controls</span><span class="p">,</span> <span class="s">&#39;,&#39;</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">gate</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gate</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">r&#39;</span><span class="si">%s</span><span class="s">_{</span><span class="si">%s</span><span class="s">}{\left(</span><span class="si">%s</span><span class="s">\right)}&#39;</span> <span class="o">%</span> \
-            <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gate_name_latex</span><span class="p">,</span> <span class="n">controls</span><span class="p">,</span> <span class="n">gate</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">plot_gate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">circ_plot</span><span class="p">,</span> <span class="n">gate_idx</span><span class="p">):</span>
-        <span class="n">min_wire</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="nb">min</span><span class="p">(</span><span class="n">chain</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">controls</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">)))</span>
-        <span class="n">max_wire</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="n">chain</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">controls</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">)))</span>
-        <span class="n">circ_plot</span><span class="o">.</span><span class="n">control_line</span><span class="p">(</span><span class="n">gate_idx</span><span class="p">,</span> <span class="n">min_wire</span><span class="p">,</span> <span class="n">max_wire</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">controls</span><span class="p">:</span>
-            <span class="n">circ_plot</span><span class="o">.</span><span class="n">control_point</span><span class="p">(</span><span class="n">gate_idx</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">c</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">gate</span><span class="o">.</span><span class="n">plot_gate</span><span class="p">(</span><span class="n">circ_plot</span><span class="p">,</span> <span class="n">gate_idx</span><span class="p">)</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Miscellaneous</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="k">def</span> <span class="nf">_eval_dagger</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gate</span><span class="p">,</span> <span class="n">HermitianOperator</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Gate</span><span class="o">.</span><span class="n">_eval_dagger</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gate</span><span class="p">,</span> <span class="n">HermitianOperator</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Gate</span><span class="o">.</span><span class="n">_eval_inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">exp</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gate</span><span class="p">,</span> <span class="n">HermitianOperator</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">exp</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Gate</span><span class="o">.</span><span class="n">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="nb">abs</span><span class="p">(</span><span class="n">exp</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">*</span><span class="p">(</span><span class="n">Gate</span><span class="o">.</span><span class="n">_eval_inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Gate</span><span class="o">.</span><span class="n">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="UGate"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.UGate">[docs]</a><span class="k">class</span> <span class="nc">UGate</span><span class="p">(</span><span class="n">Gate</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;General gate specified by a set of targets and a target matrix.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    label : tuple</span>
-<span class="sd">        A tuple of the form (targets, U), where targets is a tuple of the</span>
-<span class="sd">        target qubits and U is a unitary matrix with dimension of</span>
-<span class="sd">        len(targets).</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">gate_name</span> <span class="o">=</span> <span class="s">&#39;U&#39;</span>
-    <span class="n">gate_name_latex</span> <span class="o">=</span> <span class="s">&#39;U&#39;</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Initialization</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_args</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="n">targets</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">targets</span><span class="p">):</span>
-            <span class="n">targets</span> <span class="o">=</span> <span class="p">(</span><span class="n">targets</span><span class="p">,)</span>
-        <span class="n">targets</span> <span class="o">=</span> <span class="n">Gate</span><span class="o">.</span><span class="n">_eval_args</span><span class="p">(</span><span class="n">targets</span><span class="p">)</span>
-        <span class="n">_validate_targets_controls</span><span class="p">(</span><span class="n">targets</span><span class="p">)</span>
-        <span class="n">mat</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mat</span><span class="p">,</span> <span class="n">MatrixBase</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Matrix expected, got: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">mat</span><span class="p">)</span>
-        <span class="n">dim</span> <span class="o">=</span> <span class="mi">2</span><span class="o">**</span><span class="nb">len</span><span class="p">(</span><span class="n">targets</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">(</span><span class="n">dim</span> <span class="o">==</span> <span class="n">shape</span> <span class="k">for</span> <span class="n">shape</span> <span class="ow">in</span> <span class="n">mat</span><span class="o">.</span><span class="n">shape</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span>
-                <span class="s">&#39;Number of targets must match the matrix size: </span><span class="si">%r</span><span class="s"> </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span>
-                <span class="p">(</span><span class="n">targets</span><span class="p">,</span> <span class="n">mat</span><span class="p">)</span>
-            <span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">targets</span><span class="p">,</span> <span class="n">mat</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_hilbert_space</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;This returns the smallest possible Hilbert space.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">ComplexSpace</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Properties</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="UGate.targets"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.UGate.targets">[docs]</a>    <span class="k">def</span> <span class="nf">targets</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A tuple of target qubits.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Gate methods</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-</div>
-<div class="viewcode-block" id="UGate.get_target_matrix"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.UGate.get_target_matrix">[docs]</a>    <span class="k">def</span> <span class="nf">get_target_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="s">&#39;sympy&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The matrix rep. of the target part of the gate.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ----------</span>
-<span class="sd">        format : str</span>
-<span class="sd">            The format string (&#39;sympy&#39;,&#39;numpy&#39;, etc.)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Print methods</span>
-    <span class="c">#-------------------------------------------------------------------------</span></div>
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">targets</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_sequence_pretty</span><span class="p">(</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">,</span> <span class="s">&#39;,&#39;</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">gate_name</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gate_name</span><span class="p">))</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_subscript_pretty</span><span class="p">(</span><span class="n">gate_name</span><span class="p">,</span> <span class="n">targets</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">targets</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_sequence</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">,</span> <span class="s">&#39;,&#39;</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">r&#39;</span><span class="si">%s</span><span class="s">_{</span><span class="si">%s</span><span class="s">}&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gate_name_latex</span><span class="p">,</span> <span class="n">targets</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">plot_gate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">circ_plot</span><span class="p">,</span> <span class="n">gate_idx</span><span class="p">):</span>
-        <span class="n">circ_plot</span><span class="o">.</span><span class="n">one_qubit_box</span><span class="p">(</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">gate_name_plot</span><span class="p">,</span>
-            <span class="n">gate_idx</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="OneQubitGate"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.OneQubitGate">[docs]</a><span class="k">class</span> <span class="nc">OneQubitGate</span><span class="p">(</span><span class="n">Gate</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A single qubit unitary gate base class.&quot;&quot;&quot;</span>
-
-    <span class="n">nqubits</span> <span class="o">=</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">plot_gate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">circ_plot</span><span class="p">,</span> <span class="n">gate_idx</span><span class="p">):</span>
-        <span class="n">circ_plot</span><span class="o">.</span><span class="n">one_qubit_box</span><span class="p">(</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">gate_name_plot</span><span class="p">,</span>
-            <span class="n">gate_idx</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">OneQubitGate</span><span class="p">):</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">targets</span> <span class="o">!=</span> <span class="n">other</span><span class="o">.</span><span class="n">targets</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">__class__</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Operator</span><span class="o">.</span><span class="n">_eval_commutator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_anticommutator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">OneQubitGate</span><span class="p">):</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">targets</span> <span class="o">!=</span> <span class="n">other</span><span class="o">.</span><span class="n">targets</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">__class__</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">*</span><span class="n">other</span>
-        <span class="k">return</span> <span class="n">Operator</span><span class="o">.</span><span class="n">_eval_anticommutator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="TwoQubitGate"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.TwoQubitGate">[docs]</a><span class="k">class</span> <span class="nc">TwoQubitGate</span><span class="p">(</span><span class="n">Gate</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A two qubit unitary gate base class.&quot;&quot;&quot;</span>
-
-    <span class="n">nqubits</span> <span class="o">=</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># Single Qubit Gates</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-</div>
-<div class="viewcode-block" id="IdentityGate"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.IdentityGate">[docs]</a><span class="k">class</span> <span class="nc">IdentityGate</span><span class="p">(</span><span class="n">OneQubitGate</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The single qubit identity gate.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    target : int</span>
-<span class="sd">        The target qubit this gate will apply to.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    --------</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">gate_name</span> <span class="o">=</span> <span class="s">&#39;1&#39;</span>
-    <span class="n">gate_name_latex</span> <span class="o">=</span> <span class="s">&#39;1&#39;</span>
-
-    <span class="k">def</span> <span class="nf">get_target_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="s">&#39;sympy&#39;</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">matrix_cache</span><span class="o">.</span><span class="n">get_matrix</span><span class="p">(</span><span class="s">&#39;eye2&#39;</span><span class="p">,</span> <span class="n">format</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_anticommutator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">other</span>
-
-</div>
-<div class="viewcode-block" id="HadamardGate"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.HadamardGate">[docs]</a><span class="k">class</span> <span class="nc">HadamardGate</span><span class="p">(</span><span class="n">HermitianOperator</span><span class="p">,</span> <span class="n">OneQubitGate</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The single qubit Hadamard gate.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    target : int</span>
-<span class="sd">        The target qubit this gate will apply to.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    --------</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import sqrt</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.qubit import Qubit</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.gate import HadamardGate</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.qapply import qapply</span>
-<span class="sd">    &gt;&gt;&gt; qapply(HadamardGate(0)*Qubit(&#39;1&#39;))</span>
-<span class="sd">    sqrt(2)*|0&gt;/2 - sqrt(2)*|1&gt;/2</span>
-<span class="sd">    &gt;&gt;&gt; # Hadamard on bell state, applied on 2 qubits.</span>
-<span class="sd">    &gt;&gt;&gt; psi = 1/sqrt(2)*(Qubit(&#39;00&#39;)+Qubit(&#39;11&#39;))</span>
-<span class="sd">    &gt;&gt;&gt; qapply(HadamardGate(0)*HadamardGate(1)*psi)</span>
-<span class="sd">    sqrt(2)*|00&gt;/2 + sqrt(2)*|11&gt;/2</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">gate_name</span> <span class="o">=</span> <span class="s">&#39;H&#39;</span>
-    <span class="n">gate_name_latex</span> <span class="o">=</span> <span class="s">&#39;H&#39;</span>
-
-    <span class="k">def</span> <span class="nf">get_target_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="s">&#39;sympy&#39;</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">_normalized</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">matrix_cache</span><span class="o">.</span><span class="n">get_matrix</span><span class="p">(</span><span class="s">&#39;H&#39;</span><span class="p">,</span> <span class="n">format</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">matrix_cache</span><span class="o">.</span><span class="n">get_matrix</span><span class="p">(</span><span class="s">&#39;Hsqrt2&#39;</span><span class="p">,</span> <span class="n">format</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_XGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">I</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">YGate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_YGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">I</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">ZGate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">-</span> <span class="n">XGate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_ZGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">YGate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_anticommutator_XGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">IdentityGate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_anticommutator_YGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_anticommutator_ZGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">IdentityGate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-</div>
-<div class="viewcode-block" id="XGate"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.XGate">[docs]</a><span class="k">class</span> <span class="nc">XGate</span><span class="p">(</span><span class="n">HermitianOperator</span><span class="p">,</span> <span class="n">OneQubitGate</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The single qubit X, or NOT, gate.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    target : int</span>
-<span class="sd">        The target qubit this gate will apply to.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    --------</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">gate_name</span> <span class="o">=</span> <span class="s">&#39;X&#39;</span>
-    <span class="n">gate_name_latex</span> <span class="o">=</span> <span class="s">&#39;X&#39;</span>
-
-    <span class="k">def</span> <span class="nf">get_target_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="s">&#39;sympy&#39;</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">matrix_cache</span><span class="o">.</span><span class="n">get_matrix</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="n">format</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">plot_gate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">circ_plot</span><span class="p">,</span> <span class="n">gate_idx</span><span class="p">):</span>
-        <span class="n">circ_plot</span><span class="o">.</span><span class="n">not_point</span><span class="p">(</span>
-            <span class="n">gate_idx</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_YGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">ZGate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_anticommutator_XGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">IdentityGate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_anticommutator_YGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_anticommutator_ZGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="YGate"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.YGate">[docs]</a><span class="k">class</span> <span class="nc">YGate</span><span class="p">(</span><span class="n">HermitianOperator</span><span class="p">,</span> <span class="n">OneQubitGate</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The single qubit Y gate.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    target : int</span>
-<span class="sd">        The target qubit this gate will apply to.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    --------</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">gate_name</span> <span class="o">=</span> <span class="s">&#39;Y&#39;</span>
-    <span class="n">gate_name_latex</span> <span class="o">=</span> <span class="s">&#39;Y&#39;</span>
-
-    <span class="k">def</span> <span class="nf">get_target_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="s">&#39;sympy&#39;</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">matrix_cache</span><span class="o">.</span><span class="n">get_matrix</span><span class="p">(</span><span class="s">&#39;Y&#39;</span><span class="p">,</span> <span class="n">format</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_ZGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">XGate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_anticommutator_YGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">IdentityGate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_anticommutator_ZGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="ZGate"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.ZGate">[docs]</a><span class="k">class</span> <span class="nc">ZGate</span><span class="p">(</span><span class="n">HermitianOperator</span><span class="p">,</span> <span class="n">OneQubitGate</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The single qubit Z gate.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    target : int</span>
-<span class="sd">        The target qubit this gate will apply to.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    --------</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">gate_name</span> <span class="o">=</span> <span class="s">&#39;Z&#39;</span>
-    <span class="n">gate_name_latex</span> <span class="o">=</span> <span class="s">&#39;Z&#39;</span>
-
-    <span class="k">def</span> <span class="nf">get_target_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="s">&#39;sympy&#39;</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">matrix_cache</span><span class="o">.</span><span class="n">get_matrix</span><span class="p">(</span><span class="s">&#39;Z&#39;</span><span class="p">,</span> <span class="n">format</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_XGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">YGate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_anticommutator_YGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="PhaseGate"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.PhaseGate">[docs]</a><span class="k">class</span> <span class="nc">PhaseGate</span><span class="p">(</span><span class="n">OneQubitGate</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The single qubit phase, or S, gate.</span>
-
-<span class="sd">    This gate rotates the phase of the state by pi/2 if the state is ``|1&gt;`` and</span>
-<span class="sd">    does nothing if the state is ``|0&gt;``.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    target : int</span>
-<span class="sd">        The target qubit this gate will apply to.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    --------</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">gate_name</span> <span class="o">=</span> <span class="s">&#39;S&#39;</span>
-    <span class="n">gate_name_latex</span> <span class="o">=</span> <span class="s">&#39;S&#39;</span>
-
-    <span class="k">def</span> <span class="nf">get_target_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="s">&#39;sympy&#39;</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">matrix_cache</span><span class="o">.</span><span class="n">get_matrix</span><span class="p">(</span><span class="s">&#39;S&#39;</span><span class="p">,</span> <span class="n">format</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_ZGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_TGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="TGate"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.TGate">[docs]</a><span class="k">class</span> <span class="nc">TGate</span><span class="p">(</span><span class="n">OneQubitGate</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The single qubit pi/8 gate.</span>
-
-<span class="sd">    This gate rotates the phase of the state by pi/4 if the state is ``|1&gt;`` and</span>
-<span class="sd">    does nothing if the state is ``|0&gt;``.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    target : int</span>
-<span class="sd">        The target qubit this gate will apply to.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    --------</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">gate_name</span> <span class="o">=</span> <span class="s">&#39;T&#39;</span>
-    <span class="n">gate_name_latex</span> <span class="o">=</span> <span class="s">&#39;T&#39;</span>
-
-    <span class="k">def</span> <span class="nf">get_target_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="s">&#39;sympy&#39;</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">matrix_cache</span><span class="o">.</span><span class="n">get_matrix</span><span class="p">(</span><span class="s">&#39;T&#39;</span><span class="p">,</span> <span class="n">format</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_ZGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_PhaseGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-
-
-<span class="c"># Aliases for gate names.</span></div>
-<span class="n">H</span> <span class="o">=</span> <span class="n">HadamardGate</span>
-<span class="n">X</span> <span class="o">=</span> <span class="n">XGate</span>
-<span class="n">Y</span> <span class="o">=</span> <span class="n">YGate</span>
-<span class="n">Z</span> <span class="o">=</span> <span class="n">ZGate</span>
-<span class="n">T</span> <span class="o">=</span> <span class="n">TGate</span>
-<span class="n">Phase</span> <span class="o">=</span> <span class="n">S</span> <span class="o">=</span> <span class="n">PhaseGate</span>
-
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># 2 Qubit Gates</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-
-<div class="viewcode-block" id="CNotGate"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CNotGate">[docs]</a><span class="k">class</span> <span class="nc">CNotGate</span><span class="p">(</span><span class="n">HermitianOperator</span><span class="p">,</span> <span class="n">CGate</span><span class="p">,</span> <span class="n">TwoQubitGate</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Two qubit controlled-NOT.</span>
-
-<span class="sd">    This gate performs the NOT or X gate on the target qubit if the control</span>
-<span class="sd">    qubits all have the value 1.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    label : tuple</span>
-<span class="sd">        A tuple of the form (control, target).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    --------</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.gate import CNOT</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.qapply import qapply</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.qubit import Qubit</span>
-<span class="sd">    &gt;&gt;&gt; c = CNOT(1,0)</span>
-<span class="sd">    &gt;&gt;&gt; qapply(c*Qubit(&#39;10&#39;)) # note that qubits are indexed from right to left</span>
-<span class="sd">    |11&gt;</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">gate_name</span> <span class="o">=</span> <span class="s">&#39;CNOT&#39;</span>
-    <span class="n">gate_name_latex</span> <span class="o">=</span> <span class="s">&#39;CNOT&#39;</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Initialization</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_args</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="n">Gate</span><span class="o">.</span><span class="n">_eval_args</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">args</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_hilbert_space</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;This returns the smallest possible Hilbert space.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">ComplexSpace</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Properties</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="CNotGate.min_qubits"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CNotGate.min_qubits">[docs]</a>    <span class="k">def</span> <span class="nf">min_qubits</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The minimum number of qubits this gate needs to act on.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">max</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="CNotGate.targets"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CNotGate.targets">[docs]</a>    <span class="k">def</span> <span class="nf">targets</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A tuple of target qubits.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">1</span><span class="p">],)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="CNotGate.controls"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CNotGate.controls">[docs]</a>    <span class="k">def</span> <span class="nf">controls</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A tuple of control qubits.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">],)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="CNotGate.gate"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CNotGate.gate">[docs]</a>    <span class="k">def</span> <span class="nf">gate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The non-controlled gate that will be applied to the targets.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">XGate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Properties</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="c"># The default printing of Gate works better than those of CGate, so we</span>
-    <span class="c"># go around the overridden methods in CGate.</span>
-</div>
-    <span class="k">def</span> <span class="nf">_print_label</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Gate</span><span class="o">.</span><span class="n">_print_label</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Gate</span><span class="o">.</span><span class="n">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Gate</span><span class="o">.</span><span class="n">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Commutator/AntiCommutator</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_ZGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;[CNOT(i, j), Z(i)] == 0.&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">controls</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">targets</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;Commutator not implemented: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_TGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;[CNOT(i, j), T(i)] == 0.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_commutator_ZGate</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_PhaseGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;[CNOT(i, j), S(i)] == 0.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_commutator_ZGate</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_XGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;[CNOT(i, j), X(j)] == 0.&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">targets</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;Commutator not implemented: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_CNotGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;[CNOT(i, j), CNOT(i,k)] == 0.&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">controls</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">controls</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;Commutator not implemented: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">other</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="SwapGate"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.SwapGate">[docs]</a><span class="k">class</span> <span class="nc">SwapGate</span><span class="p">(</span><span class="n">TwoQubitGate</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Two qubit SWAP gate.</span>
-
-<span class="sd">    This gate swap the values of the two qubits.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    label : tuple</span>
-<span class="sd">        A tuple of the form (target1, target2).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    --------</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">gate_name</span> <span class="o">=</span> <span class="s">&#39;SWAP&#39;</span>
-    <span class="n">gate_name_latex</span> <span class="o">=</span> <span class="s">&#39;SWAP&#39;</span>
-
-    <span class="k">def</span> <span class="nf">get_target_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="s">&#39;sympy&#39;</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">matrix_cache</span><span class="o">.</span><span class="n">get_matrix</span><span class="p">(</span><span class="s">&#39;SWAP&#39;</span><span class="p">,</span> <span class="n">format</span><span class="p">)</span>
-
-<div class="viewcode-block" id="SwapGate.decompose"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.SwapGate.decompose">[docs]</a>    <span class="k">def</span> <span class="nf">decompose</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Decompose the SWAP gate into CNOT gates.&quot;&quot;&quot;</span>
-        <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">g1</span> <span class="o">=</span> <span class="n">CNotGate</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
-        <span class="n">g2</span> <span class="o">=</span> <span class="n">CNotGate</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">g1</span><span class="o">*</span><span class="n">g2</span><span class="o">*</span><span class="n">g1</span>
-</div>
-    <span class="k">def</span> <span class="nf">plot_gate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">circ_plot</span><span class="p">,</span> <span class="n">gate_idx</span><span class="p">):</span>
-        <span class="n">min_wire</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="nb">min</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">))</span>
-        <span class="n">max_wire</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">))</span>
-        <span class="n">circ_plot</span><span class="o">.</span><span class="n">control_line</span><span class="p">(</span><span class="n">gate_idx</span><span class="p">,</span> <span class="n">min_wire</span><span class="p">,</span> <span class="n">max_wire</span><span class="p">)</span>
-        <span class="n">circ_plot</span><span class="o">.</span><span class="n">swap_point</span><span class="p">(</span><span class="n">gate_idx</span><span class="p">,</span> <span class="n">min_wire</span><span class="p">)</span>
-        <span class="n">circ_plot</span><span class="o">.</span><span class="n">swap_point</span><span class="p">(</span><span class="n">gate_idx</span><span class="p">,</span> <span class="n">max_wire</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_ZGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Represent the SWAP gate in the computational basis.</span>
-
-<span class="sd">        The following representation is used to compute this:</span>
-
-<span class="sd">        SWAP = |1&gt;&lt;1|x|1&gt;&lt;1| + |0&gt;&lt;0|x|0&gt;&lt;0| + |1&gt;&lt;0|x|0&gt;&lt;1| + |0&gt;&lt;1|x|1&gt;&lt;0|</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">format</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;format&#39;</span><span class="p">,</span> <span class="s">&#39;sympy&#39;</span><span class="p">)</span>
-        <span class="n">targets</span> <span class="o">=</span> <span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">targets</span><span class="p">]</span>
-        <span class="n">min_target</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">targets</span><span class="p">)</span>
-        <span class="n">max_target</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">targets</span><span class="p">)</span>
-        <span class="n">nqubits</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;nqubits&#39;</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">min_qubits</span><span class="p">)</span>
-
-        <span class="n">op01</span> <span class="o">=</span> <span class="n">matrix_cache</span><span class="o">.</span><span class="n">get_matrix</span><span class="p">(</span><span class="s">&#39;op01&#39;</span><span class="p">,</span> <span class="n">format</span><span class="p">)</span>
-        <span class="n">op10</span> <span class="o">=</span> <span class="n">matrix_cache</span><span class="o">.</span><span class="n">get_matrix</span><span class="p">(</span><span class="s">&#39;op10&#39;</span><span class="p">,</span> <span class="n">format</span><span class="p">)</span>
-        <span class="n">op11</span> <span class="o">=</span> <span class="n">matrix_cache</span><span class="o">.</span><span class="n">get_matrix</span><span class="p">(</span><span class="s">&#39;op11&#39;</span><span class="p">,</span> <span class="n">format</span><span class="p">)</span>
-        <span class="n">op00</span> <span class="o">=</span> <span class="n">matrix_cache</span><span class="o">.</span><span class="n">get_matrix</span><span class="p">(</span><span class="s">&#39;op00&#39;</span><span class="p">,</span> <span class="n">format</span><span class="p">)</span>
-        <span class="n">eye2</span> <span class="o">=</span> <span class="n">matrix_cache</span><span class="o">.</span><span class="n">get_matrix</span><span class="p">(</span><span class="s">&#39;eye2&#39;</span><span class="p">,</span> <span class="n">format</span><span class="p">)</span>
-
-        <span class="n">result</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="ow">in</span> <span class="p">((</span><span class="n">op01</span><span class="p">,</span> <span class="n">op10</span><span class="p">),</span> <span class="p">(</span><span class="n">op10</span><span class="p">,</span> <span class="n">op01</span><span class="p">),</span> <span class="p">(</span><span class="n">op00</span><span class="p">,</span> <span class="n">op00</span><span class="p">),</span> <span class="p">(</span><span class="n">op11</span><span class="p">,</span> <span class="n">op11</span><span class="p">)):</span>
-            <span class="n">product</span> <span class="o">=</span> <span class="n">nqubits</span><span class="o">*</span><span class="p">[</span><span class="n">eye2</span><span class="p">]</span>
-            <span class="n">product</span><span class="p">[</span><span class="n">nqubits</span> <span class="o">-</span> <span class="n">min_target</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">i</span>
-            <span class="n">product</span><span class="p">[</span><span class="n">nqubits</span> <span class="o">-</span> <span class="n">max_target</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">j</span>
-            <span class="n">new_result</span> <span class="o">=</span> <span class="n">matrix_tensor_product</span><span class="p">(</span><span class="o">*</span><span class="n">product</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">new_result</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">result</span> <span class="o">+</span> <span class="n">new_result</span>
-
-        <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="c"># Aliases for gate names.</span></div>
-<span class="n">CNOT</span> <span class="o">=</span> <span class="n">CNotGate</span>
-<span class="n">SWAP</span> <span class="o">=</span> <span class="n">SwapGate</span>
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># Represent</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-
-<span class="k">def</span> <span class="nf">represent_zbasis</span><span class="p">(</span><span class="n">controls</span><span class="p">,</span> <span class="n">targets</span><span class="p">,</span> <span class="n">target_matrix</span><span class="p">,</span> <span class="n">nqubits</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="s">&#39;sympy&#39;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Represent a gate with controls, targets and target_matrix.</span>
-
-<span class="sd">    This function does the low-level work of representing gates as matrices</span>
-<span class="sd">    in the standard computational basis (ZGate). Currently, we support two</span>
-<span class="sd">    main cases:</span>
-
-<span class="sd">    1. One target qubit and no control qubits.</span>
-<span class="sd">    2. One target qubits and multiple control qubits.</span>
-
-<span class="sd">    For the base of multiple controls, we use the following expression [1]:</span>
-
-<span class="sd">    1_{2**n} + (|1&gt;&lt;1|)^{(n-1)} x (target-matrix - 1_{2})</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    controls : list, tuple</span>
-<span class="sd">        A sequence of control qubits.</span>
-<span class="sd">    targets : list, tuple</span>
-<span class="sd">        A sequence of target qubits.</span>
-<span class="sd">    target_matrix : sympy.Matrix, numpy.matrix, scipy.sparse</span>
-<span class="sd">        The matrix form of the transformation to be performed on the target</span>
-<span class="sd">        qubits.  The format of this matrix must match that passed into</span>
-<span class="sd">        the `format` argument.</span>
-<span class="sd">    nqubits : int</span>
-<span class="sd">        The total number of qubits used for the representation.</span>
-<span class="sd">    format : str</span>
-<span class="sd">        The format of the final matrix (&#39;sympy&#39;, &#39;numpy&#39;, &#39;scipy.sparse&#39;).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    --------</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ----------</span>
-<span class="sd">    [1] http://www.johnlapeyre.com/qinf/qinf_html/node6.html.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">controls</span> <span class="o">=</span> <span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">controls</span><span class="p">]</span>
-    <span class="n">targets</span> <span class="o">=</span> <span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">targets</span><span class="p">]</span>
-    <span class="n">nqubits</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">nqubits</span><span class="p">)</span>
-
-    <span class="c"># This checks for the format as well.</span>
-    <span class="n">op11</span> <span class="o">=</span> <span class="n">matrix_cache</span><span class="o">.</span><span class="n">get_matrix</span><span class="p">(</span><span class="s">&#39;op11&#39;</span><span class="p">,</span> <span class="n">format</span><span class="p">)</span>
-    <span class="n">eye2</span> <span class="o">=</span> <span class="n">matrix_cache</span><span class="o">.</span><span class="n">get_matrix</span><span class="p">(</span><span class="s">&#39;eye2&#39;</span><span class="p">,</span> <span class="n">format</span><span class="p">)</span>
-
-    <span class="c"># Plain single qubit case</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">controls</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">targets</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">product</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">bit</span> <span class="o">=</span> <span class="n">targets</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="c"># Fill product with [I1,Gate,I2] such that the unitaries,</span>
-        <span class="c"># I, cause the gate to be applied to the correct Qubit</span>
-        <span class="k">if</span> <span class="n">bit</span> <span class="o">!=</span> <span class="n">nqubits</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">product</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">matrix_eye</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="n">nqubits</span> <span class="o">-</span> <span class="n">bit</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">format</span><span class="o">=</span><span class="n">format</span><span class="p">))</span>
-        <span class="n">product</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">target_matrix</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">bit</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">product</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">matrix_eye</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">bit</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="n">format</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">matrix_tensor_product</span><span class="p">(</span><span class="o">*</span><span class="n">product</span><span class="p">)</span>
-
-    <span class="c"># Single target, multiple controls.</span>
-    <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">targets</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">controls</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">target</span> <span class="o">=</span> <span class="n">targets</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="c"># Build the non-trivial part.</span>
-        <span class="n">product2</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nqubits</span><span class="p">):</span>
-            <span class="n">product2</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">matrix_eye</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="n">format</span><span class="p">))</span>
-        <span class="k">for</span> <span class="n">control</span> <span class="ow">in</span> <span class="n">controls</span><span class="p">:</span>
-            <span class="n">product2</span><span class="p">[</span><span class="n">nqubits</span> <span class="o">-</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">control</span><span class="p">]</span> <span class="o">=</span> <span class="n">op11</span>
-        <span class="n">product2</span><span class="p">[</span><span class="n">nqubits</span> <span class="o">-</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">target</span><span class="p">]</span> <span class="o">=</span> <span class="n">target_matrix</span> <span class="o">-</span> <span class="n">eye2</span>
-
-        <span class="k">return</span> <span class="n">matrix_eye</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">nqubits</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="n">format</span><span class="p">)</span> <span class="o">+</span> \
-            <span class="n">matrix_tensor_product</span><span class="p">(</span><span class="o">*</span><span class="n">product2</span><span class="p">)</span>
-
-    <span class="c"># Multi-target, multi-control is not yet implemented.</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-            <span class="s">&#39;The representation of multi-target, multi-control gates &#39;</span>
-            <span class="s">&#39;is not implemented.&#39;</span>
-        <span class="p">)</span>
-
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># Gate manipulation functions.</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-
-<div class="viewcode-block" id="gate_simp"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.gate_simp">[docs]</a><span class="k">def</span> <span class="nf">gate_simp</span><span class="p">(</span><span class="n">circuit</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Simplifies gates symbolically</span>
-
-<span class="sd">    It first sorts gates using gate_sort. It then applies basic</span>
-<span class="sd">    simplification rules to the circuit, e.g., XGate**2 = Identity</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="c"># Bubble sort out gates that commute.</span>
-    <span class="n">circuit</span> <span class="o">=</span> <span class="n">gate_sort</span><span class="p">(</span><span class="n">circuit</span><span class="p">)</span>
-
-    <span class="c"># Do simplifications by subing a simplification into the first element</span>
-    <span class="c"># which can be simplified. We recursively call gate_simp with new circuit</span>
-    <span class="c"># as input more simplifications exist.</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">circuit</span><span class="p">,</span> <span class="n">Add</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="n">gate_simp</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">circuit</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">circuit</span><span class="p">,</span> <span class="n">Mul</span><span class="p">):</span>
-        <span class="n">circuit_args</span> <span class="o">=</span> <span class="n">circuit</span><span class="o">.</span><span class="n">args</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">circuit</span><span class="p">,</span> <span class="n">Pow</span><span class="p">):</span>
-        <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">circuit</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-        <span class="n">circuit_args</span> <span class="o">=</span> <span class="p">(</span><span class="n">gate_simp</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">e</span><span class="p">,)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">circuit</span>
-
-    <span class="c"># Iterate through each element in circuit, simplify if possible.</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">circuit_args</span><span class="p">)):</span>
-        <span class="c"># H,X,Y or Z squared is 1.</span>
-        <span class="c"># T**2 = S, S**2 = Z</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">circuit_args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">Pow</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">circuit_args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">base</span><span class="p">,</span>
-                <span class="p">(</span><span class="n">HadamardGate</span><span class="p">,</span> <span class="n">XGate</span><span class="p">,</span> <span class="n">YGate</span><span class="p">,</span> <span class="n">ZGate</span><span class="p">))</span> \
-                    <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">circuit_args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">exp</span><span class="p">,</span> <span class="n">Number</span><span class="p">):</span>
-                <span class="c"># Build a new circuit taking replacing the</span>
-                <span class="c"># H,X,Y,Z squared with one.</span>
-                <span class="n">newargs</span> <span class="o">=</span> <span class="p">(</span><span class="n">circuit_args</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span>
-                          <span class="p">(</span><span class="n">circuit_args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">base</span><span class="o">**</span><span class="p">(</span><span class="n">circuit_args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">exp</span> <span class="o">%</span> <span class="mi">2</span><span class="p">),)</span> <span class="o">+</span>
-                           <span class="n">circuit_args</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:])</span>
-                <span class="c"># Recursively simplify the new circuit.</span>
-                <span class="n">circuit</span> <span class="o">=</span> <span class="n">gate_simp</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">newargs</span><span class="p">))</span>
-                <span class="k">break</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">circuit_args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">PhaseGate</span><span class="p">):</span>
-                <span class="c"># Build a new circuit taking old circuit but splicing</span>
-                <span class="c"># in simplification.</span>
-                <span class="n">newargs</span> <span class="o">=</span> <span class="n">circuit_args</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span>
-                <span class="c"># Replace PhaseGate**2 with ZGate.</span>
-                <span class="n">newargs</span> <span class="o">=</span> <span class="n">newargs</span> <span class="o">+</span> <span class="p">(</span><span class="n">ZGate</span><span class="p">(</span><span class="n">circuit_args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">**</span>
-                <span class="p">(</span><span class="n">Integer</span><span class="p">(</span><span class="n">circuit_args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">exp</span><span class="o">/</span><span class="mi">2</span><span class="p">)),</span> <span class="n">circuit_args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">base</span><span class="o">**</span>
-                <span class="p">(</span><span class="n">circuit_args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">exp</span> <span class="o">%</span> <span class="mi">2</span><span class="p">))</span>
-                <span class="c"># Append the last elements.</span>
-                <span class="n">newargs</span> <span class="o">=</span> <span class="n">newargs</span> <span class="o">+</span> <span class="n">circuit_args</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-                <span class="c"># Recursively simplify the new circuit.</span>
-                <span class="n">circuit</span> <span class="o">=</span> <span class="n">gate_simp</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">newargs</span><span class="p">))</span>
-                <span class="k">break</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">circuit_args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">TGate</span><span class="p">):</span>
-                <span class="c"># Build a new circuit taking all the old elements.</span>
-                <span class="n">newargs</span> <span class="o">=</span> <span class="n">circuit_args</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span>
-
-                <span class="c"># Put an Phasegate in place of any TGate**2.</span>
-                <span class="n">newargs</span> <span class="o">=</span> <span class="n">newargs</span> <span class="o">+</span> <span class="p">(</span><span class="n">PhaseGate</span><span class="p">(</span><span class="n">circuit_args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">**</span>
-                <span class="n">Integer</span><span class="p">(</span><span class="n">circuit_args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">exp</span><span class="o">/</span><span class="mi">2</span><span class="p">),</span> <span class="n">circuit_args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">base</span><span class="o">**</span>
-                    <span class="p">(</span><span class="n">circuit_args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">exp</span> <span class="o">%</span> <span class="mi">2</span><span class="p">))</span>
-
-                <span class="c"># Append the last elements.</span>
-                <span class="n">newargs</span> <span class="o">=</span> <span class="n">newargs</span> <span class="o">+</span> <span class="n">circuit_args</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-                <span class="c"># Recursively simplify the new circuit.</span>
-                <span class="n">circuit</span> <span class="o">=</span> <span class="n">gate_simp</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">newargs</span><span class="p">))</span>
-                <span class="k">break</span>
-    <span class="k">return</span> <span class="n">circuit</span>
-
-</div>
-<div class="viewcode-block" id="gate_sort"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.gate_sort">[docs]</a><span class="k">def</span> <span class="nf">gate_sort</span><span class="p">(</span><span class="n">circuit</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Sorts the gates while keeping track of commutation relations</span>
-
-<span class="sd">    This function uses a bubble sort to rearrange the order of gate</span>
-<span class="sd">    application. Keeps track of Quantum computations special commutation</span>
-<span class="sd">    relations (e.g. things that apply to the same Qubit do not commute with</span>
-<span class="sd">    each other)</span>
-
-<span class="sd">    circuit is the Mul of gates that are to be sorted.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># Make sure we have an Add or Mul.</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">circuit</span><span class="p">,</span> <span class="n">Add</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="n">gate_sort</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">circuit</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">circuit</span><span class="p">,</span> <span class="n">Pow</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">gate_sort</span><span class="p">(</span><span class="n">circuit</span><span class="o">.</span><span class="n">base</span><span class="p">)</span><span class="o">**</span><span class="n">circuit</span><span class="o">.</span><span class="n">exp</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">circuit</span><span class="p">,</span> <span class="n">Gate</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">circuit</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">circuit</span><span class="p">,</span> <span class="n">Mul</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">circuit</span>
-
-    <span class="n">changes</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="k">while</span> <span class="n">changes</span><span class="p">:</span>
-        <span class="n">changes</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="n">circ_array</span> <span class="o">=</span> <span class="n">circuit</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">circ_array</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="c"># Go through each element and switch ones that are in wrong order</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">circ_array</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="p">(</span><span class="n">Gate</span><span class="p">,</span> <span class="n">Pow</span><span class="p">))</span> <span class="ow">and</span> \
-                    <span class="nb">isinstance</span><span class="p">(</span><span class="n">circ_array</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="p">(</span><span class="n">Gate</span><span class="p">,</span> <span class="n">Pow</span><span class="p">)):</span>
-                <span class="c"># If we have a Pow object, look at only the base</span>
-                <span class="n">first_base</span><span class="p">,</span> <span class="n">first_exp</span> <span class="o">=</span> <span class="n">circ_array</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                <span class="n">second_base</span><span class="p">,</span> <span class="n">second_exp</span> <span class="o">=</span> <span class="n">circ_array</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-
-                <span class="c"># Use sympy&#39;s hash based sorting. This is not mathematical</span>
-                <span class="c"># sorting, but is rather based on comparing hashes of objects.</span>
-                <span class="c"># See Basic.compare for details.</span>
-                <span class="k">if</span> <span class="n">first_base</span><span class="o">.</span><span class="n">compare</span><span class="p">(</span><span class="n">second_base</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">Commutator</span><span class="p">(</span><span class="n">first_base</span><span class="p">,</span> <span class="n">second_base</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">new_args</span> <span class="o">=</span> <span class="p">(</span><span class="n">circuit</span><span class="o">.</span><span class="n">args</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="p">(</span><span class="n">circuit</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],)</span> <span class="o">+</span>
-                                   <span class="p">(</span><span class="n">circuit</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],)</span> <span class="o">+</span> <span class="n">circuit</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">2</span><span class="p">:])</span>
-                        <span class="n">circuit</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">new_args</span><span class="p">)</span>
-                        <span class="n">circ_array</span> <span class="o">=</span> <span class="n">circuit</span><span class="o">.</span><span class="n">args</span>
-                        <span class="n">changes</span> <span class="o">=</span> <span class="bp">True</span>
-                        <span class="k">break</span>
-                    <span class="k">if</span> <span class="n">AntiCommutator</span><span class="p">(</span><span class="n">first_base</span><span class="p">,</span> <span class="n">second_base</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">new_args</span> <span class="o">=</span> <span class="p">(</span><span class="n">circuit</span><span class="o">.</span><span class="n">args</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="p">(</span><span class="n">circuit</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],)</span> <span class="o">+</span>
-                                   <span class="p">(</span><span class="n">circuit</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],)</span> <span class="o">+</span> <span class="n">circuit</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">2</span><span class="p">:])</span>
-                        <span class="n">sign</span> <span class="o">=</span> <span class="n">Integer</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">first_exp</span><span class="o">*</span><span class="n">second_exp</span><span class="p">)</span>
-                        <span class="n">circuit</span> <span class="o">=</span> <span class="n">sign</span><span class="o">*</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">new_args</span><span class="p">)</span>
-                        <span class="n">circ_array</span> <span class="o">=</span> <span class="n">circuit</span><span class="o">.</span><span class="n">args</span>
-                        <span class="n">changes</span> <span class="o">=</span> <span class="bp">True</span>
-                        <span class="k">break</span>
-    <span class="k">return</span> <span class="n">circuit</span>
-
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># Utility functions</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-</div>
-<div class="viewcode-block" id="random_circuit"><a class="viewcode-back" href="../../../../modules/physics/quantum/gate.html#sympy.physics.quantum.gate.random_circuit">[docs]</a><span class="k">def</span> <span class="nf">random_circuit</span><span class="p">(</span><span class="n">ngates</span><span class="p">,</span> <span class="n">nqubits</span><span class="p">,</span> <span class="n">gate_space</span><span class="o">=</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">Z</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">CNOT</span><span class="p">,</span> <span class="n">SWAP</span><span class="p">)):</span>
-    <span class="sd">&quot;&quot;&quot;Return a random circuit of ngates and nqubits.</span>
-
-<span class="sd">    This uses an equally weighted sample of (X, Y, Z, S, T, H, CNOT, SWAP)</span>
-<span class="sd">    gates.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    ngates : int</span>
-<span class="sd">        The number of gates in the circuit.</span>
-<span class="sd">    nqubits : int</span>
-<span class="sd">        The number of qubits in the circuit.</span>
-<span class="sd">    gate_space : tuple</span>
-<span class="sd">        A tuple of the gate classes that will be used in the circuit.</span>
-<span class="sd">        Repeating gate classes multiple times in this tuple will increase</span>
-<span class="sd">        the frequency they appear in the random circuit.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">qubit_space</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">nqubits</span><span class="p">))</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">ngates</span><span class="p">):</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">gate_space</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">g</span> <span class="o">==</span> <span class="n">CNotGate</span> <span class="ow">or</span> <span class="n">g</span> <span class="o">==</span> <span class="n">SwapGate</span><span class="p">:</span>
-            <span class="n">qubits</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">qubit_space</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="p">(</span><span class="o">*</span><span class="n">qubits</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">qubit</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">qubit_space</span><span class="p">)</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="p">(</span><span class="n">qubit</span><span class="p">)</span>
-        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">result</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">zx_basis_transform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="s">&#39;sympy&#39;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Transformation matrix from Z to X basis.&quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">matrix_cache</span><span class="o">.</span><span class="n">get_matrix</span><span class="p">(</span><span class="s">&#39;ZX&#39;</span><span class="p">,</span> <span class="n">format</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">zy_basis_transform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="s">&#39;sympy&#39;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Transformation matrix from Z to Y basis.&quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">matrix_cache</span><span class="o">.</span><span class="n">get_matrix</span><span class="p">(</span><span class="s">&#39;ZY&#39;</span><span class="p">,</span> <span class="n">format</span><span class="p">)</span>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.physics.quantum.grover</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Grover&#39;s algorithm and helper functions.</span>
-
-<span class="sd">Todo:</span>
-
-<span class="sd">* W gate construction (or perhaps -W gate based on Mermin&#39;s book)</span>
-<span class="sd">* Generalize the algorithm for an unknown function that returns 1 on multiple</span>
-<span class="sd">  qubit states, not just one.</span>
-<span class="sd">* Implement _represent_ZGate in OracleGate</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">floor</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="nb">callable</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.qapply</span> <span class="kn">import</span> <span class="n">qapply</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.qexpr</span> <span class="kn">import</span> <span class="n">QuantumError</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.hilbert</span> <span class="kn">import</span> <span class="n">ComplexSpace</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.operator</span> <span class="kn">import</span> <span class="n">UnitaryOperator</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.gate</span> <span class="kn">import</span> <span class="n">Gate</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.qubit</span> <span class="kn">import</span> <span class="n">IntQubit</span>
-<span class="kn">import</span> <span class="nn">collections</span>
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="s">&#39;OracleGate&#39;</span><span class="p">,</span>
-    <span class="s">&#39;WGate&#39;</span><span class="p">,</span>
-    <span class="s">&#39;superposition_basis&#39;</span><span class="p">,</span>
-    <span class="s">&#39;grover_iteration&#39;</span><span class="p">,</span>
-    <span class="s">&#39;apply_grover&#39;</span>
-<span class="p">]</span>
-
-
-<div class="viewcode-block" id="superposition_basis"><a class="viewcode-back" href="../../../../modules/physics/quantum/grover.html#sympy.physics.quantum.grover.superposition_basis">[docs]</a><span class="k">def</span> <span class="nf">superposition_basis</span><span class="p">(</span><span class="n">nqubits</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Creates an equal superposition of the computational basis.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    nqubits : int</span>
-<span class="sd">        The number of qubits.</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    state : Qubit</span>
-<span class="sd">        An equal superposition of the computational basis with nqubits.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Create an equal superposition of 2 qubits::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.grover import superposition_basis</span>
-<span class="sd">        &gt;&gt;&gt; superposition_basis(2)</span>
-<span class="sd">        |0&gt;/2 + |1&gt;/2 + |2&gt;/2 + |3&gt;/2</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">amp</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">nqubits</span><span class="p">)</span>
-    <span class="k">return</span> <span class="nb">sum</span><span class="p">([</span><span class="n">amp</span><span class="o">*</span><span class="n">IntQubit</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">nqubits</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">nqubits</span><span class="p">)])</span>
-
-</div>
-<div class="viewcode-block" id="OracleGate"><a class="viewcode-back" href="../../../../modules/physics/quantum/grover.html#sympy.physics.quantum.grover.OracleGate">[docs]</a><span class="k">class</span> <span class="nc">OracleGate</span><span class="p">(</span><span class="n">Gate</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A black box gate.</span>
-
-<span class="sd">    The gate marks the desired qubits of an unknown function by flipping</span>
-<span class="sd">    the sign of the qubits.  The unknown function returns true when it</span>
-<span class="sd">    finds its desired qubits and false otherwise.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    qubits : int</span>
-<span class="sd">        Number of qubits.</span>
-
-<span class="sd">    oracle : callable</span>
-<span class="sd">        A callable function that returns a boolean on a computational basis.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Apply an Oracle gate that flips the sign of ``|2&gt;`` on different qubits::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.qubit import IntQubit</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.qapply import qapply</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.grover import OracleGate</span>
-<span class="sd">        &gt;&gt;&gt; f = lambda qubits: qubits == IntQubit(2)</span>
-<span class="sd">        &gt;&gt;&gt; v = OracleGate(2, f)</span>
-<span class="sd">        &gt;&gt;&gt; qapply(v*IntQubit(2))</span>
-<span class="sd">        -|2&gt;</span>
-<span class="sd">        &gt;&gt;&gt; qapply(v*IntQubit(3))</span>
-<span class="sd">        |3&gt;</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">gate_name</span> <span class="o">=</span> <span class="s">&#39;V&#39;</span>
-    <span class="n">gate_name_latex</span> <span class="o">=</span> <span class="s">&#39;V&#39;</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Initialization/creation</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_args</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="c"># TODO: args[1] is not a subclass of Basic</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">QuantumError</span><span class="p">(</span>
-                <span class="s">&#39;Insufficient/excessive arguments to Oracle.  Please &#39;</span> <span class="o">+</span>
-                <span class="s">&#39;supply the number of qubits and an unknown function.&#39;</span>
-            <span class="p">)</span>
-        <span class="n">sub_args</span> <span class="o">=</span> <span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],)</span>
-        <span class="n">sub_args</span> <span class="o">=</span> <span class="n">UnitaryOperator</span><span class="o">.</span><span class="n">_eval_args</span><span class="p">(</span><span class="n">sub_args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">sub_args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Integer expected, got: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">sub_args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">collections</span><span class="o">.</span><span class="n">Callable</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Callable expected, got: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">sub_args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_hilbert_space</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;This returns the smallest possible Hilbert space.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">ComplexSpace</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Properties</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="OracleGate.search_function"><a class="viewcode-back" href="../../../../modules/physics/quantum/grover.html#sympy.physics.quantum.grover.OracleGate.search_function">[docs]</a>    <span class="k">def</span> <span class="nf">search_function</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The unknown function that helps find the sought after qubits.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="OracleGate.targets"><a class="viewcode-back" href="../../../../modules/physics/quantum/grover.html#sympy.physics.quantum.grover.OracleGate.targets">[docs]</a>    <span class="k">def</span> <span class="nf">targets</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A tuple of target qubits.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="nb">tuple</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])))</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Apply</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-</div>
-    <span class="k">def</span> <span class="nf">_apply_operator_Qubit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">qubits</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Apply this operator to a Qubit subclass.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        qubits : Qubit</span>
-<span class="sd">            The qubit subclass to apply this operator to.</span>
-
-<span class="sd">        Returns</span>
-<span class="sd">        =======</span>
-
-<span class="sd">        state : Expr</span>
-<span class="sd">            The resulting quantum state.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">qubits</span><span class="o">.</span><span class="n">nqubits</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">nqubits</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">QuantumError</span><span class="p">(</span>
-                <span class="s">&#39;OracleGate operates on </span><span class="si">%r</span><span class="s"> qubits, got: </span><span class="si">%r</span><span class="s">&#39;</span>
-                <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">nqubits</span><span class="p">,</span> <span class="n">qubits</span><span class="o">.</span><span class="n">nqubits</span><span class="p">)</span>
-            <span class="p">)</span>
-        <span class="c"># If function returns 1 on qubits</span>
-            <span class="c"># return the negative of the qubits (flip the sign)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">search_function</span><span class="p">(</span><span class="n">qubits</span><span class="p">):</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">qubits</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">qubits</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Represent</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="k">def</span> <span class="nf">_represent_ZGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-            <span class="s">&quot;Represent for the Oracle has not been implemented yet&quot;</span>
-        <span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="WGate"><a class="viewcode-back" href="../../../../modules/physics/quantum/grover.html#sympy.physics.quantum.grover.WGate">[docs]</a><span class="k">class</span> <span class="nc">WGate</span><span class="p">(</span><span class="n">Gate</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;General n qubit W Gate in Grover&#39;s algorithm.</span>
-
-<span class="sd">    The gate performs the operation ``2|phi&gt;&lt;phi| - 1`` on some qubits.</span>
-<span class="sd">    ``|phi&gt; = (tensor product of n Hadamards)*(|0&gt; with n qubits)``</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    nqubits : int</span>
-<span class="sd">        The number of qubits to operate on</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">gate_name</span> <span class="o">=</span> <span class="s">&#39;W&#39;</span>
-    <span class="n">gate_name_latex</span> <span class="o">=</span> <span class="s">&#39;W&#39;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_args</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">QuantumError</span><span class="p">(</span>
-                <span class="s">&#39;Insufficient/excessive arguments to W gate.  Please &#39;</span> <span class="o">+</span>
-                <span class="s">&#39;supply the number of qubits to operate on.&#39;</span>
-            <span class="p">)</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="n">UnitaryOperator</span><span class="o">.</span><span class="n">_eval_args</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Integer expected, got: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">args</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Properties</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">targets</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="nb">tuple</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])))))</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Apply</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_Qubit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">qubits</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        qubits: a set of qubits (Qubit)</span>
-<span class="sd">        Returns: quantum object (quantum expression - QExpr)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">qubits</span><span class="o">.</span><span class="n">nqubits</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">nqubits</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">QuantumError</span><span class="p">(</span>
-                <span class="s">&#39;WGate operates on </span><span class="si">%r</span><span class="s"> qubits, got: </span><span class="si">%r</span><span class="s">&#39;</span>
-                <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">nqubits</span><span class="p">,</span> <span class="n">qubits</span><span class="o">.</span><span class="n">nqubits</span><span class="p">)</span>
-            <span class="p">)</span>
-
-        <span class="c"># See &#39;Quantum Computer Science&#39; by David Mermin p.92 -&gt; W|a&gt; result</span>
-        <span class="c"># Return (2/(sqrt(2^n)))|phi&gt; - |a&gt; where |a&gt; is the current basis</span>
-        <span class="c"># state and phi is the superposition of basis states (see function</span>
-        <span class="c"># create_computational_basis above)</span>
-        <span class="n">basis_states</span> <span class="o">=</span> <span class="n">superposition_basis</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">nqubits</span><span class="p">)</span>
-        <span class="n">change_to_basis</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">nqubits</span><span class="p">))</span><span class="o">*</span><span class="n">basis_states</span>
-        <span class="k">return</span> <span class="n">change_to_basis</span> <span class="o">-</span> <span class="n">qubits</span>
-
-</div>
-<div class="viewcode-block" id="grover_iteration"><a class="viewcode-back" href="../../../../modules/physics/quantum/grover.html#sympy.physics.quantum.grover.grover_iteration">[docs]</a><span class="k">def</span> <span class="nf">grover_iteration</span><span class="p">(</span><span class="n">qstate</span><span class="p">,</span> <span class="n">oracle</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Applies one application of the Oracle and W Gate, WV.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    qstate : Qubit</span>
-<span class="sd">        A superposition of qubits.</span>
-<span class="sd">    oracle : OracleGate</span>
-<span class="sd">        The black box operator that flips the sign of the desired basis qubits.</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    Qubit : The qubits after applying the Oracle and W gate.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Perform one iteration of grover&#39;s algorithm to see a phase change::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.qapply import qapply</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.qubit import IntQubit</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.grover import OracleGate</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.grover import superposition_basis</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.grover import grover_iteration</span>
-<span class="sd">        &gt;&gt;&gt; numqubits = 2</span>
-<span class="sd">        &gt;&gt;&gt; basis_states = superposition_basis(numqubits)</span>
-<span class="sd">        &gt;&gt;&gt; f = lambda qubits: qubits == IntQubit(2)</span>
-<span class="sd">        &gt;&gt;&gt; v = OracleGate(numqubits, f)</span>
-<span class="sd">        &gt;&gt;&gt; qapply(grover_iteration(basis_states, v))</span>
-<span class="sd">        |2&gt;</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">wgate</span> <span class="o">=</span> <span class="n">WGate</span><span class="p">(</span><span class="n">oracle</span><span class="o">.</span><span class="n">nqubits</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">wgate</span><span class="o">*</span><span class="n">oracle</span><span class="o">*</span><span class="n">qstate</span>
-
-</div>
-<div class="viewcode-block" id="apply_grover"><a class="viewcode-back" href="../../../../modules/physics/quantum/grover.html#sympy.physics.quantum.grover.apply_grover">[docs]</a><span class="k">def</span> <span class="nf">apply_grover</span><span class="p">(</span><span class="n">oracle</span><span class="p">,</span> <span class="n">nqubits</span><span class="p">,</span> <span class="n">iterations</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Applies grover&#39;s algorithm.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    oracle : callable</span>
-<span class="sd">        The unknown callable function that returns true when applied to the</span>
-<span class="sd">        desired qubits and false otherwise.</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    state : Expr</span>
-<span class="sd">        The resulting state after Grover&#39;s algorithm has been iterated.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Apply grover&#39;s algorithm to an even superposition of 2 qubits::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.qapply import qapply</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.qubit import IntQubit</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.grover import apply_grover</span>
-<span class="sd">        &gt;&gt;&gt; f = lambda qubits: qubits == IntQubit(2)</span>
-<span class="sd">        &gt;&gt;&gt; qapply(apply_grover(f, 2))</span>
-<span class="sd">        |2&gt;</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">nqubits</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">QuantumError</span><span class="p">(</span>
-            <span class="s">&#39;Grover</span><span class="se">\&#39;</span><span class="s">s algorithm needs nqubits &gt; 0, received </span><span class="si">%r</span><span class="s"> qubits&#39;</span>
-            <span class="o">%</span> <span class="n">nqubits</span>
-        <span class="p">)</span>
-    <span class="k">if</span> <span class="n">iterations</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">iterations</span> <span class="o">=</span> <span class="n">floor</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">nqubits</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">))</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">OracleGate</span><span class="p">(</span><span class="n">nqubits</span><span class="p">,</span> <span class="n">oracle</span><span class="p">)</span>
-    <span class="n">iterated</span> <span class="o">=</span> <span class="n">superposition_basis</span><span class="p">(</span><span class="n">nqubits</span><span class="p">)</span>
-    <span class="k">for</span> <span class="nb">iter</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">iterations</span><span class="p">):</span>
-        <span class="n">iterated</span> <span class="o">=</span> <span class="n">grover_iteration</span><span class="p">(</span><span class="n">iterated</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
-        <span class="n">iterated</span> <span class="o">=</span> <span class="n">qapply</span><span class="p">(</span><span class="n">iterated</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">iterated</span></div>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../../index.html">
-              <img class="logo" src="../../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-<div id="searchbox" style="display: none">
-  <h3>Quick search</h3>
-    <form class="search" action="../../../../search.html" method="get">
-      <input type="text" name="q" />
-      <input type="submit" value="Go" />
-      <input type="hidden" name="check_keywords" value="yes" />
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diff --git a/dev-py3k/_modules/sympy/physics/quantum/hilbert.html b/dev-py3k/_modules/sympy/physics/quantum/hilbert.html
deleted file mode 100644
index 90c02819db3..00000000000
--- a/dev-py3k/_modules/sympy/physics/quantum/hilbert.html
+++ /dev/null
@@ -1,769 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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-  <h1>Source code for sympy.physics.quantum.hilbert</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Hilbert spaces for quantum mechanics.</span>
-
-<span class="sd">Authors:</span>
-<span class="sd">* Brian Granger</span>
-<span class="sd">* Matt Curry</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">Interval</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.pretty.stringpict</span> <span class="kn">import</span> <span class="n">prettyForm</span>
-
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.qexpr</span> <span class="kn">import</span> <span class="n">QuantumError</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="nb">reduce</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="s">&#39;HilbertSpaceError&#39;</span><span class="p">,</span>
-    <span class="s">&#39;HilbertSpace&#39;</span><span class="p">,</span>
-    <span class="s">&#39;ComplexSpace&#39;</span><span class="p">,</span>
-    <span class="s">&#39;L2&#39;</span><span class="p">,</span>
-    <span class="s">&#39;FockSpace&#39;</span>
-<span class="p">]</span>
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># Main objects</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">HilbertSpaceError</span><span class="p">(</span><span class="n">QuantumError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># Main objects</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-
-<div class="viewcode-block" id="HilbertSpace"><a class="viewcode-back" href="../../../../modules/physics/quantum/hilbert.html#sympy.physics.quantum.hilbert.HilbertSpace">[docs]</a><span class="k">class</span> <span class="nc">HilbertSpace</span><span class="p">(</span><span class="n">Basic</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;An abstract Hilbert space for quantum mechanics.</span>
-
-<span class="sd">    In short, a Hilbert space is an abstract vector space that is complete</span>
-<span class="sd">    with inner products defined [1]_.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.hilbert import HilbertSpace</span>
-<span class="sd">    &gt;&gt;&gt; hs = HilbertSpace()</span>
-<span class="sd">    &gt;&gt;&gt; hs</span>
-<span class="sd">    H</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Hilbert_space</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">):</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="HilbertSpace.dimension"><a class="viewcode-back" href="../../../../modules/physics/quantum/hilbert.html#sympy.physics.quantum.hilbert.HilbertSpace.dimension">[docs]</a>    <span class="k">def</span> <span class="nf">dimension</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the Hilbert dimension of the space.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;This Hilbert space has no dimension.&#39;</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">DirectSumHilbertSpace</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__radd__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">DirectSumHilbertSpace</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">TensorProductHilbertSpace</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">TensorProductHilbertSpace</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__pow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="n">mod</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">mod</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;The third argument to __pow__ is not supported </span><span class="se">\</span>
-<span class="s">            for Hilbert spaces.&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">TensorPowerHilbertSpace</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__contains__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Is the operator or state in this Hilbert space.</span>
-
-<span class="sd">        This is checked by comparing the classes of the Hilbert spaces, not</span>
-<span class="sd">        the instances. This is to allow Hilbert Spaces with symbolic</span>
-<span class="sd">        dimensions.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">hilbert_space</span><span class="o">.</span><span class="n">__class__</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">_sympystr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;H&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="c"># u = u&#39;\u2108&#39; # script</span>
-        <span class="n">u</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\u0048</span><span class="s">&#39;</span>
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&#39;\mathcal{H}&#39;</span>
-
-</div>
-<div class="viewcode-block" id="ComplexSpace"><a class="viewcode-back" href="../../../../modules/physics/quantum/hilbert.html#sympy.physics.quantum.hilbert.ComplexSpace">[docs]</a><span class="k">class</span> <span class="nc">ComplexSpace</span><span class="p">(</span><span class="n">HilbertSpace</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Finite dimensional Hilbert space of complex vectors.</span>
-
-<span class="sd">    The elements of this Hilbert space are n-dimensional complex valued</span>
-<span class="sd">    vectors with the usual inner product that takes the complex conjugate</span>
-<span class="sd">    of the vector on the right.</span>
-
-<span class="sd">    A classic example of this type of Hilbert space is spin-1/2, which is</span>
-<span class="sd">    ``ComplexSpace(2)``. Generalizing to spin-s, the space is</span>
-<span class="sd">    ``ComplexSpace(2*s+1)``.  Quantum computing with N qubits is done with the</span>
-<span class="sd">    direct product space ``ComplexSpace(2)**N``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.hilbert import ComplexSpace</span>
-<span class="sd">    &gt;&gt;&gt; c1 = ComplexSpace(2)</span>
-<span class="sd">    &gt;&gt;&gt; c1</span>
-<span class="sd">    C(2)</span>
-<span class="sd">    &gt;&gt;&gt; c1.dimension</span>
-<span class="sd">    2</span>
-
-<span class="sd">    &gt;&gt;&gt; n = symbols(&#39;n&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; c2 = ComplexSpace(n)</span>
-<span class="sd">    &gt;&gt;&gt; c2</span>
-<span class="sd">    C(n)</span>
-<span class="sd">    &gt;&gt;&gt; c2.dimension</span>
-<span class="sd">    n</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">dimension</span><span class="p">):</span>
-        <span class="n">dimension</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">dimension</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="n">dimension</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">r</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">dimension</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">dimension</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">dimension</span><span class="o">.</span><span class="n">atoms</span><span class="p">())</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">dimension</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">dimension</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">dimension</span> <span class="ow">is</span> <span class="n">oo</span>
-            <span class="ow">or</span> <span class="n">dimension</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;The dimension of a ComplexSpace can only&#39;</span>
-                                <span class="s">&#39;be a positive integer, oo, or a Symbol: </span><span class="si">%r</span><span class="s">&#39;</span>
-                                <span class="o">%</span> <span class="n">dimension</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">dim</span> <span class="ow">in</span> <span class="n">dimension</span><span class="o">.</span><span class="n">atoms</span><span class="p">():</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">dim</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">or</span> <span class="n">dim</span> <span class="ow">is</span> <span class="n">oo</span> <span class="ow">or</span> <span class="n">dim</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">):</span>
-                    <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;The dimension of a ComplexSpace can only&#39;</span>
-                                    <span class="s">&#39; contain integers, oo, or a Symbol: </span><span class="si">%r</span><span class="s">&#39;</span>
-                                    <span class="o">%</span> <span class="n">dim</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">dimension</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">_sympyrepr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span>
-                           <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dimension</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_sympystr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;C(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dimension</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="c"># u = u&#39;\u2102&#39; # script</span>
-        <span class="n">u</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\u0043</span><span class="s">&#39;</span>
-        <span class="n">pform_exp</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dimension</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">pform_base</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">pform_base</span><span class="o">**</span><span class="n">pform_exp</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&#39;\mathcal{C}^{</span><span class="si">%s</span><span class="s">}&#39;</span> <span class="o">%</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dimension</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="L2"><a class="viewcode-back" href="../../../../modules/physics/quantum/hilbert.html#sympy.physics.quantum.hilbert.L2">[docs]</a><span class="k">class</span> <span class="nc">L2</span><span class="p">(</span><span class="n">HilbertSpace</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The Hilbert space of square integrable functions on an interval.</span>
-
-<span class="sd">    An L2 object takes in a single sympy Interval argument which represents</span>
-<span class="sd">    the interval its functions (vectors) are defined on.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Interval, oo</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.hilbert import L2</span>
-<span class="sd">    &gt;&gt;&gt; hs = L2(Interval(0,oo))</span>
-<span class="sd">    &gt;&gt;&gt; hs</span>
-<span class="sd">    L2([0, oo))</span>
-<span class="sd">    &gt;&gt;&gt; hs.dimension</span>
-<span class="sd">    oo</span>
-<span class="sd">    &gt;&gt;&gt; hs.interval</span>
-<span class="sd">    [0, oo)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">interval</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">interval</span><span class="p">,</span> <span class="n">Interval</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;L2 interval must be an Interval instance: </span><span class="si">%r</span><span class="s">&#39;</span>
-            <span class="o">%</span> <span class="n">interval</span><span class="p">)</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">interval</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">dimension</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">oo</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">interval</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">_sympyrepr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;L2(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">interval</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sympystr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;L2(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">interval</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">pform_exp</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&quot;2&quot;</span><span class="p">)</span>
-        <span class="n">pform_base</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&quot;L&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">pform_base</span><span class="o">**</span><span class="n">pform_exp</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">interval</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">interval</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">r&#39;{\mathcal{L}^2}\left( </span><span class="si">%s</span><span class="s"> \right)&#39;</span> <span class="o">%</span> <span class="n">interval</span>
-
-</div>
-<div class="viewcode-block" id="FockSpace"><a class="viewcode-back" href="../../../../modules/physics/quantum/hilbert.html#sympy.physics.quantum.hilbert.FockSpace">[docs]</a><span class="k">class</span> <span class="nc">FockSpace</span><span class="p">(</span><span class="n">HilbertSpace</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The Hilbert space for second quantization.</span>
-
-<span class="sd">    Technically, this Hilbert space is a infinite direct sum of direct</span>
-<span class="sd">    products of single particle Hilbert spaces [1]_. This is a mess, so we have</span>
-<span class="sd">    a class to represent it directly.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.hilbert import FockSpace</span>
-<span class="sd">    &gt;&gt;&gt; hs = FockSpace()</span>
-<span class="sd">    &gt;&gt;&gt; hs</span>
-<span class="sd">    F</span>
-<span class="sd">    &gt;&gt;&gt; hs.dimension</span>
-<span class="sd">    oo</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Fock_space</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">):</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">dimension</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">oo</span>
-
-    <span class="k">def</span> <span class="nf">_sympyrepr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;FockSpace()&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_sympystr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;F&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="c"># u = u&#39;\u2131&#39; # script</span>
-        <span class="n">u</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\u0046</span><span class="s">&#39;</span>
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&#39;\mathcal{F}&#39;</span>
-
-</div>
-<span class="k">class</span> <span class="nc">TensorProductHilbertSpace</span><span class="p">(</span><span class="n">HilbertSpace</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A tensor product of Hilbert spaces [1]_.</span>
-
-<span class="sd">    The tensor product between Hilbert spaces is represented by the</span>
-<span class="sd">    operator ``*`` Products of the same Hilbert space will be combined into</span>
-<span class="sd">    tensor powers.</span>
-
-<span class="sd">    A ``TensorProductHilbertSpace`` object takes in an arbitrary number of</span>
-<span class="sd">    ``HilbertSpace`` objects as its arguments. In addition, multiplication of</span>
-<span class="sd">    ``HilbertSpace`` objects will automatically return this tensor product</span>
-<span class="sd">    object.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.hilbert import ComplexSpace, FockSpace</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-
-<span class="sd">    &gt;&gt;&gt; c = ComplexSpace(2)</span>
-<span class="sd">    &gt;&gt;&gt; f = FockSpace()</span>
-<span class="sd">    &gt;&gt;&gt; hs = c*f</span>
-<span class="sd">    &gt;&gt;&gt; hs</span>
-<span class="sd">    C(2)*F</span>
-<span class="sd">    &gt;&gt;&gt; hs.dimension</span>
-<span class="sd">    oo</span>
-<span class="sd">    &gt;&gt;&gt; hs.spaces</span>
-<span class="sd">    (C(2), F)</span>
-
-<span class="sd">    &gt;&gt;&gt; c1 = ComplexSpace(2)</span>
-<span class="sd">    &gt;&gt;&gt; n = symbols(&#39;n&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; c2 = ComplexSpace(n)</span>
-<span class="sd">    &gt;&gt;&gt; hs = c1*c2</span>
-<span class="sd">    &gt;&gt;&gt; hs</span>
-<span class="sd">    C(2)*C(n)</span>
-<span class="sd">    &gt;&gt;&gt; hs.dimension</span>
-<span class="sd">    2*n</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Hilbert_space#Tensor_products</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">r</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Evaluates the direct product.&quot;&quot;&quot;</span>
-        <span class="n">new_args</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">recall</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="c">#flatten arguments</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">TensorProductHilbertSpace</span><span class="p">):</span>
-                <span class="n">new_args</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-                <span class="n">recall</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="p">(</span><span class="n">HilbertSpace</span><span class="p">,</span> <span class="n">TensorPowerHilbertSpace</span><span class="p">)):</span>
-                <span class="n">new_args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Hilbert spaces can only be multiplied by </span><span class="se">\</span>
-<span class="s">                other Hilbert spaces: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">arg</span><span class="p">)</span>
-        <span class="c">#combine like arguments into direct powers</span>
-        <span class="n">comb_args</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">prev_arg</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">for</span> <span class="n">new_arg</span> <span class="ow">in</span> <span class="n">new_args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">prev_arg</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">new_arg</span><span class="p">,</span> <span class="n">TensorPowerHilbertSpace</span><span class="p">)</span> <span class="ow">and</span> \
-                    <span class="nb">isinstance</span><span class="p">(</span><span class="n">prev_arg</span><span class="p">,</span> <span class="n">TensorPowerHilbertSpace</span><span class="p">)</span> <span class="ow">and</span> \
-                        <span class="n">new_arg</span><span class="o">.</span><span class="n">base</span> <span class="o">==</span> <span class="n">prev_arg</span><span class="o">.</span><span class="n">base</span><span class="p">:</span>
-                    <span class="n">prev_arg</span> <span class="o">=</span> <span class="n">new_arg</span><span class="o">.</span><span class="n">base</span><span class="o">**</span><span class="p">(</span><span class="n">new_arg</span><span class="o">.</span><span class="n">exp</span> <span class="o">+</span> <span class="n">prev_arg</span><span class="o">.</span><span class="n">exp</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">new_arg</span><span class="p">,</span> <span class="n">TensorPowerHilbertSpace</span><span class="p">)</span> <span class="ow">and</span> \
-                        <span class="n">new_arg</span><span class="o">.</span><span class="n">base</span> <span class="o">==</span> <span class="n">prev_arg</span><span class="p">:</span>
-                    <span class="n">prev_arg</span> <span class="o">=</span> <span class="n">prev_arg</span><span class="o">**</span><span class="p">(</span><span class="n">new_arg</span><span class="o">.</span><span class="n">exp</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">prev_arg</span><span class="p">,</span> <span class="n">TensorPowerHilbertSpace</span><span class="p">)</span> <span class="ow">and</span> \
-                        <span class="n">new_arg</span> <span class="o">==</span> <span class="n">prev_arg</span><span class="o">.</span><span class="n">base</span><span class="p">:</span>
-                    <span class="n">prev_arg</span> <span class="o">=</span> <span class="n">new_arg</span><span class="o">**</span><span class="p">(</span><span class="n">prev_arg</span><span class="o">.</span><span class="n">exp</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="n">new_arg</span> <span class="o">==</span> <span class="n">prev_arg</span><span class="p">:</span>
-                    <span class="n">prev_arg</span> <span class="o">=</span> <span class="n">new_arg</span><span class="o">**</span><span class="mi">2</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">comb_args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">prev_arg</span><span class="p">)</span>
-                    <span class="n">prev_arg</span> <span class="o">=</span> <span class="n">new_arg</span>
-            <span class="k">elif</span> <span class="n">prev_arg</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">prev_arg</span> <span class="o">=</span> <span class="n">new_arg</span>
-        <span class="n">comb_args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">prev_arg</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">recall</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">TensorProductHilbertSpace</span><span class="p">(</span><span class="o">*</span><span class="n">comb_args</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">comb_args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">TensorPowerHilbertSpace</span><span class="p">(</span><span class="n">comb_args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">comb_args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">dimension</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">arg_list</span> <span class="o">=</span> <span class="p">[</span><span class="n">arg</span><span class="o">.</span><span class="n">dimension</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">oo</span> <span class="ow">in</span> <span class="n">arg_list</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">oo</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">reduce</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="n">arg_list</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">spaces</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A tuple of the Hilbert spaces in this tensor product.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-
-    <span class="k">def</span> <span class="nf">_spaces_printer</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">spaces_strs</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">DirectSumHilbertSpace</span><span class="p">):</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="s">&#39;(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="n">s</span>
-            <span class="n">spaces_strs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">spaces_strs</span>
-
-    <span class="k">def</span> <span class="nf">_sympyrepr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">spaces_reprs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_spaces_printer</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&quot;TensorProductHilbertSpace(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="s">&#39;,&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">spaces_reprs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sympystr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">spaces_strs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_spaces_printer</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&#39;*&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">spaces_strs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">length</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="s">&#39;&#39;</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">length</span><span class="p">):</span>
-            <span class="n">next_pform</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="p">(</span><span class="n">DirectSumHilbertSpace</span><span class="p">,</span>
-                          <span class="n">TensorProductHilbertSpace</span><span class="p">)):</span>
-                <span class="n">next_pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span>
-                    <span class="o">*</span><span class="n">next_pform</span><span class="o">.</span><span class="n">parens</span><span class="p">(</span><span class="n">left</span><span class="o">=</span><span class="s">&#39;(&#39;</span><span class="p">,</span> <span class="n">right</span><span class="o">=</span><span class="s">&#39;)&#39;</span><span class="p">)</span>
-                <span class="p">)</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">next_pform</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="n">length</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">printer</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-                    <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\u2a02</span><span class="s">&#39;</span> <span class="o">+</span> <span class="s">&#39; &#39;</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; x &#39;</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">length</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">length</span><span class="p">):</span>
-            <span class="n">arg_s</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="p">(</span><span class="n">DirectSumHilbertSpace</span><span class="p">,</span>
-                 <span class="n">TensorProductHilbertSpace</span><span class="p">)):</span>
-                <span class="n">arg_s</span> <span class="o">=</span> <span class="s">r&#39;\left(</span><span class="si">%s</span><span class="s">\right)&#39;</span> <span class="o">%</span> <span class="n">arg_s</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">s</span> <span class="o">+</span> <span class="n">arg_s</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="n">length</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">s</span> <span class="o">+</span> <span class="s">r&#39;\otimes &#39;</span>
-        <span class="k">return</span> <span class="n">s</span>
-
-
-<span class="k">class</span> <span class="nc">DirectSumHilbertSpace</span><span class="p">(</span><span class="n">HilbertSpace</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A direct sum of Hilbert spaces [1]_.</span>
-
-<span class="sd">    This class uses the ``+`` operator to represent direct sums between</span>
-<span class="sd">    different Hilbert spaces.</span>
-
-<span class="sd">    A ``DirectSumHilbertSpace`` object takes in an arbitrary number of</span>
-<span class="sd">    ``HilbertSpace`` objects as its arguments. Also, addition of</span>
-<span class="sd">    ``HilbertSpace`` objects will automatically return a direct sum object.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.hilbert import ComplexSpace, FockSpace</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-
-<span class="sd">    &gt;&gt;&gt; c = ComplexSpace(2)</span>
-<span class="sd">    &gt;&gt;&gt; f = FockSpace()</span>
-<span class="sd">    &gt;&gt;&gt; hs = c+f</span>
-<span class="sd">    &gt;&gt;&gt; hs</span>
-<span class="sd">    C(2)+F</span>
-<span class="sd">    &gt;&gt;&gt; hs.dimension</span>
-<span class="sd">    oo</span>
-<span class="sd">    &gt;&gt;&gt; list(hs.spaces)</span>
-<span class="sd">    [C(2), F]</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Hilbert_space#Direct_sums</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">r</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Evaluates the direct product.&quot;&quot;&quot;</span>
-        <span class="n">new_args</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">recall</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="c">#flatten arguments</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">DirectSumHilbertSpace</span><span class="p">):</span>
-                <span class="n">new_args</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-                <span class="n">recall</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">HilbertSpace</span><span class="p">):</span>
-                <span class="n">new_args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Hilbert spaces can only be summed with other </span><span class="se">\</span>
-<span class="s">                Hilbert spaces: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">arg</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">recall</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">DirectSumHilbertSpace</span><span class="p">(</span><span class="o">*</span><span class="n">new_args</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">dimension</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">arg_list</span> <span class="o">=</span> <span class="p">[</span><span class="n">arg</span><span class="o">.</span><span class="n">dimension</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">oo</span> <span class="ow">in</span> <span class="n">arg_list</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">oo</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">reduce</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">,</span> <span class="n">arg_list</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">spaces</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A tuple of the Hilbert spaces in this direct sum.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-
-    <span class="k">def</span> <span class="nf">_sympyrepr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">spaces_reprs</span> <span class="o">=</span> <span class="p">[</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="k">return</span> <span class="s">&quot;DirectSumHilbertSpace(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="s">&#39;,&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">spaces_reprs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sympystr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">spaces_strs</span> <span class="o">=</span> <span class="p">[</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="k">return</span> <span class="s">&#39;+&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">spaces_strs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">length</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="s">&#39;&#39;</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">length</span><span class="p">):</span>
-            <span class="n">next_pform</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="p">(</span><span class="n">DirectSumHilbertSpace</span><span class="p">,</span>
-                          <span class="n">TensorProductHilbertSpace</span><span class="p">)):</span>
-                <span class="n">next_pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span>
-                    <span class="o">*</span><span class="n">next_pform</span><span class="o">.</span><span class="n">parens</span><span class="p">(</span><span class="n">left</span><span class="o">=</span><span class="s">&#39;(&#39;</span><span class="p">,</span> <span class="n">right</span><span class="o">=</span><span class="s">&#39;)&#39;</span><span class="p">)</span>
-                <span class="p">)</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">next_pform</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="n">length</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">printer</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-                    <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\u2295</span><span class="s">&#39;</span> <span class="o">+</span> <span class="s">&#39; &#39;</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; + &#39;</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">length</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">length</span><span class="p">):</span>
-            <span class="n">arg_s</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="p">(</span><span class="n">DirectSumHilbertSpace</span><span class="p">,</span>
-                 <span class="n">TensorProductHilbertSpace</span><span class="p">)):</span>
-                <span class="n">arg_s</span> <span class="o">=</span> <span class="s">r&#39;\left(</span><span class="si">%s</span><span class="s">\right)&#39;</span> <span class="o">%</span> <span class="n">arg_s</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">s</span> <span class="o">+</span> <span class="n">arg_s</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="n">length</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">s</span> <span class="o">+</span> <span class="s">r&#39;\oplus &#39;</span>
-        <span class="k">return</span> <span class="n">s</span>
-
-
-<span class="k">class</span> <span class="nc">TensorPowerHilbertSpace</span><span class="p">(</span><span class="n">HilbertSpace</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;An exponentiated Hilbert space [1]_.</span>
-
-<span class="sd">    Tensor powers (repeated tensor products) are represented by the</span>
-<span class="sd">    operator ``**`` Identical Hilbert spaces that are multiplied together</span>
-<span class="sd">    will be automatically combined into a single tensor power object.</span>
-
-<span class="sd">    Any Hilbert space, product, or sum may be raised to a tensor power. The</span>
-<span class="sd">    ``TensorPowerHilbertSpace`` takes two arguments: the Hilbert space; and the</span>
-<span class="sd">    tensor power (number).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.hilbert import ComplexSpace, FockSpace</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-
-<span class="sd">    &gt;&gt;&gt; n = symbols(&#39;n&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; c = ComplexSpace(2)</span>
-<span class="sd">    &gt;&gt;&gt; hs = c**n</span>
-<span class="sd">    &gt;&gt;&gt; hs</span>
-<span class="sd">    C(2)**n</span>
-<span class="sd">    &gt;&gt;&gt; hs.dimension</span>
-<span class="sd">    2**n</span>
-
-<span class="sd">    &gt;&gt;&gt; c = ComplexSpace(2)</span>
-<span class="sd">    &gt;&gt;&gt; c*c</span>
-<span class="sd">    C(2)**2</span>
-<span class="sd">    &gt;&gt;&gt; f = FockSpace()</span>
-<span class="sd">    &gt;&gt;&gt; c*f*f</span>
-<span class="sd">    C(2)*F**2</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Hilbert_space#Tensor_products</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">r</span>
-        <span class="k">return</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">r</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="n">new_args</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sympify</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="n">exp</span> <span class="o">=</span> <span class="n">new_args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="c">#simplify hs**1 -&gt; hs</span>
-        <span class="k">if</span> <span class="n">exp</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="c">#simplify hs**0 -&gt; 1</span>
-        <span class="k">if</span> <span class="n">exp</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-        <span class="c">#check (and allow) for hs**(x+42+y...) case</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">exp</span><span class="o">.</span><span class="n">atoms</span><span class="p">())</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">exp</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">exp</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Hilbert spaces can only be raised to </span><span class="se">\</span>
-<span class="s">                positive integers or Symbols: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">power</span> <span class="ow">in</span> <span class="n">exp</span><span class="o">.</span><span class="n">atoms</span><span class="p">():</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">power</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">or</span> <span class="n">power</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">):</span>
-                    <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Tensor powers can only contain integers </span><span class="se">\</span>
-<span class="s">                    or Symbols: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">power</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">new_args</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">base</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">exp</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">dimension</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">dimension</span> <span class="o">==</span> <span class="n">oo</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">oo</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">dimension</span><span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">exp</span>
-
-    <span class="k">def</span> <span class="nf">_sympyrepr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;TensorPowerHilbertSpace(</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="p">,</span>
-        <span class="o">*</span><span class="n">args</span><span class="p">),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_sympystr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">**</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">),</span>
-        <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">pform_exp</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">printer</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="n">pform_exp</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform_exp</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\u2a02</span><span class="s">&#39;</span><span class="p">)))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">pform_exp</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform_exp</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)))</span>
-        <span class="n">pform_base</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">pform_base</span><span class="o">**</span><span class="n">pform_exp</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">base</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">exp</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">exp</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">r&#39;{</span><span class="si">%s</span><span class="s">}^{\otimes </span><span class="si">%s</span><span class="s">}&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-</pre></div>
-
-          </div>
-        </div>
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diff --git a/dev-py3k/_modules/sympy/physics/quantum/innerproduct.html b/dev-py3k/_modules/sympy/physics/quantum/innerproduct.html
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+++ /dev/null
@@ -1,254 +0,0 @@
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-            
-  <h1>Source code for sympy.physics.quantum.innerproduct</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Symbolic inner product.&quot;&quot;&quot;</span>
-
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">conjugate</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.pretty.stringpict</span> <span class="kn">import</span> <span class="n">prettyForm</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.dagger</span> <span class="kn">import</span> <span class="n">Dagger</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.state</span> <span class="kn">import</span> <span class="n">KetBase</span><span class="p">,</span> <span class="n">BraBase</span>
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="s">&#39;InnerProduct&#39;</span>
-<span class="p">]</span>
-
-
-<span class="c"># InnerProduct is not an QExpr because it is really just a regular commutative</span>
-<span class="c"># number. We have gone back and forth about this, but we gain a lot by having</span>
-<span class="c"># it subclass Expr. The main challenges were getting Dagger to work</span>
-<span class="c"># (we use _eval_conjugate) and represent (we can use atoms and subs). Having</span>
-<span class="c"># it be an Expr, mean that there are no commutative QExpr subclasses,</span>
-<span class="c"># which simplifies the design of everything.</span>
-
-<div class="viewcode-block" id="InnerProduct"><a class="viewcode-back" href="../../../../modules/physics/quantum/innerproduct.html#sympy.physics.quantum.innerproduct.InnerProduct">[docs]</a><span class="k">class</span> <span class="nc">InnerProduct</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;An unevaluated inner product between a Bra and a Ket [1].</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    bra : BraBase or subclass</span>
-<span class="sd">        The bra on the left side of the inner product.</span>
-<span class="sd">    ket : KetBase or subclass</span>
-<span class="sd">        The ket on the right side of the inner product.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Create an InnerProduct and check its properties:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum import Bra, Ket, InnerProduct</span>
-<span class="sd">        &gt;&gt;&gt; b = Bra(&#39;b&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; k = Ket(&#39;k&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; ip = b*k</span>
-<span class="sd">        &gt;&gt;&gt; ip</span>
-<span class="sd">        &lt;b|k&gt;</span>
-<span class="sd">        &gt;&gt;&gt; ip.bra</span>
-<span class="sd">        &lt;b|</span>
-<span class="sd">        &gt;&gt;&gt; ip.ket</span>
-<span class="sd">        |k&gt;</span>
-
-<span class="sd">    In simple products of kets and bras inner products will be automatically</span>
-<span class="sd">    identified and created::</span>
-
-<span class="sd">        &gt;&gt;&gt; b*k</span>
-<span class="sd">        &lt;b|k&gt;</span>
-
-<span class="sd">    But in more complex expressions, there is ambiguity in whether inner or</span>
-<span class="sd">    outer products should be created::</span>
-
-<span class="sd">        &gt;&gt;&gt; k*b*k*b</span>
-<span class="sd">        |k&gt;&lt;b|*|k&gt;*&lt;b|</span>
-
-<span class="sd">    A user can force the creation of a inner products in a complex expression</span>
-<span class="sd">    by using parentheses to group the bra and ket::</span>
-
-<span class="sd">        &gt;&gt;&gt; k*(b*k)*b</span>
-<span class="sd">        &lt;b|k&gt;*|k&gt;*&lt;b|</span>
-
-<span class="sd">    Notice how the inner product &lt;b|k&gt; moved to the left of the expression</span>
-<span class="sd">    because inner products are commutative complex numbers.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Inner_product</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_complex</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">bra</span><span class="p">,</span> <span class="n">ket</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ket</span><span class="p">,</span> <span class="n">KetBase</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;KetBase subclass expected, got: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">ket</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">bra</span><span class="p">,</span> <span class="n">BraBase</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;BraBase subclass expected, got: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">ket</span><span class="p">)</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">bra</span><span class="p">,</span> <span class="n">ket</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">bra</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">ket</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">InnerProduct</span><span class="p">(</span><span class="n">Dagger</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ket</span><span class="p">),</span> <span class="n">Dagger</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bra</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_sympyrepr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span>
-            <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bra</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ket</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_sympystr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">sbra</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bra</span><span class="p">)</span>
-        <span class="n">sket</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ket</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">|</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">sbra</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">sket</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="c"># Print state contents</span>
-        <span class="n">bra</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">bra</span><span class="o">.</span><span class="n">_print_contents_pretty</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">ket</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ket</span><span class="o">.</span><span class="n">_print_contents_pretty</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="c"># Print brackets</span>
-        <span class="n">height</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">bra</span><span class="o">.</span><span class="n">height</span><span class="p">(),</span> <span class="n">ket</span><span class="o">.</span><span class="n">height</span><span class="p">())</span>
-        <span class="n">use_unicode</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_use_unicode</span>
-        <span class="n">lbracket</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">bra</span><span class="o">.</span><span class="n">_pretty_brackets</span><span class="p">(</span><span class="n">height</span><span class="p">,</span> <span class="n">use_unicode</span><span class="p">)</span>
-        <span class="n">cbracket</span><span class="p">,</span> <span class="n">rbracket</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ket</span><span class="o">.</span><span class="n">_pretty_brackets</span><span class="p">(</span><span class="n">height</span><span class="p">,</span> <span class="n">use_unicode</span><span class="p">)</span>
-        <span class="c"># Build innerproduct</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">bra</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="n">lbracket</span><span class="p">))</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">cbracket</span><span class="p">))</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">ket</span><span class="p">))</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">rbracket</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">bra_label</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">bra</span><span class="o">.</span><span class="n">_print_contents_latex</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">ket</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ket</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">r&#39;\left\langle </span><span class="si">%s</span><span class="s"> \right. </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">bra_label</span><span class="p">,</span> <span class="n">ket</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ket</span><span class="o">.</span><span class="n">_eval_innerproduct</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bra</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">r</span> <span class="o">=</span> <span class="n">conjugate</span><span class="p">(</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">bra</span><span class="o">.</span><span class="n">dual</span><span class="o">.</span><span class="n">_eval_innerproduct</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ket</span><span class="o">.</span><span class="n">dual</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-                <span class="p">)</span>
-            <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                <span class="n">r</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">r</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">r</span>
-        <span class="k">return</span> <span class="bp">self</span></div>
-</pre></div>
-
-          </div>
-        </div>
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-            
-  <h1>Source code for sympy.physics.quantum.operator</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Quantum mechanical operators.</span>
-
-<span class="sd">TODO:</span>
-
-<span class="sd">* Fix early 0 in apply_operators.</span>
-<span class="sd">* Debug and test apply_operators.</span>
-<span class="sd">* Get cse working with classes in this file.</span>
-<span class="sd">* Doctests and documentation of special methods for InnerProduct, Commutator,</span>
-<span class="sd">  AntiCommutator, represent, apply_operators.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Derivative</span><span class="p">,</span> <span class="n">Expr</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.pretty.stringpict</span> <span class="kn">import</span> <span class="n">prettyForm</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.dagger</span> <span class="kn">import</span> <span class="n">Dagger</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.qexpr</span> <span class="kn">import</span> <span class="n">QExpr</span><span class="p">,</span> <span class="n">dispatch_method</span>
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="s">&#39;Operator&#39;</span><span class="p">,</span>
-    <span class="s">&#39;HermitianOperator&#39;</span><span class="p">,</span>
-    <span class="s">&#39;UnitaryOperator&#39;</span><span class="p">,</span>
-    <span class="s">&#39;OuterProduct&#39;</span><span class="p">,</span>
-    <span class="s">&#39;DifferentialOperator&#39;</span>
-<span class="p">]</span>
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># Operators and outer products</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-
-<div class="viewcode-block" id="Operator"><a class="viewcode-back" href="../../../../modules/physics/quantum/operator.html#sympy.physics.quantum.operator.Operator">[docs]</a><span class="k">class</span> <span class="nc">Operator</span><span class="p">(</span><span class="n">QExpr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Base class for non-commuting quantum operators.</span>
-
-<span class="sd">    An operator maps between quantum states [1]_. In quantum mechanics,</span>
-<span class="sd">    observables (including, but not limited to, measured physical values) are</span>
-<span class="sd">    represented as Hermitian operators [2]_.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    args : tuple</span>
-<span class="sd">        The list of numbers or parameters that uniquely specify the</span>
-<span class="sd">        operator. For time-dependent operators, this will include the time.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Create an operator and examine its attributes::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum import Operator</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols, I</span>
-<span class="sd">        &gt;&gt;&gt; A = Operator(&#39;A&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; A</span>
-<span class="sd">        A</span>
-<span class="sd">        &gt;&gt;&gt; A.hilbert_space</span>
-<span class="sd">        H</span>
-<span class="sd">        &gt;&gt;&gt; A.label</span>
-<span class="sd">        (A,)</span>
-<span class="sd">        &gt;&gt;&gt; A.is_commutative</span>
-<span class="sd">        False</span>
-
-<span class="sd">    Create another operator and do some arithmetic operations::</span>
-
-<span class="sd">        &gt;&gt;&gt; B = Operator(&#39;B&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; C = 2*A*A + I*B</span>
-<span class="sd">        &gt;&gt;&gt; C</span>
-<span class="sd">        2*A**2 + I*B</span>
-
-<span class="sd">    Operators don&#39;t commute::</span>
-
-<span class="sd">        &gt;&gt;&gt; A.is_commutative</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; B.is_commutative</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; A*B == B*A</span>
-<span class="sd">        False</span>
-
-<span class="sd">    Polymonials of operators respect the commutation properties::</span>
-
-<span class="sd">        &gt;&gt;&gt; e = (A+B)**3</span>
-<span class="sd">        &gt;&gt;&gt; e.expand()</span>
-<span class="sd">        A*B*A + A*B**2 + A**2*B + A**3 + B*A*B + B*A**2 + B**2*A + B**3</span>
-
-<span class="sd">    Operator inverses are handle symbolically::</span>
-
-<span class="sd">        &gt;&gt;&gt; A.inv()</span>
-<span class="sd">        A**(-1)</span>
-<span class="sd">        &gt;&gt;&gt; A*A.inv()</span>
-<span class="sd">        1</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Operator_%28physics%29</span>
-<span class="sd">    .. [2] http://en.wikipedia.org/wiki/Observable</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">default_args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="s">&quot;O&quot;</span><span class="p">,)</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Printing</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="n">_label_separator</span> <span class="o">=</span> <span class="s">&#39;,&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_print_operator_name</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="n">_print_operator_name_latex</span> <span class="o">=</span> <span class="n">_print_operator_name</span>
-
-    <span class="k">def</span> <span class="nf">_print_operator_name_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_contents</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_label</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_print_operator_name</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">),</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_print_label</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-            <span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_contents_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_label_pretty</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_operator_name_pretty</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-            <span class="n">label_pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_label_pretty</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-            <span class="n">label_pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span>
-                <span class="o">*</span><span class="n">label_pform</span><span class="o">.</span><span class="n">parens</span><span class="p">(</span><span class="n">left</span><span class="o">=</span><span class="s">&#39;(&#39;</span><span class="p">,</span> <span class="n">right</span><span class="o">=</span><span class="s">&#39;)&#39;</span><span class="p">)</span>
-            <span class="p">)</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">((</span><span class="n">label_pform</span><span class="p">)))</span>
-            <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_contents_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_label_latex</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&#39;</span><span class="si">%s</span><span class="s">\left(</span><span class="si">%s</span><span class="s">\right)&#39;</span> <span class="o">%</span> <span class="p">(</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_print_operator_name_latex</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">),</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_print_label_latex</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-            <span class="p">)</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># _eval_* methods</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Evaluate [self, other] if known, return None if not known.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dispatch_method</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s">&#39;_eval_commutator&#39;</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_anticommutator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Evaluate [self, other] if known.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dispatch_method</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s">&#39;_eval_anticommutator&#39;</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Operator application</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">dispatch_method</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s">&#39;_apply_operator&#39;</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">matrix_element</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;matrix_elements is not defined&#39;</span><span class="p">)</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Printing</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="k">def</span> <span class="nf">inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_inverse</span><span class="p">()</span>
-
-    <span class="n">inv</span> <span class="o">=</span> <span class="n">inverse</span>
-
-    <span class="k">def</span> <span class="nf">_eval_inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># TODO: make non-commutative Exprs print powers using A**-1, not 1/A.</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="HermitianOperator"><a class="viewcode-back" href="../../../../modules/physics/quantum/operator.html#sympy.physics.quantum.operator.HermitianOperator">[docs]</a><span class="k">class</span> <span class="nc">HermitianOperator</span><span class="p">(</span><span class="n">Operator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A Hermitian operator that satisfies H == Dagger(H).</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    args : tuple</span>
-<span class="sd">        The list of numbers or parameters that uniquely specify the</span>
-<span class="sd">        operator. For time-dependent operators, this will include the time.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum import Dagger, HermitianOperator</span>
-<span class="sd">    &gt;&gt;&gt; H = HermitianOperator(&#39;H&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; Dagger(H)</span>
-<span class="sd">    H</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_hermitian</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_eval_inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">UnitaryOperator</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Operator</span><span class="o">.</span><span class="n">_eval_inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">exp</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">UnitaryOperator</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">exp</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Operator</span><span class="o">.</span><span class="n">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="nb">abs</span><span class="p">(</span><span class="n">exp</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">*</span><span class="p">(</span><span class="n">Operator</span><span class="o">.</span><span class="n">_eval_inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Operator</span><span class="o">.</span><span class="n">_eval_power</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="UnitaryOperator"><a class="viewcode-back" href="../../../../modules/physics/quantum/operator.html#sympy.physics.quantum.operator.UnitaryOperator">[docs]</a><span class="k">class</span> <span class="nc">UnitaryOperator</span><span class="p">(</span><span class="n">Operator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A unitary operator that satisfies U*Dagger(U) == 1.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    args : tuple</span>
-<span class="sd">        The list of numbers or parameters that uniquely specify the</span>
-<span class="sd">        operator. For time-dependent operators, this will include the time.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum import Dagger, UnitaryOperator</span>
-<span class="sd">    &gt;&gt;&gt; U = UnitaryOperator(&#39;U&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; U*Dagger(U)</span>
-<span class="sd">    1</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_eval_adjoint</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_eval_inverse</span><span class="p">()</span>
-
-</div>
-<div class="viewcode-block" id="OuterProduct"><a class="viewcode-back" href="../../../../modules/physics/quantum/operator.html#sympy.physics.quantum.operator.OuterProduct">[docs]</a><span class="k">class</span> <span class="nc">OuterProduct</span><span class="p">(</span><span class="n">Operator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;An unevaluated outer product between a ket and bra.</span>
-
-<span class="sd">    This constructs an outer product between any subclass of ``KetBase`` and</span>
-<span class="sd">    ``BraBase`` as ``|a&gt;&lt;b|``. An ``OuterProduct`` inherits from Operator as they act as</span>
-<span class="sd">    operators in quantum expressions.  For reference see [1]_.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    ket : KetBase</span>
-<span class="sd">        The ket on the left side of the outer product.</span>
-<span class="sd">    bar : BraBase</span>
-<span class="sd">        The bra on the right side of the outer product.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Create a simple outer product by hand and take its dagger::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum import Ket, Bra, OuterProduct, Dagger</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum import Operator</span>
-
-<span class="sd">        &gt;&gt;&gt; k = Ket(&#39;k&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; b = Bra(&#39;b&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; op = OuterProduct(k, b)</span>
-<span class="sd">        &gt;&gt;&gt; op</span>
-<span class="sd">        |k&gt;&lt;b|</span>
-<span class="sd">        &gt;&gt;&gt; op.hilbert_space</span>
-<span class="sd">        H</span>
-<span class="sd">        &gt;&gt;&gt; op.ket</span>
-<span class="sd">        |k&gt;</span>
-<span class="sd">        &gt;&gt;&gt; op.bra</span>
-<span class="sd">        &lt;b|</span>
-<span class="sd">        &gt;&gt;&gt; Dagger(op)</span>
-<span class="sd">        |b&gt;&lt;k|</span>
-
-<span class="sd">    In simple products of kets and bras outer products will be automatically</span>
-<span class="sd">    identified and created::</span>
-
-<span class="sd">        &gt;&gt;&gt; k*b</span>
-<span class="sd">        |k&gt;&lt;b|</span>
-
-<span class="sd">    But in more complex expressions, outer products are not automatically</span>
-<span class="sd">    created::</span>
-
-<span class="sd">        &gt;&gt;&gt; A = Operator(&#39;A&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; A*k*b</span>
-<span class="sd">        A*|k&gt;*&lt;b|</span>
-
-<span class="sd">    A user can force the creation of an outer product in a complex expression</span>
-<span class="sd">    by using parentheses to group the ket and bra::</span>
-
-<span class="sd">        &gt;&gt;&gt; A*(k*b)</span>
-<span class="sd">        A*|k&gt;&lt;b|</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Outer_product</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">old_assumptions</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.physics.quantum.state</span> <span class="kn">import</span> <span class="n">KetBase</span><span class="p">,</span> <span class="n">BraBase</span>
-        <span class="n">ket</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">bra</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ket</span><span class="p">,</span> <span class="n">KetBase</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;KetBase subclass expected, got: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">ket</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">bra</span><span class="p">,</span> <span class="n">BraBase</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;BraBase subclass expected, got: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">ket</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">ket</span><span class="o">.</span><span class="n">dual_class</span><span class="p">()</span> <span class="o">==</span> <span class="n">bra</span><span class="o">.</span><span class="n">__class__</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span>
-                <span class="s">&#39;ket and bra are not dual classes: </span><span class="si">%r</span><span class="s">, </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span>
-                <span class="p">(</span><span class="n">ket</span><span class="o">.</span><span class="n">__class__</span><span class="p">,</span> <span class="n">bra</span><span class="o">.</span><span class="n">__class__</span><span class="p">)</span>
-            <span class="p">)</span>
-        <span class="c"># TODO: make sure the hilbert spaces of the bra and ket are compatible</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">old_assumptions</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">hilbert_space</span> <span class="o">=</span> <span class="n">ket</span><span class="o">.</span><span class="n">hilbert_space</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="OuterProduct.ket"><a class="viewcode-back" href="../../../../modules/physics/quantum/operator.html#sympy.physics.quantum.operator.OuterProduct.ket">[docs]</a>    <span class="k">def</span> <span class="nf">ket</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the ket on the left side of the outer product.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="OuterProduct.bra"><a class="viewcode-back" href="../../../../modules/physics/quantum/operator.html#sympy.physics.quantum.operator.OuterProduct.bra">[docs]</a>    <span class="k">def</span> <span class="nf">bra</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the bra on the right side of the outer product.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_adjoint</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">OuterProduct</span><span class="p">(</span><span class="n">Dagger</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bra</span><span class="p">),</span> <span class="n">Dagger</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ket</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_sympystr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ket</span><span class="p">)</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bra</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sympyrepr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span>
-            <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ket</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bra</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ket</span><span class="o">.</span><span class="n">_pretty</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bra</span><span class="o">.</span><span class="n">_pretty</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)))</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ket</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bra</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">k</span> <span class="o">+</span> <span class="n">b</span>
-
-    <span class="k">def</span> <span class="nf">_represent</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ket</span><span class="o">.</span><span class="n">_represent</span><span class="p">(</span><span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">bra</span><span class="o">.</span><span class="n">_represent</span><span class="p">(</span><span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">k</span><span class="o">*</span><span class="n">b</span>
-
-    <span class="k">def</span> <span class="nf">_eval_trace</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="c"># TODO if operands are tensorproducts this may be will be handled</span>
-        <span class="c"># differently.</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ket</span><span class="o">.</span><span class="n">_eval_trace</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bra</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="DifferentialOperator"><a class="viewcode-back" href="../../../../modules/physics/quantum/operator.html#sympy.physics.quantum.operator.DifferentialOperator">[docs]</a><span class="k">class</span> <span class="nc">DifferentialOperator</span><span class="p">(</span><span class="n">Operator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;An operator for representing the differential operator, i.e. d/dx</span>
-
-<span class="sd">    It is initialized by passing two arguments. The first is an arbitrary</span>
-<span class="sd">    expression that involves a function, such as ``Derivative(f(x), x)``. The</span>
-<span class="sd">    second is the function (e.g. ``f(x)``) which we are to replace with the</span>
-<span class="sd">    ``Wavefunction`` that this ``DifferentialOperator`` is applied to.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    expr : Expr</span>
-<span class="sd">           The arbitrary expression which the appropriate Wavefunction is to be</span>
-<span class="sd">           substituted into</span>
-
-<span class="sd">    func : Expr</span>
-<span class="sd">           A function (e.g. f(x)) which is to be replaced with the appropriate</span>
-<span class="sd">           Wavefunction when this DifferentialOperator is applied</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    You can define a completely arbitrary expression and specify where the</span>
-<span class="sd">    Wavefunction is to be substituted</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Derivative, Function, Symbol</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.operator import DifferentialOperator</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.state import Wavefunction</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.qapply import qapply</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; x = Symbol(&#39;x&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; d = DifferentialOperator(1/x*Derivative(f(x), x), f(x))</span>
-<span class="sd">    &gt;&gt;&gt; w = Wavefunction(x**2, x)</span>
-<span class="sd">    &gt;&gt;&gt; d.function</span>
-<span class="sd">    f(x)</span>
-<span class="sd">    &gt;&gt;&gt; d.variables</span>
-<span class="sd">    (x,)</span>
-<span class="sd">    &gt;&gt;&gt; qapply(d*w)</span>
-<span class="sd">    Wavefunction(2, x)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="DifferentialOperator.variables"><a class="viewcode-back" href="../../../../modules/physics/quantum/operator.html#sympy.physics.quantum.operator.DifferentialOperator.variables">[docs]</a>    <span class="k">def</span> <span class="nf">variables</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the variables with which the function in the specified</span>
-<span class="sd">        arbitrary expression is evaluated</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.operator import DifferentialOperator</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol, Function, Derivative</span>
-<span class="sd">        &gt;&gt;&gt; x = Symbol(&#39;x&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; d = DifferentialOperator(1/x*Derivative(f(x), x), f(x))</span>
-<span class="sd">        &gt;&gt;&gt; d.variables</span>
-<span class="sd">        (x,)</span>
-<span class="sd">        &gt;&gt;&gt; y = Symbol(&#39;y&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; d = DifferentialOperator(Derivative(f(x, y), x) +</span>
-<span class="sd">        ...                          Derivative(f(x, y), y), f(x, y))</span>
-<span class="sd">        &gt;&gt;&gt; d.variables</span>
-<span class="sd">        (x, y)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">args</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="DifferentialOperator.function"><a class="viewcode-back" href="../../../../modules/physics/quantum/operator.html#sympy.physics.quantum.operator.DifferentialOperator.function">[docs]</a>    <span class="k">def</span> <span class="nf">function</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the function which is to be replaced with the Wavefunction</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.operator import DifferentialOperator</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import Function, Symbol, Derivative</span>
-<span class="sd">        &gt;&gt;&gt; x = Symbol(&#39;x&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; d = DifferentialOperator(Derivative(f(x), x), f(x))</span>
-<span class="sd">        &gt;&gt;&gt; d.function</span>
-<span class="sd">        f(x)</span>
-<span class="sd">        &gt;&gt;&gt; y = Symbol(&#39;y&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; d = DifferentialOperator(Derivative(f(x, y), x) +</span>
-<span class="sd">        ...                          Derivative(f(x, y), y), f(x, y))</span>
-<span class="sd">        &gt;&gt;&gt; d.function</span>
-<span class="sd">        f(x, y)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="DifferentialOperator.expr"><a class="viewcode-back" href="../../../../modules/physics/quantum/operator.html#sympy.physics.quantum.operator.DifferentialOperator.expr">[docs]</a>    <span class="k">def</span> <span class="nf">expr</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the arbitary expression which is to have the Wavefunction</span>
-<span class="sd">        substituted into it</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.operator import DifferentialOperator</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import Function, Symbol, Derivative</span>
-<span class="sd">        &gt;&gt;&gt; x = Symbol(&#39;x&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; d = DifferentialOperator(Derivative(f(x), x), f(x))</span>
-<span class="sd">        &gt;&gt;&gt; d.expr</span>
-<span class="sd">        Derivative(f(x), x)</span>
-<span class="sd">        &gt;&gt;&gt; y = Symbol(&#39;y&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; d = DifferentialOperator(Derivative(f(x, y), x) +</span>
-<span class="sd">        ...                          Derivative(f(x, y), y), f(x, y))</span>
-<span class="sd">        &gt;&gt;&gt; d.expr</span>
-<span class="sd">        Derivative(f(x, y), x) + Derivative(f(x, y), y)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="DifferentialOperator.free_symbols"><a class="viewcode-back" href="../../../../modules/physics/quantum/operator.html#sympy.physics.quantum.operator.DifferentialOperator.free_symbols">[docs]</a>    <span class="k">def</span> <span class="nf">free_symbols</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the free symbols of the expression.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">free_symbols</span>
-</div>
-    <span class="k">def</span> <span class="nf">_apply_operator_Wavefunction</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.physics.quantum.state</span> <span class="kn">import</span> <span class="n">Wavefunction</span>
-        <span class="n">var</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">variables</span>
-        <span class="n">wf_vars</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-        <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">function</span>
-        <span class="n">new_expr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">var</span><span class="p">))</span>
-        <span class="n">new_expr</span> <span class="o">=</span> <span class="n">new_expr</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-
-        <span class="k">return</span> <span class="n">Wavefunction</span><span class="p">(</span><span class="n">new_expr</span><span class="p">,</span> <span class="o">*</span><span class="n">wf_vars</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">symbol</span><span class="p">):</span>
-        <span class="n">new_expr</span> <span class="o">=</span> <span class="n">Derivative</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="n">symbol</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">DifferentialOperator</span><span class="p">(</span><span class="n">new_expr</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Printing</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="k">def</span> <span class="nf">_print</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_print_operator_name</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">),</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_print_label</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_operator_name_pretty</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">label_pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_label_pretty</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">label_pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span>
-            <span class="o">*</span><span class="n">label_pform</span><span class="o">.</span><span class="n">parens</span><span class="p">(</span><span class="n">left</span><span class="o">=</span><span class="s">&#39;(&#39;</span><span class="p">,</span> <span class="n">right</span><span class="o">=</span><span class="s">&#39;)&#39;</span><span class="p">)</span>
-        <span class="p">)</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">((</span><span class="n">label_pform</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">pform</span></div>
-</pre></div>
-
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-  <h1>Source code for sympy.physics.quantum.operatorset</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot; A module for mapping operators to their corresponding eigenstates</span>
-<span class="sd">and vice versa</span>
-
-<span class="sd">It contains a global dictionary with eigenstate-operator pairings.</span>
-<span class="sd">If a new state-operator pair is created, this dictionary should be</span>
-<span class="sd">updated as well.</span>
-
-<span class="sd">It also contains functions operators_to_state and state_to_operators</span>
-<span class="sd">for mapping between the two. These can handle both classes and</span>
-<span class="sd">instances of operators and states. See the individual function</span>
-<span class="sd">descriptions for details.</span>
-
-<span class="sd">TODO List:</span>
-<span class="sd">- Update the dictionary with a complete list of state-operator pairs</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.cartesian</span> <span class="kn">import</span> <span class="p">(</span><span class="n">XOp</span><span class="p">,</span> <span class="n">YOp</span><span class="p">,</span> <span class="n">ZOp</span><span class="p">,</span> <span class="n">XKet</span><span class="p">,</span> <span class="n">PxOp</span><span class="p">,</span> <span class="n">PxKet</span><span class="p">,</span>
-                                             <span class="n">PositionKet3D</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.operator</span> <span class="kn">import</span> <span class="n">Operator</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.state</span> <span class="kn">import</span> <span class="n">StateBase</span><span class="p">,</span> <span class="n">BraBase</span><span class="p">,</span> <span class="n">Ket</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.spin</span> <span class="kn">import</span> <span class="p">(</span><span class="n">JxOp</span><span class="p">,</span> <span class="n">JyOp</span><span class="p">,</span> <span class="n">JzOp</span><span class="p">,</span> <span class="n">J2Op</span><span class="p">,</span> <span class="n">JxKet</span><span class="p">,</span> <span class="n">JyKet</span><span class="p">,</span>
-                                        <span class="n">JzKet</span><span class="p">)</span>
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="s">&#39;operators_to_state&#39;</span><span class="p">,</span>
-    <span class="s">&#39;state_to_operators&#39;</span>
-<span class="p">]</span>
-
-<span class="c">#state_mapping stores the mappings between states and their associated</span>
-<span class="c">#operators or tuples of operators. This should be updated when new</span>
-<span class="c">#classes are written! Entries are of the form PxKet : PxOp or</span>
-<span class="c">#something like 3DKet : (ROp, ThetaOp, PhiOp)</span>
-
-<span class="c">#frozenset is used so that the reverse mapping can be made</span>
-<span class="c">#(regular sets are not hashable because they are mutable</span>
-<span class="n">state_mapping</span> <span class="o">=</span> <span class="p">{</span> <span class="n">JxKet</span><span class="p">:</span> <span class="nb">frozenset</span><span class="p">((</span><span class="n">J2Op</span><span class="p">,</span> <span class="n">JxOp</span><span class="p">)),</span>
-                  <span class="n">JyKet</span><span class="p">:</span> <span class="nb">frozenset</span><span class="p">((</span><span class="n">J2Op</span><span class="p">,</span> <span class="n">JyOp</span><span class="p">)),</span>
-                  <span class="n">JzKet</span><span class="p">:</span> <span class="nb">frozenset</span><span class="p">((</span><span class="n">J2Op</span><span class="p">,</span> <span class="n">JzOp</span><span class="p">)),</span>
-                  <span class="n">Ket</span><span class="p">:</span> <span class="n">Operator</span><span class="p">,</span>
-                  <span class="n">PositionKet3D</span><span class="p">:</span> <span class="nb">frozenset</span><span class="p">((</span><span class="n">XOp</span><span class="p">,</span> <span class="n">YOp</span><span class="p">,</span> <span class="n">ZOp</span><span class="p">)),</span>
-                  <span class="n">PxKet</span><span class="p">:</span> <span class="n">PxOp</span><span class="p">,</span>
-                  <span class="n">XKet</span><span class="p">:</span> <span class="n">XOp</span> <span class="p">}</span>
-
-<span class="n">op_mapping</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">((</span><span class="n">v</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">state_mapping</span><span class="o">.</span><span class="n">items</span><span class="p">())</span>
-
-
-<div class="viewcode-block" id="operators_to_state"><a class="viewcode-back" href="../../../../modules/physics/quantum/operatorset.html#sympy.physics.quantum.operatorset.operators_to_state">[docs]</a><span class="k">def</span> <span class="nf">operators_to_state</span><span class="p">(</span><span class="n">operators</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Returns the eigenstate of the given operator or set of operators</span>
-
-<span class="sd">    A global function for mapping operator classes to their associated</span>
-<span class="sd">    states. It takes either an Operator or a set of operators and</span>
-<span class="sd">    returns the state associated with these.</span>
-
-<span class="sd">    This function can handle both instances of a given operator or</span>
-<span class="sd">    just the class itself (i.e. both XOp() and XOp)</span>
-
-<span class="sd">    There are multiple use cases to consider:</span>
-
-<span class="sd">    1) A class or set of classes is passed: First, we try to</span>
-<span class="sd">    instantiate default instances for these operators. If this fails,</span>
-<span class="sd">    then the class is simply returned. If we succeed in instantiating</span>
-<span class="sd">    default instances, then we try to call state._operators_to_state</span>
-<span class="sd">    on the operator instances. If this fails, the class is returned.</span>
-<span class="sd">    Otherwise, the instance returned by _operators_to_state is returned.</span>
-
-<span class="sd">    2) An instance or set of instances is passed: In this case,</span>
-<span class="sd">    state._operators_to_state is called on the instances passed. If</span>
-<span class="sd">    this fails, a state class is returned. If the method returns an</span>
-<span class="sd">    instance, that instance is returned.</span>
-
-<span class="sd">    In both cases, if the operator class or set does not exist in the</span>
-<span class="sd">    state_mapping dictionary, None is returned.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    arg: Operator or set</span>
-<span class="sd">         The class or instance of the operator or set of operators</span>
-<span class="sd">         to be mapped to a state</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.cartesian import XOp, PxOp</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.operatorset import operators_to_state</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.operator import Operator</span>
-<span class="sd">    &gt;&gt;&gt; operators_to_state(XOp)</span>
-<span class="sd">    |x&gt;</span>
-<span class="sd">    &gt;&gt;&gt; operators_to_state(XOp())</span>
-<span class="sd">    |x&gt;</span>
-<span class="sd">    &gt;&gt;&gt; operators_to_state(PxOp)</span>
-<span class="sd">    |px&gt;</span>
-<span class="sd">    &gt;&gt;&gt; operators_to_state(PxOp())</span>
-<span class="sd">    |px&gt;</span>
-<span class="sd">    &gt;&gt;&gt; operators_to_state(Operator)</span>
-<span class="sd">    |psi&gt;</span>
-<span class="sd">    &gt;&gt;&gt; operators_to_state(Operator())</span>
-<span class="sd">    |psi&gt;</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">operators</span><span class="p">,</span> <span class="n">Operator</span><span class="p">)</span>
-            <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">operators</span><span class="p">,</span> <span class="nb">set</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">operators</span><span class="p">,</span> <span class="n">Operator</span><span class="p">)):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Argument is not an Operator or a set!&quot;</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">operators</span><span class="p">,</span> <span class="nb">set</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">operators</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">Operator</span><span class="p">)</span>
-                   <span class="ow">or</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">Operator</span><span class="p">)):</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Set is not all Operators!&quot;</span><span class="p">)</span>
-
-        <span class="c">#ops = tuple(operators)</span>
-        <span class="n">ops</span> <span class="o">=</span> <span class="nb">frozenset</span><span class="p">(</span><span class="n">operators</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">ops</span> <span class="ow">in</span> <span class="n">op_mapping</span><span class="p">:</span>  <span class="c"># ops is a list of classes in this case</span>
-            <span class="c">#Try to get an object from default instances of the</span>
-            <span class="c">#operators...if this fails, return the class</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">op_instances</span> <span class="o">=</span> <span class="p">[</span><span class="n">op</span><span class="p">()</span> <span class="k">for</span> <span class="n">op</span> <span class="ow">in</span> <span class="n">ops</span><span class="p">]</span>
-                <span class="n">ret</span> <span class="o">=</span> <span class="n">_get_state</span><span class="p">(</span><span class="n">op_mapping</span><span class="p">[</span><span class="n">ops</span><span class="p">],</span> <span class="nb">set</span><span class="p">(</span><span class="n">op_instances</span><span class="p">),</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                <span class="n">ret</span> <span class="o">=</span> <span class="n">op_mapping</span><span class="p">[</span><span class="n">ops</span><span class="p">]</span>
-
-            <span class="k">return</span> <span class="n">ret</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tmp</span> <span class="o">=</span> <span class="p">[</span><span class="nb">type</span><span class="p">(</span><span class="n">o</span><span class="p">)</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">ops</span><span class="p">]</span>
-            <span class="n">classes</span> <span class="o">=</span> <span class="nb">frozenset</span><span class="p">(</span><span class="n">tmp</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">classes</span> <span class="ow">in</span> <span class="n">op_mapping</span><span class="p">:</span>
-                <span class="n">ret</span> <span class="o">=</span> <span class="n">_get_state</span><span class="p">(</span><span class="n">op_mapping</span><span class="p">[</span><span class="n">classes</span><span class="p">],</span> <span class="n">ops</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">ret</span> <span class="o">=</span> <span class="bp">None</span>
-
-            <span class="k">return</span> <span class="n">ret</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">operators</span> <span class="ow">in</span> <span class="n">op_mapping</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">op_instance</span> <span class="o">=</span> <span class="n">operators</span><span class="p">()</span>
-                <span class="n">ret</span> <span class="o">=</span> <span class="n">_get_state</span><span class="p">(</span><span class="n">op_mapping</span><span class="p">[</span><span class="n">operators</span><span class="p">],</span> <span class="n">op_instance</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                <span class="n">ret</span> <span class="o">=</span> <span class="n">op_mapping</span><span class="p">[</span><span class="n">operators</span><span class="p">]</span>
-
-            <span class="k">return</span> <span class="n">ret</span>
-        <span class="k">elif</span> <span class="nb">type</span><span class="p">(</span><span class="n">operators</span><span class="p">)</span> <span class="ow">in</span> <span class="n">op_mapping</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_get_state</span><span class="p">(</span><span class="n">op_mapping</span><span class="p">[</span><span class="nb">type</span><span class="p">(</span><span class="n">operators</span><span class="p">)],</span> <span class="n">operators</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-</div>
-<div class="viewcode-block" id="state_to_operators"><a class="viewcode-back" href="../../../../modules/physics/quantum/operatorset.html#sympy.physics.quantum.operatorset.state_to_operators">[docs]</a><span class="k">def</span> <span class="nf">state_to_operators</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Returns the operator or set of operators corresponding to the</span>
-<span class="sd">    given eigenstate</span>
-
-<span class="sd">    A global function for mapping state classes to their associated</span>
-<span class="sd">    operators or sets of operators. It takes either a state class</span>
-<span class="sd">    or instance.</span>
-
-<span class="sd">    This function can handle both instances of a given state or just</span>
-<span class="sd">    the class itself (i.e. both XKet() and XKet)</span>
-
-<span class="sd">    There are multiple use cases to consider:</span>
-
-<span class="sd">    1) A state class is passed: In this case, we first try</span>
-<span class="sd">    instantiating a default instance of the class. If this succeeds,</span>
-<span class="sd">    then we try to call state._state_to_operators on that instance.</span>
-<span class="sd">    If the creation of the default instance or if the calling of</span>
-<span class="sd">    _state_to_operators fails, then either an operator class or set of</span>
-<span class="sd">    operator classes is returned. Otherwise, the appropriate</span>
-<span class="sd">    operator instances are returned.</span>
-
-<span class="sd">    2) A state instance is returned: Here, state._state_to_operators</span>
-<span class="sd">    is called for the instance. If this fails, then a class or set of</span>
-<span class="sd">    operator classes is returned. Otherwise, the instances are returned.</span>
-
-<span class="sd">    In either case, if the state&#39;s class does not exist in</span>
-<span class="sd">    state_mapping, None is returned.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    arg: StateBase class or instance (or subclasses)</span>
-<span class="sd">         The class or instance of the state to be mapped to an</span>
-<span class="sd">         operator or set of operators</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.cartesian import XKet, PxKet, XBra, PxBra</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.operatorset import state_to_operators</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.state import Ket, Bra</span>
-<span class="sd">    &gt;&gt;&gt; state_to_operators(XKet)</span>
-<span class="sd">    X</span>
-<span class="sd">    &gt;&gt;&gt; state_to_operators(XKet())</span>
-<span class="sd">    X</span>
-<span class="sd">    &gt;&gt;&gt; state_to_operators(PxKet)</span>
-<span class="sd">    Px</span>
-<span class="sd">    &gt;&gt;&gt; state_to_operators(PxKet())</span>
-<span class="sd">    Px</span>
-<span class="sd">    &gt;&gt;&gt; state_to_operators(PxBra)</span>
-<span class="sd">    Px</span>
-<span class="sd">    &gt;&gt;&gt; state_to_operators(XBra)</span>
-<span class="sd">    X</span>
-<span class="sd">    &gt;&gt;&gt; state_to_operators(Ket)</span>
-<span class="sd">    O</span>
-<span class="sd">    &gt;&gt;&gt; state_to_operators(Bra)</span>
-<span class="sd">    O</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">StateBase</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">StateBase</span><span class="p">)):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Argument is not a state!&quot;</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">state</span> <span class="ow">in</span> <span class="n">state_mapping</span><span class="p">:</span>  <span class="c"># state is a class</span>
-        <span class="n">state_inst</span> <span class="o">=</span> <span class="n">_make_default</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">ret</span> <span class="o">=</span> <span class="n">_get_ops</span><span class="p">(</span><span class="n">state_inst</span><span class="p">,</span>
-                           <span class="n">_make_set</span><span class="p">(</span><span class="n">state_mapping</span><span class="p">[</span><span class="n">state</span><span class="p">]),</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">except</span> <span class="p">(</span><span class="ne">NotImplementedError</span><span class="p">,</span> <span class="ne">TypeError</span><span class="p">):</span>
-            <span class="n">ret</span> <span class="o">=</span> <span class="n">state_mapping</span><span class="p">[</span><span class="n">state</span><span class="p">]</span>
-    <span class="k">elif</span> <span class="nb">type</span><span class="p">(</span><span class="n">state</span><span class="p">)</span> <span class="ow">in</span> <span class="n">state_mapping</span><span class="p">:</span>
-        <span class="n">ret</span> <span class="o">=</span> <span class="n">_get_ops</span><span class="p">(</span><span class="n">state</span><span class="p">,</span>
-                       <span class="n">_make_set</span><span class="p">(</span><span class="n">state_mapping</span><span class="p">[</span><span class="nb">type</span><span class="p">(</span><span class="n">state</span><span class="p">)]),</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">BraBase</span><span class="p">)</span> <span class="ow">and</span> <span class="n">state</span><span class="o">.</span><span class="n">dual_class</span><span class="p">()</span> <span class="ow">in</span> <span class="n">state_mapping</span><span class="p">:</span>
-        <span class="n">ret</span> <span class="o">=</span> <span class="n">_get_ops</span><span class="p">(</span><span class="n">state</span><span class="p">,</span>
-                       <span class="n">_make_set</span><span class="p">(</span><span class="n">state_mapping</span><span class="p">[</span><span class="n">state</span><span class="o">.</span><span class="n">dual_class</span><span class="p">()]))</span>
-    <span class="k">elif</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">BraBase</span><span class="p">)</span> <span class="ow">and</span> <span class="n">state</span><span class="o">.</span><span class="n">dual_class</span><span class="p">()</span> <span class="ow">in</span> <span class="n">state_mapping</span><span class="p">:</span>
-        <span class="n">state_inst</span> <span class="o">=</span> <span class="n">_make_default</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">ret</span> <span class="o">=</span> <span class="n">_get_ops</span><span class="p">(</span><span class="n">state_inst</span><span class="p">,</span>
-                           <span class="n">_make_set</span><span class="p">(</span><span class="n">state_mapping</span><span class="p">[</span><span class="n">state</span><span class="o">.</span><span class="n">dual_class</span><span class="p">()]))</span>
-        <span class="k">except</span> <span class="p">(</span><span class="ne">NotImplementedError</span><span class="p">,</span> <span class="ne">TypeError</span><span class="p">):</span>
-            <span class="n">ret</span> <span class="o">=</span> <span class="n">state_mapping</span><span class="p">[</span><span class="n">state</span><span class="o">.</span><span class="n">dual_class</span><span class="p">()]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">ret</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="k">return</span> <span class="n">_make_set</span><span class="p">(</span><span class="n">ret</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_make_default</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">ret</span> <span class="o">=</span> <span class="n">expr</span><span class="p">()</span>
-    <span class="k">except</span> <span class="ne">Exception</span><span class="p">:</span>
-        <span class="n">ret</span> <span class="o">=</span> <span class="n">expr</span>
-
-    <span class="k">return</span> <span class="n">ret</span>
-
-
-<span class="k">def</span> <span class="nf">_get_state</span><span class="p">(</span><span class="n">state_class</span><span class="p">,</span> <span class="n">ops</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-    <span class="c"># Try to get a state instance from the operator INSTANCES.</span>
-    <span class="c"># If this fails, get the class</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">ret</span> <span class="o">=</span> <span class="n">state_class</span><span class="o">.</span><span class="n">_operators_to_state</span><span class="p">(</span><span class="n">ops</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-    <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-        <span class="n">ret</span> <span class="o">=</span> <span class="n">_make_default</span><span class="p">(</span><span class="n">state_class</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">ret</span>
-
-
-<span class="k">def</span> <span class="nf">_get_ops</span><span class="p">(</span><span class="n">state_inst</span><span class="p">,</span> <span class="n">op_classes</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-    <span class="c"># Try to get operator instances from the state INSTANCE.</span>
-    <span class="c"># If this fails, just return the classes</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">ret</span> <span class="o">=</span> <span class="n">state_inst</span><span class="o">.</span><span class="n">_state_to_operators</span><span class="p">(</span><span class="n">op_classes</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-    <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">op_classes</span><span class="p">,</span> <span class="p">(</span><span class="nb">set</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">,</span> <span class="nb">frozenset</span><span class="p">)):</span>
-            <span class="n">ret</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">([</span><span class="n">_make_default</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">op_classes</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">ret</span> <span class="o">=</span> <span class="n">_make_default</span><span class="p">(</span><span class="n">op_classes</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ret</span><span class="p">,</span> <span class="nb">set</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">ret</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">ret</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">return</span> <span class="n">ret</span>
-
-
-<span class="k">def</span> <span class="nf">_make_set</span><span class="p">(</span><span class="n">ops</span><span class="p">):</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ops</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="nb">list</span><span class="p">,</span> <span class="nb">frozenset</span><span class="p">)):</span>
-        <span class="k">return</span> <span class="nb">set</span><span class="p">(</span><span class="n">ops</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">ops</span>
-</pre></div>
-
-          </div>
-        </div>
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index a1ee41face0..00000000000
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-  <h1>Source code for sympy.physics.quantum.piab</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;1D quantum particle in a box.&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">Interval</span><span class="p">,</span> <span class="n">S</span>
-
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.operator</span> <span class="kn">import</span> <span class="n">HermitianOperator</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.state</span> <span class="kn">import</span> <span class="n">Ket</span><span class="p">,</span> <span class="n">Bra</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.constants</span> <span class="kn">import</span> <span class="n">hbar</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.special.tensor_functions</span> <span class="kn">import</span> <span class="n">KroneckerDelta</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.hilbert</span> <span class="kn">import</span> <span class="n">L2</span>
-
-<span class="n">m</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;m&#39;</span><span class="p">)</span>
-<span class="n">L</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;L&#39;</span><span class="p">)</span>
-
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="s">&#39;PIABHamiltonian&#39;</span><span class="p">,</span>
-    <span class="s">&#39;PIABKet&#39;</span><span class="p">,</span>
-    <span class="s">&#39;PIABBra&#39;</span>
-<span class="p">]</span>
-
-
-<div class="viewcode-block" id="PIABHamiltonian"><a class="viewcode-back" href="../../../../modules/physics/quantum/piab.html#sympy.physics.quantum.piab.PIABHamiltonian">[docs]</a><span class="k">class</span> <span class="nc">PIABHamiltonian</span><span class="p">(</span><span class="n">HermitianOperator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Particle in a box Hamiltonian operator.&quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_hilbert_space</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">label</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">L2</span><span class="p">(</span><span class="n">Interval</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_PIABKet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">ket</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">n</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">hbar</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">m</span><span class="o">*</span><span class="n">L</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">ket</span>
-
-</div>
-<div class="viewcode-block" id="PIABKet"><a class="viewcode-back" href="../../../../modules/physics/quantum/piab.html#sympy.physics.quantum.piab.PIABKet">[docs]</a><span class="k">class</span> <span class="nc">PIABKet</span><span class="p">(</span><span class="n">Ket</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Particle in a box eigenket.&quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_hilbert_space</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">L2</span><span class="p">(</span><span class="n">Interval</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">))</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">PIABBra</span>
-
-    <span class="k">def</span> <span class="nf">_represent_default_basis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_XOp</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_XOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">)</span>
-        <span class="n">subs_info</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;subs&#39;</span><span class="p">,</span> <span class="p">{})</span>
-        <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">/</span><span class="n">L</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="o">/</span><span class="n">L</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">subs_info</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_innerproduct_PIABBra</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">bra</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">bra</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-</div>
-<div class="viewcode-block" id="PIABBra"><a class="viewcode-back" href="../../../../modules/physics/quantum/piab.html#sympy.physics.quantum.piab.PIABBra">[docs]</a><span class="k">class</span> <span class="nc">PIABBra</span><span class="p">(</span><span class="n">Bra</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Particle in a box eigenbra.&quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_hilbert_space</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">label</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">L2</span><span class="p">(</span><span class="n">Interval</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">))</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">PIABKet</span></div>
-</pre></div>
-
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-        </div>
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-            
-  <h1>Source code for sympy.physics.quantum.qapply</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Logic for applying operators to states.</span>
-
-<span class="sd">Todo:</span>
-<span class="sd">* Sometimes the final result needs to be expanded, we should do this by hand.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Pow</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">S</span>
-
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.anticommutator</span> <span class="kn">import</span> <span class="n">AntiCommutator</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.commutator</span> <span class="kn">import</span> <span class="n">Commutator</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.dagger</span> <span class="kn">import</span> <span class="n">Dagger</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.innerproduct</span> <span class="kn">import</span> <span class="n">InnerProduct</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.operator</span> <span class="kn">import</span> <span class="n">OuterProduct</span><span class="p">,</span> <span class="n">Operator</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.state</span> <span class="kn">import</span> <span class="n">State</span><span class="p">,</span> <span class="n">KetBase</span><span class="p">,</span> <span class="n">BraBase</span><span class="p">,</span> <span class="n">Wavefunction</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.tensorproduct</span> <span class="kn">import</span> <span class="n">TensorProduct</span>
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="s">&#39;qapply&#39;</span>
-<span class="p">]</span>
-
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># Main code</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-<div class="viewcode-block" id="qapply"><a class="viewcode-back" href="../../../../modules/physics/quantum/qapply.html#sympy.physics.quantum.qapply.qapply">[docs]</a><span class="k">def</span> <span class="nf">qapply</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Apply operators to states in a quantum expression.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    e : Expr</span>
-<span class="sd">        The expression containing operators and states. This expression tree</span>
-<span class="sd">        will be walked to find operators acting on states symbolically.</span>
-<span class="sd">    options : dict</span>
-<span class="sd">        A dict of key/value pairs that determine how the operator actions</span>
-<span class="sd">        are carried out.</span>
-
-<span class="sd">        The following options are valid:</span>
-
-<span class="sd">        * ``dagger``: try to apply Dagger operators to the left</span>
-<span class="sd">          (default: False).</span>
-<span class="sd">        * ``ip_doit``: call ``.doit()`` in inner products when they are</span>
-<span class="sd">          encountered (default: True).</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    e : Expr</span>
-<span class="sd">        The original expression, but with the operators applied to states.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.physics.quantum.density</span> <span class="kn">import</span> <span class="n">Density</span>
-
-    <span class="n">dagger</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;dagger&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">e</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="c"># This may be a bit aggressive but ensures that everything gets expanded</span>
-    <span class="c"># to its simplest form before trying to apply operators. This includes</span>
-    <span class="c"># things like (A+B+C)*|a&gt; and A*(|a&gt;+|b&gt;) and all Commutators and</span>
-    <span class="c"># TensorProducts. The only problem with this is that if we can&#39;t apply</span>
-    <span class="c"># all the Operators, we have just expanded everything.</span>
-    <span class="c"># TODO: don&#39;t expand the scalars in front of each Mul.</span>
-    <span class="n">e</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">commutator</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">tensorproduct</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="c"># If we just have a raw ket, return it.</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">KetBase</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">e</span>
-
-    <span class="c"># We have an Add(a, b, c, ...) and compute</span>
-    <span class="c"># Add(qapply(a), qapply(b), ...)</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Add</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">+=</span> <span class="n">qapply</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="c"># For a Density operator call qapply on its state</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Density</span><span class="p">):</span>
-        <span class="n">new_args</span> <span class="o">=</span> <span class="p">[(</span><span class="n">qapply</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">),</span> <span class="n">prob</span><span class="p">)</span> <span class="k">for</span> <span class="p">(</span><span class="n">state</span><span class="p">,</span>
-                     <span class="n">prob</span><span class="p">)</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">Density</span><span class="p">(</span><span class="o">*</span><span class="n">new_args</span><span class="p">)</span>
-
-    <span class="c"># For a raw TensorProduct, call qapply on its args.</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">TensorProduct</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">TensorProduct</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">qapply</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="c"># For a Pow, call qapply on its base.</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Pow</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">qapply</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span><span class="o">**</span><span class="n">e</span><span class="o">.</span><span class="n">exp</span>
-
-    <span class="c"># We have a Mul where there might be actual operators to apply to kets.</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Mul</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">qapply_Mul</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">result</span> <span class="o">==</span> <span class="n">e</span> <span class="ow">and</span> <span class="n">dagger</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Dagger</span><span class="p">(</span><span class="n">qapply_Mul</span><span class="p">(</span><span class="n">Dagger</span><span class="p">(</span><span class="n">e</span><span class="p">),</span> <span class="o">**</span><span class="n">options</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">result</span>
-
-    <span class="c"># In all other cases (State, Operator, Pow, Commutator, InnerProduct,</span>
-    <span class="c"># OuterProduct) we won&#39;t ever have operators to apply to kets.</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">e</span>
-
-</div>
-<span class="k">def</span> <span class="nf">qapply_Mul</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-
-    <span class="n">ip_doit</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;ip_doit&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-
-    <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="c"># If we only have 0 or 1 args, we have nothing to do and return.</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">1</span> <span class="ow">or</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Mul</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">e</span>
-    <span class="n">rhs</span> <span class="o">=</span> <span class="n">args</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-    <span class="n">lhs</span> <span class="o">=</span> <span class="n">args</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-
-    <span class="c"># Make sure we have two non-commutative objects before proceeding.</span>
-    <span class="k">if</span> <span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">rhs</span><span class="p">)</span><span class="o">.</span><span class="n">is_commutative</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">rhs</span><span class="p">,</span> <span class="n">Wavefunction</span><span class="p">))</span> <span class="ow">or</span> \
-            <span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">lhs</span><span class="p">)</span><span class="o">.</span><span class="n">is_commutative</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">Wavefunction</span><span class="p">)):</span>
-        <span class="k">return</span> <span class="n">e</span>
-
-    <span class="c"># For a Pow with an integer exponent, apply one of them and reduce the</span>
-    <span class="c"># exponent by one.</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">Pow</span><span class="p">)</span> <span class="ow">and</span> <span class="n">lhs</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-        <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">lhs</span><span class="o">.</span><span class="n">base</span><span class="o">**</span><span class="p">(</span><span class="n">lhs</span><span class="o">.</span><span class="n">exp</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-        <span class="n">lhs</span> <span class="o">=</span> <span class="n">lhs</span><span class="o">.</span><span class="n">base</span>
-
-    <span class="c"># Pull OuterProduct apart</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">OuterProduct</span><span class="p">):</span>
-        <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">lhs</span><span class="o">.</span><span class="n">ket</span><span class="p">)</span>
-        <span class="n">lhs</span> <span class="o">=</span> <span class="n">lhs</span><span class="o">.</span><span class="n">bra</span>
-
-   <span class="c"># Call .doit() on Commutator/AntiCommutator.</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="p">(</span><span class="n">Commutator</span><span class="p">,</span> <span class="n">AntiCommutator</span><span class="p">)):</span>
-        <span class="n">comm</span> <span class="o">=</span> <span class="n">lhs</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">comm</span><span class="p">,</span> <span class="n">Add</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">qapply</span><span class="p">(</span>
-                <span class="n">e</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">args</span> <span class="o">+</span> <span class="p">[</span><span class="n">comm</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">rhs</span><span class="p">]))</span> <span class="o">+</span>
-                <span class="n">e</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">args</span> <span class="o">+</span> <span class="p">[</span><span class="n">comm</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">rhs</span><span class="p">])),</span>
-                <span class="o">**</span><span class="n">options</span>
-            <span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">qapply</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span><span class="o">*</span><span class="n">comm</span><span class="o">*</span><span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="c"># Apply tensor products of operators to states</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">TensorProduct</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">all</span><span class="p">([</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Operator</span><span class="p">)</span> <span class="ow">or</span> <span class="n">arg</span> <span class="o">==</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">lhs</span><span class="o">.</span><span class="n">args</span><span class="p">])</span> <span class="ow">and</span> \
-            <span class="nb">isinstance</span><span class="p">(</span><span class="n">rhs</span><span class="p">,</span> <span class="n">TensorProduct</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">all</span><span class="p">([</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">State</span><span class="p">)</span> <span class="ow">or</span> <span class="n">arg</span> <span class="o">==</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">rhs</span><span class="o">.</span><span class="n">args</span><span class="p">])</span> <span class="ow">and</span> \
-            <span class="nb">len</span><span class="p">(</span><span class="n">lhs</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">rhs</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">TensorProduct</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">qapply</span><span class="p">(</span><span class="n">lhs</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">n</span><span class="p">]</span><span class="o">*</span><span class="n">rhs</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">n</span><span class="p">],</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">lhs</span><span class="o">.</span><span class="n">args</span><span class="p">))])</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">tensorproduct</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">qapply_Mul</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">),</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span><span class="o">*</span><span class="n">result</span>
-
-    <span class="c"># Now try to actually apply the operator and build an inner product.</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">lhs</span><span class="o">.</span><span class="n">_apply_operator</span><span class="p">(</span><span class="n">rhs</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-    <span class="k">except</span> <span class="p">(</span><span class="ne">NotImplementedError</span><span class="p">,</span> <span class="ne">AttributeError</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">rhs</span><span class="o">.</span><span class="n">_apply_operator</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">except</span> <span class="p">(</span><span class="ne">NotImplementedError</span><span class="p">,</span> <span class="ne">AttributeError</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">BraBase</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">rhs</span><span class="p">,</span> <span class="n">KetBase</span><span class="p">):</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">InnerProduct</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">ip_doit</span><span class="p">:</span>
-                    <span class="n">result</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="c"># TODO: I may need to expand before returning the final result.</span>
-    <span class="k">if</span> <span class="n">result</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-    <span class="k">elif</span> <span class="n">result</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="c"># We had two args to begin with so args=[].</span>
-            <span class="k">return</span> <span class="n">e</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">qapply_Mul</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">args</span> <span class="o">+</span> <span class="p">[</span><span class="n">lhs</span><span class="p">])),</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span><span class="o">*</span><span class="n">rhs</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">InnerProduct</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">result</span><span class="o">*</span><span class="n">qapply_Mul</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">),</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>  <span class="c"># result is a scalar times a Mul, Add or TensorProduct</span>
-        <span class="k">return</span> <span class="n">qapply</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span><span class="o">*</span><span class="n">result</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-</pre></div>
-
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-  <h1>Source code for sympy.physics.quantum.qft</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;An implementation of qubits and gates acting on them.</span>
-
-<span class="sd">Todo:</span>
-
-<span class="sd">* Update docstrings.</span>
-<span class="sd">* Update tests.</span>
-<span class="sd">* Implement apply using decompose.</span>
-<span class="sd">* Implement represent using decompose or something smarter. For this to</span>
-<span class="sd">  work we first have to implement represent for SWAP.</span>
-<span class="sd">* Decide if we want upper index to be inclusive in the constructor.</span>
-<span class="sd">* Fix the printing of Rk gates in plotting.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">Integer</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">sqrt</span>
-
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.qapply</span> <span class="kn">import</span> <span class="n">qapply</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.qexpr</span> <span class="kn">import</span> <span class="n">QuantumError</span><span class="p">,</span> <span class="n">QExpr</span>
-<span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">eye</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.tensorproduct</span> <span class="kn">import</span> <span class="n">matrix_tensor_product</span>
-
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.gate</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">Gate</span><span class="p">,</span> <span class="n">HadamardGate</span><span class="p">,</span> <span class="n">SwapGate</span><span class="p">,</span> <span class="n">OneQubitGate</span><span class="p">,</span> <span class="n">CGate</span><span class="p">,</span> <span class="n">PhaseGate</span><span class="p">,</span> <span class="n">TGate</span><span class="p">,</span> <span class="n">ZGate</span>
-<span class="p">)</span>
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="s">&#39;QFT&#39;</span><span class="p">,</span>
-    <span class="s">&#39;IQFT&#39;</span><span class="p">,</span>
-    <span class="s">&#39;RkGate&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Rk&#39;</span>
-<span class="p">]</span>
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># Fourier stuff</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-
-<div class="viewcode-block" id="RkGate"><a class="viewcode-back" href="../../../../modules/physics/quantum/qft.html#sympy.physics.quantum.qft.RkGate">[docs]</a><span class="k">class</span> <span class="nc">RkGate</span><span class="p">(</span><span class="n">OneQubitGate</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;This is the R_k gate of the QTF.&quot;&quot;&quot;</span>
-    <span class="n">gate_name</span> <span class="o">=</span> <span class="s">&#39;Rk&#39;</span>
-    <span class="n">gate_name_latex</span> <span class="o">=</span> <span class="s">&#39;R&#39;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">QuantumError</span><span class="p">(</span>
-                <span class="s">&#39;Rk gates only take two arguments, got: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">args</span>
-            <span class="p">)</span>
-        <span class="c"># For small k, Rk gates simplify to other gates, using these</span>
-        <span class="c"># substitutions give us familiar results for the QFT for small numbers</span>
-        <span class="c"># of qubits.</span>
-        <span class="n">target</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">k</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">ZGate</span><span class="p">(</span><span class="n">target</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">k</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">PhaseGate</span><span class="p">(</span><span class="n">target</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">k</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">TGate</span><span class="p">(</span><span class="n">target</span><span class="p">)</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_eval_args</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">inst</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">inst</span><span class="o">.</span><span class="n">hilbert_space</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_eval_hilbert_space</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">inst</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_args</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="c"># Fall back to this, because Gate._eval_args assumes that args is</span>
-        <span class="c"># all targets and can&#39;t contain duplicates.</span>
-        <span class="k">return</span> <span class="n">QExpr</span><span class="o">.</span><span class="n">_eval_args</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">k</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">targets</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[:</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">gate_name_plot</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&#39;$</span><span class="si">%s</span><span class="s">_</span><span class="si">%s</span><span class="s">$&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gate_name_latex</span><span class="p">,</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">k</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">get_target_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="s">&#39;sympy&#39;</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">format</span> <span class="o">==</span> <span class="s">&#39;sympy&#39;</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">exp</span><span class="p">(</span><span class="n">Integer</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">/</span><span class="p">(</span><span class="n">Integer</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">k</span><span class="p">))]])</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-            <span class="s">&#39;Invalid format for the R_k gate: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">format</span><span class="p">)</span>
-
-</div>
-<span class="n">Rk</span> <span class="o">=</span> <span class="n">RkGate</span>
-
-
-<span class="k">class</span> <span class="nc">Fourier</span><span class="p">(</span><span class="n">Gate</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Superclass of Quantum Fourier and Inverse Quantum Fourier Gates.&quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_args</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">QuantumError</span><span class="p">(</span>
-                <span class="s">&#39;QFT/IQFT only takes two arguments, got: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">args</span>
-            <span class="p">)</span>
-        <span class="k">if</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&gt;=</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span>
-            <span class="k">raise</span> <span class="n">QuantumError</span><span class="p">(</span><span class="s">&quot;Start must be smaller than finish&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Gate</span><span class="o">.</span><span class="n">_eval_args</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_default_basis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_ZGate</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_ZGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            Represents the (I)QFT In the Z Basis</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">nqubits</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;nqubits&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">nqubits</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">QuantumError</span><span class="p">(</span>
-                <span class="s">&#39;The number of qubits must be given as nqubits.&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">nqubits</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">min_qubits</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">QuantumError</span><span class="p">(</span>
-                <span class="s">&#39;The number of qubits </span><span class="si">%r</span><span class="s"> is too small for the gate.&#39;</span> <span class="o">%</span> <span class="n">nqubits</span>
-            <span class="p">)</span>
-        <span class="n">size</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span>
-        <span class="n">omega</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">omega</span>
-
-        <span class="c">#Make a matrix that has the basic Fourier Transform Matrix</span>
-        <span class="n">arrayFT</span> <span class="o">=</span> <span class="p">[[</span><span class="n">omega</span><span class="o">**</span><span class="p">(</span>
-            <span class="n">i</span><span class="o">*</span><span class="n">j</span> <span class="o">%</span> <span class="n">size</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">size</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">)]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">)]</span>
-        <span class="n">matrixFT</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">arrayFT</span><span class="p">)</span>
-
-        <span class="c">#Embed the FT Matrix in a higher space, if necessary</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">matrixFT</span> <span class="o">=</span> <span class="n">matrix_tensor_product</span><span class="p">(</span><span class="n">eye</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">matrixFT</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">min_qubits</span> <span class="o">&lt;</span> <span class="n">nqubits</span><span class="p">:</span>
-            <span class="n">matrixFT</span> <span class="o">=</span> <span class="n">matrix_tensor_product</span><span class="p">(</span>
-                <span class="n">matrixFT</span><span class="p">,</span> <span class="n">eye</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="n">nqubits</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">min_qubits</span><span class="p">)))</span>
-
-        <span class="k">return</span> <span class="n">matrixFT</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">targets</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">min_qubits</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">size</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Size is the size of the QFT matrix&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">omega</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;omega&#39;</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="QFT"><a class="viewcode-back" href="../../../../modules/physics/quantum/qft.html#sympy.physics.quantum.qft.QFT">[docs]</a><span class="k">class</span> <span class="nc">QFT</span><span class="p">(</span><span class="n">Fourier</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The forward quantum Fourier transform.&quot;&quot;&quot;</span>
-
-    <span class="n">gate_name</span> <span class="o">=</span> <span class="s">&#39;QFT&#39;</span>
-    <span class="n">gate_name_latex</span> <span class="o">=</span> <span class="s">&#39;QFT&#39;</span>
-
-<div class="viewcode-block" id="QFT.decompose"><a class="viewcode-back" href="../../../../modules/physics/quantum/qft.html#sympy.physics.quantum.qft.QFT.decompose">[docs]</a>    <span class="k">def</span> <span class="nf">decompose</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Decomposes QFT into elementary gates.&quot;&quot;&quot;</span>
-        <span class="n">start</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">finish</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">circuit</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">for</span> <span class="n">level</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="n">finish</span><span class="p">))):</span>
-            <span class="n">circuit</span> <span class="o">=</span> <span class="n">HadamardGate</span><span class="p">(</span><span class="n">level</span><span class="p">)</span><span class="o">*</span><span class="n">circuit</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">level</span> <span class="o">-</span> <span class="n">start</span><span class="p">):</span>
-                <span class="n">circuit</span> <span class="o">=</span> <span class="n">CGate</span><span class="p">(</span><span class="n">level</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">RkGate</span><span class="p">(</span><span class="n">level</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">2</span><span class="p">))</span><span class="o">*</span><span class="n">circuit</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">((</span><span class="n">finish</span> <span class="o">-</span> <span class="n">start</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span><span class="p">):</span>
-            <span class="n">circuit</span> <span class="o">=</span> <span class="n">SwapGate</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="n">start</span><span class="p">,</span> <span class="n">finish</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">circuit</span>
-        <span class="k">return</span> <span class="n">circuit</span>
-</div>
-    <span class="k">def</span> <span class="nf">_apply_operator_Qubit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">qubits</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">qapply</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">decompose</span><span class="p">()</span><span class="o">*</span><span class="n">qubits</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">IQFT</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">omega</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="IQFT"><a class="viewcode-back" href="../../../../modules/physics/quantum/qft.html#sympy.physics.quantum.qft.IQFT">[docs]</a><span class="k">class</span> <span class="nc">IQFT</span><span class="p">(</span><span class="n">Fourier</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The inverse quantum Fourier transform.&quot;&quot;&quot;</span>
-
-    <span class="n">gate_name</span> <span class="o">=</span> <span class="s">&#39;IQFT&#39;</span>
-    <span class="n">gate_name_latex</span> <span class="o">=</span> <span class="s">&#39;{QFT^{-1}}&#39;</span>
-
-<div class="viewcode-block" id="IQFT.decompose"><a class="viewcode-back" href="../../../../modules/physics/quantum/qft.html#sympy.physics.quantum.qft.IQFT.decompose">[docs]</a>    <span class="k">def</span> <span class="nf">decompose</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Decomposes IQFT into elementary gates.&quot;&quot;&quot;</span>
-        <span class="n">start</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">finish</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">circuit</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">((</span><span class="n">finish</span> <span class="o">-</span> <span class="n">start</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span><span class="p">):</span>
-            <span class="n">circuit</span> <span class="o">=</span> <span class="n">SwapGate</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="n">start</span><span class="p">,</span> <span class="n">finish</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">circuit</span>
-        <span class="k">for</span> <span class="n">level</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="n">finish</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">level</span> <span class="o">-</span> <span class="n">start</span><span class="p">))):</span>
-                <span class="n">circuit</span> <span class="o">=</span> <span class="n">CGate</span><span class="p">(</span><span class="n">level</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">RkGate</span><span class="p">(</span><span class="n">level</span><span class="p">,</span> <span class="o">-</span><span class="n">i</span> <span class="o">-</span> <span class="mi">2</span><span class="p">))</span><span class="o">*</span><span class="n">circuit</span>
-            <span class="n">circuit</span> <span class="o">=</span> <span class="n">HadamardGate</span><span class="p">(</span><span class="n">level</span><span class="p">)</span><span class="o">*</span><span class="n">circuit</span>
-        <span class="k">return</span> <span class="n">circuit</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">QFT</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">omega</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">)</span></div>
-</pre></div>
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-            
-  <h1>Source code for sympy.physics.quantum.qubit</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Qubits for quantum computing.</span>
-
-<span class="sd">Todo:</span>
-<span class="sd">* Finish implementing measurement logic. This should include POVM.</span>
-<span class="sd">* Update docstrings.</span>
-<span class="sd">* Update tests.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">import</span> <span class="nn">math</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integer</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Pow</span><span class="p">,</span> <span class="n">conjugate</span>
-<span class="kn">from</span> <span class="nn">sympy.core.basic</span> <span class="kn">import</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">zeros</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.pretty.stringpict</span> <span class="kn">import</span> <span class="n">prettyForm</span>
-
-
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.hilbert</span> <span class="kn">import</span> <span class="n">ComplexSpace</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.state</span> <span class="kn">import</span> <span class="n">Ket</span><span class="p">,</span> <span class="n">Bra</span><span class="p">,</span> <span class="n">State</span>
-
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.qexpr</span> <span class="kn">import</span> <span class="n">QuantumError</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.represent</span> <span class="kn">import</span> <span class="n">represent</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.matrixutils</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">numpy_ndarray</span><span class="p">,</span> <span class="n">scipy_sparse_matrix</span>
-<span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.mpmath.libmp.libintmath</span> <span class="kn">import</span> <span class="n">bitcount</span>
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="s">&#39;Qubit&#39;</span><span class="p">,</span>
-    <span class="s">&#39;QubitBra&#39;</span><span class="p">,</span>
-    <span class="s">&#39;IntQubit&#39;</span><span class="p">,</span>
-    <span class="s">&#39;IntQubitBra&#39;</span><span class="p">,</span>
-    <span class="s">&#39;qubit_to_matrix&#39;</span><span class="p">,</span>
-    <span class="s">&#39;matrix_to_qubit&#39;</span><span class="p">,</span>
-    <span class="s">&#39;matrix_to_density&#39;</span><span class="p">,</span>
-    <span class="s">&#39;measure_all&#39;</span><span class="p">,</span>
-    <span class="s">&#39;measure_partial&#39;</span><span class="p">,</span>
-    <span class="s">&#39;measure_partial_oneshot&#39;</span><span class="p">,</span>
-    <span class="s">&#39;measure_all_oneshot&#39;</span>
-<span class="p">]</span>
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># Qubit Classes</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">QubitState</span><span class="p">(</span><span class="n">State</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Base class for Qubit and QubitBra.&quot;&quot;&quot;</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Initialization/creation</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_args</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="c"># If we are passed a QubitState or subclass, we just take its qubit</span>
-        <span class="c"># values directly.</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">QubitState</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">qubit_values</span>
-
-        <span class="c"># Turn strings into tuple of strings</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="n">args</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="c"># Validate input (must have 0 or 1 input)</span>
-        <span class="k">for</span> <span class="n">element</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">element</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">element</span> <span class="o">==</span> <span class="mi">0</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                    <span class="s">&quot;Qubit values must be 0 or 1, got: </span><span class="si">%r</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">element</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">args</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_hilbert_space</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">ComplexSpace</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Properties</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">dimension</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The number of Qubits in the state.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">qubit_values</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">nqubits</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">dimension</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">qubit_values</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the values of the qubits as a tuple.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Special methods</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="k">def</span> <span class="nf">__len__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">dimension</span>
-
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">bit</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">qubit_values</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dimension</span> <span class="o">-</span> <span class="n">bit</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)]</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Utility methods</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="k">def</span> <span class="nf">flip</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">bits</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Flip the bit(s) given.&quot;&quot;&quot;</span>
-        <span class="n">newargs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">qubit_values</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">bits</span><span class="p">:</span>
-            <span class="n">bit</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dimension</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">newargs</span><span class="p">[</span><span class="n">bit</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">newargs</span><span class="p">[</span><span class="n">bit</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">newargs</span><span class="p">[</span><span class="n">bit</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="o">*</span><span class="nb">tuple</span><span class="p">(</span><span class="n">newargs</span><span class="p">))</span>
-
-
-<div class="viewcode-block" id="Qubit"><a class="viewcode-back" href="../../../../modules/physics/quantum/qubit.html#sympy.physics.quantum.qubit.Qubit">[docs]</a><span class="k">class</span> <span class="nc">Qubit</span><span class="p">(</span><span class="n">QubitState</span><span class="p">,</span> <span class="n">Ket</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A multi-qubit ket in the computational (z) basis.</span>
-
-<span class="sd">    We use the normal convention that the least significant qubit is on the</span>
-<span class="sd">    right, so ``|00001&gt;`` has a 1 in the least significant qubit.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    values : list, str</span>
-<span class="sd">        The qubit values as a list of ints ([0,0,0,1,1,]) or a string (&#39;011&#39;).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Create a qubit in a couple of different ways and look at their attributes:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.qubit import Qubit</span>
-<span class="sd">        &gt;&gt;&gt; Qubit(0,0,0)</span>
-<span class="sd">        |000&gt;</span>
-<span class="sd">        &gt;&gt;&gt; q = Qubit(&#39;0101&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; q</span>
-<span class="sd">        |0101&gt;</span>
-
-<span class="sd">        &gt;&gt;&gt; q.nqubits</span>
-<span class="sd">        4</span>
-<span class="sd">        &gt;&gt;&gt; len(q)</span>
-<span class="sd">        4</span>
-<span class="sd">        &gt;&gt;&gt; q.dimension</span>
-<span class="sd">        4</span>
-<span class="sd">        &gt;&gt;&gt; q.qubit_values</span>
-<span class="sd">        (0, 1, 0, 1)</span>
-
-<span class="sd">    We can flip the value of an individual qubit:</span>
-
-<span class="sd">        &gt;&gt;&gt; q.flip(1)</span>
-<span class="sd">        |0111&gt;</span>
-
-<span class="sd">    We can take the dagger of a Qubit to get a bra:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.dagger import Dagger</span>
-<span class="sd">        &gt;&gt;&gt; Dagger(q)</span>
-<span class="sd">        &lt;0101|</span>
-<span class="sd">        &gt;&gt;&gt; type(Dagger(q))</span>
-<span class="sd">        &lt;class &#39;sympy.physics.quantum.qubit.QubitBra&#39;&gt;</span>
-
-<span class="sd">    Inner products work as expected:</span>
-
-<span class="sd">        &gt;&gt;&gt; ip = Dagger(q)*q</span>
-<span class="sd">        &gt;&gt;&gt; ip</span>
-<span class="sd">        &lt;0101|0101&gt;</span>
-<span class="sd">        &gt;&gt;&gt; ip.doit()</span>
-<span class="sd">        1</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">QubitBra</span>
-
-    <span class="k">def</span> <span class="nf">_eval_innerproduct_QubitBra</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">bra</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span> <span class="o">==</span> <span class="n">bra</span><span class="o">.</span><span class="n">label</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_default_basis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_ZGate</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_ZGate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Represent this qubits in the computational basis (ZGate).</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">format</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;format&#39;</span><span class="p">,</span> <span class="s">&#39;sympy&#39;</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">definite_state</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">it</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">qubit_values</span><span class="p">):</span>
-            <span class="n">definite_state</span> <span class="o">+=</span> <span class="n">n</span><span class="o">*</span><span class="n">it</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">n</span><span class="o">*</span><span class="mi">2</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">dimension</span><span class="p">)</span>
-        <span class="n">result</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">definite_state</span><span class="p">)]</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">format</span> <span class="o">==</span> <span class="s">&#39;sympy&#39;</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">format</span> <span class="o">==</span> <span class="s">&#39;numpy&#39;</span><span class="p">:</span>
-            <span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
-            <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">matrix</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="s">&#39;complex&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span>
-        <span class="k">elif</span> <span class="n">format</span> <span class="o">==</span> <span class="s">&#39;scipy.sparse&#39;</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">sparse</span>
-            <span class="k">return</span> <span class="n">sparse</span><span class="o">.</span><span class="n">csr_matrix</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="s">&#39;complex&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">_eval_trace</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">bra</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">indices</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;indices&#39;</span><span class="p">,</span> <span class="p">[])</span>
-
-        <span class="c">#sort index list to begin trace from most-significant</span>
-        <span class="c">#qubit</span>
-        <span class="n">sorted_idx</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">indices</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">sorted_idx</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">sorted_idx</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">nqubits</span><span class="p">))</span>
-        <span class="n">sorted_idx</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-
-        <span class="c">#trace out for each of index</span>
-        <span class="n">new_mat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">*</span><span class="n">bra</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">sorted_idx</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-            <span class="c"># start from tracing out from leftmost qubit</span>
-            <span class="n">new_mat</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_reduced_density</span><span class="p">(</span><span class="n">new_mat</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">sorted_idx</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span>
-
-        <span class="k">if</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">sorted_idx</span><span class="p">)</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">nqubits</span><span class="p">):</span>
-            <span class="c">#in case full trace was requested</span>
-            <span class="k">return</span> <span class="n">new_mat</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">matrix_to_density</span><span class="p">(</span><span class="n">new_mat</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_reduced_density</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">matrix</span><span class="p">,</span> <span class="n">qubit</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute the reduced density matrix by tracing out one qubit.</span>
-<span class="sd">           The qubit argument should be of type python int, since it is used</span>
-<span class="sd">           in bit operations</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">def</span> <span class="nf">find_index_that_is_projected</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">qubit</span><span class="p">):</span>
-            <span class="n">bit_mask</span> <span class="o">=</span> <span class="mi">2</span><span class="o">**</span><span class="n">qubit</span> <span class="o">-</span> <span class="mi">1</span>
-            <span class="k">return</span> <span class="p">((</span><span class="n">j</span> <span class="o">&gt;&gt;</span> <span class="n">qubit</span><span class="p">)</span> <span class="o">&lt;&lt;</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">qubit</span><span class="p">))</span> <span class="o">+</span> <span class="p">(</span><span class="n">j</span> <span class="o">&amp;</span> <span class="n">bit_mask</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">k</span> <span class="o">&lt;&lt;</span> <span class="n">qubit</span><span class="p">)</span>
-
-        <span class="n">old_matrix</span> <span class="o">=</span> <span class="n">represent</span><span class="p">(</span><span class="n">matrix</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="n">old_size</span> <span class="o">=</span> <span class="n">old_matrix</span><span class="o">.</span><span class="n">cols</span>
-        <span class="c">#we expect the old_size to be even</span>
-        <span class="n">new_size</span> <span class="o">=</span> <span class="n">old_size</span><span class="o">//</span><span class="mi">2</span>
-        <span class="n">new_matrix</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">()</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">new_size</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">new_size</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">new_size</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">):</span>
-                    <span class="n">col</span> <span class="o">=</span> <span class="n">find_index_that_is_projected</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">qubit</span><span class="p">)</span>
-                    <span class="n">row</span> <span class="o">=</span> <span class="n">find_index_that_is_projected</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">qubit</span><span class="p">)</span>
-                    <span class="n">new_matrix</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="n">old_matrix</span><span class="p">[</span><span class="n">row</span><span class="p">,</span> <span class="n">col</span><span class="p">]</span>
-
-        <span class="k">return</span> <span class="n">new_matrix</span>
-
-</div>
-<div class="viewcode-block" id="QubitBra"><a class="viewcode-back" href="../../../../modules/physics/quantum/qubit.html#sympy.physics.quantum.qubit.QubitBra">[docs]</a><span class="k">class</span> <span class="nc">QubitBra</span><span class="p">(</span><span class="n">QubitState</span><span class="p">,</span> <span class="n">Bra</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A multi-qubit bra in the computational (z) basis.</span>
-
-<span class="sd">    We use the normal convention that the least significant qubit is on the</span>
-<span class="sd">    right, so ``|00001&gt;`` has a 1 in the least significant qubit.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    values : list, str</span>
-<span class="sd">        The qubit values as a list of ints ([0,0,0,1,1,]) or a string (&#39;011&#39;).</span>
-
-<span class="sd">    See also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Qubit: Examples using qubits</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Qubit</span>
-
-</div>
-<span class="k">class</span> <span class="nc">IntQubitState</span><span class="p">(</span><span class="n">QubitState</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A base class for qubits that work with binary representations.&quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_args</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="c"># The case of a QubitState instance</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">QubitState</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">QubitState</span><span class="o">.</span><span class="n">_eval_args</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="c"># For a single argument, we construct the binary representation of</span>
-        <span class="c"># that integer with the minimal number of bits.</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c">#rvalues is the minimum number of bits needed to express the number</span>
-            <span class="n">rvalues</span> <span class="o">=</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">bitcount</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]))))</span>
-            <span class="n">qubit_values</span> <span class="o">=</span> <span class="p">[(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&gt;&gt;</span> <span class="n">i</span><span class="p">)</span> <span class="o">&amp;</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">rvalues</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">QubitState</span><span class="o">.</span><span class="n">_eval_args</span><span class="p">(</span><span class="n">qubit_values</span><span class="p">)</span>
-        <span class="c"># For two numbers, the second number is the number of bits</span>
-        <span class="c"># on which it is expressed, so IntQubit(0,5) == |00000&gt;.</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">need</span> <span class="o">=</span> <span class="n">bitcount</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-            <span class="k">if</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">need</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                    <span class="s">&#39;cannot represent </span><span class="si">%s</span><span class="s"> with </span><span class="si">%s</span><span class="s"> bits&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-            <span class="n">qubit_values</span> <span class="o">=</span> <span class="p">[(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&gt;&gt;</span> <span class="n">i</span><span class="p">)</span> <span class="o">&amp;</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])))]</span>
-            <span class="k">return</span> <span class="n">QubitState</span><span class="o">.</span><span class="n">_eval_args</span><span class="p">(</span><span class="n">qubit_values</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">QubitState</span><span class="o">.</span><span class="n">_eval_args</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">as_int</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the numerical value of the qubit.&quot;&quot;&quot;</span>
-        <span class="n">number</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">qubit_values</span><span class="p">):</span>
-            <span class="n">number</span> <span class="o">+=</span> <span class="n">n</span><span class="o">*</span><span class="n">i</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">n</span> <span class="o">&lt;&lt;</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="n">number</span>
-
-    <span class="k">def</span> <span class="nf">_print_label</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">as_int</span><span class="p">())</span>
-
-    <span class="k">def</span> <span class="nf">_print_label_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">label</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_label</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">label</span><span class="p">)</span>
-
-    <span class="n">_print_label_repr</span> <span class="o">=</span> <span class="n">_print_label</span>
-    <span class="n">_print_label_latex</span> <span class="o">=</span> <span class="n">_print_label</span>
-
-
-<div class="viewcode-block" id="IntQubit"><a class="viewcode-back" href="../../../../modules/physics/quantum/qubit.html#sympy.physics.quantum.qubit.IntQubit">[docs]</a><span class="k">class</span> <span class="nc">IntQubit</span><span class="p">(</span><span class="n">IntQubitState</span><span class="p">,</span> <span class="n">Qubit</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A qubit ket that store integers as binary numbers in qubit values.</span>
-
-<span class="sd">    The differences between this class and ``Qubit`` are:</span>
-
-<span class="sd">    * The form of the constructor.</span>
-<span class="sd">    * The qubit values are printed as their corresponding integer, rather</span>
-<span class="sd">      than the raw qubit values. The internal storage format of the qubit</span>
-<span class="sd">      values in the same as ``Qubit``.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    values : int, tuple</span>
-<span class="sd">        If a single argument, the integer we want to represent in the qubit</span>
-<span class="sd">        values. This integer will be represented using the fewest possible</span>
-<span class="sd">        number of qubits. If a pair of integers, the first integer gives the</span>
-<span class="sd">        integer to represent in binary form and the second integer gives</span>
-<span class="sd">        the number of qubits to use.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Create a qubit for the integer 5:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.qubit import IntQubit</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.qubit import Qubit</span>
-<span class="sd">        &gt;&gt;&gt; q = IntQubit(5)</span>
-<span class="sd">        &gt;&gt;&gt; q</span>
-<span class="sd">        |5&gt;</span>
-
-<span class="sd">    We can also create an ``IntQubit`` by passing a ``Qubit`` instance.</span>
-
-<span class="sd">        &gt;&gt;&gt; q = IntQubit(Qubit(&#39;101&#39;))</span>
-<span class="sd">        &gt;&gt;&gt; q</span>
-<span class="sd">        |5&gt;</span>
-<span class="sd">        &gt;&gt;&gt; q.as_int()</span>
-<span class="sd">        5</span>
-<span class="sd">        &gt;&gt;&gt; q.nqubits</span>
-<span class="sd">        3</span>
-<span class="sd">        &gt;&gt;&gt; q.qubit_values</span>
-<span class="sd">        (1, 0, 1)</span>
-
-<span class="sd">    We can go back to the regular qubit form.</span>
-
-<span class="sd">        &gt;&gt;&gt; Qubit(q)</span>
-<span class="sd">        |101&gt;</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">IntQubitBra</span>
-
-</div>
-<div class="viewcode-block" id="IntQubitBra"><a class="viewcode-back" href="../../../../modules/physics/quantum/qubit.html#sympy.physics.quantum.qubit.IntQubitBra">[docs]</a><span class="k">class</span> <span class="nc">IntQubitBra</span><span class="p">(</span><span class="n">IntQubitState</span><span class="p">,</span> <span class="n">QubitBra</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A qubit bra that store integers as binary numbers in qubit values.&quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">IntQubit</span>
-
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># Qubit &lt;---&gt; Matrix conversion functions</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-</div>
-<div class="viewcode-block" id="matrix_to_qubit"><a class="viewcode-back" href="../../../../modules/physics/quantum/qubit.html#sympy.physics.quantum.qubit.matrix_to_qubit">[docs]</a><span class="k">def</span> <span class="nf">matrix_to_qubit</span><span class="p">(</span><span class="n">matrix</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Convert from the matrix repr. to a sum of Qubit objects.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    matrix : Matrix, numpy.matrix, scipy.sparse</span>
-<span class="sd">        The matrix to build the Qubit representation of. This works with</span>
-<span class="sd">        sympy matrices, numpy matrices and scipy.sparse sparse matrices.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    --------</span>
-
-<span class="sd">    Represent a state and then go back to its qubit form:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.qubit import matrix_to_qubit, Qubit</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.gate import Z</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.represent import represent</span>
-<span class="sd">        &gt;&gt;&gt; q = Qubit(&#39;01&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; matrix_to_qubit(represent(q))</span>
-<span class="sd">        |01&gt;</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># Determine the format based on the type of the input matrix</span>
-    <span class="n">format</span> <span class="o">=</span> <span class="s">&#39;sympy&#39;</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">matrix</span><span class="p">,</span> <span class="n">numpy_ndarray</span><span class="p">):</span>
-        <span class="n">format</span> <span class="o">=</span> <span class="s">&#39;numpy&#39;</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">matrix</span><span class="p">,</span> <span class="n">scipy_sparse_matrix</span><span class="p">):</span>
-        <span class="n">format</span> <span class="o">=</span> <span class="s">&#39;scipy.sparse&#39;</span>
-
-    <span class="c"># Make sure it is of correct dimensions for a Qubit-matrix representation.</span>
-    <span class="c"># This logic should work with sympy, numpy or scipy.sparse matrices.</span>
-    <span class="k">if</span> <span class="n">matrix</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">mlistlen</span> <span class="o">=</span> <span class="n">matrix</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">nqubits</span> <span class="o">=</span> <span class="n">log</span><span class="p">(</span><span class="n">mlistlen</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-        <span class="n">ket</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="n">cls</span> <span class="o">=</span> <span class="n">QubitBra</span>
-    <span class="k">elif</span> <span class="n">matrix</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">mlistlen</span> <span class="o">=</span> <span class="n">matrix</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">nqubits</span> <span class="o">=</span> <span class="n">log</span><span class="p">(</span><span class="n">mlistlen</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-        <span class="n">ket</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="n">cls</span> <span class="o">=</span> <span class="n">Qubit</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">QuantumError</span><span class="p">(</span>
-            <span class="s">&#39;Matrix must be a row/column vector, got </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">matrix</span>
-        <span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">nqubits</span><span class="p">,</span> <span class="n">Integer</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="n">QuantumError</span><span class="p">(</span><span class="s">&#39;Matrix must be a row/column vector of size &#39;</span>
-                           <span class="s">&#39;2**nqubits, got: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">matrix</span><span class="p">)</span>
-    <span class="c"># Go through each item in matrix, if element is non-zero, make it into a</span>
-    <span class="c"># Qubit item times the element.</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">mlistlen</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">ket</span><span class="p">:</span>
-            <span class="n">element</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">element</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">format</span> <span class="o">==</span> <span class="s">&#39;numpy&#39;</span> <span class="ow">or</span> <span class="n">format</span> <span class="o">==</span> <span class="s">&#39;scipy.sparse&#39;</span><span class="p">:</span>
-            <span class="n">element</span> <span class="o">=</span> <span class="nb">complex</span><span class="p">(</span><span class="n">element</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">element</span> <span class="o">!=</span> <span class="mf">0.0</span><span class="p">:</span>
-            <span class="c"># Form Qubit array; 0 in bit-locations where i is 0, 1 in</span>
-            <span class="c"># bit-locations where i is 1</span>
-            <span class="n">qubit_array</span> <span class="o">=</span> <span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">i</span> <span class="o">&amp;</span> <span class="p">(</span><span class="mi">1</span> <span class="o">&lt;&lt;</span> <span class="n">x</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nqubits</span><span class="p">)]</span>
-            <span class="n">qubit_array</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">result</span> <span class="o">+</span> <span class="n">element</span><span class="o">*</span><span class="n">cls</span><span class="p">(</span><span class="o">*</span><span class="n">qubit_array</span><span class="p">)</span>
-
-    <span class="c"># If sympy simplified by pulling out a constant coefficient, undo that.</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="p">(</span><span class="n">Mul</span><span class="p">,</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Pow</span><span class="p">)):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="matrix_to_density"><a class="viewcode-back" href="../../../../modules/physics/quantum/qubit.html#sympy.physics.quantum.qubit.matrix_to_density">[docs]</a><span class="k">def</span> <span class="nf">matrix_to_density</span><span class="p">(</span><span class="n">mat</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Works by finding the eigenvectors and eigenvalues of the matrix.</span>
-<span class="sd">    We know we can decompose rho by doing:</span>
-<span class="sd">    sum(EigenVal*|Eigenvect&gt;&lt;Eigenvect|)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.physics.quantum.density</span> <span class="kn">import</span> <span class="n">Density</span>
-    <span class="n">eigen</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">eigenvects</span><span class="p">()</span>
-    <span class="n">args</span> <span class="o">=</span> <span class="p">[[</span><span class="n">matrix_to_qubit</span><span class="p">(</span><span class="n">Matrix</span><span class="p">(</span>
-        <span class="p">[</span><span class="n">vector</span><span class="p">,</span> <span class="p">])),</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">eigen</span> <span class="k">for</span> <span class="n">vector</span> <span class="ow">in</span> <span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="k">if</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">]</span>
-    <span class="k">if</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">Density</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="qubit_to_matrix"><a class="viewcode-back" href="../../../../modules/physics/quantum/qubit.html#sympy.physics.quantum.qubit.qubit_to_matrix">[docs]</a><span class="k">def</span> <span class="nf">qubit_to_matrix</span><span class="p">(</span><span class="n">qubit</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="s">&#39;sympy&#39;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Coverts an Add/Mul of Qubit objects into it&#39;s matrix representation</span>
-
-<span class="sd">    This function is the inverse of ``matrix_to_qubit`` and is a shorthand</span>
-<span class="sd">    for ``represent(qubit)``.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">represent</span><span class="p">(</span><span class="n">qubit</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="n">format</span><span class="p">)</span>
-
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># Measurement</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-</div>
-<div class="viewcode-block" id="measure_all"><a class="viewcode-back" href="../../../../modules/physics/quantum/qubit.html#sympy.physics.quantum.qubit.measure_all">[docs]</a><span class="k">def</span> <span class="nf">measure_all</span><span class="p">(</span><span class="n">qubit</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="s">&#39;sympy&#39;</span><span class="p">,</span> <span class="n">normalize</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Perform an ensemble measurement of all qubits.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    qubit : Qubit, Add</span>
-<span class="sd">        The qubit to measure. This can be any Qubit or a linear combination</span>
-<span class="sd">        of them.</span>
-<span class="sd">    format : str</span>
-<span class="sd">        The format of the intermediate matrices to use. Possible values are</span>
-<span class="sd">        (&#39;sympy&#39;,&#39;numpy&#39;,&#39;scipy.sparse&#39;). Currently only &#39;sympy&#39; is</span>
-<span class="sd">        implemented.</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    result : list</span>
-<span class="sd">        A list that consists of primitive states and their probabilities.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.qubit import Qubit, measure_all</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.gate import H, X, Y, Z</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.qapply import qapply</span>
-
-<span class="sd">        &gt;&gt;&gt; c = H(0)*H(1)*Qubit(&#39;00&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; c</span>
-<span class="sd">        H(0)*H(1)*|00&gt;</span>
-<span class="sd">        &gt;&gt;&gt; q = qapply(c)</span>
-<span class="sd">        &gt;&gt;&gt; measure_all(q)</span>
-<span class="sd">        [(|00&gt;, 1/4), (|01&gt;, 1/4), (|10&gt;, 1/4), (|11&gt;, 1/4)]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">qubit_to_matrix</span><span class="p">(</span><span class="n">qubit</span><span class="p">,</span> <span class="n">format</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">format</span> <span class="o">==</span> <span class="s">&#39;sympy&#39;</span><span class="p">:</span>
-        <span class="n">results</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">if</span> <span class="n">normalize</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">normalized</span><span class="p">()</span>
-
-        <span class="n">size</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>  <span class="c"># Max of shape to account for bra or ket</span>
-        <span class="n">nqubits</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">size</span><span class="p">)</span><span class="o">/</span><span class="n">math</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">m</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">!=</span> <span class="mf">0.0</span><span class="p">:</span>
-                <span class="n">results</span><span class="o">.</span><span class="n">append</span><span class="p">(</span>
-                    <span class="p">(</span><span class="n">Qubit</span><span class="p">(</span><span class="n">IntQubit</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">nqubits</span><span class="p">)),</span> <span class="n">m</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">conjugate</span><span class="p">(</span><span class="n">m</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span>
-                <span class="p">)</span>
-        <span class="k">return</span> <span class="n">results</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-            <span class="s">&quot;This function can&#39;t handle non-sympy matrix formats yet&quot;</span>
-        <span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="measure_partial"><a class="viewcode-back" href="../../../../modules/physics/quantum/qubit.html#sympy.physics.quantum.qubit.measure_partial">[docs]</a><span class="k">def</span> <span class="nf">measure_partial</span><span class="p">(</span><span class="n">qubit</span><span class="p">,</span> <span class="n">bits</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="s">&#39;sympy&#39;</span><span class="p">,</span> <span class="n">normalize</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Perform a partial ensemble measure on the specifed qubits.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    qubits : Qubit</span>
-<span class="sd">        The qubit to measure.  This can be any Qubit or a linear combination</span>
-<span class="sd">        of them.</span>
-<span class="sd">    bits : tuple</span>
-<span class="sd">        The qubits to measure.</span>
-<span class="sd">    format : str</span>
-<span class="sd">        The format of the intermediate matrices to use. Possible values are</span>
-<span class="sd">        (&#39;sympy&#39;,&#39;numpy&#39;,&#39;scipy.sparse&#39;). Currently only &#39;sympy&#39; is</span>
-<span class="sd">        implemented.</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    result : list</span>
-<span class="sd">        A list that consists of primitive states and their probabilities.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.qubit import Qubit, measure_partial</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.gate import H, X, Y, Z</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.qapply import qapply</span>
-
-<span class="sd">        &gt;&gt;&gt; c = H(0)*H(1)*Qubit(&#39;00&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; c</span>
-<span class="sd">        H(0)*H(1)*|00&gt;</span>
-<span class="sd">        &gt;&gt;&gt; q = qapply(c)</span>
-<span class="sd">        &gt;&gt;&gt; measure_partial(q, (0,))</span>
-<span class="sd">        [(sqrt(2)*|00&gt;/2 + sqrt(2)*|10&gt;/2, 1/2), (sqrt(2)*|01&gt;/2 + sqrt(2)*|11&gt;/2, 1/2)]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">qubit_to_matrix</span><span class="p">(</span><span class="n">qubit</span><span class="p">,</span> <span class="n">format</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">bits</span><span class="p">,</span> <span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="n">Integer</span><span class="p">)):</span>
-        <span class="n">bits</span> <span class="o">=</span> <span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">bits</span><span class="p">),)</span>
-
-    <span class="k">if</span> <span class="n">format</span> <span class="o">==</span> <span class="s">&#39;sympy&#39;</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">normalize</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">normalized</span><span class="p">()</span>
-
-        <span class="n">possible_outcomes</span> <span class="o">=</span> <span class="n">_get_possible_outcomes</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">bits</span><span class="p">)</span>
-
-        <span class="c"># Form output from function.</span>
-        <span class="n">output</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">outcome</span> <span class="ow">in</span> <span class="n">possible_outcomes</span><span class="p">:</span>
-            <span class="c"># Calculate probability of finding the specified bits with</span>
-            <span class="c"># given values.</span>
-            <span class="n">prob_of_outcome</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="n">prob_of_outcome</span> <span class="o">+=</span> <span class="p">(</span><span class="n">outcome</span><span class="o">.</span><span class="n">H</span><span class="o">*</span><span class="n">outcome</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-            <span class="c"># If the output has a chance, append it to output with found</span>
-            <span class="c"># probability.</span>
-            <span class="k">if</span> <span class="n">prob_of_outcome</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">normalize</span><span class="p">:</span>
-                    <span class="n">next_matrix</span> <span class="o">=</span> <span class="n">matrix_to_qubit</span><span class="p">(</span><span class="n">outcome</span><span class="o">.</span><span class="n">normalized</span><span class="p">())</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">next_matrix</span> <span class="o">=</span> <span class="n">matrix_to_qubit</span><span class="p">(</span><span class="n">outcome</span><span class="p">)</span>
-
-                <span class="n">output</span><span class="o">.</span><span class="n">append</span><span class="p">((</span>
-                    <span class="n">next_matrix</span><span class="p">,</span>
-                    <span class="n">prob_of_outcome</span>
-                <span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">output</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-            <span class="s">&quot;This function can&#39;t handle non-sympy matrix formats yet&quot;</span>
-        <span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="measure_partial_oneshot"><a class="viewcode-back" href="../../../../modules/physics/quantum/qubit.html#sympy.physics.quantum.qubit.measure_partial_oneshot">[docs]</a><span class="k">def</span> <span class="nf">measure_partial_oneshot</span><span class="p">(</span><span class="n">qubit</span><span class="p">,</span> <span class="n">bits</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="s">&#39;sympy&#39;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Perform a partial oneshot measurement on the specified qubits.</span>
-
-<span class="sd">    A oneshot measurement is equivalent to performing a measurement on a</span>
-<span class="sd">    quantum system. This type of measurement does not return the probabilities</span>
-<span class="sd">    like an ensemble measurement does, but rather returns *one* of the</span>
-<span class="sd">    possible resulting states. The exact state that is returned is determined</span>
-<span class="sd">    by picking a state randomly according to the ensemble probabilities.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    qubits : Qubit</span>
-<span class="sd">        The qubit to measure.  This can be any Qubit or a linear combination</span>
-<span class="sd">        of them.</span>
-<span class="sd">    bits : tuple</span>
-<span class="sd">        The qubits to measure.</span>
-<span class="sd">    format : str</span>
-<span class="sd">        The format of the intermediate matrices to use. Possible values are</span>
-<span class="sd">        (&#39;sympy&#39;,&#39;numpy&#39;,&#39;scipy.sparse&#39;). Currently only &#39;sympy&#39; is</span>
-<span class="sd">        implemented.</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    -------</span>
-<span class="sd">    result : Qubit</span>
-<span class="sd">        The qubit that the system collapsed to upon measurement.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">import</span> <span class="nn">random</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">qubit_to_matrix</span><span class="p">(</span><span class="n">qubit</span><span class="p">,</span> <span class="n">format</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">format</span> <span class="o">==</span> <span class="s">&#39;sympy&#39;</span><span class="p">:</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">normalized</span><span class="p">()</span>
-        <span class="n">possible_outcomes</span> <span class="o">=</span> <span class="n">_get_possible_outcomes</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">bits</span><span class="p">)</span>
-
-        <span class="c"># Form output from function</span>
-        <span class="n">random_number</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">()</span>
-        <span class="n">total_prob</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">outcome</span> <span class="ow">in</span> <span class="n">possible_outcomes</span><span class="p">:</span>
-            <span class="c"># Calculate probability of finding the specified bits</span>
-            <span class="c"># with given values</span>
-            <span class="n">total_prob</span> <span class="o">+=</span> <span class="p">(</span><span class="n">outcome</span><span class="o">.</span><span class="n">H</span><span class="o">*</span><span class="n">outcome</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">total_prob</span> <span class="o">&gt;=</span> <span class="n">random_number</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">matrix_to_qubit</span><span class="p">(</span><span class="n">outcome</span><span class="o">.</span><span class="n">normalized</span><span class="p">())</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-            <span class="s">&quot;This function can&#39;t handle non-sympy matrix formats yet&quot;</span>
-        <span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_get_possible_outcomes</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">bits</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Get the possible states that can be produced in a measurement.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    m : Matrix</span>
-<span class="sd">        The matrix representing the state of the system.</span>
-<span class="sd">    bits : tuple, list</span>
-<span class="sd">        Which bits will be measured.</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    -------</span>
-<span class="sd">    result : list</span>
-<span class="sd">        The list of possible states which can occur given this measurement.</span>
-<span class="sd">        These are un-normalized so we can derive the probability of finding</span>
-<span class="sd">        this state by taking the inner product with itself</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="c"># This is filled with loads of dirty binary tricks...You have been warned</span>
-
-    <span class="n">size</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>  <span class="c"># Max of shape to account for bra or ket</span>
-    <span class="n">nqubits</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">size</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="o">.</span><span class="mi">1</span><span class="p">)</span>  <span class="c"># Number of qubits possible</span>
-
-    <span class="c"># Make the output states and put in output_matrices, nothing in them now.</span>
-    <span class="c"># Each state will represent a possible outcome of the measurement</span>
-    <span class="c"># Thus, output_matrices[0] is the matrix which we get when all measured</span>
-    <span class="c"># bits return 0. and output_matrices[1] is the matrix for only the 0th</span>
-    <span class="c"># bit being true</span>
-    <span class="n">output_matrices</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span> <span class="o">&lt;&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">bits</span><span class="p">)):</span>
-        <span class="n">output_matrices</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">zeros</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">nqubits</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-
-    <span class="c"># Bitmasks will help sort how to determine possible outcomes.</span>
-    <span class="c"># When the bit mask is and-ed with a matrix-index,</span>
-    <span class="c"># it will determine which state that index belongs to</span>
-    <span class="n">bit_masks</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">bit</span> <span class="ow">in</span> <span class="n">bits</span><span class="p">:</span>
-        <span class="n">bit_masks</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">1</span> <span class="o">&lt;&lt;</span> <span class="n">bit</span><span class="p">)</span>
-
-    <span class="c"># Make possible outcome states</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">nqubits</span><span class="p">):</span>
-        <span class="n">trueness</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c"># This tells us to which output_matrix this value belongs</span>
-        <span class="c"># Find trueness</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">bit_masks</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">&amp;</span> <span class="n">bit_masks</span><span class="p">[</span><span class="n">j</span><span class="p">]:</span>
-                <span class="n">trueness</span> <span class="o">+=</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="c"># Put the value in the correct output matrix</span>
-        <span class="n">output_matrices</span><span class="p">[</span><span class="n">trueness</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">m</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">output_matrices</span>
-
-
-<div class="viewcode-block" id="measure_all_oneshot"><a class="viewcode-back" href="../../../../modules/physics/quantum/qubit.html#sympy.physics.quantum.qubit.measure_all_oneshot">[docs]</a><span class="k">def</span> <span class="nf">measure_all_oneshot</span><span class="p">(</span><span class="n">qubit</span><span class="p">,</span> <span class="n">format</span><span class="o">=</span><span class="s">&#39;sympy&#39;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Perform a oneshot ensemble measurement on all qubits.</span>
-
-<span class="sd">    A oneshot measurement is equivalent to performing a measurement on a</span>
-<span class="sd">    quantum system. This type of measurement does not return the probabilities</span>
-<span class="sd">    like an ensemble measurement does, but rather returns *one* of the</span>
-<span class="sd">    possible resulting states. The exact state that is returned is determined</span>
-<span class="sd">    by picking a state randomly according to the ensemble probabilities.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    qubits : Qubit</span>
-<span class="sd">        The qubit to measure.  This can be any Qubit or a linear combination</span>
-<span class="sd">        of them.</span>
-<span class="sd">    format : str</span>
-<span class="sd">        The format of the intermediate matrices to use. Possible values are</span>
-<span class="sd">        (&#39;sympy&#39;,&#39;numpy&#39;,&#39;scipy.sparse&#39;). Currently only &#39;sympy&#39; is</span>
-<span class="sd">        implemented.</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    -------</span>
-<span class="sd">    result : Qubit</span>
-<span class="sd">        The qubit that the system collapsed to upon measurement.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">import</span> <span class="nn">random</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">qubit_to_matrix</span><span class="p">(</span><span class="n">qubit</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">format</span> <span class="o">==</span> <span class="s">&#39;sympy&#39;</span><span class="p">:</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">normalized</span><span class="p">()</span>
-        <span class="n">random_number</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">()</span>
-        <span class="n">total</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">m</span><span class="p">:</span>
-            <span class="n">total</span> <span class="o">+=</span> <span class="n">i</span><span class="o">*</span><span class="n">i</span><span class="o">.</span><span class="n">conjugate</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">total</span> <span class="o">&gt;</span> <span class="n">random_number</span><span class="p">:</span>
-                <span class="k">break</span>
-            <span class="n">result</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="n">Qubit</span><span class="p">(</span><span class="n">IntQubit</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">shape</span><span class="p">),</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="o">.</span><span class="mi">1</span><span class="p">)))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-            <span class="s">&quot;This function can&#39;t handle non-sympy matrix formats yet&quot;</span>
-        <span class="p">)</span></div>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.physics.quantum.represent</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Logic for representing operators in state in various bases.</span>
-
-<span class="sd">TODO:</span>
-
-<span class="sd">* Get represent working with continuous hilbert spaces.</span>
-<span class="sd">* Document default basis functionality.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">integrate</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Pow</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.dagger</span> <span class="kn">import</span> <span class="n">Dagger</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.commutator</span> <span class="kn">import</span> <span class="n">Commutator</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.anticommutator</span> <span class="kn">import</span> <span class="n">AntiCommutator</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.innerproduct</span> <span class="kn">import</span> <span class="n">InnerProduct</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.qexpr</span> <span class="kn">import</span> <span class="n">QExpr</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.tensorproduct</span> <span class="kn">import</span> <span class="n">TensorProduct</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.matrixutils</span> <span class="kn">import</span> <span class="n">flatten_scalar</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.state</span> <span class="kn">import</span> <span class="n">KetBase</span><span class="p">,</span> <span class="n">BraBase</span><span class="p">,</span> <span class="n">StateBase</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.operator</span> <span class="kn">import</span> <span class="n">Operator</span><span class="p">,</span> <span class="n">OuterProduct</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.qapply</span> <span class="kn">import</span> <span class="n">qapply</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.operatorset</span> <span class="kn">import</span> <span class="n">operators_to_state</span><span class="p">,</span> <span class="n">state_to_operators</span>
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="s">&#39;represent&#39;</span><span class="p">,</span>
-    <span class="s">&#39;rep_innerproduct&#39;</span><span class="p">,</span>
-    <span class="s">&#39;rep_expectation&#39;</span><span class="p">,</span>
-    <span class="s">&#39;integrate_result&#39;</span><span class="p">,</span>
-    <span class="s">&#39;get_basis&#39;</span><span class="p">,</span>
-    <span class="s">&#39;enumerate_states&#39;</span>
-<span class="p">]</span>
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># Represent</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-
-<span class="k">def</span> <span class="nf">_sympy_to_scalar</span><span class="p">(</span><span class="n">e</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Convert from a sympy scalar to a Python scalar.&quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">float</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">float</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">or</span> <span class="n">e</span><span class="o">.</span><span class="n">is_NumberSymbol</span> <span class="ow">or</span> <span class="n">e</span> <span class="o">==</span> <span class="n">I</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">complex</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-    <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Expected number, got: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">e</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="represent"><a class="viewcode-back" href="../../../../modules/physics/quantum/represent.html#sympy.physics.quantum.represent.represent">[docs]</a><span class="k">def</span> <span class="nf">represent</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Represent the quantum expression in the given basis.</span>
-
-<span class="sd">    In quantum mechanics abstract states and operators can be represented in</span>
-<span class="sd">    various basis sets. Under this operation the follow transforms happen:</span>
-
-<span class="sd">    * Ket -&gt; column vector or function</span>
-<span class="sd">    * Bra -&gt; row vector of function</span>
-<span class="sd">    * Operator -&gt; matrix or differential operator</span>
-
-<span class="sd">    This function is the top-level interface for this action.</span>
-
-<span class="sd">    This function walks the sympy expression tree looking for ``QExpr``</span>
-<span class="sd">    instances that have a ``_represent`` method. This method is then called</span>
-<span class="sd">    and the object is replaced by the representation returned by this method.</span>
-<span class="sd">    By default, the ``_represent`` method will dispatch to other methods</span>
-<span class="sd">    that handle the representation logic for a particular basis set. The</span>
-<span class="sd">    naming convention for these methods is the following::</span>
-
-<span class="sd">        def _represent_FooBasis(self, e, basis, **options)</span>
-
-<span class="sd">    This function will have the logic for representing instances of its class</span>
-<span class="sd">    in the basis set having a class named ``FooBasis``.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    expr  : Expr</span>
-<span class="sd">        The expression to represent.</span>
-<span class="sd">    basis : Operator, basis set</span>
-<span class="sd">        An object that contains the information about the basis set. If an</span>
-<span class="sd">        operator is used, the basis is assumed to be the orthonormal</span>
-<span class="sd">        eigenvectors of that operator. In general though, the basis argument</span>
-<span class="sd">        can be any object that contains the basis set information.</span>
-<span class="sd">    options : dict</span>
-<span class="sd">        Key/value pairs of options that are passed to the underlying method</span>
-<span class="sd">        that does finds the representation. These options can be used to</span>
-<span class="sd">        control how the representation is done. For example, this is where</span>
-<span class="sd">        the size of the basis set would be set.</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    e : Expr</span>
-<span class="sd">        The sympy expression of the represented quantum expression.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Here we subclass ``Operator`` and ``Ket`` to create the z-spin operator</span>
-<span class="sd">    and its spin 1/2 up eigenstate. By definining the ``_represent_SzOp``</span>
-<span class="sd">    method, the ket can be represented in the z-spin basis.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum import Operator, represent, Ket</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Matrix</span>
-
-<span class="sd">    &gt;&gt;&gt; class SzUpKet(Ket):</span>
-<span class="sd">    ...     def _represent_SzOp(self, basis, **options):</span>
-<span class="sd">    ...         return Matrix([1,0])</span>
-<span class="sd">    ...</span>
-<span class="sd">    &gt;&gt;&gt; class SzOp(Operator):</span>
-<span class="sd">    ...     pass</span>
-<span class="sd">    ...</span>
-<span class="sd">    &gt;&gt;&gt; sz = SzOp(&#39;Sz&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; up = SzUpKet(&#39;up&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; represent(up, basis=sz)</span>
-<span class="sd">    [1]</span>
-<span class="sd">    [0]</span>
-
-<span class="sd">    Here we see an example of representations in a continuous</span>
-<span class="sd">    basis. We see that the result of representing various combinations</span>
-<span class="sd">    of cartesian position operators and kets give us continuous</span>
-<span class="sd">    expressions involving DiracDelta functions.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.cartesian import XOp, XKet, XBra</span>
-<span class="sd">    &gt;&gt;&gt; X = XOp()</span>
-<span class="sd">    &gt;&gt;&gt; x = XKet()</span>
-<span class="sd">    &gt;&gt;&gt; y = XBra(&#39;y&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; represent(X*x)</span>
-<span class="sd">    x*DiracDelta(x - x_2)</span>
-<span class="sd">    &gt;&gt;&gt; represent(X*x*y)</span>
-<span class="sd">    x*DiracDelta(x - x_3)*DiracDelta(x_1 - y)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">format</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;format&#39;</span><span class="p">,</span> <span class="s">&#39;sympy&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">QExpr</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">OuterProduct</span><span class="p">):</span>
-        <span class="n">options</span><span class="p">[</span><span class="s">&#39;replace_none&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="n">temp_basis</span> <span class="o">=</span> <span class="n">get_basis</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">temp_basis</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">options</span><span class="p">[</span><span class="s">&#39;basis&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">temp_basis</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">_represent</span><span class="p">(</span><span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">NotImplementedError</span> <span class="k">as</span> <span class="n">strerr</span><span class="p">:</span>
-            <span class="c">#If no _represent_FOO method exists, map to the</span>
-            <span class="c">#appropriate basis state and try</span>
-            <span class="c">#the other methods of representation</span>
-            <span class="n">options</span><span class="p">[</span><span class="s">&#39;replace_none&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">True</span>
-
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="p">(</span><span class="n">KetBase</span><span class="p">,</span> <span class="n">BraBase</span><span class="p">)):</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">rep_innerproduct</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-                <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="n">strerr</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Operator</span><span class="p">):</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">rep_expectation</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-                <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="n">strerr</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="n">strerr</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Add</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">represent</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">args</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="c"># scipy.sparse doesn&#39;t support += so we use plain = here.</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">result</span> <span class="o">+</span> <span class="n">represent</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">result</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Pow</span><span class="p">):</span>
-        <span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">format</span> <span class="o">==</span> <span class="s">&#39;numpy&#39;</span> <span class="ow">or</span> <span class="n">format</span> <span class="o">==</span> <span class="s">&#39;scipy.sparse&#39;</span><span class="p">:</span>
-            <span class="n">exp</span> <span class="o">=</span> <span class="n">_sympy_to_scalar</span><span class="p">(</span><span class="n">exp</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">represent</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span><span class="o">**</span><span class="n">exp</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">TensorProduct</span><span class="p">):</span>
-        <span class="n">new_args</span> <span class="o">=</span> <span class="p">[</span><span class="n">represent</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">TensorProduct</span><span class="p">(</span><span class="o">*</span><span class="n">new_args</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Dagger</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Dagger</span><span class="p">(</span><span class="n">represent</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="o">**</span><span class="n">options</span><span class="p">))</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Commutator</span><span class="p">):</span>
-        <span class="n">A</span> <span class="o">=</span> <span class="n">represent</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="n">B</span> <span class="o">=</span> <span class="n">represent</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">A</span><span class="o">*</span><span class="n">B</span> <span class="o">-</span> <span class="n">B</span><span class="o">*</span><span class="n">A</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">AntiCommutator</span><span class="p">):</span>
-        <span class="n">A</span> <span class="o">=</span> <span class="n">represent</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="n">B</span> <span class="o">=</span> <span class="n">represent</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">A</span><span class="o">*</span><span class="n">B</span> <span class="o">+</span> <span class="n">B</span><span class="o">*</span><span class="n">A</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">InnerProduct</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">represent</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">bra</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">ket</span><span class="p">),</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Mul</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">OuterProduct</span><span class="p">)):</span>
-        <span class="c"># For numpy and scipy.sparse, we can only handle numerical prefactors.</span>
-        <span class="k">if</span> <span class="n">format</span> <span class="o">==</span> <span class="s">&#39;numpy&#39;</span> <span class="ow">or</span> <span class="n">format</span> <span class="o">==</span> <span class="s">&#39;scipy.sparse&#39;</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_sympy_to_scalar</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Mul</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">OuterProduct</span><span class="p">)):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Mul expected, got: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="s">&quot;index&quot;</span> <span class="ow">in</span> <span class="n">options</span><span class="p">:</span>
-        <span class="n">options</span><span class="p">[</span><span class="s">&quot;index&quot;</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">options</span><span class="p">[</span><span class="s">&quot;index&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="s">&quot;unities&quot;</span> <span class="ow">in</span> <span class="n">options</span><span class="p">:</span>
-        <span class="n">options</span><span class="p">[</span><span class="s">&quot;unities&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">represent</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-    <span class="n">last_arg</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">last_arg</span><span class="p">,</span> <span class="n">Operator</span><span class="p">):</span>
-            <span class="n">options</span><span class="p">[</span><span class="s">&quot;index&quot;</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">options</span><span class="p">[</span><span class="s">&quot;unities&quot;</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">options</span><span class="p">[</span><span class="s">&quot;index&quot;</span><span class="p">])</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">last_arg</span><span class="p">,</span> <span class="n">BraBase</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">KetBase</span><span class="p">):</span>
-            <span class="n">options</span><span class="p">[</span><span class="s">&quot;index&quot;</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">last_arg</span><span class="p">,</span> <span class="n">KetBase</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Operator</span><span class="p">):</span>
-            <span class="n">options</span><span class="p">[</span><span class="s">&quot;unities&quot;</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">options</span><span class="p">[</span><span class="s">&quot;index&quot;</span><span class="p">])</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">last_arg</span><span class="p">,</span> <span class="n">KetBase</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">BraBase</span><span class="p">):</span>
-            <span class="n">options</span><span class="p">[</span><span class="s">&quot;unities&quot;</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">options</span><span class="p">[</span><span class="s">&quot;index&quot;</span><span class="p">])</span>
-
-        <span class="n">result</span> <span class="o">=</span> <span class="n">represent</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span><span class="o">*</span><span class="n">result</span>
-        <span class="n">last_arg</span> <span class="o">=</span> <span class="n">arg</span>
-
-    <span class="c"># All three matrix formats create 1 by 1 matrices when inner products of</span>
-    <span class="c"># vectors are taken. In these cases, we simply return a scalar.</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="n">flatten_scalar</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">integrate_result</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">result</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="rep_innerproduct"><a class="viewcode-back" href="../../../../modules/physics/quantum/represent.html#sympy.physics.quantum.represent.rep_innerproduct">[docs]</a><span class="k">def</span> <span class="nf">rep_innerproduct</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns an innerproduct like representation (e.g. ``&lt;x&#39;|x&gt;``) for the</span>
-<span class="sd">    given state.</span>
-
-<span class="sd">    Attempts to calculate inner product with a bra from the specified</span>
-<span class="sd">    basis. Should only be passed an instance of KetBase or BraBase</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    expr : KetBase or BraBase</span>
-<span class="sd">        The expression to be represented</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.represent import rep_innerproduct</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.cartesian import XOp, XKet, PxOp, PxKet</span>
-<span class="sd">    &gt;&gt;&gt; rep_innerproduct(XKet())</span>
-<span class="sd">    DiracDelta(x - x_1)</span>
-<span class="sd">    &gt;&gt;&gt; rep_innerproduct(XKet(), basis=PxOp())</span>
-<span class="sd">    sqrt(2)*exp(-I*px_1*x/hbar)/(2*sqrt(hbar)*sqrt(pi))</span>
-<span class="sd">    &gt;&gt;&gt; rep_innerproduct(PxKet(), basis=XOp())</span>
-<span class="sd">    sqrt(2)*exp(I*px*x_1/hbar)/(2*sqrt(hbar)*sqrt(pi))</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="p">(</span><span class="n">KetBase</span><span class="p">,</span> <span class="n">BraBase</span><span class="p">)):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;expr passed is not a Bra or Ket&quot;</span><span class="p">)</span>
-
-    <span class="n">basis</span> <span class="o">=</span> <span class="n">get_basis</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">basis</span><span class="p">,</span> <span class="n">StateBase</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Can&#39;t form this representation!&quot;</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="s">&quot;index&quot;</span> <span class="ow">in</span> <span class="n">options</span><span class="p">:</span>
-        <span class="n">options</span><span class="p">[</span><span class="s">&quot;index&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="n">basis_kets</span> <span class="o">=</span> <span class="n">enumerate_states</span><span class="p">(</span><span class="n">basis</span><span class="p">,</span> <span class="n">options</span><span class="p">[</span><span class="s">&quot;index&quot;</span><span class="p">],</span> <span class="mi">2</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">BraBase</span><span class="p">):</span>
-        <span class="n">bra</span> <span class="o">=</span> <span class="n">expr</span>
-        <span class="n">ket</span> <span class="o">=</span> <span class="p">(</span><span class="n">basis_kets</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">if</span> <span class="n">basis_kets</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">dual</span> <span class="o">==</span> <span class="n">expr</span> <span class="k">else</span> <span class="n">basis_kets</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">bra</span> <span class="o">=</span> <span class="p">(</span><span class="n">basis_kets</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">dual</span> <span class="k">if</span> <span class="n">basis_kets</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-               <span class="o">==</span> <span class="n">expr</span> <span class="k">else</span> <span class="n">basis_kets</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">dual</span><span class="p">)</span>
-        <span class="n">ket</span> <span class="o">=</span> <span class="n">expr</span>
-
-    <span class="n">prod</span> <span class="o">=</span> <span class="n">InnerProduct</span><span class="p">(</span><span class="n">bra</span><span class="p">,</span> <span class="n">ket</span><span class="p">)</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="n">prod</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-
-    <span class="n">format</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;format&#39;</span><span class="p">,</span> <span class="s">&#39;sympy&#39;</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">_format_represent</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">format</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="rep_expectation"><a class="viewcode-back" href="../../../../modules/physics/quantum/represent.html#sympy.physics.quantum.represent.rep_expectation">[docs]</a><span class="k">def</span> <span class="nf">rep_expectation</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns an ``&lt;x&#39;|A|x&gt;`` type representation for the given operator.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    expr : Operator</span>
-<span class="sd">        Operator to be represented in the specified basis</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.cartesian import XOp, XKet, PxOp, PxKet</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.represent import rep_expectation</span>
-<span class="sd">    &gt;&gt;&gt; rep_expectation(XOp())</span>
-<span class="sd">    x_1*DiracDelta(x_1 - x_2)</span>
-<span class="sd">    &gt;&gt;&gt; rep_expectation(XOp(), basis=PxOp())</span>
-<span class="sd">    &lt;px_2|*X*|px_1&gt;</span>
-<span class="sd">    &gt;&gt;&gt; rep_expectation(XOp(), basis=PxKet())</span>
-<span class="sd">    &lt;px_2|*X*|px_1&gt;</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="s">&quot;index&quot;</span> <span class="ow">in</span> <span class="n">options</span><span class="p">:</span>
-        <span class="n">options</span><span class="p">[</span><span class="s">&quot;index&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Operator</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;The passed expression is not an operator&quot;</span><span class="p">)</span>
-
-    <span class="n">basis_state</span> <span class="o">=</span> <span class="n">get_basis</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">basis_state</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">basis_state</span><span class="p">,</span> <span class="n">StateBase</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Could not get basis kets for this operator&quot;</span><span class="p">)</span>
-
-    <span class="n">basis_kets</span> <span class="o">=</span> <span class="n">enumerate_states</span><span class="p">(</span><span class="n">basis_state</span><span class="p">,</span> <span class="n">options</span><span class="p">[</span><span class="s">&quot;index&quot;</span><span class="p">],</span> <span class="mi">2</span><span class="p">)</span>
-
-    <span class="n">bra</span> <span class="o">=</span> <span class="n">basis_kets</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">dual</span>
-    <span class="n">ket</span> <span class="o">=</span> <span class="n">basis_kets</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">return</span> <span class="n">qapply</span><span class="p">(</span><span class="n">bra</span><span class="o">*</span><span class="n">expr</span><span class="o">*</span><span class="n">ket</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="integrate_result"><a class="viewcode-back" href="../../../../modules/physics/quantum/represent.html#sympy.physics.quantum.represent.integrate_result">[docs]</a><span class="k">def</span> <span class="nf">integrate_result</span><span class="p">(</span><span class="n">orig_expr</span><span class="p">,</span> <span class="n">result</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the result of integrating over any unities ``(|x&gt;&lt;x|)`` in</span>
-<span class="sd">    the given expression. Intended for integrating over the result of</span>
-<span class="sd">    representations in continuous bases.</span>
-
-<span class="sd">    This function integrates over any unities that may have been</span>
-<span class="sd">    inserted into the quantum expression and returns the result.</span>
-<span class="sd">    It uses the interval of the Hilbert space of the basis state</span>
-<span class="sd">    passed to it in order to figure out the limits of integration.</span>
-<span class="sd">    The unities option must be</span>
-<span class="sd">    specified for this to work.</span>
-
-<span class="sd">    Note: This is mostly used internally by represent(). Examples are</span>
-<span class="sd">    given merely to show the use cases.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    orig_expr : quantum expression</span>
-<span class="sd">        The original expression which was to be represented</span>
-
-<span class="sd">    result: Expr</span>
-<span class="sd">        The resulting representation that we wish to integrate over</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, DiracDelta</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.represent import integrate_result</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.cartesian import XOp, XKet</span>
-<span class="sd">    &gt;&gt;&gt; x_ket = XKet()</span>
-<span class="sd">    &gt;&gt;&gt; X_op = XOp()</span>
-<span class="sd">    &gt;&gt;&gt; x, x_1, x_2 = symbols(&#39;x, x_1, x_2&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; integrate_result(X_op*x_ket, x*DiracDelta(x-x_1)*DiracDelta(x_1-x_2))</span>
-<span class="sd">    x*DiracDelta(x - x_1)*DiracDelta(x_1 - x_2)</span>
-<span class="sd">    &gt;&gt;&gt; integrate_result(X_op*x_ket, x*DiracDelta(x-x_1)*DiracDelta(x_1-x_2),</span>
-<span class="sd">    ...     unities=[1])</span>
-<span class="sd">    x*DiracDelta(x - x_2)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">Expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="n">options</span><span class="p">[</span><span class="s">&#39;replace_none&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="s">&quot;basis&quot;</span> <span class="ow">in</span> <span class="n">options</span><span class="p">:</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">orig_expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">options</span><span class="p">[</span><span class="s">&quot;basis&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">get_basis</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">options</span><span class="p">[</span><span class="s">&quot;basis&quot;</span><span class="p">],</span> <span class="n">StateBase</span><span class="p">):</span>
-        <span class="n">options</span><span class="p">[</span><span class="s">&quot;basis&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">get_basis</span><span class="p">(</span><span class="n">orig_expr</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="n">basis</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&quot;basis&quot;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">basis</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="n">unities</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&quot;unities&quot;</span><span class="p">,</span> <span class="p">[])</span>
-
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">unities</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="n">kets</span> <span class="o">=</span> <span class="n">enumerate_states</span><span class="p">(</span><span class="n">basis</span><span class="p">,</span> <span class="n">unities</span><span class="p">)</span>
-    <span class="n">coords</span> <span class="o">=</span> <span class="p">[</span><span class="n">k</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">kets</span><span class="p">]</span>
-
-    <span class="k">for</span> <span class="n">coord</span> <span class="ow">in</span> <span class="n">coords</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">coord</span> <span class="ow">in</span> <span class="n">result</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-            <span class="c">#TODO: Add support for sets of operators</span>
-            <span class="n">basis_op</span> <span class="o">=</span> <span class="n">state_to_operators</span><span class="p">(</span><span class="n">basis</span><span class="p">)</span>
-            <span class="n">start</span> <span class="o">=</span> <span class="n">basis_op</span><span class="o">.</span><span class="n">hilbert_space</span><span class="o">.</span><span class="n">interval</span><span class="o">.</span><span class="n">start</span>
-            <span class="n">end</span> <span class="o">=</span> <span class="n">basis_op</span><span class="o">.</span><span class="n">hilbert_space</span><span class="o">.</span><span class="n">interval</span><span class="o">.</span><span class="n">end</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">integrate</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="p">(</span><span class="n">coord</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">end</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="get_basis"><a class="viewcode-back" href="../../../../modules/physics/quantum/represent.html#sympy.physics.quantum.represent.get_basis">[docs]</a><span class="k">def</span> <span class="nf">get_basis</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns a basis state instance corresponding to the basis specified in</span>
-<span class="sd">    options=s. If no basis is specified, the function tries to form a default</span>
-<span class="sd">    basis state of the given expression.</span>
-
-<span class="sd">    There are three behaviors:</span>
-
-<span class="sd">    1. The basis specified in options is already an instance of StateBase. If</span>
-<span class="sd">       this is the case, it is simply returned. If the class is specified but</span>
-<span class="sd">       not an instance, a default instance is returned.</span>
-
-<span class="sd">    2. The basis specified is an operator or set of operators. If this</span>
-<span class="sd">       is the case, the operator_to_state mapping method is used.</span>
-
-<span class="sd">    3. No basis is specified. If expr is a state, then a default instance of</span>
-<span class="sd">       its class is returned.  If expr is an operator, then it is mapped to the</span>
-<span class="sd">       corresponding state.  If it is neither, then we cannot obtain the basis</span>
-<span class="sd">       state.</span>
-
-<span class="sd">    If the basis cannot be mapped, then it is not changed.</span>
-
-<span class="sd">    This will be called from within represent, and represent will</span>
-<span class="sd">    only pass QExpr&#39;s.</span>
-
-<span class="sd">    TODO (?): Support for Muls and other types of expressions?</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    expr : Operator or StateBase</span>
-<span class="sd">        Expression whose basis is sought</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.represent import get_basis</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.cartesian import XOp, XKet, PxOp, PxKet</span>
-<span class="sd">    &gt;&gt;&gt; x = XKet()</span>
-<span class="sd">    &gt;&gt;&gt; X = XOp()</span>
-<span class="sd">    &gt;&gt;&gt; get_basis(x)</span>
-<span class="sd">    |x&gt;</span>
-<span class="sd">    &gt;&gt;&gt; get_basis(X)</span>
-<span class="sd">    |x&gt;</span>
-<span class="sd">    &gt;&gt;&gt; get_basis(x, basis=PxOp())</span>
-<span class="sd">    |px&gt;</span>
-<span class="sd">    &gt;&gt;&gt; get_basis(x, basis=PxKet)</span>
-<span class="sd">    |px&gt;</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">basis</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&quot;basis&quot;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-    <span class="n">replace_none</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&quot;replace_none&quot;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">basis</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">replace_none</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">if</span> <span class="n">basis</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">KetBase</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">_make_default</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">BraBase</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">_make_default</span><span class="p">((</span><span class="n">expr</span><span class="o">.</span><span class="n">dual_class</span><span class="p">()))</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Operator</span><span class="p">):</span>
-            <span class="n">state_inst</span> <span class="o">=</span> <span class="n">operators_to_state</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">state_inst</span> <span class="k">if</span> <span class="n">state_inst</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="k">else</span> <span class="bp">None</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-    <span class="k">elif</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">basis</span><span class="p">,</span> <span class="n">Operator</span><span class="p">)</span> <span class="ow">or</span>
-          <span class="p">(</span><span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">basis</span><span class="p">,</span> <span class="n">StateBase</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">basis</span><span class="p">,</span> <span class="n">Operator</span><span class="p">))):</span>
-        <span class="n">state</span> <span class="o">=</span> <span class="n">operators_to_state</span><span class="p">(</span><span class="n">basis</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">state</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">StateBase</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">state</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_make_default</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">basis</span><span class="p">,</span> <span class="n">StateBase</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">basis</span>
-    <span class="k">elif</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">basis</span><span class="p">,</span> <span class="n">StateBase</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_make_default</span><span class="p">(</span><span class="n">basis</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_make_default</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="p">()</span>
-    <span class="k">except</span> <span class="ne">Exception</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="k">return</span> <span class="n">expr</span>
-
-
-<div class="viewcode-block" id="enumerate_states"><a class="viewcode-back" href="../../../../modules/physics/quantum/represent.html#sympy.physics.quantum.represent.enumerate_states">[docs]</a><span class="k">def</span> <span class="nf">enumerate_states</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns instances of the given state with dummy indices appended</span>
-
-<span class="sd">    Operates in two different modes:</span>
-
-<span class="sd">    1. Two arguments are passed to it. The first is the base state which is to</span>
-<span class="sd">       be indexed, and the second argument is a list of indices to append.</span>
-
-<span class="sd">    2. Three arguments are passed. The first is again the base state to be</span>
-<span class="sd">       indexed. The second is the start index for counting.  The final argument</span>
-<span class="sd">       is the number of kets you wish to receive.</span>
-
-<span class="sd">    Tries to call state._enumerate_state. If this fails, returns an empty list</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    args : list</span>
-<span class="sd">        See list of operation modes above for explanation</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.cartesian import XBra, XKet</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.represent import enumerate_states</span>
-<span class="sd">    &gt;&gt;&gt; test = XKet(&#39;foo&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; enumerate_states(test, 1, 3)</span>
-<span class="sd">    [|foo_1&gt;, |foo_2&gt;, |foo_3&gt;]</span>
-<span class="sd">    &gt;&gt;&gt; test2 = XBra(&#39;bar&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; enumerate_states(test2, [4, 5, 10])</span>
-<span class="sd">    [&lt;bar_4|, &lt;bar_5|, &lt;bar_10|]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">state</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Wrong number of arguments!&quot;</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">StateBase</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;First argument is not a state!&quot;</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-        <span class="n">num_states</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-        <span class="n">options</span><span class="p">[</span><span class="s">&#39;start_index&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">num_states</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="n">options</span><span class="p">[</span><span class="s">&#39;index_list&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">ret</span> <span class="o">=</span> <span class="n">state</span><span class="o">.</span><span class="n">_enumerate_state</span><span class="p">(</span><span class="n">num_states</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-    <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-        <span class="n">ret</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">return</span> <span class="n">ret</span></div>
-</pre></div>
-
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-        </div>
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-            
-  <h1>Source code for sympy.physics.quantum.shor</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Shor&#39;s algorithm and helper functions.</span>
-
-<span class="sd">Todo:</span>
-
-<span class="sd">* Get the CMod gate working again using the new Gate API.</span>
-<span class="sd">* Fix everything.</span>
-<span class="sd">* Update docstrings and reformat.</span>
-<span class="sd">* Remove print statements. We may want to think about a better API for this.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">import</span> <span class="nn">math</span>
-<span class="kn">import</span> <span class="nn">random</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">S</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">log</span><span class="p">,</span> <span class="n">sqrt</span>
-<span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">igcd</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">variations</span>
-
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.gate</span> <span class="kn">import</span> <span class="n">Gate</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.qubit</span> <span class="kn">import</span> <span class="n">Qubit</span><span class="p">,</span> <span class="n">measure_partial_oneshot</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.qapply</span> <span class="kn">import</span> <span class="n">qapply</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.qft</span> <span class="kn">import</span> <span class="n">QFT</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.qexpr</span> <span class="kn">import</span> <span class="n">QuantumError</span>
-
-
-<span class="k">class</span> <span class="nc">OrderFindingException</span><span class="p">(</span><span class="n">QuantumError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-
-<div class="viewcode-block" id="CMod"><a class="viewcode-back" href="../../../../modules/physics/quantum/shor.html#sympy.physics.quantum.shor.CMod">[docs]</a><span class="k">class</span> <span class="nc">CMod</span><span class="p">(</span><span class="n">Gate</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A controlled mod gate.</span>
-
-<span class="sd">    This is black box controlled Mod function for use by shor&#39;s algorithm.</span>
-<span class="sd">    TODO implement a decompose property that returns how to do this in terms</span>
-<span class="sd">    of elementary gates</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_args</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="c"># t = args[0]</span>
-        <span class="c"># a = args[1]</span>
-        <span class="c"># N = args[2]</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;The CMod gate has not been completed.&#39;</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="CMod.t"><a class="viewcode-back" href="../../../../modules/physics/quantum/shor.html#sympy.physics.quantum.shor.CMod.t">[docs]</a>    <span class="k">def</span> <span class="nf">t</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Size of 1/2 input register.  First 1/2 holds output.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="CMod.a"><a class="viewcode-back" href="../../../../modules/physics/quantum/shor.html#sympy.physics.quantum.shor.CMod.a">[docs]</a>    <span class="k">def</span> <span class="nf">a</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Base of the controlled mod function.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="CMod.N"><a class="viewcode-back" href="../../../../modules/physics/quantum/shor.html#sympy.physics.quantum.shor.CMod.N">[docs]</a>    <span class="k">def</span> <span class="nf">N</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;N is the type of modular arithmetic we are doing.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-</div>
-    <span class="k">def</span> <span class="nf">_apply_operator_Qubit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">qubits</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            This directly calculates the controlled mod of the second half of</span>
-<span class="sd">            the register and puts it in the second</span>
-<span class="sd">            This will look pretty when we get Tensor Symbolically working</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="c"># Determine the value stored in high memory.</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">t</span><span class="p">):</span>
-            <span class="n">k</span> <span class="o">=</span> <span class="n">k</span> <span class="o">+</span> <span class="n">n</span><span class="o">*</span><span class="n">qubits</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">t</span> <span class="o">+</span> <span class="n">i</span><span class="p">]</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">n</span><span class="o">*</span><span class="mi">2</span>
-
-        <span class="c"># The value to go in low memory will be out.</span>
-        <span class="n">out</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">a</span><span class="o">**</span><span class="n">k</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">N</span><span class="p">)</span>
-
-        <span class="c"># Create array for new qbit-ket which will have high memory unaffected</span>
-        <span class="n">outarray</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">qubits</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">][:</span><span class="bp">self</span><span class="o">.</span><span class="n">t</span><span class="p">])</span>
-
-        <span class="c"># Place out in low memory</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">t</span><span class="p">))):</span>
-            <span class="n">outarray</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">out</span> <span class="o">&gt;&gt;</span> <span class="n">i</span><span class="p">)</span> <span class="o">&amp;</span> <span class="mi">1</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">Qubit</span><span class="p">(</span><span class="o">*</span><span class="n">outarray</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="shor"><a class="viewcode-back" href="../../../../modules/physics/quantum/shor.html#sympy.physics.quantum.shor.shor">[docs]</a><span class="k">def</span> <span class="nf">shor</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;This function implements Shor&#39;s factoring algorithm on the Integer N</span>
-
-<span class="sd">    The algorithm starts by picking a random number (a) and seeing if it is</span>
-<span class="sd">    coprime with N. If it isn&#39;t, then the gcd of the two numbers is a factor</span>
-<span class="sd">    and we are done. Otherwise, it begins the period_finding subroutine which</span>
-<span class="sd">    finds the period of a in modulo N arithmetic. This period, if even, can</span>
-<span class="sd">    be used to calculate factors by taking a**(r/2)-1 and a**(r/2)+1.</span>
-<span class="sd">    These values are returned.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randrange</span><span class="p">(</span><span class="n">N</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span>
-    <span class="k">if</span> <span class="n">igcd</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">print</span><span class="p">(</span><span class="s">&quot;got lucky with rand&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">igcd</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&quot;a= &quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&quot;N= &quot;</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">period_find</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&quot;r= &quot;</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">r</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">print</span><span class="p">(</span><span class="s">&quot;r is not even, begin again&quot;</span><span class="p">)</span>
-        <span class="n">shor</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-    <span class="n">answer</span> <span class="o">=</span> <span class="p">(</span><span class="n">igcd</span><span class="p">(</span><span class="n">a</span><span class="o">**</span><span class="p">(</span><span class="n">r</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="p">),</span> <span class="n">igcd</span><span class="p">(</span><span class="n">a</span><span class="o">**</span><span class="p">(</span><span class="n">r</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">answer</span>
-
-</div>
-<span class="k">def</span> <span class="nf">getr</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">N</span><span class="p">):</span>
-    <span class="n">fraction</span> <span class="o">=</span> <span class="n">continued_fraction</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-    <span class="c"># Now convert into r</span>
-    <span class="n">total</span> <span class="o">=</span> <span class="n">ratioize</span><span class="p">(</span><span class="n">fraction</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">total</span>
-
-
-<span class="k">def</span> <span class="nf">ratioize</span><span class="p">(</span><span class="nb">list</span><span class="p">,</span> <span class="n">N</span><span class="p">):</span>
-    <span class="k">if</span> <span class="nb">list</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">N</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="nb">list</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">list</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">return</span> <span class="nb">list</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">ratioize</span><span class="p">(</span><span class="nb">list</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">N</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="continued_fraction"><a class="viewcode-back" href="../../../../modules/physics/quantum/shor.html#sympy.physics.quantum.shor.continued_fraction">[docs]</a><span class="k">def</span> <span class="nf">continued_fraction</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;This applies the continued fraction expansion to two numbers x/y</span>
-
-<span class="sd">    x is the numerator and y is the denominator</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum.shor import continued_fraction</span>
-<span class="sd">    &gt;&gt;&gt; continued_fraction(3, 8)</span>
-<span class="sd">    [0, 2, 1, 2]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="n">y</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-    <span class="n">temp</span> <span class="o">=</span> <span class="n">x</span><span class="o">//</span><span class="n">y</span>
-    <span class="k">if</span> <span class="n">temp</span><span class="o">*</span><span class="n">y</span> <span class="o">==</span> <span class="n">x</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">temp</span><span class="p">,</span> <span class="p">]</span>
-
-    <span class="nb">list</span> <span class="o">=</span> <span class="n">continued_fraction</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span> <span class="o">-</span> <span class="n">temp</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
-    <span class="nb">list</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">temp</span><span class="p">)</span>
-    <span class="k">return</span> <span class="nb">list</span>
-
-</div>
-<div class="viewcode-block" id="period_find"><a class="viewcode-back" href="../../../../modules/physics/quantum/shor.html#sympy.physics.quantum.shor.period_find">[docs]</a><span class="k">def</span> <span class="nf">period_find</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">N</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Finds the period of a in modulo N arithmetic</span>
-
-<span class="sd">    This is quantum part of Shor&#39;s algorithm.It takes two registers,</span>
-<span class="sd">    puts first in superposition of states with Hadamards so: ``|k&gt;|0&gt;``</span>
-<span class="sd">    with k being all possible choices. It then does a controlled mod and</span>
-<span class="sd">    a QFT to determine the order of a.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">epsilon</span> <span class="o">=</span> <span class="o">.</span><span class="mi">5</span>
-    <span class="c">#picks out t&#39;s such that maintains accuracy within epsilon</span>
-    <span class="n">t</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">ceil</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">2</span><span class="p">)))</span>
-    <span class="c"># make the first half of register be 0&#39;s |000...000&gt;</span>
-    <span class="n">start</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">t</span><span class="p">)]</span>
-    <span class="c">#Put second half into superposition of states so we have |1&gt;x|0&gt; + |2&gt;x|0&gt; + ... |k&gt;x&gt;|0&gt; + ... + |2**n-1&gt;x|0&gt;</span>
-    <span class="n">factor</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">t</span><span class="p">)</span>
-    <span class="n">qubits</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">arr</span> <span class="ow">in</span> <span class="n">variations</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">)),</span> <span class="n">t</span><span class="p">,</span> <span class="n">repetition</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="n">qbitArray</span> <span class="o">=</span> <span class="n">arr</span> <span class="o">+</span> <span class="n">start</span>
-        <span class="n">qubits</span> <span class="o">=</span> <span class="n">qubits</span> <span class="o">+</span> <span class="n">Qubit</span><span class="p">(</span><span class="o">*</span><span class="n">qbitArray</span><span class="p">)</span>
-    <span class="n">circuit</span> <span class="o">=</span> <span class="p">(</span><span class="n">factor</span><span class="o">*</span><span class="n">qubits</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-    <span class="c">#Controlled second half of register so that we have:</span>
-    <span class="c"># |1&gt;x|a**1 %N&gt; + |2&gt;x|a**2 %N&gt; + ... + |k&gt;x|a**k %N &gt;+ ... + |2**n-1=k&gt;x|a**k % n&gt;</span>
-    <span class="n">circuit</span> <span class="o">=</span> <span class="n">CMod</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span><span class="o">*</span><span class="n">circuit</span>
-    <span class="c">#will measure first half of register giving one of the a**k%N&#39;s</span>
-    <span class="n">circuit</span> <span class="o">=</span> <span class="n">qapply</span><span class="p">(</span><span class="n">circuit</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&quot;controlled Mod&#39;d&quot;</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
-        <span class="n">circuit</span> <span class="o">=</span> <span class="n">measure_partial_oneshot</span><span class="p">(</span><span class="n">circuit</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-        <span class="c"># circuit = measure(i)*circuit</span>
-    <span class="c"># circuit = qapply(circuit)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&quot;measured 1&quot;</span><span class="p">)</span>
-    <span class="c">#Now apply Inverse Quantum Fourier Transform on the second half of the register</span>
-    <span class="n">circuit</span> <span class="o">=</span> <span class="n">qapply</span><span class="p">(</span><span class="n">QFT</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">t</span><span class="o">*</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">decompose</span><span class="p">()</span><span class="o">*</span><span class="n">circuit</span><span class="p">,</span> <span class="n">floatingPoint</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&quot;QFT&#39;d&quot;</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
-        <span class="n">circuit</span> <span class="o">=</span> <span class="n">measure_partial_oneshot</span><span class="p">(</span><span class="n">circuit</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="n">t</span><span class="p">)</span>
-        <span class="c"># circuit = measure(i+t)*circuit</span>
-    <span class="c"># circuit = qapply(circuit)</span>
-    <span class="k">print</span><span class="p">(</span><span class="n">circuit</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">circuit</span><span class="p">,</span> <span class="n">Qubit</span><span class="p">):</span>
-        <span class="n">register</span> <span class="o">=</span> <span class="n">circuit</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">circuit</span><span class="p">,</span> <span class="n">Mul</span><span class="p">):</span>
-        <span class="n">register</span> <span class="o">=</span> <span class="n">circuit</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">register</span> <span class="o">=</span> <span class="n">circuit</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="k">print</span><span class="p">(</span><span class="n">register</span><span class="p">)</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">answer</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">register</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">):</span>
-        <span class="n">answer</span> <span class="o">+=</span> <span class="n">n</span><span class="o">*</span><span class="n">register</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="n">t</span><span class="p">]</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">n</span> <span class="o">&lt;&lt;</span> <span class="mi">1</span>
-    <span class="k">if</span> <span class="n">answer</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">OrderFindingException</span><span class="p">(</span>
-            <span class="s">&quot;Order finder returned 0. Happens with chance </span><span class="si">%f</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">epsilon</span><span class="p">)</span>
-    <span class="c">#turn answer into r using continued fractions</span>
-    <span class="n">g</span> <span class="o">=</span> <span class="n">getr</span><span class="p">(</span><span class="n">answer</span><span class="p">,</span> <span class="mi">2</span><span class="o">**</span><span class="n">t</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">g</span></div>
-</pre></div>
-
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-              <img class="logo" src="../../../../_static/sympylogo.png" alt="Logo"/>
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diff --git a/dev-py3k/_modules/sympy/physics/quantum/spin.html b/dev-py3k/_modules/sympy/physics/quantum/spin.html
deleted file mode 100644
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--- a/dev-py3k/_modules/sympy/physics/quantum/spin.html
+++ /dev/null
@@ -1,2192 +0,0 @@
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-  <h1>Source code for sympy.physics.quantum.spin</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Quantum mechanical angular momemtum.&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="p">(</span><span class="n">Add</span><span class="p">,</span> <span class="n">binomial</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">Integer</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span>
-                   <span class="n">pi</span><span class="p">,</span> <span class="n">Rational</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">Sum</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span>
-                   <span class="n">Tuple</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">zeros</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.pretty.stringpict</span> <span class="kn">import</span> <span class="n">prettyForm</span><span class="p">,</span> <span class="n">stringPict</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.pretty.pretty_symbology</span> <span class="kn">import</span> <span class="n">pretty_symbol</span>
-
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.qexpr</span> <span class="kn">import</span> <span class="n">QExpr</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.operator</span> <span class="kn">import</span> <span class="p">(</span><span class="n">HermitianOperator</span><span class="p">,</span> <span class="n">Operator</span><span class="p">,</span>
-                                            <span class="n">UnitaryOperator</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.state</span> <span class="kn">import</span> <span class="n">Bra</span><span class="p">,</span> <span class="n">Ket</span><span class="p">,</span> <span class="n">State</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.special.tensor_functions</span> <span class="kn">import</span> <span class="n">KroneckerDelta</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.constants</span> <span class="kn">import</span> <span class="n">hbar</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.hilbert</span> <span class="kn">import</span> <span class="n">ComplexSpace</span><span class="p">,</span> <span class="n">DirectSumHilbertSpace</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.tensorproduct</span> <span class="kn">import</span> <span class="n">TensorProduct</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.cg</span> <span class="kn">import</span> <span class="n">CG</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.qapply</span> <span class="kn">import</span> <span class="n">qapply</span>
-
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="s">&#39;m_values&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Jplus&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Jminus&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Jx&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Jy&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Jz&#39;</span><span class="p">,</span>
-    <span class="s">&#39;J2&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Rotation&#39;</span><span class="p">,</span>
-    <span class="s">&#39;WignerD&#39;</span><span class="p">,</span>
-    <span class="s">&#39;JxKet&#39;</span><span class="p">,</span>
-    <span class="s">&#39;JxBra&#39;</span><span class="p">,</span>
-    <span class="s">&#39;JyKet&#39;</span><span class="p">,</span>
-    <span class="s">&#39;JyBra&#39;</span><span class="p">,</span>
-    <span class="s">&#39;JzKet&#39;</span><span class="p">,</span>
-    <span class="s">&#39;JzBra&#39;</span><span class="p">,</span>
-    <span class="s">&#39;JxKetCoupled&#39;</span><span class="p">,</span>
-    <span class="s">&#39;JxBraCoupled&#39;</span><span class="p">,</span>
-    <span class="s">&#39;JyKetCoupled&#39;</span><span class="p">,</span>
-    <span class="s">&#39;JyBraCoupled&#39;</span><span class="p">,</span>
-    <span class="s">&#39;JzKetCoupled&#39;</span><span class="p">,</span>
-    <span class="s">&#39;JzBraCoupled&#39;</span><span class="p">,</span>
-    <span class="s">&#39;couple&#39;</span><span class="p">,</span>
-    <span class="s">&#39;uncouple&#39;</span>
-<span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">m_values</span><span class="p">(</span><span class="n">j</span><span class="p">):</span>
-    <span class="n">j</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-    <span class="n">size</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">size</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">size</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-            <span class="s">&#39;Only integer or half-integer values allowed for j, got: : </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">j</span>
-        <span class="p">)</span>
-    <span class="k">return</span> <span class="n">size</span><span class="p">,</span> <span class="p">[</span><span class="n">j</span> <span class="o">-</span> <span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))]</span>
-
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># Spin Operators</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">SpinOpBase</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Base class for spin operators.&quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_hilbert_space</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">label</span><span class="p">):</span>
-        <span class="c"># We consider all j values so our space is infinite.</span>
-        <span class="k">return</span> <span class="n">ComplexSpace</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">name</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">_print_contents</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_coord</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_contents_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">))</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_coord</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_subscript_pretty</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_contents_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&#39;</span><span class="si">%s</span><span class="s">_</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">((</span><span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_coord</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_represent_base</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;j&#39;</span><span class="p">,</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-        <span class="n">size</span><span class="p">,</span> <span class="n">mvals</span> <span class="o">=</span> <span class="n">m_values</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">size</span><span class="p">,</span> <span class="n">size</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">):</span>
-                <span class="n">me</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">matrix_element</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">mvals</span><span class="p">[</span><span class="n">p</span><span class="p">],</span> <span class="n">j</span><span class="p">,</span> <span class="n">mvals</span><span class="p">[</span><span class="n">q</span><span class="p">])</span>
-                <span class="n">result</span><span class="p">[</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">]</span> <span class="o">=</span> <span class="n">me</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="k">def</span> <span class="nf">_apply_op</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="n">orig_basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">state</span> <span class="o">=</span> <span class="n">ket</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">basis</span><span class="p">)</span>
-        <span class="c"># If the state has only one term</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">State</span><span class="p">):</span>
-            <span class="n">ret</span> <span class="o">=</span> <span class="p">(</span><span class="n">hbar</span><span class="o">*</span><span class="n">state</span><span class="o">.</span><span class="n">m</span><span class="p">)</span> <span class="o">*</span> <span class="n">state</span>
-        <span class="c"># state is a linear combination of states</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">Sum</span><span class="p">):</span>
-            <span class="n">ret</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_apply_operator_Sum</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">ret</span> <span class="o">=</span> <span class="n">qapply</span><span class="p">(</span><span class="bp">self</span><span class="o">*</span><span class="n">state</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">ret</span> <span class="o">==</span> <span class="bp">self</span><span class="o">*</span><span class="n">state</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-        <span class="k">return</span> <span class="n">ret</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">orig_basis</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_JxKet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_apply_op</span><span class="p">(</span><span class="n">ket</span><span class="p">,</span> <span class="s">&#39;Jx&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_JxKetCoupled</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_apply_op</span><span class="p">(</span><span class="n">ket</span><span class="p">,</span> <span class="s">&#39;Jx&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_JyKet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_apply_op</span><span class="p">(</span><span class="n">ket</span><span class="p">,</span> <span class="s">&#39;Jy&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_JyKetCoupled</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_apply_op</span><span class="p">(</span><span class="n">ket</span><span class="p">,</span> <span class="s">&#39;Jy&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_JzKet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_apply_op</span><span class="p">(</span><span class="n">ket</span><span class="p">,</span> <span class="s">&#39;Jz&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_JzKetCoupled</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_apply_op</span><span class="p">(</span><span class="n">ket</span><span class="p">,</span> <span class="s">&#39;Jz&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_TensorProduct</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">tp</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="c"># Uncoupling operator is only easily found for coordinate basis spin operators</span>
-        <span class="c"># TODO: add methods for uncoupling operators</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">JxOp</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">JyOp</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">JzOp</span><span class="p">)):</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">tp</span><span class="o">.</span><span class="n">args</span><span class="p">)):</span>
-            <span class="n">arg</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">arg</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">tp</span><span class="o">.</span><span class="n">args</span><span class="p">[:</span><span class="n">n</span><span class="p">])</span>
-            <span class="n">arg</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_apply_operator</span><span class="p">(</span><span class="n">tp</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">n</span><span class="p">]))</span>
-            <span class="n">arg</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">tp</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:])</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tp</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="o">*</span><span class="n">arg</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">result</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-
-    <span class="c"># TODO: move this to qapply_Mul</span>
-    <span class="k">def</span> <span class="nf">_apply_operator_Sum</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">new_func</span> <span class="o">=</span> <span class="n">qapply</span><span class="p">(</span><span class="bp">self</span> <span class="o">*</span> <span class="n">s</span><span class="o">.</span><span class="n">function</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">new_func</span> <span class="o">==</span> <span class="bp">self</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">function</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-        <span class="k">return</span> <span class="n">Sum</span><span class="p">(</span><span class="n">new_func</span><span class="p">,</span> <span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">limits</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_trace</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="c">#TODO: use options to use different j values</span>
-        <span class="c">#For now eval at default basis</span>
-
-        <span class="c"># is it efficient to represent each time</span>
-        <span class="c"># to do a trace?</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_default_basis</span><span class="p">()</span><span class="o">.</span><span class="n">trace</span><span class="p">()</span>
-
-
-<span class="k">class</span> <span class="nc">JplusOp</span><span class="p">(</span><span class="n">SpinOpBase</span><span class="p">,</span> <span class="n">Operator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The J+ operator.&quot;&quot;&quot;</span>
-
-    <span class="n">_coord</span> <span class="o">=</span> <span class="s">&#39;+&#39;</span>
-
-    <span class="n">basis</span> <span class="o">=</span> <span class="s">&#39;Jz&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_JminusOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">hbar</span><span class="o">*</span><span class="n">JzOp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_JzKet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="n">ket</span><span class="o">.</span><span class="n">j</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">ket</span><span class="o">.</span><span class="n">m</span>
-        <span class="k">if</span> <span class="n">m</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">and</span> <span class="n">j</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">m</span> <span class="o">&gt;=</span> <span class="n">j</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">return</span> <span class="n">hbar</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">j</span><span class="o">*</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span> <span class="o">-</span> <span class="n">m</span><span class="o">*</span><span class="p">(</span><span class="n">m</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">))</span><span class="o">*</span><span class="n">JzKet</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">m</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_JzKetCoupled</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="n">ket</span><span class="o">.</span><span class="n">j</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">ket</span><span class="o">.</span><span class="n">m</span>
-        <span class="n">jn</span> <span class="o">=</span> <span class="n">ket</span><span class="o">.</span><span class="n">jn</span>
-        <span class="n">coupling</span> <span class="o">=</span> <span class="n">ket</span><span class="o">.</span><span class="n">coupling</span>
-        <span class="k">if</span> <span class="n">m</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">and</span> <span class="n">j</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">m</span> <span class="o">&gt;=</span> <span class="n">j</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">return</span> <span class="n">hbar</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">j</span><span class="o">*</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span> <span class="o">-</span> <span class="n">m</span><span class="o">*</span><span class="p">(</span><span class="n">m</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">))</span><span class="o">*</span><span class="n">JzKetCoupled</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">m</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">jn</span><span class="p">,</span> <span class="n">coupling</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">matrix_element</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">jp</span><span class="p">,</span> <span class="n">mp</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">hbar</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">j</span><span class="o">*</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span> <span class="o">-</span> <span class="n">mp</span><span class="o">*</span><span class="p">(</span><span class="n">mp</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">))</span>
-        <span class="n">result</span> <span class="o">*=</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">mp</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">*=</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">jp</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="k">def</span> <span class="nf">_represent_default_basis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_JzOp</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JzOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_base</span><span class="p">(</span><span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_xyz</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JxOp</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">JyOp</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-
-<span class="k">class</span> <span class="nc">JminusOp</span><span class="p">(</span><span class="n">SpinOpBase</span><span class="p">,</span> <span class="n">Operator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The J- operator.&quot;&quot;&quot;</span>
-
-    <span class="n">_coord</span> <span class="o">=</span> <span class="s">&#39;-&#39;</span>
-
-    <span class="n">basis</span> <span class="o">=</span> <span class="s">&#39;Jz&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_JzKet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="n">ket</span><span class="o">.</span><span class="n">j</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">ket</span><span class="o">.</span><span class="n">m</span>
-        <span class="k">if</span> <span class="n">m</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">and</span> <span class="n">j</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">m</span> <span class="o">&lt;=</span> <span class="o">-</span><span class="n">j</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">return</span> <span class="n">hbar</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">j</span><span class="o">*</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span> <span class="o">-</span> <span class="n">m</span><span class="o">*</span><span class="p">(</span><span class="n">m</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">))</span><span class="o">*</span><span class="n">JzKet</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">m</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_JzKetCoupled</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="n">ket</span><span class="o">.</span><span class="n">j</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">ket</span><span class="o">.</span><span class="n">m</span>
-        <span class="n">jn</span> <span class="o">=</span> <span class="n">ket</span><span class="o">.</span><span class="n">jn</span>
-        <span class="n">coupling</span> <span class="o">=</span> <span class="n">ket</span><span class="o">.</span><span class="n">coupling</span>
-        <span class="k">if</span> <span class="n">m</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">and</span> <span class="n">j</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">m</span> <span class="o">&lt;=</span> <span class="o">-</span><span class="n">j</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">return</span> <span class="n">hbar</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">j</span><span class="o">*</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span> <span class="o">-</span> <span class="n">m</span><span class="o">*</span><span class="p">(</span><span class="n">m</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">))</span><span class="o">*</span><span class="n">JzKetCoupled</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">m</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">jn</span><span class="p">,</span> <span class="n">coupling</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">matrix_element</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">jp</span><span class="p">,</span> <span class="n">mp</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">hbar</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">j</span><span class="o">*</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span> <span class="o">-</span> <span class="n">mp</span><span class="o">*</span><span class="p">(</span><span class="n">mp</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">))</span>
-        <span class="n">result</span> <span class="o">*=</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">mp</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">*=</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">jp</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="k">def</span> <span class="nf">_represent_default_basis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_JzOp</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JzOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_base</span><span class="p">(</span><span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_xyz</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JxOp</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">-</span> <span class="n">I</span><span class="o">*</span><span class="n">JyOp</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-
-<span class="k">class</span> <span class="nc">JxOp</span><span class="p">(</span><span class="n">SpinOpBase</span><span class="p">,</span> <span class="n">HermitianOperator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The Jx operator.&quot;&quot;&quot;</span>
-
-    <span class="n">_coord</span> <span class="o">=</span> <span class="s">&#39;x&#39;</span>
-
-    <span class="n">basis</span> <span class="o">=</span> <span class="s">&#39;Jx&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_JyOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">I</span><span class="o">*</span><span class="n">hbar</span><span class="o">*</span><span class="n">JzOp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_JzOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">hbar</span><span class="o">*</span><span class="n">JyOp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_JzKet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">jp</span> <span class="o">=</span> <span class="n">JplusOp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span><span class="o">.</span><span class="n">_apply_operator_JzKet</span><span class="p">(</span><span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="n">jm</span> <span class="o">=</span> <span class="n">JminusOp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span><span class="o">.</span><span class="n">_apply_operator_JzKet</span><span class="p">(</span><span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">jp</span> <span class="o">+</span> <span class="n">jm</span><span class="p">)</span><span class="o">/</span><span class="n">Integer</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_JzKetCoupled</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">jp</span> <span class="o">=</span> <span class="n">JplusOp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span><span class="o">.</span><span class="n">_apply_operator_JzKetCoupled</span><span class="p">(</span><span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="n">jm</span> <span class="o">=</span> <span class="n">JminusOp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span><span class="o">.</span><span class="n">_apply_operator_JzKetCoupled</span><span class="p">(</span><span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">jp</span> <span class="o">+</span> <span class="n">jm</span><span class="p">)</span><span class="o">/</span><span class="n">Integer</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_default_basis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_JzOp</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JzOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">jp</span> <span class="o">=</span> <span class="n">JplusOp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span><span class="o">.</span><span class="n">_represent_JzOp</span><span class="p">(</span><span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="n">jm</span> <span class="o">=</span> <span class="n">JminusOp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span><span class="o">.</span><span class="n">_represent_JzOp</span><span class="p">(</span><span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">jp</span> <span class="o">+</span> <span class="n">jm</span><span class="p">)</span><span class="o">/</span><span class="n">Integer</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_plusminus</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">JplusOp</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">+</span> <span class="n">JminusOp</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span><span class="o">/</span><span class="mi">2</span>
-
-
-<span class="k">class</span> <span class="nc">JyOp</span><span class="p">(</span><span class="n">SpinOpBase</span><span class="p">,</span> <span class="n">HermitianOperator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The Jy operator.&quot;&quot;&quot;</span>
-
-    <span class="n">_coord</span> <span class="o">=</span> <span class="s">&#39;y&#39;</span>
-
-    <span class="n">basis</span> <span class="o">=</span> <span class="s">&#39;Jy&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_JzOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">I</span><span class="o">*</span><span class="n">hbar</span><span class="o">*</span><span class="n">JxOp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_JxOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">hbar</span><span class="o">*</span><span class="n">J2Op</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_JzKet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">jp</span> <span class="o">=</span> <span class="n">JplusOp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span><span class="o">.</span><span class="n">_apply_operator_JzKet</span><span class="p">(</span><span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="n">jm</span> <span class="o">=</span> <span class="n">JminusOp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span><span class="o">.</span><span class="n">_apply_operator_JzKet</span><span class="p">(</span><span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">jp</span> <span class="o">-</span> <span class="n">jm</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">Integer</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">I</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_JzKetCoupled</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">jp</span> <span class="o">=</span> <span class="n">JplusOp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span><span class="o">.</span><span class="n">_apply_operator_JzKetCoupled</span><span class="p">(</span><span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="n">jm</span> <span class="o">=</span> <span class="n">JminusOp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span><span class="o">.</span><span class="n">_apply_operator_JzKetCoupled</span><span class="p">(</span><span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">jp</span> <span class="o">-</span> <span class="n">jm</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">Integer</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">I</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_default_basis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_JzOp</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JzOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">jp</span> <span class="o">=</span> <span class="n">JplusOp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span><span class="o">.</span><span class="n">_represent_JzOp</span><span class="p">(</span><span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="n">jm</span> <span class="o">=</span> <span class="n">JminusOp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span><span class="o">.</span><span class="n">_represent_JzOp</span><span class="p">(</span><span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">jp</span> <span class="o">-</span> <span class="n">jm</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">Integer</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">I</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_plusminus</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">JplusOp</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">-</span> <span class="n">JminusOp</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">I</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">JzOp</span><span class="p">(</span><span class="n">SpinOpBase</span><span class="p">,</span> <span class="n">HermitianOperator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The Jz operator.&quot;&quot;&quot;</span>
-
-    <span class="n">_coord</span> <span class="o">=</span> <span class="s">&#39;z&#39;</span>
-
-    <span class="n">basis</span> <span class="o">=</span> <span class="s">&#39;Jz&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_JxOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">I</span><span class="o">*</span><span class="n">hbar</span><span class="o">*</span><span class="n">JyOp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_JyOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">hbar</span><span class="o">*</span><span class="n">JxOp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_JplusOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">hbar</span><span class="o">*</span><span class="n">JplusOp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_JminusOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">hbar</span><span class="o">*</span><span class="n">JminusOp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">matrix_element</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">jp</span><span class="p">,</span> <span class="n">mp</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">hbar</span><span class="o">*</span><span class="n">mp</span>
-        <span class="n">result</span> <span class="o">*=</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">mp</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">*=</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">jp</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="k">def</span> <span class="nf">_represent_default_basis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_JzOp</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JzOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_base</span><span class="p">(</span><span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">J2Op</span><span class="p">(</span><span class="n">SpinOpBase</span><span class="p">,</span> <span class="n">HermitianOperator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The J^2 operator.&quot;&quot;&quot;</span>
-
-    <span class="n">_coord</span> <span class="o">=</span> <span class="s">&#39;2&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_JxOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_JyOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_JzOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_JplusOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="k">def</span> <span class="nf">_eval_commutator_JminusOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_JxKet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="n">ket</span><span class="o">.</span><span class="n">j</span>
-        <span class="k">return</span> <span class="n">hbar</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">j</span><span class="o">*</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">ket</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_JxKetCoupled</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="n">ket</span><span class="o">.</span><span class="n">j</span>
-        <span class="k">return</span> <span class="n">hbar</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">j</span><span class="o">*</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">ket</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_JyKet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="n">ket</span><span class="o">.</span><span class="n">j</span>
-        <span class="k">return</span> <span class="n">hbar</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">j</span><span class="o">*</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">ket</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_JyKetCoupled</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="n">ket</span><span class="o">.</span><span class="n">j</span>
-        <span class="k">return</span> <span class="n">hbar</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">j</span><span class="o">*</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">ket</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_JzKet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="n">ket</span><span class="o">.</span><span class="n">j</span>
-        <span class="k">return</span> <span class="n">hbar</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">j</span><span class="o">*</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">ket</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator_JzKetCoupled</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ket</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="n">ket</span><span class="o">.</span><span class="n">j</span>
-        <span class="k">return</span> <span class="n">hbar</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">j</span><span class="o">*</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">ket</span>
-
-    <span class="k">def</span> <span class="nf">matrix_element</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">jp</span><span class="p">,</span> <span class="n">mp</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">(</span><span class="n">hbar</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">j</span><span class="o">*</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">*=</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">mp</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">*=</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">jp</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="k">def</span> <span class="nf">_represent_default_basis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_JzOp</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JzOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_base</span><span class="p">(</span><span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_contents_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">))</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;2&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">a</span><span class="o">**</span><span class="n">b</span>
-
-    <span class="k">def</span> <span class="nf">_print_contents_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&#39;</span><span class="si">%s</span><span class="s">^2&#39;</span> <span class="o">%</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_xyz</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JxOp</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">JyOp</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">JzOp</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">**</span><span class="mi">2</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_plusminus</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">JzOp</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> \
-            <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">JplusOp</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">JminusOp</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">+</span> <span class="n">JminusOp</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">JplusOp</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
-
-
-<div class="viewcode-block" id="Rotation"><a class="viewcode-back" href="../../../../modules/physics/quantum/spin.html#sympy.physics.quantum.spin.Rotation">[docs]</a><span class="k">class</span> <span class="nc">Rotation</span><span class="p">(</span><span class="n">UnitaryOperator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Wigner D operator in terms of Euler angles.</span>
-
-<span class="sd">    Defines the rotation operator in terms of the Euler angles defined by</span>
-<span class="sd">    the z-y-z convention for a passive transformation. That is the coordinate</span>
-<span class="sd">    axes are rotated first about the z-axis, giving the new x&#39;-y&#39;-z&#39; axes. Then</span>
-<span class="sd">    this new coordinate system is rotated about the new y&#39;-axis, giving new</span>
-<span class="sd">    x&#39;&#39;-y&#39;&#39;-z&#39;&#39; axes. Then this new coordinate system is rotated about the</span>
-<span class="sd">    z&#39;&#39;-axis. Conventions follow those laid out in [1]_.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    alpha : Number, Symbol</span>
-<span class="sd">        First Euler Angle</span>
-<span class="sd">    beta : Number, Symbol</span>
-<span class="sd">        Second Euler angle</span>
-<span class="sd">    gamma : Number, Symbol</span>
-<span class="sd">        Third Euler angle</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    A simple example rotation operator:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import pi</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.spin import Rotation</span>
-<span class="sd">        &gt;&gt;&gt; Rotation(pi, 0, pi/2)</span>
-<span class="sd">        R(pi,0,pi/2)</span>
-
-<span class="sd">    With symbolic Euler angles and calculating the inverse rotation operator:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; a, b, c = symbols(&#39;a b c&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; Rotation(a, b, c)</span>
-<span class="sd">        R(a,b,c)</span>
-<span class="sd">        &gt;&gt;&gt; Rotation(a, b, c).inverse()</span>
-<span class="sd">        R(-c,-b,-a)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    WignerD: Symbolic Wigner-D function</span>
-<span class="sd">    D: Wigner-D function</span>
-<span class="sd">    d: Wigner small-d function</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_args</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="n">QExpr</span><span class="o">.</span><span class="n">_eval_args</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;3 Euler angles required, got: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">args</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_hilbert_space</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">label</span><span class="p">):</span>
-        <span class="c"># We consider all j values so our space is infinite.</span>
-        <span class="k">return</span> <span class="n">ComplexSpace</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">alpha</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">beta</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">gamma</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">_print_operator_name</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;R&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_print_operator_name_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">printer</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\u211B</span><span class="s">&quot;</span> <span class="o">+</span> <span class="s">&quot; &quot;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&quot;R &quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_operator_name_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&#39;\mathcal{R}&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_eval_inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Rotation</span><span class="p">(</span><span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">gamma</span><span class="p">,</span> <span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">beta</span><span class="p">,</span> <span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">alpha</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Rotation.D"><a class="viewcode-back" href="../../../../modules/physics/quantum/spin.html#sympy.physics.quantum.spin.Rotation.D">[docs]</a>    <span class="k">def</span> <span class="nf">D</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">mp</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">gamma</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Wigner D-function.</span>
-
-<span class="sd">        Returns an instance of the WignerD class corresponding to the Wigner-D</span>
-<span class="sd">        function specified by the parameters.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ===========</span>
-
-<span class="sd">        j : Number</span>
-<span class="sd">            Total angular momentum</span>
-<span class="sd">        m : Number</span>
-<span class="sd">            Eigenvalue of angular momentum along axis after rotation</span>
-<span class="sd">        mp : Number</span>
-<span class="sd">            Eigenvalue of angular momentum along rotated axis</span>
-<span class="sd">        alpha : Number, Symbol</span>
-<span class="sd">            First Euler angle of rotation</span>
-<span class="sd">        beta : Number, Symbol</span>
-<span class="sd">            Second Euler angle of rotation</span>
-<span class="sd">        gamma : Number, Symbol</span>
-<span class="sd">            Third Euler angle of rotation</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        Return the Wigner-D matrix element for a defined rotation, both</span>
-<span class="sd">        numerical and symbolic:</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy.physics.quantum.spin import Rotation</span>
-<span class="sd">            &gt;&gt;&gt; from sympy import pi, symbols</span>
-<span class="sd">            &gt;&gt;&gt; alpha, beta, gamma = symbols(&#39;alpha beta gamma&#39;)</span>
-<span class="sd">            &gt;&gt;&gt; Rotation.D(1, 1, 0,pi, pi/2,-pi)</span>
-<span class="sd">            WignerD(1, 1, 0, pi, pi/2, -pi)</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        WignerD: Symbolic Wigner-D function</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">WignerD</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">mp</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">gamma</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Rotation.d"><a class="viewcode-back" href="../../../../modules/physics/quantum/spin.html#sympy.physics.quantum.spin.Rotation.d">[docs]</a>    <span class="k">def</span> <span class="nf">d</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">mp</span><span class="p">,</span> <span class="n">beta</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Wigner small-d function.</span>
-
-<span class="sd">        Returns an instance of the WignerD class corresponding to the Wigner-D</span>
-<span class="sd">        function specified by the parameters with the alpha and gamma angles</span>
-<span class="sd">        given as 0.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ===========</span>
-
-<span class="sd">        j : Number</span>
-<span class="sd">            Total angular momentum</span>
-<span class="sd">        m : Number</span>
-<span class="sd">            Eigenvalue of angular momentum along axis after rotation</span>
-<span class="sd">        mp : Number</span>
-<span class="sd">            Eigenvalue of angular momentum along rotated axis</span>
-<span class="sd">        beta : Number, Symbol</span>
-<span class="sd">            Second Euler angle of rotation</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        Return the Wigner-D matrix element for a defined rotation, both</span>
-<span class="sd">        numerical and symbolic:</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy.physics.quantum.spin import Rotation</span>
-<span class="sd">            &gt;&gt;&gt; from sympy import pi, symbols</span>
-<span class="sd">            &gt;&gt;&gt; beta = symbols(&#39;beta&#39;)</span>
-<span class="sd">            &gt;&gt;&gt; Rotation.d(1, 1, 0, pi/2)</span>
-<span class="sd">            WignerD(1, 1, 0, 0, pi/2, 0)</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        WignerD: Symbolic Wigner-D function</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">WignerD</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">mp</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">matrix_element</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">jp</span><span class="p">,</span> <span class="n">mp</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">D</span><span class="p">(</span>
-            <span class="n">jp</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">mp</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">alpha</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">beta</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">gamma</span>
-        <span class="p">)</span>
-        <span class="n">result</span> <span class="o">*=</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">jp</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="k">def</span> <span class="nf">_represent_base</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;j&#39;</span><span class="p">,</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)))</span>
-        <span class="c"># TODO: move evaluation up to represent function/implement elsewhere</span>
-        <span class="n">evaluate</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;doit&#39;</span><span class="p">))</span>
-        <span class="n">size</span><span class="p">,</span> <span class="n">mvals</span> <span class="o">=</span> <span class="n">m_values</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">size</span><span class="p">,</span> <span class="n">size</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">):</span>
-                <span class="n">me</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">matrix_element</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">mvals</span><span class="p">[</span><span class="n">p</span><span class="p">],</span> <span class="n">j</span><span class="p">,</span> <span class="n">mvals</span><span class="p">[</span><span class="n">q</span><span class="p">])</span>
-                <span class="k">if</span> <span class="n">evaluate</span><span class="p">:</span>
-                    <span class="n">result</span><span class="p">[</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">]</span> <span class="o">=</span> <span class="n">me</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">result</span><span class="p">[</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">]</span> <span class="o">=</span> <span class="n">me</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="k">def</span> <span class="nf">_represent_default_basis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_JzOp</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JzOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_base</span><span class="p">(</span><span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="WignerD"><a class="viewcode-back" href="../../../../modules/physics/quantum/spin.html#sympy.physics.quantum.spin.WignerD">[docs]</a><span class="k">class</span> <span class="nc">WignerD</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Wigner-D function</span>
-
-<span class="sd">    The Wigner D-function gives the matrix elements of the rotation</span>
-<span class="sd">    operator in the jm-representation. For the Euler angles `\\alpha`,</span>
-<span class="sd">    `\\beta`, `\gamma`, the D-function is defined such that:</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        &lt;j,m| \mathcal{R}(\\alpha, \\beta, \gamma ) |j&#39;,m&#39;&gt; = \delta_{jj&#39;} D(j, m, m&#39;, \\alpha, \\beta, \gamma)</span>
-
-<span class="sd">    Where the rotation operator is as defined by the Rotation class [1]_.</span>
-
-<span class="sd">    The Wigner D-function defined in this way gives:</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        D(j, m, m&#39;, \\alpha, \\beta, \gamma) = e^{-i m \\alpha} d(j, m, m&#39;, \\beta) e^{-i m&#39; \gamma}</span>
-
-<span class="sd">    Where d is the Wigner small-d function, which is given by Rotation.d.</span>
-
-<span class="sd">    The Wigner small-d function gives the component of the Wigner</span>
-<span class="sd">    D-function that is determined by the second Euler angle. That is the</span>
-<span class="sd">    Wigner D-function is:</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        D(j, m, m&#39;, \\alpha, \\beta, \gamma) = e^{-i m \\alpha} d(j, m, m&#39;, \\beta) e^{-i m&#39; \gamma}</span>
-
-<span class="sd">    Where d is the small-d function. The Wigner D-function is given by</span>
-<span class="sd">    Rotation.D.</span>
-
-<span class="sd">    Note that to evaluate the D-function, the j, m and mp parameters must</span>
-<span class="sd">    be integer or half integer numbers.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    j : Number</span>
-<span class="sd">        Total angular momentum</span>
-<span class="sd">    m : Number</span>
-<span class="sd">        Eigenvalue of angular momentum along axis after rotation</span>
-<span class="sd">    mp : Number</span>
-<span class="sd">        Eigenvalue of angular momentum along rotated axis</span>
-<span class="sd">    alpha : Number, Symbol</span>
-<span class="sd">        First Euler angle of rotation</span>
-<span class="sd">    beta : Number, Symbol</span>
-<span class="sd">        Second Euler angle of rotation</span>
-<span class="sd">    gamma : Number, Symbol</span>
-<span class="sd">        Third Euler angle of rotation</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Evaluate the Wigner-D matrix elements of a simple rotation:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.spin import Rotation</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import pi</span>
-<span class="sd">        &gt;&gt;&gt; rot = Rotation.D(1, 1, 0, pi, pi/2, 0)</span>
-<span class="sd">        &gt;&gt;&gt; rot</span>
-<span class="sd">        WignerD(1, 1, 0, pi, pi/2, 0)</span>
-<span class="sd">        &gt;&gt;&gt; rot.doit()</span>
-<span class="sd">        sqrt(2)/2</span>
-
-<span class="sd">    Evaluate the Wigner-d matrix elements of a simple rotation</span>
-
-<span class="sd">        &gt;&gt;&gt; rot = Rotation.d(1, 1, 0, pi/2)</span>
-<span class="sd">        &gt;&gt;&gt; rot</span>
-<span class="sd">        WignerD(1, 1, 0, 0, pi/2, 0)</span>
-<span class="sd">        &gt;&gt;&gt; rot.doit()</span>
-<span class="sd">        -sqrt(2)/2</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Rotation: Rotation operator</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">6</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;6 parameters expected, got </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">args</span><span class="p">)</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">evaluate</span> <span class="o">=</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;evaluate&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">evaluate</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span><span class="o">.</span><span class="n">_eval_wignerd</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">j</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">m</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">mp</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">alpha</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">beta</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">4</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">gamma</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">alpha</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">gamma</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&#39;d^{</span><span class="si">%s</span><span class="s">}_{</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">}\left(</span><span class="si">%s</span><span class="s">\right)&#39;</span> <span class="o">%</span> \
-                <span class="p">(</span>
-                    <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">m</span><span class="p">),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mp</span><span class="p">),</span>
-                    <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">beta</span><span class="p">)</span> <span class="p">)</span>
-        <span class="k">return</span> <span class="s">r&#39;D^{</span><span class="si">%s</span><span class="s">}_{</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">}\left(</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">\right)&#39;</span> <span class="o">%</span> \
-            <span class="p">(</span>
-                <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">m</span><span class="p">),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mp</span><span class="p">),</span>
-                <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">alpha</span><span class="p">),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">beta</span><span class="p">),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gamma</span><span class="p">)</span> <span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">top</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">)</span>
-
-        <span class="n">bot</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">m</span><span class="p">)</span>
-        <span class="n">bot</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">bot</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39;,&#39;</span><span class="p">))</span>
-        <span class="n">bot</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">bot</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mp</span><span class="p">)))</span>
-
-        <span class="n">pad</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">top</span><span class="o">.</span><span class="n">width</span><span class="p">(),</span> <span class="n">bot</span><span class="o">.</span><span class="n">width</span><span class="p">())</span>
-        <span class="n">top</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">top</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">))</span>
-        <span class="n">bot</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">bot</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">pad</span> <span class="o">&gt;</span> <span class="n">top</span><span class="o">.</span><span class="n">width</span><span class="p">():</span>
-            <span class="n">top</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">top</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span> <span class="o">*</span> <span class="p">(</span><span class="n">pad</span> <span class="o">-</span> <span class="n">top</span><span class="o">.</span><span class="n">width</span><span class="p">())))</span>
-        <span class="k">if</span> <span class="n">pad</span> <span class="o">&gt;</span> <span class="n">bot</span><span class="o">.</span><span class="n">width</span><span class="p">():</span>
-            <span class="n">bot</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">bot</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span> <span class="o">*</span> <span class="p">(</span><span class="n">pad</span> <span class="o">-</span> <span class="n">bot</span><span class="o">.</span><span class="n">width</span><span class="p">())))</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">alpha</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">gamma</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">beta</span><span class="p">)</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="s">&#39;d&#39;</span> <span class="o">+</span> <span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">pad</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">alpha</span><span class="p">)</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39;,&#39;</span><span class="p">))</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">beta</span><span class="p">)))</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39;,&#39;</span><span class="p">))</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gamma</span><span class="p">)))</span>
-
-            <span class="n">s</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="s">&#39;D&#39;</span> <span class="o">+</span> <span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">pad</span><span class="p">)</span>
-
-        <span class="n">args</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">above</span><span class="p">(</span><span class="n">top</span><span class="p">))</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="n">bot</span><span class="p">))</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">args</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">s</span>
-
-    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="n">hints</span><span class="p">[</span><span class="s">&#39;evaluate&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="n">WignerD</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_wignerd</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">)</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">m</span><span class="p">)</span>
-        <span class="n">mp</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mp</span><span class="p">)</span>
-        <span class="n">alpha</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">alpha</span><span class="p">)</span>
-        <span class="n">beta</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">beta</span><span class="p">)</span>
-        <span class="n">gamma</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gamma</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">j</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&quot;j parameter must be numerical to evaluate, got </span><span class="si">%s</span><span class="s">&quot;</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">if</span> <span class="n">beta</span> <span class="o">==</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span>
-            <span class="c"># Varshalovich Equation (5), Section 4.16, page 113, setting</span>
-            <span class="c"># alpha=gamma=0.</span>
-            <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">k</span> <span class="o">&gt;</span> <span class="n">j</span> <span class="o">+</span> <span class="n">mp</span> <span class="ow">or</span> <span class="n">k</span> <span class="o">&gt;</span> <span class="n">j</span> <span class="o">-</span> <span class="n">m</span> <span class="ow">or</span> <span class="n">k</span> <span class="o">&lt;</span> <span class="n">mp</span> <span class="o">-</span> <span class="n">m</span><span class="p">:</span>
-                    <span class="k">continue</span>
-                <span class="n">r</span> <span class="o">+=</span> <span class="p">(</span><span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span><span class="o">**</span><span class="n">k</span> <span class="o">*</span> <span class="n">binomial</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="n">mp</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="o">*</span> <span class="n">binomial</span><span class="p">(</span><span class="n">j</span> <span class="o">-</span> <span class="n">mp</span><span class="p">,</span> <span class="n">k</span> <span class="o">+</span> <span class="n">m</span> <span class="o">-</span> <span class="n">mp</span><span class="p">)</span>
-            <span class="n">r</span> <span class="o">*=</span> <span class="p">(</span><span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span><span class="o">**</span><span class="p">(</span><span class="n">m</span> <span class="o">-</span> <span class="n">mp</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span><span class="o">**</span><span class="n">j</span> <span class="o">*</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">factorial</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="n">m</span><span class="p">)</span> <span class="o">*</span>
-                    <span class="n">factorial</span><span class="p">(</span><span class="n">j</span> <span class="o">-</span> <span class="n">m</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">factorial</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="n">mp</span><span class="p">)</span> <span class="o">*</span> <span class="n">factorial</span><span class="p">(</span><span class="n">j</span> <span class="o">-</span> <span class="n">mp</span><span class="p">)))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># Varshalovich Equation(5), Section 4.7.2, page 87, where we set</span>
-            <span class="c"># beta1=beta2=pi/2, and we get alpha=gamma=pi/2 and beta=phi+pi,</span>
-            <span class="c"># then we use the Eq. (1), Section 4.4. page 79, to simplify:</span>
-            <span class="c"># d(j, m, mp, beta+pi) = (-1)**(j-mp) * d(j, m, -mp, beta)</span>
-            <span class="c"># This happens to be almost the same as in Eq.(10), Section 4.16,</span>
-            <span class="c"># except that we need to substitute -mp for mp.</span>
-            <span class="n">size</span><span class="p">,</span> <span class="n">mvals</span> <span class="o">=</span> <span class="n">m_values</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">mpp</span> <span class="ow">in</span> <span class="n">mvals</span><span class="p">:</span>
-                <span class="n">r</span> <span class="o">+=</span> <span class="n">Rotation</span><span class="o">.</span><span class="n">d</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">mpp</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span> <span class="o">*</span> <span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="o">-</span><span class="n">mpp</span><span class="o">*</span><span class="n">beta</span><span class="p">)</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="o">-</span><span class="n">mpp</span><span class="o">*</span><span class="n">beta</span><span class="p">))</span> <span class="o">*</span> \
-                    <span class="n">Rotation</span><span class="o">.</span><span class="n">d</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">mpp</span><span class="p">,</span> <span class="o">-</span><span class="n">mp</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-            <span class="c"># Empirical normalization factor so results match Varshalovich</span>
-            <span class="c"># Tables 4.3-4.12</span>
-            <span class="c"># Note that this exact normalization does not follow from the</span>
-            <span class="c"># above equations</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">r</span> <span class="o">*</span> <span class="n">I</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">j</span> <span class="o">-</span> <span class="n">m</span> <span class="o">-</span> <span class="n">mp</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">m</span><span class="p">)</span>
-            <span class="c"># Finally, simplify the whole expression</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">*=</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">m</span><span class="o">*</span><span class="n">alpha</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">mp</span><span class="o">*</span><span class="n">gamma</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">r</span>
-
-</div>
-<span class="n">Jx</span> <span class="o">=</span> <span class="n">JxOp</span><span class="p">(</span><span class="s">&#39;J&#39;</span><span class="p">)</span>
-<span class="n">Jy</span> <span class="o">=</span> <span class="n">JyOp</span><span class="p">(</span><span class="s">&#39;J&#39;</span><span class="p">)</span>
-<span class="n">Jz</span> <span class="o">=</span> <span class="n">JzOp</span><span class="p">(</span><span class="s">&#39;J&#39;</span><span class="p">)</span>
-<span class="n">J2</span> <span class="o">=</span> <span class="n">J2Op</span><span class="p">(</span><span class="s">&#39;J&#39;</span><span class="p">)</span>
-<span class="n">Jplus</span> <span class="o">=</span> <span class="n">JplusOp</span><span class="p">(</span><span class="s">&#39;J&#39;</span><span class="p">)</span>
-<span class="n">Jminus</span> <span class="o">=</span> <span class="n">JminusOp</span><span class="p">(</span><span class="s">&#39;J&#39;</span><span class="p">)</span>
-
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># Spin States</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">SpinState</span><span class="p">(</span><span class="n">State</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Base class for angular momentum states.&quot;&quot;&quot;</span>
-
-    <span class="n">_label_separator</span> <span class="o">=</span> <span class="s">&#39;,&#39;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">j</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="mi">2</span><span class="o">*</span><span class="n">j</span> <span class="o">!=</span> <span class="nb">int</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">j</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                    <span class="s">&#39;j must be integer or half-integer, got: </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">j</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;j must be &gt;= 0, got: </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">j</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">m</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="mi">2</span><span class="o">*</span><span class="n">m</span> <span class="o">!=</span> <span class="nb">int</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">m</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                    <span class="s">&#39;m must be integer or half-integer, got: </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">m</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">j</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="n">m</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">j</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Allowed values for m are -j &lt;= m &lt;= j, got j, m: </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">))</span>
-            <span class="k">if</span> <span class="nb">int</span><span class="p">(</span><span class="n">j</span> <span class="o">-</span> <span class="n">m</span><span class="p">)</span> <span class="o">!=</span> <span class="n">j</span> <span class="o">-</span> <span class="n">m</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Both j and m must be integer or half-integer, got j, m: </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">State</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">j</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">m</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_hilbert_space</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">label</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">ComplexSpace</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">label</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_base</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">j</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">m</span>
-        <span class="n">alpha</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;alpha&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-        <span class="n">beta</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;beta&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-        <span class="n">gamma</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;gamma&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-        <span class="n">size</span><span class="p">,</span> <span class="n">mvals</span> <span class="o">=</span> <span class="n">m_values</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">size</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="c"># TODO: Use KroneckerDelta if all Euler angles == 0</span>
-        <span class="c"># breaks finding angles on L930</span>
-        <span class="k">for</span> <span class="n">p</span><span class="p">,</span> <span class="n">mval</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">mvals</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">m</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-                <span class="n">result</span><span class="p">[</span><span class="n">p</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">Rotation</span><span class="o">.</span><span class="n">D</span><span class="p">(</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">,</span> <span class="n">mval</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">m</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">gamma</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">result</span><span class="p">[</span><span class="n">p</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">Rotation</span><span class="o">.</span><span class="n">D</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">,</span> <span class="n">mval</span><span class="p">,</span>
-                                          <span class="bp">self</span><span class="o">.</span><span class="n">m</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">gamma</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_Jx</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Bra</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rewrite_basis</span><span class="p">(</span><span class="n">Jx</span><span class="p">,</span> <span class="n">JxBra</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rewrite_basis</span><span class="p">(</span><span class="n">Jx</span><span class="p">,</span> <span class="n">JxKet</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_Jy</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Bra</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rewrite_basis</span><span class="p">(</span><span class="n">Jy</span><span class="p">,</span> <span class="n">JyBra</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rewrite_basis</span><span class="p">(</span><span class="n">Jy</span><span class="p">,</span> <span class="n">JyKet</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_Jz</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Bra</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rewrite_basis</span><span class="p">(</span><span class="n">Jz</span><span class="p">,</span> <span class="n">JzBra</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rewrite_basis</span><span class="p">(</span><span class="n">Jz</span><span class="p">,</span> <span class="n">JzKet</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_rewrite_basis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="n">evect</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.physics.quantum.represent</span> <span class="kn">import</span> <span class="n">represent</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">j</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">:]</span>
-        <span class="k">if</span> <span class="n">j</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">CoupledSpinState</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">j</span> <span class="o">==</span> <span class="nb">int</span><span class="p">(</span><span class="n">j</span><span class="p">):</span>
-                    <span class="n">start</span> <span class="o">=</span> <span class="n">j</span><span class="o">**</span><span class="mi">2</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">start</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">start</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="n">vect</span> <span class="o">=</span> <span class="n">represent</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="o">=</span><span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span>
-                <span class="o">*</span><span class="p">[</span><span class="n">vect</span><span class="p">[</span><span class="n">start</span> <span class="o">+</span> <span class="n">i</span><span class="p">]</span> <span class="o">*</span> <span class="n">evect</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span> <span class="o">-</span> <span class="n">i</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)])</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">CoupledSpinState</span><span class="p">)</span> <span class="ow">and</span> <span class="n">options</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;coupled&#39;</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">uncouple</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">result</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="n">mi</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;mi&#39;</span><span class="p">)</span>
-            <span class="c"># make sure not to introduce a symbol already in the state</span>
-            <span class="k">while</span> <span class="bp">self</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">mi</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="o">!=</span> <span class="bp">self</span><span class="p">:</span>
-                <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="n">mi</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;mi</span><span class="si">%d</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">i</span><span class="p">)</span>
-                <span class="k">break</span>
-            <span class="c"># TODO: better way to get angles of rotation</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">CoupledSpinState</span><span class="p">):</span>
-                <span class="n">test_args</span> <span class="o">=</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">mi</span><span class="p">,</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">test_args</span> <span class="o">=</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">mi</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Ket</span><span class="p">):</span>
-                <span class="n">angles</span> <span class="o">=</span> <span class="n">represent</span><span class="p">(</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="o">*</span><span class="n">test_args</span><span class="p">),</span> <span class="n">basis</span><span class="o">=</span><span class="n">basis</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">3</span><span class="p">:</span><span class="mi">6</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">angles</span> <span class="o">=</span> <span class="n">represent</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span>
-                    <span class="o">*</span><span class="n">test_args</span><span class="p">),</span> <span class="n">basis</span><span class="o">=</span><span class="n">basis</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">3</span><span class="p">:</span><span class="mi">6</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">angles</span> <span class="o">==</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">self</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">state</span> <span class="o">=</span> <span class="n">evect</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">mi</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-                <span class="n">lt</span> <span class="o">=</span> <span class="n">Rotation</span><span class="o">.</span><span class="n">D</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">mi</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">m</span><span class="p">,</span> <span class="o">*</span><span class="n">angles</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">Sum</span><span class="p">(</span><span class="n">lt</span> <span class="o">*</span> <span class="n">state</span><span class="p">,</span> <span class="p">(</span><span class="n">mi</span><span class="p">,</span> <span class="o">-</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_innerproduct_JxBra</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">bra</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">,</span> <span class="n">bra</span><span class="o">.</span><span class="n">j</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">bra</span><span class="o">.</span><span class="n">dual_class</span><span class="p">()</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">*=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_JxOp</span><span class="p">(</span><span class="bp">None</span><span class="p">)[</span><span class="n">bra</span><span class="o">.</span><span class="n">j</span> <span class="o">-</span> <span class="n">bra</span><span class="o">.</span><span class="n">m</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">*=</span> <span class="n">KroneckerDelta</span><span class="p">(</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">,</span> <span class="n">bra</span><span class="o">.</span><span class="n">j</span><span class="p">)</span> <span class="o">*</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">m</span><span class="p">,</span> <span class="n">bra</span><span class="o">.</span><span class="n">m</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="k">def</span> <span class="nf">_eval_innerproduct_JyBra</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">bra</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">,</span> <span class="n">bra</span><span class="o">.</span><span class="n">j</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">bra</span><span class="o">.</span><span class="n">dual_class</span><span class="p">()</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">*=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_JyOp</span><span class="p">(</span><span class="bp">None</span><span class="p">)[</span><span class="n">bra</span><span class="o">.</span><span class="n">j</span> <span class="o">-</span> <span class="n">bra</span><span class="o">.</span><span class="n">m</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">*=</span> <span class="n">KroneckerDelta</span><span class="p">(</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">,</span> <span class="n">bra</span><span class="o">.</span><span class="n">j</span><span class="p">)</span> <span class="o">*</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">m</span><span class="p">,</span> <span class="n">bra</span><span class="o">.</span><span class="n">m</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="k">def</span> <span class="nf">_eval_innerproduct_JzBra</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">bra</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">,</span> <span class="n">bra</span><span class="o">.</span><span class="n">j</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">bra</span><span class="o">.</span><span class="n">dual_class</span><span class="p">()</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">*=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_JzOp</span><span class="p">(</span><span class="bp">None</span><span class="p">)[</span><span class="n">bra</span><span class="o">.</span><span class="n">j</span> <span class="o">-</span> <span class="n">bra</span><span class="o">.</span><span class="n">m</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">*=</span> <span class="n">KroneckerDelta</span><span class="p">(</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">,</span> <span class="n">bra</span><span class="o">.</span><span class="n">j</span><span class="p">)</span> <span class="o">*</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">m</span><span class="p">,</span> <span class="n">bra</span><span class="o">.</span><span class="n">m</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="k">def</span> <span class="nf">_eval_trace</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">bra</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-
-        <span class="c"># One way to implement this method is to assume the basis set k is</span>
-        <span class="c"># passed.</span>
-        <span class="c"># Then we can apply the discrete form of Trace formula here</span>
-        <span class="c"># Tr(|i&gt;&lt;j| ) = \Sum_k &lt;k|i&gt;&lt;j|k&gt;</span>
-        <span class="c">#then we do qapply() on each each inner product and sum over them.</span>
-
-        <span class="c"># OR</span>
-
-        <span class="c"># Inner product of |i&gt;&lt;j| = Trace(Outer Product).</span>
-        <span class="c"># we could just use this unless there are cases when this is not true</span>
-
-        <span class="k">return</span> <span class="p">(</span><span class="n">bra</span><span class="o">*</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-
-
-<div class="viewcode-block" id="JxKet"><a class="viewcode-back" href="../../../../modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JxKet">[docs]</a><span class="k">class</span> <span class="nc">JxKet</span><span class="p">(</span><span class="n">SpinState</span><span class="p">,</span> <span class="n">Ket</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Eigenket of Jx.</span>
-
-<span class="sd">    See JzKet for the usage of spin eigenstates.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    JzKet: Usage of spin states</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JxBra</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">coupled_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JxKetCoupled</span>
-
-    <span class="k">def</span> <span class="nf">_represent_default_basis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_JxOp</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JxOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_base</span><span class="p">(</span><span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JyOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_base</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JzOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_base</span><span class="p">(</span><span class="n">beta</span><span class="o">=</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="JxBra"><a class="viewcode-back" href="../../../../modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JxBra">[docs]</a><span class="k">class</span> <span class="nc">JxBra</span><span class="p">(</span><span class="n">SpinState</span><span class="p">,</span> <span class="n">Bra</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Eigenbra of Jx.</span>
-
-<span class="sd">    See JzKet for the usage of spin eigenstates.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    JzKet: Usage of spin states</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JxKet</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">coupled_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JxBraCoupled</span>
-
-</div>
-<div class="viewcode-block" id="JyKet"><a class="viewcode-back" href="../../../../modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JyKet">[docs]</a><span class="k">class</span> <span class="nc">JyKet</span><span class="p">(</span><span class="n">SpinState</span><span class="p">,</span> <span class="n">Ket</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Eigenket of Jy.</span>
-
-<span class="sd">    See JzKet for the usage of spin eigenstates.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    JzKet: Usage of spin states</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JyBra</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">coupled_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JyKetCoupled</span>
-
-    <span class="k">def</span> <span class="nf">_represent_default_basis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_JyOp</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JxOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_base</span><span class="p">(</span><span class="n">gamma</span><span class="o">=</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JyOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_base</span><span class="p">(</span><span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JzOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_base</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">beta</span><span class="o">=-</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="JyBra"><a class="viewcode-back" href="../../../../modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JyBra">[docs]</a><span class="k">class</span> <span class="nc">JyBra</span><span class="p">(</span><span class="n">SpinState</span><span class="p">,</span> <span class="n">Bra</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Eigenbra of Jy.</span>
-
-<span class="sd">    See JzKet for the usage of spin eigenstates.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    JzKet: Usage of spin states</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JyKet</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">coupled_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JyBraCoupled</span>
-
-</div>
-<div class="viewcode-block" id="JzKet"><a class="viewcode-back" href="../../../../modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JzKet">[docs]</a><span class="k">class</span> <span class="nc">JzKet</span><span class="p">(</span><span class="n">SpinState</span><span class="p">,</span> <span class="n">Ket</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Eigenket of Jz.</span>
-
-<span class="sd">    Spin state which is an eigenstate of the Jz operator. Uncoupled states,</span>
-<span class="sd">    that is states representing the interaction of multiple separate spin</span>
-<span class="sd">    states, are defined as a tensor product of states.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    j : Number, Symbol</span>
-<span class="sd">        Total spin angular momentum</span>
-<span class="sd">    m : Number, Symbol</span>
-<span class="sd">        Eigenvalue of the Jz spin operator</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    *Normal States:*</span>
-
-<span class="sd">    Defining simple spin states, both numerical and symbolic:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.spin import JzKet, JxKet</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; JzKet(1, 0)</span>
-<span class="sd">        |1,0&gt;</span>
-<span class="sd">        &gt;&gt;&gt; j, m = symbols(&#39;j m&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; JzKet(j, m)</span>
-<span class="sd">        |j,m&gt;</span>
-
-<span class="sd">    Rewriting the JzKet in terms of eigenkets of the Jx operator:</span>
-<span class="sd">    Note: that the resulting eigenstates are JxKet&#39;s</span>
-
-<span class="sd">        &gt;&gt;&gt; JzKet(1,1).rewrite(&quot;Jx&quot;)</span>
-<span class="sd">        |1,-1&gt;/2 - sqrt(2)*|1,0&gt;/2 + |1,1&gt;/2</span>
-
-<span class="sd">    Get the vector representation of a state in terms of the basis elements</span>
-<span class="sd">    of the Jx operator:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.represent import represent</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.spin import Jx, Jz</span>
-<span class="sd">        &gt;&gt;&gt; represent(JzKet(1,-1), basis=Jx)</span>
-<span class="sd">        [      1/2]</span>
-<span class="sd">        [sqrt(2)/2]</span>
-<span class="sd">        [      1/2]</span>
-
-<span class="sd">    Apply innerproducts between states:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.innerproduct import InnerProduct</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.spin import JxBra</span>
-<span class="sd">        &gt;&gt;&gt; i = InnerProduct(JxBra(1,1), JzKet(1,1))</span>
-<span class="sd">        &gt;&gt;&gt; i</span>
-<span class="sd">        &lt;1,1|1,1&gt;</span>
-<span class="sd">        &gt;&gt;&gt; i.doit()</span>
-<span class="sd">        1/2</span>
-
-<span class="sd">    *Uncoupled States:*</span>
-
-<span class="sd">    Define an uncoupled state as a TensorProduct between two Jz eigenkets:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.tensorproduct import TensorProduct</span>
-<span class="sd">        &gt;&gt;&gt; j1,m1,j2,m2 = symbols(&#39;j1 m1 j2 m2&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; TensorProduct(JzKet(1,0), JzKet(1,1))</span>
-<span class="sd">        |1,0&gt;x|1,1&gt;</span>
-<span class="sd">        &gt;&gt;&gt; TensorProduct(JzKet(j1,m1), JzKet(j2,m2))</span>
-<span class="sd">        |j1,m1&gt;x|j2,m2&gt;</span>
-
-<span class="sd">    A TensorProduct can be rewritten, in which case the eigenstates that make</span>
-<span class="sd">    up the tensor product is rewritten to the new basis:</span>
-
-<span class="sd">        &gt;&gt;&gt; TensorProduct(JzKet(1,1),JxKet(1,1)).rewrite(&#39;Jz&#39;)</span>
-<span class="sd">        |1,1&gt;x|1,-1&gt;/2 + sqrt(2)*|1,1&gt;x|1,0&gt;/2 + |1,1&gt;x|1,1&gt;/2</span>
-
-<span class="sd">    The represent method for TensorProduct&#39;s gives the vector representation of</span>
-<span class="sd">    the state. Note that the state in the product basis is the equivalent of the</span>
-<span class="sd">    tensor product of the vector representation of the component eigenstates:</span>
-
-<span class="sd">        &gt;&gt;&gt; represent(TensorProduct(JzKet(1,0),JzKet(1,1)))</span>
-<span class="sd">        [0]</span>
-<span class="sd">        [0]</span>
-<span class="sd">        [0]</span>
-<span class="sd">        [1]</span>
-<span class="sd">        [0]</span>
-<span class="sd">        [0]</span>
-<span class="sd">        [0]</span>
-<span class="sd">        [0]</span>
-<span class="sd">        [0]</span>
-<span class="sd">        &gt;&gt;&gt; represent(TensorProduct(JzKet(1,1),JxKet(1,1)), basis=Jz)</span>
-<span class="sd">        [      1/2]</span>
-<span class="sd">        [sqrt(2)/2]</span>
-<span class="sd">        [      1/2]</span>
-<span class="sd">        [        0]</span>
-<span class="sd">        [        0]</span>
-<span class="sd">        [        0]</span>
-<span class="sd">        [        0]</span>
-<span class="sd">        [        0]</span>
-<span class="sd">        [        0]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    JzKetCoupled: Coupled eigenstates</span>
-<span class="sd">    TensorProduct: Used to specify uncoupled states</span>
-<span class="sd">    uncouple: Uncouples states given coupling parameters</span>
-<span class="sd">    couple: Couples uncoupled states</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JzBra</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">coupled_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JzKetCoupled</span>
-
-    <span class="k">def</span> <span class="nf">_represent_default_basis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_JzOp</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JxOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_base</span><span class="p">(</span><span class="n">beta</span><span class="o">=</span><span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JyOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_base</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">beta</span><span class="o">=</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JzOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_base</span><span class="p">(</span><span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="JzBra"><a class="viewcode-back" href="../../../../modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JzBra">[docs]</a><span class="k">class</span> <span class="nc">JzBra</span><span class="p">(</span><span class="n">SpinState</span><span class="p">,</span> <span class="n">Bra</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Eigenbra of Jz.</span>
-
-<span class="sd">    See the JzKet for the usage of spin eigenstates.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    JzKet: Usage of spin states</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JzKet</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">coupled_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JzBraCoupled</span>
-
-
-<span class="c"># Method used primarily to create coupled_n and coupled_jn by __new__ in</span>
-<span class="c"># CoupledSpinState</span>
-<span class="c"># This same method is also used by the uncouple method, and is separated from</span>
-<span class="c"># the CoupledSpinState class to maintain consistency in defining coupling</span></div>
-<span class="k">def</span> <span class="nf">_build_coupled</span><span class="p">(</span><span class="n">jcoupling</span><span class="p">,</span> <span class="n">length</span><span class="p">):</span>
-    <span class="n">n_list</span> <span class="o">=</span> <span class="p">[</span> <span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">length</span><span class="p">)</span> <span class="p">]</span>
-    <span class="n">coupled_jn</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">coupled_n</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">,</span> <span class="n">j_new</span> <span class="ow">in</span> <span class="n">jcoupling</span><span class="p">:</span>
-        <span class="n">coupled_jn</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">j_new</span><span class="p">)</span>
-        <span class="n">coupled_n</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="p">(</span><span class="n">n_list</span><span class="p">[</span><span class="n">n1</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">n_list</span><span class="p">[</span><span class="n">n2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span> <span class="p">)</span>
-        <span class="n">n_sort</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">n_list</span><span class="p">[</span><span class="n">n1</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">n_list</span><span class="p">[</span><span class="n">n2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span>
-        <span class="n">n_list</span><span class="p">[</span><span class="n">n_sort</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">n_sort</span>
-    <span class="k">return</span> <span class="n">coupled_n</span><span class="p">,</span> <span class="n">coupled_jn</span>
-
-
-<span class="k">class</span> <span class="nc">CoupledSpinState</span><span class="p">(</span><span class="n">SpinState</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Base class for coupled angular momentum states.&quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">jn</span><span class="p">,</span> <span class="o">*</span><span class="n">jcoupling</span><span class="p">):</span>
-        <span class="c"># Check j and m values using SpinState</span>
-        <span class="n">SpinState</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-        <span class="c"># Build and check coupling scheme from arguments</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">jcoupling</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="c"># Use default coupling scheme</span>
-            <span class="n">jcoupling</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">jn</span><span class="p">)):</span>
-                <span class="n">jcoupling</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">jn</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)]))</span> <span class="p">)</span>
-            <span class="n">jcoupling</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">jn</span><span class="p">),</span> <span class="n">j</span><span class="p">)</span> <span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">jcoupling</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># Use specified coupling scheme</span>
-            <span class="n">jcoupling</span> <span class="o">=</span> <span class="n">jcoupling</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;CoupledSpinState only takes 3 or 4 arguments, got: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">jcoupling</span><span class="p">)</span> <span class="o">+</span> <span class="mi">3</span><span class="p">)</span> <span class="p">)</span>
-        <span class="c"># Check arguments have correct form</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">jn</span><span class="p">,</span> <span class="nb">list</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">jn</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">jn</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">)):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;jn must be Tuple, list or tuple, got </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span>
-                            <span class="n">jn</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">jcoupling</span><span class="p">,</span> <span class="nb">list</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">jcoupling</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">jcoupling</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">)):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;jcoupling must be Tuple, list or tuple, got </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span>
-                            <span class="n">jcoupling</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="nb">list</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">)</span> <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">jcoupling</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span>
-                <span class="s">&#39;All elements of jcoupling must be list, tuple or Tuple&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">len</span><span class="p">(</span><span class="n">jn</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">jcoupling</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;jcoupling must have length of </span><span class="si">%d</span><span class="s">, got </span><span class="si">%d</span><span class="s">&#39;</span> <span class="o">%</span>
-                             <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">jn</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">jcoupling</span><span class="p">)))</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">jcoupling</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;All elements of jcoupling must have length 3&#39;</span><span class="p">)</span>
-        <span class="c"># Build sympified args</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-        <span class="n">jn</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span> <span class="o">*</span><span class="p">[</span><span class="n">sympify</span><span class="p">(</span><span class="n">ji</span><span class="p">)</span> <span class="k">for</span> <span class="n">ji</span> <span class="ow">in</span> <span class="n">jn</span><span class="p">]</span> <span class="p">)</span>
-        <span class="n">jcoupling</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span> <span class="o">*</span><span class="p">[</span><span class="n">Tuple</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span>
-            <span class="n">n1</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">n2</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">ji</span><span class="p">))</span> <span class="k">for</span> <span class="p">(</span><span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">,</span> <span class="n">ji</span><span class="p">)</span> <span class="ow">in</span> <span class="n">jcoupling</span><span class="p">]</span> <span class="p">)</span>
-        <span class="c"># Check values in coupling scheme give physical state</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">ji</span> <span class="o">!=</span> <span class="nb">int</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">ji</span><span class="p">)</span> <span class="k">for</span> <span class="n">ji</span> <span class="ow">in</span> <span class="n">jn</span> <span class="k">if</span> <span class="n">ji</span><span class="o">.</span><span class="n">is_number</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;All elements of jn must be integer or half-integer, got: </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">jn</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">n1</span> <span class="o">!=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n1</span><span class="p">)</span> <span class="ow">or</span> <span class="n">n2</span> <span class="o">!=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n2</span><span class="p">)</span> <span class="k">for</span> <span class="p">(</span><span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">,</span> <span class="n">_</span><span class="p">)</span> <span class="ow">in</span> <span class="n">jcoupling</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Indicies in jcoupling must be integers&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">n1</span> <span class="o">&lt;</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">n2</span> <span class="o">&lt;</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">n1</span> <span class="o">&gt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">jn</span><span class="p">)</span> <span class="ow">or</span> <span class="n">n2</span> <span class="o">&gt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">jn</span><span class="p">)</span> <span class="k">for</span> <span class="p">(</span><span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">,</span> <span class="n">_</span><span class="p">)</span> <span class="ow">in</span> <span class="n">jcoupling</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Indicies must be between 1 and the number of coupled spin spaces&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">ji</span> <span class="o">!=</span> <span class="nb">int</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">ji</span><span class="p">)</span> <span class="k">for</span> <span class="p">(</span><span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">ji</span><span class="p">)</span> <span class="ow">in</span> <span class="n">jcoupling</span> <span class="k">if</span> <span class="n">ji</span><span class="o">.</span><span class="n">is_number</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;All coupled j values in coupling scheme must be integer or half-integer&#39;</span><span class="p">)</span>
-        <span class="n">coupled_n</span><span class="p">,</span> <span class="n">coupled_jn</span> <span class="o">=</span> <span class="n">_build_coupled</span><span class="p">(</span><span class="n">jcoupling</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">jn</span><span class="p">))</span>
-        <span class="n">jvals</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">jn</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="p">(</span><span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">coupled_n</span><span class="p">):</span>
-            <span class="n">j1</span> <span class="o">=</span> <span class="n">jvals</span><span class="p">[</span><span class="nb">min</span><span class="p">(</span><span class="n">n1</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-            <span class="n">j2</span> <span class="o">=</span> <span class="n">jvals</span><span class="p">[</span><span class="nb">min</span><span class="p">(</span><span class="n">n2</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-            <span class="n">j3</span> <span class="o">=</span> <span class="n">coupled_jn</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">sympify</span><span class="p">(</span><span class="n">j1</span><span class="p">)</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="n">sympify</span><span class="p">(</span><span class="n">j2</span><span class="p">)</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="n">sympify</span><span class="p">(</span><span class="n">j3</span><span class="p">)</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">j1</span> <span class="o">+</span> <span class="n">j2</span> <span class="o">&lt;</span> <span class="n">j3</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;All couplings must have j1+j2 &gt;= j3, &#39;</span>
-                        <span class="s">&#39;in coupling number </span><span class="si">%d</span><span class="s"> got j1,j2,j3: </span><span class="si">%d</span><span class="s">,</span><span class="si">%d</span><span class="s">,</span><span class="si">%d</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j1</span><span class="p">,</span> <span class="n">j2</span><span class="p">,</span> <span class="n">j3</span><span class="p">))</span>
-                <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="n">j1</span> <span class="o">-</span> <span class="n">j2</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">j3</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;All couplings must have |j1+j2| &lt;= j3, &quot;</span>
-                        <span class="s">&quot;in coupling number </span><span class="si">%d</span><span class="s"> got j1,j2,j3: </span><span class="si">%d</span><span class="s">,</span><span class="si">%d</span><span class="s">,</span><span class="si">%d</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j1</span><span class="p">,</span> <span class="n">j2</span><span class="p">,</span> <span class="n">j3</span><span class="p">))</span>
-                <span class="k">if</span> <span class="nb">int</span><span class="p">(</span><span class="n">j1</span> <span class="o">+</span> <span class="n">j2</span><span class="p">)</span> <span class="o">==</span> <span class="n">j1</span> <span class="o">+</span> <span class="n">j2</span><span class="p">:</span>
-                    <span class="k">pass</span>
-            <span class="n">jvals</span><span class="p">[</span><span class="nb">min</span><span class="p">(</span><span class="n">n1</span> <span class="o">+</span> <span class="n">n2</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">j3</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">jcoupling</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">jcoupling</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span> <span class="o">!=</span> <span class="n">j</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Last j value coupled together must be the final j of the state&#39;</span><span class="p">)</span>
-        <span class="c"># Return state</span>
-        <span class="k">return</span> <span class="n">State</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">jn</span><span class="p">,</span> <span class="n">jcoupling</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_label</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">label</span> <span class="o">=</span> <span class="p">[</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">m</span><span class="p">)]</span>
-        <span class="c"># After 2.5 is dropped:</span>
-        <span class="c">#for i, ji in enumerate(self.jn, start=1):</span>
-        <span class="c">#    label.append(&#39;j%d=%s&#39; % (i, ji) )</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">ji</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">jn</span><span class="p">):</span>
-            <span class="n">label</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;j</span><span class="si">%d</span><span class="s">=</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span>
-                <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">ji</span><span class="p">)</span>
-            <span class="p">))</span>
-        <span class="k">for</span> <span class="n">jn</span><span class="p">,</span> <span class="p">(</span><span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">coupled_jn</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">coupled_n</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]):</span>
-            <span class="n">label</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;j(</span><span class="si">%s</span><span class="s">)=</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span>
-                <span class="s">&#39;,&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">n1</span> <span class="o">+</span> <span class="n">n2</span><span class="p">)),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">jn</span><span class="p">)</span>
-            <span class="p">))</span>
-        <span class="k">return</span> <span class="s">&#39;,&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">label</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_label_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">label</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">m</span><span class="p">]</span>
-        <span class="c"># After 2.5 is dropped:</span>
-        <span class="c">#for i, ji in enumerate(self.jn, start=1):</span>
-        <span class="c">#    n = &#39;%d&#39; % (i)</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">ji</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">jn</span><span class="p">):</span>
-            <span class="n">symb</span> <span class="o">=</span> <span class="s">&#39;j</span><span class="si">%d</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="n">symb</span> <span class="o">=</span> <span class="n">pretty_symbol</span><span class="p">(</span><span class="n">symb</span><span class="p">)</span>
-            <span class="n">symb</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">symb</span> <span class="o">+</span> <span class="s">&#39;=&#39;</span><span class="p">)</span>
-            <span class="n">item</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">symb</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">ji</span><span class="p">)))</span>
-            <span class="n">label</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">item</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">jn</span><span class="p">,</span> <span class="p">(</span><span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">coupled_jn</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">coupled_n</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]):</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="s">&#39;,&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">pretty_symbol</span><span class="p">(</span><span class="s">&quot;j</span><span class="si">%d</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">i</span><span class="p">)[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">n1</span> <span class="o">+</span> <span class="n">n2</span><span class="p">))</span>
-            <span class="n">symb</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;j&#39;</span> <span class="o">+</span> <span class="n">n</span> <span class="o">+</span> <span class="s">&#39;=&#39;</span><span class="p">)</span>
-            <span class="n">item</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">symb</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">jn</span><span class="p">)))</span>
-            <span class="n">label</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">item</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_sequence_pretty</span><span class="p">(</span>
-            <span class="n">label</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_label_separator</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span>
-        <span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_label_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">label</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">m</span><span class="p">]</span>
-        <span class="c"># After 2.5 dropped</span>
-        <span class="c">#for i, ji in enumerate(self.jn, start=1):</span>
-        <span class="c">#    label.append(&#39;j_{%d}=%s&#39; % (i, printer._print(ji)) )</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">ji</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">jn</span><span class="p">):</span>
-            <span class="n">label</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;j_{</span><span class="si">%d</span><span class="s">}=</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">ji</span><span class="p">))</span> <span class="p">)</span>
-        <span class="k">for</span> <span class="n">jn</span><span class="p">,</span> <span class="p">(</span><span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">coupled_jn</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">coupled_n</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]):</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="s">&#39;,&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">n1</span> <span class="o">+</span> <span class="n">n2</span><span class="p">))</span>
-            <span class="n">label</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;j_{</span><span class="si">%s</span><span class="s">}=</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">jn</span><span class="p">))</span> <span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_sequence</span><span class="p">(</span>
-            <span class="n">label</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_label_separator</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span>
-        <span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">jn</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">coupling</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">coupled_jn</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_build_coupled</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">2</span><span class="p">]))[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">coupled_n</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_build_coupled</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">[</span><span class="mi">2</span><span class="p">]))[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_eval_hilbert_space</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">label</span><span class="p">):</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">label</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">j</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">DirectSumHilbertSpace</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">ComplexSpace</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">)</span> <span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># TODO: Need hilbert space fix, see issue 2633</span>
-            <span class="c"># Desired behavior:</span>
-            <span class="c">#ji = symbols(&#39;ji&#39;)</span>
-            <span class="c">#ret = Sum(ComplexSpace(2*ji + 1), (ji, 0, j))</span>
-            <span class="c"># Temporary fix:</span>
-            <span class="k">return</span> <span class="n">ComplexSpace</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_coupled_base</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">evect</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">uncoupled_class</span><span class="p">()</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&#39;State must not have symbolic j value to represent&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">hilbert_space</span><span class="o">.</span><span class="n">dimension</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&#39;State must not have symbolic j values to represent&#39;</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">hilbert_space</span><span class="o">.</span><span class="n">dimension</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">j</span> <span class="o">==</span> <span class="nb">int</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">):</span>
-            <span class="n">start</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="o">**</span><span class="mi">2</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">start</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span>
-        <span class="n">result</span><span class="p">[</span><span class="n">start</span><span class="p">:</span><span class="n">start</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">evect</span><span class="p">(</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">j</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">m</span><span class="p">)</span><span class="o">.</span><span class="n">_represent_base</span><span class="p">(</span><span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_Jx</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Bra</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rewrite_basis</span><span class="p">(</span><span class="n">Jx</span><span class="p">,</span> <span class="n">JxBraCoupled</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rewrite_basis</span><span class="p">(</span><span class="n">Jx</span><span class="p">,</span> <span class="n">JxKetCoupled</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_Jy</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Bra</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rewrite_basis</span><span class="p">(</span><span class="n">Jy</span><span class="p">,</span> <span class="n">JyBraCoupled</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rewrite_basis</span><span class="p">(</span><span class="n">Jy</span><span class="p">,</span> <span class="n">JyKetCoupled</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite_as_Jz</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Bra</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rewrite_basis</span><span class="p">(</span><span class="n">Jz</span><span class="p">,</span> <span class="n">JzBraCoupled</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rewrite_basis</span><span class="p">(</span><span class="n">Jz</span><span class="p">,</span> <span class="n">JzKetCoupled</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="JxKetCoupled"><a class="viewcode-back" href="../../../../modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JxKetCoupled">[docs]</a><span class="k">class</span> <span class="nc">JxKetCoupled</span><span class="p">(</span><span class="n">CoupledSpinState</span><span class="p">,</span> <span class="n">Ket</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Coupled eigenket of Jx.</span>
-
-<span class="sd">    See JzKetCoupled for the usage of coupled spin eigenstates.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    JzKetCoupled: Usage of coupled spin states</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JxBraCoupled</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">uncoupled_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JxKet</span>
-
-    <span class="k">def</span> <span class="nf">_represent_default_basis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_JzOp</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JxOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_coupled_base</span><span class="p">(</span><span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JyOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_coupled_base</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JzOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_coupled_base</span><span class="p">(</span><span class="n">beta</span><span class="o">=</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="JxBraCoupled"><a class="viewcode-back" href="../../../../modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JxBraCoupled">[docs]</a><span class="k">class</span> <span class="nc">JxBraCoupled</span><span class="p">(</span><span class="n">CoupledSpinState</span><span class="p">,</span> <span class="n">Bra</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Coupled eigenbra of Jx.</span>
-
-<span class="sd">    See JzKetCoupled for the usage of coupled spin eigenstates.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    JzKetCoupled: Usage of coupled spin states</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JxKetCoupled</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">uncoupled_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JxBra</span>
-
-</div>
-<div class="viewcode-block" id="JyKetCoupled"><a class="viewcode-back" href="../../../../modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JyKetCoupled">[docs]</a><span class="k">class</span> <span class="nc">JyKetCoupled</span><span class="p">(</span><span class="n">CoupledSpinState</span><span class="p">,</span> <span class="n">Ket</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Coupled eigenket of Jy.</span>
-
-<span class="sd">    See JzKetCoupled for the usage of coupled spin eigenstates.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    JzKetCoupled: Usage of coupled spin states</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JyBraCoupled</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">uncoupled_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JyKet</span>
-
-    <span class="k">def</span> <span class="nf">_represent_default_basis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_JzOp</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JxOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_coupled_base</span><span class="p">(</span><span class="n">gamma</span><span class="o">=</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JyOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_coupled_base</span><span class="p">(</span><span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JzOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_coupled_base</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">beta</span><span class="o">=-</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="JyBraCoupled"><a class="viewcode-back" href="../../../../modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JyBraCoupled">[docs]</a><span class="k">class</span> <span class="nc">JyBraCoupled</span><span class="p">(</span><span class="n">CoupledSpinState</span><span class="p">,</span> <span class="n">Bra</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Coupled eigenbra of Jy.</span>
-
-<span class="sd">    See JzKetCoupled for the usage of coupled spin eigenstates.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    JzKetCoupled: Usage of coupled spin states</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JyKetCoupled</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">uncoupled_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JyBra</span>
-
-</div>
-<div class="viewcode-block" id="JzKetCoupled"><a class="viewcode-back" href="../../../../modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JzKetCoupled">[docs]</a><span class="k">class</span> <span class="nc">JzKetCoupled</span><span class="p">(</span><span class="n">CoupledSpinState</span><span class="p">,</span> <span class="n">Ket</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Coupled eigenket of Jz</span>
-
-<span class="sd">    Spin state that is an eigenket of Jz which represents the coupling of</span>
-<span class="sd">    separate spin spaces.</span>
-
-<span class="sd">    The arguments for creating instances of JzKetCoupled are ``j``, ``m``,</span>
-<span class="sd">    ``jn`` and an optional ``jcoupling`` argument. The ``j`` and ``m`` options</span>
-<span class="sd">    are the total angular momentum quantum numbers, as used for normal states</span>
-<span class="sd">    (e.g. JzKet).</span>
-
-<span class="sd">    The other required parameter in ``jn``, which is a tuple defining the `j_n`</span>
-<span class="sd">    angular momentum quantum numbers of the product spaces. So for example, if</span>
-<span class="sd">    a state represented the coupling of the product basis state</span>
-<span class="sd">    `|j_1,m_1\\rangle\\times|j_2,m_2\\rangle`, the ``jn`` for this state would be</span>
-<span class="sd">    ``(j1,j2)``.</span>
-
-<span class="sd">    The final option is ``jcoupling``, which is used to define how the spaces</span>
-<span class="sd">    specified by ``jn`` are coupled, which includes both the order these spaces</span>
-<span class="sd">    are coupled together and the quantum numbers that arise from these</span>
-<span class="sd">    couplings. The ``jcoupling`` parameter itself is a list of lists, such that</span>
-<span class="sd">    each of the sublists defines a single coupling between the spin spaces. If</span>
-<span class="sd">    there are N coupled angular momentum spaces, that is ``jn`` has N elements,</span>
-<span class="sd">    then there must be N-1 sublists. Each of these sublists making up the</span>
-<span class="sd">    ``jcoupling`` parameter have length 3. The first two elements are the</span>
-<span class="sd">    indicies of the product spaces that are considered to be coupled together.</span>
-<span class="sd">    For example, if we want to couple `j_1` and `j_4`, the indicies would be 1</span>
-<span class="sd">    and 4. If a state has already been coupled, it is referenced by the</span>
-<span class="sd">    smallest index that is coupled, so if `j_2` and `j_4` has already been</span>
-<span class="sd">    coupled to some `j_{24}`, then this value can be coupled by referencing it</span>
-<span class="sd">    with index 2. The final element of the sublist is the quantum number of the</span>
-<span class="sd">    coupled state. So putting everything together, into a valid sublist for</span>
-<span class="sd">    ``jcoupling``, if `j_1` and `j_2` are coupled to an angular momentum space</span>
-<span class="sd">    with quantum number `j_{12}` with the value ``j12``, the sublist would be</span>
-<span class="sd">    ``(1,2,j12)``, N-1 of these sublists are used in the list for</span>
-<span class="sd">    ``jcoupling``.</span>
-
-<span class="sd">    Note the ``jcoupling`` parameter is optional, if it is not specified, the</span>
-<span class="sd">    default coupling is taken. This default value is to coupled the spaces in</span>
-<span class="sd">    order and take the quantum number of the coupling to be the maximum value.</span>
-<span class="sd">    For example, if the spin spaces are `j_1`, `j_2`, `j_3`, `j_4`, then the</span>
-<span class="sd">    default coupling couples `j_1` and `j_2` to `j_{12}=j_1+j_2`, then,</span>
-<span class="sd">    `j_{12}` and `j_3` are coupled to `j_{123}=j_{12}+j_3`, and finally</span>
-<span class="sd">    `j_{123}` and `j_4` to `j=j_{123}+j_4`. The jcoupling value that would</span>
-<span class="sd">    correspond to this is:</span>
-
-<span class="sd">        ``((1,2,j1+j2),(1,3,j1+j2+j3))``</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    args : tuple</span>
-<span class="sd">        The arguments that must be passed are ``j``, ``m``, ``jn``, and</span>
-<span class="sd">        ``jcoupling``. The ``j`` value is the total angular momentum. The ``m``</span>
-<span class="sd">        value is the eigenvalue of the Jz spin operator. The ``jn`` list are</span>
-<span class="sd">        the j values of argular momentum spaces coupled together. The</span>
-<span class="sd">        ``jcoupling`` parameter is an optional parameter defining how the spaces</span>
-<span class="sd">        are coupled together. See the above description for how these coupling</span>
-<span class="sd">        parameters are defined.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Defining simple spin states, both numerical and symbolic:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.spin import JzKetCoupled</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; JzKetCoupled(1, 0, (1, 1))</span>
-<span class="sd">        |1,0,j1=1,j2=1&gt;</span>
-<span class="sd">        &gt;&gt;&gt; j, m, j1, j2 = symbols(&#39;j m j1 j2&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; JzKetCoupled(j, m, (j1, j2))</span>
-<span class="sd">        |j,m,j1=j1,j2=j2&gt;</span>
-
-<span class="sd">    Defining coupled spin states for more than 2 coupled spaces with various</span>
-<span class="sd">    coupling parameters:</span>
-
-<span class="sd">        &gt;&gt;&gt; JzKetCoupled(2, 1, (1, 1, 1))</span>
-<span class="sd">        |2,1,j1=1,j2=1,j3=1,j(1,2)=2&gt;</span>
-<span class="sd">        &gt;&gt;&gt; JzKetCoupled(2, 1, (1, 1, 1), ((1,2,2),(1,3,2)) )</span>
-<span class="sd">        |2,1,j1=1,j2=1,j3=1,j(1,2)=2&gt;</span>
-<span class="sd">        &gt;&gt;&gt; JzKetCoupled(2, 1, (1, 1, 1), ((2,3,1),(1,2,2)) )</span>
-<span class="sd">        |2,1,j1=1,j2=1,j3=1,j(2,3)=1&gt;</span>
-
-<span class="sd">    Rewriting the JzKetCoupled in terms of eigenkets of the Jx operator:</span>
-<span class="sd">    Note: that the resulting eigenstates are JxKetCoupled</span>
-
-<span class="sd">        &gt;&gt;&gt; JzKetCoupled(1,1,(1,1)).rewrite(&quot;Jx&quot;)</span>
-<span class="sd">        |1,-1,j1=1,j2=1&gt;/2 - sqrt(2)*|1,0,j1=1,j2=1&gt;/2 + |1,1,j1=1,j2=1&gt;/2</span>
-
-<span class="sd">    The rewrite method can be used to convert a coupled state to an uncoupled</span>
-<span class="sd">    state. This is done by passing coupled=False to the rewrite function:</span>
-
-<span class="sd">        &gt;&gt;&gt; JzKetCoupled(1, 0, (1, 1)).rewrite(&#39;Jz&#39;, coupled=False)</span>
-<span class="sd">        -sqrt(2)*|1,-1&gt;x|1,1&gt;/2 + sqrt(2)*|1,1&gt;x|1,-1&gt;/2</span>
-
-<span class="sd">    Get the vector representation of a state in terms of the basis elements</span>
-<span class="sd">    of the Jx operator:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.represent import represent</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.spin import Jx</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import S</span>
-<span class="sd">        &gt;&gt;&gt; represent(JzKetCoupled(1,-1,(S(1)/2,S(1)/2)), basis=Jx)</span>
-<span class="sd">        [        0]</span>
-<span class="sd">        [      1/2]</span>
-<span class="sd">        [sqrt(2)/2]</span>
-<span class="sd">        [      1/2]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    JzKet: Normal spin eigenstates</span>
-<span class="sd">    uncouple: Uncoupling of coupling spin states</span>
-<span class="sd">    couple: Coupling of uncoupled spin states</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JzBraCoupled</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">uncoupled_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JzKet</span>
-
-    <span class="k">def</span> <span class="nf">_represent_default_basis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_JzOp</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JxOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_coupled_base</span><span class="p">(</span><span class="n">beta</span><span class="o">=</span><span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JyOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_coupled_base</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">beta</span><span class="o">=</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_JzOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent_coupled_base</span><span class="p">(</span><span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="JzBraCoupled"><a class="viewcode-back" href="../../../../modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JzBraCoupled">[docs]</a><span class="k">class</span> <span class="nc">JzBraCoupled</span><span class="p">(</span><span class="n">CoupledSpinState</span><span class="p">,</span> <span class="n">Bra</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Coupled eigenbra of Jz.</span>
-
-<span class="sd">    See the JzKetCoupled for the usage of coupled spin eigenstates.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    JzKetCoupled: Usage of coupled spin states</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JzKetCoupled</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">uncoupled_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">JzBra</span>
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># Coupling/uncoupling</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-</div>
-<div class="viewcode-block" id="couple"><a class="viewcode-back" href="../../../../modules/physics/quantum/spin.html#sympy.physics.quantum.spin.couple">[docs]</a><span class="k">def</span> <span class="nf">couple</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">jcoupling_list</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Couple a tensor product of spin states</span>
-
-<span class="sd">    This function can be used to couple an uncoupled tensor product of spin</span>
-<span class="sd">    states. All of the eigenstates to be coupled must be of the same class. It</span>
-<span class="sd">    will return a linear combination of eigenstates that are subclasses of</span>
-<span class="sd">    CoupledSpinState determined by Clebsch-Gordan angular momentum coupling</span>
-<span class="sd">    coefficients.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    expr : Expr</span>
-<span class="sd">        An expression involving TensorProducts of spin states to be coupled.</span>
-<span class="sd">        Each state must be a subclass of SpinState and they all must be the</span>
-<span class="sd">        same class.</span>
-
-<span class="sd">    jcoupling_list : list or tuple</span>
-<span class="sd">        Elements of this list are sub-lists of length 2 specifying the order of</span>
-<span class="sd">        the coupling of the spin spaces. The length of this must be N-1, where N</span>
-<span class="sd">        is the number of states in the tensor product to be coupled. The</span>
-<span class="sd">        elements of this sublist are the same as the first two elements of each</span>
-<span class="sd">        sublist in the ``jcoupling`` parameter defined for JzKetCoupled. If this</span>
-<span class="sd">        parameter is not specified, the default value is taken, which couples</span>
-<span class="sd">        the first and second product basis spaces, then couples this new coupled</span>
-<span class="sd">        space to the third product space, etc</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Couple a tensor product of numerical states for two spaces:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.spin import JzKet, couple</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.tensorproduct import TensorProduct</span>
-<span class="sd">        &gt;&gt;&gt; couple(TensorProduct(JzKet(1,0), JzKet(1,1)))</span>
-<span class="sd">        -sqrt(2)*|1,1,j1=1,j2=1&gt;/2 + sqrt(2)*|2,1,j1=1,j2=1&gt;/2</span>
-
-
-<span class="sd">    Numerical coupling of three spaces using the default coupling method, i.e.</span>
-<span class="sd">    first and second spaces couple, then this couples to the third space:</span>
-
-<span class="sd">        &gt;&gt;&gt; couple(TensorProduct(JzKet(1,1), JzKet(1,1), JzKet(1,0)))</span>
-<span class="sd">        sqrt(6)*|2,2,j1=1,j2=1,j3=1,j(1,2)=2&gt;/3 + sqrt(3)*|3,2,j1=1,j2=1,j3=1,j(1,2)=2&gt;/3</span>
-
-<span class="sd">    Perform this same coupling, but we define the coupling to first couple</span>
-<span class="sd">    the first and third spaces:</span>
-
-<span class="sd">        &gt;&gt;&gt; couple(TensorProduct(JzKet(1,1), JzKet(1,1), JzKet(1,0)), ((1,3),(1,2)) )</span>
-<span class="sd">        sqrt(2)*|2,2,j1=1,j2=1,j3=1,j(1,3)=1&gt;/2 - sqrt(6)*|2,2,j1=1,j2=1,j3=1,j(1,3)=2&gt;/6 + sqrt(3)*|3,2,j1=1,j2=1,j3=1,j(1,3)=2&gt;/3</span>
-
-<span class="sd">    Couple a tensor product of symbolic states:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; j1,m1,j2,m2 = symbols(&#39;j1 m1 j2 m2&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; couple(TensorProduct(JzKet(j1,m1), JzKet(j2,m2)))</span>
-<span class="sd">        Sum(CG(j1, m1, j2, m2, j, m1 + m2)*|j,m1 + m2,j1=j1,j2=j2&gt;, (j, m1 + m2, j1 + j2))</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">TensorProduct</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">tp</span> <span class="ow">in</span> <span class="n">a</span><span class="p">:</span>
-        <span class="c"># Allow other tensor products to be in expression</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">([</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">SpinState</span><span class="p">)</span> <span class="k">for</span> <span class="n">state</span> <span class="ow">in</span> <span class="n">tp</span><span class="o">.</span><span class="n">args</span><span class="p">]):</span>
-            <span class="k">continue</span>
-        <span class="c"># If tensor product has all spin states, raise error for invalid tensor product state</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">([</span><span class="n">state</span><span class="o">.</span><span class="n">__class__</span> <span class="ow">is</span> <span class="n">tp</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">__class__</span> <span class="k">for</span> <span class="n">state</span> <span class="ow">in</span> <span class="n">tp</span><span class="o">.</span><span class="n">args</span><span class="p">]):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;All states must be the same basis&#39;</span><span class="p">)</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">tp</span><span class="p">,</span> <span class="n">_couple</span><span class="p">(</span><span class="n">tp</span><span class="p">,</span> <span class="n">jcoupling_list</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">expr</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_couple</span><span class="p">(</span><span class="n">tp</span><span class="p">,</span> <span class="n">jcoupling_list</span><span class="p">):</span>
-    <span class="n">states</span> <span class="o">=</span> <span class="n">tp</span><span class="o">.</span><span class="n">args</span>
-    <span class="n">coupled_evect</span> <span class="o">=</span> <span class="n">states</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">coupled_class</span><span class="p">()</span>
-
-    <span class="c"># Define default coupling if none is specified</span>
-    <span class="k">if</span> <span class="n">jcoupling_list</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">jcoupling_list</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">states</span><span class="p">)):</span>
-            <span class="n">jcoupling_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">)</span>
-
-    <span class="c"># Check jcoupling_list valid</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">len</span><span class="p">(</span><span class="n">jcoupling_list</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">states</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;jcoupling_list must be length </span><span class="si">%d</span><span class="s">, got </span><span class="si">%d</span><span class="s">&#39;</span> <span class="o">%</span>
-                        <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">states</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">jcoupling_list</span><span class="p">)))</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">(</span> <span class="nb">len</span><span class="p">(</span><span class="n">coupling</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span> <span class="k">for</span> <span class="n">coupling</span> <span class="ow">in</span> <span class="n">jcoupling_list</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Each coupling must define 2 spaces&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">any</span><span class="p">([</span><span class="n">n1</span> <span class="o">==</span> <span class="n">n2</span> <span class="k">for</span> <span class="n">n1</span><span class="p">,</span> <span class="n">n2</span> <span class="ow">in</span> <span class="n">jcoupling_list</span><span class="p">]):</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Spin spaces cannot couple to themselves&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">all</span><span class="p">([</span><span class="n">sympify</span><span class="p">(</span><span class="n">n1</span><span class="p">)</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="n">sympify</span><span class="p">(</span><span class="n">n2</span><span class="p">)</span><span class="o">.</span><span class="n">is_number</span> <span class="k">for</span> <span class="n">n1</span><span class="p">,</span> <span class="n">n2</span> <span class="ow">in</span> <span class="n">jcoupling_list</span><span class="p">]):</span>
-        <span class="n">j_test</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="n">states</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">n1</span><span class="p">,</span> <span class="n">n2</span> <span class="ow">in</span> <span class="n">jcoupling_list</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">j_test</span><span class="p">[</span><span class="n">n1</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span> <span class="ow">or</span> <span class="n">j_test</span><span class="p">[</span><span class="n">n2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Spaces coupling j_n</span><span class="se">\&#39;</span><span class="s">s are referenced by smallest n value&#39;</span><span class="p">)</span>
-            <span class="n">j_test</span><span class="p">[</span><span class="nb">max</span><span class="p">(</span><span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-
-    <span class="c"># j values of states to be coupled together</span>
-    <span class="n">jn</span> <span class="o">=</span> <span class="p">[</span><span class="n">state</span><span class="o">.</span><span class="n">j</span> <span class="k">for</span> <span class="n">state</span> <span class="ow">in</span> <span class="n">states</span><span class="p">]</span>
-    <span class="n">mn</span> <span class="o">=</span> <span class="p">[</span><span class="n">state</span><span class="o">.</span><span class="n">m</span> <span class="k">for</span> <span class="n">state</span> <span class="ow">in</span> <span class="n">states</span><span class="p">]</span>
-
-    <span class="c"># Create coupling_list, which defines all the couplings between all</span>
-    <span class="c"># the spaces from jcoupling_list</span>
-    <span class="n">coupling_list</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">n_list</span> <span class="o">=</span> <span class="p">[</span> <span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">states</span><span class="p">))</span> <span class="p">]</span>
-    <span class="k">for</span> <span class="n">j_coupling</span> <span class="ow">in</span> <span class="n">jcoupling_list</span><span class="p">:</span>
-        <span class="c"># Least n for all j_n which is coupled as first and second spaces</span>
-        <span class="n">n1</span><span class="p">,</span> <span class="n">n2</span> <span class="o">=</span> <span class="n">j_coupling</span>
-        <span class="c"># List of all n&#39;s coupled in first and second spaces</span>
-        <span class="n">j1_n</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">n_list</span><span class="p">[</span><span class="n">n1</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span>
-        <span class="n">j2_n</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">n_list</span><span class="p">[</span><span class="n">n2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span>
-        <span class="n">coupling_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="p">(</span><span class="n">j1_n</span><span class="p">,</span> <span class="n">j2_n</span><span class="p">)</span> <span class="p">)</span>
-        <span class="c"># Set new j_n to be coupling of all j_n in both first and second spaces</span>
-        <span class="n">n_list</span><span class="p">[</span> <span class="nb">min</span><span class="p">(</span><span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="p">]</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">j1_n</span> <span class="o">+</span> <span class="n">j2_n</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">state</span><span class="o">.</span><span class="n">j</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="n">state</span><span class="o">.</span><span class="n">m</span><span class="o">.</span><span class="n">is_number</span> <span class="k">for</span> <span class="n">state</span> <span class="ow">in</span> <span class="n">states</span><span class="p">):</span>
-        <span class="c"># Numerical coupling</span>
-        <span class="c"># Iterate over difference between maximum possible j value of each coupling and the actual value</span>
-        <span class="n">diff_max</span> <span class="o">=</span> <span class="p">[</span> <span class="n">Add</span><span class="p">(</span> <span class="o">*</span><span class="p">[</span> <span class="n">jn</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">mn</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">coupling</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span>
-                         <span class="n">coupling</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="p">]</span> <span class="p">)</span> <span class="k">for</span> <span class="n">coupling</span> <span class="ow">in</span> <span class="n">coupling_list</span> <span class="p">]</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">diff</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">diff_max</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="c"># Determine available configurations</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">coupling_list</span><span class="p">)</span>
-            <span class="n">tot</span> <span class="o">=</span> <span class="n">binomial</span><span class="p">(</span><span class="n">diff</span> <span class="o">+</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">diff</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">config_num</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">tot</span><span class="p">):</span>
-                <span class="n">diff_list</span> <span class="o">=</span> <span class="n">_confignum_to_difflist</span><span class="p">(</span><span class="n">config_num</span><span class="p">,</span> <span class="n">diff</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-
-                <span class="c"># Skip the configuration if non-physical</span>
-                <span class="c"># This is a lazy check for physical states given the loose restrictions of diff_max</span>
-                <span class="k">if</span> <span class="nb">any</span><span class="p">(</span> <span class="p">[</span> <span class="n">d</span> <span class="o">&gt;</span> <span class="n">m</span> <span class="k">for</span> <span class="n">d</span><span class="p">,</span> <span class="n">m</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">diff_list</span><span class="p">,</span> <span class="n">diff_max</span><span class="p">)</span> <span class="p">]</span> <span class="p">):</span>
-                    <span class="k">continue</span>
-
-                <span class="c"># Determine term</span>
-                <span class="n">cg_terms</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="n">coupled_j</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">jn</span><span class="p">)</span>
-                <span class="n">jcoupling</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">for</span> <span class="p">(</span><span class="n">j1_n</span><span class="p">,</span> <span class="n">j2_n</span><span class="p">),</span> <span class="n">coupling_diff</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">coupling_list</span><span class="p">,</span> <span class="n">diff_list</span><span class="p">):</span>
-                    <span class="n">j1</span> <span class="o">=</span> <span class="n">coupled_j</span><span class="p">[</span> <span class="nb">min</span><span class="p">(</span><span class="n">j1_n</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="p">]</span>
-                    <span class="n">j2</span> <span class="o">=</span> <span class="n">coupled_j</span><span class="p">[</span> <span class="nb">min</span><span class="p">(</span><span class="n">j2_n</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="p">]</span>
-                    <span class="n">j3</span> <span class="o">=</span> <span class="n">j1</span> <span class="o">+</span> <span class="n">j2</span> <span class="o">-</span> <span class="n">coupling_diff</span>
-                    <span class="n">coupled_j</span><span class="p">[</span> <span class="nb">min</span><span class="p">(</span><span class="n">j1_n</span> <span class="o">+</span> <span class="n">j2_n</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="p">]</span> <span class="o">=</span> <span class="n">j3</span>
-                    <span class="n">m1</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span> <span class="o">*</span><span class="p">[</span> <span class="n">mn</span><span class="p">[</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">j1_n</span><span class="p">]</span> <span class="p">)</span>
-                    <span class="n">m2</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span> <span class="o">*</span><span class="p">[</span> <span class="n">mn</span><span class="p">[</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">j2_n</span><span class="p">]</span> <span class="p">)</span>
-                    <span class="n">m3</span> <span class="o">=</span> <span class="n">m1</span> <span class="o">+</span> <span class="n">m2</span>
-                    <span class="n">cg_terms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="p">(</span><span class="n">j1</span><span class="p">,</span> <span class="n">m1</span><span class="p">,</span> <span class="n">j2</span><span class="p">,</span> <span class="n">m2</span><span class="p">,</span> <span class="n">j3</span><span class="p">,</span> <span class="n">m3</span><span class="p">)</span> <span class="p">)</span>
-                    <span class="n">jcoupling</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="p">(</span><span class="nb">min</span><span class="p">(</span><span class="n">j1_n</span><span class="p">),</span> <span class="nb">min</span><span class="p">(</span><span class="n">j2_n</span><span class="p">),</span> <span class="n">j3</span><span class="p">)</span> <span class="p">)</span>
-                <span class="c"># Better checks that state is physical</span>
-                <span class="k">if</span> <span class="nb">any</span><span class="p">([</span> <span class="nb">abs</span><span class="p">(</span><span class="n">term</span><span class="p">[</span><span class="mi">5</span><span class="p">])</span> <span class="o">&gt;</span> <span class="n">term</span><span class="p">[</span><span class="mi">4</span><span class="p">]</span> <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">cg_terms</span> <span class="p">]):</span>
-                    <span class="k">continue</span>
-                <span class="k">if</span> <span class="nb">any</span><span class="p">([</span> <span class="n">term</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">term</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">term</span><span class="p">[</span><span class="mi">4</span><span class="p">]</span> <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">cg_terms</span> <span class="p">]):</span>
-                    <span class="k">continue</span>
-                <span class="k">if</span> <span class="nb">any</span><span class="p">([</span> <span class="nb">abs</span><span class="p">(</span><span class="n">term</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">term</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span> <span class="o">&gt;</span> <span class="n">term</span><span class="p">[</span><span class="mi">4</span><span class="p">]</span> <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">cg_terms</span> <span class="p">]):</span>
-                    <span class="k">continue</span>
-                <span class="n">coeff</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span> <span class="o">*</span><span class="p">[</span> <span class="n">CG</span><span class="p">(</span><span class="o">*</span><span class="n">term</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span> <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">cg_terms</span><span class="p">]</span> <span class="p">)</span>
-                <span class="n">state</span> <span class="o">=</span> <span class="n">coupled_evect</span><span class="p">(</span><span class="n">j3</span><span class="p">,</span> <span class="n">m3</span><span class="p">,</span> <span class="n">jn</span><span class="p">,</span> <span class="n">jcoupling</span><span class="p">)</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">coeff</span><span class="o">*</span><span class="n">state</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">result</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="c"># Symbolic coupling</span>
-        <span class="n">cg_terms</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">jcoupling</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">sum_terms</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">coupled_j</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">jn</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">j1_n</span><span class="p">,</span> <span class="n">j2_n</span> <span class="ow">in</span> <span class="n">coupling_list</span><span class="p">:</span>
-            <span class="n">j1</span> <span class="o">=</span> <span class="n">coupled_j</span><span class="p">[</span> <span class="nb">min</span><span class="p">(</span><span class="n">j1_n</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="p">]</span>
-            <span class="n">j2</span> <span class="o">=</span> <span class="n">coupled_j</span><span class="p">[</span> <span class="nb">min</span><span class="p">(</span><span class="n">j2_n</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="p">]</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">j1_n</span> <span class="o">+</span> <span class="n">j2_n</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">states</span><span class="p">):</span>
-                <span class="n">j3</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;j&#39;</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">j3_name</span> <span class="o">=</span> <span class="s">&#39;j&#39;</span> <span class="o">+</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="s">&quot;</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">n</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">j1_n</span> <span class="o">+</span> <span class="n">j2_n</span><span class="p">])</span>
-                <span class="n">j3</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="n">j3_name</span><span class="p">)</span>
-            <span class="n">coupled_j</span><span class="p">[</span> <span class="nb">min</span><span class="p">(</span><span class="n">j1_n</span> <span class="o">+</span> <span class="n">j2_n</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="p">]</span> <span class="o">=</span> <span class="n">j3</span>
-            <span class="n">m1</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span> <span class="o">*</span><span class="p">[</span> <span class="n">mn</span><span class="p">[</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">j1_n</span><span class="p">]</span> <span class="p">)</span>
-            <span class="n">m2</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span> <span class="o">*</span><span class="p">[</span> <span class="n">mn</span><span class="p">[</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">j2_n</span><span class="p">]</span> <span class="p">)</span>
-            <span class="n">m3</span> <span class="o">=</span> <span class="n">m1</span> <span class="o">+</span> <span class="n">m2</span>
-            <span class="n">cg_terms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="p">(</span><span class="n">j1</span><span class="p">,</span> <span class="n">m1</span><span class="p">,</span> <span class="n">j2</span><span class="p">,</span> <span class="n">m2</span><span class="p">,</span> <span class="n">j3</span><span class="p">,</span> <span class="n">m3</span><span class="p">)</span> <span class="p">)</span>
-            <span class="n">jcoupling</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="p">(</span><span class="nb">min</span><span class="p">(</span><span class="n">j1_n</span><span class="p">),</span> <span class="nb">min</span><span class="p">(</span><span class="n">j2_n</span><span class="p">),</span> <span class="n">j3</span><span class="p">)</span> <span class="p">)</span>
-            <span class="n">sum_terms</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">j3</span><span class="p">,</span> <span class="n">m3</span><span class="p">,</span> <span class="n">j1</span> <span class="o">+</span> <span class="n">j2</span><span class="p">))</span>
-        <span class="n">coeff</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span> <span class="o">*</span><span class="p">[</span> <span class="n">CG</span><span class="p">(</span><span class="o">*</span><span class="n">term</span><span class="p">)</span> <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">cg_terms</span><span class="p">]</span> <span class="p">)</span>
-        <span class="n">state</span> <span class="o">=</span> <span class="n">coupled_evect</span><span class="p">(</span><span class="n">j3</span><span class="p">,</span> <span class="n">m3</span><span class="p">,</span> <span class="n">jn</span><span class="p">,</span> <span class="n">jcoupling</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Sum</span><span class="p">(</span><span class="n">coeff</span><span class="o">*</span><span class="n">state</span><span class="p">,</span> <span class="o">*</span><span class="n">sum_terms</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="uncouple"><a class="viewcode-back" href="../../../../modules/physics/quantum/spin.html#sympy.physics.quantum.spin.uncouple">[docs]</a><span class="k">def</span> <span class="nf">uncouple</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">jn</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">jcoupling_list</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Uncouple a coupled spin state</span>
-
-<span class="sd">    Gives the uncoupled representation of a coupled spin state. Arguments must</span>
-<span class="sd">    be either a spin state that is a subclass of CoupledSpinState or a spin</span>
-<span class="sd">    state that is a subclass of SpinState and an array giving the j values</span>
-<span class="sd">    of the spaces that are to be coupled</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    expr : Expr</span>
-<span class="sd">        The expression containing states that are to be coupled. If the states</span>
-<span class="sd">        are a subclass of SpinState, the ``jn`` and ``jcoupling`` parameters</span>
-<span class="sd">        must be defined. If the states are a subclass of CoupledSpinState,</span>
-<span class="sd">        ``jn`` and ``jcoupling`` will be taken from the state.</span>
-
-<span class="sd">    jn : list or tuple</span>
-<span class="sd">        The list of the j-values that are coupled. If state is a</span>
-<span class="sd">        CoupledSpinState, this parameter is ignored. This must be defined if</span>
-<span class="sd">        state is not a subclass of CoupledSpinState. The syntax of this</span>
-<span class="sd">        parameter is the same as the ``jn`` parameter of JzKetCoupled.</span>
-
-<span class="sd">    jcoupling_list : list or tuple</span>
-<span class="sd">        The list defining how the j-values are coupled together. If state is a</span>
-<span class="sd">        CoupledSpinState, this parameter is ignored. This must be defined if</span>
-<span class="sd">        state is not a subclass of CoupledSpinState. The syntax of this</span>
-<span class="sd">        parameter is the same as the ``jcoupling`` parameter of JzKetCoupled.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Uncouple a numerical state using a CoupledSpinState state:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.spin import JzKetCoupled, uncouple</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import S</span>
-<span class="sd">        &gt;&gt;&gt; uncouple(JzKetCoupled(1, 0, (S(1)/2, S(1)/2)))</span>
-<span class="sd">        sqrt(2)*|1/2,-1/2&gt;x|1/2,1/2&gt;/2 + sqrt(2)*|1/2,1/2&gt;x|1/2,-1/2&gt;/2</span>
-
-<span class="sd">    Perform the same calculation using a SpinState state:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.spin import JzKet</span>
-<span class="sd">        &gt;&gt;&gt; uncouple(JzKet(1, 0), (S(1)/2, S(1)/2))</span>
-<span class="sd">        sqrt(2)*|1/2,-1/2&gt;x|1/2,1/2&gt;/2 + sqrt(2)*|1/2,1/2&gt;x|1/2,-1/2&gt;/2</span>
-
-<span class="sd">    Uncouple a numerical state of three coupled spaces using a CoupledSpinState state:</span>
-
-<span class="sd">        &gt;&gt;&gt; uncouple(JzKetCoupled(1, 1, (1, 1, 1), ((1,3,1),(1,2,1)) ))</span>
-<span class="sd">        |1,-1&gt;x|1,1&gt;x|1,1&gt;/2 - |1,0&gt;x|1,0&gt;x|1,1&gt;/2 + |1,1&gt;x|1,0&gt;x|1,0&gt;/2 - |1,1&gt;x|1,1&gt;x|1,-1&gt;/2</span>
-
-<span class="sd">    Perform the same calculation using a SpinState state:</span>
-
-<span class="sd">        &gt;&gt;&gt; uncouple(JzKet(1, 1), (1, 1, 1), ((1,3,1),(1,2,1)) )</span>
-<span class="sd">        |1,-1&gt;x|1,1&gt;x|1,1&gt;/2 - |1,0&gt;x|1,0&gt;x|1,1&gt;/2 + |1,1&gt;x|1,0&gt;x|1,0&gt;/2 - |1,1&gt;x|1,1&gt;x|1,-1&gt;/2</span>
-
-<span class="sd">    Uncouple a symbolic state using a CoupledSpinState state:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; j,m,j1,j2 = symbols(&#39;j m j1 j2&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; uncouple(JzKetCoupled(j, m, (j1, j2)))</span>
-<span class="sd">        Sum(CG(j1, m1, j2, m2, j, m)*|j1,m1&gt;x|j2,m2&gt;, (m1, -j1, j1), (m2, -j2, j2))</span>
-
-<span class="sd">    Perform the same calculation using a SpinState state</span>
-
-<span class="sd">        &gt;&gt;&gt; uncouple(JzKet(j, m), (j1, j2))</span>
-<span class="sd">        Sum(CG(j1, m1, j2, m2, j, m)*|j1,m1&gt;x|j2,m2&gt;, (m1, -j1, j1), (m2, -j2, j2))</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">SpinState</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">state</span> <span class="ow">in</span> <span class="n">a</span><span class="p">:</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">_uncouple</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">jn</span><span class="p">,</span> <span class="n">jcoupling_list</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">expr</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_uncouple</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">jn</span><span class="p">,</span> <span class="n">jcoupling_list</span><span class="p">):</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">CoupledSpinState</span><span class="p">):</span>
-        <span class="n">jn</span> <span class="o">=</span> <span class="n">state</span><span class="o">.</span><span class="n">jn</span>
-        <span class="n">coupled_n</span> <span class="o">=</span> <span class="n">state</span><span class="o">.</span><span class="n">coupled_n</span>
-        <span class="n">coupled_jn</span> <span class="o">=</span> <span class="n">state</span><span class="o">.</span><span class="n">coupled_jn</span>
-        <span class="n">evect</span> <span class="o">=</span> <span class="n">state</span><span class="o">.</span><span class="n">uncoupled_class</span><span class="p">()</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">SpinState</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">jn</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Must specify j-values for coupled state&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">jn</span><span class="p">,</span> <span class="nb">list</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">jn</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">)):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;jn must be list or tuple&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">jcoupling_list</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="c"># Use default</span>
-            <span class="n">jcoupling_list</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">jn</span><span class="p">)):</span>
-                <span class="n">jcoupling_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span>
-                    <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">i</span><span class="p">,</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">jn</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]))</span> <span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">jcoupling_list</span><span class="p">,</span> <span class="nb">list</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">jcoupling_list</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">)):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;jcoupling must be a list or tuple&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">len</span><span class="p">(</span><span class="n">jcoupling_list</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">jn</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Must specify 2 fewer coupling terms than the number of j values&quot;</span><span class="p">)</span>
-        <span class="n">coupled_n</span><span class="p">,</span> <span class="n">coupled_jn</span> <span class="o">=</span> <span class="n">_build_coupled</span><span class="p">(</span><span class="n">jcoupling_list</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">jn</span><span class="p">))</span>
-        <span class="n">evect</span> <span class="o">=</span> <span class="n">state</span><span class="o">.</span><span class="n">__class__</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;state must be a spin state&quot;</span><span class="p">)</span>
-    <span class="n">j</span> <span class="o">=</span> <span class="n">state</span><span class="o">.</span><span class="n">j</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">state</span><span class="o">.</span><span class="n">m</span>
-    <span class="n">coupling_list</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">j_list</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">jn</span><span class="p">)</span>
-
-    <span class="c"># Create coupling, which defines all the couplings between all the spaces</span>
-    <span class="k">for</span> <span class="n">j3</span><span class="p">,</span> <span class="p">(</span><span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">coupled_jn</span><span class="p">,</span> <span class="n">coupled_n</span><span class="p">):</span>
-        <span class="c"># j&#39;s which are coupled as first and second spaces</span>
-        <span class="n">j1</span> <span class="o">=</span> <span class="n">j_list</span><span class="p">[</span><span class="n">n1</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-        <span class="n">j2</span> <span class="o">=</span> <span class="n">j_list</span><span class="p">[</span><span class="n">n2</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-        <span class="c"># Build coupling list</span>
-        <span class="n">coupling_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="p">(</span><span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">,</span> <span class="n">j1</span><span class="p">,</span> <span class="n">j2</span><span class="p">,</span> <span class="n">j3</span><span class="p">)</span> <span class="p">)</span>
-        <span class="c"># Set new value in j_list</span>
-        <span class="n">j_list</span><span class="p">[</span><span class="nb">min</span><span class="p">(</span><span class="n">n1</span> <span class="o">+</span> <span class="n">n2</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">j3</span>
-
-    <span class="k">if</span> <span class="n">j</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="n">m</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-        <span class="n">diff_max</span> <span class="o">=</span> <span class="p">[</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">jn</span> <span class="p">]</span>
-        <span class="n">diff</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">jn</span><span class="p">)</span> <span class="o">-</span> <span class="n">m</span>
-
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">jn</span><span class="p">)</span>
-        <span class="n">tot</span> <span class="o">=</span> <span class="n">binomial</span><span class="p">(</span><span class="n">diff</span> <span class="o">+</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">diff</span><span class="p">)</span>
-
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">config_num</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">tot</span><span class="p">):</span>
-            <span class="n">diff_list</span> <span class="o">=</span> <span class="n">_confignum_to_difflist</span><span class="p">(</span><span class="n">config_num</span><span class="p">,</span> <span class="n">diff</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">any</span><span class="p">(</span> <span class="p">[</span> <span class="n">d</span> <span class="o">&gt;</span> <span class="n">p</span> <span class="k">for</span> <span class="n">d</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">diff_list</span><span class="p">,</span> <span class="n">diff_max</span><span class="p">)</span> <span class="p">]</span> <span class="p">):</span>
-                <span class="k">continue</span>
-
-            <span class="n">cg_terms</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">coupling</span> <span class="ow">in</span> <span class="n">coupling_list</span><span class="p">:</span>
-                <span class="n">j1_n</span><span class="p">,</span> <span class="n">j2_n</span><span class="p">,</span> <span class="n">j1</span><span class="p">,</span> <span class="n">j2</span><span class="p">,</span> <span class="n">j3</span> <span class="o">=</span> <span class="n">coupling</span>
-                <span class="n">m1</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span> <span class="o">*</span><span class="p">[</span> <span class="n">jn</span><span class="p">[</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">diff_list</span><span class="p">[</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">j1_n</span> <span class="p">]</span> <span class="p">)</span>
-                <span class="n">m2</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span> <span class="o">*</span><span class="p">[</span> <span class="n">jn</span><span class="p">[</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">diff_list</span><span class="p">[</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">j2_n</span> <span class="p">]</span> <span class="p">)</span>
-                <span class="n">m3</span> <span class="o">=</span> <span class="n">m1</span> <span class="o">+</span> <span class="n">m2</span>
-                <span class="n">cg_terms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="p">(</span><span class="n">j1</span><span class="p">,</span> <span class="n">m1</span><span class="p">,</span> <span class="n">j2</span><span class="p">,</span> <span class="n">m2</span><span class="p">,</span> <span class="n">j3</span><span class="p">,</span> <span class="n">m3</span><span class="p">)</span> <span class="p">)</span>
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span> <span class="o">*</span><span class="p">[</span> <span class="n">CG</span><span class="p">(</span><span class="o">*</span><span class="n">term</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span> <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">cg_terms</span> <span class="p">]</span> <span class="p">)</span>
-            <span class="n">state</span> <span class="o">=</span> <span class="n">TensorProduct</span><span class="p">(</span>
-                <span class="o">*</span><span class="p">[</span> <span class="n">evect</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span> <span class="o">-</span> <span class="n">d</span><span class="p">)</span> <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">jn</span><span class="p">,</span> <span class="n">diff_list</span><span class="p">)</span> <span class="p">]</span> <span class="p">)</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">coeff</span><span class="o">*</span><span class="n">state</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">result</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="c"># Symbolic coupling</span>
-        <span class="n">m_str</span> <span class="o">=</span> <span class="s">&quot;m1:</span><span class="si">%d</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">jn</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">mvals</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="n">m_str</span><span class="p">)</span>
-        <span class="n">cg_terms</span> <span class="o">=</span> <span class="p">[(</span><span class="n">j1</span><span class="p">,</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">mvals</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">j1_n</span><span class="p">]),</span>
-                     <span class="n">j2</span><span class="p">,</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">mvals</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">j2_n</span><span class="p">]),</span>
-                     <span class="n">j3</span><span class="p">,</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">mvals</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">j1_n</span> <span class="o">+</span> <span class="n">j2_n</span><span class="p">]))</span> <span class="k">for</span> <span class="n">j1_n</span><span class="p">,</span> <span class="n">j2_n</span><span class="p">,</span> <span class="n">j1</span><span class="p">,</span> <span class="n">j2</span><span class="p">,</span> <span class="n">j3</span> <span class="ow">in</span> <span class="n">coupling_list</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="p">]</span>
-        <span class="n">cg_terms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">*</span><span class="p">[(</span><span class="n">j1</span><span class="p">,</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">mvals</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">j1_n</span><span class="p">]),</span>
-                           <span class="n">j2</span><span class="p">,</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">mvals</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">j2_n</span><span class="p">]),</span>
-                           <span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span> <span class="k">for</span> <span class="n">j1_n</span><span class="p">,</span> <span class="n">j2_n</span><span class="p">,</span> <span class="n">j1</span><span class="p">,</span> <span class="n">j2</span><span class="p">,</span> <span class="n">j3</span> <span class="ow">in</span> <span class="p">[</span><span class="n">coupling_list</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span> <span class="p">])</span>
-        <span class="n">cg_coeff</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">CG</span><span class="p">(</span><span class="o">*</span><span class="n">cg_term</span><span class="p">)</span> <span class="k">for</span> <span class="n">cg_term</span> <span class="ow">in</span> <span class="n">cg_terms</span><span class="p">])</span>
-        <span class="n">sum_terms</span> <span class="o">=</span> <span class="p">[</span> <span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="o">-</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">m</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">jn</span><span class="p">,</span> <span class="n">mvals</span><span class="p">)</span> <span class="p">]</span>
-        <span class="n">state</span> <span class="o">=</span> <span class="n">TensorProduct</span><span class="p">(</span> <span class="o">*</span><span class="p">[</span> <span class="n">evect</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span> <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">m</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">jn</span><span class="p">,</span> <span class="n">mvals</span><span class="p">)</span> <span class="p">]</span> <span class="p">)</span>
-        <span class="k">return</span> <span class="n">Sum</span><span class="p">(</span><span class="n">cg_coeff</span><span class="o">*</span><span class="n">state</span><span class="p">,</span> <span class="o">*</span><span class="n">sum_terms</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">_confignum_to_difflist</span><span class="p">(</span><span class="n">config_num</span><span class="p">,</span> <span class="n">diff</span><span class="p">,</span> <span class="n">list_len</span><span class="p">):</span>
-    <span class="c"># Determines configuration of diffs into list_len number of slots</span>
-    <span class="n">diff_list</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">list_len</span><span class="p">):</span>
-        <span class="n">prev_diff</span> <span class="o">=</span> <span class="n">diff</span>
-        <span class="c"># Number of spots after current one</span>
-        <span class="n">rem_spots</span> <span class="o">=</span> <span class="n">list_len</span> <span class="o">-</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="c"># Number of configurations of distributing diff among the remaining spots</span>
-        <span class="n">rem_configs</span> <span class="o">=</span> <span class="n">binomial</span><span class="p">(</span><span class="n">diff</span> <span class="o">+</span> <span class="n">rem_spots</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">diff</span><span class="p">)</span>
-        <span class="k">while</span> <span class="n">config_num</span> <span class="o">&gt;=</span> <span class="n">rem_configs</span><span class="p">:</span>
-            <span class="n">config_num</span> <span class="o">-=</span> <span class="n">rem_configs</span>
-            <span class="n">diff</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="n">rem_configs</span> <span class="o">=</span> <span class="n">binomial</span><span class="p">(</span><span class="n">diff</span> <span class="o">+</span> <span class="n">rem_spots</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">diff</span><span class="p">)</span>
-        <span class="n">diff_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">prev_diff</span> <span class="o">-</span> <span class="n">diff</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">diff_list</span>
-</pre></div>
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-  <h1>Source code for sympy.physics.quantum.state</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Dirac notation for states.&quot;&quot;&quot;</span>
-
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="p">(</span><span class="n">cacheit</span><span class="p">,</span> <span class="n">conjugate</span><span class="p">,</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">Function</span><span class="p">,</span> <span class="n">integrate</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span>
-                   <span class="n">Tuple</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.pretty.stringpict</span> <span class="kn">import</span> <span class="n">prettyForm</span><span class="p">,</span> <span class="n">stringPict</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.qexpr</span> <span class="kn">import</span> <span class="n">QExpr</span><span class="p">,</span> <span class="n">dispatch_method</span>
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="s">&#39;KetBase&#39;</span><span class="p">,</span>
-    <span class="s">&#39;BraBase&#39;</span><span class="p">,</span>
-    <span class="s">&#39;StateBase&#39;</span><span class="p">,</span>
-    <span class="s">&#39;State&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Ket&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Bra&#39;</span><span class="p">,</span>
-    <span class="s">&#39;TimeDepState&#39;</span><span class="p">,</span>
-    <span class="s">&#39;TimeDepBra&#39;</span><span class="p">,</span>
-    <span class="s">&#39;TimeDepKet&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Wavefunction&#39;</span>
-<span class="p">]</span>
-
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># States, bras and kets.</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-<span class="c"># ASCII brackets</span>
-<span class="n">_lbracket</span> <span class="o">=</span> <span class="s">&quot;&lt;&quot;</span>
-<span class="n">_rbracket</span> <span class="o">=</span> <span class="s">&quot;&gt;&quot;</span>
-<span class="n">_straight_bracket</span> <span class="o">=</span> <span class="s">&quot;|&quot;</span>
-
-
-<span class="c"># Unicode brackets</span>
-<span class="c"># MATHEMATICAL ANGLE BRACKETS</span>
-<span class="n">_lbracket_ucode</span> <span class="o">=</span> <span class="s">&quot;</span><span class="se">\u27E8</span><span class="s">&quot;</span>
-<span class="n">_rbracket_ucode</span> <span class="o">=</span> <span class="s">&quot;</span><span class="se">\u27E9</span><span class="s">&quot;</span>
-<span class="c"># LIGHT VERTICAL BAR</span>
-<span class="n">_straight_bracket_ucode</span> <span class="o">=</span> <span class="s">&quot;</span><span class="se">\u2758</span><span class="s">&quot;</span>
-
-<span class="c"># Other options for unicode printing of &lt;, &gt; and | for Dirac notation.</span>
-
-<span class="c"># LEFT-POINTING ANGLE BRACKET</span>
-<span class="c"># _lbracket = u&quot;\u2329&quot;</span>
-<span class="c"># _rbracket = u&quot;\u232A&quot;</span>
-
-<span class="c"># LEFT ANGLE BRACKET</span>
-<span class="c"># _lbracket = u&quot;\u3008&quot;</span>
-<span class="c"># _rbracket = u&quot;\u3009&quot;</span>
-
-<span class="c"># VERTICAL LINE</span>
-<span class="c"># _straight_bracket = u&quot;\u007C&quot;</span>
-
-
-<div class="viewcode-block" id="StateBase"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.StateBase">[docs]</a><span class="k">class</span> <span class="nc">StateBase</span><span class="p">(</span><span class="n">QExpr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Abstract base class for general abstract states in quantum mechanics.</span>
-
-<span class="sd">    All other state classes defined will need to inherit from this class. It</span>
-<span class="sd">    carries the basic structure for all other states such as dual, _eval_adjoint</span>
-<span class="sd">    and label.</span>
-
-<span class="sd">    This is an abstract base class and you should not instantiate it directly,</span>
-<span class="sd">    instead use State.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_operators_to_state</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ops</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Returns the eigenstate instance for the passed operators.</span>
-
-<span class="sd">        This method should be overridden in subclasses. It will handle being</span>
-<span class="sd">        passed either an Operator instance or set of Operator instances. It</span>
-<span class="sd">        should return the corresponding state INSTANCE or simply raise a</span>
-<span class="sd">        NotImplementedError. See cartesian.py for an example.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Cannot map operators to states in this class. Method not implemented!&quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_state_to_operators</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">op_classes</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Returns the operators which this state instance is an eigenstate</span>
-<span class="sd">        of.</span>
-
-<span class="sd">        This method should be overridden in subclasses. It will be called on</span>
-<span class="sd">        state instances and be passed the operator classes that we wish to make</span>
-<span class="sd">        into instances. The state instance will then transform the classes</span>
-<span class="sd">        appropriately, or raise a NotImplementedError if it cannot return</span>
-<span class="sd">        operator instances. See cartesian.py for examples,</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-            <span class="s">&quot;Cannot map this state to operators. Method not implemented!&quot;</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="StateBase.operators"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.StateBase.operators">[docs]</a>    <span class="k">def</span> <span class="nf">operators</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the operator(s) that this state is an eigenstate of&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">.operatorset</span> <span class="kn">import</span> <span class="n">state_to_operators</span>  <span class="c"># import internally to avoid circular import errors</span>
-        <span class="k">return</span> <span class="n">state_to_operators</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_enumerate_state</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">num_states</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Cannot enumerate this state!&quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent_default_basis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_represent</span><span class="p">(</span><span class="n">basis</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">operators</span><span class="p">)</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Dagger/dual</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="StateBase.dual"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.StateBase.dual">[docs]</a>    <span class="k">def</span> <span class="nf">dual</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the dual state of this one.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">dual_class</span><span class="p">()</span><span class="o">.</span><span class="n">_new_rawargs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">hilbert_space</span><span class="p">,</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="StateBase.dual_class"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.StateBase.dual_class">[docs]</a>    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the class used to construt the dual.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-            <span class="s">&#39;dual_class must be implemented in a subclass&#39;</span>
-        <span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_adjoint</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute the dagger of this state using the dual.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">dual</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Printing</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="k">def</span> <span class="nf">_pretty_brackets</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">height</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="c"># Return pretty printed brackets for the state</span>
-        <span class="c"># Ideally, this could be done by pform.parens but it does not support the angled &lt; and &gt;</span>
-
-        <span class="c"># Setup for unicode vs ascii</span>
-        <span class="k">if</span> <span class="n">use_unicode</span><span class="p">:</span>
-            <span class="n">lbracket</span><span class="p">,</span> <span class="n">rbracket</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lbracket_ucode</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rbracket_ucode</span>
-            <span class="n">slash</span><span class="p">,</span> <span class="n">bslash</span><span class="p">,</span> <span class="n">vert</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\u2571</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;</span><span class="se">\u2572</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;</span><span class="se">\u2502</span><span class="s">&#39;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">lbracket</span><span class="p">,</span> <span class="n">rbracket</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lbracket</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rbracket</span>
-            <span class="n">slash</span><span class="p">,</span> <span class="n">bslash</span><span class="p">,</span> <span class="n">vert</span> <span class="o">=</span> <span class="s">&#39;/&#39;</span><span class="p">,</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;|&#39;</span>
-
-        <span class="c"># If height is 1, just return brackets</span>
-        <span class="k">if</span> <span class="n">height</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">stringPict</span><span class="p">(</span><span class="n">lbracket</span><span class="p">),</span> <span class="n">stringPict</span><span class="p">(</span><span class="n">rbracket</span><span class="p">)</span>
-        <span class="c"># Make height even</span>
-        <span class="n">height</span> <span class="o">+=</span> <span class="p">(</span><span class="n">height</span> <span class="o">%</span> <span class="mi">2</span><span class="p">)</span>
-
-        <span class="n">brackets</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">bracket</span> <span class="ow">in</span> <span class="n">lbracket</span><span class="p">,</span> <span class="n">rbracket</span><span class="p">:</span>
-            <span class="c"># Create left bracket</span>
-            <span class="k">if</span> <span class="n">bracket</span> <span class="ow">in</span> <span class="nb">set</span><span class="p">([</span><span class="n">_lbracket</span><span class="p">,</span> <span class="n">_lbracket_ucode</span><span class="p">]):</span>
-                <span class="n">bracket_args</span> <span class="o">=</span> <span class="p">[</span> <span class="s">&#39; &#39;</span> <span class="o">*</span> <span class="p">(</span><span class="n">height</span><span class="o">//</span><span class="mi">2</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span>
-                                 <span class="n">slash</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">height</span> <span class="o">//</span> <span class="mi">2</span><span class="p">)]</span>
-                <span class="n">bracket_args</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span>
-                    <span class="p">[</span> <span class="s">&#39; &#39;</span> <span class="o">*</span> <span class="n">i</span> <span class="o">+</span> <span class="n">bslash</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">height</span> <span class="o">//</span> <span class="mi">2</span><span class="p">)])</span>
-            <span class="c"># Create right bracket</span>
-            <span class="k">elif</span> <span class="n">bracket</span> <span class="ow">in</span> <span class="nb">set</span><span class="p">([</span><span class="n">_rbracket</span><span class="p">,</span> <span class="n">_rbracket_ucode</span><span class="p">]):</span>
-                <span class="n">bracket_args</span> <span class="o">=</span> <span class="p">[</span> <span class="s">&#39; &#39;</span> <span class="o">*</span> <span class="n">i</span> <span class="o">+</span> <span class="n">bslash</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">height</span> <span class="o">//</span> <span class="mi">2</span><span class="p">)]</span>
-                <span class="n">bracket_args</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span> <span class="s">&#39; &#39;</span> <span class="o">*</span> <span class="p">(</span>
-                    <span class="n">height</span><span class="o">//</span><span class="mi">2</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">slash</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">height</span> <span class="o">//</span> <span class="mi">2</span><span class="p">)])</span>
-            <span class="c"># Create straight bracket</span>
-            <span class="k">elif</span> <span class="n">bracket</span> <span class="ow">in</span> <span class="nb">set</span><span class="p">([</span><span class="n">_straight_bracket</span><span class="p">,</span> <span class="n">_straight_bracket_ucode</span><span class="p">]):</span>
-                <span class="n">bracket_args</span> <span class="o">=</span> <span class="p">[</span><span class="n">vert</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">height</span><span class="p">)]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="n">bracket</span><span class="p">)</span>
-            <span class="n">brackets</span><span class="o">.</span><span class="n">append</span><span class="p">(</span>
-                <span class="n">stringPict</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">bracket_args</span><span class="p">),</span> <span class="n">baseline</span><span class="o">=</span><span class="n">height</span><span class="o">//</span><span class="mi">2</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">brackets</span>
-
-    <span class="k">def</span> <span class="nf">_sympystr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">contents</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_contents</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s%s%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lbracket</span><span class="p">,</span> <span class="n">contents</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rbracket</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.printing.pretty.stringpict</span> <span class="kn">import</span> <span class="n">prettyForm</span>
-        <span class="c"># Get brackets</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_contents_pretty</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">lbracket</span><span class="p">,</span> <span class="n">rbracket</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_pretty_brackets</span><span class="p">(</span>
-            <span class="n">pform</span><span class="o">.</span><span class="n">height</span><span class="p">(),</span> <span class="n">printer</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">)</span>
-        <span class="c"># Put together state</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="n">lbracket</span><span class="p">))</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">rbracket</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">contents</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_contents_latex</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="c"># The extra {} brackets are needed to get matplotlib&#39;s latex</span>
-        <span class="c"># rendered to render this properly.</span>
-        <span class="k">return</span> <span class="s">&#39;{</span><span class="si">%s%s%s</span><span class="s">}&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lbracket_latex</span><span class="p">,</span> <span class="n">contents</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rbracket_latex</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="KetBase"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.KetBase">[docs]</a><span class="k">class</span> <span class="nc">KetBase</span><span class="p">(</span><span class="n">StateBase</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Base class for Kets.</span>
-
-<span class="sd">    This class defines the dual property and the brackets for printing. This is</span>
-<span class="sd">    an abstract base class and you should not instantiate it directly, instead</span>
-<span class="sd">    use Ket.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">lbracket</span> <span class="o">=</span> <span class="n">_straight_bracket</span>
-    <span class="n">rbracket</span> <span class="o">=</span> <span class="n">_rbracket</span>
-    <span class="n">lbracket_ucode</span> <span class="o">=</span> <span class="n">_straight_bracket_ucode</span>
-    <span class="n">rbracket_ucode</span> <span class="o">=</span> <span class="n">_rbracket_ucode</span>
-    <span class="n">lbracket_latex</span> <span class="o">=</span> <span class="s">r&#39;\left|&#39;</span>
-    <span class="n">rbracket_latex</span> <span class="o">=</span> <span class="s">r&#39;\right\rangle &#39;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">default_args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="s">&quot;psi&quot;</span><span class="p">,)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">BraBase</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;KetBase*other&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.physics.quantum.operator</span> <span class="kn">import</span> <span class="n">OuterProduct</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">BraBase</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">OuterProduct</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;other*KetBase&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.physics.quantum.innerproduct</span> <span class="kn">import</span> <span class="n">InnerProduct</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">BraBase</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">InnerProduct</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># _eval_* methods</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="k">def</span> <span class="nf">_eval_innerproduct</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">bra</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Evaluate the inner product betweeen this ket and a bra.</span>
-
-<span class="sd">        This is called to compute &lt;bra|ket&gt;, where the ket is ``self``.</span>
-
-<span class="sd">        This method will dispatch to sub-methods having the format::</span>
-
-<span class="sd">            ``def _eval_innerproduct_BraClass(self, **hints):``</span>
-
-<span class="sd">        Subclasses should define these methods (one for each BraClass) to</span>
-<span class="sd">        teach the ket how to take inner products with bras.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dispatch_method</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s">&#39;_eval_innerproduct&#39;</span><span class="p">,</span> <span class="n">bra</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_apply_operator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">op</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Apply an Operator to this Ket.</span>
-
-<span class="sd">        This method will dispatch to methods having the format::</span>
-
-<span class="sd">            ``def _apply_operator_OperatorName(op, **options):``</span>
-
-<span class="sd">        Subclasses should define these methods (one for each OperatorName) to</span>
-<span class="sd">        teach the Ket how operators act on it.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        op : Operator</span>
-<span class="sd">            The Operator that is acting on the Ket.</span>
-<span class="sd">        options : dict</span>
-<span class="sd">            A dict of key/value pairs that control how the operator is applied</span>
-<span class="sd">            to the Ket.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dispatch_method</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s">&#39;_apply_operator&#39;</span><span class="p">,</span> <span class="n">op</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="BraBase"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.BraBase">[docs]</a><span class="k">class</span> <span class="nc">BraBase</span><span class="p">(</span><span class="n">StateBase</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Base class for Bras.</span>
-
-<span class="sd">    This class defines the dual property and the brackets for printing. This</span>
-<span class="sd">    is an abstract base class and you should not instantiate it directly,</span>
-<span class="sd">    instead use Bra.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">lbracket</span> <span class="o">=</span> <span class="n">_lbracket</span>
-    <span class="n">rbracket</span> <span class="o">=</span> <span class="n">_straight_bracket</span>
-    <span class="n">lbracket_ucode</span> <span class="o">=</span> <span class="n">_lbracket_ucode</span>
-    <span class="n">rbracket_ucode</span> <span class="o">=</span> <span class="n">_straight_bracket_ucode</span>
-    <span class="n">lbracket_latex</span> <span class="o">=</span> <span class="s">r&#39;\left\langle &#39;</span>
-    <span class="n">rbracket_latex</span> <span class="o">=</span> <span class="s">r&#39;\right|&#39;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_operators_to_state</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ops</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">state</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dual_class</span><span class="p">()</span><span class="o">.</span><span class="n">operators_to_state</span><span class="p">(</span><span class="n">ops</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">state</span><span class="o">.</span><span class="n">dual</span>
-
-    <span class="k">def</span> <span class="nf">_state_to_operators</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">op_classes</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">dual</span><span class="o">.</span><span class="n">_state_to_operators</span><span class="p">(</span><span class="n">op_classes</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_enumerate_state</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">num_states</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">dual_states</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dual</span><span class="o">.</span><span class="n">_enumerate_state</span><span class="p">(</span><span class="n">num_states</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">x</span><span class="o">.</span><span class="n">dual</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">dual_states</span><span class="p">]</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">default_args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">dual_class</span><span class="p">()</span><span class="o">.</span><span class="n">default_args</span><span class="p">()</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">KetBase</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;BraBase*other&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.physics.quantum.innerproduct</span> <span class="kn">import</span> <span class="n">InnerProduct</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">KetBase</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">InnerProduct</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;other*BraBase&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.physics.quantum.operator</span> <span class="kn">import</span> <span class="n">OuterProduct</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">KetBase</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">OuterProduct</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__rmul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_represent</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A default represent that uses the Ket&#39;s version.&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.physics.quantum.dagger</span> <span class="kn">import</span> <span class="n">Dagger</span>
-        <span class="k">return</span> <span class="n">Dagger</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dual</span><span class="o">.</span><span class="n">_represent</span><span class="p">(</span><span class="o">**</span><span class="n">options</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="State"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.State">[docs]</a><span class="k">class</span> <span class="nc">State</span><span class="p">(</span><span class="n">StateBase</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;General abstract quantum state used as a base class for Ket and Bra.&quot;&quot;&quot;</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="Ket"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.Ket">[docs]</a><span class="k">class</span> <span class="nc">Ket</span><span class="p">(</span><span class="n">State</span><span class="p">,</span> <span class="n">KetBase</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A general time-independent Ket in quantum mechanics.</span>
-
-<span class="sd">    Inherits from State and KetBase. This class should be used as the base</span>
-<span class="sd">    class for all physical, time-independent Kets in a system. This class</span>
-<span class="sd">    and its subclasses will be the main classes that users will use for</span>
-<span class="sd">    expressing Kets in Dirac notation [1]_.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    args : tuple</span>
-<span class="sd">        The list of numbers or parameters that uniquely specify the</span>
-<span class="sd">        ket. This will usually be its symbol or its quantum numbers. For</span>
-<span class="sd">        time-dependent state, this will include the time.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Create a simple Ket and looking at its properties::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum import Ket, Bra</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols, I</span>
-<span class="sd">        &gt;&gt;&gt; k = Ket(&#39;psi&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; k</span>
-<span class="sd">        |psi&gt;</span>
-<span class="sd">        &gt;&gt;&gt; k.hilbert_space</span>
-<span class="sd">        H</span>
-<span class="sd">        &gt;&gt;&gt; k.is_commutative</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; k.label</span>
-<span class="sd">        (psi,)</span>
-
-<span class="sd">    Ket&#39;s know about their associated bra::</span>
-
-<span class="sd">        &gt;&gt;&gt; k.dual</span>
-<span class="sd">        &lt;psi|</span>
-<span class="sd">        &gt;&gt;&gt; k.dual_class()</span>
-<span class="sd">        &lt;class &#39;sympy.physics.quantum.state.Bra&#39;&gt;</span>
-
-<span class="sd">    Take a linear combination of two kets::</span>
-
-<span class="sd">        &gt;&gt;&gt; k0 = Ket(0)</span>
-<span class="sd">        &gt;&gt;&gt; k1 = Ket(1)</span>
-<span class="sd">        &gt;&gt;&gt; 2*I*k0 - 4*k1</span>
-<span class="sd">        2*I*|0&gt; - 4*|1&gt;</span>
-
-<span class="sd">    Compound labels are passed as tuples::</span>
-
-<span class="sd">        &gt;&gt;&gt; n, m = symbols(&#39;n,m&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; k = Ket(n,m)</span>
-<span class="sd">        &gt;&gt;&gt; k</span>
-<span class="sd">        |nm&gt;</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Bra-ket_notation</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Bra</span>
-
-</div>
-<div class="viewcode-block" id="Bra"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.Bra">[docs]</a><span class="k">class</span> <span class="nc">Bra</span><span class="p">(</span><span class="n">State</span><span class="p">,</span> <span class="n">BraBase</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A general time-independent Bra in quantum mechanics.</span>
-
-<span class="sd">    Inherits from State and BraBase. A Bra is the dual of a Ket [1]_. This</span>
-<span class="sd">    class and its subclasses will be the main classes that users will use for</span>
-<span class="sd">    expressing Bras in Dirac notation.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    args : tuple</span>
-<span class="sd">        The list of numbers or parameters that uniquely specify the</span>
-<span class="sd">        ket. This will usually be its symbol or its quantum numbers. For</span>
-<span class="sd">        time-dependent state, this will include the time.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Create a simple Bra and look at its properties::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum import Ket, Bra</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols, I</span>
-<span class="sd">        &gt;&gt;&gt; b = Bra(&#39;psi&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; b</span>
-<span class="sd">        &lt;psi|</span>
-<span class="sd">        &gt;&gt;&gt; b.hilbert_space</span>
-<span class="sd">        H</span>
-<span class="sd">        &gt;&gt;&gt; b.is_commutative</span>
-<span class="sd">        False</span>
-
-<span class="sd">    Bra&#39;s know about their dual Ket&#39;s::</span>
-
-<span class="sd">        &gt;&gt;&gt; b.dual</span>
-<span class="sd">        |psi&gt;</span>
-<span class="sd">        &gt;&gt;&gt; b.dual_class()</span>
-<span class="sd">        &lt;class &#39;sympy.physics.quantum.state.Ket&#39;&gt;</span>
-
-<span class="sd">    Like Kets, Bras can have compound labels and be manipulated in a similar</span>
-<span class="sd">    manner::</span>
-
-<span class="sd">        &gt;&gt;&gt; n, m = symbols(&#39;n,m&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; b = Bra(n,m) - I*Bra(m,n)</span>
-<span class="sd">        &gt;&gt;&gt; b</span>
-<span class="sd">        -I*&lt;mn| + &lt;nm|</span>
-
-<span class="sd">    Symbols in a Bra can be substituted using ``.subs``::</span>
-
-<span class="sd">        &gt;&gt;&gt; b.subs(n,m)</span>
-<span class="sd">        &lt;mm| - I*&lt;mm|</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] http://en.wikipedia.org/wiki/Bra-ket_notation</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Ket</span>
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># Time dependent states, bras and kets.</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-</div>
-<div class="viewcode-block" id="TimeDepState"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.TimeDepState">[docs]</a><span class="k">class</span> <span class="nc">TimeDepState</span><span class="p">(</span><span class="n">StateBase</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Base class for a general time-dependent quantum state.</span>
-
-<span class="sd">    This class is used as a base class for any time-dependent state. The main</span>
-<span class="sd">    difference between this class and the time-independent state is that this</span>
-<span class="sd">    class takes a second argument that is the time in addition to the usual</span>
-<span class="sd">    label argument.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    args : tuple</span>
-<span class="sd">        The list of numbers or parameters that uniquely specify the ket. This</span>
-<span class="sd">        will usually be its symbol or its quantum numbers. For time-dependent</span>
-<span class="sd">        state, this will include the time as the final argument.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Initialization</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">default_args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="s">&quot;psi&quot;</span><span class="p">,</span> <span class="s">&quot;t&quot;</span><span class="p">)</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Properties</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="TimeDepState.label"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.TimeDepState.label">[docs]</a>    <span class="k">def</span> <span class="nf">label</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The label of the state.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="TimeDepState.time"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.TimeDepState.time">[docs]</a>    <span class="k">def</span> <span class="nf">time</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The time of the state.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="c">#-------------------------------------------------------------------------</span>
-    <span class="c"># Printing</span>
-    <span class="c">#-------------------------------------------------------------------------</span>
-</div>
-    <span class="k">def</span> <span class="nf">_print_time</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">time</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="n">_print_time_repr</span> <span class="o">=</span> <span class="n">_print_time</span>
-    <span class="n">_print_time_latex</span> <span class="o">=</span> <span class="n">_print_time</span>
-
-    <span class="k">def</span> <span class="nf">_print_time_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">time</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_contents</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">label</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_label</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">time</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_time</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">;</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">label</span><span class="p">,</span> <span class="n">time</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_label_repr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">label</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_sequence</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">,</span> <span class="s">&#39;,&#39;</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">time</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_time_repr</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">label</span><span class="p">,</span> <span class="n">time</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_contents_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">label</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_label_pretty</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">time</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_time_pretty</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">((</span><span class="n">label</span><span class="p">,</span> <span class="n">time</span><span class="p">),</span> <span class="n">delimiter</span><span class="o">=</span><span class="s">&#39;;&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_contents_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">label</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_sequence</span><span class="p">(</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_label_separator</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">time</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_time_latex</span><span class="p">(</span><span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">;</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">label</span><span class="p">,</span> <span class="n">time</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="TimeDepKet"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.TimeDepKet">[docs]</a><span class="k">class</span> <span class="nc">TimeDepKet</span><span class="p">(</span><span class="n">TimeDepState</span><span class="p">,</span> <span class="n">KetBase</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;General time-dependent Ket in quantum mechanics.</span>
-
-<span class="sd">    This inherits from ``TimeDepState`` and ``KetBase`` and is the main class</span>
-<span class="sd">    that should be used for Kets that vary with time. Its dual is a</span>
-<span class="sd">    ``TimeDepBra``.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    args : tuple</span>
-<span class="sd">        The list of numbers or parameters that uniquely specify the ket. This</span>
-<span class="sd">        will usually be its symbol or its quantum numbers. For time-dependent</span>
-<span class="sd">        state, this will include the time as the final argument.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Create a TimeDepKet and look at its attributes::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum import TimeDepKet</span>
-<span class="sd">        &gt;&gt;&gt; k = TimeDepKet(&#39;psi&#39;, &#39;t&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; k</span>
-<span class="sd">        |psi;t&gt;</span>
-<span class="sd">        &gt;&gt;&gt; k.time</span>
-<span class="sd">        t</span>
-<span class="sd">        &gt;&gt;&gt; k.label</span>
-<span class="sd">        (psi,)</span>
-<span class="sd">        &gt;&gt;&gt; k.hilbert_space</span>
-<span class="sd">        H</span>
-
-<span class="sd">    TimeDepKets know about their dual bra::</span>
-
-<span class="sd">        &gt;&gt;&gt; k.dual</span>
-<span class="sd">        &lt;psi;t|</span>
-<span class="sd">        &gt;&gt;&gt; k.dual_class()</span>
-<span class="sd">        &lt;class &#39;sympy.physics.quantum.state.TimeDepBra&#39;&gt;</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">TimeDepBra</span>
-
-</div>
-<div class="viewcode-block" id="TimeDepBra"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.TimeDepBra">[docs]</a><span class="k">class</span> <span class="nc">TimeDepBra</span><span class="p">(</span><span class="n">TimeDepState</span><span class="p">,</span> <span class="n">BraBase</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;General time-dependent Bra in quantum mechanics.</span>
-
-<span class="sd">    This inherits from TimeDepState and BraBase and is the main class that</span>
-<span class="sd">    should be used for Bras that vary with time. Its dual is a TimeDepBra.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    args : tuple</span>
-<span class="sd">        The list of numbers or parameters that uniquely specify the ket. This</span>
-<span class="sd">        will usually be its symbol or its quantum numbers. For time-dependent</span>
-<span class="sd">        state, this will include the time as the final argument.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum import TimeDepBra</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols, I</span>
-<span class="sd">        &gt;&gt;&gt; b = TimeDepBra(&#39;psi&#39;, &#39;t&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; b</span>
-<span class="sd">        &lt;psi;t|</span>
-<span class="sd">        &gt;&gt;&gt; b.time</span>
-<span class="sd">        t</span>
-<span class="sd">        &gt;&gt;&gt; b.label</span>
-<span class="sd">        (psi,)</span>
-<span class="sd">        &gt;&gt;&gt; b.hilbert_space</span>
-<span class="sd">        H</span>
-<span class="sd">        &gt;&gt;&gt; b.dual</span>
-<span class="sd">        |psi;t&gt;</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">dual_class</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">TimeDepKet</span>
-
-</div>
-<div class="viewcode-block" id="Wavefunction"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.Wavefunction">[docs]</a><span class="k">class</span> <span class="nc">Wavefunction</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Class for representations in continuous bases</span>
-
-<span class="sd">    This class takes an expression and coordinates in its constructor. It can</span>
-<span class="sd">    be used to easily calculate normalizations and probabilities.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    expr : Expr</span>
-<span class="sd">           The expression representing the functional form of the w.f.</span>
-
-<span class="sd">    coords : Symbol or tuple</span>
-<span class="sd">           The coordinates to be integrated over, and their bounds</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Particle in a box, specifying bounds in the more primitive way of using</span>
-<span class="sd">    Piecewise:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol, Piecewise, pi, N</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.functions import sqrt, sin</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.state import Wavefunction</span>
-<span class="sd">        &gt;&gt;&gt; x = Symbol(&#39;x&#39;, real=True)</span>
-<span class="sd">        &gt;&gt;&gt; n = 1</span>
-<span class="sd">        &gt;&gt;&gt; L = 1</span>
-<span class="sd">        &gt;&gt;&gt; g = Piecewise((0, x &lt; 0), (0, x &gt; L), (sqrt(2//L)*sin(n*pi*x/L), True))</span>
-<span class="sd">        &gt;&gt;&gt; f = Wavefunction(g, x)</span>
-<span class="sd">        &gt;&gt;&gt; f.norm</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; f.is_normalized</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; p = f.prob()</span>
-<span class="sd">        &gt;&gt;&gt; p(0)</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; p(L)</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; p(0.5)</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; p(0.85*L)</span>
-<span class="sd">        2*sin(0.85*pi)**2</span>
-<span class="sd">        &gt;&gt;&gt; N(p(0.85*L))</span>
-<span class="sd">        0.412214747707527</span>
-
-<span class="sd">    Additionally, you can specify the bounds of the function and the indices in</span>
-<span class="sd">    a more compact way:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols, pi, diff</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.functions import sqrt, sin</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.state import Wavefunction</span>
-<span class="sd">        &gt;&gt;&gt; x, L = symbols(&#39;x,L&#39;, positive=True)</span>
-<span class="sd">        &gt;&gt;&gt; n = symbols(&#39;n&#39;, integer=True)</span>
-<span class="sd">        &gt;&gt;&gt; g = sqrt(2/L)*sin(n*pi*x/L)</span>
-<span class="sd">        &gt;&gt;&gt; f = Wavefunction(g, (x, 0, L))</span>
-<span class="sd">        &gt;&gt;&gt; f.norm</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; f(L+1)</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; f(L-1)</span>
-<span class="sd">        sqrt(2)*sin(pi*n*(L - 1)/L)/sqrt(L)</span>
-<span class="sd">        &gt;&gt;&gt; f(-1)</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; f(0.85)</span>
-<span class="sd">        sqrt(2)*sin(0.85*pi*n/L)/sqrt(L)</span>
-<span class="sd">        &gt;&gt;&gt; f(0.85, n=1, L=1)</span>
-<span class="sd">        sqrt(2)*sin(0.85*pi)</span>
-<span class="sd">        &gt;&gt;&gt; f.is_commutative</span>
-<span class="sd">        False</span>
-
-<span class="sd">    All arguments are automatically sympified, so you can define the variables</span>
-<span class="sd">    as strings rather than symbols:</span>
-
-<span class="sd">        &gt;&gt;&gt; expr = x**2</span>
-<span class="sd">        &gt;&gt;&gt; f = Wavefunction(expr, &#39;x&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; type(f.variables[0])</span>
-<span class="sd">        &lt;class &#39;sympy.core.symbol.Symbol&#39;&gt;</span>
-
-<span class="sd">    Derivatives of Wavefunctions will return Wavefunctions:</span>
-
-<span class="sd">        &gt;&gt;&gt; diff(f, x)</span>
-<span class="sd">        Wavefunction(2*x, x)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="c">#Any passed tuples for coordinates and their bounds need to be</span>
-    <span class="c">#converted to Tuples before Function&#39;s constructor is called, to</span>
-    <span class="c">#avoid errors from calling is_Float in the constructor</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">new_args</span> <span class="o">=</span> <span class="p">[</span><span class="bp">None</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span>
-        <span class="n">ct</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span>
-                <span class="n">new_args</span><span class="p">[</span><span class="n">ct</span><span class="p">]</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">new_args</span><span class="p">[</span><span class="n">ct</span><span class="p">]</span> <span class="o">=</span> <span class="n">arg</span>
-            <span class="n">ct</span> <span class="o">+=</span> <span class="mi">1</span>
-
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Function</span><span class="p">,</span> <span class="n">cls</span><span class="p">)</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">new_args</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="n">var</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">variables</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">var</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                <span class="s">&quot;Incorrect number of arguments to function!&quot;</span><span class="p">)</span>
-
-        <span class="n">ct</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="c">#If the passed value is outside the specified bounds, return 0</span>
-        <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">var</span><span class="p">:</span>
-            <span class="n">lower</span><span class="p">,</span> <span class="n">upper</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">limits</span><span class="p">[</span><span class="n">v</span><span class="p">]</span>
-
-            <span class="c">#Do the comparison to limits only if the passed symbol is actually</span>
-            <span class="c">#a symbol present in the limits;</span>
-            <span class="c">#Had problems with a comparison of x &gt; L</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="n">ct</span><span class="p">],</span> <span class="n">Expr</span><span class="p">)</span> <span class="ow">and</span> \
-                <span class="ow">not</span> <span class="p">(</span><span class="n">lower</span> <span class="ow">in</span> <span class="n">args</span><span class="p">[</span><span class="n">ct</span><span class="p">]</span><span class="o">.</span><span class="n">free_symbols</span>
-                     <span class="ow">or</span> <span class="n">upper</span> <span class="ow">in</span> <span class="n">args</span><span class="p">[</span><span class="n">ct</span><span class="p">]</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">):</span>
-                <span class="k">continue</span>
-
-            <span class="k">if</span> <span class="n">args</span><span class="p">[</span><span class="n">ct</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">lower</span> <span class="ow">or</span> <span class="n">args</span><span class="p">[</span><span class="n">ct</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">upper</span><span class="p">:</span>
-                <span class="k">return</span> <span class="mi">0</span>
-
-            <span class="n">ct</span> <span class="o">+=</span> <span class="mi">1</span>
-
-        <span class="n">expr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span>
-
-        <span class="c">#Allows user to make a call like f(2, 4, m=1, n=1)</span>
-        <span class="k">for</span> <span class="n">symbol</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">str</span><span class="p">(</span><span class="n">symbol</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">options</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
-                <span class="n">val</span> <span class="o">=</span> <span class="n">options</span><span class="p">[</span><span class="nb">str</span><span class="p">(</span><span class="n">symbol</span><span class="p">)]</span>
-                <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">symbol</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">var</span><span class="p">,</span> <span class="n">args</span><span class="p">)))</span>
-
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">symbol</span><span class="p">):</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span>
-        <span class="n">deriv</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">_eval_derivative</span><span class="p">(</span><span class="n">symbol</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">Wavefunction</span><span class="p">(</span><span class="n">deriv</span><span class="p">,</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Wavefunction</span><span class="p">(</span><span class="n">conjugate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">),</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_transpose</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">free_symbols</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">free_symbols</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Wavefunction.is_commutative"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.Wavefunction.is_commutative">[docs]</a>    <span class="k">def</span> <span class="nf">is_commutative</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Override Function&#39;s is_commutative so that order is preserved in</span>
-<span class="sd">        represented expressions</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">False</span>
-</div>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Wavefunction.variables"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.Wavefunction.variables">[docs]</a>    <span class="k">def</span> <span class="nf">variables</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the coordinates which the wavefunction depends on</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy.physics.quantum.state import Wavefunction</span>
-<span class="sd">            &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">            &gt;&gt;&gt; x,y = symbols(&#39;x,y&#39;)</span>
-<span class="sd">            &gt;&gt;&gt; f = Wavefunction(x*y, x, y)</span>
-<span class="sd">            &gt;&gt;&gt; f.variables</span>
-<span class="sd">            (x, y)</span>
-<span class="sd">            &gt;&gt;&gt; g = Wavefunction(x*y, x)</span>
-<span class="sd">            &gt;&gt;&gt; g.variables</span>
-<span class="sd">            (x,)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">var</span> <span class="o">=</span> <span class="p">[</span><span class="n">g</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">)</span> <span class="k">else</span> <span class="n">g</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]]</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">var</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Wavefunction.limits"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.Wavefunction.limits">[docs]</a>    <span class="k">def</span> <span class="nf">limits</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the limits of the coordinates which the w.f. depends on If no</span>
-<span class="sd">        limits are specified, defaults to ``(-oo, oo)``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy.physics.quantum.state import Wavefunction</span>
-<span class="sd">            &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">            &gt;&gt;&gt; x, y = symbols(&#39;x, y&#39;)</span>
-<span class="sd">            &gt;&gt;&gt; f = Wavefunction(x**2, (x, 0, 1))</span>
-<span class="sd">            &gt;&gt;&gt; f.limits</span>
-<span class="sd">            {x: (0, 1)}</span>
-<span class="sd">            &gt;&gt;&gt; f = Wavefunction(x**2, x)</span>
-<span class="sd">            &gt;&gt;&gt; f.limits</span>
-<span class="sd">            {x: (-oo, oo)}</span>
-<span class="sd">            &gt;&gt;&gt; f = Wavefunction(x**2 + y**2, x, (y, -1, 2))</span>
-<span class="sd">            &gt;&gt;&gt; f.limits</span>
-<span class="sd">            {x: (-oo, oo), y: (-1, 2)}</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">limits</span> <span class="o">=</span> <span class="p">[(</span><span class="n">g</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">g</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">)</span> <span class="k">else</span> <span class="p">(</span><span class="o">-</span><span class="n">oo</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-                  <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]]</span>
-        <span class="k">return</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">limits</span><span class="p">))))</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Wavefunction.expr"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.Wavefunction.expr">[docs]</a>    <span class="k">def</span> <span class="nf">expr</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the expression which is the functional form of the Wavefunction</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy.physics.quantum.state import Wavefunction</span>
-<span class="sd">            &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">            &gt;&gt;&gt; x, y = symbols(&#39;x, y&#39;)</span>
-<span class="sd">            &gt;&gt;&gt; f = Wavefunction(x**2, x)</span>
-<span class="sd">            &gt;&gt;&gt; f.expr</span>
-<span class="sd">            x**2</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Wavefunction.is_normalized"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.Wavefunction.is_normalized">[docs]</a>    <span class="k">def</span> <span class="nf">is_normalized</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns true if the Wavefunction is properly normalized</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy import symbols, pi</span>
-<span class="sd">            &gt;&gt;&gt; from sympy.functions import sqrt, sin</span>
-<span class="sd">            &gt;&gt;&gt; from sympy.physics.quantum.state import Wavefunction</span>
-<span class="sd">            &gt;&gt;&gt; x, L = symbols(&#39;x,L&#39;, positive=True)</span>
-<span class="sd">            &gt;&gt;&gt; n = symbols(&#39;n&#39;, integer=True)</span>
-<span class="sd">            &gt;&gt;&gt; g = sqrt(2/L)*sin(n*pi*x/L)</span>
-<span class="sd">            &gt;&gt;&gt; f = Wavefunction(g, (x, 0, L))</span>
-<span class="sd">            &gt;&gt;&gt; f.is_normalized</span>
-<span class="sd">            True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">norm</span> <span class="o">==</span> <span class="mf">1.0</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-    <span class="nd">@cacheit</span>
-<div class="viewcode-block" id="Wavefunction.norm"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.Wavefunction.norm">[docs]</a>    <span class="k">def</span> <span class="nf">norm</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the normalization of the specified functional form.</span>
-
-<span class="sd">        This function integrates over the coordinates of the Wavefunction, with</span>
-<span class="sd">        the bounds specified.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy import symbols, pi</span>
-<span class="sd">            &gt;&gt;&gt; from sympy.functions import sqrt, sin</span>
-<span class="sd">            &gt;&gt;&gt; from sympy.physics.quantum.state import Wavefunction</span>
-<span class="sd">            &gt;&gt;&gt; x, L = symbols(&#39;x,L&#39;, positive=True)</span>
-<span class="sd">            &gt;&gt;&gt; n = symbols(&#39;n&#39;, integer=True)</span>
-<span class="sd">            &gt;&gt;&gt; g = sqrt(2/L)*sin(n*pi*x/L)</span>
-<span class="sd">            &gt;&gt;&gt; f = Wavefunction(g, (x, 0, L))</span>
-<span class="sd">            &gt;&gt;&gt; f.norm</span>
-<span class="sd">            1</span>
-<span class="sd">            &gt;&gt;&gt; g = sin(n*pi*x/L)</span>
-<span class="sd">            &gt;&gt;&gt; f = Wavefunction(g, (x, 0, L))</span>
-<span class="sd">            &gt;&gt;&gt; f.norm</span>
-<span class="sd">            sqrt(2)*sqrt(L)/2</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">exp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">*</span><span class="n">conjugate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">)</span>
-        <span class="n">var</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">variables</span>
-        <span class="n">limits</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">limits</span>
-
-        <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">var</span><span class="p">:</span>
-            <span class="n">curr_limits</span> <span class="o">=</span> <span class="n">limits</span><span class="p">[</span><span class="n">v</span><span class="p">]</span>
-            <span class="n">exp</span> <span class="o">=</span> <span class="n">integrate</span><span class="p">(</span><span class="n">exp</span><span class="p">,</span> <span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">curr_limits</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">curr_limits</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-
-        <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">exp</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Wavefunction.normalize"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.Wavefunction.normalize">[docs]</a>    <span class="k">def</span> <span class="nf">normalize</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return a normalized version of the Wavefunction</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy import symbols, pi</span>
-<span class="sd">            &gt;&gt;&gt; from sympy.functions import sqrt, sin</span>
-<span class="sd">            &gt;&gt;&gt; from sympy.physics.quantum.state import Wavefunction</span>
-<span class="sd">            &gt;&gt;&gt; x, L = symbols(&#39;x,L&#39;, real=True)</span>
-<span class="sd">            &gt;&gt;&gt; n = symbols(&#39;n&#39;, integer=True)</span>
-<span class="sd">            &gt;&gt;&gt; g = sin(n*pi*x/L)</span>
-<span class="sd">            &gt;&gt;&gt; f = Wavefunction(g, (x, 0, L))</span>
-<span class="sd">            &gt;&gt;&gt; f.normalize()</span>
-<span class="sd">            Wavefunction(sqrt(2)*sin(pi*n*x/L)/sqrt(L), (x, 0, L))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">const</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">norm</span>
-
-        <span class="k">if</span> <span class="n">const</span> <span class="o">==</span> <span class="n">oo</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;The function is not normalizable!&quot;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Wavefunction</span><span class="p">((</span><span class="n">const</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-</div>
-<div class="viewcode-block" id="Wavefunction.prob"><a class="viewcode-back" href="../../../../modules/physics/quantum/state.html#sympy.physics.quantum.state.Wavefunction.prob">[docs]</a>    <span class="k">def</span> <span class="nf">prob</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the absolute magnitude of the w.f., `|\psi(x)|^2`</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy import symbols, pi</span>
-<span class="sd">            &gt;&gt;&gt; from sympy.functions import sqrt, sin</span>
-<span class="sd">            &gt;&gt;&gt; from sympy.physics.quantum.state import Wavefunction</span>
-<span class="sd">            &gt;&gt;&gt; x, L = symbols(&#39;x,L&#39;, real=True)</span>
-<span class="sd">            &gt;&gt;&gt; n = symbols(&#39;n&#39;, integer=True)</span>
-<span class="sd">            &gt;&gt;&gt; g = sin(n*pi*x/L)</span>
-<span class="sd">            &gt;&gt;&gt; f = Wavefunction(g, (x, 0, L))</span>
-<span class="sd">            &gt;&gt;&gt; f.prob()</span>
-<span class="sd">            Wavefunction(sin(pi*n*x/L)**2, x)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">return</span> <span class="n">Wavefunction</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">*</span><span class="n">conjugate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">),</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span></div></div>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.physics.quantum.tensorproduct</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Abstract tensor product.&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">Pow</span><span class="p">,</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.pretty.stringpict</span> <span class="kn">import</span> <span class="n">prettyForm</span>
-
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.qexpr</span> <span class="kn">import</span> <span class="n">QuantumError</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.dagger</span> <span class="kn">import</span> <span class="n">Dagger</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.commutator</span> <span class="kn">import</span> <span class="n">Commutator</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.anticommutator</span> <span class="kn">import</span> <span class="n">AntiCommutator</span>
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.matrixutils</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">numpy_ndarray</span><span class="p">,</span>
-    <span class="n">scipy_sparse_matrix</span><span class="p">,</span>
-    <span class="n">matrix_tensor_product</span>
-<span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.core.trace</span> <span class="kn">import</span> <span class="n">Tr</span>
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="s">&#39;TensorProduct&#39;</span><span class="p">,</span>
-    <span class="s">&#39;tensor_product_simp&#39;</span>
-<span class="p">]</span>
-
-<span class="c">#-----------------------------------------------------------------------------</span>
-<span class="c"># Tensor product</span>
-<span class="c">#-----------------------------------------------------------------------------</span>
-
-
-<div class="viewcode-block" id="TensorProduct"><a class="viewcode-back" href="../../../../modules/physics/quantum/tensorproduct.html#sympy.physics.quantum.tensorproduct.TensorProduct">[docs]</a><span class="k">class</span> <span class="nc">TensorProduct</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The tensor product of two or more arguments.</span>
-
-<span class="sd">    For matrices, this uses ``matrix_tensor_product`` to compute the Kronecker</span>
-<span class="sd">    or tensor product matrix. For other objects a symbolic ``TensorProduct``</span>
-<span class="sd">    instance is returned. The tensor product is a non-commutative</span>
-<span class="sd">    multiplication that is used primarily with operators and states in quantum</span>
-<span class="sd">    mechanics.</span>
-
-<span class="sd">    Currently, the tensor product distinguishes between commutative and non-</span>
-<span class="sd">    commutative arguments.  Commutative arguments are assumed to be scalars and</span>
-<span class="sd">    are pulled out in front of the ``TensorProduct``. Non-commutative arguments</span>
-<span class="sd">    remain in the resulting ``TensorProduct``.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    args : tuple</span>
-<span class="sd">        A sequence of the objects to take the tensor product of.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Start with a simple tensor product of sympy matrices::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import I, Matrix, symbols</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum import TensorProduct</span>
-
-<span class="sd">        &gt;&gt;&gt; m1 = Matrix([[1,2],[3,4]])</span>
-<span class="sd">        &gt;&gt;&gt; m2 = Matrix([[1,0],[0,1]])</span>
-<span class="sd">        &gt;&gt;&gt; TensorProduct(m1, m2)</span>
-<span class="sd">        [1, 0, 2, 0]</span>
-<span class="sd">        [0, 1, 0, 2]</span>
-<span class="sd">        [3, 0, 4, 0]</span>
-<span class="sd">        [0, 3, 0, 4]</span>
-<span class="sd">        &gt;&gt;&gt; TensorProduct(m2, m1)</span>
-<span class="sd">        [1, 2, 0, 0]</span>
-<span class="sd">        [3, 4, 0, 0]</span>
-<span class="sd">        [0, 0, 1, 2]</span>
-<span class="sd">        [0, 0, 3, 4]</span>
-
-<span class="sd">    We can also construct tensor products of non-commutative symbols:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; A = Symbol(&#39;A&#39;,commutative=False)</span>
-<span class="sd">        &gt;&gt;&gt; B = Symbol(&#39;B&#39;,commutative=False)</span>
-<span class="sd">        &gt;&gt;&gt; tp = TensorProduct(A, B)</span>
-<span class="sd">        &gt;&gt;&gt; tp</span>
-<span class="sd">        AxB</span>
-
-<span class="sd">    We can take the dagger of a tensor product (note the order does NOT reverse</span>
-<span class="sd">    like the dagger of a normal product):</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum import Dagger</span>
-<span class="sd">        &gt;&gt;&gt; Dagger(tp)</span>
-<span class="sd">        Dagger(A)xDagger(B)</span>
-
-<span class="sd">    Expand can be used to distribute a tensor product across addition:</span>
-
-<span class="sd">        &gt;&gt;&gt; C = Symbol(&#39;C&#39;,commutative=False)</span>
-<span class="sd">        &gt;&gt;&gt; tp = TensorProduct(A+B,C)</span>
-<span class="sd">        &gt;&gt;&gt; tp</span>
-<span class="sd">        (A + B)xC</span>
-<span class="sd">        &gt;&gt;&gt; tp.expand(tensorproduct=True)</span>
-<span class="sd">        AxC + BxC</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="p">(</span><span class="n">Matrix</span><span class="p">,</span> <span class="n">numpy_ndarray</span><span class="p">,</span> <span class="n">scipy_sparse_matrix</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="n">matrix_tensor_product</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">c_part</span><span class="p">,</span> <span class="n">new_args</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">flatten</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">args</span><span class="p">))</span>
-        <span class="n">c_part</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">c_part</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">new_args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">c_part</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">new_args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">c_part</span><span class="o">*</span><span class="n">new_args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tp</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">new_args</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">c_part</span><span class="o">*</span><span class="n">tp</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">flatten</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="c"># TODO: disallow nested TensorProducts.</span>
-        <span class="n">c_part</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">nc_parts</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="n">cp</span><span class="p">,</span> <span class="n">ncp</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-            <span class="n">c_part</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">cp</span><span class="p">))</span>
-            <span class="n">nc_parts</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Mul</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">ncp</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">c_part</span><span class="p">,</span> <span class="n">nc_parts</span>
-
-    <span class="k">def</span> <span class="nf">_eval_adjoint</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">TensorProduct</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">Dagger</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_rewrite</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">pattern</span><span class="p">,</span> <span class="n">rule</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="n">sargs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="n">terms</span> <span class="o">=</span> <span class="p">[</span> <span class="n">t</span><span class="o">.</span><span class="n">_eval_rewrite</span><span class="p">(</span><span class="n">pattern</span><span class="p">,</span> <span class="n">rule</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">sargs</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">TensorProduct</span><span class="p">(</span><span class="o">*</span><span class="n">terms</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">tensorproduct</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sympystr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.printing.str</span> <span class="kn">import</span> <span class="n">sstr</span>
-        <span class="n">length</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">length</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="p">(</span><span class="n">Add</span><span class="p">,</span> <span class="n">Pow</span><span class="p">,</span> <span class="n">Mul</span><span class="p">)):</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">s</span> <span class="o">+</span> <span class="s">&#39;(&#39;</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">s</span> <span class="o">+</span> <span class="n">sstr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="p">(</span><span class="n">Add</span><span class="p">,</span> <span class="n">Pow</span><span class="p">,</span> <span class="n">Mul</span><span class="p">)):</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">s</span> <span class="o">+</span> <span class="s">&#39;)&#39;</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="n">length</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">s</span> <span class="o">+</span> <span class="s">&#39;x&#39;</span>
-        <span class="k">return</span> <span class="n">s</span>
-
-    <span class="k">def</span> <span class="nf">_pretty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">length</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="s">&#39;&#39;</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">length</span><span class="p">):</span>
-            <span class="n">next_pform</span> <span class="o">=</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="p">(</span><span class="n">Add</span><span class="p">,</span> <span class="n">Mul</span><span class="p">)):</span>
-                <span class="n">next_pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span>
-                    <span class="o">*</span><span class="n">next_pform</span><span class="o">.</span><span class="n">parens</span><span class="p">(</span><span class="n">left</span><span class="o">=</span><span class="s">&#39;(&#39;</span><span class="p">,</span> <span class="n">right</span><span class="o">=</span><span class="s">&#39;)&#39;</span><span class="p">)</span>
-                <span class="p">)</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">next_pform</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="n">length</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">printer</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-                    <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\u2a02</span><span class="s">&#39;</span> <span class="o">+</span> <span class="s">&#39; &#39;</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39;x&#39;</span> <span class="o">+</span> <span class="s">&#39; &#39;</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">length</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">length</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="p">(</span><span class="n">Add</span><span class="p">,</span> <span class="n">Mul</span><span class="p">)):</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">s</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">left(&#39;</span>
-            <span class="c"># The extra {} brackets are needed to get matplotlib&#39;s latex</span>
-            <span class="c"># rendered to render this properly.</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">s</span> <span class="o">+</span> <span class="s">&#39;{&#39;</span> <span class="o">+</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39;}&#39;</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="p">(</span><span class="n">Add</span><span class="p">,</span> <span class="n">Mul</span><span class="p">)):</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">s</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">right)&#39;</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="n">length</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">s</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">otimes &#39;</span>
-        <span class="k">return</span> <span class="n">s</span>
-
-    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">TensorProduct</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">item</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span> <span class="k">for</span> <span class="n">item</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_expand_tensorproduct</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Distribute TensorProducts across addition.&quot;&quot;&quot;</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="n">add_args</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">stop</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">Add</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">aa</span> <span class="ow">in</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                    <span class="n">tp</span> <span class="o">=</span> <span class="n">TensorProduct</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="p">(</span><span class="n">aa</span><span class="p">,)</span> <span class="o">+</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:])</span>
-                    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">tp</span><span class="p">,</span> <span class="n">TensorProduct</span><span class="p">):</span>
-                        <span class="n">tp</span> <span class="o">=</span> <span class="n">tp</span><span class="o">.</span><span class="n">_eval_expand_tensorproduct</span><span class="p">()</span>
-                    <span class="n">add_args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tp</span><span class="p">)</span>
-                <span class="k">break</span>
-
-        <span class="k">if</span> <span class="n">add_args</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">add_args</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="k">def</span> <span class="nf">_eval_trace</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">indices</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;indices&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-        <span class="n">exp</span> <span class="o">=</span> <span class="n">tensor_product_simp</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">indices</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">indices</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">Tr</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">exp</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">Tr</span><span class="p">(</span><span class="n">value</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span> <span class="k">if</span> <span class="n">idx</span> <span class="ow">in</span> <span class="n">indices</span> <span class="k">else</span> <span class="n">value</span>
-                        <span class="k">for</span> <span class="n">idx</span><span class="p">,</span> <span class="n">value</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">exp</span><span class="o">.</span><span class="n">args</span><span class="p">)])</span>
-
-</div>
-<span class="k">def</span> <span class="nf">tensor_product_simp_Mul</span><span class="p">(</span><span class="n">e</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Simplify a Mul with TensorProducts.</span>
-
-<span class="sd">    Current the main use of this is to simplify a ``Mul`` of ``TensorProduct``s</span>
-<span class="sd">    to a ``TensorProduct`` of ``Muls``. It currently only works for relatively</span>
-<span class="sd">    simple cases where the initial ``Mul`` only has scalars and raw</span>
-<span class="sd">    ``TensorProduct``s, not ``Add``, ``Pow``, ``Commutator``s of</span>
-<span class="sd">    ``TensorProduct``s.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    e : Expr</span>
-<span class="sd">        A ``Mul`` of ``TensorProduct``s to be simplified.</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    e : Expr</span>
-<span class="sd">        A ``TensorProduct`` of ``Mul``s.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    This is an example of the type of simplification that this function</span>
-<span class="sd">    performs::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.quantum.tensorproduct import \</span>
-<span class="sd">                    tensor_product_simp_Mul, TensorProduct</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; A = Symbol(&#39;A&#39;,commutative=False)</span>
-<span class="sd">        &gt;&gt;&gt; B = Symbol(&#39;B&#39;,commutative=False)</span>
-<span class="sd">        &gt;&gt;&gt; C = Symbol(&#39;C&#39;,commutative=False)</span>
-<span class="sd">        &gt;&gt;&gt; D = Symbol(&#39;D&#39;,commutative=False)</span>
-<span class="sd">        &gt;&gt;&gt; e = TensorProduct(A,B)*TensorProduct(C,D)</span>
-<span class="sd">        &gt;&gt;&gt; e</span>
-<span class="sd">        AxB*CxD</span>
-<span class="sd">        &gt;&gt;&gt; tensor_product_simp_Mul(e)</span>
-<span class="sd">        (A*C)x(B*D)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># TODO: This won&#39;t work with Muls that have other composites of</span>
-    <span class="c"># TensorProducts, like an Add, Pow, Commutator, etc.</span>
-    <span class="c"># TODO: This only works for the equivalent of single Qbit gates.</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Mul</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">e</span>
-    <span class="n">c_part</span><span class="p">,</span> <span class="n">nc_part</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-    <span class="n">n_nc</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">nc_part</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n_nc</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">n_nc</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">e</span>
-    <span class="k">elif</span> <span class="n">e</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">TensorProduct</span><span class="p">):</span>
-        <span class="n">current</span> <span class="o">=</span> <span class="n">nc_part</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">current</span><span class="p">,</span> <span class="n">TensorProduct</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;TensorProduct expected, got: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">current</span><span class="p">)</span>
-        <span class="n">n_terms</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">current</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">new_args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">current</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">for</span> <span class="nb">next</span> <span class="ow">in</span> <span class="n">nc_part</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="c"># TODO: check the hilbert spaces of next and current here.</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="nb">next</span><span class="p">,</span> <span class="n">TensorProduct</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">n_terms</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="nb">next</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-                    <span class="k">raise</span> <span class="n">QuantumError</span><span class="p">(</span>
-                        <span class="s">&#39;TensorProducts of different lengths: </span><span class="si">%r</span><span class="s"> and </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span>
-                        <span class="p">(</span><span class="n">current</span><span class="p">,</span> <span class="nb">next</span><span class="p">)</span>
-                    <span class="p">)</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">new_args</span><span class="p">)):</span>
-                    <span class="n">new_args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">new_args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="nb">next</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># this won&#39;t quite work as we don&#39;t want next in the TensorProduct</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">new_args</span><span class="p">)):</span>
-                    <span class="n">new_args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">new_args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="nb">next</span>
-            <span class="n">current</span> <span class="o">=</span> <span class="nb">next</span>
-        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">c_part</span><span class="p">)</span><span class="o">*</span><span class="n">TensorProduct</span><span class="p">(</span><span class="o">*</span><span class="n">new_args</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">e</span>
-
-
-<div class="viewcode-block" id="tensor_product_simp"><a class="viewcode-back" href="../../../../modules/physics/quantum/tensorproduct.html#sympy.physics.quantum.tensorproduct.tensor_product_simp">[docs]</a><span class="k">def</span> <span class="nf">tensor_product_simp</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Try to simplify and combine TensorProducts.</span>
-
-<span class="sd">    In general this will try to pull expressions inside of ``TensorProducts``.</span>
-<span class="sd">    It currently only works for relatively simple cases where the products have</span>
-<span class="sd">    only scalars, raw ``TensorProducts``, not ``Add``, ``Pow``, ``Commutators``</span>
-<span class="sd">    of ``TensorProducts``. It is best to see what it does by showing examples.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum import tensor_product_simp</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.quantum import TensorProduct</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">    &gt;&gt;&gt; A = Symbol(&#39;A&#39;,commutative=False)</span>
-<span class="sd">    &gt;&gt;&gt; B = Symbol(&#39;B&#39;,commutative=False)</span>
-<span class="sd">    &gt;&gt;&gt; C = Symbol(&#39;C&#39;,commutative=False)</span>
-<span class="sd">    &gt;&gt;&gt; D = Symbol(&#39;D&#39;,commutative=False)</span>
-
-<span class="sd">    First see what happens to products of tensor products:</span>
-
-<span class="sd">    &gt;&gt;&gt; e = TensorProduct(A,B)*TensorProduct(C,D)</span>
-<span class="sd">    &gt;&gt;&gt; e</span>
-<span class="sd">    AxB*CxD</span>
-<span class="sd">    &gt;&gt;&gt; tensor_product_simp(e)</span>
-<span class="sd">    (A*C)x(B*D)</span>
-
-<span class="sd">    This is the core logic of this function, and it works inside, powers, sums,</span>
-<span class="sd">    commutators and anticommutators as well:</span>
-
-<span class="sd">    &gt;&gt;&gt; tensor_product_simp(e**2)</span>
-<span class="sd">    (A*C)x(B*D)**2</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Add</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">tensor_product_simp</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Pow</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">tensor_product_simp</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">base</span><span class="p">)</span><span class="o">**</span><span class="n">e</span><span class="o">.</span><span class="n">exp</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Mul</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">tensor_product_simp_Mul</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Commutator</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Commutator</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">tensor_product_simp</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">AntiCommutator</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">AntiCommutator</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">tensor_product_simp</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">e</span></div>
-</pre></div>
-
-          </div>
-        </div>
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-            
-  <h1>Source code for sympy.physics.secondquant</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Second quantization operators and states for bosons.</span>
-
-<span class="sd">This follow the formulation of Fetter and Welecka, &quot;Quantum Theory</span>
-<span class="sd">of Many-Particle Systems.&quot;</span>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="p">(</span><span class="n">Add</span><span class="p">,</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">cacheit</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">,</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">Function</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span>
-                   <span class="n">KroneckerDelta</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Pow</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">,</span>
-                   <span class="n">zeros</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="nb">reduce</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.str</span> <span class="kn">import</span> <span class="n">StrPrinter</span>
-
-<span class="kn">from</span> <span class="nn">sympy.physics.quantum.qexpr</span> <span class="kn">import</span> <span class="n">split_commutative_parts</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="nb">reduce</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">has_dups</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">default_sort_key</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="s">&#39;Dagger&#39;</span><span class="p">,</span>
-    <span class="s">&#39;KroneckerDelta&#39;</span><span class="p">,</span>
-    <span class="s">&#39;BosonicOperator&#39;</span><span class="p">,</span>
-    <span class="s">&#39;AnnihilateBoson&#39;</span><span class="p">,</span>
-    <span class="s">&#39;CreateBoson&#39;</span><span class="p">,</span>
-    <span class="s">&#39;AnnihilateFermion&#39;</span><span class="p">,</span>
-    <span class="s">&#39;CreateFermion&#39;</span><span class="p">,</span>
-    <span class="s">&#39;FockState&#39;</span><span class="p">,</span>
-    <span class="s">&#39;FockStateBra&#39;</span><span class="p">,</span>
-    <span class="s">&#39;FockStateKet&#39;</span><span class="p">,</span>
-    <span class="s">&#39;FockStateBosonKet&#39;</span><span class="p">,</span>
-    <span class="s">&#39;FockStateBosonBra&#39;</span><span class="p">,</span>
-    <span class="s">&#39;BBra&#39;</span><span class="p">,</span>
-    <span class="s">&#39;BKet&#39;</span><span class="p">,</span>
-    <span class="s">&#39;FBra&#39;</span><span class="p">,</span>
-    <span class="s">&#39;FKet&#39;</span><span class="p">,</span>
-    <span class="s">&#39;F&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Fd&#39;</span><span class="p">,</span>
-    <span class="s">&#39;B&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Bd&#39;</span><span class="p">,</span>
-    <span class="s">&#39;apply_operators&#39;</span><span class="p">,</span>
-    <span class="s">&#39;InnerProduct&#39;</span><span class="p">,</span>
-    <span class="s">&#39;BosonicBasis&#39;</span><span class="p">,</span>
-    <span class="s">&#39;VarBosonicBasis&#39;</span><span class="p">,</span>
-    <span class="s">&#39;FixedBosonicBasis&#39;</span><span class="p">,</span>
-    <span class="s">&#39;Commutator&#39;</span><span class="p">,</span>
-    <span class="s">&#39;matrix_rep&#39;</span><span class="p">,</span>
-    <span class="s">&#39;contraction&#39;</span><span class="p">,</span>
-    <span class="s">&#39;wicks&#39;</span><span class="p">,</span>
-    <span class="s">&#39;NO&#39;</span><span class="p">,</span>
-    <span class="s">&#39;evaluate_deltas&#39;</span><span class="p">,</span>
-    <span class="s">&#39;AntiSymmetricTensor&#39;</span><span class="p">,</span>
-    <span class="s">&#39;substitute_dummies&#39;</span><span class="p">,</span>
-    <span class="s">&#39;PermutationOperator&#39;</span><span class="p">,</span>
-    <span class="s">&#39;simplify_index_permutations&#39;</span><span class="p">,</span>
-<span class="p">]</span>
-
-
-<span class="k">class</span> <span class="nc">SecondQuantizationError</span><span class="p">(</span><span class="ne">Exception</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-
-<span class="k">class</span> <span class="nc">AppliesOnlyToSymbolicIndex</span><span class="p">(</span><span class="n">SecondQuantizationError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-
-<span class="k">class</span> <span class="nc">ContractionAppliesOnlyToFermions</span><span class="p">(</span><span class="n">SecondQuantizationError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-
-<span class="k">class</span> <span class="nc">ViolationOfPauliPrinciple</span><span class="p">(</span><span class="n">SecondQuantizationError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-
-<span class="k">class</span> <span class="nc">SubstitutionOfAmbigousOperatorFailed</span><span class="p">(</span><span class="n">SecondQuantizationError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-
-<span class="k">class</span> <span class="nc">WicksTheoremDoesNotApply</span><span class="p">(</span><span class="n">SecondQuantizationError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-
-<div class="viewcode-block" id="Dagger"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.Dagger">[docs]</a><span class="k">class</span> <span class="nc">Dagger</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Hermitian conjugate of creation/annihilation operators.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import I</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.secondquant import Dagger, B, Bd</span>
-<span class="sd">    &gt;&gt;&gt; Dagger(2*I)</span>
-<span class="sd">    -2*I</span>
-<span class="sd">    &gt;&gt;&gt; Dagger(B(0))</span>
-<span class="sd">    CreateBoson(0)</span>
-<span class="sd">    &gt;&gt;&gt; Dagger(Bd(0))</span>
-<span class="sd">    AnnihilateBoson(0)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">r</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Dagger.eval"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.Dagger.eval">[docs]</a>    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Evaluates the Dagger instance.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import I</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import Dagger, B, Bd</span>
-<span class="sd">        &gt;&gt;&gt; Dagger(2*I)</span>
-<span class="sd">        -2*I</span>
-<span class="sd">        &gt;&gt;&gt; Dagger(B(0))</span>
-<span class="sd">        CreateBoson(0)</span>
-<span class="sd">        &gt;&gt;&gt; Dagger(Bd(0))</span>
-<span class="sd">        AnnihilateBoson(0)</span>
-
-<span class="sd">        The eval() method is called automatically.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">_dagger_</span><span class="p">()</span>
-        <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="nb">tuple</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Dagger</span><span class="p">,</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">)))</span>
-                <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="nb">tuple</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Dagger</span><span class="p">,</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">))))</span>
-                <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">arg</span>
-                <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">Pow</span><span class="p">(</span><span class="n">Dagger</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-                <span class="k">if</span> <span class="n">arg</span> <span class="o">==</span> <span class="n">I</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="o">-</span><span class="n">arg</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">d</span>
-</div>
-    <span class="k">def</span> <span class="nf">_dagger_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-</div>
-<span class="k">class</span> <span class="nc">TensorSymbol</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">True</span>
-
-
-<div class="viewcode-block" id="AntiSymmetricTensor"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.AntiSymmetricTensor">[docs]</a><span class="k">class</span> <span class="nc">AntiSymmetricTensor</span><span class="p">(</span><span class="n">TensorSymbol</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Stores upper and lower indices in separate Tuple&#39;s.</span>
-
-<span class="sd">    Each group of indices is assumed to be antisymmetric.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.secondquant import AntiSymmetricTensor</span>
-<span class="sd">    &gt;&gt;&gt; i, j = symbols(&#39;i j&#39;, below_fermi=True)</span>
-<span class="sd">    &gt;&gt;&gt; a, b = symbols(&#39;a b&#39;, above_fermi=True)</span>
-<span class="sd">    &gt;&gt;&gt; AntiSymmetricTensor(&#39;v&#39;, (a, i), (b, j))</span>
-<span class="sd">    AntiSymmetricTensor(v, (a, i), (b, j))</span>
-<span class="sd">    &gt;&gt;&gt; AntiSymmetricTensor(&#39;v&#39;, (i, a), (b, j))</span>
-<span class="sd">    -AntiSymmetricTensor(v, (a, i), (b, j))</span>
-
-<span class="sd">    As you can see, the indices are automatically sorted to a canonical form.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">3</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">symbol</span><span class="p">,</span> <span class="n">upper</span><span class="p">,</span> <span class="n">lower</span><span class="p">):</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">upper</span><span class="p">,</span> <span class="n">signu</span> <span class="o">=</span> <span class="n">_sort_anticommuting_fermions</span><span class="p">(</span>
-                <span class="n">upper</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">cls</span><span class="o">.</span><span class="n">_sortkey</span><span class="p">)</span>
-            <span class="n">lower</span><span class="p">,</span> <span class="n">signl</span> <span class="o">=</span> <span class="n">_sort_anticommuting_fermions</span><span class="p">(</span>
-                <span class="n">lower</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">cls</span><span class="o">.</span><span class="n">_sortkey</span><span class="p">)</span>
-
-        <span class="k">except</span> <span class="n">ViolationOfPauliPrinciple</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-        <span class="n">symbol</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">symbol</span><span class="p">)</span>
-        <span class="n">upper</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">upper</span><span class="p">)</span>
-        <span class="n">lower</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">lower</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="p">(</span><span class="n">signu</span> <span class="o">+</span> <span class="n">signl</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">TensorSymbol</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">symbol</span><span class="p">,</span> <span class="n">upper</span><span class="p">,</span> <span class="n">lower</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-
-            <span class="k">return</span> <span class="n">TensorSymbol</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">symbol</span><span class="p">,</span> <span class="n">upper</span><span class="p">,</span> <span class="n">lower</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_sortkey</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">index</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Key for sorting of indices.</span>
-
-<span class="sd">        particle &lt; hole &lt; general</span>
-
-<span class="sd">        FIXME: This is a bottle-neck, can we do it faster?</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="nb">hash</span><span class="p">(</span><span class="n">index</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">index</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">index</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;above_fermi&#39;</span><span class="p">):</span>
-                <span class="k">return</span> <span class="p">(</span><span class="mi">20</span><span class="p">,</span> <span class="n">h</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">index</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;below_fermi&#39;</span><span class="p">):</span>
-                <span class="k">return</span> <span class="p">(</span><span class="mi">21</span><span class="p">,</span> <span class="n">h</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="mi">22</span><span class="p">,</span> <span class="n">h</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">index</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;above_fermi&#39;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="n">h</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">index</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;below_fermi&#39;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="n">h</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="n">h</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}_{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">symbol</span><span class="p">,</span>
-            <span class="s">&quot;&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span> <span class="n">i</span><span class="o">.</span><span class="n">name</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]]),</span>
-            <span class="s">&quot;&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span> <span class="n">i</span><span class="o">.</span><span class="n">name</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]])</span>
-        <span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="AntiSymmetricTensor.symbol"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.AntiSymmetricTensor.symbol">[docs]</a>    <span class="k">def</span> <span class="nf">symbol</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the symbol of the tensor.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import AntiSymmetricTensor</span>
-<span class="sd">        &gt;&gt;&gt; i, j = symbols(&#39;i,j&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; a, b = symbols(&#39;a,b&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; AntiSymmetricTensor(&#39;v&#39;, (a, i), (b, j))</span>
-<span class="sd">        AntiSymmetricTensor(v, (a, i), (b, j))</span>
-<span class="sd">        &gt;&gt;&gt; AntiSymmetricTensor(&#39;v&#39;, (a, i), (b, j)).symbol</span>
-<span class="sd">        v</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="AntiSymmetricTensor.upper"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.AntiSymmetricTensor.upper">[docs]</a>    <span class="k">def</span> <span class="nf">upper</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the upper indices.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import AntiSymmetricTensor</span>
-<span class="sd">        &gt;&gt;&gt; i, j = symbols(&#39;i,j&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; a, b = symbols(&#39;a,b&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; AntiSymmetricTensor(&#39;v&#39;, (a, i), (b, j))</span>
-<span class="sd">        AntiSymmetricTensor(v, (a, i), (b, j))</span>
-<span class="sd">        &gt;&gt;&gt; AntiSymmetricTensor(&#39;v&#39;, (a, i), (b, j)).upper</span>
-<span class="sd">        (a, i)</span>
-
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="AntiSymmetricTensor.lower"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.AntiSymmetricTensor.lower">[docs]</a>    <span class="k">def</span> <span class="nf">lower</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the lower indices.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import AntiSymmetricTensor</span>
-<span class="sd">        &gt;&gt;&gt; i, j = symbols(&#39;i,j&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; a, b = symbols(&#39;a,b&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; AntiSymmetricTensor(&#39;v&#39;, (a, i), (b, j))</span>
-<span class="sd">        AntiSymmetricTensor(v, (a, i), (b, j))</span>
-<span class="sd">        &gt;&gt;&gt; AntiSymmetricTensor(&#39;v&#39;, (a, i), (b, j)).lower</span>
-<span class="sd">        (b, j)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-</div>
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-
-<div class="viewcode-block" id="AntiSymmetricTensor.doit"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.AntiSymmetricTensor.doit">[docs]</a>    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns self.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import AntiSymmetricTensor</span>
-<span class="sd">        &gt;&gt;&gt; i, j = symbols(&#39;i,j&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; a, b = symbols(&#39;a,b&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; AntiSymmetricTensor(&#39;v&#39;, (a, i), (b, j)).doit()</span>
-<span class="sd">        AntiSymmetricTensor(v, (a, i), (b, j))</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-</div></div>
-<span class="k">class</span> <span class="nc">SqOperator</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Base class for Second Quantization operators.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">op_symbol</span> <span class="o">=</span> <span class="s">&#39;sq&#39;</span>
-
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">sympify</span><span class="p">(</span><span class="n">k</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">state</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the state index related to this operator.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import F, Fd, B, Bd</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; F(p).state</span>
-<span class="sd">        p</span>
-<span class="sd">        &gt;&gt;&gt; Fd(p).state</span>
-<span class="sd">        p</span>
-<span class="sd">        &gt;&gt;&gt; B(p).state</span>
-<span class="sd">        p</span>
-<span class="sd">        &gt;&gt;&gt; Bd(p).state</span>
-<span class="sd">        p</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_symbolic</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns True if the state is a symbol (as opposed to a number).</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import F</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; F(p).is_symbolic</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; F(1).is_symbolic</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">state</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        FIXME: hack to prevent crash further up...</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">(</span><span class="si">%r</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">op_symbol</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">state</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">apply_operator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Applies an operator to itself.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;implement apply_operator in a subclass&#39;</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">BosonicOperator</span><span class="p">(</span><span class="n">SqOperator</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-
-<span class="k">class</span> <span class="nc">Annihilator</span><span class="p">(</span><span class="n">SqOperator</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-
-<span class="k">class</span> <span class="nc">Creator</span><span class="p">(</span><span class="n">SqOperator</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-
-<div class="viewcode-block" id="AnnihilateBoson"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.AnnihilateBoson">[docs]</a><span class="k">class</span> <span class="nc">AnnihilateBoson</span><span class="p">(</span><span class="n">BosonicOperator</span><span class="p">,</span> <span class="n">Annihilator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Bosonic annihilation operator.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.secondquant import B</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; B(x)</span>
-<span class="sd">    AnnihilateBoson(x)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">op_symbol</span> <span class="o">=</span> <span class="s">&#39;b&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_dagger_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">CreateBoson</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">state</span><span class="p">)</span>
-
-<div class="viewcode-block" id="AnnihilateBoson.apply_operator"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.AnnihilateBoson.apply_operator">[docs]</a>    <span class="k">def</span> <span class="nf">apply_operator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Apply state to self if self is not symbolic and state is a FockStateKet, else</span>
-<span class="sd">        multiply self by state.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import B, BKet</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y, n</span>
-<span class="sd">        &gt;&gt;&gt; B(x).apply_operator(y)</span>
-<span class="sd">        y*AnnihilateBoson(x)</span>
-<span class="sd">        &gt;&gt;&gt; B(0).apply_operator(BKet((n,)))</span>
-<span class="sd">        sqrt(n)*FockStateBosonKet((n - 1,))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_symbolic</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">FockStateKet</span><span class="p">):</span>
-            <span class="n">element</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">state</span>
-            <span class="n">amp</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">state</span><span class="p">[</span><span class="n">element</span><span class="p">])</span>
-            <span class="k">return</span> <span class="n">amp</span><span class="o">*</span><span class="n">state</span><span class="o">.</span><span class="n">down</span><span class="p">(</span><span class="n">element</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;AnnihilateBoson(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">state</span>
-
-</div>
-<div class="viewcode-block" id="CreateBoson"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.CreateBoson">[docs]</a><span class="k">class</span> <span class="nc">CreateBoson</span><span class="p">(</span><span class="n">BosonicOperator</span><span class="p">,</span> <span class="n">Creator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Bosonic creation operator.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">op_symbol</span> <span class="o">=</span> <span class="s">&#39;b+&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_dagger_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">AnnihilateBoson</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">state</span><span class="p">)</span>
-
-<div class="viewcode-block" id="CreateBoson.apply_operator"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.CreateBoson.apply_operator">[docs]</a>    <span class="k">def</span> <span class="nf">apply_operator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Apply state to self if self is not symbolic and state is a FockStateKet, else</span>
-<span class="sd">        multiply self by state.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import B, Dagger, BKet</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y, n</span>
-<span class="sd">        &gt;&gt;&gt; Dagger(B(x)).apply_operator(y)</span>
-<span class="sd">        y*CreateBoson(x)</span>
-<span class="sd">        &gt;&gt;&gt; B(0).apply_operator(BKet((n,)))</span>
-<span class="sd">        sqrt(n)*FockStateBosonKet((n - 1,))</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_symbolic</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">FockStateKet</span><span class="p">):</span>
-            <span class="n">element</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">state</span>
-            <span class="n">amp</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">state</span><span class="p">[</span><span class="n">element</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">amp</span><span class="o">*</span><span class="n">state</span><span class="o">.</span><span class="n">up</span><span class="p">(</span><span class="n">element</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;CreateBoson(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">state</span>
-</div>
-<span class="n">B</span> <span class="o">=</span> <span class="n">AnnihilateBoson</span>
-<span class="n">Bd</span> <span class="o">=</span> <span class="n">CreateBoson</span>
-
-
-<span class="k">class</span> <span class="nc">FermionicOperator</span><span class="p">(</span><span class="n">SqOperator</span><span class="p">):</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_restricted</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Is this FermionicOperator restricted with respect to fermi level?</span>
-
-<span class="sd">        Return values:</span>
-<span class="sd">        1  : restricted to orbits above fermi</span>
-<span class="sd">        0  : no restriction</span>
-<span class="sd">        -1 : restricted to orbits below fermi</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import F, Fd</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = Symbol(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; F(a).is_restricted</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; Fd(a).is_restricted</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; F(i).is_restricted</span>
-<span class="sd">        -1</span>
-<span class="sd">        &gt;&gt;&gt; Fd(i).is_restricted</span>
-<span class="sd">        -1</span>
-<span class="sd">        &gt;&gt;&gt; F(p).is_restricted</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; Fd(p).is_restricted</span>
-<span class="sd">        0</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">ass</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">assumptions0</span>
-        <span class="k">if</span> <span class="n">ass</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;below_fermi&quot;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-        <span class="k">if</span> <span class="n">ass</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;above_fermi&quot;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="mi">0</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_above_fermi</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Does the index of this FermionicOperator allow values above fermi?</span>
-
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import F</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = Symbol(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; F(a).is_above_fermi</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; F(i).is_above_fermi</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; F(p).is_above_fermi</span>
-<span class="sd">        True</span>
-
-<span class="sd">        The same applies to creation operators Fd</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;below_fermi&quot;</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_below_fermi</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Does the index of this FermionicOperator allow values below fermi?</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import F</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = Symbol(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; F(a).is_below_fermi</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; F(i).is_below_fermi</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; F(p).is_below_fermi</span>
-<span class="sd">        True</span>
-
-<span class="sd">        The same applies to creation operators Fd</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;above_fermi&quot;</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_only_below_fermi</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Is the index of this FermionicOperator restricted to values below fermi?</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import F</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = Symbol(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; F(a).is_only_below_fermi</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; F(i).is_only_below_fermi</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; F(p).is_only_below_fermi</span>
-<span class="sd">        False</span>
-
-<span class="sd">        The same applies to creation operators Fd</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_below_fermi</span> <span class="ow">and</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_above_fermi</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_only_above_fermi</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Is the index of this FermionicOperator restricted to values above fermi?</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import F</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = Symbol(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; F(a).is_only_above_fermi</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; F(i).is_only_above_fermi</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; F(p).is_only_above_fermi</span>
-<span class="sd">        False</span>
-
-<span class="sd">        The same applies to creation operators Fd</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_above_fermi</span> <span class="ow">and</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_below_fermi</span>
-
-    <span class="k">def</span> <span class="nf">_sortkey</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="nb">hash</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="n">label</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_only_q_creator</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">1</span><span class="p">,</span> <span class="n">label</span><span class="p">,</span> <span class="n">h</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_only_q_annihilator</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">4</span><span class="p">,</span> <span class="n">label</span><span class="p">,</span> <span class="n">h</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Annihilator</span><span class="p">):</span>
-            <span class="k">return</span> <span class="mi">3</span><span class="p">,</span> <span class="n">label</span><span class="p">,</span> <span class="n">h</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Creator</span><span class="p">):</span>
-            <span class="k">return</span> <span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="p">,</span> <span class="n">h</span>
-
-
-<div class="viewcode-block" id="AnnihilateFermion"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.AnnihilateFermion">[docs]</a><span class="k">class</span> <span class="nc">AnnihilateFermion</span><span class="p">(</span><span class="n">FermionicOperator</span><span class="p">,</span> <span class="n">Annihilator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Fermionic annihilation operator.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">op_symbol</span> <span class="o">=</span> <span class="s">&#39;f&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_dagger_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">CreateFermion</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">state</span><span class="p">)</span>
-
-<div class="viewcode-block" id="AnnihilateFermion.apply_operator"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.AnnihilateFermion.apply_operator">[docs]</a>    <span class="k">def</span> <span class="nf">apply_operator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Apply state to self if self is not symbolic and state is a FockStateKet, else</span>
-<span class="sd">        multiply self by state.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import B, Dagger, BKet</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y, n</span>
-<span class="sd">        &gt;&gt;&gt; Dagger(B(x)).apply_operator(y)</span>
-<span class="sd">        y*CreateBoson(x)</span>
-<span class="sd">        &gt;&gt;&gt; B(0).apply_operator(BKet((n,)))</span>
-<span class="sd">        sqrt(n)*FockStateBosonKet((n - 1,))</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">FockStateFermionKet</span><span class="p">):</span>
-            <span class="n">element</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">state</span>
-            <span class="k">return</span> <span class="n">state</span><span class="o">.</span><span class="n">down</span><span class="p">(</span><span class="n">element</span><span class="p">)</span>
-
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">Mul</span><span class="p">):</span>
-            <span class="n">c_part</span><span class="p">,</span> <span class="n">nc_part</span> <span class="o">=</span> <span class="n">state</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">nc_part</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">FockStateFermionKet</span><span class="p">):</span>
-                <span class="n">element</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">state</span>
-                <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">c_part</span> <span class="o">+</span> <span class="p">[</span><span class="n">nc_part</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">down</span><span class="p">(</span><span class="n">element</span><span class="p">)]</span> <span class="o">+</span> <span class="n">nc_part</span><span class="p">[</span><span class="mi">1</span><span class="p">:]))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">)</span>
-
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="AnnihilateFermion.is_q_creator"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.AnnihilateFermion.is_q_creator">[docs]</a>    <span class="k">def</span> <span class="nf">is_q_creator</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Can we create a quasi-particle?  (create hole or create particle)</span>
-<span class="sd">        If so, would that be above or below the fermi surface?</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import F</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = Symbol(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; F(a).is_q_creator</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; F(i).is_q_creator</span>
-<span class="sd">        -1</span>
-<span class="sd">        &gt;&gt;&gt; F(p).is_q_creator</span>
-<span class="sd">        -1</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_below_fermi</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-        <span class="k">return</span> <span class="mi">0</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="AnnihilateFermion.is_q_annihilator"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.AnnihilateFermion.is_q_annihilator">[docs]</a>    <span class="k">def</span> <span class="nf">is_q_annihilator</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Can we destroy a quasi-particle?  (annihilate hole or annihilate particle)</span>
-<span class="sd">        If so, would that be above or below the fermi surface?</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import F</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, above_fermi=1)</span>
-<span class="sd">        &gt;&gt;&gt; i = Symbol(&#39;i&#39;, below_fermi=1)</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; F(a).is_q_annihilator</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; F(i).is_q_annihilator</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; F(p).is_q_annihilator</span>
-<span class="sd">        1</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_above_fermi</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="mi">0</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="AnnihilateFermion.is_only_q_creator"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.AnnihilateFermion.is_only_q_creator">[docs]</a>    <span class="k">def</span> <span class="nf">is_only_q_creator</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Always create a quasi-particle?  (create hole or create particle)</span>
-
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import F</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = Symbol(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; F(a).is_only_q_creator</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; F(i).is_only_q_creator</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; F(p).is_only_q_creator</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_only_below_fermi</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="AnnihilateFermion.is_only_q_annihilator"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.AnnihilateFermion.is_only_q_annihilator">[docs]</a>    <span class="k">def</span> <span class="nf">is_only_q_annihilator</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Always destroy a quasi-particle?  (annihilate hole or annihilate particle)</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import F</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = Symbol(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; F(a).is_only_q_annihilator</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; F(i).is_only_q_annihilator</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; F(p).is_only_q_annihilator</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_only_above_fermi</span>
-</div>
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;AnnihilateFermion(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">state</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;a_{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">state</span><span class="o">.</span><span class="n">name</span>
-
-</div>
-<div class="viewcode-block" id="CreateFermion"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.CreateFermion">[docs]</a><span class="k">class</span> <span class="nc">CreateFermion</span><span class="p">(</span><span class="n">FermionicOperator</span><span class="p">,</span> <span class="n">Creator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Fermionic creation operator.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">op_symbol</span> <span class="o">=</span> <span class="s">&#39;f+&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_dagger_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">AnnihilateFermion</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">state</span><span class="p">)</span>
-
-<div class="viewcode-block" id="CreateFermion.apply_operator"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.CreateFermion.apply_operator">[docs]</a>    <span class="k">def</span> <span class="nf">apply_operator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Apply state to self if self is not symbolic and state is a FockStateKet, else</span>
-<span class="sd">        multiply self by state.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import B, Dagger, BKet</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y, n</span>
-<span class="sd">        &gt;&gt;&gt; Dagger(B(x)).apply_operator(y)</span>
-<span class="sd">        y*CreateBoson(x)</span>
-<span class="sd">        &gt;&gt;&gt; B(0).apply_operator(BKet((n,)))</span>
-<span class="sd">        sqrt(n)*FockStateBosonKet((n - 1,))</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">FockStateFermionKet</span><span class="p">):</span>
-            <span class="n">element</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">state</span>
-            <span class="k">return</span> <span class="n">state</span><span class="o">.</span><span class="n">up</span><span class="p">(</span><span class="n">element</span><span class="p">)</span>
-
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">Mul</span><span class="p">):</span>
-            <span class="n">c_part</span><span class="p">,</span> <span class="n">nc_part</span> <span class="o">=</span> <span class="n">state</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">nc_part</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">FockStateFermionKet</span><span class="p">):</span>
-                <span class="n">element</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">state</span>
-                <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">c_part</span> <span class="o">+</span> <span class="p">[</span><span class="n">nc_part</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">up</span><span class="p">(</span><span class="n">element</span><span class="p">)]</span> <span class="o">+</span> <span class="n">nc_part</span><span class="p">[</span><span class="mi">1</span><span class="p">:]))</span>
-
-        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="CreateFermion.is_q_creator"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.CreateFermion.is_q_creator">[docs]</a>    <span class="k">def</span> <span class="nf">is_q_creator</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Can we create a quasi-particle?  (create hole or create particle)</span>
-<span class="sd">        If so, would that be above or below the fermi surface?</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import Fd</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = Symbol(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; Fd(a).is_q_creator</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; Fd(i).is_q_creator</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; Fd(p).is_q_creator</span>
-<span class="sd">        1</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_above_fermi</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="mi">0</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="CreateFermion.is_q_annihilator"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.CreateFermion.is_q_annihilator">[docs]</a>    <span class="k">def</span> <span class="nf">is_q_annihilator</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Can we destroy a quasi-particle?  (annihilate hole or annihilate particle)</span>
-<span class="sd">        If so, would that be above or below the fermi surface?</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import Fd</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, above_fermi=1)</span>
-<span class="sd">        &gt;&gt;&gt; i = Symbol(&#39;i&#39;, below_fermi=1)</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; Fd(a).is_q_annihilator</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; Fd(i).is_q_annihilator</span>
-<span class="sd">        -1</span>
-<span class="sd">        &gt;&gt;&gt; Fd(p).is_q_annihilator</span>
-<span class="sd">        -1</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_below_fermi</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-        <span class="k">return</span> <span class="mi">0</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="CreateFermion.is_only_q_creator"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.CreateFermion.is_only_q_creator">[docs]</a>    <span class="k">def</span> <span class="nf">is_only_q_creator</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Always create a quasi-particle?  (create hole or create particle)</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import Fd</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = Symbol(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; Fd(a).is_only_q_creator</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Fd(i).is_only_q_creator</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Fd(p).is_only_q_creator</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_only_above_fermi</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="CreateFermion.is_only_q_annihilator"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.CreateFermion.is_only_q_annihilator">[docs]</a>    <span class="k">def</span> <span class="nf">is_only_q_annihilator</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Always destroy a quasi-particle?  (annihilate hole or annihilate particle)</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import Fd</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = Symbol(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; Fd(a).is_only_q_annihilator</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Fd(i).is_only_q_annihilator</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Fd(p).is_only_q_annihilator</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_only_below_fermi</span>
-</div>
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;CreateFermion(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">state</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;a^</span><span class="se">\\</span><span class="s">dagger_{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">state</span><span class="o">.</span><span class="n">name</span>
-</div>
-<span class="n">Fd</span> <span class="o">=</span> <span class="n">CreateFermion</span>
-<span class="n">F</span> <span class="o">=</span> <span class="n">AnnihilateFermion</span>
-
-
-<div class="viewcode-block" id="FockState"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.FockState">[docs]</a><span class="k">class</span> <span class="nc">FockState</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Many particle Fock state with a sequence of occupation numbers.</span>
-
-<span class="sd">    Anywhere you can have a FockState, you can also have S.Zero.</span>
-<span class="sd">    All code must check for this!</span>
-
-<span class="sd">    Base class to represent FockStates.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">occupations</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        occupations is a list with two possible meanings:</span>
-
-<span class="sd">        - For bosons it is a list of occupation numbers.</span>
-<span class="sd">          Element i is the number of particles in state i.</span>
-
-<span class="sd">        - For fermions it is a list of occupied orbits.</span>
-<span class="sd">          Element 0 is the state that was occupied first, element i</span>
-<span class="sd">          is the i&#39;th occupied state.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">occupations</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="n">occupations</span><span class="p">))</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">occupations</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">i</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="s">&quot;FockState(</span><span class="si">%r</span><span class="s">)&quot;</span><span class="p">)</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s%r%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lbracket</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_labels</span><span class="p">(),</span> <span class="bp">self</span><span class="o">.</span><span class="n">rbracket</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_labels</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__len__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-</div>
-<span class="k">class</span> <span class="nc">BosonState</span><span class="p">(</span><span class="n">FockState</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Base class for FockStateBoson(Ket/Bra).</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">up</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Performs the action of a creation operator.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import BBra</span>
-<span class="sd">        &gt;&gt;&gt; b = BBra([1, 2])</span>
-<span class="sd">        &gt;&gt;&gt; b</span>
-<span class="sd">        FockStateBosonBra((1, 2))</span>
-<span class="sd">        &gt;&gt;&gt; b.up(1)</span>
-<span class="sd">        FockStateBosonBra((1, 3))</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-        <span class="n">new_occs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="n">new_occs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">new_occs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">new_occs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">down</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Performs the action of an annihilation operator.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import BBra</span>
-<span class="sd">        &gt;&gt;&gt; b = BBra([1, 2])</span>
-<span class="sd">        &gt;&gt;&gt; b</span>
-<span class="sd">        FockStateBosonBra((1, 2))</span>
-<span class="sd">        &gt;&gt;&gt; b.down(1)</span>
-<span class="sd">        FockStateBosonBra((1, 1))</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-        <span class="n">new_occs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">new_occs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">new_occs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">new_occs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">new_occs</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">FermionState</span><span class="p">(</span><span class="n">FockState</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Base class for FockStateFermion(Ket/Bra).</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">fermi_level</span> <span class="o">=</span> <span class="mi">0</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">occupations</span><span class="p">,</span> <span class="n">fermi_level</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="n">occupations</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="n">occupations</span><span class="p">))</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">occupations</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="p">(</span><span class="n">occupations</span><span class="p">,</span> <span class="n">sign</span><span class="p">)</span> <span class="o">=</span> <span class="n">_sort_anticommuting_fermions</span><span class="p">(</span>
-                    <span class="n">occupations</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="nb">hash</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">ViolationOfPauliPrinciple</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">sign</span> <span class="o">=</span> <span class="mi">0</span>
-
-        <span class="n">cls</span><span class="o">.</span><span class="n">fermi_level</span> <span class="o">=</span> <span class="n">fermi_level</span>
-
-        <span class="k">if</span> <span class="n">cls</span><span class="o">.</span><span class="n">_count_holes</span><span class="p">(</span><span class="n">occupations</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">fermi_level</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-        <span class="k">if</span> <span class="n">sign</span> <span class="o">%</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">*</span><span class="n">FockState</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">occupations</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">FockState</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">occupations</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">up</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Performs the action of a creation operator.</span>
-
-<span class="sd">        If below fermi we try to remove a hole,</span>
-<span class="sd">        if above fermi we try to create a particle.</span>
-
-<span class="sd">        if general index p we return Kronecker(p,i)*self</span>
-<span class="sd">        where i is a new symbol with restriction above or below.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import FKet</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = Symbol(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; FKet([]).up(a)</span>
-<span class="sd">        FockStateFermionKet((a,))</span>
-
-<span class="sd">        A creator acting on vacuum below fermi vanishes</span>
-<span class="sd">        &gt;&gt;&gt; FKet([]).up(i)</span>
-<span class="sd">        0</span>
-
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">present</span> <span class="o">=</span> <span class="n">i</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_only_above_fermi</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">present</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_add_orbit</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">_only_below_fermi</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">present</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_remove_orbit</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">present</span><span class="p">:</span>
-                <span class="n">hole</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;i&quot;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">hole</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">_remove_orbit</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">particle</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;a&quot;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">particle</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">_add_orbit</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">down</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Performs the action of an annihilation operator.</span>
-
-<span class="sd">        If below fermi we try to create a hole,</span>
-<span class="sd">        if above fermi we try to remove a particle.</span>
-
-<span class="sd">        if general index p we return Kronecker(p,i)*self</span>
-<span class="sd">        where i is a new symbol with restriction above or below.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import FKet</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = Symbol(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; p = Symbol(&#39;p&#39;)</span>
-
-<span class="sd">        An annihilator acting on vacuum above fermi vanishes</span>
-<span class="sd">        &gt;&gt;&gt; FKet([]).down(a)</span>
-<span class="sd">        0</span>
-
-<span class="sd">        Also below fermi, it vanishes, unless we specify a fermi level &gt; 0</span>
-<span class="sd">        &gt;&gt;&gt; FKet([]).down(i)</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; FKet([],4).down(i)</span>
-<span class="sd">        FockStateFermionKet((i,))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">present</span> <span class="o">=</span> <span class="n">i</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_only_above_fermi</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">present</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_remove_orbit</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">_only_below_fermi</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">present</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_add_orbit</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">present</span><span class="p">:</span>
-                <span class="n">hole</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;i&quot;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">hole</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">_add_orbit</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">particle</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;a&quot;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">particle</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">_remove_orbit</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_only_below_fermi</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Tests if given orbit is only below fermi surface.</span>
-
-<span class="sd">        If nothing can be concluded we return a conservative False.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">i</span> <span class="o">&lt;=</span> <span class="n">cls</span><span class="o">.</span><span class="n">fermi_level</span>
-        <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;below_fermi&#39;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_only_above_fermi</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Tests if given orbit is only above fermi surface.</span>
-
-<span class="sd">        If fermi level has not been set we return True.</span>
-<span class="sd">        If nothing can be concluded we return a conservative False.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">i</span> <span class="o">&gt;</span> <span class="n">cls</span><span class="o">.</span><span class="n">fermi_level</span>
-        <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;above_fermi&#39;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="n">cls</span><span class="o">.</span><span class="n">fermi_level</span>
-
-    <span class="k">def</span> <span class="nf">_remove_orbit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Removes particle/fills hole in orbit i. No input tests performed here.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">new_occs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="n">pos</span> <span class="o">=</span> <span class="n">new_occs</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-        <span class="k">del</span> <span class="n">new_occs</span><span class="p">[</span><span class="n">pos</span><span class="p">]</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">pos</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">new_occs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">fermi_level</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">new_occs</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">fermi_level</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_add_orbit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Adds particle/creates hole in orbit i. No input tests performed here.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">((</span><span class="n">i</span><span class="p">,)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">fermi_level</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_count_holes</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="nb">list</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        returns number of identified hole states in list.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">([</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">list</span> <span class="k">if</span> <span class="n">cls</span><span class="o">.</span><span class="n">_only_below_fermi</span><span class="p">(</span><span class="n">i</span><span class="p">)])</span>
-
-    <span class="k">def</span> <span class="nf">_negate_holes</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">list</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">([</span><span class="o">-</span><span class="n">i</span> <span class="k">if</span> <span class="n">i</span> <span class="o">&lt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">fermi_level</span> <span class="k">else</span> <span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">fermi_level</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;FockStateKet(</span><span class="si">%r</span><span class="s">, fermi_level=</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">fermi_level</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;FockStateKet(</span><span class="si">%r</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],)</span>
-
-    <span class="k">def</span> <span class="nf">_labels</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_negate_holes</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-
-<div class="viewcode-block" id="FockStateKet"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.FockStateKet">[docs]</a><span class="k">class</span> <span class="nc">FockStateKet</span><span class="p">(</span><span class="n">FockState</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Representation of a ket.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">lbracket</span> <span class="o">=</span> <span class="s">&#39;|&#39;</span>
-    <span class="n">rbracket</span> <span class="o">=</span> <span class="s">&#39;&gt;&#39;</span>
-
-</div>
-<div class="viewcode-block" id="FockStateBra"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.FockStateBra">[docs]</a><span class="k">class</span> <span class="nc">FockStateBra</span><span class="p">(</span><span class="n">FockState</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Representation of a bra.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">lbracket</span> <span class="o">=</span> <span class="s">&#39;&lt;&#39;</span>
-    <span class="n">rbracket</span> <span class="o">=</span> <span class="s">&#39;|&#39;</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">FockStateKet</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">InnerProduct</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="FockStateBosonKet"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.FockStateBosonKet">[docs]</a><span class="k">class</span> <span class="nc">FockStateBosonKet</span><span class="p">(</span><span class="n">BosonState</span><span class="p">,</span> <span class="n">FockStateKet</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Many particle Fock state with a sequence of occupation numbers.</span>
-
-<span class="sd">    Occupation numbers can be any integer &gt;= 0.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.secondquant import BKet</span>
-<span class="sd">    &gt;&gt;&gt; BKet([1, 2])</span>
-<span class="sd">    FockStateBosonKet((1, 2))</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">_dagger_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">FockStateBosonBra</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="FockStateBosonBra"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.FockStateBosonBra">[docs]</a><span class="k">class</span> <span class="nc">FockStateBosonBra</span><span class="p">(</span><span class="n">BosonState</span><span class="p">,</span> <span class="n">FockStateBra</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Describes a collection of BosonBra particles.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.secondquant import BBra</span>
-<span class="sd">    &gt;&gt;&gt; BBra([1, 2])</span>
-<span class="sd">    FockStateBosonBra((1, 2))</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">_dagger_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">FockStateBosonKet</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-</div>
-<span class="k">class</span> <span class="nc">FockStateFermionKet</span><span class="p">(</span><span class="n">FermionState</span><span class="p">,</span> <span class="n">FockStateKet</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Many-particle Fock state with a sequence of occupied orbits.</span>
-
-<span class="sd">    Each state can only have one particle, so we choose to store a list of</span>
-<span class="sd">    occupied orbits rather than a tuple with occupation numbers (zeros and ones).</span>
-
-<span class="sd">    states below fermi level are holes, and are represented by negative labels</span>
-<span class="sd">    in the occupation list.</span>
-
-<span class="sd">    For symbolic state labels, the fermi_level caps the number of allowed hole-</span>
-<span class="sd">    states.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.secondquant import FKet</span>
-<span class="sd">    &gt;&gt;&gt; FKet([1, 2]) #doctest: +SKIP</span>
-<span class="sd">    FockStateFermionKet((1, 2))</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">_dagger_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">FockStateFermionBra</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">FockStateFermionBra</span><span class="p">(</span><span class="n">FermionState</span><span class="p">,</span> <span class="n">FockStateBra</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    FockStateFermionKet</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.secondquant import FBra</span>
-<span class="sd">    &gt;&gt;&gt; FBra([1, 2]) #doctest: +SKIP</span>
-<span class="sd">    FockStateFermionBra((1, 2))</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">_dagger_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">FockStateFermionKet</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-<span class="n">BBra</span> <span class="o">=</span> <span class="n">FockStateBosonBra</span>
-<span class="n">BKet</span> <span class="o">=</span> <span class="n">FockStateBosonKet</span>
-<span class="n">FBra</span> <span class="o">=</span> <span class="n">FockStateFermionBra</span>
-<span class="n">FKet</span> <span class="o">=</span> <span class="n">FockStateFermionKet</span>
-
-
-<span class="k">def</span> <span class="nf">_apply_Mul</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Take a Mul instance with operators and apply them to states.</span>
-
-<span class="sd">    This method applies all operators with integer state labels</span>
-<span class="sd">    to the actual states.  For symbolic state labels, nothing is done.</span>
-<span class="sd">    When inner products of FockStates are encountered (like &lt;a|b&gt;),</span>
-<span class="sd">    they are converted to instances of InnerProduct.</span>
-
-<span class="sd">    This does not currently work on double inner products like,</span>
-<span class="sd">    &lt;a|b&gt;&lt;c|d&gt;.</span>
-
-<span class="sd">    If the argument is not a Mul, it is simply returned as is.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">Mul</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">m</span>
-    <span class="n">c_part</span><span class="p">,</span> <span class="n">nc_part</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-    <span class="n">n_nc</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">nc_part</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n_nc</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">n_nc</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">m</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">last</span> <span class="o">=</span> <span class="n">nc_part</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">next_to_last</span> <span class="o">=</span> <span class="n">nc_part</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">last</span><span class="p">,</span> <span class="n">FockStateKet</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">next_to_last</span><span class="p">,</span> <span class="n">SqOperator</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">next_to_last</span><span class="o">.</span><span class="n">is_symbolic</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">m</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">result</span> <span class="o">=</span> <span class="n">next_to_last</span><span class="o">.</span><span class="n">apply_operator</span><span class="p">(</span><span class="n">last</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">result</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">_apply_Mul</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">c_part</span> <span class="o">+</span> <span class="n">nc_part</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">result</span><span class="p">])))</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">next_to_last</span><span class="p">,</span> <span class="n">Pow</span><span class="p">):</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">next_to_last</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">SqOperator</span><span class="p">)</span> <span class="ow">and</span> \
-                        <span class="n">next_to_last</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">next_to_last</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_symbolic</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">m</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">result</span> <span class="o">=</span> <span class="n">last</span>
-                        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">next_to_last</span><span class="o">.</span><span class="n">exp</span><span class="p">):</span>
-                            <span class="n">result</span> <span class="o">=</span> <span class="n">next_to_last</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">apply_operator</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-                            <span class="k">if</span> <span class="n">result</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                                <span class="k">break</span>
-                        <span class="k">if</span> <span class="n">result</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="n">_apply_Mul</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">c_part</span> <span class="o">+</span> <span class="n">nc_part</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">result</span><span class="p">])))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">m</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">next_to_last</span><span class="p">,</span> <span class="n">FockStateBra</span><span class="p">):</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">InnerProduct</span><span class="p">(</span><span class="n">next_to_last</span><span class="p">,</span> <span class="n">last</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">result</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">_apply_Mul</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">c_part</span> <span class="o">+</span> <span class="n">nc_part</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">result</span><span class="p">])))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">m</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">m</span>
-
-
-<div class="viewcode-block" id="apply_operators"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.apply_operators">[docs]</a><span class="k">def</span> <span class="nf">apply_operators</span><span class="p">(</span><span class="n">e</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Take a sympy expression with operators and states and apply the operators.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.secondquant import apply_operators</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import sympify</span>
-<span class="sd">    &gt;&gt;&gt; apply_operators(sympify(3)+4)</span>
-<span class="sd">    7</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">e</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-    <span class="n">muls</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Mul</span><span class="p">)</span>
-    <span class="n">subs_list</span> <span class="o">=</span> <span class="p">[(</span><span class="n">m</span><span class="p">,</span> <span class="n">_apply_Mul</span><span class="p">(</span><span class="n">m</span><span class="p">))</span> <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="nb">iter</span><span class="p">(</span><span class="n">muls</span><span class="p">)]</span>
-    <span class="k">return</span> <span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">subs_list</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="InnerProduct"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.InnerProduct">[docs]</a><span class="k">class</span> <span class="nc">InnerProduct</span><span class="p">(</span><span class="n">Basic</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    An unevaluated inner product between a bra and ket.</span>
-
-<span class="sd">    Currently this class just reduces things to a product of</span>
-<span class="sd">    Kronecker Deltas.  In the future, we could introduce abstract</span>
-<span class="sd">    states like ``|a&gt;`` and ``|b&gt;``, and leave the inner product unevaluated as</span>
-<span class="sd">    ``&lt;a|b&gt;``.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">bra</span><span class="p">,</span> <span class="n">ket</span><span class="p">):</span>
-        <span class="k">assert</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">bra</span><span class="p">,</span> <span class="n">FockStateBra</span><span class="p">),</span> <span class="s">&#39;must be a bra&#39;</span>
-        <span class="k">assert</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ket</span><span class="p">,</span> <span class="n">FockStateKet</span><span class="p">),</span> <span class="s">&#39;must be a key&#39;</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="n">bra</span><span class="p">,</span> <span class="n">ket</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">bra</span><span class="p">,</span> <span class="n">ket</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">bra</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">ket</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
-            <span class="n">result</span> <span class="o">*=</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">result</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">break</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="InnerProduct.bra"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.InnerProduct.bra">[docs]</a>    <span class="k">def</span> <span class="nf">bra</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the bra part of the state&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="InnerProduct.ket"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.InnerProduct.ket">[docs]</a>    <span class="k">def</span> <span class="nf">ket</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the ket part of the state&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-</div>
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">sbra</span> <span class="o">=</span> <span class="nb">repr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bra</span><span class="p">)</span>
-        <span class="n">sket</span> <span class="o">=</span> <span class="nb">repr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ket</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">|</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">sbra</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">sket</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__repr__</span><span class="p">()</span>
-
-</div>
-<div class="viewcode-block" id="matrix_rep"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.matrix_rep">[docs]</a><span class="k">def</span> <span class="nf">matrix_rep</span><span class="p">(</span><span class="n">op</span><span class="p">,</span> <span class="n">basis</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Find the representation of an operator in a basis.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.secondquant import VarBosonicBasis, B, matrix_rep</span>
-<span class="sd">    &gt;&gt;&gt; b = VarBosonicBasis(5)</span>
-<span class="sd">    &gt;&gt;&gt; o = B(0)</span>
-<span class="sd">    &gt;&gt;&gt; matrix_rep(o, b)</span>
-<span class="sd">    [0, 1,       0,       0, 0]</span>
-<span class="sd">    [0, 0, sqrt(2),       0, 0]</span>
-<span class="sd">    [0, 0,       0, sqrt(3), 0]</span>
-<span class="sd">    [0, 0,       0,       0, 2]</span>
-<span class="sd">    [0, 0,       0,       0, 0]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">basis</span><span class="p">))</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">basis</span><span class="p">)):</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">basis</span><span class="p">)):</span>
-            <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">apply_operators</span><span class="p">(</span><span class="n">Dagger</span><span class="p">(</span><span class="n">basis</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="o">*</span><span class="n">op</span><span class="o">*</span><span class="n">basis</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-    <span class="k">return</span> <span class="n">a</span>
-
-</div>
-<div class="viewcode-block" id="BosonicBasis"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.BosonicBasis">[docs]</a><span class="k">class</span> <span class="nc">BosonicBasis</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Base class for a basis set of bosonic Fock states.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="VarBosonicBasis"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.VarBosonicBasis">[docs]</a><span class="k">class</span> <span class="nc">VarBosonicBasis</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A single state, variable particle number basis set.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.secondquant import VarBosonicBasis</span>
-<span class="sd">    &gt;&gt;&gt; b = VarBosonicBasis(5)</span>
-<span class="sd">    &gt;&gt;&gt; b</span>
-<span class="sd">    [FockState((0,)), FockState((1,)), FockState((2,)),</span>
-<span class="sd">     FockState((3,)), FockState((4,))]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n_max</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">n_max</span> <span class="o">=</span> <span class="n">n_max</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_build_states</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">_build_states</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">basis</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_max</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">basis</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">FockStateBosonKet</span><span class="p">([</span><span class="n">i</span><span class="p">]))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">n_basis</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">basis</span><span class="p">)</span>
-
-<div class="viewcode-block" id="VarBosonicBasis.index"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.VarBosonicBasis.index">[docs]</a>    <span class="k">def</span> <span class="nf">index</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the index of state in basis.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import VarBosonicBasis</span>
-<span class="sd">        &gt;&gt;&gt; b = VarBosonicBasis(3)</span>
-<span class="sd">        &gt;&gt;&gt; state = b.state(1)</span>
-<span class="sd">        &gt;&gt;&gt; b</span>
-<span class="sd">        [FockState((0,)), FockState((1,)), FockState((2,))]</span>
-<span class="sd">        &gt;&gt;&gt; state</span>
-<span class="sd">        FockStateBosonKet((1,))</span>
-<span class="sd">        &gt;&gt;&gt; b.index(state)</span>
-<span class="sd">        1</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">basis</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="VarBosonicBasis.state"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.VarBosonicBasis.state">[docs]</a>    <span class="k">def</span> <span class="nf">state</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The state of a single basis.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import VarBosonicBasis</span>
-<span class="sd">        &gt;&gt;&gt; b = VarBosonicBasis(5)</span>
-<span class="sd">        &gt;&gt;&gt; b.state(3)</span>
-<span class="sd">        FockStateBosonKet((3,))</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">basis</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-</div>
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">state</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__len__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">basis</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">repr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">basis</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="FixedBosonicBasis"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.FixedBosonicBasis">[docs]</a><span class="k">class</span> <span class="nc">FixedBosonicBasis</span><span class="p">(</span><span class="n">BosonicBasis</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Fixed particle number basis set.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.secondquant import FixedBosonicBasis</span>
-<span class="sd">    &gt;&gt;&gt; b = FixedBosonicBasis(2, 2)</span>
-<span class="sd">    &gt;&gt;&gt; state = b.state(1)</span>
-<span class="sd">    &gt;&gt;&gt; b</span>
-<span class="sd">    [FockState((2, 0)), FockState((1, 1)), FockState((0, 2))]</span>
-<span class="sd">    &gt;&gt;&gt; state</span>
-<span class="sd">    FockStateBosonKet((1, 1))</span>
-<span class="sd">    &gt;&gt;&gt; b.index(state)</span>
-<span class="sd">    1</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n_particles</span><span class="p">,</span> <span class="n">n_levels</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">n_particles</span> <span class="o">=</span> <span class="n">n_particles</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">n_levels</span> <span class="o">=</span> <span class="n">n_levels</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_build_particle_locations</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_build_states</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">_build_particle_locations</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">tup</span> <span class="o">=</span> <span class="p">[</span><span class="s">&quot;i</span><span class="si">%i</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_particles</span><span class="p">)]</span>
-        <span class="n">first_loop</span> <span class="o">=</span> <span class="s">&quot;for i0 in range(</span><span class="si">%i</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_levels</span>
-        <span class="n">other_loops</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-        <span class="k">for</span> <span class="n">cur</span><span class="p">,</span> <span class="n">prev</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">tup</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">tup</span><span class="p">):</span>
-            <span class="n">temp</span> <span class="o">=</span> <span class="s">&quot;for </span><span class="si">%s</span><span class="s"> in range(</span><span class="si">%s</span><span class="s"> + 1) &quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">cur</span><span class="p">,</span> <span class="n">prev</span><span class="p">)</span>
-            <span class="n">other_loops</span> <span class="o">=</span> <span class="n">other_loops</span> <span class="o">+</span> <span class="n">temp</span>
-        <span class="n">tup_string</span> <span class="o">=</span> <span class="s">&quot;(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">tup</span><span class="p">)</span>
-        <span class="n">list_comp</span> <span class="o">=</span> <span class="s">&quot;[</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">]&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tup_string</span><span class="p">,</span> <span class="n">first_loop</span><span class="p">,</span> <span class="n">other_loops</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="nb">eval</span><span class="p">(</span><span class="n">list_comp</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_particles</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="p">[(</span><span class="n">item</span><span class="p">,)</span> <span class="k">for</span> <span class="n">item</span> <span class="ow">in</span> <span class="n">result</span><span class="p">]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">particle_locations</span> <span class="o">=</span> <span class="n">result</span>
-
-    <span class="k">def</span> <span class="nf">_build_states</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">basis</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">tuple_of_indices</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">particle_locations</span><span class="p">:</span>
-            <span class="n">occ_numbers</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_levels</span><span class="o">*</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">for</span> <span class="n">level</span> <span class="ow">in</span> <span class="n">tuple_of_indices</span><span class="p">:</span>
-                <span class="n">occ_numbers</span><span class="p">[</span><span class="n">level</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">basis</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">FockStateBosonKet</span><span class="p">(</span><span class="n">occ_numbers</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">n_basis</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">basis</span><span class="p">)</span>
-
-<div class="viewcode-block" id="FixedBosonicBasis.index"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.FixedBosonicBasis.index">[docs]</a>    <span class="k">def</span> <span class="nf">index</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">state</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the index of state in basis.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import FixedBosonicBasis</span>
-<span class="sd">        &gt;&gt;&gt; b = FixedBosonicBasis(2, 3)</span>
-<span class="sd">        &gt;&gt;&gt; b.index(b.state(3))</span>
-<span class="sd">        3</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">basis</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="FixedBosonicBasis.state"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.FixedBosonicBasis.state">[docs]</a>    <span class="k">def</span> <span class="nf">state</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the state that lies at index i of the basis</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import FixedBosonicBasis</span>
-<span class="sd">        &gt;&gt;&gt; b = FixedBosonicBasis(2, 3)</span>
-<span class="sd">        &gt;&gt;&gt; b.state(3)</span>
-<span class="sd">        FockStateBosonKet((1, 0, 1))</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">basis</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-</div>
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">state</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__len__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">basis</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">repr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">basis</span><span class="p">)</span>
-
-
-<span class="c"># def move(e, i, d):</span>
-<span class="c">#     &quot;&quot;&quot;</span>
-<span class="c">#     Takes the expression &quot;e&quot; and moves the operator at the position i by &quot;d&quot;.</span>
-<span class="c">#     &quot;&quot;&quot;</span>
-<span class="c">#     if e.is_Mul:</span>
-<span class="c">#         if d == 1:</span>
-<span class="c">#             # e = a*b*c*d</span>
-<span class="c">#             a = Mul(*e.args[:i])</span>
-<span class="c">#             b = e.args[i]</span>
-<span class="c">#             c = e.args[i+1]</span>
-<span class="c">#             d = Mul(*e.args[i+2:])</span>
-<span class="c">#             if isinstance(b, Dagger) and not isinstance(c, Dagger):</span>
-<span class="c">#                 i, j = b.args[0].args[0], c.args[0]</span>
-<span class="c">#                 return a*c*b*d-a*KroneckerDelta(i, j)*d</span>
-<span class="c">#             elif not isinstance(b, Dagger) and isinstance(c, Dagger):</span>
-<span class="c">#                 i, j = b.args[0], c.args[0].args[0]</span>
-<span class="c">#                 return a*c*b*d-a*KroneckerDelta(i, j)*d</span>
-<span class="c">#             else:</span>
-<span class="c">#                 return a*c*b*d</span>
-<span class="c">#         elif d == -1:</span>
-<span class="c">#             # e = a*b*c*d</span>
-<span class="c">#             a = Mul(*e.args[:i-1])</span>
-<span class="c">#             b = e.args[i-1]</span>
-<span class="c">#             c = e.args[i]</span>
-<span class="c">#             d = Mul(*e.args[i+1:])</span>
-<span class="c">#             if isinstance(b, Dagger) and not isinstance(c, Dagger):</span>
-<span class="c">#                 i, j = b.args[0].args[0], c.args[0]</span>
-<span class="c">#                 return a*c*b*d-a*KroneckerDelta(i, j)*d</span>
-<span class="c">#             elif not isinstance(b, Dagger) and isinstance(c, Dagger):</span>
-<span class="c">#                 i, j = b.args[0], c.args[0].args[0]</span>
-<span class="c">#                 return a*c*b*d-a*KroneckerDelta(i, j)*d</span>
-<span class="c">#             else:</span>
-<span class="c">#                 return a*c*b*d</span>
-<span class="c">#         else:</span>
-<span class="c">#             if d &gt; 1:</span>
-<span class="c">#                 while d &gt;= 1:</span>
-<span class="c">#                     e = move(e, i, 1)</span>
-<span class="c">#                     d -= 1</span>
-<span class="c">#                     i += 1</span>
-<span class="c">#                 return e</span>
-<span class="c">#             elif d &lt; -1:</span>
-<span class="c">#                 while d &lt;= -1:</span>
-<span class="c">#                     e = move(e, i, -1)</span>
-<span class="c">#                     d += 1</span>
-<span class="c">#                     i -= 1</span>
-<span class="c">#                 return e</span>
-<span class="c">#     elif isinstance(e, Add):</span>
-<span class="c">#         a, b = e.as_two_terms()</span>
-<span class="c">#         return move(a, i, d) + move(b, i, d)</span>
-<span class="c">#     raise NotImplementedError()</span>
-</div>
-<div class="viewcode-block" id="Commutator"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.Commutator">[docs]</a><span class="k">class</span> <span class="nc">Commutator</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The Commutator:  [A, B] = A*B - B*A</span>
-
-<span class="sd">    The arguments are ordered according to .__cmp__()</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.secondquant import Commutator</span>
-<span class="sd">    &gt;&gt;&gt; A, B = symbols(&#39;A,B&#39;, commutative=False)</span>
-<span class="sd">    &gt;&gt;&gt; Commutator(B, A)</span>
-<span class="sd">    -Commutator(A, B)</span>
-
-<span class="sd">    Evaluate the commutator with .doit()</span>
-
-<span class="sd">    &gt;&gt;&gt; comm = Commutator(A,B); comm</span>
-<span class="sd">    Commutator(A, B)</span>
-<span class="sd">    &gt;&gt;&gt; comm.doit()</span>
-<span class="sd">    A*B - B*A</span>
-
-
-<span class="sd">    For two second quantization operators the commutator is evaluated</span>
-<span class="sd">    immediately:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.secondquant import Fd, F</span>
-<span class="sd">    &gt;&gt;&gt; a = symbols(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">    &gt;&gt;&gt; i = symbols(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">    &gt;&gt;&gt; p,q = symbols(&#39;p,q&#39;)</span>
-
-<span class="sd">    &gt;&gt;&gt; Commutator(Fd(a),Fd(i))</span>
-<span class="sd">    2*NO(CreateFermion(a)*CreateFermion(i))</span>
-
-<span class="sd">    But for more complicated expressions, the evaluation is triggered by</span>
-<span class="sd">    a call to .doit()</span>
-
-<span class="sd">    &gt;&gt;&gt; comm = Commutator(Fd(p)*Fd(q),F(i)); comm</span>
-<span class="sd">    Commutator(CreateFermion(p)*CreateFermion(q), AnnihilateFermion(i))</span>
-<span class="sd">    &gt;&gt;&gt; comm.doit(wicks=True)</span>
-<span class="sd">    -KroneckerDelta(p, i)*CreateFermion(q) +</span>
-<span class="sd">     KroneckerDelta(q, i)*CreateFermion(p)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Commutator.eval"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.Commutator.eval">[docs]</a>    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The Commutator [A,B] is on canonical form if A &lt; B.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import Commutator, F, Fd</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; c1 = Commutator(F(x), Fd(x))</span>
-<span class="sd">        &gt;&gt;&gt; c2 = Commutator(Fd(x), F(x))</span>
-<span class="sd">        &gt;&gt;&gt; Commutator.eval(c1, c2)</span>
-<span class="sd">        0</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">a</span> <span class="ow">and</span> <span class="n">b</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="n">b</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_commutative</span> <span class="ow">or</span> <span class="n">b</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-        <span class="c">#</span>
-        <span class="c"># [A+B,C]  -&gt;  [A,C] + [B,C]</span>
-        <span class="c">#</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Add</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">cls</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">a</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Add</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">cls</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">term</span><span class="p">)</span> <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">b</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-        <span class="c">#</span>
-        <span class="c"># [xA,yB]  -&gt;  xy*[A,B]</span>
-        <span class="c">#</span>
-        <span class="n">ca</span><span class="p">,</span> <span class="n">nca</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-        <span class="n">cb</span><span class="p">,</span> <span class="n">ncb</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-        <span class="n">c_part</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">ca</span><span class="p">)</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">cb</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">c_part</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">c_part</span><span class="p">),</span> <span class="n">cls</span><span class="p">(</span><span class="n">Mul</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">nca</span><span class="p">),</span> <span class="n">Mul</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">ncb</span><span class="p">)))</span>
-
-        <span class="c">#</span>
-        <span class="c"># single second quantization operators</span>
-        <span class="c">#</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">BosonicOperator</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">BosonicOperator</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">CreateBoson</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">AnnihilateBoson</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">state</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">state</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">CreateBoson</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">AnnihilateBoson</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">*</span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">state</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">state</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">FermionicOperator</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">FermionicOperator</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">wicks</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">b</span><span class="p">)</span> <span class="o">-</span> <span class="n">wicks</span><span class="p">(</span><span class="n">b</span><span class="o">*</span><span class="n">a</span><span class="p">)</span>
-
-        <span class="c">#</span>
-        <span class="c"># Canonical ordering of arguments</span>
-        <span class="c">#</span>
-        <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">sort_key</span><span class="p">()</span> <span class="o">&gt;</span> <span class="n">b</span><span class="o">.</span><span class="n">sort_key</span><span class="p">():</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">*</span><span class="n">cls</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Commutator.doit"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.Commutator.doit">[docs]</a>    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Enables the computation of complex expressions.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import Commutator, F, Fd</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; i, j = symbols(&#39;i,j&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; a, b = symbols(&#39;a,b&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; c = Commutator(Fd(a)*F(i),Fd(b)*F(j))</span>
-<span class="sd">        &gt;&gt;&gt; c.doit(wicks=True)</span>
-<span class="sd">        0</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;wicks&quot;</span><span class="p">):</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">wicks</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">b</span><span class="p">)</span> <span class="o">-</span> <span class="n">wicks</span><span class="p">(</span><span class="n">b</span><span class="o">*</span><span class="n">a</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">ContractionAppliesOnlyToFermions</span><span class="p">:</span>
-                <span class="k">pass</span>
-            <span class="k">except</span> <span class="n">WicksTheoremDoesNotApply</span><span class="p">:</span>
-                <span class="k">pass</span>
-
-        <span class="k">return</span> <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">b</span> <span class="o">-</span> <span class="n">b</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;Commutator(</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;[</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">]&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="se">\\</span><span class="s">left[</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="se">\\</span><span class="s">right]&quot;</span> <span class="o">%</span> <span class="nb">tuple</span><span class="p">([</span>
-            <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-</div>
-<div class="viewcode-block" id="NO"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.NO">[docs]</a><span class="k">class</span> <span class="nc">NO</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    This Object is used to represent normal ordering brackets.</span>
-
-<span class="sd">    i.e.  {abcd}  sometimes written  :abcd:</span>
-
-<span class="sd">    Applying the function NO(arg) to an argument means that all operators in</span>
-<span class="sd">    the argument will be assumed to anticommute, and have vanishing</span>
-<span class="sd">    contractions.  This allows an immediate reordering to canonical form</span>
-<span class="sd">    upon object creation.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.secondquant import NO, F, Fd</span>
-<span class="sd">    &gt;&gt;&gt; p,q = symbols(&#39;p,q&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; NO(Fd(p)*F(q))</span>
-<span class="sd">    NO(CreateFermion(p)*AnnihilateFermion(q))</span>
-<span class="sd">    &gt;&gt;&gt; NO(F(q)*Fd(p))</span>
-<span class="sd">    -NO(CreateFermion(p)*AnnihilateFermion(q))</span>
-
-
-<span class="sd">    Note:</span>
-<span class="sd">    If you want to generate a normal ordered equivalent of an expression, you</span>
-<span class="sd">    should use the function wicks().  This class only indicates that all</span>
-<span class="sd">    operators inside the brackets anticommute, and have vanishing contractions.</span>
-<span class="sd">    Nothing more, nothing less.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Use anticommutation to get canonical form of operators.</span>
-
-<span class="sd">        Employ associativity of normal ordered product: {ab{cd}} = {abcd}</span>
-<span class="sd">        but note that {ab}{cd} /= {abcd}.</span>
-
-<span class="sd">        We also employ distributivity: {ab + cd} = {ab} + {cd}.</span>
-
-<span class="sd">        Canonical form also implies expand() {ab(c+d)} = {abc} + {abd}.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="c"># {ab + cd} = {ab} + {cd}</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">cls</span><span class="p">(</span><span class="n">term</span><span class="p">)</span> <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-
-            <span class="c"># take coefficient outside of normal ordering brackets</span>
-            <span class="n">c_part</span><span class="p">,</span> <span class="n">seq</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">c_part</span><span class="p">:</span>
-                <span class="n">coeff</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">c_part</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">seq</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">coeff</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">coeff</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-            <span class="c"># {ab{cd}} = {abcd}</span>
-            <span class="n">newseq</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">foundit</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="k">for</span> <span class="n">fac</span> <span class="ow">in</span> <span class="n">seq</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">fac</span><span class="p">,</span> <span class="n">NO</span><span class="p">):</span>
-                    <span class="n">newseq</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">fac</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-                    <span class="n">foundit</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">newseq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">fac</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">foundit</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">coeff</span><span class="o">*</span><span class="n">cls</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">newseq</span><span class="p">))</span>
-
-            <span class="c"># We assume that the user don&#39;t mix B and F operators</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">seq</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">BosonicOperator</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">newseq</span><span class="p">,</span> <span class="n">sign</span> <span class="o">=</span> <span class="n">_sort_anticommuting_fermions</span><span class="p">(</span><span class="n">seq</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">ViolationOfPauliPrinciple</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-            <span class="k">if</span> <span class="n">sign</span> <span class="o">%</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">*</span><span class="n">coeff</span><span class="p">)</span><span class="o">*</span><span class="n">cls</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">newseq</span><span class="p">))</span>
-            <span class="k">elif</span> <span class="n">sign</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">coeff</span><span class="o">*</span><span class="n">cls</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">newseq</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">pass</span>  <span class="c"># since sign==0, no permutations was necessary</span>
-
-            <span class="c"># if we couldn&#39;t do anything with Mul object, we just</span>
-            <span class="c"># mark it as normal ordered</span>
-            <span class="k">if</span> <span class="n">coeff</span> <span class="o">!=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">coeff</span><span class="o">*</span><span class="n">cls</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">newseq</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">newseq</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">NO</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">arg</span>
-
-        <span class="c"># if object was not Mul or Add, normal ordering does not apply</span>
-        <span class="k">return</span> <span class="n">arg</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="NO.has_q_creators"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.NO.has_q_creators">[docs]</a>    <span class="k">def</span> <span class="nf">has_q_creators</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return 0 if the leftmost argument of the first argument is a not a</span>
-<span class="sd">        q_creator, else 1 if it is above fermi or -1 if it is below fermi.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import NO, F, Fd</span>
-
-<span class="sd">        &gt;&gt;&gt; a = symbols(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = symbols(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; NO(Fd(a)*Fd(i)).has_q_creators</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; NO(F(i)*F(a)).has_q_creators</span>
-<span class="sd">        -1</span>
-<span class="sd">        &gt;&gt;&gt; NO(Fd(i)*F(a)).has_q_creators           #doctest: +SKIP</span>
-<span class="sd">        0</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_q_creator</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="NO.has_q_annihilators"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.NO.has_q_annihilators">[docs]</a>    <span class="k">def</span> <span class="nf">has_q_annihilators</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return 0 if the rightmost argument of the first argument is a not a</span>
-<span class="sd">        q_annihilator, else 1 if it is above fermi or -1 if it is below fermi.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import NO, F, Fd</span>
-
-<span class="sd">        &gt;&gt;&gt; a = symbols(&#39;a&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = symbols(&#39;i&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; NO(Fd(a)*Fd(i)).has_q_annihilators</span>
-<span class="sd">        -1</span>
-<span class="sd">        &gt;&gt;&gt; NO(F(i)*F(a)).has_q_annihilators</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; NO(Fd(a)*F(i)).has_q_annihilators</span>
-<span class="sd">        0</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">is_q_annihilator</span>
-</div>
-<div class="viewcode-block" id="NO.doit"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.NO.doit">[docs]</a>    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Either removes the brackets or enables complex computations</span>
-<span class="sd">        in its arguments.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import NO, Fd, F</span>
-<span class="sd">        &gt;&gt;&gt; from textwrap import fill</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols, Dummy</span>
-<span class="sd">        &gt;&gt;&gt; p,q = symbols(&#39;p,q&#39;, cls=Dummy)</span>
-<span class="sd">        &gt;&gt;&gt; print(fill(str(NO(Fd(p)*F(q)).doit())))</span>
-<span class="sd">        KroneckerDelta(_a, _p)*KroneckerDelta(_a,</span>
-<span class="sd">        _q)*CreateFermion(_a)*AnnihilateFermion(_a) + KroneckerDelta(_a,</span>
-<span class="sd">        _p)*KroneckerDelta(_i, _q)*CreateFermion(_a)*AnnihilateFermion(_i) -</span>
-<span class="sd">        KroneckerDelta(_a, _q)*KroneckerDelta(_i,</span>
-<span class="sd">        _p)*AnnihilateFermion(_a)*CreateFermion(_i) - KroneckerDelta(_i,</span>
-<span class="sd">        _p)*KroneckerDelta(_i, _q)*AnnihilateFermion(_i)*CreateFermion(_i)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">kw_args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;remove_brackets&quot;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_remove_brackets</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">kw_args</span><span class="p">))</span>
-</div>
-    <span class="k">def</span> <span class="nf">_remove_brackets</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the sorted string without normal order brackets.</span>
-
-<span class="sd">        The returned string have the property that no nonzero</span>
-<span class="sd">        contractions exist.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="c"># check if any creator is also an annihilator</span>
-        <span class="n">subslist</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">iter_q_creators</span><span class="p">():</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">is_q_annihilator</span><span class="p">:</span>
-                <span class="n">assume</span> <span class="o">=</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">state</span><span class="o">.</span><span class="n">assumptions0</span>
-
-                <span class="c"># only operators with a dummy index can be split in two terms</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">state</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">):</span>
-
-                    <span class="c"># create indices with fermi restriction</span>
-                    <span class="n">assume</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&quot;above_fermi&quot;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-                    <span class="n">assume</span><span class="p">[</span><span class="s">&quot;below_fermi&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">True</span>
-                    <span class="n">below</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">assume</span><span class="p">)</span>
-                    <span class="n">assume</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&quot;below_fermi&quot;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-                    <span class="n">assume</span><span class="p">[</span><span class="s">&quot;above_fermi&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">True</span>
-                    <span class="n">above</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">assume</span><span class="p">)</span>
-
-                    <span class="n">cls</span> <span class="o">=</span> <span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-                    <span class="n">split</span> <span class="o">=</span> <span class="p">(</span>
-                        <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">below</span><span class="p">)</span>
-                        <span class="o">*</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">below</span><span class="p">,</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">state</span><span class="p">)</span>
-                        <span class="o">+</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">above</span><span class="p">)</span>
-                        <span class="o">*</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">above</span><span class="p">,</span> <span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">state</span><span class="p">)</span>
-                    <span class="p">)</span>
-                    <span class="n">subslist</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">split</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="n">SubstitutionOfAmbigousOperatorFailed</span><span class="p">(</span><span class="bp">self</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">subslist</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">NO</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">subslist</span><span class="p">))</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">Add</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">term</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span> <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">result</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">_expand_operators</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns a sum of NO objects that contain no ambiguous q-operators.</span>
-
-<span class="sd">        If an index q has range both above and below fermi, the operator F(q)</span>
-<span class="sd">        is ambiguous in the sense that it can be both a q-creator and a q-annihilator.</span>
-<span class="sd">        If q is dummy, it is assumed to be a summation variable and this method</span>
-<span class="sd">        rewrites it into a sum of NO terms with unambiguous operators:</span>
-
-<span class="sd">        {Fd(p)*F(q)} = {Fd(a)*F(b)} + {Fd(a)*F(i)} + {Fd(j)*F(b)} -{F(i)*Fd(j)}</span>
-
-<span class="sd">        where a,b are above and i,j are below fermi level.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">NO</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_remove_brackets</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="nb">slice</span><span class="p">):</span>
-            <span class="n">indices</span> <span class="o">=</span> <span class="n">i</span><span class="o">.</span><span class="n">indices</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="p">))</span>
-            <span class="k">return</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="o">*</span><span class="n">indices</span><span class="p">)]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__len__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-<div class="viewcode-block" id="NO.iter_q_annihilators"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.NO.iter_q_annihilators">[docs]</a>    <span class="k">def</span> <span class="nf">iter_q_annihilators</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Iterates over the annihilation operators.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; i, j = symbols(&#39;i j&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; a, b = symbols(&#39;a b&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import NO, F, Fd</span>
-<span class="sd">        &gt;&gt;&gt; no = NO(Fd(a)*F(i)*F(b)*Fd(j))</span>
-
-<span class="sd">        &gt;&gt;&gt; no.iter_q_creators()</span>
-<span class="sd">        &lt;generator object... at 0x...&gt;</span>
-<span class="sd">        &gt;&gt;&gt; list(no.iter_q_creators())</span>
-<span class="sd">        [0, 1]</span>
-<span class="sd">        &gt;&gt;&gt; list(no.iter_q_annihilators())</span>
-<span class="sd">        [3, 2]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">ops</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">args</span>
-        <span class="nb">iter</span> <span class="o">=</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">ops</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">iter</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">ops</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">is_q_annihilator</span><span class="p">:</span>
-                <span class="k">yield</span> <span class="n">i</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">break</span>
-</div>
-<div class="viewcode-block" id="NO.iter_q_creators"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.NO.iter_q_creators">[docs]</a>    <span class="k">def</span> <span class="nf">iter_q_creators</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Iterates over the creation operators.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; i, j = symbols(&#39;i j&#39;, below_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; a, b = symbols(&#39;a b&#39;, above_fermi=True)</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import NO, F, Fd</span>
-<span class="sd">        &gt;&gt;&gt; no = NO(Fd(a)*F(i)*F(b)*Fd(j))</span>
-
-<span class="sd">        &gt;&gt;&gt; no.iter_q_creators()</span>
-<span class="sd">        &lt;generator object... at 0x...&gt;</span>
-<span class="sd">        &gt;&gt;&gt; list(no.iter_q_creators())</span>
-<span class="sd">        [0, 1]</span>
-<span class="sd">        &gt;&gt;&gt; list(no.iter_q_annihilators())</span>
-<span class="sd">        [3, 2]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">ops</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">args</span>
-        <span class="nb">iter</span> <span class="o">=</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">ops</span><span class="p">))</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">iter</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">ops</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">is_q_creator</span><span class="p">:</span>
-                <span class="k">yield</span> <span class="n">i</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">break</span>
-</div>
-<div class="viewcode-block" id="NO.get_subNO"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.NO.get_subNO">[docs]</a>    <span class="k">def</span> <span class="nf">get_subNO</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns a NO() without FermionicOperator at index i.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import F, NO</span>
-<span class="sd">        &gt;&gt;&gt; p,q,r = symbols(&#39;p,q,r&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; NO(F(p)*F(q)*F(r)).get_subNO(1)  # doctest: +SKIP</span>
-<span class="sd">        NO(AnnihilateFermion(p)*AnnihilateFermion(r))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">arg0</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>  <span class="c"># it&#39;s a Mul by definition of how it&#39;s created</span>
-        <span class="n">mul</span> <span class="o">=</span> <span class="n">arg0</span><span class="o">.</span><span class="n">_new_rawargs</span><span class="p">(</span><span class="n">arg0</span><span class="o">.</span><span class="n">args</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">arg0</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:])</span>
-        <span class="k">return</span> <span class="n">NO</span><span class="p">(</span><span class="n">mul</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="se">\\</span><span class="s">left</span><span class="se">\\</span><span class="s">{</span><span class="si">%s</span><span class="se">\\</span><span class="s">right</span><span class="se">\\</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="n">printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;NO(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;:</span><span class="si">%s</span><span class="s">:&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-
-<span class="c"># @cacheit</span></div>
-<div class="viewcode-block" id="contraction"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.contraction">[docs]</a><span class="k">def</span> <span class="nf">contraction</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Calculates contraction of Fermionic operators a and b.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.secondquant import F, Fd, contraction</span>
-<span class="sd">    &gt;&gt;&gt; p, q = symbols(&#39;p,q&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; a, b = symbols(&#39;a,b&#39;, above_fermi=True)</span>
-<span class="sd">    &gt;&gt;&gt; i, j = symbols(&#39;i,j&#39;, below_fermi=True)</span>
-
-<span class="sd">    A contraction is non-zero only if a quasi-creator is to the right of a</span>
-<span class="sd">    quasi-annihilator:</span>
-
-<span class="sd">    &gt;&gt;&gt; contraction(F(a),Fd(b))</span>
-<span class="sd">    KroneckerDelta(a, b)</span>
-<span class="sd">    &gt;&gt;&gt; contraction(Fd(i),F(j))</span>
-<span class="sd">    KroneckerDelta(i, j)</span>
-
-<span class="sd">    For general indices a non-zero result restricts the indices to below/above</span>
-<span class="sd">    the fermi surface:</span>
-
-<span class="sd">    &gt;&gt;&gt; contraction(Fd(p),F(q))</span>
-<span class="sd">    KroneckerDelta(p, q)*KroneckerDelta(q, _i)</span>
-<span class="sd">    &gt;&gt;&gt; contraction(F(p),Fd(q))</span>
-<span class="sd">    KroneckerDelta(p, q)*KroneckerDelta(q, _a)</span>
-
-<span class="sd">    Two creators or two annihilators always vanishes:</span>
-
-<span class="sd">    &gt;&gt;&gt; contraction(F(p),F(q))</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; contraction(Fd(p),Fd(q))</span>
-<span class="sd">    0</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">FermionicOperator</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">FermionicOperator</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">AnnihilateFermion</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">CreateFermion</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">state</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;below_fermi&quot;</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">state</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;below_fermi&quot;</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">state</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;above_fermi&quot;</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">state</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">state</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">state</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;above_fermi&quot;</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">state</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">state</span><span class="p">)</span>
-
-            <span class="k">return</span> <span class="p">(</span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">state</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">state</span><span class="p">)</span><span class="o">*</span>
-                    <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">state</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)))</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">AnnihilateFermion</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">CreateFermion</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">state</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;above_fermi&quot;</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">state</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;above_fermi&quot;</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">state</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;below_fermi&quot;</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">state</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">state</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">state</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;below_fermi&quot;</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">state</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">state</span><span class="p">)</span>
-
-            <span class="k">return</span> <span class="p">(</span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">state</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">state</span><span class="p">)</span><span class="o">*</span>
-                    <span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">state</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)))</span>
-
-        <span class="c"># vanish if 2xAnnihilator or 2xCreator</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="k">else</span><span class="p">:</span>
-        <span class="c">#not fermion operators</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="p">(</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">FermionicOperator</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="p">)</span>
-        <span class="k">raise</span> <span class="n">ContractionAppliesOnlyToFermions</span><span class="p">(</span><span class="o">*</span><span class="n">t</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_sqkey</span><span class="p">(</span><span class="n">sq_operator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Generates key for canonical sorting of SQ operators.&quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">sq_operator</span><span class="o">.</span><span class="n">_sortkey</span><span class="p">()</span>
-
-
-<span class="k">def</span> <span class="nf">_sort_anticommuting_fermions</span><span class="p">(</span><span class="n">string1</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">_sqkey</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Sort fermionic operators to canonical order, assuming all pairs anticommute.</span>
-
-<span class="sd">    Uses a bidirectional bubble sort.  Items in string1 are not referenced</span>
-<span class="sd">    so in principle they may be any comparable objects.   The sorting depends on the</span>
-<span class="sd">    operators &#39;&gt;&#39; and &#39;==&#39;.</span>
-
-<span class="sd">    If the Pauli principle is violated, an exception is raised.</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    tuple (sorted_str, sign)</span>
-
-<span class="sd">    sorted_str: list containing the sorted operators</span>
-<span class="sd">    sign: int telling how many times the sign should be changed</span>
-<span class="sd">          (if sign==0 the string was already sorted)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">verified</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">sign</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">rng</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">string1</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-    <span class="n">rev</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">string1</span><span class="p">)</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">))</span>
-
-    <span class="n">keys</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">string1</span><span class="p">))</span>
-    <span class="n">key_val</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">keys</span><span class="p">,</span> <span class="n">string1</span><span class="p">)))</span>
-
-    <span class="k">while</span> <span class="ow">not</span> <span class="n">verified</span><span class="p">:</span>
-        <span class="n">verified</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">rng</span><span class="p">:</span>
-            <span class="n">left</span> <span class="o">=</span> <span class="n">keys</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="n">right</span> <span class="o">=</span> <span class="n">keys</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">left</span> <span class="o">==</span> <span class="n">right</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">ViolationOfPauliPrinciple</span><span class="p">([</span><span class="n">left</span><span class="p">,</span> <span class="n">right</span><span class="p">])</span>
-            <span class="k">if</span> <span class="n">left</span> <span class="o">&gt;</span> <span class="n">right</span><span class="p">:</span>
-                <span class="n">verified</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="n">keys</span><span class="p">[</span><span class="n">i</span><span class="p">:</span><span class="n">i</span> <span class="o">+</span> <span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">right</span><span class="p">,</span> <span class="n">left</span><span class="p">]</span>
-                <span class="n">sign</span> <span class="o">=</span> <span class="n">sign</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">verified</span><span class="p">:</span>
-            <span class="k">break</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">rev</span><span class="p">:</span>
-            <span class="n">left</span> <span class="o">=</span> <span class="n">keys</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="n">right</span> <span class="o">=</span> <span class="n">keys</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">left</span> <span class="o">==</span> <span class="n">right</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">ViolationOfPauliPrinciple</span><span class="p">([</span><span class="n">left</span><span class="p">,</span> <span class="n">right</span><span class="p">])</span>
-            <span class="k">if</span> <span class="n">left</span> <span class="o">&gt;</span> <span class="n">right</span><span class="p">:</span>
-                <span class="n">verified</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="n">keys</span><span class="p">[</span><span class="n">i</span><span class="p">:</span><span class="n">i</span> <span class="o">+</span> <span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">right</span><span class="p">,</span> <span class="n">left</span><span class="p">]</span>
-                <span class="n">sign</span> <span class="o">=</span> <span class="n">sign</span> <span class="o">+</span> <span class="mi">1</span>
-    <span class="n">string1</span> <span class="o">=</span> <span class="p">[</span> <span class="n">key_val</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">keys</span> <span class="p">]</span>
-    <span class="k">return</span> <span class="p">(</span><span class="n">string1</span><span class="p">,</span> <span class="n">sign</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="evaluate_deltas"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.evaluate_deltas">[docs]</a><span class="k">def</span> <span class="nf">evaluate_deltas</span><span class="p">(</span><span class="n">e</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    We evaluate KroneckerDelta symbols in the expression assuming Einstein summation.</span>
-
-<span class="sd">    If one index is repeated it is summed over and in effect substituted with</span>
-<span class="sd">    the other one. If both indices are repeated we substitute according to what</span>
-<span class="sd">    is the preferred index.  this is determined by</span>
-<span class="sd">    KroneckerDelta.preferred_index and KroneckerDelta.killable_index.</span>
-
-<span class="sd">    In case there are no possible substitutions or if a substitution would</span>
-<span class="sd">    imply a loss of information, nothing is done.</span>
-
-<span class="sd">    In case an index appears in more than one KroneckerDelta, the resulting</span>
-<span class="sd">    substitution depends on the order of the factors.  Since the ordering is platform</span>
-<span class="sd">    dependent, the literal expression resulting from this function may be hard to</span>
-<span class="sd">    predict.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    We assume the following:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, Function, Dummy, KroneckerDelta</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.secondquant import evaluate_deltas</span>
-<span class="sd">    &gt;&gt;&gt; i,j = symbols(&#39;i j&#39;, below_fermi=True, cls=Dummy)</span>
-<span class="sd">    &gt;&gt;&gt; a,b = symbols(&#39;a b&#39;, above_fermi=True, cls=Dummy)</span>
-<span class="sd">    &gt;&gt;&gt; p,q = symbols(&#39;p q&#39;, cls=Dummy)</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; t = Function(&#39;t&#39;)</span>
-
-<span class="sd">    The order of preference for these indices according to KroneckerDelta is</span>
-<span class="sd">    (a, b, i, j, p, q).</span>
-
-<span class="sd">    Trivial cases:</span>
-
-<span class="sd">    &gt;&gt;&gt; evaluate_deltas(KroneckerDelta(i,j)*f(i))       # d_ij f(i) -&gt; f(j)</span>
-<span class="sd">    f(_j)</span>
-<span class="sd">    &gt;&gt;&gt; evaluate_deltas(KroneckerDelta(i,j)*f(j))       # d_ij f(j) -&gt; f(i)</span>
-<span class="sd">    f(_i)</span>
-<span class="sd">    &gt;&gt;&gt; evaluate_deltas(KroneckerDelta(i,p)*f(p))       # d_ip f(p) -&gt; f(i)</span>
-<span class="sd">    f(_i)</span>
-<span class="sd">    &gt;&gt;&gt; evaluate_deltas(KroneckerDelta(q,p)*f(p))       # d_qp f(p) -&gt; f(q)</span>
-<span class="sd">    f(_q)</span>
-<span class="sd">    &gt;&gt;&gt; evaluate_deltas(KroneckerDelta(q,p)*f(q))       # d_qp f(q) -&gt; f(p)</span>
-<span class="sd">    f(_p)</span>
-
-<span class="sd">    More interesting cases:</span>
-
-<span class="sd">    &gt;&gt;&gt; evaluate_deltas(KroneckerDelta(i,p)*t(a,i)*f(p,q))</span>
-<span class="sd">    f(_i, _q)*t(_a, _i)</span>
-<span class="sd">    &gt;&gt;&gt; evaluate_deltas(KroneckerDelta(a,p)*t(a,i)*f(p,q))</span>
-<span class="sd">    f(_a, _q)*t(_a, _i)</span>
-<span class="sd">    &gt;&gt;&gt; evaluate_deltas(KroneckerDelta(p,q)*f(p,q))</span>
-<span class="sd">    f(_p, _p)</span>
-
-<span class="sd">    Finally, here are some cases where nothing is done, because that would</span>
-<span class="sd">    imply a loss of information:</span>
-
-<span class="sd">    &gt;&gt;&gt; evaluate_deltas(KroneckerDelta(i,p)*f(q))</span>
-<span class="sd">    f(_q)*KroneckerDelta(_i, _p)</span>
-<span class="sd">    &gt;&gt;&gt; evaluate_deltas(KroneckerDelta(i,p)*f(i))</span>
-<span class="sd">    f(_i)*KroneckerDelta(_i, _p)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="c"># We treat Deltas only in mul objects</span>
-    <span class="c"># for general function objects we don&#39;t evaluate KroneckerDeltas in arguments,</span>
-    <span class="c"># but here we hard code exceptions to this rule</span>
-    <span class="n">accepted_functions</span> <span class="o">=</span> <span class="p">(</span>
-        <span class="n">Add</span><span class="p">,</span>
-    <span class="p">)</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">accepted_functions</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">e</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">evaluate_deltas</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Mul</span><span class="p">):</span>
-        <span class="c"># find all occurences of delta function and count each index present in</span>
-        <span class="c"># expression.</span>
-        <span class="n">deltas</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">indices</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">i</span><span class="o">.</span><span class="n">atoms</span><span class="p">():</span>
-                <span class="k">if</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">indices</span><span class="p">:</span>
-                    <span class="n">indices</span><span class="p">[</span><span class="n">s</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">indices</span><span class="p">[</span><span class="n">s</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c"># geek counting simplifies logic below</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">KroneckerDelta</span><span class="p">):</span>
-                <span class="n">deltas</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">deltas</span><span class="p">:</span>
-            <span class="c"># If we do something, and there are more deltas, we should recurse</span>
-            <span class="c"># to treat the resulting expression properly</span>
-            <span class="k">if</span> <span class="n">indices</span><span class="p">[</span><span class="n">d</span><span class="o">.</span><span class="n">killable_index</span><span class="p">]:</span>
-                <span class="n">e</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">killable_index</span><span class="p">,</span> <span class="n">d</span><span class="o">.</span><span class="n">preferred_index</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">deltas</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">evaluate_deltas</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">indices</span><span class="p">[</span><span class="n">d</span><span class="o">.</span><span class="n">preferred_index</span><span class="p">]</span> <span class="ow">and</span> <span class="n">d</span><span class="o">.</span><span class="n">indices_contain_equal_information</span><span class="p">:</span>
-                <span class="n">e</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">preferred_index</span><span class="p">,</span> <span class="n">d</span><span class="o">.</span><span class="n">killable_index</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">deltas</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">evaluate_deltas</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">pass</span>
-
-        <span class="k">return</span> <span class="n">e</span>
-    <span class="c"># nothing to do, maybe we hit a Symbol or a number</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">e</span>
-
-</div>
-<div class="viewcode-block" id="substitute_dummies"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.substitute_dummies">[docs]</a><span class="k">def</span> <span class="nf">substitute_dummies</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">new_indices</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">pretty_indices</span><span class="o">=</span><span class="p">{}):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Collect terms by substitution of dummy variables.</span>
-
-<span class="sd">    This routine allows simplification of Add expressions containing terms</span>
-<span class="sd">    which differ only due to dummy variables.</span>
-
-<span class="sd">    The idea is to substitute all dummy variables consistently depending on</span>
-<span class="sd">    the structure of the term.  For each term, we obtain a sequence of all</span>
-<span class="sd">    dummy variables, where the order is determined by the index range, what</span>
-<span class="sd">    factors the index belongs to and its position in each factor.  See</span>
-<span class="sd">    _get_ordered_dummies() for more inforation about the sorting of dummies.</span>
-<span class="sd">    The index sequence is then substituted consistently in each term.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, Function, Dummy</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.secondquant import substitute_dummies</span>
-<span class="sd">    &gt;&gt;&gt; a,b,c,d = symbols(&#39;a b c d&#39;, above_fermi=True, cls=Dummy)</span>
-<span class="sd">    &gt;&gt;&gt; i,j = symbols(&#39;i j&#39;, below_fermi=True, cls=Dummy)</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-
-<span class="sd">    &gt;&gt;&gt; expr = f(a,b) + f(c,d); expr</span>
-<span class="sd">    f(_a, _b) + f(_c, _d)</span>
-
-<span class="sd">    Since a, b, c and d are equivalent summation indices, the expression can be</span>
-<span class="sd">    simplified to a single term (for which the dummy indices are still summed over)</span>
-
-<span class="sd">    &gt;&gt;&gt; substitute_dummies(expr)</span>
-<span class="sd">    2*f(_a, _b)</span>
-
-
-<span class="sd">    Controlling output:</span>
-
-<span class="sd">    By default the dummy symbols that are already present in the expression</span>
-<span class="sd">    will be reused in a different permuation.  However, if new_indices=True,</span>
-<span class="sd">    new dummies will be generated and inserted.  The keyword &#39;pretty_indices&#39;</span>
-<span class="sd">    can be used to control this generation of new symbols.</span>
-
-<span class="sd">    By default the new dummies will be generated on the form i_1, i_2, a_1,</span>
-<span class="sd">    etc.  If you supply a dictionary with key:value pairs in the form:</span>
-
-<span class="sd">        { index_group: string_of_letters }</span>
-
-<span class="sd">    The letters will be used as labels for the new dummy symbols.  The</span>
-<span class="sd">    index_groups must be one of &#39;above&#39;, &#39;below&#39; or &#39;general&#39;.</span>
-
-<span class="sd">    &gt;&gt;&gt; expr = f(a,b,i,j)</span>
-<span class="sd">    &gt;&gt;&gt; my_dummies = { &#39;above&#39;:&#39;st&#39;, &#39;below&#39;:&#39;uv&#39; }</span>
-<span class="sd">    &gt;&gt;&gt; substitute_dummies(expr, new_indices=True, pretty_indices=my_dummies)</span>
-<span class="sd">    f(_s, _t, _u, _v)</span>
-
-<span class="sd">    If we run out of letters, or if there is no keyword for some index_group</span>
-<span class="sd">    the default dummy generator will be used as a fallback:</span>
-
-<span class="sd">    &gt;&gt;&gt; p,q = symbols(&#39;p q&#39;, cls=Dummy)  # general indices</span>
-<span class="sd">    &gt;&gt;&gt; expr = f(p,q)</span>
-<span class="sd">    &gt;&gt;&gt; substitute_dummies(expr, new_indices=True, pretty_indices=my_dummies)</span>
-<span class="sd">    f(_p_0, _p_1)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="c"># setup the replacing dummies</span>
-    <span class="k">if</span> <span class="n">new_indices</span><span class="p">:</span>
-        <span class="n">letters_above</span> <span class="o">=</span> <span class="n">pretty_indices</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;above&#39;</span><span class="p">,</span> <span class="s">&quot;&quot;</span><span class="p">)</span>
-        <span class="n">letters_below</span> <span class="o">=</span> <span class="n">pretty_indices</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;below&#39;</span><span class="p">,</span> <span class="s">&quot;&quot;</span><span class="p">)</span>
-        <span class="n">letters_general</span> <span class="o">=</span> <span class="n">pretty_indices</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;general&#39;</span><span class="p">,</span> <span class="s">&quot;&quot;</span><span class="p">)</span>
-        <span class="n">len_above</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">letters_above</span><span class="p">)</span>
-        <span class="n">len_below</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">letters_below</span><span class="p">)</span>
-        <span class="n">len_general</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">letters_general</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">_i</span><span class="p">(</span><span class="n">number</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">letters_below</span><span class="p">[</span><span class="n">number</span><span class="p">]</span>
-            <span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="s">&#39;i_&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">number</span> <span class="o">-</span> <span class="n">len_below</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">_a</span><span class="p">(</span><span class="n">number</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">letters_above</span><span class="p">[</span><span class="n">number</span><span class="p">]</span>
-            <span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="s">&#39;a_&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">number</span> <span class="o">-</span> <span class="n">len_above</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">_p</span><span class="p">(</span><span class="n">number</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">letters_general</span><span class="p">[</span><span class="n">number</span><span class="p">]</span>
-            <span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="s">&#39;p_&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">number</span> <span class="o">-</span> <span class="n">len_general</span><span class="p">)</span>
-
-    <span class="n">aboves</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">belows</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">generals</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="n">dummies</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Dummy</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">new_indices</span><span class="p">:</span>
-        <span class="n">dummies</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">dummies</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-
-    <span class="c"># generate lists with the dummies we will insert</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">i</span> <span class="o">=</span> <span class="n">p</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">dummies</span><span class="p">:</span>
-        <span class="n">assum</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">assumptions0</span>
-
-        <span class="k">if</span> <span class="n">assum</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;above_fermi&quot;</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">new_indices</span><span class="p">:</span>
-                <span class="n">sym</span> <span class="o">=</span> <span class="n">_a</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-                <span class="n">a</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">l1</span> <span class="o">=</span> <span class="n">aboves</span>
-        <span class="k">elif</span> <span class="n">assum</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;below_fermi&quot;</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">new_indices</span><span class="p">:</span>
-                <span class="n">sym</span> <span class="o">=</span> <span class="n">_i</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">l1</span> <span class="o">=</span> <span class="n">belows</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">new_indices</span><span class="p">:</span>
-                <span class="n">sym</span> <span class="o">=</span> <span class="n">_p</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-                <span class="n">p</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">l1</span> <span class="o">=</span> <span class="n">generals</span>
-
-        <span class="k">if</span> <span class="n">new_indices</span><span class="p">:</span>
-            <span class="n">l1</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Dummy</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="o">**</span><span class="n">assum</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">l1</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-    <span class="n">terms</span> <span class="o">=</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="n">new_terms</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">:</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="nb">iter</span><span class="p">(</span><span class="n">belows</span><span class="p">)</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="nb">iter</span><span class="p">(</span><span class="n">aboves</span><span class="p">)</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="nb">iter</span><span class="p">(</span><span class="n">generals</span><span class="p">)</span>
-        <span class="n">ordered</span> <span class="o">=</span> <span class="n">_get_ordered_dummies</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-        <span class="n">subsdict</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">ordered</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">d</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;below_fermi&#39;</span><span class="p">):</span>
-                <span class="n">subsdict</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">d</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;above_fermi&#39;</span><span class="p">):</span>
-                <span class="n">subsdict</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">subsdict</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-        <span class="n">subslist</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">final_subs</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">subsdict</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="k">if</span> <span class="n">k</span> <span class="o">==</span> <span class="n">v</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">if</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">subsdict</span><span class="p">:</span>
-                <span class="c"># We check if the sequence of substitutions end quickly.  In</span>
-                <span class="c"># that case, we can avoid temporary symbols if we ensure the</span>
-                <span class="c"># correct substitution order.</span>
-                <span class="k">if</span> <span class="n">subsdict</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="ow">in</span> <span class="n">subsdict</span><span class="p">:</span>
-                    <span class="c"># (x, y) -&gt; (y, x),  we need a temporary variable</span>
-                    <span class="n">x</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-                    <span class="n">subslist</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">k</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-                    <span class="n">final_subs</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">v</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="c"># (x, y) -&gt; (y, a),  x-&gt;y must be done last</span>
-                    <span class="c"># but before temporary variables are resolved</span>
-                    <span class="n">final_subs</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">v</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">subslist</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">k</span><span class="p">,</span> <span class="n">v</span><span class="p">))</span>
-        <span class="n">subslist</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">final_subs</span><span class="p">)</span>
-        <span class="n">new_terms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">subslist</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">new_terms</span><span class="p">)</span>
-
-</div>
-<span class="k">class</span> <span class="nc">KeyPrinter</span><span class="p">(</span><span class="n">StrPrinter</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Printer for which only equal objects are equal in print&quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">_print_Dummy</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;(</span><span class="si">%s</span><span class="s">_</span><span class="si">%i</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">dummy_index</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">__kprint</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="n">KeyPrinter</span><span class="p">()</span>
-    <span class="k">return</span> <span class="n">p</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">_get_ordered_dummies</span><span class="p">(</span><span class="n">mul</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns all dummies in the mul sorted in canonical order</span>
-
-<span class="sd">    The purpose of the canonical ordering is that dummies can be substituted</span>
-<span class="sd">    consistently across terms with the result that equivalent terms can be</span>
-<span class="sd">    simplified.</span>
-
-<span class="sd">    It is not possible to determine if two terms are equivalent based solely on</span>
-<span class="sd">    the dummy order.  However, a consistent substitution guided by the ordered</span>
-<span class="sd">    dummies should lead to trivially (non-)equivalent terms, thereby revealing</span>
-<span class="sd">    the equivalence.  This also means that if two terms have identical sequences of</span>
-<span class="sd">    dummies, the (non-)equivalence should already be apparent.</span>
-
-<span class="sd">    Strategy</span>
-<span class="sd">    --------</span>
-
-<span class="sd">    The canoncial order is given by an arbitrary sorting rule.  A sort key</span>
-<span class="sd">    is determined for each dummy as a tuple that depends on all factors where</span>
-<span class="sd">    the index is present.  The dummies are thereby sorted according to the</span>
-<span class="sd">    contraction structure of the term, instead of sorting based solely on the</span>
-<span class="sd">    dummy symbol itself.</span>
-
-<span class="sd">    After all dummies in the term has been assigned a key, we check for identical</span>
-<span class="sd">    keys, i.e. unorderable dummies.  If any are found, we call a specialized</span>
-<span class="sd">    method, _determine_ambiguous(), that will determine a unique order based</span>
-<span class="sd">    on recursive calls to _get_ordered_dummies().</span>
-
-<span class="sd">    Key description</span>
-<span class="sd">    ---------------</span>
-
-<span class="sd">    A high level description of the sort key:</span>
-
-<span class="sd">        1. Range of the dummy index</span>
-<span class="sd">        2. Relation to external (non-dummy) indices</span>
-<span class="sd">        3. Position of the index in the first factor</span>
-<span class="sd">        4. Position of the index in the second factor</span>
-
-<span class="sd">    The sort key is a tuple with the following components:</span>
-
-<span class="sd">        1. A single character indicating the range of the dummy (above, below</span>
-<span class="sd">           or general.)</span>
-<span class="sd">        2. A list of strings with fully masked string representations of all</span>
-<span class="sd">           factors where the dummy is present.  By masked, we mean that dummies</span>
-<span class="sd">           are represented by a symbol to indicate either below fermi, above or</span>
-<span class="sd">           general.  No other information is displayed about the dummies at</span>
-<span class="sd">           this point.  The list is sorted stringwise.</span>
-<span class="sd">        3. An integer number indicating the position of the index, in the first</span>
-<span class="sd">           factor as sorted in 2.</span>
-<span class="sd">        4. An integer number indicating the position of the index, in the second</span>
-<span class="sd">           factor as sorted in 2.</span>
-
-<span class="sd">    If a factor is either of type AntiSymmetricTensor or SqOperator, the index</span>
-<span class="sd">    position in items 3 and 4 is indicated as &#39;upper&#39; or &#39;lower&#39; only.</span>
-<span class="sd">    (Creation operators are considered upper and annihilation operators lower.)</span>
-
-<span class="sd">    If the masked factors are identical, the two factors cannot be ordered</span>
-<span class="sd">    unambiguously in item 2.  In this case, items 3, 4 are left out.  If several</span>
-<span class="sd">    indices are contracted between the unorderable factors, it will be handled by</span>
-<span class="sd">    _determine_ambiguous()</span>
-
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># setup dicts to avoid repeated calculations in key()</span>
-    <span class="n">args</span> <span class="o">=</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">mul</span><span class="p">)</span>
-    <span class="n">fac_dum</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">([</span> <span class="p">(</span><span class="n">fac</span><span class="p">,</span> <span class="n">fac</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Dummy</span><span class="p">))</span> <span class="k">for</span> <span class="n">fac</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span> <span class="p">)</span>
-    <span class="n">fac_repr</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">([</span> <span class="p">(</span><span class="n">fac</span><span class="p">,</span> <span class="n">__kprint</span><span class="p">(</span><span class="n">fac</span><span class="p">))</span> <span class="k">for</span> <span class="n">fac</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span> <span class="p">)</span>
-    <span class="n">all_dums</span> <span class="o">=</span> <span class="nb">reduce</span><span class="p">(</span><span class="nb">set</span><span class="o">.</span><span class="n">union</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="n">fac_dum</span><span class="o">.</span><span class="n">values</span><span class="p">()),</span> <span class="nb">set</span><span class="p">())</span>
-    <span class="n">mask</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">all_dums</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">d</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;below_fermi&#39;</span><span class="p">):</span>
-            <span class="n">mask</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">=</span> <span class="s">&#39;0&#39;</span>
-        <span class="k">elif</span> <span class="n">d</span><span class="o">.</span><span class="n">assumptions0</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;above_fermi&#39;</span><span class="p">):</span>
-            <span class="n">mask</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">=</span> <span class="s">&#39;1&#39;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">mask</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">=</span> <span class="s">&#39;2&#39;</span>
-    <span class="n">dum_repr</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">([</span> <span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">__kprint</span><span class="p">(</span><span class="n">d</span><span class="p">))</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">all_dums</span> <span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_key</span><span class="p">(</span><span class="n">d</span><span class="p">):</span>
-        <span class="n">dumstruct</span> <span class="o">=</span> <span class="p">[</span> <span class="n">fac</span> <span class="k">for</span> <span class="n">fac</span> <span class="ow">in</span> <span class="n">fac_dum</span> <span class="k">if</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">fac_dum</span><span class="p">[</span><span class="n">fac</span><span class="p">]</span> <span class="p">]</span>
-        <span class="n">other_dums</span> <span class="o">=</span> <span class="nb">reduce</span><span class="p">(</span>
-            <span class="nb">set</span><span class="o">.</span><span class="n">union</span><span class="p">,</span> <span class="p">[</span> <span class="n">fac_dum</span><span class="p">[</span><span class="n">fac</span><span class="p">]</span> <span class="k">for</span> <span class="n">fac</span> <span class="ow">in</span> <span class="n">dumstruct</span> <span class="p">],</span> <span class="nb">set</span><span class="p">())</span>
-        <span class="n">fac</span> <span class="o">=</span> <span class="n">dumstruct</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">other_dums</span> <span class="ow">is</span> <span class="n">fac_dum</span><span class="p">[</span><span class="n">fac</span><span class="p">]:</span>
-            <span class="n">other_dums</span> <span class="o">=</span> <span class="n">fac_dum</span><span class="p">[</span><span class="n">fac</span><span class="p">]</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="n">other_dums</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-        <span class="n">masked_facs</span> <span class="o">=</span> <span class="p">[</span> <span class="n">fac_repr</span><span class="p">[</span><span class="n">fac</span><span class="p">]</span> <span class="k">for</span> <span class="n">fac</span> <span class="ow">in</span> <span class="n">dumstruct</span> <span class="p">]</span>
-        <span class="k">for</span> <span class="n">d2</span> <span class="ow">in</span> <span class="n">other_dums</span><span class="p">:</span>
-            <span class="n">masked_facs</span> <span class="o">=</span> <span class="p">[</span> <span class="n">fac</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">dum_repr</span><span class="p">[</span><span class="n">d2</span><span class="p">],</span> <span class="n">mask</span><span class="p">[</span><span class="n">d2</span><span class="p">])</span>
-                    <span class="k">for</span> <span class="n">fac</span> <span class="ow">in</span> <span class="n">masked_facs</span> <span class="p">]</span>
-        <span class="n">all_masked</span> <span class="o">=</span> <span class="p">[</span> <span class="n">fac</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">dum_repr</span><span class="p">[</span><span class="n">d</span><span class="p">],</span> <span class="n">mask</span><span class="p">[</span><span class="n">d</span><span class="p">])</span>
-                       <span class="k">for</span> <span class="n">fac</span> <span class="ow">in</span> <span class="n">masked_facs</span> <span class="p">]</span>
-        <span class="n">masked_facs</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">dumstruct</span><span class="p">,</span> <span class="n">masked_facs</span><span class="p">)))</span>
-
-        <span class="c"># dummies for which the ordering cannot be determined</span>
-        <span class="k">if</span> <span class="n">has_dups</span><span class="p">(</span><span class="n">all_masked</span><span class="p">):</span>
-            <span class="n">all_masked</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-            <span class="k">return</span> <span class="n">mask</span><span class="p">[</span><span class="n">d</span><span class="p">],</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">all_masked</span><span class="p">)</span>  <span class="c"># positions are ambiguous</span>
-
-        <span class="c"># sort factors according to fully masked strings</span>
-        <span class="n">keydict</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">dumstruct</span><span class="p">,</span> <span class="n">all_masked</span><span class="p">)))</span>
-        <span class="n">dumstruct</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">keydict</span><span class="p">[</span><span class="n">x</span><span class="p">])</span>
-        <span class="n">all_masked</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-
-        <span class="n">pos_val</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">fac</span> <span class="ow">in</span> <span class="n">dumstruct</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">fac</span><span class="p">,</span> <span class="n">AntiSymmetricTensor</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">fac</span><span class="o">.</span><span class="n">upper</span><span class="p">:</span>
-                    <span class="n">pos_val</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;u&#39;</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">fac</span><span class="o">.</span><span class="n">lower</span><span class="p">:</span>
-                    <span class="n">pos_val</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;l&#39;</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">fac</span><span class="p">,</span> <span class="n">Creator</span><span class="p">):</span>
-                <span class="n">pos_val</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;u&#39;</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">fac</span><span class="p">,</span> <span class="n">Annihilator</span><span class="p">):</span>
-                <span class="n">pos_val</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;l&#39;</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">fac</span><span class="p">,</span> <span class="n">NO</span><span class="p">):</span>
-                <span class="n">ops</span> <span class="o">=</span> <span class="p">[</span> <span class="n">op</span> <span class="k">for</span> <span class="n">op</span> <span class="ow">in</span> <span class="n">fac</span> <span class="k">if</span> <span class="n">op</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">d</span><span class="p">)</span> <span class="p">]</span>
-                <span class="k">for</span> <span class="n">op</span> <span class="ow">in</span> <span class="n">ops</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">op</span><span class="p">,</span> <span class="n">Creator</span><span class="p">):</span>
-                        <span class="n">pos_val</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;u&#39;</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">pos_val</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;l&#39;</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># fallback to position in string representation</span>
-                <span class="n">facpos</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-                <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">facpos</span> <span class="o">=</span> <span class="n">masked_facs</span><span class="p">[</span><span class="n">fac</span><span class="p">]</span><span class="o">.</span><span class="n">find</span><span class="p">(</span><span class="n">dum_repr</span><span class="p">[</span><span class="n">d</span><span class="p">],</span> <span class="n">facpos</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">facpos</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                        <span class="k">break</span>
-                    <span class="n">pos_val</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">facpos</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">mask</span><span class="p">[</span><span class="n">d</span><span class="p">],</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">all_masked</span><span class="p">),</span> <span class="n">pos_val</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">pos_val</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-    <span class="n">dumkey</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">all_dums</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">_key</span><span class="p">,</span> <span class="n">all_dums</span><span class="p">)))))</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">all_dums</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">dumkey</span><span class="p">[</span><span class="n">x</span><span class="p">])</span>
-    <span class="k">if</span> <span class="n">has_dups</span><span class="p">(</span><span class="nb">iter</span><span class="p">(</span><span class="n">dumkey</span><span class="o">.</span><span class="n">values</span><span class="p">())):</span>
-        <span class="c"># We have ambiguities</span>
-        <span class="n">unordered</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">set</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">d</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">dumkey</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">unordered</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="p">[</span> <span class="n">k</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">unordered</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">unordered</span><span class="p">[</span><span class="n">k</span><span class="p">])</span> <span class="o">&lt;</span> <span class="mi">2</span> <span class="p">]:</span>
-            <span class="k">del</span> <span class="n">unordered</span><span class="p">[</span><span class="n">k</span><span class="p">]</span>
-
-        <span class="n">unordered</span> <span class="o">=</span> <span class="p">[</span> <span class="n">unordered</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">unordered</span><span class="p">)</span> <span class="p">]</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">_determine_ambiguous</span><span class="p">(</span><span class="n">mul</span><span class="p">,</span> <span class="n">result</span><span class="p">,</span> <span class="n">unordered</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="k">def</span> <span class="nf">_determine_ambiguous</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">ordered</span><span class="p">,</span> <span class="n">ambiguous_groups</span><span class="p">):</span>
-    <span class="c"># We encountered a term for which the dummy substitution is ambiguous.</span>
-    <span class="c"># This happens for terms with 2 or more contractions between factors that</span>
-    <span class="c"># cannot be uniquely ordered independent of summation indices.  For</span>
-    <span class="c"># example:</span>
-    <span class="c">#</span>
-    <span class="c"># Sum(p, q) v^{p, .}_{q, .}v^{q, .}_{p, .}</span>
-    <span class="c">#</span>
-    <span class="c"># Assuming that the indices represented by . are dummies with the</span>
-    <span class="c"># same range, the factors cannot be ordered, and there is no</span>
-    <span class="c"># way to determine a consistent ordering of p and q.</span>
-    <span class="c">#</span>
-    <span class="c"># The strategy employed here, is to relabel all unambiguous dummies with</span>
-    <span class="c"># non-dummy symbols and call _get_ordered_dummies again.  This procedure is</span>
-    <span class="c"># applied to the entire term so there is a possibility that</span>
-    <span class="c"># _determine_ambiguous() is called again from a deeper recursion level.</span>
-
-    <span class="c"># break recursion if there are no ordered dummies</span>
-    <span class="n">all_ambiguous</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="k">for</span> <span class="n">dummies</span> <span class="ow">in</span> <span class="n">ambiguous_groups</span><span class="p">:</span>
-        <span class="n">all_ambiguous</span> <span class="o">|=</span> <span class="n">dummies</span>
-    <span class="n">all_ordered</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">ordered</span><span class="p">)</span> <span class="o">-</span> <span class="n">all_ambiguous</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">all_ordered</span><span class="p">:</span>
-        <span class="c"># FIXME: If we arrive here, there are no ordered dummies. A method to</span>
-        <span class="c"># handle this needs to be implemented.  In order to return something</span>
-        <span class="c"># useful nevertheless, we choose arbitrarily the first dummy and</span>
-        <span class="c"># determine the rest from this one.  This method is dependent on the</span>
-        <span class="c"># actual dummy labels which violates an assumption for the canonization</span>
-        <span class="c"># procedure.  A better implementation is needed.</span>
-        <span class="n">group</span> <span class="o">=</span> <span class="p">[</span> <span class="n">d</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">ordered</span> <span class="k">if</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">ambiguous_groups</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="p">]</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">group</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">all_ordered</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-        <span class="n">ambiguous_groups</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-
-    <span class="n">stored_counter</span> <span class="o">=</span> <span class="n">_symbol_factory</span><span class="o">.</span><span class="n">_counter</span>
-    <span class="n">subslist</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="p">[</span> <span class="n">d</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">ordered</span> <span class="k">if</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">all_ordered</span> <span class="p">]:</span>
-        <span class="n">nondum</span> <span class="o">=</span> <span class="n">_symbol_factory</span><span class="o">.</span><span class="n">_next</span><span class="p">()</span>
-        <span class="n">subslist</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">d</span><span class="p">,</span> <span class="n">nondum</span><span class="p">))</span>
-    <span class="n">newterm</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">subslist</span><span class="p">)</span>
-    <span class="n">neworder</span> <span class="o">=</span> <span class="n">_get_ordered_dummies</span><span class="p">(</span><span class="n">newterm</span><span class="p">)</span>
-    <span class="n">_symbol_factory</span><span class="o">.</span><span class="n">_set_counter</span><span class="p">(</span><span class="n">stored_counter</span><span class="p">)</span>
-
-    <span class="c"># update ordered list with new information</span>
-    <span class="k">for</span> <span class="n">group</span> <span class="ow">in</span> <span class="n">ambiguous_groups</span><span class="p">:</span>
-        <span class="n">ordered_group</span> <span class="o">=</span> <span class="p">[</span> <span class="n">d</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">neworder</span> <span class="k">if</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">group</span> <span class="p">]</span>
-        <span class="n">ordered_group</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">ordered</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">group</span><span class="p">:</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ordered_group</span><span class="o">.</span><span class="n">pop</span><span class="p">())</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-        <span class="n">ordered</span> <span class="o">=</span> <span class="n">result</span>
-    <span class="k">return</span> <span class="n">ordered</span>
-
-
-<span class="k">class</span> <span class="nc">_SymbolFactory</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">label</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_counterVar</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_label</span> <span class="o">=</span> <span class="n">label</span>
-
-    <span class="k">def</span> <span class="nf">_set_counter</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Sets counter to value.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_counterVar</span> <span class="o">=</span> <span class="n">value</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_counter</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        What counter is currently at.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_counterVar</span>
-
-    <span class="k">def</span> <span class="nf">_next</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Generates the next symbols and increments counter by 1.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s%i</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_label</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_counterVar</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_counterVar</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="n">s</span>
-<span class="n">_symbol_factory</span> <span class="o">=</span> <span class="n">_SymbolFactory</span><span class="p">(</span><span class="s">&#39;_]&quot;]_&#39;</span><span class="p">)</span>  <span class="c"># most certainly a unique label</span>
-
-
-<span class="nd">@cacheit</span>
-<span class="k">def</span> <span class="nf">_get_contractions</span><span class="p">(</span><span class="n">string1</span><span class="p">,</span> <span class="n">keep_only_fully_contracted</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns Add-object with contracted terms.</span>
-
-<span class="sd">    Uses recursion to find all contractions. -- Internal helper function --</span>
-
-<span class="sd">    Will find nonzero contractions in string1 between indices given in</span>
-<span class="sd">    leftrange and rightrange.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="c"># Should we store current level of contraction?</span>
-    <span class="k">if</span> <span class="n">keep_only_fully_contracted</span> <span class="ow">and</span> <span class="n">string1</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[</span><span class="n">NO</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">string1</span><span class="p">))]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">string1</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">string1</span><span class="p">)):</span>
-
-            <span class="n">c</span> <span class="o">=</span> <span class="n">contraction</span><span class="p">(</span><span class="n">string1</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">string1</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-
-            <span class="k">if</span> <span class="n">c</span><span class="p">:</span>
-                <span class="c"># print &quot;found contraction&quot;,c</span>
-
-                <span class="n">sign</span> <span class="o">=</span> <span class="p">(</span><span class="n">j</span> <span class="o">-</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span>
-                <span class="k">if</span> <span class="n">sign</span><span class="p">:</span>
-                    <span class="n">coeff</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">*</span><span class="n">c</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">coeff</span> <span class="o">=</span> <span class="n">c</span>
-
-                <span class="c">#</span>
-                <span class="c">#  Call next level of recursion</span>
-                <span class="c">#  ============================</span>
-                <span class="c">#</span>
-                <span class="c"># We now need to find more contractions among operators</span>
-                <span class="c">#</span>
-                <span class="c"># oplist = string1[:i]+ string1[i+1:j] + string1[j+1:]</span>
-                <span class="c">#</span>
-                <span class="c"># To prevent overcounting, we don&#39;t allow contractions</span>
-                <span class="c"># we have already encountered. i.e. contractions between</span>
-                <span class="c">#       string1[:i] &lt;---&gt; string1[i+1:j]</span>
-                <span class="c"># and   string1[:i] &lt;---&gt; string1[j+1:].</span>
-                <span class="c">#</span>
-                <span class="c"># This leaves the case:</span>
-                <span class="n">oplist</span> <span class="o">=</span> <span class="n">string1</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:</span><span class="n">j</span><span class="p">]</span> <span class="o">+</span> <span class="n">string1</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-
-                <span class="k">if</span> <span class="n">oplist</span><span class="p">:</span>
-
-                    <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">coeff</span><span class="o">*</span><span class="n">NO</span><span class="p">(</span>
-                        <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">string1</span><span class="p">[:</span><span class="n">i</span><span class="p">])</span><span class="o">*</span><span class="n">_get_contractions</span><span class="p">(</span> <span class="n">oplist</span><span class="p">,</span>
-                            <span class="n">keep_only_fully_contracted</span><span class="o">=</span><span class="n">keep_only_fully_contracted</span><span class="p">)))</span>
-
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">coeff</span><span class="o">*</span><span class="n">NO</span><span class="p">(</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">string1</span><span class="p">[:</span><span class="n">i</span><span class="p">])))</span>
-
-        <span class="k">if</span> <span class="n">keep_only_fully_contracted</span><span class="p">:</span>
-            <span class="k">break</span>   <span class="c"># next iteration over i leaves leftmost operator string1[0] uncontracted</span>
-
-    <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">result</span><span class="p">)</span>
-
-
-<span class="c"># @cacheit</span>
-<div class="viewcode-block" id="wicks"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.wicks">[docs]</a><span class="k">def</span> <span class="nf">wicks</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the normal ordered equivalent of an expression using Wicks Theorem.</span>
-
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, Function, Dummy</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.secondquant import wicks, F, Fd, NO</span>
-<span class="sd">    &gt;&gt;&gt; p,q,r = symbols(&#39;p,q,r&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; wicks(Fd(p)*F(q))  # doctest: +SKIP</span>
-<span class="sd">    d(p, q)*d(q, _i) + NO(CreateFermion(p)*AnnihilateFermion(q))</span>
-
-<span class="sd">    By default, the expression is expanded:</span>
-
-<span class="sd">    &gt;&gt;&gt; wicks(F(p)*(F(q)+F(r))) # doctest: +SKIP</span>
-<span class="sd">    NO(AnnihilateFermion(p)*AnnihilateFermion(q)) + NO(</span>
-<span class="sd">        AnnihilateFermion(p)*AnnihilateFermion(r))</span>
-
-<span class="sd">    With the keyword &#39;keep_only_fully_contracted=True&#39;, only fully contracted</span>
-<span class="sd">    terms are returned.</span>
-
-<span class="sd">    By request, the result can be simplified in the following order:</span>
-<span class="sd">     -- KroneckerDelta functions are evaluated</span>
-<span class="sd">     -- Dummy variables are substituted consistently across terms</span>
-
-<span class="sd">    &gt;&gt;&gt; p, q, r = symbols(&#39;p q r&#39;, cls=Dummy)</span>
-<span class="sd">    &gt;&gt;&gt; wicks(Fd(p)*(F(q)+F(r)), keep_only_fully_contracted=True) # doctest: +SKIP</span>
-<span class="sd">    KroneckerDelta(_i, _q)*KroneckerDelta(</span>
-<span class="sd">        _p, _q) + KroneckerDelta(_i, _r)*KroneckerDelta(_p, _r)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">e</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="n">opts</span> <span class="o">=</span> <span class="p">{</span>
-        <span class="s">&#39;simplify_kronecker_deltas&#39;</span><span class="p">:</span> <span class="bp">False</span><span class="p">,</span>
-        <span class="s">&#39;expand&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">,</span>
-        <span class="s">&#39;simplify_dummies&#39;</span><span class="p">:</span> <span class="bp">False</span><span class="p">,</span>
-        <span class="s">&#39;keep_only_fully_contracted&#39;</span><span class="p">:</span> <span class="bp">False</span>
-    <span class="p">}</span>
-    <span class="n">opts</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">kw_args</span><span class="p">)</span>
-
-    <span class="c"># check if we are already normally ordered</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">NO</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">opts</span><span class="p">[</span><span class="s">&#39;keep_only_fully_contracted&#39;</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">e</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">FermionicOperator</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">opts</span><span class="p">[</span><span class="s">&#39;keep_only_fully_contracted&#39;</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">e</span>
-
-    <span class="c"># break up any NO-objects, and evaluate commutators</span>
-    <span class="n">e</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="n">wicks</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="c"># make sure we have only one term to consider</span>
-    <span class="n">e</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Add</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">opts</span><span class="p">[</span><span class="s">&#39;simplify_dummies&#39;</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="n">substitute_dummies</span><span class="p">(</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">wicks</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">)</span> <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">]))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">wicks</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">)</span> <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="c"># For Mul-objects we can actually do something</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Mul</span><span class="p">):</span>
-
-        <span class="c"># we dont want to mess around with commuting part of Mul</span>
-        <span class="c"># so we factorize it out before starting recursion</span>
-        <span class="n">c_part</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">string1</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">factor</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">factor</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-                <span class="n">c_part</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">factor</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">string1</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">factor</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">string1</span><span class="p">)</span>
-
-        <span class="c"># catch trivial cases</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">e</span>
-        <span class="k">elif</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">opts</span><span class="p">[</span><span class="s">&#39;keep_only_fully_contracted&#39;</span><span class="p">]:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">e</span>
-
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># non-trivial</span>
-
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">string1</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">BosonicOperator</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-            <span class="n">string1</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">string1</span><span class="p">)</span>
-
-            <span class="c"># recursion over higher order contractions</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">_get_contractions</span><span class="p">(</span><span class="n">string1</span><span class="p">,</span>
-                <span class="n">keep_only_fully_contracted</span><span class="o">=</span><span class="n">opts</span><span class="p">[</span><span class="s">&#39;keep_only_fully_contracted&#39;</span><span class="p">]</span> <span class="p">)</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">c_part</span><span class="p">)</span><span class="o">*</span><span class="n">result</span>
-
-        <span class="k">if</span> <span class="n">opts</span><span class="p">[</span><span class="s">&#39;expand&#39;</span><span class="p">]:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">opts</span><span class="p">[</span><span class="s">&#39;simplify_kronecker_deltas&#39;</span><span class="p">]:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">evaluate_deltas</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="c"># there was nothing to do</span>
-    <span class="k">return</span> <span class="n">e</span>
-
-</div>
-<div class="viewcode-block" id="PermutationOperator"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.PermutationOperator">[docs]</a><span class="k">class</span> <span class="nc">PermutationOperator</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Represents the index permutation operator P(ij).</span>
-
-<span class="sd">    P(ij)*f(i)*g(j) = f(i)*g(j) - f(j)*g(i)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">):</span>
-        <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)))</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">i</span> <span class="o">&gt;</span> <span class="n">j</span><span class="p">):</span>
-            <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-<div class="viewcode-block" id="PermutationOperator.get_permuted"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.PermutationOperator.get_permuted">[docs]</a>    <span class="k">def</span> <span class="nf">get_permuted</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns -expr with permuted indices.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols, Function</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.secondquant import PermutationOperator</span>
-<span class="sd">        &gt;&gt;&gt; p,q = symbols(&#39;p,q&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; PermutationOperator(p,q).get_permuted(f(p,q))</span>
-<span class="sd">        -f(q, p)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">j</span><span class="p">):</span>
-            <span class="n">tmp</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">()</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">tmp</span><span class="p">)</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">tmp</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">*</span><span class="n">expr</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span>
-</div>
-    <span class="k">def</span> <span class="nf">_latex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printer</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;P(</span><span class="si">%s%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-
-</div>
-<div class="viewcode-block" id="simplify_index_permutations"><a class="viewcode-back" href="../../../modules/physics/secondquant.html#sympy.physics.secondquant.simplify_index_permutations">[docs]</a><span class="k">def</span> <span class="nf">simplify_index_permutations</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">permutation_operators</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Performs simplification by introducing PermutationOperators where appropriate.</span>
-
-<span class="sd">    Schematically:</span>
-<span class="sd">        [abij] - [abji] - [baij] + [baji] -&gt;  P(ab)*P(ij)*[abij]</span>
-
-<span class="sd">    permutation_operators is a list of PermutationOperators to consider.</span>
-
-<span class="sd">    If permutation_operators=[P(ab),P(ij)] we will try to introduce the</span>
-<span class="sd">    permutation operators P(ij) and P(ab) in the expression.  If there are other</span>
-<span class="sd">    possible simplifications, we ignore them.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, Function</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.secondquant import simplify_index_permutations</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.secondquant import PermutationOperator</span>
-<span class="sd">    &gt;&gt;&gt; p,q,r,s = symbols(&#39;p,q,r,s&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; g = Function(&#39;g&#39;)</span>
-
-<span class="sd">    &gt;&gt;&gt; expr = f(p)*g(q) - f(q)*g(p); expr</span>
-<span class="sd">    f(p)*g(q) - f(q)*g(p)</span>
-<span class="sd">    &gt;&gt;&gt; simplify_index_permutations(expr,[PermutationOperator(p,q)])</span>
-<span class="sd">    f(p)*g(q)*PermutationOperator(p, q)</span>
-
-<span class="sd">    &gt;&gt;&gt; PermutList = [PermutationOperator(p,q),PermutationOperator(r,s)]</span>
-<span class="sd">    &gt;&gt;&gt; expr = f(p,r)*g(q,s) - f(q,r)*g(p,s) + f(q,s)*g(p,r) - f(p,s)*g(q,r)</span>
-<span class="sd">    &gt;&gt;&gt; simplify_index_permutations(expr,PermutList)</span>
-<span class="sd">    f(p, r)*g(q, s)*PermutationOperator(p, q)*PermutationOperator(r, s)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_get_indices</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">ind</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Collects indices recursively in predictable order.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">ind</span><span class="p">:</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                    <span class="n">result</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">_get_indices</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">ind</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="k">def</span> <span class="nf">_choose_one_to_keep</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">ind</span><span class="p">):</span>
-        <span class="c"># we keep the one where indices in ind are in order ind[0] &lt; ind[1]</span>
-        <span class="k">if</span> <span class="n">_get_indices</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">ind</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">_get_indices</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">ind</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">a</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">b</span>
-
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Add</span><span class="p">):</span>
-        <span class="n">terms</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">P</span> <span class="ow">in</span> <span class="n">permutation_operators</span><span class="p">:</span>
-            <span class="n">new_terms</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([])</span>
-            <span class="n">on_hold</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([])</span>
-            <span class="k">while</span> <span class="n">terms</span><span class="p">:</span>
-                <span class="n">term</span> <span class="o">=</span> <span class="n">terms</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-                <span class="n">permuted</span> <span class="o">=</span> <span class="n">P</span><span class="o">.</span><span class="n">get_permuted</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">permuted</span> <span class="ow">in</span> <span class="n">terms</span> <span class="o">|</span> <span class="n">on_hold</span><span class="p">:</span>
-                    <span class="k">try</span><span class="p">:</span>
-                        <span class="n">terms</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">permuted</span><span class="p">)</span>
-                    <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-                        <span class="n">on_hold</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">permuted</span><span class="p">)</span>
-                    <span class="n">keep</span> <span class="o">=</span> <span class="n">_choose_one_to_keep</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">permuted</span><span class="p">,</span> <span class="n">P</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-                    <span class="n">new_terms</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">P</span><span class="o">*</span><span class="n">keep</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-
-                    <span class="c"># Some terms must get a second chance because the permuted</span>
-                    <span class="c"># term may already have canonical dummy ordering.  Then</span>
-                    <span class="c"># substitute_dummies() does nothing.  However, the other</span>
-                    <span class="c"># term, if it exists, will be able to match with us.</span>
-                    <span class="n">permuted1</span> <span class="o">=</span> <span class="n">permuted</span>
-                    <span class="n">permuted</span> <span class="o">=</span> <span class="n">substitute_dummies</span><span class="p">(</span><span class="n">permuted</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">permuted1</span> <span class="o">==</span> <span class="n">permuted</span><span class="p">:</span>
-                        <span class="n">on_hold</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-                    <span class="k">elif</span> <span class="n">permuted</span> <span class="ow">in</span> <span class="n">terms</span> <span class="o">|</span> <span class="n">on_hold</span><span class="p">:</span>
-                        <span class="k">try</span><span class="p">:</span>
-                            <span class="n">terms</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">permuted</span><span class="p">)</span>
-                        <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-                            <span class="n">on_hold</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">permuted</span><span class="p">)</span>
-                        <span class="n">keep</span> <span class="o">=</span> <span class="n">_choose_one_to_keep</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">permuted</span><span class="p">,</span> <span class="n">P</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-                        <span class="n">new_terms</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">P</span><span class="o">*</span><span class="n">keep</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">new_terms</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-            <span class="n">terms</span> <span class="o">=</span> <span class="n">new_terms</span> <span class="o">|</span> <span class="n">on_hold</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">terms</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">expr</span></div>
-</pre></div>
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diff --git a/dev-py3k/_modules/sympy/physics/sho.html b/dev-py3k/_modules/sympy/physics/sho.html
deleted file mode 100644
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-            
-  <h1>Source code for sympy.physics.sho</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">Rational</span>
-<span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">assoc_laguerre</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">factorial2</span>
-
-
-<div class="viewcode-block" id="R_nl"><a class="viewcode-back" href="../../../modules/physics/sho.html#sympy.physics.sho.R_nl">[docs]</a><span class="k">def</span> <span class="nf">R_nl</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="n">r</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the radial wavefunction R_{nl} for a 3d isotropic harmonic</span>
-<span class="sd">    oscillator.</span>
-
-<span class="sd">    ``n``</span>
-<span class="sd">        the &quot;nodal&quot; quantum number.  Corresponds to the number of nodes in</span>
-<span class="sd">        the wavefunction.  n &gt;= 0</span>
-<span class="sd">    ``l``</span>
-<span class="sd">        the quantum number for orbital angular momentum</span>
-<span class="sd">    ``nu``</span>
-<span class="sd">        mass-scaled frequency: nu = m*omega/(2*hbar) where `m&#39; is the mass</span>
-<span class="sd">        and `omega` the frequency of the oscillator.</span>
-<span class="sd">        (in atomic units nu == omega/2)</span>
-<span class="sd">    ``r``</span>
-<span class="sd">        Radial coordinate</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.sho import R_nl</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import var</span>
-<span class="sd">    &gt;&gt;&gt; var(&quot;r nu l&quot;)</span>
-<span class="sd">    (r, nu, l)</span>
-<span class="sd">    &gt;&gt;&gt; R_nl(0, 0, 1, r)</span>
-<span class="sd">    2*2**(3/4)*exp(-r**2)/pi**(1/4)</span>
-<span class="sd">    &gt;&gt;&gt; R_nl(1, 0, 1, r)</span>
-<span class="sd">    4*2**(1/4)*sqrt(3)*(-2*r**2 + 3/2)*exp(-r**2)/(3*pi**(1/4))</span>
-
-<span class="sd">    l, nu and r may be symbolic:</span>
-
-<span class="sd">    &gt;&gt;&gt; R_nl(0, 0, nu, r)</span>
-<span class="sd">    2*2**(3/4)*sqrt(nu**(3/2))*exp(-nu*r**2)/pi**(1/4)</span>
-<span class="sd">    &gt;&gt;&gt; R_nl(0, l, 1, r)</span>
-<span class="sd">    r**l*sqrt(2**(l + 3/2)*2**(l + 2)/(2*l + 1)!!)*exp(-r**2)/pi**(1/4)</span>
-
-<span class="sd">    The normalization of the radial wavefunction is:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Integral, oo</span>
-<span class="sd">    &gt;&gt;&gt; Integral(R_nl(0, 0, 1, r)**2 * r**2, (r, 0, oo)).n()</span>
-<span class="sd">    1.00000000000000</span>
-<span class="sd">    &gt;&gt;&gt; Integral(R_nl(1, 0, 1, r)**2 * r**2, (r, 0, oo)).n()</span>
-<span class="sd">    1.00000000000000</span>
-<span class="sd">    &gt;&gt;&gt; Integral(R_nl(1, 1, 1, r)**2 * r**2, (r, 0, oo)).n()</span>
-<span class="sd">    1.00000000000000</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="n">r</span><span class="p">]))</span>
-
-    <span class="c"># formula uses n &gt;= 1 (instead of nodal n &gt;= 0)</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span>
-    <span class="n">C</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span>
-            <span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">nu</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">l</span> <span class="o">+</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span><span class="o">*</span><span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="n">l</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span><span class="o">/</span>
-            <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">factorial2</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">l</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)))</span>
-    <span class="p">)</span>
-    <span class="k">return</span> <span class="n">C</span><span class="o">*</span><span class="n">r</span><span class="o">**</span><span class="p">(</span><span class="n">l</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">nu</span><span class="o">*</span><span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">assoc_laguerre</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">l</span> <span class="o">+</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">nu</span><span class="o">*</span><span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="E_nl"><a class="viewcode-back" href="../../../modules/physics/sho.html#sympy.physics.sho.E_nl">[docs]</a><span class="k">def</span> <span class="nf">E_nl</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="n">hw</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the Energy of an isotropic harmonic oscillator</span>
-
-<span class="sd">    ``n``</span>
-<span class="sd">        the &quot;nodal&quot; quantum number</span>
-<span class="sd">    ``l``</span>
-<span class="sd">        the orbital angular momentum</span>
-<span class="sd">    ``hw``</span>
-<span class="sd">        the harmonic oscillator parameter.</span>
-
-<span class="sd">    The unit of the returned value matches the unit of hw, since the energy is</span>
-<span class="sd">    calculated as:</span>
-
-<span class="sd">        E_nl = (2*n + l + 3/2)*hw</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.sho import E_nl</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; x, y, z = symbols(&#39;x, y, z&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; E_nl(x, y, z)</span>
-<span class="sd">    z*(2*x + y + 3/2)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span> <span class="o">+</span> <span class="n">l</span> <span class="o">+</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span><span class="o">*</span><span class="n">hw</span></div>
-</pre></div>
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diff --git a/dev-py3k/_modules/sympy/physics/units.html b/dev-py3k/_modules/sympy/physics/units.html
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-  <h1>Source code for sympy.physics.units</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Physical units and dimensions.</span>
-
-<span class="sd">The base class is Unit, where all here defined units (~200) inherit from.</span>
-
-<span class="sd">The find_unit function can help you find units for a given quantity:</span>
-
-<span class="sd">    &gt;&gt;&gt; import sympy.physics.units as u</span>
-<span class="sd">    &gt;&gt;&gt; u.find_unit(&#39;coul&#39;)</span>
-<span class="sd">    [&#39;coulomb&#39;, &#39;coulombs&#39;]</span>
-<span class="sd">    &gt;&gt;&gt; u.find_unit(u.charge)</span>
-<span class="sd">    [&#39;C&#39;, &#39;charge&#39;, &#39;coulomb&#39;, &#39;coulombs&#39;]</span>
-<span class="sd">    &gt;&gt;&gt; u.coulomb</span>
-<span class="sd">    A*s</span>
-
-<span class="sd">Units are always given in terms of base units that have a name and</span>
-<span class="sd">an abbreviation:</span>
-
-<span class="sd">    &gt;&gt;&gt; u.A.name</span>
-<span class="sd">    &#39;ampere&#39;</span>
-<span class="sd">    &gt;&gt;&gt; u.ampere.abbrev</span>
-<span class="sd">    &#39;A&#39;</span>
-
-<span class="sd">The generic name for a unit (like &#39;length&#39;, &#39;mass&#39;, etc...)</span>
-<span class="sd">can help you find units:</span>
-
-<span class="sd">    &gt;&gt;&gt; u.find_unit(&#39;magnet&#39;)</span>
-<span class="sd">    [&#39;magnetic_flux&#39;, &#39;magnetic_constant&#39;, &#39;magnetic_flux_density&#39;]</span>
-<span class="sd">    &gt;&gt;&gt; u.find_unit(u.magnetic_flux)</span>
-<span class="sd">    [&#39;Wb&#39;, &#39;wb&#39;, &#39;weber&#39;, &#39;webers&#39;, &#39;magnetic_flux&#39;]</span>
-
-<span class="sd">If, for a given session, you wish to add a unit you may do so:</span>
-
-<span class="sd">    &gt;&gt;&gt; u.find_unit(&#39;gal&#39;)</span>
-<span class="sd">    []</span>
-<span class="sd">    &gt;&gt;&gt; u.gal = 4*u.quart</span>
-<span class="sd">    &gt;&gt;&gt; u.gal/u.inch**3</span>
-<span class="sd">    924</span>
-
-<span class="sd">To see a given quantity in terms of some other unit, divide by the desired</span>
-<span class="sd">unit:</span>
-
-<span class="sd">    &gt;&gt;&gt; mph = u.miles/u.hours</span>
-<span class="sd">    &gt;&gt;&gt; (u.m/u.s/mph).n(2)</span>
-<span class="sd">    2.2</span>
-
-<span class="sd">The units are defined in terms of base units, so when you divide similar</span>
-<span class="sd">units you will obtain a pure number. This means, for example, that if you</span>
-<span class="sd">divide a real-world mass (like grams) by the atomic mass unit (amu) you</span>
-<span class="sd">will obtain Avogadro&#39;s number. To obtain the answer in moles you</span>
-<span class="sd">should divide by the unit ``avogadro``:</span>
-
-<span class="sd">    &gt;&gt;&gt; u.grams/u.amu</span>
-<span class="sd">    602214179000000000000000</span>
-<span class="sd">    &gt;&gt;&gt; _/u.avogadro</span>
-<span class="sd">    mol</span>
-
-<span class="sd">For chemical calculations the unit ``mmu`` (molar mass unit) has been</span>
-<span class="sd">defined so this conversion is handled automatically. For example, the</span>
-<span class="sd">number of moles in 1 kg of water might be calculated as:</span>
-
-<span class="sd">    &gt;&gt;&gt; u.kg/(18*u.mmu).n(3)</span>
-<span class="sd">    55.5*mol</span>
-
-<span class="sd">If you need the number of atoms in a mol as a pure number you can use</span>
-<span class="sd">``avogadro_number`` but if you need it as a dimensional quantity you should use</span>
-<span class="sd">``avogadro_constant``. (``avogadro`` is a shorthand for the dimensional</span>
-<span class="sd">quantity.)</span>
-
-<span class="sd">    &gt;&gt;&gt; u.avogadro_number</span>
-<span class="sd">    602214179000000000000000</span>
-<span class="sd">    &gt;&gt;&gt; u.avogadro_constant</span>
-<span class="sd">    602214179000000000000000/mol</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Rational</span><span class="p">,</span> <span class="n">pi</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">AtomicExpr</span>
-
-
-<div class="viewcode-block" id="Unit"><a class="viewcode-back" href="../../../modules/physics/units.html#sympy.physics.units.Unit">[docs]</a><span class="k">class</span> <span class="nc">Unit</span><span class="p">(</span><span class="n">AtomicExpr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Base class for base unit of physical units.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.units import Unit</span>
-<span class="sd">    &gt;&gt;&gt; Unit(&quot;meter&quot;, &quot;m&quot;)</span>
-<span class="sd">    m</span>
-
-<span class="sd">    Other units are derived from base units:</span>
-
-<span class="sd">    &gt;&gt;&gt; import sympy.physics.units as u</span>
-<span class="sd">    &gt;&gt;&gt; cm = u.m/100</span>
-<span class="sd">    &gt;&gt;&gt; 100*u.cm</span>
-<span class="sd">    m</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_positive</span> <span class="o">=</span> <span class="bp">True</span>    <span class="c"># make sqrt(m**2) --&gt; m</span>
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_number</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&quot;name&quot;</span><span class="p">,</span> <span class="s">&quot;abbrev&quot;</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">abbrev</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">):</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">AtomicExpr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">)</span>
-        <span class="k">assert</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="nb">str</span><span class="p">),</span> <span class="nb">repr</span><span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">name</span><span class="p">))</span>
-        <span class="k">assert</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">abbrev</span><span class="p">,</span> <span class="nb">str</span><span class="p">),</span> <span class="nb">repr</span><span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">abbrev</span><span class="p">))</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="n">name</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">abbrev</span> <span class="o">=</span> <span class="n">abbrev</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">__getnewargs__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">abbrev</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Unit</span><span class="p">)</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">name</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">name</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Unit</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">_hashable_content</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">abbrev</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">free_symbols</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">set</span><span class="p">()</span>
-
-<span class="c"># Dimensionless</span>
-</div>
-<span class="n">percent</span> <span class="o">=</span> <span class="n">percents</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
-<span class="n">permille</span> <span class="o">=</span> <span class="n">permille</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
-
-<span class="n">ten</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-
-<span class="n">yotta</span> <span class="o">=</span> <span class="n">ten</span><span class="o">**</span><span class="mi">24</span>
-<span class="n">zetta</span> <span class="o">=</span> <span class="n">ten</span><span class="o">**</span><span class="mi">21</span>
-<span class="n">exa</span> <span class="o">=</span> <span class="n">ten</span><span class="o">**</span><span class="mi">18</span>
-<span class="n">peta</span> <span class="o">=</span> <span class="n">ten</span><span class="o">**</span><span class="mi">15</span>
-<span class="n">tera</span> <span class="o">=</span> <span class="n">ten</span><span class="o">**</span><span class="mi">12</span>
-<span class="n">giga</span> <span class="o">=</span> <span class="n">ten</span><span class="o">**</span><span class="mi">9</span>
-<span class="n">mega</span> <span class="o">=</span> <span class="n">ten</span><span class="o">**</span><span class="mi">6</span>
-<span class="n">kilo</span> <span class="o">=</span> <span class="n">ten</span><span class="o">**</span><span class="mi">3</span>
-<span class="n">deca</span> <span class="o">=</span> <span class="n">ten</span><span class="o">**</span><span class="mi">1</span>
-<span class="n">deci</span> <span class="o">=</span> <span class="n">ten</span><span class="o">**-</span><span class="mi">1</span>
-<span class="n">centi</span> <span class="o">=</span> <span class="n">ten</span><span class="o">**-</span><span class="mi">2</span>
-<span class="n">milli</span> <span class="o">=</span> <span class="n">ten</span><span class="o">**-</span><span class="mi">3</span>
-<span class="n">micro</span> <span class="o">=</span> <span class="n">ten</span><span class="o">**-</span><span class="mi">6</span>
-<span class="n">nano</span> <span class="o">=</span> <span class="n">ten</span><span class="o">**-</span><span class="mi">9</span>
-<span class="n">pico</span> <span class="o">=</span> <span class="n">ten</span><span class="o">**-</span><span class="mi">12</span>
-<span class="n">femto</span> <span class="o">=</span> <span class="n">ten</span><span class="o">**-</span><span class="mi">15</span>
-<span class="n">atto</span> <span class="o">=</span> <span class="n">ten</span><span class="o">**-</span><span class="mi">18</span>
-<span class="n">zepto</span> <span class="o">=</span> <span class="n">ten</span><span class="o">**-</span><span class="mi">21</span>
-<span class="n">yocto</span> <span class="o">=</span> <span class="n">ten</span><span class="o">**-</span><span class="mi">24</span>
-
-<span class="n">rad</span> <span class="o">=</span> <span class="n">radian</span> <span class="o">=</span> <span class="n">radians</span> <span class="o">=</span> <span class="mi">1</span>
-<span class="n">deg</span> <span class="o">=</span> <span class="n">degree</span> <span class="o">=</span> <span class="n">degrees</span> <span class="o">=</span> <span class="n">pi</span><span class="o">/</span><span class="mi">180</span>
-
-
-<span class="c"># Base units</span>
-
-<span class="n">length</span> <span class="o">=</span> <span class="n">m</span> <span class="o">=</span> <span class="n">meter</span> <span class="o">=</span> <span class="n">meters</span> <span class="o">=</span> <span class="n">Unit</span><span class="p">(</span><span class="s">&#39;meter&#39;</span><span class="p">,</span> <span class="s">&#39;m&#39;</span><span class="p">)</span>
-<span class="n">mass</span> <span class="o">=</span> <span class="n">kg</span> <span class="o">=</span> <span class="n">kilogram</span> <span class="o">=</span> <span class="n">kilograms</span> <span class="o">=</span> <span class="n">Unit</span><span class="p">(</span><span class="s">&#39;kilogram&#39;</span><span class="p">,</span> <span class="s">&#39;kg&#39;</span><span class="p">)</span>
-<span class="n">time</span> <span class="o">=</span> <span class="n">s</span> <span class="o">=</span> <span class="n">second</span> <span class="o">=</span> <span class="n">seconds</span> <span class="o">=</span> <span class="n">Unit</span><span class="p">(</span><span class="s">&#39;second&#39;</span><span class="p">,</span> <span class="s">&#39;s&#39;</span><span class="p">)</span>
-<span class="n">current</span> <span class="o">=</span> <span class="n">A</span> <span class="o">=</span> <span class="n">ampere</span> <span class="o">=</span> <span class="n">amperes</span> <span class="o">=</span> <span class="n">Unit</span><span class="p">(</span><span class="s">&#39;ampere&#39;</span><span class="p">,</span> <span class="s">&#39;A&#39;</span><span class="p">)</span>
-<span class="n">temperature</span> <span class="o">=</span> <span class="n">K</span> <span class="o">=</span> <span class="n">kelvin</span> <span class="o">=</span> <span class="n">kelvins</span> <span class="o">=</span> <span class="n">Unit</span><span class="p">(</span><span class="s">&#39;kelvin&#39;</span><span class="p">,</span> <span class="s">&#39;K&#39;</span><span class="p">)</span>
-<span class="n">amount</span> <span class="o">=</span> <span class="n">mol</span> <span class="o">=</span> <span class="n">mole</span> <span class="o">=</span> <span class="n">moles</span> <span class="o">=</span> <span class="n">Unit</span><span class="p">(</span><span class="s">&#39;mole&#39;</span><span class="p">,</span> <span class="s">&#39;mol&#39;</span><span class="p">)</span>
-<span class="n">luminosity</span> <span class="o">=</span> <span class="n">cd</span> <span class="o">=</span> <span class="n">candela</span> <span class="o">=</span> <span class="n">candelas</span> <span class="o">=</span> <span class="n">Unit</span><span class="p">(</span><span class="s">&#39;candela&#39;</span><span class="p">,</span> <span class="s">&#39;cd&#39;</span><span class="p">)</span>
-
-
-<span class="c"># Derived units</span>
-<span class="n">volume</span> <span class="o">=</span> <span class="n">meter</span><span class="o">**</span><span class="mi">3</span>
-<span class="n">frequency</span> <span class="o">=</span> <span class="n">Hz</span> <span class="o">=</span> <span class="n">hz</span> <span class="o">=</span> <span class="n">hertz</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">s</span>
-<span class="n">force</span> <span class="o">=</span> <span class="n">N</span> <span class="o">=</span> <span class="n">newton</span> <span class="o">=</span> <span class="n">newtons</span> <span class="o">=</span> <span class="n">m</span><span class="o">*</span><span class="n">kg</span><span class="o">/</span><span class="n">s</span><span class="o">**</span><span class="mi">2</span>
-<span class="n">energy</span> <span class="o">=</span> <span class="n">J</span> <span class="o">=</span> <span class="n">joule</span> <span class="o">=</span> <span class="n">joules</span> <span class="o">=</span> <span class="n">N</span><span class="o">*</span><span class="n">m</span>
-<span class="n">power</span> <span class="o">=</span> <span class="n">W</span> <span class="o">=</span> <span class="n">watt</span> <span class="o">=</span> <span class="n">watts</span> <span class="o">=</span> <span class="n">J</span><span class="o">/</span><span class="n">s</span>
-<span class="n">pressure</span> <span class="o">=</span> <span class="n">Pa</span> <span class="o">=</span> <span class="n">pa</span> <span class="o">=</span> <span class="n">pascal</span> <span class="o">=</span> <span class="n">pascals</span> <span class="o">=</span> <span class="n">N</span><span class="o">/</span><span class="n">m</span><span class="o">**</span><span class="mi">2</span>
-<span class="n">charge</span> <span class="o">=</span> <span class="n">C</span> <span class="o">=</span> <span class="n">coulomb</span> <span class="o">=</span> <span class="n">coulombs</span> <span class="o">=</span> <span class="n">s</span><span class="o">*</span><span class="n">A</span>
-<span class="n">voltage</span> <span class="o">=</span> <span class="n">v</span> <span class="o">=</span> <span class="n">V</span> <span class="o">=</span> <span class="n">volt</span> <span class="o">=</span> <span class="n">volts</span> <span class="o">=</span> <span class="n">W</span><span class="o">/</span><span class="n">A</span>
-<span class="n">resistance</span> <span class="o">=</span> <span class="n">ohm</span> <span class="o">=</span> <span class="n">ohms</span> <span class="o">=</span> <span class="n">V</span><span class="o">/</span><span class="n">A</span>
-<span class="n">conductance</span> <span class="o">=</span> <span class="n">S</span> <span class="o">=</span> <span class="n">siemens</span> <span class="o">=</span> <span class="n">mho</span> <span class="o">=</span> <span class="n">mhos</span> <span class="o">=</span> <span class="n">A</span><span class="o">/</span><span class="n">V</span>
-<span class="n">capacitance</span> <span class="o">=</span> <span class="n">F</span> <span class="o">=</span> <span class="n">farad</span> <span class="o">=</span> <span class="n">farads</span> <span class="o">=</span> <span class="n">C</span><span class="o">/</span><span class="n">V</span>
-<span class="n">magnetic_flux</span> <span class="o">=</span> <span class="n">Wb</span> <span class="o">=</span> <span class="n">wb</span> <span class="o">=</span> <span class="n">weber</span> <span class="o">=</span> <span class="n">webers</span> <span class="o">=</span> <span class="n">J</span><span class="o">/</span><span class="n">A</span>
-<span class="n">magnetic_flux_density</span> <span class="o">=</span> <span class="n">T</span> <span class="o">=</span> <span class="n">tesla</span> <span class="o">=</span> <span class="n">teslas</span> <span class="o">=</span> <span class="n">V</span><span class="o">*</span><span class="n">s</span><span class="o">/</span><span class="n">m</span><span class="o">**</span><span class="mi">2</span>
-<span class="n">inductance</span> <span class="o">=</span> <span class="n">H</span> <span class="o">=</span> <span class="n">henry</span> <span class="o">=</span> <span class="n">henrys</span> <span class="o">=</span> <span class="n">V</span><span class="o">*</span><span class="n">s</span><span class="o">/</span><span class="n">A</span>
-<span class="n">speed</span> <span class="o">=</span> <span class="n">m</span><span class="o">/</span><span class="n">s</span>
-<span class="n">acceleration</span> <span class="o">=</span> <span class="n">m</span><span class="o">/</span><span class="n">s</span><span class="o">**</span><span class="mi">2</span>
-<span class="n">density</span> <span class="o">=</span> <span class="n">kg</span><span class="o">/</span><span class="n">m</span><span class="o">**</span><span class="mi">3</span>
-
-<span class="c"># Common length units</span>
-
-<span class="n">km</span> <span class="o">=</span> <span class="n">kilometer</span> <span class="o">=</span> <span class="n">kilometers</span> <span class="o">=</span> <span class="n">kilo</span><span class="o">*</span><span class="n">m</span>
-<span class="n">dm</span> <span class="o">=</span> <span class="n">decimeter</span> <span class="o">=</span> <span class="n">decimeters</span> <span class="o">=</span> <span class="n">deci</span><span class="o">*</span><span class="n">m</span>
-<span class="n">cm</span> <span class="o">=</span> <span class="n">centimeter</span> <span class="o">=</span> <span class="n">centimeters</span> <span class="o">=</span> <span class="n">centi</span><span class="o">*</span><span class="n">m</span>
-<span class="n">mm</span> <span class="o">=</span> <span class="n">millimeter</span> <span class="o">=</span> <span class="n">millimeters</span> <span class="o">=</span> <span class="n">milli</span><span class="o">*</span><span class="n">m</span>
-<span class="n">um</span> <span class="o">=</span> <span class="n">micrometer</span> <span class="o">=</span> <span class="n">micrometers</span> <span class="o">=</span> <span class="n">micron</span> <span class="o">=</span> <span class="n">microns</span> <span class="o">=</span> <span class="n">micro</span><span class="o">*</span><span class="n">m</span>
-<span class="n">nm</span> <span class="o">=</span> <span class="n">nanometer</span> <span class="o">=</span> <span class="n">nanometers</span> <span class="o">=</span> <span class="n">nano</span><span class="o">*</span><span class="n">m</span>
-<span class="n">pm</span> <span class="o">=</span> <span class="n">picometer</span> <span class="o">=</span> <span class="n">picometers</span> <span class="o">=</span> <span class="n">pico</span><span class="o">*</span><span class="n">m</span>
-
-<span class="n">ft</span> <span class="o">=</span> <span class="n">foot</span> <span class="o">=</span> <span class="n">feet</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="s">&#39;0.3048&#39;</span><span class="p">)</span><span class="o">*</span><span class="n">m</span>
-<span class="n">inch</span> <span class="o">=</span> <span class="n">inches</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="s">&#39;25.4&#39;</span><span class="p">)</span><span class="o">*</span><span class="n">mm</span>
-<span class="n">yd</span> <span class="o">=</span> <span class="n">yard</span> <span class="o">=</span> <span class="n">yards</span> <span class="o">=</span> <span class="mi">3</span><span class="o">*</span><span class="n">ft</span>
-<span class="n">mi</span> <span class="o">=</span> <span class="n">mile</span> <span class="o">=</span> <span class="n">miles</span> <span class="o">=</span> <span class="mi">5280</span><span class="o">*</span><span class="n">ft</span>
-
-
-<span class="c"># Common volume and area units</span>
-
-<span class="n">l</span> <span class="o">=</span> <span class="n">liter</span> <span class="o">=</span> <span class="n">liters</span> <span class="o">=</span> <span class="n">m</span><span class="o">**</span><span class="mi">3</span> <span class="o">/</span> <span class="mi">1000</span>
-<span class="n">dl</span> <span class="o">=</span> <span class="n">deciliter</span> <span class="o">=</span> <span class="n">deciliters</span> <span class="o">=</span> <span class="n">deci</span><span class="o">*</span><span class="n">l</span>
-<span class="n">cl</span> <span class="o">=</span> <span class="n">centiliter</span> <span class="o">=</span> <span class="n">centiliters</span> <span class="o">=</span> <span class="n">centi</span><span class="o">*</span><span class="n">l</span>
-<span class="n">ml</span> <span class="o">=</span> <span class="n">milliliter</span> <span class="o">=</span> <span class="n">milliliters</span> <span class="o">=</span> <span class="n">milli</span><span class="o">*</span><span class="n">l</span>
-
-
-<span class="c"># Common time units</span>
-
-<span class="n">ms</span> <span class="o">=</span> <span class="n">millisecond</span> <span class="o">=</span> <span class="n">milliseconds</span> <span class="o">=</span> <span class="n">milli</span><span class="o">*</span><span class="n">s</span>
-<span class="n">us</span> <span class="o">=</span> <span class="n">microsecond</span> <span class="o">=</span> <span class="n">microseconds</span> <span class="o">=</span> <span class="n">micro</span><span class="o">*</span><span class="n">s</span>
-<span class="n">ns</span> <span class="o">=</span> <span class="n">nanosecond</span> <span class="o">=</span> <span class="n">nanoseconds</span> <span class="o">=</span> <span class="n">nano</span><span class="o">*</span><span class="n">s</span>
-<span class="n">ps</span> <span class="o">=</span> <span class="n">picosecond</span> <span class="o">=</span> <span class="n">picoseconds</span> <span class="o">=</span> <span class="n">pico</span><span class="o">*</span><span class="n">s</span>
-
-<span class="n">minute</span> <span class="o">=</span> <span class="n">minutes</span> <span class="o">=</span> <span class="mi">60</span><span class="o">*</span><span class="n">s</span>
-<span class="n">h</span> <span class="o">=</span> <span class="n">hour</span> <span class="o">=</span> <span class="n">hours</span> <span class="o">=</span> <span class="mi">60</span><span class="o">*</span><span class="n">minute</span>
-<span class="n">day</span> <span class="o">=</span> <span class="n">days</span> <span class="o">=</span> <span class="mi">24</span><span class="o">*</span><span class="n">hour</span>
-
-<span class="n">sidereal_year</span> <span class="o">=</span> <span class="n">sidereal_years</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="s">&#39;31558149.540&#39;</span><span class="p">)</span><span class="o">*</span><span class="n">s</span>
-<span class="n">tropical_year</span> <span class="o">=</span> <span class="n">tropical_years</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="s">&#39;365.24219&#39;</span><span class="p">)</span><span class="o">*</span><span class="n">day</span>
-<span class="n">common_year</span> <span class="o">=</span> <span class="n">common_years</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="s">&#39;365&#39;</span><span class="p">)</span><span class="o">*</span><span class="n">day</span>
-<span class="n">julian_year</span> <span class="o">=</span> <span class="n">julian_years</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="s">&#39;365.25&#39;</span><span class="p">)</span><span class="o">*</span><span class="n">day</span>
-
-<span class="n">year</span> <span class="o">=</span> <span class="n">years</span> <span class="o">=</span> <span class="n">tropical_year</span>
-
-
-<span class="c"># Common mass units</span>
-
-<span class="n">g</span> <span class="o">=</span> <span class="n">gram</span> <span class="o">=</span> <span class="n">grams</span> <span class="o">=</span> <span class="n">kilogram</span> <span class="o">/</span> <span class="n">kilo</span>
-<span class="n">mg</span> <span class="o">=</span> <span class="n">milligram</span> <span class="o">=</span> <span class="n">milligrams</span> <span class="o">=</span> <span class="n">milli</span> <span class="o">*</span> <span class="n">g</span>
-<span class="n">ug</span> <span class="o">=</span> <span class="n">microgram</span> <span class="o">=</span> <span class="n">micrograms</span> <span class="o">=</span> <span class="n">micro</span> <span class="o">*</span> <span class="n">g</span>
-
-
-<span class="c">#----------------------------------------------------------------------------</span>
-<span class="c"># Physical constants</span>
-<span class="c">#</span>
-<span class="n">c</span> <span class="o">=</span> <span class="n">speed_of_light</span> <span class="o">=</span> <span class="mi">299792458</span> <span class="o">*</span> <span class="n">m</span><span class="o">/</span><span class="n">s</span>
-<span class="n">G</span> <span class="o">=</span> <span class="n">gravitational_constant</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="s">&#39;6.67428&#39;</span><span class="p">)</span> <span class="o">*</span> <span class="n">ten</span><span class="o">**-</span><span class="mi">11</span> <span class="o">*</span> <span class="n">m</span><span class="o">**</span><span class="mi">3</span> <span class="o">/</span> <span class="n">kg</span> <span class="o">/</span> <span class="n">s</span><span class="o">**</span><span class="mi">2</span>
-<span class="n">u0</span> <span class="o">=</span> <span class="n">magnetic_constant</span> <span class="o">=</span> <span class="mi">4</span><span class="o">*</span><span class="n">pi</span> <span class="o">*</span> <span class="n">ten</span><span class="o">**-</span><span class="mi">7</span> <span class="o">*</span> <span class="n">N</span><span class="o">/</span><span class="n">A</span><span class="o">**</span><span class="mi">2</span>
-<span class="n">e0</span> <span class="o">=</span> <span class="n">electric_constant</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">u0</span> <span class="o">*</span> <span class="n">c</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="n">Z0</span> <span class="o">=</span> <span class="n">vacuum_impedance</span> <span class="o">=</span> <span class="n">u0</span> <span class="o">*</span> <span class="n">c</span>
-
-<span class="n">planck</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="s">&#39;6.62606896&#39;</span><span class="p">)</span> <span class="o">*</span> <span class="n">ten</span><span class="o">**-</span><span class="mi">34</span> <span class="o">*</span> <span class="n">J</span><span class="o">*</span><span class="n">s</span>
-<span class="n">hbar</span> <span class="o">=</span> <span class="n">planck</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span>
-
-<span class="n">avogadro_number</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="s">&#39;6.02214179&#39;</span><span class="p">)</span> <span class="o">*</span> <span class="mi">10</span><span class="o">**</span><span class="mi">23</span>
-<span class="n">avogadro</span> <span class="o">=</span> <span class="n">avogadro_constant</span> <span class="o">=</span> <span class="n">avogadro_number</span> <span class="o">/</span> <span class="n">mol</span>
-<span class="n">boltzmann</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="s">&#39;1.3806505&#39;</span><span class="p">)</span> <span class="o">*</span> <span class="n">ten</span><span class="o">**-</span><span class="mi">23</span> <span class="o">*</span> <span class="n">J</span> <span class="o">/</span> <span class="n">K</span>
-
-<span class="n">gee</span> <span class="o">=</span> <span class="n">gees</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="s">&#39;9.80665&#39;</span><span class="p">)</span> <span class="o">*</span> <span class="n">m</span><span class="o">/</span><span class="n">s</span><span class="o">**</span><span class="mi">2</span>
-<span class="n">atmosphere</span> <span class="o">=</span> <span class="n">atmospheres</span> <span class="o">=</span> <span class="n">atm</span> <span class="o">=</span> <span class="mi">101325</span> <span class="o">*</span> <span class="n">pascal</span>
-
-<span class="n">kPa</span> <span class="o">=</span> <span class="n">kilo</span><span class="o">*</span><span class="n">Pa</span>
-<span class="n">bar</span> <span class="o">=</span> <span class="n">bars</span> <span class="o">=</span> <span class="mi">100</span><span class="o">*</span><span class="n">kPa</span>
-<span class="n">pound</span> <span class="o">=</span> <span class="n">pounds</span> <span class="o">=</span> <span class="mf">0.45359237</span> <span class="o">*</span> <span class="n">kg</span> <span class="o">*</span> <span class="n">gee</span>  <span class="c"># exact</span>
-<span class="n">psi</span> <span class="o">=</span> <span class="n">pound</span> <span class="o">/</span> <span class="n">inch</span> <span class="o">**</span> <span class="mi">2</span>
-<span class="n">dHg0</span> <span class="o">=</span> <span class="mf">13.5951</span>  <span class="c"># approx value at 0 C</span>
-<span class="n">mmHg</span> <span class="o">=</span> <span class="n">dHg0</span> <span class="o">*</span> <span class="mf">9.80665</span> <span class="o">*</span> <span class="n">Pa</span>
-<span class="n">amu</span> <span class="o">=</span> <span class="n">amus</span> <span class="o">=</span> <span class="n">gram</span> <span class="o">/</span> <span class="n">avogadro</span> <span class="o">/</span> <span class="n">mol</span>
-<span class="n">mmu</span> <span class="o">=</span> <span class="n">mmus</span> <span class="o">=</span> <span class="n">gram</span> <span class="o">/</span> <span class="n">mol</span>
-<span class="n">quart</span> <span class="o">=</span> <span class="n">quarts</span> <span class="o">=</span> <span class="mi">231</span> <span class="o">*</span> <span class="n">inch</span><span class="o">**</span><span class="mi">3</span>
-<span class="n">eV</span> <span class="o">=</span> <span class="mf">1.602176487e-19</span> <span class="o">*</span> <span class="n">J</span>
-
-<span class="c"># Other convenient units and magnitudes</span>
-
-<span class="n">ly</span> <span class="o">=</span> <span class="n">lightyear</span> <span class="o">=</span> <span class="n">lightyears</span> <span class="o">=</span> <span class="n">c</span><span class="o">*</span><span class="n">julian_year</span>
-<span class="n">au</span> <span class="o">=</span> <span class="n">astronomical_unit</span> <span class="o">=</span> <span class="n">astronomical_units</span> <span class="o">=</span> <span class="mi">149597870691</span><span class="o">*</span><span class="n">m</span>
-
-
-<div class="viewcode-block" id="find_unit"><a class="viewcode-back" href="../../../modules/physics/units.html#sympy.physics.units.find_unit">[docs]</a><span class="k">def</span> <span class="nf">find_unit</span><span class="p">(</span><span class="n">quantity</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a list of matching units names.</span>
-<span class="sd">    if quantity is a string -- units containing the string `quantity`</span>
-<span class="sd">    if quantity is a unit -- units having matching base units</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics import units as u</span>
-<span class="sd">    &gt;&gt;&gt; u.find_unit(&#39;charge&#39;)</span>
-<span class="sd">    [&#39;charge&#39;]</span>
-<span class="sd">    &gt;&gt;&gt; u.find_unit(u.charge)</span>
-<span class="sd">    [&#39;C&#39;, &#39;charge&#39;, &#39;coulomb&#39;, &#39;coulombs&#39;]</span>
-<span class="sd">    &gt;&gt;&gt; u.find_unit(&#39;volt&#39;)</span>
-<span class="sd">    [&#39;volt&#39;, &#39;volts&#39;, &#39;voltage&#39;]</span>
-<span class="sd">    &gt;&gt;&gt; u.find_unit(u.inch**3)[:5]</span>
-<span class="sd">    [&#39;l&#39;, &#39;cl&#39;, &#39;dl&#39;, &#39;ml&#39;, &#39;liter&#39;]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">import</span> <span class="nn">sympy.physics.units</span> <span class="kn">as</span> <span class="nn">u</span>
-    <span class="n">rv</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">quantity</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">dir</span><span class="p">(</span><span class="n">u</span><span class="p">)</span> <span class="k">if</span> <span class="n">quantity</span> <span class="ow">in</span> <span class="n">i</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">units</span> <span class="o">=</span> <span class="n">quantity</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">dir</span><span class="p">(</span><span class="n">u</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">units</span> <span class="o">==</span> <span class="nb">eval</span><span class="p">(</span><span class="s">&#39;u.&#39;</span> <span class="o">+</span> <span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()[</span><span class="mi">1</span><span class="p">]:</span>
-                    <span class="n">rv</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>
-            <span class="k">except</span><span class="p">:</span>
-                <span class="k">pass</span>
-    <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">rv</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="nb">len</span><span class="p">)</span>
-
-<span class="c"># Delete this so it doesn&#39;t pollute the namespace</span></div>
-<span class="k">del</span> <span class="n">Rational</span><span class="p">,</span> <span class="n">pi</span>
-</pre></div>
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-  <h1>Source code for sympy.physics.wigner</h1><div class="highlight"><pre>
-<span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">Wigner, Clebsch-Gordan, Racah, and Gaunt coefficients</span>
-
-<span class="sd">Collection of functions for calculating Wigner 3j, 6j, 9j,</span>
-<span class="sd">Clebsch-Gordan, Racah as well as Gaunt coefficients exactly, all</span>
-<span class="sd">evaluating to a rational number times the square root of a rational</span>
-<span class="sd">number [Rasch03]_.</span>
-
-<span class="sd">Please see the description of the individual functions for further</span>
-<span class="sd">details and examples.</span>
-
-<span class="sd">REFERENCES:</span>
-
-<span class="sd">.. [Rasch03] J. Rasch and A. C. H. Yu, &#39;Efficient Storage Scheme for</span>
-<span class="sd">  Pre-calculated Wigner 3j, 6j and Gaunt Coefficients&#39;, SIAM</span>
-<span class="sd">  J. Sci. Comput. Volume 25, Issue 4, pp. 1416-1428 (2003)</span>
-
-<span class="sd">This code was taken from Sage with the permission of all authors:</span>
-
-<span class="sd">http://groups.google.com/group/sage-devel/browse_thread/thread/33835976efbb3b7f</span>
-
-<span class="sd">AUTHORS:</span>
-
-<span class="sd">- Jens Rasch (2009-03-24): initial version for Sage</span>
-
-<span class="sd">- Jens Rasch (2009-05-31): updated to sage-4.0</span>
-
-<span class="sd">Copyright (C) 2008 Jens Rasch &lt;jyr2000@gmail.com&gt;</span>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integer</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">sympify</span>
-<span class="c">#from sage.rings.complex_number import ComplexNumber</span>
-<span class="c">#from sage.rings.finite_rings.integer_mod import Mod</span>
-
-<span class="c"># This list of precomputed factorials is needed to massively</span>
-<span class="c"># accelerate future calculations of the various coefficients</span>
-<span class="n">_Factlist</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">_calc_factlist</span><span class="p">(</span><span class="n">nn</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Function calculates a list of precomputed factorials in order to</span>
-<span class="sd">    massively accelerate future calculations of the various</span>
-<span class="sd">    coefficients.</span>
-
-<span class="sd">    INPUT:</span>
-
-<span class="sd">    -  ``nn`` -  integer, highest factorial to be computed</span>
-
-<span class="sd">    OUTPUT:</span>
-
-<span class="sd">    list of integers -- the list of precomputed factorials</span>
-
-<span class="sd">    EXAMPLES:</span>
-
-<span class="sd">    Calculate list of factorials::</span>
-
-<span class="sd">        sage: from sage.functions.wigner import _calc_factlist</span>
-<span class="sd">        sage: _calc_factlist(10)</span>
-<span class="sd">        [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">nn</span> <span class="o">&gt;=</span> <span class="nb">len</span><span class="p">(</span><span class="n">_Factlist</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">ii</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">_Factlist</span><span class="p">),</span> <span class="n">nn</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">_Factlist</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">_Factlist</span><span class="p">[</span><span class="n">ii</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">ii</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">_Factlist</span><span class="p">[:</span><span class="nb">int</span><span class="p">(</span><span class="n">nn</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
-
-
-<div class="viewcode-block" id="wigner_3j"><a class="viewcode-back" href="../../../modules/physics/wigner.html#sympy.physics.wigner.wigner_3j">[docs]</a><span class="k">def</span> <span class="nf">wigner_3j</span><span class="p">(</span><span class="n">j_1</span><span class="p">,</span> <span class="n">j_2</span><span class="p">,</span> <span class="n">j_3</span><span class="p">,</span> <span class="n">m_1</span><span class="p">,</span> <span class="n">m_2</span><span class="p">,</span> <span class="n">m_3</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Calculate the Wigner 3j symbol `Wigner3j(j_1,j_2,j_3,m_1,m_2,m_3)`.</span>
-
-<span class="sd">    INPUT:</span>
-
-<span class="sd">    -  ``j_1``, ``j_2``, ``j_3``, ``m_1``, ``m_2``, ``m_3`` - integer or half integer</span>
-
-<span class="sd">    -  ``prec`` - precision, default: ``None``. Providing a precision can</span>
-<span class="sd">       drastically speed up the calculation.</span>
-
-<span class="sd">    OUTPUT:</span>
-
-<span class="sd">    Rational number times the square root of a rational number</span>
-<span class="sd">    (if ``prec=None``), or real number if a precision is given.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.wigner import wigner_3j</span>
-<span class="sd">    &gt;&gt;&gt; wigner_3j(2, 6, 4, 0, 0, 0)</span>
-<span class="sd">    sqrt(715)/143</span>
-<span class="sd">    &gt;&gt;&gt; wigner_3j(2, 6, 4, 0, 0, 1)</span>
-<span class="sd">    0</span>
-
-<span class="sd">    It is an error to have arguments that are not integer or half</span>
-<span class="sd">    integer values::</span>
-
-<span class="sd">        sage: wigner_3j(2.1, 6, 4, 0, 0, 0)</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">        ...</span>
-<span class="sd">        ValueError: j values must be integer or half integer</span>
-<span class="sd">        sage: wigner_3j(2, 6, 4, 1, 0, -1.1)</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">        ...</span>
-<span class="sd">        ValueError: m values must be integer or half integer</span>
-
-<span class="sd">    NOTES:</span>
-
-<span class="sd">    The Wigner 3j symbol obeys the following symmetry rules:</span>
-
-<span class="sd">    - invariant under any permutation of the columns (with the</span>
-<span class="sd">      exception of a sign change where `J:=j_1+j_2+j_3`):</span>
-
-<span class="sd">      .. math::</span>
-
-<span class="sd">         Wigner3j(j_1,j_2,j_3,m_1,m_2,m_3)</span>
-<span class="sd">          =Wigner3j(j_3,j_1,j_2,m_3,m_1,m_2)</span>
-<span class="sd">          =Wigner3j(j_2,j_3,j_1,m_2,m_3,m_1)</span>
-<span class="sd">          =(-1)^J Wigner3j(j_3,j_2,j_1,m_3,m_2,m_1)</span>
-<span class="sd">          =(-1)^J Wigner3j(j_1,j_3,j_2,m_1,m_3,m_2)</span>
-<span class="sd">          =(-1)^J Wigner3j(j_2,j_1,j_3,m_2,m_1,m_3)</span>
-
-<span class="sd">    - invariant under space inflection, i.e.</span>
-
-<span class="sd">      .. math::</span>
-
-<span class="sd">         Wigner3j(j_1,j_2,j_3,m_1,m_2,m_3)</span>
-<span class="sd">         =(-1)^J Wigner3j(j_1,j_2,j_3,-m_1,-m_2,-m_3)</span>
-
-<span class="sd">    - symmetric with respect to the 72 additional symmetries based on</span>
-<span class="sd">      the work by [Regge58]_</span>
-
-<span class="sd">    - zero for `j_1`, `j_2`, `j_3` not fulfilling triangle relation</span>
-
-<span class="sd">    - zero for `m_1 + m_2 + m_3 \neq 0`</span>
-
-<span class="sd">    - zero for violating any one of the conditions</span>
-<span class="sd">      `j_1 \ge |m_1|`,  `j_2 \ge |m_2|`,  `j_3 \ge |m_3|`</span>
-
-<span class="sd">    ALGORITHM:</span>
-
-<span class="sd">    This function uses the algorithm of [Edmonds74]_ to calculate the</span>
-<span class="sd">    value of the 3j symbol exactly. Note that the formula contains</span>
-<span class="sd">    alternating sums over large factorials and is therefore unsuitable</span>
-<span class="sd">    for finite precision arithmetic and only useful for a computer</span>
-<span class="sd">    algebra system [Rasch03]_.</span>
-
-<span class="sd">    REFERENCES:</span>
-
-<span class="sd">    .. [Regge58] &#39;Symmetry Properties of Clebsch-Gordan Coefficients&#39;,</span>
-<span class="sd">      T. Regge, Nuovo Cimento, Volume 10, pp. 544 (1958)</span>
-
-<span class="sd">    .. [Edmonds74] &#39;Angular Momentum in Quantum Mechanics&#39;,</span>
-<span class="sd">      A. R. Edmonds, Princeton University Press (1974)</span>
-
-<span class="sd">    AUTHORS:</span>
-
-<span class="sd">    - Jens Rasch (2009-03-24): initial version</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">int</span><span class="p">(</span><span class="n">j_1</span> <span class="o">*</span> <span class="mi">2</span><span class="p">)</span> <span class="o">!=</span> <span class="n">j_1</span> <span class="o">*</span> <span class="mi">2</span> <span class="ow">or</span> <span class="nb">int</span><span class="p">(</span><span class="n">j_2</span> <span class="o">*</span> <span class="mi">2</span><span class="p">)</span> <span class="o">!=</span> <span class="n">j_2</span> <span class="o">*</span> <span class="mi">2</span> <span class="ow">or</span> \
-            <span class="nb">int</span><span class="p">(</span><span class="n">j_3</span> <span class="o">*</span> <span class="mi">2</span><span class="p">)</span> <span class="o">!=</span> <span class="n">j_3</span> <span class="o">*</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;j values must be integer or half integer&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">int</span><span class="p">(</span><span class="n">m_1</span> <span class="o">*</span> <span class="mi">2</span><span class="p">)</span> <span class="o">!=</span> <span class="n">m_1</span> <span class="o">*</span> <span class="mi">2</span> <span class="ow">or</span> <span class="nb">int</span><span class="p">(</span><span class="n">m_2</span> <span class="o">*</span> <span class="mi">2</span><span class="p">)</span> <span class="o">!=</span> <span class="n">m_2</span> <span class="o">*</span> <span class="mi">2</span> <span class="ow">or</span> \
-            <span class="nb">int</span><span class="p">(</span><span class="n">m_3</span> <span class="o">*</span> <span class="mi">2</span><span class="p">)</span> <span class="o">!=</span> <span class="n">m_3</span> <span class="o">*</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;m values must be integer or half integer&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">m_1</span> <span class="o">+</span> <span class="n">m_2</span> <span class="o">+</span> <span class="n">m_3</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="n">prefid</span> <span class="o">=</span> <span class="n">Integer</span><span class="p">((</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">**</span> <span class="nb">int</span><span class="p">(</span><span class="n">j_1</span> <span class="o">-</span> <span class="n">j_2</span> <span class="o">-</span> <span class="n">m_3</span><span class="p">))</span>
-    <span class="n">m_3</span> <span class="o">=</span> <span class="o">-</span><span class="n">m_3</span>
-    <span class="n">a1</span> <span class="o">=</span> <span class="n">j_1</span> <span class="o">+</span> <span class="n">j_2</span> <span class="o">-</span> <span class="n">j_3</span>
-    <span class="k">if</span> <span class="n">a1</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="n">a2</span> <span class="o">=</span> <span class="n">j_1</span> <span class="o">-</span> <span class="n">j_2</span> <span class="o">+</span> <span class="n">j_3</span>
-    <span class="k">if</span> <span class="n">a2</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="n">a3</span> <span class="o">=</span> <span class="o">-</span><span class="n">j_1</span> <span class="o">+</span> <span class="n">j_2</span> <span class="o">+</span> <span class="n">j_3</span>
-    <span class="k">if</span> <span class="n">a3</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="k">if</span> <span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">m_1</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">j_1</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">m_2</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">j_2</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">m_3</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">j_3</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">0</span>
-
-    <span class="n">maxfact</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">j_1</span> <span class="o">+</span> <span class="n">j_2</span> <span class="o">+</span> <span class="n">j_3</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j_1</span> <span class="o">+</span> <span class="nb">abs</span><span class="p">(</span><span class="n">m_1</span><span class="p">),</span> <span class="n">j_2</span> <span class="o">+</span> <span class="nb">abs</span><span class="p">(</span><span class="n">m_2</span><span class="p">),</span>
-                  <span class="n">j_3</span> <span class="o">+</span> <span class="nb">abs</span><span class="p">(</span><span class="n">m_3</span><span class="p">))</span>
-    <span class="n">_calc_factlist</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">maxfact</span><span class="p">))</span>
-
-    <span class="n">argsqrt</span> <span class="o">=</span> <span class="n">Integer</span><span class="p">(</span><span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">j_1</span> <span class="o">+</span> <span class="n">j_2</span> <span class="o">-</span> <span class="n">j_3</span><span class="p">)]</span> <span class="o">*</span>
-                     <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">j_1</span> <span class="o">-</span> <span class="n">j_2</span> <span class="o">+</span> <span class="n">j_3</span><span class="p">)]</span> <span class="o">*</span>
-                     <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="o">-</span><span class="n">j_1</span> <span class="o">+</span> <span class="n">j_2</span> <span class="o">+</span> <span class="n">j_3</span><span class="p">)]</span> <span class="o">*</span>
-                     <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">j_1</span> <span class="o">-</span> <span class="n">m_1</span><span class="p">)]</span> <span class="o">*</span>
-                     <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">j_1</span> <span class="o">+</span> <span class="n">m_1</span><span class="p">)]</span> <span class="o">*</span>
-                     <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">j_2</span> <span class="o">-</span> <span class="n">m_2</span><span class="p">)]</span> <span class="o">*</span>
-                     <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">j_2</span> <span class="o">+</span> <span class="n">m_2</span><span class="p">)]</span> <span class="o">*</span>
-                     <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">j_3</span> <span class="o">-</span> <span class="n">m_3</span><span class="p">)]</span> <span class="o">*</span>
-                     <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">j_3</span> <span class="o">+</span> <span class="n">m_3</span><span class="p">)])</span> <span class="o">/</span> \
-        <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">j_1</span> <span class="o">+</span> <span class="n">j_2</span> <span class="o">+</span> <span class="n">j_3</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
-
-    <span class="n">ressqrt</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">argsqrt</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">ressqrt</span><span class="o">.</span><span class="n">is_complex</span><span class="p">:</span>
-        <span class="n">ressqrt</span> <span class="o">=</span> <span class="n">ressqrt</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="n">imin</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="o">-</span><span class="n">j_3</span> <span class="o">+</span> <span class="n">j_1</span> <span class="o">+</span> <span class="n">m_2</span><span class="p">,</span> <span class="o">-</span><span class="n">j_3</span> <span class="o">+</span> <span class="n">j_2</span> <span class="o">-</span> <span class="n">m_1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-    <span class="n">imax</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">j_2</span> <span class="o">+</span> <span class="n">m_2</span><span class="p">,</span> <span class="n">j_1</span> <span class="o">-</span> <span class="n">m_1</span><span class="p">,</span> <span class="n">j_1</span> <span class="o">+</span> <span class="n">j_2</span> <span class="o">-</span> <span class="n">j_3</span><span class="p">)</span>
-    <span class="n">sumres</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">ii</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">imin</span><span class="p">),</span> <span class="nb">int</span><span class="p">(</span><span class="n">imax</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">den</span> <span class="o">=</span> <span class="n">_Factlist</span><span class="p">[</span><span class="n">ii</span><span class="p">]</span> <span class="o">*</span> \
-            <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">ii</span> <span class="o">+</span> <span class="n">j_3</span> <span class="o">-</span> <span class="n">j_1</span> <span class="o">-</span> <span class="n">m_2</span><span class="p">)]</span> <span class="o">*</span> \
-            <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">j_2</span> <span class="o">+</span> <span class="n">m_2</span> <span class="o">-</span> <span class="n">ii</span><span class="p">)]</span> <span class="o">*</span> \
-            <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">j_1</span> <span class="o">-</span> <span class="n">ii</span> <span class="o">-</span> <span class="n">m_1</span><span class="p">)]</span> <span class="o">*</span> \
-            <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">ii</span> <span class="o">+</span> <span class="n">j_3</span> <span class="o">-</span> <span class="n">j_2</span> <span class="o">+</span> <span class="n">m_1</span><span class="p">)]</span> <span class="o">*</span> \
-            <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">j_1</span> <span class="o">+</span> <span class="n">j_2</span> <span class="o">-</span> <span class="n">j_3</span> <span class="o">-</span> <span class="n">ii</span><span class="p">)]</span>
-        <span class="n">sumres</span> <span class="o">=</span> <span class="n">sumres</span> <span class="o">+</span> <span class="n">Integer</span><span class="p">((</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">**</span> <span class="n">ii</span><span class="p">)</span> <span class="o">/</span> <span class="n">den</span>
-
-    <span class="n">res</span> <span class="o">=</span> <span class="n">ressqrt</span> <span class="o">*</span> <span class="n">sumres</span> <span class="o">*</span> <span class="n">prefid</span>
-    <span class="k">return</span> <span class="n">res</span>
-
-</div>
-<div class="viewcode-block" id="clebsch_gordan"><a class="viewcode-back" href="../../../modules/physics/wigner.html#sympy.physics.wigner.clebsch_gordan">[docs]</a><span class="k">def</span> <span class="nf">clebsch_gordan</span><span class="p">(</span><span class="n">j_1</span><span class="p">,</span> <span class="n">j_2</span><span class="p">,</span> <span class="n">j_3</span><span class="p">,</span> <span class="n">m_1</span><span class="p">,</span> <span class="n">m_2</span><span class="p">,</span> <span class="n">m_3</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Calculates the Clebsch-Gordan coefficient</span>
-<span class="sd">    `\langle j_1 m_1 \; j_2 m_2 | j_3 m_3 \rangle`.</span>
-
-<span class="sd">    The reference for this function is [Edmonds74]_.</span>
-
-<span class="sd">    INPUT:</span>
-
-<span class="sd">    -  ``j_1``, ``j_2``, ``j_3``, ``m_1``, ``m_2``, ``m_3`` - integer or half integer</span>
-
-<span class="sd">    -  ``prec`` - precision, default: ``None``. Providing a precision can</span>
-<span class="sd">       drastically speed up the calculation.</span>
-
-<span class="sd">    OUTPUT:</span>
-
-<span class="sd">    Rational number times the square root of a rational number</span>
-<span class="sd">    (if ``prec=None``), or real number if a precision is given.</span>
-
-<span class="sd">    EXAMPLES::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import S</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.physics.wigner import clebsch_gordan</span>
-<span class="sd">        &gt;&gt;&gt; clebsch_gordan(S(3)/2, S(1)/2, 2, S(3)/2, S(1)/2, 2)</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; clebsch_gordan(S(3)/2, S(1)/2, 1, S(3)/2, -S(1)/2, 1)</span>
-<span class="sd">        sqrt(3)/2</span>
-<span class="sd">        &gt;&gt;&gt; clebsch_gordan(S(3)/2, S(1)/2, 1, -S(1)/2, S(1)/2, 0)</span>
-<span class="sd">        -sqrt(2)/2</span>
-
-<span class="sd">    NOTES:</span>
-
-<span class="sd">    The Clebsch-Gordan coefficient will be evaluated via its relation</span>
-<span class="sd">    to Wigner 3j symbols:</span>
-
-<span class="sd">    .. math::</span>
-
-<span class="sd">        \langle j_1 m_1 \; j_2 m_2 | j_3 m_3 \rangle</span>
-<span class="sd">        =(-1)^{j_1-j_2+m_3} \sqrt{2j_3+1} \;</span>
-<span class="sd">        Wigner3j(j_1,j_2,j_3,m_1,m_2,-m_3)</span>
-
-<span class="sd">    See also the documentation on Wigner 3j symbols which exhibit much</span>
-<span class="sd">    higher symmetry relations than the Clebsch-Gordan coefficient.</span>
-
-<span class="sd">    AUTHORS:</span>
-
-<span class="sd">    - Jens Rasch (2009-03-24): initial version</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">**</span> <span class="n">sympify</span><span class="p">(</span><span class="n">j_1</span> <span class="o">-</span> <span class="n">j_2</span> <span class="o">+</span> <span class="n">m_3</span><span class="p">)</span> <span class="o">*</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">j_3</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> \
-        <span class="n">wigner_3j</span><span class="p">(</span><span class="n">j_1</span><span class="p">,</span> <span class="n">j_2</span><span class="p">,</span> <span class="n">j_3</span><span class="p">,</span> <span class="n">m_1</span><span class="p">,</span> <span class="n">m_2</span><span class="p">,</span> <span class="o">-</span><span class="n">m_3</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">res</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_big_delta_coeff</span><span class="p">(</span><span class="n">aa</span><span class="p">,</span> <span class="n">bb</span><span class="p">,</span> <span class="n">cc</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Calculates the Delta coefficient of the 3 angular momenta for</span>
-<span class="sd">    Racah symbols. Also checks that the differences are of integer</span>
-<span class="sd">    value.</span>
-
-<span class="sd">    INPUT:</span>
-
-<span class="sd">    -  ``aa`` - first angular momentum, integer or half integer</span>
-
-<span class="sd">    -  ``bb`` - second angular momentum, integer or half integer</span>
-
-<span class="sd">    -  ``cc`` - third angular momentum, integer or half integer</span>
-
-<span class="sd">    -  ``prec`` - precision of the ``sqrt()`` calculation</span>
-
-<span class="sd">    OUTPUT:</span>
-
-<span class="sd">    double - Value of the Delta coefficient</span>
-
-<span class="sd">    EXAMPLES::</span>
-
-<span class="sd">        sage: from sage.functions.wigner import _big_delta_coeff</span>
-<span class="sd">        sage: _big_delta_coeff(1,1,1)</span>
-<span class="sd">        1/2*sqrt(1/6)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">int</span><span class="p">(</span><span class="n">aa</span> <span class="o">+</span> <span class="n">bb</span> <span class="o">-</span> <span class="n">cc</span><span class="p">)</span> <span class="o">!=</span> <span class="p">(</span><span class="n">aa</span> <span class="o">+</span> <span class="n">bb</span> <span class="o">-</span> <span class="n">cc</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;j values must be integer or half integer and fulfill the triangle relation&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">int</span><span class="p">(</span><span class="n">aa</span> <span class="o">+</span> <span class="n">cc</span> <span class="o">-</span> <span class="n">bb</span><span class="p">)</span> <span class="o">!=</span> <span class="p">(</span><span class="n">aa</span> <span class="o">+</span> <span class="n">cc</span> <span class="o">-</span> <span class="n">bb</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;j values must be integer or half integer and fulfill the triangle relation&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">int</span><span class="p">(</span><span class="n">bb</span> <span class="o">+</span> <span class="n">cc</span> <span class="o">-</span> <span class="n">aa</span><span class="p">)</span> <span class="o">!=</span> <span class="p">(</span><span class="n">bb</span> <span class="o">+</span> <span class="n">cc</span> <span class="o">-</span> <span class="n">aa</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;j values must be integer or half integer and fulfill the triangle relation&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="p">(</span><span class="n">aa</span> <span class="o">+</span> <span class="n">bb</span> <span class="o">-</span> <span class="n">cc</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="k">if</span> <span class="p">(</span><span class="n">aa</span> <span class="o">+</span> <span class="n">cc</span> <span class="o">-</span> <span class="n">bb</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="k">if</span> <span class="p">(</span><span class="n">bb</span> <span class="o">+</span> <span class="n">cc</span> <span class="o">-</span> <span class="n">aa</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-
-    <span class="n">maxfact</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">aa</span> <span class="o">+</span> <span class="n">bb</span> <span class="o">-</span> <span class="n">cc</span><span class="p">,</span> <span class="n">aa</span> <span class="o">+</span> <span class="n">cc</span> <span class="o">-</span> <span class="n">bb</span><span class="p">,</span> <span class="n">bb</span> <span class="o">+</span> <span class="n">cc</span> <span class="o">-</span> <span class="n">aa</span><span class="p">,</span> <span class="n">aa</span> <span class="o">+</span> <span class="n">bb</span> <span class="o">+</span> <span class="n">cc</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="n">_calc_factlist</span><span class="p">(</span><span class="n">maxfact</span><span class="p">)</span>
-
-    <span class="n">argsqrt</span> <span class="o">=</span> <span class="n">Integer</span><span class="p">(</span><span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">aa</span> <span class="o">+</span> <span class="n">bb</span> <span class="o">-</span> <span class="n">cc</span><span class="p">)]</span> <span class="o">*</span>
-                     <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">aa</span> <span class="o">+</span> <span class="n">cc</span> <span class="o">-</span> <span class="n">bb</span><span class="p">)]</span> <span class="o">*</span>
-                     <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">bb</span> <span class="o">+</span> <span class="n">cc</span> <span class="o">-</span> <span class="n">aa</span><span class="p">)])</span> <span class="o">/</span> \
-        <span class="n">Integer</span><span class="p">(</span><span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">aa</span> <span class="o">+</span> <span class="n">bb</span> <span class="o">+</span> <span class="n">cc</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)])</span>
-
-    <span class="n">ressqrt</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">argsqrt</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">prec</span><span class="p">:</span>
-        <span class="n">ressqrt</span> <span class="o">=</span> <span class="n">ressqrt</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">ressqrt</span>
-
-
-<div class="viewcode-block" id="racah"><a class="viewcode-back" href="../../../modules/physics/wigner.html#sympy.physics.wigner.racah">[docs]</a><span class="k">def</span> <span class="nf">racah</span><span class="p">(</span><span class="n">aa</span><span class="p">,</span> <span class="n">bb</span><span class="p">,</span> <span class="n">cc</span><span class="p">,</span> <span class="n">dd</span><span class="p">,</span> <span class="n">ee</span><span class="p">,</span> <span class="n">ff</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Calculate the Racah symbol `W(a,b,c,d;e,f)`.</span>
-
-<span class="sd">    INPUT:</span>
-
-<span class="sd">    -  ``a``, ..., ``f`` - integer or half integer</span>
-
-<span class="sd">    -  ``prec`` - precision, default: ``None``. Providing a precision can</span>
-<span class="sd">       drastically speed up the calculation.</span>
-
-<span class="sd">    OUTPUT:</span>
-
-<span class="sd">    Rational number times the square root of a rational number</span>
-<span class="sd">    (if ``prec=None``), or real number if a precision is given.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.wigner import racah</span>
-<span class="sd">    &gt;&gt;&gt; racah(3,3,3,3,3,3)</span>
-<span class="sd">    -1/14</span>
-
-<span class="sd">    NOTES:</span>
-
-<span class="sd">    The Racah symbol is related to the Wigner 6j symbol:</span>
-
-<span class="sd">    .. math::</span>
-
-<span class="sd">       Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6)</span>
-<span class="sd">       =(-1)^{j_1+j_2+j_4+j_5} W(j_1,j_2,j_5,j_4,j_3,j_6)</span>
-
-<span class="sd">    Please see the 6j symbol for its much richer symmetries and for</span>
-<span class="sd">    additional properties.</span>
-
-<span class="sd">    ALGORITHM:</span>
-
-<span class="sd">    This function uses the algorithm of [Edmonds74]_ to calculate the</span>
-<span class="sd">    value of the 6j symbol exactly. Note that the formula contains</span>
-<span class="sd">    alternating sums over large factorials and is therefore unsuitable</span>
-<span class="sd">    for finite precision arithmetic and only useful for a computer</span>
-<span class="sd">    algebra system [Rasch03]_.</span>
-
-<span class="sd">    AUTHORS:</span>
-
-<span class="sd">    - Jens Rasch (2009-03-24): initial version</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">prefac</span> <span class="o">=</span> <span class="n">_big_delta_coeff</span><span class="p">(</span><span class="n">aa</span><span class="p">,</span> <span class="n">bb</span><span class="p">,</span> <span class="n">ee</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span> <span class="o">*</span> \
-        <span class="n">_big_delta_coeff</span><span class="p">(</span><span class="n">cc</span><span class="p">,</span> <span class="n">dd</span><span class="p">,</span> <span class="n">ee</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span> <span class="o">*</span> \
-        <span class="n">_big_delta_coeff</span><span class="p">(</span><span class="n">aa</span><span class="p">,</span> <span class="n">cc</span><span class="p">,</span> <span class="n">ff</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span> <span class="o">*</span> \
-        <span class="n">_big_delta_coeff</span><span class="p">(</span><span class="n">bb</span><span class="p">,</span> <span class="n">dd</span><span class="p">,</span> <span class="n">ff</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">prefac</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="n">imin</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">aa</span> <span class="o">+</span> <span class="n">bb</span> <span class="o">+</span> <span class="n">ee</span><span class="p">,</span> <span class="n">cc</span> <span class="o">+</span> <span class="n">dd</span> <span class="o">+</span> <span class="n">ee</span><span class="p">,</span> <span class="n">aa</span> <span class="o">+</span> <span class="n">cc</span> <span class="o">+</span> <span class="n">ff</span><span class="p">,</span> <span class="n">bb</span> <span class="o">+</span> <span class="n">dd</span> <span class="o">+</span> <span class="n">ff</span><span class="p">)</span>
-    <span class="n">imax</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">aa</span> <span class="o">+</span> <span class="n">bb</span> <span class="o">+</span> <span class="n">cc</span> <span class="o">+</span> <span class="n">dd</span><span class="p">,</span> <span class="n">aa</span> <span class="o">+</span> <span class="n">dd</span> <span class="o">+</span> <span class="n">ee</span> <span class="o">+</span> <span class="n">ff</span><span class="p">,</span> <span class="n">bb</span> <span class="o">+</span> <span class="n">cc</span> <span class="o">+</span> <span class="n">ee</span> <span class="o">+</span> <span class="n">ff</span><span class="p">)</span>
-
-    <span class="n">maxfact</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">imax</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">aa</span> <span class="o">+</span> <span class="n">bb</span> <span class="o">+</span> <span class="n">cc</span> <span class="o">+</span> <span class="n">dd</span><span class="p">,</span> <span class="n">aa</span> <span class="o">+</span> <span class="n">dd</span> <span class="o">+</span> <span class="n">ee</span> <span class="o">+</span> <span class="n">ff</span><span class="p">,</span>
-                 <span class="n">bb</span> <span class="o">+</span> <span class="n">cc</span> <span class="o">+</span> <span class="n">ee</span> <span class="o">+</span> <span class="n">ff</span><span class="p">)</span>
-    <span class="n">_calc_factlist</span><span class="p">(</span><span class="n">maxfact</span><span class="p">)</span>
-
-    <span class="n">sumres</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">kk</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">imin</span><span class="p">,</span> <span class="n">imax</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">den</span> <span class="o">=</span> <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">kk</span> <span class="o">-</span> <span class="n">aa</span> <span class="o">-</span> <span class="n">bb</span> <span class="o">-</span> <span class="n">ee</span><span class="p">)]</span> <span class="o">*</span> \
-            <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">kk</span> <span class="o">-</span> <span class="n">cc</span> <span class="o">-</span> <span class="n">dd</span> <span class="o">-</span> <span class="n">ee</span><span class="p">)]</span> <span class="o">*</span> \
-            <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">kk</span> <span class="o">-</span> <span class="n">aa</span> <span class="o">-</span> <span class="n">cc</span> <span class="o">-</span> <span class="n">ff</span><span class="p">)]</span> <span class="o">*</span> \
-            <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">kk</span> <span class="o">-</span> <span class="n">bb</span> <span class="o">-</span> <span class="n">dd</span> <span class="o">-</span> <span class="n">ff</span><span class="p">)]</span> <span class="o">*</span> \
-            <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">aa</span> <span class="o">+</span> <span class="n">bb</span> <span class="o">+</span> <span class="n">cc</span> <span class="o">+</span> <span class="n">dd</span> <span class="o">-</span> <span class="n">kk</span><span class="p">)]</span> <span class="o">*</span> \
-            <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">aa</span> <span class="o">+</span> <span class="n">dd</span> <span class="o">+</span> <span class="n">ee</span> <span class="o">+</span> <span class="n">ff</span> <span class="o">-</span> <span class="n">kk</span><span class="p">)]</span> <span class="o">*</span> \
-            <span class="n">_Factlist</span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">bb</span> <span class="o">+</span> <span class="n">cc</span> <span class="o">+</span> <span class="n">ee</span> <span class="o">+</span> <span class="n">ff</span> <span class="o">-</span> <span class="n">kk</span><span class="p">)]</span>
-        <span class="n">sumres</span> <span class="o">=</span> <span class="n">sumres</span> <span class="o">+</span> <span class="n">Integer</span><span class="p">((</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">**</span> <span class="n">kk</span> <span class="o">*</span> <span class="n">_Factlist</span><span class="p">[</span><span class="n">kk</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span> <span class="o">/</span> <span class="n">den</span>
-
-    <span class="n">res</span> <span class="o">=</span> <span class="n">prefac</span> <span class="o">*</span> <span class="n">sumres</span> <span class="o">*</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">**</span> <span class="nb">int</span><span class="p">(</span><span class="n">aa</span> <span class="o">+</span> <span class="n">bb</span> <span class="o">+</span> <span class="n">cc</span> <span class="o">+</span> <span class="n">dd</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">res</span>
-
-</div>
-<div class="viewcode-block" id="wigner_6j"><a class="viewcode-back" href="../../../modules/physics/wigner.html#sympy.physics.wigner.wigner_6j">[docs]</a><span class="k">def</span> <span class="nf">wigner_6j</span><span class="p">(</span><span class="n">j_1</span><span class="p">,</span> <span class="n">j_2</span><span class="p">,</span> <span class="n">j_3</span><span class="p">,</span> <span class="n">j_4</span><span class="p">,</span> <span class="n">j_5</span><span class="p">,</span> <span class="n">j_6</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Calculate the Wigner 6j symbol `Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6)`.</span>
-
-<span class="sd">    INPUT:</span>
-
-<span class="sd">    -  ``j_1``, ..., ``j_6`` - integer or half integer</span>
-
-<span class="sd">    -  ``prec`` - precision, default: ``None``. Providing a precision can</span>
-<span class="sd">       drastically speed up the calculation.</span>
-
-<span class="sd">    OUTPUT:</span>
-
-<span class="sd">    Rational number times the square root of a rational number</span>
-<span class="sd">    (if ``prec=None``), or real number if a precision is given.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.wigner import wigner_6j</span>
-<span class="sd">    &gt;&gt;&gt; wigner_6j(3,3,3,3,3,3)</span>
-<span class="sd">    -1/14</span>
-<span class="sd">    &gt;&gt;&gt; wigner_6j(5,5,5,5,5,5)</span>
-<span class="sd">    1/52</span>
-
-<span class="sd">    It is an error to have arguments that are not integer or half</span>
-<span class="sd">    integer values or do not fulfill the triangle relation::</span>
-
-<span class="sd">        sage: wigner_6j(2.5,2.5,2.5,2.5,2.5,2.5)</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">        ...</span>
-<span class="sd">        ValueError: j values must be integer or half integer and fulfill the triangle relation</span>
-<span class="sd">        sage: wigner_6j(0.5,0.5,1.1,0.5,0.5,1.1)</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">        ...</span>
-<span class="sd">        ValueError: j values must be integer or half integer and fulfill the triangle relation</span>
-
-<span class="sd">    NOTES:</span>
-
-<span class="sd">    The Wigner 6j symbol is related to the Racah symbol but exhibits</span>
-<span class="sd">    more symmetries as detailed below.</span>
-
-<span class="sd">    .. math::</span>
-
-<span class="sd">       Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6)</span>
-<span class="sd">        =(-1)^{j_1+j_2+j_4+j_5} W(j_1,j_2,j_5,j_4,j_3,j_6)</span>
-
-<span class="sd">    The Wigner 6j symbol obeys the following symmetry rules:</span>
-
-<span class="sd">    - Wigner 6j symbols are left invariant under any permutation of</span>
-<span class="sd">      the columns:</span>
-
-<span class="sd">      .. math::</span>
-
-<span class="sd">         Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6)</span>
-<span class="sd">          =Wigner6j(j_3,j_1,j_2,j_6,j_4,j_5)</span>
-<span class="sd">          =Wigner6j(j_2,j_3,j_1,j_5,j_6,j_4)</span>
-<span class="sd">          =Wigner6j(j_3,j_2,j_1,j_6,j_5,j_4)</span>
-<span class="sd">          =Wigner6j(j_1,j_3,j_2,j_4,j_6,j_5)</span>
-<span class="sd">          =Wigner6j(j_2,j_1,j_3,j_5,j_4,j_6)</span>
-
-<span class="sd">    - They are invariant under the exchange of the upper and lower</span>
-<span class="sd">      arguments in each of any two columns, i.e.</span>
-
-<span class="sd">      .. math::</span>
-
-<span class="sd">         Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6)</span>
-<span class="sd">          =Wigner6j(j_1,j_5,j_6,j_4,j_2,j_3)</span>
-<span class="sd">          =Wigner6j(j_4,j_2,j_6,j_1,j_5,j_3)</span>
-<span class="sd">          =Wigner6j(j_4,j_5,j_3,j_1,j_2,j_6)</span>
-
-<span class="sd">    - additional 6 symmetries [Regge59]_ giving rise to 144 symmetries</span>
-<span class="sd">      in total</span>
-
-<span class="sd">    - only non-zero if any triple of `j`&#39;s fulfill a triangle relation</span>
-
-<span class="sd">    ALGORITHM:</span>
-
-<span class="sd">    This function uses the algorithm of [Edmonds74]_ to calculate the</span>
-<span class="sd">    value of the 6j symbol exactly. Note that the formula contains</span>
-<span class="sd">    alternating sums over large factorials and is therefore unsuitable</span>
-<span class="sd">    for finite precision arithmetic and only useful for a computer</span>
-<span class="sd">    algebra system [Rasch03]_.</span>
-
-<span class="sd">    REFERENCES:</span>
-
-<span class="sd">    .. [Regge59] &#39;Symmetry Properties of Racah Coefficients&#39;,</span>
-<span class="sd">      T. Regge, Nuovo Cimento, Volume 11, pp. 116 (1959)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">**</span> <span class="nb">int</span><span class="p">(</span><span class="n">j_1</span> <span class="o">+</span> <span class="n">j_2</span> <span class="o">+</span> <span class="n">j_4</span> <span class="o">+</span> <span class="n">j_5</span><span class="p">)</span> <span class="o">*</span> \
-        <span class="n">racah</span><span class="p">(</span><span class="n">j_1</span><span class="p">,</span> <span class="n">j_2</span><span class="p">,</span> <span class="n">j_5</span><span class="p">,</span> <span class="n">j_4</span><span class="p">,</span> <span class="n">j_3</span><span class="p">,</span> <span class="n">j_6</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">res</span>
-
-</div>
-<div class="viewcode-block" id="wigner_9j"><a class="viewcode-back" href="../../../modules/physics/wigner.html#sympy.physics.wigner.wigner_9j">[docs]</a><span class="k">def</span> <span class="nf">wigner_9j</span><span class="p">(</span><span class="n">j_1</span><span class="p">,</span> <span class="n">j_2</span><span class="p">,</span> <span class="n">j_3</span><span class="p">,</span> <span class="n">j_4</span><span class="p">,</span> <span class="n">j_5</span><span class="p">,</span> <span class="n">j_6</span><span class="p">,</span> <span class="n">j_7</span><span class="p">,</span> <span class="n">j_8</span><span class="p">,</span> <span class="n">j_9</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Calculate the Wigner 9j symbol</span>
-<span class="sd">    `Wigner9j(j_1,j_2,j_3,j_4,j_5,j_6,j_7,j_8,j_9)`.</span>
-
-<span class="sd">    INPUT:</span>
-
-<span class="sd">    -  ``j_1``, ..., ``j_9`` - integer or half integer</span>
-
-<span class="sd">    -  ``prec`` - precision, default: ``None``. Providing a precision can</span>
-<span class="sd">       drastically speed up the calculation.</span>
-
-<span class="sd">    OUTPUT:</span>
-
-<span class="sd">    Rational number times the square root of a rational number</span>
-<span class="sd">    (if ``prec=None``), or real number if a precision is given.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.wigner import wigner_9j</span>
-<span class="sd">    &gt;&gt;&gt; wigner_9j(1,1,1, 1,1,1, 1,1,0 ,prec=64) # ==1/18</span>
-<span class="sd">    0.05555555...</span>
-
-<span class="sd">    It is an error to have arguments that are not integer or half</span>
-<span class="sd">    integer values or do not fulfill the triangle relation::</span>
-
-<span class="sd">        sage: wigner_9j(0.5,0.5,0.5, 0.5,0.5,0.5, 0.5,0.5,0.5,prec=64)</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">        ...</span>
-<span class="sd">        ValueError: j values must be integer or half integer and fulfill the triangle relation</span>
-<span class="sd">        sage: wigner_9j(1,1,1, 0.5,1,1.5, 0.5,1,2.5,prec=64)</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">        ...</span>
-<span class="sd">        ValueError: j values must be integer or half integer and fulfill the triangle relation</span>
-
-<span class="sd">    ALGORITHM:</span>
-
-<span class="sd">    This function uses the algorithm of [Edmonds74]_ to calculate the</span>
-<span class="sd">    value of the 3j symbol exactly. Note that the formula contains</span>
-<span class="sd">    alternating sums over large factorials and is therefore unsuitable</span>
-<span class="sd">    for finite precision arithmetic and only useful for a computer</span>
-<span class="sd">    algebra system [Rasch03]_.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">imin</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">imax</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">j_1</span> <span class="o">+</span> <span class="n">j_9</span><span class="p">,</span> <span class="n">j_2</span> <span class="o">+</span> <span class="n">j_6</span><span class="p">,</span> <span class="n">j_4</span> <span class="o">+</span> <span class="n">j_8</span><span class="p">)</span>
-
-    <span class="n">sumres</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">kk</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">imin</span><span class="p">,</span> <span class="n">imax</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">sumres</span> <span class="o">=</span> <span class="n">sumres</span> <span class="o">+</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">kk</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> \
-            <span class="n">racah</span><span class="p">(</span><span class="n">j_1</span><span class="p">,</span> <span class="n">j_2</span><span class="p">,</span> <span class="n">j_9</span><span class="p">,</span> <span class="n">j_6</span><span class="p">,</span> <span class="n">j_3</span><span class="p">,</span> <span class="n">kk</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span> <span class="o">*</span> \
-            <span class="n">racah</span><span class="p">(</span><span class="n">j_4</span><span class="p">,</span> <span class="n">j_6</span><span class="p">,</span> <span class="n">j_8</span><span class="p">,</span> <span class="n">j_2</span><span class="p">,</span> <span class="n">j_5</span><span class="p">,</span> <span class="n">kk</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span> <span class="o">*</span> \
-            <span class="n">racah</span><span class="p">(</span><span class="n">j_1</span><span class="p">,</span> <span class="n">j_4</span><span class="p">,</span> <span class="n">j_9</span><span class="p">,</span> <span class="n">j_8</span><span class="p">,</span> <span class="n">j_7</span><span class="p">,</span> <span class="n">kk</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">sumres</span>
-
-</div>
-<div class="viewcode-block" id="gaunt"><a class="viewcode-back" href="../../../modules/physics/wigner.html#sympy.physics.wigner.gaunt">[docs]</a><span class="k">def</span> <span class="nf">gaunt</span><span class="p">(</span><span class="n">l_1</span><span class="p">,</span> <span class="n">l_2</span><span class="p">,</span> <span class="n">l_3</span><span class="p">,</span> <span class="n">m_1</span><span class="p">,</span> <span class="n">m_2</span><span class="p">,</span> <span class="n">m_3</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Calculate the Gaunt coefficient.</span>
-
-<span class="sd">    The Gaunt coefficient is defined as the integral over three</span>
-<span class="sd">    spherical harmonics:</span>
-
-<span class="sd">    .. math::</span>
-
-<span class="sd">        Y(j_1,j_2,j_3,m_1,m_2,m_3)</span>
-<span class="sd">        =\int Y_{l_1,m_1}(\Omega)</span>
-<span class="sd">         Y_{l_2,m_2}(\Omega) Y_{l_3,m_3}(\Omega) d\Omega</span>
-<span class="sd">        =\sqrt{(2l_1+1)(2l_2+1)(2l_3+1)/(4\pi)}</span>
-<span class="sd">         \; Y(j_1,j_2,j_3,0,0,0) \; Y(j_1,j_2,j_3,m_1,m_2,m_3)</span>
-
-<span class="sd">    INPUT:</span>
-
-<span class="sd">    -  ``l_1``, ``l_2``, ``l_3``, ``m_1``, ``m_2``, ``m_3`` - integer</span>
-
-<span class="sd">    -  ``prec`` - precision, default: ``None``. Providing a precision can</span>
-<span class="sd">       drastically speed up the calculation.</span>
-
-<span class="sd">    OUTPUT:</span>
-
-<span class="sd">    Rational number times the square root of a rational number</span>
-<span class="sd">    (if ``prec=None``), or real number if a precision is given.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.physics.wigner import gaunt</span>
-<span class="sd">    &gt;&gt;&gt; gaunt(1,0,1,1,0,-1)</span>
-<span class="sd">    -1/(2*sqrt(pi))</span>
-<span class="sd">    &gt;&gt;&gt; gaunt(1000,1000,1200,9,3,-12).n(64)</span>
-<span class="sd">    0.00689500421922113448...</span>
-
-<span class="sd">    It is an error to use non-integer values for `l` and `m`::</span>
-
-<span class="sd">        sage: gaunt(1.2,0,1.2,0,0,0)</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">        ...</span>
-<span class="sd">        ValueError: l values must be integer</span>
-<span class="sd">        sage: gaunt(1,0,1,1.1,0,-1.1)</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">        ...</span>
-<span class="sd">        ValueError: m values must be integer</span>
-
-<span class="sd">    NOTES:</span>
-
-<span class="sd">    The Gaunt coefficient obeys the following symmetry rules:</span>
-
-<span class="sd">    - invariant under any permutation of the columns</span>
-
-<span class="sd">      .. math::</span>
-<span class="sd">          Y(j_1,j_2,j_3,m_1,m_2,m_3)</span>
-<span class="sd">          =Y(j_3,j_1,j_2,m_3,m_1,m_2)</span>
-<span class="sd">          =Y(j_2,j_3,j_1,m_2,m_3,m_1)</span>
-<span class="sd">          =Y(j_3,j_2,j_1,m_3,m_2,m_1)</span>
-<span class="sd">          =Y(j_1,j_3,j_2,m_1,m_3,m_2)</span>
-<span class="sd">          =Y(j_2,j_1,j_3,m_2,m_1,m_3)</span>
-
-<span class="sd">    - invariant under space inflection, i.e.</span>
-
-<span class="sd">      .. math::</span>
-<span class="sd">          Y(j_1,j_2,j_3,m_1,m_2,m_3)</span>
-<span class="sd">          =Y(j_1,j_2,j_3,-m_1,-m_2,-m_3)</span>
-
-<span class="sd">    - symmetric with respect to the 72 Regge symmetries as inherited</span>
-<span class="sd">      for the `3j` symbols [Regge58]_</span>
-
-<span class="sd">    - zero for `l_1`, `l_2`, `l_3` not fulfilling triangle relation</span>
-
-<span class="sd">    - zero for violating any one of the conditions: `l_1 \ge |m_1|`,</span>
-<span class="sd">      `l_2 \ge |m_2|`, `l_3 \ge |m_3|`</span>
-
-<span class="sd">    - non-zero only for an even sum of the `l_i`, i.e.</span>
-<span class="sd">      `J = l_1 + l_2 + l_3 = 2n` for `n` in `\mathbb{N}`</span>
-
-<span class="sd">    ALGORITHM:</span>
-
-<span class="sd">    This function uses the algorithm of [Liberatodebrito82]_ to</span>
-<span class="sd">    calculate the value of the Gaunt coefficient exactly. Note that</span>
-<span class="sd">    the formula contains alternating sums over large factorials and is</span>
-<span class="sd">    therefore unsuitable for finite precision arithmetic and only</span>
-<span class="sd">    useful for a computer algebra system [Rasch03]_.</span>
-
-<span class="sd">    REFERENCES:</span>
-
-<span class="sd">    .. [Liberatodebrito82] &#39;FORTRAN program for the integral of three</span>
-<span class="sd">      spherical harmonics&#39;, A. Liberato de Brito,</span>
-<span class="sd">      Comput. Phys. Commun., Volume 25, pp. 81-85 (1982)</span>
-
-<span class="sd">    AUTHORS:</span>
-
-<span class="sd">    - Jens Rasch (2009-03-24): initial version for Sage</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">int</span><span class="p">(</span><span class="n">l_1</span><span class="p">)</span> <span class="o">!=</span> <span class="n">l_1</span> <span class="ow">or</span> <span class="nb">int</span><span class="p">(</span><span class="n">l_2</span><span class="p">)</span> <span class="o">!=</span> <span class="n">l_2</span> <span class="ow">or</span> <span class="nb">int</span><span class="p">(</span><span class="n">l_3</span><span class="p">)</span> <span class="o">!=</span> <span class="n">l_3</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;l values must be integer&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">int</span><span class="p">(</span><span class="n">m_1</span><span class="p">)</span> <span class="o">!=</span> <span class="n">m_1</span> <span class="ow">or</span> <span class="nb">int</span><span class="p">(</span><span class="n">m_2</span><span class="p">)</span> <span class="o">!=</span> <span class="n">m_2</span> <span class="ow">or</span> <span class="nb">int</span><span class="p">(</span><span class="n">m_3</span><span class="p">)</span> <span class="o">!=</span> <span class="n">m_3</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;m values must be integer&quot;</span><span class="p">)</span>
-
-    <span class="n">bigL</span> <span class="o">=</span> <span class="p">(</span><span class="n">l_1</span> <span class="o">+</span> <span class="n">l_2</span> <span class="o">+</span> <span class="n">l_3</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>
-    <span class="n">a1</span> <span class="o">=</span> <span class="n">l_1</span> <span class="o">+</span> <span class="n">l_2</span> <span class="o">-</span> <span class="n">l_3</span>
-    <span class="k">if</span> <span class="n">a1</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="n">a2</span> <span class="o">=</span> <span class="n">l_1</span> <span class="o">-</span> <span class="n">l_2</span> <span class="o">+</span> <span class="n">l_3</span>
-    <span class="k">if</span> <span class="n">a2</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="n">a3</span> <span class="o">=</span> <span class="o">-</span><span class="n">l_1</span> <span class="o">+</span> <span class="n">l_2</span> <span class="o">+</span> <span class="n">l_3</span>
-    <span class="k">if</span> <span class="n">a3</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="k">if</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">bigL</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="k">if</span> <span class="p">(</span><span class="n">m_1</span> <span class="o">+</span> <span class="n">m_2</span> <span class="o">+</span> <span class="n">m_3</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="k">if</span> <span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">m_1</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">l_1</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">m_2</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">l_2</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">m_3</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">l_3</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">0</span>
-
-    <span class="n">imin</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="o">-</span><span class="n">l_3</span> <span class="o">+</span> <span class="n">l_1</span> <span class="o">+</span> <span class="n">m_2</span><span class="p">,</span> <span class="o">-</span><span class="n">l_3</span> <span class="o">+</span> <span class="n">l_2</span> <span class="o">-</span> <span class="n">m_1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-    <span class="n">imax</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">l_2</span> <span class="o">+</span> <span class="n">m_2</span><span class="p">,</span> <span class="n">l_1</span> <span class="o">-</span> <span class="n">m_1</span><span class="p">,</span> <span class="n">l_1</span> <span class="o">+</span> <span class="n">l_2</span> <span class="o">-</span> <span class="n">l_3</span><span class="p">)</span>
-
-    <span class="n">maxfact</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">l_1</span> <span class="o">+</span> <span class="n">l_2</span> <span class="o">+</span> <span class="n">l_3</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">imax</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="n">_calc_factlist</span><span class="p">(</span><span class="n">maxfact</span><span class="p">)</span>
-
-    <span class="n">argsqrt</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">l_1</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">l_2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">l_3</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> \
-        <span class="n">_Factlist</span><span class="p">[</span><span class="n">l_1</span> <span class="o">-</span> <span class="n">m_1</span><span class="p">]</span> <span class="o">*</span> <span class="n">_Factlist</span><span class="p">[</span><span class="n">l_1</span> <span class="o">+</span> <span class="n">m_1</span><span class="p">]</span> <span class="o">*</span> <span class="n">_Factlist</span><span class="p">[</span><span class="n">l_2</span> <span class="o">-</span> <span class="n">m_2</span><span class="p">]</span> <span class="o">*</span> \
-        <span class="n">_Factlist</span><span class="p">[</span><span class="n">l_2</span> <span class="o">+</span> <span class="n">m_2</span><span class="p">]</span> <span class="o">*</span> <span class="n">_Factlist</span><span class="p">[</span><span class="n">l_3</span> <span class="o">-</span> <span class="n">m_3</span><span class="p">]</span> <span class="o">*</span> <span class="n">_Factlist</span><span class="p">[</span><span class="n">l_3</span> <span class="o">+</span> <span class="n">m_3</span><span class="p">]</span> <span class="o">/</span> \
-        <span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span>
-    <span class="n">ressqrt</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">argsqrt</span><span class="p">)</span>
-
-    <span class="n">prefac</span> <span class="o">=</span> <span class="n">Integer</span><span class="p">(</span><span class="n">_Factlist</span><span class="p">[</span><span class="n">bigL</span><span class="p">]</span> <span class="o">*</span> <span class="n">_Factlist</span><span class="p">[</span><span class="n">l_2</span> <span class="o">-</span> <span class="n">l_1</span> <span class="o">+</span> <span class="n">l_3</span><span class="p">]</span> <span class="o">*</span>
-                     <span class="n">_Factlist</span><span class="p">[</span><span class="n">l_1</span> <span class="o">-</span> <span class="n">l_2</span> <span class="o">+</span> <span class="n">l_3</span><span class="p">]</span> <span class="o">*</span> <span class="n">_Factlist</span><span class="p">[</span><span class="n">l_1</span> <span class="o">+</span> <span class="n">l_2</span> <span class="o">-</span> <span class="n">l_3</span><span class="p">])</span><span class="o">/</span> \
-        <span class="n">_Factlist</span><span class="p">[</span><span class="mi">2</span> <span class="o">*</span> <span class="n">bigL</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span><span class="o">/</span> \
-        <span class="p">(</span><span class="n">_Factlist</span><span class="p">[</span><span class="n">bigL</span> <span class="o">-</span> <span class="n">l_1</span><span class="p">]</span> <span class="o">*</span>
-         <span class="n">_Factlist</span><span class="p">[</span><span class="n">bigL</span> <span class="o">-</span> <span class="n">l_2</span><span class="p">]</span> <span class="o">*</span> <span class="n">_Factlist</span><span class="p">[</span><span class="n">bigL</span> <span class="o">-</span> <span class="n">l_3</span><span class="p">])</span>
-
-    <span class="n">sumres</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">ii</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">imin</span><span class="p">,</span> <span class="n">imax</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">den</span> <span class="o">=</span> <span class="n">_Factlist</span><span class="p">[</span><span class="n">ii</span><span class="p">]</span> <span class="o">*</span> <span class="n">_Factlist</span><span class="p">[</span><span class="n">ii</span> <span class="o">+</span> <span class="n">l_3</span> <span class="o">-</span> <span class="n">l_1</span> <span class="o">-</span> <span class="n">m_2</span><span class="p">]</span> <span class="o">*</span> \
-            <span class="n">_Factlist</span><span class="p">[</span><span class="n">l_2</span> <span class="o">+</span> <span class="n">m_2</span> <span class="o">-</span> <span class="n">ii</span><span class="p">]</span> <span class="o">*</span> <span class="n">_Factlist</span><span class="p">[</span><span class="n">l_1</span> <span class="o">-</span> <span class="n">ii</span> <span class="o">-</span> <span class="n">m_1</span><span class="p">]</span> <span class="o">*</span> \
-            <span class="n">_Factlist</span><span class="p">[</span><span class="n">ii</span> <span class="o">+</span> <span class="n">l_3</span> <span class="o">-</span> <span class="n">l_2</span> <span class="o">+</span> <span class="n">m_1</span><span class="p">]</span> <span class="o">*</span> <span class="n">_Factlist</span><span class="p">[</span><span class="n">l_1</span> <span class="o">+</span> <span class="n">l_2</span> <span class="o">-</span> <span class="n">l_3</span> <span class="o">-</span> <span class="n">ii</span><span class="p">]</span>
-        <span class="n">sumres</span> <span class="o">=</span> <span class="n">sumres</span> <span class="o">+</span> <span class="n">Integer</span><span class="p">((</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">**</span> <span class="n">ii</span><span class="p">)</span> <span class="o">/</span> <span class="n">den</span>
-
-    <span class="n">res</span> <span class="o">=</span> <span class="n">ressqrt</span> <span class="o">*</span> <span class="n">prefac</span> <span class="o">*</span> <span class="n">sumres</span> <span class="o">*</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">**</span> <span class="p">(</span><span class="n">bigL</span> <span class="o">+</span> <span class="n">l_3</span> <span class="o">+</span> <span class="n">m_1</span> <span class="o">-</span> <span class="n">m_2</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">prec</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="n">res</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">res</span></div>
-</pre></div>
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-            
-  <h1>Source code for sympy.plotting.plot</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Plotting module for Sympy.</span>
-
-<span class="sd">A plot is represented by the ``Plot`` class that contains a reference to the</span>
-<span class="sd">backend and a list of the data series to be plotted. The data series are</span>
-<span class="sd">instances of classes meant to simplify getting points and meshes from sympy</span>
-<span class="sd">expressions. ``plot_backends`` is a dictionary with all the backends.</span>
-
-<span class="sd">This module gives only the essential. For all the fancy stuff use directly</span>
-<span class="sd">the backend. You can get the backend wrapper for every plot from the</span>
-<span class="sd">``_backend`` attribute. Moreover the data series classes have various useful</span>
-<span class="sd">methods like ``get_points``, ``get_segments``, ``get_meshes``, etc, that may</span>
-<span class="sd">be useful if you wish to use another plotting library.</span>
-
-<span class="sd">Especially if you need publication ready graphs and this module is not enough</span>
-<span class="sd">for you - just get the ``_backend`` attribute and add whatever you want</span>
-<span class="sd">directly to it. In the case of matplotlib (the common way to graph data in</span>
-<span class="sd">python) just copy ``_backend.fig`` which is the figure and ``_backend.ax``</span>
-<span class="sd">which is the axis and work on them as you would on any other matplotlib object.</span>
-
-<span class="sd">Simplicity of code takes much greater importance than performance. Don&#39;t use it</span>
-<span class="sd">if you care at all about performance. A new backend instance is initialized</span>
-<span class="sd">every time you call ``show()`` and the old one is left to the garbage collector.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">inspect</span> <span class="kn">import</span> <span class="n">getargspec</span>
-<span class="kn">from</span> <span class="nn">itertools</span> <span class="kn">import</span> <span class="n">chain</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">,</span> <span class="n">Dummy</span>
-<span class="kn">from</span> <span class="nn">sympy.external</span> <span class="kn">import</span> <span class="n">import_module</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">set_union</span>
-<span class="kn">import</span> <span class="nn">warnings</span>
-<span class="kn">from</span> <span class="nn">.experimental_lambdify</span> <span class="kn">import</span> <span class="p">(</span><span class="n">vectorized_lambdify</span><span class="p">,</span> <span class="n">lambdify</span><span class="p">)</span>
-<span class="kn">import</span> <span class="nn">collections</span>
-
-<span class="c">#TODO probably all of the imports after this line can be put inside function to</span>
-<span class="c"># speed up the `from sympy import *` command.</span>
-<span class="n">np</span> <span class="o">=</span> <span class="n">import_module</span><span class="p">(</span><span class="s">&#39;numpy&#39;</span><span class="p">)</span>
-
-<span class="c"># Backend specific imports - matplotlib</span>
-<span class="n">matplotlib</span> <span class="o">=</span> <span class="n">import_module</span><span class="p">(</span><span class="s">&#39;matplotlib&#39;</span><span class="p">,</span>
-    <span class="n">__import__kwargs</span><span class="o">=</span><span class="p">{</span><span class="s">&#39;fromlist&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s">&#39;pyplot&#39;</span><span class="p">,</span> <span class="s">&#39;cm&#39;</span><span class="p">,</span> <span class="s">&#39;collections&#39;</span><span class="p">]},</span>
-    <span class="n">min_module_version</span><span class="o">=</span><span class="s">&#39;1.0.0&#39;</span><span class="p">,</span> <span class="n">catch</span><span class="o">=</span><span class="p">(</span><span class="ne">RuntimeError</span><span class="p">,))</span>
-<span class="k">if</span> <span class="n">matplotlib</span><span class="p">:</span>
-    <span class="n">plt</span> <span class="o">=</span> <span class="n">matplotlib</span><span class="o">.</span><span class="n">pyplot</span>
-    <span class="n">cm</span> <span class="o">=</span> <span class="n">matplotlib</span><span class="o">.</span><span class="n">cm</span>
-    <span class="n">LineCollection</span> <span class="o">=</span> <span class="n">matplotlib</span><span class="o">.</span><span class="n">collections</span><span class="o">.</span><span class="n">LineCollection</span>
-    <span class="n">mpl_toolkits</span> <span class="o">=</span> <span class="n">import_module</span><span class="p">(</span><span class="s">&#39;mpl_toolkits&#39;</span><span class="p">,</span>
-            <span class="n">__import__kwargs</span><span class="o">=</span><span class="p">{</span><span class="s">&#39;fromlist&#39;</span><span class="p">:</span> <span class="p">[</span><span class="s">&#39;mplot3d&#39;</span><span class="p">]})</span>
-    <span class="n">Axes3D</span> <span class="o">=</span> <span class="n">mpl_toolkits</span><span class="o">.</span><span class="n">mplot3d</span><span class="o">.</span><span class="n">Axes3D</span>
-    <span class="n">art3d</span> <span class="o">=</span> <span class="n">mpl_toolkits</span><span class="o">.</span><span class="n">mplot3d</span><span class="o">.</span><span class="n">art3d</span>
-    <span class="n">ListedColormap</span> <span class="o">=</span> <span class="n">matplotlib</span><span class="o">.</span><span class="n">colors</span><span class="o">.</span><span class="n">ListedColormap</span>
-
-<span class="c"># Backend specific imports - textplot</span>
-<span class="kn">from</span> <span class="nn">sympy.plotting.textplot</span> <span class="kn">import</span> <span class="n">textplot</span>
-
-<span class="c"># Global variable</span>
-<span class="c"># Set to False when running tests / doctests so that the plots don&#39;t</span>
-<span class="c"># show.</span>
-<span class="n">_show</span> <span class="o">=</span> <span class="bp">True</span>
-
-
-<span class="k">def</span> <span class="nf">unset_show</span><span class="p">():</span>
-    <span class="k">global</span> <span class="n">_show</span>
-    <span class="n">_show</span> <span class="o">=</span> <span class="bp">False</span>
-
-<span class="c">##############################################################################</span>
-<span class="c"># The public interface</span>
-<span class="c">##############################################################################</span>
-
-
-<div class="viewcode-block" id="Plot"><a class="viewcode-back" href="../../../modules/plotting.html#sympy.plotting.plot.Plot">[docs]</a><span class="k">class</span> <span class="nc">Plot</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The central class of the plotting module.</span>
-
-<span class="sd">    For interactive work the function ``plot`` is better suited.</span>
-
-<span class="sd">    This class permits the plotting of sympy expressions using numerous</span>
-<span class="sd">    backends (matplotlib, textplot, the old pyglet module for sympy, Google</span>
-<span class="sd">    charts api, etc).</span>
-
-<span class="sd">    The figure can contain an arbitrary number of plots of sympy expressions,</span>
-<span class="sd">    lists of coordinates of points, etc. Plot has a private attribute _series that</span>
-<span class="sd">    contains all data series to be plotted (expressions for lines or surfaces,</span>
-<span class="sd">    lists of points, etc (all subclasses of BaseSeries)). Those data series are</span>
-<span class="sd">    instances of classes not imported by ``from sympy import *``.</span>
-
-<span class="sd">    The customization of the figure is on two levels. Global options that</span>
-<span class="sd">    concern the figure as a whole (eg title, xlabel, scale, etc) and</span>
-<span class="sd">    per-data series options (eg name) and aesthetics (eg. color, point shape,</span>
-<span class="sd">    line type, etc.).</span>
-
-<span class="sd">    The difference between options and aesthetics is that an aesthetic can be</span>
-<span class="sd">    a function of the coordinates (or parameters in a parametric plot). The</span>
-<span class="sd">    supported values for an aesthetic are:</span>
-<span class="sd">    - None (the backend uses default values)</span>
-<span class="sd">    - a constant</span>
-<span class="sd">    - a function of one variable (the first coordinate or parameter)</span>
-<span class="sd">    - a function of two variables (the first and second coordinate or</span>
-<span class="sd">    parameters)</span>
-<span class="sd">    - a function of three variables (only in nonparametric 3D plots)</span>
-<span class="sd">    Their implementation depends on the backend so they may not work in some</span>
-<span class="sd">    backends.</span>
-
-<span class="sd">    If the plot is parametric and the arity of the aesthetic function permits</span>
-<span class="sd">    it the aesthetic is calculated over parameters and not over coordinates.</span>
-<span class="sd">    If the arity does not permit calculation over parameters the calculation is</span>
-<span class="sd">    done over coordinates.</span>
-
-<span class="sd">    Only cartesian coordinates are supported for the moment, but you can use</span>
-<span class="sd">    the parametric plots to plot in polar, spherical and cylindrical</span>
-<span class="sd">    coordinates.</span>
-
-<span class="sd">    The arguments for the constructor Plot must be subclasses of BaseSeries.</span>
-
-<span class="sd">    Any global option can be specified as a keyword argument.</span>
-
-<span class="sd">    The global options for a figure are:</span>
-
-<span class="sd">    - title : str</span>
-<span class="sd">    - xlabel : str</span>
-<span class="sd">    - ylabel : str</span>
-<span class="sd">    - legend : bool</span>
-<span class="sd">    - xscale : {&#39;linear&#39;, &#39;log&#39;}</span>
-<span class="sd">    - yscale : {&#39;linear&#39;, &#39;log&#39;}</span>
-<span class="sd">    - axis : bool</span>
-<span class="sd">    - axis_center : tuple of two floats or {&#39;center&#39;, &#39;auto&#39;}</span>
-<span class="sd">    - xlim : tuple of two floats</span>
-<span class="sd">    - ylim : tuple of two floats</span>
-<span class="sd">    - aspect_ratio : tuple of two floats or {&#39;auto&#39;}</span>
-<span class="sd">    - autoscale : bool</span>
-<span class="sd">    - margin : float in [0, 1]</span>
-
-<span class="sd">    The per data series options and aesthetics are:</span>
-<span class="sd">    There are none in the base series. See below for options for subclasses.</span>
-
-<span class="sd">    Some data series support additional aesthetics or options:</span>
-
-<span class="sd">    ListSeries, LineOver1DRangeSeries, Parametric2DLineSeries,</span>
-<span class="sd">    Parametric3DLineSeries support the following:</span>
-
-<span class="sd">    Aesthetics:</span>
-
-<span class="sd">    - line_color : function which returns a float.</span>
-
-<span class="sd">    options:</span>
-
-<span class="sd">    - label : str</span>
-<span class="sd">    - steps : bool</span>
-<span class="sd">    - integers_only : bool</span>
-
-<span class="sd">    SurfaceOver2DRangeSeries, ParametricSurfaceSeries support the following:</span>
-
-<span class="sd">    aesthetics:</span>
-
-<span class="sd">    - surface_color : function which returns a float.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="nb">super</span><span class="p">(</span><span class="n">Plot</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__init__</span><span class="p">()</span>
-
-        <span class="c">#  Options for the graph as a whole.</span>
-        <span class="c">#  The possible values for each option are described in the docstring of</span>
-        <span class="c"># Plot. They are based purely on convention, no checking is done.</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">title</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">xlabel</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ylabel</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">aspect_ratio</span> <span class="o">=</span> <span class="s">&#39;auto&#39;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">xlim</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ylim</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">axis_center</span> <span class="o">=</span> <span class="s">&#39;auto&#39;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">axis</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">xscale</span> <span class="o">=</span> <span class="s">&#39;linear&#39;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">yscale</span> <span class="o">=</span> <span class="s">&#39;linear&#39;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">legend</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">autoscale</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">margin</span> <span class="o">=</span> <span class="mi">0</span>
-
-        <span class="c"># Contains the data objects to be plotted. The backend should be smart</span>
-        <span class="c"># enough to iterate over this list.</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_series</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_series</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="c"># The backend type. On every show() a new backend instance is created</span>
-        <span class="c"># in self._backend which is tightly coupled to the Plot instance</span>
-        <span class="c"># (thanks to the parent attribute of the backend).</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">backend</span> <span class="o">=</span> <span class="n">DefaultBackend</span>
-
-        <span class="c"># The keyword arguments should only contain options for the plot.</span>
-        <span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="n">val</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">):</span>
-                <span class="nb">setattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">show</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># TODO move this to the backend (also for save)</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s">&#39;_backend&#39;</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_backend</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_backend</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">backend</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_backend</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">save</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">path</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s">&#39;_backend&#39;</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_backend</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_backend</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">backend</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_backend</span><span class="o">.</span><span class="n">save</span><span class="p">(</span><span class="n">path</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">series_strs</span> <span class="o">=</span> <span class="p">[(</span><span class="s">&#39;[</span><span class="si">%d</span><span class="s">]: &#39;</span> <span class="o">%</span> <span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-                       <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">s</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_series</span><span class="p">)]</span>
-        <span class="k">return</span> <span class="s">&#39;Plot object containing:</span><span class="se">\n</span><span class="s">&#39;</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">series_strs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">index</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_series</span><span class="p">[</span><span class="n">index</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__setitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">index</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">BaseSeries</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_series</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="o">=</span> <span class="n">args</span>
-
-    <span class="k">def</span> <span class="nf">__delitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">index</span><span class="p">):</span>
-        <span class="k">del</span> <span class="bp">self</span><span class="o">.</span><span class="n">_series</span><span class="p">[</span><span class="n">index</span><span class="p">]</span>
-
-<div class="viewcode-block" id="Plot.append"><a class="viewcode-back" href="../../../modules/plotting.html#sympy.plotting.plot.Plot.append">[docs]</a>    <span class="k">def</span> <span class="nf">append</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Adds one more graph to the figure.&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">BaseSeries</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_series</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_series</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Series</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="Plot.extend"><a class="viewcode-back" href="../../../modules/plotting.html#sympy.plotting.plot.Plot.extend">[docs]</a>    <span class="k">def</span> <span class="nf">extend</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Adds the series from another plot or a list of series.&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Plot</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_series</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">_series</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_series</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-
-<span class="c">##############################################################################</span>
-<span class="c"># Data Series</span>
-<span class="c">##############################################################################</span>
-<span class="c">#TODO more general way to calculate aesthetics (see get_color_array)</span>
-
-<span class="c">### The base class for all series</span></div></div>
-<div class="viewcode-block" id="BaseSeries"><a class="viewcode-back" href="../../../modules/plotting.html#sympy.plotting.plot.BaseSeries">[docs]</a><span class="k">class</span> <span class="nc">BaseSeries</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Base class for the data objects containing stuff to be plotted.</span>
-
-<span class="sd">    The backend should check if it supports the data series that it&#39;s given.</span>
-<span class="sd">    (eg TextBackend supports only LineOver1DRange).</span>
-<span class="sd">    It&#39;s the backend responsibility to know how to use the class of</span>
-<span class="sd">    data series that it&#39;s given.</span>
-
-<span class="sd">    Some data series classes are grouped (using a class attribute like is_2Dline)</span>
-<span class="sd">    according to the api they present (based only on convention). The backend is</span>
-<span class="sd">    not obliged to use that api (eg. The LineOver1DRange belongs to the</span>
-<span class="sd">    is_2Dline group and presents the get_points method, but the</span>
-<span class="sd">    TextBackend does not use the get_points method).</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="c"># Some flags follow. The rationale for using flags instead of checking base</span>
-    <span class="c"># classes is that setting multiple flags is simpler than multiple</span>
-    <span class="c"># inheritance.</span>
-
-    <span class="n">is_2Dline</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="c"># Some of the backends expect:</span>
-    <span class="c">#  - get_points returning 1D np.arrays list_x, list_y</span>
-    <span class="c">#  - get_segments returning np.array (done in Line2DBaseSeries)</span>
-    <span class="c">#  - get_color_array returning 1D np.array (done in Line2DBaseSeries)</span>
-    <span class="c"># with the colors calculated at the points from get_points</span>
-
-    <span class="n">is_3Dline</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="c"># Some of the backends expect:</span>
-    <span class="c">#  - get_points returning 1D np.arrays list_x, list_y, list_y</span>
-    <span class="c">#  - get_segments returning np.array (done in Line2DBaseSeries)</span>
-    <span class="c">#  - get_color_array returning 1D np.array (done in Line2DBaseSeries)</span>
-    <span class="c"># with the colors calculated at the points from get_points</span>
-
-    <span class="n">is_3Dsurface</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="c"># Some of the backends expect:</span>
-    <span class="c">#   - get_meshes returning mesh_x, mesh_y, mesh_z (2D np.arrays)</span>
-    <span class="c">#   - get_points an alias for get_meshes</span>
-
-    <span class="n">is_contour</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="c"># Some of the backends expect:</span>
-    <span class="c">#   - get_meshes returning mesh_x, mesh_y, mesh_z (2D np.arrays)</span>
-    <span class="c">#   - get_points an alias for get_meshes</span>
-
-    <span class="n">is_implicit</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="c"># Some of the backends expect:</span>
-    <span class="c">#   - get_meshes returning mesh_x (1D array), mesh_y(1D array,</span>
-    <span class="c">#     mesh_z (2D np.arrays)</span>
-    <span class="c">#   - get_points an alias for get_meshes</span>
-    <span class="c">#Different from is_contour as the colormap in backend will be</span>
-    <span class="c">#different</span>
-
-    <span class="n">is_parametric</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="c"># The calculation of aesthetics expects:</span>
-    <span class="c">#   - get_parameter_points returning one or two np.arrays (1D or 2D)</span>
-    <span class="c"># used for calculation aesthetics</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="nb">super</span><span class="p">(</span><span class="n">BaseSeries</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__init__</span><span class="p">()</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_3D</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">flags3D</span> <span class="o">=</span> <span class="p">[</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">is_3Dline</span><span class="p">,</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">is_3Dsurface</span>
-        <span class="p">]</span>
-        <span class="k">return</span> <span class="nb">any</span><span class="p">(</span><span class="n">flags3D</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_line</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">flagslines</span> <span class="o">=</span> <span class="p">[</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">is_2Dline</span><span class="p">,</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">is_3Dline</span>
-        <span class="p">]</span>
-        <span class="k">return</span> <span class="nb">any</span><span class="p">(</span><span class="n">flagslines</span><span class="p">)</span>
-
-
-<span class="c">### 2D lines</span></div>
-<div class="viewcode-block" id="Line2DBaseSeries"><a class="viewcode-back" href="../../../modules/plotting.html#sympy.plotting.plot.Line2DBaseSeries">[docs]</a><span class="k">class</span> <span class="nc">Line2DBaseSeries</span><span class="p">(</span><span class="n">BaseSeries</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A base class for 2D lines.</span>
-
-<span class="sd">    - adding the label, steps and only_integers options</span>
-<span class="sd">    - making is_2Dline true</span>
-<span class="sd">    - defining get_segments and get_color_array</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_2Dline</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="n">_dim</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="nb">super</span><span class="p">(</span><span class="n">Line2DBaseSeries</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__init__</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">label</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">steps</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">only_integers</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">line_color</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">get_segments</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">points</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_points</span><span class="p">()</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">steps</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">((</span><span class="n">points</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">points</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span><span class="o">.</span><span class="n">T</span><span class="o">.</span><span class="n">flatten</span><span class="p">()[</span><span class="mi">1</span><span class="p">:]</span>
-            <span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">((</span><span class="n">points</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">points</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span><span class="o">.</span><span class="n">T</span><span class="o">.</span><span class="n">flatten</span><span class="p">()[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-            <span class="n">points</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-        <span class="n">points</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ma</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">points</span><span class="p">)</span><span class="o">.</span><span class="n">T</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_dim</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">ma</span><span class="o">.</span><span class="n">concatenate</span><span class="p">([</span><span class="n">points</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">points</span><span class="p">[</span><span class="mi">1</span><span class="p">:]],</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">get_color_array</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">line_color</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="s">&#39;__call__&#39;</span><span class="p">):</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vectorize</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-            <span class="n">arity</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">getargspec</span><span class="p">(</span><span class="n">c</span><span class="p">)[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="k">if</span> <span class="n">arity</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_parametric</span><span class="p">:</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_parameter_points</span><span class="p">()</span>
-                <span class="k">return</span> <span class="n">f</span><span class="p">(</span><span class="n">centers_of_segments</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">variables</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">centers_of_segments</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_points</span><span class="p">()))</span>
-                <span class="k">if</span> <span class="n">arity</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">f</span><span class="p">(</span><span class="n">variables</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-                <span class="k">elif</span> <span class="n">arity</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">f</span><span class="p">(</span><span class="o">*</span><span class="n">variables</span><span class="p">[:</span><span class="mi">2</span><span class="p">])</span>
-                <span class="k">else</span><span class="p">:</span>  <span class="c"># only if the line is 3D (otherwise raises an error)</span>
-                    <span class="k">return</span> <span class="n">f</span><span class="p">(</span><span class="o">*</span><span class="n">variables</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">c</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points</span><span class="p">)</span>
-
-</div>
-<span class="k">class</span> <span class="nc">List2DSeries</span><span class="p">(</span><span class="n">Line2DBaseSeries</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Representation for a line consisting of list of points.&quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">list_x</span><span class="p">,</span> <span class="n">list_y</span><span class="p">):</span>
-        <span class="nb">super</span><span class="p">(</span><span class="n">List2DSeries</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__init__</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">list_x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">list_x</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">list_y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">list_y</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">label</span> <span class="o">=</span> <span class="s">&#39;list&#39;</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;list plot&#39;</span>
-
-    <span class="k">def</span> <span class="nf">get_points</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">list_x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">list_y</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="LineOver1DRangeSeries"><a class="viewcode-back" href="../../../modules/plotting.html#sympy.plotting.plot.LineOver1DRangeSeries">[docs]</a><span class="k">class</span> <span class="nc">LineOver1DRangeSeries</span><span class="p">(</span><span class="n">Line2DBaseSeries</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Representation for a line consisting of a sympy expression over a range.&quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">var_start_end</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="nb">super</span><span class="p">(</span><span class="n">LineOver1DRangeSeries</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__init__</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">label</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">var</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">var_start_end</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">start</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">var_start_end</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">end</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">var_start_end</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;nb_of_points&#39;</span><span class="p">,</span> <span class="mi">300</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">adaptive</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;adaptive&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">depth</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;depth&#39;</span><span class="p">,</span> <span class="mi">12</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">line_color</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;line_color&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;cartesian line: </span><span class="si">%s</span><span class="s"> for </span><span class="si">%s</span><span class="s"> over </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span>
-            <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">),</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">var</span><span class="p">),</span> <span class="nb">str</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="p">)))</span>
-
-<div class="viewcode-block" id="LineOver1DRangeSeries.get_segments"><a class="viewcode-back" href="../../../modules/plotting.html#sympy.plotting.plot.LineOver1DRangeSeries.get_segments">[docs]</a>    <span class="k">def</span> <span class="nf">get_segments</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Adaptively gets segments for plotting.</span>
-
-<span class="sd">        The adaptive sampling is done by recursively checking if three</span>
-<span class="sd">        points are almost collinear. If they are not collinear, then more</span>
-<span class="sd">        points are added between those points.</span>
-
-<span class="sd">        References</span>
-<span class="sd">        ==========</span>
-<span class="sd">        [1] Adaptive polygonal approximation of parametric curves,</span>
-<span class="sd">            Luiz Henrique de Figueiredo.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">only_integers</span> <span class="ow">or</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">adaptive</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">LineOver1DRangeSeries</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">get_segments</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">lambdify</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">var</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">)</span>
-            <span class="n">list_segments</span> <span class="o">=</span> <span class="p">[]</span>
-
-            <span class="k">def</span> <span class="nf">sample</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">depth</span><span class="p">):</span>
-                <span class="sd">&quot;&quot;&quot; Samples recursively if three points are almost collinear.</span>
-<span class="sd">                For depth &lt; 6, points are added irrespective of whether they</span>
-<span class="sd">                satisfy the collinearity condition or not. The maximum depth</span>
-<span class="sd">                allowed is 12.</span>
-<span class="sd">                &quot;&quot;&quot;</span>
-                <span class="c">#Randomly sample to avoid aliasing.</span>
-                <span class="n">random</span> <span class="o">=</span> <span class="mf">0.45</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">()</span> <span class="o">*</span> <span class="mf">0.1</span>
-                <span class="n">xnew</span> <span class="o">=</span> <span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">random</span> <span class="o">*</span> <span class="p">(</span><span class="n">q</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-                <span class="n">ynew</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">xnew</span><span class="p">)</span>
-                <span class="n">new_point</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">xnew</span><span class="p">,</span> <span class="n">ynew</span><span class="p">])</span>
-
-                <span class="c">#Maximum depth</span>
-                <span class="k">if</span> <span class="n">depth</span> <span class="o">&gt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">depth</span><span class="p">:</span>
-                    <span class="n">list_segments</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">])</span>
-
-                <span class="c">#Sample irrespective of whether the line is flat till the</span>
-                <span class="c">#depth of 6. We are not using linspace to avoid aliasing.</span>
-                <span class="k">elif</span> <span class="n">depth</span> <span class="o">&lt;</span> <span class="mi">6</span><span class="p">:</span>
-                    <span class="n">sample</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">new_point</span><span class="p">,</span> <span class="n">depth</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-                    <span class="n">sample</span><span class="p">(</span><span class="n">new_point</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">depth</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-                <span class="c">#Sample ten points if complex values are encountered</span>
-                <span class="c">#at both ends. If there is a real value in between, then</span>
-                <span class="c">#sample those points further.</span>
-                <span class="k">elif</span> <span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">q</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">xarray</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">q</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">10</span><span class="p">)</span>
-                    <span class="n">yarray</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">xarray</span><span class="p">))</span>
-                    <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">y</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">yarray</span><span class="p">):</span>
-                        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">len</span><span class="p">(</span><span class="n">yarray</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                            <span class="k">if</span> <span class="n">yarray</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">yarray</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                                <span class="n">sample</span><span class="p">([</span><span class="n">xarray</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">yarray</span><span class="p">[</span><span class="n">i</span><span class="p">]],</span>
-                                    <span class="p">[</span><span class="n">xarray</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">yarray</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]],</span> <span class="n">depth</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-                <span class="c">#Sample further if one of the end points in None( i.e. a complex</span>
-                <span class="c">#value) or the three points are not almost collinear.</span>
-                <span class="k">elif</span> <span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">q</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">new_point</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span>
-                        <span class="ow">or</span> <span class="ow">not</span> <span class="n">flat</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">new_point</span><span class="p">,</span> <span class="n">q</span><span class="p">)):</span>
-                    <span class="n">sample</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">new_point</span><span class="p">,</span> <span class="n">depth</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-                    <span class="n">sample</span><span class="p">(</span><span class="n">new_point</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">depth</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">list_segments</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">])</span>
-
-            <span class="n">f_start</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="p">)</span>
-            <span class="n">f_end</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="p">)</span>
-            <span class="n">sample</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="p">,</span> <span class="n">f_start</span><span class="p">],</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="p">,</span> <span class="n">f_end</span><span class="p">],</span> <span class="mi">0</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">list_segments</span>
-</div>
-    <span class="k">def</span> <span class="nf">get_points</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">only_integers</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">list_x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="p">),</span> <span class="nb">int</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="p">),</span>
-                    <span class="n">num</span><span class="o">=</span><span class="nb">int</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="p">)</span> <span class="o">-</span> <span class="nb">int</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">list_x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points</span><span class="p">)</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">vectorized_lambdify</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">var</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">)</span>
-        <span class="n">list_y</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">list_x</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">list_x</span><span class="p">,</span> <span class="n">list_y</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="Parametric2DLineSeries"><a class="viewcode-back" href="../../../modules/plotting.html#sympy.plotting.plot.Parametric2DLineSeries">[docs]</a><span class="k">class</span> <span class="nc">Parametric2DLineSeries</span><span class="p">(</span><span class="n">Line2DBaseSeries</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Representation for a line consisting of two parametric sympy expressions</span>
-<span class="sd">    over a range.&quot;&quot;&quot;</span>
-
-    <span class="n">is_parametric</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr_x</span><span class="p">,</span> <span class="n">expr_y</span><span class="p">,</span> <span class="n">var_start_end</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="nb">super</span><span class="p">(</span><span class="n">Parametric2DLineSeries</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__init__</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">expr_x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr_x</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">expr_y</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr_y</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">label</span> <span class="o">=</span> <span class="s">&quot;(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr_x</span><span class="p">),</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr_y</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">var</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">var_start_end</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">start</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">var_start_end</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">end</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">var_start_end</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;nb_of_points&#39;</span><span class="p">,</span> <span class="mi">300</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">adaptive</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;adaptive&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">depth</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;depth&#39;</span><span class="p">,</span> <span class="mi">12</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">line_color</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;line_color&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;parametric cartesian line: (</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">) for </span><span class="si">%s</span><span class="s"> over </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span>
-            <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr_x</span><span class="p">),</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr_y</span><span class="p">),</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">var</span><span class="p">),</span>
-            <span class="nb">str</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="p">)))</span>
-
-    <span class="k">def</span> <span class="nf">get_parameter_points</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">get_points</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">param</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_parameter_points</span><span class="p">()</span>
-        <span class="n">fx</span> <span class="o">=</span> <span class="n">vectorized_lambdify</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">var</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr_x</span><span class="p">)</span>
-        <span class="n">fy</span> <span class="o">=</span> <span class="n">vectorized_lambdify</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">var</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr_y</span><span class="p">)</span>
-        <span class="n">list_x</span> <span class="o">=</span> <span class="n">fx</span><span class="p">(</span><span class="n">param</span><span class="p">)</span>
-        <span class="n">list_y</span> <span class="o">=</span> <span class="n">fy</span><span class="p">(</span><span class="n">param</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">list_x</span><span class="p">,</span> <span class="n">list_y</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Parametric2DLineSeries.get_segments"><a class="viewcode-back" href="../../../modules/plotting.html#sympy.plotting.plot.Parametric2DLineSeries.get_segments">[docs]</a>    <span class="k">def</span> <span class="nf">get_segments</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Adaptively gets segments for plotting.</span>
-
-<span class="sd">        The adaptive sampling is done by recursively checking if three</span>
-<span class="sd">        points are almost collinear. If they are not collinear, then more</span>
-<span class="sd">        points are added between those points.</span>
-
-<span class="sd">        References</span>
-<span class="sd">        ==========</span>
-<span class="sd">        [1] Adaptive polygonal approximation of parametric curves,</span>
-<span class="sd">            Luiz Henrique de Figueiredo.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">adaptive</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Parametric2DLineSeries</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">get_segments</span><span class="p">()</span>
-
-        <span class="n">f_x</span> <span class="o">=</span> <span class="n">lambdify</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">var</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr_x</span><span class="p">)</span>
-        <span class="n">f_y</span> <span class="o">=</span> <span class="n">lambdify</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">var</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr_y</span><span class="p">)</span>
-        <span class="n">list_segments</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">def</span> <span class="nf">sample</span><span class="p">(</span><span class="n">param_p</span><span class="p">,</span> <span class="n">param_q</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">depth</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot; Samples recursively if three points are almost collinear.</span>
-<span class="sd">            For depth &lt; 6, points are added irrespective of whether they</span>
-<span class="sd">            satisfy the collinearity condition or not. The maximum depth</span>
-<span class="sd">            allowed is 12.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="c">#Randomly sample to avoid aliasing.</span>
-            <span class="n">random</span> <span class="o">=</span> <span class="mf">0.45</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">()</span> <span class="o">*</span> <span class="mf">0.1</span>
-            <span class="n">param_new</span> <span class="o">=</span> <span class="n">param_p</span> <span class="o">+</span> <span class="n">random</span> <span class="o">*</span> <span class="p">(</span><span class="n">param_q</span> <span class="o">-</span> <span class="n">param_p</span><span class="p">)</span>
-            <span class="n">xnew</span> <span class="o">=</span> <span class="n">f_x</span><span class="p">(</span><span class="n">param_new</span><span class="p">)</span>
-            <span class="n">ynew</span> <span class="o">=</span> <span class="n">f_y</span><span class="p">(</span><span class="n">param_new</span><span class="p">)</span>
-            <span class="n">new_point</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">xnew</span><span class="p">,</span> <span class="n">ynew</span><span class="p">])</span>
-
-            <span class="c">#Maximum depth</span>
-            <span class="k">if</span> <span class="n">depth</span> <span class="o">&gt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">depth</span><span class="p">:</span>
-                <span class="n">list_segments</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">])</span>
-
-            <span class="c">#Sample irrespective of whether the line is flat till the</span>
-            <span class="c">#depth of 6. We are not using linspace to avoid aliasing.</span>
-            <span class="k">elif</span> <span class="n">depth</span> <span class="o">&lt;</span> <span class="mi">6</span><span class="p">:</span>
-                <span class="n">sample</span><span class="p">(</span><span class="n">param_p</span><span class="p">,</span> <span class="n">param_new</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">new_point</span><span class="p">,</span> <span class="n">depth</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-                <span class="n">sample</span><span class="p">(</span><span class="n">param_new</span><span class="p">,</span> <span class="n">param_q</span><span class="p">,</span> <span class="n">new_point</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">depth</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-            <span class="c">#Sample ten points if complex values are encountered</span>
-            <span class="c">#at both ends. If there is a real value in between, then</span>
-            <span class="c">#sample those points further.</span>
-            <span class="k">elif</span> <span class="p">((</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">q</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">)</span> <span class="ow">or</span>
-                    <span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">q</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">)):</span>
-                <span class="n">param_array</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">param_p</span><span class="p">,</span> <span class="n">param_q</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-                <span class="n">x_array</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">f_x</span><span class="p">,</span> <span class="n">param_array</span><span class="p">))</span>
-                <span class="n">y_array</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">f_y</span><span class="p">,</span> <span class="n">param_array</span><span class="p">))</span>
-                <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">x</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">y</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span>
-                        <span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">x_array</span><span class="p">,</span> <span class="n">y_array</span><span class="p">)):</span>
-                    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">len</span><span class="p">(</span><span class="n">y_array</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="p">((</span><span class="n">x_array</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">y_array</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">)</span> <span class="ow">or</span>
-                                <span class="p">(</span><span class="n">x_array</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">y_array</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">)):</span>
-                            <span class="n">point_a</span> <span class="o">=</span> <span class="p">[</span><span class="n">x_array</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">y_array</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span>
-                            <span class="n">point_b</span> <span class="o">=</span> <span class="p">[</span><span class="n">x_array</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">y_array</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]]</span>
-                            <span class="n">sample</span><span class="p">(</span><span class="n">param_array</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">param_array</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">point_a</span><span class="p">,</span>
-                                   <span class="n">point_b</span><span class="p">,</span> <span class="n">depth</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-            <span class="c">#Sample further if one of the end points in None( ie a complex</span>
-            <span class="c">#value) or the three points are not almost collinear.</span>
-            <span class="k">elif</span> <span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span>
-                    <span class="ow">or</span> <span class="n">q</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">q</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span>
-                    <span class="ow">or</span> <span class="ow">not</span> <span class="n">flat</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">new_point</span><span class="p">,</span> <span class="n">q</span><span class="p">)):</span>
-                <span class="n">sample</span><span class="p">(</span><span class="n">param_p</span><span class="p">,</span> <span class="n">param_new</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">new_point</span><span class="p">,</span> <span class="n">depth</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-                <span class="n">sample</span><span class="p">(</span><span class="n">param_new</span><span class="p">,</span> <span class="n">param_q</span><span class="p">,</span> <span class="n">new_point</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">depth</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">list_segments</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">])</span>
-
-        <span class="n">f_start_x</span> <span class="o">=</span> <span class="n">f_x</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="p">)</span>
-        <span class="n">f_start_y</span> <span class="o">=</span> <span class="n">f_y</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="p">)</span>
-        <span class="n">start</span> <span class="o">=</span> <span class="p">[</span><span class="n">f_start_x</span><span class="p">,</span> <span class="n">f_start_y</span><span class="p">]</span>
-        <span class="n">f_end_x</span> <span class="o">=</span> <span class="n">f_x</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="p">)</span>
-        <span class="n">f_end_y</span> <span class="o">=</span> <span class="n">f_y</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="p">)</span>
-        <span class="n">end</span> <span class="o">=</span> <span class="p">[</span><span class="n">f_end_x</span><span class="p">,</span> <span class="n">f_end_y</span><span class="p">]</span>
-        <span class="n">sample</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">end</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">list_segments</span>
-
-
-<span class="c">### 3D lines</span></div></div>
-<div class="viewcode-block" id="Line3DBaseSeries"><a class="viewcode-back" href="../../../modules/plotting.html#sympy.plotting.plot.Line3DBaseSeries">[docs]</a><span class="k">class</span> <span class="nc">Line3DBaseSeries</span><span class="p">(</span><span class="n">Line2DBaseSeries</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A base class for 3D lines.</span>
-
-<span class="sd">    Most of the stuff is derived from Line2DBaseSeries.&quot;&quot;&quot;</span>
-
-    <span class="n">is_2Dline</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_3Dline</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">_dim</span> <span class="o">=</span> <span class="mi">3</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="nb">super</span><span class="p">(</span><span class="n">Line3DBaseSeries</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__init__</span><span class="p">()</span>
-
-</div>
-<div class="viewcode-block" id="Parametric3DLineSeries"><a class="viewcode-back" href="../../../modules/plotting.html#sympy.plotting.plot.Parametric3DLineSeries">[docs]</a><span class="k">class</span> <span class="nc">Parametric3DLineSeries</span><span class="p">(</span><span class="n">Line3DBaseSeries</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Representation for a 3D line consisting of two parametric sympy</span>
-<span class="sd">    expressions and a range.&quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr_x</span><span class="p">,</span> <span class="n">expr_y</span><span class="p">,</span> <span class="n">expr_z</span><span class="p">,</span> <span class="n">var_start_end</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="nb">super</span><span class="p">(</span><span class="n">Parametric3DLineSeries</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__init__</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">expr_x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr_x</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">expr_y</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr_y</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">expr_z</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr_z</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">label</span> <span class="o">=</span> <span class="s">&quot;(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr_x</span><span class="p">),</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr_y</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">var</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">var_start_end</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">start</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">var_start_end</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">end</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">var_start_end</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;nb_of_points&#39;</span><span class="p">,</span> <span class="mi">300</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">line_color</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;line_color&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;3D parametric cartesian line: (</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">) for </span><span class="si">%s</span><span class="s"> over </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span>
-            <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr_x</span><span class="p">),</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr_y</span><span class="p">),</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr_z</span><span class="p">),</span>
-            <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">var</span><span class="p">),</span> <span class="nb">str</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="p">)))</span>
-
-    <span class="k">def</span> <span class="nf">get_parameter_points</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">get_points</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">param</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_parameter_points</span><span class="p">()</span>
-        <span class="n">fx</span> <span class="o">=</span> <span class="n">vectorized_lambdify</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">var</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr_x</span><span class="p">)</span>
-        <span class="n">fy</span> <span class="o">=</span> <span class="n">vectorized_lambdify</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">var</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr_y</span><span class="p">)</span>
-        <span class="n">fz</span> <span class="o">=</span> <span class="n">vectorized_lambdify</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">var</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr_z</span><span class="p">)</span>
-        <span class="n">list_x</span> <span class="o">=</span> <span class="n">fx</span><span class="p">(</span><span class="n">param</span><span class="p">)</span>
-        <span class="n">list_y</span> <span class="o">=</span> <span class="n">fy</span><span class="p">(</span><span class="n">param</span><span class="p">)</span>
-        <span class="n">list_z</span> <span class="o">=</span> <span class="n">fz</span><span class="p">(</span><span class="n">param</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">list_x</span><span class="p">,</span> <span class="n">list_y</span><span class="p">,</span> <span class="n">list_z</span><span class="p">)</span>
-
-
-<span class="c">### Surfaces</span></div>
-<div class="viewcode-block" id="SurfaceBaseSeries"><a class="viewcode-back" href="../../../modules/plotting.html#sympy.plotting.plot.SurfaceBaseSeries">[docs]</a><span class="k">class</span> <span class="nc">SurfaceBaseSeries</span><span class="p">(</span><span class="n">BaseSeries</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A base class for 3D surfaces.&quot;&quot;&quot;</span>
-
-    <span class="n">is_3Dsurface</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="nb">super</span><span class="p">(</span><span class="n">SurfaceBaseSeries</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__init__</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">surface_color</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">get_color_array</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">surface_color</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">collections</span><span class="o">.</span><span class="n">Callable</span><span class="p">):</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vectorize</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-            <span class="n">arity</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">getargspec</span><span class="p">(</span><span class="n">c</span><span class="p">)[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_parametric</span><span class="p">:</span>
-                <span class="n">variables</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">centers_of_faces</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_parameter_meshes</span><span class="p">()))</span>
-                <span class="k">if</span> <span class="n">arity</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">f</span><span class="p">(</span><span class="n">variables</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-                <span class="k">elif</span> <span class="n">arity</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">f</span><span class="p">(</span><span class="o">*</span><span class="n">variables</span><span class="p">)</span>
-            <span class="n">variables</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">centers_of_faces</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_meshes</span><span class="p">()))</span>
-            <span class="k">if</span> <span class="n">arity</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="p">(</span><span class="n">variables</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="k">elif</span> <span class="n">arity</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="p">(</span><span class="o">*</span><span class="n">variables</span><span class="p">[:</span><span class="mi">2</span><span class="p">])</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="p">(</span><span class="o">*</span><span class="n">variables</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">c</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="SurfaceOver2DRangeSeries"><a class="viewcode-back" href="../../../modules/plotting.html#sympy.plotting.plot.SurfaceOver2DRangeSeries">[docs]</a><span class="k">class</span> <span class="nc">SurfaceOver2DRangeSeries</span><span class="p">(</span><span class="n">SurfaceBaseSeries</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Representation for a 3D surface consisting of a sympy expression and 2D</span>
-<span class="sd">    range.&quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">var_start_end_x</span><span class="p">,</span> <span class="n">var_start_end_y</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="nb">super</span><span class="p">(</span><span class="n">SurfaceOver2DRangeSeries</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__init__</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">var_x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">var_start_end_x</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">start_x</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">var_start_end_x</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">end_x</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">var_start_end_x</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">var_y</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">var_start_end_y</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">start_y</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">var_start_end_y</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">end_y</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">var_start_end_y</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points_x</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;nb_of_points_x&#39;</span><span class="p">,</span> <span class="mi">50</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points_y</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;nb_of_points_y&#39;</span><span class="p">,</span> <span class="mi">50</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">surface_color</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;surface_color&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="s">&#39;cartesian surface: </span><span class="si">%s</span><span class="s"> for&#39;</span>
-                <span class="s">&#39; </span><span class="si">%s</span><span class="s"> over </span><span class="si">%s</span><span class="s"> and </span><span class="si">%s</span><span class="s"> over </span><span class="si">%s</span><span class="s">&#39;</span><span class="p">)</span> <span class="o">%</span> <span class="p">(</span>
-                    <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">),</span>
-                    <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">var_x</span><span class="p">),</span>
-                    <span class="nb">str</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">start_x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end_x</span><span class="p">)),</span>
-                    <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">var_y</span><span class="p">),</span>
-                    <span class="nb">str</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">start_y</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end_y</span><span class="p">)))</span>
-
-    <span class="k">def</span> <span class="nf">get_meshes</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">mesh_x</span><span class="p">,</span> <span class="n">mesh_y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start_x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end_x</span><span class="p">,</span>
-                                                 <span class="n">num</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points_x</span><span class="p">),</span>
-                                     <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start_y</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end_y</span><span class="p">,</span>
-                                                 <span class="n">num</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points_y</span><span class="p">))</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">vectorized_lambdify</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">var_x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">var_y</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">mesh_x</span><span class="p">,</span> <span class="n">mesh_y</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">mesh_x</span><span class="p">,</span> <span class="n">mesh_y</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="ParametricSurfaceSeries"><a class="viewcode-back" href="../../../modules/plotting.html#sympy.plotting.plot.ParametricSurfaceSeries">[docs]</a><span class="k">class</span> <span class="nc">ParametricSurfaceSeries</span><span class="p">(</span><span class="n">SurfaceBaseSeries</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Representation for a 3D surface consisting of three parametric sympy</span>
-<span class="sd">    expressions and a range.&quot;&quot;&quot;</span>
-
-    <span class="n">is_parametric</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span>
-        <span class="bp">self</span><span class="p">,</span> <span class="n">expr_x</span><span class="p">,</span> <span class="n">expr_y</span><span class="p">,</span> <span class="n">expr_z</span><span class="p">,</span> <span class="n">var_start_end_u</span><span class="p">,</span> <span class="n">var_start_end_v</span><span class="p">,</span>
-            <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="nb">super</span><span class="p">(</span><span class="n">ParametricSurfaceSeries</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__init__</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">expr_x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr_x</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">expr_y</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr_y</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">expr_z</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr_z</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">var_u</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">var_start_end_u</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">start_u</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">var_start_end_u</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">end_u</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">var_start_end_u</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">var_v</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">var_start_end_v</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">start_v</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">var_start_end_v</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">end_v</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">var_start_end_v</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points_u</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;nb_of_points_u&#39;</span><span class="p">,</span> <span class="mi">50</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points_v</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;nb_of_points_v&#39;</span><span class="p">,</span> <span class="mi">50</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">surface_color</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;surface_color&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="s">&#39;parametric cartesian surface: (</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">) for&#39;</span>
-                <span class="s">&#39; </span><span class="si">%s</span><span class="s"> over </span><span class="si">%s</span><span class="s"> and </span><span class="si">%s</span><span class="s"> over </span><span class="si">%s</span><span class="s">&#39;</span><span class="p">)</span> <span class="o">%</span> <span class="p">(</span>
-                    <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr_x</span><span class="p">),</span>
-                    <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr_y</span><span class="p">),</span>
-                    <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr_z</span><span class="p">),</span>
-                    <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">var_u</span><span class="p">),</span>
-                    <span class="nb">str</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">start_u</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end_u</span><span class="p">)),</span>
-                    <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">var_v</span><span class="p">),</span>
-                    <span class="nb">str</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">start_v</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end_v</span><span class="p">)))</span>
-
-    <span class="k">def</span> <span class="nf">get_parameter_meshes</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start_u</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end_u</span><span class="p">,</span>
-                                       <span class="n">num</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points_u</span><span class="p">),</span>
-                           <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start_v</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end_v</span><span class="p">,</span>
-                                       <span class="n">num</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points_v</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">get_meshes</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">mesh_u</span><span class="p">,</span> <span class="n">mesh_v</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_parameter_meshes</span><span class="p">()</span>
-        <span class="n">fx</span> <span class="o">=</span> <span class="n">vectorized_lambdify</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">var_u</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">var_v</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr_x</span><span class="p">)</span>
-        <span class="n">fy</span> <span class="o">=</span> <span class="n">vectorized_lambdify</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">var_u</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">var_v</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr_y</span><span class="p">)</span>
-        <span class="n">fz</span> <span class="o">=</span> <span class="n">vectorized_lambdify</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">var_u</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">var_v</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr_z</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">fx</span><span class="p">(</span><span class="n">mesh_u</span><span class="p">,</span> <span class="n">mesh_v</span><span class="p">),</span> <span class="n">fy</span><span class="p">(</span><span class="n">mesh_u</span><span class="p">,</span> <span class="n">mesh_v</span><span class="p">),</span> <span class="n">fz</span><span class="p">(</span><span class="n">mesh_u</span><span class="p">,</span> <span class="n">mesh_v</span><span class="p">))</span>
-
-
-<span class="c">### Contours</span></div>
-<span class="k">class</span> <span class="nc">ContourSeries</span><span class="p">(</span><span class="n">BaseSeries</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Representation for a contour plot.&quot;&quot;&quot;</span>
-    <span class="c">#The code is mostly repetition of SurfaceOver2DRange.</span>
-    <span class="c">#XXX: Presently not used in any of those functions.</span>
-    <span class="c">#XXX: Add contour plot and use this seties.</span>
-
-    <span class="n">is_contour</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">var_start_end_x</span><span class="p">,</span> <span class="n">var_start_end_y</span><span class="p">):</span>
-        <span class="nb">super</span><span class="p">(</span><span class="n">ContourSeries</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__init__</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points_x</span> <span class="o">=</span> <span class="mi">50</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points_y</span> <span class="o">=</span> <span class="mi">50</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">var_x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">var_start_end_x</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">start_x</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">var_start_end_x</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">end_x</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">var_start_end_x</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">var_y</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">var_start_end_y</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">start_y</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">var_start_end_y</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">end_y</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">var_start_end_y</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">get_points</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_meshes</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="s">&#39;contour: </span><span class="si">%s</span><span class="s"> for &#39;</span>
-                <span class="s">&#39;</span><span class="si">%s</span><span class="s"> over </span><span class="si">%s</span><span class="s"> and </span><span class="si">%s</span><span class="s"> over </span><span class="si">%s</span><span class="s">&#39;</span><span class="p">)</span> <span class="o">%</span> <span class="p">(</span>
-                    <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">),</span>
-                    <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">var_x</span><span class="p">),</span>
-                    <span class="nb">str</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">start_x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end_x</span><span class="p">)),</span>
-                    <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">var_y</span><span class="p">),</span>
-                    <span class="nb">str</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">start_y</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end_y</span><span class="p">)))</span>
-
-    <span class="k">def</span> <span class="nf">get_meshes</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">mesh_x</span><span class="p">,</span> <span class="n">mesh_y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start_x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end_x</span><span class="p">,</span>
-                                                 <span class="n">num</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points_x</span><span class="p">),</span>
-                                     <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start_y</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end_y</span><span class="p">,</span>
-                                                 <span class="n">num</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points_y</span><span class="p">))</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">vectorized_lambdify</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">var_x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">var_y</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">mesh_x</span><span class="p">,</span> <span class="n">mesh_y</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">mesh_x</span><span class="p">,</span> <span class="n">mesh_y</span><span class="p">))</span>
-
-
-<span class="c">##############################################################################</span>
-<span class="c"># Backends</span>
-<span class="c">##############################################################################</span>
-
-<span class="k">class</span> <span class="nc">BaseBackend</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">parent</span><span class="p">):</span>
-        <span class="nb">super</span><span class="p">(</span><span class="n">BaseBackend</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__init__</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">parent</span> <span class="o">=</span> <span class="n">parent</span>
-
-
-<span class="k">class</span> <span class="nc">MatplotlibBackend</span><span class="p">(</span><span class="n">BaseBackend</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">parent</span><span class="p">):</span>
-        <span class="nb">super</span><span class="p">(</span><span class="n">MatplotlibBackend</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="n">parent</span><span class="p">)</span>
-        <span class="n">are_3D</span> <span class="o">=</span> <span class="p">[</span><span class="n">s</span><span class="o">.</span><span class="n">is_3D</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">parent</span><span class="o">.</span><span class="n">_series</span><span class="p">]</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">are_3D</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">(</span><span class="n">are_3D</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;The matplotlib backend can not mix 2D and 3D.&#39;</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="nb">any</span><span class="p">(</span><span class="n">are_3D</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s">&#39;left&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">set_position</span><span class="p">(</span><span class="s">&#39;zero&#39;</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s">&#39;right&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">set_color</span><span class="p">(</span><span class="s">&#39;none&#39;</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s">&#39;bottom&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">set_position</span><span class="p">(</span><span class="s">&#39;zero&#39;</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s">&#39;top&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">set_color</span><span class="p">(</span><span class="s">&#39;none&#39;</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s">&#39;left&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">set_smart_bounds</span><span class="p">(</span><span class="bp">True</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s">&#39;bottom&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">set_smart_bounds</span><span class="p">(</span><span class="bp">True</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">xaxis</span><span class="o">.</span><span class="n">set_ticks_position</span><span class="p">(</span><span class="s">&#39;bottom&#39;</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">yaxis</span><span class="o">.</span><span class="n">set_ticks_position</span><span class="p">(</span><span class="s">&#39;left&#39;</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">all</span><span class="p">(</span><span class="n">are_3D</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">,</span> <span class="n">projection</span><span class="o">=</span><span class="s">&#39;3d&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">process_series</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">parent</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">parent</span>
-
-        <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">parent</span><span class="o">.</span><span class="n">_series</span><span class="p">:</span>
-            <span class="c"># Create the collections</span>
-            <span class="k">if</span> <span class="n">s</span><span class="o">.</span><span class="n">is_2Dline</span><span class="p">:</span>
-                <span class="n">collection</span> <span class="o">=</span> <span class="n">LineCollection</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">get_segments</span><span class="p">())</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">add_collection</span><span class="p">(</span><span class="n">collection</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">s</span><span class="o">.</span><span class="n">is_contour</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">get_meshes</span><span class="p">())</span>
-            <span class="k">elif</span> <span class="n">s</span><span class="o">.</span><span class="n">is_3Dline</span><span class="p">:</span>
-                <span class="c"># TODO too complicated, I blame matplotlib</span>
-                <span class="n">collection</span> <span class="o">=</span> <span class="n">art3d</span><span class="o">.</span><span class="n">Line3DCollection</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">get_segments</span><span class="p">())</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">add_collection</span><span class="p">(</span><span class="n">collection</span><span class="p">)</span>
-                <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">get_points</span><span class="p">()</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">((</span><span class="nb">min</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="nb">max</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">((</span><span class="nb">min</span><span class="p">(</span><span class="n">y</span><span class="p">),</span> <span class="nb">max</span><span class="p">(</span><span class="n">y</span><span class="p">)))</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_zlim</span><span class="p">((</span><span class="nb">min</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="nb">max</span><span class="p">(</span><span class="n">z</span><span class="p">)))</span>
-            <span class="k">elif</span> <span class="n">s</span><span class="o">.</span><span class="n">is_3Dsurface</span><span class="p">:</span>
-                <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">get_meshes</span><span class="p">()</span>
-                <span class="n">collection</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">plot_surface</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">cm</span><span class="o">.</span><span class="n">jet</span><span class="p">,</span>
-                                                  <span class="n">rstride</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">cstride</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
-                                                  <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">s</span><span class="o">.</span><span class="n">is_implicit</span><span class="p">:</span>
-                <span class="c">#Smart bounds have to be set to False for implicit plots.</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s">&#39;left&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">set_smart_bounds</span><span class="p">(</span><span class="bp">False</span><span class="p">)</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s">&#39;bottom&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">set_smart_bounds</span><span class="p">(</span><span class="bp">False</span><span class="p">)</span>
-                <span class="n">points</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">get_raster</span><span class="p">()</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">points</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                    <span class="c">#interval math plotting</span>
-                    <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">_matplotlib_list</span><span class="p">(</span><span class="n">points</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">fill</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">facecolor</span><span class="o">=</span><span class="s">&#39;b&#39;</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s">&#39;None&#39;</span> <span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="c"># use contourf or contour depending on whether it is</span>
-                    <span class="c"># an inequality or equality.</span>
-                    <span class="c">#XXX: ``contour`` plots multiple lines. Should be fixed.</span>
-                    <span class="n">colormap</span> <span class="o">=</span> <span class="n">ListedColormap</span><span class="p">([</span><span class="s">&quot;white&quot;</span><span class="p">,</span> <span class="s">&quot;blue&quot;</span><span class="p">])</span>
-                    <span class="n">xarray</span><span class="p">,</span> <span class="n">yarray</span><span class="p">,</span> <span class="n">zarray</span><span class="p">,</span> <span class="n">plot_type</span> <span class="o">=</span> <span class="n">points</span>
-                    <span class="k">if</span> <span class="n">plot_type</span> <span class="o">==</span> <span class="s">&#39;contour&#39;</span><span class="p">:</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span><span class="n">xarray</span><span class="p">,</span> <span class="n">yarray</span><span class="p">,</span> <span class="n">zarray</span><span class="p">,</span>
-                                <span class="n">contours</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">fill</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">colormap</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">xarray</span><span class="p">,</span> <span class="n">yarray</span><span class="p">,</span> <span class="n">zarray</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">colormap</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;The matplotlib backend supports only &#39;</span>
-                                 <span class="s">&#39;is_2Dline, is_3Dline, is_3Dsurface and &#39;</span>
-                                 <span class="s">&#39;is_contour objects.&#39;</span><span class="p">)</span>
-
-            <span class="c"># Customise the collections with the corresponding per-series</span>
-            <span class="c"># options.</span>
-            <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="s">&#39;label&#39;</span><span class="p">):</span>
-                <span class="n">collection</span><span class="o">.</span><span class="n">set_label</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">label</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">s</span><span class="o">.</span><span class="n">is_line</span> <span class="ow">and</span> <span class="n">s</span><span class="o">.</span><span class="n">line_color</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">line_color</span><span class="p">,</span> <span class="p">(</span><span class="nb">float</span><span class="p">,</span> <span class="nb">int</span><span class="p">))</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">line_color</span><span class="p">,</span> <span class="n">collections</span><span class="o">.</span><span class="n">Callable</span><span class="p">):</span>
-                    <span class="n">color_array</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">get_color_array</span><span class="p">()</span>
-                    <span class="n">collection</span><span class="o">.</span><span class="n">set_array</span><span class="p">(</span><span class="n">color_array</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">collection</span><span class="o">.</span><span class="n">set_color</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">line_color</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">s</span><span class="o">.</span><span class="n">is_3Dsurface</span> <span class="ow">and</span> <span class="n">s</span><span class="o">.</span><span class="n">surface_color</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">matplotlib</span><span class="o">.</span><span class="n">__version__</span> <span class="o">&lt;</span> <span class="s">&quot;1.2.0&quot;</span><span class="p">:</span>  <span class="c"># TODO in the distant future remove this check</span>
-                    <span class="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="s">&#39;The version of matplotlib is too old to use surface coloring.&#39;</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">surface_color</span><span class="p">,</span> <span class="p">(</span><span class="nb">float</span><span class="p">,</span> <span class="nb">int</span><span class="p">))</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">surface_color</span><span class="p">,</span> <span class="n">collections</span><span class="o">.</span><span class="n">Callable</span><span class="p">):</span>
-                    <span class="n">color_array</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">get_color_array</span><span class="p">()</span>
-                    <span class="n">color_array</span> <span class="o">=</span> <span class="n">color_array</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">color_array</span><span class="o">.</span><span class="n">size</span><span class="p">)</span>
-                    <span class="n">collection</span><span class="o">.</span><span class="n">set_array</span><span class="p">(</span><span class="n">color_array</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">collection</span><span class="o">.</span><span class="n">set_color</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">surface_color</span><span class="p">)</span>
-
-        <span class="c"># Set global options.</span>
-        <span class="c"># TODO The 3D stuff</span>
-        <span class="c"># XXX The order of those is important.</span>
-        <span class="k">if</span> <span class="n">parent</span><span class="o">.</span><span class="n">xscale</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="p">,</span> <span class="n">Axes3D</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_xscale</span><span class="p">(</span><span class="n">parent</span><span class="o">.</span><span class="n">xscale</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">parent</span><span class="o">.</span><span class="n">yscale</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="p">,</span> <span class="n">Axes3D</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_yscale</span><span class="p">(</span><span class="n">parent</span><span class="o">.</span><span class="n">yscale</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">parent</span><span class="o">.</span><span class="n">xlim</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="n">parent</span><span class="o">.</span><span class="n">xlim</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">parent</span><span class="o">.</span><span class="n">ylim</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="n">parent</span><span class="o">.</span><span class="n">ylim</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="p">,</span> <span class="n">Axes3D</span><span class="p">)</span> <span class="ow">or</span> <span class="n">matplotlib</span><span class="o">.</span><span class="n">__version__</span> <span class="o">&gt;=</span> <span class="s">&#39;1.2.0&#39;</span><span class="p">:</span>  <span class="c"># XXX in the distant future remove this check</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_autoscale_on</span><span class="p">(</span><span class="n">parent</span><span class="o">.</span><span class="n">autoscale</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">parent</span><span class="o">.</span><span class="n">axis_center</span><span class="p">:</span>
-            <span class="n">val</span> <span class="o">=</span> <span class="n">parent</span><span class="o">.</span><span class="n">axis_center</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="p">,</span> <span class="n">Axes3D</span><span class="p">):</span>
-                <span class="k">pass</span>
-            <span class="k">elif</span> <span class="n">val</span> <span class="o">==</span> <span class="s">&#39;center&#39;</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s">&#39;left&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">set_position</span><span class="p">(</span><span class="s">&#39;center&#39;</span><span class="p">)</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s">&#39;bottom&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">set_position</span><span class="p">(</span><span class="s">&#39;center&#39;</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">val</span> <span class="o">==</span> <span class="s">&#39;auto&#39;</span><span class="p">:</span>
-                <span class="n">xl</span><span class="p">,</span> <span class="n">xh</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">get_xlim</span><span class="p">()</span>
-                <span class="n">yl</span><span class="p">,</span> <span class="n">yh</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">get_ylim</span><span class="p">()</span>
-                <span class="n">pos_left</span> <span class="o">=</span> <span class="p">(</span><span class="s">&#39;data&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="k">if</span> <span class="n">xl</span><span class="o">*</span><span class="n">xh</span> <span class="o">&lt;=</span> <span class="mi">0</span> <span class="k">else</span> <span class="s">&#39;center&#39;</span>
-                <span class="n">pos_bottom</span> <span class="o">=</span> <span class="p">(</span><span class="s">&#39;data&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="k">if</span> <span class="n">yl</span><span class="o">*</span><span class="n">yh</span> <span class="o">&lt;=</span> <span class="mi">0</span> <span class="k">else</span> <span class="s">&#39;center&#39;</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s">&#39;left&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">set_position</span><span class="p">(</span><span class="n">pos_left</span><span class="p">)</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s">&#39;bottom&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">set_position</span><span class="p">(</span><span class="n">pos_bottom</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s">&#39;left&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">set_position</span><span class="p">((</span><span class="s">&#39;data&#39;</span><span class="p">,</span> <span class="n">val</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">spines</span><span class="p">[</span><span class="s">&#39;bottom&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">set_position</span><span class="p">((</span><span class="s">&#39;data&#39;</span><span class="p">,</span> <span class="n">val</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">parent</span><span class="o">.</span><span class="n">axis</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_axis_off</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">parent</span><span class="o">.</span><span class="n">legend</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">legend_</span><span class="o">.</span><span class="n">set_visible</span><span class="p">(</span><span class="n">parent</span><span class="o">.</span><span class="n">legend</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">parent</span><span class="o">.</span><span class="n">margin</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_xmargin</span><span class="p">(</span><span class="n">parent</span><span class="o">.</span><span class="n">margin</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_ymargin</span><span class="p">(</span><span class="n">parent</span><span class="o">.</span><span class="n">margin</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">parent</span><span class="o">.</span><span class="n">title</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="n">parent</span><span class="o">.</span><span class="n">title</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">parent</span><span class="o">.</span><span class="n">xlabel</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="n">parent</span><span class="o">.</span><span class="n">xlabel</span><span class="p">,</span> <span class="n">position</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">parent</span><span class="o">.</span><span class="n">ylabel</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="n">parent</span><span class="o">.</span><span class="n">ylabel</span><span class="p">,</span> <span class="n">position</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">show</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">process_series</span><span class="p">()</span>
-        <span class="c">#TODO after fixing https://github.com/ipython/ipython/issues/1255</span>
-        <span class="c"># you can uncomment the next line and remove the pyplot.show() call</span>
-        <span class="c">#self.fig.show()</span>
-        <span class="k">if</span> <span class="n">_show</span><span class="p">:</span>
-            <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">save</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">path</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">process_series</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">fig</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="n">path</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">close</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">fig</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">TextBackend</span><span class="p">(</span><span class="n">BaseBackend</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">parent</span><span class="p">):</span>
-        <span class="nb">super</span><span class="p">(</span><span class="n">TextBackend</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="n">parent</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">show</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">parent</span><span class="o">.</span><span class="n">_series</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&#39;The TextBackend supports only one graph per Plot.&#39;</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">parent</span><span class="o">.</span><span class="n">_series</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">LineOver1DRangeSeries</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&#39;The TextBackend supports only expressions over a 1D range&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">ser</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">parent</span><span class="o">.</span><span class="n">_series</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">textplot</span><span class="p">(</span><span class="n">ser</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="n">ser</span><span class="o">.</span><span class="n">start</span><span class="p">,</span> <span class="n">ser</span><span class="o">.</span><span class="n">end</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">close</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">pass</span>
-
-
-<span class="k">class</span> <span class="nc">DefaultBackend</span><span class="p">(</span><span class="n">BaseBackend</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">parent</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">matplotlib</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">MatplotlibBackend</span><span class="p">(</span><span class="n">parent</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">TextBackend</span><span class="p">(</span><span class="n">parent</span><span class="p">)</span>
-
-
-<span class="n">plot_backends</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="s">&#39;matplotlib&#39;</span><span class="p">:</span> <span class="n">MatplotlibBackend</span><span class="p">,</span>
-    <span class="s">&#39;text&#39;</span><span class="p">:</span> <span class="n">TextBackend</span><span class="p">,</span>
-    <span class="s">&#39;default&#39;</span><span class="p">:</span> <span class="n">DefaultBackend</span>
-<span class="p">}</span>
-
-
-<span class="c">##############################################################################</span>
-<span class="c"># Finding the centers of line segments or mesh faces</span>
-<span class="c">##############################################################################</span>
-
-<span class="k">def</span> <span class="nf">centers_of_segments</span><span class="p">(</span><span class="n">array</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">average</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">((</span><span class="n">array</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">array</span><span class="p">[</span><span class="mi">1</span><span class="p">:])),</span> <span class="mi">0</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">centers_of_faces</span><span class="p">(</span><span class="n">array</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">average</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dstack</span><span class="p">((</span><span class="n">array</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span>
-                                 <span class="n">array</span><span class="p">[</span><span class="mi">1</span><span class="p">:,</span> <span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span>
-                                 <span class="n">array</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">:</span> <span class="p">],</span>
-                                 <span class="n">array</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span>
-                                 <span class="p">)),</span> <span class="mi">2</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">flat</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Checks whether three points are almost collinear&quot;&quot;&quot;</span>
-    <span class="n">vector_a</span> <span class="o">=</span> <span class="n">x</span> <span class="o">-</span> <span class="n">y</span>
-    <span class="n">vector_b</span> <span class="o">=</span> <span class="n">z</span> <span class="o">-</span> <span class="n">y</span>
-    <span class="n">dot_product</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">vector_a</span><span class="p">,</span> <span class="n">vector_b</span><span class="p">)</span>
-    <span class="n">vector_a_norm</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">vector_a</span><span class="p">)</span>
-    <span class="n">vector_b_norm</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">vector_b</span><span class="p">)</span>
-    <span class="n">cos_theta</span> <span class="o">=</span> <span class="n">dot_product</span> <span class="o">/</span> <span class="p">(</span><span class="n">vector_a_norm</span> <span class="o">*</span> <span class="n">vector_b_norm</span><span class="p">)</span>
-    <span class="k">return</span> <span class="nb">abs</span><span class="p">(</span><span class="n">cos_theta</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">eps</span>
-
-
-<span class="k">def</span> <span class="nf">_matplotlib_list</span><span class="p">(</span><span class="n">interval_list</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns lists for matplotlib ``fill`` command from a list of bounding</span>
-<span class="sd">    rectangular intervals</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">xlist</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">ylist</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">interval_list</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">intervals</span> <span class="ow">in</span> <span class="n">interval_list</span><span class="p">:</span>
-            <span class="n">intervalx</span> <span class="o">=</span> <span class="n">intervals</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">intervaly</span> <span class="o">=</span> <span class="n">intervals</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-            <span class="n">xlist</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">intervalx</span><span class="o">.</span><span class="n">start</span><span class="p">,</span> <span class="n">intervalx</span><span class="o">.</span><span class="n">start</span><span class="p">,</span>
-                          <span class="n">intervalx</span><span class="o">.</span><span class="n">end</span><span class="p">,</span> <span class="n">intervalx</span><span class="o">.</span><span class="n">end</span><span class="p">,</span> <span class="bp">None</span><span class="p">])</span>
-            <span class="n">ylist</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">intervaly</span><span class="o">.</span><span class="n">start</span><span class="p">,</span> <span class="n">intervaly</span><span class="o">.</span><span class="n">end</span><span class="p">,</span>
-                          <span class="n">intervaly</span><span class="o">.</span><span class="n">end</span><span class="p">,</span> <span class="n">intervaly</span><span class="o">.</span><span class="n">start</span><span class="p">,</span> <span class="bp">None</span><span class="p">])</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="c">#XXX Ugly hack. Matplotlib does not accept empty lists for ``fill``</span>
-        <span class="n">xlist</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">])</span>
-        <span class="n">ylist</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">])</span>
-    <span class="k">return</span> <span class="n">xlist</span><span class="p">,</span> <span class="n">ylist</span>
-
-
-<span class="c">####New API for plotting module ####</span>
-
-<span class="c"># TODO: Add color arrays for plots.</span>
-<span class="c"># TODO: Add more plotting options for 3d plots.</span>
-<span class="c"># TODO: Adaptive sampling for 3D plots.</span>
-
-<div class="viewcode-block" id="plot"><a class="viewcode-back" href="../../../modules/plotting.html#sympy.plotting.plot.plot">[docs]</a><span class="k">def</span> <span class="nf">plot</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Plots a function of a single variable.</span>
-
-<span class="sd">    The plotting uses an adaptive algorithm which samples recursively to</span>
-<span class="sd">    accurately plot the plot. The adaptive algorithm uses a random point near</span>
-<span class="sd">    the midpoint of two points that has to be further sampled. Hence the same</span>
-<span class="sd">    plots can appear slightly different.</span>
-
-<span class="sd">    Usage</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    Single Plot</span>
-
-<span class="sd">    ``plot(expr, range, **kwargs)``</span>
-
-<span class="sd">    If the range is not specified, then a default range of (-10, 10) is used.</span>
-
-<span class="sd">    Multiple plots with same range.</span>
-
-<span class="sd">    ``plot(expr1, expr2, ..., range, **kwargs)``</span>
-
-<span class="sd">    If the range is not specified, then a default range of (-10, 10) is used.</span>
-
-<span class="sd">    Multiple plots with different ranges.</span>
-
-<span class="sd">    ``plot((expr1, range), (expr2, range), ..., **kwargs)``</span>
-
-<span class="sd">    Range has to be specified for every expression.</span>
-
-<span class="sd">    Default range may change in the future if a more advanced default range</span>
-<span class="sd">    detection algorithm is implemented.</span>
-
-<span class="sd">    Arguments</span>
-<span class="sd">    =========</span>
-
-<span class="sd">    ``expr`` : Expression representing the function of single variable</span>
-
-<span class="sd">    ``range``: (x, 0, 5), A 3-tuple denoting the range of the free variable.</span>
-
-<span class="sd">    Keyword Arguments</span>
-<span class="sd">    =================</span>
-
-<span class="sd">    Arguments for ``LineOver1DRangeSeries`` class:</span>
-
-<span class="sd">    ``adaptive``: Boolean. The default value is set to True. Set adaptive to False and</span>
-<span class="sd">    specify ``nb_of_points`` if uniform sampling is required.</span>
-
-<span class="sd">    ``depth``: int Recursion depth of the adaptive algorithm. A depth of value ``n``</span>
-<span class="sd">    samples a maximum of `2^{n}` points.</span>
-
-<span class="sd">    ``nb_of_points``: int. Used when the ``adaptive`` is set to False. The function</span>
-<span class="sd">    is uniformly sampled at ``nb_of_points`` number of points.</span>
-
-<span class="sd">    Aesthetics options:</span>
-
-<span class="sd">    ``line_color``: float. Specifies the color for the plot.</span>
-<span class="sd">    See ``Plot`` to see how to set color for the plots.</span>
-
-<span class="sd">    If there are multiple plots, then the same series series are applied to</span>
-<span class="sd">    all the plots. If you want to set these options separately, you can index</span>
-<span class="sd">    the ``Plot`` object returned and set it.</span>
-
-<span class="sd">    Arguments for ``Plot`` class:</span>
-
-<span class="sd">    ``title`` : str. Title of the plot. It is set to the latex representation of</span>
-<span class="sd">    the expression, if the plot has only one expression.</span>
-
-<span class="sd">    ``xlabel`` : str. Label for the x - axis.</span>
-
-<span class="sd">    ``ylabel`` : str. Label for the y - axis.</span>
-
-<span class="sd">    ``xscale``: {&#39;linear&#39;, &#39;log&#39;} Sets the scaling of the x - axis.</span>
-
-<span class="sd">    ``yscale``: {&#39;linear&#39;, &#39;log&#39;} Sets the scaling if the y - axis.</span>
-
-<span class="sd">    ``axis_center``: tuple of two floats denoting the coordinates of the center or</span>
-<span class="sd">    {&#39;center&#39;, &#39;auto&#39;}</span>
-
-<span class="sd">    ``xlim`` : tuple of two floats, denoting the x - axis limits.</span>
-
-<span class="sd">    ``ylim`` : tuple of two floats, denoting the y - axis limits.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.plotting import plot</span>
-<span class="sd">    &gt;&gt;&gt; x = symbols(&#39;x&#39;)</span>
-
-<span class="sd">    Single Plot</span>
-
-<span class="sd">    &gt;&gt;&gt; plot(x**2, (x, -5, 5))# doctest: +SKIP</span>
-
-<span class="sd">    Multiple plots with single range.</span>
-
-<span class="sd">    &gt;&gt;&gt; plot(x, x**2, x**3, (x, -5, 5))# doctest: +SKIP</span>
-
-<span class="sd">    Multiple plots with different ranges.</span>
-
-<span class="sd">    &gt;&gt;&gt; plot((x**2, (x, -6, 6)), (x, (x, -5, 5)))# doctest: +SKIP</span>
-
-<span class="sd">    No adaptive sampling.</span>
-
-<span class="sd">    &gt;&gt;&gt; plot(x**2, adaptive=False, nb_of_points=400)# doctest: +SKIP</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Plot, LineOver1DRangeSeries.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="n">args</span><span class="p">))</span>
-    <span class="n">show</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;show&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-    <span class="n">series</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">plot_expr</span> <span class="o">=</span> <span class="n">check_arguments</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="n">series</span> <span class="o">=</span> <span class="p">[</span><span class="n">LineOver1DRangeSeries</span><span class="p">(</span><span class="o">*</span><span class="n">arg</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">plot_expr</span><span class="p">]</span>
-
-    <span class="n">plots</span> <span class="o">=</span> <span class="n">Plot</span><span class="p">(</span><span class="o">*</span><span class="n">series</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">show</span><span class="p">:</span>
-        <span class="n">plots</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
-    <span class="k">return</span> <span class="n">plots</span>
-
-</div>
-<div class="viewcode-block" id="plot_parametric"><a class="viewcode-back" href="../../../modules/plotting.html#sympy.plotting.plot.plot_parametric">[docs]</a><span class="k">def</span> <span class="nf">plot_parametric</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Plots a 2D parametric plot.</span>
-
-<span class="sd">    The plotting uses an adaptive algorithm which samples recursively to</span>
-<span class="sd">    accurately plot the plot. The adaptive algorithm uses a random point near</span>
-<span class="sd">    the midpoint of two points that has to be further sampled. Hence the same</span>
-<span class="sd">    plots can appear slightly different.</span>
-
-<span class="sd">    Usage</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    Single plot.</span>
-
-<span class="sd">    ``plot_parametric(expr_x, expr_y, range, **kwargs)``</span>
-
-<span class="sd">    If the range is not specified, then a default range of (-10, 10) is used.</span>
-
-<span class="sd">    Multiple plots with same range.</span>
-
-<span class="sd">    ``plot_parametric((expr1_x, expr1_y), (expr2_x, expr2_y), range, **kwargs)``</span>
-
-<span class="sd">    If the range is not specified, then a default range of (-10, 10) is used.</span>
-
-<span class="sd">    Multiple plots with different ranges.</span>
-
-<span class="sd">    ``plot_parametric((expr_x, expr_y, range), ..., **kwargs)``</span>
-
-<span class="sd">    Range has to be specified for every expression.</span>
-
-<span class="sd">    Default range may change in the future if a more advanced default range</span>
-<span class="sd">    detection algorithm is implemented.</span>
-
-<span class="sd">    Arguments</span>
-<span class="sd">    =========</span>
-
-<span class="sd">    ``expr_x`` : Expression representing the function along x.</span>
-
-<span class="sd">    ``expr_y`` : Expression representing the function along y.</span>
-
-<span class="sd">    ``range``: (u, 0, 5), A 3-tuple denoting the range of the parameter</span>
-<span class="sd">    variable.</span>
-
-<span class="sd">    Keyword Arguments</span>
-<span class="sd">    =================</span>
-
-<span class="sd">    Arguments for ``Parametric2DLineSeries`` class:</span>
-
-<span class="sd">    ``adaptive``: Boolean. The default value is set to True. Set adaptive to</span>
-<span class="sd">    False and specify ``nb_of_points`` if uniform sampling is required.</span>
-
-<span class="sd">    ``depth``: int Recursion depth of the adaptive algorithm. A depth of</span>
-<span class="sd">    value ``n`` samples a maximum of `2^{n}` points.</span>
-
-<span class="sd">    ``nb_of_points``: int. Used when the ``adaptive`` is set to False. The</span>
-<span class="sd">    function is uniformly sampled at ``nb_of_points`` number of points.</span>
-
-<span class="sd">    Aesthetics</span>
-<span class="sd">    ----------</span>
-
-<span class="sd">    ``line_color``: function which returns a float. Specifies the color for the</span>
-<span class="sd">    plot. See ``sympy.plotting.Plot`` for more details.</span>
-
-<span class="sd">    If there are multiple plots, then the same Series arguments are applied to</span>
-<span class="sd">    all the plots. If you want to set these options separately, you can index</span>
-<span class="sd">    the returned ``Plot`` object and set it.</span>
-
-<span class="sd">    Arguments for ``Plot`` class:</span>
-
-<span class="sd">    ``xlabel`` : str. Label for the x - axis.</span>
-
-<span class="sd">    ``ylabel`` : str. Label for the y - axis.</span>
-
-<span class="sd">    ``xscale``: {&#39;linear&#39;, &#39;log&#39;} Sets the scaling of the x - axis.</span>
-
-<span class="sd">    ``yscale``: {&#39;linear&#39;, &#39;log&#39;} Sets the scaling if the y - axis.</span>
-
-<span class="sd">    ``axis_center``: tuple of two floats denoting the coordinates of the center</span>
-<span class="sd">    or {&#39;center&#39;, &#39;auto&#39;}</span>
-
-<span class="sd">    ``xlim`` : tuple of two floats, denoting the x - axis limits.</span>
-
-<span class="sd">    ``ylim`` : tuple of two floats, denoting the y - axis limits.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, cos, sin</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.plotting import plot_parametric</span>
-<span class="sd">    &gt;&gt;&gt; u = symbols(&#39;u&#39;)</span>
-
-<span class="sd">    Single Parametric plot</span>
-
-<span class="sd">    &gt;&gt;&gt; plot_parametric(cos(u), sin(u), (u, -5, 5))# doctest: +SKIP</span>
-
-<span class="sd">    Multiple parametric plot with single range.</span>
-
-<span class="sd">    &gt;&gt;&gt; plot_parametric((cos(u), sin(u)), (u, cos(u)))  # doctest: +SKIP</span>
-
-<span class="sd">    Multiple parametric plots.</span>
-
-<span class="sd">    &gt;&gt;&gt; plot_parametric((cos(u), sin(u), (u, -5, 5)),</span>
-<span class="sd">    ...     (cos(u), u, (u, -5, 5)))  # doctest: +SKIP</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    Plot, Parametric2DLineSeries</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="n">args</span><span class="p">))</span>
-    <span class="n">show</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;show&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-    <span class="n">series</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">plot_expr</span> <span class="o">=</span> <span class="n">check_arguments</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="n">series</span> <span class="o">=</span> <span class="p">[</span><span class="n">Parametric2DLineSeries</span><span class="p">(</span><span class="o">*</span><span class="n">arg</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">plot_expr</span><span class="p">]</span>
-    <span class="n">plots</span> <span class="o">=</span> <span class="n">Plot</span><span class="p">(</span><span class="o">*</span><span class="n">series</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">show</span><span class="p">:</span>
-        <span class="n">plots</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
-    <span class="k">return</span> <span class="n">plots</span>
-
-</div>
-<div class="viewcode-block" id="plot3d_parametric_line"><a class="viewcode-back" href="../../../modules/plotting.html#sympy.plotting.plot.plot3d_parametric_line">[docs]</a><span class="k">def</span> <span class="nf">plot3d_parametric_line</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Plots a 3D parametric line plot.</span>
-
-<span class="sd">    Usage</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    Single plot:</span>
-
-<span class="sd">    ``plot3d_parametric_line(expr_x, expr_y, expr_z, range, **kwargs)``</span>
-
-<span class="sd">    If the range is not specified, then a default range of (-10, 10) is used.</span>
-
-<span class="sd">    Multiple plots.</span>
-
-<span class="sd">    ``plot3d_parametric_line((expr_x, expr_y, expr_z, range), ..., **kwargs)``</span>
-
-<span class="sd">    Ranges have to be specified for every expression.</span>
-
-<span class="sd">    Default range may change in the future if a more advanced default range</span>
-<span class="sd">    detection algorithm is implemented.</span>
-
-<span class="sd">    Arguments</span>
-<span class="sd">    =========</span>
-
-<span class="sd">    ``expr_x`` : Expression representing the function along x.</span>
-
-<span class="sd">    ``expr_y`` : Expression representing the function along y.</span>
-
-<span class="sd">    ``expr_z`` : Expression representing the function along z.</span>
-
-<span class="sd">    ``range``: ``(u, 0, 5)``, A 3-tuple denoting the range of the parameter</span>
-<span class="sd">    variable.</span>
-
-<span class="sd">    Keyword Arguments</span>
-<span class="sd">    =================</span>
-
-<span class="sd">    Arguments for ``Parametric3DLineSeries`` class.</span>
-
-<span class="sd">    ``nb_of_points``: The range is uniformly sampled at ``nb_of_points``</span>
-<span class="sd">    number of points.</span>
-
-<span class="sd">    Aesthetics:</span>
-
-<span class="sd">    ``line_color``: function which returns a float. Specifies the color for the</span>
-<span class="sd">    plot. See ``sympy.plotting.Plot`` for more details.</span>
-
-<span class="sd">    If there are multiple plots, then the same series arguments are applied to</span>
-<span class="sd">    all the plots. If you want to set these options separately, you can index</span>
-<span class="sd">    the returned ``Plot`` object and set it.</span>
-
-<span class="sd">    Arguments for ``Plot`` class.</span>
-
-<span class="sd">    ``title`` : str. Title of the plot.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, cos, sin</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.plotting import plot3d_parametric_line</span>
-<span class="sd">    &gt;&gt;&gt; u = symbols(&#39;u&#39;)</span>
-
-<span class="sd">    Single plot.</span>
-
-<span class="sd">    &gt;&gt;&gt; plot3d_parametric_line(cos(u), sin(u), u, (u, -5, 5))  # doctest: +SKIP</span>
-
-<span class="sd">    Multiple plots.</span>
-
-<span class="sd">    &gt;&gt;&gt; plot3d_parametric_line((cos(u), sin(u), u, (u, -5, 5)),</span>
-<span class="sd">    ...     (sin(u), u**2, u, (u, -5, 5)))  # doctest: +SKIP</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Plot, Parametric3DLineSeries</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="n">args</span><span class="p">))</span>
-    <span class="n">show</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;show&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-    <span class="n">series</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">plot_expr</span> <span class="o">=</span> <span class="n">check_arguments</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="n">series</span> <span class="o">=</span> <span class="p">[</span><span class="n">Parametric3DLineSeries</span><span class="p">(</span><span class="o">*</span><span class="n">arg</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">plot_expr</span><span class="p">]</span>
-    <span class="n">plots</span> <span class="o">=</span> <span class="n">Plot</span><span class="p">(</span><span class="o">*</span><span class="n">series</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">show</span><span class="p">:</span>
-        <span class="n">plots</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
-    <span class="k">return</span> <span class="n">plots</span>
-
-</div>
-<div class="viewcode-block" id="plot3d"><a class="viewcode-back" href="../../../modules/plotting.html#sympy.plotting.plot.plot3d">[docs]</a><span class="k">def</span> <span class="nf">plot3d</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Plots a 3D surface plot.</span>
-
-<span class="sd">    Usage</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    Single plot</span>
-
-<span class="sd">    ``plot3d(expr, range_x, range_y, **kwargs)``</span>
-
-<span class="sd">    If the ranges are not specified, then a default range of (-10, 10) is used.</span>
-
-<span class="sd">    Multiple plot with the same range.</span>
-
-<span class="sd">    ``plot3d(expr1, expr2, range_x, range_y, **kwargs)``</span>
-
-<span class="sd">    If the ranges are not specified, then a default range of (-10, 10) is used.</span>
-
-<span class="sd">    Multiple plots with different ranges.</span>
-
-<span class="sd">    ``plot3d((expr1, range_x, range_y), (expr2, range_x, range_y), ..., **kwargs)``</span>
-
-<span class="sd">    Ranges have to be specified for every expression.</span>
-
-<span class="sd">    Default range may change in the future if a more advanced default range</span>
-<span class="sd">    detection algorithm is implemented.</span>
-
-<span class="sd">    Arguments</span>
-<span class="sd">    =========</span>
-
-<span class="sd">    ``expr`` : Expression representing the function along x.</span>
-
-<span class="sd">    ``range_x``: (x, 0, 5), A 3-tuple denoting the range of the x</span>
-<span class="sd">    variable.</span>
-
-<span class="sd">    ``range_y``: (y, 0, 5), A 3-tuple denoting the range of the y</span>
-<span class="sd">     variable.</span>
-
-<span class="sd">    Keyword Arguments</span>
-<span class="sd">    =================</span>
-
-<span class="sd">    Arguments for ``SurfaceOver2DRangeSeries`` class:</span>
-
-<span class="sd">    ``nb_of_points_x``: int. The x range is sampled uniformly at</span>
-<span class="sd">    ``nb_of_points_x`` of points.</span>
-
-<span class="sd">    ``nb_of_points_y``: int. The y range is sampled uniformly at</span>
-<span class="sd">    ``nb_of_points_y`` of points.</span>
-
-<span class="sd">    Aesthetics:</span>
-
-<span class="sd">    ``surface_color``: Function which returns a float. Specifies the color for</span>
-<span class="sd">    the surface of the plot. See ``sympy.plotting.Plot`` for more details.</span>
-
-<span class="sd">    If there are multiple plots, then the same series arguments are applied to</span>
-<span class="sd">    all the plots. If you want to set these options separately, you can index</span>
-<span class="sd">    the returned ``Plot`` object and set it.</span>
-
-<span class="sd">    Arguments for ``Plot`` class:</span>
-
-<span class="sd">    ``title`` : str. Title of the plot.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.plotting import plot3d</span>
-<span class="sd">    &gt;&gt;&gt; x, y = symbols(&#39;x y&#39;)</span>
-
-<span class="sd">    Single plot</span>
-
-<span class="sd">    &gt;&gt;&gt; plot3d(x*y, (x, -5, 5), (y, -5, 5))  # doctest: +SKIP</span>
-
-<span class="sd">    Multiple plots with same range</span>
-
-<span class="sd">    &gt;&gt;&gt; plot3d(x*y, -x*y, (x, -5, 5), (y, -5, 5))  # doctest: +SKIP</span>
-
-<span class="sd">    Multiple plots with different ranges.</span>
-
-<span class="sd">    &gt;&gt;&gt; plot3d((x**2 + y**2, (x, -5, 5), (y, -5, 5)),</span>
-<span class="sd">    ...     (x*y, (x, -3, 3), (y, -3, 3)))  # doctest: +SKIP</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    Plot, SurfaceOver2DRangeSeries</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="n">args</span><span class="p">))</span>
-    <span class="n">show</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;show&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-    <span class="n">series</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">plot_expr</span> <span class="o">=</span> <span class="n">check_arguments</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-    <span class="n">series</span> <span class="o">=</span> <span class="p">[</span><span class="n">SurfaceOver2DRangeSeries</span><span class="p">(</span><span class="o">*</span><span class="n">arg</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">plot_expr</span><span class="p">]</span>
-    <span class="n">plots</span> <span class="o">=</span> <span class="n">Plot</span><span class="p">(</span><span class="o">*</span><span class="n">series</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">show</span><span class="p">:</span>
-        <span class="n">plots</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
-    <span class="k">return</span> <span class="n">plots</span>
-
-</div>
-<div class="viewcode-block" id="plot3d_parametric_surface"><a class="viewcode-back" href="../../../modules/plotting.html#sympy.plotting.plot.plot3d_parametric_surface">[docs]</a><span class="k">def</span> <span class="nf">plot3d_parametric_surface</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Plots a 3D parametric surface plot.</span>
-
-<span class="sd">    Usage</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    Single plot.</span>
-
-<span class="sd">    ``plot3d_parametric_surface(expr_x, expr_y, expr_z, range_u, range_v, **kwargs)``</span>
-
-<span class="sd">    If the ranges is not specified, then a default range of (-10, 10) is used.</span>
-
-<span class="sd">    Multiple plots.</span>
-
-<span class="sd">    ``plot3d_parametric_surface((expr_x, expr_y, expr_z, range_u, range_v), ..., **kwargs)``</span>
-
-<span class="sd">    Ranges have to be specified for every expression.</span>
-
-<span class="sd">    Default range may change in the future if a more advanced default range</span>
-<span class="sd">    detection algorithm is implemented.</span>
-
-<span class="sd">    Arguments</span>
-<span class="sd">    =========</span>
-
-<span class="sd">    ``expr_x``: Expression representing the function along ``x``.</span>
-
-<span class="sd">    ``expr_y``: Expression representing the function along ``y``.</span>
-
-<span class="sd">    ``expr_z``: Expression representing the function along ``z``.</span>
-
-<span class="sd">    ``range_u``: ``(u, 0, 5)``,  A 3-tuple denoting the range of the ``u``</span>
-<span class="sd">    variable.</span>
-
-<span class="sd">    ``range_v``: ``(v, 0, 5)``,  A 3-tuple denoting the range of the v</span>
-<span class="sd">    variable.</span>
-
-<span class="sd">    Keyword Arguments</span>
-<span class="sd">    =================</span>
-
-<span class="sd">    Arguments for ``ParametricSurfaceSeries`` class:</span>
-
-<span class="sd">    ``nb_of_points_u``: int. The ``u`` range is sampled uniformly at</span>
-<span class="sd">    ``nb_of_points_v`` of points</span>
-
-<span class="sd">    ``nb_of_points_y``: int. The ``v`` range is sampled uniformly at</span>
-<span class="sd">    ``nb_of_points_y`` of points</span>
-
-<span class="sd">    Aesthetics:</span>
-
-<span class="sd">    ``surface_color``: Function which returns a float. Specifies the color for</span>
-<span class="sd">    the surface of the plot. See ``sympy.plotting.Plot`` for more details.</span>
-
-<span class="sd">    If there are multiple plots, then the same series arguments are applied for</span>
-<span class="sd">    all the plots. If you want to set these options separately, you can index</span>
-<span class="sd">    the returned ``Plot`` object and set it.</span>
-
-
-<span class="sd">    Arguments for ``Plot`` class:</span>
-
-<span class="sd">    ``title`` : str. Title of the plot.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, cos, sin</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.plotting import plot3d_parametric_surface</span>
-<span class="sd">    &gt;&gt;&gt; u, v = symbols(&#39;u v&#39;)</span>
-
-<span class="sd">    Single plot.</span>
-
-<span class="sd">    &gt;&gt;&gt; plot3d_parametric_surface(cos(u + v), sin(u - v), u - v,</span>
-<span class="sd">    ...     (u, -5, 5), (v, -5, 5))  # doctest: +SKIP</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    Plot, ParametricSurfaceSeries</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="n">args</span><span class="p">))</span>
-    <span class="n">show</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;show&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-    <span class="n">series</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">plot_expr</span> <span class="o">=</span> <span class="n">check_arguments</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-    <span class="n">series</span> <span class="o">=</span> <span class="p">[</span><span class="n">ParametricSurfaceSeries</span><span class="p">(</span><span class="o">*</span><span class="n">arg</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">plot_expr</span><span class="p">]</span>
-    <span class="n">plots</span> <span class="o">=</span> <span class="n">Plot</span><span class="p">(</span><span class="o">*</span><span class="n">series</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">show</span><span class="p">:</span>
-        <span class="n">plots</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
-    <span class="k">return</span> <span class="n">plots</span>
-
-</div>
-<span class="k">def</span> <span class="nf">check_arguments</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="n">expr_len</span><span class="p">,</span> <span class="n">nb_of_free_symbols</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Checks the arguments and converts into tuples of the</span>
-<span class="sd">    form (exprs, ranges)</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import plot, cos, sin, symbols</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.plotting.plot import check_arguments</span>
-<span class="sd">    &gt;&gt;&gt; x,y,u,v = symbols(&#39;x y u v&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; check_arguments([cos(x), sin(x)], 2, 1)</span>
-<span class="sd">        [(cos(x), sin(x), (x, -10, 10))]</span>
-
-<span class="sd">    &gt;&gt;&gt; check_arguments([x, x**2], 1, 1)</span>
-<span class="sd">        [(x, (x, -10, 10)), (x**2, (x, -10, 10))]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">expr_len</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Expr</span><span class="p">):</span>
-        <span class="c"># Multiple expressions same range.</span>
-        <span class="c"># The arguments are tuples when the expression length is</span>
-        <span class="c"># greater than 1.</span>
-        <span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="n">expr_len</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">Tuple</span><span class="p">):</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
-
-        <span class="n">exprs</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">[:</span><span class="n">i</span><span class="p">])</span>
-        <span class="n">free_symbols</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">set_union</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">e</span><span class="o">.</span><span class="n">free_symbols</span> <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">exprs</span><span class="p">]))</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="n">expr_len</span> <span class="o">+</span> <span class="n">nb_of_free_symbols</span><span class="p">:</span>
-            <span class="c">#Ranges given</span>
-            <span class="n">plots</span> <span class="o">=</span> <span class="p">[</span><span class="n">exprs</span> <span class="o">+</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">[</span><span class="n">expr_len</span><span class="p">:])]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">default_range</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-            <span class="n">ranges</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">symbol</span> <span class="ow">in</span> <span class="n">free_symbols</span><span class="p">:</span>
-                <span class="n">ranges</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Tuple</span><span class="p">(</span><span class="n">symbol</span><span class="p">)</span> <span class="o">+</span> <span class="n">default_range</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">free_symbols</span><span class="p">)</span> <span class="o">-</span> <span class="n">nb_of_free_symbols</span><span class="p">):</span>
-                <span class="n">ranges</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Tuple</span><span class="p">(</span><span class="n">Dummy</span><span class="p">())</span> <span class="o">+</span> <span class="n">default_range</span><span class="p">)</span>
-            <span class="n">plots</span> <span class="o">=</span> <span class="p">[</span><span class="n">exprs</span> <span class="o">+</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">ranges</span><span class="p">)]</span>
-        <span class="k">return</span> <span class="n">plots</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Expr</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Tuple</span><span class="p">)</span> <span class="ow">and</span>
-                                     <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">==</span> <span class="n">expr_len</span> <span class="ow">and</span>
-                                     <span class="n">expr_len</span> <span class="o">!=</span> <span class="mi">3</span><span class="p">):</span>
-        <span class="c"># Cannot handle expressions with number of expression = 3. It is</span>
-        <span class="c"># not possible to differentiate between expressions and ranges.</span>
-        <span class="c">#Series of plots with same range</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">Tuple</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="o">!=</span> <span class="n">expr_len</span><span class="p">:</span>
-                <span class="k">break</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">Tuple</span><span class="p">):</span>
-                <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
-
-        <span class="n">exprs</span> <span class="o">=</span> <span class="n">args</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span>
-        <span class="k">assert</span> <span class="nb">all</span><span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Expr</span><span class="p">)</span> <span class="k">for</span> <span class="n">expr</span> <span class="ow">in</span> <span class="n">exprs</span> <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">expr</span><span class="p">)</span>
-        <span class="n">free_symbols</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">set_union</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">e</span><span class="o">.</span><span class="n">free_symbols</span> <span class="k">for</span> <span class="n">expr</span> <span class="ow">in</span> <span class="n">exprs</span>
-                                        <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">expr</span><span class="p">]))</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">free_symbols</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">nb_of_free_symbols</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;The number of free_symbols in the expression&quot;</span>
-                             <span class="s">&quot;is greater than </span><span class="si">%d</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">nb_of_free_symbols</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="n">i</span> <span class="o">+</span> <span class="n">nb_of_free_symbols</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">Tuple</span><span class="p">):</span>
-            <span class="n">ranges</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">range_expr</span> <span class="k">for</span> <span class="n">range_expr</span> <span class="ow">in</span> <span class="n">args</span><span class="p">[</span>
-                           <span class="n">i</span><span class="p">:</span><span class="n">i</span> <span class="o">+</span> <span class="n">nb_of_free_symbols</span><span class="p">]])</span>
-            <span class="n">plots</span> <span class="o">=</span> <span class="p">[</span><span class="n">expr</span> <span class="o">+</span> <span class="n">ranges</span> <span class="k">for</span> <span class="n">expr</span> <span class="ow">in</span> <span class="n">exprs</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">plots</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c">#Use default ranges.</span>
-            <span class="n">default_range</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-            <span class="n">ranges</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">symbol</span> <span class="ow">in</span> <span class="n">free_symbols</span><span class="p">:</span>
-                <span class="n">ranges</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Tuple</span><span class="p">(</span><span class="n">symbol</span><span class="p">)</span> <span class="o">+</span> <span class="n">default_range</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">free_symbols</span><span class="p">)</span> <span class="o">-</span> <span class="n">nb_of_free_symbols</span><span class="p">):</span>
-                <span class="n">ranges</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Tuple</span><span class="p">(</span><span class="n">Dummy</span><span class="p">())</span> <span class="o">+</span> <span class="n">default_range</span><span class="p">)</span>
-            <span class="n">ranges</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">ranges</span><span class="p">)</span>
-            <span class="n">plots</span> <span class="o">=</span> <span class="p">[</span><span class="n">expr</span> <span class="o">+</span> <span class="n">ranges</span> <span class="k">for</span> <span class="n">expr</span> <span class="ow">in</span> <span class="n">exprs</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">plots</span>
-
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Tuple</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">==</span> <span class="n">expr_len</span> <span class="o">+</span> <span class="n">nb_of_free_symbols</span><span class="p">:</span>
-        <span class="c">#Multiple plots with different ranges.</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">expr_len</span><span class="p">):</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">Expr</span><span class="p">):</span>
-                    <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Expected an expression, given </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span>
-                                     <span class="nb">str</span><span class="p">(</span><span class="n">arg</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nb_of_free_symbols</span><span class="p">):</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="nb">len</span><span class="p">(</span><span class="n">arg</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="n">expr_len</span><span class="p">])</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;The ranges should be a tuple of&quot;</span>
-                                     <span class="s">&quot;length 3, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">str</span><span class="p">(</span><span class="n">arg</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="n">expr_len</span><span class="p">]))</span>
-        <span class="k">return</span> <span class="n">args</span>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
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diff --git a/dev-py3k/_modules/sympy/plotting/plot_implicit.html b/dev-py3k/_modules/sympy/plotting/plot_implicit.html
deleted file mode 100644
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-  <h1>Source code for sympy.plotting.plot_implicit</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Implicit plotting module for SymPy</span>
-
-<span class="sd">The module implements a data series called ImplicitSeries which is used by</span>
-<span class="sd">``Plot`` class to plot implicit plots for different backends. The module,</span>
-<span class="sd">by default, implements plotting using interval arithmetic. It switches to a</span>
-<span class="sd">fall back algorithm if the expression cannot be plotted used interval</span>
-<span class="sd">interval arithmetic. It is also possible to specify to use the fall back</span>
-<span class="sd">algorithm for all plots.</span>
-
-<span class="sd">Boolean combinations of expressions cannot be plotted by the fall back</span>
-<span class="sd">algorithm.</span>
-
-<span class="sd">See Also</span>
-<span class="sd">========</span>
-<span class="sd">sympy.plotting.plot</span>
-
-<span class="sd">References</span>
-<span class="sd">==========</span>
-<span class="sd">- Jeffrey Allen Tupper. Reliable Two-Dimensional Graphing Methods for</span>
-<span class="sd">Mathematical Formulae with Two Free Variables.</span>
-
-<span class="sd">- Jeffrey Allen Tupper. Graphing Equations with Generalized Interval</span>
-<span class="sd">Arithmetic. Master&#39;s thesis. University of Toronto, 1996</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">.plot</span> <span class="kn">import</span> <span class="n">BaseSeries</span><span class="p">,</span> <span class="n">Plot</span>
-<span class="kn">from</span> <span class="nn">.experimental_lambdify</span> <span class="kn">import</span> <span class="n">experimental_lambdify</span><span class="p">,</span> <span class="n">vectorized_lambdify</span>
-<span class="kn">from</span> <span class="nn">.intervalmath</span> <span class="kn">import</span> <span class="n">interval</span>
-<span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="p">(</span><span class="n">Equality</span><span class="p">,</span> <span class="n">GreaterThan</span><span class="p">,</span> <span class="n">LessThan</span><span class="p">,</span>
-                <span class="n">Relational</span><span class="p">,</span> <span class="n">StrictLessThan</span><span class="p">,</span> <span class="n">StrictGreaterThan</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Eq</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">Dummy</span>
-<span class="kn">from</span> <span class="nn">sympy.external</span> <span class="kn">import</span> <span class="n">import_module</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">set_union</span>
-<span class="kn">from</span> <span class="nn">sympy.logic.boolalg</span> <span class="kn">import</span> <span class="n">BooleanFunction</span>
-<span class="kn">import</span> <span class="nn">warnings</span>
-
-<span class="n">np</span> <span class="o">=</span> <span class="n">import_module</span><span class="p">(</span><span class="s">&#39;numpy&#39;</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="ImplicitSeries"><a class="viewcode-back" href="../../../modules/plotting.html#sympy.plotting.plot_implicit.ImplicitSeries">[docs]</a><span class="k">class</span> <span class="nc">ImplicitSeries</span><span class="p">(</span><span class="n">BaseSeries</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Representation for Implicit plot &quot;&quot;&quot;</span>
-    <span class="n">is_implicit</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">var_start_end_x</span><span class="p">,</span> <span class="n">var_start_end_y</span><span class="p">,</span>
-            <span class="n">has_equality</span><span class="p">,</span> <span class="n">use_interval_math</span><span class="p">,</span> <span class="n">depth</span><span class="p">,</span> <span class="n">nb_of_points</span><span class="p">):</span>
-        <span class="nb">super</span><span class="p">(</span><span class="n">ImplicitSeries</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__init__</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">var_x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">var_start_end_x</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">start_x</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">var_start_end_x</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">end_x</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">var_start_end_x</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">var_y</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">var_start_end_y</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">start_y</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">var_start_end_y</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">end_y</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">var_start_end_y</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">get_points</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_raster</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">has_equality</span> <span class="o">=</span> <span class="n">has_equality</span>  <span class="c"># If the expression has equality, i.e.</span>
-                                         <span class="c">#Eq, Greaterthan, LessThan.</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points</span> <span class="o">=</span> <span class="n">nb_of_points</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">use_interval_math</span> <span class="o">=</span> <span class="n">use_interval_math</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">depth</span> <span class="o">=</span> <span class="mi">4</span> <span class="o">+</span> <span class="n">depth</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="s">&#39;Implicit equation: </span><span class="si">%s</span><span class="s"> for &#39;</span>
-                <span class="s">&#39;</span><span class="si">%s</span><span class="s"> over </span><span class="si">%s</span><span class="s"> and </span><span class="si">%s</span><span class="s"> over </span><span class="si">%s</span><span class="s">&#39;</span><span class="p">)</span> <span class="o">%</span> <span class="p">(</span>
-                    <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">),</span>
-                    <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">var_x</span><span class="p">),</span>
-                    <span class="nb">str</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">start_x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end_x</span><span class="p">)),</span>
-                    <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">var_y</span><span class="p">),</span>
-                    <span class="nb">str</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">start_y</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end_y</span><span class="p">)))</span>
-
-    <span class="k">def</span> <span class="nf">get_raster</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">func</span> <span class="o">=</span> <span class="n">experimental_lambdify</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">var_x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">var_y</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span>
-                                    <span class="n">use_interval</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="n">xinterval</span> <span class="o">=</span> <span class="n">interval</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start_x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end_x</span><span class="p">)</span>
-        <span class="n">yinterval</span> <span class="o">=</span> <span class="n">interval</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start_y</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end_y</span><span class="p">)</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">temp</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">xinterval</span><span class="p">,</span> <span class="n">yinterval</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">use_interval_math</span><span class="p">:</span>
-                <span class="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="s">&quot;Adaptive meshing could not be applied to the&quot;</span>
-                            <span class="s">&quot; expression. Using uniform meshing.&quot;</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">use_interval_math</span> <span class="o">=</span> <span class="bp">False</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">use_interval_math</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_raster_interval</span><span class="p">(</span><span class="n">func</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_meshes_grid</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">_get_raster_interval</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Uses interval math to adaptively mesh and obtain the plot&quot;&quot;&quot;</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">depth</span>
-        <span class="n">interval_list</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="c">#Create initial 32 divisions</span>
-        <span class="n">xsample</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start_x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end_x</span><span class="p">,</span> <span class="mi">33</span><span class="p">)</span>
-        <span class="n">ysample</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start_y</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end_y</span><span class="p">,</span> <span class="mi">33</span><span class="p">)</span>
-
-        <span class="c">#Add a small jitter so that there are no false positives for equality.</span>
-        <span class="c"># Ex: y==x becomes True for x interval(1, 2) and y interval(1, 2)</span>
-        <span class="c">#which will draw a rectangle.</span>
-        <span class="n">jitterx</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span>
-            <span class="nb">len</span><span class="p">(</span><span class="n">xsample</span><span class="p">))</span> <span class="o">*</span> <span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">end_x</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">start_x</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span><span class="o">**</span><span class="mi">20</span>
-        <span class="n">jittery</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span>
-            <span class="nb">len</span><span class="p">(</span><span class="n">ysample</span><span class="p">))</span> <span class="o">*</span> <span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">end_y</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">start_y</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span><span class="o">**</span><span class="mi">20</span>
-        <span class="n">xsample</span> <span class="o">+=</span> <span class="n">jitterx</span>
-        <span class="n">ysample</span> <span class="o">+=</span> <span class="n">jittery</span>
-
-        <span class="n">xinter</span> <span class="o">=</span> <span class="p">[</span><span class="n">interval</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span> <span class="n">x2</span><span class="p">)</span> <span class="k">for</span> <span class="n">x1</span><span class="p">,</span> <span class="n">x2</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">xsample</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span>
-                           <span class="n">xsample</span><span class="p">[</span><span class="mi">1</span><span class="p">:])]</span>
-        <span class="n">yinter</span> <span class="o">=</span> <span class="p">[</span><span class="n">interval</span><span class="p">(</span><span class="n">y1</span><span class="p">,</span> <span class="n">y2</span><span class="p">)</span> <span class="k">for</span> <span class="n">y1</span><span class="p">,</span> <span class="n">y2</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">ysample</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span>
-                           <span class="n">ysample</span><span class="p">[</span><span class="mi">1</span><span class="p">:])]</span>
-        <span class="n">interval_list</span> <span class="o">=</span> <span class="p">[[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xinter</span> <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">yinter</span><span class="p">]</span>
-        <span class="n">plot_list</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="c">#recursive call refinepixels which subdivides the intervals which are</span>
-        <span class="c">#neither True nor False according to the expression.</span>
-        <span class="k">def</span> <span class="nf">refine_pixels</span><span class="p">(</span><span class="n">interval_list</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot; Evaluates the intervals and subdivides the interval if the</span>
-<span class="sd">            expression is partially satisfied.&quot;&quot;&quot;</span>
-            <span class="n">temp_interval_list</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">plot_list</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">intervals</span> <span class="ow">in</span> <span class="n">interval_list</span><span class="p">:</span>
-
-                <span class="c">#Convert the array indices to x and y values</span>
-                <span class="n">intervalx</span> <span class="o">=</span> <span class="n">intervals</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="n">intervaly</span> <span class="o">=</span> <span class="n">intervals</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-                <span class="n">func_eval</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">intervalx</span><span class="p">,</span> <span class="n">intervaly</span><span class="p">)</span>
-                <span class="c">#The expression is valid in the interval. Change the contour</span>
-                <span class="c">#array values to 1.</span>
-                <span class="k">if</span> <span class="n">func_eval</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">False</span> <span class="ow">or</span> <span class="n">func_eval</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                    <span class="k">pass</span>
-                <span class="k">elif</span> <span class="n">func_eval</span> <span class="o">==</span> <span class="p">(</span><span class="bp">True</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-                    <span class="n">plot_list</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">intervalx</span><span class="p">,</span> <span class="n">intervaly</span><span class="p">])</span>
-                <span class="k">elif</span> <span class="n">func_eval</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">func_eval</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="c">#Subdivide</span>
-                    <span class="n">avgx</span> <span class="o">=</span> <span class="n">intervalx</span><span class="o">.</span><span class="n">mid</span>
-                    <span class="n">avgy</span> <span class="o">=</span> <span class="n">intervaly</span><span class="o">.</span><span class="n">mid</span>
-                    <span class="n">a</span> <span class="o">=</span> <span class="n">interval</span><span class="p">(</span><span class="n">intervalx</span><span class="o">.</span><span class="n">start</span><span class="p">,</span> <span class="n">avgx</span><span class="p">)</span>
-                    <span class="n">b</span> <span class="o">=</span> <span class="n">interval</span><span class="p">(</span><span class="n">avgx</span><span class="p">,</span> <span class="n">intervalx</span><span class="o">.</span><span class="n">end</span><span class="p">)</span>
-                    <span class="n">c</span> <span class="o">=</span> <span class="n">interval</span><span class="p">(</span><span class="n">intervaly</span><span class="o">.</span><span class="n">start</span><span class="p">,</span> <span class="n">avgy</span><span class="p">)</span>
-                    <span class="n">d</span> <span class="o">=</span> <span class="n">interval</span><span class="p">(</span><span class="n">avgy</span><span class="p">,</span> <span class="n">intervaly</span><span class="o">.</span><span class="n">end</span><span class="p">)</span>
-                    <span class="n">temp_interval_list</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">c</span><span class="p">])</span>
-                    <span class="n">temp_interval_list</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">d</span><span class="p">])</span>
-                    <span class="n">temp_interval_list</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">])</span>
-                    <span class="n">temp_interval_list</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">b</span><span class="p">,</span> <span class="n">d</span><span class="p">])</span>
-            <span class="k">return</span> <span class="n">temp_interval_list</span><span class="p">,</span> <span class="n">plot_list</span>
-
-        <span class="k">while</span> <span class="n">k</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">interval_list</span><span class="p">):</span>
-            <span class="n">interval_list</span><span class="p">,</span> <span class="n">plot_list_temp</span> <span class="o">=</span> <span class="n">refine_pixels</span><span class="p">(</span><span class="n">interval_list</span><span class="p">)</span>
-            <span class="n">plot_list</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">plot_list_temp</span><span class="p">)</span>
-            <span class="n">k</span> <span class="o">=</span> <span class="n">k</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="c">#Check whether the expression represents an equality</span>
-        <span class="c">#If it represents an equality, then none of the intervals</span>
-        <span class="c">#would have satisfied the expression due to floating point</span>
-        <span class="c">#differences. Add all the undecided values to the plot.</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">has_equality</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">intervals</span> <span class="ow">in</span> <span class="n">interval_list</span><span class="p">:</span>
-                <span class="n">intervalx</span> <span class="o">=</span> <span class="n">intervals</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="n">intervaly</span> <span class="o">=</span> <span class="n">intervals</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-                <span class="n">func_eval</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">intervalx</span><span class="p">,</span> <span class="n">intervaly</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">func_eval</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">and</span> <span class="n">func_eval</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">:</span>
-                    <span class="n">plot_list</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">intervalx</span><span class="p">,</span> <span class="n">intervaly</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">plot_list</span><span class="p">,</span> <span class="s">&#39;fill&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_get_meshes_grid</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Generates the mesh for generating a contour.</span>
-
-<span class="sd">        In the case of equality, ``contour`` function of matplotlib can</span>
-<span class="sd">        be used. In other cases, matplotlib&#39;s ``contourf`` is used.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">equal</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="n">Equality</span><span class="p">):</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">lhs</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">rhs</span>
-            <span class="n">equal</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="p">(</span><span class="n">GreaterThan</span><span class="p">,</span> <span class="n">StrictGreaterThan</span><span class="p">)):</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">lhs</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">rhs</span>
-
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="p">(</span><span class="n">LessThan</span><span class="p">,</span> <span class="n">StrictLessThan</span><span class="p">)):</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">rhs</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">lhs</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;The expression is not supported for&quot;</span>
-                                    <span class="s">&quot;plotting in uniform meshed plot.&quot;</span><span class="p">)</span>
-        <span class="n">xarray</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start_x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end_x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points</span><span class="p">)</span>
-        <span class="n">yarray</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">start_y</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">end_y</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">nb_of_points</span><span class="p">)</span>
-        <span class="n">x_grid</span><span class="p">,</span> <span class="n">y_grid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">xarray</span><span class="p">,</span> <span class="n">yarray</span><span class="p">)</span>
-
-        <span class="n">func</span> <span class="o">=</span> <span class="n">vectorized_lambdify</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">var_x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">var_y</span><span class="p">),</span> <span class="n">expr</span><span class="p">)</span>
-        <span class="n">z_grid</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">x_grid</span><span class="p">,</span> <span class="n">y_grid</span><span class="p">)</span>
-        <span class="n">z_grid</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ma</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">z_grid</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">)]</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-        <span class="n">z_grid</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">ma</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">z_grid</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">)]</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">equal</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">xarray</span><span class="p">,</span> <span class="n">yarray</span><span class="p">,</span> <span class="n">z_grid</span><span class="p">,</span> <span class="s">&#39;contour&#39;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">xarray</span><span class="p">,</span> <span class="n">yarray</span><span class="p">,</span> <span class="n">z_grid</span><span class="p">,</span> <span class="s">&#39;contourf&#39;</span>
-
-</div>
-<div class="viewcode-block" id="plot_implicit"><a class="viewcode-back" href="../../../modules/plotting.html#sympy.plotting.plot_implicit.plot_implicit">[docs]</a><span class="k">def</span> <span class="nf">plot_implicit</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A plot function to plot implicit equations / inequalities.</span>
-
-<span class="sd">    Arguments</span>
-<span class="sd">    =========</span>
-
-<span class="sd">    - ``expr`` : The equation / inequality that is to be plotted.</span>
-<span class="sd">    - ``(x, xmin, xmax)`` optional, 3-tuple denoting the range of symbol</span>
-<span class="sd">      ``x``</span>
-<span class="sd">    - ``(y, ymin, ymax)`` optional, 3-tuple denoting the range of symbol</span>
-<span class="sd">      ``y``</span>
-
-<span class="sd">    The following arguments can be passed as named parameters.</span>
-
-<span class="sd">    - ``adaptive``. Boolean. The default value is set to True. It has to be</span>
-<span class="sd">        set to False if you want to use a mesh grid.</span>
-
-<span class="sd">    - ``depth`` integer. The depth of recursion for adaptive mesh grid.</span>
-<span class="sd">        Default value is 0. Takes value in the range (0, 4).</span>
-
-<span class="sd">    - ``points`` integer. The number of points if adaptive mesh grid is not</span>
-<span class="sd">        used. Default value is 200.</span>
-
-<span class="sd">    - ``title`` string .The title for the plot.</span>
-
-<span class="sd">    - ``xlabel`` string. The label for the x - axis</span>
-
-<span class="sd">    - ``ylabel`` string. The label for the y - axis</span>
-
-<span class="sd">    plot_implicit, by default, uses interval arithmetic to plot functions. If</span>
-<span class="sd">    the expression cannot be plotted using interval arithmetic, it defaults to</span>
-<span class="sd">    a generating a contour using a mesh grid of fixed number of points. By</span>
-<span class="sd">    setting adaptive to False, you can force plot_implicit to use the mesh</span>
-<span class="sd">    grid. The mesh grid method can be effective when adaptive plotting using</span>
-<span class="sd">    interval arithmetic, fails to plot with small line width.</span>
-
-<span class="sd">    Examples:</span>
-<span class="sd">    =========</span>
-
-<span class="sd">    Plot expressions:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import plot_implicit, cos, sin, symbols, Eq</span>
-<span class="sd">    &gt;&gt;&gt; x, y = symbols(&#39;x y&#39;)</span>
-
-<span class="sd">    Without any ranges for the symbols in the expression</span>
-
-<span class="sd">    &gt;&gt;&gt; p1 = plot_implicit(Eq(x**2 + y**2, 5))  # doctest: +SKIP</span>
-
-<span class="sd">    With the range for the symbols</span>
-
-<span class="sd">    &gt;&gt;&gt; p2 = plot_implicit(Eq(x**2 + y**2, 3),</span>
-<span class="sd">    ...         (x, -3, 3), (y, -3, 3))  # doctest: +SKIP</span>
-
-<span class="sd">    With depth of recursion as argument.</span>
-
-<span class="sd">    &gt;&gt;&gt; p3 = plot_implicit(Eq(x**2 + y**2, 5),</span>
-<span class="sd">    ...         (x, -4, 4), (y, -4, 4), depth = 2)  # doctest: +SKIP</span>
-
-<span class="sd">    Using mesh grid and not using adaptive meshing.</span>
-
-<span class="sd">    &gt;&gt;&gt; p4 = plot_implicit(Eq(x**2 + y**2, 5),</span>
-<span class="sd">    ...         (x, -5, 5), (y, -2, 2), adaptive=False)  # doctest: +SKIP</span>
-
-<span class="sd">    Using mesh grid with number of points as input.</span>
-
-<span class="sd">    &gt;&gt;&gt; p5 = plot_implicit(Eq(x**2 + y**2, 5),</span>
-<span class="sd">    ...         (x, -5, 5), (y, -2, 2),</span>
-<span class="sd">    ...         adaptive=False, points=400)  # doctest: +SKIP</span>
-
-<span class="sd">    Plotting regions.</span>
-
-<span class="sd">    &gt;&gt;&gt; p6 = plot_implicit(y &gt; x**2)  # doctest: +SKIP</span>
-
-<span class="sd">    Plotting Using boolean conjunctions.</span>
-
-<span class="sd">    &gt;&gt;&gt; p7 = plot_implicit(And(y &gt; x, y &gt; -x))  # doctest: +SKIP</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">has_equality</span> <span class="o">=</span> <span class="bp">False</span>  <span class="c"># Represents whether the expression contains an Equality,</span>
-                     <span class="c">#GreaterThan or LessThan</span>
-
-    <span class="k">def</span> <span class="nf">arg_expand</span><span class="p">(</span><span class="n">bool_expr</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Recursively expands the arguments of an Boolean Function</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">bool_expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">BooleanFunction</span><span class="p">):</span>
-                <span class="n">arg_expand</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Relational</span><span class="p">):</span>
-                <span class="n">arg_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="n">arg_list</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">BooleanFunction</span><span class="p">):</span>
-        <span class="n">arg_expand</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="c">#Check whether there is an equality in the expression provided.</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="p">(</span><span class="n">Equality</span><span class="p">,</span> <span class="n">GreaterThan</span><span class="p">,</span> <span class="n">LessThan</span><span class="p">))</span>
-               <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">arg_list</span><span class="p">):</span>
-            <span class="n">has_equality</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">elif</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Relational</span><span class="p">):</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">Eq</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="n">has_equality</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="p">(</span><span class="n">Equality</span><span class="p">,</span> <span class="n">GreaterThan</span><span class="p">,</span> <span class="n">LessThan</span><span class="p">)):</span>
-        <span class="n">has_equality</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="n">free_symbols</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">)</span>
-    <span class="n">range_symbols</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span><span class="n">t</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">args</span><span class="p">])</span>
-    <span class="n">symbols</span> <span class="o">=</span> <span class="n">set_union</span><span class="p">(</span><span class="n">free_symbols</span><span class="p">,</span> <span class="n">range_symbols</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Implicit plotting is not implemented for &quot;</span>
-                                  <span class="s">&quot;more than 2 variables&quot;</span><span class="p">)</span>
-
-    <span class="c">#Create default ranges if the range is not provided.</span>
-    <span class="n">default_range</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="n">var_start_end_x</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">var_start_end_y</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-    <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">free_symbols</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">var_start_end_x</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">var_start_end_y</span><span class="p">,</span> <span class="o">=</span> <span class="p">(</span><span class="n">Tuple</span><span class="p">(</span><span class="n">e</span><span class="p">)</span> <span class="o">+</span> <span class="n">default_range</span>
-                                <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="p">(</span><span class="n">free_symbols</span> <span class="o">-</span> <span class="n">range_symbols</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">var_start_end_x</span><span class="p">,</span> <span class="o">=</span> <span class="p">(</span><span class="n">Tuple</span><span class="p">(</span><span class="n">e</span><span class="p">)</span> <span class="o">+</span> <span class="n">default_range</span> <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">free_symbols</span><span class="p">)</span>
-            <span class="c">#Create a random symbol</span>
-            <span class="n">var_start_end_y</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="n">Dummy</span><span class="p">())</span> <span class="o">+</span> <span class="n">default_range</span>
-
-    <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">free_symbols</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">var_start_end_x</span><span class="p">,</span> <span class="o">=</span> <span class="p">(</span><span class="n">Tuple</span><span class="p">(</span><span class="n">e</span><span class="p">)</span> <span class="o">+</span> <span class="n">default_range</span> <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">free_symbols</span><span class="p">)</span>
-            <span class="c">#create a random symbol</span>
-            <span class="n">var_start_end_y</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="n">Dummy</span><span class="p">())</span> <span class="o">+</span> <span class="n">default_range</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">var_start_end_x</span><span class="p">,</span> <span class="n">var_start_end_y</span> <span class="o">=</span> <span class="p">(</span><span class="n">Tuple</span><span class="p">(</span><span class="n">e</span><span class="p">)</span> <span class="o">+</span> <span class="n">default_range</span>
-                                                <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">free_symbols</span><span class="p">)</span>
-
-    <span class="n">use_interval</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;adaptive&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-    <span class="n">nb_of_points</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;points&#39;</span><span class="p">,</span> <span class="mi">300</span><span class="p">)</span>
-    <span class="n">depth</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;depth&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-    <span class="c">#Check whether the depth is greater than 4 or less than 0.</span>
-    <span class="k">if</span> <span class="n">depth</span> <span class="o">&gt;</span> <span class="mi">4</span><span class="p">:</span>
-        <span class="n">depth</span> <span class="o">=</span> <span class="mi">4</span>
-    <span class="k">elif</span> <span class="n">depth</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">depth</span> <span class="o">=</span> <span class="mi">0</span>
-
-    <span class="n">series_argument</span> <span class="o">=</span> <span class="n">ImplicitSeries</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">var_start_end_x</span><span class="p">,</span> <span class="n">var_start_end_y</span><span class="p">,</span>
-                                    <span class="n">has_equality</span><span class="p">,</span> <span class="n">use_interval</span><span class="p">,</span> <span class="n">depth</span><span class="p">,</span>
-                                    <span class="n">nb_of_points</span><span class="p">)</span>
-    <span class="n">show</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;show&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-
-    <span class="c">#set the x and y limits</span>
-    <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;xlim&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">var_start_end_x</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-    <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;ylim&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">y</span><span class="p">)</span> <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">var_start_end_y</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="n">Plot</span><span class="p">(</span><span class="n">series_argument</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">show</span><span class="p">:</span>
-        <span class="n">p</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
-    <span class="k">return</span> <span class="n">p</span></div>
-</pre></div>
-
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-  <h1>Source code for sympy.polys.agca.homomorphisms</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Computations with homomorphisms of modules and rings.</span>
-
-<span class="sd">This module implements classes for representing homomorphisms of rings and</span>
-<span class="sd">their modules. Instead of instantiating the classes directly, you should use</span>
-<span class="sd">the function ``homomorphism(from, to, matrix)`` to create homomorphism objects.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.agca.modules</span> <span class="kn">import</span> <span class="p">(</span><span class="n">Module</span><span class="p">,</span> <span class="n">FreeModule</span><span class="p">,</span> <span class="n">QuotientModule</span><span class="p">,</span>
-    <span class="n">SubModule</span><span class="p">,</span> <span class="n">SubQuotientModule</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="n">CoercionFailed</span>
-
-<span class="c"># The main computational task for module homomorphisms is kernels.</span>
-<span class="c"># For this reason, the concrete classes are organised by domain module type.</span>
-
-
-<div class="viewcode-block" id="ModuleHomomorphism"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.homomorphisms.ModuleHomomorphism">[docs]</a><span class="k">class</span> <span class="nc">ModuleHomomorphism</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Abstract base class for module homomoprhisms. Do not instantiate.</span>
-
-<span class="sd">    Instead, use the ``homomorphism`` function:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import QQ, homomorphism</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">    &gt;&gt;&gt; homomorphism(F, F, [[1, 0], [0, 1]])</span>
-<span class="sd">    [1, 0]</span>
-<span class="sd">    [0, 1] : QQ[x]**2 -&gt; QQ[x]**2</span>
-
-<span class="sd">    Attributes:</span>
-
-<span class="sd">    - ring - the ring over which we are considering modules</span>
-<span class="sd">    - domain - the domain module</span>
-<span class="sd">    - codomain - the codomain module</span>
-<span class="sd">    - _ker - cached kernel</span>
-<span class="sd">    - _img - cachd image</span>
-
-<span class="sd">    Non-implemented methods:</span>
-
-<span class="sd">    - _kernel</span>
-<span class="sd">    - _image</span>
-<span class="sd">    - _restrict_domain</span>
-<span class="sd">    - _restrict_codomain</span>
-<span class="sd">    - _quotient_domain</span>
-<span class="sd">    - _quotient_codomain</span>
-<span class="sd">    - _apply</span>
-<span class="sd">    - _mul_scalar</span>
-<span class="sd">    - _compose</span>
-<span class="sd">    - _add</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">codomain</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">Module</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Source must be a module, got </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">domain</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">codomain</span><span class="p">,</span> <span class="n">Module</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Target must be a module, got </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">codomain</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">domain</span><span class="o">.</span><span class="n">ring</span> <span class="o">!=</span> <span class="n">codomain</span><span class="o">.</span><span class="n">ring</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Source and codomain must be over same ring, &#39;</span>
-                             <span class="s">&#39;got </span><span class="si">%s</span><span class="s"> != </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">codomain</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">domain</span> <span class="o">=</span> <span class="n">domain</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">codomain</span> <span class="o">=</span> <span class="n">codomain</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ring</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">ring</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_ker</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_img</span> <span class="o">=</span> <span class="bp">None</span>
-
-<div class="viewcode-block" id="ModuleHomomorphism.kernel"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.homomorphisms.ModuleHomomorphism.kernel">[docs]</a>    <span class="k">def</span> <span class="nf">kernel</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">        Compute the kernel of ``self``.</span>
-
-<span class="sd">        That is, if ``self`` is the homomorphism `\phi: M \to N`, then compute</span>
-<span class="sd">        `ker(\phi) = \{x \in M | \phi(x) = 0\}`.  This is a submodule of `M`.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import homomorphism, QQ</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; homomorphism(F, F, [[1, 0], [x, 0]]).kernel()</span>
-<span class="sd">        &lt;[x, -1]&gt;</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ker</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_ker</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_kernel</span><span class="p">()</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ker</span>
-</div>
-<div class="viewcode-block" id="ModuleHomomorphism.image"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.homomorphisms.ModuleHomomorphism.image">[docs]</a>    <span class="k">def</span> <span class="nf">image</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">        Compute the image of ``self``.</span>
-
-<span class="sd">        That is, if ``self`` is the homomorphism `\phi: M \to N`, then compute</span>
-<span class="sd">        `im(\phi) = \{\phi(x) | x \in M \}`.  This is a submodule of `N`.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import homomorphism, QQ</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; homomorphism(F, F, [[1, 0], [x, 0]]).image() == F.submodule([1, 0])</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_img</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_img</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_image</span><span class="p">()</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_img</span>
-</div>
-    <span class="k">def</span> <span class="nf">_kernel</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute the kernel of ``self``.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">def</span> <span class="nf">_image</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute the image of ``self``.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">def</span> <span class="nf">_restrict_domain</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sm</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Implementation of domain restriction.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">def</span> <span class="nf">_restrict_codomain</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sm</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Implementation of codomain restriction.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">def</span> <span class="nf">_quotient_domain</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sm</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Implementation of domain quotienting.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">def</span> <span class="nf">_quotient_codomain</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sm</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Implementation of codomain quotienting.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-<div class="viewcode-block" id="ModuleHomomorphism.restrict_domain"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.homomorphisms.ModuleHomomorphism.restrict_domain">[docs]</a>    <span class="k">def</span> <span class="nf">restrict_domain</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sm</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return ``self``, with the domain restricted to ``sm``.</span>
-
-<span class="sd">        Here ``sm`` has to be a submodule of ``self.domain``.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import homomorphism, QQ</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; h = homomorphism(F, F, [[1, 0], [x, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; h</span>
-<span class="sd">        [1, x]</span>
-<span class="sd">        [0, 0] : QQ[x]**2 -&gt; QQ[x]**2</span>
-<span class="sd">        &gt;&gt;&gt; h.restrict_domain(F.submodule([1, 0]))</span>
-<span class="sd">        [1, x]</span>
-<span class="sd">        [0, 0] : &lt;[1, 0]&gt; -&gt; QQ[x]**2</span>
-
-<span class="sd">        This is the same as just composing on the right with the submodule</span>
-<span class="sd">        inclusion:</span>
-
-<span class="sd">        &gt;&gt;&gt; h * F.submodule([1, 0]).inclusion_hom()</span>
-<span class="sd">        [1, x]</span>
-<span class="sd">        [0, 0] : &lt;[1, 0]&gt; -&gt; QQ[x]**2</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="o">.</span><span class="n">is_submodule</span><span class="p">(</span><span class="n">sm</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;sm must be a submodule of </span><span class="si">%s</span><span class="s">, got </span><span class="si">%s</span><span class="s">&#39;</span>
-                             <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">,</span> <span class="n">sm</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">sm</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_restrict_domain</span><span class="p">(</span><span class="n">sm</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="ModuleHomomorphism.restrict_codomain"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.homomorphisms.ModuleHomomorphism.restrict_codomain">[docs]</a>    <span class="k">def</span> <span class="nf">restrict_codomain</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sm</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return ``self``, with codomain restricted to to ``sm``.</span>
-
-<span class="sd">        Here ``sm`` has to be a submodule of ``self.codomain`` containing the</span>
-<span class="sd">        image.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import homomorphism, QQ</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; h = homomorphism(F, F, [[1, 0], [x, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; h</span>
-<span class="sd">        [1, x]</span>
-<span class="sd">        [0, 0] : QQ[x]**2 -&gt; QQ[x]**2</span>
-<span class="sd">        &gt;&gt;&gt; h.restrict_codomain(F.submodule([1, 0]))</span>
-<span class="sd">        [1, x]</span>
-<span class="sd">        [0, 0] : QQ[x]**2 -&gt; &lt;[1, 0]&gt;</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">sm</span><span class="o">.</span><span class="n">is_submodule</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">image</span><span class="p">()):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;the image </span><span class="si">%s</span><span class="s"> must contain sm, got </span><span class="si">%s</span><span class="s">&#39;</span>
-                             <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">image</span><span class="p">(),</span> <span class="n">sm</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">sm</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">codomain</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_restrict_codomain</span><span class="p">(</span><span class="n">sm</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="ModuleHomomorphism.quotient_domain"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.homomorphisms.ModuleHomomorphism.quotient_domain">[docs]</a>    <span class="k">def</span> <span class="nf">quotient_domain</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sm</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return ``self`` with domain replaced by ``domain/sm``.</span>
-
-<span class="sd">        Here ``sm`` must be a submodule of ``self.kernel()``.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import homomorphism, QQ</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; h = homomorphism(F, F, [[1, 0], [x, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; h</span>
-<span class="sd">        [1, x]</span>
-<span class="sd">        [0, 0] : QQ[x]**2 -&gt; QQ[x]**2</span>
-<span class="sd">        &gt;&gt;&gt; h.quotient_domain(F.submodule([-x, 1]))</span>
-<span class="sd">        [1, x]</span>
-<span class="sd">        [0, 0] : QQ[x]**2/&lt;[-x, 1]&gt; -&gt; QQ[x]**2</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">kernel</span><span class="p">()</span><span class="o">.</span><span class="n">is_submodule</span><span class="p">(</span><span class="n">sm</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;kernel </span><span class="si">%s</span><span class="s"> must contain sm, got </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span>
-                             <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">kernel</span><span class="p">(),</span> <span class="n">sm</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">sm</span><span class="o">.</span><span class="n">is_zero</span><span class="p">():</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_quotient_domain</span><span class="p">(</span><span class="n">sm</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="ModuleHomomorphism.quotient_codomain"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.homomorphisms.ModuleHomomorphism.quotient_codomain">[docs]</a>    <span class="k">def</span> <span class="nf">quotient_codomain</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sm</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return ``self`` with codomain replaced by ``codomain/sm``.</span>
-
-<span class="sd">        Here ``sm`` must be a submodule of ``self.codomain``.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import homomorphism, QQ</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; h = homomorphism(F, F, [[1, 0], [x, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; h</span>
-<span class="sd">        [1, x]</span>
-<span class="sd">        [0, 0] : QQ[x]**2 -&gt; QQ[x]**2</span>
-<span class="sd">        &gt;&gt;&gt; h.quotient_codomain(F.submodule([1, 1]))</span>
-<span class="sd">        [1, x]</span>
-<span class="sd">        [0, 0] : QQ[x]**2 -&gt; QQ[x]**2/&lt;[1, 1]&gt;</span>
-
-<span class="sd">        This is the same as composing with the quotient map on the left:</span>
-
-<span class="sd">        &gt;&gt;&gt; (F/[(1, 1)]).quotient_hom() * h</span>
-<span class="sd">        [1, x]</span>
-<span class="sd">        [0, 0] : QQ[x]**2 -&gt; QQ[x]**2/&lt;[1, 1]&gt;</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">codomain</span><span class="o">.</span><span class="n">is_submodule</span><span class="p">(</span><span class="n">sm</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;sm must be a submodule of codomain </span><span class="si">%s</span><span class="s">, got </span><span class="si">%s</span><span class="s">&#39;</span>
-                             <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">codomain</span><span class="p">,</span> <span class="n">sm</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">sm</span><span class="o">.</span><span class="n">is_zero</span><span class="p">():</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_quotient_codomain</span><span class="p">(</span><span class="n">sm</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_apply</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">elem</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Apply ``self`` to ``elem``.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">def</span> <span class="nf">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">elem</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">codomain</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_apply</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">elem</span><span class="p">)))</span>
-
-    <span class="k">def</span> <span class="nf">_compose</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">oth</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Compose ``self`` with ``oth``, that is, return the homomorphism</span>
-<span class="sd">        obtained by first applying then ``self``, then ``oth``.</span>
-
-<span class="sd">        (This method is private since in this syntax, it is non-obvious which</span>
-<span class="sd">        homomorphism is executed first.)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">def</span> <span class="nf">_mul_scalar</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Scalar multiplication. ``c`` is guaranteed in self.ring.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">def</span> <span class="nf">_add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">oth</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Homomorphism addition.</span>
-<span class="sd">        ``oth`` is guaranteed to be a homomorphism with same domain/codomain.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">def</span> <span class="nf">_check_hom</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">oth</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Helper to check that oth is a homomorphism with same domain/codomain.&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">oth</span><span class="p">,</span> <span class="n">ModuleHomomorphism</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="n">oth</span><span class="o">.</span><span class="n">domain</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span> <span class="ow">and</span> <span class="n">oth</span><span class="o">.</span><span class="n">codomain</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">codomain</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">oth</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">oth</span><span class="p">,</span> <span class="n">ModuleHomomorphism</span><span class="p">)</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span> <span class="o">==</span> <span class="n">oth</span><span class="o">.</span><span class="n">codomain</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">oth</span><span class="o">.</span><span class="n">_compose</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mul_scalar</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">oth</span><span class="p">))</span>
-        <span class="k">except</span> <span class="n">CoercionFailed</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="c"># NOTE: _compose will never be called from rmul</span>
-    <span class="n">__rmul__</span> <span class="o">=</span> <span class="n">__mul__</span>
-
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">oth</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mul_scalar</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">oth</span><span class="p">))</span>
-        <span class="k">except</span> <span class="n">CoercionFailed</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="n">__truediv__</span> <span class="o">=</span> <span class="n">__div__</span>
-
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">oth</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_check_hom</span><span class="p">(</span><span class="n">oth</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_add</span><span class="p">(</span><span class="n">oth</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">oth</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_check_hom</span><span class="p">(</span><span class="n">oth</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_add</span><span class="p">(</span><span class="n">oth</span><span class="o">.</span><span class="n">_mul_scalar</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-<div class="viewcode-block" id="ModuleHomomorphism.is_injective"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.homomorphisms.ModuleHomomorphism.is_injective">[docs]</a>    <span class="k">def</span> <span class="nf">is_injective</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if ``self`` is injective.</span>
-
-<span class="sd">        That is, check if the elements of the domain are mapped to the same</span>
-<span class="sd">        codomain element.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import homomorphism, QQ</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; h = homomorphism(F, F, [[1, 0], [x, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; h.is_injective()</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; h.quotient_domain(h.kernel()).is_injective()</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">kernel</span><span class="p">()</span><span class="o">.</span><span class="n">is_zero</span><span class="p">()</span>
-</div>
-<div class="viewcode-block" id="ModuleHomomorphism.is_surjective"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.homomorphisms.ModuleHomomorphism.is_surjective">[docs]</a>    <span class="k">def</span> <span class="nf">is_surjective</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if ``self`` is surjective.</span>
-
-<span class="sd">        That is, check if every element of the codomain has at least one</span>
-<span class="sd">        preimage.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import homomorphism, QQ</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; h = homomorphism(F, F, [[1, 0], [x, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; h.is_surjective()</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; h.restrict_codomain(h.image()).is_surjective()</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">image</span><span class="p">()</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">codomain</span>
-</div>
-<div class="viewcode-block" id="ModuleHomomorphism.is_isomorphism"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.homomorphisms.ModuleHomomorphism.is_isomorphism">[docs]</a>    <span class="k">def</span> <span class="nf">is_isomorphism</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if ``self`` is an isomorphism.</span>
-
-<span class="sd">        That is, check if every element of the codomain has precisely one</span>
-<span class="sd">        preimage. Equivalently, ``self`` is both injective and surjective.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import homomorphism, QQ</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; h = homomorphism(F, F, [[1, 0], [x, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; h = h.restrict_codomain(h.image())</span>
-<span class="sd">        &gt;&gt;&gt; h.is_isomorphism()</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; h.quotient_domain(h.kernel()).is_isomorphism()</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_injective</span><span class="p">()</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_surjective</span><span class="p">()</span>
-</div>
-<div class="viewcode-block" id="ModuleHomomorphism.is_zero"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.homomorphisms.ModuleHomomorphism.is_zero">[docs]</a>    <span class="k">def</span> <span class="nf">is_zero</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if ``self`` is a zero morphism.</span>
-
-<span class="sd">        That is, check if every element of the domain is mapped to zero</span>
-<span class="sd">        under self.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import homomorphism, QQ</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; h = homomorphism(F, F, [[1, 0], [x, 0]])</span>
-<span class="sd">        &gt;&gt;&gt; h.is_zero()</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; h.restrict_domain(F.submodule()).is_zero()</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; h.quotient_codomain(h.image()).is_zero()</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">image</span><span class="p">()</span><span class="o">.</span><span class="n">is_zero</span><span class="p">()</span>
-</div>
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">oth</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="bp">self</span> <span class="o">-</span> <span class="n">oth</span><span class="p">)</span><span class="o">.</span><span class="n">is_zero</span><span class="p">()</span>
-        <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">oth</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="p">(</span><span class="bp">self</span> <span class="o">==</span> <span class="n">oth</span><span class="p">)</span>
-
-</div>
-<span class="k">class</span> <span class="nc">MatrixHomomorphism</span><span class="p">(</span><span class="n">ModuleHomomorphism</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Helper class for all homomoprhisms which are expressed via a matrix.</span>
-
-<span class="sd">    That is, for such homomorphisms ``domain`` is contained in a module</span>
-<span class="sd">    generated by finitely many elements `e_1, \dots, e_n`, so that the</span>
-<span class="sd">    homomorphism is determined uniquely by its action on the `e_i`. It</span>
-<span class="sd">    can thus be represented as a vector of elements of the codomain module,</span>
-<span class="sd">    or potentially a supermodule of the codomain module</span>
-<span class="sd">    (and hence conventionally as a matrix, if there is a similar interpretation</span>
-<span class="sd">    for elements of the codomain module).</span>
-
-<span class="sd">    Note that this class does *not* assume that the `e_i` freely generate a</span>
-<span class="sd">    submodule, nor that ``domain`` is even all of this submodule. It exists</span>
-<span class="sd">    only to unify the interface.</span>
-
-<span class="sd">    Do not instantiate.</span>
-
-<span class="sd">    Attributes:</span>
-
-<span class="sd">    - matrix - the list of images determining the homomorphism.</span>
-<span class="sd">    NOTE: the elements of matrix belong to either self.codomain or</span>
-<span class="sd">          self.codomain.container</span>
-
-<span class="sd">    Still non-implemented methods:</span>
-
-<span class="sd">    - kernel</span>
-<span class="sd">    - _apply</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">codomain</span><span class="p">,</span> <span class="n">matrix</span><span class="p">):</span>
-        <span class="n">ModuleHomomorphism</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">codomain</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">matrix</span><span class="p">)</span> <span class="o">!=</span> <span class="n">domain</span><span class="o">.</span><span class="n">rank</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Need to provide </span><span class="si">%s</span><span class="s"> elements, got </span><span class="si">%s</span><span class="s">&#39;</span>
-                             <span class="o">%</span> <span class="p">(</span><span class="n">domain</span><span class="o">.</span><span class="n">rank</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">matrix</span><span class="p">)))</span>
-
-        <span class="n">converter</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">codomain</span><span class="o">.</span><span class="n">convert</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">codomain</span><span class="p">,</span> <span class="p">(</span><span class="n">SubModule</span><span class="p">,</span> <span class="n">SubQuotientModule</span><span class="p">)):</span>
-            <span class="n">converter</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">codomain</span><span class="o">.</span><span class="n">container</span><span class="o">.</span><span class="n">convert</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">matrix</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">converter</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">matrix</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sympy_matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Helper function which returns a sympy matrix ``self.matrix``.&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">codomain</span><span class="p">,</span> <span class="p">(</span><span class="n">QuotientModule</span><span class="p">,</span> <span class="n">SubQuotientModule</span><span class="p">)):</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">data</span>
-        <span class="k">return</span> <span class="n">Matrix</span><span class="p">([[</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">y</span><span class="p">)</span> <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">c</span><span class="p">(</span><span class="n">x</span><span class="p">)]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">matrix</span><span class="p">])</span><span class="o">.</span><span class="n">T</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">lines</span> <span class="o">=</span> <span class="nb">repr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_sympy_matrix</span><span class="p">())</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">)</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="s">&quot; : </span><span class="si">%s</span><span class="s"> -&gt; </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">codomain</span><span class="p">)</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="s">&#39; &#39;</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">lines</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">//</span> <span class="mi">2</span><span class="p">):</span>
-            <span class="n">lines</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+=</span> <span class="n">s</span>
-        <span class="n">lines</span><span class="p">[</span><span class="n">n</span> <span class="o">//</span> <span class="mi">2</span><span class="p">]</span> <span class="o">+=</span> <span class="n">t</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="o">//</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-            <span class="n">lines</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+=</span> <span class="n">s</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">lines</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_restrict_domain</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sm</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Implementation of domain restriction.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">SubModuleHomomorphism</span><span class="p">(</span><span class="n">sm</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">codomain</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">matrix</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_restrict_codomain</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sm</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Implementation of codomain restriction.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">,</span> <span class="n">sm</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">matrix</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_quotient_domain</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sm</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Implementation of domain quotienting.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="o">/</span><span class="n">sm</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">codomain</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">matrix</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_quotient_codomain</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sm</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Implementation of codomain quotienting.&quot;&quot;&quot;</span>
-        <span class="n">Q</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">codomain</span><span class="o">/</span><span class="n">sm</span>
-        <span class="n">converter</span> <span class="o">=</span> <span class="n">Q</span><span class="o">.</span><span class="n">convert</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">codomain</span><span class="p">,</span> <span class="n">SubModule</span><span class="p">):</span>
-            <span class="n">converter</span> <span class="o">=</span> <span class="n">Q</span><span class="o">.</span><span class="n">container</span><span class="o">.</span><span class="n">convert</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">codomain</span><span class="o">/</span><span class="n">sm</span><span class="p">,</span>
-            <span class="p">[</span><span class="n">converter</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">matrix</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">oth</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">codomain</span><span class="p">,</span>
-                              <span class="p">[</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">matrix</span><span class="p">,</span> <span class="n">oth</span><span class="o">.</span><span class="n">matrix</span><span class="p">)])</span>
-
-    <span class="k">def</span> <span class="nf">_mul_scalar</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">codomain</span><span class="p">,</span> <span class="p">[</span><span class="n">c</span><span class="o">*</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">matrix</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_compose</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">oth</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">,</span> <span class="n">oth</span><span class="o">.</span><span class="n">codomain</span><span class="p">,</span> <span class="p">[</span><span class="n">oth</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">matrix</span><span class="p">])</span>
-
-
-<span class="k">class</span> <span class="nc">FreeModuleHomomorphism</span><span class="p">(</span><span class="n">MatrixHomomorphism</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Concrete class for homomorphisms with domain a free module or a quotient</span>
-<span class="sd">    thereof.</span>
-
-<span class="sd">    Do not instantiate; the constructor does not check that your data is well</span>
-<span class="sd">    defined. Use the ``homomorphism`` function instead:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import QQ, homomorphism</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">    &gt;&gt;&gt; homomorphism(F, F, [[1, 0], [0, 1]])</span>
-<span class="sd">    [1, 0]</span>
-<span class="sd">    [0, 1] : QQ[x]**2 -&gt; QQ[x]**2</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_apply</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">elem</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">,</span> <span class="n">QuotientModule</span><span class="p">):</span>
-            <span class="n">elem</span> <span class="o">=</span> <span class="n">elem</span><span class="o">.</span><span class="n">data</span>
-        <span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="n">x</span> <span class="o">*</span> <span class="n">e</span> <span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">elem</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">matrix</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_image</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">codomain</span><span class="o">.</span><span class="n">submodule</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">matrix</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_kernel</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># The domain is either a free module or a quotient thereof.</span>
-        <span class="c"># It does not matter if it is a quotient, because that won&#39;t increase</span>
-        <span class="c"># the kernel.</span>
-        <span class="c"># Our generators {e_i} are sent to the matrix entries {b_i}.</span>
-        <span class="c"># The kernel is essentially the syzygy module of these {b_i}.</span>
-        <span class="n">syz</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">image</span><span class="p">()</span><span class="o">.</span><span class="n">syzygy_module</span><span class="p">()</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="o">.</span><span class="n">submodule</span><span class="p">(</span><span class="o">*</span><span class="n">syz</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">SubModuleHomomorphism</span><span class="p">(</span><span class="n">MatrixHomomorphism</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Concrete class for homomorphism with domain a submodule of a free module</span>
-<span class="sd">    or a quotient thereof.</span>
-
-<span class="sd">    Do not instantiate; the constructor does not check that your data is well</span>
-<span class="sd">    defined. Use the ``homomorphism`` function instead:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import QQ, homomorphism</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; M = QQ[x].free_module(2)*x</span>
-<span class="sd">    &gt;&gt;&gt; homomorphism(M, M, [[1, 0], [0, 1]])</span>
-<span class="sd">    [1, 0]</span>
-<span class="sd">    [0, 1] : &lt;[x, 0], [0, x]&gt; -&gt; &lt;[x, 0], [0, x]&gt;</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_apply</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">elem</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">,</span> <span class="n">SubQuotientModule</span><span class="p">):</span>
-            <span class="n">elem</span> <span class="o">=</span> <span class="n">elem</span><span class="o">.</span><span class="n">data</span>
-        <span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="n">x</span> <span class="o">*</span> <span class="n">e</span> <span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">elem</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">matrix</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_image</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">codomain</span><span class="o">.</span><span class="n">submodule</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="bp">self</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="o">.</span><span class="n">gens</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_kernel</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">syz</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">image</span><span class="p">()</span><span class="o">.</span><span class="n">syzygy_module</span><span class="p">()</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="o">.</span><span class="n">submodule</span><span class="p">(</span>
-            <span class="o">*</span><span class="p">[</span><span class="nb">sum</span><span class="p">(</span><span class="n">xi</span><span class="o">*</span><span class="n">gi</span> <span class="k">for</span> <span class="n">xi</span><span class="p">,</span> <span class="n">gi</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="o">.</span><span class="n">gens</span><span class="p">))</span>
-              <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">syz</span><span class="o">.</span><span class="n">gens</span><span class="p">])</span>
-
-
-<div class="viewcode-block" id="homomorphism"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.homomorphisms.homomorphism">[docs]</a><span class="k">def</span> <span class="nf">homomorphism</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">codomain</span><span class="p">,</span> <span class="n">matrix</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a homomorphism object.</span>
-
-<span class="sd">    This function tries to build a homomorphism from ``domain`` to ``codomain``</span>
-<span class="sd">    via the matrix ``matrix``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import QQ, homomorphism</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; R = QQ[x]</span>
-<span class="sd">    &gt;&gt;&gt; T = R.free_module(2)</span>
-
-<span class="sd">    If ``domain`` is a free module generated by `e_1, \dots, e_n`, then</span>
-<span class="sd">    ``matrix`` should be an n-element iterable `(b_1, \dots, b_n)` where</span>
-<span class="sd">    the `b_i` are elements of ``codomain``. The constructed homomorphism is the</span>
-<span class="sd">    unique homomorphism sending `e_i` to `b_i`.</span>
-
-<span class="sd">    &gt;&gt;&gt; F = R.free_module(2)</span>
-<span class="sd">    &gt;&gt;&gt; h = homomorphism(F, T, [[1, x], [x**2, 0]])</span>
-<span class="sd">    &gt;&gt;&gt; h</span>
-<span class="sd">    [1, x**2]</span>
-<span class="sd">    [x,    0] : QQ[x]**2 -&gt; QQ[x]**2</span>
-<span class="sd">    &gt;&gt;&gt; h([1, 0])</span>
-<span class="sd">    [1, x]</span>
-<span class="sd">    &gt;&gt;&gt; h([0, 1])</span>
-<span class="sd">    [x**2, 0]</span>
-<span class="sd">    &gt;&gt;&gt; h([1, 1])</span>
-<span class="sd">    [x**2 + 1, x]</span>
-
-<span class="sd">    If ``domain`` is a submodule of a free module, them ``matrix`` determines</span>
-<span class="sd">    a homomoprhism from the containing free module to ``codomain``, and the</span>
-<span class="sd">    homomorphism returned is obtained by restriction to ``domain``.</span>
-
-<span class="sd">    &gt;&gt;&gt; S = F.submodule([1, 0], [0, x])</span>
-<span class="sd">    &gt;&gt;&gt; homomorphism(S, T, [[1, x], [x**2, 0]])</span>
-<span class="sd">    [1, x**2]</span>
-<span class="sd">    [x,    0] : &lt;[1, 0], [0, x]&gt; -&gt; QQ[x]**2</span>
-
-<span class="sd">    If ``domain`` is a (sub)quotient `N/K`, then ``matrix`` determines a</span>
-<span class="sd">    homomorphism from `N` to ``codomain``. If the kernel contains `K`, this</span>
-<span class="sd">    homomorphism descends to ``domain`` and is returned; otherwise an exception</span>
-<span class="sd">    is raised.</span>
-
-<span class="sd">    &gt;&gt;&gt; homomorphism(S/[(1, 0)], T, [0, [x**2, 0]])</span>
-<span class="sd">    [0, x**2]</span>
-<span class="sd">    [0,    0] : &lt;[1, 0] + &lt;[1, 0]&gt;, [0, x] + &lt;[1, 0]&gt;, [1, 0] + &lt;[1, 0]&gt;&gt; -&gt; QQ[x]**2</span>
-<span class="sd">    &gt;&gt;&gt; homomorphism(S/[(0, x)], T, [0, [x**2, 0]])</span>
-<span class="sd">    Traceback (most recent call last):</span>
-<span class="sd">    ...</span>
-<span class="sd">    ValueError: kernel &lt;[1, 0], [0, 0]&gt; must contain sm, got &lt;[0,x]&gt;</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">freepres</span><span class="p">(</span><span class="n">module</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return a tuple ``(F, S, Q, c)`` where ``F`` is a free module, ``S`` is a</span>
-<span class="sd">        submodule of ``F``, and ``Q`` a submodule of ``S``, such that</span>
-<span class="sd">        ``module = S/Q``, and ``c`` is a conversion function.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">module</span><span class="p">,</span> <span class="n">FreeModule</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">module</span><span class="p">,</span> <span class="n">module</span><span class="p">,</span> <span class="n">module</span><span class="o">.</span><span class="n">submodule</span><span class="p">(),</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">module</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">module</span><span class="p">,</span> <span class="n">QuotientModule</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">module</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">module</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">module</span><span class="o">.</span><span class="n">killed_module</span><span class="p">,</span>
-                    <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">module</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">data</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">module</span><span class="p">,</span> <span class="n">SubQuotientModule</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">module</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">container</span><span class="p">,</span> <span class="n">module</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">module</span><span class="o">.</span><span class="n">killed_module</span><span class="p">,</span>
-                    <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">module</span><span class="o">.</span><span class="n">container</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">data</span><span class="p">)</span>
-        <span class="c"># an ordinary submodule</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">module</span><span class="o">.</span><span class="n">container</span><span class="p">,</span> <span class="n">module</span><span class="p">,</span> <span class="n">module</span><span class="o">.</span><span class="n">submodule</span><span class="p">(),</span>
-                <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">module</span><span class="o">.</span><span class="n">container</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-
-    <span class="n">SF</span><span class="p">,</span> <span class="n">SS</span><span class="p">,</span> <span class="n">SQ</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">freepres</span><span class="p">(</span><span class="n">domain</span><span class="p">)</span>
-    <span class="n">TF</span><span class="p">,</span> <span class="n">TS</span><span class="p">,</span> <span class="n">TQ</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">freepres</span><span class="p">(</span><span class="n">codomain</span><span class="p">)</span>
-    <span class="c"># NOTE this is probably a bit inefficient (redundant checks)</span>
-    <span class="k">return</span> <span class="n">FreeModuleHomomorphism</span><span class="p">(</span><span class="n">SF</span><span class="p">,</span> <span class="n">TF</span><span class="p">,</span> <span class="p">[</span><span class="n">c</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">matrix</span><span class="p">]</span>
-         <span class="p">)</span><span class="o">.</span><span class="n">restrict_domain</span><span class="p">(</span><span class="n">SS</span><span class="p">)</span><span class="o">.</span><span class="n">restrict_codomain</span><span class="p">(</span><span class="n">TS</span>
-         <span class="p">)</span><span class="o">.</span><span class="n">quotient_codomain</span><span class="p">(</span><span class="n">TQ</span><span class="p">)</span><span class="o">.</span><span class="n">quotient_domain</span><span class="p">(</span><span class="n">SQ</span><span class="p">)</span></div>
-</pre></div>
-
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-  <h1>Source code for sympy.polys.agca.ideals</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Computations with ideals of polynomial rings.&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="n">CoercionFailed</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-
-<div class="viewcode-block" id="Ideal"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.ideals.Ideal">[docs]</a><span class="k">class</span> <span class="nc">Ideal</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Abstract base class for ideals.</span>
-
-<span class="sd">    Do not instantiate - use explicit constructors in the ring class instead:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; QQ[x].ideal(x+1)</span>
-<span class="sd">    &lt;x + 1&gt;</span>
-
-<span class="sd">    Attributes</span>
-
-<span class="sd">    - ring - the ring this ideal belongs to</span>
-
-<span class="sd">    Non-implemented methods:</span>
-
-<span class="sd">    - _contains_elem</span>
-<span class="sd">    - _contains_ideal</span>
-<span class="sd">    - _quotient</span>
-<span class="sd">    - _intersect</span>
-<span class="sd">    - _union</span>
-<span class="sd">    - _product</span>
-<span class="sd">    - is_whole_ring</span>
-<span class="sd">    - is_zero</span>
-<span class="sd">    - is_prime, is_maximal, is_primary, is_radical</span>
-<span class="sd">    - is_principal</span>
-<span class="sd">    - height, depth</span>
-<span class="sd">    - radical</span>
-
-<span class="sd">    Methods that likely should be overridden in subclasses:</span>
-
-<span class="sd">    - reduce_element</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_contains_elem</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Implementation of element containment.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">def</span> <span class="nf">_contains_ideal</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">I</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Implementation of ideal containment.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">def</span> <span class="nf">_quotient</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">J</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Implementation of ideal quotient.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">def</span> <span class="nf">_intersect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">J</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Implementation of ideal intersection.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-<div class="viewcode-block" id="Ideal.is_whole_ring"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.ideals.Ideal.is_whole_ring">[docs]</a>    <span class="k">def</span> <span class="nf">is_whole_ring</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return True if ``self`` is the whole ring.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-</div>
-<div class="viewcode-block" id="Ideal.is_zero"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.ideals.Ideal.is_zero">[docs]</a>    <span class="k">def</span> <span class="nf">is_zero</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return True if ``self`` is the zero ideal.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-</div>
-    <span class="k">def</span> <span class="nf">_equals</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">J</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Implementation of ideal equality.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_contains_ideal</span><span class="p">(</span><span class="n">J</span><span class="p">)</span> <span class="ow">and</span> <span class="n">J</span><span class="o">.</span><span class="n">_contains_ideal</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Ideal.is_prime"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.ideals.Ideal.is_prime">[docs]</a>    <span class="k">def</span> <span class="nf">is_prime</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return True if ``self`` is a prime ideal.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-</div>
-<div class="viewcode-block" id="Ideal.is_maximal"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.ideals.Ideal.is_maximal">[docs]</a>    <span class="k">def</span> <span class="nf">is_maximal</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return True if ``self`` is a maximal ideal.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-</div>
-<div class="viewcode-block" id="Ideal.is_radical"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.ideals.Ideal.is_radical">[docs]</a>    <span class="k">def</span> <span class="nf">is_radical</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return True if ``self`` is a radical ideal.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-</div>
-<div class="viewcode-block" id="Ideal.is_primary"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.ideals.Ideal.is_primary">[docs]</a>    <span class="k">def</span> <span class="nf">is_primary</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return True if ``self`` is a primary ideal.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-</div>
-<div class="viewcode-block" id="Ideal.is_principal"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.ideals.Ideal.is_principal">[docs]</a>    <span class="k">def</span> <span class="nf">is_principal</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return True if ``self`` is a principal ideal.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-</div>
-<div class="viewcode-block" id="Ideal.radical"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.ideals.Ideal.radical">[docs]</a>    <span class="k">def</span> <span class="nf">radical</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute the radical of ``self``.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-</div>
-<div class="viewcode-block" id="Ideal.depth"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.ideals.Ideal.depth">[docs]</a>    <span class="k">def</span> <span class="nf">depth</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute the depth of ``self``.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-</div>
-<div class="viewcode-block" id="Ideal.height"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.ideals.Ideal.height">[docs]</a>    <span class="k">def</span> <span class="nf">height</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute the height of ``self``.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="c"># TODO more</span>
-
-    <span class="c"># non-implemented methods end here</span>
-</div>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ring</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ring</span> <span class="o">=</span> <span class="n">ring</span>
-
-    <span class="k">def</span> <span class="nf">_check_ideal</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">J</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Helper to check ``J`` is an ideal of our ring.&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">J</span><span class="p">,</span> <span class="n">Ideal</span><span class="p">)</span> <span class="ow">or</span> <span class="n">J</span><span class="o">.</span><span class="n">ring</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&#39;J must be an ideal of </span><span class="si">%s</span><span class="s">, got </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="p">,</span> <span class="n">J</span><span class="p">))</span>
-
-<div class="viewcode-block" id="Ideal.contains"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.ideals.Ideal.contains">[docs]</a>    <span class="k">def</span> <span class="nf">contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">elem</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if ``elem`` is an element of this ideal.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; QQ[x].ideal(x+1, x-1).contains(3)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; QQ[x].ideal(x**2, x**3).contains(x)</span>
-<span class="sd">        False</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_contains_elem</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">elem</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="Ideal.subset"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.ideals.Ideal.subset">[docs]</a>    <span class="k">def</span> <span class="nf">subset</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns True if ``other`` is is a subset of ``self``.</span>
-
-<span class="sd">        Here ``other`` may be an ideal.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; I = QQ[x].ideal(x+1)</span>
-<span class="sd">        &gt;&gt;&gt; I.subset([x**2 - 1, x**2 + 2*x + 1])</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; I.subset([x**2 + 1, x + 1])</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; I.subset(QQ[x].ideal(x**2 - 1))</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Ideal</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_contains_ideal</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_contains_elem</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">other</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Ideal.quotient"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.ideals.Ideal.quotient">[docs]</a>    <span class="k">def</span> <span class="nf">quotient</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">J</span><span class="p">,</span> <span class="o">**</span><span class="n">opts</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">        Compute the ideal quotient of ``self`` by ``J``.</span>
-
-<span class="sd">        That is, if ``self`` is the ideal `I`, compute the set</span>
-<span class="sd">        `I : J = \{x \in R | xJ \subset I \}`.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; R = QQ[x, y]</span>
-<span class="sd">        &gt;&gt;&gt; R.ideal(x*y).quotient(R.ideal(x))</span>
-<span class="sd">        &lt;y&gt;</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_check_ideal</span><span class="p">(</span><span class="n">J</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_quotient</span><span class="p">(</span><span class="n">J</span><span class="p">,</span> <span class="o">**</span><span class="n">opts</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Ideal.intersect"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.ideals.Ideal.intersect">[docs]</a>    <span class="k">def</span> <span class="nf">intersect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">J</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Compute the intersection of self with ideal J.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; R = QQ[x, y]</span>
-<span class="sd">        &gt;&gt;&gt; R.ideal(x).intersect(R.ideal(y))</span>
-<span class="sd">        &lt;x*y&gt;</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_check_ideal</span><span class="p">(</span><span class="n">J</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_intersect</span><span class="p">(</span><span class="n">J</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Ideal.saturate"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.ideals.Ideal.saturate">[docs]</a>    <span class="k">def</span> <span class="nf">saturate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">J</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">        Compute the ideal saturation of ``self`` by ``J``.</span>
-
-<span class="sd">        That is, if ``self`` is the ideal `I`, compute the set</span>
-<span class="sd">        `I : J^\infty = \{x \in R | xJ^n \subset I \text{ for some } n\}`.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-        <span class="c"># Note this can be implemented using repeated quotient</span>
-</div>
-<div class="viewcode-block" id="Ideal.union"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.ideals.Ideal.union">[docs]</a>    <span class="k">def</span> <span class="nf">union</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">J</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Compute the ideal generated by the union of ``self`` and ``J``.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; QQ[x].ideal(x**2 - 1).union(QQ[x].ideal((x+1)**2)) == QQ[x].ideal(x+1)</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_check_ideal</span><span class="p">(</span><span class="n">J</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_union</span><span class="p">(</span><span class="n">J</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Ideal.product"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.ideals.Ideal.product">[docs]</a>    <span class="k">def</span> <span class="nf">product</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">J</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Compute the ideal product of ``self`` and ``J``.</span>
-
-<span class="sd">        That is, compute the ideal generated by products `xy`, for `x` an element</span>
-<span class="sd">        of ``self`` and `y \in J`.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; QQ[x, y].ideal(x).product(QQ[x, y].ideal(y))</span>
-<span class="sd">        &lt;x*y&gt;</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_check_ideal</span><span class="p">(</span><span class="n">J</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_product</span><span class="p">(</span><span class="n">J</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Ideal.reduce_element"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.ideals.Ideal.reduce_element">[docs]</a>    <span class="k">def</span> <span class="nf">reduce_element</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Reduce the element ``x`` of our ring modulo the ideal ``self``.</span>
-
-<span class="sd">        Here &quot;reduce&quot; has no specific meaning: it could return a unique normal</span>
-<span class="sd">        form, simplify the expression a bit, or just do nothing.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">x</span>
-</div>
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Ideal</span><span class="p">):</span>
-            <span class="n">R</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">quotient_ring</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">R</span><span class="o">.</span><span class="n">dtype</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">e</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">R</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">dtype</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">R</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">R</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_check_ideal</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-
-    <span class="n">__radd__</span> <span class="o">=</span> <span class="n">__add__</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Ideal</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">e</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">CoercionFailed</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">NotImplemented</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_check_ideal</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">product</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-
-    <span class="n">__rmul__</span> <span class="o">=</span> <span class="n">__mul__</span>
-
-    <span class="k">def</span> <span class="nf">__pow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">exp</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">exp</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-        <span class="c"># TODO exponentiate by squaring</span>
-        <span class="k">return</span> <span class="nb">reduce</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="p">[</span><span class="bp">self</span><span class="p">]</span><span class="o">*</span><span class="n">exp</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Ideal</span><span class="p">)</span> <span class="ow">or</span> <span class="n">e</span><span class="o">.</span><span class="n">ring</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_equals</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="p">(</span><span class="bp">self</span> <span class="o">==</span> <span class="n">e</span><span class="p">)</span>
-
-</div>
-<span class="k">class</span> <span class="nc">ModuleImplementedIdeal</span><span class="p">(</span><span class="n">Ideal</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Ideal implementation relying on the modules code.</span>
-
-<span class="sd">    Attributes:</span>
-
-<span class="sd">    - _module - the underlying module</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ring</span><span class="p">,</span> <span class="n">module</span><span class="p">):</span>
-        <span class="n">Ideal</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ring</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_module</span> <span class="o">=</span> <span class="n">module</span>
-
-    <span class="k">def</span> <span class="nf">_contains_elem</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_module</span><span class="o">.</span><span class="n">contains</span><span class="p">([</span><span class="n">x</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_contains_ideal</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">J</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">J</span><span class="p">,</span> <span class="n">ModuleImplementedIdeal</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_module</span><span class="o">.</span><span class="n">is_submodule</span><span class="p">(</span><span class="n">J</span><span class="o">.</span><span class="n">_module</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_intersect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">J</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">J</span><span class="p">,</span> <span class="n">ModuleImplementedIdeal</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_module</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="n">J</span><span class="o">.</span><span class="n">_module</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_quotient</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">J</span><span class="p">,</span> <span class="o">**</span><span class="n">opts</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">J</span><span class="p">,</span> <span class="n">ModuleImplementedIdeal</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_module</span><span class="o">.</span><span class="n">module_quotient</span><span class="p">(</span><span class="n">J</span><span class="o">.</span><span class="n">_module</span><span class="p">,</span> <span class="o">**</span><span class="n">opts</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_union</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">J</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">J</span><span class="p">,</span> <span class="n">ModuleImplementedIdeal</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_module</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="n">J</span><span class="o">.</span><span class="n">_module</span><span class="p">))</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">gens</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return generators for ``self``.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; list(QQ[x, y].ideal(x, y, x**2 + y).gens)</span>
-<span class="sd">        [x, y, x**2 + y]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_module</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">is_zero</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if ``self`` is the zero ideal.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; QQ[x].ideal(x).is_zero()</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; QQ[x].ideal().is_zero()</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_module</span><span class="o">.</span><span class="n">is_zero</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">is_whole_ring</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if ``self`` is the whole ring, i.e. one generator is a unit.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ, ilex</span>
-<span class="sd">        &gt;&gt;&gt; QQ[x].ideal(x).is_whole_ring()</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; QQ[x].ideal(3).is_whole_ring()</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; QQ.poly_ring(x, order=ilex).ideal(2 + x).is_whole_ring()</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_module</span><span class="o">.</span><span class="n">is_full_module</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sstr</span>
-        <span class="k">return</span> <span class="s">&#39;&lt;&#39;</span> <span class="o">+</span> <span class="s">&#39;,&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">sstr</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="p">[</span><span class="n">x</span><span class="p">]</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_module</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39;&gt;&#39;</span>
-
-    <span class="c"># NOTE this is the only method using the fact that the module is a SubModule</span>
-    <span class="k">def</span> <span class="nf">_product</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">J</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">J</span><span class="p">,</span> <span class="n">ModuleImplementedIdeal</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_module</span><span class="o">.</span><span class="n">submodule</span><span class="p">(</span>
-            <span class="o">*</span><span class="p">[[</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">]</span> <span class="k">for</span> <span class="p">[</span><span class="n">x</span><span class="p">]</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_module</span><span class="o">.</span><span class="n">gens</span> <span class="k">for</span> <span class="p">[</span><span class="n">y</span><span class="p">]</span> <span class="ow">in</span> <span class="n">J</span><span class="o">.</span><span class="n">_module</span><span class="o">.</span><span class="n">gens</span><span class="p">]))</span>
-
-    <span class="k">def</span> <span class="nf">in_terms_of_generators</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Express ``e`` in terms of the generators of ``self``.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; I = QQ[x].ideal(x**2 + 1, x)</span>
-<span class="sd">        &gt;&gt;&gt; I.in_terms_of_generators(1)</span>
-<span class="sd">        [1, -x]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_module</span><span class="o">.</span><span class="n">in_terms_of_generators</span><span class="p">([</span><span class="n">e</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">reduce_element</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_module</span><span class="o">.</span><span class="n">reduce_element</span><span class="p">([</span><span class="n">x</span><span class="p">],</span> <span class="o">**</span><span class="n">options</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.polys.agca.modules</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Computations with modules over polynomial rings.</span>
-
-<span class="sd">This module implements various classes that encapsulate groebner basis</span>
-<span class="sd">computations for modules. Most of them should not be instantiated by hand.</span>
-<span class="sd">Instead, use the constructing routines on objects you already have.</span>
-
-<span class="sd">For example, to construct a free module over ``QQ[x, y]``, call</span>
-<span class="sd">``QQ[x, y].free_module(rank)`` instead of the ``FreeModule`` constructor.</span>
-<span class="sd">In fact ``FreeModule`` is an abstract base class that should not be</span>
-<span class="sd">instantiated, the ``free_module`` method instead returns the implementing class</span>
-<span class="sd">``FreeModulePolyRing``.</span>
-
-<span class="sd">In general, the abstract base classes implement most functionality in terms of</span>
-<span class="sd">a few non-implemented methods. The concrete base classes supply only these</span>
-<span class="sd">non-implemented methods. They may also supply new implementations of the</span>
-<span class="sd">convenience methods, for example if there are faster algorithms available.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">copy</span> <span class="kn">import</span> <span class="n">copy</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="n">CoercionFailed</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.monomialtools</span> <span class="kn">import</span> <span class="n">ProductOrder</span><span class="p">,</span> <span class="n">monomial_key</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">Field</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.agca.ideals</span> <span class="kn">import</span> <span class="n">Ideal</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">iterable</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-<span class="c"># TODO</span>
-<span class="c"># - module saturation</span>
-<span class="c"># - module quotient/intersection for quotient rings</span>
-<span class="c"># - free resoltutions / syzygies</span>
-<span class="c"># - finding small/minimal generating sets</span>
-<span class="c"># - ...</span>
-
-<span class="c">##########################################################################</span>
-<span class="c">## Abstract base classes #################################################</span>
-<span class="c">##########################################################################</span>
-
-
-<div class="viewcode-block" id="Module"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.Module">[docs]</a><span class="k">class</span> <span class="nc">Module</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Abstract base class for modules.</span>
-
-<span class="sd">    Do not instantiate - use ring explicit constructors instead:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; QQ[x].free_module(2)</span>
-<span class="sd">    QQ[x]**2</span>
-
-<span class="sd">    Attributes:</span>
-
-<span class="sd">    - dtype - type of elements</span>
-<span class="sd">    - ring - containing ring</span>
-
-<span class="sd">    Non-implemented methods:</span>
-
-<span class="sd">    - submodule</span>
-<span class="sd">    - quotient_module</span>
-<span class="sd">    - is_zero</span>
-<span class="sd">    - is_submodule</span>
-<span class="sd">    - multiply_ideal</span>
-
-<span class="sd">    The method convert likely needs to be changed in subclasses.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ring</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ring</span> <span class="o">=</span> <span class="n">ring</span>
-
-<div class="viewcode-block" id="Module.convert"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.Module.convert">[docs]</a>    <span class="k">def</span> <span class="nf">convert</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">elem</span><span class="p">,</span> <span class="n">M</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Convert ``elem`` into internal representation of this module.</span>
-
-<span class="sd">        If ``M`` is not None, it should be a module containing it.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">elem</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">dtype</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="n">CoercionFailed</span>
-        <span class="k">return</span> <span class="n">elem</span>
-</div>
-<div class="viewcode-block" id="Module.submodule"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.Module.submodule">[docs]</a>    <span class="k">def</span> <span class="nf">submodule</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Generate a submodule.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-</div>
-<div class="viewcode-block" id="Module.quotient_module"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.Module.quotient_module">[docs]</a>    <span class="k">def</span> <span class="nf">quotient_module</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Generate a quotient module.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-</div>
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Module</span><span class="p">):</span>
-            <span class="n">e</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">submodule</span><span class="p">(</span><span class="o">*</span><span class="n">e</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">quotient_module</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-
-    <span class="n">__truediv__</span> <span class="o">=</span> <span class="n">__div__</span>
-
-<div class="viewcode-block" id="Module.contains"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.Module.contains">[docs]</a>    <span class="k">def</span> <span class="nf">contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">elem</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return True if ``elem`` is an element of this module.&quot;&quot;&quot;</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">elem</span><span class="p">)</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">except</span> <span class="n">CoercionFailed</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-</div>
-    <span class="k">def</span> <span class="nf">__contains__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">elem</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">elem</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Module.subset"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.Module.subset">[docs]</a>    <span class="k">def</span> <span class="nf">subset</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns True if ``other`` is is a subset of ``self``.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; F.subset([(1, x), (x, 2)])</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; F.subset([(1/x, x), (x, 2)])</span>
-<span class="sd">        False</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">other</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_submodule</span><span class="p">(</span><span class="n">other</span><span class="p">)</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">is_submodule</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="p">(</span><span class="bp">self</span> <span class="o">==</span> <span class="n">other</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Module.is_zero"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.Module.is_zero">[docs]</a>    <span class="k">def</span> <span class="nf">is_zero</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns True if ``self`` is a zero module.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-</div>
-<div class="viewcode-block" id="Module.is_submodule"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.Module.is_submodule">[docs]</a>    <span class="k">def</span> <span class="nf">is_submodule</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns True if ``other`` is a submodule of ``self``.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-</div>
-<div class="viewcode-block" id="Module.multiply_ideal"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.Module.multiply_ideal">[docs]</a>    <span class="k">def</span> <span class="nf">multiply_ideal</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Multiply ``self`` by the ideal ``other``.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-</div>
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Ideal</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">e</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-            <span class="k">except</span> <span class="p">(</span><span class="n">CoercionFailed</span><span class="p">,</span> <span class="ne">NotImplementedError</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">NotImplemented</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">multiply_ideal</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-
-    <span class="n">__rmul__</span> <span class="o">=</span> <span class="n">__mul__</span>
-
-<div class="viewcode-block" id="Module.identity_hom"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.Module.identity_hom">[docs]</a>    <span class="k">def</span> <span class="nf">identity_hom</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the identity homomorphism on ``self``.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-</div></div>
-<span class="k">class</span> <span class="nc">ModuleElement</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Base class for module element wrappers.</span>
-
-<span class="sd">    Use this class to wrap primitive data types as module elements. It stores</span>
-<span class="sd">    a reference to the containing module, and implements all the arithmetic</span>
-<span class="sd">    operators.</span>
-
-<span class="sd">    Attributes:</span>
-
-<span class="sd">    - module - containing module</span>
-<span class="sd">    - data - internal data</span>
-
-<span class="sd">    Methods that likely need change in subclasses:</span>
-
-<span class="sd">    - add</span>
-<span class="sd">    - mul</span>
-<span class="sd">    - div</span>
-<span class="sd">    - eq</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">module</span><span class="p">,</span> <span class="n">data</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">module</span> <span class="o">=</span> <span class="n">module</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">data</span> <span class="o">=</span> <span class="n">data</span>
-
-    <span class="k">def</span> <span class="nf">add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">d1</span><span class="p">,</span> <span class="n">d2</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Add data ``d1`` and ``d2``.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">d1</span> <span class="o">+</span> <span class="n">d2</span>
-
-    <span class="k">def</span> <span class="nf">mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">d</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Multiply module data ``m`` by coefficient d.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">m</span> <span class="o">*</span> <span class="n">d</span>
-
-    <span class="k">def</span> <span class="nf">div</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">d</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Divide module data ``m`` by coefficient d.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">m</span> <span class="o">/</span> <span class="n">d</span>
-
-    <span class="k">def</span> <span class="nf">eq</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">d1</span><span class="p">,</span> <span class="n">d2</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return true if d1 and d2 represent the same element.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">d1</span> <span class="o">==</span> <span class="n">d2</span>
-
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">om</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">om</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">)</span> <span class="ow">or</span> <span class="n">om</span><span class="o">.</span><span class="n">module</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">module</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">om</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">om</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">CoercionFailed</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">NotImplemented</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">module</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">om</span><span class="o">.</span><span class="n">data</span><span class="p">))</span>
-
-    <span class="n">__radd__</span> <span class="o">=</span> <span class="n">__add__</span>
-
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">module</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="p">,</span>
-                       <span class="bp">self</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)))</span>
-
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">om</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">om</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">)</span> <span class="ow">or</span> <span class="n">om</span><span class="o">.</span><span class="n">module</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">module</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">om</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">om</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">CoercionFailed</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">NotImplemented</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__add__</span><span class="p">(</span><span class="o">-</span><span class="n">om</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__rsub__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">om</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__add__</span><span class="p">(</span><span class="n">om</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">dtype</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">o</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">CoercionFailed</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">NotImplemented</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">module</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">o</span><span class="p">))</span>
-
-    <span class="n">__rmul__</span> <span class="o">=</span> <span class="n">__mul__</span>
-
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">dtype</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">o</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">CoercionFailed</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">NotImplemented</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">module</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">div</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">o</span><span class="p">))</span>
-
-    <span class="n">__truediv__</span> <span class="o">=</span> <span class="n">__div__</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">om</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">om</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">)</span> <span class="ow">or</span> <span class="n">om</span><span class="o">.</span><span class="n">module</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">module</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">om</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">om</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">CoercionFailed</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">eq</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">om</span><span class="o">.</span><span class="n">data</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">om</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="n">om</span><span class="p">)</span>
-
-<span class="c">##########################################################################</span>
-<span class="c">## Free Modules ##########################################################</span>
-<span class="c">##########################################################################</span>
-
-
-<span class="k">class</span> <span class="nc">FreeModuleElement</span><span class="p">(</span><span class="n">ModuleElement</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Element of a free module. Data stored as a tuple.&quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">d1</span><span class="p">,</span> <span class="n">d2</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">d1</span><span class="p">,</span> <span class="n">d2</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">x</span> <span class="o">*</span> <span class="n">p</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">d</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">div</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">x</span> <span class="o">/</span> <span class="n">p</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">d</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sstr</span>
-        <span class="k">return</span> <span class="s">&#39;[&#39;</span> <span class="o">+</span> <span class="s">&#39;, &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">sstr</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39;]&#39;</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">__iter__</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">idx</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="p">[</span><span class="n">idx</span><span class="p">]</span>
-
-
-<div class="viewcode-block" id="FreeModule"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.FreeModule">[docs]</a><span class="k">class</span> <span class="nc">FreeModule</span><span class="p">(</span><span class="n">Module</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Abstract base class for free modules.</span>
-
-<span class="sd">    Additional attributes:</span>
-
-<span class="sd">    - rank - rank of the free module</span>
-
-<span class="sd">    Non-implemented methods:</span>
-
-<span class="sd">    - submodule</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">dtype</span> <span class="o">=</span> <span class="n">FreeModuleElement</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ring</span><span class="p">,</span> <span class="n">rank</span><span class="p">):</span>
-        <span class="n">Module</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ring</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">rank</span> <span class="o">=</span> <span class="n">rank</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">repr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot;**&quot;</span> <span class="o">+</span> <span class="nb">repr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rank</span><span class="p">)</span>
-
-<div class="viewcode-block" id="FreeModule.is_submodule"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.FreeModule.is_submodule">[docs]</a>    <span class="k">def</span> <span class="nf">is_submodule</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns True if ``other`` is a submodule of ``self``.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; M = F.submodule([2, x])</span>
-<span class="sd">        &gt;&gt;&gt; F.is_submodule(F)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; F.is_submodule(M)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; M.is_submodule(F)</span>
-<span class="sd">        False</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">SubModule</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">other</span><span class="o">.</span><span class="n">container</span> <span class="o">==</span> <span class="bp">self</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">FreeModule</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">other</span><span class="o">.</span><span class="n">ring</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">rank</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">rank</span>
-        <span class="k">return</span> <span class="bp">False</span>
-</div>
-<div class="viewcode-block" id="FreeModule.convert"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.FreeModule.convert">[docs]</a>    <span class="k">def</span> <span class="nf">convert</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">elem</span><span class="p">,</span> <span class="n">M</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Convert ``elem`` into the internal representation.</span>
-
-<span class="sd">        This method is called implicitly whenever computations involve elements</span>
-<span class="sd">        not in the internal representation.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; F.convert([1, 0])</span>
-<span class="sd">        [1, 0]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">elem</span><span class="p">,</span> <span class="n">FreeModuleElement</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">elem</span><span class="o">.</span><span class="n">module</span> <span class="ow">is</span> <span class="bp">self</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">elem</span>
-            <span class="k">if</span> <span class="n">elem</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">rank</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rank</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">CoercionFailed</span>
-            <span class="k">return</span> <span class="n">FreeModuleElement</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span>
-                     <span class="nb">tuple</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">elem</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">ring</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">elem</span><span class="o">.</span><span class="n">data</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="n">iterable</span><span class="p">(</span><span class="n">elem</span><span class="p">):</span>
-            <span class="n">tpl</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">elem</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">tpl</span><span class="p">)</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rank</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">CoercionFailed</span>
-            <span class="k">return</span> <span class="n">FreeModuleElement</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">tpl</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">elem</span> <span class="ow">is</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">FreeModuleElement</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="mi">0</span><span class="p">),)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">rank</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">CoercionFailed</span>
-</div>
-<div class="viewcode-block" id="FreeModule.is_zero"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.FreeModule.is_zero">[docs]</a>    <span class="k">def</span> <span class="nf">is_zero</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns True if ``self`` is a zero module.</span>
-
-<span class="sd">        (If, as this implementation assumes, the coefficient ring is not the</span>
-<span class="sd">        zero ring, then this is equivalent to the rank being zero.)</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; QQ[x].free_module(0).is_zero()</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; QQ[x].free_module(1).is_zero()</span>
-<span class="sd">        False</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rank</span> <span class="o">==</span> <span class="mi">0</span>
-</div>
-<div class="viewcode-block" id="FreeModule.basis"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.FreeModule.basis">[docs]</a>    <span class="k">def</span> <span class="nf">basis</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return a set of basis elements.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; QQ[x].free_module(3).basis()</span>
-<span class="sd">        ([1, 0, 0], [0, 1, 0], [0, 0, 1])</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">eye</span>
-        <span class="n">M</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rank</span><span class="p">)</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">row</span><span class="p">(</span><span class="n">i</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rank</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="FreeModule.quotient_module"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.FreeModule.quotient_module">[docs]</a>    <span class="k">def</span> <span class="nf">quotient_module</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">submodule</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return a quotient module.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; M = QQ[x].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; M.quotient_module(M.submodule([1, x], [x, 2]))</span>
-<span class="sd">        QQ[x]**2/&lt;[1, x], [x, 2]&gt;</span>
-
-<span class="sd">        Or more conicisely, using the overloaded division operator:</span>
-
-<span class="sd">        &gt;&gt;&gt; QQ[x].free_module(2) / [[1, x], [x, 2]]</span>
-<span class="sd">        QQ[x]**2/&lt;[1, x], [x, 2]&gt;</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">QuotientModule</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="n">submodule</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="FreeModule.multiply_ideal"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.FreeModule.multiply_ideal">[docs]</a>    <span class="k">def</span> <span class="nf">multiply_ideal</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Multiply ``self`` by the ideal ``other``.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; I = QQ[x].ideal(x)</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; F.multiply_ideal(I)</span>
-<span class="sd">        &lt;[x, 0], [0, x]&gt;</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">submodule</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">basis</span><span class="p">())</span><span class="o">.</span><span class="n">multiply_ideal</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="FreeModule.identity_hom"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.FreeModule.identity_hom">[docs]</a>    <span class="k">def</span> <span class="nf">identity_hom</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the identity homomorphism on ``self``.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; QQ[x].free_module(2).identity_hom()</span>
-<span class="sd">        [1, 0]</span>
-<span class="sd">        [0, 1] : QQ[x]**2 -&gt; QQ[x]**2</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.polys.agca.homomorphisms</span> <span class="kn">import</span> <span class="n">homomorphism</span>
-        <span class="k">return</span> <span class="n">homomorphism</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">basis</span><span class="p">())</span>
-
-</div></div>
-<span class="k">class</span> <span class="nc">FreeModulePolyRing</span><span class="p">(</span><span class="n">FreeModule</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Free module over a generalized polynomial ring.</span>
-
-<span class="sd">    Do not instantiate this, use the constructor method of the ring instead:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">    &gt;&gt;&gt; F = QQ[x].free_module(3)</span>
-<span class="sd">    &gt;&gt;&gt; F</span>
-<span class="sd">    QQ[x]**3</span>
-<span class="sd">    &gt;&gt;&gt; F.contains([x, 1, 0])</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; F.contains([1/x, 0, 1])</span>
-<span class="sd">    False</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ring</span><span class="p">,</span> <span class="n">rank</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.polys.domains.polynomialring</span> <span class="kn">import</span> <span class="n">PolynomialRingBase</span>
-        <span class="n">FreeModule</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ring</span><span class="p">,</span> <span class="n">rank</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ring</span><span class="p">,</span> <span class="n">PolynomialRingBase</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;This implementation only works over &#39;</span>
-                                      <span class="o">+</span> <span class="s">&#39;polynomial rings, got </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">ring</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ring</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">Field</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;Ground domain must be a field, &#39;</span>
-                                      <span class="o">+</span> <span class="s">&#39;got </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">ring</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">submodule</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">opts</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Generate a submodule.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; M = QQ[x, y].free_module(2).submodule([x, x + y])</span>
-<span class="sd">        &gt;&gt;&gt; M</span>
-<span class="sd">        &lt;[x, x + y]&gt;</span>
-<span class="sd">        &gt;&gt;&gt; M.contains([2*x, 2*x + 2*y])</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; M.contains([x, y])</span>
-<span class="sd">        False</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">SubModulePolyRing</span><span class="p">(</span><span class="n">gens</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">opts</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">FreeModuleQuotientRing</span><span class="p">(</span><span class="n">FreeModule</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Free module over a quotient ring.</span>
-
-<span class="sd">    Do not instantiate this, use the constructor method of the ring instead:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">    &gt;&gt;&gt; F = (QQ[x]/[x**2 + 1]).free_module(3)</span>
-<span class="sd">    &gt;&gt;&gt; F</span>
-<span class="sd">    (QQ[x]/&lt;x**2 + 1&gt;)**3</span>
-
-<span class="sd">    Attributes</span>
-
-<span class="sd">    - quot - the quotient module `R^n / IR^n`, where `R/I` is our ring</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ring</span><span class="p">,</span> <span class="n">rank</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">QuotientRing</span>
-        <span class="n">FreeModule</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ring</span><span class="p">,</span> <span class="n">rank</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ring</span><span class="p">,</span> <span class="n">QuotientRing</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;This implementation only works over &#39;</span>
-                             <span class="o">+</span> <span class="s">&#39;quotient rings, got </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">ring</span><span class="p">)</span>
-        <span class="n">F</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rank</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">quot</span> <span class="o">=</span> <span class="n">F</span> <span class="o">/</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">base_ideal</span><span class="o">*</span><span class="n">F</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;(&quot;</span> <span class="o">+</span> <span class="nb">repr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot;)&quot;</span> <span class="o">+</span> <span class="s">&quot;**&quot;</span> <span class="o">+</span> <span class="nb">repr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rank</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">submodule</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">opts</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Generate a submodule.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; M = (QQ[x, y]/[x**2 - y**2]).free_module(2).submodule([x, x + y])</span>
-<span class="sd">        &gt;&gt;&gt; M</span>
-<span class="sd">        &lt;[x + &lt;x**2 - y**2&gt;, x + y + &lt;x**2 - y**2&gt;]&gt;</span>
-<span class="sd">        &gt;&gt;&gt; M.contains([y**2, x**2 + x*y])</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; M.contains([x, y])</span>
-<span class="sd">        False</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">SubModuleQuotientRing</span><span class="p">(</span><span class="n">gens</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">opts</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">lift</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">elem</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Lift the element ``elem`` of self to the module self.quot.</span>
-
-<span class="sd">        Note that self.quot is the same set as self, just as an R-module</span>
-<span class="sd">        and not as an R/I-module, so this makes sense.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; F = (QQ[x]/[x**2 + 1]).free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; e = F.convert([1, 0])</span>
-<span class="sd">        &gt;&gt;&gt; e</span>
-<span class="sd">        [1 + &lt;x**2 + 1&gt;, 0 + &lt;x**2 + 1&gt;]</span>
-<span class="sd">        &gt;&gt;&gt; L = F.quot</span>
-<span class="sd">        &gt;&gt;&gt; l = F.lift(e)</span>
-<span class="sd">        &gt;&gt;&gt; l</span>
-<span class="sd">        [1, 0] + &lt;[x**2 + 1, 0], [0, x**2 + 1]&gt;</span>
-<span class="sd">        &gt;&gt;&gt; L.contains(l)</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">quot</span><span class="o">.</span><span class="n">convert</span><span class="p">([</span><span class="n">x</span><span class="o">.</span><span class="n">data</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">elem</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">unlift</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">elem</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Push down an element of self.quot to self.</span>
-
-<span class="sd">        This undoes ``lift``.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; F = (QQ[x]/[x**2 + 1]).free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; e = F.convert([1, 0])</span>
-<span class="sd">        &gt;&gt;&gt; l = F.lift(e)</span>
-<span class="sd">        &gt;&gt;&gt; e == l</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; e == F.unlift(l)</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">elem</span><span class="o">.</span><span class="n">data</span><span class="p">)</span>
-
-<span class="c">##########################################################################</span>
-<span class="c">## Submodules and subquotients ###########################################</span>
-<span class="c">##########################################################################</span>
-
-
-<div class="viewcode-block" id="SubModule"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.SubModule">[docs]</a><span class="k">class</span> <span class="nc">SubModule</span><span class="p">(</span><span class="n">Module</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Base class for submodules.</span>
-
-<span class="sd">    Attributes:</span>
-
-<span class="sd">    - container - containing module</span>
-<span class="sd">    - gens - generators (subset of containing module)</span>
-<span class="sd">    - rank - rank of containing module</span>
-
-<span class="sd">    Non-implemented methods:</span>
-
-<span class="sd">    - _contains</span>
-<span class="sd">    - _syzygies</span>
-<span class="sd">    - _in_terms_of_generators</span>
-<span class="sd">    - _intersect</span>
-<span class="sd">    - _module_quotient</span>
-
-<span class="sd">    Methods that likely need change in subclasses:</span>
-
-<span class="sd">    - reduce_element</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">container</span><span class="p">):</span>
-        <span class="n">Module</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">container</span><span class="o">.</span><span class="n">ring</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">gens</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">container</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">container</span> <span class="o">=</span> <span class="n">container</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">rank</span> <span class="o">=</span> <span class="n">container</span><span class="o">.</span><span class="n">rank</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ring</span> <span class="o">=</span> <span class="n">container</span><span class="o">.</span><span class="n">ring</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">dtype</span> <span class="o">=</span> <span class="n">container</span><span class="o">.</span><span class="n">dtype</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;&lt;&quot;</span> <span class="o">+</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">repr</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot;&gt;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Implementation of containment.</span>
-<span class="sd">           Other is guaranteed to be FreeModuleElement.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">def</span> <span class="nf">_syzygies</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Implementation of syzygy computation wrt self generators.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">def</span> <span class="nf">_in_terms_of_generators</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Implementation of expression in terms of generators.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-<div class="viewcode-block" id="SubModule.convert"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.SubModule.convert">[docs]</a>    <span class="k">def</span> <span class="nf">convert</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">elem</span><span class="p">,</span> <span class="n">M</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Convert ``elem`` into the internal represantition.</span>
-
-<span class="sd">        Mostly called implicitly.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; M = QQ[x].free_module(2).submodule([1, x])</span>
-<span class="sd">        &gt;&gt;&gt; M.convert([2, 2*x])</span>
-<span class="sd">        [2, 2*x]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">elem</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">container</span><span class="o">.</span><span class="n">dtype</span><span class="p">)</span> <span class="ow">and</span> <span class="n">elem</span><span class="o">.</span><span class="n">module</span> <span class="ow">is</span> <span class="bp">self</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">elem</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">copy</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">container</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">elem</span><span class="p">,</span> <span class="n">M</span><span class="p">))</span>
-        <span class="n">r</span><span class="o">.</span><span class="n">module</span> <span class="o">=</span> <span class="bp">self</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">_contains</span><span class="p">(</span><span class="n">r</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="n">CoercionFailed</span>
-        <span class="k">return</span> <span class="n">r</span>
-</div>
-    <span class="k">def</span> <span class="nf">_intersect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Implementation of intersection.</span>
-<span class="sd">           Other is guaranteed to be a submodule of same free module.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">def</span> <span class="nf">_module_quotient</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Implementation of quotient.</span>
-<span class="sd">           Other is guaranteed to be a submodule of same free module.&quot;&quot;&quot;</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-<div class="viewcode-block" id="SubModule.intersect"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.SubModule.intersect">[docs]</a>    <span class="k">def</span> <span class="nf">intersect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the intersection of ``self`` with submodule ``other``.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x, y].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; F.submodule([x, x]).intersect(F.submodule([y, y]))</span>
-<span class="sd">        &lt;[x*y, x*y]&gt;</span>
-
-<span class="sd">        Some implementation allow further options to be passed. Currently, to</span>
-<span class="sd">        only one implemented is ``relations=True``, in which case the function</span>
-<span class="sd">        will return a triple ``(res, rela, relb)``, where ``res`` is the</span>
-<span class="sd">        intersection module, and ``rela`` and ``relb`` are lists of coefficient</span>
-<span class="sd">        vectors, expressing the generators of ``res`` in terms of the</span>
-<span class="sd">        generators of ``self`` (``rela``) and ``other`` (``relb``).</span>
-
-<span class="sd">        &gt;&gt;&gt; F.submodule([x, x]).intersect(F.submodule([y, y]), relations=True)</span>
-<span class="sd">        (&lt;[x*y, x*y]&gt;, [(y,)], [(x,)])</span>
-
-<span class="sd">        The above result says: the intersection module is generated by the</span>
-<span class="sd">        single element `(-xy, -xy) = -y (x, x) = -x (y, y)`, where</span>
-<span class="sd">        `(x, x)` and `(y, y)` respectively are the unique generators of</span>
-<span class="sd">        the two modules being intersected.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">SubModule</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> is not a SubModule&#39;</span> <span class="o">%</span> <span class="n">other</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">container</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">container</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&#39;</span><span class="si">%s</span><span class="s"> is contained in a different free module&#39;</span> <span class="o">%</span> <span class="n">other</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_intersect</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="SubModule.module_quotient"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.SubModule.module_quotient">[docs]</a>    <span class="k">def</span> <span class="nf">module_quotient</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">        Returns the module quotient of ``self`` by submodule ``other``.</span>
-
-<span class="sd">        That is, if ``self`` is the module `M` and ``other`` is `N`, then</span>
-<span class="sd">        return the ideal `\{f \in R | fN \subset M\}`.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x, y].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; S = F.submodule([x*y, x*y])</span>
-<span class="sd">        &gt;&gt;&gt; T = F.submodule([x, x])</span>
-<span class="sd">        &gt;&gt;&gt; S.module_quotient(T)</span>
-<span class="sd">        &lt;y&gt;</span>
-
-<span class="sd">        Some implementations allow further options to be passed. Currently, the</span>
-<span class="sd">        only one implemented is ``relations=True``, which may only be passed</span>
-<span class="sd">        if ``other`` is prinicipal. In this case the function</span>
-<span class="sd">        will return a pair ``(res, rel)`` where ``res`` is the ideal, and</span>
-<span class="sd">        ``rel`` is a list of coefficient vectors, expressing the generators of</span>
-<span class="sd">        the ideal, multiplied by the generator of ``other`` in terms of</span>
-<span class="sd">        generators of ``self``.</span>
-
-<span class="sd">        &gt;&gt;&gt; S.module_quotient(T, relations=True)</span>
-<span class="sd">        (&lt;y&gt;, [[1]])</span>
-
-<span class="sd">        This means that the quotient ideal is generated by the single element</span>
-<span class="sd">        `y`, and that `y (x, x) = 1 (xy, xy)`, `(x, x)` and `(xy, xy)` being</span>
-<span class="sd">        the generators of `T` and `S`, respectively.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">SubModule</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> is not a SubModule&#39;</span> <span class="o">%</span> <span class="n">other</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">container</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">container</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&#39;</span><span class="si">%s</span><span class="s"> is contained in a different free module&#39;</span> <span class="o">%</span> <span class="n">other</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_module_quotient</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="SubModule.union"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.SubModule.union">[docs]</a>    <span class="k">def</span> <span class="nf">union</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the module generated by the union of ``self`` and ``other``.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(1)</span>
-<span class="sd">        &gt;&gt;&gt; M = F.submodule([x**2 + x]) # &lt;x(x+1)&gt;</span>
-<span class="sd">        &gt;&gt;&gt; N = F.submodule([x**2 - 1]) # &lt;(x-1)(x+1)&gt;</span>
-<span class="sd">        &gt;&gt;&gt; M.union(N) == F.submodule([x+1])</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">SubModule</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> is not a SubModule&#39;</span> <span class="o">%</span> <span class="n">other</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">container</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">container</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&#39;</span><span class="si">%s</span><span class="s"> is contained in a different free module&#39;</span> <span class="o">%</span> <span class="n">other</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gens</span> <span class="o">+</span> <span class="n">other</span><span class="o">.</span><span class="n">gens</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">container</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="SubModule.is_zero"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.SubModule.is_zero">[docs]</a>    <span class="k">def</span> <span class="nf">is_zero</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if ``self`` is a zero module.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; F.submodule([x, 1]).is_zero()</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; F.submodule([0, 0]).is_zero()</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="n">x</span> <span class="o">==</span> <span class="mi">0</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="SubModule.submodule"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.SubModule.submodule">[docs]</a>    <span class="k">def</span> <span class="nf">submodule</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Generate a submodule.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; M = QQ[x].free_module(2).submodule([x, 1])</span>
-<span class="sd">        &gt;&gt;&gt; M.submodule([x**2, x])</span>
-<span class="sd">        &lt;[x**2, x]&gt;</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">subset</span><span class="p">(</span><span class="n">gens</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> not a subset of </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">gens</span><span class="p">,</span> <span class="bp">self</span><span class="p">))</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">gens</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">container</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="SubModule.is_full_module"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.SubModule.is_full_module">[docs]</a>    <span class="k">def</span> <span class="nf">is_full_module</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if ``self`` is the entire free module.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; F.submodule([x, 1]).is_full_module()</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; F.submodule([1, 1], [1, 2]).is_full_module()</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">container</span><span class="o">.</span><span class="n">basis</span><span class="p">())</span>
-</div>
-<div class="viewcode-block" id="SubModule.is_submodule"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.SubModule.is_submodule">[docs]</a>    <span class="k">def</span> <span class="nf">is_submodule</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns True if ``other`` is a submodule of ``self``.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; M = F.submodule([2, x])</span>
-<span class="sd">        &gt;&gt;&gt; N = M.submodule([2*x, x**2])</span>
-<span class="sd">        &gt;&gt;&gt; M.is_submodule(M)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; M.is_submodule(N)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; N.is_submodule(M)</span>
-<span class="sd">        False</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">SubModule</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">container</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">container</span> <span class="ow">and</span> \
-                <span class="nb">all</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">other</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="p">(</span><span class="n">FreeModule</span><span class="p">,</span> <span class="n">QuotientModule</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">container</span> <span class="o">==</span> <span class="n">other</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_full_module</span><span class="p">()</span>
-        <span class="k">return</span> <span class="bp">False</span>
-</div>
-<div class="viewcode-block" id="SubModule.syzygy_module"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.SubModule.syzygy_module">[docs]</a>    <span class="k">def</span> <span class="nf">syzygy_module</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">opts</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">        Compute the syzygy module of the generators of ``self``.</span>
-
-<span class="sd">        Suppose `M` is generated by `f_1, \dots, f_n` over the ring</span>
-<span class="sd">        `R`. Consider the homomorphism `\phi: R^n \to M`, given by</span>
-<span class="sd">        sending `(r_1, \dots, r_n) \to r_1 f_1 + \dots + r_n f_n`.</span>
-<span class="sd">        The syzygy module is defined to be the kernel of `\phi`.</span>
-
-<span class="sd">        The syzygy module is zero iff the generators generate freely a free</span>
-<span class="sd">        submodule:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; QQ[x].free_module(2).submodule([1, 0], [1, 1]).syzygy_module().is_zero()</span>
-<span class="sd">        True</span>
-
-<span class="sd">        A slightly more interesting example:</span>
-
-<span class="sd">        &gt;&gt;&gt; M = QQ[x, y].free_module(2).submodule([x, 2*x], [y, 2*y])</span>
-<span class="sd">        &gt;&gt;&gt; S = QQ[x, y].free_module(2).submodule([y, -x])</span>
-<span class="sd">        &gt;&gt;&gt; M.syzygy_module() == S</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">F</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">))</span>
-        <span class="c"># NOTE we filter out zero syzygies. This is for convenience of the</span>
-        <span class="c"># _syzygies function and not meant to replace any real &quot;generating set</span>
-        <span class="c"># reduction&quot; algorithm</span>
-        <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_syzygies</span><span class="p">()</span> <span class="k">if</span> <span class="n">F</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">],</span>
-                           <span class="o">**</span><span class="n">opts</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="SubModule.in_terms_of_generators"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.SubModule.in_terms_of_generators">[docs]</a>    <span class="k">def</span> <span class="nf">in_terms_of_generators</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Express element ``e`` of ``self`` in terms of the generators.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; M = F.submodule([1, 0], [1, 1])</span>
-<span class="sd">        &gt;&gt;&gt; M.in_terms_of_generators([x, x**2])</span>
-<span class="sd">        [-x**2 + x, x**2]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">e</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">CoercionFailed</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> is not an element of </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="bp">self</span><span class="p">))</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_in_terms_of_generators</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="SubModule.reduce_element"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.SubModule.reduce_element">[docs]</a>    <span class="k">def</span> <span class="nf">reduce_element</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Reduce the element ``x`` of our ring modulo the ideal ``self``.</span>
-
-<span class="sd">        Here &quot;reduce&quot; has no specific meaning, it could return a unique normal</span>
-<span class="sd">        form, simplify the expression a bit, or just do nothing.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">x</span>
-</div>
-<div class="viewcode-block" id="SubModule.quotient_module"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.SubModule.quotient_module">[docs]</a>    <span class="k">def</span> <span class="nf">quotient_module</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="o">**</span><span class="n">opts</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return a quotient module.</span>
-
-<span class="sd">        This is the same as taking a submodule of a quotient of the containing</span>
-<span class="sd">        module.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; S1 = F.submodule([x, 1])</span>
-<span class="sd">        &gt;&gt;&gt; S2 = F.submodule([x**2, x])</span>
-<span class="sd">        &gt;&gt;&gt; S1.quotient_module(S2)</span>
-<span class="sd">        &lt;[x, 1] + &lt;[x**2, x]&gt;&gt;</span>
-
-<span class="sd">        Or more coincisely, using the overloaded division operator:</span>
-
-<span class="sd">        &gt;&gt;&gt; F.submodule([x, 1]) / [(x**2, x)]</span>
-<span class="sd">        &lt;[x, 1] + &lt;[x**2, x]&gt;&gt;</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_submodule</span><span class="p">(</span><span class="n">other</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> not a submodule of </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">SubQuotientModule</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">,</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">container</span><span class="o">.</span><span class="n">quotient_module</span><span class="p">(</span><span class="n">other</span><span class="p">),</span> <span class="o">**</span><span class="n">opts</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">oth</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">container</span><span class="o">.</span><span class="n">quotient_module</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">oth</span><span class="p">)</span>
-
-    <span class="n">__radd__</span> <span class="o">=</span> <span class="n">__add__</span>
-
-<div class="viewcode-block" id="SubModule.multiply_ideal"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.SubModule.multiply_ideal">[docs]</a>    <span class="k">def</span> <span class="nf">multiply_ideal</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">I</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Multiply ``self`` by the ideal ``I``.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; I = QQ[x].ideal(x**2)</span>
-<span class="sd">        &gt;&gt;&gt; M = QQ[x].free_module(2).submodule([1, 1])</span>
-<span class="sd">        &gt;&gt;&gt; I*M</span>
-<span class="sd">        &lt;[x**2, x**2]&gt;</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">submodule</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">x</span><span class="o">*</span><span class="n">g</span> <span class="k">for</span> <span class="p">[</span><span class="n">x</span><span class="p">]</span> <span class="ow">in</span> <span class="n">I</span><span class="o">.</span><span class="n">_module</span><span class="o">.</span><span class="n">gens</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">])</span>
-</div>
-<div class="viewcode-block" id="SubModule.inclusion_hom"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.SubModule.inclusion_hom">[docs]</a>    <span class="k">def</span> <span class="nf">inclusion_hom</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return a homomorphism representing the inclusion map of ``self``.</span>
-
-<span class="sd">        That is, the natural map from ``self`` to ``self.container``.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; QQ[x].free_module(2).submodule([x, x]).inclusion_hom()</span>
-<span class="sd">        [1, 0]</span>
-<span class="sd">        [0, 1] : &lt;[x, x]&gt; -&gt; QQ[x]**2</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">container</span><span class="o">.</span><span class="n">identity_hom</span><span class="p">()</span><span class="o">.</span><span class="n">restrict_domain</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="SubModule.identity_hom"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.SubModule.identity_hom">[docs]</a>    <span class="k">def</span> <span class="nf">identity_hom</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the identity homomorphism on ``self``.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; QQ[x].free_module(2).submodule([x, x]).identity_hom()</span>
-<span class="sd">        [1, 0]</span>
-<span class="sd">        [0, 1] : &lt;[x, x]&gt; -&gt; &lt;[x, x]&gt;</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">container</span><span class="o">.</span><span class="n">identity_hom</span><span class="p">()</span><span class="o">.</span><span class="n">restrict_domain</span><span class="p">(</span>
-            <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">restrict_codomain</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-</div></div>
-<div class="viewcode-block" id="SubQuotientModule"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.SubQuotientModule">[docs]</a><span class="k">class</span> <span class="nc">SubQuotientModule</span><span class="p">(</span><span class="n">SubModule</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Submodule of a quotient module.</span>
-
-<span class="sd">    Equivalently, quotient module of a submodule.</span>
-
-<span class="sd">    Do not instantiate this, instead use the submodule or quotient_module</span>
-<span class="sd">    constructing methods:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">    &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">    &gt;&gt;&gt; S = F.submodule([1, 0], [1, x])</span>
-<span class="sd">    &gt;&gt;&gt; Q = F/[(1, 0)]</span>
-<span class="sd">    &gt;&gt;&gt; S/[(1, 0)] == Q.submodule([5, x])</span>
-<span class="sd">    True</span>
-
-<span class="sd">    Attributes:</span>
-
-<span class="sd">    - base - base module we are quotient of</span>
-<span class="sd">    - killed_module - submodule we are quotienting by</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">container</span><span class="p">,</span> <span class="o">**</span><span class="n">opts</span><span class="p">):</span>
-        <span class="n">SubModule</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">container</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">killed_module</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">container</span><span class="o">.</span><span class="n">killed_module</span>
-        <span class="c"># XXX it is important for some code below that the generators of base</span>
-        <span class="c">#     are in this particular order!</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">base</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">container</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">submodule</span><span class="p">(</span>
-            <span class="o">*</span><span class="p">[</span><span class="n">x</span><span class="o">.</span><span class="n">data</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">],</span> <span class="o">**</span><span class="n">opts</span><span class="p">)</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">killed_module</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">elem</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">elem</span><span class="o">.</span><span class="n">data</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_syzygies</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># let N = self.killed_module be generated by e_1, ..., e_r</span>
-        <span class="c"># let F = self.base be generated by f_1, ..., f_s and e_1, ..., e_r</span>
-        <span class="c"># Then self = F/N.</span>
-        <span class="c"># Let phi: R**s --&gt; self be the evident surjection.</span>
-        <span class="c"># Similarly psi: R**(s + r) --&gt; F.</span>
-        <span class="c"># We need to find generators for ker(phi). Let chi: R**s --&gt; F be the</span>
-        <span class="c"># evident lift of phi. For X in R**s, phi(X) = 0 iff chi(X) is</span>
-        <span class="c"># contained in N, iff there exists Y in R**r such that</span>
-        <span class="c"># psi(X, Y) = 0.</span>
-        <span class="c"># Hence if alpha: R**(s + r) --&gt; R**s is the projection map, then</span>
-        <span class="c"># ker(phi) = alpha ker(psi).</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">X</span><span class="p">[:</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">)]</span> <span class="k">for</span> <span class="n">X</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">_syzygies</span><span class="p">()]</span>
-
-    <span class="k">def</span> <span class="nf">_in_terms_of_generators</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">_in_terms_of_generators</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">data</span><span class="p">)[:</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">)]</span>
-
-<div class="viewcode-block" id="SubQuotientModule.is_full_module"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.SubQuotientModule.is_full_module">[docs]</a>    <span class="k">def</span> <span class="nf">is_full_module</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if ``self`` is the entire free module.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; F.submodule([x, 1]).is_full_module()</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; F.submodule([1, 1], [1, 2]).is_full_module()</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_full_module</span><span class="p">()</span>
-</div>
-<div class="viewcode-block" id="SubQuotientModule.quotient_hom"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.SubQuotientModule.quotient_hom">[docs]</a>    <span class="k">def</span> <span class="nf">quotient_hom</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the quotient homomorphism to self.</span>
-
-<span class="sd">        That is, return the natural map from ``self.base`` to ``self``.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; M = (QQ[x].free_module(2) / [(1, x)]).submodule([1, 0])</span>
-<span class="sd">        &gt;&gt;&gt; M.quotient_hom()</span>
-<span class="sd">        [1, 0]</span>
-<span class="sd">        [0, 1] : &lt;[1, 0], [1, x]&gt; -&gt; &lt;[1, 0] + &lt;[1, x]&gt;, [1, x] + &lt;[1, x]&gt;&gt;</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">identity_hom</span><span class="p">()</span><span class="o">.</span><span class="n">quotient_codomain</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">killed_module</span><span class="p">)</span>
-
-</div></div>
-<span class="n">_subs0</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-<span class="n">_subs1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-
-<span class="k">class</span> <span class="nc">ModuleOrder</span><span class="p">(</span><span class="n">ProductOrder</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A product monomial order with a zeroth term as module index.&quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o1</span><span class="p">,</span> <span class="n">o2</span><span class="p">,</span> <span class="n">TOP</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">TOP</span><span class="p">:</span>
-            <span class="n">ProductOrder</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="p">(</span><span class="n">o2</span><span class="p">,</span> <span class="n">_subs1</span><span class="p">),</span> <span class="p">(</span><span class="n">o1</span><span class="p">,</span> <span class="n">_subs0</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">ProductOrder</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="p">(</span><span class="n">o1</span><span class="p">,</span> <span class="n">_subs0</span><span class="p">),</span> <span class="p">(</span><span class="n">o2</span><span class="p">,</span> <span class="n">_subs1</span><span class="p">))</span>
-
-
-<span class="k">class</span> <span class="nc">SubModulePolyRing</span><span class="p">(</span><span class="n">SubModule</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Submodule of a free module over a generalized polynomial ring.</span>
-
-<span class="sd">    Do not instantiate this, use the constructor method of FreeModule instead:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">    &gt;&gt;&gt; F = QQ[x, y].free_module(2)</span>
-<span class="sd">    &gt;&gt;&gt; F.submodule([x, y], [1, 0])</span>
-<span class="sd">    &lt;[x, y], [1, 0]&gt;</span>
-
-<span class="sd">    Attributes:</span>
-
-<span class="sd">    - order - monomial order used</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="c">#self._gb - cached groebner basis</span>
-    <span class="c">#self._gbe - cached groebner basis relations</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">container</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s">&quot;lex&quot;</span><span class="p">,</span> <span class="n">TOP</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="n">SubModule</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">container</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">container</span><span class="p">,</span> <span class="n">FreeModulePolyRing</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;This implementation is for submodules of &#39;</span>
-                             <span class="o">+</span> <span class="s">&#39;FreeModulePolyRing, got </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">container</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">order</span> <span class="o">=</span> <span class="n">ModuleOrder</span><span class="p">(</span><span class="n">monomial_key</span><span class="p">(</span><span class="n">order</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">order</span><span class="p">,</span> <span class="n">TOP</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_gb</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_gbe</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">SubModulePolyRing</span><span class="p">)</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span> <span class="o">!=</span> <span class="n">other</span><span class="o">.</span><span class="n">order</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="n">SubModule</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_groebner</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">extended</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a standard basis in sdm form.&quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.polys.distributedmodules</span> <span class="kn">import</span> <span class="n">sdm_groebner</span><span class="p">,</span> <span class="n">sdm_nf_mora</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_gbe</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">extended</span><span class="p">:</span>
-            <span class="n">gb</span><span class="p">,</span> <span class="n">gbe</span> <span class="o">=</span> <span class="n">sdm_groebner</span><span class="p">(</span>
-                <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">_vector_to_sdm</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">],</span>
-                <span class="n">sdm_nf_mora</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">extended</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_gb</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_gbe</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">gb</span><span class="p">),</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">gbe</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_gb</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_gb</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">sdm_groebner</span><span class="p">(</span>
-                             <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">_vector_to_sdm</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">],</span>
-               <span class="n">sdm_nf_mora</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">extended</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_gb</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_gbe</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_gb</span>
-
-    <span class="k">def</span> <span class="nf">_groebner_vec</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">extended</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a standard basis in element form.&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">extended</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">_sdm_to_vector</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rank</span><span class="p">))</span>
-                    <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_groebner</span><span class="p">()]</span>
-        <span class="n">gb</span><span class="p">,</span> <span class="n">gbe</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_groebner</span><span class="p">(</span><span class="n">extended</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">_sdm_to_vector</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rank</span><span class="p">))</span>
-                 <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">gb</span><span class="p">],</span>
-                <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">_sdm_to_vector</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">))</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">gbe</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.polys.distributedmodules</span> <span class="kn">import</span> <span class="n">sdm_zero</span><span class="p">,</span> <span class="n">sdm_nf_mora</span>
-        <span class="k">return</span> <span class="n">sdm_nf_mora</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">_vector_to_sdm</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span><span class="p">),</span>
-                           <span class="bp">self</span><span class="o">.</span><span class="n">_groebner</span><span class="p">(),</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span> <span class="o">==</span> \
-            <span class="n">sdm_zero</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">_syzygies</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute syzygies. See [SCA, algorithm 2.5.4].&quot;&quot;&quot;</span>
-        <span class="c"># NOTE if self.gens is a standard basis, this can be done more</span>
-        <span class="c">#      efficiently using Schreyer&#39;s theorem</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">eye</span>
-
-        <span class="c"># First bullet point</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rank</span>
-        <span class="n">im</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-        <span class="n">Rkr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="n">r</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span>
-        <span class="n">newgens</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">f</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">):</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">r</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-                <span class="n">m</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
-                <span class="n">m</span><span class="p">[</span><span class="n">r</span> <span class="o">+</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">im</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span>
-            <span class="n">newgens</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Rkr</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">m</span><span class="p">))</span>
-        <span class="c"># Note: we need *descending* order on module index, and TOP=False to</span>
-        <span class="c">#       get an eliminetaion order</span>
-        <span class="n">F</span> <span class="o">=</span> <span class="n">Rkr</span><span class="o">.</span><span class="n">submodule</span><span class="p">(</span><span class="o">*</span><span class="n">newgens</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;order&#39;</span><span class="p">:</span> <span class="s">&#39;ilex&#39;</span><span class="p">,</span> <span class="s">&#39;TOP&#39;</span><span class="p">:</span> <span class="bp">False</span><span class="p">})</span>
-
-        <span class="c"># Second bullet point: standard basis of F</span>
-        <span class="n">G</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">_groebner_vec</span><span class="p">()</span>
-
-        <span class="c"># Third bullet point: G0 = G intersect the new k components</span>
-        <span class="n">G0</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="p">[</span><span class="n">r</span><span class="p">:]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">G</span> <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">y</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-                                      <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">x</span><span class="p">[:</span><span class="n">r</span><span class="p">])]</span>
-
-        <span class="c"># Fourth and fifth bullet points: we are done</span>
-        <span class="k">return</span> <span class="n">G0</span>
-
-    <span class="k">def</span> <span class="nf">_in_terms_of_generators</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Expression in terms of generators. See [SCA, 2.8.1].&quot;&quot;&quot;</span>
-        <span class="c"># NOTE: if gens is a standard basis, this can be done more efficiently</span>
-        <span class="n">M</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rank</span><span class="p">)</span><span class="o">.</span><span class="n">submodule</span><span class="p">(</span><span class="o">*</span><span class="p">((</span><span class="n">e</span><span class="p">,)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">))</span>
-        <span class="n">S</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">syzygy_module</span><span class="p">(</span>
-            <span class="n">order</span><span class="o">=</span><span class="s">&quot;ilex&quot;</span><span class="p">,</span> <span class="n">TOP</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>  <span class="c"># We want decreasing order!</span>
-        <span class="n">G</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">_groebner_vec</span><span class="p">()</span>
-        <span class="c"># This list cannot not be empty since e is an element</span>
-        <span class="n">e</span> <span class="o">=</span> <span class="nb">list</span><span class="p">([</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">G</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">is_unit</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">])])[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">return</span> <span class="p">[</span><span class="o">-</span><span class="n">x</span><span class="o">/</span><span class="n">e</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">e</span><span class="p">[</span><span class="mi">1</span><span class="p">:]]</span>
-
-    <span class="k">def</span> <span class="nf">reduce_element</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">NF</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Reduce the element ``x`` of our container modulo ``self``.</span>
-
-<span class="sd">        This applies the normal form ``NF`` to ``x``. If ``NF`` is passed</span>
-<span class="sd">        as none, the default Mora normal form is used (which is not unique!).</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.polys.distributedmodules</span> <span class="kn">import</span> <span class="n">sdm_nf_mora</span>
-        <span class="k">if</span> <span class="n">NF</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">NF</span> <span class="o">=</span> <span class="n">sdm_nf_mora</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">container</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">_sdm_to_vector</span><span class="p">(</span><span class="n">NF</span><span class="p">(</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">_vector_to_sdm</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_groebner</span><span class="p">(),</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">order</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">dom</span><span class="p">),</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">rank</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_intersect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="n">relations</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="c"># See: [SCA, section 2.8.2]</span>
-        <span class="n">fi</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">gens</span>
-        <span class="n">hi</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">gens</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rank</span>
-        <span class="n">ci</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">r</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">r</span><span class="p">)]</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">r</span><span class="p">):</span>
-            <span class="n">ci</span><span class="p">[</span><span class="n">k</span><span class="p">][</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="n">ci</span><span class="p">[</span><span class="n">k</span><span class="p">][</span><span class="n">r</span> <span class="o">+</span> <span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">di</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">+</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">r</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">fi</span><span class="p">]</span>
-        <span class="n">ei</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">r</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">h</span><span class="p">)</span> <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">hi</span><span class="p">]</span>
-        <span class="n">syz</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">r</span><span class="p">)</span><span class="o">.</span><span class="n">submodule</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">ci</span> <span class="o">+</span> <span class="n">di</span> <span class="o">+</span> <span class="n">ei</span><span class="p">))</span><span class="o">.</span><span class="n">_syzygies</span><span class="p">()</span>
-        <span class="n">nonzero</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">syz</span> <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">y</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">zero</span> <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">x</span><span class="p">[:</span><span class="n">r</span><span class="p">])]</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">container</span><span class="o">.</span><span class="n">submodule</span><span class="p">(</span><span class="o">*</span><span class="p">([</span><span class="o">-</span><span class="n">y</span> <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">x</span><span class="p">[:</span><span class="n">r</span><span class="p">]]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">nonzero</span><span class="p">))</span>
-        <span class="n">reln1</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="p">[</span><span class="n">r</span><span class="p">:</span><span class="n">r</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">fi</span><span class="p">)]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">nonzero</span><span class="p">]</span>
-        <span class="n">reln2</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="p">[</span><span class="n">r</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">fi</span><span class="p">):]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">nonzero</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">relations</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">res</span><span class="p">,</span> <span class="n">reln1</span><span class="p">,</span> <span class="n">reln2</span>
-        <span class="k">return</span> <span class="n">res</span>
-
-    <span class="k">def</span> <span class="nf">_module_quotient</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="n">relations</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="c"># See: [SCA, section 2.8.4]</span>
-        <span class="k">if</span> <span class="n">relations</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># We do some trickery. Let f be the (vector!) generating ``other``</span>
-            <span class="c"># and f1, .., fn be the (vectors) generating self.</span>
-            <span class="c"># Consider the submodule of R^{r+1} generated by (f, 1) and</span>
-            <span class="c"># {(fi, 0) | i}. Then the intersection with the last module</span>
-            <span class="c"># component yields the quotient.</span>
-            <span class="n">g1</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">gens</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">+</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-            <span class="n">gi</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">]</span>
-            <span class="c"># NOTE: We *need* to use an elimination order</span>
-            <span class="n">M</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rank</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">submodule</span><span class="p">(</span><span class="o">*</span><span class="p">([</span><span class="n">g1</span><span class="p">]</span> <span class="o">+</span> <span class="n">gi</span><span class="p">),</span>
-                                            <span class="o">**</span><span class="p">{</span><span class="s">&#39;order&#39;</span><span class="p">:</span> <span class="s">&#39;ilex&#39;</span><span class="p">,</span> <span class="s">&#39;TOP&#39;</span><span class="p">:</span> <span class="bp">False</span><span class="p">})</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">relations</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">x</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">M</span><span class="o">.</span><span class="n">_groebner_vec</span><span class="p">()</span> <span class="k">if</span>
-                                         <span class="nb">all</span><span class="p">(</span><span class="n">y</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">zero</span> <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">x</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">])])</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">G</span><span class="p">,</span> <span class="n">R</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">_groebner_vec</span><span class="p">(</span><span class="n">extended</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                <span class="n">indices</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="k">if</span>
-                           <span class="nb">all</span><span class="p">(</span><span class="n">y</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">zero</span> <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">x</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">])]</span>
-                <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">G</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">indices</span><span class="p">]),</span>
-                        <span class="p">[[</span><span class="o">-</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">R</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">:]]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">indices</span><span class="p">])</span>
-        <span class="c"># For more generators, we use I : &lt;h1, .., hn&gt; = intersection of</span>
-        <span class="c">#                                    {I : &lt;hi&gt; | i}</span>
-        <span class="c"># TODO this can be done more efficiently</span>
-        <span class="k">return</span> <span class="nb">reduce</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="n">y</span><span class="p">),</span>
-            <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_module_quotient</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">container</span><span class="o">.</span><span class="n">submodule</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">other</span><span class="o">.</span><span class="n">gens</span><span class="p">))</span>
-
-
-<span class="k">class</span> <span class="nc">SubModuleQuotientRing</span><span class="p">(</span><span class="n">SubModule</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Class for submodules of free modules over quotient rings.</span>
-
-<span class="sd">    Do not instantiate this. Instead use the submodule methods.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">    &gt;&gt;&gt; M = (QQ[x, y]/[x**2 - y**2]).free_module(2).submodule([x, x + y])</span>
-<span class="sd">    &gt;&gt;&gt; M</span>
-<span class="sd">    &lt;[x + &lt;x**2 - y**2&gt;, x + y + &lt;x**2 - y**2&gt;]&gt;</span>
-<span class="sd">    &gt;&gt;&gt; M.contains([y**2, x**2 + x*y])</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; M.contains([x, y])</span>
-<span class="sd">    False</span>
-
-<span class="sd">    Attributes:</span>
-
-<span class="sd">    - quot - the subquotient of `R^n/IR^n` generated by lifts of our generators</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">container</span><span class="p">):</span>
-        <span class="n">SubModule</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">container</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">quot</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">container</span><span class="o">.</span><span class="n">quot</span><span class="o">.</span><span class="n">submodule</span><span class="p">(</span>
-            <span class="o">*</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">container</span><span class="o">.</span><span class="n">lift</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">elem</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">quot</span><span class="o">.</span><span class="n">_contains</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">container</span><span class="o">.</span><span class="n">lift</span><span class="p">(</span><span class="n">elem</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_syzygies</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">[</span><span class="nb">tuple</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">quot</span><span class="o">.</span><span class="n">ring</span><span class="p">)</span> <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">x</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">quot</span><span class="o">.</span><span class="n">_syzygies</span><span class="p">()]</span>
-
-    <span class="k">def</span> <span class="nf">_in_terms_of_generators</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">elem</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">quot</span><span class="o">.</span><span class="n">ring</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">quot</span><span class="o">.</span><span class="n">_in_terms_of_generators</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">container</span><span class="o">.</span><span class="n">lift</span><span class="p">(</span><span class="n">elem</span><span class="p">))]</span>
-
-<span class="c">##########################################################################</span>
-<span class="c">## Quotient Modules ######################################################</span>
-<span class="c">##########################################################################</span>
-
-
-<span class="k">class</span> <span class="nc">QuotientModuleElement</span><span class="p">(</span><span class="n">ModuleElement</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Element of a quotient module.&quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">eq</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">d1</span><span class="p">,</span> <span class="n">d2</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Equality comparison.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">killed_module</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">d1</span> <span class="o">-</span> <span class="n">d2</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sstr</span>
-        <span class="k">return</span> <span class="nb">repr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot; + &quot;</span> <span class="o">+</span> <span class="nb">repr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">killed_module</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="QuotientModule"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.QuotientModule">[docs]</a><span class="k">class</span> <span class="nc">QuotientModule</span><span class="p">(</span><span class="n">Module</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Class for quotient modules.</span>
-
-<span class="sd">    Do not instantiate this directly. For subquotients, see the</span>
-<span class="sd">    SubQuotientModule class.</span>
-
-<span class="sd">    Attributes:</span>
-
-<span class="sd">    - base - the base module we are a quotient of</span>
-<span class="sd">    - killed_module - the submodule we are quotienting by</span>
-<span class="sd">    - rank of the base</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">dtype</span> <span class="o">=</span> <span class="n">QuotientModuleElement</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ring</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="n">submodule</span><span class="p">):</span>
-        <span class="n">Module</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ring</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">base</span><span class="o">.</span><span class="n">is_submodule</span><span class="p">(</span><span class="n">submodule</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> is not a submodule of </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">submodule</span><span class="p">,</span> <span class="n">base</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">base</span> <span class="o">=</span> <span class="n">base</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">killed_module</span> <span class="o">=</span> <span class="n">submodule</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">rank</span> <span class="o">=</span> <span class="n">base</span><span class="o">.</span><span class="n">rank</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">repr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot;/&quot;</span> <span class="o">+</span> <span class="nb">repr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">killed_module</span><span class="p">)</span>
-
-<div class="viewcode-block" id="QuotientModule.is_zero"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.QuotientModule.is_zero">[docs]</a>    <span class="k">def</span> <span class="nf">is_zero</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if ``self`` is a zero module.</span>
-
-<span class="sd">        This happens if and only if the base module is the same as the</span>
-<span class="sd">        submodule being killed.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(2)</span>
-<span class="sd">        &gt;&gt;&gt; (F/[(1, 0)]).is_zero()</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; (F/[(1, 0), (0, 1)]).is_zero()</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">killed_module</span>
-</div>
-<div class="viewcode-block" id="QuotientModule.is_submodule"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.QuotientModule.is_submodule">[docs]</a>    <span class="k">def</span> <span class="nf">is_submodule</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if ``other`` is a submodule of ``self``.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; Q = QQ[x].free_module(2) / [(x, x)]</span>
-<span class="sd">        &gt;&gt;&gt; S = Q.submodule([1, 0])</span>
-<span class="sd">        &gt;&gt;&gt; Q.is_submodule(S)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; S.is_submodule(Q)</span>
-<span class="sd">        False</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">QuotientModule</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">killed_module</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">killed_module</span> <span class="ow">and</span> \
-                <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_submodule</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">base</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">SubQuotientModule</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">other</span><span class="o">.</span><span class="n">container</span> <span class="o">==</span> <span class="bp">self</span>
-        <span class="k">return</span> <span class="bp">False</span>
-</div>
-<div class="viewcode-block" id="QuotientModule.submodule"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.QuotientModule.submodule">[docs]</a>    <span class="k">def</span> <span class="nf">submodule</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">opts</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Generate a submodule.</span>
-
-<span class="sd">        This is the same as taking a quotient of a submodule of the base</span>
-<span class="sd">        module.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; Q = QQ[x].free_module(2) / [(x, x)]</span>
-<span class="sd">        &gt;&gt;&gt; Q.submodule([x, 0])</span>
-<span class="sd">        &lt;[x, 0] + &lt;[x, x]&gt;&gt;</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">SubQuotientModule</span><span class="p">(</span><span class="n">gens</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">opts</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="QuotientModule.convert"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.QuotientModule.convert">[docs]</a>    <span class="k">def</span> <span class="nf">convert</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">elem</span><span class="p">,</span> <span class="n">M</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Convert ``elem`` into the internal representation.</span>
-
-<span class="sd">        This method is called implicitly whenever computations involve elements</span>
-<span class="sd">        not in the internal representation.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; F = QQ[x].free_module(2) / [(1, 2), (1, x)]</span>
-<span class="sd">        &gt;&gt;&gt; F.convert([1, 0])</span>
-<span class="sd">        [1, 0] + &lt;[1, 2], [1, x]&gt;</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">elem</span><span class="p">,</span> <span class="n">QuotientModuleElement</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">elem</span><span class="o">.</span><span class="n">module</span> <span class="ow">is</span> <span class="bp">self</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">elem</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">killed_module</span><span class="o">.</span><span class="n">is_submodule</span><span class="p">(</span><span class="n">elem</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">killed_module</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">QuotientModuleElement</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">elem</span><span class="o">.</span><span class="n">data</span><span class="p">))</span>
-            <span class="k">raise</span> <span class="n">CoercionFailed</span>
-        <span class="k">return</span> <span class="n">QuotientModuleElement</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">elem</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="QuotientModule.identity_hom"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.QuotientModule.identity_hom">[docs]</a>    <span class="k">def</span> <span class="nf">identity_hom</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the identity homomorphism on ``self``.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; M = QQ[x].free_module(2) / [(1, 2), (1, x)]</span>
-<span class="sd">        &gt;&gt;&gt; M.identity_hom()</span>
-<span class="sd">        [1, 0]</span>
-<span class="sd">        [0, 1] : QQ[x]**2/&lt;[1, 2], [1, x]&gt; -&gt; QQ[x]**2/&lt;[1, 2], [1, x]&gt;</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">identity_hom</span><span class="p">()</span><span class="o">.</span><span class="n">quotient_codomain</span><span class="p">(</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">killed_module</span><span class="p">)</span><span class="o">.</span><span class="n">quotient_domain</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">killed_module</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="QuotientModule.quotient_hom"><a class="viewcode-back" href="../../../../modules/polys/agca.html#sympy.polys.agca.modules.QuotientModule.quotient_hom">[docs]</a>    <span class="k">def</span> <span class="nf">quotient_hom</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the quotient homomorphism to ``self``.</span>
-
-<span class="sd">        That is, return a homomorphism representing the natural map from</span>
-<span class="sd">        ``self.base`` to ``self``.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import QQ</span>
-<span class="sd">        &gt;&gt;&gt; M = QQ[x].free_module(2) / [(1, 2), (1, x)]</span>
-<span class="sd">        &gt;&gt;&gt; M.quotient_hom()</span>
-<span class="sd">        [1, 0]</span>
-<span class="sd">        [0, 1] : QQ[x]**2 -&gt; QQ[x]**2/&lt;[1, 2], [1, x]&gt;</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">identity_hom</span><span class="p">()</span><span class="o">.</span><span class="n">quotient_codomain</span><span class="p">(</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">killed_module</span><span class="p">)</span></div></div>
-</pre></div>
-
-          </div>
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-  <h1>Source code for sympy.polys.constructor</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Tools for constructing domains for expressions. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyutils</span> <span class="kn">import</span> <span class="n">parallel_dict_from_basic</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polyoptions</span> <span class="kn">import</span> <span class="n">build_options</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="n">GeneratorsNeeded</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span><span class="p">,</span> <span class="n">QQ</span><span class="p">,</span> <span class="n">RR</span><span class="p">,</span> <span class="n">EX</span>
-<span class="kn">from</span> <span class="nn">sympy.assumptions</span> <span class="kn">import</span> <span class="n">ask</span><span class="p">,</span> <span class="n">Q</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">sympify</span>
-
-
-<span class="k">def</span> <span class="nf">_construct_simple</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">opt</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Handle simple domains, e.g.: ZZ, QQ, RR and algebraic domains. &quot;&quot;&quot;</span>
-    <span class="n">result</span><span class="p">,</span> <span class="n">rationals</span><span class="p">,</span> <span class="n">reals</span><span class="p">,</span> <span class="n">algebraics</span> <span class="o">=</span> <span class="p">{},</span> <span class="bp">False</span><span class="p">,</span> <span class="bp">False</span><span class="p">,</span> <span class="bp">False</span>
-
-    <span class="k">if</span> <span class="n">opt</span><span class="o">.</span><span class="n">extension</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">is_algebraic</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">coeff</span><span class="p">:</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">algebraic</span><span class="p">(</span><span class="n">coeff</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">is_algebraic</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">coeff</span><span class="p">:</span> <span class="bp">False</span>
-
-    <span class="c"># XXX: add support for a + b*I coefficients</span>
-    <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">coeffs</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="n">rationals</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">elif</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">algebraics</span><span class="p">:</span>
-                <span class="n">reals</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># there are both reals and algebraics -&gt; EX</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="n">is_algebraic</span><span class="p">(</span><span class="n">coeff</span><span class="p">):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">reals</span><span class="p">:</span>
-                <span class="n">algebraics</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># there are both algebraics and reals -&gt; EX</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># this is a composite domain, e.g. ZZ[X], EX</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">if</span> <span class="n">algebraics</span><span class="p">:</span>
-        <span class="n">domain</span><span class="p">,</span> <span class="n">result</span> <span class="o">=</span> <span class="n">_construct_algebraic</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">reals</span><span class="p">:</span>
-            <span class="n">domain</span> <span class="o">=</span> <span class="n">RR</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">opt</span><span class="o">.</span><span class="n">field</span> <span class="ow">or</span> <span class="n">rationals</span><span class="p">:</span>
-                <span class="n">domain</span> <span class="o">=</span> <span class="n">QQ</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">domain</span> <span class="o">=</span> <span class="n">ZZ</span>
-
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">coeffs</span><span class="p">:</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">domain</span><span class="o">.</span><span class="n">from_sympy</span><span class="p">(</span><span class="n">coeff</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">domain</span><span class="p">,</span> <span class="n">result</span>
-
-
-<span class="k">def</span> <span class="nf">_construct_algebraic</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">opt</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;We know that coefficients are algebraic so construct the extension. &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.polys.numberfields</span> <span class="kn">import</span> <span class="n">primitive_element</span>
-
-    <span class="n">result</span><span class="p">,</span> <span class="n">exts</span> <span class="o">=</span> <span class="p">[],</span> <span class="nb">set</span><span class="p">([])</span>
-
-    <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">coeffs</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="n">coeff</span> <span class="o">=</span> <span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">QQ</span><span class="o">.</span><span class="n">from_sympy</span><span class="p">(</span><span class="n">coeff</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">coeff</span><span class="o">.</span><span class="n">as_coeff_add</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">coeff</span> <span class="o">-=</span> <span class="n">a</span>
-
-            <span class="n">b</span> <span class="o">=</span> <span class="n">coeff</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">coeff</span> <span class="o">/=</span> <span class="n">b</span>
-
-            <span class="n">exts</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-
-            <span class="n">a</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">from_sympy</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">from_sympy</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-
-            <span class="n">coeff</span> <span class="o">=</span> <span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-
-        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-
-    <span class="n">exts</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">exts</span><span class="p">)</span>
-
-    <span class="n">g</span><span class="p">,</span> <span class="n">span</span><span class="p">,</span> <span class="n">H</span> <span class="o">=</span> <span class="n">primitive_element</span><span class="p">(</span><span class="n">exts</span><span class="p">,</span> <span class="n">ex</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">polys</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">root</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">([</span> <span class="n">s</span><span class="o">*</span><span class="n">ext</span> <span class="k">for</span> <span class="n">s</span><span class="p">,</span> <span class="n">ext</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">span</span><span class="p">,</span> <span class="n">exts</span><span class="p">)</span> <span class="p">])</span>
-
-    <span class="n">domain</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">algebraic_field</span><span class="p">((</span><span class="n">g</span><span class="p">,</span> <span class="n">root</span><span class="p">)),</span> <span class="n">g</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">rep</span>
-
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">result</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">a</span><span class="o">*</span><span class="n">domain</span><span class="o">.</span><span class="n">dtype</span><span class="o">.</span><span class="n">from_list</span><span class="p">(</span><span class="n">H</span><span class="p">[</span><span class="n">exts</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">coeff</span><span class="p">)],</span> <span class="n">g</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span> <span class="o">+</span> <span class="n">b</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">dtype</span><span class="o">.</span><span class="n">from_list</span><span class="p">([</span><span class="n">b</span><span class="p">],</span> <span class="n">g</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-
-        <span class="n">result</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">coeff</span>
-
-    <span class="k">return</span> <span class="n">domain</span><span class="p">,</span> <span class="n">result</span>
-
-
-<span class="k">def</span> <span class="nf">_construct_composite</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">opt</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Handle composite domains, e.g.: ZZ[X], QQ[X], ZZ(X), QQ(X). &quot;&quot;&quot;</span>
-    <span class="n">numers</span><span class="p">,</span> <span class="n">denoms</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">coeffs</span><span class="p">:</span>
-        <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="o">=</span> <span class="n">coeff</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-
-        <span class="n">numers</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">numer</span><span class="p">)</span>
-        <span class="n">denoms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">denom</span><span class="p">)</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">polys</span><span class="p">,</span> <span class="n">gens</span> <span class="o">=</span> <span class="n">parallel_dict_from_basic</span><span class="p">(</span><span class="n">numers</span> <span class="o">+</span> <span class="n">denoms</span><span class="p">)</span>  <span class="c"># XXX: sorting</span>
-    <span class="k">except</span> <span class="n">GeneratorsNeeded</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">gen</span><span class="o">.</span><span class="n">is_number</span> <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">None</span>  <span class="c"># generators are number-like so lets better use EX</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span>
-    <span class="n">k</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">polys</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span>
-
-    <span class="n">numers</span> <span class="o">=</span> <span class="n">polys</span><span class="p">[:</span><span class="n">k</span><span class="p">]</span>
-    <span class="n">denoms</span> <span class="o">=</span> <span class="n">polys</span><span class="p">[</span><span class="n">k</span><span class="p">:]</span>
-
-    <span class="k">if</span> <span class="n">opt</span><span class="o">.</span><span class="n">field</span><span class="p">:</span>
-        <span class="n">fractions</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">fractions</span><span class="p">,</span> <span class="n">zeros</span> <span class="o">=</span> <span class="bp">False</span><span class="p">,</span> <span class="p">(</span><span class="mi">0</span><span class="p">,)</span><span class="o">*</span><span class="n">n</span>
-
-        <span class="k">for</span> <span class="n">denom</span> <span class="ow">in</span> <span class="n">denoms</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">denom</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">zeros</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">denom</span><span class="p">:</span>
-                <span class="n">fractions</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">break</span>
-
-    <span class="n">coeffs</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([])</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">fractions</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">numers</span><span class="p">,</span> <span class="n">denoms</span><span class="p">):</span>
-            <span class="n">denom</span> <span class="o">=</span> <span class="n">denom</span><span class="p">[</span><span class="n">zeros</span><span class="p">]</span>
-
-            <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">numer</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                <span class="n">coeff</span> <span class="o">/=</span> <span class="n">denom</span>
-                <span class="n">coeffs</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-                <span class="n">numer</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span> <span class="o">=</span> <span class="n">coeff</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">numers</span><span class="p">,</span> <span class="n">denoms</span><span class="p">):</span>
-            <span class="n">coeffs</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">numer</span><span class="o">.</span><span class="n">values</span><span class="p">()))</span>
-            <span class="n">coeffs</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">denom</span><span class="o">.</span><span class="n">values</span><span class="p">()))</span>
-
-    <span class="n">rationals</span><span class="p">,</span> <span class="n">reals</span> <span class="o">=</span> <span class="bp">False</span><span class="p">,</span> <span class="bp">False</span>
-
-    <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">coeffs</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="n">rationals</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">elif</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-            <span class="n">reals</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">break</span>
-
-    <span class="k">if</span> <span class="n">reals</span><span class="p">:</span>
-        <span class="n">ground</span> <span class="o">=</span> <span class="n">RR</span>
-    <span class="k">elif</span> <span class="n">rationals</span><span class="p">:</span>
-        <span class="n">ground</span> <span class="o">=</span> <span class="n">QQ</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">ground</span> <span class="o">=</span> <span class="n">ZZ</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">fractions</span><span class="p">:</span>
-        <span class="n">domain</span> <span class="o">=</span> <span class="n">ground</span><span class="o">.</span><span class="n">poly_ring</span><span class="p">(</span><span class="o">*</span><span class="n">gens</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">numer</span> <span class="ow">in</span> <span class="n">numers</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">numer</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                <span class="n">numer</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span> <span class="o">=</span> <span class="n">ground</span><span class="o">.</span><span class="n">from_sympy</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">domain</span><span class="p">(</span><span class="n">numer</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">domain</span> <span class="o">=</span> <span class="n">ground</span><span class="o">.</span><span class="n">frac_field</span><span class="p">(</span><span class="o">*</span><span class="n">gens</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">numers</span><span class="p">,</span> <span class="n">denoms</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">numer</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                <span class="n">numer</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span> <span class="o">=</span> <span class="n">ground</span><span class="o">.</span><span class="n">from_sympy</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">denom</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                <span class="n">denom</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span> <span class="o">=</span> <span class="n">ground</span><span class="o">.</span><span class="n">from_sympy</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">domain</span><span class="p">((</span><span class="n">numer</span><span class="p">,</span> <span class="n">denom</span><span class="p">)))</span>
-
-    <span class="k">return</span> <span class="n">domain</span><span class="p">,</span> <span class="n">result</span>
-
-
-<span class="k">def</span> <span class="nf">_construct_expression</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">opt</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;The last resort case, i.e. use the expression domain. &quot;&quot;&quot;</span>
-    <span class="n">domain</span><span class="p">,</span> <span class="n">result</span> <span class="o">=</span> <span class="n">EX</span><span class="p">,</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">coeffs</span><span class="p">:</span>
-        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">domain</span><span class="o">.</span><span class="n">from_sympy</span><span class="p">(</span><span class="n">coeff</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">domain</span><span class="p">,</span> <span class="n">result</span>
-
-
-<div class="viewcode-block" id="construct_domain"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.constructor.construct_domain">[docs]</a><span class="k">def</span> <span class="nf">construct_domain</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Construct a minimal domain for the list of coefficients. &quot;&quot;&quot;</span>
-    <span class="n">opt</span> <span class="o">=</span> <span class="n">build_options</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="s">&#39;__iter__&#39;</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="nb">dict</span><span class="p">):</span>
-            <span class="n">monoms</span><span class="p">,</span> <span class="n">coeffs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="nb">list</span><span class="p">(</span><span class="n">obj</span><span class="o">.</span><span class="n">items</span><span class="p">())))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">coeffs</span> <span class="o">=</span> <span class="n">obj</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">coeffs</span> <span class="o">=</span> <span class="p">[</span><span class="n">obj</span><span class="p">]</span>
-
-    <span class="n">coeffs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="n">coeffs</span><span class="p">))</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="n">_construct_simple</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="n">domain</span><span class="p">,</span> <span class="n">coeffs</span> <span class="o">=</span> <span class="n">result</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">domain</span><span class="p">,</span> <span class="n">coeffs</span> <span class="o">=</span> <span class="n">_construct_expression</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">opt</span><span class="o">.</span><span class="n">composite</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">_construct_composite</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="bp">None</span>
-
-        <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">domain</span><span class="p">,</span> <span class="n">coeffs</span> <span class="o">=</span> <span class="n">result</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">domain</span><span class="p">,</span> <span class="n">coeffs</span> <span class="o">=</span> <span class="n">_construct_expression</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="s">&#39;__iter__&#39;</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="nb">dict</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">domain</span><span class="p">,</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">monoms</span><span class="p">,</span> <span class="n">coeffs</span><span class="p">)))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">domain</span><span class="p">,</span> <span class="n">coeffs</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">domain</span><span class="p">,</span> <span class="n">coeffs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span></div>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/polys/densearith.html b/dev-py3k/_modules/sympy/polys/densearith.html
deleted file mode 100644
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+++ /dev/null
@@ -1,2139 +0,0 @@
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-            
-  <h1>Source code for sympy.polys.densearith</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Arithmetics for dense recursive polynomials in ``K[x]`` or ``K[X]``. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dup_LC</span><span class="p">,</span> <span class="n">dmp_LC</span><span class="p">,</span>
-    <span class="n">dup_degree</span><span class="p">,</span> <span class="n">dmp_degree</span><span class="p">,</span>
-    <span class="n">dup_normal</span><span class="p">,</span>
-    <span class="n">dup_strip</span><span class="p">,</span> <span class="n">dmp_strip</span><span class="p">,</span>
-    <span class="n">dmp_zero_p</span><span class="p">,</span> <span class="n">dmp_zero</span><span class="p">,</span>
-    <span class="n">dmp_one_p</span><span class="p">,</span> <span class="n">dmp_one</span><span class="p">,</span>
-    <span class="n">dmp_ground</span><span class="p">,</span> <span class="n">dmp_zeros</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">ExactQuotientFailed</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">cythonized</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i,n,m&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_add_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Add ``c*x**i`` to ``f`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_add_term</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, -1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_add_term(f, ZZ(2), 4, ZZ)</span>
-<span class="sd">    [2, 0, 1, 0, -1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">c</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_strip</span><span class="p">([</span><span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">c</span><span class="p">]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="o">&gt;=</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[</span><span class="n">c</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="n">n</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="p">[:</span><span class="n">m</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">f</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">+</span> <span class="n">c</span><span class="p">]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i,u,v,n,m&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_add_term"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_add_term">[docs]</a><span class="k">def</span> <span class="nf">dmp_add_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Add ``c(x_2..x_u)*x_0**i`` to ``f`` in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_add_term</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1, 0], [1]])</span>
-<span class="sd">    &gt;&gt;&gt; c = ZZ.map([2])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_add_term(f, c, 2, 1, ZZ)</span>
-<span class="sd">    [[2], [1, 0], [1]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_add_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_strip</span><span class="p">([</span><span class="n">dmp_add</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">u</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="o">&gt;=</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[</span><span class="n">c</span><span class="p">]</span> <span class="o">+</span> <span class="n">dmp_zeros</span><span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="n">n</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="p">[:</span><span class="n">m</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">dmp_add</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">m</span><span class="p">],</span> <span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i,n,m&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_sub_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Subtract ``c*x**i`` from ``f`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_sub_term</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([2, 0, 1, 0, -1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_sub_term(f, ZZ(2), 4, ZZ)</span>
-<span class="sd">    [1, 0, -1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">c</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_strip</span><span class="p">([</span><span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="n">c</span><span class="p">]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="o">&gt;=</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[</span><span class="o">-</span><span class="n">c</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="n">n</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="p">[:</span><span class="n">m</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">f</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">-</span> <span class="n">c</span><span class="p">]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i,u,v,n,m&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_sub_term"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_sub_term">[docs]</a><span class="k">def</span> <span class="nf">dmp_sub_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Subtract ``c(x_2..x_u)*x_0**i`` from ``f`` in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_sub_term</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[2], [1, 0], [1]])</span>
-<span class="sd">    &gt;&gt;&gt; c = ZZ.map([2])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_sub_term(f, c, 2, 1, ZZ)</span>
-<span class="sd">    [[1, 0], [1]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_add_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">-</span><span class="n">c</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_strip</span><span class="p">([</span><span class="n">dmp_sub</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">u</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="o">&gt;=</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[</span><span class="n">dmp_neg</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)]</span> <span class="o">+</span> <span class="n">dmp_zeros</span><span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="n">n</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="p">[:</span><span class="n">m</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">dmp_sub</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">m</span><span class="p">],</span> <span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_mul_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Multiply ``f`` by ``c*x**i`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_mul_term</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, -1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_mul_term(f, ZZ(3), 2, ZZ)</span>
-<span class="sd">    [3, 0, -3, 0, 0]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">c</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="n">cf</span> <span class="o">*</span> <span class="n">c</span> <span class="k">for</span> <span class="n">cf</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span><span class="o">*</span><span class="n">i</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i,u,v&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_mul_term"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_mul_term">[docs]</a><span class="k">def</span> <span class="nf">dmp_mul_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Multiply ``f`` by ``c(x_2..x_u)*x_0**i`` in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_mul_term</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1, 0], [1], []])</span>
-<span class="sd">    &gt;&gt;&gt; c = ZZ.map([3, 0])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_mul_term(f, c, 2, 1, ZZ)</span>
-<span class="sd">    [[3, 0, 0], [3, 0], [], [], []]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_mul_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span>
-    <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">cf</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">cf</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span> <span class="o">+</span> <span class="n">dmp_zeros</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_add_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Add an element of the ground domain to ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_add_ground</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 2, 3, 4])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_add_ground(f, ZZ(4), ZZ)</span>
-<span class="sd">    [1, 2, 3, 8]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dup_add_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="dmp_add_ground"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_add_ground">[docs]</a><span class="k">def</span> <span class="nf">dmp_add_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Add an element of the ground domain to ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_add_ground</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [2], [3], [4]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_add_ground(f, ZZ(4), 1, ZZ)</span>
-<span class="sd">    [[1], [2], [3], [8]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dmp_add_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">dmp_ground</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_sub_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Subtract an element of the ground domain from ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_sub_ground</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 2, 3, 4])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_sub_ground(f, ZZ(4), ZZ)</span>
-<span class="sd">    [1, 2, 3, 0]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dup_sub_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="dmp_sub_ground"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_sub_ground">[docs]</a><span class="k">def</span> <span class="nf">dmp_sub_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Subtract an element of the ground domain from ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_sub_ground</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [2], [3], [4]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_sub_ground(f, ZZ(4), 1, ZZ)</span>
-<span class="sd">    [[1], [2], [3], []]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dmp_sub_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">dmp_ground</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_mul_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Multiply ``f`` by a constant value in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_mul_ground</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 2, -1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_mul_ground(f, ZZ(3), ZZ)</span>
-<span class="sd">    [3, 6, -3]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">c</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="n">cf</span> <span class="o">*</span> <span class="n">c</span> <span class="k">for</span> <span class="n">cf</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_mul_ground"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_mul_ground">[docs]</a><span class="k">def</span> <span class="nf">dmp_mul_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Multiply ``f`` by a constant value in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_mul_ground</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[2], [2, 0]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_mul_ground(f, ZZ(3), 1, ZZ)</span>
-<span class="sd">    [[6], [6, 0]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="p">[</span> <span class="n">dmp_mul_ground</span><span class="p">(</span><span class="n">cf</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">cf</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_quo_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Quotient by a constant in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ, QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_quo_ground</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([3, 0, 2])</span>
-<span class="sd">    &gt;&gt;&gt; g = QQ.map([3, 0, 2])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_quo_ground(f, ZZ(2), ZZ)</span>
-<span class="sd">    [1, 0, 1]</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_quo_ground(g, QQ(2), QQ)</span>
-<span class="sd">    [3/2, 0/1, 1/1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">c</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ZeroDivisionError</span><span class="p">(</span><span class="s">&#39;polynomial division&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">has_Field</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">K</span><span class="o">.</span><span class="n">is_Exact</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="n">K</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">cf</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">cf</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="n">cf</span> <span class="o">//</span> <span class="n">c</span> <span class="k">for</span> <span class="n">cf</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_quo_ground"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_quo_ground">[docs]</a><span class="k">def</span> <span class="nf">dmp_quo_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Quotient by a constant in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ, QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_quo_ground</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[2, 0], [3], []])</span>
-<span class="sd">    &gt;&gt;&gt; g = QQ.map([[2, 0], [3], []])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_quo_ground(f, ZZ(2), 1, ZZ)</span>
-<span class="sd">    [[1, 0], [1], []]</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_quo_ground(g, QQ(2), 1, QQ)</span>
-<span class="sd">    [[1/1, 0/1], [3/2], []]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_quo_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="p">[</span> <span class="n">dmp_quo_ground</span><span class="p">(</span><span class="n">cf</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">cf</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_exquo_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Exact quotient by a constant in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ, QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_exquo_ground</span>
-
-<span class="sd">    &gt;&gt;&gt; f = QQ.map([1, 0, 2])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_exquo_ground(f, QQ(2), QQ)</span>
-<span class="sd">    [1/2, 0/1, 1/1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">c</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ZeroDivisionError</span><span class="p">(</span><span class="s">&#39;polynomial division&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="k">return</span> <span class="p">[</span> <span class="n">K</span><span class="o">.</span><span class="n">exquo</span><span class="p">(</span><span class="n">cf</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">cf</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_exquo_ground"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_exquo_ground">[docs]</a><span class="k">def</span> <span class="nf">dmp_exquo_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Exact quotient by a constant in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ, QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_exquo_ground</span>
-
-<span class="sd">    &gt;&gt;&gt; f = QQ.map([[1, 0], [2], []])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_exquo_ground(f, QQ(2), 1, QQ)</span>
-<span class="sd">    [[1/2, 0/1], [1/1], []]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_exquo_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="p">[</span> <span class="n">dmp_exquo_ground</span><span class="p">(</span><span class="n">cf</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">cf</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dup_lshift"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dup_lshift">[docs]</a><span class="k">def</span> <span class="nf">dup_lshift</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Efficiently multiply ``f`` by ``x**n`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_lshift</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, 1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_lshift(f, 2, ZZ)</span>
-<span class="sd">    [1, 0, 1, 0, 0]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span> <span class="o">+</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span><span class="o">*</span><span class="n">n</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dup_rshift"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dup_rshift">[docs]</a><span class="k">def</span> <span class="nf">dup_rshift</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Efficiently divide ``f`` by ``x**n`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_rshift</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, 1, 0, 0])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([1, 0, 1, 0, 2])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_rshift(f, 2, ZZ)</span>
-<span class="sd">    [1, 0, 1]</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_rshift(g, 2, ZZ)</span>
-<span class="sd">    [1, 0, 1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">f</span><span class="p">[:</span><span class="o">-</span><span class="n">n</span><span class="p">]</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_abs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Make all coefficients positive in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_abs</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, -1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_abs(f, ZZ)</span>
-<span class="sd">    [1, 0, 1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">[</span> <span class="n">K</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span> <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_abs"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_abs">[docs]</a><span class="k">def</span> <span class="nf">dmp_abs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Make all coefficients positive in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_abs</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1, 0], [-1], []])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_abs(f, 1, ZZ)</span>
-<span class="sd">    [[1, 0], [1], []]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_abs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="p">[</span> <span class="n">dmp_abs</span><span class="p">(</span><span class="n">cf</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">cf</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_neg</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Negate a polynomial in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_neg</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, -1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_neg(f, ZZ)</span>
-<span class="sd">    [-1, 0, 1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">[</span> <span class="o">-</span><span class="n">coeff</span> <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_neg"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_neg">[docs]</a><span class="k">def</span> <span class="nf">dmp_neg</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Negate a polynomial in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_neg</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1, 0], [-1], []])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_neg(f, 1, ZZ)</span>
-<span class="sd">    [[-1, 0], [1], []]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_neg</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="p">[</span> <span class="n">dmp_neg</span><span class="p">(</span><span class="n">cf</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">cf</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;df,dg,k&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_add</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Add dense polynomials in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_add</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, -1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([1, -2])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_add(f, g, ZZ)</span>
-<span class="sd">    [1, 1, -3]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">g</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">df</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">dg</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">df</span> <span class="o">==</span> <span class="n">dg</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_strip</span><span class="p">([</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span> <span class="p">])</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">df</span> <span class="o">-</span> <span class="n">dg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">df</span> <span class="o">&gt;</span> <span class="n">dg</span><span class="p">:</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="p">[:</span><span class="n">k</span><span class="p">],</span> <span class="n">f</span><span class="p">[</span><span class="n">k</span><span class="p">:]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="p">[:</span><span class="n">k</span><span class="p">],</span> <span class="n">g</span><span class="p">[</span><span class="n">k</span><span class="p">:]</span>
-
-        <span class="k">return</span> <span class="n">h</span> <span class="o">+</span> <span class="p">[</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span> <span class="p">]</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v,df,dg,k&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_add"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_add">[docs]</a><span class="k">def</span> <span class="nf">dmp_add</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Add dense polynomials in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_add</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [], [1, 0]])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[1, 0], [1], []])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_add(f, g, 1, ZZ)</span>
-<span class="sd">    [[1, 1], [1], [1, 0]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_add</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">df</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">df</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">g</span>
-
-    <span class="n">dg</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dg</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="n">df</span> <span class="o">==</span> <span class="n">dg</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_strip</span><span class="p">([</span> <span class="n">dmp_add</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span> <span class="p">],</span> <span class="n">u</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">df</span> <span class="o">-</span> <span class="n">dg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">df</span> <span class="o">&gt;</span> <span class="n">dg</span><span class="p">:</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="p">[:</span><span class="n">k</span><span class="p">],</span> <span class="n">f</span><span class="p">[</span><span class="n">k</span><span class="p">:]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="p">[:</span><span class="n">k</span><span class="p">],</span> <span class="n">g</span><span class="p">[</span><span class="n">k</span><span class="p">:]</span>
-
-        <span class="k">return</span> <span class="n">h</span> <span class="o">+</span> <span class="p">[</span> <span class="n">dmp_add</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span> <span class="p">]</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;df,dg,k&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_sub</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Subtract dense polynomials in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_sub</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, -1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([1, -2])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_sub(f, g, ZZ)</span>
-<span class="sd">    [1, -1, 1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_neg</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">df</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">dg</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">df</span> <span class="o">==</span> <span class="n">dg</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_strip</span><span class="p">([</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span> <span class="p">])</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">df</span> <span class="o">-</span> <span class="n">dg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">df</span> <span class="o">&gt;</span> <span class="n">dg</span><span class="p">:</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="p">[:</span><span class="n">k</span><span class="p">],</span> <span class="n">f</span><span class="p">[</span><span class="n">k</span><span class="p">:]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">dup_neg</span><span class="p">(</span><span class="n">g</span><span class="p">[:</span><span class="n">k</span><span class="p">],</span> <span class="n">K</span><span class="p">),</span> <span class="n">g</span><span class="p">[</span><span class="n">k</span><span class="p">:]</span>
-
-        <span class="k">return</span> <span class="n">h</span> <span class="o">+</span> <span class="p">[</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span> <span class="p">]</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v,df,dg,k&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_sub"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_sub">[docs]</a><span class="k">def</span> <span class="nf">dmp_sub</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Subtract dense polynomials in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_sub</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [], [1, 0]])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[1, 0], [1], []])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_sub(f, g, 1, ZZ)</span>
-<span class="sd">    [[-1, 1], [-1], [1, 0]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_sub</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">df</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">df</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_neg</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">dg</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dg</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="n">df</span> <span class="o">==</span> <span class="n">dg</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_strip</span><span class="p">([</span> <span class="n">dmp_sub</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span> <span class="p">],</span> <span class="n">u</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">df</span> <span class="o">-</span> <span class="n">dg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">df</span> <span class="o">&gt;</span> <span class="n">dg</span><span class="p">:</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="p">[:</span><span class="n">k</span><span class="p">],</span> <span class="n">f</span><span class="p">[</span><span class="n">k</span><span class="p">:]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">dmp_neg</span><span class="p">(</span><span class="n">g</span><span class="p">[:</span><span class="n">k</span><span class="p">],</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">g</span><span class="p">[</span><span class="n">k</span><span class="p">:]</span>
-
-        <span class="k">return</span> <span class="n">h</span> <span class="o">+</span> <span class="p">[</span> <span class="n">dmp_sub</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span> <span class="p">]</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_add_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns ``f + g*h`` where ``f, g, h`` are in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_add_mul</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, -1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([1, -2])</span>
-<span class="sd">    &gt;&gt;&gt; h = ZZ.map([1, 2])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_add_mul(f, g, h, ZZ)</span>
-<span class="sd">    [2, 0, -5]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dup_add</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">dup_mul</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_add_mul"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_add_mul">[docs]</a><span class="k">def</span> <span class="nf">dmp_add_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns ``f + g*h`` where ``f, g, h`` are in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_add_mul</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [], [1, 0]])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[1], []])</span>
-<span class="sd">    &gt;&gt;&gt; h = ZZ.map([[1], [2]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_add_mul(f, g, h, 1, ZZ)</span>
-<span class="sd">    [[2], [2], [1, 0]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dmp_add</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_sub_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns ``f - g*h`` where ``f, g, h`` are in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_sub_mul</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, -1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([1, -2])</span>
-<span class="sd">    &gt;&gt;&gt; h = ZZ.map([1, 2])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_sub_mul(f, g, h, ZZ)</span>
-<span class="sd">    [3]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dup_sub</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">dup_mul</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_sub_mul"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_sub_mul">[docs]</a><span class="k">def</span> <span class="nf">dmp_sub_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns ``f - g*h`` where ``f, g, h`` are in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_sub_mul</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [], [1, 0]])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[1], []])</span>
-<span class="sd">    &gt;&gt;&gt; h = ZZ.map([[1], [2]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_sub_mul(f, g, h, 1, ZZ)</span>
-<span class="sd">    [[-2], [1, 0]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dmp_sub</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;df,dg,i,j&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Multiply dense polynomials in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_mul</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, -2])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([1, 2])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_mul(f, g, ZZ)</span>
-<span class="sd">    [1, 0, -4]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">f</span> <span class="o">==</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_sqr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">f</span> <span class="ow">and</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-    <span class="n">df</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">dg</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">df</span> <span class="o">+</span> <span class="n">dg</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">coeff</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">i</span> <span class="o">-</span> <span class="n">dg</span><span class="p">),</span> <span class="nb">min</span><span class="p">(</span><span class="n">df</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">coeff</span> <span class="o">+=</span> <span class="n">f</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="n">j</span><span class="p">]</span>
-
-        <span class="n">h</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">dup_strip</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v,df,dg,i,j&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_mul"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_mul">[docs]</a><span class="k">def</span> <span class="nf">dmp_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Multiply dense polynomials in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_mul</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1, 0], [1]])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[1], []])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_mul(f, g, 1, ZZ)</span>
-<span class="sd">    [[1, 0], [1], []]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">f</span> <span class="o">==</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_sqr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">df</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">df</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">dg</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dg</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">g</span>
-
-    <span class="n">h</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="p">[],</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">df</span> <span class="o">+</span> <span class="n">dg</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">coeff</span> <span class="o">=</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">i</span> <span class="o">-</span> <span class="n">dg</span><span class="p">),</span> <span class="nb">min</span><span class="p">(</span><span class="n">df</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">dmp_add</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">j</span><span class="p">],</span> <span class="n">g</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="n">j</span><span class="p">],</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">h</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">dmp_strip</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;df,jmin,jmax,n,i,j&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_sqr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Square dense polynomials in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_sqr</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, 1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_sqr(f, ZZ)</span>
-<span class="sd">    [1, 0, 2, 0, 1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">df</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">df</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-
-        <span class="n">jmin</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">i</span> <span class="o">-</span> <span class="n">df</span><span class="p">)</span>
-        <span class="n">jmax</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">df</span><span class="p">)</span>
-
-        <span class="n">n</span> <span class="o">=</span> <span class="n">jmax</span> <span class="o">-</span> <span class="n">jmin</span> <span class="o">+</span> <span class="mi">1</span>
-
-        <span class="n">jmax</span> <span class="o">=</span> <span class="n">jmin</span> <span class="o">+</span> <span class="n">n</span> <span class="o">//</span> <span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">jmin</span><span class="p">,</span> <span class="n">jmax</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">c</span> <span class="o">+=</span> <span class="n">f</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="n">j</span><span class="p">]</span>
-
-        <span class="n">c</span> <span class="o">+=</span> <span class="n">c</span>
-
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&amp;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">elem</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="n">jmax</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
-            <span class="n">c</span> <span class="o">+=</span> <span class="n">elem</span><span class="o">**</span><span class="mi">2</span>
-
-        <span class="n">h</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">dup_strip</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v,df,jmin,jmax,n,i,j&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_sqr"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_sqr">[docs]</a><span class="k">def</span> <span class="nf">dmp_sqr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Square dense polynomials in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_sqr</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [1, 0], [1, 0, 0]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_sqr(f, 1, ZZ)</span>
-<span class="sd">    [[1], [2, 0], [3, 0, 0], [2, 0, 0, 0], [1, 0, 0, 0, 0]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_sqr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">df</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">df</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">h</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="p">[],</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">df</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-
-        <span class="n">jmin</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">i</span> <span class="o">-</span> <span class="n">df</span><span class="p">)</span>
-        <span class="n">jmax</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">df</span><span class="p">)</span>
-
-        <span class="n">n</span> <span class="o">=</span> <span class="n">jmax</span> <span class="o">-</span> <span class="n">jmin</span> <span class="o">+</span> <span class="mi">1</span>
-
-        <span class="n">jmax</span> <span class="o">=</span> <span class="n">jmin</span> <span class="o">+</span> <span class="n">n</span> <span class="o">//</span> <span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">jmin</span><span class="p">,</span> <span class="n">jmax</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">dmp_add</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">j</span><span class="p">],</span> <span class="n">f</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="n">j</span><span class="p">],</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">c</span> <span class="o">=</span> <span class="n">dmp_mul_ground</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&amp;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">elem</span> <span class="o">=</span> <span class="n">dmp_sqr</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">jmax</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">dmp_add</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">elem</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">h</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">dmp_strip</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,m&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_pow</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Raise ``f`` to the ``n``-th power in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_pow</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_pow([ZZ(1), -ZZ(2)], 3, ZZ)</span>
-<span class="sd">    [1, -6, 12, -8]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;can&#39;t raise polynomial to a negative power&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">f</span> <span class="ow">or</span> <span class="n">f</span> <span class="o">==</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">g</span> <span class="o">=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">n</span><span class="o">//</span><span class="mi">2</span><span class="p">,</span> <span class="n">n</span>
-
-        <span class="k">if</span> <span class="n">m</span> <span class="o">%</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">dup_mul</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-                <span class="k">break</span>
-
-        <span class="n">f</span> <span class="o">=</span> <span class="n">dup_sqr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">g</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,n,m&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_pow"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_pow">[docs]</a><span class="k">def</span> <span class="nf">dmp_pow</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Raise ``f`` to the ``n``-th power in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_pow</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1, 0], [1]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_pow(f, 3, 1, ZZ)</span>
-<span class="sd">    [[1, 0, 0, 0], [3, 0, 0], [3, 0], [1]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_pow</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_one</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;can&#39;t raise polynomial to a negative power&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span> <span class="ow">or</span> <span class="n">dmp_one_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">g</span> <span class="o">=</span> <span class="n">dmp_one</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">n</span><span class="o">//</span><span class="mi">2</span><span class="p">,</span> <span class="n">n</span>
-
-        <span class="k">if</span> <span class="n">m</span> <span class="o">&amp;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-                <span class="k">break</span>
-
-        <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_sqr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">g</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;df,dg,dr,N,j&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_pdiv</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Polynomial pseudo-division in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_pdiv</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, 1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([2, -4])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_pdiv(f, g, ZZ)</span>
-<span class="sd">    ([2, 4], [20])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">df</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">dg</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="p">[],</span> <span class="n">f</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ZeroDivisionError</span><span class="p">(</span><span class="s">&quot;polynomial division&quot;</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">df</span> <span class="o">&lt;</span> <span class="n">dg</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">q</span><span class="p">,</span> <span class="n">r</span>
-
-    <span class="n">N</span> <span class="o">=</span> <span class="n">df</span> <span class="o">-</span> <span class="n">dg</span> <span class="o">+</span> <span class="mi">1</span>
-    <span class="n">lc_g</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">dr</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">dr</span> <span class="o">&lt;</span> <span class="n">dg</span><span class="p">:</span>
-            <span class="k">break</span>
-
-        <span class="n">lc_r</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">j</span><span class="p">,</span> <span class="n">N</span> <span class="o">=</span> <span class="n">dr</span> <span class="o">-</span> <span class="n">dg</span><span class="p">,</span> <span class="n">N</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="n">Q</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">lc_g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">q</span> <span class="o">=</span> <span class="n">dup_add_term</span><span class="p">(</span><span class="n">Q</span><span class="p">,</span> <span class="n">lc_r</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">R</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">lc_g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">G</span> <span class="o">=</span> <span class="n">dup_mul_term</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">lc_r</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">dup_sub</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">c</span> <span class="o">=</span> <span class="n">lc_g</span><span class="o">**</span><span class="n">N</span>
-
-    <span class="n">q</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">q</span><span class="p">,</span> <span class="n">r</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;df,dg,dr,N,j&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_prem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Polynomial pseudo-remainder in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_prem</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, 1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([2, -4])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_prem(f, g, ZZ)</span>
-<span class="sd">    [20]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">df</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">dg</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="n">r</span> <span class="o">=</span> <span class="n">f</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ZeroDivisionError</span><span class="p">(</span><span class="s">&quot;polynomial division&quot;</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">df</span> <span class="o">&lt;</span> <span class="n">dg</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">r</span>
-
-    <span class="n">N</span> <span class="o">=</span> <span class="n">df</span> <span class="o">-</span> <span class="n">dg</span> <span class="o">+</span> <span class="mi">1</span>
-    <span class="n">lc_g</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">dr</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">dr</span> <span class="o">&lt;</span> <span class="n">dg</span><span class="p">:</span>
-            <span class="k">break</span>
-
-        <span class="n">lc_r</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">j</span><span class="p">,</span> <span class="n">N</span> <span class="o">=</span> <span class="n">dr</span> <span class="o">-</span> <span class="n">dg</span><span class="p">,</span> <span class="n">N</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="n">R</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">lc_g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">G</span> <span class="o">=</span> <span class="n">dup_mul_term</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">lc_r</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">dup_sub</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">lc_g</span><span class="o">**</span><span class="n">N</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">dup_pquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Polynomial exact pseudo-quotient in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_pquo</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, -1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([2, -2])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_pquo(f, g, ZZ)</span>
-<span class="sd">    [2, 2]</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, 1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([2, -4])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_pquo(f, g, ZZ)</span>
-<span class="sd">    [2, 4]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dup_pdiv</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">dup_pexquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Polynomial pseudo-quotient in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_pexquo</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, -1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([2, -2])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_pexquo(f, g, ZZ)</span>
-<span class="sd">    [2, 2]</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, 1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([2, -4])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_pexquo(f, g, ZZ)</span>
-<span class="sd">    Traceback (most recent call last):</span>
-<span class="sd">    ...</span>
-<span class="sd">    ExactQuotientFailed: [2, -4] does not divide [1, 0, 1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dup_pdiv</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">q</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ExactQuotientFailed</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,df,dg,dr,N,j&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_pdiv"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_pdiv">[docs]</a><span class="k">def</span> <span class="nf">dmp_pdiv</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Polynomial pseudo-division in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_pdiv</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [1, 0], []])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[2], [2]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_pdiv(f, g, 1, ZZ)</span>
-<span class="sd">    ([[2], [2, -2]], [[-4, 4]])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_pdiv</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">df</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-    <span class="n">dg</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dg</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ZeroDivisionError</span><span class="p">(</span><span class="s">&quot;polynomial division&quot;</span><span class="p">)</span>
-
-    <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">f</span>
-
-    <span class="k">if</span> <span class="n">df</span> <span class="o">&lt;</span> <span class="n">dg</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">q</span><span class="p">,</span> <span class="n">r</span>
-
-    <span class="n">N</span> <span class="o">=</span> <span class="n">df</span> <span class="o">-</span> <span class="n">dg</span> <span class="o">+</span> <span class="mi">1</span>
-    <span class="n">lc_g</span> <span class="o">=</span> <span class="n">dmp_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">dr</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">dr</span> <span class="o">&lt;</span> <span class="n">dg</span><span class="p">:</span>
-            <span class="k">break</span>
-
-        <span class="n">lc_r</span> <span class="o">=</span> <span class="n">dmp_LC</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">j</span><span class="p">,</span> <span class="n">N</span> <span class="o">=</span> <span class="n">dr</span> <span class="o">-</span> <span class="n">dg</span><span class="p">,</span> <span class="n">N</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="n">Q</span> <span class="o">=</span> <span class="n">dmp_mul_term</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">lc_g</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">q</span> <span class="o">=</span> <span class="n">dmp_add_term</span><span class="p">(</span><span class="n">Q</span><span class="p">,</span> <span class="n">lc_r</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">R</span> <span class="o">=</span> <span class="n">dmp_mul_term</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">lc_g</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">G</span> <span class="o">=</span> <span class="n">dmp_mul_term</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">lc_r</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_sub</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">c</span> <span class="o">=</span> <span class="n">dmp_pow</span><span class="p">(</span><span class="n">lc_g</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">q</span> <span class="o">=</span> <span class="n">dmp_mul_term</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_mul_term</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">q</span><span class="p">,</span> <span class="n">r</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,df,dg,dr,N,j&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_prem"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_prem">[docs]</a><span class="k">def</span> <span class="nf">dmp_prem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Polynomial pseudo-remainder in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_prem</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [1, 0], []])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[2], [2]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_prem(f, g, 1, ZZ)</span>
-<span class="sd">    [[-4, 4]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_prem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">df</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-    <span class="n">dg</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dg</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ZeroDivisionError</span><span class="p">(</span><span class="s">&quot;polynomial division&quot;</span><span class="p">)</span>
-
-    <span class="n">r</span> <span class="o">=</span> <span class="n">f</span>
-
-    <span class="k">if</span> <span class="n">df</span> <span class="o">&lt;</span> <span class="n">dg</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">r</span>
-
-    <span class="n">N</span> <span class="o">=</span> <span class="n">df</span> <span class="o">-</span> <span class="n">dg</span> <span class="o">+</span> <span class="mi">1</span>
-    <span class="n">lc_g</span> <span class="o">=</span> <span class="n">dmp_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">dr</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">dr</span> <span class="o">&lt;</span> <span class="n">dg</span><span class="p">:</span>
-            <span class="k">break</span>
-
-        <span class="n">lc_r</span> <span class="o">=</span> <span class="n">dmp_LC</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">j</span><span class="p">,</span> <span class="n">N</span> <span class="o">=</span> <span class="n">dr</span> <span class="o">-</span> <span class="n">dg</span><span class="p">,</span> <span class="n">N</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="n">R</span> <span class="o">=</span> <span class="n">dmp_mul_term</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">lc_g</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">G</span> <span class="o">=</span> <span class="n">dmp_mul_term</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">lc_r</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_sub</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">c</span> <span class="o">=</span> <span class="n">dmp_pow</span><span class="p">(</span><span class="n">lc_g</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">dmp_mul_term</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="dmp_pquo"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_pquo">[docs]</a><span class="k">def</span> <span class="nf">dmp_pquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Polynomial exact pseudo-quotient in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_pquo</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [1, 0], []])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[2], [2, 0]])</span>
-<span class="sd">    &gt;&gt;&gt; h = ZZ.map([[2], [2]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_pquo(f, g, 1, ZZ)</span>
-<span class="sd">    [[2], []]</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_pquo(f, h, 1, ZZ)</span>
-<span class="sd">    [[2], [2, -2]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dmp_pdiv</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="dmp_pexquo"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_pexquo">[docs]</a><span class="k">def</span> <span class="nf">dmp_pexquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Polynomial pseudo-quotient in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_pexquo</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [1, 0], []])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[2], [2, 0]])</span>
-<span class="sd">    &gt;&gt;&gt; h = ZZ.map([[2], [2]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_pexquo(f, g, 1, ZZ)</span>
-<span class="sd">    [[2], []]</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_pexquo(f, h, 1, ZZ)</span>
-<span class="sd">    Traceback (most recent call last):</span>
-<span class="sd">    ...</span>
-<span class="sd">    ExactQuotientFailed: [[2], [2]] does not divide [[1], [1, 0], []]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_pdiv</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">q</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ExactQuotientFailed</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;df,dg,dr,j&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_rr_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Univariate division with remainder over a ring.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_rr_div</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, 1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([2, -4])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_rr_div(f, g, ZZ)</span>
-<span class="sd">    ([], [1, 0, 1])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">df</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">dg</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="p">[],</span> <span class="n">f</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ZeroDivisionError</span><span class="p">(</span><span class="s">&quot;polynomial division&quot;</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">df</span> <span class="o">&lt;</span> <span class="n">dg</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">q</span><span class="p">,</span> <span class="n">r</span>
-
-    <span class="n">lc_g</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">dr</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">dr</span> <span class="o">&lt;</span> <span class="n">dg</span><span class="p">:</span>
-            <span class="k">break</span>
-
-        <span class="n">lc_r</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">lc_r</span> <span class="o">%</span> <span class="n">lc_g</span><span class="p">:</span>
-            <span class="k">break</span>
-
-        <span class="n">c</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">exquo</span><span class="p">(</span><span class="n">lc_r</span><span class="p">,</span> <span class="n">lc_g</span><span class="p">)</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="n">dr</span> <span class="o">-</span> <span class="n">dg</span>
-
-        <span class="n">q</span> <span class="o">=</span> <span class="n">dup_add_term</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dup_mul_term</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">r</span> <span class="o">=</span> <span class="n">dup_sub</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">q</span><span class="p">,</span> <span class="n">r</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,df,dg,dr,j&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_rr_div"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_rr_div">[docs]</a><span class="k">def</span> <span class="nf">dmp_rr_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Multivariate division with remainder over a ring.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_rr_div</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [1, 0], []])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[2], [2]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_rr_div(f, g, 1, ZZ)</span>
-<span class="sd">    ([[]], [[1], [1, 0], []])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_rr_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">df</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-    <span class="n">dg</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dg</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ZeroDivisionError</span><span class="p">(</span><span class="s">&quot;polynomial division&quot;</span><span class="p">)</span>
-
-    <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">f</span>
-
-    <span class="k">if</span> <span class="n">df</span> <span class="o">&lt;</span> <span class="n">dg</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">q</span><span class="p">,</span> <span class="n">r</span>
-
-    <span class="n">lc_g</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">dmp_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">dr</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">dr</span> <span class="o">&lt;</span> <span class="n">dg</span><span class="p">:</span>
-            <span class="k">break</span>
-
-        <span class="n">lc_r</span> <span class="o">=</span> <span class="n">dmp_LC</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">c</span><span class="p">,</span> <span class="n">R</span> <span class="o">=</span> <span class="n">dmp_rr_div</span><span class="p">(</span><span class="n">lc_r</span><span class="p">,</span> <span class="n">lc_g</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">v</span><span class="p">):</span>
-            <span class="k">break</span>
-
-        <span class="n">j</span> <span class="o">=</span> <span class="n">dr</span> <span class="o">-</span> <span class="n">dg</span>
-
-        <span class="n">q</span> <span class="o">=</span> <span class="n">dmp_add_term</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_mul_term</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_sub</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">q</span><span class="p">,</span> <span class="n">r</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;df,dg,dr,j&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_ff_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Polynomial division with remainder over a field.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_ff_div</span>
-
-<span class="sd">    &gt;&gt;&gt; f = QQ.map([1, 0, 1])</span>
-<span class="sd">    &gt;&gt;&gt; g = QQ.map([2, -4])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_ff_div(f, g, QQ)</span>
-<span class="sd">    ([1/2, 1/1], [5/1])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">df</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">dg</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="p">[],</span> <span class="n">f</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ZeroDivisionError</span><span class="p">(</span><span class="s">&quot;polynomial division&quot;</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">df</span> <span class="o">&lt;</span> <span class="n">dg</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">q</span><span class="p">,</span> <span class="n">r</span>
-
-    <span class="n">lc_g</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">dr</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">dr</span> <span class="o">&lt;</span> <span class="n">dg</span><span class="p">:</span>
-            <span class="k">break</span>
-
-        <span class="n">lc_r</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">c</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">exquo</span><span class="p">(</span><span class="n">lc_r</span><span class="p">,</span> <span class="n">lc_g</span><span class="p">)</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="n">dr</span> <span class="o">-</span> <span class="n">dg</span>
-
-        <span class="n">q</span> <span class="o">=</span> <span class="n">dup_add_term</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dup_mul_term</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">r</span> <span class="o">=</span> <span class="n">dup_sub</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">K</span><span class="o">.</span><span class="n">is_Exact</span><span class="p">:</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">dup_normal</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">q</span><span class="p">,</span> <span class="n">r</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,df,dg,dr,j&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_ff_div"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_ff_div">[docs]</a><span class="k">def</span> <span class="nf">dmp_ff_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Polynomial division with remainder over a field.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_ff_div</span>
-
-<span class="sd">    &gt;&gt;&gt; f = QQ.map([[1], [1, 0], []])</span>
-<span class="sd">    &gt;&gt;&gt; g = QQ.map([[2], [2]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_ff_div(f, g, 1, QQ)</span>
-<span class="sd">    ([[1/2], [1/2, -1/2]], [[-1/1, 1/1]])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_ff_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">df</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-    <span class="n">dg</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dg</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ZeroDivisionError</span><span class="p">(</span><span class="s">&quot;polynomial division&quot;</span><span class="p">)</span>
-
-    <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">f</span>
-
-    <span class="k">if</span> <span class="n">df</span> <span class="o">&lt;</span> <span class="n">dg</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">q</span><span class="p">,</span> <span class="n">r</span>
-
-    <span class="n">lc_g</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">dmp_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">dr</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">dr</span> <span class="o">&lt;</span> <span class="n">dg</span><span class="p">:</span>
-            <span class="k">break</span>
-
-        <span class="n">lc_r</span> <span class="o">=</span> <span class="n">dmp_LC</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">c</span><span class="p">,</span> <span class="n">R</span> <span class="o">=</span> <span class="n">dmp_ff_div</span><span class="p">(</span><span class="n">lc_r</span><span class="p">,</span> <span class="n">lc_g</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">v</span><span class="p">):</span>
-            <span class="k">break</span>
-
-        <span class="n">j</span> <span class="o">=</span> <span class="n">dr</span> <span class="o">-</span> <span class="n">dg</span>
-
-        <span class="n">q</span> <span class="o">=</span> <span class="n">dmp_add_term</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_mul_term</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_sub</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">q</span><span class="p">,</span> <span class="n">r</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Polynomial division with remainder in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ, QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_div</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, 1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([2, -4])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_div(f, g, ZZ)</span>
-<span class="sd">    ([], [1, 0, 1])</span>
-
-<span class="sd">    &gt;&gt;&gt; f = QQ.map([1, 0, 1])</span>
-<span class="sd">    &gt;&gt;&gt; g = QQ.map([2, -4])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_div(f, g, QQ)</span>
-<span class="sd">    ([1/2, 1/1], [5/1])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">has_Field</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">K</span><span class="o">.</span><span class="n">is_Exact</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_ff_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_rr_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">dup_rem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns polynomial remainder in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ, QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_rem</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, 1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([2, -4])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_rem(f, g, ZZ)</span>
-<span class="sd">    [1, 0, 1]</span>
-
-<span class="sd">    &gt;&gt;&gt; f = QQ.map([1, 0, 1])</span>
-<span class="sd">    &gt;&gt;&gt; g = QQ.map([2, -4])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_rem(f, g, QQ)</span>
-<span class="sd">    [5/1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dup_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">dup_quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns exact polynomial quotient in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ, QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_quo</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, 1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([2, -4])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_quo(f, g, ZZ)</span>
-<span class="sd">    []</span>
-
-<span class="sd">    &gt;&gt;&gt; f = QQ.map([1, 0, 1])</span>
-<span class="sd">    &gt;&gt;&gt; g = QQ.map([2, -4])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_quo(f, g, QQ)</span>
-<span class="sd">    [1/2, 1/1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dup_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">dup_exquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns polynomial quotient in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_exquo</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, -1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([1, -1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_exquo(f, g, ZZ)</span>
-<span class="sd">    [1, 1]</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, 1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([2, -4])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_exquo(f, g, ZZ)</span>
-<span class="sd">    Traceback (most recent call last):</span>
-<span class="sd">    ...</span>
-<span class="sd">    ExactQuotientFailed: [2, -4] does not divide [1, 0, 1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dup_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">q</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ExactQuotientFailed</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_div"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_div">[docs]</a><span class="k">def</span> <span class="nf">dmp_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Polynomial division with remainder in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ, QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_div</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [1, 0], []])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[2], [2]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_div(f, g, 1, ZZ)</span>
-<span class="sd">    ([[]], [[1], [1, 0], []])</span>
-
-<span class="sd">    &gt;&gt;&gt; f = QQ.map([[1], [1, 0], []])</span>
-<span class="sd">    &gt;&gt;&gt; g = QQ.map([[2], [2]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_div(f, g, 1, QQ)</span>
-<span class="sd">    ([[1/2], [1/2, -1/2]], [[-1/1, 1/1]])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">has_Field</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">K</span><span class="o">.</span><span class="n">is_Exact</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_ff_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_rr_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_rem"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_rem">[docs]</a><span class="k">def</span> <span class="nf">dmp_rem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns polynomial remainder in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ, QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_rem</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [1, 0], []])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[2], [2]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_rem(f, g, 1, ZZ)</span>
-<span class="sd">    [[1], [1, 0], []]</span>
-
-<span class="sd">    &gt;&gt;&gt; f = QQ.map([[1], [1, 0], []])</span>
-<span class="sd">    &gt;&gt;&gt; g = QQ.map([[2], [2]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_rem(f, g, 1, QQ)</span>
-<span class="sd">    [[-1/1, 1/1]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dmp_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_quo"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_quo">[docs]</a><span class="k">def</span> <span class="nf">dmp_quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns exact polynomial quotient in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ, QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_quo</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [1, 0], []])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[2], [2]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_quo(f, g, 1, ZZ)</span>
-<span class="sd">    [[]]</span>
-
-<span class="sd">    &gt;&gt;&gt; f = QQ.map([[1], [1, 0], []])</span>
-<span class="sd">    &gt;&gt;&gt; g = QQ.map([[2], [2]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_quo(f, g, 1, QQ)</span>
-<span class="sd">    [[1/2], [1/2, -1/2]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dmp_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_exquo"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_exquo">[docs]</a><span class="k">def</span> <span class="nf">dmp_exquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns polynomial quotient in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_exquo</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [1, 0], []])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[1], [1, 0]])</span>
-<span class="sd">    &gt;&gt;&gt; h = ZZ.map([[2], [2]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_exquo(f, g, 1, ZZ)</span>
-<span class="sd">    [[1], []]</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_exquo(f, h, 1, ZZ)</span>
-<span class="sd">    Traceback (most recent call last):</span>
-<span class="sd">    ...</span>
-<span class="sd">    ExactQuotientFailed: [[2], [2]] does not divide [[1], [1, 0], []]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">q</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ExactQuotientFailed</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_max_norm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns maximum norm of a polynomial in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_max_norm</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([-1, 2, -3])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_max_norm(f, ZZ)</span>
-<span class="sd">    3</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">max</span><span class="p">(</span><span class="n">dup_abs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_max_norm"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_max_norm">[docs]</a><span class="k">def</span> <span class="nf">dmp_max_norm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns maximum norm of a polynomial in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_max_norm</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[2, -1], [-3]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_max_norm(f, 1, ZZ)</span>
-<span class="sd">    3</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_max_norm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="nb">max</span><span class="p">([</span> <span class="n">dmp_max_norm</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">])</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_l1_norm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns l1 norm of a polynomial in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_l1_norm</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([2, -3, 0, 1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_l1_norm(f, ZZ)</span>
-<span class="sd">    6</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="n">dup_abs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_l1_norm"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_l1_norm">[docs]</a><span class="k">def</span> <span class="nf">dmp_l1_norm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns l1 norm of a polynomial in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_l1_norm</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[2, -1], [-3]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_l1_norm(f, 1, ZZ)</span>
-<span class="sd">    6</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_l1_norm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="nb">sum</span><span class="p">([</span> <span class="n">dmp_l1_norm</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">])</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_expand</span><span class="p">(</span><span class="n">polys</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Multiply together several polynomials in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dup_expand</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, -1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([1, 0])</span>
-<span class="sd">    &gt;&gt;&gt; h = ZZ.map([2])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_expand([f, g, h], ZZ)</span>
-<span class="sd">    [2, 0, -2, 0]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]</span>
-
-    <span class="n">f</span> <span class="o">=</span> <span class="n">polys</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">polys</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">dup_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">f</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_expand"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densearith.dmp_expand">[docs]</a><span class="k">def</span> <span class="nf">dmp_expand</span><span class="p">(</span><span class="n">polys</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Multiply together several polynomials in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densearith import dmp_expand</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [], [1, 0, 0]])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[1], [1]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_expand([f, g], 1, ZZ)</span>
-<span class="sd">    [[1], [1], [1, 0, 0], [1, 0, 0]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_one</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">f</span> <span class="o">=</span> <span class="n">polys</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">polys</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">f</span></div>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
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diff --git a/dev-py3k/_modules/sympy/polys/densebasic.html b/dev-py3k/_modules/sympy/polys/densebasic.html
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--- a/dev-py3k/_modules/sympy/polys/densebasic.html
+++ /dev/null
@@ -1,2050 +0,0 @@
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-            
-  <h1>Source code for sympy.polys.densebasic</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Basic tools for dense recursive polynomials in ``K[x]`` or ``K[X]``. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">igcd</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.monomialtools</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">monomial_key</span><span class="p">,</span> <span class="n">monomial_min</span><span class="p">,</span> <span class="n">monomial_div</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.distributedpolys</span> <span class="kn">import</span> <span class="n">sdp_sort</span>
-
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">cythonized</span>
-
-<span class="kn">import</span> <span class="nn">random</span>
-
-
-<span class="k">def</span> <span class="nf">poly_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return leading coefficient of ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import poly_LC</span>
-
-<span class="sd">    &gt;&gt;&gt; poly_LC([], ZZ)</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; poly_LC([ZZ(1), ZZ(2), ZZ(3)], ZZ)</span>
-<span class="sd">    1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">poly_TC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return trailing coefficient of ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import poly_TC</span>
-
-<span class="sd">    &gt;&gt;&gt; poly_TC([], ZZ)</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; poly_TC([ZZ(1), ZZ(2), ZZ(3)], ZZ)</span>
-<span class="sd">    3</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-
-<span class="n">dup_LC</span> <span class="o">=</span> <span class="n">dmp_LC</span> <span class="o">=</span> <span class="n">poly_LC</span>
-<span class="n">dup_TC</span> <span class="o">=</span> <span class="n">dmp_TC</span> <span class="o">=</span> <span class="n">poly_TC</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_ground_LC"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_ground_LC">[docs]</a><span class="k">def</span> <span class="nf">dmp_ground_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the ground leading coefficient.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_ground_LC</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[[1], [2, 3]]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_ground_LC(f, 2, ZZ)</span>
-<span class="sd">    1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">while</span> <span class="n">u</span><span class="p">:</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">u</span> <span class="o">-=</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_ground_TC"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_ground_TC">[docs]</a><span class="k">def</span> <span class="nf">dmp_ground_TC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the ground trailing coefficient.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_ground_TC</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[[1], [2, 3]]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_ground_TC(f, 2, ZZ)</span>
-<span class="sd">    3</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">while</span> <span class="n">u</span><span class="p">:</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_TC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">u</span> <span class="o">-=</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="n">dup_TC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_true_LT"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_true_LT">[docs]</a><span class="k">def</span> <span class="nf">dmp_true_LT</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the leading term ``c * x_1**n_1 ... x_k**n_k``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_true_LT</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[4], [2, 0], [3, 0, 0]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_true_LT(f, 1, ZZ)</span>
-<span class="sd">    ((2, 0), 4)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">monom</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">while</span> <span class="n">u</span><span class="p">:</span>
-        <span class="n">monom</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">f</span><span class="p">,</span> <span class="n">u</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="n">monom</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">monom</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">monom</span><span class="p">),</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the leading degree of ``f`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dup_degree</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 2, 0, 3])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_degree(f)</span>
-<span class="sd">    3</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_degree"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_degree">[docs]</a><span class="k">def</span> <span class="nf">dmp_degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the leading degree of ``f`` in ``x_0`` in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_degree</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_degree([[[]]], 2)</span>
-<span class="sd">    -1</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[2], [1, 2, 3]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_degree(f, 1)</span>
-<span class="sd">    1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;v,i,j&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_rec_degree_in</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Recursive helper function for :func:`dmp_degree_in`.&quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
-
-    <span class="n">v</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="n">v</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="nb">max</span><span class="p">([</span> <span class="n">_rec_degree_in</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">g</span> <span class="p">])</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;j,u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_degree_in"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_degree_in">[docs]</a><span class="k">def</span> <span class="nf">dmp_degree_in</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the leading degree of ``f`` in ``x_j`` in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_degree_in</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[2], [1, 2, 3]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_degree_in(f, 0, 1)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; dmp_degree_in(f, 1, 1)</span>
-<span class="sd">    2</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">j</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">j</span> <span class="o">&gt;</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s">&quot;0 &lt;= j &lt;= </span><span class="si">%s</span><span class="s"> expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">_rec_degree_in</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;v,i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_rec_degree_list</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">degs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Recursive helper for :func:`dmp_degree_list`.&quot;&quot;&quot;</span>
-    <span class="n">degs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">degs</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">v</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="n">v</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">v</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="n">v</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-
-        <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">g</span><span class="p">:</span>
-            <span class="n">_rec_degree_list</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">degs</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_degree_list"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_degree_list">[docs]</a><span class="k">def</span> <span class="nf">dmp_degree_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a list of degrees of ``f`` in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_degree_list</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [1, 2, 3]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_degree_list(f, 1)</span>
-<span class="sd">    (1, 2)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">degs</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="n">_rec_degree_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">degs</span><span class="p">)</span>
-    <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">degs</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_strip</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Remove leading zeros from ``f`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dup_strip</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_strip([0, 0, 1, 2, 3, 0])</span>
-<span class="sd">    [1, 2, 3, 0]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span> <span class="ow">or</span> <span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-
-    <span class="k">for</span> <span class="n">cf</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">cf</span><span class="p">:</span>
-            <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="n">f</span><span class="p">[</span><span class="n">i</span><span class="p">:]</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v,i&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_strip"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_strip">[docs]</a><span class="k">def</span> <span class="nf">dmp_strip</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Remove leading zeros from ``f`` in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_strip</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_strip([[], [0, 1, 2], [1]], 1)</span>
-<span class="sd">    [[0, 1, 2], [1]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_strip</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">):</span>
-            <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span><span class="p">[</span><span class="n">i</span><span class="p">:]</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i,j&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_rec_validate</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Recursive helper for :func:`dmp_validate`.&quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">g</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="nb">list</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">K</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">K</span><span class="o">.</span><span class="n">of_type</span><span class="p">(</span><span class="n">g</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> in </span><span class="si">%s</span><span class="s"> in not of type </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">dtype</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="nb">set</span><span class="p">([</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span>
-    <span class="k">elif</span> <span class="ow">not</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">set</span><span class="p">([</span><span class="n">i</span><span class="p">])</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">j</span><span class="p">,</span> <span class="n">levels</span> <span class="o">=</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">set</span><span class="p">([])</span>
-
-        <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">g</span><span class="p">:</span>
-            <span class="n">levels</span> <span class="o">|=</span> <span class="n">_rec_validate</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">levels</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;v,w&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_rec_strip</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">v</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Recursive helper for :func:`_rec_strip`.&quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">v</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_strip</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="n">w</span> <span class="o">=</span> <span class="n">v</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="n">dmp_strip</span><span class="p">([</span> <span class="n">_rec_strip</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">w</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">g</span> <span class="p">],</span> <span class="n">v</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_validate"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_validate">[docs]</a><span class="k">def</span> <span class="nf">dmp_validate</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the number of levels in ``f`` and recursively strip it.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_validate</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_validate([[], [0, 1, 2], [1]])</span>
-<span class="sd">    ([[1, 2], [1]], 1)</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_validate([[1], 1])</span>
-<span class="sd">    Traceback (most recent call last):</span>
-<span class="sd">    ...</span>
-<span class="sd">    ValueError: invalid data structure for a multivariate polynomial</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">levels</span> <span class="o">=</span> <span class="n">_rec_validate</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">u</span> <span class="o">=</span> <span class="n">levels</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">levels</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">_rec_strip</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">),</span> <span class="n">u</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-            <span class="s">&quot;invalid data structure for a multivariate polynomial&quot;</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="dup_reverse"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dup_reverse">[docs]</a><span class="k">def</span> <span class="nf">dup_reverse</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute ``x**n * f(1/x)``, i.e.: reverse ``f`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dup_reverse</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 2, 3, 0])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_reverse(f)</span>
-<span class="sd">    [3, 2, 1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dup_strip</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">f</span><span class="p">)))</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_copy</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create a new copy of a polynomial ``f`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dup_copy</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 2, 3, 0])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_copy([1, 2, 3, 0])</span>
-<span class="sd">    [1, 2, 3, 0]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_copy"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_copy">[docs]</a><span class="k">def</span> <span class="nf">dmp_copy</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create a new copy of a polynomial ``f`` in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_copy</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [1, 2]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_copy(f, 1)</span>
-<span class="sd">    [[1], [1, 2]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="p">[</span> <span class="n">dmp_copy</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_to_tuple</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Convert `f` into a tuple.</span>
-
-<span class="sd">    This is needed for hashing. This is similar to dup_copy().</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dup_copy</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 2, 3, 0])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_copy([1, 2, 3, 0])</span>
-<span class="sd">    [1, 2, 3, 0]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_to_tuple"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_to_tuple">[docs]</a><span class="k">def</span> <span class="nf">dmp_to_tuple</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Convert `f` into a nested tuple of tuples.</span>
-
-<span class="sd">    This is needed for hashing.  This is similar to dmp_copy().</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_to_tuple</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [1, 2]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_to_tuple(f, 1)</span>
-<span class="sd">    ((1,), (1, 2))</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">dmp_to_tuple</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_normal</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Normalize univariate polynomial in the given domain.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dup_normal</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_normal([0, 1.5, 2, 3], ZZ)</span>
-<span class="sd">    [1, 2, 3]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dup_strip</span><span class="p">([</span> <span class="n">K</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">])</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_normal"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_normal">[docs]</a><span class="k">def</span> <span class="nf">dmp_normal</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Normalize a multivariate polynomial in the given domain.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_normal</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_normal([[], [0, 1.5, 2]], 1, ZZ)</span>
-<span class="sd">    [[1, 2]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_normal</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="n">dmp_strip</span><span class="p">([</span> <span class="n">dmp_normal</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">],</span> <span class="n">u</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_convert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Convert the ground domain of ``f`` from ``K0`` to ``K1``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.polyclasses import DMP</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dup_convert</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_convert([DMP([1], ZZ), DMP([2], ZZ)], ZZ[&#39;x&#39;], ZZ)</span>
-<span class="sd">    [1, 2]</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_convert([ZZ(1), ZZ(2)], ZZ, ZZ[&#39;x&#39;])</span>
-<span class="sd">    [1, 2]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">K0</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">K0</span> <span class="o">==</span> <span class="n">K1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_strip</span><span class="p">([</span> <span class="n">K1</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">])</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_convert"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_convert">[docs]</a><span class="k">def</span> <span class="nf">dmp_convert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Convert the ground domain of ``f`` from ``K0`` to ``K1``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.polyclasses import DMP</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_convert</span>
-
-<span class="sd">    &gt;&gt;&gt; f = [[DMP([1], ZZ)], [DMP([2], ZZ)]]</span>
-<span class="sd">    &gt;&gt;&gt; g = [[ZZ(1)], [ZZ(2)]]</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_convert(f, 1, ZZ[&#39;x&#39;], ZZ)</span>
-<span class="sd">    [[1], [2]]</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_convert(g, 1, ZZ, ZZ[&#39;x&#39;])</span>
-<span class="sd">    [[1], [2]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_convert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">K0</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">K0</span> <span class="o">==</span> <span class="n">K1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="n">dmp_strip</span><span class="p">([</span> <span class="n">dmp_convert</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">],</span> <span class="n">u</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_from_sympy</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Convert the ground domain of ``f`` from SymPy to ``K``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import S</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dup_from_sympy</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_from_sympy([S(1), S(2)], ZZ) == [ZZ(1), ZZ(2)]</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dup_strip</span><span class="p">([</span> <span class="n">K</span><span class="o">.</span><span class="n">from_sympy</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">])</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_from_sympy"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_from_sympy">[docs]</a><span class="k">def</span> <span class="nf">dmp_from_sympy</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Convert the ground domain of ``f`` from SymPy to ``K``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import S</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_from_sympy</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_from_sympy([[S(1)], [S(2)]], 1, ZZ) == [[ZZ(1)], [ZZ(2)]]</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_from_sympy</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="n">dmp_strip</span><span class="p">([</span> <span class="n">dmp_from_sympy</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">],</span> <span class="n">u</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_nth</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the ``n``-th coefficient of ``f`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dup_nth</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 2, 3])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_nth(f, 0, ZZ)</span>
-<span class="sd">    3</span>
-<span class="sd">    &gt;&gt;&gt; dup_nth(f, 4, ZZ)</span>
-<span class="sd">    0</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s">&quot;&#39;n&#39; must be non-negative, got </span><span class="si">%i</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">n</span> <span class="o">&gt;=</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span><span class="p">[</span><span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">-</span> <span class="n">n</span><span class="p">]</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_nth"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_nth">[docs]</a><span class="k">def</span> <span class="nf">dmp_nth</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the ``n``-th coefficient of ``f`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_nth</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [2], [3]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_nth(f, 0, 1, ZZ)</span>
-<span class="sd">    [3]</span>
-<span class="sd">    &gt;&gt;&gt; dmp_nth(f, 4, 1, ZZ)</span>
-<span class="sd">    []</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s">&quot;&#39;n&#39; must be non-negative, got </span><span class="si">%i</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">n</span> <span class="o">&gt;=</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span><span class="p">[</span><span class="n">dmp_degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span> <span class="o">-</span> <span class="n">n</span><span class="p">]</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,u,v&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_ground_nth"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_ground_nth">[docs]</a><span class="k">def</span> <span class="nf">dmp_ground_nth</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the ground ``n``-th coefficient of ``f`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_ground_nth</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [2, 3]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_ground_nth(f, (0, 1), 1, ZZ)</span>
-<span class="sd">    2</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span>
-
-    <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">N</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s">&quot;`n` must be non-negative, got </span><span class="si">%i</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">n</span> <span class="o">&gt;=</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">f</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="n">dmp_degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="o">-</span> <span class="n">n</span><span class="p">],</span> <span class="n">v</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="n">f</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_zero_p"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_zero_p">[docs]</a><span class="k">def</span> <span class="nf">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return ``True`` if ``f`` is zero in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_zero_p</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_zero_p([[[[[]]]]], 4)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; dmp_zero_p([[[[[1]]]]], 4)</span>
-<span class="sd">    False</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">while</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">u</span> <span class="o">-=</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="ow">not</span> <span class="n">f</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_zero"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_zero">[docs]</a><span class="k">def</span> <span class="nf">dmp_zero</span><span class="p">(</span><span class="n">u</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a multivariate zero.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_zero</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_zero(4)</span>
-<span class="sd">    [[[[[]]]]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">u</span><span class="p">):</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="p">[</span><span class="n">r</span><span class="p">]</span>
-
-    <span class="k">return</span> <span class="n">r</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_one_p"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_one_p">[docs]</a><span class="k">def</span> <span class="nf">dmp_one_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return ``True`` if ``f`` is one in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_one_p</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_one_p([[[ZZ(1)]]], 2, ZZ)</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dmp_ground_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_one"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_one">[docs]</a><span class="k">def</span> <span class="nf">dmp_one</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a multivariate one over ``K``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_one</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_one(2, ZZ)</span>
-<span class="sd">    [[[1]]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dmp_ground</span><span class="p">(</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_ground_p"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_ground_p">[docs]</a><span class="k">def</span> <span class="nf">dmp_ground_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return True if ``f`` is constant in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_ground_p</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_ground_p([[[3]]], 3, 2)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; dmp_ground_p([[[4]]], None, 2)</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">c</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="k">while</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">u</span> <span class="o">-=</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">1</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span> <span class="o">==</span> <span class="p">[</span><span class="n">c</span><span class="p">]</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i,u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_ground"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_ground">[docs]</a><span class="k">def</span> <span class="nf">dmp_ground</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a multivariate constant.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_ground</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_ground(3, 5)</span>
-<span class="sd">    [[[[[[3]]]]]]</span>
-<span class="sd">    &gt;&gt;&gt; dmp_ground(1, -1)</span>
-<span class="sd">    1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">c</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="p">[</span><span class="n">c</span><span class="p">]</span>
-
-    <span class="k">return</span> <span class="n">c</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_zeros"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_zeros">[docs]</a><span class="k">def</span> <span class="nf">dmp_zeros</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a list of multivariate zeros.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_zeros</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_zeros(3, 2, ZZ)</span>
-<span class="sd">    [[[[]]], [[[]]], [[[]]]]</span>
-<span class="sd">    &gt;&gt;&gt; dmp_zeros(3, -1, ZZ)</span>
-<span class="sd">    [0, 0, 0]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-    <span class="k">if</span> <span class="n">u</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span><span class="o">*</span><span class="n">n</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">u</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="p">]</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_grounds"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_grounds">[docs]</a><span class="k">def</span> <span class="nf">dmp_grounds</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a list of multivariate constants.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_grounds</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_grounds(ZZ(4), 3, 2)</span>
-<span class="sd">    [[[[4]]], [[[4]]], [[[4]]]]</span>
-<span class="sd">    &gt;&gt;&gt; dmp_grounds(ZZ(4), 3, -1)</span>
-<span class="sd">    [4, 4, 4]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-    <span class="k">if</span> <span class="n">u</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="o">*</span><span class="n">n</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="n">dmp_ground</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="p">]</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_negative_p"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_negative_p">[docs]</a><span class="k">def</span> <span class="nf">dmp_negative_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return ``True`` if ``LC(f)`` is negative.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_negative_p</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_negative_p([[ZZ(1)], [-ZZ(1)]], 1, ZZ)</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; dmp_negative_p([[-ZZ(1)], [ZZ(1)]], 1, ZZ)</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">is_negative</span><span class="p">(</span><span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_positive_p"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_positive_p">[docs]</a><span class="k">def</span> <span class="nf">dmp_positive_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return ``True`` if ``LC(f)`` is positive.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_positive_p</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_positive_p([[ZZ(1)], [-ZZ(1)]], 1, ZZ)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; dmp_positive_p([[-ZZ(1)], [ZZ(1)]], 1, ZZ)</span>
-<span class="sd">    False</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">is_positive</span><span class="p">(</span><span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,k&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_from_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create a ``K[x]`` polynomial from a ``dict``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dup_from_dict</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_from_dict({(0,): ZZ(7), (2,): ZZ(5), (4,): ZZ(1)}, ZZ)</span>
-<span class="sd">    [1, 0, 5, 0, 7]</span>
-<span class="sd">    &gt;&gt;&gt; dup_from_dict({}, ZZ)</span>
-<span class="sd">    []</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-    <span class="n">n</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">keys</span><span class="p">()),</span> <span class="p">[]</span>
-
-    <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">int</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-            <span class="n">h</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="p">(</span><span class="n">n</span><span class="p">,)</span> <span class="o">=</span> <span class="n">n</span>
-
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-            <span class="n">h</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">get</span><span class="p">((</span><span class="n">k</span><span class="p">,),</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">dup_strip</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,k&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_from_raw_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create a ``K[x]`` polynomial from a raw ``dict``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dup_from_raw_dict</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_from_raw_dict({0: ZZ(7), 2: ZZ(5), 4: ZZ(1)}, ZZ)</span>
-<span class="sd">    [1, 0, 5, 0, 7]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-    <span class="n">n</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">keys</span><span class="p">()),</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-        <span class="n">h</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">dup_strip</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v,n,k&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_from_dict"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_from_dict">[docs]</a><span class="k">def</span> <span class="nf">dmp_from_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create a ``K[X]`` polynomial from a ``dict``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_from_dict</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_from_dict({(0, 0): ZZ(3), (0, 1): ZZ(2), (2, 1): ZZ(1)}, 1, ZZ)</span>
-<span class="sd">    [[1, 0], [], [2, 3]]</span>
-<span class="sd">    &gt;&gt;&gt; dmp_from_dict({}, 0, ZZ)</span>
-<span class="sd">    []</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_from_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
-
-    <span class="n">coeffs</span> <span class="o">=</span> <span class="p">{}</span>
-
-    <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="n">head</span><span class="p">,</span> <span class="n">tail</span> <span class="o">=</span> <span class="n">monom</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">monom</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-        <span class="k">if</span> <span class="n">head</span> <span class="ow">in</span> <span class="n">coeffs</span><span class="p">:</span>
-            <span class="n">coeffs</span><span class="p">[</span><span class="n">head</span><span class="p">][</span><span class="n">tail</span><span class="p">]</span> <span class="o">=</span> <span class="n">coeff</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">coeffs</span><span class="p">[</span><span class="n">head</span><span class="p">]</span> <span class="o">=</span> <span class="p">{</span> <span class="n">tail</span><span class="p">:</span> <span class="n">coeff</span> <span class="p">}</span>
-
-    <span class="n">n</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">coeffs</span><span class="o">.</span><span class="n">keys</span><span class="p">()),</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-        <span class="n">coeff</span> <span class="o">=</span> <span class="n">coeffs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">h</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dmp_from_dict</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">h</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dmp_zero</span><span class="p">(</span><span class="n">v</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">dmp_strip</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,k&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_to_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">zero</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Convert ``K[x]`` polynomial to a ``dict``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dup_to_dict</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_to_dict([1, 0, 5, 0, 7])</span>
-<span class="sd">    {(0,): 7, (2,): 5, (4,): 1}</span>
-<span class="sd">    &gt;&gt;&gt; dup_to_dict([])</span>
-<span class="sd">    {}</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span> <span class="ow">and</span> <span class="n">zero</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">{(</span><span class="mi">0</span><span class="p">,):</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">}</span>
-
-    <span class="n">n</span><span class="p">,</span> <span class="n">result</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="p">{}</span>
-
-    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">f</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">]:</span>
-            <span class="n">result</span><span class="p">[(</span><span class="n">k</span><span class="p">,)]</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">]</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,k&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_to_raw_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">zero</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Convert a ``K[x]`` polynomial to a raw ``dict``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dup_to_raw_dict</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_to_raw_dict([1, 0, 5, 0, 7])</span>
-<span class="sd">    {0: 7, 2: 5, 4: 1}</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span> <span class="ow">and</span> <span class="n">zero</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">{</span><span class="mi">0</span><span class="p">:</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">}</span>
-
-    <span class="n">n</span><span class="p">,</span> <span class="n">result</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="p">{}</span>
-
-    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">f</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">]:</span>
-            <span class="n">result</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">]</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v,n,k&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_to_dict"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_to_dict">[docs]</a><span class="k">def</span> <span class="nf">dmp_to_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">zero</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Convert a ``K[X]`` polynomial to a ``dict````.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_to_dict</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_to_dict([[1, 0], [], [2, 3]], 1)</span>
-<span class="sd">    {(0, 0): 3, (0, 1): 2, (2, 1): 1}</span>
-<span class="sd">    &gt;&gt;&gt; dmp_to_dict([], 0)</span>
-<span class="sd">    {}</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_to_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">zero</span><span class="o">=</span><span class="n">zero</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span> <span class="ow">and</span> <span class="n">zero</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">{(</span><span class="mi">0</span><span class="p">,)</span><span class="o">*</span><span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">}</span>
-
-    <span class="n">n</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">result</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">),</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="p">{}</span>
-
-    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_to_dict</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">],</span> <span class="n">v</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">exp</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">h</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">result</span><span class="p">[(</span><span class="n">k</span><span class="p">,)</span> <span class="o">+</span> <span class="n">exp</span><span class="p">]</span> <span class="o">=</span> <span class="n">coeff</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,i,j&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_swap"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_swap">[docs]</a><span class="k">def</span> <span class="nf">dmp_swap</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Transform ``K[..x_i..x_j..]`` to ``K[..x_j..x_i..]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_swap</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[[2], [1, 0]], []])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_swap(f, 0, 1, 2, ZZ)</span>
-<span class="sd">    [[[2], []], [[1, 0], []]]</span>
-<span class="sd">    &gt;&gt;&gt; dmp_swap(f, 1, 2, 2, ZZ)</span>
-<span class="sd">    [[[1], [2, 0]], [[]]]</span>
-<span class="sd">    &gt;&gt;&gt; dmp_swap(f, 0, 2, 2, ZZ)</span>
-<span class="sd">    [[[1, 0]], [[2, 0], []]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">i</span> <span class="o">&gt;</span> <span class="n">u</span> <span class="ow">or</span> <span class="n">j</span> <span class="o">&gt;</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s">&quot;0 &lt;= i &lt; j &lt;= </span><span class="si">%s</span><span class="s"> expected&quot;</span> <span class="o">%</span> <span class="n">u</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">F</span><span class="p">,</span> <span class="n">H</span> <span class="o">=</span> <span class="n">dmp_to_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">),</span> <span class="p">{}</span>
-
-    <span class="k">for</span> <span class="n">exp</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">F</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="n">H</span><span class="p">[</span><span class="n">exp</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="p">(</span><span class="n">exp</span><span class="p">[</span><span class="n">j</span><span class="p">],)</span> <span class="o">+</span>
-          <span class="n">exp</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:</span><span class="n">j</span><span class="p">]</span> <span class="o">+</span>
-          <span class="p">(</span><span class="n">exp</span><span class="p">[</span><span class="n">i</span><span class="p">],)</span> <span class="o">+</span> <span class="n">exp</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]]</span> <span class="o">=</span> <span class="n">coeff</span>
-
-    <span class="k">return</span> <span class="n">dmp_from_dict</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_permute"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_permute">[docs]</a><span class="k">def</span> <span class="nf">dmp_permute</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a polynomial in ``K[x_{P(1)},..,x_{P(n)}]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_permute</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[[2], [1, 0]], []])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_permute(f, [1, 0, 2], 2, ZZ)</span>
-<span class="sd">    [[[2], []], [[1, 0], []]]</span>
-<span class="sd">    &gt;&gt;&gt; dmp_permute(f, [1, 2, 0], 2, ZZ)</span>
-<span class="sd">    [[[1], []], [[2, 0], []]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">F</span><span class="p">,</span> <span class="n">H</span> <span class="o">=</span> <span class="n">dmp_to_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">),</span> <span class="p">{}</span>
-
-    <span class="k">for</span> <span class="n">exp</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">F</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="n">new_exp</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="n">exp</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">exp</span><span class="p">,</span> <span class="n">P</span><span class="p">):</span>
-            <span class="n">new_exp</span><span class="p">[</span><span class="n">p</span><span class="p">]</span> <span class="o">=</span> <span class="n">e</span>
-
-        <span class="n">H</span><span class="p">[</span><span class="nb">tuple</span><span class="p">(</span><span class="n">new_exp</span><span class="p">)]</span> <span class="o">=</span> <span class="n">coeff</span>
-
-    <span class="k">return</span> <span class="n">dmp_from_dict</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i,l&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_nest"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_nest">[docs]</a><span class="k">def</span> <span class="nf">dmp_nest</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a multivariate value nested ``l``-levels.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_nest</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_nest([[ZZ(1)]], 2, ZZ)</span>
-<span class="sd">    [[[[1]]]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="nb">list</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">dmp_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">l</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">l</span><span class="p">):</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="p">]</span>
-
-    <span class="k">return</span> <span class="n">f</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;l,k,u,v&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_raise"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_raise">[docs]</a><span class="k">def</span> <span class="nf">dmp_raise</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a multivariate polynomial raised ``l``-levels.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_raise</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[], [1, 2]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_raise(f, 2, 1, ZZ)</span>
-<span class="sd">    [[[[]]], [[[1]], [[2]]]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">l</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">l</span><span class="p">)</span>
-
-        <span class="n">k</span> <span class="o">=</span> <span class="n">l</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="k">return</span> <span class="p">[</span> <span class="n">dmp_ground</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="p">[</span> <span class="n">dmp_raise</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;g,i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_deflate</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Map ``x**m`` to ``y`` in a polynomial in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dup_deflate</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, 0, 1, 0, 0, 1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_deflate(f, ZZ)</span>
-<span class="sd">    (3, [1, 1, 1])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">1</span><span class="p">,</span> <span class="n">f</span>
-
-    <span class="n">g</span> <span class="o">=</span> <span class="mi">0</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]:</span>
-            <span class="k">continue</span>
-
-        <span class="n">g</span> <span class="o">=</span> <span class="n">igcd</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">g</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">1</span><span class="p">,</span> <span class="n">f</span>
-
-    <span class="k">return</span> <span class="n">g</span><span class="p">,</span> <span class="n">f</span><span class="p">[::</span><span class="n">g</span><span class="p">]</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,i,m,a,b&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_deflate"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_deflate">[docs]</a><span class="k">def</span> <span class="nf">dmp_deflate</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Map ``x_i**m_i`` to ``y_i`` in a polynomial in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_deflate</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1, 0, 0, 2], [], [3, 0, 0, 4]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_deflate(f, 1, ZZ)</span>
-<span class="sd">    ((2, 3), [[1, 2], [3, 4]])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="mi">1</span><span class="p">,)</span><span class="o">*</span><span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">f</span>
-
-    <span class="n">F</span> <span class="o">=</span> <span class="n">dmp_to_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-    <span class="n">B</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">M</span> <span class="ow">in</span> <span class="n">F</span><span class="o">.</span><span class="n">keys</span><span class="p">():</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">m</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">M</span><span class="p">):</span>
-            <span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">igcd</span><span class="p">(</span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">m</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">B</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">b</span><span class="p">:</span>
-            <span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="n">B</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">b</span> <span class="o">==</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">B</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">B</span><span class="p">,</span> <span class="n">f</span>
-
-    <span class="n">H</span> <span class="o">=</span> <span class="p">{}</span>
-
-    <span class="k">for</span> <span class="n">A</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">F</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="n">N</span> <span class="o">=</span> <span class="p">[</span> <span class="n">a</span> <span class="o">//</span> <span class="n">b</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span> <span class="p">]</span>
-        <span class="n">H</span><span class="p">[</span><span class="nb">tuple</span><span class="p">(</span><span class="n">N</span><span class="p">)]</span> <span class="o">=</span> <span class="n">coeff</span>
-
-    <span class="k">return</span> <span class="n">B</span><span class="p">,</span> <span class="n">dmp_from_dict</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;G,g,i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_multi_deflate</span><span class="p">(</span><span class="n">polys</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Map ``x**m`` to ``y`` in a set of polynomials in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dup_multi_deflate</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, 2, 0, 3])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([4, 0, 0])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_multi_deflate((f, g), ZZ)</span>
-<span class="sd">    (2, ([1, 2, 3], [4, 0]))</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">G</span> <span class="o">=</span> <span class="mi">0</span>
-
-    <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">polys</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">1</span><span class="p">,</span> <span class="n">polys</span>
-
-        <span class="n">g</span> <span class="o">=</span> <span class="mi">0</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">p</span><span class="p">[</span><span class="o">-</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]:</span>
-                <span class="k">continue</span>
-
-            <span class="n">g</span> <span class="o">=</span> <span class="n">igcd</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">g</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="mi">1</span><span class="p">,</span> <span class="n">polys</span>
-
-        <span class="n">G</span> <span class="o">=</span> <span class="n">igcd</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">G</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">([</span> <span class="n">p</span><span class="p">[::</span><span class="n">G</span><span class="p">]</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">polys</span> <span class="p">])</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,G,g,m,a,b&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_multi_deflate"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_multi_deflate">[docs]</a><span class="k">def</span> <span class="nf">dmp_multi_deflate</span><span class="p">(</span><span class="n">polys</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Map ``x_i**m_i`` to ``y_i`` in a set of polynomials in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_multi_deflate</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1, 0, 0, 2], [], [3, 0, 0, 4]])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[1, 0, 2], [], [3, 0, 4]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_multi_deflate((f, g), 1, ZZ)</span>
-<span class="sd">    ((2, 1), ([[1, 0, 0, 2], [3, 0, 0, 4]], [[1, 0, 2], [3, 0, 4]]))</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="n">M</span><span class="p">,</span> <span class="n">H</span> <span class="o">=</span> <span class="n">dup_multi_deflate</span><span class="p">(</span><span class="n">polys</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">M</span><span class="p">,),</span> <span class="n">H</span>
-
-    <span class="n">F</span><span class="p">,</span> <span class="n">B</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">polys</span><span class="p">:</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_to_dict</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">M</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">keys</span><span class="p">():</span>
-                <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">m</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">M</span><span class="p">):</span>
-                    <span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">igcd</span><span class="p">(</span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">m</span><span class="p">)</span>
-
-        <span class="n">F</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">B</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">b</span><span class="p">:</span>
-            <span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="n">B</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">b</span> <span class="o">==</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">B</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">B</span><span class="p">,</span> <span class="n">polys</span>
-
-    <span class="n">H</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">F</span><span class="p">:</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="p">{}</span>
-
-        <span class="k">for</span> <span class="n">A</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">N</span> <span class="o">=</span> <span class="p">[</span> <span class="n">a</span> <span class="o">//</span> <span class="n">b</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span> <span class="p">]</span>
-            <span class="n">h</span><span class="p">[</span><span class="nb">tuple</span><span class="p">(</span><span class="n">N</span><span class="p">)]</span> <span class="o">=</span> <span class="n">coeff</span>
-
-        <span class="n">H</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dmp_from_dict</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">B</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">H</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;m&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_inflate</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Map ``y`` to ``x**m`` in a polynomial in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dup_inflate</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 1, 1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_inflate(f, 3, ZZ)</span>
-<span class="sd">    [1, 0, 0, 1, 0, 0, 1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s">&quot;&#39;m&#39; must be positive, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">m</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span>
-
-    <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-        <span class="n">result</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">m</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v,i,j&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_rec_inflate</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Recursive helper for :func:`dmp_inflate`.&quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">v</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_inflate</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s">&quot;all M[i] must be positive, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-
-    <span class="n">w</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">v</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-
-    <span class="n">g</span> <span class="o">=</span> <span class="p">[</span> <span class="n">_rec_inflate</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">g</span> <span class="p">]</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[</span><span class="n">g</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span>
-
-    <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">g</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-        <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">]):</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dmp_zero</span><span class="p">(</span><span class="n">w</span><span class="p">))</span>
-
-        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,m&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_inflate"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_inflate">[docs]</a><span class="k">def</span> <span class="nf">dmp_inflate</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Map ``y_i`` to ``x_i**k_i`` in a polynomial in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_inflate</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1, 2], [3, 4]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_inflate(f, (2, 3), 1, ZZ)</span>
-<span class="sd">    [[1, 0, 0, 2], [], [3, 0, 0, 4]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_inflate</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">m</span> <span class="o">==</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">M</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">_rec_inflate</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,j&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_exclude"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_exclude">[docs]</a><span class="k">def</span> <span class="nf">dmp_exclude</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Exclude useless levels from ``f``.</span>
-
-<span class="sd">    Return the levels excluded, the new excluded ``f``, and the new ``u``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_exclude</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[[1]], [[1], [2]]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_exclude(f, 2, ZZ)</span>
-<span class="sd">    ([2], [[1], [1, 2]], 1)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span> <span class="ow">or</span> <span class="n">dmp_ground_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">[],</span> <span class="n">f</span><span class="p">,</span> <span class="n">u</span>
-
-    <span class="n">J</span><span class="p">,</span> <span class="n">F</span> <span class="o">=</span> <span class="p">[],</span> <span class="n">dmp_to_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">monom</span> <span class="ow">in</span> <span class="n">F</span><span class="o">.</span><span class="n">keys</span><span class="p">():</span>
-            <span class="k">if</span> <span class="n">monom</span><span class="p">[</span><span class="n">j</span><span class="p">]:</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">J</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">J</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[],</span> <span class="n">f</span><span class="p">,</span> <span class="n">u</span>
-
-    <span class="n">f</span> <span class="o">=</span> <span class="p">{}</span>
-
-    <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">F</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="n">monom</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">monom</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">J</span><span class="p">):</span>
-            <span class="k">del</span> <span class="n">monom</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-
-        <span class="n">f</span><span class="p">[</span><span class="nb">tuple</span><span class="p">(</span><span class="n">monom</span><span class="p">)]</span> <span class="o">=</span> <span class="n">coeff</span>
-
-    <span class="n">u</span> <span class="o">-=</span> <span class="nb">len</span><span class="p">(</span><span class="n">J</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">J</span><span class="p">,</span> <span class="n">dmp_from_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">u</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,j&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_include"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_include">[docs]</a><span class="k">def</span> <span class="nf">dmp_include</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">J</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Include useless levels in ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_include</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [1, 2]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_include(f, [2], 1, ZZ)</span>
-<span class="sd">    [[[1]], [[1], [2]]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">J</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">F</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_to_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">),</span> <span class="p">{}</span>
-
-    <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">F</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="n">monom</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">monom</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">J</span><span class="p">:</span>
-            <span class="n">monom</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-
-        <span class="n">f</span><span class="p">[</span><span class="nb">tuple</span><span class="p">(</span><span class="n">monom</span><span class="p">)]</span> <span class="o">=</span> <span class="n">coeff</span>
-
-    <span class="n">u</span> <span class="o">+=</span> <span class="nb">len</span><span class="p">(</span><span class="n">J</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">dmp_from_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v,w&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_inject"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_inject">[docs]</a><span class="k">def</span> <span class="nf">dmp_inject</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">front</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Convert ``f`` from ``K[X][Y]`` to ``K[X,Y]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_inject</span>
-
-<span class="sd">    &gt;&gt;&gt; K = ZZ[&#39;x&#39;, &#39;y&#39;]</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_inject([K([[1]]), K([[1], [2]])], 0, K)</span>
-<span class="sd">    ([[[1]], [[1], [2]]], 2)</span>
-<span class="sd">    &gt;&gt;&gt; dmp_inject([K([[1]]), K([[1], [2]])], 0, K, front=True)</span>
-<span class="sd">    ([[[1]], [[1, 2]]], 2)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">f</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_to_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">),</span> <span class="p">{}</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">K</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">for</span> <span class="n">f_monom</span><span class="p">,</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">to_dict</span><span class="p">()</span>
-
-        <span class="k">for</span> <span class="n">g_monom</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">g</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="k">if</span> <span class="n">front</span><span class="p">:</span>
-                <span class="n">h</span><span class="p">[</span><span class="n">g_monom</span> <span class="o">+</span> <span class="n">f_monom</span><span class="p">]</span> <span class="o">=</span> <span class="n">c</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">h</span><span class="p">[</span><span class="n">f_monom</span> <span class="o">+</span> <span class="n">g_monom</span><span class="p">]</span> <span class="o">=</span> <span class="n">c</span>
-
-    <span class="n">w</span> <span class="o">=</span> <span class="n">u</span> <span class="o">+</span> <span class="n">v</span> <span class="o">+</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="n">dmp_from_dict</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">dom</span><span class="p">),</span> <span class="n">w</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_eject"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_eject">[docs]</a><span class="k">def</span> <span class="nf">dmp_eject</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">front</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Convert ``f`` from ``K[X,Y]`` to ``K[X][Y]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_eject</span>
-
-<span class="sd">    &gt;&gt;&gt; K = ZZ[&#39;x&#39;, &#39;y&#39;]</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_eject([[[1]], [[1], [2]]], 2, K)</span>
-<span class="sd">    [1, x + 2]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">f</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_to_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">),</span> <span class="p">{}</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">K</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
-
-    <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="k">if</span> <span class="n">front</span><span class="p">:</span>
-            <span class="n">g_monom</span><span class="p">,</span> <span class="n">f_monom</span> <span class="o">=</span> <span class="n">monom</span><span class="p">[:</span><span class="n">v</span><span class="p">],</span> <span class="n">monom</span><span class="p">[</span><span class="n">v</span><span class="p">:]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">f_monom</span><span class="p">,</span> <span class="n">g_monom</span> <span class="o">=</span> <span class="n">monom</span><span class="p">[:</span><span class="n">v</span><span class="p">],</span> <span class="n">monom</span><span class="p">[</span><span class="n">v</span><span class="p">:]</span>
-
-        <span class="k">if</span> <span class="n">f_monom</span> <span class="ow">in</span> <span class="n">h</span><span class="p">:</span>
-            <span class="n">h</span><span class="p">[</span><span class="n">f_monom</span><span class="p">][</span><span class="n">g_monom</span><span class="p">]</span> <span class="o">=</span> <span class="n">c</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">h</span><span class="p">[</span><span class="n">f_monom</span><span class="p">]</span> <span class="o">=</span> <span class="p">{</span><span class="n">g_monom</span><span class="p">:</span> <span class="n">c</span><span class="p">}</span>
-
-    <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">h</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="n">h</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span> <span class="o">=</span> <span class="n">K</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">dmp_from_dict</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">v</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_terms_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Remove GCD of terms from ``f`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dup_terms_gcd</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, 1, 0, 0])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_terms_gcd(f, ZZ)</span>
-<span class="sd">    (2, [1, 0, 1])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">dup_TC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span><span class="p">,</span> <span class="n">f</span>
-
-    <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-
-    <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">c</span><span class="p">:</span>
-            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">break</span>
-
-    <span class="k">return</span> <span class="n">i</span><span class="p">,</span> <span class="n">f</span><span class="p">[:</span><span class="o">-</span><span class="n">i</span><span class="p">]</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,g&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_terms_gcd"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_terms_gcd">[docs]</a><span class="k">def</span> <span class="nf">dmp_terms_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Remove GCD of terms from ``f`` in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_terms_gcd</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1, 0], [1, 0, 0], [], []])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_terms_gcd(f, 1, ZZ)</span>
-<span class="sd">    ((2, 1), [[1], [1, 0]])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">dmp_ground_TC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="ow">or</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="mi">0</span><span class="p">,)</span><span class="o">*</span><span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">f</span>
-
-    <span class="n">F</span> <span class="o">=</span> <span class="n">dmp_to_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-    <span class="n">G</span> <span class="o">=</span> <span class="n">monomial_min</span><span class="p">(</span><span class="o">*</span><span class="nb">list</span><span class="p">(</span><span class="n">F</span><span class="o">.</span><span class="n">keys</span><span class="p">()))</span>
-
-    <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">g</span> <span class="o">==</span> <span class="mi">0</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">G</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">G</span><span class="p">,</span> <span class="n">f</span>
-
-    <span class="n">f</span> <span class="o">=</span> <span class="p">{}</span>
-
-    <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">F</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="n">f</span><span class="p">[</span><span class="n">monomial_div</span><span class="p">(</span><span class="n">monom</span><span class="p">,</span> <span class="n">G</span><span class="p">)]</span> <span class="o">=</span> <span class="n">coeff</span>
-
-    <span class="k">return</span> <span class="n">G</span><span class="p">,</span> <span class="n">dmp_from_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;v,w,d,i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_rec_list_terms</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">monom</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Recursive helper for :func:`dmp_list_terms`.&quot;&quot;&quot;</span>
-    <span class="n">d</span><span class="p">,</span> <span class="n">terms</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">v</span><span class="p">),</span> <span class="p">[]</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">v</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">g</span><span class="p">):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">c</span><span class="p">:</span>
-                <span class="k">continue</span>
-
-            <span class="n">terms</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">monom</span> <span class="o">+</span> <span class="p">(</span><span class="n">d</span> <span class="o">-</span> <span class="n">i</span><span class="p">,),</span> <span class="n">c</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">w</span> <span class="o">=</span> <span class="n">v</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">g</span><span class="p">):</span>
-            <span class="n">terms</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">_rec_list_terms</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">monom</span> <span class="o">+</span> <span class="p">(</span><span class="n">d</span> <span class="o">-</span> <span class="n">i</span><span class="p">,)))</span>
-
-    <span class="k">return</span> <span class="n">terms</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_list_terms"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_list_terms">[docs]</a><span class="k">def</span> <span class="nf">dmp_list_terms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    List all non-zero terms from ``f`` in the given order ``order``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_list_terms</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1, 1], [2, 3]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_list_terms(f, 1, ZZ)</span>
-<span class="sd">    [((1, 1), 1), ((1, 0), 1), ((0, 1), 2), ((0, 0), 3)]</span>
-<span class="sd">    &gt;&gt;&gt; dmp_list_terms(f, 1, ZZ, order=&#39;grevlex&#39;)</span>
-<span class="sd">    [((1, 1), 1), ((1, 0), 1), ((0, 1), 2), ((0, 0), 3)]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">terms</span> <span class="o">=</span> <span class="n">_rec_list_terms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="p">())</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">terms</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[((</span><span class="mi">0</span><span class="p">,)</span><span class="o">*</span><span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">)]</span>
-
-    <span class="k">if</span> <span class="n">order</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">terms</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">sdp_sort</span><span class="p">(</span><span class="n">terms</span><span class="p">,</span> <span class="n">monomial_key</span><span class="p">(</span><span class="n">order</span><span class="p">))</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,m&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_apply_pairs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">args</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Apply ``h`` to pairs of coefficients of ``f`` and ``g``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dup_apply_pairs</span>
-
-<span class="sd">    &gt;&gt;&gt; h = lambda x, y, z: 2*x + y - z</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_apply_pairs([1, 2, 3], [3, 2, 1], h, (1,), ZZ)</span>
-<span class="sd">    [4, 5, 6]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">!=</span> <span class="n">m</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="n">m</span><span class="p">:</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">m</span><span class="p">)</span> <span class="o">+</span> <span class="n">g</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">m</span> <span class="o">-</span> <span class="n">n</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">h</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">dup_strip</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v,n,m&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_apply_pairs"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_apply_pairs">[docs]</a><span class="k">def</span> <span class="nf">dmp_apply_pairs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">args</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Apply ``h`` to pairs of coefficients of ``f`` and ``g``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dmp_apply_pairs</span>
-
-<span class="sd">    &gt;&gt;&gt; h = lambda x, y, z: 2*x + y - z</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_apply_pairs([[1], [2, 3]], [[3], [2, 1]], h, (1,), 1, ZZ)</span>
-<span class="sd">    [[4], [5, 6]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_apply_pairs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">args</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">!=</span> <span class="n">m</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="n">m</span><span class="p">:</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">dmp_zeros</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">m</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">+</span> <span class="n">g</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_zeros</span><span class="p">(</span><span class="n">m</span> <span class="o">-</span> <span class="n">n</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dmp_apply_pairs</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">args</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">dmp_strip</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;m,n,k,M,N&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_slice</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Take a continuous subsequence of terms of ``f`` in ``K[x]``. &quot;&quot;&quot;</span>
-    <span class="n">k</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">k</span> <span class="o">&gt;=</span> <span class="n">m</span><span class="p">:</span>
-        <span class="n">M</span> <span class="o">=</span> <span class="n">k</span> <span class="o">-</span> <span class="n">m</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">M</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">if</span> <span class="n">k</span> <span class="o">&gt;=</span> <span class="n">n</span><span class="p">:</span>
-        <span class="n">N</span> <span class="o">=</span> <span class="n">k</span> <span class="o">-</span> <span class="n">n</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">N</span> <span class="o">=</span> <span class="mi">0</span>
-
-    <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="n">N</span><span class="p">:</span><span class="n">M</span><span class="p">]</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span> <span class="o">+</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span><span class="o">*</span><span class="n">m</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;m,n,u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_slice"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dmp_slice">[docs]</a><span class="k">def</span> <span class="nf">dmp_slice</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Take a continuous subsequence of terms of ``f`` in ``K[X]``. &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dmp_slice_in</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;m,n,j,u,k&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dmp_slice_in</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Take a continuous subsequence of terms of ``f`` in ``x_j`` in ``K[X]``. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">j</span> <span class="o">&gt;</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s">&quot;-</span><span class="si">%s</span><span class="s"> &lt;= j &lt; </span><span class="si">%s</span><span class="s"> expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_slice</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">f</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">dmp_to_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">),</span> <span class="p">{}</span>
-
-    <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">monom</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">k</span> <span class="o">&lt;</span> <span class="n">m</span> <span class="ow">or</span> <span class="n">k</span> <span class="o">&gt;=</span> <span class="n">n</span><span class="p">:</span>
-            <span class="n">monom</span> <span class="o">=</span> <span class="n">monom</span><span class="p">[:</span><span class="n">j</span><span class="p">]</span> <span class="o">+</span> <span class="p">(</span><span class="mi">0</span><span class="p">,)</span> <span class="o">+</span> <span class="n">monom</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-
-        <span class="k">if</span> <span class="n">monom</span> <span class="ow">in</span> <span class="n">g</span><span class="p">:</span>
-            <span class="n">g</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span> <span class="o">+=</span> <span class="n">coeff</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">g</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span> <span class="o">=</span> <span class="n">coeff</span>
-
-    <span class="k">return</span> <span class="n">dmp_from_dict</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="dup_random"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densebasic.dup_random">[docs]</a><span class="k">def</span> <span class="nf">dup_random</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a polynomial of degree ``n`` with coefficients in ``[a, b]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densebasic import dup_random</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_random(3, -10, 10, ZZ) #doctest: +SKIP</span>
-<span class="sd">    [-2, -8, 9, -4]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="p">[</span> <span class="n">K</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">]</span>
-
-    <span class="k">while</span> <span class="ow">not</span> <span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-        <span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">f</span></div>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
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diff --git a/dev-py3k/_modules/sympy/polys/densetools.html b/dev-py3k/_modules/sympy/polys/densetools.html
deleted file mode 100644
index 2c32136fc53..00000000000
--- a/dev-py3k/_modules/sympy/polys/densetools.html
+++ /dev/null
@@ -1,1470 +0,0 @@
-
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-             >modules</a> |</li>
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-        <div class="bodywrapper">
-          <div class="body">
-            
-  <h1>Source code for sympy.polys.densetools</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Advanced tools for dense recursive polynomials in ``K[x]`` or ``K[X]``. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dup_strip</span><span class="p">,</span> <span class="n">dmp_strip</span><span class="p">,</span>
-    <span class="n">dup_convert</span><span class="p">,</span> <span class="n">dmp_convert</span><span class="p">,</span>
-    <span class="n">dup_degree</span><span class="p">,</span> <span class="n">dmp_degree</span><span class="p">,</span>
-    <span class="n">dmp_to_dict</span><span class="p">,</span>
-    <span class="n">dmp_from_dict</span><span class="p">,</span>
-    <span class="n">dup_LC</span><span class="p">,</span> <span class="n">dmp_LC</span><span class="p">,</span> <span class="n">dmp_ground_LC</span><span class="p">,</span>
-    <span class="n">dup_TC</span><span class="p">,</span> <span class="n">dmp_TC</span><span class="p">,</span>
-    <span class="n">dmp_zero</span><span class="p">,</span> <span class="n">dmp_ground</span><span class="p">,</span>
-    <span class="n">dmp_zero_p</span><span class="p">,</span>
-    <span class="n">dup_to_raw_dict</span><span class="p">,</span> <span class="n">dup_from_raw_dict</span><span class="p">,</span>
-    <span class="n">dmp_zeros</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dup_add_term</span><span class="p">,</span> <span class="n">dmp_add_term</span><span class="p">,</span>
-    <span class="n">dup_lshift</span><span class="p">,</span>
-    <span class="n">dup_add</span><span class="p">,</span> <span class="n">dmp_add</span><span class="p">,</span>
-    <span class="n">dup_sub</span><span class="p">,</span> <span class="n">dmp_sub</span><span class="p">,</span>
-    <span class="n">dup_mul</span><span class="p">,</span> <span class="n">dmp_mul</span><span class="p">,</span>
-    <span class="n">dup_sqr</span><span class="p">,</span>
-    <span class="n">dup_div</span><span class="p">,</span>
-    <span class="n">dup_rem</span><span class="p">,</span> <span class="n">dmp_rem</span><span class="p">,</span>
-    <span class="n">dmp_expand</span><span class="p">,</span>
-    <span class="n">dup_mul_ground</span><span class="p">,</span> <span class="n">dmp_mul_ground</span><span class="p">,</span>
-    <span class="n">dup_quo_ground</span><span class="p">,</span> <span class="n">dmp_quo_ground</span><span class="p">,</span>
-    <span class="n">dup_exquo_ground</span><span class="p">,</span> <span class="n">dmp_exquo_ground</span><span class="p">,</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">MultivariatePolynomialError</span><span class="p">,</span>
-    <span class="n">DomainError</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">cythonized</span><span class="p">,</span> <span class="n">variations</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">ceil</span> <span class="k">as</span> <span class="n">_ceil</span><span class="p">,</span> <span class="n">log</span> <span class="k">as</span> <span class="n">_log</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;m,n,i,j&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_integrate</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes the indefinite integral of ``f`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dup_integrate</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_integrate([QQ(1), QQ(2), QQ(0)], 1, QQ)</span>
-<span class="sd">    [1/3, 1/1, 0/1, 0/1]</span>
-<span class="sd">    &gt;&gt;&gt; dup_integrate([QQ(1), QQ(2), QQ(0)], 2, QQ)</span>
-<span class="sd">    [1/12, 1/3, 0/1, 0/1, 0/1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">&lt;=</span> <span class="mi">0</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">g</span> <span class="o">=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span><span class="o">*</span><span class="n">m</span>
-
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">f</span><span class="p">)):</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-            <span class="n">n</span> <span class="o">*=</span> <span class="n">i</span> <span class="o">+</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">1</span>
-
-        <span class="n">g</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">exquo</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">K</span><span class="p">(</span><span class="n">n</span><span class="p">)))</span>
-
-    <span class="k">return</span> <span class="n">g</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;m,u,v,n,i,j&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_integrate"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dmp_integrate">[docs]</a><span class="k">def</span> <span class="nf">dmp_integrate</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes the indefinite integral of ``f`` in ``x_0`` in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dmp_integrate</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_integrate([[QQ(1)], [QQ(2), QQ(0)]], 1, 1, QQ)</span>
-<span class="sd">    [[1/2], [2/1, 0/1], []]</span>
-<span class="sd">    &gt;&gt;&gt; dmp_integrate([[QQ(1)], [QQ(2), QQ(0)]], 2, 1, QQ)</span>
-<span class="sd">    [[1/6], [1/1, 0/1], [], []]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_integrate</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">m</span> <span class="o">&lt;=</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">g</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">dmp_zeros</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">f</span><span class="p">)):</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-            <span class="n">n</span> <span class="o">*=</span> <span class="n">i</span> <span class="o">+</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">1</span>
-
-        <span class="n">g</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">dmp_quo_ground</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">K</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">g</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;m,v,w,i,j&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_rec_integrate_in</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Recursive helper for :func:`dmp_integrate_in`.&quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_integrate</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">w</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="n">v</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="n">dmp_strip</span><span class="p">([</span> <span class="n">_rec_integrate_in</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">g</span> <span class="p">],</span> <span class="n">v</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;m,j,u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_integrate_in"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dmp_integrate_in">[docs]</a><span class="k">def</span> <span class="nf">dmp_integrate_in</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes the indefinite integral of ``f`` in ``x_j`` in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dmp_integrate_in</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_integrate_in([[QQ(1)], [QQ(2), QQ(0)]], 1, 0, 1, QQ)</span>
-<span class="sd">    [[1/2], [2/1, 0/1], []]</span>
-<span class="sd">    &gt;&gt;&gt; dmp_integrate_in([[QQ(1)], [QQ(2), QQ(0)]], 1, 1, 1, QQ)</span>
-<span class="sd">    [[1/1, 0/1], [1/1, 0/1, 0/1]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">j</span> <span class="o">&gt;</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s">&quot;0 &lt;= j &lt;= u expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">_rec_integrate_in</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;m,n,k,i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    ``m``-th order derivative of a polynomial in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dup_diff</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_diff([ZZ(1), ZZ(2), ZZ(3), ZZ(4)], 1, ZZ)</span>
-<span class="sd">    [3, 4, 3]</span>
-<span class="sd">    &gt;&gt;&gt; dup_diff([ZZ(1), ZZ(2), ZZ(3), ZZ(4)], 2, ZZ)</span>
-<span class="sd">    [6, 4]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="n">m</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-    <span class="n">deriv</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span><span class="p">[:</span><span class="o">-</span><span class="n">m</span><span class="p">]:</span>
-            <span class="n">deriv</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">K</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">coeff</span><span class="p">)</span>
-            <span class="n">n</span> <span class="o">-=</span> <span class="mi">1</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span><span class="p">[:</span><span class="o">-</span><span class="n">m</span><span class="p">]:</span>
-            <span class="n">k</span> <span class="o">=</span> <span class="n">n</span>
-
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">m</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-                <span class="n">k</span> <span class="o">*=</span> <span class="n">i</span>
-
-            <span class="n">deriv</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">K</span><span class="p">(</span><span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="n">coeff</span><span class="p">)</span>
-            <span class="n">n</span> <span class="o">-=</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="n">dup_strip</span><span class="p">(</span><span class="n">deriv</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v,m,n,k,i&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_diff"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dmp_diff">[docs]</a><span class="k">def</span> <span class="nf">dmp_diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    ``m``-th order derivative in ``x_0`` of a polynomial in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dmp_diff</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1, 2, 3], [2, 3, 1]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_diff(f, 1, 1, ZZ)</span>
-<span class="sd">    [[1, 2, 3]]</span>
-<span class="sd">    &gt;&gt;&gt; dmp_diff(f, 2, 1, ZZ)</span>
-<span class="sd">    [[]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="n">m</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
-
-    <span class="n">deriv</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="p">[],</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span><span class="p">[:</span><span class="o">-</span><span class="n">m</span><span class="p">]:</span>
-            <span class="n">deriv</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dmp_mul_ground</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">K</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-            <span class="n">n</span> <span class="o">-=</span> <span class="mi">1</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span><span class="p">[:</span><span class="o">-</span><span class="n">m</span><span class="p">]:</span>
-            <span class="n">k</span> <span class="o">=</span> <span class="n">n</span>
-
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">m</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-                <span class="n">k</span> <span class="o">*=</span> <span class="n">i</span>
-
-            <span class="n">deriv</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dmp_mul_ground</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">K</span><span class="p">(</span><span class="n">k</span><span class="p">),</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-            <span class="n">n</span> <span class="o">-=</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="n">dmp_strip</span><span class="p">(</span><span class="n">deriv</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;m,v,w,i,j&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_rec_diff_in</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Recursive helper for :func:`dmp_diff_in`.&quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_diff</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">w</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="n">v</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="n">dmp_strip</span><span class="p">([</span> <span class="n">_rec_diff_in</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">g</span> <span class="p">],</span> <span class="n">v</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;m,j,u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_diff_in"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dmp_diff_in">[docs]</a><span class="k">def</span> <span class="nf">dmp_diff_in</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    ``m``-th order derivative in ``x_j`` of a polynomial in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dmp_diff_in</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1, 2, 3], [2, 3, 1]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_diff_in(f, 1, 0, 1, ZZ)</span>
-<span class="sd">    [[1, 2, 3]]</span>
-<span class="sd">    &gt;&gt;&gt; dmp_diff_in(f, 1, 1, 1, ZZ)</span>
-<span class="sd">    [[2, 2], [4, 3]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">j</span> <span class="o">&gt;</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s">&quot;0 &lt;= j &lt;= </span><span class="si">%s</span><span class="s"> expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">_rec_diff_in</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_eval</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Evaluate a polynomial at ``x = a`` in ``K[x]`` using Horner scheme.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dup_eval</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_eval([ZZ(1), ZZ(2), ZZ(3)], 2, ZZ)</span>
-<span class="sd">    11</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">a</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_TC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-
-    <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">*=</span> <span class="n">a</span>
-        <span class="n">result</span> <span class="o">+=</span> <span class="n">c</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_eval"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dmp_eval">[docs]</a><span class="k">def</span> <span class="nf">dmp_eval</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Evaluate a polynomial at ``x_0 = a`` in ``K[X]`` using the Horner scheme.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dmp_eval</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[2, 3], [1, 2]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_eval(f, 2, 1, ZZ)</span>
-<span class="sd">    [5, 8]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_eval</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">a</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_TC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">result</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">dmp_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">dmp_mul_ground</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">dmp_add</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;v,i,j&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_rec_eval_in</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Recursive helper for :func:`dmp_eval_in`.&quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_eval</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">v</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="n">v</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="n">dmp_strip</span><span class="p">([</span> <span class="n">_rec_eval_in</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">g</span> <span class="p">],</span> <span class="n">v</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_eval_in"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dmp_eval_in">[docs]</a><span class="k">def</span> <span class="nf">dmp_eval_in</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Evaluate a polynomial at ``x_j = a`` in ``K[X]`` using the Horner scheme.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dmp_eval_in</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[2, 3], [1, 2]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_eval_in(f, 2, 0, 1, ZZ)</span>
-<span class="sd">    [5, 8]</span>
-<span class="sd">    &gt;&gt;&gt; dmp_eval_in(f, 2, 1, 1, ZZ)</span>
-<span class="sd">    [7, 4]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">j</span> <span class="o">&gt;</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s">&quot;0 &lt;= j &lt;= </span><span class="si">%s</span><span class="s"> expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">_rec_eval_in</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i,u&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_rec_eval_tail</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Recursive helper for :func:`dmp_eval_tail`.&quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_eval</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">A</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="p">[</span> <span class="n">_rec_eval_tail</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">g</span> <span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">u</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">A</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">h</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">dup_eval</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">A</span><span class="p">[</span><span class="o">-</span><span class="n">u</span> <span class="o">+</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">K</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_eval_tail"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dmp_eval_tail">[docs]</a><span class="k">def</span> <span class="nf">dmp_eval_tail</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Evaluate a polynomial at ``x_j = a_j, ...`` in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dmp_eval_tail</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[2, 3], [1, 2]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_eval_tail(f, (2, 2), 1, ZZ)</span>
-<span class="sd">    18</span>
-<span class="sd">    &gt;&gt;&gt; dmp_eval_tail(f, (2,), 1, ZZ)</span>
-<span class="sd">    [7, 4]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">A</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">A</span><span class="p">))</span>
-
-    <span class="n">e</span> <span class="o">=</span> <span class="n">_rec_eval_tail</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">u</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">A</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">e</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_strip</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">u</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">A</span><span class="p">))</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;m,v,i,j&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_rec_diff_eval</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Recursive helper for :func:`dmp_diff_eval`.&quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_eval</span><span class="p">(</span><span class="n">dmp_diff</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">a</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">v</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="n">v</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="n">dmp_strip</span><span class="p">([</span> <span class="n">_rec_diff_eval</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">g</span> <span class="p">],</span> <span class="n">v</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;m,j,u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_diff_eval_in"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dmp_diff_eval_in">[docs]</a><span class="k">def</span> <span class="nf">dmp_diff_eval_in</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Differentiate and evaluate a polynomial in ``x_j`` at ``a`` in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dmp_diff_eval_in</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1, 2, 3], [2, 3, 1]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_diff_eval_in(f, 1, 2, 0, 1, ZZ)</span>
-<span class="sd">    [1, 2, 3]</span>
-<span class="sd">    &gt;&gt;&gt; dmp_diff_eval_in(f, 1, 2, 1, 1, ZZ)</span>
-<span class="sd">    [6, 11]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">j</span> <span class="o">&gt;</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s">&quot;-</span><span class="si">%s</span><span class="s"> &lt;= j &lt; </span><span class="si">%s</span><span class="s"> expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">j</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_eval</span><span class="p">(</span><span class="n">dmp_diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">a</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">_rec_diff_eval</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_trunc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Reduce a ``K[x]`` polynomial modulo a constant ``p`` in ``K``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dup_trunc</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([2, 3, 5, 7])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_trunc(f, ZZ(3), ZZ)</span>
-<span class="sd">    [-1, 0, -1, 1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_ZZ</span><span class="p">:</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">c</span> <span class="o">%</span> <span class="n">p</span>
-
-            <span class="k">if</span> <span class="n">c</span> <span class="o">&gt;</span> <span class="n">p</span> <span class="o">//</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">g</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">c</span> <span class="o">-</span> <span class="n">p</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">g</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="p">[</span> <span class="n">c</span> <span class="o">%</span> <span class="n">p</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-    <span class="k">return</span> <span class="n">dup_strip</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_trunc"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dmp_trunc">[docs]</a><span class="k">def</span> <span class="nf">dmp_trunc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Reduce a ``K[X]`` polynomial modulo a polynomial ``p`` in ``K[Y]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dmp_trunc</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[3, 8], [5, 6], [2, 3]])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([1, -1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_trunc(f, g, 1, ZZ)</span>
-<span class="sd">    [[11], [11], [5]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dmp_strip</span><span class="p">([</span> <span class="n">dmp_rem</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">],</span> <span class="n">u</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_ground_trunc"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dmp_ground_trunc">[docs]</a><span class="k">def</span> <span class="nf">dmp_ground_trunc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Reduce a ``K[X]`` polynomial modulo a constant ``p`` in ``K``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dmp_ground_trunc</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[3, 8], [5, 6], [2, 3]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_ground_trunc(f, ZZ(3), 1, ZZ)</span>
-<span class="sd">    [[-1], [-1, 0], [-1, 0]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_trunc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="n">dmp_strip</span><span class="p">([</span> <span class="n">dmp_ground_trunc</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">],</span> <span class="n">u</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="dup_monic"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dup_monic">[docs]</a><span class="k">def</span> <span class="nf">dup_monic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Divide all coefficients by ``LC(f)`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ, QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dup_monic</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_monic([ZZ(3), ZZ(6), ZZ(9)], ZZ)</span>
-<span class="sd">    [1, 2, 3]</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_monic([QQ(3), QQ(4), QQ(2)], QQ)</span>
-<span class="sd">    [1/1, 4/3, 2/3]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">lc</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">lc</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_exquo_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">lc</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_ground_monic"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dmp_ground_monic">[docs]</a><span class="k">def</span> <span class="nf">dmp_ground_monic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Divide all coefficients by ``LC(f)`` in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ, QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dmp_ground_monic</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[3, 6], [3, 0], [9, 3]])</span>
-<span class="sd">    &gt;&gt;&gt; g = QQ.map([[3, 8], [5, 6], [2, 3]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_ground_monic(f, 1, ZZ)</span>
-<span class="sd">    [[1, 2], [1, 0], [3, 1]]</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_ground_monic(g, 1, QQ)</span>
-<span class="sd">    [[1/1, 8/3], [5/3, 2/1], [2/3, 1/1]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_monic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">lc</span> <span class="o">=</span> <span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">lc</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_exquo_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">lc</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="dup_content"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dup_content">[docs]</a><span class="k">def</span> <span class="nf">dup_content</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute the GCD of coefficients of ``f`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ, QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dup_content</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([6, 8, 12])</span>
-<span class="sd">    &gt;&gt;&gt; g = QQ.map([6, 8, 12])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_content(f, ZZ)</span>
-<span class="sd">    2</span>
-<span class="sd">    &gt;&gt;&gt; dup_content(g, QQ)</span>
-<span class="sd">    2/1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">QQ</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-
-    <span class="n">cont</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-
-    <span class="k">if</span> <span class="n">K</span> <span class="o">==</span> <span class="n">QQ</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-            <span class="n">cont</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">cont</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-            <span class="n">cont</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">cont</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">cont</span><span class="p">):</span>
-                <span class="k">break</span>
-
-    <span class="k">return</span> <span class="n">cont</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_ground_content"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dmp_ground_content">[docs]</a><span class="k">def</span> <span class="nf">dmp_ground_content</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute the GCD of coefficients of ``f`` in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ, QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dmp_ground_content</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[2, 6], [4, 12]])</span>
-<span class="sd">    &gt;&gt;&gt; g = QQ.map([[2, 6], [4, 12]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_ground_content(f, 1, ZZ)</span>
-<span class="sd">    2</span>
-<span class="sd">    &gt;&gt;&gt; dmp_ground_content(g, 1, QQ)</span>
-<span class="sd">    2/1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">QQ</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_content</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-
-    <span class="n">cont</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">,</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="n">K</span> <span class="o">==</span> <span class="n">QQ</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-            <span class="n">cont</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">cont</span><span class="p">,</span> <span class="n">dmp_ground_content</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-            <span class="n">cont</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">cont</span><span class="p">,</span> <span class="n">dmp_ground_content</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-
-            <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">cont</span><span class="p">):</span>
-                <span class="k">break</span>
-
-    <span class="k">return</span> <span class="n">cont</span>
-
-</div>
-<div class="viewcode-block" id="dup_primitive"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dup_primitive">[docs]</a><span class="k">def</span> <span class="nf">dup_primitive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute content and the primitive form of ``f`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ, QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dup_primitive</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([6, 8, 12])</span>
-<span class="sd">    &gt;&gt;&gt; g = QQ.map([6, 8, 12])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_primitive(f, ZZ)</span>
-<span class="sd">    (2, [3, 4, 6])</span>
-<span class="sd">    &gt;&gt;&gt; dup_primitive(g, QQ)</span>
-<span class="sd">    (2/1, [3/1, 4/1, 6/1])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">,</span> <span class="n">f</span>
-
-    <span class="n">cont</span> <span class="o">=</span> <span class="n">dup_content</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">cont</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">cont</span><span class="p">,</span> <span class="n">f</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">cont</span><span class="p">,</span> <span class="n">dup_quo_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">cont</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_ground_primitive"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dmp_ground_primitive">[docs]</a><span class="k">def</span> <span class="nf">dmp_ground_primitive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute content and the primitive form of ``f`` in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ, QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dmp_ground_primitive</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[2, 6], [4, 12]])</span>
-<span class="sd">    &gt;&gt;&gt; g = QQ.map([[2, 6], [4, 12]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_ground_primitive(f, 1, ZZ)</span>
-<span class="sd">    (2, [[1, 3], [2, 6]])</span>
-<span class="sd">    &gt;&gt;&gt; dmp_ground_primitive(g, 1, QQ)</span>
-<span class="sd">    (2/1, [[1/1, 3/1], [2/1, 6/1]])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_primitive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">,</span> <span class="n">f</span>
-
-    <span class="n">cont</span> <span class="o">=</span> <span class="n">dmp_ground_content</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">cont</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">cont</span><span class="p">,</span> <span class="n">f</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">cont</span><span class="p">,</span> <span class="n">dmp_quo_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">cont</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="dup_extract"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dup_extract">[docs]</a><span class="k">def</span> <span class="nf">dup_extract</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Extract common content from a pair of polynomials in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dup_extract</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([6, 12, 18])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([4, 8, 12])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_extract(f, g, ZZ)</span>
-<span class="sd">    (2, [3, 6, 9], [2, 4, 6])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">fc</span> <span class="o">=</span> <span class="n">dup_content</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">gc</span> <span class="o">=</span> <span class="n">dup_content</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">gcd</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">fc</span><span class="p">,</span> <span class="n">gc</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">K</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">gcd</span><span class="p">):</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">dup_quo_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gcd</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">dup_quo_ground</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">gcd</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">gcd</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">g</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_ground_extract"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dmp_ground_extract">[docs]</a><span class="k">def</span> <span class="nf">dmp_ground_extract</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Extract common content from a pair of polynomials in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dmp_ground_extract</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[6, 12], [18]])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[4, 8], [12]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_ground_extract(f, g, 1, ZZ)</span>
-<span class="sd">    (2, [[3, 6], [9]], [[2, 4], [6]])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">fc</span> <span class="o">=</span> <span class="n">dmp_ground_content</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">gc</span> <span class="o">=</span> <span class="n">dmp_ground_content</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">gcd</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">fc</span><span class="p">,</span> <span class="n">gc</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">K</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">gcd</span><span class="p">):</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_quo_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gcd</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">dmp_quo_ground</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">gcd</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">gcd</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">g</span>
-
-</div>
-<div class="viewcode-block" id="dup_real_imag"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dup_real_imag">[docs]</a><span class="k">def</span> <span class="nf">dup_real_imag</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return bivariate polynomials ``f1`` and ``f2``, such that ``f = f1 + f2*I``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dup_real_imag</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_real_imag([ZZ(1), ZZ(1), ZZ(1), ZZ(1)], ZZ)</span>
-<span class="sd">    ([[1], [1], [-3, 0, 1], [-1, 0, 1]], [[3, 0], [2, 0], [-1, 0, 1, 0]])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">K</span><span class="o">.</span><span class="n">is_ZZ</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">K</span><span class="o">.</span><span class="n">is_QQ</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">DomainError</span><span class="p">(</span>
-            <span class="s">&quot;computing real and imaginary parts is not supported over </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">f1</span> <span class="o">=</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-    <span class="n">f2</span> <span class="o">=</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f1</span><span class="p">,</span> <span class="n">f2</span>
-
-    <span class="n">g</span> <span class="o">=</span> <span class="p">[[[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]],</span> <span class="p">[[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">],</span> <span class="p">[]]]</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_ground</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">2</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_add_term</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">dmp_ground</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">H</span> <span class="o">=</span> <span class="n">dup_to_raw_dict</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">H</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">k</span> <span class="o">%</span> <span class="mi">4</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">m</span><span class="p">:</span>
-            <span class="n">f1</span> <span class="o">=</span> <span class="n">dmp_add</span><span class="p">(</span><span class="n">f1</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">f2</span> <span class="o">=</span> <span class="n">dmp_add</span><span class="p">(</span><span class="n">f2</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">f1</span> <span class="o">=</span> <span class="n">dmp_sub</span><span class="p">(</span><span class="n">f1</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">f2</span> <span class="o">=</span> <span class="n">dmp_sub</span><span class="p">(</span><span class="n">f2</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">f1</span><span class="p">,</span> <span class="n">f2</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&#39;i,n&#39;</span><span class="p">)</span>
-<div class="viewcode-block" id="dup_mirror"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dup_mirror">[docs]</a><span class="k">def</span> <span class="nf">dup_mirror</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Evaluate efficiently the composition ``f(-x)`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dup_mirror</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_mirror([ZZ(1), ZZ(2), -ZZ(4), ZZ(2)], ZZ)</span>
-<span class="sd">    [-1, 2, 4, 2]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">a</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="o">-</span><span class="n">K</span><span class="o">.</span><span class="n">one</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-        <span class="n">f</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="o">-</span><span class="n">a</span>
-
-    <span class="k">return</span> <span class="n">f</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&#39;i,n&#39;</span><span class="p">)</span>
-<div class="viewcode-block" id="dup_scale"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dup_scale">[docs]</a><span class="k">def</span> <span class="nf">dup_scale</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Evaluate efficiently composition ``f(a*x)`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dup_scale</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_scale([ZZ(1), -ZZ(2), ZZ(1)], ZZ(2), ZZ)</span>
-<span class="sd">    [4, -4, 1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="n">a</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-        <span class="n">f</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">b</span><span class="o">*</span><span class="n">a</span>
-
-    <span class="k">return</span> <span class="n">f</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&#39;i,j,n&#39;</span><span class="p">)</span>
-<div class="viewcode-block" id="dup_shift"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dup_shift">[docs]</a><span class="k">def</span> <span class="nf">dup_shift</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Evaluate efficiently Taylor shift ``f(x + a)`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dup_shift</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_shift([ZZ(1), -ZZ(2), ZZ(1)], ZZ(2), ZZ)</span>
-<span class="sd">    [1, 2, 1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">f</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-            <span class="n">f</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+=</span> <span class="n">a</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-
-    <span class="k">return</span> <span class="n">f</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&#39;i,n&#39;</span><span class="p">)</span>
-<div class="viewcode-block" id="dup_transform"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dup_transform">[docs]</a><span class="k">def</span> <span class="nf">dup_transform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Evaluate functional transformation ``q**n * f(p/q)`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dup_transform</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, -2, 1])</span>
-<span class="sd">    &gt;&gt;&gt; p = ZZ.map([1, 0, 1])</span>
-<span class="sd">    &gt;&gt;&gt; q = ZZ.map([1, -1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_transform(f, p, q, ZZ)</span>
-<span class="sd">    [1, -2, 5, -4, 4]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">h</span><span class="p">,</span> <span class="n">Q</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span> <span class="p">[[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="n">Q</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dup_mul</span><span class="p">(</span><span class="n">Q</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">q</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-
-    <span class="k">for</span> <span class="n">c</span><span class="p">,</span> <span class="n">q</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">Q</span><span class="p">[</span><span class="mi">1</span><span class="p">:]):</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dup_mul</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">q</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dup_add</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">h</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_compose</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Evaluate functional composition ``f(g)`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dup_compose</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 1, 0])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([1, -1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_compose(f, g, ZZ)</span>
-<span class="sd">    [1, -1, 0]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">g</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_strip</span><span class="p">([</span><span class="n">dup_eval</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">)])</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span>
-
-    <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dup_mul</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dup_add_term</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">h</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_compose"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dmp_compose">[docs]</a><span class="k">def</span> <span class="nf">dmp_compose</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Evaluate functional composition ``f(g)`` in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dmp_compose</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1, 2], [1, 0]])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[1, 0]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_compose(f, g, 1, ZZ)</span>
-<span class="sd">    [[1, 3, 0]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_compose</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span>
-
-    <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_add_term</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">h</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;s,n,r,i,j&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_dup_right_decompose</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Helper function for :func:`_dup_decompose`.&quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">lc</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">f</span> <span class="o">=</span> <span class="n">dup_to_raw_dict</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">g</span> <span class="o">=</span> <span class="p">{</span> <span class="n">s</span><span class="p">:</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span> <span class="p">}</span>
-
-    <span class="n">r</span> <span class="o">=</span> <span class="n">n</span> <span class="o">//</span> <span class="n">s</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="n">coeff</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span> <span class="o">+</span> <span class="n">j</span> <span class="o">-</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-                <span class="k">continue</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">s</span> <span class="o">-</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">g</span><span class="p">:</span>
-                <span class="k">continue</span>
-
-            <span class="n">fc</span><span class="p">,</span> <span class="n">gc</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="n">n</span> <span class="o">+</span> <span class="n">j</span> <span class="o">-</span> <span class="n">i</span><span class="p">],</span> <span class="n">g</span><span class="p">[</span><span class="n">s</span> <span class="o">-</span> <span class="n">j</span><span class="p">]</span>
-            <span class="n">coeff</span> <span class="o">+=</span> <span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="n">r</span><span class="o">*</span><span class="n">j</span><span class="p">)</span><span class="o">*</span><span class="n">fc</span><span class="o">*</span><span class="n">gc</span>
-
-        <span class="n">g</span><span class="p">[</span><span class="n">s</span> <span class="o">-</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">i</span><span class="o">*</span><span class="n">r</span><span class="o">*</span><span class="n">lc</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">dup_from_raw_dict</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_dup_left_decompose</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Helper function for :func:`_dup_decompose`.&quot;&quot;&quot;</span>
-    <span class="n">g</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="p">{},</span> <span class="mi">0</span>
-
-    <span class="k">while</span> <span class="n">f</span><span class="p">:</span>
-        <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dup_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">r</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">f</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="n">q</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="n">dup_from_raw_dict</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;df,s&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_dup_decompose</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Helper function for :func:`dup_decompose`.&quot;&quot;&quot;</span>
-    <span class="n">df</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">df</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">df</span> <span class="o">%</span> <span class="n">s</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">continue</span>
-
-        <span class="n">h</span> <span class="o">=</span> <span class="n">_dup_right_decompose</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">h</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">_dup_left_decompose</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">g</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span>
-
-    <span class="k">return</span> <span class="bp">None</span>
-
-
-<div class="viewcode-block" id="dup_decompose"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dup_decompose">[docs]</a><span class="k">def</span> <span class="nf">dup_decompose</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes functional decomposition of ``f`` in ``K[x]``.</span>
-
-<span class="sd">    Given a univariate polynomial ``f`` with coefficients in a field of</span>
-<span class="sd">    characteristic zero, returns list ``[f_1, f_2, ..., f_n]``, where::</span>
-
-<span class="sd">              f = f_1 o f_2 o ... f_n = f_1(f_2(... f_n))</span>
-
-<span class="sd">    and ``f_2, ..., f_n`` are monic and homogeneous polynomials of at</span>
-<span class="sd">    least second degree.</span>
-
-<span class="sd">    Unlike factorization, complete functional decompositions of</span>
-<span class="sd">    polynomials are not unique, consider examples:</span>
-
-<span class="sd">    1. ``f o g = f(x + b) o (g - b)``</span>
-<span class="sd">    2. ``x**n o x**m = x**m o x**n``</span>
-<span class="sd">    3. ``T_n o T_m = T_m o T_n``</span>
-
-<span class="sd">    where ``T_n`` and ``T_m`` are Chebyshev polynomials.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dup_decompose</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, -2, 1, 0, 0])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_decompose(f, ZZ)</span>
-<span class="sd">    [[1, 0, 0], [1, -1, 0]]</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Kozen89]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">F</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">_dup_decompose</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">f</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">result</span>
-            <span class="n">F</span> <span class="o">=</span> <span class="p">[</span><span class="n">h</span><span class="p">]</span> <span class="o">+</span> <span class="n">F</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">break</span>
-
-    <span class="k">return</span> <span class="p">[</span><span class="n">f</span><span class="p">]</span> <span class="o">+</span> <span class="n">F</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_lift"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dmp_lift">[docs]</a><span class="k">def</span> <span class="nf">dmp_lift</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Convert algebraic coefficients to integers in ``K[X]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import I</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dmp_lift</span>
-
-<span class="sd">    &gt;&gt;&gt; K = QQ.algebraic_field(I)</span>
-<span class="sd">    &gt;&gt;&gt; f = [K(1), K([QQ(1), QQ(0)]), K([QQ(2), QQ(0)])]</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_lift(f, 0, K)</span>
-<span class="sd">    [1/1, 0/1, 2/1, 0/1, 9/1, 0/1, -8/1, 0/1, 16/1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">K</span><span class="o">.</span><span class="n">is_Algebraic</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">DomainError</span><span class="p">(</span>
-            <span class="s">&#39;computation can be done only in an algebraic domain&#39;</span><span class="p">)</span>
-
-    <span class="n">F</span><span class="p">,</span> <span class="n">monoms</span><span class="p">,</span> <span class="n">polys</span> <span class="o">=</span> <span class="n">dmp_to_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">),</span> <span class="p">[],</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">F</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_ground</span><span class="p">:</span>
-            <span class="n">monoms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">monom</span><span class="p">)</span>
-
-    <span class="n">perms</span> <span class="o">=</span> <span class="n">variations</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="nb">len</span><span class="p">(</span><span class="n">monoms</span><span class="p">),</span> <span class="n">repetition</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">perm</span> <span class="ow">in</span> <span class="n">perms</span><span class="p">:</span>
-        <span class="n">G</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">F</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">sign</span><span class="p">,</span> <span class="n">monom</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">perm</span><span class="p">,</span> <span class="n">monoms</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">sign</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                <span class="n">G</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">G</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span>
-
-        <span class="n">polys</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dmp_from_dict</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">dmp_convert</span><span class="p">(</span><span class="n">dmp_expand</span><span class="p">(</span><span class="n">polys</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="dup_sign_variations"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dup_sign_variations">[docs]</a><span class="k">def</span> <span class="nf">dup_sign_variations</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute the number of sign variations of ``f`` in ``K[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dup_sign_variations</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, -1, -1, 1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_sign_variations(f, ZZ)</span>
-<span class="sd">    2</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">prev</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">,</span> <span class="mi">0</span>
-
-    <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_negative</span><span class="p">(</span><span class="n">coeff</span><span class="o">*</span><span class="n">prev</span><span class="p">):</span>
-            <span class="n">k</span> <span class="o">+=</span> <span class="mi">1</span>
-
-        <span class="k">if</span> <span class="n">coeff</span><span class="p">:</span>
-            <span class="n">prev</span> <span class="o">=</span> <span class="n">coeff</span>
-
-    <span class="k">return</span> <span class="n">k</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_clear_denoms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">convert</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Clear denominators, i.e. transform ``K_0`` to ``K_1``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ, ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dup_clear_denoms</span>
-
-<span class="sd">    &gt;&gt;&gt; f = [QQ(1,2), QQ(1,3)]</span>
-<span class="sd">    &gt;&gt;&gt; dup_clear_denoms(f, QQ, convert=False)</span>
-<span class="sd">    (6, [3/1, 2/1])</span>
-
-<span class="sd">    &gt;&gt;&gt; f = [QQ(1,2), QQ(1,3)]</span>
-<span class="sd">    &gt;&gt;&gt; dup_clear_denoms(f, QQ, convert=True)</span>
-<span class="sd">    (6, [3, 2])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">K1</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">K0</span><span class="o">.</span><span class="n">has_assoc_Ring</span><span class="p">:</span>
-            <span class="n">K1</span> <span class="o">=</span> <span class="n">K0</span><span class="o">.</span><span class="n">get_ring</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">K1</span> <span class="o">=</span> <span class="n">K0</span>
-
-    <span class="n">common</span> <span class="o">=</span> <span class="n">K1</span><span class="o">.</span><span class="n">one</span>
-
-    <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-        <span class="n">common</span> <span class="o">=</span> <span class="n">K1</span><span class="o">.</span><span class="n">lcm</span><span class="p">(</span><span class="n">common</span><span class="p">,</span> <span class="n">K0</span><span class="o">.</span><span class="n">denom</span><span class="p">(</span><span class="n">c</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">K1</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">common</span><span class="p">):</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">common</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">convert</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">common</span><span class="p">,</span> <span class="n">f</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">common</span><span class="p">,</span> <span class="n">dup_convert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;v,w&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_rec_clear_denoms</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Recursive helper for :func:`dmp_clear_denoms`.&quot;&quot;&quot;</span>
-    <span class="n">common</span> <span class="o">=</span> <span class="n">K1</span><span class="o">.</span><span class="n">one</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">v</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">g</span><span class="p">:</span>
-            <span class="n">common</span> <span class="o">=</span> <span class="n">K1</span><span class="o">.</span><span class="n">lcm</span><span class="p">(</span><span class="n">common</span><span class="p">,</span> <span class="n">K0</span><span class="o">.</span><span class="n">denom</span><span class="p">(</span><span class="n">c</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">w</span> <span class="o">=</span> <span class="n">v</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">g</span><span class="p">:</span>
-            <span class="n">common</span> <span class="o">=</span> <span class="n">K1</span><span class="o">.</span><span class="n">lcm</span><span class="p">(</span><span class="n">common</span><span class="p">,</span> <span class="n">_rec_clear_denoms</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">common</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_clear_denoms"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dmp_clear_denoms">[docs]</a><span class="k">def</span> <span class="nf">dmp_clear_denoms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">convert</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Clear denominators, i.e. transform ``K_0`` to ``K_1``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ, ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dmp_clear_denoms</span>
-
-<span class="sd">    &gt;&gt;&gt; f = [[QQ(1,2)], [QQ(1,3), QQ(1)]]</span>
-<span class="sd">    &gt;&gt;&gt; dmp_clear_denoms(f, 1, QQ, convert=False)</span>
-<span class="sd">    (6, [[3/1], [2/1, 6/1]])</span>
-
-<span class="sd">    &gt;&gt;&gt; f = [[QQ(1,2)], [QQ(1,3), QQ(1)]]</span>
-<span class="sd">    &gt;&gt;&gt; dmp_clear_denoms(f, 1, QQ, convert=True)</span>
-<span class="sd">    (6, [[3], [2, 6]])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_clear_denoms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">,</span> <span class="n">convert</span><span class="o">=</span><span class="n">convert</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">K1</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">K0</span><span class="o">.</span><span class="n">has_assoc_Ring</span><span class="p">:</span>
-            <span class="n">K1</span> <span class="o">=</span> <span class="n">K0</span><span class="o">.</span><span class="n">get_ring</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">K1</span> <span class="o">=</span> <span class="n">K0</span>
-
-    <span class="n">common</span> <span class="o">=</span> <span class="n">_rec_clear_denoms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">K1</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">common</span><span class="p">):</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_mul_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">common</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">convert</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">common</span><span class="p">,</span> <span class="n">f</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">common</span><span class="p">,</span> <span class="n">dmp_convert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&#39;i,n&#39;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_revert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute ``f**(-1)`` mod ``x**n`` using Newton iteration.</span>
-
-<span class="sd">    This function computes first ``2**n`` terms of a polynomial that</span>
-<span class="sd">    is a result of inversion of a polynomial modulo ``x**n``. This is</span>
-<span class="sd">    useful to efficiently compute series expansion of ``1/f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dup_revert</span>
-
-<span class="sd">    &gt;&gt;&gt; f = [-QQ(1,720), QQ(0), QQ(1,24), QQ(0), -QQ(1,2), QQ(0), QQ(1)]</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_revert(f, 8, QQ)</span>
-<span class="sd">    [61/720, 0/1, 5/24, 0/1, 1/2, 0/1, 1/1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">g</span> <span class="o">=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">revert</span><span class="p">(</span><span class="n">dup_TC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">))]</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span>
-
-    <span class="n">N</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">_ceil</span><span class="p">(</span><span class="n">_log</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">)))</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">N</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">dup_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">dup_sqr</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">dup_rem</span><span class="p">(</span><span class="n">dup_sub</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dup_lshift</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">h</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">g</span>
-
-
-<div class="viewcode-block" id="dmp_revert"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.densetools.dmp_revert">[docs]</a><span class="k">def</span> <span class="nf">dmp_revert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute ``f**(-1)`` mod ``x**n`` using Newton iteration.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.densetools import dmp_revert</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_revert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">MultivariatePolynomialError</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span></div>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.polys.distributedmodules</h1><div class="highlight"><pre>
-<span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">Sparse distributed elements of free modules over multivariate (generalized)</span>
-<span class="sd">polynomial rings.</span>
-
-<span class="sd">This code and its data structures are very much like the distributed</span>
-<span class="sd">polynomials ``sdp_...``, except that the first &quot;exponent&quot; of the monomial is</span>
-<span class="sd">a module generator index. That is, the multi-exponent ``(i, e_1, ..., e_n)``</span>
-<span class="sd">represents the &quot;monomial&quot; `x_1^{e_1} \dots x_n^{e_n} f_i` of the free module</span>
-<span class="sd">`F` generated by `f_1, \dots, f_r` over (a localization of) the ring</span>
-<span class="sd">`K[x_1, \dots, x_n]`. A module element is simply stored as a list of terms</span>
-<span class="sd">ordered by the monomial order. Here a term is a pair of a multi-exponent and a</span>
-<span class="sd">coefficient. In general, this coefficient should never be zero (since it can</span>
-<span class="sd">then be omitted). The zero module element is stored as an empty list.</span>
-
-<span class="sd">The main routines are ``sdm_nf_mora`` and ``sdm_groebner`` which can be used</span>
-<span class="sd">to compute, respectively, weak normal forms and standard bases. They work with</span>
-<span class="sd">arbitrary (not necessarily global) monomial orders.</span>
-
-<span class="sd">In general, product orders have to be used to construct valid monomial orders</span>
-<span class="sd">for modules. However, ``lex`` can be used as-is.</span>
-
-<span class="sd">Note that the &quot;level&quot; (number of variables, i.e. parameter u+1 in</span>
-<span class="sd">distributedpolys.py) is never needed in this code.</span>
-
-<span class="sd">The main reference for this file is [SCA],</span>
-<span class="sd">&quot;A Singular Introduction to Commutative Algebra&quot;.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="c"># TODO cythonize</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.monomialtools</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">monomial_mul</span><span class="p">,</span> <span class="n">monomial_lcm</span><span class="p">,</span> <span class="n">monomial_div</span><span class="p">,</span> <span class="n">monomial_deg</span><span class="p">,</span> <span class="n">monomial_divides</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.distributedpolys</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">sdp_LC</span><span class="p">,</span> <span class="n">sdp_from_dict</span><span class="p">,</span> <span class="n">sdp_to_dict</span><span class="p">,</span> <span class="n">sdp_add</span><span class="p">,</span> <span class="n">sdp_strip</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polytools</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polyutils</span> <span class="kn">import</span> <span class="n">parallel_dict_from_expr</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">permutations</span><span class="p">,</span> <span class="nb">next</span>
-
-<span class="c"># Additional monomial tools.</span>
-
-
-<div class="viewcode-block" id="sdm_monomial_mul"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedmodules.sdm_monomial_mul">[docs]</a><span class="k">def</span> <span class="nf">sdm_monomial_mul</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Multiply tuple ``X`` representing a monomial of `K[X]` into the tuple</span>
-<span class="sd">    ``M`` representing a monomial of `F`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Multiplying `xy^3` into `x f_1` yields `x^2 y^3 f_1`:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.distributedmodules import sdm_monomial_mul</span>
-<span class="sd">    &gt;&gt;&gt; sdm_monomial_mul((1, 1, 0), (1, 3))</span>
-<span class="sd">    (1, 2, 3)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">(</span><span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">],)</span> <span class="o">+</span> <span class="n">monomial_mul</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">M</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-
-</div>
-<div class="viewcode-block" id="sdm_monomial_deg"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedmodules.sdm_monomial_deg">[docs]</a><span class="k">def</span> <span class="nf">sdm_monomial_deg</span><span class="p">(</span><span class="n">M</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the total degree of ``M``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    For example, the total degree of `x^2 y f_5` is 3:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.distributedmodules import sdm_monomial_deg</span>
-<span class="sd">    &gt;&gt;&gt; sdm_monomial_deg((5, 2, 1))</span>
-<span class="sd">    3</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">monomial_deg</span><span class="p">(</span><span class="n">M</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-
-</div>
-<span class="k">def</span> <span class="nf">sdm_monomial_lcm</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the &quot;least common multiple&quot; of ``A`` and ``B``.</span>
-
-<span class="sd">    IF `A = M e_j` and `B = N e_j`, where `M` and `N` are polynomial monomials,</span>
-<span class="sd">    this returns `\lcm(M, N) e_j`. Note that ``A`` and ``B`` involve distinct</span>
-<span class="sd">    monomials.</span>
-
-<span class="sd">    Otherwise the result is undefined.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.distributedmodules import sdm_monomial_lcm</span>
-<span class="sd">    &gt;&gt;&gt; sdm_monomial_lcm((1, 2, 3), (1, 0, 5))</span>
-<span class="sd">    (1, 2, 5)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">(</span><span class="n">A</span><span class="p">[</span><span class="mi">0</span><span class="p">],)</span> <span class="o">+</span> <span class="n">monomial_lcm</span><span class="p">(</span><span class="n">A</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">B</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-
-
-<div class="viewcode-block" id="sdm_monomial_divides"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedmodules.sdm_monomial_divides">[docs]</a><span class="k">def</span> <span class="nf">sdm_monomial_divides</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Does there exist a (polynomial) monomial X such that XA = B?</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Positive examples:</span>
-
-<span class="sd">    In the following examples, the monomial is given in terms of x, y and the</span>
-<span class="sd">    generator(s), f_1, f_2 etc. The tuple form of that monomial is used in</span>
-<span class="sd">    the call to sdm_monomial_divides.</span>
-<span class="sd">    Note: the generator appears last in the expression but first in the tuple</span>
-<span class="sd">    and other factors appear in the same order that they appear in the monomial</span>
-<span class="sd">    expression.</span>
-
-<span class="sd">    `A = f_1` divides `B = f_1`</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.distributedmodules import sdm_monomial_divides</span>
-<span class="sd">    &gt;&gt;&gt; sdm_monomial_divides((1, 0, 0), (1, 0, 0))</span>
-<span class="sd">    True</span>
-
-<span class="sd">    `A = f_1` divides `B = x^2 y f_1`</span>
-
-<span class="sd">    &gt;&gt;&gt; sdm_monomial_divides((1, 0, 0), (1, 2, 1))</span>
-<span class="sd">    True</span>
-
-<span class="sd">    `A = xy f_5` divides `B = x^2 y f_5`</span>
-
-<span class="sd">    &gt;&gt;&gt; sdm_monomial_divides((5, 1, 1), (5, 2, 1))</span>
-<span class="sd">    True</span>
-
-<span class="sd">    Negative examples:</span>
-
-<span class="sd">    `A = f_1` does not divide `B = f_2`</span>
-
-<span class="sd">    &gt;&gt;&gt; sdm_monomial_divides((1, 0, 0), (2, 0, 0))</span>
-<span class="sd">    False</span>
-
-<span class="sd">    `A = x f_1` does not divide `B = f_1`</span>
-
-<span class="sd">    &gt;&gt;&gt; sdm_monomial_divides((1, 1, 0), (1, 0, 0))</span>
-<span class="sd">    False</span>
-
-<span class="sd">    `A = xy^2 f_5` does not divide `B = y f_5`</span>
-
-<span class="sd">    &gt;&gt;&gt; sdm_monomial_divides((5, 1, 2), (5, 0, 1))</span>
-<span class="sd">    False</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">A</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">B</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">and</span> <span class="nb">all</span><span class="p">(</span><span class="n">a</span> <span class="o">&lt;=</span> <span class="n">b</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">A</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">B</span><span class="p">[</span><span class="mi">1</span><span class="p">:]))</span>
-
-
-<span class="c"># The actual distributed modules code.</span>
-
-<span class="c"># These can be re-used without change:</span></div>
-<span class="n">sdm_LC</span> <span class="o">=</span> <span class="n">sdp_LC</span>
-<span class="n">sdm_to_dict</span> <span class="o">=</span> <span class="n">sdp_to_dict</span>
-
-
-<div class="viewcode-block" id="sdm_from_dict"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedmodules.sdm_from_dict">[docs]</a><span class="k">def</span> <span class="nf">sdm_from_dict</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">O</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create an sdm from a dictionary.</span>
-
-<span class="sd">    Here ``O`` is the monomial order to use.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.distributedmodules import sdm_from_dict</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys import QQ, lex</span>
-<span class="sd">    &gt;&gt;&gt; dic = {(1, 1, 0): QQ(1), (1, 0, 0): QQ(2), (0, 1, 0): QQ(0)}</span>
-<span class="sd">    &gt;&gt;&gt; sdm_from_dict(dic, lex)</span>
-<span class="sd">    [((1, 1, 0), 1/1), ((1, 0, 0), 2/1)]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">sdp_strip</span><span class="p">(</span><span class="n">sdp_from_dict</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">O</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="sdm_add"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedmodules.sdm_add">[docs]</a><span class="k">def</span> <span class="nf">sdm_add</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Add two module elements ``f``, ``g``.</span>
-
-<span class="sd">    Addition is done over the ground field ``K``, monomials are ordered</span>
-<span class="sd">    according to ``O``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    All examples use lexicographic order.</span>
-
-<span class="sd">    `(xy f_1) + (f_2) = f_2 + xy f_1`</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.distributedmodules import sdm_add</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys import lex, QQ</span>
-<span class="sd">    &gt;&gt;&gt; sdm_add([((1, 1, 1), QQ(1))], [((2, 0, 0), QQ(1))], lex, QQ)</span>
-<span class="sd">    [((2, 0, 0), 1/1), ((1, 1, 1), 1/1)]</span>
-
-<span class="sd">    `(xy f_1) + (-xy f_1)` = 0`</span>
-
-<span class="sd">    &gt;&gt;&gt; sdm_add([((1, 1, 1), QQ(1))], [((1, 1, 1), QQ(-1))], lex, QQ)</span>
-<span class="sd">    []</span>
-
-<span class="sd">    `(f_1) + (2f_1) = 3f_1`</span>
-
-<span class="sd">    &gt;&gt;&gt; sdm_add([((1, 0, 0), QQ(1))], [((1, 0, 0), QQ(2))], lex, QQ)</span>
-<span class="sd">    [((1, 0, 0), 3/1)]</span>
-
-<span class="sd">    `(yf_1) + (xf_1) = xf_1 + yf_1`</span>
-
-<span class="sd">    &gt;&gt;&gt; sdm_add([((1, 0, 1), QQ(1))], [((1, 1, 0), QQ(1))], lex, QQ)</span>
-<span class="sd">    [((1, 1, 0), 1/1), ((1, 0, 1), 1/1)]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># send 0 for u (3rd parameter) since it is not needed</span>
-    <span class="k">return</span> <span class="n">sdp_add</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="sdm_LM"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedmodules.sdm_LM">[docs]</a><span class="k">def</span> <span class="nf">sdm_LM</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Returns the leading monomial of ``f``.</span>
-
-<span class="sd">    Only valid if `f \ne 0`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.distributedmodules import sdm_LM, sdm_from_dict</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys import QQ, lex</span>
-<span class="sd">    &gt;&gt;&gt; dic = {(1, 2, 3): QQ(1), (4, 0, 0): QQ(1), (4, 0, 1): QQ(1)}</span>
-<span class="sd">    &gt;&gt;&gt; sdm_LM(sdm_from_dict(dic, lex))</span>
-<span class="sd">    (4, 0, 1)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="sdm_LT"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedmodules.sdm_LT">[docs]</a><span class="k">def</span> <span class="nf">sdm_LT</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Returns the leading term of ``f``.</span>
-
-<span class="sd">    Only valid if `f \ne 0`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.distributedmodules import sdm_LT, sdm_from_dict</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys import QQ, lex</span>
-<span class="sd">    &gt;&gt;&gt; dic = {(1, 2, 3): QQ(1), (4, 0, 0): QQ(2), (4, 0, 1): QQ(3)}</span>
-<span class="sd">    &gt;&gt;&gt; sdm_LT(sdm_from_dict(dic, lex))</span>
-<span class="sd">    ((4, 0, 1), 3/1)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="sdm_mul_term"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedmodules.sdm_mul_term">[docs]</a><span class="k">def</span> <span class="nf">sdm_mul_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">term</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Multiply a distributed module element ``f`` by a (polynomial) term ``term``.</span>
-
-<span class="sd">    Multiplication of coefficients is done over the ground field ``K``, and</span>
-<span class="sd">    monomials are ordered according to ``O``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    `0 f_1 = 0`</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.distributedmodules import sdm_mul_term</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys import lex, QQ</span>
-<span class="sd">    &gt;&gt;&gt; sdm_mul_term([((1, 0, 0), QQ(1))], ((0, 0), QQ(0)), lex, QQ)</span>
-<span class="sd">    []</span>
-
-<span class="sd">    `x 0 = 0`</span>
-
-<span class="sd">    &gt;&gt;&gt; sdm_mul_term([], ((1, 0), QQ(1)), lex, QQ)</span>
-<span class="sd">    []</span>
-
-<span class="sd">    `(x) (f_1) = xf_1`</span>
-
-<span class="sd">    &gt;&gt;&gt; sdm_mul_term([((1, 0, 0), QQ(1))], ((1, 0), QQ(1)), lex, QQ)</span>
-<span class="sd">    [((1, 1, 0), 1/1)]</span>
-
-<span class="sd">    `(2xy) (3x f_1 + 4y f_2) = 8xy^2 f_2 + 6x^2y f_1`</span>
-
-<span class="sd">    &gt;&gt;&gt; f = [((2, 0, 1), QQ(4)), ((1, 1, 0), QQ(3))]</span>
-<span class="sd">    &gt;&gt;&gt; sdm_mul_term(f, ((1, 1), QQ(2)), lex, QQ)</span>
-<span class="sd">    [((2, 1, 2), 8/1), ((1, 2, 1), 6/1)]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># Note O is not used in this implementation</span>
-    <span class="c"># This is an almost-verbatim copy of sdp_mul_term</span>
-    <span class="n">X</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">term</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">c</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">sdm_monomial_mul</span><span class="p">(</span><span class="n">f_M</span><span class="p">,</span> <span class="n">X</span><span class="p">),</span> <span class="n">f_c</span><span class="p">)</span> <span class="k">for</span> <span class="n">f_M</span><span class="p">,</span> <span class="n">f_c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">sdm_monomial_mul</span><span class="p">(</span><span class="n">f_M</span><span class="p">,</span> <span class="n">X</span><span class="p">),</span> <span class="n">f_c</span> <span class="o">*</span> <span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">f_M</span><span class="p">,</span> <span class="n">f_c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="sdm_zero"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedmodules.sdm_zero">[docs]</a><span class="k">def</span> <span class="nf">sdm_zero</span><span class="p">():</span>
-    <span class="sd">&quot;&quot;&quot;Return the zero module element.&quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">[]</span>
-
-</div>
-<div class="viewcode-block" id="sdm_deg"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedmodules.sdm_deg">[docs]</a><span class="k">def</span> <span class="nf">sdm_deg</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Degree of ``f``.</span>
-
-<span class="sd">    This is the maximum of the degrees of all its monomials.</span>
-<span class="sd">    Invalid if ``f`` is zero.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.distributedmodules import sdm_deg</span>
-<span class="sd">    &gt;&gt;&gt; sdm_deg([((1, 2, 3), 1), ((10, 0, 1), 1), ((2, 3, 4), 4)])</span>
-<span class="sd">    7</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="nb">max</span><span class="p">(</span><span class="n">sdm_monomial_deg</span><span class="p">(</span><span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="k">for</span> <span class="n">M</span> <span class="ow">in</span> <span class="n">f</span><span class="p">)</span>
-
-
-<span class="c"># Conversion</span>
-</div>
-<div class="viewcode-block" id="sdm_from_vector"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedmodules.sdm_from_vector">[docs]</a><span class="k">def</span> <span class="nf">sdm_from_vector</span><span class="p">(</span><span class="n">vec</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="o">**</span><span class="n">opts</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create an sdm from an iterable of expressions.</span>
-
-<span class="sd">    Coefficients are created in the ground field ``K``, and terms are ordered</span>
-<span class="sd">    according to monomial order ``O``. Named arguments are passed on to the</span>
-<span class="sd">    polys conversion code and can be used to specify for example generators.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.distributedmodules import sdm_from_vector</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys import QQ, lex</span>
-<span class="sd">    &gt;&gt;&gt; sdm_from_vector([x**2+y**2, 2*z], lex, QQ)</span>
-<span class="sd">    [((1, 0, 0, 1), 2/1), ((0, 2, 0, 0), 1/1), ((0, 0, 2, 0), 1/1)]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">dics</span><span class="p">,</span> <span class="n">gens</span> <span class="o">=</span> <span class="n">parallel_dict_from_expr</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">vec</span><span class="p">),</span> <span class="o">**</span><span class="n">opts</span><span class="p">)</span>
-    <span class="n">dic</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">dics</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">d</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">dic</span><span class="p">[(</span><span class="n">i</span><span class="p">,)</span> <span class="o">+</span> <span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">sdm_from_dict</span><span class="p">(</span><span class="n">dic</span><span class="p">,</span> <span class="n">O</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="sdm_to_vector"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedmodules.sdm_to_vector">[docs]</a><span class="k">def</span> <span class="nf">sdm_to_vector</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Convert sdm ``f`` into a list of polynomial expressions.</span>
-
-<span class="sd">    The generators for the polynomial ring are specified via ``gens``. The rank</span>
-<span class="sd">    of the module is guessed, or passed via ``n``. The ground field is assumed</span>
-<span class="sd">    to be ``K``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.distributedmodules import sdm_to_vector</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys import QQ, lex</span>
-<span class="sd">    &gt;&gt;&gt; f = [((1, 0, 0, 1), QQ(2)), ((0, 2, 0, 0), QQ(1)), ((0, 0, 2, 0), QQ(1))]</span>
-<span class="sd">    &gt;&gt;&gt; sdm_to_vector(f, [x, y, z], QQ)</span>
-<span class="sd">    [x**2 + y**2, 2*z]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">dic</span> <span class="o">=</span> <span class="n">sdm_to_dict</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">dics</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">dic</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="n">dics</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="n">k</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="p">[])</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">k</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">v</span><span class="p">))</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">n</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">dics</span><span class="p">)</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">dics</span><span class="p">:</span>
-            <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">dics</span><span class="p">[</span><span class="n">k</span><span class="p">]),</span> <span class="n">gens</span><span class="o">=</span><span class="n">gens</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="n">K</span><span class="p">)</span><span class="o">.</span><span class="n">as_expr</span><span class="p">())</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">res</span>
-
-<span class="c"># Algorithms.</span>
-
-</div>
-<div class="viewcode-block" id="sdm_spoly"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedmodules.sdm_spoly">[docs]</a><span class="k">def</span> <span class="nf">sdm_spoly</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">phantom</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute the generalized s-polynomial of ``f`` and ``g``.</span>
-
-<span class="sd">    The ground field is assumed to be ``K``, and monomials ordered according to</span>
-<span class="sd">    ``O``.</span>
-
-<span class="sd">    This is invalid if either of ``f`` or ``g`` is zero.</span>
-
-<span class="sd">    If the leading terms of `f` and `g` involve different basis elements of</span>
-<span class="sd">    `F`, their s-poly is defined to be zero. Otherwise it is a certain linear</span>
-<span class="sd">    combination of `f` and `g` in which the leading terms cancel.</span>
-<span class="sd">    See [SCA, defn 2.3.6] for details.</span>
-
-<span class="sd">    If ``phantom`` is not ``None``, it should be a pair of module elements on</span>
-<span class="sd">    which to perform the same operation(s) as on ``f`` and ``g``. The in this</span>
-<span class="sd">    case both results are returned.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.distributedmodules import sdm_spoly</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys import QQ, lex</span>
-<span class="sd">    &gt;&gt;&gt; f = [((2, 1, 1), QQ(1)), ((1, 0, 1), QQ(1))]</span>
-<span class="sd">    &gt;&gt;&gt; g = [((2, 3, 0), QQ(1))]</span>
-<span class="sd">    &gt;&gt;&gt; h = [((1, 2, 3), QQ(1))]</span>
-<span class="sd">    &gt;&gt;&gt; sdm_spoly(f, h, lex, QQ)</span>
-<span class="sd">    []</span>
-<span class="sd">    &gt;&gt;&gt; sdm_spoly(f, g, lex, QQ)</span>
-<span class="sd">    [((1, 2, 1), 1/1)]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">sdm_zero</span><span class="p">()</span>
-    <span class="n">LM1</span> <span class="o">=</span> <span class="n">sdm_LM</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">LM2</span> <span class="o">=</span> <span class="n">sdm_LM</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">LM1</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="n">LM2</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-        <span class="k">return</span> <span class="n">sdm_zero</span><span class="p">()</span>
-    <span class="n">LM1</span> <span class="o">=</span> <span class="n">LM1</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-    <span class="n">LM2</span> <span class="o">=</span> <span class="n">LM2</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-    <span class="n">lcm</span> <span class="o">=</span> <span class="n">monomial_lcm</span><span class="p">(</span><span class="n">LM1</span><span class="p">,</span> <span class="n">LM2</span><span class="p">)</span>
-    <span class="n">m1</span> <span class="o">=</span> <span class="n">monomial_div</span><span class="p">(</span><span class="n">lcm</span><span class="p">,</span> <span class="n">LM1</span><span class="p">)</span>
-    <span class="n">m2</span> <span class="o">=</span> <span class="n">monomial_div</span><span class="p">(</span><span class="n">lcm</span><span class="p">,</span> <span class="n">LM2</span><span class="p">)</span>
-    <span class="n">c</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="o">-</span><span class="n">sdm_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">sdm_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-    <span class="n">r1</span> <span class="o">=</span> <span class="n">sdm_add</span><span class="p">(</span><span class="n">sdm_mul_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">m1</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">),</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span>
-                 <span class="n">sdm_mul_term</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="p">(</span><span class="n">m2</span><span class="p">,</span> <span class="n">c</span><span class="p">),</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">phantom</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">r1</span>
-    <span class="n">r2</span> <span class="o">=</span> <span class="n">sdm_add</span><span class="p">(</span><span class="n">sdm_mul_term</span><span class="p">(</span><span class="n">phantom</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="p">(</span><span class="n">m1</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">),</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span>
-                 <span class="n">sdm_mul_term</span><span class="p">(</span><span class="n">phantom</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="p">(</span><span class="n">m2</span><span class="p">,</span> <span class="n">c</span><span class="p">),</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">r1</span><span class="p">,</span> <span class="n">r2</span>
-
-</div>
-<div class="viewcode-block" id="sdm_ecart"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedmodules.sdm_ecart">[docs]</a><span class="k">def</span> <span class="nf">sdm_ecart</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute the ecart of ``f``.</span>
-
-<span class="sd">    This is defined to be the difference of the total degree of `f` and the</span>
-<span class="sd">    total degree of the leading monomial of `f` [SCA, defn 2.3.7].</span>
-
-<span class="sd">    Invalid if f is zero.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.distributedmodules import sdm_ecart</span>
-<span class="sd">    &gt;&gt;&gt; sdm_ecart([((1, 2, 3), 1), ((1, 0, 1), 1)])</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; sdm_ecart([((2, 2, 1), 1), ((1, 5, 1), 1)])</span>
-<span class="sd">    3</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">sdm_deg</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">-</span> <span class="n">sdm_monomial_deg</span><span class="p">(</span><span class="n">sdm_LM</span><span class="p">(</span><span class="n">f</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="sdm_nf_mora"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedmodules.sdm_nf_mora">[docs]</a><span class="k">def</span> <span class="nf">sdm_nf_mora</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">phantom</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Compute a weak normal form of ``f`` with respect to ``G`` and order ``O``.</span>
-
-<span class="sd">    The ground field is assumed to be ``K``, and monomials ordered according to</span>
-<span class="sd">    ``O``.</span>
-
-<span class="sd">    Weak normal forms are defined in [SCA, defn 2.3.3]. They are not unique.</span>
-<span class="sd">    This function deterministically computes a weak normal form, depending on</span>
-<span class="sd">    the order of `G`.</span>
-
-<span class="sd">    The most important property of a weak normal form is the following: if</span>
-<span class="sd">    `R` is the ring associated with the monomial ordering (if the ordering is</span>
-<span class="sd">    global, we just have `R = K[x_1, \dots, x_n]`, otherwise it is a certain</span>
-<span class="sd">    localization thereof), `I` any ideal of `R` and `G` a standard basis for</span>
-<span class="sd">    `I`, then for any `f \in R`, we have `f \in I` if and only if</span>
-<span class="sd">    `NF(f | G) = 0`.</span>
-
-<span class="sd">    This is the generalized Mora algorithm for computing weak normal forms with</span>
-<span class="sd">    respect to arbitrary monomial orders [SCA, algorithm 2.3.9].</span>
-
-<span class="sd">    If ``phantom`` is not ``None``, it should be a pair of &quot;phantom&quot; arguments</span>
-<span class="sd">    on which to perform the same computations as on ``f``, ``G``, both results</span>
-<span class="sd">    are then returned.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">itertools</span> <span class="kn">import</span> <span class="n">repeat</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">f</span>
-    <span class="n">T</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">phantom</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="c"># &quot;phantom&quot; variables with suffix p</span>
-        <span class="n">hp</span> <span class="o">=</span> <span class="n">phantom</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">Tp</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">phantom</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="n">phantom</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">Tp</span> <span class="o">=</span> <span class="n">repeat</span><span class="p">([])</span>
-        <span class="n">phantom</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="k">while</span> <span class="n">h</span><span class="p">:</span>
-        <span class="c"># TODO better data structure!!!</span>
-        <span class="n">Th</span> <span class="o">=</span> <span class="p">[(</span><span class="n">g</span><span class="p">,</span> <span class="n">sdm_ecart</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">gp</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span><span class="p">,</span> <span class="n">gp</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">Tp</span><span class="p">)</span>
-              <span class="k">if</span> <span class="n">sdm_monomial_divides</span><span class="p">(</span><span class="n">sdm_LM</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">sdm_LM</span><span class="p">(</span><span class="n">h</span><span class="p">))]</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">Th</span><span class="p">:</span>
-            <span class="k">break</span>
-        <span class="n">g</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">gp</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">Th</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">sdm_ecart</span><span class="p">(</span><span class="n">g</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">sdm_ecart</span><span class="p">(</span><span class="n">h</span><span class="p">):</span>
-            <span class="n">T</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">phantom</span><span class="p">:</span>
-                <span class="n">Tp</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">hp</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">phantom</span><span class="p">:</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">hp</span> <span class="o">=</span> <span class="n">sdm_spoly</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">phantom</span><span class="o">=</span><span class="p">(</span><span class="n">hp</span><span class="p">,</span> <span class="n">gp</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">sdm_spoly</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">phantom</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">h</span><span class="p">,</span> <span class="n">hp</span>
-    <span class="k">return</span> <span class="n">h</span>
-
-</div>
-<span class="k">def</span> <span class="nf">sdm_nf_buchberger</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">phantom</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Compute a weak normal form of ``f`` with respect to ``G`` and order ``O``.</span>
-
-<span class="sd">    The ground field is assumed to be ``K``, and monomials ordered according to</span>
-<span class="sd">    ``O``.</span>
-
-<span class="sd">    This is the standard Buchberger algorithm for computing weak normal forms with</span>
-<span class="sd">    respect to *global* monomial orders [SCA, algorithm 1.6.10].</span>
-
-<span class="sd">    If ``phantom`` is not ``None``, it should be a pair of &quot;phantom&quot; arguments</span>
-<span class="sd">    on which to perform the same computations as on ``f``, ``G``, both results</span>
-<span class="sd">    are then returned.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">itertools</span> <span class="kn">import</span> <span class="n">repeat</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">f</span>
-    <span class="n">T</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">phantom</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="c"># &quot;phantom&quot; variables with suffix p</span>
-        <span class="n">hp</span> <span class="o">=</span> <span class="n">phantom</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">Tp</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">phantom</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="n">phantom</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">Tp</span> <span class="o">=</span> <span class="n">repeat</span><span class="p">([])</span>
-        <span class="n">phantom</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="k">while</span> <span class="n">h</span><span class="p">:</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">g</span><span class="p">,</span> <span class="n">gp</span> <span class="o">=</span> <span class="nb">next</span><span class="p">((</span><span class="n">g</span><span class="p">,</span> <span class="n">gp</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span><span class="p">,</span> <span class="n">gp</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">Tp</span><span class="p">)</span>
-                         <span class="k">if</span> <span class="n">sdm_monomial_divides</span><span class="p">(</span><span class="n">sdm_LM</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">sdm_LM</span><span class="p">(</span><span class="n">h</span><span class="p">)))</span>
-        <span class="k">except</span> <span class="ne">StopIteration</span><span class="p">:</span>
-            <span class="k">break</span>
-        <span class="k">if</span> <span class="n">phantom</span><span class="p">:</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">hp</span> <span class="o">=</span> <span class="n">sdm_spoly</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">phantom</span><span class="o">=</span><span class="p">(</span><span class="n">hp</span><span class="p">,</span> <span class="n">gp</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">sdm_spoly</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">phantom</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">h</span><span class="p">,</span> <span class="n">hp</span>
-    <span class="k">return</span> <span class="n">h</span>
-
-
-<span class="k">def</span> <span class="nf">sdm_nf_buchberger_reduced</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Compute a reduced normal form of ``f`` with respect to ``G`` and order ``O``.</span>
-
-<span class="sd">    The ground field is assumed to be ``K``, and monomials ordered according to</span>
-<span class="sd">    ``O``.</span>
-
-<span class="sd">    In contrast to weak normal forms, reduced normal forms *are* unique, but</span>
-<span class="sd">    their computation is more expensive.</span>
-
-<span class="sd">    This is the standard Buchberger algorithm for computing reduced normal forms</span>
-<span class="sd">    with respect to *global* monomial orders [SCA, algorithm 1.6.11].</span>
-
-<span class="sd">    The ``pantom`` option is not supported, so this normal form cannot be used</span>
-<span class="sd">    as a normal form for the &quot;extended&quot; groebner algorithm.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">sdm_zero</span><span class="p">()</span>
-    <span class="n">g</span> <span class="o">=</span> <span class="n">f</span>
-    <span class="k">while</span> <span class="n">g</span><span class="p">:</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">sdm_nf_buchberger</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">g</span><span class="p">:</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">sdm_add</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="p">[</span><span class="n">sdm_LT</span><span class="p">(</span><span class="n">g</span><span class="p">)],</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-    <span class="k">return</span> <span class="n">h</span>
-
-
-<div class="viewcode-block" id="sdm_groebner"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedmodules.sdm_groebner">[docs]</a><span class="k">def</span> <span class="nf">sdm_groebner</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">NF</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">extended</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute a minimal standard basis of ``G`` with respect to order ``O``.</span>
-
-<span class="sd">    The algorithm uses a normal form ``NF``, for example ``sdm_nf_mora``.</span>
-<span class="sd">    The ground field is assumed to be ``K``, and monomials ordered according</span>
-<span class="sd">    to ``O``.</span>
-
-<span class="sd">    Let `N` denote the submodule generated by elements of `G`. A standard</span>
-<span class="sd">    basis for `N` is a subset `S` of `N`, such that `in(S) = in(N)`, where for</span>
-<span class="sd">    any subset `X` of `F`, `in(X)` denotes the submodule generated by the</span>
-<span class="sd">    initial forms of elements of `X`. [SCA, defn 2.3.2]</span>
-
-<span class="sd">    A standard basis is called minimal if no subset of it is a standard basis.</span>
-
-<span class="sd">    One may show that standard bases are always generating sets.</span>
-
-<span class="sd">    Minimal standard bases are not unique. This algorithm computes a</span>
-<span class="sd">    deterministic result, depending on the particular order of `G`.</span>
-
-<span class="sd">    If ``extended=True``, also compute the transition matrix from the initial</span>
-<span class="sd">    generators to the groebner basis. That is, return a list of coefficient</span>
-<span class="sd">    vectors, expressing the elements of the groebner basis in terms of the</span>
-<span class="sd">    elements of ``G``.</span>
-
-<span class="sd">    This functions implements the &quot;sugar&quot; strategy, see</span>
-
-<span class="sd">    Giovini et al: &quot;One sugar cube, please&quot; OR Selection strategies in</span>
-<span class="sd">    Buchberger algorithm.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="c"># The critical pair set.</span>
-    <span class="c"># A critical pair is stored as (i, j, s, t) where (i, j) defines the pair</span>
-    <span class="c"># (by indexing S), s is the sugar of the pair, and t is the lcm of their</span>
-    <span class="c"># leading monomials.</span>
-    <span class="n">P</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="c"># The eventual standard basis.</span>
-    <span class="n">S</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">Sugars</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">Ssugar</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute the sugar of the S-poly corresponding to (i, j).&quot;&quot;&quot;</span>
-        <span class="n">LMi</span> <span class="o">=</span> <span class="n">sdm_LM</span><span class="p">(</span><span class="n">S</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-        <span class="n">LMj</span> <span class="o">=</span> <span class="n">sdm_LM</span><span class="p">(</span><span class="n">S</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-        <span class="k">return</span> <span class="nb">max</span><span class="p">(</span><span class="n">Sugars</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">sdm_monomial_deg</span><span class="p">(</span><span class="n">LMi</span><span class="p">),</span>
-                   <span class="n">Sugars</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">sdm_monomial_deg</span><span class="p">(</span><span class="n">LMj</span><span class="p">))</span> \
-            <span class="o">+</span> <span class="n">sdm_monomial_deg</span><span class="p">(</span><span class="n">sdm_monomial_lcm</span><span class="p">(</span><span class="n">LMi</span><span class="p">,</span> <span class="n">LMj</span><span class="p">))</span>
-
-    <span class="n">ourkey</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">p</span><span class="p">:</span> <span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">O</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="mi">3</span><span class="p">]),</span> <span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">update</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">sugar</span><span class="p">,</span> <span class="n">P</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Add f with sugar ``sugar`` to S, update P.&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">P</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">S</span><span class="p">)</span>
-        <span class="n">S</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-        <span class="n">Sugars</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sugar</span><span class="p">)</span>
-
-        <span class="n">LMf</span> <span class="o">=</span> <span class="n">sdm_LM</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">removethis</span><span class="p">(</span><span class="n">pair</span><span class="p">):</span>
-            <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">pair</span>
-            <span class="k">if</span> <span class="n">LMf</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="n">t</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="n">tik</span> <span class="o">=</span> <span class="n">sdm_monomial_lcm</span><span class="p">(</span><span class="n">LMf</span><span class="p">,</span> <span class="n">sdm_LM</span><span class="p">(</span><span class="n">S</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span>
-            <span class="n">tjk</span> <span class="o">=</span> <span class="n">sdm_monomial_lcm</span><span class="p">(</span><span class="n">LMf</span><span class="p">,</span> <span class="n">sdm_LM</span><span class="p">(</span><span class="n">S</span><span class="p">[</span><span class="n">j</span><span class="p">]))</span>
-            <span class="k">return</span> <span class="n">tik</span> <span class="o">!=</span> <span class="n">t</span> <span class="ow">and</span> <span class="n">tjk</span> <span class="o">!=</span> <span class="n">t</span> <span class="ow">and</span> <span class="n">sdm_monomial_divides</span><span class="p">(</span><span class="n">tik</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span> <span class="ow">and</span> \
-                <span class="n">sdm_monomial_divides</span><span class="p">(</span><span class="n">tjk</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
-        <span class="c"># apply the chain criterion</span>
-        <span class="n">P</span> <span class="o">=</span> <span class="p">[</span><span class="n">p</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">P</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">removethis</span><span class="p">(</span><span class="n">p</span><span class="p">)]</span>
-
-        <span class="c"># new-pair set</span>
-        <span class="n">N</span> <span class="o">=</span> <span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">Ssugar</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">),</span> <span class="n">sdm_monomial_lcm</span><span class="p">(</span><span class="n">LMf</span><span class="p">,</span> <span class="n">sdm_LM</span><span class="p">(</span><span class="n">S</span><span class="p">[</span><span class="n">i</span><span class="p">])))</span>
-             <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="p">)</span> <span class="k">if</span> <span class="n">LMf</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">sdm_LM</span><span class="p">(</span><span class="n">S</span><span class="p">[</span><span class="n">i</span><span class="p">])[</span><span class="mi">0</span><span class="p">]]</span>
-        <span class="c"># TODO apply the product criterion?</span>
-        <span class="n">N</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">ourkey</span><span class="p">)</span>
-        <span class="n">remove</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">N</span><span class="p">)):</span>
-                <span class="k">if</span> <span class="n">sdm_monomial_divides</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="n">N</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="mi">3</span><span class="p">]):</span>
-                    <span class="n">remove</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-
-        <span class="c"># TODO mergesort?</span>
-        <span class="n">P</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="nb">reversed</span><span class="p">([</span><span class="n">p</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">N</span><span class="p">)</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">remove</span><span class="p">]))</span>
-        <span class="n">P</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">ourkey</span><span class="p">,</span> <span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="c"># NOTE reverse-sort, because we want to pop from the end</span>
-        <span class="k">return</span> <span class="n">P</span>
-
-    <span class="c"># Figure out the number of generators in the ground ring.</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="c"># NOTE: we look for the first non-zero vector, take its first monomial</span>
-        <span class="c">#       the number of generators in the ring is one less than the length</span>
-        <span class="c">#       (since the zeroth entry is for the module generators)</span>
-        <span class="n">numgens</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="nb">next</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">G</span> <span class="k">if</span> <span class="n">x</span><span class="p">)[</span><span class="mi">0</span><span class="p">])</span> <span class="o">-</span> <span class="mi">1</span>
-    <span class="k">except</span> <span class="ne">StopIteration</span><span class="p">:</span>
-        <span class="c"># No non-zero elements in G ...</span>
-        <span class="k">if</span> <span class="n">extended</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[],</span> <span class="p">[]</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-    <span class="c"># This list will store expressions of the elements of S in terms of the</span>
-    <span class="c"># initial generators</span>
-    <span class="n">coefficients</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="c"># First add all the elements of G to S</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">f</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
-        <span class="n">P</span> <span class="o">=</span> <span class="n">update</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">sdm_deg</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="n">P</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">extended</span> <span class="ow">and</span> <span class="n">f</span><span class="p">:</span>
-            <span class="n">coefficients</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sdm_from_dict</span><span class="p">({(</span><span class="n">i</span><span class="p">,)</span> <span class="o">+</span> <span class="p">(</span><span class="mi">0</span><span class="p">,)</span><span class="o">*</span><span class="n">numgens</span><span class="p">:</span> <span class="n">K</span><span class="p">(</span><span class="mi">1</span><span class="p">)},</span> <span class="n">O</span><span class="p">))</span>
-
-    <span class="c"># Now carry out the buchberger algorithm.</span>
-    <span class="k">while</span> <span class="n">P</span><span class="p">:</span>
-        <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">P</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-        <span class="n">f</span><span class="p">,</span> <span class="n">sf</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">sg</span> <span class="o">=</span> <span class="n">S</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">Sugars</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">S</span><span class="p">[</span><span class="n">j</span><span class="p">],</span> <span class="n">Sugars</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">extended</span><span class="p">:</span>
-            <span class="n">sp</span><span class="p">,</span> <span class="n">coeff</span> <span class="o">=</span> <span class="n">sdm_spoly</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span>
-                                  <span class="n">phantom</span><span class="o">=</span><span class="p">(</span><span class="n">coefficients</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">coefficients</span><span class="p">[</span><span class="n">j</span><span class="p">]))</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">hcoeff</span> <span class="o">=</span> <span class="n">NF</span><span class="p">(</span><span class="n">sp</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">phantom</span><span class="o">=</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">coefficients</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">h</span><span class="p">:</span>
-                <span class="n">coefficients</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">hcoeff</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">NF</span><span class="p">(</span><span class="n">sdm_spoly</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">S</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">P</span> <span class="o">=</span> <span class="n">update</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">Ssugar</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">),</span> <span class="n">P</span><span class="p">)</span>
-
-    <span class="c"># Finally interreduce the standard basis.</span>
-    <span class="c"># (TODO again, better data structures)</span>
-    <span class="n">S</span> <span class="o">=</span> <span class="nb">set</span><span class="p">((</span><span class="nb">tuple</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">f</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">S</span><span class="p">))</span>
-    <span class="k">for</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">ai</span><span class="p">),</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">bi</span><span class="p">)</span> <span class="ow">in</span> <span class="n">permutations</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="mi">2</span><span class="p">):</span>
-        <span class="n">A</span> <span class="o">=</span> <span class="n">sdm_LM</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="n">B</span> <span class="o">=</span> <span class="n">sdm_LM</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">sdm_monomial_divides</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">bi</span><span class="p">)</span> <span class="ow">in</span> <span class="n">S</span> <span class="ow">and</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">ai</span><span class="p">)</span> <span class="ow">in</span> <span class="n">S</span><span class="p">:</span>
-            <span class="n">S</span><span class="o">.</span><span class="n">remove</span><span class="p">((</span><span class="n">b</span><span class="p">,</span> <span class="n">bi</span><span class="p">))</span>
-
-    <span class="n">L</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(((</span><span class="nb">list</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">f</span><span class="p">,</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">S</span><span class="p">),</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">p</span><span class="p">:</span> <span class="n">O</span><span class="p">(</span><span class="n">sdm_LM</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">])),</span>
-               <span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">L</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">extended</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">res</span><span class="p">,</span> <span class="p">[</span><span class="n">coefficients</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">_</span><span class="p">,</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">L</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">res</span></div>
-</pre></div>
-
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-  <h1>Source code for sympy.polys.distributedpolys</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Sparse distributed multivariate polynomials. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.monomialtools</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">monomial_mul</span><span class="p">,</span> <span class="n">monomial_div</span><span class="p">,</span> <span class="n">monomial_lcm</span><span class="p">,</span> <span class="n">lex</span><span class="p">,</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">ExactQuotientFailed</span><span class="p">,</span>
-<span class="p">)</span>
-
-
-<div class="viewcode-block" id="sdp_LC"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_LC">[docs]</a><span class="k">def</span> <span class="nf">sdp_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns the leading coeffcient of `f`. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="sdp_LM"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_LM">[docs]</a><span class="k">def</span> <span class="nf">sdp_LM</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns the leading monomial of `f`. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">(</span><span class="mi">0</span><span class="p">,)</span> <span class="o">*</span> <span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="sdp_LT"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_LT">[docs]</a><span class="k">def</span> <span class="nf">sdp_LT</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns the leading term of `f`. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">(</span><span class="mi">0</span><span class="p">,)</span> <span class="o">*</span> <span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-
-</div>
-<div class="viewcode-block" id="sdp_del_LT"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_del_LT">[docs]</a><span class="k">def</span> <span class="nf">sdp_del_LT</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Removes the leading from `f`. &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-</div>
-<span class="k">def</span> <span class="nf">sdp_coeffs</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns a list of monomials in `f`. &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">[</span> <span class="n">coeff</span> <span class="k">for</span> <span class="n">_</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-
-<div class="viewcode-block" id="sdp_monoms"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_monoms">[docs]</a><span class="k">def</span> <span class="nf">sdp_monoms</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns a list of monomials in `f`. &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">[</span> <span class="n">monom</span> <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="sdp_sort"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_sort">[docs]</a><span class="k">def</span> <span class="nf">sdp_sort</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">O</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Sort terms in `f` using the given monomial order `O`. &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">term</span><span class="p">:</span> <span class="n">O</span><span class="p">(</span><span class="n">term</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="sdp_strip"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_strip">[docs]</a><span class="k">def</span> <span class="nf">sdp_strip</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Remove terms with zero coefficients from `f` in `K[X]`. &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span><span class="p">)</span> <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span> <span class="k">if</span> <span class="n">coeff</span> <span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="sdp_normal"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_normal">[docs]</a><span class="k">def</span> <span class="nf">sdp_normal</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Normalize distributed polynomial in the given domain. &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">monom</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">coeff</span><span class="p">))</span> <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span> <span class="k">if</span> <span class="n">coeff</span> <span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="sdp_from_dict"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_from_dict">[docs]</a><span class="k">def</span> <span class="nf">sdp_from_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">O</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Make a distributed polynomial from a dictionary. &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">sdp_sort</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">items</span><span class="p">()),</span> <span class="n">O</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="sdp_to_dict"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_to_dict">[docs]</a><span class="k">def</span> <span class="nf">sdp_to_dict</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Make a dictionary from a distributed polynomial. &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="nb">dict</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="sdp_indep_p"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_indep_p">[docs]</a><span class="k">def</span> <span class="nf">sdp_indep_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns `True` if a polynomial is independent of `x_j`. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">j</span> <span class="o">&gt;</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s">&quot;-</span><span class="si">%s</span><span class="s"> &lt;= j &lt; </span><span class="si">%s</span><span class="s"> expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="ow">not</span> <span class="n">monom</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="k">for</span> <span class="n">monom</span> <span class="ow">in</span> <span class="n">sdp_monoms</span><span class="p">(</span><span class="n">h</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="sdp_one_p"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_one_p">[docs]</a><span class="k">def</span> <span class="nf">sdp_one_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns True if `f` is a multivariate one in `K[X]`. &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">f</span> <span class="o">==</span> <span class="n">sdp_one</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="sdp_one"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_one">[docs]</a><span class="k">def</span> <span class="nf">sdp_one</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns a multivariate one in `K[X]`. &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">(((</span><span class="mi">0</span><span class="p">,)</span> <span class="o">*</span> <span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">),)</span>
-
-</div>
-<div class="viewcode-block" id="sdp_term_p"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_term_p">[docs]</a><span class="k">def</span> <span class="nf">sdp_term_p</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns True if `f` has a single term or is zero. &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">1</span>
-
-</div>
-<div class="viewcode-block" id="sdp_abs"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_abs">[docs]</a><span class="k">def</span> <span class="nf">sdp_abs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Make all coefficients positive in `K[X]`. &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">monom</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">coeff</span><span class="p">))</span> <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="sdp_neg"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_neg">[docs]</a><span class="k">def</span> <span class="nf">sdp_neg</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Negate a polynomial in `K[X]`. &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">monom</span><span class="p">,</span> <span class="o">-</span><span class="n">coeff</span><span class="p">)</span> <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="sdp_add_term"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_add_term">[docs]</a><span class="k">def</span> <span class="nf">sdp_add_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">term</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Add a single term using bisection method. &quot;&quot;&quot;</span>
-    <span class="n">M</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">term</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">c</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[(</span><span class="n">M</span><span class="p">,</span> <span class="n">c</span><span class="p">)]</span>
-
-    <span class="n">monoms</span> <span class="o">=</span> <span class="n">sdp_monoms</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">O</span><span class="p">(</span><span class="n">M</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">O</span><span class="p">(</span><span class="n">monoms</span><span class="p">[</span> <span class="mi">0</span><span class="p">]):</span>
-        <span class="k">return</span> <span class="p">[(</span><span class="n">M</span><span class="p">,</span> <span class="n">c</span><span class="p">)]</span> <span class="o">+</span> <span class="n">f</span>
-    <span class="k">if</span> <span class="n">O</span><span class="p">(</span><span class="n">M</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">O</span><span class="p">(</span><span class="n">monoms</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]):</span>
-        <span class="k">return</span> <span class="n">f</span> <span class="o">+</span> <span class="p">[(</span><span class="n">M</span><span class="p">,</span> <span class="n">c</span><span class="p">)]</span>
-
-    <span class="n">lo</span><span class="p">,</span> <span class="n">hi</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">monoms</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">while</span> <span class="n">lo</span> <span class="o">&lt;=</span> <span class="n">hi</span><span class="p">:</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="p">(</span><span class="n">lo</span> <span class="o">+</span> <span class="n">hi</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>
-
-        <span class="k">if</span> <span class="n">O</span><span class="p">(</span><span class="n">M</span><span class="p">)</span> <span class="o">==</span> <span class="n">O</span><span class="p">(</span><span class="n">monoms</span><span class="p">[</span><span class="n">i</span><span class="p">]):</span>
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">c</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">coeff</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="p">[(</span><span class="n">M</span><span class="p">,</span> <span class="n">coeff</span><span class="p">)]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">O</span><span class="p">(</span><span class="n">M</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">O</span><span class="p">(</span><span class="n">monoms</span><span class="p">[</span><span class="n">i</span><span class="p">]):</span>
-                <span class="n">hi</span> <span class="o">=</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">lo</span> <span class="o">=</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="p">[(</span><span class="n">M</span><span class="p">,</span> <span class="n">c</span><span class="p">)]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-
-</div>
-<div class="viewcode-block" id="sdp_sub_term"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_sub_term">[docs]</a><span class="k">def</span> <span class="nf">sdp_sub_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">term</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Sub a single term using bisection method. &quot;&quot;&quot;</span>
-    <span class="n">M</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">term</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">c</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[(</span><span class="n">M</span><span class="p">,</span> <span class="o">-</span><span class="n">c</span><span class="p">)]</span>
-
-    <span class="n">monoms</span> <span class="o">=</span> <span class="n">sdp_monoms</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">O</span><span class="p">(</span><span class="n">M</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">O</span><span class="p">(</span><span class="n">monoms</span><span class="p">[</span> <span class="mi">0</span><span class="p">]):</span>
-        <span class="k">return</span> <span class="p">[(</span><span class="n">M</span><span class="p">,</span> <span class="o">-</span><span class="n">c</span><span class="p">)]</span> <span class="o">+</span> <span class="n">f</span>
-    <span class="k">if</span> <span class="n">O</span><span class="p">(</span><span class="n">M</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">O</span><span class="p">(</span><span class="n">monoms</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]):</span>
-        <span class="k">return</span> <span class="n">f</span> <span class="o">+</span> <span class="p">[(</span><span class="n">M</span><span class="p">,</span> <span class="o">-</span><span class="n">c</span><span class="p">)]</span>
-
-    <span class="n">lo</span><span class="p">,</span> <span class="n">hi</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">monoms</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">while</span> <span class="n">lo</span> <span class="o">&lt;=</span> <span class="n">hi</span><span class="p">:</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="p">(</span><span class="n">lo</span> <span class="o">+</span> <span class="n">hi</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>
-
-        <span class="k">if</span> <span class="n">O</span><span class="p">(</span><span class="n">M</span><span class="p">)</span> <span class="o">==</span> <span class="n">O</span><span class="p">(</span><span class="n">monoms</span><span class="p">[</span><span class="n">i</span><span class="p">]):</span>
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">c</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">coeff</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="p">[(</span><span class="n">M</span><span class="p">,</span> <span class="n">coeff</span><span class="p">)]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">O</span><span class="p">(</span><span class="n">M</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">O</span><span class="p">(</span><span class="n">monoms</span><span class="p">[</span><span class="n">i</span><span class="p">]):</span>
-                <span class="n">hi</span> <span class="o">=</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">lo</span> <span class="o">=</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="p">[(</span><span class="n">M</span><span class="p">,</span> <span class="o">-</span><span class="n">c</span><span class="p">)]</span> <span class="o">+</span> <span class="n">f</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-
-</div>
-<div class="viewcode-block" id="sdp_mul_term"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_mul_term">[docs]</a><span class="k">def</span> <span class="nf">sdp_mul_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">term</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Multiply a distributed polynomial by a term. &quot;&quot;&quot;</span>
-    <span class="n">M</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">term</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">c</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">monomial_mul</span><span class="p">(</span><span class="n">f_M</span><span class="p">,</span> <span class="n">M</span><span class="p">),</span> <span class="n">f_c</span><span class="p">)</span> <span class="k">for</span> <span class="n">f_M</span><span class="p">,</span> <span class="n">f_c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">monomial_mul</span><span class="p">(</span><span class="n">f_M</span><span class="p">,</span> <span class="n">M</span><span class="p">),</span> <span class="n">f_c</span> <span class="o">*</span> <span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">f_M</span><span class="p">,</span> <span class="n">f_c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="sdp_add"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_add">[docs]</a><span class="k">def</span> <span class="nf">sdp_add</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Add distributed polynomials in `K[X]`. &quot;&quot;&quot;</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">monom</span> <span class="ow">in</span> <span class="n">h</span><span class="p">:</span>
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">h</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span> <span class="o">+</span> <span class="n">c</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">coeff</span><span class="p">:</span>
-                <span class="k">del</span> <span class="n">h</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">h</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span> <span class="o">=</span> <span class="n">coeff</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">h</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span> <span class="o">=</span> <span class="n">c</span>
-
-    <span class="k">return</span> <span class="n">sdp_from_dict</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">O</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="sdp_sub"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_sub">[docs]</a><span class="k">def</span> <span class="nf">sdp_sub</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Subtract distributed polynomials in `K[X]`. &quot;&quot;&quot;</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">monom</span> <span class="ow">in</span> <span class="n">h</span><span class="p">:</span>
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">h</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span> <span class="o">-</span> <span class="n">c</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">coeff</span><span class="p">:</span>
-                <span class="k">del</span> <span class="n">h</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">h</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span> <span class="o">=</span> <span class="n">coeff</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">h</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">c</span>
-
-    <span class="k">return</span> <span class="n">sdp_from_dict</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">O</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="sdp_mul"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_mul">[docs]</a><span class="k">def</span> <span class="nf">sdp_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Multiply distributed polynomials in `K[X]`. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">sdp_term_p</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">sdp_mul_term</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">sdp_term_p</span><span class="p">(</span><span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">g</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">sdp_mul_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="p">{}</span>
-
-    <span class="k">for</span> <span class="n">fm</span><span class="p">,</span> <span class="n">fc</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">gm</span><span class="p">,</span> <span class="n">gc</span> <span class="ow">in</span> <span class="n">g</span><span class="p">:</span>
-            <span class="n">monom</span> <span class="o">=</span> <span class="n">monomial_mul</span><span class="p">(</span><span class="n">fm</span><span class="p">,</span> <span class="n">gm</span><span class="p">)</span>
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">fc</span> <span class="o">*</span> <span class="n">gc</span>
-
-            <span class="k">if</span> <span class="n">monom</span> <span class="ow">in</span> <span class="n">h</span><span class="p">:</span>
-                <span class="n">coeff</span> <span class="o">+=</span> <span class="n">h</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span>
-
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">coeff</span><span class="p">:</span>
-                    <span class="k">del</span> <span class="n">h</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span>
-                    <span class="k">continue</span>
-
-            <span class="n">h</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span> <span class="o">=</span> <span class="n">coeff</span>
-
-    <span class="k">return</span> <span class="n">sdp_from_dict</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">O</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="sdp_sqr"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_sqr">[docs]</a><span class="k">def</span> <span class="nf">sdp_sqr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Square a distributed polynomial in `K[X]`. &quot;&quot;&quot;</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="p">{}</span>
-
-    <span class="k">for</span> <span class="n">fm</span><span class="p">,</span> <span class="n">fc</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">Fm</span><span class="p">,</span> <span class="n">Fc</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-            <span class="n">monom</span> <span class="o">=</span> <span class="n">monomial_mul</span><span class="p">(</span><span class="n">fm</span><span class="p">,</span> <span class="n">Fm</span><span class="p">)</span>
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">fc</span> <span class="o">*</span> <span class="n">Fc</span>
-
-            <span class="k">if</span> <span class="n">monom</span> <span class="ow">in</span> <span class="n">h</span><span class="p">:</span>
-                <span class="n">coeff</span> <span class="o">+=</span> <span class="n">h</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span>
-
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">coeff</span><span class="p">:</span>
-                    <span class="k">del</span> <span class="n">h</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span>
-                    <span class="k">continue</span>
-
-            <span class="n">h</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span> <span class="o">=</span> <span class="n">coeff</span>
-
-    <span class="k">return</span> <span class="n">sdp_from_dict</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">O</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="sdp_pow"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_pow">[docs]</a><span class="k">def</span> <span class="nf">sdp_pow</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Raise `f` to the n-th power in `K[X]`. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">sdp_one</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;can&#39;t raise a polynomial to negative power&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">f</span> <span class="ow">or</span> <span class="n">sdp_one_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">g</span> <span class="o">=</span> <span class="n">sdp_one</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">n</span> <span class="o">//</span> <span class="mi">2</span><span class="p">,</span> <span class="n">n</span>
-
-        <span class="k">if</span> <span class="n">m</span> <span class="o">&amp;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">sdp_mul</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-                <span class="k">break</span>
-
-        <span class="n">f</span> <span class="o">=</span> <span class="n">sdp_sqr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">g</span>
-
-</div>
-<div class="viewcode-block" id="sdp_monic"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_monic">[docs]</a><span class="k">def</span> <span class="nf">sdp_monic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Divides all coefficients by `LC(f)` in `K[X]`. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">lc_f</span> <span class="o">=</span> <span class="n">sdp_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">lc_f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">lc_f</span><span class="p">))</span> <span class="k">for</span> <span class="n">m</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="sdp_content"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_content">[docs]</a><span class="k">def</span> <span class="nf">sdp_content</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns GCD of coefficients in `K[X]`. &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">QQ</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">cont</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-
-        <span class="k">if</span> <span class="n">K</span> <span class="o">==</span> <span class="n">QQ</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-                <span class="n">cont</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">cont</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-                <span class="n">cont</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">cont</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">cont</span><span class="p">)</span> <span class="ow">and</span> <span class="n">i</span><span class="p">:</span>
-                    <span class="k">break</span>
-
-        <span class="k">return</span> <span class="n">cont</span>
-
-</div>
-<div class="viewcode-block" id="sdp_primitive"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_primitive">[docs]</a><span class="k">def</span> <span class="nf">sdp_primitive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns content and a primitive polynomial in `K[X]`. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">f</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">cont</span> <span class="o">=</span> <span class="n">sdp_content</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">cont</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">cont</span><span class="p">,</span> <span class="n">f</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cont</span><span class="p">,</span> <span class="p">[</span> <span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">cont</span><span class="p">))</span> <span class="k">for</span> <span class="n">m</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_term_rr_div</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Division of two terms in over a ring. &quot;&quot;&quot;</span>
-    <span class="n">a_lm</span><span class="p">,</span> <span class="n">a_lc</span> <span class="o">=</span> <span class="n">a</span>
-    <span class="n">b_lm</span><span class="p">,</span> <span class="n">b_lc</span> <span class="o">=</span> <span class="n">b</span>
-
-    <span class="n">monom</span> <span class="o">=</span> <span class="n">monomial_div</span><span class="p">(</span><span class="n">a_lm</span><span class="p">,</span> <span class="n">b_lm</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">monom</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">a_lc</span> <span class="o">%</span> <span class="n">b_lc</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">monom</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">a_lc</span><span class="p">,</span> <span class="n">b_lc</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-
-<span class="k">def</span> <span class="nf">_term_ff_div</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Division of two terms in over a field. &quot;&quot;&quot;</span>
-    <span class="n">a_lm</span><span class="p">,</span> <span class="n">a_lc</span> <span class="o">=</span> <span class="n">a</span>
-    <span class="n">b_lm</span><span class="p">,</span> <span class="n">b_lc</span> <span class="o">=</span> <span class="n">b</span>
-
-    <span class="n">monom</span> <span class="o">=</span> <span class="n">monomial_div</span><span class="p">(</span><span class="n">a_lm</span><span class="p">,</span> <span class="n">b_lm</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">monom</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">monom</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">a_lc</span><span class="p">,</span> <span class="n">b_lc</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-
-<div class="viewcode-block" id="sdp_div"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_div">[docs]</a><span class="k">def</span> <span class="nf">sdp_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Generalized polynomial division with remainder in `K[X]`.</span>
-
-<span class="sd">    Given polynomial `f` and a set of polynomials `g = (g_1, ..., g_n)`</span>
-<span class="sd">    compute a set of quotients `q = (q_1, ..., q_n)` and remainder `r`</span>
-<span class="sd">    such that `f = q_1*f_1 + ... + q_n*f_n + r`, where `r = 0` or `r`</span>
-<span class="sd">    is a completely reduced polynomial with respect to `g`.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Cox97]_</span>
-<span class="sd">    2. [Ajwa95]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">Q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="p">[</span> <span class="p">[]</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">))</span> <span class="p">],</span> <span class="p">[]</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-        <span class="n">term_div</span> <span class="o">=</span> <span class="n">_term_ff_div</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">term_div</span> <span class="o">=</span> <span class="n">_term_rr_div</span>
-
-    <span class="k">while</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">g</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
-            <span class="n">tq</span> <span class="o">=</span> <span class="n">term_div</span><span class="p">(</span><span class="n">sdp_LT</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">sdp_LT</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">tq</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">sdp_add_term</span><span class="p">(</span><span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">tq</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-                <span class="n">f</span> <span class="o">=</span> <span class="n">sdp_sub</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">sdp_mul_term</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">tq</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">sdp_add_term</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">sdp_LT</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">sdp_del_LT</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">Q</span><span class="p">,</span> <span class="n">r</span>
-
-</div>
-<div class="viewcode-block" id="sdp_rem"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_rem">[docs]</a><span class="k">def</span> <span class="nf">sdp_rem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns polynomial remainder in `K[X]`. &quot;&quot;&quot;</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="p">{}</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-        <span class="n">term_div</span> <span class="o">=</span> <span class="n">_term_ff_div</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">term_div</span> <span class="o">=</span> <span class="n">_term_rr_div</span>
-
-    <span class="n">ltf</span> <span class="o">=</span> <span class="n">sdp_LT</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">get</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">get</span>
-    <span class="k">while</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
-            <span class="n">tq</span> <span class="o">=</span> <span class="n">term_div</span><span class="p">(</span><span class="n">ltf</span><span class="p">,</span> <span class="n">sdp_LT</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">tq</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">m</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">tq</span>
-                <span class="k">for</span> <span class="n">mg</span><span class="p">,</span> <span class="n">cg</span> <span class="ow">in</span> <span class="n">g</span><span class="p">:</span>
-                    <span class="n">m1</span> <span class="o">=</span> <span class="n">monomial_mul</span><span class="p">(</span><span class="n">mg</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-                    <span class="n">c1</span> <span class="o">=</span> <span class="n">get</span><span class="p">(</span><span class="n">m1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="o">-</span> <span class="n">c</span><span class="o">*</span><span class="n">cg</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">c1</span><span class="p">:</span>
-                        <span class="k">del</span> <span class="n">f</span><span class="p">[</span><span class="n">m1</span><span class="p">]</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">f</span><span class="p">[</span><span class="n">m1</span><span class="p">]</span> <span class="o">=</span> <span class="n">c1</span>
-                <span class="k">if</span> <span class="n">f</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">O</span> <span class="o">==</span> <span class="n">lex</span><span class="p">:</span>
-                        <span class="n">ltm</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">ltm</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">mx</span><span class="p">:</span> <span class="n">O</span><span class="p">(</span><span class="n">mx</span><span class="p">))</span>
-                    <span class="n">ltf</span> <span class="o">=</span> <span class="n">ltm</span><span class="p">,</span> <span class="n">f</span><span class="p">[</span><span class="n">ltm</span><span class="p">]</span>
-
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">ltm</span><span class="p">,</span> <span class="n">ltc</span> <span class="o">=</span> <span class="n">ltf</span>
-            <span class="k">if</span> <span class="n">ltm</span> <span class="ow">in</span> <span class="n">r</span><span class="p">:</span>
-                <span class="n">r</span><span class="p">[</span><span class="n">ltm</span><span class="p">]</span> <span class="o">+=</span> <span class="n">ltc</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">r</span><span class="p">[</span><span class="n">ltm</span><span class="p">]</span> <span class="o">=</span> <span class="n">ltc</span>
-            <span class="k">del</span> <span class="n">f</span><span class="p">[</span><span class="n">ltm</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">f</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">O</span> <span class="o">==</span> <span class="n">lex</span><span class="p">:</span>
-                    <span class="n">ltm</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">ltm</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">mx</span><span class="p">:</span> <span class="n">O</span><span class="p">(</span><span class="n">mx</span><span class="p">))</span>
-                <span class="n">ltf</span> <span class="o">=</span> <span class="n">ltm</span><span class="p">,</span> <span class="n">f</span><span class="p">[</span><span class="n">ltm</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">sdp_from_dict</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">O</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="sdp_quo"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_quo">[docs]</a><span class="k">def</span> <span class="nf">sdp_quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns polynomial quotient in `K[x]`. &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">sdp_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="sdp_exquo"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_exquo">[docs]</a><span class="k">def</span> <span class="nf">sdp_exquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns exact polynomial quotient in `K[X]`. &quot;&quot;&quot;</span>
-    <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">sdp_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">q</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ExactQuotientFailed</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="sdp_lcm"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_lcm">[docs]</a><span class="k">def</span> <span class="nf">sdp_lcm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes LCM of two polynomials in `K[X]`.</span>
-
-<span class="sd">    The LCM is computed as the unique generater of the intersection</span>
-<span class="sd">    of the two ideals generated by `f` and `g`. The approach is to</span>
-<span class="sd">    compute a Groebner basis with respect to lexicographic ordering</span>
-<span class="sd">    of `t*f` and `(1 - t)*g`, where `t` is an unrelated variable and</span>
-<span class="sd">    then filtering out the solution that doesn&#39;t contain `t`.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Cox97]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.polys.groebnertools</span> <span class="kn">import</span> <span class="n">sdp_groebner</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-    <span class="k">if</span> <span class="n">sdp_term_p</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="ow">and</span> <span class="n">sdp_term_p</span><span class="p">(</span><span class="n">g</span><span class="p">):</span>
-        <span class="n">monom</span> <span class="o">=</span> <span class="n">monomial_lcm</span><span class="p">(</span><span class="n">sdp_LM</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">),</span> <span class="n">sdp_LM</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">))</span>
-
-        <span class="n">fc</span><span class="p">,</span> <span class="n">gc</span> <span class="o">=</span> <span class="n">sdp_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">sdp_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">lcm</span><span class="p">(</span><span class="n">fc</span><span class="p">,</span> <span class="n">gc</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="p">[(</span><span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span><span class="p">)]</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">K</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-        <span class="n">lcm</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">fc</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">sdp_primitive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">gc</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">sdp_primitive</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">lcm</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">lcm</span><span class="p">(</span><span class="n">fc</span><span class="p">,</span> <span class="n">gc</span><span class="p">)</span>
-
-    <span class="n">f_terms</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span> <span class="p">((</span><span class="mi">1</span><span class="p">,)</span> <span class="o">+</span> <span class="n">m</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">m</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">)</span>
-    <span class="n">g_terms</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span> <span class="p">((</span><span class="mi">0</span><span class="p">,)</span> <span class="o">+</span> <span class="n">m</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">m</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">g</span> <span class="p">)</span> \
-        <span class="o">+</span> <span class="nb">tuple</span><span class="p">(</span> <span class="p">((</span><span class="mi">1</span><span class="p">,)</span> <span class="o">+</span> <span class="n">m</span><span class="p">,</span> <span class="o">-</span><span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">m</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">g</span> <span class="p">)</span>
-
-    <span class="n">F</span> <span class="o">=</span> <span class="n">sdp_sort</span><span class="p">(</span><span class="n">f_terms</span><span class="p">,</span> <span class="n">lex</span><span class="p">)</span>
-    <span class="n">G</span> <span class="o">=</span> <span class="n">sdp_sort</span><span class="p">(</span><span class="n">g_terms</span><span class="p">,</span> <span class="n">lex</span><span class="p">)</span>
-
-    <span class="n">basis</span> <span class="o">=</span> <span class="n">sdp_groebner</span><span class="p">([</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">],</span> <span class="n">u</span><span class="p">,</span> <span class="n">lex</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">H</span> <span class="o">=</span> <span class="p">[</span> <span class="n">h</span> <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">basis</span> <span class="k">if</span> <span class="n">sdp_indep_p</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span> <span class="p">]</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">lcm</span><span class="p">):</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="p">[</span> <span class="p">(</span><span class="n">m</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">m</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">H</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="p">[</span> <span class="p">(</span><span class="n">m</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">c</span> <span class="o">*</span> <span class="n">lcm</span><span class="p">)</span> <span class="k">for</span> <span class="n">m</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">H</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="p">]</span>
-
-    <span class="k">return</span> <span class="n">sdp_sort</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">O</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="sdp_gcd"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.distributedpolys.sdp_gcd">[docs]</a><span class="k">def</span> <span class="nf">sdp_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Compute GCD of two polynomials in `K[X]` via LCM. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">K</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-        <span class="n">fc</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">sdp_primitive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">gc</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">sdp_primitive</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">gcd</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">fc</span><span class="p">,</span> <span class="n">gc</span><span class="p">)</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="n">sdp_quo</span><span class="p">(</span><span class="n">sdp_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span>
-                <span class="n">sdp_lcm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">K</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">gcd</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">h</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">c</span> <span class="o">*</span> <span class="n">gcd</span><span class="p">)</span> <span class="k">for</span> <span class="n">m</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">h</span> <span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">sdp_monic</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span></div>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/polys/domains.html b/dev-py3k/_modules/sympy/polys/domains.html
deleted file mode 100644
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+++ /dev/null
@@ -1,207 +0,0 @@
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-            
-  <h1>Source code for sympy.polys.domains</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Implementation of mathematical domains. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">.domain</span> <span class="kn">import</span> <span class="n">Domain</span>
-<span class="kn">from</span> <span class="nn">.ring</span> <span class="kn">import</span> <span class="n">Ring</span>
-<span class="kn">from</span> <span class="nn">.field</span> <span class="kn">import</span> <span class="n">Field</span>
-
-<span class="kn">from</span> <span class="nn">.simpledomain</span> <span class="kn">import</span> <span class="n">SimpleDomain</span>
-<span class="kn">from</span> <span class="nn">.compositedomain</span> <span class="kn">import</span> <span class="n">CompositeDomain</span>
-<span class="kn">from</span> <span class="nn">.characteristiczero</span> <span class="kn">import</span> <span class="n">CharacteristicZero</span>
-
-<span class="kn">from</span> <span class="nn">.finitefield</span> <span class="kn">import</span> <span class="n">FiniteField</span>
-<span class="kn">from</span> <span class="nn">.integerring</span> <span class="kn">import</span> <span class="n">IntegerRing</span>
-<span class="kn">from</span> <span class="nn">.rationalfield</span> <span class="kn">import</span> <span class="n">RationalField</span>
-<span class="kn">from</span> <span class="nn">.realdomain</span> <span class="kn">import</span> <span class="n">RealDomain</span>
-
-<span class="kn">from</span> <span class="nn">.pythonfinitefield</span> <span class="kn">import</span> <span class="n">PythonFiniteField</span>
-<span class="kn">from</span> <span class="nn">.sympyfinitefield</span> <span class="kn">import</span> <span class="n">SymPyFiniteField</span>
-<span class="kn">from</span> <span class="nn">.gmpyfinitefield</span> <span class="kn">import</span> <span class="n">GMPYFiniteField</span>
-
-<span class="kn">from</span> <span class="nn">.pythonintegerring</span> <span class="kn">import</span> <span class="n">PythonIntegerRing</span>
-<span class="kn">from</span> <span class="nn">.sympyintegerring</span> <span class="kn">import</span> <span class="n">SymPyIntegerRing</span>
-<span class="kn">from</span> <span class="nn">.gmpyintegerring</span> <span class="kn">import</span> <span class="n">GMPYIntegerRing</span>
-
-<span class="kn">from</span> <span class="nn">.pythonrationalfield</span> <span class="kn">import</span> <span class="n">PythonRationalField</span>
-<span class="kn">from</span> <span class="nn">.sympyrationalfield</span> <span class="kn">import</span> <span class="n">SymPyRationalField</span>
-<span class="kn">from</span> <span class="nn">.gmpyrationalfield</span> <span class="kn">import</span> <span class="n">GMPYRationalField</span>
-
-<span class="kn">from</span> <span class="nn">.mpmathrealdomain</span> <span class="kn">import</span> <span class="n">MPmathRealDomain</span>
-
-<span class="kn">from</span> <span class="nn">.algebraicfield</span> <span class="kn">import</span> <span class="n">AlgebraicField</span>
-
-<span class="kn">from</span> <span class="nn">.polynomialring</span> <span class="kn">import</span> <span class="n">PolynomialRing</span>
-<span class="kn">from</span> <span class="nn">.fractionfield</span> <span class="kn">import</span> <span class="n">FractionField</span>
-
-<span class="kn">from</span> <span class="nn">.expressiondomain</span> <span class="kn">import</span> <span class="n">ExpressionDomain</span>
-
-<span class="kn">from</span> <span class="nn">.quotientring</span> <span class="kn">import</span> <span class="n">QuotientRing</span>
-
-<span class="n">FF_python</span> <span class="o">=</span> <span class="n">PythonFiniteField</span>
-<span class="n">FF_sympy</span> <span class="o">=</span> <span class="n">SymPyFiniteField</span>
-<span class="n">FF_gmpy</span> <span class="o">=</span> <span class="n">GMPYFiniteField</span>
-
-<span class="n">ZZ_python</span> <span class="o">=</span> <span class="n">PythonIntegerRing</span>
-<span class="n">ZZ_sympy</span> <span class="o">=</span> <span class="n">SymPyIntegerRing</span>
-<span class="n">ZZ_gmpy</span> <span class="o">=</span> <span class="n">GMPYIntegerRing</span>
-
-<span class="n">QQ_python</span> <span class="o">=</span> <span class="n">PythonRationalField</span>
-<span class="n">QQ_sympy</span> <span class="o">=</span> <span class="n">SymPyRationalField</span>
-<span class="n">QQ_gmpy</span> <span class="o">=</span> <span class="n">GMPYRationalField</span>
-
-<span class="n">RR_mpmath</span> <span class="o">=</span> <span class="n">MPmathRealDomain</span>
-
-<span class="kn">from</span> <span class="nn">.pythonrationaltype</span> <span class="kn">import</span> <span class="n">PythonRationalType</span>
-
-<span class="kn">from</span> <span class="nn">.groundtypes</span> <span class="kn">import</span> <span class="n">HAS_GMPY</span>
-
-
-<span class="k">def</span> <span class="nf">_getenv</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">default</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="kn">from</span> <span class="nn">os</span> <span class="kn">import</span> <span class="n">getenv</span>
-    <span class="k">return</span> <span class="n">getenv</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">default</span><span class="p">)</span>
-
-<span class="n">GROUND_TYPES</span> <span class="o">=</span> <span class="n">_getenv</span><span class="p">(</span><span class="s">&#39;SYMPY_GROUND_TYPES&#39;</span><span class="p">,</span> <span class="s">&#39;auto&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span>
-
-<span class="k">if</span> <span class="n">GROUND_TYPES</span> <span class="o">==</span> <span class="s">&#39;auto&#39;</span><span class="p">:</span>
-    <span class="k">if</span> <span class="n">HAS_GMPY</span><span class="p">:</span>
-        <span class="n">GROUND_TYPES</span> <span class="o">=</span> <span class="s">&#39;gmpy&#39;</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">GROUND_TYPES</span> <span class="o">=</span> <span class="s">&#39;python&#39;</span>
-
-<span class="k">if</span> <span class="n">GROUND_TYPES</span> <span class="o">==</span> <span class="s">&#39;gmpy&#39;</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">HAS_GMPY</span><span class="p">:</span>
-    <span class="kn">from</span> <span class="nn">warnings</span> <span class="kn">import</span> <span class="n">warn</span>
-    <span class="n">warn</span><span class="p">(</span><span class="s">&quot;gmpy library is not installed, switching to &#39;python&#39; ground types&quot;</span><span class="p">)</span>
-    <span class="n">GROUND_TYPES</span> <span class="o">=</span> <span class="s">&#39;python&#39;</span>
-
-<span class="n">_GROUND_TYPES_MAP</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="s">&#39;gmpy&#39;</span><span class="p">:</span> <span class="p">(</span><span class="n">FF_gmpy</span><span class="p">,</span> <span class="n">ZZ_gmpy</span><span class="p">(),</span> <span class="n">QQ_gmpy</span><span class="p">()),</span>
-    <span class="s">&#39;sympy&#39;</span><span class="p">:</span> <span class="p">(</span><span class="n">FF_sympy</span><span class="p">,</span> <span class="n">ZZ_sympy</span><span class="p">(),</span> <span class="n">QQ_sympy</span><span class="p">()),</span>
-    <span class="s">&#39;python&#39;</span><span class="p">:</span> <span class="p">(</span><span class="n">FF_python</span><span class="p">,</span> <span class="n">ZZ_python</span><span class="p">(),</span> <span class="n">QQ_python</span><span class="p">()),</span>
-<span class="p">}</span>
-
-<span class="k">try</span><span class="p">:</span>
-    <span class="n">FF</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">,</span> <span class="n">QQ</span> <span class="o">=</span> <span class="n">_GROUND_TYPES_MAP</span><span class="p">[</span><span class="n">GROUND_TYPES</span><span class="p">]</span>
-<span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-    <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;invalid ground types: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">GROUND_TYPES</span><span class="p">)</span>
-
-<span class="n">GF</span> <span class="o">=</span> <span class="n">FF</span>
-
-<span class="n">RR</span> <span class="o">=</span> <span class="n">RR_mpmath</span><span class="p">()</span>
-
-<span class="n">EX</span> <span class="o">=</span> <span class="n">ExpressionDomain</span><span class="p">()</span>
-</pre></div>
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diff --git a/dev-py3k/_modules/sympy/polys/euclidtools.html b/dev-py3k/_modules/sympy/polys/euclidtools.html
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-  <h1>Source code for sympy.polys.euclidtools</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Euclidean algorithms, GCDs, LCMs and polynomial remainder sequences. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dup_strip</span><span class="p">,</span> <span class="n">dmp_raise</span><span class="p">,</span>
-    <span class="n">dmp_zero</span><span class="p">,</span> <span class="n">dmp_one</span><span class="p">,</span> <span class="n">dmp_ground</span><span class="p">,</span>
-    <span class="n">dmp_one_p</span><span class="p">,</span> <span class="n">dmp_zero_p</span><span class="p">,</span>
-    <span class="n">dmp_zeros</span><span class="p">,</span>
-    <span class="n">dup_degree</span><span class="p">,</span> <span class="n">dmp_degree</span><span class="p">,</span> <span class="n">dmp_degree_in</span><span class="p">,</span>
-    <span class="n">dup_LC</span><span class="p">,</span> <span class="n">dmp_LC</span><span class="p">,</span> <span class="n">dmp_ground_LC</span><span class="p">,</span>
-    <span class="n">dmp_multi_deflate</span><span class="p">,</span> <span class="n">dmp_inflate</span><span class="p">,</span>
-    <span class="n">dup_convert</span><span class="p">,</span> <span class="n">dmp_convert</span><span class="p">,</span>
-    <span class="n">dmp_apply_pairs</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dup_sub_mul</span><span class="p">,</span>
-    <span class="n">dup_neg</span><span class="p">,</span> <span class="n">dmp_neg</span><span class="p">,</span>
-    <span class="n">dmp_add</span><span class="p">,</span>
-    <span class="n">dmp_sub</span><span class="p">,</span>
-    <span class="n">dup_mul</span><span class="p">,</span> <span class="n">dmp_mul</span><span class="p">,</span>
-    <span class="n">dmp_pow</span><span class="p">,</span>
-    <span class="n">dup_div</span><span class="p">,</span> <span class="n">dmp_div</span><span class="p">,</span>
-    <span class="n">dup_rem</span><span class="p">,</span>
-    <span class="n">dup_quo</span><span class="p">,</span> <span class="n">dmp_quo</span><span class="p">,</span>
-    <span class="n">dup_prem</span><span class="p">,</span> <span class="n">dmp_prem</span><span class="p">,</span>
-    <span class="n">dup_mul_ground</span><span class="p">,</span> <span class="n">dmp_mul_ground</span><span class="p">,</span>
-    <span class="n">dmp_mul_term</span><span class="p">,</span>
-    <span class="n">dup_quo_ground</span><span class="p">,</span> <span class="n">dmp_quo_ground</span><span class="p">,</span>
-    <span class="n">dup_max_norm</span><span class="p">,</span> <span class="n">dmp_max_norm</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dup_clear_denoms</span><span class="p">,</span> <span class="n">dmp_clear_denoms</span><span class="p">,</span>
-    <span class="n">dup_diff</span><span class="p">,</span> <span class="n">dmp_diff</span><span class="p">,</span>
-    <span class="n">dup_eval</span><span class="p">,</span> <span class="n">dmp_eval</span><span class="p">,</span> <span class="n">dmp_eval_in</span><span class="p">,</span>
-    <span class="n">dup_trunc</span><span class="p">,</span> <span class="n">dmp_ground_trunc</span><span class="p">,</span>
-    <span class="n">dup_monic</span><span class="p">,</span> <span class="n">dmp_ground_monic</span><span class="p">,</span>
-    <span class="n">dup_primitive</span><span class="p">,</span> <span class="n">dmp_ground_primitive</span><span class="p">,</span>
-    <span class="n">dup_extract</span><span class="p">,</span> <span class="n">dmp_ground_extract</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">gf_int</span><span class="p">,</span> <span class="n">gf_crt</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">MultivariatePolynomialError</span><span class="p">,</span>
-    <span class="n">HeuristicGCDFailed</span><span class="p">,</span>
-    <span class="n">HomomorphismFailed</span><span class="p">,</span>
-    <span class="n">NotInvertible</span><span class="p">,</span>
-    <span class="n">DomainError</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyconfig</span> <span class="kn">import</span> <span class="n">query</span>
-
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">cythonized</span>
-
-<span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">nextprime</span>
-
-
-<span class="k">def</span> <span class="nf">dup_half_gcdex</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Half extended Euclidean algorithm in `F[x]`.</span>
-
-<span class="sd">    Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dup_half_gcdex</span>
-
-<span class="sd">    &gt;&gt;&gt; f = QQ.map([1, -2, -6, 12, 15])</span>
-<span class="sd">    &gt;&gt;&gt; g = QQ.map([1, 1, -4, -4])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_half_gcdex(f, g, QQ)</span>
-<span class="sd">    ([-1/5, 3/5], [1/1, 1/1])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">K</span><span class="o">.</span><span class="n">has_Field</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">K</span><span class="o">.</span><span class="n">is_Exact</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="n">DomainError</span><span class="p">(</span><span class="s">&quot;can&#39;t compute half extended GCD over </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">],</span> <span class="p">[]</span>
-
-    <span class="k">while</span> <span class="n">g</span><span class="p">:</span>
-        <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dup_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">f</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="p">,</span> <span class="n">r</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="p">,</span> <span class="n">dup_sub_mul</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">a</span> <span class="o">=</span> <span class="n">dup_quo_ground</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">dup_monic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">a</span><span class="p">,</span> <span class="n">f</span>
-
-
-<div class="viewcode-block" id="dmp_half_gcdex"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_half_gcdex">[docs]</a><span class="k">def</span> <span class="nf">dmp_half_gcdex</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Half extended Euclidean algorithm in `F[X]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_half_gcdex</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_half_gcdex</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">MultivariatePolynomialError</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_gcdex</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Extended Euclidean algorithm in `F[x]`.</span>
-
-<span class="sd">    Returns ``(s, t, h)`` such that ``h = gcd(f, g)`` and ``s*f + t*g = h``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dup_gcdex</span>
-
-<span class="sd">    &gt;&gt;&gt; f = QQ.map([1, -2, -6, 12, 15])</span>
-<span class="sd">    &gt;&gt;&gt; g = QQ.map([1, 1, -4, -4])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_gcdex(f, g, QQ)</span>
-<span class="sd">    ([-1/5, 3/5], [1/5, -6/5, 2/1], [1/1, 1/1])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">s</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">dup_half_gcdex</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">F</span> <span class="o">=</span> <span class="n">dup_sub_mul</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">t</span> <span class="o">=</span> <span class="n">dup_quo</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">h</span>
-
-
-<div class="viewcode-block" id="dmp_gcdex"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_gcdex">[docs]</a><span class="k">def</span> <span class="nf">dmp_gcdex</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Extended Euclidean algorithm in `F[X]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_gcdex</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_gcdex</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">MultivariatePolynomialError</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_invert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute multiplicative inverse of `f` modulo `g` in `F[x]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dup_invert</span>
-
-<span class="sd">    &gt;&gt;&gt; f = QQ.map([1, 0, -1])</span>
-<span class="sd">    &gt;&gt;&gt; g = QQ.map([2, -1])</span>
-<span class="sd">    &gt;&gt;&gt; h = QQ.map([1, -1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_invert(f, g, QQ)</span>
-<span class="sd">    [-4/3]</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_invert(f, h, QQ)</span>
-<span class="sd">    Traceback (most recent call last):</span>
-<span class="sd">    ...</span>
-<span class="sd">    NotInvertible: zero divisor</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">s</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">dup_half_gcdex</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">h</span> <span class="o">==</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]:</span>
-        <span class="k">return</span> <span class="n">dup_rem</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">NotInvertible</span><span class="p">(</span><span class="s">&quot;zero divisor&quot;</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="dmp_invert"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_invert">[docs]</a><span class="k">def</span> <span class="nf">dmp_invert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute multiplicative inverse of `f` modulo `g` in `F[X]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_invert</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_invert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">MultivariatePolynomialError</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_euclidean_prs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Euclidean polynomial remainder sequence (PRS) in `K[x]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dup_euclidean_prs</span>
-
-<span class="sd">    &gt;&gt;&gt; f = QQ.map([1, 0, 1, 0, -3, -3, 8, 2, -5])</span>
-<span class="sd">    &gt;&gt;&gt; g = QQ.map([3, 0, 5, 0, -4, -9, 21])</span>
-
-<span class="sd">    &gt;&gt;&gt; prs = dup_euclidean_prs(f, g, QQ)</span>
-
-<span class="sd">    &gt;&gt;&gt; prs[0]</span>
-<span class="sd">    [1/1, 0/1, 1/1, 0/1, -3/1, -3/1, 8/1, 2/1, -5/1]</span>
-<span class="sd">    &gt;&gt;&gt; prs[1]</span>
-<span class="sd">    [3/1, 0/1, 5/1, 0/1, -4/1, -9/1, 21/1]</span>
-<span class="sd">    &gt;&gt;&gt; prs[2]</span>
-<span class="sd">    [-5/9, 0/1, 1/9, 0/1, -1/3]</span>
-<span class="sd">    &gt;&gt;&gt; prs[3]</span>
-<span class="sd">    [-117/25, -9/1, 441/25]</span>
-<span class="sd">    &gt;&gt;&gt; prs[4]</span>
-<span class="sd">    [233150/19773, -102500/6591]</span>
-<span class="sd">    &gt;&gt;&gt; prs[5]</span>
-<span class="sd">    [-1288744821/543589225]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">prs</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">]</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dup_rem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">while</span> <span class="n">h</span><span class="p">:</span>
-        <span class="n">prs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-        <span class="n">f</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dup_rem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">prs</span>
-
-
-<div class="viewcode-block" id="dmp_euclidean_prs"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_euclidean_prs">[docs]</a><span class="k">def</span> <span class="nf">dmp_euclidean_prs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Euclidean polynomial remainder sequence (PRS) in `K[X]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_euclidean_prs</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_euclidean_prs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">MultivariatePolynomialError</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_primitive_prs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Primitive polynomial remainder sequence (PRS) in `K[x]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dup_primitive_prs</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, 1, 0, -3, -3, 8, 2, -5])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([3, 0, 5, 0, -4, -9, 21])</span>
-
-<span class="sd">    &gt;&gt;&gt; prs = dup_primitive_prs(f, g, ZZ)</span>
-
-<span class="sd">    &gt;&gt;&gt; prs[0]</span>
-<span class="sd">    [1, 0, 1, 0, -3, -3, 8, 2, -5]</span>
-<span class="sd">    &gt;&gt;&gt; prs[1]</span>
-<span class="sd">    [3, 0, 5, 0, -4, -9, 21]</span>
-<span class="sd">    &gt;&gt;&gt; prs[2]</span>
-<span class="sd">    [-5, 0, 1, 0, -3]</span>
-<span class="sd">    &gt;&gt;&gt; prs[3]</span>
-<span class="sd">    [13, 25, -49]</span>
-<span class="sd">    &gt;&gt;&gt; prs[4]</span>
-<span class="sd">    [4663, -6150]</span>
-<span class="sd">    &gt;&gt;&gt; prs[5]</span>
-<span class="sd">    [1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">prs</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">]</span>
-    <span class="n">_</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">dup_primitive</span><span class="p">(</span><span class="n">dup_prem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">while</span> <span class="n">h</span><span class="p">:</span>
-        <span class="n">prs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-        <span class="n">f</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">dup_primitive</span><span class="p">(</span><span class="n">dup_prem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">prs</span>
-
-
-<div class="viewcode-block" id="dmp_primitive_prs"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_primitive_prs">[docs]</a><span class="k">def</span> <span class="nf">dmp_primitive_prs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Primitive polynomial remainder sequence (PRS) in `K[X]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_primitive_prs</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_primitive_prs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">MultivariatePolynomialError</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,m,d,k&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_inner_subresultants</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Subresultant PRS algorithm in `K[x]`.</span>
-
-<span class="sd">    Computes the subresultant polynomial remainder sequence (PRS) of `f`</span>
-<span class="sd">    and `g`, and the values for `\beta_i` and `\delta_i`. The last two</span>
-<span class="sd">    sequences of values are necessary for computing the resultant in</span>
-<span class="sd">    :func:`dup_prs_resultant`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dup_inner_subresultants</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, 1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([1, 0, -1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_inner_subresultants(f, g, ZZ)</span>
-<span class="sd">    ([[1, 0, 1], [1, 0, -1], [-2]], [-1, -1], [0, 2])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="n">m</span><span class="p">:</span>
-        <span class="n">f</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="p">,</span> <span class="n">f</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span>
-
-    <span class="n">R</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">]</span>
-    <span class="n">d</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="n">m</span>
-
-    <span class="n">b</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">d</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="n">c</span> <span class="o">=</span> <span class="o">-</span><span class="n">K</span><span class="o">.</span><span class="n">one</span>
-
-    <span class="n">B</span><span class="p">,</span> <span class="n">D</span> <span class="o">=</span> <span class="p">[</span><span class="n">b</span><span class="p">],</span> <span class="p">[</span><span class="n">d</span><span class="p">]</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">R</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">D</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dup_prem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">while</span> <span class="n">h</span><span class="p">:</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-        <span class="n">R</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-
-        <span class="n">lc</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">d</span><span class="p">:</span>
-            <span class="n">q</span> <span class="o">=</span> <span class="n">c</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">q</span> <span class="o">=</span> <span class="n">c</span><span class="o">**</span><span class="p">(</span><span class="n">d</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-
-        <span class="n">c</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">quo</span><span class="p">((</span><span class="o">-</span><span class="n">lc</span><span class="p">)</span><span class="o">**</span><span class="n">d</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="o">-</span><span class="n">lc</span> <span class="o">*</span> <span class="n">c</span><span class="o">**</span><span class="p">(</span><span class="n">m</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span>
-
-        <span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">m</span> <span class="o">-</span> <span class="n">k</span>
-
-        <span class="n">B</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="n">D</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dup_prem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dup_quo_ground</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">R</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">D</span>
-
-
-<span class="k">def</span> <span class="nf">dup_subresultants</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes subresultant PRS of two polynomials in `K[x]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dup_subresultants</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, 1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([1, 0, -1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_subresultants(f, g, ZZ)</span>
-<span class="sd">    [[1, 0, 1], [1, 0, -1], [-2]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dup_inner_subresultants</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;s,i,du,dv,dw&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_prs_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Resultant algorithm in `K[x]` using subresultant PRS.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dup_prs_resultant</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, 1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([1, 0, -1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_prs_resultant(f, g, ZZ)</span>
-<span class="sd">    (4, [[1, 0, 1], [1, 0, -1], [-2]])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">,</span> <span class="p">[])</span>
-
-    <span class="n">R</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">D</span> <span class="o">=</span> <span class="n">dup_inner_subresultants</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">R</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">R</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]:</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">dup_LC</span><span class="p">(</span><span class="n">R</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">K</span><span class="p">),</span> <span class="n">R</span><span class="p">)</span>
-
-    <span class="n">s</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span>
-    <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span>
-
-    <span class="k">for</span> <span class="n">b</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">D</span><span class="p">))[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-        <span class="n">du</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">R</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span>
-        <span class="n">dv</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">R</span><span class="p">[</span><span class="n">i</span>  <span class="p">])</span>
-        <span class="n">dw</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">R</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">du</span> <span class="o">%</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">dv</span> <span class="o">%</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="o">-</span><span class="n">s</span>
-
-        <span class="n">lc</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">R</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">K</span><span class="p">),</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-
-        <span class="n">p</span> <span class="o">*=</span> <span class="n">b</span><span class="o">**</span><span class="n">dv</span> <span class="o">*</span> <span class="n">lc</span><span class="o">**</span><span class="p">(</span><span class="n">du</span> <span class="o">-</span> <span class="n">dw</span><span class="p">)</span>
-        <span class="n">q</span> <span class="o">*=</span> <span class="n">lc</span><span class="o">**</span><span class="p">(</span><span class="n">dv</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">d</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="n">s</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="o">-</span><span class="n">p</span>
-
-    <span class="n">i</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">R</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">])</span>
-
-    <span class="n">res</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">R</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">K</span><span class="p">)</span><span class="o">**</span><span class="n">i</span>
-
-    <span class="n">res</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">res</span><span class="o">*</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">res</span><span class="p">,</span> <span class="n">R</span>
-
-
-<span class="k">def</span> <span class="nf">dup_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">includePRS</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes resultant of two polynomials in `K[x]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dup_resultant</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, 1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([1, 0, -1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_resultant(f, g, ZZ)</span>
-<span class="sd">    4</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">includePRS</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_prs_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">dup_prs_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v,n,m,d,k&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_inner_subresultants"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_inner_subresultants">[docs]</a><span class="k">def</span> <span class="nf">dmp_inner_subresultants</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Subresultant PRS algorithm in `K[X]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_inner_subresultants</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[3, 0], [], [-1, 0, 0, -4]])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[1], [1, 0, 0, 0], [-9]])</span>
-
-<span class="sd">    &gt;&gt;&gt; a = [[3, 0, 0, 0, 0], [1, 0, -27, 4]]</span>
-<span class="sd">    &gt;&gt;&gt; b = [[-3, 0, 0, -12, 1, 0, -54, 8, 729, -216, 16]]</span>
-
-<span class="sd">    &gt;&gt;&gt; R = ZZ.map([f, g, a, b])</span>
-<span class="sd">    &gt;&gt;&gt; B = ZZ.map([[-1], [1], [9, 0, 0, 0, 0, 0, 0, 0, 0]])</span>
-<span class="sd">    &gt;&gt;&gt; D = ZZ.map([0, 1, 1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_inner_subresultants(f, g, 1, ZZ) == (R, B, D)</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_inner_subresultants</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="n">m</span><span class="p">:</span>
-        <span class="n">f</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="p">,</span> <span class="n">f</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span>
-
-    <span class="n">R</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">]</span>
-    <span class="n">d</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="n">m</span>
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="n">b</span> <span class="o">=</span> <span class="n">dmp_pow</span><span class="p">(</span><span class="n">dmp_ground</span><span class="p">(</span><span class="o">-</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">v</span><span class="p">),</span> <span class="n">d</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">c</span> <span class="o">=</span> <span class="n">dmp_ground</span><span class="p">(</span><span class="o">-</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
-
-    <span class="n">B</span><span class="p">,</span> <span class="n">D</span> <span class="o">=</span> <span class="p">[</span><span class="n">b</span><span class="p">],</span> <span class="p">[</span><span class="n">d</span><span class="p">]</span>
-
-    <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span> <span class="ow">or</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">R</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">D</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_prem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_mul_term</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">while</span> <span class="ow">not</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-        <span class="n">R</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-
-        <span class="n">lc</span> <span class="o">=</span> <span class="n">dmp_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">p</span> <span class="o">=</span> <span class="n">dmp_pow</span><span class="p">(</span><span class="n">dmp_neg</span><span class="p">(</span><span class="n">lc</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">d</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">d</span><span class="p">:</span>
-            <span class="n">q</span> <span class="o">=</span> <span class="n">c</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">q</span> <span class="o">=</span> <span class="n">dmp_pow</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">d</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">c</span> <span class="o">=</span> <span class="n">dmp_quo</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">dmp_neg</span><span class="p">(</span><span class="n">lc</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span>
-                    <span class="n">dmp_pow</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">m</span> <span class="o">-</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">m</span> <span class="o">-</span> <span class="n">k</span>
-
-        <span class="n">B</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="n">D</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_prem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">h</span> <span class="o">=</span> <span class="p">[</span> <span class="n">dmp_quo</span><span class="p">(</span><span class="n">ch</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">ch</span> <span class="ow">in</span> <span class="n">h</span> <span class="p">]</span>
-
-    <span class="k">return</span> <span class="n">R</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">D</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_subresultants"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_subresultants">[docs]</a><span class="k">def</span> <span class="nf">dmp_subresultants</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes subresultant PRS of two polynomials in `K[X]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_subresultants</span>
-
-<span class="sd">    &gt;&gt;&gt; f = [[3, 0], [], [-1, 0, 0, -4]]</span>
-<span class="sd">    &gt;&gt;&gt; g = [[1], [1, 0, 0, 0], [-9]]</span>
-
-<span class="sd">    &gt;&gt;&gt; a = [[3, 0, 0, 0, 0], [1, 0, -27, 4]]</span>
-<span class="sd">    &gt;&gt;&gt; b = [[-3, 0, 0, -12, 1, 0, -54, 8, 729, -216, 16]]</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_subresultants(f, g, 1, ZZ) == [f, g, a, b]</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dmp_inner_subresultants</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v,s,i,d,du,dv,dw&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_prs_resultant"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_prs_resultant">[docs]</a><span class="k">def</span> <span class="nf">dmp_prs_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Resultant algorithm in `K[X]` using subresultant PRS.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_prs_resultant</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[3, 0], [], [-1, 0, 0, -4]])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[1], [1, 0, 0, 0], [-9]])</span>
-
-<span class="sd">    &gt;&gt;&gt; a = ZZ.map([[3, 0, 0, 0, 0], [1, 0, -27, 4]])</span>
-<span class="sd">    &gt;&gt;&gt; b = ZZ.map([[-3, 0, 0, -12, 1, 0, -54, 8, 729, -216, 16]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_prs_resultant(f, g, 1, ZZ) == (b[0], [f, g, a, b])</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_prs_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span> <span class="ow">or</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">dmp_zero</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="p">[])</span>
-
-    <span class="n">R</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">D</span> <span class="o">=</span> <span class="n">dmp_inner_subresultants</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">R</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">u</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">dmp_zero</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">R</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">dmp_one_p</span><span class="p">(</span><span class="n">R</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">dmp_LC</span><span class="p">(</span><span class="n">R</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">K</span><span class="p">),</span> <span class="n">R</span><span class="p">)</span>
-
-    <span class="n">s</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="n">p</span> <span class="o">=</span> <span class="n">dmp_one</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">q</span> <span class="o">=</span> <span class="n">dmp_one</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">b</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">D</span><span class="p">))[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-        <span class="n">du</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">R</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">u</span><span class="p">)</span>
-        <span class="n">dv</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">R</span><span class="p">[</span><span class="n">i</span>  <span class="p">],</span> <span class="n">u</span><span class="p">)</span>
-        <span class="n">dw</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">R</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">u</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">du</span> <span class="o">%</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">dv</span> <span class="o">%</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="o">-</span><span class="n">s</span>
-
-        <span class="n">lc</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="n">dmp_LC</span><span class="p">(</span><span class="n">R</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">K</span><span class="p">),</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-
-        <span class="n">p</span> <span class="o">=</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">dmp_mul</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">dmp_pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">dv</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span>
-                    <span class="n">dmp_pow</span><span class="p">(</span><span class="n">lc</span><span class="p">,</span> <span class="n">du</span> <span class="o">-</span> <span class="n">dw</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">q</span> <span class="o">=</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">dmp_pow</span><span class="p">(</span><span class="n">lc</span><span class="p">,</span> <span class="n">dv</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">d</span><span class="p">),</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">_</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">dmp_inner_gcd</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">s</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">dmp_neg</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">i</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">R</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="n">res</span> <span class="o">=</span> <span class="n">dmp_pow</span><span class="p">(</span><span class="n">dmp_LC</span><span class="p">(</span><span class="n">R</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">K</span><span class="p">),</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="n">dmp_quo</span><span class="p">(</span><span class="n">dmp_mul</span><span class="p">(</span><span class="n">res</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">q</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">res</span><span class="p">,</span> <span class="n">R</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v,n,m,N,M,B&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_zz_modular_resultant"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_zz_modular_resultant">[docs]</a><span class="k">def</span> <span class="nf">dmp_zz_modular_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute resultant of `f` and `g` modulo a prime `p`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_zz_modular_resultant</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [1, 2]])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[2, 1], [3]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_zz_modular_resultant(f, g, ZZ(5), 1, ZZ)</span>
-<span class="sd">    [-2, 0, 1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">gf_int</span><span class="p">(</span><span class="n">dup_prs_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span> <span class="o">%</span> <span class="n">p</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="n">N</span> <span class="o">=</span> <span class="n">dmp_degree_in</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-    <span class="n">M</span> <span class="o">=</span> <span class="n">dmp_degree_in</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="n">B</span> <span class="o">=</span> <span class="n">n</span><span class="o">*</span><span class="n">M</span> <span class="o">+</span> <span class="n">m</span><span class="o">*</span><span class="n">N</span>
-
-    <span class="n">D</span><span class="p">,</span> <span class="n">a</span> <span class="o">=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">],</span> <span class="o">-</span><span class="n">K</span><span class="o">.</span><span class="n">one</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-
-    <span class="k">while</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">D</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="n">B</span><span class="p">:</span>
-        <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">+=</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span>
-
-            <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="n">p</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">HomomorphismFailed</span><span class="p">(</span><span class="s">&#39;no luck&#39;</span><span class="p">)</span>
-
-            <span class="n">F</span> <span class="o">=</span> <span class="n">dmp_eval_in</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gf_int</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="o">==</span> <span class="n">n</span><span class="p">:</span>
-                <span class="n">G</span> <span class="o">=</span> <span class="n">dmp_eval_in</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">gf_int</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="o">==</span> <span class="n">m</span><span class="p">:</span>
-                    <span class="k">break</span>
-
-        <span class="n">R</span> <span class="o">=</span> <span class="n">dmp_zz_modular_resultant</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">e</span> <span class="o">=</span> <span class="n">dmp_eval</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">v</span><span class="p">:</span>
-            <span class="n">R</span> <span class="o">=</span> <span class="n">dup_strip</span><span class="p">([</span><span class="n">R</span><span class="p">])</span>
-            <span class="n">e</span> <span class="o">=</span> <span class="n">dup_strip</span><span class="p">([</span><span class="n">e</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">R</span> <span class="o">=</span> <span class="p">[</span><span class="n">R</span><span class="p">]</span>
-            <span class="n">e</span> <span class="o">=</span> <span class="p">[</span><span class="n">e</span><span class="p">]</span>
-
-        <span class="n">d</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">invert</span><span class="p">(</span><span class="n">dup_eval</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">p</span><span class="p">)</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">dmp_raise</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">c</span> <span class="o">=</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">dmp_sub</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_add</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_ground_trunc</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">D</span> <span class="o">=</span> <span class="n">dup_mul</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="o">-</span><span class="n">a</span><span class="p">],</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="n">dup_trunc</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">r</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_collins_crt</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Wrapper of CRT for Collins&#39;s resultant algorithm. &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">gf_int</span><span class="p">(</span><span class="n">gf_crt</span><span class="p">([</span><span class="n">r</span><span class="p">,</span> <span class="n">R</span><span class="p">],</span> <span class="p">[</span><span class="n">P</span><span class="p">,</span> <span class="n">p</span><span class="p">],</span> <span class="n">K</span><span class="p">),</span> <span class="n">P</span><span class="o">*</span><span class="n">p</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v,n,m&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_zz_collins_resultant"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_zz_collins_resultant">[docs]</a><span class="k">def</span> <span class="nf">dmp_zz_collins_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Collins&#39;s modular resultant algorithm in `Z[X]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_zz_collins_resultant</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [1, 2]])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[2, 1], [3]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_zz_collins_resultant(f, g, 1, ZZ)</span>
-<span class="sd">    [-2, -5, 1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">m</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="n">A</span> <span class="o">=</span> <span class="n">dmp_max_norm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">B</span> <span class="o">=</span> <span class="n">dmp_max_norm</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">a</span> <span class="o">=</span> <span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">b</span> <span class="o">=</span> <span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="n">B</span> <span class="o">=</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">K</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">K</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="n">m</span><span class="p">))</span><span class="o">*</span><span class="n">A</span><span class="o">**</span><span class="n">m</span><span class="o">*</span><span class="n">B</span><span class="o">**</span><span class="n">n</span>
-    <span class="n">r</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">P</span> <span class="o">=</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span>
-
-    <span class="k">while</span> <span class="n">P</span> <span class="o">&lt;=</span> <span class="n">B</span><span class="p">:</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">K</span><span class="p">(</span><span class="n">nextprime</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
-
-        <span class="k">while</span> <span class="ow">not</span> <span class="p">(</span><span class="n">a</span> <span class="o">%</span> <span class="n">p</span><span class="p">)</span> <span class="ow">or</span> <span class="ow">not</span> <span class="p">(</span><span class="n">b</span> <span class="o">%</span> <span class="n">p</span><span class="p">):</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">K</span><span class="p">(</span><span class="n">nextprime</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
-
-        <span class="n">F</span> <span class="o">=</span> <span class="n">dmp_ground_trunc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">G</span> <span class="o">=</span> <span class="n">dmp_ground_trunc</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">R</span> <span class="o">=</span> <span class="n">dmp_zz_modular_resultant</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">HomomorphismFailed</span><span class="p">:</span>
-            <span class="k">continue</span>
-
-        <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">P</span><span class="p">):</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">R</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_apply_pairs</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="n">_collins_crt</span><span class="p">,</span> <span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">P</span> <span class="o">*=</span> <span class="n">p</span>
-
-    <span class="k">return</span> <span class="n">r</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,n,m&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_qq_collins_resultant"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_qq_collins_resultant">[docs]</a><span class="k">def</span> <span class="nf">dmp_qq_collins_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Collins&#39;s modular resultant algorithm in `Q[X]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_qq_collins_resultant</span>
-
-<span class="sd">    &gt;&gt;&gt; f = [[QQ(1,2)], [QQ(1), QQ(2,3)]]</span>
-<span class="sd">    &gt;&gt;&gt; g = [[QQ(2), QQ(1)], [QQ(3)]]</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_qq_collins_resultant(f, g, 1, QQ)</span>
-<span class="sd">    [-2/1, -7/3, 5/6]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">m</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="n">K1</span> <span class="o">=</span> <span class="n">K0</span><span class="o">.</span><span class="n">get_ring</span><span class="p">()</span>
-
-    <span class="n">cf</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_clear_denoms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">)</span>
-    <span class="n">cg</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">dmp_clear_denoms</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">)</span>
-
-    <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_convert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">)</span>
-    <span class="n">g</span> <span class="o">=</span> <span class="n">dmp_convert</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">)</span>
-
-    <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_zz_collins_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K1</span><span class="p">)</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_convert</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K1</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-
-    <span class="n">c</span> <span class="o">=</span> <span class="n">K0</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">cf</span><span class="o">**</span><span class="n">m</span> <span class="o">*</span> <span class="n">cg</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="n">K1</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">dmp_quo_ground</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_resultant"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_resultant">[docs]</a><span class="k">def</span> <span class="nf">dmp_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">includePRS</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes resultant of two polynomials in `K[X]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_resultant</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[3, 0], [], [-1, 0, 0, -4]])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[1], [1, 0, 0, 0], [-9]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_resultant(f, g, 1, ZZ)</span>
-<span class="sd">    [-3, 0, 0, -12, 1, 0, -54, 8, 729, -216, 16]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">includePRS</span><span class="o">=</span><span class="n">includePRS</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">includePRS</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_prs_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_QQ</span> <span class="ow">and</span> <span class="n">query</span><span class="p">(</span><span class="s">&#39;USE_COLLINS_RESULTANT&#39;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">dmp_qq_collins_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_ZZ</span> <span class="ow">and</span> <span class="n">query</span><span class="p">(</span><span class="s">&#39;USE_COLLINS_RESULTANT&#39;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">dmp_zz_collins_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">dmp_prs_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;d,s&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_discriminant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes discriminant of a polynomial in `K[x]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dup_discriminant</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_discriminant([ZZ(1), ZZ(2), ZZ(3)], ZZ)</span>
-<span class="sd">    -8</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">d</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">d</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">((</span><span class="n">d</span><span class="o">*</span><span class="p">(</span><span class="n">d</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span> <span class="o">//</span> <span class="mi">2</span><span class="p">)</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">r</span> <span class="o">=</span> <span class="n">dup_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">dup_diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">c</span><span class="o">*</span><span class="n">K</span><span class="p">(</span><span class="n">s</span><span class="p">))</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v,d,s&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_discriminant"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_discriminant">[docs]</a><span class="k">def</span> <span class="nf">dmp_discriminant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes discriminant of a polynomial in `K[X]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_discriminant</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[[[1]], [[]]], [[[1], []]], [[[1, 0]]]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_discriminant(f, 3, ZZ)</span>
-<span class="sd">    [[[-4, 0]], [[1], [], []]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_discriminant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">d</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">),</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="n">d</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">((</span><span class="n">d</span><span class="o">*</span><span class="p">(</span><span class="n">d</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span> <span class="o">//</span> <span class="mi">2</span><span class="p">)</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">dmp_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">dmp_diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">dmp_mul_ground</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">K</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">dmp_quo</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_dup_rr_trivial_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Handle trivial cases in GCD algorithm over a ring. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">f</span> <span class="ow">or</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[]</span>
-    <span class="k">elif</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">(</span><span class="n">dup_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="n">g</span><span class="p">,</span> <span class="p">[],</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">dup_neg</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="p">[],</span> <span class="p">[</span><span class="o">-</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]</span>
-    <span class="k">elif</span> <span class="ow">not</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">(</span><span class="n">dup_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">],</span> <span class="p">[]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">dup_neg</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">],</span> <span class="p">[]</span>
-
-    <span class="k">return</span> <span class="bp">None</span>
-
-
-<span class="k">def</span> <span class="nf">_dup_ff_trivial_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Handle trivial cases in GCD algorithm over a field. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">f</span> <span class="ow">or</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[]</span>
-    <span class="k">elif</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_monic</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="p">[],</span> <span class="p">[</span><span class="n">dup_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)]</span>
-    <span class="k">elif</span> <span class="ow">not</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_monic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="p">[</span><span class="n">dup_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)],</span> <span class="p">[]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_dmp_rr_trivial_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Handle trivial cases in GCD algorithm over a ring. &quot;&quot;&quot;</span>
-    <span class="n">zero_f</span> <span class="o">=</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-    <span class="n">zero_g</span> <span class="o">=</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">zero_f</span> <span class="ow">and</span> <span class="n">zero_g</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">dmp_zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-    <span class="k">elif</span> <span class="n">zero_f</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">(</span><span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="n">g</span><span class="p">,</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">dmp_one</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">dmp_neg</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">dmp_ground</span><span class="p">(</span><span class="o">-</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">zero_g</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">(</span><span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="p">,</span> <span class="n">dmp_one</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">dmp_neg</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">dmp_ground</span><span class="p">(</span><span class="o">-</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">u</span><span class="p">),</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">query</span><span class="p">(</span><span class="s">&#39;USE_SIMPLIFY_GCD&#39;</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_dmp_simplify_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_dmp_ff_trivial_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Handle trivial cases in GCD algorithm over a field. &quot;&quot;&quot;</span>
-    <span class="n">zero_f</span> <span class="o">=</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-    <span class="n">zero_g</span> <span class="o">=</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">zero_f</span> <span class="ow">and</span> <span class="n">zero_g</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">dmp_zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-    <span class="k">elif</span> <span class="n">zero_f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">dmp_ground_monic</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span>
-                <span class="n">dmp_zero</span><span class="p">(</span><span class="n">u</span><span class="p">),</span>
-                <span class="n">dmp_ground</span><span class="p">(</span><span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">u</span><span class="p">))</span>
-    <span class="k">elif</span> <span class="n">zero_g</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">dmp_ground_monic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span>
-                <span class="n">dmp_ground</span><span class="p">(</span><span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">u</span><span class="p">),</span>
-                <span class="n">dmp_zero</span><span class="p">(</span><span class="n">u</span><span class="p">))</span>
-    <span class="k">elif</span> <span class="n">query</span><span class="p">(</span><span class="s">&#39;USE_SIMPLIFY_GCD&#39;</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_dmp_simplify_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v,df,dg&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_dmp_simplify_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Try to eliminate `x_0` from GCD computation in `K[X]`. &quot;&quot;&quot;</span>
-    <span class="n">df</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-    <span class="n">dg</span> <span class="o">=</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">df</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">dg</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">df</span> <span class="ow">or</span> <span class="n">dg</span><span class="p">):</span>
-        <span class="n">F</span> <span class="o">=</span> <span class="n">dmp_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">G</span> <span class="o">=</span> <span class="n">dmp_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">df</span><span class="p">:</span>
-            <span class="n">F</span> <span class="o">=</span> <span class="n">dmp_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">G</span> <span class="o">=</span> <span class="n">dmp_content</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">F</span> <span class="o">=</span> <span class="n">dmp_content</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">G</span> <span class="o">=</span> <span class="n">dmp_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_gcd</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">cff</span> <span class="o">=</span> <span class="p">[</span> <span class="n">dmp_quo</span><span class="p">(</span><span class="n">cf</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">cf</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-    <span class="n">cfg</span> <span class="o">=</span> <span class="p">[</span> <span class="n">dmp_quo</span><span class="p">(</span><span class="n">cg</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">cg</span> <span class="ow">in</span> <span class="n">g</span> <span class="p">]</span>
-
-    <span class="k">return</span> <span class="p">[</span><span class="n">h</span><span class="p">],</span> <span class="n">cff</span><span class="p">,</span> <span class="n">cfg</span>
-
-
-<span class="k">def</span> <span class="nf">dup_rr_prs_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes polynomial GCD using subresultants over a ring.</span>
-
-<span class="sd">    Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``,</span>
-<span class="sd">    and ``cfg = quo(g, h)``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dup_rr_prs_gcd</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, -1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([1, -3, 2])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_rr_prs_gcd(f, g, ZZ)</span>
-<span class="sd">    ([1, -1], [1, 1], [1, -2])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="n">_dup_rr_trivial_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="n">fc</span><span class="p">,</span> <span class="n">F</span> <span class="o">=</span> <span class="n">dup_primitive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">gc</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">dup_primitive</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">c</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">fc</span><span class="p">,</span> <span class="n">gc</span><span class="p">)</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dup_subresultants</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-    <span class="n">_</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">dup_primitive</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_negative</span><span class="p">(</span><span class="n">dup_LC</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)):</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="o">-</span><span class="n">c</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">cff</span> <span class="o">=</span> <span class="n">dup_quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">cfg</span> <span class="o">=</span> <span class="n">dup_quo</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">h</span><span class="p">,</span> <span class="n">cff</span><span class="p">,</span> <span class="n">cfg</span>
-
-
-<span class="k">def</span> <span class="nf">dup_ff_prs_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes polynomial GCD using subresultants over a field.</span>
-
-<span class="sd">    Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``,</span>
-<span class="sd">    and ``cfg = quo(g, h)``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dup_ff_prs_gcd</span>
-
-<span class="sd">    &gt;&gt;&gt; f = QQ.map([1, 0, -1])</span>
-<span class="sd">    &gt;&gt;&gt; g = QQ.map([1, -3, 2])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_ff_prs_gcd(f, g, QQ)</span>
-<span class="sd">    ([1/1, -1/1], [1/1, 1/1], [1/1, -2/1])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="n">_dup_ff_trivial_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dup_subresultants</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dup_monic</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">cff</span> <span class="o">=</span> <span class="n">dup_quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">cfg</span> <span class="o">=</span> <span class="n">dup_quo</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">h</span><span class="p">,</span> <span class="n">cff</span><span class="p">,</span> <span class="n">cfg</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_rr_prs_gcd"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_rr_prs_gcd">[docs]</a><span class="k">def</span> <span class="nf">dmp_rr_prs_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes polynomial GCD using subresultants over a ring.</span>
-
-<span class="sd">    Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``,</span>
-<span class="sd">    and ``cfg = quo(g, h)``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_rr_prs_gcd</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [2, 0], [1, 0, 0]])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[1], [1, 0], []])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_rr_prs_gcd(f, g, 1, ZZ)</span>
-<span class="sd">    ([[1], [1, 0]], [[1], [1, 0]], [[1], []])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_rr_prs_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">_dmp_rr_trivial_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="n">fc</span><span class="p">,</span> <span class="n">F</span> <span class="o">=</span> <span class="n">dmp_primitive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">gc</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">dmp_primitive</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_subresultants</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-    <span class="n">c</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">dmp_rr_prs_gcd</span><span class="p">(</span><span class="n">fc</span><span class="p">,</span> <span class="n">gc</span><span class="p">,</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_negative</span><span class="p">(</span><span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)):</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_neg</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">_</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_primitive</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_mul_term</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">cff</span> <span class="o">=</span> <span class="n">dmp_quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">cfg</span> <span class="o">=</span> <span class="n">dmp_quo</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">h</span><span class="p">,</span> <span class="n">cff</span><span class="p">,</span> <span class="n">cfg</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_ff_prs_gcd"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_ff_prs_gcd">[docs]</a><span class="k">def</span> <span class="nf">dmp_ff_prs_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes polynomial GCD using subresultants over a field.</span>
-
-<span class="sd">    Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``, ``cff = quo(f, h)``,</span>
-<span class="sd">    and ``cfg = quo(g, h)``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_ff_prs_gcd</span>
-
-<span class="sd">    &gt;&gt;&gt; f = [[QQ(1,2)], [QQ(1), QQ(0)], [QQ(1,2), QQ(0), QQ(0)]]</span>
-<span class="sd">    &gt;&gt;&gt; g = [[QQ(1)], [QQ(1), QQ(0)], []]</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_ff_prs_gcd(f, g, 1, QQ)</span>
-<span class="sd">    ([[1/1], [1/1, 0/1]], [[1/2], [1/2, 0/1]], [[1/1], []])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_ff_prs_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">_dmp_ff_trivial_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="n">fc</span><span class="p">,</span> <span class="n">F</span> <span class="o">=</span> <span class="n">dmp_primitive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">gc</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">dmp_primitive</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_subresultants</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-    <span class="n">c</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">dmp_ff_prs_gcd</span><span class="p">(</span><span class="n">fc</span><span class="p">,</span> <span class="n">gc</span><span class="p">,</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">_</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_primitive</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_mul_term</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_ground_monic</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">cff</span> <span class="o">=</span> <span class="n">dmp_quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">cfg</span> <span class="o">=</span> <span class="n">dmp_quo</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">h</span><span class="p">,</span> <span class="n">cff</span><span class="p">,</span> <span class="n">cfg</span>
-</div>
-<span class="n">HEU_GCD_MAX</span> <span class="o">=</span> <span class="mi">6</span>
-
-
-<span class="k">def</span> <span class="nf">_dup_zz_gcd_interpolate</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Interpolate polynomial GCD from integer GCD. &quot;&quot;&quot;</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">while</span> <span class="n">h</span><span class="p">:</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">h</span> <span class="o">%</span> <span class="n">x</span>
-
-        <span class="k">if</span> <span class="n">g</span> <span class="o">&gt;</span> <span class="n">x</span> <span class="o">//</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">g</span> <span class="o">-=</span> <span class="n">x</span>
-
-        <span class="n">f</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="p">(</span><span class="n">h</span> <span class="o">-</span> <span class="n">g</span><span class="p">)</span> <span class="o">//</span> <span class="n">x</span>
-
-    <span class="k">return</span> <span class="n">f</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i,df,dg&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_zz_heu_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Heuristic polynomial GCD in `Z[x]`.</span>
-
-<span class="sd">    Given univariate polynomials `f` and `g` in `Z[x]`, returns</span>
-<span class="sd">    their GCD and cofactors, i.e. polynomials ``h``, ``cff`` and ``cfg``</span>
-<span class="sd">    such that::</span>
-
-<span class="sd">          h = gcd(f, g), cff = quo(f, h) and cfg = quo(g, h)</span>
-
-<span class="sd">    The algorithm is purely heuristic which means it may fail to compute</span>
-<span class="sd">    the GCD. This will be signaled by raising an exception. In this case</span>
-<span class="sd">    you will need to switch to another GCD method.</span>
-
-<span class="sd">    The algorithm computes the polynomial GCD by evaluating polynomials</span>
-<span class="sd">    f and g at certain points and computing (fast) integer GCD of those</span>
-<span class="sd">    evaluations. The polynomial GCD is recovered from the integer image</span>
-<span class="sd">    by interpolation.  The final step is to verify if the result is the</span>
-<span class="sd">    correct GCD. This gives cofactors as a side effect.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dup_zz_heu_gcd</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, -1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([1, -3, 2])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_zz_heu_gcd(f, g, ZZ)</span>
-<span class="sd">    ([1, -1], [1, 1], [1, -2])</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Liao95]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="n">_dup_rr_trivial_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="n">df</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">dg</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="n">gcd</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">dup_extract</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">df</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">dg</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">gcd</span><span class="p">],</span> <span class="n">f</span><span class="p">,</span> <span class="n">g</span>
-
-    <span class="n">f_norm</span> <span class="o">=</span> <span class="n">dup_max_norm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">g_norm</span> <span class="o">=</span> <span class="n">dup_max_norm</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">B</span> <span class="o">=</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="nb">min</span><span class="p">(</span><span class="n">f_norm</span><span class="p">,</span> <span class="n">g_norm</span><span class="p">)</span> <span class="o">+</span> <span class="mi">29</span><span class="p">)</span>
-
-    <span class="n">x</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="nb">min</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="mi">99</span><span class="o">*</span><span class="n">K</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">B</span><span class="p">)),</span>
-            <span class="mi">2</span><span class="o">*</span><span class="nb">min</span><span class="p">(</span><span class="n">f_norm</span> <span class="o">//</span> <span class="nb">abs</span><span class="p">(</span><span class="n">dup_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)),</span>
-                  <span class="n">g_norm</span> <span class="o">//</span> <span class="nb">abs</span><span class="p">(</span><span class="n">dup_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)))</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">HEU_GCD_MAX</span><span class="p">):</span>
-        <span class="n">ff</span> <span class="o">=</span> <span class="n">dup_eval</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">gg</span> <span class="o">=</span> <span class="n">dup_eval</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">ff</span> <span class="ow">and</span> <span class="n">gg</span><span class="p">:</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">ff</span><span class="p">,</span> <span class="n">gg</span><span class="p">)</span>
-
-            <span class="n">cff</span> <span class="o">=</span> <span class="n">ff</span> <span class="o">//</span> <span class="n">h</span>
-            <span class="n">cfg</span> <span class="o">=</span> <span class="n">gg</span> <span class="o">//</span> <span class="n">h</span>
-
-            <span class="n">h</span> <span class="o">=</span> <span class="n">_dup_zz_gcd_interpolate</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">dup_primitive</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-
-            <span class="n">cff_</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dup_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span><span class="p">:</span>
-                <span class="n">cfg_</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dup_div</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span><span class="p">:</span>
-                    <span class="n">h</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">gcd</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-                    <span class="k">return</span> <span class="n">h</span><span class="p">,</span> <span class="n">cff_</span><span class="p">,</span> <span class="n">cfg_</span>
-
-            <span class="n">cff</span> <span class="o">=</span> <span class="n">_dup_zz_gcd_interpolate</span><span class="p">(</span><span class="n">cff</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="n">h</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dup_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">cff</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span><span class="p">:</span>
-                <span class="n">cfg_</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dup_div</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span><span class="p">:</span>
-                    <span class="n">h</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">gcd</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-                    <span class="k">return</span> <span class="n">h</span><span class="p">,</span> <span class="n">cff</span><span class="p">,</span> <span class="n">cfg_</span>
-
-            <span class="n">cfg</span> <span class="o">=</span> <span class="n">_dup_zz_gcd_interpolate</span><span class="p">(</span><span class="n">cfg</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="n">h</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dup_div</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">cfg</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span><span class="p">:</span>
-                <span class="n">cff_</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dup_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span><span class="p">:</span>
-                    <span class="n">h</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">gcd</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-                    <span class="k">return</span> <span class="n">h</span><span class="p">,</span> <span class="n">cff_</span><span class="p">,</span> <span class="n">cfg</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="mi">73794</span><span class="o">*</span><span class="n">x</span> <span class="o">*</span> <span class="n">K</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">K</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="o">//</span> <span class="mi">27011</span>
-
-    <span class="k">raise</span> <span class="n">HeuristicGCDFailed</span><span class="p">(</span><span class="s">&#39;no luck&#39;</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;v&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_dmp_zz_gcd_interpolate</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Interpolate polynomial GCD from integer GCD. &quot;&quot;&quot;</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">while</span> <span class="ow">not</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">v</span><span class="p">):</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">dmp_ground_trunc</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">f</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_sub</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_quo_ground</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_negative</span><span class="p">(</span><span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">v</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)):</span>
-        <span class="k">return</span> <span class="n">dmp_neg</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">v</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v,i,dg,df&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_zz_heu_gcd"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_zz_heu_gcd">[docs]</a><span class="k">def</span> <span class="nf">dmp_zz_heu_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Heuristic polynomial GCD in `Z[X]`.</span>
-
-<span class="sd">    Given univariate polynomials `f` and `g` in `Z[X]`, returns</span>
-<span class="sd">    their GCD and cofactors, i.e. polynomials ``h``, ``cff`` and ``cfg``</span>
-<span class="sd">    such that::</span>
-
-<span class="sd">          h = gcd(f, g), cff = quo(f, h) and cfg = quo(g, h)</span>
-
-<span class="sd">    The algorithm is purely heuristic which means it may fail to compute</span>
-<span class="sd">    the GCD. This will be signaled by raising an exception. In this case</span>
-<span class="sd">    you will need to switch to another GCD method.</span>
-
-<span class="sd">    The algorithm computes the polynomial GCD by evaluating polynomials</span>
-<span class="sd">    f and g at certain points and computing (fast) integer GCD of those</span>
-<span class="sd">    evaluations. The polynomial GCD is recovered from the integer image</span>
-<span class="sd">    by interpolation. The evaluation proces reduces f and g variable by</span>
-<span class="sd">    variable into a large integer.  The final step is to verify if the</span>
-<span class="sd">    interpolated polynomial is the correct GCD. This gives cofactors of</span>
-<span class="sd">    the input polynomials as a side effect.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_zz_heu_gcd</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [2, 0], [1, 0, 0]])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[1], [1, 0], []])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_zz_heu_gcd(f, g, 1, ZZ)</span>
-<span class="sd">    ([[1], [1, 0]], [[1], [1, 0]], [[1], []])</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Liao95]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_zz_heu_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">_dmp_rr_trivial_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="n">gcd</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">dmp_ground_extract</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">f_norm</span> <span class="o">=</span> <span class="n">dmp_max_norm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">g_norm</span> <span class="o">=</span> <span class="n">dmp_max_norm</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">B</span> <span class="o">=</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="nb">min</span><span class="p">(</span><span class="n">f_norm</span><span class="p">,</span> <span class="n">g_norm</span><span class="p">)</span> <span class="o">+</span> <span class="mi">29</span><span class="p">)</span>
-
-    <span class="n">x</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="nb">min</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="mi">99</span><span class="o">*</span><span class="n">K</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">B</span><span class="p">)),</span>
-            <span class="mi">2</span><span class="o">*</span><span class="nb">min</span><span class="p">(</span><span class="n">f_norm</span> <span class="o">//</span> <span class="nb">abs</span><span class="p">(</span><span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)),</span>
-                  <span class="n">g_norm</span> <span class="o">//</span> <span class="nb">abs</span><span class="p">(</span><span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)))</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">HEU_GCD_MAX</span><span class="p">):</span>
-        <span class="n">ff</span> <span class="o">=</span> <span class="n">dmp_eval</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">gg</span> <span class="o">=</span> <span class="n">dmp_eval</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">ff</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="ow">or</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">gg</span><span class="p">,</span> <span class="n">v</span><span class="p">)):</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">cff</span><span class="p">,</span> <span class="n">cfg</span> <span class="o">=</span> <span class="n">dmp_zz_heu_gcd</span><span class="p">(</span><span class="n">ff</span><span class="p">,</span> <span class="n">gg</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="n">h</span> <span class="o">=</span> <span class="n">_dmp_zz_gcd_interpolate</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_ground_primitive</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-
-            <span class="n">cff_</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-                <span class="n">cfg_</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_div</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-                    <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_mul_ground</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">gcd</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-                    <span class="k">return</span> <span class="n">h</span><span class="p">,</span> <span class="n">cff_</span><span class="p">,</span> <span class="n">cfg_</span>
-
-            <span class="n">cff</span> <span class="o">=</span> <span class="n">_dmp_zz_gcd_interpolate</span><span class="p">(</span><span class="n">cff</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="n">h</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">cff</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-                <span class="n">cfg_</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_div</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-                    <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_mul_ground</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">gcd</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-                    <span class="k">return</span> <span class="n">h</span><span class="p">,</span> <span class="n">cff</span><span class="p">,</span> <span class="n">cfg_</span>
-
-            <span class="n">cfg</span> <span class="o">=</span> <span class="n">_dmp_zz_gcd_interpolate</span><span class="p">(</span><span class="n">cfg</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="n">h</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_div</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">cfg</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-                <span class="n">cff_</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-                    <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_mul_ground</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">gcd</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-                    <span class="k">return</span> <span class="n">h</span><span class="p">,</span> <span class="n">cff_</span><span class="p">,</span> <span class="n">cfg</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="mi">73794</span><span class="o">*</span><span class="n">x</span> <span class="o">*</span> <span class="n">K</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">K</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="o">//</span> <span class="mi">27011</span>
-
-    <span class="k">raise</span> <span class="n">HeuristicGCDFailed</span><span class="p">(</span><span class="s">&#39;no luck&#39;</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_qq_heu_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K0</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Heuristic polynomial GCD in `Q[x]`.</span>
-
-<span class="sd">    Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``,</span>
-<span class="sd">    ``cff = quo(f, h)``, and ``cfg = quo(g, h)``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dup_qq_heu_gcd</span>
-
-<span class="sd">    &gt;&gt;&gt; f = [QQ(1,2), QQ(7,4), QQ(3,2)]</span>
-<span class="sd">    &gt;&gt;&gt; g = [QQ(1,2), QQ(1), QQ(0)]</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_qq_heu_gcd(f, g, QQ)</span>
-<span class="sd">    ([1/1, 2/1], [1/2, 3/4], [1/2, 0/1])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="n">_dup_ff_trivial_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="n">K1</span> <span class="o">=</span> <span class="n">K0</span><span class="o">.</span><span class="n">get_ring</span><span class="p">()</span>
-
-    <span class="n">cf</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">dup_clear_denoms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">)</span>
-    <span class="n">cg</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">dup_clear_denoms</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">)</span>
-
-    <span class="n">f</span> <span class="o">=</span> <span class="n">dup_convert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">)</span>
-    <span class="n">g</span> <span class="o">=</span> <span class="n">dup_convert</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">)</span>
-
-    <span class="n">h</span><span class="p">,</span> <span class="n">cff</span><span class="p">,</span> <span class="n">cfg</span> <span class="o">=</span> <span class="n">dup_zz_heu_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K1</span><span class="p">)</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dup_convert</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">K1</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-
-    <span class="n">c</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dup_monic</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-
-    <span class="n">cff</span> <span class="o">=</span> <span class="n">dup_convert</span><span class="p">(</span><span class="n">cff</span><span class="p">,</span> <span class="n">K1</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-    <span class="n">cfg</span> <span class="o">=</span> <span class="n">dup_convert</span><span class="p">(</span><span class="n">cfg</span><span class="p">,</span> <span class="n">K1</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-
-    <span class="n">cff</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">cff</span><span class="p">,</span> <span class="n">K0</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">cf</span><span class="p">),</span> <span class="n">K0</span><span class="p">)</span>
-    <span class="n">cfg</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">cfg</span><span class="p">,</span> <span class="n">K0</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">cg</span><span class="p">),</span> <span class="n">K0</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">h</span><span class="p">,</span> <span class="n">cff</span><span class="p">,</span> <span class="n">cfg</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_qq_heu_gcd"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_qq_heu_gcd">[docs]</a><span class="k">def</span> <span class="nf">dmp_qq_heu_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Heuristic polynomial GCD in `Q[X]`.</span>
-
-<span class="sd">    Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``,</span>
-<span class="sd">    ``cff = quo(f, h)``, and ``cfg = quo(g, h)``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_qq_heu_gcd</span>
-
-<span class="sd">    &gt;&gt;&gt; f = [[QQ(1,4)], [QQ(1), QQ(0)], [QQ(1), QQ(0), QQ(0)]]</span>
-<span class="sd">    &gt;&gt;&gt; g = [[QQ(1,2)], [QQ(1), QQ(0)], []]</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_qq_heu_gcd(f, g, 1, QQ)</span>
-<span class="sd">    ([[1/1], [2/1, 0/1]], [[1/4], [1/2, 0/1]], [[1/2], []])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="n">_dmp_ff_trivial_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="n">K1</span> <span class="o">=</span> <span class="n">K0</span><span class="o">.</span><span class="n">get_ring</span><span class="p">()</span>
-
-    <span class="n">cf</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_clear_denoms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">)</span>
-    <span class="n">cg</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">dmp_clear_denoms</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">)</span>
-
-    <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_convert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">)</span>
-    <span class="n">g</span> <span class="o">=</span> <span class="n">dmp_convert</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">)</span>
-
-    <span class="n">h</span><span class="p">,</span> <span class="n">cff</span><span class="p">,</span> <span class="n">cfg</span> <span class="o">=</span> <span class="n">dmp_zz_heu_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K1</span><span class="p">)</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_convert</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K1</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-
-    <span class="n">c</span> <span class="o">=</span> <span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_ground_monic</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-
-    <span class="n">cff</span> <span class="o">=</span> <span class="n">dmp_convert</span><span class="p">(</span><span class="n">cff</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K1</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-    <span class="n">cfg</span> <span class="o">=</span> <span class="n">dmp_convert</span><span class="p">(</span><span class="n">cfg</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K1</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-
-    <span class="n">cff</span> <span class="o">=</span> <span class="n">dmp_mul_ground</span><span class="p">(</span><span class="n">cff</span><span class="p">,</span> <span class="n">K0</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">cf</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-    <span class="n">cfg</span> <span class="o">=</span> <span class="n">dmp_mul_ground</span><span class="p">(</span><span class="n">cfg</span><span class="p">,</span> <span class="n">K0</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">cg</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">h</span><span class="p">,</span> <span class="n">cff</span><span class="p">,</span> <span class="n">cfg</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_inner_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes polynomial GCD and cofactors of `f` and `g` in `K[x]`.</span>
-
-<span class="sd">    Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``,</span>
-<span class="sd">    ``cff = quo(f, h)``, and ``cfg = quo(g, h)``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dup_inner_gcd</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, -1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([1, -3, 2])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_inner_gcd(f, g, ZZ)</span>
-<span class="sd">    ([1, -1], [1, 1], [1, -2])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">has_Field</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">K</span><span class="o">.</span><span class="n">is_Exact</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_QQ</span> <span class="ow">and</span> <span class="n">query</span><span class="p">(</span><span class="s">&#39;USE_HEU_GCD&#39;</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">dup_qq_heu_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">HeuristicGCDFailed</span><span class="p">:</span>
-                <span class="k">pass</span>
-
-        <span class="k">return</span> <span class="n">dup_ff_prs_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_ZZ</span> <span class="ow">and</span> <span class="n">query</span><span class="p">(</span><span class="s">&#39;USE_HEU_GCD&#39;</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">dup_zz_heu_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">HeuristicGCDFailed</span><span class="p">:</span>
-                <span class="k">pass</span>
-
-        <span class="k">return</span> <span class="n">dup_rr_prs_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_dmp_inner_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Helper function for `dmp_inner_gcd()`. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">has_Field</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">K</span><span class="o">.</span><span class="n">is_Exact</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_QQ</span> <span class="ow">and</span> <span class="n">query</span><span class="p">(</span><span class="s">&#39;USE_HEU_GCD&#39;</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">dmp_qq_heu_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">HeuristicGCDFailed</span><span class="p">:</span>
-                <span class="k">pass</span>
-
-        <span class="k">return</span> <span class="n">dmp_ff_prs_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_ZZ</span> <span class="ow">and</span> <span class="n">query</span><span class="p">(</span><span class="s">&#39;USE_HEU_GCD&#39;</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">dmp_zz_heu_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">HeuristicGCDFailed</span><span class="p">:</span>
-                <span class="k">pass</span>
-
-        <span class="k">return</span> <span class="n">dmp_rr_prs_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_inner_gcd"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_inner_gcd">[docs]</a><span class="k">def</span> <span class="nf">dmp_inner_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes polynomial GCD and cofactors of `f` and `g` in `K[X]`.</span>
-
-<span class="sd">    Returns ``(h, cff, cfg)`` such that ``a = gcd(f, g)``,</span>
-<span class="sd">    ``cff = quo(f, h)``, and ``cfg = quo(g, h)``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_inner_gcd</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [2, 0], [1, 0, 0]])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[1], [1, 0], []])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_inner_gcd(f, g, 1, ZZ)</span>
-<span class="sd">    ([[1], [1, 0]], [[1], [1, 0]], [[1], []])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_inner_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">J</span><span class="p">,</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span> <span class="o">=</span> <span class="n">dmp_multi_deflate</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">h</span><span class="p">,</span> <span class="n">cff</span><span class="p">,</span> <span class="n">cfg</span> <span class="o">=</span> <span class="n">_dmp_inner_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="p">(</span><span class="n">dmp_inflate</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">J</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span>
-            <span class="n">dmp_inflate</span><span class="p">(</span><span class="n">cff</span><span class="p">,</span> <span class="n">J</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span>
-            <span class="n">dmp_inflate</span><span class="p">(</span><span class="n">cfg</span><span class="p">,</span> <span class="n">J</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes polynomial GCD of `f` and `g` in `K[x]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dup_gcd</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, -1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([1, -3, 2])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_gcd(f, g, ZZ)</span>
-<span class="sd">    [1, -1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dup_inner_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_gcd"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_gcd">[docs]</a><span class="k">def</span> <span class="nf">dmp_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes polynomial GCD of `f` and `g` in `K[X]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_gcd</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [2, 0], [1, 0, 0]])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[1], [1, 0], []])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_gcd(f, g, 1, ZZ)</span>
-<span class="sd">    [[1], [1, 0]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dmp_inner_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_rr_lcm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes polynomial LCM over a ring in `K[x]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dup_rr_lcm</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, -1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([1, -3, 2])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_rr_lcm(f, g, ZZ)</span>
-<span class="sd">    [1, -2, -1, 2]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">fc</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">dup_primitive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">gc</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">dup_primitive</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">c</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">lcm</span><span class="p">(</span><span class="n">fc</span><span class="p">,</span> <span class="n">gc</span><span class="p">)</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dup_quo</span><span class="p">(</span><span class="n">dup_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span>
-                <span class="n">dup_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">dup_ff_lcm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes polynomial LCM over a field in `K[x]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dup_ff_lcm</span>
-
-<span class="sd">    &gt;&gt;&gt; f = [QQ(1,2), QQ(7,4), QQ(3,2)]</span>
-<span class="sd">    &gt;&gt;&gt; g = [QQ(1,2), QQ(1), QQ(0)]</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_ff_lcm(f, g, QQ)</span>
-<span class="sd">    [1/1, 7/2, 3/1, 0/1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dup_quo</span><span class="p">(</span><span class="n">dup_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span>
-                <span class="n">dup_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">dup_monic</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">dup_lcm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes polynomial LCM of `f` and `g` in `K[x]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dup_lcm</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([1, 0, -1])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([1, -3, 2])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_lcm(f, g, ZZ)</span>
-<span class="sd">    [1, -2, -1, 2]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">has_Field</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">K</span><span class="o">.</span><span class="n">is_Exact</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_ff_lcm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_rr_lcm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dmp_rr_lcm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes polynomial LCM over a ring in `K[X]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_rr_lcm</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [2, 0], [1, 0, 0]])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[1], [1, 0], []])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_rr_lcm(f, g, 1, ZZ)</span>
-<span class="sd">    [[1], [2, 0], [1, 0, 0], []]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">fc</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_ground_primitive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">gc</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">dmp_ground_primitive</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">c</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">lcm</span><span class="p">(</span><span class="n">fc</span><span class="p">,</span> <span class="n">gc</span><span class="p">)</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_quo</span><span class="p">(</span><span class="n">dmp_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span>
-                <span class="n">dmp_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">dmp_mul_ground</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dmp_ff_lcm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes polynomial LCM over a field in `K[X]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_ff_lcm</span>
-
-<span class="sd">    &gt;&gt;&gt; f = [[QQ(1,4)], [QQ(1), QQ(0)], [QQ(1), QQ(0), QQ(0)]]</span>
-<span class="sd">    &gt;&gt;&gt; g = [[QQ(1,2)], [QQ(1), QQ(0)], []]</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_ff_lcm(f, g, 1, QQ)</span>
-<span class="sd">    [[1/1], [4/1, 0/1], [4/1, 0/1, 0/1], []]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_quo</span><span class="p">(</span><span class="n">dmp_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span>
-                <span class="n">dmp_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">dmp_ground_monic</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_lcm"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_lcm">[docs]</a><span class="k">def</span> <span class="nf">dmp_lcm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes polynomial LCM of `f` and `g` in `K[X]`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_lcm</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[1], [2, 0], [1, 0, 0]])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[1], [1, 0], []])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_lcm(f, g, 1, ZZ)</span>
-<span class="sd">    [[1], [2, 0], [1, 0, 0], []]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_lcm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">has_Field</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">K</span><span class="o">.</span><span class="n">is_Exact</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_ff_lcm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dmp_rr_lcm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_content"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_content">[docs]</a><span class="k">def</span> <span class="nf">dmp_content</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns GCD of multivariate coefficients.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_content</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[2, 6], [4, 12]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_content(f, 1, ZZ)</span>
-<span class="sd">    [2, 6]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">cont</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">dmp_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">cont</span>
-
-    <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-        <span class="n">cont</span> <span class="o">=</span> <span class="n">dmp_gcd</span><span class="p">(</span><span class="n">cont</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">dmp_one_p</span><span class="p">(</span><span class="n">cont</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-            <span class="k">break</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_negative</span><span class="p">(</span><span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">cont</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)):</span>
-        <span class="k">return</span> <span class="n">dmp_neg</span><span class="p">(</span><span class="n">cont</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">cont</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_primitive"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_primitive">[docs]</a><span class="k">def</span> <span class="nf">dmp_primitive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns multivariate content and a primitive polynomial.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_primitive</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[2, 6], [4, 12]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_primitive(f, 1, ZZ)</span>
-<span class="sd">    ([2, 6], [[1], [2]])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">cont</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">dmp_content</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span> <span class="ow">or</span> <span class="n">dmp_one_p</span><span class="p">(</span><span class="n">cont</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">cont</span><span class="p">,</span> <span class="n">f</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">cont</span><span class="p">,</span> <span class="p">[</span> <span class="n">dmp_quo</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">cont</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_cancel</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">include</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Cancel common factors in a rational function `f/g`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dup_cancel</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([2, 0, -2])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([1, -2, 1])</span>
-
-<span class="sd">    &gt;&gt;&gt; dup_cancel(f, g, ZZ)</span>
-<span class="sd">    ([2, 2], [1, -1])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dmp_cancel</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">include</span><span class="o">=</span><span class="n">include</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="dmp_cancel"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.euclidtools.dmp_cancel">[docs]</a><span class="k">def</span> <span class="nf">dmp_cancel</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">include</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Cancel common factors in a rational function `f/g`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.euclidtools import dmp_cancel</span>
-
-<span class="sd">    &gt;&gt;&gt; f = ZZ.map([[2], [0], [-2]])</span>
-<span class="sd">    &gt;&gt;&gt; g = ZZ.map([[1], [-2], [1]])</span>
-
-<span class="sd">    &gt;&gt;&gt; dmp_cancel(f, g, 1, ZZ)</span>
-<span class="sd">    ([[2], [2]], [[1], [-1]])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">K0</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">has_Field</span> <span class="ow">and</span> <span class="n">K</span><span class="o">.</span><span class="n">has_assoc_Ring</span><span class="p">:</span>
-        <span class="n">K0</span><span class="p">,</span> <span class="n">K</span> <span class="o">=</span> <span class="n">K</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">get_ring</span><span class="p">()</span>
-
-        <span class="n">cq</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_clear_denoms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">convert</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="n">cp</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">dmp_clear_denoms</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">convert</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">cp</span><span class="p">,</span> <span class="n">cq</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span>
-
-    <span class="n">_</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">dmp_inner_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">K0</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">dmp_convert</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-        <span class="n">q</span> <span class="o">=</span> <span class="n">dmp_convert</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-
-        <span class="n">K</span> <span class="o">=</span> <span class="n">K0</span>
-
-    <span class="n">p_neg</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">is_negative</span><span class="p">(</span><span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-    <span class="n">q_neg</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">is_negative</span><span class="p">(</span><span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="n">p_neg</span> <span class="ow">and</span> <span class="n">q_neg</span><span class="p">:</span>
-        <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">dmp_neg</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">dmp_neg</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">p_neg</span><span class="p">:</span>
-        <span class="n">cp</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="o">-</span><span class="n">cp</span><span class="p">,</span> <span class="n">dmp_neg</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">q_neg</span><span class="p">:</span>
-        <span class="n">cp</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="o">-</span><span class="n">cp</span><span class="p">,</span> <span class="n">dmp_neg</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">include</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">cp</span><span class="p">,</span> <span class="n">cq</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span>
-
-    <span class="n">p</span> <span class="o">=</span> <span class="n">dmp_mul_ground</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">cp</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">q</span> <span class="o">=</span> <span class="n">dmp_mul_ground</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">cq</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span></div>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/polys/factortools.html b/dev-py3k/_modules/sympy/polys/factortools.html
deleted file mode 100644
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+++ /dev/null
@@ -1,1470 +0,0 @@
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-            
-  <h1>Source code for sympy.polys.factortools</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Polynomial factorization routines in characteristic zero. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">gf_from_int_poly</span><span class="p">,</span> <span class="n">gf_to_int_poly</span><span class="p">,</span>
-    <span class="n">gf_lshift</span><span class="p">,</span> <span class="n">gf_add_mul</span><span class="p">,</span> <span class="n">gf_mul</span><span class="p">,</span>
-    <span class="n">gf_div</span><span class="p">,</span> <span class="n">gf_rem</span><span class="p">,</span>
-    <span class="n">gf_gcdex</span><span class="p">,</span>
-    <span class="n">gf_sqf_p</span><span class="p">,</span>
-    <span class="n">gf_factor_sqf</span><span class="p">,</span> <span class="n">gf_factor</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dup_LC</span><span class="p">,</span> <span class="n">dmp_LC</span><span class="p">,</span> <span class="n">dmp_ground_LC</span><span class="p">,</span>
-    <span class="n">dup_TC</span><span class="p">,</span>
-    <span class="n">dup_convert</span><span class="p">,</span> <span class="n">dmp_convert</span><span class="p">,</span>
-    <span class="n">dup_degree</span><span class="p">,</span> <span class="n">dmp_degree</span><span class="p">,</span>
-    <span class="n">dmp_degree_in</span><span class="p">,</span> <span class="n">dmp_degree_list</span><span class="p">,</span>
-    <span class="n">dmp_from_dict</span><span class="p">,</span>
-    <span class="n">dmp_zero_p</span><span class="p">,</span>
-    <span class="n">dmp_one</span><span class="p">,</span>
-    <span class="n">dmp_nest</span><span class="p">,</span> <span class="n">dmp_raise</span><span class="p">,</span>
-    <span class="n">dup_strip</span><span class="p">,</span>
-    <span class="n">dmp_ground</span><span class="p">,</span>
-    <span class="n">dup_inflate</span><span class="p">,</span>
-    <span class="n">dmp_exclude</span><span class="p">,</span> <span class="n">dmp_include</span><span class="p">,</span>
-    <span class="n">dmp_inject</span><span class="p">,</span> <span class="n">dmp_eject</span><span class="p">,</span>
-    <span class="n">dup_terms_gcd</span><span class="p">,</span> <span class="n">dmp_terms_gcd</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dup_neg</span><span class="p">,</span> <span class="n">dmp_neg</span><span class="p">,</span>
-    <span class="n">dup_add</span><span class="p">,</span> <span class="n">dmp_add</span><span class="p">,</span>
-    <span class="n">dup_sub</span><span class="p">,</span> <span class="n">dmp_sub</span><span class="p">,</span>
-    <span class="n">dup_mul</span><span class="p">,</span> <span class="n">dmp_mul</span><span class="p">,</span>
-    <span class="n">dup_sqr</span><span class="p">,</span>
-    <span class="n">dmp_pow</span><span class="p">,</span>
-    <span class="n">dup_div</span><span class="p">,</span> <span class="n">dmp_div</span><span class="p">,</span>
-    <span class="n">dup_quo</span><span class="p">,</span> <span class="n">dmp_quo</span><span class="p">,</span>
-    <span class="n">dmp_expand</span><span class="p">,</span>
-    <span class="n">dmp_add_mul</span><span class="p">,</span>
-    <span class="n">dup_sub_mul</span><span class="p">,</span> <span class="n">dmp_sub_mul</span><span class="p">,</span>
-    <span class="n">dup_lshift</span><span class="p">,</span>
-    <span class="n">dup_max_norm</span><span class="p">,</span> <span class="n">dmp_max_norm</span><span class="p">,</span>
-    <span class="n">dup_l1_norm</span><span class="p">,</span>
-    <span class="n">dup_mul_ground</span><span class="p">,</span> <span class="n">dmp_mul_ground</span><span class="p">,</span>
-    <span class="n">dmp_quo_ground</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dup_clear_denoms</span><span class="p">,</span> <span class="n">dmp_clear_denoms</span><span class="p">,</span>
-    <span class="n">dup_trunc</span><span class="p">,</span> <span class="n">dmp_ground_trunc</span><span class="p">,</span>
-    <span class="n">dup_content</span><span class="p">,</span>
-    <span class="n">dup_monic</span><span class="p">,</span> <span class="n">dmp_ground_monic</span><span class="p">,</span>
-    <span class="n">dup_primitive</span><span class="p">,</span> <span class="n">dmp_ground_primitive</span><span class="p">,</span>
-    <span class="n">dmp_eval_tail</span><span class="p">,</span>
-    <span class="n">dmp_eval_in</span><span class="p">,</span> <span class="n">dmp_diff_eval_in</span><span class="p">,</span>
-    <span class="n">dmp_compose</span><span class="p">,</span>
-    <span class="n">dup_shift</span><span class="p">,</span> <span class="n">dup_mirror</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dmp_primitive</span><span class="p">,</span>
-    <span class="n">dup_inner_gcd</span><span class="p">,</span> <span class="n">dmp_inner_gcd</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.sqfreetools</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dup_sqf_p</span><span class="p">,</span>
-    <span class="n">dup_sqf_norm</span><span class="p">,</span> <span class="n">dmp_sqf_norm</span><span class="p">,</span>
-    <span class="n">dup_sqf_part</span><span class="p">,</span> <span class="n">dmp_sqf_part</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyutils</span> <span class="kn">import</span> <span class="n">_sort_factors</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polyconfig</span> <span class="kn">import</span> <span class="n">query</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">ExtraneousFactors</span><span class="p">,</span> <span class="n">DomainError</span><span class="p">,</span> <span class="n">CoercionFailed</span><span class="p">,</span> <span class="n">EvaluationFailed</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">nextprime</span><span class="p">,</span> <span class="n">isprime</span><span class="p">,</span> <span class="n">factorint</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">subsets</span><span class="p">,</span> <span class="n">cythonized</span>
-
-<span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">ceil</span> <span class="k">as</span> <span class="n">_ceil</span><span class="p">,</span> <span class="n">log</span> <span class="k">as</span> <span class="n">_log</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_trial_division</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">factors</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Determine multiplicities of factors using trial division. &quot;&quot;&quot;</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">factor</span> <span class="ow">in</span> <span class="n">factors</span><span class="p">:</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="mi">0</span>
-
-        <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dup_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">factor</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span><span class="p">:</span>
-                <span class="n">f</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="n">q</span><span class="p">,</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">break</span>
-
-        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">factor</span><span class="p">,</span> <span class="n">k</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">_sort_factors</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,k&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_trial_division"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dmp_trial_division">[docs]</a><span class="k">def</span> <span class="nf">dmp_trial_division</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">factors</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Determine multiplicities of factors using trial division. &quot;&quot;&quot;</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">factor</span> <span class="ow">in</span> <span class="n">factors</span><span class="p">:</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="mi">0</span>
-
-        <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">factor</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-                <span class="n">f</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="n">q</span><span class="p">,</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">break</span>
-
-        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">factor</span><span class="p">,</span> <span class="n">k</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">_sort_factors</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_zz_mignotte_bound</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Mignotte bound for univariate polynomials in `K[x]`. &quot;&quot;&quot;</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">dup_max_norm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">b</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">dup_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">K</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span><span class="o">*</span><span class="mi">2</span><span class="o">**</span><span class="n">n</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">b</span>
-
-
-<div class="viewcode-block" id="dmp_zz_mignotte_bound"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dmp_zz_mignotte_bound">[docs]</a><span class="k">def</span> <span class="nf">dmp_zz_mignotte_bound</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Mignotte bound for multivariate polynomials in `K[X]`. &quot;&quot;&quot;</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">dmp_max_norm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">b</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">dmp_degree_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">K</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span><span class="o">*</span><span class="mi">2</span><span class="o">**</span><span class="n">n</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">b</span>
-
-</div>
-<div class="viewcode-block" id="dup_zz_hensel_step"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dup_zz_hensel_step">[docs]</a><span class="k">def</span> <span class="nf">dup_zz_hensel_step</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    One step in Hensel lifting in `Z[x]`.</span>
-
-<span class="sd">    Given positive integer `m` and `Z[x]` polynomials `f`, `g`, `h`, `s`</span>
-<span class="sd">    and `t` such that::</span>
-
-<span class="sd">        f == g*h (mod m)</span>
-<span class="sd">        s*g + t*h == 1 (mod m)</span>
-
-<span class="sd">        lc(f) is not a zero divisor (mod m)</span>
-<span class="sd">        lc(h) == 1</span>
-
-<span class="sd">        deg(f) == deg(g) + deg(h)</span>
-<span class="sd">        deg(s) &lt; deg(h)</span>
-<span class="sd">        deg(t) &lt; deg(g)</span>
-
-<span class="sd">    returns polynomials `G`, `H`, `S` and `T`, such that::</span>
-
-<span class="sd">        f == G*H (mod m**2)</span>
-<span class="sd">        S*G + T**H == 1 (mod m**2)</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Gathen99]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">M</span> <span class="o">=</span> <span class="n">m</span><span class="o">**</span><span class="mi">2</span>
-
-    <span class="n">e</span> <span class="o">=</span> <span class="n">dup_sub_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">e</span> <span class="o">=</span> <span class="n">dup_trunc</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dup_div</span><span class="p">(</span><span class="n">dup_mul</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">q</span> <span class="o">=</span> <span class="n">dup_trunc</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">dup_trunc</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">u</span> <span class="o">=</span> <span class="n">dup_add</span><span class="p">(</span><span class="n">dup_mul</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">dup_mul</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">G</span> <span class="o">=</span> <span class="n">dup_trunc</span><span class="p">(</span><span class="n">dup_add</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">M</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">H</span> <span class="o">=</span> <span class="n">dup_trunc</span><span class="p">(</span><span class="n">dup_add</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">M</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">u</span> <span class="o">=</span> <span class="n">dup_add</span><span class="p">(</span><span class="n">dup_mul</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">dup_mul</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">b</span> <span class="o">=</span> <span class="n">dup_trunc</span><span class="p">(</span><span class="n">dup_sub</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">],</span> <span class="n">K</span><span class="p">),</span> <span class="n">M</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">c</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">dup_div</span><span class="p">(</span><span class="n">dup_mul</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">H</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">c</span> <span class="o">=</span> <span class="n">dup_trunc</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">d</span> <span class="o">=</span> <span class="n">dup_trunc</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">u</span> <span class="o">=</span> <span class="n">dup_add</span><span class="p">(</span><span class="n">dup_mul</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">dup_mul</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">S</span> <span class="o">=</span> <span class="n">dup_trunc</span><span class="p">(</span><span class="n">dup_sub</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">M</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">T</span> <span class="o">=</span> <span class="n">dup_trunc</span><span class="p">(</span><span class="n">dup_sub</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">M</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">G</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">T</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;l,r,k,d&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dup_zz_hensel_lift"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dup_zz_hensel_lift">[docs]</a><span class="k">def</span> <span class="nf">dup_zz_hensel_lift</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">f_list</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Multifactor Hensel lifting in `Z[x]`.</span>
-
-<span class="sd">    Given a prime `p`, polynomial `f` over `Z[x]` such that `lc(f)`</span>
-<span class="sd">    is a unit modulo `p`, monic pair-wise coprime polynomials `f_i`</span>
-<span class="sd">    over `Z[x]` satisfying::</span>
-
-<span class="sd">        f = lc(f) f_1 ... f_r (mod p)</span>
-
-<span class="sd">    and a positive integer `l`, returns a list of monic polynomials</span>
-<span class="sd">    `F_1`, `F_2`, ..., `F_r` satisfying::</span>
-
-<span class="sd">       f = lc(f) F_1 ... F_r (mod p**l)</span>
-
-<span class="sd">       F_i = f_i (mod p), i = 1..r</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Gathen99]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">f_list</span><span class="p">)</span>
-    <span class="n">lc</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">r</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">F</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">gcdex</span><span class="p">(</span><span class="n">lc</span><span class="p">,</span> <span class="n">p</span><span class="o">**</span><span class="n">l</span><span class="p">)[</span><span class="mi">0</span><span class="p">],</span> <span class="n">K</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="n">dup_trunc</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">p</span><span class="o">**</span><span class="n">l</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="p">]</span>
-
-    <span class="n">m</span> <span class="o">=</span> <span class="n">p</span>
-    <span class="n">k</span> <span class="o">=</span> <span class="n">r</span> <span class="o">//</span> <span class="mi">2</span>
-    <span class="n">d</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">_ceil</span><span class="p">(</span><span class="n">_log</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="mi">2</span><span class="p">)))</span>
-
-    <span class="n">g</span> <span class="o">=</span> <span class="n">gf_from_int_poly</span><span class="p">([</span><span class="n">lc</span><span class="p">],</span> <span class="n">p</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">f_i</span> <span class="ow">in</span> <span class="n">f_list</span><span class="p">[:</span><span class="n">k</span><span class="p">]:</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">gf_mul</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">gf_from_int_poly</span><span class="p">(</span><span class="n">f_i</span><span class="p">,</span> <span class="n">p</span><span class="p">),</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="n">gf_from_int_poly</span><span class="p">(</span><span class="n">f_list</span><span class="p">[</span><span class="n">k</span><span class="p">],</span> <span class="n">p</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">f_i</span> <span class="ow">in</span> <span class="n">f_list</span><span class="p">[</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]:</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">gf_mul</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">gf_from_int_poly</span><span class="p">(</span><span class="n">f_i</span><span class="p">,</span> <span class="n">p</span><span class="p">),</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">gf_gcdex</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">g</span> <span class="o">=</span> <span class="n">gf_to_int_poly</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">gf_to_int_poly</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="n">gf_to_int_poly</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-    <span class="n">t</span> <span class="o">=</span> <span class="n">gf_to_int_poly</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">d</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">),</span> <span class="n">m</span> <span class="o">=</span> <span class="n">dup_zz_hensel_step</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">m</span><span class="o">**</span><span class="mi">2</span>
-
-    <span class="k">return</span> <span class="n">dup_zz_hensel_lift</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">f_list</span><span class="p">[:</span><span class="n">k</span><span class="p">],</span> <span class="n">l</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> \
-        <span class="o">+</span> <span class="n">dup_zz_hensel_lift</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">f_list</span><span class="p">[</span><span class="n">k</span><span class="p">:],</span> <span class="n">l</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;l,s&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dup_zz_zassenhaus"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dup_zz_zassenhaus">[docs]</a><span class="k">def</span> <span class="nf">dup_zz_zassenhaus</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Factor primitive square-free polynomials in `Z[x]`. &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">f</span><span class="p">]</span>
-
-    <span class="n">A</span> <span class="o">=</span> <span class="n">dup_max_norm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">b</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">B</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">K</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">K</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span><span class="o">*</span><span class="mi">2</span><span class="o">**</span><span class="n">n</span><span class="o">*</span><span class="n">A</span><span class="o">*</span><span class="n">b</span><span class="p">))</span>
-    <span class="n">C</span> <span class="o">=</span> <span class="nb">int</span><span class="p">((</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">A</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-    <span class="n">gamma</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">_ceil</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">_log</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="mi">2</span><span class="p">)))</span>
-    <span class="n">bound</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">gamma</span><span class="o">*</span><span class="n">_log</span><span class="p">(</span><span class="n">gamma</span><span class="p">))</span>
-
-    <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">bound</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">isprime</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="ow">or</span> <span class="n">b</span> <span class="o">%</span> <span class="n">p</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">continue</span>
-
-        <span class="n">p</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-
-        <span class="n">F</span> <span class="o">=</span> <span class="n">gf_from_int_poly</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">gf_sqf_p</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-            <span class="k">break</span>
-
-    <span class="n">l</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">_ceil</span><span class="p">(</span><span class="n">_log</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">B</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">p</span><span class="p">)))</span>
-
-    <span class="n">modular</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">ff</span> <span class="ow">in</span> <span class="n">gf_factor_sqf</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">1</span><span class="p">]:</span>
-        <span class="n">modular</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">gf_to_int_poly</span><span class="p">(</span><span class="n">ff</span><span class="p">,</span> <span class="n">p</span><span class="p">))</span>
-
-    <span class="n">g</span> <span class="o">=</span> <span class="n">dup_zz_hensel_lift</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">modular</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">sorted_T</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">g</span><span class="p">)))</span>
-    <span class="n">T</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">sorted_T</span><span class="p">)</span>
-    <span class="n">factors</span><span class="p">,</span> <span class="n">s</span> <span class="o">=</span> <span class="p">[],</span> <span class="mi">1</span>
-
-    <span class="k">while</span> <span class="mi">2</span><span class="o">*</span><span class="n">s</span> <span class="o">&lt;=</span> <span class="nb">len</span><span class="p">(</span><span class="n">T</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">S</span> <span class="ow">in</span> <span class="n">subsets</span><span class="p">(</span><span class="n">sorted_T</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-            <span class="n">G</span><span class="p">,</span> <span class="n">H</span> <span class="o">=</span> <span class="p">[</span><span class="n">b</span><span class="p">],</span> <span class="p">[</span><span class="n">b</span><span class="p">]</span>
-
-            <span class="n">S</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">S</span><span class="p">)</span>
-            <span class="n">T_S</span> <span class="o">=</span> <span class="n">T</span> <span class="o">-</span> <span class="n">S</span>
-
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">S</span><span class="p">:</span>
-                <span class="n">G</span> <span class="o">=</span> <span class="n">dup_mul</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">T_S</span><span class="p">:</span>
-                <span class="n">H</span> <span class="o">=</span> <span class="n">dup_mul</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">g</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="n">G</span> <span class="o">=</span> <span class="n">dup_trunc</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">p</span><span class="o">**</span><span class="n">l</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">H</span> <span class="o">=</span> <span class="n">dup_trunc</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">p</span><span class="o">**</span><span class="n">l</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="n">G_norm</span> <span class="o">=</span> <span class="n">dup_l1_norm</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">H_norm</span> <span class="o">=</span> <span class="n">dup_l1_norm</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">G_norm</span><span class="o">*</span><span class="n">H_norm</span> <span class="o">&lt;=</span> <span class="n">B</span><span class="p">:</span>
-                <span class="n">T</span> <span class="o">=</span> <span class="n">T_S</span>
-                <span class="n">sorted_T</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">sorted_T</span> <span class="k">if</span> <span class="n">i</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">S</span><span class="p">]</span>
-
-                <span class="n">G</span> <span class="o">=</span> <span class="n">dup_primitive</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-                <span class="n">f</span> <span class="o">=</span> <span class="n">dup_primitive</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-
-                <span class="n">factors</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">s</span> <span class="o">+=</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="n">factors</span> <span class="o">+</span> <span class="p">[</span><span class="n">f</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="dup_zz_irreducible_p"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dup_zz_irreducible_p">[docs]</a><span class="k">def</span> <span class="nf">dup_zz_irreducible_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Test irreducibility using Eisenstein&#39;s criterion. &quot;&quot;&quot;</span>
-    <span class="n">lc</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">tc</span> <span class="o">=</span> <span class="n">dup_TC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">e_fc</span> <span class="o">=</span> <span class="n">dup_content</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">e_fc</span><span class="p">:</span>
-        <span class="n">e_ff</span> <span class="o">=</span> <span class="n">factorint</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">e_fc</span><span class="p">))</span>
-
-        <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">e_ff</span><span class="o">.</span><span class="n">keys</span><span class="p">():</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">lc</span> <span class="o">%</span> <span class="n">p</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">tc</span> <span class="o">%</span> <span class="n">p</span><span class="o">**</span><span class="mi">2</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">True</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,i&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dup_cyclotomic_p"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dup_cyclotomic_p">[docs]</a><span class="k">def</span> <span class="nf">dup_cyclotomic_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">irreducible</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Efficiently test if ``f`` is a cyclotomic polnomial.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.factortools import dup_cyclotomic_p</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-
-<span class="sd">    &gt;&gt;&gt; f = [1, 0, 1, 0, 0, 0,-1, 0, 1, 0,-1, 0, 0, 0, 1, 0, 1]</span>
-<span class="sd">    &gt;&gt;&gt; dup_cyclotomic_p(f, ZZ)</span>
-<span class="sd">    False</span>
-
-<span class="sd">    &gt;&gt;&gt; g = [1, 0, 1, 0, 0, 0,-1, 0,-1, 0,-1, 0, 0, 0, 1, 0, 1]</span>
-<span class="sd">    &gt;&gt;&gt; dup_cyclotomic_p(g, ZZ)</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_QQ</span><span class="p">:</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">K0</span><span class="p">,</span> <span class="n">K</span> <span class="o">=</span> <span class="n">K</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">get_ring</span><span class="p">()</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">dup_convert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">CoercionFailed</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-    <span class="k">elif</span> <span class="ow">not</span> <span class="n">K</span><span class="o">.</span><span class="n">is_ZZ</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="n">lc</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">tc</span> <span class="o">=</span> <span class="n">dup_TC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">lc</span> <span class="o">!=</span> <span class="mi">1</span> <span class="ow">or</span> <span class="p">(</span><span class="n">tc</span> <span class="o">!=</span> <span class="o">-</span><span class="mi">1</span> <span class="ow">and</span> <span class="n">tc</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">irreducible</span><span class="p">:</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">dup_factor_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">coeff</span> <span class="o">!=</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span> <span class="ow">or</span> <span class="n">factors</span> <span class="o">!=</span> <span class="p">[(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">g</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">):</span>
-        <span class="n">g</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">f</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">):</span>
-        <span class="n">h</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">f</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-
-    <span class="n">g</span> <span class="o">=</span> <span class="n">dup_sqr</span><span class="p">(</span><span class="n">dup_strip</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dup_sqr</span><span class="p">(</span><span class="n">dup_strip</span><span class="p">(</span><span class="n">h</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">F</span> <span class="o">=</span> <span class="n">dup_sub</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">dup_lshift</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_negative</span><span class="p">(</span><span class="n">dup_LC</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">K</span><span class="p">)):</span>
-        <span class="n">F</span> <span class="o">=</span> <span class="n">dup_neg</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">F</span> <span class="o">==</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="n">g</span> <span class="o">=</span> <span class="n">dup_mirror</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_negative</span><span class="p">(</span><span class="n">dup_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)):</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">dup_neg</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">F</span> <span class="o">==</span> <span class="n">g</span> <span class="ow">and</span> <span class="n">dup_cyclotomic_p</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="n">G</span> <span class="o">=</span> <span class="n">dup_sqf_part</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dup_sqr</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">==</span> <span class="n">F</span> <span class="ow">and</span> <span class="n">dup_cyclotomic_p</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">return</span> <span class="bp">False</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,p,k&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dup_zz_cyclotomic_poly"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dup_zz_cyclotomic_poly">[docs]</a><span class="k">def</span> <span class="nf">dup_zz_cyclotomic_poly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Efficiently generate n-th cyclotomic polnomial. &quot;&quot;&quot;</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="o">-</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]</span>
-
-    <span class="k">for</span> <span class="n">p</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">factorint</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dup_quo</span><span class="p">(</span><span class="n">dup_inflate</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dup_inflate</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="o">**</span><span class="p">(</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">h</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,p,k,i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">_dup_cyclotomic_decompose</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="n">H</span> <span class="o">=</span> <span class="p">[[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="o">-</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]]</span>
-
-    <span class="k">for</span> <span class="n">p</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">factorint</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="n">Q</span> <span class="o">=</span> <span class="p">[</span> <span class="n">dup_quo</span><span class="p">(</span><span class="n">dup_inflate</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">H</span> <span class="p">]</span>
-        <span class="n">H</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">Q</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-            <span class="n">Q</span> <span class="o">=</span> <span class="p">[</span> <span class="n">dup_inflate</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">Q</span> <span class="p">]</span>
-            <span class="n">H</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">Q</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">H</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dup_zz_cyclotomic_factor"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dup_zz_cyclotomic_factor">[docs]</a><span class="k">def</span> <span class="nf">dup_zz_cyclotomic_factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Efficiently factor polynomials `x**n - 1` and `x**n + 1` in `Z[x]`.</span>
-
-<span class="sd">    Given a univariate polynomial `f` in `Z[x]` returns a list of factors</span>
-<span class="sd">    of `f`, provided that `f` is in the form `x**n - 1` or `x**n + 1` for</span>
-<span class="sd">    `n &gt;= 1`. Otherwise returns None.</span>
-
-<span class="sd">    Factorization is performed using using cyclotomic decomposition of `f`,</span>
-<span class="sd">    which makes this method much faster that any other direct factorization</span>
-<span class="sd">    approach (e.g. Zassenhaus&#39;s).</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Weisstein09]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">lc_f</span><span class="p">,</span> <span class="n">tc_f</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">dup_TC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">if</span> <span class="n">lc_f</span> <span class="o">!=</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">tc_f</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="nb">bool</span><span class="p">(</span><span class="n">cf</span><span class="p">)</span> <span class="k">for</span> <span class="n">cf</span> <span class="ow">in</span> <span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]):</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">F</span> <span class="o">=</span> <span class="n">_dup_cyclotomic_decompose</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">K</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">tc_f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">F</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">H</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">_dup_cyclotomic_decompose</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">h</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">F</span><span class="p">:</span>
-                <span class="n">H</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">H</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dup_zz_factor_sqf"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dup_zz_factor_sqf">[docs]</a><span class="k">def</span> <span class="nf">dup_zz_factor_sqf</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Factor square-free (non-primitive) polyomials in `Z[x]`. &quot;&quot;&quot;</span>
-    <span class="n">cont</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">dup_primitive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">cont</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="o">-</span><span class="n">cont</span><span class="p">,</span> <span class="n">dup_neg</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">cont</span><span class="p">,</span> <span class="p">[]</span>
-    <span class="k">elif</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">cont</span><span class="p">,</span> <span class="p">[(</span><span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]</span>
-
-    <span class="k">if</span> <span class="n">query</span><span class="p">(</span><span class="s">&#39;USE_IRREDUCIBLE_IN_FACTOR&#39;</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">dup_zz_irreducible_p</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">cont</span><span class="p">,</span> <span class="p">[(</span><span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]</span>
-
-    <span class="n">factors</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="k">if</span> <span class="n">query</span><span class="p">(</span><span class="s">&#39;USE_CYCLOTOMIC_FACTOR&#39;</span><span class="p">):</span>
-        <span class="n">factors</span> <span class="o">=</span> <span class="n">dup_zz_cyclotomic_factor</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">factors</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">factors</span> <span class="o">=</span> <span class="n">dup_zz_zassenhaus</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">cont</span><span class="p">,</span> <span class="n">_sort_factors</span><span class="p">(</span><span class="n">factors</span><span class="p">,</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,k&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dup_zz_factor"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dup_zz_factor">[docs]</a><span class="k">def</span> <span class="nf">dup_zz_factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Factor (non square-free) polynomials in `Z[x]`.</span>
-
-<span class="sd">    Given a univariate polynomial `f` in `Z[x]` computes its complete</span>
-<span class="sd">    factorization `f_1, ..., f_n` into irreducibles over integers::</span>
-
-<span class="sd">                f = content(f) f_1**k_1 ... f_n**k_n</span>
-
-<span class="sd">    The factorization is computed by reducing the input polynomial</span>
-<span class="sd">    into a primitive square-free polynomial and factoring it using</span>
-<span class="sd">    Zassenhaus algorithm. Trial division is used to recover the</span>
-<span class="sd">    multiplicities of factors.</span>
-
-<span class="sd">    The result is returned as a tuple consisting of::</span>
-
-<span class="sd">              (content(f), [(f_1, k_1), ..., (f_n, k_n))</span>
-
-<span class="sd">    Consider polynomial `f = 2*x**4 - 2`::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.polys.factortools import dup_zz_factor</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-
-<span class="sd">        &gt;&gt;&gt; dup_zz_factor([2, 0, 0, 0, -2], ZZ)</span>
-<span class="sd">        (2, [([1, -1], 1), ([1, 1], 1), ([1, 0, 1], 1)])</span>
-
-<span class="sd">    In result we got the following factorization::</span>
-
-<span class="sd">                 f = 2 (x - 1) (x + 1) (x**2 + 1)</span>
-
-<span class="sd">    Note that this is a complete factorization over integers,</span>
-<span class="sd">    however over Gaussian integers we can factor the last term.</span>
-
-<span class="sd">    By default, polynomials `x**n - 1` and `x**n + 1` are factored</span>
-<span class="sd">    using cyclotomic decomposition to speedup computations. To</span>
-<span class="sd">    disable this behaviour set cyclotomic=False.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Gathen99]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">cont</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">dup_primitive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">cont</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="o">-</span><span class="n">cont</span><span class="p">,</span> <span class="n">dup_neg</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">cont</span><span class="p">,</span> <span class="p">[]</span>
-    <span class="k">elif</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">cont</span><span class="p">,</span> <span class="p">[(</span><span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]</span>
-
-    <span class="k">if</span> <span class="n">query</span><span class="p">(</span><span class="s">&#39;USE_IRREDUCIBLE_IN_FACTOR&#39;</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">dup_zz_irreducible_p</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">cont</span><span class="p">,</span> <span class="p">[(</span><span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]</span>
-
-    <span class="n">g</span> <span class="o">=</span> <span class="n">dup_sqf_part</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">H</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="bp">None</span><span class="p">,</span> <span class="p">[]</span>
-
-    <span class="k">if</span> <span class="n">query</span><span class="p">(</span><span class="s">&#39;USE_CYCLOTOMIC_FACTOR&#39;</span><span class="p">):</span>
-        <span class="n">H</span> <span class="o">=</span> <span class="n">dup_zz_cyclotomic_factor</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">H</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">H</span> <span class="o">=</span> <span class="n">dup_zz_zassenhaus</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">H</span><span class="p">:</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="mi">0</span>
-
-        <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dup_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span><span class="p">:</span>
-                <span class="n">f</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="n">q</span><span class="p">,</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">break</span>
-
-        <span class="n">factors</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">h</span><span class="p">,</span> <span class="n">k</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">cont</span><span class="p">,</span> <span class="n">_sort_factors</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="dmp_zz_wang_non_divisors"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dmp_zz_wang_non_divisors">[docs]</a><span class="k">def</span> <span class="nf">dmp_zz_wang_non_divisors</span><span class="p">(</span><span class="n">E</span><span class="p">,</span> <span class="n">cs</span><span class="p">,</span> <span class="n">ct</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Wang/EEZ: Compute a set of valid divisors.  &quot;&quot;&quot;</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[</span> <span class="n">cs</span><span class="o">*</span><span class="n">ct</span> <span class="p">]</span>
-
-    <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">E</span><span class="p">:</span>
-        <span class="n">q</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">result</span><span class="p">):</span>
-            <span class="k">while</span> <span class="n">r</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">r</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
-                <span class="n">q</span> <span class="o">=</span> <span class="n">q</span> <span class="o">//</span> <span class="n">r</span>
-
-            <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">q</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">result</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_zz_wang_test_points"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dmp_zz_wang_test_points">[docs]</a><span class="k">def</span> <span class="nf">dmp_zz_wang_test_points</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">ct</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Wang/EEZ: Test evaluation points for suitability. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">dmp_eval_tail</span><span class="p">(</span><span class="n">dmp_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">A</span><span class="p">,</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="n">EvaluationFailed</span><span class="p">(</span><span class="s">&#39;no luck&#39;</span><span class="p">)</span>
-
-    <span class="n">g</span> <span class="o">=</span> <span class="n">dmp_eval_tail</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">dup_sqf_p</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="n">EvaluationFailed</span><span class="p">(</span><span class="s">&#39;no luck&#39;</span><span class="p">)</span>
-
-    <span class="n">c</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">dup_primitive</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_negative</span><span class="p">(</span><span class="n">dup_LC</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)):</span>
-        <span class="n">c</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="o">-</span><span class="n">c</span><span class="p">,</span> <span class="n">dup_neg</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="n">E</span> <span class="o">=</span> <span class="p">[</span> <span class="n">dmp_eval_tail</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">T</span> <span class="p">]</span>
-    <span class="n">D</span> <span class="o">=</span> <span class="n">dmp_zz_wang_non_divisors</span><span class="p">(</span><span class="n">E</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">ct</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">D</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">c</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">E</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">EvaluationFailed</span><span class="p">(</span><span class="s">&#39;no luck&#39;</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v,i,j,k&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_zz_wang_lead_coeffs"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dmp_zz_wang_lead_coeffs">[docs]</a><span class="k">def</span> <span class="nf">dmp_zz_wang_lead_coeffs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">cs</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Wang/EEZ: Compute correct leading coefficients. &quot;&quot;&quot;</span>
-    <span class="n">C</span><span class="p">,</span> <span class="n">J</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="n">E</span><span class="p">),</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">H</span><span class="p">:</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">dmp_one</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span><span class="o">*</span><span class="n">cs</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">E</span><span class="p">))):</span>
-            <span class="n">k</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">_</span><span class="p">)</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="n">E</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">T</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-
-            <span class="k">while</span> <span class="ow">not</span> <span class="p">(</span><span class="n">d</span> <span class="o">%</span> <span class="n">e</span><span class="p">):</span>
-                <span class="n">d</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="n">d</span><span class="o">//</span><span class="n">e</span><span class="p">,</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span>
-
-            <span class="k">if</span> <span class="n">k</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">c</span><span class="p">,</span> <span class="n">J</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">dmp_pow</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="mi">1</span>
-
-        <span class="n">C</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="ow">not</span> <span class="n">j</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">J</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="n">ExtraneousFactors</span>  <span class="c"># pragma: no cover</span>
-
-    <span class="n">CC</span><span class="p">,</span> <span class="n">HH</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">c</span><span class="p">,</span> <span class="n">h</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">H</span><span class="p">):</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">dmp_eval_tail</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">lc</span> <span class="o">=</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">cs</span><span class="p">):</span>
-            <span class="n">cc</span> <span class="o">=</span> <span class="n">lc</span><span class="o">//</span><span class="n">d</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">lc</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
-            <span class="n">d</span><span class="p">,</span> <span class="n">cc</span> <span class="o">=</span> <span class="n">d</span><span class="o">//</span><span class="n">g</span><span class="p">,</span> <span class="n">lc</span><span class="o">//</span><span class="n">g</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">cs</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">cs</span><span class="o">//</span><span class="n">d</span>
-
-        <span class="n">c</span> <span class="o">=</span> <span class="n">dmp_mul_ground</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">cc</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">CC</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-        <span class="n">HH</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">cs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="p">,</span> <span class="n">HH</span><span class="p">,</span> <span class="n">CC</span>
-
-    <span class="n">CCC</span><span class="p">,</span> <span class="n">HHH</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">c</span><span class="p">,</span> <span class="n">h</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">CC</span><span class="p">,</span> <span class="n">HH</span><span class="p">):</span>
-        <span class="n">CCC</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dmp_mul_ground</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">cs</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-        <span class="n">HHH</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dmp_mul_ground</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">cs</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-
-    <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_mul_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">cs</span><span class="o">**</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">H</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">f</span><span class="p">,</span> <span class="n">HHH</span><span class="p">,</span> <span class="n">CCC</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;m&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_zz_diophantine</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Wang/EEZ: Solve univariate Diophantine equations. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">F</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">F</span>
-
-        <span class="n">f</span> <span class="o">=</span> <span class="n">gf_from_int_poly</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">gf_from_int_poly</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-
-        <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">gf_gcdex</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">s</span> <span class="o">=</span> <span class="n">gf_lshift</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">gf_lshift</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">q</span><span class="p">,</span> <span class="n">s</span> <span class="o">=</span> <span class="n">gf_div</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">t</span> <span class="o">=</span> <span class="n">gf_add_mul</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">s</span> <span class="o">=</span> <span class="n">gf_to_int_poly</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">gf_to_int_poly</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[</span><span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">G</span> <span class="o">=</span> <span class="p">[</span><span class="n">F</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span>
-
-        <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">F</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]):</span>
-            <span class="n">G</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">dup_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">G</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">K</span><span class="p">))</span>
-
-        <span class="n">S</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[[</span><span class="mi">1</span><span class="p">]]</span>
-
-        <span class="k">for</span> <span class="n">f</span><span class="p">,</span> <span class="n">g</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">):</span>
-            <span class="n">t</span><span class="p">,</span> <span class="n">s</span> <span class="o">=</span> <span class="n">dmp_zz_diophantine</span><span class="p">([</span><span class="n">g</span><span class="p">,</span> <span class="n">f</span><span class="p">],</span> <span class="n">T</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">[],</span> <span class="mi">0</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">T</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-            <span class="n">S</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-
-        <span class="n">result</span><span class="p">,</span> <span class="n">S</span> <span class="o">=</span> <span class="p">[],</span> <span class="n">S</span> <span class="o">+</span> <span class="p">[</span><span class="n">T</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span>
-
-        <span class="k">for</span> <span class="n">s</span><span class="p">,</span> <span class="n">f</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">F</span><span class="p">):</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">gf_from_int_poly</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">gf_from_int_poly</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-
-            <span class="n">r</span> <span class="o">=</span> <span class="n">gf_rem</span><span class="p">(</span><span class="n">gf_lshift</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">gf_to_int_poly</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v,d,n,i,j,k&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_zz_diophantine"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dmp_zz_diophantine">[docs]</a><span class="k">def</span> <span class="nf">dmp_zz_diophantine</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Wang/EEZ: Solve multivariate Diophantine equations. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">A</span><span class="p">:</span>
-        <span class="n">S</span> <span class="o">=</span> <span class="p">[</span> <span class="p">[]</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">F</span> <span class="p">]</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">coeff</span><span class="p">:</span>
-                <span class="k">continue</span>
-
-            <span class="n">T</span> <span class="o">=</span> <span class="n">dup_zz_diophantine</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">i</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">T</span><span class="p">))):</span>
-                <span class="n">t</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-                <span class="n">S</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">dup_trunc</span><span class="p">(</span><span class="n">dup_add</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-        <span class="n">e</span> <span class="o">=</span> <span class="n">dmp_expand</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">a</span><span class="p">,</span> <span class="n">A</span> <span class="o">=</span> <span class="n">A</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">A</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">B</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">F</span><span class="p">:</span>
-            <span class="n">B</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dmp_quo</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-            <span class="n">G</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dmp_eval_in</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-
-        <span class="n">C</span> <span class="o">=</span> <span class="n">dmp_eval_in</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">v</span> <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="n">S</span> <span class="o">=</span> <span class="n">dmp_zz_diophantine</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">S</span> <span class="o">=</span> <span class="p">[</span> <span class="n">dmp_raise</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">S</span> <span class="p">]</span>
-
-        <span class="k">for</span> <span class="n">s</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">B</span><span class="p">):</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">dmp_sub_mul</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">c</span> <span class="o">=</span> <span class="n">dmp_ground_trunc</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">m</span> <span class="o">=</span> <span class="n">dmp_nest</span><span class="p">([</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="o">-</span><span class="n">a</span><span class="p">],</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">M</span> <span class="o">=</span> <span class="n">dmp_one</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">K</span><span class="o">.</span><span class="n">map</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">d</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-                <span class="k">break</span>
-
-            <span class="n">M</span> <span class="o">=</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">C</span> <span class="o">=</span> <span class="n">dmp_diff_eval_in</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">v</span><span class="p">):</span>
-                <span class="n">C</span> <span class="o">=</span> <span class="n">dmp_quo_ground</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-                <span class="n">T</span> <span class="o">=</span> <span class="n">dmp_zz_diophantine</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-                <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">T</span><span class="p">):</span>
-                    <span class="n">T</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">dmp_raise</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">M</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-                <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">T</span><span class="p">))):</span>
-                    <span class="n">S</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">dmp_add</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-                <span class="k">for</span> <span class="n">t</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">B</span><span class="p">):</span>
-                    <span class="n">c</span> <span class="o">=</span> <span class="n">dmp_sub_mul</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-                <span class="n">c</span> <span class="o">=</span> <span class="n">dmp_ground_trunc</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">S</span> <span class="o">=</span> <span class="p">[</span> <span class="n">dmp_ground_trunc</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">S</span> <span class="p">]</span>
-
-    <span class="k">return</span> <span class="n">S</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v,d,dj,n,i,j,k,w&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_zz_wang_hensel_lifting"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dmp_zz_wang_hensel_lifting">[docs]</a><span class="k">def</span> <span class="nf">dmp_zz_wang_hensel_lifting</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">LC</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Wang/EEZ: Parallel Hensel lifting algorithm. &quot;&quot;&quot;</span>
-    <span class="n">S</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="p">],</span> <span class="nb">len</span><span class="p">(</span><span class="n">A</span><span class="p">),</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="n">H</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">H</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">A</span><span class="p">[</span><span class="mi">1</span><span class="p">:])):</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">dmp_eval_in</span><span class="p">(</span><span class="n">S</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">a</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">i</span><span class="p">,</span> <span class="n">u</span> <span class="o">-</span> <span class="n">i</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">S</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">dmp_ground_trunc</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">v</span> <span class="o">-</span> <span class="n">i</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-
-    <span class="n">d</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">dmp_degree_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)[</span><span class="mi">1</span><span class="p">:])</span>
-
-    <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">2</span><span class="p">),</span> <span class="n">S</span><span class="p">,</span> <span class="n">A</span><span class="p">):</span>
-        <span class="n">G</span><span class="p">,</span> <span class="n">w</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">H</span><span class="p">),</span> <span class="n">j</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="n">I</span><span class="p">,</span> <span class="n">J</span> <span class="o">=</span> <span class="n">A</span><span class="p">[:</span><span class="n">j</span> <span class="o">-</span> <span class="mi">2</span><span class="p">],</span> <span class="n">A</span><span class="p">[</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:]</span>
-
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">lc</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">LC</span><span class="p">))):</span>
-            <span class="n">lc</span> <span class="o">=</span> <span class="n">dmp_ground_trunc</span><span class="p">(</span><span class="n">dmp_eval_tail</span><span class="p">(</span><span class="n">lc</span><span class="p">,</span> <span class="n">J</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">p</span><span class="p">,</span> <span class="n">w</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">H</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">lc</span><span class="p">]</span> <span class="o">+</span> <span class="n">dmp_raise</span><span class="p">(</span><span class="n">h</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">w</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">m</span> <span class="o">=</span> <span class="n">dmp_nest</span><span class="p">([</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="o">-</span><span class="n">a</span><span class="p">],</span> <span class="n">w</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">M</span> <span class="o">=</span> <span class="n">dmp_one</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">c</span> <span class="o">=</span> <span class="n">dmp_sub</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">dmp_expand</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">w</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">dj</span> <span class="o">=</span> <span class="n">dmp_degree_in</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">w</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">K</span><span class="o">.</span><span class="n">map</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">dj</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">w</span><span class="p">):</span>
-                <span class="k">break</span>
-
-            <span class="n">M</span> <span class="o">=</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">C</span> <span class="o">=</span> <span class="n">dmp_diff_eval_in</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">w</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="n">C</span> <span class="o">=</span> <span class="n">dmp_quo_ground</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">w</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-                <span class="n">T</span> <span class="o">=</span> <span class="n">dmp_zz_diophantine</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">w</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-                <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">T</span><span class="p">))):</span>
-                    <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_add_mul</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">dmp_raise</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">w</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">M</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-                    <span class="n">H</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">dmp_ground_trunc</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-                <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_sub</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">dmp_expand</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">w</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="n">dmp_ground_trunc</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dmp_expand</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">!=</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ExtraneousFactors</span>  <span class="c"># pragma: no cover</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">H</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,mod,i,j,s_arg,negative&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_zz_wang"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dmp_zz_wang">[docs]</a><span class="k">def</span> <span class="nf">dmp_zz_wang</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">mod</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Factor primitive square-free polynomials in `Z[X]`.</span>
-
-<span class="sd">    Given a multivariate polynomial `f` in `Z[x_1,...,x_n]`, which is</span>
-<span class="sd">    primitive and square-free in `x_1`, computes factorization of `f` into</span>
-<span class="sd">    irreducibles over integers.</span>
-
-<span class="sd">    The procedure is based on Wang&#39;s Enhanced Extended Zassenhaus</span>
-<span class="sd">    algorithm. The algorithm works by viewing `f` as a univariate polynomial</span>
-<span class="sd">    in `Z[x_2,...,x_n][x_1]`, for which an evaluation mapping is computed::</span>
-
-<span class="sd">                      x_2 -&gt; a_2, ..., x_n -&gt; a_n</span>
-
-<span class="sd">    where `a_i`, for `i = 2, ..., n`, are carefully chosen integers.  The</span>
-<span class="sd">    mapping is used to transform `f` into a univariate polynomial in `Z[x_1]`,</span>
-<span class="sd">    which can be factored efficiently using Zassenhaus algorithm. The last</span>
-<span class="sd">    step is to lift univariate factors to obtain true multivariate</span>
-<span class="sd">    factors. For this purpose a parallel Hensel lifting procedure is used.</span>
-
-<span class="sd">    The parameter ``seed`` is passed to _randint and can be used to seed randint</span>
-<span class="sd">    (when an integer) or (for testing purposes) can be a sequence of numbers.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Wang78]_</span>
-<span class="sd">    2. [Geddes92]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.utilities.randtest</span> <span class="kn">import</span> <span class="n">_randint</span>
-
-    <span class="n">randint</span> <span class="o">=</span> <span class="n">_randint</span><span class="p">(</span><span class="n">seed</span><span class="p">)</span>
-
-    <span class="n">ct</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="n">dmp_zz_factor</span><span class="p">(</span><span class="n">dmp_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">b</span> <span class="o">=</span> <span class="n">dmp_zz_mignotte_bound</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="n">K</span><span class="p">(</span><span class="n">nextprime</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="n">mod</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">u</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">mod</span> <span class="o">=</span> <span class="mi">2</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">mod</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="n">history</span><span class="p">,</span> <span class="n">configs</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([]),</span> <span class="p">[],</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span><span class="o">*</span><span class="n">u</span><span class="p">,</span> <span class="bp">None</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">cs</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">E</span> <span class="o">=</span> <span class="n">dmp_zz_wang_test_points</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">ct</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">_</span><span class="p">,</span> <span class="n">H</span> <span class="o">=</span> <span class="n">dup_zz_factor_sqf</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">r</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">H</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">r</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[</span><span class="n">f</span><span class="p">]</span>
-
-        <span class="n">configs</span> <span class="o">=</span> <span class="p">[(</span><span class="n">s</span><span class="p">,</span> <span class="n">cs</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">A</span><span class="p">)]</span>
-    <span class="k">except</span> <span class="n">EvaluationFailed</span><span class="p">:</span>
-        <span class="k">pass</span>
-
-    <span class="n">eez_num_configs</span> <span class="o">=</span> <span class="n">query</span><span class="p">(</span><span class="s">&#39;EEZ_NUMBER_OF_CONFIGS&#39;</span><span class="p">)</span>
-    <span class="n">eez_num_tries</span> <span class="o">=</span> <span class="n">query</span><span class="p">(</span><span class="s">&#39;EEZ_NUMBER_OF_TRIES&#39;</span><span class="p">)</span>
-    <span class="n">eez_mod_step</span> <span class="o">=</span> <span class="n">query</span><span class="p">(</span><span class="s">&#39;EEZ_MODULUS_STEP&#39;</span><span class="p">)</span>
-
-    <span class="k">while</span> <span class="nb">len</span><span class="p">(</span><span class="n">configs</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">eez_num_configs</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">eez_num_tries</span><span class="p">):</span>
-            <span class="n">A</span> <span class="o">=</span> <span class="p">[</span> <span class="n">K</span><span class="p">(</span><span class="n">randint</span><span class="p">(</span><span class="o">-</span><span class="n">mod</span><span class="p">,</span> <span class="n">mod</span><span class="p">))</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">u</span><span class="p">)</span> <span class="p">]</span>
-
-            <span class="k">if</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">A</span><span class="p">)</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">history</span><span class="p">:</span>
-                <span class="n">history</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="nb">tuple</span><span class="p">(</span><span class="n">A</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">continue</span>
-
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">cs</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">E</span> <span class="o">=</span> <span class="n">dmp_zz_wang_test_points</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">ct</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">EvaluationFailed</span><span class="p">:</span>
-                <span class="k">continue</span>
-
-            <span class="n">_</span><span class="p">,</span> <span class="n">H</span> <span class="o">=</span> <span class="n">dup_zz_factor_sqf</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="n">rr</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">H</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">r</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">rr</span> <span class="o">!=</span> <span class="n">r</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-                    <span class="k">if</span> <span class="n">rr</span> <span class="o">&lt;</span> <span class="n">r</span><span class="p">:</span>
-                        <span class="n">configs</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="p">[],</span> <span class="n">rr</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">continue</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">r</span> <span class="o">=</span> <span class="n">rr</span>
-
-            <span class="k">if</span> <span class="n">r</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">f</span><span class="p">]</span>
-
-            <span class="n">configs</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">s</span><span class="p">,</span> <span class="n">cs</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">A</span><span class="p">))</span>
-
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">configs</span><span class="p">)</span> <span class="o">==</span> <span class="n">eez_num_configs</span><span class="p">:</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">mod</span> <span class="o">+=</span> <span class="n">eez_mod_step</span>
-
-    <span class="n">s_norm</span><span class="p">,</span> <span class="n">s_arg</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="bp">None</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span>
-
-    <span class="k">for</span> <span class="n">s</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">configs</span><span class="p">:</span>
-        <span class="n">_s_norm</span> <span class="o">=</span> <span class="n">dup_max_norm</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">s_norm</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">_s_norm</span> <span class="o">&lt;</span> <span class="n">s_norm</span><span class="p">:</span>
-                <span class="n">s_norm</span> <span class="o">=</span> <span class="n">_s_norm</span>
-                <span class="n">s_arg</span> <span class="o">=</span> <span class="n">i</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">s_norm</span> <span class="o">=</span> <span class="n">_s_norm</span>
-
-        <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-
-    <span class="n">_</span><span class="p">,</span> <span class="n">cs</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">A</span> <span class="o">=</span> <span class="n">configs</span><span class="p">[</span><span class="n">s_arg</span><span class="p">]</span>
-    <span class="n">orig_f</span> <span class="o">=</span> <span class="n">f</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">f</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">LC</span> <span class="o">=</span> <span class="n">dmp_zz_wang_lead_coeffs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">cs</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">factors</span> <span class="o">=</span> <span class="n">dmp_zz_wang_hensel_lifting</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">LC</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">ExtraneousFactors</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-        <span class="k">if</span> <span class="n">query</span><span class="p">(</span><span class="s">&#39;EEZ_RESTART_IF_NEEDED&#39;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">dmp_zz_wang</span><span class="p">(</span><span class="n">orig_f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">mod</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ExtraneousFactors</span><span class="p">(</span>
-                <span class="s">&quot;we need to restart algorithm with better parameters&quot;</span><span class="p">)</span>
-
-    <span class="n">negative</span><span class="p">,</span> <span class="n">result</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">factors</span><span class="p">:</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_ground_primitive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_negative</span><span class="p">(</span><span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)):</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_neg</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,d,k&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_zz_factor"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dmp_zz_factor">[docs]</a><span class="k">def</span> <span class="nf">dmp_zz_factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Factor (non square-free) polynomials in `Z[X]`.</span>
-
-<span class="sd">    Given a multivariate polynomial `f` in `Z[x]` computes its complete</span>
-<span class="sd">    factorization `f_1, ..., f_n` into irreducibles over integers::</span>
-
-<span class="sd">                 f = content(f) f_1**k_1 ... f_n**k_n</span>
-
-<span class="sd">    The factorization is computed by reducing the input polynomial</span>
-<span class="sd">    into a primitive square-free polynomial and factoring it using</span>
-<span class="sd">    Enhanced Extended Zassenhaus (EEZ) algorithm. Trial division</span>
-<span class="sd">    is used to recover the multiplicities of factors.</span>
-
-<span class="sd">    The result is returned as a tuple consisting of::</span>
-
-<span class="sd">             (content(f), [(f_1, k_1), ..., (f_n, k_n))</span>
-
-<span class="sd">    Consider polynomial `f = 2*(x**2 - y**2)`::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.polys.factortools import dmp_zz_factor</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-
-<span class="sd">        &gt;&gt;&gt; dmp_zz_factor([[2], [], [-2, 0, 0]], 1, ZZ)</span>
-<span class="sd">        (2, [([[1], [-1, 0]], 1), ([[1], [1, 0]], 1)])</span>
-
-<span class="sd">    In result we got the following factorization::</span>
-
-<span class="sd">                    f = 2 (x - y) (x + y)</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Gathen99]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_zz_factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">,</span> <span class="p">[]</span>
-
-    <span class="n">cont</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">dmp_ground_primitive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">cont</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="o">-</span><span class="n">cont</span><span class="p">,</span> <span class="n">dmp_neg</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">d</span> <span class="o">&lt;=</span> <span class="mi">0</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">dmp_degree_list</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">)):</span>
-        <span class="k">return</span> <span class="n">cont</span><span class="p">,</span> <span class="p">[]</span>
-
-    <span class="n">G</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">dmp_primitive</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">factors</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">if</span> <span class="n">dmp_degree</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">dmp_sqf_part</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">H</span> <span class="o">=</span> <span class="n">dmp_zz_wang</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">H</span><span class="p">:</span>
-            <span class="n">k</span> <span class="o">=</span> <span class="mi">0</span>
-
-            <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-                <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-                    <span class="n">f</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="n">q</span><span class="p">,</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">break</span>
-
-            <span class="n">factors</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">h</span><span class="p">,</span> <span class="n">k</span><span class="p">))</span>
-
-    <span class="k">for</span> <span class="n">g</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">dmp_zz_factor</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">1</span><span class="p">]:</span>
-        <span class="n">factors</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="p">([</span><span class="n">g</span><span class="p">],</span> <span class="n">k</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">cont</span><span class="p">,</span> <span class="n">_sort_factors</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_ext_factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Factor univariate polynomials over algebraic number fields. &quot;&quot;&quot;</span>
-    <span class="n">n</span><span class="p">,</span> <span class="n">lc</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">f</span> <span class="o">=</span> <span class="n">dup_monic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">lc</span><span class="p">,</span> <span class="p">[]</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">lc</span><span class="p">,</span> <span class="p">[(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]</span>
-
-    <span class="n">f</span><span class="p">,</span> <span class="n">F</span> <span class="o">=</span> <span class="n">dup_sqf_part</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">f</span>
-    <span class="n">s</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dup_sqf_norm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">factors</span> <span class="o">=</span> <span class="n">dup_factor_list_include</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">lc</span><span class="p">,</span> <span class="p">[(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="o">//</span><span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="p">))]</span>
-
-    <span class="n">H</span> <span class="o">=</span> <span class="n">s</span><span class="o">*</span><span class="n">K</span><span class="o">.</span><span class="n">unit</span>
-
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">factor</span><span class="p">,</span> <span class="n">_</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">factors</span><span class="p">):</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dup_convert</span><span class="p">(</span><span class="n">factor</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">h</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">dup_inner_gcd</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">dup_shift</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">factors</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">h</span>
-
-    <span class="n">factors</span> <span class="o">=</span> <span class="n">dup_trial_division</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">factors</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">lc</span><span class="p">,</span> <span class="n">factors</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_ext_factor"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dmp_ext_factor">[docs]</a><span class="k">def</span> <span class="nf">dmp_ext_factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Factor multivariate polynomials over algebraic number fields. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_ext_factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">lc</span> <span class="o">=</span> <span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_ground_monic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">d</span> <span class="o">&lt;=</span> <span class="mi">0</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">dmp_degree_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)):</span>
-        <span class="k">return</span> <span class="n">lc</span><span class="p">,</span> <span class="p">[]</span>
-
-    <span class="n">f</span><span class="p">,</span> <span class="n">F</span> <span class="o">=</span> <span class="n">dmp_sqf_part</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">f</span>
-    <span class="n">s</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_sqf_norm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">factors</span> <span class="o">=</span> <span class="n">dmp_factor_list_include</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">lc</span><span class="p">,</span> <span class="p">[</span><span class="n">f</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">H</span> <span class="o">=</span> <span class="n">dmp_raise</span><span class="p">([</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">s</span><span class="o">*</span><span class="n">K</span><span class="o">.</span><span class="n">unit</span><span class="p">],</span> <span class="n">u</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">factor</span><span class="p">,</span> <span class="n">_</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">factors</span><span class="p">):</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_convert</span><span class="p">(</span><span class="n">factor</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">dmp_inner_gcd</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_compose</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">factors</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">h</span>
-
-    <span class="k">return</span> <span class="n">lc</span><span class="p">,</span> <span class="n">dmp_trial_division</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">factors</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dup_gf_factor"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dup_gf_factor">[docs]</a><span class="k">def</span> <span class="nf">dup_gf_factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Factor univariate polynomials over finite fields. &quot;&quot;&quot;</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">dup_convert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-
-    <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">gf_factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">mod</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">factors</span><span class="p">):</span>
-        <span class="n">factors</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">dup_convert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">k</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">dom</span><span class="p">),</span> <span class="n">factors</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dmp_gf_factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Factor multivariate polynomials over finite fields. &quot;&quot;&quot;</span>
-    <span class="k">raise</span> <span class="n">DomainError</span><span class="p">(</span><span class="s">&#39;multivariate polynomials over </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">K</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i,k,u&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_factor_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K0</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Factor polynomials into irreducibles in `K[x]`. &quot;&quot;&quot;</span>
-    <span class="n">j</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">dup_terms_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">K0</span><span class="o">.</span><span class="n">has_CharacteristicZero</span><span class="p">:</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">dup_gf_factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">K0</span><span class="o">.</span><span class="n">is_Algebraic</span><span class="p">:</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">dup_ext_factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">K0</span><span class="o">.</span><span class="n">is_Exact</span><span class="p">:</span>
-            <span class="n">K0_inexact</span><span class="p">,</span> <span class="n">K0</span> <span class="o">=</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K0</span><span class="o">.</span><span class="n">get_exact</span><span class="p">()</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">dup_convert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K0_inexact</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">K0_inexact</span> <span class="o">=</span> <span class="bp">None</span>
-
-        <span class="k">if</span> <span class="n">K0</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-            <span class="n">K</span> <span class="o">=</span> <span class="n">K0</span><span class="o">.</span><span class="n">get_ring</span><span class="p">()</span>
-
-            <span class="n">denom</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">dup_clear_denoms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">dup_convert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">K</span> <span class="o">=</span> <span class="n">K0</span>
-
-        <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_ZZ</span><span class="p">:</span>
-            <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">dup_zz_factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">K</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="n">f</span><span class="p">,</span> <span class="n">u</span> <span class="o">=</span> <span class="n">dmp_inject</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">dmp_factor_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">factors</span><span class="p">):</span>
-                <span class="n">factors</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">dmp_eject</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">k</span><span class="p">)</span>
-
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">DomainError</span><span class="p">(</span><span class="s">&#39;factorization not supported over </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">K0</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">K0</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">factors</span><span class="p">):</span>
-                <span class="n">factors</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">dup_convert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">K0</span><span class="p">),</span> <span class="n">k</span><span class="p">)</span>
-
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">K0</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">denom</span> <span class="o">=</span> <span class="n">K0</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">denom</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">K0</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">denom</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">K0_inexact</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">factors</span><span class="p">):</span>
-                <span class="n">factors</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">dup_convert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K0_inexact</span><span class="p">),</span> <span class="n">k</span><span class="p">)</span>
-
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">K0_inexact</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">j</span><span class="p">:</span>
-        <span class="n">factors</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="p">([</span><span class="n">K0</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">K0</span><span class="o">.</span><span class="n">zero</span><span class="p">],</span> <span class="n">j</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">_sort_factors</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">dup_factor_list_include</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Factor polynomials into irreducibles in `K[x]`. &quot;&quot;&quot;</span>
-    <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">dup_factor_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">factors</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[(</span><span class="n">dup_strip</span><span class="p">([</span><span class="n">coeff</span><span class="p">]),</span> <span class="mi">1</span><span class="p">)]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">factors</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">[(</span><span class="n">g</span><span class="p">,</span> <span class="n">factors</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">])]</span> <span class="o">+</span> <span class="n">factors</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u,v,i,k&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_factor_list"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dmp_factor_list">[docs]</a><span class="k">def</span> <span class="nf">dmp_factor_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Factor polynomials into irreducibles in `K[X]`. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_factor_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-
-    <span class="n">J</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_terms_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">K0</span><span class="o">.</span><span class="n">has_CharacteristicZero</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">dmp_gf_factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">K0</span><span class="o">.</span><span class="n">is_Algebraic</span><span class="p">:</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">dmp_ext_factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">K0</span><span class="o">.</span><span class="n">is_Exact</span><span class="p">:</span>
-            <span class="n">K0_inexact</span><span class="p">,</span> <span class="n">K0</span> <span class="o">=</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K0</span><span class="o">.</span><span class="n">get_exact</span><span class="p">()</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_convert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0_inexact</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">K0_inexact</span> <span class="o">=</span> <span class="bp">None</span>
-
-        <span class="k">if</span> <span class="n">K0</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-            <span class="n">K</span> <span class="o">=</span> <span class="n">K0</span><span class="o">.</span><span class="n">get_ring</span><span class="p">()</span>
-
-            <span class="n">denom</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_clear_denoms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_convert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">K</span> <span class="o">=</span> <span class="n">K0</span>
-
-        <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_ZZ</span><span class="p">:</span>
-            <span class="n">levels</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">dmp_exclude</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">dmp_zz_factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">factors</span><span class="p">):</span>
-                <span class="n">factors</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">dmp_include</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">levels</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">k</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">K</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="n">f</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">dmp_inject</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">dmp_factor_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">factors</span><span class="p">):</span>
-                <span class="n">factors</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">dmp_eject</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">k</span><span class="p">)</span>
-
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">DomainError</span><span class="p">(</span><span class="s">&#39;factorization not supported over </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">K0</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">K0</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">factors</span><span class="p">):</span>
-                <span class="n">factors</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">dmp_convert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">K0</span><span class="p">),</span> <span class="n">k</span><span class="p">)</span>
-
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">K0</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">denom</span> <span class="o">=</span> <span class="n">K0</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">denom</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">K0</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">denom</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">K0_inexact</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">factors</span><span class="p">):</span>
-                <span class="n">factors</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">dmp_convert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K0_inexact</span><span class="p">),</span> <span class="n">k</span><span class="p">)</span>
-
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">K0_inexact</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">J</span><span class="p">)):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">j</span><span class="p">:</span>
-            <span class="k">continue</span>
-
-        <span class="n">term</span> <span class="o">=</span> <span class="p">{(</span><span class="mi">0</span><span class="p">,)</span><span class="o">*</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span><span class="p">,)</span> <span class="o">+</span> <span class="p">(</span><span class="mi">0</span><span class="p">,)</span><span class="o">*</span><span class="n">i</span><span class="p">:</span> <span class="n">K0</span><span class="o">.</span><span class="n">one</span><span class="p">}</span>
-        <span class="n">factors</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="p">(</span><span class="n">dmp_from_dict</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K0</span><span class="p">),</span> <span class="n">j</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">_sort_factors</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;u&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="dmp_factor_list_include"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dmp_factor_list_include">[docs]</a><span class="k">def</span> <span class="nf">dmp_factor_list_include</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Factor polynomials into irreducibles in `K[X]`. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">u</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dup_factor_list_include</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">dmp_factor_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">factors</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[(</span><span class="n">dmp_ground</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">u</span><span class="p">),</span> <span class="mi">1</span><span class="p">)]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">dmp_mul_ground</span><span class="p">(</span><span class="n">factors</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">[(</span><span class="n">g</span><span class="p">,</span> <span class="n">factors</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">])]</span> <span class="o">+</span> <span class="n">factors</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-</div>
-<span class="k">def</span> <span class="nf">dup_irreducible_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns ``True`` if ``f`` has no factors over its domain. &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">dmp_irreducible_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="dmp_irreducible_p"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.factortools.dmp_irreducible_p">[docs]</a><span class="k">def</span> <span class="nf">dmp_irreducible_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns ``True`` if ``f`` has no factors over its domain. &quot;&quot;&quot;</span>
-    <span class="n">_</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">dmp_factor_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">factors</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">True</span>
-    <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">False</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="n">factors</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">k</span> <span class="o">==</span> <span class="mi">1</span></div>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/polys/galoistools.html b/dev-py3k/_modules/sympy/polys/galoistools.html
deleted file mode 100644
index 8289edf25c3..00000000000
--- a/dev-py3k/_modules/sympy/polys/galoistools.html
+++ /dev/null
@@ -1,2406 +0,0 @@
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-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
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-  <h1>Source code for sympy.polys.galoistools</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Dense univariate polynomials with coefficients in Galois fields. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">uniform</span>
-<span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">ceil</span> <span class="k">as</span> <span class="n">_ceil</span><span class="p">,</span> <span class="n">sqrt</span> <span class="k">as</span> <span class="n">_sqrt</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.mul</span> <span class="kn">import</span> <span class="n">prod</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polyutils</span> <span class="kn">import</span> <span class="n">_sort_factors</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polyconfig</span> <span class="kn">import</span> <span class="n">query</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="n">ExactQuotientFailed</span>
-
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">cythonized</span>
-
-<span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">factorint</span>
-
-
-<div class="viewcode-block" id="gf_crt"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_crt">[docs]</a><span class="k">def</span> <span class="nf">gf_crt</span><span class="p">(</span><span class="n">U</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">K</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Chinese Remainder Theorem.</span>
-
-<span class="sd">    Given a set of integer residues ``u_0,...,u_n`` and a set of</span>
-<span class="sd">    co-prime integer moduli ``m_0,...,m_n``, returns an integer</span>
-<span class="sd">    ``u``, such that ``u = u_i mod m_i`` for ``i = ``0,...,n``.</span>
-
-<span class="sd">    As an example consider a set of residues ``U = [49, 76, 65]``</span>
-<span class="sd">    and a set of moduli ``M = [99, 97, 95]``. Then we have::</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">       &gt;&gt;&gt; from sympy.polys.galoistools import gf_crt</span>
-<span class="sd">       &gt;&gt;&gt; from sympy.ntheory.modular import solve_congruence</span>
-
-<span class="sd">       &gt;&gt;&gt; gf_crt([49, 76, 65], [99, 97, 95], ZZ)</span>
-<span class="sd">       639985</span>
-
-<span class="sd">    This is the correct result because::</span>
-
-<span class="sd">       &gt;&gt;&gt; [639985 % m for m in [99, 97, 95]]</span>
-<span class="sd">       [49, 76, 65]</span>
-
-<span class="sd">    Note: this is a low-level routine with no error checking.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.ntheory.modular.crt : a higher level crt routine</span>
-<span class="sd">    sympy.ntheory.modular.solve_congruence</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="n">prod</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">start</span><span class="o">=</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">)</span>
-    <span class="n">v</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-
-    <span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">m</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">U</span><span class="p">,</span> <span class="n">M</span><span class="p">):</span>
-        <span class="n">e</span> <span class="o">=</span> <span class="n">p</span> <span class="o">//</span> <span class="n">m</span>
-        <span class="n">s</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">gcdex</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-        <span class="n">v</span> <span class="o">+=</span> <span class="n">e</span><span class="o">*</span><span class="p">(</span><span class="n">u</span><span class="o">*</span><span class="n">s</span> <span class="o">%</span> <span class="n">m</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">v</span> <span class="o">%</span> <span class="n">p</span>
-
-</div>
-<div class="viewcode-block" id="gf_crt1"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_crt1">[docs]</a><span class="k">def</span> <span class="nf">gf_crt1</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    First part of the Chinese Remainder Theorem.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_crt1</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_crt1([99, 97, 95], ZZ)</span>
-<span class="sd">    (912285, [9215, 9405, 9603], [62, 24, 12])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">E</span><span class="p">,</span> <span class="n">S</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="n">prod</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">start</span><span class="o">=</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">M</span><span class="p">:</span>
-        <span class="n">E</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">p</span> <span class="o">//</span> <span class="n">m</span><span class="p">)</span>
-        <span class="n">S</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">K</span><span class="o">.</span><span class="n">gcdex</span><span class="p">(</span><span class="n">E</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">m</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span> <span class="o">%</span> <span class="n">m</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">p</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">S</span>
-
-</div>
-<div class="viewcode-block" id="gf_crt2"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_crt2">[docs]</a><span class="k">def</span> <span class="nf">gf_crt2</span><span class="p">(</span><span class="n">U</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Second part of the Chinese Remainder Theorem.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_crt2</span>
-
-<span class="sd">    &gt;&gt;&gt; U = [49, 76, 65]</span>
-<span class="sd">    &gt;&gt;&gt; M = [99, 97, 95]</span>
-<span class="sd">    &gt;&gt;&gt; p = 912285</span>
-<span class="sd">    &gt;&gt;&gt; E = [9215, 9405, 9603]</span>
-<span class="sd">    &gt;&gt;&gt; S = [62, 24, 12]</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_crt2(U, M, p, E, S, ZZ)</span>
-<span class="sd">    639985</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">v</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-
-    <span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">s</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">U</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">S</span><span class="p">):</span>
-        <span class="n">v</span> <span class="o">+=</span> <span class="n">e</span><span class="o">*</span><span class="p">(</span><span class="n">u</span><span class="o">*</span><span class="n">s</span> <span class="o">%</span> <span class="n">m</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">v</span> <span class="o">%</span> <span class="n">p</span>
-
-</div>
-<div class="viewcode-block" id="gf_int"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_int">[docs]</a><span class="k">def</span> <span class="nf">gf_int</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Coerce ``a mod p`` to an integer in the range ``[-p/2, p/2]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_int</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_int(2, 7)</span>
-<span class="sd">    2</span>
-<span class="sd">    &gt;&gt;&gt; gf_int(5, 7)</span>
-<span class="sd">    -2</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">a</span> <span class="o">&lt;=</span> <span class="n">p</span> <span class="o">//</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">a</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">a</span> <span class="o">-</span> <span class="n">p</span>
-
-</div>
-<div class="viewcode-block" id="gf_degree"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_degree">[docs]</a><span class="k">def</span> <span class="nf">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the leading degree of ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_degree</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_degree([1, 1, 2, 0])</span>
-<span class="sd">    3</span>
-<span class="sd">    &gt;&gt;&gt; gf_degree([])</span>
-<span class="sd">    -1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-
-</div>
-<div class="viewcode-block" id="gf_LC"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_LC">[docs]</a><span class="k">def</span> <span class="nf">gf_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the leading coefficient of ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_LC</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_LC([3, 0, 1], ZZ)</span>
-<span class="sd">    3</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="gf_TC"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_TC">[docs]</a><span class="k">def</span> <span class="nf">gf_TC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the trailing coefficient of ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_TC</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_TC([3, 0, 1], ZZ)</span>
-<span class="sd">    1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_strip"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_strip">[docs]</a><span class="k">def</span> <span class="nf">gf_strip</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Remove leading zeros from ``f``.</span>
-
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_strip</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_strip([0, 0, 0, 3, 0, 1])</span>
-<span class="sd">    [3, 0, 1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span> <span class="ow">or</span> <span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">k</span> <span class="o">=</span> <span class="mi">0</span>
-
-    <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">coeff</span><span class="p">:</span>
-            <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">k</span> <span class="o">+=</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="n">f</span><span class="p">[</span><span class="n">k</span><span class="p">:]</span>
-
-</div>
-<div class="viewcode-block" id="gf_trunc"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_trunc">[docs]</a><span class="k">def</span> <span class="nf">gf_trunc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Reduce all coefficients modulo ``p``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_trunc</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_trunc([7, -2, 3], 5)</span>
-<span class="sd">    [2, 3, 3]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">gf_strip</span><span class="p">([</span> <span class="n">a</span> <span class="o">%</span> <span class="n">p</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">])</span>
-
-</div>
-<div class="viewcode-block" id="gf_normal"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_normal">[docs]</a><span class="k">def</span> <span class="nf">gf_normal</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Normalize all coefficients in ``K``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_normal</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_normal([5, 10, 21, -3], 5, ZZ)</span>
-<span class="sd">    [1, 2]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">gf_trunc</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">K</span><span class="p">,</span> <span class="n">f</span><span class="p">)),</span> <span class="n">p</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="gf_convert"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_convert">[docs]</a><span class="k">def</span> <span class="nf">gf_convert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K0</span><span class="p">,</span> <span class="n">K1</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Normalize all coefficients in ``K``.</span>
-
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ, QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_convert</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_convert([QQ(1), QQ(4), QQ(0)], 3, QQ, ZZ)</span>
-<span class="sd">    [1, 1, 0]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">gf_trunc</span><span class="p">([</span> <span class="n">K1</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">K0</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">],</span> <span class="n">p</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;k,n&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_from_dict"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_from_dict">[docs]</a><span class="k">def</span> <span class="nf">gf_from_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create a ``GF(p)[x]`` polynomial from a dict.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_from_dict</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_from_dict({10: ZZ(4), 4: ZZ(33), 0: ZZ(-1)}, 5, ZZ)</span>
-<span class="sd">    [4, 0, 0, 0, 0, 0, 3, 0, 0, 0, 4]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">keys</span><span class="p">()),</span> <span class="p">[]</span>
-
-    <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">int</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-            <span class="n">h</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">)</span> <span class="o">%</span> <span class="n">p</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="p">(</span><span class="n">n</span><span class="p">,)</span> <span class="o">=</span> <span class="n">n</span>
-
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-            <span class="n">h</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">get</span><span class="p">((</span><span class="n">k</span><span class="p">,),</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">)</span> <span class="o">%</span> <span class="n">p</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">gf_trunc</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;k,n&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_to_dict"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_to_dict">[docs]</a><span class="k">def</span> <span class="nf">gf_to_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">symmetric</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Convert a ``GF(p)[x]`` polynomial to a dict.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_to_dict</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_to_dict([4, 0, 0, 0, 0, 0, 3, 0, 0, 0, 4], 5)</span>
-<span class="sd">    {0: -1, 4: -2, 10: -1}</span>
-<span class="sd">    &gt;&gt;&gt; gf_to_dict([4, 0, 0, 0, 0, 0, 3, 0, 0, 0, 4], 5, symmetric=False)</span>
-<span class="sd">    {0: 4, 4: 3, 10: 4}</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span><span class="p">,</span> <span class="n">result</span> <span class="o">=</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="p">{}</span>
-
-    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">symmetric</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">gf_int</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">],</span> <span class="n">p</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">a</span><span class="p">:</span>
-            <span class="n">result</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">a</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="gf_from_int_poly"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_from_int_poly">[docs]</a><span class="k">def</span> <span class="nf">gf_from_int_poly</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create a ``GF(p)[x]`` polynomial from ``Z[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_from_int_poly</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_from_int_poly([7, -2, 3], 5)</span>
-<span class="sd">    [2, 3, 3]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">gf_trunc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="gf_to_int_poly"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_to_int_poly">[docs]</a><span class="k">def</span> <span class="nf">gf_to_int_poly</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">symmetric</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Convert a ``GF(p)[x]`` polynomial to ``Z[x]``.</span>
-
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_to_int_poly</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_to_int_poly([2, 3, 3], 5)</span>
-<span class="sd">    [2, -2, -2]</span>
-<span class="sd">    &gt;&gt;&gt; gf_to_int_poly([2, 3, 3], 5, symmetric=False)</span>
-<span class="sd">    [2, 3, 3]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">symmetric</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="n">gf_int</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-</div>
-<div class="viewcode-block" id="gf_neg"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_neg">[docs]</a><span class="k">def</span> <span class="nf">gf_neg</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Negate a polynomial in ``GF(p)[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_neg</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_neg([3, 2, 1, 0], 5, ZZ)</span>
-<span class="sd">    [2, 3, 4, 0]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">[</span> <span class="o">-</span><span class="n">coeff</span> <span class="o">%</span> <span class="n">p</span> <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="gf_add_ground"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_add_ground">[docs]</a><span class="k">def</span> <span class="nf">gf_add_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute ``f + a`` where ``f`` in ``GF(p)[x]`` and ``a`` in ``GF(p)``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_add_ground</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_add_ground([3, 2, 4], 2, 5, ZZ)</span>
-<span class="sd">    [3, 2, 1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">a</span> <span class="o">%</span> <span class="n">p</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">a</span><span class="p">)</span> <span class="o">%</span> <span class="n">p</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">a</span><span class="p">]</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">a</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">a</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="gf_sub_ground"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_sub_ground">[docs]</a><span class="k">def</span> <span class="nf">gf_sub_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute ``f - a`` where ``f`` in ``GF(p)[x]`` and ``a`` in ``GF(p)``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_sub_ground</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_sub_ground([3, 2, 4], 2, 5, ZZ)</span>
-<span class="sd">    [3, 2, 2]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="o">-</span><span class="n">a</span> <span class="o">%</span> <span class="n">p</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">a</span><span class="p">)</span> <span class="o">%</span> <span class="n">p</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">a</span><span class="p">]</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">a</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">a</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="gf_mul_ground"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_mul_ground">[docs]</a><span class="k">def</span> <span class="nf">gf_mul_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute ``f * a`` where ``f`` in ``GF(p)[x]`` and ``a`` in ``GF(p)``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_mul_ground</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_mul_ground([3, 2, 4], 2, 5, ZZ)</span>
-<span class="sd">    [1, 4, 3]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">a</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">b</span><span class="p">)</span> <span class="o">%</span> <span class="n">p</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">f</span> <span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="gf_quo_ground"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_quo_ground">[docs]</a><span class="k">def</span> <span class="nf">gf_quo_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute ``f/a`` where ``f`` in ``GF(p)[x]`` and ``a`` in ``GF(p)``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_quo_ground</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_quo_ground(ZZ.map([3, 2, 4]), ZZ(2), 5, ZZ)</span>
-<span class="sd">    [4, 1, 2]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">gf_mul_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">invert</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">),</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;df,dg,k&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_add"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_add">[docs]</a><span class="k">def</span> <span class="nf">gf_add</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Add polynomials in ``GF(p)[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_add</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_add([3, 2, 4], [2, 2, 2], 5, ZZ)</span>
-<span class="sd">    [4, 1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">g</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="n">df</span> <span class="o">=</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">dg</span> <span class="o">=</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">df</span> <span class="o">==</span> <span class="n">dg</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">gf_strip</span><span class="p">([</span> <span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span> <span class="o">%</span> <span class="n">p</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span> <span class="p">])</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">df</span> <span class="o">-</span> <span class="n">dg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">df</span> <span class="o">&gt;</span> <span class="n">dg</span><span class="p">:</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="p">[:</span><span class="n">k</span><span class="p">],</span> <span class="n">f</span><span class="p">[</span><span class="n">k</span><span class="p">:]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="p">[:</span><span class="n">k</span><span class="p">],</span> <span class="n">g</span><span class="p">[</span><span class="n">k</span><span class="p">:]</span>
-
-        <span class="k">return</span> <span class="n">h</span> <span class="o">+</span> <span class="p">[</span> <span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span> <span class="o">%</span> <span class="n">p</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span> <span class="p">]</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;df,dg,k&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_sub"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_sub">[docs]</a><span class="k">def</span> <span class="nf">gf_sub</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Subtract polynomials in ``GF(p)[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_sub</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_sub([3, 2, 4], [2, 2, 2], 5, ZZ)</span>
-<span class="sd">    [1, 0, 2]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">gf_neg</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">df</span> <span class="o">=</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">dg</span> <span class="o">=</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">df</span> <span class="o">==</span> <span class="n">dg</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">gf_strip</span><span class="p">([</span> <span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span> <span class="o">%</span> <span class="n">p</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span> <span class="p">])</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">df</span> <span class="o">-</span> <span class="n">dg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">df</span> <span class="o">&gt;</span> <span class="n">dg</span><span class="p">:</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="p">[:</span><span class="n">k</span><span class="p">],</span> <span class="n">f</span><span class="p">[</span><span class="n">k</span><span class="p">:]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">gf_neg</span><span class="p">(</span><span class="n">g</span><span class="p">[:</span><span class="n">k</span><span class="p">],</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">g</span><span class="p">[</span><span class="n">k</span><span class="p">:]</span>
-
-        <span class="k">return</span> <span class="n">h</span> <span class="o">+</span> <span class="p">[</span> <span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span> <span class="o">%</span> <span class="n">p</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span> <span class="p">]</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;df,dg,dh,i,j&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_mul"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_mul">[docs]</a><span class="k">def</span> <span class="nf">gf_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Multiply polynomials in ``GF(p)[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_mul</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_mul([3, 2, 4], [2, 2, 2], 5, ZZ)</span>
-<span class="sd">    [1, 0, 3, 2, 3]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">df</span> <span class="o">=</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">dg</span> <span class="o">=</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="n">dh</span> <span class="o">=</span> <span class="n">df</span> <span class="o">+</span> <span class="n">dg</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">dh</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">dh</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">coeff</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">i</span> <span class="o">-</span> <span class="n">dg</span><span class="p">),</span> <span class="nb">min</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">df</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">coeff</span> <span class="o">+=</span> <span class="n">f</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="n">j</span><span class="p">]</span>
-
-        <span class="n">h</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">coeff</span> <span class="o">%</span> <span class="n">p</span>
-
-    <span class="k">return</span> <span class="n">gf_strip</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;df,dh,i,j,jmin,jmax,n&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_sqr"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_sqr">[docs]</a><span class="k">def</span> <span class="nf">gf_sqr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Square polynomials in ``GF(p)[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_sqr</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_sqr([3, 2, 4], 5, ZZ)</span>
-<span class="sd">    [4, 2, 3, 1, 1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">df</span> <span class="o">=</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="n">dh</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">df</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">dh</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">dh</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">coeff</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-
-        <span class="n">jmin</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">i</span> <span class="o">-</span> <span class="n">df</span><span class="p">)</span>
-        <span class="n">jmax</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">df</span><span class="p">)</span>
-
-        <span class="n">n</span> <span class="o">=</span> <span class="n">jmax</span> <span class="o">-</span> <span class="n">jmin</span> <span class="o">+</span> <span class="mi">1</span>
-
-        <span class="n">jmax</span> <span class="o">=</span> <span class="n">jmin</span> <span class="o">+</span> <span class="n">n</span> <span class="o">//</span> <span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">jmin</span><span class="p">,</span> <span class="n">jmax</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">coeff</span> <span class="o">+=</span> <span class="n">f</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="n">j</span><span class="p">]</span>
-
-        <span class="n">coeff</span> <span class="o">+=</span> <span class="n">coeff</span>
-
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&amp;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">elem</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="n">jmax</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
-            <span class="n">coeff</span> <span class="o">+=</span> <span class="n">elem</span><span class="o">**</span><span class="mi">2</span>
-
-        <span class="n">h</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">coeff</span> <span class="o">%</span> <span class="n">p</span>
-
-    <span class="k">return</span> <span class="n">gf_strip</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="gf_add_mul"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_add_mul">[docs]</a><span class="k">def</span> <span class="nf">gf_add_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns ``f + g*h`` where ``f``, ``g``, ``h`` in ``GF(p)[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_add_mul</span>
-<span class="sd">    &gt;&gt;&gt; gf_add_mul([3, 2, 4], [2, 2, 2], [1, 4], 5, ZZ)</span>
-<span class="sd">    [2, 3, 2, 2]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">gf_add</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gf_mul</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="gf_sub_mul"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_sub_mul">[docs]</a><span class="k">def</span> <span class="nf">gf_sub_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute ``f - g*h`` where ``f``, ``g``, ``h`` in ``GF(p)[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_sub_mul</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_sub_mul([3, 2, 4], [2, 2, 2], [1, 4], 5, ZZ)</span>
-<span class="sd">    [3, 3, 2, 1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">gf_sub</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gf_mul</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_expand"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_expand">[docs]</a><span class="k">def</span> <span class="nf">gf_expand</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Expand results of :func:`factor` in ``GF(p)[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_expand</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_expand([([3, 2, 4], 1), ([2, 2], 2), ([3, 1], 3)], 5, ZZ)</span>
-<span class="sd">    [4, 3, 0, 3, 0, 1, 4, 1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">F</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">tuple</span><span class="p">:</span>
-        <span class="n">lc</span><span class="p">,</span> <span class="n">F</span> <span class="o">=</span> <span class="n">F</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">lc</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span>
-
-    <span class="n">g</span> <span class="o">=</span> <span class="p">[</span><span class="n">lc</span><span class="p">]</span>
-
-    <span class="k">for</span> <span class="n">f</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">F</span><span class="p">:</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">gf_pow</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">gf_mul</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">g</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;df,dg,dq,dr,i,j&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_div"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_div">[docs]</a><span class="k">def</span> <span class="nf">gf_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Division with remainder in ``GF(p)[x]``.</span>
-
-<span class="sd">    Given univariate polynomials ``f`` and ``g`` with coefficients in a</span>
-<span class="sd">    finite field with ``p`` elements, returns polynomials ``q`` and ``r``</span>
-<span class="sd">    (quotient and remainder) such that ``f = q*g + r``.</span>
-
-<span class="sd">    Consider polynomials ``x**3 + x + 1`` and ``x**2 + x`` in GF(2)::</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">       &gt;&gt;&gt; from sympy.polys.galoistools import gf_div, gf_add_mul</span>
-
-<span class="sd">       &gt;&gt;&gt; gf_div(ZZ.map([1, 0, 1, 1]), ZZ.map([1, 1, 0]), 2, ZZ)</span>
-<span class="sd">       ([1, 1], [1])</span>
-
-<span class="sd">    As result we obtained quotient ``x + 1`` and remainder ``1``, thus::</span>
-
-<span class="sd">       &gt;&gt;&gt; gf_add_mul(ZZ.map([1]), ZZ.map([1, 1]), ZZ.map([1, 1, 0]), 2, ZZ)</span>
-<span class="sd">       [1, 0, 1, 1]</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Monagan93]_</span>
-<span class="sd">    2. [Gathen99]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">df</span> <span class="o">=</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">dg</span> <span class="o">=</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ZeroDivisionError</span><span class="p">(</span><span class="s">&quot;polynomial division&quot;</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">df</span> <span class="o">&lt;</span> <span class="n">dg</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[],</span> <span class="n">f</span>
-
-    <span class="n">inv</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">invert</span><span class="p">(</span><span class="n">g</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">p</span><span class="p">)</span>
-
-    <span class="n">h</span><span class="p">,</span> <span class="n">dq</span><span class="p">,</span> <span class="n">dr</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="n">df</span> <span class="o">-</span> <span class="n">dg</span><span class="p">,</span> <span class="n">dg</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">df</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">coeff</span> <span class="o">=</span> <span class="n">h</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">dg</span> <span class="o">-</span> <span class="n">i</span><span class="p">),</span> <span class="nb">min</span><span class="p">(</span><span class="n">df</span> <span class="o">-</span> <span class="n">i</span><span class="p">,</span> <span class="n">dr</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">coeff</span> <span class="o">-=</span> <span class="n">h</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span> <span class="o">-</span> <span class="n">dg</span><span class="p">]</span> <span class="o">*</span> <span class="n">g</span><span class="p">[</span><span class="n">dg</span> <span class="o">-</span> <span class="n">j</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">i</span> <span class="o">&lt;=</span> <span class="n">dq</span><span class="p">:</span>
-            <span class="n">coeff</span> <span class="o">*=</span> <span class="n">inv</span>
-
-        <span class="n">h</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">coeff</span> <span class="o">%</span> <span class="n">p</span>
-
-    <span class="k">return</span> <span class="n">h</span><span class="p">[:</span><span class="n">dq</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">gf_strip</span><span class="p">(</span><span class="n">h</span><span class="p">[</span><span class="n">dq</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:])</span>
-
-</div>
-<div class="viewcode-block" id="gf_rem"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_rem">[docs]</a><span class="k">def</span> <span class="nf">gf_rem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute polynomial remainder in ``GF(p)[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_rem</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_rem(ZZ.map([1, 0, 1, 1]), ZZ.map([1, 1, 0]), 2, ZZ)</span>
-<span class="sd">    [1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">gf_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;df,dg,dq,dr,i,j&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_quo"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_quo">[docs]</a><span class="k">def</span> <span class="nf">gf_quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute exact quotient in ``GF(p)[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_quo</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_quo(ZZ.map([1, 0, 1, 1]), ZZ.map([1, 1, 0]), 2, ZZ)</span>
-<span class="sd">    [1, 1]</span>
-<span class="sd">    &gt;&gt;&gt; gf_quo(ZZ.map([1, 0, 3, 2, 3]), ZZ.map([2, 2, 2]), 5, ZZ)</span>
-<span class="sd">    [3, 2, 4]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">df</span> <span class="o">=</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">dg</span> <span class="o">=</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ZeroDivisionError</span><span class="p">(</span><span class="s">&quot;polynomial division&quot;</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">df</span> <span class="o">&lt;</span> <span class="n">dg</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-    <span class="n">inv</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">invert</span><span class="p">(</span><span class="n">g</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">p</span><span class="p">)</span>
-
-    <span class="n">h</span><span class="p">,</span> <span class="n">dq</span><span class="p">,</span> <span class="n">dr</span> <span class="o">=</span> <span class="n">f</span><span class="p">[:],</span> <span class="n">df</span> <span class="o">-</span> <span class="n">dg</span><span class="p">,</span> <span class="n">dg</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">dq</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">coeff</span> <span class="o">=</span> <span class="n">h</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">dg</span> <span class="o">-</span> <span class="n">i</span><span class="p">),</span> <span class="nb">min</span><span class="p">(</span><span class="n">df</span> <span class="o">-</span> <span class="n">i</span><span class="p">,</span> <span class="n">dr</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">coeff</span> <span class="o">-=</span> <span class="n">h</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span> <span class="o">-</span> <span class="n">dg</span><span class="p">]</span> <span class="o">*</span> <span class="n">g</span><span class="p">[</span><span class="n">dg</span> <span class="o">-</span> <span class="n">j</span><span class="p">]</span>
-
-        <span class="n">h</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">coeff</span> <span class="o">*</span> <span class="n">inv</span><span class="p">)</span> <span class="o">%</span> <span class="n">p</span>
-
-    <span class="k">return</span> <span class="n">h</span><span class="p">[:</span><span class="n">dq</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="gf_exquo"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_exquo">[docs]</a><span class="k">def</span> <span class="nf">gf_exquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute polynomial quotient in ``GF(p)[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_exquo</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_exquo(ZZ.map([1, 0, 3, 2, 3]), ZZ.map([2, 2, 2]), 5, ZZ)</span>
-<span class="sd">    [3, 2, 4]</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_exquo(ZZ.map([1, 0, 1, 1]), ZZ.map([1, 1, 0]), 2, ZZ)</span>
-<span class="sd">    Traceback (most recent call last):</span>
-<span class="sd">    ...</span>
-<span class="sd">    ExactQuotientFailed: [1, 1, 0] does not divide [1, 0, 1, 1]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">gf_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">q</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ExactQuotientFailed</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_lshift"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_lshift">[docs]</a><span class="k">def</span> <span class="nf">gf_lshift</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Efficiently multiply ``f`` by ``x**n``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_lshift</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_lshift([3, 2, 4], 4, ZZ)</span>
-<span class="sd">    [3, 2, 4, 0, 0, 0, 0]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span> <span class="o">+</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span><span class="o">*</span><span class="n">n</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_rshift"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_rshift">[docs]</a><span class="k">def</span> <span class="nf">gf_rshift</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Efficiently divide ``f`` by ``x**n``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_rshift</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_rshift([1, 2, 3, 4, 0], 3, ZZ)</span>
-<span class="sd">    ([1, 2], [3, 4, 0])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span><span class="p">,</span> <span class="p">[]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span><span class="p">[:</span><span class="o">-</span><span class="n">n</span><span class="p">],</span> <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="n">n</span><span class="p">:]</span>
-
-</div>
-<div class="viewcode-block" id="gf_pow"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_pow">[docs]</a><span class="k">def</span> <span class="nf">gf_pow</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute ``f**n`` in ``GF(p)[x]`` using repeated squaring.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_pow</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_pow([3, 2, 4], 3, 5, ZZ)</span>
-<span class="sd">    [2, 4, 4, 2, 2, 1, 4]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]</span>
-    <span class="k">elif</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-    <span class="k">elif</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">gf_sqr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&amp;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">gf_mul</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">n</span> <span class="o">-=</span> <span class="mi">1</span>
-
-        <span class="n">n</span> <span class="o">&gt;&gt;=</span> <span class="mi">1</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">break</span>
-
-        <span class="n">f</span> <span class="o">=</span> <span class="n">gf_sqr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">h</span>
-
-</div>
-<div class="viewcode-block" id="gf_pow_mod"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_pow_mod">[docs]</a><span class="k">def</span> <span class="nf">gf_pow_mod</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute ``f**n`` in ``GF(p)[x]/(g)`` using repeated squaring.</span>
-
-<span class="sd">    Given polynomials ``f`` and ``g`` in ``GF(p)[x]`` and a non-negative</span>
-<span class="sd">    integer ``n``, efficiently computes ``f**n (mod g)`` i.e. the remainder</span>
-<span class="sd">    of ``f**n`` from division by ``g``, using the repeated squaring algorithm.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_pow_mod</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_pow_mod(ZZ.map([3, 2, 4]), 3, ZZ.map([1, 1]), 5, ZZ)</span>
-<span class="sd">    []</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Gathen99]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]</span>
-    <span class="k">elif</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">gf_rem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">gf_rem</span><span class="p">(</span><span class="n">gf_sqr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&amp;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">gf_mul</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">gf_rem</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">n</span> <span class="o">-=</span> <span class="mi">1</span>
-
-        <span class="n">n</span> <span class="o">&gt;&gt;=</span> <span class="mi">1</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">break</span>
-
-        <span class="n">f</span> <span class="o">=</span> <span class="n">gf_sqr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">gf_rem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">h</span>
-
-</div>
-<div class="viewcode-block" id="gf_gcd"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_gcd">[docs]</a><span class="k">def</span> <span class="nf">gf_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Euclidean Algorithm in ``GF(p)[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_gcd</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_gcd(ZZ.map([3, 2, 4]), ZZ.map([2, 2, 3]), 5, ZZ)</span>
-<span class="sd">    [1, 3]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">while</span> <span class="n">g</span><span class="p">:</span>
-        <span class="n">f</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="p">,</span> <span class="n">gf_rem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">gf_monic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="gf_lcm"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_lcm">[docs]</a><span class="k">def</span> <span class="nf">gf_lcm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute polynomial LCM in ``GF(p)[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_lcm</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_lcm(ZZ.map([3, 2, 4]), ZZ.map([2, 2, 3]), 5, ZZ)</span>
-<span class="sd">    [1, 2, 0, 4]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="n">gf_quo</span><span class="p">(</span><span class="n">gf_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span>
-               <span class="n">gf_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">gf_monic</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="gf_cofactors"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_cofactors">[docs]</a><span class="k">def</span> <span class="nf">gf_cofactors</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute polynomial GCD and cofactors in ``GF(p)[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_cofactors</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_cofactors(ZZ.map([3, 2, 4]), ZZ.map([2, 2, 3]), 5, ZZ)</span>
-<span class="sd">    ([1, 3], [3, 3], [2, 1])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">([],</span> <span class="p">[],</span> <span class="p">[])</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="n">gf_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">gf_quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span>
-            <span class="n">gf_quo</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="gf_gcdex"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_gcdex">[docs]</a><span class="k">def</span> <span class="nf">gf_gcdex</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Extended Euclidean Algorithm in ``GF(p)[x]``.</span>
-
-<span class="sd">    Given polynomials ``f`` and ``g`` in ``GF(p)[x]``, computes polynomials</span>
-<span class="sd">    ``s``, ``t`` and ``h``, such that ``h = gcd(f, g)`` and ``s*f + t*g = h``.</span>
-<span class="sd">    The typical application of EEA is solving polynomial diophantine equations.</span>
-
-<span class="sd">    Consider polynomials ``f = (x + 7) (x + 1)``, ``g = (x + 7) (x**2 + 1)``</span>
-<span class="sd">    in ``GF(11)[x]``. Application of Extended Euclidean Algorithm gives::</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">       &gt;&gt;&gt; from sympy.polys.galoistools import gf_gcdex, gf_mul, gf_add</span>
-
-<span class="sd">       &gt;&gt;&gt; s, t, g = gf_gcdex(ZZ.map([1, 8, 7]), ZZ.map([1, 7, 1, 7]), 11, ZZ)</span>
-<span class="sd">       &gt;&gt;&gt; s, t, g</span>
-<span class="sd">       ([5, 6], [6], [1, 7])</span>
-
-<span class="sd">    As result we obtained polynomials ``s = 5*x + 6`` and ``t = 6``, and</span>
-<span class="sd">    additionally ``gcd(f, g) = x + 7``. This is correct because::</span>
-
-<span class="sd">       &gt;&gt;&gt; S = gf_mul(s, ZZ.map([1, 8, 7]), 11, ZZ)</span>
-<span class="sd">       &gt;&gt;&gt; T = gf_mul(t, ZZ.map([1, 7, 1, 7]), 11, ZZ)</span>
-
-<span class="sd">       &gt;&gt;&gt; gf_add(S, T, 11, ZZ) == [1, 7]</span>
-<span class="sd">       True</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Gathen99]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">f</span> <span class="ow">or</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[]</span>
-
-    <span class="n">p0</span><span class="p">,</span> <span class="n">r0</span> <span class="o">=</span> <span class="n">gf_monic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">p1</span><span class="p">,</span> <span class="n">r1</span> <span class="o">=</span> <span class="n">gf_monic</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[],</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">invert</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p</span><span class="p">)],</span> <span class="n">r1</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">invert</span><span class="p">(</span><span class="n">p0</span><span class="p">,</span> <span class="n">p</span><span class="p">)],</span> <span class="p">[],</span> <span class="n">r0</span>
-
-    <span class="n">s0</span><span class="p">,</span> <span class="n">s1</span> <span class="o">=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">invert</span><span class="p">(</span><span class="n">p0</span><span class="p">,</span> <span class="n">p</span><span class="p">)],</span> <span class="p">[]</span>
-    <span class="n">t0</span><span class="p">,</span> <span class="n">t1</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">invert</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p</span><span class="p">)]</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">Q</span><span class="p">,</span> <span class="n">R</span> <span class="o">=</span> <span class="n">gf_div</span><span class="p">(</span><span class="n">r0</span><span class="p">,</span> <span class="n">r1</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">R</span><span class="p">:</span>
-            <span class="k">break</span>
-
-        <span class="p">(</span><span class="n">lc</span><span class="p">,</span> <span class="n">r1</span><span class="p">),</span> <span class="n">r0</span> <span class="o">=</span> <span class="n">gf_monic</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">r1</span>
-
-        <span class="n">inv</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">invert</span><span class="p">(</span><span class="n">lc</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-
-        <span class="n">s</span> <span class="o">=</span> <span class="n">gf_sub_mul</span><span class="p">(</span><span class="n">s0</span><span class="p">,</span> <span class="n">s1</span><span class="p">,</span> <span class="n">Q</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">gf_sub_mul</span><span class="p">(</span><span class="n">t0</span><span class="p">,</span> <span class="n">t1</span><span class="p">,</span> <span class="n">Q</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">s1</span><span class="p">,</span> <span class="n">s0</span> <span class="o">=</span> <span class="n">gf_mul_ground</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">inv</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">s1</span>
-        <span class="n">t1</span><span class="p">,</span> <span class="n">t0</span> <span class="o">=</span> <span class="n">gf_mul_ground</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">inv</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">t1</span>
-
-    <span class="k">return</span> <span class="n">s1</span><span class="p">,</span> <span class="n">t1</span><span class="p">,</span> <span class="n">r1</span>
-
-</div>
-<div class="viewcode-block" id="gf_monic"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_monic">[docs]</a><span class="k">def</span> <span class="nf">gf_monic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute LC and a monic polynomial in ``GF(p)[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_monic</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_monic(ZZ.map([3, 2, 4]), 5, ZZ)</span>
-<span class="sd">    (3, [1, 4, 3])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">,</span> <span class="p">[]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">lc</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">lc</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">lc</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">lc</span><span class="p">,</span> <span class="n">gf_quo_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">lc</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;df,n&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_diff"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_diff">[docs]</a><span class="k">def</span> <span class="nf">gf_diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Differentiate polynomial in ``GF(p)[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_diff</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_diff([3, 2, 4], 5, ZZ)</span>
-<span class="sd">    [1, 2]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">df</span> <span class="o">=</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="n">h</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span><span class="o">*</span><span class="n">df</span><span class="p">,</span> <span class="n">df</span>
-
-    <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-        <span class="n">coeff</span> <span class="o">*=</span> <span class="n">K</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="n">coeff</span> <span class="o">%=</span> <span class="n">p</span>
-
-        <span class="k">if</span> <span class="n">coeff</span><span class="p">:</span>
-            <span class="n">h</span><span class="p">[</span><span class="n">df</span> <span class="o">-</span> <span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">coeff</span>
-
-        <span class="n">n</span> <span class="o">-=</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="n">gf_strip</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="gf_eval"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_eval">[docs]</a><span class="k">def</span> <span class="nf">gf_eval</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Evaluate ``f(a)`` in ``GF(p)`` using Horner scheme.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_eval</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_eval([3, 2, 4], 2, 5, ZZ)</span>
-<span class="sd">    0</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-
-    <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">*=</span> <span class="n">a</span>
-        <span class="n">result</span> <span class="o">+=</span> <span class="n">c</span>
-        <span class="n">result</span> <span class="o">%=</span> <span class="n">p</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="gf_multi_eval"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_multi_eval">[docs]</a><span class="k">def</span> <span class="nf">gf_multi_eval</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Evaluate ``f(a)`` for ``a`` in ``[a_1, ..., a_n]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_multi_eval</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_multi_eval([3, 2, 4], [0, 1, 2, 3, 4], 5, ZZ)</span>
-<span class="sd">    [4, 4, 0, 2, 0]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">[</span> <span class="n">gf_eval</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">A</span> <span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="gf_compose"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_compose">[docs]</a><span class="k">def</span> <span class="nf">gf_compose</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute polynomial composition ``f(g)`` in ``GF(p)[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_compose</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_compose([3, 2, 4], [2, 2, 2], 5, ZZ)</span>
-<span class="sd">    [2, 4, 0, 3, 0]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">g</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">gf_strip</span><span class="p">([</span><span class="n">gf_eval</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gf_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)])</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span>
-
-    <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">gf_mul</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">gf_add_ground</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">h</span>
-
-</div>
-<div class="viewcode-block" id="gf_compose_mod"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_compose_mod">[docs]</a><span class="k">def</span> <span class="nf">gf_compose_mod</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute polynomial composition ``g(h)`` in ``GF(p)[x]/(f)``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_compose_mod</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_compose_mod(ZZ.map([3, 2, 4]), ZZ.map([2, 2, 2]), ZZ.map([4, 3]), 5, ZZ)</span>
-<span class="sd">    [4]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-    <span class="n">comp</span> <span class="o">=</span> <span class="p">[</span><span class="n">g</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span>
-
-    <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">g</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-        <span class="n">comp</span> <span class="o">=</span> <span class="n">gf_mul</span><span class="p">(</span><span class="n">comp</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">comp</span> <span class="o">=</span> <span class="n">gf_add_ground</span><span class="p">(</span><span class="n">comp</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">comp</span> <span class="o">=</span> <span class="n">gf_rem</span><span class="p">(</span><span class="n">comp</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">comp</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_trace_map"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_trace_map">[docs]</a><span class="k">def</span> <span class="nf">gf_trace_map</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute polynomial trace map in ``GF(p)[x]/(f)``.</span>
-
-<span class="sd">    Given a polynomial ``f`` in ``GF(p)[x]``, polynomials ``a``, ``b``,</span>
-<span class="sd">    ``c`` in the quotient ring ``GF(p)[x]/(f)`` such that ``b = c**t</span>
-<span class="sd">    (mod f)`` for some positive power ``t`` of ``p``, and a positive</span>
-<span class="sd">    integer ``n``, returns a mapping::</span>
-
-<span class="sd">       a -&gt; a**t**n, a + a**t + a**t**2 + ... + a**t**n (mod f)</span>
-
-<span class="sd">    In factorization context, ``b = x**p mod f`` and ``c = x mod f``.</span>
-<span class="sd">    This way we can efficiently compute trace polynomials in equal</span>
-<span class="sd">    degree factorization routine, much faster than with other methods,</span>
-<span class="sd">    like iterated Frobenius algorithm, for large degrees.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_trace_map</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_trace_map([1, 2], [4, 4], [1, 1], 4, [3, 2, 4], 5, ZZ)</span>
-<span class="sd">    ([1, 3], [1, 3])</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Gathen92]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">u</span> <span class="o">=</span> <span class="n">gf_compose_mod</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">v</span> <span class="o">=</span> <span class="n">b</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&amp;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">U</span> <span class="o">=</span> <span class="n">gf_add</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">V</span> <span class="o">=</span> <span class="n">b</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">U</span> <span class="o">=</span> <span class="n">a</span>
-        <span class="n">V</span> <span class="o">=</span> <span class="n">c</span>
-
-    <span class="n">n</span> <span class="o">&gt;&gt;=</span> <span class="mi">1</span>
-
-    <span class="k">while</span> <span class="n">n</span><span class="p">:</span>
-        <span class="n">u</span> <span class="o">=</span> <span class="n">gf_add</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">gf_compose_mod</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="n">gf_compose_mod</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&amp;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">U</span> <span class="o">=</span> <span class="n">gf_add</span><span class="p">(</span><span class="n">U</span><span class="p">,</span> <span class="n">gf_compose_mod</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">V</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">V</span> <span class="o">=</span> <span class="n">gf_compose_mod</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">V</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">n</span> <span class="o">&gt;&gt;=</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="n">gf_compose_mod</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">V</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">U</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i,n&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_random"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_random">[docs]</a><span class="k">def</span> <span class="nf">gf_random</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Generate a random polynomial in ``GF(p)[x]`` of degree ``n``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_random</span>
-<span class="sd">    &gt;&gt;&gt; gf_random(10, 5, ZZ) #doctest: +SKIP</span>
-<span class="sd">    [1, 2, 3, 2, 1, 1, 1, 2, 0, 4, 2]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span> <span class="n">K</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">p</span><span class="p">)))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="p">]</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i,n&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_irreducible"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_irreducible">[docs]</a><span class="k">def</span> <span class="nf">gf_irreducible</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Generate random irreducible polynomial of degree ``n`` in ``GF(p)[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_irreducible</span>
-<span class="sd">    &gt;&gt;&gt; gf_irreducible(10, 5, ZZ) #doctest: +SKIP</span>
-<span class="sd">    [1, 4, 2, 2, 3, 2, 4, 1, 4, 0, 4]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">gf_random</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">gf_irreducible_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">f</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i,n&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">gf_irred_p_ben_or</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Ben-Or&#39;s polynomial irreducibility test over finite fields.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_irred_p_ben_or</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_irred_p_ben_or(ZZ.map([1, 4, 2, 2, 3, 2, 4, 1, 4, 0, 4]), 5, ZZ)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; gf_irred_p_ben_or(ZZ.map([3, 2, 4]), 5, ZZ)</span>
-<span class="sd">    False</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="n">_</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">gf_monic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">H</span> <span class="o">=</span> <span class="n">h</span> <span class="o">=</span> <span class="n">gf_pow_mod</span><span class="p">([</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">],</span> <span class="n">p</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="o">//</span><span class="mi">2</span><span class="p">):</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">gf_sub</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">],</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">gf_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]:</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">gf_compose_mod</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">return</span> <span class="bp">True</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i,n,d&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">gf_irred_p_rabin</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Rabin&#39;s polynomial irreducibility test over finite fields.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_irred_p_rabin</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_irred_p_rabin(ZZ.map([1, 4, 2, 2, 3, 2, 4, 1, 4, 0, 4]), 5, ZZ)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; gf_irred_p_rabin(ZZ.map([3, 2, 4]), 5, ZZ)</span>
-<span class="sd">    False</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="n">_</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">gf_monic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span>
-
-    <span class="n">H</span> <span class="o">=</span> <span class="n">h</span> <span class="o">=</span> <span class="n">gf_pow_mod</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">indices</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span> <span class="n">n</span><span class="o">//</span><span class="n">d</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">factorint</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="p">])</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">indices</span><span class="p">:</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">gf_sub</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">gf_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">!=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="n">h</span> <span class="o">=</span> <span class="n">gf_compose_mod</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">h</span> <span class="o">==</span> <span class="n">x</span>
-
-<span class="n">_irred_methods</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="s">&#39;ben-or&#39;</span><span class="p">:</span> <span class="n">gf_irred_p_ben_or</span><span class="p">,</span>
-    <span class="s">&#39;rabin&#39;</span><span class="p">:</span> <span class="n">gf_irred_p_rabin</span><span class="p">,</span>
-<span class="p">}</span>
-
-
-<div class="viewcode-block" id="gf_irreducible_p"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_irreducible_p">[docs]</a><span class="k">def</span> <span class="nf">gf_irreducible_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Test irreducibility of a polynomial ``f`` in ``GF(p)[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_irreducible_p</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_irreducible_p(ZZ.map([1, 4, 2, 2, 3, 2, 4, 1, 4, 0, 4]), 5, ZZ)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; gf_irreducible_p(ZZ.map([3, 2, 4]), 5, ZZ)</span>
-<span class="sd">    False</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">method</span> <span class="o">=</span> <span class="n">query</span><span class="p">(</span><span class="s">&#39;GF_IRRED_METHOD&#39;</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">method</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">irred</span> <span class="o">=</span> <span class="n">_irred_methods</span><span class="p">[</span><span class="n">method</span><span class="p">](</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">irred</span> <span class="o">=</span> <span class="n">gf_irred_p_rabin</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">irred</span>
-
-</div>
-<div class="viewcode-block" id="gf_sqf_p"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_sqf_p">[docs]</a><span class="k">def</span> <span class="nf">gf_sqf_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return ``True`` if ``f`` is square-free in ``GF(p)[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_sqf_p</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_sqf_p(ZZ.map([3, 2, 4]), 5, ZZ)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; gf_sqf_p(ZZ.map([2, 4, 4, 2, 2, 1, 4]), 5, ZZ)</span>
-<span class="sd">    False</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">_</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">gf_monic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">True</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">gf_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gf_diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="gf_sqf_part"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_sqf_part">[docs]</a><span class="k">def</span> <span class="nf">gf_sqf_part</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return square-free part of a ``GF(p)[x]`` polynomial.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_sqf_part</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_sqf_part(ZZ.map([1, 1, 3, 0, 1, 0, 2, 2, 1]), 5, ZZ)</span>
-<span class="sd">    [1, 4, 3]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">_</span><span class="p">,</span> <span class="n">sqf</span> <span class="o">=</span> <span class="n">gf_sqf_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">g</span> <span class="o">=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]</span>
-
-    <span class="k">for</span> <span class="n">f</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">sqf</span><span class="p">:</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">gf_mul</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">g</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i,n,d,r&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_sqf_list"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_sqf_list">[docs]</a><span class="k">def</span> <span class="nf">gf_sqf_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="nb">all</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the square-free decomposition of a ``GF(p)[x]`` polynomial.</span>
-
-<span class="sd">    Given a polynomial ``f`` in ``GF(p)[x]``, returns the leading coefficient</span>
-<span class="sd">    of ``f`` and a square-free decomposition ``f_1**e_1 f_2**e_2 ... f_k**e_k``</span>
-<span class="sd">    such that all ``f_i`` are monic polynomials and ``(f_i, f_j)`` for ``i != j``</span>
-<span class="sd">    are co-prime and ``e_1 ... e_k`` are given in increasing order. All trivial</span>
-<span class="sd">    terms (i.e. ``f_i = 1``) aren&#39;t included in the output.</span>
-
-<span class="sd">    Consider polynomial ``f = x**11 + 1`` over ``GF(11)[x]``::</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy.polys.galoistools import (</span>
-<span class="sd">       ...     gf_from_dict, gf_diff, gf_sqf_list, gf_pow,</span>
-<span class="sd">       ... )</span>
-<span class="sd">       ... # doctest: +NORMALIZE_WHITESPACE</span>
-
-<span class="sd">       &gt;&gt;&gt; f = gf_from_dict({11: ZZ(1), 0: ZZ(1)}, 11, ZZ)</span>
-
-<span class="sd">    Note that ``f&#39;(x) = 0``::</span>
-
-<span class="sd">       &gt;&gt;&gt; gf_diff(f, 11, ZZ)</span>
-<span class="sd">       []</span>
-
-<span class="sd">    This phenomenon doesn&#39;t happen in characteristic zero. However we can</span>
-<span class="sd">    still compute square-free decomposition of ``f`` using ``gf_sqf()``::</span>
-
-<span class="sd">       &gt;&gt;&gt; gf_sqf_list(f, 11, ZZ)</span>
-<span class="sd">       (1, [([1, 1], 11)])</span>
-
-<span class="sd">    We obtained factorization ``f = (x + 1)**11``. This is correct because::</span>
-
-<span class="sd">       &gt;&gt;&gt; gf_pow([1, 1], 11, 11, ZZ) == f</span>
-<span class="sd">       True</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Geddes92]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span><span class="p">,</span> <span class="n">sqf</span><span class="p">,</span> <span class="n">factors</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="bp">False</span><span class="p">,</span> <span class="p">[],</span> <span class="nb">int</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-
-    <span class="n">lc</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">gf_monic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">lc</span><span class="p">,</span> <span class="p">[]</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">F</span> <span class="o">=</span> <span class="n">gf_diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">F</span> <span class="o">!=</span> <span class="p">[]:</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">gf_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">gf_quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="n">i</span> <span class="o">=</span> <span class="mi">1</span>
-
-            <span class="k">while</span> <span class="n">h</span> <span class="o">!=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]:</span>
-                <span class="n">G</span> <span class="o">=</span> <span class="n">gf_gcd</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-                <span class="n">H</span> <span class="o">=</span> <span class="n">gf_quo</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">H</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">factors</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">H</span><span class="p">,</span> <span class="n">i</span><span class="o">*</span><span class="n">n</span><span class="p">))</span>
-
-                <span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="n">gf_quo</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">G</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-
-            <span class="k">if</span> <span class="n">g</span> <span class="o">==</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]:</span>
-                <span class="n">sqf</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">f</span> <span class="o">=</span> <span class="n">g</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">sqf</span><span class="p">:</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">//</span> <span class="n">r</span>
-
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">d</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="n">f</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="n">r</span><span class="p">]</span>
-
-            <span class="n">f</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">f</span><span class="p">[:</span><span class="n">d</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">n</span><span class="o">*</span><span class="n">r</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">break</span>
-
-    <span class="k">if</span> <span class="nb">all</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;&#39;all=True&#39; is not supported yet&quot;</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">lc</span><span class="p">,</span> <span class="n">factors</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,i,j,r&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_Qmatrix"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_Qmatrix">[docs]</a><span class="k">def</span> <span class="nf">gf_Qmatrix</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Calculate Berlekamp&#39;s ``Q`` matrix.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_Qmatrix</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_Qmatrix([3, 2, 4], 5, ZZ)</span>
-<span class="sd">    [[1, 0],</span>
-<span class="sd">     [3, 4]]</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_Qmatrix([1, 0, 0, 0, 1], 5, ZZ)</span>
-<span class="sd">    [[1, 0, 0, 0],</span>
-<span class="sd">     [0, 4, 0, 0],</span>
-<span class="sd">     [0, 0, 1, 0],</span>
-<span class="sd">     [0, 0, 0, 4]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="nb">int</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-
-    <span class="n">q</span> <span class="o">=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="n">Q</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="n">q</span><span class="p">)]</span> <span class="o">+</span> <span class="p">[[]]</span><span class="o">*</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">r</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">qq</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="p">[(</span><span class="o">-</span><span class="n">q</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span> <span class="o">%</span> <span class="n">p</span><span class="p">],</span> <span class="n">q</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-            <span class="n">qq</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">q</span><span class="p">[</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">c</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span> <span class="o">%</span> <span class="n">p</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">i</span> <span class="o">%</span> <span class="n">r</span><span class="p">):</span>
-            <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="o">//</span><span class="n">r</span><span class="p">]</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">qq</span><span class="p">)</span>
-
-        <span class="n">q</span> <span class="o">=</span> <span class="n">qq</span>
-
-    <span class="k">return</span> <span class="n">Q</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,i,j,k&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_Qbasis"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_Qbasis">[docs]</a><span class="k">def</span> <span class="nf">gf_Qbasis</span><span class="p">(</span><span class="n">Q</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute a basis of the kernel of ``Q``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_Qmatrix, gf_Qbasis</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_Qbasis(gf_Qmatrix([1, 0, 0, 0, 1], 5, ZZ), 5, ZZ)</span>
-<span class="sd">    [[1, 0, 0, 0], [0, 0, 1, 0]]</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_Qbasis(gf_Qmatrix([3, 2, 4], 5, ZZ), 5, ZZ)</span>
-<span class="sd">    [[1, 0]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">Q</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="p">[</span> <span class="nb">list</span><span class="p">(</span><span class="n">q</span><span class="p">)</span> <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">Q</span> <span class="p">],</span> <span class="nb">len</span><span class="p">(</span><span class="n">Q</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="n">Q</span><span class="p">[</span><span class="n">k</span><span class="p">][</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">Q</span><span class="p">[</span><span class="n">k</span><span class="p">][</span><span class="n">k</span><span class="p">]</span> <span class="o">-</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">)</span> <span class="o">%</span> <span class="n">p</span>
-
-    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">Q</span><span class="p">[</span><span class="n">k</span><span class="p">][</span><span class="n">i</span><span class="p">]:</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">continue</span>
-
-        <span class="n">inv</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">invert</span><span class="p">(</span><span class="n">Q</span><span class="p">[</span><span class="n">k</span><span class="p">][</span><span class="n">i</span><span class="p">],</span> <span class="n">p</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-            <span class="n">Q</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">Q</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">inv</span><span class="p">)</span> <span class="o">%</span> <span class="n">p</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-            <span class="n">t</span> <span class="o">=</span> <span class="n">Q</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">k</span><span class="p">]</span>
-            <span class="n">Q</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">Q</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span>
-            <span class="n">Q</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">t</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="n">k</span><span class="p">:</span>
-                <span class="n">q</span> <span class="o">=</span> <span class="n">Q</span><span class="p">[</span><span class="n">k</span><span class="p">][</span><span class="n">i</span><span class="p">]</span>
-
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-                    <span class="n">Q</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">Q</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">Q</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">q</span><span class="p">)</span> <span class="o">%</span> <span class="n">p</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
-                <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">K</span><span class="o">.</span><span class="n">one</span> <span class="o">-</span> <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">])</span> <span class="o">%</span> <span class="n">p</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">])</span> <span class="o">%</span> <span class="n">p</span>
-
-    <span class="n">basis</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">Q</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">q</span><span class="p">):</span>
-            <span class="n">basis</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">basis</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i,k&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_berlekamp"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_berlekamp">[docs]</a><span class="k">def</span> <span class="nf">gf_berlekamp</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Factor a square-free ``f`` in ``GF(p)[x]`` for small ``p``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_berlekamp</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_berlekamp([1, 0, 0, 0, 1], 5, ZZ)</span>
-<span class="sd">    [[1, 0, 2], [1, 0, 3]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">Q</span> <span class="o">=</span> <span class="n">gf_Qmatrix</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">V</span> <span class="o">=</span> <span class="n">gf_Qbasis</span><span class="p">(</span><span class="n">Q</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">V</span><span class="p">):</span>
-        <span class="n">V</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">gf_strip</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">v</span><span class="p">)))</span>
-
-    <span class="n">factors</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="p">]</span>
-
-    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">V</span><span class="p">)):</span>
-        <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">factors</span><span class="p">):</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span>
-
-            <span class="k">while</span> <span class="n">s</span> <span class="o">&lt;</span> <span class="n">p</span><span class="p">:</span>
-                <span class="n">g</span> <span class="o">=</span> <span class="n">gf_sub_ground</span><span class="p">(</span><span class="n">V</span><span class="p">[</span><span class="n">k</span><span class="p">],</span> <span class="n">s</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-                <span class="n">h</span> <span class="o">=</span> <span class="n">gf_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">h</span> <span class="o">!=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]</span> <span class="ow">and</span> <span class="n">h</span> <span class="o">!=</span> <span class="n">f</span><span class="p">:</span>
-                    <span class="n">factors</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-                    <span class="n">f</span> <span class="o">=</span> <span class="n">gf_quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-                    <span class="n">factors</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">])</span>
-
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">V</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">_sort_factors</span><span class="p">(</span><span class="n">factors</span><span class="p">,</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-                <span class="n">s</span> <span class="o">+=</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span>
-
-    <span class="k">return</span> <span class="n">_sort_factors</span><span class="p">(</span><span class="n">factors</span><span class="p">,</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">gf_ddf_zassenhaus</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Cantor-Zassenhaus: Deterministic Distinct Degree Factorization</span>
-
-<span class="sd">    Given a monic square-free polynomial ``f`` in ``GF(p)[x]``, computes</span>
-<span class="sd">    partial distinct degree factorization ``f_1 ... f_d`` of ``f`` where</span>
-<span class="sd">    ``deg(f_i) != deg(f_j)`` for ``i != j``. The result is returned as a</span>
-<span class="sd">    list of pairs ``(f_i, e_i)`` where ``deg(f_i) &gt; 0`` and ``e_i &gt; 0``</span>
-<span class="sd">    is an argument to the equal degree factorization routine.</span>
-
-<span class="sd">    Consider the polynomial ``x**15 - 1`` in ``GF(11)[x]``::</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">       &gt;&gt;&gt; from sympy.polys.galoistools import gf_from_dict</span>
-
-<span class="sd">       &gt;&gt;&gt; f = gf_from_dict({15: ZZ(1), 0: ZZ(-1)}, 11, ZZ)</span>
-
-<span class="sd">    Distinct degree factorization gives::</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy.polys.galoistools import gf_ddf_zassenhaus</span>
-
-<span class="sd">       &gt;&gt;&gt; gf_ddf_zassenhaus(f, 11, ZZ)</span>
-<span class="sd">       [([1, 0, 0, 0, 0, 10], 1), ([1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1], 2)]</span>
-
-<span class="sd">    which means ``x**15 - 1 = (x**5 - 1) (x**10 + x**5 + 1)``. To obtain</span>
-<span class="sd">    factorization into irreducibles, use equal degree factorization</span>
-<span class="sd">    procedure (EDF) with each of the factors.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Gathen99]_</span>
-<span class="sd">    2. [Geddes92]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">i</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">],</span> <span class="p">[]</span>
-
-    <span class="k">while</span> <span class="mi">2</span><span class="o">*</span><span class="n">i</span> <span class="o">&lt;=</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">gf_pow_mod</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">p</span><span class="p">),</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">gf_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gf_sub</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">],</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">h</span> <span class="o">!=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]:</span>
-            <span class="n">factors</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">h</span><span class="p">,</span> <span class="n">i</span><span class="p">))</span>
-
-            <span class="n">f</span> <span class="o">=</span> <span class="n">gf_quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">gf_rem</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="n">f</span> <span class="o">!=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]:</span>
-        <span class="k">return</span> <span class="n">factors</span> <span class="o">+</span> <span class="p">[(</span><span class="n">f</span><span class="p">,</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">))]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">factors</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,N,i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">gf_edf_zassenhaus</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Cantor-Zassenhaus: Probabilistic Equal Degree Factorization</span>
-
-<span class="sd">    Given a monic square-free polynomial ``f`` in ``GF(p)[x]`` and</span>
-<span class="sd">    an integer ``n``, such that ``n`` divides ``deg(f)``, returns all</span>
-<span class="sd">    irreducible factors ``f_1,...,f_d`` of ``f``, each of degree ``n``.</span>
-<span class="sd">    EDF procedure gives complete factorization over Galois fields.</span>
-
-<span class="sd">    Consider the square-free polynomial ``f = x**3 + x**2 + x + 1`` in</span>
-<span class="sd">    ``GF(5)[x]``. Let&#39;s compute its irreducible factors of degree one::</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">       &gt;&gt;&gt; from sympy.polys.galoistools import gf_edf_zassenhaus</span>
-
-<span class="sd">       &gt;&gt;&gt; gf_edf_zassenhaus([1,1,1,1], 1, 5, ZZ)</span>
-<span class="sd">       [[1, 1], [1, 2], [1, 3]]</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Gathen99]_</span>
-<span class="sd">    2. [Geddes92]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">factors</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="p">],</span> <span class="nb">int</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="n">n</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">factors</span>
-
-    <span class="n">N</span> <span class="o">=</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">//</span> <span class="n">n</span>
-
-    <span class="k">while</span> <span class="nb">len</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">N</span><span class="p">:</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">gf_random</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">p</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">r</span>
-
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">N</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)):</span>
-                <span class="n">r</span> <span class="o">=</span> <span class="n">gf_pow_mod</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-                <span class="n">h</span> <span class="o">=</span> <span class="n">gf_add</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="n">g</span> <span class="o">=</span> <span class="n">gf_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">gf_pow_mod</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="p">(</span><span class="n">q</span><span class="o">**</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">gf_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gf_sub_ground</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">g</span> <span class="o">!=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]</span> <span class="ow">and</span> <span class="n">g</span> <span class="o">!=</span> <span class="n">f</span><span class="p">:</span>
-            <span class="n">factors</span> <span class="o">=</span> <span class="n">gf_edf_zassenhaus</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> \
-                <span class="o">+</span> <span class="n">gf_edf_zassenhaus</span><span class="p">(</span><span class="n">gf_quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">_sort_factors</span><span class="p">(</span><span class="n">factors</span><span class="p">,</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,k,i,j&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">gf_ddf_shoup</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Kaltofen-Shoup: Deterministic Distinct Degree Factorization</span>
-
-<span class="sd">    Given a monic square-free polynomial ``f`` in ``GF(p)[x]``, computes</span>
-<span class="sd">    partial distinct degree factorization ``f_1,...,f_d`` of ``f`` where</span>
-<span class="sd">    ``deg(f_i) != deg(f_j)`` for ``i != j``. The result is returned as a</span>
-<span class="sd">    list of pairs ``(f_i, e_i)`` where ``deg(f_i) &gt; 0`` and ``e_i &gt; 0``</span>
-<span class="sd">    is an argument to the equal degree factorization routine.</span>
-
-<span class="sd">    This algorithm is an improved version of Zassenhaus algorithm for</span>
-<span class="sd">    large ``deg(f)`` and modulus ``p`` (especially for ``deg(f) ~ lg(p)``).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_ddf_shoup, gf_from_dict</span>
-
-<span class="sd">    &gt;&gt;&gt; f = gf_from_dict({6: ZZ(1), 5: ZZ(-1), 4: ZZ(1), 3: ZZ(1), 1: ZZ(-1)}, 3, ZZ)</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_ddf_shoup(f, 3, ZZ)</span>
-<span class="sd">    [([1, 1, 0], 1), ([1, 1, 0, 1, 2], 2)]</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Kaltofen98]_</span>
-<span class="sd">    2. [Shoup95]_</span>
-<span class="sd">    3. [Gathen92]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">k</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">_ceil</span><span class="p">(</span><span class="n">_sqrt</span><span class="p">(</span><span class="n">n</span><span class="o">//</span><span class="mi">2</span><span class="p">)))</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="n">gf_pow_mod</span><span class="p">([</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">],</span> <span class="nb">int</span><span class="p">(</span><span class="n">p</span><span class="p">),</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">U</span> <span class="o">=</span> <span class="p">[[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">],</span> <span class="n">h</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">U</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">gf_compose_mod</span><span class="p">(</span><span class="n">U</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">h</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">h</span><span class="p">,</span> <span class="n">U</span> <span class="o">=</span> <span class="n">U</span><span class="p">[</span><span class="n">k</span><span class="p">],</span> <span class="n">U</span><span class="p">[:</span><span class="n">k</span><span class="p">]</span>
-    <span class="n">V</span> <span class="o">=</span> <span class="p">[</span><span class="n">h</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="n">V</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">gf_compose_mod</span><span class="p">(</span><span class="n">V</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">h</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">factors</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">V</span><span class="p">):</span>
-        <span class="n">h</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">],</span> <span class="n">k</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="k">for</span> <span class="n">u</span> <span class="ow">in</span> <span class="n">U</span><span class="p">:</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">gf_sub</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">gf_mul</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">gf_rem</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">g</span> <span class="o">=</span> <span class="n">gf_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">gf_quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">u</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">U</span><span class="p">):</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">gf_sub</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">F</span> <span class="o">=</span> <span class="n">gf_gcd</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">F</span> <span class="o">!=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]:</span>
-                <span class="n">factors</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">F</span><span class="p">,</span> <span class="n">k</span><span class="o">*</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">j</span><span class="p">))</span>
-
-            <span class="n">g</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">gf_quo</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">j</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="n">f</span> <span class="o">!=</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]:</span>
-        <span class="n">factors</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)))</span>
-
-    <span class="k">return</span> <span class="n">factors</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,N,q&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">gf_edf_shoup</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Gathen-Shoup: Probabilistic Equal Degree Factorization</span>
-
-<span class="sd">    Given a monic square-free polynomial ``f`` in ``GF(p)[x]`` and integer</span>
-<span class="sd">    ``n`` such that ``n`` divides ``deg(f)``, returns all irreducible factors</span>
-<span class="sd">    ``f_1,...,f_d`` of ``f``, each of degree ``n``. This is a complete</span>
-<span class="sd">    factorization over Galois fields.</span>
-
-<span class="sd">    This algorithm is an improved version of Zassenhaus algorithm for</span>
-<span class="sd">    large ``deg(f)`` and modulus ``p`` (especially for ``deg(f) ~ lg(p)``).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_edf_shoup</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_edf_shoup(ZZ.map([1, 2837, 2277]), 1, 2917, ZZ)</span>
-<span class="sd">    [[1, 852], [1, 1985]]</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Shoup91]_</span>
-<span class="sd">    2. [Gathen92]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">N</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="nb">int</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">N</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-    <span class="k">if</span> <span class="n">N</span> <span class="o">&lt;=</span> <span class="n">n</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">f</span><span class="p">]</span>
-
-    <span class="n">factors</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="p">],</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span>
-
-    <span class="n">r</span> <span class="o">=</span> <span class="n">gf_random</span><span class="p">(</span><span class="n">N</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="n">gf_pow_mod</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">H</span> <span class="o">=</span> <span class="n">gf_trace_map</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="k">if</span> <span class="n">p</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="n">h1</span> <span class="o">=</span> <span class="n">gf_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">h2</span> <span class="o">=</span> <span class="n">gf_quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h1</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">factors</span> <span class="o">=</span> <span class="n">gf_edf_shoup</span><span class="p">(</span><span class="n">h1</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> \
-            <span class="o">+</span> <span class="n">gf_edf_shoup</span><span class="p">(</span><span class="n">h2</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">gf_pow_mod</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="p">(</span><span class="n">q</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">h1</span> <span class="o">=</span> <span class="n">gf_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">h2</span> <span class="o">=</span> <span class="n">gf_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gf_sub_ground</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">h3</span> <span class="o">=</span> <span class="n">gf_quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gf_mul</span><span class="p">(</span><span class="n">h1</span><span class="p">,</span> <span class="n">h2</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">factors</span> <span class="o">=</span> <span class="n">gf_edf_shoup</span><span class="p">(</span><span class="n">h1</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> \
-            <span class="o">+</span> <span class="n">gf_edf_shoup</span><span class="p">(</span><span class="n">h2</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> \
-            <span class="o">+</span> <span class="n">gf_edf_shoup</span><span class="p">(</span><span class="n">h3</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">_sort_factors</span><span class="p">(</span><span class="n">factors</span><span class="p">,</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_zassenhaus"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_zassenhaus">[docs]</a><span class="k">def</span> <span class="nf">gf_zassenhaus</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Factor a square-free ``f`` in ``GF(p)[x]`` for medium ``p``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_zassenhaus</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_zassenhaus(ZZ.map([1, 4, 3]), 5, ZZ)</span>
-<span class="sd">    [[1, 1], [1, 3]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">factors</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">factor</span><span class="p">,</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">gf_ddf_zassenhaus</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-        <span class="n">factors</span> <span class="o">+=</span> <span class="n">gf_edf_zassenhaus</span><span class="p">(</span><span class="n">factor</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">_sort_factors</span><span class="p">(</span><span class="n">factors</span><span class="p">,</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_shoup"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_shoup">[docs]</a><span class="k">def</span> <span class="nf">gf_shoup</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Factor a square-free ``f`` in ``GF(p)[x]`` for large ``p``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_shoup</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_shoup(ZZ.map([1, 4, 3]), 5, ZZ)</span>
-<span class="sd">    [[1, 1], [1, 3]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">factors</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">factor</span><span class="p">,</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">gf_ddf_shoup</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-        <span class="n">factors</span> <span class="o">+=</span> <span class="n">gf_edf_shoup</span><span class="p">(</span><span class="n">factor</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">_sort_factors</span><span class="p">(</span><span class="n">factors</span><span class="p">,</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-</div>
-<span class="n">_factor_methods</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="s">&#39;berlekamp&#39;</span><span class="p">:</span> <span class="n">gf_berlekamp</span><span class="p">,</span>  <span class="c"># ``p`` : small</span>
-    <span class="s">&#39;zassenhaus&#39;</span><span class="p">:</span> <span class="n">gf_zassenhaus</span><span class="p">,</span>  <span class="c"># ``p`` : medium</span>
-    <span class="s">&#39;shoup&#39;</span><span class="p">:</span> <span class="n">gf_shoup</span><span class="p">,</span>      <span class="c"># ``p`` : large</span>
-<span class="p">}</span>
-
-
-<div class="viewcode-block" id="gf_factor_sqf"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_factor_sqf">[docs]</a><span class="k">def</span> <span class="nf">gf_factor_sqf</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Factor a square-free polynomial ``f`` in ``GF(p)[x]``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_factor_sqf</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_factor_sqf(ZZ.map([3, 2, 4]), 5, ZZ)</span>
-<span class="sd">    (3, [[1, 1], [1, 3]])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">lc</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">gf_monic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">lc</span><span class="p">,</span> <span class="p">[]</span>
-
-    <span class="n">method</span> <span class="o">=</span> <span class="n">method</span> <span class="ow">or</span> <span class="n">query</span><span class="p">(</span><span class="s">&#39;GF_FACTOR_METHOD&#39;</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">method</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">factors</span> <span class="o">=</span> <span class="n">_factor_methods</span><span class="p">[</span><span class="n">method</span><span class="p">](</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">factors</span> <span class="o">=</span> <span class="n">gf_zassenhaus</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">lc</span><span class="p">,</span> <span class="n">factors</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="gf_factor"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_factor">[docs]</a><span class="k">def</span> <span class="nf">gf_factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Factor (non square-free) polynomials in ``GF(p)[x]``.</span>
-
-<span class="sd">    Given a possibly non square-free polynomial ``f`` in ``GF(p)[x]``,</span>
-<span class="sd">    returns its complete factorization into irreducibles::</span>
-
-<span class="sd">                 f_1(x)**e_1 f_2(x)**e_2 ... f_d(x)**e_d</span>
-
-<span class="sd">    where each ``f_i`` is a monic polynomial and ``gcd(f_i, f_j) == 1``,</span>
-<span class="sd">    for ``i != j``.  The result is given as a tuple consisting of the</span>
-<span class="sd">    leading coefficient of ``f`` and a list of factors of ``f`` with</span>
-<span class="sd">    their multiplicities.</span>
-
-<span class="sd">    The algorithm proceeds by first computing square-free decomposition</span>
-<span class="sd">    of ``f`` and then iteratively factoring each of square-free factors.</span>
-
-<span class="sd">    Consider a non square-free polynomial ``f = (7*x + 1) (x + 2)**2`` in</span>
-<span class="sd">    ``GF(11)[x]``. We obtain its factorization into irreducibles as follows::</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-<span class="sd">       &gt;&gt;&gt; from sympy.polys.galoistools import gf_factor</span>
-
-<span class="sd">       &gt;&gt;&gt; gf_factor(ZZ.map([5, 2, 7, 2]), 11, ZZ)</span>
-<span class="sd">       (5, [([1, 2], 1), ([1, 8], 2)])</span>
-
-<span class="sd">    We arrived with factorization ``f = 5 (x + 2) (x + 8)**2``. We didn&#39;t</span>
-<span class="sd">    recover the exact form of the input polynomial because we requested to</span>
-<span class="sd">    get monic factors of ``f`` and its leading coefficient separately.</span>
-
-<span class="sd">    Square-free factors of ``f`` can be factored into irreducibles over</span>
-<span class="sd">    ``GF(p)`` using three very different methods:</span>
-
-<span class="sd">    Berlekamp</span>
-<span class="sd">        efficient for very small values of ``p`` (usually ``p &lt; 25``)</span>
-<span class="sd">    Cantor-Zassenhaus</span>
-<span class="sd">        efficient on average input and with &quot;typical&quot; ``p``</span>
-<span class="sd">    Shoup-Kaltofen-Gathen</span>
-<span class="sd">        efficient with very large inputs and modulus</span>
-
-<span class="sd">    If you want to use a specific factorization method, instead of the default</span>
-<span class="sd">    one, set ``GF_FACTOR_METHOD`` with one of ``berlekamp``, ``zassenhaus`` or</span>
-<span class="sd">    ``shoup`` values.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Gathen99]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">lc</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">gf_monic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">gf_degree</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">lc</span><span class="p">,</span> <span class="p">[]</span>
-
-    <span class="n">factors</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">g</span><span class="p">,</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">gf_sqf_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">1</span><span class="p">]:</span>
-        <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">gf_factor_sqf</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)[</span><span class="mi">1</span><span class="p">]:</span>
-            <span class="n">factors</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">h</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">lc</span><span class="p">,</span> <span class="n">_sort_factors</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="gf_value"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_value">[docs]</a><span class="k">def</span> <span class="nf">gf_value</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Value of polynomial &#39;f&#39; at &#39;a&#39; in field R.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_value</span>
-
-<span class="sd">    &gt;&gt;&gt; gf_value([1, 7, 2, 4], 11)</span>
-<span class="sd">    2204</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">*=</span> <span class="n">a</span>
-        <span class="n">result</span> <span class="o">+=</span> <span class="n">c</span>
-    <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<span class="k">def</span> <span class="nf">linear_congruence</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the values of x satisfying a*x congruent b mod(m)</span>
-
-<span class="sd">    Here m is positive integer and a, b are natural numbers.</span>
-<span class="sd">    This function returns only those values of x which are distinct mod(m).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import linear_congruence</span>
-
-<span class="sd">    &gt;&gt;&gt; linear_congruence(3, 12, 15)</span>
-<span class="sd">    [4, 9, 14]</span>
-
-<span class="sd">    There are 3 solutions distinct mod(15) since gcd(a, m) = gcd(3, 15) = 3.</span>
-
-<span class="sd">    **Reference**</span>
-<span class="sd">    1) Wikipedia http://en.wikipedia.org/wiki/Linear_congruence_theorem</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.polys.polytools</span> <span class="kn">import</span> <span class="n">gcdex</span>
-    <span class="k">if</span> <span class="n">a</span> <span class="o">%</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">b</span> <span class="o">%</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[]</span>
-    <span class="n">r</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">gcdex</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">b</span> <span class="o">%</span> <span class="n">g</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-    <span class="k">return</span> <span class="p">[(</span><span class="n">r</span> <span class="o">*</span> <span class="n">b</span> <span class="o">//</span> <span class="n">g</span> <span class="o">+</span> <span class="n">t</span> <span class="o">*</span> <span class="n">m</span> <span class="o">//</span> <span class="n">g</span><span class="p">)</span> <span class="o">%</span> <span class="n">m</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">g</span><span class="p">)]</span>
-
-
-<span class="k">def</span> <span class="nf">_raise_mod_power</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Used in gf_csolve to generate solutions of f(x) cong 0 mod(p**(s + 1))</span>
-<span class="sd">    from the solutions of f(x) cong 0 mod(p**s).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import _raise_mod_power</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import csolve_prime</span>
-
-<span class="sd">    These is the solutions of f(x) = x**2 + x + 7 cong 0 mod(3)</span>
-
-<span class="sd">    &gt;&gt;&gt; f = [1, 1, 7]</span>
-<span class="sd">    &gt;&gt;&gt; csolve_prime(f, 3)</span>
-<span class="sd">    [1]</span>
-<span class="sd">    &gt;&gt;&gt; [ i for i in range(3) if not (i**2 + i + 7) % 3]</span>
-<span class="sd">    [1]</span>
-
-<span class="sd">    The solutions of f(x) cong 0 mod(9) are constructed from the</span>
-<span class="sd">    values returned from _raise_mod_power:</span>
-
-<span class="sd">    &gt;&gt;&gt; x, s, p = 1, 1, 3</span>
-<span class="sd">    &gt;&gt;&gt; V = _raise_mod_power(x, s, p, f)</span>
-<span class="sd">    &gt;&gt;&gt; [x + v * p**s for v in V]</span>
-<span class="sd">    [1, 4, 7]</span>
-
-<span class="sd">    And these are confirmed with the following:</span>
-
-<span class="sd">    &gt;&gt;&gt; [ i for i in range(3**2) if not (i**2 + i + 7) % 3**2]</span>
-<span class="sd">    [1, 4, 7]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-    <span class="n">f_f</span> <span class="o">=</span> <span class="n">gf_diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-    <span class="n">alpha</span> <span class="o">=</span> <span class="n">gf_value</span><span class="p">(</span><span class="n">f_f</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-    <span class="n">beta</span> <span class="o">=</span> <span class="o">-</span> <span class="n">gf_value</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">//</span> <span class="n">p</span><span class="o">**</span><span class="n">s</span>
-    <span class="k">return</span> <span class="n">linear_congruence</span><span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">csolve_prime</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">e</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Solutions of f(x) congruent 0 mod(p**e).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import csolve_prime</span>
-
-<span class="sd">    &gt;&gt;&gt; csolve_prime([1, 1, 7], 3, 1)</span>
-<span class="sd">    [1]</span>
-<span class="sd">    &gt;&gt;&gt; csolve_prime([1, 1, 7], 3, 2)</span>
-<span class="sd">    [1, 4, 7]</span>
-
-<span class="sd">    Solutions [7, 4, 1] (mod 3**2) are generated by ``_raise_mod_power()``</span>
-<span class="sd">    from solution [1] (mod 3).</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-    <span class="n">X1</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="k">if</span> <span class="n">gf_eval</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">e</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">X1</span>
-    <span class="n">X</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">S</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">X1</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="n">X1</span><span class="p">)))</span>
-    <span class="k">while</span> <span class="n">S</span><span class="p">:</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">s</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">s</span> <span class="o">==</span> <span class="n">e</span><span class="p">:</span>
-            <span class="n">X</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">s1</span> <span class="o">=</span> <span class="n">s</span> <span class="o">+</span> <span class="mi">1</span>
-            <span class="n">ps</span> <span class="o">=</span> <span class="n">p</span><span class="o">**</span><span class="n">s</span>
-            <span class="n">S</span><span class="o">.</span><span class="n">extend</span><span class="p">([(</span><span class="n">x</span> <span class="o">+</span> <span class="n">v</span><span class="o">*</span><span class="n">ps</span><span class="p">,</span> <span class="n">s1</span><span class="p">)</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">_raise_mod_power</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">f</span><span class="p">)])</span>
-    <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="gf_csolve"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.galoistools.gf_csolve">[docs]</a><span class="k">def</span> <span class="nf">gf_csolve</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    To solve f(x) congruent 0 mod(n).</span>
-
-<span class="sd">    n is divided into canonical factors and f(x) cong 0 mod(p**e) will be</span>
-<span class="sd">    solved for each factor. Applying the Chinese Remainder Theorem to the</span>
-<span class="sd">    results returns the final answers.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Solve [1, 1, 7] congruent 0 mod(189):</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.galoistools import gf_csolve</span>
-<span class="sd">    &gt;&gt;&gt; gf_csolve([1, 1, 7], 189)</span>
-<span class="sd">    [13, 49, 76, 112, 139, 175]</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] &#39;An introduction to the Theory of Numbers&#39; 5th Edition by Ivan Niven,</span>
-<span class="sd">        Zuckerman and Montgomery.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-    <span class="n">P</span> <span class="o">=</span> <span class="n">factorint</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="n">X</span> <span class="o">=</span> <span class="p">[</span><span class="n">csolve_prime</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span> <span class="k">for</span> <span class="n">p</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">P</span><span class="o">.</span><span class="n">items</span><span class="p">()]</span>
-    <span class="n">pools</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="n">X</span><span class="p">))</span>
-    <span class="n">perms</span> <span class="o">=</span> <span class="p">[[]]</span>
-    <span class="k">for</span> <span class="n">pool</span> <span class="ow">in</span> <span class="n">pools</span><span class="p">:</span>
-        <span class="n">perms</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="o">+</span> <span class="p">[</span><span class="n">y</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">perms</span> <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">pool</span><span class="p">]</span>
-    <span class="n">dist_factors</span> <span class="o">=</span> <span class="p">[</span><span class="nb">pow</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span> <span class="k">for</span> <span class="n">p</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">P</span><span class="o">.</span><span class="n">items</span><span class="p">()]</span>
-    <span class="k">return</span> <span class="nb">sorted</span><span class="p">([</span><span class="n">gf_crt</span><span class="p">(</span><span class="n">per</span><span class="p">,</span> <span class="n">dist_factors</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span> <span class="k">for</span> <span class="n">per</span> <span class="ow">in</span> <span class="n">perms</span><span class="p">])</span></div>
-</pre></div>
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-            
-  <h1>Source code for sympy.polys.groebnertools</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Groebner bases algorithms. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.monomialtools</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">monomial_mul</span><span class="p">,</span> <span class="n">monomial_div</span><span class="p">,</span> <span class="n">monomial_lcm</span><span class="p">,</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.distributedpolys</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">sdp_LC</span><span class="p">,</span> <span class="n">sdp_LM</span><span class="p">,</span> <span class="n">sdp_LT</span><span class="p">,</span> <span class="n">sdp_mul_term</span><span class="p">,</span>
-    <span class="n">sdp_sub</span><span class="p">,</span> <span class="n">sdp_mul_term</span><span class="p">,</span> <span class="n">sdp_monic</span><span class="p">,</span>
-    <span class="n">sdp_rem</span><span class="p">,</span> <span class="n">sdp_strip</span><span class="p">,</span> <span class="n">sdp_sort</span><span class="p">,</span>
-    <span class="n">_term_ff_div</span><span class="p">,</span> <span class="n">_term_rr_div</span><span class="p">,</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">DomainError</span><span class="p">,</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyconfig</span> <span class="kn">import</span> <span class="n">query</span>
-
-
-<div class="viewcode-block" id="sdp_groebner"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.groebnertools.sdp_groebner">[docs]</a><span class="k">def</span> <span class="nf">sdp_groebner</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">gens</span><span class="o">=</span><span class="s">&#39;&#39;</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes Groebner basis for a set of polynomials in `K[X]`.</span>
-
-<span class="sd">    Wrapper around the (default) improved Buchberger and the other algorithms</span>
-<span class="sd">    for computing Groebner bases. The choice of algorithm can be changed via</span>
-<span class="sd">    ``method`` argument or :func:`setup` from :mod:`sympy.polys.polyconfig`,</span>
-<span class="sd">    where ``method`` can be either ``buchberger`` or ``f5b``.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">method</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">method</span> <span class="o">=</span> <span class="n">query</span><span class="p">(</span><span class="s">&#39;GB_METHOD&#39;</span><span class="p">)</span>
-
-    <span class="n">_groebner_methods</span> <span class="o">=</span> <span class="p">{</span>
-        <span class="s">&#39;buchberger&#39;</span><span class="p">:</span> <span class="n">buchberger</span><span class="p">,</span>
-        <span class="s">&#39;f5b&#39;</span><span class="p">:</span> <span class="n">f5b</span><span class="p">,</span>
-    <span class="p">}</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">func</span> <span class="o">=</span> <span class="n">_groebner_methods</span><span class="p">[</span><span class="n">method</span><span class="p">]</span>
-    <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;&#39;</span><span class="si">%s</span><span class="s">&#39; is not a valid Groebner bases algorithm (valid are &#39;buchberger&#39; and &#39;f5b&#39;)&quot;</span> <span class="o">%</span> <span class="n">method</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">func</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">verbose</span><span class="p">)</span>
-
-<span class="c"># Buchberger algorithm</span>
-
-</div>
-<div class="viewcode-block" id="buchberger"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.groebnertools.buchberger">[docs]</a><span class="k">def</span> <span class="nf">buchberger</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">gens</span><span class="o">=</span><span class="s">&#39;&#39;</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes Groebner basis for a set of polynomials in `K[X]`.</span>
-
-<span class="sd">    Given a set of multivariate polynomials `F`, finds another</span>
-<span class="sd">    set `G`, such that Ideal `F = Ideal G` and `G` is a reduced</span>
-<span class="sd">    Groebner basis.</span>
-
-<span class="sd">    The resulting basis is unique and has monic generators if the</span>
-<span class="sd">    ground domains is a field. Otherwise the result is non-unique</span>
-<span class="sd">    but Groebner bases over e.g. integers can be computed (if the</span>
-<span class="sd">    input polynomials are monic).</span>
-
-<span class="sd">    Groebner bases can be used to choose specific generators for a</span>
-<span class="sd">    polynomial ideal. Because these bases are unique you can check</span>
-<span class="sd">    for ideal equality by comparing the Groebner bases.  To see if</span>
-<span class="sd">    one polynomial lies in an ideal, divide by the elements in the</span>
-<span class="sd">    base and see if the remainder vanishes.</span>
-
-<span class="sd">    They can also be used to solve systems of polynomial equations</span>
-<span class="sd">    as,  by choosing lexicographic ordering,  you can eliminate one</span>
-<span class="sd">    variable at a time, provided that the ideal is zero-dimensional</span>
-<span class="sd">    (finite number of solutions).</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Bose03]_</span>
-<span class="sd">    2. [Giovini91]_</span>
-<span class="sd">    3. [Ajwa95]_</span>
-<span class="sd">    4. [Cox97]_</span>
-
-<span class="sd">    Algorithm used: an improved version of Buchberger&#39;s algorithm</span>
-<span class="sd">    as presented in T. Becker, V. Weispfenning, Groebner Bases: A</span>
-<span class="sd">    Computational Approach to Commutative Algebra, Springer, 1993,</span>
-<span class="sd">    page 232.</span>
-
-<span class="sd">    Added optional ``gens`` argument to apply :func:`sdp_str` for</span>
-<span class="sd">    the purpose of debugging the algorithm.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">K</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">DomainError</span><span class="p">(</span><span class="s">&quot;can&#39;t compute a Groebner basis over </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">select</span><span class="p">(</span><span class="n">P</span><span class="p">):</span>
-        <span class="c"># normal selection strategy</span>
-        <span class="c"># select the pair with minimum LCM(LM(f), LM(g))</span>
-        <span class="n">pr</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">pair</span><span class="p">:</span> <span class="n">O</span><span class="p">(</span>
-            <span class="n">monomial_lcm</span><span class="p">(</span><span class="n">sdp_LM</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">pair</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span> <span class="n">u</span><span class="p">),</span> <span class="n">sdp_LM</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">pair</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span> <span class="n">u</span><span class="p">))))</span>
-        <span class="k">return</span> <span class="n">pr</span>
-
-    <span class="k">def</span> <span class="nf">normal</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">J</span><span class="p">):</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">sdp_rem</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="p">[</span> <span class="n">f</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">J</span> <span class="p">],</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">h</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">sdp_monic</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">I</span><span class="p">:</span>
-                <span class="n">I</span><span class="p">[</span><span class="n">h</span><span class="p">]</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-                <span class="n">f</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-
-            <span class="k">return</span> <span class="n">sdp_LM</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">),</span> <span class="n">I</span><span class="p">[</span><span class="n">h</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">update</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">ih</span><span class="p">):</span>
-        <span class="c"># update G using the set of critical pairs B and h</span>
-        <span class="c"># [BW] page 230</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="n">ih</span><span class="p">]</span>
-        <span class="n">mh</span> <span class="o">=</span> <span class="n">sdp_LM</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-
-        <span class="c"># filter new pairs (h, g), g in G</span>
-        <span class="n">C</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-
-        <span class="k">while</span> <span class="n">C</span><span class="p">:</span>
-            <span class="c"># select a pair (h, g) by popping an element from C</span>
-            <span class="n">ig</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="n">ig</span><span class="p">]</span>
-            <span class="n">mg</span> <span class="o">=</span> <span class="n">sdp_LM</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-            <span class="n">LCMhg</span> <span class="o">=</span> <span class="n">monomial_lcm</span><span class="p">(</span><span class="n">mh</span><span class="p">,</span> <span class="n">mg</span><span class="p">)</span>
-
-            <span class="k">def</span> <span class="nf">lcm_divides</span><span class="p">(</span><span class="n">ip</span><span class="p">):</span>
-                <span class="c"># LCM(LM(h), LM(p)) divides LCM(LM(h), LM(g))</span>
-                <span class="n">m</span> <span class="o">=</span> <span class="n">monomial_lcm</span><span class="p">(</span><span class="n">mh</span><span class="p">,</span> <span class="n">sdp_LM</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">ip</span><span class="p">],</span> <span class="n">u</span><span class="p">))</span>
-                <span class="k">return</span> <span class="n">monomial_div</span><span class="p">(</span><span class="n">LCMhg</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-
-            <span class="c"># HT(h) and HT(g) disjoint: mh*mg == LCMhg</span>
-            <span class="k">if</span> <span class="n">monomial_mul</span><span class="p">(</span><span class="n">mh</span><span class="p">,</span> <span class="n">mg</span><span class="p">)</span> <span class="o">==</span> <span class="n">LCMhg</span> <span class="ow">or</span> <span class="p">(</span>
-                <span class="ow">not</span> <span class="nb">any</span><span class="p">(</span><span class="n">lcm_divides</span><span class="p">(</span><span class="n">ipx</span><span class="p">)</span> <span class="k">for</span> <span class="n">ipx</span> <span class="ow">in</span> <span class="n">C</span><span class="p">)</span> <span class="ow">and</span>
-                    <span class="ow">not</span> <span class="nb">any</span><span class="p">(</span><span class="n">lcm_divides</span><span class="p">(</span><span class="n">pr</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="k">for</span> <span class="n">pr</span> <span class="ow">in</span> <span class="n">D</span><span class="p">)):</span>
-                <span class="n">D</span><span class="o">.</span><span class="n">add</span><span class="p">((</span><span class="n">ih</span><span class="p">,</span> <span class="n">ig</span><span class="p">))</span>
-
-        <span class="n">E</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-
-        <span class="k">while</span> <span class="n">D</span><span class="p">:</span>
-            <span class="c"># select h, g from D (h the same as above)</span>
-            <span class="n">ih</span><span class="p">,</span> <span class="n">ig</span> <span class="o">=</span> <span class="n">D</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-            <span class="n">mg</span> <span class="o">=</span> <span class="n">sdp_LM</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">ig</span><span class="p">],</span> <span class="n">u</span><span class="p">)</span>
-            <span class="n">LCMhg</span> <span class="o">=</span> <span class="n">monomial_lcm</span><span class="p">(</span><span class="n">mh</span><span class="p">,</span> <span class="n">mg</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">monomial_mul</span><span class="p">(</span><span class="n">mh</span><span class="p">,</span> <span class="n">mg</span><span class="p">)</span> <span class="o">==</span> <span class="n">LCMhg</span><span class="p">:</span>
-                <span class="n">E</span><span class="o">.</span><span class="n">add</span><span class="p">((</span><span class="n">ih</span><span class="p">,</span> <span class="n">ig</span><span class="p">))</span>
-
-        <span class="c"># filter old pairs</span>
-        <span class="n">B_new</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-
-        <span class="k">while</span> <span class="n">B</span><span class="p">:</span>
-            <span class="c"># select g1, g2 from B (-&gt; CP)</span>
-            <span class="n">ig1</span><span class="p">,</span> <span class="n">ig2</span> <span class="o">=</span> <span class="n">B</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-            <span class="n">mg1</span> <span class="o">=</span> <span class="n">sdp_LM</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">ig1</span><span class="p">],</span> <span class="n">u</span><span class="p">)</span>
-            <span class="n">mg2</span> <span class="o">=</span> <span class="n">sdp_LM</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">ig2</span><span class="p">],</span> <span class="n">u</span><span class="p">)</span>
-            <span class="n">LCM12</span> <span class="o">=</span> <span class="n">monomial_lcm</span><span class="p">(</span><span class="n">mg1</span><span class="p">,</span> <span class="n">mg2</span><span class="p">)</span>
-
-            <span class="c"># if HT(h) does not divide lcm(HT(g1), HT(g2))</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">monomial_div</span><span class="p">(</span><span class="n">LCM12</span><span class="p">,</span> <span class="n">mh</span><span class="p">)</span> <span class="ow">or</span> \
-                <span class="n">monomial_lcm</span><span class="p">(</span><span class="n">mg1</span><span class="p">,</span> <span class="n">mh</span><span class="p">)</span> <span class="o">==</span> <span class="n">LCM12</span> <span class="ow">or</span> \
-                    <span class="n">monomial_lcm</span><span class="p">(</span><span class="n">mg2</span><span class="p">,</span> <span class="n">mh</span><span class="p">)</span> <span class="o">==</span> <span class="n">LCM12</span><span class="p">:</span>
-                <span class="n">B_new</span><span class="o">.</span><span class="n">add</span><span class="p">((</span><span class="n">ig1</span><span class="p">,</span> <span class="n">ig2</span><span class="p">))</span>
-
-        <span class="n">B_new</span> <span class="o">|=</span> <span class="n">E</span>
-
-        <span class="c"># filter polynomials</span>
-        <span class="n">G_new</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-
-        <span class="k">while</span> <span class="n">G</span><span class="p">:</span>
-            <span class="n">ig</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-            <span class="n">mg</span> <span class="o">=</span> <span class="n">sdp_LM</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">ig</span><span class="p">],</span> <span class="n">u</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">monomial_div</span><span class="p">(</span><span class="n">mg</span><span class="p">,</span> <span class="n">mh</span><span class="p">):</span>
-                <span class="n">G_new</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">ig</span><span class="p">)</span>
-
-        <span class="n">G_new</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">ih</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">G_new</span><span class="p">,</span> <span class="n">B_new</span>
-        <span class="c"># end of update ################################</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-    <span class="c"># replace f with a reduced list of initial polynomials; see [BW] page 203</span>
-    <span class="n">f1</span> <span class="o">=</span> <span class="n">f</span><span class="p">[:]</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">f1</span><span class="p">[:]</span>
-        <span class="n">f1</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)):</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">f</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">sdp_rem</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">f</span><span class="p">[:</span><span class="n">i</span><span class="p">],</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">r</span><span class="p">:</span>
-                <span class="n">f1</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sdp_monic</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">f</span> <span class="o">==</span> <span class="n">f1</span><span class="p">:</span>
-            <span class="k">break</span>
-
-    <span class="n">f</span> <span class="o">=</span> <span class="p">[</span><span class="nb">tuple</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">f</span><span class="p">]</span>
-    <span class="n">I</span> <span class="o">=</span> <span class="p">{}</span>            <span class="c"># ip = I[p]; p = f[ip]</span>
-    <span class="n">F</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>         <span class="c"># set of indices of polynomials</span>
-    <span class="n">G</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>         <span class="c"># set of indices of intermediate would-be Groebner basis</span>
-    <span class="n">CP</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>        <span class="c"># set of pairs of indices of critical pairs</span>
-
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">h</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="n">I</span><span class="p">[</span><span class="n">h</span><span class="p">]</span> <span class="o">=</span> <span class="n">i</span>
-        <span class="n">F</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-
-    <span class="c">#####################################</span>
-    <span class="c"># algorithm GROEBNERNEWS2 in [BW] page 232</span>
-    <span class="k">while</span> <span class="n">F</span><span class="p">:</span>
-        <span class="c"># select p with minimum monomial according to the monomial ordering O</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="nb">min</span><span class="p">([</span><span class="n">f</span><span class="p">[</span><span class="n">x</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">F</span><span class="p">],</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">f</span><span class="p">:</span> <span class="n">O</span><span class="p">(</span><span class="n">sdp_LM</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)))</span>
-        <span class="n">ih</span> <span class="o">=</span> <span class="n">I</span><span class="p">[</span><span class="n">h</span><span class="p">]</span>
-        <span class="n">F</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">ih</span><span class="p">)</span>
-        <span class="n">G</span><span class="p">,</span> <span class="n">CP</span> <span class="o">=</span> <span class="n">update</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">CP</span><span class="p">,</span> <span class="n">ih</span><span class="p">)</span>
-
-    <span class="c"># count the number of critical pairs which reduce to zero</span>
-    <span class="n">reductions_to_zero</span> <span class="o">=</span> <span class="mi">0</span>
-
-    <span class="k">while</span> <span class="n">CP</span><span class="p">:</span>
-        <span class="n">ig1</span><span class="p">,</span> <span class="n">ig2</span> <span class="o">=</span> <span class="n">select</span><span class="p">(</span><span class="n">CP</span><span class="p">)</span>
-        <span class="n">CP</span><span class="o">.</span><span class="n">remove</span><span class="p">((</span><span class="n">ig1</span><span class="p">,</span> <span class="n">ig2</span><span class="p">))</span>
-
-        <span class="n">h</span> <span class="o">=</span> <span class="n">sdp_spoly</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">ig1</span><span class="p">],</span> <span class="n">f</span><span class="p">[</span><span class="n">ig2</span><span class="p">],</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="c"># ordering divisors is on average more efficient [Cox] page 111</span>
-        <span class="n">G1</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">g</span><span class="p">:</span> <span class="n">O</span><span class="p">(</span><span class="n">sdp_LM</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">g</span><span class="p">],</span> <span class="n">u</span><span class="p">)))</span>
-        <span class="n">ht</span> <span class="o">=</span> <span class="n">normal</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">G1</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">ht</span><span class="p">:</span>
-            <span class="n">G</span><span class="p">,</span> <span class="n">CP</span> <span class="o">=</span> <span class="n">update</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">CP</span><span class="p">,</span> <span class="n">ht</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">reductions_to_zero</span> <span class="o">+=</span> <span class="mi">1</span>
-
-    <span class="c">######################################</span>
-    <span class="c"># now G is a Groebner basis; reduce it</span>
-    <span class="n">Gr</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-
-    <span class="k">for</span> <span class="n">ig</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
-        <span class="n">ht</span> <span class="o">=</span> <span class="n">normal</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">ig</span><span class="p">],</span> <span class="n">G</span> <span class="o">-</span> <span class="nb">set</span><span class="p">([</span><span class="n">ig</span><span class="p">]))</span>
-
-        <span class="k">if</span> <span class="n">ht</span><span class="p">:</span>
-            <span class="n">Gr</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">ht</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-
-    <span class="n">Gr</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">ig</span><span class="p">])</span> <span class="k">for</span> <span class="n">ig</span> <span class="ow">in</span> <span class="n">Gr</span><span class="p">]</span>
-
-    <span class="c"># order according to the monomial ordering</span>
-    <span class="n">Gr</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">Gr</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">f</span><span class="p">:</span> <span class="n">O</span><span class="p">(</span><span class="n">sdp_LM</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)),</span> <span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-        <span class="k">print</span><span class="p">(</span><span class="s">&#39;reductions_to_zero = </span><span class="si">%d</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">reductions_to_zero</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">Gr</span>
-
-</div>
-<span class="k">def</span> <span class="nf">sdp_str</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gens</span><span class="p">):</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">gens</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="n">gens</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;,&#39;</span><span class="p">)</span>
-    <span class="n">ngens</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-    <span class="k">for</span> <span class="n">expv</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">c</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">s</span> <span class="o">+=</span> <span class="s">&#39; +&#39;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">s</span> <span class="o">+=</span> <span class="s">&#39; -&#39;</span>
-        <span class="k">if</span> <span class="n">c</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="o">-</span><span class="n">c</span>
-        <span class="k">if</span> <span class="n">c</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>  <span class="c"># and expv != z:</span>
-            <span class="n">cnt1</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">cnt1</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-        <span class="n">sa</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">ngens</span><span class="p">):</span>
-            <span class="n">exp</span> <span class="o">=</span> <span class="n">expv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">exp</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">sa</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s">^</span><span class="si">%d</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">gens</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">exp</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">exp</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">sa</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">gens</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">cnt1</span><span class="p">:</span>
-            <span class="n">sa</span> <span class="o">=</span> <span class="p">[</span><span class="n">cnt1</span><span class="p">]</span> <span class="o">+</span> <span class="n">sa</span>
-        <span class="n">s</span> <span class="o">+=</span> <span class="s">&#39;*&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">sa</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">s</span>
-
-
-<div class="viewcode-block" id="sdp_spoly"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.groebnertools.sdp_spoly">[docs]</a><span class="k">def</span> <span class="nf">sdp_spoly</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute LCM(LM(p1), LM(p2))/LM(p1)*p1 - LCM(LM(p1), LM(p2))/LM(p2)*p2</span>
-<span class="sd">    This is the S-poly provided p1 and p2 are monic</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">LM1</span> <span class="o">=</span> <span class="n">sdp_LM</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-    <span class="n">LM2</span> <span class="o">=</span> <span class="n">sdp_LM</span><span class="p">(</span><span class="n">p2</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-    <span class="n">LCM12</span> <span class="o">=</span> <span class="n">monomial_lcm</span><span class="p">(</span><span class="n">LM1</span><span class="p">,</span> <span class="n">LM2</span><span class="p">)</span>
-    <span class="n">m1</span> <span class="o">=</span> <span class="n">monomial_div</span><span class="p">(</span><span class="n">LCM12</span><span class="p">,</span> <span class="n">LM1</span><span class="p">)</span>
-    <span class="n">m2</span> <span class="o">=</span> <span class="n">monomial_div</span><span class="p">(</span><span class="n">LCM12</span><span class="p">,</span> <span class="n">LM2</span><span class="p">)</span>
-    <span class="n">s1</span> <span class="o">=</span> <span class="n">sdp_mul_term</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="p">(</span><span class="n">m1</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">s2</span> <span class="o">=</span> <span class="n">sdp_mul_term</span><span class="p">(</span><span class="n">p2</span><span class="p">,</span> <span class="p">(</span><span class="n">m2</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="n">sdp_sub</span><span class="p">(</span><span class="n">s1</span><span class="p">,</span> <span class="n">s2</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">s</span>
-
-<span class="c"># F5B</span>
-
-<span class="c"># convenience functions</span>
-
-</div>
-<span class="k">def</span> <span class="nf">Sign</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">Polyn</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">Num</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">f</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">sig</span><span class="p">(</span><span class="n">monomial</span><span class="p">,</span> <span class="n">index</span><span class="p">):</span>
-    <span class="k">return</span> <span class="p">(</span><span class="n">monomial</span><span class="p">,</span> <span class="n">index</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">lbp</span><span class="p">(</span><span class="n">signature</span><span class="p">,</span> <span class="n">polynomial</span><span class="p">,</span> <span class="n">number</span><span class="p">):</span>
-    <span class="k">return</span> <span class="p">(</span><span class="n">signature</span><span class="p">,</span> <span class="n">polynomial</span><span class="p">,</span> <span class="n">number</span><span class="p">)</span>
-
-<span class="c"># signature functions</span>
-
-
-<span class="k">def</span> <span class="nf">sig_cmp</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">O</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compare two signatures by extending the term order to K[X]^n.</span>
-
-<span class="sd">    u &lt; v iff</span>
-<span class="sd">        - the index of v is greater than the index of u</span>
-<span class="sd">    or</span>
-<span class="sd">        - the index of v is equal to the index of u and u[0] &lt; v[0] w.r.t. O</span>
-
-<span class="sd">    u &gt; v otherwise</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span>
-        <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-    <span class="k">if</span> <span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span>
-        <span class="c">#if u[0] == v[0]:</span>
-        <span class="c">#    return 0</span>
-        <span class="k">if</span> <span class="n">O</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">&lt;</span> <span class="n">O</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
-            <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-    <span class="k">return</span> <span class="mi">1</span>
-
-
-<span class="k">def</span> <span class="nf">sig_key</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">O</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Key for comparing two signatures.</span>
-
-<span class="sd">    s = (m, k), t = (n, l)</span>
-
-<span class="sd">    s &lt; t iff [k &gt; l] or [k == l and m &lt; n]</span>
-<span class="sd">    s &gt; t otherwise</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">O</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-
-
-<span class="k">def</span> <span class="nf">sig_mult</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Multiply a signature by a monomial.</span>
-
-<span class="sd">    The product of a signature (m, i) and a monomial n is defined as</span>
-<span class="sd">    (m * t, i).</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">sig</span><span class="p">(</span><span class="n">monomial_mul</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">m</span><span class="p">),</span> <span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-
-<span class="c"># labeled polynomial functions</span>
-
-
-<span class="k">def</span> <span class="nf">lbp_sub</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Subtract labeled polynomial g from f.</span>
-
-<span class="sd">    The signature and number of the difference of f and g are signature</span>
-<span class="sd">    and number of the maximum of f and g, w.r.t. lbp_cmp.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">sig_cmp</span><span class="p">(</span><span class="n">Sign</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="n">Sign</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">O</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">max_poly</span> <span class="o">=</span> <span class="n">g</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">max_poly</span> <span class="o">=</span> <span class="n">f</span>
-
-    <span class="n">ret</span> <span class="o">=</span> <span class="n">sdp_sub</span><span class="p">(</span><span class="n">Polyn</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="n">Polyn</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">lbp</span><span class="p">(</span><span class="n">Sign</span><span class="p">(</span><span class="n">max_poly</span><span class="p">),</span> <span class="n">ret</span><span class="p">,</span> <span class="n">Num</span><span class="p">(</span><span class="n">max_poly</span><span class="p">))</span>
-
-
-<span class="k">def</span> <span class="nf">lbp_mul_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">cx</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Multiply a labeled polynomial with a term.</span>
-
-<span class="sd">    The product of a labeled polynomial (s, p, k) by a monomial is</span>
-<span class="sd">    defined as (m * s, m * p, k).</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">lbp</span><span class="p">(</span><span class="n">sig_mult</span><span class="p">(</span><span class="n">Sign</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="n">cx</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">sdp_mul_term</span><span class="p">(</span><span class="n">Polyn</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="n">cx</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">Num</span><span class="p">(</span><span class="n">f</span><span class="p">))</span>
-
-
-<span class="k">def</span> <span class="nf">lbp_cmp</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">O</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compare two labeled polynomials.</span>
-
-<span class="sd">    f &lt; g iff</span>
-<span class="sd">        - Sign(f) &lt; Sign(g)</span>
-<span class="sd">    or</span>
-<span class="sd">        - Sign(f) == Sign(g) and Num(f) &gt; Num(g)</span>
-
-<span class="sd">    f &gt; g otherwise</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">sig_cmp</span><span class="p">(</span><span class="n">Sign</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="n">Sign</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">O</span><span class="p">)</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-    <span class="k">if</span> <span class="n">Sign</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">==</span> <span class="n">Sign</span><span class="p">(</span><span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">Num</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">Num</span><span class="p">(</span><span class="n">g</span><span class="p">):</span>
-            <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-        <span class="c">#if Num(f) == Num(g):</span>
-        <span class="c">#    return 0</span>
-    <span class="k">return</span> <span class="mi">1</span>
-
-
-<span class="k">def</span> <span class="nf">lbp_key</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">O</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Key for comparing two labeled polynomials.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">(</span><span class="n">sig_key</span><span class="p">(</span><span class="n">Sign</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="n">O</span><span class="p">),</span> <span class="o">-</span><span class="n">Num</span><span class="p">(</span><span class="n">f</span><span class="p">))</span>
-
-<span class="c"># algorithm and helper functions</span>
-
-
-<span class="k">def</span> <span class="nf">critical_pair</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute the critical pair corresponding to two labeled polynomials.</span>
-
-<span class="sd">    A critical pair is a tuple (um, f, vm, g), where um and vm are</span>
-<span class="sd">    terms such that um * f - vm * g is the S-polynomial of f and g (so,</span>
-<span class="sd">    wlog assume um * f &gt; vm * g).</span>
-<span class="sd">    For performance sake, a critical pair is represented as a tuple</span>
-<span class="sd">    (Sign(um * f), um, f, Sign(vm * g), vm, g), since um * f creates</span>
-<span class="sd">    a new, relatively expensive object in memory, whereas Sign(um *</span>
-<span class="sd">    f) and um are lightweight and f (in the tuple) is a reference to</span>
-<span class="sd">    an already existing object in memory.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">ltf</span> <span class="o">=</span> <span class="n">sdp_LT</span><span class="p">(</span><span class="n">Polyn</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">ltg</span> <span class="o">=</span> <span class="n">sdp_LT</span><span class="p">(</span><span class="n">Polyn</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">lt</span> <span class="o">=</span> <span class="p">(</span><span class="n">monomial_lcm</span><span class="p">(</span><span class="n">ltf</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">ltg</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-        <span class="n">term_div</span> <span class="o">=</span> <span class="n">_term_ff_div</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">term_div</span> <span class="o">=</span> <span class="n">_term_rr_div</span>
-
-    <span class="n">um</span> <span class="o">=</span> <span class="n">term_div</span><span class="p">(</span><span class="n">lt</span><span class="p">,</span> <span class="n">ltf</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">vm</span> <span class="o">=</span> <span class="n">term_div</span><span class="p">(</span><span class="n">lt</span><span class="p">,</span> <span class="n">ltg</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="c"># The full information is not needed (now), so only the product</span>
-    <span class="c"># with the leading term is considered:</span>
-    <span class="n">fr</span> <span class="o">=</span> <span class="n">lbp_mul_term</span><span class="p">(</span>
-        <span class="n">lbp</span><span class="p">(</span><span class="n">Sign</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="p">[</span><span class="n">sdp_LT</span><span class="p">(</span><span class="n">Polyn</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)],</span> <span class="n">Num</span><span class="p">(</span><span class="n">f</span><span class="p">)),</span> <span class="n">um</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">gr</span> <span class="o">=</span> <span class="n">lbp_mul_term</span><span class="p">(</span>
-        <span class="n">lbp</span><span class="p">(</span><span class="n">Sign</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="p">[</span><span class="n">sdp_LT</span><span class="p">(</span><span class="n">Polyn</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">)],</span> <span class="n">Num</span><span class="p">(</span><span class="n">g</span><span class="p">)),</span> <span class="n">vm</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="c"># return in proper order, such that the S-polynomial is just</span>
-    <span class="c"># u_first * f_first - u_second * f_second:</span>
-    <span class="k">if</span> <span class="n">lbp_cmp</span><span class="p">(</span><span class="n">fr</span><span class="p">,</span> <span class="n">gr</span><span class="p">,</span> <span class="n">O</span><span class="p">)</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">Sign</span><span class="p">(</span><span class="n">gr</span><span class="p">),</span> <span class="n">vm</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">Sign</span><span class="p">(</span><span class="n">fr</span><span class="p">),</span> <span class="n">um</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">Sign</span><span class="p">(</span><span class="n">fr</span><span class="p">),</span> <span class="n">um</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">Sign</span><span class="p">(</span><span class="n">gr</span><span class="p">),</span> <span class="n">vm</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">cp_cmp</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">O</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compare two critical pairs c and d.</span>
-
-<span class="sd">    c &lt; d iff</span>
-<span class="sd">        - lbp(c[0], _, Num(c[2]) &lt; lbp(d[0], _, Num(d[2])) (this</span>
-<span class="sd">        corresponds to um_c * f_c and um_d * f_d)</span>
-<span class="sd">    or</span>
-<span class="sd">        - lbp(c[0], _, Num(c[2]) &gt;&lt; lbp(d[0], _, Num(d[2])) and</span>
-<span class="sd">        lbp(c[3], _, Num(c[5])) &lt; lbp(d[3], _, Num(d[5])) (this</span>
-<span class="sd">        corresponds to vm_c * g_c and vm_d * g_d)</span>
-
-<span class="sd">    c &gt; d otherwise</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">c0</span> <span class="o">=</span> <span class="n">lbp</span><span class="p">(</span><span class="n">c</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="n">Num</span><span class="p">(</span><span class="n">c</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span>
-    <span class="n">d0</span> <span class="o">=</span> <span class="n">lbp</span><span class="p">(</span><span class="n">d</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="n">Num</span><span class="p">(</span><span class="n">d</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span>
-
-    <span class="n">r</span> <span class="o">=</span> <span class="n">lbp_cmp</span><span class="p">(</span><span class="n">c0</span><span class="p">,</span> <span class="n">d0</span><span class="p">,</span> <span class="n">O</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">r</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-    <span class="k">if</span> <span class="n">r</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">c1</span> <span class="o">=</span> <span class="n">lbp</span><span class="p">(</span><span class="n">c</span><span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="p">[],</span> <span class="n">Num</span><span class="p">(</span><span class="n">c</span><span class="p">[</span><span class="mi">5</span><span class="p">]))</span>
-        <span class="n">d1</span> <span class="o">=</span> <span class="n">lbp</span><span class="p">(</span><span class="n">d</span><span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="p">[],</span> <span class="n">Num</span><span class="p">(</span><span class="n">d</span><span class="p">[</span><span class="mi">5</span><span class="p">]))</span>
-
-        <span class="n">r</span> <span class="o">=</span> <span class="n">lbp_cmp</span><span class="p">(</span><span class="n">c1</span><span class="p">,</span> <span class="n">d1</span><span class="p">,</span> <span class="n">O</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">r</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-        <span class="c">#if r == 0:</span>
-        <span class="c">#    return 0</span>
-    <span class="k">return</span> <span class="mi">1</span>
-
-
-<span class="k">def</span> <span class="nf">cp_key</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">O</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Key for comparing critical pairs.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">(</span><span class="n">lbp_key</span><span class="p">(</span><span class="n">lbp</span><span class="p">(</span><span class="n">c</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="n">Num</span><span class="p">(</span><span class="n">c</span><span class="p">[</span><span class="mi">2</span><span class="p">])),</span> <span class="n">O</span><span class="p">),</span> <span class="n">lbp_key</span><span class="p">(</span><span class="n">lbp</span><span class="p">(</span><span class="n">c</span><span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="p">[],</span> <span class="n">Num</span><span class="p">(</span><span class="n">c</span><span class="p">[</span><span class="mi">5</span><span class="p">])),</span> <span class="n">O</span><span class="p">))</span>
-
-
-<span class="k">def</span> <span class="nf">s_poly</span><span class="p">(</span><span class="n">cp</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute the S-polynomial of a critical pair.</span>
-
-<span class="sd">    The S-polynomial of a critical pair cp is cp[1] * cp[2] - cp[4] * cp[5].</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">lbp_sub</span><span class="p">(</span><span class="n">lbp_mul_term</span><span class="p">(</span><span class="n">cp</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">cp</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">lbp_mul_term</span><span class="p">(</span><span class="n">cp</span><span class="p">[</span><span class="mi">5</span><span class="p">],</span> <span class="n">cp</span><span class="p">[</span><span class="mi">4</span><span class="p">],</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">is_rewritable_or_comparable</span><span class="p">(</span><span class="n">sign</span><span class="p">,</span> <span class="n">num</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Check if a labeled polynomial is redundant by checking if its</span>
-<span class="sd">    signature and number imply rewritability or comparability.</span>
-
-<span class="sd">    (sign, num) is comparable if there exists a labeled polynomial</span>
-<span class="sd">    h in B, such that sign[1] (the index) is less than Sign(h)[1]</span>
-<span class="sd">    and sign[0] is divisible by the leading monomial of h.</span>
-
-<span class="sd">    (sign, num) is rewritable if there exists a labeled polynomial</span>
-<span class="sd">    h in B, such thatsign[1] is equal to Sign(h)[1], num &lt; Num(h)</span>
-<span class="sd">    and sign[0] is divisible by Sign(h)[0].</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">B</span><span class="p">:</span>
-        <span class="c"># comparable</span>
-        <span class="k">if</span> <span class="n">sign</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">Sign</span><span class="p">(</span><span class="n">h</span><span class="p">)[</span><span class="mi">1</span><span class="p">]:</span>
-            <span class="k">if</span> <span class="n">monomial_divides</span><span class="p">(</span><span class="n">sign</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sdp_LM</span><span class="p">(</span><span class="n">Polyn</span><span class="p">(</span><span class="n">h</span><span class="p">),</span> <span class="n">u</span><span class="p">)):</span>
-                <span class="k">return</span> <span class="bp">True</span>
-
-        <span class="c"># rewritable</span>
-        <span class="k">if</span> <span class="n">sign</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">Sign</span><span class="p">(</span><span class="n">h</span><span class="p">)[</span><span class="mi">1</span><span class="p">]:</span>
-            <span class="k">if</span> <span class="n">num</span> <span class="o">&lt;</span> <span class="n">Num</span><span class="p">(</span><span class="n">h</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">monomial_divides</span><span class="p">(</span><span class="n">sign</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Sign</span><span class="p">(</span><span class="n">h</span><span class="p">)[</span><span class="mi">0</span><span class="p">]):</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-    <span class="k">return</span> <span class="bp">False</span>
-
-
-<span class="k">def</span> <span class="nf">f5_reduce</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    F5-reduce a labeled polynomial f by B.</span>
-
-<span class="sd">    Continously searches for non-zero labeled polynomial h in B, such</span>
-<span class="sd">    that the leading term lt_h of h divides the leading term lt_f of</span>
-<span class="sd">    f and Sign(lt_h * h) &lt; Sign(f). If such a labeled polynomial h is</span>
-<span class="sd">    found, f gets replaced by f - lt_f / lt_h * h. If no such h can be</span>
-<span class="sd">    found or f is 0, f is no further F5-reducible and f gets returned.</span>
-
-<span class="sd">    A polynomial that is reducible in the usual sense (sdp_rem)</span>
-<span class="sd">    need not be F5-reducible, e.g.:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.groebnertools import lbp, sig, f5_reduce, Polyn</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.distributedpolys import sdp_rem</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.monomialtools import lex</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import QQ</span>
-
-<span class="sd">    &gt;&gt;&gt; f = lbp(sig((1, 1, 1), 4), [((1, 0, 0), QQ(1))], 3)</span>
-<span class="sd">    &gt;&gt;&gt; g = lbp(sig((0, 0, 0), 2), [((1, 0, 0), QQ(1))], 2)</span>
-
-<span class="sd">    &gt;&gt;&gt; sdp_rem(Polyn(f), [Polyn(g)], 2, lex, QQ)</span>
-<span class="sd">    []</span>
-<span class="sd">    &gt;&gt;&gt; f5_reduce(f, [g], 2, lex, QQ)</span>
-<span class="sd">    (((1, 1, 1), 4), [((1, 0, 0), 1/1)], 3)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">Polyn</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">==</span> <span class="p">[]:</span>
-        <span class="k">return</span> <span class="n">f</span>
-
-    <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-        <span class="n">term_div</span> <span class="o">=</span> <span class="n">_term_ff_div</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">term_div</span> <span class="o">=</span> <span class="n">_term_rr_div</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">f</span>
-
-        <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">B</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">Polyn</span><span class="p">(</span><span class="n">h</span><span class="p">)</span> <span class="o">!=</span> <span class="p">[]:</span>
-                <span class="k">if</span> <span class="n">monomial_divides</span><span class="p">(</span><span class="n">sdp_LM</span><span class="p">(</span><span class="n">Polyn</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="n">u</span><span class="p">),</span> <span class="n">sdp_LM</span><span class="p">(</span><span class="n">Polyn</span><span class="p">(</span><span class="n">h</span><span class="p">),</span> <span class="n">u</span><span class="p">)):</span>
-                    <span class="n">t</span> <span class="o">=</span> <span class="n">term_div</span><span class="p">(</span>
-                        <span class="n">sdp_LT</span><span class="p">(</span><span class="n">Polyn</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">sdp_LT</span><span class="p">(</span><span class="n">Polyn</span><span class="p">(</span><span class="n">h</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">sig_cmp</span><span class="p">(</span><span class="n">sig_mult</span><span class="p">(</span><span class="n">Sign</span><span class="p">(</span><span class="n">h</span><span class="p">),</span> <span class="n">t</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">Sign</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="n">O</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="c"># The following check need not be done and is in general slower than without.</span>
-                        <span class="c">#if not is_rewritable_or_comparable(Sign(gp), Num(gp), B, u, K):</span>
-                        <span class="n">hp</span> <span class="o">=</span> <span class="n">lbp_mul_term</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-                        <span class="n">f</span> <span class="o">=</span> <span class="n">lbp_sub</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">hp</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-                        <span class="k">break</span>
-
-        <span class="k">if</span> <span class="n">g</span> <span class="o">==</span> <span class="n">f</span> <span class="ow">or</span> <span class="n">Polyn</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">==</span> <span class="p">[]:</span>
-            <span class="k">return</span> <span class="n">f</span>
-
-
-<div class="viewcode-block" id="f5b"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.groebnertools.f5b">[docs]</a><span class="k">def</span> <span class="nf">f5b</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">gens</span><span class="o">=</span><span class="s">&#39;&#39;</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes a reduced Groebner basis for the ideal generated by F.</span>
-
-<span class="sd">    f5b is an implementation of the F5B algorithm by Yao Sun and</span>
-<span class="sd">    Dingkang Wang. Similarly to Buchberger&#39;s algorithm, the algorithm</span>
-<span class="sd">    proceeds by computing critical pairs, computing the S-polynomial,</span>
-<span class="sd">    reducing it and adjoining the reduced S-polynomial if it is not 0.</span>
-
-<span class="sd">    Unlike Buchberger&#39;s algorithm, each polynomial contains additional</span>
-<span class="sd">    information, namely a signature and a number. The signature</span>
-<span class="sd">    specifies the path of computation (i.e. from which polynomial in</span>
-<span class="sd">    the original basis was it derived and how), the number says when</span>
-<span class="sd">    the polynomial was added to the basis.  With this information it</span>
-<span class="sd">    is (often) possible to decide if an S-polynomial will reduce to</span>
-<span class="sd">    0 and can be discarded.</span>
-
-<span class="sd">    Optimizations include: Reducing the generators before computing</span>
-<span class="sd">    a Groebner basis, removing redundant critical pairs when a new</span>
-<span class="sd">    polynomial enters the basis and sorting the critical pairs and</span>
-<span class="sd">    the current basis.</span>
-
-<span class="sd">    Once a Groebner basis has been found, it gets reduced.</span>
-
-<span class="sd">    ** References **</span>
-<span class="sd">    Yao Sun, Dingkang Wang: &quot;A New Proof for the Correctness of F5</span>
-<span class="sd">    (F5-Like) Algorithm&quot;, http://arxiv.org/abs/1004.0084 (specifically</span>
-<span class="sd">    v4)</span>
-
-<span class="sd">    Thomas Becker, Volker Weispfenning, Groebner bases: A computational</span>
-<span class="sd">    approach to commutative algebra, 1993, p. 203, 216</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">K</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">DomainError</span><span class="p">(</span><span class="s">&quot;can&#39;t compute a Groebner basis over </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="c"># reduce polynomials (like in Mario Pernici&#39;s implementation) (Becker, Weispfenning, p. 203)</span>
-    <span class="n">B</span> <span class="o">=</span> <span class="n">F</span>
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">F</span> <span class="o">=</span> <span class="n">B</span>
-        <span class="n">B</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">F</span><span class="p">)):</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">F</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">sdp_rem</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">F</span><span class="p">[:</span><span class="n">i</span><span class="p">],</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">r</span> <span class="o">!=</span> <span class="p">[]:</span>
-                <span class="n">B</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">F</span> <span class="o">==</span> <span class="n">B</span><span class="p">:</span>
-            <span class="k">break</span>
-
-    <span class="c"># basis</span>
-    <span class="n">B</span> <span class="o">=</span> <span class="p">[</span><span class="n">lbp</span><span class="p">(</span><span class="n">sig</span><span class="p">((</span><span class="mi">0</span><span class="p">,)</span> <span class="o">*</span> <span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">F</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">F</span><span class="p">))]</span>
-    <span class="n">B</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">f</span><span class="p">:</span> <span class="n">O</span><span class="p">(</span><span class="n">sdp_LM</span><span class="p">(</span><span class="n">Polyn</span><span class="p">(</span><span class="n">f</span><span class="p">),</span> <span class="n">u</span><span class="p">)),</span> <span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="c"># critical pairs</span>
-    <span class="n">CP</span> <span class="o">=</span> <span class="p">[</span><span class="n">critical_pair</span><span class="p">(</span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">B</span><span class="p">[</span><span class="n">j</span><span class="p">],</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span>
-        <span class="nb">len</span><span class="p">(</span><span class="n">B</span><span class="p">))</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">B</span><span class="p">))]</span>
-    <span class="n">CP</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">cp</span><span class="p">:</span> <span class="n">cp_key</span><span class="p">(</span><span class="n">cp</span><span class="p">,</span> <span class="n">O</span><span class="p">),</span> <span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="n">k</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
-
-    <span class="n">reductions_to_zero</span> <span class="o">=</span> <span class="mi">0</span>
-
-    <span class="k">while</span> <span class="nb">len</span><span class="p">(</span><span class="n">CP</span><span class="p">):</span>
-        <span class="n">cp</span> <span class="o">=</span> <span class="n">CP</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-
-        <span class="c"># discard redundant critical pairs:</span>
-        <span class="k">if</span> <span class="n">is_rewritable_or_comparable</span><span class="p">(</span><span class="n">cp</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Num</span><span class="p">(</span><span class="n">cp</span><span class="p">[</span><span class="mi">2</span><span class="p">]),</span> <span class="n">B</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-            <span class="k">continue</span>
-        <span class="k">if</span> <span class="n">is_rewritable_or_comparable</span><span class="p">(</span><span class="n">cp</span><span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="n">Num</span><span class="p">(</span><span class="n">cp</span><span class="p">[</span><span class="mi">5</span><span class="p">]),</span> <span class="n">B</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-            <span class="k">continue</span>
-
-        <span class="n">s</span> <span class="o">=</span> <span class="n">s_poly</span><span class="p">(</span><span class="n">cp</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">p</span> <span class="o">=</span> <span class="n">f5_reduce</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">p</span> <span class="o">=</span> <span class="n">lbp</span><span class="p">(</span><span class="n">Sign</span><span class="p">(</span><span class="n">p</span><span class="p">),</span> <span class="n">sdp_monic</span><span class="p">(</span><span class="n">Polyn</span><span class="p">(</span><span class="n">p</span><span class="p">),</span> <span class="n">K</span><span class="p">),</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">Polyn</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="o">!=</span> <span class="p">[]:</span>
-            <span class="c"># remove old critical pairs, that become redundant when adding p:</span>
-            <span class="n">indices</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">cp</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">CP</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">is_rewritable_or_comparable</span><span class="p">(</span><span class="n">cp</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Num</span><span class="p">(</span><span class="n">cp</span><span class="p">[</span><span class="mi">2</span><span class="p">]),</span> <span class="p">[</span><span class="n">p</span><span class="p">],</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-                    <span class="n">indices</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="n">is_rewritable_or_comparable</span><span class="p">(</span><span class="n">cp</span><span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="n">Num</span><span class="p">(</span><span class="n">cp</span><span class="p">[</span><span class="mi">5</span><span class="p">]),</span> <span class="p">[</span><span class="n">p</span><span class="p">],</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-                    <span class="n">indices</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">indices</span><span class="p">):</span>
-                <span class="k">del</span> <span class="n">CP</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-
-            <span class="c"># only add new critical pairs that are not made redundant by p:</span>
-            <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">B</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">Polyn</span><span class="p">(</span><span class="n">g</span><span class="p">)</span> <span class="o">!=</span> <span class="p">[]:</span>
-                    <span class="n">cp</span> <span class="o">=</span> <span class="n">critical_pair</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">is_rewritable_or_comparable</span><span class="p">(</span><span class="n">cp</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Num</span><span class="p">(</span><span class="n">cp</span><span class="p">[</span><span class="mi">2</span><span class="p">]),</span> <span class="p">[</span><span class="n">p</span><span class="p">],</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-                        <span class="k">continue</span>
-                    <span class="k">elif</span> <span class="n">is_rewritable_or_comparable</span><span class="p">(</span><span class="n">cp</span><span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="n">Num</span><span class="p">(</span><span class="n">cp</span><span class="p">[</span><span class="mi">5</span><span class="p">]),</span> <span class="p">[</span><span class="n">p</span><span class="p">],</span> <span class="n">u</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-                        <span class="k">continue</span>
-
-                    <span class="n">CP</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">cp</span><span class="p">)</span>
-
-            <span class="c"># sort (other sorting methods/selection strategies were not as successful)</span>
-            <span class="n">CP</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">cp</span><span class="p">:</span> <span class="n">cp_key</span><span class="p">(</span><span class="n">cp</span><span class="p">,</span> <span class="n">O</span><span class="p">),</span> <span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-            <span class="c"># insert p into B:</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="n">sdp_LM</span><span class="p">(</span><span class="n">Polyn</span><span class="p">(</span><span class="n">p</span><span class="p">),</span> <span class="n">u</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">O</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="n">O</span><span class="p">(</span><span class="n">sdp_LM</span><span class="p">(</span><span class="n">Polyn</span><span class="p">(</span><span class="n">B</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]),</span> <span class="n">u</span><span class="p">)):</span>
-                <span class="n">B</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">q</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">B</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">O</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">O</span><span class="p">(</span><span class="n">sdp_LM</span><span class="p">(</span><span class="n">Polyn</span><span class="p">(</span><span class="n">q</span><span class="p">),</span> <span class="n">u</span><span class="p">)):</span>
-                        <span class="n">B</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-                        <span class="k">break</span>
-
-            <span class="n">k</span> <span class="o">+=</span> <span class="mi">1</span>
-
-            <span class="c">#print(len(B), len(CP), &quot;%d critical pairs removed&quot; % len(indices))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">reductions_to_zero</span> <span class="o">+=</span> <span class="mi">1</span>
-
-    <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-        <span class="k">print</span><span class="p">((</span><span class="s">&quot;</span><span class="si">%d</span><span class="s"> reductions to zero&quot;</span> <span class="o">%</span> <span class="n">reductions_to_zero</span><span class="p">))</span>
-
-    <span class="c"># reduce Groebner basis:</span>
-    <span class="n">H</span> <span class="o">=</span> <span class="p">[</span><span class="n">sdp_monic</span><span class="p">(</span><span class="n">Polyn</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">B</span><span class="p">]</span>
-    <span class="n">H</span> <span class="o">=</span> <span class="n">red_groebner</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">f</span><span class="p">:</span> <span class="n">O</span><span class="p">(</span><span class="n">sdp_LM</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">)),</span> <span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="red_groebner"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.groebnertools.red_groebner">[docs]</a><span class="k">def</span> <span class="nf">red_groebner</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute reduced Groebner basis, from BeckerWeispfenning93, p. 216</span>
-
-<span class="sd">    Selects a subset of generators, that already generate the ideal</span>
-<span class="sd">    and computes a reduced Groebner basis for them.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">reduction</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The actual reduction algorithm.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">Q</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">P</span><span class="p">):</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">sdp_rem</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">P</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:],</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">h</span> <span class="o">!=</span> <span class="p">[]:</span>
-                <span class="n">Q</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="p">[</span><span class="n">sdp_monic</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">Q</span><span class="p">]</span>
-
-    <span class="n">F</span> <span class="o">=</span> <span class="n">G</span>
-    <span class="n">H</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">while</span> <span class="n">F</span><span class="p">:</span>
-        <span class="n">f0</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">any</span><span class="p">(</span><span class="n">monomial_divides</span><span class="p">(</span><span class="n">sdp_LM</span><span class="p">(</span><span class="n">f0</span><span class="p">,</span> <span class="n">u</span><span class="p">),</span> <span class="n">sdp_LM</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">u</span><span class="p">))</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">F</span> <span class="o">+</span> <span class="n">H</span><span class="p">):</span>
-            <span class="n">H</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f0</span><span class="p">)</span>
-
-    <span class="c"># Becker, Weispfenning, p. 217: H is Groebner basis of the ideal generated by G.</span>
-    <span class="k">return</span> <span class="n">reduction</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="is_groebner"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.groebnertools.is_groebner">[docs]</a><span class="k">def</span> <span class="nf">is_groebner</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Check if G is a Groebner basis.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)):</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)):</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">sdp_spoly</span><span class="p">(</span><span class="n">G</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">G</span><span class="p">[</span><span class="n">j</span><span class="p">],</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">sdp_rem</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">s</span> <span class="o">!=</span> <span class="p">[]:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">return</span> <span class="bp">True</span>
-
-</div>
-<div class="viewcode-block" id="is_minimal"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.groebnertools.is_minimal">[docs]</a><span class="k">def</span> <span class="nf">is_minimal</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Checks if G is a minimal Groebner basis.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">G</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">g</span><span class="p">:</span> <span class="n">O</span><span class="p">(</span><span class="n">sdp_LM</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">)))</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">g</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">sdp_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">!=</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">G</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">G</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]:</span>
-            <span class="k">if</span> <span class="n">monomial_divides</span><span class="p">(</span><span class="n">sdp_LM</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">),</span> <span class="n">sdp_LM</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">)):</span>
-                <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">return</span> <span class="bp">True</span>
-
-</div>
-<div class="viewcode-block" id="is_reduced"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.groebnertools.is_reduced">[docs]</a><span class="k">def</span> <span class="nf">is_reduced</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Checks if G is a reduced Groebner basis.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">G</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">g</span><span class="p">:</span> <span class="n">O</span><span class="p">(</span><span class="n">sdp_LM</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">)))</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">g</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">sdp_LC</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="o">!=</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">g</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">G</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">G</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]:</span>
-                <span class="k">if</span> <span class="n">monomial_divides</span><span class="p">(</span><span class="n">term</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sdp_LM</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">u</span><span class="p">)):</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">return</span> <span class="bp">True</span>
-
-</div>
-<span class="k">def</span> <span class="nf">monomial_divides</span><span class="p">(</span><span class="n">m1</span><span class="p">,</span> <span class="n">m2</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns True if m2 divides m1, False otherwise. Does not create</span>
-<span class="sd">    the quotient. Does not check if both are have the same length.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">m1</span><span class="p">)):</span>
-        <span class="k">if</span> <span class="n">m1</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">m2</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">return</span> <span class="bp">True</span>
-
-<span class="c"># FGLM</span>
-
-
-<div class="viewcode-block" id="matrix_fglm"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.groebnertools.matrix_fglm">[docs]</a><span class="k">def</span> <span class="nf">matrix_fglm</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O_from</span><span class="p">,</span> <span class="n">O_to</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Converts the reduced Groebner basis ``F`` of a zero-dimensional</span>
-<span class="sd">    ideal w.r.t. ``O_from`` to a reduced Groebner basis</span>
-<span class="sd">    w.r.t. ``O_to``.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    J.C. Faugere, P. Gianni, D. Lazard, T. Mora (1994). Efficient</span>
-<span class="sd">    Computation of Zero-dimensional Groebner Bases by Change of</span>
-<span class="sd">    Ordering</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">old_basis</span> <span class="o">=</span> <span class="n">_basis</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O_from</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">M</span> <span class="o">=</span> <span class="n">_representing_matrices</span><span class="p">(</span><span class="n">old_basis</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O_from</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="c"># V contains the normalforms (wrt O_from) of S</span>
-    <span class="n">S</span> <span class="o">=</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,)</span> <span class="o">*</span> <span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
-    <span class="n">V</span> <span class="o">=</span> <span class="p">[[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">old_basis</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)]</span>
-    <span class="n">G</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="n">L</span> <span class="o">=</span> <span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>  <span class="c"># (i, j) corresponds to x_i * S[j]</span>
-    <span class="n">L</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">k_l1</span><span class="p">:</span> <span class="n">O_to</span><span class="p">(</span><span class="n">_incr_k</span><span class="p">(</span><span class="n">S</span><span class="p">[</span><span class="n">k_l1</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span> <span class="n">k_l1</span><span class="p">[</span><span class="mi">0</span><span class="p">])),</span> <span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">t</span> <span class="o">=</span> <span class="n">L</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-
-    <span class="n">P</span> <span class="o">=</span> <span class="n">_identity_matrix</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">old_basis</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">S</span><span class="p">)</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="n">_matrix_mul</span><span class="p">(</span><span class="n">M</span><span class="p">[</span><span class="n">t</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span> <span class="n">V</span><span class="p">[</span><span class="n">t</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">_lambda</span> <span class="o">=</span> <span class="n">_matrix_mul</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">_lambda</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">old_basis</span><span class="p">))):</span>
-            <span class="c"># there is a linear combination of v by V</span>
-
-            <span class="n">lt</span> <span class="o">=</span> <span class="p">[(</span><span class="n">_incr_k</span><span class="p">(</span><span class="n">S</span><span class="p">[</span><span class="n">t</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span> <span class="n">t</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">)]</span>
-            <span class="n">rest</span> <span class="o">=</span> <span class="n">sdp_strip</span><span class="p">(</span>
-                <span class="n">sdp_sort</span><span class="p">([(</span><span class="n">S</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">_lambda</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">s</span><span class="p">)],</span> <span class="n">O_to</span><span class="p">))</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">sdp_sub</span><span class="p">(</span><span class="n">lt</span><span class="p">,</span> <span class="n">rest</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O_to</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">g</span> <span class="o">!=</span> <span class="p">[]:</span>
-                <span class="n">G</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># v is linearly independant from V</span>
-            <span class="n">P</span> <span class="o">=</span> <span class="n">_update</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">_lambda</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-            <span class="n">S</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">_incr_k</span><span class="p">(</span><span class="n">S</span><span class="p">[</span><span class="n">t</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span> <span class="n">t</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-            <span class="n">V</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-
-            <span class="n">L</span><span class="o">.</span><span class="n">extend</span><span class="p">([(</span><span class="n">i</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)])</span>
-            <span class="n">L</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">L</span><span class="p">))</span>
-            <span class="n">L</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">k_l</span><span class="p">:</span> <span class="n">O_to</span><span class="p">(</span><span class="n">_incr_k</span><span class="p">(</span><span class="n">S</span><span class="p">[</span><span class="n">k_l</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span> <span class="n">k_l</span><span class="p">[</span><span class="mi">0</span><span class="p">])),</span> <span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-        <span class="n">L</span> <span class="o">=</span> <span class="p">[(</span><span class="n">k</span><span class="p">,</span> <span class="n">l</span><span class="p">)</span> <span class="k">for</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">l</span><span class="p">)</span> <span class="ow">in</span> <span class="n">L</span> <span class="k">if</span>
-            <span class="nb">all</span><span class="p">(</span><span class="n">monomial_div</span><span class="p">(</span><span class="n">_incr_k</span><span class="p">(</span><span class="n">S</span><span class="p">[</span><span class="n">l</span><span class="p">],</span> <span class="n">k</span><span class="p">),</span> <span class="n">sdp_LM</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">))</span> <span class="ow">is</span> <span class="bp">None</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">G</span><span class="p">)]</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">L</span><span class="p">:</span>
-            <span class="n">G</span> <span class="o">=</span> <span class="p">[</span> <span class="n">sdp_monic</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">G</span> <span class="p">]</span>
-            <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">g</span><span class="p">:</span> <span class="n">O_to</span><span class="p">(</span><span class="n">sdp_LM</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">)),</span> <span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-        <span class="n">t</span> <span class="o">=</span> <span class="n">L</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_incr_k</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-    <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">m</span><span class="p">[:</span><span class="n">k</span><span class="p">])</span> <span class="o">+</span> <span class="p">[</span><span class="n">m</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">m</span><span class="p">[</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]))</span>
-
-
-<span class="k">def</span> <span class="nf">_identity_matrix</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="n">M</span> <span class="o">=</span> <span class="p">[[</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span> <span class="o">*</span> <span class="n">n</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-        <span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span>
-
-    <span class="k">return</span> <span class="n">M</span>
-
-
-<span class="k">def</span> <span class="nf">_matrix_mul</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="k">return</span> <span class="p">[</span><span class="nb">sum</span><span class="p">([</span><span class="n">row</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">*</span> <span class="n">v</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">v</span><span class="p">))])</span> <span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="n">M</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">_update</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">_lambda</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Update ``P`` such that for the updated `P&#39;` `P&#39; v = e_{s}`.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">k</span> <span class="o">=</span> <span class="nb">min</span><span class="p">([</span><span class="n">j</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">_lambda</span><span class="p">))</span> <span class="k">if</span> <span class="n">_lambda</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">_lambda</span><span class="p">)):</span>
-        <span class="k">if</span> <span class="n">r</span> <span class="o">!=</span> <span class="n">k</span><span class="p">:</span>
-            <span class="n">P</span><span class="p">[</span><span class="n">r</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">P</span><span class="p">[</span><span class="n">r</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="p">(</span>
-                <span class="n">P</span><span class="p">[</span><span class="n">k</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">*</span> <span class="n">_lambda</span><span class="p">[</span><span class="n">r</span><span class="p">])</span> <span class="o">/</span> <span class="n">_lambda</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">P</span><span class="p">[</span><span class="n">r</span><span class="p">]))]</span>
-
-    <span class="n">P</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">P</span><span class="p">[</span><span class="n">k</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">/</span> <span class="n">_lambda</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">P</span><span class="p">[</span><span class="n">k</span><span class="p">]))]</span>
-
-    <span class="n">P</span><span class="p">[</span><span class="n">k</span><span class="p">],</span> <span class="n">P</span><span class="p">[</span><span class="n">s</span><span class="p">]</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">s</span><span class="p">],</span> <span class="n">P</span><span class="p">[</span><span class="n">k</span><span class="p">]</span>
-
-    <span class="k">return</span> <span class="n">P</span>
-
-
-<span class="k">def</span> <span class="nf">_representing_matrices</span><span class="p">(</span><span class="n">basis</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute the matrices corresponding to the linear maps `m \mapsto</span>
-<span class="sd">    x_i m` for all variables `x_i`.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">var</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">([</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">i</span> <span class="o">+</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="n">i</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">representing_matrix</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
-        <span class="n">M</span> <span class="o">=</span> <span class="p">[[</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">basis</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">basis</span><span class="p">))]</span>
-
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">basis</span><span class="p">):</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">sdp_rem</span><span class="p">([(</span><span class="n">monomial_mul</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">v</span><span class="p">),</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">)],</span> <span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">r</span><span class="p">:</span>
-                <span class="n">j</span> <span class="o">=</span> <span class="n">basis</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">term</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-                <span class="n">M</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">term</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-        <span class="k">return</span> <span class="n">M</span>
-
-    <span class="k">return</span> <span class="p">[</span><span class="n">representing_matrix</span><span class="p">(</span><span class="n">var</span><span class="p">(</span><span class="n">i</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
-
-
-<span class="k">def</span> <span class="nf">_basis</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes a list of monomials which are not divisible by the leading</span>
-<span class="sd">    monomials wrt to ``O`` of ``G``. These monomials are a basis of</span>
-<span class="sd">    `K[X_1, \ldots, X_n]/(G)`.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">leading_monomials</span> <span class="o">=</span> <span class="p">[</span><span class="n">sdp_LM</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">G</span><span class="p">]</span>
-    <span class="n">candidates</span> <span class="o">=</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,)</span> <span class="o">*</span> <span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
-    <span class="n">basis</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">while</span> <span class="n">candidates</span><span class="p">:</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">candidates</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-        <span class="n">basis</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-
-        <span class="n">new_candidates</span> <span class="o">=</span> <span class="p">[</span><span class="n">_incr_k</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">monomial_div</span><span class="p">(</span><span class="n">_incr_k</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">k</span><span class="p">),</span> <span class="n">lmg</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">None</span>
-            <span class="k">for</span> <span class="n">lmg</span> <span class="ow">in</span> <span class="n">leading_monomials</span><span class="p">)]</span>
-        <span class="n">candidates</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">new_candidates</span><span class="p">)</span>
-        <span class="n">candidates</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">m</span><span class="p">:</span> <span class="n">O</span><span class="p">(</span><span class="n">m</span><span class="p">),</span> <span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="n">basis</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">basis</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">basis</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">m</span><span class="p">:</span> <span class="n">O</span><span class="p">(</span><span class="n">m</span><span class="p">))</span>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
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-            <p class="logo"><a href="../../../index.html">
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-  <h3>Quick search</h3>
-    <form class="search" action="../../../search.html" method="get">
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diff --git a/dev-py3k/_modules/sympy/polys/monomialtools.html b/dev-py3k/_modules/sympy/polys/monomialtools.html
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+++ /dev/null
@@ -1,753 +0,0 @@
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-  <h1>Source code for sympy.polys.monomialtools</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Tools and arithmetics for monomials of distributed polynomials. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Tuple</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polyutils</span> <span class="kn">import</span> <span class="n">PicklableWithSlots</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">cythonized</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="n">ExactQuotientFailed</span>
-
-
-<div class="viewcode-block" id="monomials"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.monomialtools.monomials">[docs]</a><span class="k">def</span> <span class="nf">monomials</span><span class="p">(</span><span class="n">variables</span><span class="p">,</span> <span class="n">degree</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Generate a set of monomials of the given total degree or less.</span>
-
-<span class="sd">    Given a set of variables `V` and a total degree `N` generate</span>
-<span class="sd">    a set of monomials of degree at most `N`. The total number of</span>
-<span class="sd">    monomials is huge and is given by the following formula:</span>
-
-<span class="sd">    .. math::</span>
-
-<span class="sd">        \frac{(\#V + N)!}{\#V! N!}</span>
-
-<span class="sd">    For example if we would like to generate a dense polynomial of</span>
-<span class="sd">    a total degree `N = 50` in 5 variables, assuming that exponents</span>
-<span class="sd">    and all of coefficients are 32-bit long and stored in an array we</span>
-<span class="sd">    would need almost 80 GiB of memory! Fortunately most polynomials,</span>
-<span class="sd">    that we will encounter, are sparse.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Consider monomials in variables `x` and `y`::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import monomials</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; sorted(monomials([x, y], 2))</span>
-<span class="sd">        [1, x, y, x**2, y**2, x*y]</span>
-
-<span class="sd">        &gt;&gt;&gt; sorted(monomials([x, y], 3))</span>
-<span class="sd">        [1, x, y, x**2, x**3, y**2, y**3, x*y, x*y**2, x**2*y]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">variables</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">set</span><span class="p">([</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">])</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">tail</span> <span class="o">=</span> <span class="n">variables</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">variables</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-        <span class="n">monoms</span> <span class="o">=</span> <span class="n">monomials</span><span class="p">(</span><span class="n">tail</span><span class="p">,</span> <span class="n">degree</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">degree</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">monoms</span> <span class="o">|=</span> <span class="nb">set</span><span class="p">([</span> <span class="n">x</span><span class="o">**</span><span class="n">i</span> <span class="o">*</span> <span class="n">m</span> <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">monomials</span><span class="p">(</span><span class="n">tail</span><span class="p">,</span> <span class="n">degree</span> <span class="o">-</span> <span class="n">i</span><span class="p">)</span> <span class="p">])</span>
-
-        <span class="k">return</span> <span class="n">monoms</span>
-
-</div>
-<div class="viewcode-block" id="monomial_count"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.monomialtools.monomial_count">[docs]</a><span class="k">def</span> <span class="nf">monomial_count</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">N</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Computes the number of monomials.</span>
-
-<span class="sd">    The number of monomials is given by the following formula:</span>
-
-<span class="sd">    .. math::</span>
-
-<span class="sd">        \frac{(\#V + N)!}{\#V! N!}</span>
-
-<span class="sd">    where `N` is a total degree and `V` is a set of variables.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import monomials, monomial_count</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">    &gt;&gt;&gt; monomial_count(2, 2)</span>
-<span class="sd">    6</span>
-
-<span class="sd">    &gt;&gt;&gt; M = monomials([x, y], 2)</span>
-
-<span class="sd">    &gt;&gt;&gt; sorted(M)</span>
-<span class="sd">    [1, x, y, x**2, y**2, x*y]</span>
-<span class="sd">    &gt;&gt;&gt; len(M)</span>
-<span class="sd">    6</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">V</span> <span class="o">+</span> <span class="n">N</span><span class="p">)</span> <span class="o">/</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">V</span><span class="p">)</span> <span class="o">/</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-
-</div>
-<span class="k">class</span> <span class="nc">MonomialOrder</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Base class for monomial orderings. &quot;&quot;&quot;</span>
-
-    <span class="n">alias</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">is_global</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">key</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">monomial</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">alias</span>
-
-    <span class="k">def</span> <span class="nf">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">monomial</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">key</span><span class="p">(</span><span class="n">monomial</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">__class__</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">hash</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="p">(</span><span class="bp">self</span> <span class="o">==</span> <span class="n">other</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="LexOrder"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.monomialtools.LexOrder">[docs]</a><span class="k">class</span> <span class="nc">LexOrder</span><span class="p">(</span><span class="n">MonomialOrder</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Lexicographic order of monomials. &quot;&quot;&quot;</span>
-
-    <span class="n">alias</span> <span class="o">=</span> <span class="s">&#39;lex&#39;</span>
-    <span class="n">is_global</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">key</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">monomial</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">monomial</span>
-
-</div>
-<div class="viewcode-block" id="GradedLexOrder"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.monomialtools.GradedLexOrder">[docs]</a><span class="k">class</span> <span class="nc">GradedLexOrder</span><span class="p">(</span><span class="n">MonomialOrder</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Graded lexicographic order of monomials. &quot;&quot;&quot;</span>
-
-    <span class="n">alias</span> <span class="o">=</span> <span class="s">&#39;grlex&#39;</span>
-    <span class="n">is_global</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">key</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">monomial</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="nb">sum</span><span class="p">(</span><span class="n">monomial</span><span class="p">),</span> <span class="n">monomial</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="ReversedGradedLexOrder"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.monomialtools.ReversedGradedLexOrder">[docs]</a><span class="k">class</span> <span class="nc">ReversedGradedLexOrder</span><span class="p">(</span><span class="n">MonomialOrder</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Reversed graded lexicographic order of monomials. &quot;&quot;&quot;</span>
-
-    <span class="n">alias</span> <span class="o">=</span> <span class="s">&#39;grevlex&#39;</span>
-    <span class="n">is_global</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">key</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">monomial</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="nb">sum</span><span class="p">(</span><span class="n">monomial</span><span class="p">),</span> <span class="nb">tuple</span><span class="p">(</span><span class="nb">reversed</span><span class="p">([</span><span class="o">-</span><span class="n">m</span> <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">monomial</span><span class="p">])))</span>
-
-</div>
-<span class="k">class</span> <span class="nc">ProductOrder</span><span class="p">(</span><span class="n">MonomialOrder</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A product order built from other monomial orders.</span>
-
-<span class="sd">    Given (not necessarily total) orders O1, O2, ..., On, their product order</span>
-<span class="sd">    P is defined as M1 &gt; M2 iff there exists i such that O1(M1) = O2(M2),</span>
-<span class="sd">    ..., Oi(M1) = Oi(M2), O{i+1}(M1) &gt; O{i+1}(M2).</span>
-
-<span class="sd">    Product orders are typically built from monomial orders on different sets</span>
-<span class="sd">    of variables.</span>
-
-<span class="sd">    ProductOrder is constructed by passing a list of pairs</span>
-<span class="sd">    [(O1, L1), (O2, L2), ...] where Oi are MonomialOrders and Li are callables.</span>
-<span class="sd">    Upon comparison, the Li are passed the total monomial, and should filter</span>
-<span class="sd">    out the part of the monomial to pass to Oi.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    We can use a lexicographic order on x_1, x_2 and also on</span>
-<span class="sd">    y_1, y_2, y_3, and their product on {x_i, y_i} as follows:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.monomialtools import lex, grlex, ProductOrder</span>
-<span class="sd">    &gt;&gt;&gt; P = ProductOrder(</span>
-<span class="sd">    ...     (lex, lambda m: m[:2]), # lex order on x_1 and x_2 of monomial</span>
-<span class="sd">    ...     (grlex, lambda m: m[2:]) # grlex on y_1, y_2, y_3</span>
-<span class="sd">    ... )</span>
-<span class="sd">    &gt;&gt;&gt; P((2, 1, 1, 0, 0)) &gt; P((1, 10, 0, 2, 0))</span>
-<span class="sd">    True</span>
-
-<span class="sd">    Here the exponent `2` of `x_1` in the first monomial</span>
-<span class="sd">    (`x_1^2 x_2 y_1`) is bigger than the exponent `1` of `x_1` in the</span>
-<span class="sd">    second monomial (`x_1 x_2^10 y_2^2`), so the first monomial is greater</span>
-<span class="sd">    in the product ordering.</span>
-
-<span class="sd">    &gt;&gt;&gt; P((2, 1, 1, 0, 0)) &lt; P((2, 1, 0, 2, 0))</span>
-<span class="sd">    True</span>
-
-<span class="sd">    Here the exponents of `x_1` and `x_2` agree, so the grlex order on</span>
-<span class="sd">    `y_1, y_2, y_3` is used to decide the ordering. In this case the monomial</span>
-<span class="sd">    `y_2^2` is ordered larger than `y_1`, since for the grlex order the degree</span>
-<span class="sd">    of the monomial is most important.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">=</span> <span class="n">args</span>
-
-    <span class="k">def</span> <span class="nf">key</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">monomial</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">O</span><span class="p">(</span><span class="n">lamda</span><span class="p">(</span><span class="n">monomial</span><span class="p">))</span> <span class="k">for</span> <span class="p">(</span><span class="n">O</span><span class="p">,</span> <span class="n">lamda</span><span class="p">)</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Tuple</span>
-        <span class="k">return</span> <span class="s">&quot;ProductOrder&quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">]))</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">ProductOrder</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">args</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">hash</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_global</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">o</span><span class="o">.</span><span class="n">is_global</span> <span class="ow">is</span> <span class="bp">True</span> <span class="k">for</span> <span class="n">o</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">o</span><span class="o">.</span><span class="n">is_global</span> <span class="ow">is</span> <span class="bp">False</span> <span class="k">for</span> <span class="n">o</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-
-<span class="k">class</span> <span class="nc">InverseOrder</span><span class="p">(</span><span class="n">MonomialOrder</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The &quot;inverse&quot; of another monomial order.</span>
-
-<span class="sd">    If O is any monomial order, we can construct another monomial order iO</span>
-<span class="sd">    such that `A &gt;_{iO} B` if and only if `B &gt;_O A`. This is useful for</span>
-<span class="sd">    constructing local orders.</span>
-
-<span class="sd">    Note that many algorithms only work with *global* orders.</span>
-
-<span class="sd">    For example, in the inverse lexicographic order on a single variable `x`,</span>
-<span class="sd">    high powers of `x` count as small:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.monomialtools import lex, InverseOrder</span>
-<span class="sd">    &gt;&gt;&gt; ilex = InverseOrder(lex)</span>
-<span class="sd">    &gt;&gt;&gt; ilex((5,)) &lt; ilex((0,))</span>
-<span class="sd">    True</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">O</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">O</span> <span class="o">=</span> <span class="n">O</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;i&quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">O</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">key</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">monomial</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">iterable</span>
-
-        <span class="k">def</span> <span class="nf">inv</span><span class="p">(</span><span class="n">l</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">iterable</span><span class="p">(</span><span class="n">l</span><span class="p">):</span>
-                <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">inv</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">l</span><span class="p">)</span>
-            <span class="k">return</span> <span class="o">-</span><span class="n">l</span>
-        <span class="k">return</span> <span class="n">inv</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">O</span><span class="o">.</span><span class="n">key</span><span class="p">(</span><span class="n">monomial</span><span class="p">))</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_global</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">O</span><span class="o">.</span><span class="n">is_global</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">O</span><span class="o">.</span><span class="n">is_global</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">InverseOrder</span><span class="p">)</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">O</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">O</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">hash</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">O</span><span class="p">))</span>
-
-<span class="n">lex</span> <span class="o">=</span> <span class="n">LexOrder</span><span class="p">()</span>
-<span class="n">grlex</span> <span class="o">=</span> <span class="n">GradedLexOrder</span><span class="p">()</span>
-<span class="n">grevlex</span> <span class="o">=</span> <span class="n">ReversedGradedLexOrder</span><span class="p">()</span>
-<span class="n">ilex</span> <span class="o">=</span> <span class="n">InverseOrder</span><span class="p">(</span><span class="n">lex</span><span class="p">)</span>
-<span class="n">igrlex</span> <span class="o">=</span> <span class="n">InverseOrder</span><span class="p">(</span><span class="n">grlex</span><span class="p">)</span>
-<span class="n">igrevlex</span> <span class="o">=</span> <span class="n">InverseOrder</span><span class="p">(</span><span class="n">grevlex</span><span class="p">)</span>
-
-<span class="n">_monomial_key</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="s">&#39;lex&#39;</span><span class="p">:</span> <span class="n">lex</span><span class="p">,</span>
-    <span class="s">&#39;grlex&#39;</span><span class="p">:</span> <span class="n">grlex</span><span class="p">,</span>
-    <span class="s">&#39;grevlex&#39;</span><span class="p">:</span> <span class="n">grevlex</span><span class="p">,</span>
-    <span class="s">&#39;ilex&#39;</span><span class="p">:</span> <span class="n">ilex</span><span class="p">,</span>
-    <span class="s">&#39;igrlex&#39;</span><span class="p">:</span> <span class="n">igrlex</span><span class="p">,</span>
-    <span class="s">&#39;igrevlex&#39;</span><span class="p">:</span> <span class="n">igrevlex</span>
-<span class="p">}</span>
-
-
-<span class="k">def</span> <span class="nf">monomial_key</span><span class="p">(</span><span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a function defining admissible order on monomials.</span>
-
-<span class="sd">    The result of a call to :func:`monomial_key` is a function which should</span>
-<span class="sd">    be used as a key to :func:`sorted` built-in function, to provide order</span>
-<span class="sd">    in a set of monomials of the same length.</span>
-
-<span class="sd">    Currently supported monomial orderings are:</span>
-
-<span class="sd">    1. lex       - lexicographic order (default)</span>
-<span class="sd">    2. grlex     - graded lexicographic order</span>
-<span class="sd">    3. grevlex   - reversed graded lexicographic order</span>
-<span class="sd">    4. ilex, igrlex, igrevlex - the corresponding inverse orders</span>
-
-<span class="sd">    If the input argument is not a string but has ``__call__`` attribute,</span>
-<span class="sd">    then it will pass through with an assumption that the callable object</span>
-<span class="sd">    defines an admissible order on monomials.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">order</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">lex</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">order</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">):</span>
-        <span class="n">order</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">order</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">order</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_monomial_key</span><span class="p">[</span><span class="n">order</span><span class="p">]</span>
-        <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;supported monomial orderings are &#39;lex&#39;, &#39;grlex&#39; and &#39;grevlex&#39;, got </span><span class="si">%r</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">order</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">order</span><span class="p">,</span> <span class="s">&#39;__call__&#39;</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">order</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;monomial ordering specification must be a string or a callable, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">order</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">_ItemGetter</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Helper class to return a subsequence of values.&quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">seq</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">seq</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">seq</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">m</span><span class="p">[</span><span class="n">idx</span><span class="p">]</span> <span class="k">for</span> <span class="n">idx</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">seq</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">_ItemGetter</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">seq</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">seq</span>
-
-
-<span class="k">def</span> <span class="nf">build_product_order</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">gens</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Build a monomial order on ``gens``.</span>
-
-<span class="sd">    ``arg`` should be a tuple of iterables. The first element of each iterable</span>
-<span class="sd">    should be a string or monomial order (will be passed to monomial_key),</span>
-<span class="sd">    the others should be subsets of the generators. This function will build</span>
-<span class="sd">    the corresponding product order.</span>
-
-<span class="sd">    For example, build a product of two grlex orders:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.monomialtools import grlex, build_product_order</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z, t</span>
-<span class="sd">    &gt;&gt;&gt; O = build_product_order(((&quot;grlex&quot;, x, y), (&quot;grlex&quot;, z, t)), [x, y, z, t])</span>
-<span class="sd">    &gt;&gt;&gt; O((1, 2, 3, 4))</span>
-<span class="sd">    ((3, (1, 2)), (7, (3, 4)))</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">gens2idx</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">g</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">gens</span><span class="p">):</span>
-        <span class="n">gens2idx</span><span class="p">[</span><span class="n">g</span><span class="p">]</span> <span class="o">=</span> <span class="n">i</span>
-    <span class="n">order</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">expr</span> <span class="ow">in</span> <span class="n">arg</span><span class="p">:</span>
-        <span class="n">name</span> <span class="o">=</span> <span class="n">expr</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">var</span> <span class="o">=</span> <span class="n">expr</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-        <span class="k">def</span> <span class="nf">makelambda</span><span class="p">(</span><span class="n">var</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">_ItemGetter</span><span class="p">(</span><span class="n">gens2idx</span><span class="p">[</span><span class="n">g</span><span class="p">]</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">var</span><span class="p">)</span>
-        <span class="n">order</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">monomial_key</span><span class="p">(</span><span class="n">name</span><span class="p">),</span> <span class="n">makelambda</span><span class="p">(</span><span class="n">var</span><span class="p">)))</span>
-    <span class="k">return</span> <span class="n">ProductOrder</span><span class="p">(</span><span class="o">*</span><span class="n">order</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;a,b&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">monomial_mul</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Multiplication of tuples representing monomials.</span>
-
-<span class="sd">    Lets multiply `x**3*y**4*z` with `x*y**2`::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.polys.monomialtools import monomial_mul</span>
-
-<span class="sd">        &gt;&gt;&gt; monomial_mul((3, 4, 1), (1, 2, 0))</span>
-<span class="sd">        (4, 6, 1)</span>
-
-<span class="sd">    which gives `x**4*y**5*z`.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="nb">tuple</span><span class="p">([</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span> <span class="p">])</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;a,b,c&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">monomial_div</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Division of tuples representing monomials.</span>
-
-<span class="sd">    Lets divide `x**3*y**4*z` by `x*y**2`::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.polys.monomialtools import monomial_div</span>
-
-<span class="sd">        &gt;&gt;&gt; monomial_div((3, 4, 1), (1, 2, 0))</span>
-<span class="sd">        (2, 2, 1)</span>
-
-<span class="sd">    which gives `x**2*y**2*z`. However::</span>
-
-<span class="sd">        &gt;&gt;&gt; monomial_div((3, 4, 1), (1, 2, 2)) is None</span>
-<span class="sd">        True</span>
-
-<span class="sd">    `x*y**2*z**2` does not divide `x**3*y**4*z`.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">C</span> <span class="o">=</span> <span class="p">[</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span> <span class="p">]</span>
-
-    <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">c</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">C</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">C</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;a,b&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">monomial_gcd</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Greatest common divisor of tuples representing monomials.</span>
-
-<span class="sd">    Lets compute GCD of `x*y**4*z` and `x**3*y**2`::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.polys.monomialtools import monomial_gcd</span>
-
-<span class="sd">        &gt;&gt;&gt; monomial_gcd((1, 4, 1), (3, 2, 0))</span>
-<span class="sd">        (1, 2, 0)</span>
-
-<span class="sd">    which gives `x*y**2`.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="nb">tuple</span><span class="p">([</span> <span class="nb">min</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span> <span class="p">])</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;a,b&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">monomial_lcm</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Least common multiple of tuples representing monomials.</span>
-
-<span class="sd">    Lets compute LCM of `x*y**4*z` and `x**3*y**2`::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.polys.monomialtools import monomial_lcm</span>
-
-<span class="sd">        &gt;&gt;&gt; monomial_lcm((1, 4, 1), (3, 2, 0))</span>
-<span class="sd">        (3, 4, 1)</span>
-
-<span class="sd">    which gives `x**3*y**4*z`.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="nb">tuple</span><span class="p">([</span> <span class="nb">max</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span> <span class="p">])</span>
-
-<span class="c"># TODO cythonize</span>
-
-
-<span class="k">def</span> <span class="nf">monomial_divides</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Does there exist a monomial X such that XA == B?</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.monomialtools import monomial_divides</span>
-<span class="sd">    &gt;&gt;&gt; monomial_divides((1, 2), (3, 4))</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; monomial_divides((1, 2), (0, 2))</span>
-<span class="sd">    False</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="n">a</span> <span class="o">&lt;=</span> <span class="n">b</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">))</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i,n&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">monomial_max</span><span class="p">(</span><span class="o">*</span><span class="n">monoms</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns maximal degree for each variable in a set of monomials.</span>
-
-<span class="sd">    Consider monomials `x**3*y**4*z**5`, `y**5*z` and `x**6*y**3*z**9`.</span>
-<span class="sd">    We wish to find out what is the maximal degree for each of `x`, `y`</span>
-<span class="sd">    and `z` variables::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.polys.monomialtools import monomial_max</span>
-
-<span class="sd">        &gt;&gt;&gt; monomial_max((3,4,5), (0,5,1), (6,3,9))</span>
-<span class="sd">        (6, 5, 9)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">M</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">monoms</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">for</span> <span class="n">N</span> <span class="ow">in</span> <span class="n">monoms</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
-            <span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;i,n&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">monomial_min</span><span class="p">(</span><span class="o">*</span><span class="n">monoms</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns minimal degree for each variable in a set of monomials.</span>
-
-<span class="sd">    Consider monomials `x**3*y**4*z**5`, `y**5*z` and `x**6*y**3*z**9`.</span>
-<span class="sd">    We wish to find out what is the minimal degree for each of `x`, `y`</span>
-<span class="sd">    and `z` variables::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.polys.monomialtools import monomial_min</span>
-
-<span class="sd">        &gt;&gt;&gt; monomial_min((3,4,5), (0,5,1), (6,3,9))</span>
-<span class="sd">        (0, 3, 1)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">M</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">monoms</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">for</span> <span class="n">N</span> <span class="ow">in</span> <span class="n">monoms</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
-            <span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">monomial_deg</span><span class="p">(</span><span class="n">M</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the total degree of a monomial.</span>
-
-<span class="sd">    For example, the total degree of `xy^2` is 3:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.monomialtools import monomial_deg</span>
-<span class="sd">    &gt;&gt;&gt; monomial_deg((1, 2))</span>
-<span class="sd">    3</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="Monomial"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.monomialtools.Monomial">[docs]</a><span class="k">class</span> <span class="nc">Monomial</span><span class="p">(</span><span class="n">PicklableWithSlots</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Class representing a monomial, i.e. a product of powers. &quot;&quot;&quot;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;exponents&#39;</span><span class="p">,</span> <span class="s">&#39;gens&#39;</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">exponents</span><span class="p">,</span> <span class="n">gens</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">exponents</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">exponents</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">gens</span> <span class="o">=</span> <span class="n">gens</span>
-
-    <span class="k">def</span> <span class="nf">rebuild</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">exponents</span><span class="p">,</span> <span class="n">gens</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">exponents</span><span class="p">,</span> <span class="n">gens</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__len__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">exponents</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">iter</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">exponents</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">item</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">exponents</span><span class="p">[</span><span class="n">item</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">hash</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">exponents</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;*&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">**</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">gen</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span> <span class="k">for</span> <span class="n">gen</span><span class="p">,</span> <span class="n">exp</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">exponents</span><span class="p">)</span> <span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">exponents</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">as_expr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Convert a monomial instance to a SymPy expression. &quot;&quot;&quot;</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="n">gens</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">gens</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">gens</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&quot;can&#39;t convert </span><span class="si">%s</span><span class="s"> to an expression without generators&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">gen</span><span class="o">**</span><span class="n">exp</span> <span class="k">for</span> <span class="n">gen</span><span class="p">,</span> <span class="n">exp</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">gens</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">exponents</span><span class="p">)</span> <span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Monomial</span><span class="p">):</span>
-            <span class="n">exponents</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">exponents</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">)):</span>
-            <span class="n">exponents</span> <span class="o">=</span> <span class="n">other</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">exponents</span> <span class="o">==</span> <span class="n">exponents</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Monomial</span><span class="p">):</span>
-            <span class="n">exponents</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">exponents</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">)):</span>
-            <span class="n">exponents</span> <span class="o">=</span> <span class="n">other</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="ne">NotImplementedError</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rebuild</span><span class="p">(</span><span class="n">monomial_mul</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">exponents</span><span class="p">,</span> <span class="n">exponents</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Monomial</span><span class="p">):</span>
-            <span class="n">exponents</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">exponents</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">)):</span>
-            <span class="n">exponents</span> <span class="o">=</span> <span class="n">other</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="ne">NotImplementedError</span>
-
-        <span class="n">result</span> <span class="o">=</span> <span class="n">monomial_div</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">exponents</span><span class="p">,</span> <span class="n">exponents</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rebuild</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ExactQuotientFailed</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Monomial</span><span class="p">(</span><span class="n">other</span><span class="p">))</span>
-
-    <span class="n">__floordiv__</span> <span class="o">=</span> <span class="n">__truediv__</span> <span class="o">=</span> <span class="n">__div__</span>
-
-    <span class="k">def</span> <span class="nf">__pow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rebuild</span><span class="p">([</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">exponents</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exponents</span>
-
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-                <span class="n">exponents</span> <span class="o">=</span> <span class="n">monomial_mul</span><span class="p">(</span><span class="n">exponents</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">exponents</span><span class="p">)</span>
-
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rebuild</span><span class="p">(</span><span class="n">exponents</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;a non-negative integer expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">gcd</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Greatest common divisor of monomials. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Monomial</span><span class="p">):</span>
-            <span class="n">exponents</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">exponents</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">)):</span>
-            <span class="n">exponents</span> <span class="o">=</span> <span class="n">other</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span>
-                <span class="s">&quot;an instance of Monomial class expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">other</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rebuild</span><span class="p">(</span><span class="n">monomial_gcd</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">exponents</span><span class="p">,</span> <span class="n">exponents</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">lcm</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Least common multiple of monomials. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Monomial</span><span class="p">):</span>
-            <span class="n">exponents</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">exponents</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">)):</span>
-            <span class="n">exponents</span> <span class="o">=</span> <span class="n">other</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span>
-                <span class="s">&quot;an instance of Monomial class expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">other</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rebuild</span><span class="p">(</span><span class="n">monomial_lcm</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">exponents</span><span class="p">,</span> <span class="n">exponents</span><span class="p">))</span></div>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
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-  <h1>Source code for sympy.polys.numberfields</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Computational algebraic number field theory. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">S</span><span class="p">,</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">Rational</span><span class="p">,</span>
-    <span class="n">Symbol</span><span class="p">,</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">Q</span><span class="p">,</span> <span class="n">ask</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">,</span> <span class="n">Tuple</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polytools</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">Poly</span><span class="p">,</span> <span class="n">PurePoly</span><span class="p">,</span> <span class="n">sqf_norm</span><span class="p">,</span> <span class="n">invert</span><span class="p">,</span> <span class="n">factor_list</span><span class="p">,</span> <span class="n">groebner</span><span class="p">,</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyclasses</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">ANP</span><span class="p">,</span> <span class="n">DMP</span><span class="p">,</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">IsomorphismFailed</span><span class="p">,</span>
-    <span class="n">CoercionFailed</span><span class="p">,</span>
-    <span class="n">NotAlgebraic</span><span class="p">,</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.printing.lambdarepr</span> <span class="kn">import</span> <span class="n">LambdaPrinter</span>
-
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">numbered_symbols</span><span class="p">,</span> <span class="n">variations</span><span class="p">,</span> <span class="n">lambdify</span><span class="p">,</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">sieve</span>
-<span class="kn">from</span> <span class="nn">sympy.mpmath</span> <span class="kn">import</span> <span class="n">pslq</span><span class="p">,</span> <span class="n">mp</span>
-
-
-<div class="viewcode-block" id="minimal_polynomial"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.numberfields.minimal_polynomial">[docs]</a><span class="k">def</span> <span class="nf">minimal_polynomial</span><span class="p">(</span><span class="n">ex</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes the minimal polynomial of an algebraic number.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import minimal_polynomial, sqrt</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; minimal_polynomial(sqrt(2), x)</span>
-<span class="sd">    x**2 - 2</span>
-<span class="sd">    &gt;&gt;&gt; minimal_polynomial(sqrt(2) + sqrt(3), x)</span>
-<span class="sd">    x**4 - 10*x**2 + 1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">generator</span> <span class="o">=</span> <span class="n">numbered_symbols</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Dummy</span><span class="p">)</span>
-    <span class="n">mapping</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">replace</span> <span class="o">=</span> <span class="p">{},</span> <span class="p">{},</span> <span class="p">[]</span>
-
-    <span class="n">ex</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">ex</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">cls</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">Poly</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">cls</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">),</span> <span class="n">PurePoly</span>
-
-    <span class="k">def</span> <span class="nf">update_mapping</span><span class="p">(</span><span class="n">ex</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">base</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">generator</span><span class="p">)</span>
-        <span class="n">symbols</span><span class="p">[</span><span class="n">ex</span><span class="p">]</span> <span class="o">=</span> <span class="n">a</span>
-
-        <span class="k">if</span> <span class="n">base</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">mapping</span><span class="p">[</span><span class="n">ex</span><span class="p">]</span> <span class="o">=</span> <span class="n">a</span><span class="o">**</span><span class="n">exp</span> <span class="o">+</span> <span class="n">base</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">mapping</span><span class="p">[</span><span class="n">ex</span><span class="p">]</span> <span class="o">=</span> <span class="n">exp</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">a</span>
-
-    <span class="k">def</span> <span class="nf">bottom_up_scan</span><span class="p">(</span><span class="n">ex</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">ex</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">ex</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">ex</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">mapping</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">update_mapping</span><span class="p">(</span><span class="n">ex</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">symbols</span><span class="p">[</span><span class="n">ex</span><span class="p">]</span>
-            <span class="k">elif</span> <span class="n">ex</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">ex</span>
-        <span class="k">elif</span> <span class="n">ex</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">bottom_up_scan</span><span class="p">(</span><span class="n">g</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">ex</span><span class="o">.</span><span class="n">args</span> <span class="p">])</span>
-        <span class="k">elif</span> <span class="n">ex</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">bottom_up_scan</span><span class="p">(</span><span class="n">g</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">ex</span><span class="o">.</span><span class="n">args</span> <span class="p">])</span>
-        <span class="k">elif</span> <span class="n">ex</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">ex</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">ex</span><span class="o">.</span><span class="n">exp</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">ex</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                    <span class="n">coeff</span><span class="p">,</span> <span class="n">terms</span> <span class="o">=</span> <span class="n">ex</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">as_coeff_add</span><span class="p">()</span>
-                    <span class="n">elt</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">primitive_element</span><span class="p">(</span><span class="n">terms</span><span class="p">,</span> <span class="n">polys</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-                    <span class="n">alg</span> <span class="o">=</span> <span class="n">ex</span><span class="o">.</span><span class="n">base</span> <span class="o">-</span> <span class="n">coeff</span>
-
-                    <span class="c"># XXX: turn this into eval()</span>
-                    <span class="n">inverse</span> <span class="o">=</span> <span class="n">invert</span><span class="p">(</span><span class="n">elt</span><span class="o">.</span><span class="n">gen</span> <span class="o">+</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">elt</span><span class="p">)</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-                    <span class="n">base</span> <span class="o">=</span> <span class="n">inverse</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">elt</span><span class="o">.</span><span class="n">gen</span><span class="p">,</span> <span class="n">alg</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-
-                    <span class="k">if</span> <span class="n">ex</span><span class="o">.</span><span class="n">exp</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">bottom_up_scan</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">ex</span> <span class="o">=</span> <span class="n">base</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">ex</span><span class="o">.</span><span class="n">exp</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">ex</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                    <span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="p">(</span>
-                        <span class="n">ex</span><span class="o">.</span><span class="n">base</span><span class="o">**</span><span class="n">ex</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(),</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">ex</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="n">ex</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">ex</span><span class="o">.</span><span class="n">exp</span>
-
-                <span class="n">base</span> <span class="o">=</span> <span class="n">bottom_up_scan</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-                <span class="n">expr</span> <span class="o">=</span> <span class="n">base</span><span class="o">**</span><span class="n">exp</span>
-
-                <span class="k">if</span> <span class="n">expr</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">mapping</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">update_mapping</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">exp</span><span class="p">,</span> <span class="o">-</span><span class="n">base</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">symbols</span><span class="p">[</span><span class="n">expr</span><span class="p">]</span>
-        <span class="k">elif</span> <span class="n">ex</span><span class="o">.</span><span class="n">is_AlgebraicNumber</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">ex</span><span class="o">.</span><span class="n">root</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">mapping</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">update_mapping</span><span class="p">(</span><span class="n">ex</span><span class="o">.</span><span class="n">root</span><span class="p">,</span> <span class="n">ex</span><span class="o">.</span><span class="n">minpoly</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">symbols</span><span class="p">[</span><span class="n">ex</span><span class="o">.</span><span class="n">root</span><span class="p">]</span>
-
-        <span class="k">raise</span> <span class="n">NotAlgebraic</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> doesn&#39;t seem to be an algebraic number&quot;</span> <span class="o">%</span> <span class="n">ex</span><span class="p">)</span>
-
-    <span class="n">polys</span> <span class="o">=</span> <span class="n">args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;polys&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">ex</span><span class="o">.</span><span class="n">is_AlgebraicNumber</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">polys</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">ex</span><span class="o">.</span><span class="n">minpoly</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">ex</span><span class="o">.</span><span class="n">minpoly</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">ex</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">ex</span><span class="o">.</span><span class="n">q</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="n">ex</span><span class="o">.</span><span class="n">p</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">F</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="o">-</span> <span class="n">bottom_up_scan</span><span class="p">(</span><span class="n">ex</span><span class="p">)]</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">mapping</span><span class="o">.</span><span class="n">values</span><span class="p">())</span>
-        <span class="n">G</span> <span class="o">=</span> <span class="n">groebner</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="n">symbols</span><span class="o">.</span><span class="n">values</span><span class="p">())</span> <span class="o">+</span> <span class="p">[</span><span class="n">x</span><span class="p">],</span> <span class="n">order</span><span class="o">=</span><span class="s">&#39;lex&#39;</span><span class="p">)</span>
-
-        <span class="n">_</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">factor_list</span><span class="p">(</span><span class="n">G</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="p">((</span><span class="n">result</span><span class="p">,</span> <span class="n">_</span><span class="p">),)</span> <span class="o">=</span> <span class="n">factors</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">result</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">factors</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">result</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">ex</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">chop</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">break</span>
-            <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;multiple candidates for the minimal polynomial of </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">ex</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">field</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-</div>
-<span class="n">minpoly</span> <span class="o">=</span> <span class="n">minimal_polynomial</span>
-
-
-<span class="k">def</span> <span class="nf">_coeffs_generator</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Generate coefficients for `primitive_element()`. &quot;&quot;&quot;</span>
-    <span class="k">for</span> <span class="n">coeffs</span> <span class="ow">in</span> <span class="n">variations</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">n</span><span class="p">,</span> <span class="n">repetition</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="k">yield</span> <span class="nb">list</span><span class="p">(</span><span class="n">coeffs</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="primitive_element"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.numberfields.primitive_element">[docs]</a><span class="k">def</span> <span class="nf">primitive_element</span><span class="p">(</span><span class="n">extension</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Construct a common number field for all extensions. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">extension</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;can&#39;t compute primitive element for empty extension&quot;</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">cls</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">Poly</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">cls</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">),</span> <span class="n">PurePoly</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;ex&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="n">extension</span> <span class="o">=</span> <span class="p">[</span> <span class="n">AlgebraicNumber</span><span class="p">(</span><span class="n">ext</span><span class="p">,</span> <span class="n">gen</span><span class="o">=</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">ext</span> <span class="ow">in</span> <span class="n">extension</span> <span class="p">]</span>
-
-        <span class="n">g</span><span class="p">,</span> <span class="n">coeffs</span> <span class="o">=</span> <span class="n">extension</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">minpoly</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-        <span class="k">for</span> <span class="n">ext</span> <span class="ow">in</span> <span class="n">extension</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="n">s</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">sqf_norm</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">extension</span><span class="o">=</span><span class="n">ext</span><span class="p">)</span>
-            <span class="n">coeffs</span> <span class="o">=</span> <span class="p">[</span> <span class="n">s</span><span class="o">*</span><span class="n">c</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">coeffs</span> <span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;polys&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">g</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(),</span> <span class="n">coeffs</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">coeffs</span>
-
-    <span class="n">generator</span> <span class="o">=</span> <span class="n">numbered_symbols</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Dummy</span><span class="p">)</span>
-
-    <span class="n">F</span><span class="p">,</span> <span class="n">Y</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">ext</span> <span class="ow">in</span> <span class="n">extension</span><span class="p">:</span>
-        <span class="n">y</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">generator</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">ext</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">ext</span><span class="o">.</span><span class="n">is_univariate</span><span class="p">:</span>
-                <span class="n">f</span> <span class="o">=</span> <span class="n">ext</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;expected minimal polynomial, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">ext</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">minpoly</span><span class="p">(</span><span class="n">ext</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-
-        <span class="n">F</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-        <span class="n">Y</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-
-    <span class="n">coeffs_generator</span> <span class="o">=</span> <span class="n">args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;coeffs&#39;</span><span class="p">,</span> <span class="n">_coeffs_generator</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">coeffs</span> <span class="ow">in</span> <span class="n">coeffs_generator</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">Y</span><span class="p">)):</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">x</span> <span class="o">-</span> <span class="nb">sum</span><span class="p">([</span> <span class="n">c</span><span class="o">*</span><span class="n">y</span> <span class="k">for</span> <span class="n">c</span><span class="p">,</span> <span class="n">y</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">Y</span><span class="p">)])</span>
-        <span class="n">G</span> <span class="o">=</span> <span class="n">groebner</span><span class="p">(</span><span class="n">F</span> <span class="o">+</span> <span class="p">[</span><span class="n">f</span><span class="p">],</span> <span class="n">Y</span> <span class="o">+</span> <span class="p">[</span><span class="n">x</span><span class="p">],</span> <span class="n">order</span><span class="o">=</span><span class="s">&#39;lex&#39;</span><span class="p">,</span> <span class="n">field</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-        <span class="n">H</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">G</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">cls</span><span class="p">(</span><span class="n">G</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">x</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="s">&#39;QQ&#39;</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">Y</span><span class="p">))):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">H</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">y</span> <span class="o">-</span> <span class="n">h</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span>
-                            <span class="n">domain</span><span class="o">=</span><span class="s">&#39;QQ&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()</span>  <span class="c"># XXX: composite=False</span>
-            <span class="k">except</span> <span class="n">CoercionFailed</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-                <span class="k">break</span>  <span class="c"># G is not a triangular set</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">break</span>
-    <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-        <span class="k">raise</span> <span class="ne">RuntimeError</span><span class="p">(</span><span class="s">&quot;run out of coefficient configurations&quot;</span><span class="p">)</span>
-
-    <span class="n">_</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">clear_denoms</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;polys&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">g</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(),</span> <span class="n">coeffs</span><span class="p">,</span> <span class="n">H</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">g</span><span class="p">,</span> <span class="n">coeffs</span><span class="p">,</span> <span class="n">H</span>
-
-</div>
-<span class="k">def</span> <span class="nf">is_isomorphism_possible</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns `True` if there is a chance for isomorphism. &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">minpoly</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">minpoly</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="n">m</span> <span class="o">%</span> <span class="n">n</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="n">m</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="n">da</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">minpoly</span><span class="o">.</span><span class="n">discriminant</span><span class="p">()</span>
-    <span class="n">db</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">minpoly</span><span class="o">.</span><span class="n">discriminant</span><span class="p">()</span>
-
-    <span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">half</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="n">m</span><span class="o">//</span><span class="n">n</span><span class="p">,</span> <span class="n">db</span><span class="o">//</span><span class="mi">2</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">sieve</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-        <span class="n">P</span> <span class="o">=</span> <span class="n">p</span><span class="o">**</span><span class="n">k</span>
-
-        <span class="k">if</span> <span class="n">P</span> <span class="o">&gt;</span> <span class="n">half</span><span class="p">:</span>
-            <span class="k">break</span>
-
-        <span class="k">if</span> <span class="p">((</span><span class="n">da</span> <span class="o">%</span> <span class="n">p</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="p">(</span><span class="n">db</span> <span class="o">%</span> <span class="n">P</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-
-    <span class="k">return</span> <span class="bp">True</span>
-
-
-<span class="k">def</span> <span class="nf">field_isomorphism_pslq</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Construct field isomorphism using PSLQ algorithm. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">a</span><span class="o">.</span><span class="n">root</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">b</span><span class="o">.</span><span class="n">root</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;PSLQ doesn&#39;t support complex coefficients&quot;</span><span class="p">)</span>
-
-    <span class="n">f</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">minpoly</span>
-    <span class="n">g</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">minpoly</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">gen</span><span class="p">)</span>
-
-    <span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">prev</span> <span class="o">=</span> <span class="mi">100</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">minpoly</span><span class="o">.</span><span class="n">degree</span><span class="p">(),</span> <span class="bp">None</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">):</span>
-        <span class="n">A</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">root</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="n">B</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">root</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-
-        <span class="n">basis</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">B</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span> <span class="n">B</span><span class="o">**</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span> <span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">A</span><span class="p">]</span>
-
-        <span class="n">dps</span><span class="p">,</span> <span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">dps</span><span class="p">,</span> <span class="n">n</span>
-        <span class="n">coeffs</span> <span class="o">=</span> <span class="n">pslq</span><span class="p">(</span><span class="n">basis</span><span class="p">,</span> <span class="n">maxcoeff</span><span class="o">=</span><span class="nb">int</span><span class="p">(</span><span class="mf">1e10</span><span class="p">),</span> <span class="n">maxsteps</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
-        <span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="n">dps</span>
-
-        <span class="k">if</span> <span class="n">coeffs</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">break</span>
-
-        <span class="k">if</span> <span class="n">coeffs</span> <span class="o">!=</span> <span class="n">prev</span><span class="p">:</span>
-            <span class="n">prev</span> <span class="o">=</span> <span class="n">coeffs</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">break</span>
-
-        <span class="n">coeffs</span> <span class="o">=</span> <span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="n">c</span><span class="p">)</span><span class="o">/</span><span class="n">coeffs</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">coeffs</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span>
-
-        <span class="k">while</span> <span class="ow">not</span> <span class="n">coeffs</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-            <span class="n">coeffs</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-
-        <span class="n">coeffs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">coeffs</span><span class="p">))</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">gen</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="s">&#39;QQ&#39;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">compose</span><span class="p">(</span><span class="n">h</span><span class="p">)</span><span class="o">.</span><span class="n">rem</span><span class="p">(</span><span class="n">g</span><span class="p">)</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-            <span class="n">d</span><span class="p">,</span> <span class="n">approx</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">coeffs</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span>
-
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">coeffs</span><span class="p">):</span>
-                <span class="n">approx</span> <span class="o">+=</span> <span class="n">coeff</span><span class="o">*</span><span class="n">B</span><span class="o">**</span><span class="p">(</span><span class="n">d</span> <span class="o">-</span> <span class="n">i</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">A</span><span class="o">*</span><span class="n">approx</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span> <span class="o">-</span><span class="n">c</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">coeffs</span> <span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">coeffs</span>
-        <span class="k">elif</span> <span class="n">f</span><span class="o">.</span><span class="n">compose</span><span class="p">(</span><span class="o">-</span><span class="n">h</span><span class="p">)</span><span class="o">.</span><span class="n">rem</span><span class="p">(</span><span class="n">g</span><span class="p">)</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[</span> <span class="o">-</span><span class="n">c</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">coeffs</span> <span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">*=</span> <span class="mi">2</span>
-
-    <span class="k">return</span> <span class="bp">None</span>
-
-
-<span class="k">def</span> <span class="nf">field_isomorphism_factor</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Construct field isomorphism via factorization. &quot;&quot;&quot;</span>
-    <span class="n">_</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">factor_list</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">minpoly</span><span class="p">,</span> <span class="n">extension</span><span class="o">=</span><span class="n">b</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">f</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">factors</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">coeffs</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">TC</span><span class="p">()</span><span class="o">.</span><span class="n">to_sympy_list</span><span class="p">()</span>
-            <span class="n">d</span><span class="p">,</span> <span class="n">terms</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">coeffs</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="p">[]</span>
-
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">coeffs</span><span class="p">):</span>
-                <span class="n">terms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">coeff</span><span class="o">*</span><span class="n">b</span><span class="o">.</span><span class="n">root</span><span class="o">**</span><span class="p">(</span><span class="n">d</span> <span class="o">-</span> <span class="n">i</span><span class="p">))</span>
-
-            <span class="n">root</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">terms</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">root</span> <span class="o">-</span> <span class="n">root</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">chop</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">coeffs</span>
-
-            <span class="k">if</span> <span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">root</span> <span class="o">+</span> <span class="n">root</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">chop</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span> <span class="o">-</span><span class="n">c</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">coeffs</span> <span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-
-<div class="viewcode-block" id="field_isomorphism"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.numberfields.field_isomorphism">[docs]</a><span class="k">def</span> <span class="nf">field_isomorphism</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Construct an isomorphism between two number fields. &quot;&quot;&quot;</span>
-    <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">a</span><span class="o">.</span><span class="n">is_AlgebraicNumber</span><span class="p">:</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">AlgebraicNumber</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">b</span><span class="o">.</span><span class="n">is_AlgebraicNumber</span><span class="p">:</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">AlgebraicNumber</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="n">b</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">a</span><span class="o">.</span><span class="n">coeffs</span><span class="p">()</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">minpoly</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">minpoly</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">a</span><span class="o">.</span><span class="n">root</span><span class="p">]</span>
-
-    <span class="k">if</span> <span class="n">m</span> <span class="o">%</span> <span class="n">n</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">if</span> <span class="n">args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;fast&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">field_isomorphism_pslq</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">result</span>
-        <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-            <span class="k">pass</span>
-
-    <span class="k">return</span> <span class="n">field_isomorphism_factor</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="to_number_field"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.numberfields.to_number_field">[docs]</a><span class="k">def</span> <span class="nf">to_number_field</span><span class="p">(</span><span class="n">extension</span><span class="p">,</span> <span class="n">theta</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Express `extension` in the field generated by `theta`. &quot;&quot;&quot;</span>
-    <span class="n">gen</span> <span class="o">=</span> <span class="n">args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;gen&#39;</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">extension</span><span class="p">,</span> <span class="s">&#39;__iter__&#39;</span><span class="p">):</span>
-        <span class="n">extension</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">extension</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">extension</span> <span class="o">=</span> <span class="p">[</span><span class="n">extension</span><span class="p">]</span>
-
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">extension</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">type</span><span class="p">(</span><span class="n">extension</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="ow">is</span> <span class="nb">tuple</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">AlgebraicNumber</span><span class="p">(</span><span class="n">extension</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="n">minpoly</span><span class="p">,</span> <span class="n">coeffs</span> <span class="o">=</span> <span class="n">primitive_element</span><span class="p">(</span><span class="n">extension</span><span class="p">,</span> <span class="n">gen</span><span class="p">,</span> <span class="n">polys</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">root</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">([</span> <span class="n">coeff</span><span class="o">*</span><span class="n">ext</span> <span class="k">for</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">ext</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">extension</span><span class="p">)</span> <span class="p">])</span>
-
-    <span class="k">if</span> <span class="n">theta</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">AlgebraicNumber</span><span class="p">((</span><span class="n">minpoly</span><span class="p">,</span> <span class="n">root</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">theta</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">theta</span><span class="o">.</span><span class="n">is_AlgebraicNumber</span><span class="p">:</span>
-            <span class="n">theta</span> <span class="o">=</span> <span class="n">AlgebraicNumber</span><span class="p">(</span><span class="n">theta</span><span class="p">,</span> <span class="n">gen</span><span class="o">=</span><span class="n">gen</span><span class="p">)</span>
-
-        <span class="n">coeffs</span> <span class="o">=</span> <span class="n">field_isomorphism</span><span class="p">(</span><span class="n">root</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">coeffs</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">AlgebraicNumber</span><span class="p">(</span><span class="n">theta</span><span class="p">,</span> <span class="n">coeffs</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">IsomorphismFailed</span><span class="p">(</span>
-                <span class="s">&quot;</span><span class="si">%s</span><span class="s"> is not in a subfield of </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">root</span><span class="p">,</span> <span class="n">theta</span><span class="o">.</span><span class="n">root</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="AlgebraicNumber"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.numberfields.AlgebraicNumber">[docs]</a><span class="k">class</span> <span class="nc">AlgebraicNumber</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Class for representing algebraic numbers in SymPy. &quot;&quot;&quot;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;rep&#39;</span><span class="p">,</span> <span class="s">&#39;root&#39;</span><span class="p">,</span> <span class="s">&#39;alias&#39;</span><span class="p">,</span> <span class="s">&#39;minpoly&#39;</span><span class="p">]</span>
-
-    <span class="n">is_AlgebraicNumber</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">coeffs</span><span class="o">=</span><span class="n">Tuple</span><span class="p">(),</span> <span class="n">alias</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Construct a new algebraic number. &quot;&quot;&quot;</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">)):</span>
-            <span class="n">minpoly</span><span class="p">,</span> <span class="n">root</span> <span class="o">=</span> <span class="n">expr</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">minpoly</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-                <span class="n">minpoly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">minpoly</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_AlgebraicNumber</span><span class="p">:</span>
-            <span class="n">minpoly</span><span class="p">,</span> <span class="n">root</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">minpoly</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">root</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">minpoly</span><span class="p">,</span> <span class="n">root</span> <span class="o">=</span> <span class="n">minimal_polynomial</span><span class="p">(</span>
-                <span class="n">expr</span><span class="p">,</span> <span class="n">args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;gen&#39;</span><span class="p">),</span> <span class="n">polys</span><span class="o">=</span><span class="bp">True</span><span class="p">),</span> <span class="n">expr</span>
-
-        <span class="n">dom</span> <span class="o">=</span> <span class="n">minpoly</span><span class="o">.</span><span class="n">get_domain</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="n">coeffs</span> <span class="o">!=</span> <span class="n">Tuple</span><span class="p">():</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">ANP</span><span class="p">):</span>
-                <span class="n">rep</span> <span class="o">=</span> <span class="n">DMP</span><span class="o">.</span><span class="n">from_sympy_list</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">coeffs</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-                <span class="n">scoeffs</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">coeffs</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">rep</span> <span class="o">=</span> <span class="n">DMP</span><span class="o">.</span><span class="n">from_list</span><span class="p">(</span><span class="n">coeffs</span><span class="o">.</span><span class="n">to_list</span><span class="p">(),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-                <span class="n">scoeffs</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">coeffs</span><span class="o">.</span><span class="n">to_list</span><span class="p">())</span>
-
-            <span class="k">if</span> <span class="n">rep</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span> <span class="o">&gt;=</span> <span class="n">minpoly</span><span class="o">.</span><span class="n">degree</span><span class="p">():</span>
-                <span class="n">rep</span> <span class="o">=</span> <span class="n">rep</span><span class="o">.</span><span class="n">rem</span><span class="p">(</span><span class="n">minpoly</span><span class="o">.</span><span class="n">rep</span><span class="p">)</span>
-
-            <span class="n">sargs</span> <span class="o">=</span> <span class="p">(</span><span class="n">root</span><span class="p">,</span> <span class="n">scoeffs</span><span class="p">)</span>
-
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">rep</span> <span class="o">=</span> <span class="n">DMP</span><span class="o">.</span><span class="n">from_list</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="mi">0</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">(</span><span class="n">root</span><span class="p">)):</span>
-                <span class="n">rep</span> <span class="o">=</span> <span class="o">-</span><span class="n">rep</span>
-
-            <span class="n">sargs</span> <span class="o">=</span> <span class="p">(</span><span class="n">root</span><span class="p">,</span> <span class="n">coeffs</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">alias</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">alias</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">):</span>
-                <span class="n">alias</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">alias</span><span class="p">)</span>
-            <span class="n">sargs</span> <span class="o">=</span> <span class="n">sargs</span> <span class="o">+</span> <span class="p">(</span><span class="n">alias</span><span class="p">,)</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">sargs</span><span class="p">)</span>
-
-        <span class="n">obj</span><span class="o">.</span><span class="n">rep</span> <span class="o">=</span> <span class="n">rep</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">root</span> <span class="o">=</span> <span class="n">root</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">alias</span> <span class="o">=</span> <span class="n">alias</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">minpoly</span> <span class="o">=</span> <span class="n">minpoly</span>
-
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">AlgebraicNumber</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span><span class="o">.</span><span class="n">_evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="AlgebraicNumber.is_aliased"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.numberfields.AlgebraicNumber.is_aliased">[docs]</a>    <span class="k">def</span> <span class="nf">is_aliased</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns ``True`` if ``alias`` was set. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">alias</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span>
-</div>
-<div class="viewcode-block" id="AlgebraicNumber.as_poly"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.numberfields.AlgebraicNumber.as_poly">[docs]</a>    <span class="k">def</span> <span class="nf">as_poly</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Create a Poly instance from ``self``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Poly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">alias</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Poly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">alias</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">PurePoly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="AlgebraicNumber.as_expr"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.numberfields.AlgebraicNumber.as_expr">[docs]</a>    <span class="k">def</span> <span class="nf">as_expr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Create a Basic expression from ``self``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">as_poly</span><span class="p">(</span><span class="n">x</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">root</span><span class="p">)</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-</div>
-<div class="viewcode-block" id="AlgebraicNumber.coeffs"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.numberfields.AlgebraicNumber.coeffs">[docs]</a>    <span class="k">def</span> <span class="nf">coeffs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns all SymPy coefficients of an algebraic number. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="bp">self</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()</span> <span class="p">]</span>
-</div>
-<div class="viewcode-block" id="AlgebraicNumber.native_coeffs"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.numberfields.AlgebraicNumber.native_coeffs">[docs]</a>    <span class="k">def</span> <span class="nf">native_coeffs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns all native coefficients of an algebraic number. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()</span>
-</div>
-<div class="viewcode-block" id="AlgebraicNumber.to_algebraic_integer"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.numberfields.AlgebraicNumber.to_algebraic_integer">[docs]</a>    <span class="k">def</span> <span class="nf">to_algebraic_integer</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Convert ``self`` to an algebraic integer. &quot;&quot;&quot;</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">minpoly</span>
-
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">LC</span><span class="p">()</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-
-        <span class="n">coeff</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">LC</span><span class="p">()</span><span class="o">**</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">compose</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">gen</span><span class="o">/</span><span class="n">f</span><span class="o">.</span><span class="n">LC</span><span class="p">()))</span>
-
-        <span class="n">minpoly</span> <span class="o">=</span> <span class="n">poly</span><span class="o">*</span><span class="n">coeff</span>
-        <span class="n">root</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">LC</span><span class="p">()</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">root</span>
-
-        <span class="k">return</span> <span class="n">AlgebraicNumber</span><span class="p">((</span><span class="n">minpoly</span><span class="p">,</span> <span class="n">root</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">coeffs</span><span class="p">())</span>
-
-</div></div>
-<span class="k">class</span> <span class="nc">IntervalPrinter</span><span class="p">(</span><span class="n">LambdaPrinter</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Use ``lambda`` printer but print numbers as ``mpi`` intervals. &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Integer</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;mpi(&#39;</span><span class="si">%s</span><span class="s">&#39;)&quot;</span> <span class="o">%</span> <span class="nb">super</span><span class="p">(</span><span class="n">IntervalPrinter</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">_print_Integer</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Rational</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;mpi(&#39;</span><span class="si">%s</span><span class="s">&#39;)&quot;</span> <span class="o">%</span> <span class="nb">super</span><span class="p">(</span><span class="n">IntervalPrinter</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">_print_Rational</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Pow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">IntervalPrinter</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">_print_Pow</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">rational</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="isolate"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.numberfields.isolate">[docs]</a><span class="k">def</span> <span class="nf">isolate</span><span class="p">(</span><span class="n">alg</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">fast</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Give a rational isolating interval for an algebraic number. &quot;&quot;&quot;</span>
-    <span class="n">alg</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">alg</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">alg</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">alg</span><span class="p">,</span> <span class="n">alg</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="ow">not</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">alg</span><span class="p">)):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-            <span class="s">&quot;complex algebraic numbers are not supported&quot;</span><span class="p">)</span>
-
-    <span class="n">func</span> <span class="o">=</span> <span class="n">lambdify</span><span class="p">((),</span> <span class="n">alg</span><span class="p">,</span> <span class="n">modules</span><span class="o">=</span><span class="s">&quot;mpmath&quot;</span><span class="p">,</span> <span class="n">printer</span><span class="o">=</span><span class="n">IntervalPrinter</span><span class="p">())</span>
-
-    <span class="n">poly</span> <span class="o">=</span> <span class="n">minpoly</span><span class="p">(</span><span class="n">alg</span><span class="p">,</span> <span class="n">polys</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">intervals</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">intervals</span><span class="p">(</span><span class="n">sqf</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="n">dps</span><span class="p">,</span> <span class="n">done</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">dps</span><span class="p">,</span> <span class="bp">False</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="k">while</span> <span class="ow">not</span> <span class="n">done</span><span class="p">:</span>
-            <span class="n">alg</span> <span class="o">=</span> <span class="n">func</span><span class="p">()</span>
-
-            <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">intervals</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">a</span> <span class="o">&lt;=</span> <span class="n">alg</span><span class="o">.</span><span class="n">a</span> <span class="ow">and</span> <span class="n">alg</span><span class="o">.</span><span class="n">b</span> <span class="o">&lt;=</span> <span class="n">b</span><span class="p">:</span>
-                    <span class="n">done</span> <span class="o">=</span> <span class="bp">True</span>
-                    <span class="k">break</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">*=</span> <span class="mi">2</span>
-    <span class="k">finally</span><span class="p">:</span>
-        <span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="n">dps</span>
-
-    <span class="k">if</span> <span class="n">eps</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">refine_root</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="n">eps</span><span class="p">,</span> <span class="n">fast</span><span class="o">=</span><span class="n">fast</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span></div>
-</pre></div>
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-  <h1>Source code for sympy.polys.orthopolys</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Efficient functions for generating orthogonal polynomials. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Dummy</span>
-
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">cythonized</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.constructor</span> <span class="kn">import</span> <span class="n">construct_domain</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polytools</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">PurePoly</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polyclasses</span> <span class="kn">import</span> <span class="n">DMP</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dup_mul</span><span class="p">,</span> <span class="n">dup_mul_ground</span><span class="p">,</span> <span class="n">dup_lshift</span><span class="p">,</span> <span class="n">dup_sub</span><span class="p">,</span> <span class="n">dup_add</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span><span class="p">,</span> <span class="n">QQ</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_jacobi</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Low-level implementation of Jacobi polynomials. &quot;&quot;&quot;</span>
-    <span class="n">seq</span> <span class="o">=</span> <span class="p">[[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">],</span> <span class="p">[(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">/</span><span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span><span class="o">/</span><span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">)]]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">den</span> <span class="o">=</span> <span class="n">K</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="n">i</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">i</span> <span class="o">-</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-        <span class="n">f0</span> <span class="o">=</span> <span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">i</span> <span class="o">-</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="o">*</span><span class="n">b</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">den</span><span class="p">)</span>
-        <span class="n">f1</span> <span class="o">=</span> <span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">i</span> <span class="o">-</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">i</span> <span class="o">-</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span> <span class="o">*</span> <span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">i</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">den</span><span class="p">)</span>
-        <span class="n">f2</span> <span class="o">=</span> <span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">i</span> <span class="o">-</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">i</span> <span class="o">-</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">i</span><span class="p">)</span> <span class="o">/</span> <span class="n">den</span>
-        <span class="n">p0</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">seq</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">f0</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">p1</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">dup_lshift</span><span class="p">(</span><span class="n">seq</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">f1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">p2</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">seq</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">f2</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">seq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dup_sub</span><span class="p">(</span><span class="n">dup_add</span><span class="p">(</span><span class="n">p0</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">p2</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">seq</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-
-
-<div class="viewcode-block" id="jacobi_poly"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.orthopolys.jacobi_poly">[docs]</a><span class="k">def</span> <span class="nf">jacobi_poly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Generates Jacobi polynomial of degree `n` in `x`. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;can&#39;t generate Jacobi polynomial of degree </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="n">K</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">construct_domain</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">],</span> <span class="n">field</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">poly</span> <span class="o">=</span> <span class="n">DMP</span><span class="p">(</span><span class="n">dup_jacobi</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">PurePoly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;polys&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">poly</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">poly</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_gegenbauer</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Low-level implementation of Gegenbauer polynomials. &quot;&quot;&quot;</span>
-    <span class="n">seq</span> <span class="o">=</span> <span class="p">[[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">],</span> <span class="p">[</span><span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">f1</span> <span class="o">=</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="n">a</span> <span class="o">-</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">)</span> <span class="o">/</span> <span class="n">i</span>
-        <span class="n">f2</span> <span class="o">=</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span> <span class="o">/</span> <span class="n">i</span>
-        <span class="n">p1</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">dup_lshift</span><span class="p">(</span><span class="n">seq</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">f1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">p2</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">seq</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">f2</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">seq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dup_sub</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">seq</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-
-
-<div class="viewcode-block" id="gegenbauer_poly"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.orthopolys.gegenbauer_poly">[docs]</a><span class="k">def</span> <span class="nf">gegenbauer_poly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Generates Gegenbauer polynomial of degree `n` in `x`. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-            <span class="s">&quot;can&#39;t generate Gegenbauer polynomial of degree </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="n">K</span><span class="p">,</span> <span class="n">a</span> <span class="o">=</span> <span class="n">construct_domain</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">field</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">poly</span> <span class="o">=</span> <span class="n">DMP</span><span class="p">(</span><span class="n">dup_gegenbauer</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">a</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">PurePoly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;polys&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">poly</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">poly</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_chebyshevt</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Low-level implementation of Chebyshev polynomials of the 1st kind. &quot;&quot;&quot;</span>
-    <span class="n">seq</span> <span class="o">=</span> <span class="p">[[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">],</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">dup_lshift</span><span class="p">(</span><span class="n">seq</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">seq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dup_sub</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">seq</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">K</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">seq</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-
-
-<div class="viewcode-block" id="chebyshevt_poly"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.orthopolys.chebyshevt_poly">[docs]</a><span class="k">def</span> <span class="nf">chebyshevt_poly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Generates Chebyshev polynomial of the first kind of degree `n` in `x`. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-            <span class="s">&quot;can&#39;t generate 1st kind Chebyshev polynomial of degree </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="n">poly</span> <span class="o">=</span> <span class="n">DMP</span><span class="p">(</span><span class="n">dup_chebyshevt</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">PurePoly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;polys&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">poly</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">poly</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_chebyshevu</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Low-level implementation of Chebyshev polynomials of the 2nd kind. &quot;&quot;&quot;</span>
-    <span class="n">seq</span> <span class="o">=</span> <span class="p">[[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">],</span> <span class="p">[</span><span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">dup_lshift</span><span class="p">(</span><span class="n">seq</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">seq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dup_sub</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">seq</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">K</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">seq</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-
-
-<div class="viewcode-block" id="chebyshevu_poly"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.orthopolys.chebyshevu_poly">[docs]</a><span class="k">def</span> <span class="nf">chebyshevu_poly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Generates Chebyshev polynomial of the second kind of degree `n` in `x`. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-            <span class="s">&quot;can&#39;t generate 2nd kind Chebyshev polynomial of degree </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="n">poly</span> <span class="o">=</span> <span class="n">DMP</span><span class="p">(</span><span class="n">dup_chebyshevu</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">PurePoly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;polys&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">poly</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">poly</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_hermite</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Low-level implementation of Hermite polynomials. &quot;&quot;&quot;</span>
-    <span class="n">seq</span> <span class="o">=</span> <span class="p">[[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">],</span> <span class="p">[</span><span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">dup_lshift</span><span class="p">(</span><span class="n">seq</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">seq</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">K</span><span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">c</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">dup_sub</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">seq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">seq</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-
-
-<div class="viewcode-block" id="hermite_poly"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.orthopolys.hermite_poly">[docs]</a><span class="k">def</span> <span class="nf">hermite_poly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Generates Hermite polynomial of degree `n` in `x`. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;can&#39;t generate Hermite polynomial of degree </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="n">poly</span> <span class="o">=</span> <span class="n">DMP</span><span class="p">(</span><span class="n">dup_hermite</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">PurePoly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;polys&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">poly</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">poly</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Low-level implementation of Legendre polynomials. &quot;&quot;&quot;</span>
-    <span class="n">seq</span> <span class="o">=</span> <span class="p">[[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">],</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">dup_lshift</span><span class="p">(</span><span class="n">seq</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">i</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">seq</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">K</span><span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">i</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">seq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dup_sub</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">seq</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-
-
-<div class="viewcode-block" id="legendre_poly"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.orthopolys.legendre_poly">[docs]</a><span class="k">def</span> <span class="nf">legendre_poly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Generates Legendre polynomial of degree `n` in `x`. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;can&#39;t generate Legendre polynomial of degree </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="n">poly</span> <span class="o">=</span> <span class="n">DMP</span><span class="p">(</span><span class="n">dup_legendre</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">QQ</span><span class="p">),</span> <span class="n">QQ</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">PurePoly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;polys&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">poly</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">poly</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_laguerre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Low-level implementation of Laguerre polynomials. &quot;&quot;&quot;</span>
-    <span class="n">seq</span> <span class="o">=</span> <span class="p">[[</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">],</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">]]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">dup_mul</span><span class="p">(</span><span class="n">seq</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="o">/</span><span class="n">i</span><span class="p">,</span> <span class="n">alpha</span><span class="o">/</span><span class="n">i</span> <span class="o">+</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">i</span><span class="p">],</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">seq</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">alpha</span><span class="o">/</span><span class="n">i</span> <span class="o">+</span> <span class="n">K</span><span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">i</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-        <span class="n">seq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dup_sub</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">K</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">seq</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-
-
-<div class="viewcode-block" id="laguerre_poly"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.orthopolys.laguerre_poly">[docs]</a><span class="k">def</span> <span class="nf">laguerre_poly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Generates Laguerre polynomial of degree `n` in `x`. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;can&#39;t generate Laguerre polynomial of degree </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">alpha</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">K</span><span class="p">,</span> <span class="n">alpha</span> <span class="o">=</span> <span class="n">construct_domain</span><span class="p">(</span>
-            <span class="n">alpha</span><span class="p">,</span> <span class="n">field</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>  <span class="c"># XXX: ground_field=True</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">K</span><span class="p">,</span> <span class="n">alpha</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">,</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-
-    <span class="n">poly</span> <span class="o">=</span> <span class="n">DMP</span><span class="p">(</span><span class="n">dup_laguerre</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">PurePoly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;polys&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">poly</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">poly</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_spherical_bessel_fn</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Low-level implementation of fn(n, x) &quot;&quot;&quot;</span>
-    <span class="n">seq</span> <span class="o">=</span> <span class="p">[[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">],</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">dup_lshift</span><span class="p">(</span><span class="n">seq</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">seq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dup_sub</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">seq</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">K</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">dup_lshift</span><span class="p">(</span><span class="n">seq</span><span class="p">[</span><span class="n">n</span><span class="p">],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dup_spherical_bessel_fn_minus</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Low-level implementation of fn(-n, x) &quot;&quot;&quot;</span>
-    <span class="n">seq</span> <span class="o">=</span> <span class="p">[[</span><span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">,</span> <span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">],</span> <span class="p">[</span><span class="n">K</span><span class="o">.</span><span class="n">zero</span><span class="p">]]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">dup_mul_ground</span><span class="p">(</span><span class="n">dup_lshift</span><span class="p">(</span><span class="n">seq</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">K</span><span class="p">(</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">i</span><span class="p">),</span> <span class="n">K</span><span class="p">)</span>
-        <span class="n">seq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">dup_sub</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">seq</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">K</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">seq</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">spherical_bessel_fn</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Coefficients for the spherical Bessel functions.</span>
-
-<span class="sd">    Those are only needed in the jn() function.</span>
-
-<span class="sd">    The coefficients are calculated from:</span>
-
-<span class="sd">    fn(0, z) = 1/z</span>
-<span class="sd">    fn(1, z) = 1/z**2</span>
-<span class="sd">    fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z)</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.orthopolys import spherical_bessel_fn as fn</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">    &gt;&gt;&gt; z = Symbol(&quot;z&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; fn(1, z)</span>
-<span class="sd">    z**(-2)</span>
-<span class="sd">    &gt;&gt;&gt; fn(2, z)</span>
-<span class="sd">    -1/z + 3/z**3</span>
-<span class="sd">    &gt;&gt;&gt; fn(3, z)</span>
-<span class="sd">    -6/z**2 + 15/z**4</span>
-<span class="sd">    &gt;&gt;&gt; fn(4, z)</span>
-<span class="sd">    1/z - 45/z**3 + 105/z**5</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">dup</span> <span class="o">=</span> <span class="n">dup_spherical_bessel_fn_minus</span><span class="p">(</span><span class="o">-</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">dup</span> <span class="o">=</span> <span class="n">dup_spherical_bessel_fn</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">)</span>
-
-    <span class="n">poly</span> <span class="o">=</span> <span class="n">DMP</span><span class="p">(</span><span class="n">dup</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">PurePoly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;polys&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">poly</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">poly</span>
-</pre></div>
-
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-  <h1>Source code for sympy.polys.partfrac</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Algorithms for partial fraction decomposition of rational functions. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">RootSum</span><span class="p">,</span> <span class="n">cancel</span><span class="p">,</span> <span class="n">factor</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polytools</span> <span class="kn">import</span> <span class="n">parallel_poly_from_expr</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polyoptions</span> <span class="kn">import</span> <span class="n">allowed_flags</span><span class="p">,</span> <span class="n">set_defaults</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">Add</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Lambda</span><span class="p">,</span> <span class="n">Dummy</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">numbered_symbols</span><span class="p">,</span> <span class="n">take</span><span class="p">,</span> <span class="n">threaded</span>
-
-
-<span class="nd">@threaded</span>
-<div class="viewcode-block" id="apart"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.partfrac.apart">[docs]</a><span class="k">def</span> <span class="nf">apart</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">full</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute partial fraction decomposition of a rational function.</span>
-
-<span class="sd">    Given a rational function ``f`` compute partial fraction decomposition</span>
-<span class="sd">    of ``f``. Two algorithms are available: one is based on undetermined</span>
-<span class="sd">    coefficients method and the other is Bronstein&#39;s full partial fraction</span>
-<span class="sd">    decomposition algorithm.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.partfrac import apart</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">    &gt;&gt;&gt; apart(y/(x + 2)/(x + 1), x)</span>
-<span class="sd">    -y/(x + 2) + y/(x + 1)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">allowed_flags</span><span class="p">(</span><span class="n">options</span><span class="p">,</span> <span class="p">[])</span>
-
-    <span class="n">f</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">P</span><span class="p">,</span> <span class="n">Q</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-
-    <span class="n">options</span> <span class="o">=</span> <span class="n">set_defaults</span><span class="p">(</span><span class="n">options</span><span class="p">,</span> <span class="n">extension</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">Q</span><span class="p">),</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">((</span><span class="n">P</span><span class="p">,</span> <span class="n">Q</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">P</span><span class="o">.</span><span class="n">is_multivariate</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-            <span class="s">&quot;multivariate partial fraction decomposition&quot;</span><span class="p">)</span>
-
-    <span class="n">common</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">Q</span> <span class="o">=</span> <span class="n">P</span><span class="o">.</span><span class="n">cancel</span><span class="p">(</span><span class="n">Q</span><span class="p">)</span>
-
-    <span class="n">poly</span><span class="p">,</span> <span class="n">P</span> <span class="o">=</span> <span class="n">P</span><span class="o">.</span><span class="n">div</span><span class="p">(</span><span class="n">Q</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">P</span><span class="p">,</span> <span class="n">Q</span> <span class="o">=</span> <span class="n">P</span><span class="o">.</span><span class="n">rat_clear_denoms</span><span class="p">(</span><span class="n">Q</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">Q</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">partial</span> <span class="o">=</span> <span class="n">P</span><span class="o">/</span><span class="n">Q</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">full</span><span class="p">:</span>
-            <span class="n">partial</span> <span class="o">=</span> <span class="n">apart_undetermined_coeffs</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">Q</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">partial</span> <span class="o">=</span> <span class="n">apart_full_decomposition</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">Q</span><span class="p">)</span>
-
-    <span class="n">terms</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">partial</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">term</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">RootSum</span><span class="p">):</span>
-            <span class="n">terms</span> <span class="o">+=</span> <span class="n">term</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">terms</span> <span class="o">+=</span> <span class="n">factor</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">common</span><span class="o">*</span><span class="p">(</span><span class="n">poly</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span> <span class="o">+</span> <span class="n">terms</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">apart_undetermined_coeffs</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">Q</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Partial fractions via method of undetermined coefficients. &quot;&quot;&quot;</span>
-    <span class="n">X</span> <span class="o">=</span> <span class="n">numbered_symbols</span><span class="p">(</span><span class="n">cls</span><span class="o">=</span><span class="n">Dummy</span><span class="p">)</span>
-    <span class="n">partial</span><span class="p">,</span> <span class="n">symbols</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
-
-    <span class="n">_</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">Q</span><span class="o">.</span><span class="n">factor_list</span><span class="p">()</span>
-
-    <span class="k">for</span> <span class="n">f</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">factors</span><span class="p">:</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">degree</span><span class="p">(),</span> <span class="n">Q</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">coeffs</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">take</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">n</span><span class="p">),</span> <span class="n">q</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-            <span class="n">partial</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">i</span><span class="p">))</span>
-            <span class="n">symbols</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">coeffs</span><span class="p">)</span>
-
-    <span class="n">dom</span> <span class="o">=</span> <span class="n">Q</span><span class="o">.</span><span class="n">get_domain</span><span class="p">()</span><span class="o">.</span><span class="n">inject</span><span class="p">(</span><span class="o">*</span><span class="n">symbols</span><span class="p">)</span>
-    <span class="n">F</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">gen</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="n">dom</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">partial</span><span class="p">):</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">gen</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="n">dom</span><span class="p">)</span>
-        <span class="n">partial</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-        <span class="n">q</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">set_domain</span><span class="p">(</span><span class="n">dom</span><span class="p">)</span>
-        <span class="n">F</span> <span class="o">+=</span> <span class="n">h</span><span class="o">*</span><span class="n">q</span>
-
-    <span class="n">system</span><span class="p">,</span> <span class="n">result</span> <span class="o">=</span> <span class="p">[],</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="p">(</span><span class="n">k</span><span class="p">,),</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">F</span><span class="o">.</span><span class="n">terms</span><span class="p">():</span>
-        <span class="n">system</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">coeff</span> <span class="o">-</span> <span class="n">P</span><span class="o">.</span><span class="n">nth</span><span class="p">(</span><span class="n">k</span><span class="p">))</span>
-
-    <span class="kn">from</span> <span class="nn">sympy.solvers</span> <span class="kn">import</span> <span class="n">solve</span>
-    <span class="n">solution</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">system</span><span class="p">,</span> <span class="n">symbols</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">h</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">partial</span><span class="p">:</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="n">h</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">solution</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">+=</span> <span class="n">h</span><span class="o">/</span><span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span><span class="o">**</span><span class="n">k</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="k">def</span> <span class="nf">apart_full_decomposition</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">Q</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Bronstein&#39;s full partial fraction decomposition algorithm.</span>
-
-<span class="sd">    Given a univariate rational function ``f``, performing only GCD</span>
-<span class="sd">    operations over the algebraic closure of the initial ground domain</span>
-<span class="sd">    of definition, compute full partial fraction decomposition with</span>
-<span class="sd">    fractions having linear denominators.</span>
-
-<span class="sd">    Note that no factorization of the initial denominator of ``f`` is</span>
-<span class="sd">    performed. The final decomposition is formed in terms of a sum of</span>
-<span class="sd">    :class:`RootSum` instances.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Bronstein93]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">U</span> <span class="o">=</span> <span class="n">P</span><span class="o">/</span><span class="n">Q</span><span class="p">,</span> <span class="n">P</span><span class="o">.</span><span class="n">gen</span><span class="p">,</span> <span class="p">[]</span>
-
-    <span class="n">u</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;u&#39;</span><span class="p">)(</span><span class="n">x</span><span class="p">)</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">)</span>
-
-    <span class="n">partial</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">d</span><span class="p">,</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">Q</span><span class="o">.</span><span class="n">sqf_list_include</span><span class="p">(</span><span class="nb">all</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-        <span class="n">U</span> <span class="o">+=</span> <span class="p">[</span> <span class="n">u</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="p">]</span>
-
-        <span class="n">h</span> <span class="o">=</span> <span class="n">cancel</span><span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="n">b</span><span class="o">**</span><span class="n">n</span><span class="p">)</span> <span class="o">/</span> <span class="n">u</span><span class="o">**</span><span class="n">n</span>
-
-        <span class="n">H</span><span class="p">,</span> <span class="n">subs</span> <span class="o">=</span> <span class="p">[</span><span class="n">h</span><span class="p">],</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-            <span class="n">H</span> <span class="o">+=</span> <span class="p">[</span> <span class="n">H</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">/</span> <span class="n">j</span> <span class="p">]</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">subs</span> <span class="o">+=</span> <span class="p">[</span> <span class="p">(</span><span class="n">U</span><span class="p">[</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">b</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="o">/</span> <span class="n">j</span><span class="p">)</span> <span class="p">]</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-            <span class="n">P</span><span class="p">,</span> <span class="n">Q</span> <span class="o">=</span> <span class="n">cancel</span><span class="p">(</span><span class="n">H</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="n">P</span> <span class="o">=</span> <span class="n">P</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="o">*</span><span class="n">subs</span><span class="p">[</span><span class="n">j</span> <span class="o">-</span> <span class="n">i</span><span class="p">])</span>
-
-            <span class="n">Q</span> <span class="o">=</span> <span class="n">Q</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="o">*</span><span class="n">subs</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-            <span class="n">P</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-            <span class="n">Q</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">Q</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-            <span class="n">G</span> <span class="o">=</span> <span class="n">P</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-            <span class="n">D</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-            <span class="n">B</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">Q</span><span class="o">.</span><span class="n">half_gcdex</span><span class="p">(</span><span class="n">D</span><span class="p">)</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="p">(</span><span class="n">P</span> <span class="o">*</span> <span class="n">B</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">g</span><span class="p">))</span><span class="o">.</span><span class="n">rem</span><span class="p">(</span><span class="n">D</span><span class="p">)</span>
-
-            <span class="n">numer</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-            <span class="n">denom</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">a</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">j</span><span class="p">)</span>
-
-            <span class="n">func</span> <span class="o">=</span> <span class="n">Lambda</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">numer</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="n">denom</span><span class="p">)</span>
-            <span class="n">partial</span> <span class="o">+=</span> <span class="n">RootSum</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">partial</span>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/polys/polyclasses.html b/dev-py3k/_modules/sympy/polys/polyclasses.html
deleted file mode 100644
index 4f01ca14403..00000000000
--- a/dev-py3k/_modules/sympy/polys/polyclasses.html
+++ /dev/null
@@ -1,1822 +0,0 @@
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-            
-  <h1>Source code for sympy.polys.polyclasses</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;OO layer for several polynomial representations. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyutils</span> <span class="kn">import</span> <span class="n">PicklableWithSlots</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="n">CoercionFailed</span><span class="p">,</span> <span class="n">NotReversible</span>
-
-
-<span class="k">class</span> <span class="nc">GenericPoly</span><span class="p">(</span><span class="n">PicklableWithSlots</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Base class for low-level polynomial representations. &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">ground_to_ring</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Make the ground domain a ring. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">set_domain</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">get_ring</span><span class="p">())</span>
-
-    <span class="k">def</span> <span class="nf">ground_to_field</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Make the ground domain a field. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">set_domain</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">get_field</span><span class="p">())</span>
-
-    <span class="k">def</span> <span class="nf">ground_to_exact</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Make the ground domain exact. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">set_domain</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">get_exact</span><span class="p">())</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_perify_factors</span><span class="p">(</span><span class="n">per</span><span class="p">,</span> <span class="n">result</span><span class="p">,</span> <span class="n">include</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">include</span><span class="p">:</span>
-            <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">result</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">result</span>
-
-        <span class="n">factors</span> <span class="o">=</span> <span class="p">[</span> <span class="p">(</span><span class="n">per</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">factors</span> <span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">include</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">factors</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dmp_validate</span><span class="p">,</span>
-    <span class="n">dup_normal</span><span class="p">,</span> <span class="n">dmp_normal</span><span class="p">,</span>
-    <span class="n">dup_convert</span><span class="p">,</span> <span class="n">dmp_convert</span><span class="p">,</span>
-    <span class="n">dmp_from_sympy</span><span class="p">,</span>
-    <span class="n">dup_strip</span><span class="p">,</span>
-    <span class="n">dup_degree</span><span class="p">,</span> <span class="n">dmp_degree_in</span><span class="p">,</span>
-    <span class="n">dmp_degree_list</span><span class="p">,</span>
-    <span class="n">dmp_negative_p</span><span class="p">,</span>
-    <span class="n">dup_LC</span><span class="p">,</span> <span class="n">dmp_ground_LC</span><span class="p">,</span>
-    <span class="n">dup_TC</span><span class="p">,</span> <span class="n">dmp_ground_TC</span><span class="p">,</span>
-    <span class="n">dmp_ground_nth</span><span class="p">,</span>
-    <span class="n">dmp_one</span><span class="p">,</span> <span class="n">dmp_ground</span><span class="p">,</span>
-    <span class="n">dmp_zero_p</span><span class="p">,</span> <span class="n">dmp_one_p</span><span class="p">,</span> <span class="n">dmp_ground_p</span><span class="p">,</span>
-    <span class="n">dup_from_dict</span><span class="p">,</span> <span class="n">dmp_from_dict</span><span class="p">,</span>
-    <span class="n">dmp_to_dict</span><span class="p">,</span>
-    <span class="n">dmp_deflate</span><span class="p">,</span>
-    <span class="n">dmp_inject</span><span class="p">,</span> <span class="n">dmp_eject</span><span class="p">,</span>
-    <span class="n">dmp_terms_gcd</span><span class="p">,</span>
-    <span class="n">dmp_list_terms</span><span class="p">,</span> <span class="n">dmp_exclude</span><span class="p">,</span>
-    <span class="n">dmp_slice_in</span><span class="p">,</span> <span class="n">dmp_permute</span><span class="p">,</span>
-    <span class="n">dmp_to_tuple</span><span class="p">,)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dmp_add_ground</span><span class="p">,</span>
-    <span class="n">dmp_sub_ground</span><span class="p">,</span>
-    <span class="n">dmp_mul_ground</span><span class="p">,</span>
-    <span class="n">dmp_quo_ground</span><span class="p">,</span>
-    <span class="n">dmp_exquo_ground</span><span class="p">,</span>
-    <span class="n">dmp_abs</span><span class="p">,</span>
-    <span class="n">dup_neg</span><span class="p">,</span> <span class="n">dmp_neg</span><span class="p">,</span>
-    <span class="n">dup_add</span><span class="p">,</span> <span class="n">dmp_add</span><span class="p">,</span>
-    <span class="n">dup_sub</span><span class="p">,</span> <span class="n">dmp_sub</span><span class="p">,</span>
-    <span class="n">dup_mul</span><span class="p">,</span> <span class="n">dmp_mul</span><span class="p">,</span>
-    <span class="n">dmp_sqr</span><span class="p">,</span>
-    <span class="n">dup_pow</span><span class="p">,</span> <span class="n">dmp_pow</span><span class="p">,</span>
-    <span class="n">dmp_pdiv</span><span class="p">,</span>
-    <span class="n">dmp_prem</span><span class="p">,</span>
-    <span class="n">dmp_pquo</span><span class="p">,</span>
-    <span class="n">dmp_pexquo</span><span class="p">,</span>
-    <span class="n">dmp_div</span><span class="p">,</span>
-    <span class="n">dup_rem</span><span class="p">,</span> <span class="n">dmp_rem</span><span class="p">,</span>
-    <span class="n">dmp_quo</span><span class="p">,</span>
-    <span class="n">dmp_exquo</span><span class="p">,</span>
-    <span class="n">dmp_add_mul</span><span class="p">,</span> <span class="n">dmp_sub_mul</span><span class="p">,</span>
-    <span class="n">dmp_max_norm</span><span class="p">,</span>
-    <span class="n">dmp_l1_norm</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dmp_clear_denoms</span><span class="p">,</span>
-    <span class="n">dmp_integrate_in</span><span class="p">,</span>
-    <span class="n">dmp_diff_in</span><span class="p">,</span>
-    <span class="n">dmp_eval_in</span><span class="p">,</span>
-    <span class="n">dup_revert</span><span class="p">,</span>
-    <span class="n">dmp_ground_trunc</span><span class="p">,</span>
-    <span class="n">dmp_ground_content</span><span class="p">,</span>
-    <span class="n">dmp_ground_primitive</span><span class="p">,</span>
-    <span class="n">dmp_ground_monic</span><span class="p">,</span>
-    <span class="n">dmp_compose</span><span class="p">,</span>
-    <span class="n">dup_decompose</span><span class="p">,</span>
-    <span class="n">dup_shift</span><span class="p">,</span>
-    <span class="n">dmp_lift</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dup_half_gcdex</span><span class="p">,</span> <span class="n">dup_gcdex</span><span class="p">,</span> <span class="n">dup_invert</span><span class="p">,</span>
-    <span class="n">dmp_subresultants</span><span class="p">,</span>
-    <span class="n">dmp_resultant</span><span class="p">,</span>
-    <span class="n">dmp_discriminant</span><span class="p">,</span>
-    <span class="n">dmp_inner_gcd</span><span class="p">,</span>
-    <span class="n">dmp_gcd</span><span class="p">,</span>
-    <span class="n">dmp_lcm</span><span class="p">,</span>
-    <span class="n">dmp_cancel</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.sqfreetools</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dup_gff_list</span><span class="p">,</span>
-    <span class="n">dmp_sqf_p</span><span class="p">,</span>
-    <span class="n">dmp_sqf_norm</span><span class="p">,</span>
-    <span class="n">dmp_sqf_part</span><span class="p">,</span>
-    <span class="n">dmp_sqf_list</span><span class="p">,</span> <span class="n">dmp_sqf_list_include</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.factortools</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dup_cyclotomic_p</span><span class="p">,</span> <span class="n">dmp_irreducible_p</span><span class="p">,</span>
-    <span class="n">dmp_factor_list</span><span class="p">,</span> <span class="n">dmp_factor_list_include</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.rootisolation</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dup_isolate_real_roots_sqf</span><span class="p">,</span>
-    <span class="n">dup_isolate_real_roots</span><span class="p">,</span>
-    <span class="n">dup_isolate_all_roots_sqf</span><span class="p">,</span>
-    <span class="n">dup_isolate_all_roots</span><span class="p">,</span>
-    <span class="n">dup_refine_real_root</span><span class="p">,</span>
-    <span class="n">dup_count_real_roots</span><span class="p">,</span>
-    <span class="n">dup_count_complex_roots</span><span class="p">,</span>
-    <span class="n">dup_sturm</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">UnificationFailed</span><span class="p">,</span>
-    <span class="n">PolynomialError</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">init_normal_DMP</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">DMP</span><span class="p">(</span><span class="n">dmp_normal</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="DMP"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP">[docs]</a><span class="k">class</span> <span class="nc">DMP</span><span class="p">(</span><span class="n">PicklableWithSlots</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Dense Multivariate Polynomials over `K`. &quot;&quot;&quot;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;rep&#39;</span><span class="p">,</span> <span class="s">&#39;lev&#39;</span><span class="p">,</span> <span class="s">&#39;dom&#39;</span><span class="p">,</span> <span class="s">&#39;ring&#39;</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">ring</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">lev</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">rep</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">dict</span><span class="p">:</span>
-                <span class="n">rep</span> <span class="o">=</span> <span class="n">dmp_from_dict</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="nb">type</span><span class="p">(</span><span class="n">rep</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="nb">list</span><span class="p">:</span>
-                <span class="n">rep</span> <span class="o">=</span> <span class="n">dmp_ground</span><span class="p">(</span><span class="n">dom</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">rep</span><span class="p">),</span> <span class="n">lev</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">rep</span><span class="p">,</span> <span class="n">lev</span> <span class="o">=</span> <span class="n">dmp_validate</span><span class="p">(</span><span class="n">rep</span><span class="p">)</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">rep</span> <span class="o">=</span> <span class="n">rep</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">lev</span> <span class="o">=</span> <span class="n">lev</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">dom</span> <span class="o">=</span> <span class="n">dom</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ring</span> <span class="o">=</span> <span class="n">ring</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">hash</span><span class="p">((</span><span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">to_tuple</span><span class="p">(),</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span><span class="p">))</span>
-
-<div class="viewcode-block" id="DMP.unify"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.unify">[docs]</a>    <span class="k">def</span> <span class="nf">unify</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Unify representations of two multivariate polynomials. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">DMP</span><span class="p">)</span> <span class="ow">or</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span> <span class="o">!=</span> <span class="n">g</span><span class="o">.</span><span class="n">lev</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">UnificationFailed</span><span class="p">(</span><span class="s">&quot;can&#39;t unify </span><span class="si">%s</span><span class="s"> with </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span> <span class="o">==</span> <span class="n">g</span><span class="o">.</span><span class="n">dom</span> <span class="ow">and</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span> <span class="o">==</span> <span class="n">g</span><span class="o">.</span><span class="n">ring</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">g</span><span class="o">.</span><span class="n">rep</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-            <span class="n">ring</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span>
-            <span class="k">if</span> <span class="n">g</span><span class="o">.</span><span class="n">ring</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">ring</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">ring</span> <span class="o">=</span> <span class="n">ring</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">ring</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">ring</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">ring</span>
-
-            <span class="n">F</span> <span class="o">=</span> <span class="n">dmp_convert</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-            <span class="n">G</span> <span class="o">=</span> <span class="n">dmp_convert</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">g</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-
-            <span class="k">def</span> <span class="nf">per</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">dom</span><span class="o">=</span><span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="o">=</span><span class="n">lev</span><span class="p">,</span> <span class="n">kill</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">kill</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">lev</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">rep</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">lev</span> <span class="o">-=</span> <span class="mi">1</span>
-
-                <span class="k">return</span> <span class="n">DMP</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">ring</span><span class="p">)</span>
-
-            <span class="k">return</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span>
-</div>
-<div class="viewcode-block" id="DMP.per"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.per">[docs]</a>    <span class="k">def</span> <span class="nf">per</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="n">dom</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">kill</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">ring</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Create a DMP out of the given representation. &quot;&quot;&quot;</span>
-        <span class="n">lev</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span>
-
-        <span class="k">if</span> <span class="n">kill</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">lev</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">rep</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">lev</span> <span class="o">-=</span> <span class="mi">1</span>
-
-        <span class="k">if</span> <span class="n">dom</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">dom</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span>
-
-        <span class="k">if</span> <span class="n">ring</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">ring</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span>
-
-        <span class="k">return</span> <span class="n">DMP</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">ring</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">zero</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">ring</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">DMP</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">ring</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">one</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">ring</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">DMP</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">ring</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="DMP.from_list"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.from_list">[docs]</a>    <span class="k">def</span> <span class="nf">from_list</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Create an instance of ``cls`` given a list of native coefficients. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">dmp_convert</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="DMP.from_sympy_list"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.from_sympy_list">[docs]</a>    <span class="k">def</span> <span class="nf">from_sympy_list</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Create an instance of ``cls`` given a list of SymPy coefficients. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">dmp_from_sympy</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.to_dict"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.to_dict">[docs]</a>    <span class="k">def</span> <span class="nf">to_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">zero</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Convert ``f`` to a dict representation with native coefficients. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dmp_to_dict</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">zero</span><span class="o">=</span><span class="n">zero</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.to_sympy_dict"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.to_sympy_dict">[docs]</a>    <span class="k">def</span> <span class="nf">to_sympy_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">zero</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Convert ``f`` to a dict representation with SymPy coefficients. &quot;&quot;&quot;</span>
-        <span class="n">rep</span> <span class="o">=</span> <span class="n">dmp_to_dict</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">zero</span><span class="o">=</span><span class="n">zero</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">rep</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">rep</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">rep</span>
-</div>
-<div class="viewcode-block" id="DMP.to_tuple"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.to_tuple">[docs]</a>    <span class="k">def</span> <span class="nf">to_tuple</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Convert ``f`` to a tuple representation with native coefficients.</span>
-
-<span class="sd">        This is needed for hashing.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dmp_to_tuple</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="DMP.from_dict"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.from_dict">[docs]</a>    <span class="k">def</span> <span class="nf">from_dict</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Construct and instance of ``cls`` from a ``dict`` representation. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">dmp_from_dict</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">from_monoms_coeffs</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">monoms</span><span class="p">,</span> <span class="n">coeffs</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">ring</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">DMP</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">monoms</span><span class="p">,</span> <span class="n">coeffs</span><span class="p">))),</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">ring</span><span class="p">)</span>
-
-<div class="viewcode-block" id="DMP.to_ring"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.to_ring">[docs]</a>    <span class="k">def</span> <span class="nf">to_ring</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Make the ground domain a ring. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">get_ring</span><span class="p">())</span>
-</div>
-<div class="viewcode-block" id="DMP.to_field"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.to_field">[docs]</a>    <span class="k">def</span> <span class="nf">to_field</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Make the ground domain a field. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">get_field</span><span class="p">())</span>
-</div>
-<div class="viewcode-block" id="DMP.to_exact"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.to_exact">[docs]</a>    <span class="k">def</span> <span class="nf">to_exact</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Make the ground domain exact. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">get_exact</span><span class="p">())</span>
-</div>
-<div class="viewcode-block" id="DMP.convert"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.convert">[docs]</a>    <span class="k">def</span> <span class="nf">convert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">dom</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Convert the ground domain of ``f``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span> <span class="o">==</span> <span class="n">dom</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">DMP</span><span class="p">(</span><span class="n">dmp_convert</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span> <span class="n">dom</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.slice"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.slice">[docs]</a>    <span class="k">def</span> <span class="nf">slice</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">j</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Take a continuous subsequence of terms of ``f``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_slice_in</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.coeffs"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.coeffs">[docs]</a>    <span class="k">def</span> <span class="nf">coeffs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns all non-zero coefficients from ``f`` in lex order. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="n">c</span> <span class="k">for</span> <span class="n">_</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">dmp_list_terms</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">)</span> <span class="p">]</span>
-</div>
-<div class="viewcode-block" id="DMP.monoms"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.monoms">[docs]</a>    <span class="k">def</span> <span class="nf">monoms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns all non-zero monomials from ``f`` in lex order. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="n">m</span> <span class="k">for</span> <span class="n">m</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">dmp_list_terms</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">)</span> <span class="p">]</span>
-</div>
-<div class="viewcode-block" id="DMP.terms"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.terms">[docs]</a>    <span class="k">def</span> <span class="nf">terms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns all non-zero terms from ``f`` in lex order. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dmp_list_terms</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.all_coeffs"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.all_coeffs">[docs]</a>    <span class="k">def</span> <span class="nf">all_coeffs</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns all coefficients from ``f``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">zero</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span> <span class="n">c</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span> <span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span><span class="s">&#39;multivariate polynomials not supported&#39;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.all_monoms"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.all_monoms">[docs]</a>    <span class="k">def</span> <span class="nf">all_monoms</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns all monomials from ``f``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,)]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">i</span><span class="p">,)</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">)</span> <span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span><span class="s">&#39;multivariate polynomials not supported&#39;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.all_terms"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.all_terms">[docs]</a>    <span class="k">def</span> <span class="nf">all_terms</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns all terms from a ``f``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">dup_degree</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[((</span><span class="mi">0</span><span class="p">,),</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">zero</span><span class="p">)]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span> <span class="p">((</span><span class="n">n</span> <span class="o">-</span> <span class="n">i</span><span class="p">,),</span> <span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">)</span> <span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span><span class="s">&#39;multivariate polynomials not supported&#39;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.lift"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.lift">[docs]</a>    <span class="k">def</span> <span class="nf">lift</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Convert algebraic coefficients to rationals. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_lift</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">),</span> <span class="n">dom</span><span class="o">=</span><span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.deflate"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.deflate">[docs]</a>    <span class="k">def</span> <span class="nf">deflate</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Reduce degree of `f` by mapping `x_i^m` to `y_i`. &quot;&quot;&quot;</span>
-        <span class="n">J</span><span class="p">,</span> <span class="n">F</span> <span class="o">=</span> <span class="n">dmp_deflate</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">J</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">F</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.inject"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.inject">[docs]</a>    <span class="k">def</span> <span class="nf">inject</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">front</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Inject ground domain generators into ``f``. &quot;&quot;&quot;</span>
-        <span class="n">F</span><span class="p">,</span> <span class="n">lev</span> <span class="o">=</span> <span class="n">dmp_inject</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">front</span><span class="o">=</span><span class="n">front</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.eject"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.eject">[docs]</a>    <span class="k">def</span> <span class="nf">eject</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">front</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Eject selected generators into the ground domain. &quot;&quot;&quot;</span>
-        <span class="n">F</span> <span class="o">=</span> <span class="n">dmp_eject</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">front</span><span class="o">=</span><span class="n">front</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">dom</span><span class="o">.</span><span class="n">gens</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.exclude"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.exclude">[docs]</a>    <span class="k">def</span> <span class="nf">exclude</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">        Remove useless generators from ``f``.</span>
-
-<span class="sd">        Returns the removed generators and the new excluded ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.polys.polyclasses import DMP</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-
-<span class="sd">        &gt;&gt;&gt; DMP([[[ZZ(1)]], [[ZZ(1)], [ZZ(2)]]], ZZ).exclude()</span>
-<span class="sd">        ([2], DMP([[1], [1, 2]], ZZ, None))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">J</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">u</span> <span class="o">=</span> <span class="n">dmp_exclude</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">J</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.permute"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.permute">[docs]</a>    <span class="k">def</span> <span class="nf">permute</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">P</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">        Returns a polynomial in `K[x_{P(1)}, ..., x_{P(n)}]`.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.polys.polyclasses import DMP</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.polys.domains import ZZ</span>
-
-<span class="sd">        &gt;&gt;&gt; DMP([[[ZZ(2)], [ZZ(1), ZZ(0)]], [[]]], ZZ).permute([1, 0, 2])</span>
-<span class="sd">        DMP([[[2], []], [[1, 0], []]], ZZ, None)</span>
-
-<span class="sd">        &gt;&gt;&gt; DMP([[[ZZ(2)], [ZZ(1), ZZ(0)]], [[]]], ZZ).permute([1, 2, 0])</span>
-<span class="sd">        DMP([[[1], []], [[2, 0], []]], ZZ, None)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_permute</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.terms_gcd"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.terms_gcd">[docs]</a>    <span class="k">def</span> <span class="nf">terms_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Remove GCD of terms from the polynomial ``f``. &quot;&quot;&quot;</span>
-        <span class="n">J</span><span class="p">,</span> <span class="n">F</span> <span class="o">=</span> <span class="n">dmp_terms_gcd</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">J</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">F</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.add_ground"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.add_ground">[docs]</a>    <span class="k">def</span> <span class="nf">add_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Add an element of the ground domain to ``f``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_add_ground</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">c</span><span class="p">),</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.sub_ground"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.sub_ground">[docs]</a>    <span class="k">def</span> <span class="nf">sub_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Subtract an element of the ground domain from ``f``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_sub_ground</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">c</span><span class="p">),</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.mul_ground"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.mul_ground">[docs]</a>    <span class="k">def</span> <span class="nf">mul_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Multiply ``f`` by a an element of the ground domain. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_mul_ground</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">c</span><span class="p">),</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.quo_ground"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.quo_ground">[docs]</a>    <span class="k">def</span> <span class="nf">quo_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Quotient of ``f`` by a an element of the ground domain. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_quo_ground</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">c</span><span class="p">),</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.exquo_ground"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.exquo_ground">[docs]</a>    <span class="k">def</span> <span class="nf">exquo_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Exact quotient of ``f`` by a an element of the ground domain. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_exquo_ground</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">c</span><span class="p">),</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.abs"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.abs">[docs]</a>    <span class="k">def</span> <span class="nf">abs</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Make all coefficients in ``f`` positive. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_abs</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.neg"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.neg">[docs]</a>    <span class="k">def</span> <span class="nf">neg</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Negate all cefficients in ``f``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_neg</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.add"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.add">[docs]</a>    <span class="k">def</span> <span class="nf">add</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Add two multivariate polynomials ``f`` and ``g``. &quot;&quot;&quot;</span>
-        <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">dmp_add</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.sub"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.sub">[docs]</a>    <span class="k">def</span> <span class="nf">sub</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Subtract two multivariate polynomials ``f`` and ``g``. &quot;&quot;&quot;</span>
-        <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">dmp_sub</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.mul"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.mul">[docs]</a>    <span class="k">def</span> <span class="nf">mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Multiply two multivariate polynomials ``f`` and ``g``. &quot;&quot;&quot;</span>
-        <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">dmp_mul</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.sqr"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.sqr">[docs]</a>    <span class="k">def</span> <span class="nf">sqr</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Square a multivariate polynomial ``f``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_sqr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.pow"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.pow">[docs]</a>    <span class="k">def</span> <span class="nf">pow</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Raise ``f`` to a non-negative power ``n``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_pow</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;``int`` expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">type</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.pdiv"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.pdiv">[docs]</a>    <span class="k">def</span> <span class="nf">pdiv</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Polynomial pseudo-division of ``f`` and ``g``. &quot;&quot;&quot;</span>
-        <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_pdiv</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">q</span><span class="p">),</span> <span class="n">per</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.prem"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.prem">[docs]</a>    <span class="k">def</span> <span class="nf">prem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Polynomial pseudo-remainder of ``f`` and ``g``. &quot;&quot;&quot;</span>
-        <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">dmp_prem</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.pquo"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.pquo">[docs]</a>    <span class="k">def</span> <span class="nf">pquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Polynomial pseudo-quotient of ``f`` and ``g``. &quot;&quot;&quot;</span>
-        <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">dmp_pquo</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.pexquo"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.pexquo">[docs]</a>    <span class="k">def</span> <span class="nf">pexquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Polynomial exact pseudo-quotient of ``f`` and ``g``. &quot;&quot;&quot;</span>
-        <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">dmp_pexquo</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.div"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.div">[docs]</a>    <span class="k">def</span> <span class="nf">div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Polynomial division with remainder of ``f`` and ``g``. &quot;&quot;&quot;</span>
-        <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_div</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">q</span><span class="p">),</span> <span class="n">per</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.rem"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.rem">[docs]</a>    <span class="k">def</span> <span class="nf">rem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes polynomial remainder of ``f`` and ``g``. &quot;&quot;&quot;</span>
-        <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">dmp_rem</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.quo"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.quo">[docs]</a>    <span class="k">def</span> <span class="nf">quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes polynomial quotient of ``f`` and ``g``. &quot;&quot;&quot;</span>
-        <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">dmp_quo</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.exquo"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.exquo">[docs]</a>    <span class="k">def</span> <span class="nf">exquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes polynomial exact quotient of ``f`` and ``g``. &quot;&quot;&quot;</span>
-        <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="n">per</span><span class="p">(</span><span class="n">dmp_exquo</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span> <span class="ow">and</span> <span class="n">res</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="n">ExactQuotientFailed</span>
-            <span class="k">raise</span> <span class="n">ExactQuotientFailed</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">res</span>
-</div>
-<div class="viewcode-block" id="DMP.degree"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.degree">[docs]</a>    <span class="k">def</span> <span class="nf">degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">j</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the leading degree of ``f`` in ``x_j``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">dmp_degree_in</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;``int`` expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">type</span><span class="p">(</span><span class="n">j</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.degree_list"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.degree_list">[docs]</a>    <span class="k">def</span> <span class="nf">degree_list</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a list of degrees of ``f``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dmp_degree_list</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.total_degree"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.total_degree">[docs]</a>    <span class="k">def</span> <span class="nf">total_degree</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the total degree of ``f``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">max</span><span class="p">([</span><span class="nb">sum</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">monoms</span><span class="p">()])</span>
-</div>
-<div class="viewcode-block" id="DMP.homogeneous_order"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.homogeneous_order">[docs]</a>    <span class="k">def</span> <span class="nf">homogeneous_order</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the homogeneous order of ``f``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-
-        <span class="n">monoms</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">monoms</span><span class="p">()</span>
-        <span class="n">tdeg</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">monoms</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">for</span> <span class="n">monom</span> <span class="ow">in</span> <span class="n">monoms</span><span class="p">:</span>
-            <span class="n">_tdeg</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">monom</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">_tdeg</span> <span class="o">!=</span> <span class="n">tdeg</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="k">return</span> <span class="n">tdeg</span>
-</div>
-<div class="viewcode-block" id="DMP.LC"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.LC">[docs]</a>    <span class="k">def</span> <span class="nf">LC</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the leading coefficient of ``f``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.TC"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.TC">[docs]</a>    <span class="k">def</span> <span class="nf">TC</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the trailing coefficient of ``f``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dmp_ground_TC</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.nth"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.nth">[docs]</a>    <span class="k">def</span> <span class="nf">nth</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">N</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the ``n``-th coefficient of ``f``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="nb">int</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">N</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">dmp_ground_nth</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;a sequence of integers expected&quot;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.max_norm"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.max_norm">[docs]</a>    <span class="k">def</span> <span class="nf">max_norm</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns maximum norm of ``f``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dmp_max_norm</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.l1_norm"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.l1_norm">[docs]</a>    <span class="k">def</span> <span class="nf">l1_norm</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns l1 norm of ``f``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dmp_l1_norm</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.clear_denoms"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.clear_denoms">[docs]</a>    <span class="k">def</span> <span class="nf">clear_denoms</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Clear denominators, but keep the ground domain. &quot;&quot;&quot;</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">F</span> <span class="o">=</span> <span class="n">dmp_clear_denoms</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">F</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.integrate"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.integrate">[docs]</a>    <span class="k">def</span> <span class="nf">integrate</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">j</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes the ``m``-th order indefinite integral of ``f`` in ``x_j``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;``int`` expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">type</span><span class="p">(</span><span class="n">m</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;``int`` expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">type</span><span class="p">(</span><span class="n">j</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_integrate_in</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.diff"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.diff">[docs]</a>    <span class="k">def</span> <span class="nf">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">j</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes the ``m``-th order derivative of ``f`` in ``x_j``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;``int`` expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">type</span><span class="p">(</span><span class="n">m</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;``int`` expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">type</span><span class="p">(</span><span class="n">j</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_diff_in</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.eval"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.eval">[docs]</a>    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">j</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Evaluates ``f`` at the given point ``a`` in ``x_j``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;``int`` expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">type</span><span class="p">(</span><span class="n">j</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_eval_in</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span>
-            <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">j</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">),</span> <span class="n">kill</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.half_gcdex"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.half_gcdex">[docs]</a>    <span class="k">def</span> <span class="nf">half_gcdex</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Half extended Euclidean algorithm, if univariate. &quot;&quot;&quot;</span>
-        <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">lev</span><span class="p">:</span>
-            <span class="n">s</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">dup_half_gcdex</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="n">per</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;univariate polynomial expected&#39;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.gcdex"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.gcdex">[docs]</a>    <span class="k">def</span> <span class="nf">gcdex</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Extended Euclidean algorithm, if univariate. &quot;&quot;&quot;</span>
-        <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">lev</span><span class="p">:</span>
-            <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">dup_gcdex</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="n">per</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="n">per</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;univariate polynomial expected&#39;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.invert"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.invert">[docs]</a>    <span class="k">def</span> <span class="nf">invert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Invert ``f`` modulo ``g``, if possible. &quot;&quot;&quot;</span>
-        <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">lev</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">dup_invert</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">dom</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;univariate polynomial expected&#39;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.revert"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.revert">[docs]</a>    <span class="k">def</span> <span class="nf">revert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute ``f**(-1)`` mod ``x**n``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dup_revert</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;univariate polynomial expected&#39;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.subresultants"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.subresultants">[docs]</a>    <span class="k">def</span> <span class="nf">subresultants</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes subresultant PRS sequence of ``f`` and ``g``. &quot;&quot;&quot;</span>
-        <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="n">R</span> <span class="o">=</span> <span class="n">dmp_subresultants</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-        <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">per</span><span class="p">,</span> <span class="n">R</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.resultant"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.resultant">[docs]</a>    <span class="k">def</span> <span class="nf">resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">includePRS</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes resultant of ``f`` and ``g`` via PRS. &quot;&quot;&quot;</span>
-        <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">includePRS</span><span class="p">:</span>
-            <span class="n">res</span><span class="p">,</span> <span class="n">R</span> <span class="o">=</span> <span class="n">dmp_resultant</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">includePRS</span><span class="o">=</span><span class="n">includePRS</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">res</span><span class="p">,</span> <span class="n">kill</span><span class="o">=</span><span class="bp">True</span><span class="p">),</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">per</span><span class="p">,</span> <span class="n">R</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">dmp_resultant</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span> <span class="n">kill</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.discriminant"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.discriminant">[docs]</a>    <span class="k">def</span> <span class="nf">discriminant</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes discriminant of ``f``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_discriminant</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">),</span> <span class="n">kill</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.cofactors"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.cofactors">[docs]</a>    <span class="k">def</span> <span class="nf">cofactors</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns GCD of ``f`` and ``g`` and their cofactors. &quot;&quot;&quot;</span>
-        <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="n">h</span><span class="p">,</span> <span class="n">cff</span><span class="p">,</span> <span class="n">cfg</span> <span class="o">=</span> <span class="n">dmp_inner_gcd</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">h</span><span class="p">),</span> <span class="n">per</span><span class="p">(</span><span class="n">cff</span><span class="p">),</span> <span class="n">per</span><span class="p">(</span><span class="n">cfg</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.gcd"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.gcd">[docs]</a>    <span class="k">def</span> <span class="nf">gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns polynomial GCD of ``f`` and ``g``. &quot;&quot;&quot;</span>
-        <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">dmp_gcd</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.lcm"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.lcm">[docs]</a>    <span class="k">def</span> <span class="nf">lcm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns polynomial LCM of ``f`` and ``g``. &quot;&quot;&quot;</span>
-        <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">dmp_lcm</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.cancel"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.cancel">[docs]</a>    <span class="k">def</span> <span class="nf">cancel</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">include</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Cancel common factors in a rational function ``f/g``. &quot;&quot;&quot;</span>
-        <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">include</span><span class="p">:</span>
-            <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">dmp_cancel</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">include</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">cF</span><span class="p">,</span> <span class="n">cG</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">dmp_cancel</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">include</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-        <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">per</span><span class="p">(</span><span class="n">F</span><span class="p">),</span> <span class="n">per</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">include</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cF</span><span class="p">,</span> <span class="n">cG</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span>
-</div>
-<div class="viewcode-block" id="DMP.trunc"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.trunc">[docs]</a>    <span class="k">def</span> <span class="nf">trunc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Reduce ``f`` modulo a constant ``p``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_ground_trunc</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">p</span><span class="p">),</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.monic"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.monic">[docs]</a>    <span class="k">def</span> <span class="nf">monic</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Divides all coefficients by ``LC(f)``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_ground_monic</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.content"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.content">[docs]</a>    <span class="k">def</span> <span class="nf">content</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns GCD of polynomial coefficients. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dmp_ground_content</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.primitive"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.primitive">[docs]</a>    <span class="k">def</span> <span class="nf">primitive</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns content and a primitive form of ``f``. &quot;&quot;&quot;</span>
-        <span class="n">cont</span><span class="p">,</span> <span class="n">F</span> <span class="o">=</span> <span class="n">dmp_ground_primitive</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">cont</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">F</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.compose"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.compose">[docs]</a>    <span class="k">def</span> <span class="nf">compose</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes functional composition of ``f`` and ``g``. &quot;&quot;&quot;</span>
-        <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">dmp_compose</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.decompose"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.decompose">[docs]</a>    <span class="k">def</span> <span class="nf">decompose</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes functional decomposition of ``f``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">,</span> <span class="n">dup_decompose</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;univariate polynomial expected&#39;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.shift"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.shift">[docs]</a>    <span class="k">def</span> <span class="nf">shift</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Efficiently compute Taylor shift ``f(x + a)``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dup_shift</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;univariate polynomial expected&#39;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.sturm"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.sturm">[docs]</a>    <span class="k">def</span> <span class="nf">sturm</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes the Sturm sequence of ``f``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">,</span> <span class="n">dup_sturm</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;univariate polynomial expected&#39;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.gff_list"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.gff_list">[docs]</a>    <span class="k">def</span> <span class="nf">gff_list</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes greatest factorial factorization of ``f``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">dup_gff_list</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span> <span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;univariate polynomial expected&#39;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.sqf_norm"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.sqf_norm">[docs]</a>    <span class="k">def</span> <span class="nf">sqf_norm</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes square-free norm of ``f``. &quot;&quot;&quot;</span>
-        <span class="n">s</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dmp_sqf_norm</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">s</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">dom</span><span class="o">=</span><span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.sqf_part"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.sqf_part">[docs]</a>    <span class="k">def</span> <span class="nf">sqf_part</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes square-free part of ``f``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_sqf_part</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMP.sqf_list"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.sqf_list">[docs]</a>    <span class="k">def</span> <span class="nf">sqf_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="nb">all</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a list of square-free factors of ``f``. &quot;&quot;&quot;</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">dmp_sqf_list</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="nb">all</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">coeff</span><span class="p">,</span> <span class="p">[</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">factors</span> <span class="p">]</span>
-</div>
-<div class="viewcode-block" id="DMP.sqf_list_include"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.sqf_list_include">[docs]</a>    <span class="k">def</span> <span class="nf">sqf_list_include</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="nb">all</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a list of square-free factors of ``f``. &quot;&quot;&quot;</span>
-        <span class="n">factors</span> <span class="o">=</span> <span class="n">dmp_sqf_list_include</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="nb">all</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">factors</span> <span class="p">]</span>
-</div>
-<div class="viewcode-block" id="DMP.factor_list"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.factor_list">[docs]</a>    <span class="k">def</span> <span class="nf">factor_list</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a list of irreducible factors of ``f``. &quot;&quot;&quot;</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">dmp_factor_list</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">coeff</span><span class="p">,</span> <span class="p">[</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">factors</span> <span class="p">]</span>
-</div>
-<div class="viewcode-block" id="DMP.factor_list_include"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.factor_list_include">[docs]</a>    <span class="k">def</span> <span class="nf">factor_list_include</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a list of irreducible factors of ``f``. &quot;&quot;&quot;</span>
-        <span class="n">factors</span> <span class="o">=</span> <span class="n">dmp_factor_list_include</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">factors</span> <span class="p">]</span>
-</div>
-<div class="viewcode-block" id="DMP.intervals"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.intervals">[docs]</a>    <span class="k">def</span> <span class="nf">intervals</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="nb">all</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">inf</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">sup</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">fast</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">sqf</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute isolating intervals for roots of ``f``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">sqf</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">dup_isolate_real_roots</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="n">eps</span><span class="p">,</span> <span class="n">inf</span><span class="o">=</span><span class="n">inf</span><span class="p">,</span> <span class="n">sup</span><span class="o">=</span><span class="n">sup</span><span class="p">,</span> <span class="n">fast</span><span class="o">=</span><span class="n">fast</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">dup_isolate_real_roots_sqf</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="n">eps</span><span class="p">,</span> <span class="n">inf</span><span class="o">=</span><span class="n">inf</span><span class="p">,</span> <span class="n">sup</span><span class="o">=</span><span class="n">sup</span><span class="p">,</span> <span class="n">fast</span><span class="o">=</span><span class="n">fast</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">sqf</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">dup_isolate_all_roots</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="n">eps</span><span class="p">,</span> <span class="n">inf</span><span class="o">=</span><span class="n">inf</span><span class="p">,</span> <span class="n">sup</span><span class="o">=</span><span class="n">sup</span><span class="p">,</span> <span class="n">fast</span><span class="o">=</span><span class="n">fast</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">dup_isolate_all_roots_sqf</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="n">eps</span><span class="p">,</span> <span class="n">inf</span><span class="o">=</span><span class="n">inf</span><span class="p">,</span> <span class="n">sup</span><span class="o">=</span><span class="n">sup</span><span class="p">,</span> <span class="n">fast</span><span class="o">=</span><span class="n">fast</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span>
-                <span class="s">&quot;can&#39;t isolate roots of a multivariate polynomial&quot;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.refine_root"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.refine_root">[docs]</a>    <span class="k">def</span> <span class="nf">refine_root</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">steps</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">fast</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Refine an isolating interval to the given precision.</span>
-
-<span class="sd">        ``eps`` should be a rational number.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">dup_refine_real_root</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="n">eps</span><span class="p">,</span> <span class="n">steps</span><span class="o">=</span><span class="n">steps</span><span class="p">,</span> <span class="n">fast</span><span class="o">=</span><span class="n">fast</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span>
-                <span class="s">&quot;can&#39;t refine a root of a multivariate polynomial&quot;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.count_real_roots"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.count_real_roots">[docs]</a>    <span class="k">def</span> <span class="nf">count_real_roots</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">inf</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">sup</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the number of real roots of ``f`` in ``[inf, sup]``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dup_count_real_roots</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">inf</span><span class="o">=</span><span class="n">inf</span><span class="p">,</span> <span class="n">sup</span><span class="o">=</span><span class="n">sup</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMP.count_complex_roots"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.count_complex_roots">[docs]</a>    <span class="k">def</span> <span class="nf">count_complex_roots</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">inf</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">sup</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the number of complex roots of ``f`` in ``[inf, sup]``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dup_count_complex_roots</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">inf</span><span class="o">=</span><span class="n">inf</span><span class="p">,</span> <span class="n">sup</span><span class="o">=</span><span class="n">sup</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="DMP.is_zero"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_zero">[docs]</a>    <span class="k">def</span> <span class="nf">is_zero</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns ``True`` if ``f`` is a zero polynomial. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="DMP.is_one"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_one">[docs]</a>    <span class="k">def</span> <span class="nf">is_one</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns ``True`` if ``f`` is a unit polynomial. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dmp_one_p</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="DMP.is_ground"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_ground">[docs]</a>    <span class="k">def</span> <span class="nf">is_ground</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns ``True`` if ``f`` is an element of the ground domain. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dmp_ground_p</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="DMP.is_sqf"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_sqf">[docs]</a>    <span class="k">def</span> <span class="nf">is_sqf</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns ``True`` if ``f`` is a square-free polynomial. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dmp_sqf_p</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="DMP.is_monic"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_monic">[docs]</a>    <span class="k">def</span> <span class="nf">is_monic</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns ``True`` if the leading coefficient of ``f`` is one. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="DMP.is_primitive"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_primitive">[docs]</a>    <span class="k">def</span> <span class="nf">is_primitive</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns ``True`` if the GCD of the coefficients of ``f`` is one. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">dmp_ground_content</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="DMP.is_linear"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_linear">[docs]</a>    <span class="k">def</span> <span class="nf">is_linear</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns ``True`` if ``f`` is linear in all its variables. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="nb">sum</span><span class="p">(</span><span class="n">monom</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">monom</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">dmp_to_dict</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span><span class="o">.</span><span class="n">keys</span><span class="p">()))</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="DMP.is_quadratic"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_quadratic">[docs]</a>    <span class="k">def</span> <span class="nf">is_quadratic</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns ``True`` if ``f`` is quadratic in all its variables. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="nb">sum</span><span class="p">(</span><span class="n">monom</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">2</span> <span class="k">for</span> <span class="n">monom</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">dmp_to_dict</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span><span class="o">.</span><span class="n">keys</span><span class="p">()))</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="DMP.is_monomial"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_monomial">[docs]</a>    <span class="k">def</span> <span class="nf">is_monomial</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns ``True`` if ``f`` is zero or has only one term. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">to_dict</span><span class="p">())</span> <span class="o">&lt;=</span> <span class="mi">1</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="DMP.is_homogeneous"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_homogeneous">[docs]</a>    <span class="k">def</span> <span class="nf">is_homogeneous</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns ``True`` if ``f`` is a homogeneous polynomial. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">homogeneous_order</span><span class="p">()</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="DMP.is_irreducible"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_irreducible">[docs]</a>    <span class="k">def</span> <span class="nf">is_irreducible</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns ``True`` if ``f`` has no factors over its domain. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dmp_irreducible_p</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="DMP.is_cyclotomic"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_cyclotomic">[docs]</a>    <span class="k">def</span> <span class="nf">is_cyclotomic</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns ``True`` if ``f`` is a cyclotomic polynomial. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">dup_cyclotomic_p</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-</div>
-    <span class="k">def</span> <span class="nf">__abs__</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">abs</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">neg</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">DMP</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_ground</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">))</span>
-            <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">NotImplemented</span>
-            <span class="k">except</span> <span class="p">(</span><span class="n">CoercionFailed</span><span class="p">,</span> <span class="ne">NotImplementedError</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">try</span><span class="p">:</span>
-                        <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-                    <span class="k">except</span> <span class="p">(</span><span class="n">CoercionFailed</span><span class="p">,</span> <span class="ne">NotImplementedError</span><span class="p">):</span>
-                        <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__radd__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">__add__</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">DMP</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_ground</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">))</span>
-            <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">NotImplemented</span>
-            <span class="k">except</span> <span class="p">(</span><span class="n">CoercionFailed</span><span class="p">,</span> <span class="ne">NotImplementedError</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">try</span><span class="p">:</span>
-                        <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-                    <span class="k">except</span> <span class="p">(</span><span class="n">CoercionFailed</span><span class="p">,</span> <span class="ne">NotImplementedError</span><span class="p">):</span>
-                        <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">sub</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__rsub__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">__add__</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">DMP</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">mul_ground</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">NotImplemented</span>
-            <span class="k">except</span> <span class="p">(</span><span class="n">CoercionFailed</span><span class="p">,</span> <span class="ne">NotImplementedError</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">try</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">g</span><span class="p">))</span>
-                    <span class="k">except</span> <span class="p">(</span><span class="n">CoercionFailed</span><span class="p">,</span> <span class="ne">NotImplementedError</span><span class="p">):</span>
-                        <span class="k">pass</span>
-                <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">DMP</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">exquo</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">mul_ground</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">NotImplemented</span>
-            <span class="k">except</span> <span class="p">(</span><span class="n">CoercionFailed</span><span class="p">,</span> <span class="ne">NotImplementedError</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">try</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">exquo</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">g</span><span class="p">))</span>
-                    <span class="k">except</span> <span class="p">(</span><span class="n">CoercionFailed</span><span class="p">,</span> <span class="ne">NotImplementedError</span><span class="p">):</span>
-                        <span class="k">pass</span>
-                <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="nf">__rdiv__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">DMP</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">g</span><span class="o">.</span><span class="n">exquo</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">g</span><span class="p">)</span><span class="o">.</span><span class="n">exquo</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-            <span class="k">except</span> <span class="p">(</span><span class="n">CoercionFailed</span><span class="p">,</span> <span class="ne">NotImplementedError</span><span class="p">):</span>
-                <span class="k">pass</span>
-        <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="n">__truediv__</span> <span class="o">=</span> <span class="n">__div__</span>
-    <span class="n">__rtruediv__</span> <span class="o">=</span> <span class="n">__rdiv__</span>
-
-    <span class="k">def</span> <span class="nf">__rmul__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__pow__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__divmod__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">div</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__mod__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rem</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__floordiv__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">DMP</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">quo_ground</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span> <span class="o">==</span> <span class="n">g</span><span class="o">.</span><span class="n">lev</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">F</span> <span class="o">==</span> <span class="n">G</span>
-        <span class="k">except</span> <span class="n">UnificationFailed</span><span class="p">:</span>
-            <span class="k">pass</span>
-
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">eq</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">strict</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">_strict_eq</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">ne</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">eq</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="n">strict</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_strict_eq</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="p">)</span> <span class="ow">and</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span> <span class="o">==</span> <span class="n">g</span><span class="o">.</span><span class="n">lev</span> \
-            <span class="ow">and</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span> <span class="o">==</span> <span class="n">g</span><span class="o">.</span><span class="n">dom</span> \
-            <span class="ow">and</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span> <span class="o">==</span> <span class="n">g</span><span class="o">.</span><span class="n">rep</span>
-
-    <span class="k">def</span> <span class="nf">__lt__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">__lt__</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__le__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">__le__</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__gt__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">__gt__</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__ge__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">__ge__</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__bool__</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">init_normal_DMF</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">den</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">DMF</span><span class="p">(</span><span class="n">dmp_normal</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span>
-               <span class="n">dmp_normal</span><span class="p">(</span><span class="n">den</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="DMF"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMF">[docs]</a><span class="k">class</span> <span class="nc">DMF</span><span class="p">(</span><span class="n">PicklableWithSlots</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Dense Multivariate Fractions over `K`. &quot;&quot;&quot;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;num&#39;</span><span class="p">,</span> <span class="s">&#39;den&#39;</span><span class="p">,</span> <span class="s">&#39;lev&#39;</span><span class="p">,</span> <span class="s">&#39;dom&#39;</span><span class="p">,</span> <span class="s">&#39;ring&#39;</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">ring</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">num</span><span class="p">,</span> <span class="n">den</span><span class="p">,</span> <span class="n">lev</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_parse</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="p">)</span>
-        <span class="n">num</span><span class="p">,</span> <span class="n">den</span> <span class="o">=</span> <span class="n">dmp_cancel</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">den</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">num</span> <span class="o">=</span> <span class="n">num</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">den</span> <span class="o">=</span> <span class="n">den</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">lev</span> <span class="o">=</span> <span class="n">lev</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">dom</span> <span class="o">=</span> <span class="n">dom</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ring</span> <span class="o">=</span> <span class="n">ring</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">new</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">ring</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">num</span><span class="p">,</span> <span class="n">den</span><span class="p">,</span> <span class="n">lev</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_parse</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="p">)</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="nb">object</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-
-        <span class="n">obj</span><span class="o">.</span><span class="n">num</span> <span class="o">=</span> <span class="n">num</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">den</span> <span class="o">=</span> <span class="n">den</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">lev</span> <span class="o">=</span> <span class="n">lev</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">dom</span> <span class="o">=</span> <span class="n">dom</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">ring</span> <span class="o">=</span> <span class="n">ring</span>
-
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_parse</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">rep</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">tuple</span><span class="p">:</span>
-            <span class="n">num</span><span class="p">,</span> <span class="n">den</span> <span class="o">=</span> <span class="n">rep</span>
-
-            <span class="k">if</span> <span class="n">lev</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">num</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">dict</span><span class="p">:</span>
-                    <span class="n">num</span> <span class="o">=</span> <span class="n">dmp_from_dict</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">den</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">dict</span><span class="p">:</span>
-                    <span class="n">den</span> <span class="o">=</span> <span class="n">dmp_from_dict</span><span class="p">(</span><span class="n">den</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">num</span><span class="p">,</span> <span class="n">num_lev</span> <span class="o">=</span> <span class="n">dmp_validate</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
-                <span class="n">den</span><span class="p">,</span> <span class="n">den_lev</span> <span class="o">=</span> <span class="n">dmp_validate</span><span class="p">(</span><span class="n">den</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">num_lev</span> <span class="o">==</span> <span class="n">den_lev</span><span class="p">:</span>
-                    <span class="n">lev</span> <span class="o">=</span> <span class="n">num_lev</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;inconsistent number of levels&#39;</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">den</span><span class="p">,</span> <span class="n">lev</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">ZeroDivisionError</span><span class="p">(</span><span class="s">&#39;fraction denominator&#39;</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">lev</span><span class="p">):</span>
-                <span class="n">den</span> <span class="o">=</span> <span class="n">dmp_one</span><span class="p">(</span><span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">dmp_negative_p</span><span class="p">(</span><span class="n">den</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">):</span>
-                    <span class="n">num</span> <span class="o">=</span> <span class="n">dmp_neg</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-                    <span class="n">den</span> <span class="o">=</span> <span class="n">dmp_neg</span><span class="p">(</span><span class="n">den</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">num</span> <span class="o">=</span> <span class="n">rep</span>
-
-            <span class="k">if</span> <span class="n">lev</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">num</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">dict</span><span class="p">:</span>
-                    <span class="n">num</span> <span class="o">=</span> <span class="n">dmp_from_dict</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="nb">type</span><span class="p">(</span><span class="n">num</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="nb">list</span><span class="p">:</span>
-                    <span class="n">num</span> <span class="o">=</span> <span class="n">dmp_ground</span><span class="p">(</span><span class="n">dom</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">num</span><span class="p">),</span> <span class="n">lev</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">num</span><span class="p">,</span> <span class="n">lev</span> <span class="o">=</span> <span class="n">dmp_validate</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
-
-            <span class="n">den</span> <span class="o">=</span> <span class="n">dmp_one</span><span class="p">(</span><span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">num</span><span class="p">,</span> <span class="n">den</span><span class="p">,</span> <span class="n">lev</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">((</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">), </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">num</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">den</span><span class="p">,</span>
-                                         <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">hash</span><span class="p">((</span><span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="n">dmp_to_tuple</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">num</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">),</span>
-            <span class="n">dmp_to_tuple</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">den</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">),</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span><span class="p">))</span>
-
-<div class="viewcode-block" id="DMF.poly_unify"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMF.poly_unify">[docs]</a>    <span class="k">def</span> <span class="nf">poly_unify</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Unify a multivariate fraction and a polynomial. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">DMP</span><span class="p">)</span> <span class="ow">or</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span> <span class="o">!=</span> <span class="n">g</span><span class="o">.</span><span class="n">lev</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">UnificationFailed</span><span class="p">(</span><span class="s">&quot;can&#39;t unify </span><span class="si">%s</span><span class="s"> with </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span> <span class="o">==</span> <span class="n">g</span><span class="o">.</span><span class="n">dom</span> <span class="ow">and</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span> <span class="o">==</span> <span class="n">g</span><span class="o">.</span><span class="n">ring</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">,</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">num</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">den</span><span class="p">),</span> <span class="n">g</span><span class="o">.</span><span class="n">rep</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-            <span class="n">ring</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span>
-            <span class="k">if</span> <span class="n">g</span><span class="o">.</span><span class="n">ring</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">ring</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">ring</span> <span class="o">=</span> <span class="n">ring</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">ring</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">ring</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">ring</span>
-
-            <span class="n">F</span> <span class="o">=</span> <span class="p">(</span><span class="n">dmp_convert</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">num</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span>
-                 <span class="n">dmp_convert</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">den</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">dom</span><span class="p">))</span>
-
-            <span class="n">G</span> <span class="o">=</span> <span class="n">dmp_convert</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">g</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-
-            <span class="k">def</span> <span class="nf">per</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">den</span><span class="p">,</span> <span class="n">cancel</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">kill</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">lev</span><span class="o">=</span><span class="n">lev</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">kill</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">lev</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">num</span><span class="o">/</span><span class="n">den</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">lev</span> <span class="o">=</span> <span class="n">lev</span> <span class="o">-</span> <span class="mi">1</span>
-
-                <span class="k">if</span> <span class="n">cancel</span><span class="p">:</span>
-                    <span class="n">num</span><span class="p">,</span> <span class="n">den</span> <span class="o">=</span> <span class="n">dmp_cancel</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">den</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-
-                <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">new</span><span class="p">((</span><span class="n">num</span><span class="p">,</span> <span class="n">den</span><span class="p">),</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">ring</span><span class="o">=</span><span class="n">ring</span><span class="p">)</span>
-
-            <span class="k">return</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span>
-</div>
-<div class="viewcode-block" id="DMF.frac_unify"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMF.frac_unify">[docs]</a>    <span class="k">def</span> <span class="nf">frac_unify</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Unify representations of two multivariate fractions. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">DMF</span><span class="p">)</span> <span class="ow">or</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span> <span class="o">!=</span> <span class="n">g</span><span class="o">.</span><span class="n">lev</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">UnificationFailed</span><span class="p">(</span><span class="s">&quot;can&#39;t unify </span><span class="si">%s</span><span class="s"> with </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span> <span class="o">==</span> <span class="n">g</span><span class="o">.</span><span class="n">dom</span> <span class="ow">and</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span> <span class="o">==</span> <span class="n">g</span><span class="o">.</span><span class="n">ring</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">,</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">num</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">den</span><span class="p">),</span>
-                                         <span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">num</span><span class="p">,</span> <span class="n">g</span><span class="o">.</span><span class="n">den</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-            <span class="n">ring</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span>
-            <span class="k">if</span> <span class="n">g</span><span class="o">.</span><span class="n">ring</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">ring</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">ring</span> <span class="o">=</span> <span class="n">ring</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">ring</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">ring</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">ring</span>
-
-            <span class="n">F</span> <span class="o">=</span> <span class="p">(</span><span class="n">dmp_convert</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">num</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span>
-                 <span class="n">dmp_convert</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">den</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">dom</span><span class="p">))</span>
-
-            <span class="n">G</span> <span class="o">=</span> <span class="p">(</span><span class="n">dmp_convert</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">num</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">g</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span>
-                 <span class="n">dmp_convert</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">den</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">g</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">dom</span><span class="p">))</span>
-
-            <span class="k">def</span> <span class="nf">per</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">den</span><span class="p">,</span> <span class="n">cancel</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">kill</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">lev</span><span class="o">=</span><span class="n">lev</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">kill</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">lev</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">num</span><span class="o">/</span><span class="n">den</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">lev</span> <span class="o">=</span> <span class="n">lev</span> <span class="o">-</span> <span class="mi">1</span>
-
-                <span class="k">if</span> <span class="n">cancel</span><span class="p">:</span>
-                    <span class="n">num</span><span class="p">,</span> <span class="n">den</span> <span class="o">=</span> <span class="n">dmp_cancel</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">den</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-
-                <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">new</span><span class="p">((</span><span class="n">num</span><span class="p">,</span> <span class="n">den</span><span class="p">),</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">ring</span><span class="o">=</span><span class="n">ring</span><span class="p">)</span>
-
-            <span class="k">return</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span>
-</div>
-<div class="viewcode-block" id="DMF.per"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMF.per">[docs]</a>    <span class="k">def</span> <span class="nf">per</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">num</span><span class="p">,</span> <span class="n">den</span><span class="p">,</span> <span class="n">cancel</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">kill</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">ring</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Create a DMF out of the given representation. &quot;&quot;&quot;</span>
-        <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span>
-
-        <span class="k">if</span> <span class="n">kill</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">lev</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">num</span><span class="o">/</span><span class="n">den</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">lev</span> <span class="o">-=</span> <span class="mi">1</span>
-
-        <span class="k">if</span> <span class="n">cancel</span><span class="p">:</span>
-            <span class="n">num</span><span class="p">,</span> <span class="n">den</span> <span class="o">=</span> <span class="n">dmp_cancel</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">den</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">ring</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">ring</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">new</span><span class="p">((</span><span class="n">num</span><span class="p">,</span> <span class="n">den</span><span class="p">),</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">ring</span><span class="o">=</span><span class="n">ring</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMF.half_per"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMF.half_per">[docs]</a>    <span class="k">def</span> <span class="nf">half_per</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="n">kill</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Create a DMP out of the given representation. &quot;&quot;&quot;</span>
-        <span class="n">lev</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span>
-
-        <span class="k">if</span> <span class="n">kill</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">lev</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">rep</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">lev</span> <span class="o">-=</span> <span class="mi">1</span>
-
-        <span class="k">return</span> <span class="n">DMP</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">zero</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">ring</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">ring</span><span class="o">=</span><span class="n">ring</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">one</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">ring</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">ring</span><span class="o">=</span><span class="n">ring</span><span class="p">)</span>
-
-<div class="viewcode-block" id="DMF.numer"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMF.numer">[docs]</a>    <span class="k">def</span> <span class="nf">numer</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the numerator of ``f``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">half_per</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">num</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMF.denom"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMF.denom">[docs]</a>    <span class="k">def</span> <span class="nf">denom</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the denominator of ``f``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">half_per</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">den</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMF.cancel"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMF.cancel">[docs]</a>    <span class="k">def</span> <span class="nf">cancel</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Remove common factors from ``f.num`` and ``f.den``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">num</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">den</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMF.neg"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMF.neg">[docs]</a>    <span class="k">def</span> <span class="nf">neg</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Negate all cefficients in ``f``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_neg</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">num</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">),</span> <span class="n">f</span><span class="o">.</span><span class="n">den</span><span class="p">,</span> <span class="n">cancel</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMF.add"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMF.add">[docs]</a>    <span class="k">def</span> <span class="nf">add</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Add two multivariate fractions ``f`` and ``g``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">DMP</span><span class="p">):</span>
-            <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="p">(</span><span class="n">F_num</span><span class="p">,</span> <span class="n">F_den</span><span class="p">),</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">poly_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-            <span class="n">num</span><span class="p">,</span> <span class="n">den</span> <span class="o">=</span> <span class="n">dmp_add_mul</span><span class="p">(</span><span class="n">F_num</span><span class="p">,</span> <span class="n">F_den</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span> <span class="n">F_den</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">frac_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-            <span class="p">(</span><span class="n">F_num</span><span class="p">,</span> <span class="n">F_den</span><span class="p">),</span> <span class="p">(</span><span class="n">G_num</span><span class="p">,</span> <span class="n">G_den</span><span class="p">)</span> <span class="o">=</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span>
-
-            <span class="n">num</span> <span class="o">=</span> <span class="n">dmp_add</span><span class="p">(</span><span class="n">dmp_mul</span><span class="p">(</span><span class="n">F_num</span><span class="p">,</span> <span class="n">G_den</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span>
-                          <span class="n">dmp_mul</span><span class="p">(</span><span class="n">F_den</span><span class="p">,</span> <span class="n">G_num</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-            <span class="n">den</span> <span class="o">=</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">F_den</span><span class="p">,</span> <span class="n">G_den</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">den</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMF.sub"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMF.sub">[docs]</a>    <span class="k">def</span> <span class="nf">sub</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Subtract two multivariate fractions ``f`` and ``g``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">DMP</span><span class="p">):</span>
-            <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="p">(</span><span class="n">F_num</span><span class="p">,</span> <span class="n">F_den</span><span class="p">),</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">poly_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-            <span class="n">num</span><span class="p">,</span> <span class="n">den</span> <span class="o">=</span> <span class="n">dmp_sub_mul</span><span class="p">(</span><span class="n">F_num</span><span class="p">,</span> <span class="n">F_den</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span> <span class="n">F_den</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">frac_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-            <span class="p">(</span><span class="n">F_num</span><span class="p">,</span> <span class="n">F_den</span><span class="p">),</span> <span class="p">(</span><span class="n">G_num</span><span class="p">,</span> <span class="n">G_den</span><span class="p">)</span> <span class="o">=</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span>
-
-            <span class="n">num</span> <span class="o">=</span> <span class="n">dmp_sub</span><span class="p">(</span><span class="n">dmp_mul</span><span class="p">(</span><span class="n">F_num</span><span class="p">,</span> <span class="n">G_den</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span>
-                          <span class="n">dmp_mul</span><span class="p">(</span><span class="n">F_den</span><span class="p">,</span> <span class="n">G_num</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-            <span class="n">den</span> <span class="o">=</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">F_den</span><span class="p">,</span> <span class="n">G_den</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">den</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMF.mul"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMF.mul">[docs]</a>    <span class="k">def</span> <span class="nf">mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Multiply two multivariate fractions ``f`` and ``g``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">DMP</span><span class="p">):</span>
-            <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="p">(</span><span class="n">F_num</span><span class="p">,</span> <span class="n">F_den</span><span class="p">),</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">poly_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-            <span class="n">num</span><span class="p">,</span> <span class="n">den</span> <span class="o">=</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">F_num</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span> <span class="n">F_den</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">frac_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-            <span class="p">(</span><span class="n">F_num</span><span class="p">,</span> <span class="n">F_den</span><span class="p">),</span> <span class="p">(</span><span class="n">G_num</span><span class="p">,</span> <span class="n">G_den</span><span class="p">)</span> <span class="o">=</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span>
-
-            <span class="n">num</span> <span class="o">=</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">F_num</span><span class="p">,</span> <span class="n">G_num</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-            <span class="n">den</span> <span class="o">=</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">F_den</span><span class="p">,</span> <span class="n">G_den</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">den</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="DMF.pow"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMF.pow">[docs]</a>    <span class="k">def</span> <span class="nf">pow</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Raise ``f`` to a non-negative power ``n``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dmp_pow</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">num</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">),</span>
-                         <span class="n">dmp_pow</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">den</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">),</span> <span class="n">cancel</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;``int`` expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">type</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="DMF.quo"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMF.quo">[docs]</a>    <span class="k">def</span> <span class="nf">quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes quotient of fractions ``f`` and ``g``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">DMP</span><span class="p">):</span>
-            <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="p">(</span><span class="n">F_num</span><span class="p">,</span> <span class="n">F_den</span><span class="p">),</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">poly_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-            <span class="n">num</span><span class="p">,</span> <span class="n">den</span> <span class="o">=</span> <span class="n">F_num</span><span class="p">,</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">F_den</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">frac_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-            <span class="p">(</span><span class="n">F_num</span><span class="p">,</span> <span class="n">F_den</span><span class="p">),</span> <span class="p">(</span><span class="n">G_num</span><span class="p">,</span> <span class="n">G_den</span><span class="p">)</span> <span class="o">=</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span>
-
-            <span class="n">num</span> <span class="o">=</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">F_num</span><span class="p">,</span> <span class="n">G_den</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-            <span class="n">den</span> <span class="o">=</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">F_den</span><span class="p">,</span> <span class="n">G_num</span><span class="p">,</span> <span class="n">lev</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-
-        <span class="n">res</span> <span class="o">=</span> <span class="n">per</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">den</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">res</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="n">ExactQuotientFailed</span>
-            <span class="k">raise</span> <span class="n">ExactQuotientFailed</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">res</span>
-</div>
-    <span class="n">exquo</span> <span class="o">=</span> <span class="n">quo</span>
-
-<div class="viewcode-block" id="DMF.invert"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMF.invert">[docs]</a>    <span class="k">def</span> <span class="nf">invert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">check</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Computes inverse of a fraction ``f``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">check</span> <span class="ow">and</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">is_unit</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="n">NotReversible</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span><span class="p">)</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">den</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">num</span><span class="p">,</span> <span class="n">cancel</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">res</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="DMF.is_zero"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMF.is_zero">[docs]</a>    <span class="k">def</span> <span class="nf">is_zero</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns ``True`` if ``f`` is a zero fraction. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">num</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="DMF.is_one"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.DMF.is_one">[docs]</a>    <span class="k">def</span> <span class="nf">is_one</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns ``True`` if ``f`` is a unit fraction. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dmp_one_p</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">num</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span> <span class="ow">and</span> \
-            <span class="n">dmp_one_p</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">den</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">neg</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="p">(</span><span class="n">DMP</span><span class="p">,</span> <span class="n">DMF</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">half_per</span><span class="p">(</span><span class="n">g</span><span class="p">))</span>
-        <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-        <span class="k">except</span> <span class="p">(</span><span class="n">CoercionFailed</span><span class="p">,</span> <span class="ne">NotImplementedError</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">g</span><span class="p">))</span>
-                <span class="k">except</span> <span class="p">(</span><span class="n">CoercionFailed</span><span class="p">,</span> <span class="ne">NotImplementedError</span><span class="p">):</span>
-                    <span class="k">pass</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="nf">__radd__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">__add__</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="p">(</span><span class="n">DMP</span><span class="p">,</span> <span class="n">DMF</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">sub</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">sub</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">half_per</span><span class="p">(</span><span class="n">g</span><span class="p">))</span>
-        <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-        <span class="k">except</span> <span class="p">(</span><span class="n">CoercionFailed</span><span class="p">,</span> <span class="ne">NotImplementedError</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">sub</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">g</span><span class="p">))</span>
-                <span class="k">except</span> <span class="p">(</span><span class="n">CoercionFailed</span><span class="p">,</span> <span class="ne">NotImplementedError</span><span class="p">):</span>
-                    <span class="k">pass</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="nf">__rsub__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">__add__</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="p">(</span><span class="n">DMP</span><span class="p">,</span> <span class="n">DMF</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">half_per</span><span class="p">(</span><span class="n">g</span><span class="p">))</span>
-        <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-        <span class="k">except</span> <span class="p">(</span><span class="n">CoercionFailed</span><span class="p">,</span> <span class="ne">NotImplementedError</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">g</span><span class="p">))</span>
-                <span class="k">except</span> <span class="p">(</span><span class="n">CoercionFailed</span><span class="p">,</span> <span class="ne">NotImplementedError</span><span class="p">):</span>
-                    <span class="k">pass</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="nf">__rmul__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__pow__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="p">(</span><span class="n">DMP</span><span class="p">,</span> <span class="n">DMF</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">half_per</span><span class="p">(</span><span class="n">g</span><span class="p">))</span>
-        <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-        <span class="k">except</span> <span class="p">(</span><span class="n">CoercionFailed</span><span class="p">,</span> <span class="ne">NotImplementedError</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">ring</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">g</span><span class="p">))</span>
-                <span class="k">except</span> <span class="p">(</span><span class="n">CoercionFailed</span><span class="p">,</span> <span class="ne">NotImplementedError</span><span class="p">):</span>
-                    <span class="k">pass</span>
-            <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="nf">__rdiv__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">invert</span><span class="p">(</span><span class="n">check</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span><span class="o">*</span><span class="n">g</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span> <span class="ow">and</span> <span class="n">r</span> <span class="ow">not</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="n">ExactQuotientFailed</span>
-            <span class="k">raise</span> <span class="n">ExactQuotientFailed</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="bp">self</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">ring</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">r</span>
-
-    <span class="n">__truediv__</span> <span class="o">=</span> <span class="n">__div__</span>
-    <span class="n">__rtruediv__</span> <span class="o">=</span> <span class="n">__rdiv__</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">DMP</span><span class="p">):</span>
-                <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="p">(</span><span class="n">F_num</span><span class="p">,</span> <span class="n">F_den</span><span class="p">),</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">poly_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span> <span class="o">==</span> <span class="n">g</span><span class="o">.</span><span class="n">lev</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">dmp_one_p</span><span class="p">(</span><span class="n">F_den</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span> <span class="ow">and</span> <span class="n">F_num</span> <span class="o">==</span> <span class="n">G</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">frac_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span> <span class="o">==</span> <span class="n">g</span><span class="o">.</span><span class="n">lev</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">F</span> <span class="o">==</span> <span class="n">G</span>
-        <span class="k">except</span> <span class="n">UnificationFailed</span><span class="p">:</span>
-            <span class="k">pass</span>
-
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">DMP</span><span class="p">):</span>
-                <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="p">(</span><span class="n">F_num</span><span class="p">,</span> <span class="n">F_den</span><span class="p">),</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">poly_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span> <span class="o">==</span> <span class="n">g</span><span class="o">.</span><span class="n">lev</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="ow">not</span> <span class="p">(</span><span class="n">dmp_one_p</span><span class="p">(</span><span class="n">F_den</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span> <span class="ow">and</span> <span class="n">F_num</span> <span class="o">==</span> <span class="n">G</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">frac_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span> <span class="o">==</span> <span class="n">g</span><span class="o">.</span><span class="n">lev</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">F</span> <span class="o">!=</span> <span class="n">G</span>
-        <span class="k">except</span> <span class="n">UnificationFailed</span><span class="p">:</span>
-            <span class="k">pass</span>
-
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__lt__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">frac_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">__lt__</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__le__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">frac_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">__le__</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__gt__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">frac_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">__gt__</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__ge__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">frac_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">__ge__</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__bool__</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="n">dmp_zero_p</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">num</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">lev</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">init_normal_ANP</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">mod</span><span class="p">,</span> <span class="n">dom</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">ANP</span><span class="p">(</span><span class="n">dup_normal</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span>
-               <span class="n">dup_normal</span><span class="p">(</span><span class="n">mod</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span> <span class="n">dom</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="ANP"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.ANP">[docs]</a><span class="k">class</span> <span class="nc">ANP</span><span class="p">(</span><span class="n">PicklableWithSlots</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Dense Algebraic Number Polynomials over a field. &quot;&quot;&quot;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;rep&#39;</span><span class="p">,</span> <span class="s">&#39;mod&#39;</span><span class="p">,</span> <span class="s">&#39;dom&#39;</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="n">mod</span><span class="p">,</span> <span class="n">dom</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">rep</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">dict</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">rep</span> <span class="o">=</span> <span class="n">dup_from_dict</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">rep</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="nb">list</span><span class="p">:</span>
-                <span class="n">rep</span> <span class="o">=</span> <span class="p">[</span><span class="n">dom</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">rep</span><span class="p">)]</span>
-
-            <span class="bp">self</span><span class="o">.</span><span class="n">rep</span> <span class="o">=</span> <span class="n">dup_strip</span><span class="p">(</span><span class="n">rep</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mod</span><span class="p">,</span> <span class="n">DMP</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">mod</span> <span class="o">=</span> <span class="n">mod</span><span class="o">.</span><span class="n">rep</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">mod</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">dict</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">mod</span> <span class="o">=</span> <span class="n">dup_from_dict</span><span class="p">(</span><span class="n">mod</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">mod</span> <span class="o">=</span> <span class="n">dup_strip</span><span class="p">(</span><span class="n">mod</span><span class="p">)</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">dom</span> <span class="o">=</span> <span class="n">dom</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">mod</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">hash</span><span class="p">((</span><span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">to_tuple</span><span class="p">(),</span> <span class="n">dmp_to_tuple</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">mod</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-
-<div class="viewcode-block" id="ANP.unify"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.ANP.unify">[docs]</a>    <span class="k">def</span> <span class="nf">unify</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Unify representations of two algebraic numbers. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">ANP</span><span class="p">)</span> <span class="ow">or</span> <span class="n">f</span><span class="o">.</span><span class="n">mod</span> <span class="o">!=</span> <span class="n">g</span><span class="o">.</span><span class="n">mod</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">UnificationFailed</span><span class="p">(</span><span class="s">&quot;can&#39;t unify </span><span class="si">%s</span><span class="s"> with </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span> <span class="o">==</span> <span class="n">g</span><span class="o">.</span><span class="n">dom</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">g</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">mod</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">dom</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-
-            <span class="n">F</span> <span class="o">=</span> <span class="n">dup_convert</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-            <span class="n">G</span> <span class="o">=</span> <span class="n">dup_convert</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">g</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">dom</span> <span class="o">!=</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span> <span class="ow">and</span> <span class="n">dom</span> <span class="o">!=</span> <span class="n">g</span><span class="o">.</span><span class="n">dom</span><span class="p">:</span>
-                <span class="n">mod</span> <span class="o">=</span> <span class="n">dup_convert</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">mod</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">dom</span> <span class="o">==</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">:</span>
-                    <span class="n">mod</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">mod</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">mod</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">mod</span>
-
-            <span class="n">per</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">rep</span><span class="p">:</span> <span class="n">ANP</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">mod</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">mod</span>
-</div>
-    <span class="k">def</span> <span class="nf">per</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="n">mod</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">dom</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">ANP</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">mod</span> <span class="ow">or</span> <span class="n">f</span><span class="o">.</span><span class="n">mod</span><span class="p">,</span> <span class="n">dom</span> <span class="ow">or</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">zero</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">mod</span><span class="p">,</span> <span class="n">dom</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">ANP</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">mod</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">one</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">mod</span><span class="p">,</span> <span class="n">dom</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">ANP</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">mod</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-
-<div class="viewcode-block" id="ANP.to_dict"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.ANP.to_dict">[docs]</a>    <span class="k">def</span> <span class="nf">to_dict</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Convert ``f`` to a dict representation with native coefficients. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dmp_to_dict</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="ANP.to_sympy_dict"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.ANP.to_sympy_dict">[docs]</a>    <span class="k">def</span> <span class="nf">to_sympy_dict</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Convert ``f`` to a dict representation with SymPy coefficients. &quot;&quot;&quot;</span>
-        <span class="n">rep</span> <span class="o">=</span> <span class="n">dmp_to_dict</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">rep</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">rep</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">rep</span>
-</div>
-<div class="viewcode-block" id="ANP.to_list"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.ANP.to_list">[docs]</a>    <span class="k">def</span> <span class="nf">to_list</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Convert ``f`` to a list representation with native coefficients. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span>
-</div>
-<div class="viewcode-block" id="ANP.to_sympy_list"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.ANP.to_sympy_list">[docs]</a>    <span class="k">def</span> <span class="nf">to_sympy_list</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Convert ``f`` to a list representation with SymPy coefficients. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span> <span class="p">]</span>
-</div>
-<div class="viewcode-block" id="ANP.to_tuple"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.ANP.to_tuple">[docs]</a>    <span class="k">def</span> <span class="nf">to_tuple</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Convert ``f`` to a tuple representation with native coefficients.</span>
-
-<span class="sd">        This is needed for hashing.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dmp_to_tuple</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">from_list</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="n">mod</span><span class="p">,</span> <span class="n">dom</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">ANP</span><span class="p">(</span><span class="n">dup_strip</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">dom</span><span class="o">.</span><span class="n">convert</span><span class="p">,</span> <span class="n">rep</span><span class="p">))),</span> <span class="n">mod</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">neg</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dup_neg</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">add</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">mod</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">dup_add</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">dom</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">sub</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">mod</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">dup_sub</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">dom</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">mod</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">dup_rem</span><span class="p">(</span><span class="n">dup_mul</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span> <span class="n">mod</span><span class="p">,</span> <span class="n">dom</span><span class="p">))</span>
-
-<div class="viewcode-block" id="ANP.pow"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.ANP.pow">[docs]</a>    <span class="k">def</span> <span class="nf">pow</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Raise ``f`` to a non-negative power ``n``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">F</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">dup_invert</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">mod</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">),</span> <span class="o">-</span><span class="n">n</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">F</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span>
-
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">dup_rem</span><span class="p">(</span><span class="n">dup_pow</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">),</span> <span class="n">f</span><span class="o">.</span><span class="n">mod</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;``int`` expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">type</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-</div>
-    <span class="k">def</span> <span class="nf">div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">mod</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">per</span><span class="p">(</span><span class="n">dup_rem</span><span class="p">(</span><span class="n">dup_mul</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">dup_invert</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">mod</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span> <span class="n">dom</span><span class="p">),</span> <span class="n">mod</span><span class="p">,</span> <span class="n">dom</span><span class="p">)),</span> <span class="bp">self</span><span class="o">.</span><span class="n">zero</span><span class="p">(</span><span class="n">mod</span><span class="p">,</span> <span class="n">dom</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">rem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">dom</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">mod</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">zero</span><span class="p">(</span><span class="n">mod</span><span class="p">,</span> <span class="n">dom</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">mod</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">dup_rem</span><span class="p">(</span><span class="n">dup_mul</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">dup_invert</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">mod</span><span class="p">,</span> <span class="n">dom</span><span class="p">),</span> <span class="n">dom</span><span class="p">),</span> <span class="n">mod</span><span class="p">,</span> <span class="n">dom</span><span class="p">))</span>
-
-    <span class="n">exquo</span> <span class="o">=</span> <span class="n">quo</span>
-
-<div class="viewcode-block" id="ANP.LC"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.ANP.LC">[docs]</a>    <span class="k">def</span> <span class="nf">LC</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the leading coefficient of ``f``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dup_LC</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="ANP.TC"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.ANP.TC">[docs]</a>    <span class="k">def</span> <span class="nf">TC</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the trailing coefficient of ``f``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">dup_TC</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="ANP.is_zero"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.ANP.is_zero">[docs]</a>    <span class="k">def</span> <span class="nf">is_zero</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns ``True`` if ``f`` is a zero algebraic number. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="n">f</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="ANP.is_one"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.ANP.is_one">[docs]</a>    <span class="k">def</span> <span class="nf">is_one</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns ``True`` if ``f`` is a unit algebraic number. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span> <span class="o">==</span> <span class="p">[</span><span class="n">f</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">one</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="ANP.is_ground"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyclasses.ANP.is_ground">[docs]</a>    <span class="k">def</span> <span class="nf">is_ground</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns ``True`` if ``f`` is an element of the ground domain. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span>
-</div>
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">neg</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">ANP</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">g</span><span class="p">))</span>
-            <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="nf">__radd__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">__add__</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">ANP</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">sub</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">sub</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">g</span><span class="p">))</span>
-            <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="nf">__rsub__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">__add__</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">ANP</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">g</span><span class="p">))</span>
-            <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="k">def</span> <span class="nf">__rmul__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__pow__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__divmod__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">div</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__mod__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rem</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">ANP</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">g</span><span class="p">))</span>
-            <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">NotImplemented</span>
-
-    <span class="n">__truediv__</span> <span class="o">=</span> <span class="n">__div__</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-            <span class="k">return</span> <span class="n">F</span> <span class="o">==</span> <span class="n">G</span>
-        <span class="k">except</span> <span class="n">UnificationFailed</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-            <span class="k">return</span> <span class="n">F</span> <span class="o">!=</span> <span class="n">G</span>
-        <span class="k">except</span> <span class="n">UnificationFailed</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__lt__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">__lt__</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__le__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">__le__</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__gt__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">__gt__</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__ge__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">__ge__</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__bool__</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">)</span></div>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/polys/polyerrors.html b/dev-py3k/_modules/sympy/polys/polyerrors.html
deleted file mode 100644
index 6a19f83db42..00000000000
--- a/dev-py3k/_modules/sympy/polys/polyerrors.html
+++ /dev/null
@@ -1,261 +0,0 @@
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-            
-  <h1>Source code for sympy.polys.polyerrors</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Definitions of common exceptions for `polys` module. &quot;&quot;&quot;</span>
-
-
-<div class="viewcode-block" id="BasePolynomialError"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.BasePolynomialError">[docs]</a><span class="k">class</span> <span class="nc">BasePolynomialError</span><span class="p">(</span><span class="ne">Exception</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Base class for polynomial related exceptions. &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">new</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;abstract base class&quot;</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="ExactQuotientFailed"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.ExactQuotientFailed">[docs]</a><span class="k">class</span> <span class="nc">ExactQuotientFailed</span><span class="p">(</span><span class="n">BasePolynomialError</span><span class="p">):</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">dom</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">f</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">g</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span> <span class="o">=</span> <span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">dom</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>  <span class="c"># pragma: no cover</span>
-        <span class="kn">from</span> <span class="nn">sympy.printing.str</span> <span class="kn">import</span> <span class="n">sstr</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s"> does not divide </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">sstr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">g</span><span class="p">),</span> <span class="n">sstr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">f</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s"> does not divide </span><span class="si">%s</span><span class="s"> in </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">sstr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">g</span><span class="p">),</span> <span class="n">sstr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">f</span><span class="p">),</span> <span class="n">sstr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">new</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="OperationNotSupported"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.OperationNotSupported">[docs]</a><span class="k">class</span> <span class="nc">OperationNotSupported</span><span class="p">(</span><span class="n">BasePolynomialError</span><span class="p">):</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">poly</span><span class="p">,</span> <span class="n">func</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">poly</span> <span class="o">=</span> <span class="n">poly</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">func</span> <span class="o">=</span> <span class="n">func</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>  <span class="c"># pragma: no cover</span>
-        <span class="k">return</span> <span class="s">&quot;`</span><span class="si">%s</span><span class="s">` operation not supported by </span><span class="si">%s</span><span class="s"> representation&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">poly</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="HeuristicGCDFailed"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.HeuristicGCDFailed">[docs]</a><span class="k">class</span> <span class="nc">HeuristicGCDFailed</span><span class="p">(</span><span class="n">BasePolynomialError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="HomomorphismFailed"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.HomomorphismFailed">[docs]</a><span class="k">class</span> <span class="nc">HomomorphismFailed</span><span class="p">(</span><span class="n">BasePolynomialError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="IsomorphismFailed"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.IsomorphismFailed">[docs]</a><span class="k">class</span> <span class="nc">IsomorphismFailed</span><span class="p">(</span><span class="n">BasePolynomialError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="ExtraneousFactors"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.ExtraneousFactors">[docs]</a><span class="k">class</span> <span class="nc">ExtraneousFactors</span><span class="p">(</span><span class="n">BasePolynomialError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="EvaluationFailed"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.EvaluationFailed">[docs]</a><span class="k">class</span> <span class="nc">EvaluationFailed</span><span class="p">(</span><span class="n">BasePolynomialError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="RefinementFailed"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.RefinementFailed">[docs]</a><span class="k">class</span> <span class="nc">RefinementFailed</span><span class="p">(</span><span class="n">BasePolynomialError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="CoercionFailed"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.CoercionFailed">[docs]</a><span class="k">class</span> <span class="nc">CoercionFailed</span><span class="p">(</span><span class="n">BasePolynomialError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="NotInvertible"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.NotInvertible">[docs]</a><span class="k">class</span> <span class="nc">NotInvertible</span><span class="p">(</span><span class="n">BasePolynomialError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="NotReversible"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.NotReversible">[docs]</a><span class="k">class</span> <span class="nc">NotReversible</span><span class="p">(</span><span class="n">BasePolynomialError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="NotAlgebraic"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.NotAlgebraic">[docs]</a><span class="k">class</span> <span class="nc">NotAlgebraic</span><span class="p">(</span><span class="n">BasePolynomialError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="DomainError"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.DomainError">[docs]</a><span class="k">class</span> <span class="nc">DomainError</span><span class="p">(</span><span class="n">BasePolynomialError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="PolynomialError"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.PolynomialError">[docs]</a><span class="k">class</span> <span class="nc">PolynomialError</span><span class="p">(</span><span class="n">BasePolynomialError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="UnificationFailed"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.UnificationFailed">[docs]</a><span class="k">class</span> <span class="nc">UnificationFailed</span><span class="p">(</span><span class="n">BasePolynomialError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="GeneratorsNeeded"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.GeneratorsNeeded">[docs]</a><span class="k">class</span> <span class="nc">GeneratorsNeeded</span><span class="p">(</span><span class="n">BasePolynomialError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="ComputationFailed"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.ComputationFailed">[docs]</a><span class="k">class</span> <span class="nc">ComputationFailed</span><span class="p">(</span><span class="n">BasePolynomialError</span><span class="p">):</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">nargs</span><span class="p">,</span> <span class="n">exc</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">func</span> <span class="o">=</span> <span class="n">func</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">nargs</span> <span class="o">=</span> <span class="n">nargs</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">exc</span> <span class="o">=</span> <span class="n">exc</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">) failed without generators&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">,</span> <span class="s">&#39;, &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">str</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">exc</span><span class="o">.</span><span class="n">exprs</span><span class="p">[:</span><span class="bp">self</span><span class="o">.</span><span class="n">nargs</span><span class="p">])))</span>
-
-</div>
-<div class="viewcode-block" id="GeneratorsError"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.GeneratorsError">[docs]</a><span class="k">class</span> <span class="nc">GeneratorsError</span><span class="p">(</span><span class="n">BasePolynomialError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="UnivariatePolynomialError"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.UnivariatePolynomialError">[docs]</a><span class="k">class</span> <span class="nc">UnivariatePolynomialError</span><span class="p">(</span><span class="n">PolynomialError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="MultivariatePolynomialError"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.MultivariatePolynomialError">[docs]</a><span class="k">class</span> <span class="nc">MultivariatePolynomialError</span><span class="p">(</span><span class="n">PolynomialError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="PolificationFailed"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.PolificationFailed">[docs]</a><span class="k">class</span> <span class="nc">PolificationFailed</span><span class="p">(</span><span class="n">PolynomialError</span><span class="p">):</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">opt</span><span class="p">,</span> <span class="n">origs</span><span class="p">,</span> <span class="n">exprs</span><span class="p">,</span> <span class="n">seq</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">seq</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">orig</span> <span class="o">=</span> <span class="n">origs</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">expr</span> <span class="o">=</span> <span class="n">exprs</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">origs</span> <span class="o">=</span> <span class="p">[</span><span class="n">origs</span><span class="p">]</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">exprs</span> <span class="o">=</span> <span class="p">[</span><span class="n">exprs</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">origs</span> <span class="o">=</span> <span class="n">origs</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">exprs</span> <span class="o">=</span> <span class="n">exprs</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">opt</span> <span class="o">=</span> <span class="n">opt</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">seq</span> <span class="o">=</span> <span class="n">seq</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>  <span class="c"># pragma: no cover</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">seq</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;can&#39;t construct a polynomial from </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">orig</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;can&#39;t construct polynomials from </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="s">&#39;, &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">str</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">origs</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="OptionError"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.OptionError">[docs]</a><span class="k">class</span> <span class="nc">OptionError</span><span class="p">(</span><span class="n">BasePolynomialError</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="FlagError"><a class="viewcode-back" href="../../../modules/polys/internals.html#sympy.polys.polyerrors.FlagError">[docs]</a><span class="k">class</span> <span class="nc">FlagError</span><span class="p">(</span><span class="n">OptionError</span><span class="p">):</span>
-    <span class="k">pass</span></div>
-</pre></div>
-
-          </div>
-        </div>
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-        <div class="sphinxsidebarwrapper">
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-  <h3>Quick search</h3>
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-        <li class="right" >
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diff --git a/dev-py3k/_modules/sympy/polys/polyfuncs.html b/dev-py3k/_modules/sympy/polys/polyfuncs.html
deleted file mode 100644
index b841dd0a6f9..00000000000
--- a/dev-py3k/_modules/sympy/polys/polyfuncs.html
+++ /dev/null
@@ -1,420 +0,0 @@
-
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-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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-  <h1>Source code for sympy.polys.polyfuncs</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;High-level polynomials manipulation functions. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polytools</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">poly_from_expr</span><span class="p">,</span> <span class="n">parallel_poly_from_expr</span><span class="p">,</span> <span class="n">Poly</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polyoptions</span> <span class="kn">import</span> <span class="n">allowed_flags</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.specialpolys</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">symmetric_poly</span><span class="p">,</span> <span class="n">interpolating_poly</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">PolificationFailed</span><span class="p">,</span> <span class="n">ComputationFailed</span><span class="p">,</span>
-    <span class="n">MultivariatePolynomialError</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">numbered_symbols</span><span class="p">,</span> <span class="n">take</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Mul</span>
-
-
-<div class="viewcode-block" id="symmetrize"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polyfuncs.symmetrize">[docs]</a><span class="k">def</span> <span class="nf">symmetrize</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Rewrite a polynomial in terms of elementary symmetric polynomials.</span>
-
-<span class="sd">    A symmetric polynomial is a multivariate polynomial that remains invariant</span>
-<span class="sd">    under any variable permutation, i.e., if ``f = f(x_1, x_2, ..., x_n)``,</span>
-<span class="sd">    then ``f = f(x_{i_1}, x_{i_2}, ..., x_{i_n})``, where</span>
-<span class="sd">    ``(i_1, i_2, ..., i_n)`` is a permutation of ``(1, 2, ..., n)`` (an</span>
-<span class="sd">    element of the group ``S_n``).</span>
-
-<span class="sd">    Returns a tuple of symmetric polynomials ``(f1, f2, ..., fn)`` such that</span>
-<span class="sd">    ``f = f1 + f2 + ... + fn``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.polyfuncs import symmetrize</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">    &gt;&gt;&gt; symmetrize(x**2 + y**2)</span>
-<span class="sd">    (-2*x*y + (x + y)**2, 0)</span>
-
-<span class="sd">    &gt;&gt;&gt; symmetrize(x**2 + y**2, formal=True)</span>
-<span class="sd">    (s1**2 - 2*s2, 0, [(s1, x + y), (s2, x*y)])</span>
-
-<span class="sd">    &gt;&gt;&gt; symmetrize(x**2 - y**2)</span>
-<span class="sd">    (-2*x*y + (x + y)**2, -2*y**2)</span>
-
-<span class="sd">    &gt;&gt;&gt; symmetrize(x**2 - y**2, formal=True)</span>
-<span class="sd">    (s1**2 - 2*s2, -2*y**2, [(s1, x + y), (s2, x*y)])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;formal&#39;</span><span class="p">,</span> <span class="s">&#39;symbols&#39;</span><span class="p">])</span>
-
-    <span class="n">iterable</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="s">&#39;__iter__&#39;</span><span class="p">):</span>
-        <span class="n">iterable</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="n">F</span> <span class="o">=</span> <span class="p">[</span><span class="n">F</span><span class="p">]</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">expr</span> <span class="ow">in</span> <span class="n">exc</span><span class="o">.</span><span class="n">exprs</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">expr</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;symmetrize&#39;</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">F</span><span class="p">),</span> <span class="n">exc</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">iterable</span><span class="p">:</span>
-                <span class="n">result</span><span class="p">,</span> <span class="o">=</span> <span class="n">result</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">exc</span><span class="o">.</span><span class="n">opt</span><span class="o">.</span><span class="n">formal</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">result</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">iterable</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">result</span><span class="p">,</span> <span class="p">[]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">result</span> <span class="o">+</span> <span class="p">([],)</span>
-
-    <span class="n">polys</span><span class="p">,</span> <span class="n">symbols</span> <span class="o">=</span> <span class="p">[],</span> <span class="n">opt</span><span class="o">.</span><span class="n">symbols</span>
-    <span class="n">gens</span><span class="p">,</span> <span class="n">dom</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">)):</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">symmetric_poly</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">polys</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="n">polys</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="nb">next</span><span class="p">(</span><span class="n">symbols</span><span class="p">),</span> <span class="n">poly</span><span class="o">.</span><span class="n">set_domain</span><span class="p">(</span><span class="n">dom</span><span class="p">)))</span>
-
-    <span class="n">indices</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-    <span class="n">weights</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">))</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">F</span><span class="p">:</span>
-        <span class="n">symmetric</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">is_homogeneous</span><span class="p">:</span>
-            <span class="n">symmetric</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">TC</span><span class="p">())</span>
-            <span class="n">f</span> <span class="o">-=</span> <span class="n">f</span><span class="o">.</span><span class="n">TC</span><span class="p">()</span>
-
-        <span class="k">while</span> <span class="n">f</span><span class="p">:</span>
-            <span class="n">_height</span><span class="p">,</span> <span class="n">_monom</span><span class="p">,</span> <span class="n">_coeff</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">terms</span><span class="p">()):</span>
-                <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">monom</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&gt;=</span> <span class="n">monom</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">indices</span><span class="p">):</span>
-                    <span class="n">height</span> <span class="o">=</span> <span class="nb">max</span><span class="p">([</span> <span class="n">n</span><span class="o">*</span><span class="n">m</span> <span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">weights</span><span class="p">,</span> <span class="n">monom</span><span class="p">)</span> <span class="p">])</span>
-
-                    <span class="k">if</span> <span class="n">height</span> <span class="o">&gt;</span> <span class="n">_height</span><span class="p">:</span>
-                        <span class="n">_height</span><span class="p">,</span> <span class="n">_monom</span><span class="p">,</span> <span class="n">_coeff</span> <span class="o">=</span> <span class="n">height</span><span class="p">,</span> <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span>
-
-            <span class="k">if</span> <span class="n">_height</span> <span class="o">!=</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="o">=</span> <span class="n">_monom</span><span class="p">,</span> <span class="n">_coeff</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">break</span>
-
-            <span class="n">exponents</span> <span class="o">=</span> <span class="p">[]</span>
-
-            <span class="k">for</span> <span class="n">m1</span><span class="p">,</span> <span class="n">m2</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">monom</span><span class="p">,</span> <span class="n">monom</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span> <span class="o">+</span> <span class="p">(</span><span class="mi">0</span><span class="p">,)):</span>
-                <span class="n">exponents</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">m1</span> <span class="o">-</span> <span class="n">m2</span><span class="p">)</span>
-
-            <span class="n">term</span> <span class="o">=</span> <span class="p">[</span> <span class="n">s</span><span class="o">**</span><span class="n">n</span> <span class="k">for</span> <span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">_</span><span class="p">),</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">polys</span><span class="p">,</span> <span class="n">exponents</span><span class="p">)</span> <span class="p">]</span>
-            <span class="n">poly</span> <span class="o">=</span> <span class="p">[</span> <span class="n">p</span><span class="o">**</span><span class="n">n</span> <span class="k">for</span> <span class="p">(</span><span class="n">_</span><span class="p">,</span> <span class="n">p</span><span class="p">),</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">polys</span><span class="p">,</span> <span class="n">exponents</span><span class="p">)</span> <span class="p">]</span>
-
-            <span class="n">symmetric</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="o">*</span><span class="n">term</span><span class="p">))</span>
-            <span class="n">product</span> <span class="o">=</span> <span class="n">poly</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">poly</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-                <span class="n">product</span> <span class="o">=</span> <span class="n">product</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-
-            <span class="n">f</span> <span class="o">-=</span> <span class="n">product</span>
-
-        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">symmetric</span><span class="p">),</span> <span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()))</span>
-
-    <span class="n">polys</span> <span class="o">=</span> <span class="p">[</span> <span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">as_expr</span><span class="p">())</span> <span class="k">for</span> <span class="n">s</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">polys</span> <span class="p">]</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">formal</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">non_sym</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">result</span><span class="p">):</span>
-            <span class="n">result</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">sym</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">polys</span><span class="p">),</span> <span class="n">non_sym</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">iterable</span><span class="p">:</span>
-        <span class="n">result</span><span class="p">,</span> <span class="o">=</span> <span class="n">result</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">formal</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">iterable</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">result</span><span class="p">,</span> <span class="n">polys</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">result</span> <span class="o">+</span> <span class="p">(</span><span class="n">polys</span><span class="p">,)</span>
-
-</div>
-<div class="viewcode-block" id="horner"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polyfuncs.horner">[docs]</a><span class="k">def</span> <span class="nf">horner</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Rewrite a polynomial in Horner form.</span>
-
-<span class="sd">    Among other applications, evaluation of a polynomial at a point is optimal</span>
-<span class="sd">    when it is applied using the Horner scheme ([1]).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.polyfuncs import horner</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, a, b, c, d, e</span>
-
-<span class="sd">    &gt;&gt;&gt; horner(9*x**4 + 8*x**3 + 7*x**2 + 6*x + 5)</span>
-<span class="sd">    x*(x*(x*(9*x + 8) + 7) + 6) + 5</span>
-
-<span class="sd">    &gt;&gt;&gt; horner(a*x**4 + b*x**3 + c*x**2 + d*x + e)</span>
-<span class="sd">    e + x*(d + x*(c + x*(a*x + b)))</span>
-
-<span class="sd">    &gt;&gt;&gt; f = 4*x**2*y**2 + 2*x**2*y + 2*x*y**2 + x*y</span>
-
-<span class="sd">    &gt;&gt;&gt; horner(f, wrt=x)</span>
-<span class="sd">    x*(x*y*(4*y + 2) + y*(2*y + 1))</span>
-
-<span class="sd">    &gt;&gt;&gt; horner(f, wrt=y)</span>
-<span class="sd">    y*(x*y*(4*x + 2) + x*(2*x + 1))</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-<span class="sd">    [1] - http://en.wikipedia.org/wiki/Horner_scheme</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">poly_from_expr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">exc</span><span class="o">.</span><span class="n">expr</span>
-
-    <span class="n">form</span><span class="p">,</span> <span class="n">gen</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="n">F</span><span class="o">.</span><span class="n">gen</span>
-
-    <span class="k">if</span> <span class="n">F</span><span class="o">.</span><span class="n">is_univariate</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">F</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">():</span>
-            <span class="n">form</span> <span class="o">=</span> <span class="n">form</span><span class="o">*</span><span class="n">gen</span> <span class="o">+</span> <span class="n">coeff</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">F</span><span class="p">,</span> <span class="n">gens</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">gen</span><span class="p">),</span> <span class="n">gens</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-        <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">F</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">():</span>
-            <span class="n">form</span> <span class="o">=</span> <span class="n">form</span><span class="o">*</span><span class="n">gen</span> <span class="o">+</span> <span class="n">horner</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">form</span>
-
-</div>
-<div class="viewcode-block" id="interpolate"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polyfuncs.interpolate">[docs]</a><span class="k">def</span> <span class="nf">interpolate</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Construct an interpolating polynomial for the data points.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.polyfuncs import interpolate</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    A list is interpreted as though it were paired with a range starting</span>
-<span class="sd">    from 1:</span>
-
-<span class="sd">    &gt;&gt;&gt; interpolate([1, 4, 9, 16], x)</span>
-<span class="sd">    x**2</span>
-
-<span class="sd">    This can be made explicit by giving a list of coordinates:</span>
-
-<span class="sd">    &gt;&gt;&gt; interpolate([(1, 1), (2, 4), (3, 9)], x)</span>
-<span class="sd">    x**2</span>
-
-<span class="sd">    The (x, y) coordinates can also be given as keys and values of a</span>
-<span class="sd">    dictionary (and the points need not be equispaced):</span>
-
-<span class="sd">    &gt;&gt;&gt; interpolate([(-1, 2), (1, 2), (2, 5)], x)</span>
-<span class="sd">    x**2 + 1</span>
-<span class="sd">    &gt;&gt;&gt; interpolate({-1: 2, 1: 2, 2: 5}, x)</span>
-<span class="sd">    x**2 + 1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="nb">dict</span><span class="p">):</span>
-        <span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="nb">list</span><span class="p">(</span><span class="n">data</span><span class="o">.</span><span class="n">items</span><span class="p">())))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">data</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="nb">tuple</span><span class="p">):</span>
-            <span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">data</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">X</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-            <span class="n">Y</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
-
-    <span class="n">poly</span> <span class="o">=</span> <span class="n">interpolating_poly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">poly</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-
-</div>
-<div class="viewcode-block" id="viete"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polyfuncs.viete">[docs]</a><span class="k">def</span> <span class="nf">viete</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">roots</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Generate Viete&#39;s formulas for ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.polyfuncs import viete</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-
-<span class="sd">    &gt;&gt;&gt; x, a, b, c, r1, r2 = symbols(&#39;x,a:c,r1:3&#39;)</span>
-
-<span class="sd">    &gt;&gt;&gt; viete(a*x**2 + b*x + c, [r1, r2], x)</span>
-<span class="sd">    [(r1 + r2, -b/a), (r1*r2, c/a)]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[])</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">roots</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-        <span class="n">gens</span><span class="p">,</span> <span class="n">roots</span> <span class="o">=</span> <span class="p">(</span><span class="n">roots</span><span class="p">,)</span> <span class="o">+</span> <span class="n">gens</span><span class="p">,</span> <span class="bp">None</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">f</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">poly_from_expr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;viete&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_multivariate</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">MultivariatePolynomialError</span><span class="p">(</span>
-            <span class="s">&quot;multivariate polynomials are not allowed&quot;</span><span class="p">)</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-            <span class="s">&quot;can&#39;t derive Viete&#39;s formulas for a constant polynomial&quot;</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">roots</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">roots</span> <span class="o">=</span> <span class="n">numbered_symbols</span><span class="p">(</span><span class="s">&#39;r&#39;</span><span class="p">,</span> <span class="n">start</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-
-    <span class="n">roots</span> <span class="o">=</span> <span class="n">take</span><span class="p">(</span><span class="n">roots</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">roots</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;required </span><span class="si">%s</span><span class="s"> roots, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">roots</span><span class="p">)))</span>
-
-    <span class="n">lc</span><span class="p">,</span> <span class="n">coeffs</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">LC</span><span class="p">(),</span> <span class="n">f</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()</span>
-    <span class="n">result</span><span class="p">,</span> <span class="n">sign</span> <span class="o">=</span> <span class="p">[],</span> <span class="o">-</span><span class="mi">1</span>
-
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">coeffs</span><span class="p">[</span><span class="mi">1</span><span class="p">:]):</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">symmetric_poly</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">roots</span><span class="p">)</span>
-        <span class="n">coeff</span> <span class="o">=</span> <span class="n">sign</span><span class="o">*</span><span class="p">(</span><span class="n">coeff</span><span class="o">/</span><span class="n">lc</span><span class="p">)</span>
-        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">poly</span><span class="p">,</span> <span class="n">coeff</span><span class="p">))</span>
-        <span class="n">sign</span> <span class="o">=</span> <span class="o">-</span><span class="n">sign</span>
-
-    <span class="k">return</span> <span class="n">result</span></div>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
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diff --git a/dev-py3k/_modules/sympy/polys/polyroots.html b/dev-py3k/_modules/sympy/polys/polyroots.html
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-            
-  <h1>Source code for sympy.polys.polyroots</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Algorithms for computing symbolic roots of polynomials. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.symbol</span> <span class="kn">import</span> <span class="n">Dummy</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">I</span>
-<span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">Rational</span><span class="p">,</span> <span class="n">igcd</span>
-
-<span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">divisors</span><span class="p">,</span> <span class="n">isprime</span><span class="p">,</span> <span class="n">nextprime</span>
-<span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sqrt</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polytools</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">cancel</span><span class="p">,</span> <span class="n">factor</span><span class="p">,</span> <span class="n">gcd_list</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.specialpolys</span> <span class="kn">import</span> <span class="n">cyclotomic_poly</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="n">PolynomialError</span><span class="p">,</span> <span class="n">GeneratorsNeeded</span><span class="p">,</span> <span class="n">DomainError</span>
-
-<span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">simplify</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">default_sort_key</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-<span class="kn">import</span> <span class="nn">math</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-
-<span class="k">def</span> <span class="nf">roots_linear</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns a list of roots of a linear polynomial.&quot;&quot;&quot;</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="o">-</span><span class="n">f</span><span class="o">.</span><span class="n">nth</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="o">/</span><span class="n">f</span><span class="o">.</span><span class="n">nth</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-    <span class="n">dom</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">get_domain</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">dom</span><span class="o">.</span><span class="n">is_Numerical</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">dom</span><span class="o">.</span><span class="n">is_Composite</span><span class="p">:</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">factor</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="p">[</span><span class="n">r</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">roots_quadratic</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns a list of roots of a quadratic polynomial.&quot;&quot;&quot;</span>
-    <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()</span>
-    <span class="n">dom</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">get_domain</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">_simplify</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">dom</span><span class="o">.</span><span class="n">is_Composite</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">factor</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">simplify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-        <span class="n">r0</span><span class="p">,</span> <span class="n">r1</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="o">-</span><span class="n">b</span><span class="o">/</span><span class="n">a</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">dom</span><span class="o">.</span><span class="n">is_Numerical</span><span class="p">:</span>
-            <span class="n">r1</span> <span class="o">=</span> <span class="n">_simplify</span><span class="p">(</span><span class="n">r1</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">b</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="o">-</span><span class="n">c</span><span class="o">/</span><span class="n">a</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">dom</span><span class="o">.</span><span class="n">is_Numerical</span><span class="p">:</span>
-            <span class="n">R</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">_simplify</span><span class="p">(</span><span class="n">r</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">R</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-
-        <span class="n">r0</span> <span class="o">=</span> <span class="n">R</span>
-        <span class="n">r1</span> <span class="o">=</span> <span class="o">-</span><span class="n">R</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">b</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">4</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">c</span>
-
-        <span class="k">if</span> <span class="n">dom</span><span class="o">.</span><span class="n">is_Numerical</span><span class="p">:</span>
-            <span class="n">D</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-
-            <span class="n">r0</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="n">b</span> <span class="o">+</span> <span class="n">D</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span>
-            <span class="n">r1</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="n">b</span> <span class="o">-</span> <span class="n">D</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">D</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">_simplify</span><span class="p">(</span><span class="n">d</span><span class="p">))</span>
-            <span class="n">A</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span>
-
-            <span class="n">E</span> <span class="o">=</span> <span class="n">_simplify</span><span class="p">(</span><span class="o">-</span><span class="n">b</span><span class="o">/</span><span class="n">A</span><span class="p">)</span>
-            <span class="n">F</span> <span class="o">=</span> <span class="n">D</span><span class="o">/</span><span class="n">A</span>
-
-            <span class="n">r0</span> <span class="o">=</span> <span class="n">E</span> <span class="o">+</span> <span class="n">F</span>
-            <span class="n">r1</span> <span class="o">=</span> <span class="n">E</span> <span class="o">-</span> <span class="n">F</span>
-
-    <span class="k">return</span> <span class="nb">sorted</span><span class="p">([</span><span class="n">r0</span><span class="p">,</span> <span class="n">r1</span><span class="p">],</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">roots_cubic</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns a list of roots of a cubic polynomial.&quot;&quot;&quot;</span>
-    <span class="n">_</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">monic</span><span class="p">()</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-        <span class="n">x1</span><span class="p">,</span> <span class="n">x2</span> <span class="o">=</span> <span class="n">roots</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">],</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">x1</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="n">x2</span><span class="p">]</span>
-
-    <span class="n">p</span> <span class="o">=</span> <span class="n">b</span> <span class="o">-</span> <span class="n">a</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">3</span>
-    <span class="n">q</span> <span class="o">=</span> <span class="n">c</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="n">b</span><span class="o">/</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">**</span><span class="mi">3</span><span class="o">/</span><span class="mi">27</span>
-
-    <span class="n">pon3</span> <span class="o">=</span> <span class="n">p</span><span class="o">/</span><span class="mi">3</span>
-    <span class="n">aon3</span> <span class="o">=</span> <span class="n">a</span><span class="o">/</span><span class="mi">3</span>
-
-    <span class="k">if</span> <span class="n">p</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">q</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[</span><span class="o">-</span><span class="n">aon3</span><span class="p">]</span><span class="o">*</span><span class="mi">3</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">u1</span> <span class="o">=</span> <span class="n">q</span><span class="o">**</span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">q</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-        <span class="n">y1</span><span class="p">,</span> <span class="n">y2</span> <span class="o">=</span> <span class="n">roots</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">p</span><span class="p">],</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">tmp</span> <span class="o">-</span> <span class="n">aon3</span> <span class="k">for</span> <span class="n">tmp</span> <span class="ow">in</span> <span class="p">[</span><span class="n">y1</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="n">y2</span><span class="p">]]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">u1</span> <span class="o">=</span> <span class="p">(</span><span class="n">q</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">q</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">4</span> <span class="o">+</span> <span class="n">pon3</span><span class="o">**</span><span class="mi">3</span><span class="p">))</span><span class="o">**</span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-
-    <span class="n">coeff</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-
-    <span class="n">u2</span> <span class="o">=</span> <span class="n">u1</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="o">+</span> <span class="n">coeff</span><span class="p">)</span>
-    <span class="n">u3</span> <span class="o">=</span> <span class="n">u1</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="o">-</span> <span class="n">coeff</span><span class="p">)</span>
-
-    <span class="n">soln</span> <span class="o">=</span> <span class="p">[</span>
-        <span class="o">-</span><span class="n">u1</span> <span class="o">+</span> <span class="n">pon3</span><span class="o">/</span><span class="n">u1</span> <span class="o">-</span> <span class="n">aon3</span><span class="p">,</span>
-        <span class="o">-</span><span class="n">u2</span> <span class="o">+</span> <span class="n">pon3</span><span class="o">/</span><span class="n">u2</span> <span class="o">-</span> <span class="n">aon3</span><span class="p">,</span>
-        <span class="o">-</span><span class="n">u3</span> <span class="o">+</span> <span class="n">pon3</span><span class="o">/</span><span class="n">u3</span> <span class="o">-</span> <span class="n">aon3</span>
-    <span class="p">]</span>
-
-    <span class="k">return</span> <span class="n">soln</span>
-
-
-<span class="k">def</span> <span class="nf">roots_quartic</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Returns a list of roots of a quartic polynomial.</span>
-
-<span class="sd">    There are many references for solving quartic expressions available [1-5].</span>
-<span class="sd">    This reviewer has found that many of them require one to select from among</span>
-<span class="sd">    2 or more possible sets of solutions and that some solutions work when one</span>
-<span class="sd">    is searching for real roots but don&#39;t work when searching for complex roots</span>
-<span class="sd">    (though this is not always stated clearly). The following routine has been</span>
-<span class="sd">    tested and found to be correct for 0, 2 or 4 complex roots.</span>
-
-<span class="sd">    The quasisymmetric case solution [6] looks for quartics that have the form</span>
-<span class="sd">    `x**4 + A*x**3 + B*x**2 + C*x + D = 0` where `(C/A)**2 = D`.</span>
-
-<span class="sd">    Although there is a general solution, simpler results can be obtained for</span>
-<span class="sd">    certain values of the coefficients. In all cases, 4 roots are returned:</span>
-
-<span class="sd">      1) `f = c + a*(a**2/8 - b/2) == 0`</span>
-<span class="sd">      2) `g = d - a*(a*(3*a**2/256 - b/16) + c/4) = 0`</span>
-<span class="sd">      3) if `f != 0` and `g != 0` and `p = -d + a*c/4 - b**2/12` then</span>
-<span class="sd">        a) `p == 0`</span>
-<span class="sd">        b) `p != 0`</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly, symbols, I</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.polys.polyroots import roots_quartic</span>
-
-<span class="sd">        &gt;&gt;&gt; r = roots_quartic(Poly(&#39;x**4-6*x**3+17*x**2-26*x+20&#39;))</span>
-
-<span class="sd">        &gt;&gt;&gt; # 4 complex roots: 1+-I*sqrt(3), 2+-I</span>
-<span class="sd">        &gt;&gt;&gt; sorted(str(tmp.evalf(n=2)) for tmp in r)</span>
-<span class="sd">        [&#39;1.0 + 1.7*I&#39;, &#39;1.0 - 1.7*I&#39;, &#39;2.0 + 1.0*I&#39;, &#39;2.0 - 1.0*I&#39;]</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. http://mathforum.org/dr.math/faq/faq.cubic.equations.html</span>
-<span class="sd">    2. http://en.wikipedia.org/wiki/Quartic_function#Summary_of_Ferrari.27s_method</span>
-<span class="sd">    3. http://planetmath.org/encyclopedia/GaloisTheoreticDerivationOfTheQuarticFormula.html</span>
-<span class="sd">    4. http://staff.bath.ac.uk/masjhd/JHD-CA.pdf</span>
-<span class="sd">    5. http://www.albmath.org/files/Math_5713.pdf</span>
-<span class="sd">    6. http://www.statemaster.com/encyclopedia/Quartic-equation</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">_</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">monic</span><span class="p">()</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">d</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">]</span> <span class="o">+</span> <span class="n">roots</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">],</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="p">(</span><span class="n">c</span><span class="o">/</span><span class="n">a</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">==</span> <span class="n">d</span><span class="p">:</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">gen</span><span class="p">,</span> <span class="n">c</span><span class="o">/</span><span class="n">a</span>
-
-        <span class="n">g</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">b</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">m</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-        <span class="n">z1</span><span class="p">,</span> <span class="n">z2</span> <span class="o">=</span> <span class="n">roots_quadratic</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="n">h1</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">z1</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">m</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="n">h2</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">z2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">m</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-        <span class="n">r1</span> <span class="o">=</span> <span class="n">roots_quadratic</span><span class="p">(</span><span class="n">h1</span><span class="p">)</span>
-        <span class="n">r2</span> <span class="o">=</span> <span class="n">roots_quadratic</span><span class="p">(</span><span class="n">h2</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">r1</span> <span class="o">+</span> <span class="n">r2</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">a2</span> <span class="o">=</span> <span class="n">a</span><span class="o">**</span><span class="mi">2</span>
-        <span class="n">e</span> <span class="o">=</span> <span class="n">b</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">a2</span><span class="o">/</span><span class="mi">8</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">c</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="n">a2</span><span class="o">/</span><span class="mi">8</span> <span class="o">-</span> <span class="n">b</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">d</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">a2</span><span class="o">/</span><span class="mi">256</span> <span class="o">-</span> <span class="n">b</span><span class="o">/</span><span class="mi">16</span><span class="p">)</span> <span class="o">+</span> <span class="n">c</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span>
-        <span class="n">aon4</span> <span class="o">=</span> <span class="n">a</span><span class="o">/</span><span class="mi">4</span>
-        <span class="n">ans</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">if</span> <span class="n">f</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-            <span class="n">y1</span><span class="p">,</span> <span class="n">y2</span> <span class="o">=</span> <span class="p">[</span><span class="n">sqrt</span><span class="p">(</span><span class="n">tmp</span><span class="p">)</span> <span class="k">for</span> <span class="n">tmp</span> <span class="ow">in</span>
-                      <span class="n">roots</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">g</span><span class="p">],</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">True</span><span class="p">)]</span>
-            <span class="k">return</span> <span class="p">[</span><span class="n">tmp</span> <span class="o">-</span> <span class="n">aon4</span> <span class="k">for</span> <span class="n">tmp</span> <span class="ow">in</span> <span class="p">[</span><span class="o">-</span><span class="n">y1</span><span class="p">,</span> <span class="o">-</span><span class="n">y2</span><span class="p">,</span> <span class="n">y1</span><span class="p">,</span> <span class="n">y2</span><span class="p">]]</span>
-        <span class="k">if</span> <span class="n">g</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-            <span class="n">y</span> <span class="o">=</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">]</span> <span class="o">+</span> <span class="n">roots</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">f</span><span class="p">],</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="k">return</span> <span class="p">[</span><span class="n">tmp</span> <span class="o">-</span> <span class="n">aon4</span> <span class="k">for</span> <span class="n">tmp</span> <span class="ow">in</span> <span class="n">y</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="o">-</span><span class="n">e</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">12</span> <span class="o">-</span> <span class="n">g</span>
-            <span class="n">q</span> <span class="o">=</span> <span class="o">-</span><span class="n">e</span><span class="o">**</span><span class="mi">3</span><span class="o">/</span><span class="mi">108</span> <span class="o">+</span> <span class="n">e</span><span class="o">*</span><span class="n">g</span><span class="o">/</span><span class="mi">3</span> <span class="o">-</span> <span class="n">f</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">8</span>
-            <span class="n">TH</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">p</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="n">y</span> <span class="o">=</span> <span class="o">-</span><span class="mi">5</span><span class="o">*</span><span class="n">e</span><span class="o">/</span><span class="mi">6</span> <span class="o">-</span> <span class="n">q</span><span class="o">**</span><span class="n">TH</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># with p !=0 then u below is not 0</span>
-                <span class="n">root</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">q</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">4</span> <span class="o">+</span> <span class="n">p</span><span class="o">**</span><span class="mi">3</span><span class="o">/</span><span class="mi">27</span><span class="p">)</span>
-                <span class="n">r</span> <span class="o">=</span> <span class="o">-</span><span class="n">q</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="n">root</span>  <span class="c"># or -q/2 - root</span>
-                <span class="n">u</span> <span class="o">=</span> <span class="n">r</span><span class="o">**</span><span class="n">TH</span>  <span class="c"># primary root of solve(x**3-r, x)</span>
-                <span class="n">y</span> <span class="o">=</span> <span class="o">-</span><span class="mi">5</span><span class="o">*</span><span class="n">e</span><span class="o">/</span><span class="mi">6</span> <span class="o">+</span> <span class="n">u</span> <span class="o">-</span> <span class="n">p</span><span class="o">/</span><span class="n">u</span><span class="o">/</span><span class="mi">3</span>
-            <span class="n">w</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">e</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
-            <span class="n">arg1</span> <span class="o">=</span> <span class="mi">3</span><span class="o">*</span><span class="n">e</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span>
-            <span class="n">arg2</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">f</span><span class="o">/</span><span class="n">w</span>
-            <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]:</span>
-                <span class="n">root</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">arg1</span> <span class="o">+</span> <span class="n">s</span><span class="o">*</span><span class="n">arg2</span><span class="p">))</span>
-                <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]:</span>
-                    <span class="n">ans</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">s</span><span class="o">*</span><span class="n">w</span> <span class="o">-</span> <span class="n">t</span><span class="o">*</span><span class="n">root</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">aon4</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">ans</span>
-
-
-<span class="k">def</span> <span class="nf">roots_binomial</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns a list of roots of a binomial polynomial.&quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span>
-
-    <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">nth</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">f</span><span class="o">.</span><span class="n">nth</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-    <span class="n">alpha</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="n">cancel</span><span class="p">(</span><span class="n">b</span><span class="o">/</span><span class="n">a</span><span class="p">))</span><span class="o">**</span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">alpha</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-        <span class="n">alpha</span> <span class="o">=</span> <span class="n">alpha</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="n">roots</span><span class="p">,</span> <span class="n">I</span> <span class="o">=</span> <span class="p">[],</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-
-    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-        <span class="n">zeta</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">*</span><span class="n">I</span><span class="o">/</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="n">roots</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">alpha</span><span class="o">*</span><span class="n">zeta</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">power_base</span><span class="o">=</span><span class="bp">False</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">roots</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">_inv_totient_estimate</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Find ``(L, U)`` such that ``L &lt;= phi^-1(m) &lt;= U``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.polyroots import _inv_totient_estimate</span>
-
-<span class="sd">    &gt;&gt;&gt; _inv_totient_estimate(192)</span>
-<span class="sd">    (192, 840)</span>
-<span class="sd">    &gt;&gt;&gt; _inv_totient_estimate(400)</span>
-<span class="sd">    (400, 1750)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">primes</span> <span class="o">=</span> <span class="p">[</span> <span class="n">d</span> <span class="o">+</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">divisors</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="k">if</span> <span class="n">isprime</span><span class="p">(</span><span class="n">d</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">]</span>
-
-    <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span>
-
-    <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">primes</span><span class="p">:</span>
-        <span class="n">a</span> <span class="o">*=</span> <span class="n">p</span>
-        <span class="n">b</span> <span class="o">*=</span> <span class="n">p</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="n">L</span> <span class="o">=</span> <span class="n">m</span>
-    <span class="n">U</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">ceil</span><span class="p">(</span><span class="n">m</span><span class="o">*</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="n">b</span><span class="p">)))</span>
-
-    <span class="n">P</span> <span class="o">=</span> <span class="n">p</span> <span class="o">=</span> <span class="mi">2</span>
-    <span class="n">primes</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">while</span> <span class="n">P</span> <span class="o">&lt;=</span> <span class="n">U</span><span class="p">:</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">nextprime</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-        <span class="n">primes</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-        <span class="n">P</span> <span class="o">*=</span> <span class="n">p</span>
-
-    <span class="n">P</span> <span class="o">//=</span> <span class="n">p</span>
-    <span class="n">b</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">primes</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-        <span class="n">b</span> <span class="o">*=</span> <span class="n">p</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="n">U</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">ceil</span><span class="p">(</span><span class="n">m</span><span class="o">*</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">P</span><span class="p">)</span><span class="o">/</span><span class="n">b</span><span class="p">)))</span>
-
-    <span class="k">return</span> <span class="n">L</span><span class="p">,</span> <span class="n">U</span>
-
-
-<span class="k">def</span> <span class="nf">roots_cyclotomic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">factor</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Compute roots of cyclotomic polynomials. &quot;&quot;&quot;</span>
-    <span class="n">L</span><span class="p">,</span> <span class="n">U</span> <span class="o">=</span> <span class="n">_inv_totient_estimate</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">degree</span><span class="p">())</span>
-
-    <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="n">U</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">cyclotomic_poly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">gen</span><span class="p">,</span> <span class="n">polys</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">f</span> <span class="o">==</span> <span class="n">g</span><span class="p">:</span>
-            <span class="k">break</span>
-    <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-        <span class="k">raise</span> <span class="ne">RuntimeError</span><span class="p">(</span><span class="s">&quot;failed to find index of a cyclotomic polynomial&quot;</span><span class="p">)</span>
-
-    <span class="n">roots</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">factor</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">igcd</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">roots</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">*</span><span class="n">I</span><span class="o">/</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">extension</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-
-        <span class="k">for</span> <span class="n">h</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">g</span><span class="o">.</span><span class="n">factor_list</span><span class="p">()[</span><span class="mi">1</span><span class="p">]:</span>
-            <span class="n">roots</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">-</span><span class="n">h</span><span class="o">.</span><span class="n">TC</span><span class="p">())</span>
-
-    <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">roots</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">roots_rational</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns a list of rational roots of a polynomial.&quot;&quot;&quot;</span>
-    <span class="n">domain</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">get_domain</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="n">domain</span><span class="o">.</span><span class="n">is_QQ</span><span class="p">:</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">clear_denoms</span><span class="p">()</span>
-    <span class="k">elif</span> <span class="n">domain</span><span class="o">.</span><span class="n">is_ZZ</span><span class="p">:</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">set_domain</span><span class="p">(</span><span class="s">&#39;QQ&#39;</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-    <span class="n">LC_divs</span> <span class="o">=</span> <span class="n">divisors</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">LC</span><span class="p">()))</span>
-    <span class="n">EC_divs</span> <span class="o">=</span> <span class="n">divisors</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">EC</span><span class="p">()))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">):</span>
-        <span class="n">zeros</span> <span class="o">=</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">zeros</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">LC_divs</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">EC_divs</span><span class="p">:</span>
-            <span class="n">zero</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="n">zero</span><span class="p">):</span>
-                <span class="n">zeros</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">zero</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="o">-</span><span class="n">zero</span><span class="p">):</span>
-                <span class="n">zeros</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">-</span><span class="n">zero</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">zeros</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">_integer_basis</span><span class="p">(</span><span class="n">poly</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Compute coefficient basis for a polynomial over integers. &quot;&quot;&quot;</span>
-    <span class="n">monoms</span><span class="p">,</span> <span class="n">coeffs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">poly</span><span class="o">.</span><span class="n">terms</span><span class="p">()))</span>
-
-    <span class="n">monoms</span><span class="p">,</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">monoms</span><span class="p">))</span>
-    <span class="n">coeffs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">abs</span><span class="p">,</span> <span class="n">coeffs</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="n">coeffs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">coeffs</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-        <span class="n">coeffs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">coeffs</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="n">monoms</span> <span class="o">=</span> <span class="n">monoms</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-    <span class="n">coeffs</span> <span class="o">=</span> <span class="n">coeffs</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="n">divs</span> <span class="o">=</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">divisors</span><span class="p">(</span><span class="n">gcd_list</span><span class="p">(</span><span class="n">coeffs</span><span class="p">))[</span><span class="mi">1</span><span class="p">:])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">div</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">divs</span><span class="p">)</span>
-    <span class="k">except</span> <span class="ne">StopIteration</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">monoms</span><span class="p">,</span> <span class="n">coeffs</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">coeff</span> <span class="o">%</span> <span class="n">div</span><span class="o">**</span><span class="n">monom</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="n">div</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">divs</span><span class="p">)</span>
-                <span class="k">except</span> <span class="ne">StopIteration</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">div</span>
-
-
-<span class="k">def</span> <span class="nf">preprocess_roots</span><span class="p">(</span><span class="n">poly</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Try to get rid of symbolic coefficients from ``poly``. &quot;&quot;&quot;</span>
-    <span class="n">coeff</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">poly</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">clear_denoms</span><span class="p">(</span><span class="n">convert</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">DomainError</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">poly</span>
-
-    <span class="n">poly</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">primitive</span><span class="p">()[</span><span class="mi">1</span><span class="p">]</span>
-    <span class="n">poly</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">retract</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="n">poly</span><span class="o">.</span><span class="n">get_domain</span><span class="p">()</span><span class="o">.</span><span class="n">is_Poly</span> <span class="ow">and</span> <span class="nb">all</span><span class="p">(</span><span class="n">c</span><span class="o">.</span><span class="n">is_monomial</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">poly</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">coeffs</span><span class="p">()):</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">inject</span><span class="p">()</span>
-
-        <span class="n">strips</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">poly</span><span class="o">.</span><span class="n">monoms</span><span class="p">()))</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">poly</span><span class="o">.</span><span class="n">gens</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-
-        <span class="n">base</span><span class="p">,</span> <span class="n">strips</span> <span class="o">=</span> <span class="n">strips</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">strips</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-        <span class="k">for</span> <span class="n">gen</span><span class="p">,</span> <span class="n">strip</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">gens</span><span class="p">),</span> <span class="n">strips</span><span class="p">):</span>
-            <span class="n">reverse</span> <span class="o">=</span> <span class="bp">False</span>
-
-            <span class="k">if</span> <span class="n">strip</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">strip</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-                <span class="n">strip</span> <span class="o">=</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">strip</span><span class="p">)</span>
-                <span class="n">reverse</span> <span class="o">=</span> <span class="bp">True</span>
-
-            <span class="n">ratio</span> <span class="o">=</span> <span class="bp">None</span>
-
-            <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">strip</span><span class="p">):</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">a</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">b</span><span class="p">:</span>
-                    <span class="k">continue</span>
-                <span class="k">elif</span> <span class="ow">not</span> <span class="n">a</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">b</span><span class="p">:</span>
-                    <span class="k">break</span>
-                <span class="k">elif</span> <span class="n">b</span> <span class="o">%</span> <span class="n">a</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">break</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">_ratio</span> <span class="o">=</span> <span class="n">b</span> <span class="o">//</span> <span class="n">a</span>
-
-                    <span class="k">if</span> <span class="n">ratio</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="n">ratio</span> <span class="o">=</span> <span class="n">_ratio</span>
-                    <span class="k">elif</span> <span class="n">ratio</span> <span class="o">!=</span> <span class="n">_ratio</span><span class="p">:</span>
-                        <span class="k">break</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">reverse</span><span class="p">:</span>
-                    <span class="n">ratio</span> <span class="o">=</span> <span class="o">-</span><span class="n">ratio</span>
-
-                <span class="n">poly</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="n">gen</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-                <span class="n">coeff</span> <span class="o">*=</span> <span class="n">gen</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">ratio</span><span class="p">)</span>
-                <span class="n">gens</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">gen</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">gens</span><span class="p">:</span>
-            <span class="n">poly</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">eject</span><span class="p">(</span><span class="o">*</span><span class="n">gens</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">poly</span><span class="o">.</span><span class="n">is_univariate</span> <span class="ow">and</span> <span class="n">poly</span><span class="o">.</span><span class="n">get_domain</span><span class="p">()</span><span class="o">.</span><span class="n">is_ZZ</span><span class="p">:</span>
-        <span class="n">basis</span> <span class="o">=</span> <span class="n">_integer_basis</span><span class="p">(</span><span class="n">poly</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">basis</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span>
-
-            <span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">coeff</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">coeff</span><span class="o">//</span><span class="n">basis</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-            <span class="n">poly</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">termwise</span><span class="p">(</span><span class="n">func</span><span class="p">)</span>
-            <span class="n">coeff</span> <span class="o">*=</span> <span class="n">basis</span>
-
-    <span class="k">return</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">poly</span>
-
-
-<div class="viewcode-block" id="roots"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polyroots.roots">[docs]</a><span class="k">def</span> <span class="nf">roots</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes symbolic roots of a univariate polynomial.</span>
-
-<span class="sd">    Given a univariate polynomial f with symbolic coefficients (or</span>
-<span class="sd">    a list of the polynomial&#39;s coefficients), returns a dictionary</span>
-<span class="sd">    with its roots and their multiplicities.</span>
-
-<span class="sd">    Only roots expressible via radicals will be returned.  To get</span>
-<span class="sd">    a complete set of roots use RootOf class or numerical methods</span>
-<span class="sd">    instead. By default cubic and quartic formulas are used in</span>
-<span class="sd">    the algorithm. To disable them because of unreadable output</span>
-<span class="sd">    set ``cubics=False`` or ``quartics=False`` respectively.</span>
-
-<span class="sd">    To get roots from a specific domain set the ``filter`` flag with</span>
-<span class="sd">    one of the following specifiers: Z, Q, R, I, C. By default all</span>
-<span class="sd">    roots are returned (this is equivalent to setting ``filter=&#39;C&#39;``).</span>
-
-<span class="sd">    By default a dictionary is returned giving a compact result in</span>
-<span class="sd">    case of multiple roots.  However to get a tuple containing all</span>
-<span class="sd">    those roots set the ``multiple`` flag to True.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Poly, roots</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">    &gt;&gt;&gt; roots(x**2 - 1, x)</span>
-<span class="sd">    {-1: 1, 1: 1}</span>
-
-<span class="sd">    &gt;&gt;&gt; p = Poly(x**2-1, x)</span>
-<span class="sd">    &gt;&gt;&gt; roots(p)</span>
-<span class="sd">    {-1: 1, 1: 1}</span>
-
-<span class="sd">    &gt;&gt;&gt; p = Poly(x**2-y, x, y)</span>
-
-<span class="sd">    &gt;&gt;&gt; roots(Poly(p, x))</span>
-<span class="sd">    {-sqrt(y): 1, sqrt(y): 1}</span>
-
-<span class="sd">    &gt;&gt;&gt; roots(x**2 - y, x)</span>
-<span class="sd">    {-sqrt(y): 1, sqrt(y): 1}</span>
-
-<span class="sd">    &gt;&gt;&gt; roots([1, 0, -1])</span>
-<span class="sd">    {-1: 1, 1: 1}</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">flags</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">flags</span><span class="p">)</span>
-
-    <span class="n">auto</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;auto&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-    <span class="n">cubics</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;cubics&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-    <span class="n">quartics</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;quartics&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-    <span class="n">multiple</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;multiple&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-    <span class="nb">filter</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;filter&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-    <span class="n">predicate</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;predicate&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="nb">list</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">gens</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;redundant generators given&#39;</span><span class="p">)</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-
-        <span class="n">poly</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="p">{},</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-            <span class="n">poly</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">i</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">coeff</span><span class="p">),</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="n">f</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">field</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">GeneratorsNeeded</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">multiple</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">{}</span>
-
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_multivariate</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span><span class="s">&#39;multivariate polynomials are not supported&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_update_dict</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">root</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">root</span> <span class="ow">in</span> <span class="n">result</span><span class="p">:</span>
-            <span class="n">result</span><span class="p">[</span><span class="n">root</span><span class="p">]</span> <span class="o">+=</span> <span class="n">k</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">result</span><span class="p">[</span><span class="n">root</span><span class="p">]</span> <span class="o">=</span> <span class="n">k</span>
-
-    <span class="k">def</span> <span class="nf">_try_decompose</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Find roots using functional decomposition. &quot;&quot;&quot;</span>
-        <span class="n">factors</span><span class="p">,</span> <span class="n">roots</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">decompose</span><span class="p">(),</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">root</span> <span class="ow">in</span> <span class="n">_try_heuristics</span><span class="p">(</span><span class="n">factors</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
-            <span class="n">roots</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">factor</span> <span class="ow">in</span> <span class="n">factors</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="n">previous</span><span class="p">,</span> <span class="n">roots</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">roots</span><span class="p">),</span> <span class="p">[]</span>
-
-            <span class="k">for</span> <span class="n">root</span> <span class="ow">in</span> <span class="n">previous</span><span class="p">:</span>
-                <span class="n">g</span> <span class="o">=</span> <span class="n">factor</span> <span class="o">-</span> <span class="n">Poly</span><span class="p">(</span><span class="n">root</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">gen</span><span class="p">)</span>
-
-                <span class="k">for</span> <span class="n">root</span> <span class="ow">in</span> <span class="n">_try_heuristics</span><span class="p">(</span><span class="n">g</span><span class="p">):</span>
-                    <span class="n">roots</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">roots</span>
-
-    <span class="k">def</span> <span class="nf">_try_heuristics</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Find roots using formulas and some tricks. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_ground</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[]</span>
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_monomial</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)]</span><span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">length</span><span class="p">()</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">cancel</span><span class="p">,</span> <span class="n">roots_linear</span><span class="p">(</span><span class="n">f</span><span class="p">)))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">roots_binomial</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
-                <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">gen</span> <span class="o">-</span> <span class="n">i</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">gen</span><span class="p">))</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                <span class="k">break</span>
-
-        <span class="n">n</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">+=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">cancel</span><span class="p">,</span> <span class="n">roots_linear</span><span class="p">(</span><span class="n">f</span><span class="p">)))</span>
-        <span class="k">elif</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">+=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">cancel</span><span class="p">,</span> <span class="n">roots_quadratic</span><span class="p">(</span><span class="n">f</span><span class="p">)))</span>
-        <span class="k">elif</span> <span class="n">f</span><span class="o">.</span><span class="n">is_cyclotomic</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">+=</span> <span class="n">roots_cyclotomic</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">3</span> <span class="ow">and</span> <span class="n">cubics</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">+=</span> <span class="n">roots_cubic</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">4</span> <span class="ow">and</span> <span class="n">quartics</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">+=</span> <span class="n">roots_quartic</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="p">(</span><span class="n">k</span><span class="p">,),</span> <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">terms_gcd</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">k</span><span class="p">:</span>
-        <span class="n">zeros</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">zeros</span> <span class="o">=</span> <span class="p">{</span><span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">):</span> <span class="n">k</span><span class="p">}</span>
-
-    <span class="n">coeff</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">preprocess_roots</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">auto</span> <span class="ow">and</span> <span class="n">f</span><span class="o">.</span><span class="n">get_domain</span><span class="p">()</span><span class="o">.</span><span class="n">has_Ring</span><span class="p">:</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">to_field</span><span class="p">()</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="p">{}</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">is_ground</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">get_domain</span><span class="p">()</span><span class="o">.</span><span class="n">is_Exact</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">nroots</span><span class="p">():</span>
-                <span class="n">_update_dict</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">f</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">result</span><span class="p">[</span><span class="n">roots_linear</span><span class="p">(</span><span class="n">f</span><span class="p">)[</span><span class="mi">0</span><span class="p">]]</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">elif</span> <span class="n">f</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">roots_quadratic</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-                <span class="n">_update_dict</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">f</span><span class="o">.</span><span class="n">length</span><span class="p">()</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">roots_binomial</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-                <span class="n">_update_dict</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">_</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">())</span><span class="o">.</span><span class="n">factor_list</span><span class="p">()</span>
-
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">factors</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">root</span> <span class="ow">in</span> <span class="n">_try_decompose</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-                    <span class="n">_update_dict</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">root</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">factor</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">factors</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">_try_heuristics</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">factor</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">gen</span><span class="p">,</span> <span class="n">field</span><span class="o">=</span><span class="bp">True</span><span class="p">)):</span>
-                        <span class="n">_update_dict</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-        <span class="n">_result</span><span class="p">,</span> <span class="n">result</span><span class="p">,</span> <span class="o">=</span> <span class="n">result</span><span class="p">,</span> <span class="p">{}</span>
-
-        <span class="k">for</span> <span class="n">root</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">_result</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">result</span><span class="p">[</span><span class="n">coeff</span><span class="o">*</span><span class="n">root</span><span class="p">]</span> <span class="o">=</span> <span class="n">k</span>
-
-    <span class="n">result</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">zeros</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">filter</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">[</span><span class="bp">None</span><span class="p">,</span> <span class="s">&#39;C&#39;</span><span class="p">]:</span>
-        <span class="n">handlers</span> <span class="o">=</span> <span class="p">{</span>
-            <span class="s">&#39;Z&#39;</span><span class="p">:</span> <span class="k">lambda</span> <span class="n">r</span><span class="p">:</span> <span class="n">r</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">,</span>
-            <span class="s">&#39;Q&#39;</span><span class="p">:</span> <span class="k">lambda</span> <span class="n">r</span><span class="p">:</span> <span class="n">r</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">,</span>
-            <span class="s">&#39;R&#39;</span><span class="p">:</span> <span class="k">lambda</span> <span class="n">r</span><span class="p">:</span> <span class="n">r</span><span class="o">.</span><span class="n">is_real</span><span class="p">,</span>
-            <span class="s">&#39;I&#39;</span><span class="p">:</span> <span class="k">lambda</span> <span class="n">r</span><span class="p">:</span> <span class="n">r</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">,</span>
-        <span class="p">}</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">query</span> <span class="o">=</span> <span class="n">handlers</span><span class="p">[</span><span class="nb">filter</span><span class="p">]</span>
-        <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Invalid filter: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">filter</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">zero</span> <span class="ow">in</span> <span class="nb">dict</span><span class="p">(</span><span class="n">result</span><span class="p">)</span><span class="o">.</span><span class="n">keys</span><span class="p">():</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">query</span><span class="p">(</span><span class="n">zero</span><span class="p">):</span>
-                <span class="k">del</span> <span class="n">result</span><span class="p">[</span><span class="n">zero</span><span class="p">]</span>
-
-    <span class="k">if</span> <span class="n">predicate</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">zero</span> <span class="ow">in</span> <span class="nb">dict</span><span class="p">(</span><span class="n">result</span><span class="p">)</span><span class="o">.</span><span class="n">keys</span><span class="p">():</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">predicate</span><span class="p">(</span><span class="n">zero</span><span class="p">):</span>
-                <span class="k">del</span> <span class="n">result</span><span class="p">[</span><span class="n">zero</span><span class="p">]</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">multiple</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">zeros</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">zero</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">result</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">zeros</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">zero</span><span class="p">]</span><span class="o">*</span><span class="n">k</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">zeros</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">root_factors</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns all factors of a univariate polynomial.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.polyroots import root_factors</span>
-
-<span class="sd">    &gt;&gt;&gt; root_factors(x**2 - y, x)</span>
-<span class="sd">    [x - sqrt(y), x + sqrt(y)]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">args</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-    <span class="nb">filter</span> <span class="o">=</span> <span class="n">args</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;filter&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-
-    <span class="n">F</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">F</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">f</span><span class="p">]</span>
-
-    <span class="k">if</span> <span class="n">F</span><span class="o">.</span><span class="n">is_multivariate</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;multivariate polynomials not supported&#39;</span><span class="p">)</span>
-
-    <span class="n">x</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">gens</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="n">zeros</span> <span class="o">=</span> <span class="n">roots</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="nb">filter</span><span class="o">=</span><span class="nb">filter</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">zeros</span><span class="p">:</span>
-        <span class="n">factors</span> <span class="o">=</span> <span class="p">[</span><span class="n">F</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">factors</span><span class="p">,</span> <span class="n">N</span> <span class="o">=</span> <span class="p">[],</span> <span class="mi">0</span>
-
-        <span class="k">for</span> <span class="n">r</span><span class="p">,</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">zeros</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">factors</span><span class="p">,</span> <span class="n">N</span> <span class="o">=</span> <span class="n">factors</span> <span class="o">+</span> <span class="p">[</span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">r</span><span class="p">,</span> <span class="n">x</span><span class="p">)]</span><span class="o">*</span><span class="n">n</span><span class="p">,</span> <span class="n">N</span> <span class="o">+</span> <span class="n">n</span>
-
-        <span class="k">if</span> <span class="n">N</span> <span class="o">&lt;</span> <span class="n">F</span><span class="o">.</span><span class="n">degree</span><span class="p">():</span>
-            <span class="n">G</span> <span class="o">=</span> <span class="nb">reduce</span><span class="p">(</span><span class="k">lambda</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">:</span> <span class="n">p</span><span class="o">*</span><span class="n">q</span><span class="p">,</span> <span class="n">factors</span><span class="p">)</span>
-            <span class="n">factors</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">F</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">G</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">Poly</span><span class="p">):</span>
-        <span class="n">factors</span> <span class="o">=</span> <span class="p">[</span> <span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">factors</span> <span class="p">]</span>
-
-    <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">factors</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-</pre></div>
-
-          </div>
-        </div>
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deleted file mode 100644
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+++ /dev/null
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-            
-  <h1>Source code for sympy.polys.polytools</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;User-friendly public interface to polynomial functions. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">S</span><span class="p">,</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">Integer</span><span class="p">,</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">,</span> <span class="n">Rational</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.mul</span> <span class="kn">import</span> <span class="n">_keep_coeff</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">sympify</span><span class="p">,</span> <span class="n">SympifyError</span><span class="p">,</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.decorators</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">_sympifyit</span><span class="p">,</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyclasses</span> <span class="kn">import</span> <span class="n">DMP</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyutils</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">basic_from_dict</span><span class="p">,</span>
-    <span class="n">_sort_gens</span><span class="p">,</span>
-    <span class="n">_unify_gens</span><span class="p">,</span>
-    <span class="n">_dict_reorder</span><span class="p">,</span>
-    <span class="n">_dict_from_expr</span><span class="p">,</span>
-    <span class="n">_parallel_dict_from_expr</span><span class="p">,</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.rationaltools</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">together</span><span class="p">,</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.rootisolation</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dup_isolate_real_roots_list</span><span class="p">,</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.distributedpolys</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">sdp_from_dict</span><span class="p">,</span> <span class="n">sdp_div</span><span class="p">,</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.groebnertools</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">sdp_groebner</span><span class="p">,</span> <span class="n">matrix_fglm</span><span class="p">,</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.monomialtools</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">Monomial</span><span class="p">,</span> <span class="n">monomial_key</span><span class="p">,</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">OperationNotSupported</span><span class="p">,</span> <span class="n">DomainError</span><span class="p">,</span>
-    <span class="n">CoercionFailed</span><span class="p">,</span> <span class="n">UnificationFailed</span><span class="p">,</span>
-    <span class="n">GeneratorsNeeded</span><span class="p">,</span> <span class="n">PolynomialError</span><span class="p">,</span>
-    <span class="n">MultivariatePolynomialError</span><span class="p">,</span>
-    <span class="n">ExactQuotientFailed</span><span class="p">,</span>
-    <span class="n">PolificationFailed</span><span class="p">,</span>
-    <span class="n">ComputationFailed</span><span class="p">,</span>
-    <span class="n">GeneratorsError</span><span class="p">,</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">group</span>
-
-<span class="kn">import</span> <span class="nn">sympy.polys</span>
-<span class="kn">import</span> <span class="nn">sympy.mpmath</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">FF</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.constructor</span> <span class="kn">import</span> <span class="n">construct_domain</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">polyoptions</span> <span class="k">as</span> <span class="n">options</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">iterable</span>
-
-
-<div class="viewcode-block" id="Poly"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly">[docs]</a><span class="k">class</span> <span class="nc">Poly</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Generic class for representing polynomial expressions. &quot;&quot;&quot;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;rep&#39;</span><span class="p">,</span> <span class="s">&#39;gens&#39;</span><span class="p">]</span>
-
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_Poly</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Create a new polynomial instance out of something useful. &quot;&quot;&quot;</span>
-        <span class="n">opt</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">build_options</span><span class="p">(</span><span class="n">gens</span><span class="p">,</span> <span class="n">args</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="s">&#39;order&#39;</span> <span class="ow">in</span> <span class="n">opt</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;&#39;order&#39; keyword is not implemented yet&quot;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">iterable</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="nb">str</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="nb">dict</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_from_dict</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_from_list</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">rep</span><span class="p">),</span> <span class="n">opt</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">rep</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">rep</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">rep</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_from_poly</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_from_expr</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Poly.new"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.new">[docs]</a>    <span class="k">def</span> <span class="nf">new</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Construct :class:`Poly` instance from raw representation. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">DMP</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span>
-                <span class="s">&quot;invalid polynomial representation: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">rep</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">rep</span><span class="o">.</span><span class="n">lev</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span><span class="s">&quot;invalid arguments: </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">gens</span><span class="p">))</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-
-        <span class="n">obj</span><span class="o">.</span><span class="n">rep</span> <span class="o">=</span> <span class="n">rep</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">gens</span> <span class="o">=</span> <span class="n">gens</span>
-
-        <span class="k">return</span> <span class="n">obj</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Poly.from_dict"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.from_dict">[docs]</a>    <span class="k">def</span> <span class="nf">from_dict</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Construct a polynomial from a ``dict``. &quot;&quot;&quot;</span>
-        <span class="n">opt</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">build_options</span><span class="p">(</span><span class="n">gens</span><span class="p">,</span> <span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_from_dict</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Poly.from_list"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.from_list">[docs]</a>    <span class="k">def</span> <span class="nf">from_list</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Construct a polynomial from a ``list``. &quot;&quot;&quot;</span>
-        <span class="n">opt</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">build_options</span><span class="p">(</span><span class="n">gens</span><span class="p">,</span> <span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_from_list</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Poly.from_poly"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.from_poly">[docs]</a>    <span class="k">def</span> <span class="nf">from_poly</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Construct a polynomial from a polynomial. &quot;&quot;&quot;</span>
-        <span class="n">opt</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">build_options</span><span class="p">(</span><span class="n">gens</span><span class="p">,</span> <span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_from_poly</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Poly.from_expr"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.from_expr">[docs]</a>    <span class="k">def</span> <span class="nf">from_expr</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Construct a polynomial from an expression. &quot;&quot;&quot;</span>
-        <span class="n">opt</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">build_options</span><span class="p">(</span><span class="n">gens</span><span class="p">,</span> <span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_from_expr</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-</div>
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_from_dict</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="n">opt</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Construct a polynomial from a ``dict``. &quot;&quot;&quot;</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">gens</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">gens</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">GeneratorsNeeded</span><span class="p">(</span>
-                <span class="s">&quot;can&#39;t initialize from &#39;dict&#39; without generators&quot;</span><span class="p">)</span>
-
-        <span class="n">level</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="n">domain</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span>
-
-        <span class="k">if</span> <span class="n">domain</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">domain</span><span class="p">,</span> <span class="n">rep</span> <span class="o">=</span> <span class="n">construct_domain</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">opt</span><span class="o">=</span><span class="n">opt</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">rep</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                <span class="n">rep</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">DMP</span><span class="o">.</span><span class="n">from_dict</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">level</span><span class="p">,</span> <span class="n">domain</span><span class="p">),</span> <span class="o">*</span><span class="n">gens</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_from_list</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="n">opt</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Construct a polynomial from a ``list``. &quot;&quot;&quot;</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">gens</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">gens</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">GeneratorsNeeded</span><span class="p">(</span>
-                <span class="s">&quot;can&#39;t initialize from &#39;list&#39; without generators&quot;</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">MultivariatePolynomialError</span><span class="p">(</span>
-                <span class="s">&quot;&#39;list&#39; representation not supported&quot;</span><span class="p">)</span>
-
-        <span class="n">level</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="n">domain</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span>
-
-        <span class="k">if</span> <span class="n">domain</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">domain</span><span class="p">,</span> <span class="n">rep</span> <span class="o">=</span> <span class="n">construct_domain</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">opt</span><span class="o">=</span><span class="n">opt</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">rep</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">domain</span><span class="o">.</span><span class="n">convert</span><span class="p">,</span> <span class="n">rep</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">DMP</span><span class="o">.</span><span class="n">from_list</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">level</span><span class="p">,</span> <span class="n">domain</span><span class="p">),</span> <span class="o">*</span><span class="n">gens</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_from_poly</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="n">opt</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Construct a polynomial from a polynomial. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">cls</span> <span class="o">!=</span> <span class="n">rep</span><span class="o">.</span><span class="n">__class__</span><span class="p">:</span>
-            <span class="n">rep</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">rep</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="o">*</span><span class="n">rep</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-
-        <span class="n">gens</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">gens</span>
-        <span class="n">field</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">field</span>
-        <span class="n">domain</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span>
-
-        <span class="k">if</span> <span class="n">gens</span> <span class="ow">and</span> <span class="n">rep</span><span class="o">.</span><span class="n">gens</span> <span class="o">!=</span> <span class="n">gens</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">set</span><span class="p">(</span><span class="n">rep</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">set</span><span class="p">(</span><span class="n">gens</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_from_expr</span><span class="p">(</span><span class="n">rep</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(),</span> <span class="n">opt</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">rep</span> <span class="o">=</span> <span class="n">rep</span><span class="o">.</span><span class="n">reorder</span><span class="p">(</span><span class="o">*</span><span class="n">gens</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="s">&#39;domain&#39;</span> <span class="ow">in</span> <span class="n">opt</span> <span class="ow">and</span> <span class="n">domain</span><span class="p">:</span>
-            <span class="n">rep</span> <span class="o">=</span> <span class="n">rep</span><span class="o">.</span><span class="n">set_domain</span><span class="p">(</span><span class="n">domain</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">field</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">rep</span> <span class="o">=</span> <span class="n">rep</span><span class="o">.</span><span class="n">to_field</span><span class="p">()</span>
-
-        <span class="k">return</span> <span class="n">rep</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_from_expr</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="n">opt</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Construct a polynomial from an expression. &quot;&quot;&quot;</span>
-        <span class="n">rep</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">_dict_from_expr</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_from_dict</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_hashable_content</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Allow SymPy to hash Poly instances. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">Poly</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Poly.free_symbols"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.free_symbols">[docs]</a>    <span class="k">def</span> <span class="nf">free_symbols</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Free symbols of a polynomial expression.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1).free_symbols</span>
-<span class="sd">        set([x])</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + y).free_symbols</span>
-<span class="sd">        set([x, y])</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + y, x).free_symbols</span>
-<span class="sd">        set([x, y])</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([])</span>
-
-        <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">:</span>
-            <span class="n">symbols</span> <span class="o">|=</span> <span class="n">gen</span><span class="o">.</span><span class="n">free_symbols</span>
-
-        <span class="k">return</span> <span class="n">symbols</span> <span class="o">|</span> <span class="bp">self</span><span class="o">.</span><span class="n">free_symbols_in_domain</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Poly.free_symbols_in_domain"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.free_symbols_in_domain">[docs]</a>    <span class="k">def</span> <span class="nf">free_symbols_in_domain</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Free symbols of the domain of ``self``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1).free_symbols_in_domain</span>
-<span class="sd">        set()</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + y).free_symbols_in_domain</span>
-<span class="sd">        set()</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + y, x).free_symbols_in_domain</span>
-<span class="sd">        set([y])</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">domain</span><span class="p">,</span> <span class="n">symbols</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="nb">set</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="n">domain</span><span class="o">.</span><span class="n">is_Composite</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">domain</span><span class="o">.</span><span class="n">gens</span><span class="p">:</span>
-                <span class="n">symbols</span> <span class="o">|=</span> <span class="n">gen</span><span class="o">.</span><span class="n">free_symbols</span>
-        <span class="k">elif</span> <span class="n">domain</span><span class="o">.</span><span class="n">is_EX</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">coeffs</span><span class="p">():</span>
-                <span class="n">symbols</span> <span class="o">|=</span> <span class="n">coeff</span><span class="o">.</span><span class="n">free_symbols</span>
-
-        <span class="k">return</span> <span class="n">symbols</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Poly.args"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.args">[docs]</a>    <span class="k">def</span> <span class="nf">args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Don&#39;t mess up with the core.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, x).args</span>
-<span class="sd">        (x**2 + 1,)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(),)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Poly.gen"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.gen">[docs]</a>    <span class="k">def</span> <span class="nf">gen</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the principal generator.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, x).gen</span>
-<span class="sd">        x</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Poly.domain"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.domain">[docs]</a>    <span class="k">def</span> <span class="nf">domain</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Get the ground domain of ``self``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_domain</span><span class="p">()</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Poly.zero"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.zero">[docs]</a>    <span class="k">def</span> <span class="nf">zero</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return zero polynomial with ``self``&#39;s properties. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">zero</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="p">),</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Poly.one"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.one">[docs]</a>    <span class="k">def</span> <span class="nf">one</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return one polynomial with ``self``&#39;s properties. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">one</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="p">),</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Poly.unit"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.unit">[docs]</a>    <span class="k">def</span> <span class="nf">unit</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return unit polynomial with ``self``&#39;s properties. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">unit</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">lev</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="p">),</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.unify"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.unify">[docs]</a>    <span class="k">def</span> <span class="nf">unify</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Make ``f`` and ``g`` belong to the same domain.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; f, g = Poly(x/2 + 1), Poly(2*x + 1)</span>
-
-<span class="sd">        &gt;&gt;&gt; f</span>
-<span class="sd">        Poly(1/2*x + 1, x, domain=&#39;QQ&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; g</span>
-<span class="sd">        Poly(2*x + 1, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; F, G = f.unify(g)</span>
-
-<span class="sd">        &gt;&gt;&gt; F</span>
-<span class="sd">        Poly(1/2*x + 1, x, domain=&#39;QQ&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; G</span>
-<span class="sd">        Poly(2*x + 1, x, domain=&#39;QQ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">F</span><span class="p">),</span> <span class="n">per</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_unify</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">from_sympy</span><span class="p">(</span><span class="n">g</span><span class="p">))</span>
-            <span class="k">except</span> <span class="n">CoercionFailed</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">UnificationFailed</span><span class="p">(</span><span class="s">&quot;can&#39;t unify </span><span class="si">%s</span><span class="s"> with </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">DMP</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">DMP</span><span class="p">):</span>
-            <span class="n">gens</span> <span class="o">=</span> <span class="n">_unify_gens</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">,</span> <span class="n">g</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-
-            <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">gens</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-
-            <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span> <span class="o">!=</span> <span class="n">gens</span><span class="p">:</span>
-                <span class="n">f_monoms</span><span class="p">,</span> <span class="n">f_coeffs</span> <span class="o">=</span> <span class="n">_dict_reorder</span><span class="p">(</span>
-                    <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">to_dict</span><span class="p">(),</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">,</span> <span class="n">gens</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span> <span class="o">!=</span> <span class="n">dom</span><span class="p">:</span>
-                    <span class="n">f_coeffs</span> <span class="o">=</span> <span class="p">[</span> <span class="n">dom</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f_coeffs</span> <span class="p">]</span>
-
-                <span class="n">F</span> <span class="o">=</span> <span class="n">DMP</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">f_monoms</span><span class="p">,</span> <span class="n">f_coeffs</span><span class="p">))),</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">F</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">dom</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">g</span><span class="o">.</span><span class="n">gens</span> <span class="o">!=</span> <span class="n">gens</span><span class="p">:</span>
-                <span class="n">g_monoms</span><span class="p">,</span> <span class="n">g_coeffs</span> <span class="o">=</span> <span class="n">_dict_reorder</span><span class="p">(</span>
-                    <span class="n">g</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">to_dict</span><span class="p">(),</span> <span class="n">g</span><span class="o">.</span><span class="n">gens</span><span class="p">,</span> <span class="n">gens</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">g</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span> <span class="o">!=</span> <span class="n">dom</span><span class="p">:</span>
-                    <span class="n">g_coeffs</span> <span class="o">=</span> <span class="p">[</span> <span class="n">dom</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">g</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">g_coeffs</span> <span class="p">]</span>
-
-                <span class="n">G</span> <span class="o">=</span> <span class="n">DMP</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">g_monoms</span><span class="p">,</span> <span class="n">g_coeffs</span><span class="p">))),</span> <span class="n">dom</span><span class="p">,</span> <span class="n">lev</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">G</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">dom</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">UnificationFailed</span><span class="p">(</span><span class="s">&quot;can&#39;t unify </span><span class="si">%s</span><span class="s"> with </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">))</span>
-
-        <span class="n">cls</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span>
-
-        <span class="k">def</span> <span class="nf">per</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">dom</span><span class="o">=</span><span class="n">dom</span><span class="p">,</span> <span class="n">gens</span><span class="o">=</span><span class="n">gens</span><span class="p">,</span> <span class="n">remove</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">remove</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">gens</span> <span class="o">=</span> <span class="n">gens</span><span class="p">[:</span><span class="n">remove</span><span class="p">]</span> <span class="o">+</span> <span class="n">gens</span><span class="p">[</span><span class="n">remove</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">gens</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">rep</span><span class="p">)</span>
-
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span>
-
-<div class="viewcode-block" id="Poly.per"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.per">[docs]</a>    <span class="k">def</span> <span class="nf">per</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="n">gens</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">remove</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Create a Poly out of the given representation.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly, ZZ</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.polys.polyclasses import DMP</span>
-
-<span class="sd">        &gt;&gt;&gt; a = Poly(x**2 + 1)</span>
-
-<span class="sd">        &gt;&gt;&gt; a.per(DMP([ZZ(1), ZZ(1)], ZZ), gens=[y])</span>
-<span class="sd">        Poly(y + 1, y, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">gens</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">gens</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span>
-
-        <span class="k">if</span> <span class="n">remove</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">gens</span> <span class="o">=</span> <span class="n">gens</span><span class="p">[:</span><span class="n">remove</span><span class="p">]</span> <span class="o">+</span> <span class="n">gens</span><span class="p">[</span><span class="n">remove</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">gens</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">rep</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.set_domain"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.set_domain">[docs]</a>    <span class="k">def</span> <span class="nf">set_domain</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">domain</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Set the ground domain of ``f``. &quot;&quot;&quot;</span>
-        <span class="n">opt</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">build_options</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">,</span> <span class="p">{</span><span class="s">&#39;domain&#39;</span><span class="p">:</span> <span class="n">domain</span><span class="p">})</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">opt</span><span class="o">.</span><span class="n">domain</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="Poly.get_domain"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.get_domain">[docs]</a>    <span class="k">def</span> <span class="nf">get_domain</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Get the ground domain of ``f``. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span>
-</div>
-<div class="viewcode-block" id="Poly.set_modulus"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.set_modulus">[docs]</a>    <span class="k">def</span> <span class="nf">set_modulus</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">modulus</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Set the modulus of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(5*x**2 + 2*x - 1, x).set_modulus(2)</span>
-<span class="sd">        Poly(x**2 + 1, x, modulus=2)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">modulus</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">Modulus</span><span class="o">.</span><span class="n">preprocess</span><span class="p">(</span><span class="n">modulus</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">set_domain</span><span class="p">(</span><span class="n">FF</span><span class="p">(</span><span class="n">modulus</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="Poly.get_modulus"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.get_modulus">[docs]</a>    <span class="k">def</span> <span class="nf">get_modulus</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Get the modulus of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, modulus=2).get_modulus()</span>
-<span class="sd">        2</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">domain</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">get_domain</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">domain</span><span class="o">.</span><span class="n">has_CharacteristicZero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="n">domain</span><span class="o">.</span><span class="n">characteristic</span><span class="p">())</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span><span class="s">&quot;not a polynomial over a Galois field&quot;</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_eval_subs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Internal implementation of :func:`subs`. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">old</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">new</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">)</span>
-                <span class="k">except</span> <span class="n">PolynomialError</span><span class="p">:</span>
-                    <span class="k">pass</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Poly.exclude"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.exclude">[docs]</a>    <span class="k">def</span> <span class="nf">exclude</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Remove unnecessary generators from ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import a, b, c, d, x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(a + x, a, b, c, d, x).exclude()</span>
-<span class="sd">        Poly(a + x, a, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">J</span><span class="p">,</span> <span class="n">new</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">exclude</span><span class="p">()</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="n">j</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">J</span><span class="p">:</span>
-                <span class="n">gens</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">new</span><span class="p">,</span> <span class="n">gens</span><span class="o">=</span><span class="n">gens</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.replace"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.replace">[docs]</a>    <span class="k">def</span> <span class="nf">replace</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Replace ``x`` with ``y`` in generators list.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, x).replace(x, y)</span>
-<span class="sd">        Poly(y**2 + 1, y, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">y</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_univariate</span><span class="p">:</span>
-                <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">gen</span><span class="p">,</span> <span class="n">x</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span>
-                    <span class="s">&quot;syntax supported only in univariate case&quot;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="n">y</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span> <span class="ow">and</span> <span class="n">y</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">:</span>
-            <span class="n">dom</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">get_domain</span><span class="p">()</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">dom</span><span class="o">.</span><span class="n">is_Composite</span> <span class="ow">or</span> <span class="n">y</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">dom</span><span class="o">.</span><span class="n">gens</span><span class="p">:</span>
-                <span class="n">gens</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-                <span class="n">gens</span><span class="p">[</span><span class="n">gens</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">x</span><span class="p">)]</span> <span class="o">=</span> <span class="n">y</span>
-                <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">gens</span><span class="o">=</span><span class="n">gens</span><span class="p">)</span>
-
-        <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span><span class="s">&quot;can&#39;t replace </span><span class="si">%s</span><span class="s"> with </span><span class="si">%s</span><span class="s"> in </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">f</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="Poly.reorder"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.reorder">[docs]</a>    <span class="k">def</span> <span class="nf">reorder</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Efficiently apply new order of generators.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + x*y**2, x, y).reorder(y, x)</span>
-<span class="sd">        Poly(y**2*x + x**2, y, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">opt</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">Options</span><span class="p">((),</span> <span class="n">args</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">gens</span><span class="p">:</span>
-            <span class="n">gens</span> <span class="o">=</span> <span class="n">_sort_gens</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">,</span> <span class="n">opt</span><span class="o">=</span><span class="n">opt</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">set</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">set</span><span class="p">(</span><span class="n">gens</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span>
-                <span class="s">&quot;generators list can differ only up to order of elements&quot;</span><span class="p">)</span>
-
-        <span class="n">rep</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">_dict_reorder</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">to_dict</span><span class="p">(),</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">,</span> <span class="n">gens</span><span class="p">))))</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">DMP</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">gens</span><span class="o">=</span><span class="n">gens</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.ltrim"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.ltrim">[docs]</a>    <span class="k">def</span> <span class="nf">ltrim</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gen</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Remove dummy generators from the &quot;left&quot; of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(y**2 + y*z**2, x, y, z).ltrim(y)</span>
-<span class="sd">        Poly(y**2 + y*z**2, y, z, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">rep</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">as_dict</span><span class="p">(</span><span class="n">native</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_gen_to_level</span><span class="p">(</span><span class="n">gen</span><span class="p">)</span>
-        <span class="n">terms</span> <span class="o">=</span> <span class="p">{}</span>
-
-        <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">rep</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">monom</span> <span class="o">=</span> <span class="n">monom</span><span class="p">[</span><span class="n">j</span><span class="p">:]</span>
-
-            <span class="k">if</span> <span class="n">monom</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">:</span>
-                <span class="n">terms</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span> <span class="o">=</span> <span class="n">coeff</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span><span class="s">&quot;can&#39;t left trim </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">f</span><span class="p">)</span>
-
-        <span class="n">gens</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">[</span><span class="n">j</span><span class="p">:]</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">DMP</span><span class="o">.</span><span class="n">from_dict</span><span class="p">(</span><span class="n">terms</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="p">),</span> <span class="o">*</span><span class="n">gens</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.has_only_gens"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.has_only_gens">[docs]</a>    <span class="k">def</span> <span class="nf">has_only_gens</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return ``True`` if ``Poly(f, *gens)`` retains ground domain.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x*y + 1, x, y, z).has_only_gens(x, y)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x*y + z, x, y, z).has_only_gens(x, y)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">f_gens</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-        <span class="n">indices</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([])</span>
-
-        <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">index</span> <span class="o">=</span> <span class="n">f_gens</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">gen</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">GeneratorsError</span><span class="p">(</span>
-                    <span class="s">&quot;</span><span class="si">%s</span><span class="s"> doesn&#39;t have </span><span class="si">%s</span><span class="s"> as generator&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gen</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">indices</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">index</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">monom</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">monoms</span><span class="p">():</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">elt</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">monom</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">i</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">indices</span> <span class="ow">and</span> <span class="n">elt</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="k">return</span> <span class="bp">True</span>
-</div>
-<div class="viewcode-block" id="Poly.to_ring"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.to_ring">[docs]</a>    <span class="k">def</span> <span class="nf">to_ring</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Make the ground domain a ring.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly, QQ</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, domain=QQ).to_ring()</span>
-<span class="sd">        Poly(x**2 + 1, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;to_ring&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">to_ring</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;to_ring&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.to_field"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.to_field">[docs]</a>    <span class="k">def</span> <span class="nf">to_field</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Make the ground domain a field.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly, ZZ</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, x, domain=ZZ).to_field()</span>
-<span class="sd">        Poly(x**2 + 1, x, domain=&#39;QQ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;to_field&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">to_field</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;to_field&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.to_exact"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.to_exact">[docs]</a>    <span class="k">def</span> <span class="nf">to_exact</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Make the ground domain exact.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly, RR</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1.0, x, domain=RR).to_exact()</span>
-<span class="sd">        Poly(x**2 + 1, x, domain=&#39;QQ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;to_exact&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">to_exact</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;to_exact&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.retract"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.retract">[docs]</a>    <span class="k">def</span> <span class="nf">retract</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">field</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Recalculate the ground domain of a polynomial.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; f = Poly(x**2 + 1, x, domain=&#39;QQ[y]&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; f</span>
-<span class="sd">        Poly(x**2 + 1, x, domain=&#39;QQ[y]&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; f.retract()</span>
-<span class="sd">        Poly(x**2 + 1, x, domain=&#39;ZZ&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; f.retract(field=True)</span>
-<span class="sd">        Poly(x**2 + 1, x, domain=&#39;QQ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">dom</span><span class="p">,</span> <span class="n">rep</span> <span class="o">=</span> <span class="n">construct_domain</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">as_dict</span><span class="p">(</span><span class="n">zero</span><span class="o">=</span><span class="bp">True</span><span class="p">),</span> <span class="n">field</span><span class="o">=</span><span class="n">field</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">from_dict</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="n">dom</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.slice"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.slice">[docs]</a>    <span class="k">def</span> <span class="nf">slice</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Take a continuous subsequence of terms of ``f``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">m</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">j</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_gen_to_level</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">m</span><span class="p">),</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;slice&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">slice</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;slice&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.coeffs"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.coeffs">[docs]</a>    <span class="k">def</span> <span class="nf">coeffs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns all non-zero coefficients from ``f`` in lex order.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**3 + 2*x + 3, x).coeffs()</span>
-<span class="sd">        [1, 2, 3]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">coeffs</span><span class="p">(</span><span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">)</span> <span class="p">]</span>
-</div>
-<div class="viewcode-block" id="Poly.monoms"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.monoms">[docs]</a>    <span class="k">def</span> <span class="nf">monoms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns all non-zero monomials from ``f`` in lex order.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 2*x*y**2 + x*y + 3*y, x, y).monoms()</span>
-<span class="sd">        [(2, 0), (1, 2), (1, 1), (0, 1)]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">monoms</span><span class="p">(</span><span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.terms"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.terms">[docs]</a>    <span class="k">def</span> <span class="nf">terms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns all non-zero terms from ``f`` in lex order.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 2*x*y**2 + x*y + 3*y, x, y).terms()</span>
-<span class="sd">        [((2, 0), 1), ((1, 2), 2), ((1, 1), 1), ((0, 1), 3)]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">c</span><span class="p">))</span> <span class="k">for</span> <span class="n">m</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">terms</span><span class="p">(</span><span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">)</span> <span class="p">]</span>
-</div>
-<div class="viewcode-block" id="Poly.all_coeffs"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.all_coeffs">[docs]</a>    <span class="k">def</span> <span class="nf">all_coeffs</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns all coefficients from a univariate polynomial ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**3 + 2*x - 1, x).all_coeffs()</span>
-<span class="sd">        [1, 0, 2, -1]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()</span> <span class="p">]</span>
-</div>
-<div class="viewcode-block" id="Poly.all_monoms"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.all_monoms">[docs]</a>    <span class="k">def</span> <span class="nf">all_monoms</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns all monomials from a univariate polynomial ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**3 + 2*x - 1, x).all_monoms()</span>
-<span class="sd">        [(3,), (2,), (1,), (0,)]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">all_monoms</span><span class="p">()</span>
-</div>
-<div class="viewcode-block" id="Poly.all_terms"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.all_terms">[docs]</a>    <span class="k">def</span> <span class="nf">all_terms</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns all terms from a univariate polynomial ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**3 + 2*x - 1, x).all_terms()</span>
-<span class="sd">        [((3,), 1), ((2,), 0), ((1,), 2), ((0,), -1)]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">c</span><span class="p">))</span> <span class="k">for</span> <span class="n">m</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">all_terms</span><span class="p">()</span> <span class="p">]</span>
-</div>
-<div class="viewcode-block" id="Poly.termwise"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.termwise">[docs]</a>    <span class="k">def</span> <span class="nf">termwise</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Apply a function to all terms of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; def func(xxx_todo_changeme, coeff):</span>
-<span class="sd">        ...     (k,) = xxx_todo_changeme</span>
-<span class="sd">        ...     return coeff//10**(2-k)</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 20*x + 400).termwise(func)</span>
-<span class="sd">        Poly(x**2 + 2*x + 4, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">terms</span> <span class="o">=</span> <span class="p">{}</span>
-
-        <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">terms</span><span class="p">():</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span>
-                <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="o">=</span> <span class="n">result</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">coeff</span> <span class="o">=</span> <span class="n">result</span>
-
-            <span class="k">if</span> <span class="n">coeff</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">monom</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">:</span>
-                    <span class="n">terms</span><span class="p">[</span><span class="n">monom</span><span class="p">]</span> <span class="o">=</span> <span class="n">coeff</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span>
-                        <span class="s">&quot;</span><span class="si">%s</span><span class="s"> monomial was generated twice&quot;</span> <span class="o">%</span> <span class="n">monom</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">from_dict</span><span class="p">(</span><span class="n">terms</span><span class="p">,</span> <span class="o">*</span><span class="p">(</span><span class="n">gens</span> <span class="ow">or</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">),</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.length"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.length">[docs]</a>    <span class="k">def</span> <span class="nf">length</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the number of non-zero terms in ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 2*x - 1).length()</span>
-<span class="sd">        3</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">as_dict</span><span class="p">())</span>
-</div>
-<div class="viewcode-block" id="Poly.as_dict"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.as_dict">[docs]</a>    <span class="k">def</span> <span class="nf">as_dict</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">native</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">zero</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Switch to a ``dict`` representation.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 2*x*y**2 - y, x, y).as_dict()</span>
-<span class="sd">        {(0, 1): -1, (1, 2): 2, (2, 0): 1}</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">native</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">to_dict</span><span class="p">(</span><span class="n">zero</span><span class="o">=</span><span class="n">zero</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">to_sympy_dict</span><span class="p">(</span><span class="n">zero</span><span class="o">=</span><span class="n">zero</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.as_list"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.as_list">[docs]</a>    <span class="k">def</span> <span class="nf">as_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">native</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Switch to a ``list`` representation. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">native</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">to_list</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">to_sympy_list</span><span class="p">()</span>
-</div>
-<div class="viewcode-block" id="Poly.as_expr"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.as_expr">[docs]</a>    <span class="k">def</span> <span class="nf">as_expr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Convert a polynomial an expression.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; f = Poly(x**2 + 2*x*y**2 - y, x, y)</span>
-
-<span class="sd">        &gt;&gt;&gt; f.as_expr()</span>
-<span class="sd">        x**2 + 2*x*y**2 - y</span>
-<span class="sd">        &gt;&gt;&gt; f.as_expr({x: 5})</span>
-<span class="sd">        10*y**2 - y + 25</span>
-<span class="sd">        &gt;&gt;&gt; f.as_expr(5, 6)</span>
-<span class="sd">        379</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">gens</span><span class="p">:</span>
-            <span class="n">gens</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">gens</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="nb">dict</span><span class="p">):</span>
-            <span class="n">mapping</span> <span class="o">=</span> <span class="n">gens</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">gens</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">gen</span><span class="p">,</span> <span class="n">value</span> <span class="ow">in</span> <span class="n">mapping</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="n">index</span> <span class="o">=</span> <span class="n">gens</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">gen</span><span class="p">)</span>
-                <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="n">GeneratorsError</span><span class="p">(</span>
-                        <span class="s">&quot;</span><span class="si">%s</span><span class="s"> doesn&#39;t have </span><span class="si">%s</span><span class="s"> as generator&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gen</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">gens</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="o">=</span> <span class="n">value</span>
-
-        <span class="k">return</span> <span class="n">basic_from_dict</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">to_sympy_dict</span><span class="p">(),</span> <span class="o">*</span><span class="n">gens</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.lift"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.lift">[docs]</a>    <span class="k">def</span> <span class="nf">lift</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Convert algebraic coefficients to rationals.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly, I</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + I*x + 1, x, extension=I).lift()</span>
-<span class="sd">        Poly(x**4 + 3*x**2 + 1, x, domain=&#39;QQ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;lift&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">lift</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;lift&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.deflate"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.deflate">[docs]</a>    <span class="k">def</span> <span class="nf">deflate</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Reduce degree of ``f`` by mapping ``x_i**m`` to ``y_i``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**6*y**2 + x**3 + 1, x, y).deflate()</span>
-<span class="sd">        ((3, 2), Poly(x**2*y + x + 1, x, y, domain=&#39;ZZ&#39;))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;deflate&#39;</span><span class="p">):</span>
-            <span class="n">J</span><span class="p">,</span> <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">deflate</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;deflate&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">J</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.inject"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.inject">[docs]</a>    <span class="k">def</span> <span class="nf">inject</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">front</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Inject ground domain generators into ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; f = Poly(x**2*y + x*y**3 + x*y + 1, x)</span>
-
-<span class="sd">        &gt;&gt;&gt; f.inject()</span>
-<span class="sd">        Poly(x**2*y + x*y**3 + x*y + 1, x, y, domain=&#39;ZZ&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; f.inject(front=True)</span>
-<span class="sd">        Poly(y**3*x + y*x**2 + y*x + 1, y, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">dom</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span>
-
-        <span class="k">if</span> <span class="n">dom</span><span class="o">.</span><span class="n">is_Numerical</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="n">dom</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">DomainError</span><span class="p">(</span><span class="s">&quot;can&#39;t inject generators over </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">dom</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;inject&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">inject</span><span class="p">(</span><span class="n">front</span><span class="o">=</span><span class="n">front</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;inject&#39;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">front</span><span class="p">:</span>
-            <span class="n">gens</span> <span class="o">=</span> <span class="n">dom</span><span class="o">.</span><span class="n">gens</span> <span class="o">+</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">gens</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span> <span class="o">+</span> <span class="n">dom</span><span class="o">.</span><span class="n">gens</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.eject"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.eject">[docs]</a>    <span class="k">def</span> <span class="nf">eject</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Eject selected generators into the ground domain.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; f = Poly(x**2*y + x*y**3 + x*y + 1, x, y)</span>
-
-<span class="sd">        &gt;&gt;&gt; f.eject(x)</span>
-<span class="sd">        Poly(x*y**3 + (x**2 + x)*y + 1, y, domain=&#39;ZZ[x]&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; f.eject(y)</span>
-<span class="sd">        Poly(y*x**2 + (y**3 + y)*x + 1, x, domain=&#39;ZZ[y]&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">dom</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">dom</span><span class="o">.</span><span class="n">is_Numerical</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">DomainError</span><span class="p">(</span><span class="s">&quot;can&#39;t eject generators over </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">dom</span><span class="p">)</span>
-
-        <span class="n">n</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">[:</span><span class="n">k</span><span class="p">]</span> <span class="o">==</span> <span class="n">gens</span><span class="p">:</span>
-            <span class="n">_gens</span><span class="p">,</span> <span class="n">front</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">:],</span> <span class="bp">True</span>
-        <span class="k">elif</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">[</span><span class="o">-</span><span class="n">k</span><span class="p">:]</span> <span class="o">==</span> <span class="n">gens</span><span class="p">:</span>
-            <span class="n">_gens</span><span class="p">,</span> <span class="n">front</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">[:</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">],</span> <span class="bp">False</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                <span class="s">&quot;can only eject front or back generators&quot;</span><span class="p">)</span>
-
-        <span class="n">dom</span> <span class="o">=</span> <span class="n">dom</span><span class="o">.</span><span class="n">inject</span><span class="p">(</span><span class="o">*</span><span class="n">gens</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;eject&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">eject</span><span class="p">(</span><span class="n">dom</span><span class="p">,</span> <span class="n">front</span><span class="o">=</span><span class="n">front</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;eject&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="o">*</span><span class="n">_gens</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.terms_gcd"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.terms_gcd">[docs]</a>    <span class="k">def</span> <span class="nf">terms_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Remove GCD of terms from the polynomial ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**6*y**2 + x**3*y, x, y).terms_gcd()</span>
-<span class="sd">        ((3, 1), Poly(x**3*y + 1, x, y, domain=&#39;ZZ&#39;))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;terms_gcd&#39;</span><span class="p">):</span>
-            <span class="n">J</span><span class="p">,</span> <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">terms_gcd</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;terms_gcd&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">J</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.add_ground"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.add_ground">[docs]</a>    <span class="k">def</span> <span class="nf">add_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">coeff</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Add an element of the ground domain to ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x + 1).add_ground(2)</span>
-<span class="sd">        Poly(x + 3, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;add_ground&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">add_ground</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;add_ground&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.sub_ground"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.sub_ground">[docs]</a>    <span class="k">def</span> <span class="nf">sub_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">coeff</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Subtract an element of the ground domain from ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x + 1).sub_ground(2)</span>
-<span class="sd">        Poly(x - 1, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;sub_ground&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">sub_ground</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;sub_ground&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.mul_ground"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.mul_ground">[docs]</a>    <span class="k">def</span> <span class="nf">mul_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">coeff</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Multiply ``f`` by a an element of the ground domain.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x + 1).mul_ground(2)</span>
-<span class="sd">        Poly(2*x + 2, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;mul_ground&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">mul_ground</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;mul_ground&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.quo_ground"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.quo_ground">[docs]</a>    <span class="k">def</span> <span class="nf">quo_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">coeff</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Quotient of ``f`` by a an element of the ground domain.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(2*x + 4).quo_ground(2)</span>
-<span class="sd">        Poly(x + 2, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(2*x + 3).quo_ground(2)</span>
-<span class="sd">        Poly(x + 1, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;quo_ground&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">quo_ground</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;quo_ground&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.exquo_ground"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.exquo_ground">[docs]</a>    <span class="k">def</span> <span class="nf">exquo_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">coeff</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Exact quotient of ``f`` by a an element of the ground domain.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(2*x + 4).exquo_ground(2)</span>
-<span class="sd">        Poly(x + 2, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(2*x + 3).exquo_ground(2)</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">        ...</span>
-<span class="sd">        ExactQuotientFailed: 2 does not divide 3 in ZZ</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;exquo_ground&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">exquo_ground</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;exquo_ground&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.abs"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.abs">[docs]</a>    <span class="k">def</span> <span class="nf">abs</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Make all coefficients in ``f`` positive.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 - 1, x).abs()</span>
-<span class="sd">        Poly(x**2 + 1, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;abs&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">abs</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;abs&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.neg"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.neg">[docs]</a>    <span class="k">def</span> <span class="nf">neg</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Negate all coefficients in ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 - 1, x).neg()</span>
-<span class="sd">        Poly(-x**2 + 1, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; -Poly(x**2 - 1, x)</span>
-<span class="sd">        Poly(-x**2 + 1, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;neg&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">neg</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;neg&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.add"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.add">[docs]</a>    <span class="k">def</span> <span class="nf">add</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Add two polynomials ``f`` and ``g``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, x).add(Poly(x - 2, x))</span>
-<span class="sd">        Poly(x**2 + x - 1, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, x) + Poly(x - 2, x)</span>
-<span class="sd">        Poly(x**2 + x - 1, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">add_ground</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="n">_</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;add&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;add&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.sub"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.sub">[docs]</a>    <span class="k">def</span> <span class="nf">sub</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Subtract two polynomials ``f`` and ``g``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, x).sub(Poly(x - 2, x))</span>
-<span class="sd">        Poly(x**2 - x + 3, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, x) - Poly(x - 2, x)</span>
-<span class="sd">        Poly(x**2 - x + 3, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">sub_ground</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="n">_</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;sub&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">sub</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;sub&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.mul"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.mul">[docs]</a>    <span class="k">def</span> <span class="nf">mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Multiply two polynomials ``f`` and ``g``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, x).mul(Poly(x - 2, x))</span>
-<span class="sd">        Poly(x**3 - 2*x**2 + x - 2, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, x)*Poly(x - 2, x)</span>
-<span class="sd">        Poly(x**3 - 2*x**2 + x - 2, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">mul_ground</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="n">_</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;mul&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;mul&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.sqr"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.sqr">[docs]</a>    <span class="k">def</span> <span class="nf">sqr</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Square a polynomial ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x - 2, x).sqr()</span>
-<span class="sd">        Poly(x**2 - 4*x + 4, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x - 2, x)**2</span>
-<span class="sd">        Poly(x**2 - 4*x + 4, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;sqr&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">sqr</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;sqr&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.pow"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.pow">[docs]</a>    <span class="k">def</span> <span class="nf">pow</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Raise ``f`` to a non-negative power ``n``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x - 2, x).pow(3)</span>
-<span class="sd">        Poly(x**3 - 6*x**2 + 12*x - 8, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x - 2, x)**3</span>
-<span class="sd">        Poly(x**3 - 6*x**2 + 12*x - 8, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;pow&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;pow&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.pdiv"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.pdiv">[docs]</a>    <span class="k">def</span> <span class="nf">pdiv</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Polynomial pseudo-division of ``f`` by ``g``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, x).pdiv(Poly(2*x - 4, x))</span>
-<span class="sd">        (Poly(2*x + 4, x, domain=&#39;ZZ&#39;), Poly(20, x, domain=&#39;ZZ&#39;))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;pdiv&#39;</span><span class="p">):</span>
-            <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">pdiv</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;pdiv&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">q</span><span class="p">),</span> <span class="n">per</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.prem"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.prem">[docs]</a>    <span class="k">def</span> <span class="nf">prem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Polynomial pseudo-remainder of ``f`` by ``g``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, x).prem(Poly(2*x - 4, x))</span>
-<span class="sd">        Poly(20, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;prem&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">prem</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;prem&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.pquo"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.pquo">[docs]</a>    <span class="k">def</span> <span class="nf">pquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Polynomial pseudo-quotient of ``f`` by ``g``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, x).pquo(Poly(2*x - 4, x))</span>
-<span class="sd">        Poly(2*x + 4, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 - 1, x).pquo(Poly(2*x - 2, x))</span>
-<span class="sd">        Poly(2*x + 2, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;pquo&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">pquo</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;pquo&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.pexquo"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.pexquo">[docs]</a>    <span class="k">def</span> <span class="nf">pexquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Polynomial exact pseudo-quotient of ``f`` by ``g``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 - 1, x).pexquo(Poly(2*x - 2, x))</span>
-<span class="sd">        Poly(2*x + 2, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, x).pexquo(Poly(2*x - 4, x))</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">        ...</span>
-<span class="sd">        ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;pexquo&#39;</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">pexquo</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">ExactQuotientFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">exc</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(),</span> <span class="n">g</span><span class="o">.</span><span class="n">as_expr</span><span class="p">())</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;pexquo&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.div"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.div">[docs]</a>    <span class="k">def</span> <span class="nf">div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Polynomial division with remainder of ``f`` by ``g``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, x).div(Poly(2*x - 4, x))</span>
-<span class="sd">        (Poly(1/2*x + 1, x, domain=&#39;QQ&#39;), Poly(5, x, domain=&#39;QQ&#39;))</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, x).div(Poly(2*x - 4, x), auto=False)</span>
-<span class="sd">        (Poly(0, x, domain=&#39;ZZ&#39;), Poly(x**2 + 1, x, domain=&#39;ZZ&#39;))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="n">retract</span> <span class="o">=</span> <span class="bp">False</span>
-
-        <span class="k">if</span> <span class="n">auto</span> <span class="ow">and</span> <span class="n">dom</span><span class="o">.</span><span class="n">has_Ring</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">dom</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-            <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">to_field</span><span class="p">(),</span> <span class="n">G</span><span class="o">.</span><span class="n">to_field</span><span class="p">()</span>
-            <span class="n">retract</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;div&#39;</span><span class="p">):</span>
-            <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">div</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;div&#39;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">retract</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">Q</span><span class="p">,</span> <span class="n">R</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">to_ring</span><span class="p">(),</span> <span class="n">r</span><span class="o">.</span><span class="n">to_ring</span><span class="p">()</span>
-            <span class="k">except</span> <span class="n">CoercionFailed</span><span class="p">:</span>
-                <span class="k">pass</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">Q</span><span class="p">,</span> <span class="n">R</span>
-
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">q</span><span class="p">),</span> <span class="n">per</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.rem"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.rem">[docs]</a>    <span class="k">def</span> <span class="nf">rem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes the polynomial remainder of ``f`` by ``g``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly, ZZ, QQ</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, x).rem(Poly(2*x - 4, x))</span>
-<span class="sd">        Poly(5, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, x).rem(Poly(2*x - 4, x), auto=False)</span>
-<span class="sd">        Poly(x**2 + 1, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="n">retract</span> <span class="o">=</span> <span class="bp">False</span>
-
-        <span class="k">if</span> <span class="n">auto</span> <span class="ow">and</span> <span class="n">dom</span><span class="o">.</span><span class="n">has_Ring</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">dom</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-            <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">to_field</span><span class="p">(),</span> <span class="n">G</span><span class="o">.</span><span class="n">to_field</span><span class="p">()</span>
-            <span class="n">retract</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;rem&#39;</span><span class="p">):</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">rem</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;rem&#39;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">retract</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">r</span> <span class="o">=</span> <span class="n">r</span><span class="o">.</span><span class="n">to_ring</span><span class="p">()</span>
-            <span class="k">except</span> <span class="n">CoercionFailed</span><span class="p">:</span>
-                <span class="k">pass</span>
-
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.quo"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.quo">[docs]</a>    <span class="k">def</span> <span class="nf">quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes polynomial quotient of ``f`` by ``g``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, x).quo(Poly(2*x - 4, x))</span>
-<span class="sd">        Poly(1/2*x + 1, x, domain=&#39;QQ&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 - 1, x).quo(Poly(x - 1, x))</span>
-<span class="sd">        Poly(x + 1, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="n">retract</span> <span class="o">=</span> <span class="bp">False</span>
-
-        <span class="k">if</span> <span class="n">auto</span> <span class="ow">and</span> <span class="n">dom</span><span class="o">.</span><span class="n">has_Ring</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">dom</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-            <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">to_field</span><span class="p">(),</span> <span class="n">G</span><span class="o">.</span><span class="n">to_field</span><span class="p">()</span>
-            <span class="n">retract</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;quo&#39;</span><span class="p">):</span>
-            <span class="n">q</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;quo&#39;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">retract</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">q</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">to_ring</span><span class="p">()</span>
-            <span class="k">except</span> <span class="n">CoercionFailed</span><span class="p">:</span>
-                <span class="k">pass</span>
-
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.exquo"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.exquo">[docs]</a>    <span class="k">def</span> <span class="nf">exquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes polynomial exact quotient of ``f`` by ``g``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 - 1, x).exquo(Poly(x - 1, x))</span>
-<span class="sd">        Poly(x + 1, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, x).exquo(Poly(2*x - 4, x))</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">        ...</span>
-<span class="sd">        ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="n">retract</span> <span class="o">=</span> <span class="bp">False</span>
-
-        <span class="k">if</span> <span class="n">auto</span> <span class="ow">and</span> <span class="n">dom</span><span class="o">.</span><span class="n">has_Ring</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">dom</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-            <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">to_field</span><span class="p">(),</span> <span class="n">G</span><span class="o">.</span><span class="n">to_field</span><span class="p">()</span>
-            <span class="n">retract</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;exquo&#39;</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">q</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">exquo</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">ExactQuotientFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">exc</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(),</span> <span class="n">g</span><span class="o">.</span><span class="n">as_expr</span><span class="p">())</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;exquo&#39;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">retract</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">q</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">to_ring</span><span class="p">()</span>
-            <span class="k">except</span> <span class="n">CoercionFailed</span><span class="p">:</span>
-                <span class="k">pass</span>
-
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_gen_to_level</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gen</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns level associated with the given generator. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">gen</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-            <span class="n">length</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="o">-</span><span class="n">length</span> <span class="o">&lt;=</span> <span class="n">gen</span> <span class="o">&lt;</span> <span class="n">length</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">gen</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">length</span> <span class="o">+</span> <span class="n">gen</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">gen</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span><span class="s">&quot;-</span><span class="si">%s</span><span class="s"> &lt;= gen &lt; </span><span class="si">%s</span><span class="s"> expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span>
-                                      <span class="p">(</span><span class="n">length</span><span class="p">,</span> <span class="n">length</span><span class="p">,</span> <span class="n">gen</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">gen</span><span class="p">))</span>
-            <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span>
-                    <span class="s">&quot;a valid generator expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">gen</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Poly.degree"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.degree">[docs]</a>    <span class="k">def</span> <span class="nf">degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gen</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns degree of ``f`` in ``x_j``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + y*x + 1, x, y).degree()</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + y*x + y, x, y).degree(y)</span>
-<span class="sd">        1</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_gen_to_level</span><span class="p">(</span><span class="n">gen</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;degree&#39;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">degree</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;degree&#39;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.degree_list"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.degree_list">[docs]</a>    <span class="k">def</span> <span class="nf">degree_list</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns a list of degrees of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + y*x + 1, x, y).degree_list()</span>
-<span class="sd">        (2, 1)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;degree_list&#39;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">degree_list</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;degree_list&#39;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.total_degree"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.total_degree">[docs]</a>    <span class="k">def</span> <span class="nf">total_degree</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the total degree of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + y*x + 1, x, y).total_degree()</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x + y**5, x, y).total_degree()</span>
-<span class="sd">        5</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;total_degree&#39;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">total_degree</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;total_degree&#39;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.homogeneous_order"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.homogeneous_order">[docs]</a>    <span class="k">def</span> <span class="nf">homogeneous_order</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the homogeneous order of ``f``.</span>
-
-<span class="sd">        A homogeneous polynomial is a polynomial whose all monomials with</span>
-<span class="sd">        non-zero coefficients have the same total degree. This degree is</span>
-<span class="sd">        the homogeneous order of ``f``. If you only want to check if a</span>
-<span class="sd">        polynomial is homogeneous, then use :func:`Poly.is_homogeneous`.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; f = Poly(x**5 + 2*x**3*y**2 + 9*x*y**4)</span>
-<span class="sd">        &gt;&gt;&gt; f.homogeneous_order()</span>
-<span class="sd">        5</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;homogeneous_order&#39;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">homogeneous_order</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;homogeneous_order&#39;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.LC"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.LC">[docs]</a>    <span class="k">def</span> <span class="nf">LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the leading coefficient of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(4*x**3 + 2*x**2 + 3*x, x).LC()</span>
-<span class="sd">        4</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">order</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">coeffs</span><span class="p">(</span><span class="n">order</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;LC&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">LC</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;LC&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.TC"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.TC">[docs]</a>    <span class="k">def</span> <span class="nf">TC</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the trailing coefficient of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**3 + 2*x**2 + 3*x, x).TC()</span>
-<span class="sd">        0</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;TC&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">TC</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;TC&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.EC"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.EC">[docs]</a>    <span class="k">def</span> <span class="nf">EC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the last non-zero coefficient of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**3 + 2*x**2 + 3*x, x).EC()</span>
-<span class="sd">        3</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;coeffs&#39;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">coeffs</span><span class="p">(</span><span class="n">order</span><span class="p">)[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;EC&#39;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.nth"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.nth">[docs]</a>    <span class="k">def</span> <span class="nf">nth</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">N</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the ``n``-th coefficient of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**3 + 2*x**2 + 3*x, x).nth(2)</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x**3 + 2*x*y**2 + y**2, x, y).nth(1, 2)</span>
-<span class="sd">        2</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;nth&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">nth</span><span class="p">(</span><span class="o">*</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="n">N</span><span class="p">)))</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;nth&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.LM"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.LM">[docs]</a>    <span class="k">def</span> <span class="nf">LM</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the leading monomial of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).LM()</span>
-<span class="sd">        x**2*y**0</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Monomial</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">monoms</span><span class="p">(</span><span class="n">order</span><span class="p">)[</span><span class="mi">0</span><span class="p">],</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.EM"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.EM">[docs]</a>    <span class="k">def</span> <span class="nf">EM</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the last non-zero monomial of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).EM()</span>
-<span class="sd">        x**0*y**1</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">Monomial</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">monoms</span><span class="p">(</span><span class="n">order</span><span class="p">)[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.LT"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.LT">[docs]</a>    <span class="k">def</span> <span class="nf">LT</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the leading term of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).LT()</span>
-<span class="sd">        (x**2*y**0, 4)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">terms</span><span class="p">(</span><span class="n">order</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">Monomial</span><span class="p">(</span><span class="n">monom</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">),</span> <span class="n">coeff</span>
-</div>
-<div class="viewcode-block" id="Poly.ET"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.ET">[docs]</a>    <span class="k">def</span> <span class="nf">ET</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the last non-zero term of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(4*x**2 + 2*x*y**2 + x*y + 3*y, x, y).ET()</span>
-<span class="sd">        (x**0*y**1, 3)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">terms</span><span class="p">(</span><span class="n">order</span><span class="p">)[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">Monomial</span><span class="p">(</span><span class="n">monom</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">),</span> <span class="n">coeff</span>
-</div>
-<div class="viewcode-block" id="Poly.max_norm"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.max_norm">[docs]</a>    <span class="k">def</span> <span class="nf">max_norm</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns maximum norm of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(-x**2 + 2*x - 3, x).max_norm()</span>
-<span class="sd">        3</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;max_norm&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">max_norm</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;max_norm&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.l1_norm"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.l1_norm">[docs]</a>    <span class="k">def</span> <span class="nf">l1_norm</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns l1 norm of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(-x**2 + 2*x - 3, x).l1_norm()</span>
-<span class="sd">        6</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;l1_norm&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">l1_norm</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;l1_norm&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.clear_denoms"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.clear_denoms">[docs]</a>    <span class="k">def</span> <span class="nf">clear_denoms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">convert</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Clear denominators, but keep the ground domain.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly, S, QQ</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; f = Poly(x/2 + S(1)/3, x, domain=QQ)</span>
-
-<span class="sd">        &gt;&gt;&gt; f.clear_denoms()</span>
-<span class="sd">        (6, Poly(3*x + 2, x, domain=&#39;QQ&#39;))</span>
-<span class="sd">        &gt;&gt;&gt; f.clear_denoms(convert=True)</span>
-<span class="sd">        (6, Poly(3*x + 2, x, domain=&#39;ZZ&#39;))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">f</span>
-
-        <span class="n">dom</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">get_domain</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">dom</span><span class="o">.</span><span class="n">has_assoc_Ring</span><span class="p">:</span>
-            <span class="n">dom</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">get_ring</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;clear_denoms&#39;</span><span class="p">):</span>
-            <span class="n">coeff</span><span class="p">,</span> <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">clear_denoms</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;clear_denoms&#39;</span><span class="p">)</span>
-
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">coeff</span><span class="p">),</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">convert</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">f</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">to_ring</span><span class="p">()</span>
-</div>
-<div class="viewcode-block" id="Poly.rat_clear_denoms"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.rat_clear_denoms">[docs]</a>    <span class="k">def</span> <span class="nf">rat_clear_denoms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Clear denominators in a rational function ``f/g``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; f = Poly(x**2/y + 1, x)</span>
-<span class="sd">        &gt;&gt;&gt; g = Poly(x**3 + y, x)</span>
-
-<span class="sd">        &gt;&gt;&gt; p, q = f.rat_clear_denoms(g)</span>
-
-<span class="sd">        &gt;&gt;&gt; p</span>
-<span class="sd">        Poly(x**2 + y, x, domain=&#39;ZZ[y]&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; q</span>
-<span class="sd">        Poly(y*x**3 + y**2, x, domain=&#39;ZZ[y]&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="n">f</span> <span class="o">=</span> <span class="n">per</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">per</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">dom</span><span class="o">.</span><span class="n">has_Field</span> <span class="ow">and</span> <span class="n">dom</span><span class="o">.</span><span class="n">has_assoc_Ring</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">f</span><span class="p">,</span> <span class="n">g</span>
-
-        <span class="n">a</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">clear_denoms</span><span class="p">(</span><span class="n">convert</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="n">b</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">clear_denoms</span><span class="p">(</span><span class="n">convert</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-        <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">mul_ground</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">mul_ground</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="p">,</span> <span class="n">g</span>
-</div>
-<div class="viewcode-block" id="Poly.integrate"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.integrate">[docs]</a>    <span class="k">def</span> <span class="nf">integrate</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">specs</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes indefinite integral of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 2*x + 1, x).integrate()</span>
-<span class="sd">        Poly(1/3*x**3 + x**2 + x, x, domain=&#39;QQ&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x*y**2 + x, x, y).integrate((0, 1), (1, 0))</span>
-<span class="sd">        Poly(1/2*x**2*y**2 + 1/2*x**2, x, y, domain=&#39;QQ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;auto&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span> <span class="ow">and</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">has_Ring</span><span class="p">:</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">to_field</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;integrate&#39;</span><span class="p">):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">specs</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">integrate</span><span class="p">(</span><span class="n">m</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span>
-
-            <span class="n">rep</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span>
-
-            <span class="k">for</span> <span class="n">spec</span> <span class="ow">in</span> <span class="n">specs</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">spec</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">tuple</span><span class="p">:</span>
-                    <span class="n">gen</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">spec</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">gen</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">spec</span><span class="p">,</span> <span class="mi">1</span>
-
-                <span class="n">rep</span> <span class="o">=</span> <span class="n">rep</span><span class="o">.</span><span class="n">integrate</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">m</span><span class="p">),</span> <span class="n">f</span><span class="o">.</span><span class="n">_gen_to_level</span><span class="p">(</span><span class="n">gen</span><span class="p">))</span>
-
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">rep</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;integrate&#39;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.diff"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.diff">[docs]</a>    <span class="k">def</span> <span class="nf">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">specs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes partial derivative of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 2*x + 1, x).diff()</span>
-<span class="sd">        Poly(2*x + 2, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x*y**2 + x, x, y).diff((0, 0), (1, 1))</span>
-<span class="sd">        Poly(2*x*y, x, y, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;diff&#39;</span><span class="p">):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">specs</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">m</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span>
-
-            <span class="n">rep</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span>
-
-            <span class="k">for</span> <span class="n">spec</span> <span class="ow">in</span> <span class="n">specs</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">spec</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">tuple</span><span class="p">:</span>
-                    <span class="n">gen</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">spec</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">gen</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">spec</span><span class="p">,</span> <span class="mi">1</span>
-
-                <span class="n">rep</span> <span class="o">=</span> <span class="n">rep</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">m</span><span class="p">),</span> <span class="n">f</span><span class="o">.</span><span class="n">_gen_to_level</span><span class="p">(</span><span class="n">gen</span><span class="p">))</span>
-
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">rep</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;diff&#39;</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.eval"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.eval">[docs]</a>    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Evaluate ``f`` at ``a`` in the given variable.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 2*x + 3, x).eval(2)</span>
-<span class="sd">        11</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(2*x*y + 3*x + y + 2, x, y).eval(x, 2)</span>
-<span class="sd">        Poly(5*y + 8, y, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; f = Poly(2*x*y + 3*x + y + 2*z, x, y, z)</span>
-
-<span class="sd">        &gt;&gt;&gt; f.eval({x: 2})</span>
-<span class="sd">        Poly(5*y + 2*z + 6, y, z, domain=&#39;ZZ&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; f.eval({x: 2, y: 5})</span>
-<span class="sd">        Poly(2*z + 31, z, domain=&#39;ZZ&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; f.eval({x: 2, y: 5, z: 7})</span>
-<span class="sd">        45</span>
-
-<span class="sd">        &gt;&gt;&gt; f.eval((2, 5))</span>
-<span class="sd">        Poly(2*z + 31, z, domain=&#39;ZZ&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; f(2, 5)</span>
-<span class="sd">        Poly(2*z + 31, z, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">a</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="nb">dict</span><span class="p">):</span>
-                <span class="n">mapping</span> <span class="o">=</span> <span class="n">x</span>
-
-                <span class="k">for</span> <span class="n">gen</span><span class="p">,</span> <span class="n">value</span> <span class="ow">in</span> <span class="n">mapping</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                    <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="n">gen</span><span class="p">,</span> <span class="n">value</span><span class="p">)</span>
-
-                <span class="k">return</span> <span class="n">f</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="nb">list</span><span class="p">)):</span>
-                <span class="n">values</span> <span class="o">=</span> <span class="n">x</span>
-
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">values</span><span class="p">)</span> <span class="o">&gt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">):</span>
-                    <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;too many values provided&quot;</span><span class="p">)</span>
-
-                <span class="k">for</span> <span class="n">gen</span><span class="p">,</span> <span class="n">value</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">,</span> <span class="n">values</span><span class="p">):</span>
-                    <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="n">gen</span><span class="p">,</span> <span class="n">value</span><span class="p">)</span>
-
-                <span class="k">return</span> <span class="n">f</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">j</span><span class="p">,</span> <span class="n">a</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="n">x</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">j</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_gen_to_level</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;eval&#39;</span><span class="p">):</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;eval&#39;</span><span class="p">)</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">CoercionFailed</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">auto</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">DomainError</span><span class="p">(</span>
-                    <span class="s">&quot;can&#39;t evaluate at </span><span class="si">%s</span><span class="s"> in </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">domain</span><span class="p">,</span> <span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="n">construct_domain</span><span class="p">([</span><span class="n">a</span><span class="p">])</span>
-                <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">set_domain</span><span class="p">(</span><span class="n">domain</span><span class="p">)</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">remove</span><span class="o">=</span><span class="n">j</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">__call__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">values</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Evaluate ``f`` at the give values.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-
-<span class="sd">        &gt;&gt;&gt; f = Poly(2*x*y + 3*x + y + 2*z, x, y, z)</span>
-
-<span class="sd">        &gt;&gt;&gt; f(2)</span>
-<span class="sd">        Poly(5*y + 2*z + 6, y, z, domain=&#39;ZZ&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; f(2, 5)</span>
-<span class="sd">        Poly(2*z + 31, z, domain=&#39;ZZ&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; f(2, 5, 7)</span>
-<span class="sd">        45</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="n">values</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Poly.half_gcdex"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.half_gcdex">[docs]</a>    <span class="k">def</span> <span class="nf">half_gcdex</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Half extended Euclidean algorithm of ``f`` and ``g``.</span>
-
-<span class="sd">        Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15</span>
-<span class="sd">        &gt;&gt;&gt; g = x**3 + x**2 - 4*x - 4</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(f).half_gcdex(Poly(g))</span>
-<span class="sd">        (Poly(-1/5*x + 3/5, x, domain=&#39;QQ&#39;), Poly(x + 1, x, domain=&#39;QQ&#39;))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">auto</span> <span class="ow">and</span> <span class="n">dom</span><span class="o">.</span><span class="n">has_Ring</span><span class="p">:</span>
-            <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">to_field</span><span class="p">(),</span> <span class="n">G</span><span class="o">.</span><span class="n">to_field</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;half_gcdex&#39;</span><span class="p">):</span>
-            <span class="n">s</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">half_gcdex</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;half_gcdex&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="n">per</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.gcdex"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.gcdex">[docs]</a>    <span class="k">def</span> <span class="nf">gcdex</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Extended Euclidean algorithm of ``f`` and ``g``.</span>
-
-<span class="sd">        Returns ``(s, t, h)`` such that ``h = gcd(f, g)`` and ``s*f + t*g = h``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15</span>
-<span class="sd">        &gt;&gt;&gt; g = x**3 + x**2 - 4*x - 4</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(f).gcdex(Poly(g))</span>
-<span class="sd">        (Poly(-1/5*x + 3/5, x, domain=&#39;QQ&#39;),</span>
-<span class="sd">         Poly(1/5*x**2 - 6/5*x + 2, x, domain=&#39;QQ&#39;),</span>
-<span class="sd">         Poly(x + 1, x, domain=&#39;QQ&#39;))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">auto</span> <span class="ow">and</span> <span class="n">dom</span><span class="o">.</span><span class="n">has_Ring</span><span class="p">:</span>
-            <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">to_field</span><span class="p">(),</span> <span class="n">G</span><span class="o">.</span><span class="n">to_field</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;gcdex&#39;</span><span class="p">):</span>
-            <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">gcdex</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;gcdex&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="n">per</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="n">per</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.invert"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.invert">[docs]</a>    <span class="k">def</span> <span class="nf">invert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Invert ``f`` modulo ``g`` when possible.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 - 1, x).invert(Poly(2*x - 1, x))</span>
-<span class="sd">        Poly(-4/3, x, domain=&#39;QQ&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 - 1, x).invert(Poly(x - 1, x))</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">        ...</span>
-<span class="sd">        NotInvertible: zero divisor</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">auto</span> <span class="ow">and</span> <span class="n">dom</span><span class="o">.</span><span class="n">has_Ring</span><span class="p">:</span>
-            <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">to_field</span><span class="p">(),</span> <span class="n">G</span><span class="o">.</span><span class="n">to_field</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;invert&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">invert</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;invert&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.revert"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.revert">[docs]</a>    <span class="k">def</span> <span class="nf">revert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute ``f**(-1)`` mod ``x**n``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;revert&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">revert</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;revert&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.subresultants"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.subresultants">[docs]</a>    <span class="k">def</span> <span class="nf">subresultants</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes the subresultant PRS of ``f`` and ``g``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, x).subresultants(Poly(x**2 - 1, x))</span>
-<span class="sd">        [Poly(x**2 + 1, x, domain=&#39;ZZ&#39;),</span>
-<span class="sd">         Poly(x**2 - 1, x, domain=&#39;ZZ&#39;),</span>
-<span class="sd">         Poly(-2, x, domain=&#39;ZZ&#39;)]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;subresultants&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">subresultants</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;subresultants&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">per</span><span class="p">,</span> <span class="n">result</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="Poly.resultant"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.resultant">[docs]</a>    <span class="k">def</span> <span class="nf">resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">includePRS</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes the resultant of ``f`` and ``g`` via PRS.</span>
-
-<span class="sd">        If includePRS=True, it includes the subresultant PRS in the result.</span>
-<span class="sd">        Because the PRS is used to calculate the resultant, this is more</span>
-<span class="sd">        efficient than calling :func:`subresultants` separately.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; f = Poly(x**2 + 1, x)</span>
-
-<span class="sd">        &gt;&gt;&gt; f.resultant(Poly(x**2 - 1, x))</span>
-<span class="sd">        4</span>
-<span class="sd">        &gt;&gt;&gt; f.resultant(Poly(x**2 - 1, x), includePRS=True)</span>
-<span class="sd">        (4, [Poly(x**2 + 1, x, domain=&#39;ZZ&#39;), Poly(x**2 - 1, x, domain=&#39;ZZ&#39;),</span>
-<span class="sd">             Poly(-2, x, domain=&#39;ZZ&#39;)])</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;resultant&#39;</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">includePRS</span><span class="p">:</span>
-                <span class="n">result</span><span class="p">,</span> <span class="n">R</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">resultant</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">includePRS</span><span class="o">=</span><span class="n">includePRS</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">resultant</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;resultant&#39;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">includePRS</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">remove</span><span class="o">=</span><span class="mi">0</span><span class="p">),</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">per</span><span class="p">,</span> <span class="n">R</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">remove</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.discriminant"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.discriminant">[docs]</a>    <span class="k">def</span> <span class="nf">discriminant</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes the discriminant of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 2*x + 3, x).discriminant()</span>
-<span class="sd">        -8</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;discriminant&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">discriminant</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;discriminant&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">remove</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.cofactors"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.cofactors">[docs]</a>    <span class="k">def</span> <span class="nf">cofactors</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the GCD of ``f`` and ``g`` and their cofactors.</span>
-
-<span class="sd">        Returns polynomials ``(h, cff, cfg)`` such that ``h = gcd(f, g)``, and</span>
-<span class="sd">        ``cff = quo(f, h)`` and ``cfg = quo(g, h)`` are, so called, cofactors</span>
-<span class="sd">        of ``f`` and ``g``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 - 1, x).cofactors(Poly(x**2 - 3*x + 2, x))</span>
-<span class="sd">        (Poly(x - 1, x, domain=&#39;ZZ&#39;),</span>
-<span class="sd">         Poly(x + 1, x, domain=&#39;ZZ&#39;),</span>
-<span class="sd">         Poly(x - 2, x, domain=&#39;ZZ&#39;))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;cofactors&#39;</span><span class="p">):</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">cff</span><span class="p">,</span> <span class="n">cfg</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">cofactors</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;cofactors&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">h</span><span class="p">),</span> <span class="n">per</span><span class="p">(</span><span class="n">cff</span><span class="p">),</span> <span class="n">per</span><span class="p">(</span><span class="n">cfg</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.gcd"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.gcd">[docs]</a>    <span class="k">def</span> <span class="nf">gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the polynomial GCD of ``f`` and ``g``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 - 1, x).gcd(Poly(x**2 - 3*x + 2, x))</span>
-<span class="sd">        Poly(x - 1, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;gcd&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;gcd&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.lcm"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.lcm">[docs]</a>    <span class="k">def</span> <span class="nf">lcm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns polynomial LCM of ``f`` and ``g``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 - 1, x).lcm(Poly(x**2 - 3*x + 2, x))</span>
-<span class="sd">        Poly(x**3 - 2*x**2 - x + 2, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;lcm&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">lcm</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;lcm&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.trunc"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.trunc">[docs]</a>    <span class="k">def</span> <span class="nf">trunc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Reduce ``f`` modulo a constant ``p``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(2*x**3 + 3*x**2 + 5*x + 7, x).trunc(3)</span>
-<span class="sd">        Poly(-x**3 - x + 1, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;trunc&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">trunc</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;trunc&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.monic"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.monic">[docs]</a>    <span class="k">def</span> <span class="nf">monic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Divides all coefficients by ``LC(f)``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly, ZZ</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(3*x**2 + 6*x + 9, x, domain=ZZ).monic()</span>
-<span class="sd">        Poly(x**2 + 2*x + 3, x, domain=&#39;QQ&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(3*x**2 + 4*x + 2, x, domain=ZZ).monic()</span>
-<span class="sd">        Poly(x**2 + 4/3*x + 2/3, x, domain=&#39;QQ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">auto</span> <span class="ow">and</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">has_Ring</span><span class="p">:</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">to_field</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;monic&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">monic</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;monic&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.content"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.content">[docs]</a>    <span class="k">def</span> <span class="nf">content</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the GCD of polynomial coefficients.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(6*x**2 + 8*x + 12, x).content()</span>
-<span class="sd">        2</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;content&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">content</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;content&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.primitive"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.primitive">[docs]</a>    <span class="k">def</span> <span class="nf">primitive</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the content and a primitive form of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(2*x**2 + 8*x + 12, x).primitive()</span>
-<span class="sd">        (2, Poly(x**2 + 4*x + 6, x, domain=&#39;ZZ&#39;))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;primitive&#39;</span><span class="p">):</span>
-            <span class="n">cont</span><span class="p">,</span> <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">primitive</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;primitive&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">cont</span><span class="p">),</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.compose"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.compose">[docs]</a>    <span class="k">def</span> <span class="nf">compose</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes the functional composition of ``f`` and ``g``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + x, x).compose(Poly(x - 1, x))</span>
-<span class="sd">        Poly(x**2 - x, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;compose&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">compose</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;compose&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.decompose"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.decompose">[docs]</a>    <span class="k">def</span> <span class="nf">decompose</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes a functional decomposition of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**4 + 2*x**3 - x - 1, x, domain=&#39;ZZ&#39;).decompose()</span>
-<span class="sd">        [Poly(x**2 - x - 1, x, domain=&#39;ZZ&#39;), Poly(x**2 + x, x, domain=&#39;ZZ&#39;)]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;decompose&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">decompose</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;decompose&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">,</span> <span class="n">result</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="Poly.shift"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.shift">[docs]</a>    <span class="k">def</span> <span class="nf">shift</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Efficiently compute Taylor shift ``f(x + a)``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 - 2*x + 1, x).shift(2)</span>
-<span class="sd">        Poly(x**2 + 2*x + 1, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;shift&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">shift</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;shift&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.sturm"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.sturm">[docs]</a>    <span class="k">def</span> <span class="nf">sturm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes the Sturm sequence of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**3 - 2*x**2 + x - 3, x).sturm()</span>
-<span class="sd">        [Poly(x**3 - 2*x**2 + x - 3, x, domain=&#39;QQ&#39;),</span>
-<span class="sd">         Poly(3*x**2 - 4*x + 1, x, domain=&#39;QQ&#39;),</span>
-<span class="sd">         Poly(2/9*x + 25/9, x, domain=&#39;QQ&#39;),</span>
-<span class="sd">         Poly(-2079/4, x, domain=&#39;QQ&#39;)]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">auto</span> <span class="ow">and</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">has_Ring</span><span class="p">:</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">to_field</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;sturm&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">sturm</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;sturm&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">,</span> <span class="n">result</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="Poly.gff_list"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.gff_list">[docs]</a>    <span class="k">def</span> <span class="nf">gff_list</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes greatest factorial factorization of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; f = x**5 + 2*x**4 - x**3 - 2*x**2</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(f).gff_list()</span>
-<span class="sd">        [(Poly(x, x, domain=&#39;ZZ&#39;), 1), (Poly(x + 2, x, domain=&#39;ZZ&#39;), 4)]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;gff_list&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">gff_list</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;gff_list&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">result</span> <span class="p">]</span>
-</div>
-<div class="viewcode-block" id="Poly.sqf_norm"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.sqf_norm">[docs]</a>    <span class="k">def</span> <span class="nf">sqf_norm</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes square-free norm of ``f``.</span>
-
-<span class="sd">        Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and</span>
-<span class="sd">        ``r(x) = Norm(g(x))`` is a square-free polynomial over ``K``,</span>
-<span class="sd">        where ``a`` is the algebraic extension of the ground domain.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly, sqrt</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; s, f, r = Poly(x**2 + 1, x, extension=[sqrt(3)]).sqf_norm()</span>
-
-<span class="sd">        &gt;&gt;&gt; s</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; f</span>
-<span class="sd">        Poly(x**2 - 2*sqrt(3)*x + 4, x, domain=&#39;QQ&lt;sqrt(3)&gt;&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; r</span>
-<span class="sd">        Poly(x**4 - 4*x**2 + 16, x, domain=&#39;QQ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;sqf_norm&#39;</span><span class="p">):</span>
-            <span class="n">s</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">sqf_norm</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;sqf_norm&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">s</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.sqf_part"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.sqf_part">[docs]</a>    <span class="k">def</span> <span class="nf">sqf_part</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes square-free part of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**3 - 3*x - 2, x).sqf_part()</span>
-<span class="sd">        Poly(x**2 - x - 2, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;sqf_part&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">sqf_part</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;sqf_part&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.sqf_list"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.sqf_list">[docs]</a>    <span class="k">def</span> <span class="nf">sqf_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="nb">all</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns a list of square-free factors of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; f = 2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(f).sqf_list()</span>
-<span class="sd">        (2, [(Poly(x + 1, x, domain=&#39;ZZ&#39;), 2),</span>
-<span class="sd">             (Poly(x + 2, x, domain=&#39;ZZ&#39;), 3)])</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(f).sqf_list(all=True)</span>
-<span class="sd">        (2, [(Poly(1, x, domain=&#39;ZZ&#39;), 1),</span>
-<span class="sd">             (Poly(x + 1, x, domain=&#39;ZZ&#39;), 2),</span>
-<span class="sd">             (Poly(x + 2, x, domain=&#39;ZZ&#39;), 3)])</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;sqf_list&#39;</span><span class="p">):</span>
-            <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">sqf_list</span><span class="p">(</span><span class="nb">all</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;sqf_list&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">coeff</span><span class="p">),</span> <span class="p">[</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">factors</span> <span class="p">]</span>
-</div>
-<div class="viewcode-block" id="Poly.sqf_list_include"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.sqf_list_include">[docs]</a>    <span class="k">def</span> <span class="nf">sqf_list_include</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="nb">all</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns a list of square-free factors of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly, expand</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; f = expand(2*(x + 1)**3*x**4)</span>
-<span class="sd">        &gt;&gt;&gt; f</span>
-<span class="sd">        2*x**7 + 6*x**6 + 6*x**5 + 2*x**4</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(f).sqf_list_include()</span>
-<span class="sd">        [(Poly(2, x, domain=&#39;ZZ&#39;), 1),</span>
-<span class="sd">         (Poly(x + 1, x, domain=&#39;ZZ&#39;), 3),</span>
-<span class="sd">         (Poly(x, x, domain=&#39;ZZ&#39;), 4)]</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(f).sqf_list_include(all=True)</span>
-<span class="sd">        [(Poly(2, x, domain=&#39;ZZ&#39;), 1),</span>
-<span class="sd">         (Poly(1, x, domain=&#39;ZZ&#39;), 2),</span>
-<span class="sd">         (Poly(x + 1, x, domain=&#39;ZZ&#39;), 3),</span>
-<span class="sd">         (Poly(x, x, domain=&#39;ZZ&#39;), 4)]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;sqf_list_include&#39;</span><span class="p">):</span>
-            <span class="n">factors</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">sqf_list_include</span><span class="p">(</span><span class="nb">all</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;sqf_list_include&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">factors</span> <span class="p">]</span>
-</div>
-<div class="viewcode-block" id="Poly.factor_list"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.factor_list">[docs]</a>    <span class="k">def</span> <span class="nf">factor_list</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns a list of irreducible factors of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; f = 2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(f).factor_list()</span>
-<span class="sd">        (2, [(Poly(x + y, x, y, domain=&#39;ZZ&#39;), 1),</span>
-<span class="sd">             (Poly(x**2 + 1, x, y, domain=&#39;ZZ&#39;), 2)])</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;factor_list&#39;</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">factor_list</span><span class="p">()</span>
-            <span class="k">except</span> <span class="n">DomainError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="p">[(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;factor_list&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">coeff</span><span class="p">),</span> <span class="p">[</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">factors</span> <span class="p">]</span>
-</div>
-<div class="viewcode-block" id="Poly.factor_list_include"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.factor_list_include">[docs]</a>    <span class="k">def</span> <span class="nf">factor_list_include</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns a list of irreducible factors of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; f = 2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(f).factor_list_include()</span>
-<span class="sd">        [(Poly(2*x + 2*y, x, y, domain=&#39;ZZ&#39;), 1),</span>
-<span class="sd">         (Poly(x**2 + 1, x, y, domain=&#39;ZZ&#39;), 2)]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;factor_list_include&#39;</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">factors</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">factor_list_include</span><span class="p">()</span>
-            <span class="k">except</span> <span class="n">DomainError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;factor_list_include&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">factors</span> <span class="p">]</span>
-</div>
-<div class="viewcode-block" id="Poly.intervals"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.intervals">[docs]</a>    <span class="k">def</span> <span class="nf">intervals</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="nb">all</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">inf</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">sup</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">fast</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">sqf</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Compute isolating intervals for roots of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 - 3, x).intervals()</span>
-<span class="sd">        [((-2, -1), 1), ((1, 2), 1)]</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 - 3, x).intervals(eps=1e-2)</span>
-<span class="sd">        [((-26/15, -19/11), 1), ((19/11, 26/15), 1)]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">eps</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">eps</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">eps</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">eps</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;&#39;eps&#39; must be a positive rational&quot;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">inf</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">inf</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">sup</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">sup</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">sup</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;intervals&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">intervals</span><span class="p">(</span>
-                <span class="nb">all</span><span class="o">=</span><span class="nb">all</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="n">eps</span><span class="p">,</span> <span class="n">inf</span><span class="o">=</span><span class="n">inf</span><span class="p">,</span> <span class="n">sup</span><span class="o">=</span><span class="n">sup</span><span class="p">,</span> <span class="n">fast</span><span class="o">=</span><span class="n">fast</span><span class="p">,</span> <span class="n">sqf</span><span class="o">=</span><span class="n">sqf</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;intervals&#39;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">sqf</span><span class="p">:</span>
-            <span class="k">def</span> <span class="nf">_real</span><span class="p">(</span><span class="n">interval</span><span class="p">):</span>
-                <span class="n">s</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">interval</span>
-                <span class="k">return</span> <span class="p">(</span><span class="n">QQ</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="n">QQ</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">t</span><span class="p">))</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">:</span>
-                <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">_real</span><span class="p">,</span> <span class="n">result</span><span class="p">))</span>
-
-            <span class="k">def</span> <span class="nf">_complex</span><span class="p">(</span><span class="n">rectangle</span><span class="p">):</span>
-                <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">),</span> <span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span> <span class="o">=</span> <span class="n">rectangle</span>
-                <span class="k">return</span> <span class="p">(</span><span class="n">QQ</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">u</span><span class="p">)</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">QQ</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">v</span><span class="p">),</span>
-                        <span class="n">QQ</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">QQ</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">t</span><span class="p">))</span>
-
-            <span class="n">real_part</span><span class="p">,</span> <span class="n">complex_part</span> <span class="o">=</span> <span class="n">result</span>
-
-            <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">_real</span><span class="p">,</span> <span class="n">real_part</span><span class="p">)),</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">_complex</span><span class="p">,</span> <span class="n">complex_part</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">def</span> <span class="nf">_real</span><span class="p">(</span><span class="n">interval</span><span class="p">):</span>
-                <span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">),</span> <span class="n">k</span> <span class="o">=</span> <span class="n">interval</span>
-                <span class="k">return</span> <span class="p">((</span><span class="n">QQ</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="n">QQ</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">t</span><span class="p">)),</span> <span class="n">k</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">:</span>
-                <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">_real</span><span class="p">,</span> <span class="n">result</span><span class="p">))</span>
-
-            <span class="k">def</span> <span class="nf">_complex</span><span class="p">(</span><span class="n">rectangle</span><span class="p">):</span>
-                <span class="p">((</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">),</span> <span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">)),</span> <span class="n">k</span> <span class="o">=</span> <span class="n">rectangle</span>
-                <span class="k">return</span> <span class="p">((</span><span class="n">QQ</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">u</span><span class="p">)</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">QQ</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">v</span><span class="p">),</span>
-                         <span class="n">QQ</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">QQ</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">t</span><span class="p">)),</span> <span class="n">k</span><span class="p">)</span>
-
-            <span class="n">real_part</span><span class="p">,</span> <span class="n">complex_part</span> <span class="o">=</span> <span class="n">result</span>
-
-            <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">_real</span><span class="p">,</span> <span class="n">real_part</span><span class="p">)),</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">_complex</span><span class="p">,</span> <span class="n">complex_part</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="Poly.refine_root"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.refine_root">[docs]</a>    <span class="k">def</span> <span class="nf">refine_root</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">steps</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">fast</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">check_sqf</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Refine an isolating interval of a root to the given precision.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 - 3, x).refine_root(1, 2, eps=1e-2)</span>
-<span class="sd">        (19/11, 26/15)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">check_sqf</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">is_sqf</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span><span class="s">&quot;only square-free polynomials supported&quot;</span><span class="p">)</span>
-
-        <span class="n">s</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="n">QQ</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">eps</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">eps</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">eps</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">eps</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;&#39;eps&#39; must be a positive rational&quot;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">steps</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">steps</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">steps</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">eps</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">steps</span> <span class="o">=</span> <span class="mi">1</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;refine_root&#39;</span><span class="p">):</span>
-            <span class="n">S</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">refine_root</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="n">eps</span><span class="p">,</span> <span class="n">steps</span><span class="o">=</span><span class="n">steps</span><span class="p">,</span> <span class="n">fast</span><span class="o">=</span><span class="n">fast</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;refine_root&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">QQ</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">S</span><span class="p">),</span> <span class="n">QQ</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">T</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.count_roots"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.count_roots">[docs]</a>    <span class="k">def</span> <span class="nf">count_roots</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">inf</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">sup</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the number of roots of ``f`` in ``[inf, sup]`` interval.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly, I</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**4 - 4, x).count_roots(-3, 3)</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x**4 - 4, x).count_roots(0, 1 + 3*I)</span>
-<span class="sd">        1</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">inf_real</span><span class="p">,</span> <span class="n">sup_real</span> <span class="o">=</span> <span class="bp">True</span><span class="p">,</span> <span class="bp">True</span>
-
-        <span class="k">if</span> <span class="n">inf</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">inf</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">inf</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">:</span>
-                <span class="n">inf</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="n">inf</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">im</span><span class="p">:</span>
-                    <span class="n">inf</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">inf</span><span class="p">,</span> <span class="n">inf_real</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">QQ</span><span class="o">.</span><span class="n">convert</span><span class="p">,</span> <span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">))),</span> <span class="bp">False</span>
-
-        <span class="k">if</span> <span class="n">sup</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">sup</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">sup</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">sup</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="n">sup</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="n">sup</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">im</span><span class="p">:</span>
-                    <span class="n">sup</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">sup</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">sup</span><span class="p">,</span> <span class="n">sup_real</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">QQ</span><span class="o">.</span><span class="n">convert</span><span class="p">,</span> <span class="p">(</span><span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">))),</span> <span class="bp">False</span>
-
-        <span class="k">if</span> <span class="n">inf_real</span> <span class="ow">and</span> <span class="n">sup_real</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;count_real_roots&#39;</span><span class="p">):</span>
-                <span class="n">count</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">count_real_roots</span><span class="p">(</span><span class="n">inf</span><span class="o">=</span><span class="n">inf</span><span class="p">,</span> <span class="n">sup</span><span class="o">=</span><span class="n">sup</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-                <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;count_real_roots&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">inf_real</span> <span class="ow">and</span> <span class="n">inf</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">inf</span> <span class="o">=</span> <span class="p">(</span><span class="n">inf</span><span class="p">,</span> <span class="n">QQ</span><span class="o">.</span><span class="n">zero</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">sup_real</span> <span class="ow">and</span> <span class="n">sup</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">sup</span> <span class="o">=</span> <span class="p">(</span><span class="n">sup</span><span class="p">,</span> <span class="n">QQ</span><span class="o">.</span><span class="n">zero</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="s">&#39;count_complex_roots&#39;</span><span class="p">):</span>
-                <span class="n">count</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">count_complex_roots</span><span class="p">(</span><span class="n">inf</span><span class="o">=</span><span class="n">inf</span><span class="p">,</span> <span class="n">sup</span><span class="o">=</span><span class="n">sup</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-                <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;count_complex_roots&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="n">count</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.root"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.root">[docs]</a>    <span class="k">def</span> <span class="nf">root</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">index</span><span class="p">,</span> <span class="n">radicals</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Get an indexed root of a polynomial.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; f = Poly(2*x**3 - 7*x**2 + 4*x + 4)</span>
-
-<span class="sd">        &gt;&gt;&gt; f.root(0)</span>
-<span class="sd">        -1/2</span>
-<span class="sd">        &gt;&gt;&gt; f.root(1)</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; f.root(2)</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; f.root(3)</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">        ...</span>
-<span class="sd">        IndexError: root index out of [-3, 2] range, got 3</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**5 + x + 1).root(0)</span>
-<span class="sd">        RootOf(x**3 - x**2 + 1, 0)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">sympy</span><span class="o">.</span><span class="n">polys</span><span class="o">.</span><span class="n">rootoftools</span><span class="o">.</span><span class="n">RootOf</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">index</span><span class="p">,</span> <span class="n">radicals</span><span class="o">=</span><span class="n">radicals</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.real_roots"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.real_roots">[docs]</a>    <span class="k">def</span> <span class="nf">real_roots</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">radicals</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return a list of real roots with multiplicities.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(2*x**3 - 7*x**2 + 4*x + 4).real_roots()</span>
-<span class="sd">        [-1/2, 2, 2]</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x**3 + x + 1).real_roots()</span>
-<span class="sd">        [RootOf(x**3 + x + 1, 0)]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">reals</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">polys</span><span class="o">.</span><span class="n">rootoftools</span><span class="o">.</span><span class="n">RootOf</span><span class="o">.</span><span class="n">real_roots</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">radicals</span><span class="o">=</span><span class="n">radicals</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">multiple</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">reals</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">group</span><span class="p">(</span><span class="n">reals</span><span class="p">,</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.all_roots"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.all_roots">[docs]</a>    <span class="k">def</span> <span class="nf">all_roots</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">radicals</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return a list of real and complex roots with multiplicities.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(2*x**3 - 7*x**2 + 4*x + 4).all_roots()</span>
-<span class="sd">        [-1/2, 2, 2]</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x**3 + x + 1).all_roots()</span>
-<span class="sd">        [RootOf(x**3 + x + 1, 0),</span>
-<span class="sd">         RootOf(x**3 + x + 1, 1),</span>
-<span class="sd">         RootOf(x**3 + x + 1, 2)]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">roots</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">polys</span><span class="o">.</span><span class="n">rootoftools</span><span class="o">.</span><span class="n">RootOf</span><span class="o">.</span><span class="n">all_roots</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">radicals</span><span class="o">=</span><span class="n">radicals</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">multiple</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">roots</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">group</span><span class="p">(</span><span class="n">roots</span><span class="p">,</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.nroots"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.nroots">[docs]</a>    <span class="k">def</span> <span class="nf">nroots</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">15</span><span class="p">,</span> <span class="n">maxsteps</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span> <span class="n">cleanup</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">error</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Compute numerical approximations of roots of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 - 3).nroots(n=15)</span>
-<span class="sd">        [-1.73205080756888, 1.73205080756888]</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 - 3).nroots(n=30)</span>
-<span class="sd">        [-1.73205080756887729352744634151, 1.73205080756887729352744634151]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_multivariate</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">MultivariatePolynomialError</span><span class="p">(</span>
-                <span class="s">&quot;can&#39;t compute numerical roots of </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">f</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[]</span>
-
-        <span class="n">coeffs</span> <span class="o">=</span> <span class="p">[</span> <span class="n">coeff</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-                   <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()</span> <span class="p">]</span>
-
-        <span class="n">dps</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span>
-        <span class="n">sympy</span><span class="o">.</span><span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="n">n</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">coeffs</span> <span class="o">=</span> <span class="p">[</span> <span class="n">sympy</span><span class="o">.</span><span class="n">mpmath</span><span class="o">.</span><span class="n">mpc</span><span class="p">(</span><span class="o">*</span><span class="n">coeff</span><span class="p">)</span> <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">coeffs</span> <span class="p">]</span>
-            <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">DomainError</span><span class="p">(</span>
-                    <span class="s">&quot;numerical domain expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-
-            <span class="n">result</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">mpmath</span><span class="o">.</span><span class="n">polyroots</span><span class="p">(</span>
-                <span class="n">coeffs</span><span class="p">,</span> <span class="n">maxsteps</span><span class="o">=</span><span class="n">maxsteps</span><span class="p">,</span> <span class="n">cleanup</span><span class="o">=</span><span class="n">cleanup</span><span class="p">,</span> <span class="n">error</span><span class="o">=</span><span class="n">error</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">error</span><span class="p">:</span>
-                <span class="n">roots</span><span class="p">,</span> <span class="n">error</span> <span class="o">=</span> <span class="n">result</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">roots</span><span class="p">,</span> <span class="n">error</span> <span class="o">=</span> <span class="n">result</span><span class="p">,</span> <span class="bp">None</span>
-
-            <span class="n">roots</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">roots</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">r</span><span class="p">:</span> <span class="p">(</span><span class="n">r</span><span class="o">.</span><span class="n">real</span><span class="p">,</span> <span class="n">r</span><span class="o">.</span><span class="n">imag</span><span class="p">))))</span>
-        <span class="k">finally</span><span class="p">:</span>
-            <span class="n">sympy</span><span class="o">.</span><span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="n">dps</span>
-
-        <span class="k">if</span> <span class="n">error</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">roots</span><span class="p">,</span> <span class="n">sympify</span><span class="p">(</span><span class="n">error</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">roots</span>
-</div>
-<div class="viewcode-block" id="Poly.ground_roots"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.ground_roots">[docs]</a>    <span class="k">def</span> <span class="nf">ground_roots</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Compute roots of ``f`` by factorization in the ground domain.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**6 - 4*x**4 + 4*x**3 - x**2).ground_roots()</span>
-<span class="sd">        {0: 2, 1: 2}</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_multivariate</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">MultivariatePolynomialError</span><span class="p">(</span>
-                <span class="s">&quot;can&#39;t compute ground roots of </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">f</span><span class="p">)</span>
-
-        <span class="n">roots</span> <span class="o">=</span> <span class="p">{}</span>
-
-        <span class="k">for</span> <span class="n">factor</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">factor_list</span><span class="p">()[</span><span class="mi">1</span><span class="p">]:</span>
-            <span class="k">if</span> <span class="n">factor</span><span class="o">.</span><span class="n">is_linear</span><span class="p">:</span>
-                <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">factor</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()</span>
-                <span class="n">roots</span><span class="p">[</span><span class="o">-</span><span class="n">b</span><span class="o">/</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="n">k</span>
-
-        <span class="k">return</span> <span class="n">roots</span>
-</div>
-<div class="viewcode-block" id="Poly.nth_power_roots_poly"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.nth_power_roots_poly">[docs]</a>    <span class="k">def</span> <span class="nf">nth_power_roots_poly</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Construct a polynomial with n-th powers of roots of ``f``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; f = Poly(x**4 - x**2 + 1)</span>
-
-<span class="sd">        &gt;&gt;&gt; f.nth_power_roots_poly(2)</span>
-<span class="sd">        Poly(x**4 - 2*x**3 + 3*x**2 - 2*x + 1, x, domain=&#39;ZZ&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; f.nth_power_roots_poly(3)</span>
-<span class="sd">        Poly(x**4 + 2*x**2 + 1, x, domain=&#39;ZZ&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; f.nth_power_roots_poly(4)</span>
-<span class="sd">        Poly(x**4 + 2*x**3 + 3*x**2 + 2*x + 1, x, domain=&#39;ZZ&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; f.nth_power_roots_poly(12)</span>
-<span class="sd">        Poly(x**4 - 4*x**3 + 6*x**2 - 4*x + 1, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_multivariate</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">MultivariatePolynomialError</span><span class="p">(</span>
-                <span class="s">&quot;must be a univariate polynomial&quot;</span><span class="p">)</span>
-
-        <span class="n">N</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">N</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">N</span> <span class="o">&gt;=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;&#39;n&#39; must an integer and n &gt;= 1, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">gen</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">)</span>
-
-        <span class="n">r</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">resultant</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">from_expr</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span> <span class="o">-</span> <span class="n">t</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">r</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Poly.cancel"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.cancel">[docs]</a>    <span class="k">def</span> <span class="nf">cancel</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">include</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Cancel common factors in a rational function ``f/g``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(2*x**2 - 2, x).cancel(Poly(x**2 - 2*x + 1, x))</span>
-<span class="sd">        (1, Poly(2*x + 2, x, domain=&#39;ZZ&#39;), Poly(x - 1, x, domain=&#39;ZZ&#39;))</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(2*x**2 - 2, x).cancel(Poly(x**2 - 2*x + 1, x), include=True)</span>
-<span class="sd">        (Poly(2*x + 2, x, domain=&#39;ZZ&#39;), Poly(x - 1, x, domain=&#39;ZZ&#39;))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="s">&#39;cancel&#39;</span><span class="p">):</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">cancel</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">include</span><span class="o">=</span><span class="n">include</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>  <span class="c"># pragma: no cover</span>
-            <span class="k">raise</span> <span class="n">OperationNotSupported</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;cancel&#39;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">include</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">dom</span><span class="o">.</span><span class="n">has_assoc_Ring</span><span class="p">:</span>
-                <span class="n">dom</span> <span class="o">=</span> <span class="n">dom</span><span class="o">.</span><span class="n">get_ring</span><span class="p">()</span>
-
-            <span class="n">cp</span><span class="p">,</span> <span class="n">cq</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">result</span>
-
-            <span class="n">cp</span> <span class="o">=</span> <span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">cp</span><span class="p">)</span>
-            <span class="n">cq</span> <span class="o">=</span> <span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">cq</span><span class="p">)</span>
-
-            <span class="k">return</span> <span class="n">cp</span><span class="o">/</span><span class="n">cq</span><span class="p">,</span> <span class="n">per</span><span class="p">(</span><span class="n">p</span><span class="p">),</span> <span class="n">per</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">per</span><span class="p">,</span> <span class="n">result</span><span class="p">))</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Poly.is_zero"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.is_zero">[docs]</a>    <span class="k">def</span> <span class="nf">is_zero</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns ``True`` if ``f`` is a zero polynomial.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(0, x).is_zero</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Poly(1, x).is_zero</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">is_zero</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Poly.is_one"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.is_one">[docs]</a>    <span class="k">def</span> <span class="nf">is_one</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns ``True`` if ``f`` is a unit polynomial.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(0, x).is_one</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Poly(1, x).is_one</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">is_one</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Poly.is_sqf"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.is_sqf">[docs]</a>    <span class="k">def</span> <span class="nf">is_sqf</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns ``True`` if ``f`` is a square-free polynomial.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 - 2*x + 1, x).is_sqf</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 - 1, x).is_sqf</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">is_sqf</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Poly.is_monic"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.is_monic">[docs]</a>    <span class="k">def</span> <span class="nf">is_monic</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns ``True`` if the leading coefficient of ``f`` is one.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x + 2, x).is_monic</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Poly(2*x + 2, x).is_monic</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">is_monic</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Poly.is_primitive"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.is_primitive">[docs]</a>    <span class="k">def</span> <span class="nf">is_primitive</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns ``True`` if GCD of the coefficients of ``f`` is one.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(2*x**2 + 6*x + 12, x).is_primitive</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 3*x + 6, x).is_primitive</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">is_primitive</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Poly.is_ground"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.is_ground">[docs]</a>    <span class="k">def</span> <span class="nf">is_ground</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns ``True`` if ``f`` is an element of the ground domain.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x, x).is_ground</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Poly(2, x).is_ground</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Poly(y, x).is_ground</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">is_ground</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Poly.is_linear"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.is_linear">[docs]</a>    <span class="k">def</span> <span class="nf">is_linear</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns ``True`` if ``f`` is linear in all its variables.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x + y + 2, x, y).is_linear</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x*y + 2, x, y).is_linear</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">is_linear</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Poly.is_quadratic"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.is_quadratic">[docs]</a>    <span class="k">def</span> <span class="nf">is_quadratic</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns ``True`` if ``f`` is quadratic in all its variables.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x*y + 2, x, y).is_quadratic</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x*y**2 + 2, x, y).is_quadratic</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">is_quadratic</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Poly.is_monomial"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.is_monomial">[docs]</a>    <span class="k">def</span> <span class="nf">is_monomial</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns ``True`` if ``f`` is zero or has only one term.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(3*x**2, x).is_monomial</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Poly(3*x**2 + 1, x).is_monomial</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">is_monomial</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Poly.is_homogeneous"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.is_homogeneous">[docs]</a>    <span class="k">def</span> <span class="nf">is_homogeneous</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns ``True`` if ``f`` is a homogeneous polynomial.</span>
-
-<span class="sd">        A homogeneous polynomial is a polynomial whose all monomials with</span>
-<span class="sd">        non-zero coefficients have the same total degree. If you want not</span>
-<span class="sd">        only to check if a polynomial is homogeneous but also compute its</span>
-<span class="sd">        homogeneous order, then use :func:`Poly.homogeneous_order`.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + x*y, x, y).is_homogeneous</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x**3 + x*y, x, y).is_homogeneous</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">is_homogeneous</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Poly.is_irreducible"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.is_irreducible">[docs]</a>    <span class="k">def</span> <span class="nf">is_irreducible</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns ``True`` if ``f`` has no factors over its domain.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + x + 1, x, modulus=2).is_irreducible</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + 1, x, modulus=2).is_irreducible</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">is_irreducible</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Poly.is_univariate"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.is_univariate">[docs]</a>    <span class="k">def</span> <span class="nf">is_univariate</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns ``True`` if ``f`` is a univariate polynomial.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + x + 1, x).is_univariate</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x*y**2 + x*y + 1, x, y).is_univariate</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x*y**2 + x*y + 1, x).is_univariate</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + x + 1, x, y).is_univariate</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Poly.is_multivariate"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.is_multivariate">[docs]</a>    <span class="k">def</span> <span class="nf">is_multivariate</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns ``True`` if ``f`` is a multivariate polynomial.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + x + 1, x).is_multivariate</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x*y**2 + x*y + 1, x, y).is_multivariate</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x*y**2 + x*y + 1, x).is_multivariate</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Poly(x**2 + x + 1, x, y).is_multivariate</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Poly.is_cyclotomic"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.Poly.is_cyclotomic">[docs]</a>    <span class="k">def</span> <span class="nf">is_cyclotomic</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns ``True`` if ``f`` is a cyclotomic polnomial.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; f = x**16 + x**14 - x**10 + x**8 - x**6 + x**2 + 1</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(f).is_cyclotomic</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &gt;&gt;&gt; g = x**16 + x**14 - x**10 - x**8 - x**6 + x**2 + 1</span>
-
-<span class="sd">        &gt;&gt;&gt; Poly(g).is_cyclotomic</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">is_cyclotomic</span>
-</div>
-    <span class="k">def</span> <span class="nf">__abs__</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">abs</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__neg__</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">neg</span><span class="p">()</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">PolynomialError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span> <span class="o">+</span> <span class="n">g</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__radd__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">PolynomialError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">g</span> <span class="o">+</span> <span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-
-        <span class="k">return</span> <span class="n">g</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__sub__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">PolynomialError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span> <span class="o">-</span> <span class="n">g</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">sub</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__rsub__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">PolynomialError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">g</span> <span class="o">-</span> <span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-
-        <span class="k">return</span> <span class="n">g</span><span class="o">.</span><span class="n">sub</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">PolynomialError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span><span class="o">*</span><span class="n">g</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__rmul__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">PolynomialError</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">g</span><span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-
-        <span class="k">return</span> <span class="n">g</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__pow__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">n</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span><span class="o">**</span><span class="n">n</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__divmod__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">div</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__rdivmod__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">g</span><span class="o">.</span><span class="n">div</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__mod__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rem</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__rmod__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">g</span><span class="o">.</span><span class="n">rem</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__floordiv__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__rfloordiv__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">g</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span><span class="o">/</span><span class="n">g</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__rdiv__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">g</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span><span class="o">/</span><span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-
-    <span class="n">__truediv__</span> <span class="o">=</span> <span class="n">__div__</span>
-    <span class="n">__rtruediv__</span> <span class="o">=</span> <span class="n">__rdiv__</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="n">f</span><span class="o">.</span><span class="n">get_domain</span><span class="p">())</span>
-            <span class="k">except</span> <span class="p">(</span><span class="n">PolynomialError</span><span class="p">,</span> <span class="n">DomainError</span><span class="p">,</span> <span class="n">CoercionFailed</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span> <span class="o">!=</span> <span class="n">g</span><span class="o">.</span><span class="n">gens</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span> <span class="o">!=</span> <span class="n">g</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">dom</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">UnificationFailed</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-
-            <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">set_domain</span><span class="p">(</span><span class="n">dom</span><span class="p">)</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">set_domain</span><span class="p">(</span><span class="n">dom</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span> <span class="o">==</span> <span class="n">g</span><span class="o">.</span><span class="n">rep</span>
-
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__bool__</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">is_zero</span>
-
-    <span class="k">def</span> <span class="nf">eq</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">strict</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">_strict_eq</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">g</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">ne</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">eq</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="n">strict</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_strict_eq</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="p">)</span> <span class="ow">and</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span> <span class="o">==</span> <span class="n">g</span><span class="o">.</span><span class="n">gens</span> <span class="ow">and</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">eq</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="PurePoly"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.PurePoly">[docs]</a><span class="k">class</span> <span class="nc">PurePoly</span><span class="p">(</span><span class="n">Poly</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Class for representing pure polynomials. &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_hashable_content</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Allow SymPy to hash Poly instances. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rep</span><span class="p">,)</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">PurePoly</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="PurePoly.free_symbols"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.PurePoly.free_symbols">[docs]</a>    <span class="k">def</span> <span class="nf">free_symbols</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Free symbols of a polynomial.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import PurePoly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; PurePoly(x**2 + 1).free_symbols</span>
-<span class="sd">        set()</span>
-<span class="sd">        &gt;&gt;&gt; PurePoly(x**2 + y).free_symbols</span>
-<span class="sd">        set()</span>
-<span class="sd">        &gt;&gt;&gt; PurePoly(x**2 + y, x).free_symbols</span>
-<span class="sd">        set([y])</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">free_symbols_in_domain</span>
-</div>
-    <span class="nd">@_sympifyit</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">,</span> <span class="bp">NotImplemented</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="n">f</span><span class="o">.</span><span class="n">get_domain</span><span class="p">())</span>
-            <span class="k">except</span> <span class="p">(</span><span class="n">PolynomialError</span><span class="p">,</span> <span class="n">DomainError</span><span class="p">,</span> <span class="n">CoercionFailed</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">gens</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span> <span class="o">!=</span> <span class="n">g</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">dom</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">UnificationFailed</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-
-            <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">set_domain</span><span class="p">(</span><span class="n">dom</span><span class="p">)</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">set_domain</span><span class="p">(</span><span class="n">dom</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span> <span class="o">==</span> <span class="n">g</span><span class="o">.</span><span class="n">rep</span>
-
-    <span class="k">def</span> <span class="nf">_strict_eq</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="p">)</span> <span class="ow">and</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">eq</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_unify</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">):</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">per</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">from_sympy</span><span class="p">(</span><span class="n">g</span><span class="p">))</span>
-            <span class="k">except</span> <span class="n">CoercionFailed</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">UnificationFailed</span><span class="p">(</span><span class="s">&quot;can&#39;t unify </span><span class="si">%s</span><span class="s"> with </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">gens</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="n">UnificationFailed</span><span class="p">(</span><span class="s">&quot;can&#39;t unify </span><span class="si">%s</span><span class="s"> with </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">DMP</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">DMP</span><span class="p">)):</span>
-            <span class="k">raise</span> <span class="n">UnificationFailed</span><span class="p">(</span><span class="s">&quot;can&#39;t unify </span><span class="si">%s</span><span class="s"> with </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">))</span>
-
-        <span class="n">cls</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span>
-
-        <span class="n">dom</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">gens</span><span class="p">)</span>
-
-        <span class="n">F</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">dom</span><span class="p">)</span>
-        <span class="n">G</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">dom</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">per</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="n">dom</span><span class="o">=</span><span class="n">dom</span><span class="p">,</span> <span class="n">gens</span><span class="o">=</span><span class="n">gens</span><span class="p">,</span> <span class="n">remove</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">remove</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">gens</span> <span class="o">=</span> <span class="n">gens</span><span class="p">[:</span><span class="n">remove</span><span class="p">]</span> <span class="o">+</span> <span class="n">gens</span><span class="p">[</span><span class="n">remove</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">gens</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">dom</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">rep</span><span class="p">)</span>
-
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">dom</span><span class="p">,</span> <span class="n">per</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span>
-
-</div>
-<div class="viewcode-block" id="poly_from_expr"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.poly_from_expr">[docs]</a><span class="k">def</span> <span class="nf">poly_from_expr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Construct a polynomial from an expression. &quot;&quot;&quot;</span>
-    <span class="n">opt</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">build_options</span><span class="p">(</span><span class="n">gens</span><span class="p">,</span> <span class="n">args</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">_poly_from_expr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_poly_from_expr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">opt</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Construct a polynomial from an expression. &quot;&quot;&quot;</span>
-    <span class="n">orig</span><span class="p">,</span> <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="p">,</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="n">PolificationFailed</span><span class="p">(</span><span class="n">opt</span><span class="p">,</span> <span class="n">orig</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">_from_poly</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-
-        <span class="n">opt</span><span class="p">[</span><span class="s">&#39;gens&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">gens</span>
-        <span class="n">opt</span><span class="p">[</span><span class="s">&#39;domain&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">domain</span>
-
-        <span class="k">if</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">opt</span><span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="k">return</span> <span class="n">poly</span><span class="p">,</span> <span class="n">opt</span>
-    <span class="k">elif</span> <span class="n">opt</span><span class="o">.</span><span class="n">expand</span><span class="p">:</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">rep</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">_dict_from_expr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">GeneratorsNeeded</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">PolificationFailed</span><span class="p">(</span><span class="n">opt</span><span class="p">,</span> <span class="n">orig</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-
-    <span class="n">monoms</span><span class="p">,</span> <span class="n">coeffs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="nb">list</span><span class="p">(</span><span class="n">rep</span><span class="o">.</span><span class="n">items</span><span class="p">())))</span>
-    <span class="n">domain</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span>
-
-    <span class="k">if</span> <span class="n">domain</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">domain</span><span class="p">,</span> <span class="n">coeffs</span> <span class="o">=</span> <span class="n">construct_domain</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">opt</span><span class="o">=</span><span class="n">opt</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">coeffs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">domain</span><span class="o">.</span><span class="n">from_sympy</span><span class="p">,</span> <span class="n">coeffs</span><span class="p">))</span>
-
-    <span class="n">level</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="n">poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span>
-        <span class="n">DMP</span><span class="o">.</span><span class="n">from_monoms_coeffs</span><span class="p">(</span><span class="n">monoms</span><span class="p">,</span> <span class="n">coeffs</span><span class="p">,</span> <span class="n">level</span><span class="p">,</span> <span class="n">domain</span><span class="p">),</span> <span class="o">*</span><span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-
-    <span class="n">opt</span><span class="p">[</span><span class="s">&#39;domain&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">domain</span>
-
-    <span class="k">if</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">opt</span><span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="k">return</span> <span class="n">poly</span><span class="p">,</span> <span class="n">opt</span>
-
-
-<div class="viewcode-block" id="parallel_poly_from_expr"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.parallel_poly_from_expr">[docs]</a><span class="k">def</span> <span class="nf">parallel_poly_from_expr</span><span class="p">(</span><span class="n">exprs</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Construct polynomials from expressions. &quot;&quot;&quot;</span>
-    <span class="n">opt</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">build_options</span><span class="p">(</span><span class="n">gens</span><span class="p">,</span> <span class="n">args</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">_parallel_poly_from_expr</span><span class="p">(</span><span class="n">exprs</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_parallel_poly_from_expr</span><span class="p">(</span><span class="n">exprs</span><span class="p">,</span> <span class="n">opt</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Construct polynomials from expressions. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">exprs</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="n">f</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">exprs</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">Poly</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">Poly</span><span class="p">):</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">_from_poly</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">_from_poly</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-
-            <span class="n">f</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-            <span class="n">opt</span><span class="p">[</span><span class="s">&#39;gens&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span>
-            <span class="n">opt</span><span class="p">[</span><span class="s">&#39;domain&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">domain</span>
-
-            <span class="k">if</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">opt</span><span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">True</span>
-
-            <span class="k">return</span> <span class="p">[</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">],</span> <span class="n">opt</span>
-
-    <span class="n">origs</span><span class="p">,</span> <span class="n">exprs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">exprs</span><span class="p">),</span> <span class="p">[]</span>
-    <span class="n">_exprs</span><span class="p">,</span> <span class="n">_polys</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
-
-    <span class="n">failed</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">expr</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">origs</span><span class="p">):</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-                <span class="n">_polys</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">_exprs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">opt</span><span class="o">.</span><span class="n">expand</span><span class="p">:</span>
-                    <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">failed</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="n">exprs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">failed</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">PolificationFailed</span><span class="p">(</span><span class="n">opt</span><span class="p">,</span> <span class="n">origs</span><span class="p">,</span> <span class="n">exprs</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">_polys</span><span class="p">:</span>
-        <span class="c"># XXX: this is a temporary solution</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">_polys</span><span class="p">:</span>
-            <span class="n">exprs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">exprs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">reps</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">_parallel_dict_from_expr</span><span class="p">(</span><span class="n">exprs</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">GeneratorsNeeded</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">PolificationFailed</span><span class="p">(</span><span class="n">opt</span><span class="p">,</span> <span class="n">origs</span><span class="p">,</span> <span class="n">exprs</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-
-    <span class="n">coeffs_list</span><span class="p">,</span> <span class="n">lengths</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
-
-    <span class="n">all_monoms</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">all_coeffs</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">rep</span> <span class="ow">in</span> <span class="n">reps</span><span class="p">:</span>
-        <span class="n">monoms</span><span class="p">,</span> <span class="n">coeffs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="nb">list</span><span class="p">(</span><span class="n">rep</span><span class="o">.</span><span class="n">items</span><span class="p">())))</span>
-
-        <span class="n">coeffs_list</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">coeffs</span><span class="p">)</span>
-        <span class="n">all_monoms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">monoms</span><span class="p">)</span>
-
-        <span class="n">lengths</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">coeffs</span><span class="p">))</span>
-
-    <span class="n">domain</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span>
-
-    <span class="k">if</span> <span class="n">domain</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">domain</span><span class="p">,</span> <span class="n">coeffs_list</span> <span class="o">=</span> <span class="n">construct_domain</span><span class="p">(</span><span class="n">coeffs_list</span><span class="p">,</span> <span class="n">opt</span><span class="o">=</span><span class="n">opt</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">coeffs_list</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">domain</span><span class="o">.</span><span class="n">from_sympy</span><span class="p">,</span> <span class="n">coeffs_list</span><span class="p">))</span>
-
-    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">lengths</span><span class="p">:</span>
-        <span class="n">all_coeffs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">coeffs_list</span><span class="p">[:</span><span class="n">k</span><span class="p">])</span>
-        <span class="n">coeffs_list</span> <span class="o">=</span> <span class="n">coeffs_list</span><span class="p">[</span><span class="n">k</span><span class="p">:]</span>
-
-    <span class="n">polys</span><span class="p">,</span> <span class="n">level</span> <span class="o">=</span> <span class="p">[],</span> <span class="nb">len</span><span class="p">(</span><span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="k">for</span> <span class="n">monoms</span><span class="p">,</span> <span class="n">coeffs</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">all_monoms</span><span class="p">,</span> <span class="n">all_coeffs</span><span class="p">):</span>
-        <span class="n">rep</span> <span class="o">=</span> <span class="n">DMP</span><span class="o">.</span><span class="n">from_monoms_coeffs</span><span class="p">(</span><span class="n">monoms</span><span class="p">,</span> <span class="n">coeffs</span><span class="p">,</span> <span class="n">level</span><span class="p">,</span> <span class="n">domain</span><span class="p">)</span>
-        <span class="n">polys</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Poly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">rep</span><span class="p">,</span> <span class="o">*</span><span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">))</span>
-
-    <span class="n">opt</span><span class="p">[</span><span class="s">&#39;domain&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">domain</span>
-
-    <span class="k">if</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">opt</span><span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">bool</span><span class="p">(</span><span class="n">_polys</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">polys</span><span class="p">,</span> <span class="n">opt</span>
-
-
-<span class="k">def</span> <span class="nf">_update_args</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="p">,</span> <span class="n">value</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Add a new ``(key, value)`` pair to arguments ``dict``. &quot;&quot;&quot;</span>
-    <span class="n">args</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">key</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-        <span class="n">args</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">=</span> <span class="n">value</span>
-
-    <span class="k">return</span> <span class="n">args</span>
-
-
-<div class="viewcode-block" id="degree"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.degree">[docs]</a><span class="k">def</span> <span class="nf">degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the degree of ``f`` in the given variable.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import degree</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">    &gt;&gt;&gt; degree(x**2 + y*x + 1, gen=x)</span>
-<span class="sd">    2</span>
-<span class="sd">    &gt;&gt;&gt; degree(x**2 + y*x + 1, gen=y)</span>
-<span class="sd">    1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;gen&#39;</span><span class="p">,</span> <span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">poly_from_expr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;degree&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="n">F</span><span class="o">.</span><span class="n">degree</span><span class="p">(</span><span class="n">opt</span><span class="o">.</span><span class="n">gen</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="degree_list"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.degree_list">[docs]</a><span class="k">def</span> <span class="nf">degree_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a list of degrees of ``f`` in all variables.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import degree_list</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">    &gt;&gt;&gt; degree_list(x**2 + y*x + 1)</span>
-<span class="sd">    (2, 1)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">poly_from_expr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;degree_list&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">degrees</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">degree_list</span><span class="p">()</span>
-
-    <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Integer</span><span class="p">,</span> <span class="n">degrees</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="LC"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.LC">[docs]</a><span class="k">def</span> <span class="nf">LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the leading coefficient of ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import LC</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">    &gt;&gt;&gt; LC(4*x**2 + 2*x*y**2 + x*y + 3*y)</span>
-<span class="sd">    4</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">poly_from_expr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;LC&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">LC</span><span class="p">(</span><span class="n">order</span><span class="o">=</span><span class="n">opt</span><span class="o">.</span><span class="n">order</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="LM"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.LM">[docs]</a><span class="k">def</span> <span class="nf">LM</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the leading monomial of ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import LM</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">    &gt;&gt;&gt; LM(4*x**2 + 2*x*y**2 + x*y + 3*y)</span>
-<span class="sd">    x**2</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">poly_from_expr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;LM&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">monom</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">LM</span><span class="p">(</span><span class="n">order</span><span class="o">=</span><span class="n">opt</span><span class="o">.</span><span class="n">order</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">monom</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-
-</div>
-<div class="viewcode-block" id="LT"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.LT">[docs]</a><span class="k">def</span> <span class="nf">LT</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the leading term of ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import LT</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">    &gt;&gt;&gt; LT(4*x**2 + 2*x*y**2 + x*y + 3*y)</span>
-<span class="sd">    4*x**2</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">poly_from_expr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;LT&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">LT</span><span class="p">(</span><span class="n">order</span><span class="o">=</span><span class="n">opt</span><span class="o">.</span><span class="n">order</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">coeff</span><span class="o">*</span><span class="n">monom</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-
-</div>
-<div class="viewcode-block" id="pdiv"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.pdiv">[docs]</a><span class="k">def</span> <span class="nf">pdiv</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute polynomial pseudo-division of ``f`` and ``g``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import pdiv</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; pdiv(x**2 + 1, 2*x - 4)</span>
-<span class="sd">    (2*x + 4, 20)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">),</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">),</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;pdiv&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">pdiv</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">q</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(),</span> <span class="n">r</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">q</span><span class="p">,</span> <span class="n">r</span>
-
-</div>
-<div class="viewcode-block" id="prem"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.prem">[docs]</a><span class="k">def</span> <span class="nf">prem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute polynomial pseudo-remainder of ``f`` and ``g``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import prem</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; prem(x**2 + 1, 2*x - 4)</span>
-<span class="sd">    20</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">),</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">),</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;prem&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">r</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">prem</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">r</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">r</span>
-
-</div>
-<div class="viewcode-block" id="pquo"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.pquo">[docs]</a><span class="k">def</span> <span class="nf">pquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute polynomial pseudo-quotient of ``f`` and ``g``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import pquo</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; pquo(x**2 + 1, 2*x - 4)</span>
-<span class="sd">    2*x + 4</span>
-<span class="sd">    &gt;&gt;&gt; pquo(x**2 - 1, 2*x - 1)</span>
-<span class="sd">    2*x + 1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">),</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">),</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;pquo&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">q</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">pquo</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">ExactQuotientFailed</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ExactQuotientFailed</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">q</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">q</span>
-
-</div>
-<div class="viewcode-block" id="pexquo"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.pexquo">[docs]</a><span class="k">def</span> <span class="nf">pexquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute polynomial exact pseudo-quotient of ``f`` and ``g``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import pexquo</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; pexquo(x**2 - 1, 2*x - 2)</span>
-<span class="sd">    2*x + 2</span>
-
-<span class="sd">    &gt;&gt;&gt; pexquo(x**2 + 1, 2*x - 4)</span>
-<span class="sd">    Traceback (most recent call last):</span>
-<span class="sd">    ...</span>
-<span class="sd">    ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">),</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">),</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;pexquo&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">q</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">pexquo</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">q</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">q</span>
-
-</div>
-<div class="viewcode-block" id="div"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.div">[docs]</a><span class="k">def</span> <span class="nf">div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute polynomial division of ``f`` and ``g``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import div, ZZ, QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; div(x**2 + 1, 2*x - 4, domain=ZZ)</span>
-<span class="sd">    (0, x**2 + 1)</span>
-<span class="sd">    &gt;&gt;&gt; div(x**2 + 1, 2*x - 4, domain=QQ)</span>
-<span class="sd">    (x/2 + 1, 5)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;auto&#39;</span><span class="p">,</span> <span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">),</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">),</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;div&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">div</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="n">opt</span><span class="o">.</span><span class="n">auto</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">q</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(),</span> <span class="n">r</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">q</span><span class="p">,</span> <span class="n">r</span>
-
-</div>
-<div class="viewcode-block" id="rem"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.rem">[docs]</a><span class="k">def</span> <span class="nf">rem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute polynomial remainder of ``f`` and ``g``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import rem, ZZ, QQ</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; rem(x**2 + 1, 2*x - 4, domain=ZZ)</span>
-<span class="sd">    x**2 + 1</span>
-<span class="sd">    &gt;&gt;&gt; rem(x**2 + 1, 2*x - 4, domain=QQ)</span>
-<span class="sd">    5</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;auto&#39;</span><span class="p">,</span> <span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">),</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">),</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;rem&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">r</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">rem</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="n">opt</span><span class="o">.</span><span class="n">auto</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">r</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">r</span>
-
-</div>
-<div class="viewcode-block" id="quo"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.quo">[docs]</a><span class="k">def</span> <span class="nf">quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute polynomial quotient of ``f`` and ``g``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import quo</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; quo(x**2 + 1, 2*x - 4)</span>
-<span class="sd">    x/2 + 1</span>
-<span class="sd">    &gt;&gt;&gt; quo(x**2 - 1, x - 1)</span>
-<span class="sd">    x + 1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;auto&#39;</span><span class="p">,</span> <span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">),</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">),</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;quo&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">q</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="n">opt</span><span class="o">.</span><span class="n">auto</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">q</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">q</span>
-
-</div>
-<div class="viewcode-block" id="exquo"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.exquo">[docs]</a><span class="k">def</span> <span class="nf">exquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute polynomial exact quotient of ``f`` and ``g``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import exquo</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; exquo(x**2 - 1, x - 1)</span>
-<span class="sd">    x + 1</span>
-
-<span class="sd">    &gt;&gt;&gt; exquo(x**2 + 1, 2*x - 4)</span>
-<span class="sd">    Traceback (most recent call last):</span>
-<span class="sd">    ...</span>
-<span class="sd">    ExactQuotientFailed: 2*x - 4 does not divide x**2 + 1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;auto&#39;</span><span class="p">,</span> <span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">),</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">),</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;exquo&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">q</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">exquo</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="n">opt</span><span class="o">.</span><span class="n">auto</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">q</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">q</span>
-
-</div>
-<div class="viewcode-block" id="half_gcdex"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.half_gcdex">[docs]</a><span class="k">def</span> <span class="nf">half_gcdex</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Half extended Euclidean algorithm of ``f`` and ``g``.</span>
-
-<span class="sd">    Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import half_gcdex</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; half_gcdex(x**4 - 2*x**3 - 6*x**2 + 12*x + 15, x**3 + x**2 - 4*x - 4)</span>
-<span class="sd">    (-x/5 + 3/5, x + 1)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;auto&#39;</span><span class="p">,</span> <span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">),</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">),</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="n">domain</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">=</span> <span class="n">construct_domain</span><span class="p">(</span><span class="n">exc</span><span class="o">.</span><span class="n">exprs</span><span class="p">)</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">s</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">half_gcdex</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;half_gcdex&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">domain</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="n">domain</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-
-    <span class="n">s</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">half_gcdex</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="n">opt</span><span class="o">.</span><span class="n">auto</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">s</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(),</span> <span class="n">h</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">s</span><span class="p">,</span> <span class="n">h</span>
-
-</div>
-<div class="viewcode-block" id="gcdex"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.gcdex">[docs]</a><span class="k">def</span> <span class="nf">gcdex</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Extended Euclidean algorithm of ``f`` and ``g``.</span>
-
-<span class="sd">    Returns ``(s, t, h)`` such that ``h = gcd(f, g)`` and ``s*f + t*g = h``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import gcdex</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; gcdex(x**4 - 2*x**3 - 6*x**2 + 12*x + 15, x**3 + x**2 - 4*x - 4)</span>
-<span class="sd">    (-x/5 + 3/5, x**2/5 - 6*x/5 + 2, x + 1)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;auto&#39;</span><span class="p">,</span> <span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">),</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">),</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="n">domain</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">=</span> <span class="n">construct_domain</span><span class="p">(</span><span class="n">exc</span><span class="o">.</span><span class="n">exprs</span><span class="p">)</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">gcdex</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;gcdex&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">domain</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="n">domain</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="n">domain</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-
-    <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">gcdex</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="n">opt</span><span class="o">.</span><span class="n">auto</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">s</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(),</span> <span class="n">t</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(),</span> <span class="n">h</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">h</span>
-
-</div>
-<div class="viewcode-block" id="invert"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.invert">[docs]</a><span class="k">def</span> <span class="nf">invert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Invert ``f`` modulo ``g`` when possible.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import invert</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; invert(x**2 - 1, 2*x - 1)</span>
-<span class="sd">    -4/3</span>
-
-<span class="sd">    &gt;&gt;&gt; invert(x**2 - 1, x - 1)</span>
-<span class="sd">    Traceback (most recent call last):</span>
-<span class="sd">    ...</span>
-<span class="sd">    NotInvertible: zero divisor</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;auto&#39;</span><span class="p">,</span> <span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">),</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">),</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="n">domain</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">=</span> <span class="n">construct_domain</span><span class="p">(</span><span class="n">exc</span><span class="o">.</span><span class="n">exprs</span><span class="p">)</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">domain</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">domain</span><span class="o">.</span><span class="n">invert</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-        <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;invert&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">invert</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="n">opt</span><span class="o">.</span><span class="n">auto</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">h</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">h</span>
-
-</div>
-<div class="viewcode-block" id="subresultants"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.subresultants">[docs]</a><span class="k">def</span> <span class="nf">subresultants</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute subresultant PRS of ``f`` and ``g``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import subresultants</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; subresultants(x**2 + 1, x**2 - 1)</span>
-<span class="sd">    [x**2 + 1, x**2 - 1, -2]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">),</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">),</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;subresultants&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">subresultants</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="n">r</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">result</span> <span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="resultant"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.resultant">[docs]</a><span class="k">def</span> <span class="nf">resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute resultant of ``f`` and ``g``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import resultant</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; resultant(x**2 + 1, x**2 - 1)</span>
-<span class="sd">    4</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">includePRS</span> <span class="o">=</span> <span class="n">args</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;includePRS&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">),</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">),</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;resultant&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">includePRS</span><span class="p">:</span>
-        <span class="n">result</span><span class="p">,</span> <span class="n">R</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">resultant</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">includePRS</span><span class="o">=</span><span class="n">includePRS</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">resultant</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">includePRS</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">result</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(),</span> <span class="p">[</span><span class="n">r</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">R</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">result</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">includePRS</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">result</span><span class="p">,</span> <span class="n">R</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="discriminant"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.discriminant">[docs]</a><span class="k">def</span> <span class="nf">discriminant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute discriminant of ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import discriminant</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; discriminant(x**2 + 2*x + 3)</span>
-<span class="sd">    -8</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">poly_from_expr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;discriminant&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">discriminant</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="cofactors"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.cofactors">[docs]</a><span class="k">def</span> <span class="nf">cofactors</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute GCD and cofactors of ``f`` and ``g``.</span>
-
-<span class="sd">    Returns polynomials ``(h, cff, cfg)`` such that ``h = gcd(f, g)``, and</span>
-<span class="sd">    ``cff = quo(f, h)`` and ``cfg = quo(g, h)`` are, so called, cofactors</span>
-<span class="sd">    of ``f`` and ``g``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import cofactors</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; cofactors(x**2 - 1, x**2 - 3*x + 2)</span>
-<span class="sd">    (x - 1, x + 1, x - 2)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">),</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">),</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="n">domain</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">=</span> <span class="n">construct_domain</span><span class="p">(</span><span class="n">exc</span><span class="o">.</span><span class="n">exprs</span><span class="p">)</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">cff</span><span class="p">,</span> <span class="n">cfg</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">cofactors</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;cofactors&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">domain</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">h</span><span class="p">),</span> <span class="n">domain</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">cff</span><span class="p">),</span> <span class="n">domain</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">cfg</span><span class="p">)</span>
-
-    <span class="n">h</span><span class="p">,</span> <span class="n">cff</span><span class="p">,</span> <span class="n">cfg</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">cofactors</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">h</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(),</span> <span class="n">cff</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(),</span> <span class="n">cfg</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">h</span><span class="p">,</span> <span class="n">cff</span><span class="p">,</span> <span class="n">cfg</span>
-
-</div>
-<div class="viewcode-block" id="gcd_list"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.gcd_list">[docs]</a><span class="k">def</span> <span class="nf">gcd_list</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute GCD of a list of polynomials.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import gcd_list</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; gcd_list([x**3 - 1, x**2 - 1, x**2 - 3*x + 2])</span>
-<span class="sd">    x - 1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">seq</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">seq</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">gens</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">args</span><span class="p">:</span>
-        <span class="n">domain</span><span class="p">,</span> <span class="n">numbers</span> <span class="o">=</span> <span class="n">construct_domain</span><span class="p">(</span><span class="n">seq</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">numbers</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">domain</span><span class="o">.</span><span class="n">zero</span>
-        <span class="k">elif</span> <span class="n">domain</span><span class="o">.</span><span class="n">is_Numerical</span><span class="p">:</span>
-            <span class="n">result</span><span class="p">,</span> <span class="n">numbers</span> <span class="o">=</span> <span class="n">numbers</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">numbers</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-            <span class="k">for</span> <span class="n">number</span> <span class="ow">in</span> <span class="n">numbers</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">number</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">domain</span><span class="o">.</span><span class="n">is_one</span><span class="p">(</span><span class="n">result</span><span class="p">):</span>
-                    <span class="k">break</span>
-
-            <span class="k">return</span> <span class="n">domain</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">polys</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;gcd_list&#39;</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">seq</span><span class="p">),</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">polys</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Poly</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">opt</span><span class="o">=</span><span class="n">opt</span><span class="p">)</span>
-
-    <span class="n">result</span><span class="p">,</span> <span class="n">polys</span> <span class="o">=</span> <span class="n">polys</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">polys</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-    <span class="k">for</span> <span class="n">poly</span> <span class="ow">in</span> <span class="n">polys</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">poly</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">result</span><span class="o">.</span><span class="n">is_one</span><span class="p">:</span>
-            <span class="k">break</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="gcd"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.gcd">[docs]</a><span class="k">def</span> <span class="nf">gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute GCD of ``f`` and ``g``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import gcd</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; gcd(x**2 - 1, x**2 - 3*x + 2)</span>
-<span class="sd">    x - 1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;__iter__&#39;</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">g</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">gens</span> <span class="o">=</span> <span class="p">(</span><span class="n">g</span><span class="p">,)</span> <span class="o">+</span> <span class="n">gens</span>
-
-        <span class="k">return</span> <span class="n">gcd_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">g</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;gcd() takes 2 arguments or a sequence of arguments&quot;</span><span class="p">)</span>
-
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">),</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">),</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="n">domain</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">=</span> <span class="n">construct_domain</span><span class="p">(</span><span class="n">exc</span><span class="o">.</span><span class="n">exprs</span><span class="p">)</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">domain</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">domain</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-        <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;gcd&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="lcm_list"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.lcm_list">[docs]</a><span class="k">def</span> <span class="nf">lcm_list</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute LCM of a list of polynomials.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import lcm_list</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; lcm_list([x**3 - 1, x**2 - 1, x**2 - 3*x + 2])</span>
-<span class="sd">    x**5 - x**4 - 2*x**3 - x**2 + x + 2</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">seq</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">seq</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">gens</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">args</span><span class="p">:</span>
-        <span class="n">domain</span><span class="p">,</span> <span class="n">numbers</span> <span class="o">=</span> <span class="n">construct_domain</span><span class="p">(</span><span class="n">seq</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">numbers</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">domain</span><span class="o">.</span><span class="n">one</span>
-        <span class="k">elif</span> <span class="n">domain</span><span class="o">.</span><span class="n">is_Numerical</span><span class="p">:</span>
-            <span class="n">result</span><span class="p">,</span> <span class="n">numbers</span> <span class="o">=</span> <span class="n">numbers</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">numbers</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-            <span class="k">for</span> <span class="n">number</span> <span class="ow">in</span> <span class="n">numbers</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">lcm</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">number</span><span class="p">)</span>
-
-            <span class="k">return</span> <span class="n">domain</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">polys</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;lcm_list&#39;</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">seq</span><span class="p">),</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">polys</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Poly</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">opt</span><span class="o">=</span><span class="n">opt</span><span class="p">)</span>
-
-    <span class="n">result</span><span class="p">,</span> <span class="n">polys</span> <span class="o">=</span> <span class="n">polys</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">polys</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-    <span class="k">for</span> <span class="n">poly</span> <span class="ow">in</span> <span class="n">polys</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">lcm</span><span class="p">(</span><span class="n">poly</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="lcm"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.lcm">[docs]</a><span class="k">def</span> <span class="nf">lcm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute LCM of ``f`` and ``g``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import lcm</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; lcm(x**2 - 1, x**2 - 3*x + 2)</span>
-<span class="sd">    x**3 - 2*x**2 - x + 2</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;__iter__&#39;</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">g</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">gens</span> <span class="o">=</span> <span class="p">(</span><span class="n">g</span><span class="p">,)</span> <span class="o">+</span> <span class="n">gens</span>
-
-        <span class="k">return</span> <span class="n">lcm_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">g</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;lcm() takes 2 arguments or a sequence of arguments&quot;</span><span class="p">)</span>
-
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">),</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">),</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="n">domain</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">=</span> <span class="n">construct_domain</span><span class="p">(</span><span class="n">exc</span><span class="o">.</span><span class="n">exprs</span><span class="p">)</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">domain</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">domain</span><span class="o">.</span><span class="n">lcm</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-        <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;lcm&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">lcm</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="terms_gcd"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.terms_gcd">[docs]</a><span class="k">def</span> <span class="nf">terms_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Remove GCD of terms from ``f``.</span>
-
-<span class="sd">    If the ``deep`` flag is True, then the arguments of ``f`` will have</span>
-<span class="sd">    terms_gcd applied to them.</span>
-
-<span class="sd">    If a fraction is factored out of ``f`` and ``f`` is an Add, then</span>
-<span class="sd">    an unevaluated Mul will be returned so that automatic simplification</span>
-<span class="sd">    does not redistribute it. The hint ``clear``, when set to False, can be</span>
-<span class="sd">    used to prevent such factoring when all coefficients are not fractions.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import terms_gcd, cos, pi</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; terms_gcd(x**6*y**2 + x**3*y, x, y)</span>
-<span class="sd">    x**3*y*(x**3*y + 1)</span>
-
-<span class="sd">    The default action of polys routines is to expand the expression</span>
-<span class="sd">    given to them. terms_gcd follows this behavior:</span>
-
-<span class="sd">    &gt;&gt;&gt; terms_gcd((3+3*x)*(x+x*y))</span>
-<span class="sd">    3*x*(x*y + x + y + 1)</span>
-
-<span class="sd">    If this is not desired then the hint ``expand`` can be set to False.</span>
-<span class="sd">    In this case the expression will be treated as though it were comprised</span>
-<span class="sd">    of one or more terms:</span>
-
-<span class="sd">    &gt;&gt;&gt; terms_gcd((3+3*x)*(x+x*y), expand=False)</span>
-<span class="sd">    (3*x + 3)*(x*y + x)</span>
-
-<span class="sd">    In order to traverse factors of a Mul or the arguments of other</span>
-<span class="sd">    functions, the ``deep`` hint can be used:</span>
-
-<span class="sd">    &gt;&gt;&gt; terms_gcd((3 + 3*x)*(x + x*y), expand=False, deep=True)</span>
-<span class="sd">    3*x*(x + 1)*(y + 1)</span>
-<span class="sd">    &gt;&gt;&gt; terms_gcd(cos(x + x*y), deep=True)</span>
-<span class="sd">    cos(x*(y + 1))</span>
-
-<span class="sd">    Rationals are factored out by default:</span>
-
-<span class="sd">    &gt;&gt;&gt; terms_gcd(x + y/2)</span>
-<span class="sd">    (2*x + y)/2</span>
-
-<span class="sd">    Only the y-term had a coefficient that was a fraction; if one</span>
-<span class="sd">    does not want to factor out the 1/2 in cases like this, the</span>
-<span class="sd">    flag ``clear`` can be set to False:</span>
-
-<span class="sd">    &gt;&gt;&gt; terms_gcd(x + y/2, clear=False)</span>
-<span class="sd">    x + y/2</span>
-<span class="sd">    &gt;&gt;&gt; terms_gcd(x*y/2 + y**2, clear=False)</span>
-<span class="sd">    y*(x/2 + y)</span>
-
-<span class="sd">    The ``clear`` flag is ignored if all coefficients are fractions:</span>
-
-<span class="sd">    &gt;&gt;&gt; terms_gcd(x/3 + y/2, clear=False)</span>
-<span class="sd">    (2*x + 3*y)/6</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    sympy.core.exprtools.gcd_terms, sympy.core.exprtools.factor_terms</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">Expr</span><span class="p">)</span> <span class="ow">or</span> <span class="n">f</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;deep&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="n">new</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">terms_gcd</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-        <span class="n">args</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;deep&#39;</span><span class="p">)</span>
-        <span class="n">args</span><span class="p">[</span><span class="s">&#39;expand&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="n">terms_gcd</span><span class="p">(</span><span class="n">new</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="n">clear</span> <span class="o">=</span> <span class="n">args</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;clear&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">poly_from_expr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">exc</span><span class="o">.</span><span class="n">expr</span>
-
-    <span class="n">J</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">terms_gcd</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span><span class="o">.</span><span class="n">has_Ring</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-            <span class="n">denom</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">clear_denoms</span><span class="p">(</span><span class="n">convert</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">primitive</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-            <span class="n">coeff</span> <span class="o">/=</span> <span class="n">denom</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">coeff</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-    <span class="n">term</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">x</span><span class="o">**</span><span class="n">j</span> <span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">gens</span><span class="p">,</span> <span class="n">J</span><span class="p">)</span> <span class="p">])</span>
-
-    <span class="k">if</span> <span class="n">clear</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">term</span><span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">())</span>
-    <span class="c"># base the clearing on the form of the original expression, not</span>
-    <span class="c"># the (perhaps) Mul that we have now</span>
-    <span class="n">coeff</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(),</span> <span class="n">clear</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-    <span class="k">return</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">term</span><span class="o">*</span><span class="n">f</span><span class="p">,</span> <span class="n">clear</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="trunc"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.trunc">[docs]</a><span class="k">def</span> <span class="nf">trunc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Reduce ``f`` modulo a constant ``p``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import trunc</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; trunc(2*x**3 + 3*x**2 + 5*x + 7, 3)</span>
-<span class="sd">    -x**3 - x + 1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;auto&#39;</span><span class="p">,</span> <span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">poly_from_expr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;trunc&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">trunc</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="monic"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.monic">[docs]</a><span class="k">def</span> <span class="nf">monic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Divide all coefficients of ``f`` by ``LC(f)``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import monic</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; monic(3*x**2 + 4*x + 2)</span>
-<span class="sd">    x**2 + 4*x/3 + 2/3</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;auto&#39;</span><span class="p">,</span> <span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">poly_from_expr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;monic&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">monic</span><span class="p">(</span><span class="n">auto</span><span class="o">=</span><span class="n">opt</span><span class="o">.</span><span class="n">auto</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="content"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.content">[docs]</a><span class="k">def</span> <span class="nf">content</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute GCD of coefficients of ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import content</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; content(6*x**2 + 8*x + 12)</span>
-<span class="sd">    2</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">poly_from_expr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;content&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">content</span><span class="p">()</span>
-
-</div>
-<div class="viewcode-block" id="primitive"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.primitive">[docs]</a><span class="k">def</span> <span class="nf">primitive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute content and the primitive form of ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys.polytools import primitive</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">    &gt;&gt;&gt; primitive(6*x**2 + 8*x + 12)</span>
-<span class="sd">    (2, 3*x**2 + 4*x + 6)</span>
-
-<span class="sd">    &gt;&gt;&gt; eq = (2 + 2*x)*x + 2</span>
-
-<span class="sd">    Expansion is performed by default:</span>
-
-<span class="sd">    &gt;&gt;&gt; primitive(eq)</span>
-<span class="sd">    (2, x**2 + x + 1)</span>
-
-<span class="sd">    Set ``expand`` to False to shut this off. Note that the</span>
-<span class="sd">    extraction will not be recursive; use the as_content_primitive method</span>
-<span class="sd">    for recursive, non-destructive Rational extraction.</span>
-
-<span class="sd">    &gt;&gt;&gt; primitive(eq, expand=False)</span>
-<span class="sd">    (1, x*(2*x + 2) + 2)</span>
-
-<span class="sd">    &gt;&gt;&gt; eq.as_content_primitive()</span>
-<span class="sd">    (2, x*(x + 1) + 1)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">poly_from_expr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;primitive&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">cont</span><span class="p">,</span> <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">primitive</span><span class="p">()</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">cont</span><span class="p">,</span> <span class="n">result</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">cont</span><span class="p">,</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="compose"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.compose">[docs]</a><span class="k">def</span> <span class="nf">compose</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute functional composition ``f(g)``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import compose</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; compose(x**2 + x, x - 1)</span>
-<span class="sd">    x**2 - x</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">),</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">),</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;compose&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">compose</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="decompose"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.decompose">[docs]</a><span class="k">def</span> <span class="nf">decompose</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute functional decomposition of ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import decompose</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; decompose(x**4 + 2*x**3 - x - 1)</span>
-<span class="sd">    [x**2 - x - 1, x**2 + x]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">poly_from_expr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;decompose&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">decompose</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="n">r</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">result</span> <span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="sturm"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.sturm">[docs]</a><span class="k">def</span> <span class="nf">sturm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute Sturm sequence of ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import sturm</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; sturm(x**3 - 2*x**2 + x - 3)</span>
-<span class="sd">    [x**3 - 2*x**2 + x - 3, 3*x**2 - 4*x + 1, 2*x/9 + 25/9, -2079/4]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;auto&#39;</span><span class="p">,</span> <span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">poly_from_expr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;sturm&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">sturm</span><span class="p">(</span><span class="n">auto</span><span class="o">=</span><span class="n">opt</span><span class="o">.</span><span class="n">auto</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="n">r</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">result</span> <span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="gff_list"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.gff_list">[docs]</a><span class="k">def</span> <span class="nf">gff_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute a list of greatest factorial factors of ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import gff_list, ff</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; f = x**5 + 2*x**4 - x**3 - 2*x**2</span>
-
-<span class="sd">    &gt;&gt;&gt; gff_list(f)</span>
-<span class="sd">    [(x, 1), (x + 2, 4)]</span>
-
-<span class="sd">    &gt;&gt;&gt; (ff(x, 1)*ff(x + 2, 4)).expand() == f</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">poly_from_expr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;gff_list&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">factors</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">gff_list</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(),</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">factors</span> <span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">factors</span>
-
-</div>
-<div class="viewcode-block" id="gff"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.gff">[docs]</a><span class="k">def</span> <span class="nf">gff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Compute greatest factorial factorization of ``f``. &quot;&quot;&quot;</span>
-    <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;symbolic falling factorial&#39;</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="sqf_norm"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.sqf_norm">[docs]</a><span class="k">def</span> <span class="nf">sqf_norm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute square-free norm of ``f``.</span>
-
-<span class="sd">    Returns ``s``, ``f``, ``r``, such that ``g(x) = f(x-sa)`` and</span>
-<span class="sd">    ``r(x) = Norm(g(x))`` is a square-free polynomial over ``K``,</span>
-<span class="sd">    where ``a`` is the algebraic extension of the ground domain.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import sqf_norm, sqrt</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; sqf_norm(x**2 + 1, extension=[sqrt(3)])</span>
-<span class="sd">    (1, x**2 - 2*sqrt(3)*x + 4, x**4 - 4*x**2 + 16)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">poly_from_expr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;sqf_norm&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">s</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">sqf_norm</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="n">g</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(),</span> <span class="n">r</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">Integer</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="n">g</span><span class="p">,</span> <span class="n">r</span>
-
-</div>
-<div class="viewcode-block" id="sqf_part"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.sqf_part">[docs]</a><span class="k">def</span> <span class="nf">sqf_part</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute square-free part of ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import sqf_part</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; sqf_part(x**3 - 3*x - 2)</span>
-<span class="sd">    x**2 - x - 2</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">poly_from_expr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;sqf_part&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">sqf_part</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_sorted_factors</span><span class="p">(</span><span class="n">factors</span><span class="p">,</span> <span class="n">method</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Sort a list of ``(expr, exp)`` pairs. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">method</span> <span class="o">==</span> <span class="s">&#39;sqf&#39;</span><span class="p">:</span>
-        <span class="k">def</span> <span class="nf">key</span><span class="p">(</span><span class="n">obj</span><span class="p">):</span>
-            <span class="n">poly</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="n">obj</span>
-            <span class="n">rep</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">rep</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">exp</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">rep</span><span class="p">),</span> <span class="n">rep</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">def</span> <span class="nf">key</span><span class="p">(</span><span class="n">obj</span><span class="p">):</span>
-            <span class="n">poly</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="n">obj</span>
-            <span class="n">rep</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">rep</span>
-            <span class="k">return</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">rep</span><span class="p">),</span> <span class="n">exp</span><span class="p">,</span> <span class="n">rep</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">factors</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">key</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">_factors_product</span><span class="p">(</span><span class="n">factors</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Multiply a list of ``(expr, exp)`` pairs. &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span><span class="o">**</span><span class="n">k</span> <span class="k">for</span> <span class="n">f</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">factors</span> <span class="p">])</span>
-
-
-<span class="k">def</span> <span class="nf">_symbolic_factor_list</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">opt</span><span class="p">,</span> <span class="n">method</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Helper function for :func:`_symbolic_factor`. &quot;&quot;&quot;</span>
-    <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="n">coeff</span> <span class="o">*=</span> <span class="n">arg</span>
-            <span class="k">continue</span>
-        <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-            <span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">if</span> <span class="n">base</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                <span class="n">factors</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">base</span><span class="p">,</span> <span class="n">exp</span><span class="p">))</span>
-                <span class="k">continue</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="n">arg</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">poly</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">_poly_from_expr</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-            <span class="n">factors</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">exc</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">func</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">method</span> <span class="o">+</span> <span class="s">&#39;_list&#39;</span><span class="p">)</span>
-
-            <span class="n">_coeff</span><span class="p">,</span> <span class="n">_factors</span> <span class="o">=</span> <span class="n">func</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">_coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                    <span class="n">coeff</span> <span class="o">*=</span> <span class="n">_coeff</span><span class="o">**</span><span class="n">exp</span>
-                <span class="k">elif</span> <span class="n">_coeff</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                    <span class="n">factors</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">_coeff</span><span class="p">,</span> <span class="n">exp</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">_factors</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">_coeff</span><span class="p">,</span> <span class="bp">None</span><span class="p">))</span>
-
-            <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="n">factors</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">_factors</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">exp</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">_factors</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">factors</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="o">*</span><span class="n">exp</span><span class="p">)</span> <span class="k">for</span> <span class="n">f</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">_factors</span> <span class="p">])</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">other</span> <span class="o">=</span> <span class="p">[]</span>
-
-                <span class="k">for</span> <span class="n">f</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">_factors</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                        <span class="n">factors</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="o">*</span><span class="n">exp</span><span class="p">))</span>
-                    <span class="k">elif</span> <span class="n">k</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="n">other</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="p">))</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">other</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">))</span>
-
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">other</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">f</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="n">other</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                    <span class="n">factors</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="o">*</span><span class="n">exp</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">factors</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">_factors_product</span><span class="p">(</span><span class="n">other</span><span class="p">),</span> <span class="n">exp</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span>
-
-
-<span class="k">def</span> <span class="nf">_symbolic_factor</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">opt</span><span class="p">,</span> <span class="n">method</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Helper function for :func:`_factor`. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Expr</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Relational</span><span class="p">:</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">factors</span> <span class="o">=</span> <span class="n">_symbolic_factor_list</span><span class="p">(</span><span class="n">together</span><span class="p">(</span><span class="n">expr</span><span class="p">),</span> <span class="n">opt</span><span class="p">,</span> <span class="n">method</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">_factors_product</span><span class="p">(</span><span class="n">factors</span><span class="p">))</span>
-    <span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&#39;args&#39;</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">_symbolic_factor</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">opt</span><span class="p">,</span> <span class="n">method</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span> <span class="p">])</span>
-    <span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&#39;__iter__&#39;</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="p">([</span> <span class="n">_symbolic_factor</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">opt</span><span class="p">,</span> <span class="n">method</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span> <span class="p">])</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-
-<span class="k">def</span> <span class="nf">_generic_factor_list</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">args</span><span class="p">,</span> <span class="n">method</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Helper function for :func:`sqf_list` and :func:`factor_list`. &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;frac&#39;</span><span class="p">,</span> <span class="s">&#39;polys&#39;</span><span class="p">])</span>
-    <span class="n">opt</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">build_options</span><span class="p">(</span><span class="n">gens</span><span class="p">,</span> <span class="n">args</span><span class="p">)</span>
-
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Expr</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Relational</span><span class="p">:</span>
-        <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="o">=</span> <span class="n">together</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-
-        <span class="n">cp</span><span class="p">,</span> <span class="n">fp</span> <span class="o">=</span> <span class="n">_symbolic_factor_list</span><span class="p">(</span><span class="n">numer</span><span class="p">,</span> <span class="n">opt</span><span class="p">,</span> <span class="n">method</span><span class="p">)</span>
-        <span class="n">cq</span><span class="p">,</span> <span class="n">fq</span> <span class="o">=</span> <span class="n">_symbolic_factor_list</span><span class="p">(</span><span class="n">denom</span><span class="p">,</span> <span class="n">opt</span><span class="p">,</span> <span class="n">method</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">fq</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">frac</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span><span class="s">&quot;a polynomial expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">expr</span><span class="p">)</span>
-
-        <span class="n">_opt</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">clone</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">expand</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-
-        <span class="k">for</span> <span class="n">factors</span> <span class="ow">in</span> <span class="p">(</span><span class="n">fp</span><span class="p">,</span> <span class="n">fq</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">factors</span><span class="p">):</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-                    <span class="n">f</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">_poly_from_expr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">_opt</span><span class="p">)</span>
-                    <span class="n">factors</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-
-        <span class="n">fp</span> <span class="o">=</span> <span class="n">_sorted_factors</span><span class="p">(</span><span class="n">fp</span><span class="p">,</span> <span class="n">method</span><span class="p">)</span>
-        <span class="n">fq</span> <span class="o">=</span> <span class="n">_sorted_factors</span><span class="p">(</span><span class="n">fq</span><span class="p">,</span> <span class="n">method</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-            <span class="n">fp</span> <span class="o">=</span> <span class="p">[</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(),</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">f</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">fp</span> <span class="p">]</span>
-            <span class="n">fq</span> <span class="o">=</span> <span class="p">[</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(),</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">f</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">fq</span> <span class="p">]</span>
-
-        <span class="n">coeff</span> <span class="o">=</span> <span class="n">cp</span><span class="o">/</span><span class="n">cq</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">frac</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">fp</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">fp</span><span class="p">,</span> <span class="n">fq</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span><span class="s">&quot;a polynomial expected, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">expr</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">_generic_factor</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">args</span><span class="p">,</span> <span class="n">method</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Helper function for :func:`sqf` and :func:`factor`. &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[])</span>
-    <span class="n">opt</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">build_options</span><span class="p">(</span><span class="n">gens</span><span class="p">,</span> <span class="n">args</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">_symbolic_factor</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">),</span> <span class="n">opt</span><span class="p">,</span> <span class="n">method</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="sqf_list"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.sqf_list">[docs]</a><span class="k">def</span> <span class="nf">sqf_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute a list of square-free factors of ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import sqf_list</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; sqf_list(2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16)</span>
-<span class="sd">    (2, [(x + 1, 2), (x + 2, 3)])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">_generic_factor_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">args</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;sqf&#39;</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="sqf"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.sqf">[docs]</a><span class="k">def</span> <span class="nf">sqf</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute square-free factorization of ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import sqf</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; sqf(2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16)</span>
-<span class="sd">    2*(x + 1)**2*(x + 2)**3</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">_generic_factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">args</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;sqf&#39;</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="factor_list"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.factor_list">[docs]</a><span class="k">def</span> <span class="nf">factor_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute a list of irreducible factors of ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import factor_list</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">    &gt;&gt;&gt; factor_list(2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y)</span>
-<span class="sd">    (2, [(x + y, 1), (x**2 + 1, 2)])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">_generic_factor_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">args</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;factor&#39;</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="factor"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.factor">[docs]</a><span class="k">def</span> <span class="nf">factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute the factorization of ``f`` into irreducibles. (Use factorint to</span>
-<span class="sd">    factor an integer.)</span>
-
-<span class="sd">    There two modes implemented: symbolic and formal. If ``f`` is not an</span>
-<span class="sd">    instance of :class:`Poly` and generators are not specified, then the</span>
-<span class="sd">    former mode is used. Otherwise, the formal mode is used.</span>
-
-<span class="sd">    In symbolic mode, :func:`factor` will traverse the expression tree and</span>
-<span class="sd">    factor its components without any prior expansion, unless an instance</span>
-<span class="sd">    of :class:`Add` is encountered (in this case formal factorization is</span>
-<span class="sd">    used). This way :func:`factor` can handle large or symbolic exponents.</span>
-
-<span class="sd">    By default, the factorization is computed over the rationals. To factor</span>
-<span class="sd">    over other domain, e.g. an algebraic or finite field, use appropriate</span>
-<span class="sd">    options: ``extension``, ``modulus`` or ``domain``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import factor, sqrt</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">    &gt;&gt;&gt; factor(2*x**5 + 2*x**4*y + 4*x**3 + 4*x**2*y + 2*x + 2*y)</span>
-<span class="sd">    2*(x + y)*(x**2 + 1)**2</span>
-
-<span class="sd">    &gt;&gt;&gt; factor(x**2 + 1)</span>
-<span class="sd">    x**2 + 1</span>
-<span class="sd">    &gt;&gt;&gt; factor(x**2 + 1, modulus=2)</span>
-<span class="sd">    (x + 1)**2</span>
-<span class="sd">    &gt;&gt;&gt; factor(x**2 + 1, gaussian=True)</span>
-<span class="sd">    (x - I)*(x + I)</span>
-
-<span class="sd">    &gt;&gt;&gt; factor(x**2 - 2, extension=sqrt(2))</span>
-<span class="sd">    (x - sqrt(2))*(x + sqrt(2))</span>
-
-<span class="sd">    &gt;&gt;&gt; factor((x**2 - 1)/(x**2 + 4*x + 4))</span>
-<span class="sd">    (x - 1)*(x + 1)/(x + 2)**2</span>
-<span class="sd">    &gt;&gt;&gt; factor((x**2 + 4*x + 4)**10000000*(x**2 + 1))</span>
-<span class="sd">    (x + 2)**20000000*(x**2 + 1)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">_generic_factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">args</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;factor&#39;</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolynomialError</span> <span class="k">as</span> <span class="n">msg</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">sympy.core.exprtools</span> <span class="kn">import</span> <span class="n">factor_nc</span>
-            <span class="k">return</span> <span class="n">factor_nc</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span><span class="n">msg</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="intervals"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.intervals">[docs]</a><span class="k">def</span> <span class="nf">intervals</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="nb">all</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">inf</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">sup</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">fast</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">sqf</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute isolating intervals for roots of ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import intervals</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; intervals(x**2 - 3)</span>
-<span class="sd">    [((-2, -1), 1), ((1, 2), 1)]</span>
-<span class="sd">    &gt;&gt;&gt; intervals(x**2 - 3, eps=1e-2)</span>
-<span class="sd">    [((-26/15, -19/11), 1), ((19/11, 26/15), 1)]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="s">&#39;__iter__&#39;</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">F</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">F</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">GeneratorsNeeded</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[]</span>
-
-        <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">intervals</span><span class="p">(</span><span class="nb">all</span><span class="o">=</span><span class="nb">all</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="n">eps</span><span class="p">,</span> <span class="n">inf</span><span class="o">=</span><span class="n">inf</span><span class="p">,</span> <span class="n">sup</span><span class="o">=</span><span class="n">sup</span><span class="p">,</span> <span class="n">fast</span><span class="o">=</span><span class="n">fast</span><span class="p">,</span> <span class="n">sqf</span><span class="o">=</span><span class="n">sqf</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">polys</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="s">&#39;QQ&#39;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">MultivariatePolynomialError</span>
-
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">poly</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">polys</span><span class="p">):</span>
-            <span class="n">polys</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">rep</span>
-
-        <span class="k">if</span> <span class="n">eps</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">eps</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">eps</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">eps</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;&#39;eps&#39; must be a positive rational&quot;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">inf</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">inf</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">sup</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">sup</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">sup</span><span class="p">)</span>
-
-        <span class="n">intervals</span> <span class="o">=</span> <span class="n">dup_isolate_real_roots_list</span><span class="p">(</span><span class="n">polys</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span><span class="p">,</span>
-            <span class="n">eps</span><span class="o">=</span><span class="n">eps</span><span class="p">,</span> <span class="n">inf</span><span class="o">=</span><span class="n">inf</span><span class="p">,</span> <span class="n">sup</span><span class="o">=</span><span class="n">sup</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="n">strict</span><span class="p">,</span> <span class="n">fast</span><span class="o">=</span><span class="n">fast</span><span class="p">)</span>
-
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">),</span> <span class="n">indices</span> <span class="ow">in</span> <span class="n">intervals</span><span class="p">:</span>
-            <span class="n">s</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(((</span><span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">),</span> <span class="n">indices</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="refine_root"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.refine_root">[docs]</a><span class="k">def</span> <span class="nf">refine_root</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">steps</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">fast</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">check_sqf</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Refine an isolating interval of a root to the given precision.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import refine_root</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; refine_root(x**2 - 3, 1, 2, eps=1e-2)</span>
-<span class="sd">    (19/11, 26/15)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">GeneratorsNeeded</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span>
-            <span class="s">&quot;can&#39;t refine a root of </span><span class="si">%s</span><span class="s">, not a polynomial&quot;</span> <span class="o">%</span> <span class="n">f</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">refine_root</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="n">eps</span><span class="p">,</span> <span class="n">steps</span><span class="o">=</span><span class="n">steps</span><span class="p">,</span> <span class="n">fast</span><span class="o">=</span><span class="n">fast</span><span class="p">,</span> <span class="n">check_sqf</span><span class="o">=</span><span class="n">check_sqf</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="count_roots"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.count_roots">[docs]</a><span class="k">def</span> <span class="nf">count_roots</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">inf</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">sup</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the number of roots of ``f`` in ``[inf, sup]`` interval.</span>
-
-<span class="sd">    If one of ``inf`` or ``sup`` is complex, it will return the number of roots</span>
-<span class="sd">    in the complex rectangle with corners at ``inf`` and ``sup``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import count_roots, I</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; count_roots(x**4 - 4, -3, 3)</span>
-<span class="sd">    2</span>
-<span class="sd">    &gt;&gt;&gt; count_roots(x**4 - 4, 0, 1 + 3*I)</span>
-<span class="sd">    1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">greedy</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">GeneratorsNeeded</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span><span class="s">&quot;can&#39;t count roots of </span><span class="si">%s</span><span class="s">, not a polynomial&quot;</span> <span class="o">%</span> <span class="n">f</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">count_roots</span><span class="p">(</span><span class="n">inf</span><span class="o">=</span><span class="n">inf</span><span class="p">,</span> <span class="n">sup</span><span class="o">=</span><span class="n">sup</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="real_roots"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.real_roots">[docs]</a><span class="k">def</span> <span class="nf">real_roots</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a list of real roots with multiplicities of ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import real_roots</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; real_roots(2*x**3 - 7*x**2 + 4*x + 4)</span>
-<span class="sd">    [-1/2, 2, 2]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">greedy</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">GeneratorsNeeded</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span>
-            <span class="s">&quot;can&#39;t compute real roots of </span><span class="si">%s</span><span class="s">, not a polynomial&quot;</span> <span class="o">%</span> <span class="n">f</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">real_roots</span><span class="p">(</span><span class="n">multiple</span><span class="o">=</span><span class="n">multiple</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="nroots"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.nroots">[docs]</a><span class="k">def</span> <span class="nf">nroots</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">15</span><span class="p">,</span> <span class="n">maxsteps</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span> <span class="n">cleanup</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">error</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute numerical approximations of roots of ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import nroots</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; nroots(x**2 - 3, n=15)</span>
-<span class="sd">    [-1.73205080756888, 1.73205080756888]</span>
-<span class="sd">    &gt;&gt;&gt; nroots(x**2 - 3, n=30)</span>
-<span class="sd">    [-1.73205080756887729352744634151, 1.73205080756887729352744634151]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">greedy</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">GeneratorsNeeded</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span>
-            <span class="s">&quot;can&#39;t compute numerical roots of </span><span class="si">%s</span><span class="s">, not a polynomial&quot;</span> <span class="o">%</span> <span class="n">f</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">nroots</span><span class="p">(</span><span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">maxsteps</span><span class="o">=</span><span class="n">maxsteps</span><span class="p">,</span> <span class="n">cleanup</span><span class="o">=</span><span class="n">cleanup</span><span class="p">,</span> <span class="n">error</span><span class="o">=</span><span class="n">error</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="ground_roots"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.ground_roots">[docs]</a><span class="k">def</span> <span class="nf">ground_roots</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute roots of ``f`` by factorization in the ground domain.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import ground_roots</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; ground_roots(x**6 - 4*x**4 + 4*x**3 - x**2)</span>
-<span class="sd">    {0: 2, 1: 2}</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">poly_from_expr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;ground_roots&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">F</span><span class="o">.</span><span class="n">ground_roots</span><span class="p">()</span>
-
-</div>
-<div class="viewcode-block" id="nth_power_roots_poly"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.nth_power_roots_poly">[docs]</a><span class="k">def</span> <span class="nf">nth_power_roots_poly</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Construct a polynomial with n-th powers of roots of ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import nth_power_roots_poly, factor, roots</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; f = x**4 - x**2 + 1</span>
-<span class="sd">    &gt;&gt;&gt; g = factor(nth_power_roots_poly(f, 2))</span>
-
-<span class="sd">    &gt;&gt;&gt; g</span>
-<span class="sd">    (x**2 - x + 1)**2</span>
-
-<span class="sd">    &gt;&gt;&gt; R_f = [ (r**2).expand() for r in roots(f) ]</span>
-<span class="sd">    &gt;&gt;&gt; R_g = list(roots(g).keys())</span>
-
-<span class="sd">    &gt;&gt;&gt; set(R_f) == set(R_g)</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">F</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">poly_from_expr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;nth_power_roots_poly&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">nth_power_roots_poly</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="cancel"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.cancel">[docs]</a><span class="k">def</span> <span class="nf">cancel</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Cancel common factors in a rational function ``f``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import cancel</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; cancel((2*x**2 - 2)/(x**2 - 2*x + 1))</span>
-<span class="sd">    (2*x + 2)/(x - 1)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">])</span>
-
-    <span class="n">f</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">)):</span>
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">f</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">f</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">),</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">((</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">),</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="n">f</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span>
-
-    <span class="n">c</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">Q</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">cancel</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">)):</span>
-        <span class="k">return</span> <span class="n">c</span><span class="o">*</span><span class="p">(</span><span class="n">P</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span><span class="o">/</span><span class="n">Q</span><span class="o">.</span><span class="n">as_expr</span><span class="p">())</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">c</span><span class="p">,</span> <span class="n">P</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(),</span> <span class="n">Q</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">c</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">Q</span>
-
-</div>
-<div class="viewcode-block" id="reduced"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.reduced">[docs]</a><span class="k">def</span> <span class="nf">reduced</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Reduces a polynomial ``f`` modulo a set of polynomials ``G``.</span>
-
-<span class="sd">    Given a polynomial ``f`` and a set of polynomials ``G = (g_1, ..., g_n)``,</span>
-<span class="sd">    computes a set of quotients ``q = (q_1, ..., q_n)`` and the remainder ``r``</span>
-<span class="sd">    such that ``f = q_1*f_1 + ... + q_n*f_n + r``, where ``r`` vanishes or ``r``</span>
-<span class="sd">    is a completely reduced polynomial with respect to ``G``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import reduced</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">    &gt;&gt;&gt; reduced(2*x**4 + y**2 - x**2 + y**3, [x**3 - x, y**3 - y])</span>
-<span class="sd">    ([2*x, 1], x**2 + y**2 + y)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">,</span> <span class="s">&#39;auto&#39;</span><span class="p">])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">polys</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">([</span><span class="n">f</span><span class="p">]</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">G</span><span class="p">),</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;reduced&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="n">domain</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span>
-    <span class="n">retract</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="k">if</span> <span class="n">opt</span><span class="o">.</span><span class="n">auto</span> <span class="ow">and</span> <span class="n">domain</span><span class="o">.</span><span class="n">has_Ring</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">domain</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-        <span class="n">opt</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">clone</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">domain</span><span class="o">=</span><span class="n">domain</span><span class="o">.</span><span class="n">get_field</span><span class="p">()))</span>
-        <span class="n">retract</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">poly</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">polys</span><span class="p">):</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">set_domain</span><span class="p">(</span><span class="n">opt</span><span class="o">.</span><span class="n">domain</span><span class="p">)</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">to_dict</span><span class="p">()</span>
-        <span class="n">polys</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">sdp_from_dict</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">order</span><span class="p">)</span>
-
-    <span class="n">level</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="n">Q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">sdp_div</span><span class="p">(</span><span class="n">polys</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">polys</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">level</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">order</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span><span class="p">)</span>
-
-    <span class="n">Q</span> <span class="o">=</span> <span class="p">[</span> <span class="n">Poly</span><span class="o">.</span><span class="n">_from_dict</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">q</span><span class="p">),</span> <span class="n">opt</span><span class="p">)</span> <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">Q</span> <span class="p">]</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">Poly</span><span class="o">.</span><span class="n">_from_dict</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">r</span><span class="p">),</span> <span class="n">opt</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">retract</span><span class="p">:</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">_Q</span><span class="p">,</span> <span class="n">_r</span> <span class="o">=</span> <span class="p">[</span> <span class="n">q</span><span class="o">.</span><span class="n">to_ring</span><span class="p">()</span> <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">Q</span> <span class="p">],</span> <span class="n">r</span><span class="o">.</span><span class="n">to_ring</span><span class="p">()</span>
-        <span class="k">except</span> <span class="n">CoercionFailed</span><span class="p">:</span>
-            <span class="k">pass</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">Q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">_Q</span><span class="p">,</span> <span class="n">_r</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="n">q</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span> <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">Q</span> <span class="p">],</span> <span class="n">r</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">Q</span><span class="p">,</span> <span class="n">r</span>
-
-</div>
-<div class="viewcode-block" id="groebner"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.groebner">[docs]</a><span class="k">def</span> <span class="nf">groebner</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Computes the reduced Groebner basis for a set of polynomials.</span>
-
-<span class="sd">    Use the ``order`` argument to set the monomial ordering that will be</span>
-<span class="sd">    used to compute the basis. Allowed orders are ``lex``, ``grlex`` and</span>
-<span class="sd">    ``grevlex``. If no order is specified, it defaults to ``lex``.</span>
-
-<span class="sd">    For more information on Groebner bases, see the references and the docstring</span>
-<span class="sd">    of `solve_poly_system()`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Example taken from [1].</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import groebner</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">    &gt;&gt;&gt; F = [x*y - 2*y, 2*y**2 - x**2]</span>
-
-<span class="sd">    &gt;&gt;&gt; groebner(F, x, y, order=&#39;lex&#39;)</span>
-<span class="sd">    GroebnerBasis([x**2 - 2*y**2, x*y - 2*y, y**3 - 2*y], x, y,</span>
-<span class="sd">                  domain=&#39;ZZ&#39;, order=&#39;lex&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; groebner(F, x, y, order=&#39;grlex&#39;)</span>
-<span class="sd">    GroebnerBasis([y**3 - 2*y, x**2 - 2*y**2, x*y - 2*y], x, y,</span>
-<span class="sd">                  domain=&#39;ZZ&#39;, order=&#39;grlex&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; groebner(F, x, y, order=&#39;grevlex&#39;)</span>
-<span class="sd">    GroebnerBasis([y**3 - 2*y, x**2 - 2*y**2, x*y - 2*y], x, y,</span>
-<span class="sd">                  domain=&#39;ZZ&#39;, order=&#39;grevlex&#39;)</span>
-
-<span class="sd">    By default, an improved implementation of the Buchberger algorithm is</span>
-<span class="sd">    used. Optionally, an implementation of the F5B algorithm can be used.</span>
-<span class="sd">    The algorithm can be set using ``method`` flag or with the :func:`setup`</span>
-<span class="sd">    function from :mod:`sympy.polys.polyconfig`:</span>
-
-<span class="sd">    &gt;&gt;&gt; F = [x**2 - x - 1, (2*x - 1) * y - (x**10 - (1 - x)**10)]</span>
-
-<span class="sd">    &gt;&gt;&gt; groebner(F, x, y, method=&#39;buchberger&#39;)</span>
-<span class="sd">    GroebnerBasis([x**2 - x - 1, y - 55], x, y, domain=&#39;ZZ&#39;, order=&#39;lex&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; groebner(F, x, y, method=&#39;f5b&#39;)</span>
-<span class="sd">    GroebnerBasis([x**2 - x - 1, y - 55], x, y, domain=&#39;ZZ&#39;, order=&#39;lex&#39;)</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. [Buchberger01]_</span>
-<span class="sd">    2. [Cox97]_</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">GroebnerBasis</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="is_zero_dimensional"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.is_zero_dimensional">[docs]</a><span class="k">def</span> <span class="nf">is_zero_dimensional</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Checks if the ideal generated by a Groebner basis is zero-dimensional.</span>
-
-<span class="sd">    The algorithm checks if the set of monomials not divisible by the</span>
-<span class="sd">    leading monomial of any element of ``F`` is bounded.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    David A. Cox, John B. Little, Donal O&#39;Shea. Ideals, Varieties and</span>
-<span class="sd">    Algorithms, 3rd edition, p. 230</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">GroebnerBasis</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span><span class="o">.</span><span class="n">is_zero_dimensional</span>
-
-</div>
-<div class="viewcode-block" id="GroebnerBasis"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.GroebnerBasis">[docs]</a><span class="k">class</span> <span class="nc">GroebnerBasis</span><span class="p">(</span><span class="n">Basic</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Represents a reduced Groebner basis. &quot;&quot;&quot;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;_basis&#39;</span><span class="p">,</span> <span class="s">&#39;_options&#39;</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute a reduced Groebner basis for a system of polynomials. &quot;&quot;&quot;</span>
-        <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;polys&#39;</span><span class="p">,</span> <span class="s">&#39;method&#39;</span><span class="p">])</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">polys</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;groebner&#39;</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">F</span><span class="p">),</span> <span class="n">exc</span><span class="p">)</span>
-
-        <span class="n">domain</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span>
-
-        <span class="k">if</span> <span class="n">domain</span><span class="o">.</span><span class="n">has_assoc_Field</span><span class="p">:</span>
-            <span class="n">opt</span><span class="o">.</span><span class="n">domain</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">get_field</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">DomainError</span><span class="p">(</span>
-                <span class="s">&quot;can&#39;t compute a Groebner basis over </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">poly</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">polys</span><span class="p">):</span>
-            <span class="n">poly</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">set_domain</span><span class="p">(</span><span class="n">opt</span><span class="o">.</span><span class="n">domain</span><span class="p">)</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">to_dict</span><span class="p">()</span>
-            <span class="n">polys</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">sdp_from_dict</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">order</span><span class="p">)</span>
-
-        <span class="n">level</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="n">G</span> <span class="o">=</span> <span class="n">sdp_groebner</span><span class="p">(</span>
-            <span class="n">polys</span><span class="p">,</span> <span class="n">level</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">order</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="n">opt</span><span class="o">.</span><span class="n">method</span><span class="p">)</span>
-        <span class="n">G</span> <span class="o">=</span> <span class="p">[</span> <span class="n">Poly</span><span class="o">.</span><span class="n">_from_dict</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">opt</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">G</span> <span class="p">]</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">domain</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-            <span class="n">G</span> <span class="o">=</span> <span class="p">[</span> <span class="n">g</span><span class="o">.</span><span class="n">clear_denoms</span><span class="p">(</span><span class="n">convert</span><span class="o">=</span><span class="bp">True</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">G</span> <span class="p">]</span>
-            <span class="n">opt</span><span class="o">.</span><span class="n">domain</span> <span class="o">=</span> <span class="n">domain</span>
-
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_new</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="n">options</span><span class="p">):</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-
-        <span class="n">obj</span><span class="o">.</span><span class="n">_basis</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">basis</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_options</span> <span class="o">=</span> <span class="n">options</span>
-
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">_basis</span><span class="p">),</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">_options</span><span class="o">.</span><span class="n">gens</span><span class="p">))</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">exprs</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">[</span> <span class="n">poly</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span> <span class="k">for</span> <span class="n">poly</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_basis</span> <span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">polys</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_basis</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">gens</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_options</span><span class="o">.</span><span class="n">gens</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">domain</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_options</span><span class="o">.</span><span class="n">domain</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">order</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_options</span><span class="o">.</span><span class="n">order</span>
-
-    <span class="k">def</span> <span class="nf">__len__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_basis</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_options</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">iter</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">polys</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">iter</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">exprs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">item</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_options</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-            <span class="n">basis</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">polys</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">basis</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">exprs</span>
-
-        <span class="k">return</span> <span class="n">basis</span><span class="p">[</span><span class="n">item</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">hash</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">_basis</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_options</span><span class="o">.</span><span class="n">items</span><span class="p">())))</span>
-
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_basis</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">_basis</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">_options</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">_options</span>
-        <span class="k">elif</span> <span class="n">iterable</span><span class="p">(</span><span class="n">other</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">polys</span> <span class="o">==</span> <span class="nb">list</span><span class="p">(</span><span class="n">other</span><span class="p">)</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">exprs</span> <span class="o">==</span> <span class="nb">list</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__ne__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="GroebnerBasis.is_zero_dimensional"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.GroebnerBasis.is_zero_dimensional">[docs]</a>    <span class="k">def</span> <span class="nf">is_zero_dimensional</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Checks if the ideal generated by a Groebner basis is zero-dimensional.</span>
-
-<span class="sd">        The algorithm checks if the set of monomials not divisible by the</span>
-<span class="sd">        leading monomial of any element of ``F`` is bounded.</span>
-
-<span class="sd">        References</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        David A. Cox, John B. Little, Donal O&#39;Shea. Ideals, Varieties and</span>
-<span class="sd">        Algorithms, 3rd edition, p. 230</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">def</span> <span class="nf">single_var</span><span class="p">(</span><span class="n">monomial</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">bool</span><span class="p">,</span> <span class="n">monomial</span><span class="p">))</span> <span class="o">==</span> <span class="mi">1</span>
-
-        <span class="n">exponents</span> <span class="o">=</span> <span class="n">Monomial</span><span class="p">([</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">gens</span><span class="p">))</span>
-        <span class="n">order</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_options</span><span class="o">.</span><span class="n">order</span>
-
-        <span class="k">for</span> <span class="n">poly</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-            <span class="n">monomial</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">LM</span><span class="p">(</span><span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">single_var</span><span class="p">(</span><span class="n">monomial</span><span class="p">):</span>
-                <span class="n">exponents</span> <span class="o">*=</span> <span class="n">monomial</span>
-
-        <span class="c"># If any element of the exponents vector is zero, then there&#39;s</span>
-        <span class="c"># a variable for which there&#39;s no degree bound and the ideal</span>
-        <span class="c"># generated by this Groebner basis isn&#39;t zero-dimensional.</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="n">exponents</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="GroebnerBasis.fglm"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.GroebnerBasis.fglm">[docs]</a>    <span class="k">def</span> <span class="nf">fglm</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">order</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Convert a Groebner basis from one ordering to another.</span>
-
-<span class="sd">        The FGLM algorithm converts reduced Groebner bases of zero-dimensional</span>
-<span class="sd">        ideals from one ordering to another. This method is often used when it</span>
-<span class="sd">        is infeasible to compute a Groebner basis with respect to a particular</span>
-<span class="sd">        ordering directly.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import groebner</span>
-
-<span class="sd">        &gt;&gt;&gt; F = [x**2 - 3*y - x + 1, y**2 - 2*x + y - 1]</span>
-<span class="sd">        &gt;&gt;&gt; G = groebner(F, x, y, order=&#39;grlex&#39;)</span>
-
-<span class="sd">        &gt;&gt;&gt; list(G.fglm(&#39;lex&#39;))</span>
-<span class="sd">        [2*x - y**2 - y + 1, y**4 + 2*y**3 - 3*y**2 - 16*y + 7]</span>
-<span class="sd">        &gt;&gt;&gt; list(groebner(F, x, y, order=&#39;lex&#39;))</span>
-<span class="sd">        [2*x - y**2 - y + 1, y**4 + 2*y**3 - 3*y**2 - 16*y + 7]</span>
-
-<span class="sd">        References</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        J.C. Faugere, P. Gianni, D. Lazard, T. Mora (1994). Efficient</span>
-<span class="sd">        Computation of Zero-dimensional Groebner Bases by Change of</span>
-<span class="sd">        Ordering</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">opt</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_options</span>
-
-        <span class="n">src_order</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">order</span>
-        <span class="n">dst_order</span> <span class="o">=</span> <span class="n">monomial_key</span><span class="p">(</span><span class="n">order</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">src_order</span> <span class="o">==</span> <span class="n">dst_order</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_zero_dimensional</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;can&#39;t convert Groebner bases of ideals with positive dimension&quot;</span><span class="p">)</span>
-
-        <span class="n">polys</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_basis</span><span class="p">)</span>
-        <span class="n">domain</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span>
-
-        <span class="n">opt</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">clone</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span>
-            <span class="n">domain</span><span class="o">=</span><span class="n">domain</span><span class="o">.</span><span class="n">get_field</span><span class="p">(),</span>
-            <span class="n">order</span><span class="o">=</span><span class="n">dst_order</span><span class="p">,</span>
-        <span class="p">))</span>
-
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">poly</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">polys</span><span class="p">):</span>
-            <span class="n">poly</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">set_domain</span><span class="p">(</span><span class="n">opt</span><span class="o">.</span><span class="n">domain</span><span class="p">)</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">to_dict</span><span class="p">()</span>
-            <span class="n">polys</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">sdp_from_dict</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">src_order</span><span class="p">)</span>
-
-        <span class="n">level</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="n">G</span> <span class="o">=</span> <span class="n">matrix_fglm</span><span class="p">(</span><span class="n">polys</span><span class="p">,</span> <span class="n">level</span><span class="p">,</span> <span class="n">src_order</span><span class="p">,</span> <span class="n">dst_order</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span><span class="p">)</span>
-        <span class="n">G</span> <span class="o">=</span> <span class="p">[</span> <span class="n">Poly</span><span class="o">.</span><span class="n">_from_dict</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">g</span><span class="p">),</span> <span class="n">opt</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">G</span> <span class="p">]</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">domain</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-            <span class="n">G</span> <span class="o">=</span> <span class="p">[</span> <span class="n">g</span><span class="o">.</span><span class="n">clear_denoms</span><span class="p">(</span><span class="n">convert</span><span class="o">=</span><span class="bp">True</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">G</span> <span class="p">]</span>
-            <span class="n">opt</span><span class="o">.</span><span class="n">domain</span> <span class="o">=</span> <span class="n">domain</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="GroebnerBasis.reduce"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.GroebnerBasis.reduce">[docs]</a>    <span class="k">def</span> <span class="nf">reduce</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Reduces a polynomial modulo a Groebner basis.</span>
-
-<span class="sd">        Given a polynomial ``f`` and a set of polynomials ``G = (g_1, ..., g_n)``,</span>
-<span class="sd">        computes a set of quotients ``q = (q_1, ..., q_n)`` and the remainder ``r``</span>
-<span class="sd">        such that ``f = q_1*f_1 + ... + q_n*f_n + r``, where ``r`` vanishes or ``r``</span>
-<span class="sd">        is a completely reduced polynomial with respect to ``G``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import groebner, expand</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; f = 2*x**4 - x**2 + y**3 + y**2</span>
-<span class="sd">        &gt;&gt;&gt; G = groebner([x**3 - x, y**3 - y])</span>
-
-<span class="sd">        &gt;&gt;&gt; G.reduce(f)</span>
-<span class="sd">        ([2*x, 1], x**2 + y**2 + y)</span>
-<span class="sd">        &gt;&gt;&gt; Q, r = _</span>
-
-<span class="sd">        &gt;&gt;&gt; expand(sum(q*g for q, g in zip(Q, G)) + r)</span>
-<span class="sd">        2*x**4 - x**2 + y**3 + y**2</span>
-<span class="sd">        &gt;&gt;&gt; _ == f</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="o">.</span><span class="n">_from_expr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_options</span><span class="p">)</span>
-        <span class="n">polys</span> <span class="o">=</span> <span class="p">[</span><span class="n">poly</span><span class="p">]</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_basis</span><span class="p">)</span>
-
-        <span class="n">opt</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_options</span>
-        <span class="n">domain</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span>
-
-        <span class="n">retract</span> <span class="o">=</span> <span class="bp">False</span>
-
-        <span class="k">if</span> <span class="n">auto</span> <span class="ow">and</span> <span class="n">domain</span><span class="o">.</span><span class="n">has_Ring</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">domain</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-            <span class="n">opt</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">clone</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">domain</span><span class="o">=</span><span class="n">domain</span><span class="o">.</span><span class="n">get_field</span><span class="p">()))</span>
-            <span class="n">retract</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">poly</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">polys</span><span class="p">):</span>
-            <span class="n">poly</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">set_domain</span><span class="p">(</span><span class="n">opt</span><span class="o">.</span><span class="n">domain</span><span class="p">)</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">to_dict</span><span class="p">()</span>
-            <span class="n">polys</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">sdp_from_dict</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">order</span><span class="p">)</span>
-
-        <span class="n">level</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="n">Q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">sdp_div</span><span class="p">(</span><span class="n">polys</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">polys</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">level</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">order</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span><span class="p">)</span>
-
-        <span class="n">Q</span> <span class="o">=</span> <span class="p">[</span> <span class="n">Poly</span><span class="o">.</span><span class="n">_from_dict</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">q</span><span class="p">),</span> <span class="n">opt</span><span class="p">)</span> <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">Q</span> <span class="p">]</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">Poly</span><span class="o">.</span><span class="n">_from_dict</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">r</span><span class="p">),</span> <span class="n">opt</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">retract</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">_Q</span><span class="p">,</span> <span class="n">_r</span> <span class="o">=</span> <span class="p">[</span> <span class="n">q</span><span class="o">.</span><span class="n">to_ring</span><span class="p">()</span> <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">Q</span> <span class="p">],</span> <span class="n">r</span><span class="o">.</span><span class="n">to_ring</span><span class="p">()</span>
-            <span class="k">except</span> <span class="n">CoercionFailed</span><span class="p">:</span>
-                <span class="k">pass</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">Q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">_Q</span><span class="p">,</span> <span class="n">_r</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">opt</span><span class="o">.</span><span class="n">polys</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[</span> <span class="n">q</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span> <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">Q</span> <span class="p">],</span> <span class="n">r</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Q</span><span class="p">,</span> <span class="n">r</span>
-</div>
-<div class="viewcode-block" id="GroebnerBasis.contains"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.GroebnerBasis.contains">[docs]</a>    <span class="k">def</span> <span class="nf">contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">poly</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Check if ``poly`` belongs the ideal generated by ``self``.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import groebner</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">        &gt;&gt;&gt; f = 2*x**3 + y**3 + 3*y</span>
-<span class="sd">        &gt;&gt;&gt; G = groebner([x**2 + y**2 - 1, x*y - 2])</span>
-
-<span class="sd">        &gt;&gt;&gt; G.contains(f)</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; G.contains(f + 1)</span>
-<span class="sd">        False</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">reduce</span><span class="p">(</span><span class="n">poly</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span>
-
-</div></div>
-<div class="viewcode-block" id="poly"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.polytools.poly">[docs]</a><span class="k">def</span> <span class="nf">poly</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Efficiently transform an expression into a polynomial.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import poly</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; poly(x*(x**2 + x - 1)**2)</span>
-<span class="sd">    Poly(x**5 + 2*x**4 - x**3 - 2*x**2 + x, x, domain=&#39;ZZ&#39;)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">options</span><span class="o">.</span><span class="n">allowed_flags</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="p">[])</span>
-
-    <span class="k">def</span> <span class="nf">_poly</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">opt</span><span class="p">):</span>
-        <span class="n">terms</span><span class="p">,</span> <span class="n">poly_terms</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-            <span class="n">factors</span><span class="p">,</span> <span class="n">poly_factors</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
-
-            <span class="k">for</span> <span class="n">factor</span> <span class="ow">in</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">term</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">factor</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                    <span class="n">poly_factors</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">_poly</span><span class="p">(</span><span class="n">factor</span><span class="p">,</span> <span class="n">opt</span><span class="p">))</span>
-                <span class="k">elif</span> <span class="n">factor</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">factor</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">and</span> <span class="n">factor</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                    <span class="n">poly_factors</span><span class="o">.</span><span class="n">append</span><span class="p">(</span>
-                        <span class="n">_poly</span><span class="p">(</span><span class="n">factor</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="n">factor</span><span class="o">.</span><span class="n">exp</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">factors</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">factor</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">poly_factors</span><span class="p">:</span>
-                <span class="n">terms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">product</span> <span class="o">=</span> <span class="n">poly_factors</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-                <span class="k">for</span> <span class="n">factor</span> <span class="ow">in</span> <span class="n">poly_factors</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-                    <span class="n">product</span> <span class="o">=</span> <span class="n">product</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">factor</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">factors</span><span class="p">:</span>
-                    <span class="n">factor</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">factors</span><span class="p">)</span>
-
-                    <span class="k">if</span> <span class="n">factor</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                        <span class="n">product</span> <span class="o">=</span> <span class="n">product</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">factor</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">product</span> <span class="o">=</span> <span class="n">product</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">Poly</span><span class="o">.</span><span class="n">_from_expr</span><span class="p">(</span><span class="n">factor</span><span class="p">,</span> <span class="n">opt</span><span class="p">))</span>
-
-                <span class="n">poly_terms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">product</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">poly_terms</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">Poly</span><span class="o">.</span><span class="n">_from_expr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">poly_terms</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-            <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">poly_terms</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">terms</span><span class="p">:</span>
-                <span class="n">term</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">terms</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">term</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                    <span class="n">result</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">result</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">Poly</span><span class="o">.</span><span class="n">_from_expr</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">opt</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">result</span><span class="o">.</span><span class="n">reorder</span><span class="p">(</span><span class="o">*</span><span class="n">opt</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;gens&#39;</span><span class="p">,</span> <span class="p">()),</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">Poly</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="s">&#39;expand&#39;</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-        <span class="n">args</span><span class="p">[</span><span class="s">&#39;expand&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="n">opt</span> <span class="o">=</span> <span class="n">options</span><span class="o">.</span><span class="n">build_options</span><span class="p">(</span><span class="n">gens</span><span class="p">,</span> <span class="n">args</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">_poly</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span></div>
-</pre></div>
-
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-  <h1>Source code for sympy.polys.rationaltools</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Tools for manipulation of rational expressions. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">Add</span><span class="p">,</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.core.exprtools</span> <span class="kn">import</span> <span class="n">gcd_terms</span>
-
-
-<div class="viewcode-block" id="together"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.rationaltools.together">[docs]</a><span class="k">def</span> <span class="nf">together</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Denest and combine rational expressions using symbolic methods.</span>
-
-<span class="sd">    This function takes an expression or a container of expressions</span>
-<span class="sd">    and puts it (them) together by denesting and combining rational</span>
-<span class="sd">    subexpressions. No heroic measures are taken to minimize degree</span>
-<span class="sd">    of the resulting numerator and denominator. To obtain completely</span>
-<span class="sd">    reduced expression use :func:`cancel`. However, :func:`together`</span>
-<span class="sd">    can preserve as much as possible of the structure of the input</span>
-<span class="sd">    expression in the output (no expansion is performed).</span>
-
-<span class="sd">    A wide variety of objects can be put together including lists,</span>
-<span class="sd">    tuples, sets, relational objects, integrals and others. It is</span>
-<span class="sd">    also possible to transform interior of function applications,</span>
-<span class="sd">    by setting ``deep`` flag to ``True``.</span>
-
-<span class="sd">    By definition, :func:`together` is a complement to :func:`apart`,</span>
-<span class="sd">    so ``apart(together(expr))`` should return expr unchanged. Note</span>
-<span class="sd">    however, that :func:`together` uses only symbolic methods, so</span>
-<span class="sd">    it might be necessary to use :func:`cancel` to perform algebraic</span>
-<span class="sd">    simplification and minimise degree of the numerator and denominator.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import together, exp</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-
-<span class="sd">    &gt;&gt;&gt; together(1/x + 1/y)</span>
-<span class="sd">    (x + y)/(x*y)</span>
-<span class="sd">    &gt;&gt;&gt; together(1/x + 1/y + 1/z)</span>
-<span class="sd">    (x*y + x*z + y*z)/(x*y*z)</span>
-
-<span class="sd">    &gt;&gt;&gt; together(1/(x*y) + 1/y**2)</span>
-<span class="sd">    (x + y)/(x*y**2)</span>
-
-<span class="sd">    &gt;&gt;&gt; together(1/(1 + 1/x) + 1/(1 + 1/y))</span>
-<span class="sd">    (x*(y + 1) + y*(x + 1))/((x + 1)*(y + 1))</span>
-
-<span class="sd">    &gt;&gt;&gt; together(exp(1/x + 1/y))</span>
-<span class="sd">    exp(1/y + 1/x)</span>
-<span class="sd">    &gt;&gt;&gt; together(exp(1/x + 1/y), deep=True)</span>
-<span class="sd">    exp((x + y)/(x*y))</span>
-
-<span class="sd">    &gt;&gt;&gt; together(1/exp(x) + 1/(x*exp(x)))</span>
-<span class="sd">    (x + 1)*exp(-x)/x</span>
-
-<span class="sd">    &gt;&gt;&gt; together(1/exp(2*x) + 1/(x*exp(3*x)))</span>
-<span class="sd">    (x*exp(x) + 1)*exp(-3*x)/x</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">_together</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Atom</span> <span class="ow">or</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">is_Function</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">deep</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">expr</span>
-            <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">gcd_terms</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">_together</span><span class="p">,</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">expr</span><span class="p">))))</span>
-            <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-                <span class="n">base</span> <span class="o">=</span> <span class="n">_together</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-                    <span class="n">exp</span> <span class="o">=</span> <span class="n">_together</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">exp</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span>
-
-                <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">_together</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span> <span class="p">])</span>
-        <span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&#39;__iter__&#39;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="p">([</span> <span class="n">_together</span><span class="p">(</span><span class="n">ex</span><span class="p">)</span> <span class="k">for</span> <span class="n">ex</span> <span class="ow">in</span> <span class="n">expr</span> <span class="p">])</span>
-
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="k">return</span> <span class="n">_together</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span></div>
-</pre></div>
-
-          </div>
-        </div>
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-            <p class="logo"><a href="../../../index.html">
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-  <h3>Quick search</h3>
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diff --git a/dev-py3k/_modules/sympy/polys/rootoftools.html b/dev-py3k/_modules/sympy/polys/rootoftools.html
deleted file mode 100644
index 8a7a98602a3..00000000000
--- a/dev-py3k/_modules/sympy/polys/rootoftools.html
+++ /dev/null
@@ -1,742 +0,0 @@
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-            
-  <h1>Source code for sympy.polys.rootoftools</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Implementation of RootOf class and related tools. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">Integer</span><span class="p">,</span> <span class="n">Float</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Lambda</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">sympify</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polytools</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">PurePoly</span><span class="p">,</span> <span class="n">factor</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.rationaltools</span> <span class="kn">import</span> <span class="n">together</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polyfuncs</span> <span class="kn">import</span> <span class="n">symmetrize</span><span class="p">,</span> <span class="n">viete</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.rootisolation</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dup_isolate_complex_roots_sqf</span><span class="p">,</span>
-    <span class="n">dup_isolate_real_roots_sqf</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyroots</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">roots_linear</span><span class="p">,</span> <span class="n">roots_quadratic</span><span class="p">,</span> <span class="n">roots_binomial</span><span class="p">,</span>
-    <span class="n">preprocess_roots</span><span class="p">,</span> <span class="n">roots</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">MultivariatePolynomialError</span><span class="p">,</span>
-    <span class="n">GeneratorsNeeded</span><span class="p">,</span>
-    <span class="n">PolynomialError</span><span class="p">,</span>
-    <span class="n">DomainError</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">QQ</span>
-
-<span class="kn">from</span> <span class="nn">sympy.mpmath</span> <span class="kn">import</span> <span class="n">mp</span><span class="p">,</span> <span class="n">mpf</span><span class="p">,</span> <span class="n">mpc</span><span class="p">,</span> <span class="n">findroot</span>
-<span class="kn">from</span> <span class="nn">sympy.mpmath.libmp.libmpf</span> <span class="kn">import</span> <span class="n">prec_to_dps</span>
-
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">lambdify</span>
-
-<span class="n">_reals_cache</span> <span class="o">=</span> <span class="p">{}</span>
-<span class="n">_complexes_cache</span> <span class="o">=</span> <span class="p">{}</span>
-
-
-<div class="viewcode-block" id="RootOf"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.rootoftools.RootOf">[docs]</a><span class="k">class</span> <span class="nc">RootOf</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Represents ``k``-th root of a univariate polynomial. &quot;&quot;&quot;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;poly&#39;</span><span class="p">,</span> <span class="s">&#39;index&#39;</span><span class="p">]</span>
-    <span class="n">is_complex</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">index</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">radicals</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">expand</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Construct a new ``RootOf`` object for ``k``-th root of ``f``. &quot;&quot;&quot;</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">index</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">x</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="n">x</span><span class="p">,</span> <span class="n">index</span> <span class="o">=</span> <span class="bp">None</span><span class="p">,</span> <span class="n">x</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">index</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">index</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">index</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="n">index</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">index</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;expected an integer root index, got </span><span class="si">%d</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">index</span><span class="p">)</span>
-
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">PurePoly</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">greedy</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">expand</span><span class="o">=</span><span class="n">expand</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">poly</span><span class="o">.</span><span class="n">is_univariate</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span><span class="s">&quot;only univariate polynomials are allowed&quot;</span><span class="p">)</span>
-
-        <span class="n">degree</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="n">degree</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span><span class="s">&quot;can&#39;t construct RootOf object for </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">f</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">index</span> <span class="o">&lt;</span> <span class="o">-</span><span class="n">degree</span> <span class="ow">or</span> <span class="n">index</span> <span class="o">&gt;=</span> <span class="n">degree</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s">&quot;root index out of [</span><span class="si">%d</span><span class="s">, </span><span class="si">%d</span><span class="s">] range, got </span><span class="si">%d</span><span class="s">&quot;</span> <span class="o">%</span>
-                             <span class="p">(</span><span class="o">-</span><span class="n">degree</span><span class="p">,</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">index</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="n">index</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">index</span> <span class="o">+=</span> <span class="n">degree</span>
-
-        <span class="n">dom</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">get_domain</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">dom</span><span class="o">.</span><span class="n">is_Exact</span><span class="p">:</span>
-            <span class="n">poly</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">to_exact</span><span class="p">()</span>
-
-        <span class="n">roots</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_roots_trivial</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">radicals</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">roots</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">roots</span><span class="p">[</span><span class="n">index</span><span class="p">]</span>
-
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">poly</span> <span class="o">=</span> <span class="n">preprocess_roots</span><span class="p">(</span><span class="n">poly</span><span class="p">)</span>
-        <span class="n">dom</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">get_domain</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">dom</span><span class="o">.</span><span class="n">is_ZZ</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;RootOf is not supported over </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">dom</span><span class="p">)</span>
-
-        <span class="n">root</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_indexed_root</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">index</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">coeff</span><span class="o">*</span><span class="n">cls</span><span class="o">.</span><span class="n">_postprocess_root</span><span class="p">(</span><span class="n">root</span><span class="p">,</span> <span class="n">radicals</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_new</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">poly</span><span class="p">,</span> <span class="n">index</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Construct new ``RootOf`` object from raw data. &quot;&quot;&quot;</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-
-        <span class="n">obj</span><span class="o">.</span><span class="n">poly</span> <span class="o">=</span> <span class="n">poly</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">index</span> <span class="o">=</span> <span class="n">index</span>
-
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">_hashable_content</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">poly</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">expr</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">poly</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="n">Integer</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">))</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">free_symbols</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">poly</span><span class="o">.</span><span class="n">free_symbols</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return ``True`` if the root is real. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">_reals_cache</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">poly</span><span class="p">])</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">real_roots</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">poly</span><span class="p">,</span> <span class="n">radicals</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Get real roots of a polynomial. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_get_roots</span><span class="p">(</span><span class="s">&quot;_real_roots&quot;</span><span class="p">,</span> <span class="n">poly</span><span class="p">,</span> <span class="n">radicals</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">all_roots</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">poly</span><span class="p">,</span> <span class="n">radicals</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Get real and complex roots of a polynomial. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_get_roots</span><span class="p">(</span><span class="s">&quot;_all_roots&quot;</span><span class="p">,</span> <span class="n">poly</span><span class="p">,</span> <span class="n">radicals</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_get_reals_sqf</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">factor</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute real root isolating intervals for a square-free polynomial. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">factor</span> <span class="ow">in</span> <span class="n">_reals_cache</span><span class="p">:</span>
-            <span class="n">real_part</span> <span class="o">=</span> <span class="n">_reals_cache</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">_reals_cache</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span> <span class="o">=</span> <span class="n">real_part</span> <span class="o">=</span> \
-                <span class="n">dup_isolate_real_roots_sqf</span><span class="p">(</span>
-                    <span class="n">factor</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">factor</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">blackbox</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">real_part</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_get_complexes_sqf</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">factor</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute complex root isolating intervals for a square-free polynomial. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">factor</span> <span class="ow">in</span> <span class="n">_complexes_cache</span><span class="p">:</span>
-            <span class="n">complex_part</span> <span class="o">=</span> <span class="n">_complexes_cache</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">_complexes_cache</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span> <span class="o">=</span> <span class="n">complex_part</span> <span class="o">=</span> \
-                <span class="n">dup_isolate_complex_roots_sqf</span><span class="p">(</span>
-                    <span class="n">factor</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">rep</span><span class="p">,</span> <span class="n">factor</span><span class="o">.</span><span class="n">rep</span><span class="o">.</span><span class="n">dom</span><span class="p">,</span> <span class="n">blackbox</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">complex_part</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_get_reals</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">factors</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute real root isolating intervals for a list of factors. &quot;&quot;&quot;</span>
-        <span class="n">reals</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">factor</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">factors</span><span class="p">:</span>
-            <span class="n">real_part</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_get_reals_sqf</span><span class="p">(</span><span class="n">factor</span><span class="p">)</span>
-            <span class="n">reals</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span> <span class="p">(</span><span class="n">root</span><span class="p">,</span> <span class="n">factor</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">root</span> <span class="ow">in</span> <span class="n">real_part</span> <span class="p">])</span>
-
-        <span class="k">return</span> <span class="n">reals</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_get_complexes</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">factors</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute complex root isolating intervals for a list of factors. &quot;&quot;&quot;</span>
-        <span class="n">complexes</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">factor</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">factors</span><span class="p">:</span>
-            <span class="n">complex_part</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_get_complexes_sqf</span><span class="p">(</span><span class="n">factor</span><span class="p">)</span>
-            <span class="n">complexes</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span> <span class="p">(</span><span class="n">root</span><span class="p">,</span> <span class="n">factor</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">root</span> <span class="ow">in</span> <span class="n">complex_part</span> <span class="p">])</span>
-
-        <span class="k">return</span> <span class="n">complexes</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_reals_sorted</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">reals</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Make real isolating intervals disjoint and sort roots. &quot;&quot;&quot;</span>
-        <span class="n">cache</span> <span class="o">=</span> <span class="p">{}</span>
-
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">reals</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">reals</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]):</span>
-                <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">u</span><span class="o">.</span><span class="n">refine_disjoint</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-                <span class="n">reals</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-
-            <span class="n">reals</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-
-        <span class="n">reals</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">reals</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">r</span><span class="p">:</span> <span class="n">r</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">a</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">root</span><span class="p">,</span> <span class="n">factor</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">reals</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">factor</span> <span class="ow">in</span> <span class="n">cache</span><span class="p">:</span>
-                <span class="n">cache</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">cache</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">root</span><span class="p">]</span>
-
-        <span class="k">for</span> <span class="n">factor</span><span class="p">,</span> <span class="n">roots</span> <span class="ow">in</span> <span class="n">cache</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">_reals_cache</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span> <span class="o">=</span> <span class="n">roots</span>
-
-        <span class="k">return</span> <span class="n">reals</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_complexes_sorted</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">complexes</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Make complex isolating intervals disjoint and sort roots. &quot;&quot;&quot;</span>
-        <span class="n">cache</span> <span class="o">=</span> <span class="p">{}</span>
-
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">complexes</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">complexes</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]):</span>
-                <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">u</span><span class="o">.</span><span class="n">refine_disjoint</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-                <span class="n">complexes</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-
-            <span class="n">complexes</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-
-        <span class="n">complexes</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">complexes</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">r</span><span class="p">:</span> <span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">ax</span><span class="p">,</span> <span class="n">r</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">ay</span><span class="p">))</span>
-
-        <span class="k">for</span> <span class="n">root</span><span class="p">,</span> <span class="n">factor</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">complexes</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">factor</span> <span class="ow">in</span> <span class="n">cache</span><span class="p">:</span>
-                <span class="n">cache</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">cache</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">root</span><span class="p">]</span>
-
-        <span class="k">for</span> <span class="n">factor</span><span class="p">,</span> <span class="n">roots</span> <span class="ow">in</span> <span class="n">cache</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">_complexes_cache</span><span class="p">[</span><span class="n">factor</span><span class="p">]</span> <span class="o">=</span> <span class="n">roots</span>
-
-        <span class="k">return</span> <span class="n">complexes</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_reals_index</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">reals</span><span class="p">,</span> <span class="n">index</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Map initial real root index to an index in a factor where the root belongs. &quot;&quot;&quot;</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-
-        <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="p">(</span><span class="n">_</span><span class="p">,</span> <span class="n">factor</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">reals</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">index</span> <span class="o">&lt;</span> <span class="n">i</span> <span class="o">+</span> <span class="n">k</span><span class="p">:</span>
-                <span class="n">poly</span><span class="p">,</span> <span class="n">index</span> <span class="o">=</span> <span class="n">factor</span><span class="p">,</span> <span class="mi">0</span>
-
-                <span class="k">for</span> <span class="n">_</span><span class="p">,</span> <span class="n">factor</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">reals</span><span class="p">[:</span><span class="n">j</span><span class="p">]:</span>
-                    <span class="k">if</span> <span class="n">factor</span> <span class="o">==</span> <span class="n">poly</span><span class="p">:</span>
-                        <span class="n">index</span> <span class="o">+=</span> <span class="mi">1</span>
-
-                <span class="k">return</span> <span class="n">poly</span><span class="p">,</span> <span class="n">index</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">i</span> <span class="o">+=</span> <span class="n">k</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_complexes_index</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">complexes</span><span class="p">,</span> <span class="n">index</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Map initial complex root index to an index in a factor where the root belongs. &quot;&quot;&quot;</span>
-        <span class="n">index</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="n">index</span><span class="p">,</span> <span class="mi">0</span>
-
-        <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="p">(</span><span class="n">_</span><span class="p">,</span> <span class="n">factor</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">complexes</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">index</span> <span class="o">&lt;</span> <span class="n">i</span> <span class="o">+</span> <span class="n">k</span><span class="p">:</span>
-                <span class="n">poly</span><span class="p">,</span> <span class="n">index</span> <span class="o">=</span> <span class="n">factor</span><span class="p">,</span> <span class="mi">0</span>
-
-                <span class="k">for</span> <span class="n">_</span><span class="p">,</span> <span class="n">factor</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">complexes</span><span class="p">[:</span><span class="n">j</span><span class="p">]:</span>
-                    <span class="k">if</span> <span class="n">factor</span> <span class="o">==</span> <span class="n">poly</span><span class="p">:</span>
-                        <span class="n">index</span> <span class="o">+=</span> <span class="mi">1</span>
-
-                <span class="n">index</span> <span class="o">+=</span> <span class="nb">len</span><span class="p">(</span><span class="n">_reals_cache</span><span class="p">[</span><span class="n">poly</span><span class="p">])</span>
-
-                <span class="k">return</span> <span class="n">poly</span><span class="p">,</span> <span class="n">index</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">i</span> <span class="o">+=</span> <span class="n">k</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_count_roots</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">roots</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Count the number of real or complex roots including multiplicites. &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">sum</span><span class="p">([</span> <span class="n">k</span> <span class="k">for</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">roots</span> <span class="p">])</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_indexed_root</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">poly</span><span class="p">,</span> <span class="n">index</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Get a root of a composite polynomial by index. &quot;&quot;&quot;</span>
-        <span class="p">(</span><span class="n">_</span><span class="p">,</span> <span class="n">factors</span><span class="p">)</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">factor_list</span><span class="p">()</span>
-
-        <span class="n">reals</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_get_reals</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span>
-        <span class="n">reals_count</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_count_roots</span><span class="p">(</span><span class="n">reals</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">index</span> <span class="o">&lt;</span> <span class="n">reals_count</span><span class="p">:</span>
-            <span class="n">reals</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_reals_sorted</span><span class="p">(</span><span class="n">reals</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_reals_index</span><span class="p">(</span><span class="n">reals</span><span class="p">,</span> <span class="n">index</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">complexes</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_get_complexes</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span>
-            <span class="n">complexes</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_complexes_sorted</span><span class="p">(</span><span class="n">complexes</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_complexes_index</span><span class="p">(</span><span class="n">complexes</span><span class="p">,</span> <span class="n">index</span> <span class="o">-</span> <span class="n">reals_count</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_real_roots</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">poly</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Get real roots of a composite polynomial. &quot;&quot;&quot;</span>
-        <span class="p">(</span><span class="n">_</span><span class="p">,</span> <span class="n">factors</span><span class="p">)</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">factor_list</span><span class="p">()</span>
-
-        <span class="n">reals</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_get_reals</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span>
-        <span class="n">reals</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_reals_sorted</span><span class="p">(</span><span class="n">reals</span><span class="p">)</span>
-        <span class="n">reals_count</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_count_roots</span><span class="p">(</span><span class="n">reals</span><span class="p">)</span>
-
-        <span class="n">roots</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">index</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">reals_count</span><span class="p">):</span>
-            <span class="n">roots</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">cls</span><span class="o">.</span><span class="n">_reals_index</span><span class="p">(</span><span class="n">reals</span><span class="p">,</span> <span class="n">index</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">roots</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_all_roots</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">poly</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Get real and complex roots of a composite polynomial. &quot;&quot;&quot;</span>
-        <span class="p">(</span><span class="n">_</span><span class="p">,</span> <span class="n">factors</span><span class="p">)</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">factor_list</span><span class="p">()</span>
-
-        <span class="n">reals</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_get_reals</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span>
-        <span class="n">reals</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_reals_sorted</span><span class="p">(</span><span class="n">reals</span><span class="p">)</span>
-        <span class="n">reals_count</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_count_roots</span><span class="p">(</span><span class="n">reals</span><span class="p">)</span>
-
-        <span class="n">roots</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">index</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">reals_count</span><span class="p">):</span>
-            <span class="n">roots</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">cls</span><span class="o">.</span><span class="n">_reals_index</span><span class="p">(</span><span class="n">reals</span><span class="p">,</span> <span class="n">index</span><span class="p">))</span>
-
-        <span class="n">complexes</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_get_complexes</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span>
-        <span class="n">complexes</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_complexes_sorted</span><span class="p">(</span><span class="n">complexes</span><span class="p">)</span>
-        <span class="n">complexes_count</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_count_roots</span><span class="p">(</span><span class="n">complexes</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">index</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">complexes_count</span><span class="p">):</span>
-            <span class="n">roots</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">cls</span><span class="o">.</span><span class="n">_complexes_index</span><span class="p">(</span><span class="n">complexes</span><span class="p">,</span> <span class="n">index</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">roots</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_roots_trivial</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">poly</span><span class="p">,</span> <span class="n">radicals</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute roots in linear, quadratic and binomial cases. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">poly</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">roots_linear</span><span class="p">(</span><span class="n">poly</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">radicals</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="k">if</span> <span class="n">radicals</span> <span class="ow">and</span> <span class="n">poly</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">roots_quadratic</span><span class="p">(</span><span class="n">poly</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">radicals</span> <span class="ow">and</span> <span class="n">poly</span><span class="o">.</span><span class="n">length</span><span class="p">()</span> <span class="o">==</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">poly</span><span class="o">.</span><span class="n">TC</span><span class="p">():</span>
-            <span class="k">return</span> <span class="n">roots_binomial</span><span class="p">(</span><span class="n">poly</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_preprocess_roots</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">poly</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Take heroic measures to make ``poly`` compatible with ``RootOf``. &quot;&quot;&quot;</span>
-        <span class="n">dom</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">get_domain</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">dom</span><span class="o">.</span><span class="n">is_Exact</span><span class="p">:</span>
-            <span class="n">poly</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">to_exact</span><span class="p">()</span>
-
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">poly</span> <span class="o">=</span> <span class="n">preprocess_roots</span><span class="p">(</span><span class="n">poly</span><span class="p">)</span>
-        <span class="n">dom</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">get_domain</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">dom</span><span class="o">.</span><span class="n">is_ZZ</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;RootOf is not supported over </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">dom</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">poly</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_postprocess_root</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">root</span><span class="p">,</span> <span class="n">radicals</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the root if it is trivial or a ``RootOf`` object. &quot;&quot;&quot;</span>
-        <span class="n">poly</span><span class="p">,</span> <span class="n">index</span> <span class="o">=</span> <span class="n">root</span>
-        <span class="n">roots</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_roots_trivial</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">radicals</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">roots</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">roots</span><span class="p">[</span><span class="n">index</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">index</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_get_roots</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">method</span><span class="p">,</span> <span class="n">poly</span><span class="p">,</span> <span class="n">radicals</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return postprocessed roots of specified kind. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">poly</span><span class="o">.</span><span class="n">is_univariate</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">PolynomialError</span><span class="p">(</span><span class="s">&quot;only univariate polynomials are allowed&quot;</span><span class="p">)</span>
-
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">poly</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_preprocess_roots</span><span class="p">(</span><span class="n">poly</span><span class="p">)</span>
-        <span class="n">roots</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">root</span> <span class="ow">in</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">method</span><span class="p">)(</span><span class="n">poly</span><span class="p">):</span>
-            <span class="n">roots</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">coeff</span><span class="o">*</span><span class="n">cls</span><span class="o">.</span><span class="n">_postprocess_root</span><span class="p">(</span><span class="n">root</span><span class="p">,</span> <span class="n">radicals</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">roots</span>
-
-    <span class="k">def</span> <span class="nf">_get_interval</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Internal function for retrieving isolation interval from cache. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_reals_cache</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">poly</span><span class="p">][</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">reals_count</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">_reals_cache</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">poly</span><span class="p">])</span>
-            <span class="k">return</span> <span class="n">_complexes_cache</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">poly</span><span class="p">][</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span> <span class="o">-</span> <span class="n">reals_count</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">_set_interval</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">interval</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Internal function for updating isolation interval in cache. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-            <span class="n">_reals_cache</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">poly</span><span class="p">][</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">]</span> <span class="o">=</span> <span class="n">interval</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">reals_count</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">_reals_cache</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">poly</span><span class="p">])</span>
-            <span class="n">_complexes_cache</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">poly</span><span class="p">][</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span> <span class="o">-</span> <span class="n">reals_count</span><span class="p">]</span> <span class="o">=</span> <span class="n">interval</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Evaluate this complex root to the given precision. &quot;&quot;&quot;</span>
-        <span class="n">_prec</span><span class="p">,</span> <span class="n">mp</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">prec</span><span class="p">,</span> <span class="n">prec</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">func</span> <span class="o">=</span> <span class="n">lambdify</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">poly</span><span class="o">.</span><span class="n">gen</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">)</span>
-
-            <span class="n">interval</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_interval</span><span class="p">()</span>
-            <span class="n">refined</span> <span class="o">=</span> <span class="bp">False</span>
-
-            <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-                    <span class="n">x0</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">interval</span><span class="o">.</span><span class="n">center</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">x0</span> <span class="o">=</span> <span class="n">mpc</span><span class="p">(</span><span class="o">*</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">str</span><span class="p">,</span> <span class="n">interval</span><span class="o">.</span><span class="n">center</span><span class="p">)))</span>
-
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="n">root</span> <span class="o">=</span> <span class="n">findroot</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">x0</span><span class="p">)</span>
-                <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-                    <span class="n">interval</span> <span class="o">=</span> <span class="n">interval</span><span class="o">.</span><span class="n">refine</span><span class="p">()</span>
-                    <span class="n">refined</span> <span class="o">=</span> <span class="bp">True</span>
-                    <span class="k">continue</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">refined</span><span class="p">:</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">_set_interval</span><span class="p">(</span><span class="n">interval</span><span class="p">)</span>
-
-                    <span class="k">break</span>
-        <span class="k">finally</span><span class="p">:</span>
-            <span class="n">mp</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="n">_prec</span>
-
-        <span class="k">return</span> <span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">real</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">Float</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">imag</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="RootSum"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.rootoftools.RootSum">[docs]</a><span class="k">class</span> <span class="nc">RootSum</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Represents a sum of all roots of a univariate polynomial. &quot;&quot;&quot;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;poly&#39;</span><span class="p">,</span> <span class="s">&#39;fun&#39;</span><span class="p">,</span> <span class="s">&#39;auto&#39;</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">func</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">quadratic</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Construct a new ``RootSum`` instance carrying all roots of a polynomial. &quot;&quot;&quot;</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">poly</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_transform</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">poly</span><span class="o">.</span><span class="n">is_univariate</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">MultivariatePolynomialError</span><span class="p">(</span>
-                <span class="s">&quot;only univariate polynomials are allowed&quot;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">func</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">func</span> <span class="o">=</span> <span class="n">Lambda</span><span class="p">(</span><span class="n">poly</span><span class="o">.</span><span class="n">gen</span><span class="p">,</span> <span class="n">poly</span><span class="o">.</span><span class="n">gen</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">is_func</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">is_Function</span>
-            <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-                <span class="n">is_func</span> <span class="o">=</span> <span class="bp">False</span>
-
-            <span class="k">if</span> <span class="n">is_func</span> <span class="ow">and</span> <span class="p">(</span><span class="n">func</span><span class="o">.</span><span class="n">nargs</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="mi">1</span> <span class="ow">in</span> <span class="n">func</span><span class="o">.</span><span class="n">nargs</span><span class="p">):</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">Lambda</span><span class="p">):</span>
-                    <span class="n">func</span> <span class="o">=</span> <span class="n">Lambda</span><span class="p">(</span><span class="n">poly</span><span class="o">.</span><span class="n">gen</span><span class="p">,</span> <span class="n">func</span><span class="p">(</span><span class="n">poly</span><span class="o">.</span><span class="n">gen</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                    <span class="s">&quot;expected a univariate function, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">func</span><span class="p">)</span>
-
-        <span class="n">var</span><span class="p">,</span> <span class="n">expr</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">variables</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">func</span><span class="o">.</span><span class="n">expr</span>
-
-        <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">var</span><span class="p">,</span> <span class="n">coeff</span><span class="o">*</span><span class="n">var</span><span class="p">)</span>
-
-        <span class="n">deg</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">var</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">deg</span><span class="o">*</span><span class="n">expr</span>
-
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">add_const</span><span class="p">,</span> <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">var</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">add_const</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="n">mul_const</span><span class="p">,</span> <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">var</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">mul_const</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-        <span class="n">func</span> <span class="o">=</span> <span class="n">Lambda</span><span class="p">(</span><span class="n">var</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-
-        <span class="n">rational</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_is_func_rational</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">func</span><span class="p">)</span>
-        <span class="p">(</span><span class="n">_</span><span class="p">,</span> <span class="n">factors</span><span class="p">),</span> <span class="n">terms</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">factor_list</span><span class="p">(),</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">poly</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">factors</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">poly</span><span class="o">.</span><span class="n">is_linear</span><span class="p">:</span>
-                <span class="n">term</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">roots_linear</span><span class="p">(</span><span class="n">poly</span><span class="p">)[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="k">elif</span> <span class="n">quadratic</span> <span class="ow">and</span> <span class="n">poly</span><span class="o">.</span><span class="n">is_quadratic</span><span class="p">:</span>
-                <span class="n">term</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">roots_quadratic</span><span class="p">(</span><span class="n">poly</span><span class="p">)))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">rational</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">auto</span><span class="p">:</span>
-                    <span class="n">term</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">auto</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">term</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_rational_case</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">func</span><span class="p">)</span>
-
-            <span class="n">terms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">k</span><span class="o">*</span><span class="n">term</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">mul_const</span><span class="o">*</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">terms</span><span class="p">)</span> <span class="o">+</span> <span class="n">deg</span><span class="o">*</span><span class="n">add_const</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_new</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">poly</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Construct new raw ``RootSum`` instance. &quot;&quot;&quot;</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-
-        <span class="n">obj</span><span class="o">.</span><span class="n">poly</span> <span class="o">=</span> <span class="n">poly</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">fun</span> <span class="o">=</span> <span class="n">func</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">auto</span> <span class="o">=</span> <span class="n">auto</span>
-
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">new</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">poly</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">auto</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Construct new ``RootSum`` instance. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">func</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">func</span><span class="o">.</span><span class="n">variables</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">func</span><span class="o">.</span><span class="n">expr</span>
-
-        <span class="n">rational</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">_is_func_rational</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">func</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">rational</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">auto</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_new</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">auto</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_rational_case</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">func</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_transform</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Transform an expression to a polynomial. &quot;&quot;&quot;</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">PurePoly</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">greedy</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">preprocess_roots</span><span class="p">(</span><span class="n">poly</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_is_func_rational</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">poly</span><span class="p">,</span> <span class="n">func</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Check if a lambda is areational function. &quot;&quot;&quot;</span>
-        <span class="n">var</span><span class="p">,</span> <span class="n">expr</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">variables</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">func</span><span class="o">.</span><span class="n">expr</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_rational_function</span><span class="p">(</span><span class="n">var</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_rational_case</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">poly</span><span class="p">,</span> <span class="n">func</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Handle the rational function case. &quot;&quot;&quot;</span>
-        <span class="n">roots</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;r:</span><span class="si">%d</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">poly</span><span class="o">.</span><span class="n">degree</span><span class="p">())</span>
-        <span class="n">var</span><span class="p">,</span> <span class="n">expr</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">variables</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">func</span><span class="o">.</span><span class="n">expr</span>
-
-        <span class="n">f</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">var</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">roots</span><span class="p">)</span>
-        <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">together</span><span class="p">(</span><span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-
-        <span class="n">domain</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">roots</span><span class="p">]</span>
-
-        <span class="n">p</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-        <span class="n">q</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="n">domain</span><span class="p">,</span> <span class="n">expand</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">GeneratorsNeeded</span><span class="p">:</span>
-            <span class="n">p</span><span class="p">,</span> <span class="n">p_coeff</span> <span class="o">=</span> <span class="bp">None</span><span class="p">,</span> <span class="p">(</span><span class="n">p</span><span class="p">,)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">p_monom</span><span class="p">,</span> <span class="n">p_coeff</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">p</span><span class="o">.</span><span class="n">terms</span><span class="p">()))</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">q</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="n">domain</span><span class="p">,</span> <span class="n">expand</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">GeneratorsNeeded</span><span class="p">:</span>
-            <span class="n">q</span><span class="p">,</span> <span class="n">q_coeff</span> <span class="o">=</span> <span class="bp">None</span><span class="p">,</span> <span class="p">(</span><span class="n">q</span><span class="p">,)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">q_monom</span><span class="p">,</span> <span class="n">q_coeff</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">q</span><span class="o">.</span><span class="n">terms</span><span class="p">()))</span>
-
-        <span class="n">coeffs</span><span class="p">,</span> <span class="n">mapping</span> <span class="o">=</span> <span class="n">symmetrize</span><span class="p">(</span><span class="n">p_coeff</span> <span class="o">+</span> <span class="n">q_coeff</span><span class="p">,</span> <span class="n">formal</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="n">formulas</span><span class="p">,</span> <span class="n">values</span> <span class="o">=</span> <span class="n">viete</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">roots</span><span class="p">),</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">_</span><span class="p">),</span> <span class="p">(</span><span class="n">_</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">mapping</span><span class="p">,</span> <span class="n">formulas</span><span class="p">):</span>
-            <span class="n">values</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">sym</span><span class="p">,</span> <span class="n">val</span><span class="p">))</span>
-
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">_</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">coeffs</span><span class="p">):</span>
-            <span class="n">coeffs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">coeff</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">values</span><span class="p">)</span>
-
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">p_coeff</span><span class="p">)</span>
-
-        <span class="n">p_coeff</span> <span class="o">=</span> <span class="n">coeffs</span><span class="p">[:</span><span class="n">n</span><span class="p">]</span>
-        <span class="n">q_coeff</span> <span class="o">=</span> <span class="n">coeffs</span><span class="p">[</span><span class="n">n</span><span class="p">:]</span>
-
-        <span class="k">if</span> <span class="n">p</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">p_monom</span><span class="p">,</span> <span class="n">p_coeff</span><span class="p">))),</span> <span class="o">*</span><span class="n">p</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="p">(</span><span class="n">p</span><span class="p">,)</span> <span class="o">=</span> <span class="n">p_coeff</span>
-
-        <span class="k">if</span> <span class="n">q</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">q</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">q_monom</span><span class="p">,</span> <span class="n">q_coeff</span><span class="p">))),</span> <span class="o">*</span><span class="n">q</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="p">(</span><span class="n">q</span><span class="p">,)</span> <span class="o">=</span> <span class="n">q_coeff</span>
-
-        <span class="k">return</span> <span class="n">factor</span><span class="p">(</span><span class="n">p</span><span class="o">/</span><span class="n">q</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_hashable_content</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">poly</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">fun</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">expr</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">poly</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">fun</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">poly</span><span class="o">.</span><span class="n">gen</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">free_symbols</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">poly</span><span class="o">.</span><span class="n">free_symbols</span> <span class="o">|</span> <span class="bp">self</span><span class="o">.</span><span class="n">fun</span><span class="o">.</span><span class="n">free_symbols</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_commutative</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;roots&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span>
-
-        <span class="n">_roots</span> <span class="o">=</span> <span class="n">roots</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">poly</span><span class="p">,</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">_roots</span><span class="p">)</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">poly</span><span class="o">.</span><span class="n">degree</span><span class="p">():</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="bp">self</span><span class="o">.</span><span class="n">fun</span><span class="p">(</span><span class="n">r</span><span class="p">)</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">_roots</span> <span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_evalf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prec</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">_roots</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">poly</span><span class="o">.</span><span class="n">nroots</span><span class="p">(</span><span class="n">n</span><span class="o">=</span><span class="n">prec_to_dps</span><span class="p">(</span><span class="n">prec</span><span class="p">))</span>
-        <span class="k">except</span> <span class="p">(</span><span class="n">DomainError</span><span class="p">,</span> <span class="n">PolynomialError</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="bp">self</span><span class="o">.</span><span class="n">fun</span><span class="p">(</span><span class="n">r</span><span class="p">)</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">_roots</span> <span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="n">var</span><span class="p">,</span> <span class="n">expr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">fun</span><span class="o">.</span><span class="n">args</span>
-        <span class="n">func</span> <span class="o">=</span> <span class="n">Lambda</span><span class="p">(</span><span class="n">var</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">poly</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">auto</span><span class="p">)</span></div>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/polys/specialpolys.html b/dev-py3k/_modules/sympy/polys/specialpolys.html
deleted file mode 100644
index 4a60b66cbc7..00000000000
--- a/dev-py3k/_modules/sympy/polys/specialpolys.html
+++ /dev/null
@@ -1,479 +0,0 @@
-
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-            
-  <h1>Source code for sympy.polys.specialpolys</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Functions for generating interesting polynomials, e.g. for benchmarking. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="kn">from</span> <span class="nn">sympy.core.singleton</span> <span class="kn">import</span> <span class="n">S</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polytools</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">PurePoly</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polyutils</span> <span class="kn">import</span> <span class="n">_analyze_gens</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyclasses</span> <span class="kn">import</span> <span class="n">DMP</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dmp_zero</span><span class="p">,</span> <span class="n">dmp_one</span><span class="p">,</span> <span class="n">dmp_ground</span><span class="p">,</span> <span class="n">dmp_normal</span><span class="p">,</span>
-    <span class="n">dup_from_raw_dict</span><span class="p">,</span> <span class="n">dmp_raise</span><span class="p">,</span> <span class="n">dup_random</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dmp_add_term</span><span class="p">,</span> <span class="n">dmp_neg</span><span class="p">,</span> <span class="n">dmp_mul</span><span class="p">,</span> <span class="n">dmp_sqr</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.factortools</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">dup_zz_cyclotomic_poly</span>
-<span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-
-<span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">nextprime</span>
-
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">cythonized</span><span class="p">,</span> <span class="n">subsets</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,i&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="swinnerton_dyer_poly"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.specialpolys.swinnerton_dyer_poly">[docs]</a><span class="k">def</span> <span class="nf">swinnerton_dyer_poly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Generates n-th Swinnerton-Dyer polynomial in `x`.  &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-            <span class="s">&quot;can&#39;t generate Swinnerton-Dyer polynomial of order </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">cls</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">Poly</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">cls</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">),</span> <span class="n">PurePoly</span>
-
-    <span class="n">p</span><span class="p">,</span> <span class="n">elts</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="p">[[</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)],</span>
-                  <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)]]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">p</span><span class="p">,</span> <span class="n">_elts</span> <span class="o">=</span> <span class="n">nextprime</span><span class="p">(</span><span class="n">p</span><span class="p">),</span> <span class="p">[]</span>
-
-        <span class="n">neg_sqrt</span> <span class="o">=</span> <span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-        <span class="n">pos_sqrt</span> <span class="o">=</span> <span class="o">+</span><span class="n">sqrt</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">elt</span> <span class="ow">in</span> <span class="n">elts</span><span class="p">:</span>
-            <span class="n">_elts</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">elt</span> <span class="o">+</span> <span class="p">[</span><span class="n">neg_sqrt</span><span class="p">])</span>
-            <span class="n">_elts</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">elt</span> <span class="o">+</span> <span class="p">[</span><span class="n">pos_sqrt</span><span class="p">])</span>
-
-        <span class="n">elts</span> <span class="o">=</span> <span class="n">_elts</span>
-
-    <span class="n">poly</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">elt</span> <span class="ow">in</span> <span class="n">elts</span><span class="p">:</span>
-        <span class="n">poly</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">elt</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;polys&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">poly</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">PurePoly</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">poly</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="cyclotomic_poly"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.specialpolys.cyclotomic_poly">[docs]</a><span class="k">def</span> <span class="nf">cyclotomic_poly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Generates cyclotomic polynomial of order `n` in `x`. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-            <span class="s">&quot;can&#39;t generate cyclotomic polynomial of order </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="n">poly</span> <span class="o">=</span> <span class="n">DMP</span><span class="p">(</span><span class="n">dup_zz_cyclotomic_poly</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">PurePoly</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;polys&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">poly</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">poly</span>
-
-</div>
-<div class="viewcode-block" id="symmetric_poly"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.specialpolys.symmetric_poly">[docs]</a><span class="k">def</span> <span class="nf">symmetric_poly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Generates symmetric polynomial of order `n`. &quot;&quot;&quot;</span>
-    <span class="n">gens</span> <span class="o">=</span> <span class="n">_analyze_gens</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">gens</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;can&#39;t generate symmetric polynomial of order </span><span class="si">%s</span><span class="s"> for </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">gens</span><span class="p">))</span>
-    <span class="k">elif</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">subsets</span><span class="p">(</span><span class="n">gens</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">))</span> <span class="p">])</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;polys&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">poly</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">Poly</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="random_poly"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.specialpolys.random_poly">[docs]</a><span class="k">def</span> <span class="nf">random_poly</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">inf</span><span class="p">,</span> <span class="n">sup</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="n">ZZ</span><span class="p">,</span> <span class="n">polys</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return a polynomial of degree ``n`` with coefficients in ``[inf, sup]``. &quot;&quot;&quot;</span>
-    <span class="n">poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">dup_random</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">inf</span><span class="p">,</span> <span class="n">sup</span><span class="p">,</span> <span class="n">domain</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="n">domain</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">polys</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">poly</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">poly</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,i,j&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="interpolating_poly"><a class="viewcode-back" href="../../../modules/polys/reference.html#sympy.polys.specialpolys.interpolating_poly">[docs]</a><span class="k">def</span> <span class="nf">interpolating_poly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">X</span><span class="o">=</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="n">Y</span><span class="o">=</span><span class="s">&#39;y&#39;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Construct Lagrange interpolating polynomial for ``n`` data points. &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-        <span class="n">X</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s">:</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-        <span class="n">Y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s">:</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-
-    <span class="n">coeffs</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="n">numer</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">denom</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
-                <span class="k">continue</span>
-
-            <span class="n">numer</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">X</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-            <span class="n">denom</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">X</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-
-        <span class="n">numer</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">numer</span><span class="p">)</span>
-        <span class="n">denom</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">denom</span><span class="p">)</span>
-
-        <span class="n">coeffs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">numer</span><span class="o">/</span><span class="n">denom</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">coeff</span><span class="o">*</span><span class="n">y</span> <span class="k">for</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">y</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span> <span class="p">])</span>
-
-</div>
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">fateman_poly_F_1</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Fateman&#39;s GCD benchmark: trivial GCD &quot;&quot;&quot;</span>
-    <span class="n">Y</span> <span class="o">=</span> <span class="p">[</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y_&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">]</span>
-
-    <span class="n">y_0</span><span class="p">,</span> <span class="n">y_1</span> <span class="o">=</span> <span class="n">Y</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Y</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="n">u</span> <span class="o">=</span> <span class="n">y_0</span> <span class="o">+</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">y</span> <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">Y</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span> <span class="p">])</span>
-    <span class="n">v</span> <span class="o">=</span> <span class="n">y_0</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">Y</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span> <span class="p">])</span>
-
-    <span class="n">F</span> <span class="o">=</span> <span class="p">((</span><span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">as_poly</span><span class="p">(</span><span class="o">*</span><span class="n">Y</span><span class="p">)</span>
-    <span class="n">G</span> <span class="o">=</span> <span class="p">((</span><span class="n">v</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">y_1</span><span class="o">*</span><span class="n">y_0</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y_1</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">as_poly</span><span class="p">(</span><span class="o">*</span><span class="n">Y</span><span class="p">)</span>
-
-    <span class="n">H</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">*</span><span class="n">Y</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">H</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,m,i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dmp_fateman_poly_F_1</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Fateman&#39;s GCD benchmark: trivial GCD &quot;&quot;&quot;</span>
-    <span class="n">u</span> <span class="o">=</span> <span class="p">[</span><span class="n">K</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">K</span><span class="p">(</span><span class="mi">0</span><span class="p">)]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="n">u</span> <span class="o">=</span> <span class="p">[</span><span class="n">dmp_one</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">u</span><span class="p">]</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="n">K</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">K</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">K</span><span class="p">(</span><span class="mi">0</span><span class="p">)]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="n">dmp_one</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">dmp_zero</span><span class="p">(</span><span class="n">i</span><span class="p">),</span> <span class="n">v</span><span class="p">]</span>
-
-    <span class="n">m</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="n">U</span> <span class="o">=</span> <span class="n">dmp_add_term</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">dmp_ground</span><span class="p">(</span><span class="n">K</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">m</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">V</span> <span class="o">=</span> <span class="n">dmp_add_term</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">dmp_ground</span><span class="p">(</span><span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">m</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">f</span> <span class="o">=</span> <span class="p">[[</span><span class="o">-</span><span class="n">K</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span> <span class="n">K</span><span class="p">(</span><span class="mi">0</span><span class="p">)],</span> <span class="p">[],</span> <span class="p">[</span><span class="n">K</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">K</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="o">-</span><span class="n">K</span><span class="p">(</span><span class="mi">1</span><span class="p">)]]</span>
-
-    <span class="n">W</span> <span class="o">=</span> <span class="n">dmp_add_term</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">dmp_ground</span><span class="p">(</span><span class="n">K</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">m</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">Y</span> <span class="o">=</span> <span class="n">dmp_raise</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">F</span> <span class="o">=</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">U</span><span class="p">,</span> <span class="n">V</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">G</span> <span class="o">=</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">W</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">H</span> <span class="o">=</span> <span class="n">dmp_one</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">H</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">fateman_poly_F_2</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Fateman&#39;s GCD benchmark: linearly dense quartic inputs &quot;&quot;&quot;</span>
-    <span class="n">Y</span> <span class="o">=</span> <span class="p">[</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y_&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">]</span>
-
-    <span class="n">y_0</span> <span class="o">=</span> <span class="n">Y</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="n">u</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">y</span> <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">Y</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span> <span class="p">])</span>
-
-    <span class="n">H</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">((</span><span class="n">y_0</span> <span class="o">+</span> <span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="o">*</span><span class="n">Y</span><span class="p">)</span>
-
-    <span class="n">F</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">((</span><span class="n">y_0</span> <span class="o">-</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="o">*</span><span class="n">Y</span><span class="p">)</span>
-    <span class="n">G</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">((</span><span class="n">y_0</span> <span class="o">+</span> <span class="n">u</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="o">*</span><span class="n">Y</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">H</span><span class="o">*</span><span class="n">F</span><span class="p">,</span> <span class="n">H</span><span class="o">*</span><span class="n">G</span><span class="p">,</span> <span class="n">H</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,m,i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dmp_fateman_poly_F_2</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Fateman&#39;s GCD benchmark: linearly dense quartic inputs &quot;&quot;&quot;</span>
-    <span class="n">u</span> <span class="o">=</span> <span class="p">[</span><span class="n">K</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">K</span><span class="p">(</span><span class="mi">0</span><span class="p">)]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">u</span> <span class="o">=</span> <span class="p">[</span><span class="n">dmp_one</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">u</span><span class="p">]</span>
-
-    <span class="n">m</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">dmp_add_term</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">dmp_ground</span><span class="p">(</span><span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">m</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_sqr</span><span class="p">([</span><span class="n">dmp_one</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">dmp_neg</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">K</span><span class="p">)],</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">g</span> <span class="o">=</span> <span class="n">dmp_sqr</span><span class="p">([</span><span class="n">dmp_one</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">v</span><span class="p">],</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">dmp_add_term</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">dmp_one</span><span class="p">(</span><span class="n">m</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_sqr</span><span class="p">([</span><span class="n">dmp_one</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">v</span><span class="p">],</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">h</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">fateman_poly_F_3</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Fateman&#39;s GCD benchmark: sparse inputs (deg f ~ vars f) &quot;&quot;&quot;</span>
-    <span class="n">Y</span> <span class="o">=</span> <span class="p">[</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y_&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">]</span>
-
-    <span class="n">y_0</span> <span class="o">=</span> <span class="n">Y</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="n">u</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">y</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">Y</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span> <span class="p">])</span>
-
-    <span class="n">H</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">((</span><span class="n">y_0</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="o">*</span><span class="n">Y</span><span class="p">)</span>
-
-    <span class="n">F</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">((</span><span class="n">y_0</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="o">*</span><span class="n">Y</span><span class="p">)</span>
-    <span class="n">G</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">((</span><span class="n">y_0</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">u</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="o">*</span><span class="n">Y</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">H</span><span class="o">*</span><span class="n">F</span><span class="p">,</span> <span class="n">H</span><span class="o">*</span><span class="n">G</span><span class="p">,</span> <span class="n">H</span>
-
-
-<span class="nd">@cythonized</span><span class="p">(</span><span class="s">&quot;n,i&quot;</span><span class="p">)</span>
-<span class="k">def</span> <span class="nf">dmp_fateman_poly_F_3</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Fateman&#39;s GCD benchmark: sparse inputs (deg f ~ vars f) &quot;&quot;&quot;</span>
-    <span class="n">u</span> <span class="o">=</span> <span class="n">dup_from_raw_dict</span><span class="p">({</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:</span> <span class="n">K</span><span class="o">.</span><span class="n">one</span><span class="p">},</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">u</span> <span class="o">=</span> <span class="n">dmp_add_term</span><span class="p">([</span><span class="n">u</span><span class="p">],</span> <span class="n">dmp_one</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">dmp_add_term</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">dmp_ground</span><span class="p">(</span><span class="n">K</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">f</span> <span class="o">=</span> <span class="n">dmp_sqr</span><span class="p">(</span>
-        <span class="n">dmp_add_term</span><span class="p">([</span><span class="n">dmp_neg</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)],</span> <span class="n">dmp_one</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-    <span class="n">g</span> <span class="o">=</span> <span class="n">dmp_sqr</span><span class="p">(</span><span class="n">dmp_add_term</span><span class="p">([</span><span class="n">v</span><span class="p">],</span> <span class="n">dmp_one</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">v</span> <span class="o">=</span> <span class="n">dmp_add_term</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">dmp_one</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="n">dmp_sqr</span><span class="p">(</span><span class="n">dmp_add_term</span><span class="p">([</span><span class="n">v</span><span class="p">],</span> <span class="n">dmp_one</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">dmp_mul</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">K</span><span class="p">),</span> <span class="n">h</span>
-
-<span class="c"># A few useful polynomials from Wang&#39;s paper (&#39;78).</span>
-
-<span class="n">f_0</span> <span class="o">=</span> <span class="n">dmp_normal</span><span class="p">([</span>
-    <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]],</span>
-    <span class="p">[[</span><span class="mi">3</span><span class="p">]],</span>
-    <span class="p">[[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">]]</span>
-<span class="p">],</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-
-<span class="n">f_1</span> <span class="o">=</span> <span class="n">dmp_normal</span><span class="p">([</span>
-    <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]],</span>
-    <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">20</span><span class="p">,</span> <span class="mi">30</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">0</span><span class="p">]],</span>
-    <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">30</span><span class="p">,</span> <span class="mi">20</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">610</span><span class="p">],</span> <span class="p">[</span><span class="mi">20</span><span class="p">,</span> <span class="mi">230</span><span class="p">,</span> <span class="mi">300</span><span class="p">]],</span>
-    <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">30</span><span class="p">,</span> <span class="mi">320</span><span class="p">,</span> <span class="mi">200</span><span class="p">],</span> <span class="p">[</span><span class="mi">600</span><span class="p">,</span> <span class="mi">6000</span><span class="p">]]</span>
-<span class="p">],</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-
-<span class="n">f_2</span> <span class="o">=</span> <span class="n">dmp_normal</span><span class="p">([</span>
-    <span class="p">[[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]],</span>
-    <span class="p">[[]],</span>
-    <span class="p">[[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">90</span><span class="p">],</span> <span class="p">[</span><span class="mi">90</span><span class="p">,</span> <span class="mi">0</span><span class="p">]],</span>
-    <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">11</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">11</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]],</span>
-    <span class="p">[[]],</span>
-    <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">11</span><span class="p">],</span> <span class="p">[</span><span class="mi">90</span><span class="p">,</span> <span class="o">-</span><span class="mi">990</span><span class="p">]]</span>
-<span class="p">],</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-
-<span class="n">f_3</span> <span class="o">=</span> <span class="n">dmp_normal</span><span class="p">([</span>
-    <span class="p">[[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[]],</span>
-    <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]],</span>
-    <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]],</span>
-    <span class="p">[[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]],</span>
-    <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]],</span>
-    <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]]</span>
-<span class="p">],</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-
-<span class="n">f_4</span> <span class="o">=</span> <span class="n">dmp_normal</span><span class="p">([</span>
-    <span class="p">[[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[]],</span>
-    <span class="p">[[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[]],</span>
-    <span class="p">[[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[]],</span>
-    <span class="p">[[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
-        <span class="p">[],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[]],</span>
-    <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]],</span>
-    <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">15</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[</span><span class="o">-</span>
-        <span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[]],</span>
-    <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span>
-        <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]],</span>
-    <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]],</span>
-    <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[]],</span>
-    <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]</span>
-<span class="p">],</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-
-<span class="n">f_5</span> <span class="o">=</span> <span class="n">dmp_normal</span><span class="p">([</span>
-    <span class="p">[[</span><span class="o">-</span><span class="mi">1</span><span class="p">]],</span>
-    <span class="p">[[</span><span class="o">-</span><span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">]],</span>
-    <span class="p">[[</span><span class="o">-</span><span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">6</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]],</span>
-    <span class="p">[[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]</span>
-<span class="p">],</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-
-<span class="n">f_6</span> <span class="o">=</span> <span class="n">dmp_normal</span><span class="p">([</span>
-    <span class="p">[[[</span><span class="mi">2115</span><span class="p">]],</span> <span class="p">[[]]],</span>
-    <span class="p">[[[</span><span class="mi">45</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[</span><span class="o">-</span><span class="mi">45</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]],</span>
-    <span class="p">[[[]]],</span>
-    <span class="p">[[[</span><span class="o">-</span><span class="mi">423</span><span class="p">]],</span> <span class="p">[[</span><span class="o">-</span><span class="mi">47</span><span class="p">]],</span> <span class="p">[[]],</span> <span class="p">[[</span><span class="mi">141</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="mi">94</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]],</span> <span class="p">[[]]],</span>
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-    <span class="p">]</span>
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-
-
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-    <span class="p">[[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]]</span>
-<span class="p">],</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-
-<span class="n">w_2</span> <span class="o">=</span> <span class="n">dmp_normal</span><span class="p">([</span>
-    <span class="p">[</span><span class="mi">24</span><span class="p">,</span> <span class="mi">48</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
-    <span class="p">[</span><span class="mi">24</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">72</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
-    <span class="p">[</span><span class="mi">25</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">8</span><span class="p">],</span>
-    <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">12</span><span class="p">],</span>
-    <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">292</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
-    <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
-    <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">48</span><span class="p">],</span>
-    <span class="p">[],</span>
-    <span class="p">[</span><span class="o">-</span><span class="mi">12</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-<span class="p">],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-</pre></div>
-
-          </div>
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diff --git a/dev-py3k/_modules/sympy/printing/ccode.html b/dev-py3k/_modules/sympy/printing/ccode.html
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+++ /dev/null
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-            
-  <h1>Source code for sympy.printing.ccode</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">C code printer</span>
-
-<span class="sd">The CCodePrinter converts single sympy expressions into single C expressions,</span>
-<span class="sd">using the functions defined in math.h where possible.</span>
-
-<span class="sd">A complete code generator, which uses ccode extensively, can be found in</span>
-<span class="sd">sympy.utilities.codegen. The codegen module can be used to generate complete</span>
-<span class="sd">source code files that are compilable without further modifications.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">C</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.codeprinter</span> <span class="kn">import</span> <span class="n">CodePrinter</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.precedence</span> <span class="kn">import</span> <span class="n">precedence</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">default_sort_key</span>
-
-<span class="c"># dictionary mapping sympy function to (argument_conditions, C_function).</span>
-<span class="c"># Used in CCodePrinter._print_Function(self)</span>
-<span class="n">known_functions</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="s">&quot;ceiling&quot;</span><span class="p">:</span> <span class="p">[(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="bp">True</span><span class="p">,</span> <span class="s">&quot;ceil&quot;</span><span class="p">)],</span>
-    <span class="s">&quot;Abs&quot;</span><span class="p">:</span> <span class="p">[(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="ow">not</span> <span class="n">x</span><span class="o">.</span><span class="n">is_integer</span><span class="p">,</span> <span class="s">&quot;fabs&quot;</span><span class="p">)],</span>
-<span class="p">}</span>
-
-
-<div class="viewcode-block" id="CCodePrinter"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.ccode.CCodePrinter">[docs]</a><span class="k">class</span> <span class="nc">CCodePrinter</span><span class="p">(</span><span class="n">CodePrinter</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A printer to convert python expressions to strings of c code&quot;&quot;&quot;</span>
-    <span class="n">printmethod</span> <span class="o">=</span> <span class="s">&quot;_ccode&quot;</span>
-
-    <span class="n">_default_settings</span> <span class="o">=</span> <span class="p">{</span>
-        <span class="s">&#39;order&#39;</span><span class="p">:</span> <span class="bp">None</span><span class="p">,</span>
-        <span class="s">&#39;full_prec&#39;</span><span class="p">:</span> <span class="s">&#39;auto&#39;</span><span class="p">,</span>
-        <span class="s">&#39;precision&#39;</span><span class="p">:</span> <span class="mi">15</span><span class="p">,</span>
-        <span class="s">&#39;user_functions&#39;</span><span class="p">:</span> <span class="p">{},</span>
-        <span class="s">&#39;human&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">,</span>
-    <span class="p">}</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">settings</span><span class="o">=</span><span class="p">{}):</span>
-        <span class="sd">&quot;&quot;&quot;Register function mappings supplied by user&quot;&quot;&quot;</span>
-        <span class="n">CodePrinter</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">settings</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">known_functions</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">known_functions</span><span class="p">)</span>
-        <span class="n">userfuncs</span> <span class="o">=</span> <span class="n">settings</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;user_functions&#39;</span><span class="p">,</span> <span class="p">{})</span>
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">userfuncs</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span>
-                <span class="n">userfuncs</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="k">lambda</span> <span class="o">*</span><span class="n">x</span><span class="p">:</span> <span class="bp">True</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">known_functions</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">userfuncs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_rate_index_position</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;function to calculate score based on position among indices</span>
-
-<span class="sd">        This method is used to sort loops in an optimized order, see</span>
-<span class="sd">        CodePrinter._sort_optimized()</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">p</span><span class="o">*</span><span class="mi">5</span>
-
-    <span class="k">def</span> <span class="nf">_get_statement</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">codestring</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">;&quot;</span> <span class="o">%</span> <span class="n">codestring</span>
-
-<div class="viewcode-block" id="CCodePrinter.doprint"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.ccode.CCodePrinter.doprint">[docs]</a>    <span class="k">def</span> <span class="nf">doprint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">assign_to</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Actually format the expression as C code.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">assign_to</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="n">assign_to</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="n">assign_to</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">assign_to</span><span class="p">,</span> <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Basic</span><span class="p">,</span> <span class="nb">type</span><span class="p">(</span><span class="bp">None</span><span class="p">))):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;CCodePrinter cannot assign to object of type </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span>
-                    <span class="nb">type</span><span class="p">(</span><span class="n">assign_to</span><span class="p">))</span>
-
-        <span class="c"># keep a set of expressions that are not strictly translatable to C</span>
-        <span class="c"># and number constants that must be declared and initialized</span>
-        <span class="n">not_c</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_not_supported</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_number_symbols</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-
-        <span class="c"># We treat top level Piecewise here to get if tests outside loops</span>
-        <span class="n">lines</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Piecewise</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;if (</span><span class="si">%s</span><span class="s">) {&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">c</span><span class="p">))</span>
-                <span class="k">elif</span> <span class="n">i</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">c</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                    <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;else {&quot;</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;else if (</span><span class="si">%s</span><span class="s">) {&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">c</span><span class="p">))</span>
-                <span class="n">code0</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_doprint_a_piece</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">assign_to</span><span class="p">)</span>
-                <span class="n">lines</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">code0</span><span class="p">)</span>
-                <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;}&quot;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">code0</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_doprint_a_piece</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assign_to</span><span class="p">)</span>
-            <span class="n">lines</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">code0</span><span class="p">)</span>
-
-        <span class="c"># format the output</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&quot;human&quot;</span><span class="p">]:</span>
-            <span class="n">frontlines</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">not_c</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">frontlines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;// Not C:&quot;</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">expr</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">not_c</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="nb">str</span><span class="p">):</span>
-                    <span class="n">frontlines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;// </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">repr</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-            <span class="k">for</span> <span class="n">name</span><span class="p">,</span> <span class="n">value</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_number_symbols</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="nb">str</span><span class="p">):</span>
-                <span class="n">frontlines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;double const </span><span class="si">%s</span><span class="s"> = </span><span class="si">%s</span><span class="s">;&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">value</span><span class="p">))</span>
-            <span class="n">lines</span> <span class="o">=</span> <span class="n">frontlines</span> <span class="o">+</span> <span class="n">lines</span>
-            <span class="n">lines</span> <span class="o">=</span> <span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">lines</span><span class="p">)</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">indent_code</span><span class="p">(</span><span class="n">lines</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">lines</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">indent_code</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">lines</span><span class="p">))</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_number_symbols</span><span class="p">,</span> <span class="n">not_c</span><span class="p">,</span> <span class="n">lines</span>
-        <span class="k">del</span> <span class="bp">self</span><span class="o">.</span><span class="n">_not_supported</span>
-        <span class="k">del</span> <span class="bp">self</span><span class="o">.</span><span class="n">_number_symbols</span>
-        <span class="k">return</span> <span class="n">result</span>
-</div>
-    <span class="k">def</span> <span class="nf">_get_loop_opening_ending</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">indices</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a tuple (open_lines, close_lines) containing lists of codelines</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">open_lines</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">close_lines</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">loopstart</span> <span class="o">=</span> <span class="s">&quot;for (int </span><span class="si">%(var)s</span><span class="s">=</span><span class="si">%(start)s</span><span class="s">; </span><span class="si">%(var)s</span><span class="s">&lt;</span><span class="si">%(end)s</span><span class="s">; </span><span class="si">%(var)s</span><span class="s">++){&quot;</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">indices</span><span class="p">:</span>
-            <span class="c"># C arrays start at 0 and end at dimension-1</span>
-            <span class="n">open_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">loopstart</span> <span class="o">%</span> <span class="p">{</span>
-                <span class="s">&#39;var&#39;</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">label</span><span class="p">),</span>
-                <span class="s">&#39;start&#39;</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">lower</span><span class="p">),</span>
-                <span class="s">&#39;end&#39;</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">upper</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)})</span>
-            <span class="n">close_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;}&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">open_lines</span><span class="p">,</span> <span class="n">close_lines</span>
-
-    <span class="k">def</span> <span class="nf">_print_Pow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">PREC</span> <span class="o">=</span> <span class="n">precedence</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&#39;1.0/</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">PREC</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span> <span class="o">==</span> <span class="mf">0.5</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&#39;sqrt(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&#39;pow(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span>
-                                 <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Rational</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">p</span><span class="p">),</span> <span class="nb">int</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%d</span><span class="s">.0/</span><span class="si">%d</span><span class="s">.0&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Indexed</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="c"># calculate index for 1d array</span>
-        <span class="n">dims</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">shape</span>
-        <span class="n">inds</span> <span class="o">=</span> <span class="p">[</span> <span class="n">i</span><span class="o">.</span><span class="n">label</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">indices</span> <span class="p">]</span>
-        <span class="n">elem</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="n">offset</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">rank</span><span class="p">))):</span>
-            <span class="n">elem</span> <span class="o">+=</span> <span class="n">offset</span><span class="o">*</span><span class="n">inds</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="n">offset</span> <span class="o">*=</span> <span class="n">dims</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">[</span><span class="si">%s</span><span class="s">]&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">label</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">elem</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Exp1</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;M_E&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Pi</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;M_PI&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Infinity</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;HUGE_VAL&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_print_NegativeInfinity</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;-HUGE_VAL&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Piecewise</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="c"># This method is called only for inline if constructs</span>
-        <span class="c"># Top level piecewise is handled in doprint()</span>
-        <span class="n">ecpairs</span> <span class="o">=</span> <span class="p">[</span><span class="s">&quot;(</span><span class="si">%s</span><span class="s">) {</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="s">}</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">c</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="p">))</span>
-                   <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span>
-        <span class="n">last_line</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">cond</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">last_line</span> <span class="o">=</span> <span class="s">&quot;else {</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">ecpairs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;(</span><span class="si">%s</span><span class="s">) {</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span>
-                           <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">cond</span><span class="p">),</span>
-                            <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">expr</span><span class="p">)))</span>
-        <span class="n">code</span> <span class="o">=</span> <span class="s">&quot;if </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">+</span> <span class="n">last_line</span>
-        <span class="k">return</span> <span class="n">code</span> <span class="o">%</span> <span class="s">&quot;else if &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">ecpairs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_And</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">PREC</span> <span class="o">=</span> <span class="n">precedence</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&#39; &amp;&amp; &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">PREC</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Or</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">PREC</span> <span class="o">=</span> <span class="n">precedence</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&#39; || &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">PREC</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Not</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">PREC</span> <span class="o">=</span> <span class="n">precedence</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&#39;!&#39;</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">PREC</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Function</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="o">.</span><span class="n">__name__</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">known_functions</span><span class="p">:</span>
-            <span class="n">cond_cfunc</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">known_functions</span><span class="p">[</span><span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="o">.</span><span class="n">__name__</span><span class="p">]</span>
-            <span class="k">for</span> <span class="n">cond</span><span class="p">,</span> <span class="n">cfunc</span> <span class="ow">in</span> <span class="n">cond_cfunc</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">cond</span><span class="p">(</span><span class="o">*</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">cfunc</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">stringify</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="s">&quot;, &quot;</span><span class="p">))</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&#39;_imp_&#39;</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">_imp_</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Lambda</span><span class="p">):</span>
-            <span class="c"># inlined function</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">_imp_</span><span class="p">(</span><span class="o">*</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">CodePrinter</span><span class="o">.</span><span class="n">_print_Function</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-
-<div class="viewcode-block" id="CCodePrinter.indent_code"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.ccode.CCodePrinter.indent_code">[docs]</a>    <span class="k">def</span> <span class="nf">indent_code</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">code</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Accepts a string of code or a list of code lines&quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">code</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="n">code_lines</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">indent_code</span><span class="p">(</span><span class="n">code</span><span class="o">.</span><span class="n">splitlines</span><span class="p">(</span><span class="bp">True</span><span class="p">))</span>
-            <span class="k">return</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">code_lines</span><span class="p">)</span>
-
-        <span class="n">tab</span> <span class="o">=</span> <span class="s">&quot;   &quot;</span>
-        <span class="n">inc_token</span> <span class="o">=</span> <span class="p">(</span><span class="s">&#39;{&#39;</span><span class="p">,</span> <span class="s">&#39;(&#39;</span><span class="p">,</span> <span class="s">&#39;{</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;(</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">)</span>
-        <span class="n">dec_token</span> <span class="o">=</span> <span class="p">(</span><span class="s">&#39;}&#39;</span><span class="p">,</span> <span class="s">&#39;)&#39;</span><span class="p">)</span>
-
-        <span class="n">code</span> <span class="o">=</span> <span class="p">[</span> <span class="n">line</span><span class="o">.</span><span class="n">lstrip</span><span class="p">(</span><span class="s">&#39; </span><span class="se">\t</span><span class="s">&#39;</span><span class="p">)</span> <span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="n">code</span> <span class="p">]</span>
-
-        <span class="n">increase</span> <span class="o">=</span> <span class="p">[</span> <span class="nb">int</span><span class="p">(</span><span class="nb">any</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">line</span><span class="o">.</span><span class="n">endswith</span><span class="p">,</span> <span class="n">inc_token</span><span class="p">)))</span> <span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="n">code</span> <span class="p">]</span>
-        <span class="n">decrease</span> <span class="o">=</span> <span class="p">[</span> <span class="nb">int</span><span class="p">(</span><span class="nb">any</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">line</span><span class="o">.</span><span class="n">startswith</span><span class="p">,</span> <span class="n">dec_token</span><span class="p">)))</span>
-                     <span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="n">code</span> <span class="p">]</span>
-
-        <span class="n">pretty</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">level</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">line</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">code</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">line</span> <span class="o">==</span> <span class="s">&#39;&#39;</span> <span class="ow">or</span> <span class="n">line</span> <span class="o">==</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">:</span>
-                <span class="n">pretty</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">line</span><span class="p">)</span>
-                <span class="k">continue</span>
-            <span class="n">level</span> <span class="o">-=</span> <span class="n">decrease</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-            <span class="n">pretty</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tab</span><span class="o">*</span><span class="n">level</span><span class="p">,</span> <span class="n">line</span><span class="p">))</span>
-            <span class="n">level</span> <span class="o">+=</span> <span class="n">increase</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">pretty</span>
-
-</div></div>
-<div class="viewcode-block" id="ccode"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.ccode.ccode">[docs]</a><span class="k">def</span> <span class="nf">ccode</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assign_to</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;Converts an expr to a string of c code</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        expr : sympy.core.Expr</span>
-<span class="sd">            a sympy expression to be converted</span>
-<span class="sd">        precision : optional</span>
-<span class="sd">            the precision for numbers such as pi [default=15]</span>
-<span class="sd">        user_functions : optional</span>
-<span class="sd">            A dictionary where keys are FunctionClass instances and values</span>
-<span class="sd">            are there string representations.  Alternatively, the</span>
-<span class="sd">            dictionary value can be a list of tuples i.e. [(argument_test,</span>
-<span class="sd">            cfunction_string)].  See below for examples.</span>
-<span class="sd">        human : optional</span>
-<span class="sd">            If True, the result is a single string that may contain some</span>
-<span class="sd">            constant declarations for the number symbols. If False, the</span>
-<span class="sd">            same information is returned in a more programmer-friendly</span>
-<span class="sd">            data structure.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import ccode, symbols, Rational, sin</span>
-<span class="sd">        &gt;&gt;&gt; x, tau = symbols([&quot;x&quot;, &quot;tau&quot;])</span>
-<span class="sd">        &gt;&gt;&gt; ccode((2*tau)**Rational(7,2))</span>
-<span class="sd">        &#39;8*sqrt(2)*pow(tau, 7.0/2.0)&#39;</span>
-<span class="sd">        &gt;&gt;&gt; ccode(sin(x), assign_to=&quot;s&quot;)</span>
-<span class="sd">        &#39;s = sin(x);&#39;</span>
-
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">CCodePrinter</span><span class="p">(</span><span class="n">settings</span><span class="p">)</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assign_to</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="print_ccode"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.ccode.print_ccode">[docs]</a><span class="k">def</span> <span class="nf">print_ccode</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Prints C representation of the given expression.&quot;&quot;&quot;</span>
-    <span class="k">print</span><span class="p">(</span><span class="n">ccode</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">))</span></div>
-</pre></div>
-
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-  <h1>Source code for sympy.printing.codeprinter</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">C</span><span class="p">,</span> <span class="n">Add</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.str</span> <span class="kn">import</span> <span class="n">StrPrinter</span>
-<span class="kn">from</span> <span class="nn">sympy.tensor</span> <span class="kn">import</span> <span class="n">get_indices</span><span class="p">,</span> <span class="n">get_contraction_structure</span>
-
-
-<div class="viewcode-block" id="AssignmentError"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.codeprinter.AssignmentError">[docs]</a><span class="k">class</span> <span class="nc">AssignmentError</span><span class="p">(</span><span class="ne">Exception</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Raised if an assignment variable for a loop is missing.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="CodePrinter"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.codeprinter.CodePrinter">[docs]</a><span class="k">class</span> <span class="nc">CodePrinter</span><span class="p">(</span><span class="n">StrPrinter</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    The base class for code-printing subclasses.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_doprint_a_piece</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">assign_to</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="c"># Here we print an expression that may contain Indexed objects, they</span>
-        <span class="c"># correspond to arrays in the generated code.  The low-level implementation</span>
-        <span class="c"># involves looping over array elements and possibly storing results in temporary</span>
-        <span class="c"># variables or accumulate it in the assign_to object.</span>
-
-        <span class="n">lhs_printed</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">assign_to</span><span class="p">)</span>
-        <span class="n">lines</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="c"># Setup loops over non-dummy indices  --  all terms need these</span>
-        <span class="n">indices</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_expression_indices</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assign_to</span><span class="p">)</span>
-        <span class="n">openloop</span><span class="p">,</span> <span class="n">closeloop</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_loop_opening_ending</span><span class="p">(</span><span class="n">indices</span><span class="p">)</span>
-
-        <span class="c"># Setup loops over dummy indices  --  each term needs separate treatment</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">get_contraction_structure</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-        <span class="c"># terms with no summations first</span>
-        <span class="k">if</span> <span class="bp">None</span> <span class="ow">in</span> <span class="n">d</span><span class="p">:</span>
-            <span class="n">text</span> <span class="o">=</span> <span class="n">CodePrinter</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">d</span><span class="p">[</span><span class="bp">None</span><span class="p">]))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># If all terms have summations we must initialize array to Zero</span>
-            <span class="n">text</span> <span class="o">=</span> <span class="n">CodePrinter</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="c"># skip redundant assignments</span>
-        <span class="k">if</span> <span class="n">text</span> <span class="o">!=</span> <span class="n">lhs_printed</span><span class="p">:</span>
-            <span class="n">lines</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">openloop</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">assign_to</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">text</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_statement</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> = </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">lhs_printed</span><span class="p">,</span> <span class="n">text</span><span class="p">))</span>
-            <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">text</span><span class="p">)</span>
-            <span class="n">lines</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">closeloop</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">dummies</span> <span class="ow">in</span> <span class="n">d</span><span class="p">:</span>
-            <span class="c"># then terms with summations</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">dummies</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span>
-                <span class="n">indices</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sort_optimized</span><span class="p">(</span><span class="n">dummies</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-                <span class="n">openloop_d</span><span class="p">,</span> <span class="n">closeloop_d</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_loop_opening_ending</span><span class="p">(</span>
-                    <span class="n">indices</span><span class="p">)</span>
-
-                <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">d</span><span class="p">[</span><span class="n">dummies</span><span class="p">]:</span>
-                    <span class="k">if</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">d</span> <span class="ow">and</span> <span class="ow">not</span> <span class="p">([</span><span class="nb">list</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">d</span><span class="p">[</span><span class="n">term</span><span class="p">]]</span>
-                            <span class="o">==</span> <span class="p">[[</span><span class="bp">None</span><span class="p">]</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">d</span><span class="p">[</span><span class="n">term</span><span class="p">]]):</span>
-                        <span class="c"># If one factor in the term has it&#39;s own internal</span>
-                        <span class="c"># contractions, those must be computed first.</span>
-                        <span class="c"># (temporary variables?)</span>
-                        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                            <span class="s">&quot;FIXME: no support for contractions in factor yet&quot;</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-
-                        <span class="c"># We need the lhs expression as an accumulator for</span>
-                        <span class="c"># the loops, i.e</span>
-                        <span class="c">#</span>
-                        <span class="c"># for (int d=0; d &lt; dim; d++){</span>
-                        <span class="c">#    lhs[] = lhs[] + term[][d]</span>
-                        <span class="c"># }           ^.................. the accumulator</span>
-                        <span class="c">#</span>
-                        <span class="c"># We check if the expression already contains the</span>
-                        <span class="c"># lhs, and raise an exception if it does, as that</span>
-                        <span class="c"># syntax is currently undefined.  FIXME: What would be</span>
-                        <span class="c"># a good interpretation?</span>
-                        <span class="k">if</span> <span class="n">assign_to</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                            <span class="k">raise</span> <span class="n">AssignmentError</span><span class="p">(</span>
-                                <span class="s">&quot;need assignment variable for loops&quot;</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">term</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">assign_to</span><span class="p">):</span>
-                            <span class="k">raise</span> <span class="ne">ValueError</span>
-
-                        <span class="n">lines</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">openloop</span><span class="p">)</span>
-                        <span class="n">lines</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">openloop_d</span><span class="p">)</span>
-                        <span class="n">text</span> <span class="o">=</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s"> = </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">lhs_printed</span><span class="p">,</span> <span class="n">CodePrinter</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span>
-                            <span class="bp">self</span><span class="p">,</span> <span class="n">assign_to</span> <span class="o">+</span> <span class="n">term</span><span class="p">))</span>
-                        <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_get_statement</span><span class="p">(</span><span class="n">text</span><span class="p">))</span>
-                        <span class="n">lines</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">closeloop_d</span><span class="p">)</span>
-                        <span class="n">lines</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">closeloop</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">lines</span>
-
-    <span class="k">def</span> <span class="nf">get_expression_indices</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">assign_to</span><span class="p">):</span>
-        <span class="n">rinds</span><span class="p">,</span> <span class="n">junk</span> <span class="o">=</span> <span class="n">get_indices</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="n">linds</span><span class="p">,</span> <span class="n">junk</span> <span class="o">=</span> <span class="n">get_indices</span><span class="p">(</span><span class="n">assign_to</span><span class="p">)</span>
-
-        <span class="c"># support broadcast of scalar</span>
-        <span class="k">if</span> <span class="n">linds</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">rinds</span><span class="p">:</span>
-            <span class="n">rinds</span> <span class="o">=</span> <span class="n">linds</span>
-        <span class="k">if</span> <span class="n">rinds</span> <span class="o">!=</span> <span class="n">linds</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;lhs indices must match non-dummy&quot;</span>
-                    <span class="s">&quot; rhs indices in </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">expr</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_sort_optimized</span><span class="p">(</span><span class="n">rinds</span><span class="p">,</span> <span class="n">assign_to</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sort_optimized</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">indices</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">indices</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[]</span>
-
-        <span class="c"># determine optimized loop order by giving a score to each index</span>
-        <span class="c"># the index with the highest score are put in the innermost loop.</span>
-        <span class="n">score_table</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">indices</span><span class="p">:</span>
-            <span class="n">score_table</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-
-        <span class="n">arrays</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Indexed</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">arr</span> <span class="ow">in</span> <span class="n">arrays</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">p</span><span class="p">,</span> <span class="n">ind</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">arr</span><span class="o">.</span><span class="n">indices</span><span class="p">):</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="n">score_table</span><span class="p">[</span><span class="n">ind</span><span class="p">]</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rate_index_position</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-                <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-                    <span class="k">pass</span>
-
-        <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">indices</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">score_table</span><span class="p">[</span><span class="n">x</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_print_NumberSymbol</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="c"># A Number symbol that is not implemented here or with _printmethod</span>
-        <span class="c"># is registered and evaluated</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_number_symbols</span><span class="o">.</span><span class="n">add</span><span class="p">((</span><span class="n">expr</span><span class="p">,</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&quot;precision&quot;</span><span class="p">]))))</span>
-        <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Dummy</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="c"># dummies must be printed as unique symbols</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">_</span><span class="si">%i</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">dummy_index</span><span class="p">)</span>  <span class="c"># Dummy</span>
-
-    <span class="n">_print_Catalan</span> <span class="o">=</span> <span class="n">_print_NumberSymbol</span>
-    <span class="n">_print_EulerGamma</span> <span class="o">=</span> <span class="n">_print_NumberSymbol</span>
-    <span class="n">_print_GoldenRatio</span> <span class="o">=</span> <span class="n">_print_NumberSymbol</span>
-
-    <span class="k">def</span> <span class="nf">_print_not_supported</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_not_supported</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">emptyPrinter</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="c"># The following can not be simply translated into C or Fortran</span>
-    <span class="n">_print_Basic</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_ComplexInfinity</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_Derivative</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_dict</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_ExprCondPair</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_GeometryEntity</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_Infinity</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_Integral</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_Interval</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_Limit</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_list</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_Matrix</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_DeferredVector</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_NaN</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_NegativeInfinity</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_Normal</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_Order</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_PDF</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_RootOf</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_RootsOf</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_RootSum</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_Sample</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_SparseMatrix</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_tuple</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_Uniform</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_Unit</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_Wild</span> <span class="o">=</span> <span class="n">_print_not_supported</span>
-    <span class="n">_print_WildFunction</span> <span class="o">=</span> <span class="n">_print_not_supported</span></div>
-</pre></div>
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diff --git a/dev-py3k/_modules/sympy/printing/conventions.html b/dev-py3k/_modules/sympy/printing/conventions.html
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-  <h1>Source code for sympy.printing.conventions</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">A few practical conventions common to all printers.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-
-<span class="kn">import</span> <span class="nn">re</span>
-
-
-<div class="viewcode-block" id="split_super_sub"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.conventions.split_super_sub">[docs]</a><span class="k">def</span> <span class="nf">split_super_sub</span><span class="p">(</span><span class="n">text</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Split a symbol name into a name, superscripts and subscripts</span>
-
-<span class="sd">       The first part of the symbol name is considered to be its actual</span>
-<span class="sd">       &#39;name&#39;, followed by super- and subscripts. Each superscript is</span>
-<span class="sd">       preceded with a &quot;^&quot; character or by &quot;__&quot;. Each subscript is preceded</span>
-<span class="sd">       by a &quot;_&quot; character.  The three return values are the actual name, a</span>
-<span class="sd">       list with superscripts and a list with subscripts.</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy.printing.conventions import split_super_sub</span>
-<span class="sd">       &gt;&gt;&gt; split_super_sub(&#39;a_x^1&#39;)</span>
-<span class="sd">       (&#39;a&#39;, [&#39;1&#39;], [&#39;x&#39;])</span>
-<span class="sd">       &gt;&gt;&gt; split_super_sub(&#39;var_sub1__sup_sub2&#39;)</span>
-<span class="sd">       (&#39;var&#39;, [&#39;sup&#39;], [&#39;sub1&#39;, &#39;sub2&#39;])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">pos</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">name</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">supers</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">subs</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">while</span> <span class="n">pos</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">text</span><span class="p">):</span>
-        <span class="n">start</span> <span class="o">=</span> <span class="n">pos</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">text</span><span class="p">[</span><span class="n">pos</span><span class="p">:</span><span class="n">pos</span> <span class="o">+</span> <span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="s">&quot;__&quot;</span><span class="p">:</span>
-            <span class="n">start</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="n">pos_hat</span> <span class="o">=</span> <span class="n">text</span><span class="o">.</span><span class="n">find</span><span class="p">(</span><span class="s">&quot;^&quot;</span><span class="p">,</span> <span class="n">start</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">pos_hat</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">pos_hat</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">text</span><span class="p">)</span>
-        <span class="n">pos_usc</span> <span class="o">=</span> <span class="n">text</span><span class="o">.</span><span class="n">find</span><span class="p">(</span><span class="s">&quot;_&quot;</span><span class="p">,</span> <span class="n">start</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">pos_usc</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">pos_usc</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">text</span><span class="p">)</span>
-        <span class="n">pos_next</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">pos_hat</span><span class="p">,</span> <span class="n">pos_usc</span><span class="p">)</span>
-        <span class="n">part</span> <span class="o">=</span> <span class="n">text</span><span class="p">[</span><span class="n">pos</span><span class="p">:</span><span class="n">pos_next</span><span class="p">]</span>
-        <span class="n">pos</span> <span class="o">=</span> <span class="n">pos_next</span>
-        <span class="k">if</span> <span class="n">name</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">name</span> <span class="o">=</span> <span class="n">part</span>
-        <span class="k">elif</span> <span class="n">part</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&quot;^&quot;</span><span class="p">):</span>
-            <span class="n">supers</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">part</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-        <span class="k">elif</span> <span class="n">part</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&quot;__&quot;</span><span class="p">):</span>
-            <span class="n">supers</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">part</span><span class="p">[</span><span class="mi">2</span><span class="p">:])</span>
-        <span class="k">elif</span> <span class="n">part</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&quot;_&quot;</span><span class="p">):</span>
-            <span class="n">subs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">part</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">RuntimeError</span><span class="p">(</span><span class="s">&quot;This should never happen.&quot;</span><span class="p">)</span>
-
-    <span class="c"># make a little exception when a name ends with digits, i.e. treat them</span>
-    <span class="c"># as a subscript too.</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">re</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="s">&#39;(^[a-zA-Z]+)([0-9]+)$&#39;</span><span class="p">,</span> <span class="n">name</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">name</span><span class="p">,</span> <span class="n">sub</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">groups</span><span class="p">()</span>
-        <span class="n">subs</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">sub</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">name</span><span class="p">,</span> <span class="n">supers</span><span class="p">,</span> <span class="n">subs</span></div>
-</pre></div>
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diff --git a/dev-py3k/_modules/sympy/printing/fcode.html b/dev-py3k/_modules/sympy/printing/fcode.html
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-            
-  <h1>Source code for sympy.printing.fcode</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Fortran code printer</span>
-
-<span class="sd">The FCodePrinter converts single sympy expressions into single Fortran</span>
-<span class="sd">expressions, using the functions defined in the Fortran 77 standard where</span>
-<span class="sd">possible. Some useful pointers to Fortran can be found on wikipedia:</span>
-
-<span class="sd">http://en.wikipedia.org/wiki/Fortran</span>
-
-<span class="sd">Most of the code below is based on the &quot;Professional Programmer\&#39;s Guide to</span>
-<span class="sd">Fortran77&quot; by Clive G. Page:</span>
-
-<span class="sd">http://www.star.le.ac.uk/~cgp/prof77.html</span>
-
-<span class="sd">Fortran is a case-insensitive language. This might cause trouble because</span>
-<span class="sd">SymPy is case sensitive. The implementation below does not care and leaves</span>
-<span class="sd">the responsibility for generating properly cased Fortran code to the user.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">Add</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.codeprinter</span> <span class="kn">import</span> <span class="n">CodePrinter</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.precedence</span> <span class="kn">import</span> <span class="n">precedence</span>
-<span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">tan</span><span class="p">,</span> <span class="n">asin</span><span class="p">,</span> <span class="n">acos</span><span class="p">,</span> <span class="n">atan</span><span class="p">,</span> <span class="n">atan2</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> \
-    <span class="n">cosh</span><span class="p">,</span> <span class="n">tanh</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">Abs</span><span class="p">,</span> <span class="n">sign</span><span class="p">,</span> <span class="n">conjugate</span><span class="p">,</span> <span class="n">Piecewise</span>
-
-<span class="n">implicit_functions</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span>
-    <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">tan</span><span class="p">,</span> <span class="n">asin</span><span class="p">,</span> <span class="n">acos</span><span class="p">,</span> <span class="n">atan</span><span class="p">,</span> <span class="n">atan2</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">cosh</span><span class="p">,</span> <span class="n">tanh</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span>
-    <span class="n">Abs</span><span class="p">,</span> <span class="n">sign</span><span class="p">,</span> <span class="n">conjugate</span>
-<span class="p">])</span>
-
-
-<div class="viewcode-block" id="FCodePrinter"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.fcode.FCodePrinter">[docs]</a><span class="k">class</span> <span class="nc">FCodePrinter</span><span class="p">(</span><span class="n">CodePrinter</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A printer to convert sympy expressions to strings of Fortran code&quot;&quot;&quot;</span>
-    <span class="n">printmethod</span> <span class="o">=</span> <span class="s">&quot;_fcode&quot;</span>
-
-    <span class="n">_default_settings</span> <span class="o">=</span> <span class="p">{</span>
-        <span class="s">&#39;order&#39;</span><span class="p">:</span> <span class="bp">None</span><span class="p">,</span>
-        <span class="s">&#39;full_prec&#39;</span><span class="p">:</span> <span class="s">&#39;auto&#39;</span><span class="p">,</span>
-        <span class="s">&#39;assign_to&#39;</span><span class="p">:</span> <span class="bp">None</span><span class="p">,</span>
-        <span class="s">&#39;precision&#39;</span><span class="p">:</span> <span class="mi">15</span><span class="p">,</span>
-        <span class="s">&#39;user_functions&#39;</span><span class="p">:</span> <span class="p">{},</span>
-        <span class="s">&#39;human&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">,</span>
-        <span class="s">&#39;source_format&#39;</span><span class="p">:</span> <span class="s">&#39;fixed&#39;</span><span class="p">,</span>
-    <span class="p">}</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">settings</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">CodePrinter</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">settings</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_init_leading_padding</span><span class="p">()</span>
-        <span class="n">assign_to</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;assign_to&#39;</span><span class="p">]</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">assign_to</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;assign_to&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="n">assign_to</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">assign_to</span><span class="p">,</span> <span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Basic</span><span class="p">,</span> <span class="nb">type</span><span class="p">(</span><span class="bp">None</span><span class="p">))):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;FCodePrinter cannot assign to object of type </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span>
-                    <span class="nb">type</span><span class="p">(</span><span class="n">assign_to</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_rate_index_position</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;function to calculate score based on position among indices</span>
-
-<span class="sd">        This method is used to sort loops in an optimized order, see</span>
-<span class="sd">        CodePrinter._sort_optimized()</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">p</span><span class="o">*</span><span class="mi">5</span>
-
-    <span class="k">def</span> <span class="nf">_get_statement</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">codestring</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">codestring</span>
-
-    <span class="k">def</span> <span class="nf">_init_leading_padding</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># leading columns depend on fixed or free format</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;source_format&#39;</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;fixed&#39;</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_lead_code</span> <span class="o">=</span> <span class="s">&quot;      &quot;</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_lead_cont</span> <span class="o">=</span> <span class="s">&quot;     @ &quot;</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_lead_comment</span> <span class="o">=</span> <span class="s">&quot;C     &quot;</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;source_format&#39;</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;free&#39;</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_lead_code</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_lead_cont</span> <span class="o">=</span> <span class="s">&quot;      &quot;</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_lead_comment</span> <span class="o">=</span> <span class="s">&quot;! &quot;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&quot;Unknown source format: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span>
-                <span class="s">&#39;source_format&#39;</span><span class="p">]</span>
-            <span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_pad_leading_columns</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">lines</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="n">lines</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">line</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39;!&#39;</span><span class="p">):</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_lead_comment</span> <span class="o">+</span> <span class="n">line</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span><span class="o">.</span><span class="n">lstrip</span><span class="p">())</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_lead_code</span> <span class="o">+</span> <span class="n">line</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="k">def</span> <span class="nf">_get_loop_opening_ending</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">indices</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a tuple (open_lines, close_lines) containing lists of codelines</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">open_lines</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">close_lines</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">indices</span><span class="p">:</span>
-            <span class="c"># fortran arrays start at 1 and end at dimension</span>
-            <span class="n">var</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">stop</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span>
-                    <span class="p">[</span><span class="n">i</span><span class="o">.</span><span class="n">label</span><span class="p">,</span> <span class="n">i</span><span class="o">.</span><span class="n">lower</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">i</span><span class="o">.</span><span class="n">upper</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]))</span>
-            <span class="n">open_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;do </span><span class="si">%s</span><span class="s"> = </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">var</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">stop</span><span class="p">))</span>
-            <span class="n">close_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;end do&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">open_lines</span><span class="p">,</span> <span class="n">close_lines</span>
-
-<div class="viewcode-block" id="FCodePrinter.doprint"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.fcode.FCodePrinter.doprint">[docs]</a>    <span class="k">def</span> <span class="nf">doprint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns Fortran code for expr (as a string)&quot;&quot;&quot;</span>
-        <span class="c"># find all number symbols</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_number_symbols</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-
-        <span class="c"># keep a set of expressions that are not strictly translatable to</span>
-        <span class="c"># Fortran.</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_not_supported</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-
-        <span class="n">lines</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Piecewise</span><span class="p">):</span>
-            <span class="c"># support for top-level Piecewise function</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;if (</span><span class="si">%s</span><span class="s">) then&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">c</span><span class="p">))</span>
-                <span class="k">elif</span> <span class="n">i</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">c</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                    <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;else&quot;</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;else if (</span><span class="si">%s</span><span class="s">) then&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">c</span><span class="p">))</span>
-                <span class="n">lines</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">_doprint_a_piece</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;assign_to&#39;</span><span class="p">]))</span>
-            <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;end if&quot;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">lines</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_doprint_a_piece</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;assign_to&#39;</span><span class="p">]))</span>
-
-        <span class="c"># format the output</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&quot;human&quot;</span><span class="p">]:</span>
-            <span class="n">frontlines</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_not_supported</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">frontlines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;! Not Fortran:&quot;</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">expr</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_not_supported</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">):</span>
-                    <span class="n">frontlines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;! </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">repr</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-            <span class="k">for</span> <span class="n">name</span><span class="p">,</span> <span class="n">value</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_number_symbols</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="nb">str</span><span class="p">):</span>
-                <span class="n">frontlines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;parameter (</span><span class="si">%s</span><span class="s"> = </span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">name</span><span class="p">),</span> <span class="n">value</span><span class="p">))</span>
-            <span class="n">frontlines</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">lines</span><span class="p">)</span>
-            <span class="n">lines</span> <span class="o">=</span> <span class="n">frontlines</span>
-            <span class="n">lines</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">indent_code</span><span class="p">(</span><span class="n">lines</span><span class="p">)</span>
-            <span class="n">lines</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_wrap_fortran</span><span class="p">(</span><span class="n">lines</span><span class="p">)</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">lines</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">lines</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">indent_code</span><span class="p">(</span><span class="n">lines</span><span class="p">)</span>
-            <span class="n">lines</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_wrap_fortran</span><span class="p">(</span><span class="n">lines</span><span class="p">)</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_number_symbols</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_not_supported</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span>
-                <span class="n">lines</span><span class="p">)</span>
-
-        <span class="k">del</span> <span class="bp">self</span><span class="o">.</span><span class="n">_not_supported</span>
-        <span class="k">del</span> <span class="bp">self</span><span class="o">.</span><span class="n">_number_symbols</span>
-        <span class="k">return</span> <span class="n">result</span>
-</div>
-    <span class="k">def</span> <span class="nf">_print_Add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="c"># purpose: print complex numbers nicely in Fortran.</span>
-        <span class="c"># collect the purely real and purely imaginary parts:</span>
-        <span class="n">pure_real</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">pure_imaginary</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">mixed</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-                <span class="n">pure_real</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">:</span>
-                <span class="n">pure_imaginary</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">mixed</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">pure_imaginary</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">mixed</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">PREC</span> <span class="o">=</span> <span class="n">precedence</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-                <span class="n">term</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">mixed</span><span class="p">)</span>
-                <span class="n">t</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39;-&#39;</span><span class="p">):</span>
-                    <span class="n">sign</span> <span class="o">=</span> <span class="s">&quot;-&quot;</span>
-                    <span class="n">t</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">sign</span> <span class="o">=</span> <span class="s">&quot;+&quot;</span>
-                <span class="k">if</span> <span class="n">precedence</span><span class="p">(</span><span class="n">term</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">PREC</span><span class="p">:</span>
-                    <span class="n">t</span> <span class="o">=</span> <span class="s">&quot;(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="n">t</span>
-
-                <span class="k">return</span> <span class="s">&quot;cmplx(</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">) </span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">pure_real</span><span class="p">)),</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">*</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">pure_imaginary</span><span class="p">)),</span>
-                    <span class="n">sign</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span>
-                <span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="s">&quot;cmplx(</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">pure_real</span><span class="p">)),</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">*</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">pure_imaginary</span><span class="p">)),</span>
-                <span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">CodePrinter</span><span class="o">.</span><span class="n">_print_Add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Function</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">name</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&quot;user_functions&quot;</span><span class="p">]</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">name</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">conjugate</span><span class="p">:</span>
-                <span class="n">name</span> <span class="o">=</span> <span class="s">&quot;conjg&quot;</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">name</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="o">.</span><span class="n">__name__</span>
-            <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&#39;_imp_&#39;</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">_imp_</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Lambda</span><span class="p">):</span>
-                <span class="c"># inlined function.</span>
-                <span class="c"># the expression is printed with _print to avoid loops</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">_imp_</span><span class="p">(</span><span class="o">*</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">implicit_functions</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_not_supported</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">stringify</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="s">&quot;, &quot;</span><span class="p">))</span>
-
-    <span class="n">_print_factorial</span> <span class="o">=</span> <span class="n">_print_Function</span>
-
-    <span class="k">def</span> <span class="nf">_print_ImaginaryUnit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="c"># purpose: print complex numbers nicely in Fortran.</span>
-        <span class="k">return</span> <span class="s">&quot;cmplx(0,1)&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_int</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="c"># purpose: print complex numbers nicely in Fortran.</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;cmplx(0,</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="o">*</span><span class="n">expr</span><span class="p">)</span>
-            <span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">CodePrinter</span><span class="o">.</span><span class="n">_print_Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-
-    <span class="n">_print_Exp1</span> <span class="o">=</span> <span class="n">CodePrinter</span><span class="o">.</span><span class="n">_print_NumberSymbol</span>
-    <span class="n">_print_Pi</span> <span class="o">=</span> <span class="n">CodePrinter</span><span class="o">.</span><span class="n">_print_NumberSymbol</span>
-
-    <span class="k">def</span> <span class="nf">_print_Pow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">PREC</span> <span class="o">=</span> <span class="n">precedence</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&#39;1.0/</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">PREC</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span> <span class="o">==</span> <span class="mf">0.5</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-                <span class="c"># Fortan intrinsic sqrt() does not accept integer argument</span>
-                <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="s">&#39;sqrt(</span><span class="si">%s</span><span class="s">.0d0)&#39;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="s">&#39;sqrt(dble(</span><span class="si">%s</span><span class="s">))&#39;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="s">&#39;sqrt(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">CodePrinter</span><span class="o">.</span><span class="n">_print_Pow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Rational</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">p</span><span class="p">),</span> <span class="nb">int</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%d</span><span class="s">.0d0/</span><span class="si">%d</span><span class="s">.0d0&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Float</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">printed</span> <span class="o">=</span> <span class="n">CodePrinter</span><span class="o">.</span><span class="n">_print_Float</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-        <span class="n">e</span> <span class="o">=</span> <span class="n">printed</span><span class="o">.</span><span class="n">find</span><span class="p">(</span><span class="s">&#39;e&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">e</span> <span class="o">&gt;</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">d</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">printed</span><span class="p">[:</span><span class="n">e</span><span class="p">],</span> <span class="n">printed</span><span class="p">[</span><span class="n">e</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:])</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">d0&quot;</span> <span class="o">%</span> <span class="n">printed</span>
-
-    <span class="k">def</span> <span class="nf">_print_Indexed</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">inds</span> <span class="o">=</span> <span class="p">[</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">indices</span> <span class="p">]</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">label</span><span class="p">),</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">inds</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Idx</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">label</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_wrap_fortran</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">lines</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Wrap long Fortran lines</span>
-
-<span class="sd">           Argument:</span>
-<span class="sd">             lines  --  a list of lines (without \\n character)</span>
-
-<span class="sd">           A comment line is split at white space. Code lines are split with a more</span>
-<span class="sd">           complex rule to give nice results.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># routine to find split point in a code line</span>
-        <span class="n">my_alnum</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="s">&quot;_+-.0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ_&quot;</span><span class="p">)</span>
-        <span class="n">my_white</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="s">&quot; </span><span class="se">\t</span><span class="s">()&quot;</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">split_pos_code</span><span class="p">(</span><span class="n">line</span><span class="p">,</span> <span class="n">endpos</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">line</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="n">endpos</span><span class="p">:</span>
-                <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="n">line</span><span class="p">)</span>
-            <span class="n">pos</span> <span class="o">=</span> <span class="n">endpos</span>
-            <span class="n">split</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">pos</span><span class="p">:</span> \
-                <span class="p">(</span><span class="n">line</span><span class="p">[</span><span class="n">pos</span><span class="p">]</span> <span class="ow">in</span> <span class="n">my_alnum</span> <span class="ow">and</span> <span class="n">line</span><span class="p">[</span><span class="n">pos</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">my_alnum</span><span class="p">)</span> <span class="ow">or</span> \
-                <span class="p">(</span><span class="n">line</span><span class="p">[</span><span class="n">pos</span><span class="p">]</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">my_alnum</span> <span class="ow">and</span> <span class="n">line</span><span class="p">[</span><span class="n">pos</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="ow">in</span> <span class="n">my_alnum</span><span class="p">)</span> <span class="ow">or</span> \
-                <span class="p">(</span><span class="n">line</span><span class="p">[</span><span class="n">pos</span><span class="p">]</span> <span class="ow">in</span> <span class="n">my_white</span> <span class="ow">and</span> <span class="n">line</span><span class="p">[</span><span class="n">pos</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">my_white</span><span class="p">)</span> <span class="ow">or</span> \
-                <span class="p">(</span><span class="n">line</span><span class="p">[</span><span class="n">pos</span><span class="p">]</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">my_white</span> <span class="ow">and</span> <span class="n">line</span><span class="p">[</span><span class="n">pos</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="ow">in</span> <span class="n">my_white</span><span class="p">)</span>
-            <span class="k">while</span> <span class="ow">not</span> <span class="n">split</span><span class="p">(</span><span class="n">pos</span><span class="p">):</span>
-                <span class="n">pos</span> <span class="o">-=</span> <span class="mi">1</span>
-                <span class="k">if</span> <span class="n">pos</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">endpos</span>
-            <span class="k">return</span> <span class="n">pos</span>
-        <span class="c"># split line by line and add the splitted lines to result</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;source_format&#39;</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;free&#39;</span><span class="p">:</span>
-            <span class="n">trailing</span> <span class="o">=</span> <span class="s">&#39; &amp;&#39;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">trailing</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-        <span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="n">lines</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">line</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_lead_comment</span><span class="p">):</span>
-                <span class="c"># comment line</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">line</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">72</span><span class="p">:</span>
-                    <span class="n">pos</span> <span class="o">=</span> <span class="n">line</span><span class="o">.</span><span class="n">rfind</span><span class="p">(</span><span class="s">&quot; &quot;</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">72</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">pos</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                        <span class="n">pos</span> <span class="o">=</span> <span class="mi">72</span>
-                    <span class="n">hunk</span> <span class="o">=</span> <span class="n">line</span><span class="p">[:</span><span class="n">pos</span><span class="p">]</span>
-                    <span class="n">line</span> <span class="o">=</span> <span class="n">line</span><span class="p">[</span><span class="n">pos</span><span class="p">:]</span><span class="o">.</span><span class="n">lstrip</span><span class="p">()</span>
-                    <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">hunk</span><span class="p">)</span>
-                    <span class="k">while</span> <span class="nb">len</span><span class="p">(</span><span class="n">line</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">pos</span> <span class="o">=</span> <span class="n">line</span><span class="o">.</span><span class="n">rfind</span><span class="p">(</span><span class="s">&quot; &quot;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">66</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">pos</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">line</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">66</span><span class="p">:</span>
-                            <span class="n">pos</span> <span class="o">=</span> <span class="mi">66</span>
-                        <span class="n">hunk</span> <span class="o">=</span> <span class="n">line</span><span class="p">[:</span><span class="n">pos</span><span class="p">]</span>
-                        <span class="n">line</span> <span class="o">=</span> <span class="n">line</span><span class="p">[</span><span class="n">pos</span><span class="p">:]</span><span class="o">.</span><span class="n">lstrip</span><span class="p">()</span>
-                        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_lead_comment</span><span class="p">,</span> <span class="n">hunk</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">line</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">line</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_lead_code</span><span class="p">):</span>
-                <span class="c"># code line</span>
-                <span class="n">pos</span> <span class="o">=</span> <span class="n">split_pos_code</span><span class="p">(</span><span class="n">line</span><span class="p">,</span> <span class="mi">72</span><span class="p">)</span>
-                <span class="n">hunk</span> <span class="o">=</span> <span class="n">line</span><span class="p">[:</span><span class="n">pos</span><span class="p">]</span><span class="o">.</span><span class="n">rstrip</span><span class="p">()</span>
-                <span class="n">line</span> <span class="o">=</span> <span class="n">line</span><span class="p">[</span><span class="n">pos</span><span class="p">:]</span><span class="o">.</span><span class="n">lstrip</span><span class="p">()</span>
-                <span class="k">if</span> <span class="n">line</span><span class="p">:</span>
-                    <span class="n">hunk</span> <span class="o">+=</span> <span class="n">trailing</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">hunk</span><span class="p">)</span>
-                <span class="k">while</span> <span class="nb">len</span><span class="p">(</span><span class="n">line</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">pos</span> <span class="o">=</span> <span class="n">split_pos_code</span><span class="p">(</span><span class="n">line</span><span class="p">,</span> <span class="mi">65</span><span class="p">)</span>
-                    <span class="n">hunk</span> <span class="o">=</span> <span class="n">line</span><span class="p">[:</span><span class="n">pos</span><span class="p">]</span><span class="o">.</span><span class="n">rstrip</span><span class="p">()</span>
-                    <span class="n">line</span> <span class="o">=</span> <span class="n">line</span><span class="p">[</span><span class="n">pos</span><span class="p">:]</span><span class="o">.</span><span class="n">lstrip</span><span class="p">()</span>
-                    <span class="k">if</span> <span class="n">line</span><span class="p">:</span>
-                        <span class="n">hunk</span> <span class="o">+=</span> <span class="n">trailing</span>
-                    <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_lead_cont</span><span class="p">,</span> <span class="n">hunk</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">line</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-<div class="viewcode-block" id="FCodePrinter.indent_code"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.fcode.FCodePrinter.indent_code">[docs]</a>    <span class="k">def</span> <span class="nf">indent_code</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">code</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Accepts a string of code or a list of code lines&quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">code</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="n">code_lines</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">indent_code</span><span class="p">(</span><span class="n">code</span><span class="o">.</span><span class="n">splitlines</span><span class="p">(</span><span class="bp">True</span><span class="p">))</span>
-            <span class="k">return</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">code_lines</span><span class="p">)</span>
-
-        <span class="n">free</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;source_format&#39;</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;free&#39;</span>
-        <span class="n">code</span> <span class="o">=</span> <span class="p">[</span> <span class="n">line</span><span class="o">.</span><span class="n">lstrip</span><span class="p">(</span><span class="s">&#39; </span><span class="se">\t</span><span class="s">&#39;</span><span class="p">)</span> <span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="n">code</span> <span class="p">]</span>
-
-        <span class="n">inc_keyword</span> <span class="o">=</span> <span class="p">(</span><span class="s">&#39;do &#39;</span><span class="p">,</span> <span class="s">&#39;if(&#39;</span><span class="p">,</span> <span class="s">&#39;if &#39;</span><span class="p">,</span> <span class="s">&#39;do</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;else&#39;</span><span class="p">)</span>
-        <span class="n">dec_keyword</span> <span class="o">=</span> <span class="p">(</span><span class="s">&#39;end do&#39;</span><span class="p">,</span> <span class="s">&#39;enddo&#39;</span><span class="p">,</span> <span class="s">&#39;end if&#39;</span><span class="p">,</span> <span class="s">&#39;endif&#39;</span><span class="p">,</span> <span class="s">&#39;else&#39;</span><span class="p">)</span>
-
-        <span class="n">increase</span> <span class="o">=</span> <span class="p">[</span> <span class="nb">int</span><span class="p">(</span><span class="nb">any</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">line</span><span class="o">.</span><span class="n">startswith</span><span class="p">,</span> <span class="n">inc_keyword</span><span class="p">)))</span>
-                     <span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="n">code</span> <span class="p">]</span>
-        <span class="n">decrease</span> <span class="o">=</span> <span class="p">[</span> <span class="nb">int</span><span class="p">(</span><span class="nb">any</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">line</span><span class="o">.</span><span class="n">startswith</span><span class="p">,</span> <span class="n">dec_keyword</span><span class="p">)))</span>
-                     <span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="n">code</span> <span class="p">]</span>
-        <span class="n">continuation</span> <span class="o">=</span> <span class="p">[</span> <span class="nb">int</span><span class="p">(</span><span class="nb">any</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">line</span><span class="o">.</span><span class="n">endswith</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;&amp;&#39;</span><span class="p">,</span> <span class="s">&#39;&amp;</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">])))</span>
-                         <span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="n">code</span> <span class="p">]</span>
-
-        <span class="n">level</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">cont_padding</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">tabwidth</span> <span class="o">=</span> <span class="mi">3</span>
-        <span class="n">new_code</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">line</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">code</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">line</span> <span class="o">==</span> <span class="s">&#39;&#39;</span> <span class="ow">or</span> <span class="n">line</span> <span class="o">==</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">:</span>
-                <span class="n">new_code</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">line</span><span class="p">)</span>
-                <span class="k">continue</span>
-            <span class="n">level</span> <span class="o">-=</span> <span class="n">decrease</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-
-            <span class="k">if</span> <span class="n">free</span><span class="p">:</span>
-                <span class="n">padding</span> <span class="o">=</span> <span class="s">&quot; &quot;</span><span class="o">*</span><span class="p">(</span><span class="n">level</span><span class="o">*</span><span class="n">tabwidth</span> <span class="o">+</span> <span class="n">cont_padding</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">padding</span> <span class="o">=</span> <span class="s">&quot; &quot;</span><span class="o">*</span><span class="n">level</span><span class="o">*</span><span class="n">tabwidth</span>
-
-            <span class="n">line</span> <span class="o">=</span> <span class="s">&quot;</span><span class="si">%s%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">padding</span><span class="p">,</span> <span class="n">line</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">free</span><span class="p">:</span>
-                <span class="n">line</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_pad_leading_columns</span><span class="p">([</span><span class="n">line</span><span class="p">])[</span><span class="mi">0</span><span class="p">]</span>
-
-            <span class="n">new_code</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">line</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">continuation</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                <span class="n">cont_padding</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">tabwidth</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">cont_padding</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="n">level</span> <span class="o">+=</span> <span class="n">increase</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">free</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_wrap_fortran</span><span class="p">(</span><span class="n">new_code</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">new_code</span>
-
-</div></div>
-<div class="viewcode-block" id="fcode"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.fcode.fcode">[docs]</a><span class="k">def</span> <span class="nf">fcode</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Converts an expr to a string of Fortran 77 code</span>
-
-<span class="sd">       Parameters</span>
-<span class="sd">       ==========</span>
-
-<span class="sd">       expr : sympy.core.Expr</span>
-<span class="sd">           a sympy expression to be converted</span>
-<span class="sd">       assign_to : optional</span>
-<span class="sd">           When given, the argument is used as the name of the</span>
-<span class="sd">           variable to which the Fortran expression is assigned.</span>
-<span class="sd">           (This is helpful in case of line-wrapping.)</span>
-<span class="sd">       precision : optional</span>
-<span class="sd">           the precision for numbers such as pi [default=15]</span>
-<span class="sd">       user_functions : optional</span>
-<span class="sd">           A dictionary where keys are FunctionClass instances and values</span>
-<span class="sd">           are there string representations.</span>
-<span class="sd">       human : optional</span>
-<span class="sd">           If True, the result is a single string that may contain some</span>
-<span class="sd">           parameter statements for the number symbols. If False, the same</span>
-<span class="sd">           information is returned in a more programmer-friendly data</span>
-<span class="sd">           structure.</span>
-<span class="sd">       source_format : optional</span>
-<span class="sd">           The source format can be either &#39;fixed&#39; or &#39;free&#39;.</span>
-<span class="sd">           [default=&#39;fixed&#39;]</span>
-
-<span class="sd">       Examples</span>
-<span class="sd">       ========</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy import fcode, symbols, Rational, pi, sin</span>
-<span class="sd">       &gt;&gt;&gt; x, tau = symbols(&#39;x,tau&#39;)</span>
-<span class="sd">       &gt;&gt;&gt; fcode((2*tau)**Rational(7,2))</span>
-<span class="sd">       &#39;      8*sqrt(2.0d0)*tau**(7.0d0/2.0d0)&#39;</span>
-<span class="sd">       &gt;&gt;&gt; fcode(sin(x), assign_to=&quot;s&quot;)</span>
-<span class="sd">       &#39;      s = sin(x)&#39;</span>
-<span class="sd">       &gt;&gt;&gt; print(fcode(pi))</span>
-<span class="sd">             parameter (pi = 3.14159265358979d0)</span>
-<span class="sd">             pi</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># run the printer</span>
-    <span class="n">printer</span> <span class="o">=</span> <span class="n">FCodePrinter</span><span class="p">(</span><span class="n">settings</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">printer</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="print_fcode"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.fcode.print_fcode">[docs]</a><span class="k">def</span> <span class="nf">print_fcode</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Prints the Fortran representation of the given expression.</span>
-
-<span class="sd">       See fcode for the meaning of the optional arguments.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">print</span><span class="p">(</span><span class="n">fcode</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">))</span></div>
-</pre></div>
-
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-<div id="searchbox" style="display: none">
-  <h3>Quick search</h3>
-    <form class="search" action="../../../search.html" method="get">
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diff --git a/dev-py3k/_modules/sympy/printing/gtk.html b/dev-py3k/_modules/sympy/printing/gtk.html
deleted file mode 100644
index 1bb7e2f94da..00000000000
--- a/dev-py3k/_modules/sympy/printing/gtk.html
+++ /dev/null
@@ -1,134 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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-    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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-    
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-      };
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-             >modules</a> |</li>
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-        <div class="bodywrapper">
-          <div class="body">
-            
-  <h1>Source code for sympy.printing.gtk</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.printing.mathml</span> <span class="kn">import</span> <span class="n">mathml</span>
-<span class="kn">import</span> <span class="nn">tempfile</span>
-<span class="kn">import</span> <span class="nn">os</span>
-
-
-<span class="k">def</span> <span class="nf">print_gtk</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">start_viewer</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Print to Gtkmathview, a gtk widget capable of rendering MathML.</span>
-<div class="viewcode-block" id="print_gtk"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.gtk.print_gtk">[docs]</a>
-<span class="sd">    Needs libgtkmathview-bin&quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.utilities.mathml</span> <span class="kn">import</span> <span class="n">c2p</span>
-
-    <span class="n">tmp</span> <span class="o">=</span> <span class="n">tempfile</span><span class="o">.</span><span class="n">mktemp</span><span class="p">()</span>  <span class="c"># create a temp file to store the result</span>
-    <span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="n">tmp</span><span class="p">,</span> <span class="s">&#39;wb&#39;</span><span class="p">)</span> <span class="k">as</span> <span class="nb">file</span><span class="p">:</span>
-        <span class="nb">file</span><span class="o">.</span><span class="n">write</span><span class="p">(</span> <span class="n">c2p</span><span class="p">(</span><span class="n">mathml</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">simple</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">start_viewer</span><span class="p">:</span>
-        <span class="n">os</span><span class="o">.</span><span class="n">system</span><span class="p">(</span><span class="s">&quot;mathmlviewer &quot;</span> <span class="o">+</span> <span class="n">tmp</span><span class="p">)</span>
-</pre></div></div>
-
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-        </div>
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-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-<div id="searchbox" style="display: none">
-  <h3>Quick search</h3>
-    <form class="search" action="../../../search.html" method="get">
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-</html>
diff --git a/dev-py3k/_modules/sympy/printing/lambdarepr.html b/dev-py3k/_modules/sympy/printing/lambdarepr.html
deleted file mode 100644
index ab85e891954..00000000000
--- a/dev-py3k/_modules/sympy/printing/lambdarepr.html
+++ /dev/null
@@ -1,196 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
-
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-<html xmlns="http://www.w3.org/1999/xhtml">
-  <head>
-    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
-    
-    <title>sympy.printing.lambdarepr &mdash; SymPy 0.7.2-git documentation</title>
-    
-    <link rel="stylesheet" href="../../../_static/default.css" type="text/css" />
-    <link rel="stylesheet" href="../../../_static/pygments.css" type="text/css" />
-    <link rel="stylesheet" href="http://live.sympy.org/static/live-core.css" type="text/css" />
-    <link rel="stylesheet" href="http://live.sympy.org/static/live-autocomplete.css" type="text/css" />
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-    
-    <script type="text/javascript">
-      var DOCUMENTATION_OPTIONS = {
-        URL_ROOT:    '../../../',
-        VERSION:     '0.7.2-git',
-        COLLAPSE_INDEX: false,
-        FILE_SUFFIX: '.html',
-        HAS_SOURCE:  true
-      };
-    </script>
-    <script type="text/javascript" src="../../../_static/jquery.js"></script>
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-    <script type="text/javascript" src="../../../_static/doctools.js"></script>
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-    <link rel="up" title="sympy" href="../../sympy.html" /> 
-  </head>
-  <body>
-    <div class="related">
-      <h3>Navigation</h3>
-      <ul>
-        <li class="right" style="margin-right: 10px">
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-        <li class="right" >
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-             >modules</a> |</li>
-        <li class="right" >
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-             >modules</a> |</li>
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-          <div class="body">
-            
-  <h1>Source code for sympy.printing.lambdarepr</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">.str</span> <span class="kn">import</span> <span class="n">StrPrinter</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">default_sort_key</span>
-
-
-<span class="k">def</span> <span class="nf">_find_first_symbol</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="k">for</span> <span class="n">atom</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">atoms</span><span class="p">():</span>
-        <span class="k">if</span> <span class="n">atom</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">atom</span>
-    <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;expression must contain a Symbol: </span><span class="si">%r</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">expr</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="LambdaPrinter"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.lambdarepr.LambdaPrinter">[docs]</a><span class="k">class</span> <span class="nc">LambdaPrinter</span><span class="p">(</span><span class="n">StrPrinter</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    This printer converts expressions into strings that can be used by</span>
-<span class="sd">    lambdify.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_MatrixBase</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">([</span><span class="si">%s</span><span class="s">])&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span>
-        <span class="n">expr</span><span class="o">.</span><span class="n">_format_str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="s">&quot;,&quot;</span><span class="p">))</span>
-    <span class="n">_print_SparseMatrix</span> <span class="o">=</span> \
-        <span class="n">_print_MutableSparseMatrix</span> <span class="o">=</span> \
-        <span class="n">_print_ImmutableSparseMatrix</span> <span class="o">=</span> \
-        <span class="n">_print_Matrix</span> <span class="o">=</span> \
-        <span class="n">_print_DenseMatrix</span> <span class="o">=</span> \
-        <span class="n">_print_MutableDenseMatrix</span> <span class="o">=</span> \
-        <span class="n">_print_ImmutableMatrix</span> <span class="o">=</span> \
-        <span class="n">_print_ImmutableDenseMatrix</span> <span class="o">=</span> \
-        <span class="n">_print_MatrixBase</span>
-
-    <span class="k">def</span> <span class="nf">_print_Piecewise</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.sets</span> <span class="kn">import</span> <span class="n">Interval</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">e</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">expr</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">cond</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;((&#39;</span><span class="p">)</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="p">))</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;) if (&#39;</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">Interval</span><span class="p">):</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">c</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">_find_first_symbol</span><span class="p">(</span><span class="n">e</span><span class="p">))))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">c</span><span class="p">))</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;) else (&#39;</span><span class="p">)</span>
-            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">result</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;) else None)&#39;</span><span class="p">)</span>
-        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;)&#39;</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span> <span class="o">-</span> <span class="mi">2</span><span class="p">))</span>
-        <span class="k">return</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_And</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;(&#39;</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">):</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="s">&#39;(&#39;</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="s">&#39;)&#39;</span><span class="p">])</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39; and &#39;</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">result</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;)&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Or</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;(&#39;</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">):</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="s">&#39;(&#39;</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="s">&#39;)&#39;</span><span class="p">])</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39; or &#39;</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">result</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;)&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Not</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;(&#39;</span><span class="p">,</span> <span class="s">&#39;not (&#39;</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="s">&#39;))&#39;</span><span class="p">]</span>
-        <span class="k">return</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="lambdarepr"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.lambdarepr.lambdarepr">[docs]</a><span class="k">def</span> <span class="nf">lambdarepr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns a string usable for lambdifying.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">LambdaPrinter</span><span class="p">(</span><span class="n">settings</span><span class="p">)</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span></div>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/printing/latex.html b/dev-py3k/_modules/sympy/printing/latex.html
deleted file mode 100644
index 5df2caea254..00000000000
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+++ /dev/null
@@ -1,1675 +0,0 @@
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-            
-  <h1>Source code for sympy.printing.latex</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">A Printer which converts an expression into its LaTeX equivalent.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">_coeff_isneg</span>
-<span class="kn">from</span> <span class="nn">.printer</span> <span class="kn">import</span> <span class="n">Printer</span>
-<span class="kn">from</span> <span class="nn">.conventions</span> <span class="kn">import</span> <span class="n">split_super_sub</span>
-<span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">fraction</span>
-<span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="n">SympifyError</span>
-
-<span class="kn">import</span> <span class="nn">sympy.mpmath.libmp</span> <span class="kn">as</span> <span class="nn">mlib</span>
-<span class="kn">from</span> <span class="nn">sympy.mpmath.libmp</span> <span class="kn">import</span> <span class="n">prec_to_dps</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">default_sort_key</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">has_variety</span>
-
-<span class="kn">import</span> <span class="nn">re</span>
-
-<span class="c"># Hand-picked functions which can be used directly in both LaTeX and MathJax</span>
-<span class="c"># Complete list at http://www.mathjax.org/docs/1.1/tex.html#supported-latex-commands</span>
-<span class="c"># This variable only contains those functions which sympy uses.</span>
-<span class="n">accepted_latex_functions</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;arcsin&#39;</span><span class="p">,</span> <span class="s">&#39;arccos&#39;</span><span class="p">,</span> <span class="s">&#39;arctan&#39;</span><span class="p">,</span> <span class="s">&#39;sin&#39;</span><span class="p">,</span> <span class="s">&#39;cos&#39;</span><span class="p">,</span> <span class="s">&#39;tan&#39;</span><span class="p">,</span>
-                    <span class="s">&#39;theta&#39;</span><span class="p">,</span> <span class="s">&#39;beta&#39;</span><span class="p">,</span> <span class="s">&#39;alpha&#39;</span><span class="p">,</span> <span class="s">&#39;gamma&#39;</span><span class="p">,</span> <span class="s">&#39;sinh&#39;</span><span class="p">,</span> <span class="s">&#39;cosh&#39;</span><span class="p">,</span> <span class="s">&#39;tanh&#39;</span><span class="p">,</span> <span class="s">&#39;sqrt&#39;</span><span class="p">,</span>
-                    <span class="s">&#39;ln&#39;</span><span class="p">,</span> <span class="s">&#39;log&#39;</span><span class="p">,</span> <span class="s">&#39;sec&#39;</span><span class="p">,</span> <span class="s">&#39;csc&#39;</span><span class="p">,</span> <span class="s">&#39;cot&#39;</span><span class="p">,</span> <span class="s">&#39;coth&#39;</span><span class="p">,</span> <span class="s">&#39;re&#39;</span><span class="p">,</span> <span class="s">&#39;im&#39;</span><span class="p">,</span> <span class="s">&#39;frac&#39;</span><span class="p">,</span> <span class="s">&#39;root&#39;</span><span class="p">,</span>
-                    <span class="s">&#39;arg&#39;</span><span class="p">,</span> <span class="s">&#39;zeta&#39;</span><span class="p">,</span> <span class="s">&#39;psi&#39;</span><span class="p">]</span>
-
-
-<div class="viewcode-block" id="LatexPrinter"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.latex.LatexPrinter">[docs]</a><span class="k">class</span> <span class="nc">LatexPrinter</span><span class="p">(</span><span class="n">Printer</span><span class="p">):</span>
-    <span class="n">printmethod</span> <span class="o">=</span> <span class="s">&quot;_latex&quot;</span>
-
-    <span class="n">_default_settings</span> <span class="o">=</span> <span class="p">{</span>
-        <span class="s">&quot;order&quot;</span><span class="p">:</span> <span class="bp">None</span><span class="p">,</span>
-        <span class="s">&quot;mode&quot;</span><span class="p">:</span> <span class="s">&quot;plain&quot;</span><span class="p">,</span>
-        <span class="s">&quot;itex&quot;</span><span class="p">:</span> <span class="bp">False</span><span class="p">,</span>
-        <span class="s">&quot;fold_frac_powers&quot;</span><span class="p">:</span> <span class="bp">False</span><span class="p">,</span>
-        <span class="s">&quot;fold_func_brackets&quot;</span><span class="p">:</span> <span class="bp">False</span><span class="p">,</span>
-        <span class="s">&quot;mul_symbol&quot;</span><span class="p">:</span> <span class="bp">None</span><span class="p">,</span>
-        <span class="s">&quot;inv_trig_style&quot;</span><span class="p">:</span> <span class="s">&quot;abbreviated&quot;</span><span class="p">,</span>
-        <span class="s">&quot;mat_str&quot;</span><span class="p">:</span> <span class="s">&quot;smallmatrix&quot;</span><span class="p">,</span>
-        <span class="s">&quot;mat_delim&quot;</span><span class="p">:</span> <span class="s">&quot;[&quot;</span><span class="p">,</span>
-        <span class="s">&quot;symbol_names&quot;</span><span class="p">:</span> <span class="p">{},</span>
-    <span class="p">}</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">settings</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">settings</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="s">&#39;inline&#39;</span> <span class="ow">in</span> <span class="n">settings</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">settings</span><span class="p">[</span><span class="s">&#39;inline&#39;</span><span class="p">]:</span>
-            <span class="c"># Change to &quot;good&quot; defaults for inline=False</span>
-            <span class="n">settings</span><span class="p">[</span><span class="s">&#39;mat_str&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="s">&#39;bmatrix&#39;</span>
-            <span class="n">settings</span><span class="p">[</span><span class="s">&#39;mat_delim&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="n">Printer</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">settings</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="s">&#39;mode&#39;</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">:</span>
-            <span class="n">valid_modes</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;inline&#39;</span><span class="p">,</span> <span class="s">&#39;plain&#39;</span><span class="p">,</span> <span class="s">&#39;equation&#39;</span><span class="p">,</span>
-                           <span class="s">&#39;equation*&#39;</span><span class="p">]</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;mode&#39;</span><span class="p">]</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">valid_modes</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;&#39;mode&#39; must be one of &#39;inline&#39;, &#39;plain&#39;, &quot;</span>
-                    <span class="s">&quot;&#39;equation&#39; or &#39;equation*&#39;&quot;</span><span class="p">)</span>
-
-        <span class="n">mul_symbol_table</span> <span class="o">=</span> <span class="p">{</span>
-            <span class="bp">None</span><span class="p">:</span> <span class="s">r&quot; &quot;</span><span class="p">,</span>
-            <span class="s">&quot;ldot&quot;</span><span class="p">:</span> <span class="s">r&quot; \,.\, &quot;</span><span class="p">,</span>
-            <span class="s">&quot;dot&quot;</span><span class="p">:</span> <span class="s">r&quot; \cdot &quot;</span><span class="p">,</span>
-            <span class="s">&quot;times&quot;</span><span class="p">:</span> <span class="s">r&quot; \times &quot;</span>
-        <span class="p">}</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;mul_symbol_latex&#39;</span><span class="p">]</span> <span class="o">=</span> \
-            <span class="n">mul_symbol_table</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;mul_symbol&#39;</span><span class="p">]]</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_delim_dict</span> <span class="o">=</span> <span class="p">{</span><span class="s">&#39;(&#39;</span><span class="p">:</span> <span class="s">&#39;)&#39;</span><span class="p">,</span> <span class="s">&#39;[&#39;</span><span class="p">:</span> <span class="s">&#39;]&#39;</span><span class="p">}</span>
-
-    <span class="k">def</span> <span class="nf">doprint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="n">Printer</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;mode&#39;</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;plain&#39;</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">tex</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;mode&#39;</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;inline&#39;</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;$</span><span class="si">%s</span><span class="s">$&quot;</span> <span class="o">%</span> <span class="n">tex</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;itex&#39;</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="s">r&quot;$$</span><span class="si">%s</span><span class="s">$$&quot;</span> <span class="o">%</span> <span class="n">tex</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">env_str</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;mode&#39;</span><span class="p">]</span>
-            <span class="k">return</span> <span class="s">r&quot;\begin{</span><span class="si">%s</span><span class="s">}</span><span class="si">%s</span><span class="s">\end{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">env_str</span><span class="p">,</span> <span class="n">tex</span><span class="p">,</span> <span class="n">env_str</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_needs_brackets</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns True if the expression needs to be wrapped in brackets when</span>
-<span class="sd">        printed, False otherwise. For example: a + b =&gt; True; a =&gt; False;</span>
-<span class="sd">        10 =&gt; False; -10 =&gt; True.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="p">((</span><span class="n">expr</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">)</span>
-                <span class="ow">or</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">is_Atom</span> <span class="ow">and</span> <span class="n">expr</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_needs_function_brackets</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns True if the expression needs to be wrapped in brackets when</span>
-<span class="sd">        passed as an argument to a function, False otherwise. This is a more</span>
-<span class="sd">        liberal version of _needs_brackets, in that many expressions which need</span>
-<span class="sd">        to be wrapped in brackets when added/subtracted/raised to a power do</span>
-<span class="sd">        not need them when passed to a function. Such an example is a*b.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">_needs_brackets</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># Muls of the form a*b*c... can be folded</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span> <span class="ow">and</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mul_is_clean</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="c"># Pows which don&#39;t need brackets can be folded</span>
-            <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">_pow_is_clean</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="c"># Add and Function always need brackets</span>
-            <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">or</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Function</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">_mul_is_clean</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Function</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_pow_is_clean</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">_needs_brackets</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_do_exponent</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="k">def</span> <span class="nf">_print_Add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span> <span class="o">==</span> <span class="s">&#39;none&#39;</span><span class="p">:</span>
-            <span class="n">terms</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">terms</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_as_ordered_terms</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">)</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">terms</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">term</span><span class="p">):</span>
-                <span class="n">tex</span> <span class="o">+=</span> <span class="s">&quot; + &quot;</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">tex</span> <span class="o">+=</span> <span class="s">&quot; - &quot;</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="o">-</span><span class="n">term</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_Float</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="c"># Based off of that in StrPrinter</span>
-        <span class="n">dps</span> <span class="o">=</span> <span class="n">prec_to_dps</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">_prec</span><span class="p">)</span>
-        <span class="n">str_real</span> <span class="o">=</span> <span class="n">mlib</span><span class="o">.</span><span class="n">to_str</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="n">dps</span><span class="p">,</span> <span class="n">strip_zeros</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-        <span class="c"># Must always have a mul symbol (as 2.5 10^{20} just looks odd)</span>
-        <span class="n">separator</span> <span class="o">=</span> <span class="s">r&quot; \times &quot;</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;mul_symbol&#39;</span><span class="p">]</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">separator</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;mul_symbol_latex&#39;</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="s">&#39;e&#39;</span> <span class="ow">in</span> <span class="n">str_real</span><span class="p">:</span>
-            <span class="p">(</span><span class="n">mant</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span> <span class="o">=</span> <span class="n">str_real</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;e&#39;</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">exp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;+&#39;</span><span class="p">:</span>
-                <span class="n">exp</span> <span class="o">=</span> <span class="n">exp</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s%s</span><span class="s">10^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">mant</span><span class="p">,</span> <span class="n">separator</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">str_real</span> <span class="o">==</span> <span class="s">&quot;+inf&quot;</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\infty&quot;</span>
-        <span class="k">elif</span> <span class="n">str_real</span> <span class="o">==</span> <span class="s">&quot;-inf&quot;</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;- \infty&quot;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">str_real</span>
-
-    <span class="k">def</span> <span class="nf">_print_Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">tail</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">coeff</span> <span class="o">=</span> <span class="o">-</span><span class="n">coeff</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">&quot;- &quot;</span>
-
-        <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="o">=</span> <span class="n">fraction</span><span class="p">(</span><span class="n">tail</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="n">separator</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;mul_symbol_latex&#39;</span><span class="p">]</span>
-
-        <span class="k">def</span> <span class="nf">convert</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">_tex</span> <span class="o">=</span> <span class="n">last_term_tex</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">(</span><span class="s">&#39;old&#39;</span><span class="p">,</span> <span class="s">&#39;none&#39;</span><span class="p">):</span>
-                    <span class="n">args</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_ordered_factors</span><span class="p">()</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">args</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span>
-
-                <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">Piecewise</span><span class="p">,</span> <span class="n">Product</span><span class="p">,</span> <span class="n">Sum</span>
-
-                <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">term</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-                    <span class="n">term_tex</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-
-                    <span class="k">if</span> <span class="n">term</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">or</span> <span class="p">(</span><span class="n">i</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="ow">and</span>
-                            <span class="nb">isinstance</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="p">(</span><span class="n">Integral</span><span class="p">,</span> <span class="n">Piecewise</span><span class="p">,</span> <span class="n">Product</span><span class="p">,</span> <span class="n">Sum</span><span class="p">))):</span>
-                        <span class="n">term_tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="n">term_tex</span>
-
-                    <span class="c"># between two digits, \times must always be used,</span>
-                    <span class="c"># to avoid confusion</span>
-                    <span class="k">if</span> <span class="n">separator</span> <span class="o">==</span> <span class="s">&quot; &quot;</span> <span class="ow">and</span> \
-                            <span class="n">re</span><span class="o">.</span><span class="n">search</span><span class="p">(</span><span class="s">&quot;[0-9][} ]*$&quot;</span><span class="p">,</span> <span class="n">last_term_tex</span><span class="p">)</span> <span class="ow">and</span> \
-                            <span class="n">re</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="s">&quot;[{ ]*[-+0-9]&quot;</span><span class="p">,</span> <span class="n">term_tex</span><span class="p">):</span>
-                        <span class="n">_tex</span> <span class="o">+=</span> <span class="s">r&quot; \times &quot;</span>
-                    <span class="k">elif</span> <span class="n">_tex</span><span class="p">:</span>
-                        <span class="n">_tex</span> <span class="o">+=</span> <span class="n">separator</span>
-
-                    <span class="n">_tex</span> <span class="o">+=</span> <span class="n">term_tex</span>
-                    <span class="n">last_term_tex</span> <span class="o">=</span> <span class="n">term_tex</span>
-                <span class="k">return</span> <span class="n">_tex</span>
-
-        <span class="k">if</span> <span class="n">denom</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">numer</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="n">_tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="n">convert</span><span class="p">(</span><span class="n">numer</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">_tex</span> <span class="o">=</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">convert</span><span class="p">(</span><span class="n">numer</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="n">tex</span> <span class="o">+=</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">coeff</span><span class="p">))</span>
-
-                <span class="c"># between two digits, \times must always be used, to avoid</span>
-                <span class="c"># confusion</span>
-                <span class="k">if</span> <span class="n">separator</span> <span class="o">==</span> <span class="s">&quot; &quot;</span> <span class="ow">and</span> <span class="n">re</span><span class="o">.</span><span class="n">search</span><span class="p">(</span><span class="s">&quot;[0-9][} ]*$&quot;</span><span class="p">,</span> <span class="n">tex</span><span class="p">)</span> <span class="ow">and</span> \
-                        <span class="n">re</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="s">&quot;[{ ]*[-+0-9]&quot;</span><span class="p">,</span> <span class="n">_tex</span><span class="p">):</span>
-                    <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot; \times &quot;</span> <span class="o">+</span> <span class="n">_tex</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">tex</span> <span class="o">+=</span> <span class="n">separator</span> <span class="o">+</span> <span class="n">_tex</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">tex</span> <span class="o">+=</span> <span class="n">_tex</span>
-
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">numer</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                    <span class="n">numer</span> <span class="o">*=</span> <span class="n">coeff</span><span class="o">.</span><span class="n">p</span>
-                <span class="k">elif</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">p</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="n">numer</span> <span class="o">*=</span> <span class="n">coeff</span><span class="o">.</span><span class="n">p</span>
-
-                    <span class="n">denom</span> <span class="o">*=</span> <span class="n">coeff</span><span class="o">.</span><span class="n">q</span>
-                <span class="k">elif</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                    <span class="n">tex</span> <span class="o">+=</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">coeff</span><span class="p">))</span> <span class="o">+</span> <span class="s">&quot; &quot;</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">coeff</span><span class="o">.</span><span class="n">p</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">denom</span> <span class="o">*=</span> <span class="n">coeff</span><span class="o">.</span><span class="n">q</span>
-                <span class="k">elif</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                    <span class="n">tex</span> <span class="o">+=</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">coeff</span><span class="p">))</span> <span class="o">+</span> <span class="s">&quot; &quot;</span>
-
-            <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot;\frac{</span><span class="si">%s</span><span class="s">}{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> \
-                <span class="p">(</span><span class="n">convert</span><span class="p">(</span><span class="n">numer</span><span class="p">),</span> <span class="n">convert</span><span class="p">(</span><span class="n">denom</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_Pow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="c"># Treat x**Rational(1,n) as special case</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="nb">abs</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">p</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">q</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">base</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">)</span>
-            <span class="n">expq</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">q</span>
-
-            <span class="k">if</span> <span class="n">expq</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\sqrt{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="n">base</span>
-            <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;itex&#39;</span><span class="p">]:</span>
-                <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\root{</span><span class="si">%d</span><span class="s">}{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expq</span><span class="p">,</span> <span class="n">base</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\sqrt[</span><span class="si">%d</span><span class="s">]{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expq</span><span class="p">,</span> <span class="n">base</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="k">return</span> <span class="s">r&quot;\frac{1}{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="n">tex</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">tex</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;fold_frac_powers&#39;</span><span class="p">]</span> \
-            <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span> \
-                <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">q</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">base</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">q</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">/</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_negative</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_Function</span><span class="p">:</span>
-            <span class="c"># Things like 1/x</span>
-            <span class="k">return</span> <span class="s">r&quot;\frac{</span><span class="si">%s</span><span class="s">}{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> \
-                <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Pow</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="o">-</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">)))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_Function</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_commutative</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                    <span class="c">#solves issue 1030</span>
-                    <span class="c">#As Mul always simplify 1/x to x**-1</span>
-                    <span class="c">#The objective is achieved with this hack</span>
-                    <span class="c">#first we get the latex for -1 * expr,</span>
-                    <span class="c">#which is a Mul expression</span>
-                    <span class="n">tex</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span> <span class="o">*</span> <span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">strip</span><span class="p">()</span>
-                    <span class="c">#the result comes with a minus and a space, so we remove</span>
-                    <span class="k">if</span> <span class="n">tex</span><span class="p">[:</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="s">&quot;-&quot;</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">tex</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span><span class="o">.</span><span class="n">strip</span><span class="p">()</span>
-                <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_needs_brackets</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">):</span>
-                    <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)^{</span><span class="si">%s</span><span class="s">}&quot;</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span>
-
-                <span class="k">return</span> <span class="n">tex</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">),</span>
-                              <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Sum</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">limits</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\sum_{</span><span class="si">%s</span><span class="s">=</span><span class="si">%s</span><span class="s">}^{</span><span class="si">%s</span><span class="s">} &quot;</span> <span class="o">%</span> \
-                <span class="nb">tuple</span><span class="p">([</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">limits</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">def</span> <span class="nf">_format_ineq</span><span class="p">(</span><span class="n">l</span><span class="p">):</span>
-                <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s"> \leq </span><span class="si">%s</span><span class="s"> \leq </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> \
-                    <span class="nb">tuple</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="p">(</span><span class="n">l</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">l</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">l</span><span class="p">[</span><span class="mi">2</span><span class="p">])])</span>
-
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\sum_{\substack{</span><span class="si">%s</span><span class="s">}} &quot;</span> <span class="o">%</span> \
-                <span class="nb">str</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\\\\</span><span class="s">&#39;</span><span class="p">,</span> <span class="p">[</span> <span class="n">_format_ineq</span><span class="p">(</span><span class="n">l</span><span class="p">)</span> <span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">limits</span> <span class="p">])</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">function</span><span class="p">,</span> <span class="n">Add</span><span class="p">):</span>
-            <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">function</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">function</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_Product</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">limits</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\prod_{</span><span class="si">%s</span><span class="s">=</span><span class="si">%s</span><span class="s">}^{</span><span class="si">%s</span><span class="s">} &quot;</span> <span class="o">%</span> \
-                <span class="nb">tuple</span><span class="p">([</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">limits</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">def</span> <span class="nf">_format_ineq</span><span class="p">(</span><span class="n">l</span><span class="p">):</span>
-                <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s"> \leq </span><span class="si">%s</span><span class="s"> \leq </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> \
-                    <span class="nb">tuple</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="p">(</span><span class="n">l</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">l</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">l</span><span class="p">[</span><span class="mi">2</span><span class="p">])])</span>
-
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\prod_{\substack{</span><span class="si">%s</span><span class="s">}} &quot;</span> <span class="o">%</span> \
-                <span class="nb">str</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\\\\</span><span class="s">&#39;</span><span class="p">,</span> <span class="p">[</span> <span class="n">_format_ineq</span><span class="p">(</span><span class="n">l</span><span class="p">)</span> <span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">limits</span> <span class="p">])</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">function</span><span class="p">,</span> <span class="n">Add</span><span class="p">):</span>
-            <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">function</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">function</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_Derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">dim</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">dim</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\frac{\partial}{\partial </span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> \
-                <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">variables</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">multiplicity</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">tex</span> <span class="o">=</span> <span class="p">[],</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&quot;&quot;</span>
-            <span class="n">current</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">variables</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-            <span class="k">for</span> <span class="n">symbol</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">variables</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-                <span class="k">if</span> <span class="n">symbol</span> <span class="o">==</span> <span class="n">current</span><span class="p">:</span>
-                    <span class="n">i</span> <span class="o">=</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">multiplicity</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">current</span><span class="p">,</span> <span class="n">i</span><span class="p">))</span>
-                    <span class="n">current</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="n">symbol</span><span class="p">,</span> <span class="mi">1</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">multiplicity</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">current</span><span class="p">,</span> <span class="n">i</span><span class="p">))</span>
-
-            <span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">multiplicity</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot;\partial </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot;\partial^{</span><span class="si">%s</span><span class="s">} </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\frac{\partial^{</span><span class="si">%s</span><span class="s">}}{</span><span class="si">%s</span><span class="s">} &quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">dim</span><span class="p">,</span> <span class="n">tex</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">AssocOp</span><span class="p">):</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">expr</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">expr</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">subs</span><span class="p">):</span>
-        <span class="n">expr</span><span class="p">,</span> <span class="n">old</span><span class="p">,</span> <span class="n">new</span> <span class="o">=</span> <span class="n">subs</span><span class="o">.</span><span class="n">args</span>
-        <span class="n">latex_expr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="n">latex_old</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="p">)</span> <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">old</span><span class="p">)</span>
-        <span class="n">latex_new</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="p">)</span> <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">new</span><span class="p">)</span>
-        <span class="n">latex_subs</span> <span class="o">=</span> <span class="s">r&#39;</span><span class="se">\\</span><span class="s"> &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span>
-            <span class="n">e</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="s">&#39;=&#39;</span> <span class="o">+</span> <span class="n">e</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">latex_old</span><span class="p">,</span> <span class="n">latex_new</span><span class="p">))</span>
-        <span class="k">return</span> <span class="s">r&#39;\left. </span><span class="si">%s</span><span class="s"> \right|_{\substack{ </span><span class="si">%s</span><span class="s"> }}&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">latex_expr</span><span class="p">,</span> <span class="n">latex_subs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Integral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">tex</span><span class="p">,</span> <span class="n">symbols</span> <span class="o">=</span> <span class="s">&quot;&quot;</span><span class="p">,</span> <span class="p">[]</span>
-
-        <span class="c"># Only up to \iiiint exists</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">limits</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">4</span> <span class="ow">and</span> <span class="nb">all</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">lim</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">lim</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">limits</span><span class="p">):</span>
-            <span class="c"># Use len(expr.limits)-1 so that syntax highlighters don&#39;t think</span>
-            <span class="c"># \&quot; is an escaped quote</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\i&quot;</span> <span class="o">+</span> <span class="s">&quot;i&quot;</span><span class="o">*</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">limits</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot;nt&quot;</span>
-            <span class="n">symbols</span> <span class="o">=</span> <span class="p">[</span><span class="s">r&quot;\, d</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">symbol</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-                       <span class="k">for</span> <span class="n">symbol</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">limits</span><span class="p">]</span>
-
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">lim</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">limits</span><span class="p">):</span>
-                <span class="n">symbol</span> <span class="o">=</span> <span class="n">lim</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot;\int&quot;</span>
-
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">lim</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;mode&#39;</span><span class="p">]</span> <span class="ow">in</span> <span class="p">[</span><span class="s">&#39;equation&#39;</span><span class="p">,</span> <span class="s">&#39;equation*&#39;</span><span class="p">]</span> \
-                            <span class="ow">and</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;itex&#39;</span><span class="p">]:</span>
-                        <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot;\limits&quot;</span>
-
-                    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">lim</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-                        <span class="n">tex</span> <span class="o">+=</span> <span class="s">&quot;_{</span><span class="si">%s</span><span class="s">}^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">lim</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span>
-                                               <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">lim</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span>
-                    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">lim</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                        <span class="n">tex</span> <span class="o">+=</span> <span class="s">&quot;^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">lim</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-
-                <span class="n">symbols</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="s">r&quot;\, d</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">symbol</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span>
-            <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">function</span><span class="p">)),</span> <span class="s">&quot;&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">symbols</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Limit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">e</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="nb">dir</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span>
-
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\lim_{</span><span class="si">%s</span><span class="s"> \to </span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">z</span><span class="p">),</span>
-                                     <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">z0</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">AssocOp</span><span class="p">):</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Function</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">func</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="o">.</span><span class="n">__name__</span>
-
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s">&#39;_print_&#39;</span> <span class="o">+</span> <span class="n">func</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s">&#39;_print_&#39;</span> <span class="o">+</span> <span class="n">func</span><span class="p">)(</span><span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="p">[</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span> <span class="p">]</span>
-            <span class="c"># How inverse trig functions should be displayed, formats are:</span>
-            <span class="c"># abbreviated: asin, full: arcsin, power: sin^-1</span>
-            <span class="n">inv_trig_style</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;inv_trig_style&#39;</span><span class="p">]</span>
-            <span class="c"># If we are dealing with a power-style inverse trig function</span>
-            <span class="n">inv_trig_power_case</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="c"># If it is applicable to fold the argument brackets</span>
-            <span class="n">can_fold_brackets</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;fold_func_brackets&#39;</span><span class="p">]</span> <span class="ow">and</span> \
-                <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> \
-                <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">_needs_function_brackets</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-            <span class="n">inv_trig_table</span> <span class="o">=</span> <span class="p">[</span><span class="s">&quot;asin&quot;</span><span class="p">,</span> <span class="s">&quot;acos&quot;</span><span class="p">,</span> <span class="s">&quot;atan&quot;</span><span class="p">,</span> <span class="s">&quot;acot&quot;</span><span class="p">]</span>
-
-            <span class="c"># If the function is an inverse trig function, handle the style</span>
-            <span class="k">if</span> <span class="n">func</span> <span class="ow">in</span> <span class="n">inv_trig_table</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">inv_trig_style</span> <span class="o">==</span> <span class="s">&quot;abbreviated&quot;</span><span class="p">:</span>
-                    <span class="n">func</span> <span class="o">=</span> <span class="n">func</span>
-                <span class="k">elif</span> <span class="n">inv_trig_style</span> <span class="o">==</span> <span class="s">&quot;full&quot;</span><span class="p">:</span>
-                    <span class="n">func</span> <span class="o">=</span> <span class="s">&quot;arc&quot;</span> <span class="o">+</span> <span class="n">func</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-                <span class="k">elif</span> <span class="n">inv_trig_style</span> <span class="o">==</span> <span class="s">&quot;power&quot;</span><span class="p">:</span>
-                    <span class="n">func</span> <span class="o">=</span> <span class="n">func</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-                    <span class="n">inv_trig_power_case</span> <span class="o">=</span> <span class="bp">True</span>
-
-                    <span class="c"># Can never fold brackets if we&#39;re raised to a power</span>
-                    <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="n">can_fold_brackets</span> <span class="o">=</span> <span class="bp">False</span>
-
-            <span class="k">if</span> <span class="n">inv_trig_power_case</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">func</span> <span class="ow">in</span> <span class="n">accepted_latex_functions</span><span class="p">:</span>
-                    <span class="n">name</span> <span class="o">=</span> <span class="s">r&quot;\</span><span class="si">%s</span><span class="s">^{-1}&quot;</span> <span class="o">%</span> <span class="n">func</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">name</span> <span class="o">=</span> <span class="s">r&quot;\operatorname{</span><span class="si">%s</span><span class="s">}^{-1}&quot;</span> <span class="o">%</span> <span class="n">func</span>
-            <span class="k">elif</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">func</span> <span class="ow">in</span> <span class="n">accepted_latex_functions</span><span class="p">:</span>
-                    <span class="n">name</span> <span class="o">=</span> <span class="s">r&quot;\</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="c"># If the generic function name contains an underscore, handle it</span>
-                    <span class="n">name</span> <span class="o">=</span> <span class="s">r&quot;\operatorname{</span><span class="si">%s</span><span class="s">}^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span>
-                        <span class="n">func</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&quot;_&quot;</span><span class="p">,</span> <span class="s">r&quot;\_&quot;</span><span class="p">),</span> <span class="n">exp</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">func</span> <span class="ow">in</span> <span class="n">accepted_latex_functions</span><span class="p">:</span>
-                    <span class="n">name</span> <span class="o">=</span> <span class="s">r&quot;\</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">func</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="c"># If the generic function name contains an underscore, handle it</span>
-                    <span class="n">name</span> <span class="o">=</span> <span class="s">r&quot;\operatorname{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="n">func</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&quot;_&quot;</span><span class="p">,</span> <span class="s">r&quot;\_&quot;</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">can_fold_brackets</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">func</span> <span class="ow">in</span> <span class="n">accepted_latex_functions</span><span class="p">:</span>
-                    <span class="c"># Wrap argument safely to avoid parse-time conflicts</span>
-                    <span class="c"># with the function name itself</span>
-                    <span class="n">name</span> <span class="o">+=</span> <span class="s">r&quot; {</span><span class="si">%s</span><span class="s">}&quot;</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">name</span> <span class="o">+=</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">&quot;</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">name</span> <span class="o">+=</span> <span class="s">r&quot;{\left (</span><span class="si">%s</span><span class="s"> \right )}&quot;</span>
-
-            <span class="k">if</span> <span class="n">inv_trig_power_case</span> <span class="ow">and</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">name</span> <span class="o">+=</span> <span class="s">r&quot;^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="n">exp</span>
-
-            <span class="k">return</span> <span class="n">name</span> <span class="o">%</span> <span class="s">&quot;,&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Lambda</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">symbols</span><span class="p">,</span> <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">symbols</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">symbols</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">symbols</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="nb">tuple</span><span class="p">(</span><span class="n">symbols</span><span class="p">))</span>
-
-        <span class="n">args</span> <span class="o">=</span> <span class="p">(</span><span class="n">symbols</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\Lambda {\left (</span><span class="si">%s</span><span class="s"> \right )}&quot;</span> <span class="o">%</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_Min</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="n">texargs</span> <span class="o">=</span> <span class="p">[</span><span class="s">r&quot;</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">symbol</span><span class="p">)</span> <span class="k">for</span> <span class="n">symbol</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\min\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">texargs</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_Max</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="n">texargs</span> <span class="o">=</span> <span class="p">[</span><span class="s">r&quot;</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">symbol</span><span class="p">)</span> <span class="k">for</span> <span class="n">symbol</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\max\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">texargs</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_floor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\lfloor{</span><span class="si">%s</span><span class="s">}\rfloor&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_ceiling</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\lceil{</span><span class="si">%s</span><span class="s">}\rceil&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_Abs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\lvert{</span><span class="si">%s</span><span class="s">}\rvert&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_re</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_needs_brackets</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\Re {\left (</span><span class="si">%s</span><span class="s"> \right )}&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\Re{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_do_exponent</span><span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_im</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_needs_brackets</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\Im {\left ( </span><span class="si">%s</span><span class="s"> \right )}&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\Im{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_do_exponent</span><span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Not</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Boolean</span><span class="p">):</span>
-            <span class="k">return</span> <span class="s">r&quot;\neg (</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\neg </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_print_And</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Boolean</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Not</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Boolean</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Not</span><span class="p">:</span>
-                <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot; \wedge \left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot; \wedge </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_Or</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Boolean</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Not</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Boolean</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Not</span><span class="p">:</span>
-                <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot; \vee \left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">tex</span> <span class="o">+=</span> <span class="s">r&quot; \vee </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_Implies</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s"> \Rightarrow </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Equivalent</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s"> \Leftrightarrow </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-
-    <span class="k">def</span> <span class="nf">_print_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\overline{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_ExpBase</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="c"># TODO should exp_polar be printed differently?</span>
-        <span class="c">#      what about exp_polar(0), exp_polar(1)?</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;e^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_do_exponent</span><span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_gamma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\Gamma^{</span><span class="si">%s</span><span class="s">}</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">exp</span><span class="p">,</span> <span class="n">tex</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\Gamma</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_uppergamma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span>
-                                        <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\Gamma^{</span><span class="si">%s</span><span class="s">}</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">exp</span><span class="p">,</span> <span class="n">tex</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\Gamma</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_lowergamma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span>
-                                        <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\gamma^{</span><span class="si">%s</span><span class="s">}</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">exp</span><span class="p">,</span> <span class="n">tex</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\gamma</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_expint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="n">nu</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\operatorname{E}_{</span><span class="si">%s</span><span class="s">}^{</span><span class="si">%s</span><span class="s">}</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">tex</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\operatorname{E}_{</span><span class="si">%s</span><span class="s">}</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">tex</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_subfactorial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_needs_brackets</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;!\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">&quot;!&quot;</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_factorial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_needs_brackets</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)!&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot;!&quot;</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_factorial2</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_needs_brackets</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)!!&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot;!!&quot;</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_binomial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;{\binom{</span><span class="si">%s</span><span class="s">}{</span><span class="si">%s</span><span class="s">}}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span>
-                                     <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_RisingFactorial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;{\left(</span><span class="si">%s</span><span class="s">\right)}^{\left(</span><span class="si">%s</span><span class="s">\right)}&quot;</span> <span class="o">%</span> \
-            <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_do_exponent</span><span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_FallingFactorial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;{\left(</span><span class="si">%s</span><span class="s">\right)}_{\left(</span><span class="si">%s</span><span class="s">\right)}&quot;</span> <span class="o">%</span> \
-            <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_do_exponent</span><span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_hprint_BesselBase</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sym</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">sym</span><span class="p">)</span>
-
-        <span class="n">need_exp</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">tex</span><span class="o">.</span><span class="n">find</span><span class="p">(</span><span class="s">&#39;^&#39;</span><span class="p">)</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">exp</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">need_exp</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">_{</span><span class="si">%s</span><span class="s">}\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">order</span><span class="p">),</span>
-                                           <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">argument</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">need_exp</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_do_exponent</span><span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_hprint_vec</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">vec</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">vec</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;&quot;</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">vec</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-            <span class="n">s</span> <span class="o">+=</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">, &quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-        <span class="n">s</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">vec</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">s</span>
-
-    <span class="k">def</span> <span class="nf">_print_besselj</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_hprint_BesselBase</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="s">&#39;J&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_besseli</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_hprint_BesselBase</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="s">&#39;I&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_besselk</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_hprint_BesselBase</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="s">&#39;K&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_bessely</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_hprint_BesselBase</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="s">&#39;Y&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_yn</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_hprint_BesselBase</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="s">&#39;y&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_jn</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_hprint_BesselBase</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="s">&#39;j&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_hankel1</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_hprint_BesselBase</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="s">&#39;H^{(1)}&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_hankel2</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_hprint_BesselBase</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="s">&#39;H^{(2)}&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_fresnels</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;S^{</span><span class="si">%s</span><span class="s">}</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">exp</span><span class="p">,</span> <span class="n">tex</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;S</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_fresnelc</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;C^{</span><span class="si">%s</span><span class="s">}</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">exp</span><span class="p">,</span> <span class="n">tex</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;C</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_hyper</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;{{}_{</span><span class="si">%s</span><span class="s">}F_{</span><span class="si">%s</span><span class="s">}\left(\begin{matrix} </span><span class="si">%s</span><span class="s"> </span><span class="se">\\</span><span class="s"> </span><span class="si">%s</span><span class="s"> \end{matrix}&quot;</span> \
-              <span class="s">r&quot;\middle| {</span><span class="si">%s</span><span class="s">} \right)}&quot;</span> <span class="o">%</span> \
-            <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">ap</span><span class="p">)),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">bq</span><span class="p">)),</span>
-              <span class="bp">self</span><span class="o">.</span><span class="n">_hprint_vec</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">ap</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_hprint_vec</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">bq</span><span class="p">),</span>
-              <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">argument</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;{</span><span class="si">%s</span><span class="s">}^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">exp</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_meijerg</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;{G_{</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">}^{</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">}\left(\begin{matrix} </span><span class="si">%s</span><span class="s"> &amp; </span><span class="si">%s</span><span class="s"> </span><span class="se">\\</span><span class="s">&quot;</span> \
-              <span class="s">r&quot;</span><span class="si">%s</span><span class="s"> &amp; </span><span class="si">%s</span><span class="s"> \end{matrix} \middle| {</span><span class="si">%s</span><span class="s">} \right)}&quot;</span> <span class="o">%</span> \
-            <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">ap</span><span class="p">)),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">bq</span><span class="p">)),</span>
-              <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">bm</span><span class="p">)),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">an</span><span class="p">)),</span>
-              <span class="bp">self</span><span class="o">.</span><span class="n">_hprint_vec</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">an</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_hprint_vec</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">aother</span><span class="p">),</span>
-              <span class="bp">self</span><span class="o">.</span><span class="n">_hprint_vec</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">bm</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_hprint_vec</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">bother</span><span class="p">),</span>
-              <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">argument</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;{</span><span class="si">%s</span><span class="s">}^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">exp</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_dirichlet_eta</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\eta^{</span><span class="si">%s</span><span class="s">}</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">exp</span><span class="p">),</span> <span class="n">tex</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">r&quot;\eta</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_zeta</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="nb">tuple</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\zeta^{</span><span class="si">%s</span><span class="s">}</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">exp</span><span class="p">),</span> <span class="n">tex</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">r&quot;\zeta</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_lerchphi</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="nb">tuple</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\Phi</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">tex</span>
-        <span class="k">return</span> <span class="s">r&quot;\Phi^{</span><span class="si">%s</span><span class="s">}</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">exp</span><span class="p">),</span> <span class="n">tex</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_polylog</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">s</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="n">z</span>
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\operatorname{Li}_{</span><span class="si">%s</span><span class="s">}</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">tex</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">r&quot;\operatorname{Li}_{</span><span class="si">%s</span><span class="s">}^{</span><span class="si">%s</span><span class="s">}</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">exp</span><span class="p">),</span> <span class="n">tex</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_jacobi</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;P_{</span><span class="si">%s</span><span class="s">}^{\left(</span><span class="si">%s</span><span class="s">,</span><span class="si">%s</span><span class="s">\right)}\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(&quot;</span> <span class="o">+</span> <span class="n">tex</span> <span class="o">+</span> <span class="s">r&quot;\right)^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">exp</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_gegenbauer</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;C_{</span><span class="si">%s</span><span class="s">}^{\left(</span><span class="si">%s</span><span class="s">\right)}\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(&quot;</span> <span class="o">+</span> <span class="n">tex</span> <span class="o">+</span> <span class="s">r&quot;\right)^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">exp</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_chebyshevt</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;T_{</span><span class="si">%s</span><span class="s">}\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(&quot;</span> <span class="o">+</span> <span class="n">tex</span> <span class="o">+</span> <span class="s">r&quot;\right)^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">exp</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_chebyshevu</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;U_{</span><span class="si">%s</span><span class="s">}\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(&quot;</span> <span class="o">+</span> <span class="n">tex</span> <span class="o">+</span> <span class="s">r&quot;\right)^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">exp</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_legendre</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;P_{</span><span class="si">%s</span><span class="s">}\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(&quot;</span> <span class="o">+</span> <span class="n">tex</span> <span class="o">+</span> <span class="s">r&quot;\right)^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">exp</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_assoc_legendre</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;P_{</span><span class="si">%s</span><span class="s">}^{\left(</span><span class="si">%s</span><span class="s">\right)}\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(&quot;</span> <span class="o">+</span> <span class="n">tex</span> <span class="o">+</span> <span class="s">r&quot;\right)^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">exp</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_hermite</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;H_{</span><span class="si">%s</span><span class="s">}\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(&quot;</span> <span class="o">+</span> <span class="n">tex</span> <span class="o">+</span> <span class="s">r&quot;\right)^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">exp</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_laguerre</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;L_{</span><span class="si">%s</span><span class="s">}\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(&quot;</span> <span class="o">+</span> <span class="n">tex</span> <span class="o">+</span> <span class="s">r&quot;\right)^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">exp</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_assoc_laguerre</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;L_{</span><span class="si">%s</span><span class="s">}^{\left(</span><span class="si">%s</span><span class="s">\right)}\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">exp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(&quot;</span> <span class="o">+</span> <span class="n">tex</span> <span class="o">+</span> <span class="s">r&quot;\right)^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">exp</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_Rational</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">q</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">sign</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">p</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">p</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">sign</span> <span class="o">=</span> <span class="s">&quot;- &quot;</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="o">-</span><span class="n">p</span>
-            <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">\frac{</span><span class="si">%d</span><span class="s">}{</span><span class="si">%d</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">sign</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Infinity</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\infty&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_NegativeInfinity</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;-\infty&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_ComplexInfinity</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\tilde{\infty}&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_ImaginaryUnit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\mathbf{\imath}&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_NaN</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\bot&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Pi</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\pi&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Exp1</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;e&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_EulerGamma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\gamma&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Order</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\mathcal{O}\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> \
-            <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_print_Symbol</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expr</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;symbol_names&#39;</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;symbol_names&#39;</span><span class="p">][</span><span class="n">expr</span><span class="p">]</span>
-
-        <span class="n">name</span><span class="p">,</span> <span class="n">supers</span><span class="p">,</span> <span class="n">subs</span> <span class="o">=</span> <span class="n">split_super_sub</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-
-        <span class="c"># translate name, supers and subs to tex keywords</span>
-        <span class="n">greek</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span> <span class="s">&#39;alpha&#39;</span><span class="p">,</span> <span class="s">&#39;beta&#39;</span><span class="p">,</span> <span class="s">&#39;gamma&#39;</span><span class="p">,</span> <span class="s">&#39;delta&#39;</span><span class="p">,</span> <span class="s">&#39;epsilon&#39;</span><span class="p">,</span> <span class="s">&#39;zeta&#39;</span><span class="p">,</span>
-                      <span class="s">&#39;eta&#39;</span><span class="p">,</span> <span class="s">&#39;theta&#39;</span><span class="p">,</span> <span class="s">&#39;iota&#39;</span><span class="p">,</span> <span class="s">&#39;kappa&#39;</span><span class="p">,</span> <span class="s">&#39;lambda&#39;</span><span class="p">,</span> <span class="s">&#39;mu&#39;</span><span class="p">,</span> <span class="s">&#39;nu&#39;</span><span class="p">,</span>
-                      <span class="s">&#39;xi&#39;</span><span class="p">,</span> <span class="s">&#39;omicron&#39;</span><span class="p">,</span> <span class="s">&#39;pi&#39;</span><span class="p">,</span> <span class="s">&#39;rho&#39;</span><span class="p">,</span> <span class="s">&#39;sigma&#39;</span><span class="p">,</span> <span class="s">&#39;tau&#39;</span><span class="p">,</span> <span class="s">&#39;upsilon&#39;</span><span class="p">,</span>
-                      <span class="s">&#39;phi&#39;</span><span class="p">,</span> <span class="s">&#39;chi&#39;</span><span class="p">,</span> <span class="s">&#39;psi&#39;</span><span class="p">,</span> <span class="s">&#39;omega&#39;</span> <span class="p">])</span>
-
-        <span class="n">greek_translated</span> <span class="o">=</span> <span class="p">{</span><span class="s">&#39;lamda&#39;</span><span class="p">:</span> <span class="s">&#39;lambda&#39;</span><span class="p">,</span> <span class="s">&#39;Lamda&#39;</span><span class="p">:</span> <span class="s">&#39;Lambda&#39;</span><span class="p">}</span>
-
-        <span class="n">other</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span> <span class="p">[</span><span class="s">&#39;aleph&#39;</span><span class="p">,</span> <span class="s">&#39;beth&#39;</span><span class="p">,</span> <span class="s">&#39;daleth&#39;</span><span class="p">,</span> <span class="s">&#39;gimel&#39;</span><span class="p">,</span> <span class="s">&#39;ell&#39;</span><span class="p">,</span> <span class="s">&#39;eth&#39;</span><span class="p">,</span>
-                      <span class="s">&#39;hbar&#39;</span><span class="p">,</span> <span class="s">&#39;hslash&#39;</span><span class="p">,</span> <span class="s">&#39;mho&#39;</span> <span class="p">])</span>
-
-        <span class="k">def</span> <span class="nf">translate</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
-            <span class="n">tmp</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">tmp</span> <span class="ow">in</span> <span class="n">greek</span> <span class="ow">or</span> <span class="n">tmp</span> <span class="ow">in</span> <span class="n">other</span><span class="p">:</span>
-                <span class="k">return</span> <span class="s">&quot;</span><span class="se">\\</span><span class="s">&quot;</span> <span class="o">+</span> <span class="n">s</span>
-            <span class="k">if</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">greek_translated</span><span class="p">:</span>
-                <span class="k">return</span> <span class="s">&quot;</span><span class="se">\\</span><span class="s">&quot;</span> <span class="o">+</span> <span class="n">greek_translated</span><span class="p">[</span><span class="n">s</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">s</span>
-
-        <span class="n">name</span> <span class="o">=</span> <span class="n">translate</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">supers</span> <span class="o">=</span> <span class="p">[</span><span class="n">translate</span><span class="p">(</span><span class="n">sup</span><span class="p">)</span> <span class="k">for</span> <span class="n">sup</span> <span class="ow">in</span> <span class="n">supers</span><span class="p">]</span>
-        <span class="n">subs</span> <span class="o">=</span> <span class="p">[</span><span class="n">translate</span><span class="p">(</span><span class="n">sub</span><span class="p">)</span> <span class="k">for</span> <span class="n">sub</span> <span class="ow">in</span> <span class="n">subs</span><span class="p">]</span>
-
-        <span class="c"># glue all items together:</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">supers</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">name</span> <span class="o">+=</span> <span class="s">&quot;^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="s">&quot; &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">supers</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">subs</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">name</span> <span class="o">+=</span> <span class="s">&quot;_{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="s">&quot; &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">subs</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">name</span>
-
-    <span class="k">def</span> <span class="nf">_print_Relational</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;itex&#39;</span><span class="p">]:</span>
-            <span class="n">gt</span> <span class="o">=</span> <span class="s">r&quot;\gt&quot;</span>
-            <span class="n">lt</span> <span class="o">=</span> <span class="s">r&quot;\lt&quot;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">gt</span> <span class="o">=</span> <span class="s">&quot;&gt;&quot;</span>
-            <span class="n">lt</span> <span class="o">=</span> <span class="s">&quot;&lt;&quot;</span>
-
-        <span class="n">charmap</span> <span class="o">=</span> <span class="p">{</span>
-            <span class="s">&quot;==&quot;</span><span class="p">:</span> <span class="s">&quot;=&quot;</span><span class="p">,</span>
-            <span class="s">&quot;&gt;&quot;</span><span class="p">:</span> <span class="n">gt</span><span class="p">,</span>
-            <span class="s">&quot;&lt;&quot;</span><span class="p">:</span> <span class="n">lt</span><span class="p">,</span>
-            <span class="s">&quot;&gt;=&quot;</span><span class="p">:</span> <span class="s">r&quot;\geq&quot;</span><span class="p">,</span>
-            <span class="s">&quot;&lt;=&quot;</span><span class="p">:</span> <span class="s">r&quot;\leq&quot;</span><span class="p">,</span>
-            <span class="s">&quot;!=&quot;</span><span class="p">:</span> <span class="s">r&quot;\neq&quot;</span><span class="p">,</span>
-        <span class="p">}</span>
-
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">lhs</span><span class="p">),</span>
-            <span class="n">charmap</span><span class="p">[</span><span class="n">expr</span><span class="o">.</span><span class="n">rel_op</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">rhs</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Piecewise</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">ecpairs</span> <span class="o">=</span> <span class="p">[</span><span class="s">r&quot;</span><span class="si">%s</span><span class="s"> &amp; \text{for}\: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">c</span><span class="p">))</span>
-                   <span class="k">for</span> <span class="n">e</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">cond</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">ecpairs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">r&quot;</span><span class="si">%s</span><span class="s"> &amp; \text{otherwise}&quot;</span> <span class="o">%</span>
-                           <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">expr</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">ecpairs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">r&quot;</span><span class="si">%s</span><span class="s"> &amp; \text{for}\: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span>
-                           <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">expr</span><span class="p">),</span>
-                            <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">cond</span><span class="p">)))</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\begin{cases} </span><span class="si">%s</span><span class="s"> \end{cases}&quot;</span>
-        <span class="k">return</span> <span class="n">tex</span> <span class="o">%</span> <span class="s">r&quot; </span><span class="se">\\</span><span class="s">&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">ecpairs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_MatrixBase</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">lines</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>  <span class="c"># horrible, should be &#39;rows&#39;</span>
-            <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot; &amp; &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span><span class="p">[</span><span class="n">line</span><span class="p">,</span> <span class="p">:]</span> <span class="p">]))</span>
-
-        <span class="n">out_str</span> <span class="o">=</span> <span class="s">r&#39;\begin{%MATSTR%}</span><span class="si">%s</span><span class="s">\end{%MATSTR%}&#39;</span>
-        <span class="n">out_str</span> <span class="o">=</span> <span class="n">out_str</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&#39;%MATSTR%&#39;</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;mat_str&#39;</span><span class="p">])</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;mat_delim&#39;</span><span class="p">]:</span>
-            <span class="n">left_delim</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;mat_delim&#39;</span><span class="p">]</span>
-            <span class="n">right_delim</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_delim_dict</span><span class="p">[</span><span class="n">left_delim</span><span class="p">]</span>
-            <span class="n">out_str</span> <span class="o">=</span> <span class="s">r&#39;\left&#39;</span> <span class="o">+</span> <span class="n">left_delim</span> <span class="o">+</span> <span class="n">out_str</span> <span class="o">+</span> \
-                      <span class="s">r&#39;\right&#39;</span> <span class="o">+</span> <span class="n">right_delim</span>
-        <span class="k">return</span> <span class="n">out_str</span> <span class="o">%</span> <span class="s">r&quot;</span><span class="se">\\</span><span class="s">&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">lines</span><span class="p">)</span>
-    <span class="n">_print_ImmutableMatrix</span> <span class="o">=</span> <span class="n">_print_MatrixBase</span>
-    <span class="n">_print_Matrix</span> <span class="o">=</span> <span class="n">_print_MatrixBase</span>
-
-    <span class="k">def</span> <span class="nf">_print_BlockMatrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">blocks</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Transpose</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">mat</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">arg</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">MatrixSymbol</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mat</span><span class="p">,</span> <span class="n">MatrixSymbol</span><span class="p">):</span>
-            <span class="k">return</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)^T&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">mat</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">^T&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">mat</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Adjoint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">mat</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">arg</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">MatrixSymbol</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mat</span><span class="p">,</span> <span class="n">MatrixSymbol</span><span class="p">):</span>
-            <span class="k">return</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)^\dag&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">mat</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">^\dag&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">mat</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_MatAdd</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">terms</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">&quot; + &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="n">terms</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_MatMul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Add</span><span class="p">,</span> <span class="n">MatAdd</span><span class="p">,</span> <span class="n">HadamardProduct</span>
-
-        <span class="k">def</span> <span class="nf">parens</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">Add</span><span class="p">,</span> <span class="n">MatAdd</span><span class="p">,</span> <span class="n">HadamardProduct</span><span class="p">)):</span>
-                <span class="k">return</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&#39; &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">parens</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_HadamardProduct</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Add</span><span class="p">,</span> <span class="n">MatAdd</span><span class="p">,</span> <span class="n">MatMul</span>
-
-        <span class="k">def</span> <span class="nf">parens</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">Add</span><span class="p">,</span> <span class="n">MatAdd</span><span class="p">,</span> <span class="n">MatMul</span><span class="p">)):</span>
-                <span class="k">return</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&#39; \circ &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">parens</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_MatPow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">MatrixSymbol</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">MatrixSymbol</span><span class="p">):</span>
-            <span class="k">return</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">base</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">exp</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">base</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">exp</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_ZeroMatrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">Z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\bold{0}&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Identity</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">I</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\mathbb{I}&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_tuple</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\begin{pmatrix}</span><span class="si">%s</span><span class="s">\end{pmatrix}&quot;</span> <span class="o">%</span> \
-            <span class="s">r&quot;, &amp; &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span> <span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_print_Tuple</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_tuple</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_list</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\begin{bmatrix}</span><span class="si">%s</span><span class="s">\end{bmatrix}&quot;</span> <span class="o">%</span> \
-            <span class="s">r&quot;, &amp; &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span> <span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_print_dict</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">d</span><span class="p">):</span>
-        <span class="n">keys</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">keys</span><span class="p">()),</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="n">items</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">key</span> <span class="ow">in</span> <span class="n">keys</span><span class="p">:</span>
-            <span class="n">val</span> <span class="o">=</span> <span class="n">d</span><span class="p">[</span><span class="n">key</span><span class="p">]</span>
-            <span class="n">items</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> : </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">key</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">val</span><span class="p">)))</span>
-
-        <span class="k">return</span> <span class="s">r&quot;\begin{Bmatrix}</span><span class="si">%s</span><span class="s">\end{Bmatrix}&quot;</span> <span class="o">%</span> <span class="s">r&quot;, &amp; &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">items</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Dict</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_dict</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_DiracDelta</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\delta\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\delta^{\left( </span><span class="si">%s</span><span class="s"> \right)}\left( </span><span class="si">%s</span><span class="s"> \right)&quot;</span> <span class="o">%</span> <span class="p">(</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-        <span class="k">if</span> <span class="n">exp</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_Heaviside</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\theta\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">exp</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\left(</span><span class="si">%s</span><span class="s">\right)^{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_KroneckerDelta</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="n">j</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Atom</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&#39;\delta_{</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">}&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&#39;\delta_{</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">}&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">exp</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&#39;\left(</span><span class="si">%s</span><span class="s">\right)^{</span><span class="si">%s</span><span class="s">}&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_LeviCivita</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">indices</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-        <span class="k">if</span> <span class="nb">all</span><span class="p">([</span><span class="n">x</span><span class="o">.</span><span class="n">is_Atom</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">]):</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&#39;\varepsilon_{</span><span class="si">%s</span><span class="s">}&#39;</span> <span class="o">%</span> <span class="s">&quot; &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">indices</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&#39;\varepsilon_{</span><span class="si">%s</span><span class="s">}&#39;</span> <span class="o">%</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">indices</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">exp</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&#39;\left(</span><span class="si">%s</span><span class="s">\right)^{</span><span class="si">%s</span><span class="s">}&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tex</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_ProductSet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">sets</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">has_variety</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">sets</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">sets</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">+</span> <span class="s">&quot;^</span><span class="si">%d</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">sets</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot; \times &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="nb">set</span><span class="p">)</span> <span class="k">for</span> <span class="nb">set</span> <span class="ow">in</span> <span class="n">p</span><span class="o">.</span><span class="n">sets</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_RandomDomain</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">d</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&#39;Domain: &#39;</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">as_boolean</span><span class="p">())</span>
-        <span class="k">except</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="s">&#39;Domain: &#39;</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">symbols</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39; in &#39;</span> <span class="o">+</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">set</span><span class="p">))</span>
-            <span class="k">except</span><span class="p">:</span>
-                <span class="k">return</span> <span class="s">&#39;Domain on &#39;</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">symbols</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_FiniteSet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="n">items</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_set</span><span class="p">(</span><span class="n">items</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_set</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="n">items</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="n">items</span> <span class="o">=</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="n">items</span><span class="p">))</span>
-        <span class="k">return</span> <span class="s">r&quot;\left\{</span><span class="si">%s</span><span class="s">\right\}&quot;</span> <span class="o">%</span> <span class="n">items</span>
-
-    <span class="n">_print_frozenset</span> <span class="o">=</span> <span class="n">_print_set</span>
-
-    <span class="k">def</span> <span class="nf">_print_Range</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">4</span><span class="p">:</span>
-            <span class="n">it</span> <span class="o">=</span> <span class="nb">iter</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-            <span class="n">printset</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">it</span><span class="p">),</span> <span class="nb">next</span><span class="p">(</span><span class="n">it</span><span class="p">),</span> <span class="s">&#39;\ldots&#39;</span><span class="p">,</span> <span class="n">s</span><span class="o">.</span><span class="n">_last_element</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">printset</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="p">(</span><span class="s">r&quot;\left\{&quot;</span>
-              <span class="o">+</span> <span class="s">r&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">el</span><span class="p">)</span> <span class="k">for</span> <span class="n">el</span> <span class="ow">in</span> <span class="n">printset</span><span class="p">)</span>
-              <span class="o">+</span> <span class="s">r&quot;\right\}&quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Interval</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">start</span> <span class="o">==</span> <span class="n">i</span><span class="o">.</span><span class="n">end</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\left\{</span><span class="si">%s</span><span class="s">\right\}&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">start</span><span class="p">)</span>
-
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">left_open</span><span class="p">:</span>
-                <span class="n">left</span> <span class="o">=</span> <span class="s">&#39;(&#39;</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">left</span> <span class="o">=</span> <span class="s">&#39;[&#39;</span>
-
-            <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">right_open</span><span class="p">:</span>
-                <span class="n">right</span> <span class="o">=</span> <span class="s">&#39;)&#39;</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">right</span> <span class="o">=</span> <span class="s">&#39;]&#39;</span>
-
-            <span class="k">return</span> <span class="s">r&quot;\left</span><span class="si">%s%s</span><span class="s">, </span><span class="si">%s</span><span class="s">\right</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> \
-                   <span class="p">(</span><span class="n">left</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">start</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">end</span><span class="p">),</span> <span class="n">right</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Union</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot; \cup &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">u</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_print_Intersection</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot; \cap &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">u</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_print_EmptySet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\emptyset&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Naturals</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\mathbb{N}&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Integers</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\mathbb{Z}&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Reals</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\mathbb{R}&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_TransformationSet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\left\{</span><span class="si">%s</span><span class="s">\; |\; </span><span class="si">%s</span><span class="s"> \in </span><span class="si">%s</span><span class="s">\right\}&quot;</span> <span class="o">%</span> <span class="p">(</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">lamda</span><span class="o">.</span><span class="n">expr</span><span class="p">),</span>
-            <span class="s">&#39;, &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">var</span><span class="p">)</span> <span class="k">for</span> <span class="n">var</span> <span class="ow">in</span> <span class="n">s</span><span class="o">.</span><span class="n">lamda</span><span class="o">.</span><span class="n">variables</span><span class="p">]),</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">base_set</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_FiniteField</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\mathbb{F}_{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="n">expr</span><span class="o">.</span><span class="n">mod</span>
-
-    <span class="k">def</span> <span class="nf">_print_IntegerRing</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\mathbb{Z}&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_RationalField</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\mathbb{Q}&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_RealDomain</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\mathbb{R}&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_ComplexDomain</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\mathbb{C}&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_PolynomialRingBase</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">domain</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">gens</span><span class="p">))</span>
-        <span class="n">inv</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Poly</span><span class="p">:</span>
-            <span class="n">inv</span> <span class="o">=</span> <span class="s">r&quot;S_&lt;^{-1}&quot;</span>
-        <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s%s</span><span class="s">\left[</span><span class="si">%s</span><span class="s">\right]&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">inv</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">gens</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_FractionField</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">domain</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">gens</span><span class="p">))</span>
-        <span class="k">return</span> <span class="s">r&quot;</span><span class="si">%s</span><span class="s">\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">gens</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Poly</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">poly</span><span class="p">):</span>
-        <span class="n">cls</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">poly</span><span class="o">.</span><span class="n">as_expr</span><span class="p">())</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="n">poly</span><span class="o">.</span><span class="n">gens</span><span class="p">))</span>
-        <span class="n">domain</span> <span class="o">=</span> <span class="s">&quot;domain=</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">poly</span><span class="o">.</span><span class="n">get_domain</span><span class="p">())</span>
-
-        <span class="n">args</span> <span class="o">=</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="n">expr</span><span class="p">]</span> <span class="o">+</span> <span class="n">gens</span> <span class="o">+</span> <span class="p">[</span><span class="n">domain</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">cls</span> <span class="ow">in</span> <span class="n">accepted_latex_functions</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\</span><span class="si">%s</span><span class="s"> {\left (</span><span class="si">%s</span><span class="s"> \right )}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">tex</span> <span class="o">=</span> <span class="s">r&quot;\operatorname{</span><span class="si">%s</span><span class="s">}{\left( </span><span class="si">%s</span><span class="s"> \right)}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">args</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">tex</span>
-
-    <span class="k">def</span> <span class="nf">_print_RootOf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">root</span><span class="p">):</span>
-        <span class="n">cls</span> <span class="o">=</span> <span class="n">root</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">expr</span><span class="p">)</span>
-        <span class="n">index</span> <span class="o">=</span> <span class="n">root</span><span class="o">.</span><span class="n">index</span>
-        <span class="k">if</span> <span class="n">cls</span> <span class="ow">in</span> <span class="n">accepted_latex_functions</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\</span><span class="si">%s</span><span class="s"> {\left(</span><span class="si">%s</span><span class="s">, </span><span class="si">%d</span><span class="s">\right)}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">index</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\operatorname{</span><span class="si">%s</span><span class="s">} {\left(</span><span class="si">%s</span><span class="s">, </span><span class="si">%d</span><span class="s">\right)}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">index</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_RootSum</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">cls</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">expr</span><span class="p">)]</span>
-
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">fun</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">IdentityFunction</span><span class="p">:</span>
-            <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">fun</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">cls</span> <span class="ow">in</span> <span class="n">accepted_latex_functions</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\</span><span class="si">%s</span><span class="s"> {\left(</span><span class="si">%s</span><span class="s">\right)}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">args</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">r&quot;\operatorname{</span><span class="si">%s</span><span class="s">} {\left(</span><span class="si">%s</span><span class="s">\right)}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">args</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_euler</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;E_{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_print_catalan</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;C_{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_print_MellinTransform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\mathcal{M}_{</span><span class="si">%s</span><span class="s">}\left[</span><span class="si">%s</span><span class="s">\right]\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span>
-
-    <span class="k">def</span> <span class="nf">_print_InverseMellinTransform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\mathcal{M}^{-1}_{</span><span class="si">%s</span><span class="s">}\left[</span><span class="si">%s</span><span class="s">\right]\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span>
-
-    <span class="k">def</span> <span class="nf">_print_LaplaceTransform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\mathcal{L}_{</span><span class="si">%s</span><span class="s">}\left[</span><span class="si">%s</span><span class="s">\right]\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span>
-
-    <span class="k">def</span> <span class="nf">_print_InverseLaplaceTransform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\mathcal{L}^{-1}_{</span><span class="si">%s</span><span class="s">}\left[</span><span class="si">%s</span><span class="s">\right]\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span>
-
-    <span class="k">def</span> <span class="nf">_print_FourierTransform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\mathcal{F}_{</span><span class="si">%s</span><span class="s">}\left[</span><span class="si">%s</span><span class="s">\right]\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span>
-
-    <span class="k">def</span> <span class="nf">_print_InverseFourierTransform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\mathcal{F}^{-1}_{</span><span class="si">%s</span><span class="s">}\left[</span><span class="si">%s</span><span class="s">\right]\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span>
-
-    <span class="k">def</span> <span class="nf">_print_SineTransform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\mathcal{SIN}_{</span><span class="si">%s</span><span class="s">}\left[</span><span class="si">%s</span><span class="s">\right]\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span>
-
-    <span class="k">def</span> <span class="nf">_print_InverseSineTransform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\mathcal{SIN}^{-1}_{</span><span class="si">%s</span><span class="s">}\left[</span><span class="si">%s</span><span class="s">\right]\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span>
-
-    <span class="k">def</span> <span class="nf">_print_CosineTransform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\mathcal{COS}_{</span><span class="si">%s</span><span class="s">}\left[</span><span class="si">%s</span><span class="s">\right]\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span>
-
-    <span class="k">def</span> <span class="nf">_print_InverseCosineTransform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\mathcal{COS}^{-1}_{</span><span class="si">%s</span><span class="s">}\left[</span><span class="si">%s</span><span class="s">\right]\left(</span><span class="si">%s</span><span class="s">\right)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span>
-
-    <span class="k">def</span> <span class="nf">_print_DMP</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">ring</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="c"># TODO incorporate order</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
-        <span class="k">except</span> <span class="n">SympifyError</span><span class="p">:</span>
-            <span class="k">pass</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="nb">repr</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_DMF</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_DMP</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Object</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">object</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">Symbol</span><span class="p">(</span><span class="nb">object</span><span class="o">.</span><span class="n">name</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Morphism</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">morphism</span><span class="p">):</span>
-        <span class="n">domain</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">morphism</span><span class="o">.</span><span class="n">domain</span><span class="p">)</span>
-        <span class="n">codomain</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">morphism</span><span class="o">.</span><span class="n">codomain</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="se">\\</span><span class="s">rightarrow </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">codomain</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_NamedMorphism</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">morphism</span><span class="p">):</span>
-        <span class="n">pretty_name</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">Symbol</span><span class="p">(</span><span class="n">morphism</span><span class="o">.</span><span class="n">name</span><span class="p">))</span>
-        <span class="n">pretty_morphism</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Morphism</span><span class="p">(</span><span class="n">morphism</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">:</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">pretty_name</span><span class="p">,</span> <span class="n">pretty_morphism</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_IdentityMorphism</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">morphism</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">NamedMorphism</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_NamedMorphism</span><span class="p">(</span><span class="n">NamedMorphism</span><span class="p">(</span>
-            <span class="n">morphism</span><span class="o">.</span><span class="n">domain</span><span class="p">,</span> <span class="n">morphism</span><span class="o">.</span><span class="n">codomain</span><span class="p">,</span> <span class="s">&quot;id&quot;</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_CompositeMorphism</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">morphism</span><span class="p">):</span>
-        <span class="c"># All components of the morphism have names and it is thus</span>
-        <span class="c"># possible to build the name of the composite.</span>
-        <span class="n">component_names_list</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">Symbol</span><span class="p">(</span><span class="n">component</span><span class="o">.</span><span class="n">name</span><span class="p">))</span> <span class="k">for</span>
-                                <span class="n">component</span> <span class="ow">in</span> <span class="n">morphism</span><span class="o">.</span><span class="n">components</span><span class="p">]</span>
-        <span class="n">component_names_list</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-        <span class="n">component_names</span> <span class="o">=</span> <span class="s">&quot;</span><span class="se">\\</span><span class="s">circ &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">component_names_list</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot;:&quot;</span>
-
-        <span class="n">pretty_morphism</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Morphism</span><span class="p">(</span><span class="n">morphism</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">component_names</span> <span class="o">+</span> <span class="n">pretty_morphism</span>
-
-    <span class="k">def</span> <span class="nf">_print_Category</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">morphism</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="se">\\</span><span class="s">mathbf{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">Symbol</span><span class="p">(</span><span class="n">morphism</span><span class="o">.</span><span class="n">name</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Diagram</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">diagram</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">diagram</span><span class="o">.</span><span class="n">premises</span><span class="p">:</span>
-            <span class="c"># This is an empty diagram.</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">EmptySet</span><span class="p">)</span>
-
-        <span class="n">latex_result</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">diagram</span><span class="o">.</span><span class="n">premises</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">diagram</span><span class="o">.</span><span class="n">conclusions</span><span class="p">:</span>
-            <span class="n">latex_result</span> <span class="o">+=</span> <span class="s">&quot;</span><span class="se">\\</span><span class="s">Longrightarrow </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> \
-                            <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">diagram</span><span class="o">.</span><span class="n">conclusions</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">latex_result</span>
-
-    <span class="k">def</span> <span class="nf">_print_DiagramGrid</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">grid</span><span class="p">):</span>
-        <span class="n">latex_result</span> <span class="o">=</span> <span class="s">&quot;</span><span class="se">\\</span><span class="s">begin{array}{</span><span class="si">%s</span><span class="s">}</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="s">&quot;c&quot;</span> <span class="o">*</span> <span class="n">grid</span><span class="o">.</span><span class="n">width</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid</span><span class="o">.</span><span class="n">height</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid</span><span class="o">.</span><span class="n">width</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]:</span>
-                    <span class="n">latex_result</span> <span class="o">+=</span> <span class="n">latex</span><span class="p">(</span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">])</span>
-                <span class="n">latex_result</span> <span class="o">+=</span> <span class="s">&quot; &quot;</span>
-                <span class="k">if</span> <span class="n">j</span> <span class="o">!=</span> <span class="n">grid</span><span class="o">.</span><span class="n">width</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">latex_result</span> <span class="o">+=</span> <span class="s">&quot;&amp; &quot;</span>
-
-            <span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="n">grid</span><span class="o">.</span><span class="n">height</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">latex_result</span> <span class="o">+=</span> <span class="s">&quot;</span><span class="se">\\\\</span><span class="s">&quot;</span>
-            <span class="n">latex_result</span> <span class="o">+=</span> <span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span>
-
-        <span class="n">latex_result</span> <span class="o">+=</span> <span class="s">&quot;</span><span class="se">\\</span><span class="s">end{array}</span><span class="se">\n</span><span class="s">&quot;</span>
-        <span class="k">return</span> <span class="n">latex_result</span>
-
-    <span class="k">def</span> <span class="nf">_print_FreeModule</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">M</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;{</span><span class="si">%s</span><span class="s">}^{</span><span class="si">%s</span><span class="s">}&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">ring</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">rank</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_FreeModuleElement</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-        <span class="c"># Print as row vector for convenience, for now.</span>
-        <span class="k">return</span> <span class="s">r&quot;\left[ </span><span class="si">%s</span><span class="s"> \right]&quot;</span> <span class="o">%</span> <span class="s">&quot;,&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span>
-            <span class="s">&#39;{&#39;</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39;}&#39;</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">m</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_SubModule</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\left&lt; </span><span class="si">%s</span><span class="s"> \right&gt;&quot;</span> <span class="o">%</span> <span class="s">&quot;,&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span>
-            <span class="s">&#39;{&#39;</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39;}&#39;</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">m</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_ModuleImplementedIdeal</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;\left&lt; </span><span class="si">%s</span><span class="s"> \right&gt;&quot;</span> <span class="o">%</span> <span class="s">&quot;,&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span>
-            <span class="s">&#39;{&#39;</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39;}&#39;</span> <span class="k">for</span> <span class="p">[</span><span class="n">x</span><span class="p">]</span> <span class="ow">in</span> <span class="n">m</span><span class="o">.</span><span class="n">_module</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_QuotientRing</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">R</span><span class="p">):</span>
-        <span class="c"># TODO nicer fractions for few generators...</span>
-        <span class="k">return</span> <span class="s">r&quot;\frac{</span><span class="si">%s</span><span class="s">}{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">R</span><span class="o">.</span><span class="n">ring</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">R</span><span class="o">.</span><span class="n">base_ideal</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_QuotientRingElement</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;{</span><span class="si">%s</span><span class="s">} + {</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">data</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">base_ideal</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_QuotientModuleElement</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;{</span><span class="si">%s</span><span class="s">} + {</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">data</span><span class="p">),</span>
-                                 <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">killed_module</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_QuotientModule</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">M</span><span class="p">):</span>
-        <span class="c"># TODO nicer fractions for few generators...</span>
-        <span class="k">return</span> <span class="s">r&quot;\frac{</span><span class="si">%s</span><span class="s">}{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">base</span><span class="p">),</span>
-                                   <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">killed_module</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_MatrixHomomorphism</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">h</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">r&quot;{</span><span class="si">%s</span><span class="s">} : {</span><span class="si">%s</span><span class="s">} \to {</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">h</span><span class="o">.</span><span class="n">_sympy_matrix</span><span class="p">()),</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">h</span><span class="o">.</span><span class="n">domain</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">h</span><span class="o">.</span><span class="n">codomain</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_BaseScalarField</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">field</span><span class="p">):</span>
-        <span class="n">string</span> <span class="o">=</span> <span class="n">field</span><span class="o">.</span><span class="n">_coord_sys</span><span class="o">.</span><span class="n">_names</span><span class="p">[</span><span class="n">field</span><span class="o">.</span><span class="n">_index</span><span class="p">]</span>
-        <span class="k">return</span> <span class="s">r&#39;\boldsymbol{\mathrm{</span><span class="si">%s</span><span class="s">}}&#39;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">Symbol</span><span class="p">(</span><span class="n">string</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_BaseVectorField</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">field</span><span class="p">):</span>
-        <span class="n">string</span> <span class="o">=</span> <span class="n">field</span><span class="o">.</span><span class="n">_coord_sys</span><span class="o">.</span><span class="n">_names</span><span class="p">[</span><span class="n">field</span><span class="o">.</span><span class="n">_index</span><span class="p">]</span>
-        <span class="k">return</span> <span class="s">r&#39;\partial_{</span><span class="si">%s</span><span class="s">}&#39;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">Symbol</span><span class="p">(</span><span class="n">string</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Differential</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">diff</span><span class="p">):</span>
-        <span class="n">field</span> <span class="o">=</span> <span class="n">diff</span><span class="o">.</span><span class="n">_form_field</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">field</span><span class="p">,</span> <span class="s">&#39;_coord_sys&#39;</span><span class="p">):</span>
-            <span class="n">string</span> <span class="o">=</span> <span class="n">field</span><span class="o">.</span><span class="n">_coord_sys</span><span class="o">.</span><span class="n">_names</span><span class="p">[</span><span class="n">field</span><span class="o">.</span><span class="n">_index</span><span class="p">]</span>
-            <span class="k">return</span> <span class="s">r&#39;\mathbb{d}</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">Symbol</span><span class="p">(</span><span class="n">string</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&#39;d(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">field</span><span class="p">)</span>
-            <span class="n">string</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">field</span><span class="p">)</span>
-            <span class="k">return</span> <span class="s">r&#39;\mathbb{d}\left(</span><span class="si">%s</span><span class="s">\right)&#39;</span> <span class="o">%</span> <span class="n">string</span>
-
-    <span class="k">def</span> <span class="nf">_print_Tr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="c">#Todo: Handle indices</span>
-        <span class="n">contents</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">return</span> <span class="s">r&#39;\mbox{Tr}\left(</span><span class="si">%s</span><span class="s">\right)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">contents</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="latex"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.latex.latex">[docs]</a><span class="k">def</span> <span class="nf">latex</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Convert the given expression to LaTeX representation.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import latex, sin, asin, Matrix, Rational</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, mu, tau</span>
-
-<span class="sd">    &gt;&gt;&gt; latex((2*tau)**Rational(7,2))</span>
-<span class="sd">    &#39;8 \\sqrt{2} \\tau^{\\frac{7}{2}}&#39;</span>
-
-<span class="sd">    order: Any of the supported monomial orderings (currently &quot;lex&quot;, &quot;grlex&quot;, or</span>
-<span class="sd">    &quot;grevlex&quot;), &quot;old&quot;, and &quot;none&quot;. This parameter does nothing for Mul objects.</span>
-<span class="sd">    Setting order to &quot;old&quot; uses the compatibility ordering for Add defined in</span>
-<span class="sd">    Printer. For very large expressions, set the &#39;order&#39; keyword to &#39;none&#39; if</span>
-<span class="sd">    speed is a concern.</span>
-
-<span class="sd">    mode: Specifies how the generated code will be delimited. &#39;mode&#39; can be one</span>
-<span class="sd">    of &#39;plain&#39;, &#39;inline&#39;, &#39;equation&#39; or &#39;equation*&#39;.  If &#39;mode&#39; is set to</span>
-<span class="sd">    &#39;plain&#39;, then the resulting code will not be delimited at all (this is the</span>
-<span class="sd">    default). If &#39;mode&#39; is set to &#39;inline&#39; then inline LaTeX $ $ will be used.</span>
-<span class="sd">    If &#39;mode&#39; is set to &#39;equation&#39; or &#39;equation*&#39;, the resulting code will be</span>
-<span class="sd">    enclosed in the &#39;equation&#39; or &#39;equation*&#39; environment (remember to import</span>
-<span class="sd">    &#39;amsmath&#39; for &#39;equation*&#39;), unless the &#39;itex&#39; option is set. In the latter</span>
-<span class="sd">    case, the ``$$ $$`` syntax is used.</span>
-
-<span class="sd">    &gt;&gt;&gt; latex((2*mu)**Rational(7,2), mode=&#39;plain&#39;)</span>
-<span class="sd">    &#39;8 \\sqrt{2} \\mu^{\\frac{7}{2}}&#39;</span>
-
-<span class="sd">    &gt;&gt;&gt; latex((2*tau)**Rational(7,2), mode=&#39;inline&#39;)</span>
-<span class="sd">    &#39;$8 \\sqrt{2} \\tau^{\\frac{7}{2}}$&#39;</span>
-
-<span class="sd">    &gt;&gt;&gt; latex((2*mu)**Rational(7,2), mode=&#39;equation*&#39;)</span>
-<span class="sd">    &#39;\\begin{equation*}8 \\sqrt{2} \\mu^{\\frac{7}{2}}\\end{equation*}&#39;</span>
-
-<span class="sd">    &gt;&gt;&gt; latex((2*mu)**Rational(7,2), mode=&#39;equation&#39;)</span>
-<span class="sd">    &#39;\\begin{equation}8 \\sqrt{2} \\mu^{\\frac{7}{2}}\\end{equation}&#39;</span>
-
-<span class="sd">    itex: Specifies if itex-specific syntax is used, including emitting ``$$ $$``.</span>
-
-<span class="sd">    &gt;&gt;&gt; latex((2*mu)**Rational(7,2), mode=&#39;equation&#39;, itex=True)</span>
-<span class="sd">    &#39;$$8 \\sqrt{2} \\mu^{\\frac{7}{2}}$$&#39;</span>
-
-<span class="sd">    fold_frac_powers: Emit &quot;^{p/q}&quot; instead of &quot;^{\frac{p}{q}}&quot; for fractional</span>
-<span class="sd">    powers.</span>
-
-<span class="sd">    &gt;&gt;&gt; latex((2*tau)**Rational(7,2), fold_frac_powers=True)</span>
-<span class="sd">    &#39;8 \\sqrt{2} \\tau^{7/2}&#39;</span>
-
-<span class="sd">    fold_func_brackets: Fold function brackets where applicable.</span>
-
-<span class="sd">    &gt;&gt;&gt; latex((2*tau)**sin(Rational(7,2)))</span>
-<span class="sd">    &#39;\\left(2 \\tau\\right)^{\\sin{\\left (\\frac{7}{2} \\right )}}&#39;</span>
-<span class="sd">    &gt;&gt;&gt; latex((2*tau)**sin(Rational(7,2)), fold_func_brackets = True)</span>
-<span class="sd">    &#39;\\left(2 \\tau\\right)^{\\sin {\\frac{7}{2}}}&#39;</span>
-
-<span class="sd">    mul_symbol: The symbol to use for multiplication. Can be one of None,</span>
-<span class="sd">    &quot;ldot&quot;, &quot;dot&quot;, or &quot;times&quot;.</span>
-
-<span class="sd">    &gt;&gt;&gt; latex((2*tau)**sin(Rational(7,2)), mul_symbol=&quot;times&quot;)</span>
-<span class="sd">    &#39;\\left(2 \\times \\tau\\right)^{\\sin{\\left (\\frac{7}{2} \\right )}}&#39;</span>
-
-<span class="sd">    inv_trig_style: How inverse trig functions should be displayed. Can be one</span>
-<span class="sd">    of &quot;abbreviated&quot;, &quot;full&quot;, or &quot;power&quot;. Defaults to &quot;abbreviated&quot;.</span>
-
-<span class="sd">    &gt;&gt;&gt; latex(asin(Rational(7,2)))</span>
-<span class="sd">    &#39;\\operatorname{asin}{\\left (\\frac{7}{2} \\right )}&#39;</span>
-<span class="sd">    &gt;&gt;&gt; latex(asin(Rational(7,2)), inv_trig_style=&quot;full&quot;)</span>
-<span class="sd">    &#39;\\arcsin{\\left (\\frac{7}{2} \\right )}&#39;</span>
-<span class="sd">    &gt;&gt;&gt; latex(asin(Rational(7,2)), inv_trig_style=&quot;power&quot;)</span>
-<span class="sd">    &#39;\\sin^{-1}{\\left (\\frac{7}{2} \\right )}&#39;</span>
-
-<span class="sd">    mat_str: Which matrix environment string to emit. &quot;smallmatrix&quot;, &quot;bmatrix&quot;,</span>
-<span class="sd">    etc. Defaults to &quot;smallmatrix&quot;.</span>
-
-<span class="sd">    &gt;&gt;&gt; latex(Matrix(2, 1, [x, y]), mat_str = &quot;array&quot;)</span>
-<span class="sd">    &#39;\\left[\\begin{array}x\\\\y\\end{array}\\right]&#39;</span>
-
-<span class="sd">    mat_delim: The delimiter to wrap around matrices. Can be one of &quot;[&quot;, &quot;(&quot;,</span>
-<span class="sd">    or the empty string. Defaults to &quot;[&quot;.</span>
-
-<span class="sd">    &gt;&gt;&gt; latex(Matrix(2, 1, [x, y]), mat_delim=&quot;(&quot;)</span>
-<span class="sd">    &#39;\\left(\\begin{smallmatrix}x\\\\y\\end{smallmatrix}\\right)&#39;</span>
-
-<span class="sd">    symbol_names: Dictionary of symbols and the custom strings they should be</span>
-<span class="sd">    emitted as.</span>
-
-<span class="sd">    &gt;&gt;&gt; latex(x**2, symbol_names={x:&#39;x_i&#39;})</span>
-<span class="sd">    &#39;x_i^{2}&#39;</span>
-
-<span class="sd">    Besides all Basic based expressions, you can recursively</span>
-<span class="sd">    convert Python containers (lists, tuples and dicts) and</span>
-<span class="sd">    also SymPy matrices:</span>
-
-<span class="sd">    &gt;&gt;&gt; latex([2/x, y], mode=&#39;inline&#39;)</span>
-<span class="sd">    &#39;$\\begin{bmatrix}\\frac{2}{x}, &amp; y\\end{bmatrix}$&#39;</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">LatexPrinter</span><span class="p">(</span><span class="n">settings</span><span class="p">)</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="print_latex"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.latex.print_latex">[docs]</a><span class="k">def</span> <span class="nf">print_latex</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Prints LaTeX representation of the given expression.&quot;&quot;&quot;</span>
-    <span class="k">print</span><span class="p">(</span><span class="n">latex</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">))</span></div>
-</pre></div>
-
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-        </div>
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-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
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diff --git a/dev-py3k/_modules/sympy/printing/mathml.html b/dev-py3k/_modules/sympy/printing/mathml.html
deleted file mode 100644
index 6af0425d724..00000000000
--- a/dev-py3k/_modules/sympy/printing/mathml.html
+++ /dev/null
@@ -1,608 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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-            
-  <h1>Source code for sympy.printing.mathml</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">A MathML printer.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">Mul</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">_coeff_isneg</span>
-<span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">fraction</span>
-<span class="kn">from</span> <span class="nn">.printer</span> <span class="kn">import</span> <span class="n">Printer</span>
-<span class="kn">from</span> <span class="nn">.conventions</span> <span class="kn">import</span> <span class="n">split_super_sub</span>
-
-
-<div class="viewcode-block" id="MathMLPrinter"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.mathml.MathMLPrinter">[docs]</a><span class="k">class</span> <span class="nc">MathMLPrinter</span><span class="p">(</span><span class="n">Printer</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Prints an expression to the MathML markup language</span>
-
-<span class="sd">    Whenever possible tries to use Content markup and not Presentation markup.</span>
-
-<span class="sd">    References: http://www.w3.org/TR/MathML2/</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">printmethod</span> <span class="o">=</span> <span class="s">&quot;_mathml&quot;</span>
-    <span class="n">_default_settings</span> <span class="o">=</span> <span class="p">{</span>
-        <span class="s">&quot;order&quot;</span><span class="p">:</span> <span class="bp">None</span><span class="p">,</span>
-        <span class="s">&quot;encoding&quot;</span><span class="p">:</span> <span class="s">&quot;utf-8&quot;</span>
-    <span class="p">}</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">settings</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">Printer</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">settings</span><span class="p">)</span>
-        <span class="kn">from</span> <span class="nn">xml.dom.minidom</span> <span class="kn">import</span> <span class="n">Document</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">dom</span> <span class="o">=</span> <span class="n">Document</span><span class="p">()</span>
-
-<div class="viewcode-block" id="MathMLPrinter.doprint"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.mathml.MathMLPrinter.doprint">[docs]</a>    <span class="k">def</span> <span class="nf">doprint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Prints the expression as MathML.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">mathML</span> <span class="o">=</span> <span class="n">Printer</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-        <span class="n">unistr</span> <span class="o">=</span> <span class="n">mathML</span><span class="o">.</span><span class="n">toxml</span><span class="p">()</span>
-        <span class="n">xmlbstr</span> <span class="o">=</span> <span class="n">unistr</span><span class="o">.</span><span class="n">encode</span><span class="p">(</span><span class="s">&#39;ascii&#39;</span><span class="p">,</span> <span class="s">&#39;xmlcharrefreplace&#39;</span><span class="p">)</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="n">xmlbstr</span><span class="o">.</span><span class="n">decode</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">res</span>
-</div>
-<div class="viewcode-block" id="MathMLPrinter.mathml_tag"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.mathml.MathMLPrinter.mathml_tag">[docs]</a>    <span class="k">def</span> <span class="nf">mathml_tag</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the MathML tag for an expression.&quot;&quot;&quot;</span>
-        <span class="n">translate</span> <span class="o">=</span> <span class="p">{</span>
-            <span class="s">&#39;Add&#39;</span><span class="p">:</span> <span class="s">&#39;plus&#39;</span><span class="p">,</span>
-            <span class="s">&#39;Mul&#39;</span><span class="p">:</span> <span class="s">&#39;times&#39;</span><span class="p">,</span>
-            <span class="s">&#39;Derivative&#39;</span><span class="p">:</span> <span class="s">&#39;diff&#39;</span><span class="p">,</span>
-            <span class="s">&#39;Number&#39;</span><span class="p">:</span> <span class="s">&#39;cn&#39;</span><span class="p">,</span>
-            <span class="s">&#39;int&#39;</span><span class="p">:</span> <span class="s">&#39;cn&#39;</span><span class="p">,</span>
-            <span class="s">&#39;Pow&#39;</span><span class="p">:</span> <span class="s">&#39;power&#39;</span><span class="p">,</span>
-            <span class="s">&#39;Symbol&#39;</span><span class="p">:</span> <span class="s">&#39;ci&#39;</span><span class="p">,</span>
-            <span class="s">&#39;Integral&#39;</span><span class="p">:</span> <span class="s">&#39;int&#39;</span><span class="p">,</span>
-            <span class="s">&#39;Sum&#39;</span><span class="p">:</span> <span class="s">&#39;sum&#39;</span><span class="p">,</span>
-            <span class="s">&#39;sin&#39;</span><span class="p">:</span> <span class="s">&#39;sin&#39;</span><span class="p">,</span>
-            <span class="s">&#39;cos&#39;</span><span class="p">:</span> <span class="s">&#39;cos&#39;</span><span class="p">,</span>
-            <span class="s">&#39;tan&#39;</span><span class="p">:</span> <span class="s">&#39;tan&#39;</span><span class="p">,</span>
-            <span class="s">&#39;cot&#39;</span><span class="p">:</span> <span class="s">&#39;cot&#39;</span><span class="p">,</span>
-            <span class="s">&#39;asin&#39;</span><span class="p">:</span> <span class="s">&#39;arcsin&#39;</span><span class="p">,</span>
-            <span class="s">&#39;asinh&#39;</span><span class="p">:</span> <span class="s">&#39;arcsinh&#39;</span><span class="p">,</span>
-            <span class="s">&#39;acos&#39;</span><span class="p">:</span> <span class="s">&#39;arccos&#39;</span><span class="p">,</span>
-            <span class="s">&#39;acosh&#39;</span><span class="p">:</span> <span class="s">&#39;arccosh&#39;</span><span class="p">,</span>
-            <span class="s">&#39;atan&#39;</span><span class="p">:</span> <span class="s">&#39;arctan&#39;</span><span class="p">,</span>
-            <span class="s">&#39;atanh&#39;</span><span class="p">:</span> <span class="s">&#39;arctanh&#39;</span><span class="p">,</span>
-            <span class="s">&#39;acot&#39;</span><span class="p">:</span> <span class="s">&#39;arccot&#39;</span><span class="p">,</span>
-            <span class="s">&#39;atan2&#39;</span><span class="p">:</span> <span class="s">&#39;arctan&#39;</span><span class="p">,</span>
-            <span class="s">&#39;log&#39;</span><span class="p">:</span> <span class="s">&#39;ln&#39;</span><span class="p">,</span>
-            <span class="s">&#39;Equality&#39;</span><span class="p">:</span> <span class="s">&#39;eq&#39;</span><span class="p">,</span>
-            <span class="s">&#39;Unequality&#39;</span><span class="p">:</span> <span class="s">&#39;neq&#39;</span><span class="p">,</span>
-            <span class="s">&#39;GreaterThan&#39;</span><span class="p">:</span> <span class="s">&#39;geq&#39;</span><span class="p">,</span>
-            <span class="s">&#39;LessThan&#39;</span><span class="p">:</span> <span class="s">&#39;leq&#39;</span><span class="p">,</span>
-            <span class="s">&#39;StrictGreaterThan&#39;</span><span class="p">:</span> <span class="s">&#39;gt&#39;</span><span class="p">,</span>
-            <span class="s">&#39;StrictLessThan&#39;</span><span class="p">:</span> <span class="s">&#39;lt&#39;</span><span class="p">,</span>
-        <span class="p">}</span>
-
-        <span class="k">for</span> <span class="n">cls</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__mro__</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">__name__</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">translate</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">translate</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-        <span class="c"># Not found in the MRO set</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span>
-        <span class="k">return</span> <span class="n">n</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span>
-</div>
-    <span class="k">def</span> <span class="nf">_print_Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-
-        <span class="k">if</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;apply&#39;</span><span class="p">)</span>
-            <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;minus&#39;</span><span class="p">))</span>
-            <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print_Mul</span><span class="p">(</span><span class="o">-</span><span class="n">expr</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">x</span>
-
-        <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="o">=</span> <span class="n">fraction</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">denom</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;apply&#39;</span><span class="p">)</span>
-            <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;divide&#39;</span><span class="p">))</span>
-            <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">numer</span><span class="p">))</span>
-            <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">denom</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">x</span>
-
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">terms</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">terms</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># XXX since the negative coefficient has been handled, I don&#39;t</span>
-            <span class="c"># thing a coeff of 1 can remain</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">terms</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span> <span class="o">!=</span> <span class="s">&#39;old&#39;</span><span class="p">:</span>
-            <span class="n">terms</span> <span class="o">=</span> <span class="n">Mul</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">terms</span><span class="p">)</span><span class="o">.</span><span class="n">as_ordered_factors</span><span class="p">()</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;apply&#39;</span><span class="p">)</span>
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;times&#39;</span><span class="p">))</span>
-        <span class="k">if</span><span class="p">(</span><span class="n">coeff</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">coeff</span><span class="p">))</span>
-        <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">:</span>
-            <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">term</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">x</span>
-
-    <span class="k">def</span> <span class="nf">_print_Add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_as_ordered_terms</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">)</span>
-        <span class="n">lastProcessed</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="n">plusNodes</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="k">if</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">arg</span><span class="p">):</span>
-                <span class="c">#use minus</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;apply&#39;</span><span class="p">)</span>
-                <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;minus&#39;</span><span class="p">))</span>
-                <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">lastProcessed</span><span class="p">)</span>
-                <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="o">-</span><span class="n">arg</span><span class="p">))</span>
-                <span class="c">#invert expression since this is now minused</span>
-                <span class="n">lastProcessed</span> <span class="o">=</span> <span class="n">x</span>
-                <span class="k">if</span><span class="p">(</span><span class="n">arg</span> <span class="o">==</span> <span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]):</span>
-                    <span class="n">plusNodes</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">lastProcessed</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">plusNodes</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">lastProcessed</span><span class="p">)</span>
-                <span class="n">lastProcessed</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-                <span class="k">if</span><span class="p">(</span><span class="n">arg</span> <span class="o">==</span> <span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]):</span>
-                    <span class="n">plusNodes</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">plusNodes</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">lastProcessed</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;apply&#39;</span><span class="p">)</span>
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;plus&#39;</span><span class="p">))</span>
-        <span class="k">while</span> <span class="nb">len</span><span class="p">(</span><span class="n">plusNodes</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">plusNodes</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">0</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">x</span>
-
-    <span class="k">def</span> <span class="nf">_print_MatrixBase</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;matrix&#39;</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">lines</span><span class="p">):</span>
-            <span class="n">x_r</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;matrixrow&#39;</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-                <span class="n">x_r</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">m</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]))</span>
-            <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">x_r</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">x</span>
-
-    <span class="k">def</span> <span class="nf">_print_Rational</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">q</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c">#don&#39;t divide</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;cn&#39;</span><span class="p">)</span>
-            <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createTextNode</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">p</span><span class="p">)))</span>
-            <span class="k">return</span> <span class="n">x</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;apply&#39;</span><span class="p">)</span>
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;divide&#39;</span><span class="p">))</span>
-        <span class="c">#numerator</span>
-        <span class="n">xnum</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;cn&#39;</span><span class="p">)</span>
-        <span class="n">xnum</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createTextNode</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">p</span><span class="p">)))</span>
-        <span class="c">#denomenator</span>
-        <span class="n">xdenom</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;cn&#39;</span><span class="p">)</span>
-        <span class="n">xdenom</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createTextNode</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">q</span><span class="p">)))</span>
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">xnum</span><span class="p">)</span>
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">xdenom</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">x</span>
-
-    <span class="k">def</span> <span class="nf">_print_Limit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;apply&#39;</span><span class="p">)</span>
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mathml_tag</span><span class="p">(</span><span class="n">e</span><span class="p">)))</span>
-
-        <span class="n">x_1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;bvar&#39;</span><span class="p">)</span>
-        <span class="n">x_2</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;lowlimit&#39;</span><span class="p">)</span>
-        <span class="n">x_1</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-        <span class="n">x_2</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span>
-
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">x_1</span><span class="p">)</span>
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">x_2</span><span class="p">)</span>
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-        <span class="k">return</span> <span class="n">x</span>
-
-    <span class="k">def</span> <span class="nf">_print_ImaginaryUnit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;imaginaryi&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_EulerGamma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;eulergamma&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_GoldenRatio</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;We use unicode #x3c6 for Greek letter phi as defined here</span>
-<span class="sd">        http://www.w3.org/2003/entities/2007doc/isogrk1.html&quot;&quot;&quot;</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;cn&#39;</span><span class="p">)</span>
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createTextNode</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\u03c6</span><span class="s">&quot;</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">x</span>
-
-    <span class="k">def</span> <span class="nf">_print_Exp1</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;exponentiale&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Pi</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;pi&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Infinity</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;infinity&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Negative_Infinity</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;apply&#39;</span><span class="p">)</span>
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;minus&#39;</span><span class="p">))</span>
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;infinity&#39;</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">x</span>
-
-    <span class="k">def</span> <span class="nf">_print_Integral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">def</span> <span class="nf">lime_recur</span><span class="p">(</span><span class="n">limits</span><span class="p">):</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;apply&#39;</span><span class="p">)</span>
-            <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mathml_tag</span><span class="p">(</span><span class="n">e</span><span class="p">)))</span>
-            <span class="n">bvar_elem</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;bvar&#39;</span><span class="p">)</span>
-            <span class="n">bvar_elem</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">limits</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]))</span>
-            <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">bvar_elem</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">limits</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-                <span class="n">low_elem</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;lowlimit&#39;</span><span class="p">)</span>
-                <span class="n">low_elem</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">limits</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]))</span>
-                <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">low_elem</span><span class="p">)</span>
-                <span class="n">up_elem</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;uplimit&#39;</span><span class="p">)</span>
-                <span class="n">up_elem</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">limits</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">2</span><span class="p">]))</span>
-                <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">up_elem</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">limits</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">up_elem</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;uplimit&#39;</span><span class="p">)</span>
-                <span class="n">up_elem</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">limits</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]))</span>
-                <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">up_elem</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">limits</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">function</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">lime_recur</span><span class="p">(</span><span class="n">limits</span><span class="p">[</span><span class="mi">1</span><span class="p">:]))</span>
-            <span class="k">return</span> <span class="n">x</span>
-
-        <span class="n">limits</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">limits</span><span class="p">)</span>
-        <span class="n">limits</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">lime_recur</span><span class="p">(</span><span class="n">limits</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Sum</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="c"># Printer can be shared because Sum and Integral have the</span>
-        <span class="c"># same internal representation.</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Integral</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Symbol</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sym</span><span class="p">):</span>
-        <span class="n">ci</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mathml_tag</span><span class="p">(</span><span class="n">sym</span><span class="p">))</span>
-
-        <span class="k">def</span> <span class="nf">join</span><span class="p">(</span><span class="n">items</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">items</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">mrow</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;mml:mrow&#39;</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">item</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">items</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">i</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">mo</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;mml:mo&#39;</span><span class="p">)</span>
-                        <span class="n">mo</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createTextNode</span><span class="p">(</span><span class="s">&quot; &quot;</span><span class="p">))</span>
-                        <span class="n">mrow</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">mo</span><span class="p">)</span>
-                    <span class="n">mi</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;mml:mi&#39;</span><span class="p">)</span>
-                    <span class="n">mi</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createTextNode</span><span class="p">(</span><span class="n">item</span><span class="p">))</span>
-                    <span class="n">mrow</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">mi</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">mrow</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">mi</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;mml:mi&#39;</span><span class="p">)</span>
-                <span class="n">mi</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createTextNode</span><span class="p">(</span><span class="n">items</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-                <span class="k">return</span> <span class="n">mi</span>
-
-        <span class="c"># translate name, supers and subs to unicode characters</span>
-        <span class="c"># taken from http://www.w3.org/2003/entities/2007doc/isogrk1.html</span>
-        <span class="n">unitr</span> <span class="o">=</span> <span class="p">{</span>
-            <span class="s">&#39;Alpha&#39;</span><span class="p">:</span>    <span class="s">&#39;</span><span class="se">\u0391</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;Beta&#39;</span><span class="p">:</span>      <span class="s">&#39;</span><span class="se">\u0392</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;Gamma&#39;</span><span class="p">:</span>    <span class="s">&#39;</span><span class="se">\u0393</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;Delta&#39;</span><span class="p">:</span>     <span class="s">&#39;</span><span class="se">\u0394</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;Epsilon&#39;</span><span class="p">:</span>  <span class="s">&#39;</span><span class="se">\u0395</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;Zeta&#39;</span><span class="p">:</span>      <span class="s">&#39;</span><span class="se">\u0396</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;Eta&#39;</span><span class="p">:</span>      <span class="s">&#39;</span><span class="se">\u0397</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;Theta&#39;</span><span class="p">:</span>     <span class="s">&#39;</span><span class="se">\u0398</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;Iota&#39;</span><span class="p">:</span>     <span class="s">&#39;</span><span class="se">\u0399</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;Kappa&#39;</span><span class="p">:</span>     <span class="s">&#39;</span><span class="se">\u039A</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;Lambda&#39;</span><span class="p">:</span>   <span class="s">&#39;</span><span class="se">\u039B</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;Mu&#39;</span><span class="p">:</span>        <span class="s">&#39;</span><span class="se">\u039C</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;Nu&#39;</span><span class="p">:</span>       <span class="s">&#39;</span><span class="se">\u039D</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;Xi&#39;</span><span class="p">:</span>        <span class="s">&#39;</span><span class="se">\u039E</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;Omicron&#39;</span><span class="p">:</span>  <span class="s">&#39;</span><span class="se">\u039F</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;Pi&#39;</span><span class="p">:</span>        <span class="s">&#39;</span><span class="se">\u03A0</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;Rho&#39;</span><span class="p">:</span>      <span class="s">&#39;</span><span class="se">\u03A1</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;Sigma&#39;</span><span class="p">:</span>     <span class="s">&#39;</span><span class="se">\u03A3</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;Tau&#39;</span><span class="p">:</span>      <span class="s">&#39;</span><span class="se">\u03A4</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;Upsilon&#39;</span><span class="p">:</span>   <span class="s">&#39;</span><span class="se">\u03A5</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;Phi&#39;</span><span class="p">:</span>      <span class="s">&#39;</span><span class="se">\u03A6</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;Chi&#39;</span><span class="p">:</span>       <span class="s">&#39;</span><span class="se">\u03A7</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;Psi&#39;</span><span class="p">:</span>      <span class="s">&#39;</span><span class="se">\u03A8</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;Omega&#39;</span><span class="p">:</span>     <span class="s">&#39;</span><span class="se">\u03A9</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;alpha&#39;</span><span class="p">:</span>    <span class="s">&#39;</span><span class="se">\u03B1</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;beta&#39;</span><span class="p">:</span>      <span class="s">&#39;</span><span class="se">\u03B2</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;gamma&#39;</span><span class="p">:</span>    <span class="s">&#39;</span><span class="se">\u03B3</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;delta&#39;</span><span class="p">:</span>     <span class="s">&#39;</span><span class="se">\u03B4</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;epsilon&#39;</span><span class="p">:</span>  <span class="s">&#39;</span><span class="se">\u03B5</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;zeta&#39;</span><span class="p">:</span>      <span class="s">&#39;</span><span class="se">\u03B6</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;eta&#39;</span><span class="p">:</span>      <span class="s">&#39;</span><span class="se">\u03B7</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;theta&#39;</span><span class="p">:</span>     <span class="s">&#39;</span><span class="se">\u03B8</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;iota&#39;</span><span class="p">:</span>     <span class="s">&#39;</span><span class="se">\u03B9</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;kappa&#39;</span><span class="p">:</span>     <span class="s">&#39;</span><span class="se">\u03BA</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;lambda&#39;</span><span class="p">:</span>   <span class="s">&#39;</span><span class="se">\u03BB</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;mu&#39;</span><span class="p">:</span>        <span class="s">&#39;</span><span class="se">\u03BC</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;nu&#39;</span><span class="p">:</span>       <span class="s">&#39;</span><span class="se">\u03BD</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;xi&#39;</span><span class="p">:</span>        <span class="s">&#39;</span><span class="se">\u03BE</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;omicron&#39;</span><span class="p">:</span>  <span class="s">&#39;</span><span class="se">\u03BF</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;pi&#39;</span><span class="p">:</span>        <span class="s">&#39;</span><span class="se">\u03C0</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;rho&#39;</span><span class="p">:</span>      <span class="s">&#39;</span><span class="se">\u03C1</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;varsigma&#39;</span><span class="p">:</span>  <span class="s">&#39;</span><span class="se">\u03C2</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;sigma&#39;</span><span class="p">:</span>    <span class="s">&#39;</span><span class="se">\u03C3</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;tau&#39;</span><span class="p">:</span>       <span class="s">&#39;</span><span class="se">\u03C4</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;upsilon&#39;</span><span class="p">:</span>  <span class="s">&#39;</span><span class="se">\u03C5</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;phi&#39;</span><span class="p">:</span>       <span class="s">&#39;</span><span class="se">\u03C6</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;chi&#39;</span><span class="p">:</span>      <span class="s">&#39;</span><span class="se">\u03C7</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;psi&#39;</span><span class="p">:</span>       <span class="s">&#39;</span><span class="se">\u03C8</span><span class="s">&#39;</span><span class="p">,</span>
-            <span class="s">&#39;omega&#39;</span><span class="p">:</span>    <span class="s">&#39;</span><span class="se">\u03C9</span><span class="s">&#39;</span><span class="p">,</span>
-        <span class="p">}</span>
-
-        <span class="k">def</span> <span class="nf">translate</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">unitr</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">unitr</span><span class="p">[</span><span class="n">s</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">s</span>
-
-        <span class="n">name</span><span class="p">,</span> <span class="n">supers</span><span class="p">,</span> <span class="n">subs</span> <span class="o">=</span> <span class="n">split_super_sub</span><span class="p">(</span><span class="n">sym</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">name</span> <span class="o">=</span> <span class="n">translate</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">supers</span> <span class="o">=</span> <span class="p">[</span><span class="n">translate</span><span class="p">(</span><span class="n">sup</span><span class="p">)</span> <span class="k">for</span> <span class="n">sup</span> <span class="ow">in</span> <span class="n">supers</span><span class="p">]</span>
-        <span class="n">subs</span> <span class="o">=</span> <span class="p">[</span><span class="n">translate</span><span class="p">(</span><span class="n">sub</span><span class="p">)</span> <span class="k">for</span> <span class="n">sub</span> <span class="ow">in</span> <span class="n">subs</span><span class="p">]</span>
-
-        <span class="n">mname</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;mml:mi&#39;</span><span class="p">)</span>
-        <span class="n">mname</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createTextNode</span><span class="p">(</span><span class="n">name</span><span class="p">))</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">supers</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">subs</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">ci</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createTextNode</span><span class="p">(</span><span class="n">name</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">msub</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;mml:msub&#39;</span><span class="p">)</span>
-                <span class="n">msub</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">mname</span><span class="p">)</span>
-                <span class="n">msub</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">join</span><span class="p">(</span><span class="n">subs</span><span class="p">))</span>
-                <span class="n">ci</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">msub</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">subs</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">msup</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;mml:msup&#39;</span><span class="p">)</span>
-                <span class="n">msup</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">mname</span><span class="p">)</span>
-                <span class="n">msup</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">join</span><span class="p">(</span><span class="n">supers</span><span class="p">))</span>
-                <span class="n">ci</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">msup</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">msubsup</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;mml:msubsup&#39;</span><span class="p">)</span>
-                <span class="n">msubsup</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">mname</span><span class="p">)</span>
-                <span class="n">msubsup</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">join</span><span class="p">(</span><span class="n">subs</span><span class="p">))</span>
-                <span class="n">msubsup</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">join</span><span class="p">(</span><span class="n">supers</span><span class="p">))</span>
-                <span class="n">ci</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">msubsup</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">ci</span>
-
-    <span class="k">def</span> <span class="nf">_print_Pow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="c">#Here we use root instead of power if the exponent is the reciprocal of an integer</span>
-        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">e</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">p</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;apply&#39;</span><span class="p">)</span>
-            <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;root&#39;</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">q</span> <span class="o">!=</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">xmldeg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;degree&#39;</span><span class="p">)</span>
-                <span class="n">xmlci</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;ci&#39;</span><span class="p">)</span>
-                <span class="n">xmlci</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createTextNode</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">q</span><span class="p">)))</span>
-                <span class="n">xmldeg</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">xmlci</span><span class="p">)</span>
-                <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">xmldeg</span><span class="p">)</span>
-            <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">base</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">x</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;apply&#39;</span><span class="p">)</span>
-        <span class="n">x_1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mathml_tag</span><span class="p">(</span><span class="n">e</span><span class="p">))</span>
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">x_1</span><span class="p">)</span>
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">base</span><span class="p">))</span>
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">exp</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">x</span>
-
-    <span class="k">def</span> <span class="nf">_print_Number</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mathml_tag</span><span class="p">(</span><span class="n">e</span><span class="p">))</span>
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createTextNode</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">e</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">x</span>
-
-    <span class="k">def</span> <span class="nf">_print_Derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;apply&#39;</span><span class="p">)</span>
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mathml_tag</span><span class="p">(</span><span class="n">e</span><span class="p">)))</span>
-
-        <span class="n">x_1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;bvar&#39;</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">sym</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">variables</span><span class="p">:</span>
-            <span class="n">x_1</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">sym</span><span class="p">))</span>
-
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">x_1</span><span class="p">)</span>
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">expr</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">x</span>
-
-    <span class="k">def</span> <span class="nf">_print_Function</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&quot;apply&quot;</span><span class="p">)</span>
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mathml_tag</span><span class="p">(</span><span class="n">e</span><span class="p">)))</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">x</span>
-
-    <span class="k">def</span> <span class="nf">_print_Basic</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mathml_tag</span><span class="p">(</span><span class="n">e</span><span class="p">))</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">e</span><span class="p">:</span>
-            <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">x</span>
-
-    <span class="k">def</span> <span class="nf">_print_AssocOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;apply&#39;</span><span class="p">)</span>
-        <span class="n">x_1</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mathml_tag</span><span class="p">(</span><span class="n">e</span><span class="p">))</span>
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="n">x_1</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">x</span>
-
-    <span class="k">def</span> <span class="nf">_print_Relational</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;apply&#39;</span><span class="p">)</span>
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mathml_tag</span><span class="p">(</span><span class="n">e</span><span class="p">)))</span>
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">lhs</span><span class="p">))</span>
-        <span class="n">x</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">rhs</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">x</span>
-
-    <span class="k">def</span> <span class="nf">_print_list</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">seq</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;MathML reference for the &lt;list&gt; element:</span>
-<span class="sd">        http://www.w3.org/TR/MathML2/chapter4.html#contm.list&quot;&quot;&quot;</span>
-        <span class="n">dom_element</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="s">&#39;list&#39;</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">item</span> <span class="ow">in</span> <span class="n">seq</span><span class="p">:</span>
-            <span class="n">dom_element</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">item</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">dom_element</span>
-
-    <span class="k">def</span> <span class="nf">_print_int</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="n">dom_element</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createElement</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mathml_tag</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
-        <span class="n">dom_element</span><span class="o">.</span><span class="n">appendChild</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">dom</span><span class="o">.</span><span class="n">createTextNode</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">p</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">dom_element</span>
-
-    <span class="k">def</span> <span class="nf">apply_patch</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># Applying the patch of xml.dom.minidom bug</span>
-        <span class="c"># Date: 2011-11-18</span>
-        <span class="c"># Description: http://ronrothman.com/public/leftbraned/xml-dom-minidom-\</span>
-        <span class="c">#                   toprettyxml-and-silly-whitespace/#best-solution</span>
-        <span class="c"># Issue: http://bugs.python.org/issue4147</span>
-        <span class="c"># Patch: http://hg.python.org/cpython/rev/7262f8f276ff/</span>
-
-        <span class="kn">from</span> <span class="nn">xml.dom.minidom</span> <span class="kn">import</span> <span class="n">Element</span><span class="p">,</span> <span class="n">Text</span><span class="p">,</span> <span class="n">Node</span><span class="p">,</span> <span class="n">_write_data</span>
-
-        <span class="k">def</span> <span class="nf">writexml</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">writer</span><span class="p">,</span> <span class="n">indent</span><span class="o">=</span><span class="s">&quot;&quot;</span><span class="p">,</span> <span class="n">addindent</span><span class="o">=</span><span class="s">&quot;&quot;</span><span class="p">,</span> <span class="n">newl</span><span class="o">=</span><span class="s">&quot;&quot;</span><span class="p">):</span>
-            <span class="c"># indent = current indentation</span>
-            <span class="c"># addindent = indentation to add to higher levels</span>
-            <span class="c"># newl = newline string</span>
-            <span class="n">writer</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="n">indent</span> <span class="o">+</span> <span class="s">&quot;&lt;&quot;</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">tagName</span><span class="p">)</span>
-
-            <span class="n">attrs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_attributes</span><span class="p">()</span>
-            <span class="n">a_names</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">attrs</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-            <span class="n">a_names</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-
-            <span class="k">for</span> <span class="n">a_name</span> <span class="ow">in</span> <span class="n">a_names</span><span class="p">:</span>
-                <span class="n">writer</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot; </span><span class="si">%s</span><span class="s">=</span><span class="se">\&quot;</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">a_name</span><span class="p">)</span>
-                <span class="n">_write_data</span><span class="p">(</span><span class="n">writer</span><span class="p">,</span> <span class="n">attrs</span><span class="p">[</span><span class="n">a_name</span><span class="p">]</span><span class="o">.</span><span class="n">value</span><span class="p">)</span>
-                <span class="n">writer</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\&quot;</span><span class="s">&quot;</span><span class="p">)</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">childNodes</span><span class="p">:</span>
-                <span class="n">writer</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;&gt;&quot;</span><span class="p">)</span>
-                <span class="k">if</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">childNodes</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">childNodes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">nodeType</span> <span class="o">==</span> <span class="n">Node</span><span class="o">.</span><span class="n">TEXT_NODE</span><span class="p">):</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">childNodes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">writexml</span><span class="p">(</span><span class="n">writer</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">writer</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="n">newl</span><span class="p">)</span>
-                    <span class="k">for</span> <span class="n">node</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">childNodes</span><span class="p">:</span>
-                        <span class="n">node</span><span class="o">.</span><span class="n">writexml</span><span class="p">(</span>
-                            <span class="n">writer</span><span class="p">,</span> <span class="n">indent</span> <span class="o">+</span> <span class="n">addindent</span><span class="p">,</span> <span class="n">addindent</span><span class="p">,</span> <span class="n">newl</span><span class="p">)</span>
-                    <span class="n">writer</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="n">indent</span><span class="p">)</span>
-                <span class="n">writer</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;&lt;/</span><span class="si">%s</span><span class="s">&gt;</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">tagName</span><span class="p">,</span> <span class="n">newl</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">writer</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;/&gt;</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">newl</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_Element_writexml_old</span> <span class="o">=</span> <span class="n">Element</span><span class="o">.</span><span class="n">writexml</span>
-        <span class="n">Element</span><span class="o">.</span><span class="n">writexml</span> <span class="o">=</span> <span class="n">writexml</span>
-
-        <span class="k">def</span> <span class="nf">writexml</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">writer</span><span class="p">,</span> <span class="n">indent</span><span class="o">=</span><span class="s">&quot;&quot;</span><span class="p">,</span> <span class="n">addindent</span><span class="o">=</span><span class="s">&quot;&quot;</span><span class="p">,</span> <span class="n">newl</span><span class="o">=</span><span class="s">&quot;&quot;</span><span class="p">):</span>
-            <span class="n">_write_data</span><span class="p">(</span><span class="n">writer</span><span class="p">,</span> <span class="s">&quot;</span><span class="si">%s%s%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">indent</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">newl</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_Text_writexml_old</span> <span class="o">=</span> <span class="n">Text</span><span class="o">.</span><span class="n">writexml</span>
-        <span class="n">Text</span><span class="o">.</span><span class="n">writexml</span> <span class="o">=</span> <span class="n">writexml</span>
-
-    <span class="k">def</span> <span class="nf">restore_patch</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">xml.dom.minidom</span> <span class="kn">import</span> <span class="n">Element</span><span class="p">,</span> <span class="n">Text</span>
-        <span class="n">Element</span><span class="o">.</span><span class="n">writexml</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_Element_writexml_old</span>
-        <span class="n">Text</span><span class="o">.</span><span class="n">writexml</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_Text_writexml_old</span>
-
-</div>
-<div class="viewcode-block" id="mathml"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.mathml.mathml">[docs]</a><span class="k">def</span> <span class="nf">mathml</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns the MathML representation of expr&quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">MathMLPrinter</span><span class="p">(</span><span class="n">settings</span><span class="p">)</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="print_mathml"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.mathml.print_mathml">[docs]</a><span class="k">def</span> <span class="nf">print_mathml</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Prints a pretty representation of the MathML code for expr</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; ##</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.printing.mathml import print_mathml</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; print_mathml(x+1) #doctest: +NORMALIZE_WHITESPACE</span>
-<span class="sd">    &lt;apply&gt;</span>
-<span class="sd">        &lt;plus/&gt;</span>
-<span class="sd">        &lt;ci&gt;x&lt;/ci&gt;</span>
-<span class="sd">        &lt;cn&gt;1&lt;/cn&gt;</span>
-<span class="sd">    &lt;/apply&gt;</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="n">MathMLPrinter</span><span class="p">(</span><span class="n">settings</span><span class="p">)</span>
-    <span class="n">xml</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-    <span class="n">s</span><span class="o">.</span><span class="n">apply_patch</span><span class="p">()</span>
-    <span class="n">pretty_xml</span> <span class="o">=</span> <span class="n">xml</span><span class="o">.</span><span class="n">toprettyxml</span><span class="p">()</span>
-    <span class="n">s</span><span class="o">.</span><span class="n">restore_patch</span><span class="p">()</span>
-
-    <span class="k">print</span><span class="p">(</span><span class="n">pretty_xml</span><span class="p">)</span></div>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-<div id="searchbox" style="display: none">
-  <h3>Quick search</h3>
-    <form class="search" action="../../../search.html" method="get">
-      <input type="text" name="q" />
-      <input type="submit" value="Go" />
-      <input type="hidden" name="check_keywords" value="yes" />
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-    </p>
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-        <li class="right" >
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-      Last updated on Jan 20, 2013.
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diff --git a/dev-py3k/_modules/sympy/printing/precedence.html b/dev-py3k/_modules/sympy/printing/precedence.html
deleted file mode 100644
index f5048fb62c9..00000000000
--- a/dev-py3k/_modules/sympy/printing/precedence.html
+++ /dev/null
@@ -1,209 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
-
-
-<html xmlns="http://www.w3.org/1999/xhtml">
-  <head>
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-  <h1>Source code for sympy.printing.precedence</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;A module providing information about the necessity of brackets&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">_coeff_isneg</span>
-
-<span class="c"># Default precedence values for some basic types</span>
-<span class="n">PRECEDENCE</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="s">&quot;Lambda&quot;</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span>
-    <span class="s">&quot;Relational&quot;</span><span class="p">:</span> <span class="mi">20</span><span class="p">,</span>
-    <span class="s">&quot;Or&quot;</span><span class="p">:</span> <span class="mi">20</span><span class="p">,</span>
-    <span class="s">&quot;And&quot;</span><span class="p">:</span> <span class="mi">30</span><span class="p">,</span>
-    <span class="s">&quot;Add&quot;</span><span class="p">:</span> <span class="mi">40</span><span class="p">,</span>
-    <span class="s">&quot;Mul&quot;</span><span class="p">:</span> <span class="mi">50</span><span class="p">,</span>
-    <span class="s">&quot;Pow&quot;</span><span class="p">:</span> <span class="mi">60</span><span class="p">,</span>
-    <span class="s">&quot;Not&quot;</span><span class="p">:</span> <span class="mi">100</span><span class="p">,</span>
-    <span class="s">&quot;Atom&quot;</span><span class="p">:</span> <span class="mi">1000</span>
-<span class="p">}</span>
-
-<span class="c"># A dictionary assigning precedence values to certain classes. These values are</span>
-<span class="c"># treated like they were inherited, so not every single class has to be named</span>
-<span class="c"># here.</span>
-<span class="n">PRECEDENCE_VALUES</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="s">&quot;Or&quot;</span><span class="p">:</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Or&quot;</span><span class="p">],</span>
-    <span class="s">&quot;And&quot;</span><span class="p">:</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;And&quot;</span><span class="p">],</span>
-    <span class="s">&quot;Add&quot;</span><span class="p">:</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Add&quot;</span><span class="p">],</span>
-    <span class="s">&quot;Pow&quot;</span><span class="p">:</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Pow&quot;</span><span class="p">],</span>
-    <span class="s">&quot;Relational&quot;</span><span class="p">:</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Relational&quot;</span><span class="p">],</span>
-    <span class="s">&quot;Sub&quot;</span><span class="p">:</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Add&quot;</span><span class="p">],</span>
-    <span class="s">&quot;Not&quot;</span><span class="p">:</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Not&quot;</span><span class="p">],</span>
-    <span class="s">&quot;factorial&quot;</span><span class="p">:</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Pow&quot;</span><span class="p">],</span>
-    <span class="s">&quot;factorial2&quot;</span><span class="p">:</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Pow&quot;</span><span class="p">],</span>
-    <span class="s">&quot;NegativeInfinity&quot;</span><span class="p">:</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Add&quot;</span><span class="p">],</span>
-    <span class="s">&quot;MatAdd&quot;</span><span class="p">:</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Add&quot;</span><span class="p">],</span>
-    <span class="s">&quot;MatMul&quot;</span><span class="p">:</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Mul&quot;</span><span class="p">],</span>
-    <span class="s">&quot;HadamardProduct&quot;</span><span class="p">:</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Mul&quot;</span><span class="p">]</span>
-<span class="p">}</span>
-
-<span class="c"># Sometimes it&#39;s not enough to assign a fixed precedence value to a</span>
-<span class="c"># class. Then a function can be inserted in this dictionary that takes</span>
-<span class="c"># an instance of this class as argument and returns the appropriate</span>
-<span class="c"># precedence value.</span>
-
-<span class="c"># Precedence functions</span>
-
-
-<span class="k">def</span> <span class="nf">precedence_Mul</span><span class="p">(</span><span class="n">item</span><span class="p">):</span>
-    <span class="k">if</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">item</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Add&quot;</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Mul&quot;</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">precedence_Rational</span><span class="p">(</span><span class="n">item</span><span class="p">):</span>
-    <span class="k">if</span> <span class="n">item</span><span class="o">.</span><span class="n">p</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Add&quot;</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Mul&quot;</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">precedence_Integer</span><span class="p">(</span><span class="n">item</span><span class="p">):</span>
-    <span class="k">if</span> <span class="n">item</span><span class="o">.</span><span class="n">p</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Add&quot;</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Atom&quot;</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">precedence_Float</span><span class="p">(</span><span class="n">item</span><span class="p">):</span>
-    <span class="k">if</span> <span class="n">item</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Add&quot;</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Atom&quot;</span><span class="p">]</span>
-
-<span class="n">PRECEDENCE_FUNCTIONS</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="s">&quot;Integer&quot;</span><span class="p">:</span> <span class="n">precedence_Integer</span><span class="p">,</span>
-    <span class="s">&quot;Mul&quot;</span><span class="p">:</span> <span class="n">precedence_Mul</span><span class="p">,</span>
-    <span class="s">&quot;Rational&quot;</span><span class="p">:</span> <span class="n">precedence_Rational</span><span class="p">,</span>
-    <span class="s">&quot;Float&quot;</span><span class="p">:</span> <span class="n">precedence_Float</span><span class="p">,</span>
-<span class="p">}</span>
-
-
-<div class="viewcode-block" id="precedence"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.precedence.precedence">[docs]</a><span class="k">def</span> <span class="nf">precedence</span><span class="p">(</span><span class="n">item</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the precedence of a given object.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">item</span><span class="p">,</span> <span class="s">&quot;precedence&quot;</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">item</span><span class="o">.</span><span class="n">precedence</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">mro</span> <span class="o">=</span> <span class="n">item</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__mro__</span>
-    <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Atom&quot;</span><span class="p">]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">mro</span><span class="p">:</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">i</span><span class="o">.</span><span class="n">__name__</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">PRECEDENCE_FUNCTIONS</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">PRECEDENCE_FUNCTIONS</span><span class="p">[</span><span class="n">n</span><span class="p">](</span><span class="n">item</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">PRECEDENCE_VALUES</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">PRECEDENCE_VALUES</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Atom&quot;</span><span class="p">]</span></div>
-</pre></div>
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deleted file mode 100644
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+++ /dev/null
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-  <h1>Source code for sympy.printing.pretty.pretty</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">C</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">_coeff_isneg</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">group</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">has_variety</span>
-<span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="n">SympifyError</span>
-
-<span class="kn">from</span> <span class="nn">sympy.printing.printer</span> <span class="kn">import</span> <span class="n">Printer</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.str</span> <span class="kn">import</span> <span class="n">sstr</span>
-
-<span class="kn">from</span> <span class="nn">.stringpict</span> <span class="kn">import</span> <span class="n">prettyForm</span><span class="p">,</span> <span class="n">stringPict</span>
-<span class="kn">from</span> <span class="nn">.pretty_symbology</span> <span class="kn">import</span> <span class="n">xstr</span><span class="p">,</span> <span class="n">hobj</span><span class="p">,</span> <span class="n">vobj</span><span class="p">,</span> <span class="n">xobj</span><span class="p">,</span> <span class="n">xsym</span><span class="p">,</span> <span class="n">pretty_symbol</span><span class="p">,</span> \
-    <span class="n">pretty_atom</span><span class="p">,</span> <span class="n">pretty_use_unicode</span><span class="p">,</span> <span class="n">pretty_try_use_unicode</span><span class="p">,</span> <span class="n">greek</span><span class="p">,</span> <span class="n">U</span><span class="p">,</span> \
-    <span class="n">annotated</span>
-
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">default_sort_key</span>
-
-<span class="c"># rename for usage from outside</span>
-<span class="n">pprint_use_unicode</span> <span class="o">=</span> <span class="n">pretty_use_unicode</span>
-<span class="n">pprint_try_use_unicode</span> <span class="o">=</span> <span class="n">pretty_try_use_unicode</span>
-
-
-<div class="viewcode-block" id="PrettyPrinter"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.pretty.PrettyPrinter">[docs]</a><span class="k">class</span> <span class="nc">PrettyPrinter</span><span class="p">(</span><span class="n">Printer</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Printer, which converts an expression into 2D ASCII-art figure.&quot;&quot;&quot;</span>
-    <span class="n">printmethod</span> <span class="o">=</span> <span class="s">&quot;_pretty&quot;</span>
-
-    <span class="n">_default_settings</span> <span class="o">=</span> <span class="p">{</span>
-        <span class="s">&quot;order&quot;</span><span class="p">:</span> <span class="bp">None</span><span class="p">,</span>
-        <span class="s">&quot;full_prec&quot;</span><span class="p">:</span> <span class="s">&quot;auto&quot;</span><span class="p">,</span>
-        <span class="s">&quot;use_unicode&quot;</span><span class="p">:</span> <span class="bp">None</span><span class="p">,</span>
-        <span class="s">&quot;wrap_line&quot;</span><span class="p">:</span> <span class="bp">True</span><span class="p">,</span>
-        <span class="s">&quot;num_columns&quot;</span><span class="p">:</span> <span class="bp">None</span><span class="p">,</span>
-    <span class="p">}</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">settings</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">Printer</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">settings</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">emptyPrinter</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">xstr</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_use_unicode</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;use_unicode&#39;</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">pretty_use_unicode</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">doprint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">render</span><span class="p">(</span><span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">)</span>
-
-    <span class="c"># empty op so _print(stringPict) returns the same</span>
-    <span class="k">def</span> <span class="nf">_print_stringPict</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">e</span>
-
-    <span class="k">def</span> <span class="nf">_print_basestring</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_atan2</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">)</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39;atan2&#39;</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_Symbol</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">symb</span> <span class="o">=</span> <span class="n">pretty_symbol</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">symb</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Float</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="c"># we will use StrPrinter&#39;s Float printer, but we need to handle the</span>
-        <span class="c"># full_prec ourselves, according to the self._print_level</span>
-        <span class="n">full_prec</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&quot;full_prec&quot;</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">full_prec</span> <span class="o">==</span> <span class="s">&quot;auto&quot;</span><span class="p">:</span>
-            <span class="n">full_prec</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_level</span> <span class="o">==</span> <span class="mi">1</span>
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">sstr</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">full_prec</span><span class="o">=</span><span class="n">full_prec</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Atom</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="c"># print atoms like Exp1 or Pi</span>
-            <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">pretty_atom</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">))</span>
-        <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">emptyPrinter</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-
-    <span class="c"># Infinity inherits from Number, so we have to override _print_XXX order</span>
-    <span class="n">_print_Infinity</span> <span class="o">=</span> <span class="n">_print_Atom</span>
-    <span class="n">_print_NegativeInfinity</span> <span class="o">=</span> <span class="n">_print_Atom</span>
-    <span class="n">_print_EmptySet</span> <span class="o">=</span> <span class="n">_print_Atom</span>
-    <span class="n">_print_Naturals</span> <span class="o">=</span> <span class="n">_print_Atom</span>
-    <span class="n">_print_Integers</span> <span class="o">=</span> <span class="n">_print_Atom</span>
-    <span class="n">_print_Reals</span> <span class="o">=</span> <span class="n">_print_Atom</span>
-
-    <span class="k">def</span> <span class="nf">_print_subfactorial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="c"># Add parentheses if needed</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">((</span><span class="n">x</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">x</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">)</span> <span class="ow">or</span> <span class="n">x</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">):</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39;!&#39;</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_factorial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="c"># Add parentheses if needed</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">((</span><span class="n">x</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">x</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">)</span> <span class="ow">or</span> <span class="n">x</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">):</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39;!&#39;</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_factorial2</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="c"># Add parentheses if needed</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">((</span><span class="n">x</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">x</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">)</span> <span class="ow">or</span> <span class="n">x</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">):</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39;!!&#39;</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_binomial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span>
-
-        <span class="n">n_pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="n">k_pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-
-        <span class="n">bar</span> <span class="o">=</span> <span class="s">&#39; &#39;</span><span class="o">*</span><span class="nb">max</span><span class="p">(</span><span class="n">n_pform</span><span class="o">.</span><span class="n">width</span><span class="p">(),</span> <span class="n">k_pform</span><span class="o">.</span><span class="n">width</span><span class="p">())</span>
-
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">k_pform</span><span class="o">.</span><span class="n">above</span><span class="p">(</span><span class="n">bar</span><span class="p">))</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">above</span><span class="p">(</span><span class="n">n_pform</span><span class="p">))</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">parens</span><span class="p">(</span><span class="s">&#39;(&#39;</span><span class="p">,</span> <span class="s">&#39;)&#39;</span><span class="p">))</span>
-
-        <span class="n">pform</span><span class="o">.</span><span class="n">baseline</span> <span class="o">=</span> <span class="p">(</span><span class="n">pform</span><span class="o">.</span><span class="n">baseline</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span>
-
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_Relational</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">op</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39; &#39;</span> <span class="o">+</span> <span class="n">xsym</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">rel_op</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39; &#39;</span><span class="p">)</span>
-
-        <span class="n">l</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">lhs</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">stringPict</span><span class="o">.</span><span class="n">next</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">op</span><span class="p">,</span> <span class="n">r</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_Not</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="n">arg</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Boolean</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Not</span><span class="p">:</span>
-                <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-
-            <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\u00ac</span><span class="s"> &quot;</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Function</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__print_Boolean</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">char</span><span class="p">,</span> <span class="n">sort</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">if</span> <span class="n">sort</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Boolean</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Not</span><span class="p">:</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="n">pform_arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Boolean</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Not</span><span class="p">:</span>
-                <span class="n">pform_arg</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform_arg</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; </span><span class="si">%s</span><span class="s"> &#39;</span> <span class="o">%</span> <span class="n">char</span><span class="p">))</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">pform_arg</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_And</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__print_Boolean</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u2227</span><span class="s">&quot;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Function</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">sort</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Or</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__print_Boolean</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u2228</span><span class="s">&quot;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Function</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">sort</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Xor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__print_Boolean</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u22bb</span><span class="s">&quot;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Function</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">sort</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Nand</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__print_Boolean</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u22bc</span><span class="s">&quot;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Function</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">sort</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Nor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__print_Boolean</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u22bd</span><span class="s">&quot;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Function</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">sort</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Implies</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__print_Boolean</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u2192</span><span class="s">&quot;</span><span class="p">,</span> <span class="n">sort</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Function</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Equivalent</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__print_Boolean</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="s">&quot;</span><span class="se">\u2261</span><span class="s">&quot;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Function</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">sort</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span> <span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">above</span><span class="p">(</span> <span class="n">hobj</span><span class="p">(</span><span class="s">&#39;_&#39;</span><span class="p">,</span> <span class="n">pform</span><span class="o">.</span><span class="n">width</span><span class="p">()))</span> <span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Abs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">parens</span><span class="p">(</span><span class="s">&#39;|&#39;</span><span class="p">,</span> <span class="s">&#39;|&#39;</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_floor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">parens</span><span class="p">(</span><span class="s">&#39;lfloor&#39;</span><span class="p">,</span> <span class="s">&#39;rfloor&#39;</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">pform</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Function</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_ceiling</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">parens</span><span class="p">(</span><span class="s">&#39;lceil&#39;</span><span class="p">,</span> <span class="s">&#39;rceil&#39;</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">pform</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Function</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">deriv</span><span class="p">):</span>
-        <span class="c"># XXX use U(&#39;PARTIAL DIFFERENTIAL&#39;) here ?</span>
-        <span class="n">syms</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">deriv</span><span class="o">.</span><span class="n">variables</span><span class="p">))</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="bp">None</span>
-
-        <span class="k">for</span> <span class="n">sym</span><span class="p">,</span> <span class="n">num</span> <span class="ow">in</span> <span class="n">group</span><span class="p">(</span><span class="n">syms</span><span class="p">,</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">sym</span><span class="p">)</span>
-            <span class="n">ds</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39;d&#39;</span><span class="p">))</span>
-
-            <span class="k">if</span> <span class="n">num</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">ds</span> <span class="o">=</span> <span class="n">ds</span><span class="o">**</span><span class="n">prettyForm</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">num</span><span class="p">))</span>
-
-            <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="n">ds</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">x</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">))</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">x</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">ds</span><span class="p">))</span>
-
-        <span class="n">f</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span>
-            <span class="n">binding</span><span class="o">=</span><span class="n">prettyForm</span><span class="o">.</span><span class="n">FUNC</span><span class="p">,</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">deriv</span><span class="o">.</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;d&#39;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">syms</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">pform</span><span class="o">**</span><span class="n">prettyForm</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">syms</span><span class="p">)))</span>
-
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="n">stringPict</span><span class="o">.</span><span class="n">LINE</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-        <span class="n">pform</span><span class="o">.</span><span class="n">baseline</span> <span class="o">=</span> <span class="n">pform</span><span class="o">.</span><span class="n">baseline</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">stringPict</span><span class="o">.</span><span class="n">next</span><span class="p">(</span><span class="n">pform</span><span class="p">,</span> <span class="n">f</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_Cycle</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">dc</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_tuple</span><span class="p">(</span><span class="n">Permutation</span><span class="p">(</span><span class="n">dc</span><span class="o">.</span><span class="n">as_list</span><span class="p">())</span><span class="o">.</span><span class="n">cyclic_form</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_PDF</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">pdf</span><span class="p">):</span>
-        <span class="n">lim</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">pdf</span><span class="o">.</span><span class="n">pdf</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="n">lim</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">lim</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39;, &#39;</span><span class="p">))</span>
-        <span class="n">lim</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">lim</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">pdf</span><span class="o">.</span><span class="n">domain</span><span class="p">[</span><span class="mi">0</span><span class="p">])))</span>
-        <span class="n">lim</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">lim</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39;, &#39;</span><span class="p">))</span>
-        <span class="n">lim</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">lim</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">pdf</span><span class="o">.</span><span class="n">domain</span><span class="p">[</span><span class="mi">1</span><span class="p">])))</span>
-        <span class="n">lim</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">lim</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-
-        <span class="n">f</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">pdf</span><span class="o">.</span><span class="n">pdf</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39;, &#39;</span><span class="p">))</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">lim</span><span class="p">))</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;PDF&#39;</span><span class="p">)</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">f</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_Integral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">integral</span><span class="p">):</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">integral</span><span class="o">.</span><span class="n">function</span>
-
-        <span class="c"># Add parentheses if arg involves addition of terms and</span>
-        <span class="c"># create a pretty form for the argument</span>
-        <span class="n">prettyF</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-        <span class="c"># XXX generalize parens</span>
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">prettyF</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">prettyF</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-
-        <span class="c"># dx dy dz ...</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">prettyF</span>
-        <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">integral</span><span class="o">.</span><span class="n">limits</span><span class="p">:</span>
-            <span class="n">prettyArg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="c"># XXX qparens (parens if needs-parens)</span>
-            <span class="k">if</span> <span class="n">prettyArg</span><span class="o">.</span><span class="n">width</span><span class="p">()</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">prettyArg</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">prettyArg</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-
-            <span class="n">arg</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">arg</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; d&#39;</span><span class="p">,</span> <span class="n">prettyArg</span><span class="p">))</span>
-
-        <span class="c"># \int \int \int ...</span>
-        <span class="n">firstterm</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">for</span> <span class="n">lim</span> <span class="ow">in</span> <span class="n">integral</span><span class="o">.</span><span class="n">limits</span><span class="p">:</span>
-            <span class="n">x</span> <span class="o">=</span> <span class="n">lim</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="c"># Create bar based on the height of the argument</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">height</span><span class="p">()</span>
-            <span class="n">H</span> <span class="o">=</span> <span class="n">h</span> <span class="o">+</span> <span class="mi">2</span>
-
-            <span class="c"># XXX hack!</span>
-            <span class="n">ascii_mode</span> <span class="o">=</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span>
-            <span class="k">if</span> <span class="n">ascii_mode</span><span class="p">:</span>
-                <span class="n">H</span> <span class="o">+=</span> <span class="mi">2</span>
-
-            <span class="n">vint</span> <span class="o">=</span> <span class="n">vobj</span><span class="p">(</span><span class="s">&#39;int&#39;</span><span class="p">,</span> <span class="n">H</span><span class="p">)</span>
-
-            <span class="c"># Construct the pretty form with the integral sign and the argument</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">vint</span><span class="p">)</span>
-            <span class="c">#pform.baseline = pform.height()//2  # vcenter</span>
-            <span class="n">pform</span><span class="o">.</span><span class="n">baseline</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">baseline</span> <span class="o">+</span> <span class="p">(</span>
-                <span class="n">H</span> <span class="o">-</span> <span class="n">h</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span>    <span class="c"># covering the whole argument</span>
-
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">lim</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="c"># Create pretty forms for endpoints, if definite integral.</span>
-                <span class="c"># Do not print empty endpoints.</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">lim</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                    <span class="n">prettyA</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&quot;&quot;</span><span class="p">)</span>
-                    <span class="n">prettyB</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">lim</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">lim</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-                    <span class="n">prettyA</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">lim</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-                    <span class="n">prettyB</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">lim</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-
-                <span class="k">if</span> <span class="n">ascii_mode</span><span class="p">:</span>  <span class="c"># XXX hack</span>
-                    <span class="c"># Add spacing so that endpoint can more easily be</span>
-                    <span class="c"># identified with the correct integral sign</span>
-                    <span class="n">spc</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span> <span class="o">-</span> <span class="n">prettyB</span><span class="o">.</span><span class="n">width</span><span class="p">())</span>
-                    <span class="n">prettyB</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">prettyB</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39; &#39;</span> <span class="o">*</span> <span class="n">spc</span><span class="p">))</span>
-
-                    <span class="n">spc</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span> <span class="o">-</span> <span class="n">prettyA</span><span class="o">.</span><span class="n">width</span><span class="p">())</span>
-                    <span class="n">prettyA</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">prettyA</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span> <span class="o">*</span> <span class="n">spc</span><span class="p">))</span>
-
-                <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">above</span><span class="p">(</span><span class="n">prettyB</span><span class="p">))</span>
-                <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="n">prettyA</span><span class="p">))</span>
-
-                <span class="c">#if ascii_mode:  # XXX hack</span>
-                <span class="c">#    # too much vspace beetween \int and argument</span>
-                <span class="c">#    # but I left it as is</span>
-                <span class="c">#    pform = prettyForm(*pform.right(&#39; &#39;))</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">ascii_mode</span><span class="p">:</span>  <span class="c"># XXX hack</span>
-                <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">))</span>
-
-            <span class="k">if</span> <span class="n">firstterm</span><span class="p">:</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">pform</span>   <span class="c"># first term</span>
-                <span class="n">firstterm</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="n">pform</span><span class="p">))</span>
-
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">arg</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="n">s</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_Product</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">func</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">term</span>
-        <span class="n">pretty_func</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">func</span><span class="p">)</span>
-
-        <span class="n">horizontal_chr</span> <span class="o">=</span> <span class="n">xobj</span><span class="p">(</span><span class="s">&#39;_&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">corner_chr</span> <span class="o">=</span> <span class="n">xobj</span><span class="p">(</span><span class="s">&#39;_&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">vertical_chr</span> <span class="o">=</span> <span class="n">xobj</span><span class="p">(</span><span class="s">&#39;|&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="c"># use unicode corners</span>
-            <span class="n">horizontal_chr</span> <span class="o">=</span> <span class="n">xobj</span><span class="p">(</span><span class="s">&#39;-&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="n">corner_chr</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\u252c</span><span class="s">&#39;</span>
-
-        <span class="n">func_height</span> <span class="o">=</span> <span class="n">pretty_func</span><span class="o">.</span><span class="n">height</span><span class="p">()</span>
-
-        <span class="n">first</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="n">max_upper</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">sign_height</span> <span class="o">=</span> <span class="mi">0</span>
-
-        <span class="k">for</span> <span class="n">lim</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">limits</span><span class="p">:</span>
-            <span class="n">width</span> <span class="o">=</span> <span class="p">(</span><span class="n">func_height</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="mi">5</span> <span class="o">//</span> <span class="mi">3</span> <span class="o">-</span> <span class="mi">2</span>
-            <span class="n">sign_lines</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">sign_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">corner_chr</span> <span class="o">+</span> <span class="p">(</span><span class="n">horizontal_chr</span><span class="o">*</span><span class="n">width</span><span class="p">)</span> <span class="o">+</span> <span class="n">corner_chr</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">func_height</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="n">sign_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">vertical_chr</span> <span class="o">+</span> <span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">width</span><span class="p">)</span> <span class="o">+</span> <span class="n">vertical_chr</span><span class="p">)</span>
-
-            <span class="n">pretty_sign</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="s">&#39;&#39;</span><span class="p">)</span>
-            <span class="n">pretty_sign</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pretty_sign</span><span class="o">.</span><span class="n">stack</span><span class="p">(</span><span class="o">*</span><span class="n">sign_lines</span><span class="p">))</span>
-
-            <span class="n">pretty_upper</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">lim</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-            <span class="n">pretty_lower</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Equality</span><span class="p">(</span><span class="n">lim</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">lim</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-
-            <span class="n">max_upper</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">max_upper</span><span class="p">,</span> <span class="n">pretty_upper</span><span class="o">.</span><span class="n">height</span><span class="p">())</span>
-
-            <span class="k">if</span> <span class="n">first</span><span class="p">:</span>
-                <span class="n">sign_height</span> <span class="o">=</span> <span class="n">pretty_sign</span><span class="o">.</span><span class="n">height</span><span class="p">()</span>
-
-            <span class="n">pretty_sign</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pretty_sign</span><span class="o">.</span><span class="n">above</span><span class="p">(</span><span class="n">pretty_upper</span><span class="p">))</span>
-            <span class="n">pretty_sign</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pretty_sign</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="n">pretty_lower</span><span class="p">))</span>
-
-            <span class="k">if</span> <span class="n">first</span><span class="p">:</span>
-                <span class="n">pretty_func</span><span class="o">.</span><span class="n">baseline</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="n">first</span> <span class="o">=</span> <span class="bp">False</span>
-
-            <span class="n">height</span> <span class="o">=</span> <span class="n">pretty_sign</span><span class="o">.</span><span class="n">height</span><span class="p">()</span>
-            <span class="n">padding</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="s">&#39;&#39;</span><span class="p">)</span>
-            <span class="n">padding</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">padding</span><span class="o">.</span><span class="n">stack</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="s">&#39; &#39;</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">height</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)))</span>
-            <span class="n">pretty_sign</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pretty_sign</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">padding</span><span class="p">))</span>
-
-            <span class="n">pretty_func</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pretty_sign</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">pretty_func</span><span class="p">))</span>
-
-        <span class="c">#pretty_func.baseline = 0</span>
-
-        <span class="n">pretty_func</span><span class="o">.</span><span class="n">baseline</span> <span class="o">=</span> <span class="n">max_upper</span> <span class="o">+</span> <span class="n">sign_height</span><span class="o">//</span><span class="mi">2</span>
-        <span class="k">return</span> <span class="n">pretty_func</span>
-
-    <span class="k">def</span> <span class="nf">_print_Sum</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">ascii_mode</span> <span class="o">=</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span>
-
-        <span class="k">def</span> <span class="nf">asum</span><span class="p">(</span><span class="n">hrequired</span><span class="p">,</span> <span class="n">lower</span><span class="p">,</span> <span class="n">upper</span><span class="p">,</span> <span class="n">use_ascii</span><span class="p">):</span>
-            <span class="k">def</span> <span class="nf">adjust</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">wid</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">how</span><span class="o">=</span><span class="s">&#39;&lt;^&gt;&#39;</span><span class="p">):</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">wid</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">wid</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">s</span>
-                <span class="n">need</span> <span class="o">=</span> <span class="n">wid</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">how</span> <span class="o">==</span> <span class="s">&#39;&lt;^&gt;&#39;</span> <span class="ow">or</span> <span class="n">how</span> <span class="o">==</span> <span class="s">&quot;&lt;&quot;</span> <span class="ow">or</span> <span class="n">how</span> <span class="ow">not</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="s">&#39;&lt;^&gt;&#39;</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">s</span> <span class="o">+</span> <span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">need</span>
-                <span class="n">half</span> <span class="o">=</span> <span class="n">need</span><span class="o">//</span><span class="mi">2</span>
-                <span class="n">lead</span> <span class="o">=</span> <span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">half</span>
-                <span class="k">if</span> <span class="n">how</span> <span class="o">==</span> <span class="s">&quot;&gt;&quot;</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="s">&quot; &quot;</span><span class="o">*</span><span class="n">need</span> <span class="o">+</span> <span class="n">s</span>
-                <span class="k">return</span> <span class="n">lead</span> <span class="o">+</span> <span class="n">s</span> <span class="o">+</span> <span class="s">&#39; &#39;</span><span class="o">*</span><span class="p">(</span><span class="n">need</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">lead</span><span class="p">))</span>
-
-            <span class="n">h</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">hrequired</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="n">h</span><span class="o">//</span><span class="mi">2</span>
-            <span class="n">w</span> <span class="o">=</span> <span class="n">d</span> <span class="o">+</span> <span class="mi">1</span>
-            <span class="n">more</span> <span class="o">=</span> <span class="n">hrequired</span> <span class="o">%</span> <span class="mi">2</span>
-
-            <span class="n">lines</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">if</span> <span class="n">use_ascii</span><span class="p">:</span>
-                <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;_&quot;</span><span class="o">*</span><span class="p">(</span><span class="n">w</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39; &#39;</span><span class="p">)</span>
-                <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;\</span><span class="si">%s</span><span class="s">`&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="p">(</span><span class="n">w</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)))</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">d</span><span class="p">):</span>
-                    <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="se">\\</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">i</span><span class="p">,</span> <span class="s">&#39; &#39;</span><span class="o">*</span><span class="p">(</span><span class="n">w</span> <span class="o">-</span> <span class="n">i</span><span class="p">)))</span>
-                <span class="k">if</span> <span class="n">more</span><span class="p">:</span>
-                    <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s">)</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="p">(</span><span class="n">d</span><span class="p">),</span> <span class="s">&#39; &#39;</span><span class="o">*</span><span class="p">(</span><span class="n">w</span> <span class="o">-</span> <span class="n">d</span><span class="p">)))</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">d</span><span class="p">))):</span>
-                    <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s">/</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">i</span><span class="p">,</span> <span class="s">&#39; &#39;</span><span class="o">*</span><span class="p">(</span><span class="n">w</span> <span class="o">-</span> <span class="n">i</span><span class="p">)))</span>
-                <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;/&quot;</span> <span class="o">+</span> <span class="s">&quot;_&quot;</span><span class="o">*</span><span class="p">(</span><span class="n">w</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39;,&#39;</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">d</span><span class="p">,</span> <span class="n">h</span> <span class="o">+</span> <span class="n">more</span><span class="p">,</span> <span class="n">lines</span><span class="p">,</span> <span class="mi">0</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">w</span> <span class="o">=</span> <span class="n">w</span> <span class="o">+</span> <span class="n">more</span>
-                <span class="n">d</span> <span class="o">=</span> <span class="n">d</span> <span class="o">+</span> <span class="n">more</span>
-                <span class="n">vsum</span> <span class="o">=</span> <span class="n">vobj</span><span class="p">(</span><span class="s">&#39;sum&#39;</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-                <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;_&quot;</span><span class="o">*</span><span class="p">(</span><span class="n">w</span><span class="p">))</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">d</span><span class="p">):</span>
-                    <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s%s%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">i</span><span class="p">,</span> <span class="n">vsum</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="s">&#39; &#39;</span><span class="o">*</span><span class="p">(</span><span class="n">w</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)))</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">d</span><span class="p">))):</span>
-                    <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s%s%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">i</span><span class="p">,</span> <span class="n">vsum</span><span class="p">[</span><span class="mi">4</span><span class="p">],</span> <span class="s">&#39; &#39;</span><span class="o">*</span><span class="p">(</span><span class="n">w</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)))</span>
-                <span class="n">lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">vsum</span><span class="p">[</span><span class="mi">8</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">w</span><span class="p">))</span>
-                <span class="k">return</span> <span class="n">d</span><span class="p">,</span> <span class="n">h</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">more</span><span class="p">,</span> <span class="n">lines</span><span class="p">,</span> <span class="n">more</span>
-
-        <span class="n">f</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">function</span>
-
-        <span class="n">prettyF</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>  <span class="c"># add parens</span>
-            <span class="n">prettyF</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">prettyF</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-
-        <span class="n">H</span> <span class="o">=</span> <span class="n">prettyF</span><span class="o">.</span><span class="n">height</span><span class="p">()</span> <span class="o">+</span> <span class="mi">2</span>
-
-        <span class="c"># \sum \sum \sum ...</span>
-        <span class="n">first</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="n">max_upper</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">sign_height</span> <span class="o">=</span> <span class="mi">0</span>
-
-        <span class="k">for</span> <span class="n">lim</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">limits</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">lim</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-                <span class="n">prettyUpper</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">lim</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-                <span class="n">prettyLower</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Equality</span><span class="p">(</span><span class="n">lim</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">lim</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-            <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">lim</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="n">prettyUpper</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="s">&quot;&quot;</span><span class="p">)</span>
-                <span class="n">prettyLower</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Equality</span><span class="p">(</span><span class="n">lim</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">lim</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-            <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">lim</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">prettyUpper</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="s">&quot;&quot;</span><span class="p">)</span>
-                <span class="n">prettyLower</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">lim</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-            <span class="n">max_upper</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">max_upper</span><span class="p">,</span> <span class="n">prettyUpper</span><span class="o">.</span><span class="n">height</span><span class="p">())</span>
-
-            <span class="c"># Create sum sign based on the height of the argument</span>
-            <span class="n">d</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">slines</span><span class="p">,</span> <span class="n">adjustment</span> <span class="o">=</span> <span class="n">asum</span><span class="p">(</span>
-                <span class="n">H</span><span class="p">,</span> <span class="n">prettyLower</span><span class="o">.</span><span class="n">width</span><span class="p">(),</span> <span class="n">prettyUpper</span><span class="o">.</span><span class="n">width</span><span class="p">(),</span> <span class="n">ascii_mode</span><span class="p">)</span>
-            <span class="n">prettySign</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="s">&#39;&#39;</span><span class="p">)</span>
-            <span class="n">prettySign</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">prettySign</span><span class="o">.</span><span class="n">stack</span><span class="p">(</span><span class="o">*</span><span class="n">slines</span><span class="p">))</span>
-
-            <span class="k">if</span> <span class="n">first</span><span class="p">:</span>
-                <span class="n">sign_height</span> <span class="o">=</span> <span class="n">prettySign</span><span class="o">.</span><span class="n">height</span><span class="p">()</span>
-
-            <span class="n">prettySign</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">prettySign</span><span class="o">.</span><span class="n">above</span><span class="p">(</span><span class="n">prettyUpper</span><span class="p">))</span>
-            <span class="n">prettySign</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">prettySign</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="n">prettyLower</span><span class="p">))</span>
-
-            <span class="k">if</span> <span class="n">first</span><span class="p">:</span>
-                <span class="c"># change F baseline so it centers on the sign</span>
-                <span class="n">prettyF</span><span class="o">.</span><span class="n">baseline</span> <span class="o">-=</span> <span class="n">d</span> <span class="o">-</span> <span class="p">(</span><span class="n">prettyF</span><span class="o">.</span><span class="n">height</span><span class="p">()</span><span class="o">//</span><span class="mi">2</span> <span class="o">-</span>
-                                         <span class="n">prettyF</span><span class="o">.</span><span class="n">baseline</span><span class="p">)</span> <span class="o">-</span> <span class="n">adjustment</span>
-                <span class="n">first</span> <span class="o">=</span> <span class="bp">False</span>
-
-            <span class="c"># put padding to the right</span>
-            <span class="n">pad</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="s">&#39;&#39;</span><span class="p">)</span>
-            <span class="n">pad</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pad</span><span class="o">.</span><span class="n">stack</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="s">&#39; &#39;</span><span class="p">]</span><span class="o">*</span><span class="n">h</span><span class="p">))</span>
-            <span class="n">prettySign</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">prettySign</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">pad</span><span class="p">))</span>
-            <span class="c"># put the present prettyF to the right</span>
-            <span class="n">prettyF</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">prettySign</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">prettyF</span><span class="p">))</span>
-
-        <span class="n">prettyF</span><span class="o">.</span><span class="n">baseline</span> <span class="o">=</span> <span class="n">max_upper</span> <span class="o">+</span> <span class="n">sign_height</span><span class="o">//</span><span class="mi">2</span>
-        <span class="k">return</span> <span class="n">prettyF</span>
-
-    <span class="k">def</span> <span class="nf">_print_Limit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">l</span><span class="p">):</span>
-        <span class="c"># XXX we do not print dir ...</span>
-        <span class="n">e</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="nb">dir</span> <span class="o">=</span> <span class="n">l</span><span class="o">.</span><span class="n">args</span>
-
-        <span class="n">E</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-        <span class="n">Lim</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;lim&#39;</span><span class="p">)</span>
-
-        <span class="n">LimArg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-        <span class="n">LimArg</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">LimArg</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39;-&gt;&#39;</span><span class="p">))</span>
-        <span class="n">LimArg</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">LimArg</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">z0</span><span class="p">)))</span>
-
-        <span class="n">Lim</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">Lim</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="n">LimArg</span><span class="p">))</span>
-        <span class="n">Lim</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">Lim</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">E</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">Lim</span>
-
-    <span class="k">def</span> <span class="nf">_print_matrix_contents</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        This method factors out what is essentially grid printing.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">M</span> <span class="o">=</span> <span class="n">e</span>   <span class="c"># matrix</span>
-        <span class="n">Ms</span> <span class="o">=</span> <span class="p">{}</span>  <span class="c"># i,j -&gt; pretty(M[i,j])</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-                <span class="n">Ms</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">])</span>
-
-        <span class="c"># h- and v- spacers</span>
-        <span class="n">hsep</span> <span class="o">=</span> <span class="mi">2</span>
-        <span class="n">vsep</span> <span class="o">=</span> <span class="mi">1</span>
-
-        <span class="c"># max width for columns</span>
-        <span class="n">maxw</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">M</span><span class="o">.</span><span class="n">cols</span>
-
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-            <span class="n">maxw</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">([</span><span class="n">Ms</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">width</span><span class="p">()</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">rows</span><span class="p">)]</span> <span class="ow">or</span> <span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="c"># drawing result</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="bp">None</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-
-            <span class="n">D_row</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">Ms</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-
-                <span class="c"># reshape s to maxw</span>
-                <span class="c"># XXX this should be generalized, and go to stringPict.reshape ?</span>
-                <span class="k">assert</span> <span class="n">s</span><span class="o">.</span><span class="n">width</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="n">maxw</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-
-                <span class="c"># hcenter it, +0.5 to the right                        2</span>
-                <span class="c"># ( it&#39;s better to align formula starts for say 0 and r )</span>
-                <span class="c"># XXX this is not good in all cases -- maybe introduce vbaseline?</span>
-                <span class="n">wdelta</span> <span class="o">=</span> <span class="n">maxw</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">s</span><span class="o">.</span><span class="n">width</span><span class="p">()</span>
-                <span class="n">wleft</span> <span class="o">=</span> <span class="n">wdelta</span> <span class="o">//</span> <span class="mi">2</span>
-                <span class="n">wright</span> <span class="o">=</span> <span class="n">wdelta</span> <span class="o">-</span> <span class="n">wleft</span>
-
-                <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">wright</span><span class="p">))</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">wleft</span><span class="p">))</span>
-
-                <span class="c"># we don&#39;t need vcenter cells -- this is automatically done in</span>
-                <span class="c"># a pretty way because when their baselines are taking into</span>
-                <span class="c"># account in .right()</span>
-
-                <span class="k">if</span> <span class="n">D_row</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">D_row</span> <span class="o">=</span> <span class="n">s</span>   <span class="c"># first box in a row</span>
-                    <span class="k">continue</span>
-
-                <span class="n">D_row</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D_row</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">hsep</span><span class="p">))</span>  <span class="c"># h-spacer</span>
-                <span class="n">D_row</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D_row</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">s</span><span class="p">))</span>
-
-            <span class="k">if</span> <span class="n">D</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">D</span> <span class="o">=</span> <span class="n">D_row</span>       <span class="c"># first row in a picture</span>
-                <span class="k">continue</span>
-
-            <span class="c"># v-spacer</span>
-            <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">vsep</span><span class="p">):</span>
-                <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">))</span>
-
-            <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="n">D_row</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">D</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;&#39;</span><span class="p">)</span>  <span class="c"># Empty Matrix</span>
-
-        <span class="k">return</span> <span class="n">D</span>
-
-    <span class="k">def</span> <span class="nf">_print_MatrixBase</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_matrix_contents</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">parens</span><span class="p">(</span><span class="s">&#39;[&#39;</span><span class="p">,</span> <span class="s">&#39;]&#39;</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">D</span>
-    <span class="n">_print_ImmutableMatrix</span> <span class="o">=</span> <span class="n">_print_MatrixBase</span>
-    <span class="n">_print_Matrix</span> <span class="o">=</span> <span class="n">_print_MatrixBase</span>
-
-    <span class="k">def</span> <span class="nf">_print_Transpose</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">arg</span><span class="p">)</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">MatrixSymbol</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">arg</span><span class="p">,</span> <span class="n">MatrixSymbol</span><span class="p">):</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">pform</span><span class="o">**</span><span class="p">(</span><span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;T&#39;</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_Adjoint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="n">dag</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\u2020</span><span class="s">&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">dag</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;+&#39;</span><span class="p">)</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">MatrixSymbol</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">arg</span><span class="p">,</span> <span class="n">MatrixSymbol</span><span class="p">):</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">pform</span><span class="o">**</span><span class="n">dag</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_BlockMatrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">B</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">B</span><span class="o">.</span><span class="n">blocks</span><span class="o">.</span><span class="n">shape</span> <span class="o">==</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">B</span><span class="o">.</span><span class="n">blocks</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">B</span><span class="o">.</span><span class="n">blocks</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_MatAdd</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="s">&#39; + &#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_MatMul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Add</span><span class="p">,</span> <span class="n">MatAdd</span><span class="p">,</span> <span class="n">HadamardProduct</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-            <span class="k">if</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="p">(</span><span class="n">Add</span><span class="p">,</span> <span class="n">MatAdd</span><span class="p">,</span> <span class="n">HadamardProduct</span><span class="p">))</span>
-                    <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_MatPow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">)</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">MatrixSymbol</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">MatrixSymbol</span><span class="p">):</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">pform</span><span class="o">**</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_HadamardProduct</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">MatAdd</span><span class="p">,</span> <span class="n">MatMul</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="n">delim</span> <span class="o">=</span> <span class="n">pretty_atom</span><span class="p">(</span><span class="s">&#39;Ring&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">delim</span> <span class="o">=</span> <span class="s">&#39;.*&#39;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">delim</span><span class="p">,</span>
-                <span class="n">parenthesize</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">MatAdd</span><span class="p">,</span> <span class="n">MatMul</span><span class="p">)))</span>
-
-    <span class="n">_print_MatrixSymbol</span> <span class="o">=</span> <span class="n">_print_Symbol</span>
-
-    <span class="k">def</span> <span class="nf">_print_FunctionMatrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">lamda</span><span class="o">.</span><span class="n">expr</span><span class="p">)</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">parens</span><span class="p">(</span><span class="s">&#39;[&#39;</span><span class="p">,</span> <span class="s">&#39;]&#39;</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">D</span>
-
-    <span class="k">def</span> <span class="nf">_print_Piecewise</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">pexpr</span><span class="p">):</span>
-
-        <span class="n">P</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">ec</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">pexpr</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="n">P</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">ec</span><span class="o">.</span><span class="n">expr</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">ec</span><span class="o">.</span><span class="n">cond</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                <span class="n">P</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;otherwise&#39;</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">P</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span>
-                    <span class="o">*</span><span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;for &#39;</span><span class="p">)</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">ec</span><span class="o">.</span><span class="n">cond</span><span class="p">)))</span>
-        <span class="n">hsep</span> <span class="o">=</span> <span class="mi">2</span>
-        <span class="n">vsep</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">len_args</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">pexpr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="c"># max widths</span>
-        <span class="n">maxw</span> <span class="o">=</span> <span class="p">[</span><span class="nb">max</span><span class="p">([</span><span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">width</span><span class="p">()</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">len_args</span><span class="p">)])</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">)]</span>
-
-        <span class="c"># FIXME: Refactor this code and matrix into some tabular environment.</span>
-        <span class="c"># drawing result</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="bp">None</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">len_args</span><span class="p">):</span>
-            <span class="n">D_row</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">):</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-                <span class="k">assert</span> <span class="n">p</span><span class="o">.</span><span class="n">width</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="n">maxw</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-
-                <span class="n">wdelta</span> <span class="o">=</span> <span class="n">maxw</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">p</span><span class="o">.</span><span class="n">width</span><span class="p">()</span>
-                <span class="n">wleft</span> <span class="o">=</span> <span class="n">wdelta</span> <span class="o">//</span> <span class="mi">2</span>
-                <span class="n">wright</span> <span class="o">=</span> <span class="n">wdelta</span> <span class="o">-</span> <span class="n">wleft</span>
-
-                <span class="n">p</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">p</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">wright</span><span class="p">))</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">p</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">wleft</span><span class="p">))</span>
-
-                <span class="k">if</span> <span class="n">D_row</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">D_row</span> <span class="o">=</span> <span class="n">p</span>
-                    <span class="k">continue</span>
-
-                <span class="n">D_row</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D_row</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">hsep</span><span class="p">))</span>  <span class="c"># h-spacer</span>
-                <span class="n">D_row</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D_row</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">D</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">D</span> <span class="o">=</span> <span class="n">D_row</span>       <span class="c"># first row in a picture</span>
-                <span class="k">continue</span>
-
-            <span class="c"># v-spacer</span>
-            <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">vsep</span><span class="p">):</span>
-                <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">))</span>
-
-            <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="n">D_row</span><span class="p">))</span>
-
-        <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">parens</span><span class="p">(</span><span class="s">&#39;{&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">D</span>
-
-    <span class="k">def</span> <span class="nf">_hprint_vec</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">v</span><span class="p">):</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="bp">None</span>
-
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">v</span><span class="p">:</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">a</span>
-            <span class="k">if</span> <span class="n">D</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">D</span> <span class="o">=</span> <span class="n">p</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39;, &#39;</span><span class="p">))</span>
-                <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">D</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">D</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">D</span>
-
-    <span class="k">def</span> <span class="nf">_hprint_vseparator</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">):</span>
-        <span class="n">tmp</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">p1</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">p2</span><span class="p">))</span>
-        <span class="n">sep</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="n">vobj</span><span class="p">(</span><span class="s">&#39;|&#39;</span><span class="p">,</span> <span class="n">tmp</span><span class="o">.</span><span class="n">height</span><span class="p">()),</span> <span class="n">baseline</span><span class="o">=</span><span class="n">tmp</span><span class="o">.</span><span class="n">baseline</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">p1</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">sep</span><span class="p">,</span> <span class="n">p2</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_hyper</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="c"># FIXME refactor Matrix, Piecewise, and this into a tabular environment</span>
-        <span class="n">ap</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">ap</span><span class="p">]</span>
-        <span class="n">bq</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">bq</span><span class="p">]</span>
-
-        <span class="n">P</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">argument</span><span class="p">)</span>
-        <span class="n">P</span><span class="o">.</span><span class="n">baseline</span> <span class="o">=</span> <span class="n">P</span><span class="o">.</span><span class="n">height</span><span class="p">()</span><span class="o">//</span><span class="mi">2</span>
-
-        <span class="c"># Drawing result - first create the ap, bq vectors</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="p">[</span><span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">]:</span>
-            <span class="n">D_row</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_hprint_vec</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">D</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">D</span> <span class="o">=</span> <span class="n">D_row</span>       <span class="c"># first row in a picture</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">))</span>
-                <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="n">D_row</span><span class="p">))</span>
-
-        <span class="c"># make sure that the argument `z&#39; is centred vertically</span>
-        <span class="n">D</span><span class="o">.</span><span class="n">baseline</span> <span class="o">=</span> <span class="n">D</span><span class="o">.</span><span class="n">height</span><span class="p">()</span><span class="o">//</span><span class="mi">2</span>
-
-        <span class="c"># insert horizontal separator</span>
-        <span class="n">P</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">P</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">))</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">))</span>
-
-        <span class="c"># insert separating `|`</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_hprint_vseparator</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">P</span><span class="p">)</span>
-
-        <span class="c"># add parens</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">parens</span><span class="p">(</span><span class="s">&#39;(&#39;</span><span class="p">,</span> <span class="s">&#39;)&#39;</span><span class="p">))</span>
-
-        <span class="c"># create the F symbol</span>
-        <span class="n">above</span> <span class="o">=</span> <span class="n">D</span><span class="o">.</span><span class="n">height</span><span class="p">()</span><span class="o">//</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="n">below</span> <span class="o">=</span> <span class="n">D</span><span class="o">.</span><span class="n">height</span><span class="p">()</span> <span class="o">-</span> <span class="n">above</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="n">sz</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">add</span><span class="p">,</span> <span class="n">img</span> <span class="o">=</span> <span class="n">annotated</span><span class="p">(</span><span class="s">&#39;F&#39;</span><span class="p">)</span>
-        <span class="n">F</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span> <span class="o">*</span> <span class="p">(</span><span class="n">above</span> <span class="o">-</span> <span class="n">t</span><span class="p">)</span> <span class="o">+</span> <span class="n">img</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span> <span class="o">*</span> <span class="p">(</span><span class="n">below</span> <span class="o">-</span> <span class="n">b</span><span class="p">),</span>
-                       <span class="n">baseline</span><span class="o">=</span><span class="n">above</span> <span class="o">+</span> <span class="n">sz</span><span class="p">)</span>
-        <span class="n">add</span> <span class="o">=</span> <span class="p">(</span><span class="n">sz</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span>
-
-        <span class="n">F</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">F</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">ap</span><span class="p">))))</span>
-        <span class="n">F</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">F</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">bq</span><span class="p">))))</span>
-        <span class="n">F</span><span class="o">.</span><span class="n">baseline</span> <span class="o">=</span> <span class="n">above</span> <span class="o">+</span> <span class="n">add</span>
-
-        <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">F</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">,</span> <span class="n">D</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">D</span>
-
-    <span class="k">def</span> <span class="nf">_print_meijerg</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="c"># FIXME refactor Matrix, Piecewise, and this into a tabular environment</span>
-
-        <span class="n">v</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="n">v</span><span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)]</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">an</span><span class="p">]</span>
-        <span class="n">v</span><span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">aother</span><span class="p">]</span>
-        <span class="n">v</span><span class="p">[(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)]</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">bm</span><span class="p">]</span>
-        <span class="n">v</span><span class="p">[(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">bother</span><span class="p">]</span>
-
-        <span class="n">P</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">argument</span><span class="p">)</span>
-        <span class="n">P</span><span class="o">.</span><span class="n">baseline</span> <span class="o">=</span> <span class="n">P</span><span class="o">.</span><span class="n">height</span><span class="p">()</span><span class="o">//</span><span class="mi">2</span>
-
-        <span class="n">vp</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">idx</span> <span class="ow">in</span> <span class="n">v</span><span class="p">:</span>
-            <span class="n">vp</span><span class="p">[</span><span class="n">idx</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_hprint_vec</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="n">idx</span><span class="p">])</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">):</span>
-            <span class="n">maxw</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">vp</span><span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="n">i</span><span class="p">)]</span><span class="o">.</span><span class="n">width</span><span class="p">(),</span> <span class="n">vp</span><span class="p">[(</span><span class="mi">1</span><span class="p">,</span> <span class="n">i</span><span class="p">)]</span><span class="o">.</span><span class="n">width</span><span class="p">())</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">):</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">vp</span><span class="p">[(</span><span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">)]</span>
-                <span class="n">left</span> <span class="o">=</span> <span class="p">(</span><span class="n">maxw</span> <span class="o">-</span> <span class="n">s</span><span class="o">.</span><span class="n">width</span><span class="p">())</span> <span class="o">//</span> <span class="mi">2</span>
-                <span class="n">right</span> <span class="o">=</span> <span class="n">maxw</span> <span class="o">-</span> <span class="n">left</span> <span class="o">-</span> <span class="n">s</span><span class="o">.</span><span class="n">width</span><span class="p">()</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39; &#39;</span> <span class="o">*</span> <span class="n">left</span><span class="p">))</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span> <span class="o">*</span> <span class="n">right</span><span class="p">))</span>
-                <span class="n">vp</span><span class="p">[(</span><span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">)]</span> <span class="o">=</span> <span class="n">s</span>
-
-        <span class="n">D1</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">vp</span><span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)]</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39;  &#39;</span><span class="p">,</span> <span class="n">vp</span><span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]))</span>
-        <span class="n">D1</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D1</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">))</span>
-        <span class="n">D2</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">vp</span><span class="p">[(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)]</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39;  &#39;</span><span class="p">,</span> <span class="n">vp</span><span class="p">[(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]))</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D1</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="n">D2</span><span class="p">))</span>
-
-        <span class="c"># make sure that the argument `z&#39; is centred vertically</span>
-        <span class="n">D</span><span class="o">.</span><span class="n">baseline</span> <span class="o">=</span> <span class="n">D</span><span class="o">.</span><span class="n">height</span><span class="p">()</span><span class="o">//</span><span class="mi">2</span>
-
-        <span class="c"># insert horizontal separator</span>
-        <span class="n">P</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">P</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">))</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">))</span>
-
-        <span class="c"># insert separating `|`</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_hprint_vseparator</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">P</span><span class="p">)</span>
-
-        <span class="c"># add parens</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">D</span><span class="o">.</span><span class="n">parens</span><span class="p">(</span><span class="s">&#39;(&#39;</span><span class="p">,</span> <span class="s">&#39;)&#39;</span><span class="p">))</span>
-
-        <span class="c"># create the G symbol</span>
-        <span class="n">above</span> <span class="o">=</span> <span class="n">D</span><span class="o">.</span><span class="n">height</span><span class="p">()</span><span class="o">//</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="n">below</span> <span class="o">=</span> <span class="n">D</span><span class="o">.</span><span class="n">height</span><span class="p">()</span> <span class="o">-</span> <span class="n">above</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="n">sz</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">add</span><span class="p">,</span> <span class="n">img</span> <span class="o">=</span> <span class="n">annotated</span><span class="p">(</span><span class="s">&#39;G&#39;</span><span class="p">)</span>
-        <span class="n">F</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span> <span class="o">*</span> <span class="p">(</span><span class="n">above</span> <span class="o">-</span> <span class="n">t</span><span class="p">)</span> <span class="o">+</span> <span class="n">img</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span> <span class="o">*</span> <span class="p">(</span><span class="n">below</span> <span class="o">-</span> <span class="n">b</span><span class="p">),</span>
-                       <span class="n">baseline</span><span class="o">=</span><span class="n">above</span> <span class="o">+</span> <span class="n">sz</span><span class="p">)</span>
-
-        <span class="n">pp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">ap</span><span class="p">))</span>
-        <span class="n">pq</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">bq</span><span class="p">))</span>
-        <span class="n">pm</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">bm</span><span class="p">))</span>
-        <span class="n">pn</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">an</span><span class="p">))</span>
-
-        <span class="k">def</span> <span class="nf">adjust</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">):</span>
-            <span class="n">diff</span> <span class="o">=</span> <span class="n">p1</span><span class="o">.</span><span class="n">width</span><span class="p">()</span> <span class="o">-</span> <span class="n">p2</span><span class="o">.</span><span class="n">width</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">diff</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">p1</span><span class="p">,</span> <span class="n">p2</span>
-            <span class="k">elif</span> <span class="n">diff</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">p1</span><span class="p">,</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">p2</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">diff</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">p1</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*-</span><span class="n">diff</span><span class="p">)),</span> <span class="n">p2</span>
-        <span class="n">pp</span><span class="p">,</span> <span class="n">pm</span> <span class="o">=</span> <span class="n">adjust</span><span class="p">(</span><span class="n">pp</span><span class="p">,</span> <span class="n">pm</span><span class="p">)</span>
-        <span class="n">pq</span><span class="p">,</span> <span class="n">pn</span> <span class="o">=</span> <span class="n">adjust</span><span class="p">(</span><span class="n">pq</span><span class="p">,</span> <span class="n">pn</span><span class="p">)</span>
-        <span class="n">pu</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pm</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39;, &#39;</span><span class="p">,</span> <span class="n">pn</span><span class="p">))</span>
-        <span class="n">pl</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pp</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39;, &#39;</span><span class="p">,</span> <span class="n">pq</span><span class="p">))</span>
-
-        <span class="n">ht</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">baseline</span> <span class="o">-</span> <span class="n">above</span> <span class="o">-</span> <span class="mi">2</span>
-        <span class="k">if</span> <span class="n">ht</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">pu</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pu</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="o">*</span><span class="n">ht</span><span class="p">))</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pu</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="n">pl</span><span class="p">))</span>
-
-        <span class="n">F</span><span class="o">.</span><span class="n">baseline</span> <span class="o">=</span> <span class="n">above</span>
-        <span class="n">F</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">F</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
-
-        <span class="n">F</span><span class="o">.</span><span class="n">baseline</span> <span class="o">=</span> <span class="n">above</span> <span class="o">+</span> <span class="n">add</span>
-
-        <span class="n">D</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">F</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">,</span> <span class="n">D</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">D</span>
-
-    <span class="k">def</span> <span class="nf">_print_ExpBase</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="c"># TODO should exp_polar be printed differently?</span>
-        <span class="c">#      what about exp_polar(0), exp_polar(1)?</span>
-        <span class="n">base</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">pretty_atom</span><span class="p">(</span><span class="s">&#39;Exp1&#39;</span><span class="p">,</span> <span class="s">&#39;e&#39;</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">base</span> <span class="o">**</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_print_Function</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">sort</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="c"># XXX works only for applied functions</span>
-        <span class="n">func</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">func</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">if</span> <span class="n">sort</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-
-        <span class="n">func_name</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">__name__</span>
-
-        <span class="n">prettyFunc</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="n">func_name</span><span class="p">))</span>
-        <span class="n">prettyArgs</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">args</span><span class="p">)</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span>
-            <span class="n">binding</span><span class="o">=</span><span class="n">prettyForm</span><span class="o">.</span><span class="n">FUNC</span><span class="p">,</span> <span class="o">*</span><span class="n">stringPict</span><span class="o">.</span><span class="n">next</span><span class="p">(</span><span class="n">prettyFunc</span><span class="p">,</span> <span class="n">prettyArgs</span><span class="p">))</span>
-
-        <span class="c"># store pform parts so it can be reassembled e.g. when powered</span>
-        <span class="n">pform</span><span class="o">.</span><span class="n">prettyFunc</span> <span class="o">=</span> <span class="n">prettyFunc</span>
-        <span class="n">pform</span><span class="o">.</span><span class="n">prettyArgs</span> <span class="o">=</span> <span class="n">prettyArgs</span>
-
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_GeometryEntity</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="c"># GeometryEntity is based on Tuple but should not print like a Tuple</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">emptyPrinter</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Lambda</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">symbols</span><span class="p">,</span> <span class="n">expr</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">symbols</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">symbols</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">symbols</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="nb">tuple</span><span class="p">(</span><span class="n">symbols</span><span class="p">))</span>
-
-        <span class="n">args</span> <span class="o">=</span> <span class="p">(</span><span class="n">symbols</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-
-        <span class="n">prettyFunc</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;Lambda&quot;</span><span class="p">))</span>
-        <span class="n">prettyArgs</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">args</span><span class="p">)</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span>
-            <span class="n">binding</span><span class="o">=</span><span class="n">prettyForm</span><span class="o">.</span><span class="n">FUNC</span><span class="p">,</span> <span class="o">*</span><span class="n">stringPict</span><span class="o">.</span><span class="n">next</span><span class="p">(</span><span class="n">prettyFunc</span><span class="p">,</span> <span class="n">prettyArgs</span><span class="p">))</span>
-
-        <span class="c"># store pform parts so it can be reassembled e.g. when powered</span>
-        <span class="n">pform</span><span class="o">.</span><span class="n">prettyFunc</span> <span class="o">=</span> <span class="n">prettyFunc</span>
-        <span class="n">pform</span><span class="o">.</span><span class="n">prettyArgs</span> <span class="o">=</span> <span class="n">prettyArgs</span>
-
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_Order</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">expr</span><span class="p">)</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39;O&#39;</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_gamma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="n">greek</span><span class="p">[</span><span class="s">&#39;gamma&#39;</span><span class="p">][</span><span class="mi">1</span><span class="p">]))</span>
-            <span class="k">return</span> <span class="n">pform</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Function</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_uppergamma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39;, &#39;</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])))</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="n">greek</span><span class="p">[</span><span class="s">&#39;gamma&#39;</span><span class="p">][</span><span class="mi">1</span><span class="p">]))</span>
-            <span class="k">return</span> <span class="n">pform</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Function</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_lowergamma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39;, &#39;</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])))</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="n">greek</span><span class="p">[</span><span class="s">&#39;gamma&#39;</span><span class="p">][</span><span class="mi">0</span><span class="p">]))</span>
-            <span class="k">return</span> <span class="n">pform</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Function</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_expint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span>
-        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Function</span><span class="p">(</span><span class="n">Function</span><span class="p">(</span><span class="s">&#39;E_</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Function</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Chi</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="c"># This needs a special case since otherwise it comes out as greek</span>
-        <span class="c"># letter chi...</span>
-        <span class="n">prettyFunc</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&quot;Chi&quot;</span><span class="p">)</span>
-        <span class="n">prettyArgs</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">)</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span>
-            <span class="n">binding</span><span class="o">=</span><span class="n">prettyForm</span><span class="o">.</span><span class="n">FUNC</span><span class="p">,</span> <span class="o">*</span><span class="n">stringPict</span><span class="o">.</span><span class="n">next</span><span class="p">(</span><span class="n">prettyFunc</span><span class="p">,</span> <span class="n">prettyArgs</span><span class="p">))</span>
-
-        <span class="c"># store pform parts so it can be reassembled e.g. when powered</span>
-        <span class="n">pform</span><span class="o">.</span><span class="n">prettyFunc</span> <span class="o">=</span> <span class="n">prettyFunc</span>
-        <span class="n">pform</span><span class="o">.</span><span class="n">prettyArgs</span> <span class="o">=</span> <span class="n">prettyArgs</span>
-
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_Add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span> <span class="o">==</span> <span class="s">&#39;none&#39;</span><span class="p">:</span>
-            <span class="n">terms</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">terms</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_as_ordered_terms</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">)</span>
-        <span class="n">pforms</span><span class="p">,</span> <span class="n">indices</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
-
-        <span class="k">def</span> <span class="nf">pretty_negative</span><span class="p">(</span><span class="n">pform</span><span class="p">,</span> <span class="n">index</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;Prepend a minus sign to a pretty form. &quot;&quot;&quot;</span>
-            <span class="k">if</span> <span class="n">index</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">pform</span><span class="o">.</span><span class="n">height</span><span class="p">()</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">pform_neg</span> <span class="o">=</span> <span class="s">&#39;- &#39;</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">pform_neg</span> <span class="o">=</span> <span class="s">&#39;-&#39;</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">pform_neg</span> <span class="o">=</span> <span class="s">&#39; - &#39;</span>
-
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">stringPict</span><span class="o">.</span><span class="n">next</span><span class="p">(</span><span class="n">pform_neg</span><span class="p">,</span> <span class="n">pform</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">binding</span><span class="o">=</span><span class="n">prettyForm</span><span class="o">.</span><span class="n">NEG</span><span class="p">,</span> <span class="o">*</span><span class="n">pform</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">term</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">terms</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">term</span><span class="o">.</span><span class="n">is_Mul</span> <span class="ow">and</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">term</span><span class="p">):</span>
-                <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="o">-</span><span class="n">term</span><span class="p">)</span>
-                <span class="n">pforms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">pretty_negative</span><span class="p">(</span><span class="n">pform</span><span class="p">,</span> <span class="n">i</span><span class="p">))</span>
-            <span class="k">elif</span> <span class="n">term</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">term</span><span class="o">.</span><span class="n">q</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">pforms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">None</span><span class="p">)</span>
-                <span class="n">indices</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">term</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">and</span> <span class="n">term</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="o">-</span><span class="n">term</span><span class="p">)</span>
-                <span class="n">pforms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">pretty_negative</span><span class="p">(</span><span class="n">pform</span><span class="p">,</span> <span class="n">i</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">pforms</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">term</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">indices</span><span class="p">:</span>
-            <span class="n">large</span> <span class="o">=</span> <span class="bp">True</span>
-
-            <span class="k">for</span> <span class="n">pform</span> <span class="ow">in</span> <span class="n">pforms</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">pform</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">pform</span><span class="o">.</span><span class="n">height</span><span class="p">()</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">break</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">large</span> <span class="o">=</span> <span class="bp">False</span>
-
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">indices</span><span class="p">:</span>
-                <span class="n">term</span><span class="p">,</span> <span class="n">negative</span> <span class="o">=</span> <span class="n">terms</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="bp">False</span>
-
-                <span class="k">if</span> <span class="n">term</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">term</span><span class="p">,</span> <span class="n">negative</span> <span class="o">=</span> <span class="o">-</span><span class="n">term</span><span class="p">,</span> <span class="bp">True</span>
-
-                <span class="k">if</span> <span class="n">large</span><span class="p">:</span>
-                    <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">p</span><span class="p">))</span><span class="o">/</span><span class="n">prettyForm</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">q</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">negative</span><span class="p">:</span>
-                    <span class="n">pform</span> <span class="o">=</span> <span class="n">pretty_negative</span><span class="p">(</span><span class="n">pform</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-
-                <span class="n">pforms</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">pform</span>
-
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="o">.</span><span class="n">__add__</span><span class="p">(</span><span class="o">*</span><span class="n">pforms</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">product</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c"># items in the numerator</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c"># items that are in the denominator (if any)</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">(</span><span class="s">&#39;old&#39;</span><span class="p">,</span> <span class="s">&#39;none&#39;</span><span class="p">):</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="n">product</span><span class="o">.</span><span class="n">as_ordered_factors</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="n">product</span><span class="o">.</span><span class="n">args</span>
-
-        <span class="c"># Gather terms for numerator/denominator</span>
-        <span class="k">for</span> <span class="n">item</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">item</span><span class="o">.</span><span class="n">is_commutative</span> <span class="ow">and</span> <span class="n">item</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">item</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">item</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="n">b</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Pow</span><span class="p">(</span><span class="n">item</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="o">-</span><span class="n">item</span><span class="o">.</span><span class="n">exp</span><span class="p">))</span>
-            <span class="k">elif</span> <span class="n">item</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">item</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">item</span><span class="o">.</span><span class="n">p</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">a</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">C</span><span class="o">.</span><span class="n">Rational</span><span class="p">(</span><span class="n">item</span><span class="o">.</span><span class="n">p</span><span class="p">)</span> <span class="p">)</span>
-                <span class="k">if</span> <span class="n">item</span><span class="o">.</span><span class="n">q</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">b</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="n">C</span><span class="o">.</span><span class="n">Rational</span><span class="p">(</span><span class="n">item</span><span class="o">.</span><span class="n">q</span><span class="p">)</span> <span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">a</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">item</span><span class="p">)</span>
-
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">Piecewise</span><span class="p">,</span> <span class="n">Product</span><span class="p">,</span> <span class="n">Sum</span>
-
-        <span class="c"># Convert to pretty forms. Add parens to Add instances if there</span>
-        <span class="c"># is more than one term in the numer/denom</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="n">i</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="ow">and</span>
-                    <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="p">(</span><span class="n">Integral</span><span class="p">,</span> <span class="n">Piecewise</span><span class="p">,</span> <span class="n">Product</span><span class="p">,</span> <span class="n">Sum</span><span class="p">))):</span>
-                <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">b</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">b</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="n">i</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="ow">and</span>
-                    <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="p">(</span><span class="n">Integral</span><span class="p">,</span> <span class="n">Piecewise</span><span class="p">,</span> <span class="n">Product</span><span class="p">,</span> <span class="n">Sum</span><span class="p">))):</span>
-                <span class="n">b</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">b</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">b</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">b</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-
-        <span class="c"># Construct a pretty form</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">prettyForm</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="o">*</span><span class="n">a</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">a</span><span class="o">.</span><span class="n">append</span><span class="p">(</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span> <span class="p">)</span>
-            <span class="k">return</span> <span class="n">prettyForm</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="n">prettyForm</span><span class="o">.</span><span class="n">__mul__</span><span class="p">(</span><span class="o">*</span><span class="n">b</span><span class="p">)</span>
-
-    <span class="c"># A helper function for _print_Pow to print x**(1/n)</span>
-    <span class="k">def</span> <span class="nf">_print_nth_root</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="n">expt</span><span class="p">):</span>
-        <span class="n">bpretty</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-
-        <span class="c"># Construct root sign, start with the \/ shape</span>
-        <span class="n">_zZ</span> <span class="o">=</span> <span class="n">xobj</span><span class="p">(</span><span class="s">&#39;/&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">rootsign</span> <span class="o">=</span> <span class="n">xobj</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\\</span><span class="s">&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">_zZ</span>
-        <span class="c"># Make exponent number to put above it</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expt</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Rational</span><span class="p">):</span>
-            <span class="n">exp</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">expt</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">exp</span> <span class="o">==</span> <span class="s">&#39;2&#39;</span><span class="p">:</span>
-                <span class="n">exp</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">exp</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">expt</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="n">exp</span> <span class="o">=</span> <span class="n">exp</span><span class="o">.</span><span class="n">ljust</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">exp</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">rootsign</span> <span class="o">=</span> <span class="s">&#39; &#39;</span><span class="o">*</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">exp</span><span class="p">)</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">rootsign</span>
-        <span class="c"># Stack the exponent</span>
-        <span class="n">rootsign</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="n">exp</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span> <span class="o">+</span> <span class="n">rootsign</span><span class="p">)</span>
-        <span class="n">rootsign</span><span class="o">.</span><span class="n">baseline</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="c"># Diagonal: length is one less than height of base</span>
-        <span class="n">linelength</span> <span class="o">=</span> <span class="n">bpretty</span><span class="o">.</span><span class="n">height</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="n">diagonal</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span>
-            <span class="s">&#39; &#39;</span><span class="o">*</span><span class="p">(</span><span class="n">linelength</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">_zZ</span> <span class="o">+</span> <span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">i</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">linelength</span><span class="p">)</span>
-        <span class="p">))</span>
-        <span class="c"># Put baseline just below lowest line: next to exp</span>
-        <span class="n">diagonal</span><span class="o">.</span><span class="n">baseline</span> <span class="o">=</span> <span class="n">linelength</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="c"># Make the root symbol</span>
-        <span class="n">rootsign</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">rootsign</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">diagonal</span><span class="p">))</span>
-        <span class="c"># Det the baseline to match contents to fix the height</span>
-        <span class="c"># but if the height of bpretty is one, the rootsign must be one higher</span>
-        <span class="n">rootsign</span><span class="o">.</span><span class="n">baseline</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">bpretty</span><span class="o">.</span><span class="n">baseline</span><span class="p">)</span>
-        <span class="c">#build result</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">hobj</span><span class="p">(</span><span class="s">&#39;_&#39;</span><span class="p">,</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">bpretty</span><span class="o">.</span><span class="n">width</span><span class="p">()))</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">bpretty</span><span class="o">.</span><span class="n">above</span><span class="p">(</span><span class="n">s</span><span class="p">))</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="n">rootsign</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">s</span>
-
-    <span class="k">def</span> <span class="nf">_print_Pow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">power</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">fraction</span>
-        <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">power</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">power</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">e</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&quot;1&quot;</span><span class="p">)</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">fraction</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="ow">and</span> <span class="n">d</span><span class="o">.</span><span class="n">is_Atom</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_nth_root</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">e</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&quot;1&quot;</span><span class="p">)</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="o">-</span><span class="n">e</span><span class="p">)</span>
-
-        <span class="c"># None of the above special forms, do a standard power</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__print_numer_denom</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">q</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">p</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">p</span><span class="p">),</span> <span class="n">binding</span><span class="o">=</span><span class="n">prettyForm</span><span class="o">.</span><span class="n">NEG</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="nb">abs</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mi">10</span> <span class="ow">and</span> <span class="nb">abs</span><span class="p">(</span><span class="n">q</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mi">10</span><span class="p">:</span>
-            <span class="c"># If more than one digit in numer and denom, print larger fraction</span>
-            <span class="k">if</span> <span class="n">p</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="o">-</span><span class="n">p</span><span class="p">))</span><span class="o">/</span><span class="n">prettyForm</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">q</span><span class="p">))</span>
-                <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">binding</span><span class="o">=</span><span class="n">prettyForm</span><span class="o">.</span><span class="n">NEG</span><span class="p">,</span> <span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39;- &#39;</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">p</span><span class="p">))</span><span class="o">/</span><span class="n">prettyForm</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">q</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">_print_Rational</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">__print_numer_denom</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">result</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">emptyPrinter</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Fraction</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">__print_numer_denom</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">numerator</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">denominator</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">result</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">emptyPrinter</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_ProductSet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">sets</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">has_variety</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">sets</span><span class="p">):</span>
-            <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Pow</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">Pow</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">sets</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">sets</span><span class="p">),</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">prod_char</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\xd7</span><span class="s">&#39;</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">sets</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="s">&#39; </span><span class="si">%s</span><span class="s"> &#39;</span> <span class="o">%</span> <span class="n">prod_char</span><span class="p">,</span>
-                <span class="n">parenthesize</span><span class="o">=</span><span class="k">lambda</span> <span class="nb">set</span><span class="p">:</span> <span class="nb">set</span><span class="o">.</span><span class="n">is_Union</span> <span class="ow">or</span> <span class="nb">set</span><span class="o">.</span><span class="n">is_Intersection</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_FiniteSet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="n">items</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">items</span><span class="p">,</span> <span class="s">&#39;{&#39;</span><span class="p">,</span> <span class="s">&#39;}&#39;</span><span class="p">,</span> <span class="s">&#39;, &#39;</span> <span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Range</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="n">dots</span> <span class="o">=</span> <span class="s">&quot;</span><span class="se">\u2026</span><span class="s">&quot;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">dots</span> <span class="o">=</span> <span class="s">&#39;...&#39;</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">4</span><span class="p">:</span>
-            <span class="n">it</span> <span class="o">=</span> <span class="nb">iter</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-            <span class="n">printset</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">it</span><span class="p">),</span> <span class="nb">next</span><span class="p">(</span><span class="n">it</span><span class="p">),</span> <span class="n">dots</span><span class="p">,</span> <span class="n">s</span><span class="o">.</span><span class="n">_last_element</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">printset</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">printset</span><span class="p">,</span> <span class="s">&#39;{&#39;</span><span class="p">,</span> <span class="s">&#39;}&#39;</span><span class="p">,</span> <span class="s">&#39;, &#39;</span> <span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Interval</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">start</span> <span class="o">==</span> <span class="n">i</span><span class="o">.</span><span class="n">end</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">args</span><span class="p">[:</span><span class="mi">1</span><span class="p">],</span> <span class="s">&#39;{&#39;</span><span class="p">,</span> <span class="s">&#39;}&#39;</span><span class="p">)</span>
-
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">left_open</span><span class="p">:</span>
-                <span class="n">left</span> <span class="o">=</span> <span class="s">&#39;(&#39;</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">left</span> <span class="o">=</span> <span class="s">&#39;[&#39;</span>
-
-            <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">right_open</span><span class="p">:</span>
-                <span class="n">right</span> <span class="o">=</span> <span class="s">&#39;)&#39;</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">right</span> <span class="o">=</span> <span class="s">&#39;]&#39;</span>
-
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">args</span><span class="p">[:</span><span class="mi">2</span><span class="p">],</span> <span class="n">left</span><span class="p">,</span> <span class="n">right</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Intersection</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-
-        <span class="n">delimiter</span> <span class="o">=</span> <span class="s">&#39; </span><span class="si">%s</span><span class="s"> &#39;</span> <span class="o">%</span> <span class="n">pretty_atom</span><span class="p">(</span><span class="s">&#39;Intersection&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">u</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">delimiter</span><span class="p">,</span>
-                <span class="n">parenthesize</span><span class="o">=</span><span class="k">lambda</span> <span class="nb">set</span><span class="p">:</span> <span class="nb">set</span><span class="o">.</span><span class="n">is_ProductSet</span> <span class="ow">or</span> <span class="nb">set</span><span class="o">.</span><span class="n">is_Union</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Union</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">u</span><span class="p">):</span>
-
-        <span class="n">union_delimiter</span> <span class="o">=</span> <span class="s">&#39; </span><span class="si">%s</span><span class="s"> &#39;</span> <span class="o">%</span> <span class="n">pretty_atom</span><span class="p">(</span><span class="s">&#39;Union&#39;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">u</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">union_delimiter</span><span class="p">,</span>
-             <span class="n">parenthesize</span><span class="o">=</span><span class="k">lambda</span> <span class="nb">set</span><span class="p">:</span> <span class="nb">set</span><span class="o">.</span><span class="n">is_ProductSet</span> <span class="ow">or</span> <span class="nb">set</span><span class="o">.</span><span class="n">is_Intersection</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_TransformationSet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ts</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="n">inn</span> <span class="o">=</span> <span class="s">&quot;</span><span class="se">\u220a</span><span class="s">&quot;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">inn</span> <span class="o">=</span> <span class="s">&#39;in&#39;</span>
-        <span class="n">variables</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">ts</span><span class="o">.</span><span class="n">lamda</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">ts</span><span class="o">.</span><span class="n">lamda</span><span class="o">.</span><span class="n">expr</span><span class="p">)</span>
-        <span class="n">bar</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="s">&quot;|&quot;</span><span class="p">)</span>
-        <span class="n">base</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">ts</span><span class="o">.</span><span class="n">base_set</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">((</span><span class="n">expr</span><span class="p">,</span> <span class="n">bar</span><span class="p">,</span> <span class="n">variables</span><span class="p">,</span> <span class="n">inn</span><span class="p">,</span> <span class="n">base</span><span class="p">),</span> <span class="s">&quot;{&quot;</span><span class="p">,</span> <span class="s">&quot;}&quot;</span><span class="p">,</span> <span class="s">&#39; &#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_seq</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">seq</span><span class="p">,</span> <span class="n">left</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">right</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">delimiter</span><span class="o">=</span><span class="s">&#39;, &#39;</span><span class="p">,</span>
-            <span class="n">parenthesize</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="bp">None</span>
-
-        <span class="k">for</span> <span class="n">item</span> <span class="ow">in</span> <span class="n">seq</span><span class="p">:</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">item</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">parenthesize</span><span class="p">(</span><span class="n">item</span><span class="p">):</span>
-                <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-            <span class="k">if</span> <span class="n">s</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="c"># first element</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">pform</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">stringPict</span><span class="o">.</span><span class="n">next</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">delimiter</span><span class="p">))</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">stringPict</span><span class="o">.</span><span class="n">next</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">pform</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">s</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="s">&#39;&#39;</span><span class="p">)</span>
-
-        <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">parens</span><span class="p">(</span><span class="n">left</span><span class="p">,</span> <span class="n">right</span><span class="p">,</span> <span class="n">ifascii_nougly</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">s</span>
-
-    <span class="k">def</span> <span class="nf">join</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">delimiter</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="bp">None</span>
-
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">pform</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">pform</span> <span class="o">=</span> <span class="n">arg</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">delimiter</span><span class="p">))</span>
-                <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">pform</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&quot;&quot;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_list</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">l</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="s">&#39;[&#39;</span><span class="p">,</span> <span class="s">&#39;]&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_tuple</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">ptuple</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">stringPict</span><span class="o">.</span><span class="n">next</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">t</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="s">&#39;,&#39;</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">ptuple</span><span class="o">.</span><span class="n">parens</span><span class="p">(</span><span class="s">&#39;(&#39;</span><span class="p">,</span> <span class="s">&#39;)&#39;</span><span class="p">,</span> <span class="n">ifascii_nougly</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="s">&#39;(&#39;</span><span class="p">,</span> <span class="s">&#39;)&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Tuple</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_tuple</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_dict</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">d</span><span class="p">):</span>
-        <span class="n">keys</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">keys</span><span class="p">()),</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="n">items</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">keys</span><span class="p">:</span>
-            <span class="n">K</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-            <span class="n">V</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">d</span><span class="p">[</span><span class="n">k</span><span class="p">])</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">stringPict</span><span class="o">.</span><span class="n">next</span><span class="p">(</span><span class="n">K</span><span class="p">,</span> <span class="s">&#39;: &#39;</span><span class="p">,</span> <span class="n">V</span><span class="p">))</span>
-
-            <span class="n">items</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">items</span><span class="p">,</span> <span class="s">&#39;{&#39;</span><span class="p">,</span> <span class="s">&#39;}&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Dict</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">d</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_dict</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_set</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="n">items</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="n">pretty</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">items</span><span class="p">,</span> <span class="s">&#39;[&#39;</span><span class="p">,</span> <span class="s">&#39;]&#39;</span><span class="p">)</span>
-        <span class="n">pretty</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pretty</span><span class="o">.</span><span class="n">parens</span><span class="p">(</span><span class="s">&#39;(&#39;</span><span class="p">,</span> <span class="s">&#39;)&#39;</span><span class="p">,</span> <span class="n">ifascii_nougly</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-        <span class="n">pretty</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">stringPict</span><span class="o">.</span><span class="n">next</span><span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">s</span><span class="p">)</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="n">pretty</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pretty</span>
-
-    <span class="n">_print_frozenset</span> <span class="o">=</span> <span class="n">_print_set</span>
-
-    <span class="k">def</span> <span class="nf">_print_AlgebraicNumber</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_aliased</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">as_poly</span><span class="p">()</span><span class="o">.</span><span class="n">as_expr</span><span class="p">())</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">as_expr</span><span class="p">())</span>
-
-    <span class="k">def</span> <span class="nf">_print_RootOf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_print_Add</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s">&#39;lex&#39;</span><span class="p">),</span> <span class="n">expr</span><span class="o">.</span><span class="n">index</span><span class="p">]</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">args</span><span class="p">)</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39;RootOf&#39;</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_RootSum</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_print_Add</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s">&#39;lex&#39;</span><span class="p">)]</span>
-
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">fun</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">IdentityFunction</span><span class="p">:</span>
-            <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">fun</span><span class="p">))</span>
-
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">args</span><span class="p">)</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39;RootSum&#39;</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_FiniteField</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="n">form</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\u2124</span><span class="s">_</span><span class="si">%d</span><span class="s">&#39;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">form</span> <span class="o">=</span> <span class="s">&#39;GF(</span><span class="si">%d</span><span class="s">)&#39;</span>
-
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">pretty_symbol</span><span class="p">(</span><span class="n">form</span> <span class="o">%</span> <span class="n">expr</span><span class="o">.</span><span class="n">mod</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_IntegerRing</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\u2124</span><span class="s">&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;ZZ&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_RationalField</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\u211A</span><span class="s">&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;QQ&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_RealDomain</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\u211D</span><span class="s">&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;RR&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_ComplexDomain</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\u2102</span><span class="s">&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;CC&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_PolynomialRingBase</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">gens</span>
-        <span class="k">if</span> <span class="nb">str</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">order</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">str</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">default_order</span><span class="p">):</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">g</span> <span class="o">+</span> <span class="p">(</span><span class="s">&quot;order=&quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">order</span><span class="p">),)</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="s">&#39;[&#39;</span><span class="p">,</span> <span class="s">&#39;]&#39;</span><span class="p">)</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">dom</span><span class="p">)))</span>
-
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_FractionField</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">gens</span><span class="p">,</span> <span class="s">&#39;(&#39;</span><span class="p">,</span> <span class="s">&#39;)&#39;</span><span class="p">)</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">dom</span><span class="p">)))</span>
-
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_GroebnerBasis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">):</span>
-        <span class="n">exprs</span> <span class="o">=</span> <span class="p">[</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Add</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">basis</span><span class="o">.</span><span class="n">order</span><span class="p">)</span>
-                  <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">basis</span><span class="o">.</span><span class="n">exprs</span> <span class="p">]</span>
-        <span class="n">exprs</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="s">&quot;, &quot;</span><span class="p">,</span> <span class="n">exprs</span><span class="p">)</span><span class="o">.</span><span class="n">parens</span><span class="p">(</span><span class="n">left</span><span class="o">=</span><span class="s">&quot;[&quot;</span><span class="p">,</span> <span class="n">right</span><span class="o">=</span><span class="s">&quot;]&quot;</span><span class="p">))</span>
-
-        <span class="n">gens</span> <span class="o">=</span> <span class="p">[</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">gen</span><span class="p">)</span> <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">basis</span><span class="o">.</span><span class="n">gens</span> <span class="p">]</span>
-
-        <span class="n">domain</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span>
-            <span class="o">*</span><span class="n">prettyForm</span><span class="p">(</span><span class="s">&quot;domain=&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">basis</span><span class="o">.</span><span class="n">domain</span><span class="p">)))</span>
-        <span class="n">order</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span>
-            <span class="o">*</span><span class="n">prettyForm</span><span class="p">(</span><span class="s">&quot;order=&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">basis</span><span class="o">.</span><span class="n">order</span><span class="p">)))</span>
-
-        <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="s">&quot;, &quot;</span><span class="p">,</span> <span class="p">[</span><span class="n">exprs</span><span class="p">]</span> <span class="o">+</span> <span class="n">gens</span> <span class="o">+</span> <span class="p">[</span><span class="n">domain</span><span class="p">,</span> <span class="n">order</span><span class="p">])</span>
-
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="n">basis</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_Subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">expr</span><span class="p">)</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-
-        <span class="n">h</span> <span class="o">=</span> <span class="n">pform</span><span class="o">.</span><span class="n">height</span><span class="p">()</span> <span class="k">if</span> <span class="n">pform</span><span class="o">.</span><span class="n">height</span><span class="p">()</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="k">else</span> <span class="mi">2</span>
-        <span class="n">rvert</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="n">vobj</span><span class="p">(</span><span class="s">&#39;|&#39;</span><span class="p">,</span> <span class="n">h</span><span class="p">),</span> <span class="n">baseline</span><span class="o">=</span><span class="n">pform</span><span class="o">.</span><span class="n">baseline</span><span class="p">)</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">rvert</span><span class="p">))</span>
-
-        <span class="n">b</span> <span class="o">=</span> <span class="n">pform</span><span class="o">.</span><span class="n">baseline</span>
-        <span class="n">pform</span><span class="o">.</span><span class="n">baseline</span> <span class="o">=</span> <span class="n">pform</span><span class="o">.</span><span class="n">height</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">([</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">xsym</span><span class="p">(</span><span class="s">&#39;==&#39;</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">])),</span>
-                <span class="n">delimiter</span><span class="o">=</span><span class="s">&#39;&#39;</span><span class="p">)</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">variables</span><span class="p">,</span> <span class="n">e</span><span class="o">.</span><span class="n">point</span><span class="p">)</span> <span class="p">])))</span>
-
-        <span class="n">pform</span><span class="o">.</span><span class="n">baseline</span> <span class="o">=</span> <span class="n">b</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_euler</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&quot;E&quot;</span><span class="p">)</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="n">pform_arg</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&quot; &quot;</span><span class="o">*</span><span class="n">arg</span><span class="o">.</span><span class="n">width</span><span class="p">())</span>
-        <span class="n">pform_arg</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform_arg</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">pform_arg</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_catalan</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="n">pform_arg</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="s">&quot; &quot;</span><span class="o">*</span><span class="n">arg</span><span class="o">.</span><span class="n">width</span><span class="p">())</span>
-        <span class="n">pform_arg</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform_arg</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">pform_arg</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_KroneckerDelta</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">((</span><span class="n">prettyForm</span><span class="p">(</span><span class="s">&#39;,&#39;</span><span class="p">))))</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]))))</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_use_unicode</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="n">pretty_symbol</span><span class="p">(</span><span class="s">&#39;delta&#39;</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="s">&#39;d&#39;</span><span class="p">)</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">pform</span>
-        <span class="n">top</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="o">*</span><span class="n">b</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">a</span><span class="o">.</span><span class="n">width</span><span class="p">()))</span>
-        <span class="n">bot</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="o">*</span><span class="n">a</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">b</span><span class="o">.</span><span class="n">width</span><span class="p">()))</span>
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">binding</span><span class="o">=</span><span class="n">prettyForm</span><span class="o">.</span><span class="n">POW</span><span class="p">,</span> <span class="o">*</span><span class="n">bot</span><span class="o">.</span><span class="n">below</span><span class="p">(</span><span class="n">top</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_RandomDomain</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">d</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="s">&#39;Domain: &#39;</span><span class="p">)</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">as_boolean</span><span class="p">())))</span>
-            <span class="k">return</span> <span class="n">pform</span>
-
-        <span class="k">except</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="s">&#39;Domain: &#39;</span><span class="p">)</span>
-                <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">symbols</span><span class="p">)))</span>
-                <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="s">&#39; in &#39;</span><span class="p">)))</span>
-                <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">set</span><span class="p">)))</span>
-                <span class="k">return</span> <span class="n">pform</span>
-            <span class="k">except</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">None</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_DMP</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">ring</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="c"># TODO incorporate order</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
-        <span class="k">except</span> <span class="n">SympifyError</span><span class="p">:</span>
-            <span class="k">pass</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="nb">repr</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_DMF</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_DMP</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Object</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">object</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">pretty_symbol</span><span class="p">(</span><span class="nb">object</span><span class="o">.</span><span class="n">name</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Morphism</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">morphism</span><span class="p">):</span>
-        <span class="n">arrow</span> <span class="o">=</span> <span class="n">xsym</span><span class="p">(</span><span class="s">&quot;--&gt;&quot;</span><span class="p">)</span>
-
-        <span class="n">domain</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">morphism</span><span class="o">.</span><span class="n">domain</span><span class="p">)</span>
-        <span class="n">codomain</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">morphism</span><span class="o">.</span><span class="n">codomain</span><span class="p">)</span>
-        <span class="n">tail</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">arrow</span><span class="p">,</span> <span class="n">codomain</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">tail</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_NamedMorphism</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">morphism</span><span class="p">):</span>
-        <span class="n">pretty_name</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">pretty_symbol</span><span class="p">(</span><span class="n">morphism</span><span class="o">.</span><span class="n">name</span><span class="p">))</span>
-        <span class="n">pretty_morphism</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Morphism</span><span class="p">(</span><span class="n">morphism</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">pretty_name</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&quot;:&quot;</span><span class="p">,</span> <span class="n">pretty_morphism</span><span class="p">)[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_print_IdentityMorphism</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">morphism</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">NamedMorphism</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_NamedMorphism</span><span class="p">(</span>
-            <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">morphism</span><span class="o">.</span><span class="n">domain</span><span class="p">,</span> <span class="n">morphism</span><span class="o">.</span><span class="n">codomain</span><span class="p">,</span> <span class="s">&quot;id&quot;</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_CompositeMorphism</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">morphism</span><span class="p">):</span>
-
-        <span class="n">circle</span> <span class="o">=</span> <span class="n">xsym</span><span class="p">(</span><span class="s">&quot;.&quot;</span><span class="p">)</span>
-
-        <span class="c"># All components of the morphism have names and it is thus</span>
-        <span class="c"># possible to build the name of the composite.</span>
-        <span class="n">component_names_list</span> <span class="o">=</span> <span class="p">[</span><span class="n">pretty_symbol</span><span class="p">(</span><span class="n">component</span><span class="o">.</span><span class="n">name</span><span class="p">)</span> <span class="k">for</span>
-                                <span class="n">component</span> <span class="ow">in</span> <span class="n">morphism</span><span class="o">.</span><span class="n">components</span><span class="p">]</span>
-        <span class="n">component_names_list</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-        <span class="n">component_names</span> <span class="o">=</span> <span class="n">circle</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">component_names_list</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot;:&quot;</span>
-
-        <span class="n">pretty_name</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">component_names</span><span class="p">)</span>
-        <span class="n">pretty_morphism</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Morphism</span><span class="p">(</span><span class="n">morphism</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">pretty_name</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">pretty_morphism</span><span class="p">)[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_print_Category</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">category</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">pretty_symbol</span><span class="p">(</span><span class="n">category</span><span class="o">.</span><span class="n">name</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Diagram</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">diagram</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">diagram</span><span class="o">.</span><span class="n">premises</span><span class="p">:</span>
-            <span class="c"># This is an empty diagram.</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">EmptySet</span><span class="p">)</span>
-
-        <span class="n">pretty_result</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">diagram</span><span class="o">.</span><span class="n">premises</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">diagram</span><span class="o">.</span><span class="n">conclusions</span><span class="p">:</span>
-            <span class="n">results_arrow</span> <span class="o">=</span> <span class="s">&quot; </span><span class="si">%s</span><span class="s"> &quot;</span> <span class="o">%</span> <span class="n">xsym</span><span class="p">(</span><span class="s">&quot;==&gt;&quot;</span><span class="p">)</span>
-
-            <span class="n">pretty_conclusions</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">diagram</span><span class="o">.</span><span class="n">conclusions</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">pretty_result</span> <span class="o">=</span> <span class="n">pretty_result</span><span class="o">.</span><span class="n">right</span><span class="p">(</span>
-                <span class="n">results_arrow</span><span class="p">,</span> <span class="n">pretty_conclusions</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">pretty_result</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_print_DiagramGrid</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">grid</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-        <span class="n">matrix</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="k">if</span> <span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="k">else</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot; &quot;</span><span class="p">)</span>
-                          <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid</span><span class="o">.</span><span class="n">width</span><span class="p">)]</span>
-                         <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid</span><span class="o">.</span><span class="n">height</span><span class="p">)])</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_matrix_contents</span><span class="p">(</span><span class="n">matrix</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_FreeModuleElement</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-        <span class="c"># Print as row vector for convenience, for now.</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="s">&#39;[&#39;</span><span class="p">,</span> <span class="s">&#39;]&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_SubModule</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">M</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">gens</span><span class="p">,</span> <span class="s">&#39;&lt;&#39;</span><span class="p">,</span> <span class="s">&#39;&gt;&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_FreeModule</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">M</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">ring</span><span class="p">)</span><span class="o">**</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">rank</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_ModuleImplementedIdeal</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">M</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_seq</span><span class="p">([</span><span class="n">x</span> <span class="k">for</span> <span class="p">[</span><span class="n">x</span><span class="p">]</span> <span class="ow">in</span> <span class="n">M</span><span class="o">.</span><span class="n">_module</span><span class="o">.</span><span class="n">gens</span><span class="p">],</span> <span class="s">&#39;&lt;&#39;</span><span class="p">,</span> <span class="s">&#39;&gt;&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_QuotientRing</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">R</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">R</span><span class="o">.</span><span class="n">ring</span><span class="p">)</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">R</span><span class="o">.</span><span class="n">base_ideal</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_QuotientRingElement</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">R</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">R</span><span class="o">.</span><span class="n">data</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">R</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">base_ideal</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_QuotientModuleElement</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">data</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">module</span><span class="o">.</span><span class="n">killed_module</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_QuotientModule</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">M</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">base</span><span class="p">)</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">killed_module</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_MatrixHomomorphism</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">h</span><span class="p">):</span>
-        <span class="n">matrix</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">h</span><span class="o">.</span><span class="n">_sympy_matrix</span><span class="p">())</span>
-        <span class="n">matrix</span><span class="o">.</span><span class="n">baseline</span> <span class="o">=</span> <span class="n">matrix</span><span class="o">.</span><span class="n">height</span><span class="p">()</span> <span class="o">//</span> <span class="mi">2</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">matrix</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; : &#39;</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">h</span><span class="o">.</span><span class="n">domain</span><span class="p">),</span>
-            <span class="s">&#39; </span><span class="si">%s</span><span class="s">&gt; &#39;</span> <span class="o">%</span> <span class="n">hobj</span><span class="p">(</span><span class="s">&#39;-&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">h</span><span class="o">.</span><span class="n">codomain</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-    <span class="k">def</span> <span class="nf">_print_BaseScalarField</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">field</span><span class="p">):</span>
-        <span class="n">string</span> <span class="o">=</span> <span class="n">field</span><span class="o">.</span><span class="n">_coord_sys</span><span class="o">.</span><span class="n">_names</span><span class="p">[</span><span class="n">field</span><span class="o">.</span><span class="n">_index</span><span class="p">]</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">pretty_symbol</span><span class="p">(</span><span class="n">string</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_BaseVectorField</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">field</span><span class="p">):</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;PARTIAL DIFFERENTIAL&#39;</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39;_&#39;</span> <span class="o">+</span> <span class="n">field</span><span class="o">.</span><span class="n">_coord_sys</span><span class="o">.</span><span class="n">_names</span><span class="p">[</span><span class="n">field</span><span class="o">.</span><span class="n">_index</span><span class="p">]</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">pretty_symbol</span><span class="p">(</span><span class="n">s</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Differential</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">diff</span><span class="p">):</span>
-        <span class="n">field</span> <span class="o">=</span> <span class="n">diff</span><span class="o">.</span><span class="n">_form_field</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">field</span><span class="p">,</span> <span class="s">&#39;_coord_sys&#39;</span><span class="p">):</span>
-            <span class="n">string</span> <span class="o">=</span> <span class="n">field</span><span class="o">.</span><span class="n">_coord_sys</span><span class="o">.</span><span class="n">_names</span><span class="p">[</span><span class="n">field</span><span class="o">.</span><span class="n">_index</span><span class="p">]</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\u2146</span><span class="s"> &#39;</span> <span class="o">+</span> <span class="n">pretty_symbol</span><span class="p">(</span><span class="n">string</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">field</span><span class="p">)</span>
-            <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-            <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\u2146</span><span class="s">&quot;</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Tr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="c">#TODO: Handle indices</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s">(&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">)))</span>
-        <span class="n">pform</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="p">(</span><span class="o">*</span><span class="n">pform</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39;)&#39;</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">pform</span>
-
-</div>
-<div class="viewcode-block" id="pretty"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.pretty.pretty">[docs]</a><span class="k">def</span> <span class="nf">pretty</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns a string containing the prettified form of expr.</span>
-
-<span class="sd">    For information on keyword arguments see pretty_print function.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">pp</span> <span class="o">=</span> <span class="n">PrettyPrinter</span><span class="p">(</span><span class="n">settings</span><span class="p">)</span>
-
-    <span class="c"># XXX: this is an ugly hack, but at least it works</span>
-    <span class="n">use_unicode</span> <span class="o">=</span> <span class="n">pp</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;use_unicode&#39;</span><span class="p">]</span>
-    <span class="n">uflag</span> <span class="o">=</span> <span class="n">pretty_use_unicode</span><span class="p">(</span><span class="n">use_unicode</span><span class="p">)</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">pp</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">finally</span><span class="p">:</span>
-        <span class="n">pretty_use_unicode</span><span class="p">(</span><span class="n">uflag</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="pretty_print"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.pretty.pretty_print">[docs]</a><span class="k">def</span> <span class="nf">pretty_print</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Prints expr in pretty form.</span>
-
-<span class="sd">    pprint is just a shortcut for this function.</span>
-
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    expr : expression</span>
-<span class="sd">        the expression to print</span>
-<span class="sd">    wrap_line : bool, optional</span>
-<span class="sd">        line wrapping enabled/disabled, defaults to True</span>
-<span class="sd">    num_columns : bool, optional</span>
-<span class="sd">        number of columns before line breaking (default to None which reads</span>
-<span class="sd">        the terminal width), useful when using SymPy without terminal.</span>
-<span class="sd">    use_unicode : bool or None, optional</span>
-<span class="sd">        use unicode characters, such as the Greek letter pi instead of</span>
-<span class="sd">        the string pi.</span>
-<span class="sd">    full_prec : bool or string, optional</span>
-<span class="sd">        use full precision. Default to &quot;auto&quot;</span>
-<span class="sd">    order : bool or string, optional</span>
-<span class="sd">        set to &#39;none&#39; for long expressions if slow; default is None</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">print</span><span class="p">(</span><span class="n">pretty</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">))</span>
-</div>
-<span class="n">pprint</span> <span class="o">=</span> <span class="n">pretty_print</span>
-
-
-<span class="k">def</span> <span class="nf">pager_print</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Prints expr using the pager, in pretty form.</span>
-
-<span class="sd">    This invokes a pager command using pydoc. Lines are not wrapped</span>
-<span class="sd">    automatically. This routine is meant to be used with a pager that allows</span>
-<span class="sd">    sideways scrolling, like ``less -S``.</span>
-
-<span class="sd">    Parameters are the same as for ``pretty_print``. If you wish to wrap lines,</span>
-<span class="sd">    pass ``num_columns=None`` to auto-detect the width of the terminal.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">pydoc</span> <span class="kn">import</span> <span class="n">pager</span>
-    <span class="kn">from</span> <span class="nn">locale</span> <span class="kn">import</span> <span class="n">getpreferredencoding</span>
-    <span class="k">if</span> <span class="s">&#39;num_columns&#39;</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">settings</span><span class="p">:</span>
-        <span class="n">settings</span><span class="p">[</span><span class="s">&#39;num_columns&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="mi">500000</span>  <span class="c"># disable line wrap</span>
-    <span class="n">pager</span><span class="p">(</span><span class="n">pretty</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">)</span><span class="o">.</span><span class="n">encode</span><span class="p">(</span><span class="n">getpreferredencoding</span><span class="p">()))</span>
-</pre></div>
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-            
-  <h1>Source code for sympy.printing.pretty.pretty_symbology</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Symbolic primitives + unicode/ASCII abstraction for pretty.py&quot;&quot;&quot;</span>
-
-<span class="kn">import</span> <span class="nn">sys</span>
-<span class="n">warnings</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-
-<span class="c"># first, setup unicodedate environment</span>
-<span class="k">try</span><span class="p">:</span>
-    <span class="kn">import</span> <span class="nn">unicodedata</span>
-
-<div class="viewcode-block" id="U"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.pretty_symbology.U">[docs]</a>    <span class="k">def</span> <span class="nf">U</span><span class="p">(</span><span class="n">name</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;unicode character by name or None if not found&quot;&quot;&quot;</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">u</span> <span class="o">=</span> <span class="n">unicodedata</span><span class="o">.</span><span class="n">lookup</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-            <span class="n">u</span> <span class="o">=</span> <span class="bp">None</span>
-
-            <span class="k">global</span> <span class="n">warnings</span>
-            <span class="n">warnings</span> <span class="o">+=</span> <span class="s">&#39;W: no </span><span class="se">\&#39;</span><span class="si">%s</span><span class="se">\&#39;</span><span class="s"> in unocodedata</span><span class="se">\n</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">name</span>
-
-        <span class="k">return</span> <span class="n">u</span>
-</div>
-<span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>
-    <span class="n">warnings</span> <span class="o">+=</span> <span class="s">&#39;W: no unicodedata available</span><span class="se">\n</span><span class="s">&#39;</span>
-    <span class="n">U</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">name</span><span class="p">:</span> <span class="bp">None</span>
-
-<span class="kn">from</span> <span class="nn">sympy.printing.conventions</span> <span class="kn">import</span> <span class="n">split_super_sub</span>
-
-
-<span class="c"># prefix conventions when constructing tables</span>
-<span class="c"># L   - LATIN     i</span>
-<span class="c"># G   - GREEK     beta</span>
-<span class="c"># D   - DIGIT     0</span>
-<span class="c"># S   - SYMBOL    +</span>
-
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;greek&#39;</span><span class="p">,</span> <span class="s">&#39;sub&#39;</span><span class="p">,</span> <span class="s">&#39;sup&#39;</span><span class="p">,</span> <span class="s">&#39;xsym&#39;</span><span class="p">,</span> <span class="s">&#39;vobj&#39;</span><span class="p">,</span> <span class="s">&#39;hobj&#39;</span><span class="p">,</span> <span class="s">&#39;pretty_symbol&#39;</span><span class="p">,</span>
-           <span class="s">&#39;annotated&#39;</span><span class="p">]</span>
-
-
-<span class="n">_use_unicode</span> <span class="o">=</span> <span class="bp">False</span>
-
-
-<div class="viewcode-block" id="pretty_use_unicode"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.pretty_symbology.pretty_use_unicode">[docs]</a><span class="k">def</span> <span class="nf">pretty_use_unicode</span><span class="p">(</span><span class="n">flag</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Set whether pretty-printer should use unicode by default&quot;&quot;&quot;</span>
-    <span class="k">global</span> <span class="n">_use_unicode</span>
-    <span class="k">global</span> <span class="n">warnings</span>
-    <span class="k">if</span> <span class="n">flag</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">_use_unicode</span>
-
-    <span class="k">if</span> <span class="n">flag</span> <span class="ow">and</span> <span class="n">warnings</span><span class="p">:</span>
-        <span class="c"># print warnings (if any) on first unicode usage</span>
-        <span class="k">print</span><span class="p">(</span><span class="s">&quot;I: pprint -- we are going to use unicode, but there are following problems:&quot;</span><span class="p">)</span>
-        <span class="k">print</span><span class="p">(</span><span class="n">warnings</span><span class="p">)</span>
-        <span class="n">warnings</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-
-    <span class="n">use_unicode_prev</span> <span class="o">=</span> <span class="n">_use_unicode</span>
-    <span class="n">_use_unicode</span> <span class="o">=</span> <span class="n">flag</span>
-    <span class="k">return</span> <span class="n">use_unicode_prev</span>
-
-</div>
-<div class="viewcode-block" id="pretty_try_use_unicode"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.pretty_symbology.pretty_try_use_unicode">[docs]</a><span class="k">def</span> <span class="nf">pretty_try_use_unicode</span><span class="p">():</span>
-    <span class="sd">&quot;&quot;&quot;See if unicode output is available and leverage it if possible&quot;&quot;&quot;</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="c"># see, if we can represent greek alphabet</span>
-        <span class="k">for</span> <span class="n">g</span><span class="p">,</span> <span class="n">G</span> <span class="ow">in</span> <span class="n">greek</span><span class="o">.</span><span class="n">values</span><span class="p">():</span>
-            <span class="n">symbols</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-            <span class="n">symbols</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-
-        <span class="c"># and atoms</span>
-        <span class="n">symbols</span> <span class="o">+=</span> <span class="nb">list</span><span class="p">(</span><span class="n">atoms_table</span><span class="o">.</span><span class="n">values</span><span class="p">())</span>
-
-        <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">symbols</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">s</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span>  <span class="c"># common symbols not present!</span>
-
-            <span class="n">encoding</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">sys</span><span class="o">.</span><span class="n">stdout</span><span class="p">,</span> <span class="s">&#39;encoding&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-
-            <span class="c"># this happens when e.g. stdout is redirected through a pipe, or is</span>
-            <span class="c"># e.g. a cStringIO.StringO</span>
-            <span class="k">if</span> <span class="n">encoding</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span>  <span class="c"># sys.stdout has no encoding</span>
-
-            <span class="c"># try to encode</span>
-            <span class="n">s</span><span class="o">.</span><span class="n">encode</span><span class="p">(</span><span class="n">encoding</span><span class="p">)</span>
-
-    <span class="k">except</span> <span class="ne">UnicodeEncodeError</span><span class="p">:</span>
-        <span class="k">pass</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">pretty_use_unicode</span><span class="p">(</span><span class="bp">True</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="xstr"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.pretty_symbology.xstr">[docs]</a><span class="k">def</span> <span class="nf">xstr</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;call str or unicode depending on current mode&quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">_use_unicode</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-<span class="c"># GREEK</span></div>
-<span class="n">g</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">l</span><span class="p">:</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;GREEK SMALL LETTER </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">l</span><span class="o">.</span><span class="n">upper</span><span class="p">())</span>
-<span class="n">G</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">l</span><span class="p">:</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;GREEK CAPITAL LETTER </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">l</span><span class="o">.</span><span class="n">upper</span><span class="p">())</span>
-
-<span class="n">greek_letters</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="s">&#39;alpha&#39;</span><span class="p">,</span> <span class="s">&#39;beta&#39;</span><span class="p">,</span> <span class="s">&#39;gamma&#39;</span><span class="p">,</span> <span class="s">&#39;delta&#39;</span><span class="p">,</span> <span class="s">&#39;epsilon&#39;</span><span class="p">,</span> <span class="s">&#39;zeta&#39;</span><span class="p">,</span> <span class="s">&#39;eta&#39;</span><span class="p">,</span> <span class="s">&#39;theta&#39;</span><span class="p">,</span>
-    <span class="s">&#39;iota&#39;</span><span class="p">,</span> <span class="s">&#39;kappa&#39;</span><span class="p">,</span> <span class="s">&#39;lamda&#39;</span><span class="p">,</span> <span class="s">&#39;mu&#39;</span><span class="p">,</span> <span class="s">&#39;nu&#39;</span><span class="p">,</span> <span class="s">&#39;xi&#39;</span><span class="p">,</span> <span class="s">&#39;omicron&#39;</span><span class="p">,</span> <span class="s">&#39;pi&#39;</span><span class="p">,</span> <span class="s">&#39;rho&#39;</span><span class="p">,</span>
-    <span class="s">&#39;sigma&#39;</span><span class="p">,</span> <span class="s">&#39;tau&#39;</span><span class="p">,</span> <span class="s">&#39;upsilon&#39;</span><span class="p">,</span> <span class="s">&#39;phi&#39;</span><span class="p">,</span> <span class="s">&#39;chi&#39;</span><span class="p">,</span> <span class="s">&#39;psi&#39;</span><span class="p">,</span> <span class="s">&#39;omega&#39;</span> <span class="p">]</span>
-
-<span class="c"># {}  greek letter -&gt; (g,G)</span>
-<span class="n">greek</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">([(</span><span class="n">l</span><span class="p">,</span> <span class="p">(</span><span class="n">g</span><span class="p">(</span><span class="n">l</span><span class="p">),</span> <span class="n">G</span><span class="p">(</span><span class="n">l</span><span class="p">)))</span> <span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="n">greek_letters</span><span class="p">])</span>
-<span class="c"># aliases</span>
-<span class="n">greek</span><span class="p">[</span><span class="s">&#39;lambda&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">greek</span><span class="p">[</span><span class="s">&#39;lamda&#39;</span><span class="p">]</span>
-
-<span class="n">digit_2txt</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="s">&#39;0&#39;</span><span class="p">:</span>    <span class="s">&#39;ZERO&#39;</span><span class="p">,</span>
-    <span class="s">&#39;1&#39;</span><span class="p">:</span>    <span class="s">&#39;ONE&#39;</span><span class="p">,</span>
-    <span class="s">&#39;2&#39;</span><span class="p">:</span>    <span class="s">&#39;TWO&#39;</span><span class="p">,</span>
-    <span class="s">&#39;3&#39;</span><span class="p">:</span>    <span class="s">&#39;THREE&#39;</span><span class="p">,</span>
-    <span class="s">&#39;4&#39;</span><span class="p">:</span>    <span class="s">&#39;FOUR&#39;</span><span class="p">,</span>
-    <span class="s">&#39;5&#39;</span><span class="p">:</span>    <span class="s">&#39;FIVE&#39;</span><span class="p">,</span>
-    <span class="s">&#39;6&#39;</span><span class="p">:</span>    <span class="s">&#39;SIX&#39;</span><span class="p">,</span>
-    <span class="s">&#39;7&#39;</span><span class="p">:</span>    <span class="s">&#39;SEVEN&#39;</span><span class="p">,</span>
-    <span class="s">&#39;8&#39;</span><span class="p">:</span>    <span class="s">&#39;EIGHT&#39;</span><span class="p">,</span>
-    <span class="s">&#39;9&#39;</span><span class="p">:</span>    <span class="s">&#39;NINE&#39;</span><span class="p">,</span>
-<span class="p">}</span>
-
-<span class="n">symb_2txt</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="s">&#39;+&#39;</span><span class="p">:</span>    <span class="s">&#39;PLUS SIGN&#39;</span><span class="p">,</span>
-    <span class="s">&#39;-&#39;</span><span class="p">:</span>    <span class="s">&#39;MINUS&#39;</span><span class="p">,</span>
-    <span class="s">&#39;=&#39;</span><span class="p">:</span>    <span class="s">&#39;EQUALS SIGN&#39;</span><span class="p">,</span>
-    <span class="s">&#39;(&#39;</span><span class="p">:</span>    <span class="s">&#39;LEFT PARENTHESIS&#39;</span><span class="p">,</span>
-    <span class="s">&#39;)&#39;</span><span class="p">:</span>    <span class="s">&#39;RIGHT PARENTHESIS&#39;</span><span class="p">,</span>
-    <span class="s">&#39;[&#39;</span><span class="p">:</span>    <span class="s">&#39;LEFT SQUARE BRACKET&#39;</span><span class="p">,</span>
-    <span class="s">&#39;]&#39;</span><span class="p">:</span>    <span class="s">&#39;RIGHT SQUARE BRACKET&#39;</span><span class="p">,</span>
-    <span class="s">&#39;{&#39;</span><span class="p">:</span>    <span class="s">&#39;LEFT CURLY BRACKET&#39;</span><span class="p">,</span>
-    <span class="s">&#39;}&#39;</span><span class="p">:</span>    <span class="s">&#39;RIGHT CURLY BRACKET&#39;</span><span class="p">,</span>
-
-    <span class="c"># non-std</span>
-    <span class="s">&#39;{}&#39;</span><span class="p">:</span>   <span class="s">&#39;CURLY BRACKET&#39;</span><span class="p">,</span>
-    <span class="s">&#39;sum&#39;</span><span class="p">:</span>  <span class="s">&#39;SUMMATION&#39;</span><span class="p">,</span>
-    <span class="s">&#39;int&#39;</span><span class="p">:</span>  <span class="s">&#39;INTEGRAL&#39;</span><span class="p">,</span>
-<span class="p">}</span>
-
-<span class="c"># SUBSCRIPT &amp; SUPERSCRIPT</span>
-<span class="n">LSUB</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">letter</span><span class="p">:</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;LATIN SUBSCRIPT SMALL LETTER </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">letter</span><span class="o">.</span><span class="n">upper</span><span class="p">())</span>
-<span class="n">GSUB</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">letter</span><span class="p">:</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;GREEK SUBSCRIPT SMALL LETTER </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">letter</span><span class="o">.</span><span class="n">upper</span><span class="p">())</span>
-<span class="n">DSUB</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">digit</span><span class="p">:</span>  <span class="n">U</span><span class="p">(</span><span class="s">&#39;SUBSCRIPT </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">digit_2txt</span><span class="p">[</span><span class="n">digit</span><span class="p">])</span>
-<span class="n">SSUB</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">symb</span><span class="p">:</span>   <span class="n">U</span><span class="p">(</span><span class="s">&#39;SUBSCRIPT </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">symb_2txt</span><span class="p">[</span><span class="n">symb</span><span class="p">])</span>
-
-<span class="n">LSUP</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">letter</span><span class="p">:</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;SUPERSCRIPT LATIN SMALL LETTER </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">letter</span><span class="o">.</span><span class="n">upper</span><span class="p">())</span>
-<span class="n">DSUP</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">digit</span><span class="p">:</span>  <span class="n">U</span><span class="p">(</span><span class="s">&#39;SUPERSCRIPT </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">digit_2txt</span><span class="p">[</span><span class="n">digit</span><span class="p">])</span>
-<span class="n">SSUP</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">symb</span><span class="p">:</span>   <span class="n">U</span><span class="p">(</span><span class="s">&#39;SUPERSCRIPT </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">symb_2txt</span><span class="p">[</span><span class="n">symb</span><span class="p">])</span>
-
-<span class="n">sub</span> <span class="o">=</span> <span class="p">{}</span>    <span class="c"># symb -&gt; subscript symbol</span>
-<span class="n">sup</span> <span class="o">=</span> <span class="p">{}</span>    <span class="c"># symb -&gt; superscript symbol</span>
-
-<span class="c"># latin subscripts</span>
-<span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="s">&#39;aeioruvx&#39;</span><span class="p">:</span>
-    <span class="n">sub</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">LSUB</span><span class="p">(</span><span class="n">l</span><span class="p">)</span>
-
-<span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="s">&#39;in&#39;</span><span class="p">:</span>
-    <span class="n">sup</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">=</span> <span class="n">LSUP</span><span class="p">(</span><span class="n">l</span><span class="p">)</span>
-
-<span class="k">for</span> <span class="n">gl</span> <span class="ow">in</span> <span class="p">[</span><span class="s">&#39;beta&#39;</span><span class="p">,</span> <span class="s">&#39;gamma&#39;</span><span class="p">,</span> <span class="s">&#39;rho&#39;</span><span class="p">,</span> <span class="s">&#39;phi&#39;</span><span class="p">,</span> <span class="s">&#39;chi&#39;</span><span class="p">]:</span>
-    <span class="n">sub</span><span class="p">[</span><span class="n">gl</span><span class="p">]</span> <span class="o">=</span> <span class="n">GSUB</span><span class="p">(</span><span class="n">gl</span><span class="p">)</span>
-
-<span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="p">[</span><span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">)]:</span>
-    <span class="n">sub</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">=</span> <span class="n">DSUB</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-    <span class="n">sup</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">=</span> <span class="n">DSUP</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-
-<span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="s">&#39;+-=()&#39;</span><span class="p">:</span>
-    <span class="n">sub</span><span class="p">[</span><span class="n">s</span><span class="p">]</span> <span class="o">=</span> <span class="n">SSUB</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-    <span class="n">sup</span><span class="p">[</span><span class="n">s</span><span class="p">]</span> <span class="o">=</span> <span class="n">SSUP</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-
-
-<span class="c"># VERTICAL OBJECTS</span>
-<span class="n">HUP</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">symb</span><span class="p">:</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> UPPER HOOK&#39;</span> <span class="o">%</span> <span class="n">symb_2txt</span><span class="p">[</span><span class="n">symb</span><span class="p">])</span>
-<span class="n">CUP</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">symb</span><span class="p">:</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> UPPER CORNER&#39;</span> <span class="o">%</span> <span class="n">symb_2txt</span><span class="p">[</span><span class="n">symb</span><span class="p">])</span>
-<span class="n">MID</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">symb</span><span class="p">:</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> MIDDLE PIECE&#39;</span> <span class="o">%</span> <span class="n">symb_2txt</span><span class="p">[</span><span class="n">symb</span><span class="p">])</span>
-<span class="n">EXT</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">symb</span><span class="p">:</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> EXTENSION&#39;</span> <span class="o">%</span> <span class="n">symb_2txt</span><span class="p">[</span><span class="n">symb</span><span class="p">])</span>
-<span class="n">HLO</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">symb</span><span class="p">:</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> LOWER HOOK&#39;</span> <span class="o">%</span> <span class="n">symb_2txt</span><span class="p">[</span><span class="n">symb</span><span class="p">])</span>
-<span class="n">CLO</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">symb</span><span class="p">:</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> LOWER CORNER&#39;</span> <span class="o">%</span> <span class="n">symb_2txt</span><span class="p">[</span><span class="n">symb</span><span class="p">])</span>
-<span class="n">TOP</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">symb</span><span class="p">:</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> TOP&#39;</span> <span class="o">%</span> <span class="n">symb_2txt</span><span class="p">[</span><span class="n">symb</span><span class="p">])</span>
-<span class="n">BOT</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">symb</span><span class="p">:</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> BOTTOM&#39;</span> <span class="o">%</span> <span class="n">symb_2txt</span><span class="p">[</span><span class="n">symb</span><span class="p">])</span>
-
-<span class="c"># {} &#39;(&#39;  -&gt;  (extension, start, end, middle) 1-character</span>
-<span class="n">_xobj_unicode</span> <span class="o">=</span> <span class="p">{</span>
-
-    <span class="c"># vertical symbols</span>
-    <span class="c">#       (( ext, top, bot, mid ), c1)</span>
-    <span class="s">&#39;(&#39;</span><span class="p">:</span>    <span class="p">((</span> <span class="n">EXT</span><span class="p">(</span><span class="s">&#39;(&#39;</span><span class="p">),</span> <span class="n">HUP</span><span class="p">(</span><span class="s">&#39;(&#39;</span><span class="p">),</span> <span class="n">HLO</span><span class="p">(</span><span class="s">&#39;(&#39;</span><span class="p">)</span> <span class="p">),</span> <span class="s">&#39;(&#39;</span><span class="p">),</span>
-    <span class="s">&#39;)&#39;</span><span class="p">:</span>    <span class="p">((</span> <span class="n">EXT</span><span class="p">(</span><span class="s">&#39;)&#39;</span><span class="p">),</span> <span class="n">HUP</span><span class="p">(</span><span class="s">&#39;)&#39;</span><span class="p">),</span> <span class="n">HLO</span><span class="p">(</span><span class="s">&#39;)&#39;</span><span class="p">)</span> <span class="p">),</span> <span class="s">&#39;)&#39;</span><span class="p">),</span>
-    <span class="s">&#39;[&#39;</span><span class="p">:</span>    <span class="p">((</span> <span class="n">EXT</span><span class="p">(</span><span class="s">&#39;[&#39;</span><span class="p">),</span> <span class="n">CUP</span><span class="p">(</span><span class="s">&#39;[&#39;</span><span class="p">),</span> <span class="n">CLO</span><span class="p">(</span><span class="s">&#39;[&#39;</span><span class="p">)</span> <span class="p">),</span> <span class="s">&#39;[&#39;</span><span class="p">),</span>
-    <span class="s">&#39;]&#39;</span><span class="p">:</span>    <span class="p">((</span> <span class="n">EXT</span><span class="p">(</span><span class="s">&#39;]&#39;</span><span class="p">),</span> <span class="n">CUP</span><span class="p">(</span><span class="s">&#39;]&#39;</span><span class="p">),</span> <span class="n">CLO</span><span class="p">(</span><span class="s">&#39;]&#39;</span><span class="p">)</span> <span class="p">),</span> <span class="s">&#39;]&#39;</span><span class="p">),</span>
-    <span class="s">&#39;{&#39;</span><span class="p">:</span>    <span class="p">((</span> <span class="n">EXT</span><span class="p">(</span><span class="s">&#39;{}&#39;</span><span class="p">),</span> <span class="n">HUP</span><span class="p">(</span><span class="s">&#39;{&#39;</span><span class="p">),</span> <span class="n">HLO</span><span class="p">(</span><span class="s">&#39;{&#39;</span><span class="p">),</span> <span class="n">MID</span><span class="p">(</span><span class="s">&#39;{&#39;</span><span class="p">)</span> <span class="p">),</span> <span class="s">&#39;{&#39;</span><span class="p">),</span>
-    <span class="s">&#39;}&#39;</span><span class="p">:</span>    <span class="p">((</span> <span class="n">EXT</span><span class="p">(</span><span class="s">&#39;{}&#39;</span><span class="p">),</span> <span class="n">HUP</span><span class="p">(</span><span class="s">&#39;}&#39;</span><span class="p">),</span> <span class="n">HLO</span><span class="p">(</span><span class="s">&#39;}&#39;</span><span class="p">),</span> <span class="n">MID</span><span class="p">(</span><span class="s">&#39;}&#39;</span><span class="p">)</span> <span class="p">),</span> <span class="s">&#39;}&#39;</span><span class="p">),</span>
-    <span class="s">&#39;|&#39;</span><span class="p">:</span>    <span class="n">U</span><span class="p">(</span><span class="s">&#39;BOX DRAWINGS LIGHT VERTICAL&#39;</span><span class="p">),</span>
-
-    <span class="s">&#39;&lt;&#39;</span><span class="p">:</span>    <span class="p">((</span><span class="n">U</span><span class="p">(</span><span class="s">&#39;BOX DRAWINGS LIGHT VERTICAL&#39;</span><span class="p">),</span>
-              <span class="n">U</span><span class="p">(</span><span class="s">&#39;BOX DRAWINGS LIGHT DIAGONAL UPPER RIGHT TO LOWER LEFT&#39;</span><span class="p">),</span>
-              <span class="n">U</span><span class="p">(</span><span class="s">&#39;BOX DRAWINGS LIGHT DIAGONAL UPPER LEFT TO LOWER RIGHT&#39;</span><span class="p">)),</span> <span class="s">&#39;&lt;&#39;</span><span class="p">),</span>
-
-    <span class="s">&#39;&gt;&#39;</span><span class="p">:</span>    <span class="p">((</span><span class="n">U</span><span class="p">(</span><span class="s">&#39;BOX DRAWINGS LIGHT VERTICAL&#39;</span><span class="p">),</span>
-              <span class="n">U</span><span class="p">(</span><span class="s">&#39;BOX DRAWINGS LIGHT DIAGONAL UPPER LEFT TO LOWER RIGHT&#39;</span><span class="p">),</span>
-              <span class="n">U</span><span class="p">(</span><span class="s">&#39;BOX DRAWINGS LIGHT DIAGONAL UPPER RIGHT TO LOWER LEFT&#39;</span><span class="p">)),</span> <span class="s">&#39;&gt;&#39;</span><span class="p">),</span>
-
-    <span class="s">&#39;lfloor&#39;</span><span class="p">:</span> <span class="p">((</span> <span class="n">EXT</span><span class="p">(</span><span class="s">&#39;[&#39;</span><span class="p">),</span> <span class="n">EXT</span><span class="p">(</span><span class="s">&#39;[&#39;</span><span class="p">),</span> <span class="n">CLO</span><span class="p">(</span><span class="s">&#39;[&#39;</span><span class="p">)</span> <span class="p">),</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;LEFT FLOOR&#39;</span><span class="p">)),</span>
-    <span class="s">&#39;rfloor&#39;</span><span class="p">:</span> <span class="p">((</span> <span class="n">EXT</span><span class="p">(</span><span class="s">&#39;]&#39;</span><span class="p">),</span> <span class="n">EXT</span><span class="p">(</span><span class="s">&#39;]&#39;</span><span class="p">),</span> <span class="n">CLO</span><span class="p">(</span><span class="s">&#39;]&#39;</span><span class="p">)</span> <span class="p">),</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;RIGHT FLOOR&#39;</span><span class="p">)),</span>
-    <span class="s">&#39;lceil&#39;</span><span class="p">:</span>  <span class="p">((</span> <span class="n">EXT</span><span class="p">(</span><span class="s">&#39;[&#39;</span><span class="p">),</span> <span class="n">CUP</span><span class="p">(</span><span class="s">&#39;[&#39;</span><span class="p">),</span> <span class="n">EXT</span><span class="p">(</span><span class="s">&#39;[&#39;</span><span class="p">)</span> <span class="p">),</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;LEFT CEILING&#39;</span><span class="p">)),</span>
-    <span class="s">&#39;rceil&#39;</span><span class="p">:</span>  <span class="p">((</span> <span class="n">EXT</span><span class="p">(</span><span class="s">&#39;]&#39;</span><span class="p">),</span> <span class="n">CUP</span><span class="p">(</span><span class="s">&#39;]&#39;</span><span class="p">),</span> <span class="n">EXT</span><span class="p">(</span><span class="s">&#39;]&#39;</span><span class="p">)</span> <span class="p">),</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;RIGHT CEILING&#39;</span><span class="p">)),</span>
-
-    <span class="s">&#39;int&#39;</span><span class="p">:</span>  <span class="p">((</span> <span class="n">EXT</span><span class="p">(</span><span class="s">&#39;int&#39;</span><span class="p">),</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;TOP HALF INTEGRAL&#39;</span><span class="p">),</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;BOTTOM HALF INTEGRAL&#39;</span><span class="p">)</span> <span class="p">),</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;INTEGRAL&#39;</span><span class="p">)),</span>
-    <span class="s">&#39;sum&#39;</span><span class="p">:</span>  <span class="p">((</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;BOX DRAWINGS LIGHT DIAGONAL UPPER LEFT TO LOWER RIGHT&#39;</span><span class="p">),</span> <span class="s">&#39;_&#39;</span><span class="p">,</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;OVERLINE&#39;</span><span class="p">),</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;BOX DRAWINGS LIGHT DIAGONAL UPPER RIGHT TO LOWER LEFT&#39;</span><span class="p">)),</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;N-ARY SUMMATION&#39;</span><span class="p">)),</span>
-
-    <span class="c"># horizontal objects</span>
-    <span class="c">#&#39;-&#39;:   &#39;-&#39;,</span>
-    <span class="s">&#39;-&#39;</span><span class="p">:</span>    <span class="n">U</span><span class="p">(</span><span class="s">&#39;BOX DRAWINGS LIGHT HORIZONTAL&#39;</span><span class="p">),</span>
-    <span class="s">&#39;_&#39;</span><span class="p">:</span>    <span class="n">U</span><span class="p">(</span><span class="s">&#39;LOW LINE&#39;</span><span class="p">),</span>
-    <span class="c"># We used to use this, but LOW LINE looks better for roots, as it&#39;s a</span>
-    <span class="c"># little lower (i.e., it lines up with the / perfectly.  But perhaps this</span>
-    <span class="c"># one would still be wanted for some cases?</span>
-    <span class="c"># &#39;_&#39;:    U(&#39;HORIZONTAL SCAN LINE-9&#39;),</span>
-
-    <span class="c"># diagonal objects &#39;\&#39; &amp; &#39;/&#39; ?</span>
-    <span class="s">&#39;/&#39;</span><span class="p">:</span>    <span class="n">U</span><span class="p">(</span><span class="s">&#39;BOX DRAWINGS LIGHT DIAGONAL UPPER RIGHT TO LOWER LEFT&#39;</span><span class="p">),</span>
-    <span class="s">&#39;</span><span class="se">\\</span><span class="s">&#39;</span><span class="p">:</span>   <span class="n">U</span><span class="p">(</span><span class="s">&#39;BOX DRAWINGS LIGHT DIAGONAL UPPER LEFT TO LOWER RIGHT&#39;</span><span class="p">),</span>
-<span class="p">}</span>
-
-<span class="n">_xobj_ascii</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="c"># vertical symbols</span>
-    <span class="c">#       (( ext, top, bot, mid ), c1)</span>
-    <span class="s">&#39;(&#39;</span><span class="p">:</span>    <span class="p">((</span> <span class="s">&#39;|&#39;</span><span class="p">,</span> <span class="s">&#39;/&#39;</span><span class="p">,</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">&#39;</span> <span class="p">),</span> <span class="s">&#39;(&#39;</span><span class="p">),</span>
-    <span class="s">&#39;)&#39;</span><span class="p">:</span>    <span class="p">((</span> <span class="s">&#39;|&#39;</span><span class="p">,</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;/&#39;</span> <span class="p">),</span> <span class="s">&#39;)&#39;</span><span class="p">),</span>
-
-<span class="c"># XXX this looks ugly</span>
-<span class="c">#   &#39;[&#39;:    (( &#39;|&#39;, &#39;-&#39;, &#39;-&#39; ), &#39;[&#39;),</span>
-<span class="c">#   &#39;]&#39;:    (( &#39;|&#39;, &#39;-&#39;, &#39;-&#39; ), &#39;]&#39;),</span>
-<span class="c"># XXX not so ugly :(</span>
-    <span class="s">&#39;[&#39;</span><span class="p">:</span>    <span class="p">((</span> <span class="s">&#39;[&#39;</span><span class="p">,</span> <span class="s">&#39;[&#39;</span><span class="p">,</span> <span class="s">&#39;[&#39;</span> <span class="p">),</span> <span class="s">&#39;[&#39;</span><span class="p">),</span>
-    <span class="s">&#39;]&#39;</span><span class="p">:</span>    <span class="p">((</span> <span class="s">&#39;]&#39;</span><span class="p">,</span> <span class="s">&#39;]&#39;</span><span class="p">,</span> <span class="s">&#39;]&#39;</span> <span class="p">),</span> <span class="s">&#39;]&#39;</span><span class="p">),</span>
-
-    <span class="s">&#39;{&#39;</span><span class="p">:</span>    <span class="p">((</span> <span class="s">&#39;|&#39;</span><span class="p">,</span> <span class="s">&#39;/&#39;</span><span class="p">,</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;&lt;&#39;</span> <span class="p">),</span> <span class="s">&#39;{&#39;</span><span class="p">),</span>
-    <span class="s">&#39;}&#39;</span><span class="p">:</span>    <span class="p">((</span> <span class="s">&#39;|&#39;</span><span class="p">,</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;/&#39;</span><span class="p">,</span> <span class="s">&#39;&gt;&#39;</span> <span class="p">),</span> <span class="s">&#39;}&#39;</span><span class="p">),</span>
-    <span class="s">&#39;|&#39;</span><span class="p">:</span>    <span class="s">&#39;|&#39;</span><span class="p">,</span>
-
-    <span class="s">&#39;&lt;&#39;</span><span class="p">:</span>    <span class="p">((</span> <span class="s">&#39;|&#39;</span><span class="p">,</span> <span class="s">&#39;/&#39;</span><span class="p">,</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">&#39;</span> <span class="p">),</span> <span class="s">&#39;&lt;&#39;</span><span class="p">),</span>
-    <span class="s">&#39;&gt;&#39;</span><span class="p">:</span>    <span class="p">((</span> <span class="s">&#39;|&#39;</span><span class="p">,</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">&#39;</span><span class="p">,</span> <span class="s">&#39;/&#39;</span> <span class="p">),</span> <span class="s">&#39;&gt;&#39;</span><span class="p">),</span>
-
-    <span class="s">&#39;int&#39;</span><span class="p">:</span>  <span class="p">(</span> <span class="s">&#39; | &#39;</span><span class="p">,</span> <span class="s">&#39;  /&#39;</span><span class="p">,</span> <span class="s">&#39;/  &#39;</span> <span class="p">),</span>
-
-    <span class="c"># horizontal objects</span>
-    <span class="s">&#39;-&#39;</span><span class="p">:</span>    <span class="s">&#39;-&#39;</span><span class="p">,</span>
-    <span class="s">&#39;_&#39;</span><span class="p">:</span>    <span class="s">&#39;_&#39;</span><span class="p">,</span>
-
-    <span class="c"># diagonal objects &#39;\&#39; &amp; &#39;/&#39; ?</span>
-    <span class="s">&#39;/&#39;</span><span class="p">:</span>    <span class="s">&#39;/&#39;</span><span class="p">,</span>
-    <span class="s">&#39;</span><span class="se">\\</span><span class="s">&#39;</span><span class="p">:</span>   <span class="s">&#39;</span><span class="se">\\</span><span class="s">&#39;</span><span class="p">,</span>
-<span class="p">}</span>
-
-
-<div class="viewcode-block" id="xobj"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.pretty_symbology.xobj">[docs]</a><span class="k">def</span> <span class="nf">xobj</span><span class="p">(</span><span class="n">symb</span><span class="p">,</span> <span class="n">length</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Construct spatial object of given length.</span>
-
-<span class="sd">    return: [] of equal-length strings</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">assert</span> <span class="n">length</span> <span class="o">&gt;</span> <span class="mi">0</span>
-
-    <span class="c"># TODO robustify when no unicodedat available</span>
-    <span class="k">if</span> <span class="n">_use_unicode</span><span class="p">:</span>
-        <span class="n">_xobj</span> <span class="o">=</span> <span class="n">_xobj_unicode</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">_xobj</span> <span class="o">=</span> <span class="n">_xobj_ascii</span>
-
-    <span class="n">vinfo</span> <span class="o">=</span> <span class="n">_xobj</span><span class="p">[</span><span class="n">symb</span><span class="p">]</span>
-
-    <span class="n">c1</span> <span class="o">=</span> <span class="n">top</span> <span class="o">=</span> <span class="n">bot</span> <span class="o">=</span> <span class="n">mid</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">vinfo</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span>        <span class="c"># 1 entry</span>
-        <span class="n">ext</span> <span class="o">=</span> <span class="n">vinfo</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">vinfo</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="nb">tuple</span><span class="p">):</span>     <span class="c"># (vlong), c1</span>
-            <span class="n">vlong</span> <span class="o">=</span> <span class="n">vinfo</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">c1</span> <span class="o">=</span> <span class="n">vinfo</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>                               <span class="c"># (vlong), c1</span>
-            <span class="n">vlong</span> <span class="o">=</span> <span class="n">vinfo</span>
-
-        <span class="n">ext</span> <span class="o">=</span> <span class="n">vlong</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">top</span> <span class="o">=</span> <span class="n">vlong</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-            <span class="n">bot</span> <span class="o">=</span> <span class="n">vlong</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-            <span class="n">mid</span> <span class="o">=</span> <span class="n">vlong</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span>
-        <span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span>
-            <span class="k">pass</span>
-
-    <span class="k">if</span> <span class="n">c1</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">c1</span> <span class="o">=</span> <span class="n">ext</span>
-    <span class="k">if</span> <span class="n">top</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">top</span> <span class="o">=</span> <span class="n">ext</span>
-    <span class="k">if</span> <span class="n">bot</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">bot</span> <span class="o">=</span> <span class="n">ext</span>
-    <span class="k">if</span> <span class="n">mid</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">if</span> <span class="p">(</span><span class="n">length</span> <span class="o">%</span> <span class="mi">2</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="c"># even height, but we have to print it somehow anyway...</span>
-            <span class="c"># XXX is it ok?</span>
-            <span class="n">length</span> <span class="o">+=</span> <span class="mi">1</span>
-
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">mid</span> <span class="o">=</span> <span class="n">ext</span>
-
-    <span class="k">if</span> <span class="n">length</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">c1</span>
-
-    <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="nb">next</span> <span class="o">=</span> <span class="p">(</span><span class="n">length</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span>
-    <span class="n">nmid</span> <span class="o">=</span> <span class="p">(</span><span class="n">length</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="nb">next</span><span class="o">*</span><span class="mi">2</span>
-
-    <span class="n">res</span> <span class="o">+=</span> <span class="p">[</span><span class="n">top</span><span class="p">]</span>
-    <span class="n">res</span> <span class="o">+=</span> <span class="p">[</span><span class="n">ext</span><span class="p">]</span><span class="o">*</span><span class="nb">next</span>
-    <span class="n">res</span> <span class="o">+=</span> <span class="p">[</span><span class="n">mid</span><span class="p">]</span><span class="o">*</span><span class="n">nmid</span>
-    <span class="n">res</span> <span class="o">+=</span> <span class="p">[</span><span class="n">ext</span><span class="p">]</span><span class="o">*</span><span class="nb">next</span>
-    <span class="n">res</span> <span class="o">+=</span> <span class="p">[</span><span class="n">bot</span><span class="p">]</span>
-
-    <span class="k">return</span> <span class="n">res</span>
-
-</div>
-<div class="viewcode-block" id="vobj"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.pretty_symbology.vobj">[docs]</a><span class="k">def</span> <span class="nf">vobj</span><span class="p">(</span><span class="n">symb</span><span class="p">,</span> <span class="n">height</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Construct vertical object of a given height</span>
-
-<span class="sd">       see: xobj</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span> <span class="n">xobj</span><span class="p">(</span><span class="n">symb</span><span class="p">,</span> <span class="n">height</span><span class="p">)</span> <span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="hobj"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.pretty_symbology.hobj">[docs]</a><span class="k">def</span> <span class="nf">hobj</span><span class="p">(</span><span class="n">symb</span><span class="p">,</span> <span class="n">width</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Construct horizontal object of a given width</span>
-
-<span class="sd">       see: xobj</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span> <span class="n">xobj</span><span class="p">(</span><span class="n">symb</span><span class="p">,</span> <span class="n">width</span><span class="p">)</span> <span class="p">)</span>
-
-<span class="c"># RADICAL</span>
-<span class="c"># n -&gt; symbol</span></div>
-<span class="n">root</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="mi">2</span><span class="p">:</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;SQUARE ROOT&#39;</span><span class="p">),</span>   <span class="c"># U(&#39;RADICAL SYMBOL BOTTOM&#39;)</span>
-    <span class="mi">3</span><span class="p">:</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;CUBE ROOT&#39;</span><span class="p">),</span>
-    <span class="mi">4</span><span class="p">:</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;FOURTH ROOT&#39;</span><span class="p">),</span>
-<span class="p">}</span>
-
-
-<span class="c"># RATIONAL</span>
-<span class="n">VF</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">txt</span><span class="p">:</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;VULGAR FRACTION </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">txt</span><span class="p">)</span>
-
-<span class="c"># (p,q) -&gt; symbol</span>
-<span class="n">frac</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">):</span> <span class="n">VF</span><span class="p">(</span><span class="s">&#39;ONE HALF&#39;</span><span class="p">),</span>
-    <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">):</span> <span class="n">VF</span><span class="p">(</span><span class="s">&#39;ONE THIRD&#39;</span><span class="p">),</span>
-    <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">):</span> <span class="n">VF</span><span class="p">(</span><span class="s">&#39;TWO THIRDS&#39;</span><span class="p">),</span>
-    <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">):</span> <span class="n">VF</span><span class="p">(</span><span class="s">&#39;ONE QUARTER&#39;</span><span class="p">),</span>
-    <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">):</span> <span class="n">VF</span><span class="p">(</span><span class="s">&#39;THREE QUARTERS&#39;</span><span class="p">),</span>
-    <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">):</span> <span class="n">VF</span><span class="p">(</span><span class="s">&#39;ONE FIFTH&#39;</span><span class="p">),</span>
-    <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">):</span> <span class="n">VF</span><span class="p">(</span><span class="s">&#39;TWO FIFTHS&#39;</span><span class="p">),</span>
-    <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">):</span> <span class="n">VF</span><span class="p">(</span><span class="s">&#39;THREE FIFTHS&#39;</span><span class="p">),</span>
-    <span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">):</span> <span class="n">VF</span><span class="p">(</span><span class="s">&#39;FOUR FIFTHS&#39;</span><span class="p">),</span>
-    <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">):</span> <span class="n">VF</span><span class="p">(</span><span class="s">&#39;ONE SIXTH&#39;</span><span class="p">),</span>
-    <span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">):</span> <span class="n">VF</span><span class="p">(</span><span class="s">&#39;FIVE SIXTHS&#39;</span><span class="p">),</span>
-    <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">8</span><span class="p">):</span> <span class="n">VF</span><span class="p">(</span><span class="s">&#39;ONE EIGHTH&#39;</span><span class="p">),</span>
-    <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">8</span><span class="p">):</span> <span class="n">VF</span><span class="p">(</span><span class="s">&#39;THREE EIGHTHS&#39;</span><span class="p">),</span>
-    <span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">8</span><span class="p">):</span> <span class="n">VF</span><span class="p">(</span><span class="s">&#39;FIVE EIGHTHS&#39;</span><span class="p">),</span>
-    <span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">):</span> <span class="n">VF</span><span class="p">(</span><span class="s">&#39;SEVEN EIGHTHS&#39;</span><span class="p">),</span>
-<span class="p">}</span>
-
-
-<span class="c"># atom symbols</span>
-<span class="n">_xsym</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="s">&#39;==&#39;</span><span class="p">:</span>  <span class="p">(</span><span class="s">&#39;=&#39;</span><span class="p">,</span> <span class="s">&#39;=&#39;</span><span class="p">),</span>
-    <span class="s">&#39;&lt;&#39;</span><span class="p">:</span>   <span class="p">(</span><span class="s">&#39;&lt;&#39;</span><span class="p">,</span> <span class="s">&#39;&lt;&#39;</span><span class="p">),</span>
-    <span class="s">&#39;&gt;&#39;</span><span class="p">:</span>   <span class="p">(</span><span class="s">&#39;&gt;&#39;</span><span class="p">,</span> <span class="s">&#39;&gt;&#39;</span><span class="p">),</span>
-    <span class="s">&#39;&lt;=&#39;</span><span class="p">:</span>  <span class="p">(</span><span class="s">&#39;&lt;=&#39;</span><span class="p">,</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;LESS-THAN OR EQUAL TO&#39;</span><span class="p">)),</span>
-    <span class="s">&#39;&gt;=&#39;</span><span class="p">:</span>  <span class="p">(</span><span class="s">&#39;&gt;=&#39;</span><span class="p">,</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;GREATER-THAN OR EQUAL TO&#39;</span><span class="p">)),</span>
-    <span class="s">&#39;!=&#39;</span><span class="p">:</span>  <span class="p">(</span><span class="s">&#39;!=&#39;</span><span class="p">,</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;NOT EQUAL TO&#39;</span><span class="p">)),</span>
-    <span class="s">&#39;*&#39;</span><span class="p">:</span>   <span class="p">(</span><span class="s">&#39;*&#39;</span><span class="p">,</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;DOT OPERATOR&#39;</span><span class="p">)),</span>
-    <span class="s">&#39;--&gt;&#39;</span><span class="p">:</span> <span class="p">(</span><span class="s">&#39;--&gt;&#39;</span><span class="p">,</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;EM DASH&#39;</span><span class="p">)</span> <span class="o">+</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;EM DASH&#39;</span><span class="p">)</span> <span class="o">+</span>
-            <span class="n">U</span><span class="p">(</span><span class="s">&#39;BLACK RIGHT-POINTING TRIANGLE&#39;</span><span class="p">)),</span>
-    <span class="s">&#39;==&gt;&#39;</span><span class="p">:</span> <span class="p">(</span><span class="s">&#39;==&gt;&#39;</span><span class="p">,</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;BOX DRAWINGS DOUBLE HORIZONTAL&#39;</span><span class="p">)</span> <span class="o">+</span>
-            <span class="n">U</span><span class="p">(</span><span class="s">&#39;BOX DRAWINGS DOUBLE HORIZONTAL&#39;</span><span class="p">)</span> <span class="o">+</span>
-            <span class="n">U</span><span class="p">(</span><span class="s">&#39;BLACK RIGHT-POINTING TRIANGLE&#39;</span><span class="p">)),</span>
-    <span class="s">&#39;.&#39;</span><span class="p">:</span>   <span class="p">(</span><span class="s">&#39;*&#39;</span><span class="p">,</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;RING OPERATOR&#39;</span><span class="p">)),</span>
-<span class="p">}</span>
-
-
-<div class="viewcode-block" id="xsym"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.pretty_symbology.xsym">[docs]</a><span class="k">def</span> <span class="nf">xsym</span><span class="p">(</span><span class="n">sym</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;get symbology for a &#39;character&#39;&quot;&quot;&quot;</span>
-    <span class="n">op</span> <span class="o">=</span> <span class="n">_xsym</span><span class="p">[</span><span class="n">sym</span><span class="p">]</span>
-
-    <span class="k">if</span> <span class="n">_use_unicode</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">op</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">op</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-
-<span class="c"># SYMBOLS</span>
-</div>
-<span class="n">atoms_table</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="c"># class             how-to-display</span>
-    <span class="s">&#39;Exp1&#39;</span><span class="p">:</span>             <span class="n">U</span><span class="p">(</span><span class="s">&#39;SCRIPT SMALL E&#39;</span><span class="p">),</span>
-    <span class="s">&#39;Pi&#39;</span><span class="p">:</span>               <span class="n">U</span><span class="p">(</span><span class="s">&#39;GREEK SMALL LETTER PI&#39;</span><span class="p">),</span>
-    <span class="s">&#39;Infinity&#39;</span><span class="p">:</span>         <span class="n">U</span><span class="p">(</span><span class="s">&#39;INFINITY&#39;</span><span class="p">),</span>
-    <span class="s">&#39;NegativeInfinity&#39;</span><span class="p">:</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;INFINITY&#39;</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="s">&#39;-&#39;</span> <span class="o">+</span> <span class="n">U</span><span class="p">(</span><span class="s">&#39;INFINITY&#39;</span><span class="p">)),</span>  <span class="c"># XXX what to do here</span>
-    <span class="c">#&#39;ImaginaryUnit&#39;:    U(&#39;GREEK SMALL LETTER IOTA&#39;),</span>
-    <span class="c">#&#39;ImaginaryUnit&#39;:    U(&#39;MATHEMATICAL ITALIC SMALL I&#39;),</span>
-    <span class="s">&#39;ImaginaryUnit&#39;</span><span class="p">:</span>    <span class="n">U</span><span class="p">(</span><span class="s">&#39;DOUBLE-STRUCK ITALIC SMALL I&#39;</span><span class="p">),</span>
-    <span class="s">&#39;EmptySet&#39;</span><span class="p">:</span>         <span class="n">U</span><span class="p">(</span><span class="s">&#39;EMPTY SET&#39;</span><span class="p">),</span>
-    <span class="s">&#39;Naturals&#39;</span><span class="p">:</span>         <span class="n">U</span><span class="p">(</span><span class="s">&#39;DOUBLE-STRUCK CAPITAL N&#39;</span><span class="p">),</span>
-    <span class="s">&#39;Integers&#39;</span><span class="p">:</span>         <span class="n">U</span><span class="p">(</span><span class="s">&#39;DOUBLE-STRUCK CAPITAL Z&#39;</span><span class="p">),</span>
-    <span class="s">&#39;Reals&#39;</span><span class="p">:</span>            <span class="n">U</span><span class="p">(</span><span class="s">&#39;DOUBLE-STRUCK CAPITAL R&#39;</span><span class="p">),</span>
-    <span class="s">&#39;Union&#39;</span><span class="p">:</span>            <span class="n">U</span><span class="p">(</span><span class="s">&#39;UNION&#39;</span><span class="p">),</span>
-    <span class="s">&#39;Intersection&#39;</span><span class="p">:</span>     <span class="n">U</span><span class="p">(</span><span class="s">&#39;INTERSECTION&#39;</span><span class="p">),</span>
-    <span class="s">&#39;Ring&#39;</span><span class="p">:</span>             <span class="n">U</span><span class="p">(</span><span class="s">&#39;RING OPERATOR&#39;</span><span class="p">)</span>
-<span class="p">}</span>
-
-
-<div class="viewcode-block" id="pretty_atom"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.pretty_symbology.pretty_atom">[docs]</a><span class="k">def</span> <span class="nf">pretty_atom</span><span class="p">(</span><span class="n">atom_name</span><span class="p">,</span> <span class="n">default</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;return pretty representation of an atom&quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">_use_unicode</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">atoms_table</span><span class="p">[</span><span class="n">atom_name</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">default</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">default</span>
-
-        <span class="k">raise</span> <span class="ne">KeyError</span><span class="p">(</span><span class="s">&#39;only unicode&#39;</span><span class="p">)</span>  <span class="c"># send it default printer</span>
-
-</div>
-<div class="viewcode-block" id="pretty_symbol"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.pretty_symbology.pretty_symbol">[docs]</a><span class="k">def</span> <span class="nf">pretty_symbol</span><span class="p">(</span><span class="n">symb_name</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;return pretty representation of a symbol&quot;&quot;&quot;</span>
-    <span class="c"># let&#39;s split symb_name into symbol + index</span>
-    <span class="c"># UC: beta1</span>
-    <span class="c"># UC: f_beta</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">_use_unicode</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">symb_name</span>
-
-    <span class="n">name</span><span class="p">,</span> <span class="n">sups</span><span class="p">,</span> <span class="n">subs</span> <span class="o">=</span> <span class="n">split_super_sub</span><span class="p">(</span><span class="n">symb_name</span><span class="p">)</span>
-
-    <span class="c"># let&#39;s prettify name</span>
-    <span class="n">gG</span> <span class="o">=</span> <span class="n">greek</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">name</span><span class="o">.</span><span class="n">lower</span><span class="p">())</span>
-    <span class="k">if</span> <span class="n">gG</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">name</span><span class="o">.</span><span class="n">islower</span><span class="p">():</span>
-            <span class="n">greek_name</span> <span class="o">=</span> <span class="n">greek</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">name</span><span class="o">.</span><span class="n">lower</span><span class="p">())[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">greek_name</span> <span class="o">=</span> <span class="n">greek</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">name</span><span class="o">.</span><span class="n">lower</span><span class="p">())[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="c"># some letters may not be available</span>
-        <span class="k">if</span> <span class="n">greek_name</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">name</span> <span class="o">=</span> <span class="n">greek_name</span>
-
-    <span class="c"># Let&#39;s prettify sups/subs. If it fails at one of them, pretty sups/subs are</span>
-    <span class="c"># not used at all.</span>
-    <span class="k">def</span> <span class="nf">pretty_list</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">mapping</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">l</span><span class="p">:</span>
-            <span class="n">pretty</span> <span class="o">=</span> <span class="n">mapping</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">pretty</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">try</span><span class="p">:</span>  <span class="c"># match by separate characters</span>
-                    <span class="n">pretty</span> <span class="o">=</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="n">mapping</span><span class="p">[</span><span class="n">c</span><span class="p">]</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">s</span><span class="p">])</span>
-                <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">pretty</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="n">pretty_sups</span> <span class="o">=</span> <span class="n">pretty_list</span><span class="p">(</span><span class="n">sups</span><span class="p">,</span> <span class="n">sup</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">pretty_sups</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">pretty_subs</span> <span class="o">=</span> <span class="n">pretty_list</span><span class="p">(</span><span class="n">subs</span><span class="p">,</span> <span class="n">sub</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">pretty_subs</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="c"># glue the results into one string</span>
-    <span class="k">if</span> <span class="n">pretty_subs</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>  <span class="c"># nice formatting of sups/subs did not work</span>
-        <span class="k">return</span> <span class="n">symb_name</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">sups_result</span> <span class="o">=</span> <span class="s">&#39; &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">pretty_sups</span><span class="p">)</span>
-        <span class="n">subs_result</span> <span class="o">=</span> <span class="s">&#39; &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">pretty_subs</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="n">name</span><span class="p">,</span> <span class="n">sups_result</span><span class="p">,</span> <span class="n">subs_result</span><span class="p">])</span>
-
-</div>
-<div class="viewcode-block" id="annotated"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.pretty_symbology.annotated">[docs]</a><span class="k">def</span> <span class="nf">annotated</span><span class="p">(</span><span class="n">letter</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a stylised drawing of the letter ``letter``, together with</span>
-<span class="sd">    information on how to put annotations (super- and subscripts to the</span>
-<span class="sd">    left and to the right) on it.</span>
-
-<span class="sd">    See pretty.py functions _print_meijerg, _print_hyper on how to use this</span>
-<span class="sd">    information.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">ucode_pics</span> <span class="o">=</span> <span class="p">{</span>
-        <span class="s">&#39;F&#39;</span><span class="p">:</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&#39;</span><span class="se">\u250c\u2500\n\u251c\u2500\n\u2575</span><span class="s">&#39;</span><span class="p">),</span>
-        <span class="s">&#39;G&#39;</span><span class="p">:</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span>
-              <span class="s">&#39;</span><span class="se">\u256d\u2500\u256e\n\u2502\u2576\u2510\n\u2570\u2500\u256f</span><span class="s">&#39;</span><span class="p">)</span>
-    <span class="p">}</span>
-    <span class="n">ascii_pics</span> <span class="o">=</span> <span class="p">{</span>
-        <span class="s">&#39;F&#39;</span><span class="p">:</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&#39; _</span><span class="se">\n</span><span class="s">|_</span><span class="se">\n</span><span class="s">|</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">),</span>
-        <span class="s">&#39;G&#39;</span><span class="p">:</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39; __</span><span class="se">\n</span><span class="s">/__</span><span class="se">\n</span><span class="s">\_|&#39;</span><span class="p">)</span>
-    <span class="p">}</span>
-
-    <span class="k">if</span> <span class="n">_use_unicode</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">ucode_pics</span><span class="p">[</span><span class="n">letter</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">ascii_pics</span><span class="p">[</span><span class="n">letter</span><span class="p">]</span></div>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../../index.html">
-              <img class="logo" src="../../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-<div id="searchbox" style="display: none">
-  <h3>Quick search</h3>
-    <form class="search" action="../../../../search.html" method="get">
-      <input type="text" name="q" />
-      <input type="submit" value="Go" />
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-    Enter search terms or a module, class or function name.
-    </p>
-</div>
-<script type="text/javascript">$('#searchbox').show(0);</script>
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-  <head>
-    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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-    <link rel="stylesheet" href="../../../../_static/default.css" type="text/css" />
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-        <div class="bodywrapper">
-          <div class="body">
-            
-  <h1>Source code for sympy.printing.pretty.stringpict</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Prettyprinter by Jurjen Bos.</span>
-<span class="sd">(I hate spammers: mail me at pietjepuk314 at the reverse of ku.oc.oohay).</span>
-<span class="sd">All objects have a method that create a &quot;stringPict&quot;,</span>
-<span class="sd">that can be used in the str method for pretty printing.</span>
-
-<span class="sd">Updates by Jason Gedge (email &lt;my last name&gt; at cs mun ca)</span>
-<span class="sd">    - terminal_string() method</span>
-<span class="sd">    - minor fixes and changes (mostly to prettyForm)</span>
-
-<span class="sd">TODO:</span>
-<span class="sd">    - Allow left/center/right alignment options for above/below and</span>
-<span class="sd">      top/center/bottom alignment options for left/right</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">.pretty_symbology</span> <span class="kn">import</span> <span class="n">hobj</span><span class="p">,</span> <span class="n">vobj</span><span class="p">,</span> <span class="n">xsym</span><span class="p">,</span> <span class="n">xobj</span><span class="p">,</span> <span class="n">pretty_use_unicode</span>
-
-
-<div class="viewcode-block" id="stringPict"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.stringpict.stringPict">[docs]</a><span class="k">class</span> <span class="nc">stringPict</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;An ASCII picture.</span>
-<span class="sd">    The pictures are represented as a list of equal length strings.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c">#special value for stringPict.below</span>
-    <span class="n">LINE</span> <span class="o">=</span> <span class="s">&#39;line&#39;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">baseline</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Initialize from string.</span>
-<span class="sd">        Multiline strings are centered.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c">#picture is a string that just can be printed</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">picture</span> <span class="o">=</span> <span class="n">stringPict</span><span class="o">.</span><span class="n">equalLengths</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">splitlines</span><span class="p">())</span>
-        <span class="c">#baseline is the line number of the &quot;base line&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">baseline</span> <span class="o">=</span> <span class="n">baseline</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">binding</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="nd">@staticmethod</span>
-    <span class="k">def</span> <span class="nf">equalLengths</span><span class="p">(</span><span class="n">lines</span><span class="p">):</span>
-        <span class="c"># empty lines</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">lines</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[</span><span class="s">&#39;&#39;</span><span class="p">]</span>
-
-        <span class="n">width</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">line</span><span class="p">)</span> <span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="n">lines</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">line</span><span class="o">.</span><span class="n">center</span><span class="p">(</span><span class="n">width</span><span class="p">)</span> <span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="n">lines</span><span class="p">]</span>
-
-<div class="viewcode-block" id="stringPict.height"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.stringpict.stringPict.height">[docs]</a>    <span class="k">def</span> <span class="nf">height</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The height of the picture in characters.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">picture</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="stringPict.width"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.stringpict.stringPict.width">[docs]</a>    <span class="k">def</span> <span class="nf">width</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;The width of the picture in characters.&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">picture</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-</div>
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="stringPict.next"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.stringpict.stringPict.next">[docs]</a>    <span class="k">def</span> <span class="nf">next</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Put a string of stringPicts next to each other.</span>
-<span class="sd">        Returns string, baseline arguments for stringPict.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c">#convert everything to stringPicts</span>
-        <span class="n">objects</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-                <span class="n">arg</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="n">objects</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="c">#make a list of pictures, with equal height and baseline</span>
-        <span class="n">newBaseline</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">obj</span><span class="o">.</span><span class="n">baseline</span> <span class="k">for</span> <span class="n">obj</span> <span class="ow">in</span> <span class="n">objects</span><span class="p">)</span>
-        <span class="n">newHeightBelowBaseline</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span>
-            <span class="n">obj</span><span class="o">.</span><span class="n">height</span><span class="p">()</span> <span class="o">-</span> <span class="n">obj</span><span class="o">.</span><span class="n">baseline</span>
-            <span class="k">for</span> <span class="n">obj</span> <span class="ow">in</span> <span class="n">objects</span><span class="p">)</span>
-        <span class="n">newHeight</span> <span class="o">=</span> <span class="n">newBaseline</span> <span class="o">+</span> <span class="n">newHeightBelowBaseline</span>
-
-        <span class="n">pictures</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">obj</span> <span class="ow">in</span> <span class="n">objects</span><span class="p">:</span>
-            <span class="n">oneEmptyLine</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">obj</span><span class="o">.</span><span class="n">width</span><span class="p">()]</span>
-            <span class="n">basePadding</span> <span class="o">=</span> <span class="n">newBaseline</span> <span class="o">-</span> <span class="n">obj</span><span class="o">.</span><span class="n">baseline</span>
-            <span class="n">totalPadding</span> <span class="o">=</span> <span class="n">newHeight</span> <span class="o">-</span> <span class="n">obj</span><span class="o">.</span><span class="n">height</span><span class="p">()</span>
-            <span class="n">pictures</span><span class="o">.</span><span class="n">append</span><span class="p">(</span>
-                <span class="n">oneEmptyLine</span> <span class="o">*</span> <span class="n">basePadding</span> <span class="o">+</span>
-                <span class="n">obj</span><span class="o">.</span><span class="n">picture</span> <span class="o">+</span>
-                <span class="n">oneEmptyLine</span> <span class="o">*</span> <span class="p">(</span><span class="n">totalPadding</span> <span class="o">-</span> <span class="n">basePadding</span><span class="p">))</span>
-
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">lines</span><span class="p">)</span> <span class="k">for</span> <span class="n">lines</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">pictures</span><span class="p">)]</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">result</span><span class="p">),</span> <span class="n">newBaseline</span>
-</div>
-<div class="viewcode-block" id="stringPict.right"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.stringpict.stringPict.right">[docs]</a>    <span class="k">def</span> <span class="nf">right</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">r&quot;&quot;&quot;Put pictures next to this one.</span>
-<span class="sd">        Returns string, baseline arguments for stringPict.</span>
-<span class="sd">        (Multiline) strings are allowed, and are given a baseline of 0.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.printing.pretty.stringpict import stringPict</span>
-<span class="sd">        &gt;&gt;&gt; print(stringPict(&quot;10&quot;).right(&quot; + &quot;,stringPict(&quot;1\r-\r2&quot;,1))[0])</span>
-<span class="sd">             1</span>
-<span class="sd">        10 + -</span>
-<span class="sd">             2</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">stringPict</span><span class="o">.</span><span class="n">next</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="stringPict.left"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.stringpict.stringPict.left">[docs]</a>    <span class="k">def</span> <span class="nf">left</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Put pictures (left to right) at left.</span>
-<span class="sd">        Returns string, baseline arguments for stringPict.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">stringPict</span><span class="o">.</span><span class="n">next</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">args</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="p">,)))</span>
-</div>
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="stringPict.stack"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.stringpict.stringPict.stack">[docs]</a>    <span class="k">def</span> <span class="nf">stack</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Put pictures on top of each other,</span>
-<span class="sd">        from top to bottom.</span>
-<span class="sd">        Returns string, baseline arguments for stringPict.</span>
-<span class="sd">        The baseline is the baseline of the second picture.</span>
-<span class="sd">        Everything is centered.</span>
-<span class="sd">        Baseline is the baseline of the second picture.</span>
-<span class="sd">        Strings are allowed.</span>
-<span class="sd">        The special value stringPict.LINE is a row of &#39;-&#39; extended to the width.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c">#convert everything to stringPicts; keep LINE</span>
-        <span class="n">objects</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">arg</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">stringPict</span><span class="o">.</span><span class="n">LINE</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-                <span class="n">arg</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="n">objects</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="c">#compute new width</span>
-        <span class="n">newWidth</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span>
-            <span class="n">obj</span><span class="o">.</span><span class="n">width</span><span class="p">()</span>
-            <span class="k">for</span> <span class="n">obj</span> <span class="ow">in</span> <span class="n">objects</span>
-            <span class="k">if</span> <span class="n">obj</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">stringPict</span><span class="o">.</span><span class="n">LINE</span><span class="p">)</span>
-
-        <span class="n">lineObj</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="n">hobj</span><span class="p">(</span><span class="s">&#39;-&#39;</span><span class="p">,</span> <span class="n">newWidth</span><span class="p">))</span>
-
-        <span class="c">#replace LINE with proper lines</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">obj</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">objects</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">obj</span> <span class="ow">is</span> <span class="n">stringPict</span><span class="o">.</span><span class="n">LINE</span><span class="p">:</span>
-                <span class="n">objects</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">lineObj</span>
-
-        <span class="c">#stack the pictures, and center the result</span>
-        <span class="n">newPicture</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">obj</span> <span class="ow">in</span> <span class="n">objects</span><span class="p">:</span>
-            <span class="n">newPicture</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">obj</span><span class="o">.</span><span class="n">picture</span><span class="p">)</span>
-        <span class="n">newPicture</span> <span class="o">=</span> <span class="p">[</span><span class="n">line</span><span class="o">.</span><span class="n">center</span><span class="p">(</span><span class="n">newWidth</span><span class="p">)</span> <span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="n">newPicture</span><span class="p">]</span>
-        <span class="n">newBaseline</span> <span class="o">=</span> <span class="n">objects</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">height</span><span class="p">()</span> <span class="o">+</span> <span class="n">objects</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">baseline</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">newPicture</span><span class="p">),</span> <span class="n">newBaseline</span>
-</div>
-<div class="viewcode-block" id="stringPict.below"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.stringpict.stringPict.below">[docs]</a>    <span class="k">def</span> <span class="nf">below</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Put pictures under this picture.</span>
-<span class="sd">        Returns string, baseline arguments for stringPict.</span>
-<span class="sd">        Baseline is baseline of top picture</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.printing.pretty.stringpict import stringPict</span>
-<span class="sd">        &gt;&gt;&gt; print(stringPict(&quot;x+3&quot;).below(</span>
-<span class="sd">        ...       stringPict.LINE, &#39;3&#39;)[0]) #doctest: +NORMALIZE_WHITESPACE</span>
-<span class="sd">        x+3</span>
-<span class="sd">        ---</span>
-<span class="sd">         3</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">s</span><span class="p">,</span> <span class="n">baseline</span> <span class="o">=</span> <span class="n">stringPict</span><span class="o">.</span><span class="n">stack</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">s</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">baseline</span>
-</div>
-<div class="viewcode-block" id="stringPict.above"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.stringpict.stringPict.above">[docs]</a>    <span class="k">def</span> <span class="nf">above</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Put pictures above this picture.</span>
-<span class="sd">        Returns string, baseline arguments for stringPict.</span>
-<span class="sd">        Baseline is baseline of bottom picture.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">string</span><span class="p">,</span> <span class="n">baseline</span> <span class="o">=</span> <span class="n">stringPict</span><span class="o">.</span><span class="n">stack</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">args</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="p">,)))</span>
-        <span class="n">baseline</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">string</span><span class="o">.</span><span class="n">splitlines</span><span class="p">())</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">height</span><span class="p">()</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">baseline</span>
-        <span class="k">return</span> <span class="n">string</span><span class="p">,</span> <span class="n">baseline</span>
-</div>
-<div class="viewcode-block" id="stringPict.parens"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.stringpict.stringPict.parens">[docs]</a>    <span class="k">def</span> <span class="nf">parens</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">left</span><span class="o">=</span><span class="s">&#39;(&#39;</span><span class="p">,</span> <span class="n">right</span><span class="o">=</span><span class="s">&#39;)&#39;</span><span class="p">,</span> <span class="n">ifascii_nougly</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Put parentheses around self.</span>
-<span class="sd">        Returns string, baseline arguments for stringPict.</span>
-
-<span class="sd">        left or right can be None or empty string which means &#39;no paren from</span>
-<span class="sd">        that side&#39;</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">h</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">height</span><span class="p">()</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">baseline</span>
-
-        <span class="c"># XXX this is a hack -- ascii parens are ugly!</span>
-        <span class="k">if</span> <span class="n">ifascii_nougly</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">pretty_use_unicode</span><span class="p">():</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="mi">0</span>
-
-        <span class="n">res</span> <span class="o">=</span> <span class="bp">self</span>
-
-        <span class="k">if</span> <span class="n">left</span><span class="p">:</span>
-            <span class="n">lparen</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="n">vobj</span><span class="p">(</span><span class="n">left</span><span class="p">,</span> <span class="n">h</span><span class="p">),</span> <span class="n">baseline</span><span class="o">=</span><span class="n">b</span><span class="p">)</span>
-            <span class="n">res</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="o">*</span><span class="n">lparen</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="bp">self</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">right</span><span class="p">:</span>
-            <span class="n">rparen</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="n">vobj</span><span class="p">(</span><span class="n">right</span><span class="p">,</span> <span class="n">h</span><span class="p">),</span> <span class="n">baseline</span><span class="o">=</span><span class="n">b</span><span class="p">)</span>
-            <span class="n">res</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="o">*</span><span class="n">res</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">rparen</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="p">(</span><span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">res</span><span class="o">.</span><span class="n">picture</span><span class="p">),</span> <span class="n">res</span><span class="o">.</span><span class="n">baseline</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="stringPict.leftslash"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.stringpict.stringPict.leftslash">[docs]</a>    <span class="k">def</span> <span class="nf">leftslash</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Precede object by a slash of the proper size.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># XXX not used anywhere ?</span>
-        <span class="n">height</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">baseline</span><span class="p">,</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">height</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">baseline</span><span class="p">)</span><span class="o">*</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="n">slash</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span>
-            <span class="s">&#39; &#39;</span><span class="o">*</span><span class="p">(</span><span class="n">height</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">xobj</span><span class="p">(</span><span class="s">&#39;/&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">i</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">height</span><span class="p">)</span>
-        <span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="n">stringPict</span><span class="p">(</span><span class="n">slash</span><span class="p">,</span> <span class="n">height</span><span class="o">//</span><span class="mi">2</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="stringPict.root"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.stringpict.stringPict.root">[docs]</a>    <span class="k">def</span> <span class="nf">root</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Produce a nice root symbol.</span>
-<span class="sd">        Produces ugly results for big n inserts.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># XXX not used anywhere</span>
-        <span class="c"># XXX duplicate of root drawing in pretty.py</span>
-        <span class="c">#put line over expression</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">above</span><span class="p">(</span><span class="s">&#39;_&#39;</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">width</span><span class="p">())</span>
-        <span class="c">#construct right half of root symbol</span>
-        <span class="n">height</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">height</span><span class="p">()</span>
-        <span class="n">slash</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span>
-            <span class="s">&#39; &#39;</span> <span class="o">*</span> <span class="p">(</span><span class="n">height</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39;/&#39;</span> <span class="o">+</span> <span class="s">&#39; &#39;</span> <span class="o">*</span> <span class="n">i</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">height</span><span class="p">)</span>
-        <span class="p">)</span>
-        <span class="n">slash</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="n">slash</span><span class="p">,</span> <span class="n">height</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="c">#left half of root symbol</span>
-        <span class="k">if</span> <span class="n">height</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">downline</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\\</span><span class="s"> </span><span class="se">\n</span><span class="s"> </span><span class="se">\\</span><span class="s">&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">downline</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\\</span><span class="s">&#39;</span><span class="p">)</span>
-        <span class="c">#put n on top, as low as possible</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">n</span><span class="o">.</span><span class="n">width</span><span class="p">()</span> <span class="o">&gt;</span> <span class="n">downline</span><span class="o">.</span><span class="n">width</span><span class="p">():</span>
-            <span class="n">downline</span> <span class="o">=</span> <span class="n">downline</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">.</span><span class="n">width</span><span class="p">()</span> <span class="o">-</span> <span class="n">downline</span><span class="o">.</span><span class="n">width</span><span class="p">()))</span>
-            <span class="n">downline</span> <span class="o">=</span> <span class="n">downline</span><span class="o">.</span><span class="n">above</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="c">#build root symbol</span>
-        <span class="n">root</span> <span class="o">=</span> <span class="n">downline</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">slash</span><span class="p">)</span>
-        <span class="c">#glue it on at the proper height</span>
-        <span class="c">#normally, the root symbel is as high as self</span>
-        <span class="c">#which is one less than result</span>
-        <span class="c">#this moves the root symbol one down</span>
-        <span class="c">#if the root became higher, the baseline has to grow too</span>
-        <span class="n">root</span><span class="o">.</span><span class="n">baseline</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">baseline</span> <span class="o">-</span> <span class="n">result</span><span class="o">.</span><span class="n">height</span><span class="p">()</span> <span class="o">+</span> <span class="n">root</span><span class="o">.</span><span class="n">height</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">result</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="stringPict.render"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.stringpict.stringPict.render">[docs]</a>    <span class="k">def</span> <span class="nf">render</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span> <span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the string form of self.</span>
-
-<span class="sd">           Unless the argument line_break is set to False, it will</span>
-<span class="sd">           break the expression in a form that can be printed</span>
-<span class="sd">           on the terminal without being broken up.</span>
-<span class="sd">         &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&quot;wrap_line&quot;</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">picture</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&quot;num_columns&quot;</span><span class="p">]</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="c"># Read the argument num_columns if it is not None</span>
-            <span class="n">ncols</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&quot;num_columns&quot;</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># Attempt to get a terminal width</span>
-            <span class="n">ncols</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">terminal_width</span><span class="p">()</span>
-
-        <span class="n">ncols</span> <span class="o">-=</span> <span class="mi">2</span>
-        <span class="k">if</span> <span class="n">ncols</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">ncols</span> <span class="o">=</span> <span class="mi">78</span>
-
-        <span class="c"># If smaller than the terminal width, no need to correct</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">width</span><span class="p">()</span> <span class="o">&lt;=</span> <span class="n">ncols</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">type</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">picture</span><span class="p">[</span><span class="mi">0</span><span class="p">])(</span><span class="bp">self</span><span class="p">)</span>
-
-        <span class="c"># for one-line pictures we don&#39;t need v-spacers. on the other hand, for</span>
-        <span class="c"># multiline-pictures, we need v-spacers between blocks, compare:</span>
-        <span class="c">#</span>
-        <span class="c">#    2  2        3    | a*c*e + a*c*f + a*d  | a*c*e + a*c*f + a*d  | 3.14159265358979323</span>
-        <span class="c"># 6*x *y  + 4*x*y  +  |                      | *e + a*d*f + b*c*e   | 84626433832795</span>
-        <span class="c">#                     | *e + a*d*f + b*c*e   | + b*c*f + b*d*e + b  |</span>
-        <span class="c">#      3    4    4    |                      | *d*f                 |</span>
-        <span class="c"># 4*y*x  + x  + y     | + b*c*f + b*d*e + b  |                      |</span>
-        <span class="c">#                     |                      |                      |</span>
-        <span class="c">#                     | *d*f</span>
-
-        <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">svals</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">do_vspacers</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">height</span><span class="p">()</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">width</span><span class="p">():</span>
-            <span class="n">svals</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span> <span class="n">sval</span><span class="p">[</span><span class="n">i</span><span class="p">:</span><span class="n">i</span> <span class="o">+</span> <span class="n">ncols</span><span class="p">]</span> <span class="k">for</span> <span class="n">sval</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">picture</span> <span class="p">])</span>
-            <span class="k">if</span> <span class="n">do_vspacers</span><span class="p">:</span>
-                <span class="n">svals</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;&quot;</span><span class="p">)</span>  <span class="c"># a vertical spacer</span>
-            <span class="n">i</span> <span class="o">+=</span> <span class="n">ncols</span>
-
-        <span class="k">if</span> <span class="n">svals</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-            <span class="k">del</span> <span class="n">svals</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>  <span class="c"># Get rid of the last spacer</span>
-
-        <span class="k">return</span> <span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">svals</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="stringPict.terminal_width"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.stringpict.stringPict.terminal_width">[docs]</a>    <span class="k">def</span> <span class="nf">terminal_width</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return the terminal width if possible, otherwise return 0.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">ncols</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="kn">import</span> <span class="nn">curses</span>
-            <span class="kn">import</span> <span class="nn">io</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">curses</span><span class="o">.</span><span class="n">setupterm</span><span class="p">()</span>
-                <span class="n">ncols</span> <span class="o">=</span> <span class="n">curses</span><span class="o">.</span><span class="n">tigetnum</span><span class="p">(</span><span class="s">&#39;cols&#39;</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-                <span class="c"># windows curses doesn&#39;t implement setupterm or tigetnum</span>
-                <span class="c"># code below from</span>
-                <span class="c"># http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/440694</span>
-                <span class="kn">from</span> <span class="nn">ctypes</span> <span class="kn">import</span> <span class="n">windll</span><span class="p">,</span> <span class="n">create_string_buffer</span>
-                <span class="c"># stdin handle is -10</span>
-                <span class="c"># stdout handle is -11</span>
-                <span class="c"># stderr handle is -12</span>
-                <span class="n">h</span> <span class="o">=</span> <span class="n">windll</span><span class="o">.</span><span class="n">kernel32</span><span class="o">.</span><span class="n">GetStdHandle</span><span class="p">(</span><span class="o">-</span><span class="mi">12</span><span class="p">)</span>
-                <span class="n">csbi</span> <span class="o">=</span> <span class="n">create_string_buffer</span><span class="p">(</span><span class="mi">22</span><span class="p">)</span>
-                <span class="n">res</span> <span class="o">=</span> <span class="n">windll</span><span class="o">.</span><span class="n">kernel32</span><span class="o">.</span><span class="n">GetConsoleScreenBufferInfo</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">csbi</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">res</span><span class="p">:</span>
-                    <span class="kn">import</span> <span class="nn">struct</span>
-                    <span class="p">(</span><span class="n">bufx</span><span class="p">,</span> <span class="n">bufy</span><span class="p">,</span> <span class="n">curx</span><span class="p">,</span> <span class="n">cury</span><span class="p">,</span> <span class="n">wattr</span><span class="p">,</span>
-                     <span class="n">left</span><span class="p">,</span> <span class="n">top</span><span class="p">,</span> <span class="n">right</span><span class="p">,</span> <span class="n">bottom</span><span class="p">,</span> <span class="n">maxx</span><span class="p">,</span> <span class="n">maxy</span><span class="p">)</span> <span class="o">=</span> <span class="n">struct</span><span class="o">.</span><span class="n">unpack</span><span class="p">(</span><span class="s">&quot;hhhhHhhhhhh&quot;</span><span class="p">,</span> <span class="n">csbi</span><span class="o">.</span><span class="n">raw</span><span class="p">)</span>
-                    <span class="n">ncols</span> <span class="o">=</span> <span class="n">right</span> <span class="o">-</span> <span class="n">left</span> <span class="o">+</span> <span class="mi">1</span>
-            <span class="k">except</span> <span class="n">curses</span><span class="o">.</span><span class="n">error</span><span class="p">:</span>
-                <span class="k">pass</span>
-            <span class="k">except</span> <span class="n">io</span><span class="o">.</span><span class="n">UnsupportedOperation</span><span class="p">:</span>
-                <span class="k">pass</span>
-        <span class="k">except</span> <span class="p">(</span><span class="ne">ImportError</span><span class="p">,</span> <span class="ne">TypeError</span><span class="p">):</span>
-            <span class="k">pass</span>
-        <span class="k">return</span> <span class="n">ncols</span>
-</div>
-    <span class="k">def</span> <span class="nf">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="k">return</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">picture</span><span class="p">)</span> <span class="o">==</span> <span class="n">o</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">stringPict</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">o</span><span class="o">.</span><span class="n">picture</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">picture</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">stringPict</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">str</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">picture</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__unicode__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">str</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">picture</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;stringPict(</span><span class="si">%r</span><span class="s">,</span><span class="si">%d</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">picture</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">baseline</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">index</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">picture</span><span class="p">[</span><span class="n">index</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__len__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">s</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="prettyForm"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.stringpict.prettyForm">[docs]</a><span class="k">class</span> <span class="nc">prettyForm</span><span class="p">(</span><span class="n">stringPict</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Extension of the stringPict class that knows about</span>
-<span class="sd">    basic math applications, optimizing double minus signs.</span>
-
-<span class="sd">    &quot;Binding&quot; is interpreted as follows::</span>
-
-<span class="sd">        ATOM this is an atom: never needs to be parenthesized</span>
-<span class="sd">        FUNC this is a function application: parenthesize if added (?)</span>
-<span class="sd">        DIV  this is a division: make wider division if divided</span>
-<span class="sd">        POW  this is a power: only parenthesize if exponent</span>
-<span class="sd">        MUL  this is a multiplication: parenthesize if powered</span>
-<span class="sd">        ADD  this is an addition: parenthesize if multiplied or powered</span>
-<span class="sd">        NEG  this is a negative number: optimize if added, parenthesize if multiplied or powered</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">ATOM</span><span class="p">,</span> <span class="n">FUNC</span><span class="p">,</span> <span class="n">DIV</span><span class="p">,</span> <span class="n">POW</span><span class="p">,</span> <span class="n">MUL</span><span class="p">,</span> <span class="n">ADD</span><span class="p">,</span> <span class="n">NEG</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">7</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">baseline</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">binding</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="nb">str</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Initialize from stringPict and binding power.&quot;&quot;&quot;</span>
-        <span class="n">stringPict</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">baseline</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">binding</span> <span class="o">=</span> <span class="n">binding</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">str</span> <span class="o">=</span> <span class="nb">str</span> <span class="ow">or</span> <span class="n">s</span>
-
-    <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">others</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Make a pretty addition.</span>
-<span class="sd">        Addition of negative numbers is simplified.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="bp">self</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">binding</span> <span class="o">&gt;</span> <span class="n">prettyForm</span><span class="o">.</span><span class="n">NEG</span><span class="p">:</span>
-            <span class="n">arg</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="o">*</span><span class="n">arg</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[</span><span class="n">arg</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">others</span><span class="p">:</span>
-            <span class="c">#add parentheses for weak binders</span>
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">binding</span> <span class="o">&gt;</span> <span class="n">prettyForm</span><span class="o">.</span><span class="n">NEG</span><span class="p">:</span>
-                <span class="n">arg</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="o">*</span><span class="n">arg</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-            <span class="c">#use existing minus sign if available</span>
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">binding</span> <span class="o">!=</span> <span class="n">prettyForm</span><span class="o">.</span><span class="n">NEG</span><span class="p">:</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39; + &#39;</span><span class="p">)</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">binding</span><span class="o">=</span><span class="n">prettyForm</span><span class="o">.</span><span class="n">ADD</span><span class="p">,</span> <span class="o">*</span><span class="n">stringPict</span><span class="o">.</span><span class="n">next</span><span class="p">(</span><span class="o">*</span><span class="n">result</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__div__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">den</span><span class="p">,</span> <span class="n">slashed</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Make a pretty division; stacked or slashed.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">slashed</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Can&#39;t do slashed fraction yet&quot;</span><span class="p">)</span>
-        <span class="n">num</span> <span class="o">=</span> <span class="bp">self</span>
-        <span class="k">if</span> <span class="n">num</span><span class="o">.</span><span class="n">binding</span> <span class="o">==</span> <span class="n">prettyForm</span><span class="o">.</span><span class="n">DIV</span><span class="p">:</span>
-            <span class="n">num</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="o">*</span><span class="n">num</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-        <span class="k">if</span> <span class="n">den</span><span class="o">.</span><span class="n">binding</span> <span class="o">==</span> <span class="n">prettyForm</span><span class="o">.</span><span class="n">DIV</span><span class="p">:</span>
-            <span class="n">den</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="o">*</span><span class="n">den</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">binding</span><span class="o">=</span><span class="n">prettyForm</span><span class="o">.</span><span class="n">DIV</span><span class="p">,</span> <span class="o">*</span><span class="n">stringPict</span><span class="o">.</span><span class="n">stack</span><span class="p">(</span>
-            <span class="n">num</span><span class="p">,</span>
-            <span class="n">stringPict</span><span class="o">.</span><span class="n">LINE</span><span class="p">,</span>
-            <span class="n">den</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__truediv__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">o</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__div__</span><span class="p">(</span><span class="n">o</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">others</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Make a pretty multiplication.</span>
-<span class="sd">        Parentheses are needed around +, - and neg.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span>
-        <span class="k">if</span> <span class="n">args</span><span class="o">.</span><span class="n">binding</span> <span class="o">&gt;</span> <span class="n">prettyForm</span><span class="o">.</span><span class="n">MUL</span><span class="p">:</span>
-            <span class="n">arg</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[</span><span class="n">args</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">others</span><span class="p">:</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">xsym</span><span class="p">(</span><span class="s">&#39;*&#39;</span><span class="p">))</span>
-            <span class="c">#add parentheses for weak binders</span>
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">binding</span> <span class="o">&gt;</span> <span class="n">prettyForm</span><span class="o">.</span><span class="n">MUL</span><span class="p">:</span>
-                <span class="n">arg</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="o">*</span><span class="n">arg</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-        <span class="n">len_res</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">len_res</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">len_res</span> <span class="o">-</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">result</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;-1&#39;</span> <span class="ow">and</span> <span class="n">result</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">xsym</span><span class="p">(</span><span class="s">&#39;*&#39;</span><span class="p">):</span>
-                <span class="c"># substitute -1 by -, like in -1*x -&gt; -x</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="s">&#39;-&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">result</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;-&#39;</span><span class="p">:</span>
-            <span class="c"># if there is a - sign in front of all</span>
-            <span class="nb">bin</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="o">.</span><span class="n">NEG</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="nb">bin</span> <span class="o">=</span> <span class="n">prettyForm</span><span class="o">.</span><span class="n">MUL</span>
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">binding</span><span class="o">=</span><span class="nb">bin</span><span class="p">,</span> <span class="o">*</span><span class="n">stringPict</span><span class="o">.</span><span class="n">next</span><span class="p">(</span><span class="o">*</span><span class="n">result</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;prettyForm(</span><span class="si">%r</span><span class="s">,</span><span class="si">%d</span><span class="s">,</span><span class="si">%d</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span>
-            <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">picture</span><span class="p">),</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">baseline</span><span class="p">,</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">binding</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__pow__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Make a pretty power.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span>
-        <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">binding</span> <span class="o">&gt;</span> <span class="n">prettyForm</span><span class="o">.</span><span class="n">FUNC</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="o">*</span><span class="n">a</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-        <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">binding</span> <span class="o">==</span> <span class="n">prettyForm</span><span class="o">.</span><span class="n">POW</span><span class="p">:</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="o">*</span><span class="n">b</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-
-        <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">binding</span> <span class="o">==</span> <span class="n">prettyForm</span><span class="o">.</span><span class="n">FUNC</span><span class="p">:</span>
-            <span class="c">#     2     &lt;-- top</span>
-            <span class="c">#  sin (x)  &lt;-- bot</span>
-            <span class="n">top</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="o">*</span><span class="n">b</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">a</span><span class="o">.</span><span class="n">prettyFunc</span><span class="o">.</span><span class="n">width</span><span class="p">()))</span>
-            <span class="n">top</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="o">*</span><span class="n">top</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">a</span><span class="o">.</span><span class="n">prettyArgs</span><span class="o">.</span><span class="n">width</span><span class="p">()))</span>
-            <span class="n">bot</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="o">*</span><span class="n">a</span><span class="o">.</span><span class="n">prettyFunc</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">b</span><span class="o">.</span><span class="n">width</span><span class="p">()))</span>
-            <span class="n">bot</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="o">*</span><span class="n">bot</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">prettyArgs</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c">#      2    &lt;-- top</span>
-            <span class="c"># (x+y)     &lt;-- bot</span>
-            <span class="n">top</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="o">*</span><span class="n">b</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">a</span><span class="o">.</span><span class="n">width</span><span class="p">()))</span>
-            <span class="n">bot</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="o">*</span><span class="n">a</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">*</span><span class="n">b</span><span class="o">.</span><span class="n">width</span><span class="p">()))</span>
-
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">binding</span><span class="o">=</span><span class="n">prettyForm</span><span class="o">.</span><span class="n">POW</span><span class="p">,</span> <span class="o">*</span><span class="n">bot</span><span class="o">.</span><span class="n">above</span><span class="p">(</span><span class="n">top</span><span class="p">))</span>
-
-    <span class="n">simpleFunctions</span> <span class="o">=</span> <span class="p">[</span><span class="s">&quot;sin&quot;</span><span class="p">,</span> <span class="s">&quot;cos&quot;</span><span class="p">,</span> <span class="s">&quot;tan&quot;</span><span class="p">]</span>
-
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="prettyForm.apply"><a class="viewcode-back" href="../../../../modules/printing.html#sympy.printing.pretty.stringpict.prettyForm.apply">[docs]</a>    <span class="k">def</span> <span class="nf">apply</span><span class="p">(</span><span class="n">function</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Functions of one or more variables.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">function</span> <span class="ow">in</span> <span class="n">prettyForm</span><span class="o">.</span><span class="n">simpleFunctions</span><span class="p">:</span>
-            <span class="c">#simple function: use only space if possible</span>
-            <span class="k">assert</span> <span class="nb">len</span><span class="p">(</span>
-                <span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&quot;Simple function </span><span class="si">%s</span><span class="s"> must have 1 argument&quot;</span> <span class="o">%</span> <span class="n">function</span>
-            <span class="n">arg</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">__pretty__</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">binding</span> <span class="o">&lt;=</span> <span class="n">prettyForm</span><span class="o">.</span><span class="n">DIV</span><span class="p">:</span>
-                <span class="c">#optimization: no parentheses necessary</span>
-                <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">binding</span><span class="o">=</span><span class="n">prettyForm</span><span class="o">.</span><span class="n">FUNC</span><span class="p">,</span> <span class="o">*</span><span class="n">arg</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="n">function</span> <span class="o">+</span> <span class="s">&#39; &#39;</span><span class="p">))</span>
-        <span class="n">argumentList</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="n">argumentList</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;,&#39;</span><span class="p">)</span>
-            <span class="n">argumentList</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">__pretty__</span><span class="p">())</span>
-        <span class="n">argumentList</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="o">*</span><span class="n">stringPict</span><span class="o">.</span><span class="n">next</span><span class="p">(</span><span class="o">*</span><span class="n">argumentList</span><span class="p">[</span><span class="mi">1</span><span class="p">:]))</span>
-        <span class="n">argumentList</span> <span class="o">=</span> <span class="n">stringPict</span><span class="p">(</span><span class="o">*</span><span class="n">argumentList</span><span class="o">.</span><span class="n">parens</span><span class="p">())</span>
-        <span class="k">return</span> <span class="n">prettyForm</span><span class="p">(</span><span class="n">binding</span><span class="o">=</span><span class="n">prettyForm</span><span class="o">.</span><span class="n">ATOM</span><span class="p">,</span> <span class="o">*</span><span class="n">argumentList</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="n">function</span><span class="p">))</span></div></div>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/printing/preview.html b/dev-py3k/_modules/sympy/printing/preview.html
deleted file mode 100644
index 8aec5b81b33..00000000000
--- a/dev-py3k/_modules/sympy/printing/preview.html
+++ /dev/null
@@ -1,331 +0,0 @@
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-            
-  <h1>Source code for sympy.printing.preview</h1><div class="highlight"><pre>
-<span class="kn">import</span> <span class="nn">os</span>
-<span class="kn">import</span> <span class="nn">time</span>
-<span class="kn">import</span> <span class="nn">tempfile</span>
-
-<span class="kn">from</span> <span class="nn">.latex</span> <span class="kn">import</span> <span class="n">latex</span>
-
-
-<span class="k">def</span> <span class="nf">preview</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">output</span><span class="o">=</span><span class="s">&#39;png&#39;</span><span class="p">,</span> <span class="n">viewer</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">euler</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">packages</span><span class="o">=</span><span class="p">(),</span> <span class="o">**</span><span class="n">latex_settings</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<div class="viewcode-block" id="preview"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.preview.preview">[docs]</a><span class="sd">    View expression or LaTeX markup in PNG, DVI, PostScript or PDF form.</span>
-
-<span class="sd">    If the expr argument is an expression, it will be exported to LaTeX and</span>
-<span class="sd">    then compiled using available the TeX distribution.  The first argument,</span>
-<span class="sd">    &#39;expr&#39;, may also be a LaTeX string.  The function will then run the</span>
-<span class="sd">    appropriate viewer for the given output format or use the user defined</span>
-<span class="sd">    one. By default png output is generated.</span>
-
-<span class="sd">    By default pretty Euler fonts are used for typesetting (they were used to</span>
-<span class="sd">    typeset the well known &quot;Concrete Mathematics&quot; book). For that to work, you</span>
-<span class="sd">    need the &#39;eulervm.sty&#39; LaTeX style (in Debian/Ubuntu, install the</span>
-<span class="sd">    texlive-fonts-extra package). If you prefer default AMS fonts or your</span>
-<span class="sd">    system lacks &#39;eulervm&#39; LaTeX package then unset the &#39;euler&#39; keyword</span>
-<span class="sd">    argument.</span>
-
-<span class="sd">    To use viewer auto-detection, lets say for &#39;png&#39; output, issue</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, preview, Symbol</span>
-<span class="sd">    &gt;&gt;&gt; x, y = symbols(&quot;x,y&quot;)</span>
-
-<span class="sd">    &gt;&gt;&gt; preview(x + y, output=&#39;png&#39;) # doctest: +SKIP</span>
-
-<span class="sd">    This will choose &#39;pyglet&#39; by default. To select a different one, do</span>
-
-<span class="sd">    &gt;&gt;&gt; preview(x + y, output=&#39;png&#39;, viewer=&#39;gimp&#39;) # doctest: +SKIP</span>
-
-<span class="sd">    The &#39;png&#39; format is considered special. For all other formats the rules</span>
-<span class="sd">    are slightly different. As an example we will take &#39;dvi&#39; output format. If</span>
-<span class="sd">    you would run</span>
-
-<span class="sd">    &gt;&gt;&gt; preview(x + y, output=&#39;dvi&#39;) # doctest: +SKIP</span>
-
-<span class="sd">    then &#39;view&#39; will look for available &#39;dvi&#39; viewers on your system</span>
-<span class="sd">    (predefined in the function, so it will try evince, first, then kdvi and</span>
-<span class="sd">    xdvi). If nothing is found you will need to set the viewer explicitly.</span>
-
-<span class="sd">    &gt;&gt;&gt; preview(x + y, output=&#39;dvi&#39;, viewer=&#39;superior-dvi-viewer&#39;) # doctest: +SKIP</span>
-
-<span class="sd">    This will skip auto-detection and will run user specified</span>
-<span class="sd">    &#39;superior-dvi-viewer&#39;. If &#39;view&#39; fails to find it on your system it will</span>
-<span class="sd">    gracefully raise an exception. You may also enter &#39;file&#39; for the viewer</span>
-<span class="sd">    argument. Doing so will cause this function to return a file object in</span>
-<span class="sd">    read-only mode.</span>
-
-<span class="sd">    Currently this depends on pexpect, which is not available for windows.</span>
-
-<span class="sd">    Additional keyword args will be passed to the latex call, e.g., the</span>
-<span class="sd">    symbol_names flag.</span>
-
-<span class="sd">    &gt;&gt;&gt; phidd = Symbol(&#39;phidd&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; preview(phidd, symbol_names={phidd:r&#39;\ddot{\varphi}&#39;}) # doctest: +SKIP</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="c"># we don&#39;t want to depend on anything not in the</span>
-    <span class="c"># standard library with SymPy by default</span>
-    <span class="kn">import</span> <span class="nn">pexpect</span>
-
-    <span class="n">special</span> <span class="o">=</span> <span class="p">[</span> <span class="s">&#39;pyglet&#39;</span> <span class="p">]</span>
-
-    <span class="k">if</span> <span class="n">viewer</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">output</span> <span class="o">==</span> <span class="s">&quot;png&quot;</span><span class="p">:</span>
-            <span class="n">viewer</span> <span class="o">=</span> <span class="s">&quot;pyglet&quot;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># sorted in order from most pretty to most ugly</span>
-            <span class="c"># very discussable, but indeed &#39;gv&#39; looks awful :)</span>
-            <span class="n">candidates</span> <span class="o">=</span> <span class="p">{</span>
-                <span class="s">&quot;dvi&quot;</span><span class="p">:</span> <span class="p">[</span> <span class="s">&quot;evince&quot;</span><span class="p">,</span> <span class="s">&quot;okular&quot;</span><span class="p">,</span> <span class="s">&quot;kdvi&quot;</span><span class="p">,</span> <span class="s">&quot;xdvi&quot;</span> <span class="p">],</span>
-                <span class="s">&quot;ps&quot;</span><span class="p">:</span> <span class="p">[</span> <span class="s">&quot;evince&quot;</span><span class="p">,</span> <span class="s">&quot;okular&quot;</span><span class="p">,</span> <span class="s">&quot;gsview&quot;</span><span class="p">,</span> <span class="s">&quot;gv&quot;</span> <span class="p">],</span>
-                <span class="s">&quot;pdf&quot;</span><span class="p">:</span> <span class="p">[</span> <span class="s">&quot;evince&quot;</span><span class="p">,</span> <span class="s">&quot;okular&quot;</span><span class="p">,</span> <span class="s">&quot;kpdf&quot;</span><span class="p">,</span> <span class="s">&quot;acroread&quot;</span><span class="p">,</span> <span class="s">&quot;xpdf&quot;</span><span class="p">,</span> <span class="s">&quot;gv&quot;</span> <span class="p">],</span>
-            <span class="p">}</span>
-
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">candidate</span> <span class="ow">in</span> <span class="n">candidates</span><span class="p">[</span><span class="n">output</span><span class="p">]:</span>
-                    <span class="k">if</span> <span class="n">pexpect</span><span class="o">.</span><span class="n">which</span><span class="p">(</span><span class="n">candidate</span><span class="p">):</span>
-                        <span class="n">viewer</span> <span class="o">=</span> <span class="n">candidate</span>
-                        <span class="k">break</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">SystemError</span><span class="p">(</span>
-                        <span class="s">&quot;No viewers found for &#39;</span><span class="si">%s</span><span class="s">&#39; output format.&quot;</span> <span class="o">%</span> <span class="n">output</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">SystemError</span><span class="p">(</span><span class="s">&quot;Invalid output format: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">output</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">viewer</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">special</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">pexpect</span><span class="o">.</span><span class="n">which</span><span class="p">(</span><span class="n">viewer</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">SystemError</span><span class="p">(</span><span class="s">&quot;Unrecognized viewer: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">viewer</span><span class="p">)</span>
-
-    <span class="n">actual_packages</span> <span class="o">=</span> <span class="n">packages</span> <span class="o">+</span> <span class="p">(</span><span class="s">&quot;amsmath&quot;</span><span class="p">,</span> <span class="s">&quot;amsfonts&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">euler</span><span class="p">:</span>
-        <span class="n">actual_packages</span> <span class="o">+=</span> <span class="p">(</span><span class="s">&quot;euler&quot;</span><span class="p">,)</span>
-    <span class="n">package_includes</span> <span class="o">=</span> <span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="s">&quot;</span><span class="se">\\</span><span class="s">usepackage{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="n">p</span>
-                                  <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">actual_packages</span><span class="p">])</span>
-
-    <span class="n">format</span> <span class="o">=</span> <span class="s">r&quot;&quot;&quot;\documentclass[12pt]{article}</span>
-<span class="s">                 </span><span class="si">%s</span><span class="s"></span>
-<span class="s">                 \begin{document}</span>
-<span class="s">                 \pagestyle{empty}</span>
-<span class="s">                 </span><span class="si">%s</span><span class="s"></span>
-<span class="s">                 \vfill</span>
-<span class="s">                 \end{document}</span>
-<span class="s">              &quot;&quot;&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">package_includes</span><span class="p">,</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">&quot;</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-        <span class="n">latex_string</span> <span class="o">=</span> <span class="n">expr</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">latex_string</span> <span class="o">=</span> <span class="n">latex</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;inline&#39;</span><span class="p">,</span> <span class="o">**</span><span class="n">latex_settings</span><span class="p">)</span>
-
-    <span class="n">tmp</span> <span class="o">=</span> <span class="n">tempfile</span><span class="o">.</span><span class="n">mktemp</span><span class="p">()</span>
-
-    <span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="n">tmp</span> <span class="o">+</span> <span class="s">&quot;.tex&quot;</span><span class="p">,</span> <span class="s">&quot;w&quot;</span><span class="p">)</span> <span class="k">as</span> <span class="n">tex</span><span class="p">:</span>
-        <span class="n">tex</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="n">format</span> <span class="o">%</span> <span class="n">latex_string</span><span class="p">)</span>
-
-    <span class="n">cwd</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">getcwd</span><span class="p">()</span>
-    <span class="n">os</span><span class="o">.</span><span class="n">chdir</span><span class="p">(</span><span class="n">tempfile</span><span class="o">.</span><span class="n">gettempdir</span><span class="p">())</span>
-
-    <span class="k">if</span> <span class="n">os</span><span class="o">.</span><span class="n">system</span><span class="p">(</span><span class="s">&quot;latex -halt-on-error </span><span class="si">%s</span><span class="s">.tex&quot;</span> <span class="o">%</span> <span class="n">tmp</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">SystemError</span><span class="p">(</span><span class="s">&quot;Failed to generate DVI output.&quot;</span><span class="p">)</span>
-
-    <span class="n">os</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">tmp</span> <span class="o">+</span> <span class="s">&quot;.tex&quot;</span><span class="p">)</span>
-    <span class="n">os</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">tmp</span> <span class="o">+</span> <span class="s">&quot;.aux&quot;</span><span class="p">)</span>
-    <span class="n">os</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">tmp</span> <span class="o">+</span> <span class="s">&quot;.log&quot;</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">output</span> <span class="o">!=</span> <span class="s">&quot;dvi&quot;</span><span class="p">:</span>
-        <span class="n">command</span> <span class="o">=</span> <span class="p">{</span>
-            <span class="s">&quot;ps&quot;</span><span class="p">:</span> <span class="s">&quot;dvips -o </span><span class="si">%s</span><span class="s">.ps </span><span class="si">%s</span><span class="s">.dvi&quot;</span><span class="p">,</span>
-            <span class="s">&quot;pdf&quot;</span><span class="p">:</span> <span class="s">&quot;dvipdf </span><span class="si">%s</span><span class="s">.dvi </span><span class="si">%s</span><span class="s">.pdf&quot;</span><span class="p">,</span>
-            <span class="s">&quot;png&quot;</span><span class="p">:</span> <span class="s">&quot;dvipng -T tight -z 9 &quot;</span> <span class="o">+</span>
-            <span class="s">&quot;--truecolor -o </span><span class="si">%s</span><span class="s">.png </span><span class="si">%s</span><span class="s">.dvi&quot;</span><span class="p">,</span>
-        <span class="p">}</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">os</span><span class="o">.</span><span class="n">system</span><span class="p">(</span><span class="n">command</span><span class="p">[</span><span class="n">output</span><span class="p">]</span> <span class="o">%</span> <span class="p">(</span><span class="n">tmp</span><span class="p">,</span> <span class="n">tmp</span><span class="p">))</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">SystemError</span><span class="p">(</span><span class="s">&quot;Failed to generate &#39;</span><span class="si">%s</span><span class="s">&#39; output.&quot;</span> <span class="o">%</span> <span class="n">output</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">os</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">tmp</span> <span class="o">+</span> <span class="s">&quot;.dvi&quot;</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">SystemError</span><span class="p">(</span><span class="s">&quot;Invalid output format: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">output</span><span class="p">)</span>
-
-    <span class="n">src</span> <span class="o">=</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">.</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tmp</span><span class="p">,</span> <span class="n">output</span><span class="p">)</span>
-    <span class="n">src_file</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="k">if</span> <span class="n">viewer</span> <span class="o">==</span> <span class="s">&quot;file&quot;</span><span class="p">:</span>
-        <span class="n">src_file</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="n">src</span><span class="p">,</span> <span class="s">&#39;rb&#39;</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">viewer</span> <span class="o">==</span> <span class="s">&quot;pyglet&quot;</span><span class="p">:</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">pyglet</span> <span class="kn">import</span> <span class="n">window</span><span class="p">,</span> <span class="n">image</span><span class="p">,</span> <span class="n">gl</span>
-            <span class="kn">from</span> <span class="nn">pyglet.window</span> <span class="kn">import</span> <span class="n">key</span>
-        <span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ImportError</span><span class="p">(</span><span class="s">&quot;pyglet is required for plotting.</span><span class="se">\n</span><span class="s"> visit http://www.pyglet.org/&quot;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">output</span> <span class="o">==</span> <span class="s">&quot;png&quot;</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">pyglet.image.codecs.png</span> <span class="kn">import</span> <span class="n">PNGImageDecoder</span>
-            <span class="n">img</span> <span class="o">=</span> <span class="n">image</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">src</span><span class="p">,</span> <span class="n">decoder</span><span class="o">=</span><span class="n">PNGImageDecoder</span><span class="p">())</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">SystemError</span><span class="p">(</span><span class="s">&quot;pyglet preview works only for &#39;png&#39; files.&quot;</span><span class="p">)</span>
-
-        <span class="n">offset</span> <span class="o">=</span> <span class="mi">25</span>
-
-        <span class="n">win</span> <span class="o">=</span> <span class="n">window</span><span class="o">.</span><span class="n">Window</span><span class="p">(</span>
-            <span class="n">width</span><span class="o">=</span><span class="n">img</span><span class="o">.</span><span class="n">width</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">offset</span><span class="p">,</span>
-            <span class="n">height</span><span class="o">=</span><span class="n">img</span><span class="o">.</span><span class="n">height</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">offset</span><span class="p">,</span>
-            <span class="n">caption</span><span class="o">=</span><span class="s">&quot;sympy&quot;</span><span class="p">,</span>
-            <span class="n">resizable</span><span class="o">=</span><span class="bp">False</span>
-        <span class="p">)</span>
-
-        <span class="n">win</span><span class="o">.</span><span class="n">set_vsync</span><span class="p">(</span><span class="bp">False</span><span class="p">)</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">def</span> <span class="nf">on_close</span><span class="p">():</span>
-                <span class="n">win</span><span class="o">.</span><span class="n">has_exit</span> <span class="o">=</span> <span class="bp">True</span>
-
-            <span class="n">win</span><span class="o">.</span><span class="n">on_close</span> <span class="o">=</span> <span class="n">on_close</span>
-
-            <span class="k">def</span> <span class="nf">on_key_press</span><span class="p">(</span><span class="n">symbol</span><span class="p">,</span> <span class="n">modifiers</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">symbol</span> <span class="ow">in</span> <span class="p">[</span><span class="n">key</span><span class="o">.</span><span class="n">Q</span><span class="p">,</span> <span class="n">key</span><span class="o">.</span><span class="n">ESCAPE</span><span class="p">]:</span>
-                    <span class="n">on_close</span><span class="p">()</span>
-
-            <span class="n">win</span><span class="o">.</span><span class="n">on_key_press</span> <span class="o">=</span> <span class="n">on_key_press</span>
-
-            <span class="k">def</span> <span class="nf">on_expose</span><span class="p">():</span>
-                <span class="n">gl</span><span class="o">.</span><span class="n">glClearColor</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">)</span>
-                <span class="n">gl</span><span class="o">.</span><span class="n">glClear</span><span class="p">(</span><span class="n">gl</span><span class="o">.</span><span class="n">GL_COLOR_BUFFER_BIT</span><span class="p">)</span>
-
-                <span class="n">img</span><span class="o">.</span><span class="n">blit</span><span class="p">(</span>
-                    <span class="p">(</span><span class="n">win</span><span class="o">.</span><span class="n">width</span> <span class="o">-</span> <span class="n">img</span><span class="o">.</span><span class="n">width</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span><span class="p">,</span>
-                    <span class="p">(</span><span class="n">win</span><span class="o">.</span><span class="n">height</span> <span class="o">-</span> <span class="n">img</span><span class="o">.</span><span class="n">height</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span>
-                <span class="p">)</span>
-
-            <span class="n">win</span><span class="o">.</span><span class="n">on_expose</span> <span class="o">=</span> <span class="n">on_expose</span>
-
-            <span class="k">while</span> <span class="ow">not</span> <span class="n">win</span><span class="o">.</span><span class="n">has_exit</span><span class="p">:</span>
-                <span class="n">win</span><span class="o">.</span><span class="n">dispatch_events</span><span class="p">()</span>
-                <span class="n">win</span><span class="o">.</span><span class="n">flip</span><span class="p">()</span>
-        <span class="k">except</span> <span class="ne">KeyboardInterrupt</span><span class="p">:</span>
-            <span class="k">pass</span>
-
-        <span class="n">win</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">os</span><span class="o">.</span><span class="n">system</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s"> &amp;&gt; /dev/null &amp;&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">viewer</span><span class="p">,</span> <span class="n">src</span><span class="p">))</span>
-        <span class="n">time</span><span class="o">.</span><span class="n">sleep</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>  <span class="c"># wait for the viewer to read data</span>
-
-    <span class="n">os</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">src</span><span class="p">)</span>
-    <span class="n">os</span><span class="o">.</span><span class="n">chdir</span><span class="p">(</span><span class="n">cwd</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">src_file</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">src_file</span>
-</pre></div></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-<div id="searchbox" style="display: none">
-  <h3>Quick search</h3>
-    <form class="search" action="../../../search.html" method="get">
-      <input type="text" name="q" />
-      <input type="submit" value="Go" />
-      <input type="hidden" name="check_keywords" value="yes" />
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-    Enter search terms or a module, class or function name.
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-<script type="text/javascript">$('#searchbox').show(0);</script>
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diff --git a/dev-py3k/_modules/sympy/printing/printer.html b/dev-py3k/_modules/sympy/printing/printer.html
deleted file mode 100644
index 709b58f8816..00000000000
--- a/dev-py3k/_modules/sympy/printing/printer.html
+++ /dev/null
@@ -1,386 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
-
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-    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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-    
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-             >modules</a> |</li>
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-        <div class="bodywrapper">
-          <div class="body">
-            
-  <h1>Source code for sympy.printing.printer</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Printing subsystem driver</span>
-
-<span class="sd">SymPy&#39;s printing system works the following way: Any expression can be</span>
-<span class="sd">passed to a designated Printer who then is responsible to return an</span>
-<span class="sd">adequate representation of that expression.</span>
-
-<span class="sd">The basic concept is the following:</span>
-<span class="sd">  1. Let the object print itself if it knows how.</span>
-<span class="sd">  2. Take the best fitting method defined in the printer.</span>
-<span class="sd">  3. As fall-back use the emptyPrinter method for the printer.</span>
-
-<span class="sd">Some more information how the single concepts work and who should use which:</span>
-
-<span class="sd">1. The object prints itself</span>
-
-<span class="sd">    This was the original way of doing printing in sympy. Every class had</span>
-<span class="sd">    its own latex, mathml, str and repr methods, but it turned out that it</span>
-<span class="sd">    is hard to produce a high quality printer, if all the methods are spread</span>
-<span class="sd">    out that far. Therefor all printing code was combined into the different</span>
-<span class="sd">    printers, which works great for built-in sympy objects, but not that</span>
-<span class="sd">    good for user defined classes where it is inconvenient to patch the</span>
-<span class="sd">    printers.</span>
-
-<span class="sd">    Nevertheless, to get a fitting representation, the printers look for a</span>
-<span class="sd">    specific method in every object, that will be called if it&#39;s available</span>
-<span class="sd">    and is then responsible for the representation. The name of that method</span>
-<span class="sd">    depends on the specific printer and is defined under</span>
-<span class="sd">    Printer.printmethod.</span>
-
-<span class="sd">2. Take the best fitting method defined in the printer.</span>
-
-<span class="sd">    The printer loops through expr classes (class + its bases), and tries</span>
-<span class="sd">    to dispatch the work to _print_&lt;EXPR_CLASS&gt;</span>
-
-<span class="sd">    e.g., suppose we have the following class hierarchy::</span>
-
-<span class="sd">            Basic</span>
-<span class="sd">            |</span>
-<span class="sd">            Atom</span>
-<span class="sd">            |</span>
-<span class="sd">            Number</span>
-<span class="sd">            |</span>
-<span class="sd">        Rational</span>
-
-<span class="sd">    then, for expr=Rational(...), in order to dispatch, we will try</span>
-<span class="sd">    calling printer methods as shown in the figure below::</span>
-
-<span class="sd">        p._print(expr)</span>
-<span class="sd">        |</span>
-<span class="sd">        |-- p._print_Rational(expr)</span>
-<span class="sd">        |</span>
-<span class="sd">        |-- p._print_Number(expr)</span>
-<span class="sd">        |</span>
-<span class="sd">        |-- p._print_Atom(expr)</span>
-<span class="sd">        |</span>
-<span class="sd">        `-- p._print_Basic(expr)</span>
-
-<span class="sd">    if ._print_Rational method exists in the printer, then it is called,</span>
-<span class="sd">    and the result is returned back.</span>
-
-<span class="sd">    otherwise, we proceed with trying Rational bases in the inheritance</span>
-<span class="sd">    order.</span>
-
-<span class="sd">3. As fall-back use the emptyPrinter method for the printer.</span>
-
-<span class="sd">    As fall-back self.emptyPrinter will be called with the expression. If</span>
-<span class="sd">    not defined in the Printer subclass this will be the same as str(expr).</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">Add</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.core</span> <span class="kn">import</span> <span class="n">BasicMeta</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">cmp_to_key</span>
-
-
-<div class="viewcode-block" id="Printer"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.printer.Printer">[docs]</a><span class="k">class</span> <span class="nc">Printer</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Generic printer</span>
-
-<span class="sd">    Its job is to provide infrastructure for implementing new printers easily.</span>
-
-<span class="sd">    Basically, if you want to implement a printer, all you have to do is:</span>
-
-<span class="sd">    1. Subclass Printer.</span>
-
-<span class="sd">    2. Define Printer.printmethod in your subclass.</span>
-<span class="sd">       If a object has a method with that name, this method will be used</span>
-<span class="sd">       for printing.</span>
-
-<span class="sd">    3. In your subclass, define ``_print_&lt;CLASS&gt;`` methods</span>
-
-<span class="sd">       For each class you want to provide printing to, define an appropriate</span>
-<span class="sd">       method how to do it. For example if you want a class FOO to be printed in</span>
-<span class="sd">       its own way, define _print_FOO::</span>
-
-<span class="sd">           def _print_FOO(self, e):</span>
-<span class="sd">               ...</span>
-
-<span class="sd">       this should return how FOO instance e is printed</span>
-
-<span class="sd">       Also, if ``BAR`` is a subclass of ``FOO``, ``_print_FOO(bar)`` will</span>
-<span class="sd">       be called for instance of ``BAR``, if no ``_print_BAR`` is provided.</span>
-<span class="sd">       Thus, usually, we don&#39;t need to provide printing routines for every</span>
-<span class="sd">       class we want to support -- only generic routine has to be provided</span>
-<span class="sd">       for a set of classes.</span>
-
-<span class="sd">       A good example for this are functions - for example ``PrettyPrinter``</span>
-<span class="sd">       only defines ``_print_Function``, and there is no ``_print_sin``,</span>
-<span class="sd">       ``_print_tan``, etc...</span>
-
-<span class="sd">       On the other hand, a good printer will probably have to define</span>
-<span class="sd">       separate routines for ``Symbol``, ``Atom``, ``Number``, ``Integral``,</span>
-<span class="sd">       ``Limit``, etc...</span>
-
-<span class="sd">    4. If convenient, override ``self.emptyPrinter``</span>
-
-<span class="sd">       This callable will be called to obtain printing result as a last resort,</span>
-<span class="sd">       that is when no appropriate print method was found for an expression.</span>
-
-<span class="sd">    Examples of overloading StrPrinter::</span>
-
-<span class="sd">        from sympy import Basic, Function, Symbol</span>
-<span class="sd">        from sympy.printing.str import StrPrinter</span>
-
-<span class="sd">        class CustomStrPrinter(StrPrinter):</span>
-<span class="sd">            \&quot;\&quot;\&quot;</span>
-<span class="sd">            Examples of how to customize the StrPrinter for both a SymPy class and a</span>
-<span class="sd">            user defined class subclassed from the SymPy Basic class.</span>
-<span class="sd">            \&quot;\&quot;\&quot;</span>
-
-<span class="sd">            def _print_Derivative(self, expr):</span>
-<span class="sd">                \&quot;\&quot;\&quot;</span>
-<span class="sd">                Custom printing of the SymPy Derivative class.</span>
-
-<span class="sd">                Instead of:</span>
-
-<span class="sd">                D(x(t), t) or D(x(t), t, t)</span>
-
-<span class="sd">                We will print:</span>
-
-<span class="sd">                x&#39;     or     x&#39;&#39;</span>
-
-<span class="sd">                In this example, expr.args == (x(t), t), and expr.args[0] == x(t), and</span>
-<span class="sd">                expr.args[0].func == x</span>
-<span class="sd">                \&quot;\&quot;\&quot;</span>
-<span class="sd">                return str(expr.args[0].func) + &quot;&#39;&quot;*len(expr.args[1:])</span>
-
-<span class="sd">            def _print_MyClass(self, expr):</span>
-<span class="sd">                \&quot;\&quot;\&quot;</span>
-<span class="sd">                Print the characters of MyClass.s alternatively lower case and upper</span>
-<span class="sd">                case</span>
-<span class="sd">                \&quot;\&quot;\&quot;</span>
-<span class="sd">                s = &quot;&quot;</span>
-<span class="sd">                i = 0</span>
-<span class="sd">                for char in expr.s:</span>
-<span class="sd">                    if i % 2 == 0:</span>
-<span class="sd">                        s += char.lower()</span>
-<span class="sd">                    else:</span>
-<span class="sd">                        s += char.upper()</span>
-<span class="sd">                    i += 1</span>
-<span class="sd">                return s</span>
-
-<span class="sd">        # Override the __str__ method of to use CustromStrPrinter</span>
-<span class="sd">        Basic.__str__ = lambda self: CustomStrPrinter().doprint(self)</span>
-<span class="sd">        # Demonstration of CustomStrPrinter:</span>
-<span class="sd">        t = Symbol(&#39;t&#39;)</span>
-<span class="sd">        x = Function(&#39;x&#39;)(t)</span>
-<span class="sd">        dxdt = x.diff(t)            # dxdt is a Derivative instance</span>
-<span class="sd">        d2xdt2 = dxdt.diff(t)       # dxdt2 is a Derivative instance</span>
-<span class="sd">        ex = MyClass(&#39;I like both lowercase and upper case&#39;)</span>
-
-<span class="sd">        print dxdt</span>
-<span class="sd">        print d2xdt2</span>
-<span class="sd">        print ex</span>
-
-<span class="sd">    The output of the above code is::</span>
-
-<span class="sd">        x&#39;</span>
-<span class="sd">        x&#39;&#39;</span>
-<span class="sd">        i lIkE BoTh lOwErCaSe aNd uPpEr cAsE</span>
-
-<span class="sd">    By overriding Basic.__str__, we can customize the printing of anything that</span>
-<span class="sd">    is subclassed from Basic.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">_global_settings</span> <span class="o">=</span> <span class="p">{}</span>
-
-    <span class="n">_default_settings</span> <span class="o">=</span> <span class="p">{}</span>
-
-    <span class="n">emptyPrinter</span> <span class="o">=</span> <span class="nb">str</span>
-    <span class="n">printmethod</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">settings</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_str</span> <span class="o">=</span> <span class="nb">str</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_default_settings</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-
-        <span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="n">val</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_global_settings</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="k">if</span> <span class="n">key</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_default_settings</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">=</span> <span class="n">val</span>
-
-        <span class="k">if</span> <span class="n">settings</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">settings</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">)</span> <span class="o">&gt;</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_default_settings</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">key</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">key</span> <span class="ow">not</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_default_settings</span><span class="p">:</span>
-                        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Unknown setting &#39;</span><span class="si">%s</span><span class="s">&#39;.&quot;</span> <span class="o">%</span> <span class="n">key</span><span class="p">)</span>
-
-        <span class="c"># _print_level is the number of times self._print() was recursively</span>
-        <span class="c"># called. See StrPrinter._print_Float() for an example of usage</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_print_level</span> <span class="o">=</span> <span class="mi">0</span>
-
-    <span class="nd">@classmethod</span>
-<div class="viewcode-block" id="Printer.set_global_settings"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.printer.Printer.set_global_settings">[docs]</a>    <span class="k">def</span> <span class="nf">set_global_settings</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Set system-wide printing settings. &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="n">val</span> <span class="ow">in</span> <span class="n">settings</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="k">if</span> <span class="n">val</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">cls</span><span class="o">.</span><span class="n">_global_settings</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">=</span> <span class="n">val</span>
-</div>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">order</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="s">&#39;order&#39;</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&#39;order&#39;</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s">&quot;No order defined.&quot;</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Printer.doprint"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.printer.Printer.doprint">[docs]</a>    <span class="k">def</span> <span class="nf">doprint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns printer&#39;s representation for expr (as a string)&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="Printer._print"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.printer.Printer._print">[docs]</a>    <span class="k">def</span> <span class="nf">_print</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Internal dispatcher</span>
-
-<span class="sd">        Tries the following concepts to print an expression:</span>
-<span class="sd">            1. Let the object print itself if it knows how.</span>
-<span class="sd">            2. Take the best fitting method defined in the printer.</span>
-<span class="sd">            3. As fall-back use the emptyPrinter method for the printer.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_print_level</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="c"># If the printer defines a name for a printing method</span>
-            <span class="c"># (Printer.printmethod) and the object knows for itself how it</span>
-            <span class="c"># should be printed, use that method.</span>
-            <span class="k">if</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">printmethod</span> <span class="ow">and</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">printmethod</span><span class="p">)</span>
-                    <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">BasicMeta</span><span class="p">)):</span>
-                <span class="k">return</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">printmethod</span><span class="p">)(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-            <span class="c"># See if the class of expr is known, or if one of its super</span>
-            <span class="c"># classes is known, and use that print function</span>
-            <span class="k">for</span> <span class="n">cls</span> <span class="ow">in</span> <span class="nb">type</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">__mro__</span><span class="p">:</span>
-                <span class="n">printmethod</span> <span class="o">=</span> <span class="s">&#39;_print_&#39;</span> <span class="o">+</span> <span class="n">cls</span><span class="o">.</span><span class="n">__name__</span>
-                <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printmethod</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="nb">getattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">printmethod</span><span class="p">)(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-            <span class="c"># Unknown object, fall back to the emptyPrinter.</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">emptyPrinter</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">finally</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_print_level</span> <span class="o">-=</span> <span class="mi">1</span>
-</div>
-    <span class="k">def</span> <span class="nf">_as_ordered_terms</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;A compatibility function for ordering terms in Add. &quot;&quot;&quot;</span>
-        <span class="n">order</span> <span class="o">=</span> <span class="n">order</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span>
-
-        <span class="k">if</span> <span class="n">order</span> <span class="o">==</span> <span class="s">&#39;old&#39;</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">expr</span><span class="p">),</span> <span class="n">key</span><span class="o">=</span><span class="n">cmp_to_key</span><span class="p">(</span><span class="n">Basic</span><span class="o">.</span><span class="n">_compare_pretty</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_ordered_terms</span><span class="p">(</span><span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">)</span></div>
-</pre></div>
-
-          </div>
-        </div>
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-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
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-  <h3>Quick search</h3>
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-  <h1>Source code for sympy.printing.repr</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">A Printer for generating executable code.</span>
-
-<span class="sd">The most important function here is srepr that returns a string so that the</span>
-<span class="sd">relation eval(srepr(expr))=expr holds in an appropriate environment.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">.printer</span> <span class="kn">import</span> <span class="n">Printer</span>
-<span class="kn">import</span> <span class="nn">sympy.mpmath.libmp</span> <span class="kn">as</span> <span class="nn">mlib</span>
-<span class="kn">from</span> <span class="nn">sympy.mpmath.libmp</span> <span class="kn">import</span> <span class="n">prec_to_dps</span><span class="p">,</span> <span class="n">repr_dps</span>
-
-
-<div class="viewcode-block" id="ReprPrinter"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.repr.ReprPrinter">[docs]</a><span class="k">class</span> <span class="nc">ReprPrinter</span><span class="p">(</span><span class="n">Printer</span><span class="p">):</span>
-    <span class="n">printmethod</span> <span class="o">=</span> <span class="s">&quot;_sympyrepr&quot;</span>
-
-    <span class="n">_default_settings</span> <span class="o">=</span> <span class="p">{</span>
-        <span class="s">&quot;order&quot;</span><span class="p">:</span> <span class="bp">None</span>
-    <span class="p">}</span>
-
-<div class="viewcode-block" id="ReprPrinter.reprify"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.repr.ReprPrinter.reprify">[docs]</a>    <span class="k">def</span> <span class="nf">reprify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">args</span><span class="p">,</span> <span class="n">sep</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Prints each item in `args` and joins them with `sep`.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">sep</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">item</span><span class="p">)</span> <span class="k">for</span> <span class="n">item</span> <span class="ow">in</span> <span class="n">args</span><span class="p">])</span>
-</div>
-<div class="viewcode-block" id="ReprPrinter.emptyPrinter"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.repr.ReprPrinter.emptyPrinter">[docs]</a>    <span class="k">def</span> <span class="nf">emptyPrinter</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        The fallback printer.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">expr</span>
-        <span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&quot;__srepr__&quot;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">__srepr__</span><span class="p">()</span>
-        <span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&quot;args&quot;</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="s">&quot;__iter__&quot;</span><span class="p">):</span>
-            <span class="n">l</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="n">l</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">o</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span> <span class="o">+</span> <span class="s">&#39;(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="s">&#39;, &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">l</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&quot;__module__&quot;</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&quot;__name__&quot;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="s">&quot;&lt;&#39;</span><span class="si">%s</span><span class="s">.</span><span class="si">%s</span><span class="s">&#39;&gt;&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">__module__</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">__name__</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_print_Add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_as_ordered_terms</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">)</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="n">args</span><span class="p">))</span>
-        <span class="k">return</span> <span class="s">&quot;Add(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Function</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">+=</span> <span class="s">&#39;(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="s">&#39;, &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">r</span>
-
-    <span class="k">def</span> <span class="nf">_print_FunctionClass</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;Function(</span><span class="si">%r</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">__name__</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Half</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;Rational(1, 2)&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_print_RationalConstant</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_AtomicExpr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_NumberSymbol</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Integer</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;Integer(</span><span class="si">%i</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="n">expr</span><span class="o">.</span><span class="n">p</span>
-
-    <span class="k">def</span> <span class="nf">_print_list</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;[</span><span class="si">%s</span><span class="s">]&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">reprify</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&quot;, &quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_MatrixBase</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">l</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="n">l</span><span class="o">.</span><span class="n">append</span><span class="p">([])</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">cols</span><span class="p">):</span>
-                <span class="n">l</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">expr</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">])</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">l</span><span class="p">))</span>
-
-    <span class="n">_print_SparseMatrix</span> <span class="o">=</span> \
-        <span class="n">_print_MutableSparseMatrix</span> <span class="o">=</span> \
-        <span class="n">_print_ImmutableSparseMatrix</span> <span class="o">=</span> \
-        <span class="n">_print_Matrix</span> <span class="o">=</span> \
-        <span class="n">_print_DenseMatrix</span> <span class="o">=</span> \
-        <span class="n">_print_MutableDenseMatrix</span> <span class="o">=</span> \
-        <span class="n">_print_ImmutableMatrix</span> <span class="o">=</span> \
-        <span class="n">_print_ImmutableDenseMatrix</span> <span class="o">=</span> \
-        <span class="n">_print_MatrixBase</span>
-
-    <span class="k">def</span> <span class="nf">_print_NaN</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;nan&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Rational</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;Rational(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">p</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">q</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">terms</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span> <span class="o">!=</span> <span class="s">&#39;old&#39;</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">_new_rawargs</span><span class="p">(</span><span class="o">*</span><span class="n">terms</span><span class="p">)</span><span class="o">.</span><span class="n">as_ordered_factors</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="n">terms</span>
-
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="n">args</span><span class="p">))</span>
-        <span class="k">return</span> <span class="s">&quot;Mul(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Fraction</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;Fraction(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">numerator</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">denominator</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Float</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">dps</span> <span class="o">=</span> <span class="n">prec_to_dps</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">_prec</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">mlib</span><span class="o">.</span><span class="n">to_str</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="n">repr_dps</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">_prec</span><span class="p">))</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">(&#39;</span><span class="si">%s</span><span class="s">&#39;, prec=</span><span class="si">%i</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">dps</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Sum2</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;Sum2(</span><span class="si">%s</span><span class="s">, (</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">))&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">f</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">i</span><span class="p">),</span>
-                                           <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">a</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">b</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Symbol</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">name</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Predicate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">name</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_AppliedPredicate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_str</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">repr</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_tuple</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;(</span><span class="si">%s</span><span class="s">,)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">reprify</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&quot;, &quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_WildFunction</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">(&#39;</span><span class="si">%s</span><span class="s">&#39;)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_AlgebraicNumber</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">coeffs</span><span class="p">()),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">root</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="srepr"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.repr.srepr">[docs]</a><span class="k">def</span> <span class="nf">srepr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;return expr in repr form&quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">ReprPrinter</span><span class="p">(</span><span class="n">settings</span><span class="p">)</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span></div>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/printing/str.html b/dev-py3k/_modules/sympy/printing/str.html
deleted file mode 100644
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+++ /dev/null
@@ -1,795 +0,0 @@
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-            
-  <h1>Source code for sympy.printing.str</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">A Printer for generating readable representation of most sympy classes.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">Rational</span><span class="p">,</span> <span class="n">Pow</span><span class="p">,</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">Mul</span>
-<span class="kn">from</span> <span class="nn">sympy.core.mul</span> <span class="kn">import</span> <span class="n">_keep_coeff</span>
-<span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">Integer</span>
-<span class="kn">from</span> <span class="nn">.printer</span> <span class="kn">import</span> <span class="n">Printer</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.precedence</span> <span class="kn">import</span> <span class="n">precedence</span><span class="p">,</span> <span class="n">PRECEDENCE</span>
-
-<span class="kn">import</span> <span class="nn">sympy.mpmath.libmp</span> <span class="kn">as</span> <span class="nn">mlib</span>
-<span class="kn">from</span> <span class="nn">sympy.mpmath.libmp</span> <span class="kn">import</span> <span class="n">prec_to_dps</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="n">PolynomialError</span>
-
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">default_sort_key</span>
-
-
-<div class="viewcode-block" id="StrPrinter"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.str.StrPrinter">[docs]</a><span class="k">class</span> <span class="nc">StrPrinter</span><span class="p">(</span><span class="n">Printer</span><span class="p">):</span>
-    <span class="n">printmethod</span> <span class="o">=</span> <span class="s">&quot;_sympystr&quot;</span>
-    <span class="n">_default_settings</span> <span class="o">=</span> <span class="p">{</span>
-        <span class="s">&quot;order&quot;</span><span class="p">:</span> <span class="bp">None</span><span class="p">,</span>
-        <span class="s">&quot;full_prec&quot;</span><span class="p">:</span> <span class="s">&quot;auto&quot;</span><span class="p">,</span>
-    <span class="p">}</span>
-
-    <span class="k">def</span> <span class="nf">parenthesize</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">item</span><span class="p">,</span> <span class="n">level</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">precedence</span><span class="p">(</span><span class="n">item</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="n">level</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">item</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">item</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">stringify</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">args</span><span class="p">,</span> <span class="n">sep</span><span class="p">,</span> <span class="n">level</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sep</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">item</span><span class="p">,</span> <span class="n">level</span><span class="p">)</span> <span class="k">for</span> <span class="n">item</span> <span class="ow">in</span> <span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">emptyPrinter</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">expr</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&quot;args&quot;</span><span class="p">):</span>
-                <span class="k">return</span> <span class="nb">repr</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span> <span class="o">==</span> <span class="s">&#39;none&#39;</span><span class="p">:</span>
-            <span class="n">terms</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">terms</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_as_ordered_terms</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">)</span>
-
-        <span class="n">PREC</span> <span class="o">=</span> <span class="n">precedence</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="n">l</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">:</span>
-            <span class="n">t</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39;-&#39;</span><span class="p">):</span>
-                <span class="n">sign</span> <span class="o">=</span> <span class="s">&quot;-&quot;</span>
-                <span class="n">t</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">sign</span> <span class="o">=</span> <span class="s">&quot;+&quot;</span>
-            <span class="k">if</span> <span class="n">precedence</span><span class="p">(</span><span class="n">term</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">PREC</span><span class="p">:</span>
-                <span class="n">l</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">sign</span><span class="p">,</span> <span class="s">&quot;(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="n">t</span><span class="p">])</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">l</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">sign</span><span class="p">,</span> <span class="n">t</span><span class="p">])</span>
-        <span class="n">sign</span> <span class="o">=</span> <span class="n">l</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">sign</span> <span class="o">==</span> <span class="s">&#39;+&#39;</span><span class="p">:</span>
-            <span class="n">sign</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">sign</span> <span class="o">+</span> <span class="s">&#39; &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">l</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_And</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">,</span> <span class="s">&#39;, &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span>
-            <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Or</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">,</span> <span class="s">&#39;, &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span>
-            <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)))</span>
-
-    <span class="k">def</span> <span class="nf">_print_AppliedPredicate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">arg</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Basic</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">l</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">o</span><span class="p">)</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span> <span class="o">+</span> <span class="s">&quot;(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">l</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_BlockMatrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">B</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">B</span><span class="o">.</span><span class="n">blocks</span><span class="o">.</span><span class="n">shape</span> <span class="o">==</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">B</span><span class="o">.</span><span class="n">blocks</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">B</span><span class="o">.</span><span class="n">blocks</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Catalan</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;Catalan&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_print_ComplexInfinity</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;zoo&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;Derivative(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_dict</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">d</span><span class="p">):</span>
-        <span class="n">keys</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">keys</span><span class="p">()),</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="n">items</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">key</span> <span class="ow">in</span> <span class="n">keys</span><span class="p">:</span>
-            <span class="n">item</span> <span class="o">=</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">key</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">d</span><span class="p">[</span><span class="n">key</span><span class="p">]))</span>
-            <span class="n">items</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">item</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="s">&quot;{</span><span class="si">%s</span><span class="s">}&quot;</span> <span class="o">%</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">items</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Dict</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_dict</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_RandomDomain</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">d</span><span class="p">):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&#39;Domain: &#39;</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">as_boolean</span><span class="p">())</span>
-        <span class="k">except</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">(</span><span class="s">&#39;Domain: &#39;</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">symbols</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39; in &#39;</span> <span class="o">+</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">set</span><span class="p">))</span>
-            <span class="k">except</span><span class="p">:</span>
-                <span class="k">return</span> <span class="s">&#39;Domain on &#39;</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">symbols</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Dummy</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;_&#39;</span> <span class="o">+</span> <span class="n">expr</span><span class="o">.</span><span class="n">name</span>
-
-    <span class="k">def</span> <span class="nf">_print_EulerGamma</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;EulerGamma&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Exp1</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;E&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_print_ExprCondPair</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">cond</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_subfactorial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;!</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Pow&quot;</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_print_factorial</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">!&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Pow&quot;</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_print_FiniteSet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">10</span><span class="p">:</span>
-            <span class="n">printset</span> <span class="o">=</span> <span class="n">s</span><span class="p">[:</span><span class="mi">3</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="s">&#39;...&#39;</span><span class="p">]</span> <span class="o">+</span> <span class="n">s</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">:]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">printset</span> <span class="o">=</span> <span class="n">s</span>
-        <span class="k">return</span> <span class="s">&#39;{&#39;</span> <span class="o">+</span> <span class="s">&#39;, &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">el</span><span class="p">)</span> <span class="k">for</span> <span class="n">el</span> <span class="ow">in</span> <span class="n">printset</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39;}&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Function</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="o">.</span><span class="n">__name__</span> <span class="o">+</span> <span class="s">&quot;(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">stringify</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="s">&quot;, &quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_GeometryEntity</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="c"># GeometryEntity is special -- it&#39;s base is tuple</span>
-        <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_GoldenRatio</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;GoldenRatio&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_print_ImaginaryUnit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;I&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Infinity</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;oo&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Integral</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">def</span> <span class="nf">_xab_tostr</span><span class="p">(</span><span class="n">xab</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">xab</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">xab</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">((</span><span class="n">xab</span><span class="p">[</span><span class="mi">0</span><span class="p">],)</span> <span class="o">+</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">xab</span><span class="p">[</span><span class="mi">1</span><span class="p">:]))</span>
-        <span class="n">L</span> <span class="o">=</span> <span class="s">&#39;, &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="n">_xab_tostr</span><span class="p">(</span><span class="n">l</span><span class="p">)</span> <span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">limits</span><span class="p">])</span>
-        <span class="k">return</span> <span class="s">&#39;Integral(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">function</span><span class="p">),</span> <span class="n">L</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Interval</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">left_open</span><span class="p">:</span>
-            <span class="n">left</span> <span class="o">=</span> <span class="s">&#39;(&#39;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">left</span> <span class="o">=</span> <span class="s">&#39;[&#39;</span>
-
-        <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">right_open</span><span class="p">:</span>
-            <span class="n">right</span> <span class="o">=</span> <span class="s">&#39;)&#39;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">right</span> <span class="o">=</span> <span class="s">&#39;]&#39;</span>
-
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s%s</span><span class="s">, </span><span class="si">%s%s</span><span class="s">&quot;</span> <span class="o">%</span> \
-               <span class="p">(</span><span class="n">left</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">start</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">end</span><span class="p">),</span> <span class="n">right</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">I</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">^-1&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">I</span><span class="o">.</span><span class="n">arg</span><span class="p">,</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Pow&quot;</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_print_Lambda</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">obj</span><span class="p">):</span>
-        <span class="n">args</span><span class="p">,</span> <span class="n">expr</span> <span class="o">=</span> <span class="n">obj</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;Lambda(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">args</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">expr</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">arg_string</span> <span class="o">=</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">)</span>
-            <span class="k">return</span> <span class="s">&quot;Lambda((</span><span class="si">%s</span><span class="s">), </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">arg_string</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_LatticeOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="o">.</span><span class="n">__name__</span> <span class="o">+</span> <span class="s">&quot;(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Limit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">e</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="nb">dir</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">if</span> <span class="nb">str</span><span class="p">(</span><span class="nb">dir</span><span class="p">)</span> <span class="o">==</span> <span class="s">&quot;+&quot;</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;Limit(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;Limit(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, dir=&#39;</span><span class="si">%s</span><span class="s">&#39;)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="nb">dir</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_list</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;[</span><span class="si">%s</span><span class="s">]&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">stringify</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&quot;, &quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_MatrixBase</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">_format_str</span><span class="p">(</span><span class="k">lambda</span> <span class="n">elem</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">elem</span><span class="p">))</span>
-    <span class="n">_print_SparseMatrix</span> <span class="o">=</span> \
-        <span class="n">_print_MutableSparseMatrix</span> <span class="o">=</span> \
-        <span class="n">_print_ImmutableSparseMatrix</span> <span class="o">=</span> \
-        <span class="n">_print_Matrix</span> <span class="o">=</span> \
-        <span class="n">_print_DenseMatrix</span> <span class="o">=</span> \
-        <span class="n">_print_MutableDenseMatrix</span> <span class="o">=</span> \
-        <span class="n">_print_ImmutableMatrix</span> <span class="o">=</span> \
-        <span class="n">_print_ImmutableDenseMatrix</span> <span class="o">=</span> \
-        <span class="n">_print_MatrixBase</span>
-
-    <span class="k">def</span> <span class="nf">_print_DeferredVector</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">name</span>
-
-    <span class="k">def</span> <span class="nf">_print_Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-
-        <span class="n">prec</span> <span class="o">=</span> <span class="n">precedence</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-        <span class="n">c</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">c</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="o">-</span><span class="n">c</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span>
-            <span class="n">sign</span> <span class="o">=</span> <span class="s">&quot;-&quot;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">sign</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-
-        <span class="n">a</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c"># items in the numerator</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c"># items that are in the denominator (if any)</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">order</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">(</span><span class="s">&#39;old&#39;</span><span class="p">,</span> <span class="s">&#39;none&#39;</span><span class="p">):</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_ordered_factors</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># use make_args in case expr was something like -x -&gt; x</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-        <span class="c"># Gather args for numerator/denominator</span>
-        <span class="k">for</span> <span class="n">item</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">item</span><span class="o">.</span><span class="n">is_commutative</span> <span class="ow">and</span> <span class="n">item</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">item</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">item</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">item</span><span class="o">.</span><span class="n">exp</span> <span class="o">!=</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                    <span class="n">b</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Pow</span><span class="p">(</span><span class="n">item</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="o">-</span><span class="n">item</span><span class="o">.</span><span class="n">exp</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">b</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Pow</span><span class="p">(</span><span class="n">item</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="o">-</span><span class="n">item</span><span class="o">.</span><span class="n">exp</span><span class="p">))</span>
-            <span class="k">elif</span> <span class="n">item</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">item</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">item</span><span class="o">.</span><span class="n">p</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">a</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="n">item</span><span class="o">.</span><span class="n">p</span><span class="p">))</span>
-                <span class="k">if</span> <span class="n">item</span><span class="o">.</span><span class="n">q</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">b</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="n">item</span><span class="o">.</span><span class="n">q</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">a</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">item</span><span class="p">)</span>
-
-        <span class="n">a</span> <span class="o">=</span> <span class="n">a</span> <span class="ow">or</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">]</span>
-
-        <span class="n">a_str</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">a</span><span class="p">]</span>
-        <span class="n">b_str</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">prec</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">b</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">sign</span> <span class="o">+</span> <span class="s">&#39;*&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">a_str</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="ow">not</span> <span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Atom</span> <span class="ow">or</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Add</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">sign</span> <span class="o">+</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">/&quot;</span> <span class="o">%</span> <span class="n">a_str</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="s">&#39;*&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">b_str</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">sign</span> <span class="o">+</span> <span class="s">&#39;*&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">a_str</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot;/</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">b_str</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">sign</span> <span class="o">+</span> <span class="s">&#39;*&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">a_str</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot;/(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="s">&#39;*&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">b_str</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_MatMul</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;*&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">precedence</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-            <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_print_HadamardProduct</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;.*&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">precedence</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-            <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_print_MatAdd</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39; + &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">precedence</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-            <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_print_NaN</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;nan&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_print_NegativeInfinity</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;-oo&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Normal</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;Normal(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">mu</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">sigma</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Order</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&#39;O(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&#39;O(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">stringify</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="s">&#39;, &#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Cycle</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;We want it to print as Cycle in doctests for which a repr is required.</span>
-
-<span class="sd">        With __repr__ defined in Cycle, interactive output gives Cycle form but</span>
-<span class="sd">        during doctests, the dict&#39;s __repr__ form is used. Defining this _print</span>
-<span class="sd">        function solves that problem.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.combinatorics import Cycle</span>
-<span class="sd">        &gt;&gt;&gt; Cycle(1, 2) # will print as a dict without this method</span>
-<span class="sd">        Cycle(1, 2)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">__repr__</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">_print_Permutation</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span><span class="p">,</span> <span class="n">Cycle</span>
-        <span class="k">if</span> <span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">size</span><span class="p">:</span>
-                <span class="k">return</span> <span class="s">&#39;Permutation()&#39;</span>
-            <span class="c"># before taking Cycle notation, see if the last element is</span>
-            <span class="c"># a singleton and move it to the head of the string</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">Cycle</span><span class="p">(</span><span class="n">expr</span><span class="p">)(</span><span class="n">expr</span><span class="o">.</span><span class="n">size</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">__repr__</span><span class="p">()[</span><span class="nb">len</span><span class="p">(</span><span class="s">&#39;Cycle&#39;</span><span class="p">):]</span>
-            <span class="n">last</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">rfind</span><span class="p">(</span><span class="s">&#39;(&#39;</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">last</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="s">&#39;,&#39;</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">s</span><span class="p">[</span><span class="n">last</span><span class="p">:]:</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">s</span><span class="p">[</span><span class="n">last</span><span class="p">:]</span> <span class="o">+</span> <span class="n">s</span><span class="p">[:</span><span class="n">last</span><span class="p">]</span>
-            <span class="k">return</span> <span class="s">&#39;Permutation</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">s</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">support</span><span class="p">()</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">s</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">size</span> <span class="o">&lt;</span> <span class="mi">5</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="s">&#39;Permutation(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="nb">str</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">array_form</span><span class="p">)</span>
-                <span class="k">return</span> <span class="s">&#39;Permutation([], size=</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="n">expr</span><span class="o">.</span><span class="n">size</span>
-            <span class="n">trim</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">array_form</span><span class="p">[:</span><span class="n">s</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span> <span class="o">+</span> <span class="s">&#39;, size=</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">expr</span><span class="o">.</span><span class="n">size</span>
-            <span class="n">use</span> <span class="o">=</span> <span class="n">full</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">array_form</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">trim</span><span class="p">)</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">full</span><span class="p">):</span>
-                <span class="n">use</span> <span class="o">=</span> <span class="n">trim</span>
-            <span class="k">return</span> <span class="s">&#39;Permutation(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="n">use</span>
-
-    <span class="k">def</span> <span class="nf">_print_PermutationGroup</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;    </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="nb">str</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="k">return</span> <span class="s">&#39;PermutationGroup([</span><span class="se">\n</span><span class="si">%s</span><span class="s">])&#39;</span> <span class="o">%</span> <span class="s">&#39;,</span><span class="se">\n</span><span class="s">&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_PDF</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;PDF(</span><span class="si">%s</span><span class="s">, (</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">))&#39;</span> <span class="o">%</span> \
-            <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">pdf</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">pdf</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">domain</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">domain</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Pi</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;pi&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Poly</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">terms</span><span class="p">,</span> <span class="n">gens</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">gens</span> <span class="p">]</span>
-
-        <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">terms</span><span class="p">():</span>
-            <span class="n">s_monom</span> <span class="o">=</span> <span class="p">[]</span>
-
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">exp</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">monom</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">exp</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">exp</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="n">s_monom</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">gens</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">s_monom</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">gens</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="s">&quot;**</span><span class="si">%d</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">exp</span><span class="p">)</span>
-
-            <span class="n">s_monom</span> <span class="o">=</span> <span class="s">&quot;*&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">s_monom</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">s_monom</span><span class="p">:</span>
-                    <span class="n">s_coeff</span> <span class="o">=</span> <span class="s">&quot;(&quot;</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot;)&quot;</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">s_coeff</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">s_monom</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                        <span class="n">terms</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="s">&#39;+&#39;</span><span class="p">,</span> <span class="n">s_monom</span><span class="p">])</span>
-                        <span class="k">continue</span>
-
-                    <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-                        <span class="n">terms</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="s">&#39;-&#39;</span><span class="p">,</span> <span class="n">s_monom</span><span class="p">])</span>
-                        <span class="k">continue</span>
-
-                <span class="n">s_coeff</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">s_monom</span><span class="p">:</span>
-                <span class="n">s_term</span> <span class="o">=</span> <span class="n">s_coeff</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">s_term</span> <span class="o">=</span> <span class="n">s_coeff</span> <span class="o">+</span> <span class="s">&quot;*&quot;</span> <span class="o">+</span> <span class="n">s_monom</span>
-
-            <span class="k">if</span> <span class="n">s_term</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39;-&#39;</span><span class="p">):</span>
-                <span class="n">terms</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="s">&#39;-&#39;</span><span class="p">,</span> <span class="n">s_term</span><span class="p">[</span><span class="mi">1</span><span class="p">:]])</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">terms</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="s">&#39;+&#39;</span><span class="p">,</span> <span class="n">s_term</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">terms</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">in</span> <span class="p">[</span><span class="s">&#39;-&#39;</span><span class="p">,</span> <span class="s">&#39;+&#39;</span><span class="p">]:</span>
-            <span class="n">modifier</span> <span class="o">=</span> <span class="n">terms</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">modifier</span> <span class="o">==</span> <span class="s">&#39;-&#39;</span><span class="p">:</span>
-                <span class="n">terms</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="s">&#39;-&#39;</span> <span class="o">+</span> <span class="n">terms</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="n">format</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span> <span class="o">+</span> <span class="s">&quot;(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">&quot;</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">format</span> <span class="o">+=</span> <span class="s">&quot;, modulus=</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">expr</span><span class="o">.</span><span class="n">get_modulus</span><span class="p">()</span>
-        <span class="k">except</span> <span class="n">PolynomialError</span><span class="p">:</span>
-            <span class="n">format</span> <span class="o">+=</span> <span class="s">&quot;, domain=&#39;</span><span class="si">%s</span><span class="s">&#39;&quot;</span> <span class="o">%</span> <span class="n">expr</span><span class="o">.</span><span class="n">get_domain</span><span class="p">()</span>
-
-        <span class="n">format</span> <span class="o">+=</span> <span class="s">&quot;)&quot;</span>
-
-        <span class="k">return</span> <span class="n">format</span> <span class="o">%</span> <span class="p">(</span><span class="s">&#39; &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">terms</span><span class="p">),</span> <span class="s">&#39;, &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">gens</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_ProductSet</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39; x &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="nb">set</span><span class="p">)</span> <span class="k">for</span> <span class="nb">set</span> <span class="ow">in</span> <span class="n">p</span><span class="o">.</span><span class="n">sets</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_AlgebraicNumber</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_aliased</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">as_poly</span><span class="p">()</span><span class="o">.</span><span class="n">as_expr</span><span class="p">())</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">as_expr</span><span class="p">())</span>
-
-    <span class="k">def</span> <span class="nf">_print_Pow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">rational</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="n">PREC</span> <span class="o">=</span> <span class="n">precedence</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">rational</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;sqrt(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-            <span class="k">if</span> <span class="o">-</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">rational</span><span class="p">:</span>
-                <span class="c"># Note: Don&#39;t test &quot;expr.exp == -S.Half&quot; here, because that will</span>
-                <span class="c"># match -0.5, which we don&#39;t want.</span>
-                <span class="k">return</span> <span class="s">&quot;1/sqrt(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="s">&#39;1/</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">PREC</span><span class="p">)</span>
-
-        <span class="n">e</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">,</span> <span class="n">PREC</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">printmethod</span> <span class="o">==</span> <span class="s">&#39;_sympyrepr&#39;</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">q</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># the parenthesized exp should be &#39;(Rational(a, b))&#39; so strip parens,</span>
-            <span class="c"># but just check to be sure.</span>
-            <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39;(Rational&#39;</span><span class="p">):</span>
-                <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">**</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">PREC</span><span class="p">),</span> <span class="n">e</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">**</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">PREC</span><span class="p">),</span> <span class="n">e</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_MatPow</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">PREC</span> <span class="o">=</span> <span class="n">precedence</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">**</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">PREC</span><span class="p">),</span>
-                         <span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">,</span> <span class="n">PREC</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Integer</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_int</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_mpz</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">str</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Rational</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">/</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">p</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Fraction</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">/</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">numerator</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">denominator</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_mpq</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">/</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">numer</span><span class="p">(),</span> <span class="n">expr</span><span class="o">.</span><span class="n">denom</span><span class="p">())</span>
-
-    <span class="k">def</span> <span class="nf">_print_Float</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">prec</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">_prec</span>
-        <span class="k">if</span> <span class="n">prec</span> <span class="o">&lt;</span> <span class="mi">5</span><span class="p">:</span>
-            <span class="n">dps</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">dps</span> <span class="o">=</span> <span class="n">prec_to_dps</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">_prec</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&quot;full_prec&quot;</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">strip</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&quot;full_prec&quot;</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="n">strip</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">_settings</span><span class="p">[</span><span class="s">&quot;full_prec&quot;</span><span class="p">]</span> <span class="o">==</span> <span class="s">&quot;auto&quot;</span><span class="p">:</span>
-            <span class="n">strip</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_level</span> <span class="o">&gt;</span> <span class="mi">1</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">mlib</span><span class="o">.</span><span class="n">to_str</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="n">dps</span><span class="p">,</span> <span class="n">strip_zeros</span><span class="o">=</span><span class="n">strip</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">rv</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39;-.0&#39;</span><span class="p">):</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="s">&#39;-0.&#39;</span> <span class="o">+</span> <span class="n">rv</span><span class="p">[</span><span class="mi">3</span><span class="p">:]</span>
-        <span class="k">elif</span> <span class="n">rv</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39;.0&#39;</span><span class="p">):</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="s">&#39;0.&#39;</span> <span class="o">+</span> <span class="n">rv</span><span class="p">[</span><span class="mi">2</span><span class="p">:]</span>
-        <span class="k">return</span> <span class="n">rv</span>
-
-    <span class="k">def</span> <span class="nf">_print_Relational</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">precedence</span><span class="p">(</span><span class="n">expr</span><span class="p">)),</span>
-                           <span class="n">expr</span><span class="o">.</span><span class="n">rel_op</span><span class="p">,</span>
-                           <span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">precedence</span><span class="p">(</span><span class="n">expr</span><span class="p">)))</span>
-
-    <span class="k">def</span> <span class="nf">_print_RootOf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;RootOf(</span><span class="si">%s</span><span class="s">, </span><span class="si">%d</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print_Add</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s">&#39;lex&#39;</span><span class="p">),</span> <span class="n">expr</span><span class="o">.</span><span class="n">index</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_RootSum</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_print_Add</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s">&#39;lex&#39;</span><span class="p">)]</span>
-
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">fun</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">IdentityFunction</span><span class="p">:</span>
-            <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">fun</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="s">&quot;RootSum(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_GroebnerBasis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">):</span>
-        <span class="n">cls</span> <span class="o">=</span> <span class="n">basis</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span>
-
-        <span class="n">exprs</span> <span class="o">=</span> <span class="p">[</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_Add</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">basis</span><span class="o">.</span><span class="n">order</span><span class="p">)</span>
-                  <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">basis</span><span class="o">.</span><span class="n">exprs</span> <span class="p">]</span>
-        <span class="n">exprs</span> <span class="o">=</span> <span class="s">&quot;[</span><span class="si">%s</span><span class="s">]&quot;</span> <span class="o">%</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">exprs</span><span class="p">)</span>
-
-        <span class="n">gens</span> <span class="o">=</span> <span class="p">[</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">gen</span><span class="p">)</span> <span class="k">for</span> <span class="n">gen</span> <span class="ow">in</span> <span class="n">basis</span><span class="o">.</span><span class="n">gens</span> <span class="p">]</span>
-        <span class="n">domain</span> <span class="o">=</span> <span class="s">&quot;domain=&#39;</span><span class="si">%s</span><span class="s">&#39;&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">basis</span><span class="o">.</span><span class="n">domain</span><span class="p">)</span>
-        <span class="n">order</span> <span class="o">=</span> <span class="s">&quot;order=&#39;</span><span class="si">%s</span><span class="s">&#39;&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">basis</span><span class="o">.</span><span class="n">order</span><span class="p">)</span>
-
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">exprs</span><span class="p">]</span> <span class="o">+</span> <span class="n">gens</span> <span class="o">+</span> <span class="p">[</span><span class="n">domain</span><span class="p">,</span> <span class="n">order</span><span class="p">]</span>
-
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">args</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Sample</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;Sample([</span><span class="si">%s</span><span class="s">])&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">stringify</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&quot;, &quot;</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_set</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="n">items</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-
-        <span class="n">args</span> <span class="o">=</span> <span class="s">&#39;, &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">item</span><span class="p">)</span> <span class="k">for</span> <span class="n">item</span> <span class="ow">in</span> <span class="n">items</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">args</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="s">&#39;[</span><span class="si">%s</span><span class="s">]&#39;</span> <span class="o">%</span> <span class="n">args</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">s</span><span class="p">)</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="n">args</span><span class="p">)</span>
-
-    <span class="n">_print_frozenset</span> <span class="o">=</span> <span class="n">_print_set</span>
-
-    <span class="k">def</span> <span class="nf">_print_SparseMatrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">Matrix</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_print_Sum</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">def</span> <span class="nf">_xab_tostr</span><span class="p">(</span><span class="n">xab</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">xab</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">xab</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">((</span><span class="n">xab</span><span class="p">[</span><span class="mi">0</span><span class="p">],)</span> <span class="o">+</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">xab</span><span class="p">[</span><span class="mi">1</span><span class="p">:]))</span>
-        <span class="n">L</span> <span class="o">=</span> <span class="s">&#39;, &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="n">_xab_tostr</span><span class="p">(</span><span class="n">l</span><span class="p">)</span> <span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">limits</span><span class="p">])</span>
-        <span class="k">return</span> <span class="s">&#39;Sum(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">function</span><span class="p">),</span> <span class="n">L</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Symbol</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">name</span>
-    <span class="n">_print_MatrixSymbol</span> <span class="o">=</span> <span class="n">_print_Symbol</span>
-
-    <span class="k">def</span> <span class="nf">_print_Identity</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;I&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_ZeroMatrix</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;0&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Predicate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;Q.</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">expr</span><span class="o">.</span><span class="n">name</span>
-
-    <span class="k">def</span> <span class="nf">_print_str</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="k">def</span> <span class="nf">_print_tuple</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;(</span><span class="si">%s</span><span class="s">,)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">stringify</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&quot;, &quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Tuple</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_tuple</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Transpose</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">T</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">&#39;&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">parenthesize</span><span class="p">(</span><span class="n">T</span><span class="o">.</span><span class="n">arg</span><span class="p">,</span> <span class="n">PRECEDENCE</span><span class="p">[</span><span class="s">&quot;Pow&quot;</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_print_Uniform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;Uniform(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">a</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">b</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Union</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39; U &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="nb">set</span><span class="p">)</span> <span class="k">for</span> <span class="nb">set</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Unit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">abbrev</span>
-
-    <span class="k">def</span> <span class="nf">_print_Wild</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">name</span> <span class="o">+</span> <span class="s">&#39;_&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_print_WildFunction</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">name</span> <span class="o">+</span> <span class="s">&#39;_&#39;</span>
-
-    <span class="k">def</span> <span class="nf">_print_Zero</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;0&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_DMP</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="n">SympifyError</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">ring</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="c"># TODO incorporate order</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">ring</span><span class="o">.</span><span class="n">to_sympy</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
-        <span class="k">except</span> <span class="n">SympifyError</span><span class="p">:</span>
-            <span class="k">pass</span>
-
-        <span class="n">cls</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span>
-        <span class="n">rep</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">rep</span><span class="p">)</span>
-        <span class="n">dom</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">dom</span><span class="p">)</span>
-        <span class="n">ring</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">ring</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">rep</span><span class="p">,</span> <span class="n">dom</span><span class="p">,</span> <span class="n">ring</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_DMF</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print_DMP</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Object</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">object</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;Object(&quot;</span><span class="si">%s</span><span class="s">&quot;)&#39;</span> <span class="o">%</span> <span class="nb">object</span><span class="o">.</span><span class="n">name</span>
-
-    <span class="k">def</span> <span class="nf">_print_IdentityMorphism</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">morphism</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;IdentityMorphism(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="n">morphism</span><span class="o">.</span><span class="n">domain</span>
-
-    <span class="k">def</span> <span class="nf">_print_NamedMorphism</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">morphism</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;NamedMorphism(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, &quot;</span><span class="si">%s</span><span class="s">&quot;)&#39;</span> <span class="o">%</span> \
-               <span class="p">(</span><span class="n">morphism</span><span class="o">.</span><span class="n">domain</span><span class="p">,</span> <span class="n">morphism</span><span class="o">.</span><span class="n">codomain</span><span class="p">,</span> <span class="n">morphism</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Category</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">category</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;Category(&quot;</span><span class="si">%s</span><span class="s">&quot;)&#39;</span> <span class="o">%</span> <span class="n">category</span><span class="o">.</span><span class="n">name</span>
-
-    <span class="k">def</span> <span class="nf">_print_BaseScalarField</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">field</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">field</span><span class="o">.</span><span class="n">_coord_sys</span><span class="o">.</span><span class="n">_names</span><span class="p">[</span><span class="n">field</span><span class="o">.</span><span class="n">_index</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">_print_BaseVectorField</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">field</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;e_</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">field</span><span class="o">.</span><span class="n">_coord_sys</span><span class="o">.</span><span class="n">_names</span><span class="p">[</span><span class="n">field</span><span class="o">.</span><span class="n">_index</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">_print_Differential</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">diff</span><span class="p">):</span>
-        <span class="n">field</span> <span class="o">=</span> <span class="n">diff</span><span class="o">.</span><span class="n">_form_field</span>
-        <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">field</span><span class="p">,</span> <span class="s">&#39;_coord_sys&#39;</span><span class="p">):</span>
-            <span class="k">return</span> <span class="s">&#39;d</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">field</span><span class="o">.</span><span class="n">_coord_sys</span><span class="o">.</span><span class="n">_names</span><span class="p">[</span><span class="n">field</span><span class="o">.</span><span class="n">_index</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&#39;d(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">field</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_print_Tr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="c">#TODO : Handle indices</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="s">&quot;Tr&quot;</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_print</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-
-</div>
-<span class="k">def</span> <span class="nf">sstr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns the expression as a string.</span>
-
-<span class="sd">    For large expressions where speed is a concern, use the setting</span>
-<span class="sd">    order=&#39;none&#39;.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, Eq, sstr</span>
-<span class="sd">    &gt;&gt;&gt; a, b = symbols(&#39;a b&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; sstr(Eq(a + b, 0))</span>
-<span class="sd">    &#39;a + b == 0&#39;</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">p</span> <span class="o">=</span> <span class="n">StrPrinter</span><span class="p">(</span><span class="n">settings</span><span class="p">)</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">s</span>
-
-
-<span class="k">class</span> <span class="nc">StrReprPrinter</span><span class="p">(</span><span class="n">StrPrinter</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;(internal) -- see sstrrepr&quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_print_str</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">repr</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="sstrrepr"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.str.sstrrepr">[docs]</a><span class="k">def</span> <span class="nf">sstrrepr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">settings</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;return expr in mixed str/repr form</span>
-
-<span class="sd">       i.e. strings are returned in repr form with quotes, and everything else</span>
-<span class="sd">       is returned in str form.</span>
-
-<span class="sd">       This function could be useful for hooking into sys.displayhook</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">p</span> <span class="o">=</span> <span class="n">StrReprPrinter</span><span class="p">(</span><span class="n">settings</span><span class="p">)</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">s</span></div>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-<div id="searchbox" style="display: none">
-  <h3>Quick search</h3>
-    <form class="search" action="../../../search.html" method="get">
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-        <li class="right" >
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-      Last updated on Jan 20, 2013.
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diff --git a/dev-py3k/_modules/sympy/printing/tree.html b/dev-py3k/_modules/sympy/printing/tree.html
deleted file mode 100644
index 0c2ed0c1075..00000000000
--- a/dev-py3k/_modules/sympy/printing/tree.html
+++ /dev/null
@@ -1,214 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
-
-
-<html xmlns="http://www.w3.org/1999/xhtml">
-  <head>
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-  <h1>Source code for sympy.printing.tree</h1><div class="highlight"><pre>
-<span class="k">def</span> <span class="nf">pprint_nodes</span><span class="p">(</span><span class="n">subtrees</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<div class="viewcode-block" id="pprint_nodes"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.tree.pprint_nodes">[docs]</a><span class="sd">    Prettyprints systems of nodes.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.printing.tree import pprint_nodes</span>
-<span class="sd">    &gt;&gt;&gt; print(pprint_nodes([&quot;a&quot;, &quot;b1\\nb2&quot;, &quot;c&quot;]))</span>
-<span class="sd">    +-a</span>
-<span class="sd">    +-b1</span>
-<span class="sd">    | b2</span>
-<span class="sd">    +-c</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">indent</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="nb">type</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="s">&quot;+-</span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="s">&quot;&quot;</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">if</span> <span class="nb">type</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">r</span> <span class="o">+=</span> <span class="s">&quot;| </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">a</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">r</span> <span class="o">+=</span> <span class="s">&quot;  </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">a</span>
-        <span class="k">return</span> <span class="n">r</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">subtrees</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="s">&quot;&quot;</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-    <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">subtrees</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-        <span class="n">f</span> <span class="o">+=</span> <span class="n">indent</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-    <span class="n">f</span> <span class="o">+=</span> <span class="n">indent</span><span class="p">(</span><span class="n">subtrees</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="mi">2</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">f</span>
-
-
-<span class="k">def</span> <span class="nf">print_node</span><span class="p">(</span><span class="n">node</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span></div>
-<div class="viewcode-block" id="print_node"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.tree.print_node">[docs]</a><span class="sd">    Returns an information about the &quot;node&quot;.</span>
-
-<span class="sd">    This includes class name, string representation and assumptions.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">: </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="nb">str</span><span class="p">(</span><span class="n">node</span><span class="p">))</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">_assumptions</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">node</span><span class="o">.</span><span class="n">_assumptions</span><span class="p">:</span>
-            <span class="n">s</span> <span class="o">+=</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">: </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">node</span><span class="o">.</span><span class="n">_assumptions</span><span class="p">[</span><span class="n">a</span><span class="p">])</span>
-    <span class="k">return</span> <span class="n">s</span>
-
-
-<span class="k">def</span> <span class="nf">tree</span><span class="p">(</span><span class="n">node</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span></div>
-<div class="viewcode-block" id="tree"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.tree.tree">[docs]</a><span class="sd">    Returns a tree representation of &quot;node&quot; as a string.</span>
-
-<span class="sd">    It uses print_node() together with pprint_nodes() on node.args recursively.</span>
-
-<span class="sd">    See also: print_tree()</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">subtrees</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">node</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-        <span class="n">subtrees</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">tree</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="n">print_node</span><span class="p">(</span><span class="n">node</span><span class="p">)</span> <span class="o">+</span> <span class="n">pprint_nodes</span><span class="p">(</span><span class="n">subtrees</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">s</span>
-
-
-<span class="k">def</span> <span class="nf">print_tree</span><span class="p">(</span><span class="n">node</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span></div>
-<div class="viewcode-block" id="print_tree"><a class="viewcode-back" href="../../../modules/printing.html#sympy.printing.tree.print_tree">[docs]</a><span class="sd">    Prints a tree representation of &quot;node&quot;.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.printing import print_tree</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; print_tree(x**2) # doctest: +SKIP</span>
-<span class="sd">    Pow: x**2</span>
-<span class="sd">    +-Symbol: x</span>
-<span class="sd">    | comparable: False</span>
-<span class="sd">    +-Integer: 2</span>
-<span class="sd">      real: True</span>
-<span class="sd">      nonzero: True</span>
-<span class="sd">      comparable: True</span>
-<span class="sd">      commutative: True</span>
-<span class="sd">      infinitesimal: False</span>
-<span class="sd">      unbounded: False</span>
-<span class="sd">      noninteger: False</span>
-<span class="sd">      zero: False</span>
-<span class="sd">      complex: True</span>
-<span class="sd">      bounded: True</span>
-<span class="sd">      rational: True</span>
-<span class="sd">      integer: True</span>
-<span class="sd">      imaginary: False</span>
-<span class="sd">      finite: True</span>
-<span class="sd">      irrational: False</span>
-<span class="sd">    &lt;BLANKLINE&gt;</span>
-
-<span class="sd">    See also: tree()</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">print</span><span class="p">(</span><span class="n">tree</span><span class="p">(</span><span class="n">node</span><span class="p">))</span>
-</pre></div></div>
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-            
-  <h1>Source code for sympy.series.acceleration</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Convergence acceleration / extrapolation methods for series and</span>
-<span class="sd">sequences.</span>
-
-<span class="sd">References:</span>
-<span class="sd">Carl M. Bender &amp; Steven A. Orszag, &quot;Advanced Mathematical Methods for</span>
-<span class="sd">Scientists and Engineers: Asymptotic Methods and Perturbation Theory&quot;,</span>
-<span class="sd">Springer 1999. (Shanks transformation: pp. 368-375, Richardson</span>
-<span class="sd">extrapolation: pp. 375-377.)</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">Integer</span><span class="p">,</span> <span class="n">S</span>
-
-
-<div class="viewcode-block" id="richardson"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.acceleration.richardson">[docs]</a><span class="k">def</span> <span class="nf">richardson</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">N</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Calculate an approximation for lim k-&gt;oo A(k) using Richardson</span>
-<span class="sd">    extrapolation with the terms A(n), A(n+1), ..., A(n+N+1).</span>
-<span class="sd">    Choosing N ~= 2*n often gives good results.</span>
-
-<span class="sd">    A simple example is to calculate exp(1) using the limit definition.</span>
-<span class="sd">    This limit converges slowly; n = 100 only produces two accurate</span>
-<span class="sd">    digits:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import n</span>
-<span class="sd">        &gt;&gt;&gt; e = (1 + 1/n)**n</span>
-<span class="sd">        &gt;&gt;&gt; print(round(e.subs(n, 100).evalf(), 10))</span>
-<span class="sd">        2.7048138294</span>
-
-<span class="sd">    Richardson extrapolation with 11 appropriately chosen terms gives</span>
-<span class="sd">    a value that is accurate to the indicated precision:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import E</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.series.acceleration import richardson</span>
-<span class="sd">        &gt;&gt;&gt; print(round(richardson(e, n, 10, 20).evalf(), 10))</span>
-<span class="sd">        2.7182818285</span>
-<span class="sd">        &gt;&gt;&gt; print(round(E.evalf(), 10))</span>
-<span class="sd">        2.7182818285</span>
-
-<span class="sd">    Another useful application is to speed up convergence of series.</span>
-<span class="sd">    Computing 100 terms of the zeta(2) series 1/k**2 yields only</span>
-<span class="sd">    two accurate digits:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import k, n</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import Sum</span>
-<span class="sd">        &gt;&gt;&gt; A = Sum(k**-2, (k, 1, n))</span>
-<span class="sd">        &gt;&gt;&gt; print(round(A.subs(n, 100).evalf(), 10))</span>
-<span class="sd">        1.6349839002</span>
-
-<span class="sd">    Richardson extrapolation performs much better:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import pi</span>
-<span class="sd">        &gt;&gt;&gt; print(round(richardson(A, n, 10, 20).evalf(), 10))</span>
-<span class="sd">        1.6449340668</span>
-<span class="sd">        &gt;&gt;&gt; print(round(((pi**2)/6).evalf(), 10))     # Exact value</span>
-<span class="sd">        1.6449340668</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">N</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">s</span> <span class="o">+=</span> <span class="n">A</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">Integer</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="n">j</span><span class="p">))</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="n">j</span><span class="p">)</span><span class="o">**</span><span class="n">N</span> <span class="o">*</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="n">N</span><span class="p">)</span> <span class="o">/</span> \
-            <span class="p">(</span><span class="n">factorial</span><span class="p">(</span><span class="n">j</span><span class="p">)</span> <span class="o">*</span> <span class="n">factorial</span><span class="p">(</span><span class="n">N</span> <span class="o">-</span> <span class="n">j</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">s</span>
-
-</div>
-<div class="viewcode-block" id="shanks"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.acceleration.shanks">[docs]</a><span class="k">def</span> <span class="nf">shanks</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Calculate an approximation for lim k-&gt;oo A(k) using the n-term Shanks</span>
-<span class="sd">    transformation S(A)(n). With m &gt; 1, calculate the m-fold recursive</span>
-<span class="sd">    Shanks transformation S(S(...S(A)...))(n).</span>
-
-<span class="sd">    The Shanks transformation is useful for summing Taylor series that</span>
-<span class="sd">    converge slowly near a pole or singularity, e.g. for log(2):</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import k, n</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import Sum, Integer</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.series.acceleration import shanks</span>
-<span class="sd">        &gt;&gt;&gt; A = Sum(Integer(-1)**(k+1) / k, (k, 1, n))</span>
-<span class="sd">        &gt;&gt;&gt; print(round(A.subs(n, 100).doit().evalf(), 10))</span>
-<span class="sd">        0.6881721793</span>
-<span class="sd">        &gt;&gt;&gt; print(round(shanks(A, n, 25).evalf(), 10))</span>
-<span class="sd">        0.6931396564</span>
-<span class="sd">        &gt;&gt;&gt; print(round(shanks(A, n, 25, 5).evalf(), 10))</span>
-<span class="sd">        0.6931471806</span>
-
-<span class="sd">    The correct value is 0.6931471805599453094172321215.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">table</span> <span class="o">=</span> <span class="p">[</span><span class="n">A</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">Integer</span><span class="p">(</span><span class="n">j</span><span class="p">))</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)]</span>
-    <span class="n">table2</span> <span class="o">=</span> <span class="n">table</span><span class="p">[:]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="n">table</span><span class="p">[</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">table</span><span class="p">[</span><span class="n">j</span><span class="p">],</span> <span class="n">table</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
-            <span class="n">table2</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">z</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">z</span> <span class="o">+</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
-        <span class="n">table</span> <span class="o">=</span> <span class="n">table2</span><span class="p">[:]</span>
-    <span class="k">return</span> <span class="n">table</span><span class="p">[</span><span class="n">n</span><span class="p">]</span></div>
-</pre></div>
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-            
-  <h1>Source code for sympy.series.gruntz</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Limits</span>
-<span class="sd">======</span>
-
-<span class="sd">Implemented according to the PhD thesis</span>
-<span class="sd">http://www.cybertester.com/data/gruntz.pdf, which contains very thorough</span>
-<span class="sd">descriptions of the algorithm including many examples.  We summarize here</span>
-<span class="sd">the gist of it.</span>
-
-<span class="sd">All functions are sorted according to how rapidly varying they are at</span>
-<span class="sd">infinity using the following rules. Any two functions f and g can be</span>
-<span class="sd">compared using the properties of L:</span>
-
-<span class="sd">L=lim  log|f(x)| / log|g(x)|           (for x -&gt; oo)</span>
-
-<span class="sd">We define &gt;, &lt; ~ according to::</span>
-
-<span class="sd">    1. f &gt; g .... L=+-oo</span>
-
-<span class="sd">        we say that:</span>
-<span class="sd">        - f is greater than any power of g</span>
-<span class="sd">        - f is more rapidly varying than g</span>
-<span class="sd">        - f goes to infinity/zero faster than g</span>
-
-<span class="sd">    2. f &lt; g .... L=0</span>
-
-<span class="sd">        we say that:</span>
-<span class="sd">        - f is lower than any power of g</span>
-
-<span class="sd">    3. f ~ g .... L!=0, +-oo</span>
-
-<span class="sd">        we say that:</span>
-<span class="sd">        - both f and g are bounded from above and below by suitable integral</span>
-<span class="sd">          powers of the other</span>
-
-<span class="sd">Examples</span>
-<span class="sd">========</span>
-<span class="sd">::</span>
-<span class="sd">    2 &lt; x &lt; exp(x) &lt; exp(x**2) &lt; exp(exp(x))</span>
-<span class="sd">    2 ~ 3 ~ -5</span>
-<span class="sd">    x ~ x**2 ~ x**3 ~ 1/x ~ x**m ~ -x</span>
-<span class="sd">    exp(x) ~ exp(-x) ~ exp(2x) ~ exp(x)**2 ~ exp(x+exp(-x))</span>
-<span class="sd">    f ~ 1/f</span>
-
-<span class="sd">So we can divide all the functions into comparability classes (x and x^2</span>
-<span class="sd">belong to one class, exp(x) and exp(-x) belong to some other class). In</span>
-<span class="sd">principle, we could compare any two functions, but in our algorithm, we</span>
-<span class="sd">don&#39;t compare anything below the class 2~3~-5 (for example log(x) is</span>
-<span class="sd">below this), so we set 2~3~-5 as the lowest comparability class.</span>
-
-<span class="sd">Given the function f, we find the list of most rapidly varying (mrv set)</span>
-<span class="sd">subexpressions of it. This list belongs to the same comparability class.</span>
-<span class="sd">Let&#39;s say it is {exp(x), exp(2x)}. Using the rule f ~ 1/f we find an</span>
-<span class="sd">element &quot;w&quot; (either from the list or a new one) from the same</span>
-<span class="sd">comparability class which goes to zero at infinity. In our example we</span>
-<span class="sd">set w=exp(-x) (but we could also set w=exp(-2x) or w=exp(-3x) ...). We</span>
-<span class="sd">rewrite the mrv set using w, in our case {1/w, 1/w^2}, and substitute it</span>
-<span class="sd">into f. Then we expand f into a series in w::</span>
-
-<span class="sd">    f = c0*w^e0 + c1*w^e1 + ... + O(w^en),       where e0&lt;e1&lt;...&lt;en, c0!=0</span>
-
-<span class="sd">but for x-&gt;oo, lim f = lim c0*w^e0, because all the other terms go to zero,</span>
-<span class="sd">because w goes to zero faster than the ci and ei. So::</span>
-
-<span class="sd">    for e0&gt;0, lim f = 0</span>
-<span class="sd">    for e0&lt;0, lim f = +-oo   (the sign depends on the sign of c0)</span>
-<span class="sd">    for e0=0, lim f = lim c0</span>
-
-<span class="sd">We need to recursively compute limits at several places of the algorithm, but</span>
-<span class="sd">as is shown in the PhD thesis, it always finishes.</span>
-
-<span class="sd">Important functions from the implementation:</span>
-
-<span class="sd">compare(a, b, x) compares &quot;a&quot; and &quot;b&quot; by computing the limit L.</span>
-<span class="sd">mrv(e, x) returns list of most rapidly varying (mrv) subexpressions of &quot;e&quot;</span>
-<span class="sd">rewrite(e, Omega, x, wsym) rewrites &quot;e&quot; in terms of w</span>
-<span class="sd">leadterm(f, x) returns the lowest power term in the series of f</span>
-<span class="sd">mrv_leadterm(e, x) returns the lead term (c0, e0) for e</span>
-<span class="sd">limitinf(e, x) computes lim e  (for x-&gt;oo)</span>
-<span class="sd">limit(e, z, z0) computes any limit by converting it to the case x-&gt;oo</span>
-
-<span class="sd">All the functions are really simple and straightforward except</span>
-<span class="sd">rewrite(), which is the most difficult/complex part of the algorithm.</span>
-<span class="sd">When the algorithm fails, the bugs are usually in the series expansion</span>
-<span class="sd">(i.e. in SymPy) or in rewrite.</span>
-
-<span class="sd">This code is almost exact rewrite of the Maple code inside the Gruntz</span>
-<span class="sd">thesis.</span>
-
-<span class="sd">Debugging</span>
-<span class="sd">---------</span>
-
-<span class="sd">Because the gruntz algorithm is highly recursive, it&#39;s difficult to</span>
-<span class="sd">figure out what went wrong inside a debugger. Instead, turn on nice</span>
-<span class="sd">debug prints by defining the environment variable SYMPY_DEBUG. For</span>
-<span class="sd">example:</span>
-
-<span class="sd">[user@localhost]: SYMPY_DEBUG=True ./bin/isympy</span>
-
-<span class="sd">In [1]: limit(sin(x)/x, x, 0)</span>
-<span class="sd">limitinf(_x*sin(1/_x), _x) = 1</span>
-<span class="sd">+-mrv_leadterm(_x*sin(1/_x), _x) = (1, 0)</span>
-<span class="sd">| +-mrv(_x*sin(1/_x), _x) = set([_x])</span>
-<span class="sd">| | +-mrv(_x, _x) = set([_x])</span>
-<span class="sd">| | +-mrv(sin(1/_x), _x) = set([_x])</span>
-<span class="sd">| |   +-mrv(1/_x, _x) = set([_x])</span>
-<span class="sd">| |     +-mrv(_x, _x) = set([_x])</span>
-<span class="sd">| +-mrv_leadterm(exp(_x)*sin(exp(-_x)), _x, set([exp(_x)])) = (1, 0)</span>
-<span class="sd">|   +-rewrite(exp(_x)*sin(exp(-_x)), set([exp(_x)]), _x, _w) = (1/_w*sin(_w), -_x)</span>
-<span class="sd">|     +-sign(_x, _x) = 1</span>
-<span class="sd">|     +-mrv_leadterm(1, _x) = (1, 0)</span>
-<span class="sd">+-sign(0, _x) = 0</span>
-<span class="sd">+-limitinf(1, _x) = 1</span>
-
-<span class="sd">And check manually which line is wrong. Then go to the source code and</span>
-<span class="sd">debug this function to figure out the exact problem.</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">SYMPY_DEBUG</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">,</span> <span class="n">Wild</span>
-<span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">log</span><span class="p">,</span> <span class="n">exp</span>
-<span class="kn">from</span> <span class="nn">sympy.series.order</span> <span class="kn">import</span> <span class="n">Order</span>
-<span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">powsimp</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">cacheit</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="nb">reduce</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-<span class="n">O</span> <span class="o">=</span> <span class="n">Order</span>
-
-
-<span class="k">def</span> <span class="nf">debug</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Only for debugging purposes: prints a tree</span>
-
-<span class="sd">    It will print a nice execution tree with arguments and results</span>
-<span class="sd">    of all decorated functions.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">SYMPY_DEBUG</span><span class="p">:</span>
-        <span class="c">#normal mode - do nothing</span>
-        <span class="k">return</span> <span class="n">func</span>
-
-    <span class="c">#debug mode</span>
-    <span class="k">def</span> <span class="nf">decorated</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="c">#r = func(*args, **kwargs)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">maketree</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="c">#print &quot;%s = %s(%s, %s)&quot; % (r, func.__name__, args, kwargs)</span>
-        <span class="k">return</span> <span class="n">r</span>
-
-    <span class="k">return</span> <span class="n">decorated</span>
-
-<span class="kn">from</span> <span class="nn">sympy.utilities.timeutils</span> <span class="kn">import</span> <span class="n">timethis</span>
-<span class="n">timeit</span> <span class="o">=</span> <span class="n">timethis</span><span class="p">(</span><span class="s">&#39;gruntz&#39;</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">tree</span><span class="p">(</span><span class="n">subtrees</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Only debugging purposes: prints a tree&quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">indent</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="nb">type</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="s">&quot;+-</span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="s">&quot;&quot;</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">if</span> <span class="nb">type</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">r</span> <span class="o">+=</span> <span class="s">&quot;| </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">a</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">r</span> <span class="o">+=</span> <span class="s">&quot;  </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">a</span>
-        <span class="k">return</span> <span class="n">r</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">subtrees</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="s">&quot;&quot;</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">subtrees</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-        <span class="n">f</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">indent</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
-    <span class="n">f</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">indent</span><span class="p">(</span><span class="n">subtrees</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="mi">2</span><span class="p">))</span>
-    <span class="k">return</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-<span class="n">tmp</span> <span class="o">=</span> <span class="p">[]</span>
-<span class="nb">iter</span> <span class="o">=</span> <span class="mi">0</span>
-
-
-<span class="k">def</span> <span class="nf">maketree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kw</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Only debugging purposes: prints a tree&quot;&quot;&quot;</span>
-    <span class="k">global</span> <span class="n">tmp</span>
-    <span class="k">global</span> <span class="nb">iter</span>
-    <span class="n">oldtmp</span> <span class="o">=</span> <span class="n">tmp</span>
-    <span class="n">tmp</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="nb">iter</span> <span class="o">+=</span> <span class="mi">1</span>
-
-    <span class="c"># If there is a bug and the algorithm enters an infinite loop, enable the</span>
-    <span class="c"># following line. It will print the names and parameters of all major functions</span>
-    <span class="c"># that are called, *before* they are called</span>
-    <span class="c">#print &quot;%s%s %s%s&quot; % (iter, reduce(lambda x, y: x + y,map(lambda x: &#39;-&#39;,range(1,2+iter))), f.func_name, args)</span>
-
-    <span class="n">r</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kw</span><span class="p">)</span>
-
-    <span class="nb">iter</span> <span class="o">-=</span> <span class="mi">1</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="s">&quot;</span><span class="si">%s%s</span><span class="s"> = </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="n">args</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">tmp</span> <span class="o">!=</span> <span class="p">[]:</span>
-        <span class="n">s</span> <span class="o">+=</span> <span class="n">tree</span><span class="p">(</span><span class="n">tmp</span><span class="p">)</span>
-    <span class="n">tmp</span> <span class="o">=</span> <span class="n">oldtmp</span>
-    <span class="n">tmp</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">iter</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">print</span><span class="p">(</span><span class="n">tmp</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="n">tmp</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">return</span> <span class="n">r</span>
-
-
-<div class="viewcode-block" id="compare"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.gruntz.compare">[docs]</a><span class="k">def</span> <span class="nf">compare</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns &quot;&lt;&quot; if a&lt;b, &quot;=&quot; for a == b, &quot;&gt;&quot; for a&gt;b&quot;&quot;&quot;</span>
-    <span class="c"># log(exp(...)) must always be simplified here for termination</span>
-    <span class="n">la</span><span class="p">,</span> <span class="n">lb</span> <span class="o">=</span> <span class="n">log</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">log</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Basic</span><span class="p">)</span> <span class="ow">and</span> <span class="n">a</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">exp</span><span class="p">:</span>
-        <span class="n">la</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Basic</span><span class="p">)</span> <span class="ow">and</span> <span class="n">b</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">exp</span><span class="p">:</span>
-        <span class="n">lb</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="n">c</span> <span class="o">=</span> <span class="n">limitinf</span><span class="p">(</span><span class="n">la</span><span class="o">/</span><span class="n">lb</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">c</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="s">&quot;&lt;&quot;</span>
-    <span class="k">elif</span> <span class="n">c</span><span class="o">.</span><span class="n">is_unbounded</span><span class="p">:</span>
-        <span class="k">return</span> <span class="s">&quot;&gt;&quot;</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="s">&quot;=&quot;</span>
-
-</div>
-<div class="viewcode-block" id="SubsSet"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.gruntz.SubsSet">[docs]</a><span class="k">class</span> <span class="nc">SubsSet</span><span class="p">(</span><span class="nb">dict</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Stores (expr, dummy) pairs, and how to rewrite expr-s.</span>
-
-<span class="sd">    The gruntz algorithm needs to rewrite certain expressions in term of a new</span>
-<span class="sd">    variable w. We cannot use subs, because it is just too smart for us. For</span>
-<span class="sd">    example::</span>
-
-<span class="sd">        &gt; Omega=[exp(exp(_p - exp(-_p))/(1 - 1/_p)), exp(exp(_p))]</span>
-<span class="sd">        &gt; O2=[exp(-exp(_p) + exp(-exp(-_p))*exp(_p)/(1 - 1/_p))/_w, 1/_w]</span>
-<span class="sd">        &gt; e = exp(exp(_p - exp(-_p))/(1 - 1/_p)) - exp(exp(_p))</span>
-<span class="sd">        &gt; e.subs(Omega[0],O2[0]).subs(Omega[1],O2[1])</span>
-<span class="sd">        -1/w + exp(exp(p)*exp(-exp(-p))/(1 - 1/p))</span>
-
-<span class="sd">    is really not what we want!</span>
-
-<span class="sd">    So we do it the hard way and keep track of all the things we potentially</span>
-<span class="sd">    want to substitute by dummy variables. Consider the expression::</span>
-
-<span class="sd">        exp(x - exp(-x)) + exp(x) + x.</span>
-
-<span class="sd">    The mrv set is {exp(x), exp(-x), exp(x - exp(-x))}.</span>
-<span class="sd">    We introduce corresponding dummy variables d1, d2, d3 and rewrite::</span>
-
-<span class="sd">        d3 + d1 + x.</span>
-
-<span class="sd">    This class first of all keeps track of the mapping expr-&gt;variable, i.e.</span>
-<span class="sd">    will at this stage be a dictionary::</span>
-
-<span class="sd">        {exp(x): d1, exp(-x): d2, exp(x - exp(-x)): d3}.</span>
-
-<span class="sd">    [It turns out to be more convenient this way round.]</span>
-<span class="sd">    But sometimes expressions in the mrv set have other expressions from the</span>
-<span class="sd">    mrv set as subexpressions, and we need to keep track of that as well. In</span>
-<span class="sd">    this case, d3 is really exp(x - d2), so rewrites at this stage is::</span>
-
-<span class="sd">        {d3: exp(x-d2)}.</span>
-
-<span class="sd">    The function rewrite uses all this information to correctly rewrite our</span>
-<span class="sd">    expression in terms of w. In this case w can be choosen to be exp(-x),</span>
-<span class="sd">    i.e. d2. The correct rewriting then is::</span>
-
-<span class="sd">        exp(-w)/w + 1/w + x.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">rewrites</span> <span class="o">=</span> <span class="p">{}</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">super</span><span class="p">(</span><span class="n">SubsSet</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">__repr__</span><span class="p">()</span> <span class="o">+</span> <span class="s">&#39;, &#39;</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">rewrites</span><span class="o">.</span><span class="n">__repr__</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">key</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">:</span>
-            <span class="bp">self</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">()</span>
-        <span class="k">return</span> <span class="nb">dict</span><span class="o">.</span><span class="n">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">do_subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">expr</span><span class="p">,</span> <span class="n">var</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">e</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">var</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">e</span>
-
-<div class="viewcode-block" id="SubsSet.meets"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.gruntz.SubsSet.meets">[docs]</a>    <span class="k">def</span> <span class="nf">meets</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s2</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Tell whether or not self and s2 have non-empty intersection&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">set</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">s2</span><span class="o">.</span><span class="n">keys</span><span class="p">()))</span> <span class="o">!=</span> <span class="nb">set</span><span class="p">()</span>
-</div>
-<div class="viewcode-block" id="SubsSet.union"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.gruntz.SubsSet.union">[docs]</a>    <span class="k">def</span> <span class="nf">union</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s2</span><span class="p">,</span> <span class="n">exps</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Compute the union of self and s2, adjusting exps&quot;&quot;&quot;</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="n">tr</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">expr</span><span class="p">,</span> <span class="n">var</span> <span class="ow">in</span> <span class="n">s2</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="k">if</span> <span class="n">expr</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">exps</span><span class="p">:</span>
-                    <span class="n">exps</span> <span class="o">=</span> <span class="n">exps</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">var</span><span class="p">,</span> <span class="n">res</span><span class="p">[</span><span class="n">expr</span><span class="p">])</span>
-                <span class="n">tr</span><span class="p">[</span><span class="n">var</span><span class="p">]</span> <span class="o">=</span> <span class="n">res</span><span class="p">[</span><span class="n">expr</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">res</span><span class="p">[</span><span class="n">expr</span><span class="p">]</span> <span class="o">=</span> <span class="n">var</span>
-        <span class="k">for</span> <span class="n">var</span><span class="p">,</span> <span class="n">rewr</span> <span class="ow">in</span> <span class="n">s2</span><span class="o">.</span><span class="n">rewrites</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">res</span><span class="o">.</span><span class="n">rewrites</span><span class="p">[</span><span class="n">var</span><span class="p">]</span> <span class="o">=</span> <span class="n">rewr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">tr</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">res</span><span class="p">,</span> <span class="n">exps</span>
-</div>
-    <span class="k">def</span> <span class="nf">copy</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">SubsSet</span><span class="p">()</span>
-        <span class="n">r</span><span class="o">.</span><span class="n">rewrites</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">rewrites</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">expr</span><span class="p">,</span> <span class="n">var</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">r</span><span class="p">[</span><span class="n">expr</span><span class="p">]</span> <span class="o">=</span> <span class="n">var</span>
-        <span class="k">return</span> <span class="n">r</span>
-
-</div>
-<span class="nd">@debug</span>
-<div class="viewcode-block" id="mrv"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.gruntz.mrv">[docs]</a><span class="k">def</span> <span class="nf">mrv</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns a SubsSet of most rapidly varying (mrv) subexpressions of &#39;e&#39;,</span>
-<span class="sd">       and e rewritten in terms of these&quot;&quot;&quot;</span>
-    <span class="n">e</span> <span class="o">=</span> <span class="n">powsimp</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;exp&#39;</span><span class="p">)</span>
-    <span class="k">assert</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Basic</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">e</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">SubsSet</span><span class="p">(),</span> <span class="n">e</span>
-    <span class="k">elif</span> <span class="n">e</span> <span class="o">==</span> <span class="n">x</span><span class="p">:</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">SubsSet</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">s</span><span class="p">,</span> <span class="n">s</span><span class="p">[</span><span class="n">x</span><span class="p">]</span>
-    <span class="k">elif</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Mul</span> <span class="ow">or</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-        <span class="n">i</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>  <span class="c"># throw away x-independent terms</span>
-        <span class="k">if</span> <span class="n">d</span><span class="o">.</span><span class="n">func</span> <span class="o">!=</span> <span class="n">e</span><span class="o">.</span><span class="n">func</span><span class="p">:</span>
-            <span class="n">s</span><span class="p">,</span> <span class="n">expr</span> <span class="o">=</span> <span class="n">mrv</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">s</span><span class="p">,</span> <span class="n">e</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">as_two_terms</span><span class="p">()</span>
-        <span class="n">s1</span><span class="p">,</span> <span class="n">e1</span> <span class="o">=</span> <span class="n">mrv</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="n">s2</span><span class="p">,</span> <span class="n">e2</span> <span class="o">=</span> <span class="n">mrv</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">mrv_max1</span><span class="p">(</span><span class="n">s1</span><span class="p">,</span> <span class="n">s2</span><span class="p">,</span> <span class="n">e</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">e1</span><span class="p">,</span> <span class="n">e2</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-        <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">mrv</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">e</span> <span class="o">*</span> <span class="n">log</span><span class="p">(</span><span class="n">b</span><span class="p">)),</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">s</span><span class="p">,</span> <span class="n">expr</span> <span class="o">=</span> <span class="n">mrv</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">s</span><span class="p">,</span> <span class="n">expr</span><span class="o">**</span><span class="n">e</span>
-    <span class="k">elif</span> <span class="n">e</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">log</span><span class="p">:</span>
-        <span class="n">s</span><span class="p">,</span> <span class="n">expr</span> <span class="o">=</span> <span class="n">mrv</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">s</span><span class="p">,</span> <span class="n">log</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">e</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">exp</span><span class="p">:</span>
-        <span class="c"># We know from the theory of this algorithm that exp(log(...)) may always</span>
-        <span class="c"># be simplified here, and doing so is vital for termination.</span>
-        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">log</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">mrv</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">limitinf</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_unbounded</span><span class="p">:</span>
-            <span class="n">s1</span> <span class="o">=</span> <span class="n">SubsSet</span><span class="p">()</span>
-            <span class="n">e1</span> <span class="o">=</span> <span class="n">s1</span><span class="p">[</span><span class="n">e</span><span class="p">]</span>
-            <span class="n">s2</span><span class="p">,</span> <span class="n">e2</span> <span class="o">=</span> <span class="n">mrv</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
-            <span class="n">su</span> <span class="o">=</span> <span class="n">s1</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="n">s2</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">su</span><span class="o">.</span><span class="n">rewrites</span><span class="p">[</span><span class="n">e1</span><span class="p">]</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="n">e2</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">mrv_max3</span><span class="p">(</span><span class="n">s1</span><span class="p">,</span> <span class="n">e1</span><span class="p">,</span> <span class="n">s2</span><span class="p">,</span> <span class="n">exp</span><span class="p">(</span><span class="n">e2</span><span class="p">),</span> <span class="n">su</span><span class="p">,</span> <span class="n">e1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">s</span><span class="p">,</span> <span class="n">expr</span> <span class="o">=</span> <span class="n">mrv</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">s</span><span class="p">,</span> <span class="n">exp</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Function</span><span class="p">:</span>
-        <span class="n">l</span> <span class="o">=</span> <span class="p">[</span><span class="n">mrv</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="n">l2</span> <span class="o">=</span> <span class="p">[</span><span class="n">s</span> <span class="k">for</span> <span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">_</span><span class="p">)</span> <span class="ow">in</span> <span class="n">l</span> <span class="k">if</span> <span class="n">s</span> <span class="o">!=</span> <span class="n">SubsSet</span><span class="p">()]</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">l2</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># e.g. something like BesselJ(x, x)</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;MRV set computation for functions in&quot;</span>
-                                      <span class="s">&quot; several variables not implemented.&quot;</span><span class="p">)</span>
-        <span class="n">s</span><span class="p">,</span> <span class="n">ss</span> <span class="o">=</span> <span class="n">l2</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">SubsSet</span><span class="p">()</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">ss</span><span class="o">.</span><span class="n">do_subs</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">l</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">s</span><span class="p">,</span> <span class="n">e</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Derivative</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;MRV set computation for derviatives&quot;</span>
-                                  <span class="s">&quot; not implemented yet.&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">mrv</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
-    <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-        <span class="s">&quot;Don&#39;t know how to calculate the mrv of &#39;</span><span class="si">%s</span><span class="s">&#39;&quot;</span> <span class="o">%</span> <span class="n">e</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="mrv_max3"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.gruntz.mrv_max3">[docs]</a><span class="k">def</span> <span class="nf">mrv_max3</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">expsf</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">expsg</span><span class="p">,</span> <span class="n">union</span><span class="p">,</span> <span class="n">expsboth</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Computes the maximum of two sets of expressions f and g, which</span>
-<span class="sd">    are in the same comparability class, i.e. max() compares (two elements of)</span>
-<span class="sd">    f and g and returns either (f, expsf) [if f is larger], (g, expsg)</span>
-<span class="sd">    [if g is larger] or (union, expsboth) [if f, g are of the same class].</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">assert</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">SubsSet</span><span class="p">)</span>
-    <span class="k">assert</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">SubsSet</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">f</span> <span class="o">==</span> <span class="n">SubsSet</span><span class="p">():</span>
-        <span class="k">return</span> <span class="n">g</span><span class="p">,</span> <span class="n">expsg</span>
-    <span class="k">elif</span> <span class="n">g</span> <span class="o">==</span> <span class="n">SubsSet</span><span class="p">():</span>
-        <span class="k">return</span> <span class="n">f</span><span class="p">,</span> <span class="n">expsf</span>
-    <span class="k">elif</span> <span class="n">f</span><span class="o">.</span><span class="n">meets</span><span class="p">(</span><span class="n">g</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">union</span><span class="p">,</span> <span class="n">expsboth</span>
-
-    <span class="n">c</span> <span class="o">=</span> <span class="n">compare</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">keys</span><span class="p">())[</span><span class="mi">0</span><span class="p">],</span> <span class="nb">list</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">keys</span><span class="p">())[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">c</span> <span class="o">==</span> <span class="s">&quot;&gt;&quot;</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span><span class="p">,</span> <span class="n">expsf</span>
-    <span class="k">elif</span> <span class="n">c</span> <span class="o">==</span> <span class="s">&quot;&lt;&quot;</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">g</span><span class="p">,</span> <span class="n">expsg</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">assert</span> <span class="n">c</span> <span class="o">==</span> <span class="s">&quot;=&quot;</span>
-        <span class="k">return</span> <span class="n">union</span><span class="p">,</span> <span class="n">expsboth</span>
-
-</div>
-<div class="viewcode-block" id="mrv_max1"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.gruntz.mrv_max1">[docs]</a><span class="k">def</span> <span class="nf">mrv_max1</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">exps</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Computes the maximum of two sets of expressions f and g, which</span>
-<span class="sd">    are in the same comparability class, i.e. mrv_max1() compares (two elements of)</span>
-<span class="sd">    f and g and returns the set, which is in the higher comparability class</span>
-<span class="sd">    of the union of both, if they have the same order of variation.</span>
-<span class="sd">    Also returns exps, with the appropriate substitutions made.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">u</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">exps</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">mrv_max3</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="o">.</span><span class="n">do_subs</span><span class="p">(</span><span class="n">exps</span><span class="p">),</span> <span class="n">g</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">do_subs</span><span class="p">(</span><span class="n">exps</span><span class="p">),</span>
-                    <span class="n">u</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-</div>
-<span class="nd">@debug</span>
-<span class="nd">@cacheit</span>
-<span class="nd">@timeit</span>
-<div class="viewcode-block" id="sign"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.gruntz.sign">[docs]</a><span class="k">def</span> <span class="nf">sign</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns a sign of an expression e(x) for x-&gt;oo.</span>
-
-<span class="sd">    ::</span>
-
-<span class="sd">        e &gt;  0 for x sufficiently large ...  1</span>
-<span class="sd">        e == 0 for x sufficiently large ...  0</span>
-<span class="sd">        e &lt;  0 for x sufficiently large ... -1</span>
-
-<span class="sd">    The result of this function is currently undefined if e changes sign</span>
-<span class="sd">    arbitarily often for arbitrarily large x (e.g. sin(x)).</span>
-
-<span class="sd">    Note that this returns zero only if e is *constantly* zero</span>
-<span class="sd">    for x sufficiently large. [If e is constant, of course, this is just</span>
-<span class="sd">    the same thing as the sign of e.]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sign</span> <span class="k">as</span> <span class="n">_sign</span>
-    <span class="k">assert</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Basic</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">1</span>
-    <span class="k">elif</span> <span class="n">e</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-        <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-    <span class="k">elif</span> <span class="n">e</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-
-    <span class="k">elif</span> <span class="ow">not</span> <span class="n">e</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">_sign</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">e</span> <span class="o">==</span> <span class="n">x</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">1</span>
-    <span class="k">elif</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">as_two_terms</span><span class="p">()</span>
-        <span class="n">sa</span> <span class="o">=</span> <span class="n">sign</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">sa</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">0</span>
-        <span class="k">return</span> <span class="n">sa</span> <span class="o">*</span> <span class="n">sign</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">e</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">exp</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">1</span>
-    <span class="k">elif</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">sign</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">s</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">s</span><span class="o">**</span><span class="n">e</span><span class="o">.</span><span class="n">exp</span>
-    <span class="k">elif</span> <span class="n">e</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">log</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">sign</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="c"># if all else fails, do it the hard way</span>
-    <span class="n">c0</span><span class="p">,</span> <span class="n">e0</span> <span class="o">=</span> <span class="n">mrv_leadterm</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">sign</span><span class="p">(</span><span class="n">c0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-</div>
-<span class="nd">@debug</span>
-<span class="nd">@timeit</span>
-<span class="nd">@cacheit</span>
-<div class="viewcode-block" id="limitinf"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.gruntz.limitinf">[docs]</a><span class="k">def</span> <span class="nf">limitinf</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Limit e(x) for x-&gt; oo&quot;&quot;&quot;</span>
-    <span class="c">#rewrite e in terms of tractable functions only</span>
-    <span class="n">e</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="s">&#39;tractable&#39;</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">e</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">e</span>  <span class="c"># e is a constant</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">x</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-        <span class="c"># We make sure that x.is_positive is True so we</span>
-        <span class="c"># get all the correct mathematical bechavior from the expression.</span>
-        <span class="c"># We need a fresh variable.</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">bounded</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="n">e</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">p</span>
-    <span class="n">c0</span><span class="p">,</span> <span class="n">e0</span> <span class="o">=</span> <span class="n">mrv_leadterm</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-    <span class="n">sig</span> <span class="o">=</span> <span class="n">sign</span><span class="p">(</span><span class="n">e0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">sig</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>  <span class="c"># e0&gt;0: lim f = 0</span>
-    <span class="k">elif</span> <span class="n">sig</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>  <span class="c"># e0&lt;0: lim f = +-oo (the sign depends on the sign of c0)</span>
-        <span class="k">if</span> <span class="n">c0</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">Wild</span><span class="p">(</span><span class="s">&quot;a&quot;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">I</span><span class="p">])):</span>
-            <span class="k">return</span> <span class="n">c0</span><span class="o">*</span><span class="n">oo</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">sign</span><span class="p">(</span><span class="n">c0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="c">#the leading term shouldn&#39;t be 0:</span>
-        <span class="k">assert</span> <span class="n">s</span> <span class="o">!=</span> <span class="mi">0</span>
-        <span class="k">return</span> <span class="n">s</span><span class="o">*</span><span class="n">oo</span>
-    <span class="k">elif</span> <span class="n">sig</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">limitinf</span><span class="p">(</span><span class="n">c0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>  <span class="c"># e0=0: lim f = lim c0</span>
-
-</div>
-<span class="k">def</span> <span class="nf">moveup2</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">SubsSet</span><span class="p">()</span>
-    <span class="k">for</span> <span class="n">expr</span><span class="p">,</span> <span class="n">var</span> <span class="ow">in</span> <span class="n">s</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="n">r</span><span class="p">[</span><span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">))]</span> <span class="o">=</span> <span class="n">var</span>
-    <span class="k">for</span> <span class="n">var</span><span class="p">,</span> <span class="n">expr</span> <span class="ow">in</span> <span class="n">s</span><span class="o">.</span><span class="n">rewrites</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="n">r</span><span class="o">.</span><span class="n">rewrites</span><span class="p">[</span><span class="n">var</span><span class="p">]</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">rewrites</span><span class="p">[</span><span class="n">var</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">r</span>
-
-
-<span class="k">def</span> <span class="nf">moveup</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-    <span class="k">return</span> <span class="p">[</span><span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">l</span><span class="p">]</span>
-
-
-<span class="nd">@debug</span>
-<span class="nd">@timeit</span>
-<div class="viewcode-block" id="calculate_series"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.gruntz.calculate_series">[docs]</a><span class="k">def</span> <span class="nf">calculate_series</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">skip_abs</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Calculates at least one term of the series of &quot;e&quot; in &quot;x&quot;.</span>
-
-<span class="sd">    This is a place that fails most often, so it is in its own function.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">series</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logx</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">series</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">O</span><span class="p">):</span>
-            <span class="c"># The series expansion is locally exact.</span>
-            <span class="k">return</span> <span class="n">series</span>
-
-        <span class="n">series</span> <span class="o">=</span> <span class="n">series</span><span class="o">.</span><span class="n">removeO</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">series</span> <span class="ow">and</span> <span class="p">((</span><span class="ow">not</span> <span class="n">skip_abs</span><span class="p">)</span> <span class="ow">or</span> <span class="n">series</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="n">series</span>
-        <span class="n">n</span> <span class="o">*=</span> <span class="mi">2</span>
-
-</div>
-<span class="nd">@debug</span>
-<span class="nd">@timeit</span>
-<span class="nd">@cacheit</span>
-<div class="viewcode-block" id="mrv_leadterm"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.gruntz.mrv_leadterm">[docs]</a><span class="k">def</span> <span class="nf">mrv_leadterm</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns (c0, e0) for e.&quot;&quot;&quot;</span>
-    <span class="n">Omega</span> <span class="o">=</span> <span class="n">SubsSet</span><span class="p">()</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">e</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">Omega</span> <span class="o">==</span> <span class="n">SubsSet</span><span class="p">():</span>
-        <span class="n">Omega</span><span class="p">,</span> <span class="n">exps</span> <span class="o">=</span> <span class="n">mrv</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">Omega</span><span class="p">:</span>
-        <span class="c"># e really does not depend on x after simplification</span>
-        <span class="n">series</span> <span class="o">=</span> <span class="n">calculate_series</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="n">c0</span><span class="p">,</span> <span class="n">e0</span> <span class="o">=</span> <span class="n">series</span><span class="o">.</span><span class="n">leadterm</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">assert</span> <span class="n">e0</span> <span class="o">==</span> <span class="mi">0</span>
-        <span class="k">return</span> <span class="n">c0</span><span class="p">,</span> <span class="n">e0</span>
-    <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">Omega</span><span class="p">:</span>
-        <span class="c">#move the whole omega up (exponentiate each term):</span>
-        <span class="n">Omega_up</span> <span class="o">=</span> <span class="n">moveup2</span><span class="p">(</span><span class="n">Omega</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="n">e_up</span> <span class="o">=</span> <span class="n">moveup</span><span class="p">([</span><span class="n">e</span><span class="p">],</span> <span class="n">x</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">exps_up</span> <span class="o">=</span> <span class="n">moveup</span><span class="p">([</span><span class="n">exps</span><span class="p">],</span> <span class="n">x</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="c"># NOTE: there is no need to move this down!</span>
-        <span class="n">e</span> <span class="o">=</span> <span class="n">e_up</span>
-        <span class="n">Omega</span> <span class="o">=</span> <span class="n">Omega_up</span>
-        <span class="n">exps</span> <span class="o">=</span> <span class="n">exps_up</span>
-    <span class="c">#</span>
-    <span class="c"># The positive dummy, w, is used here so log(w*2) etc. will expand;</span>
-    <span class="c"># a unique dummy is needed in this algorithm</span>
-    <span class="c">#</span>
-    <span class="c"># For limits of complex functions, the algorithm would have to be</span>
-    <span class="c"># improved, or just find limits of Re and Im components separately.</span>
-    <span class="c">#</span>
-    <span class="n">w</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;w&quot;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">bounded</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">f</span><span class="p">,</span> <span class="n">logw</span> <span class="o">=</span> <span class="n">rewrite</span><span class="p">(</span><span class="n">exps</span><span class="p">,</span> <span class="n">Omega</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">w</span><span class="p">)</span>
-    <span class="n">series</span> <span class="o">=</span> <span class="n">calculate_series</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="n">logw</span><span class="p">)</span>
-    <span class="n">series</span> <span class="o">=</span> <span class="n">series</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">w</span><span class="p">),</span> <span class="n">logw</span><span class="p">)</span>  <span class="c"># this should not be necessary</span>
-    <span class="k">return</span> <span class="n">series</span><span class="o">.</span><span class="n">leadterm</span><span class="p">(</span><span class="n">w</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="build_expression_tree"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.gruntz.build_expression_tree">[docs]</a><span class="k">def</span> <span class="nf">build_expression_tree</span><span class="p">(</span><span class="n">Omega</span><span class="p">,</span> <span class="n">rewrites</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot; Helper function for rewrite.</span>
-
-<span class="sd">    We need to sort Omega (mrv set) so that we replace an expression before</span>
-<span class="sd">    we replace any expression in terms of which it has to be rewritten::</span>
-
-<span class="sd">        e1 ---&gt; e2 ---&gt; e3</span>
-<span class="sd">                 \</span>
-<span class="sd">                  -&gt; e4</span>
-
-<span class="sd">    Here we can do e1, e2, e3, e4 or e1, e2, e4, e3.</span>
-<span class="sd">    To do this we assemble the nodes into a tree, and sort them by height.</span>
-
-<span class="sd">    This function builds the tree, rewrites then sorts the nodes.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">class</span> <span class="nc">Node</span><span class="p">:</span>
-        <span class="k">def</span> <span class="nf">ht</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">reduce</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">,</span>
-                          <span class="p">[</span><span class="n">x</span><span class="o">.</span><span class="n">ht</span><span class="p">()</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">before</span><span class="p">],</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="n">nodes</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">for</span> <span class="n">expr</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">Omega</span><span class="p">:</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">Node</span><span class="p">()</span>
-        <span class="n">n</span><span class="o">.</span><span class="n">before</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">n</span><span class="o">.</span><span class="n">var</span> <span class="o">=</span> <span class="n">v</span>
-        <span class="n">n</span><span class="o">.</span><span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span>
-        <span class="n">nodes</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="n">n</span>
-    <span class="k">for</span> <span class="n">_</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">Omega</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">rewrites</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">nodes</span><span class="p">[</span><span class="n">v</span><span class="p">]</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">rewrites</span><span class="p">[</span><span class="n">v</span><span class="p">]</span>
-            <span class="k">for</span> <span class="n">_</span><span class="p">,</span> <span class="n">v2</span> <span class="ow">in</span> <span class="n">Omega</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">r</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">v2</span><span class="p">):</span>
-                    <span class="n">n</span><span class="o">.</span><span class="n">before</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">nodes</span><span class="p">[</span><span class="n">v2</span><span class="p">])</span>
-
-    <span class="k">return</span> <span class="n">nodes</span>
-
-</div>
-<span class="nd">@debug</span>
-<span class="nd">@timeit</span>
-<div class="viewcode-block" id="rewrite"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.gruntz.rewrite">[docs]</a><span class="k">def</span> <span class="nf">rewrite</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Omega</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">wsym</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;e(x) ... the function</span>
-<span class="sd">    Omega ... the mrv set</span>
-<span class="sd">    wsym ... the symbol which is going to be used for w</span>
-
-<span class="sd">    Returns the rewritten e in terms of w and log(w). See test_rewrite1()</span>
-<span class="sd">    for examples and correct results.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">ilcm</span>
-    <span class="k">assert</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">Omega</span><span class="p">,</span> <span class="n">SubsSet</span><span class="p">)</span>
-    <span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">Omega</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span>
-    <span class="c">#all items in Omega must be exponentials</span>
-    <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">Omega</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
-        <span class="k">assert</span> <span class="n">t</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">exp</span>
-    <span class="n">rewrites</span> <span class="o">=</span> <span class="n">Omega</span><span class="o">.</span><span class="n">rewrites</span>
-    <span class="n">Omega</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">Omega</span><span class="o">.</span><span class="n">items</span><span class="p">())</span>
-
-    <span class="n">nodes</span> <span class="o">=</span> <span class="n">build_expression_tree</span><span class="p">(</span><span class="n">Omega</span><span class="p">,</span> <span class="n">rewrites</span><span class="p">)</span>
-    <span class="n">Omega</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">nodes</span><span class="p">[</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]]</span><span class="o">.</span><span class="n">ht</span><span class="p">(),</span> <span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="n">g</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">Omega</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-        <span class="c"># g is going to be the &quot;w&quot; - the simplest one in the mrv set</span>
-    <span class="n">sig</span> <span class="o">=</span> <span class="n">sign</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">sig</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">wsym</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">wsym</span>  <span class="c"># if g goes to oo, substitute 1/w</span>
-    <span class="k">elif</span> <span class="n">sig</span> <span class="o">!=</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;Result depends on the sign of </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">sig</span><span class="p">)</span>
-    <span class="c">#O2 is a list, which results by rewriting each item in Omega using &quot;w&quot;</span>
-    <span class="n">O2</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">denominators</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">f</span><span class="p">,</span> <span class="n">var</span> <span class="ow">in</span> <span class="n">Omega</span><span class="p">:</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">limitinf</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="n">g</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">c</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="n">denominators</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">c</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-        <span class="n">arg</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">var</span> <span class="ow">in</span> <span class="n">rewrites</span><span class="p">:</span>
-            <span class="k">assert</span> <span class="n">rewrites</span><span class="p">[</span><span class="n">var</span><span class="p">]</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">exp</span>
-            <span class="n">arg</span> <span class="o">=</span> <span class="n">rewrites</span><span class="p">[</span><span class="n">var</span><span class="p">]</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">O2</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">var</span><span class="p">,</span> <span class="n">exp</span><span class="p">((</span><span class="n">arg</span> <span class="o">-</span> <span class="n">c</span><span class="o">*</span><span class="n">g</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">expand</span><span class="p">())</span><span class="o">*</span><span class="n">wsym</span><span class="o">**</span><span class="n">c</span><span class="p">))</span>
-
-    <span class="c">#Remember that Omega contains subexpressions of &quot;e&quot;. So now we find</span>
-    <span class="c">#them in &quot;e&quot; and substitute them for our rewriting, stored in O2</span>
-
-    <span class="c"># the following powsimp is necessary to automatically combine exponentials,</span>
-    <span class="c"># so that the .subs() below succeeds:</span>
-    <span class="c"># TODO this should not be necessary</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">powsimp</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;exp&#39;</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">O2</span><span class="p">:</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">_</span><span class="p">,</span> <span class="n">var</span> <span class="ow">in</span> <span class="n">Omega</span><span class="p">:</span>
-        <span class="k">assert</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">var</span><span class="p">)</span>
-
-    <span class="c">#finally compute the logarithm of w (logw).</span>
-    <span class="n">logw</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">sig</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">logw</span> <span class="o">=</span> <span class="o">-</span><span class="n">logw</span>  <span class="c"># log(w)-&gt;log(1/w)=-log(w)</span>
-
-    <span class="c"># Some parts of sympy have difficulty computing series expansions with</span>
-    <span class="c"># non-integral exponents. The following heuristic improves the situation:</span>
-    <span class="n">exponent</span> <span class="o">=</span> <span class="nb">reduce</span><span class="p">(</span><span class="n">ilcm</span><span class="p">,</span> <span class="n">denominators</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">wsym</span><span class="p">,</span> <span class="n">wsym</span><span class="o">**</span><span class="n">exponent</span><span class="p">)</span>
-    <span class="n">logw</span> <span class="o">/=</span> <span class="n">exponent</span>
-
-    <span class="k">return</span> <span class="n">f</span><span class="p">,</span> <span class="n">logw</span>
-
-</div>
-<div class="viewcode-block" id="gruntz"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.gruntz.gruntz">[docs]</a><span class="k">def</span> <span class="nf">gruntz</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="nb">dir</span><span class="o">=</span><span class="s">&quot;+&quot;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute the limit of e(z) at the point z0 using the Gruntz algorithm.</span>
-
-<span class="sd">    z0 can be any expression, including oo and -oo.</span>
-
-<span class="sd">    For dir=&quot;+&quot; (default) it calculates the limit from the right</span>
-<span class="sd">    (z-&gt;z0+) and for dir=&quot;-&quot; the limit from the left (z-&gt;z0-). For infinite z0</span>
-<span class="sd">    (oo or -oo), the dir argument doesn&#39;t matter.</span>
-
-<span class="sd">    This algorithm is fully described in the module docstring in the gruntz.py</span>
-<span class="sd">    file. It relies heavily on the series expansion. Most frequently, gruntz()</span>
-<span class="sd">    is only used if the faster limit() function (which uses heuristics) fails.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Second argument must be a Symbol&quot;</span><span class="p">)</span>
-
-    <span class="c">#convert all limits to the limit z-&gt;oo; sign of z is handled in limitinf</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="k">if</span> <span class="n">z0</span> <span class="o">==</span> <span class="n">oo</span><span class="p">:</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">limitinf</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">z0</span> <span class="o">==</span> <span class="o">-</span><span class="n">oo</span><span class="p">:</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">limitinf</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="o">-</span><span class="n">z</span><span class="p">),</span> <span class="n">z</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">dir</span> <span class="o">==</span> <span class="s">&quot;-&quot;</span><span class="p">:</span>
-            <span class="n">e0</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">z0</span> <span class="o">-</span> <span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">dir</span> <span class="o">==</span> <span class="s">&quot;+&quot;</span><span class="p">:</span>
-            <span class="n">e0</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">z0</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;dir must be &#39;+&#39; or &#39;-&#39;&quot;</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">limitinf</span><span class="p">(</span><span class="n">e0</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-
-    <span class="c"># This is a bit of a heuristic for nice results... we always rewrite</span>
-    <span class="c"># tractable functions in terms of familiar intractable ones.</span>
-    <span class="c"># It might be nicer to rewrite the exactly to what they were initially,</span>
-    <span class="c"># but that would take some work to implement.</span>
-    <span class="k">return</span> <span class="n">r</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="s">&#39;intractable&#39;</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span></div>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.series.limits</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Add</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">PoleError</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">C</span>
-<span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">tan</span><span class="p">,</span> <span class="n">cot</span>
-<span class="kn">from</span> <span class="nn">.gruntz</span> <span class="kn">import</span> <span class="n">gruntz</span>
-
-
-<div class="viewcode-block" id="limit"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.limits.limit">[docs]</a><span class="k">def</span> <span class="nf">limit</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="nb">dir</span><span class="o">=</span><span class="s">&quot;+&quot;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Compute the limit of e(z) at the point z0.</span>
-
-<span class="sd">    z0 can be any expression, including oo and -oo.</span>
-
-<span class="sd">    For dir=&quot;+&quot; (default) it calculates the limit from the right</span>
-<span class="sd">    (z-&gt;z0+) and for dir=&quot;-&quot; the limit from the left (z-&gt;z0-). For infinite z0</span>
-<span class="sd">    (oo or -oo), the dir argument doesn&#39;t matter.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import limit, sin, Symbol, oo</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; limit(sin(x)/x, x, 0)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; limit(1/x, x, 0, dir=&quot;+&quot;)</span>
-<span class="sd">    oo</span>
-<span class="sd">    &gt;&gt;&gt; limit(1/x, x, 0, dir=&quot;-&quot;)</span>
-<span class="sd">    -oo</span>
-<span class="sd">    &gt;&gt;&gt; limit(1/x, x, oo)</span>
-<span class="sd">    0</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    First we try some heuristics for easy and frequent cases like &quot;x&quot;, &quot;1/x&quot;,</span>
-<span class="sd">    &quot;x**2&quot; and similar, so that it&#39;s fast. For all other cases, we use the</span>
-<span class="sd">    Gruntz algorithm (see the gruntz() function).</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Wild</span><span class="p">,</span> <span class="n">log</span>
-
-    <span class="n">e</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-    <span class="n">z</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-    <span class="n">z0</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">z0</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">e</span> <span class="o">==</span> <span class="n">z</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">z0</span>
-
-    <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">e</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">e</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">e</span>
-
-    <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">tan</span><span class="p">:</span>
-        <span class="c"># discontinuity at odd multiples of pi/2; 0 at even</span>
-        <span class="n">disc</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">/</span><span class="mi">2</span>
-        <span class="n">sign</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="nb">dir</span> <span class="o">==</span> <span class="s">&#39;-&#39;</span><span class="p">:</span>
-            <span class="n">sign</span> <span class="o">*=</span> <span class="o">-</span><span class="mi">1</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="n">limit</span><span class="p">(</span><span class="n">sign</span><span class="o">*</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">)</span><span class="o">/</span><span class="n">disc</span>
-        <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-            <span class="k">elif</span> <span class="n">i</span><span class="o">.</span><span class="n">is_odd</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">dir</span> <span class="o">==</span> <span class="s">&#39;+&#39;</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-
-    <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">cot</span><span class="p">:</span>
-        <span class="c"># discontinuity at multiples of pi; 0 at odd pi/2 multiples</span>
-        <span class="n">disc</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Pi</span>
-        <span class="n">sign</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="nb">dir</span> <span class="o">==</span> <span class="s">&#39;-&#39;</span><span class="p">:</span>
-            <span class="n">sign</span> <span class="o">*=</span> <span class="o">-</span><span class="mi">1</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="n">limit</span><span class="p">(</span><span class="n">sign</span><span class="o">*</span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">)</span><span class="o">/</span><span class="n">disc</span>
-        <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">dir</span> <span class="o">==</span> <span class="s">&#39;-&#39;</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-        <span class="k">elif</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-        <span class="n">b</span><span class="p">,</span> <span class="n">ex</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="bp">None</span>  <span class="c"># records sign of b if b is +/-z or has a bounded value</span>
-        <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="n">c</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">as_two_terms</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span> <span class="ow">and</span> <span class="n">b</span> <span class="o">==</span> <span class="n">z</span><span class="p">:</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="s">&#39;-&#39;</span>
-        <span class="k">elif</span> <span class="n">b</span> <span class="o">==</span> <span class="n">z</span><span class="p">:</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="s">&#39;+&#39;</span>
-
-        <span class="k">if</span> <span class="n">ex</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">base</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">base</span><span class="o">.</span><span class="n">is_finite</span> <span class="ow">and</span> <span class="p">(</span><span class="n">ex</span><span class="o">.</span><span class="n">is_bounded</span> <span class="ow">or</span> <span class="n">base</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">base</span><span class="o">**</span><span class="n">ex</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">z0</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">ex</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="nb">dir</span> <span class="o">!=</span> <span class="n">c</span><span class="p">:</span>
-                        <span class="c"># integer</span>
-                        <span class="k">if</span> <span class="n">ex</span><span class="o">.</span><span class="n">is_even</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-                        <span class="k">elif</span> <span class="n">ex</span><span class="o">.</span><span class="n">is_odd</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span>
-                        <span class="c"># rational</span>
-                        <span class="k">elif</span> <span class="n">ex</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="o">**</span><span class="n">ex</span><span class="p">)</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-                <span class="k">return</span> <span class="n">z0</span><span class="o">**</span><span class="n">ex</span>
-
-    <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Mul</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">z0</span> <span class="ow">and</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">b</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">log</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="n">z0</span><span class="p">)</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-                <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-                <span class="c"># XXX todo: this should probably be stated in the</span>
-                <span class="c"># negative -- i.e. to exclude expressions that should</span>
-                <span class="c"># not be handled this way but I&#39;m not sure what that</span>
-                <span class="c"># condition is; when ok is True it means that the leading</span>
-                <span class="c"># term approach is going to succeed (hopefully)</span>
-                <span class="n">ok</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">w</span><span class="p">:</span> <span class="p">(</span><span class="n">z</span> <span class="ow">in</span> <span class="n">w</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span>
-                     <span class="nb">any</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">is_polynomial</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="ow">or</span>
-                     <span class="nb">any</span><span class="p">(</span><span class="n">z</span> <span class="ow">in</span> <span class="n">m</span><span class="o">.</span><span class="n">free_symbols</span> <span class="ow">and</span> <span class="n">m</span><span class="o">.</span><span class="n">is_polynomial</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-                     <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
-                     <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">w</span><span class="p">)))</span>
-                <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">ok</span><span class="p">(</span><span class="n">w</span><span class="p">)</span> <span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">d</span><span class="p">)):</span>
-                    <span class="n">u</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="n">positive</span><span class="o">=</span><span class="p">(</span><span class="n">z0</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">))</span>
-                    <span class="n">inve</span> <span class="o">=</span> <span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="n">d</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">u</span><span class="p">)</span>
-                    <span class="k">return</span> <span class="n">limit</span><span class="p">(</span><span class="n">inve</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span>
-                        <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="s">&quot;+&quot;</span> <span class="k">if</span> <span class="n">z0</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span> <span class="k">else</span> <span class="s">&quot;-&quot;</span><span class="p">)</span>
-
-            <span class="c"># weed out the z-independent terms</span>
-            <span class="n">i</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="ow">and</span> <span class="n">i</span><span class="o">.</span><span class="n">is_bounded</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">i</span><span class="o">*</span><span class="n">limit</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="nb">dir</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">i</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">e</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">z0</span><span class="p">:</span>
-            <span class="c"># look for log(z)**q or z**p*log(z)**q</span>
-            <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&quot;p&quot;</span><span class="p">),</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&quot;q&quot;</span><span class="p">)</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">z</span><span class="o">**</span><span class="n">p</span> <span class="o">*</span> <span class="n">log</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="n">q</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">r</span><span class="p">:</span>
-                <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="p">[</span><span class="n">r</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">w</span><span class="p">)</span> <span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="p">[</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">]]</span>
-                <span class="k">if</span> <span class="n">q</span> <span class="ow">and</span> <span class="n">q</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="n">p</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">q</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">p</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="o">-</span><span class="n">oo</span><span class="o">*</span><span class="n">i</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">p</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="o">-</span><span class="n">oo</span><span class="o">*</span><span class="n">i</span>
-
-    <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_polynomial</span><span class="p">()</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">z0</span><span class="o">.</span><span class="n">is_unbounded</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">limit</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="nb">dir</span><span class="p">)</span> <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-        <span class="c"># this is a case like limit(x*y+x*z, z, 2) == x*y+2*x</span>
-        <span class="c"># but we need to make sure, that the general gruntz() algorithm is</span>
-        <span class="c"># executed for a case like &quot;limit(sqrt(x+1)-sqrt(x),x,oo)==0&quot;</span>
-
-        <span class="n">unbounded</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">unbounded_result</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">unbounded_const</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">unknown</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">unknown_result</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">finite</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">zero</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">def</span> <span class="nf">_sift</span><span class="p">(</span><span class="n">term</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">z</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">term</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">term</span><span class="o">.</span><span class="n">is_unbounded</span><span class="p">:</span>
-                    <span class="n">unbounded_const</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">finite</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">)</span>
-                <span class="n">bounded</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">is_bounded</span>
-                <span class="k">if</span> <span class="n">bounded</span> <span class="ow">is</span> <span class="bp">False</span> <span class="ow">or</span> <span class="n">result</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                    <span class="n">unbounded</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">result</span> <span class="o">!=</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                        <span class="c"># take result from direction given</span>
-                        <span class="n">result</span> <span class="o">=</span> <span class="n">limit</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="nb">dir</span><span class="p">)</span>
-                    <span class="n">unbounded_result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="n">bounded</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">result</span><span class="p">:</span>
-                        <span class="n">finite</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">zero</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">unknown</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-                    <span class="n">unknown_result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">_sift</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-
-        <span class="n">bad</span> <span class="o">=</span> <span class="nb">bool</span><span class="p">(</span><span class="n">unknown</span> <span class="ow">and</span> <span class="n">unbounded</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">bad</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">unknown</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">unbounded</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">zero</span><span class="p">:</span>
-            <span class="n">uu</span> <span class="o">=</span> <span class="n">unknown</span> <span class="o">+</span> <span class="n">unbounded</span>
-            <span class="c"># we won&#39;t be able to resolve this with unbounded</span>
-            <span class="c"># terms, e.g. Sum(1/k, (k, 1, n)) - log(n) as n -&gt; oo:</span>
-            <span class="c"># since the Sum is unevaluated it&#39;s boundedness is</span>
-            <span class="c"># unknown and the log(n) is oo so you get Sum - oo</span>
-            <span class="c"># which is unsatisfactory. BUT...if there are both</span>
-            <span class="c"># unknown and unbounded terms (condition &#39;bad&#39;) or</span>
-            <span class="c"># there are multiple terms that are unknown, or</span>
-            <span class="c"># there are multiple symbolic unbounded terms they may</span>
-            <span class="c"># respond better if they are made into a rational</span>
-            <span class="c"># function, so give them a chance to do so before</span>
-            <span class="c"># reporting failure.</span>
-            <span class="n">u</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">uu</span><span class="p">)</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">u</span><span class="o">.</span><span class="n">normal</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">f</span> <span class="o">!=</span> <span class="n">u</span><span class="p">:</span>
-                <span class="n">unknown</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="n">unbounded</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="n">unbounded_result</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="n">unknown_result</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="n">_sift</span><span class="p">(</span><span class="n">limit</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="nb">dir</span><span class="p">))</span>
-
-            <span class="c"># We came in with a) unknown and unbounded terms or b) had multiple</span>
-            <span class="c"># unknown terms</span>
-
-            <span class="c"># At this point we&#39;ve done one of 3 things.</span>
-            <span class="c"># (1) We did nothing with f so we now report the error</span>
-            <span class="c"># showing the troublesome terms which are now in uu. OR</span>
-
-            <span class="c"># (2) We did something with f but the result came back as unknown.</span>
-            <span class="c"># Normally this wouldn&#39;t be a problem,</span>
-            <span class="c"># but we had either multiple terms that were troublesome (unk and</span>
-            <span class="c"># unbounded or multiple unknown terms) so if we</span>
-            <span class="c"># weren&#39;t able to resolve the boundedness by now, that indicates a</span>
-            <span class="c"># problem so we report the error showing the troublesome terms which are</span>
-            <span class="c"># now in uu.</span>
-            <span class="k">if</span> <span class="n">unknown</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">bad</span><span class="p">:</span>
-                    <span class="n">msg</span> <span class="o">=</span> <span class="s">&#39;unknown and unbounded terms present in </span><span class="si">%s</span><span class="s">&#39;</span>
-                <span class="k">elif</span> <span class="n">unknown</span><span class="p">:</span>
-                    <span class="n">msg</span> <span class="o">=</span> <span class="s">&#39;multiple terms with unknown boundedness in </span><span class="si">%s</span><span class="s">&#39;</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="n">msg</span> <span class="o">%</span> <span class="n">uu</span><span class="p">)</span>
-            <span class="c"># OR</span>
-            <span class="c"># (3) the troublesome terms have been identified as finite or unbounded</span>
-            <span class="c"># and we proceed with the non-error code since the lists have been updated.</span>
-
-        <span class="n">u</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">unknown_result</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">unbounded_result</span> <span class="ow">or</span> <span class="n">unbounded_const</span><span class="p">:</span>
-            <span class="n">unbounded</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">zero</span><span class="p">)</span>
-            <span class="n">inf_limit</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">unbounded_result</span> <span class="o">+</span> <span class="n">unbounded_const</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">inf_limit</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">inf_limit</span> <span class="o">+</span> <span class="n">u</span>
-            <span class="k">if</span> <span class="n">finite</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">finite</span><span class="p">)</span> <span class="o">+</span> <span class="n">limit</span><span class="p">(</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">unbounded</span><span class="p">),</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="nb">dir</span><span class="p">)</span> <span class="o">+</span> <span class="n">u</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">finite</span><span class="p">)</span> <span class="o">+</span> <span class="n">u</span>
-
-    <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Order</span><span class="p">:</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Order</span><span class="p">(</span><span class="n">limit</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">),</span> <span class="o">*</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">gruntz</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="nb">dir</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">r</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">PoleError</span><span class="p">()</span>
-    <span class="k">except</span> <span class="p">(</span><span class="n">PoleError</span><span class="p">,</span> <span class="ne">ValueError</span><span class="p">):</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">heuristics</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="nb">dir</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">r</span>
-
-</div>
-<span class="k">def</span> <span class="nf">heuristics</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="nb">dir</span><span class="p">):</span>
-    <span class="k">if</span> <span class="nb">abs</span><span class="p">(</span><span class="n">z0</span><span class="p">)</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">limit</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">),</span> <span class="n">z</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="s">&quot;+&quot;</span> <span class="k">if</span> <span class="n">z0</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span> <span class="k">else</span> <span class="s">&quot;-&quot;</span><span class="p">)</span>
-
-    <span class="n">rv</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">bad</span> <span class="o">=</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">a</span><span class="o">.</span><span class="n">is_bounded</span><span class="p">:</span>
-                <span class="n">r</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">limit</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="nb">dir</span><span class="p">))</span>
-                <span class="k">if</span> <span class="n">r</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="ow">in</span> <span class="n">bad</span><span class="p">:</span>
-                    <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">r</span><span class="p">:</span>
-                <span class="n">rv</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">r</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">rv</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">and</span> <span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">or</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">or</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Function</span><span class="p">):</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">limit</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="nb">dir</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-    <span class="k">if</span> <span class="n">rv</span> <span class="ow">in</span> <span class="n">bad</span><span class="p">:</span>
-        <span class="n">msg</span> <span class="o">=</span> <span class="s">&quot;Don&#39;t know how to calculate the limit(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, dir=</span><span class="si">%s</span><span class="s">), sorry.&quot;</span>
-        <span class="k">raise</span> <span class="n">PoleError</span><span class="p">(</span><span class="n">msg</span> <span class="o">%</span> <span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="nb">dir</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">rv</span>
-
-
-<div class="viewcode-block" id="Limit"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.limits.Limit">[docs]</a><span class="k">class</span> <span class="nc">Limit</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Represents an unevaluated limit.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Limit, sin, Symbol</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; Limit(sin(x)/x, x, 0)</span>
-<span class="sd">    Limit(sin(x)/x, x, 0)</span>
-<span class="sd">    &gt;&gt;&gt; Limit(1/x, x, 0, dir=&quot;-&quot;)</span>
-<span class="sd">    Limit(1/x, x, 0, dir=&#39;-&#39;)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="nb">dir</span><span class="o">=</span><span class="s">&quot;+&quot;</span><span class="p">):</span>
-        <span class="n">e</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-        <span class="n">z</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-        <span class="n">z0</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">z0</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="nb">dir</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="nb">dir</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="nb">dir</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="nb">dir</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;direction must be of type basestring or Symbol, not </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">type</span><span class="p">(</span><span class="nb">dir</span><span class="p">))</span>
-        <span class="k">if</span> <span class="nb">str</span><span class="p">(</span><span class="nb">dir</span><span class="p">)</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">(</span><span class="s">&#39;+&#39;</span><span class="p">,</span> <span class="s">&#39;-&#39;</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&quot;direction must be either &#39;+&#39; or &#39;-&#39;, not </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">dir</span><span class="p">)</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_args</span> <span class="o">=</span> <span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="nb">dir</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-<div class="viewcode-block" id="Limit.doit"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.limits.Limit.doit">[docs]</a>    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Evaluates limit&quot;&quot;&quot;</span>
-        <span class="n">e</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="nb">dir</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">if</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;deep&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-            <span class="n">e</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-            <span class="n">z</span> <span class="o">=</span> <span class="n">z</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-            <span class="n">z0</span> <span class="o">=</span> <span class="n">z0</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">limit</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="nb">str</span><span class="p">(</span><span class="nb">dir</span><span class="p">))</span></div></div>
-</pre></div>
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-            
-  <h1>Source code for sympy.series.order</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">Rational</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Mul</span>
-<span class="kn">from</span> <span class="nn">sympy.core.cache</span> <span class="kn">import</span> <span class="n">cacheit</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">cmp_to_key</span>
-
-
-<div class="viewcode-block" id="Order"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.order.Order">[docs]</a><span class="k">class</span> <span class="nc">Order</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Represents the limiting behavior of some function</span>
-
-<span class="sd">    The order of a function characterizes the function based on the limiting</span>
-<span class="sd">    behavior of the function as it goes to some limit. Only taking the limit</span>
-<span class="sd">    point to be 0 is currently supported. This is expressed in big O notation</span>
-<span class="sd">    [1]_.</span>
-
-<span class="sd">    The formal definition for the order of a function `g(x)` about a point `a`</span>
-<span class="sd">    is such that `g(x) = O(f(x))` as `x \\rightarrow a` if and only if for any</span>
-<span class="sd">    `\delta &gt; 0` there exists a `M &gt; 0` such that `|g(x)| \leq M|f(x)|` for</span>
-<span class="sd">    `|x-a| &lt; \delta`. This is equivalent to `\lim_{x \\rightarrow a}</span>
-<span class="sd">    |g(x)/f(x)| &lt; \infty`.</span>
-
-<span class="sd">    Let&#39;s illustrate it on the following example by taking the expansion of</span>
-<span class="sd">    `\sin(x)` about 0:</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        \sin(x) = x - x^3/3! + O(x^5)</span>
-
-<span class="sd">    where in this case `O(x^5) = x^5/5! - x^7/7! + \cdots`. By the definition</span>
-<span class="sd">    of `O`, for any `\delta &gt; 0` there is an `M` such that:</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        |x^5/5! - x^7/7! + ....| &lt;= M|x^5| \\text{ for } |x| &lt; \delta</span>
-
-<span class="sd">    or by the alternate definition:</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        \lim_{x \\rightarrow 0} | (x^5/5! - x^7/7! + ....) / x^5| &lt; \infty</span>
-
-<span class="sd">    which surely is true, because</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        \lim_{x \\rightarrow 0} | (x^5/5! - x^7/7! + ....) / x^5| = 1/5!</span>
-
-
-<span class="sd">    As it is usually used, the order of a function can be intuitively thought</span>
-<span class="sd">    of representing all terms of powers greater than the one specified. For</span>
-<span class="sd">    example, `O(x^3)` corresponds to any terms proportional to `x^3,</span>
-<span class="sd">    x^4,\ldots` and any higher power. For a polynomial, this leaves terms</span>
-<span class="sd">    proportional to `x^2`, `x` and constants.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import O</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; O(x)</span>
-<span class="sd">    O(x)</span>
-<span class="sd">    &gt;&gt;&gt; O(x)*x</span>
-<span class="sd">    O(x**2)</span>
-<span class="sd">    &gt;&gt;&gt; O(x)-O(x)</span>
-<span class="sd">    O(x)</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [1] `Big O notation &lt;http://en.wikipedia.org/wiki/Big_O_notation&gt;`_</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    In ``O(f(x), x)`` the expression ``f(x)`` is assumed to have a leading</span>
-<span class="sd">    term.  ``O(f(x), x)`` is automatically transformed to</span>
-<span class="sd">    ``O(f(x).as_leading_term(x),x)``.</span>
-
-<span class="sd">        ``O(expr*f(x), x)`` is ``O(f(x), x)``</span>
-
-<span class="sd">        ``O(expr, x)`` is ``O(1)``</span>
-
-<span class="sd">        ``O(0, x)`` is 0.</span>
-
-<span class="sd">    Multivariate O is also supported:</span>
-
-<span class="sd">        ``O(f(x, y), x, y)`` is transformed to</span>
-<span class="sd">        ``O(f(x, y).as_leading_term(x,y).as_leading_term(y), x, y)``</span>
-
-<span class="sd">    In the multivariate case, it is assumed the limits w.r.t. the various</span>
-<span class="sd">    symbols commute.</span>
-
-<span class="sd">    If no symbols are passed then all symbols in the expression are used:</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_Order</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">):</span>
-
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">expr</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-
-        <span class="k">if</span> <span class="n">symbols</span><span class="p">:</span>
-            <span class="n">symbols</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="n">symbols</span><span class="p">))</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">symbols</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                    <span class="s">&#39;Order at points other than 0 not supported.&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">symbols</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Order</span><span class="p">:</span>
-
-            <span class="n">new_symbols</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">symbols</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">s</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">new_symbols</span><span class="p">:</span>
-                    <span class="n">new_symbols</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">new_symbols</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">variables</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">expr</span>
-            <span class="n">symbols</span> <span class="o">=</span> <span class="n">new_symbols</span>
-
-        <span class="k">elif</span> <span class="n">symbols</span><span class="p">:</span>
-
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="n">lst</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">extract_leading_order</span><span class="p">(</span><span class="o">*</span><span class="n">symbols</span><span class="p">)</span>
-                <span class="n">expr</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">f</span><span class="o">.</span><span class="n">expr</span> <span class="k">for</span> <span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span> <span class="ow">in</span> <span class="n">lst</span><span class="p">])</span>
-            <span class="k">elif</span> <span class="n">expr</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_commutative</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                    <span class="c"># TODO</span>
-                    <span class="c"># We cannot use compute_leading_term because that only</span>
-                    <span class="c"># works in one symbol.</span>
-                    <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="o">*</span><span class="n">symbols</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">compute_leading_term</span><span class="p">(</span><span class="n">symbols</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-                <span class="n">terms</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">(</span><span class="o">*</span><span class="n">symbols</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span>
-                <span class="n">expr</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">t</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">terms</span> <span class="k">if</span> <span class="n">s</span> <span class="o">&amp;</span> <span class="n">t</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">expr</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">symbols</span><span class="p">):</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-        <span class="c"># create Order instance:</span>
-        <span class="n">symbols</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">cmp_to_key</span><span class="p">(</span><span class="n">Basic</span><span class="o">.</span><span class="n">compare</span><span class="p">))</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">_hashable_content</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span>
-
-    <span class="k">def</span> <span class="nf">oseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">order</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="k">def</span> <span class="nf">_eval_nseries</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">logx</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">expr</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">variables</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">free_symbols</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">free_symbols</span>
-
-    <span class="k">def</span> <span class="nf">_eval_power</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Order</span><span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">expr</span> <span class="o">**</span> <span class="n">e</span><span class="p">,</span> <span class="o">*</span><span class="n">b</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">as_expr_variables</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">order_symbols</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">order_symbols</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">order_symbols</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">variables</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">s</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">order_symbols</span><span class="p">:</span>
-                    <span class="n">order_symbols</span> <span class="o">=</span> <span class="n">order_symbols</span> <span class="o">+</span> <span class="p">(</span><span class="n">s</span><span class="p">,)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="n">order_symbols</span>
-
-    <span class="k">def</span> <span class="nf">removeO</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="k">def</span> <span class="nf">getO</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="nd">@cacheit</span>
-<div class="viewcode-block" id="Order.contains"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.order.Order.contains">[docs]</a>    <span class="k">def</span> <span class="nf">contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return True if expr belongs to Order(self.expr, \*self.variables).</span>
-<span class="sd">        Return False if self belongs to expr.</span>
-<span class="sd">        Return None if the inclusion relation cannot be determined</span>
-<span class="sd">        (e.g. when self and expr have different symbols).</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># NOTE: when multiplying out series a lot of queries like</span>
-        <span class="c">#       O(...).contains(a*x**b) with many a and few b are made.</span>
-        <span class="c">#       Separating out the independent part allows for better caching.</span>
-        <span class="n">c</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">m</span> <span class="o">!=</span> <span class="p">():</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_contains</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">m</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># Mul(*m) == 1, and O(1) treatment is somewhat peculiar ...</span>
-            <span class="c"># some day this else should not be necessary</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_contains</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-</div>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">_contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">powsimp</span><span class="p">,</span> <span class="n">limit</span>
-        <span class="k">if</span> <span class="n">expr</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="n">expr</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Order</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">variables</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">variables</span><span class="p">:</span>
-                <span class="n">common_symbols</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span>
-                    <span class="p">[</span><span class="n">s</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">variables</span> <span class="k">if</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">variables</span><span class="p">])</span>
-            <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">:</span>
-                <span class="n">common_symbols</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">variables</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">common_symbols</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">variables</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">common_symbols</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span> <span class="ow">or</span> <span class="n">expr</span><span class="o">.</span><span class="n">variables</span><span class="p">):</span>  <span class="c"># O(1),O(1)</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">common_symbols</span><span class="p">:</span>
-                <span class="n">l</span> <span class="o">=</span> <span class="n">limit</span><span class="p">(</span><span class="n">powsimp</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">/</span><span class="n">expr</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span>
-                <span class="n">combine</span><span class="o">=</span><span class="s">&#39;exp&#39;</span><span class="p">),</span> <span class="n">s</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span>
-                <span class="k">if</span> <span class="n">r</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">r</span> <span class="o">=</span> <span class="n">l</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">r</span> <span class="o">!=</span> <span class="n">l</span><span class="p">:</span>
-                        <span class="k">return</span>
-            <span class="k">return</span> <span class="n">r</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Order</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_subs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">old</span><span class="o">.</span><span class="n">is_Symbol</span> <span class="ow">and</span> <span class="n">old</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">:</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">old</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">new</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">Order</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">_subs</span><span class="p">(</span><span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">),</span> <span class="o">*</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="p">(</span><span class="n">new</span><span class="p">,)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]))</span>
-            <span class="k">return</span> <span class="n">Order</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">_subs</span><span class="p">(</span><span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">),</span> <span class="o">*</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]))</span>
-        <span class="k">return</span> <span class="n">Order</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">_subs</span><span class="p">(</span><span class="n">old</span><span class="p">,</span> <span class="n">new</span><span class="p">),</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">_eval_conjugate</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">expr</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_eval_derivative</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span> <span class="ow">or</span> <span class="bp">self</span>
-
-    <span class="k">def</span> <span class="nf">_eval_transpose</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">_eval_transpose</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">expr</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_sage_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c">#XXX: SAGE doesn&#39;t have Order yet. Let&#39;s return 0 instead.</span>
-        <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">_sage_</span><span class="p">()</span>
-</div>
-<span class="n">O</span> <span class="o">=</span> <span class="n">Order</span>
-</pre></div>
-
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-  <h1>Source code for sympy.series.residues</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">This module implements the Residue function and related tools for working</span>
-<span class="sd">with residues.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Wild</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">Integer</span><span class="p">,</span> <span class="n">Add</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.timeutils</span> <span class="kn">import</span> <span class="n">timethis</span>
-
-
-<span class="nd">@timethis</span><span class="p">(</span><span class="s">&#39;residue&#39;</span><span class="p">)</span>
-<div class="viewcode-block" id="residue"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.residues.residue">[docs]</a><span class="k">def</span> <span class="nf">residue</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">x0</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Finds the residue of ``expr`` at the point x=x0.</span>
-
-<span class="sd">    The residue is defined as the coefficient of 1/(x-x0) in the power series</span>
-<span class="sd">    expansion about x=x0.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, residue, sin</span>
-<span class="sd">    &gt;&gt;&gt; x = Symbol(&quot;x&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; residue(1/x, x, 0)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; residue(1/x**2, x, 0)</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; residue(2/sin(x), x, 0)</span>
-<span class="sd">    2</span>
-
-<span class="sd">    This function is essential for the Residue Theorem [1].</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. http://en.wikipedia.org/wiki/Residue_theorem</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># The current implementation uses series expansion to</span>
-    <span class="c"># calculate it. A more general implementation is explained in</span>
-    <span class="c"># the section 5.6 of the Bronstein&#39;s book {M. Bronstein:</span>
-    <span class="c"># Symbolic Integration I, Springer Verlag (2005)}. For purely</span>
-    <span class="c"># rational functions, the algorithm is much easier. See</span>
-    <span class="c"># sections 2.4, 2.5, and 2.7 (this section actually gives an</span>
-    <span class="c"># algorithm for computing any Laurent series coefficient for</span>
-    <span class="c"># a rational function). The theory in section 2.4 will help to</span>
-    <span class="c"># understand why the resultant works in the general algorithm.</span>
-    <span class="c"># For the definition of a resultant, see section 1.4 (and any</span>
-    <span class="c"># previous sections for more review).</span>
-
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">collect</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Order</span><span class="p">,</span> <span class="n">S</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">x0</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span> <span class="o">+</span> <span class="n">x0</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">16</span><span class="p">,</span> <span class="mi">32</span><span class="p">]:</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">nseries</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">s</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Order</span><span class="p">)</span> <span class="ow">and</span> <span class="n">s</span><span class="o">.</span><span class="n">removeO</span><span class="p">()</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="c"># bug in nseries</span>
-            <span class="k">continue</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">s</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Order</span><span class="p">)</span> <span class="ow">or</span> <span class="n">s</span><span class="o">.</span><span class="n">getn</span><span class="p">()</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">break</span>
-    <span class="k">if</span> <span class="n">s</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Order</span><span class="p">)</span> <span class="ow">and</span> <span class="n">s</span><span class="o">.</span><span class="n">getn</span><span class="p">()</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;Bug in nseries?&#39;</span><span class="p">)</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="n">collect</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">removeO</span><span class="p">(),</span> <span class="n">x</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">s</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">args</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">s</span><span class="p">]</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-        <span class="n">c</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">m</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">m</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">m</span> <span class="o">==</span> <span class="n">x</span> <span class="ow">or</span> <span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">m</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">)):</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;term of unexpected form: </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">m</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">:</span>
-            <span class="n">res</span> <span class="o">+=</span> <span class="n">c</span>
-    <span class="k">return</span> <span class="n">res</span></div>
-</pre></div>
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-  <h1>Source code for sympy.series.series</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="n">sympify</span>
-
-
-<div class="viewcode-block" id="series"><a class="viewcode-back" href="../../../modules/series.html#sympy.series.series.series">[docs]</a><span class="k">def</span> <span class="nf">series</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">x0</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">6</span><span class="p">,</span> <span class="nb">dir</span><span class="o">=</span><span class="s">&quot;+&quot;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Series expansion of expr around point `x = x0`.</span>
-
-<span class="sd">    See the doctring of Expr.series() for complete details of this wrapper.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="nb">dir</span><span class="p">)</span></div>
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diff --git a/dev-py3k/_modules/sympy/sets/fancysets.html b/dev-py3k/_modules/sympy/sets/fancysets.html
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-            
-  <h1>Source code for sympy.sets.fancysets</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="p">(</span><span class="n">Dummy</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">Lambda</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">ask</span><span class="p">,</span> <span class="n">Q</span><span class="p">,</span> <span class="n">Min</span><span class="p">,</span>
-        <span class="n">Max</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.integers</span> <span class="kn">import</span> <span class="n">floor</span><span class="p">,</span> <span class="n">ceiling</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.complexes</span> <span class="kn">import</span> <span class="n">sign</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">iterable</span><span class="p">,</span> <span class="n">as_int</span>
-<span class="kn">from</span> <span class="nn">sympy.core.sets</span> <span class="kn">import</span> <span class="n">Set</span><span class="p">,</span> <span class="n">Interval</span><span class="p">,</span> <span class="n">FiniteSet</span><span class="p">,</span> <span class="n">Intersection</span>
-<span class="kn">from</span> <span class="nn">sympy.core.singleton</span> <span class="kn">import</span> <span class="n">Singleton</span><span class="p">,</span> <span class="n">S</span>
-<span class="kn">from</span> <span class="nn">sympy.solvers</span> <span class="kn">import</span> <span class="n">solve</span>
-
-<span class="n">oo</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-
-
-<div class="viewcode-block" id="Naturals"><a class="viewcode-back" href="../../../modules/sets.html#sympy.sets.fancysets.Naturals">[docs]</a><span class="k">class</span> <span class="nc">Naturals</span><span class="p">(</span><span class="n">Set</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">Singleton</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Represents the Natural Numbers. The Naturals are available as a singleton</span>
-<span class="sd">    as S.Naturals</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import S, Interval, pprint</span>
-<span class="sd">        &gt;&gt;&gt; 5 in S.Naturals</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; iterable = iter(S.Naturals)</span>
-<span class="sd">        &gt;&gt;&gt; print(next(iterable))</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; print(next(iterable))</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; print(next(iterable))</span>
-<span class="sd">        3</span>
-<span class="sd">        &gt;&gt;&gt; pprint(S.Naturals.intersect(Interval(0, 10)))</span>
-<span class="sd">        {1, 2, ..., 10}</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_iterable</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_intersect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">is_Interval</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Intersection</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Integers</span><span class="p">,</span> <span class="n">other</span><span class="p">,</span> <span class="n">Interval</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">_contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">other</span><span class="p">))</span> <span class="ow">and</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">other</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-        <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="k">yield</span> <span class="n">i</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_inf</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_sup</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">oo</span>
-
-</div>
-<div class="viewcode-block" id="Integers"><a class="viewcode-back" href="../../../modules/sets.html#sympy.sets.fancysets.Integers">[docs]</a><span class="k">class</span> <span class="nc">Integers</span><span class="p">(</span><span class="n">Set</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">Singleton</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Represents the Integers. The Integers are available as a singleton</span>
-<span class="sd">    as S.Integers</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import S, Interval, pprint</span>
-<span class="sd">        &gt;&gt;&gt; 5 in S.Naturals</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; iterable = iter(S.Integers)</span>
-<span class="sd">        &gt;&gt;&gt; print(next(iterable))</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; print(next(iterable))</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; print(next(iterable))</span>
-<span class="sd">        -1</span>
-<span class="sd">        &gt;&gt;&gt; print(next(iterable))</span>
-<span class="sd">        2</span>
-
-<span class="sd">        &gt;&gt;&gt; pprint(S.Integers.intersect(Interval(-4, 4)))</span>
-<span class="sd">        {-4, -3, ..., 4}</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_iterable</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_intersect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">is_Interval</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">measure</span> <span class="o">&lt;</span> <span class="n">oo</span><span class="p">:</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">Range</span><span class="p">(</span><span class="n">ceiling</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">left</span><span class="p">),</span> <span class="n">floor</span><span class="p">(</span><span class="n">other</span><span class="o">.</span><span class="n">right</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">s</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>  <span class="c"># take out endpoints if open interval</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">_contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">other</span><span class="p">)):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">yield</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-        <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="k">yield</span> <span class="n">i</span>
-            <span class="k">yield</span> <span class="o">-</span><span class="n">i</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_inf</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="n">oo</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_sup</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">oo</span>
-
-</div>
-<span class="k">class</span> <span class="nc">Reals</span><span class="p">(</span><span class="n">Interval</span><span class="p">,</span> <span class="n">metaclass</span><span class="o">=</span><span class="n">Singleton</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Interval</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">-</span><span class="n">oo</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="TransformationSet"><a class="viewcode-back" href="../../../modules/sets.html#sympy.sets.fancysets.TransformationSet">[docs]</a><span class="k">class</span> <span class="nc">TransformationSet</span><span class="p">(</span><span class="n">Set</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A set that is a transformation of another through some algebraic expression</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    --------</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, S, TransformationSet, FiniteSet, Lambda</span>
-
-<span class="sd">    &gt;&gt;&gt; x = Symbol(&#39;x&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; N = S.Naturals</span>
-<span class="sd">    &gt;&gt;&gt; squares = TransformationSet(Lambda(x, x**2), N) # {x**2 for x in N}</span>
-<span class="sd">    &gt;&gt;&gt; 4 in squares</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; 5 in squares</span>
-<span class="sd">    False</span>
-
-<span class="sd">    &gt;&gt;&gt; FiniteSet(0, 1, 2, 3, 4, 5, 6, 7, 9, 10).intersect(squares)</span>
-<span class="sd">    {1, 4, 9}</span>
-
-<span class="sd">    &gt;&gt;&gt; square_iterable = iter(squares)</span>
-<span class="sd">    &gt;&gt;&gt; for i in range(4):</span>
-<span class="sd">    ...     next(square_iterable)</span>
-<span class="sd">    1</span>
-<span class="sd">    4</span>
-<span class="sd">    9</span>
-<span class="sd">    16</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">lamda</span><span class="p">,</span> <span class="n">base_set</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">lamda</span><span class="p">,</span> <span class="n">base_set</span><span class="p">)</span>
-
-    <span class="n">lamda</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-    <span class="n">base_set</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">already_seen</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">base_set</span><span class="p">:</span>
-            <span class="n">val</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lamda</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">val</span> <span class="ow">in</span> <span class="n">already_seen</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">already_seen</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
-                <span class="k">yield</span> <span class="n">val</span>
-
-    <span class="k">def</span> <span class="nf">_is_multivariate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lamda</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">_contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="n">L</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lamda</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_is_multivariate</span><span class="p">():</span>
-            <span class="n">solns</span> <span class="o">=</span> <span class="n">solve</span><span class="p">([</span><span class="n">expr</span> <span class="o">-</span> <span class="n">val</span> <span class="k">for</span> <span class="n">val</span><span class="p">,</span> <span class="n">expr</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">L</span><span class="o">.</span><span class="n">expr</span><span class="p">)],</span>
-                    <span class="n">L</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">solns</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">L</span><span class="o">.</span><span class="n">expr</span> <span class="o">-</span> <span class="n">other</span><span class="p">,</span> <span class="n">L</span><span class="o">.</span><span class="n">variables</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">for</span> <span class="n">soln</span> <span class="ow">in</span> <span class="n">solns</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">soln</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">base_set</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">soln</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">base_set</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_iterable</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">base_set</span><span class="o">.</span><span class="n">is_iterable</span>
-
-</div>
-<span class="k">class</span> <span class="nc">Range</span><span class="p">(</span><span class="n">Set</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Represents a range of integers.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Range</span>
-<span class="sd">        &gt;&gt;&gt; list(Range(5)) # 0 to 5</span>
-<span class="sd">        [0, 1, 2, 3, 4]</span>
-<span class="sd">        &gt;&gt;&gt; list(Range(10, 15)) # 10 to 15</span>
-<span class="sd">        [10, 11, 12, 13, 14]</span>
-<span class="sd">        &gt;&gt;&gt; list(Range(10, 20, 2)) # 10 to 20 in steps of 2</span>
-<span class="sd">        [10, 12, 14, 16, 18]</span>
-<span class="sd">        &gt;&gt;&gt; list(Range(20, 10, -2)) # 20 to 10 backward in steps of 2</span>
-<span class="sd">        [12, 14, 16, 18, 20]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_iterable</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="c"># expand range</span>
-        <span class="n">slc</span> <span class="o">=</span> <span class="nb">slice</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">start</span><span class="p">,</span> <span class="n">stop</span><span class="p">,</span> <span class="n">step</span> <span class="o">=</span> <span class="n">slc</span><span class="o">.</span><span class="n">start</span> <span class="ow">or</span> <span class="mi">0</span><span class="p">,</span> <span class="n">slc</span><span class="o">.</span><span class="n">stop</span><span class="p">,</span> <span class="n">slc</span><span class="o">.</span><span class="n">step</span> <span class="ow">or</span> <span class="mi">1</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">start</span><span class="p">,</span> <span class="n">stop</span><span class="p">,</span> <span class="n">step</span> <span class="o">=</span> <span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="n">as_int</span><span class="p">(</span><span class="n">w</span><span class="p">))</span> <span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="n">stop</span><span class="p">,</span> <span class="n">step</span><span class="p">)]</span>
-        <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Inputs to Range must be Integer Valued</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">+</span>
-                    <span class="s">&quot;Use TransformationSets of Ranges for other cases&quot;</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">ceiling</span><span class="p">((</span><span class="n">stop</span> <span class="o">-</span> <span class="n">start</span><span class="p">)</span><span class="o">/</span><span class="n">step</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">EmptySet</span>
-
-        <span class="c"># normalize args: regardless of how they are entered they will show</span>
-        <span class="c"># canonically as Range(inf, sup, step) with step &gt; 0</span>
-        <span class="n">start</span><span class="p">,</span> <span class="n">stop</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">((</span><span class="n">start</span><span class="p">,</span> <span class="n">start</span> <span class="o">+</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">step</span><span class="p">))</span>
-        <span class="n">step</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">step</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">stop</span> <span class="o">+</span> <span class="n">step</span><span class="p">,</span> <span class="n">step</span><span class="p">)</span>
-
-    <span class="n">start</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-    <span class="n">stop</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-    <span class="n">step</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="k">lambda</span> <span class="bp">self</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">_intersect</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">other</span><span class="o">.</span><span class="n">is_Interval</span><span class="p">:</span>
-            <span class="n">osup</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">sup</span>
-            <span class="n">oinf</span> <span class="o">=</span> <span class="n">other</span><span class="o">.</span><span class="n">inf</span>
-            <span class="c"># if other is [0, 10) we can only go up to 9</span>
-            <span class="k">if</span> <span class="n">osup</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">right_open</span><span class="p">:</span>
-                <span class="n">osup</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="n">oinf</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">and</span> <span class="n">other</span><span class="o">.</span><span class="n">left_open</span><span class="p">:</span>
-                <span class="n">oinf</span> <span class="o">+=</span> <span class="mi">1</span>
-
-            <span class="c"># Take the most restrictive of the bounds set by the two sets</span>
-            <span class="c"># round inwards</span>
-            <span class="n">inf</span> <span class="o">=</span> <span class="n">ceiling</span><span class="p">(</span><span class="n">Max</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">inf</span><span class="p">,</span> <span class="n">oinf</span><span class="p">))</span>
-            <span class="n">sup</span> <span class="o">=</span> <span class="n">floor</span><span class="p">(</span><span class="n">Min</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">sup</span><span class="p">,</span> <span class="n">osup</span><span class="p">))</span>
-            <span class="c"># if we are off the sequence, get back on</span>
-            <span class="n">off</span> <span class="o">=</span> <span class="p">(</span><span class="n">inf</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">inf</span><span class="p">)</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">step</span>
-            <span class="k">if</span> <span class="n">off</span><span class="p">:</span>
-                <span class="n">inf</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">step</span> <span class="o">-</span> <span class="n">off</span>
-
-            <span class="k">return</span> <span class="n">Range</span><span class="p">(</span><span class="n">inf</span><span class="p">,</span> <span class="n">sup</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">step</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">other</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Naturals</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_intersect</span><span class="p">(</span><span class="n">Interval</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">other</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Integers</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">def</span> <span class="nf">_contains</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">other</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">inf</span> <span class="ow">and</span> <span class="n">other</span> <span class="o">&lt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sup</span> <span class="ow">and</span>
-                <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">start</span> <span class="o">-</span> <span class="n">other</span><span class="p">)</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">step</span><span class="p">)))</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">start</span>
-        <span class="k">while</span><span class="p">(</span><span class="n">i</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">stop</span><span class="p">):</span>
-            <span class="k">yield</span> <span class="n">i</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="n">i</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">step</span>
-
-    <span class="k">def</span> <span class="nf">__len__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">stop</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">start</span><span class="p">)</span><span class="o">//</span><span class="bp">self</span><span class="o">.</span><span class="n">step</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_ith_element</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">start</span> <span class="o">+</span> <span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">step</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_last_element</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ith_element</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_inf</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">start</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">_sup</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">stop</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">step</span>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-<div id="searchbox" style="display: none">
-  <h3>Quick search</h3>
-    <form class="search" action="../../../search.html" method="get">
-      <input type="text" name="q" />
-      <input type="submit" value="Go" />
-      <input type="hidden" name="check_keywords" value="yes" />
-      <input type="hidden" name="area" value="default" />
-    </form>
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-    Enter search terms or a module, class or function name.
-    </p>
-</div>
-<script type="text/javascript">$('#searchbox').show(0);</script>
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-             >modules</a> |</li>
-        <li class="right" >
-          <a href="../../../np-modindex.html" title="Python Module Index"
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-        <li><a href="../../../index.html">SymPy 0.7.2-git documentation</a> &raquo;</li>
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diff --git a/dev-py3k/_modules/sympy/simplify/cse_main.html b/dev-py3k/_modules/sympy/simplify/cse_main.html
deleted file mode 100644
index 9be9b4d6088..00000000000
--- a/dev-py3k/_modules/sympy/simplify/cse_main.html
+++ /dev/null
@@ -1,525 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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-            
-  <h1>Source code for sympy.simplify.cse_main</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot; Tools for doing common subexpression elimination.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">import</span> <span class="nn">difflib</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Add</span><span class="p">,</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.core.basic</span> <span class="kn">import</span> <span class="n">preorder_traversal</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">_coeff_isneg</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">iterable</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">numbered_symbols</span><span class="p">,</span> \
-    <span class="n">sift</span><span class="p">,</span> <span class="n">topological_sort</span><span class="p">,</span> <span class="n">ordered</span>
-
-<span class="kn">from</span> <span class="nn">.</span> <span class="kn">import</span> <span class="n">cse_opts</span>
-
-<span class="c"># (preprocessor, postprocessor) pairs which are commonly useful. They should</span>
-<span class="c"># each take a sympy expression and return a possibly transformed expression.</span>
-<span class="c"># When used in the function ``cse()``, the target expressions will be transformed</span>
-<span class="c"># by each of the preprocessor functions in order. After the common</span>
-<span class="c"># subexpressions are eliminated, each resulting expression will have the</span>
-<span class="c"># postprocessor functions transform them in *reverse* order in order to undo the</span>
-<span class="c"># transformation if necessary. This allows the algorithm to operate on</span>
-<span class="c"># a representation of the expressions that allows for more optimization</span>
-<span class="c"># opportunities.</span>
-<span class="c"># ``None`` can be used to specify no transformation for either the preprocessor or</span>
-<span class="c"># postprocessor.</span>
-
-<span class="n">cse_optimizations</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">cse_opts</span><span class="o">.</span><span class="n">default_optimizations</span><span class="p">)</span>
-
-<span class="c"># sometimes we want the output in a different format; non-trivial</span>
-<span class="c"># transformations can be put here for users</span>
-<span class="c"># ===============================================================</span>
-
-
-<span class="k">def</span> <span class="nf">reps_toposort</span><span class="p">(</span><span class="n">r</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Sort replacements `r` so (k1, v1) appears before (k2, v2)</span>
-<span class="sd">    if k2 is in v1&#39;s free symbols. This orders items in the</span>
-<span class="sd">    way that cse returns its results (hence, in order to use the</span>
-<span class="sd">    replacements in a substitution option it would make sense</span>
-<span class="sd">    to reverse the order).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.cse_main import reps_toposort</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Eq</span>
-<span class="sd">    &gt;&gt;&gt; for l, r in reps_toposort([(x, y + 1), (y, 2)]):</span>
-<span class="sd">    ...     print(Eq(l, r))</span>
-<span class="sd">    ...</span>
-<span class="sd">    y == 2</span>
-<span class="sd">    x == y + 1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-    <span class="n">E</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">c1</span><span class="p">,</span> <span class="p">(</span><span class="n">k1</span><span class="p">,</span> <span class="n">v1</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">r</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">c2</span><span class="p">,</span> <span class="p">(</span><span class="n">k2</span><span class="p">,</span> <span class="n">v2</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">r</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">k1</span> <span class="ow">in</span> <span class="n">v2</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-                <span class="n">E</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">c1</span><span class="p">,</span> <span class="n">c2</span><span class="p">))</span>
-    <span class="k">return</span> <span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">topological_sort</span><span class="p">((</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">r</span><span class="p">))),</span> <span class="n">E</span><span class="p">))]</span>
-
-
-<span class="k">def</span> <span class="nf">cse_separate</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Move expressions that are in the form (symbol, expr) out of the</span>
-<span class="sd">    expressions and sort them into the replacements using the reps_toposort.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.cse_main import cse_separate</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import cos, exp, cse, Eq, symbols</span>
-<span class="sd">    &gt;&gt;&gt; x0, x1 = symbols(&#39;x:2&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; eq = (x + 1 + exp((x + 1)/(y + 1)) + cos(y + 1))</span>
-<span class="sd">    &gt;&gt;&gt; cse([eq, Eq(x, z + 1), z - 2], postprocess=cse_separate) in [</span>
-<span class="sd">    ... [[(x0, y + 1), (x, z + 1), (x1, x + 1)],</span>
-<span class="sd">    ...  [x1 + exp(x1/x0) + cos(x0), z - 2]],</span>
-<span class="sd">    ... [[(x1, y + 1), (x, z + 1), (x0, x + 1)],</span>
-<span class="sd">    ...  [x0 + exp(x0/x1) + cos(x1), z - 2]]]</span>
-<span class="sd">    ...</span>
-<span class="sd">    True</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">d</span> <span class="o">=</span> <span class="n">sift</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">w</span><span class="p">:</span> <span class="n">w</span><span class="o">.</span><span class="n">is_Equality</span> <span class="ow">and</span> <span class="n">w</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">)</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">r</span> <span class="o">+</span> <span class="p">[</span><span class="n">w</span><span class="o">.</span><span class="n">args</span> <span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">d</span><span class="p">[</span><span class="bp">True</span><span class="p">]]</span>
-    <span class="n">e</span> <span class="o">=</span> <span class="n">d</span><span class="p">[</span><span class="bp">False</span><span class="p">]</span>
-    <span class="k">return</span> <span class="p">[</span><span class="n">reps_toposort</span><span class="p">(</span><span class="n">r</span><span class="p">),</span> <span class="n">e</span><span class="p">]</span>
-
-<span class="c"># ====end of cse postprocess idioms===========================</span>
-
-
-<span class="k">def</span> <span class="nf">preprocess_for_cse</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">optimizations</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Preprocess an expression to optimize for common subexpression</span>
-<span class="sd">    elimination.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    expr : sympy expression</span>
-<span class="sd">        The target expression to optimize.</span>
-<span class="sd">    optimizations : list of (callable, callable) pairs</span>
-<span class="sd">        The (preprocessor, postprocessor) pairs.</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    -------</span>
-<span class="sd">    expr : sympy expression</span>
-<span class="sd">        The transformed expression.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">for</span> <span class="n">pre</span><span class="p">,</span> <span class="n">post</span> <span class="ow">in</span> <span class="n">optimizations</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">pre</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">pre</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">expr</span>
-
-
-<span class="k">def</span> <span class="nf">postprocess_for_cse</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">optimizations</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Postprocess an expression after common subexpression elimination to</span>
-<span class="sd">    return the expression to canonical sympy form.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    expr : sympy expression</span>
-<span class="sd">        The target expression to transform.</span>
-<span class="sd">    optimizations : list of (callable, callable) pairs, optional</span>
-<span class="sd">        The (preprocessor, postprocessor) pairs.  The postprocessors will be</span>
-<span class="sd">        applied in reversed order to undo the effects of the preprocessors</span>
-<span class="sd">        correctly.</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    -------</span>
-<span class="sd">    expr : sympy expression</span>
-<span class="sd">        The transformed expression.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">optimizations</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">optimizations</span> <span class="o">=</span> <span class="n">cse_optimizations</span>
-    <span class="k">for</span> <span class="n">pre</span><span class="p">,</span> <span class="n">post</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">optimizations</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">post</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">post</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">expr</span>
-
-
-<span class="k">def</span> <span class="nf">_remove_singletons</span><span class="p">(</span><span class="n">reps</span><span class="p">,</span> <span class="n">exprs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Helper function for cse that will remove expressions that weren&#39;t</span>
-<span class="sd">    used more than once.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">u_reps</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c"># the useful reps that are used more than once</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">ui</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">reps</span><span class="p">):</span>
-        <span class="n">used</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c"># where it was used</span>
-        <span class="n">ri</span><span class="p">,</span> <span class="n">ei</span> <span class="o">=</span> <span class="n">ui</span>
-
-        <span class="c"># keep track of whether the substitution was used more</span>
-        <span class="c"># than once. If used is None, it was never used (yet);</span>
-        <span class="c"># if used is an int, that is the last place where it was</span>
-        <span class="c"># used (&gt;=0 in the reps, &lt;0 in the expressions) and if</span>
-        <span class="c"># it is True, it was used more than once.</span>
-
-        <span class="n">used</span> <span class="o">=</span> <span class="bp">None</span>
-
-        <span class="n">tot</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c"># total times used so far</span>
-
-        <span class="c"># search through the reps</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">reps</span><span class="p">)):</span>
-            <span class="n">c</span> <span class="o">=</span> <span class="n">reps</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="n">ri</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">c</span><span class="p">:</span>
-                <span class="n">tot</span> <span class="o">+=</span> <span class="n">c</span>
-                <span class="k">if</span> <span class="n">tot</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">u_reps</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ui</span><span class="p">)</span>
-                    <span class="n">used</span> <span class="o">=</span> <span class="bp">True</span>
-                    <span class="k">break</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">used</span> <span class="o">=</span> <span class="n">j</span>
-
-        <span class="k">if</span> <span class="n">used</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">True</span><span class="p">:</span>
-
-            <span class="c"># then search through the expressions</span>
-
-            <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">rj</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">exprs</span><span class="p">):</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="n">rj</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="n">ri</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">c</span><span class="p">:</span>
-                    <span class="c"># append a negative so we know that it was in the</span>
-                    <span class="c"># expression that used it</span>
-                    <span class="n">tot</span> <span class="o">+=</span> <span class="n">c</span>
-                    <span class="k">if</span> <span class="n">tot</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="n">u_reps</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ui</span><span class="p">)</span>
-                        <span class="n">used</span> <span class="o">=</span> <span class="bp">True</span>
-                        <span class="k">break</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">used</span> <span class="o">=</span> <span class="n">j</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">exprs</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">used</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">int</span><span class="p">:</span>
-
-                <span class="c"># undo the change</span>
-
-                <span class="n">rep</span> <span class="o">=</span> <span class="p">{</span><span class="n">ri</span><span class="p">:</span> <span class="n">ei</span><span class="p">}</span>
-                <span class="n">j</span> <span class="o">=</span> <span class="n">used</span>
-                <span class="k">if</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">exprs</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">exprs</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">rep</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">reps</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">reps</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">reps</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">rep</span><span class="p">)</span>
-
-    <span class="c"># reuse unused symbols so a contiguous range of symbols is returned</span>
-
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">u_reps</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">reps</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">ri</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">u_reps</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">u_reps</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="n">reps</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]:</span>
-                <span class="n">rep</span> <span class="o">=</span> <span class="p">(</span><span class="n">u_reps</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">reps</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span>
-                <span class="n">u_reps</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">rep</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">u_reps</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="o">*</span><span class="n">rep</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">u_reps</span><span class="p">)):</span>
-                    <span class="n">u_reps</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">u_reps</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">u_reps</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="o">*</span><span class="n">rep</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">rj</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">exprs</span><span class="p">):</span>
-                    <span class="n">exprs</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">exprs</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="o">*</span><span class="n">rep</span><span class="p">)</span>
-
-    <span class="n">reps</span><span class="p">[:]</span> <span class="o">=</span> <span class="n">u_reps</span>  <span class="c"># change happens in-place</span>
-
-
-<div class="viewcode-block" id="cse"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.cse_main.cse">[docs]</a><span class="k">def</span> <span class="nf">cse</span><span class="p">(</span><span class="n">exprs</span><span class="p">,</span> <span class="n">symbols</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">optimizations</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">postprocess</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Perform common subexpression elimination on an expression.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    exprs : list of sympy expressions, or a single sympy expression</span>
-<span class="sd">        The expressions to reduce.</span>
-<span class="sd">    symbols : infinite iterator yielding unique Symbols</span>
-<span class="sd">        The symbols used to label the common subexpressions which are pulled</span>
-<span class="sd">        out. The ``numbered_symbols`` generator is useful. The default is a</span>
-<span class="sd">        stream of symbols of the form &quot;x0&quot;, &quot;x1&quot;, etc. This must be an infinite</span>
-<span class="sd">        iterator.</span>
-<span class="sd">    optimizations : list of (callable, callable) pairs, optional</span>
-<span class="sd">        The (preprocessor, postprocessor) pairs. If not provided,</span>
-<span class="sd">        ``sympy.simplify.cse.cse_optimizations`` is used.</span>
-<span class="sd">    postprocess : a function which accepts the two return values of cse and</span>
-<span class="sd">        returns the desired form of output from cse, e.g. if you want the</span>
-<span class="sd">        replacements reversed the function might be the following lambda:</span>
-<span class="sd">        lambda r, e: return reversed(r), e</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    replacements : list of (Symbol, expression) pairs</span>
-<span class="sd">        All of the common subexpressions that were replaced. Subexpressions</span>
-<span class="sd">        earlier in this list might show up in subexpressions later in this list.</span>
-<span class="sd">    reduced_exprs : list of sympy expressions</span>
-<span class="sd">        The reduced expressions with all of the replacements above.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span>
-
-    <span class="k">if</span> <span class="n">symbols</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="n">numbered_symbols</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="c"># In case we get passed an iterable with an __iter__ method instead of</span>
-        <span class="c"># an actual iterator.</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="nb">iter</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span>
-    <span class="n">seen_subexp</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="n">muls</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="n">adds</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="n">to_eliminate</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="n">optimizations</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="c"># Pull out the default here just in case there are some weird</span>
-        <span class="c"># manipulations of the module-level list in some other thread.</span>
-        <span class="n">optimizations</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">cse_optimizations</span><span class="p">)</span>
-
-    <span class="c"># Handle the case if just one expression was passed.</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">exprs</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-        <span class="n">exprs</span> <span class="o">=</span> <span class="p">[</span><span class="n">exprs</span><span class="p">]</span>
-
-    <span class="c"># Preprocess the expressions to give us better optimization opportunities.</span>
-    <span class="n">reduced_exprs</span> <span class="o">=</span> <span class="p">[</span><span class="n">preprocess_for_cse</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">optimizations</span><span class="p">)</span> <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">exprs</span><span class="p">]</span>
-
-    <span class="c"># Find all of the repeated subexpressions.</span>
-    <span class="k">for</span> <span class="n">expr</span> <span class="ow">in</span> <span class="n">reduced_exprs</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-            <span class="k">continue</span>
-        <span class="n">pt</span> <span class="o">=</span> <span class="n">preorder_traversal</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">subtree</span> <span class="ow">in</span> <span class="n">pt</span><span class="p">:</span>
-
-            <span class="n">inv</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">subtree</span> <span class="k">if</span> <span class="n">subtree</span><span class="o">.</span><span class="n">is_Pow</span> <span class="k">else</span> <span class="bp">None</span>
-
-            <span class="k">if</span> <span class="n">subtree</span><span class="o">.</span><span class="n">is_Atom</span> <span class="ow">or</span> <span class="n">iterable</span><span class="p">(</span><span class="n">subtree</span><span class="p">)</span> <span class="ow">or</span> <span class="n">inv</span> <span class="ow">and</span> <span class="n">inv</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-                <span class="c"># Exclude atoms, since there is no point in renaming them.</span>
-                <span class="k">continue</span>
-
-            <span class="k">if</span> <span class="n">subtree</span> <span class="ow">in</span> <span class="n">seen_subexp</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">inv</span> <span class="ow">and</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">subtree</span><span class="o">.</span><span class="n">exp</span><span class="p">):</span>
-                    <span class="c"># save the form with positive exponent</span>
-                    <span class="n">subtree</span> <span class="o">=</span> <span class="n">inv</span>
-                <span class="n">to_eliminate</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">subtree</span><span class="p">)</span>
-                <span class="n">pt</span><span class="o">.</span><span class="n">skip</span><span class="p">()</span>
-                <span class="k">continue</span>
-
-            <span class="k">if</span> <span class="n">inv</span> <span class="ow">and</span> <span class="n">inv</span> <span class="ow">in</span> <span class="n">seen_subexp</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">_coeff_isneg</span><span class="p">(</span><span class="n">subtree</span><span class="o">.</span><span class="n">exp</span><span class="p">):</span>
-                    <span class="c"># save the form with positive exponent</span>
-                    <span class="n">subtree</span> <span class="o">=</span> <span class="n">inv</span>
-                <span class="n">to_eliminate</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">subtree</span><span class="p">)</span>
-                <span class="n">pt</span><span class="o">.</span><span class="n">skip</span><span class="p">()</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="n">subtree</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                <span class="n">muls</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">subtree</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">subtree</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="n">adds</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">subtree</span><span class="p">)</span>
-
-            <span class="n">seen_subexp</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">subtree</span><span class="p">)</span>
-
-    <span class="c"># process adds - any adds that weren&#39;t repeated might contain</span>
-    <span class="c"># subpatterns that are repeated, e.g. x+y+z and x+y have x+y in common</span>
-    <span class="n">adds</span> <span class="o">=</span> <span class="p">[</span><span class="nb">set</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">ordered</span><span class="p">(</span><span class="n">adds</span><span class="p">)]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">adds</span><span class="p">)):</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">adds</span><span class="p">)):</span>
-            <span class="n">com</span> <span class="o">=</span> <span class="n">adds</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">adds</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">com</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">to_eliminate</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">com</span><span class="p">))</span>
-
-                <span class="c"># remove this set of symbols so it doesn&#39;t appear again</span>
-                <span class="n">adds</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">adds</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">difference</span><span class="p">(</span><span class="n">com</span><span class="p">)</span>
-                <span class="n">adds</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">adds</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">difference</span><span class="p">(</span><span class="n">com</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">adds</span><span class="p">)):</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">com</span><span class="o">.</span><span class="n">difference</span><span class="p">(</span><span class="n">adds</span><span class="p">[</span><span class="n">k</span><span class="p">]):</span>
-                        <span class="n">adds</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">adds</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">difference</span><span class="p">(</span><span class="n">com</span><span class="p">)</span>
-
-    <span class="c"># process muls - any muls that weren&#39;t repeated might contain</span>
-    <span class="c"># subpatterns that are repeated, e.g. x*y*z and x*y have x*y in common</span>
-
-    <span class="c"># use SequenceMatcher on the nc part to find the longest common expression</span>
-    <span class="c"># in common between the two nc parts</span>
-    <span class="n">sm</span> <span class="o">=</span> <span class="n">difflib</span><span class="o">.</span><span class="n">SequenceMatcher</span><span class="p">()</span>
-
-    <span class="n">muls</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">(</span><span class="n">cset</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">ordered</span><span class="p">(</span><span class="n">muls</span><span class="p">)]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">muls</span><span class="p">)):</span>
-        <span class="k">if</span> <span class="n">muls</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]:</span>
-            <span class="n">sm</span><span class="o">.</span><span class="n">set_seq1</span><span class="p">(</span><span class="n">muls</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">muls</span><span class="p">)):</span>
-            <span class="c"># the commutative part in common</span>
-            <span class="n">ccom</span> <span class="o">=</span> <span class="n">muls</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">muls</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span>
-
-            <span class="c"># the non-commutative part in common</span>
-            <span class="k">if</span> <span class="n">muls</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="ow">and</span> <span class="n">muls</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="mi">1</span><span class="p">]:</span>
-                <span class="c"># see if there is any chance of an nc match</span>
-                <span class="n">ncom</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">muls</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">muls</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="mi">1</span><span class="p">]))</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">ccom</span><span class="p">)</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">ncom</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span>
-                    <span class="k">continue</span>
-
-                <span class="c"># now work harder to find the match</span>
-                <span class="n">sm</span><span class="o">.</span><span class="n">set_seq2</span><span class="p">(</span><span class="n">muls</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span>
-                <span class="n">i1</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">find_longest_match</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">muls</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]),</span>
-                                                 <span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">muls</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="mi">1</span><span class="p">]))</span>
-                <span class="n">ncom</span> <span class="o">=</span> <span class="n">muls</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">][</span><span class="n">i1</span><span class="p">:</span><span class="n">i1</span> <span class="o">+</span> <span class="n">n</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">ncom</span> <span class="o">=</span> <span class="p">[]</span>
-
-            <span class="n">com</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">ccom</span><span class="p">)</span> <span class="o">+</span> <span class="n">ncom</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">com</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="k">continue</span>
-
-            <span class="n">to_eliminate</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">com</span><span class="p">))</span>
-
-            <span class="c"># remove ccom from all if there was no ncom; to update the nc part</span>
-            <span class="c"># would require finding the subexpr and then replacing it with a</span>
-            <span class="c"># dummy to keep bounding nc symbols from being identified as a</span>
-            <span class="c"># subexpr, e.g. removing B*C from A*B*C*D might allow A*D to be</span>
-            <span class="c"># identified as a subexpr which would not be right.</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">ncom</span><span class="p">:</span>
-                <span class="n">muls</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">muls</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">difference</span><span class="p">(</span><span class="n">ccom</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">muls</span><span class="p">)):</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">ccom</span><span class="o">.</span><span class="n">difference</span><span class="p">(</span><span class="n">muls</span><span class="p">[</span><span class="n">k</span><span class="p">][</span><span class="mi">0</span><span class="p">]):</span>
-                        <span class="n">muls</span><span class="p">[</span><span class="n">k</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">muls</span><span class="p">[</span><span class="n">k</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">difference</span><span class="p">(</span><span class="n">ccom</span><span class="p">)</span>
-
-    <span class="c"># make to_eliminate canonical; we will prefer non-Muls to Muls</span>
-    <span class="c"># so select them first (non-Muls will have False for is_Mul and will</span>
-    <span class="c"># be first in the ordering.</span>
-    <span class="n">to_eliminate</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">ordered</span><span class="p">(</span><span class="n">to_eliminate</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">_</span><span class="p">:</span> <span class="n">_</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">))</span>
-
-    <span class="c"># Substitute symbols for all of the repeated subexpressions.</span>
-    <span class="n">replacements</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">reduced_exprs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">reduced_exprs</span><span class="p">)</span>
-    <span class="n">hit</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">subtree</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">to_eliminate</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">hit</span><span class="p">:</span>
-            <span class="n">sym</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span>
-        <span class="n">hit</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">subtree</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">subtree</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="n">update</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">xreplace</span><span class="p">({</span><span class="n">subtree</span><span class="p">:</span> <span class="n">sym</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">subtree</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">sym</span><span class="p">})</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">update</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">subtree</span><span class="p">,</span> <span class="n">sym</span><span class="p">)</span>
-        <span class="c"># Make the substitution in all of the target expressions.</span>
-        <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">expr</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">reduced_exprs</span><span class="p">):</span>
-            <span class="n">old</span> <span class="o">=</span> <span class="n">reduced_exprs</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-            <span class="n">reduced_exprs</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">update</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-            <span class="n">hit</span> <span class="o">=</span> <span class="n">hit</span> <span class="ow">or</span> <span class="p">(</span><span class="n">old</span> <span class="o">!=</span> <span class="n">reduced_exprs</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-        <span class="c"># Make the substitution in all of the subsequent substitutions.</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">to_eliminate</span><span class="p">)):</span>
-            <span class="n">old</span> <span class="o">=</span> <span class="n">to_eliminate</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-            <span class="n">to_eliminate</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">update</span><span class="p">(</span><span class="n">to_eliminate</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-            <span class="n">hit</span> <span class="o">=</span> <span class="n">hit</span> <span class="ow">or</span> <span class="p">(</span><span class="n">old</span> <span class="o">!=</span> <span class="n">to_eliminate</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">hit</span><span class="p">:</span>
-            <span class="n">replacements</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">sym</span><span class="p">,</span> <span class="n">subtree</span><span class="p">))</span>
-
-    <span class="c"># Postprocess the expressions to return the expressions to canonical form.</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">subtree</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">replacements</span><span class="p">):</span>
-        <span class="n">subtree</span> <span class="o">=</span> <span class="n">postprocess_for_cse</span><span class="p">(</span><span class="n">subtree</span><span class="p">,</span> <span class="n">optimizations</span><span class="p">)</span>
-        <span class="n">replacements</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">subtree</span><span class="p">)</span>
-    <span class="n">reduced_exprs</span> <span class="o">=</span> <span class="p">[</span><span class="n">postprocess_for_cse</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">optimizations</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">reduced_exprs</span><span class="p">]</span>
-
-    <span class="c"># remove replacements that weren&#39;t used more than once</span>
-    <span class="n">_remove_singletons</span><span class="p">(</span><span class="n">replacements</span><span class="p">,</span> <span class="n">reduced_exprs</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">exprs</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">):</span>
-        <span class="n">reduced_exprs</span> <span class="o">=</span> <span class="p">[</span><span class="n">Matrix</span><span class="p">(</span><span class="n">exprs</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="n">exprs</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span> <span class="n">reduced_exprs</span><span class="p">)]</span>
-    <span class="k">if</span> <span class="n">postprocess</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">replacements</span><span class="p">,</span> <span class="n">reduced_exprs</span>
-    <span class="k">return</span> <span class="n">postprocess</span><span class="p">(</span><span class="n">replacements</span><span class="p">,</span> <span class="n">reduced_exprs</span><span class="p">)</span></div>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/simplify/epathtools.html b/dev-py3k/_modules/sympy/simplify/epathtools.html
deleted file mode 100644
index fda326fb79a..00000000000
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+++ /dev/null
@@ -1,470 +0,0 @@
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-            
-  <h1>Source code for sympy.simplify.epathtools</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Tools for manipulation of expressions using paths. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Basic</span>
-
-
-<div class="viewcode-block" id="EPath"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.epathtools.EPath">[docs]</a><span class="k">class</span> <span class="nc">EPath</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Manipulate expressions using paths.</span>
-
-<span class="sd">    EPath grammar in EBNF notation::</span>
-
-<span class="sd">        literal   ::= /[A-Za-z_][A-Za-z_0-9]*/</span>
-<span class="sd">        number    ::= /-?\d+/</span>
-<span class="sd">        type      ::= literal</span>
-<span class="sd">        attribute ::= literal &quot;?&quot;</span>
-<span class="sd">        all       ::= &quot;*&quot;</span>
-<span class="sd">        slice     ::= &quot;[&quot; number? (&quot;:&quot; number? (&quot;:&quot; number?)?)? &quot;]&quot;</span>
-<span class="sd">        range     ::= all | slice</span>
-<span class="sd">        query     ::= (type | attribute) (&quot;|&quot; (type | attribute))*</span>
-<span class="sd">        selector  ::= range | query range?</span>
-<span class="sd">        path      ::= &quot;/&quot; selector (&quot;/&quot; selector)*</span>
-
-<span class="sd">    See the docstring of the epath() function.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&quot;_path&quot;</span><span class="p">,</span> <span class="s">&quot;_epath&quot;</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">path</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Construct new EPath. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">path</span><span class="p">,</span> <span class="n">EPath</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">path</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">path</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;empty EPath&quot;</span><span class="p">)</span>
-
-        <span class="n">_path</span> <span class="o">=</span> <span class="n">path</span>
-
-        <span class="k">if</span> <span class="n">path</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="s">&#39;/&#39;</span><span class="p">:</span>
-            <span class="n">path</span> <span class="o">=</span> <span class="n">path</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;non-root EPath&quot;</span><span class="p">)</span>
-
-        <span class="n">epath</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">selector</span> <span class="ow">in</span> <span class="n">path</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;/&#39;</span><span class="p">):</span>
-            <span class="n">selector</span> <span class="o">=</span> <span class="n">selector</span><span class="o">.</span><span class="n">strip</span><span class="p">()</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">selector</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;empty selector&quot;</span><span class="p">)</span>
-
-            <span class="n">index</span> <span class="o">=</span> <span class="mi">0</span>
-
-            <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">selector</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">c</span><span class="o">.</span><span class="n">isalnum</span><span class="p">()</span> <span class="ow">or</span> <span class="n">c</span> <span class="o">==</span> <span class="s">&#39;_&#39;</span> <span class="ow">or</span> <span class="n">c</span> <span class="o">==</span> <span class="s">&#39;|&#39;</span> <span class="ow">or</span> <span class="n">c</span> <span class="o">==</span> <span class="s">&#39;?&#39;</span><span class="p">:</span>
-                    <span class="n">index</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">break</span>
-
-            <span class="n">attrs</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">types</span> <span class="o">=</span> <span class="p">[]</span>
-
-            <span class="k">if</span> <span class="n">index</span><span class="p">:</span>
-                <span class="n">elements</span> <span class="o">=</span> <span class="n">selector</span><span class="p">[:</span><span class="n">index</span><span class="p">]</span>
-                <span class="n">selector</span> <span class="o">=</span> <span class="n">selector</span><span class="p">[</span><span class="n">index</span><span class="p">:]</span>
-
-                <span class="k">for</span> <span class="n">element</span> <span class="ow">in</span> <span class="n">elements</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;|&#39;</span><span class="p">):</span>
-                    <span class="n">element</span> <span class="o">=</span> <span class="n">element</span><span class="o">.</span><span class="n">strip</span><span class="p">()</span>
-
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">element</span><span class="p">:</span>
-                        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;empty element&quot;</span><span class="p">)</span>
-
-                    <span class="k">if</span> <span class="n">element</span><span class="o">.</span><span class="n">endswith</span><span class="p">(</span><span class="s">&#39;?&#39;</span><span class="p">):</span>
-                        <span class="n">attrs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">element</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">types</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">element</span><span class="p">)</span>
-
-            <span class="n">span</span> <span class="o">=</span> <span class="bp">None</span>
-
-            <span class="k">if</span> <span class="n">selector</span> <span class="o">==</span> <span class="s">&#39;*&#39;</span><span class="p">:</span>
-                <span class="k">pass</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">selector</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39;[&#39;</span><span class="p">):</span>
-                    <span class="k">try</span><span class="p">:</span>
-                        <span class="n">i</span> <span class="o">=</span> <span class="n">selector</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="s">&#39;]&#39;</span><span class="p">)</span>
-                    <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-                        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;expected &#39;]&#39;, got EOL&quot;</span><span class="p">)</span>
-
-                    <span class="n">_span</span><span class="p">,</span> <span class="n">span</span> <span class="o">=</span> <span class="n">selector</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="n">i</span><span class="p">],</span> <span class="p">[]</span>
-
-                    <span class="k">if</span> <span class="s">&#39;:&#39;</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">_span</span><span class="p">:</span>
-                        <span class="n">span</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">_span</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">for</span> <span class="n">elt</span> <span class="ow">in</span> <span class="n">_span</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;:&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">):</span>
-                            <span class="k">if</span> <span class="ow">not</span> <span class="n">elt</span><span class="p">:</span>
-                                <span class="n">span</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">None</span><span class="p">)</span>
-                            <span class="k">else</span><span class="p">:</span>
-                                <span class="n">span</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">elt</span><span class="p">))</span>
-
-                        <span class="n">span</span> <span class="o">=</span> <span class="nb">slice</span><span class="p">(</span><span class="o">*</span><span class="n">span</span><span class="p">)</span>
-
-                    <span class="n">selector</span> <span class="o">=</span> <span class="n">selector</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-
-                <span class="k">if</span> <span class="n">selector</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;trailing characters in selector&quot;</span><span class="p">)</span>
-
-            <span class="n">epath</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">attrs</span><span class="p">,</span> <span class="n">types</span><span class="p">,</span> <span class="n">span</span><span class="p">))</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="nb">object</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-
-        <span class="n">obj</span><span class="o">.</span><span class="n">_path</span> <span class="o">=</span> <span class="n">_path</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_epath</span> <span class="o">=</span> <span class="n">epath</span>
-
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">(</span><span class="si">%r</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_path</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_get_ordered_args</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Sort ``expr.args`` using printing order. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_ordered_terms</span><span class="p">()</span>
-        <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_ordered_factors</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span>
-
-    <span class="k">def</span> <span class="nf">_hasattrs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">attrs</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Check if ``expr`` has any of ``attrs``. &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">attr</span> <span class="ow">in</span> <span class="n">attrs</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">attr</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_hastypes</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">types</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Check if ``expr`` is any of ``types``. &quot;&quot;&quot;</span>
-        <span class="n">_types</span> <span class="o">=</span> <span class="p">[</span> <span class="n">cls</span><span class="o">.</span><span class="n">__name__</span> <span class="k">for</span> <span class="n">cls</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="o">.</span><span class="n">mro</span><span class="p">()</span> <span class="p">]</span>
-        <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">_types</span><span class="p">)</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">types</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">_has</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">attrs</span><span class="p">,</span> <span class="n">types</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Apply ``_hasattrs`` and ``_hastypes`` to ``expr``. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">attrs</span> <span class="ow">or</span> <span class="n">types</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-        <span class="k">if</span> <span class="n">attrs</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">_hasattrs</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">attrs</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-        <span class="k">if</span> <span class="n">types</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">_hastypes</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">types</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">True</span>
-
-        <span class="k">return</span> <span class="bp">False</span>
-
-<div class="viewcode-block" id="EPath.apply"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.epathtools.EPath.apply">[docs]</a>    <span class="k">def</span> <span class="nf">apply</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">args</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">kwargs</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Modify parts of an expression selected by a path.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.simplify.epathtools import EPath</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import sin, cos, E</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y, z, t</span>
-
-<span class="sd">        &gt;&gt;&gt; path = EPath(&quot;/*/[0]/Symbol&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; expr = [((x, 1), 2), ((3, y), z)]</span>
-
-<span class="sd">        &gt;&gt;&gt; path.apply(expr, lambda expr: expr**2)</span>
-<span class="sd">        [((x**2, 1), 2), ((3, y**2), z)]</span>
-
-<span class="sd">        &gt;&gt;&gt; path = EPath(&quot;/*/*/Symbol&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; expr = t + sin(x + 1) + cos(x + y + E)</span>
-
-<span class="sd">        &gt;&gt;&gt; path.apply(expr, lambda expr: 2*expr)</span>
-<span class="sd">        t + sin(2*x + 1) + cos(2*x + 2*y + E)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">def</span> <span class="nf">_apply</span><span class="p">(</span><span class="n">path</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">func</span><span class="p">):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">path</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">func</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">selector</span><span class="p">,</span> <span class="n">path</span> <span class="o">=</span> <span class="n">path</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">path</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-                <span class="n">attrs</span><span class="p">,</span> <span class="n">types</span><span class="p">,</span> <span class="n">span</span> <span class="o">=</span> <span class="n">selector</span>
-
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-                        <span class="n">args</span><span class="p">,</span> <span class="n">basic</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_ordered_args</span><span class="p">(</span><span class="n">expr</span><span class="p">),</span> <span class="bp">True</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">expr</span>
-                <span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&#39;__iter__&#39;</span><span class="p">):</span>
-                    <span class="n">args</span><span class="p">,</span> <span class="n">basic</span> <span class="o">=</span> <span class="n">expr</span><span class="p">,</span> <span class="bp">False</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">expr</span>
-
-                <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">span</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">span</span><span class="p">)</span> <span class="o">==</span> <span class="nb">slice</span><span class="p">:</span>
-                        <span class="n">indices</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="o">*</span><span class="n">span</span><span class="o">.</span><span class="n">indices</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">))))</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">indices</span> <span class="o">=</span> <span class="p">[</span><span class="n">span</span><span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">indices</span> <span class="o">=</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">))</span>
-
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">indices</span><span class="p">:</span>
-                    <span class="k">try</span><span class="p">:</span>
-                        <span class="n">arg</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-                    <span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span>
-                        <span class="k">continue</span>
-
-                    <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_has</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">attrs</span><span class="p">,</span> <span class="n">types</span><span class="p">):</span>
-                        <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">_apply</span><span class="p">(</span><span class="n">path</span><span class="p">,</span> <span class="n">arg</span><span class="p">,</span> <span class="n">func</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">basic</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="n">_args</span><span class="p">,</span> <span class="n">_kwargs</span> <span class="o">=</span> <span class="n">args</span> <span class="ow">or</span> <span class="p">(),</span> <span class="n">kwargs</span> <span class="ow">or</span> <span class="p">{}</span>
-        <span class="n">_func</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">expr</span><span class="p">:</span> <span class="n">func</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="n">_args</span><span class="p">,</span> <span class="o">**</span><span class="n">_kwargs</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">_apply</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_epath</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">_func</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="EPath.select"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.epathtools.EPath.select">[docs]</a>    <span class="k">def</span> <span class="nf">select</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Retrieve parts of an expression selected by a path.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.simplify.epathtools import EPath</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import sin, cos, E</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y, z, t</span>
-
-<span class="sd">        &gt;&gt;&gt; path = EPath(&quot;/*/[0]/Symbol&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; expr = [((x, 1), 2), ((3, y), z)]</span>
-
-<span class="sd">        &gt;&gt;&gt; path.select(expr)</span>
-<span class="sd">        [x, y]</span>
-
-<span class="sd">        &gt;&gt;&gt; path = EPath(&quot;/*/*/Symbol&quot;)</span>
-<span class="sd">        &gt;&gt;&gt; expr = t + sin(x + 1) + cos(x + y + E)</span>
-
-<span class="sd">        &gt;&gt;&gt; path.select(expr)</span>
-<span class="sd">        [x, x, y]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">def</span> <span class="nf">_select</span><span class="p">(</span><span class="n">path</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">path</span><span class="p">:</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">selector</span><span class="p">,</span> <span class="n">path</span> <span class="o">=</span> <span class="n">path</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">path</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-                <span class="n">attrs</span><span class="p">,</span> <span class="n">types</span><span class="p">,</span> <span class="n">span</span> <span class="o">=</span> <span class="n">selector</span>
-
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-                    <span class="n">args</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_ordered_args</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&#39;__iter__&#39;</span><span class="p">):</span>
-                    <span class="n">args</span> <span class="o">=</span> <span class="n">expr</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span>
-
-                <span class="k">if</span> <span class="n">span</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">span</span><span class="p">)</span> <span class="o">==</span> <span class="nb">slice</span><span class="p">:</span>
-                        <span class="n">args</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="n">span</span><span class="p">]</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">try</span><span class="p">:</span>
-                            <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">args</span><span class="p">[</span><span class="n">span</span><span class="p">]]</span>
-                        <span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span>
-                            <span class="k">return</span>
-
-                <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_has</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">attrs</span><span class="p">,</span> <span class="n">types</span><span class="p">):</span>
-                        <span class="n">_select</span><span class="p">(</span><span class="n">path</span><span class="p">,</span> <span class="n">arg</span><span class="p">)</span>
-
-        <span class="n">_select</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_epath</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-</div></div>
-<div class="viewcode-block" id="epath"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.epathtools.epath">[docs]</a><span class="k">def</span> <span class="nf">epath</span><span class="p">(</span><span class="n">path</span><span class="p">,</span> <span class="n">expr</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">func</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">args</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">kwargs</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Manipulate parts of an expression selected by a path.</span>
-
-<span class="sd">    This function allows to manipulate large nested expressions in single</span>
-<span class="sd">    line of code, utilizing techniques to those applied in XML processing</span>
-<span class="sd">    standards (e.g. XPath).</span>
-
-<span class="sd">    If ``func`` is ``None``, :func:`epath` retrieves elements selected by</span>
-<span class="sd">    the ``path``. Otherwise it applies ``func`` to each matching element.</span>
-
-<span class="sd">    Note that it is more efficient to create an EPath object and use the select</span>
-<span class="sd">    and apply methods of that object, since this will compile the path string</span>
-<span class="sd">    only once.  This function should only be used as a convenient shortcut for</span>
-<span class="sd">    interactive use.</span>
-
-<span class="sd">    This is the supported syntax:</span>
-
-<span class="sd">    * select all: ``/*``</span>
-<span class="sd">          Equivalent of ``for arg in args:``.</span>
-<span class="sd">    * select slice: ``/[0]`` or ``/[1:5]`` or ``/[1:5:2]``</span>
-<span class="sd">          Supports standard Python&#39;s slice syntax.</span>
-<span class="sd">    * select by type: ``/list`` or ``/list|tuple``</span>
-<span class="sd">          Emulates :func:`isinstance`.</span>
-<span class="sd">    * select by attribute: ``/__iter__?``</span>
-<span class="sd">          Emulates :func:`hasattr`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    path : str | EPath</span>
-<span class="sd">        A path as a string or a compiled EPath.</span>
-<span class="sd">    expr : Basic | iterable</span>
-<span class="sd">        An expression or a container of expressions.</span>
-<span class="sd">    func : callable (optional)</span>
-<span class="sd">        A callable that will be applied to matching parts.</span>
-<span class="sd">    args : tuple (optional)</span>
-<span class="sd">        Additional positional arguments to ``func``.</span>
-<span class="sd">    kwargs : dict (optional)</span>
-<span class="sd">        Additional keyword arguments to ``func``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.epathtools import epath</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import sin, cos, E</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z, t</span>
-
-<span class="sd">    &gt;&gt;&gt; path = &quot;/*/[0]/Symbol&quot;</span>
-<span class="sd">    &gt;&gt;&gt; expr = [((x, 1), 2), ((3, y), z)]</span>
-
-<span class="sd">    &gt;&gt;&gt; epath(path, expr)</span>
-<span class="sd">    [x, y]</span>
-<span class="sd">    &gt;&gt;&gt; epath(path, expr, lambda expr: expr**2)</span>
-<span class="sd">    [((x**2, 1), 2), ((3, y**2), z)]</span>
-
-<span class="sd">    &gt;&gt;&gt; path = &quot;/*/*/Symbol&quot;</span>
-<span class="sd">    &gt;&gt;&gt; expr = t + sin(x + 1) + cos(x + y + E)</span>
-
-<span class="sd">    &gt;&gt;&gt; epath(path, expr)</span>
-<span class="sd">    [x, x, y]</span>
-<span class="sd">    &gt;&gt;&gt; epath(path, expr, lambda expr: 2*expr)</span>
-<span class="sd">    t + sin(2*x + 1) + cos(2*x + 2*y + E)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">_epath</span> <span class="o">=</span> <span class="n">EPath</span><span class="p">(</span><span class="n">path</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">expr</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">_epath</span>
-    <span class="k">if</span> <span class="n">func</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">_epath</span><span class="o">.</span><span class="n">select</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">_epath</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">args</span><span class="p">,</span> <span class="n">kwargs</span><span class="p">)</span></div>
-</pre></div>
-
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-  <h1>Source code for sympy.simplify.hyperexpand</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Expand Hypergeometric (and Meijer G) functions into named</span>
-<span class="sd">special functions.</span>
-
-<span class="sd">The algorithm for doing this uses a collection of lookup tables of</span>
-<span class="sd">hypergeometric functions, and various of their properties, to expand</span>
-<span class="sd">many hypergeometric functions in terms of special functions.</span>
-
-<span class="sd">It is based on the following paper:</span>
-<span class="sd">      Kelly B. Roach.  Meijer G Function Representations.</span>
-<span class="sd">      In: Proceedings of the 1997 International Symposium on Symbolic and</span>
-<span class="sd">      Algebraic Computation, pages 205-211, New York, 1997. ACM.</span>
-
-<span class="sd">It is described in great(er) detail in the Sphinx documentation.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="c"># SUMMARY OF EXTENSIONS FOR MEIJER G FUNCTIONS</span>
-<span class="c">#</span>
-<span class="c"># o z**rho G(ap, bq; z) = G(ap + rho, bq + rho; z)</span>
-<span class="c">#</span>
-<span class="c"># o denote z*d/dz by D</span>
-<span class="c">#</span>
-<span class="c"># o It is helpful to keep in mind that ap and bq play essentially symmetric</span>
-<span class="c">#   roles: G(1/z) has slightly altered parameters, with ap and bq interchanged.</span>
-<span class="c">#</span>
-<span class="c"># o There are four shift operators:</span>
-<span class="c">#   A_J = b_J - D,     J = 1, ..., n</span>
-<span class="c">#   B_J = 1 - a_j + D, J = 1, ..., m</span>
-<span class="c">#   C_J = -b_J + D,    J = m+1, ..., q</span>
-<span class="c">#   D_J = a_J - 1 - D, J = n+1, ..., p</span>
-<span class="c">#</span>
-<span class="c">#   A_J, C_J increment b_J</span>
-<span class="c">#   B_J, D_J decrement a_J</span>
-<span class="c">#</span>
-<span class="c"># o The corresponding four inverse-shift operators are defined if there</span>
-<span class="c">#   is no cancellation. Thus e.g. an index a_J (upper or lower) can be</span>
-<span class="c">#   incremented if a_J != b_i for i = 1, ..., q.</span>
-<span class="c">#</span>
-<span class="c"># o Order reduction: if b_j - a_i is a non-negative integer, where</span>
-<span class="c">#   j &lt;= m and i &gt; n, the corresponding quotient of gamma functions reduces</span>
-<span class="c">#   to a polynomial. Hence the G function can be expressed using a G-function</span>
-<span class="c">#   of lower order.</span>
-<span class="c">#   Similarly if j &gt; m and i &lt;= n.</span>
-<span class="c">#</span>
-<span class="c">#   Secondly, there are paired index theorems [Adamchik, The evaluation of</span>
-<span class="c">#   integrals of Bessel functions via G-function identities]. Suppose there</span>
-<span class="c">#   are three parameters a, b, c, where a is an a_i, i &lt;= n, b is a b_j,</span>
-<span class="c">#   j &lt;= m and c is a denominator parameter (i.e. a_i, i &gt; n or b_j, j &gt; m).</span>
-<span class="c">#   Suppose further all three differ by integers.</span>
-<span class="c">#   Then the order can be reduced.</span>
-<span class="c">#   TODO work this out in detail.</span>
-<span class="c">#</span>
-<span class="c"># o An index quadruple is called suitable if its order cannot be reduced.</span>
-<span class="c">#   If there exists a sequence of shift operators transforming one index</span>
-<span class="c">#   quadruple into another, we say one is reachable from the other.</span>
-<span class="c">#</span>
-<span class="c"># o Deciding if one index quadruple is reachable from another is tricky. For</span>
-<span class="c">#   this reason, we use hand-built routines to match and instantiate formulas.</span>
-<span class="c">#</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">,</span> <span class="n">expand</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">Mul</span>
-<span class="kn">from</span> <span class="nn">sympy.core.mod</span> <span class="kn">import</span> <span class="n">Mod</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.special.hyper</span> <span class="kn">import</span> <span class="n">hyper</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">default_sort_key</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">SYMPY_DEBUG</span>
-
-<span class="kn">from</span> <span class="nn">sympy.utilities.timeutils</span> <span class="kn">import</span> <span class="n">timethis</span>
-<span class="n">_timeit</span> <span class="o">=</span> <span class="n">timethis</span><span class="p">(</span><span class="s">&#39;meijerg&#39;</span><span class="p">)</span>
-
-<span class="c"># leave add formulae at the top for easy reference</span>
-
-
-<span class="k">def</span> <span class="nf">add_formulae</span><span class="p">(</span><span class="n">formulae</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Create our knowledge base. &quot;&quot;&quot;</span>
-
-    <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a b c, z&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Dummy</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">add</span><span class="p">(</span><span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">res</span><span class="p">):</span>
-        <span class="n">formulae</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Formula</span><span class="p">(</span><span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">res</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">)))</span>
-
-    <span class="k">def</span> <span class="nf">addb</span><span class="p">(</span><span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">M</span><span class="p">):</span>
-        <span class="n">formulae</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Formula</span><span class="p">(</span><span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">),</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">M</span><span class="p">))</span>
-
-    <span class="c"># Luke, Y. L. (1969), The Special Functions and Their Approximations,</span>
-    <span class="c"># Volume 1, section 6.2</span>
-
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="p">(</span>
-        <span class="n">exp</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">root</span><span class="p">,</span> <span class="n">cosh</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">asin</span><span class="p">,</span> <span class="n">atan</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">lowergamma</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span>
-        <span class="n">atanh</span><span class="p">,</span> <span class="n">besseli</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">erf</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">besselj</span><span class="p">,</span> <span class="n">Ei</span><span class="p">,</span>
-        <span class="n">EulerGamma</span><span class="p">,</span> <span class="n">Shi</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">cosh</span><span class="p">,</span> <span class="n">Chi</span><span class="p">,</span> <span class="n">diag</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">,</span>
-        <span class="n">fresnels</span><span class="p">,</span> <span class="n">fresnelc</span><span class="p">)</span>
-    <span class="kn">from</span> <span class="nn">sympy.functions.special.hyper</span> <span class="kn">import</span> <span class="p">(</span><span class="n">HyperRep_atanh</span><span class="p">,</span>
-        <span class="n">HyperRep_power1</span><span class="p">,</span> <span class="n">HyperRep_power2</span><span class="p">,</span> <span class="n">HyperRep_log1</span><span class="p">,</span> <span class="n">HyperRep_asin1</span><span class="p">,</span>
-        <span class="n">HyperRep_asin2</span><span class="p">,</span> <span class="n">HyperRep_sqrts1</span><span class="p">,</span> <span class="n">HyperRep_sqrts2</span><span class="p">,</span> <span class="n">HyperRep_log2</span><span class="p">,</span>
-        <span class="n">HyperRep_cosasin</span><span class="p">,</span> <span class="n">HyperRep_sinasin</span><span class="p">)</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">polar_lift</span><span class="p">,</span> <span class="n">exp_polar</span>
-
-    <span class="c"># 0F0</span>
-    <span class="n">add</span><span class="p">((),</span> <span class="p">(),</span> <span class="n">exp</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-
-    <span class="c"># 1F0</span>
-    <span class="n">add</span><span class="p">((</span><span class="o">-</span><span class="n">a</span><span class="p">,</span> <span class="p">),</span> <span class="p">(),</span> <span class="n">HyperRep_power1</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">))</span>
-
-    <span class="c"># 2F1</span>
-    <span class="n">addb</span><span class="p">((</span><span class="n">a</span><span class="p">,</span> <span class="n">a</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="p">),</span>
-         <span class="n">Matrix</span><span class="p">([</span><span class="n">HyperRep_power2</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span>
-                 <span class="n">HyperRep_power2</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span>
-         <span class="n">Matrix</span><span class="p">([[(</span><span class="n">a</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">z</span><span class="p">),</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="o">-</span> <span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">z</span><span class="p">)],</span>
-                 <span class="p">[</span><span class="n">a</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">z</span><span class="p">),</span> <span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="n">z</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">z</span><span class="p">)]]))</span>
-    <span class="n">addb</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">),</span>
-         <span class="n">Matrix</span><span class="p">([</span><span class="n">HyperRep_log1</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="mi">1</span><span class="p">]),</span> <span class="n">Matrix</span><span class="p">([[</span><span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="n">z</span><span class="o">/</span><span class="p">(</span><span class="n">z</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]))</span>
-    <span class="n">addb</span><span class="p">((</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="s">&#39;3/2&#39;</span><span class="p">),</span> <span class="p">),</span>
-         <span class="n">Matrix</span><span class="p">([</span><span class="n">HyperRep_atanh</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="mi">1</span><span class="p">]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]))</span>
-    <span class="n">addb</span><span class="p">((</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">),</span> <span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="s">&#39;3/2&#39;</span><span class="p">),</span> <span class="p">),</span>
-         <span class="n">Matrix</span><span class="p">([</span><span class="n">HyperRep_asin1</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="n">HyperRep_power1</span><span class="p">(</span><span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">z</span><span class="p">)]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">z</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">]]))</span>
-    <span class="n">addb</span><span class="p">((</span><span class="o">-</span><span class="n">a</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="o">-</span> <span class="n">a</span><span class="p">),</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="p">),</span>
-         <span class="n">Matrix</span><span class="p">([</span><span class="n">HyperRep_sqrts1</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span> <span class="o">-</span><span class="n">HyperRep_sqrts2</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">z</span><span class="p">)]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="n">a</span><span class="p">],</span>
-                 <span class="p">[</span><span class="n">z</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">z</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="o">-</span> <span class="n">z</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">z</span><span class="p">)]]))</span>
-
-    <span class="c"># A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev (1990).</span>
-    <span class="c"># Integrals and Series: More Special Functions, Vol. 3,.</span>
-    <span class="c"># Gordon and Breach Science Publisher</span>
-    <span class="n">addb</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="o">-</span><span class="n">a</span><span class="p">],</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">],</span>
-         <span class="n">Matrix</span><span class="p">([</span><span class="n">HyperRep_cosasin</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span> <span class="n">HyperRep_sinasin</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">)]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="n">a</span><span class="p">],</span> <span class="p">[</span><span class="n">a</span><span class="o">*</span><span class="n">z</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">z</span><span class="p">),</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">]]))</span>
-    <span class="n">addb</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">],</span>
-         <span class="n">Matrix</span><span class="p">([</span><span class="n">HyperRep_asin2</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="mi">1</span><span class="p">]),</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span>
-         <span class="n">Matrix</span><span class="p">([[(</span><span class="n">z</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">z</span><span class="p">),</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]))</span>
-
-    <span class="c"># 3F2</span>
-    <span class="n">addb</span><span class="p">([</span><span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span>
-         <span class="n">Matrix</span><span class="p">([</span><span class="n">z</span><span class="o">*</span><span class="n">HyperRep_atanh</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="n">HyperRep_log1</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="mi">1</span><span class="p">]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">z</span><span class="p">),</span> <span class="n">S</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span><span class="p">]]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">z</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">],</span>
-                 <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">z</span><span class="o">/</span><span class="p">(</span><span class="n">z</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)],</span>
-                 <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]))</span>
-    <span class="c"># actually the formula for 3/2 is much nicer ...</span>
-    <span class="n">addb</span><span class="p">([</span><span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span>
-         <span class="n">Matrix</span><span class="p">([</span><span class="n">HyperRep_power1</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span> <span class="n">HyperRep_log2</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="mi">1</span><span class="p">]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="n">S</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span><span class="o">/</span><span class="mi">9</span> <span class="o">-</span> <span class="mi">16</span><span class="o">/</span><span class="p">(</span><span class="mi">9</span><span class="o">*</span><span class="n">z</span><span class="p">),</span> <span class="mi">4</span><span class="o">/</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">z</span><span class="p">),</span> <span class="mi">16</span><span class="o">/</span><span class="p">(</span><span class="mi">9</span><span class="o">*</span><span class="n">z</span><span class="p">)]]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="n">z</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">z</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]))</span>
-
-    <span class="c"># 1F1</span>
-    <span class="n">addb</span><span class="p">([</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="n">b</span><span class="p">],</span> <span class="n">Matrix</span><span class="p">([</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span> <span class="o">*</span> <span class="n">exp</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">*</span> <span class="n">lowergamma</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span> <span class="mi">1</span><span class="p">]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="n">b</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span> <span class="o">-</span> <span class="n">b</span> <span class="o">+</span> <span class="n">z</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]))</span>
-    <span class="n">addb</span><span class="p">([</span><span class="n">a</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">],</span>
-         <span class="n">Matrix</span><span class="p">([</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="o">-</span> <span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-                 <span class="o">*</span> <span class="n">gamma</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="o">**</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="o">-</span> <span class="n">a</span><span class="p">),</span>
-                 <span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="o">-</span> <span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-                 <span class="o">*</span> <span class="n">gamma</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="o">**</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="o">-</span> <span class="n">a</span><span class="p">)]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)]]))</span>
-    <span class="n">mz</span> <span class="o">=</span> <span class="n">polar_lift</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">z</span>
-    <span class="n">addb</span><span class="p">([</span><span class="n">a</span><span class="p">],</span> <span class="p">[</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span>
-         <span class="n">Matrix</span><span class="p">([</span><span class="n">mz</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">lowergamma</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">mz</span><span class="p">),</span> <span class="n">a</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">z</span><span class="p">)]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="o">-</span><span class="n">a</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">z</span><span class="p">]]))</span>
-    <span class="c"># This one is redundant.</span>
-    <span class="n">add</span><span class="p">([</span><span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">],</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">],</span> <span class="n">exp</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="p">)</span><span class="o">*</span><span class="n">erf</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)))</span>
-
-    <span class="c"># Added to get nice results for Laplace transform of Fresnel functions</span>
-    <span class="c"># http://functions.wolfram.com/07.22.03.6437.01</span>
-    <span class="c"># Basic rule</span>
-    <span class="c">#add([1], [S(3)/4, S(5)/4],</span>
-    <span class="c">#    sqrt(pi) * (cos(2*sqrt(polar_lift(-1)*z))*fresnelc(2*root(polar_lift(-1)*z,4)/sqrt(pi)) +</span>
-    <span class="c">#                sin(2*sqrt(polar_lift(-1)*z))*fresnels(2*root(polar_lift(-1)*z,4)/sqrt(pi)))</span>
-    <span class="c">#    / (2*root(polar_lift(-1)*z,4)))</span>
-    <span class="c"># Manually tuned rule</span>
-    <span class="n">addb</span><span class="p">([</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">],</span>
-         <span class="n">Matrix</span><span class="p">([</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">*</span><span class="n">fresnels</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">root</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">))</span>
-                            <span class="o">+</span> <span class="n">cosh</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">*</span><span class="n">fresnelc</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">root</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)))</span>
-                  <span class="o">*</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">root</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="mi">4</span><span class="p">)),</span>
-                  <span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">root</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">sinh</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">*</span><span class="n">fresnelc</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">root</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">))</span>
-                                      <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">*</span><span class="n">fresnels</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">root</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)))</span>
-                  <span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span>
-                  <span class="mi">1</span> <span class="p">]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span>      <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">],</span>
-                 <span class="p">[</span> <span class="n">z</span><span class="p">,</span>      <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span>     <span class="p">],</span>
-                 <span class="p">[</span> <span class="mi">0</span><span class="p">,</span>      <span class="mi">0</span><span class="p">,</span>      <span class="mi">0</span>     <span class="p">]]))</span>
-
-    <span class="c"># 2F2</span>
-    <span class="n">addb</span><span class="p">([</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="n">a</span><span class="p">],</span> <span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span>
-         <span class="n">Matrix</span><span class="p">([</span><span class="n">a</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="p">)</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">erf</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)),</span>
-                 <span class="n">a</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">polar_lift</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="p">)</span><span class="o">*</span>
-                 <span class="n">lowergamma</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">polar_lift</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">),</span>
-                 <span class="n">a</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">z</span><span class="p">)]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="n">a</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">z</span><span class="p">]]))</span>
-    <span class="c"># We make a &quot;basis&quot; of four functions instead of three, and give EulerGamma</span>
-    <span class="c"># an extra slot (it could just be a coefficient to 1). The advantage is</span>
-    <span class="c"># that this way Polys will not see multivariate polynomials (it treats</span>
-    <span class="c"># EulerGamma as an indeterminate), which is *way* faster.</span>
-    <span class="n">addb</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span>
-         <span class="n">Matrix</span><span class="p">([</span><span class="n">Ei</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">-</span> <span class="n">log</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="n">exp</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">EulerGamma</span><span class="p">]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">]]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]))</span>
-
-    <span class="c"># 0F1</span>
-    <span class="n">add</span><span class="p">((),</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="p">),</span> <span class="n">cosh</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)))</span>
-    <span class="n">addb</span><span class="p">([],</span> <span class="p">[</span><span class="n">b</span><span class="p">],</span>
-         <span class="n">Matrix</span><span class="p">([</span><span class="n">gamma</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="p">((</span><span class="mi">1</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)),</span>
-                 <span class="n">gamma</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">b</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="n">z</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">b</span><span class="p">)]]))</span>
-
-    <span class="c"># 0F3</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="mi">4</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">fp</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">besseli</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">besselj</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">fm</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">besseli</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">besselj</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="c"># TODO branching</span>
-    <span class="n">addb</span><span class="p">([],</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">],</span>
-         <span class="n">Matrix</span><span class="p">([</span><span class="n">fp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span> <span class="n">fm</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">),</span>
-                 <span class="n">fm</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="n">fp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">)])</span>
-         <span class="o">*</span> <span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="p">((</span><span class="mi">1</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
-                 <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">a</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
-                 <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
-                 <span class="p">[</span><span class="n">z</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">a</span><span class="p">]]))</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="n">exp_polar</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span>
-    <span class="n">addb</span><span class="p">([],</span> <span class="p">[</span><span class="n">a</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">],</span>
-         <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">polar_lift</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">))</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span>
-         <span class="n">Matrix</span><span class="p">([</span><span class="n">besselj</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span>
-                 <span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="n">besseli</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">besselj</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-                    <span class="o">-</span> <span class="n">besseli</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">besselj</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)),</span>
-                 <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">besselj</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span>
-                 <span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="o">*</span><span class="p">(</span><span class="n">besseli</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">besselj</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-                       <span class="o">+</span> <span class="n">besseli</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">besselj</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">))]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
-                 <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
-                 <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">],</span>
-                 <span class="p">[</span><span class="o">-</span><span class="mi">32</span><span class="o">*</span><span class="n">z</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">a</span><span class="p">]]))</span>
-
-    <span class="c"># 1F2</span>
-    <span class="n">addb</span><span class="p">([</span><span class="n">a</span><span class="p">],</span> <span class="p">[</span><span class="n">a</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">],</span>
-         <span class="n">Matrix</span><span class="p">([</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="o">-</span> <span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span>
-                 <span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-                 <span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)),</span>
-                 <span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span><span class="p">]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="o">-</span><span class="n">gamma</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">4</span><span class="o">**</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="o">-</span> <span class="n">a</span><span class="p">),</span>
-                 <span class="mi">2</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">a</span><span class="p">),</span>
-                 <span class="mi">0</span><span class="p">]]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span> <span class="o">-</span> <span class="n">a</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]))</span>
-    <span class="n">addb</span><span class="p">([</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">],</span> <span class="p">[</span><span class="n">b</span><span class="p">,</span> <span class="mi">2</span> <span class="o">-</span> <span class="n">b</span><span class="p">],</span>
-         <span class="n">pi</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span><span class="o">/</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">b</span><span class="p">)</span><span class="o">*</span>
-         <span class="n">Matrix</span><span class="p">([</span><span class="n">besseli</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">b</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)),</span>
-                 <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">besseli</span><span class="p">(</span><span class="o">-</span><span class="n">b</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-                          <span class="o">+</span> <span class="n">besseli</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">b</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))),</span>
-                 <span class="n">besseli</span><span class="p">(</span><span class="o">-</span><span class="n">b</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="n">b</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
-                 <span class="p">[</span><span class="n">z</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">z</span><span class="p">],</span>
-                 <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="n">b</span><span class="p">]]))</span>
-    <span class="n">addb</span><span class="p">([</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">],</span>
-         <span class="n">Matrix</span><span class="p">([</span><span class="n">Shi</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="n">sinh</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">),</span>
-                 <span class="n">cosh</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]))</span>
-
-    <span class="c"># FresnelS</span>
-    <span class="c"># Basic rule</span>
-    <span class="c">#add([S(3)/4], [S(3)/2,S(7)/4], 6*fresnels( exp(pi*I/4)*root(z,4)*2/sqrt(pi) ) / ( pi * (exp(pi*I/4)*root(z,4)*2/sqrt(pi))**3 ) )</span>
-    <span class="c"># Manually tuned rule</span>
-    <span class="n">addb</span><span class="p">([</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">7</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">],</span>
-         <span class="n">Matrix</span><span class="p">(</span>
-             <span class="p">[</span> <span class="n">fresnels</span><span class="p">(</span>
-                 <span class="n">exp</span><span class="p">(</span>
-                     <span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="n">root</span><span class="p">(</span>
-                         <span class="n">z</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="mi">2</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span>
-                             <span class="n">pi</span><span class="p">)</span> <span class="p">)</span> <span class="o">/</span> <span class="p">(</span>
-                                 <span class="n">pi</span> <span class="o">*</span> <span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="n">root</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="mi">2</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">))</span><span class="o">**</span><span class="mi">3</span> <span class="p">),</span>
-               <span class="n">sinh</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">),</span>
-               <span class="n">cosh</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span> <span class="p">]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">6</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span>  <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">16</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
-                 <span class="p">[</span> <span class="mi">0</span><span class="p">,</span>      <span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span>  <span class="mi">1</span><span class="p">],</span>
-                 <span class="p">[</span> <span class="mi">0</span><span class="p">,</span>       <span class="n">z</span><span class="p">,</span>       <span class="mi">0</span><span class="p">]]))</span>
-
-    <span class="c"># FresnelC</span>
-    <span class="c"># Basic rule</span>
-    <span class="c">#add([S(1)/4], [S(1)/2,S(5)/4], fresnelc( exp(pi*I/4)*root(z,4)*2/sqrt(pi) ) / ( exp(pi*I/4)*root(z,4)*2/sqrt(pi) ) )</span>
-    <span class="c"># Manually tuned rule</span>
-    <span class="n">addb</span><span class="p">([</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">],</span>
-         <span class="n">Matrix</span><span class="p">(</span>
-             <span class="p">[</span> <span class="n">sqrt</span><span class="p">(</span>
-                 <span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span>
-                     <span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="n">fresnelc</span><span class="p">(</span>
-                         <span class="mi">2</span><span class="o">*</span><span class="n">root</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">root</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="mi">4</span><span class="p">)),</span>
-               <span class="n">cosh</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)),</span>
-               <span class="n">sinh</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="p">]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span>  <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span>     <span class="p">],</span>
-                 <span class="p">[</span> <span class="mi">0</span><span class="p">,</span>       <span class="mi">0</span><span class="p">,</span>      <span class="mi">1</span>     <span class="p">],</span>
-                 <span class="p">[</span> <span class="mi">0</span><span class="p">,</span>       <span class="n">z</span><span class="p">,</span>      <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">]]))</span>
-
-    <span class="c"># 2F3</span>
-    <span class="c"># XXX with this five-parameter formula is pretty slow with the current</span>
-    <span class="c">#     Formula.find_instantiations (creates 2!*3!*3**(2+3) ~ 3000</span>
-    <span class="c">#     instantiations ... But it&#39;s not too bad.</span>
-    <span class="n">addb</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="n">b</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span>
-         <span class="n">gamma</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="n">b</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span> <span class="o">*</span>
-         <span class="n">Matrix</span><span class="p">([</span><span class="n">besseli</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)),</span>
-                 <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)),</span>
-                 <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="n">b</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)),</span>
-                 <span class="n">besseli</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span> <span class="o">-</span> <span class="n">b</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
-                 <span class="p">[</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">b</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">],</span>
-                 <span class="p">[</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">b</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">],</span>
-                 <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">]]))</span>
-    <span class="c"># (C/f above comment about eulergamma in the basis).</span>
-    <span class="n">addb</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">],</span>
-         <span class="n">Matrix</span><span class="p">([</span><span class="n">Chi</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span> <span class="o">-</span> <span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)),</span>
-                 <span class="n">cosh</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)),</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">EulerGamma</span><span class="p">]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">]]),</span>
-         <span class="n">Matrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
-                 <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
-                 <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
-                 <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
-                 <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]))</span>
-
-
-<span class="k">def</span> <span class="nf">add_meijerg_formulae</span><span class="p">(</span><span class="n">formulae</span><span class="p">):</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">uppergamma</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">Si</span><span class="p">,</span> <span class="n">Ci</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">pi</span>
-
-    <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Dummy</span><span class="p">,</span> <span class="s">&#39;abcz&#39;</span><span class="p">))</span>
-    <span class="n">rho</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;rho&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">add</span><span class="p">(</span><span class="n">an</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">matcher</span><span class="p">):</span>
-        <span class="n">formulae</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">MeijerFormula</span><span class="p">(</span><span class="n">an</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="p">[</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">rho</span><span class="p">],</span>
-                                      <span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">matcher</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">detect_uppergamma</span><span class="p">(</span><span class="n">iq</span><span class="p">):</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">iq</span><span class="o">.</span><span class="n">an</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">y</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="n">iq</span><span class="o">.</span><span class="n">bm</span>
-        <span class="n">swapped</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">Mod</span><span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">(),</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">swapped</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">Mod</span><span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">(),</span> <span class="mi">1</span><span class="p">)</span> <span class="ow">or</span> <span class="n">x</span> <span class="o">&gt;</span> <span class="n">z</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="n">l</span> <span class="o">=</span> <span class="p">[</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">swapped</span><span class="p">:</span>
-            <span class="n">l</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">]</span>
-        <span class="k">return</span> <span class="p">{</span><span class="n">rho</span><span class="p">:</span> <span class="n">y</span><span class="p">,</span> <span class="n">a</span><span class="p">:</span> <span class="n">x</span> <span class="o">-</span> <span class="n">y</span><span class="p">},</span> <span class="n">IndexQuadruple</span><span class="p">([</span><span class="n">x</span><span class="p">],</span> <span class="p">[],</span> <span class="n">l</span><span class="p">,</span> <span class="p">[])</span>
-
-    <span class="n">add</span><span class="p">([</span><span class="n">a</span> <span class="o">+</span> <span class="n">rho</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="n">rho</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="n">rho</span><span class="p">],</span> <span class="p">[],</span>
-        <span class="n">Matrix</span><span class="p">([</span><span class="n">gamma</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="n">rho</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">uppergamma</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span>
-                <span class="n">gamma</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">rho</span><span class="p">)]),</span>
-        <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span>
-        <span class="n">Matrix</span><span class="p">([[</span><span class="n">rho</span> <span class="o">+</span> <span class="n">z</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="n">rho</span><span class="p">]]),</span>
-        <span class="n">detect_uppergamma</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">detect_3113</span><span class="p">(</span><span class="n">iq</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;http://functions.wolfram.com/07.34.03.0984.01&quot;&quot;&quot;</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">iq</span><span class="o">.</span><span class="n">an</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">w</span> <span class="o">=</span> <span class="n">iq</span><span class="o">.</span><span class="n">bm</span>
-        <span class="k">if</span> <span class="n">Mod</span><span class="p">((</span><span class="n">u</span> <span class="o">-</span> <span class="n">v</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">(),</span> <span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">Mod</span><span class="p">((</span><span class="n">v</span> <span class="o">-</span> <span class="n">w</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">(),</span> <span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span>
-            <span class="n">sig</span> <span class="o">=</span> <span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">))</span>
-            <span class="n">x1</span><span class="p">,</span> <span class="n">x2</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">w</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">Mod</span><span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="n">u</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">(),</span> <span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">sig</span> <span class="o">=</span> <span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-                <span class="n">x1</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">x2</span> <span class="o">=</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">w</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">sig</span> <span class="o">=</span> <span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-                <span class="n">y</span><span class="p">,</span> <span class="n">x1</span><span class="p">,</span> <span class="n">x2</span> <span class="o">=</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">w</span>
-
-        <span class="k">if</span> <span class="p">(</span><span class="n">Mod</span><span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="n">x1</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">(),</span> <span class="mi">1</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span> <span class="ow">or</span>
-            <span class="n">Mod</span><span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="n">x2</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">(),</span> <span class="mi">1</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span> <span class="ow">or</span>
-            <span class="n">Mod</span><span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">(),</span> <span class="mi">1</span><span class="p">)</span> <span class="o">!=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="ow">or</span>
-                <span class="n">x</span> <span class="o">&gt;</span> <span class="n">x1</span> <span class="ow">or</span> <span class="n">x</span> <span class="o">&gt;</span> <span class="n">x2</span><span class="p">):</span>
-            <span class="k">return</span>
-
-        <span class="k">return</span> <span class="p">{</span><span class="n">a</span><span class="p">:</span> <span class="n">x</span><span class="p">},</span> <span class="n">IndexQuadruple</span><span class="p">([</span><span class="n">x</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="n">x</span> <span class="o">-</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="n">t</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">sig</span><span class="p">],</span> <span class="p">[])</span>
-
-    <span class="n">s</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-    <span class="n">c_</span> <span class="o">=</span> <span class="n">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-    <span class="n">S_</span> <span class="o">=</span> <span class="n">Si</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span> <span class="o">-</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span>
-    <span class="n">C</span> <span class="o">=</span> <span class="n">Ci</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-    <span class="n">add</span><span class="p">([</span><span class="n">a</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="n">a</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">a</span> <span class="o">-</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">],</span> <span class="p">[],</span>
-        <span class="n">Matrix</span><span class="p">([</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">c_</span><span class="o">*</span><span class="n">S_</span> <span class="o">-</span> <span class="n">s</span><span class="o">*</span><span class="n">C</span><span class="p">),</span>
-                <span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="n">s</span><span class="o">*</span><span class="n">S_</span> <span class="o">+</span> <span class="n">c_</span><span class="o">*</span><span class="n">C</span><span class="p">),</span>
-                <span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="n">a</span><span class="p">]),</span>
-        <span class="n">Matrix</span><span class="p">([[</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]),</span>
-        <span class="n">Matrix</span><span class="p">([[</span><span class="n">a</span> <span class="o">-</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="n">z</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">a</span><span class="p">]]),</span>
-        <span class="n">detect_3113</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">make_simp</span><span class="p">(</span><span class="n">z</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Create a function that simplifies rational functions in ``z``. &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">simp</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Efficiently simplify the rational function ``expr``. &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">poly</span>
-        <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-        <span class="n">c</span><span class="p">,</span> <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="o">=</span> <span class="n">poly</span><span class="p">(</span><span class="n">numer</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">cancel</span><span class="p">(</span><span class="n">poly</span><span class="p">(</span><span class="n">denom</span><span class="p">,</span> <span class="n">z</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">c</span> <span class="o">*</span> <span class="n">numer</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span> <span class="o">/</span> <span class="n">denom</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-
-    <span class="k">return</span> <span class="n">simp</span>
-
-
-<span class="k">def</span> <span class="nf">debug</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">):</span>
-    <span class="k">if</span> <span class="n">SYMPY_DEBUG</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">end</span><span class="o">=</span><span class="s">&#39; &#39;</span><span class="p">)</span>
-        <span class="k">print</span><span class="p">()</span>
-
-
-<span class="k">class</span> <span class="nc">IndexPair</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Holds a pair of indices, and methods to compute their invariants. &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expand</span><span class="p">,</span> <span class="n">Tuple</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ap</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">expand</span><span class="p">,</span> <span class="n">ap</span><span class="p">)))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">bq</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">expand</span><span class="p">,</span> <span class="n">bq</span><span class="p">)))</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">sizes</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ap</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bq</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;IndexPair(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ap</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">bq</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">compute_buckets</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">oabuckets</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">obbuckets</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Partition ``ap`` and ``bq`` mod 1.</span>
-
-<span class="sd">        Partition parameters ``ap``, ``bq`` into buckets, i.e., return two</span>
-<span class="sd">        dicts abuckets, bbuckets such that every key in (ab)buckets is a</span>
-<span class="sd">        rational in the range [0, 1) [represented either by a Rational or a Mod</span>
-<span class="sd">        object], and such that (ab)buckets[key] is a tuple of all elements of</span>
-<span class="sd">        respectively ``ap`` or ``bq`` that are congruent to ``key`` modulo 1.</span>
-
-<span class="sd">        If oabuckets, obbuckets is specified, try to use the same Mod objects</span>
-<span class="sd">        for parameters where possible.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.simplify.hyperexpand import IndexPair</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import S</span>
-<span class="sd">        &gt;&gt;&gt; ap = (S(1)/2, S(1)/3, S(-1)/2, -2)</span>
-<span class="sd">        &gt;&gt;&gt; bq = (1, 2)</span>
-<span class="sd">        &gt;&gt;&gt; IndexPair(ap, bq).compute_buckets()</span>
-<span class="sd">        ({0: (-2,), 1/3: (1/3,), 1/2: (1/2, -1/2)}, {0: (1, 2)})</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</span>
-
-        <span class="c"># TODO this should probably be cached somewhere</span>
-        <span class="n">abuckets</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">tuple</span><span class="p">)</span>
-        <span class="n">bbuckets</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">tuple</span><span class="p">)</span>
-
-        <span class="c"># NOTE the new Mod object does so much canonization that we can ignore</span>
-        <span class="c">#      o(ab)buckets.</span>
-
-        <span class="k">for</span> <span class="n">params</span><span class="p">,</span> <span class="n">bucket</span> <span class="ow">in</span> <span class="p">[(</span><span class="bp">self</span><span class="o">.</span><span class="n">ap</span><span class="p">,</span> <span class="n">abuckets</span><span class="p">),</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bq</span><span class="p">,</span> <span class="n">bbuckets</span><span class="p">)]:</span>
-            <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">params</span><span class="p">:</span>
-                <span class="n">bucket</span><span class="p">[</span><span class="n">Mod</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]</span> <span class="o">+=</span> <span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="p">)</span>
-
-        <span class="k">return</span> <span class="nb">dict</span><span class="p">(</span><span class="n">abuckets</span><span class="p">),</span> <span class="nb">dict</span><span class="p">(</span><span class="n">bbuckets</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">build_invariants</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Compute the invariant vector of (``ap``, ``bq``).</span>
-
-<span class="sd">        The invariant vector is:</span>
-<span class="sd">            (gamma, ((s1, n1), ..., (sk, nk)), ((t1, m1), ..., (tr, mr)))</span>
-<span class="sd">        where gamma is the number of integer a &lt; 0,</span>
-<span class="sd">              s1 &lt; ... &lt; sk</span>
-<span class="sd">              nl is the number of parameters a_i congruent to sl mod 1</span>
-<span class="sd">              t1 &lt; ... &lt; tr</span>
-<span class="sd">              ml is the number of parameters b_i congruent to tl mod 1</span>
-
-<span class="sd">        If the index pair contains parameters, then this is not truly an</span>
-<span class="sd">        invariant, since the parameters cannot be sorted uniquely mod1.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.simplify.hyperexpand import IndexPair</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import S</span>
-<span class="sd">        &gt;&gt;&gt; ap = (S(1)/2, S(1)/3, S(-1)/2, -2)</span>
-<span class="sd">        &gt;&gt;&gt; bq = (1, 2)</span>
-
-<span class="sd">        Here gamma = 1,</span>
-<span class="sd">             k = 3, s1 = 0, s2 = 1/3, s3 = 1/2</span>
-<span class="sd">                    n1 = 1, n2 = 1,   n2 = 2</span>
-<span class="sd">             r = 1, t1 = 0</span>
-<span class="sd">                    m1 = 2:</span>
-<span class="sd">        &gt;&gt;&gt; IndexPair(ap, bq).build_invariants()</span>
-<span class="sd">        (1, ((0, 1), (1/3, 1), (1/2, 2)), ((0, 2),))</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">abuckets</span><span class="p">,</span> <span class="n">bbuckets</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">compute_buckets</span><span class="p">()</span>
-
-        <span class="n">gamma</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">if</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="ow">in</span> <span class="n">abuckets</span><span class="p">:</span>
-            <span class="n">gamma</span> <span class="o">=</span> <span class="nb">len</span><span class="p">([</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">abuckets</span><span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)]</span> <span class="k">if</span> <span class="n">x</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">])</span>
-
-        <span class="k">def</span> <span class="nf">tr</span><span class="p">(</span><span class="n">bucket</span><span class="p">):</span>
-            <span class="n">bucket</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">bucket</span><span class="o">.</span><span class="n">items</span><span class="p">())</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">any</span><span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Mod</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">bucket</span><span class="p">):</span>
-                <span class="n">bucket</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="n">bucket</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">([(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">bucket</span><span class="p">])</span>
-            <span class="k">return</span> <span class="n">bucket</span>
-
-        <span class="k">return</span> <span class="p">(</span><span class="n">gamma</span><span class="p">,</span> <span class="n">tr</span><span class="p">(</span><span class="n">abuckets</span><span class="p">),</span> <span class="n">tr</span><span class="p">(</span><span class="n">bbuckets</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">difficulty</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ip</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Estimate how many steps it takes to reach ``ip`` from self.</span>
-<span class="sd">            Return -1 if impossible. &quot;&quot;&quot;</span>
-        <span class="n">oabuckets</span><span class="p">,</span> <span class="n">obbuckets</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">compute_buckets</span><span class="p">()</span>
-        <span class="n">abuckets</span><span class="p">,</span> <span class="n">bbuckets</span> <span class="o">=</span> <span class="n">ip</span><span class="o">.</span><span class="n">compute_buckets</span><span class="p">(</span><span class="n">oabuckets</span><span class="p">,</span> <span class="n">obbuckets</span><span class="p">)</span>
-
-        <span class="n">gt0</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="p">(</span><span class="n">x</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="ow">in</span> <span class="n">abuckets</span> <span class="ow">and</span> <span class="p">(</span><span class="ow">not</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="ow">in</span> <span class="n">oabuckets</span> <span class="ow">or</span>
-            <span class="nb">len</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">filter</span><span class="p">(</span><span class="n">gt0</span><span class="p">,</span> <span class="n">abuckets</span><span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)])))</span> <span class="o">!=</span>
-                <span class="nb">len</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">filter</span><span class="p">(</span><span class="n">gt0</span><span class="p">,</span> <span class="n">oabuckets</span><span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)])))):</span>
-            <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-
-        <span class="n">diff</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">bucket</span><span class="p">,</span> <span class="n">obucket</span> <span class="ow">in</span> <span class="p">[(</span><span class="n">abuckets</span><span class="p">,</span> <span class="n">oabuckets</span><span class="p">),</span> <span class="p">(</span><span class="n">bbuckets</span><span class="p">,</span> <span class="n">obbuckets</span><span class="p">)]:</span>
-            <span class="k">for</span> <span class="n">mod</span> <span class="ow">in</span> <span class="nb">set</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">bucket</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">obucket</span><span class="o">.</span><span class="n">keys</span><span class="p">())):</span>
-                <span class="k">if</span> <span class="p">(</span><span class="ow">not</span> <span class="n">mod</span> <span class="ow">in</span> <span class="n">bucket</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="ow">not</span> <span class="n">mod</span> <span class="ow">in</span> <span class="n">obucket</span><span class="p">)</span> \
-                        <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">bucket</span><span class="p">[</span><span class="n">mod</span><span class="p">])</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">obucket</span><span class="p">[</span><span class="n">mod</span><span class="p">]):</span>
-                    <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-                <span class="n">l1</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">bucket</span><span class="p">[</span><span class="n">mod</span><span class="p">])</span>
-                <span class="n">l2</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">obucket</span><span class="p">[</span><span class="n">mod</span><span class="p">])</span>
-                <span class="n">l1</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-                <span class="n">l2</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-                <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">l1</span><span class="p">,</span> <span class="n">l2</span><span class="p">):</span>
-                    <span class="n">diff</span> <span class="o">+=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="n">j</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">diff</span>
-
-
-<span class="k">class</span> <span class="nc">IndexQuadruple</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Holds a quadruple of indices. &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">an</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">bq</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expand</span><span class="p">,</span> <span class="n">Tuple</span>
-
-        <span class="k">def</span> <span class="nf">tr</span><span class="p">(</span><span class="n">l</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">expand</span><span class="p">,</span> <span class="n">l</span><span class="p">)))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">an</span> <span class="o">=</span> <span class="n">tr</span><span class="p">(</span><span class="n">an</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">ap</span> <span class="o">=</span> <span class="n">tr</span><span class="p">(</span><span class="n">ap</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">bm</span> <span class="o">=</span> <span class="n">tr</span><span class="p">(</span><span class="n">bm</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">bq</span> <span class="o">=</span> <span class="n">tr</span><span class="p">(</span><span class="n">bq</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">compute_buckets</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Compute buckets for the fours sets of parameters.</span>
-
-<span class="sd">        We guarantee that any two equal Mod objects returned are actually the</span>
-<span class="sd">        same, and that the buckets are sorted by real part (an and bq</span>
-<span class="sd">        descendending, bm and ap ascending).</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.simplify.hyperexpand import IndexQuadruple</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import y</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import S</span>
-<span class="sd">        &gt;&gt;&gt; a, b = [1, 3, 2, S(3)/2], [1 + y, y, 2, y + 3]</span>
-<span class="sd">        &gt;&gt;&gt; IndexQuadruple(a, b, [2], [y]).compute_buckets()</span>
-<span class="sd">        ({0: [3, 2, 1], 1/2: [3/2]},</span>
-<span class="sd">        {0: [2], Mod(y, 1): [y, y + 1, y + 3]}, {0: [2]}, {Mod(y, 1): [y]})</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</span>
-        <span class="n">dicts</span> <span class="o">=</span> <span class="n">pan</span><span class="p">,</span> <span class="n">pap</span><span class="p">,</span> <span class="n">pbm</span><span class="p">,</span> <span class="n">pbq</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">list</span><span class="p">),</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">list</span><span class="p">),</span> \
-            <span class="n">defaultdict</span><span class="p">(</span><span class="nb">list</span><span class="p">),</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">list</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">dic</span><span class="p">,</span> <span class="n">lis</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">dicts</span><span class="p">,</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">an</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">ap</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">bm</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">bq</span><span class="p">)):</span>
-            <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">lis</span><span class="p">:</span>
-                <span class="n">dic</span><span class="p">[</span><span class="n">Mod</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">dic</span><span class="p">,</span> <span class="n">flip</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">dicts</span><span class="p">,</span> <span class="p">(</span><span class="bp">True</span><span class="p">,</span> <span class="bp">False</span><span class="p">,</span> <span class="bp">False</span><span class="p">,</span> <span class="bp">True</span><span class="p">)):</span>
-            <span class="k">for</span> <span class="n">m</span><span class="p">,</span> <span class="n">items</span> <span class="ow">in</span> <span class="n">dic</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                <span class="n">x0</span> <span class="o">=</span> <span class="n">items</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="n">items</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span> <span class="o">-</span> <span class="n">x0</span><span class="p">,</span> <span class="n">reverse</span><span class="o">=</span><span class="n">flip</span><span class="p">)</span>
-                <span class="n">dic</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">=</span> <span class="n">items</span>
-
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">([</span><span class="nb">dict</span><span class="p">(</span><span class="n">w</span><span class="p">)</span> <span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">dicts</span><span class="p">])</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">signature</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">an</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">ap</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bm</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">bq</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;IndexQuadruple(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">an</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">ap</span><span class="p">,</span>
-                                                   <span class="bp">self</span><span class="o">.</span><span class="n">bm</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">bq</span><span class="p">)</span>
-
-<span class="c"># Dummy variable.</span>
-<span class="n">_x</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">Formula</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    This class represents hypergeometric formulae.</span>
-
-<span class="sd">    Its data members are:</span>
-<span class="sd">    - z, the argument</span>
-<span class="sd">    - closed_form, the closed form expression</span>
-<span class="sd">    - symbols, the free symbols (parameters) in the formula</span>
-<span class="sd">    - indices, the parameters</span>
-<span class="sd">    - B, C, M (see _compute_basis)</span>
-<span class="sd">    - lcms, a dictionary which maps symbol -&gt; lcm of denominators</span>
-<span class="sd">    - isolation, a dictonary which maps symbol -&gt; (num, coeff) pairs</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import a, b, z</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.hyperexpand import Formula</span>
-<span class="sd">    &gt;&gt;&gt; f = Formula((a/2, a/3 + b, (1+a)/2), (a, b, (a+b)/7), z, None, [a, b])</span>
-
-<span class="sd">    The lcm of all denominators of coefficients of a is 2*3*7</span>
-<span class="sd">    &gt;&gt;&gt; f.lcms[a]</span>
-<span class="sd">    42</span>
-
-<span class="sd">    for b it is just 7:</span>
-<span class="sd">    &gt;&gt;&gt; f.lcms[b]</span>
-<span class="sd">    7</span>
-
-<span class="sd">    We can isolate a in the (1+a)/2 term, with denominator 2:</span>
-<span class="sd">    &gt;&gt;&gt; f.isolation[a]</span>
-<span class="sd">    (2, 2, 1)</span>
-
-<span class="sd">    b is isolated in the b term, with coefficient one:</span>
-<span class="sd">    &gt;&gt;&gt; f.isolation[b]</span>
-<span class="sd">    (4, 1, 1)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_compute_basis</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">closed_form</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Compute a set of functions B=(f1, ..., fn), a nxn matrix M</span>
-<span class="sd">        and a 1xn matrix C such that:</span>
-<span class="sd">           closed_form = C B</span>
-<span class="sd">           z d/dz B = M B.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">eye</span><span class="p">,</span> <span class="n">zeros</span>
-
-        <span class="n">afactors</span> <span class="o">=</span> <span class="p">[</span><span class="n">_x</span> <span class="o">+</span> <span class="n">a</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">indices</span><span class="o">.</span><span class="n">ap</span><span class="p">]</span>
-        <span class="n">bfactors</span> <span class="o">=</span> <span class="p">[</span><span class="n">_x</span> <span class="o">+</span> <span class="n">b</span> <span class="o">-</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">indices</span><span class="o">.</span><span class="n">bq</span><span class="p">]</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">_x</span><span class="o">*</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">bfactors</span><span class="p">)</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">z</span><span class="o">*</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">afactors</span><span class="p">)</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-
-        <span class="n">n</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="p">[</span><span class="n">closed_form</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="n">b</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">z</span><span class="o">*</span><span class="n">b</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">z</span><span class="p">))</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">B</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">C</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">n</span><span class="p">])</span>
-
-        <span class="n">m</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">col_insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-        <span class="n">l</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()[</span><span class="mi">1</span><span class="p">:]</span>
-        <span class="n">l</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">M</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">row_insert</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="n">Matrix</span><span class="p">([</span><span class="n">l</span><span class="p">])</span><span class="o">/</span><span class="n">poly</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">res</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">B</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">C</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">M</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">ap</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">expand</span><span class="p">,</span> <span class="n">ap</span><span class="p">)))</span>
-        <span class="n">bq</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">expand</span><span class="p">,</span> <span class="n">bq</span><span class="p">)))</span>
-        <span class="n">z</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">res</span><span class="p">)</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">sympify</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span> <span class="k">if</span> <span class="n">ap</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="ow">or</span> <span class="n">bq</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">)]</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">z</span> <span class="o">=</span> <span class="n">z</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">symbols</span> <span class="o">=</span> <span class="n">symbols</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">B</span> <span class="o">=</span> <span class="n">B</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">C</span> <span class="o">=</span> <span class="n">C</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">M</span> <span class="o">=</span> <span class="n">M</span>
-
-        <span class="n">params</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">ap</span><span class="p">)</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">bq</span><span class="p">)</span>
-        <span class="n">lcms</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="n">isolation</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">symbols</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">ilcm</span>
-            <span class="n">l</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="n">isolating</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">others</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">symbols</span><span class="p">[:])</span>
-            <span class="n">others</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">params</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">a</span><span class="p">):</span>
-                    <span class="n">c</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-                    <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                        <span class="n">c</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">a</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">c</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">m</span> <span class="o">!=</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="p">)</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">c</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;?&#39;</span><span class="p">)</span>
-                    <span class="n">l</span> <span class="o">=</span> <span class="n">ilcm</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">c</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">p</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">others</span><span class="p">):</span>
-                        <span class="n">isolating</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">i</span><span class="p">,</span> <span class="n">c</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="n">c</span><span class="o">.</span><span class="n">p</span><span class="p">))</span>
-                <span class="n">lcms</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="n">l</span>
-                <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">isolating</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;parameter is not isolated&#39;</span><span class="p">)</span>
-            <span class="n">isolating</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-            <span class="n">isolating</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="o">-</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-            <span class="n">isolation</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="n">isolating</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">lcms</span> <span class="o">=</span> <span class="n">lcms</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">isolation</span> <span class="o">=</span> <span class="n">isolation</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">indices</span> <span class="o">=</span> <span class="n">IndexPair</span><span class="p">(</span><span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">)</span>
-
-        <span class="c"># TODO with symbolic parameters, it could be advantageous</span>
-        <span class="c">#      (for prettier answers) to compute a basis only *after*</span>
-        <span class="c">#      instantiation</span>
-        <span class="k">if</span> <span class="n">res</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_compute_basis</span><span class="p">(</span><span class="n">res</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">closed_form</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">C</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">B</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">find_instantiations</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ip</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Try to find instantiations of the free symbols that match</span>
-<span class="sd">        ``ip.ap``, ``ip.bq``. Return the instantiated formulae as a list.</span>
-<span class="sd">        Note that the returned instantiations need not actually match,</span>
-<span class="sd">        or be valid!</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">ap</span> <span class="o">=</span> <span class="n">ip</span><span class="o">.</span><span class="n">ap</span>
-        <span class="n">bq</span> <span class="o">=</span> <span class="n">ip</span><span class="o">.</span><span class="n">bq</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">ap</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">indices</span><span class="o">.</span><span class="n">ap</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">bq</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">indices</span><span class="o">.</span><span class="n">bq</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Cannot instantiate other number of parameters&#39;</span><span class="p">)</span>
-
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">solve</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">permutations</span><span class="p">,</span> <span class="n">product</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">our_params</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">indices</span><span class="o">.</span><span class="n">ap</span><span class="p">)</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">indices</span><span class="o">.</span><span class="n">bq</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">na</span> <span class="ow">in</span> <span class="n">permutations</span><span class="p">(</span><span class="n">ap</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">nb</span> <span class="ow">in</span> <span class="n">permutations</span><span class="p">(</span><span class="n">bq</span><span class="p">):</span>
-                <span class="n">all_params</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">na</span><span class="p">)</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">nb</span><span class="p">)</span>
-                <span class="n">repl</span> <span class="o">=</span> <span class="p">{}</span>
-                <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">symbols</span><span class="p">:</span>
-                    <span class="n">i</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">isolation</span><span class="p">[</span><span class="n">a</span><span class="p">]</span>
-                    <span class="n">repl</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">solve</span><span class="p">(</span><span class="n">our_params</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">all_params</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">a</span><span class="p">)[</span><span class="mi">0</span><span class="p">],</span> <span class="n">d</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">change</span> <span class="ow">in</span> <span class="n">product</span><span class="p">(</span><span class="o">*</span><span class="p">[(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">symbols</span><span class="p">)):</span>
-                    <span class="n">rep</span> <span class="o">=</span> <span class="p">{}</span>
-                    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">change</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="n">repl</span><span class="o">.</span><span class="n">keys</span><span class="p">())):</span>
-                        <span class="n">rep</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="n">repl</span><span class="p">[</span><span class="n">a</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">i</span><span class="o">*</span><span class="n">repl</span><span class="p">[</span><span class="n">a</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
-                    <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Formula</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">indices</span><span class="o">.</span><span class="n">ap</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">rep</span><span class="p">),</span>
-                                       <span class="bp">self</span><span class="o">.</span><span class="n">indices</span><span class="o">.</span><span class="n">bq</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">rep</span><span class="p">),</span>
-                                       <span class="bp">self</span><span class="o">.</span><span class="n">z</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="p">[],</span> <span class="bp">self</span><span class="o">.</span><span class="n">B</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">rep</span><span class="p">),</span>
-                                       <span class="bp">self</span><span class="o">.</span><span class="n">C</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">rep</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">M</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">rep</span><span class="p">)))</span>
-                <span class="c"># if say a = -1/2, and there is 2*a in the formula, then</span>
-                <span class="c"># there will be a negative integer. But this origin is also</span>
-                <span class="c"># reachable from a = 1/2 ...</span>
-                <span class="c"># So throw this in as well.</span>
-                <span class="c"># The code is not as general as it could be, but good enough.</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">symbols</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">symbols</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                    <span class="n">aval</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">repl</span><span class="p">[</span><span class="n">a</span><span class="p">]</span>
-                    <span class="k">if</span> <span class="n">aval</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">d</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">ceiling</span>
-                        <span class="n">aval</span> <span class="o">-=</span> <span class="n">ceiling</span><span class="p">(</span><span class="n">aval</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-                        <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Formula</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">indices</span><span class="o">.</span><span class="n">ap</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">aval</span><span class="p">),</span>
-                                           <span class="bp">self</span><span class="o">.</span><span class="n">indices</span><span class="o">.</span><span class="n">bq</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">aval</span><span class="p">),</span>
-                                       <span class="bp">self</span><span class="o">.</span><span class="n">z</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="p">[],</span> <span class="bp">self</span><span class="o">.</span><span class="n">B</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">aval</span><span class="p">),</span>
-                                       <span class="bp">self</span><span class="o">.</span><span class="n">C</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">rep</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">M</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">aval</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">res</span>
-
-    <span class="k">def</span> <span class="nf">is_suitable</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Decide if ``self`` is a suitable origin.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.simplify.hyperexpand import Formula</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import S</span>
-
-<span class="sd">        If ai - bq in Z and bq &gt;= ai this is fine:</span>
-<span class="sd">        &gt;&gt;&gt; Formula((S(1)/2,), (S(3)/2,), None, None, []).is_suitable()</span>
-<span class="sd">        True</span>
-
-<span class="sd">        but ai = bq is not:</span>
-<span class="sd">        &gt;&gt;&gt; Formula((S(1)/2,), (S(1)/2,), None, None, []).is_suitable()</span>
-<span class="sd">        False</span>
-
-<span class="sd">        and ai &gt; bq is not either:</span>
-<span class="sd">        &gt;&gt;&gt; Formula((S(1)/2,), (-S(1)/2,), None, None, []).is_suitable()</span>
-<span class="sd">        False</span>
-
-<span class="sd">        None of the bj can be a non-positive integer:</span>
-<span class="sd">        &gt;&gt;&gt; Formula((S(1)/2,), (0,), None, None, []).is_suitable()</span>
-<span class="sd">        False</span>
-<span class="sd">        &gt;&gt;&gt; Formula((S(1)/2,), (-1, 1,), None, None, []).is_suitable()</span>
-<span class="sd">        False</span>
-
-<span class="sd">        None of the ai can be zero:</span>
-<span class="sd">        &gt;&gt;&gt; Formula((S(1)/2, 0), (1,), None, None, []).is_suitable()</span>
-<span class="sd">        False</span>
-
-
-<span class="sd">        More complicated examples:</span>
-<span class="sd">        &gt;&gt;&gt; Formula((S(1)/2, 1), (2, -S(2)/3), None, None, []).is_suitable()</span>
-<span class="sd">        True</span>
-<span class="sd">        &gt;&gt;&gt; a, b = (S(1)/2, 1), (2, -S(2)/3, S(3)/2)</span>
-<span class="sd">        &gt;&gt;&gt; Formula(a, b, None, None, []).is_suitable()</span>
-<span class="sd">        True</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">oo</span><span class="p">,</span> <span class="n">zoo</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">symbols</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">indices</span><span class="o">.</span><span class="n">ap</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">indices</span><span class="o">.</span><span class="n">bq</span><span class="p">:</span>
-                <span class="k">if</span> <span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">a</span> <span class="o">&lt;</span> <span class="n">b</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">indices</span><span class="o">.</span><span class="n">ap</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">indices</span><span class="o">.</span><span class="n">bq</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">b</span> <span class="o">&lt;=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">b</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">B</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">M</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">C</span><span class="p">]:</span>
-            <span class="k">if</span> <span class="n">e</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">)</span> <span class="ow">or</span> <span class="n">e</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">oo</span><span class="p">)</span> <span class="ow">or</span> <span class="n">e</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">-</span><span class="n">oo</span><span class="p">)</span> <span class="ow">or</span> <span class="n">e</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">zoo</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-
-<span class="k">class</span> <span class="nc">FormulaCollection</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; A collection of formulae to use as origins. &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Doing this globally at module init time is a pain ... &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">symbolic_formulae</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">concrete_formulae</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">formulae</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="n">add_formulae</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">formulae</span><span class="p">)</span>
-
-        <span class="c"># Now process the formulae into a helpful form.</span>
-        <span class="c"># These dicts are indexed by (p, q).</span>
-
-        <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">formulae</span><span class="p">:</span>
-            <span class="n">sizes</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">indices</span><span class="o">.</span><span class="n">sizes</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">symbols</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">symbolic_formulae</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="n">sizes</span><span class="p">,</span> <span class="p">[])</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">inv</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">indices</span><span class="o">.</span><span class="n">build_invariants</span><span class="p">()</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">concrete_formulae</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="n">sizes</span><span class="p">,</span> <span class="p">{})[</span><span class="n">inv</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span>
-
-    <span class="k">def</span> <span class="nf">lookup_origin</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ip</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Given the suitable parameters ``ip.ap``, ``ip.bq``, try to find an</span>
-<span class="sd">        origin in our knowledge base.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.simplify.hyperexpand import FormulaCollection, IndexPair</span>
-<span class="sd">        &gt;&gt;&gt; f = FormulaCollection()</span>
-<span class="sd">        &gt;&gt;&gt; f.lookup_origin(IndexPair((), ())).closed_form</span>
-<span class="sd">        exp(_z)</span>
-<span class="sd">        &gt;&gt;&gt; f.lookup_origin(IndexPair([1], ())).closed_form</span>
-<span class="sd">        HyperRep_power1(-1, _z)</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import S</span>
-<span class="sd">        &gt;&gt;&gt; i = IndexPair([S(&#39;1/4&#39;), S(&#39;3/4 + 4&#39;)], [S.Half])</span>
-<span class="sd">        &gt;&gt;&gt; f.lookup_origin(i).closed_form</span>
-<span class="sd">        HyperRep_sqrts1(-17/4, _z)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">inv</span> <span class="o">=</span> <span class="n">ip</span><span class="o">.</span><span class="n">build_invariants</span><span class="p">()</span>
-        <span class="n">sizes</span> <span class="o">=</span> <span class="n">ip</span><span class="o">.</span><span class="n">sizes</span>
-        <span class="k">if</span> <span class="n">sizes</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">concrete_formulae</span> <span class="ow">and</span> \
-                <span class="n">inv</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">concrete_formulae</span><span class="p">[</span><span class="n">sizes</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">concrete_formulae</span><span class="p">[</span><span class="n">sizes</span><span class="p">][</span><span class="n">inv</span><span class="p">]</span>
-
-        <span class="c"># We don&#39;t have a concrete formula. Try to instantiate.</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">sizes</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">symbolic_formulae</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>  <span class="c"># Too bad...</span>
-
-        <span class="n">possible</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">symbolic_formulae</span><span class="p">[</span><span class="n">sizes</span><span class="p">]:</span>
-            <span class="n">l</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">find_instantiations</span><span class="p">(</span><span class="n">ip</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">f2</span> <span class="ow">in</span> <span class="n">l</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">f2</span><span class="o">.</span><span class="n">is_suitable</span><span class="p">():</span>
-                    <span class="k">continue</span>
-                <span class="n">diff</span> <span class="o">=</span> <span class="n">f2</span><span class="o">.</span><span class="n">indices</span><span class="o">.</span><span class="n">difficulty</span><span class="p">(</span><span class="n">ip</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">diff</span> <span class="o">!=</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                    <span class="n">possible</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">diff</span><span class="p">,</span> <span class="n">f2</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">possible</span><span class="p">:</span>
-            <span class="c"># Give up.</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="c"># find the nearest origin</span>
-        <span class="n">possible</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-        <span class="k">return</span> <span class="n">possible</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
-
-
-<span class="k">class</span> <span class="nc">MeijerFormula</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    This class represents a Meijer G-function formula.</span>
-
-<span class="sd">    Its data members are:</span>
-<span class="sd">    - z, the argument</span>
-<span class="sd">    - symbols, the free symbols (parameters) in the formula</span>
-<span class="sd">    - indices, the parameters</span>
-<span class="sd">    - B, C, M (c/f ordinary Formula)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">an</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">matcher</span><span class="p">):</span>
-        <span class="n">an</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">bq</span> <span class="o">=</span> <span class="p">[</span><span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">expand</span><span class="p">,</span> <span class="n">w</span><span class="p">)))</span> <span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="p">[</span><span class="n">an</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">bq</span><span class="p">]]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">indices</span> <span class="o">=</span> <span class="n">IndexQuadruple</span><span class="p">(</span><span class="n">an</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">bq</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">z</span> <span class="o">=</span> <span class="n">z</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">symbols</span> <span class="o">=</span> <span class="n">symbols</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_matcher</span> <span class="o">=</span> <span class="n">matcher</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">B</span> <span class="o">=</span> <span class="n">B</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">C</span> <span class="o">=</span> <span class="n">C</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">M</span> <span class="o">=</span> <span class="n">M</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">closed_form</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">C</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">B</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">try_instantiate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">iq</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Try to instantiate the current formula to (almost) match iq.</span>
-<span class="sd">        This uses the _matcher passed on init.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">iq</span><span class="o">.</span><span class="n">signature</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">indices</span><span class="o">.</span><span class="n">signature</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_matcher</span><span class="p">(</span><span class="n">iq</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">res</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">subs</span><span class="p">,</span> <span class="n">niq</span> <span class="o">=</span> <span class="n">res</span>
-            <span class="k">return</span> <span class="n">MeijerFormula</span><span class="p">(</span><span class="n">niq</span><span class="o">.</span><span class="n">an</span><span class="p">,</span> <span class="n">niq</span><span class="o">.</span><span class="n">ap</span><span class="p">,</span> <span class="n">niq</span><span class="o">.</span><span class="n">bm</span><span class="p">,</span> <span class="n">niq</span><span class="o">.</span><span class="n">bq</span><span class="p">,</span>
-                                 <span class="bp">self</span><span class="o">.</span><span class="n">z</span><span class="p">,</span> <span class="p">[],</span>
-                                 <span class="bp">self</span><span class="o">.</span><span class="n">B</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">subs</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">C</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">subs</span><span class="p">),</span>
-                                 <span class="bp">self</span><span class="o">.</span><span class="n">M</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">subs</span><span class="p">),</span> <span class="bp">None</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">MeijerFormulaCollection</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    This class holds a collection of meijer g formulae.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</span>
-        <span class="n">formulae</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">add_meijerg_formulae</span><span class="p">(</span><span class="n">formulae</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">formulae</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">list</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">formula</span> <span class="ow">in</span> <span class="n">formulae</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">formulae</span><span class="p">[</span><span class="n">formula</span><span class="o">.</span><span class="n">indices</span><span class="o">.</span><span class="n">signature</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">formula</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">formulae</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">formulae</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">lookup_origin</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">iq</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Try to find a formula that matches iq. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">iq</span><span class="o">.</span><span class="n">signature</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">formulae</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">for</span> <span class="n">formula</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">formulae</span><span class="p">[</span><span class="n">iq</span><span class="o">.</span><span class="n">signature</span><span class="p">]:</span>
-            <span class="n">res</span> <span class="o">=</span> <span class="n">formula</span><span class="o">.</span><span class="n">try_instantiate</span><span class="p">(</span><span class="n">iq</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">res</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">res</span>
-
-
-<span class="k">class</span> <span class="nc">Operator</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Base class for operators to be applied to our functions.</span>
-
-<span class="sd">    These operators are differential operators. They are by convention</span>
-<span class="sd">    expressed in the variable D = z*d/dz (although this base class does</span>
-<span class="sd">    not actually care).</span>
-<span class="sd">    Note that when the operator is applied to an object, we typically do</span>
-<span class="sd">    *not* blindly differentiate but instead use a different representation</span>
-<span class="sd">    of the z*d/dz operator (see make_derivative_operator).</span>
-
-<span class="sd">    To subclass from this, define a __init__ method that initalises a</span>
-<span class="sd">    self._poly variable. This variable stores a polynomial. By convention</span>
-<span class="sd">    the generator is z*d/dz, and acts to the right of all coefficients.</span>
-
-<span class="sd">    Thus this poly</span>
-<span class="sd">        x**2 + 2*z*x + 1</span>
-<span class="sd">    represents the differential operator</span>
-<span class="sd">        (z*d/dz)**2 + 2*z**2*d/dz.</span>
-
-<span class="sd">    This class is used only in the implementation of the hypergeometric</span>
-<span class="sd">    function expansion algorithm.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">apply</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">obj</span><span class="p">,</span> <span class="n">op</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Apply ``self`` to the object ``obj``, where the generator is ``op``.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.simplify.hyperexpand import Operator</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.polys.polytools import Poly</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">        &gt;&gt;&gt; op = Operator()</span>
-<span class="sd">        &gt;&gt;&gt; op._poly = Poly(x**2 + z*x + y, x)</span>
-<span class="sd">        &gt;&gt;&gt; op.apply(z**7, lambda f: f.diff(z))</span>
-<span class="sd">        y*z**7 + 7*z**7 + 42*z**5</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">coeffs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_poly</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()</span>
-        <span class="n">coeffs</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-        <span class="n">diffs</span> <span class="o">=</span> <span class="p">[</span><span class="n">obj</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">coeffs</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="n">diffs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">op</span><span class="p">(</span><span class="n">diffs</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]))</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">coeffs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">diffs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">coeffs</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">diffs</span><span class="p">[</span><span class="mi">1</span><span class="p">:]):</span>
-            <span class="n">r</span> <span class="o">+=</span> <span class="n">c</span><span class="o">*</span><span class="n">d</span>
-        <span class="k">return</span> <span class="n">r</span>
-
-
-<span class="k">class</span> <span class="nc">MultOperator</span><span class="p">(</span><span class="n">Operator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Simply multiply by a &quot;constant&quot; &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">ShiftA</span><span class="p">(</span><span class="n">Operator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Increment an upper index. &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ai</span><span class="p">):</span>
-        <span class="n">ai</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">ai</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">ai</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Cannot increment zero upper index.&#39;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">_x</span><span class="o">/</span><span class="n">ai</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;&lt;Increment upper </span><span class="si">%s</span><span class="s">.&gt;&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">_poly</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()[</span><span class="mi">0</span><span class="p">])</span>
-
-
-<span class="k">class</span> <span class="nc">ShiftB</span><span class="p">(</span><span class="n">Operator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Decrement a lower index. &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">bi</span><span class="p">):</span>
-        <span class="n">bi</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">bi</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">bi</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Cannot decrement unit lower index.&#39;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">_x</span><span class="o">/</span><span class="p">(</span><span class="n">bi</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;&lt;Decrement lower </span><span class="si">%s</span><span class="s">.&gt;&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="bp">self</span><span class="o">.</span><span class="n">_poly</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">UnShiftA</span><span class="p">(</span><span class="n">Operator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Decrement an upper index. &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Note: i counts from zero! &quot;&quot;&quot;</span>
-        <span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">[</span><span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">i</span><span class="p">]))</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_ap</span> <span class="o">=</span> <span class="n">ap</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_bq</span> <span class="o">=</span> <span class="n">bq</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_i</span> <span class="o">=</span> <span class="n">i</span>
-
-        <span class="n">ap</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">ap</span><span class="p">)</span>
-        <span class="n">bq</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">bq</span><span class="p">)</span>
-        <span class="n">ai</span> <span class="o">=</span> <span class="n">ap</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="k">if</span> <span class="n">ai</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Cannot decrement unit upper index.&#39;</span><span class="p">)</span>
-
-        <span class="n">m</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">z</span><span class="o">*</span><span class="n">ai</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">ap</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">*=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">_x</span> <span class="o">+</span> <span class="n">a</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="c">#print m</span>
-
-        <span class="n">A</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">D</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">ai</span><span class="o">*</span><span class="n">A</span> <span class="o">-</span> <span class="n">ai</span><span class="p">,</span> <span class="n">A</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">bq</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">*=</span> <span class="p">(</span><span class="n">D</span> <span class="o">+</span> <span class="n">b</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="c">#print n</span>
-
-        <span class="n">b0</span> <span class="o">=</span> <span class="o">-</span><span class="n">n</span><span class="o">.</span><span class="n">nth</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">b0</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Cannot decrement upper index: &#39;</span>
-                             <span class="s">&#39;cancels with lower&#39;</span><span class="p">)</span>
-        <span class="c">#print b0</span>
-
-        <span class="n">n</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">n</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">A</span><span class="p">)</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">_x</span><span class="o">/</span><span class="n">ai</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">_x</span><span class="p">)</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">((</span><span class="n">n</span> <span class="o">-</span> <span class="n">m</span><span class="p">)</span><span class="o">/</span><span class="n">b0</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;&lt;Decrement upper index #</span><span class="si">%s</span><span class="s"> of </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">.&gt;&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_i</span><span class="p">,</span>
-                                                        <span class="bp">self</span><span class="o">.</span><span class="n">_ap</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_bq</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">UnShiftB</span><span class="p">(</span><span class="n">Operator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Increment a lower index. &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Note: i counts from zero! &quot;&quot;&quot;</span>
-        <span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">[</span><span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">i</span><span class="p">]))</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_ap</span> <span class="o">=</span> <span class="n">ap</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_bq</span> <span class="o">=</span> <span class="n">bq</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_i</span> <span class="o">=</span> <span class="n">i</span>
-
-        <span class="n">ap</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">ap</span><span class="p">)</span>
-        <span class="n">bq</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">bq</span><span class="p">)</span>
-        <span class="n">bi</span> <span class="o">=</span> <span class="n">bq</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
-
-        <span class="k">if</span> <span class="n">bi</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Cannot increment -1 lower index.&#39;</span><span class="p">)</span>
-
-        <span class="n">m</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">_x</span><span class="o">*</span><span class="p">(</span><span class="n">bi</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">bq</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">*=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">_x</span> <span class="o">+</span> <span class="n">b</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="c">#print m</span>
-
-        <span class="n">B</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">)</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">((</span><span class="n">bi</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">B</span> <span class="o">-</span> <span class="n">bi</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">ap</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">*=</span> <span class="p">(</span><span class="n">D</span> <span class="o">+</span> <span class="n">a</span><span class="p">)</span>
-        <span class="c">#print n</span>
-
-        <span class="n">b0</span> <span class="o">=</span> <span class="n">n</span><span class="o">.</span><span class="n">nth</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="c">#print b0</span>
-        <span class="k">if</span> <span class="n">b0</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Cannot increment index: cancels with upper&#39;</span><span class="p">)</span>
-        <span class="c">#print b0</span>
-
-        <span class="n">n</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">n</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">B</span><span class="p">)</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span>
-            <span class="n">B</span><span class="p">,</span> <span class="n">_x</span><span class="o">/</span><span class="p">(</span><span class="n">bi</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="c">#print n</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">((</span><span class="n">m</span> <span class="o">-</span> <span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">b0</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;&lt;Increment lower index #</span><span class="si">%s</span><span class="s"> of </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">.&gt;&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_i</span><span class="p">,</span>
-                                                        <span class="bp">self</span><span class="o">.</span><span class="n">_ap</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_bq</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">MeijerShiftA</span><span class="p">(</span><span class="n">Operator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Increment an upper b index. &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">bi</span><span class="p">):</span>
-        <span class="n">bi</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">bi</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">bi</span> <span class="o">-</span> <span class="n">_x</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;&lt;Increment upper b=</span><span class="si">%s</span><span class="s">.&gt;&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_poly</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()[</span><span class="mi">1</span><span class="p">])</span>
-
-
-<span class="k">class</span> <span class="nc">MeijerShiftB</span><span class="p">(</span><span class="n">Operator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Decrement an upper a index. &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">bi</span><span class="p">):</span>
-        <span class="n">bi</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">bi</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">bi</span> <span class="o">+</span> <span class="n">_x</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;&lt;Decrement upper a=</span><span class="si">%s</span><span class="s">.&gt;&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_poly</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()[</span><span class="mi">1</span><span class="p">])</span>
-
-
-<span class="k">class</span> <span class="nc">MeijerShiftC</span><span class="p">(</span><span class="n">Operator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Increment a lower b index. &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">bi</span><span class="p">):</span>
-        <span class="n">bi</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">bi</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="o">-</span><span class="n">bi</span> <span class="o">+</span> <span class="n">_x</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;&lt;Increment lower b=</span><span class="si">%s</span><span class="s">.&gt;&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">_poly</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()[</span><span class="mi">1</span><span class="p">])</span>
-
-
-<span class="k">class</span> <span class="nc">MeijerShiftD</span><span class="p">(</span><span class="n">Operator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Decrement a lower a index. &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">bi</span><span class="p">):</span>
-        <span class="n">bi</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">bi</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">bi</span> <span class="o">-</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">_x</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;&lt;Decrement lower a=</span><span class="si">%s</span><span class="s">.&gt;&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_poly</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">MeijerUnShiftA</span><span class="p">(</span><span class="n">Operator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Decrement an upper b index. &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">an</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Note: i counts from zero! &quot;&quot;&quot;</span>
-        <span class="n">an</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">[</span><span class="n">an</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">i</span><span class="p">]))</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_an</span> <span class="o">=</span> <span class="n">an</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_ap</span> <span class="o">=</span> <span class="n">ap</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_bm</span> <span class="o">=</span> <span class="n">bm</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_bq</span> <span class="o">=</span> <span class="n">bq</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_i</span> <span class="o">=</span> <span class="n">i</span>
-
-        <span class="n">an</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">an</span><span class="p">)</span>
-        <span class="n">ap</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">ap</span><span class="p">)</span>
-        <span class="n">bm</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">bm</span><span class="p">)</span>
-        <span class="n">bq</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">bq</span><span class="p">)</span>
-        <span class="n">bi</span> <span class="o">=</span> <span class="n">bm</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="n">m</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">bm</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">*=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="n">_x</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">bq</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">*=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">_x</span> <span class="o">-</span> <span class="n">b</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="c">#print m</span>
-
-        <span class="n">A</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">)</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">bi</span> <span class="o">-</span> <span class="n">A</span><span class="p">,</span> <span class="n">A</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">A</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">an</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">*=</span> <span class="p">(</span><span class="n">D</span> <span class="o">+</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">a</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">ap</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">*=</span> <span class="p">(</span><span class="o">-</span><span class="n">D</span> <span class="o">+</span> <span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="c">#print n</span>
-
-        <span class="n">b0</span> <span class="o">=</span> <span class="n">n</span><span class="o">.</span><span class="n">nth</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="c">#print b0</span>
-        <span class="k">if</span> <span class="n">b0</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Cannot decrement upper b index (cancels)&#39;</span><span class="p">)</span>
-        <span class="c">#print b0</span>
-
-        <span class="n">n</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">n</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">A</span><span class="p">)</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">bi</span> <span class="o">-</span> <span class="n">_x</span><span class="p">),</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="c">#print n</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">((</span><span class="n">m</span> <span class="o">-</span> <span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">b0</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;&lt;Decrement upper b index #</span><span class="si">%s</span><span class="s"> of </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">.&gt;&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_i</span><span class="p">,</span>
-                                      <span class="bp">self</span><span class="o">.</span><span class="n">_an</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ap</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_bm</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_bq</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">MeijerUnShiftB</span><span class="p">(</span><span class="n">Operator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Increment an upper a index. &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">an</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Note: i counts from zero! &quot;&quot;&quot;</span>
-        <span class="n">an</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">[</span><span class="n">an</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">i</span><span class="p">]))</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_an</span> <span class="o">=</span> <span class="n">an</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_ap</span> <span class="o">=</span> <span class="n">ap</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_bm</span> <span class="o">=</span> <span class="n">bm</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_bq</span> <span class="o">=</span> <span class="n">bq</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_i</span> <span class="o">=</span> <span class="n">i</span>
-
-        <span class="n">an</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">an</span><span class="p">)</span>
-        <span class="n">ap</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">ap</span><span class="p">)</span>
-        <span class="n">bm</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">bm</span><span class="p">)</span>
-        <span class="n">bq</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">bq</span><span class="p">)</span>
-        <span class="n">ai</span> <span class="o">=</span> <span class="n">an</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
-
-        <span class="n">m</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">an</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">*=</span> <span class="n">Poly</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">a</span> <span class="o">+</span> <span class="n">_x</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">ap</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">*=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">_x</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="c">#print m</span>
-
-        <span class="n">B</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">)</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">B</span> <span class="o">+</span> <span class="n">ai</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">bm</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">*=</span> <span class="p">(</span><span class="o">-</span><span class="n">D</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">bq</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">*=</span> <span class="p">(</span><span class="n">D</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span>
-        <span class="c">#print n</span>
-
-        <span class="n">b0</span> <span class="o">=</span> <span class="n">n</span><span class="o">.</span><span class="n">nth</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="c">#print b0</span>
-        <span class="k">if</span> <span class="n">b0</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Cannot increment upper a index (cancels)&#39;</span><span class="p">)</span>
-        <span class="c">#print b0</span>
-
-        <span class="n">n</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">n</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">B</span><span class="p">)</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span>
-            <span class="n">B</span><span class="p">,</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">ai</span> <span class="o">+</span> <span class="n">_x</span><span class="p">),</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="c">#print n</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">((</span><span class="n">m</span> <span class="o">-</span> <span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">b0</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;&lt;Increment upper a index #</span><span class="si">%s</span><span class="s"> of </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">.&gt;&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_i</span><span class="p">,</span>
-                                      <span class="bp">self</span><span class="o">.</span><span class="n">_an</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ap</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_bm</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_bq</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">MeijerUnShiftC</span><span class="p">(</span><span class="n">Operator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Decrement a lower b index. &quot;&quot;&quot;</span>
-    <span class="c"># XXX this is &quot;essentially&quot; the same as MeijerUnShiftA. This &quot;essentially&quot;</span>
-    <span class="c">#     can be made rigorous using the functional equation G(1/z) = G&#39;(z),</span>
-    <span class="c">#     where G&#39; denotes a G function of slightly altered parameters.</span>
-    <span class="c">#     However, sorting out the details seems harder than just coding it</span>
-    <span class="c">#     again.</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">an</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Note: i counts from zero! &quot;&quot;&quot;</span>
-        <span class="n">an</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">[</span><span class="n">an</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">i</span><span class="p">]))</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_an</span> <span class="o">=</span> <span class="n">an</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_ap</span> <span class="o">=</span> <span class="n">ap</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_bm</span> <span class="o">=</span> <span class="n">bm</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_bq</span> <span class="o">=</span> <span class="n">bq</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_i</span> <span class="o">=</span> <span class="n">i</span>
-
-        <span class="n">an</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">an</span><span class="p">)</span>
-        <span class="n">ap</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">ap</span><span class="p">)</span>
-        <span class="n">bm</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">bm</span><span class="p">)</span>
-        <span class="n">bq</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">bq</span><span class="p">)</span>
-        <span class="n">bi</span> <span class="o">=</span> <span class="n">bq</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-
-        <span class="n">m</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">bm</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">*=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="n">_x</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">bq</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">*=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">_x</span> <span class="o">-</span> <span class="n">b</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="c">#print m</span>
-
-        <span class="n">C</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;C&#39;</span><span class="p">)</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">bi</span> <span class="o">+</span> <span class="n">C</span><span class="p">,</span> <span class="n">C</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">C</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">an</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">*=</span> <span class="p">(</span><span class="n">D</span> <span class="o">+</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">a</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">ap</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">*=</span> <span class="p">(</span><span class="o">-</span><span class="n">D</span> <span class="o">+</span> <span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="c">#print n</span>
-
-        <span class="n">b0</span> <span class="o">=</span> <span class="n">n</span><span class="o">.</span><span class="n">nth</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="c">#print b0</span>
-        <span class="k">if</span> <span class="n">b0</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Cannot decrement lower b index (cancels)&#39;</span><span class="p">)</span>
-        <span class="c">#print b0</span>
-
-        <span class="n">n</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">n</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">C</span><span class="p">)</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">_x</span> <span class="o">-</span> <span class="n">bi</span><span class="p">),</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="c">#print n</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">((</span><span class="n">m</span> <span class="o">-</span> <span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">b0</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;&lt;Decrement lower b index #</span><span class="si">%s</span><span class="s"> of </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">.&gt;&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_i</span><span class="p">,</span>
-                                      <span class="bp">self</span><span class="o">.</span><span class="n">_an</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ap</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_bm</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_bq</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">MeijerUnShiftD</span><span class="p">(</span><span class="n">Operator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Increment a lower a index. &quot;&quot;&quot;</span>
-    <span class="c"># XXX This is essentially the same as MeijerUnShiftA.</span>
-    <span class="c">#     See comment at MeijerUnShiftC.</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">an</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Note: i counts from zero! &quot;&quot;&quot;</span>
-        <span class="n">an</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">[</span><span class="n">an</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">i</span><span class="p">]))</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_an</span> <span class="o">=</span> <span class="n">an</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_ap</span> <span class="o">=</span> <span class="n">ap</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_bm</span> <span class="o">=</span> <span class="n">bm</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_bq</span> <span class="o">=</span> <span class="n">bq</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_i</span> <span class="o">=</span> <span class="n">i</span>
-
-        <span class="n">an</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">an</span><span class="p">)</span>
-        <span class="n">ap</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">ap</span><span class="p">)</span>
-        <span class="n">bm</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">bm</span><span class="p">)</span>
-        <span class="n">bq</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">bq</span><span class="p">)</span>
-        <span class="n">ai</span> <span class="o">=</span> <span class="n">ap</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
-
-        <span class="n">m</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">an</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">*=</span> <span class="n">Poly</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">a</span> <span class="o">+</span> <span class="n">_x</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">ap</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">*=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">_x</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="c">#print m</span>
-
-        <span class="n">B</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">)</span>  <span class="c"># - this is the shift operator `D_I`</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">ai</span> <span class="o">-</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">B</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">bm</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">*=</span> <span class="p">(</span><span class="o">-</span><span class="n">D</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">bq</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">*=</span> <span class="p">(</span><span class="n">D</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span>
-        <span class="c">#print n</span>
-
-        <span class="n">b0</span> <span class="o">=</span> <span class="n">n</span><span class="o">.</span><span class="n">nth</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="c">#print b0</span>
-        <span class="k">if</span> <span class="n">b0</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Cannot increment lower a index (cancels)&#39;</span><span class="p">)</span>
-        <span class="c">#print b0</span>
-
-        <span class="n">n</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">n</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">B</span><span class="p">)</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span>
-            <span class="n">B</span><span class="p">,</span> <span class="n">ai</span> <span class="o">-</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">_x</span><span class="p">),</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="c">#print n</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">((</span><span class="n">m</span> <span class="o">-</span> <span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">b0</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;&lt;Increment lower a index #</span><span class="si">%s</span><span class="s"> of </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">.&gt;&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_i</span><span class="p">,</span>
-                                      <span class="bp">self</span><span class="o">.</span><span class="n">_an</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_ap</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_bm</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_bq</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">ReduceOrder</span><span class="p">(</span><span class="n">Operator</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Reduce Order by cancelling an upper and a lower index. &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">ai</span><span class="p">,</span> <span class="n">bj</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; For convenience if reduction is not possible, return None. &quot;&quot;&quot;</span>
-        <span class="n">ai</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">ai</span><span class="p">)</span>
-        <span class="n">bj</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">bj</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">ai</span> <span class="o">-</span> <span class="n">bj</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">bj</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">and</span> <span class="n">bj</span> <span class="o">&lt;=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">bj</span> <span class="o">+</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="bp">self</span> <span class="o">=</span> <span class="n">Operator</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-
-        <span class="n">p</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="n">p</span> <span class="o">*=</span> <span class="p">(</span><span class="n">_x</span> <span class="o">+</span> <span class="n">bj</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">bj</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_a</span> <span class="o">=</span> <span class="n">ai</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_b</span> <span class="o">=</span> <span class="n">bj</span>
-
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_meijer</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">sign</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Cancel b + sign*s and a + sign*s</span>
-<span class="sd">            This is for meijer G functions. &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Add</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">b</span> <span class="o">-</span> <span class="n">a</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="bp">self</span> <span class="o">=</span> <span class="n">Operator</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-
-        <span class="n">p</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="n">p</span> <span class="o">*=</span> <span class="p">(</span><span class="n">sign</span><span class="o">*</span><span class="n">_x</span> <span class="o">+</span> <span class="n">a</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">sign</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_a</span> <span class="o">=</span> <span class="n">b</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_b</span> <span class="o">=</span> <span class="n">a</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_b</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_a</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">b</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">meijer_minus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_meijer</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">meijer_plus</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">_meijer</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">a</span><span class="p">,</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">b</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&#39;&lt;Reduce order by cancelling upper </span><span class="si">%s</span><span class="s"> with lower </span><span class="si">%s</span><span class="s">.&gt;&#39;</span> <span class="o">%</span> \
-            <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_a</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_b</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">_reduce_order</span><span class="p">(</span><span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">gen</span><span class="p">,</span> <span class="n">key</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Order reduction algorithm used in Hypergeometric and Meijer G &quot;&quot;&quot;</span>
-    <span class="n">ap</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">ap</span><span class="p">)</span>
-    <span class="n">bq</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">bq</span><span class="p">)</span>
-
-    <span class="n">ap</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">key</span><span class="p">)</span>
-    <span class="n">bq</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">key</span><span class="p">)</span>
-
-    <span class="n">nap</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="c"># we will edit bq in place</span>
-    <span class="n">operators</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">ap</span><span class="p">:</span>
-        <span class="n">op</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">bq</span><span class="p">)):</span>
-            <span class="n">op</span> <span class="o">=</span> <span class="n">gen</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">bq</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-            <span class="k">if</span> <span class="n">op</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">bq</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                <span class="k">break</span>
-        <span class="k">if</span> <span class="n">op</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">nap</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">operators</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">op</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">nap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">operators</span>
-
-
-<span class="k">def</span> <span class="nf">reduce_order</span><span class="p">(</span><span class="n">ip</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Given the hypergeometric parameters ``ip.ap``, ``ip.bq``,</span>
-<span class="sd">    find a sequence of operators to reduces order as much as possible.</span>
-
-<span class="sd">    Return (nip, [operators]), where applying the operators to the</span>
-<span class="sd">    hypergeometric function specified by nip.ap, nip.bq yields ap, bq.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.hyperexpand import reduce_order, IndexPair</span>
-<span class="sd">    &gt;&gt;&gt; reduce_order(IndexPair((1, 2), (3, 4)))</span>
-<span class="sd">    (IndexPair((1, 2), (3, 4)), [])</span>
-<span class="sd">    &gt;&gt;&gt; reduce_order(IndexPair((1,), (1,)))</span>
-<span class="sd">    (IndexPair((), ()), [&lt;Reduce order by cancelling upper 1 with lower 1.&gt;])</span>
-<span class="sd">    &gt;&gt;&gt; reduce_order(IndexPair((2, 4), (3, 3)))</span>
-<span class="sd">    (IndexPair((2,), (3,)), [&lt;Reduce order by cancelling</span>
-<span class="sd">    upper 4 with lower 3.&gt;])</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">nap</span><span class="p">,</span> <span class="n">nbq</span><span class="p">,</span> <span class="n">operators</span> <span class="o">=</span> <span class="n">_reduce_order</span><span class="p">(</span><span class="n">ip</span><span class="o">.</span><span class="n">ap</span><span class="p">,</span> <span class="n">ip</span><span class="o">.</span><span class="n">bq</span><span class="p">,</span> <span class="n">ReduceOrder</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">IndexPair</span><span class="p">(</span><span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">nap</span><span class="p">),</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">nbq</span><span class="p">)),</span> <span class="n">operators</span>
-
-
-<span class="k">def</span> <span class="nf">reduce_order_meijer</span><span class="p">(</span><span class="n">iq</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Given the Meijer G function parameters, ``iq.am``, ``iq.ap``, ``iq.bm``,</span>
-<span class="sd">    ``iq.bq``, find a sequence of operators that reduces order as much as</span>
-<span class="sd">    possible.</span>
-
-<span class="sd">    Return niq, [operators].</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.hyperexpand import (reduce_order_meijer,</span>
-<span class="sd">    ...                                         IndexQuadruple)</span>
-<span class="sd">    &gt;&gt;&gt; reduce_order_meijer(IndexQuadruple([3, 4], [5, 6], [3, 4], [1, 2]))[0]</span>
-<span class="sd">    IndexQuadruple((4, 3), (5, 6), (3, 4), (2, 1))</span>
-<span class="sd">    &gt;&gt;&gt; reduce_order_meijer(IndexQuadruple([3, 4], [5, 6], [3, 4], [1, 8]))[0]</span>
-<span class="sd">    IndexQuadruple((3,), (5, 6), (3, 4), (1,))</span>
-<span class="sd">    &gt;&gt;&gt; reduce_order_meijer(IndexQuadruple([3, 4], [5, 6], [7, 5], [1, 5]))[0]</span>
-<span class="sd">    IndexQuadruple((3,), (), (), (1,))</span>
-<span class="sd">    &gt;&gt;&gt; reduce_order_meijer(IndexQuadruple([3, 4], [5, 6], [7, 5], [5, 3]))[0]</span>
-<span class="sd">    IndexQuadruple((), (), (), ())</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">nan</span><span class="p">,</span> <span class="n">nbq</span><span class="p">,</span> <span class="n">ops1</span> <span class="o">=</span> <span class="n">_reduce_order</span><span class="p">(</span><span class="n">iq</span><span class="o">.</span><span class="n">an</span><span class="p">,</span> <span class="n">iq</span><span class="o">.</span><span class="n">bq</span><span class="p">,</span> <span class="n">ReduceOrder</span><span class="o">.</span><span class="n">meijer_plus</span><span class="p">,</span>
-                                   <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="o">-</span><span class="n">x</span><span class="p">)</span>
-    <span class="n">nbm</span><span class="p">,</span> <span class="n">nap</span><span class="p">,</span> <span class="n">ops2</span> <span class="o">=</span> <span class="n">_reduce_order</span><span class="p">(</span><span class="n">iq</span><span class="o">.</span><span class="n">bm</span><span class="p">,</span> <span class="n">iq</span><span class="o">.</span><span class="n">ap</span><span class="p">,</span> <span class="n">ReduceOrder</span><span class="o">.</span><span class="n">meijer_minus</span><span class="p">,</span>
-                                   <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">IndexQuadruple</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">w</span><span class="p">)</span> <span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="p">[</span><span class="n">nan</span><span class="p">,</span> <span class="n">nap</span><span class="p">,</span> <span class="n">nbm</span><span class="p">,</span> <span class="n">nbq</span><span class="p">]]),</span> \
-        <span class="n">ops1</span> <span class="o">+</span> <span class="n">ops2</span>
-
-
-<span class="k">def</span> <span class="nf">make_derivative_operator</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Create a derivative operator, to be passed to Operator.apply. &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">poly</span>
-
-    <span class="k">def</span> <span class="nf">doit</span><span class="p">(</span><span class="n">C</span><span class="p">):</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">z</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="n">C</span><span class="o">*</span><span class="n">M</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">r</span><span class="o">.</span><span class="n">applyfunc</span><span class="p">(</span><span class="n">make_simp</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">r</span>
-    <span class="k">return</span> <span class="n">doit</span>
-
-
-<span class="k">def</span> <span class="nf">apply_operators</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="n">ops</span><span class="p">,</span> <span class="n">op</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Apply the list of operators ``ops`` to object ``obj``, substituting</span>
-<span class="sd">    ``op`` for the generator.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="n">obj</span>
-    <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">ops</span><span class="p">):</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">res</span><span class="p">,</span> <span class="n">op</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">res</span>
-
-
-<span class="k">def</span> <span class="nf">devise_plan</span><span class="p">(</span><span class="n">ip</span><span class="p">,</span> <span class="n">nip</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Devise a plan (consisting of shift and un-shift operators) to be applied</span>
-<span class="sd">    to the hypergeometric function (``nip.ap``, ``nip.bq``) to yield</span>
-<span class="sd">    (``ip.ap``, ``ip.bq``).</span>
-<span class="sd">    Returns a list of operators.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.hyperexpand import devise_plan, IndexPair</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import z</span>
-
-<span class="sd">    Nothing to do:</span>
-
-<span class="sd">    &gt;&gt;&gt; devise_plan(IndexPair((1, 2), ()), IndexPair((1, 2), ()), z)</span>
-<span class="sd">    []</span>
-<span class="sd">    &gt;&gt;&gt; devise_plan(IndexPair((), (1, 2)), IndexPair((), (1, 2)), z)</span>
-<span class="sd">    []</span>
-
-<span class="sd">    Very simple plans:</span>
-
-<span class="sd">    &gt;&gt;&gt; devise_plan(IndexPair((2,), ()), IndexPair((1,), ()), z)</span>
-<span class="sd">    [&lt;Increment upper 1.&gt;]</span>
-<span class="sd">    &gt;&gt;&gt; devise_plan(IndexPair((), (2,)), IndexPair((), (1,)), z)</span>
-<span class="sd">    [&lt;Increment lower index #0 of [], [1].&gt;]</span>
-
-<span class="sd">    Several buckets:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import S</span>
-<span class="sd">    &gt;&gt;&gt; devise_plan(IndexPair((1, S.Half), ()),</span>
-<span class="sd">    ...             IndexPair((2, S(&#39;3/2&#39;)), ()), z) #doctest: +NORMALIZE_WHITESPACE</span>
-<span class="sd">    [&lt;Decrement upper index #0 of [3/2, 1], [].&gt;,</span>
-<span class="sd">    &lt;Decrement upper index #0 of [2, 3/2], [].&gt;]</span>
-
-<span class="sd">    A slightly more complicated plan:</span>
-
-<span class="sd">    &gt;&gt;&gt; devise_plan(IndexPair((1, 3), ()), IndexPair((2, 2), ()), z)</span>
-<span class="sd">    [&lt;Increment upper 2.&gt;, &lt;Decrement upper index #0 of [2, 2], [].&gt;]</span>
-
-<span class="sd">    Another more complicated plan: (note that the ap have to be shifted first!)</span>
-
-<span class="sd">    &gt;&gt;&gt; devise_plan(IndexPair((1, -1), (2,)), IndexPair((3, -2), (4,)), z)</span>
-<span class="sd">    [&lt;Decrement lower 3.&gt;, &lt;Decrement lower 4.&gt;,</span>
-<span class="sd">    &lt;Decrement upper index #1 of [-1, 2], [4].&gt;,</span>
-<span class="sd">    &lt;Decrement upper index #1 of [-1, 3], [4].&gt;, &lt;Increment upper -2.&gt;]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">abuckets</span><span class="p">,</span> <span class="n">bbuckets</span> <span class="o">=</span> <span class="n">ip</span><span class="o">.</span><span class="n">compute_buckets</span><span class="p">()</span>
-    <span class="n">nabuckets</span><span class="p">,</span> <span class="n">nbbuckets</span> <span class="o">=</span> <span class="n">nip</span><span class="o">.</span><span class="n">compute_buckets</span><span class="p">(</span><span class="n">abuckets</span><span class="p">,</span> <span class="n">bbuckets</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">abuckets</span><span class="o">.</span><span class="n">keys</span><span class="p">()))</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">nabuckets</span><span class="o">.</span><span class="n">keys</span><span class="p">()))</span> <span class="ow">or</span> \
-            <span class="nb">len</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">bbuckets</span><span class="o">.</span><span class="n">keys</span><span class="p">()))</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">nbbuckets</span><span class="o">.</span><span class="n">keys</span><span class="p">())):</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> not reachable from </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">ip</span><span class="p">,</span> <span class="n">nip</span><span class="p">))</span>
-
-    <span class="n">ops</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">do_shifts</span><span class="p">(</span><span class="n">fro</span><span class="p">,</span> <span class="n">to</span><span class="p">,</span> <span class="n">inc</span><span class="p">,</span> <span class="n">dec</span><span class="p">):</span>
-        <span class="n">ops</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">fro</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="n">to</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">fro</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">sh</span> <span class="o">=</span> <span class="n">inc</span>
-                <span class="n">ch</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">sh</span> <span class="o">=</span> <span class="n">dec</span>
-                <span class="n">ch</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-
-            <span class="k">while</span> <span class="n">to</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">!=</span> <span class="n">fro</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-                <span class="n">ops</span> <span class="o">+=</span> <span class="p">[</span><span class="n">sh</span><span class="p">(</span><span class="n">fro</span><span class="p">,</span> <span class="n">i</span><span class="p">)]</span>
-                <span class="n">fro</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+=</span> <span class="n">ch</span>
-
-        <span class="k">return</span> <span class="n">ops</span>
-
-    <span class="k">def</span> <span class="nf">do_shifts_a</span><span class="p">(</span><span class="n">nal</span><span class="p">,</span> <span class="n">nbk</span><span class="p">,</span> <span class="n">al</span><span class="p">,</span> <span class="n">aother</span><span class="p">,</span> <span class="n">bother</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Shift us from (nal, nbk) to (al, nbk). &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">do_shifts</span><span class="p">(</span><span class="n">nal</span><span class="p">,</span> <span class="n">al</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">p</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="n">ShiftA</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="n">i</span><span class="p">]),</span>
-                         <span class="k">lambda</span> <span class="n">p</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="n">UnShiftA</span><span class="p">(</span><span class="n">p</span> <span class="o">+</span> <span class="n">aother</span><span class="p">,</span> <span class="n">nbk</span> <span class="o">+</span> <span class="n">bother</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">z</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">do_shifts_b</span><span class="p">(</span><span class="n">nal</span><span class="p">,</span> <span class="n">nbk</span><span class="p">,</span> <span class="n">bk</span><span class="p">,</span> <span class="n">aother</span><span class="p">,</span> <span class="n">bother</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Shift us from (nal, nbk) to (nal, bk). &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">do_shifts</span><span class="p">(</span><span class="n">nbk</span><span class="p">,</span> <span class="n">bk</span><span class="p">,</span>
-                         <span class="k">lambda</span> <span class="n">p</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="n">UnShiftB</span><span class="p">(</span><span class="n">nal</span> <span class="o">+</span> <span class="n">aother</span><span class="p">,</span> <span class="n">p</span> <span class="o">+</span> <span class="n">bother</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span>
-                         <span class="k">lambda</span> <span class="n">p</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="n">ShiftB</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span>
-
-    <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">abuckets</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">bbuckets</span><span class="o">.</span><span class="n">keys</span><span class="p">()),</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">):</span>
-        <span class="n">al</span> <span class="o">=</span> <span class="p">()</span>
-        <span class="n">nal</span> <span class="o">=</span> <span class="p">()</span>
-        <span class="n">bk</span> <span class="o">=</span> <span class="p">()</span>
-        <span class="n">nbk</span> <span class="o">=</span> <span class="p">()</span>
-        <span class="k">if</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">abuckets</span><span class="p">:</span>
-            <span class="n">al</span> <span class="o">=</span> <span class="n">abuckets</span><span class="p">[</span><span class="n">r</span><span class="p">]</span>
-            <span class="n">nal</span> <span class="o">=</span> <span class="n">nabuckets</span><span class="p">[</span><span class="n">r</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">bbuckets</span><span class="p">:</span>
-            <span class="n">bk</span> <span class="o">=</span> <span class="n">bbuckets</span><span class="p">[</span><span class="n">r</span><span class="p">]</span>
-            <span class="n">nbk</span> <span class="o">=</span> <span class="n">nbbuckets</span><span class="p">[</span><span class="n">r</span><span class="p">]</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">al</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">nal</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">bk</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">nbk</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> not reachable from </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span>
-                             <span class="p">(</span><span class="n">ip</span><span class="o">.</span><span class="n">ap</span><span class="p">,</span> <span class="n">ip</span><span class="o">.</span><span class="n">bq</span><span class="p">),</span> <span class="p">(</span><span class="n">nip</span><span class="o">.</span><span class="n">ap</span><span class="p">,</span> <span class="n">nip</span><span class="o">.</span><span class="n">bq</span><span class="p">)))</span>
-
-        <span class="n">al</span><span class="p">,</span> <span class="n">nal</span><span class="p">,</span> <span class="n">bk</span><span class="p">,</span> <span class="n">nbk</span> <span class="o">=</span> <span class="p">[</span><span class="nb">sorted</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">w</span><span class="p">),</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="p">[</span><span class="n">al</span><span class="p">,</span> <span class="n">nal</span><span class="p">,</span> <span class="n">bk</span><span class="p">,</span> <span class="n">nbk</span><span class="p">]]</span>
-
-        <span class="k">def</span> <span class="nf">others</span><span class="p">(</span><span class="n">dic</span><span class="p">,</span> <span class="n">key</span><span class="p">):</span>
-            <span class="n">l</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">value</span> <span class="ow">in</span> <span class="n">dic</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                <span class="k">if</span> <span class="n">k</span> <span class="o">!=</span> <span class="n">key</span><span class="p">:</span>
-                    <span class="n">l</span> <span class="o">+=</span> <span class="nb">list</span><span class="p">(</span><span class="n">dic</span><span class="p">[</span><span class="n">k</span><span class="p">])</span>
-            <span class="k">return</span> <span class="n">l</span>
-        <span class="n">aother</span> <span class="o">=</span> <span class="n">others</span><span class="p">(</span><span class="n">nabuckets</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span>
-        <span class="n">bother</span> <span class="o">=</span> <span class="n">others</span><span class="p">(</span><span class="n">nbbuckets</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">al</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="c"># there can be no complications, just shift the bs as we please</span>
-            <span class="n">ops</span> <span class="o">+=</span> <span class="n">do_shifts_b</span><span class="p">([],</span> <span class="n">nbk</span><span class="p">,</span> <span class="n">bk</span><span class="p">,</span> <span class="n">aother</span><span class="p">,</span> <span class="n">bother</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">bk</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="c"># there can be no complications, just shift the as as we please</span>
-            <span class="n">ops</span> <span class="o">+=</span> <span class="n">do_shifts_a</span><span class="p">(</span><span class="n">nal</span><span class="p">,</span> <span class="p">[],</span> <span class="n">al</span><span class="p">,</span> <span class="n">aother</span><span class="p">,</span> <span class="n">bother</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">namax</span> <span class="o">=</span> <span class="n">nal</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-            <span class="n">amax</span> <span class="o">=</span> <span class="n">al</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-
-            <span class="k">if</span> <span class="n">nbk</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&lt;=</span> <span class="n">namax</span> <span class="ow">or</span> <span class="n">bk</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&lt;=</span> <span class="n">amax</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Non-suitable parameters.&#39;</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">namax</span> <span class="o">&gt;</span> <span class="n">amax</span><span class="p">:</span>
-                <span class="c"># we are going to shift down - first do the as, then the bs</span>
-                <span class="n">ops</span> <span class="o">+=</span> <span class="n">do_shifts_a</span><span class="p">(</span><span class="n">nal</span><span class="p">,</span> <span class="n">nbk</span><span class="p">,</span> <span class="n">al</span><span class="p">,</span> <span class="n">aother</span><span class="p">,</span> <span class="n">bother</span><span class="p">)</span>
-                <span class="n">ops</span> <span class="o">+=</span> <span class="n">do_shifts_b</span><span class="p">(</span><span class="n">al</span><span class="p">,</span> <span class="n">nbk</span><span class="p">,</span> <span class="n">bk</span><span class="p">,</span> <span class="n">aother</span><span class="p">,</span> <span class="n">bother</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># we are going to shift up - first do the bs, then the as</span>
-                <span class="n">ops</span> <span class="o">+=</span> <span class="n">do_shifts_b</span><span class="p">(</span><span class="n">nal</span><span class="p">,</span> <span class="n">nbk</span><span class="p">,</span> <span class="n">bk</span><span class="p">,</span> <span class="n">aother</span><span class="p">,</span> <span class="n">bother</span><span class="p">)</span>
-                <span class="n">ops</span> <span class="o">+=</span> <span class="n">do_shifts_a</span><span class="p">(</span><span class="n">nal</span><span class="p">,</span> <span class="n">bk</span><span class="p">,</span> <span class="n">al</span><span class="p">,</span> <span class="n">aother</span><span class="p">,</span> <span class="n">bother</span><span class="p">)</span>
-
-        <span class="n">nabuckets</span><span class="p">[</span><span class="n">r</span><span class="p">]</span> <span class="o">=</span> <span class="n">al</span>
-        <span class="n">nbbuckets</span><span class="p">[</span><span class="n">r</span><span class="p">]</span> <span class="o">=</span> <span class="n">bk</span>
-
-    <span class="n">ops</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-    <span class="k">return</span> <span class="n">ops</span>
-
-
-<span class="k">def</span> <span class="nf">try_shifted_sum</span><span class="p">(</span><span class="n">ip</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Try to recognise a hypergeometric sum that starts from k &gt; 0. &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">rf</span><span class="p">,</span> <span class="n">factorial</span>
-    <span class="n">abuckets</span><span class="p">,</span> <span class="n">bbuckets</span> <span class="o">=</span> <span class="n">ip</span><span class="o">.</span><span class="n">compute_buckets</span><span class="p">()</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="ow">in</span> <span class="n">abuckets</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">abuckets</span><span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)])</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">abuckets</span><span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)][</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">r</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="ow">in</span> <span class="n">bbuckets</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-    <span class="n">l</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">bbuckets</span><span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)])</span>
-    <span class="n">l</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-    <span class="n">k</span> <span class="o">=</span> <span class="n">l</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">k</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="n">nap</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">ip</span><span class="o">.</span><span class="n">ap</span><span class="p">)</span>
-    <span class="n">nap</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-    <span class="n">nbq</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">ip</span><span class="o">.</span><span class="n">bq</span><span class="p">)</span>
-    <span class="n">nbq</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-    <span class="n">k</span> <span class="o">-=</span> <span class="mi">1</span>
-    <span class="n">nap</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="o">-</span> <span class="n">k</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">nap</span><span class="p">]</span>
-    <span class="n">nbq</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="o">-</span> <span class="n">k</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">nbq</span><span class="p">]</span>
-
-    <span class="n">ops</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">r</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">ops</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ShiftA</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-    <span class="n">ops</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-
-    <span class="n">fac</span> <span class="o">=</span> <span class="n">factorial</span><span class="p">(</span><span class="n">k</span><span class="p">)</span><span class="o">/</span><span class="n">z</span><span class="o">**</span><span class="n">k</span>
-    <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">nap</span><span class="p">:</span>
-        <span class="n">fac</span> <span class="o">/=</span> <span class="n">rf</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">nbq</span><span class="p">:</span>
-        <span class="n">fac</span> <span class="o">*=</span> <span class="n">rf</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-
-    <span class="n">ops</span> <span class="o">+=</span> <span class="p">[</span><span class="n">MultOperator</span><span class="p">(</span><span class="n">fac</span><span class="p">)]</span>
-
-    <span class="n">p</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">z</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">nap</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">*=</span> <span class="n">rf</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">nbq</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">/=</span> <span class="n">rf</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-        <span class="n">p</span> <span class="o">+=</span> <span class="n">m</span>
-
-    <span class="k">return</span> <span class="n">IndexPair</span><span class="p">(</span><span class="n">nap</span><span class="p">,</span> <span class="n">nbq</span><span class="p">),</span> <span class="n">ops</span><span class="p">,</span> <span class="o">-</span><span class="n">p</span>
-
-
-<span class="k">def</span> <span class="nf">try_polynomial</span><span class="p">(</span><span class="n">ip</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Recognise polynomial cases. Returns None if not such a case.</span>
-<span class="sd">        Requires order to be fully reduced. &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">oo</span><span class="p">,</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">rf</span><span class="p">,</span> <span class="n">Expr</span>
-    <span class="n">abuckets</span><span class="p">,</span> <span class="n">bbuckets</span> <span class="o">=</span> <span class="n">ip</span><span class="o">.</span><span class="n">compute_buckets</span><span class="p">()</span>
-    <span class="n">a0</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">abuckets</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="p">[]))</span>
-    <span class="n">b0</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">bbuckets</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="p">[]))</span>
-    <span class="n">a0</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-    <span class="n">b0</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-    <span class="n">al0</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">a0</span> <span class="k">if</span> <span class="n">x</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">]</span>
-    <span class="n">bl0</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">b0</span> <span class="k">if</span> <span class="n">x</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">if</span> <span class="n">bl0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">oo</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">al0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="n">a</span> <span class="o">=</span> <span class="n">al0</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-    <span class="n">fac</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="p">))):</span>
-        <span class="n">fac</span> <span class="o">*=</span> <span class="n">z</span>
-        <span class="n">fac</span> <span class="o">/=</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">ip</span><span class="o">.</span><span class="n">ap</span><span class="p">:</span>
-            <span class="n">fac</span> <span class="o">*=</span> <span class="n">a</span> <span class="o">+</span> <span class="n">n</span>
-        <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">ip</span><span class="o">.</span><span class="n">bq</span><span class="p">:</span>
-            <span class="n">fac</span> <span class="o">/=</span> <span class="n">b</span> <span class="o">+</span> <span class="n">n</span>
-        <span class="n">res</span> <span class="o">+=</span> <span class="n">fac</span>
-    <span class="k">return</span> <span class="n">res</span>
-
-
-<span class="k">def</span> <span class="nf">try_lerchphi</span><span class="p">(</span><span class="n">nip</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Try to find an expression for IndexPair ``nip`` in terms of Lerch</span>
-<span class="sd">    Transcendents.</span>
-
-<span class="sd">    Return None if no such expression can be found.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># This is actually quite simple, and is described in Roach&#39;s paper,</span>
-    <span class="c"># section 18.</span>
-    <span class="c"># We don&#39;t need to implement the reduction to polylog here, this</span>
-    <span class="c"># is handled by expand_func.</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="p">(</span><span class="n">expand_func</span><span class="p">,</span> <span class="n">lerchphi</span><span class="p">,</span> <span class="n">apart</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">,</span> <span class="n">rf</span><span class="p">,</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">,</span>
-                       <span class="n">zeros</span><span class="p">,</span> <span class="n">Add</span><span class="p">)</span>
-
-    <span class="c"># First we need to figure out if the summation coefficient is a rational</span>
-    <span class="c"># function of the summation index, and construct that rational function.</span>
-    <span class="n">abuckets</span><span class="p">,</span> <span class="n">bbuckets</span> <span class="o">=</span> <span class="n">nip</span><span class="o">.</span><span class="n">compute_buckets</span><span class="p">()</span>
-    <span class="c"># Update all the keys in bbuckets to be the same Mod objects as abuckets.</span>
-    <span class="n">akeys</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">abuckets</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-    <span class="n">bkeys</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">key</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">bbuckets</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
-        <span class="n">new</span> <span class="o">=</span> <span class="n">key</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">akeys</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="n">key</span><span class="p">:</span>
-                <span class="n">new</span> <span class="o">=</span> <span class="n">a</span>
-                <span class="k">break</span>
-        <span class="n">bkeys</span> <span class="o">+=</span> <span class="p">[(</span><span class="n">new</span><span class="p">,</span> <span class="n">key</span><span class="p">)]</span>
-    <span class="n">bb</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">for</span> <span class="n">new</span><span class="p">,</span> <span class="n">key</span> <span class="ow">in</span> <span class="n">bkeys</span><span class="p">:</span>
-        <span class="n">bb</span><span class="p">[</span><span class="n">new</span><span class="p">]</span> <span class="o">=</span> <span class="n">bbuckets</span><span class="p">[</span><span class="n">key</span><span class="p">]</span>
-    <span class="n">bbuckets</span> <span class="o">=</span> <span class="n">bb</span>
-
-    <span class="n">paired</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="n">value</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">abuckets</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-        <span class="k">if</span> <span class="n">key</span> <span class="o">!=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">key</span> <span class="ow">in</span> <span class="n">bbuckets</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="n">bvalue</span> <span class="o">=</span> <span class="n">bbuckets</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="p">[])</span>
-        <span class="n">paired</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">value</span><span class="p">),</span> <span class="nb">list</span><span class="p">(</span><span class="n">bvalue</span><span class="p">))</span>
-        <span class="n">bbuckets</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">bbuckets</span> <span class="o">!=</span> <span class="p">{}:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="ow">in</span> <span class="n">abuckets</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-    <span class="n">aints</span><span class="p">,</span> <span class="n">bints</span> <span class="o">=</span> <span class="n">paired</span><span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)]</span>
-    <span class="c"># Account for the additional n! in denominator</span>
-    <span class="n">paired</span><span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)]</span> <span class="o">=</span> <span class="p">(</span><span class="n">aints</span><span class="p">,</span> <span class="n">bints</span> <span class="o">+</span> <span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-
-    <span class="n">t</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">)</span>
-    <span class="n">numer</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-    <span class="n">denom</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="p">(</span><span class="n">avalue</span><span class="p">,</span> <span class="n">bvalue</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">paired</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">avalue</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">bvalue</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="c"># Note that since order has been reduced fully, all the b are</span>
-        <span class="c"># bigger than all the a they differ from by an integer. In particular</span>
-        <span class="c"># if there are any negative b left, this function is not well-defined.</span>
-        <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">avalue</span><span class="p">,</span> <span class="n">bvalue</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="o">&gt;</span> <span class="n">b</span><span class="p">:</span>
-                <span class="n">k</span> <span class="o">=</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span>
-                <span class="n">numer</span> <span class="o">*=</span> <span class="n">rf</span><span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">t</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-                <span class="n">denom</span> <span class="o">*=</span> <span class="n">rf</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">k</span> <span class="o">=</span> <span class="n">b</span> <span class="o">-</span> <span class="n">a</span>
-                <span class="n">numer</span> <span class="o">*=</span> <span class="n">rf</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-                <span class="n">denom</span> <span class="o">*=</span> <span class="n">rf</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">t</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-
-    <span class="c"># Now do a partial fraction decomposition.</span>
-    <span class="c"># We assemble two structures: a list monomials of pairs (a, b) representing</span>
-    <span class="c"># a*t**b (b a non-negative integer), and a dict terms, where</span>
-    <span class="c"># terms[a] = [(b, c)] means that there is a term b/(t-a)**c.</span>
-    <span class="n">part</span> <span class="o">=</span> <span class="n">apart</span><span class="p">(</span><span class="n">numer</span><span class="o">/</span><span class="n">denom</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
-    <span class="n">args</span> <span class="o">=</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">part</span><span class="p">)</span>
-    <span class="n">monomials</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">terms</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-        <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">denom</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">numer</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
-            <span class="k">assert</span> <span class="n">p</span><span class="o">.</span><span class="n">is_monomial</span>
-            <span class="p">((</span><span class="n">b</span><span class="p">,</span> <span class="p">),</span> <span class="n">a</span><span class="p">)</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">LT</span><span class="p">()</span>
-            <span class="n">monomials</span> <span class="o">+=</span> <span class="p">[(</span><span class="n">a</span><span class="o">/</span><span class="n">denom</span><span class="p">,</span> <span class="n">b</span><span class="p">)]</span>
-            <span class="k">continue</span>
-        <span class="k">if</span> <span class="n">numer</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;Need partial fraction decomposition&#39;</span>
-                                      <span class="s">&#39; with linear denominators&#39;</span><span class="p">)</span>
-        <span class="n">indep</span><span class="p">,</span> <span class="p">[</span><span class="n">dep</span><span class="p">]</span> <span class="o">=</span> <span class="n">denom</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">dep</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">dep</span><span class="o">.</span><span class="n">exp</span>
-            <span class="n">dep</span> <span class="o">=</span> <span class="n">dep</span><span class="o">.</span><span class="n">base</span>
-        <span class="k">if</span> <span class="n">dep</span> <span class="o">==</span> <span class="n">t</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">==</span> <span class="mi">0</span>
-        <span class="k">elif</span> <span class="n">dep</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">tmp</span> <span class="o">=</span> <span class="n">dep</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="n">tmp</span> <span class="o">!=</span> <span class="n">t</span><span class="p">:</span>
-                <span class="n">b</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">tmp</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">dep</span> <span class="o">!=</span> <span class="n">b</span><span class="o">*</span><span class="n">t</span> <span class="o">+</span> <span class="n">a</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;unrecognised form </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">dep</span><span class="p">)</span>
-            <span class="n">a</span> <span class="o">/=</span> <span class="n">b</span>
-            <span class="n">indep</span> <span class="o">*=</span> <span class="n">b</span><span class="o">**</span><span class="n">n</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;unrecognised form of partial fraction&#39;</span><span class="p">)</span>
-        <span class="n">terms</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="p">[])</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">numer</span><span class="o">/</span><span class="n">indep</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-
-    <span class="c"># Now that we have this information, assemble our formula. All the</span>
-    <span class="c"># monomials yield rational functions and go into one basis element.</span>
-    <span class="c"># The terms[a] are related by differentiation. If the largest exponent is</span>
-    <span class="c"># n, we need lerchphi(z, k, a) for k = 1, 2, ..., n.</span>
-    <span class="c"># deriv maps a basis to its derivative, expressed as a C(z)-linear</span>
-    <span class="c"># combination of other basis elements.</span>
-    <span class="n">deriv</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="n">coeffs</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="n">z</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;z&#39;</span><span class="p">)</span>
-    <span class="n">monomials</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-    <span class="n">mon</span> <span class="o">=</span> <span class="p">{</span><span class="mi">0</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">z</span><span class="p">)}</span>
-    <span class="k">if</span> <span class="n">monomials</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">monomials</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">]):</span>
-            <span class="n">mon</span><span class="p">[</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">z</span><span class="o">*</span><span class="n">mon</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">monomials</span><span class="p">:</span>
-        <span class="n">coeffs</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="p">[])</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">mon</span><span class="p">[</span><span class="n">n</span><span class="p">])</span>
-    <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">terms</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-        <span class="k">for</span> <span class="n">c</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">l</span><span class="p">:</span>
-            <span class="n">coeffs</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="n">lerchphi</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">a</span><span class="p">),</span> <span class="p">[])</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-        <span class="n">l</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">l</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">deriv</span><span class="p">[</span><span class="n">lerchphi</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">a</span><span class="p">)]</span> <span class="o">=</span> <span class="p">[(</span><span class="o">-</span><span class="n">a</span><span class="p">,</span> <span class="n">lerchphi</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">a</span><span class="p">)),</span>
-                                        <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">lerchphi</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="p">))]</span>
-        <span class="n">deriv</span><span class="p">[</span><span class="n">lerchphi</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="p">)]</span> <span class="o">=</span> <span class="p">[(</span><span class="o">-</span><span class="n">a</span><span class="p">,</span> <span class="n">lerchphi</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="p">)),</span>
-                                    <span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">z</span><span class="p">),</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">))]</span>
-    <span class="n">trans</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">([</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)]</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">deriv</span><span class="o">.</span><span class="n">keys</span><span class="p">())):</span>
-        <span class="n">trans</span><span class="p">[</span><span class="n">b</span><span class="p">]</span> <span class="o">=</span> <span class="n">n</span>
-    <span class="n">basis</span> <span class="o">=</span> <span class="p">[</span><span class="n">expand_func</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="k">for</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">_</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">trans</span><span class="o">.</span><span class="n">items</span><span class="p">()),</span>
-                                                 <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">])]</span>
-    <span class="n">B</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">basis</span><span class="p">)</span>
-    <span class="n">C</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="n">B</span><span class="p">)])</span>
-    <span class="k">for</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">coeffs</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-        <span class="n">C</span><span class="p">[</span><span class="n">trans</span><span class="p">[</span><span class="n">b</span><span class="p">]]</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">c</span><span class="p">)</span>
-    <span class="n">M</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">B</span><span class="p">))</span>
-    <span class="k">for</span> <span class="n">b</span><span class="p">,</span> <span class="n">l</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">deriv</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-        <span class="k">for</span> <span class="n">c</span><span class="p">,</span> <span class="n">b2</span> <span class="ow">in</span> <span class="n">l</span><span class="p">:</span>
-            <span class="n">M</span><span class="p">[</span><span class="n">trans</span><span class="p">[</span><span class="n">b</span><span class="p">],</span> <span class="n">trans</span><span class="p">[</span><span class="n">b2</span><span class="p">]]</span> <span class="o">=</span> <span class="n">c</span>
-    <span class="k">return</span> <span class="n">Formula</span><span class="p">(</span><span class="n">nip</span><span class="o">.</span><span class="n">ap</span><span class="p">,</span> <span class="n">nip</span><span class="o">.</span><span class="n">bq</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="p">[],</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">M</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">build_hypergeometric_formula</span><span class="p">(</span><span class="n">nip</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Create a formula object representing the hypergeometric function</span>
-<span class="sd">        Corresponding to nip. &quot;&quot;&quot;</span>
-    <span class="c"># We know that no `ap` are negative integers, otherwise &quot;detect poly&quot;</span>
-    <span class="c"># would have kicked in. However, `ap` could be empty. In this case we can</span>
-    <span class="c"># use a different basis.</span>
-    <span class="c"># I&#39;m not aware of a basis that works in all cases.</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">zeros</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">hyper</span><span class="p">,</span> <span class="n">eye</span><span class="p">,</span> <span class="n">Mul</span>
-    <span class="n">z</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;z&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">nip</span><span class="o">.</span><span class="n">ap</span><span class="p">:</span>
-        <span class="n">afactors</span> <span class="o">=</span> <span class="p">[</span><span class="n">_x</span> <span class="o">+</span> <span class="n">a</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">nip</span><span class="o">.</span><span class="n">ap</span><span class="p">]</span>
-        <span class="n">bfactors</span> <span class="o">=</span> <span class="p">[</span><span class="n">_x</span> <span class="o">+</span> <span class="n">b</span> <span class="o">-</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">nip</span><span class="o">.</span><span class="n">bq</span><span class="p">]</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">_x</span><span class="o">*</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">bfactors</span><span class="p">)</span> <span class="o">-</span> <span class="n">z</span><span class="o">*</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">afactors</span><span class="p">)</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span>
-        <span class="n">basis</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">M</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">nip</span><span class="o">.</span><span class="n">ap</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">k</span>
-            <span class="n">basis</span> <span class="o">+=</span> <span class="p">[</span><span class="n">hyper</span><span class="p">([</span><span class="n">a</span><span class="p">]</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">nip</span><span class="o">.</span><span class="n">ap</span><span class="p">[</span><span class="mi">1</span><span class="p">:]),</span> <span class="n">nip</span><span class="o">.</span><span class="n">bq</span><span class="p">,</span> <span class="n">z</span><span class="p">)]</span>
-            <span class="k">if</span> <span class="n">k</span> <span class="o">&lt;</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">M</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">a</span>
-                <span class="n">M</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">a</span>
-        <span class="n">B</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">basis</span><span class="p">)</span>
-        <span class="n">C</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)])</span>
-        <span class="n">derivs</span> <span class="o">=</span> <span class="p">[</span><span class="n">eye</span><span class="p">(</span><span class="n">n</span><span class="p">)]</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="n">derivs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">M</span><span class="o">*</span><span class="n">derivs</span><span class="p">[</span><span class="n">k</span><span class="p">])</span>
-        <span class="n">l</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()</span>
-        <span class="n">l</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">n</span>
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">l</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">r</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">C</span><span class="o">*</span><span class="n">derivs</span><span class="p">[</span><span class="n">k</span><span class="p">]):</span>
-                <span class="n">res</span><span class="p">[</span><span class="n">r</span><span class="p">]</span> <span class="o">+=</span> <span class="n">c</span><span class="o">*</span><span class="n">d</span>
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">res</span><span class="p">):</span>
-            <span class="n">M</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">c</span><span class="o">/</span><span class="n">derivs</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span><span class="o">/</span><span class="n">poly</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">Formula</span><span class="p">(</span><span class="n">nip</span><span class="o">.</span><span class="n">ap</span><span class="p">,</span> <span class="n">nip</span><span class="o">.</span><span class="n">bq</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="p">[],</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">M</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="c"># Since there are no `ap`, none of the `bq` can be non-positive</span>
-        <span class="c"># integers.</span>
-        <span class="n">basis</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">bq</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">nip</span><span class="o">.</span><span class="n">bq</span><span class="p">[:])</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">bq</span><span class="p">)):</span>
-            <span class="n">basis</span> <span class="o">+=</span> <span class="p">[</span><span class="n">hyper</span><span class="p">([],</span> <span class="n">bq</span><span class="p">,</span> <span class="n">z</span><span class="p">)]</span>
-            <span class="n">bq</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="n">basis</span> <span class="o">+=</span> <span class="p">[</span><span class="n">hyper</span><span class="p">([],</span> <span class="n">bq</span><span class="p">,</span> <span class="n">z</span><span class="p">)]</span>
-        <span class="n">B</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">basis</span><span class="p">)</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
-        <span class="n">C</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)])</span>
-        <span class="n">M</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-        <span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">z</span><span class="o">/</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">nip</span><span class="o">.</span><span class="n">bq</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-            <span class="n">M</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">nip</span><span class="o">.</span><span class="n">bq</span><span class="p">[</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-            <span class="n">M</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">nip</span><span class="o">.</span><span class="n">bq</span><span class="p">[</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">Formula</span><span class="p">(</span><span class="n">nip</span><span class="o">.</span><span class="n">ap</span><span class="p">,</span> <span class="n">nip</span><span class="o">.</span><span class="n">bq</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="p">[],</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">M</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">hyperexpand_special</span><span class="p">(</span><span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Try to find a closed-form expression for hyper(ap, bq, z), where ``z``</span>
-<span class="sd">    is supposed to be a &quot;special&quot; value, e.g. 1.</span>
-
-<span class="sd">    This function tries various of the classical summation formulae</span>
-<span class="sd">    (Gauss, Saalschuetz, etc).</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># This code is very ad-hoc. There are many clever algorithms</span>
-    <span class="c"># (notably Zeilberger&#39;s) related to this problem.</span>
-    <span class="c"># For now we just want a few simple cases to work.</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">unpolarify</span><span class="p">,</span> <span class="n">pi</span>
-    <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">ap</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">bq</span><span class="p">)</span>
-    <span class="n">z_</span> <span class="o">=</span> <span class="n">z</span>
-    <span class="n">z</span> <span class="o">=</span> <span class="n">unpolarify</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">z</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-    <span class="k">if</span> <span class="n">p</span> <span class="o">==</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">q</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="c"># 2F1</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">ap</span> <span class="o">+</span> <span class="n">bq</span>
-        <span class="k">if</span> <span class="n">z</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># Gauss</span>
-            <span class="k">return</span> <span class="n">gamma</span><span class="p">(</span><span class="n">c</span> <span class="o">-</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="n">c</span><span class="p">)</span><span class="o">/</span><span class="n">gamma</span><span class="p">(</span><span class="n">c</span> <span class="o">-</span> <span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="n">gamma</span><span class="p">(</span><span class="n">c</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">z</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span> <span class="ow">and</span> <span class="n">simplify</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="n">a</span> <span class="o">+</span> <span class="n">c</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">b</span><span class="p">,</span> <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span>
-        <span class="k">if</span> <span class="n">z</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span> <span class="ow">and</span> <span class="n">simplify</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># Kummer</span>
-            <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">and</span> <span class="n">b</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">b</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="o">-</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> \
-                    <span class="o">/</span><span class="n">gamma</span><span class="p">(</span><span class="o">-</span><span class="n">b</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">gamma</span><span class="p">(</span><span class="n">b</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">gamma</span><span class="p">(</span><span class="n">b</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> \
-                    <span class="o">/</span><span class="n">gamma</span><span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">gamma</span><span class="p">(</span><span class="n">b</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="c"># TODO tons of more formulae</span>
-    <span class="c">#      investigate what algorithms exist</span>
-    <span class="k">return</span> <span class="n">hyper</span><span class="p">(</span><span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">z_</span><span class="p">)</span>
-
-<span class="n">_collection</span> <span class="o">=</span> <span class="bp">None</span>
-
-
-<span class="nd">@_timeit</span>
-<span class="k">def</span> <span class="nf">_hyperexpand</span><span class="p">(</span><span class="n">ip</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">ops0</span><span class="o">=</span><span class="p">[],</span> <span class="n">z0</span><span class="o">=</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;z0&#39;</span><span class="p">),</span> <span class="n">premult</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">prem</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span>
-                 <span class="n">rewrite</span><span class="o">=</span><span class="s">&#39;default&#39;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Try to find an expression for the hypergeometric function</span>
-<span class="sd">    ``ip.ap``, ``ip.bq``.</span>
-
-<span class="sd">    The result is expressed in terms of a dummy variable z0. Then it</span>
-<span class="sd">    is multiplied by premult. Then ops0 is applied.</span>
-<span class="sd">    premult must be a*z**prem for some a independent of z.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">powdenest</span><span class="p">,</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">polarify</span><span class="p">,</span> <span class="n">unpolarify</span>
-    <span class="n">z</span> <span class="o">=</span> <span class="n">polarify</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">subs</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">rewrite</span> <span class="o">==</span> <span class="s">&#39;default&#39;</span><span class="p">:</span>
-        <span class="n">rewrite</span> <span class="o">=</span> <span class="s">&#39;nonrepsmall&#39;</span>
-
-    <span class="k">def</span> <span class="nf">carryout_plan</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">ops</span><span class="p">):</span>
-        <span class="n">C</span> <span class="o">=</span> <span class="n">apply_operators</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">C</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">),</span> <span class="n">ops</span><span class="p">,</span>
-                            <span class="n">make_derivative_operator</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">M</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">),</span> <span class="n">z0</span><span class="p">))</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">eye</span>
-        <span class="n">C</span> <span class="o">=</span> <span class="n">apply_operators</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">ops0</span><span class="p">,</span>
-                            <span class="n">make_derivative_operator</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">M</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">)</span>
-                                         <span class="o">+</span> <span class="n">prem</span><span class="o">*</span><span class="n">eye</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">M</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">z0</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">premult</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">C</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">applyfunc</span><span class="p">(</span><span class="n">make_simp</span><span class="p">(</span><span class="n">z0</span><span class="p">))</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">C</span><span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">B</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">)</span><span class="o">*</span><span class="n">premult</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="n">r</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">z0</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">rewrite</span><span class="p">:</span>
-            <span class="n">res</span> <span class="o">=</span> <span class="n">res</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">rewrite</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">res</span>
-
-    <span class="c"># TODO</span>
-    <span class="c"># The following would be possible:</span>
-    <span class="c"># *) PFD Duplication (see Kelly Roach&#39;s paper)</span>
-    <span class="c"># *) In a similar spirit, try_lerchphi() can be generalised considerably.</span>
-
-    <span class="k">global</span> <span class="n">_collection</span>
-    <span class="k">if</span> <span class="n">_collection</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">_collection</span> <span class="o">=</span> <span class="n">FormulaCollection</span><span class="p">()</span>
-
-    <span class="n">debug</span><span class="p">(</span><span class="s">&#39;Trying to expand hypergeometric function corresponding to&#39;</span><span class="p">,</span> <span class="n">ip</span><span class="p">)</span>
-
-    <span class="c"># First reduce order as much as possible.</span>
-    <span class="n">nip</span><span class="p">,</span> <span class="n">ops</span> <span class="o">=</span> <span class="n">reduce_order</span><span class="p">(</span><span class="n">ip</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">ops</span><span class="p">:</span>
-        <span class="n">debug</span><span class="p">(</span><span class="s">&#39;  Reduced order to&#39;</span><span class="p">,</span> <span class="n">nip</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">debug</span><span class="p">(</span><span class="s">&#39;  Could not reduce order.&#39;</span><span class="p">)</span>
-
-    <span class="c"># Now try polynomial cases</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="n">try_polynomial</span><span class="p">(</span><span class="n">nip</span><span class="p">,</span> <span class="n">z0</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">res</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">debug</span><span class="p">(</span><span class="s">&#39;  Recognised polynomial.&#39;</span><span class="p">)</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">apply_operators</span><span class="p">(</span><span class="n">res</span><span class="p">,</span> <span class="n">ops</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">f</span><span class="p">:</span> <span class="n">z0</span><span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">z0</span><span class="p">))</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">apply_operators</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">premult</span><span class="p">,</span> <span class="n">ops0</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">f</span><span class="p">:</span> <span class="n">z0</span><span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">z0</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">unpolarify</span><span class="p">(</span><span class="n">simplify</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">z0</span><span class="p">,</span> <span class="n">z</span><span class="p">))</span>
-
-    <span class="c"># Try to recognise a shifted sum.</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="n">try_shifted_sum</span><span class="p">(</span><span class="n">nip</span><span class="p">,</span> <span class="n">z0</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">res</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">nip</span><span class="p">,</span> <span class="n">nops</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="n">res</span>
-        <span class="n">debug</span><span class="p">(</span><span class="s">&#39;  Recognised shifted sum, reducerd order to&#39;</span><span class="p">,</span> <span class="n">nip</span><span class="p">)</span>
-        <span class="n">ops</span> <span class="o">+=</span> <span class="n">nops</span>
-
-    <span class="c"># apply the plan for poly</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="n">apply_operators</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">ops</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">f</span><span class="p">:</span> <span class="n">z0</span><span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">z0</span><span class="p">))</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="n">apply_operators</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">premult</span><span class="p">,</span> <span class="n">ops0</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">f</span><span class="p">:</span> <span class="n">z0</span><span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">z0</span><span class="p">))</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">z0</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-
-    <span class="c"># Try special expansions early.</span>
-    <span class="k">if</span> <span class="n">unpolarify</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="ow">and</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nip</span><span class="o">.</span><span class="n">ap</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">nip</span><span class="o">.</span><span class="n">bq</span><span class="p">))</span> <span class="o">==</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">build_hypergeometric_formula</span><span class="p">(</span><span class="n">nip</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">carryout_plan</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">ops</span><span class="p">)</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">hyper</span><span class="p">,</span> <span class="n">hyperexpand_special</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">hyper</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">r</span> <span class="o">+</span> <span class="n">p</span>
-
-    <span class="c"># Try to find a formula in our collection</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">_collection</span><span class="o">.</span><span class="n">lookup_origin</span><span class="p">(</span><span class="n">nip</span><span class="p">)</span>
-
-    <span class="c"># Now try a lerch phi formula</span>
-    <span class="k">if</span> <span class="n">f</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">try_lerchphi</span><span class="p">(</span><span class="n">nip</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">f</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">debug</span><span class="p">(</span><span class="s">&#39;  Could not find an origin.&#39;</span><span class="p">,</span>
-              <span class="s">&#39;Will return answer in terms of &#39;</span> <span class="o">+</span>
-              <span class="s">&#39;simpler hypergeometric functions.&#39;</span><span class="p">)</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">build_hypergeometric_formula</span><span class="p">(</span><span class="n">nip</span><span class="p">)</span>
-
-    <span class="n">debug</span><span class="p">(</span><span class="s">&#39;  Found an origin:&#39;</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">closed_form</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">indices</span><span class="p">)</span>
-
-    <span class="c"># We need to find the operators that convert f into (nap, nbq).</span>
-    <span class="n">ops</span> <span class="o">+=</span> <span class="n">devise_plan</span><span class="p">(</span><span class="n">nip</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">indices</span><span class="p">,</span> <span class="n">z0</span><span class="p">)</span>
-
-    <span class="c"># Now carry out the plan.</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">carryout_plan</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">ops</span><span class="p">)</span> <span class="o">+</span> <span class="n">p</span>
-
-    <span class="k">return</span> <span class="n">powdenest</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">polar</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">hyper</span><span class="p">,</span> <span class="n">hyperexpand_special</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">devise_plan_meijer</span><span class="p">(</span><span class="n">fro</span><span class="p">,</span> <span class="n">to</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Find a sequence of operators to convert index quadruple ``fro`` into</span>
-<span class="sd">    index quadruple ``to``. It is assumed that fro and to have the same</span>
-<span class="sd">    signatures, and that in fact any corresponding pair of parameters differs</span>
-<span class="sd">    by integers, and a direct path is possible. I.e. if there are parameters</span>
-<span class="sd">       a1 b1 c1  and a2 b2 c2</span>
-<span class="sd">    it is assumed that a1 can be shifted to a2, etc.</span>
-<span class="sd">    The only thing this routine determines is the order of shifts to apply,</span>
-<span class="sd">    nothing clever will be tried.</span>
-<span class="sd">    It is also assumed that fro is suitable.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.hyperexpand import (devise_plan_meijer,</span>
-<span class="sd">    ...                                         IndexQuadruple)</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import z</span>
-
-<span class="sd">    Empty plan:</span>
-
-<span class="sd">    &gt;&gt;&gt; devise_plan_meijer(IndexQuadruple([1], [2], [3], [4]),</span>
-<span class="sd">    ...                    IndexQuadruple([1], [2], [3], [4]), z)</span>
-<span class="sd">    []</span>
-
-<span class="sd">    Very simple plans:</span>
-
-<span class="sd">    &gt;&gt;&gt; devise_plan_meijer(IndexQuadruple([0], [], [], []),</span>
-<span class="sd">    ...                    IndexQuadruple([1], [], [], []), z)</span>
-<span class="sd">    [&lt;Increment upper a index #0 of [0], [], [], [].&gt;]</span>
-<span class="sd">    &gt;&gt;&gt; devise_plan_meijer(IndexQuadruple([0], [], [], []),</span>
-<span class="sd">    ...                    IndexQuadruple([-1], [], [], []), z)</span>
-<span class="sd">    [&lt;Decrement upper a=0.&gt;]</span>
-<span class="sd">    &gt;&gt;&gt; devise_plan_meijer(IndexQuadruple([], [1], [], []),</span>
-<span class="sd">    ...                    IndexQuadruple([], [2], [], []), z)</span>
-<span class="sd">    [&lt;Increment lower a index #0 of [], [1], [], [].&gt;]</span>
-
-<span class="sd">    Slightly more complicated plans:</span>
-
-<span class="sd">    &gt;&gt;&gt; devise_plan_meijer(IndexQuadruple([0], [], [], []),</span>
-<span class="sd">    ...                    IndexQuadruple([2], [], [], []), z)</span>
-<span class="sd">    [&lt;Increment upper a index #0 of [1], [], [], [].&gt;,</span>
-<span class="sd">    &lt;Increment upper a index #0 of [0], [], [], [].&gt;]</span>
-<span class="sd">    &gt;&gt;&gt; devise_plan_meijer(IndexQuadruple([0], [], [0], []),</span>
-<span class="sd">    ...                    IndexQuadruple([-1], [], [1], []), z)</span>
-<span class="sd">    [&lt;Increment upper b=0.&gt;, &lt;Decrement upper a=0.&gt;]</span>
-
-<span class="sd">    Order matters:</span>
-
-<span class="sd">    &gt;&gt;&gt; devise_plan_meijer(IndexQuadruple([0], [], [0], []),</span>
-<span class="sd">    ...                    IndexQuadruple([1], [], [1], []), z)</span>
-<span class="sd">    [&lt;Increment upper a index #0 of [0], [], [1], [].&gt;, &lt;Increment upper b=0.&gt;]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># TODO for now, we use the following simple heuristic: inverse-shift</span>
-    <span class="c">#      when possible, shift otherwise. Give up if we cannot make progress.</span>
-
-    <span class="k">def</span> <span class="nf">try_shift</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">shifter</span><span class="p">,</span> <span class="n">diff</span><span class="p">,</span> <span class="n">counter</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Try to apply ``shifter`` in order to bring some element in ``f``</span>
-<span class="sd">            nearer to its counterpart in ``to``. ``diff`` is +/- 1 and</span>
-<span class="sd">            determines the effect of ``shifter``. Counter is a list of elements</span>
-<span class="sd">            blocking the shift.</span>
-
-<span class="sd">            Return an operator if change was possible, else None.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">idx</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">t</span><span class="p">))):</span>
-            <span class="k">if</span> <span class="p">(</span>
-                <span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">and</span> <span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="n">diff</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="ow">and</span>
-                    <span class="nb">all</span><span class="p">(</span><span class="n">a</span> <span class="o">!=</span> <span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">counter</span><span class="p">)):</span>
-                <span class="n">sh</span> <span class="o">=</span> <span class="n">shifter</span><span class="p">(</span><span class="n">idx</span><span class="p">)</span>
-                <span class="n">f</span><span class="p">[</span><span class="n">idx</span><span class="p">]</span> <span class="o">+=</span> <span class="n">diff</span>
-                <span class="k">return</span> <span class="n">sh</span>
-    <span class="n">fan</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">fro</span><span class="o">.</span><span class="n">an</span><span class="p">)</span>
-    <span class="n">fap</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">fro</span><span class="o">.</span><span class="n">ap</span><span class="p">)</span>
-    <span class="n">fbm</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">fro</span><span class="o">.</span><span class="n">bm</span><span class="p">)</span>
-    <span class="n">fbq</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">fro</span><span class="o">.</span><span class="n">bq</span><span class="p">)</span>
-    <span class="n">ops</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">change</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="k">while</span> <span class="n">change</span><span class="p">:</span>
-        <span class="n">change</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="n">op</span> <span class="o">=</span> <span class="n">try_shift</span><span class="p">(</span><span class="n">fan</span><span class="p">,</span> <span class="n">to</span><span class="o">.</span><span class="n">an</span><span class="p">,</span>
-                       <span class="k">lambda</span> <span class="n">i</span><span class="p">:</span> <span class="n">MeijerUnShiftB</span><span class="p">(</span><span class="n">fan</span><span class="p">,</span> <span class="n">fap</span><span class="p">,</span> <span class="n">fbm</span><span class="p">,</span> <span class="n">fbq</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span>
-                       <span class="mi">1</span><span class="p">,</span> <span class="n">fbm</span> <span class="o">+</span> <span class="n">fbq</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">op</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">ops</span> <span class="o">+=</span> <span class="p">[</span><span class="n">op</span><span class="p">]</span>
-            <span class="n">change</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">continue</span>
-        <span class="n">op</span> <span class="o">=</span> <span class="n">try_shift</span><span class="p">(</span><span class="n">fap</span><span class="p">,</span> <span class="n">to</span><span class="o">.</span><span class="n">ap</span><span class="p">,</span>
-                       <span class="k">lambda</span> <span class="n">i</span><span class="p">:</span> <span class="n">MeijerUnShiftD</span><span class="p">(</span><span class="n">fan</span><span class="p">,</span> <span class="n">fap</span><span class="p">,</span> <span class="n">fbm</span><span class="p">,</span> <span class="n">fbq</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span>
-                       <span class="mi">1</span><span class="p">,</span> <span class="n">fbm</span> <span class="o">+</span> <span class="n">fbq</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">op</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">ops</span> <span class="o">+=</span> <span class="p">[</span><span class="n">op</span><span class="p">]</span>
-            <span class="n">change</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">continue</span>
-        <span class="n">op</span> <span class="o">=</span> <span class="n">try_shift</span><span class="p">(</span><span class="n">fbm</span><span class="p">,</span> <span class="n">to</span><span class="o">.</span><span class="n">bm</span><span class="p">,</span>
-                       <span class="k">lambda</span> <span class="n">i</span><span class="p">:</span> <span class="n">MeijerUnShiftA</span><span class="p">(</span><span class="n">fan</span><span class="p">,</span> <span class="n">fap</span><span class="p">,</span> <span class="n">fbm</span><span class="p">,</span> <span class="n">fbq</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span>
-                       <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">fan</span> <span class="o">+</span> <span class="n">fap</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">op</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">ops</span> <span class="o">+=</span> <span class="p">[</span><span class="n">op</span><span class="p">]</span>
-            <span class="n">change</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">continue</span>
-        <span class="n">op</span> <span class="o">=</span> <span class="n">try_shift</span><span class="p">(</span><span class="n">fbq</span><span class="p">,</span> <span class="n">to</span><span class="o">.</span><span class="n">bq</span><span class="p">,</span>
-                       <span class="k">lambda</span> <span class="n">i</span><span class="p">:</span> <span class="n">MeijerUnShiftC</span><span class="p">(</span><span class="n">fan</span><span class="p">,</span> <span class="n">fap</span><span class="p">,</span> <span class="n">fbm</span><span class="p">,</span> <span class="n">fbq</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span>
-                       <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">fan</span> <span class="o">+</span> <span class="n">fap</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">op</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">ops</span> <span class="o">+=</span> <span class="p">[</span><span class="n">op</span><span class="p">]</span>
-            <span class="n">change</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">continue</span>
-        <span class="n">op</span> <span class="o">=</span> <span class="n">try_shift</span><span class="p">(</span><span class="n">fan</span><span class="p">,</span> <span class="n">to</span><span class="o">.</span><span class="n">an</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">:</span> <span class="n">MeijerShiftB</span><span class="p">(</span><span class="n">fan</span><span class="p">[</span><span class="n">i</span><span class="p">]),</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">[])</span>
-        <span class="k">if</span> <span class="n">op</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">ops</span> <span class="o">+=</span> <span class="p">[</span><span class="n">op</span><span class="p">]</span>
-            <span class="n">change</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">continue</span>
-        <span class="n">op</span> <span class="o">=</span> <span class="n">try_shift</span><span class="p">(</span><span class="n">fap</span><span class="p">,</span> <span class="n">to</span><span class="o">.</span><span class="n">ap</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">:</span> <span class="n">MeijerShiftD</span><span class="p">(</span><span class="n">fap</span><span class="p">[</span><span class="n">i</span><span class="p">]),</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">[])</span>
-        <span class="k">if</span> <span class="n">op</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">ops</span> <span class="o">+=</span> <span class="p">[</span><span class="n">op</span><span class="p">]</span>
-            <span class="n">change</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">continue</span>
-        <span class="n">op</span> <span class="o">=</span> <span class="n">try_shift</span><span class="p">(</span><span class="n">fbm</span><span class="p">,</span> <span class="n">to</span><span class="o">.</span><span class="n">bm</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">:</span> <span class="n">MeijerShiftA</span><span class="p">(</span><span class="n">fbm</span><span class="p">[</span><span class="n">i</span><span class="p">]),</span> <span class="mi">1</span><span class="p">,</span> <span class="p">[])</span>
-        <span class="k">if</span> <span class="n">op</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">ops</span> <span class="o">+=</span> <span class="p">[</span><span class="n">op</span><span class="p">]</span>
-            <span class="n">change</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">continue</span>
-        <span class="n">op</span> <span class="o">=</span> <span class="n">try_shift</span><span class="p">(</span><span class="n">fbq</span><span class="p">,</span> <span class="n">to</span><span class="o">.</span><span class="n">bq</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">:</span> <span class="n">MeijerShiftC</span><span class="p">(</span><span class="n">fbq</span><span class="p">[</span><span class="n">i</span><span class="p">]),</span> <span class="mi">1</span><span class="p">,</span> <span class="p">[])</span>
-        <span class="k">if</span> <span class="n">op</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">ops</span> <span class="o">+=</span> <span class="p">[</span><span class="n">op</span><span class="p">]</span>
-            <span class="n">change</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">continue</span>
-    <span class="k">if</span> <span class="n">fan</span> <span class="o">!=</span> <span class="nb">list</span><span class="p">(</span><span class="n">to</span><span class="o">.</span><span class="n">an</span><span class="p">)</span> <span class="ow">or</span> <span class="n">fap</span> <span class="o">!=</span> <span class="nb">list</span><span class="p">(</span><span class="n">to</span><span class="o">.</span><span class="n">ap</span><span class="p">)</span> <span class="ow">or</span> <span class="n">fbm</span> <span class="o">!=</span> <span class="nb">list</span><span class="p">(</span><span class="n">to</span><span class="o">.</span><span class="n">bm</span><span class="p">)</span> <span class="ow">or</span> \
-            <span class="n">fbq</span> <span class="o">!=</span> <span class="nb">list</span><span class="p">(</span><span class="n">to</span><span class="o">.</span><span class="n">bq</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;Could not devise plan.&#39;</span><span class="p">)</span>
-    <span class="n">ops</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-    <span class="k">return</span> <span class="n">ops</span>
-
-<span class="n">_meijercollection</span> <span class="o">=</span> <span class="bp">None</span>
-
-
-<span class="nd">@_timeit</span>
-<span class="k">def</span> <span class="nf">_meijergexpand</span><span class="p">(</span><span class="n">iq</span><span class="p">,</span> <span class="n">z0</span><span class="p">,</span> <span class="n">allow_hyper</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">rewrite</span><span class="o">=</span><span class="s">&#39;default&#39;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Try to find an expression for the Meijer G function specified</span>
-<span class="sd">    by the IndexQuadruple ``iq``. If ``allow_hyper`` is True, then returning</span>
-<span class="sd">    an expression in terms of hypergeometric functions is allowed.</span>
-
-<span class="sd">    Currently this just does slater&#39;s theorem.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">hyper</span><span class="p">,</span> <span class="n">Piecewise</span><span class="p">,</span> <span class="n">meijerg</span><span class="p">,</span> <span class="n">powdenest</span><span class="p">,</span> <span class="n">re</span><span class="p">,</span> <span class="n">polar_lift</span><span class="p">,</span> <span class="n">oo</span>
-    <span class="k">global</span> <span class="n">_meijercollection</span>
-    <span class="k">if</span> <span class="n">_meijercollection</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">_meijercollection</span> <span class="o">=</span> <span class="n">MeijerFormulaCollection</span><span class="p">()</span>
-    <span class="k">if</span> <span class="n">rewrite</span> <span class="o">==</span> <span class="s">&#39;default&#39;</span><span class="p">:</span>
-        <span class="n">rewrite</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="n">iq_</span> <span class="o">=</span> <span class="n">iq</span>
-    <span class="n">debug</span><span class="p">(</span><span class="s">&#39;Try to expand meijer G function corresponding to&#39;</span><span class="p">,</span> <span class="n">iq</span><span class="p">)</span>
-
-    <span class="c"># We will play games with analytic continuation - rather use a fresh symbol</span>
-    <span class="n">z</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;z&#39;</span><span class="p">)</span>
-
-    <span class="n">iq</span><span class="p">,</span> <span class="n">ops</span> <span class="o">=</span> <span class="n">reduce_order_meijer</span><span class="p">(</span><span class="n">iq</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">ops</span><span class="p">:</span>
-        <span class="n">debug</span><span class="p">(</span><span class="s">&#39;  Reduced order to&#39;</span><span class="p">,</span> <span class="n">iq</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">debug</span><span class="p">(</span><span class="s">&#39;  Could not reduce order.&#39;</span><span class="p">)</span>
-
-    <span class="c"># Try to find a direct formula</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">_meijercollection</span><span class="o">.</span><span class="n">lookup_origin</span><span class="p">(</span><span class="n">iq</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">f</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">debug</span><span class="p">(</span><span class="s">&#39;  Found a Meijer G formula:&#39;</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">indices</span><span class="p">)</span>
-        <span class="n">ops</span> <span class="o">+=</span> <span class="n">devise_plan_meijer</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">indices</span><span class="p">,</span> <span class="n">iq</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-
-        <span class="c"># Now carry out the plan.</span>
-        <span class="n">C</span> <span class="o">=</span> <span class="n">apply_operators</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">C</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">z</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span> <span class="n">ops</span><span class="p">,</span>
-                            <span class="n">make_derivative_operator</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">M</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">z</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span> <span class="n">z</span><span class="p">))</span>
-
-        <span class="n">C</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">applyfunc</span><span class="p">(</span><span class="n">make_simp</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">C</span><span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">B</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">z</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">r</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">powdenest</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">polar</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="n">debug</span><span class="p">(</span><span class="s">&quot;  Could not find a direct formula. Trying slater&#39;s theorem.&quot;</span><span class="p">)</span>
-
-    <span class="c"># TODO the following would be possible:</span>
-    <span class="c"># *) Paired Index Theorems</span>
-    <span class="c"># *) PFD Duplication</span>
-    <span class="c">#    (See Kelly Roach&#39;s paper for details on either.)</span>
-    <span class="c">#</span>
-    <span class="c"># TODO Also, we tend to create combinations of gamma functions that can be</span>
-    <span class="c">#      simplified.</span>
-
-    <span class="k">def</span> <span class="nf">can_do</span><span class="p">(</span><span class="n">pbm</span><span class="p">,</span> <span class="n">pap</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Test if slater applies. &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">pbm</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">pbm</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">l</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="k">if</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">pap</span><span class="p">:</span>
-                    <span class="n">l</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">pap</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-                <span class="k">if</span> <span class="n">l</span> <span class="o">+</span> <span class="mi">1</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">pbm</span><span class="p">[</span><span class="n">i</span><span class="p">]):</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">do_slater</span><span class="p">(</span><span class="n">an</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">zfinal</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">residue</span><span class="p">,</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">rf</span><span class="p">,</span> <span class="n">expand_func</span><span class="p">,</span> \
-            <span class="n">polar_lift</span>
-        <span class="c"># zfinal is the value that will eventually be substituted for z.</span>
-        <span class="c"># We pass it to _hyperexpand to improve performance.</span>
-        <span class="n">iq</span> <span class="o">=</span> <span class="n">IndexQuadruple</span><span class="p">(</span><span class="n">an</span><span class="p">,</span> <span class="n">bm</span><span class="p">,</span> <span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">)</span>
-        <span class="n">_</span><span class="p">,</span> <span class="n">pbm</span><span class="p">,</span> <span class="n">pap</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">iq</span><span class="o">.</span><span class="n">compute_buckets</span><span class="p">()</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">can_do</span><span class="p">(</span><span class="n">pbm</span><span class="p">,</span> <span class="n">pap</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="bp">False</span>
-
-        <span class="n">cond</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">an</span><span class="p">)</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">ap</span><span class="p">)</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">bm</span><span class="p">)</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">bq</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">an</span><span class="p">)</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">ap</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">bm</span><span class="p">)</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">bq</span><span class="p">):</span>
-            <span class="n">cond</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">cond</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="bp">False</span>
-
-        <span class="n">res</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">pbm</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">pbm</span><span class="p">[</span><span class="n">m</span><span class="p">])</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">bh</span> <span class="o">=</span> <span class="n">pbm</span><span class="p">[</span><span class="n">m</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-                <span class="n">fac</span> <span class="o">=</span> <span class="mi">1</span>
-                <span class="n">bo</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">bm</span><span class="p">)</span>
-                <span class="n">bo</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">bh</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">bj</span> <span class="ow">in</span> <span class="n">bo</span><span class="p">:</span>
-                    <span class="n">fac</span> <span class="o">*=</span> <span class="n">gamma</span><span class="p">(</span><span class="n">bj</span> <span class="o">-</span> <span class="n">bh</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">aj</span> <span class="ow">in</span> <span class="n">an</span><span class="p">:</span>
-                    <span class="n">fac</span> <span class="o">*=</span> <span class="n">gamma</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">bh</span> <span class="o">-</span> <span class="n">aj</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">bj</span> <span class="ow">in</span> <span class="n">bq</span><span class="p">:</span>
-                    <span class="n">fac</span> <span class="o">/=</span> <span class="n">gamma</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">bh</span> <span class="o">-</span> <span class="n">bj</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">aj</span> <span class="ow">in</span> <span class="n">ap</span><span class="p">:</span>
-                    <span class="n">fac</span> <span class="o">/=</span> <span class="n">gamma</span><span class="p">(</span><span class="n">aj</span> <span class="o">-</span> <span class="n">bh</span><span class="p">)</span>
-                <span class="n">nap</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span> <span class="o">+</span> <span class="n">bh</span> <span class="o">-</span> <span class="n">a</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">an</span><span class="p">)</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">ap</span><span class="p">)]</span>
-                <span class="n">nbq</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span> <span class="o">+</span> <span class="n">bh</span> <span class="o">-</span> <span class="n">b</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">bo</span><span class="p">)</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">bq</span><span class="p">)]</span>
-
-                <span class="n">k</span> <span class="o">=</span> <span class="n">polar_lift</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">ap</span><span class="p">)</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">bm</span><span class="p">)))</span>
-                <span class="n">harg</span> <span class="o">=</span> <span class="n">k</span><span class="o">*</span><span class="n">zfinal</span>
-                <span class="c"># NOTE even though k &quot;is&quot; +-1, this has to be t/k instead of</span>
-                <span class="c">#      t*k ... we are using polar numbers for consistency!</span>
-                <span class="n">premult</span> <span class="o">=</span> <span class="p">(</span><span class="n">t</span><span class="o">/</span><span class="n">k</span><span class="p">)</span><span class="o">**</span><span class="n">bh</span>
-                <span class="n">hyp</span> <span class="o">=</span> <span class="n">_hyperexpand</span><span class="p">(</span><span class="n">IndexPair</span><span class="p">(</span><span class="n">nap</span><span class="p">,</span> <span class="n">nbq</span><span class="p">),</span> <span class="n">harg</span><span class="p">,</span> <span class="n">ops</span><span class="p">,</span>
-                                   <span class="n">t</span><span class="p">,</span> <span class="n">premult</span><span class="p">,</span> <span class="n">bh</span><span class="p">,</span> <span class="n">rewrite</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span>
-                <span class="n">res</span> <span class="o">+=</span> <span class="n">fac</span> <span class="o">*</span> <span class="n">hyp</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">b_</span> <span class="o">=</span> <span class="n">pbm</span><span class="p">[</span><span class="n">m</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-                <span class="n">ki</span> <span class="o">=</span> <span class="p">[</span><span class="n">bi</span> <span class="o">-</span> <span class="n">b_</span> <span class="k">for</span> <span class="n">bi</span> <span class="ow">in</span> <span class="n">pbm</span><span class="p">[</span><span class="n">m</span><span class="p">][</span><span class="mi">1</span><span class="p">:]]</span>
-                <span class="n">u</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">ki</span><span class="p">)</span>
-                <span class="n">li</span> <span class="o">=</span> <span class="p">[</span><span class="n">ai</span> <span class="o">-</span> <span class="n">b_</span> <span class="k">for</span> <span class="n">ai</span> <span class="ow">in</span> <span class="n">pap</span><span class="p">[</span><span class="n">m</span><span class="p">][:</span><span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]]</span>
-                <span class="n">bo</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">bm</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">pbm</span><span class="p">[</span><span class="n">m</span><span class="p">]:</span>
-                    <span class="n">bo</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-                <span class="n">ao</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">ap</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">pap</span><span class="p">[</span><span class="n">m</span><span class="p">][:</span><span class="n">u</span><span class="p">]:</span>
-                    <span class="n">ao</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-                <span class="n">lu</span> <span class="o">=</span> <span class="n">li</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-                <span class="n">di</span> <span class="o">=</span> <span class="p">[</span><span class="n">l</span> <span class="o">-</span> <span class="n">k</span> <span class="k">for</span> <span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">li</span><span class="p">,</span> <span class="n">ki</span><span class="p">)]</span>
-
-                <span class="c"># We first work out the integrand:</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;s&#39;</span><span class="p">)</span>
-                <span class="n">integrand</span> <span class="o">=</span> <span class="n">z</span><span class="o">**</span><span class="n">s</span>
-                <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">bm</span><span class="p">:</span>
-                    <span class="n">integrand</span> <span class="o">*=</span> <span class="n">gamma</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="n">s</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">an</span><span class="p">:</span>
-                    <span class="n">integrand</span> <span class="o">*=</span> <span class="n">gamma</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">a</span> <span class="o">+</span> <span class="n">s</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">bq</span><span class="p">:</span>
-                    <span class="n">integrand</span> <span class="o">/=</span> <span class="n">gamma</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">b</span> <span class="o">+</span> <span class="n">s</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">ap</span><span class="p">:</span>
-                    <span class="n">integrand</span> <span class="o">/=</span> <span class="n">gamma</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">s</span><span class="p">)</span>
-
-                <span class="c"># Now sum the finitely many residues:</span>
-                <span class="c"># XXX This speeds up some cases - is it a good idea?</span>
-                <span class="n">integrand</span> <span class="o">=</span> <span class="n">expand_func</span><span class="p">(</span><span class="n">integrand</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">lu</span><span class="p">):</span>
-                    <span class="n">resid</span> <span class="o">=</span> <span class="n">residue</span><span class="p">(</span><span class="n">integrand</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">b_</span> <span class="o">+</span> <span class="n">r</span><span class="p">)</span>
-                    <span class="n">resid</span> <span class="o">=</span> <span class="n">apply_operators</span><span class="p">(</span><span class="n">resid</span><span class="p">,</span> <span class="n">ops</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">f</span><span class="p">:</span> <span class="n">z</span><span class="o">*</span><span class="n">f</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-                    <span class="n">res</span> <span class="o">-=</span> <span class="n">resid</span>
-
-                <span class="c"># Now the hypergeometric term.</span>
-                <span class="n">au</span> <span class="o">=</span> <span class="n">b_</span> <span class="o">+</span> <span class="n">lu</span>
-                <span class="n">k</span> <span class="o">=</span> <span class="n">polar_lift</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">ao</span><span class="p">)</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">bo</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-                <span class="n">harg</span> <span class="o">=</span> <span class="n">k</span><span class="o">*</span><span class="n">zfinal</span>
-                <span class="n">premult</span> <span class="o">=</span> <span class="p">(</span><span class="n">t</span><span class="o">/</span><span class="n">k</span><span class="p">)</span><span class="o">**</span><span class="n">au</span>
-                <span class="n">nap</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span> <span class="o">+</span> <span class="n">au</span> <span class="o">-</span> <span class="n">a</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">an</span><span class="p">)</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">ap</span><span class="p">)]</span> <span class="o">+</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-                <span class="n">nbq</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span> <span class="o">+</span> <span class="n">au</span> <span class="o">-</span> <span class="n">b</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">bm</span><span class="p">)</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">bq</span><span class="p">)]</span>
-
-                <span class="n">hyp</span> <span class="o">=</span> <span class="n">_hyperexpand</span><span class="p">(</span><span class="n">IndexPair</span><span class="p">(</span><span class="n">nap</span><span class="p">,</span> <span class="n">nbq</span><span class="p">),</span> <span class="n">harg</span><span class="p">,</span> <span class="n">ops</span><span class="p">,</span>
-                                   <span class="n">t</span><span class="p">,</span> <span class="n">premult</span><span class="p">,</span> <span class="n">au</span><span class="p">,</span> <span class="n">rewrite</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span>
-
-                <span class="n">C</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">lu</span><span class="p">)</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span><span class="n">lu</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">u</span><span class="p">):</span>
-                    <span class="n">C</span> <span class="o">*=</span> <span class="n">S</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">di</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">/</span><span class="n">rf</span><span class="p">(</span><span class="n">lu</span> <span class="o">-</span> <span class="n">li</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">di</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-                <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">an</span><span class="p">:</span>
-                    <span class="n">C</span> <span class="o">*=</span> <span class="n">gamma</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">a</span> <span class="o">+</span> <span class="n">au</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">bo</span><span class="p">:</span>
-                    <span class="n">C</span> <span class="o">*=</span> <span class="n">gamma</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="n">au</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">ao</span><span class="p">:</span>
-                    <span class="n">C</span> <span class="o">/=</span> <span class="n">gamma</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">au</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">bq</span><span class="p">:</span>
-                    <span class="n">C</span> <span class="o">/=</span> <span class="n">gamma</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">b</span> <span class="o">+</span> <span class="n">au</span><span class="p">)</span>
-
-                <span class="n">res</span> <span class="o">+=</span> <span class="n">C</span><span class="o">*</span><span class="n">hyp</span>
-
-        <span class="k">return</span> <span class="n">res</span><span class="p">,</span> <span class="n">cond</span>
-
-    <span class="n">t</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">)</span>
-    <span class="n">slater1</span><span class="p">,</span> <span class="n">cond1</span> <span class="o">=</span> <span class="n">do_slater</span><span class="p">(</span><span class="n">iq</span><span class="o">.</span><span class="n">an</span><span class="p">,</span> <span class="n">iq</span><span class="o">.</span><span class="n">bm</span><span class="p">,</span> <span class="n">iq</span><span class="o">.</span><span class="n">ap</span><span class="p">,</span> <span class="n">iq</span><span class="o">.</span><span class="n">bq</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">tr</span><span class="p">(</span><span class="n">l</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">[</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">l</span><span class="p">]</span>
-
-    <span class="k">for</span> <span class="n">op</span> <span class="ow">in</span> <span class="n">ops</span><span class="p">:</span>
-        <span class="n">op</span><span class="o">.</span><span class="n">_poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">op</span><span class="o">.</span><span class="n">_poly</span><span class="o">.</span><span class="n">subs</span><span class="p">({</span><span class="n">z</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">t</span><span class="p">,</span> <span class="n">_x</span><span class="p">:</span> <span class="o">-</span><span class="n">_x</span><span class="p">}),</span> <span class="n">_x</span><span class="p">)</span>
-    <span class="n">slater2</span><span class="p">,</span> <span class="n">cond2</span> <span class="o">=</span> <span class="n">do_slater</span><span class="p">(</span><span class="n">tr</span><span class="p">(</span><span class="n">iq</span><span class="o">.</span><span class="n">bm</span><span class="p">),</span> <span class="n">tr</span><span class="p">(</span><span class="n">iq</span><span class="o">.</span><span class="n">an</span><span class="p">),</span> <span class="n">tr</span><span class="p">(</span><span class="n">iq</span><span class="o">.</span><span class="n">bq</span><span class="p">),</span> <span class="n">tr</span><span class="p">(</span><span class="n">iq</span><span class="o">.</span><span class="n">ap</span><span class="p">),</span>
-                               <span class="n">t</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">z0</span><span class="p">)</span>
-
-    <span class="n">slater1</span> <span class="o">=</span> <span class="n">powdenest</span><span class="p">(</span><span class="n">slater1</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">),</span> <span class="n">polar</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">slater2</span> <span class="o">=</span> <span class="n">powdenest</span><span class="p">(</span><span class="n">slater2</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">z0</span><span class="p">),</span> <span class="n">polar</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">cond2</span><span class="p">,</span> <span class="nb">bool</span><span class="p">):</span>
-        <span class="n">cond2</span> <span class="o">=</span> <span class="n">cond2</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">)</span>
-
-    <span class="n">m</span> <span class="o">=</span> <span class="n">meijerg</span><span class="p">(</span><span class="n">iq</span><span class="o">.</span><span class="n">an</span><span class="p">,</span> <span class="n">iq</span><span class="o">.</span><span class="n">ap</span><span class="p">,</span> <span class="n">iq</span><span class="o">.</span><span class="n">bm</span><span class="p">,</span> <span class="n">iq</span><span class="o">.</span><span class="n">bq</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">m</span><span class="o">.</span><span class="n">delta</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="ow">or</span> \
-        <span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">delta</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">ap</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">bq</span><span class="p">)</span> <span class="ow">and</span>
-            <span class="p">(</span><span class="n">re</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">nu</span><span class="p">)</span> <span class="o">&lt;</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span> <span class="ow">and</span> <span class="n">polar_lift</span><span class="p">(</span><span class="n">z0</span><span class="p">)</span> <span class="o">==</span> <span class="n">polar_lift</span><span class="p">(</span><span class="mi">1</span><span class="p">)):</span>
-        <span class="c"># The condition delta &gt; 0 means that the convergence region is</span>
-        <span class="c"># connected. Any expression we find can be continued analytically</span>
-        <span class="c"># to the entire convergence region.</span>
-        <span class="c"># The conditions delta==0, p==q, re(nu) &lt; -1 imply that G is continuous</span>
-        <span class="c"># on the positive reals, so the values at z=1 agree.</span>
-        <span class="k">if</span> <span class="n">cond1</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="n">cond1</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="n">cond2</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="n">cond2</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">if</span> <span class="n">cond1</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">slater1</span> <span class="o">=</span> <span class="n">slater1</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">rewrite</span> <span class="ow">or</span> <span class="s">&#39;nonrep&#39;</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">slater1</span> <span class="o">=</span> <span class="n">slater1</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">rewrite</span> <span class="ow">or</span> <span class="s">&#39;nonrepsmall&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">cond2</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">slater2</span> <span class="o">=</span> <span class="n">slater2</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">rewrite</span> <span class="ow">or</span> <span class="s">&#39;nonrep&#39;</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">slater2</span> <span class="o">=</span> <span class="n">slater2</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">rewrite</span> <span class="ow">or</span> <span class="s">&#39;nonrepsmall&#39;</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">cond1</span><span class="p">,</span> <span class="nb">bool</span><span class="p">):</span>
-        <span class="n">cond1</span> <span class="o">=</span> <span class="n">cond1</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">cond2</span><span class="p">,</span> <span class="nb">bool</span><span class="p">):</span>
-        <span class="n">cond2</span> <span class="o">=</span> <span class="n">cond2</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">weight</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">cond</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">oo</span><span class="p">,</span> <span class="n">zoo</span><span class="p">,</span> <span class="n">nan</span>
-        <span class="k">if</span> <span class="n">cond</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">c0</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">elif</span> <span class="n">cond</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="n">c0</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">c0</span> <span class="o">=</span> <span class="mi">2</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">oo</span><span class="p">,</span> <span class="n">zoo</span><span class="p">,</span> <span class="o">-</span><span class="n">oo</span><span class="p">,</span> <span class="n">nan</span><span class="p">):</span>
-            <span class="c"># XXX this actually should not happen, but consider</span>
-            <span class="c"># S(&#39;meijerg(((0, -1/2, 0, -1/2, 1/2), ()), ((0,),</span>
-            <span class="c">#   (-1/2, -1/2, -1/2, -1)), exp_polar(I*pi))/4&#39;)</span>
-            <span class="n">c0</span> <span class="o">=</span> <span class="mi">3</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">c0</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="n">hyper</span><span class="p">),</span> <span class="n">expr</span><span class="o">.</span><span class="n">count_ops</span><span class="p">())</span>
-
-    <span class="n">w1</span> <span class="o">=</span> <span class="n">weight</span><span class="p">(</span><span class="n">slater1</span><span class="p">,</span> <span class="n">cond1</span><span class="p">)</span>
-    <span class="n">w2</span> <span class="o">=</span> <span class="n">weight</span><span class="p">(</span><span class="n">slater2</span><span class="p">,</span> <span class="n">cond2</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">min</span><span class="p">(</span><span class="n">w1</span><span class="p">,</span> <span class="n">w2</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">oo</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">w1</span> <span class="o">&lt;</span> <span class="n">w2</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">slater1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">slater2</span>
-    <span class="k">if</span> <span class="nb">max</span><span class="p">(</span><span class="n">w1</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">w2</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">&lt;=</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">max</span><span class="p">(</span><span class="n">w1</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">w2</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">Piecewise</span><span class="p">((</span><span class="n">slater1</span><span class="p">,</span> <span class="n">cond1</span><span class="p">),</span> <span class="p">(</span><span class="n">slater2</span><span class="p">,</span> <span class="n">cond2</span><span class="p">),</span>
-                   <span class="p">(</span><span class="n">meijerg</span><span class="p">(</span><span class="n">iq_</span><span class="o">.</span><span class="n">an</span><span class="p">,</span> <span class="n">iq_</span><span class="o">.</span><span class="n">ap</span><span class="p">,</span> <span class="n">iq_</span><span class="o">.</span><span class="n">bm</span><span class="p">,</span> <span class="n">iq_</span><span class="o">.</span><span class="n">bq</span><span class="p">,</span> <span class="n">z0</span><span class="p">),</span> <span class="bp">True</span><span class="p">))</span>
-
-    <span class="c"># We couldn&#39;t find an expression without hypergeometric functions.</span>
-    <span class="c"># TODO it would be helpful to give conditions under which the integral</span>
-    <span class="c">#      is known to diverge.</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">Piecewise</span><span class="p">((</span><span class="n">slater1</span><span class="p">,</span> <span class="n">cond1</span><span class="p">),</span> <span class="p">(</span><span class="n">slater2</span><span class="p">,</span> <span class="n">cond2</span><span class="p">),</span>
-                  <span class="p">(</span><span class="n">meijerg</span><span class="p">(</span><span class="n">iq_</span><span class="o">.</span><span class="n">an</span><span class="p">,</span> <span class="n">iq_</span><span class="o">.</span><span class="n">ap</span><span class="p">,</span> <span class="n">iq_</span><span class="o">.</span><span class="n">bm</span><span class="p">,</span> <span class="n">iq_</span><span class="o">.</span><span class="n">bq</span><span class="p">,</span> <span class="n">z0</span><span class="p">),</span> <span class="bp">True</span><span class="p">))</span>
-    <span class="k">if</span> <span class="n">r</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">hyper</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">allow_hyper</span><span class="p">:</span>
-        <span class="n">debug</span><span class="p">(</span><span class="s">&#39;  Could express using hypergeometric functions, &#39;</span> <span class="o">+</span>
-              <span class="s">&#39;but not allowed.&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">hyper</span><span class="p">)</span> <span class="ow">or</span> <span class="n">allow_hyper</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">r</span>
-
-    <span class="k">return</span> <span class="n">meijerg</span><span class="p">(</span><span class="n">iq_</span><span class="o">.</span><span class="n">an</span><span class="p">,</span> <span class="n">iq_</span><span class="o">.</span><span class="n">ap</span><span class="p">,</span> <span class="n">iq_</span><span class="o">.</span><span class="n">bm</span><span class="p">,</span> <span class="n">iq_</span><span class="o">.</span><span class="n">bq</span><span class="p">,</span> <span class="n">z0</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="hyperexpand"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.hyperexpand.hyperexpand">[docs]</a><span class="k">def</span> <span class="nf">hyperexpand</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">allow_hyper</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">rewrite</span><span class="o">=</span><span class="s">&#39;default&#39;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Expand hypergeometric functions. If allow_hyper is True, allow partial</span>
-<span class="sd">    simplification (that is a result different from input,</span>
-<span class="sd">    but still containing hypergeometric functions).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.hyperexpand import hyperexpand</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.functions import hyper</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import z</span>
-<span class="sd">    &gt;&gt;&gt; hyperexpand(hyper([], [], z))</span>
-<span class="sd">    exp(z)</span>
-
-<span class="sd">    Non-hyperegeometric parts of the expression and hypergeometric expressions</span>
-<span class="sd">    that are not recognised are left unchanged:</span>
-
-<span class="sd">    &gt;&gt;&gt; hyperexpand(1 + hyper([1, 1, 1], [], z))</span>
-<span class="sd">    hyper((1, 1, 1), (), z) + 1</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">hyper</span><span class="p">,</span> <span class="n">meijerg</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">nan</span><span class="p">,</span> <span class="n">zoo</span><span class="p">,</span> <span class="n">oo</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">do_replace</span><span class="p">(</span><span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">_hyperexpand</span><span class="p">(</span><span class="n">IndexPair</span><span class="p">(</span><span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">),</span> <span class="n">z</span><span class="p">,</span> <span class="n">rewrite</span><span class="o">=</span><span class="n">rewrite</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">r</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">hyper</span><span class="p">(</span><span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">r</span>
-
-    <span class="k">def</span> <span class="nf">do_meijer</span><span class="p">(</span><span class="n">ap</span><span class="p">,</span> <span class="n">bq</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">_meijergexpand</span><span class="p">(</span><span class="n">IndexQuadruple</span><span class="p">(</span><span class="n">ap</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">ap</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">bq</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">bq</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="n">z</span><span class="p">,</span>
-                           <span class="n">allow_hyper</span><span class="p">,</span> <span class="n">rewrite</span><span class="o">=</span><span class="n">rewrite</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">nan</span><span class="p">,</span> <span class="n">zoo</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="o">-</span><span class="n">oo</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">r</span>
-    <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">hyper</span><span class="p">,</span> <span class="n">do_replace</span><span class="p">)</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">meijerg</span><span class="p">,</span> <span class="n">do_meijer</span><span class="p">)</span>
-</div>
-<span class="kn">from</span> <span class="nn">sympy.polys.polytools</span> <span class="kn">import</span> <span class="n">Poly</span>
-</pre></div>
-
-          </div>
-        </div>
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diff --git a/dev-py3k/_modules/sympy/simplify/simplify.html b/dev-py3k/_modules/sympy/simplify/simplify.html
deleted file mode 100644
index 25cc9fc4123..00000000000
--- a/dev-py3k/_modules/sympy/simplify/simplify.html
+++ /dev/null
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-            
-  <h1>Source code for sympy.simplify.simplify</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">SYMPY_DEBUG</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="p">(</span><span class="n">Basic</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Pow</span><span class="p">,</span> <span class="n">Rational</span><span class="p">,</span> <span class="n">Integer</span><span class="p">,</span>
-    <span class="n">Derivative</span><span class="p">,</span> <span class="n">Wild</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">expand</span><span class="p">,</span> <span class="n">expand_mul</span><span class="p">,</span> <span class="n">expand_func</span><span class="p">,</span>
-    <span class="n">Function</span><span class="p">,</span> <span class="n">Equality</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">,</span> <span class="n">Atom</span><span class="p">,</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">factor_terms</span><span class="p">,</span>
-    <span class="n">expand_multinomial</span><span class="p">,</span> <span class="n">expand_power_base</span><span class="p">,</span> <span class="n">symbols</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">iterable</span><span class="p">,</span> <span class="nb">reduce</span><span class="p">,</span> <span class="n">default_sort_key</span>
-<span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">Float</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">expand_log</span><span class="p">,</span> <span class="n">count_ops</span>
-<span class="kn">from</span> <span class="nn">sympy.core.mul</span> <span class="kn">import</span> <span class="n">_keep_coeff</span><span class="p">,</span> <span class="n">prod</span>
-<span class="kn">from</span> <span class="nn">sympy.core.rules</span> <span class="kn">import</span> <span class="n">Transform</span>
-
-<span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">root</span><span class="p">,</span> <span class="n">exp_polar</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">flatten</span><span class="p">,</span> <span class="n">has_variety</span>
-
-<span class="kn">from</span> <span class="nn">sympy.simplify.cse_main</span> <span class="kn">import</span> <span class="n">cse</span>
-<span class="kn">from</span> <span class="nn">sympy.simplify.cse_opts</span> <span class="kn">import</span> <span class="n">sub_pre</span><span class="p">,</span> <span class="n">sub_post</span>
-<span class="kn">from</span> <span class="nn">sympy.simplify.sqrtdenest</span> <span class="kn">import</span> <span class="n">sqrtdenest</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="p">(</span><span class="n">Poly</span><span class="p">,</span> <span class="n">together</span><span class="p">,</span> <span class="n">reduced</span><span class="p">,</span> <span class="n">cancel</span><span class="p">,</span> <span class="n">factor</span><span class="p">,</span>
-    <span class="n">ComputationFailed</span><span class="p">,</span> <span class="n">lcm</span><span class="p">,</span> <span class="n">gcd</span><span class="p">)</span>
-
-<span class="kn">import</span> <span class="nn">sympy.mpmath</span> <span class="kn">as</span> <span class="nn">mpmath</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-
-<span class="k">def</span> <span class="nf">_mexpand</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">expand_multinomial</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-
-
-<div class="viewcode-block" id="fraction"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.simplify.fraction">[docs]</a><span class="k">def</span> <span class="nf">fraction</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns a pair with expression&#39;s numerator and denominator.</span>
-<span class="sd">       If the given expression is not a fraction then this function</span>
-<span class="sd">       will return the tuple (expr, 1).</span>
-
-<span class="sd">       This function will not make any attempt to simplify nested</span>
-<span class="sd">       fractions or to do any term rewriting at all.</span>
-
-<span class="sd">       If only one of the numerator/denominator pair is needed then</span>
-<span class="sd">       use numer(expr) or denom(expr) functions respectively.</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy import fraction, Rational, Symbol</span>
-<span class="sd">       &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">       &gt;&gt;&gt; fraction(x/y)</span>
-<span class="sd">       (x, y)</span>
-<span class="sd">       &gt;&gt;&gt; fraction(x)</span>
-<span class="sd">       (x, 1)</span>
-
-<span class="sd">       &gt;&gt;&gt; fraction(1/y**2)</span>
-<span class="sd">       (1, y**2)</span>
-
-<span class="sd">       &gt;&gt;&gt; fraction(x*y/2)</span>
-<span class="sd">       (x*y, 2)</span>
-<span class="sd">       &gt;&gt;&gt; fraction(Rational(1, 2))</span>
-<span class="sd">       (1, 2)</span>
-
-<span class="sd">       This function will also work fine with assumptions:</span>
-
-<span class="sd">       &gt;&gt;&gt; k = Symbol(&#39;k&#39;, negative=True)</span>
-<span class="sd">       &gt;&gt;&gt; fraction(x * y**k)</span>
-<span class="sd">       (x, y**(-k))</span>
-
-<span class="sd">       If we know nothing about sign of some exponent and &#39;exact&#39;</span>
-<span class="sd">       flag is unset, then structure this exponent&#39;s structure will</span>
-<span class="sd">       be analyzed and pretty fraction will be returned:</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy import exp</span>
-<span class="sd">       &gt;&gt;&gt; fraction(2*x**(-y))</span>
-<span class="sd">       (2, x**y)</span>
-
-<span class="sd">       &gt;&gt;&gt; fraction(exp(-x))</span>
-<span class="sd">       (1, exp(x))</span>
-
-<span class="sd">       &gt;&gt;&gt; fraction(exp(-x), exact=True)</span>
-<span class="sd">       (exp(-x), 1)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">term</span><span class="o">.</span><span class="n">is_commutative</span> <span class="ow">and</span> <span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">or</span> <span class="n">term</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">exp</span><span class="p">):</span>
-            <span class="n">b</span><span class="p">,</span> <span class="n">ex</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">ex</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">ex</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">:</span>
-                    <span class="n">denom</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">denom</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="o">-</span><span class="n">ex</span><span class="p">))</span>
-            <span class="k">elif</span> <span class="n">ex</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                <span class="n">numer</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="ow">not</span> <span class="n">exact</span> <span class="ow">and</span> <span class="n">ex</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-                <span class="n">numer</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-                <span class="n">denom</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">numer</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">term</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-            <span class="n">numer</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-            <span class="n">denom</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">numer</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">numer</span><span class="p">),</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">denom</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">numer</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">fraction</span><span class="p">(</span><span class="n">expr</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">denom</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">fraction</span><span class="p">(</span><span class="n">expr</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">fraction_expand</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">frac</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">numer_expand</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">fraction</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">a</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">numer</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span> <span class="o">/</span> <span class="n">b</span>
-
-
-<span class="k">def</span> <span class="nf">denom_expand</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">fraction</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">a</span> <span class="o">/</span> <span class="n">b</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">denom</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-<span class="n">expand_numer</span> <span class="o">=</span> <span class="n">numer_expand</span>
-<span class="n">expand_denom</span> <span class="o">=</span> <span class="n">denom_expand</span>
-<span class="n">expand_fraction</span> <span class="o">=</span> <span class="n">fraction_expand</span>
-
-
-<div class="viewcode-block" id="separate"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.simplify.separate">[docs]</a><span class="k">def</span> <span class="nf">separate</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Deprecated wrapper around ``expand_power_base()``.  Use that function instead.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.utilities.exceptions</span> <span class="kn">import</span> <span class="n">SymPyDeprecationWarning</span>
-    <span class="n">SymPyDeprecationWarning</span><span class="p">(</span>
-        <span class="n">feature</span><span class="o">=</span><span class="s">&quot;separate()&quot;</span><span class="p">,</span> <span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;expand_power_base()&quot;</span><span class="p">,</span> <span class="n">issue</span><span class="o">=</span><span class="mi">3383</span><span class="p">,</span>
-        <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.2&quot;</span><span class="p">,</span> <span class="n">value</span><span class="o">=</span><span class="s">&quot;Note: in separate() deep &quot;</span>
-        <span class="s">&quot;defaults to False, whereas in expand_power_base(), deep defaults to True.&quot;</span><span class="p">,</span>
-    <span class="p">)</span><span class="o">.</span><span class="n">warn</span><span class="p">()</span>
-    <span class="k">return</span> <span class="n">expand_power_base</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">),</span> <span class="n">deep</span><span class="o">=</span><span class="n">deep</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="n">force</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="collect"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.simplify.collect">[docs]</a><span class="k">def</span> <span class="nf">collect</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">syms</span><span class="p">,</span> <span class="n">func</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">distribute_order_term</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Collect additive terms of an expression.</span>
-
-<span class="sd">    This function collects additive terms of an expression with respect</span>
-<span class="sd">    to a list of expression up to powers with rational exponents. By the</span>
-<span class="sd">    term symbol here are meant arbitrary expressions, which can contain</span>
-<span class="sd">    powers, products, sums etc. In other words symbol is a pattern which</span>
-<span class="sd">    will be searched for in the expression&#39;s terms.</span>
-
-<span class="sd">    The input expression is not expanded by :func:`collect`, so user is</span>
-<span class="sd">    expected to provide an expression is an appropriate form. This makes</span>
-<span class="sd">    :func:`collect` more predictable as there is no magic happening behind the</span>
-<span class="sd">    scenes. However, it is important to note, that powers of products are</span>
-<span class="sd">    converted to products of powers using the :func:`expand_power_base`</span>
-<span class="sd">    function.</span>
-
-<span class="sd">    There are two possible types of output. First, if ``evaluate`` flag is</span>
-<span class="sd">    set, this function will return an expression with collected terms or</span>
-<span class="sd">    else it will return a dictionary with expressions up to rational powers</span>
-<span class="sd">    as keys and collected coefficients as values.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import S, collect, expand, factor, Wild</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import a, b, c, x, y, z</span>
-
-<span class="sd">    This function can collect symbolic coefficients in polynomials or</span>
-<span class="sd">    rational expressions. It will manage to find all integer or rational</span>
-<span class="sd">    powers of collection variable::</span>
-
-<span class="sd">        &gt;&gt;&gt; collect(a*x**2 + b*x**2 + a*x - b*x + c, x)</span>
-<span class="sd">        c + x**2*(a + b) + x*(a - b)</span>
-
-<span class="sd">    The same result can be achieved in dictionary form::</span>
-
-<span class="sd">        &gt;&gt;&gt; d = collect(a*x**2 + b*x**2 + a*x - b*x + c, x, evaluate=False)</span>
-<span class="sd">        &gt;&gt;&gt; d[x**2]</span>
-<span class="sd">        a + b</span>
-<span class="sd">        &gt;&gt;&gt; d[x]</span>
-<span class="sd">        a - b</span>
-<span class="sd">        &gt;&gt;&gt; d[S.One]</span>
-<span class="sd">        c</span>
-
-<span class="sd">    You can also work with multivariate polynomials. However, remember that</span>
-<span class="sd">    this function is greedy so it will care only about a single symbol at time,</span>
-<span class="sd">    in specification order::</span>
-
-<span class="sd">        &gt;&gt;&gt; collect(x**2 + y*x**2 + x*y + y + a*y, [x, y])</span>
-<span class="sd">        x**2*(y + 1) + x*y + y*(a + 1)</span>
-
-<span class="sd">    Also more complicated expressions can be used as patterns::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sin, log</span>
-<span class="sd">        &gt;&gt;&gt; collect(a*sin(2*x) + b*sin(2*x), sin(2*x))</span>
-<span class="sd">        (a + b)*sin(2*x)</span>
-
-<span class="sd">        &gt;&gt;&gt; collect(a*x*log(x) + b*(x*log(x)), x*log(x))</span>
-<span class="sd">        x*(a + b)*log(x)</span>
-
-<span class="sd">    You can use wildcards in the pattern::</span>
-
-<span class="sd">        &gt;&gt;&gt; w = Wild(&#39;w1&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; collect(a*x**y - b*x**y, w**y)</span>
-<span class="sd">        x**y*(a - b)</span>
-
-<span class="sd">    It is also possible to work with symbolic powers, although it has more</span>
-<span class="sd">    complicated behavior, because in this case power&#39;s base and symbolic part</span>
-<span class="sd">    of the exponent are treated as a single symbol::</span>
-
-<span class="sd">        &gt;&gt;&gt; collect(a*x**c + b*x**c, x)</span>
-<span class="sd">        a*x**c + b*x**c</span>
-<span class="sd">        &gt;&gt;&gt; collect(a*x**c + b*x**c, x**c)</span>
-<span class="sd">        x**c*(a + b)</span>
-
-<span class="sd">    However if you incorporate rationals to the exponents, then you will get</span>
-<span class="sd">    well known behavior::</span>
-
-<span class="sd">        &gt;&gt;&gt; collect(a*x**(2*c) + b*x**(2*c), x**c)</span>
-<span class="sd">        x**(2*c)*(a + b)</span>
-
-<span class="sd">    Note also that all previously stated facts about :func:`collect` function</span>
-<span class="sd">    apply to the exponential function, so you can get::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import exp</span>
-<span class="sd">        &gt;&gt;&gt; collect(a*exp(2*x) + b*exp(2*x), exp(x))</span>
-<span class="sd">        (a + b)*exp(2*x)</span>
-
-<span class="sd">    If you are interested only in collecting specific powers of some symbols</span>
-<span class="sd">    then set ``exact`` flag in arguments::</span>
-
-<span class="sd">        &gt;&gt;&gt; collect(a*x**7 + b*x**7, x, exact=True)</span>
-<span class="sd">        a*x**7 + b*x**7</span>
-<span class="sd">        &gt;&gt;&gt; collect(a*x**7 + b*x**7, x**7, exact=True)</span>
-<span class="sd">        x**7*(a + b)</span>
-
-<span class="sd">    You can also apply this function to differential equations, where derivatives</span>
-<span class="sd">    of arbitrary order can be collected. Note that if you collect with respect</span>
-<span class="sd">    to a function or a derivative of a function, all derivatives of that function</span>
-<span class="sd">    will also be collected. Use ``exact=True`` to prevent this from happening::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Derivative as D, collect, Function</span>
-<span class="sd">        &gt;&gt;&gt; f = Function(&#39;f&#39;) (x)</span>
-
-<span class="sd">        &gt;&gt;&gt; collect(a*D(f,x) + b*D(f,x), D(f,x))</span>
-<span class="sd">        (a + b)*Derivative(f(x), x)</span>
-
-<span class="sd">        &gt;&gt;&gt; collect(a*D(D(f,x),x) + b*D(D(f,x),x), f)</span>
-<span class="sd">        (a + b)*Derivative(f(x), x, x)</span>
-
-<span class="sd">        &gt;&gt;&gt; collect(a*D(D(f,x),x) + b*D(D(f,x),x), D(f,x), exact=True)</span>
-<span class="sd">        a*Derivative(f(x), x, x) + b*Derivative(f(x), x, x)</span>
-
-<span class="sd">        &gt;&gt;&gt; collect(a*D(f,x) + b*D(f,x) + a*f + b*f, f)</span>
-<span class="sd">        (a + b)*f(x) + (a + b)*Derivative(f(x), x)</span>
-
-<span class="sd">    Or you can even match both derivative order and exponent at the same time::</span>
-
-<span class="sd">        &gt;&gt;&gt; collect(a*D(D(f,x),x)**2 + b*D(D(f,x),x)**2, D(f,x))</span>
-<span class="sd">        (a + b)*Derivative(f(x), x, x)**2</span>
-
-<span class="sd">    Finally, you can apply a function to each of the collected coefficients.</span>
-<span class="sd">    For example you can factorize symbolic coefficients of polynomial::</span>
-
-<span class="sd">        &gt;&gt;&gt; f = expand((x + a + 1)**3)</span>
-
-<span class="sd">        &gt;&gt;&gt; collect(f, x, factor)</span>
-<span class="sd">        x**3 + 3*x**2*(a + 1) + 3*x*(a + 1)**2 + (a + 1)**3</span>
-
-<span class="sd">    .. note:: Arguments are expected to be in expanded form, so you might have</span>
-<span class="sd">              to call :func:`expand` prior to calling this function.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">make_expression</span><span class="p">(</span><span class="n">terms</span><span class="p">):</span>
-        <span class="n">product</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">term</span><span class="p">,</span> <span class="n">rat</span><span class="p">,</span> <span class="n">sym</span><span class="p">,</span> <span class="n">deriv</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">deriv</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">var</span><span class="p">,</span> <span class="n">order</span> <span class="o">=</span> <span class="n">deriv</span>
-
-                <span class="k">while</span> <span class="n">order</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">term</span><span class="p">,</span> <span class="n">order</span> <span class="o">=</span> <span class="n">Derivative</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">var</span><span class="p">),</span> <span class="n">order</span> <span class="o">-</span> <span class="mi">1</span>
-
-            <span class="k">if</span> <span class="n">sym</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">rat</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                    <span class="n">product</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">product</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Pow</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">rat</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">product</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Pow</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">rat</span><span class="o">*</span><span class="n">sym</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">product</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">parse_derivative</span><span class="p">(</span><span class="n">deriv</span><span class="p">):</span>
-        <span class="c"># scan derivatives tower in the input expression and return</span>
-        <span class="c"># underlying function and maximal differentiation order</span>
-        <span class="n">expr</span><span class="p">,</span> <span class="n">sym</span><span class="p">,</span> <span class="n">order</span> <span class="o">=</span> <span class="n">deriv</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="n">deriv</span><span class="o">.</span><span class="n">variables</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">1</span>
-
-        <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">deriv</span><span class="o">.</span><span class="n">variables</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="k">if</span> <span class="n">s</span> <span class="o">==</span> <span class="n">sym</span><span class="p">:</span>
-                <span class="n">order</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                    <span class="s">&#39;Improve MV Derivative support in collect&#39;</span><span class="p">)</span>
-
-        <span class="k">while</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">):</span>
-            <span class="n">s0</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">variables</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-            <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">variables</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">s</span> <span class="o">!=</span> <span class="n">s0</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                        <span class="s">&#39;Improve MV Derivative support in collect&#39;</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">s0</span> <span class="o">==</span> <span class="n">sym</span><span class="p">:</span>
-                <span class="n">expr</span><span class="p">,</span> <span class="n">order</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="n">order</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">break</span>
-
-        <span class="k">return</span> <span class="n">expr</span><span class="p">,</span> <span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">Rational</span><span class="p">(</span><span class="n">order</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">parse_term</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Parses expression expr and outputs tuple (sexpr, rat_expo,</span>
-<span class="sd">        sym_expo, deriv)</span>
-<span class="sd">        where:</span>
-<span class="sd">         - sexpr is the base expression</span>
-<span class="sd">         - rat_expo is the rational exponent that sexpr is raised to</span>
-<span class="sd">         - sym_expo is the symbolic exponent that sexpr is raised to</span>
-<span class="sd">         - deriv contains the derivatives the the expression</span>
-
-<span class="sd">         for example, the output of x would be (x, 1, None, None)</span>
-<span class="sd">         the output of 2**x would be (2, 1, x, None)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">rat_expo</span><span class="p">,</span> <span class="n">sym_expo</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="bp">None</span>
-        <span class="n">sexpr</span><span class="p">,</span> <span class="n">deriv</span> <span class="o">=</span> <span class="n">expr</span><span class="p">,</span> <span class="bp">None</span>
-
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">):</span>
-                <span class="n">sexpr</span><span class="p">,</span> <span class="n">deriv</span> <span class="o">=</span> <span class="n">parse_derivative</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">sexpr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">base</span>
-
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                <span class="n">rat_expo</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">coeff</span><span class="p">,</span> <span class="n">tail</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-
-                <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                    <span class="n">rat_expo</span><span class="p">,</span> <span class="n">sym_expo</span> <span class="o">=</span> <span class="n">coeff</span><span class="p">,</span> <span class="n">tail</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">sym_expo</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span>
-        <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">:</span>
-            <span class="n">arg</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="n">sexpr</span><span class="p">,</span> <span class="n">rat_expo</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Exp1</span><span class="p">,</span> <span class="n">arg</span>
-            <span class="k">elif</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                <span class="n">coeff</span><span class="p">,</span> <span class="n">tail</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">(</span><span class="n">rational</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                <span class="n">sexpr</span><span class="p">,</span> <span class="n">rat_expo</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">tail</span><span class="p">),</span> <span class="n">coeff</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">):</span>
-            <span class="n">sexpr</span><span class="p">,</span> <span class="n">deriv</span> <span class="o">=</span> <span class="n">parse_derivative</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">sexpr</span><span class="p">,</span> <span class="n">rat_expo</span><span class="p">,</span> <span class="n">sym_expo</span><span class="p">,</span> <span class="n">deriv</span>
-
-    <span class="k">def</span> <span class="nf">parse_expression</span><span class="p">(</span><span class="n">terms</span><span class="p">,</span> <span class="n">pattern</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Parse terms searching for a pattern.</span>
-<span class="sd">        terms is a list of tuples as returned by parse_terms;</span>
-<span class="sd">        pattern is an expression treated as a product of factors</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">pattern</span> <span class="o">=</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">pattern</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">terms</span><span class="p">)</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">pattern</span><span class="p">):</span>
-            <span class="c"># pattern is longer than matched product</span>
-            <span class="c"># so no chance for positive parsing result</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">pattern</span> <span class="o">=</span> <span class="p">[</span><span class="n">parse_term</span><span class="p">(</span><span class="n">elem</span><span class="p">)</span> <span class="k">for</span> <span class="n">elem</span> <span class="ow">in</span> <span class="n">pattern</span><span class="p">]</span>
-
-            <span class="n">terms</span> <span class="o">=</span> <span class="n">terms</span><span class="p">[:]</span>  <span class="c"># need a copy</span>
-            <span class="n">elems</span><span class="p">,</span> <span class="n">common_expo</span><span class="p">,</span> <span class="n">has_deriv</span> <span class="o">=</span> <span class="p">[],</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">False</span>
-
-            <span class="k">for</span> <span class="n">elem</span><span class="p">,</span> <span class="n">e_rat</span><span class="p">,</span> <span class="n">e_sym</span><span class="p">,</span> <span class="n">e_ord</span> <span class="ow">in</span> <span class="n">pattern</span><span class="p">:</span>
-
-                <span class="k">if</span> <span class="n">elem</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">and</span> <span class="n">e_rat</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">e_sym</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="c"># a constant is a match for everything</span>
-                    <span class="k">continue</span>
-
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">terms</span><span class="p">)):</span>
-                    <span class="k">if</span> <span class="n">terms</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="k">continue</span>
-
-                    <span class="n">term</span><span class="p">,</span> <span class="n">t_rat</span><span class="p">,</span> <span class="n">t_sym</span><span class="p">,</span> <span class="n">t_ord</span> <span class="o">=</span> <span class="n">terms</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-
-                    <span class="c"># keeping track of whether one of the terms had</span>
-                    <span class="c"># a derivative or not as this will require rebuilding</span>
-                    <span class="c"># the expression later</span>
-                    <span class="k">if</span> <span class="n">t_ord</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="n">has_deriv</span> <span class="o">=</span> <span class="bp">True</span>
-
-                    <span class="k">if</span> <span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">elem</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span>
-                            <span class="p">(</span><span class="n">t_sym</span> <span class="o">==</span> <span class="n">e_sym</span> <span class="ow">or</span> <span class="n">t_sym</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span>
-                            <span class="n">e_sym</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span>
-                            <span class="n">t_sym</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">e_sym</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">)):</span>
-                        <span class="k">if</span> <span class="n">exact</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                            <span class="c"># we don&#39;t have to be exact so find common exponent</span>
-                            <span class="c"># for both expression&#39;s term and pattern&#39;s element</span>
-                            <span class="n">expo</span> <span class="o">=</span> <span class="n">t_rat</span> <span class="o">/</span> <span class="n">e_rat</span>
-
-                            <span class="k">if</span> <span class="n">common_expo</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                                <span class="c"># first time</span>
-                                <span class="n">common_expo</span> <span class="o">=</span> <span class="n">expo</span>
-                            <span class="k">else</span><span class="p">:</span>
-                                <span class="c"># common exponent was negotiated before so</span>
-                                <span class="c"># there is no chance for a pattern match unless</span>
-                                <span class="c"># common and current exponents are equal</span>
-                                <span class="k">if</span> <span class="n">common_expo</span> <span class="o">!=</span> <span class="n">expo</span><span class="p">:</span>
-                                    <span class="n">common_expo</span> <span class="o">=</span> <span class="mi">1</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="c"># we ought to be exact so all fields of</span>
-                            <span class="c"># interest must match in every details</span>
-                            <span class="k">if</span> <span class="n">e_rat</span> <span class="o">!=</span> <span class="n">t_rat</span> <span class="ow">or</span> <span class="n">e_ord</span> <span class="o">!=</span> <span class="n">t_ord</span><span class="p">:</span>
-                                <span class="k">continue</span>
-
-                        <span class="c"># found common term so remove it from the expression</span>
-                        <span class="c"># and try to match next element in the pattern</span>
-                        <span class="n">elems</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">terms</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-                        <span class="n">terms</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="bp">None</span>
-
-                        <span class="k">break</span>
-
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="c"># pattern element not found</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-
-            <span class="k">return</span> <span class="p">[</span><span class="n">_f</span> <span class="k">for</span> <span class="n">_f</span> <span class="ow">in</span> <span class="n">terms</span> <span class="k">if</span> <span class="n">_f</span><span class="p">],</span> <span class="n">elems</span><span class="p">,</span> <span class="n">common_expo</span><span class="p">,</span> <span class="n">has_deriv</span>
-
-    <span class="k">if</span> <span class="n">evaluate</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">collect</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">syms</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="bp">True</span><span class="p">,</span> <span class="n">exact</span><span class="p">,</span> <span class="n">distribute_order_term</span><span class="p">)</span> <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span> <span class="p">])</span>
-        <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="n">collect</span><span class="p">(</span>
-                <span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">syms</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="bp">True</span><span class="p">,</span> <span class="n">exact</span><span class="p">,</span> <span class="n">distribute_order_term</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">iterable</span><span class="p">(</span><span class="n">syms</span><span class="p">):</span>
-        <span class="n">syms</span> <span class="o">=</span> <span class="p">[</span><span class="n">expand_power_base</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">syms</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">syms</span> <span class="o">=</span> <span class="p">[</span> <span class="n">expand_power_base</span><span class="p">(</span><span class="n">syms</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span> <span class="p">]</span>
-
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="n">order_term</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="k">if</span> <span class="n">distribute_order_term</span><span class="p">:</span>
-        <span class="n">order_term</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">getO</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="n">order_term</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">order_term</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">syms</span><span class="p">):</span>
-                <span class="n">order_term</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">removeO</span><span class="p">()</span>
-
-    <span class="n">summa</span> <span class="o">=</span> <span class="p">[</span><span class="n">expand_power_base</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">expr</span><span class="p">)]</span>
-
-    <span class="n">collected</span><span class="p">,</span> <span class="n">disliked</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">list</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-    <span class="k">for</span> <span class="n">product</span> <span class="ow">in</span> <span class="n">summa</span><span class="p">:</span>
-        <span class="n">terms</span> <span class="o">=</span> <span class="p">[</span><span class="n">parse_term</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">product</span><span class="p">)]</span>
-
-        <span class="k">for</span> <span class="n">symbol</span> <span class="ow">in</span> <span class="n">syms</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">SYMPY_DEBUG</span><span class="p">:</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&quot;DEBUG: parsing of expression </span><span class="si">%s</span><span class="s"> with symbol </span><span class="si">%s</span><span class="s"> &quot;</span> <span class="o">%</span> <span class="p">(</span>
-                    <span class="nb">str</span><span class="p">(</span><span class="n">terms</span><span class="p">),</span> <span class="nb">str</span><span class="p">(</span><span class="n">symbol</span><span class="p">)))</span>
-
-            <span class="n">result</span> <span class="o">=</span> <span class="n">parse_expression</span><span class="p">(</span><span class="n">terms</span><span class="p">,</span> <span class="n">symbol</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">SYMPY_DEBUG</span><span class="p">:</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&quot;DEBUG: returned </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">str</span><span class="p">(</span><span class="n">result</span><span class="p">))</span>
-
-            <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">terms</span><span class="p">,</span> <span class="n">elems</span><span class="p">,</span> <span class="n">common_expo</span><span class="p">,</span> <span class="n">has_deriv</span> <span class="o">=</span> <span class="n">result</span>
-
-                <span class="c"># when there was derivative in current pattern we</span>
-                <span class="c"># will need to rebuild its expression from scratch</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">has_deriv</span><span class="p">:</span>
-                    <span class="n">index</span> <span class="o">=</span> <span class="mi">1</span>
-                    <span class="k">for</span> <span class="n">elem</span> <span class="ow">in</span> <span class="n">elems</span><span class="p">:</span>
-                        <span class="n">e</span> <span class="o">=</span> <span class="n">elem</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-                        <span class="k">if</span> <span class="n">elem</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                            <span class="n">e</span> <span class="o">*=</span> <span class="n">elem</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-                        <span class="n">index</span> <span class="o">*=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">elem</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">e</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">index</span> <span class="o">=</span> <span class="n">make_expression</span><span class="p">(</span><span class="n">elems</span><span class="p">)</span>
-                <span class="n">terms</span> <span class="o">=</span> <span class="n">expand_power_base</span><span class="p">(</span><span class="n">make_expression</span><span class="p">(</span><span class="n">terms</span><span class="p">),</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-                <span class="n">index</span> <span class="o">=</span> <span class="n">expand_power_base</span><span class="p">(</span><span class="n">index</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-                <span class="n">collected</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">terms</span><span class="p">)</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># none of the patterns matched</span>
-            <span class="n">disliked</span> <span class="o">+=</span> <span class="n">product</span>
-    <span class="c"># add terms now for each key</span>
-    <span class="n">collected</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">([(</span><span class="n">k</span><span class="p">,</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">v</span><span class="p">))</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">collected</span><span class="o">.</span><span class="n">items</span><span class="p">()])</span>
-
-    <span class="k">if</span> <span class="n">disliked</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-        <span class="n">collected</span><span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">]</span> <span class="o">=</span> <span class="n">disliked</span>
-
-    <span class="k">if</span> <span class="n">order_term</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="n">val</span> <span class="ow">in</span> <span class="n">collected</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">collected</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">=</span> <span class="n">val</span> <span class="o">+</span> <span class="n">order_term</span>
-
-    <span class="k">if</span> <span class="n">func</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">collected</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span>
-            <span class="p">[</span> <span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">func</span><span class="p">(</span><span class="n">val</span><span class="p">))</span> <span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="n">val</span> <span class="ow">in</span> <span class="n">collected</span><span class="o">.</span><span class="n">items</span><span class="p">()</span> <span class="p">])</span>
-
-    <span class="k">if</span> <span class="n">evaluate</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">key</span><span class="o">*</span><span class="n">val</span> <span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="n">val</span> <span class="ow">in</span> <span class="n">collected</span><span class="o">.</span><span class="n">items</span><span class="p">()])</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">collected</span>
-
-</div>
-<div class="viewcode-block" id="rcollect"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.simplify.rcollect">[docs]</a><span class="k">def</span> <span class="nf">rcollect</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="nb">vars</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Recursively collect sums in an expression.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify import rcollect</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">    &gt;&gt;&gt; expr = (x**2*y + x*y + x + y)/(x + y)</span>
-
-<span class="sd">    &gt;&gt;&gt; rcollect(expr, y)</span>
-<span class="sd">    (x + y*(x**2 + x + 1))/(x + y)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Atom</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="nb">vars</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">rcollect</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="o">*</span><span class="nb">vars</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span> <span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">collect</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="nb">vars</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span>
-
-</div>
-<div class="viewcode-block" id="separatevars"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.simplify.separatevars">[docs]</a><span class="k">def</span> <span class="nf">separatevars</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">symbols</span><span class="o">=</span><span class="p">[],</span> <span class="nb">dict</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Separates variables in an expression, if possible.  By</span>
-<span class="sd">    default, it separates with respect to all symbols in an</span>
-<span class="sd">    expression and collects constant coefficients that are</span>
-<span class="sd">    independent of symbols.</span>
-
-<span class="sd">    If dict=True then the separated terms will be returned</span>
-<span class="sd">    in a dictionary keyed to their corresponding symbols.</span>
-<span class="sd">    By default, all symbols in the expression will appear as</span>
-<span class="sd">    keys; if symbols are provided, then all those symbols will</span>
-<span class="sd">    be used as keys, and any terms in the expression containing</span>
-<span class="sd">    other symbols or non-symbols will be returned keyed to the</span>
-<span class="sd">    string &#39;coeff&#39;. (Passing None for symbols will return the</span>
-<span class="sd">    expression in a dictionary keyed to &#39;coeff&#39;.)</span>
-
-<span class="sd">    If force=True, then bases of powers will be separated regardless</span>
-<span class="sd">    of assumptions on the symbols involved.</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-<span class="sd">    The order of the factors is determined by Mul, so that the</span>
-<span class="sd">    separated expressions may not necessarily be grouped together.</span>
-
-<span class="sd">    Although factoring is necessary to separate variables in some</span>
-<span class="sd">    expressions, it is not necessary in all cases, so one should not</span>
-<span class="sd">    count on the returned factors being factored.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z, alpha</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import separatevars, sin</span>
-<span class="sd">    &gt;&gt;&gt; separatevars((x*y)**y)</span>
-<span class="sd">    (x*y)**y</span>
-<span class="sd">    &gt;&gt;&gt; separatevars((x*y)**y, force=True)</span>
-<span class="sd">    x**y*y**y</span>
-
-<span class="sd">    &gt;&gt;&gt; e = 2*x**2*z*sin(y)+2*z*x**2</span>
-<span class="sd">    &gt;&gt;&gt; separatevars(e)</span>
-<span class="sd">    2*x**2*z*(sin(y) + 1)</span>
-<span class="sd">    &gt;&gt;&gt; separatevars(e, symbols=(x, y), dict=True)</span>
-<span class="sd">    {&#39;coeff&#39;: 2*z, x: x**2, y: sin(y) + 1}</span>
-<span class="sd">    &gt;&gt;&gt; separatevars(e, [x, y, alpha], dict=True)</span>
-<span class="sd">    {&#39;coeff&#39;: 2*z, alpha: 1, x: x**2, y: sin(y) + 1}</span>
-
-<span class="sd">    If the expression is not really separable, or is only partially</span>
-<span class="sd">    separable, separatevars will do the best it can to separate it</span>
-<span class="sd">    by using factoring.</span>
-
-<span class="sd">    &gt;&gt;&gt; separatevars(x + x*y - 3*x**2)</span>
-<span class="sd">    -x*(3*x - y - 1)</span>
-
-<span class="sd">    If the expression is not separable then expr is returned unchanged</span>
-<span class="sd">    or (if dict=True) then None is returned.</span>
-
-<span class="sd">    &gt;&gt;&gt; eq = 2*x + y*sin(x)</span>
-<span class="sd">    &gt;&gt;&gt; separatevars(eq) == eq</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; separatevars(2*x + y*sin(x), symbols=(x, y), dict=True) == None</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">dict</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">_separatevars_dict</span><span class="p">(</span><span class="n">_separatevars</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">force</span><span class="p">),</span> <span class="n">symbols</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">_separatevars</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">force</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_separatevars</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">force</span><span class="p">):</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span>
-    <span class="c"># don&#39;t destroy a Mul since much of the work may already be done</span>
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="n">changed</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-            <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">separatevars</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">force</span><span class="p">)</span>
-            <span class="n">changed</span> <span class="o">=</span> <span class="n">changed</span> <span class="ow">or</span> <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">!=</span> <span class="n">a</span>
-        <span class="k">if</span> <span class="n">changed</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="c"># get a Pow ready for expansion</span>
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">separatevars</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="n">force</span><span class="p">),</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">)</span>
-
-    <span class="c"># First try other expansion methods</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">mul</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">multinomial</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="n">force</span><span class="p">)</span>
-
-    <span class="n">_expr</span> <span class="o">=</span> <span class="n">expr</span>
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>  <span class="c"># factor fails for nc</span>
-        <span class="n">_expr</span><span class="p">,</span> <span class="n">reps</span> <span class="o">=</span> <span class="n">posify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span> <span class="k">if</span> <span class="n">force</span> <span class="k">else</span> <span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="p">{})</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">factor</span><span class="p">(</span><span class="n">_expr</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">reps</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="c"># Find any common coefficients to pull out</span>
-    <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-    <span class="n">commonc</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">(</span><span class="n">cset</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">warn</span><span class="o">=</span><span class="bp">False</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-        <span class="n">commonc</span> <span class="o">&amp;=</span> <span class="n">i</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">(</span><span class="n">cset</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">warn</span><span class="o">=</span><span class="bp">False</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">commonc</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">commonc</span><span class="p">)</span>
-    <span class="n">commonc</span> <span class="o">=</span> <span class="n">commonc</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()[</span><span class="mi">1</span><span class="p">]</span>  <span class="c"># ignore constants</span>
-    <span class="n">commonc_set</span> <span class="o">=</span> <span class="n">commonc</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">(</span><span class="n">cset</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">warn</span><span class="o">=</span><span class="bp">False</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="c"># remove them</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">c</span><span class="p">,</span> <span class="n">nc</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">(</span><span class="n">cset</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">warn</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">c</span> <span class="o">-</span> <span class="n">commonc_set</span>
-        <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">c</span><span class="p">)</span><span class="o">*</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">nc</span><span class="p">)</span>
-    <span class="n">nonsepar</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">nonsepar</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">_expr</span> <span class="o">=</span> <span class="n">nonsepar</span>
-        <span class="n">_expr</span><span class="p">,</span> <span class="n">reps</span> <span class="o">=</span> <span class="n">posify</span><span class="p">(</span><span class="n">_expr</span><span class="p">)</span> <span class="k">if</span> <span class="n">force</span> <span class="k">else</span> <span class="p">(</span><span class="n">_expr</span><span class="p">,</span> <span class="p">{})</span>
-        <span class="n">_expr</span> <span class="o">=</span> <span class="p">(</span><span class="n">factor</span><span class="p">(</span><span class="n">_expr</span><span class="p">))</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">reps</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">_expr</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">nonsepar</span> <span class="o">=</span> <span class="n">_expr</span>
-
-    <span class="k">return</span> <span class="n">commonc</span><span class="o">*</span><span class="n">nonsepar</span>
-
-
-<span class="k">def</span> <span class="nf">_separatevars_dict</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">symbols</span><span class="p">):</span>
-    <span class="k">if</span> <span class="n">symbols</span><span class="p">:</span>
-        <span class="k">assert</span> <span class="nb">all</span><span class="p">((</span><span class="n">t</span><span class="o">.</span><span class="n">is_Atom</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">symbols</span><span class="p">)),</span> <span class="s">&quot;symbols must be Atoms.&quot;</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">symbols</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">{</span><span class="s">&#39;coeff&#39;</span><span class="p">:</span> <span class="n">expr</span><span class="p">}</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">symbols</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="n">ret</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(((</span><span class="n">i</span><span class="p">,</span> <span class="p">[])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">symbols</span> <span class="o">+</span> <span class="p">[</span><span class="s">&#39;coeff&#39;</span><span class="p">]))</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="n">expsym</span> <span class="o">=</span> <span class="n">i</span><span class="o">.</span><span class="n">free_symbols</span>
-        <span class="n">intersection</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">expsym</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">intersection</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">intersection</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="c"># There are no symbols, so it is part of the coefficient</span>
-            <span class="n">ret</span><span class="p">[</span><span class="s">&#39;coeff&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">ret</span><span class="p">[</span><span class="n">intersection</span><span class="o">.</span><span class="n">pop</span><span class="p">()]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-
-    <span class="c"># rebuild</span>
-    <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">ret</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-        <span class="n">ret</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">v</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">ret</span>
-
-
-<div class="viewcode-block" id="ratsimp"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.simplify.ratsimp">[docs]</a><span class="k">def</span> <span class="nf">ratsimp</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Put an expression over a common denominator, cancel and reduce.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import ratsimp</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; ratsimp(1/x + 1/y)</span>
-<span class="sd">    (x + y)/(x*y)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">f</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">cancel</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">Q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">reduced</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="n">g</span><span class="p">],</span> <span class="n">field</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">expand</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">ComputationFailed</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">/</span><span class="n">g</span>
-
-    <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">Q</span><span class="p">)</span> <span class="o">+</span> <span class="n">cancel</span><span class="p">(</span><span class="n">r</span><span class="o">/</span><span class="n">g</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">ratsimpmodprime</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Simplifies a rational expression ``expr`` modulo the prime ideal</span>
-<span class="sd">    generated by ``G``.  ``G`` should be a Groebner basis of the</span>
-<span class="sd">    ideal.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.simplify import ratsimpmodprime</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; ratsimpmodprime((x + y**5 + y)/(x - y), [x*y**5 - x - y], x, y, order=&#39;lex&#39;)</span>
-<span class="sd">    (x**2 + x*y + x + y)/(x**2 - x*y)</span>
-
-<span class="sd">    The algorithm computes a rational simplification which minimizes</span>
-<span class="sd">    the sum of the total degrees of the numerator and the denominator.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    M. Monagan, R. Pearce, Rational Simplification Modulo a Polynomial</span>
-<span class="sd">    Ideal,</span>
-<span class="sd">    http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.163.6984</span>
-<span class="sd">    (specifically, the second algorithm)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">parallel_poly_from_expr</span>
-    <span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="n">PolificationFailed</span><span class="p">,</span> <span class="n">DomainError</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">solve</span><span class="p">,</span> <span class="n">Monomial</span>
-    <span class="kn">from</span> <span class="nn">sympy.polys.monomialtools</span> <span class="kn">import</span> <span class="n">monomial_div</span>
-    <span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">product</span>
-
-    <span class="c"># usual preparation of polynomials:</span>
-
-    <span class="n">num</span><span class="p">,</span> <span class="n">denom</span> <span class="o">=</span> <span class="n">cancel</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">polys</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">([</span><span class="n">num</span><span class="p">,</span> <span class="n">denom</span><span class="p">]</span> <span class="o">+</span> <span class="n">G</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="n">domain</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">domain</span>
-
-    <span class="k">if</span> <span class="n">domain</span><span class="o">.</span><span class="n">has_assoc_Field</span><span class="p">:</span>
-        <span class="n">opt</span><span class="o">.</span><span class="n">domain</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">get_field</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">DomainError</span><span class="p">(</span>
-            <span class="s">&quot;can&#39;t compute rational simplification over </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">domain</span><span class="p">)</span>
-
-    <span class="c"># compute only once</span>
-    <span class="n">leading_monomials</span> <span class="o">=</span> <span class="p">[</span><span class="n">g</span><span class="o">.</span><span class="n">LM</span><span class="p">(</span><span class="n">opt</span><span class="o">.</span><span class="n">order</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">polys</span><span class="p">[</span><span class="mi">2</span><span class="p">:]]</span>
-
-    <span class="k">def</span> <span class="nf">staircase</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Compute all monomials with degree less than ``n`` that are</span>
-<span class="sd">        not divisible by any element of ``leading_monomials``.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">S</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">product</span><span class="p">(</span><span class="o">*</span><span class="p">([</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span> <span class="o">*</span> <span class="nb">len</span><span class="p">(</span><span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">))):</span>
-            <span class="k">if</span> <span class="nb">sum</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="n">n</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">all</span><span class="p">([</span><span class="n">monomial_div</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">lmg</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">None</span> <span class="k">for</span> <span class="n">lmg</span> <span class="ow">in</span> <span class="n">leading_monomials</span><span class="p">]):</span>
-                    <span class="n">S</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="p">[</span><span class="n">Monomial</span><span class="p">(</span><span class="n">s</span><span class="p">)</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(</span><span class="o">*</span><span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">S</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">_ratsimpmodprime</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">N</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">D</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Computes a rational simplification of ``a/b`` which minimizes</span>
-<span class="sd">        the sum of the total degrees of the numerator and the denominator.</span>
-
-<span class="sd">        The algorithm proceeds by looking at ``a * d - b * c`` modulo</span>
-<span class="sd">        the ideal generated by ``G`` for some ``c`` and ``d`` with degree</span>
-<span class="sd">        less than ``a`` and ``b`` respectively.</span>
-<span class="sd">        The coefficients of ``c`` and ``d`` are indeterminates and thus</span>
-<span class="sd">        the coefficients of the normalform of ``a * d - b * c`` are</span>
-<span class="sd">        linear polynomials in these indeterminates.</span>
-<span class="sd">        If these linear polynomials, considered as system of</span>
-<span class="sd">        equations, have a nontrivial solution, then `\frac{a}{b}</span>
-<span class="sd">        \equiv \frac{c}{d}` modulo the ideal generated by ``G``. So,</span>
-<span class="sd">        by construction, the degree of ``c`` and ``d`` is less than</span>
-<span class="sd">        the degree of ``a`` and ``b``, so a simpler representation</span>
-<span class="sd">        has been found.</span>
-<span class="sd">        After a simpler representation has been found, the algorithm</span>
-<span class="sd">        tries to reduce the degree of the numerator and denominator</span>
-<span class="sd">        and returns the result afterwards.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">c</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span>
-        <span class="n">steps</span> <span class="o">=</span> <span class="mi">0</span>
-
-        <span class="k">while</span> <span class="n">N</span> <span class="o">+</span> <span class="n">D</span> <span class="o">&lt;</span> <span class="n">a</span><span class="o">.</span><span class="n">total_degree</span><span class="p">()</span> <span class="o">+</span> <span class="n">b</span><span class="o">.</span><span class="n">total_degree</span><span class="p">():</span>
-            <span class="n">M1</span> <span class="o">=</span> <span class="n">staircase</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-            <span class="n">M2</span> <span class="o">=</span> <span class="n">staircase</span><span class="p">(</span><span class="n">D</span><span class="p">)</span>
-
-            <span class="n">Cs</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&quot;c:</span><span class="si">%d</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">M1</span><span class="p">))</span>
-            <span class="n">Ds</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&quot;d:</span><span class="si">%d</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">M2</span><span class="p">))</span>
-
-            <span class="n">c_hat</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span>
-                <span class="nb">sum</span><span class="p">([</span><span class="n">Cs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">*</span> <span class="n">M1</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">M1</span><span class="p">))]),</span> <span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-            <span class="n">d_hat</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span>
-                <span class="nb">sum</span><span class="p">([</span><span class="n">Ds</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">*</span> <span class="n">M2</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">M2</span><span class="p">))]),</span> <span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-
-            <span class="n">r</span> <span class="o">=</span> <span class="n">reduced</span><span class="p">(</span><span class="n">a</span> <span class="o">*</span> <span class="n">d_hat</span> <span class="o">-</span> <span class="n">b</span> <span class="o">*</span> <span class="n">c_hat</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">,</span>
-                        <span class="n">order</span><span class="o">=</span><span class="n">opt</span><span class="o">.</span><span class="n">order</span><span class="p">,</span> <span class="n">polys</span><span class="o">=</span><span class="bp">True</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-
-            <span class="n">S</span> <span class="o">=</span> <span class="n">r</span><span class="o">.</span><span class="n">coeffs</span><span class="p">()</span>
-            <span class="n">sol</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">Cs</span> <span class="o">+</span> <span class="n">Ds</span><span class="p">)</span>
-
-            <span class="c"># If nontrivial solutions exist, solve will give them</span>
-            <span class="c"># parametrized, i.e. the values of some keys will be</span>
-            <span class="c"># exprs. Set these to any value different from 0 to obtain</span>
-            <span class="c"># one nontrivial solution:</span>
-            <span class="k">for</span> <span class="n">key</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">sol</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
-                <span class="n">sol</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">=</span> <span class="n">sol</span><span class="p">[</span><span class="n">key</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">Cs</span> <span class="o">+</span> <span class="n">Ds</span><span class="p">,</span>
-                                         <span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">Cs</span><span class="p">)</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">Ds</span><span class="p">))))))</span>
-
-            <span class="k">if</span> <span class="n">sol</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">([</span><span class="n">s</span> <span class="o">==</span> <span class="mi">0</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">sol</span><span class="o">.</span><span class="n">values</span><span class="p">()]):</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="n">c_hat</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sol</span><span class="p">)</span>
-                <span class="n">d</span> <span class="o">=</span> <span class="n">d_hat</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sol</span><span class="p">)</span>
-
-                <span class="c"># The &quot;free&quot; variables occuring before as parameters</span>
-                <span class="c"># might still be in the substituted c, d, so set them</span>
-                <span class="c"># to the value chosen before:</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="n">c</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">Cs</span> <span class="o">+</span> <span class="n">Ds</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">Cs</span><span class="p">)</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">Ds</span><span class="p">))))))</span>
-                <span class="n">d</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">Cs</span> <span class="o">+</span> <span class="n">Ds</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">Cs</span><span class="p">)</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">Ds</span><span class="p">))))))</span>
-
-                <span class="n">c</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-                <span class="n">d</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span>
-
-                <span class="k">break</span>
-
-            <span class="n">N</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">D</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">steps</span> <span class="o">+=</span> <span class="mi">1</span>
-
-        <span class="k">if</span> <span class="n">steps</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">c</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">_ratsimpmodprime</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">D</span> <span class="o">-</span> <span class="n">steps</span><span class="p">)</span>
-            <span class="n">c</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">_ratsimpmodprime</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">N</span> <span class="o">-</span> <span class="n">steps</span><span class="p">,</span> <span class="n">D</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span>
-
-    <span class="c"># preprocessing. this improves performance a bit when deg(num)</span>
-    <span class="c"># and deg(denom) are large:</span>
-    <span class="n">num</span> <span class="o">=</span> <span class="n">reduced</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">opt</span><span class="o">.</span><span class="n">order</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-    <span class="n">denom</span> <span class="o">=</span> <span class="n">reduced</span><span class="p">(</span><span class="n">denom</span><span class="p">,</span> <span class="n">G</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">opt</span><span class="o">.</span><span class="n">order</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="n">c</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">_ratsimpmodprime</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">),</span> <span class="n">Poly</span><span class="p">(</span><span class="n">denom</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">domain</span><span class="o">.</span><span class="n">has_Field</span><span class="p">:</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">c</span><span class="o">.</span><span class="n">clear_denoms</span><span class="p">(</span><span class="n">convert</span><span class="o">=</span><span class="bp">True</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">clear_denoms</span><span class="p">(</span><span class="n">convert</span><span class="o">=</span><span class="bp">True</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="k">return</span> <span class="n">c</span><span class="o">/</span><span class="n">d</span>
-
-
-<div class="viewcode-block" id="trigsimp"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.simplify.trigsimp">[docs]</a><span class="k">def</span> <span class="nf">trigsimp</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">recursive</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    reduces expression by using known trig identities</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    deep:</span>
-<span class="sd">    - Apply trigsimp inside all objects with arguments</span>
-
-<span class="sd">    recursive:</span>
-<span class="sd">    - Use common subexpression elimination (cse()) and apply</span>
-<span class="sd">    trigsimp recursively (this is quite expensive if the</span>
-<span class="sd">    expression is large)</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import trigsimp, sin, cos, log, cosh, sinh</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; e = 2*sin(x)**2 + 2*cos(x)**2</span>
-<span class="sd">    &gt;&gt;&gt; trigsimp(e)</span>
-<span class="sd">    2</span>
-<span class="sd">    &gt;&gt;&gt; trigsimp(log(e))</span>
-<span class="sd">    log(2*sin(x)**2 + 2*cos(x)**2)</span>
-<span class="sd">    &gt;&gt;&gt; trigsimp(log(e), deep=True)</span>
-<span class="sd">    log(2)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">TrigonometricFunction</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">HyperbolicFunction</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="k">if</span> <span class="n">recursive</span><span class="p">:</span>
-        <span class="n">w</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">cse</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">_trigsimp</span><span class="p">(</span><span class="n">g</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">deep</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">sub</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">w</span><span class="p">):</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sub</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">sub</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">_trigsimp</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">deep</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">g</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">_trigsimp</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">deep</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_trigsimp</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;recursive helper for trigsimp&quot;&quot;&quot;</span>
-    <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a b c&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Wild</span><span class="p">)</span>
-    <span class="n">d</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;d&#39;</span><span class="p">,</span> <span class="n">commutative</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-    <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">tan</span><span class="p">,</span> <span class="n">cot</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">sin</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">cos</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">tan</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">cot</span>
-    <span class="n">sinh</span><span class="p">,</span> <span class="n">cosh</span><span class="p">,</span> <span class="n">tanh</span><span class="p">,</span> <span class="n">coth</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">sinh</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">cosh</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">tanh</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">coth</span>
-    <span class="c"># for the simplifications like sinh/cosh -&gt; tanh:</span>
-    <span class="n">matchers_division</span> <span class="o">=</span> <span class="p">(</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">cot</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">/</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">cot</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">/</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">cot</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="p">),</span>
-
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">/</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">tanh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">tanh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">coth</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">tanh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">/</span><span class="n">sinh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">/</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">coth</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">/</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">/</span><span class="n">sinh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">coth</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">*</span><span class="n">tanh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="p">),</span>
-
-        <span class="c"># same as above but with noncommutative prefactor</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">cot</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">/</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">cot</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">/</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">cot</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="p">),</span>
-
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">/</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">tanh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">tanh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">coth</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">tanh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">/</span><span class="n">sinh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">/</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">coth</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">/</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">/</span><span class="n">sinh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">coth</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="o">*</span><span class="n">tanh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="p">),</span>
-    <span class="p">)</span>
-    <span class="c"># for cos(x)**2 + sin(x)**2 -&gt; 1</span>
-    <span class="n">matchers_identity</span> <span class="o">=</span> <span class="p">(</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">a</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">a</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">cot</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">a</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">),</span> <span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="o">+</span> <span class="n">sin</span><span class="p">(</span><span class="n">c</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span><span class="p">))),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">),</span> <span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="o">-</span> <span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">c</span><span class="p">))),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">),</span> <span class="n">a</span><span class="o">*</span><span class="p">((</span><span class="n">tan</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">+</span> <span class="n">tan</span><span class="p">(</span><span class="n">c</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">tan</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="n">c</span><span class="p">)))),</span>
-
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">a</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">tanh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">a</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">coth</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">sinh</span><span class="p">(</span><span class="n">b</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">),</span> <span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="n">sinh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="o">+</span> <span class="n">sinh</span><span class="p">(</span><span class="n">c</span><span class="p">)</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span><span class="p">))),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">),</span> <span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="o">+</span> <span class="n">sinh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="n">c</span><span class="p">))),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">tanh</span><span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">),</span> <span class="n">a</span><span class="o">*</span><span class="p">((</span><span class="n">tanh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">+</span> <span class="n">tanh</span><span class="p">(</span><span class="n">c</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">tanh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">tanh</span><span class="p">(</span><span class="n">c</span><span class="p">)))),</span>
-
-        <span class="c"># same as above but with noncommutative prefactor</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">cot</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">),</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="o">+</span> <span class="n">sin</span><span class="p">(</span><span class="n">c</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span><span class="p">))),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">),</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="o">-</span> <span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">c</span><span class="p">))),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">),</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="p">((</span><span class="n">tan</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">+</span> <span class="n">tan</span><span class="p">(</span><span class="n">c</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">tan</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="n">c</span><span class="p">)))),</span>
-
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">tanh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">coth</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">sinh</span><span class="p">(</span><span class="n">b</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">),</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="p">(</span><span class="n">sinh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="o">+</span> <span class="n">sinh</span><span class="p">(</span><span class="n">c</span><span class="p">)</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span><span class="p">))),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">),</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="p">(</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="o">+</span> <span class="n">sinh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="n">c</span><span class="p">))),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">tanh</span><span class="p">(</span><span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">),</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="p">((</span><span class="n">tanh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">+</span> <span class="n">tanh</span><span class="p">(</span><span class="n">c</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">tanh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">tanh</span><span class="p">(</span><span class="n">c</span><span class="p">)))),</span>
-    <span class="p">)</span>
-
-    <span class="c"># Reduce any lingering artefacts, such as sin(x)**2 changing</span>
-    <span class="c"># to 1-cos(x)**2 when sin(x)**2 was &quot;simpler&quot;</span>
-    <span class="n">artifacts</span> <span class="o">=</span> <span class="p">(</span>
-        <span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">cos</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="o">-</span><span class="n">a</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">cos</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="o">-</span><span class="n">a</span><span class="o">*</span><span class="n">cot</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">sin</span><span class="p">),</span>
-
-        <span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="o">-</span><span class="n">a</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">cosh</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">tanh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">cosh</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">sinh</span><span class="p">(</span><span class="n">b</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">coth</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">sinh</span><span class="p">),</span>
-
-        <span class="c"># same as above but with noncommutative prefactor</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">cos</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">b</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="o">-</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">cos</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">sin</span><span class="p">(</span><span class="n">b</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="o">-</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">cot</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">sin</span><span class="p">),</span>
-
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="o">-</span><span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">cosh</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">cosh</span><span class="p">(</span><span class="n">b</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">tanh</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">cosh</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">d</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">sinh</span><span class="p">(</span><span class="n">b</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">d</span><span class="o">*</span><span class="n">coth</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">sinh</span><span class="p">),</span>
-    <span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-        <span class="c"># do some simplifications like sin/cos -&gt; tan:</span>
-        <span class="k">for</span> <span class="n">pattern</span><span class="p">,</span> <span class="n">simp</span> <span class="ow">in</span> <span class="n">matchers_division</span><span class="p">:</span>
-            <span class="n">res</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">pattern</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">res</span> <span class="ow">and</span> <span class="n">res</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="mi">0</span><span class="p">):</span>
-                <span class="c"># if &quot;a&quot; contains any of sin(&quot;b&quot;), cos(&quot;b&quot;), tan(&quot;b&quot;), cot(&quot;b&quot;),</span>
-                <span class="c"># sinh(&quot;b&quot;), cosh(&quot;b&quot;), tanh(&quot;b&quot;) or coth(&quot;b),</span>
-                <span class="c"># skip the simplification:</span>
-                <span class="k">if</span> <span class="n">res</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">TrigonometricFunction</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">HyperbolicFunction</span><span class="p">):</span>
-                    <span class="k">continue</span>
-                <span class="c"># simplify and finish:</span>
-                <span class="n">expr</span> <span class="o">=</span> <span class="n">simp</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">res</span><span class="p">)</span>
-                <span class="k">break</span>  <span class="c"># process below</span>
-
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-        <span class="c"># The types of hyper functions we are looking for</span>
-        <span class="c"># Scan for the terms we need</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">term</span> <span class="o">=</span> <span class="n">_trigsimp</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">deep</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">pattern</span><span class="p">,</span> <span class="n">result</span> <span class="ow">in</span> <span class="n">matchers_identity</span><span class="p">:</span>
-                <span class="n">res</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">pattern</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">res</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">term</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">res</span><span class="p">)</span>
-                    <span class="k">break</span>
-            <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">args</span> <span class="o">!=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">expand</span><span class="p">(</span><span class="n">expr</span><span class="p">),</span> <span class="n">key</span><span class="o">=</span><span class="n">count_ops</span><span class="p">)</span>
-
-        <span class="c"># Reduce any lingering artifacts, such as sin(x)**2 changing</span>
-        <span class="c"># to 1 - cos(x)**2 when sin(x)**2 was &quot;simpler&quot;</span>
-        <span class="k">for</span> <span class="n">pattern</span><span class="p">,</span> <span class="n">result</span><span class="p">,</span> <span class="n">ex</span> <span class="ow">in</span> <span class="n">artifacts</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-                <span class="k">break</span>
-            <span class="c"># Substitute a new wild that excludes some function(s)</span>
-            <span class="c"># to help influence a better match. This is because</span>
-            <span class="c"># sometimes, for example, &#39;a&#39; would match sec(x)**2</span>
-            <span class="n">a_t</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">ex</span><span class="p">])</span>
-            <span class="n">pattern</span> <span class="o">=</span> <span class="n">pattern</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">a_t</span><span class="p">)</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">a_t</span><span class="p">)</span>
-
-            <span class="n">m</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">pattern</span><span class="p">)</span>
-            <span class="k">while</span> <span class="n">m</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">m</span><span class="p">[</span><span class="n">a_t</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">or</span> <span class="o">-</span><span class="n">m</span><span class="p">[</span><span class="n">a_t</span><span class="p">]</span> <span class="ow">in</span> <span class="n">m</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">args</span> <span class="ow">or</span> <span class="n">m</span><span class="p">[</span><span class="n">a_t</span><span class="p">]</span> <span class="o">+</span> <span class="n">m</span><span class="p">[</span><span class="n">c</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">break</span>
-                <span class="k">if</span> <span class="n">d</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span> <span class="ow">and</span> <span class="n">m</span><span class="p">[</span><span class="n">a_t</span><span class="p">]</span><span class="o">*</span><span class="n">m</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">+</span> <span class="n">m</span><span class="p">[</span><span class="n">c</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">break</span>
-                <span class="n">expr</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-                <span class="n">m</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">pattern</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span> <span class="ow">or</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">or</span> <span class="n">deep</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">_trigsimp</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">deep</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">return</span> <span class="n">expr</span>
-
-
-<div class="viewcode-block" id="collect_sqrt"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.simplify.collect_sqrt">[docs]</a><span class="k">def</span> <span class="nf">collect_sqrt</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return expr with terms having common square roots collected together.</span>
-<span class="sd">    If ``evaluate`` is False a count indicating the number of sqrt-containing</span>
-<span class="sd">    terms will be returned and the returned expression will be an unevaluated</span>
-<span class="sd">    Add with args ordered by default_sort_key.</span>
-
-<span class="sd">    Note: since I = sqrt(-1), it is collected, too.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import sqrt</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.simplify import collect_sqrt</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import a, b</span>
-
-<span class="sd">    &gt;&gt;&gt; r2, r3, r5 = [sqrt(i) for i in [2, 3, 5]]</span>
-<span class="sd">    &gt;&gt;&gt; collect_sqrt(a*r2 + b*r2)</span>
-<span class="sd">    sqrt(2)*(a + b)</span>
-<span class="sd">    &gt;&gt;&gt; collect_sqrt(a*r2 + b*r2 + a*r3 + b*r3)</span>
-<span class="sd">    sqrt(2)*(a + b) + sqrt(3)*(a + b)</span>
-<span class="sd">    &gt;&gt;&gt; collect_sqrt(a*r2 + b*r2 + a*r3 + b*r5)</span>
-<span class="sd">    sqrt(3)*a + sqrt(5)*b + sqrt(2)*(a + b)</span>
-
-<span class="sd">    If evaluate is False then the arguments will be sorted and</span>
-<span class="sd">    returned as a list and a count of the number of sqrt-containing</span>
-<span class="sd">    terms will be returned:</span>
-
-<span class="sd">    &gt;&gt;&gt; collect_sqrt(a*r2 + b*r2 + a*r3 + b*r5, evaluate=False)</span>
-<span class="sd">    ((sqrt(2)*(a + b), sqrt(3)*a, sqrt(5)*b), 3)</span>
-<span class="sd">    &gt;&gt;&gt; collect_sqrt(a*sqrt(2) + b, evaluate=False)</span>
-<span class="sd">    ((b, sqrt(2)*a), 1)</span>
-<span class="sd">    &gt;&gt;&gt; collect_sqrt(a + b, evaluate=False)</span>
-<span class="sd">    ((a + b,), 0)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">coeff</span><span class="p">,</span> <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-    <span class="nb">vars</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">a</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()[</span><span class="mi">0</span><span class="p">]:</span>
-            <span class="k">if</span> <span class="n">m</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">m</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">m</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">q</span> <span class="o">==</span> <span class="mi">2</span> <span class="ow">or</span>
-                                <span class="n">m</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">):</span>
-                <span class="nb">vars</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-    <span class="nb">vars</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">vars</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">evaluate</span><span class="p">:</span>
-        <span class="nb">vars</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="nb">vars</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>  <span class="c"># since it will be reversed below</span>
-    <span class="nb">vars</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">count_ops</span><span class="p">)</span>
-    <span class="nb">vars</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-    <span class="n">d</span> <span class="o">=</span> <span class="n">collect_const</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="nb">vars</span><span class="p">,</span> <span class="o">**</span><span class="nb">dict</span><span class="p">(</span><span class="n">first</span><span class="o">=</span><span class="bp">False</span><span class="p">))</span>
-    <span class="n">hit</span> <span class="o">=</span> <span class="n">expr</span> <span class="o">!=</span> <span class="n">d</span>
-    <span class="n">d</span> <span class="o">*=</span> <span class="n">coeff</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">evaluate</span><span class="p">:</span>
-        <span class="n">nrad</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">d</span><span class="p">))</span>
-        <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="n">c</span><span class="p">,</span> <span class="n">nc</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-            <span class="k">for</span> <span class="n">ci</span> <span class="ow">in</span> <span class="n">c</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">ci</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">ci</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">ci</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">q</span> <span class="o">==</span> <span class="mi">2</span> <span class="ow">or</span> \
-                        <span class="n">ci</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span><span class="p">:</span>
-                    <span class="n">nrad</span> <span class="o">+=</span> <span class="mi">1</span>
-                    <span class="k">break</span>
-        <span class="k">if</span> <span class="n">hit</span> <span class="ow">or</span> <span class="n">nrad</span><span class="p">:</span>
-            <span class="n">args</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)]</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">args</span><span class="p">),</span> <span class="n">nrad</span>
-
-    <span class="k">return</span> <span class="n">d</span>
-
-</div>
-<div class="viewcode-block" id="collect_const"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.simplify.collect_const">[docs]</a><span class="k">def</span> <span class="nf">collect_const</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="nb">vars</span><span class="p">,</span> <span class="o">**</span><span class="n">first</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A non-greedy collection of terms with similar number coefficients in</span>
-<span class="sd">    an Add expr. If ``vars`` is given then only those constants will be</span>
-<span class="sd">    targeted.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import sqrt</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import a, s</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.simplify import collect_const</span>
-<span class="sd">    &gt;&gt;&gt; collect_const(sqrt(3) + sqrt(3)*(1 + sqrt(2)))</span>
-<span class="sd">    sqrt(3)*(sqrt(2) + 2)</span>
-<span class="sd">    &gt;&gt;&gt; collect_const(sqrt(3)*s + sqrt(7)*s + sqrt(3) + sqrt(7))</span>
-<span class="sd">    (sqrt(3) + sqrt(7))*(s + 1)</span>
-<span class="sd">    &gt;&gt;&gt; s = sqrt(2) + 2</span>
-<span class="sd">    &gt;&gt;&gt; collect_const(sqrt(3)*s + sqrt(3) + sqrt(7)*s + sqrt(7))</span>
-<span class="sd">    (sqrt(2) + 3)*(sqrt(3) + sqrt(7))</span>
-<span class="sd">    &gt;&gt;&gt; collect_const(sqrt(3)*s + sqrt(3) + sqrt(7)*s + sqrt(7), sqrt(3))</span>
-<span class="sd">    sqrt(7) + sqrt(3)*(sqrt(2) + 3) + sqrt(7)*(sqrt(2) + 2)</span>
-
-<span class="sd">    If no constants are provided then a leading Rational might be returned:</span>
-
-<span class="sd">    &gt;&gt;&gt; collect_const(2*sqrt(3) + 4*a*sqrt(5))</span>
-<span class="sd">    2*(2*sqrt(5)*a + sqrt(3))</span>
-<span class="sd">    &gt;&gt;&gt; collect_const(2*sqrt(3) + 4*a*sqrt(5), sqrt(3))</span>
-<span class="sd">    4*sqrt(5)*a + 2*sqrt(3)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="n">first</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;first&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-        <span class="n">c</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">c</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">expr</span>
-    <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">vars</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">collect_const</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="o">*</span><span class="nb">vars</span><span class="p">,</span> <span class="o">**</span><span class="nb">dict</span><span class="p">(</span><span class="n">first</span><span class="o">=</span><span class="bp">False</span><span class="p">)))</span>
-        <span class="c"># else don&#39;t leave the Rational on the outside</span>
-        <span class="k">return</span> <span class="n">c</span><span class="o">*</span><span class="n">collect_const</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="o">*</span><span class="nb">vars</span><span class="p">,</span> <span class="o">**</span><span class="nb">dict</span><span class="p">(</span><span class="n">first</span><span class="o">=</span><span class="bp">False</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">or</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span>
-    <span class="n">recurse</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">vars</span><span class="p">:</span>
-        <span class="n">recurse</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="nb">vars</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">a</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">m</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-                    <span class="nb">vars</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-        <span class="nb">vars</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="nb">vars</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">count_ops</span><span class="p">)</span>
-    <span class="c"># Rationals get autodistributed on Add so don&#39;t bother with them</span>
-    <span class="nb">vars</span> <span class="o">=</span> <span class="p">[</span><span class="n">v</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">vars</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">v</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">]</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">vars</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">vars</span><span class="p">:</span>
-        <span class="n">terms</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">list</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="n">v</span><span class="p">:</span>
-                    <span class="n">d</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">i</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-            <span class="n">ai</span><span class="p">,</span> <span class="n">ad</span> <span class="o">=</span> <span class="p">[</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">w</span><span class="p">)</span> <span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">d</span><span class="p">]]</span>
-            <span class="n">terms</span><span class="p">[</span><span class="n">ad</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ai</span><span class="p">)</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">hit</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">terms</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">v</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">v</span><span class="p">)</span>
-                <span class="n">hit</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">if</span> <span class="n">recurse</span> <span class="ow">and</span> <span class="n">v</span> <span class="o">!=</span> <span class="n">expr</span><span class="p">:</span>
-                    <span class="nb">vars</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">v</span> <span class="o">=</span> <span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">k</span><span class="o">*</span><span class="n">v</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">hit</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="k">break</span>
-    <span class="k">return</span> <span class="n">expr</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_split_gcd</span><span class="p">(</span><span class="o">*</span><span class="n">a</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    split the list of integers `a` into a list of integers a1 having</span>
-<span class="sd">    g = gcd(a1) and a list a2 whose elements are not divisible by g</span>
-<span class="sd">    Returns g, a1, a2</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.simplify import _split_gcd</span>
-<span class="sd">    &gt;&gt;&gt; _split_gcd(55,35,22,14,77,10)</span>
-<span class="sd">    (5, [55, 35, 10], [22, 14, 77])</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">g</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">b1</span> <span class="o">=</span> <span class="p">[</span><span class="n">g</span><span class="p">]</span>
-    <span class="n">b2</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-        <span class="n">g1</span> <span class="o">=</span> <span class="n">gcd</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">g1</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">b2</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">g1</span>
-            <span class="n">b1</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">g</span><span class="p">,</span> <span class="n">b1</span><span class="p">,</span> <span class="n">b2</span>
-
-
-<span class="k">def</span> <span class="nf">split_surds</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    split an expression with terms whose squares are rationals</span>
-<span class="sd">    into a sum of terms whose surds squared have gcd equal to g</span>
-<span class="sd">    and a sum of terms with surds squared prime with g</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import sqrt</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.simplify import split_surds</span>
-<span class="sd">    &gt;&gt;&gt; split_surds(3*sqrt(3) + sqrt(5)/7 + sqrt(6) + sqrt(10) + sqrt(15))</span>
-<span class="sd">    (3, sqrt(2) + sqrt(5) + 3, sqrt(5)/7 + sqrt(10))</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">args</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-    <span class="n">coeff_muls</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span>
-    <span class="n">surds</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">coeff_muls</span> <span class="k">if</span> <span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">]</span>
-    <span class="n">surds</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-    <span class="n">g</span><span class="p">,</span> <span class="n">b1</span><span class="p">,</span> <span class="n">b2</span> <span class="o">=</span> <span class="n">_split_gcd</span><span class="p">(</span><span class="o">*</span><span class="n">surds</span><span class="p">)</span>
-    <span class="n">g2</span> <span class="o">=</span> <span class="n">g</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">b2</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">b1</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="n">b1n</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="o">/</span><span class="n">g</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">b1</span><span class="p">]</span>
-        <span class="n">b1n</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">b1n</span> <span class="k">if</span> <span class="n">x</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">]</span>
-        <span class="c"># only a common factor has been factored; split again</span>
-        <span class="n">g1</span><span class="p">,</span> <span class="n">b1n</span><span class="p">,</span> <span class="n">b2</span> <span class="o">=</span> <span class="n">_split_gcd</span><span class="p">(</span><span class="o">*</span><span class="n">b1n</span><span class="p">)</span>
-        <span class="n">g2</span> <span class="o">=</span> <span class="n">g</span><span class="o">*</span><span class="n">g1</span>
-    <span class="n">a1v</span><span class="p">,</span> <span class="n">a2v</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">c</span><span class="p">,</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">coeff_muls</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">s</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">s</span><span class="o">.</span><span class="n">exp</span> <span class="o">==</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">:</span>
-            <span class="n">s1</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">base</span>
-            <span class="k">if</span> <span class="n">s1</span> <span class="ow">in</span> <span class="n">b1</span><span class="p">:</span>
-                <span class="n">a1v</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">c</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">s1</span><span class="o">/</span><span class="n">g2</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">a2v</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">c</span><span class="o">*</span><span class="n">s</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">a2v</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">c</span><span class="o">*</span><span class="n">s</span><span class="p">)</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">a1v</span><span class="p">)</span>
-    <span class="n">b</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">a2v</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">g2</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span>
-
-
-<span class="k">def</span> <span class="nf">rad_rationalize</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">den</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Rationalize num/den by removing square roots in the denominator;</span>
-<span class="sd">    num and den are sum of terms whose squares are rationals</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import sqrt</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.simplify import rad_rationalize</span>
-<span class="sd">    &gt;&gt;&gt; rad_rationalize(sqrt(3), 1 + sqrt(2)/3)</span>
-<span class="sd">    (-sqrt(3) + sqrt(6)/3, -7/9)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">den</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">num</span><span class="p">,</span> <span class="n">den</span>
-    <span class="n">g</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">split_surds</span><span class="p">(</span><span class="n">den</span><span class="p">)</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-    <span class="n">num</span> <span class="o">=</span> <span class="n">_mexpand</span><span class="p">((</span><span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">num</span><span class="p">)</span>
-    <span class="n">den</span> <span class="o">=</span> <span class="n">_mexpand</span><span class="p">(</span><span class="n">a</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">b</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">rad_rationalize</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="n">den</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="radsimp"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.simplify.radsimp">[docs]</a><span class="k">def</span> <span class="nf">radsimp</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">symbolic</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">max_terms</span><span class="o">=</span><span class="mi">4</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Rationalize the denominator by removing square roots.</span>
-
-<span class="sd">    Note: the expression returned from radsimp must be used with caution</span>
-<span class="sd">    since if the denominator contains symbols, it will be possible to make</span>
-<span class="sd">    substitutions that violate the assumptions of the simplification process:</span>
-<span class="sd">    that for a denominator matching a + b*sqrt(c), a != +/-b*sqrt(c). (If</span>
-<span class="sd">    there are no symbols, this assumptions is made valid by collecting terms</span>
-<span class="sd">    of sqrt(c) so the match variable ``a`` does not contain ``sqrt(c)``.) If</span>
-<span class="sd">    you do not want the simplification to occur for symbolic denominators, set</span>
-<span class="sd">    ``symbolic`` to False.</span>
-
-<span class="sd">    If there are more than ``max_terms`` radical terms do not simplify.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import radsimp, sqrt, Symbol, denom, pprint, I</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import a, b, c</span>
-
-<span class="sd">    &gt;&gt;&gt; radsimp(1/(I + 1))</span>
-<span class="sd">    (1 - I)/2</span>
-<span class="sd">    &gt;&gt;&gt; radsimp(1/(2 + sqrt(2)))</span>
-<span class="sd">    (-sqrt(2) + 2)/2</span>
-<span class="sd">    &gt;&gt;&gt; x,y = list(map(Symbol, &#39;xy&#39;))</span>
-<span class="sd">    &gt;&gt;&gt; e = ((2 + 2*sqrt(2))*x + (2 + sqrt(8))*y)/(2 + sqrt(2))</span>
-<span class="sd">    &gt;&gt;&gt; radsimp(e)</span>
-<span class="sd">    sqrt(2)*(x + y)</span>
-
-<span class="sd">    Terms are collected automatically:</span>
-
-<span class="sd">    &gt;&gt;&gt; r2 = sqrt(2)</span>
-<span class="sd">    &gt;&gt;&gt; r5 = sqrt(5)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(radsimp(1/(y*r2 + x*r2 + a*r5 + b*r5)))</span>
-<span class="sd">             ___              ___</span>
-<span class="sd">           \/ 5 *(-a - b) + \/ 2 *(x + y)</span>
-<span class="sd">    --------------------------------------------</span>
-<span class="sd">         2               2      2              2</span>
-<span class="sd">    - 5*a  - 10*a*b - 5*b  + 2*x  + 4*x*y + 2*y</span>
-
-<span class="sd">    If radicals in the denominator cannot be removed, the original expression</span>
-<span class="sd">    will be returned. If the denominator was 1 then any square roots will also</span>
-<span class="sd">    be collected:</span>
-
-<span class="sd">    &gt;&gt;&gt; radsimp(sqrt(2)*x + sqrt(2))</span>
-<span class="sd">    sqrt(2)*(x + 1)</span>
-
-<span class="sd">    Results with symbols will not always be valid for all substitutions:</span>
-
-<span class="sd">    &gt;&gt;&gt; eq = 1/(a + b*sqrt(c))</span>
-<span class="sd">    &gt;&gt;&gt; eq.subs(a, b*sqrt(c))</span>
-<span class="sd">    1/(2*b*sqrt(c))</span>
-<span class="sd">    &gt;&gt;&gt; radsimp(eq).subs(a, b*sqrt(c))</span>
-<span class="sd">    nan</span>
-
-<span class="sd">    If symbolic=False, symbolic denominators will not be transformed (but</span>
-<span class="sd">    numeric denominators will still be processed):</span>
-
-<span class="sd">    &gt;&gt;&gt; radsimp(eq, symbolic=False)</span>
-<span class="sd">    1/(a + b*sqrt(c))</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">handle</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Atom</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">symbolic</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">fraction</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">d</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-            <span class="n">nexpr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">handle</span><span class="p">(</span><span class="n">ai</span><span class="p">)</span> <span class="k">for</span> <span class="n">ai</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-            <span class="k">return</span> <span class="n">nexpr</span>
-        <span class="k">elif</span> <span class="n">d</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="n">nargs</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">dargs</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">di</span> <span class="ow">in</span> <span class="n">d</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="n">ni</span><span class="p">,</span> <span class="n">di</span> <span class="o">=</span> <span class="n">fraction</span><span class="p">(</span><span class="n">handle</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">di</span><span class="p">))</span>
-                <span class="n">nargs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ni</span><span class="p">)</span>
-                <span class="n">dargs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">di</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">n</span><span class="o">*</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">nargs</span><span class="p">)</span><span class="o">/</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">dargs</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">d</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="n">radsimp</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">d</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">d</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">d</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">q</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="n">sqrtdenest</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">base</span><span class="p">))</span><span class="o">**</span><span class="n">d</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">p</span>
-
-        <span class="n">changed</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># collect similar terms</span>
-            <span class="n">d</span><span class="p">,</span> <span class="n">nterms</span> <span class="o">=</span> <span class="n">collect_sqrt</span><span class="p">(</span><span class="n">_mexpand</span><span class="p">(</span><span class="n">d</span><span class="p">),</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="n">Add</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">nterms</span> <span class="o">&gt;</span> <span class="n">max_terms</span><span class="p">:</span>
-                <span class="k">break</span>
-
-            <span class="c"># check to see if we are done:</span>
-            <span class="c"># - no radical terms</span>
-            <span class="c"># - if there are more than 3 radical terms, or</span>
-            <span class="c">#   there 3 radical terms and a constant, use rad_rationalize</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">nterms</span><span class="p">:</span>
-                <span class="k">break</span>
-            <span class="k">if</span> <span class="n">nterms</span> <span class="o">&gt;</span> <span class="mi">3</span> <span class="ow">or</span> <span class="n">nterms</span> <span class="o">==</span> <span class="mi">3</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">4</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">all</span><span class="p">([(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">is_Integer</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">d</span><span class="o">.</span><span class="n">args</span><span class="p">]):</span>
-                    <span class="n">nd</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">rad_rationalize</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
-                    <span class="n">n</span> <span class="o">=</span> <span class="n">_mexpand</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">nd</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">fraction</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-                <span class="k">break</span>
-            <span class="n">changed</span> <span class="o">=</span> <span class="bp">True</span>
-
-            <span class="c"># now match for a radical</span>
-            <span class="k">if</span> <span class="n">d</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">4</span><span class="p">:</span>
-                <span class="n">r</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="o">+</span> <span class="n">D</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">E</span><span class="p">))</span>
-                <span class="n">nmul</span> <span class="o">=</span> <span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="n">b</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="o">-</span> <span class="n">D</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">E</span><span class="p">))</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-                <span class="n">d</span> <span class="o">=</span> <span class="p">(</span><span class="n">a</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">c</span><span class="o">*</span><span class="n">b</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">E</span><span class="o">*</span><span class="n">D</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">b</span><span class="o">*</span><span class="n">D</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">c</span><span class="o">*</span><span class="n">E</span><span class="p">))</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-                <span class="n">n1</span> <span class="o">=</span> <span class="n">n</span><span class="o">/</span><span class="n">d</span>
-                <span class="k">if</span> <span class="n">denom</span><span class="p">(</span><span class="n">n1</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                    <span class="n">n</span> <span class="o">=</span> <span class="o">-</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="o">/</span><span class="n">d</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">n</span> <span class="o">=</span> <span class="n">n1</span>
-                <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">fraction</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">nmul</span><span class="p">)</span>
-
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">r</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">c</span><span class="p">))</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span> <span class="ow">or</span> <span class="n">r</span><span class="p">[</span><span class="n">b</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">r</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">b</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">c</span><span class="p">))</span>
-                    <span class="k">if</span> <span class="n">r</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="k">break</span>
-                    <span class="n">r</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-                <span class="n">va</span><span class="p">,</span> <span class="n">vb</span><span class="p">,</span> <span class="n">vc</span> <span class="o">=</span> <span class="n">r</span><span class="p">[</span><span class="n">a</span><span class="p">],</span> <span class="n">r</span><span class="p">[</span><span class="n">b</span><span class="p">],</span> <span class="n">r</span><span class="p">[</span><span class="n">c</span><span class="p">]</span>
-
-                <span class="n">dold</span> <span class="o">=</span> <span class="n">d</span>
-                <span class="n">d</span> <span class="o">=</span> <span class="n">va</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">vc</span><span class="o">*</span><span class="n">vb</span><span class="o">**</span><span class="mi">2</span>
-                <span class="n">nmul</span> <span class="o">=</span> <span class="n">va</span> <span class="o">-</span> <span class="n">vb</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">vc</span><span class="p">)</span>
-                <span class="n">n1</span> <span class="o">=</span> <span class="n">n</span><span class="o">/</span><span class="n">d</span>
-                <span class="k">if</span> <span class="n">denom</span><span class="p">(</span><span class="n">n1</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                    <span class="n">n1</span> <span class="o">=</span> <span class="o">-</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="o">/</span><span class="n">d</span><span class="p">)</span>
-                <span class="n">n1</span><span class="p">,</span> <span class="n">d1</span> <span class="o">=</span> <span class="n">fraction</span><span class="p">(</span><span class="n">n1</span><span class="o">*</span><span class="n">nmul</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">d1</span> <span class="o">!=</span> <span class="n">dold</span><span class="p">:</span>
-                    <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">n1</span><span class="p">,</span> <span class="n">d1</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">n</span> <span class="o">*=</span> <span class="n">nmul</span>
-
-        <span class="n">nexpr</span> <span class="o">=</span> <span class="n">collect_sqrt</span><span class="p">(</span><span class="n">expand_mul</span><span class="p">(</span><span class="n">n</span><span class="p">))</span><span class="o">/</span><span class="n">d</span>
-        <span class="k">if</span> <span class="n">changed</span> <span class="ow">or</span> <span class="n">nexpr</span> <span class="o">!=</span> <span class="n">expr</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">nexpr</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Wild</span><span class="p">,</span> <span class="s">&#39;abcDEFG&#39;</span><span class="p">))</span>
-    <span class="c"># do this at the start in case no other change is made since</span>
-    <span class="c"># it is done if a change is made</span>
-    <span class="n">coeff</span><span class="p">,</span> <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-
-    <span class="n">newe</span> <span class="o">=</span> <span class="n">handle</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">newe</span> <span class="o">!=</span> <span class="n">expr</span><span class="p">:</span>
-        <span class="n">co</span><span class="p">,</span> <span class="n">expr</span> <span class="o">=</span> <span class="n">newe</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-        <span class="n">coeff</span> <span class="o">*=</span> <span class="n">co</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">nexpr</span><span class="p">,</span> <span class="n">hit</span> <span class="o">=</span> <span class="n">collect_sqrt</span><span class="p">(</span><span class="n">expand_mul</span><span class="p">(</span><span class="n">expr</span><span class="p">),</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-        <span class="n">nexpr</span> <span class="o">=</span> <span class="n">Add</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">nexpr</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">hit</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">count_ops</span><span class="p">()</span> <span class="o">&gt;=</span> <span class="n">nexpr</span><span class="o">.</span><span class="n">count_ops</span><span class="p">():</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">nexpr</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="posify"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.simplify.posify">[docs]</a><span class="k">def</span> <span class="nf">posify</span><span class="p">(</span><span class="n">eq</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return eq (with generic symbols made positive) and a restore</span>
-<span class="sd">    dictionary.</span>
-
-<span class="sd">    Any symbol that has positive=None will be replaced with a positive dummy</span>
-<span class="sd">    symbol having the same name. This replacement will allow more symbolic</span>
-<span class="sd">    processing of expressions, especially those involving powers and</span>
-<span class="sd">    logarithms.</span>
-
-<span class="sd">    A dictionary that can be sent to subs to restore eq to its original</span>
-<span class="sd">    symbols is also returned.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import posify, Symbol, log</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; posify(x + Symbol(&#39;p&#39;, positive=True) + Symbol(&#39;n&#39;, negative=True))</span>
-<span class="sd">    (_x + n + p, {_x: x})</span>
-
-<span class="sd">    &gt;&gt; log(1/x).expand() # should be log(1/x) but it comes back as -log(x)</span>
-<span class="sd">    log(1/x)</span>
-
-<span class="sd">    &gt;&gt;&gt; log(posify(1/x)[0]).expand() # take [0] and ignore replacements</span>
-<span class="sd">    -log(_x)</span>
-<span class="sd">    &gt;&gt;&gt; eq, rep = posify(1/x)</span>
-<span class="sd">    &gt;&gt;&gt; log(eq).expand().subs(rep)</span>
-<span class="sd">    -log(x)</span>
-<span class="sd">    &gt;&gt;&gt; posify([x, 1 + x])</span>
-<span class="sd">    ([_x, _x + 1], {_x: x})</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">eq</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">iterable</span><span class="p">(</span><span class="n">eq</span><span class="p">):</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="nb">type</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span>
-        <span class="n">eq</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span>
-        <span class="n">syms</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">eq</span><span class="p">:</span>
-            <span class="n">syms</span> <span class="o">=</span> <span class="n">syms</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">))</span>
-        <span class="n">reps</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">syms</span><span class="p">:</span>
-            <span class="n">reps</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="nb">dict</span><span class="p">((</span><span class="n">v</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">posify</span><span class="p">(</span><span class="n">s</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">items</span><span class="p">())))</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">eq</span><span class="p">):</span>
-            <span class="n">eq</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">reps</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">f</span><span class="p">(</span><span class="n">eq</span><span class="p">),</span> <span class="nb">dict</span><span class="p">([(</span><span class="n">r</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span><span class="p">,</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">reps</span><span class="o">.</span><span class="n">items</span><span class="p">()])</span>
-
-    <span class="n">reps</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">([(</span><span class="n">s</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-                 <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">eq</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Symbol</span><span class="p">)</span> <span class="k">if</span> <span class="n">s</span><span class="o">.</span><span class="n">is_positive</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">])</span>
-    <span class="n">eq</span> <span class="o">=</span> <span class="n">eq</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">reps</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">eq</span><span class="p">,</span> <span class="nb">dict</span><span class="p">([(</span><span class="n">r</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span><span class="p">,</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">reps</span><span class="o">.</span><span class="n">items</span><span class="p">()])</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_polarify</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">lift</span><span class="p">,</span> <span class="n">pause</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">polar_lift</span><span class="p">,</span> <span class="n">Integral</span>
-    <span class="k">if</span> <span class="n">eq</span><span class="o">.</span><span class="n">is_polar</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">eq</span>
-    <span class="k">if</span> <span class="n">eq</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">pause</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">polar_lift</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">pause</span> <span class="ow">and</span> <span class="n">lift</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">polar_lift</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">eq</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">eq</span>
-    <span class="k">elif</span> <span class="n">eq</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">eq</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">_polarify</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">lift</span><span class="p">,</span> <span class="n">pause</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">eq</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">lift</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">polar_lift</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">r</span>
-    <span class="k">elif</span> <span class="n">eq</span><span class="o">.</span><span class="n">is_Function</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">eq</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">_polarify</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">lift</span><span class="p">,</span> <span class="n">pause</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">eq</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">Integral</span><span class="p">):</span>
-        <span class="c"># Don&#39;t lift the integration variable</span>
-        <span class="n">func</span> <span class="o">=</span> <span class="n">_polarify</span><span class="p">(</span><span class="n">eq</span><span class="o">.</span><span class="n">function</span><span class="p">,</span> <span class="n">lift</span><span class="p">,</span> <span class="n">pause</span><span class="o">=</span><span class="n">pause</span><span class="p">)</span>
-        <span class="n">limits</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">limit</span> <span class="ow">in</span> <span class="n">eq</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-            <span class="n">var</span> <span class="o">=</span> <span class="n">_polarify</span><span class="p">(</span><span class="n">limit</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">lift</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">pause</span><span class="o">=</span><span class="n">pause</span><span class="p">)</span>
-            <span class="n">rest</span> <span class="o">=</span> <span class="n">_polarify</span><span class="p">(</span><span class="n">limit</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">lift</span><span class="o">=</span><span class="n">lift</span><span class="p">,</span> <span class="n">pause</span><span class="o">=</span><span class="n">pause</span><span class="p">)</span>
-            <span class="n">limits</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">var</span><span class="p">,)</span> <span class="o">+</span> <span class="n">rest</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Integral</span><span class="p">(</span><span class="o">*</span><span class="p">((</span><span class="n">func</span><span class="p">,)</span> <span class="o">+</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">limits</span><span class="p">)))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">eq</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">_polarify</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">lift</span><span class="p">,</span> <span class="n">pause</span><span class="o">=</span><span class="n">pause</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">eq</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-
-<span class="k">def</span> <span class="nf">polarify</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">subs</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">lift</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Turn all numbers in eq into their polar equivalents (under the standard</span>
-<span class="sd">    choice of argument).</span>
-
-<span class="sd">    Note that no attempt is made to guess a formal convention of adding</span>
-<span class="sd">    polar numbers, expressions like 1 + x will generally not be altered.</span>
-
-<span class="sd">    Note also that this function does not promote exp(x) to exp_polar(x).</span>
-
-<span class="sd">    If ``subs`` is True, all symbols which are not already polar will be</span>
-<span class="sd">    substituted for polar dummies; in this case the function behaves much</span>
-<span class="sd">    like posify.</span>
-
-<span class="sd">    If ``lift`` is True, both addition statements and non-polar symbols are</span>
-<span class="sd">    changed to their polar_lift()ed versions.</span>
-<span class="sd">    Note that lift=True implies subs=False.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import polarify, sin, I</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; expr = (-x)**y</span>
-<span class="sd">    &gt;&gt;&gt; expr.expand()</span>
-<span class="sd">    (-x)**y</span>
-<span class="sd">    &gt;&gt;&gt; polarify(expr)</span>
-<span class="sd">    ((_x*exp_polar(I*pi))**_y, {_x: x, _y: y})</span>
-<span class="sd">    &gt;&gt;&gt; polarify(expr)[0].expand()</span>
-<span class="sd">    _x**_y*exp_polar(_y*I*pi)</span>
-<span class="sd">    &gt;&gt;&gt; polarify(x, lift=True)</span>
-<span class="sd">    polar_lift(x)</span>
-<span class="sd">    &gt;&gt;&gt; polarify(x*(1+y), lift=True)</span>
-<span class="sd">    polar_lift(x)*polar_lift(y + 1)</span>
-
-<span class="sd">    Adds are treated carefully:</span>
-
-<span class="sd">    &gt;&gt;&gt; polarify(1 + sin((1 + I)*x))</span>
-<span class="sd">    (sin(_x*polar_lift(1 + I)) + 1, {_x: x})</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">lift</span><span class="p">:</span>
-        <span class="n">subs</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">eq</span> <span class="o">=</span> <span class="n">_polarify</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">eq</span><span class="p">),</span> <span class="n">lift</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">subs</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">eq</span>
-    <span class="n">reps</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">([(</span><span class="n">s</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="n">polar</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">eq</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Symbol</span><span class="p">)])</span>
-    <span class="n">eq</span> <span class="o">=</span> <span class="n">eq</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">reps</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">eq</span><span class="p">,</span> <span class="nb">dict</span><span class="p">([(</span><span class="n">r</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span><span class="p">,</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">reps</span><span class="o">.</span><span class="n">items</span><span class="p">()])</span>
-
-
-<span class="k">def</span> <span class="nf">_unpolarify</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">exponents_only</span><span class="p">,</span> <span class="n">pause</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">polar_lift</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">principal_branch</span><span class="p">,</span> <span class="n">pi</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="nb">bool</span><span class="p">)</span> <span class="ow">or</span> <span class="n">eq</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">eq</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">pause</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">eq</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">exp_polar</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">exp</span><span class="p">(</span><span class="n">_unpolarify</span><span class="p">(</span><span class="n">eq</span><span class="o">.</span><span class="n">exp</span><span class="p">,</span> <span class="n">exponents_only</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">eq</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">principal_branch</span> <span class="ow">and</span> <span class="n">eq</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_unpolarify</span><span class="p">(</span><span class="n">eq</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">exponents_only</span><span class="p">)</span>
-        <span class="k">if</span> <span class="p">(</span>
-            <span class="n">eq</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">or</span> <span class="n">eq</span><span class="o">.</span><span class="n">is_Mul</span> <span class="ow">or</span> <span class="n">eq</span><span class="o">.</span><span class="n">is_Boolean</span> <span class="ow">or</span>
-            <span class="n">eq</span><span class="o">.</span><span class="n">is_Relational</span> <span class="ow">and</span> <span class="p">(</span>
-                <span class="n">eq</span><span class="o">.</span><span class="n">rel_op</span> <span class="ow">in</span> <span class="p">(</span><span class="s">&#39;==&#39;</span><span class="p">,</span> <span class="s">&#39;!=&#39;</span><span class="p">)</span> <span class="ow">and</span> <span class="mi">0</span> <span class="ow">in</span> <span class="n">eq</span><span class="o">.</span><span class="n">args</span> <span class="ow">or</span>
-                <span class="n">eq</span><span class="o">.</span><span class="n">rel_op</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">(</span><span class="s">&#39;==&#39;</span><span class="p">,</span> <span class="s">&#39;!=&#39;</span><span class="p">))</span>
-        <span class="p">):</span>
-            <span class="k">return</span> <span class="n">eq</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">_unpolarify</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">exponents_only</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">eq</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">eq</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">polar_lift</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_unpolarify</span><span class="p">(</span><span class="n">eq</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">exponents_only</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">eq</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-        <span class="n">expo</span> <span class="o">=</span> <span class="n">_unpolarify</span><span class="p">(</span><span class="n">eq</span><span class="o">.</span><span class="n">exp</span><span class="p">,</span> <span class="n">exponents_only</span><span class="p">)</span>
-        <span class="n">base</span> <span class="o">=</span> <span class="n">_unpolarify</span><span class="p">(</span><span class="n">eq</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">exponents_only</span><span class="p">,</span>
-            <span class="ow">not</span> <span class="p">(</span><span class="n">expo</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">pause</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">base</span><span class="o">**</span><span class="n">expo</span>
-
-    <span class="k">if</span> <span class="n">eq</span><span class="o">.</span><span class="n">is_Function</span> <span class="ow">and</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">eq</span><span class="o">.</span><span class="n">func</span><span class="p">,</span> <span class="s">&#39;unbranched&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">eq</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">_unpolarify</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">exponents_only</span><span class="p">,</span> <span class="n">exponents_only</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">eq</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">return</span> <span class="n">eq</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">_unpolarify</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">exponents_only</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">eq</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-
-<span class="k">def</span> <span class="nf">unpolarify</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">subs</span><span class="o">=</span><span class="p">{},</span> <span class="n">exponents_only</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    If p denotes the projection from the Riemann surface of the logarithm to</span>
-<span class="sd">    the complex line, return a simplified version eq&#39; of `eq` such that</span>
-<span class="sd">    p(eq&#39;) == p(eq).</span>
-<span class="sd">    Also apply the substitution subs in the end. (This is a convenience, since</span>
-<span class="sd">    ``unpolarify``, in a certain sense, undoes polarify.)</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import unpolarify, polar_lift, sin, I</span>
-<span class="sd">    &gt;&gt;&gt; unpolarify(polar_lift(I + 2))</span>
-<span class="sd">    2 + I</span>
-<span class="sd">    &gt;&gt;&gt; unpolarify(sin(polar_lift(I + 7)))</span>
-<span class="sd">    sin(7 + I)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exp_polar</span><span class="p">,</span> <span class="n">polar_lift</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="nb">bool</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">eq</span>
-
-    <span class="n">eq</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">subs</span> <span class="o">!=</span> <span class="p">{}:</span>
-        <span class="k">return</span> <span class="n">unpolarify</span><span class="p">(</span><span class="n">eq</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">subs</span><span class="p">))</span>
-    <span class="n">changed</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">pause</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="k">if</span> <span class="n">exponents_only</span><span class="p">:</span>
-        <span class="n">pause</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="k">while</span> <span class="n">changed</span><span class="p">:</span>
-        <span class="n">changed</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="n">_unpolarify</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">exponents_only</span><span class="p">,</span> <span class="n">pause</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">res</span> <span class="o">!=</span> <span class="n">eq</span><span class="p">:</span>
-            <span class="n">changed</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="n">eq</span> <span class="o">=</span> <span class="n">res</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">res</span><span class="p">,</span> <span class="nb">bool</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">res</span>
-    <span class="c"># Finally, replacing Exp(0) by 1 is always correct.</span>
-    <span class="c"># So is polar_lift(0) -&gt; 0.</span>
-    <span class="k">return</span> <span class="n">res</span><span class="o">.</span><span class="n">subs</span><span class="p">({</span><span class="n">exp_polar</span><span class="p">(</span><span class="mi">0</span><span class="p">):</span> <span class="mi">1</span><span class="p">,</span> <span class="n">polar_lift</span><span class="p">(</span><span class="mi">0</span><span class="p">):</span> <span class="mi">0</span><span class="p">})</span>
-
-
-<span class="k">def</span> <span class="nf">_denest_pow</span><span class="p">(</span><span class="n">eq</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Denest powers.</span>
-
-<span class="sd">    This is a helper function for powdenest that performs the actual</span>
-<span class="sd">    transformation.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">eq</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-
-    <span class="c"># denest exp with log terms in exponent</span>
-    <span class="k">if</span> <span class="n">b</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Exp1</span> <span class="ow">and</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-        <span class="n">logs</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">other</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">ei</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">ai</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">C</span><span class="o">.</span><span class="n">log</span> <span class="k">for</span> <span class="n">ai</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">ei</span><span class="p">)):</span>
-                <span class="n">logs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ei</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">other</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ei</span><span class="p">)</span>
-        <span class="n">logs</span> <span class="o">=</span> <span class="n">logcombine</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">logs</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">Pow</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">logs</span><span class="p">),</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">other</span><span class="p">))</span>
-
-    <span class="n">_</span><span class="p">,</span> <span class="n">be</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-    <span class="k">if</span> <span class="n">be</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="ow">and</span> <span class="ow">not</span> <span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">is_Mul</span> <span class="ow">or</span>
-                            <span class="n">b</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">b</span><span class="o">.</span><span class="n">q</span> <span class="o">!=</span> <span class="mi">1</span> <span class="ow">or</span>
-                            <span class="n">b</span><span class="o">.</span><span class="n">is_positive</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">eq</span>
-
-    <span class="c"># denest eq which is either pos**e or Pow**e or Mul**e or Mul(b1**e1, b2**e2)</span>
-
-    <span class="c"># handle polar numbers specially</span>
-    <span class="n">polars</span><span class="p">,</span> <span class="n">nonpolars</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">bb</span> <span class="ow">in</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">b</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">bb</span><span class="o">.</span><span class="n">is_polar</span><span class="p">:</span>
-            <span class="n">polars</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">bb</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">())</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">nonpolars</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">bb</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">polars</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">polars</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">Pow</span><span class="p">(</span><span class="n">polars</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span> <span class="n">polars</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">e</span><span class="p">)</span><span class="o">*</span><span class="n">powdenest</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">nonpolars</span><span class="p">)</span><span class="o">**</span><span class="n">e</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">polars</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">powdenest</span><span class="p">(</span><span class="n">bb</span><span class="o">**</span><span class="p">(</span><span class="n">ee</span><span class="o">*</span><span class="n">e</span><span class="p">))</span> <span class="k">for</span> <span class="p">(</span><span class="n">bb</span><span class="p">,</span> <span class="n">ee</span><span class="p">)</span> <span class="ow">in</span> <span class="n">polars</span><span class="p">])</span> \
-            <span class="o">*</span><span class="n">powdenest</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">nonpolars</span><span class="p">)</span><span class="o">**</span><span class="n">e</span><span class="p">)</span>
-
-    <span class="c"># see if there is a positive, non-Mul base at the very bottom</span>
-    <span class="n">exponents</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">kernel</span> <span class="o">=</span> <span class="n">eq</span>
-    <span class="k">while</span> <span class="n">kernel</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-        <span class="n">kernel</span><span class="p">,</span> <span class="n">ex</span> <span class="o">=</span> <span class="n">kernel</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-        <span class="n">exponents</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ex</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">kernel</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-        <span class="n">e</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">exponents</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">kernel</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="n">kernel</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">kernel</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="c"># use log to see if there is a power here</span>
-                <span class="n">logkernel</span> <span class="o">=</span> <span class="n">log</span><span class="p">(</span><span class="n">kernel</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">logkernel</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                    <span class="n">c</span><span class="p">,</span> <span class="n">logk</span> <span class="o">=</span> <span class="n">logkernel</span><span class="o">.</span><span class="n">args</span>
-                    <span class="n">e</span> <span class="o">*=</span> <span class="n">c</span>
-                    <span class="n">kernel</span> <span class="o">=</span> <span class="n">logk</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">Pow</span><span class="p">(</span><span class="n">kernel</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span>
-
-    <span class="c"># if any factor is an atom then there is nothing to be done</span>
-    <span class="c"># but the kernel check may have created a new exponent</span>
-    <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">is_Atom</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">b</span><span class="p">)):</span>
-        <span class="k">if</span> <span class="n">exponents</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">b</span><span class="o">**</span><span class="n">e</span>
-        <span class="k">return</span> <span class="n">eq</span>
-
-    <span class="c"># let log handle the case of the base of the argument being a mul, e.g.</span>
-    <span class="c"># sqrt(x**(2*i)*y**(6*i)) -&gt; x**i*y**(3**i) if x and y are positive; we</span>
-    <span class="c"># will take the log, expand it, and then factor out the common powers that</span>
-    <span class="c"># now appear as coefficient. We do this manually since terms_gcd pulls out</span>
-    <span class="c"># fractions, terms_gcd(x+x*y/2) -&gt; x*(y + 2)/2 and we don&#39;t want the 1/2;</span>
-    <span class="c"># gcd won&#39;t pull out numerators from a fraction: gcd(3*x, 9*x/2) -&gt; x but</span>
-    <span class="c"># we want 3*x. Neither work with noncommutatives.</span>
-    <span class="k">def</span> <span class="nf">nc_gcd</span><span class="p">(</span><span class="n">aa</span><span class="p">,</span> <span class="n">bb</span><span class="p">):</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="n">aa</span><span class="p">,</span> <span class="n">bb</span><span class="p">]]</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">gcd</span><span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">(</span><span class="n">cset</span><span class="o">=</span><span class="bp">True</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&amp;</span> <span class="n">b</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">(</span><span class="n">cset</span><span class="o">=</span><span class="bp">True</span><span class="p">)[</span><span class="mi">0</span><span class="p">]))</span>
-        <span class="k">return</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-
-    <span class="n">glogb</span> <span class="o">=</span> <span class="n">expand_log</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
-    <span class="k">if</span> <span class="n">glogb</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="n">glogb</span><span class="o">.</span><span class="n">args</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="nb">reduce</span><span class="p">(</span><span class="n">nc_gcd</span><span class="p">,</span> <span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">g</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">cg</span><span class="p">,</span> <span class="n">rg</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-            <span class="n">glogb</span> <span class="o">=</span> <span class="n">_keep_coeff</span><span class="p">(</span><span class="n">cg</span><span class="p">,</span> <span class="n">rg</span><span class="o">*</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">a</span><span class="o">/</span><span class="n">g</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]))</span>
-
-    <span class="c"># now put the log back together again</span>
-    <span class="k">if</span> <span class="n">glogb</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">C</span><span class="o">.</span><span class="n">log</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">glogb</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">glogb</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">or</span> <span class="n">glogb</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">exp</span><span class="p">:</span>
-            <span class="n">glogb</span> <span class="o">=</span> <span class="n">_denest_pow</span><span class="p">(</span><span class="n">glogb</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="k">if</span> <span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">glogb</span><span class="o">.</span><span class="n">exp</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Pow</span><span class="p">(</span><span class="n">glogb</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">glogb</span><span class="o">.</span><span class="n">exp</span><span class="o">*</span><span class="n">e</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">eq</span>
-
-    <span class="c"># the log(b) was a Mul so join any adds with logcombine</span>
-    <span class="n">add</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">other</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">glogb</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">add</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">other</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">Pow</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">logcombine</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">add</span><span class="p">))),</span> <span class="n">e</span><span class="o">*</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">other</span><span class="p">))</span>
-
-
-<div class="viewcode-block" id="powdenest"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.simplify.powdenest">[docs]</a><span class="k">def</span> <span class="nf">powdenest</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">polar</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Collect exponents on powers as assumptions allow.</span>
-
-<span class="sd">    Given ``(bb**be)**e``, this can be simplified as follows:</span>
-<span class="sd">        * if ``bb`` is positive, or</span>
-<span class="sd">        * ``e`` is an integer, or</span>
-<span class="sd">        * ``|be| &lt; 1`` then this simplifies to ``bb**(be*e)``</span>
-
-<span class="sd">    Given a product of powers raised to a power, ``(bb1**be1 *</span>
-<span class="sd">    bb2**be2...)**e``, simplification can be done as follows:</span>
-
-<span class="sd">    - if e is positive, the gcd of all bei can be joined with e;</span>
-<span class="sd">    - all non-negative bb can be separated from those that are negative</span>
-<span class="sd">      and their gcd can be joined with e; autosimplification already</span>
-<span class="sd">      handles this separation.</span>
-<span class="sd">    - integer factors from powers that have integers in the denominator</span>
-<span class="sd">      of the exponent can be removed from any term and the gcd of such</span>
-<span class="sd">      integers can be joined with e</span>
-
-<span class="sd">    Setting ``force`` to True will make symbols that are not explicitly</span>
-<span class="sd">    negative behave as though they are positive, resulting in more</span>
-<span class="sd">    denesting.</span>
-
-<span class="sd">    Setting ``polar`` to True will do simplifications on the riemann surface of</span>
-<span class="sd">    the logarithm, also resulting in more denestings.</span>
-
-<span class="sd">    When there are sums of logs in exp() then a product of powers may be</span>
-<span class="sd">    obtained e.g. ``exp(3*(log(a) + 2*log(b)))`` - &gt; ``a**3*b**6``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import a, b, x, y, z</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, exp, log, sqrt, symbols, powdenest</span>
-
-<span class="sd">    &gt;&gt;&gt; powdenest((x**(2*a/3))**(3*x))</span>
-<span class="sd">    (x**(2*a/3))**(3*x)</span>
-<span class="sd">    &gt;&gt;&gt; powdenest(exp(3*x*log(2)))</span>
-<span class="sd">    2**(3*x)</span>
-
-<span class="sd">    Assumptions may prevent expansion:</span>
-
-<span class="sd">    &gt;&gt;&gt; powdenest(sqrt(x**2))</span>
-<span class="sd">    sqrt(x**2)</span>
-
-<span class="sd">    &gt;&gt;&gt; p = symbols(&#39;p&#39;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; powdenest(sqrt(p**2))</span>
-<span class="sd">    p</span>
-
-<span class="sd">    No other expansion is done.</span>
-
-<span class="sd">    &gt;&gt;&gt; i, j = symbols(&#39;i,j&#39;, integer=True)</span>
-<span class="sd">    &gt;&gt;&gt; powdenest((x**x)**(i + j)) # -X-&gt; (x**x)**i*(x**x)**j</span>
-<span class="sd">    x**(x*(i + j))</span>
-
-<span class="sd">    But exp() will be denested by moving all non-log terms outside of</span>
-<span class="sd">    the function; this may result in the collapsing of the exp to a power</span>
-<span class="sd">    with a different base:</span>
-
-<span class="sd">    &gt;&gt;&gt; powdenest(exp(3*y*log(x)))</span>
-<span class="sd">    x**(3*y)</span>
-<span class="sd">    &gt;&gt;&gt; powdenest(exp(y*(log(a) + log(b))))</span>
-<span class="sd">    (a*b)**y</span>
-<span class="sd">    &gt;&gt;&gt; powdenest(exp(3*(log(a) + log(b))))</span>
-<span class="sd">    a**3*b**3</span>
-
-<span class="sd">    If assumptions allow, symbols can also be moved to the outermost exponent:</span>
-
-<span class="sd">    &gt;&gt;&gt; i = Symbol(&#39;i&#39;, integer=True)</span>
-<span class="sd">    &gt;&gt;&gt; p = Symbol(&#39;p&#39;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; powdenest(((x**(2*i))**(3*y))**x)</span>
-<span class="sd">    ((x**(2*i))**(3*y))**x</span>
-<span class="sd">    &gt;&gt;&gt; powdenest(((x**(2*i))**(3*y))**x, force=True)</span>
-<span class="sd">    x**(6*i*x*y)</span>
-
-<span class="sd">    &gt;&gt;&gt; powdenest(((p**(2*a))**(3*y))**x)</span>
-<span class="sd">    p**(6*a*x*y)</span>
-
-<span class="sd">    &gt;&gt;&gt; powdenest(((x**(2*a/3))**(3*y/i))**x)</span>
-<span class="sd">    ((x**(2*a/3))**(3*y/i))**x</span>
-<span class="sd">    &gt;&gt;&gt; powdenest((x**(2*i)*y**(4*i))**z, force=True)</span>
-<span class="sd">    (x*y**2)**(2*i*z)</span>
-
-<span class="sd">    &gt;&gt;&gt; n = Symbol(&#39;n&#39;, negative=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; powdenest((x**i)**y, force=True)</span>
-<span class="sd">    x**(i*y)</span>
-<span class="sd">    &gt;&gt;&gt; powdenest((n**i)**x, force=True)</span>
-<span class="sd">    (n**i)**x</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="n">force</span><span class="p">:</span>
-        <span class="n">eq</span><span class="p">,</span> <span class="n">rep</span> <span class="o">=</span> <span class="n">posify</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">powdenest</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="n">rep</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">polar</span><span class="p">:</span>
-        <span class="n">eq</span><span class="p">,</span> <span class="n">rep</span> <span class="o">=</span> <span class="n">polarify</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">unpolarify</span><span class="p">(</span><span class="n">powdenest</span><span class="p">(</span><span class="n">unpolarify</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">exponents_only</span><span class="o">=</span><span class="bp">True</span><span class="p">)),</span> <span class="n">rep</span><span class="p">)</span>
-
-    <span class="n">new</span> <span class="o">=</span> <span class="n">powsimp</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">eq</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">new</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="n">Transform</span><span class="p">(</span><span class="n">_denest_pow</span><span class="p">,</span> <span class="nb">filter</span><span class="o">=</span><span class="k">lambda</span> <span class="n">m</span><span class="p">:</span> <span class="n">m</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">or</span> <span class="n">m</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">exp</span><span class="p">))</span>
-</div>
-<span class="n">_y</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="powsimp"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.simplify.powsimp">[docs]</a><span class="k">def</span> <span class="nf">powsimp</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;all&#39;</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">measure</span><span class="o">=</span><span class="n">count_ops</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    reduces expression by combining powers with similar bases and exponents.</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    If deep is True then powsimp() will also simplify arguments of</span>
-<span class="sd">    functions. By default deep is set to False.</span>
-
-<span class="sd">    If force is True then bases will be combined without checking for</span>
-<span class="sd">    assumptions, e.g. sqrt(x)*sqrt(y) -&gt; sqrt(x*y) which is not true</span>
-<span class="sd">    if x and y are both negative.</span>
-
-<span class="sd">    You can make powsimp() only combine bases or only combine exponents by</span>
-<span class="sd">    changing combine=&#39;base&#39; or combine=&#39;exp&#39;.  By default, combine=&#39;all&#39;,</span>
-<span class="sd">    which does both.  combine=&#39;base&#39; will only combine::</span>
-
-<span class="sd">         a   a          a                          2x      x</span>
-<span class="sd">        x * y  =&gt;  (x*y)   as well as things like 2   =&gt;  4</span>
-
-<span class="sd">    and combine=&#39;exp&#39; will only combine</span>
-<span class="sd">    ::</span>
-
-<span class="sd">         a   b      (a + b)</span>
-<span class="sd">        x * x  =&gt;  x</span>
-
-<span class="sd">    combine=&#39;exp&#39; will strictly only combine exponents in the way that used</span>
-<span class="sd">    to be automatic.  Also use deep=True if you need the old behavior.</span>
-
-<span class="sd">    When combine=&#39;all&#39;, &#39;exp&#39; is evaluated first.  Consider the first</span>
-<span class="sd">    example below for when there could be an ambiguity relating to this.</span>
-<span class="sd">    This is done so things like the second example can be completely</span>
-<span class="sd">    combined.  If you want &#39;base&#39; combined first, do something like</span>
-<span class="sd">    powsimp(powsimp(expr, combine=&#39;base&#39;), combine=&#39;exp&#39;).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import powsimp, exp, log, symbols</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z, n</span>
-<span class="sd">    &gt;&gt;&gt; powsimp(x**y*x**z*y**z, combine=&#39;all&#39;)</span>
-<span class="sd">    x**(y + z)*y**z</span>
-<span class="sd">    &gt;&gt;&gt; powsimp(x**y*x**z*y**z, combine=&#39;exp&#39;)</span>
-<span class="sd">    x**(y + z)*y**z</span>
-<span class="sd">    &gt;&gt;&gt; powsimp(x**y*x**z*y**z, combine=&#39;base&#39;, force=True)</span>
-<span class="sd">    x**y*(x*y)**z</span>
-
-<span class="sd">    &gt;&gt;&gt; powsimp(x**z*x**y*n**z*n**y, combine=&#39;all&#39;, force=True)</span>
-<span class="sd">    (n*x)**(y + z)</span>
-<span class="sd">    &gt;&gt;&gt; powsimp(x**z*x**y*n**z*n**y, combine=&#39;exp&#39;)</span>
-<span class="sd">    n**(y + z)*x**(y + z)</span>
-<span class="sd">    &gt;&gt;&gt; powsimp(x**z*x**y*n**z*n**y, combine=&#39;base&#39;, force=True)</span>
-<span class="sd">    (n*x)**y*(n*x)**z</span>
-
-<span class="sd">    &gt;&gt;&gt; x, y = symbols(&#39;x y&#39;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; powsimp(log(exp(x)*exp(y)))</span>
-<span class="sd">    log(exp(x)*exp(y))</span>
-<span class="sd">    &gt;&gt;&gt; powsimp(log(exp(x)*exp(y)), deep=True)</span>
-<span class="sd">    x + y</span>
-
-<span class="sd">    Radicals with Mul bases will be combined if combine=&#39;exp&#39;</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import sqrt, Mul</span>
-<span class="sd">    &gt;&gt;&gt; x, y = symbols(&#39;x y&#39;)</span>
-
-<span class="sd">    Two radicals are automatically joined through Mul:</span>
-<span class="sd">    &gt;&gt;&gt; a=sqrt(x*sqrt(y))</span>
-<span class="sd">    &gt;&gt;&gt; a*a**3 == a**4</span>
-<span class="sd">    True</span>
-
-<span class="sd">    But if an integer power of that radical has been</span>
-<span class="sd">    autoexpanded then Mul does not join the resulting factors:</span>
-<span class="sd">    &gt;&gt;&gt; a**4 # auto expands to a Mul, no longer a Pow</span>
-<span class="sd">    x**2*y</span>
-<span class="sd">    &gt;&gt;&gt; _*a # so Mul doesn&#39;t combine them</span>
-<span class="sd">    x**2*y*sqrt(x*sqrt(y))</span>
-<span class="sd">    &gt;&gt;&gt; powsimp(_) # but powsimp will</span>
-<span class="sd">    (x*sqrt(y))**(5/2)</span>
-<span class="sd">    &gt;&gt;&gt; powsimp(x*y*a) # but won&#39;t when doing so would violate assumptions</span>
-<span class="sd">    x*y*sqrt(x*sqrt(y))</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">recurse</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">_deep</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;deep&#39;</span><span class="p">,</span> <span class="n">deep</span><span class="p">)</span>
-        <span class="n">_combine</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;combine&#39;</span><span class="p">,</span> <span class="n">combine</span><span class="p">)</span>
-        <span class="n">_force</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;force&#39;</span><span class="p">,</span> <span class="n">force</span><span class="p">)</span>
-        <span class="n">_measure</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;measure&#39;</span><span class="p">,</span> <span class="n">measure</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">powsimp</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">_deep</span><span class="p">,</span> <span class="n">_combine</span><span class="p">,</span> <span class="n">_force</span><span class="p">,</span> <span class="n">_measure</span><span class="p">)</span>
-
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Basic</span><span class="p">)</span> <span class="ow">or</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Atom</span> <span class="ow">or</span> <span class="n">expr</span> <span class="ow">in</span> <span class="p">(</span><span class="n">exp_polar</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">exp_polar</span><span class="p">(</span><span class="mi">1</span><span class="p">)):</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="k">if</span> <span class="n">deep</span> <span class="ow">or</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">or</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span> <span class="ow">and</span> <span class="n">_y</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">recurse</span><span class="p">(</span><span class="n">w</span><span class="p">)</span> <span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">recurse</span><span class="p">(</span><span class="n">expr</span><span class="o">*</span><span class="n">_y</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span><span class="o">/</span><span class="n">_y</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="c"># handle the Mul</span>
-
-    <span class="k">if</span> <span class="n">combine</span> <span class="ow">in</span> <span class="p">(</span><span class="s">&#39;exp&#39;</span><span class="p">,</span> <span class="s">&#39;all&#39;</span><span class="p">):</span>
-        <span class="c"># Collect base/exp data, while maintaining order in the</span>
-        <span class="c"># non-commutative parts of the product</span>
-        <span class="n">c_powers</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">list</span><span class="p">)</span>
-        <span class="n">nc_part</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">newexpr</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">term</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-                <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-                    <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="p">[</span><span class="n">recurse</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">]]</span>
-                <span class="n">c_powers</span><span class="p">[</span><span class="n">b</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># This is the logic that combines exponents for equal,</span>
-                <span class="c"># but non-commutative bases: A**x*A**y == A**(x+y).</span>
-                <span class="k">if</span> <span class="n">nc_part</span><span class="p">:</span>
-                    <span class="n">b1</span><span class="p">,</span> <span class="n">e1</span> <span class="o">=</span> <span class="n">nc_part</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                    <span class="n">b2</span><span class="p">,</span> <span class="n">e2</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                    <span class="k">if</span> <span class="p">(</span><span class="n">b1</span> <span class="o">==</span> <span class="n">b2</span> <span class="ow">and</span>
-                            <span class="n">e1</span><span class="o">.</span><span class="n">is_commutative</span> <span class="ow">and</span> <span class="n">e2</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">):</span>
-                        <span class="n">nc_part</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">b1</span><span class="p">,</span> <span class="n">Add</span><span class="p">(</span><span class="n">e1</span><span class="p">,</span> <span class="n">e2</span><span class="p">))</span>
-                        <span class="k">continue</span>
-                <span class="n">nc_part</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-
-        <span class="c"># add up exponents of common bases</span>
-        <span class="k">for</span> <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">c_powers</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">c_powers</span><span class="p">[</span><span class="n">b</span><span class="p">]</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">e</span><span class="p">)</span>
-
-        <span class="c"># check for base and inverted base pairs</span>
-        <span class="n">be</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">c_powers</span><span class="o">.</span><span class="n">items</span><span class="p">())</span>
-        <span class="n">skip</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>  <span class="c"># skip if we already saw them</span>
-        <span class="k">for</span> <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">be</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">skip</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="n">bpos</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">is_positive</span> <span class="ow">or</span> <span class="n">b</span><span class="o">.</span><span class="n">is_polar</span>
-            <span class="k">if</span> <span class="n">bpos</span><span class="p">:</span>
-                <span class="n">binv</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">b</span>
-                <span class="k">if</span> <span class="n">b</span> <span class="o">!=</span> <span class="n">binv</span> <span class="ow">and</span> <span class="n">binv</span> <span class="ow">in</span> <span class="n">c_powers</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                        <span class="n">c_powers</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-                        <span class="n">c_powers</span><span class="p">[</span><span class="n">binv</span><span class="p">]</span> <span class="o">-=</span> <span class="n">e</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">skip</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">binv</span><span class="p">)</span>
-                        <span class="n">e</span> <span class="o">=</span> <span class="n">c_powers</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">binv</span><span class="p">)</span>
-                        <span class="n">c_powers</span><span class="p">[</span><span class="n">b</span><span class="p">]</span> <span class="o">-=</span> <span class="n">e</span>
-
-        <span class="c"># filter c_powers and convert to a list</span>
-        <span class="n">c_powers</span> <span class="o">=</span> <span class="p">[(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span> <span class="k">for</span> <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">c_powers</span><span class="o">.</span><span class="n">items</span><span class="p">()</span> <span class="k">if</span> <span class="n">e</span><span class="p">]</span>
-
-        <span class="c"># ==============================================================</span>
-        <span class="c"># check for Mul bases of Rational powers that can be combined with</span>
-        <span class="c"># separated bases, e.g. x*sqrt(x*y)*sqrt(x*sqrt(x*y)) -&gt; (x*sqrt(x*y))**(3/2)</span>
-        <span class="c"># ---------------- helper functions</span>
-        <span class="k">def</span> <span class="nf">ratq</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-            <span class="sd">&#39;&#39;&#39;Return Rational part of x&#39;s exponent as it appears in the bkey.</span>
-<span class="sd">            &#39;&#39;&#39;</span>
-            <span class="k">return</span> <span class="n">bkey</span><span class="p">(</span><span class="n">x</span><span class="p">)[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
-
-        <span class="k">def</span> <span class="nf">bkey</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-            <span class="sd">&#39;&#39;&#39;Return (b**s, c.q), c.p where e -&gt; c*s. If e is not given then</span>
-<span class="sd">            it will be taken by using as_base_exp() on the input b.</span>
-<span class="sd">            e.g.</span>
-<span class="sd">                x**3/2 -&gt; (x, 2), 3</span>
-<span class="sd">                x**y -&gt; (x**y, 1), 1</span>
-<span class="sd">                x**(2*y/3) -&gt; (x**y, 3), 2</span>
-<span class="sd">                exp(x/2) -&gt; (exp(a), 2), 1</span>
-
-<span class="sd">            &#39;&#39;&#39;</span>
-            <span class="k">if</span> <span class="n">e</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>  <span class="c"># coming from c_powers or from below</span>
-                <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">),</span> <span class="n">e</span>
-                <span class="k">elif</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Integer</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">q</span><span class="p">)),</span> <span class="n">Integer</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">c</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">(</span><span class="n">rational</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">c</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="p">(</span><span class="n">b</span><span class="o">**</span><span class="n">m</span><span class="p">,</span> <span class="n">Integer</span><span class="p">(</span><span class="n">c</span><span class="o">.</span><span class="n">q</span><span class="p">)),</span> <span class="n">Integer</span><span class="p">(</span><span class="n">c</span><span class="o">.</span><span class="n">p</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="p">(</span><span class="n">b</span><span class="o">**</span><span class="n">e</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">bkey</span><span class="p">(</span><span class="o">*</span><span class="n">b</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">())</span>
-
-        <span class="k">def</span> <span class="nf">update</span><span class="p">(</span><span class="n">b</span><span class="p">):</span>
-            <span class="sd">&#39;&#39;&#39;Decide what to do with base, b. If its exponent is now an</span>
-<span class="sd">            integer multiple of the Rational denominator, then remove it</span>
-<span class="sd">            and put the factors of its base in the common_b dictionary or</span>
-<span class="sd">            update the existing bases if necessary. If it has been zeroed</span>
-<span class="sd">            out, simply remove the base.</span>
-<span class="sd">            &#39;&#39;&#39;</span>
-            <span class="n">newe</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="nb">divmod</span><span class="p">(</span><span class="n">common_b</span><span class="p">[</span><span class="n">b</span><span class="p">],</span> <span class="n">b</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span><span class="p">:</span>
-                <span class="n">common_b</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">newe</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="n">newe</span><span class="p">):</span>
-                        <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">bkey</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">b</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">common_b</span><span class="p">:</span>
-                            <span class="n">common_b</span><span class="p">[</span><span class="n">b</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-                        <span class="n">common_b</span><span class="p">[</span><span class="n">b</span><span class="p">]</span> <span class="o">+=</span> <span class="n">e</span>
-                        <span class="k">if</span> <span class="n">b</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                            <span class="n">bases</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="c"># ---------------- end of helper functions</span>
-
-        <span class="c"># assemble a dictionary of the factors having a Rational power</span>
-        <span class="n">common_b</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="n">done</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">bases</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">c_powers</span><span class="p">:</span>
-            <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">bkey</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span>
-            <span class="n">common_b</span><span class="p">[</span><span class="n">b</span><span class="p">]</span> <span class="o">=</span> <span class="n">e</span>
-            <span class="k">if</span> <span class="n">b</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                <span class="n">bases</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="n">bases</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>  <span class="c"># this makes tie-breaking canonical</span>
-        <span class="n">bases</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">measure</span><span class="p">,</span> <span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>  <span class="c"># handle longest first</span>
-        <span class="k">for</span> <span class="n">base</span> <span class="ow">in</span> <span class="n">bases</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">base</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">common_b</span><span class="p">:</span>  <span class="c"># it may have been removed already</span>
-                <span class="k">continue</span>
-            <span class="n">b</span><span class="p">,</span> <span class="n">exponent</span> <span class="o">=</span> <span class="n">base</span>
-            <span class="n">last</span> <span class="o">=</span> <span class="bp">False</span>  <span class="c"># True when no factor of base is a radical</span>
-            <span class="n">qlcm</span> <span class="o">=</span> <span class="mi">1</span>  <span class="c"># the lcm of the radical denominators</span>
-            <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-                <span class="n">bstart</span> <span class="o">=</span> <span class="n">b</span>
-                <span class="n">qstart</span> <span class="o">=</span> <span class="n">qlcm</span>
-
-                <span class="n">bb</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c"># list of factors</span>
-                <span class="n">ee</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c"># (factor&#39;s exponent, current value of that exponent in common_b)</span>
-                <span class="k">for</span> <span class="n">bi</span> <span class="ow">in</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">b</span><span class="p">):</span>
-                    <span class="n">bib</span><span class="p">,</span> <span class="n">bie</span> <span class="o">=</span> <span class="n">bkey</span><span class="p">(</span><span class="n">bi</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">bib</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">common_b</span> <span class="ow">or</span> <span class="n">common_b</span><span class="p">[</span><span class="n">bib</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">bie</span><span class="p">:</span>
-                        <span class="n">ee</span> <span class="o">=</span> <span class="n">bb</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c"># failed</span>
-                        <span class="k">break</span>
-                    <span class="n">ee</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">bie</span><span class="p">,</span> <span class="n">common_b</span><span class="p">[</span><span class="n">bib</span><span class="p">]])</span>
-                    <span class="n">bb</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">bib</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">ee</span><span class="p">:</span>
-                    <span class="c"># find the number of extractions possible</span>
-                    <span class="c"># e.g. [(1, 2), (2, 2)] -&gt; min(2/1, 2/2) -&gt; 1</span>
-                    <span class="n">min1</span> <span class="o">=</span> <span class="n">ee</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">/</span><span class="n">ee</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-                    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">ee</span><span class="p">)):</span>
-                        <span class="n">rat</span> <span class="o">=</span> <span class="n">ee</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="o">/</span><span class="n">ee</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-                        <span class="k">if</span> <span class="n">rat</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-                            <span class="k">break</span>
-                        <span class="n">min1</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">min1</span><span class="p">,</span> <span class="n">rat</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="c"># update base factor counts</span>
-                        <span class="c"># e.g. if ee = [(2, 5), (3, 6)] then min1 = 2</span>
-                        <span class="c"># and the new base counts will be 5-2*2 and 6-2*3</span>
-                        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">bb</span><span class="p">)):</span>
-                            <span class="n">common_b</span><span class="p">[</span><span class="n">bb</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span> <span class="o">-=</span> <span class="n">min1</span><span class="o">*</span><span class="n">ee</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-                            <span class="n">update</span><span class="p">(</span><span class="n">bb</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-                        <span class="c"># update the count of the base</span>
-                        <span class="c"># e.g. x**2*y*sqrt(x*sqrt(y)) the count of x*sqrt(y)</span>
-                        <span class="c"># will increase by 4 to give bkey (x*sqrt(y), 2, 5)</span>
-                        <span class="n">common_b</span><span class="p">[</span><span class="n">base</span><span class="p">]</span> <span class="o">+=</span> <span class="n">min1</span><span class="o">*</span><span class="n">qstart</span><span class="o">*</span><span class="n">exponent</span>
-                <span class="k">if</span> <span class="p">(</span><span class="n">last</span>  <span class="c"># no more radicals in base</span>
-                    <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">common_b</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span>  <span class="c"># nothing left to join with</span>
-                    <span class="ow">or</span> <span class="nb">all</span><span class="p">(</span><span class="n">k</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">common_b</span><span class="p">)</span>  <span class="c"># no radicals left in common_b</span>
-                        <span class="p">):</span>
-                    <span class="k">break</span>
-                <span class="c"># see what we can exponentiate base by to remove any radicals</span>
-                <span class="c"># so we know what to search for</span>
-                <span class="c"># e.g. if base were x**(1/2)*y**(1/3) then we should exponentiate</span>
-                <span class="c"># by 6 and look for powers of x and y in the ratio of 2 to 3</span>
-                <span class="n">qlcm</span> <span class="o">=</span> <span class="n">lcm</span><span class="p">([</span><span class="n">ratq</span><span class="p">(</span><span class="n">bi</span><span class="p">)</span> <span class="k">for</span> <span class="n">bi</span> <span class="ow">in</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">bstart</span><span class="p">)])</span>
-                <span class="k">if</span> <span class="n">qlcm</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">break</span>  <span class="c"># we are done</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="n">bstart</span><span class="o">**</span><span class="n">qlcm</span>
-                <span class="n">qlcm</span> <span class="o">*=</span> <span class="n">qstart</span>
-                <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">ratq</span><span class="p">(</span><span class="n">bi</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">bi</span> <span class="ow">in</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">b</span><span class="p">)):</span>
-                    <span class="n">last</span> <span class="o">=</span> <span class="bp">True</span>  <span class="c"># we are going to be done after this next pass</span>
-            <span class="c"># this base no longer can find anything to join with and</span>
-            <span class="c"># since it was longer than any other we are done with it</span>
-            <span class="n">b</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">base</span>
-            <span class="n">done</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">b</span><span class="p">,</span> <span class="n">common_b</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">base</span><span class="p">)</span><span class="o">*</span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">q</span><span class="p">)))</span>
-
-        <span class="c"># update c_powers and get ready to continue with powsimp</span>
-        <span class="n">c_powers</span> <span class="o">=</span> <span class="n">done</span>
-        <span class="c"># there may be terms still in common_b that were bases that were</span>
-        <span class="c"># identified as needing processing, so remove those, too</span>
-        <span class="k">for</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">q</span><span class="p">),</span> <span class="n">e</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">common_b</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">or</span> <span class="n">b</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">exp</span><span class="p">)</span> <span class="ow">and</span> \
-                    <span class="n">q</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">b</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="n">b</span><span class="p">,</span> <span class="n">be</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="o">**</span><span class="p">(</span><span class="n">be</span><span class="o">/</span><span class="n">q</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="n">root</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
-            <span class="n">c_powers</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">))</span>
-        <span class="n">check</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">c_powers</span><span class="p">)</span>
-        <span class="n">c_powers</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">c_powers</span><span class="p">)</span>
-        <span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">c_powers</span><span class="p">)</span> <span class="o">==</span> <span class="n">check</span>  <span class="c"># there should have been no duplicates</span>
-        <span class="c"># ==============================================================</span>
-
-        <span class="c"># rebuild the expression</span>
-        <span class="n">newexpr</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span>
-            <span class="o">*</span><span class="p">(</span><span class="n">newexpr</span> <span class="o">+</span> <span class="p">[</span><span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span> <span class="k">for</span> <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">c_powers</span><span class="o">.</span><span class="n">items</span><span class="p">()]))</span>
-        <span class="k">if</span> <span class="n">combine</span> <span class="o">==</span> <span class="s">&#39;exp&#39;</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="n">newexpr</span><span class="p">,</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">nc_part</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">recurse</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">nc_part</span><span class="p">),</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;base&#39;</span><span class="p">)</span> <span class="o">*</span> \
-                <span class="n">recurse</span><span class="p">(</span><span class="n">newexpr</span><span class="p">,</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;base&#39;</span><span class="p">)</span>
-
-    <span class="k">elif</span> <span class="n">combine</span> <span class="o">==</span> <span class="s">&#39;base&#39;</span><span class="p">:</span>
-
-        <span class="c"># Build c_powers and nc_part.  These must both be lists not</span>
-        <span class="c"># dicts because exp&#39;s are not combined.</span>
-        <span class="n">c_powers</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">nc_part</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">term</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">:</span>
-                <span class="n">c_powers</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">term</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># This is the logic that combines bases that are</span>
-                <span class="c"># different and non-commutative, but with equal and</span>
-                <span class="c"># commutative exponents: A**x*B**x == (A*B)**x.</span>
-                <span class="k">if</span> <span class="n">nc_part</span><span class="p">:</span>
-                    <span class="n">b1</span><span class="p">,</span> <span class="n">e1</span> <span class="o">=</span> <span class="n">nc_part</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                    <span class="n">b2</span><span class="p">,</span> <span class="n">e2</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                    <span class="k">if</span> <span class="p">(</span><span class="n">e1</span> <span class="o">==</span> <span class="n">e2</span> <span class="ow">and</span> <span class="n">e2</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">):</span>
-                        <span class="n">nc_part</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="n">b1</span><span class="p">,</span> <span class="n">b2</span><span class="p">),</span> <span class="n">e1</span><span class="p">)</span>
-                        <span class="k">continue</span>
-                <span class="n">nc_part</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-
-        <span class="c"># Pull out numerical coefficients from exponent if assumptions allow</span>
-        <span class="c"># e.g., 2**(2*x) =&gt; 4**x</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">c_powers</span><span class="p">)):</span>
-            <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">c_powers</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">is_nonnegative</span> <span class="ow">or</span> <span class="n">e</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">or</span> <span class="n">force</span> <span class="ow">or</span> <span class="n">b</span><span class="o">.</span><span class="n">is_polar</span><span class="p">):</span>
-                <span class="k">continue</span>
-            <span class="n">exp_c</span><span class="p">,</span> <span class="n">exp_t</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">(</span><span class="n">rational</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">exp_c</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="ow">and</span> <span class="n">exp_t</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                <span class="n">c_powers</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">exp_c</span><span class="p">),</span> <span class="n">exp_t</span><span class="p">]</span>
-
-        <span class="c"># Combine bases whenever they have the same exponent and</span>
-        <span class="c"># assumptions allow</span>
-        <span class="c"># first gather the potential bases under the common exponent</span>
-        <span class="n">c_exp</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">list</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">c_powers</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">deep</span><span class="p">:</span>
-                <span class="n">e</span> <span class="o">=</span> <span class="n">recurse</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-            <span class="n">c_exp</span><span class="p">[</span><span class="n">e</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="k">del</span> <span class="n">c_powers</span>
-
-        <span class="c"># Merge back in the results of the above to form a new product</span>
-        <span class="n">c_powers</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">list</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">c_exp</span><span class="p">:</span>
-            <span class="n">bases</span> <span class="o">=</span> <span class="n">c_exp</span><span class="p">[</span><span class="n">e</span><span class="p">]</span>
-
-            <span class="c"># calculate the new base for e</span>
-
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">bases</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">new_base</span> <span class="o">=</span> <span class="n">bases</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">elif</span> <span class="n">e</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">or</span> <span class="n">force</span><span class="p">:</span>
-                <span class="n">new_base</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">bases</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># see which ones can be joined</span>
-                <span class="n">unk</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="n">nonneg</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="n">neg</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">for</span> <span class="n">bi</span> <span class="ow">in</span> <span class="n">bases</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">bi</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-                        <span class="n">neg</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">bi</span><span class="p">)</span>
-                    <span class="k">elif</span> <span class="n">bi</span><span class="o">.</span><span class="n">is_nonnegative</span><span class="p">:</span>
-                        <span class="n">nonneg</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">bi</span><span class="p">)</span>
-                    <span class="k">elif</span> <span class="n">bi</span><span class="o">.</span><span class="n">is_polar</span><span class="p">:</span>
-                        <span class="n">nonneg</span><span class="o">.</span><span class="n">append</span><span class="p">(</span>
-                            <span class="n">bi</span><span class="p">)</span>  <span class="c"># polar can be treated like non-negative</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">unk</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">bi</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">unk</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">neg</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">neg</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">unk</span><span class="p">:</span>
-                    <span class="c"># a single neg or a single unk can join the rest</span>
-                    <span class="n">nonneg</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">unk</span> <span class="o">+</span> <span class="n">neg</span><span class="p">)</span>
-                    <span class="n">unk</span> <span class="o">=</span> <span class="n">neg</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">elif</span> <span class="n">neg</span><span class="p">:</span>
-                    <span class="c"># their negative signs cancel in groups of 2*q if we know</span>
-                    <span class="c"># that e = p/q else we have to treat them as unknown</span>
-                    <span class="n">israt</span> <span class="o">=</span> <span class="bp">False</span>
-                    <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                        <span class="n">israt</span> <span class="o">=</span> <span class="bp">True</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">p</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-                        <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">and</span> <span class="n">d</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-                            <span class="n">israt</span> <span class="o">=</span> <span class="bp">True</span>
-                    <span class="k">if</span> <span class="n">israt</span><span class="p">:</span>
-                        <span class="n">neg</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="n">w</span> <span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">neg</span><span class="p">]</span>
-                        <span class="n">unk</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">]</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="n">neg</span><span class="p">))</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">unk</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">neg</span><span class="p">)</span>
-                        <span class="n">neg</span> <span class="o">=</span> <span class="p">[]</span>
-                    <span class="k">del</span> <span class="n">israt</span>
-
-                <span class="c"># these shouldn&#39;t be joined</span>
-                <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">unk</span><span class="p">:</span>
-                    <span class="n">c_powers</span><span class="p">[</span><span class="n">b</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-                <span class="c"># here is a new joined base</span>
-                <span class="n">new_base</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">nonneg</span> <span class="o">+</span> <span class="n">neg</span><span class="p">))</span>
-                <span class="c"># if there are positive parts they will just get separated again</span>
-                <span class="c"># unless some change is made</span>
-
-                <span class="k">def</span> <span class="nf">_terms</span><span class="p">(</span><span class="n">e</span><span class="p">):</span>
-                    <span class="c"># return the number of terms of this expression</span>
-                    <span class="c"># when multiplied out -- assuming no joining of terms</span>
-                    <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="nb">sum</span><span class="p">([</span><span class="n">_terms</span><span class="p">(</span><span class="n">ai</span><span class="p">)</span> <span class="k">for</span> <span class="n">ai</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-                    <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">prod</span><span class="p">([</span><span class="n">_terms</span><span class="p">(</span><span class="n">mi</span><span class="p">)</span> <span class="k">for</span> <span class="n">mi</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-                    <span class="k">return</span> <span class="mi">1</span>
-                <span class="n">xnew_base</span> <span class="o">=</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">new_base</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">xnew_base</span><span class="p">))</span> <span class="o">&lt;</span> <span class="n">_terms</span><span class="p">(</span><span class="n">new_base</span><span class="p">):</span>
-                    <span class="n">new_base</span> <span class="o">=</span> <span class="n">factor_terms</span><span class="p">(</span><span class="n">xnew_base</span><span class="p">)</span>
-
-            <span class="n">c_powers</span><span class="p">[</span><span class="n">new_base</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-
-        <span class="c"># break out the powers from c_powers now</span>
-        <span class="n">c_part</span> <span class="o">=</span> <span class="p">[</span><span class="n">Pow</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">ei</span><span class="p">)</span> <span class="k">for</span> <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">c_powers</span><span class="o">.</span><span class="n">items</span><span class="p">()</span> <span class="k">for</span> <span class="n">ei</span> <span class="ow">in</span> <span class="n">e</span><span class="p">]</span>
-
-        <span class="c"># we&#39;re done</span>
-        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">c_part</span> <span class="o">+</span> <span class="n">nc_part</span><span class="p">))</span>
-
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;combine must be one of (&#39;all&#39;, &#39;exp&#39;, &#39;base&#39;).&quot;</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="hypersimp"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.simplify.hypersimp">[docs]</a><span class="k">def</span> <span class="nf">hypersimp</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Given combinatorial term f(k) simplify its consecutive term ratio</span>
-<span class="sd">       i.e. f(k+1)/f(k).  The input term can be composed of functions and</span>
-<span class="sd">       integer sequences which have equivalent representation in terms</span>
-<span class="sd">       of gamma special function.</span>
-
-<span class="sd">       The algorithm performs three basic steps:</span>
-
-<span class="sd">       1. Rewrite all functions in terms of gamma, if possible.</span>
-
-<span class="sd">       2. Rewrite all occurrences of gamma in terms of products</span>
-<span class="sd">          of gamma and rising factorial with integer,  absolute</span>
-<span class="sd">          constant exponent.</span>
-
-<span class="sd">       3. Perform simplification of nested fractions, powers</span>
-<span class="sd">          and if the resulting expression is a quotient of</span>
-<span class="sd">          polynomials, reduce their total degree.</span>
-
-<span class="sd">       If f(k) is hypergeometric then as result we arrive with a</span>
-<span class="sd">       quotient of polynomials of minimal degree. Otherwise None</span>
-<span class="sd">       is returned.</span>
-
-<span class="sd">       For more information on the implemented algorithm refer to:</span>
-
-<span class="sd">       1. W. Koepf, Algorithms for m-fold Hypergeometric Summation,</span>
-<span class="sd">          Journal of Symbolic Computation (1995) 20, 399-417</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="n">g</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="n">f</span>
-
-    <span class="n">g</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">gamma</span><span class="p">)</span>
-    <span class="n">g</span> <span class="o">=</span> <span class="n">expand_func</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-    <span class="n">g</span> <span class="o">=</span> <span class="n">powsimp</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;exp&#39;</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">g</span><span class="o">.</span><span class="n">is_rational_function</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">simplify</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">ratio</span><span class="o">=</span><span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-</div>
-<div class="viewcode-block" id="hypersimilar"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.simplify.hypersimilar">[docs]</a><span class="k">def</span> <span class="nf">hypersimilar</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns True if &#39;f&#39; and &#39;g&#39; are hyper-similar.</span>
-
-<span class="sd">       Similarity in hypergeometric sense means that a quotient of</span>
-<span class="sd">       f(k) and g(k) is a rational function in k.  This procedure</span>
-<span class="sd">       is useful in solving recurrence relations.</span>
-
-<span class="sd">       For more information see hypersimp().</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">f</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)))</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="p">(</span><span class="n">f</span><span class="o">/</span><span class="n">g</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">gamma</span><span class="p">)</span>
-    <span class="n">h</span> <span class="o">=</span> <span class="n">h</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">func</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">basic</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">h</span><span class="o">.</span><span class="n">is_rational_function</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-</div>
-<span class="kn">from</span> <span class="nn">sympy.utilities.timeutils</span> <span class="kn">import</span> <span class="n">timethis</span>
-
-
-<span class="nd">@timethis</span><span class="p">(</span><span class="s">&#39;combsimp&#39;</span><span class="p">)</span>
-<div class="viewcode-block" id="combsimp"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.simplify.combsimp">[docs]</a><span class="k">def</span> <span class="nf">combsimp</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Simplify combinatorial expressions.</span>
-
-<span class="sd">    This function takes as input an expression containing factorials,</span>
-<span class="sd">    binomials, Pochhammer symbol and other &quot;combinatorial&quot; functions,</span>
-<span class="sd">    and tries to minimize the number of those functions and reduce</span>
-<span class="sd">    the size of their arguments. The result is be given in terms of</span>
-<span class="sd">    binomials and factorials.</span>
-
-<span class="sd">    The algorithm works by rewriting all combinatorial functions as</span>
-<span class="sd">    expressions involving rising factorials (Pochhammer symbols) and</span>
-<span class="sd">    applies recurrence relations and other transformations applicable</span>
-<span class="sd">    to rising factorials, to reduce their arguments, possibly letting</span>
-<span class="sd">    the resulting rising factorial to cancel. Rising factorials with</span>
-<span class="sd">    the second argument being an integer are expanded into polynomial</span>
-<span class="sd">    forms and finally all other rising factorial are rewritten in terms</span>
-<span class="sd">    more familiar functions. If the initial expression contained any</span>
-<span class="sd">    combinatorial functions, the result is expressed using binomial</span>
-<span class="sd">    coefficients and gamma functions. If the initial expression consisted</span>
-<span class="sd">    of gamma functions alone, the result is expressed in terms of gamma</span>
-<span class="sd">    functions.</span>
-
-<span class="sd">    If the result is expressed using gamma functions, the following three</span>
-<span class="sd">    additional steps are performed:</span>
-
-<span class="sd">    1. Reduce the number of gammas by applying the reflection theorem</span>
-<span class="sd">       gamma(x)*gamma(1-x) == pi/sin(pi*x).</span>
-<span class="sd">    2. Reduce the number of gammas by applying the multiplication theorem</span>
-<span class="sd">       gamma(x)*gamma(x+1/n)*...*gamma(x+(n-1)/n) == C*gamma(n*x).</span>
-<span class="sd">    3. Reduce the number of prefactors by absorbing them into gammas, where</span>
-<span class="sd">       possible.</span>
-
-<span class="sd">    All transformation rules can be found (or was derived from) here:</span>
-
-<span class="sd">    1. http://functions.wolfram.com/GammaBetaErf/Pochhammer/17/01/02/</span>
-<span class="sd">    2. http://functions.wolfram.com/GammaBetaErf/Pochhammer/27/01/0005/</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify import combsimp</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import factorial, binomial</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import n, k</span>
-
-<span class="sd">    &gt;&gt;&gt; combsimp(factorial(n)/factorial(n - 3))</span>
-<span class="sd">    n*(n - 2)*(n - 1)</span>
-<span class="sd">    &gt;&gt;&gt; combsimp(binomial(n+1, k+1)/binomial(n, k))</span>
-<span class="sd">    (n + 1)/(k + 1)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">factorial</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">factorial</span>
-    <span class="n">binomial</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">binomial</span>
-    <span class="n">gamma</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">gamma</span>
-
-    <span class="c"># as a rule of thumb, if the expression contained gammas initially, it</span>
-    <span class="c"># probably makes sense to retain them</span>
-    <span class="n">as_gamma</span> <span class="o">=</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">factorial</span><span class="p">,</span> <span class="n">binomial</span><span class="p">)</span>
-
-    <span class="k">class</span> <span class="nc">rf</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-        <span class="nd">@classmethod</span>
-        <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">b</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-                <span class="n">n</span><span class="p">,</span> <span class="n">result</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">b</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-                <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-                        <span class="n">result</span> <span class="o">*=</span> <span class="n">a</span> <span class="o">+</span> <span class="n">i</span>
-
-                    <span class="k">return</span> <span class="n">result</span>
-                <span class="k">elif</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                        <span class="n">result</span> <span class="o">*=</span> <span class="n">a</span> <span class="o">-</span> <span class="n">i</span>
-
-                    <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="n">result</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                    <span class="n">c</span><span class="p">,</span> <span class="n">_b</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">as_coeff_Add</span><span class="p">()</span>
-
-                    <span class="k">if</span> <span class="n">c</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">c</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="n">rf</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">_b</span><span class="p">)</span><span class="o">*</span><span class="n">rf</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">_b</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
-                        <span class="k">elif</span> <span class="n">c</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="n">rf</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">_b</span><span class="p">)</span><span class="o">/</span><span class="n">rf</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">_b</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="o">-</span><span class="n">c</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                    <span class="n">c</span><span class="p">,</span> <span class="n">_a</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">as_coeff_Add</span><span class="p">()</span>
-
-                    <span class="k">if</span> <span class="n">c</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">c</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="n">rf</span><span class="p">(</span><span class="n">_a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">rf</span><span class="p">(</span><span class="n">_a</span> <span class="o">+</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span><span class="o">/</span><span class="n">rf</span><span class="p">(</span><span class="n">_a</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
-                        <span class="k">elif</span> <span class="n">c</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="n">rf</span><span class="p">(</span><span class="n">_a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">rf</span><span class="p">(</span><span class="n">_a</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="o">-</span><span class="n">c</span><span class="p">)</span><span class="o">/</span><span class="n">rf</span><span class="p">(</span><span class="n">_a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="o">-</span><span class="n">c</span><span class="p">)</span>
-
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">binomial</span><span class="p">,</span>
-        <span class="k">lambda</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">:</span> <span class="n">rf</span><span class="p">((</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(),</span> <span class="n">k</span><span class="o">.</span><span class="n">expand</span><span class="p">())</span><span class="o">/</span><span class="n">rf</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">k</span><span class="o">.</span><span class="n">expand</span><span class="p">()))</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">factorial</span><span class="p">,</span>
-        <span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="n">rf</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">.</span><span class="n">expand</span><span class="p">()))</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">gamma</span><span class="p">)</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">gamma</span><span class="p">,</span>
-        <span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="n">rf</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()))</span>
-
-    <span class="k">if</span> <span class="n">as_gamma</span><span class="p">:</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">rf</span><span class="p">,</span>
-            <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">gamma</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span><span class="o">/</span><span class="n">gamma</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">rf</span><span class="p">,</span>
-            <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="n">binomial</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">factorial</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">rule</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="n">coeff</span><span class="p">,</span> <span class="n">rewrite</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="bp">False</span>
-
-        <span class="n">cn</span><span class="p">,</span> <span class="n">_n</span> <span class="o">=</span> <span class="n">n</span><span class="o">.</span><span class="n">as_coeff_Add</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="n">_n</span> <span class="ow">and</span> <span class="n">cn</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">cn</span><span class="p">:</span>
-            <span class="n">coeff</span> <span class="o">*=</span> <span class="n">rf</span><span class="p">(</span><span class="n">_n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">cn</span><span class="p">)</span><span class="o">/</span><span class="n">rf</span><span class="p">(</span><span class="n">_n</span> <span class="o">-</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">cn</span><span class="p">)</span>
-            <span class="n">rewrite</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="n">_n</span>
-
-        <span class="c"># this sort of binomial has already been removed by</span>
-        <span class="c"># rising factorials but is left here in case the order</span>
-        <span class="c"># of rule application is changed</span>
-        <span class="k">if</span> <span class="n">k</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">ck</span><span class="p">,</span> <span class="n">_k</span> <span class="o">=</span> <span class="n">k</span><span class="o">.</span><span class="n">as_coeff_Add</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">_k</span> <span class="ow">and</span> <span class="n">ck</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">ck</span><span class="p">:</span>
-                <span class="n">coeff</span> <span class="o">*=</span> <span class="n">rf</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">ck</span> <span class="o">-</span> <span class="n">_k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ck</span><span class="p">)</span><span class="o">/</span><span class="n">rf</span><span class="p">(</span><span class="n">_k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ck</span><span class="p">)</span>
-                <span class="n">rewrite</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="n">k</span> <span class="o">=</span> <span class="n">_k</span>
-
-        <span class="k">if</span> <span class="n">rewrite</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">coeff</span><span class="o">*</span><span class="n">binomial</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">binomial</span><span class="p">,</span> <span class="n">rule</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">rule_gamma</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; Simplify products of gamma functions further. &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span>
-
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">rule_gamma</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span>
-
-        <span class="n">numer_gammas</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">denom_gammas</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">denom_others</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">newargs</span><span class="p">,</span> <span class="n">numer_others</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-
-        <span class="c"># order newargs canonically</span>
-        <span class="n">cexpr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">newargs</span><span class="p">)</span>
-        <span class="n">newargs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">cexpr</span><span class="o">.</span><span class="n">_sorted_args</span><span class="p">)</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">cexpr</span><span class="o">.</span><span class="n">is_Atom</span> <span class="k">else</span> <span class="p">[</span><span class="n">cexpr</span><span class="p">]</span>
-        <span class="k">del</span> <span class="n">cexpr</span>
-
-        <span class="k">while</span> <span class="n">newargs</span><span class="p">:</span>
-            <span class="n">arg</span> <span class="o">=</span> <span class="n">newargs</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-            <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                <span class="n">n</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">gamma</span><span class="p">):</span>
-                    <span class="n">barg</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                    <span class="k">if</span> <span class="n">e</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">numer_gammas</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">barg</span><span class="p">]</span><span class="o">*</span><span class="n">n</span><span class="p">)</span>
-                    <span class="k">elif</span> <span class="n">e</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">denom_gammas</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">barg</span><span class="p">]</span><span class="o">*</span><span class="n">n</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">e</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">numer_others</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">b</span><span class="p">]</span><span class="o">*</span><span class="n">n</span><span class="p">)</span>
-                    <span class="k">elif</span> <span class="n">e</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="n">denom_others</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">b</span><span class="p">]</span><span class="o">*</span><span class="n">n</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">numer_others</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-
-        <span class="c"># Try to reduce the number of gamma factors by applying the</span>
-        <span class="c"># reflection formula gamma(x)*gamma(1-x) = pi/sin(pi*x)</span>
-        <span class="k">for</span> <span class="n">gammas</span><span class="p">,</span> <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="ow">in</span> <span class="p">[(</span>
-            <span class="n">numer_gammas</span><span class="p">,</span> <span class="n">numer_others</span><span class="p">,</span> <span class="n">denom_others</span><span class="p">),</span>
-                <span class="p">(</span><span class="n">denom_gammas</span><span class="p">,</span> <span class="n">denom_others</span><span class="p">,</span> <span class="n">numer_others</span><span class="p">)]:</span>
-            <span class="n">new</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">while</span> <span class="n">gammas</span><span class="p">:</span>
-                <span class="n">g1</span> <span class="o">=</span> <span class="n">gammas</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-                <span class="k">if</span> <span class="n">g1</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-                    <span class="n">new</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">g1</span><span class="p">)</span>
-                    <span class="k">continue</span>
-                <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">g2</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">gammas</span><span class="p">):</span>
-                    <span class="n">n</span> <span class="o">=</span> <span class="n">g1</span> <span class="o">+</span> <span class="n">g2</span> <span class="o">-</span> <span class="mi">1</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                        <span class="k">continue</span>
-                    <span class="n">numer</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">)</span>
-                    <span class="n">denom</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">*</span><span class="n">g1</span><span class="p">))</span>
-                    <span class="n">gammas</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-                            <span class="n">numer</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">g1</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span>
-                    <span class="k">elif</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">):</span>
-                            <span class="n">denom</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">-</span><span class="n">g1</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span>
-                    <span class="k">break</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">new</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">g1</span><span class="p">)</span>
-            <span class="c"># /!\ updating IN PLACE</span>
-            <span class="n">gammas</span><span class="p">[:]</span> <span class="o">=</span> <span class="n">new</span>
-
-        <span class="c"># Try to reduce the number of gamma factors by applying the</span>
-        <span class="c"># multiplication theorem.</span>
-
-        <span class="k">def</span> <span class="nf">_run</span><span class="p">(</span><span class="n">coeffs</span><span class="p">):</span>
-            <span class="c"># find runs in coeffs such that the difference in terms (mod 1)</span>
-            <span class="c"># of t1, t2, ..., tn is 1/n</span>
-            <span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">uniq</span>
-            <span class="n">u</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">uniq</span><span class="p">(</span><span class="n">coeffs</span><span class="p">))</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">u</span><span class="p">)):</span>
-                <span class="n">dj</span> <span class="o">=</span> <span class="p">([((</span><span class="n">u</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">u</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="o">%</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">u</span><span class="p">))])</span>
-                <span class="k">for</span> <span class="n">one</span><span class="p">,</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">dj</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">one</span><span class="o">.</span><span class="n">p</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">one</span><span class="o">.</span><span class="n">q</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="n">n</span> <span class="o">=</span> <span class="n">one</span><span class="o">.</span><span class="n">q</span>
-                        <span class="n">got</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span><span class="p">]</span>
-                        <span class="n">get</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-                        <span class="k">for</span> <span class="n">d</span><span class="p">,</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">dj</span><span class="p">:</span>
-                            <span class="n">m</span> <span class="o">=</span> <span class="n">n</span><span class="o">*</span><span class="n">d</span>
-                            <span class="k">if</span> <span class="n">m</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">get</span><span class="p">:</span>
-                                <span class="n">get</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-                                <span class="n">got</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-                                <span class="k">if</span> <span class="ow">not</span> <span class="n">get</span><span class="p">:</span>
-                                    <span class="k">break</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="k">continue</span>
-                        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">got</span><span class="p">):</span>
-                            <span class="n">c</span> <span class="o">=</span> <span class="n">u</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-                            <span class="n">coeffs</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-                            <span class="n">got</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">c</span>
-                        <span class="k">return</span> <span class="n">one</span><span class="o">.</span><span class="n">q</span><span class="p">,</span> <span class="n">got</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">got</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-
-        <span class="k">def</span> <span class="nf">_mult_thm</span><span class="p">(</span><span class="n">gammas</span><span class="p">,</span> <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span><span class="p">):</span>
-            <span class="c"># pull off and analyze the leading coefficient from each gamma arg</span>
-            <span class="c"># looking for runs in those Rationals</span>
-
-            <span class="c"># expr -&gt; coeff + resid -&gt; rats[resid] = coeff</span>
-            <span class="n">rats</span> <span class="o">=</span> <span class="p">{}</span>
-            <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">gammas</span><span class="p">:</span>
-                <span class="n">c</span><span class="p">,</span> <span class="n">resid</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">as_coeff_Add</span><span class="p">()</span>
-                <span class="n">rats</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="n">resid</span><span class="p">,</span> <span class="p">[])</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-
-            <span class="c"># look for runs in Rationals for each resid</span>
-            <span class="n">keys</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">rats</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">resid</span> <span class="ow">in</span> <span class="n">keys</span><span class="p">:</span>
-                <span class="n">coeffs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="n">rats</span><span class="p">[</span><span class="n">resid</span><span class="p">]))</span>
-                <span class="n">new</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-                    <span class="n">run</span> <span class="o">=</span> <span class="n">_run</span><span class="p">(</span><span class="n">coeffs</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">run</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="k">break</span>
-
-                    <span class="c"># process the sequence that was found:</span>
-                    <span class="c"># 1) convert all the gamma functions to have the right</span>
-                    <span class="c">#    argument (could be off by an integer)</span>
-                    <span class="c"># 2) append the factors corresponding to the theorem</span>
-                    <span class="c"># 3) append the new gamma function</span>
-
-                    <span class="n">n</span><span class="p">,</span> <span class="n">ui</span><span class="p">,</span> <span class="n">other</span> <span class="o">=</span> <span class="n">run</span>
-
-                    <span class="c"># (1)</span>
-                    <span class="k">for</span> <span class="n">u</span> <span class="ow">in</span> <span class="n">other</span><span class="p">:</span>
-                        <span class="n">con</span> <span class="o">=</span> <span class="n">resid</span> <span class="o">+</span> <span class="n">u</span> <span class="o">-</span> <span class="mi">1</span>
-                        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="n">ui</span><span class="p">)):</span>
-                            <span class="n">numer</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">con</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span>
-
-                    <span class="n">con</span> <span class="o">=</span> <span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">resid</span> <span class="o">+</span> <span class="n">ui</span><span class="p">)</span>  <span class="c"># for (2) and (3)</span>
-
-                    <span class="c"># (2)</span>
-                    <span class="n">numer</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span>
-                                 <span class="n">n</span><span class="o">**</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">con</span><span class="p">))</span>
-                    <span class="c"># (3)</span>
-                    <span class="n">new</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">con</span><span class="p">)</span>
-
-                <span class="c"># restore resid to coeffs</span>
-                <span class="n">rats</span><span class="p">[</span><span class="n">resid</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">resid</span> <span class="o">+</span> <span class="n">c</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">coeffs</span><span class="p">]</span> <span class="o">+</span> <span class="n">new</span>
-
-            <span class="c"># rebuild the gamma arguments</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">resid</span> <span class="ow">in</span> <span class="n">keys</span><span class="p">:</span>
-                <span class="n">g</span> <span class="o">+=</span> <span class="n">rats</span><span class="p">[</span><span class="n">resid</span><span class="p">]</span>
-            <span class="c"># /!\ updating IN PLACE</span>
-            <span class="n">gammas</span><span class="p">[:]</span> <span class="o">=</span> <span class="n">g</span>
-
-        <span class="k">for</span> <span class="n">l</span><span class="p">,</span> <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="ow">in</span> <span class="p">[(</span><span class="n">numer_gammas</span><span class="p">,</span> <span class="n">numer_others</span><span class="p">,</span> <span class="n">denom_others</span><span class="p">),</span>
-                                <span class="p">(</span><span class="n">denom_gammas</span><span class="p">,</span> <span class="n">denom_others</span><span class="p">,</span> <span class="n">numer_others</span><span class="p">)]:</span>
-            <span class="n">_mult_thm</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span><span class="p">)</span>
-
-        <span class="c"># Try to reduce the number of gammas by using the duplication</span>
-        <span class="c"># theorem to cancel an upper and lower.</span>
-        <span class="c"># e.g. gamma(2*s)/gamma(s) = gamma(s)*gamma(s+1/2)*C/gamma(s)</span>
-        <span class="c"># (in principle this can also be done with with factors other than two,</span>
-        <span class="c">#  but two is special in that we need only matching numer and denom, not</span>
-        <span class="c">#  several in numer).</span>
-        <span class="k">for</span> <span class="n">ng</span><span class="p">,</span> <span class="n">dg</span><span class="p">,</span> <span class="n">no</span><span class="p">,</span> <span class="n">do</span> <span class="ow">in</span> <span class="p">[(</span><span class="n">numer_gammas</span><span class="p">,</span> <span class="n">denom_gammas</span><span class="p">,</span> <span class="n">numer_others</span><span class="p">,</span>
-                                <span class="n">denom_others</span><span class="p">),</span>
-                               <span class="p">(</span><span class="n">denom_gammas</span><span class="p">,</span> <span class="n">numer_gammas</span><span class="p">,</span> <span class="n">denom_others</span><span class="p">,</span>
-                                <span class="n">numer_others</span><span class="p">)]:</span>
-
-            <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">ng</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">dg</span><span class="p">:</span>
-                        <span class="n">n</span> <span class="o">=</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span>
-                        <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                            <span class="k">break</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">continue</span>
-                    <span class="k">break</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">break</span>
-                <span class="n">ng</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-                <span class="n">dg</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-                        <span class="n">no</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">):</span>
-                        <span class="n">do</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span>
-                <span class="n">ng</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-                <span class="n">no</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-                <span class="n">do</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="p">))</span>
-
-        <span class="c"># Try to absorb factors into the gammas.</span>
-        <span class="c"># This code (in particular repeated calls to find_fuzzy) can be very</span>
-        <span class="c"># slow.</span>
-        <span class="k">def</span> <span class="nf">find_fuzzy</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-            <span class="n">S1</span><span class="p">,</span> <span class="n">T1</span> <span class="o">=</span> <span class="n">compute_ST</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">l</span><span class="p">:</span>
-                <span class="n">S2</span><span class="p">,</span> <span class="n">T2</span> <span class="o">=</span> <span class="n">inv</span><span class="p">[</span><span class="n">y</span><span class="p">]</span>
-                <span class="k">if</span> <span class="n">T1</span> <span class="o">!=</span> <span class="n">T2</span> <span class="ow">or</span> <span class="p">(</span><span class="ow">not</span> <span class="n">S1</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">S2</span><span class="p">)</span> <span class="ow">and</span>
-                                <span class="p">(</span><span class="n">S1</span> <span class="o">!=</span> <span class="nb">set</span><span class="p">()</span> <span class="ow">or</span> <span class="n">S2</span> <span class="o">!=</span> <span class="nb">set</span><span class="p">())):</span>
-                    <span class="k">continue</span>
-                <span class="c"># XXX we want some simplification (e.g. cancel or</span>
-                <span class="c"># simplify) but no matter what it&#39;s slow.</span>
-                <span class="n">a</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">cancel</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">)</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">)</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">y</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">)</span>
-                <span class="c"># TODO is there a better heuristic?</span>
-                <span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="p">(</span><span class="n">b</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">c</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="n">y</span>
-
-        <span class="c"># We thus try to avoid expensive calls by building the following</span>
-        <span class="c"># &quot;invariants&quot;: For every factor or gamma function argument</span>
-        <span class="c">#   - the set of free symbols S</span>
-        <span class="c">#   - the set of functional components T</span>
-        <span class="c"># We will only try to absorb if T1==T2 and (S1 intersect S2 != emptyset</span>
-        <span class="c"># or S1 == S2 == emptyset)</span>
-        <span class="n">inv</span> <span class="o">=</span> <span class="p">{}</span>
-
-        <span class="k">def</span> <span class="nf">compute_ST</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-            <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Pow</span>
-            <span class="k">if</span> <span class="n">expr</span> <span class="ow">in</span> <span class="n">inv</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">inv</span><span class="p">[</span><span class="n">expr</span><span class="p">]</span>
-            <span class="k">return</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Function</span><span class="p">)</span><span class="o">.</span><span class="n">union</span><span class="p">(</span>
-                    <span class="nb">set</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">exp</span> <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Pow</span><span class="p">))))</span>
-
-        <span class="k">def</span> <span class="nf">update_ST</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-            <span class="n">inv</span><span class="p">[</span><span class="n">expr</span><span class="p">]</span> <span class="o">=</span> <span class="n">compute_ST</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">expr</span> <span class="ow">in</span> <span class="n">numer_gammas</span> <span class="o">+</span> <span class="n">denom_gammas</span> <span class="o">+</span> <span class="n">numer_others</span> <span class="o">+</span> <span class="n">denom_others</span><span class="p">:</span>
-            <span class="n">update_ST</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">gammas</span><span class="p">,</span> <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="ow">in</span> <span class="p">[(</span>
-            <span class="n">numer_gammas</span><span class="p">,</span> <span class="n">numer_others</span><span class="p">,</span> <span class="n">denom_others</span><span class="p">),</span>
-                <span class="p">(</span><span class="n">denom_gammas</span><span class="p">,</span> <span class="n">denom_others</span><span class="p">,</span> <span class="n">numer_others</span><span class="p">)]:</span>
-            <span class="n">new</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">while</span> <span class="n">gammas</span><span class="p">:</span>
-                <span class="n">g</span> <span class="o">=</span> <span class="n">gammas</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-                <span class="n">cont</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">while</span> <span class="n">cont</span><span class="p">:</span>
-                    <span class="n">cont</span> <span class="o">=</span> <span class="bp">False</span>
-                    <span class="n">y</span> <span class="o">=</span> <span class="n">find_fuzzy</span><span class="p">(</span><span class="n">numer</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">y</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="n">numer</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">y</span> <span class="o">!=</span> <span class="n">g</span><span class="p">:</span>
-                            <span class="n">numer</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">y</span><span class="o">/</span><span class="n">g</span><span class="p">)</span>
-                            <span class="n">update_ST</span><span class="p">(</span><span class="n">y</span><span class="o">/</span><span class="n">g</span><span class="p">)</span>
-                        <span class="n">g</span> <span class="o">+=</span> <span class="mi">1</span>
-                        <span class="n">cont</span> <span class="o">=</span> <span class="bp">True</span>
-                    <span class="n">y</span> <span class="o">=</span> <span class="n">find_fuzzy</span><span class="p">(</span><span class="n">numer</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">g</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-                    <span class="k">if</span> <span class="n">y</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="n">numer</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">y</span> <span class="o">!=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">g</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-                            <span class="n">numer</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">g</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
-                            <span class="n">update_ST</span><span class="p">((</span><span class="n">g</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
-                        <span class="n">g</span> <span class="o">-=</span> <span class="mi">1</span>
-                        <span class="n">cont</span> <span class="o">=</span> <span class="bp">True</span>
-                    <span class="n">y</span> <span class="o">=</span> <span class="n">find_fuzzy</span><span class="p">(</span><span class="n">denom</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">g</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">y</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="n">denom</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">y</span> <span class="o">!=</span> <span class="mi">1</span><span class="o">/</span><span class="n">g</span><span class="p">:</span>
-                            <span class="n">denom</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">y</span><span class="o">*</span><span class="n">g</span><span class="p">)</span>
-                            <span class="n">update_ST</span><span class="p">(</span><span class="n">y</span><span class="o">*</span><span class="n">g</span><span class="p">)</span>
-                        <span class="n">g</span> <span class="o">+=</span> <span class="mi">1</span>
-                        <span class="n">cont</span> <span class="o">=</span> <span class="bp">True</span>
-                    <span class="n">y</span> <span class="o">=</span> <span class="n">find_fuzzy</span><span class="p">(</span><span class="n">denom</span><span class="p">,</span> <span class="n">g</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">y</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="n">denom</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">y</span> <span class="o">!=</span> <span class="n">g</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                            <span class="n">numer</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">g</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">y</span><span class="p">)</span>
-                            <span class="n">update_ST</span><span class="p">((</span><span class="n">g</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">y</span><span class="p">)</span>
-                        <span class="n">g</span> <span class="o">-=</span> <span class="mi">1</span>
-                        <span class="n">cont</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="n">new</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-            <span class="c"># /!\ updating IN PLACE</span>
-            <span class="n">gammas</span><span class="p">[:]</span> <span class="o">=</span> <span class="n">new</span>
-
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">gamma</span><span class="p">(</span><span class="n">g</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">numer_gammas</span><span class="p">])</span> \
-            <span class="o">/</span> <span class="n">C</span><span class="o">.</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">gamma</span><span class="p">(</span><span class="n">g</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">denom_gammas</span><span class="p">])</span> \
-            <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">numer_others</span><span class="p">)</span> <span class="o">/</span> <span class="n">C</span><span class="o">.</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">denom_others</span><span class="p">)</span>
-
-    <span class="c"># (for some reason we cannot use Basic.replace in this case)</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">rule_gamma</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">factor</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">signsimp</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Make all Add sub-expressions canonical wrt sign.</span>
-
-<span class="sd">    If an Add subexpression, ``a``, can have a sign extracted,</span>
-<span class="sd">    as determined by could_extract_minus_sign, it is replaced</span>
-<span class="sd">    with Mul(-1, a, evaluate=False). This allows signs to be</span>
-<span class="sd">    extracted from powers and products.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import signsimp, exp</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; n = -1 + 1/x</span>
-<span class="sd">    &gt;&gt;&gt; n/x/(-n)**2 - 1/n/x</span>
-<span class="sd">    (-1 + 1/x)/(x*(1 - 1/x)**2) - 1/(x*(-1 + 1/x))</span>
-<span class="sd">    &gt;&gt;&gt; signsimp(_)</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; x*n + x*-n</span>
-<span class="sd">    x*(-1 + 1/x) + x*(1 - 1/x)</span>
-<span class="sd">    &gt;&gt;&gt; signsimp(_)</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; n**3</span>
-<span class="sd">    (-1 + 1/x)**3</span>
-<span class="sd">    &gt;&gt;&gt; signsimp(_)</span>
-<span class="sd">    -(1 - 1/x)**3</span>
-
-<span class="sd">    By default, signsimp doesn&#39;t leave behind any hollow simplification:</span>
-<span class="sd">    if making an Add canonical wrt sign didn&#39;t change the expression, the</span>
-<span class="sd">    original Add is restored. If this is not desired then the keyword</span>
-<span class="sd">    ``evaluate`` can be set to False:</span>
-
-<span class="sd">    &gt;&gt;&gt; e = exp(y - x)</span>
-<span class="sd">    &gt;&gt;&gt; signsimp(e) == e</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; signsimp(e, evaluate=False)</span>
-<span class="sd">    exp(-(x - y))</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Expr</span><span class="p">)</span> <span class="ow">or</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span>
-    <span class="n">e</span> <span class="o">=</span> <span class="n">sub_post</span><span class="p">(</span><span class="n">sub_pre</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Expr</span><span class="p">)</span> <span class="ow">or</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">e</span>
-    <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">signsimp</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-    <span class="k">if</span> <span class="n">evaluate</span><span class="p">:</span>
-        <span class="n">e</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="nb">dict</span><span class="p">([(</span><span class="n">m</span><span class="p">,</span> <span class="o">-</span><span class="p">(</span><span class="o">-</span><span class="n">m</span><span class="p">))</span> <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Mul</span><span class="p">)</span> <span class="k">if</span> <span class="o">-</span><span class="p">(</span><span class="o">-</span><span class="n">m</span><span class="p">)</span> <span class="o">!=</span> <span class="n">m</span><span class="p">]))</span>
-    <span class="k">return</span> <span class="n">e</span>
-
-
-<div class="viewcode-block" id="simplify"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.simplify.simplify">[docs]</a><span class="k">def</span> <span class="nf">simplify</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">ratio</span><span class="o">=</span><span class="mf">1.7</span><span class="p">,</span> <span class="n">measure</span><span class="o">=</span><span class="n">count_ops</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Simplifies the given expression.</span>
-
-<span class="sd">    Simplification is not a well defined term and the exact strategies</span>
-<span class="sd">    this function tries can change in the future versions of SymPy. If</span>
-<span class="sd">    your algorithm relies on &quot;simplification&quot; (whatever it is), try to</span>
-<span class="sd">    determine what you need exactly  -  is it powsimp()?, radsimp()?,</span>
-<span class="sd">    together()?, logcombine()?, or something else? And use this particular</span>
-<span class="sd">    function directly, because those are well defined and thus your algorithm</span>
-<span class="sd">    will be robust.</span>
-
-<span class="sd">    Nonetheless, especially for interactive use, or when you don&#39;t know</span>
-<span class="sd">    anything about the structure of the expression, simplify() tries to apply</span>
-<span class="sd">    intelligent heuristics to make the input expression &quot;simpler&quot;.  For</span>
-<span class="sd">    example:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import simplify, cos, sin</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; a = (x + x**2)/(x*sin(y)**2 + x*cos(y)**2)</span>
-<span class="sd">    &gt;&gt;&gt; a</span>
-<span class="sd">    (x**2 + x)/(x*sin(y)**2 + x*cos(y)**2)</span>
-<span class="sd">    &gt;&gt;&gt; simplify(a)</span>
-<span class="sd">    x + 1</span>
-
-<span class="sd">    Note that we could have obtained the same result by using specific</span>
-<span class="sd">    simplification functions:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import trigsimp, cancel</span>
-<span class="sd">    &gt;&gt;&gt; b = trigsimp(a)</span>
-<span class="sd">    &gt;&gt;&gt; b</span>
-<span class="sd">    (x**2 + x)/x</span>
-<span class="sd">    &gt;&gt;&gt; c = cancel(b)</span>
-<span class="sd">    &gt;&gt;&gt; c</span>
-<span class="sd">    x + 1</span>
-
-<span class="sd">    In some cases, applying :func:`simplify` may actually result in some more</span>
-<span class="sd">    complicated expression. The default ``ratio=1.7`` prevents more extreme</span>
-<span class="sd">    cases: if (result length)/(input length) &gt; ratio, then input is returned</span>
-<span class="sd">    unmodified.  The ``measure`` parameter lets you specify the function used</span>
-<span class="sd">    to determine how complex an expression is.  The function should take a</span>
-<span class="sd">    single argument as an expression and return a number such that if</span>
-<span class="sd">    expression ``a`` is more complex than expression ``b``, then</span>
-<span class="sd">    ``measure(a) &gt; measure(b)``.  The default measure function is</span>
-<span class="sd">    :func:`count_ops`, which returns the total number of operations in the</span>
-<span class="sd">    expression.</span>
-
-<span class="sd">    For example, if ``ratio=1``, ``simplify`` output can&#39;t be longer</span>
-<span class="sd">    than input.</span>
-
-<span class="sd">    ::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sqrt, simplify, count_ops, oo</span>
-<span class="sd">        &gt;&gt;&gt; root = 1/(sqrt(2)+3)</span>
-
-<span class="sd">    Since ``simplify(root)`` would result in a slightly longer expression,</span>
-<span class="sd">    root is returned unchanged instead::</span>
-
-<span class="sd">       &gt;&gt;&gt; simplify(root, ratio=1) == root</span>
-<span class="sd">       True</span>
-
-<span class="sd">    If ``ratio=oo``, simplify will be applied anyway::</span>
-
-<span class="sd">        &gt;&gt;&gt; count_ops(simplify(root, ratio=oo)) &gt; count_ops(root)</span>
-<span class="sd">        True</span>
-
-<span class="sd">    Note that the shortest expression is not necessary the simplest, so</span>
-<span class="sd">    setting ``ratio`` to 1 may not be a good idea.</span>
-<span class="sd">    Heuristically, the default value ``ratio=1.7`` seems like a reasonable</span>
-<span class="sd">    choice.</span>
-
-<span class="sd">    You can easily define your own measure function based on what you feel</span>
-<span class="sd">    should represent the &quot;size&quot; or &quot;complexity&quot; of the input expression.  Note</span>
-<span class="sd">    that some choices, such as ``lambda expr: len(str(expr))`` may appear to be</span>
-<span class="sd">    good metrics, but have other problems (in this case, the measure function</span>
-<span class="sd">    may slow down simplify too much for very large expressions).  If you don&#39;t</span>
-<span class="sd">    know what a good metric would be, the default, ``count_ops``, is a good one.</span>
-
-<span class="sd">    For example:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, log</span>
-<span class="sd">    &gt;&gt;&gt; a, b = symbols(&#39;a b&#39;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; g = log(a) + log(b) + log(a)*log(1/b)</span>
-<span class="sd">    &gt;&gt;&gt; h = simplify(g)</span>
-<span class="sd">    &gt;&gt;&gt; h</span>
-<span class="sd">    log(a*b**(log(1/a) + 1))</span>
-<span class="sd">    &gt;&gt;&gt; count_ops(g)</span>
-<span class="sd">    8</span>
-<span class="sd">    &gt;&gt;&gt; count_ops(h)</span>
-<span class="sd">    6</span>
-
-<span class="sd">    So you can see that ``h`` is simpler than ``g`` using the count_ops metric.</span>
-<span class="sd">    However, we may not like how ``simplify`` (in this case, using</span>
-<span class="sd">    ``logcombine``) has created the ``b**(log(1/a) + 1)`` term.  A simple way to</span>
-<span class="sd">    reduce this would be to give more weight to powers as operations in</span>
-<span class="sd">    ``count_ops``.  We can do this by using the ``visual=True`` option:</span>
-
-<span class="sd">    &gt;&gt;&gt; print(count_ops(g, visual=True))</span>
-<span class="sd">    2*ADD + DIV + 4*LOG + MUL</span>
-<span class="sd">    &gt;&gt;&gt; print(count_ops(h, visual=True))</span>
-<span class="sd">    ADD + DIV + 2*LOG + MUL + POW</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, S</span>
-<span class="sd">    &gt;&gt;&gt; def my_measure(expr):</span>
-<span class="sd">    ...     POW = Symbol(&#39;POW&#39;)</span>
-<span class="sd">    ...     # Discourage powers by giving POW a weight of 10</span>
-<span class="sd">    ...     count = count_ops(expr, visual=True).subs(POW, 10)</span>
-<span class="sd">    ...     # Every other operation gets a weight of 1 (the default)</span>
-<span class="sd">    ...     count = count.replace(Symbol, type(S.One))</span>
-<span class="sd">    ...     return count</span>
-<span class="sd">    &gt;&gt;&gt; my_measure(g)</span>
-<span class="sd">    8</span>
-<span class="sd">    &gt;&gt;&gt; my_measure(h)</span>
-<span class="sd">    15</span>
-<span class="sd">    &gt;&gt;&gt; 15./8 &gt; 1.7 # 1.7 is the default ratio</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; simplify(g, measure=my_measure)</span>
-<span class="sd">    -log(a)*log(b) + log(a) + log(b)</span>
-
-<span class="sd">    Note that because ``simplify()`` internally tries many different</span>
-<span class="sd">    simplification strategies and then compares them using the measure</span>
-<span class="sd">    function, we get a completely different result that is still different</span>
-<span class="sd">    from the input expression by doing this.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">original_expr</span> <span class="o">=</span> <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">_eval_simplify</span><span class="p">(</span><span class="n">ratio</span><span class="o">=</span><span class="n">ratio</span><span class="p">,</span> <span class="n">measure</span><span class="o">=</span><span class="n">measure</span><span class="p">)</span>
-    <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-        <span class="k">pass</span>
-
-    <span class="kn">from</span> <span class="nn">sympy.simplify.hyperexpand</span> <span class="kn">import</span> <span class="n">hyperexpand</span>
-    <span class="kn">from</span> <span class="nn">sympy.functions.special.bessel</span> <span class="kn">import</span> <span class="n">BesselBase</span>
-
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">signsimp</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>  <span class="c"># XXX: temporary hack</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="c"># TODO: Apply different strategies, considering expression pattern:</span>
-    <span class="c"># is it a purely rational function? Is there any trigonometric function?...</span>
-    <span class="c"># See also https://github.com/sympy/sympy/pull/185.</span>
-
-    <span class="k">def</span> <span class="nf">shorter</span><span class="p">(</span><span class="o">*</span><span class="n">choices</span><span class="p">):</span>
-        <span class="sd">&#39;&#39;&#39;Return the choice that has the fewest ops. In case of a tie,</span>
-<span class="sd">        the expression listed first is selected.&#39;&#39;&#39;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">has_variety</span><span class="p">(</span><span class="n">choices</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">choices</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">return</span> <span class="nb">min</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">measure</span><span class="p">)</span>
-
-    <span class="n">expr0</span> <span class="o">=</span> <span class="n">powsimp</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_commutative</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-        <span class="n">expr1</span> <span class="o">=</span> <span class="n">together</span><span class="p">(</span><span class="n">expr0</span><span class="p">)</span>
-        <span class="n">expr2</span> <span class="o">=</span> <span class="n">factor_terms</span><span class="p">(</span><span class="n">expr1</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">expr1</span> <span class="o">=</span> <span class="n">cancel</span><span class="p">(</span><span class="n">expr0</span><span class="p">)</span>
-        <span class="n">expr2</span> <span class="o">=</span> <span class="n">together</span><span class="p">(</span><span class="n">expr1</span><span class="o">.</span><span class="n">expand</span><span class="p">(),</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="c"># sometimes factors in the denominators need to be allowed to join</span>
-    <span class="c"># factors in numerators (see issue 3270)</span>
-    <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-    <span class="k">if</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span> <span class="o">!=</span> <span class="n">fraction</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="n">expr0b</span> <span class="o">=</span> <span class="n">powsimp</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">powsimp</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">expr0b</span> <span class="o">!=</span> <span class="n">expr0</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_commutative</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                <span class="n">expr1b</span> <span class="o">=</span> <span class="n">together</span><span class="p">(</span><span class="n">expr0b</span><span class="p">)</span>
-                <span class="n">expr2b</span> <span class="o">=</span> <span class="n">factor_terms</span><span class="p">(</span><span class="n">expr1b</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">expr1b</span> <span class="o">=</span> <span class="n">cancel</span><span class="p">(</span><span class="n">expr0b</span><span class="p">)</span>
-                <span class="n">expr2b</span> <span class="o">=</span> <span class="n">together</span><span class="p">(</span><span class="n">expr1b</span><span class="o">.</span><span class="n">expand</span><span class="p">(),</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">shorter</span><span class="p">(</span><span class="n">expr2b</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span> <span class="o">==</span> <span class="n">expr2b</span><span class="p">:</span>
-                <span class="n">expr1</span><span class="p">,</span> <span class="n">expr2</span> <span class="o">=</span> <span class="n">expr1b</span><span class="p">,</span> <span class="n">expr2b</span>
-
-    <span class="k">if</span> <span class="n">ratio</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">expr2</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">shorter</span><span class="p">(</span><span class="n">expr2</span><span class="p">,</span> <span class="n">expr1</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>  <span class="c"># XXX: temporary hack</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="c"># hyperexpand automatically only works on hypergeometric terms</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">hyperexpand</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">BesselBase</span><span class="p">):</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">besselsimp</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">TrigonometricFunction</span><span class="p">)</span> <span class="ow">or</span> <span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">HyperbolicFunction</span><span class="p">):</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">trigsimp</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">log</span><span class="p">):</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">shorter</span><span class="p">(</span><span class="n">expand_log</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">),</span> <span class="n">logcombine</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">CombinatorialFunction</span><span class="p">,</span> <span class="n">gamma</span><span class="p">):</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">combsimp</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">powsimp</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;exp&#39;</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">short</span> <span class="o">=</span> <span class="n">shorter</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">powsimp</span><span class="p">(</span><span class="n">factor_terms</span><span class="p">(</span><span class="n">expr</span><span class="p">)))</span>
-    <span class="k">if</span> <span class="n">short</span> <span class="o">!=</span> <span class="n">expr</span><span class="p">:</span>
-        <span class="c"># get rid of hollow 2-arg Mul factorization</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.rules</span> <span class="kn">import</span> <span class="n">Transform</span>
-        <span class="n">hollow_mul</span> <span class="o">=</span> <span class="n">Transform</span><span class="p">(</span>
-            <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">x</span><span class="o">.</span><span class="n">args</span><span class="p">),</span>
-            <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span>
-            <span class="n">x</span><span class="o">.</span><span class="n">is_Mul</span> <span class="ow">and</span>
-            <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span> <span class="ow">and</span>
-            <span class="n">x</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">and</span>
-            <span class="n">x</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">and</span>
-            <span class="n">x</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">)</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">shorter</span><span class="p">(</span><span class="n">short</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="n">hollow_mul</span><span class="p">),</span> <span class="n">expr</span><span class="p">)</span>
-    <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-    <span class="k">if</span> <span class="n">denom</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">fraction</span><span class="p">(</span><span class="n">radsimp</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">denom</span><span class="p">,</span> <span class="n">symbolic</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">max_terms</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="p">(</span><span class="n">numer</span><span class="o">*</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span><span class="o">/</span><span class="n">d</span>
-
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">():</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">d</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="o">-</span><span class="n">n</span><span class="o">/</span><span class="p">(</span><span class="o">-</span><span class="n">d</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">measure</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">ratio</span><span class="o">*</span><span class="n">measure</span><span class="p">(</span><span class="n">original_expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">original_expr</span>
-
-    <span class="k">return</span> <span class="n">expr</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_real_to_rational</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Replace all reals in expr with rationals.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import nsimplify</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; nsimplify(.76 + .1*x**.5, rational=True)</span>
-<span class="sd">    sqrt(x)/10 + 19/25</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="n">expr</span>
-    <span class="n">reps</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">p</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Float</span><span class="p">):</span>
-        <span class="n">newr</span> <span class="o">=</span> <span class="n">nsimplify</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">rational</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">newr</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">or</span> <span class="n">r</span><span class="o">.</span><span class="n">is_finite</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">newr</span><span class="o">.</span><span class="n">is_finite</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">newr</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">newr</span> <span class="o">=</span> <span class="o">-</span><span class="n">r</span>
-                <span class="n">d</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="nb">int</span><span class="p">((</span><span class="n">mpmath</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">newr</span><span class="p">)</span><span class="o">/</span><span class="n">mpmath</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mi">10</span><span class="p">))))</span>
-                <span class="n">newr</span> <span class="o">=</span> <span class="o">-</span><span class="n">Rational</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">newr</span><span class="o">/</span><span class="n">d</span><span class="p">))</span><span class="o">*</span><span class="n">d</span>
-            <span class="k">elif</span> <span class="n">newr</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">d</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="nb">int</span><span class="p">((</span><span class="n">mpmath</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">r</span><span class="p">)</span><span class="o">/</span><span class="n">mpmath</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mi">10</span><span class="p">))))</span>
-                <span class="n">newr</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">r</span><span class="o">/</span><span class="n">d</span><span class="p">))</span><span class="o">*</span><span class="n">d</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">newr</span> <span class="o">=</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="n">reps</span><span class="p">[</span><span class="n">r</span><span class="p">]</span> <span class="o">=</span> <span class="n">newr</span>
-    <span class="k">return</span> <span class="n">p</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">reps</span><span class="p">,</span> <span class="n">simultaneous</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="nsimplify"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.simplify.nsimplify">[docs]</a><span class="k">def</span> <span class="nf">nsimplify</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">constants</span><span class="o">=</span><span class="p">[],</span> <span class="n">tolerance</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">full</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">rational</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Find a simple representation for a number or, if there are free symbols or</span>
-<span class="sd">    if rational=True, then replace Floats with their Rational equivalents. If</span>
-<span class="sd">    no change is made and rational is not False then Floats will at least be</span>
-<span class="sd">    converted to Rationals.</span>
-
-<span class="sd">    For numerical expressions, a simple formula that numerically matches the</span>
-<span class="sd">    given numerical expression is sought (and the input should be possible</span>
-<span class="sd">    to evalf to a precision of at least 30 digits).</span>
-
-<span class="sd">    Optionally, a list of (rationally independent) constants to</span>
-<span class="sd">    include in the formula may be given.</span>
-
-<span class="sd">    A lower tolerance may be set to find less exact matches. If no tolerance</span>
-<span class="sd">    is given then the least precise value will set the tolerance (e.g. Floats</span>
-<span class="sd">    default to 15 digits of precision, so would be tolerance=10**-15).</span>
-
-<span class="sd">    With full=True, a more extensive search is performed</span>
-<span class="sd">    (this is useful to find simpler numbers when the tolerance</span>
-<span class="sd">    is set low).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import nsimplify, sqrt, GoldenRatio, exp, I, exp, pi</span>
-<span class="sd">        &gt;&gt;&gt; nsimplify(4/(1+sqrt(5)), [GoldenRatio])</span>
-<span class="sd">        -2 + 2*GoldenRatio</span>
-<span class="sd">        &gt;&gt;&gt; nsimplify((1/(exp(3*pi*I/5)+1)))</span>
-<span class="sd">        1/2 - I*sqrt(sqrt(5)/10 + 1/4)</span>
-<span class="sd">        &gt;&gt;&gt; nsimplify(I**I, [pi])</span>
-<span class="sd">        exp(-pi/2)</span>
-<span class="sd">        &gt;&gt;&gt; nsimplify(pi, tolerance=0.01)</span>
-<span class="sd">        22/7</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    sympy.core.function.nfloat</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">rational</span> <span class="ow">or</span> <span class="n">expr</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">_real_to_rational</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="c"># sympy&#39;s default tolarance for Rationals is 15; other numbers may have</span>
-    <span class="c"># lower tolerances set, so use them to pick the largest tolerance if none</span>
-    <span class="c"># was given</span>
-    <span class="n">tolerance</span> <span class="o">=</span> <span class="n">tolerance</span> <span class="ow">or</span> <span class="mi">10</span><span class="o">**-</span><span class="nb">min</span><span class="p">([</span><span class="mi">15</span><span class="p">]</span> <span class="o">+</span>
-                                     <span class="p">[</span><span class="n">mpmath</span><span class="o">.</span><span class="n">libmp</span><span class="o">.</span><span class="n">libmpf</span><span class="o">.</span><span class="n">prec_to_dps</span><span class="p">(</span><span class="n">n</span><span class="o">.</span><span class="n">_prec</span><span class="p">)</span>
-                                     <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Float</span><span class="p">)])</span>
-
-    <span class="n">prec</span> <span class="o">=</span> <span class="mi">30</span>
-    <span class="n">bprec</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">prec</span><span class="o">*</span><span class="mf">3.33</span><span class="p">)</span>
-
-    <span class="n">constants_dict</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">for</span> <span class="n">constant</span> <span class="ow">in</span> <span class="n">constants</span><span class="p">:</span>
-        <span class="n">constant</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">constant</span><span class="p">)</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="n">constant</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">v</span><span class="o">.</span><span class="n">is_Float</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;constants must be real-valued&quot;</span><span class="p">)</span>
-        <span class="n">constants_dict</span><span class="p">[</span><span class="nb">str</span><span class="p">(</span><span class="n">constant</span><span class="p">)]</span> <span class="o">=</span> <span class="n">v</span><span class="o">.</span><span class="n">_to_mpmath</span><span class="p">(</span><span class="n">bprec</span><span class="p">)</span>
-
-    <span class="n">exprval</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">prec</span><span class="p">,</span> <span class="n">chop</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">re</span><span class="p">,</span> <span class="n">im</span> <span class="o">=</span> <span class="n">exprval</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-
-    <span class="c"># Must be numerical</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="p">((</span><span class="n">re</span><span class="o">.</span><span class="n">is_Float</span> <span class="ow">or</span> <span class="n">re</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">im</span><span class="o">.</span><span class="n">is_Float</span> <span class="ow">or</span> <span class="n">im</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">)):</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="k">def</span> <span class="nf">nsimplify_real</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-        <span class="n">orig</span> <span class="o">=</span> <span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span>
-        <span class="n">xv</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">_to_mpmath</span><span class="p">(</span><span class="n">bprec</span><span class="p">)</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="c"># We&#39;ll be happy with low precision if a simple fraction</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">tolerance</span> <span class="ow">or</span> <span class="n">full</span><span class="p">):</span>
-                <span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-                <span class="n">rat</span> <span class="o">=</span> <span class="n">mpmath</span><span class="o">.</span><span class="n">findpoly</span><span class="p">(</span><span class="n">xv</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">rat</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="o">-</span><span class="nb">int</span><span class="p">(</span><span class="n">rat</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="nb">int</span><span class="p">(</span><span class="n">rat</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-            <span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="n">prec</span>
-            <span class="n">newexpr</span> <span class="o">=</span> <span class="n">mpmath</span><span class="o">.</span><span class="n">identify</span><span class="p">(</span><span class="n">xv</span><span class="p">,</span> <span class="n">constants</span><span class="o">=</span><span class="n">constants_dict</span><span class="p">,</span>
-                <span class="n">tol</span><span class="o">=</span><span class="n">tolerance</span><span class="p">,</span> <span class="n">full</span><span class="o">=</span><span class="n">full</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">newexpr</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span>
-            <span class="k">if</span> <span class="n">full</span><span class="p">:</span>
-                <span class="n">newexpr</span> <span class="o">=</span> <span class="n">newexpr</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">newexpr</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_finite</span> <span class="ow">is</span> <span class="bp">False</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">xv</span> <span class="ow">in</span> <span class="p">[</span><span class="n">mpmath</span><span class="o">.</span><span class="n">inf</span><span class="p">,</span> <span class="n">mpmath</span><span class="o">.</span><span class="n">ninf</span><span class="p">]:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span>
-            <span class="k">return</span> <span class="n">expr</span>
-        <span class="k">finally</span><span class="p">:</span>
-            <span class="c"># even though there are returns above, this is executed</span>
-            <span class="c"># before leaving</span>
-            <span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="n">orig</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">re</span><span class="p">:</span>
-            <span class="n">re</span> <span class="o">=</span> <span class="n">nsimplify_real</span><span class="p">(</span><span class="n">re</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">im</span><span class="p">:</span>
-            <span class="n">im</span> <span class="o">=</span> <span class="n">nsimplify_real</span><span class="p">(</span><span class="n">im</span><span class="p">)</span>
-    <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">rational</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_real_to_rational</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="n">rv</span> <span class="o">=</span> <span class="n">re</span> <span class="o">+</span> <span class="n">im</span><span class="o">*</span><span class="n">S</span><span class="o">.</span><span class="n">ImaginaryUnit</span>
-    <span class="c"># if there was a change or rational is explicitly not wanted</span>
-    <span class="c"># return the value, else return the Rational representation</span>
-    <span class="k">if</span> <span class="n">rv</span> <span class="o">!=</span> <span class="n">expr</span> <span class="ow">or</span> <span class="n">rational</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">rv</span>
-    <span class="k">return</span> <span class="n">_real_to_rational</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="logcombine"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.simplify.logcombine">[docs]</a><span class="k">def</span> <span class="nf">logcombine</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Takes logarithms and combines them using the following rules:</span>
-
-<span class="sd">    - log(x)+log(y) == log(x*y)</span>
-<span class="sd">    - a*log(x) == log(x**a)</span>
-
-<span class="sd">    These identities are only valid if x and y are positive and if a is real,</span>
-<span class="sd">    so the function will not combine the terms unless the arguments have the</span>
-<span class="sd">    proper assumptions on them.  Use logcombine(func, force=True) to</span>
-<span class="sd">    automatically assume that the arguments of logs are positive and that</span>
-<span class="sd">    coefficients are real.  Note that this will not change any assumptions</span>
-<span class="sd">    already in place, so if the coefficient is imaginary or the argument</span>
-<span class="sd">    negative, combine will still not combine the equations.  Change the</span>
-<span class="sd">    assumptions on the variables to make them combine.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, symbols, log, logcombine</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import a, x, y, z</span>
-<span class="sd">    &gt;&gt;&gt; logcombine(a*log(x)+log(y)-log(z))</span>
-<span class="sd">    a*log(x) + log(y) - log(z)</span>
-<span class="sd">    &gt;&gt;&gt; logcombine(a*log(x)+log(y)-log(z), force=True)</span>
-<span class="sd">    log(x**a*y/z)</span>
-<span class="sd">    &gt;&gt;&gt; x,y,z = symbols(&#39;x,y,z&#39;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; a = Symbol(&#39;a&#39;, real=True)</span>
-<span class="sd">    &gt;&gt;&gt; logcombine(a*log(x)+log(y)-log(z))</span>
-<span class="sd">    log(x**a*y/z)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># Try to make (a+bi)*log(x) == a*log(x)+bi*log(x).  This needs to be a</span>
-    <span class="c"># separate function call to avoid infinite recursion.</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">_logcombine</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">force</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_logcombine</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Does the main work for logcombine, it&#39;s a separate function to avoid an</span>
-<span class="sd">    infinite recursion. See the docstrings of logcombine() for help.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">_getlogargs</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the arguments of the logarithm in an expression.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        _getlogargs(a*log(x*y))</span>
-<span class="sd">        x*y</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">log</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">log</span><span class="p">:</span>
-                    <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">_getlogargs</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">flatten</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">or</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_NumberSymbol</span> <span class="ow">or</span> <span class="nb">type</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span> <span class="o">==</span> <span class="n">C</span><span class="o">.</span><span class="n">Integral</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Equality</span><span class="p">):</span>
-        <span class="n">retval</span> <span class="o">=</span> <span class="n">Equality</span><span class="p">(</span><span class="n">_logcombine</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">lhs</span> <span class="o">-</span> <span class="n">expr</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">force</span><span class="p">),</span>
-        <span class="n">Integer</span><span class="p">(</span><span class="mi">0</span><span class="p">))</span>
-        <span class="c"># If logcombine couldn&#39;t do much with the equality, try to make it like</span>
-        <span class="c"># it was.  Hopefully extract_additively won&#39;t become smart enough to</span>
-        <span class="c"># take logs apart :)</span>
-        <span class="n">right</span> <span class="o">=</span> <span class="n">retval</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">extract_additively</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">lhs</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">right</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Equality</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">_logcombine</span><span class="p">(</span><span class="o">-</span><span class="n">right</span><span class="p">,</span> <span class="n">force</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">retval</span>
-
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-        <span class="n">argslist</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">notlogs</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">coeflogs</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">log</span><span class="p">:</span>
-                <span class="k">if</span> <span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_positive</span> <span class="ow">or</span> <span class="p">(</span><span class="n">force</span> <span class="ow">and</span> <span class="ow">not</span>
-                <span class="n">i</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_nonpositive</span><span class="p">)):</span>
-                    <span class="n">argslist</span> <span class="o">*=</span> <span class="n">_logcombine</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">force</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">notlogs</span> <span class="o">+=</span> <span class="n">i</span>
-            <span class="k">elif</span> <span class="n">i</span><span class="o">.</span><span class="n">is_Mul</span> <span class="ow">and</span> <span class="nb">any</span><span class="p">(</span>
-                <span class="p">[</span><span class="nb">getattr</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="s">&#39;func&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span> <span class="o">==</span> <span class="n">log</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">i</span><span class="o">.</span><span class="n">args</span><span class="p">]):</span>
-                <span class="n">largs</span> <span class="o">=</span> <span class="n">_getlogargs</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                <span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">largs</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span>
-                <span class="n">loglargs</span> <span class="o">=</span> <span class="mi">1</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">largs</span><span class="p">:</span>
-                    <span class="n">loglargs</span> <span class="o">*=</span> <span class="n">log</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="nb">getattr</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="s">&#39;is_positive&#39;</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">largs</span><span class="p">)</span> \
-                    <span class="ow">and</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="n">loglargs</span><span class="p">),</span> <span class="s">&#39;is_real&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span> \
-                    <span class="ow">or</span> <span class="p">(</span><span class="n">force</span>
-                        <span class="ow">and</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">(</span><span class="nb">getattr</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="s">&#39;is_nonpositive&#39;</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">largs</span><span class="p">)</span>
-                        <span class="ow">and</span> <span class="ow">not</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="n">loglargs</span><span class="p">),</span>
-                        <span class="s">&#39;is_real&#39;</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">):</span>
-
-                    <span class="n">coeflogs</span> <span class="o">+=</span> <span class="n">_logcombine</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">force</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">notlogs</span> <span class="o">+=</span> <span class="n">i</span>
-            <span class="k">elif</span> <span class="n">i</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">log</span><span class="p">):</span>
-                <span class="n">notlogs</span> <span class="o">+=</span> <span class="n">_logcombine</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">force</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">notlogs</span> <span class="o">+=</span> <span class="n">i</span>
-        <span class="k">if</span> <span class="n">notlogs</span> <span class="o">+</span> <span class="n">log</span><span class="p">(</span><span class="n">argslist</span><span class="p">)</span> <span class="o">+</span> <span class="n">coeflogs</span> <span class="o">==</span> <span class="n">expr</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">alllogs</span> <span class="o">=</span> <span class="n">_logcombine</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">argslist</span><span class="p">)</span> <span class="o">+</span> <span class="n">coeflogs</span><span class="p">,</span> <span class="n">force</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">notlogs</span> <span class="o">+</span> <span class="n">alllogs</span>
-
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-        <span class="n">coef</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">coef</span> \
-            <span class="ow">and</span> <span class="p">(</span><span class="n">coef</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span>
-                <span class="ow">or</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Number</span>
-                <span class="ow">or</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_NumberSymbol</span>
-                <span class="ow">or</span> <span class="nb">type</span><span class="p">(</span><span class="n">coef</span><span class="p">[</span><span class="n">a</span><span class="p">])</span> <span class="ow">in</span> <span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="nb">float</span><span class="p">)</span>
-                <span class="ow">or</span> <span class="p">(</span><span class="n">force</span>
-                <span class="ow">and</span> <span class="ow">not</span> <span class="n">coef</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="o">.</span><span class="n">is_imaginary</span><span class="p">))</span> \
-            <span class="ow">and</span> <span class="p">(</span><span class="n">coef</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="o">.</span><span class="n">func</span> <span class="o">!=</span> <span class="n">log</span>
-                <span class="ow">or</span> <span class="n">force</span>
-                <span class="ow">or</span> <span class="p">(</span><span class="ow">not</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">coef</span><span class="p">[</span><span class="n">a</span><span class="p">],</span> <span class="s">&#39;is_real&#39;</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">False</span>
-                    <span class="ow">and</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="s">&#39;is_positive&#39;</span><span class="p">))):</span>
-
-            <span class="k">return</span> <span class="n">log</span><span class="p">(</span><span class="n">coef</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">**</span><span class="n">coef</span><span class="p">[</span><span class="n">a</span><span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_logcombine</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">force</span><span class="p">)</span><span class="o">*</span><span class="nb">reduce</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span>
-             <span class="n">_logcombine</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">force</span><span class="p">)</span><span class="o">*</span><span class="n">_logcombine</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">force</span><span class="p">),</span>
-                <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Function</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">_logcombine</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">force</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">_logcombine</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">force</span><span class="p">)</span> <span class="o">**</span> \
-            <span class="n">_logcombine</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">force</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">expr</span>
-
-
-<div class="viewcode-block" id="besselsimp"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.simplify.besselsimp">[docs]</a><span class="k">def</span> <span class="nf">besselsimp</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Simplify bessel-type functions.</span>
-
-<span class="sd">    This routine tries to simplify bessel-type functions. Currently it only</span>
-<span class="sd">    works on the Bessel J and I functions, however. It works by looking at all</span>
-<span class="sd">    such functions in turn, and eliminating factors of &quot;I&quot; and &quot;-1&quot; (actually</span>
-<span class="sd">    their polar equivalents) in front of the argument. After that, functions of</span>
-<span class="sd">    half-integer order are rewritten using trigonometric functions.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import besselj, besseli, besselsimp, polar_lift, I, S</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import z, nu</span>
-<span class="sd">    &gt;&gt;&gt; besselsimp(besselj(nu, z*polar_lift(-1)))</span>
-<span class="sd">    exp(I*pi*nu)*besselj(nu, z)</span>
-<span class="sd">    &gt;&gt;&gt; besselsimp(besseli(nu, z*polar_lift(-I)))</span>
-<span class="sd">    exp(-I*pi*nu/2)*besselj(nu, z)</span>
-<span class="sd">    &gt;&gt;&gt; besselsimp(besseli(S(-1)/2, z))</span>
-<span class="sd">    sqrt(2)*cosh(z)/(sqrt(pi)*sqrt(z))</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">besselj</span><span class="p">,</span> <span class="n">besseli</span><span class="p">,</span> <span class="n">jn</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">Dummy</span>
-    <span class="c"># TODO</span>
-    <span class="c"># - extension to more types of functions</span>
-    <span class="c">#   (at least rewriting functions of half integer order should be straight</span>
-    <span class="c">#    forward also for Y and K)</span>
-    <span class="c"># - better algorithm?</span>
-    <span class="c"># - simplify (cos(pi*b)*besselj(b,z) - besselj(-b,z))/sin(pi*b) ...</span>
-    <span class="c"># - use contiguity relations?</span>
-
-    <span class="k">def</span> <span class="nf">replacer</span><span class="p">(</span><span class="n">fro</span><span class="p">,</span> <span class="n">to</span><span class="p">,</span> <span class="n">factors</span><span class="p">):</span>
-        <span class="n">factors</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">factors</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">repl</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">factors</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">z</span><span class="p">)):</span>
-                <span class="k">return</span> <span class="n">to</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">fro</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">repl</span>
-
-    <span class="k">def</span> <span class="nf">torewrite</span><span class="p">(</span><span class="n">fro</span><span class="p">,</span> <span class="n">to</span><span class="p">):</span>
-        <span class="k">def</span> <span class="nf">tofunc</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">fro</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">to</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">tofunc</span>
-
-    <span class="k">def</span> <span class="nf">tominus</span><span class="p">(</span><span class="n">fro</span><span class="p">):</span>
-        <span class="k">def</span> <span class="nf">tofunc</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">nu</span><span class="p">)</span><span class="o">*</span><span class="n">fro</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">exp_polar</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">tofunc</span>
-
-    <span class="n">ifactors</span> <span class="o">=</span> <span class="p">[</span><span class="n">I</span><span class="p">,</span> <span class="n">exp_polar</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">),</span> <span class="n">exp_polar</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)]</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">besselj</span><span class="p">,</span> <span class="n">replacer</span><span class="p">(</span><span class="n">besselj</span><span class="p">,</span>
-                                          <span class="n">torewrite</span><span class="p">(</span><span class="n">besselj</span><span class="p">,</span> <span class="n">besseli</span><span class="p">),</span> <span class="n">ifactors</span><span class="p">))</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">besseli</span><span class="p">,</span> <span class="n">replacer</span><span class="p">(</span><span class="n">besseli</span><span class="p">,</span>
-                                          <span class="n">torewrite</span><span class="p">(</span><span class="n">besseli</span><span class="p">,</span> <span class="n">besselj</span><span class="p">),</span> <span class="n">ifactors</span><span class="p">))</span>
-
-    <span class="n">minusfactors</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">exp_polar</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">)]</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span>
-        <span class="n">besselj</span><span class="p">,</span> <span class="n">replacer</span><span class="p">(</span><span class="n">besselj</span><span class="p">,</span> <span class="n">tominus</span><span class="p">(</span><span class="n">besselj</span><span class="p">),</span> <span class="n">minusfactors</span><span class="p">))</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span>
-        <span class="n">besseli</span><span class="p">,</span> <span class="n">replacer</span><span class="p">(</span><span class="n">besseli</span><span class="p">,</span> <span class="n">tominus</span><span class="p">(</span><span class="n">besseli</span><span class="p">),</span> <span class="n">minusfactors</span><span class="p">))</span>
-
-    <span class="n">z0</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;z&#39;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">expander</span><span class="p">(</span><span class="n">fro</span><span class="p">):</span>
-        <span class="k">def</span> <span class="nf">repl</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">):</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">nu</span> <span class="o">%</span> <span class="mi">1</span><span class="p">)</span> <span class="o">!=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">fro</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">unpolarify</span><span class="p">(</span><span class="n">fro</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">z0</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">besselj</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">jn</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">func</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">z0</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">repl</span>
-
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">besselj</span><span class="p">,</span> <span class="n">expander</span><span class="p">(</span><span class="n">besselj</span><span class="p">))</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">besseli</span><span class="p">,</span> <span class="n">expander</span><span class="p">(</span><span class="n">besseli</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">expr</span></div>
-</pre></div>
-
-          </div>
-        </div>
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-  <h1>Source code for sympy.simplify.sqrtdenest</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">sign</span><span class="p">,</span> <span class="n">root</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">Wild</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Expr</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">expand_multinomial</span><span class="p">,</span> <span class="n">expand_mul</span>
-<span class="kn">from</span> <span class="nn">sympy.core.symbol</span> <span class="kn">import</span> <span class="n">Dummy</span>
-<span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">PolynomialError</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">count_ops</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">default_sort_key</span>
-
-
-<span class="k">def</span> <span class="nf">_mexpand</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">expand_multinomial</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-
-
-<span class="k">def</span> <span class="nf">is_sqrt</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return True if expr is a sqrt, otherwise False.&quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="nb">abs</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">)</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span>
-
-
-<span class="k">def</span> <span class="nf">sqrt_depth</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return the maximum depth of any square root argument of p.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.functions.elementary.miscellaneous import sqrt</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.sqrtdenest import sqrt_depth</span>
-
-<span class="sd">    Neither of these square roots contains any other square roots</span>
-<span class="sd">    so the depth is 1:</span>
-
-<span class="sd">    &gt;&gt;&gt; sqrt_depth(1 + sqrt(2)*(1 + sqrt(3)))</span>
-<span class="sd">    1</span>
-
-<span class="sd">    The sqrt(3) is contained within a square root so the depth is</span>
-<span class="sd">    2:</span>
-
-<span class="sd">    &gt;&gt;&gt; sqrt_depth(1 + sqrt(2)*sqrt(1 + sqrt(3)))</span>
-<span class="sd">    2</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-    <span class="k">elif</span> <span class="n">p</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">or</span> <span class="n">p</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">max</span><span class="p">([</span><span class="n">sqrt_depth</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">p</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-    <span class="k">elif</span> <span class="n">is_sqrt</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sqrt_depth</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">base</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="mi">0</span>
-
-
-<span class="k">def</span> <span class="nf">is_algebraic</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return True if p is comprised of only Rationals or square roots</span>
-<span class="sd">    of Rationals and algebraic operations.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.functions.elementary.miscellaneous import sqrt</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.sqrtdenest import is_algebraic</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import cos</span>
-<span class="sd">    &gt;&gt;&gt; is_algebraic(sqrt(2)*(3/(sqrt(7) + sqrt(5)*sqrt(2))))</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; is_algebraic(sqrt(2)*(3/(sqrt(7) + sqrt(5)*cos(2))))</span>
-<span class="sd">    False</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">True</span>
-    <span class="k">elif</span> <span class="n">p</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">False</span>
-    <span class="k">elif</span> <span class="n">is_sqrt</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="ow">or</span> <span class="n">p</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">p</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">is_algebraic</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">base</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">p</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">or</span> <span class="n">p</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="n">is_algebraic</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">p</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-
-<span class="k">def</span> <span class="nf">_subsets</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns all possible subsets of the set (0, 1, ..., n-1) except the</span>
-<span class="sd">    empty set, listed in reversed lexicographical order according to binary</span>
-<span class="sd">    representation, so that the case of the fourth root is treated last.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.sqrtdenest import _subsets</span>
-<span class="sd">    &gt;&gt;&gt; _subsets(2)</span>
-<span class="sd">    [[1, 0], [0, 1], [1, 1]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">1</span><span class="p">]]</span>
-    <span class="k">elif</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]]</span>
-    <span class="k">elif</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
-             <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">_subsets</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="n">a0</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="o">+</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">b</span><span class="p">]</span>
-        <span class="n">a1</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="o">+</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">b</span><span class="p">]</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">a0</span> <span class="o">+</span> <span class="p">[[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="p">[</span><span class="mi">1</span><span class="p">]]</span> <span class="o">+</span> <span class="n">a1</span>
-    <span class="k">return</span> <span class="n">a</span>
-
-
-<div class="viewcode-block" id="sqrtdenest"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.sqrtdenest.sqrtdenest">[docs]</a><span class="k">def</span> <span class="nf">sqrtdenest</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">max_iter</span><span class="o">=</span><span class="mi">3</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Denests sqrts in an expression that contain other square roots</span>
-<span class="sd">    if possible, otherwise returns the expr unchanged. This is based on the</span>
-<span class="sd">    algorithms of [1].</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.sqrtdenest import sqrtdenest</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import sqrt</span>
-<span class="sd">    &gt;&gt;&gt; sqrtdenest(sqrt(5 + 2 * sqrt(6)))</span>
-<span class="sd">    sqrt(2) + sqrt(3)</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    sympy.solvers.solvers.unrad</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-<span class="sd">    [1] http://www.almaden.ibm.com/cs/people/fagin/symb85.pdf</span>
-
-<span class="sd">    [2] D. J. Jeffrey and A. D. Rich, &#39;Symplifying Square Roots of Square Roots</span>
-<span class="sd">    by Denesting&#39; (available at http://www.cybertester.com/data/denest.pdf)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">max_iter</span><span class="p">):</span>
-        <span class="n">z</span> <span class="o">=</span> <span class="n">_sqrtdenest0</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">expr</span> <span class="o">==</span> <span class="n">z</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">z</span>
-    <span class="k">return</span> <span class="n">expr</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_sqrt_match</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return [a, b, r] for p.match(a + b*sqrt(r)) where, in addition to</span>
-<span class="sd">    matching, sqrt(r) also has then maximal sqrt_depth among addends of p.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.functions.elementary.miscellaneous import sqrt</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.sqrtdenest import _sqrt_match</span>
-<span class="sd">    &gt;&gt;&gt; _sqrt_match(1 + sqrt(2) + sqrt(2)*sqrt(3) +  2*sqrt(1+sqrt(5)))</span>
-<span class="sd">    [1 + sqrt(2) + sqrt(6), 2, 1 + sqrt(5)]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.simplify.simplify</span> <span class="kn">import</span> <span class="n">split_surds</span>
-
-    <span class="n">p</span> <span class="o">=</span> <span class="n">_mexpand</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">p</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-        <span class="n">pargs</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">all</span><span class="p">((</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">is_Rational</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">pargs</span><span class="p">):</span>
-            <span class="n">r</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">a</span> <span class="o">=</span> <span class="n">split_surds</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-            <span class="n">res</span> <span class="o">=</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">r</span>
-            <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="n">res</span><span class="p">)</span>
-        <span class="c"># to make the process canonical, the argument is included in the tuple</span>
-        <span class="c"># so when the max is selected, it will be the largest arg having a</span>
-        <span class="c"># given depth</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="p">[(</span><span class="n">sqrt_depth</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">pargs</span><span class="p">)]</span>
-        <span class="n">nmax</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">nmax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># select r</span>
-            <span class="n">depth</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">i</span> <span class="o">=</span> <span class="n">nmax</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">pargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="n">v</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">if</span> <span class="n">r</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                <span class="n">bv</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="n">rv</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">r</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">sqrt_depth</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">depth</span><span class="p">:</span>
-                        <span class="n">bv</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">rv</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-                <span class="n">b</span> <span class="o">=</span> <span class="n">Mul</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">bv</span><span class="p">)</span>
-                <span class="n">r</span> <span class="o">=</span> <span class="n">Mul</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">rv</span><span class="p">)</span>
-            <span class="c"># collect terms comtaining r</span>
-            <span class="n">a1</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">b1</span> <span class="o">=</span> <span class="p">[</span><span class="n">b</span><span class="p">]</span>
-            <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">v</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">depth</span><span class="p">:</span>
-                    <span class="n">a1</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">x1</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-                    <span class="k">if</span> <span class="n">x1</span> <span class="o">==</span> <span class="n">r</span><span class="p">:</span>
-                        <span class="n">b1</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">x1</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                            <span class="n">x1args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">x1</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-                            <span class="k">if</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">x1args</span><span class="p">:</span>
-                                <span class="n">x1args</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-                                <span class="n">b1</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">x1args</span><span class="p">))</span>
-                            <span class="k">else</span><span class="p">:</span>
-                                <span class="n">a1</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="n">a1</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">a1</span><span class="p">)</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">b1</span><span class="p">)</span>
-            <span class="c">#a = Add._from_args(pargs)</span>
-            <span class="n">res</span> <span class="o">=</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">b</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">is_sqrt</span><span class="p">(</span><span class="n">r</span><span class="p">):</span>
-            <span class="n">res</span> <span class="o">=</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="n">res</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">SqrtdenestStopIteration</span><span class="p">(</span><span class="ne">StopIteration</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-
-<span class="k">def</span> <span class="nf">_sqrtdenest0</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns expr after denesting its arguments.&quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="n">is_sqrt</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">d</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>  <span class="c"># n is a square root</span>
-            <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="n">args</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">n</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">2</span> <span class="ow">and</span> <span class="nb">all</span><span class="p">((</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">is_Integer</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">args</span><span class="p">):</span>
-                    <span class="k">try</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">_sqrtdenest_rec</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-                    <span class="k">except</span> <span class="n">SqrtdenestStopIteration</span><span class="p">:</span>
-                        <span class="k">pass</span>
-                <span class="n">expr</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">_mexpand</span><span class="p">(</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">_sqrtdenest0</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">args</span><span class="p">])))</span>
-            <span class="k">return</span> <span class="n">_sqrtdenest1</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="p">[</span><span class="n">_sqrtdenest0</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">d</span><span class="p">)]</span>
-            <span class="k">return</span> <span class="n">n</span><span class="o">/</span><span class="n">d</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Expr</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">if</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">_sqrtdenest0</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">args</span><span class="p">])</span>
-    <span class="k">return</span> <span class="n">expr</span>
-
-
-<span class="k">def</span> <span class="nf">_sqrtdenest_rec</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Helper that denests the square root of three or more surds.</span>
-
-<span class="sd">    It returns the denested expression; if it cannot be denested it</span>
-<span class="sd">    throws SqrtdenestStopIteration</span>
-
-<span class="sd">    Algorithm: expr.base is in the extension Q_m = Q(sqrt(r_1),..,sqrt(r_k));</span>
-<span class="sd">    split expr.base = a + b*sqrt(r_k), where `a` and `b` are on</span>
-<span class="sd">    Q_(m-1) = Q(sqrt(r_1),..,sqrt(r_(k-1))); then a**2 - b**2*r_k is</span>
-<span class="sd">    on Q_(m-1); denest sqrt(a**2 - b**2*r_k) and so on.</span>
-<span class="sd">    See [1], section 6.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import sqrt</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.sqrtdenest import _sqrtdenest_rec</span>
-<span class="sd">    &gt;&gt;&gt; _sqrtdenest_rec(sqrt(-72*sqrt(2) + 158*sqrt(5) + 498))</span>
-<span class="sd">    -sqrt(10) + sqrt(2) + 9 + 9*sqrt(5)</span>
-<span class="sd">    &gt;&gt;&gt; w=-6*sqrt(55)-6*sqrt(35)-2*sqrt(22)-2*sqrt(14)+2*sqrt(77)+6*sqrt(10)+65</span>
-<span class="sd">    &gt;&gt;&gt; _sqrtdenest_rec(sqrt(w))</span>
-<span class="sd">    -sqrt(11) - sqrt(7) + sqrt(2) + 3*sqrt(5)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.simplify.simplify</span> <span class="kn">import</span> <span class="n">radsimp</span><span class="p">,</span> <span class="n">split_surds</span><span class="p">,</span> <span class="n">rad_rationalize</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">sqrtdenest</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">base</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">_sqrtdenest_rec</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="o">-</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">))</span>
-    <span class="n">g</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">split_surds</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">)</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">a</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">a</span> <span class="o">&lt;</span> <span class="n">b</span><span class="p">:</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">b</span><span class="p">,</span> <span class="n">a</span>
-    <span class="n">c2</span> <span class="o">=</span> <span class="n">_mexpand</span><span class="p">(</span><span class="n">a</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">b</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">c2</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="n">g</span><span class="p">,</span> <span class="n">a1</span><span class="p">,</span> <span class="n">b1</span> <span class="o">=</span> <span class="n">split_surds</span><span class="p">(</span><span class="n">c2</span><span class="p">)</span>
-        <span class="n">a1</span> <span class="o">=</span> <span class="n">a1</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">a1</span> <span class="o">&lt;</span> <span class="n">b1</span><span class="p">:</span>
-            <span class="n">a1</span><span class="p">,</span> <span class="n">b1</span> <span class="o">=</span> <span class="n">b1</span><span class="p">,</span> <span class="n">a1</span>
-        <span class="n">c2_1</span> <span class="o">=</span> <span class="n">_mexpand</span><span class="p">(</span><span class="n">a1</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">b1</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-        <span class="n">c_1</span> <span class="o">=</span> <span class="n">_sqrtdenest_rec</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">c2_1</span><span class="p">))</span>
-        <span class="n">d_1</span> <span class="o">=</span> <span class="n">_sqrtdenest_rec</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">a1</span> <span class="o">+</span> <span class="n">c_1</span><span class="p">))</span>
-        <span class="n">num</span><span class="p">,</span> <span class="n">den</span> <span class="o">=</span> <span class="n">rad_rationalize</span><span class="p">(</span><span class="n">b1</span><span class="p">,</span> <span class="n">d_1</span><span class="p">)</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">_mexpand</span><span class="p">(</span><span class="n">d_1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">num</span><span class="o">/</span><span class="p">(</span><span class="n">den</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">_sqrtdenest1</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">c2</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="n">sqrt_depth</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">SqrtdenestStopIteration</span>
-    <span class="n">ac</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="n">c</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">ac</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">count_ops</span><span class="p">(</span><span class="n">ac</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="n">count_ops</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="n">SqrtdenestStopIteration</span>
-    <span class="n">d</span> <span class="o">=</span> <span class="n">sqrtdenest</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">ac</span><span class="p">))</span>
-    <span class="k">if</span> <span class="n">sqrt_depth</span><span class="p">(</span><span class="n">d</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">SqrtdenestStopIteration</span>
-    <span class="n">num</span><span class="p">,</span> <span class="n">den</span> <span class="o">=</span> <span class="n">rad_rationalize</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">d</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">num</span><span class="o">/</span><span class="p">(</span><span class="n">den</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">radsimp</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">_mexpand</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">_sqrtdenest1</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">denester</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return denested expr after denesting with simpler methods or, that</span>
-<span class="sd">    failing, using the denester.&quot;&quot;&quot;</span>
-
-    <span class="kn">from</span> <span class="nn">sympy.simplify.simplify</span> <span class="kn">import</span> <span class="n">radsimp</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">is_sqrt</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="n">a</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">base</span>
-    <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span>
-    <span class="n">val</span> <span class="o">=</span> <span class="n">_sqrt_match</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">val</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">val</span>
-    <span class="c"># try a quick numeric denesting</span>
-    <span class="n">d2</span> <span class="o">=</span> <span class="n">_mexpand</span><span class="p">(</span><span class="n">a</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">b</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">r</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">d2</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">d2</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-            <span class="n">z</span> <span class="o">=</span> <span class="n">_sqrt_numeric_denest</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">d2</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">z</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">z</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># fourth root case</span>
-            <span class="c"># sqrtdenest(sqrt(3 + 2*sqrt(3))) =</span>
-            <span class="c"># sqrt(2)*3**(1/4)/2 + sqrt(2)*3**(3/4)/2</span>
-            <span class="n">dr2</span> <span class="o">=</span> <span class="n">_mexpand</span><span class="p">(</span><span class="o">-</span><span class="n">d2</span><span class="o">*</span><span class="n">r</span><span class="p">)</span>
-            <span class="n">dr</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">dr2</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">dr</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="n">z</span> <span class="o">=</span> <span class="n">_sqrt_numeric_denest</span><span class="p">(</span><span class="n">_mexpand</span><span class="p">(</span><span class="n">b</span><span class="o">*</span><span class="n">r</span><span class="p">),</span> <span class="n">a</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">dr2</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">z</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">z</span><span class="o">/</span><span class="n">root</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">z</span> <span class="o">=</span> <span class="n">_sqrt_symbolic_denest</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">z</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">z</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">denester</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">is_algebraic</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="n">res</span> <span class="o">=</span> <span class="n">sqrt_biquadratic_denest</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">d2</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">res</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">res</span>
-
-    <span class="c"># now call to the denester</span>
-    <span class="n">av0</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">d2</span><span class="p">]</span>
-    <span class="n">z</span> <span class="o">=</span> <span class="n">_denester</span><span class="p">([</span><span class="n">radsimp</span><span class="p">(</span><span class="n">expr</span><span class="o">**</span><span class="mi">2</span><span class="p">)],</span> <span class="n">av0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">sqrt_depth</span><span class="p">(</span><span class="n">expr</span><span class="p">))[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">av0</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span>
-    <span class="k">if</span> <span class="n">z</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">sqrt_depth</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">==</span> <span class="n">sqrt_depth</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span> <span class="ow">and</span> <span class="n">count_ops</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">count_ops</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">expr</span>
-        <span class="k">return</span> <span class="n">z</span>
-    <span class="k">return</span> <span class="n">expr</span>
-
-
-<span class="k">def</span> <span class="nf">_sqrt_symbolic_denest</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">r</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Given an expression, sqrt(a + b*sqrt(b)), return the denested</span>
-<span class="sd">    expression or None.</span>
-
-<span class="sd">    Algorithm:</span>
-<span class="sd">    If r = ra + rb*sqrt(rr), try replacing sqrt(rr) in ``a`` with</span>
-<span class="sd">    (y**2 - ra)/rb, and if the result is a quadratic, ca*y**2 + cb*y + cc, and</span>
-<span class="sd">    (cb + b)**2 - 4*ca*cc is 0, then sqrt(a + b*sqrt(r)) can be rewritten as</span>
-<span class="sd">    sqrt(ca*(sqrt(r) + (cb + b)/(2*ca))**2).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.sqrtdenest import _sqrt_symbolic_denest, sqrtdenest</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import sqrt, Symbol</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">    &gt;&gt;&gt; a, b, r = 16 - 2*sqrt(29), 2, -10*sqrt(29) + 55</span>
-<span class="sd">    &gt;&gt;&gt; _sqrt_symbolic_denest(a, b, r)</span>
-<span class="sd">    sqrt(-2*sqrt(29) + 11) + sqrt(5)</span>
-
-<span class="sd">    If the expression is numeric, it will be simplified:</span>
-
-<span class="sd">    &gt;&gt;&gt; w = sqrt(sqrt(sqrt(3) + 1) + 1) + 1 + sqrt(2)</span>
-<span class="sd">    &gt;&gt;&gt; sqrtdenest(sqrt((w**2).expand()))</span>
-<span class="sd">    1 + sqrt(2) + sqrt(1 + sqrt(1 + sqrt(3)))</span>
-
-<span class="sd">    Otherwise, it will only be simplified if assumptions allow:</span>
-
-<span class="sd">    &gt;&gt;&gt; w = w.subs(sqrt(3), sqrt(x + 3))</span>
-<span class="sd">    &gt;&gt;&gt; sqrtdenest(sqrt((w**2).expand()))</span>
-<span class="sd">    sqrt((sqrt(sqrt(sqrt(x + 3) + 1) + 1) + 1 + sqrt(2))**2)</span>
-
-<span class="sd">    Notice that the argument of the sqrt is a square. If x is made positive</span>
-<span class="sd">    then the sqrt of the square is resolved:</span>
-
-<span class="sd">    &gt;&gt;&gt; _.subs(x, Symbol(&#39;x&#39;, positive=True))</span>
-<span class="sd">    sqrt(sqrt(sqrt(x + 3) + 1) + 1) + 1 + sqrt(2)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">r</span><span class="p">)))</span>
-    <span class="n">rval</span> <span class="o">=</span> <span class="n">_sqrt_match</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">rval</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-    <span class="n">ra</span><span class="p">,</span> <span class="n">rb</span><span class="p">,</span> <span class="n">rr</span> <span class="o">=</span> <span class="n">rval</span>
-    <span class="k">if</span> <span class="n">rb</span><span class="p">:</span>
-        <span class="n">y</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">newa</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">rr</span><span class="p">),</span> <span class="p">(</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">ra</span><span class="p">)</span><span class="o">/</span><span class="n">rb</span><span class="p">),</span> <span class="n">y</span><span class="p">)</span>
-        <span class="k">except</span> <span class="n">PolynomialError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">newa</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">ca</span><span class="p">,</span> <span class="n">cb</span><span class="p">,</span> <span class="n">cc</span> <span class="o">=</span> <span class="n">newa</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()</span>
-            <span class="n">cb</span> <span class="o">+=</span> <span class="n">b</span>
-            <span class="k">if</span> <span class="n">_mexpand</span><span class="p">(</span><span class="n">cb</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">4</span><span class="o">*</span><span class="n">ca</span><span class="o">*</span><span class="n">cc</span><span class="p">)</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="mi">0</span><span class="p">):</span>
-                <span class="n">z</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">ca</span><span class="o">*</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">r</span><span class="p">)</span> <span class="o">+</span> <span class="n">cb</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">ca</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">z</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-                    <span class="n">z</span> <span class="o">=</span> <span class="n">_mexpand</span><span class="p">(</span><span class="n">Mul</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">z</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()))</span>
-                <span class="k">return</span> <span class="n">z</span>
-
-
-<span class="k">def</span> <span class="nf">_sqrt_numeric_denest</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">d2</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Helper that denest expr = a + b*sqrt(r), with d2 = a**2 - b**2*r &gt; 0</span>
-<span class="sd">    or returns None if not denested.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.simplify.simplify</span> <span class="kn">import</span> <span class="n">radsimp</span>
-    <span class="n">depthr</span> <span class="o">=</span> <span class="n">sqrt_depth</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-    <span class="n">d</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">d2</span><span class="p">)</span>
-    <span class="n">vad</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="n">d</span>
-    <span class="c"># sqrt_depth(res) &lt;= sqrt_depth(vad) + 1</span>
-    <span class="c"># sqrt_depth(expr) = depthr + 2</span>
-    <span class="c"># there is denesting if sqrt_depth(vad)+1 &lt; depthr + 2</span>
-    <span class="c"># if vad**2 is Number there is a fourth root</span>
-    <span class="k">if</span> <span class="n">sqrt_depth</span><span class="p">(</span><span class="n">vad</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">depthr</span> <span class="o">+</span> <span class="mi">1</span> <span class="ow">or</span> <span class="p">(</span><span class="n">vad</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-        <span class="n">vad1</span> <span class="o">=</span> <span class="n">radsimp</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">vad</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">vad</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">sign</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">sqrt</span><span class="p">((</span><span class="n">b</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">r</span><span class="o">*</span><span class="n">vad1</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()))</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-
-
-<span class="k">def</span> <span class="nf">sqrt_biquadratic_denest</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">d2</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;denest expr = sqrt(a + b*sqrt(r))</span>
-<span class="sd">    where a, b, r are linear combinations of square roots of</span>
-<span class="sd">    positive rationals on the rationals (SQRR) and r &gt; 0, b != 0,</span>
-<span class="sd">    d2 = a**2 - b**2*r &gt; 0</span>
-
-<span class="sd">    If it cannot denest it returns None.</span>
-
-<span class="sd">    ALGORITHM</span>
-<span class="sd">    Search for a solution A of type SQRR of the biquadratic equation</span>
-<span class="sd">    4*A**4 - 4*a*A**2 + b**2*r = 0                               (1)</span>
-<span class="sd">    sqd = sqrt(a**2 - b**2*r)</span>
-<span class="sd">    Choosing the sqrt to be positive, the possible solutions are</span>
-<span class="sd">    A = sqrt(a/2 +/- sqd/2)</span>
-<span class="sd">    Since a, b, r are SQRR, then a**2 - b**2*r is a SQRR,</span>
-<span class="sd">    so if sqd can be denested, it is done by</span>
-<span class="sd">    _sqrtdenest_rec, and the result is a SQRR.</span>
-<span class="sd">    Similarly for A.</span>
-<span class="sd">    Examples of solutions (in both cases a and sqd are positive):</span>
-
-<span class="sd">      Example of expr with solution sqrt(a/2 + sqd/2) but not</span>
-<span class="sd">      solution sqrt(a/2 - sqd/2):</span>
-<span class="sd">      expr = sqrt(-sqrt(15) - sqrt(2)*sqrt(-sqrt(5) + 5) - sqrt(3) + 8)</span>
-<span class="sd">      a = -sqrt(15) - sqrt(3) + 8; sqd = -2*sqrt(5) - 2 + 4*sqrt(3)</span>
-
-<span class="sd">      Example of expr with solution sqrt(a/2 - sqd/2) but not</span>
-<span class="sd">      solution sqrt(a/2 + sqd/2):</span>
-<span class="sd">      w = 2 + r2 + r3 + (1 + r3)*sqrt(2 + r2 + 5*r3)</span>
-<span class="sd">      expr = sqrt((w**2).expand())</span>
-<span class="sd">      a = 4*sqrt(6) + 8*sqrt(2) + 47 + 28*sqrt(3)</span>
-<span class="sd">      sqd = 29 + 20*sqrt(3)</span>
-
-<span class="sd">    Define B = b/2*A; eq.(1) implies a = A**2 + B**2*r; then</span>
-<span class="sd">    expr**2 = a + b*sqrt(r) = (A + B*sqrt(r))**2</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import sqrt</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.simplify.sqrtdenest import _sqrt_match, sqrt_biquadratic_denest</span>
-<span class="sd">    &gt;&gt;&gt; z = sqrt((2*sqrt(2) + 4)*sqrt(2 + sqrt(2)) + 5*sqrt(2) + 8)</span>
-<span class="sd">    &gt;&gt;&gt; a, b, r = _sqrt_match(z**2)</span>
-<span class="sd">    &gt;&gt;&gt; d2 = a**2 - b**2*r</span>
-<span class="sd">    &gt;&gt;&gt; sqrt_biquadratic_denest(z, a, b, r, d2)</span>
-<span class="sd">    sqrt(2) + sqrt(sqrt(2) + 2) + 2</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.simplify.simplify</span> <span class="kn">import</span> <span class="n">radsimp</span><span class="p">,</span> <span class="n">rad_rationalize</span>
-    <span class="k">if</span> <span class="n">r</span> <span class="o">&lt;=</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">d2</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">b</span> <span class="ow">or</span> <span class="n">sqrt_depth</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-    <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">r</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">y</span> <span class="ow">in</span> <span class="n">x</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">y2</span> <span class="o">=</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">y2</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">y2</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-    <span class="n">sqd</span> <span class="o">=</span> <span class="n">_mexpand</span><span class="p">(</span><span class="n">sqrtdenest</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">radsimp</span><span class="p">(</span><span class="n">d2</span><span class="p">))))</span>
-    <span class="k">if</span> <span class="n">sqrt_depth</span><span class="p">(</span><span class="n">sqd</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-    <span class="n">x1</span><span class="p">,</span> <span class="n">x2</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="n">sqd</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">a</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">sqd</span><span class="o">/</span><span class="mi">2</span><span class="p">]</span>
-    <span class="c"># look for a solution A with depth 1</span>
-    <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="p">(</span><span class="n">x1</span><span class="p">,</span> <span class="n">x2</span><span class="p">):</span>
-        <span class="n">A</span> <span class="o">=</span> <span class="n">sqrtdenest</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">sqrt_depth</span><span class="p">(</span><span class="n">A</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">continue</span>
-        <span class="n">Bn</span><span class="p">,</span> <span class="n">Bd</span> <span class="o">=</span> <span class="n">rad_rationalize</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">_mexpand</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">A</span><span class="p">))</span>
-        <span class="n">B</span> <span class="o">=</span> <span class="n">Bn</span><span class="o">/</span><span class="n">Bd</span>
-        <span class="n">z</span> <span class="o">=</span> <span class="n">A</span> <span class="o">+</span> <span class="n">B</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">z</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">z</span> <span class="o">=</span> <span class="o">-</span><span class="n">z</span>
-        <span class="k">return</span> <span class="n">_mexpand</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-    <span class="k">return</span> <span class="bp">None</span>
-
-
-<span class="k">def</span> <span class="nf">_denester</span><span class="p">(</span><span class="n">nested</span><span class="p">,</span> <span class="n">av0</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">max_depth_level</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Denests a list of expressions that contain nested square roots.</span>
-
-<span class="sd">    Algorithm based on &lt;http://www.almaden.ibm.com/cs/people/fagin/symb85.pdf&gt;.</span>
-
-<span class="sd">    It is assumed that all of the elements of &#39;nested&#39; share the same</span>
-<span class="sd">    bottom-level radicand. (This is stated in the paper, on page 177, in</span>
-<span class="sd">    the paragraph immediately preceding the algorithm.)</span>
-
-<span class="sd">    When evaluating all of the arguments in parallel, the bottom-level</span>
-<span class="sd">    radicand only needs to be denested once. This means that calling</span>
-<span class="sd">    _denester with x arguments results in a recursive invocation with x+1</span>
-<span class="sd">    arguments; hence _denester has polynomial complexity.</span>
-
-<span class="sd">    However, if the arguments were evaluated separately, each call would</span>
-<span class="sd">    result in two recursive invocations, and the algorithm would have</span>
-<span class="sd">    exponential complexity.</span>
-
-<span class="sd">    This is discussed in the paper in the middle paragraph of page 179.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.simplify.simplify</span> <span class="kn">import</span> <span class="n">radsimp</span>
-    <span class="k">if</span> <span class="n">h</span> <span class="o">&gt;</span> <span class="n">max_depth_level</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-    <span class="k">if</span> <span class="n">av0</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-    <span class="k">if</span> <span class="p">(</span><span class="n">av0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">and</span>
-            <span class="nb">all</span><span class="p">(</span><span class="n">n</span><span class="o">.</span><span class="n">is_Number</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">nested</span><span class="p">)):</span>  <span class="c"># no arguments are nested</span>
-        <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">_subsets</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nested</span><span class="p">)):</span>  <span class="c"># test subset &#39;f&#39; of nested</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">_mexpand</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">nested</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">))</span> <span class="k">if</span> <span class="n">f</span><span class="p">[</span><span class="n">i</span><span class="p">]]))</span>
-            <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">f</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="o">-</span><span class="n">p</span>
-            <span class="n">sqp</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">sqp</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">sqp</span><span class="p">,</span> <span class="n">f</span>  <span class="c"># got a perfect square so return its square root.</span>
-        <span class="c"># Otherwise, return the radicand from the previous invocation.</span>
-        <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">nested</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]),</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="n">nested</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">R</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="n">av0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">values</span> <span class="o">=</span> <span class="p">[</span><span class="n">av0</span><span class="p">[:</span><span class="mi">2</span><span class="p">]]</span>
-            <span class="n">R</span> <span class="o">=</span> <span class="n">av0</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-            <span class="n">nested2</span> <span class="o">=</span> <span class="p">[</span><span class="n">av0</span><span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="n">R</span><span class="p">]</span>
-            <span class="n">av0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">values</span> <span class="o">=</span> <span class="p">[</span><span class="n">_f</span> <span class="k">for</span> <span class="n">_f</span> <span class="ow">in</span> <span class="p">[</span><span class="n">_sqrt_match</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span> <span class="k">for</span> <span class="n">expr</span> <span class="ow">in</span> <span class="n">nested</span><span class="p">]</span> <span class="k">if</span> <span class="n">_f</span><span class="p">]</span>
-            <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">values</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">v</span><span class="p">[</span><span class="mi">2</span><span class="p">]:</span>  <span class="c"># Since if b=0, r is not defined</span>
-                    <span class="k">if</span> <span class="n">R</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">R</span> <span class="o">!=</span> <span class="n">v</span><span class="p">[</span><span class="mi">2</span><span class="p">]:</span>
-                            <span class="n">av0</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="bp">None</span>
-                            <span class="k">return</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">R</span> <span class="o">=</span> <span class="n">v</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">R</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="c"># return the radicand from the previous invocation</span>
-                <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">nested</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]),</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="n">nested</span><span class="p">)</span>
-            <span class="n">nested2</span> <span class="o">=</span> <span class="p">[</span><span class="n">_mexpand</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span>
-                       <span class="n">_mexpand</span><span class="p">(</span><span class="n">R</span><span class="o">*</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">values</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">R</span><span class="p">]</span>
-        <span class="n">d</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">_denester</span><span class="p">(</span><span class="n">nested2</span><span class="p">,</span> <span class="n">av0</span><span class="p">,</span> <span class="n">h</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">max_depth_level</span><span class="p">)</span>
-        <span class="c">#d = sqrtdenest(d)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">any</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nested</span><span class="p">))):</span>
-            <span class="n">v</span> <span class="o">=</span> <span class="n">values</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-            <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">_mexpand</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">d</span><span class="p">)),</span> <span class="n">f</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">nested</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nested</span><span class="p">))</span> <span class="k">if</span> <span class="n">f</span><span class="p">[</span><span class="n">i</span><span class="p">]])</span>
-            <span class="n">v</span> <span class="o">=</span> <span class="n">_sqrt_match</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-            <span class="k">if</span> <span class="mi">1</span> <span class="ow">in</span> <span class="n">f</span> <span class="ow">and</span> <span class="n">f</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">nested</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">f</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="n">nested</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]:</span>
-                <span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="n">nested</span><span class="p">)]:</span>  <span class="c"># Solution denests with square roots</span>
-                <span class="n">vad</span> <span class="o">=</span> <span class="n">_mexpand</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">d</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">vad</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="c"># return the radicand from the previous invocation.</span>
-                    <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">nested</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]),</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="n">nested</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span><span class="p">(</span><span class="n">sqrt_depth</span><span class="p">(</span><span class="n">vad</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="n">sqrt_depth</span><span class="p">(</span><span class="n">R</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span> <span class="ow">or</span>
-                       <span class="p">(</span><span class="n">vad</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">is_Number</span><span class="p">):</span>
-                    <span class="n">av0</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="bp">None</span>
-                    <span class="k">return</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-
-                <span class="n">sqvad</span> <span class="o">=</span> <span class="n">_sqrtdenest1</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">vad</span><span class="p">),</span> <span class="n">denester</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">sqrt_depth</span><span class="p">(</span><span class="n">sqvad</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="n">sqrt_depth</span><span class="p">(</span><span class="n">R</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                    <span class="n">av0</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="bp">None</span>
-                    <span class="k">return</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span>
-                <span class="n">sqvad1</span> <span class="o">=</span> <span class="n">radsimp</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">sqvad</span><span class="p">)</span>
-                <span class="n">res</span> <span class="o">=</span> <span class="n">_mexpand</span><span class="p">(</span><span class="n">sqvad</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">R</span><span class="p">)</span><span class="o">*</span><span class="n">sqvad1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)))</span>
-                <span class="k">return</span> <span class="n">res</span><span class="p">,</span> <span class="n">f</span>
-
-                      <span class="c">#          sign(v[1])*sqrt(_mexpand(v[1]**2*R*vad1/2))), f</span>
-            <span class="k">else</span><span class="p">:</span>  <span class="c"># Solution requires a fourth root</span>
-                <span class="n">s2</span> <span class="o">=</span> <span class="n">_mexpand</span><span class="p">(</span><span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">R</span><span class="p">)</span> <span class="o">+</span> <span class="n">d</span>
-                <span class="k">if</span> <span class="n">s2</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">nested</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]),</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="nb">len</span><span class="p">(</span><span class="n">nested</span><span class="p">)</span>
-                <span class="n">FR</span><span class="p">,</span> <span class="n">s</span> <span class="o">=</span> <span class="n">root</span><span class="p">(</span><span class="n">_mexpand</span><span class="p">(</span><span class="n">R</span><span class="p">),</span> <span class="mi">4</span><span class="p">),</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">s2</span><span class="p">)</span>
-                <span class="k">return</span> <span class="n">_mexpand</span><span class="p">(</span><span class="n">s</span><span class="o">/</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">FR</span><span class="p">)</span> <span class="o">+</span> <span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">FR</span><span class="o">/</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">s</span><span class="p">)),</span> <span class="n">f</span>
-</pre></div>
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diff --git a/dev-py3k/_modules/sympy/simplify/traversaltools.html b/dev-py3k/_modules/sympy/simplify/traversaltools.html
deleted file mode 100644
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-            
-  <h1>Source code for sympy.simplify.traversaltools</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Tools for applying functions to specified parts of expressions. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">sympify</span>
-
-
-<div class="viewcode-block" id="use"><a class="viewcode-back" href="../../../modules/simplify/simplify.html#sympy.simplify.traversaltools.use">[docs]</a><span class="k">def</span> <span class="nf">use</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">level</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">args</span><span class="o">=</span><span class="p">(),</span> <span class="n">kwargs</span><span class="o">=</span><span class="p">{}):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Use ``func`` to transform ``expr`` at the given level.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import use, expand</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">    &gt;&gt;&gt; f = (x + y)**2*x + 1</span>
-
-<span class="sd">    &gt;&gt;&gt; use(f, expand, level=2)</span>
-<span class="sd">    x*(x**2 + 2*x*y + y**2) + 1</span>
-<span class="sd">    &gt;&gt;&gt; expand(f)</span>
-<span class="sd">    x**3 + 2*x**2*y + x*y**2 + 1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">_use</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">level</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">level</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">func</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">expr</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">level</span> <span class="o">-=</span> <span class="mi">1</span>
-                <span class="n">_args</span> <span class="o">=</span> <span class="p">[]</span>
-
-                <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                    <span class="n">_args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">_use</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">level</span><span class="p">))</span>
-
-                <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="o">*</span><span class="n">_args</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">_use</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">),</span> <span class="n">level</span><span class="p">)</span></div>
-</pre></div>
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diff --git a/dev-py3k/_modules/sympy/solvers/ode.html b/dev-py3k/_modules/sympy/solvers/ode.html
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-  <h1>Source code for sympy.solvers.ode</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">This module contains dsolve() and different helper functions that it</span>
-<span class="sd">uses.</span>
-
-<span class="sd">dsolve() solves ordinary differential equations. See the docstring on</span>
-<span class="sd">the various functions for their uses. Note that partial differential</span>
-<span class="sd">equations support is in pde.py.  Note that ode_hint() functions have</span>
-<span class="sd">docstrings describing their various methods, but they are intended for</span>
-<span class="sd">internal use.  Use dsolve(ode, func, hint=hint) to solve an ode using a</span>
-<span class="sd">specific hint.  See also the docstring on dsolve().</span>
-
-<span class="sd">**Functions in this module**</span>
-
-<span class="sd">    These are the user functions in this module:</span>
-
-<span class="sd">    - dsolve() - Solves ODEs.</span>
-<span class="sd">    - classify_ode() - Classifies ODEs into possible hints for dsolve().</span>
-<span class="sd">    - checkodesol() - Checks if an equation is the solution to an ODE.</span>
-<span class="sd">    - ode_order() - Returns the order (degree) of an ODE.</span>
-<span class="sd">    - homogeneous_order() - Returns the homogeneous order of an</span>
-<span class="sd">      expression.</span>
-
-<span class="sd">    These are the non-solver helper functions that are for internal use.</span>
-<span class="sd">    The user should use the various options to dsolve() to obtain the</span>
-<span class="sd">    functionality provided by these functions:</span>
-
-<span class="sd">    - odesimp() - Does all forms of ODE simplification.</span>
-<span class="sd">    - ode_sol_simplicity() - A key function for comparing solutions by</span>
-<span class="sd">      simplicity.</span>
-<span class="sd">    - constantsimp() - Simplifies arbitrary constants.</span>
-<span class="sd">    - constant_renumber() - Renumber arbitrary constants</span>
-<span class="sd">    - _handle_Integral() - Evaluate unevaluated Integrals.</span>
-<span class="sd">    - preprocess - prepare the equation and detect function to solve for</span>
-
-<span class="sd">    See also the docstrings of these functions.</span>
-
-<span class="sd">**Currently implemented solver methods**</span>
-
-<span class="sd">The following methods are implemented for solving ordinary differential</span>
-<span class="sd">equations.  See the docstrings of the various ode_hint() functions for</span>
-<span class="sd">more information on each (run help(ode)):</span>
-
-<span class="sd">  - 1st order separable differential equations</span>
-<span class="sd">  - 1st order differential equations whose coefficients or dx and dy</span>
-<span class="sd">    are functions homogeneous of the same order.</span>
-<span class="sd">  - 1st order exact differential equations.</span>
-<span class="sd">  - 1st order linear differential equations</span>
-<span class="sd">  - 1st order Bernoulli differential equations.</span>
-<span class="sd">  - 2nd order Liouville differential equations.</span>
-<span class="sd">  - nth order linear homogeneous differential equation with constant</span>
-<span class="sd">    coefficients.</span>
-<span class="sd">  - nth order linear inhomogeneous differential equation with constant</span>
-<span class="sd">    coefficients using the method of undetermined coefficients.</span>
-<span class="sd">  - nth order linear inhomogeneous differential equation with constant</span>
-<span class="sd">    coefficients using the method of variation of parameters.</span>
-
-<span class="sd">**Philosophy behind this module**</span>
-
-<span class="sd">This module is designed to make it easy to add new ODE solving methods</span>
-<span class="sd">without having to mess with the solving code for other methods.  The</span>
-<span class="sd">idea is that there is a classify_ode() function, which takes in an ODE</span>
-<span class="sd">and tells you what hints, if any, will solve the ODE.  It does this</span>
-<span class="sd">without attempting to solve the ODE, so it is fast.  Each solving method</span>
-<span class="sd">is a hint, and it has its own function, named ode_hint.  That function</span>
-<span class="sd">takes in the ODE and any match expression gathered by classify_ode and</span>
-<span class="sd">returns a solved result.  If this result has any integrals in it, the</span>
-<span class="sd">ode_hint function will return an unevaluated Integral class. dsolve(),</span>
-<span class="sd">which is the user wrapper function around all of this, will then call</span>
-<span class="sd">odesimp() on the result, which, among other things, will attempt to</span>
-<span class="sd">solve the equation for the dependent variable (the function we are</span>
-<span class="sd">solving for), simplify the arbitrary constants in the expression, and</span>
-<span class="sd">evaluate any integrals, if the hint allows it.</span>
-
-<span class="sd">**How to add new solution methods**</span>
-
-<span class="sd">If you have an ODE that you want dsolve() to be able to solve, try to</span>
-<span class="sd">avoid adding special case code here.  Instead, try finding a general</span>
-<span class="sd">method that will solve your ODE, as well as others.  This way, the ode</span>
-<span class="sd">module will become more robust, and unhindered by special case hacks.</span>
-<span class="sd">WolphramAlpha and Maple&#39;s DETools[odeadvisor] function are two resources</span>
-<span class="sd">you can use to classify a specific ODE.  It is also better for a method</span>
-<span class="sd">to work with an nth order ODE instead of only with specific orders, if</span>
-<span class="sd">possible.</span>
-
-<span class="sd">To add a new method, there are a few things that you need to do.  First,</span>
-<span class="sd">you need a hint name for your method.  Try to name your hint so that it</span>
-<span class="sd">is unambiguous with all other methods, including ones that may not be</span>
-<span class="sd">implemented yet.  If your method uses integrals, also include a</span>
-<span class="sd">&quot;hint_Integral&quot; hint.  If there is more than one way to solve ODEs with</span>
-<span class="sd">your method, include a hint for each one, as well as a &quot;hint_best&quot; hint.</span>
-<span class="sd">Your ode_hint_best() function should choose the best using min with</span>
-<span class="sd">ode_sol_simplicity as the key argument.  See</span>
-<span class="sd">ode_1st_homogeneous_coeff_best(), for example. The function that uses</span>
-<span class="sd">your method will be called ode_hint(), so the hint must only use</span>
-<span class="sd">characters that are allowed in a Python function name (alphanumeric</span>
-<span class="sd">characters and the underscore &#39;_&#39; character).  Include a function for</span>
-<span class="sd">every hint, except for &quot;_Integral&quot; hints (dsolve() takes care of those</span>
-<span class="sd">automatically).  Hint names should be all lowercase, unless a word is</span>
-<span class="sd">commonly capitalized (such as Integral or Bernoulli). If you have a hint</span>
-<span class="sd">that you do not want to run with &quot;all_Integral&quot; that doesn&#39;t have an</span>
-<span class="sd">&quot;_Integral&quot; counterpart (such as a best hint that would defeat the</span>
-<span class="sd">purpose of &quot;all_Integral&quot;), you will need to remove it manually in the</span>
-<span class="sd">dsolve() code.  See also the classify_ode() docstring for guidelines on</span>
-<span class="sd">writing a hint name.</span>
-
-<span class="sd">Determine *in general* how the solutions returned by your method</span>
-<span class="sd">compare with other methods that can potentially solve the same ODEs.</span>
-<span class="sd">Then, put your hints in the allhints tuple in the order that they should</span>
-<span class="sd">be called.  The ordering of this tuple determines which hints are</span>
-<span class="sd">default. Note that exceptions are ok, because it is easy for the user to</span>
-<span class="sd">choose individual hints with dsolve().  In general, &quot;_Integral&quot; variants</span>
-<span class="sd">should go at the end of the list, and &quot;_best&quot; variants should go before</span>
-<span class="sd">the various hints they apply to.  For example, the</span>
-<span class="sd">&quot;undetermined_coefficients&quot; hint comes before the</span>
-<span class="sd">&quot;variation_of_parameters&quot; hint because, even though variation of</span>
-<span class="sd">parameters is more general than undetermined coefficients, undetermined</span>
-<span class="sd">coefficients generally returns cleaner results for the ODEs that it can</span>
-<span class="sd">solve than variation of parameters does, and it does not require</span>
-<span class="sd">integration, so it is much faster.</span>
-
-<span class="sd">Next, you need to have a match expression or a function that matches the</span>
-<span class="sd">type of the ODE, which you should put in classify_ode() (if the match</span>
-<span class="sd">function is more than just a few lines, like</span>
-<span class="sd">_undetermined_coefficients_match(), it should go outside of</span>
-<span class="sd">classify_ode()).  It should match the ODE without solving for it as much</span>
-<span class="sd">as possible, so that classify_ode() remains fast and is not hindered by</span>
-<span class="sd">bugs in solving code.  Be sure to consider corner cases. For example, if</span>
-<span class="sd">your solution method involves dividing by something, make sure you</span>
-<span class="sd">exclude the case where that division will be 0.</span>
-
-<span class="sd">In most cases, the matching of the ODE will also give you the various</span>
-<span class="sd">parts that you need to solve it. You should put that in a dictionary</span>
-<span class="sd">(.match() will do this for you), and add that as matching_hints[&#39;hint&#39;]</span>
-<span class="sd">= matchdict in the relevant part of classify_ode.  classify_ode will</span>
-<span class="sd">then send this to dsolve(), which will send it to your function as the</span>
-<span class="sd">match argument. Your function should be named ode_hint(eq, func, order,</span>
-<span class="sd">match). If you need to send more information, put it in the match</span>
-<span class="sd">dictionary.  For example, if you had to substitute in a dummy variable</span>
-<span class="sd">in classify_ode to match the ODE, you will need to pass it to your</span>
-<span class="sd">function using the match dict to access it.  You can access the</span>
-<span class="sd">independent variable using func.args[0], and the dependent variable (the</span>
-<span class="sd">function you are trying to solve for) as func.func.  If, while trying to</span>
-<span class="sd">solve the ODE, you find that you cannot, raise NotImplementedError.</span>
-<span class="sd">dsolve() will catch this error with the &quot;all&quot; meta-hint, rather than</span>
-<span class="sd">causing the whole routine to fail.</span>
-
-<span class="sd">Add a docstring to your function that describes the method employed.</span>
-<span class="sd">Like with anything else in SymPy, you will need to add a doctest to the</span>
-<span class="sd">docstring, in addition to real tests in test_ode.py.  Try to maintain</span>
-<span class="sd">consistency with the other hint functions&#39; docstrings.  Add your method</span>
-<span class="sd">to the list at the top of this docstring.  Also, add your method to</span>
-<span class="sd">ode.rst in the docs/src directory, so that the Sphinx docs will pull its</span>
-<span class="sd">docstring into the main SymPy documentation.  Be sure to make the Sphinx</span>
-<span class="sd">documentation by running &quot;make html&quot; from within the doc directory to</span>
-<span class="sd">verify that the docstring formats correctly.</span>
-
-<span class="sd">If your solution method involves integrating, use C.Integral() instead</span>
-<span class="sd">of integrate().  This allows the user to bypass hard/slow integration by</span>
-<span class="sd">using the &quot;_Integral&quot; variant of your hint.  In most cases, calling</span>
-<span class="sd">.doit() will integrate your solution.  If this is not the case, you will</span>
-<span class="sd">need to write special code in _handle_Integral().  Arbitrary constants</span>
-<span class="sd">should be symbols named C1, C2, and so on.  All solution methods should</span>
-<span class="sd">return an equality instance.  If you need an arbitrary number of</span>
-<span class="sd">arbitrary constants, you can use constants =</span>
-<span class="sd">numbered_symbols(prefix=&#39;C&#39;, cls=Symbol, start=1).  If it is</span>
-<span class="sd">possible to solve for the dependent function in a general way, do so.</span>
-<span class="sd">Otherwise, do as best as you can, but do not call solve in your</span>
-<span class="sd">ode_hint() function.  odesimp() will attempt to solve the solution for</span>
-<span class="sd">you, so you do not need to do that. Lastly, if your ODE has a common</span>
-<span class="sd">simplification that can be applied to your solutions, you can add a</span>
-<span class="sd">special case in odesimp() for it.  For example, solutions returned from</span>
-<span class="sd">the &quot;1st_homogeneous_coeff&quot; hints often have many log() terms, so</span>
-<span class="sd">odesimp() calls logcombine() on them (it also helps to write the</span>
-<span class="sd">arbitrary constant as log(C1) instead of C1 in this case).  Also</span>
-<span class="sd">consider common ways that you can rearrange your solution to have</span>
-<span class="sd">constantsimp() take better advantage of it.  It is better to put</span>
-<span class="sd">simplification in odesimp() than in your method, because it can then be</span>
-<span class="sd">turned off with the simplify flag in dsolve(). If you have any</span>
-<span class="sd">extraneous simplification in your function, be sure to only run it using</span>
-<span class="sd">&quot;if match.get(&#39;simplify&#39;, True):&quot;, especially if it can be slow or if it</span>
-<span class="sd">can reduce the domain of the solution.</span>
-
-<span class="sd">Finally, as with every contribution to SymPy, your method will need to</span>
-<span class="sd">be tested. Add a test for each method in test_ode.py.  Follow the</span>
-<span class="sd">conventions there, i.e., test the solver using dsolve(eq, f(x),</span>
-<span class="sd">hint=your_hint), and also test the solution using checkodesol (you can</span>
-<span class="sd">put these in a separate tests and skip/XFAIL if it runs too slow/doesn&#39;t</span>
-<span class="sd">work).  Be sure to call your hint specifically in dsolve, that way the</span>
-<span class="sd">test won&#39;t be broken simply by the introduction of another matching</span>
-<span class="sd">hint. If your method works for higher order (&gt;1) ODEs, you will need to</span>
-<span class="sd">run sol = constant_renumber(sol, &#39;C&#39;, 1, order), for each solution, where</span>
-<span class="sd">order is the order of the ODE. This is because constant_renumber renumbers</span>
-<span class="sd">the arbitrary constants by printing order, which is platform dependent.</span>
-<span class="sd">Try to test every corner case of your solver, including a range of</span>
-<span class="sd">orders if it is a nth order solver, but if your solver is slow, such as</span>
-<span class="sd">if it involves hard integration, try to keep the test run time down.</span>
-
-<span class="sd">Feel free to refactor existing hints to avoid duplicating code or</span>
-<span class="sd">creating inconsistencies.  If you can show that your method exactly</span>
-<span class="sd">duplicates an existing method, including in the simplicity and speed of</span>
-<span class="sd">obtaining the solutions, then you can remove the old, less general</span>
-<span class="sd">method. The existing code is tested extensively in test_ode.py, so if</span>
-<span class="sd">anything is broken, one of those tests will surely fail.</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Add</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Pow</span><span class="p">,</span> <span class="n">oo</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">iterable</span><span class="p">,</span> <span class="n">is_sequence</span><span class="p">,</span> <span class="n">set_union</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.exceptions</span> <span class="kn">import</span> <span class="n">SymPyDeprecationWarning</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Derivative</span><span class="p">,</span> <span class="n">AppliedUndef</span><span class="p">,</span> <span class="n">diff</span><span class="p">,</span> <span class="n">expand_mul</span>
-<span class="kn">from</span> <span class="nn">sympy.core.multidimensional</span> <span class="kn">import</span> <span class="n">vectorize</span>
-<span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span><span class="p">,</span> <span class="n">Eq</span>
-<span class="kn">from</span> <span class="nn">sympy.core.symbol</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Wild</span><span class="p">,</span> <span class="n">Dummy</span>
-<span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="n">sympify</span>
-
-<span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">cos</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">re</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">tan</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">sign</span>
-<span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">wronskian</span>
-<span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">RootOf</span><span class="p">,</span> <span class="n">terms_gcd</span>
-<span class="kn">from</span> <span class="nn">sympy.series</span> <span class="kn">import</span> <span class="n">Order</span>
-<span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">collect</span><span class="p">,</span> <span class="n">logcombine</span><span class="p">,</span> <span class="n">powsimp</span><span class="p">,</span> <span class="n">separatevars</span><span class="p">,</span> \
-    <span class="n">simplify</span><span class="p">,</span> <span class="n">trigsimp</span><span class="p">,</span> <span class="n">denom</span>
-<span class="kn">from</span> <span class="nn">sympy.solvers</span> <span class="kn">import</span> <span class="n">solve</span>
-
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">numbered_symbols</span><span class="p">,</span> <span class="n">default_sort_key</span><span class="p">,</span> <span class="n">sift</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-<span class="c"># This is a list of hints in the order that they should be applied.  That means</span>
-<span class="c"># that, in general, hints earlier in the list should produce simpler results</span>
-<span class="c"># than those later for ODEs that fit both.  This is just based on my own</span>
-<span class="c"># empirical observations, so if you find that *in general*, a hint later in</span>
-<span class="c"># the list is better than one before it, fell free to modify the list.  Note</span>
-<span class="c"># however that you can easily override the hint used in dsolve() for a specific ODE</span>
-<span class="c"># (see the docstring).  In general, &quot;_Integral&quot; hints should be grouped</span>
-<span class="c"># at the end of the list, unless there is a method that returns an unevaluable</span>
-<span class="c"># integral most of the time (which should surely go near the end of the list</span>
-<span class="c"># anyway).</span>
-<span class="c"># &quot;default&quot;, &quot;all&quot;, &quot;best&quot;, and &quot;all_Integral&quot; meta-hints should not be</span>
-<span class="c"># included in this list, but &quot;_best&quot; and &quot;_Integral&quot; hints should be included.</span>
-<span class="n">allhints</span> <span class="o">=</span> <span class="p">(</span>
-    <span class="s">&quot;separable&quot;</span><span class="p">,</span> <span class="s">&quot;1st_exact&quot;</span><span class="p">,</span> <span class="s">&quot;1st_linear&quot;</span><span class="p">,</span> <span class="s">&quot;Bernoulli&quot;</span><span class="p">,</span> <span class="s">&quot;Riccati_special_minus2&quot;</span><span class="p">,</span>
-    <span class="s">&quot;1st_homogeneous_coeff_best&quot;</span><span class="p">,</span> <span class="s">&quot;1st_homogeneous_coeff_subs_indep_div_dep&quot;</span><span class="p">,</span>
-    <span class="s">&quot;1st_homogeneous_coeff_subs_dep_div_indep&quot;</span><span class="p">,</span> <span class="s">&quot;nth_linear_constant_coeff_homogeneous&quot;</span><span class="p">,</span>
-    <span class="s">&quot;nth_linear_euler_eq_homogeneous&quot;</span><span class="p">,</span>
-    <span class="s">&quot;nth_linear_constant_coeff_undetermined_coefficients&quot;</span><span class="p">,</span>
-    <span class="s">&quot;nth_linear_constant_coeff_variation_of_parameters&quot;</span><span class="p">,</span>
-    <span class="s">&quot;Liouville&quot;</span><span class="p">,</span> <span class="s">&quot;separable_Integral&quot;</span><span class="p">,</span> <span class="s">&quot;1st_exact_Integral&quot;</span><span class="p">,</span> <span class="s">&quot;1st_linear_Integral&quot;</span><span class="p">,</span>
-    <span class="s">&quot;Bernoulli_Integral&quot;</span><span class="p">,</span> <span class="s">&quot;1st_homogeneous_coeff_subs_indep_div_dep_Integral&quot;</span><span class="p">,</span>
-    <span class="s">&quot;1st_homogeneous_coeff_subs_dep_div_indep_Integral&quot;</span><span class="p">,</span>
-    <span class="s">&quot;nth_linear_constant_coeff_variation_of_parameters_Integral&quot;</span><span class="p">,</span>
-    <span class="s">&quot;Liouville_Integral&quot;</span>
-<span class="p">)</span>
-
-
-<div class="viewcode-block" id="preprocess"><a class="viewcode-back" href="../../../modules/solvers/ode.html#sympy.solvers.ode.preprocess">[docs]</a><span class="k">def</span> <span class="nf">preprocess</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">func</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">hint</span><span class="o">=</span><span class="s">&#39;_Integral&#39;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Prepare expr for solving by making sure that differentiation</span>
-<span class="sd">    is done so that only func remains in unevaluated derivatives and</span>
-<span class="sd">    (if hint doesn&#39;t end with _Integral) that doit is applied to all</span>
-<span class="sd">    other derivatives. If hint is None, don&#39;t do any differentiation.</span>
-<span class="sd">    (Currently this may cause some simple differential equations to</span>
-<span class="sd">    fail.)</span>
-
-<span class="sd">    In case func is None, an attempt will be made to autodetect the</span>
-<span class="sd">    function to be solved for.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.solvers.ode import preprocess</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Derivative, Function, Integral, sin</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">    &gt;&gt;&gt; f, g = list(map(Function, &#39;fg&#39;))</span>
-
-<span class="sd">    Apply doit to derivatives that contain more than the function</span>
-<span class="sd">    of interest:</span>
-
-<span class="sd">    &gt;&gt;&gt; preprocess(Derivative(f(x) + x, x))</span>
-<span class="sd">    (Derivative(f(x), x) + 1, f(x))</span>
-
-<span class="sd">    Do others if the differentiation variable(s) intersect with those</span>
-<span class="sd">    of the function of interest or contain the function of interest:</span>
-
-<span class="sd">    &gt;&gt;&gt; preprocess(Derivative(g(x), y, z), f(y))</span>
-<span class="sd">    (0, f(y))</span>
-<span class="sd">    &gt;&gt;&gt; preprocess(Derivative(f(y), z), f(y))</span>
-<span class="sd">    (0, f(y))</span>
-
-<span class="sd">    Do others if the hint doesn&#39;t end in &#39;_Integral&#39; (the default</span>
-<span class="sd">    assumes that it does):</span>
-
-<span class="sd">    &gt;&gt;&gt; preprocess(Derivative(g(x), y), f(x))</span>
-<span class="sd">    (Derivative(g(x), y), f(x))</span>
-<span class="sd">    &gt;&gt;&gt; preprocess(Derivative(f(x), y), f(x), hint=&#39;&#39;)</span>
-<span class="sd">    (0, f(x))</span>
-
-<span class="sd">    Don&#39;t do any derivatives if hint is None:</span>
-
-<span class="sd">    &gt;&gt;&gt; preprocess(Derivative(f(x) + 1, x) + Derivative(f(x), y), f(x), hint=None)</span>
-<span class="sd">    (Derivative(f(x) + 1, x) + Derivative(f(x), y), f(x))</span>
-
-<span class="sd">    If it&#39;s not clear what the function of interest is, it must be given:</span>
-
-<span class="sd">    &gt;&gt;&gt; eq = Derivative(f(x) + g(x), x)</span>
-<span class="sd">    &gt;&gt;&gt; preprocess(eq, g(x))</span>
-<span class="sd">    (Derivative(f(x), x) + Derivative(g(x), x), g(x))</span>
-<span class="sd">    &gt;&gt;&gt; try: preprocess(eq)</span>
-<span class="sd">    ... except ValueError: print(&quot;A ValueError was raised.&quot;)</span>
-<span class="sd">    A ValueError was raised.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">derivs</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Derivative</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">func</span><span class="p">:</span>
-        <span class="n">funcs</span> <span class="o">=</span> <span class="n">set_union</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">d</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">AppliedUndef</span><span class="p">)</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">derivs</span><span class="p">])</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">funcs</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;The function cannot be automatically detected for </span><span class="si">%s</span><span class="s">.&#39;</span> <span class="o">%</span> <span class="n">expr</span><span class="p">)</span>
-        <span class="n">func</span> <span class="o">=</span> <span class="n">funcs</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-    <span class="n">fvars</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">func</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">hint</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">expr</span><span class="p">,</span> <span class="n">func</span>
-    <span class="n">reps</span> <span class="o">=</span> <span class="p">[(</span><span class="n">d</span><span class="p">,</span> <span class="n">d</span><span class="o">.</span><span class="n">doit</span><span class="p">())</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">derivs</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">hint</span><span class="o">.</span><span class="n">endswith</span><span class="p">(</span><span class="s">&#39;_Integral&#39;</span><span class="p">)</span> <span class="ow">or</span>
-            <span class="n">d</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">func</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">set</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span> <span class="o">&amp;</span> <span class="n">fvars</span><span class="p">]</span>
-    <span class="n">eq</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">reps</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">eq</span><span class="p">,</span> <span class="n">func</span>
-
-</div>
-<span class="k">def</span> <span class="nf">sub_func_doit</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">new</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;When replacing the func with something else, we usually</span>
-<span class="sd">    want the derivative evaluated, so this function helps in</span>
-<span class="sd">    making that happen.</span>
-
-<span class="sd">    To keep subs from having to look through all derivatives, we</span>
-<span class="sd">    mask them off with dummy variables, do the func sub, and then</span>
-<span class="sd">    replace masked off derivatives with their doit values.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Derivative, symbols, Function</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.solvers.ode import sub_func_doit</span>
-<span class="sd">    &gt;&gt;&gt; x, z = symbols(&#39;x, z&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; y = Function(&#39;y&#39;)</span>
-
-<span class="sd">    &gt;&gt;&gt; sub_func_doit(3*Derivative(y(x), x) - 1, y(x), x)</span>
-<span class="sd">    2</span>
-
-<span class="sd">    &gt;&gt;&gt; sub_func_doit(x*Derivative(y(x), x) - y(x)**2 + y(x), y(x),</span>
-<span class="sd">    ... 1/(x*(z + 1/x)))</span>
-<span class="sd">    x*(-1/(x**2*(z + 1/x)) + 1/(x**3*(z + 1/x)**2)) + 1/(x*(z + 1/x))</span>
-<span class="sd">    ...- 1/(x**2*(z + 1/x)**2)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">reps</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="n">repu</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">eq</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Derivative</span><span class="p">):</span>
-        <span class="n">u</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;u&#39;</span><span class="p">)</span>
-        <span class="n">repu</span><span class="p">[</span><span class="n">u</span><span class="p">]</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">new</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-        <span class="n">reps</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">=</span> <span class="n">u</span>
-
-    <span class="k">return</span> <span class="n">eq</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">reps</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">new</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">repu</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="dsolve"><a class="viewcode-back" href="../../../modules/solvers/ode.html#sympy.solvers.ode.dsolve">[docs]</a><span class="k">def</span> <span class="nf">dsolve</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">hint</span><span class="o">=</span><span class="s">&quot;default&quot;</span><span class="p">,</span> <span class="n">simplify</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">prep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Solves any (supported) kind of ordinary differential equation.</span>
-
-<span class="sd">    **Usage**</span>
-
-<span class="sd">        dsolve(eq, f(x), hint) -&gt; Solve ordinary differential equation</span>
-<span class="sd">        eq for function f(x), using method hint.</span>
-
-
-<span class="sd">    **Details**</span>
-
-<span class="sd">        ``eq`` can be any supported ordinary differential equation (see</span>
-<span class="sd">            the ode docstring for supported methods).  This can either</span>
-<span class="sd">            be an Equality, or an expression, which is assumed to be</span>
-<span class="sd">            equal to 0.</span>
-
-<span class="sd">        ``f(x)`` is a function of one variable whose derivatives in that</span>
-<span class="sd">            variable make up the ordinary differential equation eq. In many</span>
-<span class="sd">            cases it is not necessary to provide this; it will be autodetected</span>
-<span class="sd">            (and an error raised if it couldn&#39;t be detected).</span>
-
-<span class="sd">        ``hint`` is the solving method that you want dsolve to use.  Use</span>
-<span class="sd">            classify_ode(eq, f(x)) to get all of the possible hints for</span>
-<span class="sd">            an ODE.  The default hint, &#39;default&#39;, will use whatever hint</span>
-<span class="sd">            is returned first by classify_ode().  See Hints below for</span>
-<span class="sd">            more options that you can use for hint.</span>
-
-<span class="sd">        ``simplify`` enables simplification by odesimp().  See its</span>
-<span class="sd">            docstring for more information.  Turn this off, for example,</span>
-<span class="sd">            to disable solving of solutions for func or simplification</span>
-<span class="sd">            of arbitrary constants.  It will still integrate with this</span>
-<span class="sd">            hint. Note that the solution may contain more arbitrary</span>
-<span class="sd">            constants than the order of the ODE with this option</span>
-<span class="sd">            enabled.</span>
-
-<span class="sd">        ``prep``, when False and when ``func`` is given, will skip the</span>
-<span class="sd">            preprocessing step where the equation is cleaned up so it</span>
-<span class="sd">            is ready for solving.</span>
-
-<span class="sd">    **Hints**</span>
-
-<span class="sd">        Aside from the various solving methods, there are also some</span>
-<span class="sd">        meta-hints that you can pass to dsolve():</span>
-
-<span class="sd">        &quot;default&quot;:</span>
-<span class="sd">                This uses whatever hint is returned first by</span>
-<span class="sd">                classify_ode(). This is the default argument to</span>
-<span class="sd">                dsolve().</span>
-
-<span class="sd">        &quot;all&quot;:</span>
-<span class="sd">                To make dsolve apply all relevant classification hints,</span>
-<span class="sd">                use dsolve(ODE, func, hint=&quot;all&quot;).  This will return a</span>
-<span class="sd">                dictionary of hint:solution terms.  If a hint causes</span>
-<span class="sd">                dsolve to raise the NotImplementedError, value of that</span>
-<span class="sd">                hint&#39;s key will be the exception object raised.  The</span>
-<span class="sd">                dictionary will also include some special keys:</span>
-
-<span class="sd">                - order: The order of the ODE.  See also ode_order().</span>
-<span class="sd">                - best: The simplest hint; what would be returned by</span>
-<span class="sd">                  &quot;best&quot; below.</span>
-<span class="sd">                - best_hint: The hint that would produce the solution</span>
-<span class="sd">                  given by &#39;best&#39;.  If more than one hint produces the</span>
-<span class="sd">                  best solution, the first one in the tuple returned by</span>
-<span class="sd">                  classify_ode() is chosen.</span>
-<span class="sd">                - default: The solution that would be returned by</span>
-<span class="sd">                  default.  This is the one produced by the hint that</span>
-<span class="sd">                  appears first in the tuple returned by classify_ode().</span>
-
-<span class="sd">        &quot;all_Integral&quot;:</span>
-<span class="sd">                This is the same as &quot;all&quot;, except if a hint also has a</span>
-<span class="sd">                corresponding &quot;_Integral&quot; hint, it only returns the</span>
-<span class="sd">                &quot;_Integral&quot; hint.  This is useful if &quot;all&quot; causes</span>
-<span class="sd">                dsolve() to hang because of a difficult or impossible</span>
-<span class="sd">                integral.  This meta-hint will also be much faster than</span>
-<span class="sd">                &quot;all&quot;, because integrate() is an expensive routine.</span>
-
-<span class="sd">        &quot;best&quot;:</span>
-<span class="sd">                To have dsolve() try all methods and return the simplest</span>
-<span class="sd">                one.  This takes into account whether the solution is</span>
-<span class="sd">                solvable in the function, whether it contains any</span>
-<span class="sd">                Integral classes (i.e. unevaluatable integrals), and</span>
-<span class="sd">                which one is the shortest in size.</span>
-
-<span class="sd">        See also the classify_ode() docstring for more info on hints,</span>
-<span class="sd">        and the ode docstring for a list of all supported hints.</span>
-
-
-<span class="sd">    **Tips**</span>
-<span class="sd">        - You can declare the derivative of an unknown function this way:</span>
-<span class="sd">            &gt;&gt;&gt; from sympy import Function, Derivative</span>
-<span class="sd">            &gt;&gt;&gt; from sympy.abc import x # x is the independent variable</span>
-<span class="sd">            &gt;&gt;&gt; f = Function(&quot;f&quot;)(x) # f is a function of x</span>
-<span class="sd">            &gt;&gt;&gt; # f_ will be the derivative of f with respect to x</span>
-<span class="sd">            &gt;&gt;&gt; f_ = Derivative(f, x)</span>
-
-<span class="sd">        - See test_ode.py for many tests, which serves also as a set of</span>
-<span class="sd">          examples for how to use dsolve().</span>
-<span class="sd">        - dsolve always returns an Equality class (except for the case</span>
-<span class="sd">          when the hint is &quot;all&quot; or &quot;all_Integral&quot;).  If possible, it</span>
-<span class="sd">          solves the solution explicitly for the function being solved</span>
-<span class="sd">          for. Otherwise, it returns an implicit solution.</span>
-<span class="sd">        - Arbitrary constants are symbols named C1, C2, and so on.</span>
-<span class="sd">        - Because all solutions should be mathematically equivalent,</span>
-<span class="sd">          some hints may return the exact same result for an ODE. Often,</span>
-<span class="sd">          though, two different hints will return the same solution</span>
-<span class="sd">          formatted differently.  The two should be equivalent. Also</span>
-<span class="sd">          note that sometimes the values of the arbitrary constants in</span>
-<span class="sd">          two different solutions may not be the same, because one</span>
-<span class="sd">          constant may have &quot;absorbed&quot; other constants into it.</span>
-<span class="sd">        - Do help(ode.ode_hintname) to get help more information on a</span>
-<span class="sd">          specific hint, where hintname is the name of a hint without</span>
-<span class="sd">          &quot;_Integral&quot;.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, dsolve, Eq, Derivative, sin, cos</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; dsolve(Derivative(f(x),x,x)+9*f(x), f(x))</span>
-<span class="sd">    f(x) == C1*sin(3*x) + C2*cos(3*x)</span>
-<span class="sd">    &gt;&gt;&gt; dsolve(sin(x)*cos(f(x)) + cos(x)*sin(f(x))*f(x).diff(x), f(x),</span>
-<span class="sd">    ...     hint=&#39;separable&#39;, simplify=False)</span>
-<span class="sd">    -log(sin(f(x))**2 - 1)/2 == C1 + log(sin(x)**2 - 1)/2</span>
-<span class="sd">    &gt;&gt;&gt; dsolve(sin(x)*cos(f(x)) + cos(x)*sin(f(x))*f(x).diff(x), f(x),</span>
-<span class="sd">    ...     hint=&#39;1st_exact&#39;)</span>
-<span class="sd">    f(x) == acos(C1/cos(x))</span>
-<span class="sd">    &gt;&gt;&gt; dsolve(sin(x)*cos(f(x)) + cos(x)*sin(f(x))*f(x).diff(x), f(x),</span>
-<span class="sd">    ... hint=&#39;best&#39;)</span>
-<span class="sd">    f(x) == acos(C1/cos(x))</span>
-<span class="sd">    &gt;&gt;&gt; # Note that even though separable is the default, 1st_exact produces</span>
-<span class="sd">    &gt;&gt;&gt; # a simpler result in this case.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># TODO: Implement initial conditions</span>
-    <span class="c"># See issue 1621.  We first need a way to represent things like f&#39;(0).</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">Equality</span><span class="p">):</span>
-        <span class="n">eq</span> <span class="o">=</span> <span class="n">eq</span><span class="o">.</span><span class="n">lhs</span> <span class="o">-</span> <span class="n">eq</span><span class="o">.</span><span class="n">rhs</span>
-
-    <span class="c"># preprocess the equation and find func if not given</span>
-    <span class="k">if</span> <span class="n">prep</span> <span class="ow">or</span> <span class="n">func</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">eq</span><span class="p">,</span> <span class="n">func</span> <span class="o">=</span> <span class="n">preprocess</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">)</span>
-        <span class="n">prep</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="c"># Magic that should only be used internally.  Prevents classify_ode from</span>
-    <span class="c"># being called more than it needs to be by passing its results through</span>
-    <span class="c"># recursive calls.</span>
-    <span class="k">if</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;classify&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-        <span class="n">hints</span> <span class="o">=</span> <span class="n">classify_ode</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="nb">dict</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">prep</span><span class="o">=</span><span class="n">prep</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="c"># Here is what all this means:</span>
-        <span class="c">#</span>
-        <span class="c"># hint:    The hint method given to dsolve() by the user.</span>
-        <span class="c"># hints:   The dictionary of hints that match the ODE, along with</span>
-        <span class="c">#          other information (including the internal pass-through magic).</span>
-        <span class="c"># default: The default hint to return, the first hint from allhints</span>
-        <span class="c">#          that matches the hint.  This is obtained from classify_ode().</span>
-        <span class="c"># match:   The hints dictionary contains a match dictionary for each hint</span>
-        <span class="c">#          (the parts of the ODE for solving).  When going through the</span>
-        <span class="c">#          hints in &quot;all&quot;, this holds the match string for the current</span>
-        <span class="c">#          hint.</span>
-        <span class="c"># order:   The order of the ODE, as determined by ode_order().</span>
-        <span class="n">hints</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;hint&#39;</span><span class="p">,</span>
-                           <span class="p">{</span><span class="s">&#39;default&#39;</span><span class="p">:</span> <span class="n">hint</span><span class="p">,</span>
-                            <span class="n">hint</span><span class="p">:</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;match&#39;</span><span class="p">],</span>
-                            <span class="s">&#39;order&#39;</span><span class="p">:</span> <span class="n">kwargs</span><span class="p">[</span><span class="s">&#39;order&#39;</span><span class="p">]})</span>
-
-    <span class="k">if</span> <span class="n">hints</span><span class="p">[</span><span class="s">&#39;order&#39;</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-            <span class="nb">str</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot; is not a differential equation in &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">func</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">hints</span><span class="p">[</span><span class="s">&#39;default&#39;</span><span class="p">]:</span>
-        <span class="c"># classify_ode will set hints[&#39;default&#39;] to None if no hints match.</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;dsolve: Cannot solve &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">eq</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="n">hint</span> <span class="o">==</span> <span class="s">&#39;default&#39;</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">dsolve</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">hint</span><span class="o">=</span><span class="n">hints</span><span class="p">[</span><span class="s">&#39;default&#39;</span><span class="p">],</span> <span class="n">simplify</span><span class="o">=</span><span class="n">simplify</span><span class="p">,</span>
-                      <span class="n">prep</span><span class="o">=</span><span class="n">prep</span><span class="p">,</span> <span class="n">classify</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">hints</span><span class="p">[</span><span class="s">&#39;order&#39;</span><span class="p">],</span>
-                      <span class="n">match</span><span class="o">=</span><span class="n">hints</span><span class="p">[</span><span class="n">hints</span><span class="p">[</span><span class="s">&#39;default&#39;</span><span class="p">]])</span>
-    <span class="k">elif</span> <span class="n">hint</span> <span class="ow">in</span> <span class="p">(</span><span class="s">&#39;all&#39;</span><span class="p">,</span> <span class="s">&#39;all_Integral&#39;</span><span class="p">,</span> <span class="s">&#39;best&#39;</span><span class="p">):</span>
-        <span class="n">retdict</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="n">failedhints</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="n">gethints</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">hints</span><span class="p">)</span> <span class="o">-</span> <span class="nb">set</span><span class="p">([</span><span class="s">&#39;order&#39;</span><span class="p">,</span> <span class="s">&#39;default&#39;</span><span class="p">,</span> <span class="s">&#39;ordered_hints&#39;</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">hint</span> <span class="o">==</span> <span class="s">&#39;all_Integral&#39;</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">hints</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">endswith</span><span class="p">(</span><span class="s">&#39;_Integral&#39;</span><span class="p">):</span>
-                    <span class="n">gethints</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">i</span><span class="p">[:</span><span class="o">-</span><span class="nb">len</span><span class="p">(</span><span class="s">&#39;_Integral&#39;</span><span class="p">)])</span>
-            <span class="c"># special case</span>
-            <span class="k">if</span> <span class="s">&quot;1st_homogeneous_coeff_best&quot;</span> <span class="ow">in</span> <span class="n">gethints</span><span class="p">:</span>
-                <span class="n">gethints</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="s">&quot;1st_homogeneous_coeff_best&quot;</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">gethints</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">sol</span> <span class="o">=</span> <span class="n">dsolve</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">hint</span><span class="o">=</span><span class="n">i</span><span class="p">,</span> <span class="n">simplify</span><span class="o">=</span><span class="n">simplify</span><span class="p">,</span> <span class="n">prep</span><span class="o">=</span><span class="n">prep</span><span class="p">,</span>
-                   <span class="n">classify</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">hints</span><span class="p">[</span><span class="s">&#39;order&#39;</span><span class="p">],</span> <span class="n">match</span><span class="o">=</span><span class="n">hints</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-            <span class="k">except</span> <span class="ne">NotImplementedError</span> <span class="k">as</span> <span class="n">detail</span><span class="p">:</span>  <span class="c"># except NotImplementedError as detail:</span>
-                <span class="n">failedhints</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">detail</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">retdict</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">sol</span>
-        <span class="n">retdict</span><span class="p">[</span><span class="s">&#39;best&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">retdict</span><span class="o">.</span><span class="n">values</span><span class="p">()),</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span>
-            <span class="n">ode_sol_simplicity</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">trysolving</span><span class="o">=</span><span class="ow">not</span> <span class="n">simplify</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">hint</span> <span class="o">==</span> <span class="s">&#39;best&#39;</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">retdict</span><span class="p">[</span><span class="s">&#39;best&#39;</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">hints</span><span class="p">[</span><span class="s">&#39;ordered_hints&#39;</span><span class="p">]:</span>
-            <span class="k">if</span> <span class="n">retdict</span><span class="p">[</span><span class="s">&#39;best&#39;</span><span class="p">]</span> <span class="o">==</span> <span class="n">retdict</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="bp">None</span><span class="p">):</span>
-                <span class="n">retdict</span><span class="p">[</span><span class="s">&#39;best_hint&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">i</span>
-                <span class="k">break</span>
-        <span class="n">retdict</span><span class="p">[</span><span class="s">&#39;default&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">hints</span><span class="p">[</span><span class="s">&#39;default&#39;</span><span class="p">]</span>
-        <span class="n">retdict</span><span class="p">[</span><span class="s">&#39;order&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">hints</span><span class="p">[</span><span class="s">&#39;order&#39;</span><span class="p">])</span>
-        <span class="n">retdict</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">failedhints</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">retdict</span>
-    <span class="k">elif</span> <span class="n">hint</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">allhints</span><span class="p">:</span>  <span class="c"># and hint not in (&#39;default&#39;, &#39;ordered_hints&#39;):</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Hint not recognized: &quot;</span> <span class="o">+</span> <span class="n">hint</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">hint</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">hints</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;ODE &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot; does not match hint &quot;</span> <span class="o">+</span> <span class="n">hint</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">hint</span><span class="o">.</span><span class="n">endswith</span><span class="p">(</span><span class="s">&#39;_Integral&#39;</span><span class="p">):</span>
-        <span class="n">solvefunc</span> <span class="o">=</span> <span class="nb">globals</span><span class="p">()[</span><span class="s">&#39;ode_&#39;</span> <span class="o">+</span> <span class="n">hint</span><span class="p">[:</span><span class="o">-</span><span class="nb">len</span><span class="p">(</span><span class="s">&#39;_Integral&#39;</span><span class="p">)]]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="c"># convert the string into a function</span>
-        <span class="n">solvefunc</span> <span class="o">=</span> <span class="nb">globals</span><span class="p">()[</span><span class="s">&#39;ode_&#39;</span> <span class="o">+</span> <span class="n">hint</span><span class="p">]</span>
-    <span class="c"># odesimp() will attempt to integrate, if necessary, apply constantsimp(),</span>
-    <span class="c"># attempt to solve for func, and apply any other hint specific simplifications</span>
-    <span class="k">if</span> <span class="n">simplify</span><span class="p">:</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">odesimp</span><span class="p">(</span><span class="n">solvefunc</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">hints</span><span class="p">[</span><span class="s">&#39;order&#39;</span><span class="p">],</span>
-            <span class="n">match</span><span class="o">=</span><span class="n">hints</span><span class="p">[</span><span class="n">hint</span><span class="p">]),</span> <span class="n">func</span><span class="p">,</span> <span class="n">hints</span><span class="p">[</span><span class="s">&#39;order&#39;</span><span class="p">],</span> <span class="n">hint</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="c"># We still want to integrate (you can disable it separately with the hint)</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">hints</span><span class="p">[</span><span class="n">hint</span><span class="p">]</span>
-        <span class="n">r</span><span class="p">[</span><span class="s">&#39;simplify&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>  <span class="c"># Some hints can take advantage of this option</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">_handle_Integral</span><span class="p">(</span><span class="n">solvefunc</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">hints</span><span class="p">[</span><span class="s">&#39;order&#39;</span><span class="p">],</span>
-            <span class="n">match</span><span class="o">=</span><span class="n">hints</span><span class="p">[</span><span class="n">hint</span><span class="p">]),</span> <span class="n">func</span><span class="p">,</span> <span class="n">hints</span><span class="p">[</span><span class="s">&#39;order&#39;</span><span class="p">],</span> <span class="n">hint</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">rv</span>
-
-</div>
-<div class="viewcode-block" id="classify_ode"><a class="viewcode-back" href="../../../modules/solvers/ode.html#sympy.solvers.ode.classify_ode">[docs]</a><span class="k">def</span> <span class="nf">classify_ode</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="nb">dict</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">prep</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns a tuple of possible dsolve() classifications for an ODE.</span>
-
-<span class="sd">    The tuple is ordered so that first item is the classification that</span>
-<span class="sd">    dsolve() uses to solve the ODE by default.  In general,</span>
-<span class="sd">    classifications at the near the beginning of the list will produce</span>
-<span class="sd">    better solutions faster than those near the end, thought there are</span>
-<span class="sd">    always exceptions.  To make dsolve use a different classification,</span>
-<span class="sd">    use dsolve(ODE, func, hint=&lt;classification&gt;).  See also the dsolve()</span>
-<span class="sd">    docstring for different meta-hints you can use.</span>
-
-<span class="sd">    If ``dict`` is true, classify_ode() will return a dictionary of</span>
-<span class="sd">    hint:match expression terms. This is intended for internal use by</span>
-<span class="sd">    dsolve().  Note that because dictionaries are ordered arbitrarily,</span>
-<span class="sd">    this will most likely not be in the same order as the tuple.</span>
-
-<span class="sd">    If ``prep`` is False or ``func`` is None then the equation</span>
-<span class="sd">    will be preprocessed to put it in standard form for classification.</span>
-
-<span class="sd">    You can get help on different hints by doing help(ode.ode_hintname),</span>
-<span class="sd">    where hintname is the name of the hint without &quot;_Integral&quot;.</span>
-
-<span class="sd">    See sympy.ode.allhints or the sympy.ode docstring for a list of all</span>
-<span class="sd">    supported hints that can be returned from classify_ode.</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    These are remarks on hint names.</span>
-
-<span class="sd">    *&quot;_Integral&quot;*</span>
-
-<span class="sd">        If a classification has &quot;_Integral&quot; at the end, it will return</span>
-<span class="sd">        the expression with an unevaluated Integral class in it.  Note</span>
-<span class="sd">        that a hint may do this anyway if integrate() cannot do the</span>
-<span class="sd">        integral, though just using an &quot;_Integral&quot; will do so much</span>
-<span class="sd">        faster.  Indeed, an &quot;_Integral&quot; hint will always be faster than</span>
-<span class="sd">        its corresponding hint without &quot;_Integral&quot; because integrate()</span>
-<span class="sd">        is an expensive routine.  If dsolve() hangs, it is probably</span>
-<span class="sd">        because integrate() is hanging on a tough or impossible</span>
-<span class="sd">        integral.  Try using an &quot;_Integral&quot; hint or &quot;all_Integral&quot; to</span>
-<span class="sd">        get it return something.</span>
-
-<span class="sd">        Note that some hints do not have &quot;_Integral&quot; counterparts.  This</span>
-<span class="sd">        is because integrate() is not used in solving the ODE for those</span>
-<span class="sd">        method. For example, nth order linear homogeneous ODEs with</span>
-<span class="sd">        constant coefficients do not require integration to solve, so</span>
-<span class="sd">        there is no &quot;nth_linear_homogeneous_constant_coeff_Integrate&quot;</span>
-<span class="sd">        hint. You can easily evaluate any unevaluated Integrals in an</span>
-<span class="sd">        expression by doing expr.doit().</span>
-
-<span class="sd">    *Ordinals*</span>
-
-<span class="sd">        Some hints contain an ordinal such as &quot;1st_linear&quot;.  This is to</span>
-<span class="sd">        help differentiate them from other hints, as well as from other</span>
-<span class="sd">        methods that may not be implemented yet. If a hint has &quot;nth&quot; in</span>
-<span class="sd">        it, such as the &quot;nth_linear&quot; hints, this means that the method</span>
-<span class="sd">        used to applies to ODEs of any order.</span>
-
-<span class="sd">    *&quot;indep&quot; and &quot;dep&quot;*</span>
-
-<span class="sd">        Some hints contain the words &quot;indep&quot; or &quot;dep&quot;.  These reference</span>
-<span class="sd">        the independent variable and the dependent function,</span>
-<span class="sd">        respectively. For example, if an ODE is in terms of f(x), then</span>
-<span class="sd">        &quot;indep&quot; will refer to x and &quot;dep&quot; will refer to f.</span>
-
-<span class="sd">    *&quot;subs&quot;*</span>
-
-<span class="sd">        If a hints has the word &quot;subs&quot; in it, it means the the ODE is</span>
-<span class="sd">        solved by substituting the expression given after the word</span>
-<span class="sd">        &quot;subs&quot; for a single dummy variable.  This is usually in terms of</span>
-<span class="sd">        &quot;indep&quot; and &quot;dep&quot; as above.  The substituted expression will be</span>
-<span class="sd">        written only in characters allowed for names of Python objects,</span>
-<span class="sd">        meaning operators will be spelled out.  For example, indep/dep</span>
-<span class="sd">        will be written as indep_div_dep.</span>
-
-<span class="sd">    *&quot;coeff&quot;*</span>
-
-<span class="sd">        The word &quot;coeff&quot; in a hint refers to the coefficients of</span>
-<span class="sd">        something in the ODE, usually of the derivative terms.  See the</span>
-<span class="sd">        docstring for the individual methods for more info (help(ode)).</span>
-<span class="sd">        This is contrast to &quot;coefficients&quot;, as in</span>
-<span class="sd">        &quot;undetermined_coefficients&quot;, which refers to the common name of</span>
-<span class="sd">        a method.</span>
-
-<span class="sd">    *&quot;_best&quot;*</span>
-
-<span class="sd">        Methods that have more than one fundamental way to solve will</span>
-<span class="sd">        have a hint for each sub-method and a &quot;_best&quot;</span>
-<span class="sd">        meta-classification. This will evaluate all hints and return the</span>
-<span class="sd">        best, using the same considerations as the normal &quot;best&quot;</span>
-<span class="sd">        meta-hint.</span>
-
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, classify_ode, Eq</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; classify_ode(Eq(f(x).diff(x), 0), f(x))</span>
-<span class="sd">    (&#39;separable&#39;, &#39;1st_linear&#39;, &#39;1st_homogeneous_coeff_best&#39;,</span>
-<span class="sd">    &#39;1st_homogeneous_coeff_subs_indep_div_dep&#39;,</span>
-<span class="sd">    &#39;1st_homogeneous_coeff_subs_dep_div_indep&#39;,</span>
-<span class="sd">    &#39;nth_linear_constant_coeff_homogeneous&#39;, &#39;separable_Integral&#39;,</span>
-<span class="sd">    &#39;1st_linear_Integral&#39;,</span>
-<span class="sd">    &#39;1st_homogeneous_coeff_subs_indep_div_dep_Integral&#39;,</span>
-<span class="sd">    &#39;1st_homogeneous_coeff_subs_dep_div_indep_Integral&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; classify_ode(f(x).diff(x, 2) + 3*f(x).diff(x) + 2*f(x) - 4)</span>
-<span class="sd">    (&#39;nth_linear_constant_coeff_undetermined_coefficients&#39;,</span>
-<span class="sd">    &#39;nth_linear_constant_coeff_variation_of_parameters&#39;,</span>
-<span class="sd">    &#39;nth_linear_constant_coeff_variation_of_parameters_Integral&#39;)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expand</span>
-
-    <span class="k">if</span> <span class="n">func</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">func</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;dsolve() and classify_ode() only work with functions &quot;</span> <span class="o">+</span>
-            <span class="s">&quot;of one variable&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">prep</span> <span class="ow">or</span> <span class="n">func</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">eq</span><span class="p">,</span> <span class="n">func_</span> <span class="o">=</span> <span class="n">preprocess</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">func</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">func</span> <span class="o">=</span> <span class="n">func_</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">func</span>
-    <span class="n">y</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">Equality</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">eq</span><span class="o">.</span><span class="n">rhs</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">classify_ode</span><span class="p">(</span><span class="n">eq</span><span class="o">.</span><span class="n">lhs</span> <span class="o">-</span> <span class="n">eq</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">prep</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-        <span class="n">eq</span> <span class="o">=</span> <span class="n">eq</span><span class="o">.</span><span class="n">lhs</span>
-    <span class="n">order</span> <span class="o">=</span> <span class="n">ode_order</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-    <span class="c"># hint:matchdict or hint:(tuple of matchdicts)</span>
-    <span class="c"># Also will contain &quot;default&quot;:&lt;default hint&gt; and &quot;order&quot;:order items.</span>
-    <span class="n">matching_hints</span> <span class="o">=</span> <span class="p">{</span><span class="s">&quot;order&quot;</span><span class="p">:</span> <span class="n">order</span><span class="p">}</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">order</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">dict</span><span class="p">:</span>
-            <span class="n">matching_hints</span><span class="p">[</span><span class="s">&quot;default&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">return</span> <span class="n">matching_hints</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">()</span>
-
-    <span class="n">df</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)])</span>
-    <span class="n">b</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;b&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)])</span>
-    <span class="n">c</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;c&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)])</span>
-    <span class="n">d</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;d&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">df</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)])</span>
-    <span class="n">e</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;e&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">df</span><span class="p">])</span>
-    <span class="n">k</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;k&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">df</span><span class="p">])</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)])</span>
-    <span class="n">c1</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;c1&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">x</span><span class="p">])</span>
-    <span class="n">a2</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;a2&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">df</span><span class="p">])</span>
-    <span class="n">b2</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;b2&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">df</span><span class="p">])</span>
-    <span class="n">c2</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;c2&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">df</span><span class="p">])</span>
-    <span class="n">d2</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;d2&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">df</span><span class="p">])</span>
-
-    <span class="n">eq</span> <span class="o">=</span> <span class="n">expand</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span>
-
-    <span class="c"># Precondition to try remove f(x) from highest order derivative</span>
-    <span class="n">reduced_eq</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="k">if</span> <span class="n">eq</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-        <span class="n">deriv_coef</span> <span class="o">=</span> <span class="n">eq</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">order</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">deriv_coef</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">deriv_coef</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="n">c1</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">r</span> <span class="ow">and</span> <span class="n">r</span><span class="p">[</span><span class="n">c1</span><span class="p">]:</span>
-                <span class="n">den</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="n">r</span><span class="p">[</span><span class="n">c1</span><span class="p">]</span>
-                <span class="n">reduced_eq</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">arg</span><span class="o">/</span><span class="n">den</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">eq</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">reduced_eq</span><span class="p">:</span>
-        <span class="n">reduced_eq</span> <span class="o">=</span> <span class="n">eq</span>
-
-    <span class="k">if</span> <span class="n">order</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-
-        <span class="c"># Linear case: a(x)*y&#39;+b(x)*y+c(x) == 0</span>
-        <span class="k">if</span> <span class="n">eq</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">ind</span><span class="p">,</span> <span class="n">dep</span> <span class="o">=</span> <span class="n">reduced_eq</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">u</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;u&#39;</span><span class="p">)</span>
-            <span class="n">ind</span><span class="p">,</span> <span class="n">dep</span> <span class="o">=</span> <span class="p">(</span><span class="n">reduced_eq</span> <span class="o">+</span> <span class="n">u</span><span class="p">)</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-            <span class="n">ind</span><span class="p">,</span> <span class="n">dep</span> <span class="o">=</span> <span class="p">[</span><span class="n">tmp</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="k">for</span> <span class="n">tmp</span> <span class="ow">in</span> <span class="p">[</span><span class="n">ind</span><span class="p">,</span> <span class="n">dep</span><span class="p">]]</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="p">{</span><span class="n">a</span><span class="p">:</span> <span class="n">dep</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">df</span><span class="p">),</span>
-             <span class="n">b</span><span class="p">:</span> <span class="n">dep</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span>
-             <span class="n">c</span><span class="p">:</span> <span class="n">ind</span><span class="p">}</span>
-        <span class="c"># double check f[a] since the preconditioning may have failed</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="o">*</span><span class="n">df</span> <span class="o">+</span> <span class="n">r</span><span class="p">[</span><span class="n">b</span><span class="p">]</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">r</span><span class="p">[</span><span class="n">c</span><span class="p">])</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span> <span class="o">-</span> <span class="n">reduced_eq</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">r</span><span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">a</span>
-            <span class="n">r</span><span class="p">[</span><span class="s">&#39;b&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">b</span>
-            <span class="n">r</span><span class="p">[</span><span class="s">&#39;c&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">c</span>
-            <span class="n">matching_hints</span><span class="p">[</span><span class="s">&quot;1st_linear&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span>
-            <span class="n">matching_hints</span><span class="p">[</span><span class="s">&quot;1st_linear_Integral&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span>
-
-        <span class="c"># Bernoulli case: a(x)*y&#39;+b(x)*y+c(x)*y**n == 0</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">collect</span><span class="p">(</span>
-            <span class="n">reduced_eq</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">df</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">c</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">r</span> <span class="ow">and</span> <span class="n">r</span><span class="p">[</span><span class="n">c</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">r</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>  <span class="c"># See issue 1577</span>
-            <span class="n">r</span><span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">a</span>
-            <span class="n">r</span><span class="p">[</span><span class="s">&#39;b&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">b</span>
-            <span class="n">r</span><span class="p">[</span><span class="s">&#39;c&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">c</span>
-            <span class="n">r</span><span class="p">[</span><span class="s">&#39;n&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">n</span>
-            <span class="n">matching_hints</span><span class="p">[</span><span class="s">&quot;Bernoulli&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span>
-            <span class="n">matching_hints</span><span class="p">[</span><span class="s">&quot;Bernoulli_Integral&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span>
-
-        <span class="c"># Riccati special n == -2 case: a2*y&#39;+b2*y**2+c2*y/x+d2/x**2 == 0</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">collect</span><span class="p">(</span><span class="n">reduced_eq</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span>
-            <span class="n">x</span><span class="p">),</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">a2</span><span class="o">*</span><span class="n">df</span> <span class="o">+</span> <span class="n">b2</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c2</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span> <span class="o">+</span> <span class="n">d2</span><span class="o">/</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">r</span> <span class="ow">and</span> <span class="n">r</span><span class="p">[</span><span class="n">b2</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">c2</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">r</span><span class="p">[</span><span class="n">d2</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">):</span>
-            <span class="n">r</span><span class="p">[</span><span class="s">&#39;a2&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">a2</span>
-            <span class="n">r</span><span class="p">[</span><span class="s">&#39;b2&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">b2</span>
-            <span class="n">r</span><span class="p">[</span><span class="s">&#39;c2&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">c2</span>
-            <span class="n">r</span><span class="p">[</span><span class="s">&#39;d2&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">d2</span>
-            <span class="n">matching_hints</span><span class="p">[</span><span class="s">&quot;Riccati_special_minus2&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span>
-
-        <span class="c"># Exact Differential Equation: P(x,y)+Q(x,y)*y&#39;=0 where dP/dy == dQ/dx</span>
-        <span class="c"># WITH NON-REDUCED FORM OF EQUATION</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">collect</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">df</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">d</span> <span class="o">+</span> <span class="n">e</span> <span class="o">*</span> <span class="n">df</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">r</span><span class="p">:</span>
-            <span class="n">r</span><span class="p">[</span><span class="s">&#39;d&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">d</span>
-            <span class="n">r</span><span class="p">[</span><span class="s">&#39;e&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">e</span>
-            <span class="n">r</span><span class="p">[</span><span class="s">&#39;y&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">y</span>
-            <span class="n">r</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">y</span><span class="p">)</span>
-            <span class="n">r</span><span class="p">[</span><span class="n">e</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span><span class="p">[</span><span class="n">e</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">y</span><span class="p">)</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">r</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">simplify</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">y</span><span class="p">))</span> <span class="o">==</span> <span class="n">simplify</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">e</span><span class="p">]</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)):</span>
-                    <span class="n">matching_hints</span><span class="p">[</span><span class="s">&quot;1st_exact&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span>
-                    <span class="n">matching_hints</span><span class="p">[</span><span class="s">&quot;1st_exact_Integral&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span>
-            <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                <span class="c"># Differentiating the coefficients might fail because of things</span>
-                <span class="c"># like f(2*x).diff(x).  See issue 1525 and issue 1620.</span>
-                <span class="k">pass</span>
-
-        <span class="c"># This match is used for several cases below; we now collect on</span>
-        <span class="c"># f(x) so the matching works.</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">collect</span><span class="p">(</span><span class="n">reduced_eq</span><span class="p">,</span> <span class="n">df</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">d</span> <span class="o">+</span> <span class="n">e</span><span class="o">*</span><span class="n">df</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">r</span><span class="p">:</span>
-            <span class="n">r</span><span class="p">[</span><span class="s">&#39;d&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">d</span>
-            <span class="n">r</span><span class="p">[</span><span class="s">&#39;e&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">e</span>
-            <span class="n">r</span><span class="p">[</span><span class="s">&#39;y&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">y</span>
-            <span class="n">r</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">y</span><span class="p">)</span>
-            <span class="n">r</span><span class="p">[</span><span class="n">e</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span><span class="p">[</span><span class="n">e</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">y</span><span class="p">)</span>
-
-            <span class="c"># Separable Case: y&#39; == P(y)*Q(x)</span>
-            <span class="n">r</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">=</span> <span class="n">separatevars</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">d</span><span class="p">])</span>
-            <span class="n">r</span><span class="p">[</span><span class="n">e</span><span class="p">]</span> <span class="o">=</span> <span class="n">separatevars</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">e</span><span class="p">])</span>
-            <span class="c"># m1[coeff]*m1[x]*m1[y] + m2[coeff]*m2[x]*m2[y]*y&#39;</span>
-            <span class="n">m1</span> <span class="o">=</span> <span class="n">separatevars</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">d</span><span class="p">],</span> <span class="nb">dict</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">symbols</span><span class="o">=</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span>
-            <span class="n">m2</span> <span class="o">=</span> <span class="n">separatevars</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">e</span><span class="p">],</span> <span class="nb">dict</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">symbols</span><span class="o">=</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">m1</span> <span class="ow">and</span> <span class="n">m2</span><span class="p">:</span>
-                <span class="n">r1</span> <span class="o">=</span> <span class="p">{</span><span class="s">&#39;m1&#39;</span><span class="p">:</span> <span class="n">m1</span><span class="p">,</span> <span class="s">&#39;m2&#39;</span><span class="p">:</span> <span class="n">m2</span><span class="p">,</span> <span class="s">&#39;y&#39;</span><span class="p">:</span> <span class="n">y</span><span class="p">}</span>
-                <span class="n">matching_hints</span><span class="p">[</span><span class="s">&quot;separable&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">r1</span>
-                <span class="n">matching_hints</span><span class="p">[</span><span class="s">&quot;separable_Integral&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">r1</span>
-
-            <span class="c"># First order equation with homogeneous coefficients:</span>
-            <span class="c"># dy/dx == F(y/x) or dy/dx == F(x/y)</span>
-            <span class="n">ordera</span> <span class="o">=</span> <span class="n">homogeneous_order</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">d</span><span class="p">],</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-            <span class="n">orderb</span> <span class="o">=</span> <span class="n">homogeneous_order</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">e</span><span class="p">],</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">ordera</span> <span class="o">==</span> <span class="n">orderb</span> <span class="ow">and</span> <span class="n">ordera</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="c"># u1=y/x and u2=x/y</span>
-                <span class="n">u1</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;u1&#39;</span><span class="p">)</span>
-                <span class="n">u2</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;u2&#39;</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">simplify</span><span class="p">((</span><span class="n">r</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">+</span> <span class="n">u1</span><span class="o">*</span><span class="n">r</span><span class="p">[</span><span class="n">e</span><span class="p">])</span><span class="o">.</span><span class="n">subs</span><span class="p">({</span><span class="n">x</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">u1</span><span class="p">}))</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">matching_hints</span><span class="p">[</span>
-                        <span class="s">&quot;1st_homogeneous_coeff_subs_dep_div_indep&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span>
-                    <span class="n">matching_hints</span><span class="p">[</span><span class="s">&quot;1st_homogeneous_coeff_subs_dep_div_indep_Integral&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span>
-                <span class="k">if</span> <span class="n">simplify</span><span class="p">((</span><span class="n">r</span><span class="p">[</span><span class="n">e</span><span class="p">]</span> <span class="o">+</span> <span class="n">u2</span><span class="o">*</span><span class="n">r</span><span class="p">[</span><span class="n">d</span><span class="p">])</span><span class="o">.</span><span class="n">subs</span><span class="p">({</span><span class="n">x</span><span class="p">:</span> <span class="n">u2</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="mi">1</span><span class="p">}))</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">matching_hints</span><span class="p">[</span>
-                        <span class="s">&quot;1st_homogeneous_coeff_subs_indep_div_dep&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span>
-                    <span class="n">matching_hints</span><span class="p">[</span><span class="s">&quot;1st_homogeneous_coeff_subs_indep_div_dep_Integral&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span>
-                <span class="k">if</span> <span class="s">&quot;1st_homogeneous_coeff_subs_dep_div_indep&quot;</span> <span class="ow">in</span> <span class="n">matching_hints</span> \
-                        <span class="ow">and</span> <span class="s">&quot;1st_homogeneous_coeff_subs_indep_div_dep&quot;</span> <span class="ow">in</span> <span class="n">matching_hints</span><span class="p">:</span>
-                    <span class="n">matching_hints</span><span class="p">[</span><span class="s">&quot;1st_homogeneous_coeff_best&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span>
-
-    <span class="k">if</span> <span class="n">order</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="c"># Liouville ODE f(x).diff(x, 2) + g(f(x))*(f(x).diff(x))**2 + h(x)*f(x).diff(x)</span>
-        <span class="c"># See Goldstein and Braun, &quot;Advanced Methods for the Solution of</span>
-        <span class="c"># Differential Equations&quot;, pg. 98</span>
-
-        <span class="n">s</span> <span class="o">=</span> <span class="n">d</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">e</span><span class="o">*</span><span class="n">df</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">k</span><span class="o">*</span><span class="n">df</span>
-        <span class="n">r</span> <span class="o">=</span> <span class="n">reduced_eq</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">r</span> <span class="ow">and</span> <span class="n">r</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">y</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">e</span><span class="p">]</span><span class="o">/</span><span class="n">r</span><span class="p">[</span><span class="n">d</span><span class="p">])</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">y</span><span class="p">)</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">/</span><span class="n">r</span><span class="p">[</span><span class="n">d</span><span class="p">])</span>
-            <span class="k">if</span> <span class="n">h</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="ow">or</span> <span class="n">g</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-                <span class="k">pass</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">r</span> <span class="o">=</span> <span class="p">{</span><span class="s">&#39;g&#39;</span><span class="p">:</span> <span class="n">g</span><span class="p">,</span> <span class="s">&#39;h&#39;</span><span class="p">:</span> <span class="n">h</span><span class="p">,</span> <span class="s">&#39;y&#39;</span><span class="p">:</span> <span class="n">y</span><span class="p">}</span>
-                <span class="n">matching_hints</span><span class="p">[</span><span class="s">&quot;Liouville&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span>
-                <span class="n">matching_hints</span><span class="p">[</span><span class="s">&quot;Liouville_Integral&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span>
-
-    <span class="k">if</span> <span class="n">order</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="c"># nth order linear ODE</span>
-        <span class="c"># a_n(x)y^(n) + ... + a_1(x)y&#39; + a_0(x)y = F(x) = b</span>
-
-        <span class="n">r</span> <span class="o">=</span> <span class="n">_nth_linear_match</span><span class="p">(</span><span class="n">reduced_eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">)</span>
-
-        <span class="c"># Constant coefficient case (a_i is constant for all i)</span>
-        <span class="k">if</span> <span class="n">r</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">any</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">r</span> <span class="k">if</span> <span class="n">i</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">):</span>
-            <span class="c"># Inhomogeneous case: F(x) is not identically 0</span>
-            <span class="k">if</span> <span class="n">r</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-                <span class="n">undetcoeff</span> <span class="o">=</span> <span class="n">_undetermined_coefficients_match</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
-                <span class="n">matching_hints</span><span class="p">[</span>
-                    <span class="s">&quot;nth_linear_constant_coeff_variation_of_parameters&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span>
-                <span class="n">matching_hints</span><span class="p">[</span><span class="s">&quot;nth_linear_constant_coeff_variation_of_parameters&quot;</span> <span class="o">+</span>
-                    <span class="s">&quot;_Integral&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span>
-                <span class="k">if</span> <span class="n">undetcoeff</span><span class="p">[</span><span class="s">&#39;test&#39;</span><span class="p">]:</span>
-                    <span class="n">r</span><span class="p">[</span><span class="s">&#39;trialset&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">undetcoeff</span><span class="p">[</span><span class="s">&#39;trialset&#39;</span><span class="p">]</span>
-                    <span class="n">matching_hints</span><span class="p">[</span><span class="s">&quot;nth_linear_constant_coeff_undetermined_&quot;</span> <span class="o">+</span>
-                        <span class="s">&quot;coefficients&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span>
-            <span class="c"># Homogeneous case: F(x) is identically 0</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">matching_hints</span><span class="p">[</span><span class="s">&quot;nth_linear_constant_coeff_homogeneous&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span>
-
-        <span class="c"># Euler equation case (a_i * x**i for all i)</span>
-        <span class="k">def</span> <span class="nf">_test_term</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">order</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            Linear Euler ODEs have the form  K*x**order*diff(y(x),x,order),</span>
-<span class="sd">            where K is independent of x and y(x), order&gt;= 0.</span>
-<span class="sd">            So we need to check that for each term, coeff == K*x**order from</span>
-<span class="sd">            some K.  We have a few cases, since coeff may have several</span>
-<span class="sd">            different types.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="k">assert</span> <span class="n">order</span> <span class="o">&gt;=</span> <span class="mi">0</span>
-            <span class="k">if</span> <span class="n">coeff</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">if</span> <span class="n">order</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">coeff</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-                <span class="k">return</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">coeff</span><span class="o">.</span><span class="n">atoms</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)):</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-                <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="n">order</span> <span class="ow">in</span> <span class="n">coeff</span><span class="o">.</span><span class="n">args</span>
-            <span class="k">elif</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">coeff</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span> <span class="o">==</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">order</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">order</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">x</span> <span class="o">==</span> <span class="n">coeff</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">r</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">any</span><span class="p">(</span><span class="ow">not</span> <span class="n">_test_term</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">r</span> <span class="k">if</span> <span class="n">i</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-                <span class="n">matching_hints</span><span class="p">[</span><span class="s">&quot;nth_linear_euler_eq_homogeneous&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">r</span>
-
-    <span class="c"># Order keys based on allhints.</span>
-    <span class="n">retlist</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">allhints</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">matching_hints</span><span class="p">:</span>
-            <span class="n">retlist</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">dict</span><span class="p">:</span>
-        <span class="c"># Dictionaries are ordered arbitrarily, so we need to make note of which</span>
-        <span class="c"># hint would come first for dsolve().  In Python 3, this should be replaced</span>
-        <span class="c"># with an ordered dictionary.</span>
-        <span class="n">matching_hints</span><span class="p">[</span><span class="s">&quot;default&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="n">matching_hints</span><span class="p">[</span><span class="s">&quot;ordered_hints&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">retlist</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">allhints</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">matching_hints</span><span class="p">:</span>
-                <span class="n">matching_hints</span><span class="p">[</span><span class="s">&quot;default&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">i</span>
-                <span class="k">break</span>
-        <span class="k">return</span> <span class="n">matching_hints</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">retlist</span><span class="p">)</span>
-
-</div>
-<span class="nd">@vectorize</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<div class="viewcode-block" id="odesimp"><a class="viewcode-back" href="../../../modules/solvers/ode.html#sympy.solvers.ode.odesimp">[docs]</a><span class="k">def</span> <span class="nf">odesimp</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">hint</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Simplifies ODEs, including trying to solve for func and running</span>
-<span class="sd">    constantsimp().</span>
-
-<span class="sd">    It may use knowledge of the type of solution that that hint returns</span>
-<span class="sd">    to apply additional simplifications.</span>
-
-<span class="sd">    It also attempts to integrate any Integrals in the expression, if</span>
-<span class="sd">    the hint is not an &quot;_Integral&quot; hint.</span>
-
-<span class="sd">    This function should have no effect on expressions returned by</span>
-<span class="sd">    dsolve(), as dsolve already calls odesimp(), but the individual hint</span>
-<span class="sd">    functions do not call odesimp (because the dsolve() wrapper does).</span>
-<span class="sd">    Therefore, this function is designed for mainly internal use.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import sin, symbols, dsolve, pprint, Function</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.solvers.ode import odesimp</span>
-<span class="sd">    &gt;&gt;&gt; x , u2, C1= symbols(&#39;x,u2,C1&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-
-<span class="sd">    &gt;&gt;&gt; eq = dsolve(x*f(x).diff(x) - f(x) - x*sin(f(x)/x), f(x),</span>
-<span class="sd">    ... hint=&#39;1st_homogeneous_coeff_subs_indep_div_dep_Integral&#39;,</span>
-<span class="sd">    ... simplify=False)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(eq)</span>
-<span class="sd">                  x</span>
-<span class="sd">                     ----</span>
-<span class="sd">                     f(x)</span>
-<span class="sd">                       /</span>
-<span class="sd">                      |</span>
-<span class="sd">           /f(x)\     |  /  1         1     \</span>
-<span class="sd">        log|----| -   |  |- -- - -----------| d(u2) = 0</span>
-<span class="sd">           \ C1 /     |  |  u2     2    /1 \|</span>
-<span class="sd">                      |  |       u2 *sin|--||</span>
-<span class="sd">                      |  \              \u2//</span>
-<span class="sd">                      |</span>
-<span class="sd">                     /</span>
-
-<span class="sd">    &gt;&gt;&gt; pprint(odesimp(eq, f(x), 1,</span>
-<span class="sd">    ... hint=&#39;1st_homogeneous_coeff_subs_indep_div_dep&#39;</span>
-<span class="sd">    ... )) #doctest: +SKIP</span>
-<span class="sd">        x</span>
-<span class="sd">    --------- = C1</span>
-<span class="sd">       /f(x)\</span>
-<span class="sd">    tan|----|</span>
-<span class="sd">       \2*x /</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">func</span>
-    <span class="n">C1</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;C1&#39;</span><span class="p">)</span>
-
-    <span class="c"># First, integrate if the hint allows it.</span>
-    <span class="n">eq</span> <span class="o">=</span> <span class="n">_handle_Integral</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">hint</span><span class="p">)</span>
-    <span class="k">assert</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">Equality</span><span class="p">)</span>
-
-    <span class="c"># Second, clean up the arbitrary constants.</span>
-    <span class="c"># Right now, nth linear hints can put as many as 2*order constants in an</span>
-    <span class="c"># expression.  If that number grows with another hint, the third argument</span>
-    <span class="c"># here should be raised accordingly, or constantsimp() rewritten to handle</span>
-    <span class="c"># an arbitrary number of constants.</span>
-    <span class="n">eq</span> <span class="o">=</span> <span class="n">constantsimp</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">order</span><span class="p">)</span>
-
-    <span class="c"># Lastly, now that we have cleaned up the expression, try solving for func.</span>
-    <span class="c"># When RootOf is implemented in solve(), we will want to return a RootOf</span>
-    <span class="c"># everytime instead of an Equality.</span>
-
-    <span class="c"># Get the f(x) on the left if possible.</span>
-    <span class="k">if</span> <span class="n">eq</span><span class="o">.</span><span class="n">rhs</span> <span class="o">==</span> <span class="n">func</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">eq</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
-        <span class="n">eq</span> <span class="o">=</span> <span class="p">[</span><span class="n">Eq</span><span class="p">(</span><span class="n">eq</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">eq</span><span class="o">.</span><span class="n">lhs</span><span class="p">)]</span>
-
-    <span class="c"># make sure we are working with lists of solutions in simplified form.</span>
-    <span class="k">if</span> <span class="n">eq</span><span class="o">.</span><span class="n">lhs</span> <span class="o">==</span> <span class="n">func</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">eq</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
-        <span class="c"># The solution is already solved</span>
-        <span class="n">eq</span> <span class="o">=</span> <span class="p">[</span><span class="n">eq</span><span class="p">]</span>
-
-        <span class="c"># special simplification of the rhs</span>
-        <span class="k">if</span> <span class="n">hint</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&quot;nth_linear_constant_coeff&quot;</span><span class="p">):</span>
-            <span class="c"># Collect terms to make the solution look nice.</span>
-            <span class="c"># This is also necessary for constantsimp to remove unnecessary terms</span>
-            <span class="c"># from the particular solution from variation of parameters</span>
-            <span class="k">global</span> <span class="n">collectterms</span>
-            <span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">eq</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">lhs</span> <span class="o">==</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-            <span class="n">sol</span> <span class="o">=</span> <span class="n">eq</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">rhs</span>
-            <span class="n">sol</span> <span class="o">=</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">sol</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">reroot</span><span class="p">,</span> <span class="n">imroot</span> <span class="ow">in</span> <span class="n">collectterms</span><span class="p">:</span>
-                <span class="n">sol</span> <span class="o">=</span> <span class="n">collect</span><span class="p">(</span><span class="n">sol</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="n">i</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">reroot</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">imroot</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="p">))</span>
-                <span class="n">sol</span> <span class="o">=</span> <span class="n">collect</span><span class="p">(</span><span class="n">sol</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="n">i</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">reroot</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">imroot</span><span class="o">*</span><span class="n">x</span><span class="p">))</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">reroot</span><span class="p">,</span> <span class="n">imroot</span> <span class="ow">in</span> <span class="n">collectterms</span><span class="p">:</span>
-                <span class="n">sol</span> <span class="o">=</span> <span class="n">collect</span><span class="p">(</span><span class="n">sol</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="n">i</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">reroot</span><span class="o">*</span><span class="n">x</span><span class="p">))</span>
-            <span class="k">del</span> <span class="n">collectterms</span>
-            <span class="n">eq</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">Eq</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">sol</span><span class="p">)</span>
-
-    <span class="k">else</span><span class="p">:</span>
-        <span class="c"># The solution is not solved, so try to solve it</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">eqsol</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">eqsol</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-        <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-            <span class="n">eq</span> <span class="o">=</span> <span class="p">[</span><span class="n">eq</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">def</span> <span class="nf">_expand</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-                <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-
-                <span class="k">if</span> <span class="n">denom</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">expr</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">powsimp</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">expand</span><span class="p">(),</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;exp&#39;</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-            <span class="c"># XXX: the rest of odesimp() expects each ``t`` to be in a</span>
-            <span class="c"># specific normal form: rational expression with numerator</span>
-            <span class="c"># expanded, but with combined exponential functions (at</span>
-            <span class="c"># least in this setup all tests pass).</span>
-            <span class="n">eq</span> <span class="o">=</span> <span class="p">[</span><span class="n">Eq</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">_expand</span><span class="p">(</span><span class="n">t</span><span class="p">))</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">eqsol</span><span class="p">]</span>
-
-        <span class="c"># special simplification of the lhs.</span>
-        <span class="k">if</span> <span class="n">hint</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&quot;1st_homogeneous_coeff&quot;</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">eqi</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">eq</span><span class="p">):</span>
-                <span class="n">newi</span> <span class="o">=</span> <span class="n">logcombine</span><span class="p">(</span><span class="n">eqi</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">newi</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">is_Function</span> <span class="ow">and</span> <span class="n">newi</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">log</span> <span class="ow">and</span> <span class="n">newi</span><span class="o">.</span><span class="n">rhs</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">newi</span> <span class="o">=</span> <span class="n">Eq</span><span class="p">(</span><span class="n">newi</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="n">C1</span><span class="p">,</span> <span class="n">C1</span><span class="p">)</span>
-                <span class="n">eq</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">newi</span>
-
-    <span class="c"># We cleaned up the costants before solving to help the solve engine with</span>
-    <span class="c"># a simpler expression, but the solved expression could have introduced</span>
-    <span class="c"># things like -C1, so rerun constantsimp() one last time before returning.</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">eqi</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">eq</span><span class="p">):</span>
-        <span class="n">eq</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">constant_renumber</span><span class="p">(</span>
-            <span class="n">constantsimp</span><span class="p">(</span><span class="n">eqi</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">order</span><span class="p">),</span> <span class="s">&#39;C&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">order</span><span class="p">)</span>
-
-    <span class="c"># If there is only 1 solution, return it;</span>
-    <span class="c"># otherwise return the list of solutions.</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">eq</span> <span class="o">=</span> <span class="n">eq</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">return</span> <span class="n">eq</span>
-
-</div>
-<div class="viewcode-block" id="checkodesol"><a class="viewcode-back" href="../../../modules/solvers/ode.html#sympy.solvers.ode.checkodesol">[docs]</a><span class="k">def</span> <span class="nf">checkodesol</span><span class="p">(</span><span class="n">ode</span><span class="p">,</span> <span class="n">sol</span><span class="p">,</span> <span class="n">func</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s">&#39;auto&#39;</span><span class="p">,</span> <span class="n">solve_for_func</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Substitutes sol into the ode and checks that the result is 0.</span>
-
-<span class="sd">    This only works when func is one function, like f(x).  sol can be a</span>
-<span class="sd">    single solution or a list of solutions.  Each solution may be an Equality</span>
-<span class="sd">    that the solution satisfies, e.g. Eq(f(x), C1), Eq(f(x) + C1, 0); or simply</span>
-<span class="sd">    an Expr, e.g. f(x) - C1. In most cases it will not be necessary to</span>
-<span class="sd">    explicitly identify the function, but if the function cannot be inferred</span>
-<span class="sd">    from the original equation it can be supplied through the &#39;func&#39; argument.</span>
-
-<span class="sd">    If a sequence of solutions is passed, the same sort of container will be used</span>
-<span class="sd">    to return the result for each solution.</span>
-
-<span class="sd">    It tries the following methods, in order, until it finds zero</span>
-<span class="sd">    equivalence:</span>
-
-<span class="sd">        1. Substitute the solution for f in the original equation.  This</span>
-<span class="sd">           only works if the ode is solved for f.  It will attempt to solve</span>
-<span class="sd">           it first unless solve_for_func == False</span>
-<span class="sd">        2. Take n derivatives of the solution, where n is the order of</span>
-<span class="sd">           ode, and check to see if that is equal to the solution.  This</span>
-<span class="sd">           only works on exact odes.</span>
-<span class="sd">        3. Take the 1st, 2nd, ..., nth derivatives of the solution, each</span>
-<span class="sd">           time solving for the derivative of f of that order (this will</span>
-<span class="sd">           always be possible because f is a linear operator).  Then back</span>
-<span class="sd">           substitute each derivative into ode in reverse order.</span>
-
-<span class="sd">    This function returns a tuple.  The first item in the tuple is True</span>
-<span class="sd">    if the substitution results in 0, and False otherwise. The second</span>
-<span class="sd">    item in the tuple is what the substitution results in.  It should</span>
-<span class="sd">    always be 0 if the first item is True. Note that sometimes this</span>
-<span class="sd">    function will False, but with an expression that is identically</span>
-<span class="sd">    equal to 0, instead of returning True.  This is because simplify()</span>
-<span class="sd">    cannot reduce the expression to 0.  If an expression returned by</span>
-<span class="sd">    this function vanishes identically, then sol really is a solution to</span>
-<span class="sd">    ode.</span>
-
-<span class="sd">    If this function seems to hang, it is probably because of a hard</span>
-<span class="sd">    simplification.</span>
-
-<span class="sd">    To use this function to test, test the first item of the tuple.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Eq, Function, checkodesol, symbols</span>
-<span class="sd">    &gt;&gt;&gt; x, C1 = symbols(&#39;x,C1&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; checkodesol(f(x).diff(x), Eq(f(x), C1))</span>
-<span class="sd">    (True, 0)</span>
-<span class="sd">    &gt;&gt;&gt; assert checkodesol(f(x).diff(x), C1)[0]</span>
-<span class="sd">    &gt;&gt;&gt; assert not checkodesol(f(x).diff(x), x)[0]</span>
-<span class="sd">    &gt;&gt;&gt; checkodesol(f(x).diff(x, 2), x**2)</span>
-<span class="sd">    (False, 2)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ode</span><span class="p">,</span> <span class="n">Equality</span><span class="p">):</span>
-        <span class="n">ode</span> <span class="o">=</span> <span class="n">Eq</span><span class="p">(</span><span class="n">ode</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">func</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">_</span><span class="p">,</span> <span class="n">func</span> <span class="o">=</span> <span class="n">preprocess</span><span class="p">(</span><span class="n">ode</span><span class="o">.</span><span class="n">lhs</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-            <span class="n">funcs</span> <span class="o">=</span> <span class="p">[</span><span class="n">s</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">AppliedUndef</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="p">(</span>
-                <span class="n">sol</span> <span class="k">if</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">sol</span><span class="p">,</span> <span class="nb">set</span><span class="p">)</span> <span class="k">else</span> <span class="p">[</span><span class="n">sol</span><span class="p">])]</span>
-            <span class="n">funcs</span> <span class="o">=</span> <span class="nb">reduce</span><span class="p">(</span><span class="nb">set</span><span class="o">.</span><span class="n">union</span><span class="p">,</span> <span class="n">funcs</span><span class="p">,</span> <span class="nb">set</span><span class="p">())</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">funcs</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                    <span class="s">&#39;must pass func arg to checkodesol for this case.&#39;</span><span class="p">)</span>
-            <span class="n">func</span> <span class="o">=</span> <span class="n">funcs</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-    <span class="c"># ========== deprecation handling</span>
-    <span class="c"># After the deprecation period this handling section becomes:</span>
-    <span class="c"># ----------</span>
-    <span class="c"># if not is_unfunc(func) or len(func.args) != 1:</span>
-    <span class="c">#     raise ValueError(&quot;func must be a function of one variable, not %s&quot; % func)</span>
-    <span class="c"># ----------</span>
-    <span class="c"># assume, during deprecation that sol and func are reversed</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">sol</span><span class="p">,</span> <span class="n">AppliedUndef</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">sol</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">AppliedUndef</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">func</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">msg</span> <span class="o">=</span> <span class="s">&quot;If you really do want sol to be just </span><span class="si">%s</span><span class="s">, use Eq(</span><span class="si">%s</span><span class="s">, 0) &quot;</span> <span class="o">%</span> \
-                <span class="p">(</span><span class="n">sol</span><span class="p">,</span> <span class="n">sol</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot;instead.&quot;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">msg</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-        <span class="n">SymPyDeprecationWarning</span><span class="p">(</span><span class="n">msg</span><span class="p">,</span> <span class="n">feature</span><span class="o">=</span><span class="s">&quot;The order of the &quot;</span>
-            <span class="s">&quot;arguments sol and func to checkodesol()&quot;</span><span class="p">,</span>
-            <span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;checkodesol(ode, sol, func)&quot;</span><span class="p">,</span> <span class="n">issue</span><span class="o">=</span><span class="mi">3384</span><span class="p">,</span>
-        <span class="p">)</span><span class="o">.</span><span class="n">warn</span><span class="p">()</span>
-        <span class="n">sol</span><span class="p">,</span> <span class="n">func</span> <span class="o">=</span> <span class="n">func</span><span class="p">,</span> <span class="n">sol</span>
-    <span class="k">elif</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">AppliedUndef</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">func</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.utilities.misc</span> <span class="kn">import</span> <span class="n">filldedent</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&#39;&#39;&#39;</span>
-<span class="s">        func (or sol, during deprecation) must be a function</span>
-<span class="s">        of one variable. Got sol = </span><span class="si">%s</span><span class="s">, func = </span><span class="si">%s</span><span class="s">&#39;&#39;&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">sol</span><span class="p">,</span> <span class="n">func</span><span class="p">)))</span>
-    <span class="c"># ========== end of deprecation handling</span>
-    <span class="k">if</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">sol</span><span class="p">,</span> <span class="nb">set</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">type</span><span class="p">(</span><span class="n">sol</span><span class="p">)([</span><span class="n">checkodesol</span><span class="p">(</span><span class="n">ode</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">,</span>
-            <span class="n">solve_for_func</span><span class="o">=</span><span class="n">solve_for_func</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">sol</span><span class="p">])</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">sol</span><span class="p">,</span> <span class="n">Equality</span><span class="p">):</span>
-        <span class="n">sol</span> <span class="o">=</span> <span class="n">Eq</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">sol</span><span class="p">)</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">testnum</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">if</span> <span class="n">order</span> <span class="o">==</span> <span class="s">&#39;auto&#39;</span><span class="p">:</span>
-        <span class="n">order</span> <span class="o">=</span> <span class="n">ode_order</span><span class="p">(</span><span class="n">ode</span><span class="p">,</span> <span class="n">func</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">solve_for_func</span> <span class="ow">and</span> <span class="ow">not</span> <span class="p">(</span><span class="n">sol</span><span class="o">.</span><span class="n">lhs</span> <span class="o">==</span> <span class="n">func</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">sol</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">func</span><span class="p">))</span> <span class="ow">and</span> <span class="ow">not</span> \
-            <span class="p">(</span><span class="n">sol</span><span class="o">.</span><span class="n">rhs</span> <span class="o">==</span> <span class="n">func</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">sol</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">func</span><span class="p">)):</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">solved</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">sol</span><span class="p">,</span> <span class="n">func</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">solved</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-        <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-            <span class="k">pass</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">solved</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">checkodesol</span><span class="p">(</span><span class="n">ode</span><span class="p">,</span> <span class="n">Eq</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">solved</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span>
-                    <span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">,</span> <span class="n">solve_for_func</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">checkodesol</span><span class="p">(</span><span class="n">ode</span><span class="p">,</span> <span class="p">[</span><span class="n">Eq</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">solved</span><span class="p">],</span>
-                <span class="n">order</span><span class="o">=</span><span class="n">order</span><span class="p">,</span> <span class="n">solve_for_func</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-            <span class="k">return</span> <span class="n">result</span>
-
-    <span class="k">while</span> <span class="n">s</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">testnum</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="c"># First pass, try substituting a solved solution directly into the ode</span>
-            <span class="c"># This has the highest chance of succeeding.</span>
-            <span class="n">ode_diff</span> <span class="o">=</span> <span class="n">ode</span><span class="o">.</span><span class="n">lhs</span> <span class="o">-</span> <span class="n">ode</span><span class="o">.</span><span class="n">rhs</span>
-
-            <span class="k">if</span> <span class="n">sol</span><span class="o">.</span><span class="n">lhs</span> <span class="o">==</span> <span class="n">func</span><span class="p">:</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">sub_func_doit</span><span class="p">(</span><span class="n">ode_diff</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">sol</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">sol</span><span class="o">.</span><span class="n">rhs</span> <span class="o">==</span> <span class="n">func</span><span class="p">:</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">sub_func_doit</span><span class="p">(</span><span class="n">ode_diff</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">sol</span><span class="o">.</span><span class="n">lhs</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">testnum</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">continue</span>
-            <span class="n">ss</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">ss</span><span class="p">:</span>
-                <span class="c"># with the new numer_denom in power.py, if we do a simple</span>
-                <span class="c"># expansion then testnum == 0 verifies all solutions.</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">ss</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="n">testnum</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">elif</span> <span class="n">testnum</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># Second pass. If we cannot substitute f, try seeing if the nth</span>
-            <span class="c"># derivative is equal, this will only work for odes that are exact,</span>
-            <span class="c"># by definition.</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">trigsimp</span><span class="p">(</span><span class="n">diff</span><span class="p">(</span><span class="n">sol</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">order</span><span class="p">)</span> <span class="o">-</span> <span class="n">diff</span><span class="p">(</span><span class="n">sol</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">order</span><span class="p">))</span> <span class="o">-</span>
-                <span class="n">trigsimp</span><span class="p">(</span><span class="n">ode</span><span class="o">.</span><span class="n">lhs</span><span class="p">)</span> <span class="o">+</span> <span class="n">trigsimp</span><span class="p">(</span><span class="n">ode</span><span class="o">.</span><span class="n">rhs</span><span class="p">))</span>
-<span class="c">#            s2 = simplify(diff(sol.lhs, x, order) - diff(sol.rhs, x, order) - \</span>
-<span class="c">#                ode.lhs + ode.rhs)</span>
-            <span class="n">testnum</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">elif</span> <span class="n">testnum</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="c"># Third pass. Try solving for df/dx and substituting that into the ode.</span>
-            <span class="c"># Thanks to Chris Smith for suggesting this method.  Many of the</span>
-            <span class="c"># comments below are his too.</span>
-            <span class="c"># The method:</span>
-            <span class="c"># - Take each of 1..n derivatives of the solution.</span>
-            <span class="c"># - Solve each nth derivative for d^(n)f/dx^(n)</span>
-            <span class="c">#   (the differential of that order)</span>
-            <span class="c"># - Back substitute into the ode in decreasing order</span>
-            <span class="c">#   (i.e., n, n-1, ...)</span>
-            <span class="c"># - Check the result for zero equivalence</span>
-            <span class="k">if</span> <span class="n">sol</span><span class="o">.</span><span class="n">lhs</span> <span class="o">==</span> <span class="n">func</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">sol</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
-                <span class="n">diffsols</span> <span class="o">=</span> <span class="p">{</span><span class="mi">0</span><span class="p">:</span> <span class="n">sol</span><span class="o">.</span><span class="n">rhs</span><span class="p">}</span>
-            <span class="k">elif</span> <span class="n">sol</span><span class="o">.</span><span class="n">rhs</span> <span class="o">==</span> <span class="n">func</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">sol</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
-                <span class="n">diffsols</span> <span class="o">=</span> <span class="p">{</span><span class="mi">0</span><span class="p">:</span> <span class="n">sol</span><span class="o">.</span><span class="n">lhs</span><span class="p">}</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">diffsols</span> <span class="o">=</span> <span class="p">{}</span>
-            <span class="n">sol</span> <span class="o">=</span> <span class="n">sol</span><span class="o">.</span><span class="n">lhs</span> <span class="o">-</span> <span class="n">sol</span><span class="o">.</span><span class="n">rhs</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">order</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="c"># Differentiation is a linear operator, so there should always</span>
-                <span class="c"># be 1 solution. Nonetheless, we test just to make sure.</span>
-                <span class="c"># We only need to solve once.  After that, we will automatically</span>
-                <span class="c"># have the solution to the differential in the order we want.</span>
-                <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">ds</span> <span class="o">=</span> <span class="n">sol</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-                    <span class="k">try</span><span class="p">:</span>
-                        <span class="n">sdf</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">ds</span><span class="p">,</span> <span class="n">func</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">i</span><span class="p">))</span>
-                        <span class="k">if</span> <span class="ow">not</span> <span class="n">sdf</span><span class="p">:</span>
-                            <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-                    <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                        <span class="n">testnum</span> <span class="o">+=</span> <span class="mi">1</span>
-                        <span class="k">break</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">diffsols</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">sdf</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="c"># This is what the solution says df/dx should be.</span>
-                    <span class="n">diffsols</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">diffsols</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-            <span class="c"># Make sure the above didn&#39;t fail.</span>
-            <span class="k">if</span> <span class="n">testnum</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># Substitute it into ode to check for self consistency.</span>
-                <span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span> <span class="o">=</span> <span class="n">ode</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">ode</span><span class="o">.</span><span class="n">rhs</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">order</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="mi">0</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">diffsols</span><span class="p">:</span>
-                        <span class="c"># We can only substitute f(x) if the solution was</span>
-                        <span class="c"># solved for f(x).</span>
-                        <span class="k">break</span>
-                    <span class="n">lhs</span> <span class="o">=</span> <span class="n">sub_func_doit</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">func</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">i</span><span class="p">),</span> <span class="n">diffsols</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-                    <span class="n">rhs</span> <span class="o">=</span> <span class="n">sub_func_doit</span><span class="p">(</span><span class="n">rhs</span><span class="p">,</span> <span class="n">func</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">i</span><span class="p">),</span> <span class="n">diffsols</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-                    <span class="n">ode_or_bool</span> <span class="o">=</span> <span class="n">Eq</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">)</span>
-                    <span class="n">ode_or_bool</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">ode_or_bool</span><span class="p">)</span>
-
-                    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">ode_or_bool</span><span class="p">,</span> <span class="nb">bool</span><span class="p">):</span>
-                        <span class="k">if</span> <span class="n">ode_or_bool</span><span class="p">:</span>
-                            <span class="n">lhs</span> <span class="o">=</span> <span class="n">rhs</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">lhs</span> <span class="o">=</span> <span class="n">ode_or_bool</span><span class="o">.</span><span class="n">lhs</span>
-                        <span class="n">rhs</span> <span class="o">=</span> <span class="n">ode_or_bool</span><span class="o">.</span><span class="n">rhs</span>
-                <span class="c"># No sense in overworking simplify--just prove the numerator goes to zero</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">trigsimp</span><span class="p">((</span><span class="n">lhs</span> <span class="o">-</span> <span class="n">rhs</span><span class="p">)</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()[</span><span class="mi">0</span><span class="p">]))</span>
-                <span class="n">testnum</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">break</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">s</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">True</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">s</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>  <span class="c"># The code above never was able to change s</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Unable to test if &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">sol</span><span class="p">)</span> <span class="o">+</span>
-            <span class="s">&quot; is a solution to &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">ode</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot;.&quot;</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">(</span><span class="bp">False</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="ode_sol_simplicity"><a class="viewcode-back" href="../../../modules/solvers/ode.html#sympy.solvers.ode.ode_sol_simplicity">[docs]</a><span class="k">def</span> <span class="nf">ode_sol_simplicity</span><span class="p">(</span><span class="n">sol</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">trysolving</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns an extended integer representing how simple a solution to an</span>
-<span class="sd">    ODE is.</span>
-
-<span class="sd">    The following things are considered, in order from most simple to</span>
-<span class="sd">    least:</span>
-<span class="sd">    - sol is solved for func.</span>
-<span class="sd">    - sol is not solved for func, but can be if passed to solve (e.g.,</span>
-<span class="sd">    a solution returned by dsolve(ode, func, simplify=False)</span>
-<span class="sd">    - If sol is not solved for func, then base the result on the length</span>
-<span class="sd">    of sol, as computed by len(str(sol)).</span>
-<span class="sd">    - If sol has any unevaluated Integrals, this will automatically be</span>
-<span class="sd">    considered less simple than any of the above.</span>
-
-<span class="sd">    This function returns an integer such that if solution A is simpler</span>
-<span class="sd">    than solution B by above metric, then ode_sol_simplicity(sola, func)</span>
-<span class="sd">    &lt; ode_sol_simplicity(solb, func).</span>
-
-<span class="sd">    Currently, the following are the numbers returned, but if the</span>
-<span class="sd">    heuristic is ever improved, this may change.  Only the ordering is</span>
-<span class="sd">    guaranteed.</span>
-
-<span class="sd">    sol solved for func                        -2</span>
-<span class="sd">    sol not solved for func but can be         -1</span>
-<span class="sd">    sol is not solved or solvable for func     len(str(sol))</span>
-<span class="sd">    sol contains an Integral                   oo</span>
-
-<span class="sd">    oo here means the SymPy infinity, which should compare greater than</span>
-<span class="sd">    any integer.</span>
-
-<span class="sd">    If you already know solve() cannot solve sol, you can use</span>
-<span class="sd">    trysolving=False to skip that step, which is the only potentially</span>
-<span class="sd">    slow step.  For example, dsolve with the simplify=False flag should</span>
-<span class="sd">    do this.</span>
-
-<span class="sd">    If sol is a list of solutions, if the worst solution in the list</span>
-<span class="sd">    returns oo it returns that, otherwise it returns len(str(sol)), that</span>
-<span class="sd">    is, the length of the string representation of the whole list.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    This function is designed to be passed to min as the key argument,</span>
-<span class="sd">    such as min(listofsolutions, key=lambda i: ode_sol_simplicity(i, f(x))).</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, Function, Eq, tan, cos, sqrt, Integral</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.solvers.ode import ode_sol_simplicity</span>
-<span class="sd">    &gt;&gt;&gt; x, C1, C2 = symbols(&#39;x, C1, C2&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-
-<span class="sd">    &gt;&gt;&gt; ode_sol_simplicity(Eq(f(x), C1*x**2), f(x))</span>
-<span class="sd">    -2</span>
-<span class="sd">    &gt;&gt;&gt; ode_sol_simplicity(Eq(x**2 + f(x), C1), f(x))</span>
-<span class="sd">    -1</span>
-<span class="sd">    &gt;&gt;&gt; ode_sol_simplicity(Eq(f(x), C1*Integral(2*x, x)), f(x))</span>
-<span class="sd">    oo</span>
-<span class="sd">    &gt;&gt;&gt; eq1 = Eq(f(x)/tan(f(x)/(2*x)), C1)</span>
-<span class="sd">    &gt;&gt;&gt; eq2 = Eq(f(x)/tan(f(x)/(2*x) + f(x)), C2)</span>
-<span class="sd">    &gt;&gt;&gt; [ode_sol_simplicity(eq, f(x)) for eq in [eq1, eq2]]</span>
-<span class="sd">    [26, 33]</span>
-<span class="sd">    &gt;&gt;&gt; min([eq1, eq2], key=lambda i: ode_sol_simplicity(i, f(x)))</span>
-<span class="sd">    f(x)/tan(f(x)/(2*x)) == C1</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># TODO: if two solutions are solved for f(x), we still want to be</span>
-    <span class="c"># able to get the simpler of the two</span>
-
-    <span class="c"># See the docstring for the coercion rules.  We check easier (faster)</span>
-    <span class="c"># things here first, to save time.</span>
-
-    <span class="k">if</span> <span class="n">iterable</span><span class="p">(</span><span class="n">sol</span><span class="p">):</span>
-        <span class="c"># See if there are Integrals</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">sol</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">ode_sol_simplicity</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">trysolving</span><span class="o">=</span><span class="n">trysolving</span><span class="p">)</span> <span class="o">==</span> <span class="n">oo</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">oo</span>
-
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">sol</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="n">sol</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Integral</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">oo</span>
-
-    <span class="c"># Next, try to solve for func.  This code will change slightly when RootOf</span>
-    <span class="c"># is implemented in solve().  Probably a RootOf solution should fall somewhere</span>
-    <span class="c"># between a normal solution and an unsolvable expression.</span>
-
-    <span class="c"># First, see if they are already solved</span>
-    <span class="k">if</span> <span class="n">sol</span><span class="o">.</span><span class="n">lhs</span> <span class="o">==</span> <span class="n">func</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">sol</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">func</span><span class="p">)</span> <span class="ow">or</span> \
-            <span class="n">sol</span><span class="o">.</span><span class="n">rhs</span> <span class="o">==</span> <span class="n">func</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">sol</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
-        <span class="k">return</span> <span class="o">-</span><span class="mi">2</span>
-    <span class="c"># We are not so lucky, try solving manually</span>
-    <span class="k">if</span> <span class="n">trysolving</span><span class="p">:</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">sols</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">sol</span><span class="p">,</span> <span class="n">func</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">sols</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-        <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-            <span class="k">pass</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-
-    <span class="c"># Finally, a naive computation based on the length of the string version</span>
-    <span class="c"># of the expression.  This may favor combined fractions because they</span>
-    <span class="c"># will not have duplicate denominators, and may slightly favor expressions</span>
-    <span class="c"># with fewer additions and subtractions, as those are separated by spaces</span>
-    <span class="c"># by the printer.</span>
-
-    <span class="c"># Additional ideas for simplicity heuristics are welcome, like maybe</span>
-    <span class="c"># checking if a equation has a larger domain, or if constantsimp has</span>
-    <span class="c"># introduced arbitrary constants numbered higher than the order of a</span>
-    <span class="c"># given ode that sol is a solution of.</span>
-    <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">sol</span><span class="p">))</span>
-
-</div>
-<span class="nd">@vectorize</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<div class="viewcode-block" id="constantsimp"><a class="viewcode-back" href="../../../modules/solvers/ode.html#sympy.solvers.ode.constantsimp">[docs]</a><span class="k">def</span> <span class="nf">constantsimp</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">independentsymbol</span><span class="p">,</span> <span class="n">endnumber</span><span class="p">,</span> <span class="n">startnumber</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
-                 <span class="n">symbolname</span><span class="o">=</span><span class="s">&#39;C&#39;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Simplifies an expression with arbitrary constants in it.</span>
-
-<span class="sd">    This function is written specifically to work with dsolve(), and is</span>
-<span class="sd">    not intended for general use.</span>
-
-<span class="sd">    Simplification is done by &quot;absorbing&quot; the arbitrary constants in to</span>
-<span class="sd">    other arbitrary constants, numbers, and symbols that they are not</span>
-<span class="sd">    independent of.</span>
-
-<span class="sd">    The symbols must all have the same name with numbers after it, for</span>
-<span class="sd">    example, C1, C2, C3.  The symbolname here would be &#39;C&#39;, the</span>
-<span class="sd">    startnumber would be 1, and the end number would be 3.  If the</span>
-<span class="sd">    arbitrary constants are independent of the variable x, then the</span>
-<span class="sd">    independent symbol would be x.  There is no need to specify the</span>
-<span class="sd">    dependent function, such as f(x), because it already has the</span>
-<span class="sd">    independent symbol, x, in it.</span>
-
-<span class="sd">    Because terms are &quot;absorbed&quot; into arbitrary constants and because</span>
-<span class="sd">    constants are renumbered after simplifying, the arbitrary constants</span>
-<span class="sd">    in expr are not necessarily equal to the ones of the same name in</span>
-<span class="sd">    the returned result.</span>
-
-<span class="sd">    If two or more arbitrary constants are added, multiplied, or raised</span>
-<span class="sd">    to the power of each other, they are first absorbed together into a</span>
-<span class="sd">    single arbitrary constant.  Then the new constant is combined into</span>
-<span class="sd">    other terms if necessary.</span>
-
-<span class="sd">    Absorption is done with limited assistance: terms of Adds are collected</span>
-<span class="sd">    to try join constants and powers with exponents that are Adds are expanded</span>
-<span class="sd">    so (C1*cos(x) + C2*cos(x))*exp(x) will simplify to C1*cos(x)*exp(x) and</span>
-<span class="sd">    exp(C1 + x) will be simplified to C1*exp(x).</span>
-
-<span class="sd">    Use constant_renumber() to renumber constants after simplification or else</span>
-<span class="sd">    arbitrary numbers on constants may appear, e.g. C1 + C3*x.</span>
-
-<span class="sd">    In rare cases, a single constant can be &quot;simplified&quot; into two</span>
-<span class="sd">    constants.  Every differential equation solution should have as many</span>
-<span class="sd">    arbitrary constants as the order of the differential equation.  The</span>
-<span class="sd">    result here will be technically correct, but it may, for example,</span>
-<span class="sd">    have C1 and C2 in an expression, when C1 is actually equal to C2.</span>
-<span class="sd">    Use your discretion in such situations, and also take advantage of</span>
-<span class="sd">    the ability to use hints in dsolve().</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.solvers.ode import constantsimp</span>
-<span class="sd">    &gt;&gt;&gt; C1, C2, C3, x, y = symbols(&#39;C1,C2,C3,x,y&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; constantsimp(2*C1*x, x, 3)</span>
-<span class="sd">    C1*x</span>
-<span class="sd">    &gt;&gt;&gt; constantsimp(C1 + 2 + x + y, x, 3)</span>
-<span class="sd">    C1 + x</span>
-<span class="sd">    &gt;&gt;&gt; constantsimp(C1*C2 + 2 + x + y + C3*x, x, 3)</span>
-<span class="sd">    C1 + C3*x</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># This function works recursively.  The idea is that, for Mul,</span>
-    <span class="c"># Add, Pow, and Function, if the class has a constant in it, then</span>
-    <span class="c"># we can simplify it, which we do by recursing down and</span>
-    <span class="c"># simplifying up.  Otherwise, we can skip that part of the</span>
-    <span class="c"># expression.</span>
-
-    <span class="n">constant_iter</span> <span class="o">=</span> <span class="n">numbered_symbols</span><span class="p">(</span><span class="s">&#39;C&#39;</span><span class="p">,</span> <span class="n">start</span><span class="o">=</span><span class="n">startnumber</span><span class="p">)</span>
-    <span class="n">constantsymbols</span> <span class="o">=</span> <span class="p">[</span><span class="nb">next</span><span class="p">(</span><span class="n">constant_iter</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">startnumber</span><span class="p">,</span>
-                                          <span class="n">endnumber</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
-    <span class="n">constantsymbols_set</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">constantsymbols</span><span class="p">)</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">independentsymbol</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Equality</span><span class="p">):</span>
-        <span class="c"># For now, only treat the special case where one side of the equation</span>
-        <span class="c"># is a constant</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">lhs</span> <span class="ow">in</span> <span class="n">constantsymbols_set</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Eq</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">constantsimp</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">rhs</span> <span class="o">+</span> <span class="n">expr</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">endnumber</span><span class="p">,</span>
-            <span class="n">startnumber</span><span class="p">,</span> <span class="n">symbolname</span><span class="p">)</span> <span class="o">-</span> <span class="n">expr</span><span class="o">.</span><span class="n">lhs</span><span class="p">)</span>
-            <span class="c"># this could break if expr.lhs is absorbed into another constant,</span>
-            <span class="c"># but for now, the only solutions that return Eq&#39;s with a constant</span>
-            <span class="c"># on one side are first order.  At any rate, it will still be</span>
-            <span class="c"># technically correct.  The expression will just have too many</span>
-            <span class="c"># constants in it</span>
-        <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">rhs</span> <span class="ow">in</span> <span class="n">constantsymbols_set</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Eq</span><span class="p">(</span><span class="n">constantsimp</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">lhs</span> <span class="o">+</span> <span class="n">expr</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">endnumber</span><span class="p">,</span>
-            <span class="n">startnumber</span><span class="p">,</span> <span class="n">symbolname</span><span class="p">)</span> <span class="o">-</span> <span class="n">expr</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Eq</span><span class="p">(</span><span class="n">constantsimp</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">endnumber</span><span class="p">,</span> <span class="n">startnumber</span><span class="p">,</span>
-                <span class="n">symbolname</span><span class="p">),</span> <span class="n">constantsimp</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">endnumber</span><span class="p">,</span>
-                <span class="n">startnumber</span><span class="p">,</span> <span class="n">symbolname</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">constantsymbols</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="c"># ================ pre-processing ================</span>
-        <span class="c"># collect terms to get constants together</span>
-        <span class="k">def</span> <span class="nf">_take</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
-            <span class="n">t</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">([</span><span class="n">s</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">i</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Symbol</span><span class="p">)</span> <span class="k">if</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">constantsymbols</span><span class="p">])</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">t</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">i</span>
-            <span class="k">return</span> <span class="n">t</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="ow">and</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">constantsymbols</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">new_expr</span> <span class="o">=</span> <span class="n">terms_gcd</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">clear</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">new_expr</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="c"># don&#39;t let C1*exp(x) + C2*exp(2*x) become exp(x)*(C1 + C2*exp(x))</span>
-            <span class="n">infac</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="n">asfac</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">new_expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">m</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">exp</span><span class="p">:</span>
-                    <span class="n">asfac</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">elif</span> <span class="n">m</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                    <span class="n">infac</span> <span class="o">=</span> <span class="nb">any</span><span class="p">(</span><span class="n">fi</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">exp</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">m</span><span class="o">.</span><span class="n">args</span> <span class="k">for</span> <span class="n">fi</span> <span class="ow">in</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">t</span><span class="p">))</span>
-                <span class="k">if</span> <span class="n">asfac</span> <span class="ow">and</span> <span class="n">infac</span><span class="p">:</span>
-                    <span class="n">new_expr</span> <span class="o">=</span> <span class="n">expr</span>
-                    <span class="k">break</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">new_expr</span>
-        <span class="c"># don&#39;t allow a number to be factored out of an expression</span>
-        <span class="c"># that has no denominator</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="n">h</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">h</span> <span class="o">!=</span> <span class="mi">1</span> <span class="ow">and</span> <span class="p">(</span><span class="n">t</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">or</span> <span class="n">denom</span><span class="p">(</span><span class="n">t</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">t</span><span class="p">))</span>
-                <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">a</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                        <span class="n">args</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">h</span><span class="o">*</span><span class="n">a</span>
-                        <span class="n">expr</span> <span class="o">=</span> <span class="n">Mul</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-                        <span class="k">break</span>
-            <span class="c"># let numbers absorb into constants of an Add, perhaps</span>
-            <span class="c"># in the base of a power, if all its terms have a constant</span>
-            <span class="c"># symbol in them, e.g. sqrt(2)*(C1 + C2*x) -&gt; C1 + C2*x</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                <span class="n">d</span> <span class="o">=</span> <span class="n">sift</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">m</span><span class="p">:</span> <span class="n">m</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">)</span>
-                <span class="n">num</span> <span class="o">=</span> <span class="n">d</span><span class="p">[</span><span class="bp">True</span><span class="p">]</span>
-                <span class="n">other</span> <span class="o">=</span> <span class="n">d</span><span class="p">[</span><span class="bp">False</span><span class="p">]</span>
-                <span class="n">con_set</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">constantsymbols</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">num</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">other</span><span class="p">:</span>
-                        <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">o</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                        <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">and</span> \
-                            <span class="nb">all</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">(</span><span class="n">cset</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">warn</span><span class="o">=</span><span class="bp">False</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&amp;</span>
-                        <span class="n">con_set</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">b</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-                            <span class="n">expr</span> <span class="o">=</span> <span class="n">sign</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">num</span><span class="p">))</span><span class="o">*</span><span class="n">Mul</span><span class="o">.</span><span class="n">_from_args</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-                            <span class="k">break</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>  <span class="c"># check again that it&#39;s still a Mul</span>
-            <span class="n">i</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="n">newi</span> <span class="o">=</span> <span class="n">_take</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">newi</span> <span class="o">!=</span> <span class="n">i</span><span class="p">:</span>
-                <span class="n">expr</span> <span class="o">=</span> <span class="n">newi</span><span class="o">*</span><span class="n">d</span>
-        <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">i</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">_take</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="n">d</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="n">terms</span> <span class="o">=</span> <span class="p">{}</span>
-                <span class="k">for</span> <span class="n">ai</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                    <span class="n">i</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">ai</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">as_Add</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-                    <span class="n">terms</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="p">[])</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                <span class="n">expr</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">k</span><span class="o">*</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">v</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">terms</span><span class="o">.</span><span class="n">items</span><span class="p">())])</span>
-        <span class="c"># handle powers like exp(C0 + g(x)) -&gt; C0*exp(g(x))</span>
-        <span class="n">pows</span> <span class="o">=</span> <span class="p">[</span><span class="n">p</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Function</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Pow</span><span class="p">)</span> <span class="k">if</span>
-                <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">or</span> <span class="n">p</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">exp</span><span class="p">)</span> <span class="ow">and</span>
-                <span class="n">p</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">and</span>
-                <span class="n">p</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="bp">True</span><span class="p">)[</span><span class="mi">1</span><span class="p">]]</span>
-        <span class="k">if</span> <span class="n">pows</span><span class="p">:</span>
-            <span class="n">reps</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">pows</span><span class="p">:</span>
-                <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                <span class="n">ei</span><span class="p">,</span> <span class="n">ed</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                <span class="n">e</span> <span class="o">=</span> <span class="n">_take</span><span class="p">(</span><span class="n">ei</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">e</span> <span class="o">!=</span> <span class="n">ei</span> <span class="ow">or</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">constantsymbols</span><span class="p">:</span>
-                    <span class="n">reps</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">p</span><span class="p">,</span> <span class="n">e</span><span class="o">*</span><span class="n">b</span><span class="o">**</span><span class="n">ed</span><span class="p">))</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">reps</span><span class="p">)</span>
-            <span class="c"># a C1*C2 may have been introduced and the code below won&#39;t</span>
-            <span class="c"># handle that so handle it now: once to handle the C1*C2</span>
-            <span class="c"># and once to handle any C0*f(x) + C0*f(x)</span>
-            <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">):</span>
-                <span class="n">muls</span> <span class="o">=</span> <span class="p">[</span><span class="n">m</span> <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Mul</span><span class="p">)</span> <span class="k">if</span> <span class="n">m</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">constantsymbols</span><span class="p">)]</span>
-                <span class="n">reps</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">muls</span><span class="p">:</span>
-                    <span class="n">i</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                    <span class="n">newi</span> <span class="o">=</span> <span class="n">_take</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">newi</span> <span class="o">!=</span> <span class="n">i</span><span class="p">:</span>
-                        <span class="n">reps</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">m</span><span class="p">,</span> <span class="n">_take</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">*</span><span class="n">d</span><span class="p">))</span>
-                <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">reps</span><span class="p">)</span>
-        <span class="c"># ================ end of pre-processing ================</span>
-        <span class="n">newargs</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">hasconst</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="n">isPowExp</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="n">reeval</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">constantsymbols</span><span class="p">:</span>
-                <span class="n">newargs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">newconst</span> <span class="o">=</span> <span class="n">i</span>
-                <span class="n">hasconst</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">i</span> <span class="o">==</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="p">:</span>
-                    <span class="n">isPowExp</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">newargs</span><span class="p">)):</span>
-            <span class="n">isimp</span> <span class="o">=</span> <span class="n">constantsimp</span><span class="p">(</span><span class="n">newargs</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">x</span><span class="p">,</span> <span class="n">endnumber</span><span class="p">,</span> <span class="n">startnumber</span><span class="p">,</span>
-            <span class="n">symbolname</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">isimp</span> <span class="ow">in</span> <span class="n">constantsymbols</span><span class="p">:</span>
-                <span class="n">reeval</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="n">hasconst</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="n">newconst</span> <span class="o">=</span> <span class="n">isimp</span>
-                <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="n">isPowExp</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="n">newargs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">isimp</span>
-        <span class="k">if</span> <span class="n">hasconst</span><span class="p">:</span>
-            <span class="n">newargs</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">newargs</span> <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">)]</span>
-            <span class="k">if</span> <span class="n">isPowExp</span><span class="p">:</span>
-                <span class="n">newargs</span> <span class="o">=</span> <span class="n">newargs</span> <span class="o">+</span> <span class="p">[</span><span class="n">newconst</span><span class="p">]</span>  <span class="c"># Order matters in this case</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">newargs</span> <span class="o">=</span> <span class="p">[</span><span class="n">newconst</span><span class="p">]</span> <span class="o">+</span> <span class="n">newargs</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">newargs</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">newargs</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Function</span><span class="p">:</span>
-            <span class="k">if</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">newargs</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">hasconst</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">newargs</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">newconst</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">newfuncargs</span> <span class="o">=</span> <span class="p">[</span><span class="n">constantsimp</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">endnumber</span><span class="p">,</span> <span class="n">startnumber</span><span class="p">,</span>
-                <span class="n">symbolname</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-                <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">newfuncargs</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">newexpr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">newargs</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">reeval</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">constantsimp</span><span class="p">(</span><span class="n">newexpr</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">endnumber</span><span class="p">,</span> <span class="n">startnumber</span><span class="p">,</span>
-                <span class="n">symbolname</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">newexpr</span>
-
-</div>
-<div class="viewcode-block" id="constant_renumber"><a class="viewcode-back" href="../../../modules/solvers/ode.html#sympy.solvers.ode.constant_renumber">[docs]</a><span class="k">def</span> <span class="nf">constant_renumber</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">symbolname</span><span class="p">,</span> <span class="n">startnumber</span><span class="p">,</span> <span class="n">endnumber</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Renumber arbitrary constants in expr to have numbers 1 through N</span>
-<span class="sd">    where N is ``endnumber`` - ``startnumber`` + 1 at most.</span>
-
-<span class="sd">    This is a simple function that goes through and renumbers any Symbol</span>
-<span class="sd">    with a name in the form symbolname + num where num is in the range</span>
-<span class="sd">    from startnumber to endnumber.</span>
-
-<span class="sd">    Symbols are renumbered based on ``.sort_key()``, so they should be</span>
-<span class="sd">    numbered roughly in the order that they appear in the final, printed</span>
-<span class="sd">    expression.  Note that this ordering is based in part on hashes, so</span>
-<span class="sd">    it can produce different results on different machines.</span>
-
-<span class="sd">    The structure of this function is very similar to that of</span>
-<span class="sd">    constantsimp().</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, Eq, pprint</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.solvers.ode import constant_renumber</span>
-<span class="sd">    &gt;&gt;&gt; x, C0, C1, C2, C3, C4 = symbols(&#39;x,C:5&#39;)</span>
-
-<span class="sd">    Only constants in the given range (inclusive) are renumbered;</span>
-<span class="sd">    the renumbering always starts from 1:</span>
-
-<span class="sd">    &gt;&gt;&gt; constant_renumber(C1 + C3 + C4, &#39;C&#39;, 1, 3)</span>
-<span class="sd">    C1 + C2 + C4</span>
-<span class="sd">    &gt;&gt;&gt; constant_renumber(C0 + C1 + C3 + C4, &#39;C&#39;, 2, 4)</span>
-<span class="sd">    C0 + 2*C1 + C2</span>
-<span class="sd">    &gt;&gt;&gt; constant_renumber(C0 + 2*C1 + C2, &#39;C&#39;, 0, 1)</span>
-<span class="sd">    C1 + 3*C2</span>
-<span class="sd">    &gt;&gt;&gt; pprint(C2 + C1*x + C3*x**2)</span>
-<span class="sd">                    2</span>
-<span class="sd">    C1*x + C2 + C3*x</span>
-<span class="sd">    &gt;&gt;&gt; pprint(constant_renumber(C2 + C1*x + C3*x**2, &#39;C&#39;, 1, 3))</span>
-<span class="sd">                    2</span>
-<span class="sd">    C1 + C2*x + C3*x</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span> <span class="ow">in</span> <span class="p">(</span><span class="nb">set</span><span class="p">,</span> <span class="nb">list</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">type</span><span class="p">(</span>
-            <span class="n">expr</span><span class="p">)([</span><span class="n">constant_renumber</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">symbolname</span><span class="o">=</span><span class="n">symbolname</span><span class="p">,</span>
-            <span class="n">startnumber</span><span class="o">=</span><span class="n">startnumber</span><span class="p">,</span> <span class="n">endnumber</span><span class="o">=</span><span class="n">endnumber</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span><span class="p">])</span>
-    <span class="k">global</span> <span class="n">newstartnumber</span>
-    <span class="n">newstartnumber</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">_constant_renumber</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">symbolname</span><span class="p">,</span> <span class="n">startnumber</span><span class="p">,</span> <span class="n">endnumber</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        We need to have an internal recursive function so that</span>
-<span class="sd">        newstartnumber maintains its values throughout recursive calls.</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">constantsymbols</span> <span class="o">=</span> <span class="p">[</span><span class="n">Symbol</span><span class="p">(</span>
-            <span class="n">symbolname</span> <span class="o">+</span> <span class="s">&quot;</span><span class="si">%d</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">t</span><span class="p">)</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">startnumber</span><span class="p">,</span>
-        <span class="n">endnumber</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
-        <span class="k">global</span> <span class="n">newstartnumber</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Equality</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">Eq</span><span class="p">(</span>
-                <span class="n">_constant_renumber</span><span class="p">(</span>
-                    <span class="n">expr</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="n">symbolname</span><span class="p">,</span> <span class="n">startnumber</span><span class="p">,</span> <span class="n">endnumber</span><span class="p">),</span>
-                <span class="n">_constant_renumber</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="n">symbolname</span><span class="p">,</span> <span class="n">startnumber</span><span class="p">,</span> <span class="n">endnumber</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">(</span><span class="n">Mul</span><span class="p">,</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Pow</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Function</span> <span class="ow">and</span> \
-                <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">constantsymbols</span><span class="p">):</span>
-            <span class="c"># Base case, as above.  We better hope there aren&#39;t constants inside</span>
-            <span class="c"># of some other class, because they won&#39;t be renumbered.</span>
-            <span class="k">return</span> <span class="n">expr</span>
-        <span class="k">elif</span> <span class="n">expr</span> <span class="ow">in</span> <span class="n">constantsymbols</span><span class="p">:</span>
-            <span class="c"># Renumbering happens here</span>
-            <span class="n">newconst</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">symbolname</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">newstartnumber</span><span class="p">))</span>
-            <span class="n">newstartnumber</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="k">return</span> <span class="n">newconst</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Function</span> <span class="ow">or</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">(</span>
-                    <span class="o">*</span><span class="p">[</span><span class="n">_constant_renumber</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">symbolname</span><span class="p">,</span> <span class="n">startnumber</span><span class="p">,</span>
-                <span class="n">endnumber</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">sortedargs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-                <span class="c"># make a mapping to send all constantsymbols to S.One and use</span>
-                <span class="c"># that to make sure that term ordering is not dependent on</span>
-                <span class="c"># the indexed value of C</span>
-                <span class="n">C_1</span> <span class="o">=</span> <span class="p">[(</span><span class="n">ci</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span> <span class="k">for</span> <span class="n">ci</span> <span class="ow">in</span> <span class="n">constantsymbols</span><span class="p">]</span>
-                <span class="n">sortedargs</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span>
-                    <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">arg</span><span class="p">:</span> <span class="n">default_sort_key</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">C_1</span><span class="p">)))</span>
-                <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="p">(</span>
-                    <span class="o">*</span><span class="p">[</span><span class="n">_constant_renumber</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">symbolname</span><span class="p">,</span> <span class="n">startnumber</span><span class="p">,</span>
-                    <span class="n">endnumber</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">sortedargs</span><span class="p">])</span>
-
-    <span class="k">return</span> <span class="n">_constant_renumber</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">symbolname</span><span class="p">,</span> <span class="n">startnumber</span><span class="p">,</span> <span class="n">endnumber</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_handle_Integral</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">hint</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Converts a solution with Integrals in it into an actual solution.</span>
-
-<span class="sd">    For most hints, this simply runs expr.doit()</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">func</span>
-    <span class="k">if</span> <span class="n">hint</span> <span class="o">==</span> <span class="s">&quot;1st_exact&quot;</span><span class="p">:</span>
-        <span class="k">global</span> <span class="n">exactvars</span>
-        <span class="n">x0</span> <span class="o">=</span> <span class="n">exactvars</span><span class="p">[</span><span class="s">&#39;x0&#39;</span><span class="p">]</span>
-        <span class="n">y0</span> <span class="o">=</span> <span class="n">exactvars</span><span class="p">[</span><span class="s">&#39;y0&#39;</span><span class="p">]</span>
-        <span class="n">y</span> <span class="o">=</span> <span class="n">exactvars</span><span class="p">[</span><span class="s">&#39;y&#39;</span><span class="p">]</span>
-        <span class="n">tmpsol</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-        <span class="n">sol</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">assert</span> <span class="n">tmpsol</span><span class="o">.</span><span class="n">is_Add</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">tmpsol</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">i</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">i</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">y0</span><span class="p">):</span>
-                <span class="n">sol</span> <span class="o">+=</span> <span class="n">i</span>
-        <span class="k">assert</span> <span class="n">sol</span> <span class="o">!=</span> <span class="mi">0</span>
-        <span class="n">sol</span> <span class="o">=</span> <span class="n">Eq</span><span class="p">(</span><span class="n">sol</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="n">expr</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>  <span class="c"># expr.rhs == C1</span>
-        <span class="k">del</span> <span class="n">exactvars</span>
-    <span class="k">elif</span> <span class="n">hint</span> <span class="o">==</span> <span class="s">&quot;1st_exact_Integral&quot;</span><span class="p">:</span>
-        <span class="c"># FIXME: We still need to back substitute y</span>
-        <span class="c"># y = exactvars[&#39;y&#39;]</span>
-        <span class="c"># sol = expr.subs(y, f(x))</span>
-        <span class="c"># For now, we are going to have to return an expression with f(x) replaced</span>
-        <span class="c"># with y.  Substituting results in the y&#39;s in the second integral</span>
-        <span class="c"># becoming f(x), which prevents the integral from being evaluatable.</span>
-        <span class="c"># For example, Integral(cos(f(x)), (x, x0, x)).  If there were a way to</span>
-        <span class="c"># do inert substitution, that could maybe be used here instead.</span>
-        <span class="k">del</span> <span class="n">exactvars</span>
-        <span class="n">sol</span> <span class="o">=</span> <span class="n">expr</span>
-    <span class="k">elif</span> <span class="n">hint</span> <span class="o">==</span> <span class="s">&quot;nth_linear_constant_coeff_homogeneous&quot;</span><span class="p">:</span>
-        <span class="n">sol</span> <span class="o">=</span> <span class="n">expr</span>
-    <span class="k">elif</span> <span class="ow">not</span> <span class="n">hint</span><span class="o">.</span><span class="n">endswith</span><span class="p">(</span><span class="s">&quot;_Integral&quot;</span><span class="p">):</span>
-        <span class="n">sol</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">sol</span> <span class="o">=</span> <span class="n">expr</span>
-    <span class="k">return</span> <span class="n">sol</span>
-
-
-<div class="viewcode-block" id="ode_order"><a class="viewcode-back" href="../../../modules/solvers/ode.html#sympy.solvers.ode.ode_order">[docs]</a><span class="k">def</span> <span class="nf">ode_order</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">func</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the order of a given ODE with respect to func.</span>
-
-<span class="sd">    This function is implemented recursively.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, ode_order</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; f, g = list(map(Function, [&#39;f&#39;, &#39;g&#39;]))</span>
-<span class="sd">    &gt;&gt;&gt; ode_order(f(x).diff(x, 2) + f(x).diff(x)**2 +</span>
-<span class="sd">    ... f(x).diff(x), f(x))</span>
-<span class="sd">    2</span>
-<span class="sd">    &gt;&gt;&gt; ode_order(f(x).diff(x, 2) + g(x).diff(x, 3), f(x))</span>
-<span class="sd">    2</span>
-<span class="sd">    &gt;&gt;&gt; ode_order(f(x).diff(x, 2) + g(x).diff(x, 3), g(x))</span>
-<span class="sd">    3</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">func</span><span class="p">])</span>
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">a</span><span class="p">):</span>
-        <span class="k">return</span> <span class="mi">0</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">func</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">order</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="n">order</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">order</span><span class="p">,</span> <span class="n">ode_order</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">func</span><span class="p">)</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">variables</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">order</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">order</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">order</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">order</span><span class="p">,</span> <span class="n">ode_order</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">func</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">order</span>
-
-
-<span class="c"># FIXME: replace the general solution in the docstring with</span>
-<span class="c"># dsolve(equation, hint=&#39;1st_exact_Integral&#39;).  You will need to be able</span>
-<span class="c"># to have assumptions on P and Q that dP/dy = dQ/dx.</span></div>
-<div class="viewcode-block" id="ode_1st_exact"><a class="viewcode-back" href="../../../modules/solvers/ode.html#sympy.solvers.ode.ode_1st_exact">[docs]</a><span class="k">def</span> <span class="nf">ode_1st_exact</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">match</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Solves 1st order exact ordinary differential equations.</span>
-
-<span class="sd">    A 1st order differential equation is called exact if it is the total</span>
-<span class="sd">    differential of a function. That is, the differential equation</span>
-<span class="sd">    P(x, y)dx + Q(x, y)dy = 0 is exact if there is some function F(x, y)</span>
-<span class="sd">    such that P(x, y) = dF/dx and Q(x, y) = dF/dy (d here refers to the</span>
-<span class="sd">    partial derivative).  It can be shown that a necessary and</span>
-<span class="sd">    sufficient condition for a first order ODE to be exact is that</span>
-<span class="sd">    dP/dy = dQ/dx.  Then, the solution will be as given below::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Function, Eq, Integral, symbols, pprint</span>
-<span class="sd">        &gt;&gt;&gt; x, y, t, x0, y0, C1= symbols(&#39;x,y,t,x0,y0,C1&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; P, Q, F= list(map(Function, [&#39;P&#39;, &#39;Q&#39;, &#39;F&#39;]))</span>
-<span class="sd">        &gt;&gt;&gt; pprint(Eq(Eq(F(x, y), Integral(P(t, y), (t, x0, x)) +</span>
-<span class="sd">        ... Integral(Q(x0, t), (t, y0, y))), C1))</span>
-<span class="sd">                    x                y</span>
-<span class="sd">                    /                /</span>
-<span class="sd">                   |                |</span>
-<span class="sd">        F(x, y) =  |  P(t, y) dt +  |  Q(x0, t) dt = C1</span>
-<span class="sd">                   |                |</span>
-<span class="sd">                  /                /</span>
-<span class="sd">                  x0               y0</span>
-
-<span class="sd">    Where the first partials of P and Q exist and are continuous in a</span>
-<span class="sd">    simply connected region.</span>
-
-<span class="sd">    A note: SymPy currently has no way to represent inert substitution on</span>
-<span class="sd">    an expression, so the hint &#39;1st_exact_Integral&#39; will return an integral</span>
-<span class="sd">    with dy.  This is supposed to represent the function that you are</span>
-<span class="sd">    solving for.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, dsolve, cos, sin</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; dsolve(cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x),</span>
-<span class="sd">    ... f(x), hint=&#39;1st_exact&#39;)</span>
-<span class="sd">    x*cos(f(x)) + f(x)**3/3 == C1</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - http://en.wikipedia.org/wiki/Exact_differential_equation</span>
-<span class="sd">    - M. Tenenbaum &amp; H. Pollard, &quot;Ordinary Differential Equations&quot;,</span>
-<span class="sd">      Dover 1963, pp. 73</span>
-
-<span class="sd">    # indirect doctest</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">func</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">match</span>  <span class="c"># d+e*diff(f(x),x)</span>
-    <span class="n">C1</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;C1&#39;</span><span class="p">)</span>
-    <span class="n">x0</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x0&#39;</span><span class="p">)</span>
-    <span class="n">y0</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;y0&#39;</span><span class="p">)</span>
-    <span class="k">global</span> <span class="n">exactvars</span>  <span class="c"># This is the only way to pass these dummy variables to</span>
-    <span class="c"># _handle_Integral</span>
-    <span class="n">exactvars</span> <span class="o">=</span> <span class="p">{</span><span class="s">&#39;y0&#39;</span><span class="p">:</span> <span class="n">y0</span><span class="p">,</span> <span class="s">&#39;x0&#39;</span><span class="p">:</span> <span class="n">x0</span><span class="p">,</span> <span class="s">&#39;y&#39;</span><span class="p">:</span> <span class="n">r</span><span class="p">[</span><span class="s">&#39;y&#39;</span><span class="p">]}</span>
-    <span class="c"># If we ever get a Constant class, x0 and y0 should be constants, I think</span>
-    <span class="n">sol</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Integral</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;e&#39;</span><span class="p">]]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">x0</span><span class="p">),</span> <span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;y&#39;</span><span class="p">],</span> <span class="n">y0</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span> <span class="o">+</span> <span class="n">C</span><span class="o">.</span><span class="n">Integral</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;d&#39;</span><span class="p">]],</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">Eq</span><span class="p">(</span><span class="n">sol</span><span class="p">,</span> <span class="n">C1</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="ode_1st_homogeneous_coeff_best"><a class="viewcode-back" href="../../../modules/solvers/ode.html#sympy.solvers.ode.ode_1st_homogeneous_coeff_best">[docs]</a><span class="k">def</span> <span class="nf">ode_1st_homogeneous_coeff_best</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">match</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Returns the best solution to an ODE from the two hints</span>
-<span class="sd">    &#39;1st_homogeneous_coeff_subs_dep_div_indep&#39; and</span>
-<span class="sd">    &#39;1st_homogeneous_coeff_subs_indep_div_dep&#39;.</span>
-
-<span class="sd">    This is as determined by ode_sol_simplicity().</span>
-
-<span class="sd">    See the ode_1st_homogeneous_coeff_subs_indep_div_dep() and</span>
-<span class="sd">    ode_1st_homogeneous_coeff_subs_dep_div_indep() docstrings for more</span>
-<span class="sd">    information on these hints.  Note that there is no</span>
-<span class="sd">    &#39;1st_homogeneous_coeff_best_Integral&#39; hint.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, dsolve, pprint</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(dsolve(2*x*f(x) + (x**2 + f(x)**2)*f(x).diff(x), f(x),</span>
-<span class="sd">    ... hint=&#39;1st_homogeneous_coeff_best&#39;, simplify=False))</span>
-<span class="sd">                   /    2    \</span>
-<span class="sd">                   | 3*x     |</span>
-<span class="sd">                log|----- + 1|</span>
-<span class="sd">                   | 2       |</span>
-<span class="sd">       /f(x)\      \f (x)    /</span>
-<span class="sd">    log|----| + -------------- = 0</span>
-<span class="sd">       \ C1 /         3</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - http://en.wikipedia.org/wiki/Homogeneous_differential_equation</span>
-<span class="sd">    - M. Tenenbaum &amp; H. Pollard, &quot;Ordinary Differential Equations&quot;,</span>
-<span class="sd">      Dover 1963, pp. 59</span>
-
-<span class="sd">    # indirect doctest</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># There are two substitutions that solve the equation, u1=y/x and u2=x/y</span>
-    <span class="c"># They produce different integrals, so try them both and see which</span>
-    <span class="c"># one is easier.</span>
-    <span class="n">sol1</span> <span class="o">=</span> <span class="n">ode_1st_homogeneous_coeff_subs_indep_div_dep</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span>
-    <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">match</span><span class="p">)</span>
-    <span class="n">sol2</span> <span class="o">=</span> <span class="n">ode_1st_homogeneous_coeff_subs_dep_div_indep</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span>
-    <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">match</span><span class="p">)</span>
-    <span class="n">simplify</span> <span class="o">=</span> <span class="n">match</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;simplify&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">simplify</span><span class="p">:</span>
-        <span class="n">sol1</span> <span class="o">=</span> <span class="n">odesimp</span><span class="p">(</span>
-            <span class="n">sol1</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="s">&quot;1st_homogeneous_coeff_subs_indep_div_dep&quot;</span><span class="p">)</span>
-        <span class="n">sol2</span> <span class="o">=</span> <span class="n">odesimp</span><span class="p">(</span>
-            <span class="n">sol2</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="s">&quot;1st_homogeneous_coeff_subs_dep_div_indep&quot;</span><span class="p">)</span>
-    <span class="k">return</span> <span class="nb">min</span><span class="p">([</span><span class="n">sol1</span><span class="p">,</span> <span class="n">sol2</span><span class="p">],</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">ode_sol_simplicity</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span>
-        <span class="n">trysolving</span><span class="o">=</span><span class="ow">not</span> <span class="n">simplify</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="ode_1st_homogeneous_coeff_subs_dep_div_indep"><a class="viewcode-back" href="../../../modules/solvers/ode.html#sympy.solvers.ode.ode_1st_homogeneous_coeff_subs_dep_div_indep">[docs]</a><span class="k">def</span> <span class="nf">ode_1st_homogeneous_coeff_subs_dep_div_indep</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">match</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Solves a 1st order differential equation with homogeneous coefficients</span>
-<span class="sd">    using the substitution</span>
-<span class="sd">    u1 = &lt;dependent variable&gt;/&lt;independent variable&gt;.</span>
-
-<span class="sd">    This is a differential equation P(x, y) + Q(x, y)dy/dx = 0, that P</span>
-<span class="sd">    and Q are homogeneous of the same order.  A function F(x, y) is</span>
-<span class="sd">    homogeneous of order n if F(xt, yt) = t**n*F(x, y).  Equivalently,</span>
-<span class="sd">    F(x, y) can be rewritten as G(y/x) or H(x/y).  See also the</span>
-<span class="sd">    docstring of homogeneous_order().</span>
-
-<span class="sd">    If the coefficients P and Q in the differential equation above are</span>
-<span class="sd">    homogeneous functions of the same order, then it can be shown that</span>
-<span class="sd">    the substitution y = u1*x (u1 = y/x) will turn the differential</span>
-<span class="sd">    equation into an equation separable in the variables x and u.  If</span>
-<span class="sd">    h(u1) is the function that results from making the substitution</span>
-<span class="sd">    u1 = f(x)/x on P(x, f(x)) and g(u2) is the function that results</span>
-<span class="sd">    from the substitution on Q(x, f(x)) in the differential equation</span>
-<span class="sd">    P(x, f(x)) + Q(x, f(x))*diff(f(x), x) = 0, then the general solution</span>
-<span class="sd">    is::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Function, dsolve, pprint</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; f, g, h = list(map(Function, [&#39;f&#39;, &#39;g&#39;, &#39;h&#39;]))</span>
-<span class="sd">        &gt;&gt;&gt; genform = g(f(x)/x) + h(f(x)/x)*f(x).diff(x)</span>
-<span class="sd">        &gt;&gt;&gt; pprint(genform)</span>
-<span class="sd">         /f(x)\    /f(x)\ d</span>
-<span class="sd">        g|----| + h|----|*--(f(x))</span>
-<span class="sd">         \ x  /    \ x  / dx</span>
-<span class="sd">        &gt;&gt;&gt; pprint(dsolve(genform, f(x),</span>
-<span class="sd">        ... hint=&#39;1st_homogeneous_coeff_subs_dep_div_indep_Integral&#39;))</span>
-<span class="sd">                     f(x)</span>
-<span class="sd">                     ----</span>
-<span class="sd">                      x</span>
-<span class="sd">                       /</span>
-<span class="sd">                      |</span>
-<span class="sd">                      |       -h(u1)</span>
-<span class="sd">        log(C1*x) -   |  ---------------- d(u1) = 0</span>
-<span class="sd">                      |  u1*h(u1) + g(u1)</span>
-<span class="sd">                      |</span>
-<span class="sd">                     /</span>
-
-<span class="sd">    Where u1*h(u1) + g(u1) != 0 and x != 0.</span>
-
-<span class="sd">    See also the docstrings of ode_1st_homogeneous_coeff_best() and</span>
-<span class="sd">    ode_1st_homogeneous_coeff_subs_indep_div_dep().</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, dsolve</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(dsolve(2*x*f(x) + (x**2 + f(x)**2)*f(x).diff(x), f(x),</span>
-<span class="sd">    ... hint=&#39;1st_homogeneous_coeff_subs_dep_div_indep&#39;, simplify=False))</span>
-<span class="sd">                     /          3   \</span>
-<span class="sd">                     |3*f(x)   f (x)|</span>
-<span class="sd">                  log|------ + -----|</span>
-<span class="sd">                     |  x         3 |</span>
-<span class="sd">           /x \      \           x  /</span>
-<span class="sd">        log|--| + ------------------- = 0</span>
-<span class="sd">           \C1/            3</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - http://en.wikipedia.org/wiki/Homogeneous_differential_equation</span>
-<span class="sd">    - M. Tenenbaum &amp; H. Pollard, &quot;Ordinary Differential Equations&quot;,</span>
-<span class="sd">      Dover 1963, pp. 59</span>
-
-<span class="sd">    # indirect doctest</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">func</span>
-    <span class="n">u1</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;u1&#39;</span><span class="p">)</span>  <span class="c"># u1 == f(x)/x</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">match</span>  <span class="c"># d+e*diff(f(x),x)</span>
-    <span class="n">C1</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;C1&#39;</span><span class="p">)</span>
-    <span class="nb">int</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Integral</span><span class="p">(</span>
-        <span class="p">(</span><span class="o">-</span><span class="n">r</span><span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;e&#39;</span><span class="p">]]</span><span class="o">/</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;d&#39;</span><span class="p">]]</span> <span class="o">+</span> <span class="n">u1</span><span class="o">*</span><span class="n">r</span><span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;e&#39;</span><span class="p">]]))</span><span class="o">.</span><span class="n">subs</span><span class="p">({</span><span class="n">x</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span> <span class="n">r</span><span class="p">[</span><span class="s">&#39;y&#39;</span><span class="p">]:</span> <span class="n">u1</span><span class="p">}),</span>
-        <span class="p">(</span><span class="n">u1</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">))</span>
-    <span class="n">sol</span> <span class="o">=</span> <span class="n">logcombine</span><span class="p">(</span><span class="n">Eq</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="nb">int</span> <span class="o">+</span> <span class="n">log</span><span class="p">(</span><span class="n">C1</span><span class="p">)),</span> <span class="n">force</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">sol</span>
-
-</div>
-<div class="viewcode-block" id="ode_1st_homogeneous_coeff_subs_indep_div_dep"><a class="viewcode-back" href="../../../modules/solvers/ode.html#sympy.solvers.ode.ode_1st_homogeneous_coeff_subs_indep_div_dep">[docs]</a><span class="k">def</span> <span class="nf">ode_1st_homogeneous_coeff_subs_indep_div_dep</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">match</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Solves a 1st order differential equation with homogeneous coefficients</span>
-<span class="sd">    using the substitution</span>
-<span class="sd">    u2 = &lt;independent variable&gt;/&lt;dependent variable&gt;.</span>
-
-<span class="sd">    This is a differential equation P(x, y) + Q(x, y)dy/dx = 0, that P</span>
-<span class="sd">    and Q are homogeneous of the same order.  A function F(x, y) is</span>
-<span class="sd">    homogeneous of order n if F(xt, yt) = t**n*F(x, y).  Equivalently,</span>
-<span class="sd">    F(x, y) can be rewritten as G(y/x) or H(x/y).  See also the</span>
-<span class="sd">    docstring of homogeneous_order().</span>
-
-<span class="sd">    If the coefficients P and Q in the differential equation above are</span>
-<span class="sd">    homogeneous functions of the same order, then it can be shown that</span>
-<span class="sd">    the substitution x = u2*y (u2 = x/y) will turn the differential</span>
-<span class="sd">    equation into an equation separable in the variables y and u2.  If</span>
-<span class="sd">    h(u2) is the function that results from making the substitution</span>
-<span class="sd">    u2 = x/f(x) on P(x, f(x)) and g(u2) is the function that results</span>
-<span class="sd">    from the substitution on Q(x, f(x)) in the differential equation</span>
-<span class="sd">    P(x, f(x)) + Q(x, f(x))*diff(f(x), x) = 0, then the general solution</span>
-<span class="sd">    is:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, dsolve, pprint</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; f, g, h = list(map(Function, [&#39;f&#39;, &#39;g&#39;, &#39;h&#39;]))</span>
-<span class="sd">    &gt;&gt;&gt; genform = g(x/f(x)) + h(x/f(x))*f(x).diff(x)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(genform)</span>
-<span class="sd">     / x  \    / x  \ d</span>
-<span class="sd">    g|----| + h|----|*--(f(x))</span>
-<span class="sd">     \f(x)/    \f(x)/ dx</span>
-<span class="sd">    &gt;&gt;&gt; pprint(dsolve(genform, f(x),</span>
-<span class="sd">    ... hint=&#39;1st_homogeneous_coeff_subs_indep_div_dep_Integral&#39;))</span>
-<span class="sd">                 x</span>
-<span class="sd">                ----</span>
-<span class="sd">                f(x)</span>
-<span class="sd">                  /</span>
-<span class="sd">                 |</span>
-<span class="sd">                 |        g(u2)</span>
-<span class="sd">                 |  ----------------- d(u2)</span>
-<span class="sd">                 |  -u2*g(u2) - h(u2)</span>
-<span class="sd">                 |</span>
-<span class="sd">                /</span>
-<span class="sd">    &lt;BLANKLINE&gt;</span>
-<span class="sd">    f(x) = C1*e</span>
-
-<span class="sd">    Where u2*g(u2) + h(u2) != 0 and f(x) != 0.</span>
-
-<span class="sd">    See also the docstrings of ode_1st_homogeneous_coeff_best() and</span>
-<span class="sd">    ode_1st_homogeneous_coeff_subs_dep_div_indep().</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, pprint</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(dsolve(2*x*f(x) + (x**2 + f(x)**2)*f(x).diff(x), f(x),</span>
-<span class="sd">    ... hint=&#39;1st_homogeneous_coeff_subs_indep_div_dep&#39;))</span>
-<span class="sd">          ___________</span>
-<span class="sd">         /     2</span>
-<span class="sd">        /   3*x</span>
-<span class="sd">       /   ----- + 1 *f(x) = C1</span>
-<span class="sd">    3 /     2</span>
-<span class="sd">    \/     f (x)</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - http://en.wikipedia.org/wiki/Homogeneous_differential_equation</span>
-<span class="sd">    - M. Tenenbaum &amp; H. Pollard, &quot;Ordinary Differential Equations&quot;,</span>
-<span class="sd">      Dover 1963, pp. 59</span>
-
-<span class="sd">    # indirect doctest</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">func</span>
-    <span class="n">u2</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;u2&#39;</span><span class="p">)</span>  <span class="c"># u2 == x/f(x)</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">match</span>  <span class="c"># d+e*diff(f(x),x)</span>
-    <span class="n">C1</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;C1&#39;</span><span class="p">)</span>
-    <span class="nb">int</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Integral</span><span class="p">(</span>
-        <span class="n">simplify</span><span class="p">(</span>
-            <span class="p">(</span><span class="o">-</span><span class="n">r</span><span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;d&#39;</span><span class="p">]]</span><span class="o">/</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;e&#39;</span><span class="p">]]</span> <span class="o">+</span> <span class="n">u2</span><span class="o">*</span><span class="n">r</span><span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;d&#39;</span><span class="p">]]))</span><span class="o">.</span><span class="n">subs</span><span class="p">({</span><span class="n">x</span><span class="p">:</span> <span class="n">u2</span><span class="p">,</span> <span class="n">r</span><span class="p">[</span><span class="s">&#39;y&#39;</span><span class="p">]:</span> <span class="mi">1</span><span class="p">})),</span>
-        <span class="p">(</span><span class="n">u2</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">x</span><span class="o">/</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
-    <span class="n">sol</span> <span class="o">=</span> <span class="n">logcombine</span><span class="p">(</span><span class="n">Eq</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="nb">int</span> <span class="o">+</span> <span class="n">log</span><span class="p">(</span><span class="n">C1</span><span class="p">)),</span> <span class="n">force</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">sol</span>
-
-<span class="c"># XXX: Should this function maybe go somewhere else?</span>
-
-</div>
-<div class="viewcode-block" id="homogeneous_order"><a class="viewcode-back" href="../../../modules/solvers/ode.html#sympy.solvers.ode.homogeneous_order">[docs]</a><span class="k">def</span> <span class="nf">homogeneous_order</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the order n if g is homogeneous and None if it is not</span>
-<span class="sd">    homogeneous.</span>
-
-<span class="sd">    Determines if a function is homogeneous and if so of what order.</span>
-<span class="sd">    A function f(x,y,...) is homogeneous of order n if</span>
-<span class="sd">    f(t*x,t*y,t*...) == t**n*f(x,y,...).</span>
-
-<span class="sd">    If the function is of two variables, F(x, y), then f being</span>
-<span class="sd">    homogeneous of any order is equivalent to being able to rewrite</span>
-<span class="sd">    F(x, y) as G(x/y) or H(y/x).  This fact is used to solve 1st order</span>
-<span class="sd">    ordinary differential equations whose coefficients are homogeneous</span>
-<span class="sd">    of the same order (see the docstrings of</span>
-<span class="sd">    ode.ode_1st_homogeneous_coeff_subs_indep_div_dep() and</span>
-<span class="sd">    ode.ode_1st_homogeneous_coeff_subs_indep_div_dep()</span>
-
-<span class="sd">    Symbols can be functions, but every argument of the function must be</span>
-<span class="sd">    a symbol, and the arguments of the function that appear in the</span>
-<span class="sd">    expression must match those given in the list of symbols.  If a</span>
-<span class="sd">    declared function appears with different arguments than given in the</span>
-<span class="sd">    list of symbols, None is returned.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, homogeneous_order, sqrt</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; homogeneous_order(f(x), f(x)) is None</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; homogeneous_order(f(x,y), f(y, x), x, y) is None</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; homogeneous_order(f(x), f(x), x)</span>
-<span class="sd">    1</span>
-<span class="sd">    &gt;&gt;&gt; homogeneous_order(x**2*f(x)/sqrt(x**2+f(x)**2), x, f(x))</span>
-<span class="sd">    2</span>
-<span class="sd">    &gt;&gt;&gt; homogeneous_order(x**2+f(x), x, f(x)) is None</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.simplify.simplify</span> <span class="kn">import</span> <span class="n">separatevars</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">symbols</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;homogeneous_order: no symbols were given.&quot;</span><span class="p">)</span>
-    <span class="n">symset</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span>
-    <span class="n">eq</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span>
-
-    <span class="c"># The following are not supported</span>
-    <span class="k">if</span> <span class="n">eq</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Order</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="c"># These are all constants</span>
-    <span class="k">if</span> <span class="p">(</span><span class="n">eq</span><span class="o">.</span><span class="n">is_Number</span> <span class="ow">or</span>
-        <span class="n">eq</span><span class="o">.</span><span class="n">is_NumberSymbol</span> <span class="ow">or</span>
-        <span class="n">eq</span><span class="o">.</span><span class="n">is_number</span>
-            <span class="p">):</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="c"># Replace all functions with dummy variables</span>
-    <span class="n">dum</span> <span class="o">=</span> <span class="n">numbered_symbols</span><span class="p">(</span><span class="n">prefix</span><span class="o">=</span><span class="s">&#39;d&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Dummy</span><span class="p">)</span>
-    <span class="n">newsyms</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="n">j</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">symset</span> <span class="k">if</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="s">&#39;is_Function&#39;</span><span class="p">)]:</span>
-        <span class="n">iargs</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">iargs</span><span class="o">.</span><span class="n">difference</span><span class="p">(</span><span class="n">symset</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">dummyvar</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">dum</span><span class="p">)</span>
-            <span class="n">eq</span> <span class="o">=</span> <span class="n">eq</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">dummyvar</span><span class="p">)</span>
-            <span class="n">symset</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="n">newsyms</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">dummyvar</span><span class="p">)</span>
-    <span class="n">symset</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">newsyms</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">eq</span><span class="o">.</span><span class="n">free_symbols</span> <span class="o">&amp;</span> <span class="n">symset</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="c"># make the replacement of x with x*t and see if t can be factored out</span>
-    <span class="n">t</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>  <span class="c"># It is sufficient that t &gt; 0</span>
-    <span class="n">eqs</span> <span class="o">=</span> <span class="n">separatevars</span><span class="p">(</span><span class="n">eq</span><span class="o">.</span><span class="n">subs</span><span class="p">([(</span><span class="n">i</span><span class="p">,</span> <span class="n">t</span><span class="o">*</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">symset</span><span class="p">]),</span> <span class="p">[</span><span class="n">t</span><span class="p">],</span> <span class="nb">dict</span><span class="o">=</span><span class="bp">True</span><span class="p">)[</span><span class="n">t</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">eqs</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>  <span class="c"># there was no term with only t</span>
-    <span class="n">i</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">eqs</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">as_Add</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-    <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-    <span class="k">if</span> <span class="n">b</span> <span class="o">==</span> <span class="n">t</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">e</span>
-
-</div>
-<div class="viewcode-block" id="ode_1st_linear"><a class="viewcode-back" href="../../../modules/solvers/ode.html#sympy.solvers.ode.ode_1st_linear">[docs]</a><span class="k">def</span> <span class="nf">ode_1st_linear</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">match</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Solves 1st order linear differential equations.</span>
-
-<span class="sd">    These are differential equations of the form dy/dx _ P(x)*y = Q(x).</span>
-<span class="sd">    These kinds of differential equations can be solved in a general</span>
-<span class="sd">    way.  The integrating factor exp(Integral(P(x), x)) will turn the</span>
-<span class="sd">    equation into a separable equation.  The general solution is::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Function, dsolve, Eq, pprint, diff, sin</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; f, P, Q = list(map(Function, [&#39;f&#39;, &#39;P&#39;, &#39;Q&#39;]))</span>
-<span class="sd">        &gt;&gt;&gt; genform = Eq(f(x).diff(x) + P(x)*f(x), Q(x))</span>
-<span class="sd">        &gt;&gt;&gt; pprint(genform)</span>
-<span class="sd">                    d</span>
-<span class="sd">        P(x)*f(x) + --(f(x)) = Q(x)</span>
-<span class="sd">                    dx</span>
-<span class="sd">        &gt;&gt;&gt; pprint(dsolve(genform, f(x), hint=&#39;1st_linear_Integral&#39;))</span>
-<span class="sd">               /       /                   \</span>
-<span class="sd">               |      |                    |</span>
-<span class="sd">               |      |         /          |     /</span>
-<span class="sd">               |      |        |           |    |</span>
-<span class="sd">               |      |        | P(x) dx   |  - | P(x) dx</span>
-<span class="sd">               |      |        |           |    |</span>
-<span class="sd">               |      |       /            |   /</span>
-<span class="sd">        f(x) = |C1 +  | Q(x)*e           dx|*e</span>
-<span class="sd">               |      |                    |</span>
-<span class="sd">               \     /                     /</span>
-
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(dsolve(Eq(x*diff(f(x), x) - f(x), x**2*sin(x)),</span>
-<span class="sd">    ... f(x), &#39;1st_linear&#39;))</span>
-<span class="sd">    f(x) = x*(C1 - cos(x))</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - http://en.wikipedia.org/wiki/Linear_differential_equation#First_order_equation</span>
-<span class="sd">    - M. Tenenbaum &amp; H. Pollard, &quot;Ordinary Differential Equations&quot;,</span>
-<span class="sd">      Dover 1963, pp. 92</span>
-
-<span class="sd">    # indirect doctest</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">func</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">match</span>  <span class="c"># a*diff(f(x),x) + b*f(x) + c</span>
-    <span class="n">C1</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;C1&#39;</span><span class="p">)</span>
-    <span class="n">t</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Integral</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;b&#39;</span><span class="p">]]</span><span class="o">/</span><span class="n">r</span><span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">]],</span> <span class="n">x</span><span class="p">))</span>
-    <span class="n">tt</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Integral</span><span class="p">(</span><span class="n">t</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">r</span><span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;c&#39;</span><span class="p">]]</span><span class="o">/</span><span class="n">r</span><span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">]]),</span> <span class="n">x</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">Eq</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">tt</span> <span class="o">+</span> <span class="n">C1</span><span class="p">)</span><span class="o">/</span><span class="n">t</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="ode_Bernoulli"><a class="viewcode-back" href="../../../modules/solvers/ode.html#sympy.solvers.ode.ode_Bernoulli">[docs]</a><span class="k">def</span> <span class="nf">ode_Bernoulli</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">match</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Solves Bernoulli differential equations.</span>
-
-<span class="sd">    These are equations of the form dy/dx + P(x)*y = Q(x)*y**n, n != 1.</span>
-<span class="sd">    The substitution w = 1/y**(1-n) will transform an equation of this</span>
-<span class="sd">    form into one that is linear (see the docstring of</span>
-<span class="sd">    ode_1st_linear()).  The general solution is::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Function, dsolve, Eq, pprint</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, n</span>
-<span class="sd">        &gt;&gt;&gt; f, P, Q = list(map(Function, [&#39;f&#39;, &#39;P&#39;, &#39;Q&#39;]))</span>
-<span class="sd">        &gt;&gt;&gt; genform = Eq(f(x).diff(x) + P(x)*f(x), Q(x)*f(x)**n)</span>
-<span class="sd">        &gt;&gt;&gt; pprint(genform)</span>
-<span class="sd">                    d                n</span>
-<span class="sd">        P(x)*f(x) + --(f(x)) = Q(x)*f (x)</span>
-<span class="sd">                    dx</span>
-<span class="sd">        &gt;&gt;&gt; pprint(dsolve(genform, f(x), hint=&#39;Bernoulli_Integral&#39;)) #doctest: +SKIP</span>
-<span class="sd">                                                                                       1</span>
-<span class="sd">                                                                                      ----</span>
-<span class="sd">                                                                                     1 - n</span>
-<span class="sd">               //                /                            \                     \</span>
-<span class="sd">               ||               |                             |                     |</span>
-<span class="sd">               ||               |                  /          |             /       |</span>
-<span class="sd">               ||               |                 |           |            |        |</span>
-<span class="sd">               ||               |        (1 - n)* | P(x) dx   |  (-1 + n)* | P(x) dx|</span>
-<span class="sd">               ||               |                 |           |            |        |</span>
-<span class="sd">               ||               |                /            |           /         |</span>
-<span class="sd">        f(x) = ||C1 + (-1 + n)* | -Q(x)*e                   dx|*e                   |</span>
-<span class="sd">               ||               |                             |                     |</span>
-<span class="sd">               \\               /                            /                     /</span>
-
-
-<span class="sd">    Note that when n = 1, then the equation is separable (see the</span>
-<span class="sd">    docstring of ode_separable()).</span>
-
-<span class="sd">    &gt;&gt;&gt; pprint(dsolve(Eq(f(x).diff(x) + P(x)*f(x), Q(x)*f(x)), f(x),</span>
-<span class="sd">    ... hint=&#39;separable_Integral&#39;))</span>
-<span class="sd">     f(x)</span>
-<span class="sd">       /</span>
-<span class="sd">      |                /</span>
-<span class="sd">      |  1            |</span>
-<span class="sd">      |  - dy = C1 +  | (-P(x) + Q(x)) dx</span>
-<span class="sd">      |  y            |</span>
-<span class="sd">      |              /</span>
-<span class="sd">     /</span>
-
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, dsolve, Eq, pprint, log</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-
-<span class="sd">    &gt;&gt;&gt; pprint(dsolve(Eq(x*f(x).diff(x) + f(x), log(x)*f(x)**2),</span>
-<span class="sd">    ... f(x), hint=&#39;Bernoulli&#39;))</span>
-<span class="sd">                    1</span>
-<span class="sd">    f(x) = -------------------</span>
-<span class="sd">             /     log(x)   1\</span>
-<span class="sd">           x*|C1 + ------ + -|</span>
-<span class="sd">             \       x      x/</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - http://en.wikipedia.org/wiki/Bernoulli_differential_equation</span>
-<span class="sd">    - M. Tenenbaum &amp; H. Pollard, &quot;Ordinary Differential Equations&quot;,</span>
-<span class="sd">      Dover 1963, pp. 95</span>
-
-<span class="sd">    # indirect doctest</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">func</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">match</span>  <span class="c"># a*diff(f(x),x) + b*f(x) + c*f(x)**n, n != 1</span>
-    <span class="n">C1</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;C1&#39;</span><span class="p">)</span>
-    <span class="n">t</span> <span class="o">=</span> <span class="n">exp</span><span class="p">((</span><span class="mi">1</span> <span class="o">-</span> <span class="n">r</span><span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;n&#39;</span><span class="p">]])</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">Integral</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;b&#39;</span><span class="p">]]</span><span class="o">/</span><span class="n">r</span><span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">]],</span> <span class="n">x</span><span class="p">))</span>
-    <span class="n">tt</span> <span class="o">=</span> <span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;n&#39;</span><span class="p">]]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">Integral</span><span class="p">(</span><span class="n">t</span><span class="o">*</span><span class="n">r</span><span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;c&#39;</span><span class="p">]]</span><span class="o">/</span><span class="n">r</span><span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">]],</span> <span class="n">x</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">Eq</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">((</span><span class="n">tt</span> <span class="o">+</span> <span class="n">C1</span><span class="p">)</span><span class="o">/</span><span class="n">t</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">r</span><span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;n&#39;</span><span class="p">]])))</span>
-
-</div>
-<div class="viewcode-block" id="ode_Riccati_special_minus2"><a class="viewcode-back" href="../../../modules/solvers/ode.html#sympy.solvers.ode.ode_Riccati_special_minus2">[docs]</a><span class="k">def</span> <span class="nf">ode_Riccati_special_minus2</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">match</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    The general Riccati equation has the form dy/dx = f(x)*y**2 + g(x)*y + h(x).</span>
-<span class="sd">    While it does not have a general solution [1], the &quot;special&quot; form,</span>
-<span class="sd">    dy/dx = a*y**2 - b*x**c, does have solutions in many cases [2]. This routine</span>
-<span class="sd">    returns a solution for a*dy/dx = b*y**2 + c*y/x + d/x**2 that is obtained by</span>
-<span class="sd">    using a suitable change of variables to reduce it to the special form and is</span>
-<span class="sd">    valid when neither a nor b are zero and either c or d is zero.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, a, b, c, d</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.solvers.ode import dsolve, checkodesol</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import pprint, Function</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; y = f(x)</span>
-<span class="sd">    &gt;&gt;&gt; genform = a*y.diff(x) - (b*y**2 + c*y/x + d/x**2)</span>
-<span class="sd">    &gt;&gt;&gt; sol = dsolve(genform, y)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(sol)</span>
-<span class="sd">             /                                 /        __________________      \\</span>
-<span class="sd">            |           __________________    |       /                2        ||</span>
-<span class="sd">            |          /                2     |     \/  4*b*d - (a + c)  *log(x)||</span>
-<span class="sd">           -|a + c - \/  4*b*d - (a + c)  *tan|C1 + ----------------------------||</span>
-<span class="sd">            \                                 \                 2*a             //</span>
-<span class="sd">    f(x) = -----------------------------------------------------------------------</span>
-<span class="sd">                                            2*b*x</span>
-
-<span class="sd">    &gt;&gt;&gt; checkodesol(genform, sol, order=1)[0]</span>
-<span class="sd">    True</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. http://www.maplesoft.com/support/help/Maple/view.aspx?path=odeadvisor/Riccati</span>
-<span class="sd">    2. http://eqworld.ipmnet.ru/en/solutions/ode/ode0106.pdf -</span>
-<span class="sd">       http://eqworld.ipmnet.ru/en/solutions/ode/ode0123.pdf</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">x</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">func</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">match</span>  <span class="c"># a2*diff(f(x),x) + b2*f(x) + c2*f(x)/x + d2/x**2</span>
-    <span class="n">a2</span><span class="p">,</span> <span class="n">b2</span><span class="p">,</span> <span class="n">c2</span><span class="p">,</span> <span class="n">d2</span> <span class="o">=</span> <span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="n">r</span><span class="p">[</span><span class="n">s</span><span class="p">]]</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="s">&#39;a2 b2 c2 d2&#39;</span><span class="o">.</span><span class="n">split</span><span class="p">()]</span>
-    <span class="n">C1</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;C1&#39;</span><span class="p">)</span>
-    <span class="n">mu</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">d2</span><span class="o">*</span><span class="n">b2</span> <span class="o">-</span> <span class="p">(</span><span class="n">a2</span> <span class="o">-</span> <span class="n">c2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">Eq</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">a2</span> <span class="o">-</span> <span class="n">c2</span> <span class="o">-</span> <span class="n">mu</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="n">mu</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a2</span><span class="p">)</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">C1</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">b2</span><span class="o">*</span><span class="n">x</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="ode_Liouville"><a class="viewcode-back" href="../../../modules/solvers/ode.html#sympy.solvers.ode.ode_Liouville">[docs]</a><span class="k">def</span> <span class="nf">ode_Liouville</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">match</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Solves 2nd order Liouville differential equations.</span>
-
-<span class="sd">    The general form of a Liouville ODE is</span>
-<span class="sd">    d^2y/dx^2 + g(y)*(dy/dx)**2 + h(x)*dy/dx.  The general solution is:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Function, dsolve, Eq, pprint, diff</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; f, g, h = list(map(Function, [&#39;f&#39;, &#39;g&#39;, &#39;h&#39;]))</span>
-<span class="sd">        &gt;&gt;&gt; genform = Eq(diff(f(x),x,x) + g(f(x))*diff(f(x),x)**2 +</span>
-<span class="sd">        ... h(x)*diff(f(x),x), 0)</span>
-<span class="sd">        &gt;&gt;&gt; pprint(genform)</span>
-<span class="sd">                        2                    2</span>
-<span class="sd">                d                d          d</span>
-<span class="sd">        g(f(x))*--(f(x))  + h(x)*--(f(x)) + ---(f(x)) = 0</span>
-<span class="sd">                dx               dx           2</span>
-<span class="sd">                                            dx</span>
-<span class="sd">        &gt;&gt;&gt; pprint(dsolve(genform, f(x), hint=&#39;Liouville_Integral&#39;))</span>
-<span class="sd">                                          f(x)</span>
-<span class="sd">                  /                     /</span>
-<span class="sd">                 |                     |</span>
-<span class="sd">                 |     /               |     /</span>
-<span class="sd">                 |    |                |    |</span>
-<span class="sd">                 |  - | h(x) dx        |    | g(y) dy</span>
-<span class="sd">                 |    |                |    |</span>
-<span class="sd">                 |   /                 |   /</span>
-<span class="sd">        C1 + C2* | e            dx +   |  e           dy = 0</span>
-<span class="sd">                 |                     |</span>
-<span class="sd">                /                     /</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, dsolve, Eq, pprint</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(dsolve(diff(f(x), x, x) + diff(f(x), x)**2/f(x) +</span>
-<span class="sd">    ... diff(f(x), x)/x, f(x), hint=&#39;Liouville&#39;))</span>
-<span class="sd">               ________________           ________________</span>
-<span class="sd">    [f(x) = -\/ C1 + C2*log(x) , f(x) = \/ C1 + C2*log(x) ]</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - Goldstein and Braun, &quot;Advanced Methods for the Solution of</span>
-<span class="sd">      Differential Equations&quot;, pp. 98</span>
-<span class="sd">    - http://www.maplesoft.com/support/help/Maple/view.aspx?path=odeadvisor/Liouville</span>
-
-<span class="sd">    # indirect doctest</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># Liouville ODE f(x).diff(x, 2) + g(f(x))*(f(x).diff(x, 2))**2 + h(x)*f(x).diff(x)</span>
-    <span class="c"># See Goldstein and Braun, &quot;Advanced Methods for the Solution of</span>
-    <span class="c"># Differential Equations&quot;, pg. 98, as well as</span>
-    <span class="c"># http://www.maplesoft.com/support/help/view.aspx?path=odeadvisor/Liouville</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">func</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">match</span>  <span class="c"># f(x).diff(x, 2) + g*f(x).diff(x)**2 + h*f(x).diff(x)</span>
-    <span class="n">y</span> <span class="o">=</span> <span class="n">r</span><span class="p">[</span><span class="s">&#39;y&#39;</span><span class="p">]</span>
-    <span class="n">C1</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;C1&#39;</span><span class="p">)</span>
-    <span class="n">C2</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;C2&#39;</span><span class="p">)</span>
-    <span class="nb">int</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Integral</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Integral</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;g&#39;</span><span class="p">],</span> <span class="n">y</span><span class="p">)),</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
-    <span class="n">sol</span> <span class="o">=</span> <span class="n">Eq</span><span class="p">(</span><span class="nb">int</span> <span class="o">+</span> <span class="n">C1</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">Integral</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">C</span><span class="o">.</span><span class="n">Integral</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;h&#39;</span><span class="p">],</span> <span class="n">x</span><span class="p">)),</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">C2</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">sol</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_nth_linear_match</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Matches a differential equation to the linear form:</span>
-
-<span class="sd">    a_n(x)y^(n) + ... + a_1(x)y&#39; + a_0(x)y + B(x) = 0</span>
-
-<span class="sd">    Returns a dict of order:coeff terms, where order is the order of the</span>
-<span class="sd">    derivative on each term, and coeff is the coefficient of that</span>
-<span class="sd">    derivative.  The key -1 holds the function B(x). Returns None if</span>
-<span class="sd">    the ode is not linear.  This function assumes that func has already</span>
-<span class="sd">    been checked to be good.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, cos, sin</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.solvers.ode import _nth_linear_match</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; _nth_linear_match(f(x).diff(x, 3) + 2*f(x).diff(x) +</span>
-<span class="sd">    ... x*f(x).diff(x, 2) + cos(x)*f(x).diff(x) + x - f(x) -</span>
-<span class="sd">    ... sin(x), f(x), 3)</span>
-<span class="sd">    {-1: x - sin(x), 0: -1, 1: cos(x) + 2, 2: x, 3: 1}</span>
-<span class="sd">    &gt;&gt;&gt; _nth_linear_match(f(x).diff(x, 3) + 2*f(x).diff(x) +</span>
-<span class="sd">    ... x*f(x).diff(x, 2) + cos(x)*f(x).diff(x) + x - f(x) -</span>
-<span class="sd">    ... sin(f(x)), f(x), 3) == None</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">one_x</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span><span class="n">x</span><span class="p">])</span>
-    <span class="n">terms</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">([(</span><span class="n">i</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">order</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)])</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">eq</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">i</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
-            <span class="n">terms</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+=</span> <span class="n">i</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">c</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">i</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">func</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="p">((</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">set</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">variables</span><span class="p">)</span> <span class="o">==</span> <span class="n">one_x</span><span class="p">)</span> <span class="ow">or</span>
-                    <span class="n">f</span> <span class="o">==</span> <span class="n">func</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">None</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">terms</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:])]</span> <span class="o">+=</span> <span class="n">c</span>
-    <span class="k">return</span> <span class="n">terms</span>
-
-
-<span class="k">def</span> <span class="nf">ode_nth_linear_euler_eq_homogeneous</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">match</span><span class="p">,</span> <span class="n">returns</span><span class="o">=</span><span class="s">&#39;sol&#39;</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Solves an nth order linear homogeneous variable-coefficient</span>
-<span class="sd">    Cauchy-Euler equidimensional ordinary differential equation.</span>
-
-<span class="sd">    This is an equation with form 0 = a0*f(x) + a1*x*f&#39;(x) + a2*x**2*f&quot;(x)...</span>
-
-<span class="sd">    These equations can be solved in a general manner, by substituting</span>
-<span class="sd">    solutions of the form f(x) = x**r, and deriving a characteristic equation</span>
-<span class="sd">    for r.  When there are repeated roots, we include extra terms of the form</span>
-<span class="sd">    Crk*ln(x)**k*x**r, where Cnk is an arbitrary integration constant, r is a</span>
-<span class="sd">    root of the characteristic equation, and k ranges over the multiplicity of</span>
-<span class="sd">    r.  In the cases where the roots are complex, solutions of the form</span>
-<span class="sd">    C1*x**a*sin(b*log(x)) + C2*x**a*cos(b*log(x)) are returned, based on</span>
-<span class="sd">    expansions with Eulers formula.  The general solution is the sum of the</span>
-<span class="sd">    terms found.  If SymPy cannot find exact roots to the characteristic</span>
-<span class="sd">    equation, a RootOf instance will be return in its stead.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, dsolve, Eq</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; dsolve(4*x**2*f(x).diff(x, 2) + f(x), f(x),</span>
-<span class="sd">    ... hint=&#39;nth_linear_euler_eq_homogeneous&#39;)</span>
-<span class="sd">    ... # doctest: +NORMALIZE_WHITESPACE</span>
-<span class="sd">    f(x) == sqrt(x)*(C1 + C2*log(x))</span>
-
-<span class="sd">    Note that because this method does not involve integration, there is</span>
-<span class="sd">    no &#39;nth_linear_euler_eq_homogeneous_Integral&#39; hint.</span>
-
-<span class="sd">    The following is for internal use:</span>
-
-<span class="sd">    - returns = &#39;sol&#39; returns the solution to the ODE.</span>
-<span class="sd">    - returns = &#39;list&#39; returns a list of linearly independent</span>
-<span class="sd">      solutions, corresponding to the fundamental solution set, for use with</span>
-<span class="sd">      non homogeneous solution methods like variation of parameters and</span>
-<span class="sd">      undetermined coefficients.  Note that, though the solutions should be</span>
-<span class="sd">      linearly independent, this function does not explicitly check that.  You</span>
-<span class="sd">      can do &quot;assert simplify(wronskian(sollist)) != 0&quot; to check for linear</span>
-<span class="sd">      independence.  Also, &quot;assert len(sollist) == order&quot; will need to pass.</span>
-<span class="sd">    - returns = &#39;both&#39;, return a dictionary {&#39;sol&#39;:solution to ODE,</span>
-<span class="sd">      &#39;list&#39;: list of linearly independent solutions}.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, dsolve, pprint</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; eq = f(x).diff(x, 2)*x**2 - 4*f(x).diff(x)*x + 6*f(x)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(dsolve(eq, f(x),</span>
-<span class="sd">    ... hint=&#39;nth_linear_euler_eq_homogeneous&#39;))</span>
-<span class="sd">            2</span>
-<span class="sd">    f(x) = x *(C1 + C2*x)</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - http://en.wikipedia.org/wiki/Cauchy%E2%80%93Euler_equation</span>
-<span class="sd">    - C. Bender &amp; S. Orszag, &quot;Advanced Mathematical Methods for</span>
-<span class="sd">        Scientists and Engineers&quot;, Springer 1999, pp. 12</span>
-<span class="sd">    # indirect doctest</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">global</span> <span class="n">collectterms</span>
-    <span class="n">collectterms</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="n">x</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">func</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">match</span>
-
-    <span class="c"># A generator of constants</span>
-    <span class="n">constants</span> <span class="o">=</span> <span class="n">numbered_symbols</span><span class="p">(</span><span class="n">prefix</span><span class="o">=</span><span class="s">&#39;C&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Symbol</span><span class="p">,</span> <span class="n">start</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-
-    <span class="c"># First, set up characteristic equation.</span>
-    <span class="n">chareq</span><span class="p">,</span> <span class="n">symbol</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">r</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="nb">str</span><span class="p">)</span> <span class="ow">and</span> <span class="n">i</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">chareq</span> <span class="o">+=</span> <span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">symbol</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="o">**-</span><span class="n">symbol</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-
-    <span class="n">chareq</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">chareq</span><span class="p">,</span> <span class="n">symbol</span><span class="p">)</span>
-    <span class="n">chareqroots</span> <span class="o">=</span> <span class="p">[</span><span class="n">RootOf</span><span class="p">(</span><span class="n">chareq</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">chareq</span><span class="o">.</span><span class="n">degree</span><span class="p">())]</span>
-
-    <span class="c"># Create a dict root: multiplicity or charroots</span>
-    <span class="n">charroots</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">root</span> <span class="ow">in</span> <span class="n">chareqroots</span><span class="p">:</span>
-        <span class="n">charroots</span><span class="p">[</span><span class="n">root</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-    <span class="n">gsol</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-    <span class="c"># We need keep track of terms so we can run collect() at the end.</span>
-    <span class="c"># This is necessary for constantsimp to work properly.</span>
-    <span class="n">ln</span> <span class="o">=</span> <span class="n">log</span>
-    <span class="k">for</span> <span class="n">root</span><span class="p">,</span> <span class="n">multiplicity</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">charroots</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">multiplicity</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">root</span><span class="p">,</span> <span class="n">RootOf</span><span class="p">):</span>
-                <span class="n">gsol</span> <span class="o">+=</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">root</span><span class="p">)</span><span class="o">*</span><span class="nb">next</span><span class="p">(</span><span class="n">constants</span><span class="p">)</span>
-                <span class="k">assert</span> <span class="n">multiplicity</span> <span class="o">==</span> <span class="mi">1</span>
-                <span class="n">collectterms</span> <span class="o">=</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="n">root</span><span class="p">,</span> <span class="mi">0</span><span class="p">)]</span> <span class="o">+</span> <span class="n">collectterms</span>
-            <span class="k">elif</span> <span class="n">root</span><span class="o">.</span><span class="n">is_real</span><span class="p">:</span>
-                <span class="n">gsol</span> <span class="o">+=</span> <span class="n">ln</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="n">i</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">root</span><span class="p">)</span><span class="o">*</span><span class="nb">next</span><span class="p">(</span><span class="n">constants</span><span class="p">)</span>
-                <span class="n">collectterms</span> <span class="o">=</span> <span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="n">root</span><span class="p">,</span> <span class="mi">0</span><span class="p">)]</span> <span class="o">+</span> <span class="n">collectterms</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">reroot</span> <span class="o">=</span> <span class="n">re</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
-                <span class="n">imroot</span> <span class="o">=</span> <span class="n">im</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
-                <span class="n">gsol</span> <span class="o">+=</span> <span class="n">ln</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="n">i</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">reroot</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="nb">next</span><span class="p">(</span><span class="n">constants</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">imroot</span><span class="p">)</span><span class="o">*</span><span class="n">ln</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-                    <span class="o">+</span> <span class="nb">next</span><span class="p">(</span><span class="n">constants</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">imroot</span><span class="o">*</span><span class="n">ln</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
-                <span class="c"># Preserve ordering (multiplicity, real part, imaginary part)</span>
-                <span class="c"># It will be assumed implicitly when constructing</span>
-                <span class="c"># fundamental solution sets.</span>
-                <span class="n">collectterms</span> <span class="o">=</span> <span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="n">reroot</span><span class="p">,</span> <span class="n">imroot</span><span class="p">)]</span> <span class="o">+</span> <span class="n">collectterms</span>
-    <span class="k">if</span> <span class="n">returns</span> <span class="o">==</span> <span class="s">&#39;sol&#39;</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">Eq</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">gsol</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">returns</span> <span class="ow">in</span> <span class="p">(</span><span class="s">&#39;list&#39;</span> <span class="s">&#39;both&#39;</span><span class="p">):</span>
-        <span class="c"># HOW TO TEST THIS CODE? (dsolve does not pass &#39;returns&#39; through)</span>
-        <span class="c"># Create a list of (hopefully) linearly independent solutions</span>
-        <span class="n">gensols</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="c"># Keep track of when to use sin or cos for nonzero imroot</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">reroot</span><span class="p">,</span> <span class="n">imroot</span> <span class="ow">in</span> <span class="n">collectterms</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">imroot</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">gensols</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ln</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="n">i</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">reroot</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">sin_form</span> <span class="o">=</span> <span class="n">ln</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="n">i</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">reroot</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">imroot</span><span class="p">)</span><span class="o">*</span><span class="n">ln</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-                <span class="k">if</span> <span class="n">sin_form</span> <span class="ow">in</span> <span class="n">gensols</span><span class="p">:</span>
-                    <span class="n">cos_form</span> <span class="o">=</span> <span class="n">ln</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="n">i</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">reroot</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">imroot</span><span class="o">*</span><span class="n">ln</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-                    <span class="n">gensols</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">cos_form</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">gensols</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sin_form</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">returns</span> <span class="o">==</span> <span class="s">&#39;list&#39;</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">gensols</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">{</span><span class="s">&#39;sol&#39;</span><span class="p">:</span> <span class="n">Eq</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">gsol</span><span class="p">),</span> <span class="s">&#39;list&#39;</span><span class="p">:</span> <span class="n">gensols</span><span class="p">}</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Unknown value for key &quot;returns&quot;.&#39;</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="ode_nth_linear_constant_coeff_homogeneous"><a class="viewcode-back" href="../../../modules/solvers/ode.html#sympy.solvers.ode.ode_nth_linear_constant_coeff_homogeneous">[docs]</a><span class="k">def</span> <span class="nf">ode_nth_linear_constant_coeff_homogeneous</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">match</span><span class="p">,</span> <span class="n">returns</span><span class="o">=</span><span class="s">&#39;sol&#39;</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Solves an nth order linear homogeneous differential equation with</span>
-<span class="sd">    constant coefficients.</span>
-
-<span class="sd">    This is an equation of the form a_n*f(x)^(n) + a_(n-1)*f(x)^(n-1) +</span>
-<span class="sd">    ... + a1*f&#39;(x) + a0*f(x) = 0</span>
-
-<span class="sd">    These equations can be solved in a general manner, by taking the</span>
-<span class="sd">    roots of the characteristic equation a_n*m**n + a_(n-1)*m**(n-1) +</span>
-<span class="sd">    ... + a1*m + a0 = 0.  The solution will then be the sum of</span>
-<span class="sd">    Cn*x**i*exp(r*x) terms, for each where Cn is an arbitrary constant,</span>
-<span class="sd">    r is a root of the characteristic equation and i is is one of each</span>
-<span class="sd">    from 0 to the multiplicity of the root - 1 (for example, a root 3 of</span>
-<span class="sd">    multiplicity 2 would create the terms C1*exp(3*x) + C2*x*exp(3*x)).</span>
-<span class="sd">    The exponential is usually expanded for complex roots using Euler&#39;s</span>
-<span class="sd">    equation exp(I*x) = cos(x) + I*sin(x).  Complex roots always come in</span>
-<span class="sd">    conjugate pars in polynomials with real coefficients, so the two</span>
-<span class="sd">    roots will be represented (after simplifying the constants) as</span>
-<span class="sd">    exp(a*x)*(C1*cos(b*x) + C2*sin(b*x)).</span>
-
-<span class="sd">    If SymPy cannot find exact roots to the characteristic equation, a</span>
-<span class="sd">    RootOf instance will be return in its stead.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, dsolve, Eq</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; dsolve(f(x).diff(x, 5) + 10*f(x).diff(x) - 2*f(x), f(x),</span>
-<span class="sd">    ... hint=&#39;nth_linear_constant_coeff_homogeneous&#39;)</span>
-<span class="sd">    ... # doctest: +NORMALIZE_WHITESPACE</span>
-<span class="sd">    f(x) == C1*exp(x*RootOf(_x**5 + 10*_x - 2, 0)) +</span>
-<span class="sd">    C2*exp(x*RootOf(_x**5 + 10*_x - 2, 1)) +</span>
-<span class="sd">    C3*exp(x*RootOf(_x**5 + 10*_x - 2, 2)) +</span>
-<span class="sd">    C4*exp(x*RootOf(_x**5 + 10*_x - 2, 3)) +</span>
-<span class="sd">    C5*exp(x*RootOf(_x**5 + 10*_x - 2, 4))</span>
-
-<span class="sd">    Note that because this method does not involve integration, there is</span>
-<span class="sd">    no &#39;nth_linear_constant_coeff_homogeneous_Integral&#39; hint.</span>
-
-<span class="sd">    The following is for internal use:</span>
-
-<span class="sd">    - returns = &#39;sol&#39; returns the solution to the ODE.</span>
-<span class="sd">    - returns = &#39;list&#39; returns a list of linearly independent</span>
-<span class="sd">      solutions, for use with non homogeneous solution methods like</span>
-<span class="sd">      variation of parameters and undetermined coefficients.  Note that,</span>
-<span class="sd">      though the solutions should be linearly independent, this function</span>
-<span class="sd">      does not explicitly check that.  You can do &quot;assert</span>
-<span class="sd">      simplify(wronskian(sollist)) != 0&quot; to check for linear independence.</span>
-<span class="sd">      Also, &quot;assert len(sollist) == order&quot; will need to pass.</span>
-<span class="sd">    - returns = &#39;both&#39;, return a dictionary {&#39;sol&#39;:solution to ODE,</span>
-<span class="sd">      &#39;list&#39;: list of linearly independent solutions}.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, dsolve, pprint</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(dsolve(f(x).diff(x, 4) + 2*f(x).diff(x, 3) -</span>
-<span class="sd">    ... 2*f(x).diff(x, 2) - 6*f(x).diff(x) + 5*f(x), f(x),</span>
-<span class="sd">    ... hint=&#39;nth_linear_constant_coeff_homogeneous&#39;))</span>
-<span class="sd">                        x                            -2*x</span>
-<span class="sd">    f(x) = (C1 + C2*x)*e  + (C3*sin(x) + C4*cos(x))*e</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - http://en.wikipedia.org/wiki/Linear_differential_equation</span>
-<span class="sd">        section: Nonhomogeneous_equation_with_constant_coefficients</span>
-<span class="sd">    - M. Tenenbaum &amp; H. Pollard, &quot;Ordinary Differential Equations&quot;,</span>
-<span class="sd">      Dover 1963, pp. 211</span>
-
-<span class="sd">    # indirect doctest</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">func</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">match</span>
-
-    <span class="c"># A generator of constants</span>
-    <span class="n">constants</span> <span class="o">=</span> <span class="n">numbered_symbols</span><span class="p">(</span><span class="n">prefix</span><span class="o">=</span><span class="s">&#39;C&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Symbol</span><span class="p">,</span> <span class="n">start</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-
-    <span class="c"># First, set up characteristic equation.</span>
-    <span class="n">chareq</span><span class="p">,</span> <span class="n">symbol</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">r</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">==</span> <span class="nb">str</span> <span class="ow">or</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">pass</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">chareq</span> <span class="o">+=</span> <span class="n">r</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">symbol</span><span class="o">**</span><span class="n">i</span>
-
-    <span class="n">chareq</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">chareq</span><span class="p">,</span> <span class="n">symbol</span><span class="p">)</span>
-    <span class="n">chareqroots</span> <span class="o">=</span> <span class="p">[</span> <span class="n">RootOf</span><span class="p">(</span><span class="n">chareq</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">chareq</span><span class="o">.</span><span class="n">degree</span><span class="p">())</span> <span class="p">]</span>
-
-    <span class="c"># Create a dict root: multiplicity or charroots</span>
-    <span class="n">charroots</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">root</span> <span class="ow">in</span> <span class="n">chareqroots</span><span class="p">:</span>
-        <span class="n">charroots</span><span class="p">[</span><span class="n">root</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-    <span class="n">gsol</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-    <span class="c"># We need keep track of terms so we can run collect() at the end.</span>
-    <span class="c"># This is necessary for constantsimp to work properly.</span>
-    <span class="k">global</span> <span class="n">collectterms</span>
-    <span class="n">collectterms</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">root</span><span class="p">,</span> <span class="n">multiplicity</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">charroots</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">multiplicity</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">root</span><span class="p">,</span> <span class="n">RootOf</span><span class="p">):</span>
-                <span class="n">gsol</span> <span class="o">+=</span> <span class="n">exp</span><span class="p">(</span><span class="n">root</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="nb">next</span><span class="p">(</span><span class="n">constants</span><span class="p">)</span>
-                <span class="k">assert</span> <span class="n">multiplicity</span> <span class="o">==</span> <span class="mi">1</span>
-                <span class="n">collectterms</span> <span class="o">=</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="n">root</span><span class="p">,</span> <span class="mi">0</span><span class="p">)]</span> <span class="o">+</span> <span class="n">collectterms</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">reroot</span> <span class="o">=</span> <span class="n">re</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
-                <span class="n">imroot</span> <span class="o">=</span> <span class="n">im</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
-                <span class="n">gsol</span> <span class="o">+=</span> <span class="n">x</span><span class="o">**</span><span class="n">i</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">reroot</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="nb">next</span><span class="p">(</span><span class="n">constants</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">imroot</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-                <span class="o">+</span> <span class="nb">next</span><span class="p">(</span><span class="n">constants</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">imroot</span><span class="o">*</span><span class="n">x</span><span class="p">))</span>
-                <span class="c"># This ordering is important</span>
-                <span class="n">collectterms</span> <span class="o">=</span> <span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="n">reroot</span><span class="p">,</span> <span class="n">imroot</span><span class="p">)]</span> <span class="o">+</span> <span class="n">collectterms</span>
-    <span class="k">if</span> <span class="n">returns</span> <span class="o">==</span> <span class="s">&#39;sol&#39;</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">Eq</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">gsol</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">returns</span> <span class="ow">in</span> <span class="p">(</span><span class="s">&#39;list&#39;</span> <span class="s">&#39;both&#39;</span><span class="p">):</span>
-        <span class="c"># Create a list of (hopefully) linearly independent solutions</span>
-        <span class="n">gensols</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="c"># Keep track of when to use sin or cos for nonzero imroot</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">reroot</span><span class="p">,</span> <span class="n">imroot</span> <span class="ow">in</span> <span class="n">collectterms</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">imroot</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="n">gensols</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">i</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">reroot</span><span class="o">*</span><span class="n">x</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">x</span><span class="o">**</span><span class="n">i</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">reroot</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">imroot</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="p">)</span> <span class="ow">in</span> <span class="n">gensols</span><span class="p">:</span>
-                    <span class="n">gensols</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">i</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">reroot</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">imroot</span><span class="o">*</span><span class="n">x</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">gensols</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">i</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">reroot</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">imroot</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">returns</span> <span class="o">==</span> <span class="s">&#39;list&#39;</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">gensols</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">{</span><span class="s">&#39;sol&#39;</span><span class="p">:</span> <span class="n">Eq</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">gsol</span><span class="p">),</span> <span class="s">&#39;list&#39;</span><span class="p">:</span> <span class="n">gensols</span><span class="p">}</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Unknown value for key &quot;returns&quot;.&#39;</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="ode_nth_linear_constant_coeff_undetermined_coefficients"><a class="viewcode-back" href="../../../modules/solvers/ode.html#sympy.solvers.ode.ode_nth_linear_constant_coeff_undetermined_coefficients">[docs]</a><span class="k">def</span> <span class="nf">ode_nth_linear_constant_coeff_undetermined_coefficients</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">match</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Solves an nth order linear differential equation with constant</span>
-<span class="sd">    coefficients using the method of undetermined coefficients.</span>
-
-<span class="sd">    This method works on differential equations of the form a_n*f(x)^(n)</span>
-<span class="sd">    + a_(n-1)*f(x)^(n-1) + ... + a1*f&#39;(x) + a0*f(x) = P(x), where P(x)</span>
-<span class="sd">    is a function that has a finite number of linearly independent</span>
-<span class="sd">    derivatives.</span>
-
-<span class="sd">    Functions that fit this requirement are finite sums functions of the</span>
-<span class="sd">    form a*x**i*exp(b*x)*sin(c*x + d) or a*x**i*exp(b*x)*cos(c*x + d),</span>
-<span class="sd">    where i is a non-negative integer and a, b, c, and d are constants.</span>
-<span class="sd">    For example any polynomial in x, functions like x**2*exp(2*x),</span>
-<span class="sd">    x*sin(x), and exp(x)*cos(x) can all be used.  Products of sin&#39;s and</span>
-<span class="sd">    cos&#39;s have a finite number of derivatives, because they can be</span>
-<span class="sd">    expanded into sin(a*x) and cos(b*x) terms.  However, SymPy currently</span>
-<span class="sd">    cannot do that expansion, so you will need to manually rewrite the</span>
-<span class="sd">    expression in terms of the above to use this method.  So, for example,</span>
-<span class="sd">    you will need to manually convert sin(x)**2 into (1 + cos(2*x))/2 to</span>
-<span class="sd">    properly apply the method of undetermined coefficients on it.</span>
-
-<span class="sd">    This method works by creating a trial function from the expression</span>
-<span class="sd">    and all of its linear independent derivatives and substituting them</span>
-<span class="sd">    into the original ODE.  The coefficients for each term will be a</span>
-<span class="sd">    system of linear equations, which are be solved for and substituted,</span>
-<span class="sd">    giving the solution.  If any of the trial functions are linearly</span>
-<span class="sd">    dependent on the solution to the homogeneous equation, they are</span>
-<span class="sd">    multiplied by sufficient x to make them linearly independent.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, dsolve, pprint, exp, cos</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(dsolve(f(x).diff(x, 2) + 2*f(x).diff(x) + f(x) -</span>
-<span class="sd">    ... 4*exp(-x)*x**2 + cos(2*x), f(x),</span>
-<span class="sd">    ... hint=&#39;nth_linear_constant_coeff_undetermined_coefficients&#39;))</span>
-<span class="sd">           /             4\</span>
-<span class="sd">           |            x |  -x   4*sin(2*x)   3*cos(2*x)</span>
-<span class="sd">    f(x) = |C1 + C2*x + --|*e   - ---------- + ----------</span>
-<span class="sd">           \            3 /           25           25</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - http://en.wikipedia.org/wiki/Method_of_undetermined_coefficients</span>
-<span class="sd">    - M. Tenenbaum &amp; H. Pollard, &quot;Ordinary Differential Equations&quot;,</span>
-<span class="sd">      Dover 1963, pp. 221</span>
-
-<span class="sd">    # indirect doctest</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">gensol</span> <span class="o">=</span> <span class="n">ode_nth_linear_constant_coeff_homogeneous</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">match</span><span class="p">,</span>
-        <span class="n">returns</span><span class="o">=</span><span class="s">&#39;both&#39;</span><span class="p">)</span>
-    <span class="n">match</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">gensol</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">_solve_undetermined_coefficients</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">match</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_solve_undetermined_coefficients</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">match</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Helper function for the method of undetermined coefficients.</span>
-
-<span class="sd">    See the ode_nth_linear_constant_coeff_undetermined_coefficients()</span>
-<span class="sd">    docstring for more information on this method.</span>
-
-<span class="sd">    match should be a dictionary that has the following keys:</span>
-<span class="sd">    &#39;list&#39; - A list of solutions to the homogeneous equation, such as</span>
-<span class="sd">         the list returned by</span>
-<span class="sd">         ode_nth_linear_constant_coeff_homogeneous(returns=&#39;list&#39;)</span>
-<span class="sd">    &#39;sol&#39; - The general solution, such as the solution returned by</span>
-<span class="sd">        ode_nth_linear_constant_coeff_homogeneous(returns=&#39;sol&#39;)</span>
-<span class="sd">    &#39;trialset&#39; - The set of trial functions as returned by</span>
-<span class="sd">        _undetermined_coefficients_match()[&#39;trialset&#39;]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">func</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">match</span>
-    <span class="n">coeffs</span> <span class="o">=</span> <span class="n">numbered_symbols</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Dummy</span><span class="p">)</span>
-    <span class="n">coefflist</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">gensols</span> <span class="o">=</span> <span class="n">r</span><span class="p">[</span><span class="s">&#39;list&#39;</span><span class="p">]</span>
-    <span class="n">gsol</span> <span class="o">=</span> <span class="n">r</span><span class="p">[</span><span class="s">&#39;sol&#39;</span><span class="p">]</span>
-    <span class="n">trialset</span> <span class="o">=</span> <span class="n">r</span><span class="p">[</span><span class="s">&#39;trialset&#39;</span><span class="p">]</span>
-    <span class="n">notneedset</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([])</span>
-    <span class="n">newtrialset</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([])</span>
-    <span class="k">global</span> <span class="n">collectterms</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">gensols</span><span class="p">)</span> <span class="o">!=</span> <span class="n">order</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Cannot find &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">order</span><span class="p">)</span> <span class="o">+</span>
-        <span class="s">&quot; solutions to the homogeneous equation nessesary to apply &quot;</span> <span class="o">+</span>
-        <span class="s">&quot;undetermined coefficients to &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot; (number of terms != order)&quot;</span><span class="p">)</span>
-    <span class="n">usedsin</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([])</span>
-    <span class="n">mult</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c"># The multiplicity of the root</span>
-    <span class="n">getmult</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">reroot</span><span class="p">,</span> <span class="n">imroot</span> <span class="ow">in</span> <span class="n">collectterms</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">getmult</span><span class="p">:</span>
-            <span class="n">mult</span> <span class="o">=</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-            <span class="n">getmult</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">getmult</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="n">imroot</span><span class="p">:</span>
-            <span class="c"># Alternate between sin and cos</span>
-            <span class="k">if</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">reroot</span><span class="p">)</span> <span class="ow">in</span> <span class="n">usedsin</span><span class="p">:</span>
-                <span class="n">check</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="n">i</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">reroot</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">imroot</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">check</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="n">i</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">reroot</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">imroot</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-                <span class="n">usedsin</span><span class="o">.</span><span class="n">add</span><span class="p">((</span><span class="n">i</span><span class="p">,</span> <span class="n">reroot</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">check</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="n">i</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">reroot</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">check</span> <span class="ow">in</span> <span class="n">trialset</span><span class="p">:</span>
-            <span class="c"># If an element of the trial function is already part of the homogeneous</span>
-            <span class="c"># solution, we need to multiply by sufficient x to make it linearly</span>
-            <span class="c"># independent.  We also don&#39;t need to bother checking for the coefficients</span>
-            <span class="c"># on those elements, since we already know it will be 0.</span>
-            <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">check</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">mult</span> <span class="ow">in</span> <span class="n">trialset</span><span class="p">:</span>
-                    <span class="n">mult</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">break</span>
-            <span class="n">trialset</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">check</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">mult</span><span class="p">)</span>
-            <span class="n">notneedset</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">check</span><span class="p">)</span>
-
-    <span class="n">newtrialset</span> <span class="o">=</span> <span class="n">trialset</span> <span class="o">-</span> <span class="n">notneedset</span>
-
-    <span class="n">trialfunc</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">newtrialset</span><span class="p">:</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">coeffs</span><span class="p">)</span>
-        <span class="n">coefflist</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-        <span class="n">trialfunc</span> <span class="o">+=</span> <span class="n">c</span><span class="o">*</span><span class="n">i</span>
-
-    <span class="n">eqs</span> <span class="o">=</span> <span class="n">sub_func_doit</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">trialfunc</span><span class="p">)</span>
-
-    <span class="n">coeffsdict</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">trialset</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">trialset</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))))</span>
-
-    <span class="n">eqs</span> <span class="o">=</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">eqs</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">eqs</span><span class="p">):</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">separatevars</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="nb">dict</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">symbols</span><span class="o">=</span><span class="p">[</span><span class="n">x</span><span class="p">])</span>
-        <span class="n">coeffsdict</span><span class="p">[</span><span class="n">s</span><span class="p">[</span><span class="n">x</span><span class="p">]]</span> <span class="o">+=</span> <span class="n">s</span><span class="p">[</span><span class="s">&#39;coeff&#39;</span><span class="p">]</span>
-
-    <span class="n">coeffvals</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">coeffsdict</span><span class="o">.</span><span class="n">values</span><span class="p">()),</span> <span class="n">coefflist</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">coeffvals</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Could not solve &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot; using the &quot;</span> <span class="o">+</span>
-            <span class="s">&quot; method of undetermined coefficients (unable to solve for coefficients).&quot;</span><span class="p">)</span>
-
-    <span class="n">psol</span> <span class="o">=</span> <span class="n">trialfunc</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">coeffvals</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">Eq</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">gsol</span><span class="o">.</span><span class="n">rhs</span> <span class="o">+</span> <span class="n">psol</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">_undetermined_coefficients_match</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns a trial function match if undetermined coefficients can be</span>
-<span class="sd">    applied to expr, and None otherwise.</span>
-
-<span class="sd">    A trial expression can be found for an expression for use with the</span>
-<span class="sd">    method of undetermined coefficients if the expression is an</span>
-<span class="sd">    additive/multiplicative combination of constants, polynomials in x</span>
-<span class="sd">    (the independent variable of expr), sin(a*x + b), cos(a*x + b), and</span>
-<span class="sd">    exp(a*x) terms (in other words, it has a finite number of linearly</span>
-<span class="sd">    independent derivatives).</span>
-
-<span class="sd">    Note that you may still need to multiply each term returned here by</span>
-<span class="sd">    sufficient x to make it linearly independent with the solutions to</span>
-<span class="sd">    the homogeneous equation.</span>
-
-<span class="sd">    This is intended for internal use by undetermined_coefficients</span>
-<span class="sd">    hints.</span>
-
-<span class="sd">    SymPy currently has no way to convert sin(x)**n*cos(y)**m into a sum</span>
-<span class="sd">    of only sin(a*x) and cos(b*x) terms, so these are not implemented.</span>
-<span class="sd">    So, for example, you will need to manually convert sin(x)**2 into</span>
-<span class="sd">    (1 + cos(2*x))/2 to properly apply the method of undetermined</span>
-<span class="sd">    coefficients on it.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import log, exp</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.solvers.ode import _undetermined_coefficients_match</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; _undetermined_coefficients_match(9*x*exp(x) + exp(-x), x)</span>
-<span class="sd">    {&#39;test&#39;: True, &#39;trialset&#39;: set([x*exp(x), exp(-x), exp(x)])}</span>
-<span class="sd">    &gt;&gt;&gt; _undetermined_coefficients_match(log(x), x)</span>
-<span class="sd">    {&#39;test&#39;: False}</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span>
-    <span class="n">a</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">x</span><span class="p">])</span>
-    <span class="n">b</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;b&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">x</span><span class="p">])</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">powsimp</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;exp&#39;</span><span class="p">)</span>  <span class="c"># exp(x)*exp(2*x + 1) =&gt; exp(3*x + 1)</span>
-    <span class="n">retdict</span> <span class="o">=</span> <span class="p">{}</span>
-
-    <span class="k">def</span> <span class="nf">_test_term</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Test if expr fits the proper form for undetermined coefficients.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="n">_test_term</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">):</span>
-                <span class="n">foundtrig</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="c"># Make sure that there is only one trig function in the args.</span>
-                <span class="c"># See the docstring.</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">):</span>
-                        <span class="k">if</span> <span class="n">foundtrig</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="bp">False</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="n">foundtrig</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="n">_test_term</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Function</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span> <span class="ow">in</span> <span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">exp</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">b</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="bp">True</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_Symbol</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> \
-                <span class="n">expr</span><span class="o">.</span><span class="n">exp</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_number</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">b</span><span class="p">):</span>
-                <span class="k">return</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Symbol</span> <span class="ow">or</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">_get_trial_set</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">exprs</span><span class="o">=</span><span class="nb">set</span><span class="p">([])):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns a set of trial terms for undetermined coefficients.</span>
-
-<span class="sd">        The idea behind undetermined coefficients is that the terms</span>
-<span class="sd">        expression repeat themselves after a finite number of</span>
-<span class="sd">        derivatives, except for the coefficients (they are linearly</span>
-<span class="sd">        dependent).  So if we collect these, we should have the terms of</span>
-<span class="sd">        our trial function.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">def</span> <span class="nf">_remove_coefficient</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            Returns the expression without a coefficient.</span>
-
-<span class="sd">            Similar to expr.as_independent(x)[1], except it only works</span>
-<span class="sd">            multiplicatively.</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="c"># I was using the below match, but it doesn&#39;t always put all of the</span>
-            <span class="c"># coefficient in c.  c.f. 2**x*6*exp(x)*log(2)</span>
-            <span class="c"># The below code is probably cleaner anyway.</span>
-<span class="c">#            c = Wild(&#39;c&#39;, exclude=[x])</span>
-<span class="c">#            t = Wild(&#39;t&#39;)</span>
-<span class="c">#            r = expr.match(c*t)</span>
-            <span class="n">term</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-                        <span class="n">term</span> <span class="o">*=</span> <span class="n">i</span>
-            <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-                <span class="n">term</span> <span class="o">=</span> <span class="n">expr</span>
-            <span class="k">return</span> <span class="n">term</span>
-
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">_remove_coefficient</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="ow">in</span> <span class="n">exprs</span><span class="p">:</span>
-                    <span class="k">pass</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">exprs</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">_remove_coefficient</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-                    <span class="n">exprs</span> <span class="o">=</span> <span class="n">exprs</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="n">_get_trial_set</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">exprs</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">term</span> <span class="o">=</span> <span class="n">_remove_coefficient</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-            <span class="n">tmpset</span> <span class="o">=</span> <span class="n">exprs</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="nb">set</span><span class="p">([</span><span class="n">term</span><span class="p">]))</span>
-            <span class="n">oldset</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([])</span>
-            <span class="k">while</span> <span class="n">tmpset</span> <span class="o">!=</span> <span class="n">oldset</span><span class="p">:</span>
-                <span class="c"># If you get stuck in this loop, then _test_term is probably broken</span>
-                <span class="n">oldset</span> <span class="o">=</span> <span class="n">tmpset</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-                <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-                <span class="n">term</span> <span class="o">=</span> <span class="n">_remove_coefficient</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">term</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                    <span class="n">tmpset</span> <span class="o">=</span> <span class="n">tmpset</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="n">_get_trial_set</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">tmpset</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">tmpset</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-            <span class="n">exprs</span> <span class="o">=</span> <span class="n">tmpset</span>
-        <span class="k">return</span> <span class="n">exprs</span>
-
-    <span class="n">retdict</span><span class="p">[</span><span class="s">&#39;test&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">_test_term</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">retdict</span><span class="p">[</span><span class="s">&#39;test&#39;</span><span class="p">]:</span>
-        <span class="c"># Try to generate a list of trial solutions that will have the undetermined</span>
-        <span class="c"># coefficients.  Note that if any of these are not linearly independent</span>
-        <span class="c"># with any of the solutions to the homogeneous equation, then they will</span>
-        <span class="c"># need to be multiplied by sufficient x to make them so.  This function</span>
-        <span class="c"># DOES NOT do that (it doesn&#39;t even look at the homogeneous equation).</span>
-        <span class="n">retdict</span><span class="p">[</span><span class="s">&#39;trialset&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">_get_trial_set</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">retdict</span>
-
-
-<div class="viewcode-block" id="ode_nth_linear_constant_coeff_variation_of_parameters"><a class="viewcode-back" href="../../../modules/solvers/ode.html#sympy.solvers.ode.ode_nth_linear_constant_coeff_variation_of_parameters">[docs]</a><span class="k">def</span> <span class="nf">ode_nth_linear_constant_coeff_variation_of_parameters</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">match</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Solves an nth order linear differential equation with constant</span>
-<span class="sd">    coefficients using the method of variation of parameters.</span>
-
-<span class="sd">    This method works on any differential equations of the form</span>
-<span class="sd">    f(x)^(n) + a_(n-1)*f(x)^(n-1) + ... + a1*f&#39;(x) + a0*f(x) = P(x).</span>
-
-<span class="sd">    This method works by assuming that the particular solution takes the</span>
-<span class="sd">    form Sum(c_i(x)*y_i(x), (x, 1, n)), where y_i is the ith solution to</span>
-<span class="sd">    the homogeneous equation.  The solution is then solved using</span>
-<span class="sd">    Wronskian&#39;s and Cramer&#39;s Rule.  The particular solution is given by</span>
-<span class="sd">    Sum(Integral(W_i(x)/W(x), x)*y_i(x), (x, 1, n)), where W(x) is the</span>
-<span class="sd">    Wronskian of the fundamental system (the system of n linearly</span>
-<span class="sd">    independent solutions to the homogeneous equation), and W_i(x) is</span>
-<span class="sd">    the Wronskian of the fundamental system with the ith column replaced</span>
-<span class="sd">    with [0, 0, ..., 0, P(x)].</span>
-
-<span class="sd">    This method is general enough to solve any nth order inhomogeneous</span>
-<span class="sd">    linear differential equation with constant coefficients, but</span>
-<span class="sd">    sometimes SymPy cannot simplify the Wronskian well enough to</span>
-<span class="sd">    integrate it.  If this method hangs, try using the</span>
-<span class="sd">    &#39;nth_linear_constant_coeff_variation_of_parameters_Integral&#39; hint</span>
-<span class="sd">    and simplifying the integrals manually.  Also, prefer using</span>
-<span class="sd">    &#39;nth_linear_constant_coeff_undetermined_coefficients&#39; when it</span>
-<span class="sd">    applies, because it doesn&#39;t use integration, making it faster and</span>
-<span class="sd">    more reliable.</span>
-
-<span class="sd">    Warning, using simplify=False with</span>
-<span class="sd">    &#39;nth_linear_constant_coeff_variation_of_parameters&#39; in dsolve()</span>
-<span class="sd">    may cause it to hang, because it will not attempt to simplify</span>
-<span class="sd">    the Wronskian before integrating.  It is recommended that you only</span>
-<span class="sd">    use simplify=False with</span>
-<span class="sd">    &#39;nth_linear_constant_coeff_variation_of_parameters_Integral&#39; for</span>
-<span class="sd">    this method, especially if the solution to the homogeneous</span>
-<span class="sd">    equation has trigonometric functions in it.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, dsolve, pprint, exp, log</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(dsolve(f(x).diff(x, 3) - 3*f(x).diff(x, 2) +</span>
-<span class="sd">    ... 3*f(x).diff(x) - f(x) - exp(x)*log(x), f(x),</span>
-<span class="sd">    ... hint=&#39;nth_linear_constant_coeff_variation_of_parameters&#39;))</span>
-<span class="sd">           /                     3                \</span>
-<span class="sd">           |                2   x *(6*log(x) - 11)|  x</span>
-<span class="sd">    f(x) = |C1 + C2*x + C3*x  + ------------------|*e</span>
-<span class="sd">           \                            36        /</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - http://en.wikipedia.org/wiki/Variation_of_parameters</span>
-<span class="sd">    - http://planetmath.org/encyclopedia/VariationOfParameters.html</span>
-<span class="sd">    - M. Tenenbaum &amp; H. Pollard, &quot;Ordinary Differential Equations&quot;,</span>
-<span class="sd">      Dover 1963, pp. 233</span>
-
-<span class="sd">    # indirect doctest</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">gensol</span> <span class="o">=</span> <span class="n">ode_nth_linear_constant_coeff_homogeneous</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">match</span><span class="p">,</span>
-        <span class="n">returns</span><span class="o">=</span><span class="s">&#39;both&#39;</span><span class="p">)</span>
-    <span class="n">match</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">gensol</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">_solve_variation_of_parameters</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">match</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_solve_variation_of_parameters</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">match</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Helper function for the method of variation of parameters.</span>
-
-<span class="sd">    See the ode_nth_linear_constant_coeff_variation_of_parameters()</span>
-<span class="sd">    docstring for more information on this method.</span>
-
-<span class="sd">    match should be a dictionary that has the following keys:</span>
-<span class="sd">    &#39;list&#39; - A list of solutions to the homogeneous equation, such as</span>
-<span class="sd">         the list returned by</span>
-<span class="sd">         ode_nth_linear_constant_coeff_homogeneous(returns=&#39;list&#39;)</span>
-<span class="sd">    &#39;sol&#39; - The general solution, such as the solution returned by</span>
-<span class="sd">        ode_nth_linear_constant_coeff_homogeneous(returns=&#39;sol&#39;)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">x</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">func</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">match</span>
-    <span class="n">psol</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">gensols</span> <span class="o">=</span> <span class="n">r</span><span class="p">[</span><span class="s">&#39;list&#39;</span><span class="p">]</span>
-    <span class="n">gsol</span> <span class="o">=</span> <span class="n">r</span><span class="p">[</span><span class="s">&#39;sol&#39;</span><span class="p">]</span>
-    <span class="n">wr</span> <span class="o">=</span> <span class="n">wronskian</span><span class="p">(</span><span class="n">gensols</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">r</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;simplify&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-        <span class="n">wr</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">wr</span><span class="p">)</span>  <span class="c"># We need much better simplification for some ODEs.</span>
-        <span class="c">#                   See issue 1563, for example.</span>
-
-        <span class="c"># To reduce commonly occuring sin(x)**2 + cos(x)**2 to 1</span>
-        <span class="n">wr</span> <span class="o">=</span> <span class="n">trigsimp</span><span class="p">(</span><span class="n">wr</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">recursive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">wr</span><span class="p">:</span>
-        <span class="c"># The wronskian will be 0 iff the solutions are not linearly independent.</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Cannot find &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">order</span><span class="p">)</span> <span class="o">+</span>
-        <span class="s">&quot; solutions to the homogeneous equation nessesary to apply &quot;</span> <span class="o">+</span>
-        <span class="s">&quot;variation of parameters to &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot; (Wronskian == 0)&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">gensols</span><span class="p">)</span> <span class="o">!=</span> <span class="n">order</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Cannot find &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">order</span><span class="p">)</span> <span class="o">+</span>
-        <span class="s">&quot; solutions to the homogeneous equation nessesary to apply &quot;</span> <span class="o">+</span>
-        <span class="s">&quot;variation of parameters to &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span> <span class="o">+</span> <span class="s">&quot; (number of terms != order)&quot;</span><span class="p">)</span>
-    <span class="n">negoneterm</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">order</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">gensols</span><span class="p">:</span>
-        <span class="n">psol</span> <span class="o">+=</span> <span class="n">negoneterm</span><span class="o">*</span><span class="n">C</span><span class="o">.</span><span class="n">Integral</span><span class="p">(</span><span class="n">wronskian</span><span class="p">([</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">gensols</span> <span class="k">if</span> <span class="n">x</span> <span class="o">!=</span> <span class="n">i</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">r</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">/</span><span class="n">wr</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">i</span><span class="o">/</span><span class="n">r</span><span class="p">[</span><span class="n">order</span><span class="p">]</span>
-        <span class="n">negoneterm</span> <span class="o">*=</span> <span class="o">-</span><span class="mi">1</span>
-
-    <span class="k">if</span> <span class="n">r</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;simplify&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-        <span class="n">psol</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">psol</span><span class="p">)</span>
-        <span class="n">psol</span> <span class="o">=</span> <span class="n">trigsimp</span><span class="p">(</span><span class="n">psol</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">Eq</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">gsol</span><span class="o">.</span><span class="n">rhs</span> <span class="o">+</span> <span class="n">psol</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="ode_separable"><a class="viewcode-back" href="../../../modules/solvers/ode.html#sympy.solvers.ode.ode_separable">[docs]</a><span class="k">def</span> <span class="nf">ode_separable</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">order</span><span class="p">,</span> <span class="n">match</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-
-<span class="sd">    Solves separable 1st order differential equations.</span>
-
-<span class="sd">    This is any differential equation that can be written as</span>
-<span class="sd">    P(y)*dy/dx = Q(x). The solution can then just be found by</span>
-<span class="sd">    rearranging terms and integrating:</span>
-<span class="sd">    Integral(P(y), y) = Integral(Q(x), x). This hint uses separatevars()</span>
-<span class="sd">    as its back end, so if a separable equation is not caught by this</span>
-<span class="sd">    solver, it is most likely the fault of that function. separatevars()</span>
-<span class="sd">    is smart enough to do most expansion and factoring necessary to</span>
-<span class="sd">    convert a separable equation F(x, y) into the proper form P(x)*Q(y).</span>
-<span class="sd">    The general solution is::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Function, dsolve, Eq, pprint</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; a, b, c, d, f = list(map(Function, [&#39;a&#39;, &#39;b&#39;, &#39;c&#39;, &#39;d&#39;, &#39;f&#39;]))</span>
-<span class="sd">        &gt;&gt;&gt; genform = Eq(a(x)*b(f(x))*f(x).diff(x), c(x)*d(f(x)))</span>
-<span class="sd">        &gt;&gt;&gt; pprint(genform)</span>
-<span class="sd">                     d</span>
-<span class="sd">        a(x)*b(f(x))*--(f(x)) = c(x)*d(f(x))</span>
-<span class="sd">                     dx</span>
-<span class="sd">        &gt;&gt;&gt; pprint(dsolve(genform, f(x), hint=&#39;separable_Integral&#39;))</span>
-<span class="sd">             f(x)</span>
-<span class="sd">           /                  /</span>
-<span class="sd">          |                  |</span>
-<span class="sd">          |  b(y)            | c(x)</span>
-<span class="sd">          |  ---- dy = C1 +  | ---- dx</span>
-<span class="sd">          |  d(y)            | a(x)</span>
-<span class="sd">          |                  |</span>
-<span class="sd">         /                  /</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, dsolve, Eq</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(dsolve(Eq(f(x)*f(x).diff(x) + x, 3*x*f(x)**2), f(x),</span>
-<span class="sd">    ... hint=&#39;separable&#39;, simplify=False))</span>
-<span class="sd">       /   2       \         2</span>
-<span class="sd">    log\3*f (x) - 1/        x</span>
-<span class="sd">    ---------------- = C1 + --</span>
-<span class="sd">           6                2</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    - M. Tenenbaum &amp; H. Pollard, &quot;Ordinary Differential Equations&quot;,</span>
-<span class="sd">      Dover 1963, pp. 52</span>
-
-<span class="sd">    # indirect doctest</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">func</span>
-    <span class="n">C1</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;C1&#39;</span><span class="p">)</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">match</span>  <span class="c"># {&#39;m1&#39;:m1, &#39;m2&#39;:m2, &#39;y&#39;:y}</span>
-    <span class="k">return</span> <span class="n">Eq</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Integral</span><span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;m2&#39;</span><span class="p">][</span><span class="s">&#39;coeff&#39;</span><span class="p">]</span><span class="o">*</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;m2&#39;</span><span class="p">][</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;y&#39;</span><span class="p">]]</span><span class="o">/</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;m1&#39;</span><span class="p">][</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;y&#39;</span><span class="p">]],</span>
-        <span class="p">(</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;y&#39;</span><span class="p">],</span> <span class="bp">None</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))),</span> <span class="n">C</span><span class="o">.</span><span class="n">Integral</span><span class="p">(</span><span class="o">-</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;m1&#39;</span><span class="p">][</span><span class="s">&#39;coeff&#39;</span><span class="p">]</span><span class="o">*</span><span class="n">r</span><span class="p">[</span><span class="s">&#39;m1&#39;</span><span class="p">][</span><span class="n">x</span><span class="p">]</span><span class="o">/</span>
-        <span class="n">r</span><span class="p">[</span><span class="s">&#39;m2&#39;</span><span class="p">][</span><span class="n">x</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">C1</span><span class="p">)</span></div>
-</pre></div>
-
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-  <h1>Source code for sympy.solvers.pde</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Analytical methods for solving Partial Differential Equations</span>
-<span class="sd">Currently implemented methods:</span>
-<span class="sd">    - separation of variables - pde_separate</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Eq</span><span class="p">,</span> <span class="n">Equality</span>
-<span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">simplify</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="nb">reduce</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">has_dups</span>
-
-<span class="kn">import</span> <span class="nn">operator</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-
-<div class="viewcode-block" id="pde_separate"><a class="viewcode-back" href="../../../modules/solvers/solvers.html#sympy.solvers.pde.pde_separate">[docs]</a><span class="k">def</span> <span class="nf">pde_separate</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">fun</span><span class="p">,</span> <span class="n">sep</span><span class="p">,</span> <span class="n">strategy</span><span class="o">=</span><span class="s">&#39;mul&#39;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Separate variables in partial differential equation either by additive</span>
-<span class="sd">    or multiplicative separation approach. It tries to rewrite an equation so</span>
-<span class="sd">    that one of the specified variables occurs on a different side of the</span>
-<span class="sd">    equation than the others.</span>
-
-<span class="sd">    :param eq: Partial differential equation</span>
-
-<span class="sd">    :param fun: Original function F(x, y, z)</span>
-
-<span class="sd">    :param sep: List of separated functions [X(x), u(y, z)]</span>
-
-<span class="sd">    :param strategy: Separation strategy. You can choose between additive</span>
-<span class="sd">        separation (&#39;add&#39;) and multiplicative separation (&#39;mul&#39;) which is</span>
-<span class="sd">        default.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import E, Eq, Function, pde_separate, Derivative as D</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, t</span>
-<span class="sd">    &gt;&gt;&gt; u, X, T = list(map(Function, &#39;uXT&#39;))</span>
-
-<span class="sd">    &gt;&gt;&gt; eq = Eq(D(u(x, t), x), E**(u(x, t))*D(u(x, t), t))</span>
-<span class="sd">    &gt;&gt;&gt; pde_separate(eq, u(x, t), [X(x), T(t)], strategy=&#39;add&#39;)</span>
-<span class="sd">    [exp(-X(x))*Derivative(X(x), x), exp(T(t))*Derivative(T(t), t)]</span>
-
-<span class="sd">    &gt;&gt;&gt; eq = Eq(D(u(x, t), x, 2), D(u(x, t), t, 2))</span>
-<span class="sd">    &gt;&gt;&gt; pde_separate(eq, u(x, t), [X(x), T(t)], strategy=&#39;mul&#39;)</span>
-<span class="sd">    [Derivative(X(x), x, x)/X(x), Derivative(T(t), t, t)/T(t)]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    pde_separate_add, pde_separate_mul</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">do_add</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="k">if</span> <span class="n">strategy</span> <span class="o">==</span> <span class="s">&#39;add&#39;</span><span class="p">:</span>
-        <span class="n">do_add</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="k">elif</span> <span class="n">strategy</span> <span class="o">==</span> <span class="s">&#39;mul&#39;</span><span class="p">:</span>
-        <span class="n">do_add</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">assert</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Unknown strategy: </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">strategy</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">Equality</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">eq</span><span class="o">.</span><span class="n">rhs</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">pde_separate</span><span class="p">(</span><span class="n">Eq</span><span class="p">(</span><span class="n">eq</span><span class="o">.</span><span class="n">lhs</span> <span class="o">-</span> <span class="n">eq</span><span class="o">.</span><span class="n">rhs</span><span class="p">),</span> <span class="n">fun</span><span class="p">,</span> <span class="n">sep</span><span class="p">,</span> <span class="n">strategy</span><span class="p">)</span>
-    <span class="k">assert</span> <span class="n">eq</span><span class="o">.</span><span class="n">rhs</span> <span class="o">==</span> <span class="mi">0</span>
-
-    <span class="c"># Handle arguments</span>
-    <span class="n">orig_args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">fun</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-    <span class="n">subs_args</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">sep</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">args</span><span class="p">)):</span>
-            <span class="n">subs_args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-
-    <span class="k">if</span> <span class="n">do_add</span><span class="p">:</span>
-        <span class="n">functions</span> <span class="o">=</span> <span class="nb">reduce</span><span class="p">(</span><span class="n">operator</span><span class="o">.</span><span class="n">add</span><span class="p">,</span> <span class="n">sep</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">functions</span> <span class="o">=</span> <span class="nb">reduce</span><span class="p">(</span><span class="n">operator</span><span class="o">.</span><span class="n">mul</span><span class="p">,</span> <span class="n">sep</span><span class="p">)</span>
-
-    <span class="c"># Check whether variables match</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">subs_args</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">orig_args</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Variable counts do not match&quot;</span><span class="p">)</span>
-    <span class="c"># Check for duplicate arguments like  [X(x), u(x, y)]</span>
-    <span class="k">if</span> <span class="n">has_dups</span><span class="p">(</span><span class="n">subs_args</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Duplicate substitution arguments detected&quot;</span><span class="p">)</span>
-    <span class="c"># Check whether the variables match</span>
-    <span class="k">if</span> <span class="nb">set</span><span class="p">(</span><span class="n">orig_args</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">set</span><span class="p">(</span><span class="n">subs_args</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Arguments do not match&quot;</span><span class="p">)</span>
-
-    <span class="c"># Substitute original function with separated...</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="n">eq</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">fun</span><span class="p">,</span> <span class="n">functions</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-
-    <span class="c"># Divide by terms when doing multiplicative separation</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">do_add</span><span class="p">:</span>
-        <span class="n">eq</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">result</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">eq</span> <span class="o">+=</span> <span class="n">i</span><span class="o">/</span><span class="n">functions</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">eq</span>
-
-    <span class="n">svar</span> <span class="o">=</span> <span class="n">subs_args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">dvar</span> <span class="o">=</span> <span class="n">subs_args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-    <span class="k">return</span> <span class="n">_separate</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">svar</span><span class="p">,</span> <span class="n">dvar</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="pde_separate_add"><a class="viewcode-back" href="../../../modules/solvers/solvers.html#sympy.solvers.pde.pde_separate_add">[docs]</a><span class="k">def</span> <span class="nf">pde_separate_add</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">fun</span><span class="p">,</span> <span class="n">sep</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Helper function for searching additive separable solutions.</span>
-
-<span class="sd">    Consider an equation of two independent variables x, y and a dependent</span>
-<span class="sd">    variable w, we look for the product of two functions depending on different</span>
-<span class="sd">    arguments:</span>
-
-<span class="sd">    `w(x, y, z) = X(x) + y(y, z)`</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import E, Eq, Function, pde_separate_add, Derivative as D</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, t</span>
-<span class="sd">    &gt;&gt;&gt; u, X, T = list(map(Function, &#39;uXT&#39;))</span>
-
-<span class="sd">    &gt;&gt;&gt; eq = Eq(D(u(x, t), x), E**(u(x, t))*D(u(x, t), t))</span>
-<span class="sd">    &gt;&gt;&gt; pde_separate_add(eq, u(x, t), [X(x), T(t)])</span>
-<span class="sd">    [exp(-X(x))*Derivative(X(x), x), exp(T(t))*Derivative(T(t), t)]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">pde_separate</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">fun</span><span class="p">,</span> <span class="n">sep</span><span class="p">,</span> <span class="n">strategy</span><span class="o">=</span><span class="s">&#39;add&#39;</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="pde_separate_mul"><a class="viewcode-back" href="../../../modules/solvers/solvers.html#sympy.solvers.pde.pde_separate_mul">[docs]</a><span class="k">def</span> <span class="nf">pde_separate_mul</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">fun</span><span class="p">,</span> <span class="n">sep</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Helper function for searching multiplicative separable solutions.</span>
-
-<span class="sd">    Consider an equation of two independent variables x, y and a dependent</span>
-<span class="sd">    variable w, we look for the product of two functions depending on different</span>
-<span class="sd">    arguments:</span>
-
-<span class="sd">    `w(x, y, z) = X(x)*u(y, z)`</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, Eq, pde_separate_mul, Derivative as D</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; u, X, Y = list(map(Function, &#39;uXY&#39;))</span>
-
-<span class="sd">    &gt;&gt;&gt; eq = Eq(D(u(x, y), x, 2), D(u(x, y), y, 2))</span>
-<span class="sd">    &gt;&gt;&gt; pde_separate_mul(eq, u(x, y), [X(x), Y(y)])</span>
-<span class="sd">    [Derivative(X(x), x, x)/X(x), Derivative(Y(y), y, y)/Y(y)]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">pde_separate</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">fun</span><span class="p">,</span> <span class="n">sep</span><span class="p">,</span> <span class="n">strategy</span><span class="o">=</span><span class="s">&#39;mul&#39;</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_separate</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">dep</span><span class="p">,</span> <span class="n">others</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Separate expression into two parts based on dependencies of variables.&quot;&quot;&quot;</span>
-
-    <span class="c"># FIRST PASS</span>
-    <span class="c"># Extract derivatives depending our separable variable...</span>
-    <span class="n">terms</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">eq</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">term</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">term</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">i</span><span class="o">.</span><span class="n">is_Derivative</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">i</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">others</span><span class="p">):</span>
-                    <span class="n">terms</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-                    <span class="k">continue</span>
-        <span class="k">elif</span> <span class="n">term</span><span class="o">.</span><span class="n">is_Derivative</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">term</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">others</span><span class="p">):</span>
-            <span class="n">terms</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-    <span class="c"># Find the factor that we need to divide by</span>
-    <span class="n">div</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">terms</span><span class="p">:</span>
-        <span class="n">ext</span><span class="p">,</span> <span class="n">sep</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">dep</span><span class="p">)</span>
-        <span class="c"># Failed?</span>
-        <span class="k">if</span> <span class="n">sep</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">others</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="n">div</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">ext</span><span class="p">)</span>
-    <span class="c"># FIXME: Find lcm() of all the divisors and divide with it, instead of</span>
-    <span class="c"># current hack :(</span>
-    <span class="c"># http://code.google.com/p/sympy/issues/detail?id=1498</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">div</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">final</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">eq</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">eqn</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">div</span><span class="p">:</span>
-                <span class="n">eqn</span> <span class="o">+=</span> <span class="n">term</span> <span class="o">/</span> <span class="n">i</span>
-            <span class="n">final</span> <span class="o">+=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">eqn</span><span class="p">)</span>
-        <span class="n">eq</span> <span class="o">=</span> <span class="n">final</span>
-
-    <span class="c"># SECOND PASS - separate the derivatives</span>
-    <span class="n">div</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="n">lhs</span> <span class="o">=</span> <span class="n">rhs</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">eq</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-        <span class="c"># Check, whether we have already term with independent variable...</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">term</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">others</span><span class="p">):</span>
-            <span class="n">lhs</span> <span class="o">+=</span> <span class="n">term</span>
-            <span class="k">continue</span>
-        <span class="c"># ...otherwise, try to separate</span>
-        <span class="n">temp</span><span class="p">,</span> <span class="n">sep</span> <span class="o">=</span> <span class="n">term</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">dep</span><span class="p">)</span>
-        <span class="c"># Failed?</span>
-        <span class="k">if</span> <span class="n">sep</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">others</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="c"># Extract the divisors</span>
-        <span class="n">div</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">sep</span><span class="p">)</span>
-        <span class="n">rhs</span> <span class="o">-=</span> <span class="n">term</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-    <span class="c"># Do the division</span>
-    <span class="n">fulldiv</span> <span class="o">=</span> <span class="nb">reduce</span><span class="p">(</span><span class="n">operator</span><span class="o">.</span><span class="n">add</span><span class="p">,</span> <span class="n">div</span><span class="p">)</span>
-    <span class="n">lhs</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">lhs</span><span class="o">/</span><span class="n">fulldiv</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-    <span class="n">rhs</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">rhs</span><span class="o">/</span><span class="n">fulldiv</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-    <span class="c"># ...and check whether we were successful :)</span>
-    <span class="k">if</span> <span class="n">lhs</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">others</span><span class="p">)</span> <span class="ow">or</span> <span class="n">rhs</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">dep</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">None</span>
-    <span class="k">return</span> <span class="p">[</span><span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="p">]</span>
-</pre></div>
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-  <h1>Source code for sympy.solvers.polysys</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Solvers of systems of polynomial equations. &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">groebner</span><span class="p">,</span> <span class="n">roots</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polytools</span> <span class="kn">import</span> <span class="n">parallel_poly_from_expr</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="p">(</span><span class="n">ComputationFailed</span><span class="p">,</span>
-    <span class="n">PolificationFailed</span><span class="p">,</span> <span class="n">CoercionFailed</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">postfixes</span>
-<span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">rcollect</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">S</span>
-
-
-<span class="k">class</span> <span class="nc">SolveFailed</span><span class="p">(</span><span class="ne">Exception</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Raised when solver&#39;s conditions weren&#39;t met. &quot;&quot;&quot;</span>
-
-
-<div class="viewcode-block" id="solve_poly_system"><a class="viewcode-back" href="../../../modules/solvers/solvers.html#sympy.solvers.polysys.solve_poly_system">[docs]</a><span class="k">def</span> <span class="nf">solve_poly_system</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Solve a system of polynomial equations.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import solve_poly_system</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">    &gt;&gt;&gt; solve_poly_system([x*y - 2*y, 2*y**2 - x**2], x, y)</span>
-<span class="sd">    [(0, 0), (2, -sqrt(2)), (2, sqrt(2))]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">polys</span><span class="p">,</span> <span class="n">opt</span> <span class="o">=</span> <span class="n">parallel_poly_from_expr</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolificationFailed</span> <span class="k">as</span> <span class="n">exc</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">ComputationFailed</span><span class="p">(</span><span class="s">&#39;solve_poly_system&#39;</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">seq</span><span class="p">),</span> <span class="n">exc</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">polys</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="n">f</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">polys</span>
-
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">degree_list</span><span class="p">()</span>
-        <span class="n">c</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">degree_list</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="n">a</span> <span class="o">&lt;=</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">b</span> <span class="o">&lt;=</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">c</span> <span class="o">&lt;=</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">d</span> <span class="o">&lt;=</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">solve_biquadratic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">SolveFailed</span><span class="p">:</span>
-                <span class="k">pass</span>
-
-    <span class="k">return</span> <span class="n">solve_generic</span><span class="p">(</span><span class="n">polys</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">solve_biquadratic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">opt</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Solve a system of two bivariate quadratic polynomial equations.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys import Options, Poly</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.solvers.polysys import solve_biquadratic</span>
-<span class="sd">    &gt;&gt;&gt; NewOption = Options((x, y), {&#39;domain&#39;: &#39;ZZ&#39;})</span>
-
-<span class="sd">    &gt;&gt;&gt; a = Poly(y**2 - 4 + x, y, x, domain=&#39;ZZ&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; b = Poly(y*2 + 3*x - 7, y, x, domain=&#39;ZZ&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; solve_biquadratic(a, b, NewOption)</span>
-<span class="sd">    [(1/3, 3), (41/27, 11/9)]</span>
-
-<span class="sd">    &gt;&gt;&gt; a = Poly(y + x**2 - 3, y, x, domain=&#39;ZZ&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; b = Poly(-y + x - 4, y, x, domain=&#39;ZZ&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; solve_biquadratic(a, b, NewOption)</span>
-<span class="sd">    [(-sqrt(29)/2 + 7/2, -sqrt(29)/2 - 1/2), (sqrt(29)/2 + 7/2, -1/2 + \</span>
-<span class="sd">      sqrt(29)/2)]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">G</span> <span class="o">=</span> <span class="n">groebner</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">])</span>
-
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">G</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_ground</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="n">SolveFailed</span>
-
-    <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">G</span>
-    <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">opt</span><span class="o">.</span><span class="n">gens</span>
-
-    <span class="n">p</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">expand</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-    <span class="n">q</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">ltrim</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-
-    <span class="n">p_roots</span> <span class="o">=</span> <span class="p">[</span> <span class="n">rcollect</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span> <span class="k">for</span> <span class="n">expr</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">roots</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span> <span class="p">]</span>
-    <span class="n">q_roots</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">roots</span><span class="p">(</span><span class="n">q</span><span class="p">)</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-
-    <span class="n">solutions</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">q_root</span> <span class="ow">in</span> <span class="n">q_roots</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">p_root</span> <span class="ow">in</span> <span class="n">p_roots</span><span class="p">:</span>
-            <span class="n">solution</span> <span class="o">=</span> <span class="p">(</span><span class="n">p_root</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">q_root</span><span class="p">),</span> <span class="n">q_root</span><span class="p">)</span>
-            <span class="n">solutions</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">solution</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">solutions</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">solve_generic</span><span class="p">(</span><span class="n">polys</span><span class="p">,</span> <span class="n">opt</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Solve a generic system of polynomial equations.</span>
-
-<span class="sd">    Returns all possible solutions over C[x_1, x_2, ..., x_m] of a</span>
-<span class="sd">    set F = { f_1, f_2, ..., f_n } of polynomial equations,  using</span>
-<span class="sd">    Groebner basis approach. For now only zero-dimensional systems</span>
-<span class="sd">    are supported, which means F can have at most a finite number</span>
-<span class="sd">    of solutions.</span>
-
-<span class="sd">    The algorithm works by the fact that, supposing G is the basis</span>
-<span class="sd">    of F with respect to an elimination order  (here lexicographic</span>
-<span class="sd">    order is used), G and F generate the same ideal, they have the</span>
-<span class="sd">    same set of solutions. By the elimination property,  if G is a</span>
-<span class="sd">    reduced, zero-dimensional Groebner basis, then there exists an</span>
-<span class="sd">    univariate polynomial in G (in its last variable). This can be</span>
-<span class="sd">    solved by computing its roots. Substituting all computed roots</span>
-<span class="sd">    for the last (eliminated) variable in other elements of G, new</span>
-<span class="sd">    polynomial system is generated. Applying the above procedure</span>
-<span class="sd">    recursively, a finite number of solutions can be found.</span>
-
-<span class="sd">    The ability of finding all solutions by this procedure depends</span>
-<span class="sd">    on the root finding algorithms. If no solutions were found, it</span>
-<span class="sd">    means only that roots() failed, but the system is solvable. To</span>
-<span class="sd">    overcome this difficulty use numerical algorithms instead.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    .. [Buchberger01] B. Buchberger, Groebner Bases: A Short</span>
-<span class="sd">    Introduction for Systems Theorists, In: R. Moreno-Diaz,</span>
-<span class="sd">    B. Buchberger, J.L. Freire, Proceedings of EUROCAST&#39;01,</span>
-<span class="sd">    February, 2001</span>
-
-<span class="sd">    .. [Cox97] D. Cox, J. Little, D. O&#39;Shea, Ideals, Varieties</span>
-<span class="sd">    and Algorithms, Springer, Second Edition, 1997, pp. 112</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.polys import Poly, Options</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.solvers.polysys import solve_generic</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; NewOption = Options((x, y), {&#39;domain&#39;: &#39;ZZ&#39;})</span>
-
-<span class="sd">    &gt;&gt;&gt; a = Poly(x - y + 5, x, y, domain=&#39;ZZ&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; b = Poly(x + y - 3, x, y, domain=&#39;ZZ&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; solve_generic([a, b], NewOption)</span>
-<span class="sd">    [(-1, 4)]</span>
-
-<span class="sd">    &gt;&gt;&gt; a = Poly(x - 2*y + 5, x, y, domain=&#39;ZZ&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; b = Poly(2*x - y - 3, x, y, domain=&#39;ZZ&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; solve_generic([a, b], NewOption)</span>
-<span class="sd">    [(11/3, 13/3)]</span>
-
-<span class="sd">    &gt;&gt;&gt; a = Poly(x**2 + y, x, y, domain=&#39;ZZ&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; b = Poly(x + y*4, x, y, domain=&#39;ZZ&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; solve_generic([a, b], NewOption)</span>
-<span class="sd">    [(0, 0), (1/4, -1/16)]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">_is_univariate</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns True if &#39;f&#39; is univariate in its last variable. &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">monom</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">monoms</span><span class="p">():</span>
-            <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">m</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">monom</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]):</span>
-                <span class="k">return</span> <span class="bp">False</span>
-
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_subs_root</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gen</span><span class="p">,</span> <span class="n">zero</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Replace generator with a root so that the result is nice. &quot;&quot;&quot;</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">({</span><span class="n">gen</span><span class="p">:</span> <span class="n">zero</span><span class="p">})</span>
-
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">degree</span><span class="p">(</span><span class="n">gen</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">p</span>
-
-    <span class="k">def</span> <span class="nf">_solve_reduced_system</span><span class="p">(</span><span class="n">system</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">entry</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Recursively solves reduced polynomial systems. &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">system</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">zeros</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">roots</span><span class="p">(</span><span class="n">system</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">gens</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-            <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">zero</span><span class="p">,)</span> <span class="k">for</span> <span class="n">zero</span> <span class="ow">in</span> <span class="n">zeros</span> <span class="p">]</span>
-
-        <span class="n">basis</span> <span class="o">=</span> <span class="n">groebner</span><span class="p">(</span><span class="n">system</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">polys</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">basis</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">basis</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_ground</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">entry</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="n">univariate</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">filter</span><span class="p">(</span><span class="n">_is_univariate</span><span class="p">,</span> <span class="n">basis</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">univariate</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">univariate</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;only zero-dimensional systems supported (finite number of solutions)&quot;</span><span class="p">)</span>
-
-        <span class="n">gens</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">gens</span>
-        <span class="n">gen</span> <span class="o">=</span> <span class="n">gens</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-
-        <span class="n">zeros</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">roots</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">ltrim</span><span class="p">(</span><span class="n">gen</span><span class="p">))</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">zeros</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[]</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">basis</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[</span> <span class="p">(</span><span class="n">zero</span><span class="p">,)</span> <span class="k">for</span> <span class="n">zero</span> <span class="ow">in</span> <span class="n">zeros</span> <span class="p">]</span>
-
-        <span class="n">solutions</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">for</span> <span class="n">zero</span> <span class="ow">in</span> <span class="n">zeros</span><span class="p">:</span>
-            <span class="n">new_system</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">new_gens</span> <span class="o">=</span> <span class="n">gens</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-
-            <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">basis</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-                <span class="n">eq</span> <span class="o">=</span> <span class="n">_subs_root</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">gen</span><span class="p">,</span> <span class="n">zero</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">eq</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                    <span class="n">new_system</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">solution</span> <span class="ow">in</span> <span class="n">_solve_reduced_system</span><span class="p">(</span><span class="n">new_system</span><span class="p">,</span> <span class="n">new_gens</span><span class="p">):</span>
-                <span class="n">solutions</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">solution</span> <span class="o">+</span> <span class="p">(</span><span class="n">zero</span><span class="p">,))</span>
-
-        <span class="k">return</span> <span class="n">solutions</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">_solve_reduced_system</span><span class="p">(</span><span class="n">polys</span><span class="p">,</span> <span class="n">opt</span><span class="o">.</span><span class="n">gens</span><span class="p">,</span> <span class="n">entry</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">CoercionFailed</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-    <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-
-<div class="viewcode-block" id="solve_triangulated"><a class="viewcode-back" href="../../../modules/solvers/solvers.html#sympy.solvers.polysys.solve_triangulated">[docs]</a><span class="k">def</span> <span class="nf">solve_triangulated</span><span class="p">(</span><span class="n">polys</span><span class="p">,</span> <span class="o">*</span><span class="n">gens</span><span class="p">,</span> <span class="o">**</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Solve a polynomial system using Gianni-Kalkbrenner algorithm.</span>
-
-<span class="sd">    The algorithm proceeds by computing one Groebner basis in the ground</span>
-<span class="sd">    domain and then by iteratively computing polynomial factorizations in</span>
-<span class="sd">    appropriately constructed algebraic extensions of the ground domain.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.solvers.polysys import solve_triangulated</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-
-<span class="sd">    &gt;&gt;&gt; F = [x**2 + y + z - 1, x + y**2 + z - 1, x + y + z**2 - 1]</span>
-
-<span class="sd">    &gt;&gt;&gt; solve_triangulated(F, x, y, z)</span>
-<span class="sd">    [(0, 0, 1), (0, 1, 0), (1, 0, 0)]</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    1. Patrizia Gianni, Teo Mora, Algebraic Solution of System of</span>
-<span class="sd">    Polynomial Equations using Groebner Bases, AAECC-5 on Applied Algebra,</span>
-<span class="sd">    Algebraic Algorithms and Error-Correcting Codes, LNCS 356 247--257, 1989</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">G</span> <span class="o">=</span> <span class="n">groebner</span><span class="p">(</span><span class="n">polys</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">polys</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">G</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">G</span><span class="p">))</span>
-
-    <span class="n">domain</span> <span class="o">=</span> <span class="n">args</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;domain&#39;</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">domain</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">g</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
-            <span class="n">G</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">set_domain</span><span class="p">(</span><span class="n">domain</span><span class="p">)</span>
-
-    <span class="n">f</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">G</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">ltrim</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">G</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-    <span class="n">dom</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">get_domain</span><span class="p">()</span>
-
-    <span class="n">zeros</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">ground_roots</span><span class="p">()</span>
-    <span class="n">solutions</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([])</span>
-
-    <span class="k">for</span> <span class="n">zero</span> <span class="ow">in</span> <span class="n">zeros</span><span class="p">:</span>
-        <span class="n">solutions</span><span class="o">.</span><span class="n">add</span><span class="p">(((</span><span class="n">zero</span><span class="p">,),</span> <span class="n">dom</span><span class="p">))</span>
-
-    <span class="n">var_seq</span> <span class="o">=</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">gens</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-    <span class="n">vars_seq</span> <span class="o">=</span> <span class="n">postfixes</span><span class="p">(</span><span class="n">gens</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-
-    <span class="k">for</span> <span class="n">var</span><span class="p">,</span> <span class="nb">vars</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">var_seq</span><span class="p">,</span> <span class="n">vars_seq</span><span class="p">):</span>
-        <span class="n">_solutions</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([])</span>
-
-        <span class="k">for</span> <span class="n">values</span><span class="p">,</span> <span class="n">dom</span> <span class="ow">in</span> <span class="n">solutions</span><span class="p">:</span>
-            <span class="n">H</span><span class="p">,</span> <span class="n">mapping</span> <span class="o">=</span> <span class="p">[],</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="nb">vars</span><span class="p">,</span> <span class="n">values</span><span class="p">))</span>
-
-            <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
-                <span class="n">_vars</span> <span class="o">=</span> <span class="p">(</span><span class="n">var</span><span class="p">,)</span> <span class="o">+</span> <span class="nb">vars</span>
-
-                <span class="k">if</span> <span class="n">g</span><span class="o">.</span><span class="n">has_only_gens</span><span class="p">(</span><span class="o">*</span><span class="n">_vars</span><span class="p">)</span> <span class="ow">and</span> <span class="n">g</span><span class="o">.</span><span class="n">degree</span><span class="p">(</span><span class="n">var</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">h</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">ltrim</span><span class="p">(</span><span class="n">var</span><span class="p">)</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">mapping</span><span class="p">))</span>
-
-                    <span class="k">if</span> <span class="n">g</span><span class="o">.</span><span class="n">degree</span><span class="p">(</span><span class="n">var</span><span class="p">)</span> <span class="o">==</span> <span class="n">h</span><span class="o">.</span><span class="n">degree</span><span class="p">():</span>
-                        <span class="n">H</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-
-            <span class="n">p</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">H</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">h</span><span class="p">:</span> <span class="n">h</span><span class="o">.</span><span class="n">degree</span><span class="p">())</span>
-            <span class="n">zeros</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">ground_roots</span><span class="p">()</span>
-
-            <span class="k">for</span> <span class="n">zero</span> <span class="ow">in</span> <span class="n">zeros</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">zero</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                    <span class="n">dom_zero</span> <span class="o">=</span> <span class="n">dom</span><span class="o">.</span><span class="n">algebraic_field</span><span class="p">(</span><span class="n">zero</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">dom_zero</span> <span class="o">=</span> <span class="n">dom</span>
-
-                <span class="n">_solutions</span><span class="o">.</span><span class="n">add</span><span class="p">(((</span><span class="n">zero</span><span class="p">,)</span> <span class="o">+</span> <span class="n">values</span><span class="p">,</span> <span class="n">dom_zero</span><span class="p">))</span>
-
-        <span class="n">solutions</span> <span class="o">=</span> <span class="n">_solutions</span>
-
-    <span class="n">solutions</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">solutions</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">solution</span><span class="p">,</span> <span class="n">_</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">solutions</span><span class="p">):</span>
-        <span class="n">solutions</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">solution</span>
-
-    <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">solutions</span><span class="p">)</span></div>
-</pre></div>
-
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-  <h1>Source code for sympy.solvers.recurr</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;This module is intended for solving recurrences or, in other words,</span>
-<span class="sd">   difference equations. Currently supported are linear, inhomogeneous</span>
-<span class="sd">   equations with polynomial or rational coefficients.</span>
-
-<span class="sd">   The solutions are obtained among polynomials, rational functions,</span>
-<span class="sd">   hypergeometric terms, or combinations of hypergeometric term which</span>
-<span class="sd">   are pairwise dissimilar.</span>
-
-<span class="sd">   rsolve_X functions were meant as a low level interface for rsolve()</span>
-<span class="sd">   which would use Mathematica&#39;s syntax.</span>
-
-<span class="sd">   Given a recurrence relation:</span>
-
-<span class="sd">      a_{k}(n) y(n+k) + a_{k-1}(n) y(n+k-1) + ... + a_{0}(n) y(n) = f(n)</span>
-
-<span class="sd">   where k &gt; 0 and a_{i}(n) are polynomials in n. To use rsolve_X we need</span>
-<span class="sd">   to put all coefficients in to a list L of k+1 elements the following</span>
-<span class="sd">   way:</span>
-
-<span class="sd">      L = [ a_{0}(n), ..., a_{k-1}(n), a_{k}(n) ]</span>
-
-<span class="sd">   where L[i], for i=0..k, maps to a_{i}(n) y(n+i) (y(n+i) is implicit).</span>
-
-<span class="sd">   For example if we would like to compute m-th Bernoulli polynomial up to</span>
-<span class="sd">   a constant (example was taken from rsolve_poly docstring), then we would</span>
-<span class="sd">   use b(n+1) - b(n) == m*n**(m-1) recurrence, which has solution b(n) =</span>
-<span class="sd">   B_m + C.</span>
-
-<span class="sd">   Then L = [-1, 1] and f(n) = m*n**(m-1) and finally for m=4:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, bernoulli, rsolve_poly</span>
-<span class="sd">    &gt;&gt;&gt; n = Symbol(&#39;n&#39;, integer=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; rsolve_poly([-1, 1], 4*n**3, n)</span>
-<span class="sd">    C0 + n**4 - 2*n**3 + n**2</span>
-
-<span class="sd">    &gt;&gt;&gt; bernoulli(4, n)</span>
-<span class="sd">    n**4 - 2*n**3 + n**2 - 1/30</span>
-
-<span class="sd">   For the sake of completeness, f(n) can be:</span>
-
-<span class="sd">    [1] a polynomial              -&gt; rsolve_poly</span>
-<span class="sd">    [2] a rational function       -&gt; rsolve_ratio</span>
-<span class="sd">    [3] a hypergeometric function  -&gt; rsolve_hyper</span>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.singleton</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">Rational</span>
-<span class="kn">from</span> <span class="nn">sympy.core.symbol</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Wild</span><span class="p">,</span> <span class="n">Dummy</span>
-<span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Equality</span>
-<span class="kn">from</span> <span class="nn">sympy.core.add</span> <span class="kn">import</span> <span class="n">Add</span>
-<span class="kn">from</span> <span class="nn">sympy.core.mul</span> <span class="kn">import</span> <span class="n">Mul</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">sympify</span>
-
-<span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">hypersimp</span><span class="p">,</span> <span class="n">hypersimilar</span>
-<span class="kn">from</span> <span class="nn">sympy.solvers</span> <span class="kn">import</span> <span class="n">solve</span><span class="p">,</span> <span class="n">solve_undetermined_coeffs</span>
-<span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">quo</span><span class="p">,</span> <span class="n">gcd</span><span class="p">,</span> <span class="n">lcm</span><span class="p">,</span> <span class="n">roots</span><span class="p">,</span> <span class="n">resultant</span>
-<span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">binomial</span><span class="p">,</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">FallingFactorial</span><span class="p">,</span> <span class="n">RisingFactorial</span>
-<span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">casoratian</span>
-<span class="kn">from</span> <span class="nn">sympy.concrete</span> <span class="kn">import</span> <span class="n">product</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">default_sort_key</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">numbered_symbols</span>
-
-
-<div class="viewcode-block" id="rsolve_poly"><a class="viewcode-back" href="../../../modules/solvers/solvers.html#sympy.solvers.recurr.rsolve_poly">[docs]</a><span class="k">def</span> <span class="nf">rsolve_poly</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Given linear recurrence operator L of order &#39;k&#39; with polynomial</span>
-<span class="sd">       coefficients and inhomogeneous equation Ly = f, where &#39;f&#39; is a</span>
-<span class="sd">       polynomial, we seek for all polynomial solutions over field K</span>
-<span class="sd">       of characteristic zero.</span>
-
-<span class="sd">       The algorithm performs two basic steps:</span>
-
-<span class="sd">           (1) Compute degree N of the general polynomial solution.</span>
-<span class="sd">           (2) Find all polynomials of degree N or less of Ly = f.</span>
-
-<span class="sd">       There are two methods for computing the polynomial solutions.</span>
-<span class="sd">       If the degree bound is relatively small, i.e. it&#39;s smaller than</span>
-<span class="sd">       or equal to the order of the recurrence, then naive method of</span>
-<span class="sd">       undetermined coefficients is being used. This gives system</span>
-<span class="sd">       of algebraic equations with N+1 unknowns.</span>
-
-<span class="sd">       In the other case, the algorithm performs transformation of the</span>
-<span class="sd">       initial equation to an equivalent one, for which the system of</span>
-<span class="sd">       algebraic equations has only &#39;r&#39; indeterminates. This method is</span>
-<span class="sd">       quite sophisticated (in comparison with the naive one) and was</span>
-<span class="sd">       invented together by Abramov, Bronstein and Petkovsek.</span>
-
-<span class="sd">       It is possible to generalize the algorithm implemented here to</span>
-<span class="sd">       the case of linear q-difference and differential equations.</span>
-
-<span class="sd">       Lets say that we would like to compute m-th Bernoulli polynomial</span>
-<span class="sd">       up to a constant. For this we can use b(n+1) - b(n) == m*n**(m-1)</span>
-<span class="sd">       recurrence, which has solution b(n) = B_m + C. For example:</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy import Symbol, rsolve_poly</span>
-<span class="sd">       &gt;&gt;&gt; n = Symbol(&#39;n&#39;, integer=True)</span>
-
-<span class="sd">       &gt;&gt;&gt; rsolve_poly([-1, 1], 4*n**3, n)</span>
-<span class="sd">       C0 + n**4 - 2*n**3 + n**2</span>
-
-<span class="sd">       For more information on implemented algorithms refer to:</span>
-
-<span class="sd">       [1] S. A. Abramov, M. Bronstein and M. Petkovsek, On polynomial</span>
-<span class="sd">           solutions of linear operator equations, in: T. Levelt, ed.,</span>
-<span class="sd">           Proc. ISSAC &#39;95, ACM Press, New York, 1995, 290-296.</span>
-
-<span class="sd">       [2] M. Petkovsek, Hypergeometric solutions of linear recurrences</span>
-<span class="sd">           with polynomial coefficients, J. Symbolic Computation,</span>
-<span class="sd">           14 (1992), 243-264.</span>
-
-<span class="sd">       [3] M. Petkovsek, H. S. Wilf, D. Zeilberger, A = B, 1996.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">is_polynomial</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="n">homogeneous</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">is_zero</span>
-
-    <span class="n">r</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">coeffs</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="n">coeffs</span> <span class="o">=</span> <span class="p">[</span> <span class="n">Poly</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">coeffs</span> <span class="p">]</span>
-
-    <span class="n">polys</span> <span class="o">=</span> <span class="p">[</span> <span class="n">Poly</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">r</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="n">terms</span> <span class="o">=</span> <span class="p">[</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">)</span> <span class="p">]</span> <span class="o">*</span><span class="p">(</span><span class="n">r</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">r</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">r</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">polys</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+=</span> <span class="n">coeffs</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">*</span><span class="n">binomial</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">polys</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-            <span class="p">(</span><span class="n">exp</span><span class="p">,),</span> <span class="n">coeff</span> <span class="o">=</span> <span class="n">polys</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">LT</span><span class="p">()</span>
-            <span class="n">terms</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-
-    <span class="n">d</span> <span class="o">=</span> <span class="n">b</span> <span class="o">=</span> <span class="n">terms</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">r</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">terms</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">d</span><span class="p">:</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="n">terms</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">terms</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">i</span> <span class="o">&gt;</span> <span class="n">b</span><span class="p">:</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="n">terms</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">i</span>
-
-    <span class="n">d</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">d</span><span class="p">),</span> <span class="nb">int</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-
-    <span class="n">x</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-
-    <span class="n">degree_poly</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">r</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">terms</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">i</span> <span class="o">==</span> <span class="n">b</span><span class="p">:</span>
-            <span class="n">degree_poly</span> <span class="o">+=</span> <span class="n">terms</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">FallingFactorial</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-
-    <span class="n">nni_roots</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">roots</span><span class="p">(</span><span class="n">degree_poly</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="nb">filter</span><span class="o">=</span><span class="s">&#39;Z&#39;</span><span class="p">,</span>
-        <span class="n">predicate</span><span class="o">=</span><span class="k">lambda</span> <span class="n">r</span><span class="p">:</span> <span class="n">r</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-
-    <span class="k">if</span> <span class="n">nni_roots</span><span class="p">:</span>
-        <span class="n">N</span> <span class="o">=</span> <span class="p">[</span><span class="nb">max</span><span class="p">(</span><span class="n">nni_roots</span><span class="p">)]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">N</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">if</span> <span class="n">homogeneous</span><span class="p">:</span>
-        <span class="n">N</span> <span class="o">+=</span> <span class="p">[</span><span class="o">-</span><span class="n">b</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">N</span> <span class="o">+=</span> <span class="p">[</span><span class="n">f</span><span class="o">.</span><span class="n">as_poly</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span> <span class="o">-</span> <span class="n">b</span><span class="p">,</span> <span class="o">-</span><span class="n">b</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-
-    <span class="n">N</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="n">N</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="n">N</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">homogeneous</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;symbols&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-                <span class="k">return</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="p">[])</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="k">if</span> <span class="n">N</span> <span class="o">&lt;=</span> <span class="n">r</span><span class="p">:</span>
-        <span class="n">C</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">y</span> <span class="o">=</span> <span class="n">E</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">N</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">C</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;C&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)))</span>
-            <span class="n">y</span> <span class="o">+=</span> <span class="n">C</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">*</span> <span class="n">n</span><span class="o">**</span><span class="n">i</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">r</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">E</span> <span class="o">+=</span> <span class="n">coeffs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span><span class="o">*</span><span class="n">y</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="n">i</span><span class="p">)</span>
-
-        <span class="n">solutions</span> <span class="o">=</span> <span class="n">solve_undetermined_coeffs</span><span class="p">(</span><span class="n">E</span> <span class="o">-</span> <span class="n">f</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">solutions</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">C</span> <span class="o">=</span> <span class="p">[</span> <span class="n">c</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">C</span> <span class="k">if</span> <span class="p">(</span><span class="n">c</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">solutions</span><span class="p">)</span> <span class="p">]</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">y</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">solutions</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>  <span class="c"># TBD</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">A</span> <span class="o">=</span> <span class="n">r</span>
-        <span class="n">U</span> <span class="o">=</span> <span class="n">N</span> <span class="o">+</span> <span class="n">A</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="mi">1</span>
-
-        <span class="n">nni_roots</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">roots</span><span class="p">(</span><span class="n">polys</span><span class="p">[</span><span class="n">r</span><span class="p">],</span> <span class="nb">filter</span><span class="o">=</span><span class="s">&#39;Z&#39;</span><span class="p">,</span>
-            <span class="n">predicate</span><span class="o">=</span><span class="k">lambda</span> <span class="n">r</span><span class="p">:</span> <span class="n">r</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-
-        <span class="k">if</span> <span class="n">nni_roots</span> <span class="o">!=</span> <span class="p">[]:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">nni_roots</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-        <span class="k">def</span> <span class="nf">_zero_vector</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">]</span> <span class="o">*</span> <span class="n">k</span>
-
-        <span class="k">def</span> <span class="nf">_one_vector</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
-            <span class="k">return</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">]</span> <span class="o">*</span> <span class="n">k</span>
-
-        <span class="k">def</span> <span class="nf">_delta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-            <span class="n">B</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-            <span class="n">D</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="n">B</span> <span class="o">*=</span> <span class="o">-</span><span class="n">Rational</span><span class="p">(</span><span class="n">k</span> <span class="o">-</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-                <span class="n">D</span> <span class="o">+=</span> <span class="n">B</span> <span class="o">*</span> <span class="n">p</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="n">k</span> <span class="o">-</span> <span class="n">i</span><span class="p">)</span>
-
-            <span class="k">return</span> <span class="n">D</span>
-
-        <span class="n">alpha</span> <span class="o">=</span> <span class="p">{}</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="n">A</span><span class="p">,</span> <span class="n">d</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">I</span> <span class="o">=</span> <span class="n">_one_vector</span><span class="p">(</span><span class="n">d</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">d</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="n">I</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">I</span><span class="p">[</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">i</span> <span class="o">-</span> <span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">k</span>
-
-            <span class="n">alpha</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">A</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">d</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                    <span class="n">B</span> <span class="o">=</span> <span class="n">binomial</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">)</span>
-                    <span class="n">D</span> <span class="o">=</span> <span class="n">_delta</span><span class="p">(</span><span class="n">polys</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(),</span> <span class="n">k</span><span class="p">)</span>
-
-                    <span class="n">alpha</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+=</span> <span class="n">I</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">B</span><span class="o">*</span><span class="n">D</span>
-
-        <span class="n">V</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">U</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="nb">int</span><span class="p">(</span><span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">homogeneous</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">U</span><span class="p">):</span>
-                <span class="n">v</span> <span class="o">=</span> <span class="n">_zero_vector</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-
-                <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">A</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">i</span> <span class="o">-</span> <span class="n">k</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="k">break</span>
-
-                    <span class="n">B</span> <span class="o">=</span> <span class="n">alpha</span><span class="p">[</span><span class="n">k</span> <span class="o">-</span> <span class="n">A</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">i</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span>
-
-                    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">A</span><span class="p">):</span>
-                        <span class="n">v</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="n">B</span> <span class="o">*</span> <span class="n">V</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="n">k</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-
-                <span class="n">denom</span> <span class="o">=</span> <span class="n">alpha</span><span class="p">[</span><span class="o">-</span><span class="n">A</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">A</span><span class="p">):</span>
-                    <span class="n">V</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">v</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">/</span> <span class="n">denom</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">G</span> <span class="o">=</span> <span class="n">_zero_vector</span><span class="p">(</span><span class="n">U</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">U</span><span class="p">):</span>
-                <span class="n">v</span> <span class="o">=</span> <span class="n">_zero_vector</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-                <span class="n">g</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-                <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">A</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="n">i</span> <span class="o">-</span> <span class="n">k</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="k">break</span>
-
-                    <span class="n">B</span> <span class="o">=</span> <span class="n">alpha</span><span class="p">[</span><span class="n">k</span> <span class="o">-</span> <span class="n">A</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">i</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span>
-
-                    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">A</span><span class="p">):</span>
-                        <span class="n">v</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+=</span> <span class="n">B</span> <span class="o">*</span> <span class="n">V</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="n">k</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-
-                    <span class="n">g</span> <span class="o">+=</span> <span class="n">B</span> <span class="o">*</span> <span class="n">G</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="n">k</span><span class="p">]</span>
-
-                <span class="n">denom</span> <span class="o">=</span> <span class="n">alpha</span><span class="p">[</span><span class="o">-</span><span class="n">A</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">A</span><span class="p">):</span>
-                    <span class="n">V</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">v</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">/</span> <span class="n">denom</span>
-
-                <span class="n">G</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">_delta</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">i</span> <span class="o">-</span> <span class="n">A</span><span class="p">)</span> <span class="o">-</span> <span class="n">g</span><span class="p">)</span> <span class="o">/</span> <span class="n">denom</span>
-
-        <span class="n">P</span><span class="p">,</span> <span class="n">Q</span> <span class="o">=</span> <span class="n">_one_vector</span><span class="p">(</span><span class="n">U</span><span class="p">),</span> <span class="n">_zero_vector</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">U</span><span class="p">):</span>
-            <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">P</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">a</span> <span class="o">-</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">A</span><span class="p">):</span>
-            <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="p">(</span><span class="n">v</span><span class="o">*</span><span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span> <span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">V</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span> <span class="n">P</span><span class="p">)</span> <span class="p">])</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">homogeneous</span><span class="p">:</span>
-            <span class="n">h</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="p">(</span><span class="n">g</span><span class="o">*</span><span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span> <span class="k">for</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">P</span><span class="p">)</span> <span class="p">])</span>
-
-        <span class="n">C</span> <span class="o">=</span> <span class="p">[</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;C&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">A</span><span class="p">)</span> <span class="p">]</span>
-
-        <span class="n">g</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">:</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">c</span><span class="o">*</span><span class="n">_delta</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span><span class="p">,</span> <span class="n">q</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">Q</span><span class="p">)</span> <span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">homogeneous</span><span class="p">:</span>
-            <span class="n">E</span> <span class="o">=</span> <span class="p">[</span> <span class="n">g</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">U</span><span class="p">)</span> <span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">E</span> <span class="o">=</span> <span class="p">[</span> <span class="n">g</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="n">_delta</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">U</span><span class="p">)</span> <span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">E</span> <span class="o">!=</span> <span class="p">[]:</span>
-            <span class="n">solutions</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">E</span><span class="p">,</span> <span class="o">*</span><span class="n">C</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">solutions</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">homogeneous</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;symbols&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-                        <span class="k">return</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="p">[])</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">solutions</span> <span class="o">=</span> <span class="p">{}</span>
-
-        <span class="k">if</span> <span class="n">homogeneous</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">h</span>
-
-        <span class="k">for</span> <span class="n">c</span><span class="p">,</span> <span class="n">q</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">Q</span><span class="p">)):</span>
-            <span class="k">if</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">solutions</span><span class="p">:</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">solutions</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="o">*</span><span class="n">q</span>
-                <span class="n">C</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">s</span> <span class="o">=</span> <span class="n">c</span><span class="o">*</span><span class="n">q</span>
-
-            <span class="n">result</span> <span class="o">+=</span> <span class="n">s</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;symbols&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">C</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="rsolve_ratio"><a class="viewcode-back" href="../../../modules/solvers/solvers.html#sympy.solvers.recurr.rsolve_ratio">[docs]</a><span class="k">def</span> <span class="nf">rsolve_ratio</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Given linear recurrence operator L of order &#39;k&#39; with polynomial</span>
-<span class="sd">       coefficients and inhomogeneous equation Ly = f, where &#39;f&#39; is a</span>
-<span class="sd">       polynomial, we seek for all rational solutions over field K of</span>
-<span class="sd">       characteristic zero.</span>
-
-<span class="sd">       This procedure accepts only polynomials, however if you are</span>
-<span class="sd">       interested in solving recurrence with rational coefficients</span>
-<span class="sd">       then use rsolve() which will pre-process the given equation</span>
-<span class="sd">       and run this procedure with polynomial arguments.</span>
-
-<span class="sd">       The algorithm performs two basic steps:</span>
-
-<span class="sd">           (1) Compute polynomial v(n) which can be used as universal</span>
-<span class="sd">               denominator of any rational solution of equation Ly = f.</span>
-
-<span class="sd">           (2) Construct new linear difference equation by substitution</span>
-<span class="sd">               y(n) = u(n)/v(n) and solve it for u(n) finding all its</span>
-<span class="sd">               polynomial solutions. Return None if none were found.</span>
-
-<span class="sd">       Algorithm implemented here is a revised version of the original</span>
-<span class="sd">       Abramov&#39;s algorithm, developed in 1989. The new approach is much</span>
-<span class="sd">       simpler to implement and has better overall efficiency. This</span>
-<span class="sd">       method can be easily adapted to q-difference equations case.</span>
-
-<span class="sd">       Besides finding rational solutions alone, this functions is</span>
-<span class="sd">       an important part of Hyper algorithm were it is used to find</span>
-<span class="sd">       particular solution of inhomogeneous part of a recurrence.</span>
-
-<span class="sd">       For more information on the implemented algorithm refer to:</span>
-
-<span class="sd">       [1] S. A. Abramov, Rational solutions of linear difference</span>
-<span class="sd">           and q-difference equations with polynomial coefficients,</span>
-<span class="sd">           in: T. Levelt, ed., Proc. ISSAC &#39;95, ACM Press, New York,</span>
-<span class="sd">           1995, 285-289</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.solvers.recurr import rsolve_ratio</span>
-<span class="sd">        &gt;&gt;&gt; rsolve_ratio([ - 2*x**3 + x**2 + 2*x - 1, 2*x**3 + x**2 - 6*x,</span>
-<span class="sd">        ... - 2*x**3 - 11*x**2 - 18*x - 9, 2*x**3 + 13*x**2 + 22*x + 8], 0, x)</span>
-<span class="sd">        C2*(2*x - 3)/(2*(x**2 - 1))</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">is_polynomial</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="n">coeffs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="n">coeffs</span><span class="p">))</span>
-
-    <span class="n">r</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">coeffs</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-
-    <span class="n">A</span><span class="p">,</span> <span class="n">B</span> <span class="o">=</span> <span class="n">coeffs</span><span class="p">[</span><span class="n">r</span><span class="p">],</span> <span class="n">coeffs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">A</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">r</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-
-    <span class="n">h</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;h&#39;</span><span class="p">)</span>
-
-    <span class="n">res</span> <span class="o">=</span> <span class="n">resultant</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="n">h</span><span class="p">),</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">res</span><span class="o">.</span><span class="n">is_polynomial</span><span class="p">(</span><span class="n">h</span><span class="p">):</span>
-        <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">res</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-        <span class="n">res</span> <span class="o">=</span> <span class="n">quo</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">h</span><span class="p">)</span>
-
-    <span class="n">nni_roots</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">roots</span><span class="p">(</span><span class="n">res</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="nb">filter</span><span class="o">=</span><span class="s">&#39;Z&#39;</span><span class="p">,</span>
-        <span class="n">predicate</span><span class="o">=</span><span class="k">lambda</span> <span class="n">r</span><span class="p">:</span> <span class="n">r</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">nni_roots</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">rsolve_poly</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">C</span><span class="p">,</span> <span class="n">numers</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">r</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="n">nni_roots</span><span class="p">)),</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="n">gcd</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="n">i</span><span class="p">),</span> <span class="n">n</span><span class="p">)</span>
-
-            <span class="n">A</span> <span class="o">=</span> <span class="n">quo</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-            <span class="n">B</span> <span class="o">=</span> <span class="n">quo</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">d</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">i</span><span class="p">),</span> <span class="n">n</span><span class="p">)</span>
-
-            <span class="n">C</span> <span class="o">*=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">d</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">j</span><span class="p">)</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">])</span>
-
-        <span class="n">denoms</span> <span class="o">=</span> <span class="p">[</span> <span class="n">C</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">r</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">]</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">r</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="n">gcd</span><span class="p">(</span><span class="n">coeffs</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">denoms</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">n</span><span class="p">)</span>
-
-            <span class="n">numers</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">quo</span><span class="p">(</span><span class="n">coeffs</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">g</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-            <span class="n">denoms</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">quo</span><span class="p">(</span><span class="n">denoms</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">g</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">r</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">numers</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">*=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">denoms</span><span class="p">[:</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">denoms</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]))</span>
-
-        <span class="n">result</span> <span class="o">=</span> <span class="n">rsolve_poly</span><span class="p">(</span><span class="n">numers</span><span class="p">,</span> <span class="n">f</span> <span class="o">*</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">denoms</span><span class="p">),</span> <span class="n">n</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;symbols&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-                <span class="k">return</span> <span class="p">(</span><span class="n">simplify</span><span class="p">(</span><span class="n">result</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">/</span> <span class="n">C</span><span class="p">),</span> <span class="n">result</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">simplify</span><span class="p">(</span><span class="n">result</span> <span class="o">/</span> <span class="n">C</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-</div>
-<div class="viewcode-block" id="rsolve_hyper"><a class="viewcode-back" href="../../../modules/solvers/solvers.html#sympy.solvers.recurr.rsolve_hyper">[docs]</a><span class="k">def</span> <span class="nf">rsolve_hyper</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Given linear recurrence operator L of order &#39;k&#39; with polynomial</span>
-<span class="sd">       coefficients and inhomogeneous equation Ly = f we seek for all</span>
-<span class="sd">       hypergeometric solutions over field K of characteristic zero.</span>
-
-<span class="sd">       The inhomogeneous part can be either hypergeometric or a sum</span>
-<span class="sd">       of a fixed number of pairwise dissimilar hypergeometric terms.</span>
-
-<span class="sd">       The algorithm performs three basic steps:</span>
-
-<span class="sd">           (1) Group together similar hypergeometric terms in the</span>
-<span class="sd">               inhomogeneous part of Ly = f, and find particular</span>
-<span class="sd">               solution using Abramov&#39;s algorithm.</span>
-
-<span class="sd">           (2) Compute generating set of L and find basis in it,</span>
-<span class="sd">               so that all solutions are linearly independent.</span>
-
-<span class="sd">           (3) Form final solution with the number of arbitrary</span>
-<span class="sd">               constants equal to dimension of basis of L.</span>
-
-<span class="sd">       Term a(n) is hypergeometric if it is annihilated by first order</span>
-<span class="sd">       linear difference equations with polynomial coefficients or, in</span>
-<span class="sd">       simpler words, if consecutive term ratio is a rational function.</span>
-
-<span class="sd">       The output of this procedure is a linear combination of fixed</span>
-<span class="sd">       number of hypergeometric terms. However the underlying method</span>
-<span class="sd">       can generate larger class of solutions - D&#39;Alembertian terms.</span>
-
-<span class="sd">       Note also that this method not only computes the kernel of the</span>
-<span class="sd">       inhomogeneous equation, but also reduces in to a basis so that</span>
-<span class="sd">       solutions generated by this procedure are linearly independent</span>
-
-<span class="sd">       For more information on the implemented algorithm refer to:</span>
-
-<span class="sd">       [1] M. Petkovsek, Hypergeometric solutions of linear recurrences</span>
-<span class="sd">           with polynomial coefficients, J. Symbolic Computation,</span>
-<span class="sd">           14 (1992), 243-264.</span>
-
-<span class="sd">       [2] M. Petkovsek, H. S. Wilf, D. Zeilberger, A = B, 1996.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.solvers import rsolve_hyper</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; rsolve_hyper([-1, -1, 1], 0, x)</span>
-<span class="sd">        C0*(1/2 + sqrt(5)/2)**x + C1*(-sqrt(5)/2 + 1/2)**x</span>
-
-<span class="sd">        &gt;&gt;&gt; rsolve_hyper([-1, 1], 1+x, x)</span>
-<span class="sd">        C0 + x*(x + 1)/2</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">coeffs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="n">coeffs</span><span class="p">))</span>
-
-    <span class="n">f</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="n">r</span><span class="p">,</span> <span class="n">kernel</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">coeffs</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="p">[]</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">similar</span> <span class="o">=</span> <span class="p">{}</span>
-
-            <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">is_hypergeometric</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-
-                <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">similar</span><span class="o">.</span><span class="n">keys</span><span class="p">():</span>
-                    <span class="k">if</span> <span class="n">hypersimilar</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-                        <span class="n">similar</span><span class="p">[</span><span class="n">h</span><span class="p">]</span> <span class="o">+=</span> <span class="n">g</span>
-                        <span class="k">break</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">similar</span><span class="p">[</span><span class="n">g</span><span class="p">]</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-            <span class="n">inhomogeneous</span> <span class="o">=</span> <span class="p">[]</span>
-
-            <span class="k">for</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">similar</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                <span class="n">inhomogeneous</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">g</span> <span class="o">+</span> <span class="n">h</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">f</span><span class="o">.</span><span class="n">is_hypergeometric</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="n">inhomogeneous</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">g</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">inhomogeneous</span><span class="p">):</span>
-            <span class="n">coeff</span><span class="p">,</span> <span class="n">polys</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">coeffs</span><span class="p">[:]</span>
-            <span class="n">denoms</span> <span class="o">=</span> <span class="p">[</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span> <span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">r</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-            <span class="n">s</span> <span class="o">=</span> <span class="n">hypersimp</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">r</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="n">coeff</span> <span class="o">*=</span> <span class="n">s</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-
-                <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">coeff</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-
-                <span class="n">polys</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">*=</span> <span class="n">p</span>
-                <span class="n">denoms</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">q</span>
-
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">r</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="n">polys</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">*=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="n">denoms</span><span class="p">[:</span><span class="n">j</span><span class="p">]</span> <span class="o">+</span> <span class="n">denoms</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]))</span>
-
-            <span class="n">R</span> <span class="o">=</span> <span class="n">rsolve_poly</span><span class="p">(</span><span class="n">polys</span><span class="p">,</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">denoms</span><span class="p">),</span> <span class="n">n</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">R</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">R</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">):</span>
-                <span class="n">inhomogeneous</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">*=</span> <span class="n">R</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">None</span>
-
-            <span class="n">result</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">inhomogeneous</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="n">Z</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;Z&#39;</span><span class="p">)</span>
-
-    <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">coeffs</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">coeffs</span><span class="p">[</span><span class="n">r</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">r</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-    <span class="n">p_factors</span> <span class="o">=</span> <span class="p">[</span> <span class="n">z</span> <span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="n">roots</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">keys</span><span class="p">()</span> <span class="p">]</span>
-    <span class="n">q_factors</span> <span class="o">=</span> <span class="p">[</span> <span class="n">z</span> <span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="n">roots</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">keys</span><span class="p">()</span> <span class="p">]</span>
-
-    <span class="n">factors</span> <span class="o">=</span> <span class="p">[</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span> <span class="p">]</span>
-
-    <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">p_factors</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">q_factors</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">and</span> <span class="n">q</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">and</span> <span class="n">p</span> <span class="o">&lt;=</span> <span class="n">q</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">factors</span> <span class="o">+=</span> <span class="p">[(</span><span class="n">n</span> <span class="o">-</span> <span class="n">p</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">q</span><span class="p">)]</span>
-
-    <span class="n">p</span> <span class="o">=</span> <span class="p">[</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">p</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">)</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">p_factors</span> <span class="p">]</span>
-    <span class="n">q</span> <span class="o">=</span> <span class="p">[</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">q</span><span class="p">)</span> <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">q_factors</span> <span class="p">]</span>
-
-    <span class="n">factors</span> <span class="o">=</span> <span class="n">p</span> <span class="o">+</span> <span class="n">factors</span> <span class="o">+</span> <span class="n">q</span>
-
-    <span class="k">for</span> <span class="n">A</span><span class="p">,</span> <span class="n">B</span> <span class="ow">in</span> <span class="n">factors</span><span class="p">:</span>
-        <span class="n">polys</span><span class="p">,</span> <span class="n">degrees</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
-        <span class="n">D</span> <span class="o">=</span> <span class="n">A</span><span class="o">*</span><span class="n">B</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="n">r</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">r</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">a</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">A</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="n">j</span><span class="p">)</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="p">])</span>
-            <span class="n">b</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">B</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="n">j</span><span class="p">)</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span> <span class="p">])</span>
-
-            <span class="n">poly</span> <span class="o">=</span> <span class="n">quo</span><span class="p">(</span><span class="n">coeffs</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">b</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-            <span class="n">polys</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">poly</span><span class="o">.</span><span class="n">as_poly</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">poly</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-                <span class="n">degrees</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">polys</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">degree</span><span class="p">())</span>
-
-        <span class="n">d</span><span class="p">,</span> <span class="n">poly</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">degrees</span><span class="p">),</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">r</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">coeff</span> <span class="o">=</span> <span class="n">polys</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">nth</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">coeff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="n">poly</span> <span class="o">+=</span> <span class="n">coeff</span> <span class="o">*</span> <span class="n">Z</span><span class="o">**</span><span class="n">i</span>
-
-        <span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="n">roots</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">Z</span><span class="p">)</span><span class="o">.</span><span class="n">keys</span><span class="p">():</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">z</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">or</span> <span class="n">z</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-                <span class="k">continue</span>
-
-            <span class="n">C</span> <span class="o">=</span> <span class="n">rsolve_poly</span><span class="p">([</span> <span class="n">polys</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">r</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">],</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">C</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="n">C</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                <span class="n">ratio</span> <span class="o">=</span> <span class="n">z</span> <span class="o">*</span> <span class="n">A</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="n">B</span> <span class="o">/</span> <span class="n">C</span>
-                <span class="n">ratio</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">ratio</span><span class="p">)</span>
-                <span class="c"># If there is a nonnegative root in the denominator of the ratio,</span>
-                <span class="c"># this indicates that the term y(n_root) is zero, and one should</span>
-                <span class="c"># start the product with the term y(n_root + 1).</span>
-                <span class="n">n0</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="k">for</span> <span class="n">n_root</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">roots</span><span class="p">(</span><span class="n">ratio</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()[</span><span class="mi">1</span><span class="p">],</span> <span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">keys</span><span class="p">()):</span>
-                    <span class="n">n0</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">n0</span><span class="p">,</span> <span class="n">n_root</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-                <span class="n">K</span> <span class="o">=</span> <span class="n">product</span><span class="p">(</span><span class="n">ratio</span><span class="p">,</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n0</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-                <span class="k">if</span> <span class="n">K</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">factorial</span><span class="p">,</span> <span class="n">FallingFactorial</span><span class="p">,</span> <span class="n">RisingFactorial</span><span class="p">):</span>
-                    <span class="n">K</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">K</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">casoratian</span><span class="p">(</span><span class="n">kernel</span> <span class="o">+</span> <span class="p">[</span><span class="n">K</span><span class="p">],</span> <span class="n">n</span><span class="p">,</span> <span class="n">zero</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">kernel</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">K</span><span class="p">)</span>
-
-    <span class="n">symbols</span> <span class="o">=</span> <span class="n">numbered_symbols</span><span class="p">(</span><span class="s">&#39;C&#39;</span><span class="p">)</span>
-    <span class="n">kernel</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-    <span class="n">sk</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">symbols</span><span class="p">,</span> <span class="n">kernel</span><span class="p">))</span>
-
-    <span class="k">for</span> <span class="n">C</span><span class="p">,</span> <span class="n">ker</span> <span class="ow">in</span> <span class="n">sk</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">+=</span> <span class="n">C</span> <span class="o">*</span> <span class="n">ker</span>
-
-    <span class="k">if</span> <span class="n">hints</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;symbols&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="p">[</span><span class="n">s</span> <span class="k">for</span> <span class="n">s</span><span class="p">,</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">sk</span><span class="p">])</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="rsolve"><a class="viewcode-back" href="../../../modules/solvers/solvers.html#sympy.solvers.recurr.rsolve">[docs]</a><span class="k">def</span> <span class="nf">rsolve</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">init</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Solve univariate recurrence with rational coefficients.</span>
-
-<span class="sd">    Given k-th order linear recurrence Ly = f, or equivalently:</span>
-
-<span class="sd">     a_{k}(n) y(n+k) + a_{k-1}(n) y(n+k-1) + ... + a_{0}(n) y(n) = f</span>
-
-<span class="sd">    where a_{i}(n), for i=0..k, are polynomials or rational functions</span>
-<span class="sd">    in n, and f is a hypergeometric function or a sum of a fixed number</span>
-<span class="sd">    of pairwise dissimilar hypergeometric terms in n, finds all solutions</span>
-<span class="sd">    or returns None, if none were found.</span>
-
-<span class="sd">    Initial conditions can be given as a dictionary in two forms:</span>
-
-<span class="sd">      [1] {   n_0  : v_0,   n_1  : v_1, ...,   n_m  : v_m }</span>
-<span class="sd">      [2] { y(n_0) : v_0, y(n_1) : v_1, ..., y(n_m) : v_m }</span>
-
-<span class="sd">    or as a list L of values:</span>
-
-<span class="sd">      L = [ v_0, v_1, ..., v_m ]</span>
-
-<span class="sd">    where L[i] = v_i, for i=0..m, maps to y(n_i).</span>
-
-<span class="sd">    As an example lets consider the following recurrence:</span>
-
-<span class="sd">     (n - 1) y(n + 2) - (n**2 + 3 n - 2) y(n + 1) + 2 n (n + 1) y(n) == 0</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Function, rsolve</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import n</span>
-<span class="sd">    &gt;&gt;&gt; y = Function(&#39;y&#39;)</span>
-
-<span class="sd">    &gt;&gt;&gt; f = (n-1)*y(n+2) - (n**2+3*n-2)*y(n+1) + 2*n*(n+1)*y(n)</span>
-
-<span class="sd">    &gt;&gt;&gt; rsolve(f, y(n))</span>
-<span class="sd">    2**n*C0 + C1*n!</span>
-
-<span class="sd">    &gt;&gt;&gt; rsolve(f, y(n), { y(0):0, y(1):3 })</span>
-<span class="sd">    3*2**n - 3*n!</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">Equality</span><span class="p">):</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">lhs</span> <span class="o">-</span> <span class="n">f</span><span class="o">.</span><span class="n">rhs</span>
-
-    <span class="n">n</span> <span class="o">=</span> <span class="n">y</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">k</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;k&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">(</span><span class="n">n</span><span class="p">,))</span>
-
-    <span class="n">h_part</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="k">lambda</span><span class="p">:</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-    <span class="n">i_part</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-    <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="n">coeff</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="n">kspec</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">for</span> <span class="n">h</span> <span class="ow">in</span> <span class="n">Mul</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">g</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">h</span><span class="o">.</span><span class="n">is_Function</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">h</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">y</span><span class="o">.</span><span class="n">func</span><span class="p">:</span>
-                    <span class="n">result</span> <span class="o">=</span> <span class="n">h</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span>
-
-                    <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="n">kspec</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">result</span><span class="p">[</span><span class="n">k</span><span class="p">])</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                            <span class="s">&quot;&#39;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">+k)&#39; expected, got &#39;</span><span class="si">%s</span><span class="s">&#39;&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">y</span><span class="o">.</span><span class="n">func</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">h</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                        <span class="s">&quot;&#39;</span><span class="si">%s</span><span class="s">&#39; expected, got &#39;</span><span class="si">%s</span><span class="s">&#39;&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">y</span><span class="o">.</span><span class="n">func</span><span class="p">,</span> <span class="n">h</span><span class="o">.</span><span class="n">func</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">coeff</span> <span class="o">*=</span> <span class="n">h</span>
-
-        <span class="k">if</span> <span class="n">kspec</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">h_part</span><span class="p">[</span><span class="n">kspec</span><span class="p">]</span> <span class="o">+=</span> <span class="n">coeff</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">i_part</span> <span class="o">+=</span> <span class="n">coeff</span>
-
-    <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">h_part</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="n">h_part</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span>
-
-    <span class="n">common</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-    <span class="k">for</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">h_part</span><span class="o">.</span><span class="n">values</span><span class="p">():</span>
-        <span class="k">if</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_rational_function</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">coeff</span><span class="o">.</span><span class="n">is_polynomial</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-                <span class="n">common</span> <span class="o">=</span> <span class="n">lcm</span><span class="p">(</span><span class="n">common</span><span class="p">,</span> <span class="n">coeff</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()[</span><span class="mi">1</span><span class="p">],</span> <span class="n">n</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&quot;Polynomial or rational function expected, got &#39;</span><span class="si">%s</span><span class="s">&#39;&quot;</span> <span class="o">%</span> <span class="n">coeff</span><span class="p">)</span>
-
-    <span class="n">i_numer</span><span class="p">,</span> <span class="n">i_denom</span> <span class="o">=</span> <span class="n">i_part</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="n">i_denom</span><span class="o">.</span><span class="n">is_polynomial</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-        <span class="n">common</span> <span class="o">=</span> <span class="n">lcm</span><span class="p">(</span><span class="n">common</span><span class="p">,</span> <span class="n">i_denom</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">common</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">h_part</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">numer</span><span class="p">,</span> <span class="n">denom</span> <span class="o">=</span> <span class="n">coeff</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-            <span class="n">h_part</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">numer</span><span class="o">*</span><span class="n">quo</span><span class="p">(</span><span class="n">common</span><span class="p">,</span> <span class="n">denom</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-
-        <span class="n">i_part</span> <span class="o">=</span> <span class="n">i_numer</span><span class="o">*</span><span class="n">quo</span><span class="p">(</span><span class="n">common</span><span class="p">,</span> <span class="n">i_denom</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="n">K_min</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">h_part</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-
-    <span class="k">if</span> <span class="n">K_min</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">K</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">K_min</span><span class="p">)</span>
-
-        <span class="n">H_part</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="k">lambda</span><span class="p">:</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-        <span class="n">i_part</span> <span class="o">=</span> <span class="n">i_part</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="n">K</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-        <span class="n">common</span> <span class="o">=</span> <span class="n">common</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="n">K</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">h_part</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="n">H_part</span><span class="p">[</span><span class="n">k</span> <span class="o">+</span> <span class="n">K</span><span class="p">]</span> <span class="o">=</span> <span class="n">coeff</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="n">K</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">H_part</span> <span class="o">=</span> <span class="n">h_part</span>
-
-    <span class="n">K_max</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">H_part</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-    <span class="n">coeffs</span> <span class="o">=</span> <span class="p">[</span><span class="n">H_part</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">K_max</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">rsolve_hyper</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="o">-</span><span class="n">i_part</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">symbols</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="n">solution</span><span class="p">,</span> <span class="n">symbols</span> <span class="o">=</span> <span class="n">result</span>
-
-    <span class="k">if</span> <span class="n">init</span> <span class="o">==</span> <span class="p">{}</span> <span class="ow">or</span> <span class="n">init</span> <span class="o">==</span> <span class="p">[]:</span>
-        <span class="n">init</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="k">if</span> <span class="n">symbols</span> <span class="ow">and</span> <span class="n">init</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">equations</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">init</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">list</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">init</span><span class="p">)):</span>
-                <span class="n">eq</span> <span class="o">=</span> <span class="n">solution</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="o">-</span> <span class="n">init</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-                <span class="n">equations</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">init</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="n">i</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-                <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">k</span><span class="o">.</span><span class="n">is_Function</span> <span class="ow">and</span> <span class="n">k</span><span class="o">.</span><span class="n">func</span> <span class="o">==</span> <span class="n">y</span><span class="o">.</span><span class="n">func</span><span class="p">:</span>
-                        <span class="n">i</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">k</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                            <span class="s">&quot;Integer or term expected, got &#39;</span><span class="si">%s</span><span class="s">&#39;&quot;</span> <span class="o">%</span> <span class="n">k</span><span class="p">)</span>
-
-                <span class="n">eq</span> <span class="o">=</span> <span class="n">solution</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="o">-</span> <span class="n">v</span>
-                <span class="n">equations</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span>
-
-        <span class="n">result</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">equations</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">result</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">result</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                <span class="n">solution</span> <span class="o">=</span> <span class="n">solution</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">solution</span></div>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.solvers.solvers</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">This module contain solvers for all kinds of equations:</span>
-
-<span class="sd">    - algebraic or transcendental, use solve()</span>
-
-<span class="sd">    - recurrence, use rsolve()</span>
-
-<span class="sd">    - differential, use dsolve()</span>
-
-<span class="sd">    - nonlinear (numerically), use nsolve()</span>
-<span class="sd">      (you will need a good starting point)</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="p">(</span><span class="n">iterable</span><span class="p">,</span> <span class="n">is_sequence</span><span class="p">,</span> <span class="n">ordered</span><span class="p">,</span>
-    <span class="n">default_sort_key</span><span class="p">,</span> <span class="nb">reduce</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.exceptions</span> <span class="kn">import</span> <span class="n">SymPyDeprecationWarning</span>
-<span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="n">sympify</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Wild</span><span class="p">,</span> <span class="n">Equality</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">,</span> <span class="n">Basic</span><span class="p">,</span>
-    <span class="n">Expr</span><span class="p">,</span> <span class="n">Mul</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="p">(</span><span class="n">expand_mul</span><span class="p">,</span> <span class="n">expand_multinomial</span><span class="p">,</span> <span class="n">expand_log</span><span class="p">,</span>
-                          <span class="n">Derivative</span><span class="p">,</span> <span class="n">AppliedUndef</span><span class="p">,</span> <span class="n">UndefinedFunction</span><span class="p">,</span> <span class="n">nfloat</span><span class="p">,</span>
-                          <span class="n">count_ops</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">ilcm</span><span class="p">,</span> <span class="n">Float</span>
-<span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Relational</span>
-<span class="kn">from</span> <span class="nn">sympy.logic.boolalg</span> <span class="kn">import</span> <span class="n">And</span><span class="p">,</span> <span class="n">Or</span>
-<span class="kn">from</span> <span class="nn">sympy.core.basic</span> <span class="kn">import</span> <span class="n">preorder_traversal</span>
-
-<span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="p">(</span><span class="n">log</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">LambertW</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">tan</span><span class="p">,</span> <span class="n">cot</span><span class="p">,</span> <span class="n">cosh</span><span class="p">,</span>
-                             <span class="n">sinh</span><span class="p">,</span> <span class="n">tanh</span><span class="p">,</span> <span class="n">coth</span><span class="p">,</span> <span class="n">acos</span><span class="p">,</span> <span class="n">asin</span><span class="p">,</span> <span class="n">atan</span><span class="p">,</span> <span class="n">acot</span><span class="p">,</span> <span class="n">acosh</span><span class="p">,</span>
-                             <span class="n">asinh</span><span class="p">,</span> <span class="n">atanh</span><span class="p">,</span> <span class="n">acoth</span><span class="p">,</span> <span class="n">Abs</span><span class="p">,</span> <span class="n">sign</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.miscellaneous</span> <span class="kn">import</span> <span class="n">real_root</span>
-<span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="p">(</span><span class="n">simplify</span><span class="p">,</span> <span class="n">collect</span><span class="p">,</span> <span class="n">powsimp</span><span class="p">,</span> <span class="n">posify</span><span class="p">,</span> <span class="n">powdenest</span><span class="p">,</span>
-                            <span class="n">nsimplify</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.simplify.sqrtdenest</span> <span class="kn">import</span> <span class="n">sqrt_depth</span><span class="p">,</span> <span class="n">_mexpand</span>
-<span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">zeros</span>
-<span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">roots</span><span class="p">,</span> <span class="n">cancel</span><span class="p">,</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">together</span><span class="p">,</span> <span class="n">factor</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.elementary.piecewise</span> <span class="kn">import</span> <span class="n">piecewise_fold</span><span class="p">,</span> <span class="n">Piecewise</span>
-
-<span class="kn">from</span> <span class="nn">sympy.utilities.lambdify</span> <span class="kn">import</span> <span class="n">lambdify</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.misc</span> <span class="kn">import</span> <span class="n">filldedent</span>
-<span class="kn">from</span> <span class="nn">sympy.mpmath</span> <span class="kn">import</span> <span class="n">findroot</span>
-
-<span class="kn">from</span> <span class="nn">sympy.solvers.polysys</span> <span class="kn">import</span> <span class="n">solve_poly_system</span>
-<span class="kn">from</span> <span class="nn">sympy.solvers.inequalities</span> <span class="kn">import</span> <span class="n">reduce_inequalities</span>
-
-<span class="kn">from</span> <span class="nn">sympy.assumptions</span> <span class="kn">import</span> <span class="n">Q</span><span class="p">,</span> <span class="n">ask</span>
-
-<span class="kn">from</span> <span class="nn">types</span> <span class="kn">import</span> <span class="n">GeneratorType</span>
-<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-
-<span class="k">def</span> <span class="nf">_ispow</span><span class="p">(</span><span class="n">e</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return True if e is a Pow or is exp.&quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">or</span> <span class="n">e</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">exp</span>
-
-
-<span class="k">def</span> <span class="nf">denoms</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">symbols</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return (recursively) set of all denominators that appear in eq</span>
-<span class="sd">    that contain any symbol in iterable ``symbols``; if ``symbols`` is</span>
-<span class="sd">    None (default) then all denominators with symbols will be returned.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.solvers.solvers import denoms</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-
-<span class="sd">    &gt;&gt;&gt; denoms(x/y)</span>
-<span class="sd">    set([y])</span>
-
-<span class="sd">    &gt;&gt;&gt; denoms(x/(y*z))</span>
-<span class="sd">    set([y, z])</span>
-
-<span class="sd">    &gt;&gt;&gt; denoms(3/x + y/z)</span>
-<span class="sd">    set([x, z])</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">symbols</span> <span class="o">=</span> <span class="n">symbols</span> <span class="ow">or</span> <span class="n">eq</span><span class="o">.</span><span class="n">free_symbols</span>
-    <span class="n">dens</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">symbols</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">eq</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">symbols</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">dens</span>
-    <span class="n">pt</span> <span class="o">=</span> <span class="n">preorder_traversal</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">pt</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">_ispow</span><span class="p">(</span><span class="n">e</span><span class="p">):</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">dens</span><span class="p">:</span>
-                <span class="n">pt</span><span class="o">.</span><span class="n">skip</span><span class="p">()</span>
-            <span class="k">elif</span> <span class="n">d</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">symbols</span><span class="p">):</span>
-                <span class="n">dens</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()[</span><span class="mi">0</span><span class="p">])</span>
-    <span class="k">return</span> <span class="n">dens</span>
-
-
-<div class="viewcode-block" id="checksol"><a class="viewcode-back" href="../../../modules/solvers/solvers.html#sympy.solvers.solvers.checksol">[docs]</a><span class="k">def</span> <span class="nf">checksol</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">symbol</span><span class="p">,</span> <span class="n">sol</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Checks whether sol is a solution of equation f == 0.</span>
-
-<span class="sd">    Input can be either a single symbol and corresponding value</span>
-<span class="sd">    or a dictionary of symbols and values. ``f`` can be a single</span>
-<span class="sd">    equation or an iterable of equations. A solution must satisfy</span>
-<span class="sd">    all equations in ``f`` to be considered valid; if a solution</span>
-<span class="sd">    does not satisfy any equation, False is returned; if one or</span>
-<span class="sd">    more checks are inconclusive (and none are False) then None</span>
-<span class="sd">    is returned.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.solvers import checksol</span>
-<span class="sd">    &gt;&gt;&gt; x, y = symbols(&#39;x,y&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; checksol(x**4 - 1, x, 1)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; checksol(x**4 - 1, x, 0)</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; checksol(x**2 + y**2 - 5**2, {x: 3, y: 4})</span>
-<span class="sd">    True</span>
-
-<span class="sd">    To check if an expression is zero using checksol, pass it</span>
-<span class="sd">    as ``f`` and send an empty dictionary for ``symbol``:</span>
-
-<span class="sd">    &gt;&gt;&gt; checksol(x**2 + x - x*(x + 1), {})</span>
-<span class="sd">    True</span>
-
-<span class="sd">    None is returned if checksol() could not conclude.</span>
-
-<span class="sd">    flags:</span>
-<span class="sd">        &#39;numerical=True (default)&#39;</span>
-<span class="sd">           do a fast numerical check if ``f`` has only one symbol.</span>
-<span class="sd">        &#39;minimal=True (default is False)&#39;</span>
-<span class="sd">           a very fast, minimal testing.</span>
-<span class="sd">        &#39;warn=True (default is False)&#39;</span>
-<span class="sd">           print a warning if checksol() could not conclude.</span>
-<span class="sd">        &#39;simplify=True (default)&#39;</span>
-<span class="sd">           simplify solution before substituting into function and</span>
-<span class="sd">           simplify the function before trying specific simplifications</span>
-<span class="sd">        &#39;force=True (default is False)&#39;</span>
-<span class="sd">           make positive all symbols without assumptions regarding sign.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="n">sol</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">sol</span> <span class="o">=</span> <span class="p">{</span><span class="n">symbol</span><span class="p">:</span> <span class="n">sol</span><span class="p">}</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">symbol</span><span class="p">,</span> <span class="nb">dict</span><span class="p">):</span>
-        <span class="n">sol</span> <span class="o">=</span> <span class="n">symbol</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">msg</span> <span class="o">=</span> <span class="s">&#39;Expecting sym, val or {sym: val}, None but got </span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">&#39;</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="n">msg</span> <span class="o">%</span> <span class="p">(</span><span class="n">symbol</span><span class="p">,</span> <span class="n">sol</span><span class="p">))</span>
-
-    <span class="k">if</span> <span class="n">iterable</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;no functions to check&#39;</span><span class="p">)</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">for</span> <span class="n">fi</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-            <span class="n">check</span> <span class="o">=</span> <span class="n">checksol</span><span class="p">(</span><span class="n">fi</span><span class="p">,</span> <span class="n">sol</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">check</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">if</span> <span class="n">check</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="bp">None</span>  <span class="c"># don&#39;t return, wait to see if there&#39;s a False</span>
-        <span class="k">return</span> <span class="n">rv</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">Poly</span><span class="p">):</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">Equality</span><span class="p">):</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">lhs</span> <span class="o">-</span> <span class="n">f</span><span class="o">.</span><span class="n">rhs</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">if</span> <span class="n">sol</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="nb">list</span><span class="p">(</span><span class="n">sol</span><span class="o">.</span><span class="n">keys</span><span class="p">())):</span>
-        <span class="c"># if f(y) == 0, x=3 does not set f(y) to zero...nor does it not</span>
-        <span class="k">return</span> <span class="bp">None</span>
-
-    <span class="n">illegal</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span><span class="n">S</span><span class="o">.</span><span class="n">NaN</span><span class="p">,</span>
-               <span class="n">S</span><span class="o">.</span><span class="n">ComplexInfinity</span><span class="p">,</span>
-               <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">,</span>
-               <span class="n">S</span><span class="o">.</span><span class="n">NegativeInfinity</span><span class="p">])</span>
-    <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">v</span><span class="p">)</span><span class="o">.</span><span class="n">atoms</span><span class="p">()</span> <span class="o">&amp;</span> <span class="n">illegal</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">sol</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="n">was</span> <span class="o">=</span> <span class="n">f</span>
-    <span class="n">attempt</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-    <span class="n">numerical</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;numerical&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-    <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">attempt</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">attempt</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">val</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sol</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">val</span><span class="o">.</span><span class="n">atoms</span><span class="p">()</span> <span class="o">&amp;</span> <span class="n">illegal</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">elif</span> <span class="n">attempt</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">val</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">val</span><span class="o">.</span><span class="n">is_constant</span><span class="p">(</span><span class="o">*</span><span class="nb">list</span><span class="p">(</span><span class="n">sol</span><span class="o">.</span><span class="n">keys</span><span class="p">())):</span>
-                    <span class="k">return</span> <span class="bp">False</span>
-                <span class="c"># there are free symbols -- simple expansion might work</span>
-                <span class="n">_</span><span class="p">,</span> <span class="n">val</span> <span class="o">=</span> <span class="n">val</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-                <span class="n">val</span> <span class="o">=</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">expand_multinomial</span><span class="p">(</span><span class="n">val</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="n">attempt</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;minimal&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-                <span class="k">return</span>
-            <span class="k">if</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;simplify&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">sol</span><span class="p">:</span>
-                    <span class="n">sol</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">sol</span><span class="p">[</span><span class="n">k</span><span class="p">])</span>
-            <span class="c"># start over without the failed expanded form, possibly</span>
-            <span class="c"># with a simplified solution</span>
-            <span class="n">val</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sol</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;force&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-                <span class="n">val</span><span class="p">,</span> <span class="n">reps</span> <span class="o">=</span> <span class="n">posify</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
-                <span class="c"># expansion may work now, so try again and check</span>
-                <span class="n">exval</span> <span class="o">=</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">expand_multinomial</span><span class="p">(</span><span class="n">val</span><span class="p">))</span>
-                <span class="k">if</span> <span class="n">exval</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">exval</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-                    <span class="c"># we can decide now</span>
-                    <span class="n">val</span> <span class="o">=</span> <span class="n">exval</span>
-        <span class="k">elif</span> <span class="n">attempt</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="n">val</span> <span class="o">=</span> <span class="n">powsimp</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">attempt</span> <span class="o">==</span> <span class="mi">4</span><span class="p">:</span>
-            <span class="n">val</span> <span class="o">=</span> <span class="n">cancel</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">attempt</span> <span class="o">==</span> <span class="mi">5</span><span class="p">:</span>
-            <span class="n">val</span> <span class="o">=</span> <span class="n">val</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-        <span class="k">elif</span> <span class="n">attempt</span> <span class="o">==</span> <span class="mi">6</span><span class="p">:</span>
-            <span class="n">val</span> <span class="o">=</span> <span class="n">together</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">attempt</span> <span class="o">==</span> <span class="mi">7</span><span class="p">:</span>
-            <span class="n">val</span> <span class="o">=</span> <span class="n">powsimp</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># if there are no radicals and no functions then this can&#39;t be</span>
-            <span class="c"># zero anymore -- can it?</span>
-            <span class="n">pot</span> <span class="o">=</span> <span class="n">preorder_traversal</span><span class="p">(</span><span class="n">expand_mul</span><span class="p">(</span><span class="n">val</span><span class="p">))</span>
-            <span class="n">seen</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-            <span class="n">saw_pow_func</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">pot</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">seen</span><span class="p">:</span>
-                    <span class="k">continue</span>
-                <span class="n">seen</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">p</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-                    <span class="n">saw_pow_func</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">elif</span> <span class="n">p</span><span class="o">.</span><span class="n">is_Function</span><span class="p">:</span>
-                    <span class="n">saw_pow_func</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">UndefinedFunction</span><span class="p">):</span>
-                    <span class="n">saw_pow_func</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="k">if</span> <span class="n">saw_pow_func</span><span class="p">:</span>
-                    <span class="k">break</span>
-            <span class="k">if</span> <span class="n">saw_pow_func</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-            <span class="k">if</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;force&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-                <span class="c"># don&#39;t do a zero check with the positive assumptions in place</span>
-                <span class="n">val</span> <span class="o">=</span> <span class="n">val</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">reps</span><span class="p">)</span>
-            <span class="n">nz</span> <span class="o">=</span> <span class="n">val</span><span class="o">.</span><span class="n">is_nonzero</span>
-            <span class="k">if</span> <span class="n">nz</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="c"># issue 2574: nz may be True even when False</span>
-                <span class="c"># so these are just hacks to keep a false positive</span>
-                <span class="c"># from being returned</span>
-
-                <span class="c"># HACK 1: LambertW (issue 2574)</span>
-                <span class="k">if</span> <span class="n">val</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">and</span> <span class="n">val</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">LambertW</span><span class="p">):</span>
-                    <span class="c"># don&#39;t eval this to verify solution since if we got here,</span>
-                    <span class="c"># numerical must be False</span>
-                    <span class="k">return</span> <span class="bp">None</span>
-
-                <span class="c"># add other HACKs here if necessary, otherwise we assume</span>
-                <span class="c"># the nz value is correct</span>
-                <span class="k">return</span> <span class="ow">not</span> <span class="n">nz</span>
-            <span class="k">break</span>
-
-        <span class="k">if</span> <span class="n">val</span> <span class="o">==</span> <span class="n">was</span><span class="p">:</span>
-            <span class="k">continue</span>
-        <span class="k">elif</span> <span class="n">val</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">val</span> <span class="o">==</span> <span class="mi">0</span>
-        <span class="k">if</span> <span class="n">numerical</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">val</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-            <span class="k">return</span> <span class="nb">abs</span><span class="p">(</span><span class="n">val</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="mi">18</span><span class="p">)</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="n">chop</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span> <span class="o">&lt;</span> <span class="mf">1e-9</span>
-        <span class="n">was</span> <span class="o">=</span> <span class="n">val</span>
-
-    <span class="k">if</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;warn&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-        <span class="k">print</span><span class="p">((</span><span class="s">&quot;</span><span class="se">\n\t</span><span class="s">Warning: could not verify solution </span><span class="si">%s</span><span class="s">.&quot;</span> <span class="o">%</span> <span class="n">sol</span><span class="p">))</span>
-    <span class="c"># returns None if it can&#39;t conclude</span>
-    <span class="c"># TODO: improve solution testing</span>
-
-</div>
-<div class="viewcode-block" id="check_assumptions"><a class="viewcode-back" href="../../../modules/solvers/solvers.html#sympy.solvers.solvers.check_assumptions">[docs]</a><span class="k">def</span> <span class="nf">check_assumptions</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Checks whether expression `expr` satisfies all assumptions.</span>
-
-<span class="sd">    `assumptions` is a dict of assumptions: {&#39;assumption&#39;: True|False, ...}.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy import Symbol, pi, I, exp</span>
-<span class="sd">       &gt;&gt;&gt; from sympy.solvers.solvers import check_assumptions</span>
-
-<span class="sd">       &gt;&gt;&gt; check_assumptions(-5, integer=True)</span>
-<span class="sd">       True</span>
-<span class="sd">       &gt;&gt;&gt; check_assumptions(pi, real=True, integer=False)</span>
-<span class="sd">       True</span>
-<span class="sd">       &gt;&gt;&gt; check_assumptions(pi, real=True, negative=True)</span>
-<span class="sd">       False</span>
-<span class="sd">       &gt;&gt;&gt; check_assumptions(exp(I*pi/7), real=False)</span>
-<span class="sd">       True</span>
-
-<span class="sd">       &gt;&gt;&gt; x = Symbol(&#39;x&#39;, real=True, positive=True)</span>
-<span class="sd">       &gt;&gt;&gt; check_assumptions(2*x + 1, real=True, positive=True)</span>
-<span class="sd">       True</span>
-<span class="sd">       &gt;&gt;&gt; check_assumptions(-2*x - 5, real=True, positive=True)</span>
-<span class="sd">       False</span>
-
-<span class="sd">       `None` is returned if check_assumptions() could not conclude.</span>
-
-<span class="sd">       &gt;&gt;&gt; check_assumptions(2*x - 1, real=True, positive=True)</span>
-<span class="sd">       &gt;&gt;&gt; z = Symbol(&#39;z&#39;)</span>
-<span class="sd">       &gt;&gt;&gt; check_assumptions(z, real=True)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="n">expected</span> <span class="ow">in</span> <span class="n">assumptions</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="k">if</span> <span class="n">expected</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">continue</span>
-        <span class="n">test</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&#39;is_&#39;</span> <span class="o">+</span> <span class="n">key</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">test</span> <span class="ow">is</span> <span class="n">expected</span><span class="p">:</span>
-            <span class="k">continue</span>
-        <span class="k">elif</span> <span class="n">test</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="bp">None</span>  <span class="c"># Can&#39;t conclude, unless an other test fails.</span>
-    <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="solve"><a class="viewcode-back" href="../../../modules/solvers/solvers.html#sympy.solvers.solvers.solve">[docs]</a><span class="k">def</span> <span class="nf">solve</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Algebraically solves equations and systems of equations.</span>
-
-<span class="sd">    Currently supported are:</span>
-<span class="sd">        - univariate polynomial,</span>
-<span class="sd">        - transcendental</span>
-<span class="sd">        - piecewise combinations of the above</span>
-<span class="sd">        - systems of linear and polynomial equations</span>
-<span class="sd">        - sytems containing relational expressions.</span>
-
-<span class="sd">    Input is formed as:</span>
-
-<span class="sd">    * f</span>
-<span class="sd">        - a single Expr or Poly that must be zero,</span>
-<span class="sd">        - an Equality</span>
-<span class="sd">        - a Relational expression or boolean</span>
-<span class="sd">        - iterable of one or more of the above</span>
-
-<span class="sd">    * symbols (object(s) to solve for) specified as</span>
-<span class="sd">        - none given (other non-numeric objects will be used)</span>
-<span class="sd">        - single symbol</span>
-<span class="sd">        - denested list of symbols</span>
-<span class="sd">          e.g. solve(f, x, y)</span>
-<span class="sd">        - ordered iterable of symbols</span>
-<span class="sd">          e.g. solve(f, [x, y])</span>
-
-<span class="sd">    * flags</span>
-<span class="sd">        &#39;dict&#39;=True (default is False)</span>
-<span class="sd">            return list (perhaps empty) of solution mappings</span>
-<span class="sd">        &#39;set&#39;=True (default is False)</span>
-<span class="sd">            return list of symbols and set of tuple(s) of solution(s)</span>
-<span class="sd">        &#39;exclude=[] (default)&#39;</span>
-<span class="sd">            don&#39;t try to solve for any of the free symbols in exclude;</span>
-<span class="sd">            if expressions are given, the free symbols in them will</span>
-<span class="sd">            be extracted automatically.</span>
-<span class="sd">        &#39;check=True (default)&#39;</span>
-<span class="sd">            If False, don&#39;t do any testing of solutions. This can be</span>
-<span class="sd">            useful if one wants to include solutions that make any</span>
-<span class="sd">            denominator zero.</span>
-<span class="sd">        &#39;numerical=True (default)&#39;</span>
-<span class="sd">            do a fast numerical check if ``f`` has only one symbol.</span>
-<span class="sd">        &#39;minimal=True (default is False)&#39;</span>
-<span class="sd">            a very fast, minimal testing.</span>
-<span class="sd">        &#39;warning=True (default is False)&#39;</span>
-<span class="sd">            print a warning if checksol() could not conclude.</span>
-<span class="sd">        &#39;simplify=True (default)&#39;</span>
-<span class="sd">            simplify all but cubic and quartic solutions before</span>
-<span class="sd">            returning them and (if check is not False) use the</span>
-<span class="sd">            general simplify function on the solutions and the</span>
-<span class="sd">            expression obtained when they are substituted into the</span>
-<span class="sd">            function which should be zero</span>
-<span class="sd">        &#39;force=True (default is False)&#39;</span>
-<span class="sd">            make positive all symbols without assumptions regarding sign.</span>
-<span class="sd">        &#39;rational=True (default)&#39;</span>
-<span class="sd">            recast Floats as Rational; if this option is not used, the</span>
-<span class="sd">            system containing floats may fail to solve because of issues</span>
-<span class="sd">            with polys. If rational=None, Floats will be recast as</span>
-<span class="sd">            rationals but the answer will be recast as Floats. If the</span>
-<span class="sd">            flag is False then nothing will be done to the Floats.</span>
-<span class="sd">        &#39;manual=True (default is False)&#39;</span>
-<span class="sd">            do not use the polys/matrix method to solve a system of</span>
-<span class="sd">            equations, solve them one at a time as you might &quot;manually&quot;.</span>
-<span class="sd">        &#39;implicit=True (default is False)&#39;</span>
-<span class="sd">            allows solve to return a solution for a pattern in terms of</span>
-<span class="sd">            other functions that contain that pattern; this is only</span>
-<span class="sd">            needed if the pattern is inside of some invertible function</span>
-<span class="sd">            like cos, exp, ....</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    The output varies according to the input and can be seen by example::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import solve, Poly, Eq, Function, exp</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y, z, a, b</span>
-<span class="sd">        &gt;&gt;&gt; f = Function(&#39;f&#39;)</span>
-
-<span class="sd">    * boolean or univariate Relational</span>
-
-<span class="sd">        &gt;&gt;&gt; solve(x &lt; 3)</span>
-<span class="sd">        And(im(x) == 0, re(x) &lt; 3)</span>
-
-<span class="sd">    * to always get a list of solution mappings, use flag dict=True</span>
-
-<span class="sd">        &gt;&gt;&gt; solve(x - 3, dict=True)</span>
-<span class="sd">        [{x: 3}]</span>
-<span class="sd">        &gt;&gt;&gt; solve([x - 3, y - 1], dict=True)</span>
-<span class="sd">        [{x: 3, y: 1}]</span>
-
-<span class="sd">    * to get a list of symbols and set of solution(s) use flag set=True</span>
-
-<span class="sd">        &gt;&gt;&gt; solve([x**2 - 3, y - 1], set=True)</span>
-<span class="sd">        ([x, y], set([(-sqrt(3), 1), (sqrt(3), 1)]))</span>
-
-<span class="sd">    * single expression and single symbol that is in the expression</span>
-
-<span class="sd">        &gt;&gt;&gt; solve(x - y, x)</span>
-<span class="sd">        [y]</span>
-<span class="sd">        &gt;&gt;&gt; solve(x - 3, x)</span>
-<span class="sd">        [3]</span>
-<span class="sd">        &gt;&gt;&gt; solve(Eq(x, 3), x)</span>
-<span class="sd">        [3]</span>
-<span class="sd">        &gt;&gt;&gt; solve(Poly(x - 3), x)</span>
-<span class="sd">        [3]</span>
-<span class="sd">        &gt;&gt;&gt; set(solve(x**2 - y**2, x))</span>
-<span class="sd">        set([-y, y])</span>
-<span class="sd">        &gt;&gt;&gt; set(solve(x**4 - 1, x))</span>
-<span class="sd">        set([-1, 1, -I, I])</span>
-
-<span class="sd">    * single expression with no symbol that is in the expression</span>
-
-<span class="sd">        &gt;&gt;&gt; solve(3, x)</span>
-<span class="sd">        []</span>
-<span class="sd">        &gt;&gt;&gt; solve(x - 3, y)</span>
-<span class="sd">        []</span>
-
-<span class="sd">    * single expression with no symbol given</span>
-
-<span class="sd">          In this case, all free symbols will be selected as potential</span>
-<span class="sd">          symbols to solve for. If the equation is univariate then a list</span>
-<span class="sd">          of solutionsis returned; otherwise -- as is the case when symbols are</span>
-<span class="sd">          given as an iterable of length &gt; 1 -- a list of mappings will be returned.</span>
-
-<span class="sd">            &gt;&gt;&gt; solve(x - 3)</span>
-<span class="sd">            [3]</span>
-<span class="sd">            &gt;&gt;&gt; solve(x**2 - y**2) # doctest: +SKIP</span>
-<span class="sd">            [{x: -y}, {x: y}]</span>
-<span class="sd">            &gt;&gt;&gt; solve(z**2*x**2 - z**2*y**2) # doctest: +SKIP</span>
-<span class="sd">            [{x: -y}, {x: y}]</span>
-<span class="sd">            &gt;&gt;&gt; solve(z**2*x - z**2*y**2)</span>
-<span class="sd">            [{x: y**2}]</span>
-
-<span class="sd">    * when an object other than a Symbol is given as a symbol, it is</span>
-<span class="sd">      isolated algebraically and an implicit solution may be obtained.</span>
-<span class="sd">      This is mostly provided as a convenience to save one from replacing</span>
-<span class="sd">      the object with a Symbol and solving for that Symbol. It will only</span>
-<span class="sd">      work if the specified object can be replaced with a Symbol using the</span>
-<span class="sd">      subs method.</span>
-
-<span class="sd">          &gt;&gt;&gt; solve(f(x) - x, f(x))</span>
-<span class="sd">          [x]</span>
-<span class="sd">          &gt;&gt;&gt; solve(f(x).diff(x) - f(x) - x, f(x).diff(x))</span>
-<span class="sd">          [x + f(x)]</span>
-<span class="sd">          &gt;&gt;&gt; solve(f(x).diff(x) - f(x) - x, f(x))</span>
-<span class="sd">          [-x + Derivative(f(x), x)]</span>
-<span class="sd">          &gt;&gt;&gt; set(solve(x + exp(x)**2, exp(x)))</span>
-<span class="sd">          set([-sqrt(-x), sqrt(-x)])</span>
-
-<span class="sd">          &gt;&gt;&gt; from sympy import Indexed, IndexedBase, Tuple, sqrt</span>
-<span class="sd">          &gt;&gt;&gt; A = IndexedBase(&#39;A&#39;)</span>
-<span class="sd">          &gt;&gt;&gt; eqs = Tuple(A[1] + A[2] - 3, A[1] - A[2] + 1)</span>
-<span class="sd">          &gt;&gt;&gt; solve(eqs, eqs.atoms(Indexed))</span>
-<span class="sd">          {A[1]: 1, A[2]: 2}</span>
-
-<span class="sd">        * To solve for a *symbol* implicitly, use &#39;implicit=True&#39;:</span>
-
-<span class="sd">            &gt;&gt;&gt; solve(x + exp(x), x)</span>
-<span class="sd">            [-LambertW(1)]</span>
-<span class="sd">            &gt;&gt;&gt; solve(x + exp(x), x, implicit=True)</span>
-<span class="sd">            [-exp(x)]</span>
-
-<span class="sd">        * It is possible to solve for anything that can be targeted with</span>
-<span class="sd">          subs:</span>
-
-<span class="sd">            &gt;&gt;&gt; solve(x + 2 + sqrt(3), x + 2)</span>
-<span class="sd">            [-sqrt(3)]</span>
-<span class="sd">            &gt;&gt;&gt; solve((x + 2 + sqrt(3), x + 4 + y), y, x + 2)</span>
-<span class="sd">            {y: -2 + sqrt(3), x + 2: -sqrt(3)}</span>
-
-<span class="sd">        * Nothing heroic is done in this implicit solving so you may end up</span>
-<span class="sd">          with a symbol still in the solution:</span>
-
-<span class="sd">            &gt;&gt;&gt; eqs = (x*y + 3*y + sqrt(3), x + 4 + y)</span>
-<span class="sd">            &gt;&gt;&gt; solve(eqs, y, x + 2)</span>
-<span class="sd">            {y: sqrt(3)/(-x - 3), x + 2: (-2*x - 6 + sqrt(3))/(x + 3)}</span>
-<span class="sd">            &gt;&gt;&gt; solve(eqs, y*x, x)</span>
-<span class="sd">            {x: -y - 4, x*y: -3*y - sqrt(3)}</span>
-
-<span class="sd">        * if you attempt to solve for a number remember that the number</span>
-<span class="sd">          you have obtained does not necessarily mean that the value is</span>
-<span class="sd">          equivalent to the expression obtained:</span>
-
-<span class="sd">            &gt;&gt;&gt; solve(sqrt(2) - 1, 1)</span>
-<span class="sd">            [sqrt(2)]</span>
-<span class="sd">            &gt;&gt;&gt; solve(x - y + 1, 1)  # /!\ -1 is targeted, too</span>
-<span class="sd">            [x/(y - 1)]</span>
-<span class="sd">            &gt;&gt;&gt; [_.subs(z, -1) for _ in solve((x - y + 1).subs(-1, z), 1)]</span>
-<span class="sd">            [-x + y]</span>
-
-<span class="sd">        * To solve for a function within a derivative, use dsolve.</span>
-
-<span class="sd">    * single expression and more than 1 symbol</span>
-
-<span class="sd">        * when there is a linear solution</span>
-
-<span class="sd">            &gt;&gt;&gt; solve(x - y**2, x, y)</span>
-<span class="sd">            [{x: y**2}]</span>
-<span class="sd">            &gt;&gt;&gt; solve(x**2 - y, x, y)</span>
-<span class="sd">            [{y: x**2}]</span>
-
-<span class="sd">        * when undetermined coefficients are identified</span>
-
-<span class="sd">            * that are linear</span>
-
-<span class="sd">                &gt;&gt;&gt; solve((a + b)*x - b + 2, a, b)</span>
-<span class="sd">                {a: -2, b: 2}</span>
-
-<span class="sd">            * that are nonlinear</span>
-
-<span class="sd">                &gt;&gt;&gt; set(solve((a + b)*x - b**2 + 2, a, b))</span>
-<span class="sd">                set([(-sqrt(2), sqrt(2)), (sqrt(2), -sqrt(2))])</span>
-
-<span class="sd">        * if there is no linear solution then the first successful</span>
-<span class="sd">          attempt for a nonlinear solution will be returned</span>
-
-<span class="sd">            &gt;&gt;&gt; solve(x**2 - y**2, x, y) # doctest: +SKIP</span>
-<span class="sd">            [{x: -y}, {x: y}]</span>
-<span class="sd">            &gt;&gt;&gt; solve(x**2 - y**2/exp(x), x, y)</span>
-<span class="sd">            [{x: 2*LambertW(y/2)}]</span>
-<span class="sd">            &gt;&gt;&gt; solve(x**2 - y**2/exp(x), y, x) # doctest: +SKIP</span>
-<span class="sd">            [{y: -x*exp(x/2)}, {y: x*exp(x/2)}]</span>
-
-<span class="sd">    * iterable of one or more of the above</span>
-
-<span class="sd">        * involving relationals or bools</span>
-
-<span class="sd">            &gt;&gt;&gt; solve([x &lt; 3, x - 2])</span>
-<span class="sd">            And(im(x) == 0, re(x) == 2)</span>
-<span class="sd">            &gt;&gt;&gt; solve([x &gt; 3, x - 2])</span>
-<span class="sd">            False</span>
-
-<span class="sd">        * when the system is linear</span>
-
-<span class="sd">            * with a solution</span>
-
-<span class="sd">                &gt;&gt;&gt; solve([x - 3], x)</span>
-<span class="sd">                {x: 3}</span>
-<span class="sd">                &gt;&gt;&gt; solve((x + 5*y - 2, -3*x + 6*y - 15), x, y)</span>
-<span class="sd">                {x: -3, y: 1}</span>
-<span class="sd">                &gt;&gt;&gt; solve((x + 5*y - 2, -3*x + 6*y - 15), x, y, z)</span>
-<span class="sd">                {x: -3, y: 1}</span>
-<span class="sd">                &gt;&gt;&gt; solve((x + 5*y - 2, -3*x + 6*y - z), z, x, y)</span>
-<span class="sd">                {x: -5*y + 2, z: 21*y - 6}</span>
-
-<span class="sd">            * without a solution</span>
-
-<span class="sd">                &gt;&gt;&gt; solve([x + 3, x - 3])</span>
-<span class="sd">                []</span>
-
-<span class="sd">        * when the system is not linear</span>
-
-<span class="sd">            &gt;&gt;&gt; set(solve([x**2 + y -2, y**2 - 4], x, y))</span>
-<span class="sd">            set([(-2, -2), (0, 2), (2, -2)])</span>
-
-<span class="sd">        * if no symbols are given, all free symbols will be selected and a list</span>
-<span class="sd">          of mappings returned</span>
-
-<span class="sd">            &gt;&gt;&gt; solve([x - 2, x**2 + y])</span>
-<span class="sd">            [{x: 2, y: -4}]</span>
-<span class="sd">            &gt;&gt;&gt; solve([x - 2, x**2 + f(x)], set([f(x), x]))</span>
-<span class="sd">            [{x: 2, f(x): -4}]</span>
-
-<span class="sd">        * if any equation doesn&#39;t depend on the symbol(s) given it will be</span>
-<span class="sd">          eliminated from the equation set and an answer may be given</span>
-<span class="sd">          implicitly in terms of variables that were not of interest</span>
-
-<span class="sd">            &gt;&gt;&gt; solve([x - y, y - 3], x)</span>
-<span class="sd">            {x: y}</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    assumptions aren&#39;t checked when `solve()` input involves</span>
-<span class="sd">    relationals or bools.</span>
-
-<span class="sd">    When the solutions are checked, those that make any denominator zero</span>
-<span class="sd">    are automatically excluded. If you do not want to exclude such solutions</span>
-<span class="sd">    then use the check=False option:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import sin, limit</span>
-<span class="sd">        &gt;&gt;&gt; solve(sin(x)/x)</span>
-<span class="sd">        []</span>
-
-<span class="sd">    If check=False then a solution to the numerator being zero is found: x = 0.</span>
-<span class="sd">    In this case, this is a spurious solution since sin(x)/x has the well known</span>
-<span class="sd">    limit (without dicontinuity) of 1 at x = 0:</span>
-
-<span class="sd">        &gt;&gt;&gt; solve(sin(x)/x, check=False)</span>
-<span class="sd">        [0]</span>
-
-<span class="sd">    In the following case, however, the limit exists and is equal to the the</span>
-<span class="sd">    value of x = 0 that is excluded when check=True:</span>
-
-<span class="sd">        &gt;&gt;&gt; eq = x**2*(1/x - z**2/x)</span>
-<span class="sd">        &gt;&gt;&gt; solve(eq, x)</span>
-<span class="sd">        []</span>
-<span class="sd">        &gt;&gt;&gt; solve(eq, x, check=False)</span>
-<span class="sd">        [0]</span>
-<span class="sd">        &gt;&gt;&gt; limit(eq, x, 0, &#39;-&#39;)</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; limit(eq, x, 0, &#39;+&#39;)</span>
-<span class="sd">        0</span>
-
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">        - rsolve() for solving recurrence relationships</span>
-<span class="sd">        - dsolve() for solving differential equations</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># make f and symbols into lists of sympified quantities</span>
-    <span class="c"># keeping track of how f was passed since if it is a list</span>
-    <span class="c"># a dictionary of results will be returned.</span>
-    <span class="c">###########################################################################</span>
-
-    <span class="k">def</span> <span class="nf">_sympified_list</span><span class="p">(</span><span class="n">w</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="n">w</span> <span class="k">if</span> <span class="n">iterable</span><span class="p">(</span><span class="n">w</span><span class="p">)</span> <span class="k">else</span> <span class="p">[</span><span class="n">w</span><span class="p">]))</span>
-    <span class="n">bare_f</span> <span class="o">=</span> <span class="ow">not</span> <span class="n">iterable</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="n">ordered_symbols</span> <span class="o">=</span> <span class="p">(</span><span class="n">symbols</span> <span class="ow">and</span>
-                       <span class="n">symbols</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">and</span>
-                       <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">symbols</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Symbol</span><span class="p">)</span> <span class="ow">or</span>
-                        <span class="n">is_sequence</span><span class="p">(</span><span class="n">symbols</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span>
-                        <span class="n">include</span><span class="o">=</span><span class="n">GeneratorType</span><span class="p">)</span>
-                       <span class="p">)</span>
-                      <span class="p">)</span>
-    <span class="n">f</span><span class="p">,</span> <span class="n">symbols</span> <span class="o">=</span> <span class="p">(</span><span class="n">_sympified_list</span><span class="p">(</span><span class="n">w</span><span class="p">)</span> <span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="p">[</span><span class="n">f</span><span class="p">,</span> <span class="n">symbols</span><span class="p">])</span>
-
-    <span class="n">implicit</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;implicit&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-
-    <span class="c"># preprocess equation(s)</span>
-    <span class="c">###########################################################################</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">fi</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">fi</span><span class="p">,</span> <span class="n">Equality</span><span class="p">):</span>
-            <span class="n">f</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">fi</span><span class="o">.</span><span class="n">lhs</span> <span class="o">-</span> <span class="n">fi</span><span class="o">.</span><span class="n">rhs</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">fi</span><span class="p">,</span> <span class="n">Poly</span><span class="p">):</span>
-            <span class="n">f</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">fi</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">fi</span><span class="p">,</span> <span class="nb">bool</span><span class="p">)</span> <span class="ow">or</span> <span class="n">fi</span><span class="o">.</span><span class="n">is_Relational</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">reduce_inequalities</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">assume</span><span class="o">=</span><span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;assume&#39;</span><span class="p">),</span>
-                                       <span class="n">symbols</span><span class="o">=</span><span class="n">symbols</span><span class="p">)</span>
-        <span class="c"># Any embedded piecewise functions need to be brought out to the</span>
-        <span class="c"># top level so that the appropriate strategy gets selected.</span>
-        <span class="n">f</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">piecewise_fold</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-
-    <span class="c"># preprocess symbol(s)</span>
-    <span class="c">###########################################################################</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">symbols</span><span class="p">:</span>
-        <span class="c"># get symbols from equations</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="nb">reduce</span><span class="p">(</span><span class="nb">set</span><span class="o">.</span><span class="n">union</span><span class="p">,</span> <span class="p">[</span><span class="n">fi</span><span class="o">.</span><span class="n">free_symbols</span>
-                                     <span class="k">for</span> <span class="n">fi</span> <span class="ow">in</span> <span class="n">f</span><span class="p">],</span> <span class="nb">set</span><span class="p">())</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">fi</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-                <span class="n">pot</span> <span class="o">=</span> <span class="n">preorder_traversal</span><span class="p">(</span><span class="n">fi</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">pot</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">is_number</span> <span class="ow">or</span> <span class="n">p</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">or</span> <span class="n">p</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">)</span> <span class="ow">or</span> \
-                            <span class="nb">isinstance</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">AppliedUndef</span><span class="p">):</span>
-                        <span class="n">flags</span><span class="p">[</span><span class="s">&#39;dict&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">True</span>  <span class="c"># better show symbols</span>
-                        <span class="n">symbols</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-                        <span class="n">pot</span><span class="o">.</span><span class="n">skip</span><span class="p">()</span>  <span class="c"># don&#39;t go any deeper</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span>
-        <span class="c"># supply dummy symbols so solve(3) behaves like solve(3, x)</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">symbols</span><span class="p">)):</span>
-            <span class="n">symbols</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Dummy</span><span class="p">())</span>
-
-        <span class="n">ordered_symbols</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">iterable</span><span class="p">(</span><span class="n">symbols</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="c"># remove symbols the user is not interested in</span>
-    <span class="n">exclude</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;exclude&#39;</span><span class="p">,</span> <span class="nb">set</span><span class="p">())</span>
-    <span class="k">if</span> <span class="n">exclude</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">exclude</span><span class="p">,</span> <span class="n">Expr</span><span class="p">):</span>
-            <span class="n">exclude</span> <span class="o">=</span> <span class="p">[</span><span class="n">exclude</span><span class="p">]</span>
-        <span class="n">exclude</span> <span class="o">=</span> <span class="nb">reduce</span><span class="p">(</span><span class="nb">set</span><span class="o">.</span><span class="n">union</span><span class="p">,</span> <span class="p">[</span><span class="n">e</span><span class="o">.</span><span class="n">free_symbols</span> <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">sympify</span><span class="p">(</span><span class="n">exclude</span><span class="p">)])</span>
-    <span class="n">symbols</span> <span class="o">=</span> <span class="p">[</span><span class="n">s</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">symbols</span> <span class="k">if</span> <span class="n">s</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">exclude</span><span class="p">]</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">ordered_symbols</span><span class="p">:</span>
-        <span class="c"># we do this to make the results returned canonical in case f</span>
-        <span class="c"># contains a system of nonlinear equations; all other cases should</span>
-        <span class="c"># be unambiguous</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">symbols</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-
-    <span class="c"># we can solve for non-symbol entities by replacing them with Dummy symbols</span>
-    <span class="n">symbols_new</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">symbol_swapped</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">s</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">symbols</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">s</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">:</span>
-            <span class="n">s_new</span> <span class="o">=</span> <span class="n">s</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">symbol_swapped</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="n">s_new</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;X</span><span class="si">%d</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">i</span><span class="p">)</span>
-        <span class="n">symbols_new</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">s_new</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">symbol_swapped</span><span class="p">:</span>
-        <span class="n">swap_sym</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">symbols</span><span class="p">,</span> <span class="n">symbols_new</span><span class="p">))</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="p">[</span><span class="n">fi</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">swap_sym</span><span class="p">)</span> <span class="k">for</span> <span class="n">fi</span> <span class="ow">in</span> <span class="n">f</span><span class="p">]</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="n">symbols_new</span>
-        <span class="n">swap_sym</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">([(</span><span class="n">v</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">swap_sym</span><span class="p">])</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">swap_sym</span> <span class="o">=</span> <span class="p">{}</span>
-
-    <span class="c"># this is needed in the next two events</span>
-    <span class="n">symset</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span>
-
-    <span class="c"># get rid of equations that have no symbols of interest; we don&#39;t</span>
-    <span class="c"># try to solve them because the user didn&#39;t ask and they might be</span>
-    <span class="c"># hard to solve; this means that solutions may be given in terms</span>
-    <span class="c"># of the eliminated equations e.g. solve((x-y, y-3), x) -&gt; {x: y}</span>
-    <span class="n">newf</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">fi</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-        <span class="c"># let the solver handle equations that..</span>
-        <span class="c"># - have no symbols but are expressions</span>
-        <span class="c"># - have symbols of interest</span>
-        <span class="c"># - have no symbols of interest but are constant</span>
-        <span class="c"># but when an expression is not constant and has no symbols of</span>
-        <span class="c"># interest, it can&#39;t change what we obtain for a solution from</span>
-        <span class="c"># the remaining equations so we don&#39;t include it; and if it&#39;s</span>
-        <span class="c"># zero it can be removed and if it&#39;s not zero, there is no</span>
-        <span class="c"># solution for the equation set as a whole</span>
-        <span class="c">#</span>
-        <span class="c"># The reason for doing this filtering is to allow an answer</span>
-        <span class="c"># to be obtained to queries like solve((x - y, y), x); without</span>
-        <span class="c"># this mod the return value is []</span>
-        <span class="n">ok</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="k">if</span> <span class="n">fi</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">symset</span><span class="p">):</span>
-            <span class="n">ok</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">free</span> <span class="o">=</span> <span class="n">fi</span><span class="o">.</span><span class="n">free_symbols</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">free</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">fi</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">fi</span><span class="o">.</span><span class="n">is_zero</span><span class="p">:</span>
-                        <span class="k">continue</span>
-                    <span class="k">return</span> <span class="p">[]</span>
-                <span class="n">ok</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">fi</span><span class="o">.</span><span class="n">is_constant</span><span class="p">():</span>
-                    <span class="n">ok</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">if</span> <span class="n">ok</span><span class="p">:</span>
-            <span class="n">newf</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">fi</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">newf</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">newf</span>
-    <span class="k">del</span> <span class="n">newf</span>
-
-    <span class="c"># mask off any Object that we aren&#39;t going to invert: Derivative,</span>
-    <span class="c"># Integral, etc... so that solving for anything that they contain will</span>
-    <span class="c"># give an implicit solution</span>
-    <span class="n">seen</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="n">non_inverts</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="k">for</span> <span class="n">fi</span> <span class="ow">in</span> <span class="n">f</span><span class="p">:</span>
-        <span class="n">pot</span> <span class="o">=</span> <span class="n">preorder_traversal</span><span class="p">(</span><span class="n">fi</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">pot</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="nb">bool</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">Piecewise</span><span class="p">):</span>
-                <span class="k">pass</span>
-            <span class="k">elif</span> <span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="nb">bool</span><span class="p">)</span> <span class="ow">or</span>
-                <span class="ow">not</span> <span class="n">p</span><span class="o">.</span><span class="n">args</span> <span class="ow">or</span>
-                <span class="n">p</span> <span class="ow">in</span> <span class="n">symset</span> <span class="ow">or</span>
-                <span class="n">p</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">or</span> <span class="n">p</span><span class="o">.</span><span class="n">is_Mul</span> <span class="ow">or</span>
-                <span class="n">p</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">implicit</span> <span class="ow">or</span>
-                <span class="n">p</span><span class="o">.</span><span class="n">is_Function</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">AppliedUndef</span><span class="p">)</span> <span class="ow">and</span>
-                  <span class="ow">not</span> <span class="n">implicit</span><span class="p">):</span>
-                <span class="k">continue</span>
-            <span class="k">elif</span> <span class="ow">not</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">seen</span><span class="p">:</span>
-                <span class="n">seen</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">free_symbols</span> <span class="o">&amp;</span> <span class="n">symset</span><span class="p">:</span>
-                    <span class="n">non_inverts</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">continue</span>
-            <span class="n">pot</span><span class="o">.</span><span class="n">skip</span><span class="p">()</span>
-    <span class="k">del</span> <span class="n">seen</span>
-    <span class="n">non_inverts</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">non_inverts</span><span class="p">,</span> <span class="p">[</span><span class="n">Dummy</span><span class="p">()</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">non_inverts</span><span class="p">])))</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="p">[</span><span class="n">fi</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">non_inverts</span><span class="p">)</span> <span class="k">for</span> <span class="n">fi</span> <span class="ow">in</span> <span class="n">f</span><span class="p">]</span>
-    <span class="n">non_inverts</span> <span class="o">=</span> <span class="p">[(</span><span class="n">v</span><span class="p">,</span> <span class="n">k</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">swap_sym</span><span class="p">))</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">non_inverts</span><span class="o">.</span><span class="n">items</span><span class="p">()]</span>
-
-    <span class="c"># rationalize Floats</span>
-    <span class="n">floats</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="k">if</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;rational&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">fi</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">fi</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Float</span><span class="p">):</span>
-                <span class="n">floats</span> <span class="o">=</span> <span class="bp">True</span>
-                <span class="n">f</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">nsimplify</span><span class="p">(</span><span class="n">fi</span><span class="p">,</span> <span class="n">rational</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="c">#</span>
-    <span class="c"># try to get a solution</span>
-    <span class="c">###########################################################################</span>
-    <span class="k">if</span> <span class="n">bare_f</span><span class="p">:</span>
-        <span class="n">solution</span> <span class="o">=</span> <span class="n">_solve</span><span class="p">(</span><span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">solution</span> <span class="o">=</span> <span class="n">_solve_system</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-
-    <span class="c">#</span>
-    <span class="c"># postprocessing</span>
-    <span class="c">###########################################################################</span>
-    <span class="c"># Restore masked-off objects</span>
-    <span class="k">if</span> <span class="n">non_inverts</span><span class="p">:</span>
-
-        <span class="k">def</span> <span class="nf">_do_dict</span><span class="p">(</span><span class="n">solution</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">dict</span><span class="p">([(</span><span class="n">k</span><span class="p">,</span> <span class="n">v</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">non_inverts</span><span class="p">))</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span>
-                         <span class="n">solution</span><span class="o">.</span><span class="n">items</span><span class="p">()])</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">solution</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">dict</span><span class="p">:</span>
-                <span class="n">solution</span> <span class="o">=</span> <span class="n">_do_dict</span><span class="p">(</span><span class="n">solution</span><span class="p">)</span>
-                <span class="k">break</span>
-            <span class="k">elif</span> <span class="n">solution</span> <span class="ow">and</span> <span class="nb">type</span><span class="p">(</span><span class="n">solution</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">list</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">solution</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="ow">is</span> <span class="nb">dict</span><span class="p">:</span>
-                    <span class="n">solution</span> <span class="o">=</span> <span class="p">[</span><span class="n">_do_dict</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">solution</span><span class="p">]</span>
-                    <span class="k">break</span>
-                <span class="k">elif</span> <span class="nb">type</span><span class="p">(</span><span class="n">solution</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="ow">is</span> <span class="nb">tuple</span><span class="p">:</span>
-                    <span class="n">solution</span> <span class="o">=</span> <span class="p">[</span><span class="nb">tuple</span><span class="p">([</span><span class="n">v</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">non_inverts</span><span class="p">)</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">s</span><span class="p">])</span> <span class="k">for</span> <span class="n">s</span>
-                                <span class="ow">in</span> <span class="n">solution</span><span class="p">]</span>
-                    <span class="k">break</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">solution</span> <span class="o">=</span> <span class="p">[</span><span class="n">v</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">non_inverts</span><span class="p">)</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">solution</span><span class="p">]</span>
-                    <span class="k">break</span>
-            <span class="k">elif</span> <span class="ow">not</span> <span class="n">solution</span><span class="p">:</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&#39;&#39;&#39;</span>
-<span class="s">                            no handling of </span><span class="si">%s</span><span class="s"> was implemented&#39;&#39;&#39;</span> <span class="o">%</span> <span class="n">solution</span><span class="p">))</span>
-
-    <span class="c"># Restore original &quot;symbols&quot; if a dictionary is returned.</span>
-    <span class="c"># This is not necessary for</span>
-    <span class="c">#   - the single univariate equation case</span>
-    <span class="c">#     since the symbol will have been removed from the solution;</span>
-    <span class="c">#   - the nonlinear poly_system since that only supports zero-dimensional</span>
-    <span class="c">#     systems and those results come back as a list</span>
-    <span class="c">#</span>
-    <span class="c"># ** unless there were Derivatives with the symbols, but those were handled</span>
-    <span class="c">#    above.</span>
-    <span class="k">if</span> <span class="n">symbol_swapped</span><span class="p">:</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="p">[</span><span class="n">swap_sym</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">symbols</span><span class="p">]</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">solution</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">dict</span><span class="p">:</span>
-            <span class="n">solution</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">([(</span><span class="n">swap_sym</span><span class="p">[</span><span class="n">k</span><span class="p">],</span> <span class="n">v</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">swap_sym</span><span class="p">))</span>
-                             <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">solution</span><span class="o">.</span><span class="n">items</span><span class="p">()])</span>
-        <span class="k">elif</span> <span class="n">solution</span> <span class="ow">and</span> <span class="nb">type</span><span class="p">(</span><span class="n">solution</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">list</span> <span class="ow">and</span> <span class="nb">type</span><span class="p">(</span><span class="n">solution</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="ow">is</span> <span class="nb">dict</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">sol</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">solution</span><span class="p">):</span>
-                <span class="n">solution</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">([(</span><span class="n">swap_sym</span><span class="p">[</span><span class="n">k</span><span class="p">],</span> <span class="n">v</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">swap_sym</span><span class="p">))</span>
-                              <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">sol</span><span class="o">.</span><span class="n">items</span><span class="p">()])</span>
-
-    <span class="c"># undo the dictionary solutions returned when the system was only partially</span>
-    <span class="c"># solved with poly-system if all symbols are present</span>
-    <span class="k">if</span> <span class="p">(</span>
-            <span class="n">solution</span> <span class="ow">and</span>
-            <span class="n">ordered_symbols</span> <span class="ow">and</span>
-            <span class="nb">type</span><span class="p">(</span><span class="n">solution</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="nb">dict</span> <span class="ow">and</span>
-            <span class="nb">type</span><span class="p">(</span><span class="n">solution</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="ow">is</span> <span class="nb">dict</span> <span class="ow">and</span>
-            <span class="nb">all</span><span class="p">(</span><span class="n">s</span> <span class="ow">in</span> <span class="n">solution</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">symbols</span><span class="p">)</span>
-    <span class="p">):</span>
-        <span class="n">solution</span> <span class="o">=</span> <span class="p">[</span><span class="nb">tuple</span><span class="p">([</span><span class="n">r</span><span class="p">[</span><span class="n">s</span><span class="p">]</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">r</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">symbols</span><span class="p">])</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">solution</span><span class="p">]</span>
-
-    <span class="c"># Get assumptions about symbols, to filter solutions.</span>
-    <span class="c"># Note that if assumptions about a solution can&#39;t be verified, it is still</span>
-    <span class="c"># returned.</span>
-    <span class="n">check</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;check&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-
-    <span class="c"># restore floats</span>
-    <span class="k">if</span> <span class="n">floats</span> <span class="ow">and</span> <span class="n">solution</span> <span class="ow">and</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;rational&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">solution</span> <span class="o">=</span> <span class="n">nfloat</span><span class="p">(</span><span class="n">solution</span><span class="p">,</span> <span class="n">exponent</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">check</span> <span class="ow">and</span> <span class="n">solution</span><span class="p">:</span>
-
-        <span class="n">warning</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;warn&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-        <span class="n">got_None</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c"># solutions for which one or more symbols gave None</span>
-        <span class="n">no_False</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c"># solutions for which no symbols gave False</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">solution</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">list</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">solution</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="ow">is</span> <span class="nb">tuple</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">sol</span> <span class="ow">in</span> <span class="n">solution</span><span class="p">:</span>
-                    <span class="k">for</span> <span class="n">symb</span><span class="p">,</span> <span class="n">val</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">symbols</span><span class="p">,</span> <span class="n">sol</span><span class="p">):</span>
-                        <span class="n">test</span> <span class="o">=</span> <span class="n">check_assumptions</span><span class="p">(</span><span class="n">val</span><span class="p">,</span> <span class="o">**</span><span class="n">symb</span><span class="o">.</span><span class="n">assumptions0</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">test</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                            <span class="k">break</span>
-                        <span class="k">if</span> <span class="n">test</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                            <span class="n">got_None</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sol</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">no_False</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sol</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="nb">type</span><span class="p">(</span><span class="n">solution</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="ow">is</span> <span class="nb">dict</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">sol</span> <span class="ow">in</span> <span class="n">solution</span><span class="p">:</span>
-                    <span class="n">a_None</span> <span class="o">=</span> <span class="bp">False</span>
-                    <span class="k">for</span> <span class="n">symb</span><span class="p">,</span> <span class="n">val</span> <span class="ow">in</span> <span class="n">sol</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                        <span class="n">test</span> <span class="o">=</span> <span class="n">check_assumptions</span><span class="p">(</span><span class="n">val</span><span class="p">,</span> <span class="o">**</span><span class="n">symb</span><span class="o">.</span><span class="n">assumptions0</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">test</span><span class="p">:</span>
-                            <span class="k">continue</span>
-                        <span class="k">if</span> <span class="n">test</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                            <span class="k">break</span>
-                        <span class="n">a_None</span> <span class="o">=</span> <span class="bp">True</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">no_False</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sol</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">a_None</span><span class="p">:</span>
-                            <span class="n">got_None</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sol</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>  <span class="c"># list of expressions</span>
-                <span class="k">for</span> <span class="n">sol</span> <span class="ow">in</span> <span class="n">solution</span><span class="p">:</span>
-                    <span class="n">test</span> <span class="o">=</span> <span class="n">check_assumptions</span><span class="p">(</span><span class="n">sol</span><span class="p">,</span> <span class="o">**</span><span class="n">symbols</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">assumptions0</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">test</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                        <span class="k">continue</span>
-                    <span class="n">no_False</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sol</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">test</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="n">got_None</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sol</span><span class="p">)</span>
-
-        <span class="k">elif</span> <span class="nb">type</span><span class="p">(</span><span class="n">solution</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">dict</span><span class="p">:</span>
-            <span class="n">a_None</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="k">for</span> <span class="n">symb</span><span class="p">,</span> <span class="n">val</span> <span class="ow">in</span> <span class="n">solution</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                <span class="n">test</span> <span class="o">=</span> <span class="n">check_assumptions</span><span class="p">(</span><span class="n">val</span><span class="p">,</span> <span class="o">**</span><span class="n">symb</span><span class="o">.</span><span class="n">assumptions0</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">test</span><span class="p">:</span>
-                    <span class="k">continue</span>
-                <span class="k">if</span> <span class="n">test</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                    <span class="n">no_False</span> <span class="o">=</span> <span class="bp">None</span>
-                    <span class="k">break</span>
-                <span class="n">a_None</span> <span class="o">=</span> <span class="bp">True</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">no_False</span> <span class="o">=</span> <span class="n">solution</span>
-                <span class="k">if</span> <span class="n">a_None</span><span class="p">:</span>
-                    <span class="n">got_None</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">solution</span><span class="p">)</span>
-
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">solution</span><span class="p">,</span> <span class="p">(</span><span class="n">Relational</span><span class="p">,</span> <span class="n">And</span><span class="p">,</span> <span class="n">Or</span><span class="p">)):</span>
-            <span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span>
-            <span class="k">if</span> <span class="n">warning</span> <span class="ow">and</span> <span class="n">symbols</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">assumptions0</span><span class="p">:</span>
-                <span class="k">print</span><span class="p">((</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&quot;&quot;&quot;</span>
-<span class="s">                    </span><span class="se">\t</span><span class="s">Warning: assumptions about variable &#39;</span><span class="si">%s</span><span class="s">&#39; are</span>
-<span class="s">                    not handled currently.&quot;&quot;&quot;</span> <span class="o">%</span> <span class="n">symbols</span><span class="p">[</span><span class="mi">0</span><span class="p">])))</span>
-            <span class="c"># TODO: check also variable assumptions for inequalities</span>
-
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;Unrecognized solution&#39;</span><span class="p">)</span>  <span class="c"># improve the checker</span>
-
-        <span class="n">solution</span> <span class="o">=</span> <span class="n">no_False</span>
-        <span class="k">if</span> <span class="n">warning</span> <span class="ow">and</span> <span class="n">got_None</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">((</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&quot;&quot;&quot;</span>
-<span class="s">                </span><span class="se">\t</span><span class="s">Warning: assumptions concerning following solution(s)</span>
-<span class="s">                can&#39;t be checked:&quot;&quot;&quot;</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\n\t</span><span class="s">&#39;</span> <span class="o">+</span>
-                <span class="s">&#39;, &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">got_None</span><span class="p">))))</span>
-
-    <span class="c">#</span>
-    <span class="c"># done</span>
-    <span class="c">###########################################################################</span>
-
-    <span class="n">as_dict</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;dict&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-    <span class="n">as_set</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;set&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">as_set</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">solution</span><span class="p">,</span> <span class="nb">list</span><span class="p">):</span>
-        <span class="c"># Make sure that a list of solutions is ordered in a canonical way.</span>
-        <span class="n">solution</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">as_dict</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">as_set</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">solution</span> <span class="ow">or</span> <span class="p">[]</span>
-
-    <span class="c"># make return a list of mappings or []</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">solution</span><span class="p">:</span>
-        <span class="n">solution</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">solution</span><span class="p">,</span> <span class="nb">dict</span><span class="p">):</span>
-            <span class="n">solution</span> <span class="o">=</span> <span class="p">[</span><span class="n">solution</span><span class="p">]</span>
-        <span class="k">elif</span> <span class="n">iterable</span><span class="p">(</span><span class="n">solution</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
-            <span class="n">solution</span> <span class="o">=</span> <span class="p">[</span><span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">symbols</span><span class="p">,</span> <span class="n">s</span><span class="p">)))</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">solution</span><span class="p">]</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">solution</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="nb">dict</span><span class="p">):</span>
-            <span class="k">pass</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span>
-            <span class="n">solution</span> <span class="o">=</span> <span class="p">[{</span><span class="n">symbols</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span> <span class="n">s</span><span class="p">}</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">solution</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">as_dict</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">solution</span>
-    <span class="k">assert</span> <span class="n">as_set</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">solution</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[],</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="n">k</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">solution</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">keys</span><span class="p">()),</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">i</span><span class="p">:</span> <span class="n">i</span><span class="o">.</span><span class="n">sort_key</span><span class="p">())</span>
-    <span class="k">return</span> <span class="n">k</span><span class="p">,</span> <span class="nb">set</span><span class="p">([</span><span class="nb">tuple</span><span class="p">([</span><span class="n">s</span><span class="p">[</span><span class="n">ki</span><span class="p">]</span> <span class="k">for</span> <span class="n">ki</span> <span class="ow">in</span> <span class="n">k</span><span class="p">])</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">solution</span><span class="p">])</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_solve</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return a checked solution for f in terms of one or more of the</span>
-<span class="sd">    symbols.&quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">soln</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="n">free</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">free_symbols</span>
-        <span class="n">ex</span> <span class="o">=</span> <span class="n">free</span> <span class="o">-</span> <span class="nb">set</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">ex</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">ind</span><span class="p">,</span> <span class="n">dep</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="o">*</span><span class="n">symbols</span><span class="p">)</span>
-            <span class="n">ex</span> <span class="o">=</span> <span class="n">ind</span><span class="o">.</span><span class="n">free_symbols</span> <span class="o">&amp;</span> <span class="n">dep</span><span class="o">.</span><span class="n">free_symbols</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">ex</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">ex</span> <span class="o">=</span> <span class="n">ex</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="c"># may come back as dict or list (if non-linear)</span>
-                <span class="n">soln</span> <span class="o">=</span> <span class="n">solve_undetermined_coeffs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">ex</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                <span class="k">pass</span>
-        <span class="k">if</span> <span class="n">soln</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">soln</span>
-        <span class="c"># find first successful solution</span>
-        <span class="n">failed</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">symbols</span><span class="p">:</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">solve_linear</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">symbols</span><span class="o">=</span><span class="p">[</span><span class="n">s</span><span class="p">])</span>
-            <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">:</span>
-                <span class="c"># no need to check but we should simplify if desired</span>
-                <span class="k">if</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;simplify&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-                    <span class="n">d</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-                <span class="k">return</span> <span class="p">[{</span><span class="n">n</span><span class="p">:</span> <span class="n">d</span><span class="p">}]</span>
-            <span class="k">elif</span> <span class="n">n</span> <span class="ow">and</span> <span class="n">d</span><span class="p">:</span>  <span class="c"># otherwise there was no solution for s</span>
-                <span class="n">failed</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">failed</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">failed</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">soln</span> <span class="o">=</span> <span class="n">_solve</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-                <span class="k">return</span> <span class="p">[{</span><span class="n">s</span><span class="p">:</span> <span class="n">sol</span><span class="p">}</span> <span class="k">for</span> <span class="n">sol</span> <span class="ow">in</span> <span class="n">soln</span><span class="p">]</span>
-            <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                <span class="k">continue</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">msg</span> <span class="o">=</span> <span class="s">&quot;No algorithms are implemented to solve equation </span><span class="si">%s</span><span class="s">&quot;</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="n">msg</span> <span class="o">%</span> <span class="n">f</span><span class="p">)</span>
-    <span class="n">symbol</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="n">check</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;check&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-    <span class="c"># build up solutions if f is a Mul</span>
-    <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="n">dens</span> <span class="o">=</span> <span class="n">denoms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">symbols</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">soln</span> <span class="o">=</span> <span class="n">_solve</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">symbol</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">soln</span><span class="p">))</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">result</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">check</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="p">[</span><span class="n">s</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">result</span> <span class="k">if</span>
-                <span class="nb">all</span><span class="p">(</span><span class="ow">not</span> <span class="n">checksol</span><span class="p">(</span><span class="n">den</span><span class="p">,</span> <span class="p">{</span><span class="n">symbol</span><span class="p">:</span> <span class="n">s</span><span class="p">},</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span> <span class="k">for</span> <span class="n">den</span> <span class="ow">in</span> <span class="n">dens</span><span class="p">)]</span>
-        <span class="c"># set flags for quick exit at end</span>
-        <span class="n">check</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="n">flags</span><span class="p">[</span><span class="s">&#39;simplify&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="k">elif</span> <span class="n">f</span><span class="o">.</span><span class="n">is_Piecewise</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">expr</span><span class="p">,</span> <span class="n">cond</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">candidates</span> <span class="o">=</span> <span class="n">_solve</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">cond</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                <span class="c"># Only include solutions that do not match the condition</span>
-                <span class="c"># of any of the other pieces.</span>
-                <span class="k">for</span> <span class="n">candidate</span> <span class="ow">in</span> <span class="n">candidates</span><span class="p">:</span>
-                    <span class="n">matches_other_piece</span> <span class="o">=</span> <span class="bp">False</span>
-                    <span class="k">for</span> <span class="n">other_expr</span><span class="p">,</span> <span class="n">other_cond</span> <span class="ow">in</span> <span class="n">f</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">other_cond</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-                            <span class="k">continue</span>
-                        <span class="k">if</span> <span class="nb">bool</span><span class="p">(</span><span class="n">other_cond</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">symbol</span><span class="p">,</span> <span class="n">candidate</span><span class="p">)):</span>
-                            <span class="n">matches_other_piece</span> <span class="o">=</span> <span class="bp">True</span>
-                            <span class="k">break</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">matches_other_piece</span><span class="p">:</span>
-                        <span class="n">result</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">candidate</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">candidate</span> <span class="ow">in</span> <span class="n">candidates</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="nb">bool</span><span class="p">(</span><span class="n">cond</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">symbol</span><span class="p">,</span> <span class="n">candidate</span><span class="p">)):</span>
-                        <span class="n">result</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">candidate</span><span class="p">)</span>
-        <span class="n">check</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="c"># first see if it really depends on symbol and whether there</span>
-        <span class="c"># is a linear solution</span>
-        <span class="n">f_num</span><span class="p">,</span> <span class="n">sol</span> <span class="o">=</span> <span class="n">solve_linear</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">symbols</span><span class="o">=</span><span class="n">symbols</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">symbol</span> <span class="ow">in</span> <span class="n">f_num</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[]</span>
-        <span class="k">elif</span> <span class="n">f_num</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">:</span>
-            <span class="c"># no need to check but simplify if desired</span>
-            <span class="k">if</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;simplify&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-                <span class="n">sol</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">sol</span><span class="p">)</span>
-            <span class="k">return</span> <span class="p">[</span><span class="n">sol</span><span class="p">]</span>
-
-        <span class="n">result</span> <span class="o">=</span> <span class="bp">False</span>  <span class="c"># no solution was obtained</span>
-        <span class="n">msg</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>  <span class="c"># there is no failure message</span>
-        <span class="n">dens</span> <span class="o">=</span> <span class="n">denoms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">symbols</span><span class="p">)</span>  <span class="c"># store these for checking later</span>
-
-        <span class="c"># Poly is generally robust enough to convert anything to</span>
-        <span class="c"># a polynomial and tell us the different generators that it</span>
-        <span class="c"># contains, so we will inspect the generators identified by</span>
-        <span class="c"># polys to figure out what to do.</span>
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">f_num</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">poly</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;could not convert </span><span class="si">%s</span><span class="s"> to Poly&#39;</span> <span class="o">%</span> <span class="n">f_num</span><span class="p">)</span>
-        <span class="n">gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">g</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">poly</span><span class="o">.</span><span class="n">gens</span> <span class="k">if</span> <span class="n">g</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">symbol</span><span class="p">)]</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># If there is more than one generator, it could be that the</span>
-            <span class="c"># generators have the same base but different powers, e.g.</span>
-            <span class="c">#   &gt;&gt;&gt; Poly(exp(x)+1/exp(x))</span>
-            <span class="c">#   Poly(exp(-x) + exp(x), exp(-x), exp(x), domain=&#39;ZZ&#39;)</span>
-            <span class="c">#   &gt;&gt;&gt; Poly(sqrt(x)+sqrt(sqrt(x)))</span>
-            <span class="c">#   Poly(sqrt(x) + x**(1/4), sqrt(x), x**(1/4), domain=&#39;ZZ&#39;)</span>
-            <span class="c"># If the exponents are Rational then a change of variables</span>
-            <span class="c"># will make this a polynomial equation in a single base.</span>
-
-            <span class="k">def</span> <span class="nf">_as_base_q</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-                <span class="sd">&quot;&quot;&quot;Return (b**e, q) for x = b**(p*e/q) where p/q is the leading</span>
-<span class="sd">                Rational of the exponent of x, e.g. exp(-2*x/3) -&gt; (exp(x), 3)</span>
-<span class="sd">                &quot;&quot;&quot;</span>
-                <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                <span class="k">if</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Rational</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">b</span><span class="p">,</span> <span class="n">e</span><span class="o">.</span><span class="n">q</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">e</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">x</span><span class="p">,</span> <span class="mi">1</span>
-                <span class="n">c</span><span class="p">,</span> <span class="n">ee</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">as_coeff_Mul</span><span class="p">()</span>
-                <span class="k">if</span> <span class="n">c</span><span class="o">.</span><span class="n">is_Rational</span> <span class="ow">and</span> <span class="n">c</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>  <span class="c"># c could be a Float</span>
-                    <span class="k">return</span> <span class="n">b</span><span class="o">**</span><span class="n">ee</span><span class="p">,</span> <span class="n">c</span><span class="o">.</span><span class="n">q</span>
-                <span class="k">return</span> <span class="n">x</span><span class="p">,</span> <span class="mi">1</span>
-
-            <span class="n">bases</span><span class="p">,</span> <span class="n">qs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">_as_base_q</span><span class="p">(</span><span class="n">g</span><span class="p">)</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">gens</span><span class="p">]))</span>
-            <span class="n">bases</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">bases</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">bases</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">funcs</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">func</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">bases</span> <span class="k">if</span> <span class="n">b</span><span class="o">.</span><span class="n">is_Function</span><span class="p">)</span>
-
-                <span class="n">trig</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span><span class="n">cos</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">tan</span><span class="p">,</span> <span class="n">cot</span><span class="p">])</span>
-                <span class="n">other</span> <span class="o">=</span> <span class="n">funcs</span> <span class="o">-</span> <span class="n">trig</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">other</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">funcs</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">trig</span><span class="p">))</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">_solve</span><span class="p">(</span><span class="n">f_num</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">tan</span><span class="p">),</span> <span class="n">symbol</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-
-                <span class="n">trigh</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span><span class="n">cosh</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">tanh</span><span class="p">,</span> <span class="n">coth</span><span class="p">])</span>
-                <span class="n">other</span> <span class="o">=</span> <span class="n">funcs</span> <span class="o">-</span> <span class="n">trigh</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">other</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">funcs</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">trigh</span><span class="p">))</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="n">_solve</span><span class="p">(</span><span class="n">f_num</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">tanh</span><span class="p">),</span> <span class="n">symbol</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-
-                <span class="c"># just a simple case - see if replacement of single function</span>
-                <span class="c"># clears all symbol-dependent functions, e.g.</span>
-                <span class="c"># log(x) - log(log(x) - 1) - 3 can be solved even though it has</span>
-                <span class="c"># two generators.</span>
-
-                <span class="n">funcs</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">bases</span> <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">is_Function</span><span class="p">]</span>
-                <span class="k">if</span> <span class="n">funcs</span><span class="p">:</span>
-                    <span class="n">funcs</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">count_ops</span><span class="p">)</span>  <span class="c"># put shallowest function first</span>
-                    <span class="n">f1</span> <span class="o">=</span> <span class="n">funcs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                    <span class="n">t</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">)</span>
-                    <span class="c"># perform the substitution</span>
-                    <span class="n">ftry</span> <span class="o">=</span> <span class="n">f_num</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">f1</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
-
-                    <span class="c"># if no Functions left, we can proceed with usual solve</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">ftry</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">symbol</span><span class="p">):</span>
-                        <span class="n">cv_sols</span> <span class="o">=</span> <span class="n">_solve</span><span class="p">(</span><span class="n">ftry</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
-                        <span class="n">cv_inv</span> <span class="o">=</span> <span class="n">_solve</span><span class="p">(</span><span class="n">t</span> <span class="o">-</span> <span class="n">f1</span><span class="p">,</span> <span class="n">symbol</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-                        <span class="n">sols</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
-                        <span class="k">for</span> <span class="n">sol</span> <span class="ow">in</span> <span class="n">cv_sols</span><span class="p">:</span>
-                            <span class="n">sols</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">cv_inv</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">sol</span><span class="p">))</span>
-                        <span class="k">return</span> <span class="n">sols</span>
-
-                <span class="n">msg</span> <span class="o">=</span> <span class="s">&#39;multiple generators </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">gens</span>
-
-            <span class="k">elif</span> <span class="nb">any</span><span class="p">(</span><span class="n">q</span> <span class="o">!=</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">qs</span><span class="p">):</span>
-                <span class="c"># e.g. for x**(1/2) + x**(1/4) a change of variables</span>
-                <span class="c"># can be made using p**4 to give p**2 + p</span>
-                <span class="n">base</span> <span class="o">=</span> <span class="n">bases</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-                <span class="n">m</span> <span class="o">=</span> <span class="nb">reduce</span><span class="p">(</span><span class="n">ilcm</span><span class="p">,</span> <span class="n">qs</span><span class="p">)</span>
-                <span class="n">p</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                <span class="n">cov</span> <span class="o">=</span> <span class="n">p</span><span class="o">**</span><span class="n">m</span>
-                <span class="n">fnew</span> <span class="o">=</span> <span class="n">f_num</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">cov</span><span class="p">)</span>
-                <span class="n">poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">fnew</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>  <span class="c"># we now have a single generator, p</span>
-
-                <span class="c"># for cubics and quartics, if the flag wasn&#39;t set, DON&#39;T do it</span>
-                <span class="c"># by default since the results are quite long. Perhaps one</span>
-                <span class="c"># could base this decision on a certain critical length of the</span>
-                <span class="c"># roots.</span>
-                <span class="k">if</span> <span class="n">poly</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-                    <span class="n">flags</span><span class="p">[</span><span class="s">&#39;simplify&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;simplify&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-
-                <span class="n">soln</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">roots</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">cubics</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">quartics</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">soln</span><span class="p">:</span>
-                    <span class="n">soln</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">all_roots</span><span class="p">()</span>
-                    <span class="n">check</span> <span class="o">=</span> <span class="bp">False</span>  <span class="c"># RootOf instances can not be checked</span>
-
-                <span class="c"># We now know what the values of p are equal to. Now find out</span>
-                <span class="c"># how they are related to the original x, e.g. if p**2 = cos(x)</span>
-                <span class="c"># then x = acos(p**2)</span>
-                <span class="c">#</span>
-                <span class="n">inversion</span> <span class="o">=</span> <span class="n">_solve</span><span class="p">(</span><span class="n">cov</span> <span class="o">-</span> <span class="n">base</span><span class="p">,</span> <span class="n">symbol</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">inversion</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">soln</span><span class="p">]</span>
-
-            <span class="k">else</span><span class="p">:</span>  <span class="c"># len(bases) == 1 and all(q == 1 for q in qs):</span>
-                <span class="c"># e.g. case where gens are exp(x), exp(-x)</span>
-                <span class="n">u</span> <span class="o">=</span> <span class="n">bases</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-                <span class="n">t</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">)</span>
-                <span class="n">inv</span> <span class="o">=</span> <span class="n">_solve</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="n">t</span><span class="p">,</span> <span class="n">symbol</span><span class="p">)</span>
-                <span class="n">ftry</span> <span class="o">=</span> <span class="n">f_num</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">ftry</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">symbol</span><span class="p">):</span>
-                    <span class="n">soln</span> <span class="o">=</span> <span class="n">_solve</span><span class="p">(</span><span class="n">ftry</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
-                    <span class="n">sols</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
-                    <span class="k">for</span> <span class="n">sol</span> <span class="ow">in</span> <span class="n">soln</span><span class="p">:</span>
-                        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">inv</span><span class="p">:</span>
-                            <span class="n">sols</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">sol</span><span class="p">))</span>
-                    <span class="k">return</span> <span class="n">sols</span>
-
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-
-            <span class="c"># There is only one generator that we are interested in, but there</span>
-            <span class="c"># may have been more than one generator identified by polys (e.g.</span>
-            <span class="c"># for symbols other than the one we are interested in) so recast</span>
-            <span class="c"># the poly in terms of our generator of interest.</span>
-
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">poly</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">poly</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">gens</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-            <span class="c"># if we haven&#39;t tried tsolve yet, do so now</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">flags</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;tsolve&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-                <span class="c"># for cubics and quartics, if the flag wasn&#39;t set, DON&#39;T do it</span>
-                <span class="c"># by default since the results are quite long. Perhaps one</span>
-                <span class="c"># could base this decision on a certain critical length of the</span>
-                <span class="c"># roots.</span>
-                <span class="k">if</span> <span class="n">poly</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-                    <span class="n">flags</span><span class="p">[</span><span class="s">&#39;simplify&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;simplify&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-                <span class="n">soln</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">roots</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">cubics</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">quartics</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">soln</span><span class="p">:</span>
-                    <span class="n">soln</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">all_roots</span><span class="p">()</span>
-                    <span class="n">check</span> <span class="o">=</span> <span class="bp">False</span>  <span class="c"># RootOf instances can not be checked</span>
-                <span class="n">gen</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">gen</span>
-                <span class="k">if</span> <span class="n">gen</span> <span class="o">!=</span> <span class="n">symbol</span><span class="p">:</span>
-                    <span class="n">u</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">()</span>
-                    <span class="n">flags</span><span class="p">[</span><span class="s">&#39;tsolve&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">True</span>
-                    <span class="n">inversion</span> <span class="o">=</span> <span class="n">_solve</span><span class="p">(</span><span class="n">gen</span> <span class="o">-</span> <span class="n">u</span><span class="p">,</span> <span class="n">symbol</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-                    <span class="n">soln</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">set</span><span class="p">([</span><span class="n">i</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span>
-                                <span class="n">inversion</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">soln</span><span class="p">]))</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">soln</span>
-
-    <span class="c"># fallback if above fails</span>
-    <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">_tsolve</span><span class="p">(</span><span class="n">f_num</span><span class="p">,</span> <span class="n">symbol</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="k">if</span> <span class="n">result</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="n">msg</span> <span class="o">+</span>
-        <span class="s">&quot;</span><span class="se">\n</span><span class="s">No algorithms are implemented to solve equation </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">f</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;simplify&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">simplify</span><span class="p">,</span> <span class="n">result</span><span class="p">))</span>
-        <span class="c"># we just simplified the solution so we now set the flag to</span>
-        <span class="c"># False so the simplification doesn&#39;t happen again in checksol()</span>
-        <span class="n">flags</span><span class="p">[</span><span class="s">&#39;simplify&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="k">if</span> <span class="n">check</span><span class="p">:</span>
-        <span class="c"># reject any result that makes any denom. affirmatively 0;</span>
-        <span class="c"># if in doubt, keep it</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[</span><span class="n">s</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">result</span> <span class="k">if</span>
-                  <span class="nb">all</span><span class="p">(</span><span class="ow">not</span> <span class="n">checksol</span><span class="p">(</span><span class="n">den</span><span class="p">,</span> <span class="p">{</span><span class="n">symbol</span><span class="p">:</span> <span class="n">s</span><span class="p">},</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-                    <span class="k">for</span> <span class="n">den</span> <span class="ow">in</span> <span class="n">dens</span><span class="p">)]</span>
-        <span class="c"># keep only results if the check is not False</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[</span><span class="n">r</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">result</span> <span class="k">if</span>
-                  <span class="n">checksol</span><span class="p">(</span><span class="n">f_num</span><span class="p">,</span> <span class="p">{</span><span class="n">symbol</span><span class="p">:</span> <span class="n">r</span><span class="p">},</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="k">def</span> <span class="nf">_solve_system</span><span class="p">(</span><span class="n">exprs</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">):</span>
-    <span class="n">check</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;check&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">exprs</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-    <span class="n">polys</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">dens</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="n">failed</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">result</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">manual</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;manual&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">g</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">exprs</span><span class="p">):</span>
-        <span class="n">dens</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">denoms</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">symbols</span><span class="p">))</span>
-        <span class="n">i</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">_invert</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">)</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">d</span> <span class="o">-</span> <span class="n">i</span>
-        <span class="n">g</span> <span class="o">=</span> <span class="n">exprs</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">manual</span><span class="p">:</span>
-            <span class="n">failed</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-            <span class="k">continue</span>
-
-        <span class="n">poly</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">as_poly</span><span class="p">(</span><span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="p">{</span><span class="s">&#39;extension&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">})</span>
-
-        <span class="k">if</span> <span class="n">poly</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">polys</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">poly</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">failed</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">polys</span><span class="p">:</span>
-        <span class="n">solved_syms</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">is_linear</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">polys</span><span class="p">):</span>
-            <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">polys</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span>
-            <span class="n">matrix</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">poly</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">polys</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">monom</span><span class="p">,</span> <span class="n">coeff</span> <span class="ow">in</span> <span class="n">poly</span><span class="o">.</span><span class="n">terms</span><span class="p">():</span>
-                    <span class="k">try</span><span class="p">:</span>
-                        <span class="n">j</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">monom</span><span class="p">)</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-                        <span class="n">matrix</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">coeff</span>
-                    <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-                        <span class="n">matrix</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">m</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">coeff</span>
-
-            <span class="c"># returns a dictionary ({symbols: values}) or None</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">solve_linear_system</span><span class="p">(</span><span class="n">matrix</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">result</span><span class="p">:</span>
-                <span class="c"># it doesn&#39;t need to be checked but we need to see</span>
-                <span class="c"># that it didn&#39;t set any denominators to 0</span>
-                <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">checksol</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">result</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">dens</span><span class="p">):</span>
-                    <span class="n">result</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">if</span> <span class="n">failed</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">result</span><span class="p">:</span>
-                    <span class="n">solved_syms</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">result</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">solved_syms</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span> <span class="o">&gt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">polys</span><span class="p">):</span>
-                <span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">subsets</span>
-                <span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">set_union</span>
-
-                <span class="n">free</span> <span class="o">=</span> <span class="n">set_union</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">p</span><span class="o">.</span><span class="n">free_symbols</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">polys</span><span class="p">])</span>
-                <span class="n">free</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">free</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">symbols</span><span class="p">))</span>
-                <span class="n">free</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">syms</span> <span class="ow">in</span> <span class="n">subsets</span><span class="p">(</span><span class="n">free</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">polys</span><span class="p">)):</span>
-                    <span class="k">try</span><span class="p">:</span>
-                        <span class="c"># returns [] or list of tuples of solutions for syms</span>
-                        <span class="n">result</span> <span class="o">=</span> <span class="n">solve_poly_system</span><span class="p">(</span><span class="n">polys</span><span class="p">,</span> <span class="o">*</span><span class="n">syms</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">result</span><span class="p">:</span>
-                            <span class="n">solved_syms</span> <span class="o">=</span> <span class="n">syms</span>
-                            <span class="k">break</span>
-                    <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                        <span class="k">pass</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;no valid subset found&#39;</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="n">result</span> <span class="o">=</span> <span class="n">solve_poly_system</span><span class="p">(</span><span class="n">polys</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">)</span>
-                    <span class="n">solved_syms</span> <span class="o">=</span> <span class="n">symbols</span>
-                <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                    <span class="n">failed</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">g</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">polys</span><span class="p">])</span>
-                    <span class="n">solved_syms</span> <span class="o">=</span> <span class="p">[]</span>
-
-            <span class="k">if</span> <span class="n">result</span><span class="p">:</span>
-                <span class="c"># we don&#39;t know here if the symbols provided were given</span>
-                <span class="c"># or not, so let solve resolve that. A list of dictionaries</span>
-                <span class="c"># is going to always be returned from here.</span>
-                <span class="c">#</span>
-                <span class="c"># We do not check the solution obtained from polys, either.</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="p">[</span><span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">solved_syms</span><span class="p">,</span> <span class="n">r</span><span class="p">)))</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">result</span><span class="p">]</span>
-
-    <span class="k">if</span> <span class="n">failed</span><span class="p">:</span>
-        <span class="c"># For each failed equation, see if we can solve for one of the</span>
-        <span class="c"># remaining symbols from that equation. If so, we update the</span>
-        <span class="c"># solution set and continue with the next failed equation,</span>
-        <span class="c"># repeating until we are done or we get an equation that can&#39;t</span>
-        <span class="c"># be solved.</span>
-        <span class="k">if</span> <span class="n">result</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">result</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">dict</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="p">[</span><span class="n">result</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="p">[{}]</span>
-
-        <span class="k">def</span> <span class="nf">_ok_syms</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">sort</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">free_symbols</span> <span class="o">-</span> <span class="n">solved_syms</span><span class="p">)</span> <span class="o">&amp;</span> <span class="n">legal</span>
-            <span class="k">if</span> <span class="n">sort</span><span class="p">:</span>
-                <span class="n">rv</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">rv</span><span class="p">)</span>
-                <span class="n">rv</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">rv</span>
-
-        <span class="n">solved_syms</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">solved_syms</span><span class="p">)</span>  <span class="c"># set of symbols we have solved for</span>
-        <span class="n">legal</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span>  <span class="c"># what we are interested in</span>
-        <span class="n">simplify_flag</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;simplify&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-        <span class="n">do_simplify</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;simplify&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-        <span class="c"># sort so equation with the fewest potential symbols is first</span>
-        <span class="k">for</span> <span class="n">eq</span> <span class="ow">in</span> <span class="n">ordered</span><span class="p">(</span><span class="n">failed</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">_</span><span class="p">:</span> <span class="nb">len</span><span class="p">(</span><span class="n">_ok_syms</span><span class="p">(</span><span class="n">_</span><span class="p">))):</span>
-            <span class="n">newresult</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">bad_results</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="n">got_s</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="n">u</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">()</span>
-            <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">result</span><span class="p">:</span>
-                <span class="c"># update eq with everything that is known so far</span>
-                <span class="n">eq2</span> <span class="o">=</span> <span class="n">eq</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-                <span class="c"># if check is True then we see if it satisfies this</span>
-                <span class="c"># equation, otherwise we just accept it</span>
-                <span class="k">if</span> <span class="n">check</span> <span class="ow">and</span> <span class="n">r</span><span class="p">:</span>
-                    <span class="n">b</span> <span class="o">=</span> <span class="n">checksol</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">eq2</span><span class="p">,</span> <span class="n">minimal</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">b</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="c"># this solution is sufficient to know whether</span>
-                        <span class="c"># it is valid or not so we either accept or</span>
-                        <span class="c"># reject it, then continue</span>
-                        <span class="k">if</span> <span class="n">b</span><span class="p">:</span>
-                            <span class="n">newresult</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-                        <span class="k">else</span><span class="p">:</span>
-                            <span class="n">bad_results</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-                        <span class="k">continue</span>
-                <span class="c"># search for a symbol amongst those available that</span>
-                <span class="c"># can be solved for</span>
-                <span class="n">ok_syms</span> <span class="o">=</span> <span class="n">_ok_syms</span><span class="p">(</span><span class="n">eq2</span><span class="p">,</span> <span class="n">sort</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">ok_syms</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">r</span><span class="p">:</span>
-                        <span class="n">newresult</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-                    <span class="k">break</span>  <span class="c"># skip as it&#39;s independent of desired symbols</span>
-                <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">ok_syms</span><span class="p">:</span>
-                    <span class="k">try</span><span class="p">:</span>
-                        <span class="n">soln</span> <span class="o">=</span> <span class="n">_solve</span><span class="p">(</span><span class="n">eq2</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-                    <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-                        <span class="k">continue</span>
-                    <span class="c"># put each solution in r and append the now-expanded</span>
-                    <span class="c"># result in the new result list; use copy since the</span>
-                    <span class="c"># solution for s in being added in-place</span>
-                    <span class="k">if</span> <span class="n">do_simplify</span><span class="p">:</span>
-                        <span class="n">flags</span><span class="p">[</span><span class="s">&#39;simplify&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>  <span class="c"># for checksol&#39;s sake</span>
-                    <span class="k">for</span> <span class="n">sol</span> <span class="ow">in</span> <span class="n">soln</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">check</span><span class="p">:</span>
-                            <span class="c"># check that it satisfies *other* equations</span>
-                            <span class="n">ok</span> <span class="o">=</span> <span class="bp">False</span>
-                            <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">polys</span><span class="p">:</span>
-                                <span class="k">if</span> <span class="n">checksol</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">sol</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>
-                                    <span class="k">break</span>
-                            <span class="k">else</span><span class="p">:</span>
-                                <span class="n">ok</span> <span class="o">=</span> <span class="bp">True</span>
-                            <span class="k">if</span> <span class="ow">not</span> <span class="n">ok</span><span class="p">:</span>
-                                <span class="k">continue</span>
-                            <span class="c"># check that it doesn&#39;t set any denominator to 0</span>
-                            <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">checksol</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">sol</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">dens</span><span class="p">):</span>
-                                <span class="k">continue</span>
-                        <span class="c"># update existing solutions with this new one</span>
-                        <span class="n">rnew</span> <span class="o">=</span> <span class="n">r</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-                        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">r</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                            <span class="n">rnew</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">v</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">sol</span><span class="p">)</span>
-                        <span class="c"># and add this new solution</span>
-                        <span class="n">rnew</span><span class="p">[</span><span class="n">s</span><span class="p">]</span> <span class="o">=</span> <span class="n">sol</span>
-                        <span class="n">newresult</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">rnew</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">simplify_flag</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="n">flags</span><span class="p">[</span><span class="s">&#39;simplify&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">simplify_flag</span>
-                    <span class="n">got_s</span> <span class="o">=</span> <span class="n">s</span>
-                    <span class="k">break</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;could not solve </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">eq2</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">got_s</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">newresult</span>
-                <span class="n">solved_syms</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">got_s</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">bad_results</span><span class="p">:</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="c"># if there is only one result should we return just the dictionary?</span>
-    <span class="k">return</span> <span class="n">result</span>
-
-
-<div class="viewcode-block" id="solve_linear"><a class="viewcode-back" href="../../../modules/solvers/solvers.html#sympy.solvers.solvers.solve_linear">[docs]</a><span class="k">def</span> <span class="nf">solve_linear</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">rhs</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">symbols</span><span class="o">=</span><span class="p">[],</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[]):</span>
-    <span class="sd">r&quot;&quot;&quot; Return a tuple derived from f = lhs - rhs that is either:</span>
-
-<span class="sd">        (numerator, denominator) of ``f``</span>
-<span class="sd">            If this comes back as (0, 1) it means</span>
-<span class="sd">            that ``f`` is independent of the symbols in ``symbols``, e.g::</span>
-
-<span class="sd">                y*cos(x)**2 + y*sin(x)**2 - y = y*(0) = 0</span>
-<span class="sd">                cos(x)**2 + sin(x)**2 = 1</span>
-
-<span class="sd">            If it comes back as (0, 0) there is no solution to the equation</span>
-<span class="sd">            amongst the symbols given.</span>
-
-<span class="sd">            If the numerator is not zero then the function is guaranteed</span>
-<span class="sd">            to be dependent on a symbol in ``symbols``.</span>
-
-<span class="sd">        or</span>
-
-<span class="sd">        (symbol, solution) where symbol appears linearly in the numerator of</span>
-<span class="sd">        ``f``, is in ``symbols`` (if given) and is not in ``exclude`` (if given).</span>
-
-<span class="sd">        No simplification is done to ``f`` other than and mul=True expansion,</span>
-<span class="sd">        so the solution will correspond strictly to a unique solution.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.solvers.solvers import solve_linear</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-
-<span class="sd">    These are linear in x and 1/x:</span>
-
-<span class="sd">    &gt;&gt;&gt; solve_linear(x + y**2)</span>
-<span class="sd">    (x, -y**2)</span>
-<span class="sd">    &gt;&gt;&gt; solve_linear(1/x - y**2)</span>
-<span class="sd">    (x, y**(-2))</span>
-
-<span class="sd">    When not linear in x or y then the numerator and denominator are returned.</span>
-
-<span class="sd">    &gt;&gt;&gt; solve_linear(x**2/y**2 - 3)</span>
-<span class="sd">    (x**2 - 3*y**2, y**2)</span>
-
-<span class="sd">    If the numerator is a symbol then (0, 0) is returned if the solution for</span>
-<span class="sd">    that symbol would have set any denominator to 0:</span>
-
-<span class="sd">    &gt;&gt;&gt; solve_linear(1/(1/x - 2))</span>
-<span class="sd">    (0, 0)</span>
-<span class="sd">    &gt;&gt;&gt; 1/(1/x) # to SymPy, this looks like x ...</span>
-<span class="sd">    x</span>
-<span class="sd">    &gt;&gt;&gt; solve_linear(1/(1/x)) # so a solution is given</span>
-<span class="sd">    (x, 0)</span>
-
-<span class="sd">    If x is allowed to cancel, then this appears linear, but this sort of</span>
-<span class="sd">    cancellation is not done so the solution will always satisfy the original</span>
-<span class="sd">    expression without causing a division by zero error.</span>
-
-<span class="sd">    &gt;&gt;&gt; solve_linear(x**2*(1/x - z**2/x))</span>
-<span class="sd">    (x**2*(-z**2 + 1), x)</span>
-
-<span class="sd">    You can give a list of what you prefer for x candidates:</span>
-
-<span class="sd">    &gt;&gt;&gt; solve_linear(x + y + z, symbols=[y])</span>
-<span class="sd">    (y, -x - z)</span>
-
-<span class="sd">    You can also indicate what variables you don&#39;t want to consider:</span>
-
-<span class="sd">    &gt;&gt;&gt; solve_linear(x + y + z, exclude=[x, z])</span>
-<span class="sd">    (y, -x - z)</span>
-
-<span class="sd">    If only x was excluded then a solution for y or z might be obtained.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Equality</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="n">Equality</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">rhs</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&#39;&#39;&#39;</span>
-<span class="s">            If lhs is an Equality, rhs must be 0 but was </span><span class="si">%s</span><span class="s">&#39;&#39;&#39;</span> <span class="o">%</span> <span class="n">rhs</span><span class="p">))</span>
-        <span class="n">rhs</span> <span class="o">=</span> <span class="n">lhs</span><span class="o">.</span><span class="n">rhs</span>
-        <span class="n">lhs</span> <span class="o">=</span> <span class="n">lhs</span><span class="o">.</span><span class="n">lhs</span>
-    <span class="n">dens</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">eq</span> <span class="o">=</span> <span class="n">lhs</span> <span class="o">-</span> <span class="n">rhs</span>
-    <span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">eq</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-
-    <span class="n">free</span> <span class="o">=</span> <span class="n">n</span><span class="o">.</span><span class="n">free_symbols</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">symbols</span><span class="p">:</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="n">free</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">bad</span> <span class="o">=</span> <span class="p">[</span><span class="n">s</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">symbols</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">s</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">]</span>
-        <span class="k">if</span> <span class="n">bad</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">bad</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">bad</span> <span class="o">=</span> <span class="n">bad</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="n">eg</span> <span class="o">=</span> <span class="s">&#39;solve(</span><span class="si">%s</span><span class="s">, </span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">symbols</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">eg</span> <span class="o">=</span> <span class="s">&#39;solve(</span><span class="si">%s</span><span class="s">, *</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="n">symbols</span><span class="p">))</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&#39;&#39;&#39;</span>
-<span class="s">                solve_linear only handles symbols, not </span><span class="si">%s</span><span class="s">. To isolate</span>
-<span class="s">                non-symbols use solve, e.g. &gt;&gt;&gt; </span><span class="si">%s</span><span class="s"> &lt;&lt;&lt;.</span>
-<span class="s">                             &#39;&#39;&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">bad</span><span class="p">,</span> <span class="n">eg</span><span class="p">)))</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="n">free</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span>
-    <span class="n">symbols</span> <span class="o">=</span> <span class="n">symbols</span><span class="o">.</span><span class="n">difference</span><span class="p">(</span><span class="n">exclude</span><span class="p">)</span>
-    <span class="n">dfree</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">free_symbols</span>
-
-    <span class="c"># derivatives are easy to do but tricky to analyze to see if they are going</span>
-    <span class="c"># to disallow a linear solution, so for simplicity we just evaluate the</span>
-    <span class="c"># ones that have the symbols of interest</span>
-    <span class="n">derivs</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">list</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">der</span> <span class="ow">in</span> <span class="n">n</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Derivative</span><span class="p">):</span>
-        <span class="n">csym</span> <span class="o">=</span> <span class="n">der</span><span class="o">.</span><span class="n">free_symbols</span> <span class="o">&amp;</span> <span class="n">symbols</span>
-        <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">csym</span><span class="p">:</span>
-            <span class="n">derivs</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">der</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">symbols</span><span class="p">:</span>
-        <span class="n">all_zero</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">for</span> <span class="n">xi</span> <span class="ow">in</span> <span class="n">symbols</span><span class="p">:</span>
-            <span class="c"># if there are derivatives in this var, calculate them now</span>
-            <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">derivs</span><span class="p">[</span><span class="n">xi</span><span class="p">])</span> <span class="ow">is</span> <span class="nb">list</span><span class="p">:</span>
-                <span class="n">derivs</span><span class="p">[</span><span class="n">xi</span><span class="p">]</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">([(</span><span class="n">der</span><span class="p">,</span> <span class="n">der</span><span class="o">.</span><span class="n">doit</span><span class="p">())</span> <span class="k">for</span> <span class="n">der</span> <span class="ow">in</span> <span class="n">derivs</span><span class="p">[</span><span class="n">xi</span><span class="p">]])</span>
-            <span class="n">nn</span> <span class="o">=</span> <span class="n">n</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">derivs</span><span class="p">[</span><span class="n">xi</span><span class="p">])</span>
-            <span class="n">dn</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">xi</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">dn</span><span class="p">:</span>
-                <span class="n">all_zero</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">xi</span> <span class="ow">in</span> <span class="n">dn</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-                    <span class="n">vi</span> <span class="o">=</span> <span class="o">-</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">xi</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span><span class="o">/</span><span class="n">dn</span>
-                    <span class="k">if</span> <span class="n">dens</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="n">dens</span> <span class="o">=</span> <span class="n">denoms</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">symbols</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="nb">any</span><span class="p">(</span><span class="n">checksol</span><span class="p">(</span><span class="n">di</span><span class="p">,</span> <span class="p">{</span><span class="n">xi</span><span class="p">:</span> <span class="n">vi</span><span class="p">},</span> <span class="n">minimal</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">True</span>
-                              <span class="k">for</span> <span class="n">di</span> <span class="ow">in</span> <span class="n">dens</span><span class="p">):</span>
-                        <span class="c"># simplify any trivial integral</span>
-                        <span class="n">irep</span> <span class="o">=</span> <span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="o">.</span><span class="n">doit</span><span class="p">())</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">vi</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Integral</span><span class="p">)</span> <span class="k">if</span>
-                                <span class="n">i</span><span class="o">.</span><span class="n">function</span><span class="o">.</span><span class="n">is_number</span><span class="p">]</span>
-                        <span class="c"># do a slight bit of simplification</span>
-                        <span class="n">vi</span> <span class="o">=</span> <span class="n">expand_mul</span><span class="p">(</span><span class="n">vi</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">irep</span><span class="p">))</span>
-                        <span class="k">if</span> <span class="ow">not</span> <span class="n">d</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">xi</span><span class="p">)</span> <span class="ow">or</span> <span class="ow">not</span> <span class="p">(</span><span class="n">d</span><span class="o">/</span><span class="n">xi</span><span class="p">)</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">xi</span><span class="p">):</span>
-                            <span class="k">return</span> <span class="n">xi</span><span class="p">,</span> <span class="n">vi</span>
-
-        <span class="k">if</span> <span class="n">all_zero</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-    <span class="k">if</span> <span class="n">n</span><span class="o">.</span><span class="n">is_Symbol</span><span class="p">:</span>  <span class="c"># there was no valid solution</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">d</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-    <span class="k">return</span> <span class="n">n</span><span class="p">,</span> <span class="n">d</span>  <span class="c"># should we cancel now?</span>
-
-</div>
-<div class="viewcode-block" id="solve_linear_system"><a class="viewcode-back" href="../../../modules/solvers/solvers.html#sympy.solvers.solvers.solve_linear_system">[docs]</a><span class="k">def</span> <span class="nf">solve_linear_system</span><span class="p">(</span><span class="n">system</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Solve system of N linear equations with M variables, which means</span>
-<span class="sd">    both under- and overdetermined systems are supported. The possible</span>
-<span class="sd">    number of solutions is zero, one or infinite. Respectively, this</span>
-<span class="sd">    procedure will return None or a dictionary with solutions. In the</span>
-<span class="sd">    case of underdetermined systems, all arbitrary parameters are skipped.</span>
-<span class="sd">    This may cause a situation in which an empty dictionary is returned.</span>
-<span class="sd">    In that case, all symbols can be assigned arbitrary values.</span>
-
-<span class="sd">    Input to this functions is a Nx(M+1) matrix, which means it has</span>
-<span class="sd">    to be in augmented form. If you prefer to enter N equations and M</span>
-<span class="sd">    unknowns then use `solve(Neqs, *Msymbols)` instead. Note: a local</span>
-<span class="sd">    copy of the matrix is made by this routine so the matrix that is</span>
-<span class="sd">    passed will not be modified.</span>
-
-<span class="sd">    The algorithm used here is fraction-free Gaussian elimination,</span>
-<span class="sd">    which results, after elimination, in an upper-triangular matrix.</span>
-<span class="sd">    Then solutions are found using back-substitution. This approach</span>
-<span class="sd">    is more efficient and compact than the Gauss-Jordan method.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Matrix, solve_linear_system</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-
-<span class="sd">    Solve the following system::</span>
-
-<span class="sd">           x + 4 y ==  2</span>
-<span class="sd">        -2 x +   y == 14</span>
-
-<span class="sd">    &gt;&gt;&gt; system = Matrix(( (1, 4, 2), (-2, 1, 14)))</span>
-<span class="sd">    &gt;&gt;&gt; solve_linear_system(system, x, y)</span>
-<span class="sd">    {x: -6, y: 2}</span>
-
-<span class="sd">    A degenerate system returns an empty dictionary.</span>
-
-<span class="sd">    &gt;&gt;&gt; system = Matrix(( (0,0,0), (0,0,0) ))</span>
-<span class="sd">    &gt;&gt;&gt; solve_linear_system(system, x, y)</span>
-<span class="sd">    {}</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">matrix</span> <span class="o">=</span> <span class="n">system</span><span class="p">[:,</span> <span class="p">:]</span>
-    <span class="n">syms</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span>
-
-    <span class="n">i</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="n">matrix</span><span class="o">.</span><span class="n">cols</span> <span class="o">-</span> <span class="mi">1</span>  <span class="c"># don&#39;t count augmentation</span>
-
-    <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">matrix</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">m</span><span class="p">:</span>
-            <span class="c"># an overdetermined system</span>
-            <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">matrix</span><span class="p">[</span><span class="n">i</span><span class="p">:,</span> <span class="n">m</span><span class="p">]):</span>
-                <span class="k">return</span> <span class="bp">None</span>   <span class="c"># no solutions</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># remove trailing rows</span>
-                <span class="n">matrix</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">[:</span><span class="n">i</span><span class="p">,</span> <span class="p">:]</span>
-                <span class="k">break</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">matrix</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]:</span>
-            <span class="c"># there is no pivot in current column</span>
-            <span class="c"># so try to find one in other columns</span>
-            <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">matrix</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">]:</span>
-                    <span class="k">break</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">matrix</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">m</span><span class="p">]:</span>
-                    <span class="c"># we need to know if this is always zero or not. We</span>
-                    <span class="c"># assume that if there are free symbols that it is not</span>
-                    <span class="c"># identically zero (or that there is more than one way</span>
-                    <span class="c"># to make this zero. Otherwise, if there are none, this</span>
-                    <span class="c"># is a constant and we assume that it does not simplify</span>
-                    <span class="c"># to zero XXX are there better ways to test this?</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">matrix</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">m</span><span class="p">]</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-                        <span class="k">return</span> <span class="bp">None</span>  <span class="c"># no solution</span>
-
-                    <span class="c"># zero row with non-zero rhs can only be accepted</span>
-                    <span class="c"># if there is another equivalent row, so look for</span>
-                    <span class="c"># them and delete them</span>
-                    <span class="n">nrows</span> <span class="o">=</span> <span class="n">matrix</span><span class="o">.</span><span class="n">rows</span>
-                    <span class="n">rowi</span> <span class="o">=</span> <span class="n">matrix</span><span class="o">.</span><span class="n">row</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                    <span class="n">ip</span> <span class="o">=</span> <span class="bp">None</span>
-                    <span class="n">j</span> <span class="o">=</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
-                    <span class="k">while</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="n">matrix</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-                        <span class="c"># do we need to see if the rhs of j</span>
-                        <span class="c"># is a constant multiple of i&#39;s rhs?</span>
-                        <span class="n">rowj</span> <span class="o">=</span> <span class="n">matrix</span><span class="o">.</span><span class="n">row</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-                        <span class="k">if</span> <span class="n">rowj</span> <span class="o">==</span> <span class="n">rowi</span><span class="p">:</span>
-                            <span class="n">matrix</span><span class="o">.</span><span class="n">row_del</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-                        <span class="k">elif</span> <span class="n">rowj</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">rowi</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-                            <span class="k">if</span> <span class="n">ip</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                                <span class="n">_</span><span class="p">,</span> <span class="n">ip</span> <span class="o">=</span> <span class="n">rowi</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-                            <span class="n">_</span><span class="p">,</span> <span class="n">jp</span> <span class="o">=</span> <span class="n">rowj</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-                            <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">simplify</span><span class="p">(</span><span class="n">jp</span> <span class="o">-</span> <span class="n">ip</span><span class="p">)</span> <span class="ow">or</span> <span class="n">simplify</span><span class="p">(</span><span class="n">jp</span> <span class="o">+</span> <span class="n">ip</span><span class="p">)):</span>
-                                <span class="n">matrix</span><span class="o">.</span><span class="n">row_del</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-
-                        <span class="n">j</span> <span class="o">+=</span> <span class="mi">1</span>
-
-                    <span class="k">if</span> <span class="n">nrows</span> <span class="o">==</span> <span class="n">matrix</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-                        <span class="c"># no solution</span>
-                        <span class="k">return</span> <span class="bp">None</span>
-                <span class="c"># zero row or was a linear combination of</span>
-                <span class="c"># other rows or was a row with a symbolic</span>
-                <span class="c"># expression that matched other rows, e.g. [0, 0, x - y]</span>
-                <span class="c"># so now we can safely skip it</span>
-                <span class="n">matrix</span><span class="o">.</span><span class="n">row_del</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">matrix</span><span class="p">:</span>
-                    <span class="c"># every choice of variable values is a solution</span>
-                    <span class="c"># so we return an empty dict instead of None</span>
-                    <span class="k">return</span> <span class="nb">dict</span><span class="p">()</span>
-                <span class="k">continue</span>
-
-            <span class="c"># we want to change the order of colums so</span>
-            <span class="c"># the order of variables must also change</span>
-            <span class="n">syms</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">syms</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">syms</span><span class="p">[</span><span class="n">k</span><span class="p">],</span> <span class="n">syms</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="n">matrix</span><span class="o">.</span><span class="n">col_swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-
-        <span class="n">pivot_inv</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="o">/</span><span class="n">matrix</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span>
-
-        <span class="c"># divide all elements in the current row by the pivot</span>
-        <span class="n">matrix</span><span class="o">.</span><span class="n">row_op</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">_</span><span class="p">:</span> <span class="n">x</span> <span class="o">*</span> <span class="n">pivot_inv</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">matrix</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">matrix</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">i</span><span class="p">]:</span>
-                <span class="n">coeff</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span>
-
-                <span class="c"># subtract from the current row the row containing</span>
-                <span class="c"># pivot and multiplied by extracted coefficient</span>
-                <span class="n">matrix</span><span class="o">.</span><span class="n">row_op</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="n">simplify</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">matrix</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">*</span><span class="n">coeff</span><span class="p">))</span>
-
-        <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-
-    <span class="c"># if there weren&#39;t any problems, augmented matrix is now</span>
-    <span class="c"># in row-echelon form so we can check how many solutions</span>
-    <span class="c"># there are and extract them using back substitution</span>
-
-    <span class="n">do_simplify</span> <span class="o">=</span> <span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;simplify&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">syms</span><span class="p">)</span> <span class="o">==</span> <span class="n">matrix</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-        <span class="c"># this system is Cramer equivalent so there is</span>
-        <span class="c"># exactly one solution to this system of equations</span>
-        <span class="n">k</span><span class="p">,</span> <span class="n">solutions</span> <span class="o">=</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="p">{}</span>
-
-        <span class="k">while</span> <span class="n">k</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">content</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">m</span><span class="p">]</span>
-
-            <span class="c"># run back-substitution for variables</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-                <span class="n">content</span> <span class="o">-=</span> <span class="n">matrix</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">*</span><span class="n">solutions</span><span class="p">[</span><span class="n">syms</span><span class="p">[</span><span class="n">j</span><span class="p">]]</span>
-
-            <span class="k">if</span> <span class="n">do_simplify</span><span class="p">:</span>
-                <span class="n">solutions</span><span class="p">[</span><span class="n">syms</span><span class="p">[</span><span class="n">k</span><span class="p">]]</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">content</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">solutions</span><span class="p">[</span><span class="n">syms</span><span class="p">[</span><span class="n">k</span><span class="p">]]</span> <span class="o">=</span> <span class="n">content</span>
-
-            <span class="n">k</span> <span class="o">-=</span> <span class="mi">1</span>
-
-        <span class="k">return</span> <span class="n">solutions</span>
-    <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">syms</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">matrix</span><span class="o">.</span><span class="n">rows</span><span class="p">:</span>
-        <span class="c"># this system will have infinite number of solutions</span>
-        <span class="c"># dependent on exactly len(syms) - i parameters</span>
-        <span class="n">k</span><span class="p">,</span> <span class="n">solutions</span> <span class="o">=</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="p">{}</span>
-
-        <span class="k">while</span> <span class="n">k</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">content</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">m</span><span class="p">]</span>
-
-            <span class="c"># run back-substitution for variables</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-                <span class="n">content</span> <span class="o">-=</span> <span class="n">matrix</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">*</span><span class="n">solutions</span><span class="p">[</span><span class="n">syms</span><span class="p">[</span><span class="n">j</span><span class="p">]]</span>
-
-            <span class="c"># run back-substitution for parameters</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-                <span class="n">content</span> <span class="o">-=</span> <span class="n">matrix</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">*</span><span class="n">syms</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-
-            <span class="k">if</span> <span class="n">do_simplify</span><span class="p">:</span>
-                <span class="n">solutions</span><span class="p">[</span><span class="n">syms</span><span class="p">[</span><span class="n">k</span><span class="p">]]</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">content</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">solutions</span><span class="p">[</span><span class="n">syms</span><span class="p">[</span><span class="n">k</span><span class="p">]]</span> <span class="o">=</span> <span class="n">content</span>
-
-            <span class="n">k</span> <span class="o">-=</span> <span class="mi">1</span>
-
-        <span class="k">return</span> <span class="n">solutions</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>   <span class="c"># no solutions</span>
-
-</div>
-<div class="viewcode-block" id="solve_undetermined_coeffs"><a class="viewcode-back" href="../../../modules/solvers/solvers.html#sympy.solvers.solvers.solve_undetermined_coeffs">[docs]</a><span class="k">def</span> <span class="nf">solve_undetermined_coeffs</span><span class="p">(</span><span class="n">equ</span><span class="p">,</span> <span class="n">coeffs</span><span class="p">,</span> <span class="n">sym</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Solve equation of a type p(x; a_1, ..., a_k) == q(x) where both</span>
-<span class="sd">       p, q are univariate polynomials and f depends on k parameters.</span>
-<span class="sd">       The result of this functions is a dictionary with symbolic</span>
-<span class="sd">       values of those parameters with respect to coefficients in q.</span>
-
-<span class="sd">       This functions accepts both Equations class instances and ordinary</span>
-<span class="sd">       SymPy expressions. Specification of parameters and variable is</span>
-<span class="sd">       obligatory for efficiency and simplicity reason.</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy import Eq</span>
-<span class="sd">       &gt;&gt;&gt; from sympy.abc import a, b, c, x</span>
-<span class="sd">       &gt;&gt;&gt; from sympy.solvers import solve_undetermined_coeffs</span>
-
-<span class="sd">       &gt;&gt;&gt; solve_undetermined_coeffs(Eq(2*a*x + a+b, x), [a, b], x)</span>
-<span class="sd">       {a: 1/2, b: -1/2}</span>
-
-<span class="sd">       &gt;&gt;&gt; solve_undetermined_coeffs(Eq(a*c*x + a+b, x), [a, b], x)</span>
-<span class="sd">       {a: 1/c, b: -1/c}</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">equ</span><span class="p">,</span> <span class="n">Equality</span><span class="p">):</span>
-        <span class="c"># got equation, so move all the</span>
-        <span class="c"># terms to the left hand side</span>
-        <span class="n">equ</span> <span class="o">=</span> <span class="n">equ</span><span class="o">.</span><span class="n">lhs</span> <span class="o">-</span> <span class="n">equ</span><span class="o">.</span><span class="n">rhs</span>
-
-    <span class="n">equ</span> <span class="o">=</span> <span class="n">cancel</span><span class="p">(</span><span class="n">equ</span><span class="p">)</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="n">system</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">collect</span><span class="p">(</span><span class="n">equ</span><span class="o">.</span><span class="n">expand</span><span class="p">(),</span> <span class="n">sym</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span><span class="o">.</span><span class="n">values</span><span class="p">())</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">any</span><span class="p">(</span><span class="n">equ</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">sym</span><span class="p">)</span> <span class="k">for</span> <span class="n">equ</span> <span class="ow">in</span> <span class="n">system</span><span class="p">):</span>
-        <span class="c"># consecutive powers in the input expressions have</span>
-        <span class="c"># been successfully collected, so solve remaining</span>
-        <span class="c"># system using Gaussian elimination algorithm</span>
-        <span class="k">return</span> <span class="n">solve</span><span class="p">(</span><span class="n">system</span><span class="p">,</span> <span class="o">*</span><span class="n">coeffs</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>  <span class="c"># no solutions</span>
-
-</div>
-<div class="viewcode-block" id="solve_linear_system_LU"><a class="viewcode-back" href="../../../modules/solvers/solvers.html#sympy.solvers.solvers.solve_linear_system_LU">[docs]</a><span class="k">def</span> <span class="nf">solve_linear_system_LU</span><span class="p">(</span><span class="n">matrix</span><span class="p">,</span> <span class="n">syms</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Solves the augmented matrix system using LUsolve and returns a dictionary</span>
-<span class="sd">    in which solutions are keyed to the symbols of syms *as ordered*.</span>
-
-<span class="sd">    The matrix must be invertible.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Matrix</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.solvers.solvers import solve_linear_system_LU</span>
-
-<span class="sd">    &gt;&gt;&gt; solve_linear_system_LU(Matrix([</span>
-<span class="sd">    ... [1, 2, 0, 1],</span>
-<span class="sd">    ... [3, 2, 2, 1],</span>
-<span class="sd">    ... [2, 0, 0, 1]]), [x, y, z])</span>
-<span class="sd">    {x: 1/2, y: 1/4, z: -1/2}</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.matrices.LUsolve</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">assert</span> <span class="n">matrix</span><span class="o">.</span><span class="n">rows</span> <span class="o">==</span> <span class="n">matrix</span><span class="o">.</span><span class="n">cols</span> <span class="o">-</span> <span class="mi">1</span>
-    <span class="n">A</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">[:</span><span class="n">matrix</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="p">:</span><span class="n">matrix</span><span class="o">.</span><span class="n">rows</span><span class="p">]</span>
-    <span class="n">b</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">[:,</span> <span class="n">matrix</span><span class="o">.</span><span class="n">cols</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:]</span>
-    <span class="n">soln</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">LUsolve</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-    <span class="n">solutions</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">soln</span><span class="o">.</span><span class="n">rows</span><span class="p">):</span>
-        <span class="n">solutions</span><span class="p">[</span><span class="n">syms</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span> <span class="o">=</span> <span class="n">soln</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">solutions</span>
-</div>
-<span class="n">_x</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="n">_a</span><span class="p">,</span> <span class="n">_b</span><span class="p">,</span> <span class="n">_c</span><span class="p">,</span> <span class="n">_d</span><span class="p">,</span> <span class="n">_e</span><span class="p">,</span> <span class="n">_f</span><span class="p">,</span> <span class="n">_g</span><span class="p">,</span> <span class="n">_h</span> <span class="o">=</span> <span class="p">[</span><span class="n">Wild</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">_x</span><span class="p">])</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="s">&#39;abcdefgh&#39;</span><span class="p">]</span>
-<span class="n">_patterns</span> <span class="o">=</span> <span class="bp">None</span>
-
-
-<span class="k">def</span> <span class="nf">_generate_patterns</span><span class="p">():</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Generates patterns for transcendental equations.</span>
-
-<span class="sd">    This is lazily calculated (called) in the tsolve() function and stored in</span>
-<span class="sd">    the patterns global variable.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">tmp1</span> <span class="o">=</span> <span class="n">_f</span> <span class="o">**</span> <span class="p">(</span><span class="n">_h</span> <span class="o">-</span> <span class="p">(</span><span class="n">_c</span><span class="o">*</span><span class="n">_g</span><span class="o">/</span><span class="n">_b</span><span class="p">))</span>
-    <span class="n">tmp2</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="n">_e</span><span class="o">*</span><span class="n">tmp1</span><span class="o">/</span><span class="n">_a</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">_d</span><span class="p">)</span>
-    <span class="k">global</span> <span class="n">_patterns</span>
-    <span class="n">_patterns</span> <span class="o">=</span> <span class="p">[</span>
-        <span class="p">(</span><span class="n">_a</span><span class="o">*</span><span class="n">_x</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">_b</span><span class="o">*</span><span class="n">_x</span><span class="p">)</span> <span class="o">-</span> <span class="n">_c</span><span class="p">,</span>
-            <span class="n">LambertW</span><span class="p">(</span><span class="n">_b</span><span class="o">*</span><span class="n">_c</span><span class="o">/</span><span class="n">_a</span><span class="p">)</span><span class="o">/</span><span class="n">_b</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">_a</span><span class="o">*</span><span class="n">_x</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">_b</span><span class="o">*</span><span class="n">_x</span><span class="p">)</span> <span class="o">-</span> <span class="n">_c</span><span class="p">,</span>
-            <span class="n">exp</span><span class="p">(</span><span class="n">LambertW</span><span class="p">(</span><span class="n">_b</span><span class="o">*</span><span class="n">_c</span><span class="o">/</span><span class="n">_a</span><span class="p">))</span><span class="o">/</span><span class="n">_b</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">_a</span><span class="o">*</span><span class="p">(</span><span class="n">_b</span><span class="o">*</span><span class="n">_x</span> <span class="o">+</span> <span class="n">_c</span><span class="p">)</span><span class="o">**</span><span class="n">_d</span> <span class="o">+</span> <span class="n">_e</span><span class="p">,</span>
-            <span class="p">((</span><span class="o">-</span><span class="p">(</span><span class="n">_e</span><span class="o">/</span><span class="n">_a</span><span class="p">))</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">_d</span><span class="p">)</span> <span class="o">-</span> <span class="n">_c</span><span class="p">)</span><span class="o">/</span><span class="n">_b</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">_b</span> <span class="o">+</span> <span class="n">_c</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">_d</span><span class="o">*</span><span class="n">_x</span> <span class="o">+</span> <span class="n">_e</span><span class="p">),</span>
-            <span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="o">-</span><span class="n">_b</span><span class="o">/</span><span class="n">_c</span><span class="p">)</span> <span class="o">-</span> <span class="n">_e</span><span class="p">)</span><span class="o">/</span><span class="n">_d</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">_a</span><span class="o">*</span><span class="n">_x</span> <span class="o">+</span> <span class="n">_b</span> <span class="o">+</span> <span class="n">_c</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">_d</span><span class="o">*</span><span class="n">_x</span> <span class="o">+</span> <span class="n">_e</span><span class="p">),</span>
-            <span class="o">-</span><span class="n">_b</span><span class="o">/</span><span class="n">_a</span> <span class="o">-</span> <span class="n">LambertW</span><span class="p">(</span><span class="n">_c</span><span class="o">*</span><span class="n">_d</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">_e</span> <span class="o">-</span> <span class="n">_b</span><span class="o">*</span><span class="n">_d</span><span class="o">/</span><span class="n">_a</span><span class="p">)</span><span class="o">/</span><span class="n">_a</span><span class="p">)</span><span class="o">/</span><span class="n">_d</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">_b</span> <span class="o">+</span> <span class="n">_c</span><span class="o">*</span><span class="n">_f</span><span class="o">**</span><span class="p">(</span><span class="n">_d</span><span class="o">*</span><span class="n">_x</span> <span class="o">+</span> <span class="n">_e</span><span class="p">),</span>
-            <span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="o">-</span><span class="n">_b</span><span class="o">/</span><span class="n">_c</span><span class="p">)</span> <span class="o">-</span> <span class="n">_e</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">_f</span><span class="p">))</span><span class="o">/</span><span class="n">_d</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="n">_f</span><span class="p">)),</span>
-        <span class="p">(</span><span class="n">_a</span><span class="o">*</span><span class="n">_x</span> <span class="o">+</span> <span class="n">_b</span> <span class="o">+</span> <span class="n">_c</span><span class="o">*</span><span class="n">_f</span><span class="o">**</span><span class="p">(</span><span class="n">_d</span><span class="o">*</span><span class="n">_x</span> <span class="o">+</span> <span class="n">_e</span><span class="p">),</span>
-            <span class="o">-</span><span class="n">_b</span><span class="o">/</span><span class="n">_a</span> <span class="o">-</span> <span class="n">LambertW</span><span class="p">(</span><span class="n">_c</span><span class="o">*</span><span class="n">_d</span><span class="o">*</span><span class="n">_f</span><span class="o">**</span><span class="p">(</span><span class="n">_e</span> <span class="o">-</span> <span class="n">_b</span><span class="o">*</span><span class="n">_d</span><span class="o">/</span><span class="n">_a</span><span class="p">)</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">_f</span><span class="p">)</span><span class="o">/</span><span class="n">_a</span><span class="p">)</span><span class="o">/</span><span class="n">_d</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="n">_f</span><span class="p">)),</span>
-        <span class="p">(</span><span class="n">_b</span> <span class="o">+</span> <span class="n">_c</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">_d</span><span class="o">*</span><span class="n">_x</span> <span class="o">+</span> <span class="n">_e</span><span class="p">),</span>
-            <span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">_b</span><span class="o">/</span><span class="n">_c</span><span class="p">)</span> <span class="o">-</span> <span class="n">_e</span><span class="p">)</span><span class="o">/</span><span class="n">_d</span><span class="p">),</span>
-        <span class="p">(</span><span class="n">_a</span><span class="o">*</span><span class="n">_x</span> <span class="o">+</span> <span class="n">_b</span> <span class="o">+</span> <span class="n">_c</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">_d</span><span class="o">*</span><span class="n">_x</span> <span class="o">+</span> <span class="n">_e</span><span class="p">),</span>
-            <span class="o">-</span><span class="n">_e</span><span class="o">/</span><span class="n">_d</span> <span class="o">+</span> <span class="n">_c</span><span class="o">/</span><span class="n">_a</span><span class="o">*</span><span class="n">LambertW</span><span class="p">(</span><span class="n">_a</span><span class="o">/</span><span class="n">_c</span><span class="o">/</span><span class="n">_d</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">_b</span><span class="o">/</span><span class="n">_c</span> <span class="o">+</span> <span class="n">_a</span><span class="o">*</span><span class="n">_e</span><span class="o">/</span><span class="n">_c</span><span class="o">/</span><span class="n">_d</span><span class="p">))),</span>
-        <span class="p">(</span><span class="n">_a</span><span class="o">*</span><span class="p">(</span><span class="n">_b</span><span class="o">*</span><span class="n">_x</span> <span class="o">+</span> <span class="n">_c</span><span class="p">)</span><span class="o">**</span><span class="n">_d</span> <span class="o">+</span> <span class="n">_e</span><span class="o">*</span><span class="n">_f</span><span class="o">**</span><span class="p">(</span><span class="n">_g</span><span class="o">*</span><span class="n">_x</span> <span class="o">+</span> <span class="n">_h</span><span class="p">),</span>
-            <span class="o">-</span><span class="n">_c</span><span class="o">/</span><span class="n">_b</span> <span class="o">-</span> <span class="n">_d</span><span class="o">*</span><span class="n">LambertW</span><span class="p">(</span><span class="o">-</span><span class="n">tmp2</span><span class="o">*</span><span class="n">_g</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">_f</span><span class="p">)</span><span class="o">/</span><span class="n">_b</span><span class="o">/</span><span class="n">_d</span><span class="p">)</span><span class="o">/</span><span class="n">_g</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="n">_f</span><span class="p">))</span>
-    <span class="p">]</span>
-
-
-<div class="viewcode-block" id="tsolve"><a class="viewcode-back" href="../../../modules/solvers/solvers.html#sympy.solvers.solvers.tsolve">[docs]</a><span class="k">def</span> <span class="nf">tsolve</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">sym</span><span class="p">):</span>
-    <span class="n">SymPyDeprecationWarning</span><span class="p">(</span>
-        <span class="n">feature</span><span class="o">=</span><span class="s">&quot;tsolve()&quot;</span><span class="p">,</span>
-        <span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;solve()&quot;</span><span class="p">,</span>
-        <span class="n">issue</span><span class="o">=</span><span class="mi">3385</span><span class="p">,</span>
-        <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.2&quot;</span><span class="p">,</span>
-    <span class="p">)</span><span class="o">.</span><span class="n">warn</span><span class="p">()</span>
-    <span class="k">return</span> <span class="n">_tsolve</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">sym</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_tsolve</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">sym</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Helper for _solve that solves a transcendental equation with respect</span>
-<span class="sd">    to the given symbol. Various equations containing powers and logarithms,</span>
-<span class="sd">    can be solved.</span>
-
-<span class="sd">    Only a single solution is returned. This solution is generally</span>
-<span class="sd">    not unique. In some cases, a complex solution may be returned</span>
-<span class="sd">    even though a real solution exists.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import log</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.solvers.solvers import _tsolve as tsolve</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-
-<span class="sd">        &gt;&gt;&gt; tsolve(3**(2*x+5)-4, x)</span>
-<span class="sd">        [(-5*log(3) + 2*log(2))/(2*log(3))]</span>
-
-<span class="sd">        &gt;&gt;&gt; tsolve(log(x) + 2*x, x)</span>
-<span class="sd">        [LambertW(2)/2]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">_patterns</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">_generate_patterns</span><span class="p">()</span>
-    <span class="n">eq2</span> <span class="o">=</span> <span class="n">eq</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">_x</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">p</span><span class="p">,</span> <span class="n">sol</span> <span class="ow">in</span> <span class="n">_patterns</span><span class="p">:</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">eq2</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">m</span><span class="p">:</span>
-            <span class="n">soln</span> <span class="o">=</span> <span class="n">sol</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">m</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">_x</span><span class="p">,</span> <span class="n">sym</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">sym</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">soln</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-                <span class="k">return</span> <span class="p">[</span><span class="n">soln</span><span class="p">]</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">u</span> <span class="o">=</span> <span class="n">unrad</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">sym</span><span class="p">)</span>
-    <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;Radicals cannot be cleared from </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">eq</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">u</span><span class="p">:</span>
-        <span class="n">eq</span><span class="p">,</span> <span class="n">cov</span><span class="p">,</span> <span class="n">dens</span> <span class="o">=</span> <span class="n">u</span>
-        <span class="k">if</span> <span class="n">cov</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">cov</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&#39;Not sure how to handle this.&#39;</span><span class="p">)</span>
-            <span class="n">isym</span><span class="p">,</span> <span class="n">ieq</span> <span class="o">=</span> <span class="n">cov</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="c"># since cov is written in terms of positive symbols, set</span>
-            <span class="c"># check to False or else 0 would be excluded; _solve will check</span>
-            <span class="c"># the results</span>
-            <span class="n">flags</span><span class="p">[</span><span class="s">&#39;check&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>
-            <span class="n">sol</span> <span class="o">=</span> <span class="n">_solve</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">isym</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-            <span class="n">inv</span> <span class="o">=</span> <span class="n">_solve</span><span class="p">(</span><span class="n">ieq</span><span class="p">,</span> <span class="n">sym</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">sol</span><span class="p">:</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">inv</span><span class="p">:</span>
-                    <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">isym</span><span class="p">,</span> <span class="n">s</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">result</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_solve</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">sym</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-
-    <span class="n">rhs</span><span class="p">,</span> <span class="n">lhs</span> <span class="o">=</span> <span class="n">_invert</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">sym</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">lhs</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-        <span class="c"># it&#39;s time to try factoring</span>
-        <span class="n">fac</span> <span class="o">=</span> <span class="n">factor</span><span class="p">(</span><span class="n">lhs</span> <span class="o">-</span> <span class="n">rhs</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">fac</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_solve</span><span class="p">(</span><span class="n">fac</span><span class="p">,</span> <span class="n">sym</span><span class="p">)</span>
-
-    <span class="k">elif</span> <span class="n">lhs</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">lhs</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">lhs</span> <span class="o">-</span> <span class="n">rhs</span> <span class="o">!=</span> <span class="n">eq</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">_solve</span><span class="p">(</span><span class="n">lhs</span> <span class="o">-</span> <span class="n">rhs</span><span class="p">,</span> <span class="n">sym</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="ow">not</span> <span class="n">rhs</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">_solve</span><span class="p">(</span><span class="n">lhs</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">sym</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">sym</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">lhs</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_solve</span><span class="p">(</span><span class="n">lhs</span><span class="o">.</span><span class="n">base</span> <span class="o">-</span> <span class="n">rhs</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">lhs</span><span class="o">.</span><span class="n">exp</span><span class="p">),</span> <span class="n">sym</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="n">rhs</span> <span class="ow">and</span> <span class="n">sym</span> <span class="ow">in</span> <span class="n">lhs</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-            <span class="c"># f(x)**g(x) only has solutions where f(x) == 0 and g(x) != 0 at</span>
-            <span class="c"># the same place</span>
-            <span class="n">sol_base</span> <span class="o">=</span> <span class="n">_solve</span><span class="p">(</span><span class="n">lhs</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">sym</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">sol_base</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">sol_base</span>
-            <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">sol_base</span><span class="p">)</span> <span class="o">-</span> <span class="nb">set</span><span class="p">(</span><span class="n">_solve</span><span class="p">(</span><span class="n">lhs</span><span class="o">.</span><span class="n">exp</span><span class="p">,</span> <span class="n">sym</span><span class="p">)))</span>
-        <span class="k">elif</span> <span class="p">(</span><span class="n">rhs</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span> <span class="ow">and</span>
-              <span class="n">lhs</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">is_positive</span> <span class="ow">and</span>
-              <span class="n">lhs</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_real</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">_solve</span><span class="p">(</span><span class="n">lhs</span><span class="o">.</span><span class="n">exp</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">lhs</span><span class="o">.</span><span class="n">base</span><span class="p">)</span> <span class="o">-</span> <span class="n">log</span><span class="p">(</span><span class="n">rhs</span><span class="p">),</span> <span class="n">sym</span><span class="p">)</span>
-
-    <span class="k">elif</span> <span class="n">lhs</span><span class="o">.</span><span class="n">is_Mul</span> <span class="ow">and</span> <span class="n">rhs</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-        <span class="n">llhs</span> <span class="o">=</span> <span class="n">expand_log</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">lhs</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">llhs</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_solve</span><span class="p">(</span><span class="n">llhs</span> <span class="o">-</span> <span class="n">log</span><span class="p">(</span><span class="n">rhs</span><span class="p">),</span> <span class="n">sym</span><span class="p">)</span>
-
-    <span class="n">rewrite</span> <span class="o">=</span> <span class="n">lhs</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">exp</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">rewrite</span> <span class="o">!=</span> <span class="n">lhs</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">_solve</span><span class="p">(</span><span class="n">rewrite</span> <span class="o">-</span> <span class="n">rhs</span><span class="p">,</span> <span class="n">sym</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">flags</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;force&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">):</span>
-        <span class="n">flags</span><span class="p">[</span><span class="s">&#39;force&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="n">pos</span><span class="p">,</span> <span class="n">reps</span> <span class="o">=</span> <span class="n">posify</span><span class="p">(</span><span class="n">lhs</span> <span class="o">-</span> <span class="n">rhs</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">reps</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-            <span class="k">if</span> <span class="n">s</span> <span class="o">==</span> <span class="n">sym</span><span class="p">:</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">u</span> <span class="o">=</span> <span class="n">sym</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">soln</span> <span class="o">=</span> <span class="n">_solve</span><span class="p">(</span><span class="n">pos</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">s</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">reps</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">soln</span><span class="p">]</span>
-
-<span class="c"># TODO: option for calculating J numerically</span>
-
-
-<div class="viewcode-block" id="nsolve"><a class="viewcode-back" href="../../../modules/solvers/solvers.html#sympy.solvers.solvers.nsolve">[docs]</a><span class="k">def</span> <span class="nf">nsolve</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Solve a nonlinear equation system numerically::</span>
-
-<span class="sd">        nsolve(f, [args,] x0, modules=[&#39;mpmath&#39;], **kwargs)</span>
-
-<span class="sd">    f is a vector function of symbolic expressions representing the system.</span>
-<span class="sd">    args are the variables. If there is only one variable, this argument can</span>
-<span class="sd">    be omitted.</span>
-<span class="sd">    x0 is a starting vector close to a solution.</span>
-
-<span class="sd">    Use the modules keyword to specify which modules should be used to</span>
-<span class="sd">    evaluate the function and the Jacobian matrix. Make sure to use a module</span>
-<span class="sd">    that supports matrices. For more information on the syntax, please see the</span>
-<span class="sd">    docstring of lambdify.</span>
-
-<span class="sd">    Overdetermined systems are supported.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, nsolve</span>
-<span class="sd">    &gt;&gt;&gt; import sympy</span>
-<span class="sd">    &gt;&gt;&gt; sympy.mpmath.mp.dps = 15</span>
-<span class="sd">    &gt;&gt;&gt; x1 = Symbol(&#39;x1&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; x2 = Symbol(&#39;x2&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; f1 = 3 * x1**2 - 2 * x2**2 - 1</span>
-<span class="sd">    &gt;&gt;&gt; f2 = x1**2 - 2 * x1 + x2**2 + 2 * x2 - 8</span>
-<span class="sd">    &gt;&gt;&gt; print(nsolve((f1, f2), (x1, x2), (-1, 1)))</span>
-<span class="sd">    [-1.19287309935246]</span>
-<span class="sd">    [ 1.27844411169911]</span>
-
-<span class="sd">    For one-dimensional functions the syntax is simplified:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import sin, nsolve</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; nsolve(sin(x), x, 2)</span>
-<span class="sd">    3.14159265358979</span>
-<span class="sd">    &gt;&gt;&gt; nsolve(sin(x), 2)</span>
-<span class="sd">    3.14159265358979</span>
-
-<span class="sd">    mpmath.findroot is used, you can find there more extensive documentation,</span>
-<span class="sd">    especially concerning keyword parameters and available solvers.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># interpret arguments</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">fargs</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="n">x0</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-    <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">fargs</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="n">x0</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-    <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;nsolve expected at least 2 arguments, got </span><span class="si">%i</span><span class="s">&#39;</span>
-                        <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&#39;nsolve expected at most 3 arguments, got </span><span class="si">%i</span><span class="s">&#39;</span>
-                        <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">))</span>
-    <span class="n">modules</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;modules&#39;</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;mpmath&#39;</span><span class="p">])</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="nb">list</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">)):</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">T</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">):</span>
-        <span class="c"># assume it&#39;s a sympy expression</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">Equality</span><span class="p">):</span>
-            <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">lhs</span> <span class="o">-</span> <span class="n">f</span><span class="o">.</span><span class="n">rhs</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-        <span class="n">syms</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">free_symbols</span>
-        <span class="k">if</span> <span class="n">fargs</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">fargs</span> <span class="o">=</span> <span class="n">syms</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">syms</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="p">(</span><span class="n">fargs</span> <span class="ow">in</span> <span class="n">syms</span> <span class="ow">or</span> <span class="n">fargs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">in</span> <span class="n">syms</span><span class="p">)):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&#39;&#39;&#39;</span>
-<span class="s">                expected a one-dimensional and numerical function&#39;&#39;&#39;</span><span class="p">))</span>
-
-        <span class="c"># the function is much better behaved if there is no denominator</span>
-        <span class="n">f</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="n">f</span> <span class="o">=</span> <span class="n">lambdify</span><span class="p">(</span><span class="n">fargs</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">modules</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">findroot</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">fargs</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">f</span><span class="o">.</span><span class="n">cols</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&#39;&#39;&#39;</span>
-<span class="s">            need at least as many equations as variables&#39;&#39;&#39;</span><span class="p">))</span>
-    <span class="n">verbose</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;verbose&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-        <span class="k">print</span><span class="p">(</span><span class="s">&#39;f(x):&#39;</span><span class="p">)</span>
-        <span class="k">print</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-    <span class="c"># derive Jacobian</span>
-    <span class="n">J</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">fargs</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">verbose</span><span class="p">:</span>
-        <span class="k">print</span><span class="p">(</span><span class="s">&#39;J(x):&#39;</span><span class="p">)</span>
-        <span class="k">print</span><span class="p">(</span><span class="n">J</span><span class="p">)</span>
-    <span class="c"># create functions</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">lambdify</span><span class="p">(</span><span class="n">fargs</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">modules</span><span class="p">)</span>
-    <span class="n">J</span> <span class="o">=</span> <span class="n">lambdify</span><span class="p">(</span><span class="n">fargs</span><span class="p">,</span> <span class="n">J</span><span class="p">,</span> <span class="n">modules</span><span class="p">)</span>
-    <span class="c"># solve the system numerically</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">findroot</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">J</span><span class="o">=</span><span class="n">J</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">x</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_invert</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="o">*</span><span class="n">symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return tuple (i, d) where ``i`` is independent of ``symbols`` and ``d``</span>
-<span class="sd">    contains symbols. ``i`` and ``d`` are obtained after recursively using</span>
-<span class="sd">    algebraic inversion until an uninvertible ``d`` remains. If there are no</span>
-<span class="sd">    free symbols then ``d`` will be zero. Some (but not necessarily all)</span>
-<span class="sd">    solutions to the expression ``i - d`` will be related the solutions of the</span>
-<span class="sd">    original expression.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.solvers.solvers import _invert as invert</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import sqrt, cos</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; invert(x - 3)</span>
-<span class="sd">    (3, x)</span>
-<span class="sd">    &gt;&gt;&gt; invert(3)</span>
-<span class="sd">    (3, 0)</span>
-<span class="sd">    &gt;&gt;&gt; invert(2*cos(x) - 1)</span>
-<span class="sd">    (pi/3, x)</span>
-<span class="sd">    &gt;&gt;&gt; invert(sqrt(x) - 3)</span>
-<span class="sd">    (3, sqrt(x))</span>
-<span class="sd">    &gt;&gt;&gt; invert(sqrt(x) + y, x)</span>
-<span class="sd">    (-y, sqrt(x))</span>
-<span class="sd">    &gt;&gt;&gt; invert(sqrt(x) + y, y)</span>
-<span class="sd">    (-sqrt(x), y)</span>
-<span class="sd">    &gt;&gt;&gt; invert(sqrt(x) + y, x, y)</span>
-<span class="sd">    (0, sqrt(x) + y)</span>
-
-<span class="sd">    If there is more than one symbol in a power&#39;s base and the exponent</span>
-<span class="sd">    is not an Integer, then the principal root will be used for the</span>
-<span class="sd">    inversion:</span>
-
-<span class="sd">    &gt;&gt;&gt; invert(sqrt(x + y) - 2)</span>
-<span class="sd">    (4, x + y)</span>
-<span class="sd">    &gt;&gt;&gt; invert(sqrt(x + y) - 2)</span>
-<span class="sd">    (4, x + y)</span>
-
-<span class="sd">    If the exponent is an integer, setting ``integer_power`` to True</span>
-<span class="sd">    will force the principal root to be selected:</span>
-
-<span class="sd">    &gt;&gt;&gt; invert(x**2 - 4, integer_power=True)</span>
-<span class="sd">    (2, x)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">eq</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span>
-    <span class="n">free</span> <span class="o">=</span> <span class="n">eq</span><span class="o">.</span><span class="n">free_symbols</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">symbols</span><span class="p">:</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="n">free</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">free</span> <span class="o">&amp;</span> <span class="nb">set</span><span class="p">(</span><span class="n">symbols</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">eq</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-
-    <span class="n">dointpow</span> <span class="o">=</span> <span class="nb">bool</span><span class="p">(</span><span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&#39;integer_power&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">))</span>
-
-    <span class="n">inverses</span> <span class="o">=</span> <span class="p">{</span>
-        <span class="n">asin</span><span class="p">:</span> <span class="n">sin</span><span class="p">,</span>
-        <span class="n">acos</span><span class="p">:</span> <span class="n">cos</span><span class="p">,</span>
-        <span class="n">atan</span><span class="p">:</span> <span class="n">tan</span><span class="p">,</span>
-        <span class="n">acot</span><span class="p">:</span> <span class="n">cot</span><span class="p">,</span>
-        <span class="n">asinh</span><span class="p">:</span> <span class="n">sinh</span><span class="p">,</span>
-        <span class="n">acosh</span><span class="p">:</span> <span class="n">cosh</span><span class="p">,</span>
-        <span class="n">atanh</span><span class="p">:</span> <span class="n">tanh</span><span class="p">,</span>
-        <span class="n">acoth</span><span class="p">:</span> <span class="n">coth</span><span class="p">,</span>
-    <span class="p">}</span>
-
-    <span class="n">lhs</span> <span class="o">=</span> <span class="n">eq</span>
-    <span class="n">rhs</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">was</span> <span class="o">=</span> <span class="n">lhs</span>
-        <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">indep</span><span class="p">,</span> <span class="n">dep</span> <span class="o">=</span> <span class="n">lhs</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="o">*</span><span class="n">symbols</span><span class="p">)</span>
-
-            <span class="c"># dep + indep == rhs</span>
-            <span class="k">if</span> <span class="n">lhs</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="c"># this indicates we have done it all</span>
-                <span class="k">if</span> <span class="n">indep</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-                    <span class="k">break</span>
-
-                <span class="n">lhs</span> <span class="o">=</span> <span class="n">dep</span>
-                <span class="n">rhs</span> <span class="o">-=</span> <span class="n">indep</span>
-
-            <span class="c"># dep * indep == rhs</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># this indicates we have done it all</span>
-                <span class="k">if</span> <span class="n">indep</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">:</span>
-                    <span class="k">break</span>
-
-                <span class="n">lhs</span> <span class="o">=</span> <span class="n">dep</span>
-                <span class="n">rhs</span> <span class="o">/=</span> <span class="n">indep</span>
-
-        <span class="c"># collect like-terms in symbols</span>
-        <span class="k">if</span> <span class="n">lhs</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="n">terms</span> <span class="o">=</span> <span class="p">{}</span>
-            <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">lhs</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="n">i</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="o">*</span><span class="n">symbols</span><span class="p">)</span>
-                <span class="n">terms</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="p">[])</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">terms</span><span class="o">.</span><span class="n">values</span><span class="p">())):</span>
-                <span class="n">args</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">for</span> <span class="n">d</span><span class="p">,</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">terms</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-                    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-                        <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">i</span><span class="p">)</span><span class="o">*</span><span class="n">d</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">d</span><span class="p">)</span>
-                <span class="n">lhs</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="c"># if it&#39;s a two-term Add with rhs = 0 and two powers we can get the</span>
-        <span class="c"># dependent terms together, e.g. 3*f(x) + 2*g(x) -&gt; f(x)/g(x) = -2/3</span>
-        <span class="k">if</span> <span class="n">lhs</span><span class="o">.</span><span class="n">is_Add</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">rhs</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">lhs</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">lhs</span><span class="o">.</span><span class="n">as_two_terms</span><span class="p">()</span>
-            <span class="n">ai</span><span class="p">,</span> <span class="n">ad</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="o">*</span><span class="n">symbols</span><span class="p">)</span>
-            <span class="n">bi</span><span class="p">,</span> <span class="n">bd</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="o">*</span><span class="n">symbols</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">_ispow</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">(</span><span class="n">ad</span><span class="p">,</span> <span class="n">bd</span><span class="p">)):</span>
-                <span class="n">a_base</span><span class="p">,</span> <span class="n">a_exp</span> <span class="o">=</span> <span class="n">ad</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                <span class="n">b_base</span><span class="p">,</span> <span class="n">b_exp</span> <span class="o">=</span> <span class="n">bd</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                <span class="k">if</span> <span class="n">a_base</span> <span class="o">==</span> <span class="n">b_base</span><span class="p">:</span>
-                    <span class="c"># a = -b</span>
-                    <span class="n">lhs</span> <span class="o">=</span> <span class="n">powsimp</span><span class="p">(</span><span class="n">powdenest</span><span class="p">(</span><span class="n">ad</span><span class="o">/</span><span class="n">bd</span><span class="p">))</span>
-                    <span class="n">rhs</span> <span class="o">=</span> <span class="o">-</span><span class="n">bi</span><span class="o">/</span><span class="n">ai</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">bases</span> <span class="o">=</span> <span class="p">(</span><span class="n">b_base</span><span class="p">,</span> <span class="n">a_base</span><span class="p">)</span>
-                    <span class="n">expos</span> <span class="o">=</span> <span class="p">(</span><span class="n">b_exp</span><span class="p">,</span> <span class="n">a_exp</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="p">(</span><span class="nb">all</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">is_real</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">bases</span> <span class="o">+</span> <span class="n">expos</span><span class="p">)</span> <span class="ow">and</span>
-                            <span class="nb">all</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">is_positive</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">bases</span><span class="p">)</span> <span class="ow">and</span>
-                            <span class="nb">any</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">symbols</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expos</span><span class="p">)</span> <span class="ow">and</span>
-                            <span class="ow">not</span> <span class="nb">any</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="o">*</span><span class="n">symbols</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">bases</span><span class="p">)</span> <span class="ow">and</span>
-                            <span class="n">sign</span><span class="p">(</span><span class="n">ai</span><span class="o">*</span><span class="n">bi</span><span class="p">)</span><span class="o">.</span><span class="n">is_negative</span><span class="p">):</span>
-                        <span class="c"># ai*a_base**a_exp = -bi*b_base**b_exp</span>
-                        <span class="n">lhs</span> <span class="o">=</span> <span class="n">a_exp</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">a_base</span><span class="p">)</span> <span class="o">-</span> <span class="n">b_exp</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">b_base</span><span class="p">)</span>
-                        <span class="n">rhs</span> <span class="o">=</span> <span class="o">-</span><span class="n">log</span><span class="p">(</span><span class="n">ai</span><span class="o">/-</span><span class="n">bi</span><span class="p">)</span>
-
-        <span class="k">elif</span> <span class="n">lhs</span><span class="o">.</span><span class="n">is_Mul</span> <span class="ow">and</span> <span class="nb">any</span><span class="p">(</span><span class="n">_ispow</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">lhs</span><span class="o">.</span><span class="n">args</span><span class="p">):</span>
-            <span class="n">lhs</span> <span class="o">=</span> <span class="n">powsimp</span><span class="p">(</span><span class="n">powdenest</span><span class="p">(</span><span class="n">lhs</span><span class="p">))</span>
-
-        <span class="c">#                    -1</span>
-        <span class="c"># f(x) = g  -&gt;  x = f  (g)</span>
-        <span class="k">elif</span> <span class="n">lhs</span><span class="o">.</span><span class="n">is_Function</span> <span class="ow">and</span> <span class="p">(</span><span class="n">lhs</span><span class="o">.</span><span class="n">nargs</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">lhs</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">)</span> <span class="ow">and</span> \
-                                 <span class="p">(</span><span class="nb">hasattr</span><span class="p">(</span><span class="n">lhs</span><span class="p">,</span> <span class="s">&#39;inverse&#39;</span><span class="p">)</span> <span class="ow">or</span>
-                                  <span class="n">lhs</span><span class="o">.</span><span class="n">func</span> <span class="ow">in</span> <span class="n">inverses</span> <span class="ow">or</span>
-                                  <span class="n">lhs</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Abs</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">lhs</span><span class="o">.</span><span class="n">func</span> <span class="ow">in</span> <span class="n">inverses</span><span class="p">:</span>
-                <span class="n">inv</span> <span class="o">=</span> <span class="n">inverses</span><span class="p">[</span><span class="n">lhs</span><span class="o">.</span><span class="n">func</span><span class="p">]</span>
-            <span class="k">elif</span> <span class="n">lhs</span><span class="o">.</span><span class="n">func</span> <span class="ow">is</span> <span class="n">Abs</span><span class="p">:</span>
-                <span class="n">inv</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">w</span><span class="p">:</span> <span class="n">w</span><span class="o">**</span><span class="mi">2</span>
-                <span class="n">lhs</span> <span class="o">=</span> <span class="n">Basic</span><span class="p">(</span><span class="n">lhs</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>  <span class="c"># get it ready to remove the args</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">inv</span> <span class="o">=</span> <span class="n">lhs</span><span class="o">.</span><span class="n">inverse</span><span class="p">()</span>
-            <span class="n">rhs</span> <span class="o">=</span> <span class="n">inv</span><span class="p">(</span><span class="n">rhs</span><span class="p">)</span>
-            <span class="n">lhs</span> <span class="o">=</span> <span class="n">lhs</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">rhs</span> <span class="ow">and</span> <span class="n">lhs</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="n">lhs</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">lhs</span><span class="o">.</span><span class="n">exp</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">lhs</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">lhs</span>
-            <span class="n">rhs</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">rhs</span>
-
-        <span class="c"># base**a = b -&gt; base = b**(1/a) if</span>
-        <span class="c">#    a is an Integer and dointpow=True (this gives real branch of root)</span>
-        <span class="c">#    a is not an Integer and the equation is multivariate and the</span>
-        <span class="c">#      base has more than 1 symbol in it</span>
-        <span class="c"># The rationale for this is that right now the multi-system solvers</span>
-        <span class="c"># doesn&#39;t try to resolve generators to see, for example, if the whole</span>
-        <span class="c"># system is written in terms of sqrt(x + y) so it will just fail, so we</span>
-        <span class="c"># do that step here.</span>
-        <span class="k">if</span> <span class="n">lhs</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">and</span> <span class="p">(</span>
-            <span class="n">lhs</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="n">dointpow</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">lhs</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span>
-                <span class="nb">len</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">lhs</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">free_symbols</span> <span class="o">&amp;</span> <span class="nb">set</span><span class="p">(</span><span class="n">symbols</span><span class="p">))</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="n">rhs</span> <span class="o">=</span> <span class="n">rhs</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">lhs</span><span class="o">.</span><span class="n">exp</span><span class="p">)</span>
-            <span class="n">lhs</span> <span class="o">=</span> <span class="n">lhs</span><span class="o">.</span><span class="n">base</span>
-
-        <span class="k">if</span> <span class="n">lhs</span> <span class="o">==</span> <span class="n">was</span><span class="p">:</span>
-            <span class="k">break</span>
-    <span class="k">return</span> <span class="n">rhs</span><span class="p">,</span> <span class="n">lhs</span>
-
-
-<span class="k">def</span> <span class="nf">unrad</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="o">*</span><span class="n">syms</span><span class="p">,</span> <span class="o">**</span><span class="n">flags</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Remove radicals with symbolic arguments and return (eq, cov, dens),</span>
-<span class="sd">    None or raise an error:</span>
-
-<span class="sd">    None is returned if there are no radicals to remove.</span>
-
-<span class="sd">    ValueError is raised if there are radicals and they cannot be removed.</span>
-
-<span class="sd">    Otherwise</span>
-
-<span class="sd">        ``eq``, ``cov``</span>
-<span class="sd">            equation without radicals, perhaps written in terms of</span>
-<span class="sd">            change variables; the relationship to the original variables</span>
-<span class="sd">            is given by the expressions in list (``cov``) whose tuples,</span>
-<span class="sd">            (``v``, ``expr``) give the change variable introduced (``v``)</span>
-<span class="sd">            and the expression (``expr``) which equates the base of the radical</span>
-<span class="sd">            to the power of the change variable needed to clear the radical.</span>
-<span class="sd">            For example, for sqrt(2 - x) the tuple (_p, -_p**2 - x + 2), would</span>
-<span class="sd">            be obtained.</span>
-<span class="sd">        ``dens``</span>
-<span class="sd">            A set containing all denominators encountered while removing</span>
-<span class="sd">            radicals. This may be of interest since any solution obtained in</span>
-<span class="sd">            the modified expression should not set any denominator to zero.</span>
-<span class="sd">        ``syms``</span>
-<span class="sd">            an iterable of symbols which, if provided, will limit the focus of</span>
-<span class="sd">            radical removal: only radicals with one or more of the symbols of</span>
-<span class="sd">            interest will be cleared.</span>
-
-<span class="sd">    ``flags`` are used internally for communication during recursive calls.</span>
-
-<span class="sd">    Radicals can be removed from an expression if:</span>
-<span class="sd">        *   all bases of the radicals are the same; a change of variables is</span>
-<span class="sd">            done in this case.</span>
-<span class="sd">        *   if all radicals appear in one term of the expression</span>
-<span class="sd">        *   there are only 4 terms with sqrt() factors or there are less than</span>
-<span class="sd">            four terms having sqrt() factors</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.solvers.solvers import unrad</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import sqrt, Rational</span>
-<span class="sd">        &gt;&gt;&gt; unrad(sqrt(x)*x**Rational(1,3) + 2)</span>
-<span class="sd">        (x**5 - 64, [], [])</span>
-<span class="sd">        &gt;&gt;&gt; unrad(sqrt(x) + (x + 1)**Rational(1,3))</span>
-<span class="sd">        (x**3 - x**2 - 2*x - 1, [], [])</span>
-<span class="sd">        &gt;&gt;&gt; unrad(sqrt(x) + x**Rational(1,3) + 2)</span>
-<span class="sd">        (_p**3 + _p**2 + 2, [(_p, -_p**6 + x)], [])</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">eq</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-        <span class="k">return</span>
-    <span class="n">cov</span><span class="p">,</span> <span class="n">dens</span><span class="p">,</span> <span class="n">nwas</span> <span class="o">=</span> <span class="p">[</span><span class="n">flags</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span>
-                       <span class="nb">sorted</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">dens</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">cov</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span><span class="o">.</span><span class="n">items</span><span class="p">())]</span>
-
-    <span class="k">def</span> <span class="nf">_take</span><span class="p">(</span><span class="n">d</span><span class="p">):</span>
-        <span class="c"># see if this is a term that has symbols of interest</span>
-        <span class="c"># and merits further processing</span>
-        <span class="n">free</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">free_symbols</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">free</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="k">return</span> <span class="ow">not</span> <span class="n">syms</span> <span class="ow">or</span> <span class="n">free</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">syms</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">dens</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">dens</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="k">if</span> <span class="n">cov</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">cov</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="n">eq</span> <span class="o">=</span> <span class="n">powdenest</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span>
-    <span class="n">eq</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">eq</span><span class="o">.</span><span class="n">as_numer_denom</span><span class="p">()</span>
-    <span class="n">eq</span> <span class="o">=</span> <span class="n">_mexpand</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">_take</span><span class="p">(</span><span class="n">d</span><span class="p">):</span>
-        <span class="n">dens</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">eq</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">eq</span><span class="p">,</span> <span class="n">cov</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="n">dens</span><span class="p">)</span>
-
-    <span class="n">poly</span> <span class="o">=</span> <span class="n">eq</span><span class="o">.</span><span class="n">as_poly</span><span class="p">()</span>
-
-    <span class="c"># if all the bases are the same or all the radicals are in one</span>
-    <span class="c"># term, `lcm` will be the lcm of the radical&#39;s exponent</span>
-    <span class="c"># denominators</span>
-    <span class="n">lcm</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">rads</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="n">bases</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">poly</span><span class="o">.</span><span class="n">gens</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">_take</span><span class="p">(</span><span class="n">g</span><span class="p">)</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">g</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-            <span class="k">continue</span>
-        <span class="n">ecoeff</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">exp</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>  <span class="c"># a Rational</span>
-        <span class="k">if</span> <span class="n">ecoeff</span><span class="o">.</span><span class="n">q</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">rads</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-            <span class="n">lcm</span> <span class="o">=</span> <span class="n">ilcm</span><span class="p">(</span><span class="n">lcm</span><span class="p">,</span> <span class="n">ecoeff</span><span class="o">.</span><span class="n">q</span><span class="p">)</span>
-            <span class="n">bases</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">base</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">rads</span><span class="p">:</span>
-        <span class="k">return</span>
-
-    <span class="n">depth</span> <span class="o">=</span> <span class="n">sqrt_depth</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span>
-
-    <span class="c"># get terms together that have common generators</span>
-    <span class="n">drad</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">rads</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">rads</span><span class="p">))))))</span>
-    <span class="n">rterms</span> <span class="o">=</span> <span class="p">{():</span> <span class="p">[]}</span>
-    <span class="n">args</span> <span class="o">=</span> <span class="n">Add</span><span class="o">.</span><span class="n">make_args</span><span class="p">(</span><span class="n">poly</span><span class="o">.</span><span class="n">as_expr</span><span class="p">())</span>
-    <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">_take</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
-            <span class="n">common</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">t</span><span class="o">.</span><span class="n">as_poly</span><span class="p">()</span><span class="o">.</span><span class="n">gens</span><span class="p">)</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">rads</span><span class="p">)</span>
-            <span class="n">key</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="nb">sorted</span><span class="p">([</span><span class="n">drad</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">common</span><span class="p">]))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">key</span> <span class="o">=</span> <span class="p">()</span>
-        <span class="n">rterms</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="p">[])</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-    <span class="n">args</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">rterms</span><span class="o">.</span><span class="n">pop</span><span class="p">(()))</span>
-    <span class="n">rterms</span> <span class="o">=</span> <span class="p">[</span><span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="n">rterms</span><span class="p">[</span><span class="n">k</span><span class="p">])</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">rterms</span><span class="o">.</span><span class="n">keys</span><span class="p">())]</span>
-    <span class="c"># the output will depend on the order terms are processed, so</span>
-    <span class="c"># make it canonical quickly</span>
-    <span class="n">rterms</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-
-    <span class="c"># continue handling</span>
-    <span class="n">ok</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">rterms</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">eq</span> <span class="o">=</span> <span class="n">rterms</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="n">lcm</span> <span class="o">-</span> <span class="p">(</span><span class="o">-</span><span class="n">args</span><span class="p">)</span><span class="o">**</span><span class="n">lcm</span>
-
-    <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">rterms</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">args</span><span class="p">:</span>
-        <span class="n">eq</span> <span class="o">=</span> <span class="n">rterms</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="n">lcm</span> <span class="o">-</span> <span class="n">rterms</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">**</span><span class="n">lcm</span>
-
-    <span class="k">elif</span> <span class="n">log</span><span class="p">(</span><span class="n">lcm</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">is_Integer</span> <span class="ow">and</span> <span class="p">(</span><span class="ow">not</span> <span class="n">args</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">rterms</span><span class="p">)</span> <span class="o">==</span> <span class="mi">4</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">rterms</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">4</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">rterms</span><span class="p">)</span> <span class="o">==</span> <span class="mi">4</span><span class="p">:</span>
-            <span class="c"># (r0+r1)**2 - (r2+r3)**2</span>
-            <span class="n">t1</span><span class="p">,</span> <span class="n">t2</span><span class="p">,</span> <span class="n">t3</span><span class="p">,</span> <span class="n">t4</span> <span class="o">=</span> <span class="p">[</span><span class="n">t</span><span class="o">**</span><span class="mi">2</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">rterms</span><span class="p">]</span>
-            <span class="n">eq</span> <span class="o">=</span> <span class="n">t1</span> <span class="o">+</span> <span class="n">t2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">rterms</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">rterms</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> \
-                <span class="p">(</span><span class="n">t3</span> <span class="o">+</span> <span class="n">t4</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">rterms</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">rterms</span><span class="p">[</span><span class="mi">3</span><span class="p">])</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">rterms</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
-            <span class="c"># (r0+r1)**2 - (r2+a)**2</span>
-            <span class="n">t1</span><span class="p">,</span> <span class="n">t2</span><span class="p">,</span> <span class="n">t3</span> <span class="o">=</span> <span class="p">[</span><span class="n">t</span><span class="o">**</span><span class="mi">2</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">rterms</span><span class="p">]</span>
-            <span class="n">eq</span> <span class="o">=</span> <span class="n">t1</span> <span class="o">+</span> <span class="n">t2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">rterms</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">rterms</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> \
-                <span class="p">(</span><span class="n">t3</span> <span class="o">+</span> <span class="n">args</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">args</span><span class="o">*</span><span class="n">rterms</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">rterms</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">t1</span><span class="p">,</span> <span class="n">t2</span> <span class="o">=</span> <span class="p">[</span><span class="n">t</span><span class="o">**</span><span class="mi">2</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">rterms</span><span class="p">[:</span><span class="mi">2</span><span class="p">]]</span>
-            <span class="c"># r0**2 - (r1+a)**2</span>
-            <span class="n">eq</span> <span class="o">=</span> <span class="n">t1</span> <span class="o">-</span> <span class="p">(</span><span class="n">t2</span> <span class="o">+</span> <span class="n">args</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">args</span><span class="o">*</span><span class="n">rterms</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-
-    <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">bases</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>  <span class="c"># change of variables may work</span>
-        <span class="n">ok</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="n">covwas</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">cov</span><span class="p">)</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">bases</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">p</span><span class="p">,</span> <span class="n">bexpr</span> <span class="ow">in</span> <span class="n">cov</span><span class="p">:</span>
-            <span class="nb">pow</span> <span class="o">=</span> <span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="n">bexpr</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">pow</span><span class="o">.</span><span class="n">is_Pow</span><span class="p">:</span>
-                <span class="n">pb</span><span class="p">,</span> <span class="n">pe</span> <span class="o">=</span> <span class="nb">pow</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-                <span class="k">if</span> <span class="n">pe</span> <span class="o">==</span> <span class="n">lcm</span> <span class="ow">and</span> <span class="n">pb</span> <span class="o">==</span> <span class="n">p</span><span class="p">:</span>
-                    <span class="n">p</span> <span class="o">=</span> <span class="n">pb</span>
-                    <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="n">cov</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">p</span><span class="p">,</span> <span class="n">b</span> <span class="o">-</span> <span class="n">p</span><span class="o">**</span><span class="n">lcm</span><span class="p">))</span>
-        <span class="n">eq</span> <span class="o">=</span> <span class="n">poly</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">p</span><span class="o">**</span><span class="n">lcm</span><span class="p">)</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">eq</span><span class="o">.</span><span class="n">free_symbols</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">syms</span><span class="p">):</span>
-            <span class="n">ok</span> <span class="o">=</span> <span class="bp">True</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">cov</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">covwas</span><span class="p">:</span>
-                <span class="n">cov</span> <span class="o">=</span> <span class="n">cov</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">ok</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="n">new_depth</span> <span class="o">=</span> <span class="n">sqrt_depth</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">ok</span> <span class="ow">or</span> <span class="p">(</span><span class="n">nwas</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">rterms</span><span class="p">)</span> <span class="o">==</span> <span class="n">nwas</span> <span class="ow">and</span> <span class="n">new_depth</span> <span class="ow">and</span> <span class="n">new_depth</span> <span class="o">==</span> <span class="n">depth</span><span class="p">):</span>
-        <span class="c"># XXX: XFAIL tests indicate other cases that should be handled.</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Cannot remove all radicals from </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">eq</span><span class="p">)</span>
-
-    <span class="n">neq</span> <span class="o">=</span> <span class="n">unrad</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="o">*</span><span class="n">syms</span><span class="p">,</span> <span class="o">**</span><span class="nb">dict</span><span class="p">(</span><span class="n">cov</span><span class="o">=</span><span class="n">cov</span><span class="p">,</span> <span class="n">dens</span><span class="o">=</span><span class="n">dens</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">rterms</span><span class="p">)))</span>
-    <span class="k">if</span> <span class="n">neq</span><span class="p">:</span>
-        <span class="n">eq</span> <span class="o">=</span> <span class="n">neq</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">eq</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">():</span>
-        <span class="n">eq</span> <span class="o">=</span> <span class="o">-</span><span class="n">eq</span>
-    <span class="k">return</span> <span class="p">(</span><span class="n">_mexpand</span><span class="p">(</span><span class="n">eq</span><span class="p">),</span> <span class="n">cov</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="n">dens</span><span class="p">))</span>
-</pre></div>
-
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-            <p class="logo"><a href="../../../index.html">
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+++ /dev/null
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-            
-  <h1>Source code for sympy.statistics.distributions</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">Lambda</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">,</span> <span class="n">Integer</span><span class="p">,</span> <span class="n">Rational</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">Float</span><span class="p">,</span> <span class="n">pi</span>
-<span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">erf</span>
-<span class="kn">from</span> <span class="nn">sympy.printing</span> <span class="kn">import</span> <span class="n">sstr</span>
-<span class="kn">import</span> <span class="nn">random</span>
-
-
-<div class="viewcode-block" id="Sample"><a class="viewcode-back" href="../../../modules/statistics.html#sympy.statistics.distributions.Sample">[docs]</a><span class="k">class</span> <span class="nc">Sample</span><span class="p">(</span><span class="nb">tuple</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Sample([x1, x2, x3, ...]) represents a collection of samples.</span>
-<span class="sd">    Sample parameters like mean, variance and stddev can be accessed as</span>
-<span class="sd">    properties.</span>
-<span class="sd">    The sample will be sorted.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.statistics.distributions import Sample</span>
-<span class="sd">        &gt;&gt;&gt; Sample([0, 1, 2, 3])</span>
-<span class="sd">        Sample([0, 1, 2, 3])</span>
-<span class="sd">        &gt;&gt;&gt; Sample([8, 3, 2, 4, 1, 6, 9, 2])</span>
-<span class="sd">        Sample([1, 2, 2, 3, 4, 6, 8, 9])</span>
-<span class="sd">        &gt;&gt;&gt; s = Sample([1, 2, 3, 4, 5])</span>
-<span class="sd">        &gt;&gt;&gt; s.mean</span>
-<span class="sd">        3</span>
-<span class="sd">        &gt;&gt;&gt; s.stddev</span>
-<span class="sd">        sqrt(2)</span>
-<span class="sd">        &gt;&gt;&gt; s.median</span>
-<span class="sd">        3</span>
-<span class="sd">        &gt;&gt;&gt; s.variance</span>
-<span class="sd">        2</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">sample</span><span class="p">):</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="nb">tuple</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">sample</span><span class="p">))</span>
-        <span class="n">s</span><span class="o">.</span><span class="n">mean</span> <span class="o">=</span> <span class="n">mean</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="o">/</span> <span class="n">Integer</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">))</span>
-        <span class="n">s</span><span class="o">.</span><span class="n">variance</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">([(</span><span class="n">x</span> <span class="o">-</span> <span class="n">mean</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">s</span><span class="p">])</span> <span class="o">/</span> <span class="n">Integer</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">))</span>
-        <span class="n">s</span><span class="o">.</span><span class="n">stddev</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">variance</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">s</span><span class="o">.</span><span class="n">median</span> <span class="o">=</span> <span class="n">s</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">s</span><span class="o">.</span><span class="n">median</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span><span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span> <span class="o">/</span> <span class="n">Integer</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">s</span>
-
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sstr</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sstr</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="ContinuousProbability"><a class="viewcode-back" href="../../../modules/statistics.html#sympy.statistics.distributions.ContinuousProbability">[docs]</a><span class="k">class</span> <span class="nc">ContinuousProbability</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Base class for continuous probability distributions&quot;&quot;&quot;</span>
-
-<div class="viewcode-block" id="ContinuousProbability.probability"><a class="viewcode-back" href="../../../modules/statistics.html#sympy.statistics.distributions.ContinuousProbability.probability">[docs]</a>    <span class="k">def</span> <span class="nf">probability</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Calculate the probability that a random number x generated</span>
-<span class="sd">        from the distribution satisfies a &lt;= x &lt;= b</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy.statistics import Normal</span>
-<span class="sd">            &gt;&gt;&gt; from sympy.core import oo</span>
-<span class="sd">            &gt;&gt;&gt; Normal(0, 1).probability(-1, 1)</span>
-<span class="sd">            erf(sqrt(2)/2)</span>
-<span class="sd">            &gt;&gt;&gt; Normal(0, 1).probability(1, oo)</span>
-<span class="sd">            -erf(sqrt(2)/2)/2 + 1/2</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">s</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">-</span> <span class="n">s</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="ContinuousProbability.random"><a class="viewcode-back" href="../../../modules/statistics.html#sympy.statistics.distributions.ContinuousProbability.random">[docs]</a>    <span class="k">def</span> <span class="nf">random</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        random() -- generate a random number from the distribution.</span>
-<span class="sd">        random(n) -- generate a Sample of n random numbers.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy.statistics import Uniform</span>
-<span class="sd">            &gt;&gt;&gt; x = Uniform(1, 5).random()</span>
-<span class="sd">            &gt;&gt;&gt; x &lt; 5 and x &gt; 1</span>
-<span class="sd">            True</span>
-<span class="sd">            &gt;&gt;&gt; x = Uniform(-4, 2).random()</span>
-<span class="sd">            &gt;&gt;&gt; x &lt; 2 and x &gt; -4</span>
-<span class="sd">            True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">s</span><span class="o">.</span><span class="n">_random</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Sample</span><span class="p">([</span><span class="n">s</span><span class="o">.</span><span class="n">_random</span><span class="p">()</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)])</span>
-</div>
-    <span class="k">def</span> <span class="nf">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sstr</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sstr</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="Normal"><a class="viewcode-back" href="../../../modules/statistics.html#sympy.statistics.distributions.Normal">[docs]</a><span class="k">class</span> <span class="nc">Normal</span><span class="p">(</span><span class="n">ContinuousProbability</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Normal(mu, sigma) represents the normal or Gaussian distribution</span>
-<span class="sd">    with mean value mu and standard deviation sigma.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.statistics import Normal</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import oo</span>
-<span class="sd">        &gt;&gt;&gt; N = Normal(1, 2)</span>
-<span class="sd">        &gt;&gt;&gt; N.mean</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; N.variance</span>
-<span class="sd">        4</span>
-<span class="sd">        &gt;&gt;&gt; N.probability(-oo, 1)   # probability on an interval</span>
-<span class="sd">        1/2</span>
-<span class="sd">        &gt;&gt;&gt; N.probability(1, oo)</span>
-<span class="sd">        1/2</span>
-<span class="sd">        &gt;&gt;&gt; N.probability(-oo, oo)</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; N.probability(-1, 3)</span>
-<span class="sd">        erf(sqrt(2)/2)</span>
-<span class="sd">        &gt;&gt;&gt; _.evalf()</span>
-<span class="sd">        0.682689492137086</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">sigma</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">mu</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">mu</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">sigma</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">sigma</span><span class="p">)</span>
-
-    <span class="n">mean</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="k">lambda</span> <span class="n">s</span><span class="p">:</span> <span class="n">s</span><span class="o">.</span><span class="n">mu</span><span class="p">)</span>
-    <span class="n">median</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="k">lambda</span> <span class="n">s</span><span class="p">:</span> <span class="n">s</span><span class="o">.</span><span class="n">mu</span><span class="p">)</span>
-    <span class="n">mode</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="k">lambda</span> <span class="n">s</span><span class="p">:</span> <span class="n">s</span><span class="o">.</span><span class="n">mu</span><span class="p">)</span>
-    <span class="n">stddev</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="k">lambda</span> <span class="n">s</span><span class="p">:</span> <span class="n">s</span><span class="o">.</span><span class="n">sigma</span><span class="p">)</span>
-    <span class="n">variance</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="k">lambda</span> <span class="n">s</span><span class="p">:</span> <span class="n">s</span><span class="o">.</span><span class="n">sigma</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-<div class="viewcode-block" id="Normal.pdf"><a class="viewcode-back" href="../../../modules/statistics.html#sympy.statistics.distributions.Normal.pdf">[docs]</a>    <span class="k">def</span> <span class="nf">pdf</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the probability density function as an expression in x</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy.statistics import Normal</span>
-<span class="sd">            &gt;&gt;&gt; Normal(1, 2).pdf(0)</span>
-<span class="sd">            sqrt(2)*exp(-1/8)/(4*sqrt(pi))</span>
-<span class="sd">            &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">            &gt;&gt;&gt; Normal(1, 2).pdf(x)</span>
-<span class="sd">            sqrt(2)*exp(-(x - 1)**2/8)/(4*sqrt(pi))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">sigma</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span> <span class="o">*</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">s</span><span class="o">.</span><span class="n">mu</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">sigma</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-</div>
-<div class="viewcode-block" id="Normal.cdf"><a class="viewcode-back" href="../../../modules/statistics.html#sympy.statistics.distributions.Normal.cdf">[docs]</a>    <span class="k">def</span> <span class="nf">cdf</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the cumulative density function as an expression in x</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy.statistics import Normal</span>
-<span class="sd">            &gt;&gt;&gt; Normal(1, 2).cdf(0)</span>
-<span class="sd">            -erf(sqrt(2)/4)/2 + 1/2</span>
-<span class="sd">            &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">            &gt;&gt;&gt; Normal(1, 2).cdf(x)</span>
-<span class="sd">            erf(sqrt(2)*(x - 1)/4)/2 + 1/2</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">erf</span><span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="n">s</span><span class="o">.</span><span class="n">mu</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">sigma</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))))</span><span class="o">/</span><span class="mi">2</span>
-</div>
-    <span class="k">def</span> <span class="nf">_random</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">random</span><span class="o">.</span><span class="n">gauss</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">mu</span><span class="p">),</span> <span class="nb">float</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">sigma</span><span class="p">))</span>
-
-<div class="viewcode-block" id="Normal.confidence"><a class="viewcode-back" href="../../../modules/statistics.html#sympy.statistics.distributions.Normal.confidence">[docs]</a>    <span class="k">def</span> <span class="nf">confidence</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Return a symmetric (p*100)% confidence interval. For example,</span>
-<span class="sd">        p=0.95 gives a 95% confidence interval. Currently this function</span>
-<span class="sd">        only handles numerical values except in the trivial case p=1.</span>
-
-<span class="sd">        For example, one standard deviation:</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy.statistics import Normal</span>
-<span class="sd">            &gt;&gt;&gt; N = Normal(0, 1)</span>
-<span class="sd">            &gt;&gt;&gt; N.confidence(0.68)</span>
-<span class="sd">            (-0.994457883209753, 0.994457883209753)</span>
-<span class="sd">            &gt;&gt;&gt; N.probability(*_).evalf()</span>
-<span class="sd">            0.680000000000000</span>
-
-<span class="sd">        Two standard deviations:</span>
-
-<span class="sd">            &gt;&gt;&gt; N = Normal(0, 1)</span>
-<span class="sd">            &gt;&gt;&gt; N.confidence(0.95)</span>
-<span class="sd">            (-1.95996398454005, 1.95996398454005)</span>
-<span class="sd">            &gt;&gt;&gt; N.probability(*_).evalf()</span>
-<span class="sd">            0.950000000000000</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="n">p</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="o">-</span><span class="n">oo</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-
-        <span class="k">assert</span> <span class="n">p</span> <span class="o">&lt;=</span> <span class="mi">1</span>
-
-        <span class="c"># In terms of n*sigma, we have n = sqrt(2)*ierf(p). The inverse</span>
-        <span class="c"># error function is not yet implemented in SymPy but can easily be</span>
-        <span class="c"># computed numerically</span>
-
-        <span class="kn">from</span> <span class="nn">sympy.mpmath</span> <span class="kn">import</span> <span class="n">mpf</span><span class="p">,</span> <span class="n">erfinv</span>
-
-        <span class="c"># calculate y = ierf(p) by solving erf(y) - p = 0</span>
-        <span class="n">y</span> <span class="o">=</span> <span class="n">erfinv</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">Float</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">sigma</span><span class="p">))</span> <span class="o">*</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">y</span><span class="p">))</span>
-        <span class="n">mu</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">mu</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">mu</span> <span class="o">-</span> <span class="n">t</span><span class="p">,</span> <span class="n">mu</span> <span class="o">+</span> <span class="n">t</span><span class="p">)</span>
-</div>
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="Normal.fit"><a class="viewcode-back" href="../../../modules/statistics.html#sympy.statistics.distributions.Normal.fit">[docs]</a>    <span class="k">def</span> <span class="nf">fit</span><span class="p">(</span><span class="n">sample</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Create a normal distribution fit to the mean and standard</span>
-<span class="sd">        deviation of the given distribution or sample.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy.statistics import Normal</span>
-<span class="sd">            &gt;&gt;&gt; Normal.fit([1,2,3,4,5])</span>
-<span class="sd">            Normal(3, sqrt(2))</span>
-<span class="sd">            &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">            &gt;&gt;&gt; Normal.fit([x, y])</span>
-<span class="sd">            Normal(x/2 + y/2, sqrt((-x/2 + y/2)**2/2 + (x/2 - y/2)**2/2))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">sample</span><span class="p">,</span> <span class="s">&quot;stddev&quot;</span><span class="p">):</span>
-            <span class="n">sample</span> <span class="o">=</span> <span class="n">Sample</span><span class="p">(</span><span class="n">sample</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Normal</span><span class="p">(</span><span class="n">sample</span><span class="o">.</span><span class="n">mean</span><span class="p">,</span> <span class="n">sample</span><span class="o">.</span><span class="n">stddev</span><span class="p">)</span>
-
-</div></div>
-<div class="viewcode-block" id="Uniform"><a class="viewcode-back" href="../../../modules/statistics.html#sympy.statistics.distributions.Uniform">[docs]</a><span class="k">class</span> <span class="nc">Uniform</span><span class="p">(</span><span class="n">ContinuousProbability</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Uniform(a, b) represents a probability distribution with uniform</span>
-<span class="sd">    probability density on the interval [a, b] and zero density</span>
-<span class="sd">    everywhere else.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">a</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">b</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-
-    <span class="n">mean</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="k">lambda</span> <span class="n">s</span><span class="p">:</span> <span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">a</span> <span class="o">+</span> <span class="n">s</span><span class="o">.</span><span class="n">b</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-    <span class="n">median</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="k">lambda</span> <span class="n">s</span><span class="p">:</span> <span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">a</span> <span class="o">+</span> <span class="n">s</span><span class="o">.</span><span class="n">b</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-    <span class="n">mode</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="k">lambda</span> <span class="n">s</span><span class="p">:</span> <span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">a</span> <span class="o">+</span> <span class="n">s</span><span class="o">.</span><span class="n">b</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>  <span class="c"># arbitrary</span>
-    <span class="n">variance</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="k">lambda</span> <span class="n">s</span><span class="p">:</span> <span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">b</span> <span class="o">-</span> <span class="n">s</span><span class="o">.</span><span class="n">a</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span> <span class="mi">12</span><span class="p">)</span>
-    <span class="n">stddev</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="k">lambda</span> <span class="n">s</span><span class="p">:</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">variance</span><span class="p">))</span>
-
-<div class="viewcode-block" id="Uniform.pdf"><a class="viewcode-back" href="../../../modules/statistics.html#sympy.statistics.distributions.Uniform.pdf">[docs]</a>    <span class="k">def</span> <span class="nf">pdf</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the probability density function as an expression in x</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy.statistics import Uniform</span>
-<span class="sd">            &gt;&gt;&gt; Uniform(1, 5).pdf(1)</span>
-<span class="sd">            1/4</span>
-<span class="sd">            &gt;&gt;&gt; Uniform(2, 4).pdf(2)</span>
-<span class="sd">            1/2</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">x</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;SymPy does not yet support&quot;</span>
-                <span class="s">&quot;piecewise functions&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="o">&lt;</span> <span class="n">s</span><span class="o">.</span><span class="n">a</span> <span class="ow">or</span> <span class="n">x</span> <span class="o">&gt;</span> <span class="n">s</span><span class="o">.</span><span class="n">b</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">b</span> <span class="o">-</span> <span class="n">s</span><span class="o">.</span><span class="n">a</span><span class="p">)</span>
-</div>
-<div class="viewcode-block" id="Uniform.cdf"><a class="viewcode-back" href="../../../modules/statistics.html#sympy.statistics.distributions.Uniform.cdf">[docs]</a>    <span class="k">def</span> <span class="nf">cdf</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the cumulative density function as an expression in x</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy.statistics import Uniform</span>
-<span class="sd">            &gt;&gt;&gt; Uniform(1, 5).cdf(2)</span>
-<span class="sd">            1/4</span>
-<span class="sd">            &gt;&gt;&gt; Uniform(1, 5).cdf(4)</span>
-<span class="sd">            3/4</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">x</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;SymPy does not yet support&quot;</span>
-                <span class="s">&quot;piecewise functions&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="o">&lt;=</span> <span class="n">s</span><span class="o">.</span><span class="n">a</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">x</span> <span class="o">&gt;=</span> <span class="n">s</span><span class="o">.</span><span class="n">b</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">s</span><span class="o">.</span><span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">b</span> <span class="o">-</span> <span class="n">s</span><span class="o">.</span><span class="n">a</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_random</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Float</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">a</span><span class="p">),</span> <span class="nb">float</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">b</span><span class="p">)))</span>
-
-<div class="viewcode-block" id="Uniform.confidence"><a class="viewcode-back" href="../../../modules/statistics.html#sympy.statistics.distributions.Uniform.confidence">[docs]</a>    <span class="k">def</span> <span class="nf">confidence</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Generate a symmetric (p*100)% confidence interval.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Rational</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.statistics import Uniform</span>
-<span class="sd">        &gt;&gt;&gt; U = Uniform(1, 2)</span>
-<span class="sd">        &gt;&gt;&gt; U.confidence(1)</span>
-<span class="sd">        (1, 2)</span>
-<span class="sd">        &gt;&gt;&gt; U.confidence(Rational(1,2))</span>
-<span class="sd">        (5/4, 7/4)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-        <span class="k">assert</span> <span class="n">p</span> <span class="o">&lt;=</span> <span class="mi">1</span>
-
-        <span class="n">d</span> <span class="o">=</span> <span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">b</span> <span class="o">-</span> <span class="n">s</span><span class="o">.</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">p</span> <span class="o">/</span> <span class="mi">2</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">mean</span> <span class="o">-</span> <span class="n">d</span><span class="p">,</span> <span class="n">s</span><span class="o">.</span><span class="n">mean</span> <span class="o">+</span> <span class="n">d</span><span class="p">)</span>
-</div>
-    <span class="nd">@staticmethod</span>
-<div class="viewcode-block" id="Uniform.fit"><a class="viewcode-back" href="../../../modules/statistics.html#sympy.statistics.distributions.Uniform.fit">[docs]</a>    <span class="k">def</span> <span class="nf">fit</span><span class="p">(</span><span class="n">sample</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Create a uniform distribution fit to the mean and standard</span>
-<span class="sd">        deviation of the given distribution or sample.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy.statistics import Uniform</span>
-<span class="sd">            &gt;&gt;&gt; Uniform.fit([1, 2, 3, 4, 5])</span>
-<span class="sd">            Uniform(-sqrt(6) + 3, sqrt(6) + 3)</span>
-<span class="sd">            &gt;&gt;&gt; Uniform.fit([1, 2])</span>
-<span class="sd">            Uniform(-sqrt(3)/2 + 3/2, sqrt(3)/2 + 3/2)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">sample</span><span class="p">,</span> <span class="s">&quot;stddev&quot;</span><span class="p">):</span>
-            <span class="n">sample</span> <span class="o">=</span> <span class="n">Sample</span><span class="p">(</span><span class="n">sample</span><span class="p">)</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">sample</span><span class="o">.</span><span class="n">mean</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">12</span><span class="o">*</span><span class="n">sample</span><span class="o">.</span><span class="n">variance</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-        <span class="k">return</span> <span class="n">Uniform</span><span class="p">(</span><span class="n">m</span> <span class="o">-</span> <span class="n">d</span><span class="p">,</span> <span class="n">m</span> <span class="o">+</span> <span class="n">d</span><span class="p">)</span>
-
-</div></div>
-<div class="viewcode-block" id="PDF"><a class="viewcode-back" href="../../../modules/statistics.html#sympy.statistics.distributions.PDF">[docs]</a><span class="k">class</span> <span class="nc">PDF</span><span class="p">(</span><span class="n">ContinuousProbability</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    PDF(func, (x, a, b)) represents continuous probability distribution</span>
-<span class="sd">    with probability distribution function func(x) on interval (a, b)</span>
-
-<span class="sd">    If func is not normalized so that integrate(func, (x, a, b)) == 1,</span>
-<span class="sd">    it can be normalized using PDF.normalize() method</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Symbol, exp, oo</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.statistics.distributions import PDF</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">        &gt;&gt;&gt; a = Symbol(&#39;a&#39;, positive=True)</span>
-
-<span class="sd">        &gt;&gt;&gt; exponential = PDF(exp(-x/a)/a, (x,0,oo))</span>
-<span class="sd">        &gt;&gt;&gt; exponential.pdf(x)</span>
-<span class="sd">        exp(-x/a)/a</span>
-<span class="sd">        &gt;&gt;&gt; exponential.cdf(x)</span>
-<span class="sd">        1 - exp(-x/a)</span>
-<span class="sd">        &gt;&gt;&gt; exponential.mean</span>
-<span class="sd">        a</span>
-<span class="sd">        &gt;&gt;&gt; exponential.variance</span>
-<span class="sd">        a**2</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="n">var</span><span class="p">):</span>
-        <span class="c">#XXX maybe add some checking of parameters</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">var</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="nb">list</span><span class="p">)):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">pdf</span> <span class="o">=</span> <span class="n">Lambda</span><span class="p">(</span><span class="n">var</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">pdf</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">domain</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">var</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">pdf</span> <span class="o">=</span> <span class="n">Lambda</span><span class="p">(</span><span class="n">var</span><span class="p">,</span> <span class="n">pdf</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">domain</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="n">oo</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_cdf</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_mean</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_variance</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_stddev</span> <span class="o">=</span> <span class="bp">None</span>
-
-<div class="viewcode-block" id="PDF.normalize"><a class="viewcode-back" href="../../../modules/statistics.html#sympy.statistics.distributions.PDF.normalize">[docs]</a>    <span class="k">def</span> <span class="nf">normalize</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Normalize the probability distribution function so that</span>
-<span class="sd">        integrate(self.pdf(x), (x, a, b)) == 1</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy import Symbol, exp, oo</span>
-<span class="sd">            &gt;&gt;&gt; from sympy.statistics.distributions import PDF</span>
-<span class="sd">            &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">            &gt;&gt;&gt; a = Symbol(&#39;a&#39;, positive=True)</span>
-
-<span class="sd">            &gt;&gt;&gt; exponential = PDF(exp(-x/a), (x,0,oo))</span>
-<span class="sd">            &gt;&gt;&gt; exponential.normalize().pdf(x)</span>
-<span class="sd">            exp(-x/a)/a</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">norm</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">probability</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">norm</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">w</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;w&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">w</span><span class="p">)</span><span class="o">/</span><span class="n">norm</span><span class="p">,</span> <span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-            <span class="c">#self._cdf = Lambda(w, (self.cdf(w) - self.cdf(self.domain[0]))/norm)</span>
-            <span class="c">#if self._mean is not None:</span>
-            <span class="c">#    self._mean /= norm</span>
-            <span class="c">#if self._variance is not None:</span>
-            <span class="c">#    self._variance = (self._variance + (self._mean*norm)**2)/norm - self.mean**2</span>
-            <span class="c">#if self._stddev is not None:</span>
-            <span class="c">#    self._stddev = sqrt(self._variance)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span>
-</div>
-<div class="viewcode-block" id="PDF.cdf"><a class="viewcode-back" href="../../../modules/statistics.html#sympy.statistics.distributions.PDF.cdf">[docs]</a>    <span class="k">def</span> <span class="nf">cdf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return the cumulative density function as an expression in x</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy.statistics.distributions import PDF</span>
-<span class="sd">            &gt;&gt;&gt; from sympy import exp, oo</span>
-<span class="sd">            &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">            &gt;&gt;&gt; PDF(exp(-x/y), (x,0,oo)).cdf(4)</span>
-<span class="sd">            y - y*exp(-4/y)</span>
-<span class="sd">            &gt;&gt;&gt; PDF(2*x + y, (x, 10, oo)).cdf(0)</span>
-<span class="sd">            -10*y - 100</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_cdf</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_cdf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">integrate</span>
-            <span class="n">w</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;w&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_cdf</span> <span class="o">=</span> <span class="n">integrate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">w</span><span class="p">),</span> <span class="n">w</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_cdf</span> <span class="o">=</span> <span class="n">Lambda</span><span class="p">(</span>
-                <span class="n">w</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_cdf</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_cdf</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_cdf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_get_mean</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mean</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mean</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">integrate</span>
-            <span class="n">w</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;w&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_mean</span> <span class="o">=</span> <span class="n">integrate</span><span class="p">(</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">w</span><span class="p">)</span><span class="o">*</span><span class="n">w</span><span class="p">,</span> <span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_mean</span>
-
-    <span class="k">def</span> <span class="nf">_get_variance</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_variance</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_variance</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">integrate</span><span class="p">,</span> <span class="n">simplify</span>
-            <span class="n">w</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;w&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_variance</span> <span class="o">=</span> <span class="n">integrate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span>
-                <span class="n">w</span><span class="p">)</span><span class="o">*</span><span class="n">w</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">mean</span><span class="o">**</span><span class="mi">2</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_variance</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_variance</span><span class="p">)</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_variance</span>
-
-    <span class="k">def</span> <span class="nf">_get_stddev</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_stddev</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_stddev</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_stddev</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">variance</span><span class="p">)</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_stddev</span>
-
-    <span class="n">mean</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="n">_get_mean</span><span class="p">)</span>
-    <span class="n">variance</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="n">_get_variance</span><span class="p">)</span>
-    <span class="n">stddev</span> <span class="o">=</span> <span class="nb">property</span><span class="p">(</span><span class="n">_get_stddev</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_random</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span>
-
-<div class="viewcode-block" id="PDF.transform"><a class="viewcode-back" href="../../../modules/statistics.html#sympy.statistics.distributions.PDF.transform">[docs]</a>    <span class="k">def</span> <span class="nf">transform</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">var</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return a probability distribution of random variable func(x)</span>
-<span class="sd">        currently only some simple injective functions are supported</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy.statistics.distributions import PDF</span>
-<span class="sd">            &gt;&gt;&gt; from sympy import oo</span>
-<span class="sd">            &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">            &gt;&gt;&gt; PDF(2*x + y, (x, 10, oo)).transform(x, y)</span>
-<span class="sd">            PDF(0, ((_w,), x, x))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">w</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;w&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">solve</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span>
-        <span class="n">inverse</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">func</span> <span class="o">-</span> <span class="n">w</span><span class="p">,</span> <span class="n">var</span><span class="p">)</span>
-        <span class="n">newPdf</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="n">funcdiff</span> <span class="o">=</span> <span class="n">func</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">var</span><span class="p">)</span>
-        <span class="c">#TODO check if x is in domain</span>
-        <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">inverse</span><span class="p">:</span>
-            <span class="c"># this assignment holds only for x in domain</span>
-            <span class="c"># in general it would require implementing</span>
-            <span class="c"># piecewise defined functions in sympy</span>
-            <span class="n">newPdf</span> <span class="o">+=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">var</span><span class="p">)</span><span class="o">/</span><span class="nb">abs</span><span class="p">(</span><span class="n">funcdiff</span><span class="p">))</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">var</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">PDF</span><span class="p">(</span><span class="n">newPdf</span><span class="p">,</span> <span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">func</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">var</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">func</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">var</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">[</span><span class="mi">1</span><span class="p">])))</span></div></div>
-</pre></div>
-
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-  <h1>Source code for sympy.stats</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">SymPy statistics module</span>
-
-<span class="sd">Introduces a random variable type into the SymPy language.</span>
-
-<span class="sd">Random variables may be declared using prebuilt functions such as</span>
-<span class="sd">Normal, Exponential, Coin, Die, etc...  or built with functions like FiniteRV.</span>
-
-<span class="sd">Queries on random expressions can be made using the functions</span>
-
-<span class="sd">===================== =============================</span>
-<span class="sd">    Expression                Meaning</span>
-<span class="sd">--------------------- -----------------------------</span>
-<span class="sd"> P(condition)          Probability</span>
-<span class="sd"> E(expression)         Expectation value</span>
-<span class="sd"> variance(expression)  Variance</span>
-<span class="sd"> density(expression)   Probability Density Function</span>
-<span class="sd"> sample(expression)    Produce a realization</span>
-<span class="sd"> where(condition)      Where the condition is true</span>
-<span class="sd">===================== =============================</span>
-
-<span class="sd">Examples</span>
-<span class="sd">========</span>
-
-<span class="sd">&gt;&gt;&gt; from sympy.stats import P, E, variance, Die, Normal</span>
-<span class="sd">&gt;&gt;&gt; from sympy import Eq, simplify</span>
-<span class="sd">&gt;&gt;&gt; X, Y = Die(&#39;X&#39;, 6), Die(&#39;Y&#39;, 6) # Define two six sided dice</span>
-<span class="sd">&gt;&gt;&gt; Z = Normal(&#39;Z&#39;, 0, 1) # Declare a Normal random variable with mean 0, std 1</span>
-<span class="sd">&gt;&gt;&gt; P(X&gt;3) # Probability X is greater than 3</span>
-<span class="sd">1/2</span>
-<span class="sd">&gt;&gt;&gt; E(X+Y) # Expectation of the sum of two dice</span>
-<span class="sd">7</span>
-<span class="sd">&gt;&gt;&gt; variance(X+Y) # Variance of the sum of two dice</span>
-<span class="sd">35/6</span>
-<span class="sd">&gt;&gt;&gt; simplify(P(Z&gt;1)) # Probability of Z being greater than 1</span>
-<span class="sd">-erf(sqrt(2)/2)/2 + 1/2</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[]</span>
-
-<span class="kn">from</span> <span class="nn">.</span> <span class="kn">import</span> <span class="n">rv_interface</span>
-<span class="kn">from</span> <span class="nn">.rv_interface</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">cdf</span><span class="p">,</span> <span class="n">covariance</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">dependent</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">given</span><span class="p">,</span> <span class="n">independent</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">pspace</span><span class="p">,</span>
-    <span class="n">random_symbols</span><span class="p">,</span> <span class="n">sample</span><span class="p">,</span> <span class="n">sample_iter</span><span class="p">,</span> <span class="n">skewness</span><span class="p">,</span> <span class="n">std</span><span class="p">,</span> <span class="n">variance</span><span class="p">,</span> <span class="n">where</span><span class="p">,</span>
-<span class="p">)</span>
-<span class="n">__all__</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">rv_interface</span><span class="o">.</span><span class="n">__all__</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">.</span> <span class="kn">import</span> <span class="n">frv_types</span>
-<span class="kn">from</span> <span class="nn">.frv_types</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">Bernoulli</span><span class="p">,</span> <span class="n">Binomial</span><span class="p">,</span> <span class="n">Coin</span><span class="p">,</span> <span class="n">Die</span><span class="p">,</span> <span class="n">DiscreteUniform</span><span class="p">,</span> <span class="n">FiniteRV</span><span class="p">,</span> <span class="n">Hypergeometric</span><span class="p">,</span>
-<span class="p">)</span>
-<span class="n">__all__</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">frv_types</span><span class="o">.</span><span class="n">__all__</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">.</span> <span class="kn">import</span> <span class="n">crv_types</span>
-<span class="kn">from</span> <span class="nn">.crv_types</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">ContinuousRV</span><span class="p">,</span>
-    <span class="n">Arcsin</span><span class="p">,</span> <span class="n">Benini</span><span class="p">,</span> <span class="n">Beta</span><span class="p">,</span> <span class="n">BetaPrime</span><span class="p">,</span> <span class="n">Cauchy</span><span class="p">,</span> <span class="n">Chi</span><span class="p">,</span> <span class="n">ChiNoncentral</span><span class="p">,</span> <span class="n">ChiSquared</span><span class="p">,</span>
-    <span class="n">Dagum</span><span class="p">,</span> <span class="n">Erlang</span><span class="p">,</span> <span class="n">Exponential</span><span class="p">,</span> <span class="n">FDistribution</span><span class="p">,</span> <span class="n">FisherZ</span><span class="p">,</span> <span class="n">Frechet</span><span class="p">,</span> <span class="n">Gamma</span><span class="p">,</span>
-    <span class="n">GammaInverse</span><span class="p">,</span> <span class="n">Kumaraswamy</span><span class="p">,</span> <span class="n">Laplace</span><span class="p">,</span> <span class="n">Logistic</span><span class="p">,</span> <span class="n">LogNormal</span><span class="p">,</span> <span class="n">Maxwell</span><span class="p">,</span>
-    <span class="n">Nakagami</span><span class="p">,</span> <span class="n">Normal</span><span class="p">,</span> <span class="n">Pareto</span><span class="p">,</span> <span class="n">QuadraticU</span><span class="p">,</span> <span class="n">RaisedCosine</span><span class="p">,</span> <span class="n">Rayleigh</span><span class="p">,</span>
-    <span class="n">StudentT</span><span class="p">,</span> <span class="n">Triangular</span><span class="p">,</span> <span class="n">Uniform</span><span class="p">,</span> <span class="n">UniformSum</span><span class="p">,</span> <span class="n">VonMises</span><span class="p">,</span> <span class="n">Weibull</span><span class="p">,</span>
-    <span class="n">WignerSemicircle</span>
-<span class="p">)</span>
-<span class="n">__all__</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">crv_types</span><span class="o">.</span><span class="n">__all__</span><span class="p">)</span>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/stats/crv.html b/dev-py3k/_modules/sympy/stats/crv.html
deleted file mode 100644
index ec85c6f7149..00000000000
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-        <div class="bodywrapper">
-          <div class="body">
-            
-  <h1>Source code for sympy.stats.crv</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Continuous Random Variables Module</span>
-
-<span class="sd">See Also</span>
-<span class="sd">========</span>
-<span class="sd">sympy.stats.crv_types</span>
-<span class="sd">sympy.stats.rv</span>
-<span class="sd">sympy.stats.frv</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.stats.rv</span> <span class="kn">import</span> <span class="p">(</span><span class="n">RandomDomain</span><span class="p">,</span> <span class="n">SingleDomain</span><span class="p">,</span> <span class="n">ConditionalDomain</span><span class="p">,</span>
-        <span class="n">ProductDomain</span><span class="p">,</span> <span class="n">PSpace</span><span class="p">,</span> <span class="n">SinglePSpace</span><span class="p">,</span> <span class="n">random_symbols</span><span class="p">,</span> <span class="n">ProductPSpace</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.functions.special.delta_functions</span> <span class="kn">import</span> <span class="n">DiracDelta</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">Interval</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">,</span> <span class="n">FiniteSet</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">,</span>
-        <span class="n">Integral</span><span class="p">,</span> <span class="n">And</span><span class="p">,</span> <span class="n">Or</span><span class="p">,</span> <span class="n">Piecewise</span><span class="p">,</span> <span class="n">solve</span><span class="p">,</span> <span class="n">cacheit</span><span class="p">,</span> <span class="n">integrate</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">Lambda</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.solvers.inequalities</span> <span class="kn">import</span> <span class="n">reduce_poly_inequalities</span>
-<span class="kn">from</span> <span class="nn">sympy.polys.polyerrors</span> <span class="kn">import</span> <span class="n">PolynomialError</span>
-<span class="kn">import</span> <span class="nn">random</span>
-
-
-<div class="viewcode-block" id="ContinuousDomain"><a class="viewcode-back" href="../../../modules/stats.html#sympy.stats.crv.ContinuousDomain">[docs]</a><span class="k">class</span> <span class="nc">ContinuousDomain</span><span class="p">(</span><span class="n">RandomDomain</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A domain with continuous support</span>
-
-<span class="sd">    Represented using symbols and Intervals.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_Continuous</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">as_boolean</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Not Implemented for generic Domains&quot;</span><span class="p">)</span>
-
-</div>
-<span class="k">class</span> <span class="nc">SingleContinuousDomain</span><span class="p">(</span><span class="n">ContinuousDomain</span><span class="p">,</span> <span class="n">SingleDomain</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A univariate domain with continuous support</span>
-
-<span class="sd">    Represented using a single symbol and interval.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">symbol</span><span class="p">,</span> <span class="nb">set</span><span class="p">):</span>
-        <span class="k">assert</span> <span class="n">symbol</span><span class="o">.</span><span class="n">is_Symbol</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="n">symbol</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">RandomDomain</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="nb">set</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">integrate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">variables</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">variables</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">variables</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">symbols</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">variables</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span>
-        <span class="k">assert</span> <span class="nb">frozenset</span><span class="p">(</span><span class="n">variables</span><span class="p">)</span> <span class="o">==</span> <span class="nb">frozenset</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">symbols</span><span class="p">)</span>
-        <span class="c"># assumes only intervals</span>
-        <span class="n">evaluate</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;evaluate&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">evaluate</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">integrate</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">symbol</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">set</span><span class="p">),</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Integral</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">symbol</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">set</span><span class="p">),</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">as_boolean</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">set</span><span class="o">.</span><span class="n">as_relational</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">symbol</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">ProductContinuousDomain</span><span class="p">(</span><span class="n">ProductDomain</span><span class="p">,</span> <span class="n">ContinuousDomain</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A collection of independent domains with continuous support</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">integrate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">variables</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">variables</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">variables</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">symbols</span>
-        <span class="k">for</span> <span class="n">domain</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">domains</span><span class="p">:</span>
-            <span class="n">domain_vars</span> <span class="o">=</span> <span class="nb">frozenset</span><span class="p">(</span><span class="n">variables</span><span class="p">)</span> <span class="o">&amp;</span> <span class="nb">frozenset</span><span class="p">(</span><span class="n">domain</span><span class="o">.</span><span class="n">symbols</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">domain_vars</span><span class="p">:</span>
-                <span class="n">expr</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">integrate</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">domain_vars</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="k">def</span> <span class="nf">as_boolean</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">And</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">domain</span><span class="o">.</span><span class="n">as_boolean</span><span class="p">()</span> <span class="k">for</span> <span class="n">domain</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">domains</span><span class="p">])</span>
-
-
-<span class="k">class</span> <span class="nc">ConditionalContinuousDomain</span><span class="p">(</span><span class="n">ContinuousDomain</span><span class="p">,</span> <span class="n">ConditionalDomain</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A domain with continuous support that has been further restricted by a</span>
-<span class="sd">    condition such as x &gt; 3</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">integrate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">variables</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">variables</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">variables</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">symbols</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">variables</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span>
-        <span class="c"># Extract the full integral</span>
-        <span class="n">fullintgrl</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">fulldomain</span><span class="o">.</span><span class="n">integrate</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">variables</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-        <span class="c"># separate into integrand and limits</span>
-        <span class="n">integrand</span><span class="p">,</span> <span class="n">limits</span> <span class="o">=</span> <span class="n">fullintgrl</span><span class="o">.</span><span class="n">function</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="n">fullintgrl</span><span class="o">.</span><span class="n">limits</span><span class="p">)</span>
-
-        <span class="n">conditions</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">condition</span><span class="p">]</span>
-        <span class="k">while</span> <span class="n">conditions</span><span class="p">:</span>
-            <span class="n">cond</span> <span class="o">=</span> <span class="n">conditions</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-            <span class="k">if</span> <span class="n">cond</span><span class="o">.</span><span class="n">is_Boolean</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">cond</span><span class="p">,</span> <span class="n">And</span><span class="p">):</span>
-                    <span class="n">conditions</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">cond</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">cond</span><span class="p">,</span> <span class="n">Or</span><span class="p">):</span>
-                    <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Or not implemented here&quot;</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">cond</span><span class="o">.</span><span class="n">is_Relational</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">cond</span><span class="o">.</span><span class="n">is_Equality</span><span class="p">:</span>
-                    <span class="c"># Add the appropriate Delta to the integrand</span>
-                    <span class="n">integrand</span> <span class="o">*=</span> <span class="n">DiracDelta</span><span class="p">(</span><span class="n">cond</span><span class="o">.</span><span class="n">lhs</span> <span class="o">-</span> <span class="n">cond</span><span class="o">.</span><span class="n">rhs</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">symbols</span> <span class="o">=</span> <span class="n">cond</span><span class="o">.</span><span class="n">free_symbols</span> <span class="o">&amp;</span> <span class="nb">set</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">symbols</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>  <span class="c"># Can&#39;t handle x &gt; y</span>
-                        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                            <span class="s">&quot;Multivariate Inequalities not yet implemented&quot;</span><span class="p">)</span>
-                    <span class="c"># Can handle x &gt; 0</span>
-                    <span class="n">symbol</span> <span class="o">=</span> <span class="n">symbols</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-                    <span class="c"># Find the limit with x, such as (x, -oo, oo)</span>
-                    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">limit</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">limits</span><span class="p">):</span>
-                        <span class="k">if</span> <span class="n">limit</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">symbol</span><span class="p">:</span>
-                            <span class="c"># Make condition into an Interval like [0, oo]</span>
-                            <span class="n">cintvl</span> <span class="o">=</span> <span class="n">reduce_poly_inequalities_wrap</span><span class="p">(</span>
-                                <span class="n">cond</span><span class="p">,</span> <span class="n">symbol</span><span class="p">)</span>
-                            <span class="c"># Make limit into an Interval like [-oo, oo]</span>
-                            <span class="n">lintvl</span> <span class="o">=</span> <span class="n">Interval</span><span class="p">(</span><span class="n">limit</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">limit</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-                            <span class="c"># Intersect them to get [0, oo]</span>
-                            <span class="n">intvl</span> <span class="o">=</span> <span class="n">cintvl</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="n">lintvl</span><span class="p">)</span>
-                            <span class="c"># Put back into limits list</span>
-                            <span class="n">limits</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">symbol</span><span class="p">,</span> <span class="n">intvl</span><span class="o">.</span><span class="n">left</span><span class="p">,</span> <span class="n">intvl</span><span class="o">.</span><span class="n">right</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span>
-                    <span class="s">&quot;Condition </span><span class="si">%s</span><span class="s"> is not a relational or Boolean&quot;</span> <span class="o">%</span> <span class="n">cond</span><span class="p">)</span>
-
-        <span class="n">evaluate</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;evaluate&#39;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">evaluate</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">integrate</span><span class="p">(</span><span class="n">integrand</span><span class="p">,</span> <span class="o">*</span><span class="n">limits</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Integral</span><span class="p">(</span><span class="n">integrand</span><span class="p">,</span> <span class="o">*</span><span class="n">limits</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">as_boolean</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">And</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">fulldomain</span><span class="o">.</span><span class="n">as_boolean</span><span class="p">(),</span> <span class="bp">self</span><span class="o">.</span><span class="n">condition</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">set</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">symbols</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">fulldomain</span><span class="o">.</span><span class="n">set</span> <span class="o">&amp;</span> <span class="n">reduce_poly_inequalities_wrap</span><span class="p">(</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">condition</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">symbols</span><span class="p">)[</span><span class="mi">0</span><span class="p">]))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                <span class="s">&quot;Set of Conditional Domain not Implemented&quot;</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="ContinuousPSpace"><a class="viewcode-back" href="../../../modules/stats.html#sympy.stats.crv.ContinuousPSpace">[docs]</a><span class="k">class</span> <span class="nc">ContinuousPSpace</span><span class="p">(</span><span class="n">PSpace</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A Continuous Probability Space</span>
-
-<span class="sd">    Represents the likelihood of an event space defined over a continuum.</span>
-
-<span class="sd">    Represented with a set of symbols and a probability density function.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_Continuous</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">integrate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">rvs</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">rvs</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">rvs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">values</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">rvs</span> <span class="o">=</span> <span class="nb">frozenset</span><span class="p">(</span><span class="n">rvs</span><span class="p">)</span>
-
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="nb">dict</span><span class="p">((</span><span class="n">rv</span><span class="p">,</span> <span class="n">rv</span><span class="o">.</span><span class="n">symbol</span><span class="p">)</span> <span class="k">for</span> <span class="n">rv</span> <span class="ow">in</span> <span class="n">rvs</span><span class="p">))</span>
-
-        <span class="n">domain_symbols</span> <span class="o">=</span> <span class="nb">frozenset</span><span class="p">(</span><span class="n">rv</span><span class="o">.</span><span class="n">symbol</span> <span class="k">for</span> <span class="n">rv</span> <span class="ow">in</span> <span class="n">rvs</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="o">.</span><span class="n">integrate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">density</span> <span class="o">*</span> <span class="n">expr</span><span class="p">,</span>
-                <span class="n">domain_symbols</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">compute_density</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="c"># Common case Density(X) where X in self.values</span>
-        <span class="k">if</span> <span class="n">expr</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">values</span><span class="p">:</span>
-            <span class="c"># Marginalize all other random symbols out of the density</span>
-            <span class="n">density</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="o">.</span><span class="n">integrate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">density</span><span class="p">,</span> <span class="nb">set</span><span class="p">(</span><span class="n">rs</span><span class="o">.</span><span class="n">symbol</span>
-                <span class="k">for</span> <span class="n">rs</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">values</span> <span class="o">-</span> <span class="nb">frozenset</span><span class="p">((</span><span class="n">expr</span><span class="p">,))),</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Lambda</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">symbol</span><span class="p">,</span> <span class="n">density</span><span class="p">)</span>
-
-        <span class="n">z</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;z&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">bounded</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Lambda</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">integrate</span><span class="p">(</span><span class="n">DiracDelta</span><span class="p">(</span><span class="n">expr</span> <span class="o">-</span> <span class="n">z</span><span class="p">),</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">compute_cdf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="o">.</span><span class="n">set</span><span class="o">.</span><span class="n">is_Interval</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&quot;CDF not well defined on multivariate expressions&quot;</span><span class="p">)</span>
-
-        <span class="n">d</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">compute_density</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x, z&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">bounded</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Dummy</span><span class="p">)</span>
-        <span class="n">left_bound</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="o">.</span><span class="n">set</span><span class="o">.</span><span class="n">start</span>
-
-        <span class="c"># CDF is integral of PDF from left bound to z</span>
-        <span class="n">cdf</span> <span class="o">=</span> <span class="n">integrate</span><span class="p">(</span><span class="n">d</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">left_bound</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="c"># CDF Ensure that CDF left of left_bound is zero</span>
-        <span class="n">cdf</span> <span class="o">=</span> <span class="n">Piecewise</span><span class="p">((</span><span class="n">cdf</span><span class="p">,</span> <span class="n">z</span> <span class="o">&gt;=</span> <span class="n">left_bound</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="bp">True</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">Lambda</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">cdf</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">probability</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">condition</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">z</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;z&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">bounded</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="c"># Univariate case can be handled by where</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">domain</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">condition</span><span class="p">)</span>
-            <span class="n">rv</span> <span class="o">=</span> <span class="p">[</span><span class="n">rv</span> <span class="k">for</span> <span class="n">rv</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">values</span> <span class="k">if</span> <span class="n">rv</span><span class="o">.</span><span class="n">symbol</span> <span class="o">==</span> <span class="n">domain</span><span class="o">.</span><span class="n">symbol</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-            <span class="c"># Integrate out all other random variables</span>
-            <span class="n">pdf</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">compute_density</span><span class="p">(</span><span class="n">rv</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-            <span class="c"># Integrate out the last variable over the special domain</span>
-            <span class="n">evaluate</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&quot;evaluate&quot;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">evaluate</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">integrate</span><span class="p">(</span><span class="n">pdf</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">domain</span><span class="o">.</span><span class="n">set</span><span class="p">),</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">Integral</span><span class="p">(</span><span class="n">pdf</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">domain</span><span class="o">.</span><span class="n">set</span><span class="p">),</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-        <span class="c"># Other cases can be turned into univariate case</span>
-        <span class="c"># by computing a density handled by density computation</span>
-        <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">condition</span><span class="o">.</span><span class="n">lhs</span> <span class="o">-</span> <span class="n">condition</span><span class="o">.</span><span class="n">rhs</span>
-            <span class="n">density</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">compute_density</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-            <span class="c"># Turn problem into univariate case</span>
-            <span class="n">space</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">density</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">space</span><span class="o">.</span><span class="n">probability</span><span class="p">(</span><span class="n">condition</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">space</span><span class="o">.</span><span class="n">value</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">where</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">condition</span><span class="p">):</span>
-        <span class="n">rvs</span> <span class="o">=</span> <span class="nb">frozenset</span><span class="p">(</span><span class="n">random_symbols</span><span class="p">(</span><span class="n">condition</span><span class="p">))</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">rvs</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">rvs</span><span class="o">.</span><span class="n">issubset</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">values</span><span class="p">)):</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                <span class="s">&quot;Multiple continuous random variables not supported&quot;</span><span class="p">)</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">rvs</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">interval</span> <span class="o">=</span> <span class="n">reduce_poly_inequalities_wrap</span><span class="p">(</span><span class="n">condition</span><span class="p">,</span> <span class="n">rv</span><span class="p">)</span>
-        <span class="n">interval</span> <span class="o">=</span> <span class="n">interval</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="o">.</span><span class="n">set</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">SingleContinuousDomain</span><span class="p">(</span><span class="n">rv</span><span class="o">.</span><span class="n">symbol</span><span class="p">,</span> <span class="n">interval</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">conditional_space</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">condition</span><span class="p">,</span> <span class="n">normalize</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-
-        <span class="n">condition</span> <span class="o">=</span> <span class="n">condition</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="nb">dict</span><span class="p">((</span><span class="n">rv</span><span class="p">,</span> <span class="n">rv</span><span class="o">.</span><span class="n">symbol</span><span class="p">)</span> <span class="k">for</span> <span class="n">rv</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">values</span><span class="p">))</span>
-
-        <span class="n">domain</span> <span class="o">=</span> <span class="n">ConditionalContinuousDomain</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">,</span> <span class="n">condition</span><span class="p">)</span>
-        <span class="n">density</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">density</span>
-        <span class="k">if</span> <span class="n">normalize</span><span class="p">:</span>
-            <span class="n">density</span> <span class="o">=</span> <span class="n">density</span> <span class="o">/</span> <span class="n">domain</span><span class="o">.</span><span class="n">integrate</span><span class="p">(</span><span class="n">density</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">ContinuousPSpace</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">density</span><span class="p">)</span>
-
-</div>
-<span class="k">class</span> <span class="nc">SingleContinuousPSpace</span><span class="p">(</span><span class="n">ContinuousPSpace</span><span class="p">,</span> <span class="n">SinglePSpace</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A continuous probability space over a single univariate domain</span>
-
-<span class="sd">    This class is commonly implemented by the various ContinuousRV types</span>
-<span class="sd">    such as Normal, Exponential, Uniform, etc....</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">symbol</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="o">-</span><span class="n">oo</span><span class="p">,</span> <span class="n">oo</span><span class="p">)):</span>
-        <span class="n">domain</span> <span class="o">=</span> <span class="n">SingleContinuousDomain</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="n">symbol</span><span class="p">),</span> <span class="nb">set</span><span class="p">)</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">ContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">density</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_cdf</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">_inverse_cdf_expression</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Inverse of the CDF</span>
-
-<span class="sd">        See Also</span>
-<span class="sd">        ========</span>
-<span class="sd">        compute_cdf</span>
-<span class="sd">        sample</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">compute_cdf</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">value</span><span class="p">)</span>
-        <span class="n">x</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x, z&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Dummy</span><span class="p">)</span>
-        <span class="c"># Invert CDF</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">inverse_cdf</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">d</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">z</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">NotImplementedError</span><span class="p">:</span>
-            <span class="n">inverse_cdf</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">inverse_cdf</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="n">inverse_cdf</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Could not invert CDF&quot;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">Lambda</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">inverse_cdf</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-    <span class="k">def</span> <span class="nf">sample</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Internal sample method</span>
-
-<span class="sd">        Returns dictionary mapping RandomSymbol to realization value.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">icdf</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_inverse_cdf_expression</span><span class="p">()</span>
-        <span class="k">return</span> <span class="p">{</span><span class="bp">self</span><span class="o">.</span><span class="n">value</span><span class="p">:</span> <span class="n">icdf</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))}</span>
-
-
-<span class="k">class</span> <span class="nc">ProductContinuousPSpace</span><span class="p">(</span><span class="n">ProductPSpace</span><span class="p">,</span> <span class="n">ContinuousPSpace</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A collection of independent continuous probability spaces</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">density</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">space</span><span class="o">.</span><span class="n">density</span> <span class="k">for</span> <span class="n">space</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">spaces</span><span class="p">])</span>
-
-
-<span class="k">def</span> <span class="nf">_reduce_inequalities</span><span class="p">(</span><span class="n">conditions</span><span class="p">,</span> <span class="n">var</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">reduce_poly_inequalities</span><span class="p">(</span><span class="n">conditions</span><span class="p">,</span> <span class="n">var</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">PolynomialError</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Reduction of condition failed </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">conditions</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-
-
-<span class="k">def</span> <span class="nf">reduce_poly_inequalities_wrap</span><span class="p">(</span><span class="n">condition</span><span class="p">,</span> <span class="n">var</span><span class="p">):</span>
-    <span class="k">if</span> <span class="n">condition</span><span class="o">.</span><span class="n">is_Relational</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">_reduce_inequalities</span><span class="p">([[</span><span class="n">condition</span><span class="p">]],</span> <span class="n">var</span><span class="p">,</span> <span class="n">relational</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">condition</span><span class="o">.</span><span class="n">__class__</span> <span class="ow">is</span> <span class="n">Or</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">_reduce_inequalities</span><span class="p">([</span><span class="nb">list</span><span class="p">(</span><span class="n">condition</span><span class="o">.</span><span class="n">args</span><span class="p">)],</span>
-                <span class="n">var</span><span class="p">,</span> <span class="n">relational</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">condition</span><span class="o">.</span><span class="n">__class__</span> <span class="ow">is</span> <span class="n">And</span><span class="p">:</span>
-        <span class="n">intervals</span> <span class="o">=</span> <span class="p">[</span><span class="n">_reduce_inequalities</span><span class="p">([[</span><span class="n">arg</span><span class="p">]],</span> <span class="n">var</span><span class="p">,</span> <span class="n">relational</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">condition</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-        <span class="n">I</span> <span class="o">=</span> <span class="n">intervals</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">intervals</span><span class="p">:</span>
-            <span class="n">I</span> <span class="o">=</span> <span class="n">I</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">I</span>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-<div id="searchbox" style="display: none">
-  <h3>Quick search</h3>
-    <form class="search" action="../../../search.html" method="get">
-      <input type="text" name="q" />
-      <input type="submit" value="Go" />
-      <input type="hidden" name="check_keywords" value="yes" />
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-        <li class="right" >
-          <a href="../../../np-modindex.html" title="Python Module Index"
-             >modules</a> |</li>
-        <li><a href="../../../index.html">SymPy 0.7.2-git documentation</a> &raquo;</li>
-          <li><a href="../../index.html" >Module code</a> &raquo;</li>
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-      Last updated on Jan 20, 2013.
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diff --git a/dev-py3k/_modules/sympy/stats/crv_types.html b/dev-py3k/_modules/sympy/stats/crv_types.html
deleted file mode 100644
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--- a/dev-py3k/_modules/sympy/stats/crv_types.html
+++ /dev/null
@@ -1,2601 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
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-        <div class="bodywrapper">
-          <div class="body">
-            
-  <h1>Source code for sympy.stats.crv_types</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Continuous Random Variables - Prebuilt variables</span>
-
-<span class="sd">Contains</span>
-<span class="sd">========</span>
-<span class="sd">Arcsin</span>
-<span class="sd">Benini</span>
-<span class="sd">Beta</span>
-<span class="sd">BetaPrime</span>
-<span class="sd">Cauchy</span>
-<span class="sd">Chi</span>
-<span class="sd">ChiNoncentral</span>
-<span class="sd">ChiSquared</span>
-<span class="sd">Dagum</span>
-<span class="sd">Erlang</span>
-<span class="sd">Exponential</span>
-<span class="sd">FDistribution</span>
-<span class="sd">FisherZ</span>
-<span class="sd">Frechet</span>
-<span class="sd">Gamma</span>
-<span class="sd">GammaInverse</span>
-<span class="sd">Kumaraswamy</span>
-<span class="sd">Laplace</span>
-<span class="sd">Logistic</span>
-<span class="sd">LogNormal</span>
-<span class="sd">Maxwell</span>
-<span class="sd">Nakagami</span>
-<span class="sd">Normal</span>
-<span class="sd">Pareto</span>
-<span class="sd">QuadraticU</span>
-<span class="sd">RaisedCosine</span>
-<span class="sd">Rayleigh</span>
-<span class="sd">StudentT</span>
-<span class="sd">Triangular</span>
-<span class="sd">Uniform</span>
-<span class="sd">UniformSum</span>
-<span class="sd">VonMises</span>
-<span class="sd">Weibull</span>
-<span class="sd">WignerSemicircle</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="p">(</span><span class="n">exp</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">,</span> <span class="n">Interval</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span>
-                   <span class="n">Piecewise</span><span class="p">,</span> <span class="n">And</span><span class="p">,</span> <span class="n">Eq</span><span class="p">,</span> <span class="n">binomial</span><span class="p">,</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">Sum</span><span class="p">,</span> <span class="n">floor</span><span class="p">,</span> <span class="n">Abs</span><span class="p">,</span>
-                   <span class="n">Symbol</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">besseli</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">beta</span> <span class="k">as</span> <span class="n">beta_fn</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">cos</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">besseli</span>
-<span class="kn">from</span> <span class="nn">.crv</span> <span class="kn">import</span> <span class="n">SingleContinuousPSpace</span>
-<span class="kn">from</span> <span class="nn">sympy.core.decorators</span> <span class="kn">import</span> <span class="n">_sympifyit</span>
-<span class="kn">import</span> <span class="nn">random</span>
-
-<span class="n">oo</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span>
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;ContinuousRV&#39;</span><span class="p">,</span>
-<span class="s">&#39;Arcsin&#39;</span><span class="p">,</span>
-<span class="s">&#39;Benini&#39;</span><span class="p">,</span>
-<span class="s">&#39;Beta&#39;</span><span class="p">,</span>
-<span class="s">&#39;BetaPrime&#39;</span><span class="p">,</span>
-<span class="s">&#39;Cauchy&#39;</span><span class="p">,</span>
-<span class="s">&#39;Chi&#39;</span><span class="p">,</span>
-<span class="s">&#39;ChiNoncentral&#39;</span><span class="p">,</span>
-<span class="s">&#39;ChiSquared&#39;</span><span class="p">,</span>
-<span class="s">&#39;Dagum&#39;</span><span class="p">,</span>
-<span class="s">&#39;Erlang&#39;</span><span class="p">,</span>
-<span class="s">&#39;Exponential&#39;</span><span class="p">,</span>
-<span class="s">&#39;FDistribution&#39;</span><span class="p">,</span>
-<span class="s">&#39;FisherZ&#39;</span><span class="p">,</span>
-<span class="s">&#39;Frechet&#39;</span><span class="p">,</span>
-<span class="s">&#39;Gamma&#39;</span><span class="p">,</span>
-<span class="s">&#39;GammaInverse&#39;</span><span class="p">,</span>
-<span class="s">&#39;Kumaraswamy&#39;</span><span class="p">,</span>
-<span class="s">&#39;Laplace&#39;</span><span class="p">,</span>
-<span class="s">&#39;Logistic&#39;</span><span class="p">,</span>
-<span class="s">&#39;LogNormal&#39;</span><span class="p">,</span>
-<span class="s">&#39;Maxwell&#39;</span><span class="p">,</span>
-<span class="s">&#39;Nakagami&#39;</span><span class="p">,</span>
-<span class="s">&#39;Normal&#39;</span><span class="p">,</span>
-<span class="s">&#39;Pareto&#39;</span><span class="p">,</span>
-<span class="s">&#39;QuadraticU&#39;</span><span class="p">,</span>
-<span class="s">&#39;RaisedCosine&#39;</span><span class="p">,</span>
-<span class="s">&#39;Rayleigh&#39;</span><span class="p">,</span>
-<span class="s">&#39;StudentT&#39;</span><span class="p">,</span>
-<span class="s">&#39;Triangular&#39;</span><span class="p">,</span>
-<span class="s">&#39;Uniform&#39;</span><span class="p">,</span>
-<span class="s">&#39;UniformSum&#39;</span><span class="p">,</span>
-<span class="s">&#39;VonMises&#39;</span><span class="p">,</span>
-<span class="s">&#39;Weibull&#39;</span><span class="p">,</span>
-<span class="s">&#39;WignerSemicircle&#39;</span>
-<span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">_value_check</span><span class="p">(</span><span class="n">condition</span><span class="p">,</span> <span class="n">message</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Check a condition on input value.</span>
-
-<span class="sd">    Raises ValueError with message if condition is not True</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">condition</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="n">message</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">ContinuousRV</span><span class="p">(</span><span class="n">symbol</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="o">-</span><span class="n">oo</span><span class="p">,</span> <span class="n">oo</span><span class="p">)):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create a Continuous Random Variable given the following:</span>
-
-<span class="sd">    -- a symbol</span>
-<span class="sd">    -- a probability density function</span>
-<span class="sd">    -- set on which the pdf is valid (defaults to entire real line)</span>
-
-<span class="sd">    Returns a RandomSymbol.</span>
-
-<span class="sd">    Many common continuous random variable types are already implemented.</span>
-<span class="sd">    This function should be necessary only very rarely.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, sqrt, exp, pi</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import ContinuousRV, P, E</span>
-
-<span class="sd">    &gt;&gt;&gt; x = Symbol(&quot;x&quot;)</span>
-
-<span class="sd">    &gt;&gt;&gt; pdf = sqrt(2)*exp(-x**2/2)/(2*sqrt(pi)) # Normal distribution</span>
-<span class="sd">    &gt;&gt;&gt; X = ContinuousRV(x, pdf)</span>
-
-<span class="sd">    &gt;&gt;&gt; E(X)</span>
-<span class="sd">    0</span>
-<span class="sd">    &gt;&gt;&gt; P(X&gt;0)</span>
-<span class="sd">    1/2</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">SingleContinuousPSpace</span><span class="p">(</span><span class="n">symbol</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="nb">set</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">########################################</span>
-<span class="c"># Continuous Probability Distributions #</span>
-<span class="c">########################################</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Arcsin distribution ----------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">ArcsinPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">sqrt</span><span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="n">x</span><span class="p">)))</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-
-<span class="k">def</span> <span class="nf">Arcsin</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a Continuous Random Variable with an arcsin distribution.</span>
-
-<span class="sd">    The density of the arcsin distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{1}{\pi\sqrt{(x-a)(b-x)}}</span>
-
-<span class="sd">    with :math:`x \in [a,b]`. It must hold that :math:`-\infty &lt; a &lt; b &lt; \infty`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    a : Real number, the left interval boundary</span>
-<span class="sd">    b : Real number, the right interval boundary</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Arcsin, density</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify</span>
-
-<span class="sd">    &gt;&gt;&gt; a = Symbol(&quot;a&quot;, real=True)</span>
-<span class="sd">    &gt;&gt;&gt; b = Symbol(&quot;b&quot;, real=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Arcsin(&quot;x&quot;, a, b)</span>
-
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    Lambda(_x, 1/(pi*sqrt((-_x + b)*(_x - a))))</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Arcsine_distribution</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">ArcsinPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Benini distribution ----------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">BeniniPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">sigma</span><span class="p">):</span>
-        <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">sigma</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">alpha</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">beta</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">sigma</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">alpha</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="n">sigma</span><span class="p">)</span> <span class="o">-</span> <span class="n">beta</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="n">sigma</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-               <span class="o">*</span><span class="p">(</span><span class="n">alpha</span><span class="o">/</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">beta</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="n">sigma</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">))</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span>
-                                             <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="n">sigma</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-
-<span class="k">def</span> <span class="nf">Benini</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">sigma</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a Continuous Random Variable with a Benini distribution.</span>
-
-<span class="sd">    The density of the Benini distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := e^{-\alpha\log{\frac{x}{\sigma}}</span>
-<span class="sd">                -\beta\log\left[{\frac{x}{\sigma}}\right]^2}</span>
-<span class="sd">                \left(\frac{\alpha}{x}+\frac{2\beta\log{\frac{x}{\sigma}}}{x}\right)</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    alpha : Real number, `alpha` &gt; 0 a shape</span>
-<span class="sd">    beta : Real number, `beta` &gt; 0 a shape</span>
-<span class="sd">    sigma : Real number, `sigma` &gt; 0 a scale</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Benini, density</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify, pprint</span>
-
-<span class="sd">    &gt;&gt;&gt; alpha = Symbol(&quot;alpha&quot;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; beta = Symbol(&quot;beta&quot;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; sigma = Symbol(&quot;sigma&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Benini(&quot;x&quot;, alpha, beta, sigma)</span>
-
-<span class="sd">    &gt;&gt;&gt; D = density(X)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(D, use_unicode=False)</span>
-<span class="sd">          /                                                             2       \</span>
-<span class="sd">          |   /                  /  x  \\             /  x  \            /  x  \|</span>
-<span class="sd">          |   |        2*beta*log|-----||  - alpha*log|-----| - beta*log |-----||</span>
-<span class="sd">          |   |alpha             \sigma/|             \sigma/            \sigma/|</span>
-<span class="sd">    Lambda|x, |----- + -----------------|*e                                     |</span>
-<span class="sd">          \   \  x             x        /                                       /</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Benini_distribution</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">BeniniPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">sigma</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Beta distribution ------------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">BetaPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">):</span>
-        <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">alpha</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span>
-
-        <span class="n">_value_check</span><span class="p">(</span><span class="n">alpha</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&quot;Alpha must be positive&quot;</span><span class="p">)</span>
-        <span class="n">_value_check</span><span class="p">(</span><span class="n">beta</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&quot;Beta must be positive&quot;</span><span class="p">)</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">alpha</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">beta</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="n">beta_fn</span><span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">)</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">alpha</span> <span class="o">=</span> <span class="n">alpha</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">beta</span> <span class="o">=</span> <span class="n">beta</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">sample</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">{</span><span class="bp">self</span><span class="o">.</span><span class="n">value</span><span class="p">:</span> <span class="n">random</span><span class="o">.</span><span class="n">betavariate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">alpha</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">beta</span><span class="p">)}</span>
-
-
-<span class="k">def</span> <span class="nf">Beta</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a Continuous Random Variable with a Beta distribution.</span>
-
-<span class="sd">    The density of the Beta distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{x^{\alpha-1}(1-x)^{\beta-1}} {\mathrm{B}(\alpha,\beta)}</span>
-
-<span class="sd">    with :math:`x \in [0,1]`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    alpha : Real number, `alpha` &gt; 0 a shape</span>
-<span class="sd">    beta : Real number, `beta` &gt; 0 a shape</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Beta, density, E, variance</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify, pprint</span>
-
-<span class="sd">    &gt;&gt;&gt; alpha = Symbol(&quot;alpha&quot;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; beta = Symbol(&quot;beta&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Beta(&quot;x&quot;, alpha, beta)</span>
-
-<span class="sd">    &gt;&gt;&gt; D = density(X)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(D, use_unicode=False)</span>
-<span class="sd">          /    alpha - 1         beta - 1                    \</span>
-<span class="sd">          |   x         *(-x + 1)        *gamma(alpha + beta)|</span>
-<span class="sd">    Lambda|x, -----------------------------------------------|</span>
-<span class="sd">          \               gamma(alpha)*gamma(beta)           /</span>
-
-<span class="sd">    &gt;&gt;&gt; simplify(E(X, meijerg=True))</span>
-<span class="sd">    alpha/(alpha + beta)</span>
-
-<span class="sd">    &gt;&gt;&gt; simplify(variance(X, meijerg=True))</span>
-<span class="sd">    alpha*beta/((alpha + beta)**2*(alpha + beta + 1))</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Beta_distribution</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/BetaDistribution.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">BetaPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Beta prime distribution ------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">BetaPrimePSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">):</span>
-        <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">alpha</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">alpha</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">alpha</span> <span class="o">-</span> <span class="n">beta</span><span class="p">)</span><span class="o">/</span><span class="n">beta_fn</span><span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">)</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-
-<span class="k">def</span> <span class="nf">BetaPrime</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with a Beta prime distribution.</span>
-
-<span class="sd">    The density of the Beta prime distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)}</span>
-
-<span class="sd">    with :math:`x &gt; 0`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    alpha : Real number, `alpha` &gt; 0 a shape</span>
-<span class="sd">    beta : Real number, `beta` &gt; 0 a shape</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import BetaPrime, density</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, pprint</span>
-
-<span class="sd">    &gt;&gt;&gt; alpha = Symbol(&quot;alpha&quot;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; beta = Symbol(&quot;beta&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = BetaPrime(&quot;x&quot;, alpha, beta)</span>
-
-<span class="sd">    &gt;&gt;&gt; D = density(X)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(D, use_unicode=False)</span>
-<span class="sd">          /    alpha - 1        -alpha - beta                    \</span>
-<span class="sd">          |   x         *(x + 1)             *gamma(alpha + beta)|</span>
-<span class="sd">    Lambda|x, ---------------------------------------------------|</span>
-<span class="sd">          \                 gamma(alpha)*gamma(beta)             /</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Beta_prime_distribution</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/BetaPrimeDistribution.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">BetaPrimePSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Cauchy distribution ----------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">CauchyPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">gamma</span><span class="p">):</span>
-        <span class="n">x0</span><span class="p">,</span> <span class="n">gamma</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">x0</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">gamma</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">gamma</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="n">x0</span><span class="p">)</span><span class="o">/</span><span class="n">gamma</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-
-<span class="k">def</span> <span class="nf">Cauchy</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">gamma</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with a Cauchy distribution.</span>
-
-<span class="sd">    The density of the Cauchy distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{1}{\pi} \arctan\left(\frac{x-x_0}{\gamma}\right)</span>
-<span class="sd">                +\frac{1}{2}</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    x0 : Real number, the location</span>
-<span class="sd">    gamma : Real number, `gamma` &gt; 0 the scale</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Cauchy, density</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol</span>
-
-<span class="sd">    &gt;&gt;&gt; x0 = Symbol(&quot;x0&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; gamma = Symbol(&quot;gamma&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Cauchy(&quot;x&quot;, x0, gamma)</span>
-
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    Lambda(_x, 1/(pi*gamma*(1 + (_x - x0)**2/gamma**2)))</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Cauchy_distribution</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/CauchyDistribution.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">CauchyPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">gamma</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Chi distribution -------------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">ChiPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">k</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">gamma</span><span class="p">(</span><span class="n">k</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-
-<span class="k">def</span> <span class="nf">Chi</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with a Chi distribution.</span>
-
-<span class="sd">    The density of the Chi distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}</span>
-
-<span class="sd">    with :math:`x \geq 0`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    k : Integer, `k` &gt; 0 the number of degrees of freedom</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Chi, density, E, std</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify</span>
-
-<span class="sd">    &gt;&gt;&gt; k = Symbol(&quot;k&quot;, integer=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Chi(&quot;x&quot;, k)</span>
-
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    Lambda(_x, 2**(-k/2 + 1)*_x**(k - 1)*exp(-_x**2/2)/gamma(k/2))</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Chi_distribution</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/ChiDistribution.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">ChiPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Non-central Chi distribution -------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">ChiNoncentralPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">l</span><span class="p">):</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-        <span class="n">l</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">l</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">l</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">k</span><span class="o">*</span><span class="n">l</span> <span class="o">/</span> <span class="p">(</span><span class="n">l</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">k</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">besseli</span><span class="p">(</span><span class="n">k</span><span class="o">/</span><span class="mi">2</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">l</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span> <span class="o">=</span> <span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-
-<span class="k">def</span> <span class="nf">ChiNoncentral</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">l</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with a non-central Chi distribution.</span>
-
-<span class="sd">    The density of the non-central Chi distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{e^{-(x^2+\lambda^2)/2} x^k\lambda}</span>
-<span class="sd">                {(\lambda x)^{k/2}} I_{k/2-1}(\lambda x)</span>
-
-<span class="sd">    with :math:`x \geq 0`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    k : `k` &gt; 0 the number of degrees of freedom</span>
-<span class="sd">    l : Shift parameter</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import ChiNoncentral, density, E, std</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify</span>
-
-<span class="sd">    &gt;&gt;&gt; k = Symbol(&quot;k&quot;, integer=True)</span>
-<span class="sd">    &gt;&gt;&gt; l = Symbol(&quot;l&quot;)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = ChiNoncentral(&quot;x&quot;, k, l)</span>
-
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    Lambda(_x, _x**k*l*(_x*l)**(-k/2)*exp(-_x**2/2 - l**2/2)*besseli(k/2 - 1, _x*l))</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Noncentral_chi_distribution</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">ChiNoncentralPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">l</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Chi squared distribution -----------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">ChiSquaredPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="n">k</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="n">k</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">k</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-
-<span class="k">def</span> <span class="nf">ChiSquared</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with a Chi-squared distribution.</span>
-
-<span class="sd">    The density of the Chi-squared distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{1}{2^{\frac{k}{2}}\Gamma\left(\frac{k}{2}\right)}</span>
-<span class="sd">                x^{\frac{k}{2}-1} e^{-\frac{x}{2}}</span>
-
-<span class="sd">    with :math:`x \geq 0`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    k : Integer, `k` &gt; 0 the number of degrees of freedom</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import ChiSquared, density, E, variance</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify, combsimp, expand_func</span>
-
-<span class="sd">    &gt;&gt;&gt; k = Symbol(&quot;k&quot;, integer=True, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = ChiSquared(&quot;x&quot;, k)</span>
-
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    Lambda(_x, 2**(-k/2)*_x**(k/2 - 1)*exp(-_x/2)/gamma(k/2))</span>
-
-<span class="sd">    &gt;&gt;&gt; combsimp(E(X))</span>
-<span class="sd">    k</span>
-
-<span class="sd">    &gt;&gt;&gt; simplify(expand_func(variance(X)))</span>
-<span class="sd">    2*k</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Chi_squared_distribution</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/Chi-SquaredDistribution.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">ChiSquaredPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Dagum distribution -----------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">DagumPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="n">p</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">p</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="n">a</span><span class="o">*</span><span class="n">p</span><span class="o">/</span><span class="n">x</span><span class="o">*</span><span class="p">((</span><span class="n">x</span><span class="o">/</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">p</span><span class="p">)</span><span class="o">/</span><span class="p">(((</span><span class="n">x</span><span class="o">/</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">p</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)))</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-
-<span class="k">def</span> <span class="nf">Dagum</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with a Dagum distribution.</span>
-
-<span class="sd">    The density of the Dagum distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{a p}{x} \left( \frac{(\tfrac{x}{b})^{a p}}</span>
-<span class="sd">                {\left((\tfrac{x}{b})^a + 1 \right)^{p+1}} \right)</span>
-
-<span class="sd">    with :math:`x &gt; 0`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    p : Real number, `p` &gt; 0 a shape</span>
-<span class="sd">    a : Real number, `a` &gt; 0 a shape</span>
-<span class="sd">    b : Real number, `b` &gt; 0 a scale</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Dagum, density</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify</span>
-
-<span class="sd">    &gt;&gt;&gt; p = Symbol(&quot;p&quot;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; b = Symbol(&quot;b&quot;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; a = Symbol(&quot;a&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Dagum(&quot;x&quot;, p, a, b)</span>
-
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    Lambda(_x, a*p*(_x/b)**(a*p)*((_x/b)**a + 1)**(-p - 1)/_x)</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Dagum_distribution</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">DagumPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Erlang distribution ----------------------------------------------------------</span>
-
-<span class="k">def</span> <span class="nf">Erlang</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">l</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with an Erlang distribution.</span>
-
-<span class="sd">    The density of the Erlang distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{\lambda^k x^{k-1} e^{-\lambda x}}{(k-1)!}</span>
-
-<span class="sd">    with :math:`x \in [0,\infty]`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    k : Integer</span>
-<span class="sd">    l : Real number, :math:`\lambda` &gt; 0 the rate</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Erlang, density, cdf, E, variance</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify, pprint</span>
-
-<span class="sd">    &gt;&gt;&gt; k = Symbol(&quot;k&quot;, integer=True, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; l = Symbol(&quot;l&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Erlang(&quot;x&quot;, k, l)</span>
-
-<span class="sd">    &gt;&gt;&gt; D = density(X)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(D, use_unicode=False)</span>
-<span class="sd">          /    k - 1  k  -x*l\</span>
-<span class="sd">          |   x     *l *e    |</span>
-<span class="sd">    Lambda|x, ---------------|</span>
-<span class="sd">          \       gamma(k)   /</span>
-
-<span class="sd">    &gt;&gt;&gt; C = cdf(X, meijerg=True)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(C, use_unicode=False)</span>
-<span class="sd">          /   /  k*lowergamma(k, 0)   k*lowergamma(k, z*l)            \</span>
-<span class="sd">    Lambda|z, |- ------------------ + --------------------  for z &gt;= 0|</span>
-<span class="sd">          |   &lt;     gamma(k + 1)          gamma(k + 1)                |</span>
-<span class="sd">          |   |                                                       |</span>
-<span class="sd">          \   \                     0                       otherwise /</span>
-
-<span class="sd">    &gt;&gt;&gt; simplify(E(X))</span>
-<span class="sd">    k/l</span>
-
-<span class="sd">    &gt;&gt;&gt; simplify(variance(X))</span>
-<span class="sd">    (gamma(k)*gamma(k + 2) - gamma(k + 1)**2)/(l**2*gamma(k)**2)</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Erlang_distribution</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/ErlangDistribution.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">GammaPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">l</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Exponential distribution -----------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">ExponentialPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">rate</span><span class="p">):</span>
-        <span class="n">rate</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">rate</span><span class="p">)</span>
-
-        <span class="n">_value_check</span><span class="p">(</span><span class="n">rate</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&quot;Rate must be positive.&quot;</span><span class="p">)</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="n">rate</span> <span class="o">*</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">rate</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">rate</span> <span class="o">=</span> <span class="n">rate</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">sample</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">{</span><span class="bp">self</span><span class="o">.</span><span class="n">value</span><span class="p">:</span> <span class="n">random</span><span class="o">.</span><span class="n">expovariate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">rate</span><span class="p">)}</span>
-
-
-<span class="k">def</span> <span class="nf">Exponential</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">rate</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with an Exponential distribution.</span>
-
-<span class="sd">    The density of the exponential distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \lambda \exp(-\lambda x)</span>
-
-<span class="sd">    with :math:`x &gt; 0`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    rate : Real number, `rate` &gt; 0 the rate or inverse scale</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Exponential, density, cdf, E</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import variance, std, skewness</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol</span>
-
-<span class="sd">    &gt;&gt;&gt; l = Symbol(&quot;lambda&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Exponential(&quot;x&quot;, l)</span>
-
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    Lambda(_x, lambda*exp(-_x*lambda))</span>
-
-<span class="sd">    &gt;&gt;&gt; cdf(X)</span>
-<span class="sd">    Lambda(_z, Piecewise((1 - exp(-_z*lambda), _z &gt;= 0), (0, True)))</span>
-
-<span class="sd">    &gt;&gt;&gt; E(X)</span>
-<span class="sd">    1/lambda</span>
-
-<span class="sd">    &gt;&gt;&gt; variance(X)</span>
-<span class="sd">    lambda**(-2)</span>
-
-<span class="sd">    &gt;&gt;&gt; skewness(X)</span>
-<span class="sd">    2</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Exponential(&#39;x&#39;, 10)</span>
-
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    Lambda(_x, 10*exp(-10*_x))</span>
-
-<span class="sd">    &gt;&gt;&gt; E(X)</span>
-<span class="sd">    1/10</span>
-
-<span class="sd">    &gt;&gt;&gt; std(X)</span>
-<span class="sd">    1/10</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Exponential_distribution</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/ExponentialDistribution.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">ExponentialPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">rate</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># F distribution ---------------------------------------------------------------</span>
-
-<span class="k">class</span> <span class="nc">FDistributionPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">d1</span><span class="p">,</span> <span class="n">d2</span><span class="p">):</span>
-        <span class="n">d1</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">d1</span><span class="p">)</span>
-        <span class="n">d2</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">d2</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="p">(</span><span class="n">sqrt</span><span class="p">((</span><span class="n">d1</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="n">d1</span><span class="o">*</span><span class="n">d2</span><span class="o">**</span><span class="n">d2</span> <span class="o">/</span> <span class="p">(</span><span class="n">d1</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="n">d2</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">d1</span><span class="o">+</span><span class="n">d2</span><span class="p">))</span>
-               <span class="o">/</span> <span class="p">(</span><span class="n">x</span> <span class="o">*</span> <span class="n">beta_fn</span><span class="p">(</span><span class="n">d1</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">d2</span><span class="o">/</span><span class="mi">2</span><span class="p">)))</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-<span class="k">def</span> <span class="nf">FDistribution</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">d1</span><span class="p">,</span> <span class="n">d2</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with a F distribution.</span>
-
-<span class="sd">    The density of the F distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{\sqrt{\frac{(d_1 x)^{d_1} d_2^{d_2}}</span>
-<span class="sd">                {(d_1 x + d_2)^{d_1 + d_2}}}}</span>
-<span class="sd">                {x \mathrm{B} \left(\frac{d_1}{2}, \frac{d_2}{2}\right)}</span>
-
-<span class="sd">    with :math:`x &gt; 0`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    d1 : `d1` &gt; 0 a parameter</span>
-<span class="sd">    d2 : `d2` &gt; 0 a parameter</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import FDistribution, density</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify, pprint</span>
-
-<span class="sd">    &gt;&gt;&gt; d1 = Symbol(&quot;d1&quot;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; d2 = Symbol(&quot;d2&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = FDistribution(&quot;x&quot;, d1, d2)</span>
-
-<span class="sd">    &gt;&gt;&gt; D = density(X)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(D, use_unicode=False)</span>
-<span class="sd">          /     d2                                                 \</span>
-<span class="sd">          |     --    ______________________________               |</span>
-<span class="sd">          |     2    /       d1            -d1 - d2       /d1   d2\|</span>
-<span class="sd">          |   d2  *\/  (x*d1)  *(x*d1 + d2)         *gamma|-- + --||</span>
-<span class="sd">          |                                               \2    2 /|</span>
-<span class="sd">    Lambda|x, -----------------------------------------------------|</span>
-<span class="sd">          |                          /d1\      /d2\                |</span>
-<span class="sd">          |                   x*gamma|--|*gamma|--|                |</span>
-<span class="sd">          \                          \2 /      \2 /                /</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/F-distribution</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/F-Distribution.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">FDistributionPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">d1</span><span class="p">,</span> <span class="n">d2</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Fisher Z distribution --------------------------------------------------------</span>
-
-<span class="k">class</span> <span class="nc">FisherZPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">d1</span><span class="p">,</span> <span class="n">d2</span><span class="p">):</span>
-        <span class="n">d1</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">d1</span><span class="p">)</span>
-        <span class="n">d2</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">d2</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">d1</span><span class="o">**</span><span class="p">(</span><span class="n">d1</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">d2</span><span class="o">**</span><span class="p">(</span><span class="n">d2</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="n">beta_fn</span><span class="p">(</span><span class="n">d1</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">d2</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span>
-               <span class="n">exp</span><span class="p">(</span><span class="n">d1</span><span class="o">*</span><span class="n">x</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">d1</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">+</span><span class="n">d2</span><span class="p">)</span><span class="o">**</span><span class="p">((</span><span class="n">d1</span><span class="o">+</span><span class="n">d2</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-<span class="k">def</span> <span class="nf">FisherZ</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">d1</span><span class="p">,</span> <span class="n">d2</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a Continuous Random Variable with an Fisher&#39;s Z distribution.</span>
-
-<span class="sd">    The density of the Fisher&#39;s Z distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{2d_1^{d_1/2} d_2^{d_2/2}} {\mathrm{B}(d_1/2, d_2/2)}</span>
-<span class="sd">                \frac{e^{d_1z}}{\left(d_1e^{2z}+d_2\right)^{\left(d_1+d_2\right)/2}}</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    d1 : `d1` &gt; 0, degree of freedom</span>
-<span class="sd">    d2 : `d2` &gt; 0, degree of freedom</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import FisherZ, density</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify, pprint</span>
-
-<span class="sd">    &gt;&gt;&gt; d1 = Symbol(&quot;d1&quot;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; d2 = Symbol(&quot;d2&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = FisherZ(&quot;x&quot;, d1, d2)</span>
-
-<span class="sd">    &gt;&gt;&gt; D = density(X)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(D, use_unicode=False)</span>
-<span class="sd">          /                               d1   d2                     \</span>
-<span class="sd">          |       d1   d2               - -- - --                     |</span>
-<span class="sd">          |       --   --                 2    2                      |</span>
-<span class="sd">          |       2    2  /    2*x     \           x*d1      /d1   d2\|</span>
-<span class="sd">          |   2*d1  *d2  *\d1*e    + d2/         *e    *gamma|-- + --||</span>
-<span class="sd">          |                                                  \2    2 /|</span>
-<span class="sd">    Lambda|x, --------------------------------------------------------|</span>
-<span class="sd">          |                          /d1\      /d2\                   |</span>
-<span class="sd">          |                     gamma|--|*gamma|--|                   |</span>
-<span class="sd">          \                          \2 /      \2 /                   /</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Fisher%27s_z-distribution</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/Fishersz-Distribution.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">FisherZPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">d1</span><span class="p">,</span> <span class="n">d2</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Frechet distribution ---------------------------------------------------------</span>
-
-<span class="k">class</span> <span class="nc">FrechetPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">m</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="n">s</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="n">a</span><span class="o">/</span><span class="n">s</span> <span class="o">*</span> <span class="p">((</span><span class="n">x</span><span class="o">-</span><span class="n">m</span><span class="p">)</span><span class="o">/</span><span class="n">s</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="o">-</span><span class="n">a</span><span class="p">)</span> <span class="o">*</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">((</span><span class="n">x</span><span class="o">-</span><span class="n">m</span><span class="p">)</span><span class="o">/</span><span class="n">s</span><span class="p">)</span><span class="o">-</span><span class="n">a</span><span class="p">)</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span> <span class="o">=</span> <span class="n">Interval</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-<span class="k">def</span> <span class="nf">Frechet</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with a Frechet distribution.</span>
-
-<span class="sd">    The density of the Frechet distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{\alpha}{s} \left(\frac{x-m}{s}\right)^{-1-\alpha}</span>
-<span class="sd">                 e^{-(\frac{x-m}{s})^{-\alpha}}</span>
-
-<span class="sd">    with :math:`x \geq m`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    a : Real number, :math:`a \in \left(0, \infty\right)` the shape</span>
-<span class="sd">    s : Real number, :math:`s \in \left(0, \infty\right)` the scale</span>
-<span class="sd">    m : Real number, :math:`m \in \left(-\infty, \infty\right)` the minimum</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Frechet, density, E, std</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify</span>
-
-<span class="sd">    &gt;&gt;&gt; a = Symbol(&quot;a&quot;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; s = Symbol(&quot;s&quot;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; m = Symbol(&quot;m&quot;, real=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Frechet(&quot;x&quot;, a, s, m)</span>
-
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    Lambda(_x, a*((_x - m)/s)**(-a - 1)*exp(-a - (_x - m)/s)/s)</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">FrechetPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Gamma distribution -----------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">GammaPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">theta</span><span class="p">):</span>
-        <span class="n">k</span><span class="p">,</span> <span class="n">theta</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">k</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
-
-        <span class="n">_value_check</span><span class="p">(</span><span class="n">k</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&quot;k must be positive&quot;</span><span class="p">)</span>
-        <span class="n">_value_check</span><span class="p">(</span><span class="n">theta</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&quot;Theta must be positive&quot;</span><span class="p">)</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">k</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">/</span><span class="n">theta</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">gamma</span><span class="p">(</span><span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="n">theta</span><span class="o">**</span><span class="n">k</span><span class="p">)</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">k</span> <span class="o">=</span> <span class="n">k</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">theta</span> <span class="o">=</span> <span class="n">theta</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">sample</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">{</span><span class="bp">self</span><span class="o">.</span><span class="n">value</span><span class="p">:</span> <span class="n">random</span><span class="o">.</span><span class="n">gammavariate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">k</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">theta</span><span class="p">)}</span>
-
-
-<span class="k">def</span> <span class="nf">Gamma</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">theta</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with a Gamma distribution.</span>
-
-<span class="sd">    The density of the Gamma distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{1}{\Gamma(k) \theta^k} x^{k - 1} e^{-\frac{x}{\theta}}</span>
-
-<span class="sd">    with :math:`x \in [0,1]`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    k : Real number, `k` &gt; 0 a shape</span>
-<span class="sd">    theta : Real number, `theta` &gt; 0 a scale</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Gamma, density, cdf, E, variance</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, pprint</span>
-
-<span class="sd">    &gt;&gt;&gt; k = Symbol(&quot;k&quot;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; theta = Symbol(&quot;theta&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Gamma(&quot;x&quot;, k, theta)</span>
-
-<span class="sd">    &gt;&gt;&gt; D = density(X)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(D, use_unicode=False)</span>
-<span class="sd">          /                     -x \</span>
-<span class="sd">          |                   -----|</span>
-<span class="sd">          |    k - 1      -k  theta|</span>
-<span class="sd">          |   x     *theta  *e     |</span>
-<span class="sd">    Lambda|x, ---------------------|</span>
-<span class="sd">          \          gamma(k)      /</span>
-
-<span class="sd">    &gt;&gt;&gt; C = cdf(X, meijerg=True)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(C, use_unicode=False)</span>
-<span class="sd">          /   /                                   /     z  \            \</span>
-<span class="sd">          |   |                       k*lowergamma|k, -----|            |</span>
-<span class="sd">          |   |  k*lowergamma(k, 0)               \   theta/            |</span>
-<span class="sd">    Lambda|z, &lt;- ------------------ + ----------------------  for z &gt;= 0|</span>
-<span class="sd">          |   |     gamma(k + 1)           gamma(k + 1)                 |</span>
-<span class="sd">          |   |                                                         |</span>
-<span class="sd">          \   \                      0                        otherwise /</span>
-
-<span class="sd">    &gt;&gt;&gt; E(X)</span>
-<span class="sd">    theta*gamma(k + 1)/gamma(k)</span>
-
-<span class="sd">    &gt;&gt;&gt; V = variance(X)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(V, use_unicode=False)</span>
-<span class="sd">           2      2                     -k      k + 1</span>
-<span class="sd">      theta *gamma (k + 1)   theta*theta  *theta     *gamma(k + 2)</span>
-<span class="sd">    - -------------------- + -------------------------------------</span>
-<span class="sd">                2                           gamma(k)</span>
-<span class="sd">           gamma (k)</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Gamma_distribution</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/GammaDistribution.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">GammaPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Inverse Gamma distribution ---------------------------------------------------</span>
-
-<span class="k">class</span> <span class="nc">GammaInversePSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-        <span class="n">b</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="n">_value_check</span><span class="p">(</span><span class="n">a</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&quot;alpha must be positive&quot;</span><span class="p">)</span>
-        <span class="n">_value_check</span><span class="p">(</span><span class="n">b</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&quot;beta must be positive&quot;</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="n">b</span><span class="o">**</span><span class="n">a</span><span class="o">/</span><span class="n">gamma</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">b</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">a</span> <span class="o">=</span> <span class="n">a</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">b</span> <span class="o">=</span> <span class="n">b</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-<span class="k">def</span> <span class="nf">GammaInverse</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with an inverse Gamma distribution.</span>
-
-<span class="sd">    The density of the inverse Gamma distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1}</span>
-<span class="sd">                \exp\left(\frac{-\beta}{x}\right)</span>
-
-<span class="sd">    with :math:`x &gt; 0`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    a : Real number, `a` &gt; 0 a shape</span>
-<span class="sd">    b : Real number, `b` &gt; 0 a scale</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import GammaInverse, density, cdf, E, variance</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, pprint</span>
-
-<span class="sd">    &gt;&gt;&gt; a = Symbol(&quot;a&quot;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; b = Symbol(&quot;b&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = GammaInverse(&quot;x&quot;, a, b)</span>
-
-<span class="sd">    &gt;&gt;&gt; D = density(X)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(D, use_unicode=False)</span>
-<span class="sd">          /               -b\</span>
-<span class="sd">          |               --|</span>
-<span class="sd">          |    -a - 1  a  x |</span>
-<span class="sd">          |   x      *b *e  |</span>
-<span class="sd">    Lambda|x, --------------|</span>
-<span class="sd">          \      gamma(a)   /</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Inverse-gamma_distribution</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">GammaInversePSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Kumaraswamy distribution -----------------------------------------------------</span>
-
-<span class="k">class</span> <span class="nc">KumaraswamyPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-
-        <span class="n">_value_check</span><span class="p">(</span><span class="n">a</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&quot;a must be positive&quot;</span><span class="p">)</span>
-        <span class="n">_value_check</span><span class="p">(</span><span class="n">b</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&quot;b must be positive&quot;</span><span class="p">)</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="n">a</span> <span class="o">*</span> <span class="n">b</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">a</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="n">a</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">a</span> <span class="o">=</span> <span class="n">a</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">b</span> <span class="o">=</span> <span class="n">b</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-<span class="k">def</span> <span class="nf">Kumaraswamy</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a Continuous Random Variable with a Kumaraswamy distribution.</span>
-
-<span class="sd">    The density of the Kumaraswamy distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := a b x^{a-1} (1-x^a)^{b-1}</span>
-
-<span class="sd">    with :math:`x \in [0,1]`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    a : Real number, `a` &gt; 0 a shape</span>
-<span class="sd">    b : Real number, `b` &gt; 0 a shape</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Kumaraswamy, density, E, variance</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify, pprint</span>
-
-<span class="sd">    &gt;&gt;&gt; a = Symbol(&quot;a&quot;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; b = Symbol(&quot;b&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Kumaraswamy(&quot;x&quot;, a, b)</span>
-
-<span class="sd">    &gt;&gt;&gt; D = density(X)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(D, use_unicode=False)</span>
-<span class="sd">          /                        b - 1\</span>
-<span class="sd">          |    a - 1     /   a    \     |</span>
-<span class="sd">    Lambda\x, x     *a*b*\- x  + 1/     /</span>
-
-<span class="sd">    &gt;&gt;&gt; simplify(E(X, meijerg=True))</span>
-<span class="sd">    gamma(1 + 1/a)*gamma(b + 1)/gamma(b + 1 + 1/a)</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Kumaraswamy_distribution</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">KumaraswamyPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Laplace distribution ---------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">LaplacePSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="n">mu</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">mu</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">Abs</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">mu</span><span class="p">)</span><span class="o">/</span><span class="n">b</span><span class="p">)</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-
-<span class="k">def</span> <span class="nf">Laplace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with a Laplace distribution.</span>
-
-<span class="sd">    The density of the Laplace distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{1}{2 b} \exp \left(-\frac{|x-\mu|}b \right)</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    mu : Real number, the location</span>
-<span class="sd">    b : Real number, `b` &gt; 0 a scale</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Laplace, density</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol</span>
-
-<span class="sd">    &gt;&gt;&gt; mu = Symbol(&quot;mu&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; b = Symbol(&quot;b&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Laplace(&quot;x&quot;, mu, b)</span>
-
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    Lambda(_x, exp(-Abs(_x - mu)/b)/(2*b))</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Laplace_distribution</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/LaplaceDistribution.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">LaplacePSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Logistic distribution --------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">LogisticPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="n">mu</span><span class="p">,</span> <span class="n">s</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">mu</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">mu</span><span class="p">)</span><span class="o">/</span><span class="n">s</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">s</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">mu</span><span class="p">)</span><span class="o">/</span><span class="n">s</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-
-<span class="k">def</span> <span class="nf">Logistic</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with a logistic distribution.</span>
-
-<span class="sd">    The density of the logistic distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{e^{-(x-\mu)/s}} {s\left(1+e^{-(x-\mu)/s}\right)^2}</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    mu : Real number, the location</span>
-<span class="sd">    s : Real number, `s` &gt; 0 a scale</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Logistic, density</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol</span>
-
-<span class="sd">    &gt;&gt;&gt; mu = Symbol(&quot;mu&quot;, real=True)</span>
-<span class="sd">    &gt;&gt;&gt; s = Symbol(&quot;s&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Logistic(&quot;x&quot;, mu, s)</span>
-
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    Lambda(_x, exp((-_x + mu)/s)/(s*(exp((-_x + mu)/s) + 1)**2))</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Logistic_distribution</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/LogisticDistribution.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">LogisticPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Log Normal distribution ------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">LogNormalPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">mean</span><span class="p">,</span> <span class="n">std</span><span class="p">):</span>
-        <span class="n">mean</span><span class="p">,</span> <span class="n">std</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">mean</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">std</span><span class="p">)</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">mean</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">std</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span> <span class="o">/</span> <span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">std</span><span class="p">)</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">mean</span> <span class="o">=</span> <span class="n">mean</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">std</span> <span class="o">=</span> <span class="n">std</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">sample</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">{</span><span class="bp">self</span><span class="o">.</span><span class="n">value</span><span class="p">:</span> <span class="n">random</span><span class="o">.</span><span class="n">lognormvariate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mean</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">std</span><span class="p">)}</span>
-
-
-<span class="k">def</span> <span class="nf">LogNormal</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">mean</span><span class="p">,</span> <span class="n">std</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with a log-normal distribution.</span>
-
-<span class="sd">    The density of the log-normal distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{1}{x\sqrt{2\pi\sigma^2}}</span>
-<span class="sd">                e^{-\frac{\left(\ln x-\mu\right)^2}{2\sigma^2}}</span>
-
-<span class="sd">    with :math:`x \geq 0`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    mu : Real number, the log-scale</span>
-<span class="sd">    sigma : Real number, :math:`\sigma^2 &gt; 0` a shape</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import LogNormal, density</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify, pprint</span>
-
-<span class="sd">    &gt;&gt;&gt; mu = Symbol(&quot;mu&quot;, real=True)</span>
-<span class="sd">    &gt;&gt;&gt; sigma = Symbol(&quot;sigma&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = LogNormal(&quot;x&quot;, mu, sigma)</span>
-
-<span class="sd">    &gt;&gt;&gt; D = density(X)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(D, use_unicode=False)</span>
-<span class="sd">          /                         2\</span>
-<span class="sd">          |          -(-mu + log(x)) |</span>
-<span class="sd">          |          ----------------|</span>
-<span class="sd">          |                     2    |</span>
-<span class="sd">          |     ___      2*sigma     |</span>
-<span class="sd">          |   \/ 2 *e                |</span>
-<span class="sd">    Lambda|x, -----------------------|</span>
-<span class="sd">          |             ____         |</span>
-<span class="sd">          \       2*x*\/ pi *sigma   /</span>
-
-<span class="sd">    &gt;&gt;&gt; X = LogNormal(&#39;x&#39;, 0, 1) # Mean 0, standard deviation 1</span>
-
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    Lambda(_x, sqrt(2)*exp(-log(_x)**2/2)/(2*_x*sqrt(pi)))</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Lognormal</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/LogNormalDistribution.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">LogNormalPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">mean</span><span class="p">,</span> <span class="n">std</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Maxwell distribution ---------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">MaxwellPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
-        <span class="n">a</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-
-        <span class="n">pdf</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">/</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span><span class="o">/</span><span class="n">a</span><span class="o">**</span><span class="mi">3</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-
-<span class="k">def</span> <span class="nf">Maxwell</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with a Maxwell distribution.</span>
-
-<span class="sd">    The density of the Maxwell distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \sqrt{\frac{2}{\pi}} \frac{x^2 e^{-x^2/(2a^2)}}{a^3}</span>
-
-<span class="sd">    with :math:`x \geq 0`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    a : Real number, `a` &gt; 0</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Maxwell, density, E, variance</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify</span>
-
-<span class="sd">    &gt;&gt;&gt; a = Symbol(&quot;a&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Maxwell(&quot;x&quot;, a)</span>
-
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    Lambda(_x, sqrt(2)*_x**2*exp(-_x**2/(2*a**2))/(sqrt(pi)*a**3))</span>
-
-<span class="sd">    &gt;&gt;&gt; E(X)</span>
-<span class="sd">    2*sqrt(2)*a/sqrt(pi)</span>
-
-<span class="sd">    &gt;&gt;&gt; simplify(variance(X))</span>
-<span class="sd">    a**2*(-8 + 3*pi)/pi</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Maxwell_distribution</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/MaxwellDistribution.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">MaxwellPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Nakagami distribution --------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">NakagamiPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">omega</span><span class="p">):</span>
-        <span class="n">mu</span><span class="p">,</span> <span class="n">omega</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">mu</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">omega</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">mu</span><span class="o">**</span><span class="n">mu</span><span class="o">/</span><span class="p">(</span><span class="n">gamma</span><span class="p">(</span><span class="n">mu</span><span class="p">)</span><span class="o">*</span><span class="n">omega</span><span class="o">**</span><span class="n">mu</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">mu</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">mu</span><span class="o">/</span><span class="n">omega</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-
-<span class="k">def</span> <span class="nf">Nakagami</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">omega</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with a Nakagami distribution.</span>
-
-<span class="sd">    The density of the Nakagami distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{2\mu^\mu}{\Gamma(\mu)\omega^\mu} x^{2\mu-1}</span>
-<span class="sd">                \exp\left(-\frac{\mu}{\omega}x^2 \right)</span>
-
-<span class="sd">    with :math:`x &gt; 0`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    mu : Real number, :math:`mu \geq \frac{1}{2}` a shape</span>
-<span class="sd">    omega : Real number, `omega` &gt; 0 the spread</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Nakagami, density, E, variance</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify, pprint</span>
-
-<span class="sd">    &gt;&gt;&gt; mu = Symbol(&quot;mu&quot;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; omega = Symbol(&quot;omega&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Nakagami(&quot;x&quot;, mu, omega)</span>
-
-<span class="sd">    &gt;&gt;&gt; D = density(X)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(D, use_unicode=False)</span>
-<span class="sd">          /                                2   \</span>
-<span class="sd">          |                              -x *mu|</span>
-<span class="sd">          |                              ------|</span>
-<span class="sd">          |      2*mu - 1   mu      -mu  omega |</span>
-<span class="sd">          |   2*x        *mu  *omega   *e      |</span>
-<span class="sd">    Lambda|x, ---------------------------------|</span>
-<span class="sd">          \               gamma(mu)            /</span>
-
-<span class="sd">    &gt;&gt;&gt; simplify(E(X, meijerg=True))</span>
-<span class="sd">    sqrt(mu)*sqrt(omega)*gamma(mu + 1/2)/gamma(mu + 1)</span>
-
-<span class="sd">    &gt;&gt;&gt; V = simplify(variance(X, meijerg=True))</span>
-<span class="sd">    &gt;&gt;&gt; pprint(V, use_unicode=False)</span>
-<span class="sd">          /                               2          \</span>
-<span class="sd">    omega*\gamma(mu)*gamma(mu + 1) - gamma (mu + 1/2)/</span>
-<span class="sd">    --------------------------------------------------</span>
-<span class="sd">                 gamma(mu)*gamma(mu + 1)</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Nakagami_distribution</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">NakagamiPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">omega</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Normal distribution ----------------------------------------------------------</span>
-
-
-<div class="viewcode-block" id="NormalPSpace"><a class="viewcode-back" href="../../../modules/stats.html#sympy.stats.crv_types.NormalPSpace">[docs]</a><span class="k">class</span> <span class="nc">NormalPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">mean</span><span class="p">,</span> <span class="n">std</span><span class="p">):</span>
-        <span class="n">mean</span><span class="p">,</span> <span class="n">std</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">mean</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">std</span><span class="p">)</span>
-
-        <span class="n">_value_check</span><span class="p">(</span><span class="n">std</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&quot;Standard deviation must be positive&quot;</span><span class="p">)</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">mean</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">std</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span> <span class="o">/</span> <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">std</span><span class="p">)</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">mean</span> <span class="o">=</span> <span class="n">mean</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">std</span> <span class="o">=</span> <span class="n">std</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">variance</span> <span class="o">=</span> <span class="n">std</span><span class="o">**</span><span class="mi">2</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">sample</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">{</span><span class="bp">self</span><span class="o">.</span><span class="n">value</span><span class="p">:</span> <span class="n">random</span><span class="o">.</span><span class="n">normalvariate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">mean</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">std</span><span class="p">)}</span>
-
-</div>
-<span class="k">def</span> <span class="nf">Normal</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">mean</span><span class="p">,</span> <span class="n">std</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with a Normal distribution.</span>
-
-<span class="sd">    The density of the Normal distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{(x-\mu)^2}{2\sigma^2} }</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    mu : Real number, the mean</span>
-<span class="sd">    sigma : Real number, :math:`\sigma^2 &gt; 0` the variance</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Normal, density, E, std, cdf, skewness</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify, pprint, factor, together</span>
-
-<span class="sd">    &gt;&gt;&gt; mu = Symbol(&quot;mu&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; sigma = Symbol(&quot;sigma&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Normal(&quot;x&quot;, mu, sigma)</span>
-
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    Lambda(_x, sqrt(2)*exp(-(_x - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma))</span>
-
-<span class="sd">    &gt;&gt;&gt; C = simplify(cdf(X)) # it needs a little more help...</span>
-<span class="sd">    &gt;&gt;&gt; C = C.func(C.args[0], together(factor(C.args[1]), deep=True))</span>
-<span class="sd">    &gt;&gt;&gt; pprint(C, use_unicode=False)</span>
-<span class="sd">              /      /  ___         \    \</span>
-<span class="sd">              |      |\/ 2 *(z - mu)|    |</span>
-<span class="sd">              |   erf|--------------|    |</span>
-<span class="sd">              |      \   2*sigma    /   1|</span>
-<span class="sd">        Lambda|z, ------------------- + -|</span>
-<span class="sd">              \            2            2/</span>
-
-<span class="sd">    &gt;&gt;&gt; simplify(skewness(X))</span>
-<span class="sd">    0</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Normal(&quot;x&quot;, 0, 1) # Mean 0, standard deviation 1</span>
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    Lambda(_x, sqrt(2)*exp(-_x**2/2)/(2*sqrt(pi)))</span>
-
-<span class="sd">    &gt;&gt;&gt; E(2*X + 1)</span>
-<span class="sd">    1</span>
-
-<span class="sd">    &gt;&gt;&gt; simplify(std(2*X + 1))</span>
-<span class="sd">    2</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Normal_distribution</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/NormalDistributionFunction.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">NormalPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">mean</span><span class="p">,</span> <span class="n">std</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Pareto distribution ----------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">ParetoPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">xm</span><span class="p">,</span> <span class="n">alpha</span><span class="p">):</span>
-        <span class="n">xm</span><span class="p">,</span> <span class="n">alpha</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">xm</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">alpha</span><span class="p">)</span>
-
-        <span class="n">_value_check</span><span class="p">(</span><span class="n">xm</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&quot;Xm must be positive&quot;</span><span class="p">)</span>
-        <span class="n">_value_check</span><span class="p">(</span><span class="n">alpha</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&quot;Alpha must be positive&quot;</span><span class="p">)</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="n">alpha</span> <span class="o">*</span> <span class="n">xm</span><span class="o">**</span><span class="n">alpha</span> <span class="o">/</span> <span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">alpha</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="n">xm</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">xm</span> <span class="o">=</span> <span class="n">xm</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">alpha</span> <span class="o">=</span> <span class="n">alpha</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">sample</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">{</span><span class="bp">self</span><span class="o">.</span><span class="n">value</span><span class="p">:</span> <span class="n">random</span><span class="o">.</span><span class="n">paretovariate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">alpha</span><span class="p">)}</span>
-
-
-<span class="k">def</span> <span class="nf">Pareto</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">xm</span><span class="p">,</span> <span class="n">alpha</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with the Pareto distribution.</span>
-
-<span class="sd">    The density of the Pareto distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{\alpha\,x_\mathrm{m}^\alpha}{x^{\alpha+1}}</span>
-
-<span class="sd">    with :math:`x \in [x_m,\infty]`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    xm : Real number, `xm` &gt; 0 a scale</span>
-<span class="sd">    alpha : Real number, `alpha` &gt; 0 a shape</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Pareto, density</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol</span>
-
-<span class="sd">    &gt;&gt;&gt; xm = Symbol(&quot;xm&quot;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; beta = Symbol(&quot;beta&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Pareto(&quot;x&quot;, xm, beta)</span>
-
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    Lambda(_x, _x**(-beta - 1)*beta*xm**beta)</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Pareto_distribution</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/ParetoDistribution.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">ParetoPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">xm</span><span class="p">,</span> <span class="n">alpha</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># QuadraticU distribution ------------------------------------------------------</span>
-
-<span class="k">class</span> <span class="nc">QuadraticUPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="n">alpha</span> <span class="o">=</span> <span class="mi">12</span> <span class="o">/</span> <span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="p">)</span><span class="o">**</span><span class="mi">3</span>
-        <span class="n">beta</span> <span class="o">=</span> <span class="p">(</span><span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="n">Piecewise</span><span class="p">(</span>
-                <span class="p">(</span><span class="n">alpha</span> <span class="o">*</span> <span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="n">beta</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">And</span><span class="p">(</span><span class="n">a</span><span class="o">&lt;=</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="o">&lt;=</span><span class="n">b</span><span class="p">)),</span>
-                <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="bp">True</span><span class="p">))</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">a</span> <span class="o">=</span> <span class="n">a</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">b</span> <span class="o">=</span> <span class="n">b</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-<span class="k">def</span> <span class="nf">QuadraticU</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a Continuous Random Variable with a U-quadratic distribution.</span>
-
-<span class="sd">    The density of the U-quadratic distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \alpha (x-\beta)^2</span>
-
-<span class="sd">    with :math:`x \in [a,b]`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    a : Real number</span>
-<span class="sd">    b : Real number, :math:`a &lt; b`</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import QuadraticU, density, E, variance</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify, factor, pprint</span>
-
-<span class="sd">    &gt;&gt;&gt; a = Symbol(&quot;a&quot;, real=True)</span>
-<span class="sd">    &gt;&gt;&gt; b = Symbol(&quot;b&quot;, real=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = QuadraticU(&quot;x&quot;, a, b)</span>
-
-<span class="sd">    &gt;&gt;&gt; D = density(X)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(D, use_unicode=False)</span>
-<span class="sd">          /   /              2                         \</span>
-<span class="sd">          |   |   /    a   b\                          |</span>
-<span class="sd">          |   |12*|x - - - -|                          |</span>
-<span class="sd">          |   |   \    2   2/                          |</span>
-<span class="sd">    Lambda|x, &lt;---------------  for And(x &lt;= b, a &lt;= x)|</span>
-<span class="sd">          |   |           3                            |</span>
-<span class="sd">          |   |   (-a + b)                             |</span>
-<span class="sd">          |   |                                        |</span>
-<span class="sd">          \   \       0                otherwise       /</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/U-quadratic_distribution</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">QuadraticUPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># RaisedCosine distribution ----------------------------------------------------</span>
-
-<span class="k">class</span> <span class="nc">RaisedCosinePSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="n">mu</span><span class="p">,</span> <span class="n">s</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">mu</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-
-        <span class="n">_value_check</span><span class="p">(</span><span class="n">s</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&quot;s must be positive&quot;</span><span class="p">)</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="n">Piecewise</span><span class="p">(</span>
-                <span class="p">((</span><span class="mi">1</span><span class="o">+</span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="n">mu</span><span class="p">)</span><span class="o">/</span><span class="n">s</span><span class="p">))</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">s</span><span class="p">),</span> <span class="n">And</span><span class="p">(</span><span class="n">mu</span><span class="o">-</span><span class="n">s</span><span class="o">&lt;=</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="o">&lt;=</span><span class="n">mu</span><span class="o">+</span><span class="n">s</span><span class="p">)),</span>
-                <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="bp">True</span><span class="p">))</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="n">mu</span><span class="o">-</span><span class="n">s</span><span class="p">,</span> <span class="n">mu</span><span class="o">+</span><span class="n">s</span><span class="p">))</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">mu</span> <span class="o">=</span> <span class="n">mu</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">s</span> <span class="o">=</span> <span class="n">s</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-<span class="k">def</span> <span class="nf">RaisedCosine</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a Continuous Random Variable with a raised cosine distribution.</span>
-
-<span class="sd">    The density of the raised cosine distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{1}{2s}\left(1+\cos\left(\frac{x-\mu}{s}\pi\right)\right)</span>
-
-<span class="sd">    with :math:`x \in [\mu-s,\mu+s]`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    mu : Real number</span>
-<span class="sd">    s : Real number, `s` &gt; 0</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import RaisedCosine, density, E, variance</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify, pprint</span>
-
-<span class="sd">    &gt;&gt;&gt; mu = Symbol(&quot;mu&quot;, real=True)</span>
-<span class="sd">    &gt;&gt;&gt; s = Symbol(&quot;s&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = RaisedCosine(&quot;x&quot;, mu, s)</span>
-
-<span class="sd">    &gt;&gt;&gt; D = density(X)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(D, use_unicode=False)</span>
-<span class="sd">          /   /   /pi*(x - mu)\                                       \</span>
-<span class="sd">          |   |cos|-----------| + 1                                   |</span>
-<span class="sd">          |   |   \     s     /                                       |</span>
-<span class="sd">    Lambda|x, &lt;--------------------  for And(x &lt;= mu + s, mu - s &lt;= x)|</span>
-<span class="sd">          |   |        2*s                                            |</span>
-<span class="sd">          |   |                                                       |</span>
-<span class="sd">          \   \         0                        otherwise            /</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Raised_cosine_distribution</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">RaisedCosinePSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Rayleigh distribution --------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">RayleighPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">sigma</span><span class="p">):</span>
-        <span class="n">sigma</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">sigma</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="n">x</span><span class="o">/</span><span class="n">sigma</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sigma</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-
-<span class="k">def</span> <span class="nf">Rayleigh</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">sigma</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with a Rayleigh distribution.</span>
-
-<span class="sd">    The density of the Rayleigh distribution is given by</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        f(x) := \frac{x}{\sigma^2} e^{-x^2/2\sigma^2}</span>
-
-<span class="sd">    with :math:`x &gt; 0`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    sigma : Real number, `sigma` &gt; 0</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Rayleigh, density, E, variance</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify</span>
-
-<span class="sd">    &gt;&gt;&gt; sigma = Symbol(&quot;sigma&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Rayleigh(&quot;x&quot;, sigma)</span>
-
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    Lambda(_x, _x*exp(-_x**2/(2*sigma**2))/sigma**2)</span>
-
-<span class="sd">    &gt;&gt;&gt; E(X)</span>
-<span class="sd">    sqrt(2)*sqrt(pi)*sigma/2</span>
-
-<span class="sd">    &gt;&gt;&gt; variance(X)</span>
-<span class="sd">    -pi*sigma**2/2 + 2*sigma**2</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Rayleigh_distribution</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/RayleighDistribution.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">RayleighPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">sigma</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># StudentT distribution --------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">StudentTPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">nu</span><span class="p">):</span>
-        <span class="n">nu</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">nu</span><span class="p">)</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">nu</span><span class="p">)</span><span class="o">*</span><span class="n">beta_fn</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">nu</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="n">nu</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">nu</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-
-<span class="k">def</span> <span class="nf">StudentT</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">nu</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with a student&#39;s t distribution.</span>
-
-<span class="sd">    The density of the student&#39;s t distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{\Gamma \left(\frac{\nu+1}{2} \right)}</span>
-<span class="sd">                {\sqrt{\nu\pi}\Gamma \left(\frac{\nu}{2} \right)}</span>
-<span class="sd">                \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    nu : Real number, `nu` &gt; 0, the degrees of freedom</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import StudentT, density, E, variance</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify, pprint</span>
-
-<span class="sd">    &gt;&gt;&gt; nu = Symbol(&quot;nu&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = StudentT(&quot;x&quot;, nu)</span>
-
-<span class="sd">    &gt;&gt;&gt; D = density(X)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(D, use_unicode=False)</span>
-<span class="sd">          /             nu   1              \</span>
-<span class="sd">          |           - -- - -              |</span>
-<span class="sd">          |             2    2              |</span>
-<span class="sd">          |   / 2    \                      |</span>
-<span class="sd">          |   |x     |              /nu   1\|</span>
-<span class="sd">          |   |-- + 1|        *gamma|-- + -||</span>
-<span class="sd">          |   \nu    /              \2    2/|</span>
-<span class="sd">    Lambda|x, ------------------------------|</span>
-<span class="sd">          |        ____   ____      /nu\    |</span>
-<span class="sd">          |      \/ pi *\/ nu *gamma|--|    |</span>
-<span class="sd">          \                         \2 /    /</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Student_t-distribution</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/Studentst-Distribution.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">StudentTPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">nu</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Triangular distribution ------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">TriangularPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
-        <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">b</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="n">Piecewise</span><span class="p">(</span>
-            <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="p">((</span><span class="n">b</span> <span class="o">-</span> <span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">c</span> <span class="o">-</span> <span class="n">a</span><span class="p">)),</span> <span class="n">And</span><span class="p">(</span><span class="n">a</span> <span class="o">&lt;=</span> <span class="n">x</span><span class="p">,</span> <span class="n">x</span> <span class="o">&lt;</span> <span class="n">c</span><span class="p">)),</span>
-            <span class="p">(</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="n">a</span><span class="p">),</span> <span class="n">Eq</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">c</span><span class="p">)),</span>
-            <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="p">((</span><span class="n">b</span> <span class="o">-</span> <span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">b</span> <span class="o">-</span> <span class="n">c</span><span class="p">)),</span> <span class="n">And</span><span class="p">(</span><span class="n">c</span> <span class="o">&lt;</span> <span class="n">x</span><span class="p">,</span> <span class="n">x</span> <span class="o">&lt;=</span> <span class="n">b</span><span class="p">)),</span>
-            <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="bp">True</span><span class="p">))</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-
-<span class="k">def</span> <span class="nf">Triangular</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with a triangular distribution.</span>
-
-<span class="sd">    The density of the triangular distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \begin{cases}</span>
-<span class="sd">                  0 &amp; \mathrm{for\ } x &lt; a, \\</span>
-<span class="sd">                  \frac{2(x-a)}{(b-a)(c-a)} &amp; \mathrm{for\ } a \le x &lt; c, \\</span>
-<span class="sd">                  \frac{2}{b-a} &amp; \mathrm{for\ } x = c, \\</span>
-<span class="sd">                  \frac{2(b-x)}{(b-a)(b-c)} &amp; \mathrm{for\ } c &lt; x \le b, \\</span>
-<span class="sd">                  0 &amp; \mathrm{for\ } b &lt; x.</span>
-<span class="sd">                \end{cases}</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    a : Real number, :math:`a \in \left(-\infty, \infty\right)`</span>
-<span class="sd">    b : Real number, :math:`a &lt; b`</span>
-<span class="sd">    c : Real number, :math:`a \leq c \leq b`</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Triangular, density, E</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol</span>
-
-<span class="sd">    &gt;&gt;&gt; a = Symbol(&quot;a&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; b = Symbol(&quot;b&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; c = Symbol(&quot;c&quot;)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Triangular(&quot;x&quot;, a,b,c)</span>
-
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    Lambda(_x, Piecewise(((2*_x - 2*a)/((-a + b)*(-a + c)),</span>
-<span class="sd">                        And(_x &lt; c, a &lt;= _x)),</span>
-<span class="sd">                        (2/(-a + b), _x == c),</span>
-<span class="sd">                        ((-2*_x + 2*b)/((-a + b)*(b - c)),</span>
-<span class="sd">                        And(_x &lt;= b, c &lt; _x)),</span>
-<span class="sd">                        (0, True)))</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Triangular_distribution</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/TriangularDistribution.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">TriangularPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Uniform distribution ---------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">UniformPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">left</span><span class="p">,</span> <span class="n">right</span><span class="p">):</span>
-        <span class="n">left</span><span class="p">,</span> <span class="n">right</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">left</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">right</span><span class="p">)</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="n">Piecewise</span><span class="p">(</span>
-            <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="o">/</span><span class="p">(</span><span class="n">right</span> <span class="o">-</span> <span class="n">left</span><span class="p">),</span> <span class="n">And</span><span class="p">(</span><span class="n">left</span> <span class="o">&lt;=</span> <span class="n">x</span><span class="p">,</span> <span class="n">x</span> <span class="o">&lt;=</span> <span class="n">right</span><span class="p">)),</span>
-            <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="bp">True</span><span class="p">))</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="n">left</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">right</span> <span class="o">=</span> <span class="n">right</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">compute_cdf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Lambda</span><span class="p">,</span> <span class="n">Min</span>
-        <span class="n">z</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;z&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">bounded</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">compute_cdf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">result</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">({</span><span class="n">Min</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">right</span><span class="p">):</span> <span class="n">z</span><span class="p">,</span>
-                                 <span class="n">Min</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">left</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">right</span><span class="p">):</span> <span class="bp">self</span><span class="o">.</span><span class="n">left</span><span class="p">})</span>
-        <span class="k">return</span> <span class="n">Lambda</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">result</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">integrate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">rvs</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Max</span><span class="p">,</span> <span class="n">Min</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">integrate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">rvs</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">subs</span><span class="p">({</span><span class="n">Max</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">left</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">right</span><span class="p">):</span> <span class="bp">self</span><span class="o">.</span><span class="n">right</span><span class="p">,</span>
-                              <span class="n">Min</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">left</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">right</span><span class="p">):</span> <span class="bp">self</span><span class="o">.</span><span class="n">left</span><span class="p">})</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="k">def</span> <span class="nf">sample</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">{</span><span class="bp">self</span><span class="o">.</span><span class="n">value</span><span class="p">:</span> <span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">left</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">right</span><span class="p">)}</span>
-
-
-<span class="k">def</span> <span class="nf">Uniform</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">left</span><span class="p">,</span> <span class="n">right</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with a uniform distribution.</span>
-
-<span class="sd">    The density of the uniform distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \begin{cases}</span>
-<span class="sd">                  \frac{1}{b - a} &amp; \text{for } x \in [a,b]  \\</span>
-<span class="sd">                  0               &amp; \text{otherwise}</span>
-<span class="sd">                \end{cases}</span>
-
-<span class="sd">    with :math:`x \in [a,b]`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    a : Real number, :math:`-\infty &lt; a` the left boundary</span>
-<span class="sd">    b : Real number, :math:`a &lt; b &lt; \infty` the right boundary</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Uniform, density, cdf, E, variance, skewness</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify</span>
-
-<span class="sd">    &gt;&gt;&gt; a = Symbol(&quot;a&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; b = Symbol(&quot;b&quot;)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Uniform(&quot;x&quot;, a, b)</span>
-
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    Lambda(_x, Piecewise((1/(-a + b), And(_x &lt;= b, a &lt;= _x)), (0, True)))</span>
-
-<span class="sd">    &gt;&gt;&gt; cdf(X)</span>
-<span class="sd">    Lambda(_z, _z/(-a + b) - a/(-a + b))</span>
-
-<span class="sd">    &gt;&gt;&gt; simplify(E(X))</span>
-<span class="sd">    a/2 + b/2</span>
-
-<span class="sd">    &gt;&gt;&gt; simplify(variance(X))</span>
-<span class="sd">    a**2/12 - a*b/6 + b**2/12</span>
-
-<span class="sd">    &gt;&gt;&gt; simplify(skewness(X))</span>
-<span class="sd">    0</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/UniformDistribution.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">UniformPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">left</span><span class="p">,</span> <span class="n">right</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># UniformSum distribution ------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">UniformSumPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">k</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span>
-            <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">Sum</span><span class="p">((</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">k</span><span class="o">*</span><span class="n">binomial</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">floor</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-
-<span class="k">def</span> <span class="nf">UniformSum</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with an Irwin-Hall distribution.</span>
-
-<span class="sd">    The probability distribution function depends on a single parameter</span>
-<span class="sd">    `n` which is an integer.</span>
-
-<span class="sd">    The density of the Irwin-Hall distribution is given by</span>
-
-<span class="sd">    .. math ::</span>
-<span class="sd">        f(x) := \frac{1}{(n-1)!}\sum_{k=0}^{\lfloor x\rfloor}(-1)^k</span>
-<span class="sd">                \binom{n}{k}(x-k)^{n-1}</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    n : Integral number, `n` &gt; 0</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import UniformSum, density</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, pprint</span>
-
-<span class="sd">    &gt;&gt;&gt; n = Symbol(&quot;n&quot;, integer=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = UniformSum(&quot;x&quot;, n)</span>
-
-<span class="sd">    &gt;&gt;&gt; D = density(X)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(D, use_unicode=False)</span>
-<span class="sd">          /   floor(x)                        \</span>
-<span class="sd">          |     ___                           |</span>
-<span class="sd">          |     \  `                          |</span>
-<span class="sd">          |      \         k         n - 1 /n\|</span>
-<span class="sd">          |       )    (-1) *(-k + x)     *| ||</span>
-<span class="sd">          |      /                         \k/|</span>
-<span class="sd">          |     /__,                          |</span>
-<span class="sd">          |    k = 0                          |</span>
-<span class="sd">    Lambda|x, --------------------------------|</span>
-<span class="sd">          \               (n - 1)!            /</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Uniform_sum_distribution</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/UniformSumDistribution.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">UniformSumPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># VonMises distribution --------------------------------------------------------</span>
-
-<span class="k">class</span> <span class="nc">VonMisesPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="n">mu</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">mu</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-
-        <span class="n">_value_check</span><span class="p">(</span><span class="n">k</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&quot;k must be positive&quot;</span><span class="p">)</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="n">k</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="n">mu</span><span class="p">))</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">besseli</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">k</span><span class="p">))</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">mu</span> <span class="o">=</span> <span class="n">mu</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">k</span> <span class="o">=</span> <span class="n">k</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-<span class="k">def</span> <span class="nf">VonMises</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a Continuous Random Variable with a von Mises distribution.</span>
-
-<span class="sd">    The density of the von Mises distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac{e^{\kappa\cos(x-\mu)}}{2\pi I_0(\kappa)}</span>
-
-<span class="sd">    with :math:`x \in [0,2\pi]`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    mu : Real number, measure of location</span>
-<span class="sd">    k : Real number, measure of concentration</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import VonMises, density, E, variance</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify, pprint</span>
-
-<span class="sd">    &gt;&gt;&gt; mu = Symbol(&quot;mu&quot;)</span>
-<span class="sd">    &gt;&gt;&gt; k = Symbol(&quot;k&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = VonMises(&quot;x&quot;, mu, k)</span>
-
-<span class="sd">    &gt;&gt;&gt; D = density(X)</span>
-<span class="sd">    &gt;&gt;&gt; pprint(D, use_unicode=False)</span>
-<span class="sd">          /      k*cos(x - mu)  \</span>
-<span class="sd">          |     e               |</span>
-<span class="sd">    Lambda|x, ------------------|</span>
-<span class="sd">          \   2*pi*besseli(0, k)/</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Von_Mises_distribution</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/vonMisesDistribution.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">VonMisesPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Weibull distribution ---------------------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">WeibullPSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">):</span>
-        <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">alpha</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span>
-
-        <span class="n">_value_check</span><span class="p">(</span><span class="n">alpha</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&quot;Alpha must be positive&quot;</span><span class="p">)</span>
-        <span class="n">_value_check</span><span class="p">(</span><span class="n">beta</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&quot;Beta must be positive&quot;</span><span class="p">)</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="n">beta</span> <span class="o">*</span> <span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="n">alpha</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">beta</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="n">alpha</span><span class="p">)</span><span class="o">**</span><span class="n">beta</span><span class="p">)</span> <span class="o">/</span> <span class="n">alpha</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">alpha</span> <span class="o">=</span> <span class="n">alpha</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">beta</span> <span class="o">=</span> <span class="n">beta</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">sample</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">{</span><span class="bp">self</span><span class="o">.</span><span class="n">value</span><span class="p">:</span> <span class="n">random</span><span class="o">.</span><span class="n">weibullvariate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">alpha</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">beta</span><span class="p">)}</span>
-
-
-<span class="k">def</span> <span class="nf">Weibull</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with a Weibull distribution.</span>
-
-<span class="sd">    The density of the Weibull distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \begin{cases}</span>
-<span class="sd">                  \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}</span>
-<span class="sd">                  e^{-(x/\lambda)^{k}} &amp; x\geq0\\</span>
-<span class="sd">                  0 &amp; x&lt;0</span>
-<span class="sd">                \end{cases}</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    lambda : Real number, :math:`\lambda &gt; 0` a scale</span>
-<span class="sd">    k : Real number, `k` &gt; 0 a shape</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Weibull, density, E, variance</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify</span>
-
-<span class="sd">    &gt;&gt;&gt; l = Symbol(&quot;lambda&quot;, positive=True)</span>
-<span class="sd">    &gt;&gt;&gt; k = Symbol(&quot;k&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Weibull(&quot;x&quot;, l, k)</span>
-
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    Lambda(_x, k*(_x/lambda)**(k - 1)*exp(-(_x/lambda)**k)/lambda)</span>
-
-<span class="sd">    &gt;&gt;&gt; simplify(E(X))</span>
-<span class="sd">    lambda*gamma(1 + 1/k)</span>
-
-<span class="sd">    &gt;&gt;&gt; simplify(variance(X))</span>
-<span class="sd">    lambda**2*(-gamma(1 + 1/k)**2 + gamma(1 + 2/k))</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Weibull_distribution</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/WeibullDistribution.html</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">WeibullPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-<span class="c">#-------------------------------------------------------------------------------</span>
-<span class="c"># Wigner semicircle distribution -----------------------------------------------</span>
-
-
-<span class="k">class</span> <span class="nc">WignerSemicirclePSpace</span><span class="p">(</span><span class="n">SingleContinuousPSpace</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">R</span><span class="p">):</span>
-        <span class="n">R</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">R</span><span class="p">)</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">pdf</span> <span class="o">=</span> <span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">R</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">R</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">SingleContinuousPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="nb">set</span><span class="o">=</span><span class="n">Interval</span><span class="p">(</span><span class="o">-</span><span class="n">R</span><span class="p">,</span> <span class="n">R</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-
-<span class="k">def</span> <span class="nf">WignerSemicircle</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">R</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Create a continuous random variable with a Wigner semicircle distribution.</span>
-
-<span class="sd">    The density of the Wigner semicircle distribution is given by</span>
-
-<span class="sd">    .. math::</span>
-<span class="sd">        f(x) := \frac2{\pi R^2}\,\sqrt{R^2-x^2}</span>
-
-<span class="sd">    with :math:`x \in [-R,R]`.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    R : Real number, `R` &gt; 0 the radius</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    A `RandomSymbol`.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import WignerSemicircle, density, E</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol, simplify</span>
-
-<span class="sd">    &gt;&gt;&gt; R = Symbol(&quot;R&quot;, positive=True)</span>
-
-<span class="sd">    &gt;&gt;&gt; X = WignerSemicircle(&quot;x&quot;, R)</span>
-
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    Lambda(_x, 2*sqrt(-_x**2 + R**2)/(pi*R**2))</span>
-
-<span class="sd">    &gt;&gt;&gt; E(X)</span>
-<span class="sd">    0</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    [1] http://en.wikipedia.org/wiki/Wigner_semicircle_distribution</span>
-<span class="sd">    [2] http://mathworld.wolfram.com/WignersSemicircleLaw.html</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">WignerSemicirclePSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-</pre></div>
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-            
-  <h1>Source code for sympy.stats.frv</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Finite Discrete Random Variables Module</span>
-
-<span class="sd">See Also</span>
-<span class="sd">========</span>
-<span class="sd">sympy.stats.frv_types</span>
-<span class="sd">sympy.stats.rv</span>
-<span class="sd">sympy.stats.crv</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="p">(</span><span class="n">And</span><span class="p">,</span> <span class="n">Eq</span><span class="p">,</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">cacheit</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Add</span><span class="p">,</span>
-        <span class="n">And</span><span class="p">,</span> <span class="n">Or</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.core.sets</span> <span class="kn">import</span> <span class="n">FiniteSet</span>
-<span class="kn">from</span> <span class="nn">sympy.stats.rv</span> <span class="kn">import</span> <span class="p">(</span><span class="n">RandomDomain</span><span class="p">,</span> <span class="n">ProductDomain</span><span class="p">,</span> <span class="n">ConditionalDomain</span><span class="p">,</span>
-        <span class="n">PSpace</span><span class="p">,</span> <span class="n">ProductPSpace</span><span class="p">,</span> <span class="n">SinglePSpace</span><span class="p">,</span> <span class="n">random_symbols</span><span class="p">,</span> <span class="n">sumsets</span><span class="p">,</span> <span class="n">rv_subs</span><span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">product</span>
-<span class="kn">from</span> <span class="nn">sympy.core.containers</span> <span class="kn">import</span> <span class="n">Dict</span>
-<span class="kn">import</span> <span class="nn">random</span>
-
-
-<div class="viewcode-block" id="FiniteDomain"><a class="viewcode-back" href="../../../modules/stats.html#sympy.stats.frv.FiniteDomain">[docs]</a><span class="k">class</span> <span class="nc">FiniteDomain</span><span class="p">(</span><span class="n">RandomDomain</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A domain with discrete finite support</span>
-
-<span class="sd">    Represented using a FiniteSet.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_Finite</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">elements</span><span class="p">):</span>
-        <span class="n">elements</span> <span class="o">=</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="o">*</span><span class="n">elements</span><span class="p">)</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="n">sym</span> <span class="k">for</span> <span class="n">sym</span><span class="p">,</span> <span class="n">val</span> <span class="ow">in</span> <span class="n">elements</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">RandomDomain</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">elements</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">elements</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">dict</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="n">Dict</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">el</span><span class="p">))</span> <span class="k">for</span> <span class="n">el</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">elements</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__contains__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">other</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">elements</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">elements</span><span class="o">.</span><span class="n">__iter__</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">as_boolean</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Or</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">And</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">Eq</span><span class="p">(</span><span class="n">sym</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span> <span class="k">for</span> <span class="n">sym</span><span class="p">,</span> <span class="n">val</span> <span class="ow">in</span> <span class="n">item</span><span class="p">])</span> <span class="k">for</span> <span class="n">item</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">])</span>
-
-</div>
-<span class="k">class</span> <span class="nc">SingleFiniteDomain</span><span class="p">(</span><span class="n">FiniteDomain</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A FiniteDomain over a single symbol/set</span>
-
-<span class="sd">    Example: The possibilities of a *single* die roll.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">symbol</span><span class="p">,</span> <span class="nb">set</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">RandomDomain</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="p">(</span><span class="n">symbol</span><span class="p">,</span> <span class="p">),</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="o">*</span><span class="nb">set</span><span class="p">))</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">symbol</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">symbols</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">elements</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="nb">frozenset</span><span class="p">(((</span><span class="bp">self</span><span class="o">.</span><span class="n">symbol</span><span class="p">,</span> <span class="n">elem</span><span class="p">),</span> <span class="p">))</span> <span class="k">for</span> <span class="n">elem</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">set</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">set</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="nb">frozenset</span><span class="p">(((</span><span class="bp">self</span><span class="o">.</span><span class="n">symbol</span><span class="p">,</span> <span class="n">elem</span><span class="p">),))</span> <span class="k">for</span> <span class="n">elem</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">set</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__contains__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="n">sym</span><span class="p">,</span> <span class="n">val</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">other</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">sym</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">symbol</span> <span class="ow">and</span> <span class="n">val</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">set</span>
-
-
-<span class="k">class</span> <span class="nc">ProductFiniteDomain</span><span class="p">(</span><span class="n">ProductDomain</span><span class="p">,</span> <span class="n">FiniteDomain</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A Finite domain consisting of several other FiniteDomains</span>
-
-<span class="sd">    Example: The possibilities of the rolls of three independent dice</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">proditer</span> <span class="o">=</span> <span class="n">product</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">domains</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">sumsets</span><span class="p">(</span><span class="n">items</span><span class="p">)</span> <span class="k">for</span> <span class="n">items</span> <span class="ow">in</span> <span class="n">proditer</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">elements</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="nb">iter</span><span class="p">(</span><span class="bp">self</span><span class="p">))</span>
-
-
-<span class="k">class</span> <span class="nc">ConditionalFiniteDomain</span><span class="p">(</span><span class="n">ConditionalDomain</span><span class="p">,</span> <span class="n">ProductFiniteDomain</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A FiniteDomain that has been restricted by a condition</span>
-
-<span class="sd">    Example: The possibilities of a die roll under the condition that the</span>
-<span class="sd">    roll is even.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">condition</span><span class="p">):</span>
-        <span class="n">cond</span> <span class="o">=</span> <span class="n">rv_subs</span><span class="p">(</span><span class="n">condition</span><span class="p">)</span>
-        <span class="c"># Check that we aren&#39;t passed a condition like die1 == z</span>
-        <span class="c"># where &#39;z&#39; is a symbol that we don&#39;t know about</span>
-        <span class="c"># We will never be able to test this equality through iteration</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">cond</span><span class="o">.</span><span class="n">free_symbols</span><span class="o">.</span><span class="n">issubset</span><span class="p">(</span><span class="n">domain</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;Condition &quot;</span><span class="si">%s</span><span class="s">&quot; contains foreign symbols </span><span class="se">\n</span><span class="si">%s</span><span class="s">.</span><span class="se">\n</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span>
-                <span class="n">condition</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">cond</span><span class="o">.</span><span class="n">free_symbols</span> <span class="o">-</span> <span class="n">domain</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">))</span> <span class="o">+</span>
-                <span class="s">&quot;Will be unable to iterate using this condition&quot;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">ConditionalDomain</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">condition</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_test</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">elem</span><span class="p">):</span>
-        <span class="n">val</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">condition</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">elem</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">val</span> <span class="ow">in</span> <span class="p">[</span><span class="bp">True</span><span class="p">,</span> <span class="bp">False</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="n">val</span>
-        <span class="k">elif</span> <span class="n">val</span><span class="o">.</span><span class="n">is_Equality</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">val</span><span class="o">.</span><span class="n">lhs</span> <span class="o">==</span> <span class="n">val</span><span class="o">.</span><span class="n">rhs</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Undeciable if </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">str</span><span class="p">(</span><span class="n">val</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">__contains__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">other</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">fulldomain</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">_test</span><span class="p">(</span><span class="n">other</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">(</span><span class="n">elem</span> <span class="k">for</span> <span class="n">elem</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">fulldomain</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_test</span><span class="p">(</span><span class="n">elem</span><span class="p">))</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">set</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">fulldomain</span><span class="o">.</span><span class="n">__class__</span> <span class="ow">is</span> <span class="n">SingleFiniteDomain</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="n">elem</span> <span class="k">for</span> <span class="n">elem</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">fulldomain</span><span class="o">.</span><span class="n">set</span>
-                    <span class="k">if</span> <span class="nb">frozenset</span><span class="p">(((</span><span class="bp">self</span><span class="o">.</span><span class="n">fulldomain</span><span class="o">.</span><span class="n">symbol</span><span class="p">,</span> <span class="n">elem</span><span class="p">),))</span> <span class="ow">in</span> <span class="bp">self</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-                <span class="s">&quot;Not implemented on multi-dimensional conditional domain&quot;</span><span class="p">)</span>
-        <span class="c">#return FiniteSet(elem for elem in self.fulldomain if elem in self)</span>
-
-    <span class="k">def</span> <span class="nf">as_boolean</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">FiniteDomain</span><span class="o">.</span><span class="n">as_boolean</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-
-<span class="c">#=============================================</span>
-<span class="c">#=========  Probability Space  ===============</span>
-<span class="c">#=============================================</span>
-
-
-<div class="viewcode-block" id="FinitePSpace"><a class="viewcode-back" href="../../../modules/stats.html#sympy.stats.frv.FinitePSpace">[docs]</a><span class="k">class</span> <span class="nc">FinitePSpace</span><span class="p">(</span><span class="n">PSpace</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A Finite Probability Space</span>
-
-<span class="sd">    Represents the probabilities of a finite number of events.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_Finite</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">density</span><span class="p">):</span>
-        <span class="n">density</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">((</span><span class="n">sympify</span><span class="p">(</span><span class="n">key</span><span class="p">),</span> <span class="n">sympify</span><span class="p">(</span><span class="n">val</span><span class="p">))</span>
-                <span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="n">val</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">density</span><span class="o">.</span><span class="n">items</span><span class="p">()))</span>
-        <span class="n">public_density</span> <span class="o">=</span> <span class="n">Dict</span><span class="p">(</span><span class="n">density</span><span class="p">)</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">PSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">public_density</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">_density</span> <span class="o">=</span> <span class="n">density</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="k">def</span> <span class="nf">prob_of</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">elem</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_density</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">elem</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">where</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">condition</span><span class="p">):</span>
-        <span class="k">assert</span> <span class="nb">all</span><span class="p">(</span><span class="n">r</span><span class="o">.</span><span class="n">symbol</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">symbols</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">random_symbols</span><span class="p">(</span><span class="n">condition</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">ConditionalFiniteDomain</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">,</span> <span class="n">condition</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">compute_density</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="nb">dict</span><span class="p">(((</span><span class="n">rs</span><span class="p">,</span> <span class="n">rs</span><span class="o">.</span><span class="n">symbol</span><span class="p">)</span> <span class="k">for</span> <span class="n">rs</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">values</span><span class="p">)))</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">elem</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">:</span>
-            <span class="n">val</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">elem</span><span class="p">))</span>
-            <span class="n">prob</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">prob_of</span><span class="p">(</span><span class="n">elem</span><span class="p">)</span>
-            <span class="n">d</span><span class="p">[</span><span class="n">val</span><span class="p">]</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">val</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="o">+</span> <span class="n">prob</span>
-        <span class="k">return</span> <span class="n">d</span>
-
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">compute_cdf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">compute_density</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="n">cum_prob</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">cdf</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">key</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">d</span><span class="p">):</span>
-            <span class="n">prob</span> <span class="o">=</span> <span class="n">d</span><span class="p">[</span><span class="n">key</span><span class="p">]</span>
-            <span class="n">cum_prob</span> <span class="o">+=</span> <span class="n">prob</span>
-            <span class="n">cdf</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">key</span><span class="p">,</span> <span class="n">cum_prob</span><span class="p">))</span>
-
-        <span class="k">return</span> <span class="nb">dict</span><span class="p">(</span><span class="n">cdf</span><span class="p">)</span>
-
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">sorted_cdf</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">python_float</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="n">cdf</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">compute_cdf</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="n">items</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">cdf</span><span class="o">.</span><span class="n">items</span><span class="p">())</span>
-        <span class="n">sorted_items</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">items</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">val_cum_prob</span><span class="p">:</span> <span class="n">val_cum_prob</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-        <span class="k">if</span> <span class="n">python_float</span><span class="p">:</span>
-            <span class="n">sorted_items</span> <span class="o">=</span> <span class="p">[(</span><span class="n">v</span><span class="p">,</span> <span class="nb">float</span><span class="p">(</span><span class="n">cum_prob</span><span class="p">))</span>
-                    <span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">cum_prob</span> <span class="ow">in</span> <span class="n">sorted_items</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">sorted_items</span>
-
-    <span class="k">def</span> <span class="nf">integrate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">rvs</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="n">rvs</span> <span class="o">=</span> <span class="n">rvs</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">values</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="nb">dict</span><span class="p">((</span><span class="n">rs</span><span class="p">,</span> <span class="n">rs</span><span class="o">.</span><span class="n">symbol</span><span class="p">)</span> <span class="k">for</span> <span class="n">rs</span> <span class="ow">in</span> <span class="n">rvs</span><span class="p">))</span>
-        <span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">elem</span><span class="p">))</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">prob_of</span><span class="p">(</span><span class="n">elem</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">elem</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">probability</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">condition</span><span class="p">):</span>
-        <span class="n">cond_symbols</span> <span class="o">=</span> <span class="nb">frozenset</span><span class="p">(</span><span class="n">rs</span><span class="o">.</span><span class="n">symbol</span> <span class="k">for</span> <span class="n">rs</span> <span class="ow">in</span> <span class="n">random_symbols</span><span class="p">(</span><span class="n">condition</span><span class="p">))</span>
-        <span class="k">assert</span> <span class="n">cond_symbols</span><span class="o">.</span><span class="n">issubset</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">symbols</span><span class="p">)</span>
-        <span class="k">return</span> <span class="nb">sum</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">prob_of</span><span class="p">(</span><span class="n">elem</span><span class="p">)</span> <span class="k">for</span> <span class="n">elem</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">condition</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">conditional_space</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">condition</span><span class="p">):</span>
-        <span class="n">domain</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">condition</span><span class="p">)</span>
-        <span class="n">prob</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">probability</span><span class="p">(</span><span class="n">condition</span><span class="p">)</span>
-        <span class="n">density</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">((</span><span class="n">key</span><span class="p">,</span> <span class="n">val</span> <span class="o">/</span> <span class="n">prob</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="n">val</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_density</span><span class="o">.</span><span class="n">items</span><span class="p">())</span> <span class="k">if</span> <span class="n">key</span> <span class="ow">in</span> <span class="n">domain</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">FinitePSpace</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">density</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">sample</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Internal sample method</span>
-
-<span class="sd">        Returns dictionary mapping RandomSymbol to realization value.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">expr</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">values</span><span class="p">)</span>
-        <span class="n">cdf</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">sorted_cdf</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">python_float</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-        <span class="n">x</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="c"># Find first occurence with cumulative probability less than x</span>
-        <span class="c"># This should be replaced with binary search</span>
-        <span class="k">for</span> <span class="n">value</span><span class="p">,</span> <span class="n">cum_prob</span> <span class="ow">in</span> <span class="n">cdf</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">x</span> <span class="o">&lt;</span> <span class="n">cum_prob</span><span class="p">:</span>
-                <span class="c"># return dictionary mapping RandomSymbols to values</span>
-                <span class="k">return</span> <span class="nb">dict</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">value</span><span class="p">)))</span>
-
-        <span class="k">assert</span> <span class="bp">False</span><span class="p">,</span> <span class="s">&quot;We should never have gotten to this point&quot;</span>
-
-</div>
-<span class="k">class</span> <span class="nc">SingleFinitePSpace</span><span class="p">(</span><span class="n">FinitePSpace</span><span class="p">,</span> <span class="n">SinglePSpace</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A single finite probability space</span>
-
-<span class="sd">    Represents the probabilities of a set of random events that can be</span>
-<span class="sd">    attributed to a single variable/symbol.</span>
-
-<span class="sd">    This class is implemented by many of the standard FiniteRV types such as</span>
-<span class="sd">    Die, Bernoulli, Coin, etc....</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">fromdict</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">density</span><span class="p">):</span>
-        <span class="n">symbol</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">domain</span> <span class="o">=</span> <span class="n">SingleFiniteDomain</span><span class="p">(</span><span class="n">symbol</span><span class="p">,</span> <span class="nb">frozenset</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">density</span><span class="o">.</span><span class="n">keys</span><span class="p">())))</span>
-        <span class="n">density</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">((</span><span class="nb">frozenset</span><span class="p">(((</span><span class="n">symbol</span><span class="p">,</span> <span class="n">val</span><span class="p">),)),</span> <span class="n">prob</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">val</span><span class="p">,</span> <span class="n">prob</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">density</span><span class="o">.</span><span class="n">items</span><span class="p">()))</span>
-        <span class="n">density</span> <span class="o">=</span> <span class="n">Dict</span><span class="p">(</span><span class="n">density</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">FinitePSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">density</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">ProductFinitePSpace</span><span class="p">(</span><span class="n">ProductPSpace</span><span class="p">,</span> <span class="n">FinitePSpace</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A collection of several independent finite probability spaces</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">domain</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">ProductFiniteDomain</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">space</span><span class="o">.</span><span class="n">domain</span> <span class="k">for</span> <span class="n">space</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">spaces</span><span class="p">])</span>
-
-    <span class="nd">@property</span>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">_density</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">proditer</span> <span class="o">=</span> <span class="n">product</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="nb">iter</span><span class="p">(</span><span class="n">space</span><span class="o">.</span><span class="n">_density</span><span class="o">.</span><span class="n">items</span><span class="p">())</span>
-            <span class="k">for</span> <span class="n">space</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">spaces</span><span class="p">])</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">items</span> <span class="ow">in</span> <span class="n">proditer</span><span class="p">:</span>
-            <span class="n">elems</span><span class="p">,</span> <span class="n">probs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">items</span><span class="p">))</span>
-            <span class="n">elem</span> <span class="o">=</span> <span class="n">sumsets</span><span class="p">(</span><span class="n">elems</span><span class="p">)</span>
-            <span class="n">prob</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">probs</span><span class="p">)</span>
-            <span class="n">d</span><span class="p">[</span><span class="n">elem</span><span class="p">]</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">elem</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="o">+</span> <span class="n">prob</span>
-        <span class="k">return</span> <span class="n">Dict</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="nd">@cacheit</span>
-    <span class="k">def</span> <span class="nf">density</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Dict</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_density</span><span class="p">)</span>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/stats/frv_types.html b/dev-py3k/_modules/sympy/stats/frv_types.html
deleted file mode 100644
index 87c0340eb77..00000000000
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+++ /dev/null
@@ -1,441 +0,0 @@
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-            
-  <h1>Source code for sympy.stats.frv_types</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Finite Discrete Random Variables - Prebuilt variable types</span>
-
-<span class="sd">Contains</span>
-<span class="sd">========</span>
-<span class="sd">FiniteRV</span>
-<span class="sd">DiscreteUniform</span>
-<span class="sd">Die</span>
-<span class="sd">Bernoulli</span>
-<span class="sd">Coin</span>
-<span class="sd">Binomial</span>
-<span class="sd">Hypergeometric</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.stats.frv</span> <span class="kn">import</span> <span class="n">SingleFinitePSpace</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">Rational</span><span class="p">,</span> <span class="n">binomial</span>
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;FiniteRV&#39;</span><span class="p">,</span> <span class="s">&#39;DiscreteUniform&#39;</span><span class="p">,</span> <span class="s">&#39;Die&#39;</span><span class="p">,</span> <span class="s">&#39;Bernoulli&#39;</span><span class="p">,</span> <span class="s">&#39;Coin&#39;</span><span class="p">,</span>
-        <span class="s">&#39;Binomial&#39;</span><span class="p">,</span> <span class="s">&#39;Hypergeometric&#39;</span><span class="p">]</span>
-
-
-<span class="k">def</span> <span class="nf">FiniteRV</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">density</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create a Finite Random Variable given a dict representing the density.</span>
-
-<span class="sd">    Returns a RandomSymbol.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import FiniteRV, P, E</span>
-
-<span class="sd">    &gt;&gt;&gt; density = {0: .1, 1: .2, 2: .3, 3: .4}</span>
-<span class="sd">    &gt;&gt;&gt; X = FiniteRV(&#39;X&#39;, density)</span>
-
-<span class="sd">    &gt;&gt;&gt; E(X)</span>
-<span class="sd">    2.00000000000000</span>
-<span class="sd">    &gt;&gt;&gt; P(X&gt;=2)</span>
-<span class="sd">    0.700000000000000</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">SingleFinitePSpace</span><span class="o">.</span><span class="n">fromdict</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">density</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-
-<span class="k">class</span> <span class="nc">DiscreteUniformPSpace</span><span class="p">(</span><span class="n">SingleFinitePSpace</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create a Finite Random Variable representing a discrete uniform</span>
-<span class="sd">    distribution.</span>
-
-<span class="sd">    This class is for internal use.</span>
-
-<span class="sd">    Create DiscreteUniform Random Symbols using DiscreteUniform function</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import DiscreteUniform, density</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-
-<span class="sd">    &gt;&gt;&gt; X = DiscreteUniform(&#39;X&#39;, symbols(&#39;a b c&#39;)) # equally likely over a, b, c</span>
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    {a: 1/3, b: 1/3, c: 1/3}</span>
-
-<span class="sd">    &gt;&gt;&gt; Y = DiscreteUniform(&#39;Y&#39;, list(range(5))) # distribution over a range</span>
-<span class="sd">    &gt;&gt;&gt; density(Y)</span>
-<span class="sd">    {0: 1/5, 1: 1/5, 2: 1/5, 3: 1/5, 4: 1/5}</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">items</span><span class="p">):</span>
-        <span class="n">density</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">((</span><span class="n">sympify</span><span class="p">(</span><span class="n">item</span><span class="p">),</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">items</span><span class="p">)))</span>
-                       <span class="k">for</span> <span class="n">item</span> <span class="ow">in</span> <span class="n">items</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">fromdict</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">density</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">DiscreteUniform</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">items</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create a Finite Random Variable representing a uniform distribution over</span>
-<span class="sd">    the input set.</span>
-
-<span class="sd">    Returns a RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import DiscreteUniform, density</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-
-<span class="sd">    &gt;&gt;&gt; X = DiscreteUniform(&#39;X&#39;, symbols(&#39;a b c&#39;)) # equally likely over a, b, c</span>
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    {a: 1/3, b: 1/3, c: 1/3}</span>
-
-<span class="sd">    &gt;&gt;&gt; Y = DiscreteUniform(&#39;Y&#39;, list(range(5))) # distribution over a range</span>
-<span class="sd">    &gt;&gt;&gt; density(Y)</span>
-<span class="sd">    {0: 1/5, 1: 1/5, 2: 1/5, 3: 1/5, 4: 1/5}</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">DiscreteUniformPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">items</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-
-<div class="viewcode-block" id="DiePSpace"><a class="viewcode-back" href="../../../modules/stats.html#sympy.stats.frv_types.DiePSpace">[docs]</a><span class="k">class</span> <span class="nc">DiePSpace</span><span class="p">(</span><span class="n">DiscreteUniformPSpace</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create a Finite Random Variable representing a fair die.</span>
-
-<span class="sd">    This class is for internal use.</span>
-
-<span class="sd">    Create Dice Random Symbols using Die function</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Die, density</span>
-
-<span class="sd">    &gt;&gt;&gt; D6 = Die(&#39;D6&#39;, 6) # Six sided Die</span>
-<span class="sd">    &gt;&gt;&gt; density(D6)</span>
-<span class="sd">    {1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6}</span>
-
-<span class="sd">    &gt;&gt;&gt; D4 = Die(&#39;D4&#39;, 4) # Four sided Die</span>
-<span class="sd">    &gt;&gt;&gt; density(D4)</span>
-<span class="sd">    {1: 1/4, 2: 1/4, 3: 1/4, 4: 1/4}</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">sides</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">DiscreteUniformPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">sides</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)))</span>
-
-</div>
-<span class="k">def</span> <span class="nf">Die</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">sides</span><span class="o">=</span><span class="mi">6</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create a Finite Random Variable representing a fair die.</span>
-
-<span class="sd">    Returns a RandomSymbol.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Die, density</span>
-
-<span class="sd">    &gt;&gt;&gt; D6 = Die(&#39;D6&#39;, 6) # Six sided Die</span>
-<span class="sd">    &gt;&gt;&gt; density(D6)</span>
-<span class="sd">    {1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6}</span>
-
-<span class="sd">    &gt;&gt;&gt; D4 = Die(&#39;D4&#39;, 4) # Four sided Die</span>
-<span class="sd">    &gt;&gt;&gt; density(D4)</span>
-<span class="sd">    {1: 1/4, 2: 1/4, 3: 1/4, 4: 1/4}</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">DiePSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">sides</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-
-<span class="k">class</span> <span class="nc">BernoulliPSpace</span><span class="p">(</span><span class="n">SingleFinitePSpace</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create a Finite Random Variable representing a Bernoulli process.</span>
-
-<span class="sd">    Returns a RandomSymbol.</span>
-
-<span class="sd">    This class is for internal use.</span>
-
-<span class="sd">    Create Bernoulli Random Symbols using Bernoulli function.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Bernoulli, density</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import S</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Bernoulli(&#39;X&#39;, S(3)/4) # 1-0 Bernoulli variable, probability = 3/4</span>
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    {0: 1/4, 1: 3/4}</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Bernoulli(&#39;X&#39;, S.Half, &#39;Heads&#39;, &#39;Tails&#39;) # A fair coin toss</span>
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    {Heads: 1/2, Tails: 1/2}</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">succ</span><span class="p">,</span> <span class="n">fail</span><span class="p">):</span>
-        <span class="n">succ</span><span class="p">,</span> <span class="n">fail</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">succ</span><span class="p">,</span> <span class="n">fail</span><span class="p">,</span> <span class="n">p</span><span class="p">)))</span>
-        <span class="n">density</span> <span class="o">=</span> <span class="p">{</span><span class="n">succ</span><span class="p">:</span> <span class="n">p</span><span class="p">,</span> <span class="n">fail</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">p</span><span class="p">)}</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">fromdict</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">density</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">Bernoulli</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">succ</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">fail</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create a Finite Random Variable representing a Bernoulli process.</span>
-
-<span class="sd">    Returns a RandomSymbol</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Bernoulli, density</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import S</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Bernoulli(&#39;X&#39;, S(3)/4) # 1-0 Bernoulli variable, probability = 3/4</span>
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    {0: 1/4, 1: 3/4}</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Bernoulli(&#39;X&#39;, S.Half, &#39;Heads&#39;, &#39;Tails&#39;) # A fair coin toss</span>
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    {Heads: 1/2, Tails: 1/2}</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">BernoulliPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">succ</span><span class="p">,</span> <span class="n">fail</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-
-<span class="k">class</span> <span class="nc">CoinPSpace</span><span class="p">(</span><span class="n">BernoulliPSpace</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A probability space representing a coin toss.</span>
-
-<span class="sd">    Probability p is the chance of gettings &quot;Heads.&quot; Half by default</span>
-
-<span class="sd">    This class is for internal use.</span>
-
-<span class="sd">    Create Coin&#39;s using Coin function</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Coin, density</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Rational</span>
-
-<span class="sd">    &gt;&gt;&gt; C = Coin(&#39;C&#39;) # A fair coin toss</span>
-<span class="sd">    &gt;&gt;&gt; density(C)</span>
-<span class="sd">    {H: 1/2, T: 1/2}</span>
-
-<span class="sd">    &gt;&gt;&gt; C2 = Coin(&#39;C2&#39;, Rational(3, 5)) # An unfair coin</span>
-<span class="sd">    &gt;&gt;&gt; density(C2)</span>
-<span class="sd">    {H: 3/5, T: 2/5}</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">BernoulliPSpace</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="s">&#39;H&#39;</span><span class="p">,</span> <span class="s">&#39;T&#39;</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">Coin</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create a Finite Random Variable representing a Coin toss.</span>
-
-<span class="sd">    Probability p is the chance of gettings &quot;Heads.&quot; Half by default</span>
-
-<span class="sd">    Returns a RandomSymbol.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Coin, density</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Rational</span>
-
-<span class="sd">    &gt;&gt;&gt; C = Coin(&#39;C&#39;) # A fair coin toss</span>
-<span class="sd">    &gt;&gt;&gt; density(C)</span>
-<span class="sd">    {H: 1/2, T: 1/2}</span>
-
-<span class="sd">    &gt;&gt;&gt; C2 = Coin(&#39;C2&#39;, Rational(3, 5)) # An unfair coin</span>
-<span class="sd">    &gt;&gt;&gt; density(C2)</span>
-<span class="sd">    {H: 3/5, T: 2/5}</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">CoinPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-
-<span class="k">class</span> <span class="nc">BinomialPSpace</span><span class="p">(</span><span class="n">SingleFinitePSpace</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create a Finite Random Variable representing a binomial distribution.</span>
-
-<span class="sd">    This class is for internal use.</span>
-
-<span class="sd">    Create Binomial Random Symbols using Binomial function.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Binomial, density</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import S</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Binomial(&#39;X&#39;, 4, S.Half) # Four &quot;coin flips&quot;</span>
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    {0: 1/16, 1: 1/4, 2: 3/8, 3: 1/4, 4: 1/16}</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">succ</span><span class="p">,</span> <span class="n">fail</span><span class="p">):</span>
-        <span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">succ</span><span class="p">,</span> <span class="n">fail</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">succ</span><span class="p">,</span> <span class="n">fail</span><span class="p">)))</span>
-        <span class="n">density</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">((</span><span class="n">k</span><span class="o">*</span><span class="n">succ</span> <span class="o">+</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="n">fail</span><span class="p">,</span>
-                <span class="n">binomial</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="o">*</span> <span class="n">p</span><span class="o">**</span><span class="n">k</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">p</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">))</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">fromdict</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">density</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">Binomial</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">succ</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">fail</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create a Finite Random Variable representing a binomial distribution.</span>
-
-<span class="sd">    Returns a RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Binomial, density</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import S</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Binomial(&#39;X&#39;, 4, S.Half) # Four &quot;coin flips&quot;</span>
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    {0: 1/16, 1: 1/4, 2: 3/8, 3: 1/4, 4: 1/16}</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">BinomialPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">succ</span><span class="p">,</span> <span class="n">fail</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-
-
-<span class="k">class</span> <span class="nc">HypergeometricPSpace</span><span class="p">(</span><span class="n">SingleFinitePSpace</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create a Finite Random Variable representing a hypergeometric distribution.</span>
-
-<span class="sd">    This class is for internal use.</span>
-
-<span class="sd">    Create Hypergeometric Random Symbols using Hypergeometric function.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Hypergeometric, density</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import S</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Hypergeometric(&#39;X&#39;, 10, 5, 3) # 10 marbles, 5 white (success), 3 draws</span>
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    {0: 1/12, 1: 5/12, 2: 5/12, 3: 1/12}</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="n">N</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">)))</span>
-        <span class="n">density</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">((</span><span class="n">k</span><span class="p">,</span> <span class="n">binomial</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="o">*</span> <span class="n">binomial</span><span class="p">(</span><span class="n">N</span> <span class="o">-</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">k</span><span class="p">)</span> <span class="o">/</span> <span class="n">binomial</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-                <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="n">m</span> <span class="o">-</span> <span class="n">N</span><span class="p">),</span> <span class="nb">min</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">cls</span><span class="o">.</span><span class="n">fromdict</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">density</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">Hypergeometric</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Create a Finite Random Variable representing a hypergeometric distribution.</span>
-
-<span class="sd">    Returns a RandomSymbol.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Hypergeometric, density</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import S</span>
-
-<span class="sd">    &gt;&gt;&gt; X = Hypergeometric(&#39;X&#39;, 10, 5, 3) # 10 marbles, 5 white (success), 3 draws</span>
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    {0: 1/12, 1: 5/12, 2: 5/12, 3: 1/12}</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">return</span> <span class="n">HypergeometricPSpace</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">value</span>
-</pre></div>
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-            
-  <h1>Source code for sympy.stats.rv</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">Main Random Variables Module</span>
-
-<span class="sd">Defines abstract random variable type.</span>
-<span class="sd">Contains interfaces for probability space object (PSpace) as well as standard</span>
-<span class="sd">operators, P, E, sample, density, where</span>
-
-<span class="sd">See Also</span>
-<span class="sd">========</span>
-<span class="sd">sympy.stats.crv</span>
-<span class="sd">sympy.stats.frv</span>
-<span class="sd">sympy.stats.rv_interface</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">,</span> <span class="n">And</span><span class="p">,</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Eq</span><span class="p">,</span> <span class="n">lambdify</span>
-<span class="kn">from</span> <span class="nn">sympy.core.sets</span> <span class="kn">import</span> <span class="n">FiniteSet</span><span class="p">,</span> <span class="n">ProductSet</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-
-<div class="viewcode-block" id="RandomDomain"><a class="viewcode-back" href="../../../modules/stats.html#sympy.stats.rv.RandomDomain">[docs]</a><span class="k">class</span> <span class="nc">RandomDomain</span><span class="p">(</span><span class="n">Basic</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Represents a set of variables and the values which they can take</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    sympy.stats.crv.ContinuousDomain</span>
-<span class="sd">    sympy.stats.frv.FiniteDomain</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_ProductDomain</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Finite</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">is_Continuous</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="o">*</span><span class="n">symbols</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">symbols</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">set</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__contains__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">integrate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-
-</div>
-<div class="viewcode-block" id="SingleDomain"><a class="viewcode-back" href="../../../modules/stats.html#sympy.stats.rv.SingleDomain">[docs]</a><span class="k">class</span> <span class="nc">SingleDomain</span><span class="p">(</span><span class="n">RandomDomain</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A single variable and its domain</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    sympy.stats.crv.SingleContinuousDomain</span>
-<span class="sd">    sympy.stats.frv.SingleFiniteDomain</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">symbol</span><span class="p">,</span> <span class="nb">set</span><span class="p">):</span>
-        <span class="k">assert</span> <span class="n">symbol</span><span class="o">.</span><span class="n">is_Symbol</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="n">symbol</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">RandomDomain</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="nb">set</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">symbol</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">symbols</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">__contains__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">other</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-        <span class="n">sym</span><span class="p">,</span> <span class="n">val</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">other</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">symbol</span> <span class="o">==</span> <span class="n">sym</span> <span class="ow">and</span> <span class="n">val</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">set</span>
-
-</div>
-<div class="viewcode-block" id="ConditionalDomain"><a class="viewcode-back" href="../../../modules/stats.html#sympy.stats.rv.ConditionalDomain">[docs]</a><span class="k">class</span> <span class="nc">ConditionalDomain</span><span class="p">(</span><span class="n">RandomDomain</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A RandomDomain with an attached condition</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    sympy.stats.crv.ConditionalContinuousDomain</span>
-<span class="sd">    sympy.stats.frv.ConditionalFiniteDomain</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">fulldomain</span><span class="p">,</span> <span class="n">condition</span><span class="p">):</span>
-        <span class="n">condition</span> <span class="o">=</span> <span class="n">condition</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="nb">dict</span><span class="p">((</span><span class="n">rs</span><span class="p">,</span> <span class="n">rs</span><span class="o">.</span><span class="n">symbol</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">rs</span> <span class="ow">in</span> <span class="n">random_symbols</span><span class="p">(</span><span class="n">condition</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">RandomDomain</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span>
-            <span class="n">cls</span><span class="p">,</span> <span class="n">fulldomain</span><span class="o">.</span><span class="n">symbols</span><span class="p">,</span> <span class="n">fulldomain</span><span class="p">,</span> <span class="n">condition</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">fulldomain</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">condition</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">set</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Set of Conditional Domain not Implemented&quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">as_boolean</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">And</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">fulldomain</span><span class="o">.</span><span class="n">as_boolean</span><span class="p">(),</span> <span class="bp">self</span><span class="o">.</span><span class="n">condition</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="PSpace"><a class="viewcode-back" href="../../../modules/stats.html#sympy.stats.rv.PSpace">[docs]</a><span class="k">class</span> <span class="nc">PSpace</span><span class="p">(</span><span class="n">Basic</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A Probability Space</span>
-
-<span class="sd">    Probability Spaces encode processes that equal different values</span>
-<span class="sd">    probabalistically. These underly Random Symbols which occur in SymPy</span>
-<span class="sd">    expressions and contain the mechanics to evaluate statistical statements.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    sympy.stats.crv.ContinuousPSpace</span>
-<span class="sd">    sympy.stats.frv.FinitePSpace</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_Finite</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">is_Continuous</span> <span class="o">=</span> <span class="bp">None</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">domain</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">density</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">values</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">frozenset</span><span class="p">(</span><span class="n">RandomSymbol</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sym</span><span class="p">)</span> <span class="k">for</span> <span class="n">sym</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="o">.</span><span class="n">symbols</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">symbols</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">domain</span><span class="o">.</span><span class="n">symbols</span>
-
-    <span class="k">def</span> <span class="nf">where</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">condition</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">compute_density</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">sample</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">probability</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">condition</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">integrate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
-
-</div>
-<div class="viewcode-block" id="SinglePSpace"><a class="viewcode-back" href="../../../modules/stats.html#sympy.stats.rv.SinglePSpace">[docs]</a><span class="k">class</span> <span class="nc">SinglePSpace</span><span class="p">(</span><span class="n">PSpace</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Represents the probabilities of a set of random events that can be</span>
-<span class="sd">    attributed to a single variable/symbol.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">value</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">values</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="RandomSymbol"><a class="viewcode-back" href="../../../modules/stats.html#sympy.stats.rv.RandomSymbol">[docs]</a><span class="k">class</span> <span class="nc">RandomSymbol</span><span class="p">(</span><span class="n">Symbol</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Random Symbols represent ProbabilitySpaces in SymPy Expressions</span>
-<span class="sd">    In principle they can take on any value that their symbol can take on</span>
-<span class="sd">    within the associated PSpace with probability determined by the PSpace</span>
-<span class="sd">    Density.</span>
-
-<span class="sd">    Random Symbols contain pspace and symbol properties.</span>
-<span class="sd">    The pspace property points to the represented Probability Space</span>
-<span class="sd">    The symbol is a standard SymPy Symbol that is used in that probability space</span>
-<span class="sd">    for example in defining a density.</span>
-
-<span class="sd">    You can form normal SymPy expressions using RandomSymbols and operate on</span>
-<span class="sd">    those expressions with the Functions</span>
-
-<span class="sd">    E - Expectation of a random expression</span>
-<span class="sd">    P - Probability of a condition</span>
-<span class="sd">    density - Probability Density of an expression</span>
-<span class="sd">    given - A new random expression (with new random symbols) given a condition</span>
-
-<span class="sd">    An object of the RandomSymbol type should almost never be created by the</span>
-<span class="sd">    user. They tend to be created instead by the PSpace class&#39;s value method.</span>
-<span class="sd">    Traditionally a user doesn&#39;t even do this but instead calls one of the</span>
-<span class="sd">    convenience functions Normal, Exponential, Coin, Die, FiniteRV, etc....</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_bounded</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="n">is_finite</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">pspace</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">symbol</span> <span class="o">=</span> <span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">name</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">name</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">is_commutative</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">symbol</span><span class="o">.</span><span class="n">is_commutative</span>
-
-    <span class="k">def</span> <span class="nf">_hashable_content</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">pspace</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">symbol</span>
-
-</div>
-<div class="viewcode-block" id="ProductPSpace"><a class="viewcode-back" href="../../../modules/stats.html#sympy.stats.rv.ProductPSpace">[docs]</a><span class="k">class</span> <span class="nc">ProductPSpace</span><span class="p">(</span><span class="n">PSpace</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A probability space resulting from the merger of two independent probability</span>
-<span class="sd">    spaces.</span>
-
-<span class="sd">    Often created using the function, pspace</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">spaces</span><span class="p">):</span>
-        <span class="n">rs_space_dict</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">space</span> <span class="ow">in</span> <span class="n">spaces</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">value</span> <span class="ow">in</span> <span class="n">space</span><span class="o">.</span><span class="n">values</span><span class="p">:</span>
-                <span class="n">rs_space_dict</span><span class="p">[</span><span class="n">value</span><span class="p">]</span> <span class="o">=</span> <span class="n">space</span>
-
-        <span class="n">symbols</span> <span class="o">=</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="n">val</span><span class="o">.</span><span class="n">symbol</span> <span class="k">for</span> <span class="n">val</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">rs_space_dict</span><span class="o">.</span><span class="n">keys</span><span class="p">()))</span>
-
-        <span class="c"># Overlapping symbols</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">symbols</span><span class="p">)</span> <span class="o">&lt;</span> <span class="nb">sum</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">space</span><span class="o">.</span><span class="n">symbols</span><span class="p">)</span> <span class="k">for</span> <span class="n">space</span> <span class="ow">in</span> <span class="n">spaces</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Overlapping Random Variables&quot;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">space</span><span class="o">.</span><span class="n">is_Finite</span> <span class="k">for</span> <span class="n">space</span> <span class="ow">in</span> <span class="n">spaces</span><span class="p">):</span>
-            <span class="kn">from</span> <span class="nn">sympy.stats.frv</span> <span class="kn">import</span> <span class="n">ProductFinitePSpace</span>
-            <span class="n">cls</span> <span class="o">=</span> <span class="n">ProductFinitePSpace</span>
-        <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">space</span><span class="o">.</span><span class="n">is_Continuous</span> <span class="k">for</span> <span class="n">space</span> <span class="ow">in</span> <span class="n">spaces</span><span class="p">):</span>
-            <span class="kn">from</span> <span class="nn">sympy.stats.crv</span> <span class="kn">import</span> <span class="n">ProductContinuousPSpace</span>
-            <span class="n">cls</span> <span class="o">=</span> <span class="n">ProductContinuousPSpace</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Basic</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="o">*</span><span class="n">spaces</span><span class="p">))</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">rs_space_dict</span> <span class="o">=</span> <span class="n">rs_space_dict</span>
-
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">spaces</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">values</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">sumsets</span><span class="p">(</span><span class="n">space</span><span class="o">.</span><span class="n">values</span> <span class="k">for</span> <span class="n">space</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">spaces</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">integrate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">rvs</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="n">rvs</span> <span class="o">=</span> <span class="n">rvs</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">values</span>
-        <span class="n">rvs</span> <span class="o">=</span> <span class="nb">frozenset</span><span class="p">(</span><span class="n">rvs</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">space</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">spaces</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">space</span><span class="o">.</span><span class="n">integrate</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">rvs</span> <span class="o">&amp;</span> <span class="n">space</span><span class="o">.</span><span class="n">values</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">domain</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">ProductDomain</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">space</span><span class="o">.</span><span class="n">domain</span> <span class="k">for</span> <span class="n">space</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">spaces</span><span class="p">])</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">density</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s">&quot;Density not available for ProductSpaces&quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">sample</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">dict</span><span class="p">([(</span><span class="n">k</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="k">for</span> <span class="n">space</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">spaces</span>
-            <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">space</span><span class="o">.</span><span class="n">sample</span><span class="p">()</span><span class="o">.</span><span class="n">items</span><span class="p">())])</span>
-
-</div>
-<div class="viewcode-block" id="ProductDomain"><a class="viewcode-back" href="../../../modules/stats.html#sympy.stats.rv.ProductDomain">[docs]</a><span class="k">class</span> <span class="nc">ProductDomain</span><span class="p">(</span><span class="n">RandomDomain</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A domain resulting from the merger of two independent domains</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    sympy.stats.crv.ProductContinuousDomain</span>
-<span class="sd">    sympy.stats.frv.ProductFiniteDomain</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_ProductDomain</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">domains</span><span class="p">):</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="n">sumsets</span><span class="p">([</span><span class="n">domain</span><span class="o">.</span><span class="n">symbols</span> <span class="k">for</span> <span class="n">domain</span> <span class="ow">in</span> <span class="n">domains</span><span class="p">])</span>
-
-        <span class="c"># Flatten any product of products</span>
-        <span class="n">domains2</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">domain</span> <span class="ow">in</span> <span class="n">domains</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">domain</span><span class="o">.</span><span class="n">is_ProductDomain</span><span class="p">:</span>
-                <span class="n">domains2</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">domain</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">domains2</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">domain</span><span class="o">.</span><span class="n">domains</span><span class="p">)</span>
-        <span class="n">domains2</span> <span class="o">=</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="n">domains2</span><span class="p">)</span>
-
-        <span class="n">sym_domain_dict</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">domain</span> <span class="ow">in</span> <span class="n">domains2</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">symbol</span> <span class="ow">in</span> <span class="n">domain</span><span class="o">.</span><span class="n">symbols</span><span class="p">:</span>
-                <span class="n">sym_domain_dict</span><span class="p">[</span><span class="n">symbol</span><span class="p">]</span> <span class="o">=</span> <span class="n">domain</span>
-
-        <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">domain</span><span class="o">.</span><span class="n">is_Finite</span> <span class="k">for</span> <span class="n">domain</span> <span class="ow">in</span> <span class="n">domains2</span><span class="p">):</span>
-            <span class="kn">from</span> <span class="nn">sympy.stats.frv</span> <span class="kn">import</span> <span class="n">ProductFiniteDomain</span>
-            <span class="n">cls</span> <span class="o">=</span> <span class="n">ProductFiniteDomain</span>
-        <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">domain</span><span class="o">.</span><span class="n">is_Continuous</span> <span class="k">for</span> <span class="n">domain</span> <span class="ow">in</span> <span class="n">domains2</span><span class="p">):</span>
-            <span class="kn">from</span> <span class="nn">sympy.stats.crv</span> <span class="kn">import</span> <span class="n">ProductContinuousDomain</span>
-            <span class="n">cls</span> <span class="o">=</span> <span class="n">ProductContinuousDomain</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">RandomDomain</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">domains2</span><span class="p">)</span>
-        <span class="n">obj</span><span class="o">.</span><span class="n">sym_domain_dict</span> <span class="o">=</span> <span class="n">sym_domain_dict</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">domains</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">set</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">ProductSet</span><span class="p">(</span><span class="n">domain</span><span class="o">.</span><span class="n">set</span> <span class="k">for</span> <span class="n">domain</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">domains</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__contains__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
-        <span class="c"># Split event into each subdomain</span>
-        <span class="k">for</span> <span class="n">domain</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">domains</span><span class="p">:</span>
-            <span class="c"># Collect the parts of this event which associate to this domain</span>
-            <span class="n">elem</span> <span class="o">=</span> <span class="nb">frozenset</span><span class="p">([</span><span class="n">item</span> <span class="k">for</span> <span class="n">item</span> <span class="ow">in</span> <span class="n">other</span>
-                <span class="k">if</span> <span class="n">item</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">in</span> <span class="n">domain</span><span class="o">.</span><span class="n">symbols</span><span class="p">])</span>
-            <span class="c"># Test this sub-event</span>
-            <span class="k">if</span> <span class="n">elem</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">domain</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">False</span>
-        <span class="c"># All subevents passed</span>
-        <span class="k">return</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">as_boolean</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">And</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">domain</span><span class="o">.</span><span class="n">as_boolean</span><span class="p">()</span> <span class="k">for</span> <span class="n">domain</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">domains</span><span class="p">])</span>
-
-</div>
-<div class="viewcode-block" id="random_symbols"><a class="viewcode-back" href="../../../modules/stats.html#sympy.stats.rv.random_symbols">[docs]</a><span class="k">def</span> <span class="nf">random_symbols</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns all RandomSymbols within a SymPy Expression.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">RandomSymbol</span><span class="p">))</span>
-    <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-</div>
-<div class="viewcode-block" id="pspace"><a class="viewcode-back" href="../../../modules/stats.html#sympy.stats.rv.pspace">[docs]</a><span class="k">def</span> <span class="nf">pspace</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the underlying Probability Space of a random expression.</span>
-
-<span class="sd">    For internal use.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import pspace, Normal</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.stats.rv import ProductPSpace</span>
-<span class="sd">    &gt;&gt;&gt; X = Normal(&#39;X&#39;, 0, 1)</span>
-<span class="sd">    &gt;&gt;&gt; pspace(2*X + 1) == X.pspace</span>
-<span class="sd">    True</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">rvs</span> <span class="o">=</span> <span class="n">random_symbols</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">rvs</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">None</span>
-    <span class="c"># If only one space present</span>
-    <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">rv</span><span class="o">.</span><span class="n">pspace</span> <span class="o">==</span> <span class="n">rvs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">pspace</span> <span class="k">for</span> <span class="n">rv</span> <span class="ow">in</span> <span class="n">rvs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">rvs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">pspace</span>
-    <span class="c"># Otherwise make a product space</span>
-    <span class="k">return</span> <span class="n">ProductPSpace</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">rv</span><span class="o">.</span><span class="n">pspace</span> <span class="k">for</span> <span class="n">rv</span> <span class="ow">in</span> <span class="n">rvs</span><span class="p">])</span>
-
-</div>
-<span class="k">def</span> <span class="nf">sumsets</span><span class="p">(</span><span class="n">sets</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Union of sets</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="nb">reduce</span><span class="p">(</span><span class="nb">frozenset</span><span class="o">.</span><span class="n">union</span><span class="p">,</span> <span class="n">sets</span><span class="p">,</span> <span class="nb">frozenset</span><span class="p">())</span>
-
-
-<div class="viewcode-block" id="rs_swap"><a class="viewcode-back" href="../../../modules/stats.html#sympy.stats.rv.rs_swap">[docs]</a><span class="k">def</span> <span class="nf">rs_swap</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Build a dictionary to swap RandomSymbols based on their underlying symbol.</span>
-
-<span class="sd">    i.e.</span>
-<span class="sd">    if    X = (&#39;x&#39;, pspace1)</span>
-<span class="sd">    and   Y = (&#39;x&#39;, pspace2)</span>
-<span class="sd">    then X and Y match and the key, value pair</span>
-<span class="sd">    {X:Y} will appear in the result</span>
-
-<span class="sd">    Inputs: collections a and b of random variables which share common symbols</span>
-<span class="sd">    Output: dict mapping RVs in a to RVs in b</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">d</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">for</span> <span class="n">rsa</span> <span class="ow">in</span> <span class="n">a</span><span class="p">:</span>
-        <span class="n">d</span><span class="p">[</span><span class="n">rsa</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">rsb</span> <span class="k">for</span> <span class="n">rsb</span> <span class="ow">in</span> <span class="n">b</span> <span class="k">if</span> <span class="n">rsa</span><span class="o">.</span><span class="n">symbol</span> <span class="o">==</span> <span class="n">rsb</span><span class="o">.</span><span class="n">symbol</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">d</span>
-
-</div>
-<span class="k">def</span> <span class="nf">given</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">condition</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    From a random expression and a condition on that expression creates a new</span>
-<span class="sd">    probability space from the condition and returns the same expression on that</span>
-<span class="sd">    conditional probability space.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import given, density, Die</span>
-<span class="sd">    &gt;&gt;&gt; X = Die(&#39;X&#39;, 6)</span>
-<span class="sd">    &gt;&gt;&gt; Y = given(X, X&gt;3)</span>
-<span class="sd">    &gt;&gt;&gt; density(Y)</span>
-<span class="sd">    {4: 1/3, 5: 1/3, 6: 1/3}</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">random_symbols</span><span class="p">(</span><span class="n">condition</span><span class="p">)</span> <span class="ow">or</span> <span class="n">pspace_independent</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">condition</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">expr</span>
-
-    <span class="c"># Get full probability space of both the expression and the condition</span>
-    <span class="n">fullspace</span> <span class="o">=</span> <span class="n">pspace</span><span class="p">(</span><span class="n">Tuple</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">condition</span><span class="p">))</span>
-    <span class="c"># Build new space given the condition</span>
-    <span class="n">space</span> <span class="o">=</span> <span class="n">fullspace</span><span class="o">.</span><span class="n">conditional_space</span><span class="p">(</span><span class="n">condition</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-    <span class="c"># Dictionary to swap out RandomSymbols in expr with new RandomSymbols</span>
-    <span class="c"># That point to the new conditional space</span>
-    <span class="n">swapdict</span> <span class="o">=</span> <span class="n">rs_swap</span><span class="p">(</span><span class="n">fullspace</span><span class="o">.</span><span class="n">values</span><span class="p">,</span> <span class="n">space</span><span class="o">.</span><span class="n">values</span><span class="p">)</span>
-    <span class="c"># Swap random variables in the expression</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">swapdict</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">expr</span>
-
-
-<span class="k">def</span> <span class="nf">expectation</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">condition</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">numsamples</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the expected value of a random expression</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    expr : Expr containing RandomSymbols</span>
-<span class="sd">        The expression of which you want to compute the expectation value</span>
-<span class="sd">    given : Expr containing RandomSymbols</span>
-<span class="sd">        A conditional expression. E(X, X&gt;0) is expectation of X given X &gt; 0</span>
-<span class="sd">    numsamples : int</span>
-<span class="sd">        Enables sampling and approximates the expectation with this many samples</span>
-<span class="sd">    evalf : Bool (defaults to True)</span>
-<span class="sd">        If sampling return a number rather than a complex expression</span>
-<span class="sd">    evaluate : Bool (defaults to True)</span>
-<span class="sd">        In case of continuous systems return unevaluated integral</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import E, Die</span>
-<span class="sd">    &gt;&gt;&gt; X = Die(&#39;X&#39;, 6)</span>
-<span class="sd">    &gt;&gt;&gt; E(X)</span>
-<span class="sd">    7/2</span>
-<span class="sd">    &gt;&gt;&gt; E(2*X + 1)</span>
-<span class="sd">    8</span>
-
-<span class="sd">    &gt;&gt;&gt; E(X, X&gt;3) # Expectation of X given that it is above 3</span>
-<span class="sd">    5</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">random_symbols</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>  <span class="c"># expr isn&#39;t random?</span>
-        <span class="k">return</span> <span class="n">expr</span>
-    <span class="k">if</span> <span class="n">numsamples</span><span class="p">:</span>  <span class="c"># Computing by monte carlo sampling?</span>
-        <span class="k">return</span> <span class="n">sampling_E</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">condition</span><span class="p">,</span> <span class="n">numsamples</span><span class="o">=</span><span class="n">numsamples</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="c"># Create new expr and recompute E</span>
-    <span class="k">if</span> <span class="n">condition</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>  <span class="c"># If there is a condition</span>
-        <span class="k">return</span> <span class="n">expectation</span><span class="p">(</span><span class="n">given</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">condition</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="c"># A few known statements for efficiency</span>
-
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>  <span class="c"># We know that E is Linear</span>
-        <span class="k">return</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">expectation</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">])</span>
-
-    <span class="c"># Otherwise case is simple, pass work off to the ProbabilitySpace</span>
-    <span class="k">return</span> <span class="n">pspace</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">integrate</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">probability</span><span class="p">(</span><span class="n">condition</span><span class="p">,</span> <span class="n">given_condition</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">numsamples</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Probability that a condition is true, optionally given a second condition</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    expr : Relational containing RandomSymbols</span>
-<span class="sd">        The condition of which you want to compute the probability</span>
-<span class="sd">    given_condition : Relational containing RandomSymbols</span>
-<span class="sd">        A conditional expression. P(X&gt;1, X&gt;0) is expectation of X&gt;1 given X&gt;0</span>
-<span class="sd">    numsamples : int</span>
-<span class="sd">        Enables sampling and approximates the probability with this many samples</span>
-<span class="sd">    evalf : Bool (defaults to True)</span>
-<span class="sd">        If sampling return a number rather than a complex expression</span>
-<span class="sd">    evaluate : Bool (defaults to True)</span>
-<span class="sd">        In case of continuous systems return unevaluated integral</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import P, Die</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Eq</span>
-<span class="sd">    &gt;&gt;&gt; X, Y = Die(&#39;X&#39;, 6), Die(&#39;Y&#39;, 6)</span>
-<span class="sd">    &gt;&gt;&gt; P(X&gt;3)</span>
-<span class="sd">    1/2</span>
-<span class="sd">    &gt;&gt;&gt; P(Eq(X, 5), X&gt;2) # Probability that X == 5 given that X &gt; 2</span>
-<span class="sd">    1/4</span>
-<span class="sd">    &gt;&gt;&gt; P(X&gt;Y)</span>
-<span class="sd">    5/12</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="n">numsamples</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">sampling_P</span><span class="p">(</span><span class="n">condition</span><span class="p">,</span> <span class="n">given_condition</span><span class="p">,</span> <span class="n">numsamples</span><span class="o">=</span><span class="n">numsamples</span><span class="p">,</span>
-                <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">given_condition</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>  <span class="c"># If there is a condition</span>
-        <span class="c"># Recompute on new conditional expr</span>
-        <span class="k">return</span> <span class="n">probability</span><span class="p">(</span><span class="n">given</span><span class="p">(</span><span class="n">condition</span><span class="p">,</span> <span class="n">given_condition</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="c"># Otherwise pass work off to the ProbabilitySpace</span>
-    <span class="k">return</span> <span class="n">pspace</span><span class="p">(</span><span class="n">condition</span><span class="p">)</span><span class="o">.</span><span class="n">probability</span><span class="p">(</span><span class="n">condition</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">density</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">condition</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Probability density of a random expression</span>
-
-<span class="sd">    Optionally given a second condition</span>
-
-<span class="sd">    This density will take on different forms for different types of</span>
-<span class="sd">    probability spaces.</span>
-<span class="sd">    Discrete variables produce Dicts.</span>
-<span class="sd">    Continuous variables produce Lambdas.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import density, Die, Normal</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol</span>
-
-<span class="sd">    &gt;&gt;&gt; D = Die(&#39;D&#39;, 6)</span>
-<span class="sd">    &gt;&gt;&gt; X = Normal(&#39;x&#39;, 0, 1)</span>
-
-<span class="sd">    &gt;&gt;&gt; density(D)</span>
-<span class="sd">    {1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6}</span>
-<span class="sd">    &gt;&gt;&gt; density(2*D)</span>
-<span class="sd">    {2: 1/6, 4: 1/6, 6: 1/6, 8: 1/6, 10: 1/6, 12: 1/6}</span>
-<span class="sd">    &gt;&gt;&gt; density(X)</span>
-<span class="sd">    Lambda(_x, sqrt(2)*exp(-_x**2/2)/(2*sqrt(pi)))</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">condition</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>  <span class="c"># If there is a condition</span>
-        <span class="c"># Recompute on new conditional expr</span>
-        <span class="k">return</span> <span class="n">density</span><span class="p">(</span><span class="n">given</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">condition</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="c"># Otherwise pass work off to the ProbabilitySpace</span>
-    <span class="k">return</span> <span class="n">pspace</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">compute_density</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">cdf</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">condition</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Cumulative Distribution Function of a random expression.</span>
-
-<span class="sd">    optionally given a second condition</span>
-
-<span class="sd">    This density will take on different forms for different types of</span>
-<span class="sd">    probability spaces.</span>
-<span class="sd">    Discrete variables produce Dicts.</span>
-<span class="sd">    Continuous variables produce Lambdas.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import density, Die, Normal, cdf</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Symbol</span>
-
-<span class="sd">    &gt;&gt;&gt; D = Die(&#39;D&#39;, 6)</span>
-<span class="sd">    &gt;&gt;&gt; X = Normal(&#39;X&#39;, 0, 1)</span>
-
-<span class="sd">    &gt;&gt;&gt; density(D)</span>
-<span class="sd">    {1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6}</span>
-<span class="sd">    &gt;&gt;&gt; cdf(D)</span>
-<span class="sd">    {1: 1/6, 2: 1/3, 3: 1/2, 4: 2/3, 5: 5/6, 6: 1}</span>
-<span class="sd">    &gt;&gt;&gt; cdf(3*D, D&gt;2)</span>
-<span class="sd">    {9: 1/4, 12: 1/2, 15: 3/4, 18: 1}</span>
-
-<span class="sd">    &gt;&gt;&gt; cdf(X)</span>
-<span class="sd">    Lambda(_z, erf(sqrt(2)*_z/2)/2 + 1/2)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">condition</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>  <span class="c"># If there is a condition</span>
-        <span class="c"># Recompute on new conditional expr</span>
-        <span class="k">return</span> <span class="n">cdf</span><span class="p">(</span><span class="n">given</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">condition</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="c"># Otherwise pass work off to the ProbabilitySpace</span>
-    <span class="k">return</span> <span class="n">pspace</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span><span class="o">.</span><span class="n">compute_cdf</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">where</span><span class="p">(</span><span class="n">condition</span><span class="p">,</span> <span class="n">given_condition</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the domain where a condition is True.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import where, Die, Normal</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, And</span>
-
-<span class="sd">    &gt;&gt;&gt; D1, D2 = Die(&#39;a&#39;, 6), Die(&#39;b&#39;, 6)</span>
-<span class="sd">    &gt;&gt;&gt; a, b = D1.symbol, D2.symbol</span>
-<span class="sd">    &gt;&gt;&gt; X = Normal(&#39;x&#39;, 0, 1)</span>
-
-<span class="sd">    &gt;&gt;&gt; where(X**2&lt;1)</span>
-<span class="sd">    Domain: And(-1 &lt; x, x &lt; 1)</span>
-
-<span class="sd">    &gt;&gt;&gt; where(X**2&lt;1).set</span>
-<span class="sd">    (-1, 1)</span>
-
-<span class="sd">    &gt;&gt;&gt; where(And(D1&lt;=D2 , D2&lt;3))</span>
-<span class="sd">    Domain: Or(And(a == 1, b == 1), And(a == 1, b == 2), And(a == 2, b == 2))</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">given_condition</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>  <span class="c"># If there is a condition</span>
-        <span class="c"># Recompute on new conditional expr</span>
-        <span class="k">return</span> <span class="n">where</span><span class="p">(</span><span class="n">given</span><span class="p">(</span><span class="n">condition</span><span class="p">,</span> <span class="n">given_condition</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="c"># Otherwise pass work off to the ProbabilitySpace</span>
-    <span class="k">return</span> <span class="n">pspace</span><span class="p">(</span><span class="n">condition</span><span class="p">)</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">condition</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">sample</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">condition</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A realization of the random expression</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Die, sample</span>
-<span class="sd">    &gt;&gt;&gt; X, Y, Z = Die(&#39;X&#39;, 6), Die(&#39;Y&#39;, 6), Die(&#39;Z&#39;, 6)</span>
-
-<span class="sd">    &gt;&gt;&gt; die_roll = sample(X+Y+Z) # A random realization of three dice</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="nb">next</span><span class="p">(</span><span class="n">sample_iter</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">condition</span><span class="p">,</span> <span class="n">numsamples</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span>
-
-
-<span class="k">def</span> <span class="nf">sample_iter</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">condition</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">numsamples</span><span class="o">=</span><span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns an iterator of realizations from the expression given a condition</span>
-
-<span class="sd">    expr: Random expression to be realized</span>
-<span class="sd">    condition: A conditional expression (optional)</span>
-<span class="sd">    numsamples: Length of the iterator (defaults to infinity)</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    --------</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Normal, sample_iter</span>
-<span class="sd">    &gt;&gt;&gt; X = Normal(&#39;X&#39;, 0, 1)</span>
-<span class="sd">    &gt;&gt;&gt; expr = X*X + 3</span>
-<span class="sd">    &gt;&gt;&gt; iterator = sample_iter(expr, numsamples=3)</span>
-<span class="sd">    &gt;&gt;&gt; list(iterator) # doctest: +SKIP</span>
-<span class="sd">    [12, 4, 7]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    Sample</span>
-<span class="sd">    sampling_P</span>
-<span class="sd">    sampling_E</span>
-<span class="sd">    sample_iter_lambdify</span>
-<span class="sd">    sample_iter_subs</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># lambdify is much faster but not as robust</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">sample_iter_lambdify</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">condition</span><span class="p">,</span> <span class="n">numsamples</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-    <span class="c"># use subs when lambdify fails</span>
-    <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">sample_iter_subs</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">condition</span><span class="p">,</span> <span class="n">numsamples</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">sample_iter_lambdify</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">condition</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">numsamples</span><span class="o">=</span><span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    See sample_iter</span>
-
-<span class="sd">    Uses lambdify for computation. This is fast but does not always work.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">condition</span><span class="p">:</span>
-        <span class="n">ps</span> <span class="o">=</span> <span class="n">pspace</span><span class="p">(</span><span class="n">Tuple</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">condition</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">ps</span> <span class="o">=</span> <span class="n">pspace</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="n">rvs</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">ps</span><span class="o">.</span><span class="n">values</span><span class="p">)</span>
-    <span class="n">fn</span> <span class="o">=</span> <span class="n">lambdify</span><span class="p">(</span><span class="n">rvs</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">condition</span><span class="p">:</span>
-        <span class="n">given_fn</span> <span class="o">=</span> <span class="n">lambdify</span><span class="p">(</span><span class="n">rvs</span><span class="p">,</span> <span class="n">condition</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="c"># Check that lambdify can handle the expression</span>
-    <span class="c"># Some operations like Sum can prove difficult</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">ps</span><span class="o">.</span><span class="n">sample</span><span class="p">()</span>  <span class="c"># a dictionary that maps RVs to values</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">d</span><span class="p">[</span><span class="n">rv</span><span class="p">]</span> <span class="k">for</span> <span class="n">rv</span> <span class="ow">in</span> <span class="n">rvs</span><span class="p">]</span>
-        <span class="n">fn</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">condition</span><span class="p">:</span>
-            <span class="n">given_fn</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Expr/condition too complex for lambdify&quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">return_generator</span><span class="p">():</span>
-        <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="k">while</span> <span class="n">count</span> <span class="o">&lt;</span> <span class="n">numsamples</span><span class="p">:</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="n">ps</span><span class="o">.</span><span class="n">sample</span><span class="p">()</span>  <span class="c"># a dictionary that maps RVs to values</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">d</span><span class="p">[</span><span class="n">rv</span><span class="p">]</span> <span class="k">for</span> <span class="n">rv</span> <span class="ow">in</span> <span class="n">rvs</span><span class="p">]</span>
-
-            <span class="k">if</span> <span class="n">condition</span><span class="p">:</span>  <span class="c"># Check that these values satisfy the condition</span>
-                <span class="n">gd</span> <span class="o">=</span> <span class="n">given_fn</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">gd</span><span class="p">,</span> <span class="nb">bool</span><span class="p">):</span>
-                    <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                        <span class="s">&quot;Conditions must not contain free symbols&quot;</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">gd</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>  <span class="c"># If the values don&#39;t satisfy then try again</span>
-                    <span class="k">continue</span>
-
-            <span class="k">yield</span> <span class="n">fn</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-            <span class="n">count</span> <span class="o">+=</span> <span class="mi">1</span>
-    <span class="k">return</span> <span class="n">return_generator</span><span class="p">()</span>
-
-
-<span class="k">def</span> <span class="nf">sample_iter_subs</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">condition</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">numsamples</span><span class="o">=</span><span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    See sample_iter</span>
-
-<span class="sd">    Uses subs for computation. This is slow but almost always works.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">condition</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">ps</span> <span class="o">=</span> <span class="n">pspace</span><span class="p">(</span><span class="n">Tuple</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">condition</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">ps</span> <span class="o">=</span> <span class="n">pspace</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-    <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
-
-    <span class="k">while</span> <span class="n">count</span> <span class="o">&lt;</span> <span class="n">numsamples</span><span class="p">:</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="n">ps</span><span class="o">.</span><span class="n">sample</span><span class="p">()</span>  <span class="c"># a dictionary that maps RVs to values</span>
-
-        <span class="k">if</span> <span class="n">condition</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>  <span class="c"># Check that these values satisfy the condition</span>
-            <span class="n">gd</span> <span class="o">=</span> <span class="n">condition</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">gd</span><span class="p">,</span> <span class="nb">bool</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Conditions must not contain free symbols&quot;</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">gd</span> <span class="ow">is</span> <span class="bp">False</span><span class="p">:</span>  <span class="c"># If the values don&#39;t satisfy then try again</span>
-                <span class="k">continue</span>
-
-        <span class="k">yield</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-
-        <span class="n">count</span> <span class="o">+=</span> <span class="mi">1</span>
-
-
-<span class="k">def</span> <span class="nf">sampling_P</span><span class="p">(</span><span class="n">condition</span><span class="p">,</span> <span class="n">given_condition</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">numsamples</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
-               <span class="n">evalf</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Sampling version of P</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    P</span>
-<span class="sd">    sampling_E</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">count_true</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">count_false</span> <span class="o">=</span> <span class="mi">0</span>
-
-    <span class="n">samples</span> <span class="o">=</span> <span class="n">sample_iter</span><span class="p">(</span><span class="n">condition</span><span class="p">,</span> <span class="n">given_condition</span><span class="p">,</span>
-                          <span class="n">numsamples</span><span class="o">=</span><span class="n">numsamples</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">samples</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="nb">bool</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Conditions must not contain free symbols&quot;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="bp">True</span><span class="p">:</span>
-            <span class="n">count_true</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">count_false</span> <span class="o">+=</span> <span class="mi">1</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="n">count_true</span><span class="p">)</span> <span class="o">/</span> <span class="n">numsamples</span>
-    <span class="k">if</span> <span class="n">evalf</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="k">def</span> <span class="nf">sampling_E</span><span class="p">(</span><span class="n">condition</span><span class="p">,</span> <span class="n">given_condition</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">numsamples</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
-               <span class="n">evalf</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Sampling version of E</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    P</span>
-<span class="sd">    sampling_P</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">samples</span> <span class="o">=</span> <span class="n">sample_iter</span><span class="p">(</span><span class="n">condition</span><span class="p">,</span> <span class="n">given_condition</span><span class="p">,</span>
-                          <span class="n">numsamples</span><span class="o">=</span><span class="n">numsamples</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="o">*</span><span class="nb">list</span><span class="p">(</span><span class="n">samples</span><span class="p">))</span> <span class="o">/</span> <span class="n">numsamples</span>
-    <span class="k">if</span> <span class="n">evalf</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-
-<span class="k">def</span> <span class="nf">dependent</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Dependence of two random expressions</span>
-
-<span class="sd">    Two expressions are independent if knowledge of one does not change</span>
-<span class="sd">    computations on the other.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Normal, dependent, given</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Tuple, Eq</span>
-
-<span class="sd">    &gt;&gt;&gt; X, Y = Normal(&#39;X&#39;, 0, 1), Normal(&#39;Y&#39;, 0, 1)</span>
-<span class="sd">    &gt;&gt;&gt; dependent(X, Y)</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; dependent(2*X + Y, -Y)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; X, Y = given(Tuple(X, Y), Eq(X+Y,3))</span>
-<span class="sd">    &gt;&gt;&gt; dependent(X, Y)</span>
-<span class="sd">    True</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    independent</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">pspace_independent</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="n">z</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;z&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="c"># Dependent if density is unchanged when one is given information about</span>
-    <span class="c"># the other</span>
-    <span class="k">return</span> <span class="p">(</span><span class="n">density</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Eq</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">z</span><span class="p">))</span> <span class="o">!=</span> <span class="n">density</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="ow">or</span>
-            <span class="n">density</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">Eq</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">))</span> <span class="o">!=</span> <span class="n">density</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
-
-
-<span class="k">def</span> <span class="nf">independent</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Independence of two random expressions</span>
-
-<span class="sd">    Two expressions are independent if knowledge of one does not change</span>
-<span class="sd">    computations on the other.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.stats import Normal, independent, given</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Tuple, Eq</span>
-
-<span class="sd">    &gt;&gt;&gt; X, Y = Normal(&#39;X&#39;, 0, 1), Normal(&#39;Y&#39;, 0, 1)</span>
-<span class="sd">    &gt;&gt;&gt; independent(X, Y)</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; independent(2*X + Y, -Y)</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; X, Y = given(Tuple(X, Y), Eq(X+Y,3))</span>
-<span class="sd">    &gt;&gt;&gt; independent(X, Y)</span>
-<span class="sd">    False</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    dependent</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="ow">not</span> <span class="n">dependent</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">pspace_independent</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Tests for independence between a and b by checking if their PSpaces have</span>
-<span class="sd">    overlapping symbols. This is a sufficient but not necessary condition for</span>
-<span class="sd">    independence and is intended to be used internally.</span>
-<span class="sd">    Note:</span>
-<span class="sd">    pspace_independent(a,b) implies independent(a,b)</span>
-<span class="sd">    independent(a,b) does not imply pspace_independent(a,b)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">a_symbols</span> <span class="o">=</span> <span class="n">pspace</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">.</span><span class="n">symbols</span>
-    <span class="n">b_symbols</span> <span class="o">=</span> <span class="n">pspace</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">symbols</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">a_symbols</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="n">b_symbols</span><span class="p">))</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">True</span>
-    <span class="k">return</span> <span class="bp">None</span>
-
-
-<span class="k">def</span> <span class="nf">rv_subs</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">symbols</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Given a random expression replace all random variables with their symbols.</span>
-
-<span class="sd">    If symbols keyword is given restrict the swap to only the symbols listed.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">symbols</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="n">random_symbols</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="n">swapdict</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">([(</span><span class="n">rv</span><span class="p">,</span> <span class="n">rv</span><span class="o">.</span><span class="n">symbol</span><span class="p">)</span> <span class="k">for</span> <span class="n">rv</span> <span class="ow">in</span> <span class="n">symbols</span><span class="p">])</span>
-    <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">swapdict</span><span class="p">)</span>
-</pre></div>
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-            
-  <h1>Source code for sympy.tensor.index_methods</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Module with functions operating on IndexedBase, Indexed and Idx objects</span>
-
-<span class="sd">    - Check shape conformance</span>
-<span class="sd">    - Determine indices in resulting expression</span>
-
-<span class="sd">    etc.</span>
-
-<span class="sd">    Methods in this module could be implemented by calling methods on Expr</span>
-<span class="sd">    objects instead.  When things stabilize this could be a useful</span>
-<span class="sd">    refactoring.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">sympy.tensor.indexed</span> <span class="kn">import</span> <span class="n">Idx</span><span class="p">,</span> <span class="n">Indexed</span>
-<span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">exp</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">C</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="nb">reduce</span>
-<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="nb">reduce</span>
-
-
-<span class="k">class</span> <span class="nc">IndexConformanceException</span><span class="p">(</span><span class="ne">Exception</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-
-<span class="k">def</span> <span class="nf">_remove_repeated</span><span class="p">(</span><span class="n">inds</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Removes repeated objects from sequences</span>
-
-<span class="sd">    Returns a set of the unique objects and a tuple of all that have been</span>
-<span class="sd">    removed.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.tensor.index_methods import _remove_repeated</span>
-<span class="sd">    &gt;&gt;&gt; l1 = [1, 2, 3, 2]</span>
-<span class="sd">    &gt;&gt;&gt; _remove_repeated(l1)</span>
-<span class="sd">    (set([1, 3]), (2,))</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">sum_index</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">inds</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">sum_index</span><span class="p">:</span>
-            <span class="n">sum_index</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">sum_index</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">inds</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">inds</span> <span class="k">if</span> <span class="ow">not</span> <span class="n">sum_index</span><span class="p">[</span><span class="n">x</span><span class="p">]]</span>
-    <span class="k">return</span> <span class="nb">set</span><span class="p">(</span><span class="n">inds</span><span class="p">),</span> <span class="nb">tuple</span><span class="p">([</span> <span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">sum_index</span> <span class="k">if</span> <span class="n">sum_index</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="p">])</span>
-
-
-<span class="k">def</span> <span class="nf">_get_indices_Mul</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">return_dummies</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Determine the outer indices of a Mul object.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.tensor.index_methods import _get_indices_Mul</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.tensor.indexed import IndexedBase, Idx</span>
-<span class="sd">    &gt;&gt;&gt; i, j, k = list(map(Idx, [&#39;i&#39;, &#39;j&#39;, &#39;k&#39;]))</span>
-<span class="sd">    &gt;&gt;&gt; x = IndexedBase(&#39;x&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; y = IndexedBase(&#39;y&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; _get_indices_Mul(x[i, k]*y[j, k])</span>
-<span class="sd">    (set([i, j]), {})</span>
-<span class="sd">    &gt;&gt;&gt; _get_indices_Mul(x[i, k]*y[j, k], return_dummies=True)</span>
-<span class="sd">    (set([i, j]), {}, (k,))</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">inds</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">get_indices</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-    <span class="n">inds</span><span class="p">,</span> <span class="n">syms</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">inds</span><span class="p">))</span>
-
-    <span class="n">inds</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">list</span><span class="p">,</span> <span class="n">inds</span><span class="p">))</span>
-    <span class="n">inds</span> <span class="o">=</span> <span class="nb">reduce</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">,</span> <span class="n">inds</span><span class="p">)</span>
-    <span class="n">inds</span><span class="p">,</span> <span class="n">dummies</span> <span class="o">=</span> <span class="n">_remove_repeated</span><span class="p">(</span><span class="n">inds</span><span class="p">)</span>
-
-    <span class="n">symmetry</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">syms</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">pair</span> <span class="ow">in</span> <span class="n">s</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">pair</span> <span class="ow">in</span> <span class="n">symmetry</span><span class="p">:</span>
-                <span class="n">symmetry</span><span class="p">[</span><span class="n">pair</span><span class="p">]</span> <span class="o">*=</span> <span class="n">s</span><span class="p">[</span><span class="n">pair</span><span class="p">]</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">symmetry</span><span class="p">[</span><span class="n">pair</span><span class="p">]</span> <span class="o">=</span> <span class="n">s</span><span class="p">[</span><span class="n">pair</span><span class="p">]</span>
-
-    <span class="k">if</span> <span class="n">return_dummies</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">inds</span><span class="p">,</span> <span class="n">symmetry</span><span class="p">,</span> <span class="n">dummies</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">inds</span><span class="p">,</span> <span class="n">symmetry</span>
-
-
-<span class="k">def</span> <span class="nf">_get_indices_Pow</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Determine outer indices of a power or an exponential.</span>
-
-<span class="sd">    A power is considered a universal function, so that the indices of a Pow is</span>
-<span class="sd">    just the collection of indices present in the expression.  This may be</span>
-<span class="sd">    viewed as a bit inconsistent in the special case:</span>
-
-<span class="sd">        x[i]**2 = x[i]*x[i]                                                      (1)</span>
-
-<span class="sd">    The above expression could have been interpreted as the contraction of x[i]</span>
-<span class="sd">    with itself, but we choose instead to interpret it as a function</span>
-
-<span class="sd">        lambda y: y**2</span>
-
-<span class="sd">    applied to each element of x (a universal function in numpy terms).  In</span>
-<span class="sd">    order to allow an interpretation of (1) as a contraction, we need</span>
-<span class="sd">    contravariant and covariant Idx subclasses.  (FIXME: this is not yet</span>
-<span class="sd">    implemented)</span>
-
-<span class="sd">    Expressions in the base or exponent are subject to contraction as usual,</span>
-<span class="sd">    but an index that is present in the exponent, will not be considered</span>
-<span class="sd">    contractable with its own base.  Note however, that indices in the same</span>
-<span class="sd">    exponent can be contracted with each other.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.tensor.index_methods import _get_indices_Pow</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Pow, exp, IndexedBase, Idx</span>
-<span class="sd">    &gt;&gt;&gt; A = IndexedBase(&#39;A&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; x = IndexedBase(&#39;x&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; i, j, k = list(map(Idx, [&#39;i&#39;, &#39;j&#39;, &#39;k&#39;]))</span>
-<span class="sd">    &gt;&gt;&gt; _get_indices_Pow(exp(A[i, j]*x[j]))</span>
-<span class="sd">    (set([i]), {})</span>
-<span class="sd">    &gt;&gt;&gt; _get_indices_Pow(Pow(x[i], x[i]))</span>
-<span class="sd">    (set([i]), {})</span>
-<span class="sd">    &gt;&gt;&gt; _get_indices_Pow(Pow(A[i, j]*x[j], x[i]))</span>
-<span class="sd">    (set([i]), {})</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-    <span class="n">binds</span><span class="p">,</span> <span class="n">bsyms</span> <span class="o">=</span> <span class="n">get_indices</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-    <span class="n">einds</span><span class="p">,</span> <span class="n">esyms</span> <span class="o">=</span> <span class="n">get_indices</span><span class="p">(</span><span class="n">exp</span><span class="p">)</span>
-
-    <span class="n">inds</span> <span class="o">=</span> <span class="n">binds</span> <span class="o">|</span> <span class="n">einds</span>
-
-    <span class="c"># FIXME: symmetries from power needs to check special cases, else nothing</span>
-    <span class="n">symmetries</span> <span class="o">=</span> <span class="p">{}</span>
-
-    <span class="k">return</span> <span class="n">inds</span><span class="p">,</span> <span class="n">symmetries</span>
-
-
-<span class="k">def</span> <span class="nf">_get_indices_Add</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Determine outer indices of an Add object.</span>
-
-<span class="sd">    In a sum, each term must have the same set of outer indices.  A valid</span>
-<span class="sd">    expression could be</span>
-
-<span class="sd">        x(i)*y(j) - x(j)*y(i)</span>
-
-<span class="sd">    But we do not allow expressions like:</span>
-
-<span class="sd">        x(i)*y(j) - z(j)*z(j)</span>
-
-<span class="sd">    FIXME: Add support for Numpy broadcasting</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.tensor.index_methods import _get_indices_Add</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.tensor.indexed import IndexedBase, Idx</span>
-<span class="sd">    &gt;&gt;&gt; i, j, k = list(map(Idx, [&#39;i&#39;, &#39;j&#39;, &#39;k&#39;]))</span>
-<span class="sd">    &gt;&gt;&gt; x = IndexedBase(&#39;x&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; y = IndexedBase(&#39;y&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; _get_indices_Add(x[i] + x[k]*y[i, k])</span>
-<span class="sd">    (set([i]), {})</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">inds</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">get_indices</span><span class="p">,</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-    <span class="n">inds</span><span class="p">,</span> <span class="n">syms</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">inds</span><span class="p">))</span>
-
-    <span class="c"># allow broadcast of scalars</span>
-    <span class="n">non_scalars</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">inds</span> <span class="k">if</span> <span class="n">x</span> <span class="o">!=</span> <span class="nb">set</span><span class="p">()]</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">non_scalars</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">set</span><span class="p">(),</span> <span class="p">{}</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">([</span><span class="n">x</span> <span class="o">==</span> <span class="n">non_scalars</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">non_scalars</span><span class="p">[</span><span class="mi">1</span><span class="p">:]]):</span>
-        <span class="k">raise</span> <span class="n">IndexConformanceException</span><span class="p">(</span><span class="s">&quot;Indices are not consistent: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">expr</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">reduce</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">x</span> <span class="o">!=</span> <span class="n">y</span> <span class="ow">or</span> <span class="n">y</span><span class="p">,</span> <span class="n">syms</span><span class="p">):</span>
-        <span class="n">symmetries</span> <span class="o">=</span> <span class="n">syms</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="c"># FIXME: search for symmetries</span>
-        <span class="n">symmetries</span> <span class="o">=</span> <span class="p">{}</span>
-
-    <span class="k">return</span> <span class="n">non_scalars</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">symmetries</span>
-
-
-<div class="viewcode-block" id="get_indices"><a class="viewcode-back" href="../../../modules/tensor/index_methods.html#sympy.tensor.index_methods.get_indices">[docs]</a><span class="k">def</span> <span class="nf">get_indices</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Determine the outer indices of expression ``expr``</span>
-
-<span class="sd">    By *outer* we mean indices that are not summation indices.  Returns a set</span>
-<span class="sd">    and a dict.  The set contains outer indices and the dict contains</span>
-<span class="sd">    information about index symmetries.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.tensor.index_methods import get_indices</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.tensor import IndexedBase, Idx</span>
-<span class="sd">    &gt;&gt;&gt; x, y, A = list(map(IndexedBase, [&#39;x&#39;, &#39;y&#39;, &#39;A&#39;]))</span>
-<span class="sd">    &gt;&gt;&gt; i, j, a, z = symbols(&#39;i j a z&#39;, integer=True)</span>
-
-<span class="sd">    The indices of the total expression is determined, Repeated indices imply a</span>
-<span class="sd">    summation, for instance the trace of a matrix A:</span>
-
-<span class="sd">    &gt;&gt;&gt; get_indices(A[i, i])</span>
-<span class="sd">    (set(), {})</span>
-
-<span class="sd">    In the case of many terms, the terms are required to have identical</span>
-<span class="sd">    outer indices.  Else an IndexConformanceException is raised.</span>
-
-<span class="sd">    &gt;&gt;&gt; get_indices(x[i] + A[i, j]*y[j])</span>
-<span class="sd">    (set([i]), {})</span>
-
-<span class="sd">    :Exceptions:</span>
-
-<span class="sd">    An IndexConformanceException means that the terms ar not compatible, e.g.</span>
-
-<span class="sd">    &gt;&gt;&gt; get_indices(x[i] + y[j])                #doctest: +SKIP</span>
-<span class="sd">            (...)</span>
-<span class="sd">    IndexConformanceException: Indices are not consistent: x(i) + y(j)</span>
-
-<span class="sd">    .. warning::</span>
-<span class="sd">       The concept of *outer* indices applies recursively, starting on the deepest</span>
-<span class="sd">       level.  This implies that dummies inside parenthesis are assumed to be</span>
-<span class="sd">       summed first, so that the following expression is handled gracefully:</span>
-
-<span class="sd">       &gt;&gt;&gt; get_indices((x[i] + A[i, j]*y[j])*x[j])</span>
-<span class="sd">       (set([i, j]), {})</span>
-
-<span class="sd">       This is correct and may appear convenient, but you need to be careful</span>
-<span class="sd">       with this as SymPy will happily .expand() the product, if requested.  The</span>
-<span class="sd">       resulting expression would mix the outer ``j`` with the dummies inside</span>
-<span class="sd">       the parenthesis, which makes it a different expression.  To be on the</span>
-<span class="sd">       safe side, it is best to avoid such ambiguities by using unique indices</span>
-<span class="sd">       for all contractions that should be held separate.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># We call ourself recursively to determine indices of sub expressions.</span>
-
-    <span class="c"># break recursion</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Indexed</span><span class="p">):</span>
-        <span class="n">c</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">indices</span>
-        <span class="n">inds</span><span class="p">,</span> <span class="n">dummies</span> <span class="o">=</span> <span class="n">_remove_repeated</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">inds</span><span class="p">,</span> <span class="p">{}</span>
-    <span class="k">elif</span> <span class="n">expr</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">set</span><span class="p">(),</span> <span class="p">{}</span>
-    <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">set</span><span class="p">(),</span> <span class="p">{}</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Idx</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">set</span><span class="p">([</span><span class="n">expr</span><span class="p">]),</span> <span class="p">{}</span>
-
-    <span class="c"># recurse via specialized functions</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_get_indices_Mul</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">_get_indices_Add</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">_get_indices_Pow</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Piecewise</span><span class="p">):</span>
-            <span class="c"># FIXME:  No support for Piecewise yet</span>
-            <span class="k">return</span> <span class="nb">set</span><span class="p">(),</span> <span class="p">{}</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Function</span><span class="p">):</span>
-            <span class="c"># Support ufunc like behaviour by returning indices from arguments.</span>
-            <span class="c"># Functions do not interpret repeated indices across argumnts</span>
-            <span class="c"># as summation</span>
-            <span class="n">ind0</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-            <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-                <span class="n">ind</span><span class="p">,</span> <span class="n">sym</span> <span class="o">=</span> <span class="n">get_indices</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-                <span class="n">ind0</span> <span class="o">|=</span> <span class="n">ind</span>
-            <span class="k">return</span> <span class="n">ind0</span><span class="p">,</span> <span class="n">sym</span>
-
-        <span class="c"># this test is expensive, so it should be at the end</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Indexed</span><span class="p">):</span>
-            <span class="k">return</span> <span class="nb">set</span><span class="p">(),</span> <span class="p">{}</span>
-        <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-            <span class="s">&quot;FIXME: No specialized handling of type </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">type</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="get_contraction_structure"><a class="viewcode-back" href="../../../modules/tensor/index_methods.html#sympy.tensor.index_methods.get_contraction_structure">[docs]</a><span class="k">def</span> <span class="nf">get_contraction_structure</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Determine dummy indices of ``expr`` and describe it&#39;s structure</span>
-
-<span class="sd">    By *dummy* we mean indices that are summation indices.</span>
-
-<span class="sd">    The stucture of the expression is determined and described as follows:</span>
-
-<span class="sd">    1) A conforming summation of Indexed objects is described with a dict where</span>
-<span class="sd">       the keys are summation indices and the corresponding values are sets</span>
-<span class="sd">       containing all terms for which the summation applies.  All Add objects</span>
-<span class="sd">       in the SymPy expression tree are described like this.</span>
-
-<span class="sd">    2) For all nodes in the SymPy expression tree that are *not* of type Add, the</span>
-<span class="sd">       following applies:</span>
-
-<span class="sd">       If a node discovers contractions in one of it&#39;s arguments, the node</span>
-<span class="sd">       itself will be stored as a key in the dict.  For that key, the</span>
-<span class="sd">       corresponding value is a list of dicts, each of which is the result of a</span>
-<span class="sd">       recursive call to get_contraction_structure().  The list contains only</span>
-<span class="sd">       dicts for the non-trivial deeper contractions, ommitting dicts with None</span>
-<span class="sd">       as the one and only key.</span>
-
-<span class="sd">    .. Note:: The presence of expressions among the dictinary keys indicates</span>
-<span class="sd">       multiple levels of index contractions.  A nested dict displays nested</span>
-<span class="sd">       contractions and may itself contain dicts from a deeper level.  In</span>
-<span class="sd">       practical calculations the summation in the deepest nested level must be</span>
-<span class="sd">       calculated first so that the outer expression can access the resulting</span>
-<span class="sd">       indexed object.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.tensor.index_methods import get_contraction_structure</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, default_sort_key</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.tensor import IndexedBase, Idx</span>
-<span class="sd">    &gt;&gt;&gt; x, y, A = list(map(IndexedBase, [&#39;x&#39;, &#39;y&#39;, &#39;A&#39;]))</span>
-<span class="sd">    &gt;&gt;&gt; i, j, k, l = list(map(Idx, [&#39;i&#39;, &#39;j&#39;, &#39;k&#39;, &#39;l&#39;]))</span>
-<span class="sd">    &gt;&gt;&gt; get_contraction_structure(x[i]*y[i] + A[j, j])</span>
-<span class="sd">    {(i,): set([x[i]*y[i]]), (j,): set([A[j, j]])}</span>
-<span class="sd">    &gt;&gt;&gt; get_contraction_structure(x[i]*y[j])</span>
-<span class="sd">    {None: set([x[i]*y[j]])}</span>
-
-<span class="sd">    A multiplication of contracted factors results in nested dicts representing</span>
-<span class="sd">    the internal contractions.</span>
-
-<span class="sd">    &gt;&gt;&gt; d = get_contraction_structure(x[i, i]*y[j, j])</span>
-<span class="sd">    &gt;&gt;&gt; sorted(list(d.keys()), key=default_sort_key)</span>
-<span class="sd">    [None, x[i, i]*y[j, j]]</span>
-
-<span class="sd">    In this case, the product has no contractions:</span>
-
-<span class="sd">    &gt;&gt;&gt; d[None]</span>
-<span class="sd">    set([x[i, i]*y[j, j]])</span>
-
-<span class="sd">    Factors are contracted &quot;first&quot;:</span>
-
-<span class="sd">    &gt;&gt;&gt; sorted(d[x[i, i]*y[j, j]], key=default_sort_key)</span>
-<span class="sd">    [{(i,): set([x[i, i]])}, {(j,): set([y[j, j]])}]</span>
-
-<span class="sd">    A parenthesized Add object is also returned as a nested dictionary.  The</span>
-<span class="sd">    term containing the parenthesis is a Mul with a contraction among the</span>
-<span class="sd">    arguments, so it will be found as a key in the result.  It stores the</span>
-<span class="sd">    dictionary resulting from a recursive call on the Add expression.</span>
-
-<span class="sd">    &gt;&gt;&gt; d = get_contraction_structure(x[i]*(y[i] + A[i, j]*x[j]))</span>
-<span class="sd">    &gt;&gt;&gt; sorted(list(d.keys()), key=default_sort_key)</span>
-<span class="sd">    [x[i]*(y[i] + A[i, j]*x[j]), (i,)]</span>
-<span class="sd">    &gt;&gt;&gt; d[(i,)]</span>
-<span class="sd">    set([x[i]*(y[i] + A[i, j]*x[j])])</span>
-<span class="sd">    &gt;&gt;&gt; d[x[i]*(A[i, j]*x[j] + y[i])]</span>
-<span class="sd">    [{None: set([y[i]]), (j,): set([A[i, j]*x[j]])}]</span>
-
-<span class="sd">    Powers with contractions in either base or exponent will also be found as</span>
-<span class="sd">    keys in the dictionary, mapping to a list of results from recursive calls:</span>
-
-<span class="sd">    &gt;&gt;&gt; d = get_contraction_structure(A[j, j]**A[i, i])</span>
-<span class="sd">    &gt;&gt;&gt; d[None]</span>
-<span class="sd">    set([A[j, j]**A[i, i]])</span>
-<span class="sd">    &gt;&gt;&gt; nested_contractions = d[A[j, j]**A[i, i]]</span>
-<span class="sd">    &gt;&gt;&gt; nested_contractions[0]</span>
-<span class="sd">    {(j,): set([A[j, j]])}</span>
-<span class="sd">    &gt;&gt;&gt; nested_contractions[1]</span>
-<span class="sd">    {(i,): set([A[i, i]])}</span>
-
-<span class="sd">    The description of the contraction structure may appear complicated when</span>
-<span class="sd">    represented with a string in the above examples, but it is easy to iterate</span>
-<span class="sd">    over:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import Expr</span>
-<span class="sd">    &gt;&gt;&gt; for key in d:</span>
-<span class="sd">    ...     if isinstance(key, Expr):</span>
-<span class="sd">    ...         continue</span>
-<span class="sd">    ...     for term in d[key]:</span>
-<span class="sd">    ...         if term in d:</span>
-<span class="sd">    ...             # treat deepest contraction first</span>
-<span class="sd">    ...             pass</span>
-<span class="sd">    ...     # treat outermost contactions here</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="c"># We call ourself recursively to inspect sub expressions.</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Indexed</span><span class="p">):</span>
-        <span class="n">junk</span><span class="p">,</span> <span class="n">key</span> <span class="o">=</span> <span class="n">_remove_repeated</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">indices</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">{</span><span class="n">key</span> <span class="ow">or</span> <span class="bp">None</span><span class="p">:</span> <span class="nb">set</span><span class="p">([</span><span class="n">expr</span><span class="p">])}</span>
-    <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Atom</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">{</span><span class="bp">None</span><span class="p">:</span> <span class="nb">set</span><span class="p">([</span><span class="n">expr</span><span class="p">])}</span>
-    <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-        <span class="n">junk</span><span class="p">,</span> <span class="n">junk</span><span class="p">,</span> <span class="n">key</span> <span class="o">=</span> <span class="n">_get_indices_Mul</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">return_dummies</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">{</span><span class="n">key</span> <span class="ow">or</span> <span class="bp">None</span><span class="p">:</span> <span class="nb">set</span><span class="p">([</span><span class="n">expr</span><span class="p">])}</span>
-        <span class="c"># recurse on every factor</span>
-        <span class="n">nested</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">fac</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">facd</span> <span class="o">=</span> <span class="n">get_contraction_structure</span><span class="p">(</span><span class="n">fac</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="bp">None</span> <span class="ow">in</span> <span class="n">facd</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">facd</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="n">nested</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">facd</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">nested</span><span class="p">:</span>
-            <span class="n">result</span><span class="p">[</span><span class="n">expr</span><span class="p">]</span> <span class="o">=</span> <span class="n">nested</span>
-        <span class="k">return</span> <span class="n">result</span>
-    <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Pow</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">exp</span><span class="p">):</span>
-        <span class="c"># recurse in base and exp separately.  If either has internal</span>
-        <span class="c"># contractions we must include ourselves as a key in the returned dict</span>
-        <span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-        <span class="n">dbase</span> <span class="o">=</span> <span class="n">get_contraction_structure</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-        <span class="n">dexp</span> <span class="o">=</span> <span class="n">get_contraction_structure</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-
-        <span class="n">dicts</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">dbase</span><span class="p">,</span> <span class="n">dexp</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="bp">None</span> <span class="ow">in</span> <span class="n">d</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">d</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="n">dicts</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">{</span><span class="bp">None</span><span class="p">:</span> <span class="nb">set</span><span class="p">([</span><span class="n">expr</span><span class="p">])}</span>
-        <span class="k">if</span> <span class="n">dicts</span><span class="p">:</span>
-            <span class="n">result</span><span class="p">[</span><span class="n">expr</span><span class="p">]</span> <span class="o">=</span> <span class="n">dicts</span>
-        <span class="k">return</span> <span class="n">result</span>
-    <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-        <span class="c"># Note: we just collect all terms with identical summation indices, We</span>
-        <span class="c"># do nothing to identify equivalent terms here, as this would require</span>
-        <span class="c"># substitutions or pattern matching in expressions of unknown</span>
-        <span class="c"># complexity.</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="c"># recurse on every term</span>
-            <span class="n">d</span> <span class="o">=</span> <span class="n">get_contraction_structure</span><span class="p">(</span><span class="n">term</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">key</span> <span class="ow">in</span> <span class="n">d</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">key</span> <span class="ow">in</span> <span class="n">result</span><span class="p">:</span>
-                    <span class="n">result</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">|=</span> <span class="n">d</span><span class="p">[</span><span class="n">key</span><span class="p">]</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">result</span><span class="p">[</span><span class="n">key</span><span class="p">]</span> <span class="o">=</span> <span class="n">d</span><span class="p">[</span><span class="n">key</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Piecewise</span><span class="p">):</span>
-        <span class="c"># FIXME:  No support for Piecewise yet</span>
-        <span class="k">return</span> <span class="p">{</span><span class="bp">None</span><span class="p">:</span> <span class="n">expr</span><span class="p">}</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Function</span><span class="p">):</span>
-        <span class="c"># Collect non-trivial contraction structures in each argument</span>
-        <span class="c"># We do not report repeated indices in separate arguments as a</span>
-        <span class="c"># contraction</span>
-        <span class="n">deeplist</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">deep</span> <span class="o">=</span> <span class="n">get_contraction_structure</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="bp">None</span> <span class="ow">in</span> <span class="n">deep</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">deep</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="n">deeplist</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">deep</span><span class="p">)</span>
-        <span class="n">d</span> <span class="o">=</span> <span class="p">{</span><span class="bp">None</span><span class="p">:</span> <span class="nb">set</span><span class="p">([</span><span class="n">expr</span><span class="p">])}</span>
-        <span class="k">if</span> <span class="n">deeplist</span><span class="p">:</span>
-            <span class="n">d</span><span class="p">[</span><span class="n">expr</span><span class="p">]</span> <span class="o">=</span> <span class="n">deeplist</span>
-        <span class="k">return</span> <span class="n">d</span>
-
-    <span class="c"># this test is expensive, so it should be at the end</span>
-    <span class="k">elif</span> <span class="ow">not</span> <span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Indexed</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">{</span><span class="bp">None</span><span class="p">:</span> <span class="nb">set</span><span class="p">([</span><span class="n">expr</span><span class="p">])}</span>
-    <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span>
-        <span class="s">&quot;FIXME: No specialized handling of type </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">type</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span></div>
-</pre></div>
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deleted file mode 100644
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-            
-  <h1>Source code for sympy.tensor.indexed</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Module that defines indexed objects</span>
-
-<span class="sd">    The classes IndexedBase, Indexed and Idx would represent a matrix element</span>
-<span class="sd">    M[i, j] as in the following graph::</span>
-
-<span class="sd">       1) The Indexed class represents the entire indexed object.</span>
-<span class="sd">                  |</span>
-<span class="sd">               ___|___</span>
-<span class="sd">              &#39;       &#39;</span>
-<span class="sd">               M[i, j]</span>
-<span class="sd">              /   \__\______</span>
-<span class="sd">              |             |</span>
-<span class="sd">              |             |</span>
-<span class="sd">              |     2) The Idx class represent indices and each Idx can</span>
-<span class="sd">              |        optionally contain information about its range.</span>
-<span class="sd">              |</span>
-<span class="sd">        3) IndexedBase represents the `stem&#39; of an indexed object, here `M&#39;.</span>
-<span class="sd">           The stem used by itself is usually taken to represent the entire</span>
-<span class="sd">           array.</span>
-
-<span class="sd">    There can be any number of indices on an Indexed object.  No</span>
-<span class="sd">    transformation properties are implemented in these Base objects, but</span>
-<span class="sd">    implicit contraction of repeated indices is supported.</span>
-
-<span class="sd">    Note that the support for complicated (i.e. non-atomic) integer</span>
-<span class="sd">    expressions as indices is limited.  (This should be improved in</span>
-<span class="sd">    future releases.)</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    To express the above matrix element example you would write:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.tensor import IndexedBase, Idx</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; M = IndexedBase(&#39;M&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; i, j = symbols(&#39;i j&#39;, cls=Idx)</span>
-<span class="sd">    &gt;&gt;&gt; M[i, j]</span>
-<span class="sd">    M[i, j]</span>
-
-<span class="sd">    Repeated indices in a product implies a summation, so to express a</span>
-<span class="sd">    matrix-vector product in terms of Indexed objects:</span>
-
-<span class="sd">    &gt;&gt;&gt; x = IndexedBase(&#39;x&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; M[i, j]*x[j]</span>
-<span class="sd">    M[i, j]*x[j]</span>
-
-<span class="sd">    If the indexed objects will be converted to component based arrays, e.g.</span>
-<span class="sd">    with the code printers or the autowrap framework, you also need to provide</span>
-<span class="sd">    (symbolic or numerical) dimensions.  This can be done by passing an</span>
-<span class="sd">    optional shape parameter to IndexedBase upon construction:</span>
-
-<span class="sd">    &gt;&gt;&gt; dim1, dim2 = symbols(&#39;dim1 dim2&#39;, integer=True)</span>
-<span class="sd">    &gt;&gt;&gt; A = IndexedBase(&#39;A&#39;, shape=(dim1, 2*dim1, dim2))</span>
-<span class="sd">    &gt;&gt;&gt; A.shape</span>
-<span class="sd">    (dim1, 2*dim1, dim2)</span>
-<span class="sd">    &gt;&gt;&gt; A[i, j, 3].shape</span>
-<span class="sd">    (dim1, 2*dim1, dim2)</span>
-
-<span class="sd">    If an IndexedBase object has no shape information, it is assumed that the</span>
-<span class="sd">    array is as large as the ranges of it&#39;s indices:</span>
-
-<span class="sd">    &gt;&gt;&gt; n, m = symbols(&#39;n m&#39;, integer=True)</span>
-<span class="sd">    &gt;&gt;&gt; i = Idx(&#39;i&#39;, m)</span>
-<span class="sd">    &gt;&gt;&gt; j = Idx(&#39;j&#39;, n)</span>
-<span class="sd">    &gt;&gt;&gt; M[i, j].shape</span>
-<span class="sd">    (m, n)</span>
-<span class="sd">    &gt;&gt;&gt; M[i, j].ranges</span>
-<span class="sd">    [(0, m - 1), (0, n - 1)]</span>
-
-<span class="sd">    The above can be compared with the following:</span>
-
-<span class="sd">    &gt;&gt;&gt; A[i, 2, j].shape</span>
-<span class="sd">    (dim1, 2*dim1, dim2)</span>
-<span class="sd">    &gt;&gt;&gt; A[i, 2, j].ranges</span>
-<span class="sd">    [(0, m - 1), None, (0, n - 1)]</span>
-
-<span class="sd">    To analyze the structure of indexed expressions, you can use the methods</span>
-<span class="sd">    get_indices() and get_contraction_structure():</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.tensor import get_indices, get_contraction_structure</span>
-<span class="sd">    &gt;&gt;&gt; get_indices(A[i, j, j])</span>
-<span class="sd">    (set([i]), {})</span>
-<span class="sd">    &gt;&gt;&gt; get_contraction_structure(A[i, j, j])</span>
-<span class="sd">    {(j,): set([A[i, j, j]])}</span>
-
-<span class="sd">    See the appropriate docstrings for a detailed explanation of the output.</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="c">#   TODO:  (some ideas for improvement)</span>
-<span class="c">#</span>
-<span class="c">#   o test and guarantee numpy compatibility</span>
-<span class="c">#      - implement full support for broadcasting</span>
-<span class="c">#      - strided arrays</span>
-<span class="c">#</span>
-<span class="c">#   o more functions to analyze indexed expressions</span>
-<span class="c">#      - identify standard constructs, e.g matrix-vector product in a subexpression</span>
-<span class="c">#</span>
-<span class="c">#   o functions to generate component based arrays (numpy and sympy.Matrix)</span>
-<span class="c">#      - generate a single array directly from Indexed</span>
-<span class="c">#      - convert simple sub-expressions</span>
-<span class="c">#</span>
-<span class="c">#   o sophisticated indexing (possibly in subclasses to preserve simplicity)</span>
-<span class="c">#      - Idx with range smaller than dimension of Indexed</span>
-<span class="c">#      - Idx with stepsize != 1</span>
-<span class="c">#      - Idx with step determined by function call</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">S</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">is_sequence</span>
-
-
-<span class="k">class</span> <span class="nc">IndexException</span><span class="p">(</span><span class="ne">Exception</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-
-<div class="viewcode-block" id="Indexed"><a class="viewcode-back" href="../../../modules/tensor/indexed.html#sympy.tensor.indexed.Indexed">[docs]</a><span class="k">class</span> <span class="nc">Indexed</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Represents a mathematical object with indices.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.tensor import Indexed, IndexedBase, Idx</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; i, j = symbols(&#39;i j&#39;, cls=Idx)</span>
-<span class="sd">    &gt;&gt;&gt; Indexed(&#39;A&#39;, i, j)</span>
-<span class="sd">    A[i, j]</span>
-
-<span class="sd">    It is recommended that Indexed objects are created via IndexedBase:</span>
-
-<span class="sd">    &gt;&gt;&gt; A = IndexedBase(&#39;A&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; Indexed(&#39;A&#39;, i, j) == A[i, j]</span>
-<span class="sd">    True</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.utilities.misc</span> <span class="kn">import</span> <span class="n">filldedent</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">IndexException</span><span class="p">(</span><span class="s">&quot;Indexed needs at least one index.&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="p">(</span><span class="nb">str</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">)):</span>
-            <span class="n">base</span> <span class="o">=</span> <span class="n">IndexedBase</span><span class="p">(</span><span class="n">base</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">IndexedBase</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&quot;&quot;&quot;</span>
-<span class="s">                Indexed expects string, Symbol or IndexedBase as base.&quot;&quot;&quot;</span><span class="p">))</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="n">args</span><span class="p">))</span>
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Indexed.base"><a class="viewcode-back" href="../../../modules/tensor/indexed.html#sympy.tensor.indexed.Indexed.base">[docs]</a>    <span class="k">def</span> <span class="nf">base</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the IndexedBase of the Indexed object.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.tensor import Indexed, IndexedBase, Idx</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; i, j = symbols(&#39;i j&#39;, cls=Idx)</span>
-<span class="sd">        &gt;&gt;&gt; Indexed(&#39;A&#39;, i, j).base</span>
-<span class="sd">        A</span>
-<span class="sd">        &gt;&gt;&gt; B = IndexedBase(&#39;B&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; B == B[i, j].base</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Indexed.indices"><a class="viewcode-back" href="../../../modules/tensor/indexed.html#sympy.tensor.indexed.Indexed.indices">[docs]</a>    <span class="k">def</span> <span class="nf">indices</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the indices of the Indexed object.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.tensor import Indexed, Idx</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; i, j = symbols(&#39;i j&#39;, cls=Idx)</span>
-<span class="sd">        &gt;&gt;&gt; Indexed(&#39;A&#39;, i, j).indices</span>
-<span class="sd">        (i, j)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Indexed.rank"><a class="viewcode-back" href="../../../modules/tensor/indexed.html#sympy.tensor.indexed.Indexed.rank">[docs]</a>    <span class="k">def</span> <span class="nf">rank</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the rank of the Indexed object.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.tensor import Indexed, Idx</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; i, j, k, l, m = symbols(&#39;i:m&#39;, cls=Idx)</span>
-<span class="sd">        &gt;&gt;&gt; Indexed(&#39;A&#39;, i, j).rank</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; q = Indexed(&#39;A&#39;, i, j, k, l, m)</span>
-<span class="sd">        &gt;&gt;&gt; q.rank</span>
-<span class="sd">        5</span>
-<span class="sd">        &gt;&gt;&gt; q.rank == len(q.indices)</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Indexed.shape"><a class="viewcode-back" href="../../../modules/tensor/indexed.html#sympy.tensor.indexed.Indexed.shape">[docs]</a>    <span class="k">def</span> <span class="nf">shape</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a list with dimensions of each index.</span>
-
-<span class="sd">        Dimensions is a property of the array, not of the indices.  Still, if</span>
-<span class="sd">        the IndexedBase does not define a shape attribute, it is assumed that</span>
-<span class="sd">        the ranges of the indices correspond to the shape of the array.</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.tensor.indexed import IndexedBase, Idx</span>
-<span class="sd">        &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">        &gt;&gt;&gt; n, m = symbols(&#39;n m&#39;, integer=True)</span>
-<span class="sd">        &gt;&gt;&gt; i = Idx(&#39;i&#39;, m)</span>
-<span class="sd">        &gt;&gt;&gt; j = Idx(&#39;j&#39;, m)</span>
-<span class="sd">        &gt;&gt;&gt; A = IndexedBase(&#39;A&#39;, shape=(n, n))</span>
-<span class="sd">        &gt;&gt;&gt; B = IndexedBase(&#39;B&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; A[i, j].shape</span>
-<span class="sd">        (n, n)</span>
-<span class="sd">        &gt;&gt;&gt; B[i, j].shape</span>
-<span class="sd">        (m, m)</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="kn">from</span> <span class="nn">sympy.utilities.misc</span> <span class="kn">import</span> <span class="n">filldedent</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">shape</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">shape</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">i</span><span class="o">.</span><span class="n">upper</span> <span class="o">-</span> <span class="n">i</span><span class="o">.</span><span class="n">lower</span> <span class="o">+</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">indices</span><span class="p">])</span>
-        <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">IndexException</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&quot;&quot;&quot;</span>
-<span class="s">                Range is not defined for all indices in: </span><span class="si">%s</span><span class="s">&quot;&quot;&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="p">))</span>
-        <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">IndexException</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&quot;&quot;&quot;</span>
-<span class="s">                Shape cannot be inferred from Idx with</span>
-<span class="s">                undefined range: </span><span class="si">%s</span><span class="s">&quot;&quot;&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="p">))</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Indexed.ranges"><a class="viewcode-back" href="../../../modules/tensor/indexed.html#sympy.tensor.indexed.Indexed.ranges">[docs]</a>    <span class="k">def</span> <span class="nf">ranges</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a list of tuples with lower and upper range of each index.</span>
-
-<span class="sd">        If an index does not define the data members upper and lower, the</span>
-<span class="sd">        corresponding slot in the list contains ``None`` instead of a tuple.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Indexed,Idx, symbols</span>
-<span class="sd">        &gt;&gt;&gt; Indexed(&#39;A&#39;, Idx(&#39;i&#39;, 2), Idx(&#39;j&#39;, 4), Idx(&#39;k&#39;, 8)).ranges</span>
-<span class="sd">        [(0, 1), (0, 3), (0, 7)]</span>
-<span class="sd">        &gt;&gt;&gt; Indexed(&#39;A&#39;, Idx(&#39;i&#39;, 3), Idx(&#39;j&#39;, 3), Idx(&#39;k&#39;, 3)).ranges</span>
-<span class="sd">        [(0, 2), (0, 2), (0, 2)]</span>
-<span class="sd">        &gt;&gt;&gt; x, y, z = symbols(&#39;x y z&#39;, integer=True)</span>
-<span class="sd">        &gt;&gt;&gt; Indexed(&#39;A&#39;, x, y, z).ranges</span>
-<span class="sd">        [None, None, None]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">ranges</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">indices</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">ranges</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Tuple</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">lower</span><span class="p">,</span> <span class="n">i</span><span class="o">.</span><span class="n">upper</span><span class="p">))</span>
-            <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-                <span class="n">ranges</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">None</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">ranges</span>
-</div>
-    <span class="k">def</span> <span class="nf">_sympystr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="n">indices</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">doprint</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">indices</span><span class="p">))</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">[</span><span class="si">%s</span><span class="s">]&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">base</span><span class="p">),</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">indices</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="IndexedBase"><a class="viewcode-back" href="../../../modules/tensor/indexed.html#sympy.tensor.indexed.IndexedBase">[docs]</a><span class="k">class</span> <span class="nc">IndexedBase</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Represent the base or stem of an indexed object</span>
-
-<span class="sd">    The IndexedBase class represent an array that contains elements. The main purpose</span>
-<span class="sd">    of this class is to allow the convenient creation of objects of the Indexed</span>
-<span class="sd">    class.  The __getitem__ method of IndexedBase returns an instance of</span>
-<span class="sd">    Indexed.  Alone, without indices, the IndexedBase class can be used as a</span>
-<span class="sd">    notation for e.g. matrix equations, resembling what you could do with the</span>
-<span class="sd">    Symbol class.  But, the IndexedBase class adds functionality that is not</span>
-<span class="sd">    available for Symbol instances:</span>
-
-<span class="sd">      -  An IndexedBase object can optionally store shape information.  This can</span>
-<span class="sd">         be used in to check array conformance and conditions for numpy</span>
-<span class="sd">         broadcasting.  (TODO)</span>
-<span class="sd">      -  An IndexedBase object implements syntactic sugar that allows easy symbolic</span>
-<span class="sd">         representation of array operations, using implicit summation of</span>
-<span class="sd">         repeated indices.</span>
-<span class="sd">      -  The IndexedBase object symbolizes a mathematical structure equivalent</span>
-<span class="sd">         to arrays, and is recognized as such for code generation and automatic</span>
-<span class="sd">         compilation and wrapping.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.tensor import IndexedBase, Idx</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; A = IndexedBase(&#39;A&#39;); A</span>
-<span class="sd">    A</span>
-<span class="sd">    &gt;&gt;&gt; type(A)</span>
-<span class="sd">    &lt;class &#39;sympy.tensor.indexed.IndexedBase&#39;&gt;</span>
-
-<span class="sd">    When an IndexedBase object receives indices, it returns an array with named</span>
-<span class="sd">    axes, represented by an Indexed object:</span>
-
-<span class="sd">    &gt;&gt;&gt; i, j = symbols(&#39;i j&#39;, integer=True)</span>
-<span class="sd">    &gt;&gt;&gt; A[i, j, 2]</span>
-<span class="sd">    A[i, j, 2]</span>
-<span class="sd">    &gt;&gt;&gt; type(A[i, j, 2])</span>
-<span class="sd">    &lt;class &#39;sympy.tensor.indexed.Indexed&#39;&gt;</span>
-
-<span class="sd">    The IndexedBase constructor takes an optional shape argument.  If given,</span>
-<span class="sd">    it overrides any shape information in the indices. (But not the index</span>
-<span class="sd">    ranges!)</span>
-
-<span class="sd">    &gt;&gt;&gt; m, n, o, p = symbols(&#39;m n o p&#39;, integer=True)</span>
-<span class="sd">    &gt;&gt;&gt; i = Idx(&#39;i&#39;, m)</span>
-<span class="sd">    &gt;&gt;&gt; j = Idx(&#39;j&#39;, n)</span>
-<span class="sd">    &gt;&gt;&gt; A[i, j].shape</span>
-<span class="sd">    (m, n)</span>
-<span class="sd">    &gt;&gt;&gt; B = IndexedBase(&#39;B&#39;, shape=(o, p))</span>
-<span class="sd">    &gt;&gt;&gt; B[i, j].shape</span>
-<span class="sd">    (o, p)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_commutative</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">label</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">label</span><span class="p">,</span> <span class="p">(</span><span class="nb">str</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">)):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Base label should be a string or Symbol.&quot;</span><span class="p">)</span>
-
-        <span class="n">label</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">label</span><span class="p">)</span>
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">label</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">shape</span><span class="p">):</span>
-            <span class="n">obj</span><span class="o">.</span><span class="n">_shape</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">shape</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">obj</span><span class="o">.</span><span class="n">_shape</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">shape</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="IndexedBase.args"><a class="viewcode-back" href="../../../modules/tensor/indexed.html#sympy.tensor.indexed.IndexedBase.args">[docs]</a>    <span class="k">def</span> <span class="nf">args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the arguments used to create this IndexedBase object.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import IndexedBase</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; IndexedBase(&#39;A&#39;, shape=(x, y)).args</span>
-<span class="sd">        (A, (x, y))</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_shape</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_shape</span><span class="p">,)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_args</span>
-</div>
-    <span class="k">def</span> <span class="nf">_hashable_content</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">_hashable_content</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_shape</span><span class="p">,)</span>
-
-    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">indices</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">indices</span><span class="p">):</span>
-            <span class="c"># Special case needed because M[*my_tuple] is a syntax error.</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">shape</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">indices</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="n">IndexException</span><span class="p">(</span><span class="s">&quot;Rank mismatch.&quot;</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Indexed</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">indices</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">shape</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">IndexException</span><span class="p">(</span><span class="s">&quot;Rank mismatch.&quot;</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">Indexed</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">indices</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="IndexedBase.shape"><a class="viewcode-back" href="../../../modules/tensor/indexed.html#sympy.tensor.indexed.IndexedBase.shape">[docs]</a>    <span class="k">def</span> <span class="nf">shape</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the shape of the IndexedBase object.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import IndexedBase, Idx, Symbol</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; IndexedBase(&#39;A&#39;, shape=(x, y)).shape</span>
-<span class="sd">        (x, y)</span>
-
-<span class="sd">        Note: If the shape of the IndexedBase is specified, it will override</span>
-<span class="sd">        any shape information given by the indices.</span>
-
-<span class="sd">        &gt;&gt;&gt; A = IndexedBase(&#39;A&#39;, shape=(x, y))</span>
-<span class="sd">        &gt;&gt;&gt; B = IndexedBase(&#39;B&#39;)</span>
-<span class="sd">        &gt;&gt;&gt; i = Idx(&#39;i&#39;, 2)</span>
-<span class="sd">        &gt;&gt;&gt; j = Idx(&#39;j&#39;, 1)</span>
-<span class="sd">        &gt;&gt;&gt; A[i, j].shape</span>
-<span class="sd">        (x, y)</span>
-<span class="sd">        &gt;&gt;&gt; B[i, j].shape</span>
-<span class="sd">        (2, 1)</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_shape</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="IndexedBase.label"><a class="viewcode-back" href="../../../modules/tensor/indexed.html#sympy.tensor.indexed.IndexedBase.label">[docs]</a>    <span class="k">def</span> <span class="nf">label</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the label of the IndexedBase object.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import IndexedBase</span>
-<span class="sd">        &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">        &gt;&gt;&gt; IndexedBase(&#39;A&#39;, shape=(x, y)).label</span>
-<span class="sd">        A</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="k">def</span> <span class="nf">_sympystr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">p</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="Idx"><a class="viewcode-back" href="../../../modules/tensor/indexed.html#sympy.tensor.indexed.Idx">[docs]</a><span class="k">class</span> <span class="nc">Idx</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Represents an integer index as an Integer or integer expression.</span>
-
-<span class="sd">    There are a number of ways to create an Idx object.  The constructor</span>
-<span class="sd">    takes two arguments:</span>
-
-<span class="sd">    ``label``</span>
-<span class="sd">        An integer or a symbol that labels the index.</span>
-<span class="sd">    ``range``</span>
-<span class="sd">        Optionally you can specify a range as either</span>
-
-<span class="sd">    - Symbol or integer: This is interpreted as a dimension. Lower and</span>
-<span class="sd">      upper bounds are set to 0 and range - 1, respectively.</span>
-<span class="sd">    - tuple: The two elements are interpreted as the lower and upper</span>
-<span class="sd">      bounds of the range, respectively.</span>
-
-<span class="sd">    Note: the Idx constructor is rather pedantic in that it only accepts</span>
-<span class="sd">    integer arguments.  The only exception is that you can use oo and -oo to</span>
-<span class="sd">    specify an unbounded range.  For all other cases, both label and bounds</span>
-<span class="sd">    must be declared as integers, e.g. if n is given as an argument then</span>
-<span class="sd">    n.is_integer must return True.</span>
-
-<span class="sd">    For convenience, if the label is given as a string it is automatically</span>
-<span class="sd">    converted to an integer symbol.  (Note: this conversion is not done for</span>
-<span class="sd">    range or dimension arguments.)</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.tensor import IndexedBase, Idx</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols, oo</span>
-<span class="sd">    &gt;&gt;&gt; n, i, L, U = symbols(&#39;n i L U&#39;, integer=True)</span>
-
-<span class="sd">    If a string is given for the label an integer Symbol is created and the</span>
-<span class="sd">    bounds are both None:</span>
-
-<span class="sd">    &gt;&gt;&gt; idx = Idx(&#39;qwerty&#39;); idx</span>
-<span class="sd">    qwerty</span>
-<span class="sd">    &gt;&gt;&gt; idx.lower, idx.upper</span>
-<span class="sd">    (None, None)</span>
-
-<span class="sd">    Both upper and lower bounds can be specified:</span>
-
-<span class="sd">    &gt;&gt;&gt; idx = Idx(i, (L, U)); idx</span>
-<span class="sd">    i</span>
-<span class="sd">    &gt;&gt;&gt; idx.lower, idx.upper</span>
-<span class="sd">    (L, U)</span>
-
-<span class="sd">    When only a single bound is given it is interpreted as the dimension</span>
-<span class="sd">    and the lower bound defaults to 0:</span>
-
-<span class="sd">    &gt;&gt;&gt; idx = Idx(i, n); idx.lower, idx.upper</span>
-<span class="sd">    (0, n - 1)</span>
-<span class="sd">    &gt;&gt;&gt; idx = Idx(i, 4); idx.lower, idx.upper</span>
-<span class="sd">    (0, 3)</span>
-<span class="sd">    &gt;&gt;&gt; idx = Idx(i, oo); idx.lower, idx.upper</span>
-<span class="sd">    (0, oo)</span>
-
-<span class="sd">    The label can be a literal integer instead of a string/Symbol:</span>
-
-<span class="sd">    &gt;&gt;&gt; idx = Idx(2, n); idx.lower, idx.upper</span>
-<span class="sd">    (0, n - 1)</span>
-<span class="sd">    &gt;&gt;&gt; idx.label</span>
-<span class="sd">    2</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">is_integer</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">label</span><span class="p">,</span> <span class="nb">range</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">):</span>
-        <span class="kn">from</span> <span class="nn">sympy.utilities.misc</span> <span class="kn">import</span> <span class="n">filldedent</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">label</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="n">label</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="n">label</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-        <span class="n">label</span><span class="p">,</span> <span class="nb">range</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">sympify</span><span class="p">,</span> <span class="p">(</span><span class="n">label</span><span class="p">,</span> <span class="nb">range</span><span class="p">)))</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">label</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Idx object requires an integer label.&quot;</span><span class="p">)</span>
-
-        <span class="k">elif</span> <span class="n">is_sequence</span><span class="p">(</span><span class="nb">range</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="nb">range</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">2</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&quot;&quot;&quot;</span>
-<span class="s">                    Idx range tuple must have length 2, but got </span><span class="si">%s</span><span class="s">&quot;&quot;&quot;</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="nb">range</span><span class="p">)))</span>
-            <span class="k">for</span> <span class="n">bound</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">bound</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">or</span> <span class="nb">abs</span><span class="p">(</span><span class="n">bound</span><span class="p">)</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">):</span>
-                    <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Idx object requires integer bounds.&quot;</span><span class="p">)</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="n">label</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="nb">range</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="nb">range</span><span class="p">,</span> <span class="n">Expr</span><span class="p">):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="nb">range</span><span class="o">.</span><span class="n">is_integer</span> <span class="ow">or</span> <span class="nb">range</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Idx object requires an integer dimension.&quot;</span><span class="p">)</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="n">label</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">range</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">range</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="n">filldedent</span><span class="p">(</span><span class="s">&quot;&quot;&quot;</span>
-<span class="s">                The range must be an ordered iterable or</span>
-<span class="s">                integer SymPy expression.&quot;&quot;&quot;</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="n">label</span><span class="p">,</span>
-
-        <span class="n">obj</span> <span class="o">=</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kw_args</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">obj</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Idx.label"><a class="viewcode-back" href="../../../modules/tensor/indexed.html#sympy.tensor.indexed.Idx.label">[docs]</a>    <span class="k">def</span> <span class="nf">label</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the label (Integer or integer expression) of the Idx object.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Idx, Symbol</span>
-<span class="sd">        &gt;&gt;&gt; Idx(2).label</span>
-<span class="sd">        2</span>
-<span class="sd">        &gt;&gt;&gt; j = Symbol(&#39;j&#39;, integer=True)</span>
-<span class="sd">        &gt;&gt;&gt; Idx(j).label</span>
-<span class="sd">        j</span>
-<span class="sd">        &gt;&gt;&gt; Idx(j + 1).label</span>
-<span class="sd">        j + 1</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Idx.lower"><a class="viewcode-back" href="../../../modules/tensor/indexed.html#sympy.tensor.indexed.Idx.lower">[docs]</a>    <span class="k">def</span> <span class="nf">lower</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the lower bound of the Index.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Idx</span>
-<span class="sd">        &gt;&gt;&gt; Idx(&#39;j&#39;, 2).lower</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; Idx(&#39;j&#39;, 5).lower</span>
-<span class="sd">        0</span>
-<span class="sd">        &gt;&gt;&gt; Idx(&#39;j&#39;).lower is None</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span>
-            <span class="k">return</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Idx.upper"><a class="viewcode-back" href="../../../modules/tensor/indexed.html#sympy.tensor.indexed.Idx.upper">[docs]</a>    <span class="k">def</span> <span class="nf">upper</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the upper bound of the Index.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy import Idx</span>
-<span class="sd">        &gt;&gt;&gt; Idx(&#39;j&#39;, 2).upper</span>
-<span class="sd">        1</span>
-<span class="sd">        &gt;&gt;&gt; Idx(&#39;j&#39;, 5).upper</span>
-<span class="sd">        4</span>
-<span class="sd">        &gt;&gt;&gt; Idx(&#39;j&#39;).upper is None</span>
-<span class="sd">        True</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span>
-            <span class="k">return</span>
-</div>
-    <span class="k">def</span> <span class="nf">_sympystr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">p</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">label</span><span class="p">)</span></div>
-</pre></div>
-
-          </div>
-        </div>
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-        <div class="sphinxsidebarwrapper">
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-  <h1>Source code for sympy.utilities.autowrap</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Module for compiling codegen output, and wrap the binary for use in</span>
-<span class="sd">python.</span>
-
-<span class="sd">.. note:: To use the autowrap module it must first be imported</span>
-
-<span class="sd">   &gt;&gt;&gt; from sympy.utilities.autowrap import autowrap</span>
-
-<span class="sd">This module provides a common interface for different external backends, such</span>
-<span class="sd">as f2py, fwrap, Cython, SWIG(?) etc. (Currently only f2py and Cython are</span>
-<span class="sd">implemented) The goal is to provide access to compiled binaries of acceptable</span>
-<span class="sd">performance with a one-button user interface, i.e.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x,y</span>
-<span class="sd">    &gt;&gt;&gt; expr = ((x - y)**(25)).expand()</span>
-<span class="sd">    &gt;&gt;&gt; binary_callable = autowrap(expr)           # doctest: +SKIP</span>
-<span class="sd">    &gt;&gt;&gt; binary_callable(1, 2)                      # doctest: +SKIP</span>
-<span class="sd">    -1.0</span>
-
-<span class="sd">The callable returned from autowrap() is a binary python function, not a</span>
-<span class="sd">SymPy object.  If it is desired to use the compiled function in symbolic</span>
-<span class="sd">expressions, it is better to use binary_function() which returns a SymPy</span>
-<span class="sd">Function object.  The binary callable is attached as the _imp_ attribute and</span>
-<span class="sd">invoked when a numerical evaluation is requested with evalf(), or with</span>
-<span class="sd">lambdify().</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.autowrap import binary_function</span>
-<span class="sd">    &gt;&gt;&gt; f = binary_function(&#39;f&#39;, expr)             # doctest: +SKIP</span>
-<span class="sd">    &gt;&gt;&gt; 2*f(x, y) + y                              # doctest: +SKIP</span>
-<span class="sd">    y + 2*f(x, y)</span>
-<span class="sd">    &gt;&gt;&gt; (2*f(x, y) + y).evalf(2, subs={x: 1, y:2}) # doctest: +SKIP</span>
-<span class="sd">    0.0</span>
-
-<span class="sd">The idea is that a SymPy user will primarily be interested in working with</span>
-<span class="sd">mathematical expressions, and should not have to learn details about wrapping</span>
-<span class="sd">tools in order to evaluate expressions numerically, even if they are</span>
-<span class="sd">computationally expensive.</span>
-
-<span class="sd">When is this useful?</span>
-
-<span class="sd">    1) For computations on large arrays, Python iterations may be too slow,</span>
-<span class="sd">       and depending on the mathematical expression, it may be difficult to</span>
-<span class="sd">       exploit the advanced index operations provided by NumPy.</span>
-
-<span class="sd">    2) For *really* long expressions that will be called repeatedly, the</span>
-<span class="sd">       compiled binary should be significantly faster than SymPy&#39;s .evalf()</span>
-
-<span class="sd">    3) If you are generating code with the codegen utility in order to use</span>
-<span class="sd">       it in another project, the automatic python wrappers let you test the</span>
-<span class="sd">       binaries immediately from within SymPy.</span>
-
-<span class="sd">    4) To create customized ufuncs for use with numpy arrays.</span>
-<span class="sd">       See *ufuncify*.</span>
-
-<span class="sd">When is this module NOT the best approach?</span>
-
-<span class="sd">    1) If you are really concerned about speed or memory optimizations,</span>
-<span class="sd">       you will probably get better results by working directly with the</span>
-<span class="sd">       wrapper tools and the low level code.  However, the files generated</span>
-<span class="sd">       by this utility may provide a useful starting point and reference</span>
-<span class="sd">       code. Temporary files will be left intact if you supply the keyword</span>
-<span class="sd">       tempdir=&quot;path/to/files/&quot;.</span>
-
-<span class="sd">    2) If the array computation can be handled easily by numpy, and you</span>
-<span class="sd">       don&#39;t need the binaries for another project.</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-
-
-<span class="kn">import</span> <span class="nn">sys</span>
-<span class="kn">import</span> <span class="nn">os</span>
-<span class="kn">import</span> <span class="nn">shutil</span>
-<span class="kn">import</span> <span class="nn">tempfile</span>
-<span class="kn">import</span> <span class="nn">subprocess</span>
-
-<span class="kn">from</span> <span class="nn">sympy.utilities.codegen</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">get_code_generator</span><span class="p">,</span> <span class="n">Routine</span><span class="p">,</span> <span class="n">OutputArgument</span><span class="p">,</span> <span class="n">InOutArgument</span><span class="p">,</span>
-    <span class="n">CodeGenArgumentListError</span><span class="p">,</span> <span class="n">Result</span>
-<span class="p">)</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.lambdify</span> <span class="kn">import</span> <span class="n">implemented_function</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">C</span>
-
-
-<span class="k">class</span> <span class="nc">CodeWrapError</span><span class="p">(</span><span class="ne">Exception</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-
-<div class="viewcode-block" id="CodeWrapper"><a class="viewcode-back" href="../../../modules/utilities/autowrap.html#sympy.utilities.autowrap.CodeWrapper">[docs]</a><span class="k">class</span> <span class="nc">CodeWrapper</span><span class="p">:</span>
-    <span class="sd">&quot;&quot;&quot;Base Class for code wrappers&quot;&quot;&quot;</span>
-    <span class="n">_filename</span> <span class="o">=</span> <span class="s">&quot;wrapped_code&quot;</span>
-    <span class="n">_module_basename</span> <span class="o">=</span> <span class="s">&quot;wrapper_module&quot;</span>
-    <span class="n">_module_counter</span> <span class="o">=</span> <span class="mi">0</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">filename</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">_</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_filename</span><span class="p">,</span> <span class="n">CodeWrapper</span><span class="o">.</span><span class="n">_module_counter</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">module_name</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">_</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_module_basename</span><span class="p">,</span> <span class="n">CodeWrapper</span><span class="o">.</span><span class="n">_module_counter</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">generator</span><span class="p">,</span> <span class="n">filepath</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">flags</span><span class="o">=</span><span class="p">[],</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        generator -- the code generator to use</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">generator</span> <span class="o">=</span> <span class="n">generator</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">filepath</span> <span class="o">=</span> <span class="n">filepath</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">flags</span> <span class="o">=</span> <span class="n">flags</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">quiet</span> <span class="o">=</span> <span class="ow">not</span> <span class="n">verbose</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">include_header</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">filepath</span><span class="p">)</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">include_empty</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">bool</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">filepath</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_generate_code</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">main_routine</span><span class="p">,</span> <span class="n">routines</span><span class="p">):</span>
-        <span class="n">routines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">main_routine</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">generator</span><span class="o">.</span><span class="n">write</span><span class="p">(</span>
-            <span class="n">routines</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">filename</span><span class="p">,</span> <span class="bp">True</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">include_header</span><span class="p">,</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">include_empty</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">wrap_code</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routine</span><span class="p">,</span> <span class="n">helpers</span><span class="o">=</span><span class="p">[]):</span>
-
-        <span class="n">workdir</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">filepath</span> <span class="ow">or</span> <span class="n">tempfile</span><span class="o">.</span><span class="n">mkdtemp</span><span class="p">(</span><span class="s">&quot;_sympy_compile&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">access</span><span class="p">(</span><span class="n">workdir</span><span class="p">,</span> <span class="n">os</span><span class="o">.</span><span class="n">F_OK</span><span class="p">):</span>
-            <span class="n">os</span><span class="o">.</span><span class="n">mkdir</span><span class="p">(</span><span class="n">workdir</span><span class="p">)</span>
-        <span class="n">oldwork</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">getcwd</span><span class="p">()</span>
-        <span class="n">os</span><span class="o">.</span><span class="n">chdir</span><span class="p">(</span><span class="n">workdir</span><span class="p">)</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">sys</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">workdir</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_generate_code</span><span class="p">(</span><span class="n">routine</span><span class="p">,</span> <span class="n">helpers</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_prepare_files</span><span class="p">(</span><span class="n">routine</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_process_files</span><span class="p">(</span><span class="n">routine</span><span class="p">)</span>
-            <span class="n">mod</span> <span class="o">=</span> <span class="nb">__import__</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">module_name</span><span class="p">)</span>
-        <span class="k">finally</span><span class="p">:</span>
-            <span class="n">sys</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">workdir</span><span class="p">)</span>
-            <span class="n">CodeWrapper</span><span class="o">.</span><span class="n">_module_counter</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="n">os</span><span class="o">.</span><span class="n">chdir</span><span class="p">(</span><span class="n">oldwork</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">filepath</span><span class="p">:</span>
-                <span class="n">shutil</span><span class="o">.</span><span class="n">rmtree</span><span class="p">(</span><span class="n">workdir</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_wrapped_function</span><span class="p">(</span><span class="n">mod</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_process_files</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routine</span><span class="p">):</span>
-        <span class="n">command</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">command</span>
-        <span class="n">command</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">flags</span><span class="p">)</span>
-        <span class="n">null</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">devnull</span><span class="p">,</span> <span class="s">&#39;w&#39;</span><span class="p">)</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">quiet</span><span class="p">:</span>
-                <span class="n">retcode</span> <span class="o">=</span> <span class="n">subprocess</span><span class="o">.</span><span class="n">call</span><span class="p">(</span>
-                    <span class="n">command</span><span class="p">,</span> <span class="n">stdout</span><span class="o">=</span><span class="n">null</span><span class="p">,</span> <span class="n">stderr</span><span class="o">=</span><span class="n">subprocess</span><span class="o">.</span><span class="n">STDOUT</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">retcode</span> <span class="o">=</span> <span class="n">subprocess</span><span class="o">.</span><span class="n">call</span><span class="p">(</span><span class="n">command</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">OSError</span><span class="p">:</span>
-            <span class="n">retcode</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">retcode</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">CodeWrapError</span><span class="p">(</span>
-                <span class="s">&quot;Error while executing command: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="s">&quot; &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">command</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="DummyWrapper"><a class="viewcode-back" href="../../../modules/utilities/autowrap.html#sympy.utilities.autowrap.DummyWrapper">[docs]</a><span class="k">class</span> <span class="nc">DummyWrapper</span><span class="p">(</span><span class="n">CodeWrapper</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Class used for testing independent of backends &quot;&quot;&quot;</span>
-
-    <span class="n">template</span> <span class="o">=</span> <span class="s">&quot;&quot;&quot;# dummy module for testing of SymPy</span>
-<span class="s">def </span><span class="si">%(name)s</span><span class="s">():</span>
-<span class="s">    return &quot;</span><span class="si">%(expr)s</span><span class="s">&quot;</span>
-<span class="si">%(name)s</span><span class="s">.args = &quot;</span><span class="si">%(args)s</span><span class="s">&quot;</span>
-<span class="si">%(name)s</span><span class="s">.returns = &quot;</span><span class="si">%(retvals)s</span><span class="s">&quot;</span>
-<span class="s">&quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_prepare_files</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routine</span><span class="p">):</span>
-        <span class="k">return</span>
-
-    <span class="k">def</span> <span class="nf">_generate_code</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routine</span><span class="p">,</span> <span class="n">helpers</span><span class="p">):</span>
-        <span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s">.py&#39;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">module_name</span><span class="p">,</span> <span class="s">&#39;w&#39;</span><span class="p">)</span> <span class="k">as</span> <span class="n">f</span><span class="p">:</span>
-            <span class="n">printed</span> <span class="o">=</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span>
-                <span class="p">[</span><span class="nb">str</span><span class="p">(</span><span class="n">res</span><span class="o">.</span><span class="n">expr</span><span class="p">)</span> <span class="k">for</span> <span class="n">res</span> <span class="ow">in</span> <span class="n">routine</span><span class="o">.</span><span class="n">result_variables</span><span class="p">])</span>
-            <span class="c"># convert OutputArguments to return value like f2py</span>
-            <span class="n">inargs</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">routine</span><span class="o">.</span><span class="n">arguments</span> <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span>
-                <span class="n">x</span><span class="p">,</span> <span class="n">OutputArgument</span><span class="p">)]</span>
-            <span class="n">retvals</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">val</span> <span class="ow">in</span> <span class="n">routine</span><span class="o">.</span><span class="n">result_variables</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">val</span><span class="p">,</span> <span class="n">Result</span><span class="p">):</span>
-                    <span class="n">retvals</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;nameless&#39;</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">retvals</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">val</span><span class="o">.</span><span class="n">result_var</span><span class="p">)</span>
-
-            <span class="k">print</span><span class="p">(</span><span class="n">DummyWrapper</span><span class="o">.</span><span class="n">template</span> <span class="o">%</span> <span class="p">{</span>
-                <span class="s">&#39;name&#39;</span><span class="p">:</span> <span class="n">routine</span><span class="o">.</span><span class="n">name</span><span class="p">,</span>
-                <span class="s">&#39;expr&#39;</span><span class="p">:</span> <span class="n">printed</span><span class="p">,</span>
-                <span class="s">&#39;args&#39;</span><span class="p">:</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="nb">str</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">name</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">inargs</span><span class="p">]),</span>
-                <span class="s">&#39;retvals&#39;</span><span class="p">:</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="nb">str</span><span class="p">(</span><span class="n">val</span><span class="p">)</span> <span class="k">for</span> <span class="n">val</span> <span class="ow">in</span> <span class="n">retvals</span><span class="p">])</span>
-            <span class="p">},</span> <span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_process_files</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routine</span><span class="p">):</span>
-        <span class="k">return</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_get_wrapped_function</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">mod</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">mod</span><span class="o">.</span><span class="n">autofunc</span>
-
-</div>
-<div class="viewcode-block" id="CythonCodeWrapper"><a class="viewcode-back" href="../../../modules/utilities/autowrap.html#sympy.utilities.autowrap.CythonCodeWrapper">[docs]</a><span class="k">class</span> <span class="nc">CythonCodeWrapper</span><span class="p">(</span><span class="n">CodeWrapper</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Wrapper that uses Cython&quot;&quot;&quot;</span>
-
-    <span class="n">setup_template</span> <span class="o">=</span> <span class="s">&quot;&quot;&quot;</span>
-<span class="s">from distutils.core import setup</span>
-<span class="s">from distutils.extension import Extension</span>
-<span class="s">from Cython.Distutils import build_ext</span>
-
-<span class="s">setup(</span>
-<span class="s">    cmdclass = {&#39;build_ext&#39;: build_ext},</span>
-<span class="s">    ext_modules = [Extension(</span><span class="si">%(args)s</span><span class="s">)]</span>
-<span class="s">        )</span>
-<span class="s">&quot;&quot;&quot;</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">command</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">command</span> <span class="o">=</span> <span class="p">[</span><span class="n">sys</span><span class="o">.</span><span class="n">executable</span><span class="p">,</span> <span class="s">&quot;setup.py&quot;</span><span class="p">,</span> <span class="s">&quot;build_ext&quot;</span><span class="p">,</span> <span class="s">&quot;--inplace&quot;</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">command</span>
-
-    <span class="k">def</span> <span class="nf">_prepare_files</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routine</span><span class="p">):</span>
-        <span class="n">pyxfilename</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">module_name</span> <span class="o">+</span> <span class="s">&#39;.pyx&#39;</span>
-        <span class="n">codefilename</span> <span class="o">=</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">.</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">filename</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">generator</span><span class="o">.</span><span class="n">code_extension</span><span class="p">)</span>
-
-        <span class="c"># pyx</span>
-        <span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="n">pyxfilename</span><span class="p">,</span> <span class="s">&#39;w&#39;</span><span class="p">)</span> <span class="k">as</span> <span class="n">f</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">dump_pyx</span><span class="p">([</span><span class="n">routine</span><span class="p">],</span> <span class="n">f</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">filename</span><span class="p">,</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">include_header</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">include_empty</span><span class="p">)</span>
-
-        <span class="c"># setup.py</span>
-        <span class="n">ext_args</span> <span class="o">=</span> <span class="p">[</span><span class="nb">repr</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">module_name</span><span class="p">),</span> <span class="nb">repr</span><span class="p">([</span><span class="n">pyxfilename</span><span class="p">,</span> <span class="n">codefilename</span><span class="p">])]</span>
-        <span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="s">&#39;setup.py&#39;</span><span class="p">,</span> <span class="s">&#39;w&#39;</span><span class="p">)</span> <span class="k">as</span> <span class="n">f</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="n">CythonCodeWrapper</span><span class="o">.</span><span class="n">setup_template</span> <span class="o">%</span> <span class="p">{</span>
-                <span class="s">&#39;args&#39;</span><span class="p">:</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">ext_args</span><span class="p">)},</span> <span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_get_wrapped_function</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">mod</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">mod</span><span class="o">.</span><span class="n">autofunc_c</span>
-
-<div class="viewcode-block" id="CythonCodeWrapper.dump_pyx"><a class="viewcode-back" href="../../../modules/utilities/autowrap.html#sympy.utilities.autowrap.CythonCodeWrapper.dump_pyx">[docs]</a>    <span class="k">def</span> <span class="nf">dump_pyx</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routines</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">prefix</span><span class="p">,</span> <span class="n">header</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">empty</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Write a Cython file with python wrappers</span>
-
-<span class="sd">           This file contains all the definitions of the routines in c code and</span>
-<span class="sd">           refers to the header file.</span>
-
-<span class="sd">           :Arguments:</span>
-
-<span class="sd">           routines</span>
-<span class="sd">                List of Routine instances</span>
-<span class="sd">           f</span>
-<span class="sd">                File-like object to write the file to</span>
-<span class="sd">           prefix</span>
-<span class="sd">                The filename prefix, used to refer to the proper header file.</span>
-<span class="sd">                Only the basename of the prefix is used.</span>
-<span class="sd">           empty</span>
-<span class="sd">                Optional. When True, empty lines are included to structure the</span>
-<span class="sd">                source files. [DEFAULT=True]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">for</span> <span class="n">routine</span> <span class="ow">in</span> <span class="n">routines</span><span class="p">:</span>
-            <span class="n">prototype</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">generator</span><span class="o">.</span><span class="n">get_prototype</span><span class="p">(</span><span class="n">routine</span><span class="p">)</span>
-            <span class="n">origname</span> <span class="o">=</span> <span class="n">routine</span><span class="o">.</span><span class="n">name</span>
-            <span class="n">routine</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">_c&quot;</span> <span class="o">%</span> <span class="n">origname</span>
-            <span class="n">prototype_c</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">generator</span><span class="o">.</span><span class="n">get_prototype</span><span class="p">(</span><span class="n">routine</span><span class="p">)</span>
-            <span class="n">routine</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="n">origname</span>
-
-            <span class="c"># declare</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;cdef extern from &quot;</span><span class="si">%s</span><span class="s">.h&quot;:&#39;</span> <span class="o">%</span> <span class="n">prefix</span><span class="p">,</span> <span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;   </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">prototype</span><span class="p">,</span> <span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">empty</span><span class="p">:</span>
-                <span class="k">print</span><span class="p">(</span><span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span>
-
-            <span class="c"># wrap</span>
-            <span class="n">ret</span><span class="p">,</span> <span class="n">args_py</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_split_retvals_inargs</span><span class="p">(</span><span class="n">routine</span><span class="o">.</span><span class="n">arguments</span><span class="p">)</span>
-            <span class="n">args_c</span> <span class="o">=</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="nb">str</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">name</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">routine</span><span class="o">.</span><span class="n">arguments</span><span class="p">])</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&quot;def </span><span class="si">%s</span><span class="s">_c(</span><span class="si">%s</span><span class="s">):&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">routine</span><span class="o">.</span><span class="n">name</span><span class="p">,</span>
-                    <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_declare_arg</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args_py</span><span class="p">)),</span> <span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">ret</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">args_py</span><span class="p">:</span>
-                    <span class="k">print</span><span class="p">(</span><span class="s">&quot;   cdef </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_declare_arg</span><span class="p">(</span><span class="n">r</span><span class="p">),</span> <span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span>
-            <span class="n">rets</span> <span class="o">=</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="nb">str</span><span class="p">(</span><span class="n">r</span><span class="o">.</span><span class="n">name</span><span class="p">)</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">ret</span><span class="p">])</span>
-            <span class="k">if</span> <span class="n">routine</span><span class="o">.</span><span class="n">results</span><span class="p">:</span>
-                <span class="n">call</span> <span class="o">=</span> <span class="s">&#39;   return </span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">routine</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="n">args_c</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">rets</span><span class="p">:</span>
-                    <span class="k">print</span><span class="p">(</span><span class="n">call</span> <span class="o">+</span> <span class="s">&#39;, &#39;</span> <span class="o">+</span> <span class="n">rets</span><span class="p">,</span> <span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">print</span><span class="p">(</span><span class="n">call</span><span class="p">,</span> <span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&#39;   </span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">)&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">routine</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="n">args_c</span><span class="p">),</span> <span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&#39;   return </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">rets</span><span class="p">,</span> <span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">empty</span><span class="p">:</span>
-                <span class="k">print</span><span class="p">(</span><span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span></div>
-    <span class="n">dump_pyx</span><span class="o">.</span><span class="n">extension</span> <span class="o">=</span> <span class="s">&quot;pyx&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_split_retvals_inargs</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Determines arguments and return values for python wrapper&quot;&quot;&quot;</span>
-        <span class="n">py_args</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">py_returns</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">OutputArgument</span><span class="p">):</span>
-                <span class="n">py_returns</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">InOutArgument</span><span class="p">):</span>
-                <span class="n">py_returns</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-                <span class="n">py_args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">py_args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">py_returns</span><span class="p">,</span> <span class="n">py_args</span>
-
-    <span class="k">def</span> <span class="nf">_declare_arg</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">get_datatype</span><span class="p">(</span><span class="s">&#39;c&#39;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">dimensions</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s"> *</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="nb">str</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">name</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="nb">str</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">name</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="F2PyCodeWrapper"><a class="viewcode-back" href="../../../modules/utilities/autowrap.html#sympy.utilities.autowrap.F2PyCodeWrapper">[docs]</a><span class="k">class</span> <span class="nc">F2PyCodeWrapper</span><span class="p">(</span><span class="n">CodeWrapper</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Wrapper that uses f2py&quot;&quot;&quot;</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">command</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="n">filename</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">filename</span> <span class="o">+</span> <span class="s">&#39;.&#39;</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">generator</span><span class="o">.</span><span class="n">code_extension</span>
-        <span class="n">command</span> <span class="o">=</span> <span class="p">[</span><span class="s">&quot;f2py&quot;</span><span class="p">,</span> <span class="s">&quot;-m&quot;</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">module_name</span><span class="p">,</span> <span class="s">&quot;-c&quot;</span><span class="p">,</span> <span class="n">filename</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">command</span>
-
-    <span class="k">def</span> <span class="nf">_prepare_files</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routine</span><span class="p">):</span>
-        <span class="k">pass</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">_get_wrapped_function</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">mod</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">mod</span><span class="o">.</span><span class="n">autofunc</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_get_code_wrapper_class</span><span class="p">(</span><span class="n">backend</span><span class="p">):</span>
-    <span class="n">wrappers</span> <span class="o">=</span> <span class="p">{</span> <span class="s">&#39;F2PY&#39;</span><span class="p">:</span> <span class="n">F2PyCodeWrapper</span><span class="p">,</span> <span class="s">&#39;CYTHON&#39;</span><span class="p">:</span> <span class="n">CythonCodeWrapper</span><span class="p">,</span>
-        <span class="s">&#39;DUMMY&#39;</span><span class="p">:</span> <span class="n">DummyWrapper</span><span class="p">}</span>
-    <span class="k">return</span> <span class="n">wrappers</span><span class="p">[</span><span class="n">backend</span><span class="o">.</span><span class="n">upper</span><span class="p">()]</span>
-
-
-<div class="viewcode-block" id="autowrap"><a class="viewcode-back" href="../../../modules/utilities/autowrap.html#sympy.utilities.autowrap.autowrap">[docs]</a><span class="k">def</span> <span class="nf">autowrap</span><span class="p">(</span>
-    <span class="n">expr</span><span class="p">,</span> <span class="n">language</span><span class="o">=</span><span class="s">&#39;F95&#39;</span><span class="p">,</span> <span class="n">backend</span><span class="o">=</span><span class="s">&#39;f2py&#39;</span><span class="p">,</span> <span class="n">tempdir</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">args</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">flags</span><span class="o">=</span><span class="p">[],</span>
-        <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">helpers</span><span class="o">=</span><span class="p">[]):</span>
-    <span class="sd">&quot;&quot;&quot;Generates python callable binaries based on the math expression.</span>
-
-<span class="sd">    expr</span>
-<span class="sd">        The SymPy expression that should be wrapped as a binary routine</span>
-
-<span class="sd">    :Optional arguments:</span>
-
-<span class="sd">    language</span>
-<span class="sd">        The programming language to use, currently &#39;C&#39; or &#39;F95&#39;</span>
-<span class="sd">    backend</span>
-<span class="sd">        The wrapper backend to use, currently f2py or Cython</span>
-<span class="sd">    tempdir</span>
-<span class="sd">        Path to directory for temporary files.  If this argument is supplied,</span>
-<span class="sd">        the generated code and the wrapper input files are left intact in the</span>
-<span class="sd">        specified path.</span>
-<span class="sd">    args</span>
-<span class="sd">        Sequence of the formal parameters of the generated code, if ommited the</span>
-<span class="sd">        function signature is determined by the code generator.</span>
-<span class="sd">    flags</span>
-<span class="sd">        Additional option flags that will be passed to the backend</span>
-<span class="sd">    verbose</span>
-<span class="sd">        If True, autowrap will not mute the command line backends.  This can be</span>
-<span class="sd">        helpful for debugging.</span>
-<span class="sd">    helpers</span>
-<span class="sd">        Used to define auxillary expressions needed for the main expr.  If the</span>
-<span class="sd">        main expression need to do call a specialized function it should be put</span>
-<span class="sd">        in the ``helpers`` list.  Autowrap will then make sure that the compiled</span>
-<span class="sd">        main expression can link to the helper routine.  Items should be tuples</span>
-<span class="sd">        with (&lt;funtion_name&gt;, &lt;sympy_expression&gt;, &lt;arguments&gt;).  It is</span>
-<span class="sd">        mandatory to supply an argument sequence to helper routines.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.autowrap import autowrap</span>
-<span class="sd">    &gt;&gt;&gt; expr = ((x - y + z)**(13)).expand()</span>
-<span class="sd">    &gt;&gt;&gt; binary_func = autowrap(expr)               # doctest: +SKIP</span>
-<span class="sd">    &gt;&gt;&gt; binary_func(1, 4, 2)                       # doctest: +SKIP</span>
-<span class="sd">    -1.0</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">code_generator</span> <span class="o">=</span> <span class="n">get_code_generator</span><span class="p">(</span><span class="n">language</span><span class="p">,</span> <span class="s">&quot;autowrap&quot;</span><span class="p">)</span>
-    <span class="n">CodeWrapperClass</span> <span class="o">=</span> <span class="n">_get_code_wrapper_class</span><span class="p">(</span><span class="n">backend</span><span class="p">)</span>
-    <span class="n">code_wrapper</span> <span class="o">=</span> <span class="n">CodeWrapperClass</span><span class="p">(</span><span class="n">code_generator</span><span class="p">,</span> <span class="n">tempdir</span><span class="p">,</span> <span class="n">flags</span><span class="p">,</span> <span class="n">verbose</span><span class="p">)</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">routine</span> <span class="o">=</span> <span class="n">Routine</span><span class="p">(</span><span class="s">&#39;autofunc&#39;</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">args</span><span class="p">)</span>
-    <span class="k">except</span> <span class="n">CodeGenArgumentListError</span> <span class="k">as</span> <span class="n">e</span><span class="p">:</span>
-        <span class="c"># if all missing arguments are for pure output, we simply attach them</span>
-        <span class="c"># at the end and try again, because the wrappers will silently convert</span>
-        <span class="c"># them to return values anyway.</span>
-        <span class="n">new_args</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">missing</span> <span class="ow">in</span> <span class="n">e</span><span class="o">.</span><span class="n">missing_args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">missing</span><span class="p">,</span> <span class="n">OutputArgument</span><span class="p">):</span>
-                <span class="k">raise</span>
-            <span class="n">new_args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">missing</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-        <span class="n">routine</span> <span class="o">=</span> <span class="n">Routine</span><span class="p">(</span><span class="s">&#39;autofunc&#39;</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">args</span> <span class="o">+</span> <span class="n">new_args</span><span class="p">)</span>
-
-    <span class="n">helps</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">name</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">args</span> <span class="ow">in</span> <span class="n">helpers</span><span class="p">:</span>
-        <span class="n">helps</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Routine</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">args</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">code_wrapper</span><span class="o">.</span><span class="n">wrap_code</span><span class="p">(</span><span class="n">routine</span><span class="p">,</span> <span class="n">helpers</span><span class="o">=</span><span class="n">helps</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="binary_function"><a class="viewcode-back" href="../../../modules/utilities/autowrap.html#sympy.utilities.autowrap.binary_function">[docs]</a><span class="k">def</span> <span class="nf">binary_function</span><span class="p">(</span><span class="n">symfunc</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns a sympy function with expr as binary implementation</span>
-
-<span class="sd">    This is a convenience function that automates the steps needed to</span>
-<span class="sd">    autowrap the SymPy expression and attaching it to a Function object</span>
-<span class="sd">    with implemented_function().</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.autowrap import binary_function</span>
-<span class="sd">    &gt;&gt;&gt; expr = ((x - y)**(25)).expand()</span>
-<span class="sd">    &gt;&gt;&gt; f = binary_function(&#39;f&#39;, expr)             # doctest: +SKIP</span>
-<span class="sd">    &gt;&gt;&gt; type(f)                                    # doctest: +SKIP</span>
-<span class="sd">    &lt;class &#39;sympy.core.function.FunctionClass&#39;&gt;</span>
-<span class="sd">    &gt;&gt;&gt; 2*f(x, y)                                  # doctest: +SKIP</span>
-<span class="sd">    2*f(x, y)</span>
-<span class="sd">    &gt;&gt;&gt; f(x, y).evalf(2, subs={x: 1, y: 2})        # doctest: +SKIP</span>
-<span class="sd">    -1.0</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">binary</span> <span class="o">=</span> <span class="n">autowrap</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">implemented_function</span><span class="p">(</span><span class="n">symfunc</span><span class="p">,</span> <span class="n">binary</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="ufuncify"><a class="viewcode-back" href="../../../modules/utilities/autowrap.html#sympy.utilities.autowrap.ufuncify">[docs]</a><span class="k">def</span> <span class="nf">ufuncify</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Generates a binary ufunc-like lambda function for numpy arrays</span>
-
-<span class="sd">    ``args``</span>
-<span class="sd">        Either a Symbol or a tuple of symbols. Specifies the argument sequence</span>
-<span class="sd">        for the ufunc-like function.</span>
-
-<span class="sd">    ``expr``</span>
-<span class="sd">        A SymPy expression that defines the element wise operation</span>
-
-<span class="sd">    ``kwargs``</span>
-<span class="sd">        Optional keyword arguments are forwarded to autowrap().</span>
-
-<span class="sd">    The returned function can only act on one array at a time, as only the</span>
-<span class="sd">    first argument accept arrays as input.</span>
-
-<span class="sd">    .. Note:: a *proper* numpy ufunc is required to support broadcasting, type</span>
-<span class="sd">       casting and more.  The function returned here, may not qualify for</span>
-<span class="sd">       numpy&#39;s definition of a ufunc.  That why we use the term ufunc-like.</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-<span class="sd">    [1] http://docs.scipy.org/doc/numpy/reference/ufuncs.html</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.autowrap import ufuncify</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">    &gt;&gt;&gt; f = ufuncify([x, y], y + x**2)             # doctest: +SKIP</span>
-<span class="sd">    &gt;&gt;&gt; f([1, 2, 3], 2)                            # doctest: +SKIP</span>
-<span class="sd">    [2.  5.  10.]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">y</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">IndexedBase</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">))</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">IndexedBase</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">))</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;m&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">i</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">i</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Idx</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-    <span class="n">l</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Lambda</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-    <span class="n">f</span> <span class="o">=</span> <span class="n">implemented_function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">,</span> <span class="n">l</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Symbol</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">args</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="c"># first argument accepts an array</span>
-    <span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">autowrap</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">Equality</span><span class="p">(</span><span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">f</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)),</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span></div>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.utilities.codegen</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">module for generating C, C++, Fortran77, Fortran90 and python routines that</span>
-<span class="sd">evaluate sympy expressions. This module is work in progress. Only the</span>
-<span class="sd">milestones with a &#39;+&#39; character in the list below have been completed.</span>
-
-<span class="sd">--- How is sympy.utilities.codegen different from sympy.printing.ccode? ---</span>
-
-<span class="sd">We considered the idea to extend the printing routines for sympy functions in</span>
-<span class="sd">such a way that it prints complete compilable code, but this leads to a few</span>
-<span class="sd">unsurmountable issues that can only be tackled with dedicated code generator:</span>
-
-<span class="sd">- For C, one needs both a code and a header file, while the printing routines</span>
-<span class="sd">  generate just one string. This code generator can be extended to support</span>
-<span class="sd">  .pyf files for f2py.</span>
-
-<span class="sd">- SymPy functions are not concerned with programming-technical issues, such</span>
-<span class="sd">  as input, output and input-output arguments. Other examples are contiguous</span>
-<span class="sd">  or non-contiguous arrays, including headers of other libraries such as gsl</span>
-<span class="sd">  or others.</span>
-
-<span class="sd">- It is highly interesting to evaluate several sympy functions in one C</span>
-<span class="sd">  routine, eventually sharing common intermediate results with the help</span>
-<span class="sd">  of the cse routine. This is more than just printing.</span>
-
-<span class="sd">- From the programming perspective, expressions with constants should be</span>
-<span class="sd">  evaluated in the code generator as much as possible. This is different</span>
-<span class="sd">  for printing.</span>
-
-<span class="sd">--- Basic assumptions ---</span>
-
-<span class="sd">* A generic Routine data structure describes the routine that must be</span>
-<span class="sd">  translated into C/Fortran/... code. This data structure covers all</span>
-<span class="sd">  features present in one or more of the supported languages.</span>
-
-<span class="sd">* Descendants from the CodeGen class transform multiple Routine instances</span>
-<span class="sd">  into compilable code. Each derived class translates into a specific</span>
-<span class="sd">  language.</span>
-
-<span class="sd">* In many cases, one wants a simple workflow. The friendly functions in the</span>
-<span class="sd">  last part are a simple api on top of the Routine/CodeGen stuff. They are</span>
-<span class="sd">  easier to use, but are less powerful.</span>
-
-<span class="sd">--- Milestones ---</span>
-
-<span class="sd">+ First working version with scalar input arguments, generating C code,</span>
-<span class="sd">  tests</span>
-<span class="sd">+ Friendly functions that are easier to use than the rigorous</span>
-<span class="sd">  Routine/CodeGen workflow.</span>
-<span class="sd">+ Integer and Real numbers as input and output</span>
-<span class="sd">+ Output arguments</span>
-<span class="sd">+ InputOutput arguments</span>
-<span class="sd">+ Sort input/output arguments properly</span>
-<span class="sd">+ Contiguous array arguments (numpy matrices)</span>
-<span class="sd">+ Also generate .pyf code for f2py (in autowrap module)</span>
-<span class="sd">+ Isolate constants and evaluate them beforehand in double precision</span>
-<span class="sd">+ Fortran 90</span>
-
-<span class="sd">- Common Subexpression Elimination</span>
-<span class="sd">- User defined comments in the generated code</span>
-<span class="sd">- Optional extra include lines for libraries/objects that can eval special</span>
-<span class="sd">  functions</span>
-<span class="sd">- Test other C compilers and libraries: gcc, tcc, libtcc, gcc+gsl, ...</span>
-<span class="sd">- Contiguous array arguments (sympy matrices)</span>
-<span class="sd">- Non-contiguous array arguments (sympy matrices)</span>
-<span class="sd">- ccode must raise an error when it encounters something that can not be</span>
-<span class="sd">  translated into c. ccode(integrate(sin(x)/x, x)) does not make sense.</span>
-<span class="sd">- Complex numbers as input and output</span>
-<span class="sd">- A default complex datatype</span>
-<span class="sd">- Include extra information in the header: date, user, hostname, sha1</span>
-<span class="sd">  hash, ...</span>
-<span class="sd">- Fortran 77</span>
-<span class="sd">- C++</span>
-<span class="sd">- Python</span>
-<span class="sd">- ...</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-
-
-<span class="kn">import</span> <span class="nn">os</span>
-<span class="kn">from</span> <span class="nn">io</span> <span class="kn">import</span> <span class="n">StringIO</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">__version__</span> <span class="k">as</span> <span class="n">sympy_version</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">,</span> <span class="n">Equality</span><span class="p">,</span> <span class="n">Function</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">is_sequence</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.codeprinter</span> <span class="kn">import</span> <span class="n">AssignmentError</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.ccode</span> <span class="kn">import</span> <span class="n">ccode</span><span class="p">,</span> <span class="n">CCodePrinter</span>
-<span class="kn">from</span> <span class="nn">sympy.printing.fcode</span> <span class="kn">import</span> <span class="n">fcode</span><span class="p">,</span> <span class="n">FCodePrinter</span>
-<span class="kn">from</span> <span class="nn">sympy.tensor</span> <span class="kn">import</span> <span class="n">Idx</span><span class="p">,</span> <span class="n">Indexed</span><span class="p">,</span> <span class="n">IndexedBase</span>
-
-
-<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
-    <span class="c"># description of routines</span>
-    <span class="s">&quot;Routine&quot;</span><span class="p">,</span> <span class="s">&quot;DataType&quot;</span><span class="p">,</span> <span class="s">&quot;default_datatypes&quot;</span><span class="p">,</span> <span class="s">&quot;get_default_datatype&quot;</span><span class="p">,</span>
-    <span class="s">&quot;Argument&quot;</span><span class="p">,</span> <span class="s">&quot;InputArgument&quot;</span><span class="p">,</span> <span class="s">&quot;Result&quot;</span><span class="p">,</span>
-    <span class="c"># routines -&gt; code</span>
-    <span class="s">&quot;CodeGen&quot;</span><span class="p">,</span> <span class="s">&quot;CCodeGen&quot;</span><span class="p">,</span> <span class="s">&quot;FCodeGen&quot;</span><span class="p">,</span>
-    <span class="c"># friendly functions</span>
-    <span class="s">&quot;codegen&quot;</span><span class="p">,</span>
-<span class="p">]</span>
-
-
-<span class="c">#</span>
-<span class="c"># Description of routines</span>
-<span class="c">#</span>
-
-
-<div class="viewcode-block" id="Routine"><a class="viewcode-back" href="../../../modules/utilities/codegen.html#sympy.utilities.codegen.Routine">[docs]</a><span class="k">class</span> <span class="nc">Routine</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Generic description of an evaluation routine for a set of sympy expressions.</span>
-
-<span class="sd">       A CodeGen class can translate instances of this class into C/Fortran/...</span>
-<span class="sd">       code. The routine specification covers all the features present in these</span>
-<span class="sd">       languages. The CodeGen part must raise an exception when certain features</span>
-<span class="sd">       are not present in the target language. For example, multiple return</span>
-<span class="sd">       values are possible in Python, but not in C or Fortran. Another example:</span>
-<span class="sd">       Fortran and Python support complex numbers, while C does not.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">argument_sequence</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Initialize a Routine instance.</span>
-
-<span class="sd">        ``name``</span>
-<span class="sd">            A string with the name of this routine in the generated code</span>
-<span class="sd">        ``expr``</span>
-<span class="sd">            The sympy expression that the Routine instance will represent.  If</span>
-<span class="sd">            given a list or tuple of expressions, the routine will be</span>
-<span class="sd">            considered to have multiple return values.</span>
-<span class="sd">        ``argument_sequence``</span>
-<span class="sd">            Optional list/tuple containing arguments for the routine in a</span>
-<span class="sd">            preferred order.  If omitted, arguments will be ordered</span>
-<span class="sd">            alphabetically, but with all input aguments first, and then output</span>
-<span class="sd">            or in-out arguments.</span>
-
-<span class="sd">        A decision about whether to use output arguments or return values,</span>
-<span class="sd">        is made depending on the mathematical expressions.  For an expression</span>
-<span class="sd">        of type Equality, the left hand side is made into an OutputArgument</span>
-<span class="sd">        (or an InOutArgument if appropriate).  Else, the calculated</span>
-<span class="sd">        expression is the return values of the routine.</span>
-
-<span class="sd">        A tuple of exressions can be used to create a routine with both</span>
-<span class="sd">        return value(s) and output argument(s).</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">arg_list</span> <span class="o">=</span> <span class="p">[]</span>
-
-        <span class="k">if</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">expr</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;No expression given&quot;</span><span class="p">)</span>
-            <span class="n">expressions</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">*</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">expressions</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-        <span class="c"># local variables</span>
-        <span class="n">local_vars</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span><span class="n">i</span><span class="o">.</span><span class="n">label</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">expressions</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Idx</span><span class="p">)])</span>
-
-        <span class="c"># symbols that should be arguments</span>
-        <span class="n">symbols</span> <span class="o">=</span> <span class="n">expressions</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Symbol</span><span class="p">)</span> <span class="o">-</span> <span class="n">local_vars</span>
-
-        <span class="c"># Decide whether to use output argument or return value</span>
-        <span class="n">return_val</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">output_args</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">expr</span> <span class="ow">in</span> <span class="n">expressions</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Equality</span><span class="p">):</span>
-                <span class="n">out_arg</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">lhs</span>
-                <span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">rhs</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">out_arg</span><span class="p">,</span> <span class="n">Indexed</span><span class="p">):</span>
-                    <span class="n">dims</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">([</span> <span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="n">dim</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">dim</span> <span class="ow">in</span> <span class="n">out_arg</span><span class="o">.</span><span class="n">shape</span><span class="p">])</span>
-                    <span class="n">symbol</span> <span class="o">=</span> <span class="n">out_arg</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">label</span>
-                <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">out_arg</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">):</span>
-                    <span class="n">dims</span> <span class="o">=</span> <span class="p">[]</span>
-                    <span class="n">symbol</span> <span class="o">=</span> <span class="n">out_arg</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="n">CodeGenError</span><span class="p">(</span>
-                        <span class="s">&quot;Only Indexed or Symbol can define output arguments&quot;</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">symbol</span><span class="p">):</span>
-                    <span class="n">output_args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span>
-                        <span class="n">InOutArgument</span><span class="p">(</span><span class="n">symbol</span><span class="p">,</span> <span class="n">out_arg</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">dimensions</span><span class="o">=</span><span class="n">dims</span><span class="p">))</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">output_args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">OutputArgument</span><span class="p">(</span>
-                        <span class="n">symbol</span><span class="p">,</span> <span class="n">out_arg</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">dimensions</span><span class="o">=</span><span class="n">dims</span><span class="p">))</span>
-
-                <span class="c"># avoid duplicate arguments</span>
-                <span class="n">symbols</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">symbol</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">return_val</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Result</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-
-        <span class="c"># setup input argument list</span>
-        <span class="n">array_symbols</span> <span class="o">=</span> <span class="p">{}</span>
-        <span class="k">for</span> <span class="n">array</span> <span class="ow">in</span> <span class="n">expressions</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Indexed</span><span class="p">):</span>
-            <span class="n">array_symbols</span><span class="p">[</span><span class="n">array</span><span class="o">.</span><span class="n">base</span><span class="o">.</span><span class="n">label</span><span class="p">]</span> <span class="o">=</span> <span class="n">array</span>
-
-        <span class="k">for</span> <span class="n">symbol</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">symbols</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="nb">str</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">symbol</span> <span class="ow">in</span> <span class="n">array_symbols</span><span class="p">:</span>
-                <span class="n">dims</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="n">array</span> <span class="o">=</span> <span class="n">array_symbols</span><span class="p">[</span><span class="n">symbol</span><span class="p">]</span>
-                <span class="k">for</span> <span class="n">dim</span> <span class="ow">in</span> <span class="n">array</span><span class="o">.</span><span class="n">shape</span><span class="p">:</span>
-                    <span class="n">dims</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">,</span> <span class="n">dim</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-                <span class="n">metadata</span> <span class="o">=</span> <span class="p">{</span><span class="s">&#39;dimensions&#39;</span><span class="p">:</span> <span class="n">dims</span><span class="p">}</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">metadata</span> <span class="o">=</span> <span class="p">{}</span>
-
-            <span class="n">arg_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">InputArgument</span><span class="p">(</span><span class="n">symbol</span><span class="p">,</span> <span class="o">**</span><span class="n">metadata</span><span class="p">))</span>
-
-        <span class="n">output_args</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="nb">str</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">name</span><span class="p">))</span>
-        <span class="n">arg_list</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">output_args</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">argument_sequence</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="c"># if the user has supplied IndexedBase instances, we&#39;ll accept that</span>
-            <span class="n">new_sequence</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">argument_sequence</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">IndexedBase</span><span class="p">):</span>
-                    <span class="n">new_sequence</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">label</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">new_sequence</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="n">argument_sequence</span> <span class="o">=</span> <span class="n">new_sequence</span>
-
-            <span class="n">missing</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg_list</span> <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">name</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">argument_sequence</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">missing</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">CodeGenArgumentListError</span><span class="p">(</span><span class="s">&quot;Argument list didn&#39;t specify: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span>
-                        <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="nb">str</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">name</span><span class="p">)</span> <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">missing</span><span class="p">]),</span> <span class="n">missing</span><span class="p">)</span>
-
-            <span class="c"># create redundant arguments to produce the requested sequence</span>
-            <span class="n">name_arg_dict</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">([(</span><span class="n">x</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">arg_list</span><span class="p">])</span>
-            <span class="n">new_args</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">symbol</span> <span class="ow">in</span> <span class="n">argument_sequence</span><span class="p">:</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="n">new_args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">name_arg_dict</span><span class="p">[</span><span class="n">symbol</span><span class="p">])</span>
-                <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-                    <span class="n">new_args</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">InputArgument</span><span class="p">(</span><span class="n">symbol</span><span class="p">))</span>
-            <span class="n">arg_list</span> <span class="o">=</span> <span class="n">new_args</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">name</span> <span class="o">=</span> <span class="n">name</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">arguments</span> <span class="o">=</span> <span class="n">arg_list</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">results</span> <span class="o">=</span> <span class="n">return_val</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">local_vars</span> <span class="o">=</span> <span class="n">local_vars</span>
-
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Routine.variables"><a class="viewcode-back" href="../../../modules/utilities/codegen.html#sympy.utilities.codegen.Routine.variables">[docs]</a>    <span class="k">def</span> <span class="nf">variables</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a set containing all variables possibly used in this routine.</span>
-
-<span class="sd">        For routines with unnamed return values, the dummies that may or may</span>
-<span class="sd">        not be used will be included in the set.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">local_vars</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">arguments</span><span class="p">:</span>
-            <span class="n">v</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">res</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">results</span><span class="p">:</span>
-            <span class="n">v</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">res</span><span class="o">.</span><span class="n">result_var</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">v</span>
-</div>
-    <span class="nd">@property</span>
-<div class="viewcode-block" id="Routine.result_variables"><a class="viewcode-back" href="../../../modules/utilities/codegen.html#sympy.utilities.codegen.Routine.result_variables">[docs]</a>    <span class="k">def</span> <span class="nf">result_variables</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a list of OutputArgument, InOutArgument and Result.</span>
-
-<span class="sd">        If return values are present, they are at the end ot the list.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="p">[</span><span class="n">arg</span> <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">arguments</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span>
-            <span class="n">arg</span><span class="p">,</span> <span class="p">(</span><span class="n">OutputArgument</span><span class="p">,</span> <span class="n">InOutArgument</span><span class="p">))]</span>
-        <span class="n">args</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">results</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">args</span>
-
-</div></div>
-<div class="viewcode-block" id="DataType"><a class="viewcode-back" href="../../../modules/utilities/codegen.html#sympy.utilities.codegen.DataType">[docs]</a><span class="k">class</span> <span class="nc">DataType</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Holds strings for a certain datatype in different programming languages.&quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">cname</span><span class="p">,</span> <span class="n">fname</span><span class="p">,</span> <span class="n">pyname</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">cname</span> <span class="o">=</span> <span class="n">cname</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">fname</span> <span class="o">=</span> <span class="n">fname</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">pyname</span> <span class="o">=</span> <span class="n">pyname</span>
-
-</div>
-<span class="n">default_datatypes</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="s">&quot;int&quot;</span><span class="p">:</span> <span class="n">DataType</span><span class="p">(</span><span class="s">&quot;int&quot;</span><span class="p">,</span> <span class="s">&quot;INTEGER*4&quot;</span><span class="p">,</span> <span class="s">&quot;int&quot;</span><span class="p">),</span>
-    <span class="s">&quot;float&quot;</span><span class="p">:</span> <span class="n">DataType</span><span class="p">(</span><span class="s">&quot;double&quot;</span><span class="p">,</span> <span class="s">&quot;REAL*8&quot;</span><span class="p">,</span> <span class="s">&quot;float&quot;</span><span class="p">)</span>
-<span class="p">}</span>
-
-
-<div class="viewcode-block" id="get_default_datatype"><a class="viewcode-back" href="../../../modules/utilities/codegen.html#sympy.utilities.codegen.get_default_datatype">[docs]</a><span class="k">def</span> <span class="nf">get_default_datatype</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Derives a decent data type based on the assumptions on the expression.&quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_integer</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">default_datatypes</span><span class="p">[</span><span class="s">&quot;int&quot;</span><span class="p">]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">default_datatypes</span><span class="p">[</span><span class="s">&quot;float&quot;</span><span class="p">]</span>
-
-</div>
-<span class="k">class</span> <span class="nc">Variable</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Represents a typed variable.&quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">datatype</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">dimensions</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">precision</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Initializes a Variable instance</span>
-
-<span class="sd">           name  --  must be of class Symbol</span>
-<span class="sd">           datatype  --  When not given, the data type will be guessed based</span>
-<span class="sd">                         on the assumptions on the symbol argument.</span>
-<span class="sd">           dimension  --  If present, the argument is interpreted as an array.</span>
-<span class="sd">                          Dimensions must be a sequence containing tuples, i.e.</span>
-<span class="sd">                          (lower, upper) bounds for each index of the array</span>
-<span class="sd">           precision  --  FIXME</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;The first argument must be a sympy symbol.&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">datatype</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">datatype</span> <span class="o">=</span> <span class="n">get_default_datatype</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">datatype</span><span class="p">,</span> <span class="n">DataType</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;The (optional) `datatype&#39; argument must be an instance of the DataType class.&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">dimensions</span> <span class="ow">and</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">dimensions</span><span class="p">,</span> <span class="p">(</span><span class="nb">tuple</span><span class="p">,</span> <span class="nb">list</span><span class="p">)):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span>
-                <span class="s">&quot;The dimension argument must be a sequence of tuples&quot;</span><span class="p">)</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_name</span> <span class="o">=</span> <span class="n">name</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_datatype</span> <span class="o">=</span> <span class="p">{</span>
-            <span class="s">&#39;C&#39;</span><span class="p">:</span> <span class="n">datatype</span><span class="o">.</span><span class="n">cname</span><span class="p">,</span>
-            <span class="s">&#39;FORTRAN&#39;</span><span class="p">:</span> <span class="n">datatype</span><span class="o">.</span><span class="n">fname</span><span class="p">,</span>
-            <span class="s">&#39;PYTHON&#39;</span><span class="p">:</span> <span class="n">datatype</span><span class="o">.</span><span class="n">pyname</span>
-        <span class="p">}</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">dimensions</span> <span class="o">=</span> <span class="n">dimensions</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">precision</span> <span class="o">=</span> <span class="n">precision</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">name</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_name</span>
-
-    <span class="k">def</span> <span class="nf">get_datatype</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">language</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns the datatype string for the requested langage.</span>
-
-<span class="sd">            &gt;&gt;&gt; from sympy import Symbol</span>
-<span class="sd">            &gt;&gt;&gt; from sympy.utilities.codegen import Variable</span>
-<span class="sd">            &gt;&gt;&gt; x = Variable(Symbol(&#39;x&#39;))</span>
-<span class="sd">            &gt;&gt;&gt; x.get_datatype(&#39;c&#39;)</span>
-<span class="sd">            &#39;double&#39;</span>
-<span class="sd">            &gt;&gt;&gt; x.get_datatype(&#39;fortran&#39;)</span>
-<span class="sd">            &#39;REAL*8&#39;</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_datatype</span><span class="p">[</span><span class="n">language</span><span class="o">.</span><span class="n">upper</span><span class="p">()]</span>
-        <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">CodeGenError</span><span class="p">(</span><span class="s">&quot;Has datatypes for languages: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span>
-                    <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_datatype</span><span class="p">))</span>
-
-
-<div class="viewcode-block" id="Argument"><a class="viewcode-back" href="../../../modules/utilities/codegen.html#sympy.utilities.codegen.Argument">[docs]</a><span class="k">class</span> <span class="nc">Argument</span><span class="p">(</span><span class="n">Variable</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;An abstract Argument data structure: a name and a data type.</span>
-
-<span class="sd">       This structure is refined in the descendants below.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">datatype</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">dimensions</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">precision</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; See docstring of Variable.__init__</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">Variable</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">datatype</span><span class="p">,</span> <span class="n">dimensions</span><span class="p">,</span> <span class="n">precision</span><span class="p">)</span>
-
-</div>
-<span class="k">class</span> <span class="nc">InputArgument</span><span class="p">(</span><span class="n">Argument</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-
-<span class="k">class</span> <span class="nc">ResultBase</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Base class for all ``outgoing&#39;&#39; information from a routine</span>
-
-<span class="sd">       Objects of this class stores a sympy expression, and a sympy object</span>
-<span class="sd">       representing a result variable that will be used in the generated code</span>
-<span class="sd">       only if necessary.</span>
-<span class="sd">   &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">result_var</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">expr</span> <span class="o">=</span> <span class="n">expr</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">result_var</span> <span class="o">=</span> <span class="n">result_var</span>
-
-
-<span class="k">class</span> <span class="nc">OutputArgument</span><span class="p">(</span><span class="n">Argument</span><span class="p">,</span> <span class="n">ResultBase</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;OutputArgument are always initialized in the routine</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">result_var</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">datatype</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">dimensions</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">precision</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; See docstring of Variable.__init__</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">Argument</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">datatype</span><span class="p">,</span> <span class="n">dimensions</span><span class="p">,</span> <span class="n">precision</span><span class="p">)</span>
-        <span class="n">ResultBase</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">result_var</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">InOutArgument</span><span class="p">(</span><span class="n">Argument</span><span class="p">,</span> <span class="n">ResultBase</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;InOutArgument are never initialized in the routine</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">result_var</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">datatype</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">dimensions</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">precision</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot; See docstring of Variable.__init__</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">Argument</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">datatype</span><span class="p">,</span> <span class="n">dimensions</span><span class="p">,</span> <span class="n">precision</span><span class="p">)</span>
-        <span class="n">ResultBase</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">result_var</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="Result"><a class="viewcode-back" href="../../../modules/utilities/codegen.html#sympy.utilities.codegen.Result">[docs]</a><span class="k">class</span> <span class="nc">Result</span><span class="p">(</span><span class="n">ResultBase</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;An expression for a scalar return value.</span>
-
-<span class="sd">       The name result is used to avoid conflicts with the reserved word</span>
-<span class="sd">       &#39;return&#39; in the python language. It is also shorter than ReturnValue.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">datatype</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">precision</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Initialize a (scalar) return value.</span>
-
-<span class="sd">           The second argument is optional. When not given, the data type will</span>
-<span class="sd">           be guessed based on the assumptions on the expression argument.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Expr</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;The first argument must be a sympy expression.&quot;</span><span class="p">)</span>
-
-        <span class="n">temp_var</span> <span class="o">=</span> <span class="n">Variable</span><span class="p">(</span><span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;result_</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="nb">hash</span><span class="p">(</span><span class="n">expr</span><span class="p">)),</span>
-                <span class="n">datatype</span><span class="o">=</span><span class="n">datatype</span><span class="p">,</span> <span class="n">dimensions</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">precision</span><span class="o">=</span><span class="n">precision</span><span class="p">)</span>
-        <span class="n">ResultBase</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">temp_var</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_temp_variable</span> <span class="o">=</span> <span class="n">temp_var</span>
-
-    <span class="k">def</span> <span class="nf">get_datatype</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">language</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_temp_variable</span><span class="o">.</span><span class="n">get_datatype</span><span class="p">(</span><span class="n">language</span><span class="p">)</span>
-
-
-<span class="c">#</span>
-<span class="c"># Transformation of routine objects into code</span>
-<span class="c">#</span>
-</div>
-<div class="viewcode-block" id="CodeGen"><a class="viewcode-back" href="../../../modules/utilities/codegen.html#sympy.utilities.codegen.CodeGen">[docs]</a><span class="k">class</span> <span class="nc">CodeGen</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Abstract class for the code generators.&quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">project</span><span class="o">=</span><span class="s">&quot;project&quot;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Initialize a code generator.</span>
-
-<span class="sd">           Derived classes will offer more options that affect the generated</span>
-<span class="sd">           code.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">project</span> <span class="o">=</span> <span class="n">project</span>
-
-<div class="viewcode-block" id="CodeGen.write"><a class="viewcode-back" href="../../../modules/utilities/codegen.html#sympy.utilities.codegen.CodeGen.write">[docs]</a>    <span class="k">def</span> <span class="nf">write</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routines</span><span class="p">,</span> <span class="n">prefix</span><span class="p">,</span> <span class="n">to_files</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">header</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">empty</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Writes all the source code files for the given routines.</span>
-
-<span class="sd">            The generate source is returned as a list of (filename, contents)</span>
-<span class="sd">            tuples, or is written to files (see options). Each filename consists</span>
-<span class="sd">            of the given prefix, appended with an appropriate extension.</span>
-
-<span class="sd">            ``routines``</span>
-<span class="sd">                A list of Routine instances to be written</span>
-<span class="sd">            ``prefix``</span>
-<span class="sd">                The prefix for the output files</span>
-<span class="sd">            ``to_files``</span>
-<span class="sd">                When True, the output is effectively written to files.</span>
-<span class="sd">                [DEFAULT=False] Otherwise, a list of (filename, contents)</span>
-<span class="sd">                tuples is returned.</span>
-<span class="sd">            ``header``</span>
-<span class="sd">                When True, a header comment is included on top of each source</span>
-<span class="sd">                file. [DEFAULT=True]</span>
-<span class="sd">            ``empty``</span>
-<span class="sd">                When True, empty lines are included to structure the source</span>
-<span class="sd">                files. [DEFAULT=True]</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">to_files</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">dump_fn</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">dump_fns</span><span class="p">:</span>
-                <span class="n">filename</span> <span class="o">=</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">.</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">prefix</span><span class="p">,</span> <span class="n">dump_fn</span><span class="o">.</span><span class="n">extension</span><span class="p">)</span>
-                <span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="n">filename</span><span class="p">,</span> <span class="s">&quot;w&quot;</span><span class="p">)</span> <span class="k">as</span> <span class="n">f</span><span class="p">:</span>
-                    <span class="n">dump_fn</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routines</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">prefix</span><span class="p">,</span> <span class="n">header</span><span class="p">,</span> <span class="n">empty</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-            <span class="k">for</span> <span class="n">dump_fn</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">dump_fns</span><span class="p">:</span>
-                <span class="n">filename</span> <span class="o">=</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">.</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">prefix</span><span class="p">,</span> <span class="n">dump_fn</span><span class="o">.</span><span class="n">extension</span><span class="p">)</span>
-                <span class="n">contents</span> <span class="o">=</span> <span class="n">StringIO</span><span class="p">()</span>
-                <span class="n">dump_fn</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routines</span><span class="p">,</span> <span class="n">contents</span><span class="p">,</span> <span class="n">prefix</span><span class="p">,</span> <span class="n">header</span><span class="p">,</span> <span class="n">empty</span><span class="p">)</span>
-                <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">filename</span><span class="p">,</span> <span class="n">contents</span><span class="o">.</span><span class="n">getvalue</span><span class="p">()))</span>
-            <span class="k">return</span> <span class="n">result</span>
-</div>
-<div class="viewcode-block" id="CodeGen.dump_code"><a class="viewcode-back" href="../../../modules/utilities/codegen.html#sympy.utilities.codegen.CodeGen.dump_code">[docs]</a>    <span class="k">def</span> <span class="nf">dump_code</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routines</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">prefix</span><span class="p">,</span> <span class="n">header</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">empty</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Write the code file by calling language specific methods in correct order</span>
-
-<span class="sd">        The generated file contains all the definitions of the routines in</span>
-<span class="sd">        low-level code and refers to the header file if appropriate.</span>
-
-<span class="sd">        :Arguments:</span>
-
-<span class="sd">        routines</span>
-<span class="sd">            A list of Routine instances</span>
-<span class="sd">        f</span>
-<span class="sd">            A file-like object to write the file to</span>
-<span class="sd">        prefix</span>
-<span class="sd">            The filename prefix, used to refer to the proper header file. Only</span>
-<span class="sd">            the basename of the prefix is used.</span>
-
-<span class="sd">        :Optional arguments:</span>
-
-<span class="sd">        header</span>
-<span class="sd">            When True, a header comment is included on top of each source file.</span>
-<span class="sd">            [DEFAULT=True]</span>
-<span class="sd">        empty</span>
-<span class="sd">            When True, empty lines are included to structure the source files.</span>
-<span class="sd">            [DEFAULT=True]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-
-        <span class="n">code_lines</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_preprocessor_statements</span><span class="p">(</span><span class="n">prefix</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">routine</span> <span class="ow">in</span> <span class="n">routines</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">empty</span><span class="p">:</span>
-                <span class="n">code_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-            <span class="n">code_lines</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_get_routine_opening</span><span class="p">(</span><span class="n">routine</span><span class="p">))</span>
-            <span class="n">code_lines</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_declare_arguments</span><span class="p">(</span><span class="n">routine</span><span class="p">))</span>
-            <span class="n">code_lines</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_declare_locals</span><span class="p">(</span><span class="n">routine</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">empty</span><span class="p">:</span>
-                <span class="n">code_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-            <span class="n">code_lines</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_call_printer</span><span class="p">(</span><span class="n">routine</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">empty</span><span class="p">:</span>
-                <span class="n">code_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-            <span class="n">code_lines</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_get_routine_ending</span><span class="p">(</span><span class="n">routine</span><span class="p">))</span>
-
-        <span class="n">code_lines</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_indent_code</span><span class="p">(</span><span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">code_lines</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="n">header</span><span class="p">:</span>
-            <span class="n">code_lines</span> <span class="o">=</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_get_header</span><span class="p">()</span> <span class="o">+</span> <span class="p">[</span><span class="n">code_lines</span><span class="p">])</span>
-
-        <span class="k">if</span> <span class="n">code_lines</span><span class="p">:</span>
-            <span class="n">f</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="n">code_lines</span><span class="p">)</span>
-
-</div></div>
-<span class="k">class</span> <span class="nc">CodeGenError</span><span class="p">(</span><span class="ne">Exception</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-
-<span class="k">class</span> <span class="nc">CodeGenArgumentListError</span><span class="p">(</span><span class="ne">Exception</span><span class="p">):</span>
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">missing_args</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-
-
-<span class="n">header_comment</span> <span class="o">=</span> <span class="s">&quot;&quot;&quot;Code generated with sympy </span><span class="si">%(version)s</span><span class="s"></span>
-
-<span class="s">See http://www.sympy.org/ for more information.</span>
-
-<span class="s">This file is part of &#39;</span><span class="si">%(project)s</span><span class="s">&#39;</span>
-<span class="s">&quot;&quot;&quot;</span>
-
-
-<div class="viewcode-block" id="CCodeGen"><a class="viewcode-back" href="../../../modules/utilities/codegen.html#sympy.utilities.codegen.CCodeGen">[docs]</a><span class="k">class</span> <span class="nc">CCodeGen</span><span class="p">(</span><span class="n">CodeGen</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Generator for C code</span>
-
-<span class="sd">    The .write() method inherited from CodeGen will output a code file and an</span>
-<span class="sd">    inteface file, &lt;prefix&gt;.c and &lt;prefix&gt;.h respectively.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">code_extension</span> <span class="o">=</span> <span class="s">&quot;c&quot;</span>
-    <span class="n">interface_extension</span> <span class="o">=</span> <span class="s">&quot;h&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_get_header</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Writes a common header for the generated files.&quot;&quot;&quot;</span>
-        <span class="n">code_lines</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">code_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;/&quot;</span> <span class="o">+</span> <span class="s">&quot;*&quot;</span><span class="o">*</span><span class="mi">78</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">)</span>
-        <span class="n">tmp</span> <span class="o">=</span> <span class="n">header_comment</span> <span class="o">%</span> <span class="p">{</span><span class="s">&quot;version&quot;</span><span class="p">:</span> <span class="n">sympy_version</span><span class="p">,</span>
-            <span class="s">&quot;project&quot;</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">project</span><span class="p">}</span>
-        <span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="n">tmp</span><span class="o">.</span><span class="n">splitlines</span><span class="p">():</span>
-            <span class="n">code_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot; *</span><span class="si">%s</span><span class="s">*</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">line</span><span class="o">.</span><span class="n">center</span><span class="p">(</span><span class="mi">76</span><span class="p">))</span>
-        <span class="n">code_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot; &quot;</span> <span class="o">+</span> <span class="s">&quot;*&quot;</span><span class="o">*</span><span class="mi">78</span> <span class="o">+</span> <span class="s">&quot;/</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">code_lines</span>
-
-<div class="viewcode-block" id="CCodeGen.get_prototype"><a class="viewcode-back" href="../../../modules/utilities/codegen.html#sympy.utilities.codegen.CCodeGen.get_prototype">[docs]</a>    <span class="k">def</span> <span class="nf">get_prototype</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routine</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a string for the function prototype for the given routine.</span>
-
-<span class="sd">           If the routine has multiple result objects, an CodeGenError is</span>
-<span class="sd">           raised.</span>
-
-<span class="sd">           See: http://en.wikipedia.org/wiki/Function_prototype</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">routine</span><span class="o">.</span><span class="n">results</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">CodeGenError</span><span class="p">(</span><span class="s">&quot;C only supports a single or no return value.&quot;</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">routine</span><span class="o">.</span><span class="n">results</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">ctype</span> <span class="o">=</span> <span class="n">routine</span><span class="o">.</span><span class="n">results</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">get_datatype</span><span class="p">(</span><span class="s">&#39;C&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">ctype</span> <span class="o">=</span> <span class="s">&quot;void&quot;</span>
-
-        <span class="n">type_args</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">routine</span><span class="o">.</span><span class="n">arguments</span><span class="p">:</span>
-            <span class="n">name</span> <span class="o">=</span> <span class="n">ccode</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">dimensions</span><span class="p">:</span>
-                <span class="n">type_args</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">arg</span><span class="o">.</span><span class="n">get_datatype</span><span class="p">(</span><span class="s">&#39;C&#39;</span><span class="p">),</span> <span class="s">&quot;*</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">name</span><span class="p">))</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">ResultBase</span><span class="p">):</span>
-                <span class="n">type_args</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">arg</span><span class="o">.</span><span class="n">get_datatype</span><span class="p">(</span><span class="s">&#39;C&#39;</span><span class="p">),</span> <span class="s">&quot;&amp;</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">name</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">type_args</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">arg</span><span class="o">.</span><span class="n">get_datatype</span><span class="p">(</span><span class="s">&#39;C&#39;</span><span class="p">),</span> <span class="n">name</span><span class="p">))</span>
-        <span class="n">arguments</span> <span class="o">=</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">t</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">type_args</span><span class="p">])</span>
-        <span class="k">return</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">ctype</span><span class="p">,</span> <span class="n">routine</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="n">arguments</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_preprocessor_statements</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prefix</span><span class="p">):</span>
-        <span class="n">code_lines</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">code_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;#include </span><span class="se">\&quot;</span><span class="si">%s</span><span class="s">.h</span><span class="se">\&quot;\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">basename</span><span class="p">(</span><span class="n">prefix</span><span class="p">))</span>
-        <span class="n">code_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;#include &lt;math.h&gt;</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">code_lines</span>
-
-    <span class="k">def</span> <span class="nf">_get_routine_opening</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routine</span><span class="p">):</span>
-        <span class="n">prototype</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_prototype</span><span class="p">(</span><span class="n">routine</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">[</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> {</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">prototype</span><span class="p">]</span>
-
-    <span class="k">def</span> <span class="nf">_declare_arguments</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routine</span><span class="p">):</span>
-        <span class="c"># arguments are declared in prototype</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">_declare_locals</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routine</span><span class="p">):</span>
-        <span class="c"># loop variables are declared in loop statement</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">_call_printer</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routine</span><span class="p">):</span>
-        <span class="n">code_lines</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">result</span> <span class="ow">in</span> <span class="n">routine</span><span class="o">.</span><span class="n">result_variables</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">Result</span><span class="p">):</span>
-                <span class="n">assign_to</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="p">(</span><span class="n">OutputArgument</span><span class="p">,</span> <span class="n">InOutArgument</span><span class="p">)):</span>
-                <span class="n">assign_to</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">result_var</span>
-
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">constants</span><span class="p">,</span> <span class="n">not_c</span><span class="p">,</span> <span class="n">c_expr</span> <span class="o">=</span> <span class="n">ccode</span><span class="p">(</span>
-                    <span class="n">result</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="n">assign_to</span><span class="o">=</span><span class="n">assign_to</span><span class="p">,</span> <span class="n">human</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-            <span class="k">except</span> <span class="n">AssignmentError</span><span class="p">:</span>
-                <span class="n">assign_to</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">result_var</span>
-                <span class="n">code_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span>
-                    <span class="s">&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">;</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">result</span><span class="o">.</span><span class="n">get_datatype</span><span class="p">(</span><span class="s">&#39;c&#39;</span><span class="p">),</span> <span class="nb">str</span><span class="p">(</span><span class="n">assign_to</span><span class="p">)))</span>
-                <span class="n">constants</span><span class="p">,</span> <span class="n">not_c</span><span class="p">,</span> <span class="n">c_expr</span> <span class="o">=</span> <span class="n">ccode</span><span class="p">(</span>
-                    <span class="n">result</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span> <span class="n">assign_to</span><span class="o">=</span><span class="n">assign_to</span><span class="p">,</span> <span class="n">human</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">name</span><span class="p">,</span> <span class="n">value</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">constants</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="nb">str</span><span class="p">):</span>
-                <span class="n">code_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;double const </span><span class="si">%s</span><span class="s"> = </span><span class="si">%s</span><span class="s">;</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">value</span><span class="p">))</span>
-            <span class="k">if</span> <span class="n">assign_to</span><span class="p">:</span>
-                <span class="n">code_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">c_expr</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">code_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;   return </span><span class="si">%s</span><span class="s">;</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">c_expr</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">code_lines</span>
-
-    <span class="k">def</span> <span class="nf">_indent_code</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">codelines</span><span class="p">):</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">CCodePrinter</span><span class="p">()</span>
-        <span class="k">return</span> <span class="n">p</span><span class="o">.</span><span class="n">indent_code</span><span class="p">(</span><span class="n">codelines</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_get_routine_ending</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routine</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">[</span><span class="s">&quot;}</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">]</span>
-
-<div class="viewcode-block" id="CCodeGen.dump_c"><a class="viewcode-back" href="../../../modules/utilities/codegen.html#sympy.utilities.codegen.CCodeGen.dump_c">[docs]</a>    <span class="k">def</span> <span class="nf">dump_c</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routines</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">prefix</span><span class="p">,</span> <span class="n">header</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">empty</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">dump_code</span><span class="p">(</span><span class="n">routines</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">prefix</span><span class="p">,</span> <span class="n">header</span><span class="p">,</span> <span class="n">empty</span><span class="p">)</span></div>
-    <span class="n">dump_c</span><span class="o">.</span><span class="n">extension</span> <span class="o">=</span> <span class="n">code_extension</span>
-    <span class="n">dump_c</span><span class="o">.</span><span class="n">__doc__</span> <span class="o">=</span> <span class="n">CodeGen</span><span class="o">.</span><span class="n">dump_code</span><span class="o">.</span><span class="n">__doc__</span>
-
-<div class="viewcode-block" id="CCodeGen.dump_h"><a class="viewcode-back" href="../../../modules/utilities/codegen.html#sympy.utilities.codegen.CCodeGen.dump_h">[docs]</a>    <span class="k">def</span> <span class="nf">dump_h</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routines</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">prefix</span><span class="p">,</span> <span class="n">header</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">empty</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Writes the C header file.</span>
-
-<span class="sd">           This file contains all the function declarations.</span>
-
-<span class="sd">           :Arguments:</span>
-
-<span class="sd">           routines</span>
-<span class="sd">                A list of Routine instances</span>
-<span class="sd">           f</span>
-<span class="sd">                A file-like object to write the file to</span>
-<span class="sd">           prefix</span>
-<span class="sd">                The filename prefix, used to construct the include guards.</span>
-
-<span class="sd">           :Optional arguments:</span>
-
-<span class="sd">           header</span>
-<span class="sd">                When True, a header comment is included on top of each source</span>
-<span class="sd">                file. [DEFAULT=True]</span>
-<span class="sd">           empty</span>
-<span class="sd">                When True, empty lines are included to structure the source</span>
-<span class="sd">                files. [DEFAULT=True]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">header</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_get_header</span><span class="p">()),</span> <span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span>
-        <span class="n">guard_name</span> <span class="o">=</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">__</span><span class="si">%s</span><span class="s">__H&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">project</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span>
-            <span class="s">&quot; &quot;</span><span class="p">,</span> <span class="s">&quot;_&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">upper</span><span class="p">(),</span> <span class="n">prefix</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="s">&quot;/&quot;</span><span class="p">,</span> <span class="s">&quot;_&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">upper</span><span class="p">())</span>
-        <span class="c"># include guards</span>
-        <span class="k">if</span> <span class="n">empty</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span>
-        <span class="k">print</span><span class="p">(</span><span class="s">&quot;#ifndef </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">guard_name</span><span class="p">,</span> <span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span>
-        <span class="k">print</span><span class="p">(</span><span class="s">&quot;#define </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">guard_name</span><span class="p">,</span> <span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">empty</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span>
-        <span class="c"># declaration of the function prototypes</span>
-        <span class="k">for</span> <span class="n">routine</span> <span class="ow">in</span> <span class="n">routines</span><span class="p">:</span>
-            <span class="n">prototype</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_prototype</span><span class="p">(</span><span class="n">routine</span><span class="p">)</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s">;&quot;</span> <span class="o">%</span> <span class="n">prototype</span><span class="p">,</span> <span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span>
-        <span class="c"># end if include guards</span>
-        <span class="k">if</span> <span class="n">empty</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span>
-        <span class="k">print</span><span class="p">(</span><span class="s">&quot;#endif&quot;</span><span class="p">,</span> <span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">empty</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span></div>
-    <span class="n">dump_h</span><span class="o">.</span><span class="n">extension</span> <span class="o">=</span> <span class="n">interface_extension</span>
-
-    <span class="c"># This list of dump functions is used by CodeGen.write to know which dump</span>
-    <span class="c"># functions it has to call.</span>
-    <span class="n">dump_fns</span> <span class="o">=</span> <span class="p">[</span><span class="n">dump_c</span><span class="p">,</span> <span class="n">dump_h</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="FCodeGen"><a class="viewcode-back" href="../../../modules/utilities/codegen.html#sympy.utilities.codegen.FCodeGen">[docs]</a><span class="k">class</span> <span class="nc">FCodeGen</span><span class="p">(</span><span class="n">CodeGen</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Generator for Fortran 95 code</span>
-
-<span class="sd">    The .write() method inherited from CodeGen will output a code file and an</span>
-<span class="sd">    inteface file, &lt;prefix&gt;.f90 and &lt;prefix&gt;.h respectively.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">code_extension</span> <span class="o">=</span> <span class="s">&quot;f90&quot;</span>
-    <span class="n">interface_extension</span> <span class="o">=</span> <span class="s">&quot;h&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">project</span><span class="o">=</span><span class="s">&#39;project&#39;</span><span class="p">):</span>
-        <span class="n">CodeGen</span><span class="o">.</span><span class="n">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">project</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_get_symbol</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;returns the symbol as fcode print it&quot;&quot;&quot;</span>
-        <span class="k">return</span> <span class="n">fcode</span><span class="p">(</span><span class="n">s</span><span class="p">)</span><span class="o">.</span><span class="n">strip</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">_get_header</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Writes a common header for the generated files.&quot;&quot;&quot;</span>
-        <span class="n">code_lines</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">code_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;!&quot;</span> <span class="o">+</span> <span class="s">&quot;*&quot;</span><span class="o">*</span><span class="mi">78</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">)</span>
-        <span class="n">tmp</span> <span class="o">=</span> <span class="n">header_comment</span> <span class="o">%</span> <span class="p">{</span><span class="s">&quot;version&quot;</span><span class="p">:</span> <span class="n">sympy_version</span><span class="p">,</span>
-            <span class="s">&quot;project&quot;</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">project</span><span class="p">}</span>
-        <span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="n">tmp</span><span class="o">.</span><span class="n">splitlines</span><span class="p">():</span>
-            <span class="n">code_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;!*</span><span class="si">%s</span><span class="s">*</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">line</span><span class="o">.</span><span class="n">center</span><span class="p">(</span><span class="mi">76</span><span class="p">))</span>
-        <span class="n">code_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;!&quot;</span> <span class="o">+</span> <span class="s">&quot;*&quot;</span><span class="o">*</span><span class="mi">78</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">code_lines</span>
-
-    <span class="k">def</span> <span class="nf">_preprocessor_statements</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">prefix</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">_get_routine_opening</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routine</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the opening statements of the fortran routine</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">code_list</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">routine</span><span class="o">.</span><span class="n">results</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="n">CodeGenError</span><span class="p">(</span>
-                <span class="s">&quot;Fortran only supports a single or no return value.&quot;</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">routine</span><span class="o">.</span><span class="n">results</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">routine</span><span class="o">.</span><span class="n">results</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-            <span class="n">code_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">result</span><span class="o">.</span><span class="n">get_datatype</span><span class="p">(</span><span class="s">&#39;fortran&#39;</span><span class="p">))</span>
-            <span class="n">code_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;function&quot;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">code_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;subroutine&quot;</span><span class="p">)</span>
-
-        <span class="n">args</span> <span class="o">=</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_symbol</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">name</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">routine</span><span class="o">.</span><span class="n">arguments</span><span class="p">)</span>
-
-        <span class="c"># name of the routine + arguments</span>
-        <span class="n">code_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">)</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">routine</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="n">args</span><span class="p">))</span>
-        <span class="n">code_list</span> <span class="o">=</span> <span class="p">[</span> <span class="s">&quot; &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">code_list</span><span class="p">)</span> <span class="p">]</span>
-
-        <span class="n">code_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;implicit none</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">code_list</span>
-
-    <span class="k">def</span> <span class="nf">_declare_arguments</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routine</span><span class="p">):</span>
-        <span class="c"># argument type declarations</span>
-        <span class="n">code_list</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">array_list</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">scalar_list</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">routine</span><span class="o">.</span><span class="n">arguments</span><span class="p">:</span>
-
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">InputArgument</span><span class="p">):</span>
-                <span class="n">typeinfo</span> <span class="o">=</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">, intent(in)&quot;</span> <span class="o">%</span> <span class="n">arg</span><span class="o">.</span><span class="n">get_datatype</span><span class="p">(</span><span class="s">&#39;fortran&#39;</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">InOutArgument</span><span class="p">):</span>
-                <span class="n">typeinfo</span> <span class="o">=</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">, intent(inout)&quot;</span> <span class="o">%</span> <span class="n">arg</span><span class="o">.</span><span class="n">get_datatype</span><span class="p">(</span><span class="s">&#39;fortran&#39;</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">OutputArgument</span><span class="p">):</span>
-                <span class="n">typeinfo</span> <span class="o">=</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">, intent(out)&quot;</span> <span class="o">%</span> <span class="n">arg</span><span class="o">.</span><span class="n">get_datatype</span><span class="p">(</span><span class="s">&#39;fortran&#39;</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="n">CodeGenError</span><span class="p">(</span><span class="s">&quot;Unkown Argument type: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="nb">type</span><span class="p">(</span><span class="n">arg</span><span class="p">))</span>
-
-            <span class="n">fprint</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_symbol</span>
-
-            <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">dimensions</span><span class="p">:</span>
-                <span class="c"># fortran arrays start at 1</span>
-                <span class="n">dimstr</span> <span class="o">=</span> <span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="s">&quot;</span><span class="si">%s</span><span class="s">:</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span>
-                    <span class="n">fprint</span><span class="p">(</span><span class="n">dim</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">fprint</span><span class="p">(</span><span class="n">dim</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-                    <span class="k">for</span> <span class="n">dim</span> <span class="ow">in</span> <span class="n">arg</span><span class="o">.</span><span class="n">dimensions</span><span class="p">])</span>
-                <span class="n">typeinfo</span> <span class="o">+=</span> <span class="s">&quot;, dimension(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="n">dimstr</span>
-                <span class="n">array_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> :: </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">typeinfo</span><span class="p">,</span> <span class="n">fprint</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">name</span><span class="p">)))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">scalar_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> :: </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">typeinfo</span><span class="p">,</span> <span class="n">fprint</span><span class="p">(</span><span class="n">arg</span><span class="o">.</span><span class="n">name</span><span class="p">)))</span>
-
-        <span class="c"># scalars first, because they can be used in array declarations</span>
-        <span class="n">code_list</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">scalar_list</span><span class="p">)</span>
-        <span class="n">code_list</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">array_list</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">code_list</span>
-
-    <span class="k">def</span> <span class="nf">_declare_locals</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routine</span><span class="p">):</span>
-        <span class="n">code_list</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">var</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">routine</span><span class="o">.</span><span class="n">local_vars</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="nb">str</span><span class="p">):</span>
-            <span class="n">typeinfo</span> <span class="o">=</span> <span class="n">get_default_datatype</span><span class="p">(</span><span class="n">var</span><span class="p">)</span>
-            <span class="n">code_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> :: </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span>
-                <span class="n">typeinfo</span><span class="o">.</span><span class="n">fname</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_symbol</span><span class="p">(</span><span class="n">var</span><span class="p">)))</span>
-        <span class="k">return</span> <span class="n">code_list</span>
-
-    <span class="k">def</span> <span class="nf">_get_routine_ending</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routine</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the closing statements of the fortran routine</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">routine</span><span class="o">.</span><span class="n">results</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[</span><span class="s">&quot;end function</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">[</span><span class="s">&quot;end subroutine</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">]</span>
-
-<div class="viewcode-block" id="FCodeGen.get_interface"><a class="viewcode-back" href="../../../modules/utilities/codegen.html#sympy.utilities.codegen.FCodeGen.get_interface">[docs]</a>    <span class="k">def</span> <span class="nf">get_interface</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routine</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Returns a string for the function interface for the given routine and</span>
-<span class="sd">           a single result object, which can be None.</span>
-
-<span class="sd">           If the routine has multiple result objects, a CodeGenError is</span>
-<span class="sd">           raised.</span>
-
-<span class="sd">           See: http://en.wikipedia.org/wiki/Function_prototype</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">prototype</span> <span class="o">=</span> <span class="p">[</span> <span class="s">&quot;interface</span><span class="se">\n</span><span class="s">&quot;</span> <span class="p">]</span>
-        <span class="n">prototype</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_get_routine_opening</span><span class="p">(</span><span class="n">routine</span><span class="p">))</span>
-        <span class="n">prototype</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_declare_arguments</span><span class="p">(</span><span class="n">routine</span><span class="p">))</span>
-        <span class="n">prototype</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_get_routine_ending</span><span class="p">(</span><span class="n">routine</span><span class="p">))</span>
-        <span class="n">prototype</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;end interface</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="s">&quot;&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">prototype</span><span class="p">)</span>
-</div>
-    <span class="k">def</span> <span class="nf">_call_printer</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routine</span><span class="p">):</span>
-        <span class="n">declarations</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="n">code_lines</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">result</span> <span class="ow">in</span> <span class="n">routine</span><span class="o">.</span><span class="n">result_variables</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="n">Result</span><span class="p">):</span>
-                <span class="n">assign_to</span> <span class="o">=</span> <span class="n">routine</span><span class="o">.</span><span class="n">name</span>
-            <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">result</span><span class="p">,</span> <span class="p">(</span><span class="n">OutputArgument</span><span class="p">,</span> <span class="n">InOutArgument</span><span class="p">)):</span>
-                <span class="n">assign_to</span> <span class="o">=</span> <span class="n">result</span><span class="o">.</span><span class="n">result_var</span>
-
-            <span class="n">constants</span><span class="p">,</span> <span class="n">not_fortran</span><span class="p">,</span> <span class="n">f_expr</span> <span class="o">=</span> <span class="n">fcode</span><span class="p">(</span><span class="n">result</span><span class="o">.</span><span class="n">expr</span><span class="p">,</span>
-                <span class="n">assign_to</span><span class="o">=</span><span class="n">assign_to</span><span class="p">,</span> <span class="n">source_format</span><span class="o">=</span><span class="s">&#39;free&#39;</span><span class="p">,</span> <span class="n">human</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-            <span class="k">for</span> <span class="n">obj</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">constants</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="nb">str</span><span class="p">):</span>
-                <span class="n">t</span> <span class="o">=</span> <span class="n">get_default_datatype</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span>
-                <span class="n">declarations</span><span class="o">.</span><span class="n">append</span><span class="p">(</span>
-                    <span class="s">&quot;</span><span class="si">%s</span><span class="s">, parameter :: </span><span class="si">%s</span><span class="s"> = </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">t</span><span class="o">.</span><span class="n">fname</span><span class="p">,</span> <span class="n">obj</span><span class="p">,</span> <span class="n">v</span><span class="p">))</span>
-            <span class="k">for</span> <span class="n">obj</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">not_fortran</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="nb">str</span><span class="p">):</span>
-                <span class="n">t</span> <span class="o">=</span> <span class="n">get_default_datatype</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="n">Function</span><span class="p">):</span>
-                    <span class="n">name</span> <span class="o">=</span> <span class="n">obj</span><span class="o">.</span><span class="n">func</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">name</span> <span class="o">=</span> <span class="n">obj</span>
-                <span class="n">declarations</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> :: </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">t</span><span class="o">.</span><span class="n">fname</span><span class="p">,</span> <span class="n">name</span><span class="p">))</span>
-
-            <span class="n">code_lines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">f_expr</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">declarations</span> <span class="o">+</span> <span class="n">code_lines</span>
-
-    <span class="k">def</span> <span class="nf">_indent_code</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">codelines</span><span class="p">):</span>
-        <span class="n">p</span> <span class="o">=</span> <span class="n">FCodePrinter</span><span class="p">({</span><span class="s">&#39;source_format&#39;</span><span class="p">:</span> <span class="s">&#39;free&#39;</span><span class="p">,</span> <span class="s">&#39;human&#39;</span><span class="p">:</span> <span class="bp">False</span><span class="p">})</span>
-        <span class="k">return</span> <span class="n">p</span><span class="o">.</span><span class="n">indent_code</span><span class="p">(</span><span class="n">codelines</span><span class="p">)</span>
-
-<div class="viewcode-block" id="FCodeGen.dump_f95"><a class="viewcode-back" href="../../../modules/utilities/codegen.html#sympy.utilities.codegen.FCodeGen.dump_f95">[docs]</a>    <span class="k">def</span> <span class="nf">dump_f95</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routines</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">prefix</span><span class="p">,</span> <span class="n">header</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">empty</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="c"># check that symbols are unique with ignorecase</span>
-        <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">routines</span><span class="p">:</span>
-            <span class="n">lowercase</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span><span class="nb">str</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">r</span><span class="o">.</span><span class="n">variables</span><span class="p">])</span>
-            <span class="n">orig_case</span> <span class="o">=</span> <span class="nb">set</span><span class="p">([</span><span class="nb">str</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">r</span><span class="o">.</span><span class="n">variables</span><span class="p">])</span>
-            <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">lowercase</span><span class="p">)</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">orig_case</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="n">CodeGenError</span><span class="p">(</span><span class="s">&quot;Fortran ignores case. Got symbols: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span>
-                        <span class="p">(</span><span class="s">&quot;, &quot;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="nb">str</span><span class="p">(</span><span class="n">var</span><span class="p">)</span> <span class="k">for</span> <span class="n">var</span> <span class="ow">in</span> <span class="n">r</span><span class="o">.</span><span class="n">variables</span><span class="p">])))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">dump_code</span><span class="p">(</span><span class="n">routines</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">prefix</span><span class="p">,</span> <span class="n">header</span><span class="p">,</span> <span class="n">empty</span><span class="p">)</span></div>
-    <span class="n">dump_f95</span><span class="o">.</span><span class="n">extension</span> <span class="o">=</span> <span class="n">code_extension</span>
-    <span class="n">dump_f95</span><span class="o">.</span><span class="n">__doc__</span> <span class="o">=</span> <span class="n">CodeGen</span><span class="o">.</span><span class="n">dump_code</span><span class="o">.</span><span class="n">__doc__</span>
-
-<div class="viewcode-block" id="FCodeGen.dump_h"><a class="viewcode-back" href="../../../modules/utilities/codegen.html#sympy.utilities.codegen.FCodeGen.dump_h">[docs]</a>    <span class="k">def</span> <span class="nf">dump_h</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">routines</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">prefix</span><span class="p">,</span> <span class="n">header</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">empty</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Writes the interface to a header file.</span>
-
-<span class="sd">           This file contains all the function declarations.</span>
-
-<span class="sd">           :Arguments:</span>
-
-<span class="sd">           routines</span>
-<span class="sd">                A list of Routine instances</span>
-<span class="sd">           f</span>
-<span class="sd">                A file-like object to write the file to</span>
-<span class="sd">           prefix</span>
-<span class="sd">                The filename prefix</span>
-
-<span class="sd">           :Optional arguments:</span>
-
-<span class="sd">           header</span>
-<span class="sd">                When True, a header comment is included on top of each source</span>
-<span class="sd">                file. [DEFAULT=True]</span>
-<span class="sd">           empty</span>
-<span class="sd">                When True, empty lines are included to structure the source</span>
-<span class="sd">                files. [DEFAULT=True]</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">header</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_get_header</span><span class="p">()),</span> <span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">empty</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span>
-        <span class="c"># declaration of the function prototypes</span>
-        <span class="k">for</span> <span class="n">routine</span> <span class="ow">in</span> <span class="n">routines</span><span class="p">:</span>
-            <span class="n">prototype</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">get_interface</span><span class="p">(</span><span class="n">routine</span><span class="p">)</span>
-            <span class="n">f</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="n">prototype</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">empty</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="nb">file</span><span class="o">=</span><span class="n">f</span><span class="p">)</span></div>
-    <span class="n">dump_h</span><span class="o">.</span><span class="n">extension</span> <span class="o">=</span> <span class="n">interface_extension</span>
-
-    <span class="c"># This list of dump functions is used by CodeGen.write to know which dump</span>
-    <span class="c"># functions it has to call.</span>
-    <span class="n">dump_fns</span> <span class="o">=</span> <span class="p">[</span><span class="n">dump_f95</span><span class="p">,</span> <span class="n">dump_h</span><span class="p">]</span>
-
-</div>
-<span class="k">def</span> <span class="nf">get_code_generator</span><span class="p">(</span><span class="n">language</span><span class="p">,</span> <span class="n">project</span><span class="p">):</span>
-    <span class="n">CodeGenClass</span> <span class="o">=</span> <span class="p">{</span><span class="s">&quot;C&quot;</span><span class="p">:</span> <span class="n">CCodeGen</span><span class="p">,</span> <span class="s">&quot;F95&quot;</span><span class="p">:</span> <span class="n">FCodeGen</span><span class="p">}</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">language</span><span class="o">.</span><span class="n">upper</span><span class="p">())</span>
-    <span class="k">if</span> <span class="n">CodeGenClass</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Language &#39;</span><span class="si">%s</span><span class="s">&#39; is not supported.&quot;</span> <span class="o">%</span> <span class="n">language</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">CodeGenClass</span><span class="p">(</span><span class="n">project</span><span class="p">)</span>
-
-
-<span class="c">#</span>
-<span class="c"># Friendly functions</span>
-<span class="c">#</span>
-
-
-<div class="viewcode-block" id="codegen"><a class="viewcode-back" href="../../../modules/utilities/codegen.html#sympy.utilities.codegen.codegen">[docs]</a><span class="k">def</span> <span class="nf">codegen</span><span class="p">(</span>
-    <span class="n">name_expr</span><span class="p">,</span> <span class="n">language</span><span class="p">,</span> <span class="n">prefix</span><span class="p">,</span> <span class="n">project</span><span class="o">=</span><span class="s">&quot;project&quot;</span><span class="p">,</span> <span class="n">to_files</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">header</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">empty</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span>
-        <span class="n">argument_sequence</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Write source code for the given expressions in the given language.</span>
-
-<span class="sd">    :Mandatory Arguments:</span>
-
-<span class="sd">    ``name_expr``</span>
-<span class="sd">        A single (name, expression) tuple or a list of (name, expression)</span>
-<span class="sd">        tuples. Each tuple corresponds to a routine.  If the expression is an</span>
-<span class="sd">        equality (an instance of class Equality) the left hand side is</span>
-<span class="sd">        considered an output argument.</span>
-<span class="sd">    ``language``</span>
-<span class="sd">            A string that indicates the source code language. This is case</span>
-<span class="sd">            insensitive. For the moment, only &#39;C&#39; and &#39;F95&#39; is supported.</span>
-<span class="sd">    ``prefix``</span>
-<span class="sd">            A prefix for the names of the files that contain the source code.</span>
-<span class="sd">            Proper (language dependent) suffixes will be appended.</span>
-
-<span class="sd">    :Optional Arguments:</span>
-
-<span class="sd">    ``project``</span>
-<span class="sd">        A project name, used for making unique preprocessor instructions.</span>
-<span class="sd">        [DEFAULT=&quot;project&quot;]</span>
-<span class="sd">    ``to_files``</span>
-<span class="sd">        When True, the code will be written to one or more files with the given</span>
-<span class="sd">        prefix, otherwise strings with the names and contents of these files</span>
-<span class="sd">        are returned. [DEFAULT=False]</span>
-<span class="sd">    ``header``</span>
-<span class="sd">        When True, a header is written on top of each source file.</span>
-<span class="sd">        [DEFAULT=True]</span>
-<span class="sd">    ``empty``</span>
-<span class="sd">        When True, empty lines are used to structure the code.  [DEFAULT=True]</span>
-<span class="sd">    ``argument_sequence``</span>
-<span class="sd">        sequence of arguments for the routine in a preferred order.  A</span>
-<span class="sd">        CodeGenError is raised if required arguments are missing.  Redundant</span>
-<span class="sd">        arguments are used without warning.</span>
-
-<span class="sd">        If omitted, arguments will be ordered alphabetically, but with all</span>
-<span class="sd">        input aguments first, and then output or in-out arguments.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import symbols</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.codegen import codegen</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">    &gt;&gt;&gt; [(c_name, c_code), (h_name, c_header)] = codegen(</span>
-<span class="sd">    ...     (&quot;f&quot;, x+y*z), &quot;C&quot;, &quot;test&quot;, header=False, empty=False)</span>
-<span class="sd">    &gt;&gt;&gt; print(c_name)</span>
-<span class="sd">    test.c</span>
-<span class="sd">    &gt;&gt;&gt; print(c_code, end=&#39; &#39;)</span>
-<span class="sd">    #include &quot;test.h&quot;</span>
-<span class="sd">    #include &lt;math.h&gt;</span>
-<span class="sd">    double f(double x, double y, double z) {</span>
-<span class="sd">      return x + y*z;</span>
-<span class="sd">    }</span>
-<span class="sd">    &gt;&gt;&gt; print(h_name)</span>
-<span class="sd">    test.h</span>
-<span class="sd">    &gt;&gt;&gt; print(c_header, end=&#39; &#39;)</span>
-<span class="sd">    #ifndef PROJECT__TEST__H</span>
-<span class="sd">    #define PROJECT__TEST__H</span>
-<span class="sd">    double f(double x, double y, double z);</span>
-<span class="sd">    #endif</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="c"># Initialize the code generator.</span>
-    <span class="n">code_gen</span> <span class="o">=</span> <span class="n">get_code_generator</span><span class="p">(</span><span class="n">language</span><span class="p">,</span> <span class="n">project</span><span class="p">)</span>
-
-    <span class="c"># Construct the routines based on the name_expression pairs.</span>
-    <span class="c">#  mainly the input arguments require some work</span>
-    <span class="n">routines</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">name_expr</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="nb">str</span><span class="p">):</span>
-        <span class="c"># single tuple is given, turn it into a singleton list with a tuple.</span>
-        <span class="n">name_expr</span> <span class="o">=</span> <span class="p">[</span><span class="n">name_expr</span><span class="p">]</span>
-
-    <span class="k">for</span> <span class="n">name</span><span class="p">,</span> <span class="n">expr</span> <span class="ow">in</span> <span class="n">name_expr</span><span class="p">:</span>
-        <span class="n">routines</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Routine</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">argument_sequence</span><span class="p">))</span>
-
-    <span class="c"># Write the code.</span>
-    <span class="k">return</span> <span class="n">code_gen</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="n">routines</span><span class="p">,</span> <span class="n">prefix</span><span class="p">,</span> <span class="n">to_files</span><span class="p">,</span> <span class="n">header</span><span class="p">,</span> <span class="n">empty</span><span class="p">)</span></div>
-</pre></div>
-
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-        </div>
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-            
-  <h1>Source code for sympy.utilities.decorator</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">sympy.core.decorators</span> <span class="kn">import</span> <span class="n">wraps</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">iterable</span>
-
-
-<div class="viewcode-block" id="threaded_factory"><a class="viewcode-back" href="../../../modules/utilities/decorator.html#sympy.utilities.decorator.threaded_factory">[docs]</a><span class="k">def</span> <span class="nf">threaded_factory</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">use_add</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;A factory for ``threaded`` decorators. &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">sympify</span>
-    <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span>
-
-    <span class="nd">@wraps</span><span class="p">(</span><span class="n">func</span><span class="p">)</span>
-    <span class="k">def</span> <span class="nf">threaded_func</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">applyfunc</span><span class="p">(</span><span class="k">lambda</span> <span class="n">f</span><span class="p">:</span> <span class="n">func</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-        <span class="k">elif</span> <span class="n">iterable</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="p">([</span> <span class="n">func</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">expr</span> <span class="p">])</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">expr</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">use_add</span> <span class="ow">and</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="o">*</span><span class="p">[</span> <span class="n">func</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span> <span class="p">])</span>
-            <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Relational</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">expr</span><span class="o">.</span><span class="n">__class__</span><span class="p">(</span><span class="n">func</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">lhs</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">),</span>
-                                      <span class="n">func</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">rhs</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">func</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">threaded_func</span>
-
-</div>
-<div class="viewcode-block" id="threaded"><a class="viewcode-back" href="../../../modules/utilities/decorator.html#sympy.utilities.decorator.threaded">[docs]</a><span class="k">def</span> <span class="nf">threaded</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Apply ``func`` to sub--elements of an object, including :class:`Add`.</span>
-
-<span class="sd">    This decorator is intended to make it uniformly possible to apply a</span>
-<span class="sd">    function to all elements of composite objects, e.g. matrices, lists, tuples</span>
-<span class="sd">    and other iterable containers, or just expressions.</span>
-
-<span class="sd">    This version of :func:`threaded` decorator allows threading over</span>
-<span class="sd">    elements of :class:`Add` class. If this behavior is not desirable</span>
-<span class="sd">    use :func:`xthreaded` decorator.</span>
-
-<span class="sd">    Functions using this decorator must have the following signature::</span>
-
-<span class="sd">      @threaded</span>
-<span class="sd">      def function(expr, *args, **kwargs):</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">threaded_factory</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="xthreaded"><a class="viewcode-back" href="../../../modules/utilities/decorator.html#sympy.utilities.decorator.xthreaded">[docs]</a><span class="k">def</span> <span class="nf">xthreaded</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Apply ``func`` to sub--elements of an object, excluding :class:`Add`.</span>
-
-<span class="sd">    This decorator is intended to make it uniformly possible to apply a</span>
-<span class="sd">    function to all elements of composite objects, e.g. matrices, lists, tuples</span>
-<span class="sd">    and other iterable containers, or just expressions.</span>
-
-<span class="sd">    This version of :func:`threaded` decorator disallows threading over</span>
-<span class="sd">    elements of :class:`Add` class. If this behavior is not desirable</span>
-<span class="sd">    use :func:`threaded` decorator.</span>
-
-<span class="sd">    Functions using this decorator must have the following signature::</span>
-
-<span class="sd">      @xthreaded</span>
-<span class="sd">      def function(expr, *args, **kwargs):</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">threaded_factory</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="conserve_mpmath_dps"><a class="viewcode-back" href="../../../modules/utilities/decorator.html#sympy.utilities.decorator.conserve_mpmath_dps">[docs]</a><span class="k">def</span> <span class="nf">conserve_mpmath_dps</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;After the function finishes, resets the value of mpmath.mp.dps to</span>
-<span class="sd">    the value it had before the function was run.&quot;&quot;&quot;</span>
-    <span class="kn">import</span> <span class="nn">functools</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">mpmath</span>
-
-    <span class="k">def</span> <span class="nf">func_wrapper</span><span class="p">():</span>
-        <span class="n">dps</span> <span class="o">=</span> <span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">func</span><span class="p">()</span>
-        <span class="k">finally</span><span class="p">:</span>
-            <span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="n">dps</span>
-
-    <span class="n">func_wrapper</span> <span class="o">=</span> <span class="n">functools</span><span class="o">.</span><span class="n">update_wrapper</span><span class="p">(</span><span class="n">func_wrapper</span><span class="p">,</span> <span class="n">func</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">func_wrapper</span></div>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/utilities/iterables.html b/dev-py3k/_modules/sympy/utilities/iterables.html
deleted file mode 100644
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+++ /dev/null
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-            
-  <h1>Source code for sympy.utilities.iterables</h1><div class="highlight"><pre>
-<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">defaultdict</span>
-<span class="kn">import</span> <span class="nn">random</span>
-<span class="kn">from</span> <span class="nn">operator</span> <span class="kn">import</span> <span class="n">gt</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.decorators</span> <span class="kn">import</span> <span class="n">deprecated</span>
-<span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">C</span>
-<span class="kn">from</span> <span class="nn">sympy.utilities.exceptions</span> <span class="kn">import</span> <span class="n">SymPyDeprecationWarning</span>
-
-<span class="c"># this is the logical location of these functions</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="p">(</span>
-    <span class="n">as_int</span><span class="p">,</span> <span class="n">combinations</span><span class="p">,</span> <span class="n">combinations_with_replacement</span><span class="p">,</span>
-    <span class="n">default_sort_key</span><span class="p">,</span> <span class="n">is_sequence</span><span class="p">,</span> <span class="n">iterable</span><span class="p">,</span> <span class="n">permutations</span><span class="p">,</span>
-    <span class="n">product</span> <span class="k">as</span> <span class="n">cartes</span><span class="p">,</span> <span class="n">ordered</span><span class="p">,</span> <span class="nb">next</span><span class="p">,</span> <span class="nb">bin</span>
-<span class="p">)</span>
-
-
-<div class="viewcode-block" id="flatten"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.flatten">[docs]</a><span class="k">def</span> <span class="nf">flatten</span><span class="p">(</span><span class="n">iterable</span><span class="p">,</span> <span class="n">levels</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Recursively denest iterable containers.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import flatten</span>
-
-<span class="sd">    &gt;&gt;&gt; flatten([1, 2, 3])</span>
-<span class="sd">    [1, 2, 3]</span>
-<span class="sd">    &gt;&gt;&gt; flatten([1, 2, [3]])</span>
-<span class="sd">    [1, 2, 3]</span>
-<span class="sd">    &gt;&gt;&gt; flatten([1, [2, 3], [4, 5]])</span>
-<span class="sd">    [1, 2, 3, 4, 5]</span>
-<span class="sd">    &gt;&gt;&gt; flatten([1.0, 2, (1, None)])</span>
-<span class="sd">    [1.0, 2, 1, None]</span>
-
-<span class="sd">    If you want to denest only a specified number of levels of</span>
-<span class="sd">    nested containers, then set ``levels`` flag to the desired</span>
-<span class="sd">    number of levels::</span>
-
-<span class="sd">    &gt;&gt;&gt; ls = [[(-2, -1), (1, 2)], [(0, 0)]]</span>
-
-<span class="sd">    &gt;&gt;&gt; flatten(ls, levels=1)</span>
-<span class="sd">    [(-2, -1), (1, 2), (0, 0)]</span>
-
-<span class="sd">    If cls argument is specified, it will only flatten instances of that</span>
-<span class="sd">    class, for example:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.core import Basic</span>
-<span class="sd">    &gt;&gt;&gt; class MyOp(Basic):</span>
-<span class="sd">    ...     pass</span>
-<span class="sd">    ...</span>
-<span class="sd">    &gt;&gt;&gt; flatten([MyOp(1, MyOp(2, 3))], cls=MyOp)</span>
-<span class="sd">    [1, 2, 3]</span>
-
-<span class="sd">    adapted from http://kogs-www.informatik.uni-hamburg.de/~meine/python_tricks</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">levels</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">levels</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">iterable</span>
-        <span class="k">elif</span> <span class="n">levels</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">levels</span> <span class="o">-=</span> <span class="mi">1</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-                <span class="s">&quot;expected non-negative number of levels, got </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">levels</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">cls</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">reducible</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="nb">set</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">reducible</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">cls</span><span class="p">)</span>
-
-    <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">el</span> <span class="ow">in</span> <span class="n">iterable</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">reducible</span><span class="p">(</span><span class="n">el</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">el</span><span class="p">,</span> <span class="s">&#39;args&#39;</span><span class="p">):</span>
-                <span class="n">el</span> <span class="o">=</span> <span class="n">el</span><span class="o">.</span><span class="n">args</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">flatten</span><span class="p">(</span><span class="n">el</span><span class="p">,</span> <span class="n">levels</span><span class="o">=</span><span class="n">levels</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">cls</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">el</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">result</span>
-
-</div>
-<div class="viewcode-block" id="unflatten"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.unflatten">[docs]</a><span class="k">def</span> <span class="nf">unflatten</span><span class="p">(</span><span class="nb">iter</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">2</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Group ``iter`` into tuples of length ``n``. Raise an error if</span>
-<span class="sd">    the length of ``iter`` is not a multiple of ``n``.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">1</span> <span class="ow">or</span> <span class="nb">len</span><span class="p">(</span><span class="nb">iter</span><span class="p">)</span> <span class="o">%</span> <span class="n">n</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;iter length is not a multiple of </span><span class="si">%i</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-    <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="nb">iter</span><span class="p">[</span><span class="n">i</span><span class="p">::</span><span class="n">n</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">))))</span>
-
-</div>
-<div class="viewcode-block" id="reshape"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.reshape">[docs]</a><span class="k">def</span> <span class="nf">reshape</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="n">how</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Reshape the sequence according to the template in ``how``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities import reshape</span>
-<span class="sd">    &gt;&gt;&gt; seq = list(range(1, 9))</span>
-
-<span class="sd">    &gt;&gt;&gt; reshape(seq, [4]) # lists of 4</span>
-<span class="sd">    [[1, 2, 3, 4], [5, 6, 7, 8]]</span>
-
-<span class="sd">    &gt;&gt;&gt; reshape(seq, (4,)) # tuples of 4</span>
-<span class="sd">    [(1, 2, 3, 4), (5, 6, 7, 8)]</span>
-
-<span class="sd">    &gt;&gt;&gt; reshape(seq, (2, 2)) # tuples of 4</span>
-<span class="sd">    [(1, 2, 3, 4), (5, 6, 7, 8)]</span>
-
-<span class="sd">    &gt;&gt;&gt; reshape(seq, (2, [2])) # (i, i, [i, i])</span>
-<span class="sd">    [(1, 2, [3, 4]), (5, 6, [7, 8])]</span>
-
-<span class="sd">    &gt;&gt;&gt; reshape(seq, ((2,), [2])) # etc....</span>
-<span class="sd">    [((1, 2), [3, 4]), ((5, 6), [7, 8])]</span>
-
-<span class="sd">    &gt;&gt;&gt; reshape(seq, (1, [2], 1))</span>
-<span class="sd">    [(1, [2, 3], 4), (5, [6, 7], 8)]</span>
-
-<span class="sd">    &gt;&gt;&gt; reshape(tuple(seq), ([[1], 1, (2,)],))</span>
-<span class="sd">    (([[1], 2, (3, 4)],), ([[5], 6, (7, 8)],))</span>
-
-<span class="sd">    &gt;&gt;&gt; reshape(tuple(seq), ([1], 1, (2,)))</span>
-<span class="sd">    (([1], 2, (3, 4)), ([5], 6, (7, 8)))</span>
-
-<span class="sd">    &gt;&gt;&gt; reshape(list(range(12)), [2, [3], set([2]), (1, (3,), 1)])</span>
-<span class="sd">    [[0, 1, [2, 3, 4], set([5, 6]), (7, (8, 9, 10), 11)]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">flatten</span><span class="p">(</span><span class="n">how</span><span class="p">))</span>
-    <span class="n">n</span><span class="p">,</span> <span class="n">rem</span> <span class="o">=</span> <span class="nb">divmod</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">seq</span><span class="p">),</span> <span class="n">m</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">rem</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;template must sum to positive number &#39;</span>
-        <span class="s">&#39;that divides the length of the sequence&#39;</span><span class="p">)</span>
-    <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">container</span> <span class="o">=</span> <span class="nb">type</span><span class="p">(</span><span class="n">how</span><span class="p">)</span>
-    <span class="n">rv</span> <span class="o">=</span> <span class="p">[</span><span class="bp">None</span><span class="p">]</span><span class="o">*</span><span class="n">n</span>
-    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">rv</span><span class="p">)):</span>
-        <span class="n">rv</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">hi</span> <span class="ow">in</span> <span class="n">how</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">hi</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">int</span><span class="p">:</span>
-                <span class="n">rv</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">seq</span><span class="p">[</span><span class="n">i</span><span class="p">:</span> <span class="n">i</span> <span class="o">+</span> <span class="n">hi</span><span class="p">])</span>
-                <span class="n">i</span> <span class="o">+=</span> <span class="n">hi</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">n</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">flatten</span><span class="p">(</span><span class="n">hi</span><span class="p">))</span>
-                <span class="n">hi_type</span> <span class="o">=</span> <span class="nb">type</span><span class="p">(</span><span class="n">hi</span><span class="p">)</span>
-                <span class="n">rv</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">hi_type</span><span class="p">(</span><span class="n">reshape</span><span class="p">(</span><span class="n">seq</span><span class="p">[</span><span class="n">i</span><span class="p">:</span> <span class="n">i</span> <span class="o">+</span> <span class="n">n</span><span class="p">],</span> <span class="n">hi</span><span class="p">)[</span><span class="mi">0</span><span class="p">]))</span>
-                <span class="n">i</span> <span class="o">+=</span> <span class="n">n</span>
-        <span class="n">rv</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">container</span><span class="p">(</span><span class="n">rv</span><span class="p">[</span><span class="n">k</span><span class="p">])</span>
-    <span class="k">return</span> <span class="nb">type</span><span class="p">(</span><span class="n">seq</span><span class="p">)(</span><span class="n">rv</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="group"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.group">[docs]</a><span class="k">def</span> <span class="nf">group</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Splits a sequence into a list of lists of equal, adjacent elements.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import group</span>
-
-<span class="sd">    &gt;&gt;&gt; group([1, 1, 1, 2, 2, 3])</span>
-<span class="sd">    [[1, 1, 1], [2, 2], [3]]</span>
-<span class="sd">    &gt;&gt;&gt; group([1, 1, 1, 2, 2, 3], multiple=False)</span>
-<span class="sd">    [(1, 3), (2, 2), (3, 1)]</span>
-<span class="sd">    &gt;&gt;&gt; group([1, 1, 3, 2, 2, 1], multiple=False)</span>
-<span class="sd">    [(1, 2), (3, 1), (2, 2), (1, 1)]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    multiset</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">seq</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-
-    <span class="n">current</span><span class="p">,</span> <span class="n">groups</span> <span class="o">=</span> <span class="p">[</span><span class="n">seq</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span> <span class="p">[]</span>
-
-    <span class="k">for</span> <span class="n">elem</span> <span class="ow">in</span> <span class="n">seq</span><span class="p">[</span><span class="mi">1</span><span class="p">:]:</span>
-        <span class="k">if</span> <span class="n">elem</span> <span class="o">==</span> <span class="n">current</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-            <span class="n">current</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">elem</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">groups</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">current</span><span class="p">)</span>
-            <span class="n">current</span> <span class="o">=</span> <span class="p">[</span><span class="n">elem</span><span class="p">]</span>
-
-    <span class="n">groups</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">current</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">multiple</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">groups</span>
-
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">current</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">groups</span><span class="p">):</span>
-        <span class="n">groups</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">current</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="nb">len</span><span class="p">(</span><span class="n">current</span><span class="p">))</span>
-
-    <span class="k">return</span> <span class="n">groups</span>
-
-</div>
-<div class="viewcode-block" id="multiset"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.multiset">[docs]</a><span class="k">def</span> <span class="nf">multiset</span><span class="p">(</span><span class="n">seq</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return the hashable sequence in multiset form with values being the</span>
-<span class="sd">    multiplicity of the item in the sequence.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import multiset, group</span>
-<span class="sd">    &gt;&gt;&gt; multiset(&#39;mississippi&#39;)</span>
-<span class="sd">    {&#39;i&#39;: 4, &#39;m&#39;: 1, &#39;p&#39;: 2, &#39;s&#39;: 4}</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    group</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">rv</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">seq</span><span class="p">:</span>
-        <span class="n">rv</span><span class="p">[</span><span class="n">s</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-    <span class="k">return</span> <span class="nb">dict</span><span class="p">(</span><span class="n">rv</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="postorder_traversal"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.postorder_traversal">[docs]</a><span class="k">def</span> <span class="nf">postorder_traversal</span><span class="p">(</span><span class="n">node</span><span class="p">,</span> <span class="n">keys</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Do a postorder traversal of a tree.</span>
-
-<span class="sd">    This generator recursively yields nodes that it has visited in a postorder</span>
-<span class="sd">    fashion. That is, it descends through the tree depth-first to yield all of</span>
-<span class="sd">    a node&#39;s children&#39;s postorder traversal before yielding the node itself.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    node : sympy expression</span>
-<span class="sd">        The expression to traverse.</span>
-<span class="sd">    keys : (default None) sort key(s)</span>
-<span class="sd">        The key(s) used to sort args of Basic objects. When None, args of Basic</span>
-<span class="sd">        objects are processed in arbitrary order. If key is defined, it will</span>
-<span class="sd">        be passed along to ordered() as the only key(s) to use to sort the</span>
-<span class="sd">        arguments; if ``key`` is simply True then the default keys of</span>
-<span class="sd">        ``ordered`` will be used (node count and default_sort_key).</span>
-
-<span class="sd">    Yields</span>
-<span class="sd">    ======</span>
-<span class="sd">    subtree : sympy expression</span>
-<span class="sd">        All of the subtrees in the tree.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import postorder_traversal</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import w, x, y, z</span>
-
-<span class="sd">    The nodes are returned in the order that they are encountered unless key</span>
-<span class="sd">    is given; simply passing key=True will guarantee that the traversal is</span>
-<span class="sd">    unique.</span>
-
-<span class="sd">    &gt;&gt;&gt; list(postorder_traversal(w + (x + y)*z)) # doctest: +SKIP</span>
-<span class="sd">    [z, y, x, x + y, z*(x + y), w, w + z*(x + y)]</span>
-<span class="sd">    &gt;&gt;&gt; list(postorder_traversal(w + (x + y)*z, keys=True))</span>
-<span class="sd">    [w, z, x, y, x + y, z*(x + y), w + z*(x + y)]</span>
-
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">node</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="n">node</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">if</span> <span class="n">keys</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">keys</span> <span class="o">!=</span> <span class="bp">True</span><span class="p">:</span>
-                <span class="n">args</span> <span class="o">=</span> <span class="n">ordered</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="n">keys</span><span class="p">,</span> <span class="n">default</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">args</span> <span class="o">=</span> <span class="n">ordered</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">subtree</span> <span class="ow">in</span> <span class="n">postorder_traversal</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">keys</span><span class="p">):</span>
-                <span class="k">yield</span> <span class="n">subtree</span>
-    <span class="k">elif</span> <span class="n">iterable</span><span class="p">(</span><span class="n">node</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">item</span> <span class="ow">in</span> <span class="n">node</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">subtree</span> <span class="ow">in</span> <span class="n">postorder_traversal</span><span class="p">(</span><span class="n">item</span><span class="p">,</span> <span class="n">keys</span><span class="p">):</span>
-                <span class="k">yield</span> <span class="n">subtree</span>
-    <span class="k">yield</span> <span class="n">node</span>
-
-</div>
-<div class="viewcode-block" id="interactive_traversal"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.interactive_traversal">[docs]</a><span class="k">def</span> <span class="nf">interactive_traversal</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Traverse a tree asking a user which branch to choose. &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.printing</span> <span class="kn">import</span> <span class="n">pprint</span>
-
-    <span class="n">RED</span><span class="p">,</span> <span class="n">BRED</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\033</span><span class="s">[0;31m&#39;</span><span class="p">,</span> <span class="s">&#39;</span><span class="se">\033</span><span class="s">[1;31m&#39;</span>
-    <span class="n">GREEN</span><span class="p">,</span> <span class="n">BGREEN</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\033</span><span class="s">[0;32m&#39;</span><span class="p">,</span> <span class="s">&#39;</span><span class="se">\033</span><span class="s">[1;32m&#39;</span>
-    <span class="n">YELLOW</span><span class="p">,</span> <span class="n">BYELLOW</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\033</span><span class="s">[0;33m&#39;</span><span class="p">,</span> <span class="s">&#39;</span><span class="se">\033</span><span class="s">[1;33m&#39;</span>
-    <span class="n">BLUE</span><span class="p">,</span> <span class="n">BBLUE</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\033</span><span class="s">[0;34m&#39;</span><span class="p">,</span> <span class="s">&#39;</span><span class="se">\033</span><span class="s">[1;34m&#39;</span>
-    <span class="n">MAGENTA</span><span class="p">,</span> <span class="n">BMAGENTA</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\033</span><span class="s">[0;35m&#39;</span><span class="p">,</span> <span class="s">&#39;</span><span class="se">\033</span><span class="s">[1;35m&#39;</span>
-    <span class="n">CYAN</span><span class="p">,</span> <span class="n">BCYAN</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\033</span><span class="s">[0;36m&#39;</span><span class="p">,</span> <span class="s">&#39;</span><span class="se">\033</span><span class="s">[1;36m&#39;</span>
-    <span class="n">END</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\033</span><span class="s">[0m&#39;</span>
-
-    <span class="k">def</span> <span class="nf">cprint</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">):</span>
-        <span class="k">print</span><span class="p">(</span><span class="s">&quot;&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">str</span><span class="p">,</span> <span class="n">args</span><span class="p">))</span> <span class="o">+</span> <span class="n">END</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_interactive_traversal</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">stage</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">stage</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">()</span>
-
-        <span class="n">cprint</span><span class="p">(</span><span class="s">&quot;Current expression (stage &quot;</span><span class="p">,</span> <span class="n">BYELLOW</span><span class="p">,</span> <span class="n">stage</span><span class="p">,</span> <span class="n">END</span><span class="p">,</span> <span class="s">&quot;):&quot;</span><span class="p">)</span>
-        <span class="k">print</span><span class="p">(</span><span class="n">BCYAN</span><span class="p">)</span>
-        <span class="n">pprint</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">print</span><span class="p">(</span><span class="n">END</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">Basic</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Add</span><span class="p">:</span>
-                <span class="n">args</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_ordered_terms</span><span class="p">()</span>
-            <span class="k">elif</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">:</span>
-                <span class="n">args</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">as_ordered_factors</span><span class="p">()</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">args</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span>
-        <span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&quot;__iter__&quot;</span><span class="p">):</span>
-            <span class="n">args</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span>
-
-        <span class="n">n_args</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">n_args</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">expr</span>
-
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">arg</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">args</span><span class="p">):</span>
-            <span class="n">cprint</span><span class="p">(</span><span class="n">GREEN</span><span class="p">,</span> <span class="s">&quot;[&quot;</span><span class="p">,</span> <span class="n">BGREEN</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">GREEN</span><span class="p">,</span> <span class="s">&quot;] &quot;</span><span class="p">,</span> <span class="n">BLUE</span><span class="p">,</span> <span class="nb">type</span><span class="p">(</span><span class="n">arg</span><span class="p">),</span> <span class="n">END</span><span class="p">)</span>
-            <span class="n">pprint</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-            <span class="k">print</span><span class="p">()</span>
-
-        <span class="k">if</span> <span class="n">n_args</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">choices</span> <span class="o">=</span> <span class="s">&#39;0&#39;</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">choices</span> <span class="o">=</span> <span class="s">&#39;0-</span><span class="si">%d</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n_args</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">choice</span> <span class="o">=</span> <span class="nb">input</span><span class="p">(</span><span class="s">&quot;Your choice [</span><span class="si">%s</span><span class="s">,f,l,r,d,?]: &quot;</span> <span class="o">%</span> <span class="n">choices</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">EOFError</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="n">expr</span>
-            <span class="k">print</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">choice</span> <span class="o">==</span> <span class="s">&#39;?&#39;</span><span class="p">:</span>
-                <span class="n">cprint</span><span class="p">(</span><span class="n">RED</span><span class="p">,</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s"> - select subexpression with the given index&quot;</span> <span class="o">%</span>
-                       <span class="n">choices</span><span class="p">)</span>
-                <span class="n">cprint</span><span class="p">(</span><span class="n">RED</span><span class="p">,</span> <span class="s">&quot;f - select the first subexpression&quot;</span><span class="p">)</span>
-                <span class="n">cprint</span><span class="p">(</span><span class="n">RED</span><span class="p">,</span> <span class="s">&quot;l - select the last subexpression&quot;</span><span class="p">)</span>
-                <span class="n">cprint</span><span class="p">(</span><span class="n">RED</span><span class="p">,</span> <span class="s">&quot;r - select a random subexpression&quot;</span><span class="p">)</span>
-                <span class="n">cprint</span><span class="p">(</span><span class="n">RED</span><span class="p">,</span> <span class="s">&quot;d - done</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-
-                <span class="n">result</span> <span class="o">=</span> <span class="n">_interactive_traversal</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">stage</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">choice</span> <span class="ow">in</span> <span class="p">[</span><span class="s">&#39;d&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">]:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">expr</span>
-            <span class="k">elif</span> <span class="n">choice</span> <span class="o">==</span> <span class="s">&#39;f&#39;</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">_interactive_traversal</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">stage</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">choice</span> <span class="o">==</span> <span class="s">&#39;l&#39;</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">_interactive_traversal</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">stage</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="k">elif</span> <span class="n">choice</span> <span class="o">==</span> <span class="s">&#39;r&#39;</span><span class="p">:</span>
-                <span class="n">result</span> <span class="o">=</span> <span class="n">_interactive_traversal</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">args</span><span class="p">),</span> <span class="n">stage</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">try</span><span class="p">:</span>
-                    <span class="n">choice</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span>
-                <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-                    <span class="n">cprint</span><span class="p">(</span><span class="n">BRED</span><span class="p">,</span>
-                           <span class="s">&quot;Choice must be a number in </span><span class="si">%s</span><span class="s"> range</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">choices</span><span class="p">)</span>
-                    <span class="n">result</span> <span class="o">=</span> <span class="n">_interactive_traversal</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">stage</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">choice</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">choice</span> <span class="o">&gt;=</span> <span class="n">n_args</span><span class="p">:</span>
-                        <span class="n">cprint</span><span class="p">(</span><span class="n">BRED</span><span class="p">,</span> <span class="s">&quot;Choice must be in </span><span class="si">%s</span><span class="s"> range</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">choices</span><span class="p">)</span>
-                        <span class="n">result</span> <span class="o">=</span> <span class="n">_interactive_traversal</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">stage</span><span class="p">)</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">result</span> <span class="o">=</span> <span class="n">_interactive_traversal</span><span class="p">(</span><span class="n">args</span><span class="p">[</span><span class="n">choice</span><span class="p">],</span> <span class="n">stage</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-        <span class="k">return</span> <span class="n">result</span>
-
-    <span class="k">return</span> <span class="n">_interactive_traversal</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="ibin"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.ibin">[docs]</a><span class="k">def</span> <span class="nf">ibin</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">bits</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="nb">str</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return a list of length ``bits`` corresponding to the binary value</span>
-<span class="sd">    of ``n`` with small bits to the right (last). If bits is omitted, the</span>
-<span class="sd">    length will be the number required to represent ``n``. If the bits are</span>
-<span class="sd">    desired in reversed order, use the [::-1] slice of the returned list.</span>
-
-<span class="sd">    If a sequence of all bits-length lists starting from [0, 0,..., 0]</span>
-<span class="sd">    through [1, 1, ..., 1] are desired, pass a non-integer for bits, e.g.</span>
-<span class="sd">    &#39;all&#39;.</span>
-
-<span class="sd">    If the bit *string* is desired pass ``str=True``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import ibin, variations</span>
-<span class="sd">    &gt;&gt;&gt; ibin(2)</span>
-<span class="sd">    [1, 0]</span>
-<span class="sd">    &gt;&gt;&gt; ibin(2, 4)</span>
-<span class="sd">    [0, 0, 1, 0]</span>
-<span class="sd">    &gt;&gt;&gt; ibin(2, 4)[::-1]</span>
-<span class="sd">    [0, 1, 0, 0]</span>
-
-<span class="sd">    If all lists corresponding to 0 to 2**n - 1, pass a non-integer</span>
-<span class="sd">    for bits:</span>
-
-<span class="sd">    &gt;&gt;&gt; bits = 2</span>
-<span class="sd">    &gt;&gt;&gt; for i in ibin(2, &#39;all&#39;):</span>
-<span class="sd">    ...     print(i)</span>
-<span class="sd">    (0, 0)</span>
-<span class="sd">    (0, 1)</span>
-<span class="sd">    (1, 0)</span>
-<span class="sd">    (1, 1)</span>
-
-<span class="sd">    If a bit string is desired of a given length, use str=True:</span>
-
-<span class="sd">    &gt;&gt;&gt; n = 123</span>
-<span class="sd">    &gt;&gt;&gt; bits = 10</span>
-<span class="sd">    &gt;&gt;&gt; ibin(n, bits, str=True)</span>
-<span class="sd">    &#39;0001111011&#39;</span>
-<span class="sd">    &gt;&gt;&gt; ibin(n, bits, str=True)[::-1]  # small bits left</span>
-<span class="sd">    &#39;1101111000&#39;</span>
-<span class="sd">    &gt;&gt;&gt; list(ibin(3, &#39;all&#39;, str=True))</span>
-<span class="sd">    [&#39;000&#39;, &#39;001&#39;, &#39;010&#39;, &#39;011&#39;, &#39;100&#39;, &#39;101&#39;, &#39;110&#39;, &#39;111&#39;]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">str</span><span class="p">:</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">bits</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">bits</span><span class="p">)</span>
-            <span class="k">return</span> <span class="p">[</span><span class="mi">1</span> <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="s">&quot;1&quot;</span> <span class="k">else</span> <span class="mi">0</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">bin</span><span class="p">(</span><span class="n">n</span><span class="p">)[</span><span class="mi">2</span><span class="p">:]</span><span class="o">.</span><span class="n">rjust</span><span class="p">(</span><span class="n">bits</span><span class="p">,</span> <span class="s">&quot;0&quot;</span><span class="p">)]</span>
-        <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">variations</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">)),</span> <span class="n">n</span><span class="p">,</span> <span class="n">repetition</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">bits</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">bits</span><span class="p">)</span>
-            <span class="k">return</span> <span class="nb">bin</span><span class="p">(</span><span class="n">n</span><span class="p">)[</span><span class="mi">2</span><span class="p">:]</span><span class="o">.</span><span class="n">rjust</span><span class="p">(</span><span class="n">bits</span><span class="p">,</span> <span class="s">&quot;0&quot;</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-            <span class="k">return</span> <span class="p">(</span><span class="nb">bin</span><span class="p">(</span><span class="n">i</span><span class="p">)[</span><span class="mi">2</span><span class="p">:]</span><span class="o">.</span><span class="n">rjust</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="s">&quot;0&quot;</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">n</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="variations"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.variations">[docs]</a><span class="k">def</span> <span class="nf">variations</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">repetition</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Returns a generator of the n-sized variations of ``seq`` (size N).</span>
-<span class="sd">    ``repetition`` controls whether items in ``seq`` can appear more than once;</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    variations(seq, n) will return N! / (N - n)! permutations without</span>
-<span class="sd">    repetition of seq&#39;s elements:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.utilities.iterables import variations</span>
-<span class="sd">        &gt;&gt;&gt; list(variations([1, 2], 2))</span>
-<span class="sd">        [(1, 2), (2, 1)]</span>
-
-<span class="sd">    variations(seq, n, True) will return the N**n permutations obtained</span>
-<span class="sd">    by allowing repetition of elements:</span>
-
-<span class="sd">        &gt;&gt;&gt; list(variations([1, 2], 2, repetition=True))</span>
-<span class="sd">        [(1, 1), (1, 2), (2, 1), (2, 2)]</span>
-
-<span class="sd">    If you ask for more items than are in the set you get the empty set unless</span>
-<span class="sd">    you allow repetitions:</span>
-
-<span class="sd">        &gt;&gt;&gt; list(variations([0, 1], 3, repetition=False))</span>
-<span class="sd">        []</span>
-<span class="sd">        &gt;&gt;&gt; list(variations([0, 1], 3, repetition=True))[:4]</span>
-<span class="sd">        [(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1)]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    sympy.core.compatibility.permutations</span>
-<span class="sd">    sympy.core.compatibility.product</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">permutations</span><span class="p">,</span> <span class="n">product</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">repetition</span><span class="p">:</span>
-        <span class="n">seq</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">seq</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">seq</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">permutations</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-            <span class="k">yield</span> <span class="n">i</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="k">yield</span> <span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">product</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="n">repeat</span><span class="o">=</span><span class="n">n</span><span class="p">):</span>
-                <span class="k">yield</span> <span class="n">i</span>
-
-</div>
-<div class="viewcode-block" id="subsets"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.subsets">[docs]</a><span class="k">def</span> <span class="nf">subsets</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">repetition</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Generates all k-subsets (combinations) from an n-element set, seq.</span>
-
-<span class="sd">       A k-subset of an n-element set is any subset of length exactly k. The</span>
-<span class="sd">       number of k-subsets of an n-element set is given by binomial(n, k),</span>
-<span class="sd">       whereas there are 2**n subsets all together. If k is None then all</span>
-<span class="sd">       2**n subsets will be returned from shortest to longest.</span>
-
-<span class="sd">       Examples</span>
-<span class="sd">       ========</span>
-
-<span class="sd">       &gt;&gt;&gt; from sympy.utilities.iterables import subsets</span>
-
-<span class="sd">       subsets(seq, k) will return the n!/k!/(n - k)! k-subsets (combinations)</span>
-<span class="sd">       without repetition, i.e. once an item has been removed, it can no</span>
-<span class="sd">       longer be &quot;taken&quot;:</span>
-
-<span class="sd">           &gt;&gt;&gt; list(subsets([1, 2], 2))</span>
-<span class="sd">           [(1, 2)]</span>
-<span class="sd">           &gt;&gt;&gt; list(subsets([1, 2]))</span>
-<span class="sd">           [(), (1,), (2,), (1, 2)]</span>
-<span class="sd">           &gt;&gt;&gt; list(subsets([1, 2, 3], 2))</span>
-<span class="sd">           [(1, 2), (1, 3), (2, 3)]</span>
-
-
-<span class="sd">       subsets(seq, k, repetition=True) will return the (n - 1 + k)!/k!/(n - 1)!</span>
-<span class="sd">       combinations *with* repetition:</span>
-
-<span class="sd">           &gt;&gt;&gt; list(subsets([1, 2], 2, repetition=True))</span>
-<span class="sd">           [(1, 1), (1, 2), (2, 2)]</span>
-
-<span class="sd">       If you ask for more items than are in the set you get the empty set unless</span>
-<span class="sd">       you allow repetitions:</span>
-
-<span class="sd">           &gt;&gt;&gt; list(subsets([0, 1], 3, repetition=False))</span>
-<span class="sd">           []</span>
-<span class="sd">           &gt;&gt;&gt; list(subsets([0, 1], 3, repetition=True))</span>
-<span class="sd">           [(0, 0, 0), (0, 0, 1), (0, 1, 1), (1, 1, 1)]</span>
-
-<span class="sd">       &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">k</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">seq</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">subsets</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">repetition</span><span class="p">):</span>
-                <span class="k">yield</span> <span class="n">i</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">repetition</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">combinations</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-                <span class="k">yield</span> <span class="n">i</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">combinations_with_replacement</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-                <span class="k">yield</span> <span class="n">i</span>
-
-</div>
-<div class="viewcode-block" id="numbered_symbols"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.numbered_symbols">[docs]</a><span class="k">def</span> <span class="nf">numbered_symbols</span><span class="p">(</span><span class="n">prefix</span><span class="o">=</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">start</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Generate an infinite stream of Symbols consisting of a prefix and</span>
-<span class="sd">    increasing subscripts.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    prefix : str, optional</span>
-<span class="sd">        The prefix to use. By default, this function will generate symbols of</span>
-<span class="sd">        the form &quot;x0&quot;, &quot;x1&quot;, etc.</span>
-
-<span class="sd">    cls : class, optional</span>
-<span class="sd">        The class to use. By default, it uses Symbol, but you can also use Wild or Dummy.</span>
-
-<span class="sd">    start : int, optional</span>
-<span class="sd">        The start number.  By default, it is 0.</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    =======</span>
-
-<span class="sd">    sym : Symbol</span>
-<span class="sd">        The subscripted symbols.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="s">&#39;dummy&#39;</span> <span class="ow">in</span> <span class="n">assumptions</span><span class="p">:</span>
-        <span class="n">SymPyDeprecationWarning</span><span class="p">(</span>
-            <span class="n">feature</span><span class="o">=</span><span class="s">&quot;&#39;dummy=True&#39; to create numbered Dummy symbols&quot;</span><span class="p">,</span>
-            <span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;cls=Dummy&quot;</span><span class="p">,</span>
-            <span class="n">issue</span><span class="o">=</span><span class="mi">3378</span><span class="p">,</span> <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.0&quot;</span><span class="p">,</span>
-        <span class="p">)</span><span class="o">.</span><span class="n">warn</span><span class="p">()</span>
-        <span class="k">if</span> <span class="n">assumptions</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;dummy&#39;</span><span class="p">):</span>
-            <span class="n">cls</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Dummy</span>
-
-    <span class="k">if</span> <span class="n">cls</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="c"># We can&#39;t just make the default cls=C.Symbol because it isn&#39;t</span>
-        <span class="c"># imported yet.</span>
-        <span class="n">cls</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">Symbol</span>
-
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="n">name</span> <span class="o">=</span> <span class="s">&#39;</span><span class="si">%s%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">prefix</span><span class="p">,</span> <span class="n">start</span><span class="p">)</span>
-        <span class="k">yield</span> <span class="n">cls</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">assumptions</span><span class="p">)</span>
-        <span class="n">start</span> <span class="o">+=</span> <span class="mi">1</span>
-
-</div>
-<div class="viewcode-block" id="capture"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.capture">[docs]</a><span class="k">def</span> <span class="nf">capture</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return the printed output of func().</span>
-
-<span class="sd">    `func` should be a function without arguments that produces output with</span>
-<span class="sd">    print statements.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import capture</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import pprint</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; def foo():</span>
-<span class="sd">    ...     print(&#39;hello world!&#39;)</span>
-<span class="sd">    ...</span>
-<span class="sd">    &gt;&gt;&gt; &#39;hello&#39; in capture(foo) # foo, not foo()</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; capture(lambda: pprint(2/x))</span>
-<span class="sd">    &#39;2\\n-\\nx\\n&#39;</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">import</span> <span class="nn">io</span>
-    <span class="kn">import</span> <span class="nn">sys</span>
-
-    <span class="n">stdout</span> <span class="o">=</span> <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span>
-    <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span> <span class="o">=</span> <span class="nb">file</span> <span class="o">=</span> <span class="n">io</span><span class="o">.</span><span class="n">StringIO</span><span class="p">()</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">func</span><span class="p">()</span>
-    <span class="k">finally</span><span class="p">:</span>
-        <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span> <span class="o">=</span> <span class="n">stdout</span>
-    <span class="k">return</span> <span class="nb">file</span><span class="o">.</span><span class="n">getvalue</span><span class="p">()</span>
-
-</div>
-<div class="viewcode-block" id="sift"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.sift">[docs]</a><span class="k">def</span> <span class="nf">sift</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="n">keyfunc</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Sift the sequence, ``seq`` into a dictionary according to keyfunc.</span>
-
-<span class="sd">    OUTPUT: each element in expr is stored in a list keyed to the value</span>
-<span class="sd">    of keyfunc for the element.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities import sift</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import sqrt, exp</span>
-
-<span class="sd">    &gt;&gt;&gt; sift(list(range(5)), lambda x: x % 2)</span>
-<span class="sd">    {0: [0, 2, 4], 1: [1, 3]}</span>
-
-<span class="sd">    sift() returns a defaultdict() object, so any key that has no matches will</span>
-<span class="sd">    give [].</span>
-
-<span class="sd">    &gt;&gt;&gt; sift([x], lambda x: x.is_commutative)</span>
-<span class="sd">    {True: [x]}</span>
-<span class="sd">    &gt;&gt;&gt; _[False]</span>
-<span class="sd">    []</span>
-
-<span class="sd">    Sometimes you won&#39;t know how many keys you will get:</span>
-
-<span class="sd">    &gt;&gt;&gt; sift([sqrt(x), exp(x), (y**x)**2],</span>
-<span class="sd">    ...      lambda x: x.as_base_exp()[0])</span>
-<span class="sd">    {E: [exp(x)], x: [sqrt(x)], y: [y**(2*x)]}</span>
-
-<span class="sd">    If you need to sort the sifted items it might be better to use</span>
-<span class="sd">    ``ordered`` which can economically apply multiple sort keys</span>
-<span class="sd">    to a squence while sorting.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    ordered</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="n">defaultdict</span><span class="p">(</span><span class="nb">list</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">seq</span><span class="p">:</span>
-        <span class="n">m</span><span class="p">[</span><span class="n">keyfunc</span><span class="p">(</span><span class="n">i</span><span class="p">)]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">m</span>
-
-</div>
-<div class="viewcode-block" id="take"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.take">[docs]</a><span class="k">def</span> <span class="nf">take</span><span class="p">(</span><span class="nb">iter</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return ``n`` items from ``iter`` iterator. &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="p">[</span> <span class="n">value</span> <span class="k">for</span> <span class="n">_</span><span class="p">,</span> <span class="n">value</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="nb">iter</span><span class="p">)</span> <span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="dict_merge"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.dict_merge">[docs]</a><span class="k">def</span> <span class="nf">dict_merge</span><span class="p">(</span><span class="o">*</span><span class="n">dicts</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Merge dictionaries into a single dictionary. &quot;&quot;&quot;</span>
-    <span class="n">merged</span> <span class="o">=</span> <span class="p">{}</span>
-
-    <span class="k">for</span> <span class="nb">dict</span> <span class="ow">in</span> <span class="n">dicts</span><span class="p">:</span>
-        <span class="n">merged</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="nb">dict</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">merged</span>
-
-</div>
-<div class="viewcode-block" id="common_prefix"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.common_prefix">[docs]</a><span class="k">def</span> <span class="nf">common_prefix</span><span class="p">(</span><span class="o">*</span><span class="n">seqs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return the subsequence that is a common start of sequences in ``seqs``.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import common_prefix</span>
-<span class="sd">    &gt;&gt;&gt; common_prefix(list(range(3)))</span>
-<span class="sd">    [0, 1, 2]</span>
-<span class="sd">    &gt;&gt;&gt; common_prefix(list(range(3)), list(range(4)))</span>
-<span class="sd">    [0, 1, 2]</span>
-<span class="sd">    &gt;&gt;&gt; common_prefix([1, 2, 3], [1, 2, 5])</span>
-<span class="sd">    [1, 2]</span>
-<span class="sd">    &gt;&gt;&gt; common_prefix([1, 2, 3], [1, 3, 5])</span>
-<span class="sd">    [1]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="ow">not</span> <span class="n">s</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">seqs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">[]</span>
-    <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">seqs</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">seqs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">min</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">seqs</span><span class="p">)):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">(</span><span class="n">seqs</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="n">seqs</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">seqs</span><span class="p">))):</span>
-            <span class="k">break</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-    <span class="k">return</span> <span class="n">seqs</span><span class="p">[</span><span class="mi">0</span><span class="p">][:</span><span class="n">i</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="common_suffix"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.common_suffix">[docs]</a><span class="k">def</span> <span class="nf">common_suffix</span><span class="p">(</span><span class="o">*</span><span class="n">seqs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return the subsequence that is a common ending of sequences in ``seqs``.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import common_suffix</span>
-<span class="sd">    &gt;&gt;&gt; common_suffix(list(range(3)))</span>
-<span class="sd">    [0, 1, 2]</span>
-<span class="sd">    &gt;&gt;&gt; common_suffix(list(range(3)), list(range(4)))</span>
-<span class="sd">    []</span>
-<span class="sd">    &gt;&gt;&gt; common_suffix([1, 2, 3], [9, 2, 3])</span>
-<span class="sd">    [2, 3]</span>
-<span class="sd">    &gt;&gt;&gt; common_suffix([1, 2, 3], [9, 7, 3])</span>
-<span class="sd">    [3]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="ow">not</span> <span class="n">s</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">seqs</span><span class="p">):</span>
-        <span class="k">return</span> <span class="p">[]</span>
-    <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">seqs</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">seqs</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-    <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="nb">min</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">seqs</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">all</span><span class="p">(</span><span class="n">seqs</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="n">seqs</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">seqs</span><span class="p">))):</span>
-            <span class="k">break</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">i</span> <span class="o">-=</span> <span class="mi">1</span>
-    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">seqs</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-
-</div>
-<div class="viewcode-block" id="prefixes"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.prefixes">[docs]</a><span class="k">def</span> <span class="nf">prefixes</span><span class="p">(</span><span class="n">seq</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Generate all prefixes of a sequence.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import prefixes</span>
-
-<span class="sd">    &gt;&gt;&gt; list(prefixes([1,2,3,4]))</span>
-<span class="sd">    [[1], [1, 2], [1, 2, 3], [1, 2, 3, 4]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">seq</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-        <span class="k">yield</span> <span class="n">seq</span><span class="p">[:</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="postfixes"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.postfixes">[docs]</a><span class="k">def</span> <span class="nf">postfixes</span><span class="p">(</span><span class="n">seq</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Generate all postfixes of a sequence.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import postfixes</span>
-
-<span class="sd">    &gt;&gt;&gt; list(postfixes([1,2,3,4]))</span>
-<span class="sd">    [[4], [3, 4], [2, 3, 4], [1, 2, 3, 4]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">seq</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-        <span class="k">yield</span> <span class="n">seq</span><span class="p">[</span><span class="n">n</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:]</span>
-
-</div>
-<div class="viewcode-block" id="topological_sort"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.topological_sort">[docs]</a><span class="k">def</span> <span class="nf">topological_sort</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">r&quot;&quot;&quot;</span>
-<span class="sd">    Topological sort of graph&#39;s vertices.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    ``graph`` : ``tuple[list, list[tuple[T, T]]``</span>
-<span class="sd">        A tuple consisting of a list of vertices and a list of edges of</span>
-<span class="sd">        a graph to be sorted topologically.</span>
-
-<span class="sd">    ``key`` : ``callable[T]`` (optional)</span>
-<span class="sd">        Ordering key for vertices on the same level. By default the natural</span>
-<span class="sd">        (e.g. lexicographic) ordering is used (in this case the base type</span>
-<span class="sd">        must implement ordering relations).</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Consider a graph::</span>
-
-<span class="sd">        +---+     +---+     +---+</span>
-<span class="sd">        | 7 |\    | 5 |     | 3 |</span>
-<span class="sd">        +---+ \   +---+     +---+</span>
-<span class="sd">          |   _\___/ ____   _/ |</span>
-<span class="sd">          |  /  \___/    \ /   |</span>
-<span class="sd">          V  V           V V   |</span>
-<span class="sd">         +----+         +---+  |</span>
-<span class="sd">         | 11 |         | 8 |  |</span>
-<span class="sd">         +----+         +---+  |</span>
-<span class="sd">          | | \____   ___/ _   |</span>
-<span class="sd">          | \      \ /    / \  |</span>
-<span class="sd">          V  \     V V   /  V  V</span>
-<span class="sd">        +---+ \   +---+ |  +----+</span>
-<span class="sd">        | 2 |  |  | 9 | |  | 10 |</span>
-<span class="sd">        +---+  |  +---+ |  +----+</span>
-<span class="sd">               \________/</span>
-
-<span class="sd">    where vertices are integers. This graph can be encoded using</span>
-<span class="sd">    elementary Python&#39;s data structures as follows::</span>
-
-<span class="sd">        &gt;&gt;&gt; V = [2, 3, 5, 7, 8, 9, 10, 11]</span>
-<span class="sd">        &gt;&gt;&gt; E = [(7, 11), (7, 8), (5, 11), (3, 8), (3, 10),</span>
-<span class="sd">        ...      (11, 2), (11, 9), (11, 10), (8, 9)]</span>
-
-<span class="sd">    To compute a topological sort for graph ``(V, E)`` issue::</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.utilities.iterables import topological_sort</span>
-
-<span class="sd">        &gt;&gt;&gt; topological_sort((V, E))</span>
-<span class="sd">        [3, 5, 7, 8, 11, 2, 9, 10]</span>
-
-<span class="sd">    If specific tie breaking approach is needed, use ``key`` parameter::</span>
-
-<span class="sd">        &gt;&gt;&gt; topological_sort((V, E), key=lambda v: -v)</span>
-<span class="sd">        [7, 5, 11, 3, 10, 8, 9, 2]</span>
-
-<span class="sd">    Only acyclic graphs can be sorted. If the input graph has a cycle,</span>
-<span class="sd">    then :py:exc:`ValueError` will be raised::</span>
-
-<span class="sd">        &gt;&gt;&gt; topological_sort((V, E + [(10, 7)]))</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">        ...</span>
-<span class="sd">        ValueError: cycle detected</span>
-
-<span class="sd">    .. seealso:: http://en.wikipedia.org/wiki/Topological_sorting</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">V</span><span class="p">,</span> <span class="n">E</span> <span class="o">=</span> <span class="n">graph</span>
-
-    <span class="n">L</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">S</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">V</span><span class="p">)</span>
-    <span class="n">E</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">E</span><span class="p">)</span>
-
-    <span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">u</span> <span class="ow">in</span> <span class="n">E</span><span class="p">:</span>
-        <span class="n">S</span><span class="o">.</span><span class="n">discard</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">key</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">key</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">value</span><span class="p">:</span> <span class="n">value</span>
-
-    <span class="n">S</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="n">key</span><span class="p">,</span> <span class="n">reverse</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-    <span class="k">while</span> <span class="n">S</span><span class="p">:</span>
-        <span class="n">node</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-        <span class="n">L</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
-
-        <span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">E</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">u</span> <span class="o">==</span> <span class="n">node</span><span class="p">:</span>
-                <span class="n">E</span><span class="o">.</span><span class="n">remove</span><span class="p">((</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">))</span>
-
-                <span class="k">for</span> <span class="n">_u</span><span class="p">,</span> <span class="n">_v</span> <span class="ow">in</span> <span class="n">E</span><span class="p">:</span>
-                    <span class="k">if</span> <span class="n">v</span> <span class="o">==</span> <span class="n">_v</span><span class="p">:</span>
-                        <span class="k">break</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">kv</span> <span class="o">=</span> <span class="n">key</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-
-                    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">s</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">S</span><span class="p">):</span>
-                        <span class="n">ks</span> <span class="o">=</span> <span class="n">key</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-
-                        <span class="k">if</span> <span class="n">kv</span> <span class="o">&gt;</span> <span class="n">ks</span><span class="p">:</span>
-                            <span class="n">S</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
-                            <span class="k">break</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="n">S</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">E</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;cycle detected&quot;</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">L</span>
-
-</div>
-<div class="viewcode-block" id="rotate_left"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.rotate_left">[docs]</a><span class="k">def</span> <span class="nf">rotate_left</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Left rotates a list x by the number of steps specified</span>
-<span class="sd">    in y.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import rotate_left</span>
-<span class="sd">    &gt;&gt;&gt; a = [0, 1, 2]</span>
-<span class="sd">    &gt;&gt;&gt; rotate_left(a, 1)</span>
-<span class="sd">    [1, 2, 0]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-    <span class="n">y</span> <span class="o">=</span> <span class="n">y</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">x</span><span class="p">[</span><span class="n">y</span><span class="p">:]</span> <span class="o">+</span> <span class="n">x</span><span class="p">[:</span><span class="n">y</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="rotate_right"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.rotate_right">[docs]</a><span class="k">def</span> <span class="nf">rotate_right</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Left rotates a list x by the number of steps specified</span>
-<span class="sd">    in y.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import rotate_right</span>
-<span class="sd">    &gt;&gt;&gt; a = [0, 1, 2]</span>
-<span class="sd">    &gt;&gt;&gt; rotate_right(a, 1)</span>
-<span class="sd">    [2, 0, 1]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-    <span class="n">y</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">y</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">x</span><span class="p">[</span><span class="n">y</span><span class="p">:]</span> <span class="o">+</span> <span class="n">x</span><span class="p">[:</span><span class="n">y</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="multiset_combinations"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.multiset_combinations">[docs]</a><span class="k">def</span> <span class="nf">multiset_combinations</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">g</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the unique combinations of size ``n`` from multiset ``m``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import multiset_combinations</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.core.compatibility import combinations</span>
-<span class="sd">    &gt;&gt;&gt; [&#39;&#39;.join(i) for i in  multiset_combinations(&#39;baby&#39;, 3)]</span>
-<span class="sd">    [&#39;abb&#39;, &#39;aby&#39;, &#39;bby&#39;]</span>
-
-<span class="sd">    &gt;&gt;&gt; def count(f, s): return len(list(f(s, 3)))</span>
-
-<span class="sd">    The number of combinations depends on the number of letters; the</span>
-<span class="sd">    number of unique combinations depends on how the letters are</span>
-<span class="sd">    repeated.</span>
-
-<span class="sd">    &gt;&gt;&gt; s1 = &#39;abracadabra&#39;</span>
-<span class="sd">    &gt;&gt;&gt; s2 = &#39;banana tree&#39;</span>
-<span class="sd">    &gt;&gt;&gt; count(combinations, s1), count(multiset_combinations, s1)</span>
-<span class="sd">    (165, 23)</span>
-<span class="sd">    &gt;&gt;&gt; count(combinations, s2), count(multiset_combinations, s2)</span>
-<span class="sd">    (165, 54)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">g</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">dict</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="nb">sum</span><span class="p">(</span><span class="n">m</span><span class="o">.</span><span class="n">values</span><span class="p">()):</span>
-                <span class="k">return</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="p">[[</span><span class="n">k</span><span class="p">,</span> <span class="n">m</span><span class="p">[</span><span class="n">k</span><span class="p">]]</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">ordered</span><span class="p">(</span><span class="n">m</span><span class="p">)]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
-                <span class="k">return</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">m</span> <span class="o">=</span> <span class="n">multiset</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-                <span class="n">g</span> <span class="o">=</span> <span class="p">[(</span><span class="n">k</span><span class="p">,</span> <span class="n">m</span><span class="p">[</span><span class="n">k</span><span class="p">])</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">ordered</span><span class="p">(</span><span class="n">m</span><span class="p">)]</span>
-            <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-                <span class="n">m</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">ordered</span><span class="p">(</span><span class="n">m</span><span class="p">))</span>
-                <span class="n">g</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">group</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">False</span><span class="p">)]</span>
-        <span class="k">del</span> <span class="n">m</span>
-    <span class="k">if</span> <span class="nb">sum</span><span class="p">(</span><span class="n">v</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">g</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">n</span> <span class="ow">or</span> <span class="ow">not</span> <span class="n">n</span><span class="p">:</span>
-        <span class="k">yield</span> <span class="p">[]</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">g</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">v</span> <span class="o">&gt;=</span> <span class="n">n</span><span class="p">:</span>
-                <span class="k">yield</span> <span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">n</span>
-                <span class="n">v</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span>
-            <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">min</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">v</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">multiset_combinations</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="n">v</span><span class="p">,</span> <span class="n">g</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]):</span>
-                    <span class="n">rv</span> <span class="o">=</span> <span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">v</span> <span class="o">+</span> <span class="n">j</span>
-                    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">rv</span><span class="p">)</span> <span class="o">==</span> <span class="n">n</span><span class="p">:</span>
-                        <span class="k">yield</span> <span class="n">rv</span>
-
-</div>
-<div class="viewcode-block" id="multiset_permutations"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.multiset_permutations">[docs]</a><span class="k">def</span> <span class="nf">multiset_permutations</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">g</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the unique permutations of multiset ``m``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import multiset_permutations</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import factorial</span>
-<span class="sd">    &gt;&gt;&gt; [&#39;&#39;.join(i) for i in multiset_permutations(&#39;aab&#39;)]</span>
-<span class="sd">    [&#39;aab&#39;, &#39;aba&#39;, &#39;baa&#39;]</span>
-<span class="sd">    &gt;&gt;&gt; factorial(len(&#39;banana&#39;))</span>
-<span class="sd">    720</span>
-<span class="sd">    &gt;&gt;&gt; len(list(multiset_permutations(&#39;banana&#39;)))</span>
-<span class="sd">    60</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">size</span> <span class="o">=</span> <span class="n">k</span>
-    <span class="k">if</span> <span class="n">g</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">dict</span><span class="p">:</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="p">[[</span><span class="n">k</span><span class="p">,</span> <span class="n">m</span><span class="p">[</span><span class="n">k</span><span class="p">]]</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">ordered</span><span class="p">(</span><span class="n">m</span><span class="p">)]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">ordered</span><span class="p">(</span><span class="n">m</span><span class="p">))</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">group</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">False</span><span class="p">)]</span>
-        <span class="k">del</span> <span class="n">m</span>
-    <span class="n">do</span> <span class="o">=</span> <span class="p">[</span><span class="n">gi</span> <span class="k">for</span> <span class="n">gi</span> <span class="ow">in</span> <span class="n">g</span> <span class="k">if</span> <span class="n">gi</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">]</span>
-    <span class="n">SUM</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">([</span><span class="n">gi</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">gi</span> <span class="ow">in</span> <span class="n">do</span><span class="p">])</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">do</span> <span class="ow">or</span> <span class="n">size</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="ow">and</span> <span class="p">(</span><span class="n">size</span> <span class="o">&gt;</span> <span class="n">SUM</span> <span class="ow">or</span> <span class="n">size</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">):</span>
-        <span class="k">return</span>
-    <span class="k">elif</span> <span class="n">size</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">do</span><span class="p">:</span>
-            <span class="k">yield</span> <span class="p">[</span><span class="n">k</span><span class="p">]</span>
-    <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">do</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">do</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="n">v</span> <span class="k">if</span> <span class="n">size</span> <span class="ow">is</span> <span class="bp">None</span> <span class="k">else</span> <span class="p">(</span><span class="n">size</span> <span class="k">if</span> <span class="n">size</span> <span class="o">&lt;=</span> <span class="n">v</span> <span class="k">else</span> <span class="mi">0</span><span class="p">)</span>
-        <span class="k">yield</span> <span class="p">[</span><span class="n">k</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">v</span><span class="p">)]</span>
-    <span class="k">elif</span> <span class="nb">all</span><span class="p">(</span><span class="n">v</span> <span class="o">==</span> <span class="mi">1</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">do</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">permutations</span><span class="p">([</span><span class="n">k</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">do</span><span class="p">],</span> <span class="n">size</span><span class="p">):</span>
-            <span class="k">yield</span> <span class="nb">list</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">size</span> <span class="o">=</span> <span class="n">size</span> <span class="k">if</span> <span class="n">size</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span> <span class="k">else</span> <span class="n">SUM</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">do</span><span class="p">):</span>
-            <span class="n">do</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">multiset_permutations</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="n">size</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">do</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">j</span><span class="p">:</span>
-                    <span class="k">yield</span> <span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">+</span> <span class="n">j</span>
-            <span class="n">do</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_partition</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="n">vector</span><span class="p">,</span> <span class="n">m</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return the partion of seq as specified by the partition vector.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import _partition</span>
-<span class="sd">    &gt;&gt;&gt; _partition(&#39;abcde&#39;, [1, 0, 1, 2, 0])</span>
-<span class="sd">    [[&#39;b&#39;, &#39;e&#39;], [&#39;a&#39;, &#39;c&#39;], [&#39;d&#39;]]</span>
-
-<span class="sd">    Specifying the number of bins in the partition is optional:</span>
-
-<span class="sd">    &gt;&gt;&gt; _partition(&#39;abcde&#39;, [1, 0, 1, 2, 0], 3)</span>
-<span class="sd">    [[&#39;b&#39;, &#39;e&#39;], [&#39;a&#39;, &#39;c&#39;], [&#39;d&#39;]]</span>
-
-<span class="sd">    The output of _set_partitions can be passed as follows:</span>
-
-<span class="sd">    &gt;&gt;&gt; output = (3, [1, 0, 1, 2, 0])</span>
-<span class="sd">    &gt;&gt;&gt; _partition(&#39;abcde&#39;, *output)</span>
-<span class="sd">    [[&#39;b&#39;, &#39;e&#39;], [&#39;a&#39;, &#39;c&#39;], [&#39;d&#39;]]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    combinatorics.partitions.Partition.from_rgs()</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">vector</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
-    <span class="k">elif</span> <span class="nb">type</span><span class="p">(</span><span class="n">vector</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">int</span><span class="p">:</span>  <span class="c"># entered as m, vector</span>
-        <span class="n">vector</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">m</span><span class="p">,</span> <span class="n">vector</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="p">[[]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="p">)]</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">vector</span><span class="p">):</span>
-        <span class="n">p</span><span class="p">[</span><span class="n">v</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">seq</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-    <span class="k">return</span> <span class="n">p</span>
-
-
-<span class="k">def</span> <span class="nf">_set_partitions</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Cycle through all partions of n elements, yielding the</span>
-<span class="sd">    current number of partitions, ``m``, and a mutable list, ``q``</span>
-<span class="sd">    such that element[i] is in part q[i] of the partition.</span>
-
-<span class="sd">    NOTE: ``q`` is modified in place and generally should not be changed</span>
-<span class="sd">    between function calls.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import _set_partitions, _partition</span>
-<span class="sd">    &gt;&gt;&gt; for m, q in _set_partitions(3):</span>
-<span class="sd">    ...     print(m, q, _partition(&#39;abc&#39;, q, m))</span>
-<span class="sd">    1 [0, 0, 0] [[&#39;a&#39;, &#39;b&#39;, &#39;c&#39;]]</span>
-<span class="sd">    2 [0, 0, 1] [[&#39;a&#39;, &#39;b&#39;], [&#39;c&#39;]]</span>
-<span class="sd">    2 [0, 1, 0] [[&#39;a&#39;, &#39;c&#39;], [&#39;b&#39;]]</span>
-<span class="sd">    2 [0, 1, 1] [[&#39;a&#39;], [&#39;b&#39;, &#39;c&#39;]]</span>
-<span class="sd">    3 [0, 1, 2] [[&#39;a&#39;], [&#39;b&#39;], [&#39;c&#39;]]</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    This algorithm is similar to, and solves the same problem as,</span>
-<span class="sd">    Algorithm 7.2.1.5H, from volume 4A of Knuth&#39;s The Art of Computer</span>
-<span class="sd">    Programming.  Knuth uses the term &quot;restricted growth string&quot; where</span>
-<span class="sd">    this code refers to a &quot;partition vector&quot;. In each case, the meaning is</span>
-<span class="sd">    the same: the value in the ith element of the vector specifies to</span>
-<span class="sd">    which part the ith set element is to be assigned.</span>
-
-<span class="sd">    At the lowest level, this code implements an n-digit big-endian</span>
-<span class="sd">    counter (stored in the array q) which is incremented (with carries) to</span>
-<span class="sd">    get the next partition in the sequence.  A special twist is that a</span>
-<span class="sd">    digit is constrained to be at most one greater than the maximum of all</span>
-<span class="sd">    the digits to the left of it.  The array p maintains this maximum, so</span>
-<span class="sd">    that the code can efficiently decide when a digit can be incremented</span>
-<span class="sd">    in place or whether it needs to be reset to 0 and trigger a carry to</span>
-<span class="sd">    the next digit.  The enumeration starts with all the digits 0 (which</span>
-<span class="sd">    corresponds to all the set elements being assigned to the same 0th</span>
-<span class="sd">    part), and ends with 0123...n, which corresponds to each set element</span>
-<span class="sd">    being assigned to a different, singleton, part.</span>
-
-<span class="sd">    This routine was rewritten to use 0-based lists while trying to</span>
-<span class="sd">    preserve the beauty and efficiency of the original algorithm.</span>
-
-<span class="sd">    Reference</span>
-<span class="sd">    =========</span>
-
-<span class="sd">    Nijenhuis, Albert and Wilf, Herbert. (1978) Combinatorial Algorithms,</span>
-<span class="sd">    2nd Ed, p 91, algorithm &quot;nexequ&quot;. Available online from</span>
-<span class="sd">    http://www.math.upenn.edu/~wilf/website/CombAlgDownld.html (viewed</span>
-<span class="sd">    November 17, 2012).</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">n</span>
-    <span class="n">q</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">n</span>
-    <span class="n">nc</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="k">yield</span> <span class="n">nc</span><span class="p">,</span> <span class="n">q</span>
-    <span class="k">while</span> <span class="n">nc</span> <span class="o">!=</span> <span class="n">n</span><span class="p">:</span>
-        <span class="n">m</span> <span class="o">=</span> <span class="n">n</span>
-        <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="n">m</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="n">q</span><span class="p">[</span><span class="n">m</span><span class="p">]</span>
-            <span class="k">if</span> <span class="n">p</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">break</span>
-            <span class="n">q</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="n">q</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">=</span> <span class="n">i</span>
-        <span class="n">m</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="n">nc</span> <span class="o">+=</span> <span class="n">m</span> <span class="o">-</span> <span class="n">n</span>
-        <span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+=</span> <span class="n">n</span> <span class="o">-</span> <span class="n">m</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">nc</span><span class="p">:</span>
-            <span class="n">p</span><span class="p">[</span><span class="n">nc</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="n">nc</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="n">p</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">1</span>
-        <span class="n">p</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">yield</span> <span class="n">nc</span><span class="p">,</span> <span class="n">q</span>
-
-
-<div class="viewcode-block" id="multiset_partitions"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.multiset_partitions">[docs]</a><span class="k">def</span> <span class="nf">multiset_partitions</span><span class="p">(</span><span class="n">multiset</span><span class="p">,</span> <span class="n">m</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return unique partitions of the given multiset (in list form).</span>
-<span class="sd">    If ``m`` is None, all multisets will be returned, otherwise only</span>
-<span class="sd">    partitions with ``m`` parts will be returned.</span>
-
-<span class="sd">    If ``multiset`` is an integer, a range [0, 1, ..., multiset - 1]</span>
-<span class="sd">    will be supplied.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import multiset_partitions</span>
-<span class="sd">    &gt;&gt;&gt; list(multiset_partitions([1, 2, 3, 4], 2))</span>
-<span class="sd">    [[[1, 2, 3], [4]], [[1, 2, 4], [3]], [[1, 2], [3, 4]],</span>
-<span class="sd">    [[1, 3, 4], [2]], [[1, 3], [2, 4]], [[1, 4], [2, 3]],</span>
-<span class="sd">    [[1], [2, 3, 4]]]</span>
-<span class="sd">    &gt;&gt;&gt; list(multiset_partitions([1, 2, 3, 4], 1))</span>
-<span class="sd">    [[[1, 2, 3, 4]]]</span>
-
-<span class="sd">    Only unique partitions are returned and these will be returned in a</span>
-<span class="sd">    canonical order regardless of the order of the input:</span>
-
-<span class="sd">    &gt;&gt;&gt; a = [1, 2, 2, 1]</span>
-<span class="sd">    &gt;&gt;&gt; ans = list(multiset_partitions(a, 2))</span>
-<span class="sd">    &gt;&gt;&gt; a.sort()</span>
-<span class="sd">    &gt;&gt;&gt; list(multiset_partitions(a, 2)) == ans</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; a = list(range(3, 1, -1))</span>
-<span class="sd">    &gt;&gt;&gt; (list(multiset_partitions(a)) ==</span>
-<span class="sd">    ...  list(multiset_partitions(sorted(a))))</span>
-<span class="sd">    True</span>
-
-<span class="sd">    If m is omitted then all partitions will be returned:</span>
-
-<span class="sd">    &gt;&gt;&gt; list(multiset_partitions([1, 1, 2]))</span>
-<span class="sd">    [[[1, 1, 2]], [[1, 1], [2]], [[1, 2], [1]], [[1], [1], [2]]]</span>
-<span class="sd">    &gt;&gt;&gt; list(multiset_partitions([1]*3))</span>
-<span class="sd">    [[[1, 1, 1]], [[1], [1, 1]], [[1], [1], [1]]]</span>
-
-<span class="sd">    Counting</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    The number of partitions of a set is given by the bell number:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import bell</span>
-<span class="sd">    &gt;&gt;&gt; len(list(multiset_partitions(5))) == bell(5) == 52</span>
-<span class="sd">    True</span>
-
-<span class="sd">    The number of partitions of length k from a set of size n is given by the</span>
-<span class="sd">    Stirling Number of the 2nd kind:</span>
-
-<span class="sd">    &gt;&gt;&gt; def S2(n, k):</span>
-<span class="sd">    ...     from sympy import Dummy, binomial, factorial, Sum</span>
-<span class="sd">    ...     if k &gt; n:</span>
-<span class="sd">    ...         return 0</span>
-<span class="sd">    ...     j = Dummy()</span>
-<span class="sd">    ...     arg = (-1)**(k-j)*j**n*binomial(k,j)</span>
-<span class="sd">    ...     return 1/factorial(k)*Sum(arg,(j,0,k)).doit()</span>
-<span class="sd">    ...</span>
-<span class="sd">    &gt;&gt;&gt; S2(5, 2) == len(list(multiset_partitions(5, 2))) == 15</span>
-<span class="sd">    True</span>
-
-<span class="sd">    These comments on counting apply to *sets*, not multisets.</span>
-
-<span class="sd">    Notes</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    When all the elements are the same in the multiset, the order</span>
-<span class="sd">    of the returned partitions is determined by the ``partitions``</span>
-<span class="sd">    routine.</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    partitions</span>
-<span class="sd">    sympy.combinatorics.partitions.Partition</span>
-<span class="sd">    sympy.combinatorics.partitions.IntegerPartition</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">multiset</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">int</span><span class="p">:</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="n">multiset</span>
-        <span class="k">if</span> <span class="n">m</span> <span class="ow">and</span> <span class="n">m</span> <span class="o">&gt;</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="n">multiset</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">multiset</span><span class="p">))</span>
-        <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">yield</span> <span class="p">[</span><span class="n">multiset</span><span class="p">[:]]</span>
-            <span class="k">return</span>
-        <span class="k">for</span> <span class="n">nc</span><span class="p">,</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">_set_partitions</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">m</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">nc</span> <span class="o">==</span> <span class="n">m</span><span class="p">:</span>
-                <span class="n">rv</span> <span class="o">=</span> <span class="p">[[]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nc</span><span class="p">)]</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-                    <span class="n">rv</span><span class="p">[</span><span class="n">q</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">multiset</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-                <span class="k">yield</span> <span class="n">rv</span>
-        <span class="k">return</span>
-
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">multiset</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">type</span><span class="p">(</span><span class="n">multiset</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">str</span><span class="p">:</span>
-        <span class="n">multiset</span> <span class="o">=</span> <span class="p">[</span><span class="n">multiset</span><span class="p">]</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">has_variety</span><span class="p">(</span><span class="n">multiset</span><span class="p">):</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">multiset</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">m</span> <span class="ow">and</span> <span class="n">m</span> <span class="o">&gt;</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">yield</span> <span class="p">[</span><span class="n">multiset</span><span class="p">[:]]</span>
-            <span class="k">return</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">multiset</span><span class="p">[:</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">for</span> <span class="n">size</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">partitions</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">m</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">size</span> <span class="o">==</span> <span class="n">m</span><span class="p">:</span>
-                <span class="n">rv</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
-                    <span class="n">rv</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">x</span><span class="o">*</span><span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="n">k</span><span class="p">])</span>
-                <span class="k">yield</span> <span class="n">rv</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">multiset</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">ordered</span><span class="p">(</span><span class="n">multiset</span><span class="p">))</span>
-        <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">multiset</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">m</span> <span class="ow">and</span> <span class="n">m</span> <span class="o">&gt;</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">yield</span> <span class="p">[</span><span class="n">multiset</span><span class="p">[:]]</span>
-            <span class="k">return</span>
-
-        <span class="c"># if there are repeated elements, sort them and define the</span>
-        <span class="c"># canon dictionary that will be used to create the cache key</span>
-        <span class="c"># in case elements of the multiset are not hashable</span>
-        <span class="n">cache</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="n">canon</span> <span class="o">=</span> <span class="p">{}</span>  <span class="c"># {physical position: position where it appeared first}</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">mi</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">multiset</span><span class="p">):</span>
-            <span class="n">canon</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">canon</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">multiset</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">mi</span><span class="p">)))</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">canon</span><span class="o">.</span><span class="n">values</span><span class="p">()))</span> <span class="o">!=</span> <span class="n">n</span><span class="p">:</span>
-            <span class="n">canon</span> <span class="o">=</span> <span class="p">{}</span>
-            <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">mi</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">multiset</span><span class="p">):</span>
-                <span class="n">canon</span><span class="o">.</span><span class="n">setdefault</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">canon</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">multiset</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">mi</span><span class="p">)))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">canon</span> <span class="o">=</span> <span class="bp">None</span>
-
-        <span class="k">for</span> <span class="n">nc</span><span class="p">,</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">_set_partitions</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">m</span> <span class="ow">is</span> <span class="bp">None</span> <span class="ow">or</span> <span class="n">nc</span> <span class="o">==</span> <span class="n">m</span><span class="p">:</span>
-                <span class="n">rv</span> <span class="o">=</span> <span class="p">[[]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nc</span><span class="p">)]</span>
-                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-                    <span class="n">rv</span><span class="p">[</span><span class="n">q</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">canon</span><span class="p">:</span>
-                    <span class="n">canonical</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span>
-                        <span class="nb">sorted</span><span class="p">([</span><span class="nb">tuple</span><span class="p">([</span><span class="n">canon</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">j</span><span class="p">])</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">rv</span><span class="p">]))</span>
-                    <span class="k">if</span> <span class="n">canonical</span> <span class="ow">in</span> <span class="n">cache</span><span class="p">:</span>
-                        <span class="k">continue</span>
-                    <span class="n">cache</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">canonical</span><span class="p">)</span>
-
-                <span class="k">yield</span> <span class="p">[[</span><span class="n">multiset</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">rv</span><span class="p">]</span>
-
-</div>
-<div class="viewcode-block" id="partitions"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.partitions">[docs]</a><span class="k">def</span> <span class="nf">partitions</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Generate all partitions of integer n (&gt;= 0).</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    ``m`` : integer (default gives partitions of all sizes)</span>
-<span class="sd">        limits number of parts in parition (mnemonic: m, maximum parts)</span>
-<span class="sd">    ``k`` : integer (default gives partitions number from 1 through n)</span>
-<span class="sd">        limits the numbers that are kept in the partition (mnemonic: k, keys)</span>
-<span class="sd">    ``size`` : bool (default False, only partition is returned)</span>
-<span class="sd">        when ``True`` then (M, P) is returned where M is the sum of the</span>
-<span class="sd">        multiplicities and P is the generated partition.</span>
-
-<span class="sd">    Each partition is represented as a dictionary, mapping an integer</span>
-<span class="sd">    to the number of copies of that integer in the partition.  For example,</span>
-<span class="sd">    the first partition of 4 returned is {4: 1}, &quot;4: one of them&quot;.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import partitions</span>
-
-<span class="sd">    The numbers appearing in the partition (the key of the returned dict)</span>
-<span class="sd">    are limited with k:</span>
-
-<span class="sd">    &gt;&gt;&gt; for p in partitions(6, k=2):</span>
-<span class="sd">    ...     print(p)</span>
-<span class="sd">    {2: 3}</span>
-<span class="sd">    {1: 2, 2: 2}</span>
-<span class="sd">    {1: 4, 2: 1}</span>
-<span class="sd">    {1: 6}</span>
-
-<span class="sd">    The maximum number of parts in the partion (the sum of the values in</span>
-<span class="sd">    the returned dict) are limited with m:</span>
-
-<span class="sd">    &gt;&gt;&gt; for p in partitions(6, m=2):</span>
-<span class="sd">    ...     print(p)</span>
-<span class="sd">    ...</span>
-<span class="sd">    {6: 1}</span>
-<span class="sd">    {1: 1, 5: 1}</span>
-<span class="sd">    {2: 1, 4: 1}</span>
-<span class="sd">    {3: 2}</span>
-
-<span class="sd">    Note that the _same_ dictionary object is returned each time.</span>
-<span class="sd">    This is for speed:  generating each partition goes quickly,</span>
-<span class="sd">    taking constant time, independent of n.</span>
-
-<span class="sd">    &gt;&gt;&gt; [p for p in partitions(6, k=2)]</span>
-<span class="sd">    [{1: 6}, {1: 6}, {1: 6}, {1: 6}]</span>
-
-<span class="sd">    If you want to build a list of the returned dictionaries then</span>
-<span class="sd">    make a copy of them:</span>
-
-<span class="sd">    &gt;&gt;&gt; [p.copy() for p in partitions(6, k=2)]</span>
-<span class="sd">    [{2: 3}, {1: 2, 2: 2}, {1: 4, 2: 1}, {1: 6}]</span>
-<span class="sd">    &gt;&gt;&gt; [(M, p.copy()) for M, p in partitions(6, k=2, size=True)]</span>
-<span class="sd">    [(3, {2: 3}), (4, {1: 2, 2: 2}), (5, {1: 4, 2: 1}), (6, {1: 6})]</span>
-
-<span class="sd">    Reference:</span>
-<span class="sd">        modified from Tim Peter&#39;s version to allow for k and m values:</span>
-<span class="sd">        code.activestate.com/recipes/218332-generator-for-integer-partitions/</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    sympy.combinatorics.partitions.Partition</span>
-<span class="sd">    sympy.combinatorics.partitions.IntegerPartition</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;n must be &gt;= 0&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;m must be &gt; 0&quot;</span><span class="p">)</span>
-    <span class="n">m</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">m</span> <span class="ow">or</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">m</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;maximum numbers in partition, m, must be &gt; 0&quot;</span><span class="p">)</span>
-    <span class="n">k</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">k</span> <span class="ow">or</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">k</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;maximum value in partition, k, must be &gt; 0&quot;</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">m</span><span class="o">*</span><span class="n">k</span> <span class="o">&lt;</span> <span class="n">n</span><span class="p">:</span>
-        <span class="k">return</span>
-
-    <span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">as_int</span><span class="p">(</span><span class="n">m</span><span class="p">),</span> <span class="n">as_int</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-    <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="nb">divmod</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-    <span class="n">ms</span> <span class="o">=</span> <span class="p">{</span><span class="n">k</span><span class="p">:</span> <span class="n">q</span><span class="p">}</span>
-    <span class="n">keys</span> <span class="o">=</span> <span class="p">[</span><span class="n">k</span><span class="p">]</span>  <span class="c"># ms.keys(), from largest to smallest</span>
-    <span class="k">if</span> <span class="n">r</span><span class="p">:</span>
-        <span class="n">ms</span><span class="p">[</span><span class="n">r</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">keys</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-    <span class="n">room</span> <span class="o">=</span> <span class="n">m</span> <span class="o">-</span> <span class="n">q</span> <span class="o">-</span> <span class="nb">bool</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">size</span><span class="p">:</span>
-        <span class="k">yield</span> <span class="nb">sum</span><span class="p">(</span><span class="n">ms</span><span class="o">.</span><span class="n">values</span><span class="p">()),</span> <span class="n">ms</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">yield</span> <span class="n">ms</span>
-
-    <span class="k">while</span> <span class="n">keys</span> <span class="o">!=</span> <span class="p">[</span><span class="mi">1</span><span class="p">]:</span>
-        <span class="c"># Reuse any 1&#39;s.</span>
-        <span class="k">if</span> <span class="n">keys</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">del</span> <span class="n">keys</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-            <span class="n">reuse</span> <span class="o">=</span> <span class="n">ms</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-            <span class="n">room</span> <span class="o">+=</span> <span class="n">reuse</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">reuse</span> <span class="o">=</span> <span class="mi">0</span>
-
-        <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="c"># Let i be the smallest key larger than 1.  Reuse one</span>
-            <span class="c"># instance of i.</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="n">keys</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-            <span class="n">newcount</span> <span class="o">=</span> <span class="n">ms</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">ms</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span>
-            <span class="n">reuse</span> <span class="o">+=</span> <span class="n">i</span>
-            <span class="k">if</span> <span class="n">newcount</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">del</span> <span class="n">keys</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">ms</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-            <span class="n">room</span> <span class="o">+=</span> <span class="mi">1</span>
-
-            <span class="c"># Break the remainder into pieces of size i-1.</span>
-            <span class="n">i</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="nb">divmod</span><span class="p">(</span><span class="n">reuse</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-            <span class="n">need</span> <span class="o">=</span> <span class="n">q</span> <span class="o">+</span> <span class="nb">bool</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">need</span> <span class="o">&gt;</span> <span class="n">room</span><span class="p">:</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">keys</span><span class="p">:</span>
-                    <span class="k">return</span>
-                <span class="k">continue</span>
-
-            <span class="n">ms</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">q</span>
-            <span class="n">keys</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">r</span><span class="p">:</span>
-                <span class="n">ms</span><span class="p">[</span><span class="n">r</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-                <span class="n">keys</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-            <span class="k">break</span>
-        <span class="n">room</span> <span class="o">-=</span> <span class="n">need</span>
-        <span class="k">if</span> <span class="n">size</span><span class="p">:</span>
-            <span class="k">yield</span> <span class="nb">sum</span><span class="p">(</span><span class="n">ms</span><span class="o">.</span><span class="n">values</span><span class="p">()),</span> <span class="n">ms</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">yield</span> <span class="n">ms</span>
-
-</div>
-<div class="viewcode-block" id="binary_partitions"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.binary_partitions">[docs]</a><span class="k">def</span> <span class="nf">binary_partitions</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Generates the binary partition of n.</span>
-
-<span class="sd">    A binary partition consists only of numbers that are</span>
-<span class="sd">    powers of two. Each step reduces a 2**(k+1) to 2**k and</span>
-<span class="sd">    2**k. Thus 16 is converted to 8 and 8.</span>
-
-<span class="sd">    Reference: TAOCP 4, section 7.2.1.5, problem 64</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import binary_partitions</span>
-<span class="sd">    &gt;&gt;&gt; for i in binary_partitions(5):</span>
-<span class="sd">    ...     print(i)</span>
-<span class="sd">    ...</span>
-<span class="sd">    [4, 1]</span>
-<span class="sd">    [2, 2, 1]</span>
-<span class="sd">    [2, 1, 1, 1]</span>
-<span class="sd">    [1, 1, 1, 1, 1]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">ceil</span><span class="p">,</span> <span class="n">log</span>
-    <span class="nb">pow</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="n">ceil</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))))</span>
-    <span class="nb">sum</span> <span class="o">=</span> <span class="mi">0</span>
-    <span class="n">partition</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">while</span> <span class="nb">pow</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">sum</span> <span class="o">+</span> <span class="nb">pow</span> <span class="o">&lt;=</span> <span class="n">n</span><span class="p">:</span>
-            <span class="n">partition</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">pow</span><span class="p">)</span>
-            <span class="nb">sum</span> <span class="o">+=</span> <span class="nb">pow</span>
-        <span class="nb">pow</span> <span class="o">&gt;&gt;=</span> <span class="mi">1</span>
-
-    <span class="n">last_num</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">partition</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span> <span class="o">-</span> <span class="p">(</span><span class="n">n</span> <span class="o">&amp;</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="k">while</span> <span class="n">last_num</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">yield</span> <span class="n">partition</span>
-        <span class="k">if</span> <span class="n">partition</span><span class="p">[</span><span class="n">last_num</span><span class="p">]</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-            <span class="n">partition</span><span class="p">[</span><span class="n">last_num</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-            <span class="n">partition</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-            <span class="n">last_num</span> <span class="o">-=</span> <span class="mi">1</span>
-            <span class="k">continue</span>
-        <span class="n">partition</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-        <span class="n">partition</span><span class="p">[</span><span class="n">last_num</span><span class="p">]</span> <span class="o">&gt;&gt;=</span> <span class="mi">1</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">partition</span><span class="p">[</span><span class="n">last_num</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">partition</span><span class="p">[</span><span class="n">last_num</span><span class="p">]</span>
-        <span class="n">last_num</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">while</span> <span class="n">x</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">x</span> <span class="o">&lt;=</span> <span class="nb">len</span><span class="p">(</span><span class="n">partition</span><span class="p">)</span> <span class="o">-</span> <span class="n">last_num</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">del</span> <span class="n">partition</span><span class="p">[</span><span class="o">-</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-                <span class="n">last_num</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="n">partition</span><span class="p">[</span><span class="n">last_num</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">x</span> <span class="o">&gt;&gt;=</span> <span class="mi">1</span>
-    <span class="k">yield</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">n</span>
-
-</div>
-<div class="viewcode-block" id="has_dups"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.has_dups">[docs]</a><span class="k">def</span> <span class="nf">has_dups</span><span class="p">(</span><span class="n">seq</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return True if there are any duplicate elements in ``seq``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import has_dups</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Dict, Set</span>
-
-<span class="sd">    &gt;&gt;&gt; has_dups((1, 2, 1))</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; has_dups(list(range(3)))</span>
-<span class="sd">    False</span>
-<span class="sd">    &gt;&gt;&gt; all(has_dups(c) is False for c in (set(), Set(), dict(), Dict()))</span>
-<span class="sd">    True</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="p">(</span><span class="nb">dict</span><span class="p">,</span> <span class="nb">set</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Dict</span><span class="p">,</span> <span class="n">C</span><span class="o">.</span><span class="n">Set</span><span class="p">)):</span>
-        <span class="k">return</span> <span class="bp">False</span>
-    <span class="n">uniq</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-    <span class="k">return</span> <span class="nb">any</span><span class="p">(</span><span class="bp">True</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">seq</span> <span class="k">if</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">uniq</span> <span class="ow">or</span> <span class="n">uniq</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">s</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="has_variety"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.has_variety">[docs]</a><span class="k">def</span> <span class="nf">has_variety</span><span class="p">(</span><span class="n">seq</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return True if there are any different elements in ``seq``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import has_variety</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Dict, Set</span>
-
-<span class="sd">    &gt;&gt;&gt; has_variety((1, 2, 1))</span>
-<span class="sd">    True</span>
-<span class="sd">    &gt;&gt;&gt; has_variety((1, 1, 1))</span>
-<span class="sd">    False</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">s</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">seq</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">sentinel</span> <span class="o">=</span> <span class="n">s</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">s</span> <span class="o">!=</span> <span class="n">sentinel</span><span class="p">:</span>
-                <span class="k">return</span> <span class="bp">True</span>
-    <span class="k">return</span> <span class="bp">False</span>
-
-</div>
-<div class="viewcode-block" id="uniq"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.uniq">[docs]</a><span class="k">def</span> <span class="nf">uniq</span><span class="p">(</span><span class="n">seq</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Yield unique elements from ``seq`` as an iterator.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import uniq</span>
-<span class="sd">    &gt;&gt;&gt; dat = [1, 4, 1, 5, 4, 2, 1, 2]</span>
-<span class="sd">    &gt;&gt;&gt; type(uniq(dat)) in (list, tuple)</span>
-<span class="sd">    False</span>
-
-<span class="sd">    &gt;&gt;&gt; list(uniq(dat))</span>
-<span class="sd">    [1, 4, 5, 2]</span>
-<span class="sd">    &gt;&gt;&gt; list(uniq(x for x in dat))</span>
-<span class="sd">    [1, 4, 5, 2]</span>
-<span class="sd">    &gt;&gt;&gt; list(uniq([[1], [2, 1], [1]]))</span>
-<span class="sd">    [[1], [2, 1], [1]]</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Tuple</span>
-
-    <span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="s">&#39;__getitem__&#39;</span><span class="p">):</span>
-        <span class="n">container</span> <span class="o">=</span> <span class="nb">list</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">container</span> <span class="o">=</span> <span class="nb">type</span><span class="p">(</span><span class="n">seq</span><span class="p">)</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">seen</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-        <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">seq</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">s</span> <span class="ow">in</span> <span class="n">seen</span> <span class="ow">or</span> <span class="n">seen</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">s</span><span class="p">)):</span>
-                <span class="k">yield</span> <span class="n">s</span>
-    <span class="k">except</span> <span class="ne">TypeError</span><span class="p">:</span>
-        <span class="c"># something was unhashable</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="s">&#39;__getitem__&#39;</span><span class="p">):</span>
-            <span class="k">yield</span> <span class="n">s</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="p">[</span><span class="n">s</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">result</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">seq</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">result</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">s</span> <span class="o">==</span> <span class="n">r</span><span class="p">:</span>
-                    <span class="k">break</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">yield</span> <span class="n">s</span>
-
-</div>
-<div class="viewcode-block" id="generate_bell"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.generate_bell">[docs]</a><span class="k">def</span> <span class="nf">generate_bell</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Generates the bell permutations of length ``n``.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import generate_bell</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import zeros, Matrix, factorial, pprint</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.core.compatibility import permutations</span>
-
-<span class="sd">    This is the sort of permutation used in the ringing of physical bells,</span>
-<span class="sd">    and does not produce permutations in lexicographical order. Rather, the</span>
-<span class="sd">    permutations differ from each other by exactly one inversion, and the</span>
-<span class="sd">    position at which the swapping occurs varies periodically in a simple</span>
-<span class="sd">    fashion. Consider the first few permutations of 4 elements generated</span>
-<span class="sd">    by ``permutations`` and ``generate_bell``:</span>
-
-<span class="sd">    &gt;&gt;&gt; list(permutations(list(range(4))))[:5]</span>
-<span class="sd">    [(0, 1, 2, 3), (0, 1, 3, 2), (0, 2, 1, 3), (0, 2, 3, 1), (0, 3, 1, 2)]</span>
-<span class="sd">    &gt;&gt;&gt; list(generate_bell(4))[:5]</span>
-<span class="sd">    [(0, 1, 2, 3), (1, 0, 2, 3), (1, 2, 0, 3), (1, 2, 3, 0), (2, 1, 3, 0)]</span>
-
-<span class="sd">    Notice how the 2nd and 3rd lexicographical permutations have 3 elements</span>
-<span class="sd">    out of place whereas each bell permutations always has only two</span>
-<span class="sd">    elements out of place relative to the previous permutation.</span>
-
-<span class="sd">    How the position of inversion varies across the elements can be seen</span>
-<span class="sd">    by tracing out where the 0 appears in the permutations:</span>
-
-<span class="sd">    &gt;&gt;&gt; m = zeros(4, 24)</span>
-<span class="sd">    &gt;&gt;&gt; for i, p in enumerate(generate_bell(4)):</span>
-<span class="sd">    ...     m[:, i] = Matrix(list(p))</span>
-<span class="sd">    &gt;&gt;&gt; m.print_nonzero(&#39;X&#39;)</span>
-<span class="sd">    [ XXXXXX  XXXXXX  XXXXXX ]</span>
-<span class="sd">    [X XXXX XX XXXX XX XXXX X]</span>
-<span class="sd">    [XX XX XXXX XX XXXX XX XX]</span>
-<span class="sd">    [XXX  XXXXXX  XXXXXX  XXX]</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    * http://en.wikipedia.org/wiki/Method_ringing</span>
-<span class="sd">    * http://stackoverflow.com/questions/4856615/recursive-permutation/4857018</span>
-<span class="sd">    * http://programminggeeks.com/bell-algorithm-for-permutation/</span>
-<span class="sd">    * http://en.wikipedia.org/wiki/</span>
-<span class="sd">      Steinhaus%E2%80%93Johnson%E2%80%93Trotter_algorithm</span>
-<span class="sd">    * Generating involutions, derangements, and relatives by ECO</span>
-<span class="sd">      Vincent Vajnovszki, DMTCS vol 1 issue 12, 2010</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.functions.combinatorial.factorials</span> <span class="kn">import</span> <span class="n">factorial</span>
-    <span class="n">pos</span> <span class="o">=</span> <span class="nb">dir</span> <span class="o">=</span> <span class="mi">1</span>
-    <span class="n">do</span> <span class="o">=</span> <span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-    <span class="k">yield</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-    <span class="n">do</span> <span class="o">-=</span> <span class="mi">1</span>
-    <span class="k">while</span> <span class="n">do</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">pos</span> <span class="o">&gt;=</span> <span class="n">n</span><span class="p">:</span>
-            <span class="nb">dir</span> <span class="o">=</span> <span class="o">-</span><span class="nb">dir</span>
-            <span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-        <span class="k">elif</span> <span class="n">pos</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="nb">dir</span> <span class="o">=</span> <span class="o">-</span><span class="nb">dir</span>
-            <span class="n">p</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">p</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">p</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">p</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">p</span><span class="p">[</span><span class="n">pos</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">p</span><span class="p">[</span><span class="n">pos</span><span class="p">]</span> <span class="o">=</span> <span class="n">p</span><span class="p">[</span><span class="n">pos</span><span class="p">],</span> <span class="n">p</span><span class="p">[</span><span class="n">pos</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-        <span class="n">pos</span> <span class="o">+=</span> <span class="nb">dir</span>
-        <span class="k">yield</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-        <span class="n">do</span> <span class="o">-=</span> <span class="mi">1</span>
-
-</div>
-<div class="viewcode-block" id="generate_involutions"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.generate_involutions">[docs]</a><span class="k">def</span> <span class="nf">generate_involutions</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Generates involutions.</span>
-
-<span class="sd">    An involution is a permutation that when multiplied</span>
-<span class="sd">    by itself equals the identity permutation. In this</span>
-<span class="sd">    implementation the involutions are generated using</span>
-<span class="sd">    Fixed Points.</span>
-
-<span class="sd">    Alternatively, an involution can be considered as</span>
-<span class="sd">    a permutation that does not contain any cycles with</span>
-<span class="sd">    a length that is greater than two.</span>
-
-<span class="sd">    Reference:</span>
-<span class="sd">    http://mathworld.wolfram.com/PermutationInvolution.html</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import generate_involutions</span>
-<span class="sd">    &gt;&gt;&gt; list(generate_involutions(3))</span>
-<span class="sd">    [(0, 1, 2), (0, 2, 1), (1, 0, 2), (2, 1, 0)]</span>
-<span class="sd">    &gt;&gt;&gt; len(list(generate_involutions(4)))</span>
-<span class="sd">    10</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">idx</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-    <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">permutations</span><span class="p">(</span><span class="n">idx</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">idx</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">p</span><span class="p">[</span><span class="n">p</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span> <span class="o">!=</span> <span class="n">i</span><span class="p">:</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">yield</span> <span class="n">p</span>
-
-</div>
-<div class="viewcode-block" id="generate_derangements"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.generate_derangements">[docs]</a><span class="k">def</span> <span class="nf">generate_derangements</span><span class="p">(</span><span class="n">perm</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Routine to generate unique derangements.</span>
-
-<span class="sd">    TODO: This will be rewritten to use the</span>
-<span class="sd">    ECO operator approach once the permutations</span>
-<span class="sd">    branch is in master.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import generate_derangements</span>
-<span class="sd">    &gt;&gt;&gt; list(generate_derangements([0, 1, 2]))</span>
-<span class="sd">    [[1, 2, 0], [2, 0, 1]]</span>
-<span class="sd">    &gt;&gt;&gt; list(generate_derangements([0, 1, 2, 3]))</span>
-<span class="sd">    [[1, 0, 3, 2], [1, 2, 3, 0], [1, 3, 0, 2], [2, 0, 3, 1], \</span>
-<span class="sd">    [2, 3, 0, 1], [2, 3, 1, 0], [3, 0, 1, 2], [3, 2, 0, 1], \</span>
-<span class="sd">    [3, 2, 1, 0]]</span>
-<span class="sd">    &gt;&gt;&gt; list(generate_derangements([0, 1, 1]))</span>
-<span class="sd">    []</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    sympy.functions.combinatorial.factorials.subfactorial</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="n">multiset_permutations</span><span class="p">(</span><span class="n">perm</span><span class="p">)</span>
-    <span class="n">indices</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">perm</span><span class="p">)))</span>
-    <span class="n">p0</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-    <span class="k">for</span> <span class="n">pi</span> <span class="ow">in</span> <span class="n">p</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">pi</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">!=</span> <span class="n">p0</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">indices</span><span class="p">):</span>
-            <span class="k">yield</span> <span class="n">pi</span>
-
-</div>
-<span class="nd">@deprecated</span><span class="p">(</span>
-    <span class="n">useinstead</span><span class="o">=</span><span class="s">&quot;bracelets&quot;</span><span class="p">,</span> <span class="n">deprecated_since_version</span><span class="o">=</span><span class="s">&quot;0.7.3&quot;</span><span class="p">)</span>
-<div class="viewcode-block" id="unrestricted_necklace"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.unrestricted_necklace">[docs]</a><span class="k">def</span> <span class="nf">unrestricted_necklace</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Wrapper to necklaces to return a free (unrestricted) necklace.&quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">necklaces</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">free</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="necklaces"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.necklaces">[docs]</a><span class="k">def</span> <span class="nf">necklaces</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">free</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A routine to generate necklaces that may (free=True) or may not</span>
-<span class="sd">    (free=False) be turned over to be viewed. The &quot;necklaces&quot; returned</span>
-<span class="sd">    are comprised of ``n`` integers (beads) with ``k`` different</span>
-<span class="sd">    values (colors). Only unique necklaces are returned.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import necklaces, bracelets</span>
-<span class="sd">    &gt;&gt;&gt; def show(s, i):</span>
-<span class="sd">    ...     return &#39;&#39;.join(s[j] for j in i)</span>
-
-<span class="sd">    The &quot;unrestricted necklace&quot; is sometimes also referred to as a</span>
-<span class="sd">    &quot;bracelet&quot; (an object that can be turned over, a sequence that can</span>
-<span class="sd">    be reversed) and the term &quot;necklace&quot; is used to imply a sequence</span>
-<span class="sd">    that cannot be reversed. So ACB == ABC for a bracelet (rotate and</span>
-<span class="sd">    reverse) while the two are different for a necklace since rotation</span>
-<span class="sd">    alone cannot make the two sequences the same.</span>
-
-<span class="sd">    (mnemonic: Bracelets can be viewed Backwards, but Not Necklaces.)</span>
-
-<span class="sd">    &gt;&gt;&gt; B = [show(&#39;ABC&#39;, i) for i in bracelets(3, 3)]</span>
-<span class="sd">    &gt;&gt;&gt; N = [show(&#39;ABC&#39;, i) for i in necklaces(3, 3)]</span>
-<span class="sd">    &gt;&gt;&gt; set(N) - set(B)</span>
-<span class="sd">    set([&#39;ACB&#39;])</span>
-
-<span class="sd">    &gt;&gt;&gt; list(necklaces(4, 2))</span>
-<span class="sd">    [(0, 0, 0, 0), (0, 0, 0, 1), (0, 0, 1, 1),</span>
-<span class="sd">     (0, 1, 0, 1), (0, 1, 1, 1), (1, 1, 1, 1)]</span>
-
-<span class="sd">    &gt;&gt;&gt; [show(&#39;.o&#39;, i) for i in bracelets(4, 2)]</span>
-<span class="sd">    [&#39;....&#39;, &#39;...o&#39;, &#39;..oo&#39;, &#39;.o.o&#39;, &#39;.ooo&#39;, &#39;oooo&#39;]</span>
-
-<span class="sd">    References</span>
-<span class="sd">    ==========</span>
-
-<span class="sd">    http://mathworld.wolfram.com/Necklace.html</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">uniq</span><span class="p">(</span><span class="n">minlex</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">directed</span><span class="o">=</span><span class="ow">not</span> <span class="n">free</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span>
-        <span class="n">variations</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="p">)),</span> <span class="n">n</span><span class="p">,</span> <span class="n">repetition</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="bracelets"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.bracelets">[docs]</a><span class="k">def</span> <span class="nf">bracelets</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Wrapper to necklaces to return a free (unrestricted) necklace.&quot;&quot;&quot;</span>
-    <span class="k">return</span> <span class="n">necklaces</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">free</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="generate_oriented_forest"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.generate_oriented_forest">[docs]</a><span class="k">def</span> <span class="nf">generate_oriented_forest</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    This algorithm generates oriented forests.</span>
-
-<span class="sd">    An oriented graph is a directed graph having no symmetric pair of directed</span>
-<span class="sd">    edges. A forest is an acyclic graph, i.e., it has no cycles. A forest can</span>
-<span class="sd">    also be described as a disjoint union of trees, which are graphs in which</span>
-<span class="sd">    any two vertices are connected by exactly one simple path.</span>
-
-<span class="sd">    Reference:</span>
-<span class="sd">    [1] T. Beyer and S.M. Hedetniemi: constant time generation of \</span>
-<span class="sd">        rooted trees, SIAM J. Computing Vol. 9, No. 4, November 1980</span>
-<span class="sd">    [2] http://stackoverflow.com/questions/1633833/oriented-forest-taocp-algorithm-in-python</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import generate_oriented_forest</span>
-<span class="sd">    &gt;&gt;&gt; list(generate_oriented_forest(4))</span>
-<span class="sd">    [[0, 1, 2, 3], [0, 1, 2, 2], [0, 1, 2, 1], [0, 1, 2, 0], \</span>
-<span class="sd">    [0, 1, 1, 1], [0, 1, 1, 0], [0, 1, 0, 1], [0, 1, 0, 0], [0, 0, 0, 0]]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">P</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="k">yield</span> <span class="n">P</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-        <span class="k">if</span> <span class="n">P</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">P</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">P</span><span class="p">[</span><span class="n">n</span><span class="p">]]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">P</span><span class="p">[</span><span class="n">p</span><span class="p">]</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">target</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">p</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span>
-                    <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">p</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-                        <span class="k">if</span> <span class="n">P</span><span class="p">[</span><span class="n">q</span><span class="p">]</span> <span class="o">==</span> <span class="n">target</span><span class="p">:</span>
-                            <span class="k">break</span>
-                    <span class="n">offset</span> <span class="o">=</span> <span class="n">p</span> <span class="o">-</span> <span class="n">q</span>
-                    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                        <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="n">offset</span><span class="p">]</span>
-                    <span class="k">break</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">break</span>
-
-</div>
-<div class="viewcode-block" id="minlex"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.minlex">[docs]</a><span class="k">def</span> <span class="nf">minlex</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="n">directed</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">is_set</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">small</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a tuple where the smallest element appears first; if</span>
-<span class="sd">    ``directed`` is True (default) then the order is preserved, otherwise</span>
-<span class="sd">    the sequence will be reversed if that gives a smaller ordering.</span>
-
-<span class="sd">    If every element appears only once then is_set can be set to True</span>
-<span class="sd">    for more efficient processing.</span>
-
-<span class="sd">    If the smallest element is known at the time of calling, it can be</span>
-<span class="sd">    passed and the calculation of the smallest element will be omitted.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.combinatorics.polyhedron import minlex</span>
-<span class="sd">    &gt;&gt;&gt; minlex((1, 2, 0))</span>
-<span class="sd">    (0, 1, 2)</span>
-<span class="sd">    &gt;&gt;&gt; minlex((1, 0, 2))</span>
-<span class="sd">    (0, 2, 1)</span>
-<span class="sd">    &gt;&gt;&gt; minlex((1, 0, 2), directed=False)</span>
-<span class="sd">    (0, 1, 2)</span>
-
-<span class="sd">    &gt;&gt;&gt; minlex(&#39;11010011000&#39;, directed=True)</span>
-<span class="sd">    &#39;00011010011&#39;</span>
-<span class="sd">    &gt;&gt;&gt; minlex(&#39;11010011000&#39;, directed=False)</span>
-<span class="sd">    &#39;00011001011&#39;</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">is_str</span> <span class="o">=</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="nb">str</span><span class="p">)</span>
-    <span class="n">seq</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">seq</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">small</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">small</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">seq</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">is_set</span><span class="p">:</span>
-        <span class="n">i</span> <span class="o">=</span> <span class="n">seq</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">small</span><span class="p">)</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">directed</span><span class="p">:</span>
-            <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">seq</span><span class="p">)</span>
-            <span class="n">p</span> <span class="o">=</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="n">n</span>
-            <span class="n">m</span> <span class="o">=</span> <span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="n">n</span>
-            <span class="k">if</span> <span class="n">seq</span><span class="p">[</span><span class="n">p</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">seq</span><span class="p">[</span><span class="n">m</span><span class="p">]:</span>
-                <span class="n">seq</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">seq</span><span class="p">))</span>
-                <span class="n">i</span> <span class="o">=</span> <span class="n">n</span> <span class="o">-</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="n">i</span><span class="p">:</span>
-            <span class="n">seq</span> <span class="o">=</span> <span class="n">rotate_left</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-        <span class="n">best</span> <span class="o">=</span> <span class="n">seq</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">count</span> <span class="o">=</span> <span class="n">seq</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="n">small</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">count</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">directed</span><span class="p">:</span>
-            <span class="n">best</span> <span class="o">=</span> <span class="n">rotate_left</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="n">seq</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">small</span><span class="p">))</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># if not directed, and not a set, we can&#39;t just</span>
-            <span class="c"># pass this off to minlex with is_set True since</span>
-            <span class="c"># peeking at the neighbor may not be sufficient to</span>
-            <span class="c"># make the decision so we continue...</span>
-            <span class="n">best</span> <span class="o">=</span> <span class="n">seq</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">count</span><span class="p">):</span>
-                <span class="n">seq</span> <span class="o">=</span> <span class="n">rotate_left</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="n">seq</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">small</span><span class="p">,</span> <span class="n">count</span> <span class="o">!=</span> <span class="mi">1</span><span class="p">))</span>
-                <span class="k">if</span> <span class="n">seq</span> <span class="o">&lt;</span> <span class="n">best</span><span class="p">:</span>
-                    <span class="n">best</span> <span class="o">=</span> <span class="n">seq</span>
-                <span class="c"># it&#39;s cheaper to rotate now rather than search</span>
-                <span class="c"># again for these in reversed order so we test</span>
-                <span class="c"># the reverse now</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">directed</span><span class="p">:</span>
-                    <span class="n">seq</span> <span class="o">=</span> <span class="n">rotate_left</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-                    <span class="n">seq</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">seq</span><span class="p">))</span>
-                    <span class="k">if</span> <span class="n">seq</span> <span class="o">&lt;</span> <span class="n">best</span><span class="p">:</span>
-                        <span class="n">best</span> <span class="o">=</span> <span class="n">seq</span>
-                    <span class="n">seq</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">seq</span><span class="p">))</span>
-                    <span class="n">seq</span> <span class="o">=</span> <span class="n">rotate_right</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="c"># common return</span>
-    <span class="k">if</span> <span class="n">is_str</span><span class="p">:</span>
-        <span class="k">return</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">best</span><span class="p">)</span>
-    <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">best</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="runs"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.runs">[docs]</a><span class="k">def</span> <span class="nf">runs</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="n">op</span><span class="o">=</span><span class="n">gt</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Group the sequence into lists in which successive elements</span>
-<span class="sd">    all compare the same with the comparison operator, ``op``:</span>
-<span class="sd">    op(seq[i + 1], seq[i]) is True from all elements in a run.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import runs</span>
-<span class="sd">    &gt;&gt;&gt; from operator import ge</span>
-<span class="sd">    &gt;&gt;&gt; runs([0, 1, 2, 2, 1, 4, 3, 2, 2])</span>
-<span class="sd">    [[0, 1, 2], [2], [1, 4], [3], [2], [2]]</span>
-<span class="sd">    &gt;&gt;&gt; runs([0, 1, 2, 2, 1, 4, 3, 2, 2], op=ge)</span>
-<span class="sd">    [[0, 1, 2, 2], [1, 4], [3], [2, 2]]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">cycles</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="n">seq</span> <span class="o">=</span> <span class="nb">iter</span><span class="p">(</span><span class="n">seq</span><span class="p">)</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">run</span> <span class="o">=</span> <span class="p">[</span><span class="nb">next</span><span class="p">(</span><span class="n">seq</span><span class="p">)]</span>
-    <span class="k">except</span> <span class="ne">StopIteration</span><span class="p">:</span>
-        <span class="k">return</span> <span class="p">[]</span>
-    <span class="k">while</span> <span class="bp">True</span><span class="p">:</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">ei</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">seq</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">StopIteration</span><span class="p">:</span>
-            <span class="k">break</span>
-        <span class="k">if</span> <span class="n">op</span><span class="p">(</span><span class="n">ei</span><span class="p">,</span> <span class="n">run</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]):</span>
-            <span class="n">run</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">ei</span><span class="p">)</span>
-            <span class="k">continue</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">cycles</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">run</span><span class="p">)</span>
-            <span class="n">run</span> <span class="o">=</span> <span class="p">[</span><span class="n">ei</span><span class="p">]</span>
-    <span class="k">if</span> <span class="n">run</span><span class="p">:</span>
-        <span class="n">cycles</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">run</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">cycles</span>
-
-</div>
-<div class="viewcode-block" id="kbins"><a class="viewcode-back" href="../../../modules/utilities/iterables.html#sympy.utilities.iterables.kbins">[docs]</a><span class="k">def</span> <span class="nf">kbins</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">ordered</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return sequence ``l`` partitioned into ``k`` bins.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.iterables import kbins</span>
-
-<span class="sd">    The default is to give the items in the same order, but grouped</span>
-<span class="sd">    into k partitions without any reordering:</span>
-
-<span class="sd">    &gt;&gt;&gt; for p in kbins(list(range(5)), 2):</span>
-<span class="sd">    ...     print(p)</span>
-<span class="sd">    ...</span>
-<span class="sd">    [[0], [1, 2, 3, 4]]</span>
-<span class="sd">    [[0, 1], [2, 3, 4]]</span>
-<span class="sd">    [[0, 1, 2], [3, 4]]</span>
-<span class="sd">    [[0, 1, 2, 3], [4]]</span>
-
-<span class="sd">    The ``ordered`` flag which is either None (to give the simple partition</span>
-<span class="sd">    of the the elements) or is a 2 digit integer indicating whether the order of</span>
-<span class="sd">    the bins and the order of the items in the bins matters. Given::</span>
-
-<span class="sd">        A = [[0], [1, 2]]</span>
-<span class="sd">        B = [[1, 2], [0]]</span>
-<span class="sd">        C = [[2, 1], [0]]</span>
-<span class="sd">        D = [[0], [2, 1]]</span>
-
-<span class="sd">    the following values for ``ordered`` have the shown meanings::</span>
-
-<span class="sd">        00 means A == B == C == D</span>
-<span class="sd">        01 means A == B</span>
-<span class="sd">        10 means A == D</span>
-<span class="sd">        11 means A == A</span>
-
-<span class="sd">    &gt;&gt;&gt; for ordered in [None, 0, 1, 10, 11]:</span>
-<span class="sd">    ...     print(&#39;ordered =&#39;, ordered)</span>
-<span class="sd">    ...     for p in kbins(list(range(3)), 2, ordered=ordered):</span>
-<span class="sd">    ...         print(&#39;    &#39;, p)</span>
-<span class="sd">    ...</span>
-<span class="sd">    ordered = None</span>
-<span class="sd">         [[0], [1, 2]]</span>
-<span class="sd">         [[0, 1], [2]]</span>
-<span class="sd">    ordered = 0</span>
-<span class="sd">         [[0, 1], [2]]</span>
-<span class="sd">         [[0, 2], [1]]</span>
-<span class="sd">         [[0], [1, 2]]</span>
-<span class="sd">    ordered = 1</span>
-<span class="sd">         [[0], [1, 2]]</span>
-<span class="sd">         [[0], [2, 1]]</span>
-<span class="sd">         [[1], [0, 2]]</span>
-<span class="sd">         [[1], [2, 0]]</span>
-<span class="sd">         [[2], [0, 1]]</span>
-<span class="sd">         [[2], [1, 0]]</span>
-<span class="sd">    ordered = 10</span>
-<span class="sd">         [[0, 1], [2]]</span>
-<span class="sd">         [[2], [0, 1]]</span>
-<span class="sd">         [[0, 2], [1]]</span>
-<span class="sd">         [[1], [0, 2]]</span>
-<span class="sd">         [[0], [1, 2]]</span>
-<span class="sd">         [[1, 2], [0]]</span>
-<span class="sd">    ordered = 11</span>
-<span class="sd">         [[0], [1, 2]]</span>
-<span class="sd">         [[0, 1], [2]]</span>
-<span class="sd">         [[0], [2, 1]]</span>
-<span class="sd">         [[0, 2], [1]]</span>
-<span class="sd">         [[1], [0, 2]]</span>
-<span class="sd">         [[1, 0], [2]]</span>
-<span class="sd">         [[1], [2, 0]]</span>
-<span class="sd">         [[1, 2], [0]]</span>
-<span class="sd">         [[2], [0, 1]]</span>
-<span class="sd">         [[2, 0], [1]]</span>
-<span class="sd">         [[2], [1, 0]]</span>
-<span class="sd">         [[2, 1], [0]]</span>
-
-<span class="sd">    See Also</span>
-<span class="sd">    ========</span>
-<span class="sd">    partitions, multiset_partitions</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">def</span> <span class="nf">partition</span><span class="p">(</span><span class="n">lista</span><span class="p">,</span> <span class="n">bins</span><span class="p">):</span>
-        <span class="c">#  EnricoGiampieri&#39;s partition generator from</span>
-        <span class="c">#  http://stackoverflow.com/questions/13131491/</span>
-        <span class="c">#  partition-n-items-into-k-bins-in-python-lazily</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">lista</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">bins</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">yield</span> <span class="p">[</span><span class="n">lista</span><span class="p">]</span>
-        <span class="k">elif</span> <span class="nb">len</span><span class="p">(</span><span class="n">lista</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">bins</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">lista</span><span class="p">)):</span>
-                <span class="k">for</span> <span class="n">part</span> <span class="ow">in</span> <span class="n">partition</span><span class="p">(</span><span class="n">lista</span><span class="p">[</span><span class="n">i</span><span class="p">:],</span> <span class="n">bins</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-                    <span class="k">if</span> <span class="nb">len</span><span class="p">([</span><span class="n">lista</span><span class="p">[:</span><span class="n">i</span><span class="p">]]</span> <span class="o">+</span> <span class="n">part</span><span class="p">)</span> <span class="o">==</span> <span class="n">bins</span><span class="p">:</span>
-                        <span class="k">yield</span> <span class="p">[</span><span class="n">lista</span><span class="p">[:</span><span class="n">i</span><span class="p">]]</span> <span class="o">+</span> <span class="n">part</span>
-
-    <span class="k">if</span> <span class="n">ordered</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">partition</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-            <span class="k">yield</span> <span class="n">p</span>
-    <span class="k">elif</span> <span class="n">ordered</span> <span class="o">==</span> <span class="mi">11</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">pl</span> <span class="ow">in</span> <span class="n">multiset_permutations</span><span class="p">(</span><span class="n">l</span><span class="p">):</span>
-            <span class="n">pl</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">pl</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">partition</span><span class="p">(</span><span class="n">pl</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-                <span class="k">yield</span> <span class="n">p</span>
-    <span class="k">elif</span> <span class="n">ordered</span> <span class="o">==</span> <span class="mo">00</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">multiset_partitions</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-            <span class="k">yield</span> <span class="n">p</span>
-    <span class="k">elif</span> <span class="n">ordered</span> <span class="o">==</span> <span class="mi">10</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">multiset_partitions</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-            <span class="k">for</span> <span class="n">perm</span> <span class="ow">in</span> <span class="n">permutations</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
-                <span class="k">yield</span> <span class="nb">list</span><span class="p">(</span><span class="n">perm</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="n">ordered</span> <span class="o">==</span> <span class="mi">0</span><span class="n">o1</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">kgot</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">partitions</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">l</span><span class="p">),</span> <span class="n">k</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">kgot</span> <span class="o">!=</span> <span class="n">k</span><span class="p">:</span>
-                <span class="k">continue</span>
-            <span class="k">for</span> <span class="n">li</span> <span class="ow">in</span> <span class="n">multiset_permutations</span><span class="p">(</span><span class="n">l</span><span class="p">):</span>
-                <span class="n">rv</span> <span class="o">=</span> <span class="p">[]</span>
-                <span class="n">i</span> <span class="o">=</span> <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span>
-                <span class="n">li</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">li</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">size</span><span class="p">,</span> <span class="n">multiplicity</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-                    <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">multiplicity</span><span class="p">):</span>
-                        <span class="n">j</span> <span class="o">=</span> <span class="n">i</span> <span class="o">+</span> <span class="n">size</span>
-                        <span class="n">rv</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">li</span><span class="p">[</span><span class="n">i</span><span class="p">:</span> <span class="n">j</span><span class="p">])</span>
-                        <span class="n">i</span> <span class="o">=</span> <span class="n">j</span>
-                <span class="k">yield</span> <span class="n">rv</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
-            <span class="s">&#39;ordered must be one of 00, 01, 10 or 11, not </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">ordered</span><span class="p">)</span></div>
-</pre></div>
-
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-  <h1>Source code for sympy.utilities.lambdify</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">This module provides convenient functions to transform sympy expressions to</span>
-<span class="sd">lambda functions which can be used to calculate numerical values very fast.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-
-<span class="kn">from</span> <span class="nn">sympy.external</span> <span class="kn">import</span> <span class="n">import_module</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">is_sequence</span><span class="p">,</span> <span class="n">iterable</span>
-
-<span class="kn">import</span> <span class="nn">inspect</span>
-
-<span class="c"># These are the namespaces the lambda functions will use.</span>
-<span class="n">MATH</span> <span class="o">=</span> <span class="p">{}</span>
-<span class="n">MPMATH</span> <span class="o">=</span> <span class="p">{}</span>
-<span class="n">NUMPY</span> <span class="o">=</span> <span class="p">{}</span>
-<span class="n">SYMPY</span> <span class="o">=</span> <span class="p">{}</span>
-
-<span class="c"># Default namespaces, letting us define translations that can&#39;t be defined</span>
-<span class="c"># by simple variable maps, like I =&gt; 1j</span>
-<span class="c"># These are separate from the names above because the above names are modified</span>
-<span class="c"># throughout this file, whereas these should remain unmodified.</span>
-<span class="n">MATH_DEFAULT</span> <span class="o">=</span> <span class="p">{}</span>
-<span class="n">MPMATH_DEFAULT</span> <span class="o">=</span> <span class="p">{}</span>
-<span class="n">NUMPY_DEFAULT</span> <span class="o">=</span> <span class="p">{</span><span class="s">&quot;I&quot;</span><span class="p">:</span> <span class="mi">1j</span><span class="p">}</span>
-<span class="n">SYMPY_DEFAULT</span> <span class="o">=</span> <span class="p">{}</span>
-
-<span class="c"># Mappings between sympy and other modules function names.</span>
-<span class="n">MATH_TRANSLATIONS</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="s">&quot;Abs&quot;</span><span class="p">:</span> <span class="s">&quot;fabs&quot;</span><span class="p">,</span>
-    <span class="s">&quot;ceiling&quot;</span><span class="p">:</span> <span class="s">&quot;ceil&quot;</span><span class="p">,</span>
-    <span class="s">&quot;E&quot;</span><span class="p">:</span> <span class="s">&quot;e&quot;</span><span class="p">,</span>
-    <span class="s">&quot;ln&quot;</span><span class="p">:</span> <span class="s">&quot;log&quot;</span><span class="p">,</span>
-<span class="p">}</span>
-
-<span class="n">MPMATH_TRANSLATIONS</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="s">&quot;ceiling&quot;</span><span class="p">:</span> <span class="s">&quot;ceil&quot;</span><span class="p">,</span>
-    <span class="s">&quot;chebyshevt&quot;</span><span class="p">:</span> <span class="s">&quot;chebyt&quot;</span><span class="p">,</span>
-    <span class="s">&quot;chebyshevu&quot;</span><span class="p">:</span> <span class="s">&quot;chebyu&quot;</span><span class="p">,</span>
-    <span class="s">&quot;E&quot;</span><span class="p">:</span> <span class="s">&quot;e&quot;</span><span class="p">,</span>
-    <span class="s">&quot;I&quot;</span><span class="p">:</span> <span class="s">&quot;j&quot;</span><span class="p">,</span>
-    <span class="s">&quot;ln&quot;</span><span class="p">:</span> <span class="s">&quot;log&quot;</span><span class="p">,</span>
-    <span class="c">#&quot;lowergamma&quot;:&quot;lower_gamma&quot;,</span>
-    <span class="s">&quot;oo&quot;</span><span class="p">:</span> <span class="s">&quot;inf&quot;</span><span class="p">,</span>
-    <span class="c">#&quot;uppergamma&quot;:&quot;upper_gamma&quot;,</span>
-    <span class="s">&quot;LambertW&quot;</span><span class="p">:</span> <span class="s">&quot;lambertw&quot;</span><span class="p">,</span>
-    <span class="s">&quot;Matrix&quot;</span><span class="p">:</span> <span class="s">&quot;matrix&quot;</span><span class="p">,</span>
-    <span class="s">&quot;MutableDenseMatrix&quot;</span><span class="p">:</span> <span class="s">&quot;matrix&quot;</span><span class="p">,</span>
-    <span class="s">&quot;conjugate&quot;</span><span class="p">:</span> <span class="s">&quot;conj&quot;</span><span class="p">,</span>
-    <span class="s">&quot;dirichlet_eta&quot;</span><span class="p">:</span> <span class="s">&quot;altzeta&quot;</span><span class="p">,</span>
-    <span class="s">&quot;Ei&quot;</span><span class="p">:</span> <span class="s">&quot;ei&quot;</span><span class="p">,</span>
-    <span class="s">&quot;Shi&quot;</span><span class="p">:</span> <span class="s">&quot;shi&quot;</span><span class="p">,</span>
-    <span class="s">&quot;Chi&quot;</span><span class="p">:</span> <span class="s">&quot;chi&quot;</span><span class="p">,</span>
-    <span class="s">&quot;Si&quot;</span><span class="p">:</span> <span class="s">&quot;si&quot;</span><span class="p">,</span>
-    <span class="s">&quot;Ci&quot;</span><span class="p">:</span> <span class="s">&quot;ci&quot;</span>
-<span class="p">}</span>
-
-<span class="n">NUMPY_TRANSLATIONS</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="s">&quot;Abs&quot;</span><span class="p">:</span> <span class="s">&quot;abs&quot;</span><span class="p">,</span>
-    <span class="s">&quot;acos&quot;</span><span class="p">:</span> <span class="s">&quot;arccos&quot;</span><span class="p">,</span>
-    <span class="s">&quot;acosh&quot;</span><span class="p">:</span> <span class="s">&quot;arccosh&quot;</span><span class="p">,</span>
-    <span class="s">&quot;arg&quot;</span><span class="p">:</span> <span class="s">&quot;angle&quot;</span><span class="p">,</span>
-    <span class="s">&quot;asin&quot;</span><span class="p">:</span> <span class="s">&quot;arcsin&quot;</span><span class="p">,</span>
-    <span class="s">&quot;asinh&quot;</span><span class="p">:</span> <span class="s">&quot;arcsinh&quot;</span><span class="p">,</span>
-    <span class="s">&quot;atan&quot;</span><span class="p">:</span> <span class="s">&quot;arctan&quot;</span><span class="p">,</span>
-    <span class="s">&quot;atan2&quot;</span><span class="p">:</span> <span class="s">&quot;arctan2&quot;</span><span class="p">,</span>
-    <span class="s">&quot;atanh&quot;</span><span class="p">:</span> <span class="s">&quot;arctanh&quot;</span><span class="p">,</span>
-    <span class="s">&quot;ceiling&quot;</span><span class="p">:</span> <span class="s">&quot;ceil&quot;</span><span class="p">,</span>
-    <span class="s">&quot;E&quot;</span><span class="p">:</span> <span class="s">&quot;e&quot;</span><span class="p">,</span>
-    <span class="s">&quot;im&quot;</span><span class="p">:</span> <span class="s">&quot;imag&quot;</span><span class="p">,</span>
-    <span class="s">&quot;ln&quot;</span><span class="p">:</span> <span class="s">&quot;log&quot;</span><span class="p">,</span>
-    <span class="s">&quot;Matrix&quot;</span><span class="p">:</span> <span class="s">&quot;matrix&quot;</span><span class="p">,</span>
-    <span class="s">&quot;MutableDenseMatrix&quot;</span><span class="p">:</span> <span class="s">&quot;matrix&quot;</span><span class="p">,</span>
-    <span class="s">&quot;Max&quot;</span><span class="p">:</span> <span class="s">&quot;amax&quot;</span><span class="p">,</span>
-    <span class="s">&quot;Min&quot;</span><span class="p">:</span> <span class="s">&quot;amin&quot;</span><span class="p">,</span>
-    <span class="s">&quot;oo&quot;</span><span class="p">:</span> <span class="s">&quot;inf&quot;</span><span class="p">,</span>
-    <span class="s">&quot;re&quot;</span><span class="p">:</span> <span class="s">&quot;real&quot;</span><span class="p">,</span>
-<span class="p">}</span>
-
-<span class="c"># Available modules:</span>
-<span class="n">MODULES</span> <span class="o">=</span> <span class="p">{</span>
-    <span class="s">&quot;math&quot;</span><span class="p">:</span> <span class="p">(</span><span class="n">MATH</span><span class="p">,</span> <span class="n">MATH_DEFAULT</span><span class="p">,</span> <span class="n">MATH_TRANSLATIONS</span><span class="p">,</span> <span class="p">(</span><span class="s">&quot;from math import *&quot;</span><span class="p">,)),</span>
-    <span class="s">&quot;mpmath&quot;</span><span class="p">:</span> <span class="p">(</span><span class="n">MPMATH</span><span class="p">,</span> <span class="n">MPMATH_DEFAULT</span><span class="p">,</span> <span class="n">MPMATH_TRANSLATIONS</span><span class="p">,</span> <span class="p">(</span><span class="s">&quot;from sympy.mpmath import *&quot;</span><span class="p">,)),</span>
-    <span class="s">&quot;numpy&quot;</span><span class="p">:</span> <span class="p">(</span><span class="n">NUMPY</span><span class="p">,</span> <span class="n">NUMPY_DEFAULT</span><span class="p">,</span> <span class="n">NUMPY_TRANSLATIONS</span><span class="p">,</span> <span class="p">(</span><span class="s">&quot;import_module(&#39;numpy&#39;)&quot;</span><span class="p">,)),</span>
-    <span class="s">&quot;sympy&quot;</span><span class="p">:</span> <span class="p">(</span><span class="n">SYMPY</span><span class="p">,</span> <span class="n">SYMPY_DEFAULT</span><span class="p">,</span> <span class="p">{},</span> <span class="p">(</span>
-        <span class="s">&quot;from sympy.functions import *&quot;</span><span class="p">,</span>
-        <span class="s">&quot;from sympy.matrices import *&quot;</span><span class="p">,</span>
-        <span class="s">&quot;from sympy import Integral, pi, oo, nan, zoo, E, I&quot;</span><span class="p">,)),</span>
-<span class="p">}</span>
-
-
-<span class="k">def</span> <span class="nf">_import</span><span class="p">(</span><span class="n">module</span><span class="p">,</span> <span class="nb">reload</span><span class="o">=</span><span class="s">&quot;False&quot;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Creates a global translation dictionary for module.</span>
-
-<span class="sd">    The argument module has to be one of the following strings: &quot;math&quot;,</span>
-<span class="sd">    &quot;mpmath&quot;, &quot;numpy&quot;, &quot;sympy&quot;.</span>
-<span class="sd">    These dictionaries map names of python functions to their equivalent in</span>
-<span class="sd">    other modules.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">try</span><span class="p">:</span>
-        <span class="n">namespace</span><span class="p">,</span> <span class="n">namespace_default</span><span class="p">,</span> <span class="n">translations</span><span class="p">,</span> <span class="n">import_commands</span> <span class="o">=</span> <span class="n">MODULES</span><span class="p">[</span>
-            <span class="n">module</span><span class="p">]</span>
-    <span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">NameError</span><span class="p">(</span>
-            <span class="s">&quot;&#39;</span><span class="si">%s</span><span class="s">&#39; module can&#39;t be used for lambdification&quot;</span> <span class="o">%</span> <span class="n">module</span><span class="p">)</span>
-
-    <span class="c"># Clear namespace or exit</span>
-    <span class="k">if</span> <span class="n">namespace</span> <span class="o">!=</span> <span class="n">namespace_default</span><span class="p">:</span>
-        <span class="c"># The namespace was already generated, don&#39;t do it again if not forced.</span>
-        <span class="k">if</span> <span class="nb">reload</span><span class="p">:</span>
-            <span class="n">namespace</span><span class="o">.</span><span class="n">clear</span><span class="p">()</span>
-            <span class="n">namespace</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">namespace_default</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span>
-
-    <span class="k">for</span> <span class="n">import_command</span> <span class="ow">in</span> <span class="n">import_commands</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">import_command</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39;import_module&#39;</span><span class="p">):</span>
-            <span class="n">module</span> <span class="o">=</span> <span class="nb">eval</span><span class="p">(</span><span class="n">import_command</span><span class="p">)</span>
-
-            <span class="k">if</span> <span class="n">module</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">namespace</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">module</span><span class="o">.</span><span class="n">__dict__</span><span class="p">)</span>
-                <span class="k">continue</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">exec</span><span class="p">(</span><span class="n">import_command</span><span class="p">,</span> <span class="p">{},</span> <span class="n">namespace</span><span class="p">)</span>
-                <span class="k">continue</span>
-            <span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>
-                <span class="k">pass</span>
-
-        <span class="k">raise</span> <span class="ne">ImportError</span><span class="p">(</span>
-            <span class="s">&quot;can&#39;t import &#39;</span><span class="si">%s</span><span class="s">&#39; with &#39;</span><span class="si">%s</span><span class="s">&#39; command&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">module</span><span class="p">,</span> <span class="n">import_command</span><span class="p">))</span>
-
-    <span class="c"># Add translated names to namespace</span>
-    <span class="k">for</span> <span class="n">sympyname</span><span class="p">,</span> <span class="n">translation</span> <span class="ow">in</span> <span class="n">translations</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
-        <span class="n">namespace</span><span class="p">[</span><span class="n">sympyname</span><span class="p">]</span> <span class="o">=</span> <span class="n">namespace</span><span class="p">[</span><span class="n">translation</span><span class="p">]</span>
-
-
-<div class="viewcode-block" id="lambdify"><a class="viewcode-back" href="../../../modules/utilities/lambdify.html#sympy.utilities.lambdify.lambdify">[docs]</a><span class="k">def</span> <span class="nf">lambdify</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">modules</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">printer</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">use_imps</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns a lambda function for fast calculation of numerical values.</span>
-
-<span class="sd">    If not specified differently by the user, SymPy functions are replaced as</span>
-<span class="sd">    far as possible by either python-math, numpy (if available) or mpmath</span>
-<span class="sd">    functions - exactly in this order. To change this behavior, the &quot;modules&quot;</span>
-<span class="sd">    argument can be used. It accepts:</span>
-
-<span class="sd">     - the strings &quot;math&quot;, &quot;mpmath&quot;, &quot;numpy&quot;, &quot;sympy&quot;</span>
-<span class="sd">     - any modules (e.g. math)</span>
-<span class="sd">     - dictionaries that map names of sympy functions to arbitrary functions</span>
-<span class="sd">     - lists that contain a mix of the arguments above, with higher priority</span>
-<span class="sd">       given to entries appearing first.</span>
-
-<span class="sd">    Usage</span>
-<span class="sd">    =====</span>
-
-<span class="sd">    (1) Use one of the provided modules:</span>
-
-<span class="sd">        &gt;&gt; f = lambdify(x, sin(x), &quot;math&quot;)</span>
-
-<span class="sd">        Attention: Functions that are not in the math module will throw a name</span>
-<span class="sd">                   error when the lambda function is evaluated! So this would</span>
-<span class="sd">                   be better:</span>
-
-<span class="sd">        &gt;&gt; f = lambdify(x, sin(x)*gamma(x), (&quot;math&quot;, &quot;mpmath&quot;, &quot;sympy&quot;))</span>
-
-<span class="sd">    (2) Use some other module:</span>
-
-<span class="sd">        &gt;&gt; import numpy</span>
-<span class="sd">        &gt;&gt; f = lambdify((x,y), tan(x*y), numpy)</span>
-
-<span class="sd">        Attention: There are naming differences between numpy and sympy. So if</span>
-<span class="sd">                   you simply take the numpy module, e.g. sympy.atan will not be</span>
-<span class="sd">                   translated to numpy.arctan. Use the modified module instead</span>
-<span class="sd">                   by passing the string &quot;numpy&quot;:</span>
-
-<span class="sd">        &gt;&gt; f = lambdify((x,y), tan(x*y), &quot;numpy&quot;)</span>
-<span class="sd">        &gt;&gt; f(1, 2)</span>
-<span class="sd">        -2.18503986326</span>
-<span class="sd">        &gt;&gt; from numpy import array</span>
-<span class="sd">        &gt;&gt; f(array([1, 2, 3]), array([2, 3, 5]))</span>
-<span class="sd">        [-2.18503986 -0.29100619 -0.8559934 ]</span>
-
-<span class="sd">    (3) Use own dictionaries:</span>
-
-<span class="sd">        &gt;&gt; def my_cool_function(x): ...</span>
-<span class="sd">        &gt;&gt; dic = {&quot;sin&quot; : my_cool_function}</span>
-<span class="sd">        &gt;&gt; f = lambdify(x, sin(x), dic)</span>
-
-<span class="sd">        Now f would look like:</span>
-
-<span class="sd">        &gt;&gt; lambda x: my_cool_function(x)</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.lambdify import implemented_function, lambdify</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import sqrt, sin, Matrix</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Function</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-
-<span class="sd">    &gt;&gt;&gt; f = lambdify(x, x**2)</span>
-<span class="sd">    &gt;&gt;&gt; f(2)</span>
-<span class="sd">    4</span>
-<span class="sd">    &gt;&gt;&gt; f = lambdify((x, y, z), [z, y, x])</span>
-<span class="sd">    &gt;&gt;&gt; f(1,2,3)</span>
-<span class="sd">    [3, 2, 1]</span>
-<span class="sd">    &gt;&gt;&gt; f = lambdify(x, sqrt(x))</span>
-<span class="sd">    &gt;&gt;&gt; f(4)</span>
-<span class="sd">    2.0</span>
-<span class="sd">    &gt;&gt;&gt; f = lambdify((x, y), sin(x*y)**2)</span>
-<span class="sd">    &gt;&gt;&gt; f(0, 5)</span>
-<span class="sd">    0.0</span>
-<span class="sd">    &gt;&gt;&gt; f = lambdify((x, y), Matrix((x, x + y)).T, modules=&#39;sympy&#39;)</span>
-<span class="sd">    &gt;&gt;&gt; f(1, 2)</span>
-<span class="sd">    [1, 3]</span>
-
-<span class="sd">    Functions present in `expr` can also carry their own numerical</span>
-<span class="sd">    implementations, in a callable attached to the ``_imp_``</span>
-<span class="sd">    attribute.  Usually you attach this using the</span>
-<span class="sd">    ``implemented_function`` factory:</span>
-
-<span class="sd">    &gt;&gt;&gt; f = implemented_function(Function(&#39;f&#39;), lambda x: x+1)</span>
-<span class="sd">    &gt;&gt;&gt; func = lambdify(x, f(x))</span>
-<span class="sd">    &gt;&gt;&gt; func(4)</span>
-<span class="sd">    5</span>
-
-<span class="sd">    ``lambdify`` always prefers ``_imp_`` implementations to implementations</span>
-<span class="sd">    in other namespaces, unless the ``use_imps`` input parameter is False.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.core.symbol</span> <span class="kn">import</span> <span class="n">Symbol</span>
-
-    <span class="c"># If the user hasn&#39;t specified any modules, use what is available.</span>
-    <span class="k">if</span> <span class="n">modules</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="c"># Use either numpy (if available) or python.math where possible.</span>
-        <span class="c"># XXX: This leads to different behaviour on different systems and</span>
-        <span class="c">#      might be the reason for irreproducible errors.</span>
-        <span class="n">modules</span> <span class="o">=</span> <span class="p">[</span><span class="s">&quot;math&quot;</span><span class="p">,</span> <span class="s">&quot;mpmath&quot;</span><span class="p">,</span> <span class="s">&quot;sympy&quot;</span><span class="p">]</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">_import</span><span class="p">(</span><span class="s">&quot;numpy&quot;</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>
-            <span class="k">pass</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">modules</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="s">&quot;numpy&quot;</span><span class="p">)</span>
-
-    <span class="c"># Get the needed namespaces.</span>
-    <span class="n">namespaces</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="c"># First find any function implementations</span>
-    <span class="k">if</span> <span class="n">use_imps</span><span class="p">:</span>
-        <span class="n">namespaces</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">_imp_namespace</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-    <span class="c"># Check for dict before iterating</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">modules</span><span class="p">,</span> <span class="p">(</span><span class="nb">dict</span><span class="p">,</span> <span class="nb">str</span><span class="p">))</span> <span class="ow">or</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">modules</span><span class="p">,</span> <span class="s">&#39;__iter__&#39;</span><span class="p">):</span>
-        <span class="n">namespaces</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">modules</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">namespaces</span> <span class="o">+=</span> <span class="nb">list</span><span class="p">(</span><span class="n">modules</span><span class="p">)</span>
-    <span class="c"># fill namespace with first having highest priority</span>
-    <span class="n">namespace</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">namespaces</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
-        <span class="n">buf</span> <span class="o">=</span> <span class="n">_get_namespace</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-        <span class="n">namespace</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">buf</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&quot;atoms&quot;</span><span class="p">):</span>
-        <span class="c">#Try if you can extract symbols from the expression.</span>
-        <span class="c">#Move on if expr.atoms in not implemented.</span>
-        <span class="n">syms</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Symbol</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">syms</span><span class="p">:</span>
-            <span class="n">namespace</span><span class="o">.</span><span class="n">update</span><span class="p">({</span><span class="nb">str</span><span class="p">(</span><span class="n">term</span><span class="p">):</span> <span class="n">term</span><span class="p">})</span>
-
-    <span class="c"># Create lambda function.</span>
-    <span class="n">lstr</span> <span class="o">=</span> <span class="n">lambdastr</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">printer</span><span class="o">=</span><span class="n">printer</span><span class="p">)</span>
-    <span class="k">return</span> <span class="nb">eval</span><span class="p">(</span><span class="n">lstr</span><span class="p">,</span> <span class="n">namespace</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_get_namespace</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    This is used by _lambdify to parse its arguments.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-        <span class="n">_import</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">MODULES</span><span class="p">[</span><span class="n">m</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="nb">dict</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">m</span>
-    <span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="s">&quot;__dict__&quot;</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">m</span><span class="o">.</span><span class="n">__dict__</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;Argument must be either a string, dict or module but it is: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">m</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="lambdastr"><a class="viewcode-back" href="../../../modules/utilities/lambdify.html#sympy.utilities.lambdify.lambdastr">[docs]</a><span class="k">def</span> <span class="nf">lambdastr</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="n">printer</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns a string that can be evaluated to a lambda function.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.lambdify import lambdastr</span>
-<span class="sd">    &gt;&gt;&gt; lambdastr(x, x**2)</span>
-<span class="sd">    &#39;lambda x: (x**2)&#39;</span>
-<span class="sd">    &gt;&gt;&gt; lambdastr((x,y,z), [z,y,x])</span>
-<span class="sd">    &#39;lambda x,y,z: ([z, y, x])&#39;</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">printer</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">inspect</span><span class="o">.</span><span class="n">isfunction</span><span class="p">(</span><span class="n">printer</span><span class="p">):</span>
-            <span class="n">lambdarepr</span> <span class="o">=</span> <span class="n">printer</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">inspect</span><span class="o">.</span><span class="n">isclass</span><span class="p">(</span><span class="n">printer</span><span class="p">):</span>
-                <span class="n">lambdarepr</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">expr</span><span class="p">:</span> <span class="n">printer</span><span class="p">()</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">lambdarepr</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">expr</span><span class="p">:</span> <span class="n">printer</span><span class="o">.</span><span class="n">doprint</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="c">#XXX: This has to be done here because of circular imports</span>
-        <span class="kn">from</span> <span class="nn">sympy.printing.lambdarepr</span> <span class="kn">import</span> <span class="n">lambdarepr</span>
-
-    <span class="c"># Transform everything to strings.</span>
-    <span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">DeferredVector</span>
-    <span class="n">expr</span> <span class="o">=</span> <span class="n">lambdarepr</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-        <span class="k">pass</span>
-    <span class="k">elif</span> <span class="n">iterable</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="n">DeferredVector</span><span class="p">):</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="s">&quot;,&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">args</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">args</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="s">&quot;lambda </span><span class="si">%s</span><span class="s">: (</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_imp_namespace</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">namespace</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Return namespace dict with function implementations</span>
-
-<span class="sd">    We need to search for functions in anything that can be thrown at</span>
-<span class="sd">    us - that is - anything that could be passed as `expr`.  Examples</span>
-<span class="sd">    include sympy expressions, as well as tuples, lists and dicts that may</span>
-<span class="sd">    contain sympy expressions.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    expr : object</span>
-<span class="sd">       Something passed to lambdify, that will generate valid code from</span>
-<span class="sd">       ``str(expr)``.</span>
-<span class="sd">    namespace : None or mapping</span>
-<span class="sd">       Namespace to fill.  None results in new empty dict</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    -------</span>
-<span class="sd">    namespace : dict</span>
-<span class="sd">       dict with keys of implemented function names within `expr` and</span>
-<span class="sd">       corresponding values being the numerical implementation of</span>
-<span class="sd">       function</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    --------</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.lambdify import implemented_function, _imp_namespace</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Function</span>
-<span class="sd">    &gt;&gt;&gt; f = implemented_function(Function(&#39;f&#39;), lambda x: x+1)</span>
-<span class="sd">    &gt;&gt;&gt; g = implemented_function(Function(&#39;g&#39;), lambda x: x*10)</span>
-<span class="sd">    &gt;&gt;&gt; namespace = _imp_namespace(f(g(x)))</span>
-<span class="sd">    &gt;&gt;&gt; sorted(namespace.keys())</span>
-<span class="sd">    [&#39;f&#39;, &#39;g&#39;]</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># Delayed import to avoid circular imports</span>
-    <span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">FunctionClass</span>
-    <span class="k">if</span> <span class="n">namespace</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">namespace</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="c"># tuples, lists, dicts are valid expressions</span>
-    <span class="k">if</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="p">:</span>
-            <span class="n">_imp_namespace</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">namespace</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">namespace</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="nb">dict</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="n">val</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">expr</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-            <span class="c"># functions can be in dictionary keys</span>
-            <span class="n">_imp_namespace</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">namespace</span><span class="p">)</span>
-            <span class="n">_imp_namespace</span><span class="p">(</span><span class="n">val</span><span class="p">,</span> <span class="n">namespace</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">namespace</span>
-    <span class="c"># sympy expressions may be Functions themselves</span>
-    <span class="n">func</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&#39;func&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">FunctionClass</span><span class="p">):</span>
-        <span class="n">imp</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="s">&#39;_imp_&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">imp</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">name</span> <span class="o">=</span> <span class="n">expr</span><span class="o">.</span><span class="n">func</span><span class="o">.</span><span class="n">__name__</span>
-            <span class="k">if</span> <span class="n">name</span> <span class="ow">in</span> <span class="n">namespace</span> <span class="ow">and</span> <span class="n">namespace</span><span class="p">[</span><span class="n">name</span><span class="p">]</span> <span class="o">!=</span> <span class="n">imp</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;We found more than one &#39;</span>
-                                 <span class="s">&#39;implementation with name &#39;</span>
-                                 <span class="s">&#39;&quot;</span><span class="si">%s</span><span class="s">&quot;&#39;</span> <span class="o">%</span> <span class="n">name</span><span class="p">)</span>
-            <span class="n">namespace</span><span class="p">[</span><span class="n">name</span><span class="p">]</span> <span class="o">=</span> <span class="n">imp</span>
-    <span class="c"># and / or they may take Functions as arguments</span>
-    <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&#39;args&#39;</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="n">_imp_namespace</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">namespace</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">namespace</span>
-
-
-<div class="viewcode-block" id="implemented_function"><a class="viewcode-back" href="../../../modules/utilities/lambdify.html#sympy.utilities.lambdify.implemented_function">[docs]</a><span class="k">def</span> <span class="nf">implemented_function</span><span class="p">(</span><span class="n">symfunc</span><span class="p">,</span> <span class="n">implementation</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot; Add numerical ``implementation`` to function ``symfunc``.</span>
-
-<span class="sd">    ``symfunc`` can be an ``UndefinedFunction`` instance, or a name string.</span>
-<span class="sd">    In the latter case we create an ``UndefinedFunction`` instance with that</span>
-<span class="sd">    name.</span>
-
-<span class="sd">    Be aware that this is a quick workaround, not a general method to create</span>
-<span class="sd">    special symbolic functions. If you want to create a symbolic function to be</span>
-<span class="sd">    used by all the machinery of sympy you should subclass the ``Function``</span>
-<span class="sd">    class.</span>
-
-<span class="sd">    Parameters</span>
-<span class="sd">    ----------</span>
-<span class="sd">    symfunc : ``str`` or ``UndefinedFunction`` instance</span>
-<span class="sd">       If ``str``, then create new ``UndefinedFunction`` with this as</span>
-<span class="sd">       name.  If `symfunc` is a sympy function, attach implementation to it.</span>
-<span class="sd">    implementation : callable</span>
-<span class="sd">       numerical implementation to be called by ``evalf()`` or ``lambdify``</span>
-
-<span class="sd">    Returns</span>
-<span class="sd">    -------</span>
-<span class="sd">    afunc : sympy.FunctionClass instance</span>
-<span class="sd">       function with attached implementation</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    --------</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x, y, z</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.lambdify import lambdify, implemented_function</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import Function</span>
-<span class="sd">    &gt;&gt;&gt; f = implemented_function(Function(&#39;f&#39;), lambda x: x+1)</span>
-<span class="sd">    &gt;&gt;&gt; lam_f = lambdify(x, f(x))</span>
-<span class="sd">    &gt;&gt;&gt; lam_f(4)</span>
-<span class="sd">    5</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># Delayed import to avoid circular imports</span>
-    <span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">UndefinedFunction</span>
-    <span class="c"># if name, create function to hold implementation</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">symfunc</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-        <span class="n">symfunc</span> <span class="o">=</span> <span class="n">UndefinedFunction</span><span class="p">(</span><span class="n">symfunc</span><span class="p">)</span>
-    <span class="k">elif</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">symfunc</span><span class="p">,</span> <span class="n">UndefinedFunction</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;symfunc should be either a string or&#39;</span>
-                         <span class="s">&#39; an UndefinedFunction instance.&#39;</span><span class="p">)</span>
-    <span class="c"># We need to attach as a method because symfunc will be a class</span>
-    <span class="n">symfunc</span><span class="o">.</span><span class="n">_imp_</span> <span class="o">=</span> <span class="nb">staticmethod</span><span class="p">(</span><span class="n">implementation</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">symfunc</span></div>
-</pre></div>
-
-          </div>
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diff --git a/dev-py3k/_modules/sympy/utilities/memoization.html b/dev-py3k/_modules/sympy/utilities/memoization.html
deleted file mode 100644
index 71d5a1cd380..00000000000
--- a/dev-py3k/_modules/sympy/utilities/memoization.html
+++ /dev/null
@@ -1,171 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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-      };
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-             >modules</a> |</li>
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-        <div class="bodywrapper">
-          <div class="body">
-            
-  <h1>Source code for sympy.utilities.memoization</h1><div class="highlight"><pre>
-<div class="viewcode-block" id="recurrence_memo"><a class="viewcode-back" href="../../../modules/utilities/memoization.html#sympy.utilities.memoization.recurrence_memo">[docs]</a><span class="k">def</span> <span class="nf">recurrence_memo</span><span class="p">(</span><span class="n">initial</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Memo decorator for sequences defined by recurrence</span>
-
-<span class="sd">    See usage examples e.g. in the specfun/combinatorial module</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">cache</span> <span class="o">=</span> <span class="n">initial</span>
-
-    <span class="k">def</span> <span class="nf">decorator</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">def</span> <span class="nf">g</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-            <span class="n">L</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">cache</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;=</span> <span class="n">L</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">cache</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="n">cache</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">cache</span><span class="p">))</span>
-            <span class="k">return</span> <span class="n">cache</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">return</span> <span class="n">g</span>
-    <span class="k">return</span> <span class="n">decorator</span>
-
-</div>
-<div class="viewcode-block" id="assoc_recurrence_memo"><a class="viewcode-back" href="../../../modules/utilities/memoization.html#sympy.utilities.memoization.assoc_recurrence_memo">[docs]</a><span class="k">def</span> <span class="nf">assoc_recurrence_memo</span><span class="p">(</span><span class="n">base_seq</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Memo decorator for associated sequences defined by recurrence starting from base</span>
-
-<span class="sd">    base_seq(n) -- callable to get base sequence elements</span>
-
-<span class="sd">    XXX works only for Pn0 = base_seq(0) cases</span>
-<span class="sd">    XXX works only for m &lt;= n cases</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">cache</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">decorator</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-        <span class="k">def</span> <span class="nf">g</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-            <span class="n">L</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">cache</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;</span> <span class="n">L</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">cache</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="n">m</span><span class="p">]</span>
-
-            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                <span class="c"># get base sequence</span>
-                <span class="n">F_i0</span> <span class="o">=</span> <span class="n">base_seq</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                <span class="n">F_i_cache</span> <span class="o">=</span> <span class="p">[</span><span class="n">F_i0</span><span class="p">]</span>
-                <span class="n">cache</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">F_i_cache</span><span class="p">)</span>
-
-                <span class="c"># XXX only works for m &lt;= n cases</span>
-                <span class="c"># generate assoc sequence</span>
-                <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-                    <span class="n">F_ij</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">cache</span><span class="p">)</span>
-                    <span class="n">F_i_cache</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">F_ij</span><span class="p">)</span>
-
-            <span class="k">return</span> <span class="n">cache</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="n">m</span><span class="p">]</span>
-
-        <span class="k">return</span> <span class="n">g</span>
-    <span class="k">return</span> <span class="n">decorator</span></div>
-</pre></div>
-
-          </div>
-        </div>
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diff --git a/dev-py3k/_modules/sympy/utilities/misc.html b/dev-py3k/_modules/sympy/utilities/misc.html
deleted file mode 100644
index 7da9cf3cca6..00000000000
--- a/dev-py3k/_modules/sympy/utilities/misc.html
+++ /dev/null
@@ -1,232 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
-
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-    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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-    <title>sympy.utilities.misc &mdash; SymPy 0.7.2-git documentation</title>
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-             >modules</a> |</li>
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-             >modules</a> |</li>
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-        <div class="bodywrapper">
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-            
-  <h1>Source code for sympy.utilities.misc</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Miscellaneous stuff that doesn&#39;t really fit anywhere else.&quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">textwrap</span> <span class="kn">import</span> <span class="n">fill</span><span class="p">,</span> <span class="n">dedent</span>
-
-<span class="c"># if you use</span>
-<span class="c"># filldedent(&#39;&#39;&#39;</span>
-<span class="c">#             the text&#39;&#39;&#39;)</span>
-<span class="c"># a space will be put before the first line because dedent will</span>
-<span class="c"># put a \n as the first line and fill replaces \n with spaces</span>
-<span class="c"># so we strip off any leading and trailing \n since printed wrapped</span>
-<span class="c"># text should not have leading or trailing spaces.</span>
-<span class="n">filldedent</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">s</span><span class="p">,</span> <span class="n">w</span><span class="o">=</span><span class="mi">70</span><span class="p">:</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span> <span class="o">+</span> <span class="n">fill</span><span class="p">(</span><span class="n">dedent</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">s</span><span class="p">))</span><span class="o">.</span><span class="n">strip</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">),</span> <span class="n">width</span><span class="o">=</span><span class="n">w</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="rawlines"><a class="viewcode-back" href="../../../modules/utilities/misc.html#sympy.utilities.misc.rawlines">[docs]</a><span class="k">def</span> <span class="nf">rawlines</span><span class="p">(</span><span class="n">s</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return a cut-and-pastable string that, when printed, is equivalent</span>
-<span class="sd">    to the input. The string returned is formatted so it can be indented</span>
-<span class="sd">    nicely within tests; in some cases it is wrapped in the dedent</span>
-<span class="sd">    function which has to be imported from textwrap.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    Note: because there are characters in the examples below that need</span>
-<span class="sd">    to be escaped because they are themselves within a triple quoted</span>
-<span class="sd">    docstring, expressions below look more complicated than they would</span>
-<span class="sd">    be if they were printed in an interpreter window.</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.misc import rawlines</span>
-<span class="sd">    &gt;&gt;&gt; from sympy import TableForm</span>
-<span class="sd">    &gt;&gt;&gt; s = str(TableForm([[1, 10]], headings=(None, [&#39;a&#39;, &#39;bee&#39;])))</span>
-<span class="sd">    &gt;&gt;&gt; print(rawlines(s)) # the \\ appears as \ when printed</span>
-<span class="sd">    (</span>
-<span class="sd">        &#39;a bee\\n&#39;</span>
-<span class="sd">        &#39;-----\\n&#39;</span>
-<span class="sd">        &#39;1 10 &#39;</span>
-<span class="sd">    )</span>
-<span class="sd">    &gt;&gt;&gt; print(rawlines(&#39;&#39;&#39;this</span>
-<span class="sd">    ... that&#39;&#39;&#39;))</span>
-<span class="sd">    dedent(&#39;&#39;&#39;\\</span>
-<span class="sd">        this</span>
-<span class="sd">        that&#39;&#39;&#39;)</span>
-
-<span class="sd">    &gt;&gt;&gt; print(rawlines(&#39;&#39;&#39;this</span>
-<span class="sd">    ... that</span>
-<span class="sd">    ... &#39;&#39;&#39;))</span>
-<span class="sd">    dedent(&#39;&#39;&#39;\\</span>
-<span class="sd">        this</span>
-<span class="sd">        that</span>
-<span class="sd">        &#39;&#39;&#39;)</span>
-
-<span class="sd">    &gt;&gt;&gt; s = \&quot;\&quot;\&quot;this</span>
-<span class="sd">    ... is a triple &#39;&#39;&#39;</span>
-<span class="sd">    ... \&quot;\&quot;\&quot;</span>
-<span class="sd">    &gt;&gt;&gt; print(rawlines(s))</span>
-<span class="sd">    dedent(\&quot;\&quot;\&quot;\\</span>
-<span class="sd">        this</span>
-<span class="sd">        is a triple &#39;&#39;&#39;</span>
-<span class="sd">        \&quot;\&quot;\&quot;)</span>
-
-<span class="sd">    &gt;&gt;&gt; print(rawlines(&#39;&#39;&#39;this</span>
-<span class="sd">    ... that</span>
-<span class="sd">    ...     &#39;&#39;&#39;))</span>
-<span class="sd">    (</span>
-<span class="sd">        &#39;this\\n&#39;</span>
-<span class="sd">        &#39;that\\n&#39;</span>
-<span class="sd">        &#39;    &#39;</span>
-<span class="sd">    )</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">lines</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">lines</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">repr</span><span class="p">(</span><span class="n">lines</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-    <span class="n">triple</span> <span class="o">=</span> <span class="p">[</span><span class="s">&quot;&#39;&#39;&#39;&quot;</span> <span class="ow">in</span> <span class="n">s</span><span class="p">,</span> <span class="s">&#39;&quot;&quot;&quot;&#39;</span> <span class="ow">in</span> <span class="n">s</span><span class="p">]</span>
-    <span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">li</span><span class="o">.</span><span class="n">endswith</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">)</span> <span class="k">for</span> <span class="n">li</span> <span class="ow">in</span> <span class="n">lines</span><span class="p">)</span> <span class="ow">or</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">&#39;</span> <span class="ow">in</span> <span class="n">s</span> <span class="ow">or</span> <span class="nb">all</span><span class="p">(</span><span class="n">triple</span><span class="p">):</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="p">[</span><span class="s">&quot;(&quot;</span><span class="p">]</span>
-        <span class="c"># add on the newlines</span>
-        <span class="n">trailing</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">endswith</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">)</span>
-        <span class="n">last</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">lines</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">li</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">lines</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">i</span> <span class="o">!=</span> <span class="n">last</span> <span class="ow">or</span> <span class="n">trailing</span><span class="p">:</span>
-                <span class="n">rv</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">repr</span><span class="p">(</span><span class="n">li</span><span class="p">)[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">n</span><span class="se">\&#39;</span><span class="s">&#39;</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">rv</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">repr</span><span class="p">(</span><span class="n">li</span><span class="p">))</span>
-        <span class="k">return</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">    &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">rv</span><span class="p">)</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">)&#39;</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">    &#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">lines</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">triple</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
-            <span class="k">return</span> <span class="s">&#39;dedent(&quot;&quot;&quot;</span><span class="se">\\\n</span><span class="s">    </span><span class="si">%s</span><span class="s">&quot;&quot;&quot;)&#39;</span> <span class="o">%</span> <span class="n">rv</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="s">&quot;dedent(&#39;&#39;&#39;</span><span class="se">\\\n</span><span class="s">    </span><span class="si">%s</span><span class="s">&#39;&#39;&#39;)&quot;</span> <span class="o">%</span> <span class="n">rv</span>
-</div>
-<span class="kn">import</span> <span class="nn">sys</span>
-<span class="n">size</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">sys</span><span class="p">,</span> <span class="s">&quot;maxint&quot;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-<span class="k">if</span> <span class="n">size</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>  <span class="c"># Python 3 doesn&#39;t have maxint</span>
-    <span class="n">size</span> <span class="o">=</span> <span class="n">sys</span><span class="o">.</span><span class="n">maxsize</span>
-<span class="k">if</span> <span class="n">size</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="o">**</span><span class="mi">32</span><span class="p">:</span>
-    <span class="n">ARCH</span> <span class="o">=</span> <span class="s">&quot;64-bit&quot;</span>
-<span class="k">else</span><span class="p">:</span>
-    <span class="n">ARCH</span> <span class="o">=</span> <span class="s">&quot;32-bit&quot;</span>
-
-<span class="c"># Python 2.5 does not have sys.flags (it doesn&#39;t have hash randomization either)</span>
-<span class="n">HASH_RANDOMIZATION</span> <span class="o">=</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">sys</span><span class="p">,</span> <span class="s">&#39;flags&#39;</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">sys</span><span class="o">.</span><span class="n">flags</span><span class="p">,</span>
-                                                       <span class="s">&#39;hash_randomization&#39;</span><span class="p">,</span>
-                                                       <span class="bp">False</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="debug"><a class="viewcode-back" href="../../../modules/utilities/misc.html#sympy.utilities.misc.debug">[docs]</a><span class="k">def</span> <span class="nf">debug</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Print ``*args`` if SYMPY_DEBUG is True, else do nothing.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">SYMPY_DEBUG</span>
-    <span class="k">if</span> <span class="n">SYMPY_DEBUG</span><span class="p">:</span>
-        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">args</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">end</span><span class="o">=</span><span class="s">&#39; &#39;</span><span class="p">)</span>
-        <span class="k">print</span><span class="p">()</span></div>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
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-<div id="searchbox" style="display: none">
-  <h3>Quick search</h3>
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diff --git a/dev-py3k/_modules/sympy/utilities/pkgdata.html b/dev-py3k/_modules/sympy/utilities/pkgdata.html
deleted file mode 100644
index 136e8913db7..00000000000
--- a/dev-py3k/_modules/sympy/utilities/pkgdata.html
+++ /dev/null
@@ -1,173 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
-
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-<html xmlns="http://www.w3.org/1999/xhtml">
-  <head>
-    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
-    
-    <title>sympy.utilities.pkgdata &mdash; SymPy 0.7.2-git documentation</title>
-    
-    <link rel="stylesheet" href="../../../_static/default.css" type="text/css" />
-    <link rel="stylesheet" href="../../../_static/pygments.css" type="text/css" />
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-    
-    <script type="text/javascript">
-      var DOCUMENTATION_OPTIONS = {
-        URL_ROOT:    '../../../',
-        VERSION:     '0.7.2-git',
-        COLLAPSE_INDEX: false,
-        FILE_SUFFIX: '.html',
-        HAS_SOURCE:  true
-      };
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-    <link rel="top" title="SymPy 0.7.2-git documentation" href="../../../index.html" />
-    <link rel="up" title="sympy" href="../../sympy.html" /> 
-  </head>
-  <body>
-    <div class="related">
-      <h3>Navigation</h3>
-      <ul>
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-        <li class="right" >
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-             >modules</a> |</li>
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-             >modules</a> |</li>
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-        <div class="bodywrapper">
-          <div class="body">
-            
-  <h1>Source code for sympy.utilities.pkgdata</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">pkgdata is a simple, extensible way for a package to acquire data file</span>
-<span class="sd">resources.</span>
-
-<span class="sd">The getResource function is equivalent to the standard idioms, such as</span>
-<span class="sd">the following minimal implementation::</span>
-
-<span class="sd">    import sys, os</span>
-
-<span class="sd">    def getResource(identifier, pkgname=__name__):</span>
-<span class="sd">        pkgpath = os.path.dirname(sys.modules[pkgname].__file__)</span>
-<span class="sd">        path = os.path.join(pkgpath, identifier)</span>
-<span class="sd">        return file(os.path.normpath(path), mode=&#39;rb&#39;)</span>
-
-<span class="sd">When a __loader__ is present on the module given by __name__, it will defer</span>
-<span class="sd">getResource to its get_data implementation and return it as a file-like</span>
-<span class="sd">object (such as StringIO).</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">import</span> <span class="nn">sys</span>
-<span class="kn">import</span> <span class="nn">os</span>
-<span class="kn">from</span> <span class="nn">io</span> <span class="kn">import</span> <span class="n">StringIO</span>
-
-
-<div class="viewcode-block" id="get_resource"><a class="viewcode-back" href="../../../modules/utilities/pkgdata.html#sympy.utilities.pkgdata.get_resource">[docs]</a><span class="k">def</span> <span class="nf">get_resource</span><span class="p">(</span><span class="n">identifier</span><span class="p">,</span> <span class="n">pkgname</span><span class="o">=</span><span class="n">__name__</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Acquire a readable object for a given package name and identifier.</span>
-<span class="sd">    An IOError will be raised if the resource can not be found.</span>
-
-<span class="sd">    For example::</span>
-
-<span class="sd">        mydata = get_esource(&#39;mypkgdata.jpg&#39;).read()</span>
-
-<span class="sd">    Note that the package name must be fully qualified, if given, such</span>
-<span class="sd">    that it would be found in sys.modules.</span>
-
-<span class="sd">    In some cases, getResource will return a real file object.  In that</span>
-<span class="sd">    case, it may be useful to use its name attribute to get the path</span>
-<span class="sd">    rather than use it as a file-like object.  For example, you may</span>
-<span class="sd">    be handing data off to a C API.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="n">mod</span> <span class="o">=</span> <span class="n">sys</span><span class="o">.</span><span class="n">modules</span><span class="p">[</span><span class="n">pkgname</span><span class="p">]</span>
-    <span class="n">fn</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">mod</span><span class="p">,</span> <span class="s">&#39;__file__&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">fn</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">IOError</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%r</span><span class="s"> has no __file__!&quot;</span><span class="p">)</span>
-    <span class="n">path</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">dirname</span><span class="p">(</span><span class="n">fn</span><span class="p">),</span> <span class="n">identifier</span><span class="p">)</span>
-    <span class="n">loader</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">mod</span><span class="p">,</span> <span class="s">&#39;__loader__&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">loader</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">data</span> <span class="o">=</span> <span class="n">loader</span><span class="o">.</span><span class="n">get_data</span><span class="p">(</span><span class="n">path</span><span class="p">)</span>
-        <span class="k">except</span> <span class="ne">IOError</span><span class="p">:</span>
-            <span class="k">pass</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">StringIO</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
-    <span class="k">return</span> <span class="nb">file</span><span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">normpath</span><span class="p">(</span><span class="n">path</span><span class="p">),</span> <span class="s">&#39;rb&#39;</span><span class="p">)</span></div>
-</pre></div>
-
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-        <div class="sphinxsidebarwrapper">
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-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
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-  <h3>Quick search</h3>
-    <form class="search" action="../../../search.html" method="get">
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diff --git a/dev-py3k/_modules/sympy/utilities/pytest.html b/dev-py3k/_modules/sympy/utilities/pytest.html
deleted file mode 100644
index 4475bb475ee..00000000000
--- a/dev-py3k/_modules/sympy/utilities/pytest.html
+++ /dev/null
@@ -1,284 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
-
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-<html xmlns="http://www.w3.org/1999/xhtml">
-  <head>
-    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
-    
-    <title>sympy.utilities.pytest &mdash; SymPy 0.7.2-git documentation</title>
-    
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-  <h1>Source code for sympy.utilities.pytest</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;py.test hacks to support XFAIL/XPASS&quot;&quot;&quot;</span>
-
-<span class="kn">import</span> <span class="nn">sys</span>
-<span class="kn">import</span> <span class="nn">functools</span>
-<span class="kn">import</span> <span class="nn">collections</span>
-
-<span class="k">try</span><span class="p">:</span>
-    <span class="kn">import</span> <span class="nn">py</span>
-    <span class="kn">from</span> <span class="nn">py.test</span> <span class="kn">import</span> <span class="n">skip</span><span class="p">,</span> <span class="n">raises</span>
-    <span class="n">USE_PYTEST</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">sys</span><span class="p">,</span> <span class="s">&#39;_running_pytest&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-<span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>
-    <span class="n">USE_PYTEST</span> <span class="o">=</span> <span class="bp">False</span>
-
-<span class="k">if</span> <span class="ow">not</span> <span class="n">USE_PYTEST</span><span class="p">:</span>
-<div class="viewcode-block" id="raises"><a class="viewcode-back" href="../../../modules/utilities/pytest.html#sympy.utilities.pytest.raises">[docs]</a>    <span class="k">def</span> <span class="nf">raises</span><span class="p">(</span><span class="n">expectedException</span><span class="p">,</span> <span class="n">code</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Tests that ``code`` raises the exception ``expectedException``.</span>
-
-<span class="sd">        ``code`` may be a callable, such as a lambda expression or function</span>
-<span class="sd">        name.</span>
-
-<span class="sd">        If ``code`` is not given or None, ``raises`` will return a context</span>
-<span class="sd">        manager for use in ``with`` statements; the code to execute then</span>
-<span class="sd">        comes from the scope of the ``with``. The calling module must have</span>
-<span class="sd">        ``from __future__ import with_statement`` as its first code line</span>
-<span class="sd">        for compatibility with Python 2.5 in that case.</span>
-
-<span class="sd">        raises does nothing if the callable raises the expected exception,</span>
-<span class="sd">        otherwise it raises an AssertionError.</span>
-
-<span class="sd">        Examples</span>
-<span class="sd">        ========</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.utilities.pytest import raises</span>
-
-<span class="sd">        &gt;&gt;&gt; raises(ZeroDivisionError, lambda: 1/0)</span>
-<span class="sd">        &gt;&gt;&gt; raises(ZeroDivisionError, lambda: 1/2)</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">        ...</span>
-<span class="sd">        AssertionError: DID NOT RAISE</span>
-
-<span class="sd">        (Python 2.5&#39;s doctest cannot support with statements due to a bug in</span>
-<span class="sd">        its __import__ handling. That&#39;s why the following examples are</span>
-<span class="sd">        excluded from doctesting; the doctest: annotations can go once SymPy</span>
-<span class="sd">        stops Python 2.5 support.)</span>
-
-<span class="sd">        &gt;&gt;&gt; with raises(ZeroDivisionError): # doctest: +SKIP</span>
-<span class="sd">        ...     n = 1/0</span>
-<span class="sd">        &gt;&gt;&gt; with raises(ZeroDivisionError): # doctest: +SKIP</span>
-<span class="sd">        ...     n = 1/2</span>
-<span class="sd">        Traceback (most recent call last):</span>
-<span class="sd">        ...</span>
-<span class="sd">        AssertionError: DID NOT RAISE</span>
-
-<span class="sd">        Note that you cannot test multiple statements via</span>
-<span class="sd">        ``with raises``:</span>
-
-<span class="sd">        &gt;&gt;&gt; with raises(ZeroDivisionError): # doctest: +SKIP</span>
-<span class="sd">        ...     n = 1/0    # will execute and raise, aborting the ``with``</span>
-<span class="sd">        ...     n = 9999/0 # never executed</span>
-
-<span class="sd">        This is just what ``with`` is supposed to do: abort the</span>
-<span class="sd">        contained statement sequence at the first exception and let</span>
-<span class="sd">        the context manager deal with the exception.</span>
-
-<span class="sd">        To test multiple statements, you&#39;ll need a separate ``with``</span>
-<span class="sd">        for each:</span>
-
-<span class="sd">        &gt;&gt;&gt; with raises(ZeroDivisionError): # doctest: +SKIP</span>
-<span class="sd">        ...     n = 1/0    # will execute and raise</span>
-<span class="sd">        ... with raises(ZeroDivisionError):</span>
-<span class="sd">        ...     n = 9999/0 # will also execute and raise</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">code</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">RaisesContext</span><span class="p">(</span><span class="n">expectedException</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">code</span><span class="p">,</span> <span class="n">collections</span><span class="o">.</span><span class="n">Callable</span><span class="p">):</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">code</span><span class="p">()</span>
-            <span class="k">except</span> <span class="n">expectedException</span><span class="p">:</span>
-                <span class="k">return</span>
-            <span class="k">raise</span> <span class="ne">AssertionError</span><span class="p">(</span><span class="s">&quot;DID NOT RAISE&quot;</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">code</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span>
-                <span class="s">&#39;</span><span class="se">\&#39;</span><span class="s">raises(xxx, &quot;code&quot;)</span><span class="se">\&#39;</span><span class="s"> has been phased out; &#39;</span>
-                <span class="s">&#39;change </span><span class="se">\&#39;</span><span class="s">raises(xxx, &quot;expression&quot;)</span><span class="se">\&#39;</span><span class="s"> &#39;</span>
-                <span class="s">&#39;to </span><span class="se">\&#39;</span><span class="s">raises(xxx, lambda: expression)</span><span class="se">\&#39;</span><span class="s">, &#39;</span>
-                <span class="s">&#39;</span><span class="se">\&#39;</span><span class="s">raises(xxx, &quot;statement&quot;)</span><span class="se">\&#39;</span><span class="s"> &#39;</span>
-                <span class="s">&#39;to </span><span class="se">\&#39;</span><span class="s">with raises(xxx): statement</span><span class="se">\&#39;</span><span class="s">&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span>
-                <span class="s">&#39;raises() expects a callable for the 2nd argument.&#39;</span><span class="p">)</span>
-</div>
-    <span class="k">class</span> <span class="nc">RaisesContext</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-        <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">expectedException</span><span class="p">):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">expectedException</span> <span class="o">=</span> <span class="n">expectedException</span>
-
-        <span class="k">def</span> <span class="nf">__enter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="k">def</span> <span class="nf">__exit__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">exc_type</span><span class="p">,</span> <span class="n">exc_value</span><span class="p">,</span> <span class="n">traceback</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">exc_type</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">AssertionError</span><span class="p">(</span><span class="s">&quot;DID NOT RAISE&quot;</span><span class="p">)</span>
-            <span class="k">return</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">exc_type</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">expectedException</span><span class="p">)</span>
-
-    <span class="k">class</span> <span class="nc">XFail</span><span class="p">(</span><span class="ne">Exception</span><span class="p">):</span>
-        <span class="k">pass</span>
-
-    <span class="k">class</span> <span class="nc">XPass</span><span class="p">(</span><span class="ne">Exception</span><span class="p">):</span>
-        <span class="k">pass</span>
-
-    <span class="k">class</span> <span class="nc">Skipped</span><span class="p">(</span><span class="ne">Exception</span><span class="p">):</span>
-        <span class="k">pass</span>
-
-    <span class="k">def</span> <span class="nf">XFAIL</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
-        <span class="k">def</span> <span class="nf">wrapper</span><span class="p">():</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">func</span><span class="p">()</span>
-            <span class="k">except</span> <span class="ne">Exception</span> <span class="k">as</span> <span class="n">e</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">sys</span><span class="o">.</span><span class="n">version_info</span><span class="p">[:</span><span class="mi">2</span><span class="p">]</span> <span class="o">&lt;</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">6</span><span class="p">):</span>
-                    <span class="n">message</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="s">&#39;message&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">message</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">message</span> <span class="o">!=</span> <span class="s">&quot;Timeout&quot;</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="n">XFail</span><span class="p">(</span><span class="n">func</span><span class="o">.</span><span class="n">__name__</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="n">Skipped</span><span class="p">(</span><span class="s">&quot;Timeout&quot;</span><span class="p">)</span>
-            <span class="k">raise</span> <span class="n">XPass</span><span class="p">(</span><span class="n">func</span><span class="o">.</span><span class="n">__name__</span><span class="p">)</span>
-
-        <span class="n">wrapper</span> <span class="o">=</span> <span class="n">functools</span><span class="o">.</span><span class="n">update_wrapper</span><span class="p">(</span><span class="n">wrapper</span><span class="p">,</span> <span class="n">func</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">wrapper</span>
-
-    <span class="k">def</span> <span class="nf">skip</span><span class="p">(</span><span class="nb">str</span><span class="p">):</span>
-        <span class="k">raise</span> <span class="n">Skipped</span><span class="p">(</span><span class="nb">str</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">SKIP</span><span class="p">(</span><span class="n">reason</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;Similar to :func:`skip`, but this is a decorator. &quot;&quot;&quot;</span>
-        <span class="k">def</span> <span class="nf">wrapper</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
-            <span class="k">def</span> <span class="nf">func_wrapper</span><span class="p">():</span>
-                <span class="k">raise</span> <span class="n">Skipped</span><span class="p">(</span><span class="n">reason</span><span class="p">)</span>
-
-            <span class="n">func_wrapper</span> <span class="o">=</span> <span class="n">functools</span><span class="o">.</span><span class="n">update_wrapper</span><span class="p">(</span><span class="n">func_wrapper</span><span class="p">,</span> <span class="n">func</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">func_wrapper</span>
-
-        <span class="k">return</span> <span class="n">wrapper</span>
-
-    <span class="k">def</span> <span class="nf">slow</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
-        <span class="n">func</span><span class="o">.</span><span class="n">_slow</span> <span class="o">=</span> <span class="bp">True</span>
-
-        <span class="k">def</span> <span class="nf">func_wrapper</span><span class="p">():</span>
-            <span class="n">func</span><span class="p">()</span>
-
-        <span class="n">func_wrapper</span> <span class="o">=</span> <span class="n">functools</span><span class="o">.</span><span class="n">update_wrapper</span><span class="p">(</span><span class="n">func_wrapper</span><span class="p">,</span> <span class="n">func</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">func_wrapper</span>
-
-<span class="k">else</span><span class="p">:</span>
-    <span class="n">XFAIL</span> <span class="o">=</span> <span class="n">py</span><span class="o">.</span><span class="n">test</span><span class="o">.</span><span class="n">mark</span><span class="o">.</span><span class="n">xfail</span>
-    <span class="n">slow</span> <span class="o">=</span> <span class="n">py</span><span class="o">.</span><span class="n">test</span><span class="o">.</span><span class="n">mark</span><span class="o">.</span><span class="n">slow</span>
-
-<div class="viewcode-block" id="SKIP"><a class="viewcode-back" href="../../../modules/utilities/pytest.html#sympy.utilities.pytest.SKIP">[docs]</a>    <span class="k">def</span> <span class="nf">SKIP</span><span class="p">(</span><span class="n">reason</span><span class="p">):</span>
-        <span class="k">def</span> <span class="nf">skipping</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
-            <span class="nd">@functools.wraps</span><span class="p">(</span><span class="n">func</span><span class="p">)</span>
-            <span class="k">def</span> <span class="nf">inner</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-                <span class="n">skip</span><span class="p">(</span><span class="n">reason</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">inner</span>
-
-        <span class="k">return</span> <span class="n">skipping</span></div>
-</pre></div>
-
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diff --git a/dev-py3k/_modules/sympy/utilities/randtest.html b/dev-py3k/_modules/sympy/utilities/randtest.html
deleted file mode 100644
index ff4b6d838bd..00000000000
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+++ /dev/null
@@ -1,308 +0,0 @@
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-  <h1>Source code for sympy.utilities.randtest</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot; Helpers for randomized testing &quot;&quot;&quot;</span>
-
-<span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">uniform</span>
-<span class="kn">import</span> <span class="nn">random</span>
-
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">I</span><span class="p">,</span> <span class="n">nsimplify</span><span class="p">,</span> <span class="n">Tuple</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">is_sequence</span><span class="p">,</span> <span class="n">as_int</span>
-
-
-<div class="viewcode-block" id="random_complex_number"><a class="viewcode-back" href="../../../modules/utilities/randtest.html#sympy.utilities.randtest.random_complex_number">[docs]</a><span class="k">def</span> <span class="nf">random_complex_number</span><span class="p">(</span><span class="n">a</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">b</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">d</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">rational</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return a random complex number.</span>
-
-<span class="sd">    To reduce chance of hitting branch cuts or anything, we guarantee</span>
-<span class="sd">    b &lt;= Im z &lt;= d, a &lt;= Re z &lt;= c</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">A</span><span class="p">,</span> <span class="n">B</span> <span class="o">=</span> <span class="n">uniform</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">c</span><span class="p">),</span> <span class="n">uniform</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">rational</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">A</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">B</span>
-    <span class="k">return</span> <span class="n">nsimplify</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">rational</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">nsimplify</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">rational</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="comp"><a class="viewcode-back" href="../../../modules/utilities/randtest.html#sympy.utilities.randtest.comp">[docs]</a><span class="k">def</span> <span class="nf">comp</span><span class="p">(</span><span class="n">z1</span><span class="p">,</span> <span class="n">z2</span><span class="p">,</span> <span class="n">tol</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return a bool indicating whether the error between z1 and z2 is &lt;= tol.</span>
-
-<span class="sd">    If z2 is non-zero and ``|z1| &gt; 1`` the error is normalized by ``|z1|``, so</span>
-<span class="sd">    if you want the absolute error, call this as ``comp(z1 - z2, 0, tol)``.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">z1</span><span class="p">:</span>
-        <span class="n">z1</span><span class="p">,</span> <span class="n">z2</span> <span class="o">=</span> <span class="n">z2</span><span class="p">,</span> <span class="n">z1</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">z1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">True</span>
-    <span class="n">diff</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">z1</span> <span class="o">-</span> <span class="n">z2</span><span class="p">)</span>
-    <span class="n">az1</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">z1</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">z2</span> <span class="ow">and</span> <span class="n">az1</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">diff</span><span class="o">/</span><span class="n">az1</span> <span class="o">&lt;=</span> <span class="n">tol</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">diff</span> <span class="o">&lt;=</span> <span class="n">tol</span>
-
-</div>
-<div class="viewcode-block" id="test_numerically"><a class="viewcode-back" href="../../../modules/utilities/randtest.html#sympy.utilities.randtest.test_numerically">[docs]</a><span class="k">def</span> <span class="nf">test_numerically</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">z</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1.0e-6</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">b</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">d</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Test numerically that f and g agree when evaluated in the argument z.</span>
-
-<span class="sd">    If z is None, all symbols will be tested. This routine does not test</span>
-<span class="sd">    whether there are Floats present with precision higher than 15 digits</span>
-<span class="sd">    so if there are, your results may not be what you expect due to round-</span>
-<span class="sd">    off errors.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import sin, cos, S</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.randtest import test_numerically as tn</span>
-<span class="sd">    &gt;&gt;&gt; tn(sin(x)**2 + cos(x)**2, 1, x)</span>
-<span class="sd">    True</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-    <span class="n">z</span> <span class="o">=</span> <span class="p">[</span><span class="n">z</span><span class="p">]</span> <span class="k">if</span> <span class="n">z</span> <span class="k">else</span> <span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">free_symbols</span> <span class="o">|</span> <span class="n">g</span><span class="o">.</span><span class="n">free_symbols</span><span class="p">)</span>
-    <span class="n">reps</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="p">[</span><span class="n">random_complex_number</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span> <span class="k">for</span> <span class="n">zi</span> <span class="ow">in</span> <span class="n">z</span><span class="p">]))</span>
-    <span class="n">z1</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">reps</span><span class="p">)</span><span class="o">.</span><span class="n">n</span><span class="p">()</span>
-    <span class="n">z2</span> <span class="o">=</span> <span class="n">g</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">reps</span><span class="p">)</span><span class="o">.</span><span class="n">n</span><span class="p">()</span>
-    <span class="k">return</span> <span class="n">comp</span><span class="p">(</span><span class="n">z1</span><span class="p">,</span> <span class="n">z2</span><span class="p">,</span> <span class="n">tol</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="test_derivative_numerically"><a class="viewcode-back" href="../../../modules/utilities/randtest.html#sympy.utilities.randtest.test_derivative_numerically">[docs]</a><span class="k">def</span> <span class="nf">test_derivative_numerically</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1.0e-6</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">b</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">d</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Test numerically that the symbolically computed derivative of f</span>
-<span class="sd">    with respect to z is correct.</span>
-
-<span class="sd">    This routine does not test whether there are Floats present with</span>
-<span class="sd">    precision higher than 15 digits so if there are, your results may</span>
-<span class="sd">    not be what you expect due to round-off errors.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy import sin, cos</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.abc import x</span>
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.randtest import test_derivative_numerically as td</span>
-<span class="sd">    &gt;&gt;&gt; td(sin(x), x)</span>
-<span class="sd">    True</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">Derivative</span>
-    <span class="n">z0</span> <span class="o">=</span> <span class="n">random_complex_number</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
-    <span class="n">f1</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">z0</span><span class="p">)</span>
-    <span class="n">f2</span> <span class="o">=</span> <span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">doit_numerically</span><span class="p">(</span><span class="n">z0</span><span class="p">)</span>
-    <span class="k">return</span> <span class="n">comp</span><span class="p">(</span><span class="n">f1</span><span class="o">.</span><span class="n">n</span><span class="p">(),</span> <span class="n">f2</span><span class="o">.</span><span class="n">n</span><span class="p">(),</span> <span class="n">tol</span><span class="p">)</span>
-</div>
-<span class="kn">import</span> <span class="nn">random</span>
-
-
-<span class="k">def</span> <span class="nf">_randrange</span><span class="p">(</span><span class="n">seed</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return a randrange generator. ``seed`` can be</span>
-<span class="sd">        o None - return randomly seeded generator</span>
-<span class="sd">        o int - return a generator seeded with the int</span>
-<span class="sd">        o list - the values to be returned will be taken from the list</span>
-<span class="sd">          in the order given; the provided list is not modified.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.randtest import _randrange</span>
-<span class="sd">    &gt;&gt;&gt; rr = _randrange()</span>
-<span class="sd">    &gt;&gt;&gt; rr(1000) # doctest: +SKIP</span>
-<span class="sd">    999</span>
-<span class="sd">    &gt;&gt;&gt; rr = _randrange(3)</span>
-<span class="sd">    &gt;&gt;&gt; rr(1000) # doctest: +SKIP</span>
-<span class="sd">    238</span>
-<span class="sd">    &gt;&gt;&gt; rr = _randrange([0, 5, 1, 3, 4])</span>
-<span class="sd">    &gt;&gt;&gt; rr(3), rr(3)</span>
-<span class="sd">    (0, 1)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">seed</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">random</span><span class="o">.</span><span class="n">randrange</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">seed</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">random</span><span class="o">.</span><span class="n">Random</span><span class="p">(</span><span class="n">seed</span><span class="p">)</span><span class="o">.</span><span class="n">randrange</span>
-    <span class="k">elif</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">seed</span><span class="p">):</span>
-        <span class="n">seed</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">seed</span><span class="p">)</span>  <span class="c"># make a copy</span>
-        <span class="n">seed</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-
-        <span class="k">def</span> <span class="nf">give</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">seq</span><span class="o">=</span><span class="n">seed</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">b</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="n">a</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">as_int</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-            <span class="n">w</span> <span class="o">=</span> <span class="n">b</span> <span class="o">-</span> <span class="n">a</span>
-            <span class="k">if</span> <span class="n">w</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;_randrange got empty range&#39;</span><span class="p">)</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="n">seq</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-            <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;_randrange expects a list-like sequence&#39;</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;_randrange sequence was too short&#39;</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="o">&lt;=</span> <span class="n">x</span> <span class="o">&lt;</span> <span class="n">b</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">x</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">give</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">seq</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">give</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;_randrange got an unexpected seed&#39;</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">_randint</span><span class="p">(</span><span class="n">seed</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Return a randint generator. ``seed`` can be</span>
-<span class="sd">        o None - return randomly seeded generator</span>
-<span class="sd">        o int - return a generator seeded with the int</span>
-<span class="sd">        o list - the values to be returned will be taken from the list</span>
-<span class="sd">          in the order given; the provided list is not modified.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.randtest import _randint</span>
-<span class="sd">    &gt;&gt;&gt; ri = _randint()</span>
-<span class="sd">    &gt;&gt;&gt; ri(1, 1000) # doctest: +SKIP</span>
-<span class="sd">    999</span>
-<span class="sd">    &gt;&gt;&gt; ri = _randint(3)</span>
-<span class="sd">    &gt;&gt;&gt; ri(1, 1000) # doctest: +SKIP</span>
-<span class="sd">    238</span>
-<span class="sd">    &gt;&gt;&gt; ri = _randint([0, 5, 1, 2, 4])</span>
-<span class="sd">    &gt;&gt;&gt; ri(1, 3), ri(1, 3)</span>
-<span class="sd">    (1, 2)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">seed</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span>
-    <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">seed</span><span class="p">,</span> <span class="nb">int</span><span class="p">):</span>
-        <span class="k">return</span> <span class="n">random</span><span class="o">.</span><span class="n">Random</span><span class="p">(</span><span class="n">seed</span><span class="p">)</span><span class="o">.</span><span class="n">randint</span>
-    <span class="k">elif</span> <span class="n">is_sequence</span><span class="p">(</span><span class="n">seed</span><span class="p">):</span>
-        <span class="n">seed</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">seed</span><span class="p">)</span>  <span class="c"># make a copy</span>
-        <span class="n">seed</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
-
-        <span class="k">def</span> <span class="nf">give</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">seq</span><span class="o">=</span><span class="n">seed</span><span class="p">):</span>
-            <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">as_int</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">as_int</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-            <span class="n">w</span> <span class="o">=</span> <span class="n">b</span> <span class="o">-</span> <span class="n">a</span>
-            <span class="k">if</span> <span class="n">w</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;_randint got empty range&#39;</span><span class="p">)</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">x</span> <span class="o">=</span> <span class="n">seq</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-            <span class="k">except</span> <span class="ne">AttributeError</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;_randint expects a list-like sequence&#39;</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">IndexError</span><span class="p">:</span>
-                <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;_randint sequence was too short&#39;</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">a</span> <span class="o">&lt;=</span> <span class="n">x</span> <span class="o">&lt;=</span> <span class="n">b</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">x</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">return</span> <span class="n">give</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">seq</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">give</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;_randint got an unexpected seed&#39;</span><span class="p">)</span>
-</pre></div>
-
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-            
-  <h1>Source code for sympy.utilities.runtests</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">This is our testing framework.</span>
-
-<span class="sd">Goals:</span>
-
-<span class="sd">* it should be compatible with py.test and operate very similarly</span>
-<span class="sd">  (or identically)</span>
-<span class="sd">* doesn&#39;t require any external dependencies</span>
-<span class="sd">* preferably all the functionality should be in this file only</span>
-<span class="sd">* no magic, just import the test file and execute the test functions, that&#39;s it</span>
-<span class="sd">* portable</span>
-
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">import</span> <span class="nn">os</span>
-<span class="kn">import</span> <span class="nn">sys</span>
-<span class="kn">import</span> <span class="nn">inspect</span>
-<span class="kn">import</span> <span class="nn">traceback</span>
-<span class="kn">import</span> <span class="nn">pdb</span>
-<span class="kn">import</span> <span class="nn">re</span>
-<span class="kn">import</span> <span class="nn">linecache</span>
-<span class="kn">from</span> <span class="nn">fnmatch</span> <span class="kn">import</span> <span class="n">fnmatch</span>
-<span class="kn">from</span> <span class="nn">timeit</span> <span class="kn">import</span> <span class="n">default_timer</span> <span class="k">as</span> <span class="n">clock</span>
-<span class="kn">import</span> <span class="nn">doctest</span> <span class="kn">as</span> <span class="nn">pdoctest</span>  <span class="c"># avoid clashing with our doctest() function</span>
-<span class="kn">from</span> <span class="nn">doctest</span> <span class="kn">import</span> <span class="n">DocTestFinder</span><span class="p">,</span> <span class="n">DocTestRunner</span>
-<span class="kn">import</span> <span class="nn">re</span> <span class="kn">as</span> <span class="nn">pre</span>
-<span class="kn">import</span> <span class="nn">random</span>
-<span class="kn">import</span> <span class="nn">subprocess</span>
-<span class="kn">import</span> <span class="nn">signal</span>
-
-<span class="kn">from</span> <span class="nn">sympy.core.cache</span> <span class="kn">import</span> <span class="n">clear_cache</span>
-
-<span class="c"># Use sys.stdout encoding for ouput.</span>
-<span class="c"># This was only added to Python&#39;s doctest in Python 2.6, so we must duplicate</span>
-<span class="c"># it here to make utf8 files work in Python 2.5.</span>
-<span class="n">pdoctest</span><span class="o">.</span><span class="n">_encoding</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">sys</span><span class="o">.</span><span class="n">__stdout__</span><span class="p">,</span> <span class="s">&#39;encoding&#39;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span> <span class="ow">or</span> <span class="s">&#39;utf-8&#39;</span>
-
-<span class="n">IS_PYTHON_3</span> <span class="o">=</span> <span class="p">(</span><span class="n">sys</span><span class="o">.</span><span class="n">version_info</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="mi">3</span><span class="p">)</span>
-<span class="n">IS_WINDOWS</span> <span class="o">=</span> <span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">name</span> <span class="o">==</span> <span class="s">&#39;nt&#39;</span><span class="p">)</span>
-
-
-<span class="k">class</span> <span class="nc">Skipped</span><span class="p">(</span><span class="ne">Exception</span><span class="p">):</span>
-    <span class="k">pass</span>
-
-
-<span class="k">def</span> <span class="nf">_indent</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">indent</span><span class="o">=</span><span class="mi">4</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Add the given number of space characters to the beginning of</span>
-<span class="sd">    every non-blank line in ``s``, and return the result.</span>
-<span class="sd">    If the string ``s`` is Unicode, it is encoded using the stdout</span>
-<span class="sd">    encoding and the ``backslashreplace`` error handler.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># After a 2to3 run the below code is bogus, so wrap it with a version check</span>
-    <span class="k">if</span> <span class="n">sys</span><span class="o">.</span><span class="n">version_info</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&lt;</span> <span class="mi">3</span><span class="p">:</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="n">s</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">encode</span><span class="p">(</span><span class="n">pdoctest</span><span class="o">.</span><span class="n">_encoding</span><span class="p">,</span> <span class="s">&#39;backslashreplace&#39;</span><span class="p">)</span>
-    <span class="c"># This regexp matches the start of non-blank lines:</span>
-    <span class="k">return</span> <span class="n">re</span><span class="o">.</span><span class="n">sub</span><span class="p">(</span><span class="s">&#39;(?m)^(?!$)&#39;</span><span class="p">,</span> <span class="n">indent</span><span class="o">*</span><span class="s">&#39; &#39;</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span>
-
-<span class="n">pdoctest</span><span class="o">.</span><span class="n">_indent</span> <span class="o">=</span> <span class="n">_indent</span>
-
-<span class="c"># ovverride reporter to maintain windows and python3</span>
-
-
-<span class="k">def</span> <span class="nf">_report_failure</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">out</span><span class="p">,</span> <span class="n">test</span><span class="p">,</span> <span class="n">example</span><span class="p">,</span> <span class="n">got</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Report that the given example failed.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_checker</span><span class="o">.</span><span class="n">output_difference</span><span class="p">(</span><span class="n">example</span><span class="p">,</span> <span class="n">got</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">optionflags</span><span class="p">)</span>
-    <span class="n">s</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">encode</span><span class="p">(</span><span class="s">&#39;raw_unicode_escape&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">decode</span><span class="p">(</span><span class="s">&#39;utf8&#39;</span><span class="p">,</span> <span class="s">&#39;ignore&#39;</span><span class="p">)</span>
-    <span class="n">out</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_failure_header</span><span class="p">(</span><span class="n">test</span><span class="p">,</span> <span class="n">example</span><span class="p">)</span> <span class="o">+</span> <span class="n">s</span><span class="p">)</span>
-
-<span class="k">if</span> <span class="n">IS_PYTHON_3</span> <span class="ow">and</span> <span class="n">IS_WINDOWS</span><span class="p">:</span>
-    <span class="n">DocTestRunner</span><span class="o">.</span><span class="n">report_failure</span> <span class="o">=</span> <span class="n">_report_failure</span>
-
-
-<div class="viewcode-block" id="convert_to_native_paths"><a class="viewcode-back" href="../../../modules/utilities/runtests.html#sympy.utilities.runtests.convert_to_native_paths">[docs]</a><span class="k">def</span> <span class="nf">convert_to_native_paths</span><span class="p">(</span><span class="n">lst</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Converts a list of &#39;/&#39; separated paths into a list of</span>
-<span class="sd">    native (os.sep separated) paths and converts to lowercase</span>
-<span class="sd">    if the system is case insensitive.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">newlst</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">rv</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">lst</span><span class="p">):</span>
-        <span class="n">rv</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="o">*</span><span class="n">rv</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&quot;/&quot;</span><span class="p">))</span>
-        <span class="c"># on windows the slash after the colon is dropped</span>
-        <span class="k">if</span> <span class="n">sys</span><span class="o">.</span><span class="n">platform</span> <span class="o">==</span> <span class="s">&quot;win32&quot;</span><span class="p">:</span>
-            <span class="n">pos</span> <span class="o">=</span> <span class="n">rv</span><span class="o">.</span><span class="n">find</span><span class="p">(</span><span class="s">&#39;:&#39;</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">pos</span> <span class="o">!=</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">rv</span><span class="p">[</span><span class="n">pos</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">!=</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">&#39;</span><span class="p">:</span>
-                    <span class="n">rv</span> <span class="o">=</span> <span class="n">rv</span><span class="p">[:</span><span class="n">pos</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="s">&#39;</span><span class="se">\\</span><span class="s">&#39;</span> <span class="o">+</span> <span class="n">rv</span><span class="p">[</span><span class="n">pos</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-        <span class="n">newlst</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sys_normcase</span><span class="p">(</span><span class="n">rv</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">newlst</span>
-
-</div>
-<div class="viewcode-block" id="get_sympy_dir"><a class="viewcode-back" href="../../../modules/utilities/runtests.html#sympy.utilities.runtests.get_sympy_dir">[docs]</a><span class="k">def</span> <span class="nf">get_sympy_dir</span><span class="p">():</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Returns the root sympy directory and set the global value</span>
-<span class="sd">    indicating whether the system is case sensitive or not.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">global</span> <span class="n">sys_case_insensitive</span>
-
-    <span class="n">this_file</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">abspath</span><span class="p">(</span><span class="n">__file__</span><span class="p">)</span>
-    <span class="n">sympy_dir</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">dirname</span><span class="p">(</span><span class="n">this_file</span><span class="p">),</span> <span class="s">&quot;..&quot;</span><span class="p">,</span> <span class="s">&quot;..&quot;</span><span class="p">)</span>
-    <span class="n">sympy_dir</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">normpath</span><span class="p">(</span><span class="n">sympy_dir</span><span class="p">)</span>
-    <span class="n">sys_case_insensitive</span> <span class="o">=</span> <span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">isdir</span><span class="p">(</span><span class="n">sympy_dir</span><span class="p">)</span> <span class="ow">and</span>
-                        <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">isdir</span><span class="p">(</span><span class="n">sympy_dir</span><span class="o">.</span><span class="n">lower</span><span class="p">())</span> <span class="ow">and</span>
-                        <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">isdir</span><span class="p">(</span><span class="n">sympy_dir</span><span class="o">.</span><span class="n">upper</span><span class="p">()))</span>
-    <span class="k">return</span> <span class="n">sys_normcase</span><span class="p">(</span><span class="n">sympy_dir</span><span class="p">)</span>
-
-</div>
-<span class="k">def</span> <span class="nf">sys_normcase</span><span class="p">(</span><span class="n">f</span><span class="p">):</span>
-    <span class="k">if</span> <span class="n">sys_case_insensitive</span><span class="p">:</span>  <span class="c"># global defined after call to get_sympy_dir()</span>
-        <span class="k">return</span> <span class="n">f</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span>
-    <span class="k">return</span> <span class="n">f</span>
-
-
-<div class="viewcode-block" id="isgeneratorfunction"><a class="viewcode-back" href="../../../modules/utilities/runtests.html#sympy.utilities.runtests.isgeneratorfunction">[docs]</a><span class="k">def</span> <span class="nf">isgeneratorfunction</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Return True if the object is a user-defined generator function.</span>
-
-<span class="sd">    Generator function objects provides same attributes as functions.</span>
-
-<span class="sd">    See isfunction.__doc__ for attributes listing.</span>
-
-<span class="sd">    Adapted from Python 2.6.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">CO_GENERATOR</span> <span class="o">=</span> <span class="mh">0x20</span>
-    <span class="k">if</span> <span class="p">(</span><span class="n">inspect</span><span class="o">.</span><span class="n">isfunction</span><span class="p">(</span><span class="nb">object</span><span class="p">)</span> <span class="ow">or</span> <span class="n">inspect</span><span class="o">.</span><span class="n">ismethod</span><span class="p">(</span><span class="nb">object</span><span class="p">))</span> <span class="ow">and</span> \
-            <span class="nb">object</span><span class="o">.</span><span class="n">__code__</span><span class="o">.</span><span class="n">co_flags</span> <span class="o">&amp;</span> <span class="n">CO_GENERATOR</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">True</span>
-    <span class="k">return</span> <span class="bp">False</span>
-
-</div>
-<span class="k">def</span> <span class="nf">setup_pprint</span><span class="p">():</span>
-    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pprint_use_unicode</span><span class="p">,</span> <span class="n">init_printing</span>
-
-    <span class="c"># force pprint to be in ascii mode in doctests</span>
-    <span class="n">pprint_use_unicode</span><span class="p">(</span><span class="bp">False</span><span class="p">)</span>
-
-    <span class="c"># hook our nice, hash-stable strprinter</span>
-    <span class="n">init_printing</span><span class="p">(</span><span class="n">pretty_print</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="run_in_subprocess_with_hash_randomization"><a class="viewcode-back" href="../../../modules/utilities/runtests.html#sympy.utilities.runtests.run_in_subprocess_with_hash_randomization">[docs]</a><span class="k">def</span> <span class="nf">run_in_subprocess_with_hash_randomization</span><span class="p">(</span><span class="n">function</span><span class="p">,</span> <span class="n">function_args</span><span class="o">=</span><span class="p">(),</span>
-    <span class="n">function_kwargs</span><span class="o">=</span><span class="p">{},</span> <span class="n">command</span><span class="o">=</span><span class="n">sys</span><span class="o">.</span><span class="n">executable</span><span class="p">,</span>
-        <span class="n">module</span><span class="o">=</span><span class="s">&#39;sympy.utilities.runtests&#39;</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Run a function in a Python subprocess with hash randomization enabled.</span>
-
-<span class="sd">    If hash randomization is not supported by the version of Python given, it</span>
-<span class="sd">    returns False.  Otherwise, it returns the exit value of the command.  The</span>
-<span class="sd">    function is passed to sys.exit(), so the return value of the function will</span>
-<span class="sd">    be the return value.</span>
-
-<span class="sd">    The environment variable PYTHONHASHSEED is used to seed Python&#39;s hash</span>
-<span class="sd">    randomization.  If it is set, this function will return False, because</span>
-<span class="sd">    starting a new subprocess is unnecessary in that case.  If it is not set,</span>
-<span class="sd">    one is set at random, and the tests are run.  Note that if this</span>
-<span class="sd">    environment variable is set when Python starts, hash randomization is</span>
-<span class="sd">    automatically enabled.  To force a subprocess to be created even if</span>
-<span class="sd">    PYTHONHASHSEED is set, pass ``force=True``.  This flag will not force a</span>
-<span class="sd">    subprocess in Python versions that do not support hash randomization (see</span>
-<span class="sd">    below), because those versions of Python do not support the ``-R`` flag.</span>
-
-<span class="sd">    ``function`` should be a string name of a function that is importable from</span>
-<span class="sd">    the module ``module``, like &quot;_test&quot;.  The default for ``module`` is</span>
-<span class="sd">    &quot;sympy.utilities.runtests&quot;.  ``function_args`` and ``function_kwargs``</span>
-<span class="sd">    should be a repr-able tuple and dict, respectively.  The default Python</span>
-<span class="sd">    command is sys.executable, which is the currently running Python command.</span>
-
-<span class="sd">    This function is necessary because the seed for hash randomization must be</span>
-<span class="sd">    set by the environment variable before Python starts.  Hence, in order to</span>
-<span class="sd">    use a predetermined seed for tests, we must start Python in a separate</span>
-<span class="sd">    subprocess.</span>
-
-<span class="sd">    Hash randomization was added in the minor Python versions 2.6.8, 2.7.3,</span>
-<span class="sd">    3.1.5, and 3.2.3, and is enabled by default in all Python versions after</span>
-<span class="sd">    and including 3.3.0.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.runtests import (</span>
-<span class="sd">    ... run_in_subprocess_with_hash_randomization)</span>
-<span class="sd">    &gt;&gt;&gt; # run the core tests in verbose mode</span>
-<span class="sd">    &gt;&gt;&gt; run_in_subprocess_with_hash_randomization(&quot;_test&quot;,</span>
-<span class="sd">    ... function_args=(&quot;core&quot;,),</span>
-<span class="sd">    ... function_kwargs={&#39;verbose&#39;: True}) # doctest: +SKIP</span>
-<span class="sd">    # Will return 0 if sys.executable supports hash randomization and tests</span>
-<span class="sd">    # pass, 1 if they fail, and False if it does not support hash</span>
-<span class="sd">    # randomization.</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="c"># Note, we must return False everywhere, not None, as subprocess.call will</span>
-    <span class="c"># sometimes return None.</span>
-
-    <span class="c"># First check if the Python version supports hash randomization</span>
-    <span class="c"># If it doesn&#39;t have this support, it won&#39;t reconize the -R flag</span>
-    <span class="n">p</span> <span class="o">=</span> <span class="n">subprocess</span><span class="o">.</span><span class="n">Popen</span><span class="p">([</span><span class="n">command</span><span class="p">,</span> <span class="s">&quot;-RV&quot;</span><span class="p">],</span> <span class="n">stdout</span><span class="o">=</span><span class="n">subprocess</span><span class="o">.</span><span class="n">PIPE</span><span class="p">,</span>
-                         <span class="n">stderr</span><span class="o">=</span><span class="n">subprocess</span><span class="o">.</span><span class="n">STDOUT</span><span class="p">)</span>
-    <span class="n">p</span><span class="o">.</span><span class="n">communicate</span><span class="p">()</span>
-    <span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">returncode</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="k">return</span> <span class="bp">False</span>
-
-    <span class="n">hash_seed</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">getenv</span><span class="p">(</span><span class="s">&quot;PYTHONHASHSEED&quot;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">hash_seed</span><span class="p">:</span>
-        <span class="n">os</span><span class="o">.</span><span class="n">environ</span><span class="p">[</span><span class="s">&quot;PYTHONHASHSEED&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">randrange</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="mi">32</span><span class="p">))</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">force</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">False</span>
-    <span class="c"># Now run the command</span>
-    <span class="n">commandstring</span> <span class="o">=</span> <span class="p">(</span><span class="s">&quot;import sys; from </span><span class="si">%s</span><span class="s"> import </span><span class="si">%s</span><span class="s">;sys.exit(</span><span class="si">%s</span><span class="s">(*</span><span class="si">%s</span><span class="s">, **</span><span class="si">%s</span><span class="s">))&quot;</span> <span class="o">%</span>
-                     <span class="p">(</span><span class="n">module</span><span class="p">,</span> <span class="n">function</span><span class="p">,</span> <span class="n">function</span><span class="p">,</span> <span class="nb">repr</span><span class="p">(</span><span class="n">function_args</span><span class="p">),</span>
-                     <span class="nb">repr</span><span class="p">(</span><span class="n">function_kwargs</span><span class="p">)))</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">subprocess</span><span class="o">.</span><span class="n">call</span><span class="p">([</span><span class="n">command</span><span class="p">,</span> <span class="s">&quot;-R&quot;</span><span class="p">,</span> <span class="s">&quot;-c&quot;</span><span class="p">,</span> <span class="n">commandstring</span><span class="p">])</span>
-    <span class="k">finally</span><span class="p">:</span>
-        <span class="c"># Put the environment variable back, so that it reads correctly for</span>
-        <span class="c"># the current Python process.</span>
-        <span class="k">if</span> <span class="n">hash_seed</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">del</span> <span class="n">os</span><span class="o">.</span><span class="n">environ</span><span class="p">[</span><span class="s">&quot;PYTHONHASHSEED&quot;</span><span class="p">]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">os</span><span class="o">.</span><span class="n">environ</span><span class="p">[</span><span class="s">&quot;PYTHONHASHSEED&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">hash_seed</span>
-
-</div>
-<div class="viewcode-block" id="run_all_tests"><a class="viewcode-back" href="../../../modules/utilities/runtests.html#sympy.utilities.runtests.run_all_tests">[docs]</a><span class="k">def</span> <span class="nf">run_all_tests</span><span class="p">(</span><span class="n">test_args</span><span class="o">=</span><span class="p">(),</span> <span class="n">test_kwargs</span><span class="o">=</span><span class="p">{},</span> <span class="n">doctest_args</span><span class="o">=</span><span class="p">(),</span>
-                  <span class="n">doctest_kwargs</span><span class="o">=</span><span class="p">{},</span> <span class="n">examples_args</span><span class="o">=</span><span class="p">(),</span> <span class="n">examples_kwargs</span><span class="o">=</span><span class="p">{</span><span class="s">&#39;quiet&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">}):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Run all tests.</span>
-
-<span class="sd">    Right now, this runs the regular tests (bin/test), the doctests</span>
-<span class="sd">    (bin/doctest), the examples (examples/all.py), and the sage tests (see</span>
-<span class="sd">    sympy/external/tests/test_sage.py).</span>
-
-<span class="sd">    This is what ``setup.py test`` uses.</span>
-
-<span class="sd">    You can pass arguments and keyword arguments to the test functions that</span>
-<span class="sd">    support them (for now, test,  doctest, and the examples). See the</span>
-<span class="sd">    docstrings of those functions for a description of the available options.</span>
-
-<span class="sd">    For example, to run the solvers tests with colors turned off:</span>
-
-<span class="sd">    &gt;&gt;&gt; from sympy.utilities.runtests import run_all_tests</span>
-<span class="sd">    &gt;&gt;&gt; run_all_tests(test_args=(&quot;solvers&quot;,),</span>
-<span class="sd">    ... test_kwargs={&quot;colors:False&quot;}) # doctest: +SKIP</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">tests_successful</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">try</span><span class="p">:</span>
-        <span class="c"># Regular tests</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">test</span><span class="p">(</span><span class="o">*</span><span class="n">test_args</span><span class="p">,</span> <span class="o">**</span><span class="n">test_kwargs</span><span class="p">):</span>
-            <span class="c"># some regular test fails, so set the tests_successful</span>
-            <span class="c"># flag to false and continue running the doctests</span>
-            <span class="n">tests_successful</span> <span class="o">=</span> <span class="bp">False</span>
-
-        <span class="c"># Doctests</span>
-        <span class="k">print</span><span class="p">()</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">doctest</span><span class="p">(</span><span class="o">*</span><span class="n">doctest_args</span><span class="p">,</span> <span class="o">**</span><span class="n">doctest_kwargs</span><span class="p">):</span>
-            <span class="n">tests_successful</span> <span class="o">=</span> <span class="bp">False</span>
-
-        <span class="c"># Examples</span>
-        <span class="k">print</span><span class="p">()</span>
-        <span class="n">sys</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&quot;examples&quot;</span><span class="p">)</span>
-        <span class="kn">from</span> <span class="nn">all</span> <span class="kn">import</span> <span class="n">run_examples</span>  <span class="c"># examples/all.py</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">run_examples</span><span class="p">(</span><span class="o">*</span><span class="n">examples_args</span><span class="p">,</span> <span class="o">**</span><span class="n">examples_kwargs</span><span class="p">):</span>
-            <span class="n">tests_successful</span> <span class="o">=</span> <span class="bp">False</span>
-
-        <span class="c"># Sage tests</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">sys</span><span class="o">.</span><span class="n">platform</span> <span class="o">==</span> <span class="s">&quot;win32&quot;</span> <span class="ow">or</span> <span class="n">sys</span><span class="o">.</span><span class="n">version_info</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="mi">3</span><span class="p">):</span>
-            <span class="c"># run Sage tests; Sage currently doesn&#39;t support Windows or Python 3</span>
-            <span class="n">dev_null</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">devnull</span><span class="p">,</span> <span class="s">&#39;w&#39;</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">subprocess</span><span class="o">.</span><span class="n">call</span><span class="p">(</span><span class="s">&quot;sage -v&quot;</span><span class="p">,</span> <span class="n">shell</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">stdout</span><span class="o">=</span><span class="n">dev_null</span><span class="p">,</span>
-                               <span class="n">stderr</span><span class="o">=</span><span class="n">dev_null</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">subprocess</span><span class="o">.</span><span class="n">call</span><span class="p">(</span><span class="s">&quot;sage -python bin/test &quot;</span>
-                                   <span class="s">&quot;sympy/external/tests/test_sage.py&quot;</span><span class="p">,</span> <span class="n">shell</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span><span class="p">:</span>
-                    <span class="n">tests_successful</span> <span class="o">=</span> <span class="bp">False</span>
-
-        <span class="k">if</span> <span class="n">tests_successful</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># Return nonzero exit code</span>
-            <span class="n">sys</span><span class="o">.</span><span class="n">exit</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-    <span class="k">except</span> <span class="ne">KeyboardInterrupt</span><span class="p">:</span>
-        <span class="k">print</span><span class="p">()</span>
-        <span class="k">print</span><span class="p">(</span><span class="s">&quot;DO *NOT* COMMIT!&quot;</span><span class="p">)</span>
-        <span class="n">sys</span><span class="o">.</span><span class="n">exit</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="test"><a class="viewcode-back" href="../../../modules/utilities/runtests.html#sympy.utilities.runtests.test">[docs]</a><span class="k">def</span> <span class="nf">test</span><span class="p">(</span><span class="o">*</span><span class="n">paths</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Run tests in the specified test_*.py files.</span>
-
-<span class="sd">    Tests in a particular test_*.py file are run if any of the given strings</span>
-<span class="sd">    in ``paths`` matches a part of the test file&#39;s path. If ``paths=[]``,</span>
-<span class="sd">    tests in all test_*.py files are run.</span>
-
-<span class="sd">    Notes:</span>
-
-<span class="sd">    - If sort=False, tests are run in random order (not default).</span>
-<span class="sd">    - Paths can be entered in native system format or in unix,</span>
-<span class="sd">      forward-slash format.</span>
-
-<span class="sd">    **Explanation of test results**</span>
-
-<span class="sd">    ======  ===============================================================</span>
-<span class="sd">    Output  Meaning</span>
-<span class="sd">    ======  ===============================================================</span>
-<span class="sd">    .       passed</span>
-<span class="sd">    F       failed</span>
-<span class="sd">    X       XPassed (expected to fail but passed)</span>
-<span class="sd">    f       XFAILed (expected to fail and indeed failed)</span>
-<span class="sd">    s       skipped</span>
-<span class="sd">    w       slow</span>
-<span class="sd">    T       timeout (e.g., when ``--timeout`` is used)</span>
-<span class="sd">    K       KeyboardInterrupt (when running the slow tests with ``--slow``,</span>
-<span class="sd">            you can interrupt one of them without killing the test runner)</span>
-<span class="sd">    ======  ===============================================================</span>
-
-
-<span class="sd">    Colors have no additional meaning and are used just to facilitate</span>
-<span class="sd">    interpreting the output.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; import sympy</span>
-
-<span class="sd">    Run all tests:</span>
-
-<span class="sd">    &gt;&gt;&gt; sympy.test()    # doctest: +SKIP</span>
-
-<span class="sd">    Run one file:</span>
-
-<span class="sd">    &gt;&gt;&gt; sympy.test(&quot;sympy/core/tests/test_basic.py&quot;)    # doctest: +SKIP</span>
-<span class="sd">    &gt;&gt;&gt; sympy.test(&quot;_basic&quot;)    # doctest: +SKIP</span>
-
-<span class="sd">    Run all tests in sympy/functions/ and some particular file:</span>
-
-<span class="sd">    &gt;&gt;&gt; sympy.test(&quot;sympy/core/tests/test_basic.py&quot;,</span>
-<span class="sd">    ...        &quot;sympy/functions&quot;)    # doctest: +SKIP</span>
-
-<span class="sd">    Run all tests in sympy/core and sympy/utilities:</span>
-
-<span class="sd">    &gt;&gt;&gt; sympy.test(&quot;/core&quot;, &quot;/util&quot;)    # doctest: +SKIP</span>
-
-<span class="sd">    Run specific test from a file:</span>
-
-<span class="sd">    &gt;&gt;&gt; sympy.test(&quot;sympy/core/tests/test_basic.py&quot;,</span>
-<span class="sd">    ...        kw=&quot;test_equality&quot;)    # doctest: +SKIP</span>
-
-<span class="sd">    Run specific test from any file:</span>
-
-<span class="sd">    &gt;&gt;&gt; sympy.test(kw=&quot;subs&quot;)    # doctest: +SKIP</span>
-
-<span class="sd">    Run the tests with verbose mode on:</span>
-
-<span class="sd">    &gt;&gt;&gt; sympy.test(verbose=True)    # doctest: +SKIP</span>
-
-<span class="sd">    Don&#39;t sort the test output:</span>
-
-<span class="sd">    &gt;&gt;&gt; sympy.test(sort=False)    # doctest: +SKIP</span>
-
-<span class="sd">    Turn on post-mortem pdb:</span>
-
-<span class="sd">    &gt;&gt;&gt; sympy.test(pdb=True)    # doctest: +SKIP</span>
-
-<span class="sd">    Turn off colors:</span>
-
-<span class="sd">    &gt;&gt;&gt; sympy.test(colors=False)    # doctest: +SKIP</span>
-
-<span class="sd">    Force colors, even when the output is not to a terminal (this is useful,</span>
-<span class="sd">    e.g., if you are piping to ``less -r`` and you still want colors)</span>
-
-<span class="sd">    &gt;&gt;&gt; sympy.test(force_colors=False)    # doctest: +SKIP</span>
-
-<span class="sd">    The traceback verboseness can be set to &quot;short&quot; or &quot;no&quot; (default is</span>
-<span class="sd">    &quot;short&quot;)</span>
-
-<span class="sd">    &gt;&gt;&gt; sympy.test(tb=&#39;no&#39;)    # doctest: +SKIP</span>
-
-<span class="sd">    You can disable running the tests in a separate subprocess using</span>
-<span class="sd">    ``subprocess=False``.  This is done to support seeding hash randomization,</span>
-<span class="sd">    which is enabled by default in the Python versions where it is supported.</span>
-<span class="sd">    If subprocess=False, hash randomization is enabled/disabled according to</span>
-<span class="sd">    whether it has been enabled or not in the calling Python process.</span>
-<span class="sd">    However, even if it is enabled, the seed cannot be printed unless it is</span>
-<span class="sd">    called from a new Python process.</span>
-
-<span class="sd">    Hash randomization was added in the minor Python versions 2.6.8, 2.7.3,</span>
-<span class="sd">    3.1.5, and 3.2.3, and is enabled by default in all Python versions after</span>
-<span class="sd">    and including 3.3.0.</span>
-
-<span class="sd">    If hash randomization is not supported ``subprocess=False`` is used</span>
-<span class="sd">    automatically.</span>
-
-<span class="sd">    &gt;&gt;&gt; sympy.test(subprocess=False)     # doctest: +SKIP</span>
-
-<span class="sd">    To set the hash randomization seed, set the environment variable</span>
-<span class="sd">    ``PYTHONHASHSEED`` before running the tests.  This can be done from within</span>
-<span class="sd">    Python using</span>
-
-<span class="sd">    &gt;&gt;&gt; import os</span>
-<span class="sd">    &gt;&gt;&gt; os.environ[&#39;PYTHONHASHSEED&#39;] = 42 # doctest: +SKIP</span>
-
-<span class="sd">    Or from the command line using</span>
-
-<span class="sd">    $ PYTHONHASHSEED=42 ./bin/test</span>
-
-<span class="sd">    If the seed is not set, a random seed will be chosen.</span>
-
-<span class="sd">    Note that to reproduce the same hash values, you must use both the same as</span>
-<span class="sd">    well as the same architecture (32-bit vs. 64-bit).</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">subprocess</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&quot;subprocess&quot;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">subprocess</span><span class="p">:</span>
-        <span class="n">ret</span> <span class="o">=</span> <span class="n">run_in_subprocess_with_hash_randomization</span><span class="p">(</span><span class="s">&quot;_test&quot;</span><span class="p">,</span>
-            <span class="n">function_args</span><span class="o">=</span><span class="n">paths</span><span class="p">,</span> <span class="n">function_kwargs</span><span class="o">=</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">ret</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="k">return</span> <span class="ow">not</span> <span class="nb">bool</span><span class="p">(</span><span class="n">ret</span><span class="p">)</span>
-    <span class="k">return</span> <span class="ow">not</span> <span class="nb">bool</span><span class="p">(</span><span class="n">_test</span><span class="p">(</span><span class="o">*</span><span class="n">paths</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_test</span><span class="p">(</span><span class="o">*</span><span class="n">paths</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Internal function that actually runs the tests.</span>
-
-<span class="sd">    All keyword arguments from ``test()`` are passed to this function except for</span>
-<span class="sd">    ``subprocess``.</span>
-
-<span class="sd">    Returns 0 if tests passed and 1 if they failed.  See the docstring of</span>
-<span class="sd">    ``test()`` for more information.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">verbose</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;verbose&quot;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-    <span class="n">tb</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;tb&quot;</span><span class="p">,</span> <span class="s">&quot;short&quot;</span><span class="p">)</span>
-    <span class="n">kw</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;kw&quot;</span><span class="p">,</span> <span class="s">&quot;&quot;</span><span class="p">)</span>
-    <span class="n">post_mortem</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;pdb&quot;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-    <span class="n">colors</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;colors&quot;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-    <span class="n">force_colors</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;force_colors&quot;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-    <span class="n">sort</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;sort&quot;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-    <span class="n">seed</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;seed&quot;</span><span class="p">,</span> <span class="bp">None</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">seed</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">seed</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randrange</span><span class="p">(</span><span class="mi">100000000</span><span class="p">)</span>
-    <span class="n">timeout</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;timeout&quot;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-    <span class="n">slow</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;slow&quot;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-    <span class="n">r</span> <span class="o">=</span> <span class="n">PyTestReporter</span><span class="p">(</span><span class="n">verbose</span><span class="o">=</span><span class="n">verbose</span><span class="p">,</span> <span class="n">tb</span><span class="o">=</span><span class="n">tb</span><span class="p">,</span> <span class="n">colors</span><span class="o">=</span><span class="n">colors</span><span class="p">,</span>
-        <span class="n">force_colors</span><span class="o">=</span><span class="n">force_colors</span><span class="p">)</span>
-    <span class="n">t</span> <span class="o">=</span> <span class="n">SymPyTests</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">kw</span><span class="p">,</span> <span class="n">post_mortem</span><span class="p">,</span> <span class="n">seed</span><span class="p">)</span>
-
-    <span class="c"># Disable warnings for external modules</span>
-    <span class="kn">import</span> <span class="nn">sympy.external</span>
-    <span class="n">sympy</span><span class="o">.</span><span class="n">external</span><span class="o">.</span><span class="n">importtools</span><span class="o">.</span><span class="n">WARN_OLD_VERSION</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">sympy</span><span class="o">.</span><span class="n">external</span><span class="o">.</span><span class="n">importtools</span><span class="o">.</span><span class="n">WARN_NOT_INSTALLED</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="n">test_files</span> <span class="o">=</span> <span class="n">t</span><span class="o">.</span><span class="n">get_test_files</span><span class="p">(</span><span class="s">&#39;sympy&#39;</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">paths</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">t</span><span class="o">.</span><span class="n">_testfiles</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">test_files</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">paths</span> <span class="o">=</span> <span class="n">convert_to_native_paths</span><span class="p">(</span><span class="n">paths</span><span class="p">)</span>
-        <span class="n">matched</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">test_files</span><span class="p">:</span>
-            <span class="n">basename</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">basename</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">paths</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">f</span> <span class="ow">or</span> <span class="n">fnmatch</span><span class="p">(</span><span class="n">basename</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-                    <span class="n">matched</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-                    <span class="k">break</span>
-        <span class="n">t</span><span class="o">.</span><span class="n">_testfiles</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">matched</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="ow">not</span> <span class="n">t</span><span class="o">.</span><span class="n">test</span><span class="p">(</span><span class="n">sort</span><span class="o">=</span><span class="n">sort</span><span class="p">,</span> <span class="n">timeout</span><span class="o">=</span><span class="n">timeout</span><span class="p">,</span> <span class="n">slow</span><span class="o">=</span><span class="n">slow</span><span class="p">))</span>
-
-
-<div class="viewcode-block" id="doctest"><a class="viewcode-back" href="../../../modules/utilities/runtests.html#sympy.utilities.runtests.doctest">[docs]</a><span class="k">def</span> <span class="nf">doctest</span><span class="p">(</span><span class="o">*</span><span class="n">paths</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Runs doctests in all \*.py files in the sympy directory which match</span>
-<span class="sd">    any of the given strings in ``paths`` or all tests if paths=[].</span>
-
-<span class="sd">    Notes:</span>
-
-<span class="sd">    - Paths can be entered in native system format or in unix,</span>
-<span class="sd">      forward-slash format.</span>
-<span class="sd">    - Files that are on the blacklist can be tested by providing</span>
-<span class="sd">      their path; they are only excluded if no paths are given.</span>
-
-<span class="sd">    Examples</span>
-<span class="sd">    ========</span>
-
-<span class="sd">    &gt;&gt;&gt; import sympy</span>
-
-<span class="sd">    Run all tests:</span>
-
-<span class="sd">    &gt;&gt;&gt; sympy.doctest() # doctest: +SKIP</span>
-
-<span class="sd">    Run one file:</span>
-
-<span class="sd">    &gt;&gt;&gt; sympy.doctest(&quot;sympy/core/basic.py&quot;) # doctest: +SKIP</span>
-<span class="sd">    &gt;&gt;&gt; sympy.doctest(&quot;polynomial.rst&quot;) # doctest: +SKIP</span>
-
-<span class="sd">    Run all tests in sympy/functions/ and some particular file:</span>
-
-<span class="sd">    &gt;&gt;&gt; sympy.doctest(&quot;/functions&quot;, &quot;basic.py&quot;) # doctest: +SKIP</span>
-
-<span class="sd">    Run any file having polynomial in its name, doc/src/modules/polynomial.rst,</span>
-<span class="sd">    sympy/functions/special/polynomials.py, and sympy/polys/polynomial.py:</span>
-
-<span class="sd">    &gt;&gt;&gt; sympy.doctest(&quot;polynomial&quot;) # doctest: +SKIP</span>
-
-<span class="sd">    The ``subprocess`` and ``verbose`` options are the same as with the function</span>
-<span class="sd">    ``test()``.  See the docstring of that function for more information.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">subprocess</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&quot;subprocess&quot;</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">subprocess</span><span class="p">:</span>
-        <span class="n">ret</span> <span class="o">=</span> <span class="n">run_in_subprocess_with_hash_randomization</span><span class="p">(</span><span class="s">&quot;_doctest&quot;</span><span class="p">,</span>
-            <span class="n">function_args</span><span class="o">=</span><span class="n">paths</span><span class="p">,</span> <span class="n">function_kwargs</span><span class="o">=</span><span class="n">kwargs</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">ret</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">False</span><span class="p">:</span>
-            <span class="k">return</span> <span class="ow">not</span> <span class="nb">bool</span><span class="p">(</span><span class="n">ret</span><span class="p">)</span>
-    <span class="k">return</span> <span class="ow">not</span> <span class="nb">bool</span><span class="p">(</span><span class="n">_doctest</span><span class="p">(</span><span class="o">*</span><span class="n">paths</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">))</span>
-
-</div>
-<span class="k">def</span> <span class="nf">_doctest</span><span class="p">(</span><span class="o">*</span><span class="n">paths</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Internal function that actually runs the doctests.</span>
-
-<span class="sd">    All keyword arguments from ``doctest()`` are passed to this function</span>
-<span class="sd">    except for ``subprocess``.</span>
-
-<span class="sd">    Returns 0 if tests passed and 1 if they failed.  See the docstrings of</span>
-<span class="sd">    ``doctest()`` and ``test()`` for more information.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">normal</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;normal&quot;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-    <span class="n">verbose</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;verbose&quot;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-    <span class="n">blacklist</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;blacklist&quot;</span><span class="p">,</span> <span class="p">[])</span>
-    <span class="n">blacklist</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span>
-        <span class="s">&quot;doc/src/modules/mpmath&quot;</span><span class="p">,</span>  <span class="c"># needs to be fixed upstream</span>
-        <span class="s">&quot;sympy/mpmath&quot;</span><span class="p">,</span>  <span class="c"># needs to be fixed upstream</span>
-        <span class="s">&quot;doc/src/modules/plotting.rst&quot;</span><span class="p">,</span>  <span class="c"># generates live plots</span>
-        <span class="c"># &quot;sympy/plotting&quot;, # generates live plots</span>
-        <span class="s">&quot;sympy/plotting/pygletplot&quot;</span><span class="p">,</span>  <span class="c"># generates live plots</span>
-        <span class="s">&quot;sympy/statistics&quot;</span><span class="p">,</span>                <span class="c"># prints a deprecation</span>
-        <span class="s">&quot;doc/src/modules/statistics.rst&quot;</span><span class="p">,</span>  <span class="c"># warning (the module is deprecated)</span>
-        <span class="s">&quot;sympy/utilities/compilef.py&quot;</span><span class="p">,</span>  <span class="c"># needs tcc</span>
-        <span class="s">&quot;sympy/utilities/autowrap.py&quot;</span><span class="p">,</span>  <span class="c"># needs installed compiler</span>
-        <span class="s">&quot;sympy/galgebra/GA.py&quot;</span><span class="p">,</span>  <span class="c"># needs numpy</span>
-        <span class="s">&quot;sympy/galgebra/latex_ex.py&quot;</span><span class="p">,</span>  <span class="c"># needs numpy</span>
-        <span class="s">&quot;sympy/conftest.py&quot;</span><span class="p">,</span>  <span class="c"># needs py.test</span>
-        <span class="s">&quot;sympy/utilities/benchmarking.py&quot;</span><span class="p">,</span>  <span class="c"># needs py.test</span>
-        <span class="s">&quot;examples/advanced/autowrap_integrators.py&quot;</span><span class="p">,</span>  <span class="c"># needs numpy</span>
-        <span class="s">&quot;examples/advanced/autowrap_ufuncify.py&quot;</span><span class="p">,</span>  <span class="c"># needs numpy</span>
-        <span class="s">&quot;examples/intermediate/mplot2d.py&quot;</span><span class="p">,</span>
-        <span class="c"># needs numpy and matplotlib</span>
-        <span class="s">&quot;examples/intermediate/mplot3d.py&quot;</span><span class="p">,</span>
-        <span class="c"># needs numpy and matplotlib</span>
-        <span class="s">&quot;examples/intermediate/sample.py&quot;</span><span class="p">,</span>  <span class="c"># needs numpy</span>
-    <span class="p">])</span>
-    <span class="n">blacklist</span> <span class="o">=</span> <span class="n">convert_to_native_paths</span><span class="p">(</span><span class="n">blacklist</span><span class="p">)</span>
-
-    <span class="c"># Disable warnings for external modules</span>
-    <span class="kn">import</span> <span class="nn">sympy.external</span>
-    <span class="n">sympy</span><span class="o">.</span><span class="n">external</span><span class="o">.</span><span class="n">importtools</span><span class="o">.</span><span class="n">WARN_OLD_VERSION</span> <span class="o">=</span> <span class="bp">False</span>
-    <span class="n">sympy</span><span class="o">.</span><span class="n">external</span><span class="o">.</span><span class="n">importtools</span><span class="o">.</span><span class="n">WARN_NOT_INSTALLED</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="n">r</span> <span class="o">=</span> <span class="n">PyTestReporter</span><span class="p">(</span><span class="n">verbose</span><span class="p">)</span>
-    <span class="n">t</span> <span class="o">=</span> <span class="n">SymPyDocTests</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">normal</span><span class="p">)</span>
-
-    <span class="n">test_files</span> <span class="o">=</span> <span class="n">t</span><span class="o">.</span><span class="n">get_test_files</span><span class="p">(</span><span class="s">&#39;sympy&#39;</span><span class="p">)</span>
-    <span class="n">test_files</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">t</span><span class="o">.</span><span class="n">get_test_files</span><span class="p">(</span><span class="s">&#39;examples&#39;</span><span class="p">,</span> <span class="n">init_only</span><span class="o">=</span><span class="bp">False</span><span class="p">))</span>
-
-    <span class="n">not_blacklisted</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">test_files</span>
-                       <span class="k">if</span> <span class="ow">not</span> <span class="nb">any</span><span class="p">(</span><span class="n">b</span> <span class="ow">in</span> <span class="n">f</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">blacklist</span><span class="p">)]</span>
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">paths</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">t</span><span class="o">.</span><span class="n">_testfiles</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">not_blacklisted</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="c"># take only what was requested...but not blacklisted items</span>
-        <span class="c"># and allow for partial match anywhere or fnmatch of name</span>
-        <span class="n">paths</span> <span class="o">=</span> <span class="n">convert_to_native_paths</span><span class="p">(</span><span class="n">paths</span><span class="p">)</span>
-        <span class="n">matched</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">not_blacklisted</span><span class="p">:</span>
-            <span class="n">basename</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">basename</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">paths</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">f</span> <span class="ow">or</span> <span class="n">fnmatch</span><span class="p">(</span><span class="n">basename</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-                    <span class="n">matched</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-                    <span class="k">break</span>
-        <span class="n">t</span><span class="o">.</span><span class="n">_testfiles</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">matched</span><span class="p">)</span>
-
-    <span class="c"># run the tests and record the result for this *py portion of the tests</span>
-    <span class="k">if</span> <span class="n">t</span><span class="o">.</span><span class="n">_testfiles</span><span class="p">:</span>
-        <span class="n">failed</span> <span class="o">=</span> <span class="ow">not</span> <span class="n">t</span><span class="o">.</span><span class="n">test</span><span class="p">()</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">failed</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="c"># N.B.</span>
-    <span class="c"># --------------------------------------------------------------------</span>
-    <span class="c"># Here we test *.rst files at or below doc/src. Code from these must</span>
-    <span class="c"># be self supporting in terms of imports since there is no importing</span>
-    <span class="c"># of necessary modules by doctest.testfile. If you try to pass *.py</span>
-    <span class="c"># files through this they might fail because they will lack the needed</span>
-    <span class="c"># imports and smarter parsing that can be done with source code.</span>
-    <span class="c">#</span>
-    <span class="n">test_files</span> <span class="o">=</span> <span class="n">t</span><span class="o">.</span><span class="n">get_test_files</span><span class="p">(</span><span class="s">&#39;doc/src&#39;</span><span class="p">,</span> <span class="s">&#39;*.rst&#39;</span><span class="p">,</span> <span class="n">init_only</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-    <span class="n">test_files</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-
-    <span class="n">not_blacklisted</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">test_files</span>
-                       <span class="k">if</span> <span class="ow">not</span> <span class="nb">any</span><span class="p">(</span><span class="n">b</span> <span class="ow">in</span> <span class="n">f</span> <span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">blacklist</span><span class="p">)]</span>
-
-    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">paths</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-        <span class="n">matched</span> <span class="o">=</span> <span class="n">not_blacklisted</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="c"># Take only what was requested as long as it&#39;s not on the blacklist.</span>
-        <span class="c"># Paths were already made native in *py tests so don&#39;t repeat here.</span>
-        <span class="c"># There&#39;s no chance of having a *py file slip through since we</span>
-        <span class="c"># only have *rst files in test_files.</span>
-        <span class="n">matched</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">not_blacklisted</span><span class="p">:</span>
-            <span class="n">basename</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">basename</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">paths</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">f</span> <span class="ow">or</span> <span class="n">fnmatch</span><span class="p">(</span><span class="n">basename</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
-                    <span class="n">matched</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-                    <span class="k">break</span>
-
-    <span class="n">setup_pprint</span><span class="p">()</span>
-    <span class="n">first_report</span> <span class="o">=</span> <span class="bp">True</span>
-    <span class="k">for</span> <span class="n">rst_file</span> <span class="ow">in</span> <span class="n">matched</span><span class="p">:</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">isfile</span><span class="p">(</span><span class="n">rst_file</span><span class="p">):</span>
-            <span class="k">continue</span>
-        <span class="n">old_displayhook</span> <span class="o">=</span> <span class="n">sys</span><span class="o">.</span><span class="n">displayhook</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="c"># out = pdoctest.testfile(</span>
-            <span class="c">#    rst_file, module_relative=False, encoding=&#39;utf-8&#39;,</span>
-            <span class="c">#    optionflags=pdoctest.ELLIPSIS | pdoctest.NORMALIZE_WHITESPACE)</span>
-            <span class="n">out</span> <span class="o">=</span> <span class="n">sympytestfile</span><span class="p">(</span>
-                <span class="n">rst_file</span><span class="p">,</span> <span class="n">module_relative</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">encoding</span><span class="o">=</span><span class="s">&#39;utf-8&#39;</span><span class="p">,</span>
-                <span class="n">optionflags</span><span class="o">=</span><span class="n">pdoctest</span><span class="o">.</span><span class="n">ELLIPSIS</span> <span class="o">|</span> <span class="n">pdoctest</span><span class="o">.</span><span class="n">NORMALIZE_WHITESPACE</span> <span class="o">|</span>
-                <span class="n">pdoctest</span><span class="o">.</span><span class="n">IGNORE_EXCEPTION_DETAIL</span><span class="p">)</span>
-        <span class="k">finally</span><span class="p">:</span>
-            <span class="c"># make sure we return to the original displayhook in case some</span>
-            <span class="c"># doctest has changed that</span>
-            <span class="n">sys</span><span class="o">.</span><span class="n">displayhook</span> <span class="o">=</span> <span class="n">old_displayhook</span>
-
-        <span class="n">rstfailed</span><span class="p">,</span> <span class="n">tested</span> <span class="o">=</span> <span class="n">out</span>
-        <span class="k">if</span> <span class="n">tested</span><span class="p">:</span>
-            <span class="n">failed</span> <span class="o">=</span> <span class="n">rstfailed</span> <span class="ow">or</span> <span class="n">failed</span>
-            <span class="k">if</span> <span class="n">first_report</span><span class="p">:</span>
-                <span class="n">first_report</span> <span class="o">=</span> <span class="bp">False</span>
-                <span class="n">msg</span> <span class="o">=</span> <span class="s">&#39;rst doctests start&#39;</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="n">t</span><span class="o">.</span><span class="n">_testfiles</span><span class="p">:</span>
-                    <span class="n">r</span><span class="o">.</span><span class="n">start</span><span class="p">(</span><span class="n">msg</span><span class="o">=</span><span class="n">msg</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">r</span><span class="o">.</span><span class="n">write_center</span><span class="p">(</span><span class="n">msg</span><span class="p">)</span>
-                    <span class="k">print</span><span class="p">()</span>
-            <span class="c"># use as the id, everything past the first &#39;sympy&#39;</span>
-            <span class="n">file_id</span> <span class="o">=</span> <span class="n">rst_file</span><span class="p">[</span><span class="n">rst_file</span><span class="o">.</span><span class="n">find</span><span class="p">(</span><span class="s">&#39;sympy&#39;</span><span class="p">)</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="s">&#39;sympy&#39;</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-            <span class="k">print</span><span class="p">(</span><span class="n">file_id</span><span class="p">,</span> <span class="n">end</span><span class="o">=</span><span class="s">&#39; &#39;</span><span class="p">)</span>
-                <span class="c"># get at least the name out so it is know who is being tested</span>
-            <span class="n">wid</span> <span class="o">=</span> <span class="n">r</span><span class="o">.</span><span class="n">terminal_width</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">file_id</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>  <span class="c"># update width</span>
-            <span class="n">test_file</span> <span class="o">=</span> <span class="s">&#39;[</span><span class="si">%s</span><span class="s">]&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tested</span><span class="p">)</span>
-            <span class="n">report</span> <span class="o">=</span> <span class="s">&#39;[</span><span class="si">%s</span><span class="s">]&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">rstfailed</span> <span class="ow">or</span> <span class="s">&#39;OK&#39;</span><span class="p">)</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span>
-                <span class="p">[</span><span class="n">test_file</span><span class="p">,</span> <span class="s">&#39; &#39;</span><span class="o">*</span><span class="p">(</span><span class="n">wid</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">test_file</span><span class="p">)</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">report</span><span class="p">)),</span> <span class="n">report</span><span class="p">]))</span>
-
-    <span class="c"># the doctests for *py will have printed this message already if there was</span>
-    <span class="c"># a failure, so now only print it if there was intervening reporting by</span>
-    <span class="c"># testing the *rst as evidenced by first_report no longer being True.</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">first_report</span> <span class="ow">and</span> <span class="n">failed</span><span class="p">:</span>
-        <span class="k">print</span><span class="p">()</span>
-        <span class="k">print</span><span class="p">(</span><span class="s">&quot;DO *NOT* COMMIT!&quot;</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">failed</span><span class="p">)</span>
-
-<span class="c"># The Python 2.5 doctest runner uses a tuple, but in 2.6+, it uses a namedtuple</span>
-<span class="c"># (which doesn&#39;t exist in 2.5-)</span>
-<span class="k">if</span> <span class="n">sys</span><span class="o">.</span><span class="n">version_info</span><span class="p">[:</span><span class="mi">2</span><span class="p">]</span> <span class="o">&gt;</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">):</span>
-    <span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">namedtuple</span>
-    <span class="n">SymPyTestResults</span> <span class="o">=</span> <span class="n">namedtuple</span><span class="p">(</span><span class="s">&#39;TestResults&#39;</span><span class="p">,</span> <span class="s">&#39;failed attempted&#39;</span><span class="p">)</span>
-<span class="k">else</span><span class="p">:</span>
-    <span class="n">SymPyTestResults</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">:</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-
-
-<div class="viewcode-block" id="sympytestfile"><a class="viewcode-back" href="../../../modules/utilities/runtests.html#sympy.utilities.runtests.sympytestfile">[docs]</a><span class="k">def</span> <span class="nf">sympytestfile</span><span class="p">(</span><span class="n">filename</span><span class="p">,</span> <span class="n">module_relative</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">package</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span>
-             <span class="n">globs</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">report</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">optionflags</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span>
-             <span class="n">extraglobs</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">raise_on_error</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span>
-             <span class="n">parser</span><span class="o">=</span><span class="n">pdoctest</span><span class="o">.</span><span class="n">DocTestParser</span><span class="p">(),</span> <span class="n">encoding</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Test examples in the given file.  Return (#failures, #tests).</span>
-
-<span class="sd">    Optional keyword arg ``module_relative`` specifies how filenames</span>
-<span class="sd">    should be interpreted:</span>
-
-<span class="sd">    - If ``module_relative`` is True (the default), then ``filename``</span>
-<span class="sd">      specifies a module-relative path.  By default, this path is</span>
-<span class="sd">      relative to the calling module&#39;s directory; but if the</span>
-<span class="sd">      ``package`` argument is specified, then it is relative to that</span>
-<span class="sd">      package.  To ensure os-independence, ``filename`` should use</span>
-<span class="sd">      &quot;/&quot; characters to separate path segments, and should not</span>
-<span class="sd">      be an absolute path (i.e., it may not begin with &quot;/&quot;).</span>
-
-<span class="sd">    - If ``module_relative`` is False, then ``filename`` specifies an</span>
-<span class="sd">      os-specific path.  The path may be absolute or relative (to</span>
-<span class="sd">      the current working directory).</span>
-
-<span class="sd">    Optional keyword arg ``name`` gives the name of the test; by default</span>
-<span class="sd">    use the file&#39;s basename.</span>
-
-<span class="sd">    Optional keyword argument ``package`` is a Python package or the</span>
-<span class="sd">    name of a Python package whose directory should be used as the</span>
-<span class="sd">    base directory for a module relative filename.  If no package is</span>
-<span class="sd">    specified, then the calling module&#39;s directory is used as the base</span>
-<span class="sd">    directory for module relative filenames.  It is an error to</span>
-<span class="sd">    specify ``package`` if ``module_relative`` is False.</span>
-
-<span class="sd">    Optional keyword arg ``globs`` gives a dict to be used as the globals</span>
-<span class="sd">    when executing examples; by default, use {}.  A copy of this dict</span>
-<span class="sd">    is actually used for each docstring, so that each docstring&#39;s</span>
-<span class="sd">    examples start with a clean slate.</span>
-
-<span class="sd">    Optional keyword arg ``extraglobs`` gives a dictionary that should be</span>
-<span class="sd">    merged into the globals that are used to execute examples.  By</span>
-<span class="sd">    default, no extra globals are used.</span>
-
-<span class="sd">    Optional keyword arg ``verbose`` prints lots of stuff if true, prints</span>
-<span class="sd">    only failures if false; by default, it&#39;s true iff &quot;-v&quot; is in sys.argv.</span>
-
-<span class="sd">    Optional keyword arg ``report`` prints a summary at the end when true,</span>
-<span class="sd">    else prints nothing at the end.  In verbose mode, the summary is</span>
-<span class="sd">    detailed, else very brief (in fact, empty if all tests passed).</span>
-
-<span class="sd">    Optional keyword arg ``optionflags`` or&#39;s together module constants,</span>
-<span class="sd">    and defaults to 0.  Possible values (see the docs for details):</span>
-
-<span class="sd">    - DONT_ACCEPT_TRUE_FOR_1</span>
-<span class="sd">    - DONT_ACCEPT_BLANKLINE</span>
-<span class="sd">    - NORMALIZE_WHITESPACE</span>
-<span class="sd">    - ELLIPSIS</span>
-<span class="sd">    - SKIP</span>
-<span class="sd">    - IGNORE_EXCEPTION_DETAIL</span>
-<span class="sd">    - REPORT_UDIFF</span>
-<span class="sd">    - REPORT_CDIFF</span>
-<span class="sd">    - REPORT_NDIFF</span>
-<span class="sd">    - REPORT_ONLY_FIRST_FAILURE</span>
-
-<span class="sd">    Optional keyword arg ``raise_on_error`` raises an exception on the</span>
-<span class="sd">    first unexpected exception or failure. This allows failures to be</span>
-<span class="sd">    post-mortem debugged.</span>
-
-<span class="sd">    Optional keyword arg ``parser`` specifies a DocTestParser (or</span>
-<span class="sd">    subclass) that should be used to extract tests from the files.</span>
-
-<span class="sd">    Optional keyword arg ``encoding`` specifies an encoding that should</span>
-<span class="sd">    be used to convert the file to unicode.</span>
-
-<span class="sd">    Advanced tomfoolery:  testmod runs methods of a local instance of</span>
-<span class="sd">    class doctest.Tester, then merges the results into (or creates)</span>
-<span class="sd">    global Tester instance doctest.master.  Methods of doctest.master</span>
-<span class="sd">    can be called directly too, if you want to do something unusual.</span>
-<span class="sd">    Passing report=0 to testmod is especially useful then, to delay</span>
-<span class="sd">    displaying a summary.  Invoke doctest.master.summarize(verbose)</span>
-<span class="sd">    when you&#39;re done fiddling.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="n">package</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">module_relative</span><span class="p">:</span>
-        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;Package may only be specified for module-&quot;</span>
-                         <span class="s">&quot;relative paths.&quot;</span><span class="p">)</span>
-
-    <span class="c"># Relativize the path</span>
-    <span class="k">if</span> <span class="ow">not</span> <span class="n">IS_PYTHON_3</span><span class="p">:</span>
-        <span class="n">text</span><span class="p">,</span> <span class="n">filename</span> <span class="o">=</span> <span class="n">pdoctest</span><span class="o">.</span><span class="n">_load_testfile</span><span class="p">(</span>
-            <span class="n">filename</span><span class="p">,</span> <span class="n">package</span><span class="p">,</span> <span class="n">module_relative</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">encoding</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">text</span> <span class="o">=</span> <span class="n">text</span><span class="o">.</span><span class="n">decode</span><span class="p">(</span><span class="n">encoding</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">text</span><span class="p">,</span> <span class="n">filename</span> <span class="o">=</span> <span class="n">pdoctest</span><span class="o">.</span><span class="n">_load_testfile</span><span class="p">(</span>
-            <span class="n">filename</span><span class="p">,</span> <span class="n">package</span><span class="p">,</span> <span class="n">module_relative</span><span class="p">,</span> <span class="n">encoding</span><span class="p">)</span>
-
-    <span class="c"># If no name was given, then use the file&#39;s name.</span>
-    <span class="k">if</span> <span class="n">name</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">name</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">basename</span><span class="p">(</span><span class="n">filename</span><span class="p">)</span>
-
-    <span class="c"># Assemble the globals.</span>
-    <span class="k">if</span> <span class="n">globs</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">globs</span> <span class="o">=</span> <span class="p">{}</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">globs</span> <span class="o">=</span> <span class="n">globs</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-    <span class="k">if</span> <span class="n">extraglobs</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">globs</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">extraglobs</span><span class="p">)</span>
-    <span class="k">if</span> <span class="s">&#39;__name__&#39;</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">globs</span><span class="p">:</span>
-        <span class="n">globs</span><span class="p">[</span><span class="s">&#39;__name__&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="s">&#39;__main__&#39;</span>
-
-    <span class="k">if</span> <span class="n">raise_on_error</span><span class="p">:</span>
-        <span class="n">runner</span> <span class="o">=</span> <span class="n">pdoctest</span><span class="o">.</span><span class="n">DebugRunner</span><span class="p">(</span><span class="n">verbose</span><span class="o">=</span><span class="n">verbose</span><span class="p">,</span> <span class="n">optionflags</span><span class="o">=</span><span class="n">optionflags</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">runner</span> <span class="o">=</span> <span class="n">SymPyDocTestRunner</span><span class="p">(</span><span class="n">verbose</span><span class="o">=</span><span class="n">verbose</span><span class="p">,</span> <span class="n">optionflags</span><span class="o">=</span><span class="n">optionflags</span><span class="p">)</span>
-
-    <span class="c"># Read the file, convert it to a test, and run it.</span>
-    <span class="n">test</span> <span class="o">=</span> <span class="n">parser</span><span class="o">.</span><span class="n">get_doctest</span><span class="p">(</span><span class="n">text</span><span class="p">,</span> <span class="n">globs</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">filename</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-    <span class="n">runner</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">test</span><span class="p">)</span>
-
-    <span class="k">if</span> <span class="n">report</span><span class="p">:</span>
-        <span class="n">runner</span><span class="o">.</span><span class="n">summarize</span><span class="p">()</span>
-
-    <span class="k">if</span> <span class="n">pdoctest</span><span class="o">.</span><span class="n">master</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-        <span class="n">pdoctest</span><span class="o">.</span><span class="n">master</span> <span class="o">=</span> <span class="n">runner</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">pdoctest</span><span class="o">.</span><span class="n">master</span><span class="o">.</span><span class="n">merge</span><span class="p">(</span><span class="n">runner</span><span class="p">)</span>
-
-    <span class="k">return</span> <span class="n">SymPyTestResults</span><span class="p">(</span><span class="n">runner</span><span class="o">.</span><span class="n">failures</span><span class="p">,</span> <span class="n">runner</span><span class="o">.</span><span class="n">tries</span><span class="p">)</span>
-
-</div>
-<span class="k">class</span> <span class="nc">SymPyTests</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">reporter</span><span class="p">,</span> <span class="n">kw</span><span class="o">=</span><span class="s">&quot;&quot;</span><span class="p">,</span> <span class="n">post_mortem</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span>
-                 <span class="n">seed</span><span class="o">=</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">()):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_post_mortem</span> <span class="o">=</span> <span class="n">post_mortem</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_kw</span> <span class="o">=</span> <span class="n">kw</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_count</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_root_dir</span> <span class="o">=</span> <span class="n">sympy_dir</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span> <span class="o">=</span> <span class="n">reporter</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">root_dir</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_root_dir</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_testfiles</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_seed</span> <span class="o">=</span> <span class="n">seed</span>
-
-    <span class="k">def</span> <span class="nf">test</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sort</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">timeout</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">slow</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Runs the tests returning True if all tests pass, otherwise False.</span>
-
-<span class="sd">        If sort=False run tests in random order.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="n">sort</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_testfiles</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">shuffle</span>
-            <span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_seed</span><span class="p">)</span>
-            <span class="n">shuffle</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_testfiles</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">start</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_seed</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_testfiles</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">test_file</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">sort</span><span class="p">,</span> <span class="n">timeout</span><span class="p">,</span> <span class="n">slow</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">KeyboardInterrupt</span><span class="p">:</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&quot; interrupted by user&quot;</span><span class="p">)</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">finish</span><span class="p">()</span>
-                <span class="k">raise</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">finish</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">test_file</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">filename</span><span class="p">,</span> <span class="n">sort</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">timeout</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">slow</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="n">clear_cache</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_count</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="n">gl</span> <span class="o">=</span> <span class="p">{</span><span class="s">&#39;__file__&#39;</span><span class="p">:</span> <span class="n">filename</span><span class="p">}</span>
-        <span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_seed</span><span class="p">)</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">IS_PYTHON_3</span><span class="p">:</span>
-                <span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="n">filename</span><span class="p">,</span> <span class="n">encoding</span><span class="o">=</span><span class="s">&quot;utf8&quot;</span><span class="p">)</span> <span class="k">as</span> <span class="n">f</span><span class="p">:</span>
-                    <span class="n">source</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">read</span><span class="p">()</span>
-                <span class="n">c</span> <span class="o">=</span> <span class="nb">compile</span><span class="p">(</span><span class="n">source</span><span class="p">,</span> <span class="n">filename</span><span class="p">,</span> <span class="s">&#39;exec&#39;</span><span class="p">)</span>
-                <span class="k">exec</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">gl</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="k">exec</span><span class="p">(</span><span class="nb">compile</span><span class="p">(</span><span class="nb">open</span><span class="p">(</span><span class="n">filename</span><span class="p">)</span><span class="o">.</span><span class="n">read</span><span class="p">(),</span> <span class="n">filename</span><span class="p">,</span> <span class="s">&#39;exec&#39;</span><span class="p">),</span> <span class="n">gl</span><span class="p">)</span>
-        <span class="k">except</span> <span class="p">(</span><span class="ne">SystemExit</span><span class="p">,</span> <span class="ne">KeyboardInterrupt</span><span class="p">):</span>
-            <span class="k">raise</span>
-        <span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">import_error</span><span class="p">(</span><span class="n">filename</span><span class="p">,</span> <span class="n">sys</span><span class="o">.</span><span class="n">exc_info</span><span class="p">())</span>
-            <span class="k">return</span>
-        <span class="n">pytestfile</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-        <span class="k">if</span> <span class="s">&quot;XFAIL&quot;</span> <span class="ow">in</span> <span class="n">gl</span><span class="p">:</span>
-            <span class="n">pytestfile</span> <span class="o">=</span> <span class="n">inspect</span><span class="o">.</span><span class="n">getsourcefile</span><span class="p">(</span><span class="n">gl</span><span class="p">[</span><span class="s">&quot;XFAIL&quot;</span><span class="p">])</span>
-        <span class="n">pytestfile2</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-        <span class="k">if</span> <span class="s">&quot;SLOW&quot;</span> <span class="ow">in</span> <span class="n">gl</span><span class="p">:</span>
-            <span class="n">pytestfile2</span> <span class="o">=</span> <span class="n">inspect</span><span class="o">.</span><span class="n">getsourcefile</span><span class="p">(</span><span class="n">gl</span><span class="p">[</span><span class="s">&quot;SLOW&quot;</span><span class="p">])</span>
-        <span class="n">disabled</span> <span class="o">=</span> <span class="n">gl</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s">&quot;disabled&quot;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">disabled</span><span class="p">:</span>
-            <span class="n">funcs</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># we need to filter only those functions that begin with &#39;test_&#39;</span>
-            <span class="c"># that are defined in the testing file or in the file where</span>
-            <span class="c"># is defined the XFAIL decorator</span>
-            <span class="n">funcs</span> <span class="o">=</span> <span class="p">[</span><span class="n">gl</span><span class="p">[</span><span class="n">f</span><span class="p">]</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">gl</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span> <span class="k">if</span> <span class="n">f</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&quot;test_&quot;</span><span class="p">)</span> <span class="ow">and</span>
-                <span class="p">(</span><span class="n">inspect</span><span class="o">.</span><span class="n">isfunction</span><span class="p">(</span><span class="n">gl</span><span class="p">[</span><span class="n">f</span><span class="p">])</span> <span class="ow">or</span> <span class="n">inspect</span><span class="o">.</span><span class="n">ismethod</span><span class="p">(</span><span class="n">gl</span><span class="p">[</span><span class="n">f</span><span class="p">]))</span> <span class="ow">and</span>
-                <span class="p">(</span><span class="n">inspect</span><span class="o">.</span><span class="n">getsourcefile</span><span class="p">(</span><span class="n">gl</span><span class="p">[</span><span class="n">f</span><span class="p">])</span> <span class="o">==</span> <span class="n">filename</span> <span class="ow">or</span>
-                 <span class="n">inspect</span><span class="o">.</span><span class="n">getsourcefile</span><span class="p">(</span><span class="n">gl</span><span class="p">[</span><span class="n">f</span><span class="p">])</span> <span class="o">==</span> <span class="n">pytestfile</span> <span class="ow">or</span>
-                 <span class="n">inspect</span><span class="o">.</span><span class="n">getsourcefile</span><span class="p">(</span><span class="n">gl</span><span class="p">[</span><span class="n">f</span><span class="p">])</span> <span class="o">==</span> <span class="n">pytestfile2</span><span class="p">)]</span>
-            <span class="k">if</span> <span class="n">slow</span><span class="p">:</span>
-                <span class="n">funcs</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">funcs</span> <span class="k">if</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;_slow&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)]</span>
-            <span class="c"># Sorting of XFAILed functions isn&#39;t fixed yet :-(</span>
-            <span class="n">funcs</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">inspect</span><span class="o">.</span><span class="n">getsourcelines</span><span class="p">(</span><span class="n">x</span><span class="p">)[</span><span class="mi">1</span><span class="p">])</span>
-            <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="nb">len</span><span class="p">(</span><span class="n">funcs</span><span class="p">):</span>
-                <span class="k">if</span> <span class="n">isgeneratorfunction</span><span class="p">(</span><span class="n">funcs</span><span class="p">[</span><span class="n">i</span><span class="p">]):</span>
-                <span class="c"># some tests can be generators, that return the actual</span>
-                <span class="c"># test functions. We unpack it below:</span>
-                    <span class="n">f</span> <span class="o">=</span> <span class="n">funcs</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-                    <span class="k">for</span> <span class="n">fg</span> <span class="ow">in</span> <span class="n">f</span><span class="p">():</span>
-                        <span class="n">func</span> <span class="o">=</span> <span class="n">fg</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-                        <span class="n">args</span> <span class="o">=</span> <span class="n">fg</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-                        <span class="n">fgw</span> <span class="o">=</span> <span class="k">lambda</span><span class="p">:</span> <span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">)</span>
-                        <span class="n">funcs</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">fgw</span><span class="p">)</span>
-                        <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-            <span class="c"># drop functions that are not selected with the keyword expression:</span>
-            <span class="n">funcs</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">funcs</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">matches</span><span class="p">(</span><span class="n">x</span><span class="p">)]</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">funcs</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">entering_filename</span><span class="p">(</span><span class="n">filename</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">funcs</span><span class="p">))</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">sort</span><span class="p">:</span>
-            <span class="n">random</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="n">funcs</span><span class="p">)</span>
-        <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">funcs</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">entering_test</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;_slow&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">slow</span><span class="p">:</span>
-                    <span class="k">raise</span> <span class="n">Skipped</span><span class="p">(</span><span class="s">&quot;Slow&quot;</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">timeout</span><span class="p">:</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">_timeout</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">timeout</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_seed</span><span class="p">)</span>
-                    <span class="n">f</span><span class="p">()</span>
-            <span class="k">except</span> <span class="ne">KeyboardInterrupt</span><span class="p">:</span>
-                <span class="k">if</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="s">&#39;_slow&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">):</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">test_skip</span><span class="p">(</span><span class="s">&quot;KeyboardInterrupt&quot;</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">raise</span>
-            <span class="k">except</span> <span class="ne">Exception</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">timeout</span><span class="p">:</span>
-                    <span class="n">signal</span><span class="o">.</span><span class="n">alarm</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>  <span class="c"># Disable the alarm. It could not be handled before.</span>
-                <span class="n">t</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">tr</span> <span class="o">=</span> <span class="n">sys</span><span class="o">.</span><span class="n">exc_info</span><span class="p">()</span>
-                <span class="k">if</span> <span class="n">t</span> <span class="ow">is</span> <span class="ne">AssertionError</span><span class="p">:</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">test_fail</span><span class="p">((</span><span class="n">t</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">tr</span><span class="p">))</span>
-                    <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_post_mortem</span><span class="p">:</span>
-                        <span class="n">pdb</span><span class="o">.</span><span class="n">post_mortem</span><span class="p">(</span><span class="n">tr</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="n">t</span><span class="o">.</span><span class="n">__name__</span> <span class="o">==</span> <span class="s">&quot;Skipped&quot;</span><span class="p">:</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">test_skip</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-                <span class="k">elif</span> <span class="n">t</span><span class="o">.</span><span class="n">__name__</span> <span class="o">==</span> <span class="s">&quot;XFail&quot;</span><span class="p">:</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">test_xfail</span><span class="p">()</span>
-                <span class="k">elif</span> <span class="n">t</span><span class="o">.</span><span class="n">__name__</span> <span class="o">==</span> <span class="s">&quot;XPass&quot;</span><span class="p">:</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">test_xpass</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">test_exception</span><span class="p">((</span><span class="n">t</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">tr</span><span class="p">))</span>
-                    <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_post_mortem</span><span class="p">:</span>
-                        <span class="n">pdb</span><span class="o">.</span><span class="n">post_mortem</span><span class="p">(</span><span class="n">tr</span><span class="p">)</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">test_pass</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">leaving_filename</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">_timeout</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">function</span><span class="p">,</span> <span class="n">timeout</span><span class="p">):</span>
-        <span class="k">def</span> <span class="nf">callback</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
-            <span class="n">signal</span><span class="o">.</span><span class="n">alarm</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-            <span class="k">raise</span> <span class="n">Skipped</span><span class="p">(</span><span class="s">&quot;Timeout&quot;</span><span class="p">)</span>
-        <span class="n">signal</span><span class="o">.</span><span class="n">signal</span><span class="p">(</span><span class="n">signal</span><span class="o">.</span><span class="n">SIGALRM</span><span class="p">,</span> <span class="n">callback</span><span class="p">)</span>
-        <span class="n">signal</span><span class="o">.</span><span class="n">alarm</span><span class="p">(</span><span class="n">timeout</span><span class="p">)</span>  <span class="c"># Set an alarm with a given timeout</span>
-        <span class="n">function</span><span class="p">()</span>
-        <span class="n">signal</span><span class="o">.</span><span class="n">alarm</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>  <span class="c"># Disable the alarm</span>
-
-    <span class="k">def</span> <span class="nf">matches</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Does the keyword expression self._kw match &quot;x&quot;? Returns True/False.</span>
-
-<span class="sd">        Always returns True if self._kw is &quot;&quot;.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_kw</span> <span class="o">==</span> <span class="s">&quot;&quot;</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">True</span>
-        <span class="k">return</span> <span class="n">x</span><span class="o">.</span><span class="n">__name__</span><span class="o">.</span><span class="n">find</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_kw</span><span class="p">)</span> <span class="o">!=</span> <span class="o">-</span><span class="mi">1</span>
-
-    <span class="k">def</span> <span class="nf">get_test_files</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">dir</span><span class="p">,</span> <span class="n">pat</span><span class="o">=</span><span class="s">&#39;test_*.py&#39;</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the list of test_*.py (default) files at or below directory</span>
-<span class="sd">        ``dir`` relative to the sympy home directory.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="nb">dir</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_root_dir</span><span class="p">,</span> <span class="n">convert_to_native_paths</span><span class="p">([</span><span class="nb">dir</span><span class="p">])[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="n">g</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">path</span><span class="p">,</span> <span class="n">folders</span><span class="p">,</span> <span class="n">files</span> <span class="ow">in</span> <span class="n">os</span><span class="o">.</span><span class="n">walk</span><span class="p">(</span><span class="nb">dir</span><span class="p">):</span>
-            <span class="n">g</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">path</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">files</span> <span class="k">if</span> <span class="n">fnmatch</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">pat</span><span class="p">)])</span>
-
-        <span class="k">return</span> <span class="p">[</span><span class="n">sys_normcase</span><span class="p">(</span><span class="n">gi</span><span class="p">)</span> <span class="k">for</span> <span class="n">gi</span> <span class="ow">in</span> <span class="n">g</span><span class="p">]</span>
-
-
-<span class="k">class</span> <span class="nc">SymPyDocTests</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">reporter</span><span class="p">,</span> <span class="n">normal</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_count</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_root_dir</span> <span class="o">=</span> <span class="n">sympy_dir</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span> <span class="o">=</span> <span class="n">reporter</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">root_dir</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_root_dir</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_normal</span> <span class="o">=</span> <span class="n">normal</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">_testfiles</span> <span class="o">=</span> <span class="p">[]</span>
-
-    <span class="k">def</span> <span class="nf">test</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Runs the tests and returns True if all tests pass, otherwise False.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">start</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_testfiles</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">test_file</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">KeyboardInterrupt</span><span class="p">:</span>
-                <span class="k">print</span><span class="p">(</span><span class="s">&quot; interrupted by user&quot;</span><span class="p">)</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">finish</span><span class="p">()</span>
-                <span class="k">raise</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">finish</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">test_file</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">filename</span><span class="p">):</span>
-        <span class="n">clear_cache</span><span class="p">()</span>
-
-        <span class="kn">from</span> <span class="nn">io</span> <span class="kn">import</span> <span class="n">StringIO</span>
-
-        <span class="n">rel_name</span> <span class="o">=</span> <span class="n">filename</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_root_dir</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-        <span class="n">dirname</span><span class="p">,</span> <span class="nb">file</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="n">filename</span><span class="p">)</span>
-        <span class="n">module</span> <span class="o">=</span> <span class="n">rel_name</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">sep</span><span class="p">,</span> <span class="s">&#39;.&#39;</span><span class="p">)[:</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span>
-
-        <span class="k">if</span> <span class="n">rel_name</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&quot;examples&quot;</span><span class="p">):</span>
-            <span class="c"># Examples files do not have __init__.py files,</span>
-            <span class="c"># So we have to temporarily extend sys.path to import them</span>
-            <span class="n">sys</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">dirname</span><span class="p">)</span>
-            <span class="n">module</span> <span class="o">=</span> <span class="nb">file</span><span class="p">[:</span><span class="o">-</span><span class="mi">3</span><span class="p">]</span>  <span class="c"># remove &quot;.py&quot;</span>
-        <span class="n">setup_pprint</span><span class="p">()</span>
-        <span class="k">try</span><span class="p">:</span>
-            <span class="n">module</span> <span class="o">=</span> <span class="n">pdoctest</span><span class="o">.</span><span class="n">_normalize_module</span><span class="p">(</span><span class="n">module</span><span class="p">)</span>
-            <span class="n">tests</span> <span class="o">=</span> <span class="n">SymPyDocTestFinder</span><span class="p">()</span><span class="o">.</span><span class="n">find</span><span class="p">(</span><span class="n">module</span><span class="p">)</span>
-        <span class="k">except</span> <span class="p">(</span><span class="ne">SystemExit</span><span class="p">,</span> <span class="ne">KeyboardInterrupt</span><span class="p">):</span>
-            <span class="k">raise</span>
-        <span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">import_error</span><span class="p">(</span><span class="n">filename</span><span class="p">,</span> <span class="n">sys</span><span class="o">.</span><span class="n">exc_info</span><span class="p">())</span>
-            <span class="k">return</span>
-        <span class="k">finally</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">rel_name</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&quot;examples&quot;</span><span class="p">):</span>
-                <span class="k">del</span> <span class="n">sys</span><span class="o">.</span><span class="n">path</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-
-        <span class="n">tests</span> <span class="o">=</span> <span class="p">[</span><span class="n">test</span> <span class="k">for</span> <span class="n">test</span> <span class="ow">in</span> <span class="n">tests</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">test</span><span class="o">.</span><span class="n">examples</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">]</span>
-        <span class="c"># By default tests are sorted by alphabetical order by function name.</span>
-        <span class="c"># We sort by line number so one can edit the file sequentially from</span>
-        <span class="c"># bottom to top. However, if there are decorated functions, their line</span>
-        <span class="c"># numbers will be too large and for now one must just search for these</span>
-        <span class="c"># by text and function name.</span>
-        <span class="n">tests</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="o">-</span><span class="n">x</span><span class="o">.</span><span class="n">lineno</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">tests</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">entering_filename</span><span class="p">(</span><span class="n">filename</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">tests</span><span class="p">))</span>
-        <span class="k">for</span> <span class="n">test</span> <span class="ow">in</span> <span class="n">tests</span><span class="p">:</span>
-            <span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">test</span><span class="o">.</span><span class="n">examples</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span>
-            <span class="n">runner</span> <span class="o">=</span> <span class="n">SymPyDocTestRunner</span><span class="p">(</span><span class="n">optionflags</span><span class="o">=</span><span class="n">pdoctest</span><span class="o">.</span><span class="n">ELLIPSIS</span> <span class="o">|</span>
-                    <span class="n">pdoctest</span><span class="o">.</span><span class="n">NORMALIZE_WHITESPACE</span> <span class="o">|</span>
-                    <span class="n">pdoctest</span><span class="o">.</span><span class="n">IGNORE_EXCEPTION_DETAIL</span><span class="p">)</span>
-            <span class="n">old</span> <span class="o">=</span> <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span>
-            <span class="n">new</span> <span class="o">=</span> <span class="n">StringIO</span><span class="p">()</span>
-            <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span> <span class="o">=</span> <span class="n">new</span>
-            <span class="c"># If the testing is normal, the doctests get importing magic to</span>
-            <span class="c"># provide the global namespace. If not normal (the default) then</span>
-            <span class="c"># then must run on their own; all imports must be explicit within</span>
-            <span class="c"># a function&#39;s docstring. Once imported that import will be</span>
-            <span class="c"># available to the rest of the tests in a given function&#39;s</span>
-            <span class="c"># docstring (unless clear_globs=True below).</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">_normal</span><span class="p">:</span>
-                <span class="n">test</span><span class="o">.</span><span class="n">globs</span> <span class="o">=</span> <span class="p">{}</span>
-                <span class="c"># if this is uncommented then all the test would get is what</span>
-                <span class="c"># comes by default with a &quot;from sympy import *&quot;</span>
-                <span class="c">#exec(&#39;from sympy import *&#39;) in test.globs</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">f</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">runner</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">test</span><span class="p">,</span> <span class="n">out</span><span class="o">=</span><span class="n">new</span><span class="o">.</span><span class="n">write</span><span class="p">,</span> <span class="n">clear_globs</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-            <span class="k">except</span> <span class="ne">KeyboardInterrupt</span><span class="p">:</span>
-                <span class="k">raise</span>
-            <span class="k">finally</span><span class="p">:</span>
-                <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span> <span class="o">=</span> <span class="n">old</span>
-            <span class="k">if</span> <span class="n">f</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">doctest_fail</span><span class="p">(</span><span class="n">test</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="n">new</span><span class="o">.</span><span class="n">getvalue</span><span class="p">())</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">test_pass</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_reporter</span><span class="o">.</span><span class="n">leaving_filename</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">get_test_files</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">dir</span><span class="p">,</span> <span class="n">pat</span><span class="o">=</span><span class="s">&#39;*.py&#39;</span><span class="p">,</span> <span class="n">init_only</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Returns the list of \*.py files (default) from which docstrings</span>
-<span class="sd">        will be tested which are at or below directory ``dir``. By default,</span>
-<span class="sd">        only those that have an __init__.py in their parent directory</span>
-<span class="sd">        and do not start with ``test_`` will be included.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">def</span> <span class="nf">importable</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-            <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">            Checks if given pathname x is an importable module by checking for</span>
-<span class="sd">            __init__.py file.</span>
-
-<span class="sd">            Returns True/False.</span>
-
-<span class="sd">            Currently we only test if the __init__.py file exists in the</span>
-<span class="sd">            directory with the file &quot;x&quot; (in theory we should also test all the</span>
-<span class="sd">            parent dirs).</span>
-<span class="sd">            &quot;&quot;&quot;</span>
-            <span class="n">init_py</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">dirname</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="s">&quot;__init__.py&quot;</span><span class="p">)</span>
-            <span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">init_py</span><span class="p">)</span>
-
-        <span class="nb">dir</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_root_dir</span><span class="p">,</span> <span class="n">convert_to_native_paths</span><span class="p">([</span><span class="nb">dir</span><span class="p">])[</span><span class="mi">0</span><span class="p">])</span>
-
-        <span class="n">g</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="k">for</span> <span class="n">path</span><span class="p">,</span> <span class="n">folders</span><span class="p">,</span> <span class="n">files</span> <span class="ow">in</span> <span class="n">os</span><span class="o">.</span><span class="n">walk</span><span class="p">(</span><span class="nb">dir</span><span class="p">):</span>
-            <span class="n">g</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">path</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">files</span>
-                      <span class="k">if</span> <span class="ow">not</span> <span class="n">f</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39;test_&#39;</span><span class="p">)</span> <span class="ow">and</span> <span class="n">fnmatch</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">pat</span><span class="p">)])</span>
-        <span class="k">if</span> <span class="n">init_only</span><span class="p">:</span>
-            <span class="c"># skip files that are not importable (i.e. missing __init__.py)</span>
-            <span class="n">g</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">g</span> <span class="k">if</span> <span class="n">importable</span><span class="p">(</span><span class="n">x</span><span class="p">)]</span>
-
-        <span class="k">return</span> <span class="p">[</span><span class="n">sys_normcase</span><span class="p">(</span><span class="n">gi</span><span class="p">)</span> <span class="k">for</span> <span class="n">gi</span> <span class="ow">in</span> <span class="n">g</span><span class="p">]</span>
-
-
-<div class="viewcode-block" id="SymPyDocTestFinder"><a class="viewcode-back" href="../../../modules/utilities/runtests.html#sympy.utilities.runtests.SymPyDocTestFinder">[docs]</a><span class="k">class</span> <span class="nc">SymPyDocTestFinder</span><span class="p">(</span><span class="n">DocTestFinder</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A class used to extract the DocTests that are relevant to a given</span>
-<span class="sd">    object, from its docstring and the docstrings of its contained</span>
-<span class="sd">    objects.  Doctests can currently be extracted from the following</span>
-<span class="sd">    object types: modules, functions, classes, methods, staticmethods,</span>
-<span class="sd">    classmethods, and properties.</span>
-
-<span class="sd">    Modified from doctest&#39;s version by looking harder for code in the</span>
-<span class="sd">    case that it looks like the the code comes from a different module.</span>
-<span class="sd">    In the case of decorated functions (e.g. @vectorize) they appear</span>
-<span class="sd">    to come from a different module (e.g. multidemensional) even though</span>
-<span class="sd">    their code is not there.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">_find</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">tests</span><span class="p">,</span> <span class="n">obj</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">module</span><span class="p">,</span> <span class="n">source_lines</span><span class="p">,</span> <span class="n">globs</span><span class="p">,</span> <span class="n">seen</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Find tests for the given object and any contained objects, and</span>
-<span class="sd">        add them to ``tests``.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_verbose</span><span class="p">:</span>
-            <span class="k">print</span><span class="p">(</span><span class="s">&#39;Finding tests in </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">name</span><span class="p">)</span>
-
-        <span class="c"># If we&#39;ve already processed this object, then ignore it.</span>
-        <span class="k">if</span> <span class="nb">id</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span> <span class="ow">in</span> <span class="n">seen</span><span class="p">:</span>
-            <span class="k">return</span>
-        <span class="n">seen</span><span class="p">[</span><span class="nb">id</span><span class="p">(</span><span class="n">obj</span><span class="p">)]</span> <span class="o">=</span> <span class="mi">1</span>
-
-        <span class="c"># Make sure we don&#39;t run doctests for classes outside of sympy, such</span>
-        <span class="c"># as in numpy or scipy.</span>
-        <span class="k">if</span> <span class="n">inspect</span><span class="o">.</span><span class="n">isclass</span><span class="p">(</span><span class="n">obj</span><span class="p">):</span>
-            <span class="k">if</span> <span class="n">obj</span><span class="o">.</span><span class="n">__module__</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;.&#39;</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="s">&#39;sympy&#39;</span><span class="p">:</span>
-                <span class="k">return</span>
-
-        <span class="c"># Find a test for this object, and add it to the list of tests.</span>
-        <span class="n">test</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_test</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">module</span><span class="p">,</span> <span class="n">globs</span><span class="p">,</span> <span class="n">source_lines</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">test</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">tests</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">test</span><span class="p">)</span>
-
-        <span class="c"># Look for tests in a module&#39;s contained objects.</span>
-        <span class="k">if</span> <span class="n">inspect</span><span class="o">.</span><span class="n">ismodule</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">_recurse</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">rawname</span><span class="p">,</span> <span class="n">val</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">obj</span><span class="o">.</span><span class="n">__dict__</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-                <span class="c"># Recurse to functions &amp; classes.</span>
-                <span class="k">if</span> <span class="n">inspect</span><span class="o">.</span><span class="n">isfunction</span><span class="p">(</span><span class="n">val</span><span class="p">)</span> <span class="ow">or</span> <span class="n">inspect</span><span class="o">.</span><span class="n">isclass</span><span class="p">(</span><span class="n">val</span><span class="p">):</span>
-                    <span class="n">in_module</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_from_module</span><span class="p">(</span><span class="n">module</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">in_module</span><span class="p">:</span>
-                        <span class="c"># double check in case this function is decorated</span>
-                        <span class="c"># and just appears to come from a different module.</span>
-                        <span class="n">pat</span> <span class="o">=</span> <span class="s">r&#39;\s*(def|class)\s+</span><span class="si">%s</span><span class="s">\s*\(&#39;</span> <span class="o">%</span> <span class="n">rawname</span>
-                        <span class="n">PAT</span> <span class="o">=</span> <span class="n">pre</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="n">pat</span><span class="p">)</span>
-                        <span class="n">in_module</span> <span class="o">=</span> <span class="nb">any</span><span class="p">(</span>
-                            <span class="n">PAT</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">line</span><span class="p">)</span> <span class="k">for</span> <span class="n">line</span> <span class="ow">in</span> <span class="n">source_lines</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">in_module</span><span class="p">:</span>
-                        <span class="k">try</span><span class="p">:</span>
-                            <span class="n">valname</span> <span class="o">=</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">.</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">rawname</span><span class="p">)</span>
-                            <span class="bp">self</span><span class="o">.</span><span class="n">_find</span><span class="p">(</span><span class="n">tests</span><span class="p">,</span> <span class="n">val</span><span class="p">,</span> <span class="n">valname</span><span class="p">,</span> <span class="n">module</span><span class="p">,</span>
-                                <span class="n">source_lines</span><span class="p">,</span> <span class="n">globs</span><span class="p">,</span> <span class="n">seen</span><span class="p">)</span>
-                        <span class="k">except</span> <span class="ne">KeyboardInterrupt</span><span class="p">:</span>
-                            <span class="k">raise</span>
-                        <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-                            <span class="k">raise</span>
-                        <span class="k">except</span> <span class="ne">Exception</span><span class="p">:</span>
-                            <span class="k">pass</span>
-
-        <span class="c"># Look for tests in a module&#39;s __test__ dictionary.</span>
-        <span class="k">if</span> <span class="n">inspect</span><span class="o">.</span><span class="n">ismodule</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">_recurse</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">valname</span><span class="p">,</span> <span class="n">val</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="nb">getattr</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="s">&#39;__test__&#39;</span><span class="p">,</span> <span class="p">{})</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">valname</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-                    <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;SymPyDocTestFinder.find: __test__ keys &quot;</span>
-                                     <span class="s">&quot;must be strings: </span><span class="si">%r</span><span class="s">&quot;</span> <span class="o">%</span>
-                                     <span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">valname</span><span class="p">),))</span>
-                <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">inspect</span><span class="o">.</span><span class="n">isfunction</span><span class="p">(</span><span class="n">val</span><span class="p">)</span> <span class="ow">or</span> <span class="n">inspect</span><span class="o">.</span><span class="n">isclass</span><span class="p">(</span><span class="n">val</span><span class="p">)</span> <span class="ow">or</span>
-                        <span class="n">inspect</span><span class="o">.</span><span class="n">ismethod</span><span class="p">(</span><span class="n">val</span><span class="p">)</span> <span class="ow">or</span> <span class="n">inspect</span><span class="o">.</span><span class="n">ismodule</span><span class="p">(</span><span class="n">val</span><span class="p">)</span> <span class="ow">or</span>
-                        <span class="nb">isinstance</span><span class="p">(</span><span class="n">val</span><span class="p">,</span> <span class="nb">str</span><span class="p">)):</span>
-                    <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;SymPyDocTestFinder.find: __test__ values &quot;</span>
-                                     <span class="s">&quot;must be strings, functions, methods, &quot;</span>
-                                     <span class="s">&quot;classes, or modules: </span><span class="si">%r</span><span class="s">&quot;</span> <span class="o">%</span>
-                                     <span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">val</span><span class="p">),))</span>
-                <span class="n">valname</span> <span class="o">=</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">.__test__.</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">valname</span><span class="p">)</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">_find</span><span class="p">(</span><span class="n">tests</span><span class="p">,</span> <span class="n">val</span><span class="p">,</span> <span class="n">valname</span><span class="p">,</span> <span class="n">module</span><span class="p">,</span> <span class="n">source_lines</span><span class="p">,</span>
-                           <span class="n">globs</span><span class="p">,</span> <span class="n">seen</span><span class="p">)</span>
-
-        <span class="c"># Look for tests in a class&#39;s contained objects.</span>
-        <span class="k">if</span> <span class="n">inspect</span><span class="o">.</span><span class="n">isclass</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">_recurse</span><span class="p">:</span>
-            <span class="k">for</span> <span class="n">valname</span><span class="p">,</span> <span class="n">val</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">obj</span><span class="o">.</span><span class="n">__dict__</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-                <span class="c"># Special handling for staticmethod/classmethod.</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">val</span><span class="p">,</span> <span class="nb">staticmethod</span><span class="p">):</span>
-                    <span class="n">val</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="n">valname</span><span class="p">)</span>
-                <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">val</span><span class="p">,</span> <span class="nb">classmethod</span><span class="p">):</span>
-                    <span class="n">val</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="n">valname</span><span class="p">)</span><span class="o">.</span><span class="n">__func__</span>
-
-                <span class="c"># Recurse to methods, properties, and nested classes.</span>
-                <span class="k">if</span> <span class="p">(</span><span class="n">inspect</span><span class="o">.</span><span class="n">isfunction</span><span class="p">(</span><span class="n">val</span><span class="p">)</span> <span class="ow">or</span>
-                    <span class="n">inspect</span><span class="o">.</span><span class="n">isclass</span><span class="p">(</span><span class="n">val</span><span class="p">)</span> <span class="ow">or</span>
-                        <span class="nb">isinstance</span><span class="p">(</span><span class="n">val</span><span class="p">,</span> <span class="nb">property</span><span class="p">)):</span>
-                    <span class="n">in_module</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_from_module</span><span class="p">(</span><span class="n">module</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="n">in_module</span><span class="p">:</span>
-                        <span class="c"># &quot;double check&quot; again</span>
-                        <span class="n">pat</span> <span class="o">=</span> <span class="s">r&#39;\s*(def|class)\s+</span><span class="si">%s</span><span class="s">\s*\(&#39;</span> <span class="o">%</span> <span class="n">valname</span>
-                        <span class="n">PAT</span> <span class="o">=</span> <span class="n">pre</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="n">pat</span><span class="p">)</span>
-                        <span class="n">in_module</span> <span class="o">=</span> <span class="nb">any</span><span class="p">(</span><span class="n">PAT</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">line</span><span class="p">)</span> <span class="k">for</span> <span class="n">line</span> <span class="ow">in</span>
-                            <span class="n">source_lines</span><span class="p">)</span>
-                    <span class="k">if</span> <span class="n">in_module</span><span class="p">:</span>
-                        <span class="n">valname</span> <span class="o">=</span> <span class="s">&#39;</span><span class="si">%s</span><span class="s">.</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">valname</span><span class="p">)</span>
-                        <span class="bp">self</span><span class="o">.</span><span class="n">_find</span><span class="p">(</span><span class="n">tests</span><span class="p">,</span> <span class="n">val</span><span class="p">,</span> <span class="n">valname</span><span class="p">,</span> <span class="n">module</span><span class="p">,</span> <span class="n">source_lines</span><span class="p">,</span>
-                                   <span class="n">globs</span><span class="p">,</span> <span class="n">seen</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">_get_test</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">obj</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">module</span><span class="p">,</span> <span class="n">globs</span><span class="p">,</span> <span class="n">source_lines</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Return a DocTest for the given object, if it defines a docstring;</span>
-<span class="sd">        otherwise, return None.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="c"># Extract the object&#39;s docstring.  If it doesn&#39;t have one,</span>
-        <span class="c"># then return None (no test for this object).</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-            <span class="n">docstring</span> <span class="o">=</span> <span class="n">obj</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="k">try</span><span class="p">:</span>
-                <span class="k">if</span> <span class="n">obj</span><span class="o">.</span><span class="n">__doc__</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">docstring</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">docstring</span> <span class="o">=</span> <span class="n">obj</span><span class="o">.</span><span class="n">__doc__</span>
-                    <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">docstring</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-                        <span class="n">docstring</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">docstring</span><span class="p">)</span>
-            <span class="k">except</span> <span class="p">(</span><span class="ne">TypeError</span><span class="p">,</span> <span class="ne">AttributeError</span><span class="p">):</span>
-                <span class="n">docstring</span> <span class="o">=</span> <span class="s">&#39;&#39;</span>
-
-        <span class="c"># Find the docstring&#39;s location in the file.</span>
-        <span class="n">lineno</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_find_lineno</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="n">source_lines</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">lineno</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="c"># if None, then _find_lineno couldn&#39;t find the docstring.</span>
-            <span class="c"># But IT IS STILL THERE.  Likely it was decorated or something</span>
-            <span class="c"># (i.e., @property docstrings have lineno == None)</span>
-            <span class="c"># TODO: Write our own _find_lineno that is smarter in this regard.</span>
-            <span class="c"># Until then, just give it a dummy lineno.  This is just used for</span>
-            <span class="c"># sorting the tests, so the only bad effect is that they will</span>
-            <span class="c"># appear last instead of in the order that they appear in the file.</span>
-            <span class="c"># lineno is also used to report the offending line of a failing</span>
-            <span class="c"># doctest, which is another reason to fix this.  See issue 1947.</span>
-            <span class="n">lineno</span> <span class="o">=</span> <span class="mi">0</span>
-
-        <span class="c"># Don&#39;t bother if the docstring is empty.</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_exclude_empty</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">docstring</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">None</span>
-
-        <span class="c"># Return a DocTest for this object.</span>
-        <span class="k">if</span> <span class="n">module</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">filename</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">filename</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">module</span><span class="p">,</span> <span class="s">&#39;__file__&#39;</span><span class="p">,</span> <span class="n">module</span><span class="o">.</span><span class="n">__name__</span><span class="p">)</span>
-            <span class="k">if</span> <span class="n">filename</span><span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">:]</span> <span class="ow">in</span> <span class="p">(</span><span class="s">&quot;.pyc&quot;</span><span class="p">,</span> <span class="s">&quot;.pyo&quot;</span><span class="p">):</span>
-                <span class="n">filename</span> <span class="o">=</span> <span class="n">filename</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_parser</span><span class="o">.</span><span class="n">get_doctest</span><span class="p">(</span><span class="n">docstring</span><span class="p">,</span> <span class="n">globs</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span>
-                                        <span class="n">filename</span><span class="p">,</span> <span class="n">lineno</span><span class="p">)</span>
-
-</div>
-<div class="viewcode-block" id="SymPyDocTestRunner"><a class="viewcode-back" href="../../../modules/utilities/runtests.html#sympy.utilities.runtests.SymPyDocTestRunner">[docs]</a><span class="k">class</span> <span class="nc">SymPyDocTestRunner</span><span class="p">(</span><span class="n">DocTestRunner</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    A class used to run DocTest test cases, and accumulate statistics.</span>
-<span class="sd">    The ``run`` method is used to process a single DocTest case.  It</span>
-<span class="sd">    returns a tuple ``(f, t)``, where ``t`` is the number of test cases</span>
-<span class="sd">    tried, and ``f`` is the number of test cases that failed.</span>
-
-<span class="sd">    Modified from the doctest version to not reset the sys.displayhook (see</span>
-<span class="sd">    issue 2041).</span>
-
-<span class="sd">    See the docstring of the original DocTestRunner for more information.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-<div class="viewcode-block" id="SymPyDocTestRunner.run"><a class="viewcode-back" href="../../../modules/utilities/runtests.html#sympy.utilities.runtests.SymPyDocTestRunner.run">[docs]</a>    <span class="k">def</span> <span class="nf">run</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">test</span><span class="p">,</span> <span class="n">compileflags</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">out</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">clear_globs</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Run the examples in ``test``, and display the results using the</span>
-<span class="sd">        writer function ``out``.</span>
-
-<span class="sd">        The examples are run in the namespace ``test.globs``.  If</span>
-<span class="sd">        ``clear_globs`` is true (the default), then this namespace will</span>
-<span class="sd">        be cleared after the test runs, to help with garbage</span>
-<span class="sd">        collection.  If you would like to examine the namespace after</span>
-<span class="sd">        the test completes, then use ``clear_globs=False``.</span>
-
-<span class="sd">        ``compileflags`` gives the set of flags that should be used by</span>
-<span class="sd">        the Python compiler when running the examples.  If not</span>
-<span class="sd">        specified, then it will default to the set of future-import</span>
-<span class="sd">        flags that apply to ``globs``.</span>
-
-<span class="sd">        The output of each example is checked using</span>
-<span class="sd">        ``SymPyDocTestRunner.check_output``, and the results are</span>
-<span class="sd">        formatted by the ``SymPyDocTestRunner.report_*`` methods.</span>
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">test</span> <span class="o">=</span> <span class="n">test</span>
-
-        <span class="k">if</span> <span class="n">compileflags</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">compileflags</span> <span class="o">=</span> <span class="n">pdoctest</span><span class="o">.</span><span class="n">_extract_future_flags</span><span class="p">(</span><span class="n">test</span><span class="o">.</span><span class="n">globs</span><span class="p">)</span>
-
-        <span class="n">save_stdout</span> <span class="o">=</span> <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span>
-        <span class="k">if</span> <span class="n">out</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">out</span> <span class="o">=</span> <span class="n">save_stdout</span><span class="o">.</span><span class="n">write</span>
-        <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_fakeout</span>
-
-        <span class="c"># Patch pdb.set_trace to restore sys.stdout during interactive</span>
-        <span class="c"># debugging (so it&#39;s not still redirected to self._fakeout).</span>
-        <span class="c"># Note that the interactive output will go to *our*</span>
-        <span class="c"># save_stdout, even if that&#39;s not the real sys.stdout; this</span>
-        <span class="c"># allows us to write test cases for the set_trace behavior.</span>
-        <span class="n">save_set_trace</span> <span class="o">=</span> <span class="n">pdb</span><span class="o">.</span><span class="n">set_trace</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">debugger</span> <span class="o">=</span> <span class="n">pdoctest</span><span class="o">.</span><span class="n">_OutputRedirectingPdb</span><span class="p">(</span><span class="n">save_stdout</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">debugger</span><span class="o">.</span><span class="n">reset</span><span class="p">()</span>
-        <span class="n">pdb</span><span class="o">.</span><span class="n">set_trace</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">debugger</span><span class="o">.</span><span class="n">set_trace</span>
-
-        <span class="c"># Patch linecache.getlines, so we can see the example&#39;s source</span>
-        <span class="c"># when we&#39;re inside the debugger.</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">save_linecache_getlines</span> <span class="o">=</span> <span class="n">pdoctest</span><span class="o">.</span><span class="n">linecache</span><span class="o">.</span><span class="n">getlines</span>
-        <span class="n">linecache</span><span class="o">.</span><span class="n">getlines</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">__patched_linecache_getlines</span>
-
-        <span class="k">try</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">__run</span><span class="p">(</span><span class="n">test</span><span class="p">,</span> <span class="n">compileflags</span><span class="p">,</span> <span class="n">out</span><span class="p">)</span>
-        <span class="k">finally</span><span class="p">:</span>
-            <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span> <span class="o">=</span> <span class="n">save_stdout</span>
-            <span class="n">pdb</span><span class="o">.</span><span class="n">set_trace</span> <span class="o">=</span> <span class="n">save_set_trace</span>
-            <span class="n">linecache</span><span class="o">.</span><span class="n">getlines</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">save_linecache_getlines</span>
-            <span class="k">if</span> <span class="n">clear_globs</span><span class="p">:</span>
-                <span class="n">test</span><span class="o">.</span><span class="n">globs</span><span class="o">.</span><span class="n">clear</span><span class="p">()</span>
-
-<span class="c"># We have to override the name mangled methods.</span></div></div>
-<span class="n">SymPyDocTestRunner</span><span class="o">.</span><span class="n">_SymPyDocTestRunner__patched_linecache_getlines</span> <span class="o">=</span> \
-    <span class="n">DocTestRunner</span><span class="o">.</span><span class="n">_DocTestRunner__patched_linecache_getlines</span>
-<span class="n">SymPyDocTestRunner</span><span class="o">.</span><span class="n">_SymPyDocTestRunner__run</span> <span class="o">=</span> <span class="n">DocTestRunner</span><span class="o">.</span><span class="n">_DocTestRunner__run</span>
-<span class="n">SymPyDocTestRunner</span><span class="o">.</span><span class="n">_SymPyDocTestRunner__record_outcome</span> <span class="o">=</span> \
-    <span class="n">DocTestRunner</span><span class="o">.</span><span class="n">_DocTestRunner__record_outcome</span>
-
-
-<div class="viewcode-block" id="Reporter"><a class="viewcode-back" href="../../../modules/utilities/runtests.html#sympy.utilities.runtests.Reporter">[docs]</a><span class="k">class</span> <span class="nc">Reporter</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Parent class for all reporters.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">pass</span>
-
-</div>
-<div class="viewcode-block" id="PyTestReporter"><a class="viewcode-back" href="../../../modules/utilities/runtests.html#sympy.utilities.runtests.PyTestReporter">[docs]</a><span class="k">class</span> <span class="nc">PyTestReporter</span><span class="p">(</span><span class="n">Reporter</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Py.test like reporter. Should produce output identical to py.test.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-
-    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">tb</span><span class="o">=</span><span class="s">&quot;short&quot;</span><span class="p">,</span> <span class="n">colors</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span>
-                 <span class="n">force_colors</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_verbose</span> <span class="o">=</span> <span class="n">verbose</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_tb_style</span> <span class="o">=</span> <span class="n">tb</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_colors</span> <span class="o">=</span> <span class="n">colors</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_force_colors</span> <span class="o">=</span> <span class="n">force_colors</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_xfailed</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_xpassed</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_failed</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_failed_doctest</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_passed</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_skipped</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_exceptions</span> <span class="o">=</span> <span class="p">[]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_terminal_width</span> <span class="o">=</span> <span class="bp">None</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_default_width</span> <span class="o">=</span> <span class="mi">80</span>
-
-        <span class="c"># this tracks the x-position of the cursor (useful for positioning</span>
-        <span class="c"># things on the screen), without the need for any readline library:</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_write_pos</span> <span class="o">=</span> <span class="mi">0</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_line_wrap</span> <span class="o">=</span> <span class="bp">False</span>
-
-    <span class="k">def</span> <span class="nf">root_dir</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="nb">dir</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_root_dir</span> <span class="o">=</span> <span class="nb">dir</span>
-
-    <span class="nd">@property</span>
-    <span class="k">def</span> <span class="nf">terminal_width</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_terminal_width</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_terminal_width</span>
-
-        <span class="k">def</span> <span class="nf">findout_terminal_width</span><span class="p">():</span>
-            <span class="k">if</span> <span class="n">sys</span><span class="o">.</span><span class="n">platform</span> <span class="o">==</span> <span class="s">&quot;win32&quot;</span><span class="p">:</span>
-                <span class="c"># Windows support is based on:</span>
-                <span class="c">#</span>
-                <span class="c">#  http://code.activestate.com/recipes/</span>
-                <span class="c">#  440694-determine-size-of-console-window-on-windows/</span>
-
-                <span class="kn">from</span> <span class="nn">ctypes</span> <span class="kn">import</span> <span class="n">windll</span><span class="p">,</span> <span class="n">create_string_buffer</span>
-
-                <span class="n">h</span> <span class="o">=</span> <span class="n">windll</span><span class="o">.</span><span class="n">kernel32</span><span class="o">.</span><span class="n">GetStdHandle</span><span class="p">(</span><span class="o">-</span><span class="mi">12</span><span class="p">)</span>
-                <span class="n">csbi</span> <span class="o">=</span> <span class="n">create_string_buffer</span><span class="p">(</span><span class="mi">22</span><span class="p">)</span>
-                <span class="n">res</span> <span class="o">=</span> <span class="n">windll</span><span class="o">.</span><span class="n">kernel32</span><span class="o">.</span><span class="n">GetConsoleScreenBufferInfo</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">csbi</span><span class="p">)</span>
-
-                <span class="k">if</span> <span class="n">res</span><span class="p">:</span>
-                    <span class="kn">import</span> <span class="nn">struct</span>
-                    <span class="p">(</span><span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">left</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">right</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span><span class="p">)</span> <span class="o">=</span> \
-                        <span class="n">struct</span><span class="o">.</span><span class="n">unpack</span><span class="p">(</span><span class="s">&quot;hhhhHhhhhhh&quot;</span><span class="p">,</span> <span class="n">csbi</span><span class="o">.</span><span class="n">raw</span><span class="p">)</span>
-                    <span class="k">return</span> <span class="n">right</span> <span class="o">-</span> <span class="n">left</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_default_width</span>
-
-            <span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">sys</span><span class="o">.</span><span class="n">stdout</span><span class="p">,</span> <span class="s">&#39;isatty&#39;</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span><span class="o">.</span><span class="n">isatty</span><span class="p">():</span>
-                <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_default_width</span>  <span class="c"># leave PIPEs alone</span>
-
-            <span class="k">try</span><span class="p">:</span>
-                <span class="n">process</span> <span class="o">=</span> <span class="n">subprocess</span><span class="o">.</span><span class="n">Popen</span><span class="p">([</span><span class="s">&#39;stty&#39;</span><span class="p">,</span> <span class="s">&#39;-a&#39;</span><span class="p">],</span>
-                    <span class="n">stdout</span><span class="o">=</span><span class="n">subprocess</span><span class="o">.</span><span class="n">PIPE</span><span class="p">,</span>
-                    <span class="n">stderr</span><span class="o">=</span><span class="n">subprocess</span><span class="o">.</span><span class="n">PIPE</span><span class="p">)</span>
-                <span class="n">stdout</span> <span class="o">=</span> <span class="n">process</span><span class="o">.</span><span class="n">stdout</span><span class="o">.</span><span class="n">read</span><span class="p">()</span>
-                <span class="k">if</span> <span class="n">sys</span><span class="o">.</span><span class="n">version_info</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-                    <span class="n">stdout</span> <span class="o">=</span> <span class="n">stdout</span><span class="o">.</span><span class="n">decode</span><span class="p">(</span><span class="s">&quot;utf-8&quot;</span><span class="p">)</span>
-            <span class="k">except</span> <span class="p">(</span><span class="ne">OSError</span><span class="p">,</span> <span class="ne">IOError</span><span class="p">):</span>
-                <span class="k">pass</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="c"># We support the following output formats from stty:</span>
-                <span class="c">#</span>
-                <span class="c"># 1) Linux   -&gt; columns 80</span>
-                <span class="c"># 2) OS X    -&gt; 80 columns</span>
-                <span class="c"># 3) Solaris -&gt; columns = 80</span>
-
-                <span class="n">re_linux</span> <span class="o">=</span> <span class="s">r&quot;columns\s+(?P&lt;columns&gt;\d+);&quot;</span>
-                <span class="n">re_osx</span> <span class="o">=</span> <span class="s">r&quot;(?P&lt;columns&gt;\d+)\s*columns;&quot;</span>
-                <span class="n">re_solaris</span> <span class="o">=</span> <span class="s">r&quot;columns\s+=\s+(?P&lt;columns&gt;\d+);&quot;</span>
-
-                <span class="k">for</span> <span class="n">regex</span> <span class="ow">in</span> <span class="p">(</span><span class="n">re_linux</span><span class="p">,</span> <span class="n">re_osx</span><span class="p">,</span> <span class="n">re_solaris</span><span class="p">):</span>
-                    <span class="n">match</span> <span class="o">=</span> <span class="n">re</span><span class="o">.</span><span class="n">search</span><span class="p">(</span><span class="n">regex</span><span class="p">,</span> <span class="n">stdout</span><span class="p">)</span>
-
-                    <span class="k">if</span> <span class="n">match</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                        <span class="n">columns</span> <span class="o">=</span> <span class="n">match</span><span class="o">.</span><span class="n">group</span><span class="p">(</span><span class="s">&#39;columns&#39;</span><span class="p">)</span>
-
-                        <span class="k">try</span><span class="p">:</span>
-                            <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">columns</span><span class="p">)</span>
-                        <span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
-                            <span class="k">pass</span>
-
-            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_default_width</span>
-
-        <span class="n">width</span> <span class="o">=</span> <span class="n">findout_terminal_width</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_terminal_width</span> <span class="o">=</span> <span class="n">width</span>
-
-        <span class="k">return</span> <span class="n">width</span>
-
-<div class="viewcode-block" id="PyTestReporter.write"><a class="viewcode-back" href="../../../modules/utilities/runtests.html#sympy.utilities.runtests.PyTestReporter.write">[docs]</a>    <span class="k">def</span> <span class="nf">write</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">text</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s">&quot;&quot;</span><span class="p">,</span> <span class="n">align</span><span class="o">=</span><span class="s">&quot;left&quot;</span><span class="p">,</span> <span class="n">width</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span>
-              <span class="n">force_colors</span><span class="o">=</span><span class="bp">False</span><span class="p">):</span>
-        <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">        Prints a text on the screen.</span>
-
-<span class="sd">        It uses sys.stdout.write(), so no readline library is necessary.</span>
-
-<span class="sd">        Parameters</span>
-<span class="sd">        ==========</span>
-
-<span class="sd">        color : choose from the colors below, &quot;&quot; means default color</span>
-<span class="sd">        align : &quot;left&quot;/&quot;right&quot;, &quot;left&quot; is a normal print, &quot;right&quot; is aligned on</span>
-<span class="sd">                the right-hand side of the screen, filled with spaces if</span>
-<span class="sd">                necessary</span>
-<span class="sd">        width : the screen width</span>
-
-<span class="sd">        &quot;&quot;&quot;</span>
-        <span class="n">color_templates</span> <span class="o">=</span> <span class="p">(</span>
-            <span class="p">(</span><span class="s">&quot;Black&quot;</span><span class="p">,</span> <span class="s">&quot;0;30&quot;</span><span class="p">),</span>
-            <span class="p">(</span><span class="s">&quot;Red&quot;</span><span class="p">,</span> <span class="s">&quot;0;31&quot;</span><span class="p">),</span>
-            <span class="p">(</span><span class="s">&quot;Green&quot;</span><span class="p">,</span> <span class="s">&quot;0;32&quot;</span><span class="p">),</span>
-            <span class="p">(</span><span class="s">&quot;Brown&quot;</span><span class="p">,</span> <span class="s">&quot;0;33&quot;</span><span class="p">),</span>
-            <span class="p">(</span><span class="s">&quot;Blue&quot;</span><span class="p">,</span> <span class="s">&quot;0;34&quot;</span><span class="p">),</span>
-            <span class="p">(</span><span class="s">&quot;Purple&quot;</span><span class="p">,</span> <span class="s">&quot;0;35&quot;</span><span class="p">),</span>
-            <span class="p">(</span><span class="s">&quot;Cyan&quot;</span><span class="p">,</span> <span class="s">&quot;0;36&quot;</span><span class="p">),</span>
-            <span class="p">(</span><span class="s">&quot;LightGray&quot;</span><span class="p">,</span> <span class="s">&quot;0;37&quot;</span><span class="p">),</span>
-            <span class="p">(</span><span class="s">&quot;DarkGray&quot;</span><span class="p">,</span> <span class="s">&quot;1;30&quot;</span><span class="p">),</span>
-            <span class="p">(</span><span class="s">&quot;LightRed&quot;</span><span class="p">,</span> <span class="s">&quot;1;31&quot;</span><span class="p">),</span>
-            <span class="p">(</span><span class="s">&quot;LightGreen&quot;</span><span class="p">,</span> <span class="s">&quot;1;32&quot;</span><span class="p">),</span>
-            <span class="p">(</span><span class="s">&quot;Yellow&quot;</span><span class="p">,</span> <span class="s">&quot;1;33&quot;</span><span class="p">),</span>
-            <span class="p">(</span><span class="s">&quot;LightBlue&quot;</span><span class="p">,</span> <span class="s">&quot;1;34&quot;</span><span class="p">),</span>
-            <span class="p">(</span><span class="s">&quot;LightPurple&quot;</span><span class="p">,</span> <span class="s">&quot;1;35&quot;</span><span class="p">),</span>
-            <span class="p">(</span><span class="s">&quot;LightCyan&quot;</span><span class="p">,</span> <span class="s">&quot;1;36&quot;</span><span class="p">),</span>
-            <span class="p">(</span><span class="s">&quot;White&quot;</span><span class="p">,</span> <span class="s">&quot;1;37&quot;</span><span class="p">),</span>
-        <span class="p">)</span>
-
-        <span class="n">colors</span> <span class="o">=</span> <span class="p">{}</span>
-
-        <span class="k">for</span> <span class="n">name</span><span class="p">,</span> <span class="n">value</span> <span class="ow">in</span> <span class="n">color_templates</span><span class="p">:</span>
-            <span class="n">colors</span><span class="p">[</span><span class="n">name</span><span class="p">]</span> <span class="o">=</span> <span class="n">value</span>
-        <span class="n">c_normal</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\033</span><span class="s">[0m&#39;</span>
-        <span class="n">c_color</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\033</span><span class="s">[</span><span class="si">%s</span><span class="s">m&#39;</span>
-
-        <span class="k">if</span> <span class="n">width</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="n">width</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">terminal_width</span>
-
-        <span class="k">if</span> <span class="n">align</span> <span class="o">==</span> <span class="s">&quot;right&quot;</span><span class="p">:</span>
-            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_write_pos</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">text</span><span class="p">)</span> <span class="o">&gt;</span> <span class="n">width</span><span class="p">:</span>
-                <span class="c"># we don&#39;t fit on the current line, create a new line</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot; &quot;</span><span class="o">*</span><span class="p">(</span><span class="n">width</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_write_pos</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">text</span><span class="p">)))</span>
-
-        <span class="k">if</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">_force_colors</span> <span class="ow">and</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">sys</span><span class="o">.</span><span class="n">stdout</span><span class="p">,</span> <span class="s">&#39;isatty&#39;</span><span class="p">)</span> <span class="ow">and</span> <span class="ow">not</span> \
-                <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span><span class="o">.</span><span class="n">isatty</span><span class="p">():</span>
-            <span class="c"># the stdout is not a terminal, this for example happens if the</span>
-            <span class="c"># output is piped to less, e.g. &quot;bin/test | less&quot;. In this case,</span>
-            <span class="c"># the terminal control sequences would be printed verbatim, so</span>
-            <span class="c"># don&#39;t use any colors.</span>
-            <span class="n">color</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-        <span class="k">elif</span> <span class="n">sys</span><span class="o">.</span><span class="n">platform</span> <span class="o">==</span> <span class="s">&quot;win32&quot;</span><span class="p">:</span>
-            <span class="c"># Windows consoles don&#39;t support ANSI escape sequences</span>
-            <span class="n">color</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-        <span class="k">elif</span> <span class="ow">not</span> <span class="bp">self</span><span class="o">.</span><span class="n">_colors</span><span class="p">:</span>
-            <span class="n">color</span> <span class="o">=</span> <span class="s">&quot;&quot;</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_line_wrap</span><span class="p">:</span>
-            <span class="k">if</span> <span class="n">text</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">!=</span> <span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">:</span>
-                <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-
-        <span class="c"># Avoid UnicodeEncodeError when printing out test failures</span>
-        <span class="k">if</span> <span class="n">IS_PYTHON_3</span> <span class="ow">and</span> <span class="n">IS_WINDOWS</span><span class="p">:</span>
-            <span class="n">text</span> <span class="o">=</span> <span class="n">text</span><span class="o">.</span><span class="n">encode</span><span class="p">(</span><span class="s">&#39;raw_unicode_escape&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">decode</span><span class="p">(</span><span class="s">&#39;utf8&#39;</span><span class="p">,</span> <span class="s">&#39;ignore&#39;</span><span class="p">)</span>
-        <span class="k">elif</span> <span class="n">IS_PYTHON_3</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span><span class="o">.</span><span class="n">encoding</span><span class="o">.</span><span class="n">lower</span><span class="p">()</span><span class="o">.</span><span class="n">startswith</span><span class="p">(</span><span class="s">&#39;utf&#39;</span><span class="p">):</span>
-            <span class="n">text</span> <span class="o">=</span> <span class="n">text</span><span class="o">.</span><span class="n">encode</span><span class="p">(</span><span class="n">sys</span><span class="o">.</span><span class="n">stdout</span><span class="o">.</span><span class="n">encoding</span><span class="p">,</span> <span class="s">&#39;backslashreplace&#39;</span>
-                              <span class="p">)</span><span class="o">.</span><span class="n">decode</span><span class="p">(</span><span class="n">sys</span><span class="o">.</span><span class="n">stdout</span><span class="o">.</span><span class="n">encoding</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="n">color</span> <span class="o">==</span> <span class="s">&quot;&quot;</span><span class="p">:</span>
-            <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="n">text</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s%s%s</span><span class="s">&quot;</span> <span class="o">%</span>
-                <span class="p">(</span><span class="n">c_color</span> <span class="o">%</span> <span class="n">colors</span><span class="p">[</span><span class="n">color</span><span class="p">],</span> <span class="n">text</span><span class="p">,</span> <span class="n">c_normal</span><span class="p">))</span>
-        <span class="n">sys</span><span class="o">.</span><span class="n">stdout</span><span class="o">.</span><span class="n">flush</span><span class="p">()</span>
-        <span class="n">l</span> <span class="o">=</span> <span class="n">text</span><span class="o">.</span><span class="n">rfind</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">l</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_write_pos</span> <span class="o">+=</span> <span class="nb">len</span><span class="p">(</span><span class="n">text</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">_write_pos</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">text</span><span class="p">)</span> <span class="o">-</span> <span class="n">l</span> <span class="o">-</span> <span class="mi">1</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_line_wrap</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_write_pos</span> <span class="o">&gt;=</span> <span class="n">width</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_write_pos</span> <span class="o">%=</span> <span class="n">width</span>
-</div>
-    <span class="k">def</span> <span class="nf">write_center</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">text</span><span class="p">,</span> <span class="n">delim</span><span class="o">=</span><span class="s">&quot;=&quot;</span><span class="p">):</span>
-        <span class="n">width</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">terminal_width</span>
-        <span class="k">if</span> <span class="n">text</span> <span class="o">!=</span> <span class="s">&quot;&quot;</span><span class="p">:</span>
-            <span class="n">text</span> <span class="o">=</span> <span class="s">&quot; </span><span class="si">%s</span><span class="s"> &quot;</span> <span class="o">%</span> <span class="n">text</span>
-        <span class="n">idx</span> <span class="o">=</span> <span class="p">(</span><span class="n">width</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">text</span><span class="p">))</span> <span class="o">//</span> <span class="mi">2</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">delim</span><span class="o">*</span><span class="n">idx</span> <span class="o">+</span> <span class="n">text</span> <span class="o">+</span> <span class="n">delim</span><span class="o">*</span><span class="p">(</span><span class="n">width</span> <span class="o">-</span> <span class="n">idx</span> <span class="o">-</span> <span class="nb">len</span><span class="p">(</span><span class="n">text</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="n">t</span> <span class="o">+</span> <span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">write_exception</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">val</span><span class="p">,</span> <span class="n">tb</span><span class="p">):</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">traceback</span><span class="o">.</span><span class="n">extract_tb</span><span class="p">(</span><span class="n">tb</span><span class="p">)</span>
-        <span class="c"># remove the first item, as that is always runtests.py</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">t</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">traceback</span><span class="o">.</span><span class="n">format_list</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">t</span><span class="p">))</span>
-        <span class="n">t</span> <span class="o">=</span> <span class="n">traceback</span><span class="o">.</span><span class="n">format_exception_only</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">t</span><span class="p">))</span>
-
-    <span class="k">def</span> <span class="nf">start</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">msg</span><span class="o">=</span><span class="s">&quot;test process starts&quot;</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write_center</span><span class="p">(</span><span class="n">msg</span><span class="p">)</span>
-        <span class="n">executable</span> <span class="o">=</span> <span class="n">sys</span><span class="o">.</span><span class="n">executable</span>
-        <span class="n">v</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">sys</span><span class="o">.</span><span class="n">version_info</span><span class="p">)</span>
-        <span class="n">python_version</span> <span class="o">=</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">.</span><span class="si">%s</span><span class="s">.</span><span class="si">%s</span><span class="s">-</span><span class="si">%s</span><span class="s">-</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">v</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;executable:         </span><span class="si">%s</span><span class="s">  (</span><span class="si">%s</span><span class="s">)</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span>
-            <span class="p">(</span><span class="n">executable</span><span class="p">,</span> <span class="n">python_version</span><span class="p">))</span>
-        <span class="kn">from</span> <span class="nn">.misc</span> <span class="kn">import</span> <span class="n">ARCH</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;architecture:       </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">ARCH</span><span class="p">)</span>
-        <span class="kn">from</span> <span class="nn">sympy.core.cache</span> <span class="kn">import</span> <span class="n">USE_CACHE</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;cache:              </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">USE_CACHE</span><span class="p">)</span>
-        <span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">GROUND_TYPES</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;ground types:       </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">GROUND_TYPES</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">seed</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;random seed:        </span><span class="si">%d</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">seed</span><span class="p">)</span>
-        <span class="kn">from</span> <span class="nn">.misc</span> <span class="kn">import</span> <span class="n">HASH_RANDOMIZATION</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;hash randomization: &quot;</span><span class="p">)</span>
-        <span class="n">hash_seed</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">getenv</span><span class="p">(</span><span class="s">&quot;PYTHONHASHSEED&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">HASH_RANDOMIZATION</span> <span class="ow">and</span> <span class="p">(</span><span class="n">hash_seed</span> <span class="o">==</span> <span class="s">&quot;random&quot;</span> <span class="ow">or</span> <span class="nb">int</span><span class="p">(</span><span class="n">hash_seed</span><span class="p">)):</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;on (PYTHONHASHSEED=</span><span class="si">%s</span><span class="s">)</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">hash_seed</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;off</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_t_start</span> <span class="o">=</span> <span class="n">clock</span><span class="p">()</span>
-
-    <span class="k">def</span> <span class="nf">finish</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_t_end</span> <span class="o">=</span> <span class="n">clock</span><span class="p">()</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-        <span class="k">global</span> <span class="n">text</span><span class="p">,</span> <span class="n">linelen</span>
-        <span class="n">text</span> <span class="o">=</span> <span class="s">&quot;tests finished: </span><span class="si">%d</span><span class="s"> passed, &quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_passed</span>
-        <span class="n">linelen</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">text</span><span class="p">)</span>
-
-        <span class="k">def</span> <span class="nf">add_text</span><span class="p">(</span><span class="n">mytext</span><span class="p">):</span>
-            <span class="k">global</span> <span class="n">text</span><span class="p">,</span> <span class="n">linelen</span>
-            <span class="sd">&quot;&quot;&quot;Break new text if too long.&quot;&quot;&quot;</span>
-            <span class="k">if</span> <span class="n">linelen</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">mytext</span><span class="p">)</span> <span class="o">&gt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">terminal_width</span><span class="p">:</span>
-                <span class="n">text</span> <span class="o">+=</span> <span class="s">&#39;</span><span class="se">\n</span><span class="s">&#39;</span>
-                <span class="n">linelen</span> <span class="o">=</span> <span class="mi">0</span>
-            <span class="n">text</span> <span class="o">+=</span> <span class="n">mytext</span>
-            <span class="n">linelen</span> <span class="o">+=</span> <span class="nb">len</span><span class="p">(</span><span class="n">mytext</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_failed</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">add_text</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%d</span><span class="s"> failed, &quot;</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_failed</span><span class="p">))</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_failed_doctest</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">add_text</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%d</span><span class="s"> failed, &quot;</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_failed_doctest</span><span class="p">))</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_skipped</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">add_text</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%d</span><span class="s"> skipped, &quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_skipped</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_xfailed</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">add_text</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%d</span><span class="s"> expected to fail, &quot;</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">_xfailed</span><span class="p">)</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_xpassed</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">add_text</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%d</span><span class="s"> expected to fail but passed, &quot;</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_xpassed</span><span class="p">))</span>
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_exceptions</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="n">add_text</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%d</span><span class="s"> exceptions, &quot;</span> <span class="o">%</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_exceptions</span><span class="p">))</span>
-        <span class="n">add_text</span><span class="p">(</span><span class="s">&quot;in </span><span class="si">%.2f</span><span class="s"> seconds&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_t_end</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">_t_start</span><span class="p">))</span>
-
-        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_xpassed</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">write_center</span><span class="p">(</span><span class="s">&quot;xpassed tests&quot;</span><span class="p">,</span> <span class="s">&quot;_&quot;</span><span class="p">)</span>
-            <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_xpassed</span><span class="p">:</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s">: </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">e</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">e</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_tb_style</span> <span class="o">!=</span> <span class="s">&quot;no&quot;</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_exceptions</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="c">#self.write_center(&quot;These tests raised an exception&quot;, &quot;_&quot;)</span>
-            <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_exceptions</span><span class="p">:</span>
-                <span class="n">filename</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">val</span><span class="p">,</span> <span class="n">tb</span><span class="p">)</span> <span class="o">=</span> <span class="n">e</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">write_center</span><span class="p">(</span><span class="s">&quot;&quot;</span><span class="p">,</span> <span class="s">&quot;_&quot;</span><span class="p">)</span>
-                <span class="k">if</span> <span class="n">f</span> <span class="ow">is</span> <span class="bp">None</span><span class="p">:</span>
-                    <span class="n">s</span> <span class="o">=</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">filename</span>
-                <span class="k">else</span><span class="p">:</span>
-                    <span class="n">s</span> <span class="o">=</span> <span class="s">&quot;</span><span class="si">%s</span><span class="s">:</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">filename</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">__name__</span><span class="p">)</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">write_center</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="s">&quot;_&quot;</span><span class="p">)</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">write_exception</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">val</span><span class="p">,</span> <span class="n">tb</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_tb_style</span> <span class="o">!=</span> <span class="s">&quot;no&quot;</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_failed</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="c">#self.write_center(&quot;Failed&quot;, &quot;_&quot;)</span>
-            <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_failed</span><span class="p">:</span>
-                <span class="n">filename</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">val</span><span class="p">,</span> <span class="n">tb</span><span class="p">)</span> <span class="o">=</span> <span class="n">e</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">write_center</span><span class="p">(</span><span class="s">&quot;&quot;</span><span class="p">,</span> <span class="s">&quot;_&quot;</span><span class="p">)</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">write_center</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s">:</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">filename</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">__name__</span><span class="p">),</span> <span class="s">&quot;_&quot;</span><span class="p">)</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">write_exception</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">val</span><span class="p">,</span> <span class="n">tb</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_tb_style</span> <span class="o">!=</span> <span class="s">&quot;no&quot;</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_failed_doctest</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-            <span class="c">#self.write_center(&quot;Failed&quot;, &quot;_&quot;)</span>
-            <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_failed_doctest</span><span class="p">:</span>
-                <span class="n">filename</span><span class="p">,</span> <span class="n">msg</span> <span class="o">=</span> <span class="n">e</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">write_center</span><span class="p">(</span><span class="s">&quot;&quot;</span><span class="p">,</span> <span class="s">&quot;_&quot;</span><span class="p">)</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">write_center</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">filename</span><span class="p">,</span> <span class="s">&quot;_&quot;</span><span class="p">)</span>
-                <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="n">msg</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-
-        <span class="bp">self</span><span class="o">.</span><span class="n">write_center</span><span class="p">(</span><span class="n">text</span><span class="p">)</span>
-        <span class="n">ok</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_failed</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_exceptions</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> \
-            <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_failed_doctest</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">ok</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;DO *NOT* COMMIT!</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">ok</span>
-
-    <span class="k">def</span> <span class="nf">entering_filename</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">filename</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-        <span class="n">rel_name</span> <span class="o">=</span> <span class="n">filename</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_root_dir</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_active_file</span> <span class="o">=</span> <span class="n">rel_name</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_active_file_error</span> <span class="o">=</span> <span class="bp">False</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="n">rel_name</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;[</span><span class="si">%d</span><span class="s">] &quot;</span> <span class="o">%</span> <span class="n">n</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">leaving_filename</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot; &quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_active_file_error</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;[FAIL]&quot;</span><span class="p">,</span> <span class="s">&quot;Red&quot;</span><span class="p">,</span> <span class="n">align</span><span class="o">=</span><span class="s">&quot;right&quot;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;[OK]&quot;</span><span class="p">,</span> <span class="s">&quot;Green&quot;</span><span class="p">,</span> <span class="n">align</span><span class="o">=</span><span class="s">&quot;right&quot;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_verbose</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">entering_test</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">f</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_active_f</span> <span class="o">=</span> <span class="n">f</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_verbose</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span> <span class="o">+</span> <span class="n">f</span><span class="o">.</span><span class="n">__name__</span> <span class="o">+</span> <span class="s">&quot; &quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">test_xfail</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_xfailed</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;f&quot;</span><span class="p">,</span> <span class="s">&quot;Green&quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">test_xpass</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">v</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">sys</span><span class="o">.</span><span class="n">version_info</span><span class="p">[:</span><span class="mi">2</span><span class="p">]</span> <span class="o">&lt;</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">6</span><span class="p">):</span>
-            <span class="n">message</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="s">&#39;message&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">message</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_xpassed</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">_active_file</span><span class="p">,</span> <span class="n">message</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;X&quot;</span><span class="p">,</span> <span class="s">&quot;Green&quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">test_fail</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">exc_info</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_failed</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">_active_file</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_active_f</span><span class="p">,</span> <span class="n">exc_info</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;F&quot;</span><span class="p">,</span> <span class="s">&quot;Red&quot;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_active_file_error</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">doctest_fail</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">name</span><span class="p">,</span> <span class="n">error_msg</span><span class="p">):</span>
-        <span class="c"># the first line contains &quot;******&quot;, remove it:</span>
-        <span class="n">error_msg</span> <span class="o">=</span> <span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">error_msg</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)[</span><span class="mi">1</span><span class="p">:])</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_failed_doctest</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">name</span><span class="p">,</span> <span class="n">error_msg</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;F&quot;</span><span class="p">,</span> <span class="s">&quot;Red&quot;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_active_file_error</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">test_pass</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">char</span><span class="o">=</span><span class="s">&quot;.&quot;</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_passed</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_verbose</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;ok&quot;</span><span class="p">,</span> <span class="s">&quot;Green&quot;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="n">char</span><span class="p">,</span> <span class="s">&quot;Green&quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">test_skip</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">v</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">sys</span><span class="o">.</span><span class="n">version_info</span><span class="p">[:</span><span class="mi">2</span><span class="p">]</span> <span class="o">&lt;</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">6</span><span class="p">):</span>
-            <span class="n">message</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="s">&#39;message&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">message</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_skipped</span> <span class="o">+=</span> <span class="mi">1</span>
-        <span class="n">char</span> <span class="o">=</span> <span class="s">&quot;s&quot;</span>
-        <span class="k">if</span> <span class="n">message</span> <span class="o">==</span> <span class="s">&quot;KeyboardInterrupt&quot;</span><span class="p">:</span>
-            <span class="n">char</span> <span class="o">=</span> <span class="s">&quot;K&quot;</span>
-        <span class="k">elif</span> <span class="n">message</span> <span class="o">==</span> <span class="s">&quot;Timeout&quot;</span><span class="p">:</span>
-            <span class="n">char</span> <span class="o">=</span> <span class="s">&quot;T&quot;</span>
-        <span class="k">elif</span> <span class="n">message</span> <span class="o">==</span> <span class="s">&quot;Slow&quot;</span><span class="p">:</span>
-            <span class="n">char</span> <span class="o">=</span> <span class="s">&quot;w&quot;</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="n">char</span><span class="p">,</span> <span class="s">&quot;Blue&quot;</span><span class="p">)</span>
-        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_verbose</span><span class="p">:</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot; - &quot;</span><span class="p">,</span> <span class="s">&quot;Blue&quot;</span><span class="p">)</span>
-            <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="n">message</span><span class="p">,</span> <span class="s">&quot;Blue&quot;</span><span class="p">)</span>
-
-    <span class="k">def</span> <span class="nf">test_exception</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">exc_info</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_exceptions</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">_active_file</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_active_f</span><span class="p">,</span> <span class="n">exc_info</span><span class="p">))</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;E&quot;</span><span class="p">,</span> <span class="s">&quot;Red&quot;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_active_file_error</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">import_error</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">filename</span><span class="p">,</span> <span class="n">exc_info</span><span class="p">):</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">_exceptions</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">filename</span><span class="p">,</span> <span class="bp">None</span><span class="p">,</span> <span class="n">exc_info</span><span class="p">))</span>
-        <span class="n">rel_name</span> <span class="o">=</span> <span class="n">filename</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_root_dir</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="n">rel_name</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;[?]   Failed to import&quot;</span><span class="p">,</span> <span class="s">&quot;Red&quot;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot; &quot;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;[FAIL]&quot;</span><span class="p">,</span> <span class="s">&quot;Red&quot;</span><span class="p">,</span> <span class="n">align</span><span class="o">=</span><span class="s">&quot;right&quot;</span><span class="p">)</span>
-        <span class="bp">self</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">)</span>
-</div>
-<span class="n">sympy_dir</span> <span class="o">=</span> <span class="n">get_sympy_dir</span><span class="p">()</span>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-<div id="searchbox" style="display: none">
-  <h3>Quick search</h3>
-    <form class="search" action="../../../search.html" method="get">
-      <input type="text" name="q" />
-      <input type="submit" value="Go" />
-      <input type="hidden" name="check_keywords" value="yes" />
-      <input type="hidden" name="area" value="default" />
-    </form>
-    <p class="searchtip" style="font-size: 90%">
-    Enter search terms or a module, class or function name.
-    </p>
-</div>
-<script type="text/javascript">$('#searchbox').show(0);</script>
-        </div>
-      </div>
-      <div class="clearer"></div>
-    </div>
-    <div class="related">
-      <h3>Navigation</h3>
-      <ul>
-        <li class="right" style="margin-right: 10px">
-          <a href="../../../genindex.html" title="General Index"
-             >index</a></li>
-        <li class="right" >
-          <a href="../../../py-modindex.html" title="Python Module Index"
-             >modules</a> |</li>
-        <li class="right" >
-          <a href="../../../np-modindex.html" title="Python Module Index"
-             >modules</a> |</li>
-        <li><a href="../../../index.html">SymPy 0.7.2-git documentation</a> &raquo;</li>
-          <li><a href="../../index.html" >Module code</a> &raquo;</li>
-          <li><a href="../../sympy.html" >sympy</a> &raquo;</li> 
-      </ul>
-    </div>
-    <div class="footer">
-        &copy; Copyright 2013 SymPy Development Team.
-      Last updated on Jan 20, 2013.
-      Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.1.3.
-    </div>
-  </body>
-</html>
diff --git a/dev-py3k/_modules/sympy/utilities/source.html b/dev-py3k/_modules/sympy/utilities/source.html
deleted file mode 100644
index 84022447734..00000000000
--- a/dev-py3k/_modules/sympy/utilities/source.html
+++ /dev/null
@@ -1,167 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
-
-
-<html xmlns="http://www.w3.org/1999/xhtml">
-  <head>
-    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
-    
-    <title>sympy.utilities.source &mdash; SymPy 0.7.2-git documentation</title>
-    
-    <link rel="stylesheet" href="../../../_static/default.css" type="text/css" />
-    <link rel="stylesheet" href="../../../_static/pygments.css" type="text/css" />
-    <link rel="stylesheet" href="http://live.sympy.org/static/live-core.css" type="text/css" />
-    <link rel="stylesheet" href="http://live.sympy.org/static/live-autocomplete.css" type="text/css" />
-    <link rel="stylesheet" href="http://live.sympy.org/static/live-sphinx.css" type="text/css" />
-    
-    <script type="text/javascript">
-      var DOCUMENTATION_OPTIONS = {
-        URL_ROOT:    '../../../',
-        VERSION:     '0.7.2-git',
-        COLLAPSE_INDEX: false,
-        FILE_SUFFIX: '.html',
-        HAS_SOURCE:  true
-      };
-    </script>
-    <script type="text/javascript" src="../../../_static/jquery.js"></script>
-    <script type="text/javascript" src="../../../_static/underscore.js"></script>
-    <script type="text/javascript" src="../../../_static/doctools.js"></script>
-    <script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_HTML-full"></script>
-    <script type="text/javascript" src="http://live.sympy.org/static/utilities.js"></script>
-    <script type="text/javascript" src="http://live.sympy.org/static/external/classy.js"></script>
-    <script type="text/javascript" src="http://live.sympy.org/static/live-core.js"></script>
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-  <h1>Source code for sympy.utilities.source</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">This module adds several functions for interactive source code inspection.</span>
-<span class="sd">&quot;&quot;&quot;</span>
-
-<span class="kn">import</span> <span class="nn">inspect</span>
-<span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="nb">callable</span>
-<span class="kn">import</span> <span class="nn">collections</span>
-
-
-<div class="viewcode-block" id="source"><a class="viewcode-back" href="../../../modules/utilities/source.html#sympy.utilities.source.source">[docs]</a><span class="k">def</span> <span class="nf">source</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Prints the source code of a given object.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;In file: </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">inspect</span><span class="o">.</span><span class="n">getsourcefile</span><span class="p">(</span><span class="nb">object</span><span class="p">))</span>
-    <span class="k">print</span><span class="p">(</span><span class="n">inspect</span><span class="o">.</span><span class="n">getsource</span><span class="p">(</span><span class="nb">object</span><span class="p">))</span>
-
-</div>
-<div class="viewcode-block" id="get_class"><a class="viewcode-back" href="../../../modules/utilities/source.html#sympy.utilities.source.get_class">[docs]</a><span class="k">def</span> <span class="nf">get_class</span><span class="p">(</span><span class="n">lookup_view</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Convert a string version of a class name to the object.</span>
-
-<span class="sd">    For example, get_class(&#39;sympy.core.Basic&#39;) will return</span>
-<span class="sd">    class Basic located in module sympy.core</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">lookup_view</span><span class="p">,</span> <span class="nb">str</span><span class="p">):</span>
-        <span class="n">lookup_view</span> <span class="o">=</span> <span class="n">lookup_view</span>
-        <span class="n">mod_name</span><span class="p">,</span> <span class="n">func_name</span> <span class="o">=</span> <span class="n">get_mod_func</span><span class="p">(</span><span class="n">lookup_view</span><span class="p">)</span>
-        <span class="k">if</span> <span class="n">func_name</span> <span class="o">!=</span> <span class="s">&#39;&#39;</span><span class="p">:</span>
-            <span class="n">lookup_view</span> <span class="o">=</span> <span class="nb">getattr</span><span class="p">(</span>
-                <span class="nb">__import__</span><span class="p">(</span><span class="n">mod_name</span><span class="p">,</span> <span class="p">{},</span> <span class="p">{},</span> <span class="p">[</span><span class="s">&#39;*&#39;</span><span class="p">]),</span> <span class="n">func_name</span><span class="p">)</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">lookup_view</span><span class="p">,</span> <span class="n">collections</span><span class="o">.</span><span class="n">Callable</span><span class="p">):</span>
-                <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span>
-                    <span class="s">&quot;&#39;</span><span class="si">%s</span><span class="s">.</span><span class="si">%s</span><span class="s">&#39; is not a callable.&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">mod_name</span><span class="p">,</span> <span class="n">func_name</span><span class="p">))</span>
-    <span class="k">return</span> <span class="n">lookup_view</span>
-
-</div>
-<div class="viewcode-block" id="get_mod_func"><a class="viewcode-back" href="../../../modules/utilities/source.html#sympy.utilities.source.get_mod_func">[docs]</a><span class="k">def</span> <span class="nf">get_mod_func</span><span class="p">(</span><span class="n">callback</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    splits the string path to a class into a string path to the module</span>
-<span class="sd">    and the name of the class. For example:</span>
-
-<span class="sd">        &gt;&gt;&gt; from sympy.utilities.source import get_mod_func</span>
-<span class="sd">        &gt;&gt;&gt; get_mod_func(&#39;sympy.core.basic.Basic&#39;)</span>
-<span class="sd">        (&#39;sympy.core.basic&#39;, &#39;Basic&#39;)</span>
-
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">dot</span> <span class="o">=</span> <span class="n">callback</span><span class="o">.</span><span class="n">rfind</span><span class="p">(</span><span class="s">&#39;.&#39;</span><span class="p">)</span>
-    <span class="k">if</span> <span class="n">dot</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
-        <span class="k">return</span> <span class="n">callback</span><span class="p">,</span> <span class="s">&#39;&#39;</span>
-    <span class="k">return</span> <span class="n">callback</span><span class="p">[:</span><span class="n">dot</span><span class="p">],</span> <span class="n">callback</span><span class="p">[</span><span class="n">dot</span> <span class="o">+</span> <span class="mi">1</span><span class="p">:]</span></div>
-</pre></div>
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diff --git a/dev-py3k/_modules/sympy/utilities/timeutils.html b/dev-py3k/_modules/sympy/utilities/timeutils.html
deleted file mode 100644
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+++ /dev/null
@@ -1,191 +0,0 @@
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-            
-  <h1>Source code for sympy.utilities.timeutils</h1><div class="highlight"><pre>
-<span class="sd">&quot;&quot;&quot;Simple tools for timing functions execution, when IPython is not</span>
-<span class="sd">   available. &quot;&quot;&quot;</span>
-
-<span class="kn">import</span> <span class="nn">timeit</span>
-<span class="kn">import</span> <span class="nn">math</span>
-
-<span class="n">_scales</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1e0</span><span class="p">,</span> <span class="mf">1e3</span><span class="p">,</span> <span class="mf">1e6</span><span class="p">,</span> <span class="mf">1e9</span><span class="p">]</span>
-<span class="n">_units</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;s&#39;</span><span class="p">,</span> <span class="s">&#39;ms&#39;</span><span class="p">,</span> <span class="s">&#39;</span><span class="se">\u03bc</span><span class="s">s&#39;</span><span class="p">,</span> <span class="s">&#39;ns&#39;</span><span class="p">]</span>
-
-
-<div class="viewcode-block" id="timed"><a class="viewcode-back" href="../../../modules/utilities/timeutils.html#sympy.utilities.timeutils.timed">[docs]</a><span class="k">def</span> <span class="nf">timed</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">setup</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;Adaptively measure execution time of a function. &quot;&quot;&quot;</span>
-    <span class="n">timer</span> <span class="o">=</span> <span class="n">timeit</span><span class="o">.</span><span class="n">Timer</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">setup</span><span class="o">=</span><span class="n">setup</span><span class="p">)</span>
-
-    <span class="n">repeat</span><span class="p">,</span> <span class="n">number</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span>
-
-    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">):</span>
-        <span class="k">if</span> <span class="n">timer</span><span class="o">.</span><span class="n">timeit</span><span class="p">(</span><span class="n">number</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mf">0.2</span><span class="p">:</span>
-            <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">number</span> <span class="o">*=</span> <span class="mi">10</span>
-
-    <span class="n">time</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">timer</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="n">repeat</span><span class="p">,</span> <span class="n">number</span><span class="p">))</span> <span class="o">/</span> <span class="n">number</span>
-
-    <span class="k">if</span> <span class="n">time</span> <span class="o">&gt;</span> <span class="mf">0.0</span><span class="p">:</span>
-        <span class="n">order</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="o">-</span><span class="nb">int</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">floor</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">time</span><span class="p">))</span> <span class="o">//</span> <span class="mi">3</span><span class="p">),</span> <span class="mi">3</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="n">order</span> <span class="o">=</span> <span class="mi">3</span>
-
-    <span class="k">return</span> <span class="p">(</span><span class="n">number</span><span class="p">,</span> <span class="n">time</span><span class="p">,</span> <span class="n">time</span><span class="o">*</span><span class="n">_scales</span><span class="p">[</span><span class="n">order</span><span class="p">],</span> <span class="n">_units</span><span class="p">[</span><span class="n">order</span><span class="p">])</span>
-
-
-<span class="c"># Code for doing inline timings of recursive algorithms.</span>
-</div>
-<span class="k">def</span> <span class="nf">__do_timings</span><span class="p">():</span>
-    <span class="kn">import</span> <span class="nn">os</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">getenv</span><span class="p">(</span><span class="s">&#39;SYMPY_TIMINGS&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-    <span class="n">res</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="o">.</span><span class="n">strip</span><span class="p">()</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">res</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s">&#39;,&#39;</span><span class="p">)]</span>
-    <span class="k">return</span> <span class="nb">set</span><span class="p">(</span><span class="n">res</span><span class="p">)</span>
-
-<span class="n">_do_timings</span> <span class="o">=</span> <span class="n">__do_timings</span><span class="p">()</span>
-<span class="n">_timestack</span> <span class="o">=</span> <span class="bp">None</span>
-
-
-<span class="k">def</span> <span class="nf">_print_timestack</span><span class="p">(</span><span class="n">stack</span><span class="p">,</span> <span class="n">level</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;-&#39;</span><span class="o">*</span><span class="n">level</span><span class="p">,</span> <span class="s">&#39;</span><span class="si">%.2f</span><span class="s"> </span><span class="si">%s%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">stack</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">stack</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">stack</span><span class="p">[</span><span class="mi">3</span><span class="p">]))</span>
-    <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">stack</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span>
-        <span class="n">_print_timestack</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">level</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-
-
-<span class="k">def</span> <span class="nf">timethis</span><span class="p">(</span><span class="n">name</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">decorator</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
-        <span class="k">global</span> <span class="n">_do_timings</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="n">name</span> <span class="ow">in</span> <span class="n">_do_timings</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">func</span>
-
-        <span class="k">def</span> <span class="nf">wrapper</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
-            <span class="kn">from</span> <span class="nn">time</span> <span class="kn">import</span> <span class="n">time</span>
-            <span class="k">global</span> <span class="n">_timestack</span>
-            <span class="n">oldtimestack</span> <span class="o">=</span> <span class="n">_timestack</span>
-            <span class="n">_timestack</span> <span class="o">=</span> <span class="p">[</span><span class="n">func</span><span class="o">.</span><span class="n">__name__</span><span class="p">,</span> <span class="p">[],</span> <span class="mi">0</span><span class="p">,</span> <span class="n">args</span><span class="p">]</span>
-            <span class="n">t1</span> <span class="o">=</span> <span class="n">time</span><span class="p">()</span>
-            <span class="n">r</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
-            <span class="n">t2</span> <span class="o">=</span> <span class="n">time</span><span class="p">()</span>
-            <span class="n">_timestack</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">t2</span> <span class="o">-</span> <span class="n">t1</span>
-            <span class="k">if</span> <span class="n">oldtimestack</span> <span class="ow">is</span> <span class="ow">not</span> <span class="bp">None</span><span class="p">:</span>
-                <span class="n">oldtimestack</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">_timestack</span><span class="p">)</span>
-                <span class="n">_timestack</span> <span class="o">=</span> <span class="n">oldtimestack</span>
-            <span class="k">else</span><span class="p">:</span>
-                <span class="n">_print_timestack</span><span class="p">(</span><span class="n">_timestack</span><span class="p">)</span>
-                <span class="n">_timestack</span> <span class="o">=</span> <span class="bp">None</span>
-            <span class="k">return</span> <span class="n">r</span>
-        <span class="k">return</span> <span class="n">wrapper</span>
-    <span class="k">return</span> <span class="n">decorator</span>
-</pre></div>
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-<div id="searchbox" style="display: none">
-  <h3>Quick search</h3>
-    <form class="search" action="../../../search.html" method="get">
-      <input type="text" name="q" />
-      <input type="submit" value="Go" />
-      <input type="hidden" name="check_keywords" value="yes" />
-      <input type="hidden" name="area" value="default" />
-    </form>
-    <p class="searchtip" style="font-size: 90%">
-    Enter search terms or a module, class or function name.
-    </p>
-</div>
-<script type="text/javascript">$('#searchbox').show(0);</script>
-        </div>
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-    </div>
-    <div class="related">
-      <h3>Navigation</h3>
-      <ul>
-        <li class="right" style="margin-right: 10px">
-          <a href="../../../genindex.html" title="General Index"
-             >index</a></li>
-        <li class="right" >
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-             >modules</a> |</li>
-        <li class="right" >
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-             >modules</a> |</li>
-        <li><a href="../../../index.html">SymPy 0.7.2-git documentation</a> &raquo;</li>
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-    <div class="footer">
-        &copy; Copyright 2013 SymPy Development Team.
-      Last updated on Jan 20, 2013.
-      Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.1.3.
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diff --git a/dev-py3k/_sources/aboutus.txt b/dev-py3k/_sources/aboutus.txt
deleted file mode 100644
index 9e724e1836f..00000000000
--- a/dev-py3k/_sources/aboutus.txt
+++ /dev/null
@@ -1,292 +0,0 @@
-About
-=====
-
-SymPy Development Team
-----------------------
-
-SymPy is a team project and it was developed by a lot of people.
-
-Here is a list of contributors together with what they do, (and in some cases links to their
-wiki pages), where they describe in
-more details what they do and what they are interested in (some people didn't
-want to be mentioned here, so see our repository history for a full list).
-
-#. Ondřej Čertík: started the project in 2006, on Jan 4, 2011 passed the project leadership to Aaron Meurer
-#. Fabian Pedregosa: everything, reviewing patches, releases, general advice (issues and mailinglist), GSoC 2009
-#. Jurjen N.E. Bos: pretty printing and other patches
-#. Mateusz Paprocki: GSoC 2007, concrete math module, integration module, new core integration, a lot of patches, general advice, new polynomial module, improvements to solvers, simplifications, patch review
-#. Marc-Etienne M.Leveille: matrix patch
-#. Brian Jorgensen: GSoC 2007, plotting module and related things, patches
-#. Jason Gedge: GSoC 2007, geometry module, a lot of patches and fixes, new core integration
-#. Robert Schwarz: GSoC 2007, polynomials module, patches
-#. `Pearu Peterson <https://github.com/sympy/sympy/wiki/Pearu-Peterson>`_: new core, sympycore project, general advice (issues and mailinglist)
-#. `Fredrik Johansson <https://github.com/sympy/sympy/wiki/Fredrik-Johansson>`_: mpmath project and its integration in SymPy, number theory, combinatorial functions, products & summation, statistics, units, patches, documentation, general advice (issues and mailinglist)
-#. Chris Wu: GSoC 2007, linear algebra module
-#. Ulrich Hecht: pattern matching and other patches
-#. Goutham Lakshminarayan: number theory functions
-#. David Lawrence: GHOP, Mathematica parser, square root denesting
-#. Jaroslaw Tworek: GHOP, sympify AST implementation, sqrt() refactoring, maxima parser and other patches
-#. David Marek: GHOP, derivative evaluation patch, int(NumberSymbol) fix
-#. Bernhard R. Link: documentation patch
-#. Andrej Tokarčík: GHOP, python printer
-#. Or Dvory: GHOP, documentation
-#. Saroj Adhikari: bug fixes
-#. Pauli Virtanen: bug fix
-#. Robert Kern: bug fix, common subexpression elimination
-#. James Aspnes: bug fixes
-#. Nimish Telang: multivariate lambdas
-#. Abderrahim Kitouni: pretty printing + complex expansion bug fixes
-#. Pan Peng: ode solvers patch
-#. Friedrich Hagedorn: many bug fixes, refactorings and new features added
-#. Elrond der Elbenfuerst: pretty printing fix
-#. Rizgar Mella: BBP formula for pi calculating algorithm
-#. Felix Kaiser: documentation + whitespace testing patches
-#. Roberto Nobrega: several pretty printing patches
-#. David Roberts: latex printing patches
-#. Sebastian Krämer: implemented lambdify/numpy/mpmath cooperation, bug fixes, refactoring, lambdifying of matrices, large printing refactoring and bugfixes
-#. Vinzent Steinberg: docstring patches, a lot of bug fixes, nsolve (nonlinear equation systems solver), compiling functions to machine code, patches review
-#. Riccardo Gori: improvements and speedups to matrices, many bug fixes
-#. Case Van Horsen: implemented optional support for gmpy in mpmath
-#. Štěpán Roučka: a lot of bug fixes all over SymPy (matrix, simplification, limits, series, ...)
-#. Ali Raza Syed: pretty printing/isympy on windows fix
-#. Stefano Maggiolo: many bug fixes, polishings and improvements
-#. Robert Cimrman: matrix patches
-#. Bastian Weber: latex printing patches
-#. Sebastian Krause: match patches
-#. Sebastian Kreft: latex printing patches, Dirac delta function, other fixes
-#. Dan (coolg49964): documentation fixes
-#. Alan Bromborsky: geometric algebra modules
-#. Boris Timokhin: matrix fixes
-#. Robert (average.programmer): initial piecewise function patch
-#. Andy R. Terrel: piecewise function polishing and other patches
-#. Hubert Tsang: LaTeX printing fixes
-#. Konrad Meyer: policy patch
-#. Henrik Johansson: matrix zero finder patch
-#. Priit Laes: line integrals, cleanups in ODE, tests added
-#. Freddie Witherden: trigonometric simplifications fixes, LaTeX printer improvements
-#. Brian E. Granger: second quantization physics module
-#. Andrew Straw: lambdify() improvements
-#. Kaifeng Zhu: factorint() fix
-#. Ted Horst: Basic.__call__() fix, pretty printing fix
-#. Andrew Docherty: Fix to series
-#. Akshay Srinivasan: printing fix, improvements to integration
-#. Aaron Meurer: ODE solvers (GSoC 2009), The Risch Algorithm for integration (GSoC 2010), other fixes, project leader as of Jan 4, 2011
-#. Barry Wardell: implement series as function, several tests added
-#. Tomasz Buchert: ccode printing fixes, code quality concerning exceptions, documentation
-#. Vinay Kumar: polygamma tests
-#. Johann Cohen-Tanugi: commutative diff tests
-#. Jochen Voss: a test for the @cachit decorator and several other fixes
-#. Luke Peterson: improve solve() to handle Derivatives
-#. Chris Smith: improvements to solvers, many bug fixes, documentation and test improvements
-#. Thomas Sidoti: MathML printer improvements
-#. Florian Mickler: reimplementation of convex_hull, several geometry module fixes
-#. Nicolas Pourcelot: Infinity comparison fixes, latex fixes
-#. Ben Goodrich: Matrix.jacobian() function improvements and fixes
-#. Toon Verstraelen: code generation module, latex printing improvements
-#. Ronan Lamy: test coverage script; limit, expansion and other fixes and improvements; cleanup
-#. James Abbatiello: fixes tests on windows
-#. Ryan Krauss: fixes could_extract_minus_sign(), latex fixes and improvements
-#. Bill Flynn: substitution fixes
-#. Kevin Goodsell: Fix to Integer/Rational
-#. Jorn Baayen: improvements to piecewise functions and latex printing, bug fixes
-#. Eh Tan: improve trigonometric simplification
-#. Renato Coutinho: derivative improvements
-#. Oscar Benjamin: latex printer fix, gcd bug fix
-#. Øyvind Jensen: implemented coupled cluster expansion and wick theorem, improvements to assumptions, bugfixes
-#. Julio Idichekop Filho: indentation fixes, docstring improvements
-#. Łukasz Pankowski: fix matrix multiplication with numpy scalars
-#. Chu-Ching Huang: fix 3d plotting example
-#. Fernando Perez: symarray() implemented
-#. Raffaele De Feo: fix non-commutative expansion
-#. Christian Muise: fixes to logic module
-#. Matt Curry: GSoC 2010 project (symbolic quantum mechanics)
-#. Kazuo Thow: cleanup pretty printing tests
-#. Christian Schubert: Fix to sympify()
-#. Jezreel Ng: fix hyperbolic functions rewrite
-#. James Pearson: Py3k related fixes
-#. Matthew Brett: fixes to lambdify
-#. Addison Cugini: GSoC 2010 project (quantum computing)
-#. Nicholas J.S. Kinar: improved documentation about "Immutability of Expressions"
-#. Harold Erbin: Geometry related work
-#. Thomas Dixon: fix a limit bug
-#. Cristóvão Sousa: implements _sage_ method for sign function.
-#. Andre de Fortier Smit: doctests in matrices improved
-#. Mark Dewing: Fixes to Integral/Sum, MathML work
-#. Alexey U. Gudchenko: various work in matrices
-#. Gary Kerr: fix examples, docs
-#. Sherjil Ozair: fixes to SparseMatrix
-#. Oleksandr Gituliar: fixes to Matrix
-#. Sean Vig: Quantum improvement
-#. Prafullkumar P. Tale: fixed plotting documentation
-#. Vladimir Perić: fix some Python 3 issues
-#. Tom Bachmann: fixes to Real
-#. Yuri Karadzhov: improvements to hyperbolic identities
-#. Vladimir Lagunov: improvements to the geometry module
-#. Matthew Rocklin: stats, sets, matrix expressions
-#. Saptarshi Mandal: Test for an integral
-#. Gilbert Gede: Tests for solvers
-#. Anatolii Koval: Infinite 1D box example
-#. Tomo Lazovich: fixes to quantum operators
-#. Pavel Fedotov: fix to is_number
-#. Kibeom Kim: fixes to cse function and preorder_traversal
-#. Gregory Ksionda: fixes to Integral instantiation
-#. Tomáš Bambas: prettier printing of examples
-#. Jeremias Yehdegho: fixes to the nthoery module
-#. Jack McCaffery: fixes to asin and acos
-#. Raymond Wong: Quantum work
-#. Luca Weihs: improvements to the geometry module
-#. Shai 'Deshe' Wyborski: fix to numeric evaluation of hypergeometric sums
-#. Thomas Wiecki: Fix Sum.diff
-#. Óscar Nájera: better Laguerre polynomial generator
-#. Mario Pernici: faster algorithm for computing Groebner bases
-#. Benjamin McDonald: Fix bug in geometry
-#. Sam Magura: Improvements to Plot.saveimage
-#. Stefan Krastanov: Make Pyglet an external dependency
-#. Bradley Froehle: Fix shell command to generate modules in setup.py
-#. Min Ragan-Kelley: Fix isympy to work with IPython 0.11
-#. Nikhil Sarda: Fix to combinatorics/prufer
-#. Emma Hogan: Fixes to the documentation
-#. Jason Moore: Fixes to the mechanics module
-#. Julien Rioux: Fix behavior of some deprecated methods
-#. Roberto Colistete, Jr.: Add num_columns option to the pretty printer
-#. Raoul Bourquin: Implement Euler numbers
-#. Gert-Ludwig Ingold: Correct derivative of coth
-#. Srinivas Vasudevan: Implementation of Catalan numbers
-#. Miha Marolt: Add numpydoc extension to the Sphinx docs, fix ode_order
-#. Tim Lahey: Rotation matrices
-#. Luis Garcia: update examples in symarry
-#. Matt Rajca: Code quality fixes
-#. David Li: Documentation fixes
-#. David Ju: Increase test coverage, fixes to KroneckerDelta
-#. Alexandr Gudulin: Code quality fixes
-#. Bilal Akhtar: isympy man page
-#. Grzegorz Świrski: Fix to latex(), MacPorts portfile
-#. Matt Habel: SymPy-wide pyflakes editing
-#. Nikolay Lazarov: Translation of the tutorial to Bulgarian
-#. Nichita Utiu: Add pretty printing to Product()
-#. Tristan Hume: Fixes to test_lambdify.py
-#. Imran Ahmed Manzoor: Fixes to the test runner
-#. Steve Anton: pyflakes cleanup of various modules, documentation fixes for logic
-#. Sam Sleight: Fixes to geometry documentation
-#. tsmars15: Fixes to code quality
-#. Chancellor Arkantos: Fixes to the logo
-#. Stepan Simsa: Translation of the tutorial to Czech
-#. Tobias Lenz: Unicode pretty printer for Sum
-#. Siddhanathan Shanmugam: Documentation fixes for the Physics module
-#. Tiffany Zhu: Improved the latex() docstring
-#. Alexey Subach: Translation of the tutorial to Russian
-#. Joan Creus: Improvements to the test runner
-#. Geoffry Song: Improve code coverage
-#. Puneeth Chaganti: Fix for the tutorial
-#. Marcin Kostrzewa: SymPy Cheatsheet
-#. Jim Zhang: Fixes to AUTHORS and .mailmap
-#. Natalia Nawara: Fixes to Product()
-#. vishal: Update the Czech tutorial translation to use a .po file
-#. Shruti Mangipudi: added See Also documentation to Matrices
-#. Davy Mao: Documentation
-#. Swapnil Agarwal: added See Also documentation to ntheory, functions
-#. Kendhia: Add XFAIL tests
-#. jerryma1121: See Also documentation for geometry
-#. Joachim Durchholz: corrected spacing issues identified by PyDev
-#. Martin Povišer: fix problem with rationaltools
-#. Siddhant Jain: See Also documentation for functions/special
-#. Kevin Hunter: Improvements to the inequality classes
-#. Michael Mayorov: Improvement to series
-#. Nathan Alison: Additions to the stats module
-#. Christian Bühler: remove use of set_main() in GA
-#. Carsten Knoll: improvement to preview()
-#. M R Bharath: modified use of int_tested, improvement to Permutation
-#. Matthias Toews: File permissions
-#. Sergiu Ivanov: Fixes to zoo, SymPyDeprecationWarning
-#. Jorge E. Cardona: Cleanup in polys
-#. Sanket Agarwal: Rewrite coverage_doctest.py script
-#. Manoj Babu K.: Improve gamma function
-#. Sai Nikhil: Fix to Heavyside with complex arguments
-#. Aleksandar Makelov: Fixes regarding the dihedral group generator
-#. Raphael Michel: German translation of the tutorial
-#. Sachin Irukula: Changes to allow Dict sorting
-#. Ashwini Oruganti: Changes to Pow printing
-#. Andreas Kloeckner: Fix to cse()
-#. Prateek Papriwal: improve summation documentation
-#. Arpit Goyal: Improvements to Integral and Sum
-#. Angadh Nanjangud: in physics.mechanics, added a function and tests for the parallel axis theorem
-#. Comer Duncan: added dual, is_antisymmetric, and det_lu_decomposition to matrices.py
-#. Jens H. Nielsen: added sets to modules listing, update IPython printing extension
-#. Joseph Dougherty: modified whitespace cleaning to remove multiple newlines at eof
-#. marshall2389: Spelling correction
-#. Guru Devanla: Implemented quantum density operator
-#. George Waksman: Implemented JavaScript code printer and MathML printer
-#. Angus Griffith: Fix bug in rsolve
-#. Timothy Reluga: Rewrite trigonometric functions as rationals
-#. Brian Stephanik: Test for a bug in fcode
-#. Ljubiša Moćić: Serbian translation of the tutorial
-#. Piotr Korgul: Polish translation of the tutorial
-#. Rom le Clair: French translation of the tutorial
-#. Alexandr Popov: Fixes to Pauli algebra
-#. Saurabh Jha: Work on Kauers algorithm
-#. Tarun Gaba: Implemented some trigonometric integrals
-#. Takafumi Arakaki: Add info target to the doc Makefile
-#. Alexander Eberspächer: correct typo in aboutus.rst
-#. Sachin Joglekar: Simplification of logic expressions to SOP and POS forms
-#. Tyler Pirtle: Fix improperly formatted error message
-#. Vasily Povalyaev: Fix latex(Min)
-#. Colleen Lee: replace uses of fnan with S.NaN
-#. Niklas Thörne: Fix links in the docs
-#. Huijun Mai: Chinese translation of the tutorial
-#. Marek Šuppa: Improvements to symbols, tests
-#. Prasoon Shukla: Bug fixes
-
-Up-to-date list in the order of the first contribution is given in the `AUTHORS
-<https://github.com/sympy/sympy/blob/master/AUTHORS>`_ file.
-
-You can find a brief history of SymPy in the README.
-
-Financial and Infrastructure Support
-------------------------------------
-
-* `Google <http://www.google.com/corporate/>`_: SymPy has received generous
-  financial support from Google in various years through the `Google Summer of
-  Code <http://www.google-melange.com/>`_ program by providing stipends:
-
-  * in 2007 for 5 students (`GSoC 2007 <http://code.google.com/p/sympy/wiki/GSoC2007>`_)
-  * in 2008 for 1 student (`GSoC 2008 <http://code.google.com/p/sympy/wiki/GSoC2008>`_)
-  * in 2009 for 5 students (`GSoC 2009 <http://code.google.com/p/sympy/wiki/GSoC2010>`_)
-  * in 2010 for 5 students (`GSoC 2010 <http://code.google.com/p/sympy/wiki/GSoC2010>`_)
-  * in 2011 for 9 students (`GSoC 2011 <https://github.com/sympy/sympy/wiki/Gsoc-2011-report>`_)
-  * in 2012 for 6 students (`GSoC 2012 <https://github.com/sympy/sympy/wiki/Gsoc-2012-report>`_)
-
-* `Python Software Foundation (PSF) <http://www.python.org/psf/>`_ has hosted
-  various GSoC students over the years:
-
-  * 3 GSoC 2007 students (Brian, Robert and Jason)
-  * 1 GSoC 2008 student (Fredrik)
-  * 2 GSoC 2009 students (Freddie and Priit)
-  * 4 GSoC 2010 students (Aaron, Christian, Matthew and Øyvind)
-
-* `Portland State University (PSU) <http://www.pdx.edu/>`_ has hosted following
-  GSoC students:
-
-  * 1 student (Chris) in 2007
-  * 3 students (Aaron, Dale and Fabian) in 2009
-  * 1 student (Addison) in 2010
-
-* `The Space Telescope Science Institute <http://www.stsci.edu/portal/>`_: STScI hosted 1 GSoC 2007 student (Mateusz)
-
-* Several 13-17 year old pre-university students contributed as part of
-  Google's `Code-In
-  <http://www.google-melange.com/gci/homepage/google/gci2011>`_ 2011.  (`GCI
-  2011 <http://www.google-melange.com/gci/org/google/gci2011/sympy>`_)
-
-* `Simula Research Laboratory <http://www.simula.no/>`_: supports Pearu Peterson work in SymPy/SymPy Core projects
-
-* `GitHub <https://github.com/about>`_ is providing us with development
-  and collaboration tools
-
-License
--------
-
-Unless stated otherwise, all files in the SymPy project, SymPy's webpage (and
-wiki), all images and all documentation including this User's Guide are licensed
-using the new BSD license:
-
-.. literalinclude:: ../../LICENSE
diff --git a/dev-py3k/_sources/gotchas.txt b/dev-py3k/_sources/gotchas.txt
deleted file mode 100644
index ddd647865ea..00000000000
--- a/dev-py3k/_sources/gotchas.txt
+++ /dev/null
@@ -1,783 +0,0 @@
-.. _gotchas:
-
-====================
-Gotchas and Pitfalls
-====================
-
-.. role:: input(strong)
-
-Introduction
-============
-
-SymPy runs under the `Python Programming Language
-<http://www.python.org/>`_, so there are some things that may behave
-differently than they do in other, independent computer algebra systems
-like Maple or Mathematica.  These are some of the gotchas and pitfalls
-that you may encounter when using SymPy.  See also the `FAQ
-<https://github.com/sympy/sympy/wiki/Faq>`_, the :ref:`Tutorial<tutorial>`, the
-remainder of the SymPy Docs, and the `official Python Tutorial
-<http://docs.python.org/tutorial/>`_.
-
-If you are already familiar with C or Java, you might also want to look
-this `4 minute Python tutorial
-<http://www.nerdparadise.com/tech/python/4minutecrashcourse/>`_.
-
-Ignore ``#doctest: +SKIP`` in the examples.  That has to do with
-internal testing of the examples.
-
-.. _equals-signs:
-
-Equals Signs (=)
-================
-
-Single Equals Sign
-------------------
-
-The equals sign (``=``) is the assignment operator, not an equality.  If
-you want to do :math:`x = y`, use ``Eq(x, y)`` for equality.
-Alternatively, all expressions are assumed to equal zero, so you can
-just subtract one side and use ``x - y``.
-
-The proper use of the equals sign is to assign expressions to variables.
-
-For example:
-
-    >>> from sympy.abc import x, y
-    >>> a = x - y
-    >>> print(a)
-    x - y
-
-Double Equals Signs
--------------------
-
-Double equals signs (``==``) are used to test equality.  However, this
-tests expressions exactly, not symbolically.  For example:
-
-    >>> (x + 1)**2 == x**2 + 2*x + 1
-    False
-    >>> (x + 1)**2 == (x + 1)**2
-    True
-
-If you want to test for symbolic equality, one way is to subtract one
-expression from the other and run it through functions like
-:func:`expand`, :func:`simplify`, and :func:`trigsimp` and see if the
-equation reduces to 0.
-
-    >>> from sympy import simplify, cos, sin, expand
-    >>> simplify((x + 1)**2 - (x**2 + 2*x + 1))
-    0
-    >>> simplify(sin(2*x) - 2*sin(x)*cos(x))
-    -2*sin(x)*cos(x) + sin(2*x)
-    >>> expand(sin(2*x) - 2*sin(x)*cos(x), trig=True)
-    0
-
-.. note::
-
-    See also `Why does SymPy say that two equal expressions are unequal?
-    <https://github.com/sympy/sympy/wiki/Faq>`_ in the FAQ.
-
-
-Variables
-=========
-
-Variables Assignment does not Create a Relation Between Expressions
--------------------------------------------------------------------
-
-When you use ``=`` to do assignment, remember that in Python, as in most
-programming languages, the variable does not change if you change the
-value you assigned to it.  The equations you are typing use the values
-present at the time of creation to "fill in" values, just like regular
-Python definitions. They are not altered by changes made afterwards.
-Consider the following:
-
-    >>> from sympy import Symbol
-    >>> a = Symbol('a')  # Symbol, `a`, stored as variable "a"
-    >>> b = a + 1        # an expression involving `a` stored as variable "b"
-    >>> print(b)
-    a + 1
-    >>> a = 4            # "a" now points to literal integer 4, not Symbol('a')
-    >>> print(a)
-    4
-    >>> b                # "b" is still pointing at the expression involving `a`
-    a + 1
-
-Changing quantity :obj:`a` does not change :obj:`b`; you are not working
-with a set of simultaneous equations. It might be helpful to remember
-that the string that gets printed when you print a variable referring to
-a SymPy object is the string that was given to it when it was created;
-that string does not have to be the same as the variable that you assign
-it to.
-
-    >>> from sympy import var
-    >>> r, t, d = var('rate time short_life')
-    >>> d = r*t
-    >>> print(d)
-    rate*time
-    >>> r = 80
-    >>> t = 2
-    >>> print(d)         # We haven't changed d, only r and t
-    rate*time
-    >>> d = r*t
-    >>> print(d)         # Now d is using the current values of r and t
-    160
-
-
-If you need variables that have dependence on each other, you can define
-functions.  Use the ``def`` operator.  Indent the body of the function.
-See the Python docs for more information on defining functions.
-
-    >>> c, d = var('c d')
-    >>> print(c)
-    c
-    >>> print(d)
-    d
-    >>> def ctimesd():
-    ...     """
-    ...     This function returns whatever c is times whatever d is.
-    ...     """
-    ...     return c*d
-    ...
-    >>> ctimesd()
-    c*d
-    >>> c = 2
-    >>> print(c)
-    2
-    >>> ctimesd()
-    2*d
-
-
-If you define a circular relationship, you will get a
-:exc:`RuntimeError`.
-
-    >>> def a():
-    ...     return b()
-    ...
-    >>> def b():
-    ...     return a()
-    ...
-    >>> a()
-    Traceback (most recent call last):
-      File "...", line ..., in ...
-        compileflags, 1) in test.globs
-      File "<...>", line 1, in <module>
-        a()
-      File "<...>", line 2, in a
-        return b()
-      File "<...>", line 2, in b
-        return a()
-      File "<...>", line 2, in a
-        return b()
-    ...
-    RuntimeError: maximum recursion depth exceeded
-
-
-.. note::
-    See also `Why doesn't changing one variable change another that depends on it?
-    <https://github.com/sympy/sympy/wiki/Faq>`_ in the FAQ.
-
-.. _symbols:
-
-Symbols
--------
-
-Symbols are variables, and like all other variables, they need to be
-assigned before you can use them.  For example:
-
-    >>> import sympy
-    >>> z**2  # z is not defined yet #doctest: +SKIP
-    Traceback (most recent call last):
-      File "<stdin>", line 1, in <module>
-    NameError: name 'z' is not defined
-    >>> sympy.var('z')  # This is the easiest way to define z as a standard symbol
-    z
-    >>> z**2
-    z**2
-
-
-If you use :command:`isympy`, it runs the following commands for you,
-giving you some default Symbols and Functions.
-
-    >>>
-    >>> from sympy import *
-    >>> x, y, z, t = symbols('x y z t')
-    >>> k, m, n = symbols('k m n', integer=True)
-    >>> f, g, h = symbols('f g h', cls=Function)
-
-You can also import common symbol names from :mod:`sympy.abc`.
-
-    >>> from sympy.abc import w
-    >>> w
-    w
-    >>> import sympy
-    >>> dir(sympy.abc)  #doctest: +SKIP
-    ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O',
-    'P', 'Q', 'R', 'S', 'Symbol', 'T', 'U', 'V', 'W', 'X', 'Y', 'Z',
-    '__builtins__', '__doc__', '__file__', '__name__', '__package__', '_greek',
-    '_latin', 'a', 'alpha', 'b', 'beta', 'c', 'chi', 'd', 'delta', 'e',
-    'epsilon', 'eta', 'f', 'g', 'gamma', 'h', 'i', 'iota', 'j', 'k', 'kappa',
-    'l', 'm', 'mu', 'n', 'nu', 'o', 'omega', 'omicron', 'p', 'phi', 'pi',
-    'psi', 'q', 'r', 'rho', 's', 'sigma', 't', 'tau', 'theta', 'u', 'upsilon',
-    'v', 'w', 'x', 'xi', 'y', 'z', 'zeta']
-
-If you want control over the assumptions of the variables, use
-:func:`Symbol` and :func:`symbols`.  See :ref:`Keyword
-Arguments<keyword-arguments>` below.
-
-Lastly, it is recommended that you not use :obj:`I`, :obj:`E`, :obj:`S`,
-:obj:`N`, :obj:`C`, :obj:`O`, or :obj:`Q` for variable or symbol names, as those
-are used for the imaginary unit (:math:`i`), the base of the natural
-logarithm (:math:`e`), the :func:`sympify` function (see :ref:`Symbolic
-Expressions<symbolic-expressions>` below), numeric evaluation (:func:`N`
-is equivalent to :ref:`evalf()<evalf-label>` ), the class registry (for
-things like :func:`C.cos`, to prevent cyclic imports in some code),
-the `big O <http://en.wikipedia.org/wiki/Big_O_notation>`_ order symbol
-(as in :math:`O(n\log{n})`), and the assumptions object that holds a list of
-supported ask keys (such as :obj:`Q.real`), respectively.  You can use the
-mnemonic ``QCOSINE`` to remember what Symbols are defined by default in SymPy.
-Or better yet, always use lowercase letters for Symbol names.  Python will
-not prevent you from overriding default SymPy names or functions, so be
-careful.
-
-    >>> cos(pi)  # cos and pi are a built-in sympy names.
-    -1
-    >>> pi = 3   # Notice that there is no warning for overriding pi.
-    >>> cos(pi)
-    cos(3)
-    >>> def cos(x):  # No warning for overriding built-in functions either.
-    ...     return 5*x
-    ...
-    >>> cos(pi)
-    15
-    >>> from sympy import cos  # reimport to restore normal behavior
-
-
-To get a full list of all default names in SymPy do:
-
-    >>> import sympy
-    >>> dir(sympy)  #doctest: +SKIP
-    # A big list of all default sympy names and functions follows.
-    # Ignore everything that starts and ends with __.
-
-If you have `iPython <http://ipython.scipy.org/moin/>`_ installed and
-use :command:`isympy`, you can also press the TAB key to get a list of
-all built-in names and to autocomplete.  Also, see `this page
-<http://kogs-www.informatik.uni-hamburg.de/~meine/python_tricks>`_ for a
-trick for getting tab completion in the regular Python console.
-
-.. note::
-    See also `What is the best way to create symbols?
-    <https://github.com/sympy/sympy/wiki/Faq>`_ in the FAQ.
-
-.. _symbolic-expressions:
-
-Symbolic Expressions
-====================
-
-.. _python-vs-sympy-numbers:
-
-Python numbers vs. SymPy Numbers
---------------------------------
-
-SymPy uses its own classes for integers, rational numbers, and floating
-point numbers instead of the default Python :obj:`int` and :obj:`float`
-types because it allows for more control.  But you have to be careful.
-If you type an expression that just has numbers in it, it will default
-to a Python expression.  Use the :func:`sympify` function, or just
-:func:`S`, to ensure that something is a SymPy expression.
-
-    >>> 6.2  # Python float. Notice the floating point accuracy problems. #doctest: +SKIP
-    6.2000000000000002
-    >>> type(6.2)
-    <... 'float'>
-    >>> S(6.2)  # SymPy Float has no such problems because of arbitrary precision.
-    6.20000000000000
-    >>> type(S(6.2))
-    <class 'sympy.core.numbers.Float'>
-
-If you include numbers in a SymPy expression, they will be sympified
-automatically, but there is one gotcha you should be aware of.  If you
-do ``<number>/<number>`` inside of a SymPy expression, Python will
-evaluate the two numbers before SymPy has a chance to get
-to them.  The solution is to :func:`sympify` one of the numbers, or use
-:mod:`Rational`.
-
-    >>> x**(1/2)  # evaluates to x**0 or x**0.5 #doctest: +SKIP
-    x**0.5
-    >>> x**(S(1)/2)  # sympyify one of the ints
-    sqrt(x)
-    >>> x**Rational(1, 2)  # use the Rational class
-    sqrt(x)
-
-With a power of ``1/2`` you can also use ``sqrt`` shorthand:
-
-    >>> sqrt(x) == x**Rational(1, 2)
-    True
-
-If the two integers are not directly separated by a division sign then
-you don't have to worry about this problem:
-
-    >>> x**(2*x/3)
-    x**(2*x/3)
-
-.. note::
-
-    A common mistake is copying an expression that is printed and
-    reusing it.  If the expression has a :mod:`Rational` (i.e.,
-    ``<number>/<number>``) in it, you will not get the same result,
-    obtaining the Python result for the division rather than a SymPy
-    Rational.
-
-    >>> x = Symbol('x')
-    >>> print(solve(7*x -22, x))
-    [22/7]
-    >>> 22/7  # If we just copy and paste we get int 3 or a float #doctest: +SKIP
-    3.142857142857143
-    >>> # One solution is to just assign the expression to a variable
-    >>> # if we need to use it again.
-    >>> a = solve(7*x - 22, x)
-    >>> a
-    [22/7]
-
-    The other solution is to put quotes around the expression
-    and run it through S() (i.e., sympify it):
-
-    >>> S("22/7")
-    22/7
-
-Also, if you do not use :command:`isympy`, you could use ``from
-__future__ import division`` to prevent the ``/`` sign from performing
-`integer division <http://en.wikipedia.org/wiki/Integer_division>`_.
-
-    >>>
-    >>> 1/2   # With division imported it evaluates to a python float #doctest: +SKIP
-    0.5
-    >>> 1//2  # You can still achieve integer division with //
-    0
-
-    But be careful: you will now receive floats where you might have desired
-    a Rational:
-
-    >>> x**(1/2)  #doctest: +SKIP
-    x**0.5
-
-:mod:`Rational` only works for number/number and is only meant for
-rational numbers.  If you want a fraction with symbols or expressions in
-it, just use ``/``.  If you do number/expression or expression/number,
-then the number will automatically be converted into a SymPy Number.
-You only need to be careful with number/number.
-
-    >>> Rational(2, x)
-    Traceback (most recent call last):
-    ...
-    TypeError: invalid input: x
-    >>> 2/x
-    2/x
-
-Evaluating Expressions with Floats and Rationals
-------------------------------------------------
-
-SymPy keeps track of the precision of Floats. The default precision is
-15 digits. When expressions involving Floats are evaluated, the result
-will be expressed to 15 digits of precision but those digits (depending
-on the numbers involved with the calculation) may not all be significant.
-
-The first issue to keep in mind is how the Float is created: it is created
-with a value and a precision. The precision indicates how precise of a value
-to use when that Float (or an expression it appears in) is evaluated.
-
-The values can be given as strings, integers, floats, or Rationals.
-
-    - strings and integers are interpreted as exact
-
-    >>> Float(100)
-    100.000000000000
-    >>> Float('100', 5)
-    100.00
-
-    - to have the precision match the number of digits, the null string
-      can be used for the precision
-
-    >>> Float(100, '')
-    100.
-    >>> Float('12.34')
-    12.3400000000000
-    >>> Float('12.34', '')
-    12.34
-
-    >>> s, r = [Float(j, 3) for j in ('0.25', Rational(1, 7))]
-    >>> for f in [s, r]:
-    ...     print(f)
-    0.250
-    0.143
-
-Next, notice that each of those values looks correct to 3 digits. But if we try
-to evaluate them to 20 digits, a difference will become apparent:
-
-    The 0.25 (with precision of 3) represents a number that has a non-repeating
-    binary decimal; 1/7 is repeating in binary and decimal -- it cannot be
-    represented accurately too far past those first 3 digits (the correct
-    decimal is a repeating 142857):
-
-    >>> s.n(20)
-    0.25000000000000000000
-    >>> r.n(20)
-    0.14285278320312500000
-
-    It is important to realize that although a Float is being displayed in
-    decimal at aritrary precision, it is actually stored in binary. Once the
-    Float is created, its binary information is set at the given precision.
-    The accuracy of that value cannot be subsequently changed; so 1/7, at a
-    precision of 3 digits, can be padded with binary zeros, but these will
-    not make it a more accurate value of 1/7.
-
-If inexact, low-precision numbers are involved in a calculation with
-with higher precision values, the evalf engine will increase the precision
-of the low precision values and inexact results will be obtained. This is
-feature of calculations with limited precision:
-
-    >>> Float('0.1', 10) + Float('0.1', 3)
-    0.2000061035
-
-Although the evalf engine tried to maintain 10 digits of precision (since
-that was the highest precision represented) the 3-digit precision used
-limits the accuracy to about 4 digits -- not all the digits you see
-are significant. evalf doesn't try to keep track of the number of
-significant digits.
-
-That very simple expression involving the addition of two numbers with
-different precisions will hopefully be instructive in helping you
-understand why more complicated expressions (like trig expressions that
-may not be simplified) will not evaluate to an exact zero even though,
-with the right simplification, they should be zero. Consider this
-unsimplified trig identity, multiplied by a big number:
-
-    >>> big = 12345678901234567890
-    >>> big_trig_identity = big*cos(x)**2 + big*sin(x)**2 - big*1
-    >>> abs(big_trig_identity.subs(x, .1).n(2)) > 1000
-    True
-
-When the cos and sin terms were evaluated to 15 digits of precision and
-multiplied by the big number, they gave a large number that was only
-precise to 15 digits (approximately) and when the 20 digit big number
-was subtracted the result was not zero.
-
-There are three things that will help you obtain more precise numerical
-values for expressions:
-
-    1) Pass the desired substitutions with the call to evaluate. By doing
-    the subs first, the Float values can not be updated as necessary. By
-    passing the desired substitutions with the call to evalf the ability
-    to re-evaluate as necessary is gained and the results are impressively
-    better:
-
-    >>> big_trig_identity.n(2, {x: 0.1})
-    -0.e-91
-
-    2) Use Rationals, not Floats. During the evaluation process, the
-    Rational can be computed to an arbitrary precision while the Float,
-    once created -- at a default of 15 digits -- cannot. Compare the
-    value of -1.4e+3 above with the nearly zero value obtained when
-    replacing x with a Rational representing 1/10 -- before the call
-    to evaluate:
-
-    >>> big_trig_identity.subs(x, S('1/10')).n(2)
-    0.e-91
-
-    3) Try to simplify the expression. In this case, SymPy will recognize
-    the trig identity and simplify it to zero so you don't even have to
-    evaluate it numerically:
-
-    >>> big_trig_identity.simplify()
-    0
-
-
-.. _Immutability-of-Expressions:
-
-Immutability of Expressions
----------------------------
-
-Expressions in SymPy are immutable, and cannot be modified by an in-place
-operation.  This means that a function will always return an object, and the
-original expression will not be modified. The following example snippet
-demonstrates how this works::
-
-	def main():
-	    var('x y a b')
-	    expr = 3*x + 4*y
-	    print 'original =', expr
-	    expr_modified = expr.subs({x: a, y: b})
-	    print 'modified =', expr_modified
-
-	if __name__ == "__main__":
-	    main()
-
-The output shows that the :func:`subs` function has replaced variable
-:obj:`x` with variable :obj:`a`, and variable :obj:`y` with variable :obj:`b`::
-
-	original = 3*x + 4*y
-	modified = 3*a + 4*b
-
-The :func:`subs` function does not modify the original expression :obj:`expr`.
-Rather, a modified copy of the expression is returned. This returned object
-is stored in the variable :obj:`expr_modified`. Note that unlike C/C++ and
-other high-level languages, Python does not require you to declare a variable
-before it is used.
-
-
-Mathematical Operators
-----------------------
-
-SymPy uses the same default operators as Python.  Most of these, like
-``*/+-``, are standard.  Aside from integer division discussed in
-:ref:`Python numbers vs. SymPy Numbers <python-vs-sympy-numbers>` above,
-you should also be aware that implied multiplication is not allowed. You
-need to use ``*`` whenever you wish to multiply something.  Also, to
-raise something to a power, use ``**``, not ``^`` as many computer
-algebra systems use.  Parentheses ``()`` change operator precedence as
-you would normally expect.
-
-In :command:`isympy`, with the :command:`ipython` shell::
-
-    >>> 2x
-    Traceback (most recent call last):
-    ...
-    SyntaxError: invalid syntax
-    >>> 2*x
-    2*x
-    >>> (x + 1)^2  # This is not power.  Use ** instead.
-    Traceback (most recent call last):
-    ...
-    TypeError: unsupported operand type(s) for ^: 'Add' and 'int'
-    >>> (x + 1)**2
-    (x + 1)**2
-    >>> pprint(3 - x**(2*x)/(x + 1))
-        2*x
-       x
-    - ----- + 3
-      x + 1
-
-
-Inverse Trig Functions
-----------------------
-
-SymPy uses different names for some functions than most computer algebra
-systems.  In particular, the inverse trig functions use the python names
-of :func:`asin`, :func:`acos` and so on instead of the usual ``arcsin``
-and ``arccos``.  Use the methods described in :ref:`Symbols <symbols>`
-above to see the names of all SymPy functions.
-
-Special Symbols
-===============
-
-The symbols ``[]``, ``{}``, ``=``, and ``()`` have special meanings in
-Python, and thus in SymPy.  See the Python docs linked to above for
-additional information.
-
-.. _lists:
-
-Lists
------
-
-Square brackets ``[]`` denote a list.  A list is a container that holds
-any number of different objects.  A list can contain anything, including
-items of different types.  Lists are mutable, which means that you can
-change the elements of a list after it has been created.  You access the
-items of a list also using square brackets, placing them after the list
-or list variable.  Items are numbered using the space before the item.
-
-.. note::
-
-    List indexes begin at 0.
-
-Example:
-
-    >>> a = [x, 1]  # A simple list of two items
-    >>> a
-    [x, 1]
-    >>> a[0]  # This is the first item
-    x
-    >>> a[0] = 2  # You can change values of lists after they have been created
-    >>> print(a)
-    [2, 1]
-    >>> print(solve(x**2 + 2*x - 1, x))  # Some functions return lists
-    [-1 + sqrt(2), -sqrt(2) - 1]
-
-
-.. note::
-    See the Python docs for more information on lists and the square
-    bracket notation for accessing elements of a list.
-
-Dictionaries
-------------
-
-Curly brackets ``{}`` denote a dictionary, or a dict for short.  A
-dictionary is an unordered list of non-duplicate keys and values.  The
-syntax is ``{key: value}``.  You can access values of keys using square
-bracket notation.
-
-    >>> d = {'a': 1, 'b': 2}  # A dictionary.
-    >>> d
-    {'a': 1, 'b': 2}
-    >>> d['a']  # How to access items in a dict
-    1
-    >>> roots((x - 1)**2*(x - 2), x)  # Some functions return dicts
-    {1: 2, 2: 1}
-    >>> # Some SymPy functions return dictionaries.  For example,
-    >>> # roots returns a dictionary of root:multiplicity items.
-    >>> roots((x - 5)**2*(x + 3), x)
-    {-3: 1, 5: 2}
-    >>> # This means that the root -3 occurs once and the root 5 occurs twice.
-
-.. note::
-
-    See the Python docs for more information on dictionaries.
-
-Tuples
-------
-
-Parentheses ``()``, aside from changing operator precedence and their
-use in function calls, (like ``cos(x)``), are also used for tuples.  A
-``tuple`` is identical to a :ref:`list <lists>`, except that it is not
-mutable.  That means that you can not change their values after they
-have been created.  In general, you will not need tuples in SymPy, but
-sometimes it can be more convenient to type parentheses instead of
-square brackets.
-
-    >>> t = (1, 2, x)  # Tuples are like lists
-    >>> t
-    (1, 2, x)
-    >>> t[0]
-    1
-    >>> t[0] = 4  # Except you can not change them after they have been created
-    Traceback (most recent call last):
-      File "<console>", line 1, in <module>
-    TypeError: 'tuple' object does not support item assignment
-
-    Single element tuples, unlike lists, must have a comma in them:
-
-    >>> (x,)
-    (x,)
-
-    Without the comma, a single expression without a comma is not a tuple:
-
-    >>> (x)
-    x
-
-    integrate takes a sequence as the second argument if you want to integrate
-    with limits (and a tuple or list will work):
-
-    >>> integrate(x**2, (x, 0, 1))
-    1/3
-    >>> integrate(x**2, [x, 0, 1])
-    1/3
-
-
-.. note::
-
-    See the Python docs for more information on tuples.
-
-.. _keyword-arguments:
-
-Keyword Arguments
------------------
-
-Aside from the usage described :ref:`above <equals-signs>`, equals signs
-(``=``) are also used to give named arguments to functions.  Any
-function that has ``key=value`` in its parameters list (see below on how
-to find this out), then ``key`` is set to ``value`` by default.  You can
-change the value of the key by supplying your own value using the equals
-sign in the function call.  Also, functions that have ``**`` followed by
-a name in the parameters list (usually ``**kwargs`` or
-``**assumptions``) allow you to add any number of ``key=value`` pairs
-that you want, and they will all be evaluated according to the function.
-
-    sqrt(x**2) doesn't auto simplify to x because x is assumed to be
-    complex by default, and, for example, sqrt((-1)**2) == sqrt(1) == 1 != -1:
-
-    >>> sqrt(x**2)
-    sqrt(x**2)
-
-    Giving assumptions to Symbols is an example of using the keyword argument:
-
-    >>> x = Symbol('x', positive=True)
-
-    The square root will now simplify since it knows that x >= 0:
-
-    >>> sqrt(x**2)
-    x
-
-    powsimp has a default argument of combine='all':
-
-    >>> pprint(powsimp(x**n*x**m*y**n*y**m))
-         m + n
-    (x*y)
-
-    Setting combine to the default value is the same as not setting it.
-
-    >>> pprint(powsimp(x**n*x**m*y**n*y**m, combine='all'))
-         m + n
-    (x*y)
-
-    The non-default options are 'exp', which combines exponents...
-
-    >>> pprint(powsimp(x**n*x**m*y**n*y**m, combine='exp'))
-     m + n  m + n
-    x     *y
-
-    ...and 'base', which combines bases.
-
-    >>> pprint(powsimp(x**n*x**m*y**n*y**m, combine='base'))
-         m      n
-    (x*y) *(x*y)
-
-.. note::
-
-    See the Python docs for more information on function parameters.
-
-Getting help from within SymPy
-==============================
-
-help()
-------
-
-Although all docs are available at `docs.sympy.org <http://docs.sympy.org/>`_ or on the
-`SymPy Wiki <http://wiki.sympy.org/>`_, you can also get info on functions from within the
-Python interpreter that runs SymPy.  The easiest way to do this is to do
-``help(function)``, or ``function?`` if you are using :command:`ipython`::
-
-    In [1]: help(powsimp)  # help() works everywhere
-
-    In [2]: # But in ipython, you can also use ?, which is better because it
-    In [3]: # it gives you more information
-    In [4]: powsimp?
-
-These will give you the function parameters and docstring for
-:func:`powsimp`.  The output will look something like this:
-
-.. module:: sympy.simplify.simplify
-.. autofunction:noindex: powsimp
-
-source()
---------
-
-Another useful option is the :func:`source` function.  This will print
-the source code of a function, including any docstring that it may have.
-You can also do ``function??`` in :command:`ipython`.  For example,
-from SymPy 0.6.5:
-
-    >>> source(simplify)  # simplify() is actually only 2 lines of code. #doctest: +SKIP
-    In file: ./sympy/simplify/simplify.py
-    def simplify(expr):
-        """Naively simplifies the given expression.
-           ...
-           Simplification is not a well defined term and the exact strategies
-           this function tries can change in the future versions of SymPy. If
-           your algorithm relies on "simplification" (whatever it is), try to
-           determine what you need exactly  -  is it powsimp()? radsimp()?
-           together()?, logcombine()?, or something else? And use this particular
-           function directly, because those are well defined and thus your algorithm
-           will be robust.
-           ...
-        """
-        expr = Poly.cancel(powsimp(expr))
-        return powsimp(together(expr.expand()), combine='exp', deep=True)
diff --git a/dev-py3k/_sources/guide.txt b/dev-py3k/_sources/guide.txt
deleted file mode 100644
index 3dc1f67d8b4..00000000000
--- a/dev-py3k/_sources/guide.txt
+++ /dev/null
@@ -1,538 +0,0 @@
-.. _guide:
-
-==================
-SymPy User's Guide
-==================
-
-.. role:: input(strong)
-
-Introduction
-============
-
-If you are new to SymPy, start with the :ref:`Tutorial <tutorial>`. If you went
-through it, now
-it's time to learn how SymPy works internally and this is what this guide is
-about. Once you grasp the idea behind SymPy, you will be able to use it
-effectively and also know how to extend it and fix it.
-You may also be just interested in :ref:`SymPy Modules Reference <module-docs>`.
-
-Learning SymPy
-==============
-
-Everyone has different ways of understanding the code written by others.
-
-Ondřej's approach
------------------
-
-Let's say I'd like to understand how ``x + y + x`` works and how it is possible
-that it gets simplified to ``2*x + y``.
-
-I write a simple script, I usually call it ``t.py`` (I don't remember anymore
-why I call it that way)::
-
-    from sympy.abc import x, y
-
-    e = x + y + x
-
-    print e
-
-And I try if it works
-
-.. parsed-literal::
-
-    $ :input:`python t.py`
-    y + 2*x
-
-Now I start `winpdb <http://winpdb.org/>`_ on it (if you've never used winpdb
--- it's an excellent multiplatform debugger, works on Linux, Windows and Mac OS
-X):
-
-.. parsed-literal::
-
-    $ :input:`winpdb t.py`
-    y + 2*x
-
-and a winpdb window will popup, I move to the next line using F6:
-
-.. image:: pics/winpdb1.png
-
-Then I step into (F7) and after a little debugging I get for example:
-
-.. image:: pics/winpdb2.png
-
-.. tip:: Make the winpdb window larger on your screen, it was just made smaller
-         to fit in this guide.
-
-I see values of all local variables in the left panel, so it's very easy to see
-what's happening. You can see, that the ``y + 2*x`` is emerging in the ``obj``
-variable. Observing that ``obj`` is constructed from ``c_part`` and ``nc_part``
-and seeing what ``c_part`` contains (``y`` and ``2*x``). So looking at the line
-28 (the whole line is not visible on the screenshot, so here it is)::
-
-    c_part, nc_part, lambda_args, order_symbols = cls.flatten(map(_sympify, args))
-
-you can see that the simplification happens in ``cls.flatten``. Now you can set
-the breakpoint on the line 28, quit winpdb (it will remember the breakpoint),
-start it again, hit F5, this will stop at this breakpoint, hit F7, this will go
-into the function ``Add.flatten()``::
-
-    @classmethod
-    def flatten(cls, seq):
-        """
-        Takes the sequence "seq" of nested Adds and returns a flatten list.
-
-        Returns: (commutative_part, noncommutative_part, lambda_args,
-            order_symbols)
-
-        Applies associativity, all terms are commutable with respect to
-        addition.
-        """
-        terms = {}      # term -> coeff
-                        # e.g. x**2 -> 5   for ... + 5*x**2 + ...
-
-        coeff = S.Zero  # standalone term
-                        # e.g. 3 + ...
-        lambda_args = None
-        order_factors = []
-        while seq:
-            o = seq.pop(0)
-
-and then you can study how it works. I am going to stop here, this should be
-enough to get you going -- with the above technique, I am able to understand
-almost any Python code.
-
-
-SymPy's Architecture
-====================
-
-We try to make the sources easily understandable, so you can look into the
-sources and read the doctests, it should be well documented and if you don't
-understand something, ask on the mailinglist_.
-
-You can find all the decisions archived in the issues_, to see rationale for
-doing this and that.
-
-Basics
-------
-
-All symbolic things are implemented using subclasses of the ``Basic`` class.
-First, you need to create symbols using ``Symbol("x")`` or numbers using
-``Integer(5)`` or ``Float(34.3)``. Then you construct the expression using any
-class from SymPy.  For example ``Add(Symbol("a"), Symbol("b"))`` gives an
-instance of the ``Add`` class.  You can call all methods, which the particular
-class supports.
-
-For easier use, there is a syntactic sugar for expressions like:
-
-``cos(x) + 1`` is equal to ``cos(x).__add__(1)`` is equal to
-``Add(cos(x), Integer(1))``
-
-or
-
-``2/cos(x)`` is equal to ``cos(x).__rdiv__(2)`` is equal to
-``Mul(Rational(2), Pow(cos(x), Rational(-1)))``.
-
-So, you can write normal expressions using python arithmetics like this::
-
-    a = Symbol("a")
-    b = Symbol("b")
-    e = (a + b)**2
-    print e
-
-but from the sympy point of view, we just need the classes ``Add``, ``Mul``,
-``Pow``, ``Rational``, ``Integer``.
-
-Automatic evaluation to canonical form
---------------------------------------
-
-For computation, all expressions need to be in a
-canonical form, this is done during the creation of the particular instance
-and only inexpensive operations are performed, necessary to put the expression
-in the
-canonical form.  So the canonical form doesn't mean the simplest possible
-expression. The exact list of operations performed depend on the
-implementation.  Obviously, the definition of the canonical form is arbitrary,
-the only requirement is that all equivalent expressions must have the same
-canonical form.  We tried the conversion to a canonical (standard) form to be
-as fast as possible and also in a way so that the result is what you would
-write by hand - so for example ``b*a + -4 + b + a*b + 4 + (a + b)**2`` becomes
-``2*a*b + b + (a + b)**2``.
-
-Whenever you construct an expression, for example ``Add(x, x)``, the
-``Add.__new__()`` is called and it determines what to return. In this case::
-
-    >>> from sympy import Add
-    >>> from sympy.abc import x
-    >>> e = Add(x, x)
-    >>> e
-    2*x
-
-    >>> type(e)
-    <class 'sympy.core.mul.Mul'>
-
-``e`` is actually an instance of ``Mul(2, x)``, because ``Add.__new__()``
-returned ``Mul``.
-
-Comparisons
------------
-
-Expressions can be compared using a regular python syntax::
-
-    >>> from sympy.abc import x, y
-    >>> x + y == y + x
-    True
-
-    >>> x + y == y - x
-    False
-
-We made the following decision in SymPy: ``a = Symbol("x")`` and another
-``b = Symbol("x")`` (with the same string "x") is the same thing, i.e ``a == b`` is
-``True``. We chose ``a == b``, because it is more natural - ``exp(x) == exp(x)`` is
-also ``True`` for the same instance of ``x`` but different instances of ``exp``,
-so we chose to have ``exp(x) == exp(x)`` even for different instances of ``x``.
-
-Sometimes, you need to have a unique symbol, for example as a temporary one in
-some calculation, which is going to be substituted for something else at the
-end anyway. This is achieved using ``Dummy("x")``. So, to sum it
-up::
-
-    >>> from sympy import Symbol, Dummy
-    >>> Symbol("x") == Symbol("x")
-    True
-
-    >>> Dummy("x") == Dummy("x")
-    False
-
-
-Debugging
----------
-
-Starting with 0.6.4, you can turn on/off debug messages with the environment
-variable SYMPY_DEBUG, which is expected to have the values True or False. For
-example, to turn on debugging, you would issue::
-
-    [user@localhost]: SYMPY_DEBUG=True ./bin/isympy
-
-Functionality
--------------
-
-There are no given requirements on classes in the library. For example, if they
-don't implement the ``fdiff()`` method and you construct an expression using
-such a class, then trying to use the ``Basic.series()`` method will raise an
-exception of not finding the ``fdiff()`` method in your class.  This "duck
-typing" has an advantage that you just implement the functionality which you
-need.
-
-You can define the ``cos`` class like this::
-
-    class cos(Function):
-        pass
-
-and use it like ``1 + cos(x)``, but if you don't implement the ``fdiff()`` method,
-you will not be able to call ``(1 + cos(x)).series()``.
-
-The symbolic object is characterized (defined) by the things which it can do,
-so implementing more methods like ``fdiff()``, ``subs()`` etc., you are creating
-a "shape" of the symbolic object. Useful things to implement in new classes are:
-``hash()`` (to use the class in comparisons), ``fdiff()`` (to use it in series
-expansion), ``subs()`` (to use it in expressions, where some parts are being
-substituted) and ``series()`` (if the series cannot be computed using the
-general ``Basic.series()`` method). When you create a new class, don't worry
-about this too much - just try to use it in your code and you will realize
-immediately which methods need to be implemented in each situation.
-
-All objects in sympy are immutable - in the sense that any operation just
-returns a new instance (it can return the same instance only if it didn't
-change). This is a common mistake to change the current instance, like
-``self.arg = self.arg + 1`` (wrong!). Use ``arg = self.arg + 1; return arg`` instead.
-The object is immutable in the
-sense of the symbolic expression it represents. It can modify itself to keep
-track of, for example, its hash. Or it can recalculate anything regarding the
-expression it contains. But the expression cannot be changed. So you can pass
-any instance to other objects, because you don't have to worry that it will
-change, or that this would break anything.
-
-Conclusion
-----------
-
-Above are the main ideas behind SymPy that we try to obey. The rest
-depends on the current implementation and may possibly change in the future.
-The point of all of this is that the interdependencies inside SymPy should be
-kept to a minimum. If one wants to add new functionality to SymPy, all that is
-necessary is to create a subclass of ``Basic`` and implement what you want.
-
-Functions
----------
-
-How to create a new function with one variable::
-
-    class sign(Function):
-
-        nargs = 1
-
-        @classmethod
-        def eval(cls, arg):
-            if isinstance(arg, Basic.NaN):
-                return S.NaN
-            if isinstance(arg, Basic.Zero):
-                return S.Zero
-            if arg.is_positive:
-                return S.One
-            if arg.is_negative:
-                return S.NegativeOne
-            if isinstance(arg, Basic.Mul):
-                coeff, terms = arg.as_coeff_terms()
-                if not isinstance(coeff, Basic.One):
-                    return cls(coeff) * cls(Basic.Mul(*terms))
-
-        is_bounded = True
-
-        def _eval_conjugate(self):
-            return self
-
-        def _eval_is_zero(self):
-            return isinstance(self[0], Basic.Zero)
-
-and that's it. The ``_eval_*`` functions are called when something is needed.
-The ``eval`` is called when the class is about to be instantiated and it
-should return either some simplified instance of some other class or if the
-class should be unmodified, return ``None`` (see ``core/function.py`` in
-``Function.__new__`` for implementation details). See also tests in
-`sympy/functions/elementary/tests/test_interface.py
-<https://github.com/sympy/sympy/blob/master/sympy/functions/elementary/tests/test_interface.py>`_ that test this interface. You can use them to create your own new functions.
-
-The applied function ``sign(x)`` is constructed using
-::
-
-    sign(x)
-
-both inside and outside of SymPy. Unapplied functions ``sign`` is just the class
-itself::
-
-    sign
-
-both inside and outside of SymPy. This is the current structure of classes in
-SymPy::
-
-    class BasicType(type):
-        pass
-    class MetaBasicMeths(BasicType):
-        ...
-    class BasicMeths(AssumeMeths):
-        __metaclass__ = MetaBasicMeths
-        ...
-    class Basic(BasicMeths):
-        ...
-    class FunctionClass(MetaBasicMeths):
-        ...
-    class Function(Basic, RelMeths, ArithMeths):
-        __metaclass__ = FunctionClass
-        ...
-
-The exact names of the classes and the names of the methods and how they work
-can be changed in the future.
-
-This is how to create a function with two variables::
-
-    class chebyshevt_root(Function):
-        nargs = 2
-
-        @classmethod
-        def eval(cls, n, k):
-            if not 0 <= k < n:
-                raise ValueError("must have 0 <= k < n")
-            return C.cos(S.Pi*(2*k + 1)/(2*n))
-
-
-.. note:: the first argument of a @classmethod should be ``cls`` (i.e. not
-          ``self``).
-
-Here it's how to define a derivative of the function::
-
-    >>> from sympy import Function, sympify, cos
-    >>> class my_function(Function):
-    ...     nargs = 1
-    ...
-    ...     def fdiff(self, argindex = 1):
-    ...         return cos(self.args[0])
-    ...
-    ...     @classmethod
-    ...     def eval(cls, arg):
-    ...         arg = sympify(arg)
-    ...         if arg == 0:
-    ...             return sympify(0)
-
-So guess what this ``my_function`` is going to be? Well, it's derivative is
-``cos`` and the function value at 0 is 0, but let's pretend we don't know::
-
-    >>> from sympy import pprint
-    >>> pprint(my_function(x).series(x, 0, 10))
-         3     5     7       9
-        x     x     x       x       / 10\
-    x - -- + --- - ---- + ------ + O\x  /
-        6    120   5040   362880
-
-Looks familiar indeed::
-
-    >>> from sympy import sin
-    >>> pprint(sin(x).series(x, 0, 10))
-         3     5     7       9
-        x     x     x       x       / 10\
-    x - -- + --- - ---- + ------ + O\x  /
-        6    120   5040   362880
-
-Let's try a more complicated example. Let's define the derivative in terms of
-the function itself::
-
-    >>> class what_am_i(Function):
-    ...     nargs = 1
-    ...
-    ...     def fdiff(self, argindex = 1):
-    ...         return 1 - what_am_i(self.args[0])**2
-    ...
-    ...     @classmethod
-    ...     def eval(cls, arg):
-    ...         arg = sympify(arg)
-    ...         if arg == 0:
-    ...             return sympify(0)
-
-So what is ``what_am_i``?  Let's try it::
-
-    >>> pprint(what_am_i(x).series(x, 0, 10))
-         3      5       7       9
-        x    2*x    17*x    62*x     / 10\
-    x - -- + ---- - ----- + ----- + O\x  /
-        3     15     315     2835
-
-Well, it's ``tanh``::
-
-    >>> from sympy import tanh
-    >>> pprint(tanh(x).series(x, 0, 10))
-         3      5       7       9
-        x    2*x    17*x    62*x     / 10\
-    x - -- + ---- - ----- + ----- + O\x  /
-        3     15     315     2835
-
-The new functions we just defined are regular SymPy objects, you
-can use them all over SymPy, e.g.::
-
-    >>> from sympy import limit
-    >>> limit(what_am_i(x)/x, x, 0)
-    1
-
-
-Common tasks
-------------
-
-Please use the same way as is shown below all across SymPy.
-
-**accessing parameters**::
-
-    >>> from sympy import sign, sin
-    >>> from sympy.abc import x, y, z
-
-    >>> e = sign(x**2)
-    >>> e.args
-    (x**2,)
-
-    >>> e.args[0]
-    x**2
-
-    Number arguments (in Adds and Muls) will always be the first argument;
-    other arguments might be in arbitrary order:
-    >>> (1 + x + y*z).args[0]
-    1
-    >>> (1 + x + y*z).args[1] in (x, y*z)
-    True
-
-    >>> (y*z).args
-    (y, z)
-
-    >>> sin(y*z).args
-    (y*z,)
-
-Never use internal methods or variables, prefixed with "``_``" (example: don't
-use ``_args``, use ``.args`` instead).
-
-**testing the structure of a SymPy expression**
-
-Applied functions::
-
-    >>> from sympy import sign, exp, Function
-    >>> e = sign(x**2)
-
-    >>> isinstance(e, sign)
-    True
-
-    >>> isinstance(e, exp)
-    False
-
-    >>> isinstance(e, Function)
-    True
-
-So ``e`` is a ``sign(z)`` function, but not ``exp(z)`` function.
-
-Unapplied functions::
-
-    >>> from sympy import sign, exp, FunctionClass
-    >>> e = sign
-
-    >>> f = exp
-
-    >>> g = Add
-
-    >>> isinstance(e, FunctionClass)
-    True
-
-    >>> isinstance(f, FunctionClass)
-    True
-
-    >>> isinstance(g, FunctionClass)
-    False
-
-    >>> g is Add
-    True
-
-So ``e`` and ``f`` are functions, ``g`` is not a function.
-
-Contributing
-============
-
-We welcome every SymPy user to participate in it's development. Don't worry if
-you've never contributed to any open source project, we'll help you learn
-anything necessary, just ask on our mailinglist_.
-
-Don't be afraid to ask anything and don't worry that you are wasting our time
-if you are new to SymPy and ask questions that maybe most of the people know
-the answer to -- you are not, because that's exactly what the mailinglist_ is
-for and people answer your emails because they want to. Also we try hard to
-answer every email, so you'll always get some feedback and pointers what to do
-next.
-
-Improving the code
-------------------
-
-Go to issues_ that are sorted by priority and simply find something that you
-would like to get fixed and fix it. If you find something odd, please report it
-into issues_ first before fixing it. Feel free to consult with us on the
-mailinglist_.  Then send your patch either to the issues_ or the mailinglist_.
-See the `SymPy Development
-<https://github.com/sympy/sympy/wiki/old-wiki-Sympy-Development>` wiki page,
-but don't worry about it too much if you find it too formal - simply get in touch
-with us on the mailinglist_ and we'll help you get your patch accepted.
-
-.. _issues:             http://code.google.com/p/sympy/issues/list
-.. _mailinglist:        http://groups.google.com/group/sympy
-.. _SymPyDevelopment:   http://code.google.com/p/sympy/wiki/SymPyDevelopment
-
-Please read our excellent `SymPy Patches Tutorial
-<https://github.com/sympy/sympy/wiki/Development-workflow>`_ at our
-wiki for a guide on how to write patches to SymPy, how to work with Git,
-and how to make your life easier as you get started with SymPy.
-
-
-Improving the docs
-------------------
-
-Please see :ref:`the documentation <module-docs>` how to fix and improve
-SymPy's documentation. All contribution is very welcome.
diff --git a/dev-py3k/_sources/index.txt b/dev-py3k/_sources/index.txt
deleted file mode 100644
index f1f3168df58..00000000000
--- a/dev-py3k/_sources/index.txt
+++ /dev/null
@@ -1,34 +0,0 @@
-.. SymPy documentation master file, created by sphinx-quickstart.py on Sat Mar 22 19:34:32 2008.
-   You can adapt this file completely to your liking, but it should at least
-   contain the root `toctree` directive.
-
-Welcome to SymPy's documentation!
-=================================
-
-`SymPy <http://sympy.org>`_ is a Python library for symbolic mathematics.
-If you are new to SymPy, start with the Tutorial.
-
-This is the central page for all of SymPy's documentation.
-
-
-Contents:
-
-.. toctree::
-   :maxdepth: 2
-
-   install.rst
-   tutorial/tutorial.en.rst
-   gotchas.rst
-   guide.rst
-   modules/index.rst
-   python-comparisons.rst
-   wiki.rst
-   outreach.rst
-   aboutus.rst
-
-If something cannot be easily accessed from this page, it's a bug (`please
-report it`_).
-
-This documentation is maintained with docutils, so you might see some comments in the form #doctest:... . You can safely ignore them.
-
-.. _please report it: http://code.google.com/p/sympy/issues/list
diff --git a/dev-py3k/_sources/install.txt b/dev-py3k/_sources/install.txt
deleted file mode 100644
index e6207a0edf3..00000000000
--- a/dev-py3k/_sources/install.txt
+++ /dev/null
@@ -1,113 +0,0 @@
-Installation
-------------
-
-The SymPy CAS can be installed on virtually any computer with Python 2.5 or
-above. SymPy does not require any special Python modules: let us know if you
-have any problems with SymPy on a standard Python install. The current
-recommended method of installation is directly from the source files.
-Alternatively, executables are available for Windows, and some Linux
-distributions have SymPy packages available.
-
-Source
-======
-
-SymPy currently recommends that users install directly from the source files.
-You will first have to download the source files via the archive. Download the
-latest release (tar.gz) from the `downloads site`_ and open it with your
-operating system's standard decompression utility.
-
-When downloading the archive, make sure to get the correct version (Python 2 or
-Python 3). If you're not sure which one to use, you probably want the Python 2
-version. Note that you can install both if you want.
-
-After the download is complete, you should have a folder called "sympy". From
-your favorite command line terminal, change directory into that folder and
-execute the following::
-
-    $ python setup.py install
-
-Alternatively, if you don't want to install the package onto your computer, you
-may run SymPy with the "isympy" console (which automatically imports SymPy
-packages and defines common symbols) by executing within the "sympy" folder::
-
-    $ ./bin/isympy
-
-You may now run SymPy statements directly within the Python shell::
-
-    >>>
-    >>> from sympy import *
-    >>> x, y, z, t = symbols('x y z t')
-    >>> k, m, n = symbols('k m n', integer=True)
-    >>> f, g, h = symbols('f g h', cls=Function)
-    >>> print(diff(x**2/2, x))
-    x
-
-Git
-===
-
-If you are a developer or like to get the latest updates as they come, be sure
-to install from git. To download the repository, execute the following from the
-command line::
-
-    $ git clone git://github.com/sympy/sympy.git
-
-Then, execute either the setup.py or the bin/isympy scripts as demonstrated
-above.
-
-To update to the latest version, go into your repository and execute::
-
-    $ git pull origin master
-
-If you want to install SymPy, but still want to use the git version, you can run
-from your repository::
-
-    $ setupegg.py develop
-
-This will cause the installed version to always point to the version in the git
-directory.
-
-If you're using the git repository with Python 3, you have to use the
-``./bin/use2to3`` script to build the Python 3 version of SymPy. This will put
-everything in the py3k-sympy directory.
-
-Other Methods
-=============
-
-An installation executable is available for Windows users at the
-`downloads site`_ (.exe). In addition, various Linux distributions have SymPy
-available as a package. Others are strongly encouraged to download from source
-(details above).
-
-Run SymPy
-=========
-
-After installation, it is best to verify that your freshly-installed SymPy
-works. To do this, start up Python and import the SymPy libraries::
-
-    $ python
-    >>> from sympy import *
-
-From here, execute some simple SymPy statements like the ones below::
-
-    >>> x = Symbol('x')
-    >>> limit(sin(x)/x, x, 0)
-    1
-    >>> integrate(1/x, x)
-    log(x)
-
-For a starter guide on using SymPy effectively, refer to the :ref:`tutorial`.
-
-Questions
-=========
-
-If you have a question about installation or SymPy in general, feel free to
-visit the IRC channel at irc.freenode.net, channel `#sympy`_. In addition,
-our `mailing list`_ is an excellent source of community support.
-
-If you think there's a bug or you would like to request a feature, please open
-an `issue ticket`_.
-
-.. _downloads site: https://code.google.com/p/sympy/downloads/list
-.. _#sympy: irc://irc.freenode.net/sympy
-.. _issue ticket: http://code.google.com/p/sympy/issues/list
-.. _mailing list: http://groups.google.com/group/sympy
diff --git a/dev-py3k/_sources/modules/assumptions/ask.txt b/dev-py3k/_sources/modules/assumptions/ask.txt
deleted file mode 100644
index a0642e6aacb..00000000000
--- a/dev-py3k/_sources/modules/assumptions/ask.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-===
-Ask
-===
-
-.. automodule:: sympy.assumptions.ask
-   :members:
diff --git a/dev-py3k/_sources/modules/assumptions/assume.txt b/dev-py3k/_sources/modules/assumptions/assume.txt
deleted file mode 100644
index eeb09bf1f24..00000000000
--- a/dev-py3k/_sources/modules/assumptions/assume.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-======
-Assume
-======
-
-.. automodule:: sympy.assumptions.assume
-   :members:
diff --git a/dev-py3k/_sources/modules/assumptions/handlers/calculus.txt b/dev-py3k/_sources/modules/assumptions/handlers/calculus.txt
deleted file mode 100644
index 2a668aa9f9d..00000000000
--- a/dev-py3k/_sources/modules/assumptions/handlers/calculus.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-========
-Calculus
-========
-
-.. automodule:: sympy.assumptions.handlers.calculus
-   :members:
diff --git a/dev-py3k/_sources/modules/assumptions/handlers/index.txt b/dev-py3k/_sources/modules/assumptions/handlers/index.txt
deleted file mode 100644
index a446fbdcdd2..00000000000
--- a/dev-py3k/_sources/modules/assumptions/handlers/index.txt
+++ /dev/null
@@ -1,16 +0,0 @@
-========
-Handlers
-========
-
-.. automodule:: sympy.assumptions.handlers
-
-Contents
-========
-
-.. toctree::
-    :maxdepth: 3
-
-    calculus.rst
-    ntheory.rst
-    order.rst
-    sets.rst
diff --git a/dev-py3k/_sources/modules/assumptions/handlers/ntheory.txt b/dev-py3k/_sources/modules/assumptions/handlers/ntheory.txt
deleted file mode 100644
index 63c2b176e66..00000000000
--- a/dev-py3k/_sources/modules/assumptions/handlers/ntheory.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-=======
-nTheory
-=======
-
-.. automodule:: sympy.assumptions.handlers.ntheory
-   :members:
diff --git a/dev-py3k/_sources/modules/assumptions/handlers/order.txt b/dev-py3k/_sources/modules/assumptions/handlers/order.txt
deleted file mode 100644
index 040cacc8131..00000000000
--- a/dev-py3k/_sources/modules/assumptions/handlers/order.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-=====
-Order
-=====
-
-.. automodule:: sympy.assumptions.handlers.order
-   :members:
diff --git a/dev-py3k/_sources/modules/assumptions/handlers/sets.txt b/dev-py3k/_sources/modules/assumptions/handlers/sets.txt
deleted file mode 100644
index ac464967902..00000000000
--- a/dev-py3k/_sources/modules/assumptions/handlers/sets.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-====
-Sets
-====
-
-.. automodule:: sympy.assumptions.handlers.sets
-   :members:
diff --git a/dev-py3k/_sources/modules/assumptions/index.txt b/dev-py3k/_sources/modules/assumptions/index.txt
deleted file mode 100644
index 2b52d7e261f..00000000000
--- a/dev-py3k/_sources/modules/assumptions/index.txt
+++ /dev/null
@@ -1,402 +0,0 @@
-==================
-Assumptions module
-==================
-
-.. automodule:: sympy.assumptions
-
-Contents
-========
-
-.. toctree::
-    :maxdepth: 3
-
-    ask.rst
-    assume.rst
-    refine.rst
-    handlers/index.rst
-
-Queries are used to ask information about expressions. Main method for this
-is ask():
-
-.. autofunction:: sympy.assumptions.ask.ask
-   :noindex:
-
-Querying
-========
-
-ask's optional second argument should be a boolean expression involving
-assumptions about objects in expr. Valid values include:
-
-    * Q.integer(x)
-    * Q.positive(x)
-    * Q.integer(x) & Q.positive(x)
-    * etc.
-
-Q is a class in sympy.assumptions holding known predicates.
-
-See documentation for the logic module for a complete list of valid boolean
-expressions.
-
-You can also define global assumptions so you don't have to pass that argument
-each time to function ask(). This is done by using the global_assumptions
-object from module sympy.assumptions. You can then clear global assumptions
-with global_assumptions.clear()::
-
-     >>> from sympy import *
-     >>> x = Symbol('x')
-     >>> global_assumptions.add(Q.positive(x))
-     >>> ask(Q.positive(x))
-     True
-     >>> global_assumptions.clear()
-
-
-Supported predicates
-====================
-
-bounded
--------
-
-Test that a function is bounded with respect to its variables. For example,
-sin(x) is a bounded functions, but exp(x) is not.
-
-Examples::
-
-    >>> from sympy import *
-    >>> x = Symbol('x')
-    >>> ask(Q.bounded(exp(x)), ~Q.bounded(x))
-    False
-    >>> ask(Q.bounded(exp(x)) , Q.bounded(x))
-    True
-    >>> ask(Q.bounded(sin(x)), ~Q.bounded(x))
-    True
-
-
-commutative
------------
-
-Test that objects are commutative. By default, symbols in SymPy are considered
-commutative except otherwise stated.
-
-Examples::
-
-    >>> from sympy import *
-    >>> x, y = symbols('x,y')
-    >>> ask(Q.commutative(x))
-    True
-    >>> ask(Q.commutative(x), ~Q.commutative(x))
-    False
-    >>> ask(Q.commutative(x*y), ~Q.commutative(x))
-    False
-
-
-complex
--------
-
-Test that expression belongs to the field of complex numbers.
-
-Examples::
-
-    >>> from sympy import *
-    >>> ask(Q.complex(2))
-    True
-    >>> ask(Q.complex(I))
-    True
-    >>> x, y = symbols('x,y')
-    >>> ask(Q.complex(x+I*y), Q.real(x) & Q.real(y))
-    True
-
-
-even
-----
-
-Test that expression represents an even number, that is, an number that
-can be written in the form 2*n, n integer.
-
-Examples::
-
-    >>> from sympy import *
-    >>> ask(Q.even(2))
-    True
-    >>> n = Symbol('n')
-    >>> ask(Q.even(2*n), Q.integer(n))
-    True
-
-
-extended_real
--------------
-
-Test that an expression belongs to the field of extended real numbers, that is, real
-numbers union {Infinity, -Infinity}.
-
-Examples::
-
-    >>> from sympy import *
-    >>> ask(Q.extended_real(oo))
-    True
-    >>> ask(Q.extended_real(2))
-    True
-    >>> ask(Q.extended_real(x), Q.real(x))
-    True
-
-
-imaginary
----------
-
-Test that an expression belongs to the set of imaginary numbers, that is,
- it can be written as x*I, where x is real and I is the imaginary unit.
-
-Examples::
-
-    >>> from sympy import *
-    >>> ask(Q.imaginary(2*I))
-    True
-    >>> x = Symbol('x')
-    >>> ask(Q.imaginary(x*I), Q.real(x))
-    True
-
-
-infinitesimal
--------------
-
-Test that an expression is equivalent to an infinitesimal number.
-
-Examples::
-
-    >>> from sympy import *
-    >>> ask(Q.infinitesimal(1/oo))
-    True
-    >>> x, y = symbols('x,y')
-    >>> ask(Q.infinitesimal(2*x), Q.infinitesimal(x))
-    True
-    >>> ask(Q.infinitesimal(x*y), Q.infinitesimal(x) & Q.bounded(y))
-    True
-
-
-integer
--------
-
-Test that an expression belongs to the set of integer numbers.
-
-Examples::
-
-    >>> from sympy import *
-    >>> ask(Q.integer(2))
-    True
-    >>> ask(Q.integer(sqrt(2)))
-    False
-    >>> x = Symbol('x')
-    >>> ask(Q.integer(x/2), Q.even(x))
-    True
-
-
-irrational
-----------
-
-Test that an expression represents an irrational number.
-
-Examples::
-
-     >>> from sympy import *
-     >>> ask(Q.irrational(pi))
-     True
-     >>> ask(Q.irrational(sqrt(2)))
-     True
-     >>> ask(Q.irrational(x*sqrt(2)), Q.rational(x))
-     True
-
-
-rational
---------
-
-Test that an expression represents a rational number.
-
-Examples::
-
-     >>> from sympy import *
-     >>> ask(Q.rational(Rational(3, 4)))
-     True
-     >>> x, y = symbols('x,y')
-     >>> ask(Q.rational(x/2), Q.integer(x))
-     True
-     >>> ask(Q.rational(x/y), Q.integer(x) & Q.integer(y))
-     True
-
-
-negative
---------
-
-Test that an expression is less (strict) than zero.
-
-Examples::
-
-     >>> from sympy import *
-     >>> ask(Q.negative(0.3))
-     False
-     >>> x = Symbol('x')
-     >>> ask(Q.negative(-x), Q.positive(x))
-     True
-
-Remarks
-^^^^^^^
-negative numbers are defined as real numbers that are not zero nor positive, so
-complex numbers (with nontrivial imaginary coefficients) will return False
-for this predicate. The same applies to Q.positive.
-
-
-positive
---------
-
-Test that a given expression is greater (strict) than zero.
-
-Examples::
-
-     >>> from sympy import *
-     >>> ask(Q.positive(0.3))
-     True
-     >>> x = Symbol('x')
-     >>> ask(Q.positive(-x), Q.negative(x))
-     True
-
-Remarks
-^^^^^^^
-see Remarks for negative
-
-
-prime
------
-
-Test that an expression represents a prime number.
-
-Examples::
-
-    >>> from sympy import *
-    >>> ask(Q.prime(13))
-    True
-
-Remarks: Use sympy.ntheory.isprime to test numeric values efficiently.
-
-
-real
-----
-
-Test that an expression belongs to the field of real numbers.
-
-Examples::
-
-    >>> from sympy import *
-    >>> ask(Q.real(sqrt(2)))
-    True
-    >>> x, y = symbols('x,y')
-    >>> ask(Q.real(x*y), Q.real(x) & Q.real(y))
-    True
-
-
-odd
----
-
-Test that an expression represents an odd number.
-
-Examples::
-
-    >>> from sympy import *
-    >>> ask(Q.odd(3))
-    True
-    >>> n = Symbol('n')
-    >>> ask(Q.odd(2*n + 1), Q.integer(n))
-    True
-
-
-nonzero
--------
-
-Test that an expression is not zero.
-
-Examples::
-
-     >>> from sympy import *
-     >>> x = Symbol('x')
-     >>> ask(Q.nonzero(x), Q.positive(x) | Q.negative(x))
-     True
-
-
-Design
-======
-
-Each time ask is called, the appropriate Handler for the current key is called. This is
-always a subclass of sympy.assumptions.AskHandler. It's classmethods have the name's of the classes
-it supports. For example, a (simplified) AskHandler for the ask 'positive' would
-look like this::
-
-    class AskPositiveHandler(CommonHandler):
-
-        def Mul(self):
-            # return True if all argument's in self.expr.args are positive
-            ...
-
-        def Add(self):
-            for arg in self.expr.args:
-                if not ask(arg, positive, self.assumptions):
-                    break
-            else:
-                # if all argument's are positive
-                return True
-        ...
-
-The .Mul() method is called when self.expr is an instance of Mul, the Add method
-would be called when self.expr is an instance of Add and so on.
-
-
-Extensibility
-=============
-
-You can define new queries or support new types by subclassing sympy.assumptions.AskHandler
- and registering that handler for a particular key by calling register_handler:
-
-.. autofunction:: sympy.assumptions.ask.register_handler
-                  :noindex:
-
-You can undo this operation by calling remove_handler.
-
-.. autofunction:: sympy.assumptions.ask.remove_handler
-                  :noindex:
-
-You can support new types [1]_ by adding a handler to an existing key. In the
-following example, we will create a new type MyType and extend the key 'prime'
-to accept this type (and return True)
-
-.. parsed-literal::
-
-    >>> from sympy.core import Basic
-    >>> from sympy.assumptions import register_handler
-    >>> from sympy.assumptions.handlers import AskHandler
-    >>> class MyType(Basic):
-    ...     pass
-    >>> class MyAskHandler(AskHandler):
-    ...     @staticmethod
-    ...     def MyType(expr, assumptions):
-    ...         return True
-    >>> a = MyType()
-    >>> register_handler('prime', MyAskHandler)
-    >>> ask(Q.prime(a))
-    True
-
-
-Performance improvements
-========================
-
-On queries that involve symbolic coefficients, logical inference is used. Work on
-improving satisfiable function (sympy.logic.inference.satisfiable) should result
-in notable speed improvements.
-
-Logic inference used in one ask could be used to speed up further queries, and
-current system does not take advantage of this. For example, a truth maintenance
-system (http://en.wikipedia.org/wiki/Truth_maintenance_system) could be implemented.
-
-Misc
-====
-
-You can find more examples in the in the form of test under directory
-sympy/assumptions/tests/
-
-.. [1] New type must inherit from Basic, otherwise an exception will be raised.
-   This is a bug and should be fixed.
diff --git a/dev-py3k/_sources/modules/assumptions/refine.txt b/dev-py3k/_sources/modules/assumptions/refine.txt
deleted file mode 100644
index eb4a7173a61..00000000000
--- a/dev-py3k/_sources/modules/assumptions/refine.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-======
-Refine
-======
-
-.. automodule:: sympy.assumptions.refine
-   :members:
diff --git a/dev-py3k/_sources/modules/categories.txt b/dev-py3k/_sources/modules/categories.txt
deleted file mode 100644
index cba475621cc..00000000000
--- a/dev-py3k/_sources/modules/categories.txt
+++ /dev/null
@@ -1,71 +0,0 @@
-Category Theory Module
-======================
-
-Introduction
-------------
-
-The category theory module for SymPy will allow manipulating diagrams
-within a single category, including drawing them in TikZ and deciding
-whether they are commutative or not.
-
-The general reference work this module tries to follow is
-
-  [JoyOfCats] J. Adamek, H. Herrlich. G. E. Strecker: Abstract and
-              Concrete Categories. The Joy of Cats.
-
-The latest version of this book should be available for free download
-from
-
-   katmat.math.uni-bremen.de/acc/acc.pdf
-
-The module is still in its pre-embryonic stage.
-
-Base Class Reference
---------------------
-
-.. module:: sympy.categories
-
-This section lists the classes which implement some of the basic
-notions in category theory: objects, morphisms, categories, and
-diagrams.
-
-.. autoclass:: Object
-   :members:
-
-.. autoclass:: Morphism
-   :members:
-
-.. autoclass:: NamedMorphism
-   :members:
-
-.. autoclass:: CompositeMorphism
-   :members:
-
-.. autoclass:: IdentityMorphism
-   :members:
-
-.. autoclass:: Category
-   :members:
-
-.. autoclass:: Diagram
-   :members:
-
-Diagram Drawing
----------------
-
-.. module:: sympy.categories.diagram_drawing
-
-This section lists the classes which allow automatic drawing of
-diagrams.
-
-.. autoclass:: DiagramGrid
-   :members:
-
-.. autoclass:: ArrowStringDescription
-
-.. autoclass:: XypicDiagramDrawer
-   :members:
-
-.. autofunction:: xypic_draw_diagram
-
-.. autofunction:: preview_diagram
diff --git a/dev-py3k/_sources/modules/combinatorics/graycode.txt b/dev-py3k/_sources/modules/combinatorics/graycode.txt
deleted file mode 100644
index 80ca0810d2f..00000000000
--- a/dev-py3k/_sources/modules/combinatorics/graycode.txt
+++ /dev/null
@@ -1,19 +0,0 @@
-.. _combinatorics-graycode:
-
-Gray Code
-=========
-
-.. module:: sympy.combinatorics.graycode
-
-.. autoclass:: GrayCode
-   :members:
-
-.. automethod:: sympy.combinatorics.graycode.random_bitstring
-
-.. automethod:: sympy.combinatorics.graycode.gray_to_bin
-
-.. automethod:: sympy.combinatorics.graycode.bin_to_gray
-
-.. automethod:: sympy.combinatorics.graycode.get_subset_from_bitstring
-
-.. automethod:: sympy.combinatorics.graycode.graycode_subsets
diff --git a/dev-py3k/_sources/modules/combinatorics/group_constructs.txt b/dev-py3k/_sources/modules/combinatorics/group_constructs.txt
deleted file mode 100644
index 3861622cdd3..00000000000
--- a/dev-py3k/_sources/modules/combinatorics/group_constructs.txt
+++ /dev/null
@@ -1,8 +0,0 @@
-.. _combinatorics-group_constructs:
-
-Group constructors
-==================
-
-.. module:: sympy.combinatorics.group_constructs
-
-.. autofunction:: DirectProduct
diff --git a/dev-py3k/_sources/modules/combinatorics/index.txt b/dev-py3k/_sources/modules/combinatorics/index.txt
deleted file mode 100644
index 2834e376f1d..00000000000
--- a/dev-py3k/_sources/modules/combinatorics/index.txt
+++ /dev/null
@@ -1,24 +0,0 @@
-.. _combinatiorics-docs:
-
-====================
-Combinatorics Module
-====================
-
-Contents
-========
-
-.. toctree::
-   :maxdepth: 2
-
-   partitions.rst
-   permutations.rst
-   perm_groups.rst
-   polyhedron.rst
-   prufer.rst
-   subsets.rst
-   graycode.rst
-   named_groups.rst
-   util.rst
-   group_constructs.rst
-   testutil.rst
-   tensor_can.rst
diff --git a/dev-py3k/_sources/modules/combinatorics/named_groups.txt b/dev-py3k/_sources/modules/combinatorics/named_groups.txt
deleted file mode 100644
index 7ad565d40d6..00000000000
--- a/dev-py3k/_sources/modules/combinatorics/named_groups.txt
+++ /dev/null
@@ -1,16 +0,0 @@
-.. _combinatorics-named_groups:
-
-Named Groups
-============
-
-.. module:: sympy.combinatorics.named_groups
-
-.. autofunction:: SymmetricGroup
-
-.. autofunction:: CyclicGroup
-
-.. autofunction:: DihedralGroup
-
-.. autofunction:: AlternatingGroup
-
-.. autofunction:: AbelianGroup
diff --git a/dev-py3k/_sources/modules/combinatorics/partitions.txt b/dev-py3k/_sources/modules/combinatorics/partitions.txt
deleted file mode 100644
index 8935babe493..00000000000
--- a/dev-py3k/_sources/modules/combinatorics/partitions.txt
+++ /dev/null
@@ -1,22 +0,0 @@
-.. _combinatorics-partitions:
-
-Partitions
-==========
-
-.. module:: sympy.combinatorics.partitions
-
-.. autoclass:: Partition
-   :members:
-
-.. autoclass:: IntegerPartition
-   :members:
-
-.. autofunction:: random_integer_partition
-
-.. autofunction:: RGS_generalized
-
-.. autofunction:: RGS_enum
-
-.. autofunction:: RGS_unrank
-
-.. autofunction:: RGS_rank
diff --git a/dev-py3k/_sources/modules/combinatorics/perm_groups.txt b/dev-py3k/_sources/modules/combinatorics/perm_groups.txt
deleted file mode 100644
index 6f15f263227..00000000000
--- a/dev-py3k/_sources/modules/combinatorics/perm_groups.txt
+++ /dev/null
@@ -1,9 +0,0 @@
-.. _combinatorics-perm_groups:
-
-Permutation Groups
-==================
-
-.. module:: sympy.combinatorics.perm_groups
-
-.. autoclass:: PermutationGroup
-   :members:
diff --git a/dev-py3k/_sources/modules/combinatorics/permutations.txt b/dev-py3k/_sources/modules/combinatorics/permutations.txt
deleted file mode 100644
index c518e2b2ee4..00000000000
--- a/dev-py3k/_sources/modules/combinatorics/permutations.txt
+++ /dev/null
@@ -1,27 +0,0 @@
-.. _combinatorics-permutations:
-
-Permutations
-============
-
-.. module:: sympy.combinatorics.permutations
-
-.. autoclass:: Permutation
-   :members:
-
-.. autoclass:: Cycle
-   :members:
-
-.. _combinatorics-generators:
-
-Generators
-----------
-
-.. module:: sympy.combinatorics.generators
-
-.. automethod:: sympy.combinatorics.generators.symmetric
-
-.. automethod:: sympy.combinatorics.generators.cyclic
-
-.. automethod:: sympy.combinatorics.generators.alternating
-
-.. automethod:: sympy.combinatorics.generators.dihedral
diff --git a/dev-py3k/_sources/modules/combinatorics/polyhedron.txt b/dev-py3k/_sources/modules/combinatorics/polyhedron.txt
deleted file mode 100644
index 0f4d03f8d3b..00000000000
--- a/dev-py3k/_sources/modules/combinatorics/polyhedron.txt
+++ /dev/null
@@ -1,9 +0,0 @@
-.. _combinatorics-polyhedron:
-
-Polyhedron
-==========
-
-.. module:: sympy.combinatorics.polyhedron
-
-.. autoclass:: Polyhedron
-   :members:
diff --git a/dev-py3k/_sources/modules/combinatorics/prufer.txt b/dev-py3k/_sources/modules/combinatorics/prufer.txt
deleted file mode 100644
index 36b6c22fb14..00000000000
--- a/dev-py3k/_sources/modules/combinatorics/prufer.txt
+++ /dev/null
@@ -1,9 +0,0 @@
-.. _combinatorics-prufer:
-
-Prufer Sequences
-================
-
-.. module:: sympy.combinatorics.prufer
-
-.. autoclass:: Prufer
-   :members:
diff --git a/dev-py3k/_sources/modules/combinatorics/subsets.txt b/dev-py3k/_sources/modules/combinatorics/subsets.txt
deleted file mode 100644
index ec91e71c674..00000000000
--- a/dev-py3k/_sources/modules/combinatorics/subsets.txt
+++ /dev/null
@@ -1,11 +0,0 @@
-.. _combinatorics-subsets:
-
-Subsets
-=======
-
-.. module:: sympy.combinatorics.subsets
-
-.. autoclass:: Subset
-   :members:
-
-.. automethod:: sympy.combinatorics.subsets.ksubsets
diff --git a/dev-py3k/_sources/modules/combinatorics/tensor_can.txt b/dev-py3k/_sources/modules/combinatorics/tensor_can.txt
deleted file mode 100644
index 8e2e9a67657..00000000000
--- a/dev-py3k/_sources/modules/combinatorics/tensor_can.txt
+++ /dev/null
@@ -1,14 +0,0 @@
-.. _combinatorics-tensor_can:
-
-Tensor Canonicalization
-=======================
-
-.. module:: sympy.combinatorics.tensor_can
-
-.. autofunction:: canonicalize
-
-.. autofunction:: double_coset_can_rep
-
-.. autofunction:: get_symmetric_group_sgs
-
-.. autofunction:: bsgs_direct_product
diff --git a/dev-py3k/_sources/modules/combinatorics/testutil.txt b/dev-py3k/_sources/modules/combinatorics/testutil.txt
deleted file mode 100644
index 507510b9a2d..00000000000
--- a/dev-py3k/_sources/modules/combinatorics/testutil.txt
+++ /dev/null
@@ -1,16 +0,0 @@
-.. _combinatorics-testutil:
-
-Test Utilities
-==============
-
-.. module:: sympy.combinatorics.testutil
-
-.. autofunction:: _cmp_perm_lists
-
-.. autofunction:: _naive_list_centralizer
-
-.. autofunction:: _verify_bsgs
-
-.. autofunction:: _verify_centralizer
-
-.. autofunction:: _verify_normal_closure
diff --git a/dev-py3k/_sources/modules/combinatorics/util.txt b/dev-py3k/_sources/modules/combinatorics/util.txt
deleted file mode 100644
index 2c49bcbac10..00000000000
--- a/dev-py3k/_sources/modules/combinatorics/util.txt
+++ /dev/null
@@ -1,22 +0,0 @@
-.. _combinatorics-util:
-
-Utilities
-============
-
-.. module:: sympy.combinatorics.util
-
-.. autofunction:: _base_ordering
-
-.. autofunction:: _check_cycles_alt_sym
-
-.. autofunction:: _distribute_gens_by_base
-
-.. autofunction:: _handle_precomputed_bsgs
-
-.. autofunction:: _orbits_transversals_from_bsgs
-
-.. autofunction:: _remove_gens
-
-.. autofunction:: _strip
-
-.. autofunction:: _strong_gens_from_distr
diff --git a/dev-py3k/_sources/modules/concrete.txt b/dev-py3k/_sources/modules/concrete.txt
deleted file mode 100644
index 159fefb8853..00000000000
--- a/dev-py3k/_sources/modules/concrete.txt
+++ /dev/null
@@ -1,100 +0,0 @@
-Concrete Mathematics
-====================
-
-Hypergeometric terms
---------------------
-
-The center stage, in recurrence solving and summations, play hypergeometric
-terms. Formally these are sequences annihilated by first order linear
-recurrence operators. In simple words if we are given term `a(n)` then it is
-hypergeometric if its consecutive term ratio is a rational function in `n`.
-
-To check if a sequence is of this type you can use the ``is_hypergeometric``
-method which is available in Basic class. Here is simple example involving a
-polynomial:
-
-    >>> from sympy import *
-    >>> n, k = symbols('n,k')
-    >>> (n**2 + 1).is_hypergeometric(n)
-    True
-
-Of course polynomials are hypergeometric but are there any more complicated
-sequences of this type? Here are some trivial examples:
-
-    >>> factorial(n).is_hypergeometric(n)
-    True
-    >>> binomial(n, k).is_hypergeometric(n)
-    True
-    >>> rf(n, k).is_hypergeometric(n)
-    True
-    >>> ff(n, k).is_hypergeometric(n)
-    True
-    >>> gamma(n).is_hypergeometric(n)
-    True
-    >>> (2**n).is_hypergeometric(n)
-    True
-
-We see that all species used in summations and other parts of concrete
-mathematics are hypergeometric. Note also that binomial coefficients and both
-rising and falling factorials are hypergeometric in both their arguments:
-
-    >>> binomial(n, k).is_hypergeometric(k)
-    True
-    >>> rf(n, k).is_hypergeometric(k)
-    True
-    >>> ff(n, k).is_hypergeometric(k)
-    True
-
-To say more, all previously shown examples are valid for integer linear
-arguments:
-
-    >>> factorial(2*n).is_hypergeometric(n)
-    True
-    >>> binomial(3*n+1, k).is_hypergeometric(n)
-    True
-    >>> rf(n+1, k-1).is_hypergeometric(n)
-    True
-    >>> ff(n-1, k+1).is_hypergeometric(n)
-    True
-    >>> gamma(5*n).is_hypergeometric(n)
-    True
-    >>> (2**(n-7)).is_hypergeometric(n)
-    True
-
-However nonlinear arguments make those sequences fail to be hypergeometric:
-
-    >>> factorial(n**2).is_hypergeometric(n)
-    False
-    >>> (2**(n**3 + 1)).is_hypergeometric(n)
-    False
-
-If not only the knowledge of being hypergeometric or not is needed, you can use
-``hypersimp()`` function. It will try to simplify combinatorial expression and
-if the term given is hypergeometric it will return a quotient of polynomials of
-minimal degree. Otherwise is will return `None` to say that sequence is not
-hypergeometric:
-
-    >>> hypersimp(factorial(2*n), n)
-    4*n**2 + 6*n + 2
-    >>> hypersimp(factorial(n**2), n)
-
-Concrete Class Reference
-------------------------
-.. autoclass:: sympy.concrete.summations.Sum
-   :members:
-
-.. autoclass:: sympy.concrete.products.Product
-   :members:
-
-Concrete Functions Reference
-----------------------------
-
-.. autofunction:: sympy.concrete.summations.summation
-
-.. autofunction:: sympy.concrete.products.product
-
-.. autofunction:: sympy.concrete.gosper.gosper_normal
-
-.. autofunction:: sympy.concrete.gosper.gosper_term
-
-.. autofunction:: sympy.concrete.gosper.gosper_sum
diff --git a/dev-py3k/_sources/modules/core.txt b/dev-py3k/_sources/modules/core.txt
deleted file mode 100644
index 43efb10e835..00000000000
--- a/dev-py3k/_sources/modules/core.txt
+++ /dev/null
@@ -1,463 +0,0 @@
-SymPy Core
-==========
-
-sympify
--------
-.. module:: sympy.core.sympify
-
-sympify
-^^^^^^^
-.. autofunction:: sympify
-
-cache
--------
-.. module:: sympy.core.cache
-
-cacheit
-^^^^^^^
-.. autofunction:: cacheit
-
-basic
------
-.. module:: sympy.core.basic
-
-Basic
-^^^^^
-.. autoclass:: Basic
-   :members:
-
-Atom
-^^^^
-.. autoclass:: Atom
-   :members:
-
-C
-^
-.. autoclass:: C
-   :members:
-
-singleton
----------
-.. module:: sympy.core.singleton
-
-S
-^
-.. autoclass:: S
-   :members:
-
-expr
-----
-.. module:: sympy.core.expr
-
-Expr
-----
-.. autoclass:: Expr
-   :members:
-
-AtomicExpr
-----------
-.. autoclass:: AtomicExpr
-   :members:
-
-symbol
-------
-.. module:: sympy.core.symbol
-
-Symbol
-^^^^^^
-.. autoclass:: Symbol
-   :members:
-
-Wild
-^^^^
-.. autoclass:: Wild
-   :members:
-
-Dummy
-^^^^^
-.. autoclass:: Dummy
-   :members:
-
-symbols
-^^^^^^^
-.. autofunction:: symbols
-
-var
-^^^
-.. autofunction:: var
-
-numbers
--------
-.. module:: sympy.core.numbers
-
-Number
-^^^^^^
-.. autoclass:: Number
-   :members:
-
-Float
-^^^^^
-.. autoclass:: Float
-   :members:
-
-Rational
-^^^^^^^^
-.. autoclass:: Rational
-   :members:
-
-Integer
-^^^^^^^
-.. autoclass:: Integer
-   :members:
-
-NumberSymbol
-^^^^^^^^^^^^
-.. autoclass:: NumberSymbol
-   :members:
-
-RealNumber
-^^^^^^^^^^
-.. autoclass:: RealNumber
-   :members:
-
-Real
-^^^^
-.. autoclass:: Real
-   :members:
-
-igcd
-^^^^
-.. autofunction:: igcd
-
-ilcm
-^^^^
-.. autofunction:: ilcm
-
-seterr
-^^^^^^
-.. autofunction:: seterr
-
-E
-^
-.. autoclass:: E
-   :members:
-
-I
-^
-.. autoclass:: I
-   :members:
-
-nan
-^^^
-.. autofunction:: nan
-
-oo
-^^
-.. autofunction:: oo
-
-pi
-^^
-.. autofunction:: pi
-
-zoo
-^^^
-.. autofunction:: zoo
-
-power
------
-.. module:: sympy.core.power
-
-Pow
-^^^
-.. autoclass:: Pow
-   :members:
-
-integer_nthroot
-^^^^^^^^^^^^^^^
-.. autofunction:: integer_nthroot
-
-mul
----
-.. module:: sympy.core.mul
-
-Mul
-^^^
-.. autoclass:: Mul
-   :members:
-
-prod
-^^^^
-.. autofunction:: prod
-
-add
----
-.. module:: sympy.core.add
-
-Add
-^^^
-.. autoclass:: Add
-   :members:
-
-mod
----
-.. module:: sympy.core.mod
-
-Mod
-^^^
-.. autoclass:: Mod
-   :members:
-
-relational
-----------
-.. module:: sympy.core.relational
-
-Rel
-^^^
-.. autoclass:: Rel
-   :members:
-
-Eq
-^^
-.. autoclass:: Eq
-   :members:
-
-Ne
-^^
-.. autoclass:: Ne
-   :members:
-
-Lt
-^^
-.. autoclass:: Lt
-   :members:
-
-Le
-^^
-.. autoclass:: Le
-   :members:
-
-Gt
-^^
-.. autoclass:: Gt
-   :members:
-
-Ge
-^^
-.. autoclass:: Ge
-   :members:
-
-Equality
-^^^^^^^^
-.. autoclass:: Equality
-   :members:
-
-GreaterThan
-^^^^^^^^^^^
-.. autoclass:: GreaterThan
-   :members:
-
-LessThan
-^^^^^^^^
-.. autoclass:: LessThan
-   :members:
-
-Unequality
-^^^^^^^^^^
-.. autoclass:: Unequality
-   :members:
-
-StrictGreaterThan
-^^^^^^^^^^^^^^^^^
-.. autoclass:: StrictGreaterThan
-   :members:
-
-StrictLessThan
-^^^^^^^^^^^^^^
-.. autoclass:: StrictLessThan
-   :members:
-
-multidimensional
-----------------
-.. module:: sympy.core.multidimensional
-
-vectorize
-^^^^^^^^^
-.. autoclass:: vectorize
-   :members:
-
-function
---------
-.. module:: sympy.core.function
-
-Lambda
-^^^^^^
-.. autoclass:: Lambda
-   :members:
-
-WildFunction
-^^^^^^^^^^^^
-.. autoclass:: WildFunction
-   :members:
-
-Derivative
-^^^^^^^^^^
-.. autoclass:: Derivative
-   :members:
-
-diff
-^^^^
-.. autofunction:: diff
-
-FunctionClass
-^^^^^^^^^^^^^
-.. autoclass:: FunctionClass
-   :members:
-
-Function
-^^^^^^^^
-.. autoclass:: Function
-   :members:
-
-.. note:: Not all functions are the same
-
-   SymPy defines many functions (like ``cos`` and ``factorial``). It also
-   allows the user to create generic functions which act as argument
-   holders. Such functions are created just like symbols:
-
-   >>> from sympy import Function, cos
-   >>> from sympy.abc import x
-   >>> f = Function('f')
-   >>> f(2) + f(x)
-   f(2) + f(x)
-
-   If you want to see which functions appear in an expression you can use
-   the atoms method:
-
-   >>> e = (f(x) + cos(x) + 2)
-   >>> e.atoms(Function)
-   set([f(x), cos(x)])
-
-   If you just want the function you defined, not SymPy functions, the
-   thing to search for is AppliedUndef:
-
-   >>> from sympy.core.function import AppliedUndef
-   >>> e.atoms(AppliedUndef)
-   set([f(x)])
-
-Subs
-^^^^
-.. autoclass:: Subs
-   :members:
-
-expand
-^^^^^^
-.. autofunction:: expand
-
-PoleError
-^^^^^^^^^
-.. autoclass:: PoleError
-   :members:
-
-count_ops
-^^^^^^^^^
-.. autofunction:: count_ops
-
-expand_mul
-^^^^^^^^^^
-.. autofunction:: expand_mul
-
-expand_log
-^^^^^^^^^^
-.. autofunction:: expand_log
-
-expand_func
-^^^^^^^^^^^
-.. autofunction:: expand_func
-
-expand_trig
-^^^^^^^^^^^
-.. autofunction:: expand_trig
-
-expand_complex
-^^^^^^^^^^^^^^
-.. autofunction:: expand_complex
-
-expand_multinomial
-^^^^^^^^^^^^^^^^^^
-.. autofunction:: expand_multinomial
-
-expand_power_exp
-^^^^^^^^^^^^^^^^
-.. autofunction:: expand_power_exp
-
-expand_power_base
-^^^^^^^^^^^^^^^^^
-.. autofunction:: expand_power_base
-
-nfloat
-^^^^^^
-.. autofunction:: nfloat
-
-evalf
------
-.. module:: sympy.core.evalf
-
-PrecisionExhausted
-^^^^^^^^^^^^^^^^^^
-.. autoclass:: PrecisionExhausted
-   :members:
-
-N
-^
-.. autoclass:: N
-   :members:
-
-containers
-----------
-.. module:: sympy.core.containers
-
-Tuple
-^^^^^
-.. autoclass:: Tuple
-   :members:
-
-Dict
-^^^^
-.. autoclass:: Dict
-   :members:
-
-compatibility
--------------
-.. module:: sympy.core.compatibility
-
-iterable
-^^^^^^^^
-.. autofunction:: iterable
-
-is_sequence
-^^^^^^^^^^^
-.. autofunction:: is_sequence
-
-set_intersection
-^^^^^^^^^^^^^^^^
-.. autofunction:: set_intersection
-
-set_union
-^^^^^^^^^
-.. autofunction:: set_union
-
-as_int
-^^^^^^
-.. autofunction:: as_int
-
-exprtools
----------
-.. module:: sympy.core.exprtools
-
-gcd_terms
-^^^^^^^^^
-.. autofunction:: gcd_terms
-
-factor_terms
-^^^^^^^^^^^^
-.. autofunction:: factor_terms
diff --git a/dev-py3k/_sources/modules/diffgeom.txt b/dev-py3k/_sources/modules/diffgeom.txt
deleted file mode 100644
index 4379d426cf5..00000000000
--- a/dev-py3k/_sources/modules/diffgeom.txt
+++ /dev/null
@@ -1,33 +0,0 @@
-Differential Geometry Module
-============================
-
-.. module:: sympy.diffgeom
-
-Introduction
-------------
-
-Base Class Reference
---------------------
-.. autoclass:: Manifold
-   :members:
-
-.. autoclass:: Patch
-   :members:
-
-.. autoclass:: CoordSystem
-   :members:
-
-.. autoclass:: BaseScalarField
-   :members:
-
-.. autoclass:: BaseVectorField
-   :members:
-
-.. autoclass:: Differential
-   :members:
-
-.. autoclass:: BaseCovarDerivativeOp
-   :members:
-
-.. autoclass:: CovarDerivativeOp
-   :members:
diff --git a/dev-py3k/_sources/modules/evalf.txt b/dev-py3k/_sources/modules/evalf.txt
deleted file mode 100644
index 67f6f340285..00000000000
--- a/dev-py3k/_sources/modules/evalf.txt
+++ /dev/null
@@ -1,424 +0,0 @@
-.. _evalf-label:
-
-Numerical evaluation
-====================
-
-Basics
-------
-
-Exact SymPy expressions can be converted to floating-point approximations
-(decimal numbers) using either the ``.evalf()`` method or the ``N()`` function.
-``N(expr, <args>)`` is equivalent to ``sympify(expr).evalf(<args>)``.
-
-    >>> from sympy import *
-    >>> N(sqrt(2)*pi)
-    4.44288293815837
-    >>> (sqrt(2)*pi).evalf()
-    4.44288293815837
-
-
-By default, numerical evaluation is performed to an accuracy of 15 decimal
-digits. You can optionally pass a desired accuracy (which should be a positive
-integer) as an argument to ``evalf`` or ``N``:
-
-    >>> N(sqrt(2)*pi, 5)
-    4.4429
-    >>> N(sqrt(2)*pi, 50)
-    4.4428829381583662470158809900606936986146216893757
-
-
-Complex numbers are supported:
-
-    >>> N(1/(pi + I), 20)
-    0.28902548222223624241 - 0.091999668350375232456*I
-
-
-If the expression contains symbols or for some other reason cannot be evaluated
-numerically, calling ``.evalf()`` or ``N()`` returns the original expression, or
-in some cases a partially evaluated expression. For example, when the
-expression is a polynomial in expanded form, the coefficients are evaluated:
-
-    >>> x = Symbol('x')
-    >>> (pi*x**2 + x/3).evalf()
-    3.14159265358979*x**2 + 0.333333333333333*x
-
-
-You can also use the standard Python functions ``float()``, ``complex()`` to
-convert SymPy expressions to regular Python numbers:
-
-    >>> float(pi) #doctest: +SKIP
-    3.1415926535...
-    >>> complex(pi+E*I) #doctest: +SKIP
-    (3.1415926535...+2.7182818284...j)
-
-
-If these functions are used, failure to evaluate the expression to an explicit
-number (for example if the expression contains symbols) will raise an exception.
-
-There is essentially no upper precision limit. The following command, for
-example, computes the first 100,000 digits of π/e:
-
-    >>> N(pi/E, 100000) #doctest: +SKIP
-    ...
-
-
-This shows digits 999,951 through 1,000,000 of pi:
-
-    >>> str(N(pi, 10**6))[-50:] #doctest: +SKIP
-    '95678796130331164628399634646042209010610577945815'
-
-
-High-precision calculations can be slow. It is recommended (but entirely
-optional) to install gmpy (http://code.google.com/p/gmpy/), which will
-significantly speed up computations such as the one above.
-
-Floating-point numbers
-----------------------
-
-Floating-point numbers in SymPy are instances of the class ``Float``. A ``Float``
-can be created with a custom precision as second argument:
-
-    >>> Float(0.1)
-    0.100000000000000
-    >>> Float(0.1, 10)
-    0.1000000000
-    >>> Float(0.125, 30)
-    0.125000000000000000000000000000
-    >>> Float(0.1, 30)
-    0.100000000000000005551115123126
-
-As the last example shows, some Python floats are only accurate to about 15
-digits as inputs, while others (those that have a denominator that is a
-power of 2, like .125 = 1/4) are exact. To create a ``Float`` from a
-high-precision decimal number, it is better to pass a string, ``Rational``,
-or ``evalf`` a ``Rational``:
-
-    >>> Float('0.1', 30)
-    0.100000000000000000000000000000
-    >>> Float(Rational(1, 10), 30)
-    0.100000000000000000000000000000
-    >>> Rational(1, 10).evalf(30)
-    0.100000000000000000000000000000
-
-
-The precision of a number determines 1) the precision to use when performing
-arithmetic with the number, and 2) the number of digits to display when printing
-the number. When two numbers with different precision are used together in an
-arithmetic operation, the higher of the precisions is used for the result. The
-product of 0.1 +/- 0.001 and 3.1415 +/- 0.0001 has an uncertainty of about 0.003
-and yet 5 digits of precision are shown.
-
-    >>> Float(0.1, 3)*Float(3.1415, 5)
-    0.31417
-
-So the displayed precision should not be used as a model of error propagation or
-significance arithmetic; rather, this scheme is employed to ensure stability of
-numerical algorithms.
-
-``N`` and ``evalf`` can be used to change the precision of existing
-floating-point numbers:
-
-    >>> N(3.5)
-    3.50000000000000
-    >>> N(3.5, 5)
-    3.5000
-    >>> N(3.5, 30)
-    3.50000000000000000000000000000
-
-
-Accuracy and error handling
----------------------------
-
-When the input to ``N`` or ``evalf`` is a complicated expression, numerical
-error propagation becomes a concern. As an example, consider the 100'th
-Fibonacci number and the excellent (but not exact) approximation `\varphi^{100} / \sqrt{5}`
-where `\varphi` is the golden ratio. With ordinary floating-point arithmetic,
-subtracting these numbers from each other erroneously results in a complete
-cancellation:
-
-    >>> a, b = GoldenRatio**1000/sqrt(5), fibonacci(1000)
-    >>> float(a) #doctest: +SKIP
-    4.34665576869...e+208
-    >>> float(b) #doctest: +SKIP
-    4.34665576869...e+208
-    >>> float(a) - float(b)
-    0.0
-
-``N`` and ``evalf`` keep track of errors and automatically increase the
-precision used internally in order to obtain a correct result:
-
-    >>> N(fibonacci(100) - GoldenRatio**100/sqrt(5))
-    -5.64613129282185e-22
-
-
-Unfortunately, numerical evaluation cannot tell an expression that is exactly
-zero apart from one that is merely very small. The working precision is
-therefore capped, by default to around 100 digits. If we try with the 1000'th
-Fibonacci number, the following happens:
-
-    >>> N(fibonacci(1000) - (GoldenRatio)**1000/sqrt(5))
-    0.e+85
-
-
-The lack of digits in the returned number indicates that ``N`` failed to achieve
-full accuracy. The result indicates that the magnitude of the expression is something less than 10^84, but that is not a particularly good answer. To force a higher working precision, the ``maxn`` keyword argument can be used:
-
-    >>> N(fibonacci(1000) - (GoldenRatio)**1000/sqrt(5), maxn=500)
-    -4.60123853010113e-210
-
-
-Normally, ``maxn`` can be set very high (thousands of digits), but be aware that
-this may cause significant slowdown in extreme cases. Alternatively, the
-``strict=True`` option can be set to force an exception instead of silently
-returning a value with less than the requested accuracy:
-
-    >>> N(fibonacci(1000) - (GoldenRatio)**1000/sqrt(5), strict=True)
-    Traceback (most recent call last):
-    ...
-    PrecisionExhausted: Failed to distinguish the expression:
-    <BLANKLINE>
-    -sqrt(5)*GoldenRatio**1000/5 + 43466557686937456435688527675040625802564660517371780402481729089536555417949051890403879840079255169295922593080322634775209689623239873322471161642996440906533187938298969649928516003704476137795166849228875
-    <BLANKLINE>
-    from zero. Try simplifying the input, using chop=True, or providing a higher maxn for evalf
-
-
-If we add a term so that the Fibonacci approximation becomes exact (the full
-form of Binet's formula), we get an expression that is exactly zero, but ``N``
-does not know this:
-
-    >>> f = fibonacci(100) - (GoldenRatio**100 - (GoldenRatio-1)**100)/sqrt(5)
-    >>> N(f)
-    0.e-104
-    >>> N(f, maxn=1000)
-    0.e-1336
-
-
-In situations where such cancellations are known to occur, the ``chop`` options
-is useful. This basically replaces very small numbers in the real or
-imaginary portions of a number with exact zeros:
-
-    >>> N(f, chop=True)
-    0
-    >>> N(3 + I*f, chop=True)
-    3.00000000000000
-
-
-In situations where you wish to remove meaningless digits, re-evaluation or
-the use of the ``round`` method are useful:
-
-    >>> Float('.1', '')*Float('.12345', '')
-    0.012297
-    >>> ans = _
-    >>> N(ans, 1)
-    0.01
-    >>> ans.round(2)
-    0.01
-
-
-If you are dealing with a numeric expression that contains no floats, it
-can be evaluated to arbitrary precision. To round the result relative to
-a given decimal, the round method is useful:
-
-    >>> v = 10*pi + cos(1)
-    >>> N(v)
-    31.9562288417661
-    >>> v.round(3)
-    31.956
-
-
-Sums and integrals
-------------------
-
-Sums (in particular, infinite series) and integrals can be used like regular
-closed-form expressions, and support arbitrary-precision evaluation:
-
-    >>> var('n x')
-    (n, x)
-    >>> Sum(1/n**n, (n, 1, oo)).evalf()
-    1.29128599706266
-    >>> Integral(x**(-x), (x, 0, 1)).evalf()
-    1.29128599706266
-    >>> Sum(1/n**n, (n, 1, oo)).evalf(50)
-    1.2912859970626635404072825905956005414986193682745
-    >>> Integral(x**(-x), (x, 0, 1)).evalf(50)
-    1.2912859970626635404072825905956005414986193682745
-    >>> (Integral(exp(-x**2), (x, -oo, oo)) ** 2).evalf(30)
-    3.14159265358979323846264338328
-
-
-By default, the tanh-sinh quadrature algorithm is used to evaluate integrals.
-This algorithm is very efficient and robust for smooth integrands (and even
-integrals with endpoint singularities), but may struggle with integrals that
-are highly oscillatory or have mid-interval discontinuities. In many cases,
-``evalf``/``N`` will correctly estimate the error. With the following integral,
-the result is accurate but only good to four digits:
-
-    >>> f = abs(sin(x))
-    >>> Integral(abs(sin(x)), (x, 0, 4)).evalf()
-    2.346
-
-
-It is better to split this integral into two pieces:
-
-    >>> (Integral(f, (x, 0, pi)) + Integral(f, (x, pi, 4))).evalf()
-    2.34635637913639
-
-
-A similar example is the following oscillatory integral:
-
-
-    >>> Integral(sin(x)/x**2, (x, 1, oo)).evalf(maxn=20)
-    0.5
-
-
-It can be dealt with much more efficiently by telling ``evalf`` or ``N`` to
-use an oscillatory quadrature algorithm:
-
-    >>> Integral(sin(x)/x**2, (x, 1, oo)).evalf(quad='osc')
-    0.504067061906928
-    >>> Integral(sin(x)/x**2, (x, 1, oo)).evalf(20, quad='osc')
-    0.50406706190692837199
-
-
-Oscillatory quadrature requires an integrand containing a factor cos(ax+b) or
-sin(ax+b). Note that many other oscillatory integrals can be transformed to
-this form with a change of variables:
-
-    >>> init_printing(use_unicode=False, wrap_line=False, no_global=True)
-    >>> intgrl = Integral(sin(1/x), (x, 0, 1)).transform(x, 1/x)
-    >>> intgrl
-     oo
-      /
-     |
-     |  sin(x)
-     |  ------ dx
-     |     2
-     |    x
-     |
-    /
-    1
-    >>> N(intgrl, quad='osc')
-    0.504067061906928
-
-
-Infinite series use direct summation if the series converges quickly enough.
-Otherwise, extrapolation methods (generally the Euler-Maclaurin formula but
-also Richardson extrapolation) are used to speed up convergence. This allows
-high-precision evaluation of slowly convergent series:
-
-    >>> var('k')
-    k
-    >>> Sum(1/k**2, (k, 1, oo)).evalf()
-    1.64493406684823
-    >>> zeta(2).evalf()
-    1.64493406684823
-    >>> Sum(1/k-log(1+1/k), (k, 1, oo)).evalf()
-    0.577215664901533
-    >>> Sum(1/k-log(1+1/k), (k, 1, oo)).evalf(50)
-    0.57721566490153286060651209008240243104215933593992
-    >>> EulerGamma.evalf(50)
-    0.57721566490153286060651209008240243104215933593992
-
-
-The Euler-Maclaurin formula is also used for finite series, allowing them to
-be approximated quickly without evaluating all terms:
-
-    >>> Sum(1/k, (k, 10000000, 20000000)).evalf()
-    0.693147255559946
-
-
-Note that ``evalf`` makes some assumptions that are not always optimal. For
-fine-tuned control over numerical summation, it might be worthwhile to manually
-use the method ``Sum.euler_maclaurin``.
-
-Special optimizations are used for rational hypergeometric series (where the
-term is a product of polynomials, powers, factorials, binomial coefficients and
-the like). ``N``/``evalf`` sum series of this type very rapidly to high
-precision. For example, this Ramanujan formula for pi can be summed to 10,000
-digits in a fraction of a second with a simple command:
-
-    >>> f = factorial
-    >>> n = Symbol('n', integer=True)
-    >>> R = 9801/sqrt(8)/Sum(f(4*n)*(1103+26390*n)/f(n)**4/396**(4*n),
-    ...                         (n, 0, oo)) #doctest: +SKIP
-    >>> N(R, 10000) #doctest: +SKIP
-    3.141592653589793238462643383279502884197169399375105820974944592307816406286208
-    99862803482534211706798214808651328230664709384460955058223172535940812848111745
-    02841027019385211055596446229489549303819644288109756659334461284756482337867831
-    ...
-
-
-Numerical simplification
-------------------------
-
-The function ``nsimplify`` attempts to find a formula that is numerically equal
-to the given input. This feature can be used to guess an exact formula for an
-approximate floating-point input, or to guess a simpler formula for a
-complicated symbolic input. The algorithm used by ``nsimplify`` is capable of
-identifying simple fractions, simple algebraic expressions, linear combinations
-of given constants, and certain elementary functional transformations of any of
-the preceding.
-
-Optionally, ``nsimplify`` can be passed a list of constants to include (e.g. pi)
-and a minimum numerical tolerance. Here are some elementary examples:
-
-    >>> nsimplify(0.1)
-    1/10
-    >>> nsimplify(6.28, [pi], tolerance=0.01)
-    2*pi
-    >>> nsimplify(pi, tolerance=0.01)
-    22/7
-    >>> nsimplify(pi, tolerance=0.001)
-    355
-    ---
-    113
-    >>> nsimplify(0.33333, tolerance=1e-4)
-    1/3
-    >>> nsimplify(2.0**(1/3.), tolerance=0.001)
-    635
-    ---
-    504
-    >>> nsimplify(2.0**(1/3.), tolerance=0.001, full=True)
-    3 ___
-    \/ 2
-
-
-Here are several more advanced examples:
-
-    >>> nsimplify(Float('0.130198866629986772369127970337',30), [pi, E])
-        1
-    ----------
-    5*pi
-    ---- + 2*E
-     7
-    >>> nsimplify(cos(atan('1/3')))
-        ____
-    3*\/ 10
-    --------
-       10
-    >>> nsimplify(4/(1+sqrt(5)), [GoldenRatio])
-    -2 + 2*GoldenRatio
-    >>> nsimplify(2 + exp(2*atan('1/4')*I))
-    49   8*I
-    -- + ---
-    17    17
-    >>> nsimplify((1/(exp(3*pi*I/5)+1)))
-               ___________
-              /   ___
-    1        /  \/ 5    1
-    - - I*  /   ----- + -
-    2     \/      10    4
-    >>> nsimplify(I**I, [pi])
-     -pi
-     ---
-      2
-    e
-    >>> n = Symbol('n')
-    >>> nsimplify(Sum(1/n**2, (n, 1, oo)), [pi])
-      2
-    pi
-    ---
-     6
-    >>> nsimplify(gamma('1/4')*gamma('3/4'), [pi])
-      ___
-    \/ 2 *pi
diff --git a/dev-py3k/_sources/modules/functions/combinatorial.txt b/dev-py3k/_sources/modules/functions/combinatorial.txt
deleted file mode 100644
index efa0c405708..00000000000
--- a/dev-py3k/_sources/modules/functions/combinatorial.txt
+++ /dev/null
@@ -1,88 +0,0 @@
-Combinatorial
-=============
-
-This module implements various combinatorial functions.
-
-bell
-----
-
-.. autoclass:: sympy.functions.combinatorial.numbers.bell
-   :members:
-
-bernoulli
----------
-
-.. autoclass:: sympy.functions.combinatorial.numbers.bernoulli
-   :members:
-
-binomial
---------
-
-.. autoclass:: sympy.functions.combinatorial.factorials.binomial
-   :members:
-
-catalan
--------
-
-.. autoclass:: sympy.functions.combinatorial.numbers.catalan
-   :members:
-
-
-euler
------
-
-.. autoclass:: sympy.functions.combinatorial.numbers.euler
-   :members:
-
-
-factorial
----------
-
-.. autoclass:: sympy.functions.combinatorial.factorials.factorial
-   :members:
-
-factorial2 / double factorial
------------------------------
-
-.. autoclass:: sympy.functions.combinatorial.factorials.factorial2
-   :members:
-
-
-FallingFactorial
-----------------
-
-.. autoclass:: sympy.functions.combinatorial.factorials.FallingFactorial
-   :members:
-
-fibonacci
----------
-
-.. autoclass:: sympy.functions.combinatorial.numbers.fibonacci
-   :members:
-
-harmonic
---------
-
-.. autoclass:: sympy.functions.combinatorial.numbers.harmonic
-   :members:
-
-
-lucas
------
-
-.. autoclass:: sympy.functions.combinatorial.numbers.lucas
-   :members:
-
-
-MultiFactorial
---------------
-
-.. autoclass:: sympy.functions.combinatorial.factorials.MultiFactorial
-   :members:
-
-
-RisingFactorial
----------------
-
-.. autoclass:: sympy.functions.combinatorial.factorials.RisingFactorial
-   :members:
diff --git a/dev-py3k/_sources/modules/functions/elementary.txt b/dev-py3k/_sources/modules/functions/elementary.txt
deleted file mode 100644
index a6c2ee71a2c..00000000000
--- a/dev-py3k/_sources/modules/functions/elementary.txt
+++ /dev/null
@@ -1,326 +0,0 @@
-Elementary
-==========
-
-This module implements elementary functions, as well as functions like Abs,
-Max, etc.
-
-
-Abs
----
-
-Returns the absolute value of the argument.
-
-Examples::
-
-    >>> from sympy.functions import Abs
-    >>> Abs(-1)
-    1
-
-.. autoclass:: sympy.functions.elementary.complexes.Abs
-   :members:
-
-acos
-----
-
-.. autoclass:: sympy.functions.elementary.trigonometric.acos
-   :members:
-
-acosh
------
-
-.. autoclass:: sympy.functions.elementary.hyperbolic.acosh
-   :members:
-
-acot
-----
-
-.. autoclass:: sympy.functions.elementary.trigonometric.acot
-   :members:
-
-acoth
------
-
-.. autoclass:: sympy.functions.elementary.hyperbolic.acoth
-   :members:
-
-arg
----
-
-Returns the argument (in radians) of a complex number. For a real
-number, the argument is always 0.
-
-Examples::
-
-    >>> from sympy.functions import arg
-    >>> from sympy import I, sqrt
-    >>> arg(2.0)
-    0
-    >>> arg(I)
-    pi/2
-    >>> arg(sqrt(2) + I*sqrt(2))
-    pi/4
-
-.. autoclass:: sympy.functions.elementary.complexes.arg
-   :members:
-
-asin
-----
-
-.. autoclass:: sympy.functions.elementary.trigonometric.asin
-   :members:
-
-asinh
------
-
-.. autoclass:: sympy.functions.elementary.hyperbolic.asinh
-   :members:
-
-atan
-----
-
-.. autoclass:: sympy.functions.elementary.trigonometric.atan
-   :members:
-
-atan2
------
-
-This function is like `atan`, but considers the sign of both arguments in
-order to correctly determine the quadrant of its result.
-
-.. autoclass:: sympy.functions.elementary.trigonometric.atan2
-   :members:
-
-atanh
------
-
-.. autoclass:: sympy.functions.elementary.hyperbolic.atanh
-   :members:
-
-ceiling
--------
-
-.. autoclass:: sympy.functions.elementary.integers.ceiling
-   :members:
-
-conjugate
----------
-
-Returns the `complex conjugate <http://en.wikipedia.org/wiki/Complex_conjugation>`_
-of an argument. In mathematics, the complex conjugate of a complex number is given
-by changing the sign of the imaginary part. Thus, the conjugate of the complex number
-
-    :math:`a + ib`
-
-(where a and b are real numbers) is
-
-    :math:`a - ib`
-
-Examples::
-
-    >>> from sympy.functions import conjugate
-    >>> from sympy import I
-    >>> conjugate(2)
-    2
-    >>> conjugate(I)
-    -I
-
-.. autoclass:: sympy.functions.elementary.complexes.conjugate
-   :members:
-
-cos
----
-
-.. autoclass:: sympy.functions.elementary.trigonometric.cos
-   :members:
-
-cosh
-----
-.. autoclass:: sympy.functions.elementary.hyperbolic.cosh
-   :members:
-
-cot
----
-
-.. autoclass:: sympy.functions.elementary.trigonometric.cot
-   :members:
-
-coth
-----
-
-.. autoclass:: sympy.functions.elementary.hyperbolic.coth
-   :members:
-
-exp
----
-
-.. autoclass:: sympy.functions.elementary.exponential.exp
-   :members:
-
-.. seealso:: classes :py:class:`sympy.functions.elementary.exponential.log`
-
-ExprCondPair
-------------
-
-.. autoclass:: sympy.functions.elementary.piecewise.ExprCondPair
-   :members:
-
-floor
------
-
-.. autoclass:: sympy.functions.elementary.integers.floor
-   :members:
-
-HyperbolicFunction
-------------------
-
-.. autoclass:: sympy.functions.elementary.hyperbolic.HyperbolicFunction
-   :members:
-
-IdentityFunction
-----------------
-
-.. autoclass:: sympy.functions.elementary.miscellaneous.IdentityFunction
-   :members:
-
-im
---
-
-Returns the imaginary part of an expression.
-
-Examples::
-
-
-    >>> from sympy.functions import im
-    >>> from sympy import I
-    >>> im(2+3*I)
-    3
-
-.. autoclass: sympy.functions.elementary.im
-   :members:
-
-.. seealso::
-   :py:class:`sympy.functions.elementary.complexes.re`
-
-LambertW
---------
-
-.. autoclass:: sympy.functions.elementary.exponential.LambertW
-   :members:
-
-log
----
-
-.. autoclass:: sympy.functions.elementary.exponential.log
-   :members:
-
-.. seealso:: classes :py:class:`sympy.functions.elementary.exponential.exp`
-
-Min
----
-
-Returns the minimum of two (comparable) expressions.
-
-Examples::
-
-    >>> from sympy.functions import Min
-    >>> Min(1,2)
-    1
-    >>> from sympy.abc import x
-    >>> Min(1, x)
-    Min(1, x)
-
-It is named Min and not min to avoid conflicts with the built-in function min.
-
-.. autoclass:: sympy.functions.elementary.miscellaneous.Min
-   :members:
-
-
-Max
----
-
-Returns the maximum of two (comparable) expressions
-
-It is named Max and not max to avoid conflicts with the built-in function max.
-
-.. autoclass:: sympy.functions.elementary.miscellaneous.Max
-   :members:
-
-Piecewise
----------
-
-.. autoclass:: sympy.functions.elementary.piecewise.Piecewise
-   :members:
-
-.. autofunction:: sympy.functions.elementary.piecewise.piecewise_fold
-
-re
---
-
-Return the real part of an expression.
-
-Examples::
-
-    >>> from sympy.functions import re
-    >>> from sympy import I
-    >>> re(2+3*I)
-    2
-
-.. autoclass:: sympy.functions.elementary.complexes.re
-   :members:
-
-.. seealso::
-   :py:class:`sympy.functions.elementary.complexes.im`
-
-root
-----
-
-.. autofunction:: sympy.functions.elementary.miscellaneous.root
-
-RoundFunction
--------------
-
-.. autoclass:: sympy.functions.elementary.integers.RoundFunction
-
-sin
----
-
-.. autoclass:: sympy.functions.elementary.trigonometric.sin
-   :members:
-
-sinh
-----
-
-.. autoclass:: sympy.functions.elementary.hyperbolic.sinh
-   :members:
-
-sqrt
-----
-
-Returns the square root of an expression. It is equivalent to raise to Rational(1,2)
-
-    >>> from sympy.functions import sqrt
-    >>> from sympy import Rational
-    >>> sqrt(2) == 2**Rational(1,2)
-    True
-
-.. autoclass:: sympy.functions.elementary.miscellaneous.sqrt
-   :members:
-
-
-sign
-----
-
-.. autoclass:: sympy.functions.elementary.complexes.sign
-   :members:
-
-tan
----
-
-.. autoclass:: sympy.functions.elementary.trigonometric.tan
-   :members:
-
-tanh
-----
-
-.. autoclass:: sympy.functions.elementary.hyperbolic.tanh
-   :members:
diff --git a/dev-py3k/_sources/modules/functions/index.txt b/dev-py3k/_sources/modules/functions/index.txt
deleted file mode 100644
index 2f3c0e4791f..00000000000
--- a/dev-py3k/_sources/modules/functions/index.txt
+++ /dev/null
@@ -1,21 +0,0 @@
-Functions Module
-****************
-
-.. module:: sympy.functions
-
-All functions support the methods documented below, inherited from
-:py:class:`sympy.core.function.Function`.
-
-.. autoclass:: sympy.core.function.Function
-   :noindex:
-   :members:
-
-Contents
-========
-
-.. toctree::
-   :maxdepth: 2
-
-   elementary.rst
-   combinatorial.rst
-   special.rst
diff --git a/dev-py3k/_sources/modules/functions/special.txt b/dev-py3k/_sources/modules/functions/special.txt
deleted file mode 100644
index a8ce407a16b..00000000000
--- a/dev-py3k/_sources/modules/functions/special.txt
+++ /dev/null
@@ -1,176 +0,0 @@
-Special
-=======
-
-DiracDelta
-----------
-.. autoclass:: sympy.functions.special.delta_functions.DiracDelta
-   :members:
-
-Heaviside
----------
-.. autoclass:: sympy.functions.special.delta_functions.Heaviside
-   :members:
-
-beta
-----
-
-.. autofunction:: sympy.functions.special.gamma_functions.beta
-
-erf
----
-
-.. autoclass:: sympy.functions.special.error_functions.erf
-   :members:
-
-Gamma and Related Functions
----------------------------
-.. autoclass:: sympy.functions.special.gamma_functions.gamma
-   :members:
-.. autoclass:: sympy.functions.special.gamma_functions.loggamma
-   :members:
-.. autoclass:: sympy.functions.special.gamma_functions.polygamma
-   :members:
-.. autofunction:: sympy.functions.special.gamma_functions.digamma
-.. autofunction:: sympy.functions.special.gamma_functions.trigamma
-.. autoclass:: sympy.functions.special.gamma_functions.uppergamma
-   :members:
-.. autoclass:: sympy.functions.special.gamma_functions.lowergamma
-   :members:
-
-Special Cases of the Incomplete Gamma Functions
------------------------------------------------
-.. module:: sympy.functions.special.error_functions
-
-.. autoclass:: erf
-.. autoclass:: Ei
-.. autoclass:: expint
-.. autofunction:: E1
-.. autoclass:: Si
-.. autoclass:: Ci
-.. autoclass:: Shi
-.. autoclass:: Chi
-
-.. autoclass:: sympy.functions.special.error_functions.FresnelIntegral
-   :members:
-
-.. autoclass:: fresnels
-.. autoclass:: fresnelc
-
-Bessel Type Functions
----------------------
-
-.. autoclass:: sympy.functions.special.bessel.BesselBase
-   :members:
-
-.. autoclass:: sympy.functions.special.bessel.besselj
-.. autoclass:: sympy.functions.special.bessel.bessely
-.. autoclass:: sympy.functions.special.bessel.besseli
-.. autoclass:: sympy.functions.special.bessel.besselk
-.. autoclass:: sympy.functions.special.bessel.hankel1
-.. autoclass:: sympy.functions.special.bessel.hankel2
-.. autoclass:: sympy.functions.special.bessel.jn
-.. autoclass:: sympy.functions.special.bessel.yn
-
-.. autofunction:: sympy.functions.special.bessel.jn_zeros
-
-B-Splines
----------
-
-.. autofunction:: sympy.functions.special.bsplines.bspline_basis
-.. autofunction:: sympy.functions.special.bsplines.bspline_basis_set
-
-Riemann Zeta and Related Functions
-----------------------------------
-.. module:: sympy.functions.special.zeta_functions
-
-.. autoclass:: zeta
-.. autoclass:: dirichlet_eta
-.. autoclass:: polylog
-.. autoclass:: lerchphi
-
-Hypergeometric Functions
-------------------------
-.. autoclass:: sympy.functions.special.hyper.hyper
-   :members:
-
-.. autoclass:: sympy.functions.special.hyper.meijerg
-   :members:
-
-Orthogonal Polynomials
-----------------------
-
-.. automodule:: sympy.functions.special.polynomials
-
-Jacobi Polynomials
-++++++++++++++++++
-
-.. autoclass:: sympy.functions.special.polynomials.jacobi
-   :members:
-
-Gegenbauer Polynomials
-++++++++++++++++++++++
-
-.. autoclass:: sympy.functions.special.polynomials.gegenbauer
-   :members:
-
-Chebyshev Polynomials
-+++++++++++++++++++++
-
-.. autoclass:: sympy.functions.special.polynomials.chebyshevt
-   :members:
-
-.. autoclass:: sympy.functions.special.polynomials.chebyshevu
-   :members:
-
-.. autoclass:: sympy.functions.special.polynomials.chebyshevt_root
-   :members:
-
-.. autoclass:: sympy.functions.special.polynomials.chebyshevu_root
-   :members:
-
-Legendre Polynomials
-++++++++++++++++++++
-
-.. autoclass:: sympy.functions.special.polynomials.legendre
-   :members:
-
-.. autoclass:: sympy.functions.special.polynomials.assoc_legendre
-   :members:
-
-Hermite Polynomials
-+++++++++++++++++++
-
-.. autoclass:: sympy.functions.special.polynomials.hermite
-   :members:
-
-Laguerre Polynomials
-++++++++++++++++++++
-
-.. autoclass:: sympy.functions.special.polynomials.laguerre
-   :members:
-.. autoclass:: sympy.functions.special.polynomials.assoc_laguerre
-   :members:
-
-Spherical Harmonics
--------------------
-
-.. autofunction:: sympy.functions.special.spherical_harmonics.Plmcos
-
-.. autofunction:: sympy.functions.special.spherical_harmonics.Ylm
-
-.. autofunction:: sympy.functions.special.spherical_harmonics.Ylm_c
-
-.. autofunction:: sympy.functions.special.spherical_harmonics.Zlm
-
-Tensor Functions
-----------------
-
-.. autofunction:: sympy.functions.special.tensor_functions.Eijk
-
-.. autofunction:: sympy.functions.special.tensor_functions.eval_levicivita
-
-.. autoclass:: sympy.functions.special.tensor_functions.LeviCivita
-   :members:
-
-.. autoclass:: sympy.functions.special.tensor_functions.KroneckerDelta
-   :members:
diff --git a/dev-py3k/_sources/modules/galgebra/GA.txt b/dev-py3k/_sources/modules/galgebra/GA.txt
deleted file mode 100644
index 51e542b22e2..00000000000
--- a/dev-py3k/_sources/modules/galgebra/GA.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-========
-GAlgebra
-========
-
-.. automodule:: sympy.galgebra.GA
-   :members:
diff --git a/dev-py3k/_sources/modules/galgebra/GA/GAsympy.txt b/dev-py3k/_sources/modules/galgebra/GA/GAsympy.txt
deleted file mode 100644
index bbc694c2651..00000000000
--- a/dev-py3k/_sources/modules/galgebra/GA/GAsympy.txt
+++ /dev/null
@@ -1,1729 +0,0 @@
-**************************************
-  Geometric Algebra Module for SymPy
-**************************************
-
-:Author: Alan Bromborsky
-
-.. |release| replace:: 0.10
-
-.. % Complete documentation on the extended LaTeX markup used for Python
-.. % documentation is available in ``Documenting Python'', which is part
-.. % of the standard documentation for Python.  It may be found online
-.. % at:
-.. %
-.. % http://www.python.org/doc/current/doc/doc.html
-.. % \lstset{language=Python}
-.. % \input{macros}
-.. % This is a template for short or medium-size Python-related documents,
-.. % mostly notably the series of HOWTOs, but it can be used for any
-.. % document you like.
-.. % The title should be descriptive enough for people to be able to find
-.. % the relevant document.
-
-.. % Increment the release number whenever significant changes are made.
-.. % The author and/or editor can define 'significant' however they like.
-
-.. % At minimum, give your name and an email address.  You can include a
-.. % snail-mail address if you like.
-
-.. % This makes the Abstract go on a separate page in the HTML version;
-.. % if a copyright notice is used, it should go immediately after this.
-.. %
-.. % \ifhtml
-.. % \chapter*{Front Matter\label{front}}
-.. % \fi
-.. % Copyright statement should go here, if needed.
-.. % ...
-.. % The abstract should be a paragraph or two long, and describe the
-
-.. % scope of the document.
-
-.. topic:: Abstract
-
-   This document describes the implementation of a geometric algebra module in
-   python that utilizes the :mod:`sympy` symbolic algebra library.  The python
-   module :mod:`GA` has been developed for coordinate free calculations using
-   the operations (geometric, outer, and inner products etc.) of geometric algebra.
-   The operations can be defined using a completely arbitrary metric defined
-   by the inner products of a set of arbitrary vectors or the metric can be
-   restricted to enforce orthogonality and signature constraints on the set of
-   vectors.  In addition the module includes the geometric, outer (curl) and inner
-   (div) derivatives and the ability to define a curvilinear coordinate system.
-   The module requires the numpy and the sympy modules.
-
-
-What is Geometric Algebra?
-==========================
-
-Geometric algebra is the Clifford algebra of a real finite dimensional vector
-space or the algebra that results when a real finite dimensional vector space
-is extended with a product of vectors (geometric product) that is associative,
-left and right distributive, and yields a real number for the square (geometric
-product) of any vector [Hestenes,Lasenby].  The elements of the geometric
-algebra are called multivectors and consist of the linear combination of
-scalars, vectros, and the geometric product of two or more vectors. The
-additional axioms for the geometric algebra are that for any vectors :math:`a`,
-:math:`b`, and :math:`c` in the base vector space:
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  \begin{array}{c}
-  a \left( bc \right) = \left( ab \right) c \\
-  a \left( b+c \right) = ab+ac \\
-  \left( a + b \right) c = ac+bc \\
-  aa = a^{2} \in \Re
-  \end{array}
-  \end{equation*}
-
-
-By induction these also apply to any multivectors.
-
-Several software packages for numerical geometric algebra calculations are
-available from Doran-Lazenby group and the Dorst group. Symbolic packages for
-Clifford algebra using orthongonal bases such as
-:math:`e_{i}e_{j}+e_{j}e_{i} = 2\eta_{ij}`, where :math:`\eta_{ij}` is a numeric
-array are available in Maple and Mathematica. The symbolic algebra module,
-:mod:`GA`, developed for python does not depend on an orthogonal basis
-representation, but rather is generated from a set of :math:`n` arbitrary
-symbolic vectors,  :math:`a_{1},a_{2},\dots,a_{n}` and a symbolic metric
-tensor :math:`g_{ij} = a_{i}\cdot a_{j}`.
-
-In order not to reinvent the wheel all scalar symbolic algebra is handled by the
-python module  :mod:`sympy`.
-
-The basic geometic algebra operations will be implemented in python by defining
-a multivector class, MV, and overloading the python operators in Table
-:ref:`1 <table1>` where ``A`` and ``B``  are any two multivectors (In the case of
-``+``, ``-``, ``*``, ``^``, ``|``, ``<<``, and ``>>`` the operation is also defined if ``A`` or
-``B`` is a sympy symbol or a sympy real number).
-
-.. _table1:
-
-.. csv-table::
-    :header: Operation, Result
-    :widths: 10, 40
-
-    ``A+B``,             Sum of multivectors
-    ``A-B``,             Difference of multivectors
-    ``A*B``,             Geometric product
-    ``A^B``,             Outer product of multivectors
-    ``A|B``,             Inner product of multivectors
-    ``A<B`` or ``A<<B``, Left contraction of multivectors
-    ``A>B`` or ``A>>B``, Right contraction of multivectors
-
-Table :ref:`1 <table1>`. Multivector operations for symbolicGA
-
-The option to use ``<`` or ``<<`` for left contraction and ``>`` or ``>>`` is given since the ``<`` and
-``>`` operators do not have r-forms (there are no :func:`__rlt__` and :func:`__rgt__` functions to overload) while
-``<<`` and ``>>`` do have r-forms so that ``x << A`` and ``x >> A`` are allowed where ``x`` is a scalar (symbol or integer)
-and ``A`` is a multivector. With ``<`` and ``>`` we can only have mixed modes (scalars and multivectors) if the
-multivector is the first operand.
-
-.. note::
-
-    Except for ``<`` and ``>`` all the multivector operators have r-forms so that as long as one of the
-    operands, left or right, is a multivector the other can be a multivector or a scalar (sympy symbol or integer).
-
-.. warning::
-
-    Note that the operator order precedence is determined by python and is not
-    neccessarily that used by geometric algebra. It is **absolutely essential** to
-    use parenthesis in multivector
-    expressions containing ``^``, ``|``, ``<``, ``>``, ``<<`` and/or ``>>``.  As an example let
-    ``A`` and ``B`` be any two multivectors. Then ``A + A*B = A +(A*B)``, but
-    ``A+A^B = (2*A)^B`` since in python the ``^`` operator has a lower precedence
-    than the '+' operator.  In geometric algebra the outer and inner products and
-    the left and right contractions have a higher precedence than the geometric
-    product and the geometric product has a higher precedence than addition and
-    subtraction.  In python the ``^``, ``|``, ``<``, ``>``, ``<<`` and ``>>`` all have a lower
-    precedence than ``+`` and ``-`` while ``*`` has a higher precedence than
-    ``+`` and ``-``.
-
-
-.. _vbm:
-
-Vector Basis and Metric
-=======================
-
-The two structures that define the :class:`MV` (multivector) class are the
-symbolic basis vectors and the symbolic metric.  The symbolic basis
-vectors are input as a string with the symbol name separated by spaces.  For
-example if we are calculating the geometric algebra of a system with three
-vectors that we wish to denote as ``a0``, ``a1``, and ``a2`` we would define the
-string variable:
-
-``basis = 'a0 a1 a2'``
-
-that would be input into the multivector setup function.  The next step would be
-to define the symbolic metric for the geometric algebra of the basis we
-have defined. The default metric is the most general and is the matrix of
-the following symbols
-
-.. _eq1:
-
-
-
-.. math::
-  :label: 1
-  :nowrap:
-
-  \begin{equation*}
-  g = \left[
-  \begin{array}{ccc}
-    a0**2 & (a0.a1) & (a0.a2) \\
-    (a0.a1) & a1**2 & (a1.a2) \\
-    (a0.a2) & (a1.a2) & a2**2 \\
-  \end{array}
-  \right]
-  \end{equation*}
-
-
-where each of the :math:`g_{ij}` is a symbol representing all of the dot
-products of the basis vectors. Note that the symbols are named so that
-:math:`g_{ij} = g_{ji}` since for the symbol function
-:math:`(a0.a1) \ne (a1.a0)`.
-
-Note that the strings shown in equation :ref:`1 <eq1>` are only used when the values
-of :math:`g_{ij}` are output (printed).   In the :mod:`GA` module (library)
-the :math:`g_{ij}` symbols are stored in a static member list of the multivector
-class :class:`MV` as the double list ``MV.metric`` (:math:`g_{ij}` = ``MV.metric[i][j]``).
-
-The default definition of :math:`g` can be overwritten by specifying a string
-that will define :math:`g`. As an example consider a symbolic representation
-for conformal geometry. Define for a basis
-
-``basis = 'a0 a1 a2 n nbar'``
-
-and for a metric
-
-``metric = '# # # 0 0, # # # 0 0, # # # 0 0, 0 0 0 0 2, 0 0 0 2 0'``
-
-then calling `MV.setup(basis,metric)` would initialize
-
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  g = \left[
-  \begin{array}{ccccc}
-    a0**2 & (a0.a1)  & (a0.a2) & 0 & 0\\
-    (a0.a1) & a1**2  & (a1.a2) & 0 & 0\\
-    (a0.a2) & (a1.a2) & a2**2 & 0 & 0 \\
-    0 & 0 & 0 & 0 & 2 \\
-    0 & 0 & 0 & 2 & 0
-  \end{array}
-  \right]
-  \end{equation*}
-
-
-Here we have specified that :math:`n` and :math:`nbar` are orthonal to all the
-:math:`a`'s, :math:`n**2 = nbar**2 = 0`, and :math:`(n.nbar) = 2`. Using
-:math:`\#` in the metric definition string just tells the program to use the
-default symbol for that value.
-
-When ``MV.setup`` is called multivector representations of the basis local to
-the program are instantiated.  For our first example that means that the
-symbolic vectors named ``a0``, ``a1``, and ``a2`` are created and made available
-to the programmer for future calculations.
-
-In addition to the basis vectors the :math:`g_{ij}` are also made available to
-the programer with the following convention. If ``a0`` and ``a1`` are basis
-vectors, then their dot products are denoted by  ``a0sq``, ``a2sq``, and
-``a0dota1`` for use as python program varibles. If you print  ``a0sq`` the
-output would be ``a0**2`` and the output for ``a0dota1`` would be ``(a0.a1)`` as
-shown in equation :ref:`1 <eq1>`.  If the default value are overridden the new
-values are output by print.  For examle if :math:`g_{00} = 0` then "``print
-a0sq``" would output "0."
-
-More generally, if ``metric`` is not a string, but a list of lists or a two
-dimension numpy array, it is assumed that each element of ``metric`` is symbolic
-variable so that the :math:`g_{ij}` could be defined as symbolic functions as
-well as variables. For example instead of letting :math:`g_{01} = (a0.a1)` we
-could have  :math:`g_{01} = cos(theta)` where we use a symbolic :math:`\cos`
-function.
-
-.. note::
-
-    Additionally ``MV.setup`` has an option for an othogonal basis where the signature
-    of the metric space is defined by a string.  For example if the signature of the
-    vector space is :math:`(1,1,1)` (Euclidian 3-space) set
-
-    ``metric = '[1,1,1]'``
-
-    Likewise if the signature is that of spacetime, :math:`(1,-1,-1,-1)` then define
-
-    ``metric = '[1,-1,-1,-1]'``.
-
-
-Representation and Reduction of Multivector Bases
-=================================================
-
-In our symbolic geometric algebra we assume that all multivectors of interest to
-us can be obtained from the symbolic basis vectors we have input, via the
-different operations available to geometric algebra. The first problem we have
-is representing the general multivector in terms of the basis vectors.  To
-do this we form the ordered geometric products of the basis vectors and develop
-an internal representation of these products in terms of python classes.  The
-ordered geometric products are all multivectors of the form
-:math:`a_{i_{1}}a_{i_{2}}\dots a_{i_{r}}` where :math:`i_{1}<i_{2}<\dots <i_{r}`
-and :math:`r \le n`. We call these multivectors bases and represent them
-internally with the list of integers
-:math:`\left[ i_{1},i_{2},\dots,i_{r}\right]`. The bases are labeled, for the
-purpose of output display, with strings that are concatenations of the strings
-representing the basis vectors.  So that in our example  ``[1,2]`` would be
-labeled with the string ``'a1a2'`` and represents the geometric product
-``a1*a2``. Thus the list ``[0,1,2]`` represents ``a0*a1*a2``.For our example the
-complete set of bases and labels are shown in Table :ref:`2 <table2>`
-
-.. note::
-
-    The empty list, ``[]``, represents the scalar 1.
-
-.. _table2:
-
-::
-
-   MV.basislabel = ['1', ['a0', 'a1', 'a2'], ['a0a1', 'a0a2', 'a1a2'],
-                    ['a0a1a2']]
-   MV.basis      = [[], [[0], [1], [2]], [[0, 1], [0, 2], [1, 2]],
-                    [[0, 1, 2]]]
-
-Table :ref:`2 <table2>`. Multivector basis labels and internal basis representation.
-
-Since there are :math:`2^{n}` bases and the number of bases with equal list
-lengths is the same as for the grade decomposition of a dimension :math:`n`
-geometric algebra we will call the collections of bases of equal length
-``psuedogrades``.
-
-The critical operation in setting up the geometric algebra module is reducing
-the geomertric product of any two bases to a linear combination of bases so that
-we can calculate a multiplication table for the bases.  First we represent the
-product as the concatenation of two base lists.  For example ``a1a2*a0a1`` is
-represented by the list ``[1,2]+[0,1] = [1,2,0,1]``. The representation of the
-product is reduced via two operations, contraction and revision. The state of
-the reduction is saved in two lists of equal length.  The first list contains
-symbolic scale factors (symbol or numeric types) for the corresponding interger
-list representing the product of bases.  If we wish to reduce
-:math:`\left[ i_{1},\dots,i_{r}\right]` the starting point is the coefficient list
-:math:`C = \left[ 1 \right]` and the bases list
-:math:`B = \left[ \left[ i_{1},\dots,i_{r}\right] \right]`.  We now operate on each
-element of the lists as follows:
-
-* ``contraction``: Consider a basis list :math:`B` with element
-  :math:`B[j] = \left[ i_{1},\dots,i_{l},i_{l+1},\dots,i_{s}\right]` where
-  :math:`i_{l} = i_{l+1}`. Then the product of the :math:`l` and :math:`l+1` terms
-  result in a scalar and :math:`B[j]` is replaced by the new list representation
-  :math:`\left[ i_{1},\dots,i_{l-1},i_{l+2},\dots,i_{r}\right]` which is of psuedo
-  grade :math:`r-2` and :math:`C[j]` is replaced by the symbol
-  :math:`g_{i_{l}i_{l}}C[j]`.
-
-* ``revision``: Consider a basis list :math:`B` with element
-  :math:`B[j] = \left[ i_{1},\dots,i_{l},i_{l+1},\dots,i_{s}\right]` where
-  :math:`i_{l} > i_{l+1}`. Then the :math:`l` and :math:`l+1` elements must be
-  reversed to be put in normal order, but we have :math:`a_{i_{l}}a_{i_{l+1}} = 2g_{i_{l}i_{l+1}}-a_{i_{l+1}}a_{i_{l}}`
-  (From the geometric algebra definition of the dot product of two vectors). Thus
-  we append the list representing the reduced element,
-  :math:`\left[ i_{1},\dots,i_{l-1},i_{l+2},\dots,i_{s}\right]`, to the pseudo bases
-  list, :math:`B`, and append :math:`2g_{i{l}i_{l+1}}C[j]` to the coefficients
-  list, then we replace :math:`B[j]` with
-  :math:`\left[ i_{1},\dots,i_{l+1},i_{l},\dots,i_{s}\right]`  and :math:`C[j]` with
-  :math:`-C[j]`. Both lists are increased by one element if
-  :math:`g_{i_{l}i_{l+1}} \ne 0`.
-
-These processes are repeated untill every basis list in :math:`B` is in normal
-(ascending) order with no repeated elements.  Then the coefficents of equivalent
-bases are summed and the bases sorted according to psuedograde and ascending
-order.  We now have a way of calculating the geometric product of any two bases
-as a symbolic linear combination of all the bases with the coefficients
-determined by :math:`g`. The base multiplication table for our simple example of
-three vectors is given by (the coefficient of each psuedo base is enclosed with
-{} for clarity):
-
-
-.. literalinclude:: multable.dat
-
-
-Base Representation of Multivectors
-===================================
-
-In terms of the bases defined an arbitrary multivector can be represented as a
-list of arrays  (we use the numpy python module to implement arrays).   If we
-have :math:`n` basis vectors we initialize the list ``self.mv = [0,0,...,0]``
-with :math:`n+1` integers all zero.  Each zero is a placeholder for an array of
-python objects (in this case the objects will be sympy symbol objects). If
-``self.mv[r] = numpy.array([list of symbol objects])`` each entry in the
-``numpy.array`` will be a coefficient of the corresponding psuedo base.
-``self.mv[r] = 0`` indicates that the coefficients of every base of psuedo grade
-:math:`r` are 0.  The length of the array ``self.mv[r]`` is :math:`n \choose r`
-the binomial coefficient. For example the psuedo basis vector ``a1`` would be
-represented as a multivector by the list:
-
-``a1.mv = [0,numpy.array([numeric(0),numeric(1),numeric(0)]),0,0]``
-
-and ``a0a1a2`` by:
-
-``a0a1a2.mv = [0,0,0,numpy.array([numeric(1)])]``
-
-The array is stuffed with sympy numeric objects instead of python integers so
-that we can perform symbolically manipulate sympy expressions that consist of
-scalar algebraic symbols and exact rational numbers which sympy can also
-represent.
-
-The ``numpy.array`` is used because operations of addition, substraction, and
-multiplication by an object are defined for the array if they are defined for
-the objects making up the array, which they are by sympy.  We call this
-representation a base type because the ``r`` index is not a grade index since
-the bases we are using are not blades. In a blade representation the structure
-would be identical, but the bases would be replaced by blades and
-``self.mv[r]`` would represent the ``r`` grade components of the multivector.
-The first use of the base representation is to store the results of the
-multiplication tabel for the bases in the class variable ``MV.mtabel``.  This
-variable is a group of nested lists so that the geometric product of the
-``igrade`` and  ``ibase`` with the ``jgrade`` and ``jbase`` is
-``MV.mtabel[igrade][ibase][jgrade][jbase]``.  We can then use this table to
-calculate the geometric product of any two multivectors.
-
-
-Blade Representation of Multivectors
-====================================
-
-Since we can now calculate the symbolic geometric product of any two
-multivectors we can also calculate the blades corresponding to the product of
-the symbolic basis vectors using the formula
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-    A_{r}\wedge b = \frac{1}{2}\left( A_{r}b-\left( -1 \right)^{r}bA_{r} \right),
-  \end{equation*}
-
-
-where :math:`A_{r}` is a multivector of grade :math:`r` and :math:`b` is a
-vector.  For our example basis the result is shown in Table :ref:`3 <table3>`.
-
-.. _table3:
-
-::
-
-   1 = 1
-   a0 = a0
-   a1 = a1
-   a2 = a2
-   a0^a1 = {-(a0.a1)}1+a0a1
-   a0^a2 = {-(a0.a2)}1+a0a2
-   a1^a2 = {-(a1.a2)}1+a1a2
-   a0^a1^a2 = {-(a1.a2)}a0+{(a0.a2)}a1+{-(a0.a1)}a2+a0a1a2
-
-Table :ref:`3 <table3>`. Bases blades in terms of bases.
-
-The important thing to notice about Table :ref:`3 <table3>` is that it is a
-triagonal (lower triangular) system of equations so that using a simple back
-substitution algorithym we can solve for the psuedo bases in terms of the blades
-giving Table :ref:`4 <table4>`.
-
-.. _table4:
-
-::
-
-   1 = 1
-   a0 = a0
-   a1 = a1
-   a2 = a2
-   a0a1 = {(a0.a1)}1+a0^a1
-   a0a2 = {(a0.a2)}1+a0^a2
-   a1a2 = {(a1.a2)}1+a1^a2
-   a0a1a2 = {(a1.a2)}a0+{-(a0.a2)}a1+{(a0.a1)}a2+a0^a1^a2
-
-Table :ref:`4 <table4>`. Bases in terms of basis blades.
-
-Using Table :ref:`4 <table4>` and simple substitution we can convert from a base
-multivector representation to a blade representation.  Likewise, using Table
-:ref:`3 <table3>` we can convert from blades to bases.
-
-Using the blade representation it becomes simple to program functions that will
-calculate the grade projection, reverse, even, and odd multivector functions.
-
-Note that in the multivector class ``MV`` there is a class variable for each
-instantiation, ``self.bladeflg``, that is set to zero for a base representation
-and 1 for a blade representation.  One needs to keep track of which
-representation is in use since various multivector operations require conversion
-from one representation to the other.
-
-.. warning::
-
-    When the geometric product of two multivectors is calculated the module looks to
-    see if either multivector is in blade representation.  If either is the result of
-    the geometric product is converted to a blade representation.  One result of this
-    is that if either of the multivectors is a simple vector (which is automatically a
-    blade) the result will be in a blade representation.  If ``a`` and ``b`` are vectors
-    then the result ``a*b`` will be ``(a.b)+a^b`` or simply ``a^b`` if ``(a.b) = 0``.
-
-
-Outer and Inner Products, Left and Right Contractions
-=====================================================
-
-In geometric algebra any general multivector :math:`A` can be decomposed into
-pure grade multivectors (a linear combination of blades of all the same order)
-so that in a :math:`n`-dimensional vector space
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  A = \sum_{r = 0}^{n}A_{r}
-  \end{equation*}
-
-
-The geometric product of two pure grade multivectors :math:`A_{r}` and
-:math:`B_{s}` has the form
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  A_{r}B_{s} = \left\langle A_{r}B_{s} \right\rangle_{\left| r-s \right|} + \left\langle A_{r}B_{s} \right\rangle_{\left| r-s \right|+2} + \cdots + \left\langle A_{r}B_{s} \right\rangle_{r+s}
-  \end{equation*}
-
-
-where :math:`\left\langle  \right\rangle_{t}` projects the :math:`t` grade components of the
-multivector argument.  The inner and outer products of :math:`A_{r}` and
-:math:`B_{s}` are then defined to be
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  A_{r}\cdot B_{s} = \left\langle A_{r}B_{s} \right\rangle_{\left| r-s \right|}
-  \end{equation*}
-
-
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  A_{r}\wedge B_{s} = \left\langle A_{r}B_{s} \right\rangle_{r+s}
-  \end{equation*}
-
-
-and
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  A\cdot B = \sum_{r,s}A_{r}\cdot B_{s}
-  \end{equation*}
-
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  A\wedge B = \sum_{r,s}A_{r}\wedge B_{s}
-  \end{equation*}
-
-
-Likewise the right (:math:`\lfloor`) and left (:math:`\rfloor`) contractions are defined as
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  A_{r}\lfloor B_{s} = \left \{ \begin{array}{cc}
-    \left\langle A_{r}B_{s} \right\rangle_{r-s} &  r \ge s \\
-                     0                          &  r < s \end{array} \right \}
-  \end{equation*}
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  A_{r}\rfloor B_{s} = \left \{ \begin{array}{cc}
-     \left\langle A_{r}B_{s} \right\rangle_{s-r} &  s \ge r \\
-                      0                          &  s < r \end{array} \right \}
-  \end{equation*}
-
-
-and
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  A\lfloor B = \sum_{r,s}A_{r}\lfloor B_{s}
-  \end{equation*}
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  A\rfloor B = \sum_{r,s}A_{r}\rfloor B_{s}
-  \end{equation*}
-
-
-The ``MV`` class function for the outer product of the multivectors ``mv1`` and
-``mv2`` is ::
-
-   @staticmethod
-   def outer_product(mv1,mv2):
-        product = MV()
-        product.bladeflg = 1
-        mv1.convert_to_blades()
-        mv2.convert_to_blades()
-        for igrade1 in MV.n1rg:
-            if not isint(mv1.mv[igrade1]):
-                pg1 = mv1.project(igrade1)
-                for igrade2 in MV.n1rg:
-                    igrade = igrade1+igrade2
-                    if igrade <= MV.n:
-                        if not isint(mv2.mv[igrade2]):
-                            pg2 = mv2.project(igrade2)
-                            pg1pg2 = pg1*pg2
-                            product.add_in_place(pg1pg2.project(igrade))
-        return(product)
-
-
-The steps for calculating the outer product are:
-
-#. Convert ``mv1`` and ``mv2`` to blade representation if they are not already
-   in that form.
-
-#. Project and loop through each grade ``mv1.mv[i1]`` and ``mv2.mv[i2]``.
-
-#. Calculate the geometric product ``pg1*pg2``.
-
-#. Project the ``i1+i2`` grade from ``pg1*pg2``.
-
-#. Accumulate the results for each pair of grades in the input multivectors.
-
-.. warning::
-
-    In the  ``MV`` class we have overloaded the ``^`` operator to represent the outer
-    product so that instead of calling the outer product function we can write ``mv1^ mv2``.
-    Due to the precedence rules for python it is **absolutely essential** to enclose outer products
-    in parenthesis.
-
-For the inner product of the multivectors ``mv1`` and ``mv2`` the ``MV`` class
-function is  ::
-
-    @staticmethod
-    def inner_product(mv1,mv2,mode='s'):
-        """
-        MV.inner_product(mv1,mv2) calculates the inner
-
-        mode = 's' - symmetic (Doran & Lasenby)
-        mode = 'l' - left contraction (Dorst)
-        mode = 'r' - right contraction (Dorst)
-        """
-        if isinstance(mv1,MV) and isinstance(mv2,MV):
-            product = MV()
-            product.bladeflg = 1
-            mv1.convert_to_blades()
-            mv2.convert_to_blades()
-            for igrade1 in range(MV.n1):
-                if isinstance(mv1.mv[igrade1],numpy.ndarray):
-                    pg1 = mv1.project(igrade1)
-                    for igrade2 in range(MV.n1):
-                        igrade = igrade1-igrade2
-                        if mode == 's':
-                            igrade = igrade.__abs__()
-                        else:
-                            if mode == 'l':
-                                igrade = -igrade
-                        if igrade >= 0:
-                            if isinstance(mv2.mv[igrade2],numpy.ndarray):
-                                pg2 = mv2.project(igrade2)
-                                pg1pg2 = pg1*pg2
-                                product.add_in_place(pg1pg2.project(igrade))
-            return(product)
-        else:
-            if mode == 's':
-                if isinstance(mv1,MV):
-                    product = mv1.scalar_mul(mv2)
-                if isinstance(mv2,MV):
-                    product = mv2.scalar_mul(mv1)
-            else:
-                product = None
-        return(product)
-
-
-The inner product is calculated the same way as the outer product except that in
-step 4, ``i1+i2`` is replaced by ``abs(i1-i2)`` or ``i1-i2`` for the right contraction or
-``i2-i1`` for the left contraction.  If ``i1-i2`` is less than zero there is no contribution
-to the right contraction.  If ``i2-i1`` is less than zero there is no contribution to the
-left contraction.
-
-.. warning::
-
-    In the ``MV`` class we have overloaded the ``|`` operator for the inner product,
-    ``>`` operator for the right contraction, and ``<`` operator for the left contraction.
-    Instead of calling the inner product function we can write ``mv1|mv2``, ``mv1>mv2``, or
-    ``mv1<mv2`` respectively for the inner product, right contraction, or left contraction.
-    Again, due to the precedence rules for python it is **absolutely essential** to enclose inner
-    products and/or contractions in parenthesis.
-
-
-.. _reverse:
-
-Reverse of Multivector
-======================
-
-If :math:`A` is the geometric product of :math:`r` vectors
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-    A = a_{1}\dots a_{r}
-  \end{equation*}
-
-
-then the reverse of :math:`A` designated :math:`A^{\dagger}` is defined by
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-    A^{\dagger} \equiv a_{r}\dots a_{1}.
-  \end{equation*}
-
-
-The reverse is simply the product with the order of terms reversed.  The reverse
-of a sum of products is defined as the sum of the reverses so that for a general
-multivector A we have
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-    A^{\dagger} = \sum_{i=0}^{N} \left\langle A \right\rangle_{i}^{\dagger}
-  \end{equation*}
-
-
-but
-
-.. _eq14:
-
-
-
-.. math::
-  :label: 2
-  :nowrap:
-
-  \begin{equation*}
-    \left\langle A \right\rangle_{i}^{\dagger} = \left( -1\right)^{\frac{i\left( i-1\right)}{2}}\left\langle A \right\rangle_{i}
-  \end{equation*}
-
-
-which is proved by expanding the blade bases in terms of orthogonal vectors and
-showing that equation :ref:`2 <eq14>` holds for the geometric product of orthogonal
-vectors.
-
-The reverse is important in the theory of rotations in :math:`n`-dimensions.  If
-:math:`R` is the product of an even number of vectors and :math:`RR^{\dagger} = 1`
-then :math:`RaR^{\dagger}` is a composition of rotations of the vector :math:`a`.
-If :math:`R` is the product of two vectors then the plane that :math:`R` defines
-is the plane of the rotation.  That is to say that :math:`RaR^{\dagger}` rotates the
-component of :math:`a` that is projected into the plane defined by :math:`a` and
-:math:`b` where :math:`R=ab`.  :math:`R` may be written
-:math:`R = e^{\frac{\theta}{2}U}`, where :math:`\theta` is the angle of rotation
-and :math:`u` is a unit blade :math:`\left( u^{2} = \pm 1\right)` that defines the
-plane of rotation.
-
-
-.. _recframe:
-
-Reciprocal Frames
-=================
-
-If we have :math:`M` linearly independent vectors (a frame),
-:math:`a_{1},\dots,a_{M}`, then the reciprocal frame is
-:math:`a^{1},\dots,a^{M}` where :math:`a_{i}\cdot a^{j} = \delta_{i}^{j}`,
-:math:`\delta_{i}^{j}` is the Kronecker delta (zero if :math:`i \ne j` and one
-if :math:`i = j`). The reciprocal frame is constructed as follows:
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-    E_{M} = a_{1}\wedge\dots\wedge a_{M}
-  \end{equation*}
-
-
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-    E_{M}^{-1} = \displaystyle\frac{E_{M}}{E_{M}^{2}}
-  \end{equation*}
-
-
-Then
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-    a^{i} = \left( -1\right)^{i-1}\left( a_{1}\wedge\dots\wedge \breve{a}_{i} \wedge\dots\wedge a_{M}\right) E_{M}^{-1}
-  \end{equation*}
-
-
-where :math:`\breve{a}_{i}` indicates that :math:`a_{i}` is to be deleted from
-the product.  In the standard notation if a vector is denoted with a subscript
-the reciprocal vector is denoted with a superscript. The multivector setup
-function ``MV.setup(basis,metric,rframe)`` has the argument ``rframe`` with a
-default value of ``False``.  If it is set to ``True`` the reciprocal frame of
-the basis vectors is calculated. Additionaly there is the function
-``reciprocal_frame(vlst,names='')`` external to the ``MV`` class that will
-calculate the reciprocal frame of a list, ``vlst``, of vectors.  If the argument
-``names`` is set to a space delimited string of names for the vectors the
-reciprocal vectors will be given these names.
-
-
-.. _deriv:
-
-Geometric Derivative
-====================
-
-If :math:`F` is a multivector field that is a function of a vector
-:math:`x = x^{i}\boldsymbol{\gamma}_{i}` (we are using the summation convention that
-pairs of subscripts and superscripts are summed over the dimension of the vector
-space) then the geometric derivative :math:`\nabla F` is given by (in this
-section the summation convention is used):
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-    \nabla F = \boldsymbol{\gamma}^{i}\displaystyle\frac{\partial F}{\partial x^{i}}
-  \end{equation*}
-
-
-If :math:`F_{R}` is a grade-:math:`R` multivector and
-:math:`F_{R} = F_{R}^{i_{1}\dots i_{R}}\boldsymbol{\gamma}_{i_{1}}\wedge\dots\wedge \boldsymbol{\gamma}_{i_{R}}`
-then
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-    \nabla F_{R} = \displaystyle\frac{\partial F_{R}^{i_{1}\dots i_{R}}}{\partial x^{j}}\boldsymbol{\gamma}^{j}\left(\boldsymbol{\gamma}_{i_{1}}\wedge
-                   \dots\wedge \boldsymbol{\gamma}_{i_{R}} \right)
-  \end{equation*}
-
-
-Note that
-:math:`\boldsymbol{\gamma}^{j}\left(\boldsymbol{\gamma}_{i_{1}}\wedge\dots\wedge \boldsymbol{\gamma}_{i_{R}} \right)`
-can only contain grades :math:`R-1` and :math:`R+1` so that :math:`\nabla F_{R}`
-also can only contain those grades. For a grade-:math:`R` multivector
-:math:`F_{R}` the inner (div) and outer (curl) derivatives are defined as
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  \nabla\cdot F_{R} = \left < \nabla F_{R}\right >_{R-1}
-  \end{equation*}
-
-
-and
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  \nabla\wedge F_{R} = \left < \nabla F_{R}\right >_{R+1}
-  \end{equation*}
-
-
-For a general multivector function :math:`F` the inner and outer derivatives are
-just the sum of the inner and outer dervatives of each grade of the multivector
-function.
-
-Curvilinear coordinates are derived from a vector function
-:math:`x(\boldsymbol{\theta})` where
-:math:`\boldsymbol{\theta} = \left(\theta_{1},\dots,\theta_{N}\right)` where the number of
-coordinates is equal to the dimension of the vector space.  In the case of
-3-dimensional spherical coordinates :math:`\boldsymbol{\theta} = \left( r,\theta,\phi \right)`
-and the coordinate generating function :math:`x(\boldsymbol{\theta})` is
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  x =  r \cos\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+ r \sin\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{y}}}+ r \cos\left({\theta}\right){\boldsymbol{{\gamma}_{z}}}
-  \end{equation*}
-
-
-A coordinate frame is derived from :math:`x` by
-:math:`\boldsymbol{e}_{i} = \displaystyle\frac{\partial x}{\partial \theta^{i}}`.  The following show the frame for
-spherical coordinates.
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  \boldsymbol{e}_{r} = \cos\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+\sin\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{y}}}+\cos\left({\theta}\right){\boldsymbol{{\gamma}_{z}}}
-  \end{equation*}
-
-
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  \boldsymbol{e}_{{\theta}} = \cos\left({\phi}\right) \cos\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+r \cos\left({\theta}\right) \sin\left({\phi}\right){\boldsymbol{{\gamma}_{y}}} - r \sin\left({\theta}\right){\boldsymbol{{\gamma}_{z}}}
-  \end{equation*}
-
-
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  \boldsymbol{e}_{{\phi}} = - r \sin\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+r \cos\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{y}}}
-  \end{equation*}
-
-
-The coordinate frame generated in this manner is not necessarily normalized so
-define a normalized frame by
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  \boldsymbol{\hat{e}}_{i} = \displaystyle\frac{\boldsymbol{e}_{i}}{\sqrt{\left| \boldsymbol{e}_{i}^{2} \right|}} = \displaystyle\frac{\boldsymbol{e}_{i}}{\left| \boldsymbol{e}_{i} \right|}
-  \end{equation*}
-
-
-This works for all :math:`\boldsymbol{e}_{i}^{2} \neq 0` since we have defined
-:math:`\left| \boldsymbol{e}_{i} \right| = \sqrt{\left| \boldsymbol{e}_{i}^{2} \right|}`.   For spherical
-coordinates the normalized frame vectors are
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  \boldsymbol{\hat{e}}_{r} = \cos\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+\sin\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{y}}}+\cos\left({\theta}\right){\boldsymbol{{\gamma}_{z}}}
-  \end{equation*}
-
-
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  \boldsymbol{\hat{e}}_{{\theta}} = \cos\left({\phi}\right) \cos\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+\cos\left({\theta}\right) \sin\left({\phi}\right){\boldsymbol{{\gamma}_{y}}}- \sin\left({\theta}\right){\boldsymbol{{\gamma}_{z}}}
-  \end{equation*}
-
-
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  \boldsymbol{\hat{e}}_{{\phi}} = - \sin\left({\phi}\right){\boldsymbol{{\gamma}_{x}}}+\cos\left({\phi}\right){\boldsymbol{{\gamma}_{y}}}
-  \end{equation*}
-
-
-The geometric derivative in curvilinear coordinates is given by
-
-.. math::
-  :nowrap:
-
-  \begin{align*}
-    \nabla F_{R} & = \boldsymbol{\gamma}^{i}\displaystyle\frac{\partial }{\partial x^{i}}\left( F_{R}^{i_{1}\dots i_{R}}
-                     \boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right)  \\
-                 & = \boldsymbol{e^{j}}\displaystyle\frac{\partial }{\partial \theta^{j}}\left( F_{R}^{i_{1}\dots i_{R}}
-                     \boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right)  \\
-                 & = \left(\displaystyle\frac{\partial }{\partial \theta^{j}} F_{R}^{i_{1}\dots i_{R}}\right)
-                     \boldsymbol{e^{j}}\left(\boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right)+
-                     F_{R}^{i_{1}\dots i_{R}}\boldsymbol{e^{j}}
-                     \displaystyle\frac{\partial }{\partial \theta^{j}}\left(\boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right) \\
-                 & = \left(\displaystyle\frac{\partial }{\partial \theta^{j}} F_{R}^{i_{1}\dots i_{R}}\right)
-                     \boldsymbol{e^{j}}\left(\boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right)+
-                     F_{R}^{i_{1}\dots i_{R}}C\left\{ \boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right\}
-  \end{align*}
-
-
-where
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  C\left\{ \boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right\} = \boldsymbol{e^{j}}\displaystyle\frac{\partial }{\partial \theta^{j}}
-                                                                                               \left(\boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right)
-  \end{equation*}
-
-
-are the connection multivectors for the curvilinear coordinate system. For a
-spherical coordinate system they are
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  C\left\{\boldsymbol{\hat{e}}_{r}\right\} =  \frac{2}{r}
-  \end{equation*}
-
-
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  C\left\{\boldsymbol{\hat{e}}_{\theta}\right\} = \frac{\cos\left({\theta}\right)}{r \sin\left({\theta}\right)}
-                                                + \frac{1}{r}\boldsymbol{\hat{e}}_{r}\wedge\boldsymbol{\hat{e}}_{\theta}
-  \end{equation*}
-
-
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  C\left\{\boldsymbol{\hat{e}}_{\phi}\right\} = \frac{1}{r}{\boldsymbol{\boldsymbol{\hat{e}}_{r}}}\wedge\boldsymbol{\hat{e}}_{{\phi}}+ \frac{\cos\left({\theta}\right)}{r \sin\left({\theta}\right)}\boldsymbol{\hat{e}}_{{\theta}}\wedge\boldsymbol{\hat{e}}_{{\phi}}
-  \end{equation*}
-
-
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  C\left\{\hat{e}_{r}\wedge\hat{e}_{\theta}\right\} = - \frac{\cos\left({\theta}\right)}{r \sin\left({\theta}\right)}
-                                                        \boldsymbol{\hat{e}}_{r}+\frac{1}{r}\boldsymbol{\hat{e}}_{{\theta}}
-  \end{equation*}
-
-
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  C\left\{\boldsymbol{\hat{e}}_{r}\wedge\boldsymbol{\hat{e}}_{\phi}\right\} = \frac{1}{r}\boldsymbol{\hat{e}}_{{\phi}}
-                                                                            - \frac{\cos\left({\theta}\right)}{r \sin\left({\theta}\right)}\boldsymbol{\hat{e}}_{r}\wedge\boldsymbol{\hat{e}}_{{\theta}}\wedge\boldsymbol{\hat{e}}_{{\phi}}
-  \end{equation*}
-
-
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  C\left\{\boldsymbol{\hat{e}}_{\theta}\wedge\boldsymbol{\hat{e}}_{\phi}\right\} = \frac{2}{r}\boldsymbol{\hat{e}}_{r}\wedge
-                                                                                   \boldsymbol{\hat{e}}_{\theta}\wedge\boldsymbol{\hat{e}}_{\phi}
-  \end{equation*}
-
-
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  C\left\{\boldsymbol{\hat{e}}_r\wedge\boldsymbol{\hat{e}}_{\theta}\wedge\boldsymbol{\hat{e}}_{\phi}\right\} = 0
-  \end{equation*}
-
-
-
-Module Components
-=================
-
-
-Initializing Multivector Class
-------------------------------
-
-The multivector class is initialized with:
-
-
-.. function:: MV.setup(basis,metric='',rframe=False,coords=None,debug=False,offset=0)
-
-   The ``basis`` and ``metric`` parameters were described in section :ref:`vbm`. If
-   ``rframe=True`` the reciprocal frame of the symbolic bases vectors is calculated.
-   If ``debug=True`` the data structure required to initialize the :class:`MV` class
-   are printer out. ``coords`` is a list of :class:`sympy` symbols equal in length to
-   the number of basis vectors.  These symbols are used as the arguments of a
-   multivector field as a function of position and for calculating the derivatives
-   of a multivector field (if ``coords`` is defined then ``rframe`` is automatically
-   set equal to ``True``). ``offset`` is an integer that is added to the multivector
-   coefficient index. For example if one wishes to start labeling vector coefficient
-   indexes at one instead of zero then set ``offset=1``.  Additionally, :func:`MV.setup`
-   calculates the pseudo scalar, :math:`I` and its inverse, :math:`I^{-1}` and makes
-   them available to the programmer as ``MV.I`` and ``MV.Iinv``.
-
-After :func:`MV.setup` is run one can reinialize the :class:`MV` class with
-curvilinear coordinates using:
-
-
-.. function:: MV.rebase(x,coords,base_name,debug=False,debug_level=0)
-
-   A typical usage of ``MV.rebase`` for generating spherical curvilinear coordinate
-   is::
-
-      metric = '1 0 0,0 1 0,0 0 1'
-      gamma_x,gamma_y,gamma_z = MV.setup('gamma_x gamma_y gamma_z',metric,True)
-      coords = r,theta,phi = sympy.symbols('r theta phi')
-      x = r*(sympy.cos(theta)*gamma_z+sympy.sin(theta)*\
-          (sympy.cos(phi)*gamma_x+sympy.sin(phi)*gamma_y))
-      x.set_name('x')
-      MV.rebase(x,coords,'e',True)
-
-   The input parameters for ``MV.rebase`` are
-
-*  ``x``: Vector function of coordinates (derivatives define curvilinear basis)
-
-*  ``coords``: List of sympy symbols for curvilinear coordinates
-
-*  ``debug``: If ``True`` printout (LaTeX) all quantities required for derivative calculation
-
-*  ``debug_level``: Set to 0,1,2, or 3 to stop curvilinear calculation before all quatities are
-   calculated.  This is done when debugging new curvilinear coordinate systems since simplification
-   of expressions is not sufficiently automated to insure success of process of any coordinate system
-   defined by vector function ``x``
-
-
-
-   To date ``MV.rebase`` works for cylindrical and spherical coordinate systems in
-   any number of dimensions  (until the execution time becomes too long).  To make
-   it work for these systems required creating some hacks for expression
-   simplification since both trigsimp and simplify were not general enough to
-   perform the required simplification.
-
-
-.. function:: MV.set_str_format(str_mode=0)
-
-   If ``str_mode=0`` the string representation of the multivector contains no
-   newline characters (prints on one line).   If ``str_mode=1`` the string
-   representation of the multivector places a newline after each grade of the
-   multivector  (prints one grade per line).  If ``str_mode=2`` the string
-   representation of the multivector places a newline after each base of the
-   multivector  (prints one base per line). In both cases bases with zero
-   coefficients are not printed.
-
-    .. note::
-
-        This function directly affects the way multivectors are printed with the
-        print command since it interacts with the :func:`__str__` function for the
-        multivector class which is used by the ``print`` command.
-
-
-        .. csv-table::
-            :header: ``str_mode``, Effect on print
-            :widths: 10, 50
-
-            0, One multivector per line
-            1, One grade per line
-            2, One base per line
-
-Instantiating a Multivector
----------------------------
-
-Now that grades and bases have been described we can show all the ways that a
-multivector can be instantiated. As an example assume that the multivector space
-is initialized with ``e1,e2,e3 = MV.setup('e1 e2 e3')``.
-
-So that multivectors
-could be instantiated with statements such as (``a1``, ``a2``, and ``a3`` are
-``sympy`` symbols)::
-
-   x = a1*e1+a2*e2+a3*e3
-   y = x*e1*e2
-   z = x|y
-   w = x^y
-
-or with the multivector class constructor:
-
-
-.. class:: MV(value='',mvtype='',mvname='',fct=False)
-
-   ``mvname`` is a string that defines the name of the multivector for output
-   purposes. ``value`` and  ``type`` are defined by the following table and ``fct`` is a
-   switch that will convert the symbolic coefficients of a multivector to functions
-   if coordinate variables have been defined when :func:`MV.setup` is called:
-
-    .. csv-table::
-        :delim: ;
-        :header: ``mvtype``, ``value``, Result
-        :widths: 10, 30, 30
-
-        default;           default;                       Zero multivector
-        'basisvector';     int i;                         :math:`\mbox{i}^{th}` basis vector
-        'basisbivector';   int i;                         :math:`\mbox{i}^{th}` basis bivector
-        'scalar';          symbol x;                      Symbolic scalar of value x
-        ;                  string s;                      Symbolic scalar of value Symbol(s)
-        ;                  int i;                         SymPy integer of value i
-        'grade';           [int r, 1-D symbol array A];   X.grade(r) = A
-        ;                  [int r, string s];             Symbolic grade r multivector
-        'vector';          1-D symbol array A;            X.grade(1) = A
-        ;                  string s;                      Symbolic vector
-        'grade2';          1-D symbol array A;            X.grade(2) = A
-        'pseudo';          symbol x;                      X.grade(n) = x
-        'spinor';          string s;                      Symbolic even multivector
-
-
-        default;           string s;                      Symbolic general multivector
-
-
-   If the ``value`` argument has the option of being a string s then a general symbolic
-   multivector will be constructed constrained by the value of ``mvtype``.  The string
-   s will be the base name of the multivector symbolic coefficients.  If ``coords`` is
-   not defined in :func:`MV.setup` the indices of the multivector bases are appended to
-   the base name with a double underscore (superscript notation).  If ``coords`` is defined
-   the coordinate names will replace the indices in the coefficient names.  For example if
-   the base string is *A* and the coordinates *(x,y,z)* then the
-   coefficients of a spinor in 3d space would be *A*, *A__xy*, *A__xz*, and *A__yz*.  If
-   the :mod:`latex_ex` is used to print the multivector the coefficients would print as
-   :math:`A`, :math:`A^{xy}`, :math:`A^{xz}`, and :math:`A^{yz}`.
-
-   If the ``fct`` argrument of :func:`MV` is set to ``True`` and the ``coords`` argument in
-   :func:`MV.setup` is defined the symbolic coefficients of the multivector are functions
-   of the coordinates.
-
-
-Basic Multivector Class Functions
----------------------------------
-
-
-.. function:: __call__(self,igrade=0,ibase=0)
-
-   ``__call__`` returns the the ``igrade``, ``ibase`` coefficient of the
-   multivector.  The defaults return the scalar component of the multivector.
-
-
-.. function:: convert_to_blades(self)
-
-   Convert multivector from the base representation to the blade representation.
-   If multivector is already in blade representation nothing is done.
-
-
-.. function:: convert_from_blades(self)
-
-   Convert multivector from the blade representation to the base representation.
-   If multivector is already in base representation nothing is done.
-
-
-.. function:: project(self,r)
-
-   If r is a integer return the grade-:math:`r` components of the multivector. If
-   r is a multivector return the grades of the multivector that correspond to the
-   non-zero grades of r. For example if one is projecting a general multivector and
-   r is a spinor, ``A.project(r)`` will return only the even grades of the multivector
-   A since a spinor only has even grades that are non-zero.
-
-
-.. function:: even(self)
-
-   Return the even grade components of the multivector.
-
-
-.. function:: odd(self)
-
-   Return the odd grade components of the multivector.
-
-
-.. function:: rev(self)
-
-   Return the reverse of the multivector.  See section :ref:`reverse`.
-
-
-.. function:: is_pure(self)
-
-   Return true if multivector is pure grade (all grades but one are zero).
-
-
-.. function:: diff(self,x)
-
-   Return the partial derivative of the multivector function with respect to
-   variable :math:`x`.
-
-
-.. function:: grad(self)
-
-   Return the geometric derivative of the multivector function.
-
-
-.. function:: grad_ext(self)
-
-   Return the outer (curl) derivative of the multivector function. Equivalent to
-   :func:`curl`.
-
-
-.. function:: grad_int(self)
-
-   Return the inner (div) derivative of the multivector function. Equivalent to
-   :func:`div`.
-
-.. warning::
-
-    If *A* is a vector field in three dimensions :math:`\nabla\cdot {\bf A}` = A.grad_int() = A.div(), but
-    :math:`\nabla\times {\bf A}` = -MV.I*A.grad_ext() = -MV.I*A.curl(). Note that grad_int() lowers the grade
-    of all blades by one grade and grad_ext() raises the grade of all blades by one.
-
-.. function:: set_coef(self,grade,base,value)
-
-   Set the multivector coefficient of index ``(grade,base)`` to ``value``.
-
-
-SymPy Functions Applied Inplace to Multivector Coefficients
------------------------------------------------------------
-
-All the following fuctions belong to the :class:`MV` class and apply the
-corresponding :mod:`sympy` function to each component of a multivector.  All
-these functions perform the operations inplace (``None`` is returned)  on each
-coefficient.  For example if you wished to simplify all the components of the
-multivector ``A`` you would invoke ``A.simplify()``.  The argument list for each
-function is the same as for the corresponding :mod:`sympy` function.  The only
-function that differs in its action from the :mod:`sympy` version is
-:func:`trigsimp` which is applied using the recursive option.
-
-
-.. function:: collect(self,lst)
-
-
-.. function:: sqrfree(self,lst)
-
-
-.. function:: subs(self,*args)
-
-
-.. function:: simplify(self)
-
-
-.. function:: trigsimp(self)
-
-
-.. function:: cancel(self)
-
-
-.. function:: expand(self)
-
-
-Helper Functions
-----------------
-
-These are functions in :mod:`GA`, but not in the multivector (:class:`MV`)
-class.
-
-
-.. function:: set_names(var_lst,var_str)
-
-    :func:`set_names` allows one to name a list, ``var_lst``, of multivectors enmass.
-    The names are in ``var_str``, a blank separated string of names.  An error is
-    generated if the number of name is not equal to the length of ``var_lst``.
-
-
-.. function:: reciprocal_frame(vlst,names='')
-
-    :func:`reciprocal_frame` implements the proceedure described in section
-    :ref:`recframe`.  ``vlst`` is a list of independent vectors that you wish the
-    reciprocal frame calculated for. ``names`` is a blank separated string of names
-    for the reciprocal vectors if names are required by you application.  The
-    function returns a list containing the reciprocal vectors.
-
-
-.. function:: trigsimp(f)
-
-    In general :func:`sympy.trigsimp` will not catch all the trigonometric
-    simplifications in an :mod:`sympy` expression, but by using the recursive
-    option, more of them are caught. So :func:`trigsimp` uses this option by
-    default.
-
-.. function:: S(x)
-
-    :func:`S` instanciates a scaler multivector of value ``x``, where ``x`` can be a
-    :mod:`sympy` variable or integer.  This is just a shorthand method for
-    constructing scalar multivectors and can be used when there is any ambiguity
-    in a multivector expression as to whether a symbol or constant should be
-    treated as a scalar multivector or not.
-
-Examples
-========
-
-
-Algebra
--------
-
-The following examples of geometric algebra (not calculus) are all in the file
-:program:`testsymbolicGA.py` which is included in the sympy distribution
-examples under the galbebra directory.  The section of code in the program for
-each example is show with the respective output following the code section.
-
-
-Example Header
-^^^^^^^^^^^^^^
-
-This is the header of :program:`testsymbolicGA.py` that allows access to the
-required modules and also allow variables and certain multivectors to be
-broadcast from the :mod:`GA` module to the main program.
-
-
-.. literalinclude:: headerGAtest.py
-
-
-Basic Geometric Algebra Operations
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-Example of basic geometric algebra operation of geometric, outer, and inner
-products.
-
-
-.. literalinclude:: BasicGAtest.py
-
-
-Examples of Conformal Geometry
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-Examples of comformal geometry [Lasenby,Chapter 10]. The examples show that
-basic geometric entities (lines, circles, planes, and spheres) in three
-dimensions can be represented by blades in a five dimensional (conformal) space.
-
-
-.. literalinclude:: conformalgeometryGAtest.py
-
-
-Calculation of Reciprocal Frame
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-Example shows the calculation of the reciprocal frame for three arbitrary
-vectors and verifies that the calculated reciprocal vectors have the correct
-properties.
-
-
-.. literalinclude:: reciprocalframeGAtest.py
-
-
-Hyperbolic Geometry
-^^^^^^^^^^^^^^^^^^^
-
-Examples of calculation of distance in hyperbolic geometry [Lasenby,pp373-375].
-This is a good example of the utility of not restricting the basis vector to be
-orthogonal. Note that most of the calculation is simplifying a scalar
-expression.
-
-
-.. literalinclude:: hyperbolicGAtest.py
-
-
-Calculus
---------
-
-The calculus examples all use the extened LaTeXoutput module, ``latex_ex``, for
-clarity.
-
-
-Maxwell's Equations
-^^^^^^^^^^^^^^^^^^^
-
-In geometric calculus the equivalent of the electromagnetic tensor is
-:math:`F = E\gamma_{0}+IB\gamma_{0}` where a spacetime vector is given by
-:math:`x = x^{0}\gamma_{0}+x^{1}\gamma_{1}+x^{2}\gamma_{2}+x^{3}\gamma_{3}`
-where :math:`x^{0} = ct`, the pseudoscalar
-:math:`I = \gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}`, :math:`E` and :math:`B`
-are four vectors where the time component is zero and the spatial components
-equal to the electric and magnetic field components.  Then Maxwell's equations
-can be all written as :math:`\nabla F = J` with :math:`J` the four current.
-This example shows that this equations generates all of Maxwell's equations
-correctly  (in our units:math:`c=1`) [Lasenby,pp229-231].
-
-``Begin Program Maxwell.py``
-
-
-.. literalinclude:: Maxwell.py
-
-``End Program Maxwell.py``
-
-``Begin Program Output``
-
-:math:`I` Pseudo-Scalar
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  I = {\gamma}_{t}{\gamma}_{x}{\gamma}_{y}{\gamma}_{z}
-  \end{equation*}
-
-
-:math:`B` Magnetic Field Bi-Vector
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  B = - {B^{x}}{\gamma}_{t}{\gamma}_{x}- {B^{y}}{\gamma}_{t}{\gamma}_{y}- {B^{z}}{\gamma}_{t}{\gamma}_{z}
-  \end{equation*}
-
-
-:math:`F` Electric Field Bi-Vector
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  E = - {E^{x}}{\gamma}_{t}{\gamma}_{x}- {E^{y}}{\gamma}_{t}{\gamma}_{y}- {E^{z}}{\gamma}_{t}{\gamma}_{z}
-  \end{equation*}
-
-
-:math:`E+IB` Electo-Magnetic Field Bi-Vector
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  F = - {E^{x}}{\gamma}_{t}{\gamma}_{x}- {E^{y}}{\gamma}_{t}{\gamma}_{y}- {B^{z}}{\gamma}_{x}{\gamma}_{y}- {E^{z}}{\gamma}_{t}{\gamma}_{z}+ {B^{y}}{\gamma}_{x}{\gamma}_{z}- {B^{x}}{\gamma}_{y}{\gamma}_{z}
-  \end{equation*}
-
-
-:math:`J` Four Current
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  J =  {J^{t}}{\gamma}_{t}+ {J^{x}}{\gamma}_{x}+ {J^{y}}{\gamma}_{y}+ {J^{z}}{\gamma}_{z}
-  \end{equation*}
-
-
-Geometric Derivative of Electo-Magnetic Field Bi-Vector
-
-.. math::
-  :nowrap:
-
-  \begin{align*}
-  \nabla F & = \left( \partial_{z} {E^{z}} + \partial_{y} {E^{y}} + \partial_{x} {E^{x}}\right){\gamma}_{t} \\
-           & + \left(-\partial_{t} {E^{x}} + \partial_{y} {B^{z}} - \partial_{z} {B^{y}}\right){\gamma}_{x} \\
-           & + \left( \partial_{z} {B^{x}} - \partial_{t} {E^{y}} - \partial_{x} {B^{z}}\right){\gamma}_{y} \\
-           & + \left(-\partial_{y} {B^{x}} - \partial_{t} {E^{z}} + \partial_{x} {B^{y}}\right){\gamma}_{z} \\
-           & + \left(-\partial_{x} {E^{y}} - \partial_{t} {B^{z}} + \partial_{y} {E^{x}}\right){\gamma}_{t}{\gamma}_{x}{\gamma}_{y} \\
-           & + \left(-\partial_{x} {E^{z}} + \partial_{t} {B^{y}} + \partial_{z} {E^{x}}\right){\gamma}_{t}{\gamma}_{x}{\gamma}_{z} \\
-           & + \left(-\partial_{t} {B^{x}} - \partial_{y} {E^{z}} + \partial_{z} {E^{y}}\right){\gamma}_{t}{\gamma}_{y}{\gamma}_{z} \\
-           & + \left( \partial_{y} {B^{y}} + \partial_{z} {B^{z}} + \partial_{x} {B^{x}}\right){\gamma}_{x}{\gamma}_{y}{\gamma}_{z}
-  \end{align*}
-
-
-All Maxwell Equations are
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  \nabla F = J
-  \end{equation*}
-
-
-Div :math:`E` and Curl :math:`H` Equations
-
-.. math::
-  :nowrap:
-
-  \begin{align*}
-  <\nabla F>_1 -J & = \left(-{J^{t}} + \partial_{z} {E^{z}} + \partial_{y} {E^{y}} + \partial_{x} {E^{x}}\right){\gamma}_{t} \\
-                  & + \left(-{J^{x}} - \partial_{t} {E^{x}} + \partial_{y} {B^{z}} - \partial_{z} {B^{y}}\right){\gamma}_{x} \\
-                  & + \left( \partial_{z} {B^{x}} - \partial_{t} {E^{y}} -{J^{y}} - \partial_{x} {B^{z}}\right){\gamma}_{y} \\
-                  & + \left(-\partial_{y} {B^{x}} - \partial_{t} {E^{z}} -{J^{z}} + \partial_{x} {B^{y}}\right){\gamma}_{z}
-  \end{align*}
-
-
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-    = 0
-  \end{equation*}
-
-
-Curl :math:`E` and Div :math:`B` equations
-
-.. math::
-  :nowrap:
-
-  \begin{align*}
-  <\nabla F>_3 & = \left(-\partial_{x} {E^{y}} - \partial_{t} {B^{z}} + \partial_{y} {E^{x}}\right){\gamma}_{t}\wedge {\gamma}_{x}\wedge {\gamma}_{y} \\
-               & + \left(-\partial_{x} {E^{z}} + \partial_{t} {B^{y}} + \partial_{z} {E^{x}}\right){\gamma}_{t}\wedge {\gamma}_{x}\wedge {\gamma}_{z} \\
-               & + \left(-\partial_{t} {B^{x}} - \partial_{y} {E^{z}} + \partial_{z} {E^{y}}\right){\gamma}_{t}\wedge {\gamma}_{y}\wedge {\gamma}_{z} \\
-               & + \left( \partial_{y} {B^{y}} + \partial_{z} {B^{z}} + \partial_{x} {B^{x}}\right){\gamma}_{x}\wedge {\gamma}_{y}\wedge {\gamma}_{z}
-  \end{align*}
-
-
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-    = 0
-  \end{equation*}
-
-
-``End Program Output``
-
-
-Dirac's Equation
-^^^^^^^^^^^^^^^^
-
-The geometric algebra/calculus allows one to formulate the Dirac equation in
-real terms (no :math:`\sqrt{-1}`).  Spinors  :math:`\left(\Psi\right)` are even
-multivectors in space time (Minkowski space with signature (1,-1,-1,-1)) and the
-Dirac equation becomes
-:math:`\nabla \Psi I \sigma_{z}-eA\Psi = m\Psi\gamma_{t}`.  All the terms in the
-real equation are explined in Doran and Lasenby [Lasenby,pp281-283].
-
-``Begin Program Dirac.py``
-
-
-.. literalinclude:: Dirac.py
-
-``End Program Dirac.py``
-
-``Begin Program Output``
-
-:math:`A` is 4-vector potential
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  A =  {A^{t}}{\gamma}_{t}+ {A^{x}}{\gamma}_{x}+ {A^{y}}{\gamma}_{y}+ {A^{z}}{\gamma}_{z}
-  \end{equation*}
-
-
-:math:`\boldsymbol{\psi}` is 8-component real spinor (even multi-vector)
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  {}\boldsymbol{{\psi}} = {{\psi}}+ {{\psi}^{tx}}{\gamma}_{t}{\gamma}_{x}+ {{\psi}^{ty}}{\gamma}_{t}{\gamma}_{y}+ {{\psi}^{xy}}{\gamma}_{x}{\gamma}_{y}+ {{\psi}^{tz}}{\gamma}_{t}{\gamma}_{z}+ {{\psi}^{xz}}{\gamma}_{x}{\gamma}_{z}+ {{\psi}^{yz}}{\gamma}_{y}{\gamma}_{z}+ {{\psi}^{txyz}}{\gamma}_{t}{\gamma}_{x}{\gamma}_{y}{\gamma}_{z}
-  \end{equation*}
-
-
-Dirac equation in terms of real geometric algebra/calculus
-:math:`\left(\nabla \boldsymbol{\psi} I \sigma_{z}-eA\boldsymbol{\psi} = m\boldsymbol{\psi}\gamma_{t}\right)`
-Spin measured with respect to :math:`z` axis
-
-.. math::
-  :nowrap:
-
-  \begin{align*}
-  \nabla \boldsymbol{\psi} I \sigma_{z}-eA\boldsymbol{\psi}-m\boldsymbol{\psi}\gamma_{t}
-    & = \left(-e {A^{y}} {{\psi}^{ty}} - m {{\psi}} - \partial_{z} {{\psi}^{txyz}} - \partial_{y} {{\psi}^{tx}} - e {A^{z}} {{\psi}^{tz}} - e {A^{x}} {{\psi}^{tx}} + \partial_{x} {{\psi}^{ty}} - e {A^{t}} {{\psi}} + \partial_{t} {{\psi}^{xy}}\right){\gamma}_{t} \\
-    & + \left(-e {A^{y}} {{\psi}^{xy}} + \partial_{z} {{\psi}^{yz}} - e {A^{z}} {{\psi}^{xz}} - e {A^{t}} {{\psi}^{tx}} + m {{\psi}^{tx}} - \partial_{x} {{\psi}^{xy}} + \partial_{y} {{\psi}} - e {A^{x}} {{\psi}} - \partial_{t} {{\psi}^{ty}}\right){\gamma}_{x} \\
-    & + \left(-e {A^{y}} {{\psi}} + e {A^{x}} {{\psi}^{xy}} - e {A^{t}} {{\psi}^{ty}} - \partial_{x} {{\psi}} + \partial_{t} {{\psi}^{tx}} - \partial_{z} {{\psi}^{xz}} - e {A^{z}} {{\psi}^{yz}} + m {{\psi}^{ty}} - \partial_{y} {{\psi}^{xy}}\right){\gamma}_{y} \\
-    & + \left(-\partial_{z} {{\psi}^{xy}} + e {A^{x}} {{\psi}^{xz}} - e {A^{t}} {{\psi}^{tz}} - \partial_{x} {{\psi}^{yz}} + \partial_{y} {{\psi}^{xz}} + e {A^{y}} {{\psi}^{yz}} + \partial_{t} {{\psi}^{txyz}} + m {{\psi}^{tz}} - e {A^{z}} {{\psi}}\right){\gamma}_{z} \\
-    & + \left( \partial_{y} {{\psi}^{ty}} - e {A^{y}} {{\psi}^{tx}} + e {A^{x}} {{\psi}^{ty}} + \partial_{z} {{\psi}^{tz}} - e {A^{z}} {{\psi}^{txyz}} + \partial_{x} {{\psi}^{tx}} - e {A^{t}} {{\psi}^{xy}} - \partial_{t} {{\psi}} - m {{\psi}^{xy}}\right){\gamma}_{t}{\gamma}_{x}{\gamma}_{y} \\
-    & + \left(-\partial_{y} {{\psi}^{tz}} + \partial_{x} {{\psi}^{txyz}} - e {A^{t}} {{\psi}^{xz}} + e {A^{x}} {{\psi}^{tz}} + e {A^{y}} {{\psi}^{txyz}} + \partial_{z} {{\psi}^{ty}} - m {{\psi}^{xz}} - e {A^{z}} {{\psi}^{tx}} - \partial_{t} {{\psi}^{yz}}\right){\gamma}_{t}{\gamma}_{x}{\gamma}_{z} \\
-    & + \left(-e {A^{t}} {{\psi}^{yz}} - \partial_{z} {{\psi}^{tx}} + \partial_{x} {{\psi}^{tz}} - m {{\psi}^{yz}} + \partial_{y} {{\psi}^{txyz}} + \partial_{t} {{\psi}^{xz}} - e {A^{z}} {{\psi}^{ty}} - e {A^{x}} {{\psi}^{txyz}} + e {A^{y}} {{\psi}^{tz}}\right){\gamma}_{t}{\gamma}_{y}{\gamma}_{z} \\
-    & + \left(\partial_{z} {{\psi}} + e {A^{y}} {{\psi}^{xz}} + m {{\psi}^{txyz}} -\partial_{t} {{\psi}^{tz}} -\partial_{x} {{\psi}^{xz}} -e {A^{x}} {{\psi}^{yz}} -e {A^{t}} {{\psi}^{txyz}} -\partial_{y} {{\psi}^{yz}} -e {A^{z}} {{\psi}^{xy}}\right){\gamma}_{x}{\gamma}_{y}{\gamma}_{z}
-  \end{align*}
-
-
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-    = 0
-  \end{equation*}
-
-
-``End Program Output``
-
-
-Spherical Coordinates
-^^^^^^^^^^^^^^^^^^^^^
-
-Curvilinear coodinates are implemented as shown in section :ref:`deriv`.  The
-gradient of a scalar function and the divergence and curl of a vector function
-(:math:`-I\left(\nabla\wedge A\right)` is the curl in three dimensions in the notation of
-geometric algebra) to demonstrate the formulas derived in section :ref:`deriv`.
-
-``Begin Program coords.py``
-
-
-.. literalinclude:: coords.py
-
-``End Program coords.py``
-
-``Begin Program Output``
-
-Gradient of Scalar Function :math:`\psi`
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  \nabla\psi =  \partial_{r} {{\psi}}{}\boldsymbol{e}_{r}+\frac{1 \partial_{{\theta}} {{\psi}}}{r}{}\boldsymbol{e}_{{\theta}}+\frac{1 \partial_{{\phi}} {{\psi}}}{r \operatorname{sin}\left({\theta}\right)}{}\boldsymbol{e}_{{\phi}}
-  \end{equation*}
-
-
-Div and Curl of Vector Function :math:`A`
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  A =  {A^{r}}{}\boldsymbol{e}_{r}+ {A^{{\theta}}}{}\boldsymbol{e}_{{\theta}}+ {A^{{\phi}}}{}\boldsymbol{e}_{{\phi}}
-  \end{equation*}
-
-
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  \nabla \cdot A =  \left(\frac{{A^{{\theta}}} \operatorname{cos}\left({\theta}\right)}{r \operatorname{sin}\left({\theta}\right)} + 2 \frac{{A^{r}}}{r} + \partial_{r} {A^{r}} + \frac{\partial_{{\theta}} {A^{{\theta}}}}{r} + \frac{\partial_{{\phi}} {A^{{\phi}}}}{r \operatorname{sin}\left({\theta}\right)}\right)
-  \end{equation*}
-
-
-
-
-.. math::
-  :nowrap:
-
-  \begin{align*}
-  -I\left(\nabla \wedge A\right) & = \left(\frac{\partial_{{\theta}} {A^{{\phi}}}}{r} + \frac{{A^{{\phi}}} \operatorname{cos}\left({\theta}\right)}{r \operatorname{sin}\left({\theta}\right)} -\frac{\partial_{{\phi}} {A^{{\theta}}}}{r \operatorname{sin}\left({\theta}\right)}\right){}\boldsymbol{e}_{r} \\
-                                 & - \left(\frac{{A^{{\phi}}}}{r} + \partial_{r} {A^{{\phi}}} -\frac{\partial_{{\phi}} {A^{r}}}{r \operatorname{sin}\left({\theta}\right)}\right){}\boldsymbol{e}_{{\theta}} \\
-                                 & + \left(\frac{{A^{{\theta}}}}{r} + \partial_{r} {A^{{\theta}}} -\frac{\partial_{{\theta}} {A^{r}}}{r}\right){}\boldsymbol{e}_{{\phi}}
-  \end{align*}
-
-
-``End Program Output``
-
-
-.. seealso::
-
-   `Hestenes <http://geocalc.clas.asu.edu/html/CA_to_GC.html>`_
-      ``Clifford Algebra to Geometric Calculus`` by D.Hestenes and G. Sobczyk, Kluwer
-      Academic Publishers, 1984.
-
-   `Lasenby <http://www.mrao.cam.ac.uk/~cjld1/pages/book.htm>`_
-      ``Geometric Algebra for Physicists`` by C. Doran and A. Lasenby, Cambridge
-      University Press, 2003.
diff --git a/dev-py3k/_sources/modules/galgebra/index.txt b/dev-py3k/_sources/modules/galgebra/index.txt
deleted file mode 100644
index 0debcb88ced..00000000000
--- a/dev-py3k/_sources/modules/galgebra/index.txt
+++ /dev/null
@@ -1,15 +0,0 @@
-==================================
-Geometric Algebra Module Docstring
-==================================
-
-.. automodule: sympy.galgebra
-
-Documentation for Geometric Algebra module
-
-Contents:
-
-.. toctree::
-   :maxdepth: 2
-
-   GA.rst
-   latex_ex.rst
diff --git a/dev-py3k/_sources/modules/galgebra/latex_ex.txt b/dev-py3k/_sources/modules/galgebra/latex_ex.txt
deleted file mode 100644
index 0a096443870..00000000000
--- a/dev-py3k/_sources/modules/galgebra/latex_ex.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-========
-Latex_ex
-========
-
-.. automodule:: sympy.galgebra.latex_ex
-   :members:
diff --git a/dev-py3k/_sources/modules/galgebra/latex_ex/latex_ex.txt b/dev-py3k/_sources/modules/galgebra/latex_ex/latex_ex.txt
deleted file mode 100644
index 7c0a429ffd6..00000000000
--- a/dev-py3k/_sources/modules/galgebra/latex_ex/latex_ex.txt
+++ /dev/null
@@ -1,417 +0,0 @@
-.. _extended-latex:
-
-**********************************
-  Extended LaTeXModule for SymPy
-**********************************
-
-:Author: Alan Bromborsky
-
-.. |release| replace:: 0.10
-
-.. % .. math::
-.. % :nowrap:
-
-.. % Complete documentation on the extended LaTeX markup used for Python
-.. % documentation is available in ``Documenting Python'', which is part
-.. % of the standard documentation for Python.  It may be found online
-.. % at:
-.. %
-.. % http://www.python.org/doc/current/doc/doc.html
-
-.. % \input{macros}
-.. % This is a template for short or medium-size Python-related documents,
-.. % mostly notably the series of HOWTOs, but it can be used for any
-.. % document you like.
-.. % The title should be descriptive enough for people to be able to find
-.. % the relevant document.
-
-.. % Increment the release number whenever significant changes are made.
-.. % The author and/or editor can define 'significant' however they like.
-
-.. % At minimum, give your name and an email address.  You can include a
-.. % snail-mail address if you like.
-
-.. % This makes the Abstract go on a separate page in the HTML version;
-.. % if a copyright notice is used, it should go immediately after this.
-.. %
-.. % \ifhtml
-.. % \chapter*{Front Matter\label{front}}
-.. % \fi
-.. % Copyright statement should go here, if needed.
-.. % ...
-.. % The abstract should be a paragraph or two long, and describe the
-.. % scope of the document.
-
-.. topic:: Abstract
-
-   This document describes the extension of the latex module for  :mod:`sympy`. The
-   python module :mod:`latex_ex` extends the capabilities of the current
-   :mod:`latex` (while preserving the current capabilities) to geometric algebra
-   multivectors, :mod:`numpy` array's, and extends the ascii formatting of greek
-   symbols, accents, and subscripts and superscripts.  Additionally the module is
-   configured to use the print command to generate a LaTeX output file and display
-   it using xdvi in Linux and yap in Windows.  To get LaTeX displayed text latex and
-   xdvi must be installed on a Linux system and MikTex on a Windows system.
-
-
-Extended Symbol Coding
-======================
-
-One of the main extensions in :mod:`latex_ex` is the ability to encode complex
-symbols (multiple greek letters with accents and superscripts and subscripts) is
-ascii strings containing only letters, numbers, and underscores.  These
-restrictions allow :mod:`sympy` variable names to represent complex symbols. For
-example if we use the function :func:`symbols` as
-follows::
-
-   xalpha,Gammavec__1_rho,delta__j_k = symbols('xalpha Gammavec__1_rho delta__j_k')
-
-``symbols`` creates the three :mod:`sympy` symbols *xalpha*,
-*Gammavec__1_rho*, and *delta__j_k*.  If these symbols are printed with the
-:mod:`latex_ex` modules the results are
-
-
-.. csv-table::
-	:header: Ascii String, LaTeX Output
-	:widths: 15, 15
-
-	``xalpha``,            :math:`x\alpha`
-	``Gammavec__1_rho``,   :math:`\vec{\Gamma}^{1}_{\rho}`
-	``delta__j_k``,        :math:`\delta^{j}_{k}`
-
-
-A single underscore denotes a subscript and a double undscore a superscript.
-
-In addition to all normal LaTeX accents boldmath is supported so that
-``omegaomegabm``:math:`\rightarrow \Omega\boldsymbol{\Omega}` so an accent (or boldmath)
-only applies to the character (characters in the case of the string representing
-a greek letter) immediately preceeding the accent command.
-
-
-How LatexPrinter Works
-======================
-
-The actual :class:`LatexPrinter` class is hidden with the helper functions
-:func:`Format`,  :func:`LaTeX`, and :func:`xdvi`.  :class:`LatexPrinter` is
-setup when :func:`Format` is called.  In addition to setting format switches in
-:class:`LatexPrinter`, :func:`Format` does two other critical tasks. Firstly,
-the :mod:`sympy` function :func:`Basic.__str__` is redirected to the
-:class:`LatexPrinter` helper function :func:`LaTeX`.  If nothing more that this
-were done the python print command would output LaTeX code.  Secondly,
-*sys.stdout* is redirected to a file until :func:`xdvi` is called.  This file
-is then compiled with the :program:`latex` program (if present) and the dvi
-output file is displayed with the :program:`xdvi` program (if present).  Thus
-for :class:`LatexPrinter` to display the output in a window both
-:program:`latex` and :program:`xdvi` must be installed on your system and in the
-execution path.
-
-One problem for :class:`LatexPrinter` is determining when equation mode should be
-use in the output formatting.  To allow for printing output not in equation mode the
-program attemps to determine from the output string context when to use equation mode.
-The current test is to use equation mode if the string contains an =, \_, \^, or \\.
-This is not a foolproof method.  The safest thing to do if you wish to print an object, *X*,
-in math mode is to use *print 'X =',X* so the = sign triggers math mode.
-
-
-LatexPrinter Functions
-======================
-
-
-LatexPrinter Class Functions for Extending LatexPrinter Class
--------------------------------------------------------------
-
-The :class:`LatexPrinter` class functions are not called directly, but rather
-are called when :func:`print`, :func:`LaTeX`, or :func:`str` are called.  The
-two new functions for extending the :mod:`latex` module are
-:func:`_print_ndarray` and :func:`_print_MV`.  Some other functions in
-:class:`LatexPrinter` have been modified to increase their utility.
-
-
-.. function:: _print_ndarray(self,expr)
-
-   :func:`_print_ndarray` returns a latex formatted string for the *expr* equal to
-   a :class:`numpy` array with elements that can be :class:`sympy` expressions.
-   The :class:`numpy` array's can have up to three dimensions.
-
-
-.. function:: _print_MV(self,expr)
-
-   :func:`_print_MV` returns a latex formatted string for the *expr* equal to a
-   :class:`GA` multivector.
-
-
-.. function:: str_basic(in_str)
-
-   :func:`str_basic` returns a string without the latex formatting provided by
-   :class:`LatexPrinter`.  This is needed since :class:`LatexPrinter` takes over
-   the :func:`str` fuction and there are instances when the unformatted string is
-   needed such as during the automatic generation of multivector coefficients and
-   the reduction of multivector coefficients for printing.
-
-
-Helper Functions for Extending LatexPrinter Class
--------------------------------------------------
-
-
-.. function:: Format(fmt='1 1 1 1')
-
-   ``Format`` iniailizes ``LatexPrinter`` and set the format for ``sympy`` symbols,
-   functions, and derivatives and for ``GA`` multivectors. The switch are encoded
-   in the text string argument of ``Format`` as follows.  It is assumed that the
-   text string ``fmt`` always contains four integers separated by blanks.
-
-   .. csv-table::
-        :delim: ;
-	:header: Position, Switch, Values
-	:widths: 4, 6, 40
-
-	:math:`1^{st}`;   symbol;               0: Use symbol encoding in ``latex.py``
-	; ;                                     1: Use extended symbol encoding in ``latex_ex.py``
-	:math:`2^{nd}`;   function;             0: Use symbol encoding in  ``latex.py``. Print functions args, use ``\operator{ }`` format.
-	; ;                                     1: Do not print function args. Do not use ``\operator{}`` format. Suppress printing of function arguments.
-	:math:`3^{rd}`;   partial derivative;   0: Use partial derivative format in ``latex.py``.
-	; ;                                     1: Use format :math:`\partial_{x}` instead of :math:`\partial/\partial x`.
-	:math:`4^{th}`;   multivector;          1: Print entire multivector on one line.
-	; ;                                     2: Print each grade of multivector on one line.
-	; ;                                     3: Print each base of multivector on one line.
-
-
-.. function:: LaTeX(expr, inline=True)
-
-   :func:`LaTeX` returns the latex formatted string for the :class:`sympy`,
-   :class:`GA`, or :class:`numpy`  expression *expr*.  This is needed since
-   :class:`numpy` cannot be subclassed and hence cannot be used with the
-   :class:`LatexPrinter` modified :func:`print` command. Thus is *A* is a
-   :class:`numpy` array containing :class:`sympy` expressions one cannot simply
-   code  ::
-
-      print A
-
-   but rather must use  ::
-
-      print LaTeX(A)
-
-
-.. function:: xdvi(filename='tmplatex.tex',debug=False)
-
-   :func:`xdvi` postprocesses the output of the print statements and generates the
-   latex file with name *filename*.  If the :program:`latex` and :program:`xdvi`
-   programs are present on the system they are invoked to display the latex file in
-   a window.  If *debug=True* the associated output of :program:`latex` is sent to
-   *stdout*, otherwise it is sent to  */dev/null* for linux and *NUL* for Windows.
-   If :class:`LatexPrinter` has not been initialized :func:`xdvi` does nothing.  After the
-   .dvi file is generated it is displayed with :program:`xdvi` for linux (if latex and xdvi
-   are installed ) and :program:`yap` for Windows (if MikTex is installed).
-
-The functions :func:`sym_format`, :func:`fct_format`, :func:`pdiff_format`, and
-:func:`MV_format` allow one to change various formatting aspects of the
-:class:`LatexPrinter`.  They do not initialize the class and if they are called
-with the class not initialized they have no effect.  These functions and the
-function :func:`xdvi` are designed so that if the :class:`LatexPrinter` class is
-not initialized the program output is as if the :class:`LatexPrinter` class is
-not used. Thus all one needs to do to get simple ascii output (possibly for
-program debugging) is to comment out the one function call that initializes the
-:class:`LatexPrinter` class.  All other :mod:`latex_ex` function calls can
-remain in the program and have no effect on program output.
-
-
-.. function:: sym_format(sym_fmt)
-
-   :func:`sym_format` allows one to change the latex format options for
-   :class:`sympy` symbol output independent of other format switches (see
-   :math:`1^{st}` switch in Table I).
-
-
-.. function:: fct_format(fct_fmt)
-
-   :func:`fct_format` allows one to change the latex format options for
-   :class:`sympy` function output independent of other format switches (see
-   :math:`2^{nd}` switch in Table I).
-
-
-.. function:: pdiff_format(pdiff_fmt)
-
-   :func:`pdiff_format` allows one to change the latex format options for
-   :class:`sympy` partial derivative output independent of other format switches
-   (see :math:`3^{rd}` switch in Table I).
-
-
-.. function:: MV_format(mv_fmt)
-
-   :func:`MV_format` allows one to change the latex format options for
-   :class:`sympy` partial derivative output independent of other format switches
-   (see :math:`3^{rd}` switch in Table I).
-
-
-Examples
-========
-
-
-:program:`latexdemo.py` a simple example
-----------------------------------------
-
-:program:`latexdemo.py` example of using :mod:`latex_ex` with :mod:`sympy`
-
-
-.. include:: latexdemo.py
-   :literal:
-
-Start of Program Output
-
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  x = \frac{{\alpha}_{1} {}\boldsymbol{x}}{{\delta}^{{\nu}{\gamma}}_{r}}
-  \end{equation*}
-
-
-End of Program Output
-
-The program :program:`latexdemo.py` demonstrates the extended symbol naming
-conventions in :mod:`latex_ex`.   the statment ``Format()`` starts the
-:class:`LatexPrinter` driver with default formatting. Note that on the right
-hand side of the output that *xbm* gives :math:`\boldsymbol{x}`, *alpha_1* gives
-:math:`\alpha_{1}` and  *delta__nugamma_r* gives :math:`\delta^{\nu\gamma}_{r}`.
-Also the fraction is printed correctly.  The statment ``print 'x =',x`` sends
-the string ``'x = '+str(x)`` to the output processor (:func:`xdvi`).  Because
-the string contains an :math:`=` sign the processor treats the string as an
-LaTeX equation (unnumbered).  If ``'x ='``  was not in the print statment a
-LaTeX error would be generated.  In the case of a :class:`GA` multivector one
-does not need the ``'x ='`` if the multivector has been given a name.
-
-
-:program:`Maxwell.py` a multivector example
--------------------------------------------
-
-:program:`Maxwell.py` example of using :mod:`latex_ex` with :mod:`GA`
-
-
-.. include:: Maxwell.py
-   :literal:
-
-Start of Program Output
-
-:math:`I` Pseudo-Scalar
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  I = {\gamma}_{t}{\gamma}_{x}{\gamma}_{y}{\gamma}_{z}
-  \end{equation*}
-
-
-:math:`B` Magnetic Field Bi-Vector
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  B = - {B^{x}}{\gamma}_{t}{\gamma}_{x}- {B^{y}}{\gamma}_{t}{\gamma}_{y}- {B^{z}}{\gamma}_{t}{\gamma}_{z}
-  \end{equation*}
-
-
-:math:`F` Electric Field Bi-Vector
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  E = - {E^{x}}{\gamma}_{t}{\gamma}_{x}- {E^{y}}{\gamma}_{t}{\gamma}_{y}- {E^{z}}{\gamma}_{t}{\gamma}_{z}
-  \end{equation*}
-
-
-:math:`E+IB` Electo-Magnetic Field Bi-Vector
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  F = - {E^{x}}{\gamma}_{t}{\gamma}_{x}- {E^{y}}{\gamma}_{t}{\gamma}_{y}- {B^{z}}{\gamma}_{x}{\gamma}_{y}- {E^{z}}{\gamma}_{t}{\gamma}_{z}+ {B^{y}}{\gamma}_{x}{\gamma}_{z}- {B^{x}}{\gamma}_{y}{\gamma}_{z}
-  \end{equation*}
-
-
-:math:`J` Four Current
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  J =  {J^{t}}{\gamma}_{t}+ {J^{x}}{\gamma}_{x}+ {J^{y}}{\gamma}_{y}+ {J^{z}}{\gamma}_{z}
-  \end{equation*}
-
-
-Geometric Derivative of Electo-Magnetic Field Bi-Vector
-
-.. math::
-  :nowrap:
-
-  \begin{align*}
-  \nabla F & =   \left(\partial_{z} {E^{z}} + \partial_{y} {E^{y}} + \partial_{x} {E^{x}}\right){\gamma}_{t} \\ & + \left(-\partial_{t} {E^{x}} + \partial_{y} {B^{z}} -\partial_{z} {B^{y}}\right){\gamma}_{x} \\ & + \left(\partial_{z} {B^{x}} -\partial_{t} {E^{y}} -\partial_{x} {B^{z}}\right){\gamma}_{y} \\ & + \left(-\partial_{y} {B^{x}} -\partial_{t} {E^{z}} + \partial_{x} {B^{y}}\right){\gamma}_{z} \\ & + \left(-\partial_{x} {E^{y}} -\partial_{t} {B^{z}} + \partial_{y} {E^{x}}\right){\gamma}_{t}{\gamma}_{x}{\gamma}_{y} \\ & + \left(-\partial_{x} {E^{z}} + \partial_{t} {B^{y}} + \partial_{z} {E^{x}}\right){\gamma}_{t}{\gamma}_{x}{\gamma}_{z} \\ & + \left(-\partial_{t} {B^{x}} -\partial_{y} {E^{z}} + \partial_{z} {E^{y}}\right){\gamma}_{t}{\gamma}_{y}{\gamma}_{z} \\ & + \left(\partial_{y} {B^{y}} + \partial_{z} {B^{z}} + \partial_{x} {B^{x}}\right){\gamma}_{x}{\gamma}_{y}{\gamma}_{z}\end{align*}
-
-
-All Maxwell Equations are
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-  \nabla F = J
-  \end{equation*}
-
-
-Div :math:`E` and Curl :math:`H` Equations
-
-.. math::
-  :nowrap:
-
-  \begin{align*}
-  <\nabla F>_1 -J & =   \left(-{J^{t}} + \partial_{z} {E^{z}} + \partial_{y} {E^{y}} + \partial_{x} {E^{x}}\right){\gamma}_{t} \\ & + \left(-{J^{x}} -\partial_{t} {E^{x}} + \partial_{y} {B^{z}} -\partial_{z} {B^{y}}\right){\gamma}_{x} \\ & + \left(\partial_{z} {B^{x}} -\partial_{t} {E^{y}} -{J^{y}} -\partial_{x} {B^{z}}\right){\gamma}_{y} \\ & + \left(-\partial_{y} {B^{x}} -\partial_{t} {E^{z}} -{J^{z}} + \partial_{x} {B^{y}}\right){\gamma}_{z}\end{align*}
-
-
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-    = 0
-  \end{equation*}
-
-
-Curl :math:`E` and Div :math:`B` equations
-
-.. math::
-  :nowrap:
-
-  \begin{align*}
-  <\nabla F>_3 & =   \left( -\partial_{x} {E^{y}} -\partial_{t} {B^{z}} + \partial_{y} {E^{x}}\right){\gamma}_{t}\wedge {\gamma}_{x}\wedge {\gamma}_{y} \\ & + \left( -\partial_{x} {E^{z}} + \partial_{t} {B^{y}} + \partial_{z} {E^{x}}\right){\gamma}_{t}\wedge {\gamma}_{x}\wedge {\gamma}_{z} \\ & + \left( -\partial_{t} {B^{x}} -\partial_{y} {E^{z}} + \partial_{z} {E^{y}}\right){\gamma}_{t}\wedge {\gamma}_{y}\wedge {\gamma}_{z} \\ & + \left( \partial_{y} {B^{y}} + \partial_{z} {B^{z}} + \partial_{x} {B^{x}}\right){\gamma}_{x}\wedge {\gamma}_{y}\wedge {\gamma}_{z}\end{align*}
-
-
-
-
-.. math::
-  :nowrap:
-
-  \begin{equation*}
-    = 0
-  \end{equation*}
-
-
-End of Program Output
-
-The program :program:`Maxwell.py` demonstrates the use of the
-:class:`LatexPrinter` class with the :mod:`GA` module multivector class,
-:class:`MV`.  The :func:`Format` call initializes :class:`LatexPrinter`.  The
-only other explicit :mod:`latex_x`  module formatting statement used is
-``MV_format(3)``.  This statment changes the multivector latex format so that
-instead of printing the entire multivector on one line, which would run off the
-page, each multivector base and its coefficient are printed on individual lines
-using the latex align environment.  Another option used is that the printing of
-function arguments is suppressed since :math:`E`, :math:`B`, :math:`J`, and
-:math:`F` are multivector fields and printing out the argument,
-:math:`(t,x,y,z)`, for every field component would greatly lengthen the output
-and make it more difficult to format in a pleasing way.
diff --git a/dev-py3k/_sources/modules/geometry.txt b/dev-py3k/_sources/modules/geometry.txt
deleted file mode 100644
index 4c89fd049dd..00000000000
--- a/dev-py3k/_sources/modules/geometry.txt
+++ /dev/null
@@ -1,268 +0,0 @@
-Geometry Module
-===============
-
-.. module:: sympy.geometry
-
-Introduction
-------------
-
-The geometry module for SymPy allows one to create two-dimensional geometrical
-entities, such as lines and circles, and query information about these
-entities. This could include asking the area of an ellipse, checking for
-collinearity of a set of points, or finding the intersection between two lines.
-The primary use case of the module involves entities with numerical values, but
-it is possible to also use symbolic representations.
-
-Available Entities
-------------------
-
-The following entities are currently available in the geometry module:
-
-* Point
-* Line, Ray, Segment
-* Ellipse, Circle
-* Polygon, RegularPolygon, Triangle
-
-Most of the work one will do will be through the properties and methods of
-these entities, but several global methods exist for one's usage:
-
-* intersection(entity1, entity2)
-* are_similar(entity1, entity2)
-* convex_hull(points)
-
-For a full API listing and an explanation of the methods and their return
-values please see the list of classes at the end of this document.
-
-Example Usage
--------------
-
-The following Python session gives one an idea of how to work with some of the
-geometry module.
-
-    >>> from sympy import *
-    >>> from sympy.geometry import *
-    >>> x = Point(0, 0)
-    >>> y = Point(1, 1)
-    >>> z = Point(2, 2)
-    >>> zp = Point(1, 0)
-    >>> Point.is_collinear(x, y, z)
-    True
-    >>> Point.is_collinear(x, y, zp)
-    False
-    >>> t = Triangle(zp, y, x)
-    >>> t.area
-    1/2
-    >>> t.medians[x]
-    Segment(Point(0, 0), Point(1, 1/2))
-    >>> Segment(Point(1, S(1)/2), Point(0, 0))
-    Segment(Point(0, 0), Point(1, 1/2))
-    >>> m = t.medians
-    >>> intersection(m[x], m[y], m[zp])
-    [Point(2/3, 1/3)]
-    >>> c = Circle(x, 5)
-    >>> l = Line(Point(5, -5), Point(5, 5))
-    >>> c.is_tangent(l) # is l tangent to c?
-    True
-    >>> l = Line(x, y)
-    >>> c.is_tangent(l) # is l tangent to c?
-    False
-    >>> intersection(c, l)
-    [Point(-5*sqrt(2)/2, -5*sqrt(2)/2), Point(5*sqrt(2)/2, 5*sqrt(2)/2)]
-
-Intersection of medians
------------------------
-::
-
-    >>> from sympy import symbols
-    >>> from sympy.geometry import Point, Triangle, intersection
-
-    >>> a, b = symbols("a,b", positive=True)
-
-    >>> x = Point(0, 0)
-    >>> y = Point(a, 0)
-    >>> z = Point(2*a, b)
-    >>> t = Triangle(x, y, z)
-
-    >>> t.area
-    a*b/2
-
-    >>> t.medians[x]
-    Segment(Point(0, 0), Point(3*a/2, b/2))
-
-    >>> intersection(t.medians[x], t.medians[y], t.medians[z])
-    [Point(a, b/3)]
-
-An in-depth example: Pappus' Theorem
-------------------------------------
-::
-
-    >>> from sympy import *
-    >>> from sympy.geometry import *
-    >>>
-    >>> l1 = Line(Point(0, 0), Point(5, 6))
-    >>> l2 = Line(Point(0, 0), Point(2, -2))
-    >>>
-    >>> def subs_point(l, val):
-    ...    """Take an arbitrary point and make it a fixed point."""
-    ...    t = Symbol('t', real=True)
-    ...    ap = l.arbitrary_point()
-    ...    return Point(ap.x.subs(t, val), ap.y.subs(t, val))
-    ...
-    >>> p11 = subs_point(l1, 5)
-    >>> p12 = subs_point(l1, 6)
-    >>> p13 = subs_point(l1, 11)
-    >>>
-    >>> p21 = subs_point(l2, -1)
-    >>> p22 = subs_point(l2, 2)
-    >>> p23 = subs_point(l2, 13)
-    >>>
-    >>> ll1 = Line(p11, p22)
-    >>> ll2 = Line(p11, p23)
-    >>> ll3 = Line(p12, p21)
-    >>> ll4 = Line(p12, p23)
-    >>> ll5 = Line(p13, p21)
-    >>> ll6 = Line(p13, p22)
-    >>>
-    >>> pp1 = intersection(ll1, ll3)[0]
-    >>> pp2 = intersection(ll2, ll5)[0]
-    >>> pp3 = intersection(ll4, ll6)[0]
-    >>>
-    >>> Point.is_collinear(pp1, pp2, pp3)
-    True
-
-Miscellaneous Notes
--------------------
-
-* The area property of Polygon and Triangle may return a positive or
-  negative value, depending on whether or not the points are oriented
-  counter-clockwise or clockwise, respectively. If you always want a
-  positive value be sure to use the ``abs`` function.
-* Although Polygon can refer to any type of polygon, the code has been
-  written for simple polygons. Hence, expect potential problems if dealing
-  with complex polygons (overlapping sides).
-* Since !SymPy is still in its infancy some things may not simplify
-  properly and hence some things that should return True (e.g.,
-  Point.is_collinear) may not actually do so. Similarly, attempting to find
-  the intersection of entities that do intersect may result in an empty
-  result.
-
-Future Work
------------
-
-Truth Setting Expressions
-~~~~~~~~~~~~~~~~~~~~~~~~~
-
-When one deals with symbolic entities, it often happens that an assertion
-cannot be guaranteed. For example, consider the following code:
-
-    >>> from sympy import *
-    >>> from sympy.geometry import *
-    >>> x,y,z = list(map(Symbol, 'xyz'))
-    >>> p1,p2,p3 = Point(x, y), Point(y, z), Point(2*x*y, y)
-    >>> Point.is_collinear(p1, p2, p3)
-    False
-
-Even though the result is currently ``False``, this is not *always* true. If the
-quantity `z - y - 2*y*z + 2*y**2 == 0` then the points will be collinear. It
-would be really nice to inform the user of this because such a quantity may be
-useful to a user for further calculation and, at the very least, being nice to
-know. This could be potentially done by returning an object (e.g.,
-GeometryResult) that the user could use. This actually would not involve an
-extensive amount of work.
-
-Three Dimensions and Beyond
-~~~~~~~~~~~~~~~~~~~~~~~~~~~
-
-Currently there are no plans for extending the module to three dimensions, but
-it certainly would be a good addition. This would probably involve a fair
-amount of work since many of the algorithms used are specific to two
-dimensions.
-
-Geometry Visualization
-~~~~~~~~~~~~~~~~~~~~~~
-
-The plotting module is capable of plotting geometric entities. See
-`Plotting Geometric Entities <https://github.com/sympy/sympy/wiki/Plotting-Module>`_ in
-the plotting module entry.
-
-API Reference
--------------
-
-Entities
-~~~~~~~~
-
-.. module:: sympy.geometry.entity
-
-.. autoclass:: sympy.geometry.entity.GeometryEntity
-   :members:
-
-Utils
-~~~~~
-
-.. module:: sympy.geometry.util
-
-.. autofunction:: intersection
-
-.. autofunction:: convex_hull
-
-.. autofunction:: are_similar
-
-.. autofunction:: centroid
-
-Points
-~~~~~~
-
-.. module:: sympy.geometry.point
-
-.. autoclass:: Point
-   :members:
-
-Lines
-~~~~~
-
-.. module:: sympy.geometry.line
-
-.. autoclass:: LinearEntity
-   :members:
-
-.. autoclass:: Line
-   :members:
-
-.. autoclass:: Ray
-   :members:
-
-.. autoclass:: Segment
-   :members:
-
-Curves
-~~~~~~
-
-.. module:: sympy.geometry.curve
-
-.. autoclass:: Curve
-   :members:
-
-Ellipses
-~~~~~~~~
-
-.. module:: sympy.geometry.ellipse
-
-.. autoclass:: Ellipse
-   :members:
-
-.. autoclass:: Circle
-   :members:
-
-Polygons
-~~~~~~~~
-
-.. module:: sympy.geometry.polygon
-
-.. autoclass:: Polygon
-   :members:
-
-.. autoclass:: RegularPolygon
-   :members:
-
-.. autoclass:: Triangle
-   :members:
diff --git a/dev-py3k/_sources/modules/index.txt b/dev-py3k/_sources/modules/index.txt
deleted file mode 100644
index 5d5396db2bb..00000000000
--- a/dev-py3k/_sources/modules/index.txt
+++ /dev/null
@@ -1,59 +0,0 @@
-.. _module-docs:
-
-SymPy Modules Reference
-=======================
-
-Because every feature of SymPy must have a test case, when you are not sure how
-to use something, just look into the ``tests/`` directories, find that feature
-and read the tests for it, that will tell you everything you need to know.
-
-Most of the things are already documented though in this document, that is
-automatically generated using SymPy's docstrings.
-
-Click the  "modules" (:ref:`modindex`) link in the top right corner to easily
-access any SymPy module, or use this contens:
-
-.. toctree::
-   :maxdepth: 2
-
-   core.rst
-   combinatorics/index.rst
-   ntheory.rst
-   concrete.rst
-   evalf.rst
-   functions/index.rst
-   geometry.rst
-   galgebra/index.rst
-   galgebra/GA/GAsympy.rst
-   galgebra/latex_ex/latex_ex.rst
-   integrals/integrals.rst
-   logic.rst
-   matrices/index.rst
-   mpmath/index.rst
-   polys/index.rst
-   printing.rst
-   plotting.rst
-   assumptions/index.rst
-   rewriting.rst
-   series.rst
-   sets.rst
-   simplify/simplify.rst
-   simplify/hyperexpand.rst
-   statistics.rst
-   stats.rst
-   solvers/ode.rst
-   solvers/solvers.rst
-   tensor/index.rst
-   utilities/index.rst
-   parsing.rst
-   physics/index.rst
-   categories.rst
-   diffgeom.rst
-
-Contributions to docs
----------------------
-
-All contributions are welcome. If you'd like to improve something, look into
-the sources if they contain the information you need (if not, please fix them),
-otherwise the documentation generation needs to be improved (look in the
-``doc/`` directory).
diff --git a/dev-py3k/_sources/modules/integrals/g-functions.txt b/dev-py3k/_sources/modules/integrals/g-functions.txt
deleted file mode 100644
index 999af68a21a..00000000000
--- a/dev-py3k/_sources/modules/integrals/g-functions.txt
+++ /dev/null
@@ -1,508 +0,0 @@
-Computing Integrals using Meijer G-Functions
-********************************************
-
-This text aims do describe in some detail the steps (and subtleties) involved
-in using Meijer G-functions for computing definite and indefinite integrals.
-We shall ignore proofs completely.
-
-Overview
-========
-
-The algorithm to compute `\int f(x) \mathrm{d}x` or
-`\int_0^\infty f(x) \mathrm{d}x` generally consists of three steps:
-
-1. Rewrite the integrand using Meijer G-functions (one or sometimes two).
-2. Apply an integration theorem, to get the answer (usually expressed as another
-   G-function).
-3. Expand the result in named special functions.
-
-Step (3) is implemented in the function hyperexpand (q.v.). Steps (1) and (2)
-are described below. Morever, G-functions are usually branched. Thus our treatment
-of branched functions is described first.
-
-Some other integrals (e.g. `\int_{-\infty}^\infty`) can also be computed by first
-recasting them into one of the above forms. There is a lot of choice involved
-here, and the algorithm is heuristic at best.
-
-Polar Numbers and Branched Functions
-====================================
-
-Both Meijer G-Functions and Hypergeometric functions are typically branched
-(possible branchpoints being `0`, `\pm 1`, `\infty`). This is not very important
-when e.g. expanding a single hypergeometric function into named special functions,
-since sorting out the branches can be left to the human user. However this
-algorithm manipulates and transforms G-functions, and to do this correctly it needs
-at least some crude understanding of the branchings involved.
-
-To begin, we consider the set
-`\mathcal{S} = \{(r, \theta) : r > 0, \theta \in \mathbb{R}\}`. We have a map
-`p: \mathcal{S}: \rightarrow \mathbb{C}-\{0\}, (r, \theta) \mapsto r e^{i \theta}`.
-Decreeing this to be a local biholomorphism gives `\mathcal{S}` both a topology
-and a complex structure. This Riemann Surface is usually referred to as the
-Riemann Surface of the logarithm, for the following reason:
-We can define maps
-`Exp: \mathbb{C} \rightarrow \mathcal{S}, Exp(x + i y) = (exp(x), y)` and
-`Log: \mathcal{S} \rightarrow \mathbb{C}, (e^x, y) \mapsto x + iy`.
-These can be shown to be both holomorphic, and are indeed mutual inverses.
-
-We also sometimes formally attach a point "zero" to `\mathcal{S}` and denote the
-resulting object `\mathcal{S}_0`. Notably there is no complex structure
-defined near `0`. A fundamental system of neighbourhoods is given by
-`\{Exp(z) : Re(z) < k\}`, this at least defines a topology. Elements of
-`\mathcal{S}_0` shall be called polar numbers.
-We further define functions
-`Arg: \mathcal{S} \rightarrow \mathbb{R}, (r, \theta) \mapsto \theta` and
-`|.|: \mathcal{S}_0 \rightarrow \mathbb{R}_{>0}, (r, \theta) \mapsto r`.
-These have evident meaning and are both continuous everywhere.
-
-Using these maps many operations can be extended from `\mathbb{C}` to
-`\mathcal{S}`. We define `Exp(a) Exp(b) = Exp(a + b)` for `a, b \in \mathbb{C}`,
-also for `a \in \mathcal{S}` and `b \in \mathbb{C}` we define
-`a^b = Exp(b Log(a))`.
-It can be checked easily that using these definitions, many algebraic properties
-holding for positive reals (e.g. `(ab)^c = a^c b^c`) which hold in `\mathbb{C}`
-only for some numbers (because of branch cuts) hold indeed for all polar numbers.
-
-As one peculiarity it should be mentioned that addition of polar numbers is not
-usually defined. However, formal sums of polar numbers can be used to express
-branching behaviour. For example, consider the functions `F(z) = \sqrt{1 + z}`
-and `G(a, b) = \sqrt{a + b}`, where `a, b, z` are polar numbers.
-The general rule is that functions of a single polar variable are defined in
-such a way that they are continuous on circles, and agree with the usual
-definition for positive reals. Thus if `S(z)` denotes the standard branch of
-the square root function on `\mathbb{C}`, we are forced to define
-
-.. math:: F(z) = \begin{cases}
-    S(p(z)) &: |z| < 1 \\
-    S(p(z)) &: -\pi < Arg(z) + 4\pi n \le \pi \text{ for some } n \in \mathbb{Z} \\
-    -S(p(z)) &: \text{else}
-   \end{cases}.
-
-(We are omitting `|z| = 1` here, this does not matter for integration.)
-Finally we define `G(a, b) = \sqrt{a}F(b/a)`.
-
-Representing Branched Functions on the Argand Plane
-===================================================
-
-Suppose `f: \mathcal{S} \to \mathbb{C}` is a holomorphic function. We wish to
-define a function `F` on (part of) the complex numbers `\mathbb{C}` that
-represents `f` as closely as possible. This process is knows as "introducing
-branch cuts". In our situation, there is actually a canonical way of doing this
-(which is adhered to in all of SymPy), as follows: Introduce the "cut complex
-plane"
-`C = \mathbb{C} \setminus \mathbb{R}_{\le 0}`. Define a function
-`l: C \to \mathcal{S}` via `re^{i\theta} \mapsto r Exp(i\theta)`. Here `r > 0`
-and `-\pi < \theta \le \pi`. Then `l` is holomorphic, and we define
-`G = f \circ l`. This called "lifting to the principal branch" throughout the
-SymPy documentation.
-
-Table Lookups and Inverse Mellin Transforms
-===========================================
-
-Suppose we are given an integrand `f(x)` and are trying to rewrite it as a
-single G-function. To do this, we first split `f(x)` into the form `x^s g(x)`
-(where `g(x)` is supposed to be simpler than `f(x)`). This is because multiplicative
-powers can be absorbed into the G-function later. This splitting is done by
-``_split_mul(f, x)``. Then we assemble a tuple of functions that occur in
-`f` (e.g. if `f(x) = e^x \cos{x}`, we would assemble the tuple `(\cos, \exp)`).
-This is done by the function ``_mytype(f, x)``. Next we index a lookup table
-(created using ``_create_lookup_table()``) with this tuple. This (hopefully)
-yields a list of Meijer G-function formulae involving these functions, we then
-pattern-match all of them. If one fits, we were successful, otherwise not and we
-have to try something else.
-
-Suppose now we want to rewrite as a product of two G-functions. To do this,
-we (try to) find all inequivalent ways of splitting `f(x)` into a product
-`f_1(x) f_2(x)`.
-We could try these splittings in any order, but it is often a good idea to
-minimise (a) the number of powers occuring in `f_i(x)` and (b) the number of
-different functions occuring in `f_i(x)`. Thus given e.g.
-`f(x) = \sin{x}\, e^{x} \sin{2x}` we should try `f_1(x) = \sin{x}\, \sin{2x}`,
-`f_2(x) = e^{x}` first.
-All of this is done by the function ``_mul_as_two_parts(f)``.
-
-Finally, we can try a recursive Mellin transform technique. Since the Meijer
-G-function is defined essentially as a certain inverse mellin transform,
-if we want to write a function `f(x)` as a G-function, we can compute its mellin
-transform `F(s)`. If `F(s)` is in the right form, the G-function expression
-can be read off. This technique generalises many standard rewritings, e.g.
-`e^{ax} e^{bx} = e^{(a + b) x}`.
-
-One twist is that some functions don't have mellin transforms, even though they
-can be written as G-functions. This is true for example for `f(x) = e^x \sin{x}`
-(the function grows too rapidly to have a mellin transform). However if the function
-is recognised to be analytic, then we can try to compute the mellin-transform of
-`f(ax)` for a parameter `a`, and deduce the G-function expression by analytic
-continuation. (Checking for analyticity is easy. Since we can only deal with a
-certain subset of functions anyway, we only have to filter out those which are
-not analyitc.)
-
-The function ``_rewrite_single`` does the table lookup and recursive mellin
-transform. The functions ``_rewrite1`` and ``_rewrite2`` respectively use
-above-mentioned helpers and ``_rewrite_single`` to rewrite their argument as
-respectively one or two G-functions.
-
-Applying the Integral Theorems
-==============================
-
-If the integrand has been recast into G-functions, evaluating the integral is
-relatively easy. We first do some substitutions to reduce e.g. the exponent
-of the argument of the G-function to unity (see ``_rewrite_saxena_1`` and
-``_rewrite_saxena``, respectively, for one or two G-functions). Next we go through
-a list of conditions under which the integral theorem applies. It can fail for
-basically two reasons: either the integral does not exist, or the manipulations
-in deriving the theorem may not be allowed (for more details, see this [BlogPost]_).
-
-Sometimes this can be remedied by reducing the argument of the G-functions
-involved. For example it is clear that the G-function representing `e^z`
-is satisfies `G(Exp(2 \pi i)z) = G(z)` for all `z \in \mathcal{S}`. The function
-``meijerg.get_period()`` can be used to discover this, and the function
-``principal_branch(z, period)`` in ``functions/elementary/complexes.py`` can
-be used to exploit the information. This is done transparently by the
-integration code.
-
-.. [BlogPost] http://nessgrh.wordpress.com/2011/07/07/tricky-branch-cuts/
-
-The G-Function Integration Theorems
-***********************************
-
-This section intends to display in detail the definite integration theorems
-used in the code. The following two formulae go back to Meijer (In fact he
-proved more general formulae; indeed in the literature formulae are usually
-staded in more general form. However it is very easy to deduce the general
-formulae from the ones we give here. It seemed best to keep the theorems as
-simple as possible, since they are very complicated anyway.):
-
-1. .. math:: \int_0^\infty
-    G_{p, q}^{m, n} \left.\left(\begin{matrix} a_1, \dots, a_p \\
-                                               b_1, \dots, b_q \end{matrix}
-            \right| \eta x \right) \mathrm{d}x =
-     \frac{\prod_{j=1}^m \Gamma(b_j + 1) \prod_{j=1}^n \Gamma(-a_j)}{\eta
-           \prod_{j=m+1}^q \Gamma(-b_j) \prod_{j=n+1}^p \Gamma(a_j + 1)}
-
-2. .. math:: \int_0^\infty
-    G_{u, v}^{s, t} \left.\left(\begin{matrix} c_1, \dots, c_u \\
-                                               d_1, \dots, d_v \end{matrix}
-            \right| \sigma x \right)
-    G_{p, q}^{m, n} \left.\left(\begin{matrix} a_1, \dots, a_p \\
-                                               b_1, \dots, b_q \end{matrix}
-            \right| \omega x \right)
-    \mathrm{d}x =
-    G_{v+p, u+q}^{m+t, n+s} \left.\left(
-          \begin{matrix} a_1, \dots, a_n, -d_1, \dots, -d_v, a_{n+1}, \dots, a_p \\
-                         b_1, \dots, b_m, -c_1, \dots, -c_u, b_{m+1}, \dots, b_q
-          \end{matrix}
-            \right| \frac{\omega}{\sigma} \right)
-
-The more interesting question is under what conditions these formulae are
-valid. Below we detail the conditions implemented in SymPy. They are an
-amalgamation of conditions found in [Prudnikov1990]_ and [Luke1969]_; please
-let us know if you find any errors.
-
-Conditions of Convergence for Integral (1)
-==========================================
-.. TODO: Formatting could be improved.
-
-We can without loss of generality assume `p \le q`, since the G-functions
-of indices `m, n, p, q` and of indices `n, m, q, p` can be related easily
-(see e.g. [Luke1969]_, section 5.3). We introduce the following notation:
-
-.. math:: \xi = m + n - p \\
-          \delta = m + n - \frac{p + q}{2}
-
-.. math:: C_3: -\Re(b_j) < 1 \text{ for } j=1, \dots, m \\
-               0 < -\Re(a_j) \text{ for } j=1, \dots, n
-
-.. math:: C_3^*: -\Re(b_j) < 1 \text{ for } j=1, \dots, q \\
-               0 < -\Re(a_j) \text{ for } j=1, \dots, p
-
-.. math:: C_4: -\Re(\delta) + \frac{q + 1 - p}{2} > q - p
-
-The convergence conditions will be detailed in several "cases", numbered one
-to five. For later use it will be helpful to separate conditions "at infinity"
-from conditions "at zero". By conditions "at infinity" we mean conditions that
-only depend on the behaviour of the integrand for large, positive values
-of `x`, whereas by conditions "at zero" we mean conditions that only depend on
-the behaviour of the integrand on `(0, \epsilon)` for any `\epsilon > 0`.
-Since all our conditions are specified in terms of parameters of the
-G-functions, this distinction is not immediately visible. They are, however, of
-very distinct character mathematically; the conditions at infinity being in
-particular much harder to control.
-
-In order for the integral theorem to be valid, conditions
-`n` "at zero" and "at infinity" both have to be fulfilled, for some `n`.
-
-These are the conditions "at infinity":
-
-1. .. math:: \delta > 0 \wedge |\arg(\eta)| < \delta \pi \wedge (A \vee B \vee C),
-
-   where
-
-   .. math::
-      A = 1 \le n \wedge p < q \wedge 1 \le m
-
-   .. math::
-      B = 1 \le p \wedge 1 \le m \wedge q = p+1 \wedge
-                  \neg (n = 0 \wedge m = p + 1 )
-
-   .. math::
-      C = 1 \le n \wedge q = p \wedge |\arg(\eta)| \ne (\delta - 2k)\pi
-             \text{ for } k = 0, 1, \dots
-               \left\lceil \frac{\delta}{2} \right\rceil.
-2. .. math:: n = 0 \wedge p + 1 \le m \wedge |\arg(\eta)| < \delta \pi
-3. .. math:: (p < q \wedge 1 \le m \wedge \delta > 0 \wedge |\arg(\eta)| = \delta \pi)
-              \vee (p \le q - 2 \wedge \delta = 0 \wedge \arg(\eta) = 0)
-4. .. math:: p = q \wedge \delta = 0 \wedge \arg(\eta) = 0 \wedge \eta \ne 0
-        \wedge \Re\left(\sum_{j=1}^p b_j - a_j \right) < 0
-5. .. math:: \delta > 0 \wedge |\arg(\eta)| < \delta \pi
-
-And these are the conditions "at zero":
-
-1. .. math:: \eta \ne 0 \wedge C_3
-2. .. math:: C_3
-3. .. math:: C_3 \wedge C_4
-4. .. math:: C_3
-5. .. math:: C_3
-
-Conditions of Convergence for Integral (2)
-==========================================
-
-We introduce the following notation:
-
-.. many of the latex expressions below were generated semi-automatically
-
-.. math:: b^* = s + t - \frac{u + v}{2}
-.. math:: c^* = m + n - \frac{p + q}{2}
-.. math:: \rho = \sum_{j=1}^v d_j - \sum_{j=1}^u c_j + \frac{u - v}{2} + 1
-.. math:: \mu = \sum_{j=1}^q b_j - \sum_{j=1}^p a_j + \frac{p - q}{2} + 1
-.. math:: \phi = q - p - \frac{u - v}{2} + 1
-.. math:: \eta = 1 - (v - u) - \mu - \rho
-.. math:: \psi = \frac{\pi(q - m - n) + |\arg(\omega)|}{q - p}
-.. math:: \theta = \frac{\pi(v - s - t) + |\arg(\sigma)|)}{v - u}
-.. math:: \lambda_c = (q - p)|\omega|^{1/(q - p)} \cos{\psi}
-                    + (v - u)|\sigma|^{1/(v - u)} \cos{\theta}
-.. math:: \lambda_{s0}(c_1, c_2) = c_1 (q - p)|\omega|^{1/(q - p)} \sin{\psi}
-                    + c_2 (v - u)|\sigma|^{1/(v - u)} \sin{\theta}
-.. math:: \lambda_s =
-  \begin{cases} \operatorname{\lambda_{s0}}\left(-1,-1\right) \operatorname{\lambda_{s0}}\left(1,1\right) & \text{for}\: \arg(\omega) = 0 \wedge \arg(\sigma) = 0 \\\operatorname{\lambda_{s0}}\left(\operatorname{sign}\left(\operatorname{\arg}\left(\omega\right)\right),-1\right) \operatorname{\lambda_{s0}}\left(\operatorname{sign}\left(\operatorname{\arg}\left(\omega\right)\right),1\right) & \text{for}\: \arg(\omega) \ne 0 \wedge \arg(\sigma) = 0 \\\operatorname{\lambda_{s0}}\left(-1,\operatorname{sign}\left(\operatorname{\arg}\left(\sigma\right)\right)\right) \operatorname{\lambda_{s0}}\left(1,\operatorname{sign}\left(\operatorname{\arg}\left(\sigma\right)\right)\right) & \text{for}\: \arg(\omega) = 0 \wedge \arg(\sigma) \ne 0) \\\operatorname{\lambda_{s0}}\left(\operatorname{sign}\left(\operatorname{\arg}\left(\omega\right)\right),\operatorname{sign}\left(\operatorname{\arg}\left(\sigma\right)\right)\right) & \text{otherwise} \end{cases}
-.. math:: z_0 = \frac{\omega}{\sigma} e^{-i\pi (b^* + c^*)}
-.. math:: z_1 = \frac{\sigma}{\omega} e^{-i\pi (b^* + c^*)}
-
-The following conditions will be helpful:
-
-.. math:: C_1: (a_i - b_j \notin \mathbb{Z}_{>0} \text{ for } i = 1, \dots, n, j = 1, \dots, m) \\
-               \wedge
-               (c_i - d_j \notin \mathbb{Z}_{>0} \text{ for } i = 1, \dots, t, j = 1, \dots, s)
-.. math:: C_2:
-    \Re(1 + b_i + d_j) > 0 \text{ for } i = 1, \dots, m, j = 1, \dots, s
-.. math:: C_3:
-    \Re(a_i + c_j) < 1 \text{ for } i = 1, \dots, n, j = 1, \dots, t
-.. math:: C_4:
-    (p - q)\Re(c_i) - \Re(\mu) > -\frac{3}{2} \text{ for } i=1, \dots, t
-.. math:: C_5:
-    (p - q)\Re(1 + d_i) - \Re(\mu) > -\frac{3}{2} \text{ for } i=1, \dots, s
-.. math:: C_6:
-    (u - v)\Re(a_i) - \Re(\rho) > -\frac{3}{2} \text{ for } i=1, \dots, n
-.. math:: C_7:
-    (u - v)\Re(1 + b_i) - \Re(\rho) > -\frac{3}{2} \text{ for } i=1, \dots, m
-.. math:: C_8:
-    0 < \lvert{\phi}\rvert + 2 \Re\left(\left(\mu -1\right) \left(- u + v\right) + \left(- p + q\right) \left(\rho -1\right) + \left(- p + q\right) \left(- u + v\right)\right)
-.. math:: C_9:
-    0 < \lvert{\phi}\rvert - 2 \Re\left(\left(\mu -1\right) \left(- u + v\right) + \left(- p + q\right) \left(\rho -1\right) + \left(- p + q\right) \left(- u + v\right)\right)
-.. math:: C_{10}:
-    \lvert{\operatorname{arg}\left(\sigma\right)}\rvert < \pi b^{*}
-.. math:: C_{11}:
-    \lvert{\operatorname{arg}\left(\sigma\right)}\rvert = \pi b^{*}
-.. math:: C_{12}:
-    |\arg(\omega)| < c^*\pi
-.. math:: C_{13}:
-    |\arg(\omega)| = c^*\pi
-.. math:: C_{14}^1:
-    \left(z_0 \ne 1 \wedge |\arg(1 - z_0)| < \pi \right) \vee
-    \left(z_0 = 1 \wedge \Re(\mu + \rho - u + v) < 1 \right)
-.. math:: C_{14}^2:
-    \left(z_1 \ne 1 \wedge |\arg(1 - z_1)| < \pi \right) \vee
-    \left(z_1 = 1 \wedge \Re(\mu + \rho - p + q) < 1 \right)
-.. math:: C_{14}:
-    \phi = 0 \wedge b^* + c^* \le 1 \wedge (C_{14}^1 \vee C_{14}^2)
-.. math:: C_{15}:
-    \lambda_c > 0 \vee (\lambda_c = 0 \wedge \lambda_s \ne 0 \wedge \Re(\eta) > -1)
-                  \vee (\lambda_c = 0 \wedge \lambda_s = 0 \wedge \Re(\eta) > 0)
-.. math:: C_{16}: \int_0^\infty G_{u, v}^{s, t}(\sigma x) \mathrm{d} x
-                     \text{ converges at infinity }
-.. math:: C_{17}: \int_0^\infty G_{p, q}^{m, n}(\omega x) \mathrm{d} x
-                     \text{ converges at infinity }
-
-Note that `C_{16}` and `C_{17}` are the reason we split the convergence conditions for
-integral (1).
-
-With this notation established, the implemented convergence conditions can be enumerated
-as follows:
-
-1. .. math:: m n s t \neq 0 \wedge 0 < b^{*} \wedge 0 < c^{*} \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{10} \wedge C_{12}
-2. .. math:: u = v \wedge b^{*} = 0 \wedge 0 < c^{*} \wedge 0 < \sigma \wedge \Re{\rho} < 1 \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{12}
-3. .. math:: p = q \wedge u = v \wedge b^{*} = 0 \wedge c^{*} = 0 \wedge 0 < \sigma \wedge 0 < \omega \wedge \Re{\mu} < 1 \wedge \Re{\rho} < 1 \wedge \sigma \neq \omega \wedge C_{1} \wedge C_{2} \wedge C_{3}
-4. .. math:: p = q \wedge u = v \wedge b^{*} = 0 \wedge c^{*} = 0 \wedge 0 < \sigma \wedge 0 < \omega \wedge \Re\left(\mu + \rho\right) < 1 \wedge \omega \neq \sigma \wedge C_{1} \wedge C_{2} \wedge C_{3}
-5. .. math:: p = q \wedge u = v \wedge b^{*} = 0 \wedge c^{*} = 0 \wedge 0 < \sigma \wedge 0 < \omega \wedge \Re\left(\mu + \rho\right) < 1 \wedge \omega \neq \sigma \wedge C_{1} \wedge C_{2} \wedge C_{3}
-6. .. math:: q < p \wedge 0 < s \wedge 0 < b^{*} \wedge 0 \leq c^{*} \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{5} \wedge C_{10} \wedge C_{13}
-7. .. math:: p < q \wedge 0 < t \wedge 0 < b^{*} \wedge 0 \leq c^{*} \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{4} \wedge C_{10} \wedge C_{13}
-8. .. math:: v < u \wedge 0 < m \wedge 0 < c^{*} \wedge 0 \leq b^{*} \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{7} \wedge C_{11} \wedge C_{12}
-9. .. math:: u < v \wedge 0 < n \wedge 0 < c^{*} \wedge 0 \leq b^{*} \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{6} \wedge C_{11} \wedge C_{12}
-10. .. math:: q < p \wedge u = v \wedge b^{*} = 0 \wedge 0 \leq c^{*} \wedge 0 < \sigma \wedge \Re{\rho} < 1 \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{5} \wedge C_{13}
-11. .. math:: p < q \wedge u = v \wedge b^{*} = 0 \wedge 0 \leq c^{*} \wedge 0 < \sigma \wedge \Re{\rho} < 1 \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{4} \wedge C_{13}
-12. .. math:: p = q \wedge v < u \wedge 0 \leq b^{*} \wedge c^{*} = 0 \wedge 0 < \omega \wedge \Re{\mu} < 1 \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{7} \wedge C_{11}
-13. .. math:: p = q \wedge u < v \wedge 0 \leq b^{*} \wedge c^{*} = 0 \wedge 0 < \omega \wedge \Re{\mu} < 1 \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{6} \wedge C_{11}
-14. .. math:: p < q \wedge v < u \wedge 0 \leq b^{*} \wedge 0 \leq c^{*} \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{4} \wedge C_{7} \wedge C_{11} \wedge C_{13}
-15. .. math:: q < p \wedge u < v \wedge 0 \leq b^{*} \wedge 0 \leq c^{*} \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{5} \wedge C_{6} \wedge C_{11} \wedge C_{13}
-16. .. math:: q < p \wedge v < u \wedge 0 \leq b^{*} \wedge 0 \leq c^{*} \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{5} \wedge C_{7} \wedge C_{8} \wedge C_{11} \wedge C_{13} \wedge C_{14}
-17. .. math:: p < q \wedge u < v \wedge 0 \leq b^{*} \wedge 0 \leq c^{*} \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{4} \wedge C_{6} \wedge C_{9} \wedge C_{11} \wedge C_{13} \wedge C_{14}
-18. .. math:: t = 0 \wedge 0 < s \wedge 0 < b^{*} \wedge 0 < \phi \wedge C_{1} \wedge C_{2} \wedge C_{10}
-19. .. math:: s = 0 \wedge 0 < t \wedge 0 < b^{*} \wedge \phi < 0 \wedge C_{1} \wedge C_{3} \wedge C_{10}
-20. .. math::  n = 0 \wedge 0 < m \wedge 0 < c^{*} \wedge \phi < 0 \wedge C_{1} \wedge C_{2} \wedge C_{12}
-21. .. math:: m = 0 \wedge 0 < n \wedge 0 < c^{*} \wedge 0 < \phi \wedge C_{1} \wedge C_{3} \wedge C_{12}
-22. .. math:: s t = 0 \wedge 0 < b^{*} \wedge 0 < c^{*} \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{10} \wedge C_{12}
-23. .. math:: m n = 0 \wedge 0 < b^{*} \wedge 0 < c^{*} \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{10} \wedge C_{12}
-24. .. math:: p < m + n \wedge t = 0 \wedge \phi = 0 \wedge 0 < s \wedge 0 < b^{*} \wedge c^{*} < 0 \wedge \lvert{\operatorname{arg}\left(\omega\right)}\rvert < \pi \left(m + n - p + 1\right) \wedge C_{1} \wedge C_{2} \wedge C_{10} \wedge C_{14} \wedge C_{15}
-25. .. math:: q < m + n \wedge s = 0 \wedge \phi = 0 \wedge 0 < t \wedge 0 < b^{*} \wedge c^{*} < 0 \wedge \lvert{\operatorname{arg}\left(\omega\right)}\rvert < \pi \left(m + n - q + 1\right) \wedge C_{1} \wedge C_{3} \wedge C_{10} \wedge C_{14} \wedge C_{15}
-26. .. math:: p = q -1 \wedge t = 0 \wedge \phi = 0 \wedge 0 < s \wedge 0 < b^{*} \wedge 0 \leq c^{*} \wedge \pi c^{*} < \lvert{\operatorname{arg}\left(\omega\right)}\rvert \wedge C_{1} \wedge C_{2} \wedge C_{10} \wedge C_{14} \wedge C_{15}
-27. .. math:: p = q + 1 \wedge s = 0 \wedge \phi = 0 \wedge 0 < t \wedge 0 < b^{*} \wedge 0 \leq c^{*} \wedge \pi c^{*} < \lvert{\operatorname{arg}\left(\omega\right)}\rvert \wedge C_{1} \wedge C_{3} \wedge C_{10} \wedge C_{14} \wedge C_{15}
-28. .. math:: p < q -1 \wedge t = 0 \wedge \phi = 0 \wedge 0 < s \wedge 0 < b^{*} \wedge 0 \leq c^{*} \wedge \pi c^{*} < \lvert{\operatorname{arg}\left(\omega\right)}\rvert \wedge \lvert{\operatorname{arg}\left(\omega\right)}\rvert < \pi \left(m + n - p + 1\right) \wedge C_{1} \wedge C_{2} \wedge C_{10} \wedge C_{14} \wedge C_{15}
-29. .. math:: q + 1 < p \wedge s = 0 \wedge \phi = 0 \wedge 0 < t \wedge 0 < b^{*} \wedge 0 \leq c^{*} \wedge \pi c^{*} < \lvert{\operatorname{arg}\left(\omega\right)}\rvert \wedge \lvert{\operatorname{arg}\left(\omega\right)}\rvert < \pi \left(m + n - q + 1 \right) \wedge C_{1} \wedge C_{3} \wedge C_{10} \wedge C_{14} \wedge C_{15}
-30. .. math:: n = 0 \wedge \phi = 0 \wedge 0 < s + t \wedge 0 < m \wedge 0 < c^{*} \wedge b^{*} < 0 \wedge \lvert{\operatorname{arg}\left(\sigma\right)}\rvert < \pi \left(s + t - u + 1\right) \wedge C_{1} \wedge C_{2} \wedge C_{12} \wedge C_{14} \wedge C_{15}
-31. .. math:: m = 0 \wedge \phi = 0 \wedge v < s + t \wedge 0 < n \wedge 0 < c^{*} \wedge b^{*} < 0 \wedge \lvert{\operatorname{arg}\left(\sigma\right)}\rvert < \pi \left(s + t - v + 1\right) \wedge C_{1} \wedge C_{3} \wedge C_{12} \wedge C_{14} \wedge C_{15}
-32. .. math:: n = 0 \wedge \phi = 0 \wedge u = v -1 \wedge 0 < m \wedge 0 < c^{*} \wedge 0 \leq b^{*} \wedge \pi b^{*} < \lvert{\operatorname{arg}\left(\sigma\right)}\rvert \wedge \lvert{\operatorname{arg}\left(\sigma\right)}\rvert < \pi \left(b^{*} + 1\right) \wedge C_{1} \wedge C_{2} \wedge C_{12} \wedge C_{14} \wedge C_{15}
-33. .. math:: m = 0 \wedge \phi = 0 \wedge u = v + 1 \wedge 0 < n \wedge 0 < c^{*} \wedge 0 \leq b^{*} \wedge \pi b^{*} < \lvert{\operatorname{arg}\left(\sigma\right)}\rvert \wedge \lvert{\operatorname{arg}\left(\sigma\right)}\rvert < \pi \left(b^{*} + 1\right) \wedge C_{1} \wedge C_{3} \wedge C_{12} \wedge C_{14} \wedge C_{15}
-34. .. math:: n = 0 \wedge \phi = 0 \wedge u < v -1 \wedge 0 < m \wedge 0 < c^{*} \wedge 0 \leq b^{*} \wedge \pi b^{*} < \lvert{\operatorname{arg}\left(\sigma\right)}\rvert \wedge \lvert{\operatorname{arg}\left(\sigma\right)}\rvert < \pi \left(s + t - u + 1\right) \wedge C_{1} \wedge C_{2} \wedge C_{12} \wedge C_{14} \wedge C_{15}
-35. .. math:: m = 0 \wedge \phi = 0 \wedge v + 1 < u \wedge 0 < n \wedge 0 < c^{*} \wedge 0 \leq b^{*} \wedge \pi b^{*} < \lvert{\operatorname{arg}\left(\sigma\right)}\rvert \wedge \lvert{\operatorname{arg}\left(\sigma\right)}\rvert < \pi \left(s + t - v + 1 \right) \wedge C_{1} \wedge C_{3} \wedge C_{12} \wedge C_{14} \wedge C_{15}
-36. .. math:: C_{17} \wedge t = 0 \wedge u < s \wedge 0 < b^{*} \wedge C_{10} \wedge C_{1} \wedge C_{2} \wedge C_{3}
-37. .. math:: C_{17} \wedge s = 0 \wedge v < t \wedge 0 < b^{*} \wedge C_{10} \wedge C_{1} \wedge C_{2} \wedge C_{3}
-38. .. math:: C_{16} \wedge n = 0 \wedge p < m \wedge 0 < c^{*} \wedge C_{12} \wedge C_{1} \wedge C_{2} \wedge C_{3}
-39. .. math:: C_{16} \wedge m = 0 \wedge q < n \wedge 0 < c^{*} \wedge C_{12} \wedge C_{1} \wedge C_{2} \wedge C_{3}
-
-
-The Inverse Laplace Transform of a G-function
-*********************************************
-
-The inverse laplace transform of a Meijer G-function can be expressed as
-another G-function. This is a fairly versatile method for computing this
-transform. However, I could not find the details in the literature, so I work
-them out here. In [Luke1969]_, section 5.6.3, there is a formula for the inverse
-Laplace transform of a G-function of argument `bz`, and convergence conditions
-are also given. However, we need a formula for argument `bz^a` for rational `a`.
-
-We are asked to compute
-
-.. math ::
-    f(t) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} e^{zt} G(bz^a) \mathrm{d}z,
-
-for positive real `t`. Three questions arise:
-
-1. When does this integral converge?
-2. How can we compute the integral?
-3. When is our computation valid?
-
-
-How to compute the integral
-===========================
-
-We shall work formally for now. Denote by `\Delta(s)` the product of gamma
-functions appearing in the definition of `G`, so that
-
-.. math :: G(z) = \frac{1}{2\pi i} \int_L \Delta(s) z^s \mathrm{d}s.
-
-Thus
-
-.. math ::
-    f(t) = \frac{1}{(2\pi i)^2} \int_{c - i\infty}^{c + i\infty} \int_L
-                  e^{zt} \Delta(s) b^s z^{as} \mathrm{d}s \mathrm{d}z.
-
-We interchange the order of integration to get
-
-.. math ::
-    f(t) = \frac{1}{2\pi i} \int_L b^s \Delta(s)
-          \int_{c-i\infty}^{c+i\infty} e^{zt} z^{as} \frac{\mathrm{d}z}{2\pi i}
-                \mathrm{d}s.
-
-The inner integral is easily seen to be
-`\frac{1}{\Gamma(-as)} \frac{1}{t^{1+as}}`. (Using Cauchy's theorem and Jordan's
-lemma deform the contour to run from `-\infty` to `-\infty`, encircling `0` once
-in the negative sense. For `as` real and greater than one,
-this contour can be pushed onto
-the negative real axis and the integral is recognised as a product of a sine and
-a gamma function. The formula is then proved using the functional equation of the
-gamma function, and extended to the entire domain of convergence of the original
-integral by appealing to analytic continuation.)
-Hence we find
-
-.. math ::
-  f(t) = \frac{1}{t} \frac{1}{2\pi i} \int_L \Delta(s) \frac{1}{\Gamma(-as)}
-                \left(\frac{b}{t^a}\right)^s \mathrm{d}s,
-
-which is a so-called Fox H function (of argument `\frac{b}{t^a}`). For rational
-`a`, this can be expressed as a Meijer G-function using the gamma function
-multiplication theorem.
-
-When this computation is valid
-==============================
-
-There are a number of obstacles in this computation. Interchange of integrals
-is only valid if all integrals involved are absolutely convergent. In
-particular the inner integral has to converge. Also, for our identification of
-the final integral as a Fox H / Meijer G-function to be correct, the poles of
-the newly obtained gamma function must be separated properly.
-
-It is easy to check that the inner integal converges absolutely for
-`Re(as) < -1`. Thus the contour `L` has to run left of the line `Re(as) = -1`.
-Under this condition, the poles of the newly-introduced gamma function are
-separated properly.
-
-It remains to observe that the Meijer G-function is an analytic, unbranched
-function of its parameters, and of the coefficient `b`. Hence so is `f(t)`.
-Thus the final computation remains valid as long as the initial integral
-converges, and if there exists a changed set of parameters where the computation
-is valid. If we assume w.l.o.g. that `a > 0`, then the latter condition is
-fulfilled if `G` converges along contours (2) or (3) of [Luke1969]_,
-section 5.2, i.e. either `\delta >= \frac{a}{2}` or `p \ge 1, p \ge q`.
-
-When the integral exists
-========================
-
-Using [Luke1969]_, section 5.10, for any given meijer G-function we can find a
-dominant term of the form `z^a e^{bz^c}` (although this expression might not be
-the best possible, because of cancellation).
-
-We must thus investigate
-
-.. math :: \lim_{T \to \infty} \int_{c-iT}^{c+iT}
-                     e^{zt} z^a e^{bz^c} \mathrm{d}z.
-
-(This principal value integral is the exact statement used in the Laplace
-inversion theorem.) We write `z = c + i \tau`. Then
-`arg(z) \to \pm \frac{\pi}{2}`, and so `e^{zt} \sim e^{it \tau}` (where `\sim`
-shall always mean "asymptotically equivalent up to a positive real
-multiplicative constant"). Also
-`z^{x + iy} \sim |\tau|^x e^{i y \log{|\tau|}} e^{\pm x i \frac{\pi}{2}}.`
-
-Set `\omega_{\pm} = b e^{\pm i Re(c) \frac{\pi}{2}}`. We have three cases:
-
-1. `b=0` or `Re(c) \le 0`.
-   In this case the integral converges if `Re(a) \le -1`.
-2. `b \ne 0`, `Im(c) = 0`, `Re(c) > 0`.
-   In this case the integral converges if `Re(\omega_{\pm}) < 0`.
-3. `b \ne 0`, `Im(c) = 0`, `Re(c) > 0`, `Re(\omega_{\pm}) \le 0`, and at least
-   one of `Re(\omega_{\pm}) = 0`.
-   Here the same condition as in (1) applies.
-
-Implemented G-Function Formulae
-*******************************
-
-An important part of the algorithm is a table expressing various functions
-as Meijer G-functions. This is essentially a table of Mellin Transforms in
-disguise. The following automatically generated table shows the formulae
-currently implemented in SymPy. An entry "generated" means that the
-corresponding G-function has a variable number of parameters.
-This table is intended to shrink in future, when the algorithm's capabilities
-of deriving new formulae improve. Of course it has to grow whenever a new class
-of special functions is to be dealt with.
-
-.. automodule:: sympy.integrals.meijerint_doc
diff --git a/dev-py3k/_sources/modules/integrals/integrals.txt b/dev-py3k/_sources/modules/integrals/integrals.txt
deleted file mode 100644
index a6f9b0d5a1c..00000000000
--- a/dev-py3k/_sources/modules/integrals/integrals.txt
+++ /dev/null
@@ -1,117 +0,0 @@
-Integrals
-==========
-
-.. module:: sympy.integrals
-
-The *integrals* module in SymPy implements methods to calculate definite and indefinite integrals of expressions.
-
-Principal method in this module is integrate()
-
-  - integrate(f, x) returns the indefinite integral :math:`\int f\,dx`
-  - integrate(f, (x, a, b)) returns the definite integral :math:`\int_{a}^{b} f\,dx`
-
-Examples
---------
-SymPy can integrate a vast array of functions. It can integrate polynomial functions::
-
-    >>> from sympy import *
-    >>> init_printing(use_unicode=False, wrap_line=False, no_global=True)
-    >>> x = Symbol('x')
-    >>> integrate(x**2 + x + 1, x)
-     3    2
-    x    x
-    -- + -- + x
-    3    2
-
-Rational functions::
-
-	>>> integrate(x/(x**2+2*x+1), x)
-	               1
-	log(x + 1) + -----
-	             x + 1
-
-
-Exponential-polynomial functions. Multiplicative combinations of polynomials and the functions exp, cos and sin can be integrated by hand using repeated integration by parts, which is an extremely tedious process. Happily, SymPy will deal with these integrals.
-
-::
-
-    >>> integrate(x**2 * exp(x) * cos(x), x)
-     2  x           2  x                         x           x
-    x *e *sin(x)   x *e *cos(x)      x          e *sin(x)   e *cos(x)
-    ------------ + ------------ - x*e *sin(x) + --------- - ---------
-         2              2                           2           2
-
-
-
-even a few nonelementary integrals (in particular, some integrals involving the error function) can be evaluated::
-
-	>>> integrate(exp(-x**2)*erf(x), x)
-	  ____    2
-	\/ pi *erf (x)
-	--------------
-	      4
-
-
-Integral Transforms
--------------------
-
-.. module:: sympy.integrals.transforms
-
-SymPy has special support for definite integrals, and integral transforms.
-
-.. autofunction:: mellin_transform
-.. autofunction:: inverse_mellin_transform
-.. autofunction:: laplace_transform
-.. autofunction:: inverse_laplace_transform
-.. autofunction:: fourier_transform
-.. autofunction:: inverse_fourier_transform
-.. autofunction:: sine_transform
-.. autofunction:: inverse_sine_transform
-.. autofunction:: cosine_transform
-.. autofunction:: inverse_cosine_transform
-.. autofunction:: hankel_transform
-.. autofunction:: inverse_hankel_transform
-
-
-Internals
----------
-
-There is a general method for calculating antiderivatives of elementary functions, called the Risch algorithm. The Risch algorithm is a decision procedure that can determine whether an elementary solution exists, and in that case calculate it. It can be extended to handle many nonelementary functions in addition to the elementary ones.
-
-SymPy currently uses a simplified version of the Risch algorithm, called the Risch-Norman algorithm. This algorithm is much faster, but may fail to find an antiderivative, although it is still very powerful. SymPy also uses pattern matching and heuristics to speed up evaluation of some types of integrals, e.g. polynomials.
-
-For non-elementary definite integrals, sympy uses so-called Meijer G-functions.
-Details are described here:
-
-.. toctree::
-   :maxdepth: 1
-
-   g-functions.rst
-
-API reference
--------------
-
-.. automethod:: sympy.integrals.integrate
-
-.. automethod:: sympy.integrals.line_integrate
-
-.. automethod:: sympy.integrals.deltaintegrate
-
-.. automethod:: sympy.integrals.ratint
-
-.. automethod:: sympy.integrals.heurisch
-
-.. automethod:: sympy.integrals.trigintegrate
-
-Class Integral represents an unevaluated integral and has some methods that help in the integration of an expression.
-
-.. autoclass:: sympy.integrals.Integral
-   :members:
-
-   .. data:: is_commutative
-
-      Returns whether all the free symbols in the integral are commutative.
-
-TODO and Bugs
--------------
-There are still lots of functions that sympy does not know how to integrate. For bugs related to this module, see http://code.google.com/p/sympy/issues/list?q=label:Integration
diff --git a/dev-py3k/_sources/modules/logic.txt b/dev-py3k/_sources/modules/logic.txt
deleted file mode 100644
index 0750839dc1c..00000000000
--- a/dev-py3k/_sources/modules/logic.txt
+++ /dev/null
@@ -1,92 +0,0 @@
-Logic Module
-===============
-
-.. module:: sympy.logic
-
-Introduction
-------------
-
-The logic module for SymPy allows to form and manipulate logic expressions
-using symbolic and boolean values.
-
-Forming logical expressions
----------------------------
-
-You can build boolean expressions with the standard python operators ``&``
-(:class:`And`), ``|`` (:class:`Or`), ``~`` (:class:`Not`)::
-
-    >>> from sympy import *
-    >>> x, y = symbols('x,y')
-    >>> y | (x & y)
-    Or(And(x, y), y)
-    >>> x | y
-    Or(x, y)
-    >>> ~x
-    Not(x)
-
-You can also form implications with ``>>`` and ``<<``::
-
-    >>> x >> y
-    Implies(x, y)
-    >>> x << y
-    Implies(y, x)
-
-Like most types in SymPy, Boolean expressions inherit from :class:`Basic`::
-
-    >>> (y & x).subs({x: True, y: True})
-    True
-    >>> (x | y).atoms()
-    set([x, y])
-
-Boolean functions
------------------------
-
-.. autofunction:: sympy.logic.boolalg.to_cnf
-
-.. autoclass:: sympy.logic.boolalg.And
-
-.. autoclass:: sympy.logic.boolalg.Or
-
-.. autoclass:: sympy.logic.boolalg.Not
-
-.. autoclass:: sympy.logic.boolalg.Xor
-
-.. autoclass:: sympy.logic.boolalg.Nand
-
-.. autoclass:: sympy.logic.boolalg.Nor
-
-.. autoclass:: sympy.logic.boolalg.Implies
-
-.. autoclass:: sympy.logic.boolalg.Equivalent
-
-.. autoclass:: sympy.logic.boolalg.ITE
-
-Inference
----------
-
-.. module: sympy.logic.inference
-
-This module implements some inference routines in propositional logic.
-
-The function satisfiable will test that a given boolean expression is satisfiable,
-that is, you can assign values to the variables to make the sentence True.
-
-For example, the expression x & ~x is not satisfiable, since there are no values
-for x that make this sentence True. On the other hand, (x | y) & (x | ~y) & (~x | y)
-is satisfiable with both x and y being True.
-
-    >>> from sympy.logic.inference import satisfiable
-    >>> from sympy import Symbol
-    >>> x = Symbol('x')
-    >>> y = Symbol('y')
-    >>> satisfiable(x & ~x)
-    False
-    >>> satisfiable((x | y) & (x | ~y) & (~x | y))
-    {x: True, y: True}
-
-As you see, when a sentence is satisfiable, it returns a model that makes that
-sentence True. If it is not satisfiable it will return False
-
-.. autofunction:: sympy.logic.inference.satisfiable
-
-.. TODO: write about CNF file format
diff --git a/dev-py3k/_sources/modules/matrices/dense.txt b/dev-py3k/_sources/modules/matrices/dense.txt
deleted file mode 100644
index f3d9cb22647..00000000000
--- a/dev-py3k/_sources/modules/matrices/dense.txt
+++ /dev/null
@@ -1,15 +0,0 @@
-Dense Matrices
-==============
-
-Matrix Class Reference
-----------------------
-
-.. autoclass:: sympy.matrices.dense.MutableDenseMatrix
-   :members:
-
-ImmutableMatrix Class Reference
--------------------------------
-
-.. autoclass:: sympy.matrices.immutable.ImmutableMatrix
-   :members:
-   :noindex:
diff --git a/dev-py3k/_sources/modules/matrices/expressions.txt b/dev-py3k/_sources/modules/matrices/expressions.txt
deleted file mode 100644
index c2ab11ed1b0..00000000000
--- a/dev-py3k/_sources/modules/matrices/expressions.txt
+++ /dev/null
@@ -1,62 +0,0 @@
-Matrix Expressions
-==================
-
-.. module:: sympy.matrices.expressions
-
-The Matrix expression module allows users to write down statements like
-
-    >>> from sympy import MatrixSymbol, Matrix
-    >>> X = MatrixSymbol('X', 3, 3)
-    >>> Y = MatrixSymbol('Y', 3, 3)
-    >>> (X.T*X).I*Y
-    X^-1*X'^-1*Y
-
-    >>> Matrix(X)
-    [X_00, X_01, X_02]
-    [X_10, X_11, X_12]
-    [X_20, X_21, X_22]
-
-    >>> (X*Y)[1, 2]
-    X_10*Y_02 + X_11*Y_12 + X_12*Y_22
-
-where ``X`` and ``Y`` are :class:`MatrixSymbol`'s rather than scalar symbols.
-
-Matrix Expressions Core Reference
----------------------------------
-.. autoclass:: MatrixExpr
-   :members:
-.. autoclass:: MatrixSymbol
-   :members:
-.. autoclass:: MatAdd
-   :members:
-.. autoclass:: MatMul
-   :members:
-.. autoclass:: MatPow
-   :members:
-.. autoclass:: Inverse
-   :members:
-.. autoclass:: Transpose
-   :members:
-.. autoclass:: Trace
-   :members:
-.. autoclass:: FunctionMatrix
-   :members:
-.. autoclass:: Identity
-   :members:
-.. autoclass:: ZeroMatrix
-   :members:
-
-Block Matrices
---------------
-
-Block matrices allow you to construct larger matrices out of smaller
-sub-blocks. They can work with :class:`MatrixExpr` or
-:class:`ImmutableMatrix` objects.
-
-.. module:: sympy.matrices.expressions.blockmatrix
-
-.. autoclass:: BlockMatrix
-   :members:
-.. autoclass:: BlockDiagMatrix
-   :members:
-.. autofunction:: block_collapse
diff --git a/dev-py3k/_sources/modules/matrices/immutablematrices.txt b/dev-py3k/_sources/modules/matrices/immutablematrices.txt
deleted file mode 100644
index bfe57172703..00000000000
--- a/dev-py3k/_sources/modules/matrices/immutablematrices.txt
+++ /dev/null
@@ -1,41 +0,0 @@
-Immutable Matrices
-==================
-
-.. module:: sympy
-
-The standard :class:`Matrix` class in SymPy is mutable. This is important for
-performance reasons but means that standard matrices can not interact well with
-the rest of SymPy. This is because the :class:`Basic` object, from which most
-SymPy classes inherit, is immutable.
-
-The mission of the :class:`ImmutableMatrix` class is to bridge the tension
-between performance/mutability and safety/immutability. Immutable matrices can
-do almost everything that normal matrices can do but they inherit from
-:class:`Basic` and can thus interact more naturally with the rest of SymPy.
-:class:`ImmutableMatrix` also inherits from :class:`MatrixExpr`, allowing it to
-interact freely with SymPy's Matrix Expression module.
-
-You can turn any Matrix-like object into an :class:`ImmutableMatrix` by calling
-the constructor
-
-    >>> from sympy import Matrix, ImmutableMatrix
-    >>> M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
-    >>> M[1, 1] = 0
-    >>> IM = ImmutableMatrix(M)
-    >>> IM
-    [1, 2, 3]
-    [4, 0, 6]
-    [7, 8, 9]
-    >>> IM[1, 1] = 5
-    Traceback (most recent call last):
-    ...
-    TypeError: Can not set values in Immutable Matrix. Use Matrix instead.
-
-
-ImmutableMatrix Class Reference
--------------------------------
-
-.. module:: sympy.matrices.immutable
-
-.. autoclass:: ImmutableMatrix
-   :members:
diff --git a/dev-py3k/_sources/modules/matrices/index.txt b/dev-py3k/_sources/modules/matrices/index.txt
deleted file mode 100644
index c3c7ff76363..00000000000
--- a/dev-py3k/_sources/modules/matrices/index.txt
+++ /dev/null
@@ -1,18 +0,0 @@
-.. _matrices-docs:
-
-========
-Matrices
-========
-
-.. automodule:: sympy.matrices
-
-Contents:
-
-.. toctree::
-   :maxdepth: 2
-
-   matrices.rst
-   dense.rst
-   sparse.rst
-   immutablematrices.rst
-   expressions.rst
diff --git a/dev-py3k/_sources/modules/matrices/matrices.txt b/dev-py3k/_sources/modules/matrices/matrices.txt
deleted file mode 100644
index aedd22e5027..00000000000
--- a/dev-py3k/_sources/modules/matrices/matrices.txt
+++ /dev/null
@@ -1,570 +0,0 @@
-Matrices (linear algebra)
-=========================
-
-.. module:: sympy.matrices.matrices
-
-Creating Matrices
------------------
-
-The linear algebra module is designed to be as simple as possible. First, we
-import and declare our first Matrix object:
-
-    >>> from sympy.interactive.printing import init_printing
-    >>> init_printing(use_unicode=False, wrap_line=False, no_global=True)
-    >>> from sympy.matrices import Matrix, eye, zeros, ones, diag, GramSchmidt
-    >>> M = Matrix([[1,0,0], [0,0,0]]); M
-    [1  0  0]
-    [       ]
-    [0  0  0]
-    >>> Matrix([M, (0, 0, -1)])
-    [1  0  0 ]
-    [        ]
-    [0  0  0 ]
-    [        ]
-    [0  0  -1]
-    >>> Matrix([[1, 2, 3]])
-    [1 2 3]
-    >>> Matrix([1, 2, 3])
-    [1]
-    [ ]
-    [2]
-    [ ]
-    [3]
-
-In addition to creating a matrix from a list of appropriately-sized lists
-and/or matrices, SymPy also supports more advanced methods of matrix creation
-including a single list of values and dimension inputs:
-
-    >>> Matrix(2, 3, [1, 2, 3, 4, 5, 6])
-    [1  2  3]
-    [       ]
-    [4  5  6]
-
-More interesting (and useful), is the ability to use a 2-variable function
-(or lambda) to create a matrix. Here we create an indicator function which
-is 1 on the diagonal and then use it to make the identity matrix:
-
-    >>> def f(i,j):
-    ...     if i == j:
-    ...         return 1
-    ...     else:
-    ...         return 0
-    ...
-    >>> Matrix(4, 4, f)
-    [1  0  0  0]
-    [          ]
-    [0  1  0  0]
-    [          ]
-    [0  0  1  0]
-    [          ]
-    [0  0  0  1]
-
-Finally let's use lambda to create a 1-line matrix with 1's in the even
-permutation entries:
-
-    >>> Matrix(3, 4, lambda i,j: 1 - (i+j) % 2)
-    [1  0  1  0]
-    [          ]
-    [0  1  0  1]
-    [          ]
-    [1  0  1  0]
-
-There are also a couple of special constructors for quick matrix construction:
-``eye`` is the identity matrix, ``zeros`` and ``ones`` for matrices of all
-zeros and ones, respectively, and ``diag`` to put matrices or elements along
-the diagonal:
-
-    >>> eye(4)
-    [1  0  0  0]
-    [          ]
-    [0  1  0  0]
-    [          ]
-    [0  0  1  0]
-    [          ]
-    [0  0  0  1]
-    >>> zeros(2)
-    [0  0]
-    [    ]
-    [0  0]
-    >>> zeros(2, 5)
-    [0  0  0  0  0]
-    [             ]
-    [0  0  0  0  0]
-    >>> ones(3)
-    [1  1  1]
-    [       ]
-    [1  1  1]
-    [       ]
-    [1  1  1]
-    >>> ones(1, 3)
-    [1  1  1]
-    >>> diag(1, Matrix([[1, 2], [3, 4]]))
-    [1  0  0]
-    [       ]
-    [0  1  2]
-    [       ]
-    [0  3  4]
-
-
-Basic Manipulation
-------------------
-
-While learning to work with matrices, let's choose one where the entries are
-readily identifiable. One useful thing to know is that while matrices are
-2-dimensional, the storage is not and so it is allowable - though one should be
-careful - to access the entries as if they were a 1-d list.
-
-    >>> M = Matrix(2, 3, [1, 2, 3, 4, 5, 6])
-    >>> M[4]
-    5
-
-Now, the more standard entry access is a pair of indices which will always
-return the value at the corresponding row and column of the matrix:
-
-    >>> M[1, 2]
-    6
-    >>> M[0, 0]
-    1
-    >>> M[1, 1]
-    5
-
-Since this is Python we're also able to slice submatrices; slices always
-give a matrix in return, even if the dimension is 1 x 1::
-
-    >>> M[0:2, 0:2]
-    [1  2]
-    [    ]
-    [4  5]
-    >>> M[2:2, 2]
-    []
-    >>> M[:, 2]
-    [3]
-    [ ]
-    [6]
-    >>> M[:1, 2]
-    [3]
-
-In the 2nd example above notice that the slice 2:2 gives an empty range. Note
-also (in keeping with 0-based indexing of Python) the first row/column is 0.
-
-You cannot access rows or columns that are not present unless they
-are in a slice:
-
-    >>> M[:, 10] # the 10-th column (not there)
-    Traceback (most recent call last):
-    ...
-    IndexError: Index out of range: a[[0, 10]]
-    >>> M[:, 10:11] # the 10-th column (if there)
-    []
-    >>> M[:, :10] # all columns up to the 10-th
-    [1  2  3]
-    [       ]
-    [4  5  6]
-
-Slicing an empty matrix works as long as you use a slice for the coordinate
-that has no size:
-
-    >>> Matrix(0, 3, [])[:, 1]
-    []
-
-Slicing gives a copy of what is sliced, so modifications of one object
-do not affect the other:
-
-    >>> M2 = M[:, :]
-    >>> M2[0, 0] = 100
-    >>> M[0, 0] == 100
-    False
-
-Notice that changing M2 didn't change M. Since we can slice, we can also assign
-entries:
-
-    >>> M = Matrix(([1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]))
-    >>> M
-    [1   2   3   4 ]
-    [              ]
-    [5   6   7   8 ]
-    [              ]
-    [9   10  11  12]
-    [              ]
-    [13  14  15  16]
-    >>> M[2,2] = M[0,3] = 0
-    >>> M
-    [1   2   3   0 ]
-    [              ]
-    [5   6   7   8 ]
-    [              ]
-    [9   10  0   12]
-    [              ]
-    [13  14  15  16]
-
-as well as assign slices:
-
-    >>> M = Matrix(([1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]))
-    >>> M[2:,2:] = Matrix(2,2,lambda i,j: 0)
-    >>> M
-    [1   2   3  4]
-    [            ]
-    [5   6   7  8]
-    [            ]
-    [9   10  0  0]
-    [            ]
-    [13  14  0  0]
-
-All the standard arithmetic operations are supported:
-
-    >>> M = Matrix(([1,2,3],[4,5,6],[7,8,9]))
-    >>> M - M
-    [0  0  0]
-    [       ]
-    [0  0  0]
-    [       ]
-    [0  0  0]
-    >>> M + M
-    [2   4   6 ]
-    [          ]
-    [8   10  12]
-    [          ]
-    [14  16  18]
-    >>> M * M
-    [30   36   42 ]
-    [             ]
-    [66   81   96 ]
-    [             ]
-    [102  126  150]
-    >>> M2 = Matrix(3,1,[1,5,0])
-    >>> M*M2
-    [11]
-    [  ]
-    [29]
-    [  ]
-    [47]
-    >>> M**2
-    [30   36   42 ]
-    [             ]
-    [66   81   96 ]
-    [             ]
-    [102  126  150]
-
-As well as some useful vector operations:
-
-    >>> M.row_del(0)
-    >>> M
-    [4  5  6]
-    [       ]
-    [7  8  9]
-    >>> M.col_del(1)
-    >>> M
-    [4  6]
-    [    ]
-    [7  9]
-    >>> v1 = Matrix([1,2,3])
-    >>> v2 = Matrix([4,5,6])
-    >>> v3 = v1.cross(v2)
-    >>> v1.dot(v2)
-    32
-    >>> v2.dot(v3)
-    0
-    >>> v1.dot(v3)
-    0
-
-Recall that the row_del() and col_del() operations don't return a value - they
-simply change the matrix object. We can also ''glue'' together matrices of the
-appropriate size:
-
-    >>> M1 = eye(3)
-    >>> M2 = zeros(3, 4)
-    >>> M1.row_join(M2)
-    [1  0  0  0  0  0  0]
-    [                   ]
-    [0  1  0  0  0  0  0]
-    [                   ]
-    [0  0  1  0  0  0  0]
-    >>> M3 = zeros(4, 3)
-    >>> M1.col_join(M3)
-    [1  0  0]
-    [       ]
-    [0  1  0]
-    [       ]
-    [0  0  1]
-    [       ]
-    [0  0  0]
-    [       ]
-    [0  0  0]
-    [       ]
-    [0  0  0]
-    [       ]
-    [0  0  0]
-
-
-Operations on entries
----------------------
-
-We are not restricted to having multiplication between two matrices:
-
-    >>> M = eye(3)
-    >>> 2*M
-    [2  0  0]
-    [       ]
-    [0  2  0]
-    [       ]
-    [0  0  2]
-    >>> 3*M
-    [3  0  0]
-    [       ]
-    [0  3  0]
-    [       ]
-    [0  0  3]
-
-but we can also apply functions to our matrix entries using applyfunc(). Here we'll declare a function that double any input number. Then we apply it to the 3x3 identity matrix:
-
-    >>> f = lambda x: 2*x
-    >>> eye(3).applyfunc(f)
-    [2  0  0]
-    [       ]
-    [0  2  0]
-    [       ]
-    [0  0  2]
-
-One more useful matrix-wide entry application function is the substitution function. Let's declare a matrix with symbolic entries then substitute a value. Remember we can substitute anything - even another symbol!:
-
-    >>> from sympy import Symbol
-    >>> x = Symbol('x')
-    >>> M = eye(3) * x
-    >>> M
-    [x  0  0]
-    [       ]
-    [0  x  0]
-    [       ]
-    [0  0  x]
-    >>> M.subs(x, 4)
-    [4  0  0]
-    [       ]
-    [0  4  0]
-    [       ]
-    [0  0  4]
-    >>> y = Symbol('y')
-    >>> M.subs(x, y)
-    [y  0  0]
-    [       ]
-    [0  y  0]
-    [       ]
-    [0  0  y]
-
-
-Linear algebra
---------------
-
-Now that we have the basics out of the way, let's see what we can do with the
-actual matrices. Of course, one of the first things that comes to mind is the
-determinant:
-
-    >>> M = Matrix(( [1, 2, 3], [3, 6, 2], [2, 0, 1] ))
-    >>> M.det()
-    -28
-    >>> M2 = eye(3)
-    >>> M2.det()
-    1
-    >>> M3 = Matrix(( [1, 0, 0], [1, 0, 0], [1, 0, 0] ))
-    >>> M3.det()
-    0
-
-Another common operation is the inevers: In SymPy, this is computed by Gaussian
-elimination by default (for dense matrices) but we can specify it be done by LU
-decomposition as well:
-
-    >>> M2.inv()
-    [1  0  0]
-    [       ]
-    [0  1  0]
-    [       ]
-    [0  0  1]
-    >>> M2.inv(method="LU")
-    [1  0  0]
-    [       ]
-    [0  1  0]
-    [       ]
-    [0  0  1]
-    >>> M.inv(method="LU")
-    [-3/14  1/14  1/2 ]
-    [                 ]
-    [-1/28  5/28  -1/4]
-    [                 ]
-    [ 3/7   -1/7   0  ]
-    >>> M * M.inv(method="LU")
-    [1  0  0]
-    [       ]
-    [0  1  0]
-    [       ]
-    [0  0  1]
-
-We can perform a QR factorization which is handy for solving systems:
-
-    >>> A = Matrix([[1,1,1],[1,1,3],[2,3,4]])
-    >>> Q, R = A.QRdecomposition()
-    >>> Q
-    [  ___     ___     ___]
-    [\/ 6   -\/ 3   -\/ 2 ]
-    [-----  ------  ------]
-    [  6      3       2   ]
-    [                     ]
-    [  ___     ___    ___ ]
-    [\/ 6   -\/ 3   \/ 2  ]
-    [-----  ------  ----- ]
-    [  6      3       2   ]
-    [                     ]
-    [  ___    ___         ]
-    [\/ 6   \/ 3          ]
-    [-----  -----     0   ]
-    [  3      3           ]
-    >>> R
-    [           ___         ]
-    [  ___  4*\/ 6       ___]
-    [\/ 6   -------  2*\/ 6 ]
-    [          3            ]
-    [                       ]
-    [          ___          ]
-    [        \/ 3           ]
-    [  0     -----      0   ]
-    [          3            ]
-    [                       ]
-    [                   ___ ]
-    [  0       0      \/ 2  ]
-    >>> Q*R
-    [1  1  1]
-    [       ]
-    [1  1  3]
-    [       ]
-    [2  3  4]
-
-
-In addition to the solvers in the solver.py file, we can solve the system Ax=b
-by passing the b vector to the matrix A's LUsolve function. Here we'll cheat a
-little choose A and x then multiply to get b. Then we can solve for x and check
-that it's correct:
-
-    >>> A = Matrix([ [2, 3, 5], [3, 6, 2], [8, 3, 6] ])
-    >>> x = Matrix(3,1,[3,7,5])
-    >>> b = A*x
-    >>> soln = A.LUsolve(b)
-    >>> soln
-    [3]
-    [ ]
-    [7]
-    [ ]
-    [5]
-
-There's also a nice Gram-Schmidt orthogonalizer which will take a set of
-vectors and orthogonalize then with respect to another another. There is an
-optional argument which specifies whether or not the output should also be
-normalized, it defaults to False. Let's take some vectors and orthogonalize
-them - one normalized and one not:
-
-    >>> L = [Matrix([2,3,5]), Matrix([3,6,2]), Matrix([8,3,6])]
-    >>> out1 = GramSchmidt(L)
-    >>> out2 = GramSchmidt(L, True)
-
-Let's take a look at the vectors:
-
-    >>> for i in out1:
-    ...     print(i)
-    ...
-    [2]
-    [3]
-    [5]
-    [ 23/19]
-    [ 63/19]
-    [-47/19]
-    [ 1692/353]
-    [-1551/706]
-    [ -423/706]
-    >>> for i in out2:
-    ...      print(i)
-    ...
-    [  sqrt(38)/19]
-    [3*sqrt(38)/38]
-    [5*sqrt(38)/38]
-    [ 23*sqrt(6707)/6707]
-    [ 63*sqrt(6707)/6707]
-    [-47*sqrt(6707)/6707]
-    [ 12*sqrt(706)/353]
-    [-11*sqrt(706)/706]
-    [ -3*sqrt(706)/706]
-
-We can spot-check their orthogonality with dot() and their normality with
-norm():
-
-    >>> out1[0].dot(out1[1])
-    0
-    >>> out1[0].dot(out1[2])
-    0
-    >>> out1[1].dot(out1[2])
-    0
-    >>> out2[0].norm()
-    1
-    >>> out2[1].norm()
-    1
-    >>> out2[2].norm()
-    1
-
-So there is quite a bit that can be done with the module including eigenvalues,
-eigenvectors, nullspace calculation, cofactor expansion tools, and so on. From
-here one might want to look over the matrices.py file for all functionality.
-
-MatrixBase Class Reference
---------------------------
-.. autoclass:: MatrixBase
-   :members:
-
-Matrix Exceptions Reference
----------------------------
-
-.. autoclass:: MatrixError
-
-.. autoclass:: ShapeError
-
-.. autoclass:: NonSquareMatrixError
-
-
-Matrix Functions Reference
---------------------------
-
-.. autofunction:: classof
-
-.. autofunction:: sympy.matrices.dense.matrix_multiply_elementwise
-
-.. autofunction:: sympy.matrices.dense.zeros
-
-.. autofunction:: sympy.matrices.dense.ones
-
-.. autofunction:: sympy.matrices.dense.eye
-
-.. autofunction:: sympy.matrices.dense.diag
-
-.. autofunction:: sympy.matrices.dense.jordan_cell
-
-.. autofunction:: sympy.matrices.dense.hessian
-
-.. autofunction:: sympy.matrices.dense.GramSchmidt
-
-.. autofunction:: sympy.matrices.dense.wronskian
-
-.. autofunction:: sympy.matrices.dense.casoratian
-
-.. autofunction:: sympy.matrices.dense.randMatrix
-
-Numpy Utility Functions Reference
----------------------------------
-
-.. autofunction:: sympy.matrices.dense.list2numpy
-
-.. autofunction:: sympy.matrices.dense.matrix2numpy
-
-.. autofunction:: sympy.matrices.dense.symarray
-
-.. autofunction:: sympy.matrices.dense.rot_axis1
-
-.. autofunction:: sympy.matrices.dense.rot_axis2
-
-.. autofunction:: sympy.matrices.dense.rot_axis3
-
-.. autofunction:: a2idx
diff --git a/dev-py3k/_sources/modules/matrices/sparse.txt b/dev-py3k/_sources/modules/matrices/sparse.txt
deleted file mode 100644
index 734476fa1c9..00000000000
--- a/dev-py3k/_sources/modules/matrices/sparse.txt
+++ /dev/null
@@ -1,16 +0,0 @@
-Sparse Matrices
-===============
-
-.. module:: sympy.matrices.sparse
-
-SparseMatrix Class Reference
-----------------------------
-.. autoclass:: SparseMatrix
-   :members:
-.. autoclass:: MutableSparseMatrix
-   :members:
-
-ImmutableSparseMatrix Class Reference
--------------------------------------
-.. autoclass:: sympy.matrices.immutable.ImmutableSparseMatrix
-   :members:
diff --git a/dev-py3k/_sources/modules/mpmath/basics.txt b/dev-py3k/_sources/modules/mpmath/basics.txt
deleted file mode 100644
index 18cab996a74..00000000000
--- a/dev-py3k/_sources/modules/mpmath/basics.txt
+++ /dev/null
@@ -1,227 +0,0 @@
-Basic usage
-===========================
-
-In interactive code examples that follow, it will be assumed that
-all items in the ``mpmath`` namespace have been imported::
-
-    >>> from mpmath import *
-
-Importing everything can be convenient, especially when using mpmath interactively, but be
-careful when mixing mpmath with other libraries! To avoid inadvertently overriding
-other functions or objects, explicitly import only the needed objects, or use
-the ``mpmath.`` or ``mp.`` namespaces::
-
-    from mpmath import sin, cos
-    sin(1), cos(1)
-
-    import mpmath
-    mpmath.sin(1), mpmath.cos(1)
-
-    from mpmath import mp    # mp context object -- to be explained
-    mp.sin(1), mp.cos(1)
-
-
-Number types
-------------
-
-Mpmath provides the following numerical types:
-
-    +------------+----------------+
-    | Class      | Description    |
-    +============+================+
-    | ``mpf``    | Real float     |
-    +------------+----------------+
-    | ``mpc``    | Complex float  |
-    +------------+----------------+
-    | ``matrix`` | Matrix         |
-    +------------+----------------+
-
-The following section will provide a very short introduction to the types ``mpf`` and ``mpc``. Intervals and matrices are described further in the documentation chapters on interval arithmetic and matrices / linear algebra.
-
-The ``mpf`` type is analogous to Python's built-in ``float``. It holds a real number or one of the special values ``inf`` (positive infinity), ``-inf`` (negative infinity) and ``nan`` (not-a-number, indicating an indeterminate result). You can create ``mpf`` instances from strings, integers, floats, and other ``mpf`` instances:
-
-    >>> mpf(4)
-    mpf('4.0')
-    >>> mpf(2.5)
-    mpf('2.5')
-    >>> mpf("1.25e6")
-    mpf('1250000.0')
-    >>> mpf(mpf(2))
-    mpf('2.0')
-    >>> mpf("inf")
-    mpf('+inf')
-
-The ``mpc`` type represents a complex number in rectangular form as a pair of ``mpf`` instances. It can be constructed from a Python ``complex``, a real number, or a pair of real numbers:
-
-    >>> mpc(2,3)
-    mpc(real='2.0', imag='3.0')
-    >>> mpc(complex(2,3)).imag
-    mpf('3.0')
-
-You can mix ``mpf`` and ``mpc`` instances with each other and with Python numbers:
-
-    >>> mpf(3) + 2*mpf('2.5') + 1.0
-    mpf('9.0')
-    >>> mp.dps = 15      # Set precision (see below)
-    >>> mpc(1j)**0.5
-    mpc(real='0.70710678118654757', imag='0.70710678118654757')
-
-
-Setting the precision
----------------------
-
-Mpmath uses a global working precision; it does not keep track of the precision or accuracy of individual numbers. Performing an arithmetic operation or calling ``mpf()`` rounds the result to the current working precision. The working precision is controlled by a context object called ``mp``, which has the following default state:
-
-    >>> print mp
-    Mpmath settings:
-      mp.prec = 53                [default: 53]
-      mp.dps = 15                 [default: 15]
-      mp.trap_complex = False     [default: False]
-
-The term **prec** denotes the binary precision (measured in bits) while **dps** (short for *decimal places*) is the decimal precision. Binary and decimal precision are related roughly according to the formula ``prec = 3.33*dps``. For example, it takes a precision of roughly 333 bits to hold an approximation of pi that is accurate to 100 decimal places (actually slightly more than 333 bits is used).
-
-Changing either precision property of the ``mp`` object automatically updates the other; usually you just want to change the ``dps`` value:
-
-    >>> mp.dps = 100
-    >>> mp.dps
-    100
-    >>> mp.prec
-    336
-
-When the precision has been set, all ``mpf`` operations are carried out at that precision::
-
-    >>> mp.dps = 50
-    >>> mpf(1) / 6
-    mpf('0.16666666666666666666666666666666666666666666666666656')
-    >>> mp.dps = 25
-    >>> mpf(2) ** mpf('0.5')
-    mpf('1.414213562373095048801688713')
-
-The precision of complex arithmetic is also controlled by the ``mp`` object:
-
-    >>> mp.dps = 10
-    >>> mpc(1,2) / 3
-    mpc(real='0.3333333333321', imag='0.6666666666642')
-
-There is no restriction on the magnitude of numbers. An ``mpf`` can for example hold an approximation of a large Mersenne prime:
-
-    >>> mp.dps = 15
-    >>> print mpf(2)**32582657 - 1
-    1.24575026015369e+9808357
-
-Or why not 1 googolplex:
-
-    >>> print mpf(10) ** (10**100)  # doctest:+ELLIPSIS
-    1.0e+100000000000000000000000000000000000000000000000000...
-
-The (binary) exponent is stored exactly and is independent of the precision.
-
-Temporarily changing the precision
-..................................
-
-It is often useful to change the precision during only part of a calculation. A way to temporarily increase the precision and then restore it is as follows:
-
-    >>> mp.prec += 2
-    >>> # do_something()
-    >>> mp.prec -= 2
-
-In Python 2.5, the ``with`` statement along with the mpmath functions ``workprec``, ``workdps``, ``extraprec`` and ``extradps`` can be used to temporarily change precision in a more safe manner:
-
-    >>> from __future__ import with_statement
-    >>> with workdps(20):  # doctest: +SKIP
-    ...     print mpf(1)/7
-    ...     with extradps(10):
-    ...         print mpf(1)/7
-    ...
-    0.14285714285714285714
-    0.142857142857142857142857142857
-    >>> mp.dps
-    15
-
-The ``with`` statement ensures that the precision gets reset when exiting the block, even in the case that an exception is raised. (The effect of the ``with`` statement can be emulated in Python 2.4 by using a ``try/finally`` block.)
-
-The ``workprec`` family of functions can also be used as function decorators:
-
-    >>> @workdps(6)
-    ... def f():
-    ...     return mpf(1)/3
-    ...
-    >>> f()
-    mpf('0.33333331346511841')
-
-
-Some functions accept the ``prec`` and ``dps`` keyword arguments and this will override the global working precision. Note that this will not affect the precision at which the result is printed, so to get all digits, you must either use increase precision afterward when printing or use ``nstr``/``nprint``:
-
-    >>> mp.dps = 15
-    >>> print exp(1)
-    2.71828182845905
-    >>> print exp(1, dps=50)      # Extra digits won't be printed
-    2.71828182845905
-    >>> nprint(exp(1, dps=50), 50)
-    2.7182818284590452353602874713526624977572470937
-
-Finally, instead of using the global context object ``mp``, you can create custom contexts and work with methods of those instances instead of global functions. The working precision will be local to each context object:
-
-    >>> mp2 = mp.clone()
-    >>> mp.dps = 10
-    >>> mp2.dps = 20
-    >>> print mp.mpf(1) / 3
-    0.3333333333
-    >>> print mp2.mpf(1) / 3
-    0.33333333333333333333
-
-**Note**: the ability to create multiple contexts is a new feature that is only partially implemented. Not all mpmath functions are yet available as context-local methods. In the present version, you are likely to encounter bugs if you try mixing different contexts.
-
-Providing correct input
------------------------
-
-Note that when creating a new ``mpf``, the value will at most be as accurate as the input. *Be careful when mixing mpmath numbers with Python floats*. When working at high precision, fractional ``mpf`` values should be created from strings or integers:
-
-    >>> mp.dps = 30
-    >>> mpf(10.9)   # bad
-    mpf('10.9000000000000003552713678800501')
-    >>> mpf('10.9')  # good
-    mpf('10.8999999999999999999999999999997')
-    >>> mpf(109) / mpf(10)   # also good
-    mpf('10.8999999999999999999999999999997')
-    >>> mp.dps = 15
-
-(Binary fractions such as 0.5, 1.5, 0.75, 0.125, etc, are generally safe as input, however, since those can be represented exactly by Python floats.)
-
-Printing
---------
-
-By default, the ``repr()`` of a number includes its type signature. This way ``eval`` can be used to recreate a number from its string representation:
-
-    >>> eval(repr(mpf(2.5)))
-    mpf('2.5')
-
-Prettier output can be obtained by using ``str()`` or ``print``, which hide the ``mpf`` and ``mpc`` signatures and also suppress rounding artifacts in the last few digits:
-
-    >>> mpf("3.14159")
-    mpf('3.1415899999999999')
-    >>> print mpf("3.14159")
-    3.14159
-    >>> print mpc(1j)**0.5
-    (0.707106781186548 + 0.707106781186548j)
-
-Setting the ``mp.pretty`` option will use the ``str()``-style output for ``repr()`` as well:
-
-    >>> mp.pretty = True
-    >>> mpf(0.6)
-    0.6
-    >>> mp.pretty = False
-    >>> mpf(0.6)
-    mpf('0.59999999999999998')
-
-The number of digits with which numbers are printed by default is determined by the working precision. To specify the number of digits to show without changing the working precision, use :func:`mpmath.nstr` and :func:`mpmath.nprint`:
-
-    >>> a = mpf(1) / 6
-    >>> a
-    mpf('0.16666666666666666')
-    >>> nstr(a, 8)
-    '0.16666667'
-    >>> nprint(a, 8)
-    0.16666667
-    >>> nstr(a, 50)
-    '0.16666666666666665741480812812369549646973609924316'
diff --git a/dev-py3k/_sources/modules/mpmath/calculus/approximation.txt b/dev-py3k/_sources/modules/mpmath/calculus/approximation.txt
deleted file mode 100644
index c6b530969d4..00000000000
--- a/dev-py3k/_sources/modules/mpmath/calculus/approximation.txt
+++ /dev/null
@@ -1,23 +0,0 @@
-Function approximation
-----------------------
-
-Taylor series (``taylor``)
-..........................
-
-.. autofunction:: mpmath.taylor
-
-Pade approximation (``pade``)
-.............................
-
-.. autofunction:: mpmath.pade
-
-Chebyshev approximation (``chebyfit``)
-......................................
-
-.. autofunction:: mpmath.chebyfit
-
-Fourier series (``fourier``, ``fourierval``)
-............................................
-
-.. autofunction:: mpmath.fourier
-.. autofunction:: mpmath.fourierval
diff --git a/dev-py3k/_sources/modules/mpmath/calculus/differentiation.txt b/dev-py3k/_sources/modules/mpmath/calculus/differentiation.txt
deleted file mode 100644
index 5660cf16977..00000000000
--- a/dev-py3k/_sources/modules/mpmath/calculus/differentiation.txt
+++ /dev/null
@@ -1,19 +0,0 @@
-Differentiation
----------------
-
-Numerical derivatives (``diff``, ``diffs``)
-...........................................
-
-.. autofunction:: mpmath.diff
-.. autofunction:: mpmath.diffs
-
-Composition of derivatives (``diffs_prod``, ``diffs_exp``)
-..........................................................
-
-.. autofunction:: mpmath.diffs_prod
-.. autofunction:: mpmath.diffs_exp
-
-Fractional derivatives / differintegration (``differint``)
-............................................................
-
-.. autofunction:: mpmath.differint
diff --git a/dev-py3k/_sources/modules/mpmath/calculus/index.txt b/dev-py3k/_sources/modules/mpmath/calculus/index.txt
deleted file mode 100644
index d491dbb0fad..00000000000
--- a/dev-py3k/_sources/modules/mpmath/calculus/index.txt
+++ /dev/null
@@ -1,13 +0,0 @@
-Numerical calculus
-==================
-
-.. toctree::
-   :maxdepth: 2
-
-   polynomials.rst
-   optimization.rst
-   sums_limits.rst
-   differentiation.rst
-   integration.rst
-   odes.rst
-   approximation.rst
diff --git a/dev-py3k/_sources/modules/mpmath/calculus/integration.txt b/dev-py3k/_sources/modules/mpmath/calculus/integration.txt
deleted file mode 100644
index f260b3f60e2..00000000000
--- a/dev-py3k/_sources/modules/mpmath/calculus/integration.txt
+++ /dev/null
@@ -1,31 +0,0 @@
-Numerical integration (quadrature)
-----------------------------------
-
-Standard quadrature (``quad``)
-..............................
-
-.. autofunction:: mpmath.quad
-
-Oscillatory quadrature (``quadosc``)
-....................................
-
-.. autofunction:: mpmath.quadosc
-
-Quadrature rules
-................
-
-.. autoclass:: mpmath.calculus.quadrature.QuadratureRule
-   :members:
-
-Tanh-sinh rule
-~~~~~~~~~~~~~~
-
-.. autoclass:: mpmath.calculus.quadrature.TanhSinh
-   :members:
-
-
-Gauss-Legendre rule
-~~~~~~~~~~~~~~~~~~~
-
-.. autoclass:: mpmath.calculus.quadrature.GaussLegendre
-   :members:
diff --git a/dev-py3k/_sources/modules/mpmath/calculus/odes.txt b/dev-py3k/_sources/modules/mpmath/calculus/odes.txt
deleted file mode 100644
index 7c2ae83b090..00000000000
--- a/dev-py3k/_sources/modules/mpmath/calculus/odes.txt
+++ /dev/null
@@ -1,7 +0,0 @@
-Ordinary differential equations
--------------------------------
-
-Solving the ODE initial value problem (``odefun``)
-..................................................
-
-.. autofunction:: mpmath.odefun
diff --git a/dev-py3k/_sources/modules/mpmath/calculus/optimization.txt b/dev-py3k/_sources/modules/mpmath/calculus/optimization.txt
deleted file mode 100644
index b42b8efae8c..00000000000
--- a/dev-py3k/_sources/modules/mpmath/calculus/optimization.txt
+++ /dev/null
@@ -1,29 +0,0 @@
-Root-finding and optimization
------------------------------
-
-Root-finding (``findroot``)
-...........................
-
-.. autofunction:: mpmath.findroot(f, x0, solver=Secant, tol=None, verbose=False, verify=True, **kwargs)
-
-Solvers
-^^^^^^^
-
-.. autoclass:: mpmath.calculus.optimization.Secant
-.. autoclass:: mpmath.calculus.optimization.Newton
-.. autoclass:: mpmath.calculus.optimization.MNewton
-.. autoclass:: mpmath.calculus.optimization.Halley
-.. autoclass:: mpmath.calculus.optimization.Muller
-.. autoclass:: mpmath.calculus.optimization.Bisection
-.. autoclass:: mpmath.calculus.optimization.Illinois
-.. autoclass:: mpmath.calculus.optimization.Pegasus
-.. autoclass:: mpmath.calculus.optimization.Anderson
-.. autoclass:: mpmath.calculus.optimization.Ridder
-.. autoclass:: mpmath.calculus.optimization.ANewton
-.. autoclass:: mpmath.calculus.optimization.MDNewton
-
-
-.. Minimization and maximization (``findmin``, ``findmax``)
-.. ........................................................
-
-.. (To be added.)
diff --git a/dev-py3k/_sources/modules/mpmath/calculus/polynomials.txt b/dev-py3k/_sources/modules/mpmath/calculus/polynomials.txt
deleted file mode 100644
index 304fdd99775..00000000000
--- a/dev-py3k/_sources/modules/mpmath/calculus/polynomials.txt
+++ /dev/null
@@ -1,15 +0,0 @@
-Polynomials
------------
-
-See also :func:`taylor` and :func:`chebyfit` for
-approximation of functions by polynomials.
-
-Polynomial evaluation (``polyval``)
-...................................
-
-.. autofunction:: mpmath.polyval
-
-Polynomial roots (``polyroots``)
-................................
-
-.. autofunction:: mpmath.polyroots
diff --git a/dev-py3k/_sources/modules/mpmath/calculus/sums_limits.txt b/dev-py3k/_sources/modules/mpmath/calculus/sums_limits.txt
deleted file mode 100644
index f8c932bccc5..00000000000
--- a/dev-py3k/_sources/modules/mpmath/calculus/sums_limits.txt
+++ /dev/null
@@ -1,58 +0,0 @@
-Sums, products, limits and extrapolation
-----------------------------------------
-
-The functions listed here permit approximation of infinite
-sums, products, and other sequence limits.
-Use :func:`mpmath.fsum` and :func:`mpmath.fprod`
-for summation and multiplication of finite sequences.
-
-Summation
-..........................................
-
-:func:`nsum`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.nsum
-
-:func:`sumem`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.sumem
-
-:func:`sumap`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.sumap
-
-Products
-...............................
-
-:func:`nprod`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.nprod
-
-Limits (``limit``)
-..................
-
-:func:`limit`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.limit
-
-Extrapolation
-..........................................
-
-The following functions provide a direct interface to
-extrapolation algorithms. :func:`nsum` and :func:`limit` essentially
-work by calling the following functions with an increasing
-number of terms until the extrapolated limit is accurate enough.
-
-The following functions may be useful to call directly if the
-precise number of terms needed to achieve a desired accuracy is
-known in advance, or if one wishes to study the convergence
-properties of the algorithms.
-
-
-:func:`richardson`
-^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.richardson
-
-:func:`shanks`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.shanks
diff --git a/dev-py3k/_sources/modules/mpmath/contexts.txt b/dev-py3k/_sources/modules/mpmath/contexts.txt
deleted file mode 100644
index 7a84b2bd6aa..00000000000
--- a/dev-py3k/_sources/modules/mpmath/contexts.txt
+++ /dev/null
@@ -1,306 +0,0 @@
-Contexts
-========
-
-High-level code in mpmath is implemented as methods on a "context object". The context implements arithmetic, type conversions and other fundamental operations. The context also holds settings such as precision, and stores cache data. A few different contexts (with a mostly compatible interface) are provided so that the high-level algorithms can be used with different implementations of the underlying arithmetic, allowing different features and speed-accuracy tradeoffs. Currently, mpmath provides the following contexts:
-
-  * Arbitrary-precision arithmetic (``mp``)
-  * A faster Cython-based version of ``mp`` (used by default in Sage, and currently only available there)
-  * Arbitrary-precision interval arithmetic (``iv``)
-  * Double-precision arithmetic using Python's builtin ``float`` and ``complex`` types (``fp``)
-
-Most global functions in the global mpmath namespace are actually methods of the ``mp``
-context. This fact is usually transparent to the user, but sometimes shows up in the
-form of an initial parameter called "ctx" visible in the help for the function::
-
-    >>> import mpmath
-    >>> help(mpmath.fsum)   # doctest:+SKIP
-    Help on method fsum in module mpmath.ctx_mp_python:
-
-    fsum(ctx, terms, absolute=False, squared=False) method of mpmath.ctx_mp.MPContext instance
-        Calculates a sum containing a finite number of terms (for infinite
-        series, see :func:`~mpmath.nsum`). The terms will be converted to
-    ...
-
-The following operations are equivalent::
-
-    >>> mpmath.mp.dps = 15; mpmath.mp.pretty = False
-    >>> mpmath.fsum([1,2,3])
-    mpf('6.0')
-    >>> mpmath.mp.fsum([1,2,3])
-    mpf('6.0')
-
-The corresponding operation using the ``fp`` context::
-
-    >>> mpmath.fp.fsum([1,2,3])
-    6.0
-
-Common interface
-----------------
-
-``ctx.mpf`` creates a real number::
-
-    >>> from mpmath import mp, fp
-    >>> mp.mpf(3)
-    mpf('3.0')
-    >>> fp.mpf(3)
-    3.0
-
-``ctx.mpc`` creates a complex number::
-
-    >>> mp.mpc(2,3)
-    mpc(real='2.0', imag='3.0')
-    >>> fp.mpc(2,3)
-    (2+3j)
-
-``ctx.matrix`` creates a matrix::
-
-    >>> mp.matrix([[1,0],[0,1]])
-    matrix(
-    [['1.0', '0.0'],
-     ['0.0', '1.0']])
-    >>> _[0,0]
-    mpf('1.0')
-    >>> fp.matrix([[1,0],[0,1]])
-    matrix(
-    [['1.0', '0.0'],
-     ['0.0', '1.0']])
-    >>> _[0,0]
-    1.0
-
-``ctx.prec`` holds the current precision (in bits)::
-
-    >>> mp.prec
-    53
-    >>> fp.prec
-    53
-
-``ctx.dps`` holds the current precision (in digits)::
-
-    >>> mp.dps
-    15
-    >>> fp.dps
-    15
-
-``ctx.pretty`` controls whether objects should be pretty-printed automatically by :func:`repr`. Pretty-printing for ``mp`` numbers is disabled by default so that they can clearly be distinguished from Python numbers and so that ``eval(repr(x)) == x`` works::
-
-    >>> mp.mpf(3)
-    mpf('3.0')
-    >>> mpf = mp.mpf
-    >>> eval(repr(mp.mpf(3)))
-    mpf('3.0')
-    >>> mp.pretty = True
-    >>> mp.mpf(3)
-    3.0
-    >>> fp.matrix([[1,0],[0,1]])
-    matrix(
-    [['1.0', '0.0'],
-     ['0.0', '1.0']])
-    >>> fp.pretty = True
-    >>> fp.matrix([[1,0],[0,1]])
-    [1.0  0.0]
-    [0.0  1.0]
-    >>> fp.pretty = False
-    >>> mp.pretty = False
-
-
-Arbitrary-precision floating-point (``mp``)
----------------------------------------------
-
-The ``mp`` context is what most users probably want to use most of the time, as it supports the most functions, is most well-tested, and is implemented with a high level of optimization. Nearly all examples in this documentation use ``mp`` functions.
-
-See :doc:`basics` for a description of basic usage.
-
-Arbitrary-precision interval arithmetic (``iv``)
-------------------------------------------------
-
-The ``iv.mpf`` type represents a closed interval `[a,b]`; that is, the set `\{x : a \le x \le b\}`, where `a` and `b` are arbitrary-precision floating-point values, possibly `\pm \infty`. The ``iv.mpc`` type represents a rectangular complex interval `[a,b] + [c,d]i`; that is, the set `\{z = x+iy : a \le x \le b \land c \le y \le d\}`.
-
-Interval arithmetic provides rigorous error tracking. If `f` is a mathematical function and `\hat f` is its interval arithmetic version, then the basic guarantee of interval arithmetic is that `f(v) \subseteq \hat f(v)` for any input interval `v`. Put differently, if an interval represents the known uncertainty for a fixed number, any sequence of interval operations will produce an interval that contains what would be the result of applying the same sequence of operations to the exact number. The principal drawbacks of interval arithmetic are speed (``iv`` arithmetic is typically at least two times slower than ``mp`` arithmetic) and that it sometimes provides far too pessimistic bounds.
-
-.. note ::
-
-    The support for interval arithmetic in mpmath is still experimental, and many functions
-    do not yet properly support intervals. Please use this feature with caution.
-
-Intervals can be created from single numbers (treated as zero-width intervals) or pairs of endpoint numbers. Strings are treated as exact decimal numbers. Note that a Python float like ``0.1`` generally does not represent the same number as its literal; use ``'0.1'`` instead::
-
-    >>> from mpmath import iv
-    >>> iv.dps = 15; iv.pretty = False
-    >>> iv.mpf(3)
-    mpi('3.0', '3.0')
-    >>> print iv.mpf(3)
-    [3.0, 3.0]
-    >>> iv.pretty = True
-    >>> iv.mpf([2,3])
-    [2.0, 3.0]
-    >>> iv.mpf(0.1)   # probably not intended
-    [0.10000000000000000555, 0.10000000000000000555]
-    >>> iv.mpf('0.1')   # good, gives a containing interval
-    [0.099999999999999991673, 0.10000000000000000555]
-    >>> iv.mpf(['0.1', '0.2'])
-    [0.099999999999999991673, 0.2000000000000000111]
-
-The fact that ``'0.1'`` results in an interval of nonzero width indicates that 1/10 cannot be represented using binary floating-point numbers at this precision level (in fact, it cannot be represented exactly at any precision).
-
-Intervals may be infinite or half-infinite::
-
-    >>> print 1 / iv.mpf([2, 'inf'])
-    [0.0, 0.5]
-
-The equality testing operators ``==`` and ``!=`` check whether their operands are identical as intervals; that is, have the same endpoints. The ordering operators ``< <= > >=`` permit inequality testing using triple-valued logic: a guaranteed inequality returns ``True`` or ``False`` while an indeterminate inequality returns ``None``::
-
-    >>> iv.mpf([1,2]) == iv.mpf([1,2])
-    True
-    >>> iv.mpf([1,2]) != iv.mpf([1,2])
-    False
-    >>> iv.mpf([1,2]) <= 2
-    True
-    >>> iv.mpf([1,2]) > 0
-    True
-    >>> iv.mpf([1,2]) < 1
-    False
-    >>> iv.mpf([1,2]) < 2    # returns None
-    >>> iv.mpf([2,2]) < 2
-    False
-    >>> iv.mpf([1,2]) <= iv.mpf([2,3])
-    True
-    >>> iv.mpf([1,2]) < iv.mpf([2,3])  # returns None
-    >>> iv.mpf([1,2]) < iv.mpf([-1,0])
-    False
-
-The ``in`` operator tests whether a number or interval is contained in another interval::
-
-    >>> iv.mpf([0,2]) in iv.mpf([0,10])
-    True
-    >>> 3 in iv.mpf(['-inf', 0])
-    False
-
-Intervals have the properties ``.a``, ``.b`` (endpoints), ``.mid``, and ``.delta`` (width)::
-
-    >>> x = iv.mpf([2, 5])
-    >>> x.a
-    [2.0, 2.0]
-    >>> x.b
-    [5.0, 5.0]
-    >>> x.mid
-    [3.5, 3.5]
-    >>> x.delta
-    [3.0, 3.0]
-
-Some transcendental functions are supported::
-
-    >>> iv.dps = 15
-    >>> mp.dps = 15
-    >>> iv.mpf([0.5,1.5]) ** iv.mpf([0.5, 1.5])
-    [0.35355339059327373086, 1.837117307087383633]
-    >>> iv.exp(0)
-    [1.0, 1.0]
-    >>> iv.exp(['-inf','inf'])
-    [0.0, +inf]
-    >>>
-    >>> iv.exp(['-inf',0])
-    [0.0, 1.0]
-    >>> iv.exp([0,'inf'])
-    [1.0, +inf]
-    >>> iv.exp([0,1])
-    [1.0, 2.7182818284590455349]
-    >>>
-    >>> iv.log(1)
-    [0.0, 0.0]
-    >>> iv.log([0,1])
-    [-inf, 0.0]
-    >>> iv.log([0,'inf'])
-    [-inf, +inf]
-    >>> iv.log(2)
-    [0.69314718055994528623, 0.69314718055994539725]
-    >>>
-    >>> iv.sin([100,'inf'])
-    [-1.0, 1.0]
-    >>> iv.cos(['-0.1','0.1'])
-    [0.99500416527802570954, 1.0]
-
-Interval arithmetic is useful for proving inequalities involving irrational numbers.
-Naive use of ``mp`` arithmetic may result in wrong conclusions, such as the following::
-
-    >>> mp.dps = 25
-    >>> x = mp.exp(mp.pi*mp.sqrt(163))
-    >>> y = mp.mpf(640320**3+744)
-    >>> print x
-    262537412640768744.0000001
-    >>> print y
-    262537412640768744.0
-    >>> x > y
-    True
-
-But the correct result is `e^{\pi \sqrt{163}} < 262537412640768744`, as can be
-seen by increasing the precision::
-
-    >>> mp.dps = 50
-    >>> print mp.exp(mp.pi*mp.sqrt(163))
-    262537412640768743.99999999999925007259719818568888
-
-With interval arithmetic, the comparison returns ``None`` until the precision
-is large enough for `x-y` to have a definite sign::
-
-    >>> iv.dps = 15
-    >>> iv.exp(iv.pi*iv.sqrt(163)) > (640320**3+744)
-    >>> iv.dps = 30
-    >>> iv.exp(iv.pi*iv.sqrt(163)) > (640320**3+744)
-    >>> iv.dps = 60
-    >>> iv.exp(iv.pi*iv.sqrt(163)) > (640320**3+744)
-    False
-    >>> iv.dps = 15
-
-Fast low-precision arithmetic (``fp``)
----------------------------------------------
-
-Although mpmath is generally designed for arbitrary-precision arithmetic, many of the high-level algorithms work perfectly well with ordinary Python ``float`` and ``complex`` numbers, which use hardware double precision (on most systems, this corresponds to 53 bits of precision). Whereas the global functions (which are methods of the ``mp`` object) always convert inputs to mpmath numbers, the ``fp`` object instead converts them to ``float`` or ``complex``, and in some cases employs basic functions optimized for double precision. When large amounts of function evaluations (numerical integration, plotting, etc) are required, and when ``fp`` arithmetic provides sufficient accuracy, this can give a significant speedup over ``mp`` arithmetic.
-
-To take advantage of this feature, simply use the ``fp`` prefix, i.e. write ``fp.func`` instead of ``func`` or ``mp.func``::
-
-    >>> u = fp.erfc(2.5)
-    >>> print u
-    0.000406952017445
-    >>> type(u)
-    <type 'float'>
-    >>> mp.dps = 15
-    >>> print mp.erfc(2.5)
-    0.000406952017444959
-    >>> fp.matrix([[1,2],[3,4]]) ** 2
-    matrix(
-    [['7.0', '10.0'],
-     ['15.0', '22.0']])
-    >>>
-    >>> type(_[0,0])
-    <type 'float'>
-    >>> print fp.quad(fp.sin, [0, fp.pi])    # numerical integration
-    2.0
-
-The ``fp`` context wraps Python's ``math`` and ``cmath`` modules for elementary functions. It supports both real and complex numbers and automatically generates complex results for real inputs (``math`` raises an exception)::
-
-    >>> fp.sqrt(5)
-    2.2360679774997898
-    >>> fp.sqrt(-5)
-    2.2360679774997898j
-    >>> fp.sin(10)
-    -0.54402111088936977
-    >>> fp.power(-1, 0.25)
-    (0.70710678118654757+0.70710678118654746j)
-    >>> (-1) ** 0.25
-    Traceback (most recent call last):
-      ...
-    ValueError: negative number cannot be raised to a fractional power
-
-The ``prec`` and ``dps`` attributes can be changed (for interface compatibility with the ``mp`` context) but this has no effect::
-
-    >>> fp.prec
-    53
-    >>> fp.dps
-    15
-    >>> fp.prec = 80
-    >>> fp.prec
-    53
-    >>> fp.dps
-    15
-
-Due to intermediate rounding and cancellation errors, results computed with ``fp`` arithmetic may be much less accurate than those computed with ``mp`` using an equivalent precision (``mp.prec = 53``), since the latter often uses increased internal precision. The accuracy is highly problem-dependent: for some functions, ``fp`` almost always gives 14-15 correct digits; for others, results can be accurate to only 2-3 digits or even completely wrong. The recommended use for ``fp`` is therefore to speed up large-scale computations where accuracy can be verified in advance on a subset of the input set, or where results can be verified afterwards.
diff --git a/dev-py3k/_sources/modules/mpmath/functions/bessel.txt b/dev-py3k/_sources/modules/mpmath/functions/bessel.txt
deleted file mode 100644
index 19def1b1793..00000000000
--- a/dev-py3k/_sources/modules/mpmath/functions/bessel.txt
+++ /dev/null
@@ -1,185 +0,0 @@
-Bessel functions and related functions
---------------------------------------
-
-The functions in this section arise as solutions to various differential equations in physics, typically describing wavelike oscillatory behavior or a combination of oscillation and exponential decay or growth. Mathematically, they are special cases of the confluent hypergeometric functions `\,_0F_1`, `\,_1F_1` and `\,_1F_2` (see :doc:`hypergeometric`).
-
-
-Bessel functions
-...................................................
-
-:func:`besselj`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-.. autofunction:: mpmath.besselj(n,x,derivative=0)
-.. autofunction:: mpmath.j0(x)
-.. autofunction:: mpmath.j1(x)
-
-:func:`bessely`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.bessely(n,x,derivative=0)
-
-:func:`besseli`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.besseli(n,x,derivative=0)
-
-:func:`besselk`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.besselk(n,x)
-
-
-Bessel function zeros
-...............................
-
-:func:`besseljzero`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.besseljzero(v,m,derivative=0)
-
-:func:`besselyzero`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.besselyzero(v,m,derivative=0)
-
-
-Hankel functions
-................
-
-:func:`hankel1`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.hankel1(n,x)
-
-:func:`hankel2`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.hankel2(n,x)
-
-
-Kelvin functions
-................
-
-:func:`ber`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.ber
-
-:func:`bei`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-.. autofunction:: mpmath.bei
-
-:func:`ker`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.ker
-
-:func:`kei`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.kei
-
-
-Struve functions
-...................................................
-
-:func:`struveh`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.struveh
-
-:func:`struvel`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.struvel
-
-
-Anger-Weber functions
-...................................................
-
-:func:`angerj`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.angerj
-
-:func:`webere`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.webere
-
-
-Lommel functions
-...................................................
-
-:func:`lommels1`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.lommels1
-
-:func:`lommels2`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.lommels2
-
-Airy and Scorer functions
-...............................................
-
-:func:`airyai`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.airyai(z, derivative=0, **kwargs)
-
-:func:`airybi`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.airybi(z, derivative=0, **kwargs)
-
-:func:`airyaizero`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.airyaizero(k, derivative=0)
-
-:func:`airybizero`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.airybizero(k, derivative=0, complex=0)
-
-:func:`scorergi`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.scorergi(z, **kwargs)
-
-:func:`scorerhi`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.scorerhi(z, **kwargs)
-
-
-Coulomb wave functions
-...............................................
-
-:func:`coulombf`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.coulombf(l,eta,z)
-
-:func:`coulombg`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.coulombg(l,eta,z)
-
-:func:`coulombc`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.coulombc(l,eta)
-
-Confluent U and Whittaker functions
-...................................
-
-:func:`hyperu`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.hyperu(a, b, z)
-
-:func:`whitm`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.whitm(k,m,z)
-
-:func:`whitw`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.whitw(k,m,z)
-
-Parabolic cylinder functions
-.................................
-
-:func:`pcfd`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.pcfd(n,z,**kwargs)
-
-:func:`pcfu`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.pcfu(a,z,**kwargs)
-
-:func:`pcfv`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.pcfv(a,z,**kwargs)
-
-:func:`pcfw`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.pcfw(a,z,**kwargs)
diff --git a/dev-py3k/_sources/modules/mpmath/functions/constants.txt b/dev-py3k/_sources/modules/mpmath/functions/constants.txt
deleted file mode 100644
index 42c5de7ac7c..00000000000
--- a/dev-py3k/_sources/modules/mpmath/functions/constants.txt
+++ /dev/null
@@ -1,85 +0,0 @@
-Mathematical constants
-----------------------
-
-Mpmath supports arbitrary-precision computation of various common (and less common) mathematical constants. These constants are implemented as lazy objects that can evaluate to any precision. Whenever the objects are used as function arguments or as operands in arithmetic operations, they automagically evaluate to the current working precision. A lazy number can be converted to a regular ``mpf`` using the unary ``+`` operator, or by calling it as a function::
-
-    >>> from mpmath import *
-    >>> mp.dps = 15
-    >>> pi
-    <pi: 3.14159~>
-    >>> 2*pi
-    mpf('6.2831853071795862')
-    >>> +pi
-    mpf('3.1415926535897931')
-    >>> pi()
-    mpf('3.1415926535897931')
-    >>> mp.dps = 40
-    >>> pi
-    <pi: 3.14159~>
-    >>> 2*pi
-    mpf('6.283185307179586476925286766559005768394338')
-    >>> +pi
-    mpf('3.141592653589793238462643383279502884197169')
-    >>> pi()
-    mpf('3.141592653589793238462643383279502884197169')
-
-Exact constants
-...............
-
-The predefined objects :data:`j` (imaginary unit), :data:`inf` (positive infinity) and :data:`nan` (not-a-number) are shortcuts to :class:`mpc` and :class:`mpf` instances with these fixed values.
-
-Pi (``pi``)
-....................................
-
-.. autoattribute:: mpmath.mp.pi
-
-Degree (``degree``)
-....................................
-
-.. autoattribute:: mpmath.mp.degree
-
-Base of the natural logarithm (``e``)
-.....................................
-
-.. autoattribute:: mpmath.mp.e
-
-Golden ratio (``phi``)
-......................
-
-.. autoattribute:: mpmath.mp.phi
-
-
-Euler's constant (``euler``)
-............................
-
-.. autoattribute:: mpmath.mp.euler
-
-Catalan's constant (``catalan``)
-................................
-
-.. autoattribute:: mpmath.mp.catalan
-
-Apery's constant (``apery``)
-............................
-
-.. autoattribute:: mpmath.mp.apery
-
-Khinchin's constant (``khinchin``)
-..................................
-
-.. autoattribute:: mpmath.mp.khinchin
-
-Glaisher's constant (``glaisher``)
-..................................
-
-.. autoattribute:: mpmath.mp.glaisher
-
-Mertens constant (``mertens``)
-..................................
-
-.. autoattribute:: mpmath.mp.mertens
-
-Twin prime constant (``twinprime``)
-...................................
-
-.. autoattribute:: mpmath.mp.twinprime
diff --git a/dev-py3k/_sources/modules/mpmath/functions/elliptic.txt b/dev-py3k/_sources/modules/mpmath/functions/elliptic.txt
deleted file mode 100644
index f8d2e1d6d2e..00000000000
--- a/dev-py3k/_sources/modules/mpmath/functions/elliptic.txt
+++ /dev/null
@@ -1,96 +0,0 @@
-Elliptic functions
-------------------
-
-.. automodule :: mpmath.functions.elliptic
-
-
-Elliptic arguments
-...................................................
-
-:func:`qfrom`
-^^^^^^^^^^^^^
-.. autofunction:: mpmath.qfrom(**kwargs)
-
-:func:`qbarfrom`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.qbarfrom(**kwargs)
-
-:func:`mfrom`
-^^^^^^^^^^^^^
-.. autofunction:: mpmath.mfrom(**kwargs)
-
-:func:`kfrom`
-^^^^^^^^^^^^^
-.. autofunction:: mpmath.kfrom(**kwargs)
-
-:func:`taufrom`
-^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.taufrom(**kwargs)
-
-
-Legendre elliptic integrals
-...................................................
-
-:func:`ellipk`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.ellipk(m, **kwargs)
-
-:func:`ellipf`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.ellipf(phi, m)
-
-:func:`ellipe`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.ellipe(*args)
-
-:func:`ellippi`
-^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.ellippi(*args)
-
-
-Carlson symmetric elliptic integrals
-...................................................
-
-:func:`elliprf`
-^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.elliprf(x, y, z)
-
-:func:`elliprc`
-^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.elliprc(x, y, pv=True)
-
-:func:`elliprj`
-^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.elliprj(x, y, z, p)
-
-:func:`elliprd`
-^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.elliprd(x, y, z)
-
-:func:`elliprg`
-^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.elliprg(x, y, z)
-
-
-Jacobi theta functions
-......................
-
-:func:`jtheta`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.jtheta(n,z,q,derivative=0)
-
-
-Jacobi elliptic functions
-.................................................................
-
-:func:`ellipfun`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.ellipfun(kind,u=None,m=None,q=None,k=None,tau=None)
-
-
-Modular functions
-......................
-
-:func:`kleinj`
-^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.kleinj(tau=None, **kwargs)
diff --git a/dev-py3k/_sources/modules/mpmath/functions/expintegrals.txt b/dev-py3k/_sources/modules/mpmath/functions/expintegrals.txt
deleted file mode 100644
index fe01ca4e4b7..00000000000
--- a/dev-py3k/_sources/modules/mpmath/functions/expintegrals.txt
+++ /dev/null
@@ -1,104 +0,0 @@
-Exponential integrals and error functions
------------------------------------------
-
-Exponential integrals give closed-form solutions to a large class of commonly occurring transcendental integrals that cannot be evaluated using elementary functions. Integrals of this type include those with an integrand of the form `t^a e^{t}` or `e^{-x^2}`, the latter giving rise to the Gaussian (or normal) probability distribution.
-
-The most general function in this section is the incomplete gamma function, to which all others can be reduced. The incomplete gamma function, in turn, can be expressed using hypergeometric functions (see :doc:`hypergeometric`).
-
-Incomplete gamma functions
-..........................
-
-:func:`gammainc`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.gammainc(z, a=0, b=inf, regularized=False)
-
-
-Exponential integrals
-.....................
-
-:func:`ei`
-^^^^^^^^^^
-.. autofunction:: mpmath.ei(x, **kwargs)
-
-:func:`e1`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.e1(x, **kwargs)
-
-:func:`expint`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.expint(*args)
-
-
-Logarithmic integral
-....................
-
-:func:`li`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.li(x, **kwargs)
-
-
-Trigonometric integrals
-.......................
-
-:func:`ci`
-^^^^^^^^^^
-.. autofunction:: mpmath.ci(x, **kwargs)
-
-:func:`si`
-^^^^^^^^^^
-.. autofunction:: mpmath.si(x, **kwargs)
-
-
-Hyperbolic integrals
-....................
-
-:func:`chi`
-^^^^^^^^^^^
-.. autofunction:: mpmath.chi(x, **kwargs)
-
-:func:`shi`
-^^^^^^^^^^^
-.. autofunction:: mpmath.shi(x, **kwargs)
-
-
-Error functions
-...............
-
-:func:`erf`
-^^^^^^^^^^^
-.. autofunction:: mpmath.erf(x, **kwargs)
-
-:func:`erfc`
-^^^^^^^^^^^^
-.. autofunction:: mpmath.erfc(x, **kwargs)
-
-:func:`erfi`
-^^^^^^^^^^^^
-.. autofunction:: mpmath.erfi(x)
-
-:func:`erfinv`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.erfinv(x)
-
-The normal distribution
-....................................................
-
-:func:`npdf`
-^^^^^^^^^^^^
-.. autofunction:: mpmath.npdf(x, mu=0, sigma=1)
-
-:func:`ncdf`
-^^^^^^^^^^^^
-.. autofunction:: mpmath.ncdf(x, mu=0, sigma=1)
-
-
-Fresnel integrals
-......................................................
-
-:func:`fresnels`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.fresnels(x)
-
-:func:`fresnelc`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.fresnelc(x)
diff --git a/dev-py3k/_sources/modules/mpmath/functions/gamma.txt b/dev-py3k/_sources/modules/mpmath/functions/gamma.txt
deleted file mode 100644
index db4a679fa53..00000000000
--- a/dev-py3k/_sources/modules/mpmath/functions/gamma.txt
+++ /dev/null
@@ -1,112 +0,0 @@
-Factorials and gamma functions
-------------------------------
-
-Factorials and factorial-like sums and products are basic tools of combinatorics and number theory. Much like the exponential function is fundamental to differential equations and analysis in general, the factorial function (and its extension to complex numbers, the gamma function) is fundamental to difference equations and functional equations.
-
-A large selection of factorial-like functions is implemented in mpmath. All functions support complex arguments, and arguments may be arbitrarily large. Results are numerical approximations, so to compute *exact* values a high enough precision must be set manually::
-
-    >>> mp.dps = 15; mp.pretty = True
-    >>> fac(100)
-    9.33262154439442e+157
-    >>> print int(_)    # most digits are wrong
-    93326215443944150965646704795953882578400970373184098831012889540582227238570431
-    295066113089288327277825849664006524270554535976289719382852181865895959724032
-    >>> mp.dps = 160
-    >>> fac(100)
-    93326215443944152681699238856266700490715968264381621468592963895217599993229915
-    608941463976156518286253697920827223758251185210916864000000000000000000000000.0
-
-The gamma and polygamma functions are closely related to :doc:`zeta`. See also :doc:`qfunctions` for q-analogs of factorial-like functions.
-
-
-Factorials
-..........
-
-:func:`factorial`/:func:`fac`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-.. autofunction:: mpmath.factorial(x, **kwargs)
-
-:func:`fac2`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.fac2(x)
-
-Binomial coefficients
-....................................................
-
-:func:`binomial`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.binomial(n,k)
-
-
-Gamma function
-..............
-
-:func:`gamma`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.gamma(x, **kwargs)
-
-:func:`rgamma`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.rgamma(x, **kwargs)
-
-:func:`gammaprod`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.gammaprod(a, b)
-
-:func:`loggamma`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.loggamma(x)
-
-
-Rising and falling factorials
-.............................
-
-:func:`rf`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.rf(x,n)
-
-:func:`ff`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.ff(x,n)
-
-Beta function
-.............
-
-:func:`beta`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.beta(x,y)
-
-:func:`betainc`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.betainc(a,b,x1=0,x2=1,regularized=False)
-
-
-Super- and hyperfactorials
-..........................
-
-:func:`superfac`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.superfac(z)
-
-:func:`hyperfac`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.hyperfac(z)
-
-:func:`barnesg`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.barnesg(z)
-
-
-Polygamma functions and harmonic numbers
-........................................
-
-:func:`psi`/:func:`digamma`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.psi(m, z)
-
-.. autofunction:: mpmath.digamma(z)
-
-:func:`harmonic`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.harmonic(z)
diff --git a/dev-py3k/_sources/modules/mpmath/functions/hyperbolic.txt b/dev-py3k/_sources/modules/mpmath/functions/hyperbolic.txt
deleted file mode 100644
index e48b7e795a7..00000000000
--- a/dev-py3k/_sources/modules/mpmath/functions/hyperbolic.txt
+++ /dev/null
@@ -1,56 +0,0 @@
-Hyperbolic functions
---------------------
-
-Hyperbolic functions
-....................
-
-:func:`cosh`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.cosh(x, **kwargs)
-
-:func:`sinh`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.sinh(x, **kwargs)
-
-:func:`tanh`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.tanh(x, **kwargs)
-
-:func:`sech`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.sech(x)
-
-:func:`csch`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.csch(x)
-
-:func:`coth`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.coth(x)
-
-Inverse hyperbolic functions
-............................
-
-:func:`acosh`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.acosh(x, **kwargs)
-
-:func:`asinh`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.asinh(x, **kwargs)
-
-:func:`atanh`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.atanh(x, **kwargs)
-
-:func:`asech`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.asech(x)
-
-:func:`acsch`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.acsch(x)
-
-:func:`acoth`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.acoth(x)
diff --git a/dev-py3k/_sources/modules/mpmath/functions/hypergeometric.txt b/dev-py3k/_sources/modules/mpmath/functions/hypergeometric.txt
deleted file mode 100644
index 4ac3a721adf..00000000000
--- a/dev-py3k/_sources/modules/mpmath/functions/hypergeometric.txt
+++ /dev/null
@@ -1,115 +0,0 @@
-Hypergeometric functions
-------------------------
-
-The functions listed in :doc:`expintegrals`, :doc:`bessel` and
-:doc:`orthogonal`, and many other functions as well, are merely
-particular instances of the generalized hypergeometric function `\,_pF_q`.
-The functions listed in the following section enable efficient
-direct evaluation of the underlying hypergeometric series, as
-well as linear combinations, limits with respect to parameters,
-and analytic continuations thereof. Extensions to twodimensional
-series are also provided. See also the basic or q-analog of
-the hypergeometric series in :doc:`qfunctions`.
-
-For convenience, most of the hypergeometric series of low order are
-provided as standalone functions. They can equivalently be evaluated using
-:func:`~mpmath.hyper`. As will be demonstrated in the respective docstrings,
-all the ``hyp#f#`` functions implement analytic continuations and/or asymptotic
-expansions with respect to the argument `z`, thereby permitting evaluation
-for `z` anywhere in the complex plane. Functions of higher degree can be
-computed via :func:`~mpmath.hyper`, but generally only in rapidly convergent
-instances.
-
-Most hypergeometric and hypergeometric-derived functions accept optional
-keyword arguments to specify options for :func:`hypercomb` or
-:func:`hyper`. Some useful options are *maxprec*, *maxterms*,
-*zeroprec*, *accurate_small*, *hmag*, *force_series*,
-*asymp_tol* and *eliminate*. These options give control over what to
-do in case of slow convergence, extreme loss of accuracy or
-evaluation at zeros (these two cases cannot generally be
-distinguished from each other automatically),
-and singular parameter combinations.
-
-Common hypergeometric series
-............................
-
-:func:`hyp0f1`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.hyp0f1(a, z)
-
-:func:`hyp1f1`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.hyp1f1(a, b, z)
-
-:func:`hyp1f2`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.hyp1f2(a1, b1, b2, z)
-
-:func:`hyp2f0`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.hyp2f0(a, b, z)
-
-:func:`hyp2f1`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.hyp2f1(a, b, c, z)
-
-:func:`hyp2f2`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.hyp2f2(a1, a2, b1, b2, z)
-
-:func:`hyp2f3`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.hyp2f3(a1, a2, b1, b2, b3, z)
-
-:func:`hyp3f2`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.hyp3f2(a1, a2, a3, b1, b2, z)
-
-Generalized hypergeometric functions
-....................................
-
-:func:`hyper`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.hyper(a_s, b_s, z)
-
-:func:`hypercomb`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.hypercomb
-
-Meijer G-function
-...................................
-
-:func:`meijerg`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.meijerg(a_s,b_s,z,r=1,**kwargs)
-
-Bilateral hypergeometric series
-...............................
-
-:func:`bihyper`
-^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.bihyper(a_s,b_s,z,**kwargs)
-
-Hypergeometric functions of two variables
-...............................................
-
-:func:`hyper2d`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.hyper2d(a,b,x,y,**kwargs)
-
-:func:`appellf1`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.appellf1(a,b1,b2,c,x,y,**kwargs)
-
-:func:`appellf2`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.appellf2(a,b1,b2,c1,c2,x,y,**kwargs)
-
-:func:`appellf3`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.appellf3(a1,a2,b1,b2,c,x,y,**kwargs)
-
-:func:`appellf4`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.appellf4(a,b,c1,c2,x,y,**kwargs)
-
diff --git a/dev-py3k/_sources/modules/mpmath/functions/index.txt b/dev-py3k/_sources/modules/mpmath/functions/index.txt
deleted file mode 100644
index 9e637a8c1f5..00000000000
--- a/dev-py3k/_sources/modules/mpmath/functions/index.txt
+++ /dev/null
@@ -1,21 +0,0 @@
-Mathematical functions
-======================
-
-Mpmath implements the standard functions from Python's ``math`` and ``cmath`` modules, for both real and complex numbers and with arbitrary precision. Many other functions are also available in mpmath, including commonly-used variants of standard functions (such as the alternative trigonometric functions sec, csc, cot), but also a large number of "special functions" such as the gamma function, the Riemann zeta function, error functions, Bessel functions, etc.
-
-.. toctree::
-   :maxdepth: 2
-
-   constants.rst
-   powers.rst
-   trigonometric.rst
-   hyperbolic.rst
-   gamma.rst
-   expintegrals.rst
-   bessel.rst
-   orthogonal.rst
-   hypergeometric.rst
-   elliptic.rst
-   zeta.rst
-   numtheory.rst
-   qfunctions.rst
diff --git a/dev-py3k/_sources/modules/mpmath/functions/numtheory.txt b/dev-py3k/_sources/modules/mpmath/functions/numtheory.txt
deleted file mode 100644
index 65c628ecdd8..00000000000
--- a/dev-py3k/_sources/modules/mpmath/functions/numtheory.txt
+++ /dev/null
@@ -1,80 +0,0 @@
-Number-theoretical, combinatorial and integer functions
--------------------------------------------------------
-
-For factorial-type functions, including binomial coefficients,
-double factorials, etc., see the separate
-section :doc:`gamma`.
-
-Fibonacci numbers
-.................
-
-:func:`fibonacci`/:func:`fib`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.fibonacci(n, **kwargs)
-
-
-Bernoulli numbers and polynomials
-.................................
-
-:func:`bernoulli`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.bernoulli(n)
-
-:func:`bernfrac`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.bernfrac(n)
-
-:func:`bernpoly`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.bernpoly(n,z)
-
-Euler numbers and polynomials
-.................................
-
-:func:`eulernum`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.eulernum(n)
-
-:func:`eulerpoly`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.eulerpoly(n,z)
-
-
-Bell numbers and polynomials
-...........................................
-
-:func:`bell`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.bell(n,x)
-
-
-Prime counting functions
-........................
-
-:func:`primepi`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.primepi(x)
-
-:func:`primepi2`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.primepi2(x)
-
-:func:`riemannr`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.riemannr(x)
-
-
-Cyclotomic polynomials
-......................
-
-:func:`cyclotomic`
-^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.cyclotomic(n,x)
-
-
-Arithmetic functions
-......................
-
-:func:`mangoldt`
-^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.mangoldt(n)
diff --git a/dev-py3k/_sources/modules/mpmath/functions/orthogonal.txt b/dev-py3k/_sources/modules/mpmath/functions/orthogonal.txt
deleted file mode 100644
index 27791ca54b8..00000000000
--- a/dev-py3k/_sources/modules/mpmath/functions/orthogonal.txt
+++ /dev/null
@@ -1,79 +0,0 @@
-Orthogonal polynomials
-----------------------
-
-An orthogonal polynomial sequence is a sequence of polynomials `P_0(x), P_1(x), \ldots` of degree `0, 1, \ldots`, which are mutually orthogonal in the sense that
-
-.. math ::
-
-    \int_S P_n(x) P_m(x) w(x) dx =
-    \begin{cases}
-    c_n \ne 0 & \text{if $m = n$} \\
-    0         & \text{if $m \ne n$}
-    \end{cases}
-
-where `S` is some domain (e.g. an interval `[a,b] \in \mathbb{R}`) and `w(x)` is a fixed *weight function*. A sequence of orthogonal polynomials is determined completely by `w`, `S`, and a normalization convention (e.g. `c_n = 1`). Applications of orthogonal polynomials include function approximation and solution of differential equations.
-
-Orthogonal polynomials are sometimes defined using the differential equations they satisfy (as functions of `x`) or the recurrence relations they satisfy with respect to the order `n`. Other ways of defining orthogonal polynomials include differentiation formulas and generating functions. The standard orthogonal polynomials can also be represented as hypergeometric series (see :doc:`hypergeometric`), more specifically using the Gauss hypergeometric function `\,_2F_1` in most cases. The following functions are generally implemented using hypergeometric functions since this is computationally efficient and easily generalizes.
-
-For more information, see the `Wikipedia article on orthogonal polynomials <http://en.wikipedia.org/wiki/Orthogonal_polynomials>`_.
-
-Legendre functions
-.......................................
-
-:func:`legendre`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.legendre(n, x)
-
-:func:`legenp`
-^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.legenp(n, m, z, type=2)
-
-:func:`legenq`
-^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.legenq(n, m, z, type=2)
-
-Chebyshev polynomials
-.....................
-
-:func:`chebyt`
-^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.chebyt(n, x)
-
-:func:`chebyu`
-^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.chebyu(n, x)
-
-Jacobi polynomials
-..................
-
-:func:`jacobi`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.jacobi(n, a, b, z)
-
-Gegenbauer polynomials
-.....................................
-
-:func:`gegenbauer`
-^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.gegenbauer(n, a, z)
-
-Hermite polynomials
-.....................................
-
-:func:`hermite`
-^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.hermite(n, z)
-
-Laguerre polynomials
-.......................................
-
-:func:`laguerre`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.laguerre(n, a, z)
-
-Spherical harmonics
-.....................................
-
-:func:`spherharm`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.spherharm(l, m, theta, phi)
diff --git a/dev-py3k/_sources/modules/mpmath/functions/powers.txt b/dev-py3k/_sources/modules/mpmath/functions/powers.txt
deleted file mode 100644
index a5820d2b0cc..00000000000
--- a/dev-py3k/_sources/modules/mpmath/functions/powers.txt
+++ /dev/null
@@ -1,85 +0,0 @@
-Powers and logarithms
----------------------
-
-Nth roots
-.........
-
-:func:`sqrt`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.sqrt(x, **kwargs)
-
-:func:`hypot`
-^^^^^^^^^^^^^
-.. autofunction:: mpmath.hypot(x, y)
-
-:func:`cbrt`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.cbrt(x, **kwargs)
-
-:func:`root`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.root(z, n, k=0)
-
-:func:`unitroots`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.unitroots(n, primitive=False)
-
-
-Exponentiation
-..............
-
-:func:`exp`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.exp(x, **kwargs)
-
-:func:`power`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.power(x, y)
-
-:func:`expj`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.expj(x, **kwargs)
-
-:func:`expjpi`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.expjpi(x, **kwargs)
-
-:func:`expm1`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.expm1(x)
-
-:func:`powm1`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.powm1(x, y)
-
-
-Logarithms
-..........
-
-:func:`log`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.log(x, b=None)
-
-:func:`ln`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.ln(x, **kwargs)
-
-:func:`log10`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.log10(x)
-
-
-Lambert W function
-...................................................
-
-:func:`lambertw`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.lambertw(z, k=0)
-
-
-Arithmetic-geometric mean
-.......................................
-
-:func:`agm`
-^^^^^^^^^^^^^^
-.. autofunction:: mpmath.agm(a, b=1)
diff --git a/dev-py3k/_sources/modules/mpmath/functions/qfunctions.txt b/dev-py3k/_sources/modules/mpmath/functions/qfunctions.txt
deleted file mode 100644
index 4b1ae05ef21..00000000000
--- a/dev-py3k/_sources/modules/mpmath/functions/qfunctions.txt
+++ /dev/null
@@ -1,29 +0,0 @@
-q-functions
--------------------------------------------
-
-q-Pochhammer symbol
-..................................................
-
-:func:`qp`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.qp(a, q=None, n=None, **kwargs)
-
-
-q-gamma and factorial
-..................................................
-
-:func:`qgamma`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.qgamma(z, q, **kwargs)
-
-:func:`qfac`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.qfac(z, q, **kwargs)
-
-Hypergeometric q-series
-..................................................
-
-:func:`qhyper`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.qhyper(a_s, b_s, q, z, **kwargs)
-
diff --git a/dev-py3k/_sources/modules/mpmath/functions/trigonometric.txt b/dev-py3k/_sources/modules/mpmath/functions/trigonometric.txt
deleted file mode 100644
index 1c2b87e3a97..00000000000
--- a/dev-py3k/_sources/modules/mpmath/functions/trigonometric.txt
+++ /dev/null
@@ -1,117 +0,0 @@
-Trigonometric functions
------------------------
-
-Except where otherwise noted, the trigonometric functions
-take a radian angle as input and the inverse trigonometric
-functions return radian angles.
-
-The ordinary trigonometric functions are single-valued
-functions defined everywhere in the complex plane
-(except at the poles of tan, sec, csc, and cot).
-They are defined generally via the exponential function,
-e.g.
-
-.. math ::
-
-    \cos(x) = \frac{e^{ix} + e^{-ix}}{2}.
-
-The inverse trigonometric functions are multivalued,
-thus requiring branch cuts, and are generally real-valued
-only on a part of the real line. Definitions and branch cuts
-are given in the documentation of each function.
-The branch cut conventions used by mpmath are essentially
-the same as those found in most standard mathematical software,
-such as Mathematica and Python's own ``cmath`` libary (as of Python 2.6;
-earlier Python versions implement some functions
-erroneously).
-
-Degree-radian conversion
-...........................................................
-
-:func:`degrees`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.degrees(x)
-
-:func:`radians`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.radians(x)
-
-Trigonometric functions
-.......................
-
-:func:`cos`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.cos(x, **kwargs)
-
-:func:`sin`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.sin(x, **kwargs)
-
-:func:`tan`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.tan(x, **kwargs)
-
-:func:`sec`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.sec(x)
-
-:func:`csc`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.csc(x)
-
-:func:`cot`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.cot(x)
-
-Trigonometric functions with modified argument
-........................................................
-
-:func:`cospi`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.cospi(x, **kwargs)
-
-:func:`sinpi`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.sinpi(x, **kwargs)
-
-Inverse trigonometric functions
-................................................
-
-:func:`acos`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.acos(x, **kwargs)
-
-:func:`asin`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.asin(x, **kwargs)
-
-:func:`atan`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.atan(x, **kwargs)
-
-:func:`atan2`
-^^^^^^^^^^^^^
-.. autofunction:: mpmath.atan2(y, x)
-
-:func:`asec`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.asec(x)
-
-:func:`acsc`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.acsc(x)
-
-:func:`acot`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.acot(x)
-
-Sinc function
-.............
-
-:func:`sinc`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.sinc(x)
-
-:func:`sincpi`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.sincpi(x)
diff --git a/dev-py3k/_sources/modules/mpmath/functions/zeta.txt b/dev-py3k/_sources/modules/mpmath/functions/zeta.txt
deleted file mode 100644
index d797ce9ec1e..00000000000
--- a/dev-py3k/_sources/modules/mpmath/functions/zeta.txt
+++ /dev/null
@@ -1,104 +0,0 @@
-Zeta functions, L-series and polylogarithms
--------------------------------------------
-
-This section includes the Riemann zeta functions
-and associated functions pertaining to analytic number theory.
-
-
-Riemann and Hurwitz zeta functions
-..................................................
-
-:func:`zeta`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.zeta(s,a=1,derivative=0)
-
-
-Dirichlet L-series
-..................................................
-
-:func:`altzeta`
-^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.altzeta(s)
-
-:func:`dirichlet`
-^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.dirichlet(s,chi,derivative=0)
-
-
-Stieltjes constants
-...................
-
-:func:`stieltjes`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.stieltjes(n,a=1)
-
-
-Zeta function zeros
-......................................
-
-These functions are used for the study of the Riemann zeta function
-in the critical strip.
-
-:func:`zetazero`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.zetazero(n, verbose=False)
-
-:func:`nzeros`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.nzeros(t)
-
-:func:`siegelz`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.siegelz(t)
-
-:func:`siegeltheta`
-^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.siegeltheta(t)
-
-:func:`grampoint`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.grampoint(n)
-
-:func:`backlunds`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.backlunds(t)
-
-
-Lerch transcendent
-................................
-
-:func:`lerchphi`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.lerchphi(z,s,a)
-
-
-Polylogarithms and Clausen functions
-.......................................
-
-:func:`polylog`
-^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.polylog(s,z)
-
-:func:`clsin`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.clsin(s, z)
-
-:func:`clcos`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.clcos(s, z)
-
-:func:`polyexp`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.polyexp(s,z)
-
-
-Zeta function variants
-..........................
-
-:func:`primezeta`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.primezeta(s)
-
-:func:`secondzeta`
-^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.secondzeta(s, a=0.015, **kwargs)
diff --git a/dev-py3k/_sources/modules/mpmath/general.txt b/dev-py3k/_sources/modules/mpmath/general.txt
deleted file mode 100644
index 2d7e930926b..00000000000
--- a/dev-py3k/_sources/modules/mpmath/general.txt
+++ /dev/null
@@ -1,228 +0,0 @@
-Utility functions
-===============================================
-
-This page lists functions that perform basic operations
-on numbers or aid general programming.
-
-Conversion and printing
------------------------
-
-:func:`mpmathify` / :func:`convert`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.mpmathify(x, strings=True)
-
-:func:`nstr`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.nstr(x, n=6, **kwargs)
-
-:func:`nprint`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.nprint(x, n=6, **kwargs)
-
-Arithmetic operations
----------------------
-
-See also :func:`mpmath.sqrt`, :func:`mpmath.exp` etc., listed
-in :doc:`functions/powers`
-
-:func:`fadd`
-^^^^^^^^^^^^^
-.. autofunction:: mpmath.fadd
-
-:func:`fsub`
-^^^^^^^^^^^^^
-.. autofunction:: mpmath.fsub
-
-:func:`fneg`
-^^^^^^^^^^^^^
-.. autofunction:: mpmath.fneg
-
-:func:`fmul`
-^^^^^^^^^^^^^
-.. autofunction:: mpmath.fmul
-
-:func:`fdiv`
-^^^^^^^^^^^^^
-.. autofunction:: mpmath.fdiv
-
-:func:`fmod`
-^^^^^^^^^^^^^
-.. autofunction:: mpmath.fmod(x, y)
-
-:func:`fsum`
-^^^^^^^^^^^^^
-.. autofunction:: mpmath.fsum(terms, absolute=False, squared=False)
-
-:func:`fprod`
-^^^^^^^^^^^^^
-.. autofunction:: mpmath.fprod(factors)
-
-:func:`fdot`
-^^^^^^^^^^^^^
-.. autofunction:: mpmath.fdot(A, B=None, conjugate=False)
-
-Complex components
-------------------
-
-:func:`fabs`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.fabs(x)
-
-:func:`sign`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.sign(x)
-
-:func:`re`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.re(x)
-
-:func:`im`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.im(x)
-
-:func:`arg`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.arg(x)
-
-:func:`conj`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.conj(x)
-
-:func:`polar`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.polar(x)
-
-:func:`rect`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.rect(x)
-
-Integer and fractional parts
------------------------------
-
-:func:`floor`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.floor(x)
-
-:func:`ceil`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.ceil(x)
-
-:func:`nint`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.nint(x)
-
-:func:`frac`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.frac(x)
-
-Tolerances and approximate comparisons
---------------------------------------
-
-:func:`chop`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.chop(x, tol=None)
-
-:func:`almosteq`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.almosteq(s, t, rel_eps=None, abs_eps=None)
-
-Properties of numbers
--------------------------------------
-
-:func:`isinf`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.isinf(x)
-
-:func:`isnan`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.isnan(x)
-
-:func:`isnormal`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.isnormal(x)
-
-:func:`isint`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.isint(x, gaussian=False)
-
-:func:`ldexp`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.ldexp(x, n)
-
-:func:`frexp`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.frexp(x, n)
-
-:func:`mag`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.mag(x)
-
-:func:`nint_distance`
-^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.nint_distance(x)
-
-.. :func:`absmin`
-.. ^^^^^^^^^^^^^^^^^^^^
-.. .. autofunction:: mpmath.absmin(x)
-.. .. autofunction:: mpmath.absmax(x)
-
-Number generation
------------------
-
-:func:`fraction`
-^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.fraction(p,q)
-
-:func:`rand`
-^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.rand()
-
-:func:`arange`
-^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.arange(*args)
-
-:func:`linspace`
-^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.linspace(*args, **kwargs)
-
-Precision management
---------------------
-
-:func:`autoprec`
-^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.autoprec
-
-:func:`workprec`
-^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.workprec
-
-:func:`workdps`
-^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.workdps
-
-:func:`extraprec`
-^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.extraprec
-
-:func:`extradps`
-^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.extradps
-
-Performance and debugging
-------------------------------------
-
-:func:`memoize`
-^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.memoize
-
-:func:`maxcalls`
-^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.maxcalls
-
-:func:`monitor`
-^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.monitor
-
-:func:`timing`
-^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.timing
diff --git a/dev-py3k/_sources/modules/mpmath/identification.txt b/dev-py3k/_sources/modules/mpmath/identification.txt
deleted file mode 100644
index a66bfa9cd73..00000000000
--- a/dev-py3k/_sources/modules/mpmath/identification.txt
+++ /dev/null
@@ -1,31 +0,0 @@
-Number identification
-=====================
-
-Most function in mpmath are concerned with producing approximations from exact mathematical formulas. It is also useful to consider the inverse problem: given only a decimal approximation for a number, such as 0.7320508075688772935274463, is it possible to find an exact formula?
-
-Subject to certain restrictions, such "reverse engineering" is indeed possible thanks to the existence of *integer relation algorithms*. Mpmath implements the PSLQ algorithm (developed by H. Ferguson), which is one such algorithm.
-
-Automated number recognition based on PSLQ is not a silver bullet. Any occurring transcendental constants (`\pi`, `e`, etc) must be guessed by the user, and the relation between those constants in the formula must be linear (such as `x = 3 \pi + 4 e`). More complex formulas can be found by combining PSLQ with functional transformations; however, this is only feasible to a limited extent since the computation time grows exponentially with the number of operations that need to be combined.
-
-The number identification facilities in mpmath are inspired by the `Inverse Symbolic Calculator <http://oldweb.cecm.sfu.ca/projects/ISC/ISCmain.html>`_ (ISC). The ISC is more powerful than mpmath, as it uses a lookup table of millions of precomputed constants (thereby mitigating the problem with exponential complexity).
-
-Constant recognition
------------------------------------
-
-:func:`identify`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.identify
-
-Algebraic identification
----------------------------------------
-
-:func:`findpoly`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.findpoly
-
-Integer relations (PSLQ)
-----------------------------
-
-:func:`pslq`
-^^^^^^^^^^^^^^^^
-.. autofunction:: mpmath.pslq
diff --git a/dev-py3k/_sources/modules/mpmath/index.txt b/dev-py3k/_sources/modules/mpmath/index.txt
deleted file mode 100644
index 7f2e6748635..00000000000
--- a/dev-py3k/_sources/modules/mpmath/index.txt
+++ /dev/null
@@ -1,53 +0,0 @@
-.. mpmath documentation master file, created by sphinx-quickstart on Fri Mar 28 13:50:14 2008.
-   You can adapt this file completely to your liking, but it should at least
-   contain the root `toctree` directive.
-
-Welcome to mpmath's documentation!
-==================================
-
-Mpmath is a Python library for arbitrary-precision floating-point arithmetic.
-For general information about mpmath, see the project website http://code.google.com/p/mpmath/
-
-These documentation pages include general information as well as docstring listing with extensive use of examples that can be run in the interactive Python interpreter. For quick access to the docstrings of individual functions, use the :ref:`genindex`, or type ``help(mpmath.function_name)`` in the Python interactive prompt.
-
-Introduction
-------------
-
-.. toctree ::
-   :maxdepth: 2
-
-   setup.rst
-   basics.rst
-
-Basic features
-----------------
-
-.. toctree ::
-   :maxdepth: 2
-
-   contexts.rst
-   general.rst
-   plotting.rst
-
-Advanced mathematics
---------------------
-
-On top of its support for arbitrary-precision arithmetic, mpmath
-provides extensive support for transcendental functions, evaluation of sums, integrals, limits, roots, and so on.
-
-.. toctree ::
-   :maxdepth: 2
-
-   functions/index.rst
-   calculus/index.rst
-   matrices.rst
-   identification.rst
-
-End matter
-----------
-
-.. toctree ::
-   :maxdepth: 2
-
-   technical.rst
-   references.rst
diff --git a/dev-py3k/_sources/modules/mpmath/matrices.txt b/dev-py3k/_sources/modules/mpmath/matrices.txt
deleted file mode 100644
index a64e6d0be80..00000000000
--- a/dev-py3k/_sources/modules/mpmath/matrices.txt
+++ /dev/null
@@ -1,375 +0,0 @@
-Matrices
-========
-
-Creating matrices
------------------
-
-Basic methods
-.............
-
-Matrices in mpmath are implemented using dictionaries. Only non-zero values are
-stored, so it is cheap to represent sparse matrices.
-
-The most basic way to create one is to use the ``matrix`` class directly. You
-can create an empty matrix specifying the dimensions::
-
-    >>> from mpmath import *
-    >>> mp.dps = 15; mp.pretty = False
-    >>> matrix(2)
-    matrix(
-    [['0.0', '0.0'],
-     ['0.0', '0.0']])
-    >>> matrix(2, 3)
-    matrix(
-    [['0.0', '0.0', '0.0'],
-     ['0.0', '0.0', '0.0']])
-
-Calling ``matrix`` with one dimension will create a square matrix.
-
-To access the dimensions of a matrix, use the ``rows`` or ``cols`` keyword::
-
-    >>> A = matrix(3, 2)
-    >>> A
-    matrix(
-    [['0.0', '0.0'],
-     ['0.0', '0.0'],
-     ['0.0', '0.0']])
-    >>> A.rows
-    3
-    >>> A.cols
-    2
-
-You can also change the dimension of an existing matrix. This will set the
-new elements to 0. If the new dimension is smaller than before, the
-concerning elements are discarded::
-
-    >>> A.rows = 2
-    >>> A
-    matrix(
-    [['0.0', '0.0'],
-     ['0.0', '0.0']])
-
-Internally ``convert`` is applied every time an element is set. This is
-done using the syntax A[row,column], counting from 0::
-
-    >>> A = matrix(2)
-    >>> A[1,1] = 1 + 1j
-    >>> print A
-    [0.0           0.0]
-    [0.0  (1.0 + 1.0j)]
-
-A more comfortable way to create a matrix lets you use nested lists::
-
-    >>> matrix([[1, 2], [3, 4]])
-    matrix(
-    [['1.0', '2.0'],
-     ['3.0', '4.0']])
-
-Advanced methods
-................
-
-Convenient functions are available for creating various standard matrices::
-
-    >>> zeros(2)
-    matrix(
-    [['0.0', '0.0'],
-     ['0.0', '0.0']])
-    >>> ones(2)
-    matrix(
-    [['1.0', '1.0'],
-     ['1.0', '1.0']])
-    >>> diag([1, 2, 3]) # diagonal matrix
-    matrix(
-    [['1.0', '0.0', '0.0'],
-     ['0.0', '2.0', '0.0'],
-     ['0.0', '0.0', '3.0']])
-    >>> eye(2) # identity matrix
-    matrix(
-    [['1.0', '0.0'],
-     ['0.0', '1.0']])
-
-You can even create random matrices::
-
-    >>> randmatrix(2) # doctest:+SKIP
-    matrix(
-    [['0.53491598236191806', '0.57195669543302752'],
-     ['0.85589992269513615', '0.82444367501382143']])
-
-Vectors
-.......
-
-Vectors may also be represented by the ``matrix`` class (with rows = 1 or cols = 1).
-For vectors there are some things which make life easier. A column vector can
-be created using a flat list, a row vectors using an almost flat nested list::
-
-    >>> matrix([1, 2, 3])
-    matrix(
-    [['1.0'],
-     ['2.0'],
-     ['3.0']])
-    >>> matrix([[1, 2, 3]])
-    matrix(
-    [['1.0', '2.0', '3.0']])
-
-Optionally vectors can be accessed like lists, using only a single index::
-
-    >>> x = matrix([1, 2, 3])
-    >>> x[1]
-    mpf('2.0')
-    >>> x[1,0]
-    mpf('2.0')
-
-Other
-.....
-
-Like you probably expected, matrices can be printed::
-
-    >>> print randmatrix(3) # doctest:+SKIP
-    [ 0.782963853573023  0.802057689719883  0.427895717335467]
-    [0.0541876859348597  0.708243266653103  0.615134039977379]
-    [ 0.856151514955773  0.544759264818486  0.686210904770947]
-
-Use ``nstr`` or ``nprint`` to specify the number of digits to print::
-
-    >>> nprint(randmatrix(5), 3) # doctest:+SKIP
-    [2.07e-1  1.66e-1  5.06e-1  1.89e-1  8.29e-1]
-    [6.62e-1  6.55e-1  4.47e-1  4.82e-1  2.06e-2]
-    [4.33e-1  7.75e-1  6.93e-2  2.86e-1  5.71e-1]
-    [1.01e-1  2.53e-1  6.13e-1  3.32e-1  2.59e-1]
-    [1.56e-1  7.27e-2  6.05e-1  6.67e-2  2.79e-1]
-
-As matrices are mutable, you will need to copy them sometimes::
-
-    >>> A = matrix(2)
-    >>> A
-    matrix(
-    [['0.0', '0.0'],
-     ['0.0', '0.0']])
-    >>> B = A.copy()
-    >>> B[0,0] = 1
-    >>> B
-    matrix(
-    [['1.0', '0.0'],
-     ['0.0', '0.0']])
-    >>> A
-    matrix(
-    [['0.0', '0.0'],
-     ['0.0', '0.0']])
-
-Finally, it is possible to convert a matrix to a nested list. This is very useful,
-as most Python libraries involving matrices or arrays (namely NumPy or SymPy)
-support this format::
-
-    >>> B.tolist()
-    [[mpf('1.0'), mpf('0.0')], [mpf('0.0'), mpf('0.0')]]
-
-
-Matrix operations
------------------
-
-You can add and substract matrices of compatible dimensions::
-
-    >>> A = matrix([[1, 2], [3, 4]])
-    >>> B = matrix([[-2, 4], [5, 9]])
-    >>> A + B
-    matrix(
-    [['-1.0', '6.0'],
-     ['8.0', '13.0']])
-    >>> A - B
-    matrix(
-    [['3.0', '-2.0'],
-     ['-2.0', '-5.0']])
-    >>> A + ones(3) # doctest:+ELLIPSIS
-    Traceback (most recent call last):
-      File "<stdin>", line 1, in <module>
-      File "...", line 238, in __add__
-        raise ValueError('incompatible dimensions for addition')
-    ValueError: incompatible dimensions for addition
-
-It is possible to multiply or add matrices and scalars. In the latter case the
-operation will be done element-wise::
-
-    >>> A * 2
-    matrix(
-    [['2.0', '4.0'],
-     ['6.0', '8.0']])
-    >>> A / 4
-    matrix(
-    [['0.25', '0.5'],
-     ['0.75', '1.0']])
-    >>> A - 1
-    matrix(
-    [['0.0', '1.0'],
-     ['2.0', '3.0']])
-
-Of course you can perform matrix multiplication, if the dimensions are
-compatible::
-
-    >>> A * B
-    matrix(
-    [['8.0', '22.0'],
-     ['14.0', '48.0']])
-    >>> matrix([[1, 2, 3]]) * matrix([[-6], [7], [-2]])
-    matrix(
-    [['2.0']])
-
-You can raise powers of square matrices::
-
-    >>> A**2
-    matrix(
-    [['7.0', '10.0'],
-     ['15.0', '22.0']])
-
-Negative powers will calculate the inverse::
-
-    >>> A**-1
-    matrix(
-    [['-2.0', '1.0'],
-     ['1.5', '-0.5']])
-    >>> nprint(A * A**-1, 3)
-    [      1.0  1.08e-19]
-    [-2.17e-19       1.0]
-
-Matrix transposition is straightforward::
-
-    >>> A = ones(2, 3)
-    >>> A
-    matrix(
-    [['1.0', '1.0', '1.0'],
-     ['1.0', '1.0', '1.0']])
-    >>> A.T
-    matrix(
-    [['1.0', '1.0'],
-     ['1.0', '1.0'],
-     ['1.0', '1.0']])
-
-
-Norms
-.....
-
-Sometimes you need to know how "large" a matrix or vector is. Due to their
-multidimensional nature it's not possible to compare them, but there are
-several functions to map a matrix or a vector to a positive real number, the
-so called norms.
-
-.. autofunction :: mpmath.norm
-
-.. autofunction :: mpmath.mnorm
-
-
-Linear algebra
---------------
-
-Decompositions
-..............
-
-.. autofunction :: mpmath.cholesky
-
-
-Linear equations
-................
-
-Basic linear algebra is implemented; you can for example solve the linear
-equation system::
-
-      x + 2*y = -10
-    3*x + 4*y =  10
-
-using ``lu_solve``::
-
-    >>> A = matrix([[1, 2], [3, 4]])
-    >>> b = matrix([-10, 10])
-    >>> x = lu_solve(A, b)
-    >>> x
-    matrix(
-    [['30.0'],
-     ['-20.0']])
-
-If you don't trust the result, use ``residual`` to calculate the residual ||A*x-b||::
-
-    >>> residual(A, x, b)
-    matrix(
-    [['3.46944695195361e-18'],
-     ['3.46944695195361e-18']])
-    >>> str(eps)
-    '2.22044604925031e-16'
-
-As you can see, the solution is quite accurate. The error is caused by the
-inaccuracy of the internal floating point arithmetic. Though, it's even smaller
-than the current machine epsilon, which basically means you can trust the
-result.
-
-If you need more speed, use NumPy, or use ``fp`` instead ``mp`` matrices
-and methods::
-
-    >>> A = fp.matrix([[1, 2], [3, 4]])
-    >>> b = fp.matrix([-10, 10])
-    >>> fp.lu_solve(A, b)
-    matrix(
-    [['30.0'],
-     ['-20.0']])
-
-``lu_solve`` accepts overdetermined systems. It is usually not possible to solve
-such systems, so the residual is minimized instead. Internally this is done
-using Cholesky decomposition to compute a least squares approximation. This means
-that that ``lu_solve`` will square the errors. If you can't afford this, use
-``qr_solve`` instead. It is twice as slow but more accurate, and it calculates
-the residual automatically.
-
-
-Matrix factorization
-....................
-
-The function ``lu`` computes an explicit LU factorization of a matrix::
-
-    >>> P, L, U = lu(matrix([[0,2,3],[4,5,6],[7,8,9]]))
-    >>> print P
-    [0.0  0.0  1.0]
-    [1.0  0.0  0.0]
-    [0.0  1.0  0.0]
-    >>> print L
-    [              1.0                0.0  0.0]
-    [              0.0                1.0  0.0]
-    [0.571428571428571  0.214285714285714  1.0]
-    >>> print U
-    [7.0  8.0                9.0]
-    [0.0  2.0                3.0]
-    [0.0  0.0  0.214285714285714]
-    >>> print P.T*L*U
-    [0.0  2.0  3.0]
-    [4.0  5.0  6.0]
-    [7.0  8.0  9.0]
-
-Interval and double-precision matrices
---------------------------------------
-
-The ``iv.matrix`` and ``fp.matrix`` classes convert inputs
-to intervals and Python floating-point numbers respectively.
-
-Interval matrices can be used to perform linear algebra operations
-with rigorous error tracking::
-
-    >>> a = iv.matrix([['0.1','0.3','1.0'],
-    ...                ['7.1','5.5','4.8'],
-    ...                ['3.2','4.4','5.6']])
-    >>>
-    >>> b = iv.matrix(['4','0.6','0.5'])
-    >>> c = iv.lu_solve(a, b)
-    >>> print c
-    [  [5.2582327113062393041, 5.2582327113062749951]]
-    [[-13.155049396267856583, -13.155049396267821167]]
-    [  [7.4206915477497212555, 7.4206915477497310922]]
-    >>> print a*c
-    [  [3.9999999999999866773, 4.0000000000000133227]]
-    [[0.59999999999972430942, 0.60000000000027142733]]
-    [[0.49999999999982236432, 0.50000000000018474111]]
-
-Matrix functions
-----------------
-
-.. autofunction :: mpmath.expm
-.. autofunction :: mpmath.cosm
-.. autofunction :: mpmath.sinm
-.. autofunction :: mpmath.sqrtm
-.. autofunction :: mpmath.logm
-.. autofunction :: mpmath.powm
diff --git a/dev-py3k/_sources/modules/mpmath/plotting.txt b/dev-py3k/_sources/modules/mpmath/plotting.txt
deleted file mode 100644
index 2e68307cbd3..00000000000
--- a/dev-py3k/_sources/modules/mpmath/plotting.txt
+++ /dev/null
@@ -1,32 +0,0 @@
-Plotting
-========
-
-If `matplotlib <http://matplotlib.sourceforge.net/>`_ is available, the functions ``plot`` and ``cplot`` in mpmath can be used to plot functions respectively as x-y graphs and in the complex plane. Also, ``splot`` can be used to produce 3D surface plots.
-
-Function curve plots
------------------------
-
-.. image:: plot.png
-
-Output of ``plot([cos, sin], [-4, 4])``
-
-.. autofunction:: mpmath.plot
-
-Complex function plots
--------------------------
-
-.. image:: cplot.png
-
-Output of ``fp.cplot(fp.gamma, points=100000)``
-
-.. autofunction:: mpmath.cplot
-
-3D surface plots
-----------------
-
-.. image:: splot.png
-
-Output of ``splot`` for the donut example.
-
-.. autofunction:: mpmath.splot
-
diff --git a/dev-py3k/_sources/modules/mpmath/references.txt b/dev-py3k/_sources/modules/mpmath/references.txt
deleted file mode 100644
index e65239cc863..00000000000
--- a/dev-py3k/_sources/modules/mpmath/references.txt
+++ /dev/null
@@ -1,50 +0,0 @@
-References
-===================
-
-The following is a non-comprehensive list of works used in the development of mpmath
-or cited for examples or mathematical definitions used in this documentation.
-References not listed here can be found in the source code.
-
-.. [AbramowitzStegun] M Abramowitz & I Stegun. *Handbook of Mathematical Functions, 9th Ed.*, Tenth Printing, December 1972, with corrections
-
-.. [Bailey] D H Bailey. "Tanh-Sinh High-Precision Quadrature", http://crd.lbl.gov/~dhbailey/dhbpapers/dhb-tanh-sinh.pdf
-
-.. [BenderOrszag] C M Bender & S A Orszag. *Advanced Mathematical Methods for
-    Scientists and Engineers*, Springer 1999
-
-.. [BorweinBailey] J Borwein, D H Bailey & R Girgensohn. *Experimentation in Mathematics - Computational Paths to Discovery*, A K Peters, 2003
-
-.. [BorweinBorwein] J Borwein & P B Borwein. *Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity*, Wiley 1987
-
-.. [BorweinZeta] P Borwein. "An Efficient Algorithm for the Riemann Zeta Function", http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P115.pdf
-
-.. [CabralRosetti] L G Cabral-Rosetti & M A Sanchis-Lozano. "Appell Functions and the Scalar One-Loop Three-point Integrals in Feynman Diagrams". http://arxiv.org/abs/hep-ph/0206081
-
-.. [Carlson] B C Carlson. "Numerical computation of real or complex elliptic integrals". http://arxiv.org/abs/math/9409227v1
-
-.. [Corless] R M Corless et al. "On the Lambert W function", Adv. Comp. Math. 5 (1996) 329-359. http://www.apmaths.uwo.ca/~djeffrey/Offprints/W-adv-cm.pdf
-
-.. [DLMF] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/
-
-.. [GradshteynRyzhik] I S Gradshteyn & I M Ryzhik, A Jeffrey & D Zwillinger (eds.), *Table of Integrals, Series and Products*, Seventh edition (2007), Elsevier
-
-.. [GravesMorris] P R Graves-Morris, D E Roberts & A Salam. "The epsilon algorithm and related topics", *Journal of Computational and Applied Mathematics*, Volume 122, Issue 1-2  (October 2000)
-
-.. [MPFR] The MPFR team. "The MPFR Library: Algorithms and Proofs", http://www.mpfr.org/algorithms.pdf
-
-.. [Slater] L J Slater. *Generalized Hypergeometric Functions*. Cambridge University Press, 1966
-
-.. [Spouge] J L Spouge. "Computation of the gamma, digamma, and trigamma functions", SIAM J. Numer. Anal. Vol. 31, No. 3, pp. 931-944, June 1994.
-
-.. [SrivastavaKarlsson] H M Srivastava & P W Karlsson. *Multiple Gaussian Hypergeometric Series*. Ellis Horwood, 1985.
-
-.. [Vidunas] R Vidunas. "Identities between Appell's and hypergeometric functions". http://arxiv.org/abs/0804.0655
-
-.. [Weisstein] E W Weisstein. *MathWorld*. http://mathworld.wolfram.com/
-
-.. [WhittakerWatson] E T Whittaker & G N Watson. *A Course of Modern Analysis*. 4th Ed. 1946
-    Cambridge University Press
-
-.. [Wikipedia] *Wikipedia, the free encyclopedia*. http://en.wikipedia.org
-
-.. [WolframFunctions] Wolfram Research, Inc. *The Wolfram Functions Site*. http://functions.wolfram.com/
diff --git a/dev-py3k/_sources/modules/mpmath/setup.txt b/dev-py3k/_sources/modules/mpmath/setup.txt
deleted file mode 100644
index e8c1d4ec2a7..00000000000
--- a/dev-py3k/_sources/modules/mpmath/setup.txt
+++ /dev/null
@@ -1,178 +0,0 @@
-Setting up mpmath
-=================
-
-Download and installation
--------------------------
-
-Installer
-.........
-
-The mpmath setup files can be downloaded from the `mpmath download page <http://code.google.com/p/mpmath/downloads/list>`_ or the `Python Package Index <http://pypi.python.org/pypi/mpmath/>`_. Download the source package (available as both .zip and .tar.gz), extract it, open the extracted directory, and run
-
-    ``python setup.py install``
-
-If you are using Windows, you can download the binary installer
-
-    ``mpmath-(version).win32.exe``
-
-from the mpmath website or the Python Package Index. Run the installer and follow the instructions.
-
-Using setuptools
-................
-
-If you have `setuptools <http://pypi.python.org/pypi/setuptools>`_ installed, you can download and install mpmath in one step by running:
-
-    ``easy_install mpmath``
-
-or
-
-    ``python -m easy_install mpmath``
-
-If you have an old version of mpmath installed already, you may have to pass ``easy_install`` the ``-U`` flag to force an upgrade.
-
-
-Debian/Ubuntu
-.............
-
-Debian and Ubuntu users can install mpmath with
-
-    ``sudo apt-get install python-mpmath``
-
-See `debian <http://packages.debian.org/python-mpmath>`_ and `ubuntu <https://launchpad.net/ubuntu/+source/mpmath>`_ package information; please verify that you are getting the latest version.
-
-OpenSUSE
-........
-
-Mpmath is provided in the "Science" repository for all recent versions of `openSUSE <http://www.opensuse.org/>`_. To add this repository to the YAST software management tool, see http://en.opensuse.org/SDB:Add_package_repositories
-
-Look up http://download.opensuse.org/repositories/science/ for a list
-of supported OpenSUSE versions and use http://download.opensuse.org/repositories/science/openSUSE_12.2/
-(or accordingly for your OpenSUSE version) as the repository URL for YAST.
-
-Current development version
-...........................
-
-See http://code.google.com/p/mpmath/source/checkout for instructions on how to check out the mpmath Subversion repository. The source code can also be browsed online from the Google Code page.
-
-Checking that it works
-......................
-
-After the setup has completed, you should be able to fire up the interactive Python interpreter and do the following::
-
-    >>> from mpmath import *
-    >>> mp.dps = 50
-    >>> print mpf(2) ** mpf('0.5')
-    1.4142135623730950488016887242096980785696718753769
-    >>> print 2*pi
-    6.2831853071795864769252867665590057683943387987502
-
-*Note: if you have are upgrading mpmath from an earlier version, you may have to manually uninstall the old version or remove the old files.*
-
-Using gmpy (optional)
----------------------
-
-By default, mpmath uses Python integers internally. If `gmpy <http://code.google.com/p/gmpy/>`_ version 1.03 or later is installed on your system, mpmath will automatically detect it and transparently use gmpy integers intead. This makes mpmath much faster, especially at high precision (approximately above 100 digits).
-
-To verify that mpmath uses gmpy, check the internal variable ``BACKEND`` is not equal to 'python':
-
-    >>> import mpmath.libmp
-    >>> mpmath.libmp.BACKEND
-    'gmpy'
-
-The gmpy mode can be disabled by setting the MPMATH_NOGMPY environment variable. Note that the mode cannot be switched during runtime; mpmath must be re-imported for this change to take effect.
-
-Running tests
--------------
-
-It is recommended that you run mpmath's full set of unit tests to make sure everything works. The tests are located in the ``tests`` subdirectory of the main mpmath directory. They can be run in the interactive interpreter using the ``runtests()`` function::
-
-    import mpmath
-    mpmath.runtests()
-
-Alternatively, they can be run from the ``tests`` directory via
-
-    ``python runtests.py``
-
-The tests should finish in about a minute. If you have `psyco <http://psyco.sourceforge.net/>`_ installed, the tests can also be run with
-
-    ``python runtests.py -psyco``
-
-which will cut the running time in half.
-
-If any test fails, please send a detailed bug report to the `mpmath issue tracker <http://code.google.com/p/sympy/issues/list>`_. The tests can also be run with `py.test <http://codespeak.net/py/dist/>`_. This will sometimes generate more useful information in case of a failure.
-
-To run the tests with support for gmpy disabled, use
-
-    ``python runtests.py -nogmpy``
-
-To enable extra diagnostics, use
-
-    ``python runtests.py -strict``
-
-Compiling the documentation
----------------------------
-
-If you downloaded the source package, the text source for these documentation pages is included in the ``doc`` directory. The documentation can be compiled to pretty HTML using `Sphinx <http://sphinx.pocoo.org/>`_. Go to the ``doc`` directory and run
-
-    ``python build.py``
-
-You can also test that all the interactive examples in the documentation work by running
-
-    ``python run_doctest.py``
-
-and by running the individual ``.py`` files in the mpmath source.
-
-(The doctests may take several minutes.)
-
-Finally, some additional demo scripts are available in the ``demo`` directory included in the source package.
-
-Mpmath under SymPy
---------------------
-
-Mpmath is available as a subpackage of `SymPy <http://sympy.org>`_. With SymPy installed, you can just do
-
-    ``import sympy.mpmath as mpmath``
-
-instead of ``import mpmath``. Note that the SymPy version of mpmath might not be the most recent. You can make a separate mpmath installation even if SymPy is installed; the two mpmath packages will not interfere with each other.
-
-Mpmath under Sage
--------------------
-
-Mpmath is a standard package in `Sage <http://sagemath.org/>`_, in version 4.1 or later of Sage.
-Mpmath is preinstalled a regular Python module, and can be imported as usual within Sage::
-
-    ----------------------------------------------------------------------
-    | Sage Version 4.1, Release Date: 2009-07-09                         |
-    | Type notebook() for the GUI, and license() for information.        |
-    ----------------------------------------------------------------------
-    sage: import mpmath
-    sage: mpmath.mp.dps = 50
-    sage: print mpmath.mpf(2) ** 0.5
-    1.4142135623730950488016887242096980785696718753769
-
-The mpmath installation under Sage automatically use Sage integers for asymptotically fast arithmetic,
-so there is no need to install GMPY::
-
-    sage: mpmath.libmp.BACKEND
-    'sage'
-
-In Sage, mpmath can alternatively be imported via the interface library
-``sage.libs.mpmath.all``. For example::
-
-    sage: import sage.libs.mpmath.all as mpmath
-
-This module provides a few extra conversion functions, including :func:`call`
-which permits calling any mpmath function with Sage numbers as input, and getting 
-Sage ``RealNumber`` or ``ComplexNumber`` instances
-with the appropriate precision back::
-
-    sage: w = mpmath.call(mpmath.erf, 2+3*I, prec=100)   
-    sage: w
-    -20.829461427614568389103088452 + 8.6873182714701631444280787545*I
-    sage: type(w)
-    <type 'sage.rings.complex_number.ComplexNumber'>
-    sage: w.prec()
-    100
-
-See the help for ``sage.libs.mpmath.all`` for further information.
-
diff --git a/dev-py3k/_sources/modules/mpmath/technical.txt b/dev-py3k/_sources/modules/mpmath/technical.txt
deleted file mode 100644
index c36c6cc92de..00000000000
--- a/dev-py3k/_sources/modules/mpmath/technical.txt
+++ /dev/null
@@ -1,159 +0,0 @@
-Precision and representation issues
-===================================
-
-Most of the time, using mpmath is simply a matter of setting the desired precision and entering a formula. For verification purposes, a quite (but not always!) reliable technique is to calculate the same thing a second time at a higher precision and verifying that the results agree.
-
-To perform more advanced calculations, it is important to have some understanding of how mpmath works internally and what the possible sources of error are. This section gives an overview of arbitrary-precision binary floating-point arithmetic and some concepts from numerical analysis.
-
-The following concepts are important to understand:
-
-* The main sources of numerical errors are rounding and cancellation, which are due to the use of finite-precision arithmetic, and truncation or approximation errors, which are due to approximating infinite sequences or continuous functions by a finite number of samples.
-* Errors propagate between calculations. A small error in the input may result in a large error in the output.
-* Most numerical algorithms for complex problems (e.g. integrals, derivatives) give wrong answers for sufficiently ill-behaved input. Sometimes virtually the only way to get a wrong answer is to design some very contrived input, but at other times the chance of accidentally obtaining a wrong result even for reasonable-looking input is quite high.
-* Like any complex numerical software, mpmath has implementation bugs. You should be reasonably suspicious about any results computed by mpmath, even those it claims to be able to compute correctly! If possible, verify results analytically, try different algorithms, and cross-compare with other software.
-
-Precision, error and tolerance
-------------------------------
-
-The following terms are common in this documentation:
-
-- *Precision* (or *working precision*) is the precision at which floating-point arithmetic operations are performed.
-- *Error* is the difference between a computed approximation and the exact result.
-- *Accuracy* is the inverse of error.
-- *Tolerance* is the maximum error (or minimum accuracy) desired in a result.
-
-Error and accuracy can be measured either directly, or logarithmically in bits or digits. Specifically, if a `\hat y` is an approximation for `y`, then 
-
-- (Direct) absolute error = `|\hat y - y|`
-- (Direct) relative error = `|\hat y - y| |y|^{-1}`
-- (Direct) absolute accuracy = `|\hat y - y|^{-1}`
-- (Direct) relative accuracy = `|\hat y - y|^{-1} |y|`
-- (Logarithmic) absolute error = `\log_b |\hat y - y|`
-- (Logarithmic) relative error = `\log_b |\hat y - y| - \log_b |y|`
-- (Logarithmic) absolute accuracy = `-\log_b |\hat y - y|`
-- (Logarithmic) relative accuracy = `-\log_b |\hat y - y| + \log_b |y|`
-
-where `b = 2` and `b = 10` for bits and digits respectively. Note that:
-
-- The logarithmic error roughly equals the position of the first incorrect bit or digit
-- The logarithmic accuracy roughly equals the number of correct bits or digits in the result
-
-These definitions also hold for complex numbers, using `|a+bi| = \sqrt{a^2+b^2}`.
-
-*Full accuracy* means that the accuracy of a result at least equals *prec*-1, i.e. it is correct except possibly for the last bit.
-
-Representation of numbers
--------------------------
-
-Mpmath uses binary arithmetic. A binary floating-point number is a number of the form `man \times 2^{exp}` where both *man* (the *mantissa*) and *exp* (the *exponent*) are integers. Some examples of floating-point numbers are given in the following table.
-
-  +--------+----------+----------+
-  | Number | Mantissa | Exponent |
-  +========+==========+==========+
-  |    3   |    3     |     0    |
-  +--------+----------+----------+
-  |   10   |    5     |     1    |
-  +--------+----------+----------+
-  |  -16   |   -1     |     4    |
-  +--------+----------+----------+
-  |  1.25  |    5     |    -2    |
-  +--------+----------+----------+
-
-The representation as defined so far is not unique; one can always multiply the mantissa by 2 and subtract 1 from the exponent with no change in the numerical value. In mpmath, numbers are always normalized so that *man* is an odd number, with the exception of zero which is always taken to have *man = exp = 0*. With these conventions, every representable number has a unique representation. (Mpmath does not currently distinguish between positive and negative zero.)
-
-Simple mathematical operations are now easy to define. Due to uniqueness, equality testing of two numbers simply amounts to separately checking equality of the mantissas and the exponents. Multiplication of nonzero numbers is straightforward: `(m 2^e) \times (n 2^f) = (m n) \times 2^{e+f}`. Addition is a bit more involved: we first need to multiply the mantissa of one of the operands by a suitable power of 2 to obtain equal exponents.
-
-More technically, mpmath represents a floating-point number as a 4-tuple *(sign, man, exp, bc)* where *sign* is 0 or 1 (indicating positive vs negative) and the mantissa is nonnegative; *bc* (*bitcount*) is the size of the absolute value of the mantissa as measured in bits. Though redundant, keeping a separate sign field and explicitly keeping track of the bitcount significantly speeds up arithmetic (the bitcount, especially, is frequently needed but slow to compute from scratch due to the lack of a Python built-in function for the purpose).
-
-Contrary to popular belief, floating-point *numbers* do not come with an inherent "small uncertainty", although floating-point *arithmetic* generally is inexact. Every binary floating-point number is an exact rational number. With arbitrary-precision integers used for the mantissa and exponent, floating-point numbers can be added, subtracted and multiplied *exactly*. In particular, integers and integer multiples of 1/2, 1/4, 1/8, 1/16, etc. can be represented, added and multiplied exactly in binary floating-point arithmetic.
-
-Floating-point arithmetic is generally approximate because the size of the mantissa must be limited for efficiency reasons. The maximum allowed width (bitcount) of the mantissa is called the precision or *prec* for short. Sums and products of floating-point numbers are exact as long as the absolute value of the mantissa is smaller than `2^{prec}`. As soon as the mantissa becomes larger than this, it is truncated to contain at most *prec* bits (the exponent is incremented accordingly to preserve the magnitude of the number), and this operation introduces a rounding error. Division is also generally inexact; although we can add and multiply exactly by setting the precision high enough, no precision is high enough to represent for example 1/3 exactly (the same obviously applies for roots, trigonometric functions, etc).
-
-The special numbers ``+inf``, ``-inf`` and ``nan`` are represented internally by a zero mantissa and a nonzero exponent.
-
-Mpmath uses arbitrary precision integers for both the mantissa and the exponent, so numbers can be as large in magnitude as permitted by the computer's memory. Some care may be necessary when working with extremely large numbers. Although standard arithmetic operators are safe, it is for example futile to attempt to compute the exponential function of of `10^{100000}`. Mpmath does not complain when asked to perform such a calculation, but instead chugs away on the problem to the best of its ability, assuming that computer resources are infinite. In the worst case, this will be slow and allocate a huge amount of memory; if entirely impossible Python will at some point raise ``OverflowError: long int too large to convert to int``.
-
-For further details on how the arithmetic is implemented, refer to the mpmath source code. The basic arithmetic operations are found in the ``libmp`` directory; many functions there are commented extensively.
-
-Decimal issues
---------------
-
-Mpmath uses binary arithmetic internally, while most interaction with the user is done via the decimal number system. Translating between binary and decimal numbers is a somewhat subtle matter; many Python novices run into the following "bug" (addressed in the `General Python FAQ <http://www.python.org/doc/faq/general/#why-are-floating-point-calculations-so-inaccurate>`_)::
-
-    >>> 0.1
-    0.10000000000000001
-
-Decimal fractions fall into the category of numbers that generally cannot be represented exactly in binary floating-point form. For example, none of the numbers 0.1, 0.01, 0.001 has an exact representation as a binary floating-point number. Although mpmath can approximate decimal fractions with any accuracy, it does not solve this problem for all uses; users who need *exact* decimal fractions should look at the ``decimal`` module in Python's standard library (or perhaps use fractions, which are much faster).
-
-With *prec* bits of precision, an arbitrary number can be approximated relatively to within `2^{-prec}`, or within `10^{-dps}` for *dps* decimal digits. The equivalent values for *prec* and *dps* are therefore related proportionally via the factor `C = \log(10)/\log(2)`, or roughly 3.32. For example, the standard (binary) precision in mpmath is 53 bits, which corresponds to a decimal precision of 15.95 digits.
-
-More precisely, mpmath uses the following formulas to translate between *prec* and *dps*::
-
-  dps(prec) = max(1, int(round(int(prec) / C - 1)))
-
-  prec(dps) = max(1, int(round((int(dps) + 1) * C)))
-
-Note that the dps is set 1 decimal digit lower than the corresponding binary precision. This is done to hide minor rounding errors and artifacts resulting from binary-decimal conversion. As a result, mpmath interprets 53 bits as giving 15 digits of decimal precision, not 16.
-
-The *dps* value controls the number of digits to display when printing numbers with :func:`str`, while the decimal precision used by :func:`repr` is set two or three digits higher. For example, with 15 dps we have::
-
-    >>> from mpmath import *
-    >>> mp.dps = 15
-    >>> str(pi)
-    '3.14159265358979'
-    >>> repr(+pi)
-    "mpf('3.1415926535897931')"
-
-The extra digits in the output from ``repr`` ensure that ``x == eval(repr(x))`` holds, i.e. that numbers can be converted to strings and back losslessly.
-
-It should be noted that precision and accuracy do not always correlate when translating between binary and decimal. As a simple example, the number 0.1 has a decimal precision of 1 digit but is an infinitely accurate representation of 1/10. Conversely, the number `2^{-50}` has a binary representation with 1 bit of precision that is infinitely accurate; the same number can actually be represented exactly as a decimal, but doing so requires 35 significant digits::
-
-    0.00000000000000088817841970012523233890533447265625
-
-All binary floating-point numbers can be represented exactly as decimals (possibly requiring many digits), but the converse is false.
-
-Correctness guarantees
-----------------------
-
-Basic arithmetic operations (with the ``mp`` context) are always performed with correct rounding. Results that can be represented exactly are guranteed to be exact, and results from single inexact operations are guaranteed to be the best possible rounded values. For higher-level operations, mpmath does not generally guarantee correct rounding. In general, mpmath only guarantees that it will use at least the user-set precision to perform a given calculation. *The user may have to manually set the working precision higher than the desired accuracy for the result, possibly much higher.*
-
-Functions for evaluation of transcendental functions, linear algebra operations, numerical integration, etc., usually automatically increase the working precision and use a stricter tolerance to give a correctly rounded result with high probability: for example, at 50 bits the temporary precision might be set to 70 bits and the tolerance might be set to 60 bits. It can often be assumed that such functions return values that have full accuracy, given inputs that are exact (or sufficiently precise approximations of exact values), but the user must exercise judgement about whether to trust mpmath.
-
-The level of rigor in mpmath covers the entire spectrum from "always correct by design" through "nearly always correct" and "handling the most common errors" to "just computing blindly and hoping for the best". Of course, a long-term development goal is to successively increase the rigor where possible. The following list might give an idea of the current state.
-
-Operations that are correctly rounded:
-
-* Addition, subtraction and multiplication of real and complex numbers.
-* Division and square roots of real numbers.
-* Powers of real numbers, assuming sufficiently small integer exponents (huge powers are rounded in the right direction, but possibly farther than necessary).
-* Conversion from decimal to binary, for reasonably sized numbers (roughly between `10^{-100}` and `10^{100}`).
-* Typically, transcendental functions for exact input-output pairs.
-
-Operations that should be fully accurate (however, the current implementation may be based on a heuristic error analysis):
-
-* Radix conversion (large or small numbers).
-* Mathematical constants like `\pi`.
-* Both real and imaginary parts of exp, cos, sin, cosh, sinh, log.
-* Other elementary functions (the largest of the real and imaginary part).
-* The gamma and log-gamma functions (the largest of the real and the imaginary part; both, when close to real axis).
-* Some functions based on hypergeometric series (the largest of the real and imaginary part).
-
-Correctness of root-finding, numerical integration, etc. largely depends on the well-behavedness of the input functions. Specific limitations are sometimes noted in the respective sections of the documentation.
-
-Double precision emulation
---------------------------
-
-On most systems, Python's ``float`` type represents an IEEE 754 *double precision* number, with a precision of 53 bits and rounding-to-nearest. With default precision (``mp.prec = 53``), the mpmath ``mpf`` type roughly emulates the behavior of the ``float`` type. Sources of incompatibility include the following:
-
-* In hardware floating-point arithmetic, the size of the exponent is restricted to a fixed range: regular Python floats have a range between roughly `10^{-300}` and `10^{300}`. Mpmath does not emulate overflow or underflow when exponents fall outside this range.
-* On some systems, Python uses 80-bit (extended double) registers for floating-point operations. Due to double rounding, this makes the ``float`` type less accurate. This problem is only known to occur with Python versions compiled with GCC on 32-bit systems.
-* Machine floats very close to the exponent limit round subnormally, meaning that they lose accuracy (Python may raise an exception instead of rounding a ``float`` subnormally).
-* Mpmath is able to produce more accurate results for transcendental functions.
-
-Further reading
----------------
-
-There are many excellent textbooks on numerical analysis and floating-point arithmetic. Some good web resources are:
-
-* `David Goldberg, What Every Computer Scientist Should Know About Floating-Point Arithmetic <http://docs.sun.com/source/806-3568/ncg_goldberg.html>`_
-* `The Wikipedia article about numerical analysis  <http://en.wikipedia.org/wiki/Numerical_analysis>`_
diff --git a/dev-py3k/_sources/modules/ntheory.txt b/dev-py3k/_sources/modules/ntheory.txt
deleted file mode 100644
index 2f1c3a8d78e..00000000000
--- a/dev-py3k/_sources/modules/ntheory.txt
+++ /dev/null
@@ -1,98 +0,0 @@
-Number Theory
-====================
-
-.. module:: sympy.ntheory.generate
-
-Ntheory Class Reference
------------------------
-.. autoclass:: Sieve
-   :members:
-
-Ntheory Functions Reference
----------------------------
-
-.. autofunction:: prime
-
-.. autofunction:: primepi
-
-.. autofunction:: nextprime
-
-.. autofunction:: prevprime
-
-.. autofunction:: primerange
-
-.. autofunction:: randprime
-
-.. autofunction:: primorial
-
-.. autofunction:: cycle_length
-
-.. module:: sympy.ntheory.factor_
-
-.. autofunction:: smoothness
-
-.. autofunction:: smoothness_p
-
-.. autofunction:: trailing
-
-.. autofunction:: multiplicity
-
-.. autofunction:: perfect_power
-
-.. autofunction:: pollard_rho
-
-.. autofunction:: pollard_pm1
-
-.. autofunction:: factorint
-
-.. autofunction:: primefactors
-
-.. autofunction:: divisors
-
-.. autofunction:: divisor_count
-
-.. autofunction:: totient
-
-.. module:: sympy.ntheory.modular
-
-.. autofunction:: symmetric_residue
-
-.. autofunction:: crt
-
-.. autofunction:: crt1
-
-.. autofunction:: crt2
-
-.. autofunction:: solve_congruence
-
-.. module:: sympy.ntheory.multinomial
-
-.. autofunction:: binomial_coefficients
-
-.. autofunction:: binomial_coefficients_list
-
-.. autofunction:: multinomial_coefficients
-
-.. autofunction:: multinomial_coefficients_iterator
-
-.. module:: sympy.ntheory.partitions_
-
-.. autofunction:: npartitions
-
-.. module:: sympy.ntheory.primetest
-
-.. autofunction:: mr
-
-.. autofunction:: isprime
-
-.. module:: sympy.ntheory.residue_ntheory
-
-.. autofunction:: n_order
-
-.. autofunction:: is_primitive_root
-
-.. autofunction:: is_quad_residue
-
-.. autofunction:: legendre_symbol
-
-.. autofunction:: jacobi_symbol
diff --git a/dev-py3k/_sources/modules/parsing.txt b/dev-py3k/_sources/modules/parsing.txt
deleted file mode 100644
index 55e5e72d13a..00000000000
--- a/dev-py3k/_sources/modules/parsing.txt
+++ /dev/null
@@ -1,32 +0,0 @@
-Parsing input
-=============
-
-Parsing Functions Reference
----------------------------
-
-.. autofunction:: sympy.parsing.sympy_parser.parse_expr
-
-.. autofunction:: sympy.parsing.sympy_tokenize.printtoken
-
-.. autofunction:: sympy.parsing.sympy_tokenize.tokenize
-
-.. autofunction:: sympy.parsing.sympy_tokenize.untokenize
-
-.. autofunction:: sympy.parsing.sympy_tokenize.generate_tokens
-
-.. autofunction:: sympy.parsing.sympy_tokenize.group
-
-.. autofunction:: sympy.parsing.sympy_tokenize.any
-
-.. autofunction:: sympy.parsing.sympy_tokenize.maybe
-
-.. autofunction:: sympy.parsing.maxima.parse_maxima
-
-.. autofunction:: sympy.parsing.mathematica.mathematica
-
-Parsing Exceptions Reference
-----------------------------
-
-.. autoclass:: sympy.parsing.sympy_tokenize.TokenError
-
-.. autoclass:: sympy.parsing.sympy_tokenize.StopTokenizing
diff --git a/dev-py3k/_sources/modules/physics/gaussopt.txt b/dev-py3k/_sources/modules/physics/gaussopt.txt
deleted file mode 100644
index 36743fa0418..00000000000
--- a/dev-py3k/_sources/modules/physics/gaussopt.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-===============
-Gaussian Optics
-===============
-
-.. automodule:: sympy.physics.gaussopt
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/hydrogen.txt b/dev-py3k/_sources/modules/physics/hydrogen.txt
deleted file mode 100644
index 368d3dfdf47..00000000000
--- a/dev-py3k/_sources/modules/physics/hydrogen.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-======================
-Hydrogen Wavefunctions
-======================
-
-.. automodule:: sympy.physics.hydrogen
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/index.txt b/dev-py3k/_sources/modules/physics/index.txt
deleted file mode 100644
index bfefeea8f65..00000000000
--- a/dev-py3k/_sources/modules/physics/index.txt
+++ /dev/null
@@ -1,25 +0,0 @@
-.. _physics-docs:
-
-==============
-Physics Module
-==============
-
-.. automodule:: sympy.physics
-
-Contents
-========
-
-.. toctree::
-    :maxdepth: 3
-
-    gaussopt.rst
-    hydrogen.rst
-    matrices.rst
-    paulialgebra.rst
-    qho_1d.rst
-    sho.rst
-    secondquant.rst
-    wigner.rst
-    units.rst
-    mechanics/index.rst
-    quantum/index.rst
diff --git a/dev-py3k/_sources/modules/physics/matrices.txt b/dev-py3k/_sources/modules/physics/matrices.txt
deleted file mode 100644
index 8d70f309946..00000000000
--- a/dev-py3k/_sources/modules/physics/matrices.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-========
-Matrices
-========
-
-.. automodule:: sympy.physics.matrices
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/mechanics/advanced.txt b/dev-py3k/_sources/modules/physics/mechanics/advanced.txt
deleted file mode 100644
index cc27e88f8ce..00000000000
--- a/dev-py3k/_sources/modules/physics/mechanics/advanced.txt
+++ /dev/null
@@ -1,206 +0,0 @@
-============================================================================
-Potential Issues/Advanced Topics/Future Features in Physics/Mechanics Module
-============================================================================
-
-This document will describe some of the more advanced functionality that this
-module offers but which is not part of the "official" interface. Here, some of
-the features that will be implemented in the future will also be covered, along
-with unanswered questions about proper functionality. Also, common problems
-will be discussed, along with some solutions.
-
-Common Issues
-=============
-Here issues with numerically integrating code, choice of `dynamicsymbols` for
-coordinate and speed representation, printing, differentiating, and
-substitution will occur.
-
-Numerically Integrating Code
-----------------------------
-See Future Features: Code Output
-
-Choice of Coordinates and Speeds
---------------------------------
-The Kane object is set up with the assumption that the generalized speeds are
-not the same symbol as the time derivatives of the generalized coordinates.
-This isn't to say that they can't be the same, just that they have to have a
-different symbol. If you did this: ::
-
-  >> KM.coords([q1, q2, q3])
-  >> KM.speeds([q1d, q2d, q3d])
-
-Your code would not work. Currently, kinematic differential equations are
-required to be provided. It is at this point that we hope the user will
-discover they should not attempt the behavior shown in the code above.
-
-This behavior might not be true for other methods of forming the equations of
-motion though.
-
-Printing
---------
-The default printing options are to use sorting for ``Vector`` and ``Dyad``
-measure numbers, and have unsorted output from the ``mprint``, ``mpprint``, and
-``mlatex`` functions. If you are printing something large, please use one of
-those functions, as the sorting can increase printing time from seconds to
-minutes.
-
-Differentiating
----------------
-Differentiation of very large expressions can take some time in SymPy; it is
-possible for large expressions to take minutes for the derivative to be
-evaluated. This will most commonly come up in linearization.
-
-Substitution
-------------
-Substitution into large expressions can be slow, and take a few minutes.
-
-Linearization
--------------
-Currently, the ``Kane`` object's ``linearize`` method doesn't support cases
-where there are non-coordinate, non-speed dynamic symbols outside of the
-"dynamic equations". It also does not support cases where time derivatives of
-these types of dynamic symbols show up. This means if you have kinematic
-differential equations which have a non-coordinate, non-speed dynamic symbol,
-it will not work. It also means if you have defined a system parameter (say a
-length or distance or mass) as a dynamic symbol, its time derivative is likely
-to show up in the dynamic equations, and this will prevent linearization.
-
-Acceleration of Points
-----------------------
-At a minimum, points need to have their velocities defined, as the acceleration
-can be calculated by taking the time derivative of the velocity in the same
-frame. If the 1 point or 2 point theorems were used to compute the velocity,
-the time derivative of the velocity expression will most likely be more complex
-than if you were to use the acceleration level 1 point and 2 point theorems.
-Using the acceleration level methods can result in shorted expressions at this
-point, which will result in shorter expressions later (such as when forming
-Kane's equations).
-
-
-Advanced Interfaces
-===================
-
-Here we will cover advanced options in: ``ReferenceFrame``, ``dynamicsymbols``,
-and some associated functionality.
-
-ReferenceFrame
---------------
-``ReferenceFrame`` is shown as having a ``.name`` attribute and ``.x``, ``.y``,
-and ``.z`` attributes for accessing the basis vectors, as well as a fairly
-rigidly defined print output. If you wish to have a different set of indices
-defined, there is an option for this. This will also require a different
-interface for accessing the basis vectors. ::
-
-  >>> from sympy.physics.mechanics import ReferenceFrame, mprint, mpprint, mlatex
-  >>> N = ReferenceFrame('N', indices=['i', 'j', 'k'])
-  >>> N['i']
-  N['i']
-  >>> N.x
-  N['i']
-  >>> mlatex(N.x)
-  '\\mathbf{\\hat{n}_{i}}'
-
-Also, the latex output can have custom strings; rather than just indices
-though, the entirety of each basis vector can be specified. The custom latex
-strings can occur without custom indices, and also overwrites the latex string
-that would be used if there were custom indices. ::
-
-  >>> from sympy.physics.mechanics import ReferenceFrame, mlatex
-  >>> N = ReferenceFrame('N', latexs=['n1','\mathbf{n}_2','cat'])
-  >>> mlatex(N.x)
-  'n1'
-  >>> mlatex(N.y)
-  '\\mathbf{n}_2'
-  >>> mlatex(N.z)
-  'cat'
-
-dynamicsymbols
---------------
-The ``dynamicsymbols`` function also has 'hidden' functionality; the variable
-which is associated with time can be changed, as well as the notation for
-printing derivatives. ::
-
-  >>> from sympy import symbols
-  >>> from sympy.physics.mechanics import dynamicsymbols, mprint
-  >>> q1 = dynamicsymbols('q1')
-  >>> q1
-  q1(t)
-  >>> dynamicsymbols._t = symbols('T')
-  >>> q2 = dynamicsymbols('q2')
-  >>> q2
-  q2(T)
-  >>> q1
-  q1(t)
-  >>> q1d = dynamicsymbols('q1', 1)
-  >>> mprint(q1d)
-  q1'
-  >>> dynamicsymbols._str = 'd'
-  >>> mprint(q1d)
-  q1d
-  >>> dynamicsymbols._str = '\''
-  >>> dynamicsymbols._t = symbols('t')
-
-
-Note that only dynamic symbols created after the change are different. The same
-is not true for the `._str` attribute; this affects the printing output only,
-so dynamic symbols created before or after will print the same way.
-
-Also note that ``Vector``'s ``.dt`` method uses the ``._t`` attribute of
-``dynamicsymbols``, along with a number of other important functions and
-methods. Don't mix and match symbols representing time.
-
-Advanced Functionality
-----------------------
-Remember that the ``Kane`` object supports bodies which have time-varying
-masses and inertias, although this functionality isn't completely compatible
-with the linearization method.
-
-Operators were discussed earlier as a potential way to do mathematical
-operations on ``Vector`` and ``Dyad`` objects. The majority of the code in this
-module is actually coded with them, as it can (subjectively) result in cleaner,
-shorter, more readable code. If using this interface in your code, remember to
-take care and use parentheses; the default order of operations in Python
-results in addition occurring before some of the vector products, so use
-parentheses liberally.
-
-
-Future Features
-===============
-
-This will cover the planned features to be added to this submodule.
-
-Code Output
------------
-A function for generating code output for numerical integration is the highest
-priority feature to implement next. There are a number of considerations here.
-
-Code output for C (using the GSL libraries), Fortran 90 (using LSODA), MATLAB,
-and SciPy is the goal. Things to be considered include: use of ``cse`` on large
-expressions for MATLAB and SciPy, which are interpretive. It is currently unclear
-whether compiled languages will benefit from common subexpression elimination,
-especially considering that it is a common part of compiler optimization, and
-there can be a significant time penalty when calling ``cse``.
-
-Care needs to be taken when constructing the strings for these expressions, as
-well as handling of input parameters, and other dynamic symbols. How to deal
-with output quantities when integrating also needs to be decided, with the
-potential for multiple options being considered.
-
-Additional Options on Initialization of Kane, RigidBody, and Particle
----------------------------------------------------------------------
-This would allow a user to specify all relevant information using keyword
-arguments when creating these objects. This is fairly clear for ``RigidBody``
-and ``Point``. For ``Kane``, everything but the force and body lists will be
-able to be entered, as computation of Fr and Fr* can take a while, and produce
-an output.
-
-Additional Methods for RigidBody and Particle
----------------------------------------------
-For ``RigidBody`` and ``Particle`` (not all methods for ``Particle`` though),
-add methods for getting: momentum, angular momentum, and kinetic energy.
-Additionally, adding a attribute and method for defining potential energy would
-allow for a total energy method/property.
-
-Also possible is including the method which creates a transformation matrix for
-3D animations; this would require a "reference orientation" for a camera as
-well as a "reference point" for distance to the camera. Development of this
-could also be tied into code output.
diff --git a/dev-py3k/_sources/modules/physics/mechanics/api/essential.txt b/dev-py3k/_sources/modules/physics/mechanics/api/essential.txt
deleted file mode 100644
index 4828a1bdc37..00000000000
--- a/dev-py3k/_sources/modules/physics/mechanics/api/essential.txt
+++ /dev/null
@@ -1,23 +0,0 @@
-=================================
-Essential Components (Docstrings)
-=================================
-
-ReferenceFrame
-==============
-
-.. autoclass:: sympy.physics.mechanics.essential.ReferenceFrame
-   :members:
-
-
-Vector
-======
-
-.. autoclass:: sympy.physics.mechanics.essential.Vector
-   :members:
-
-
-Dyadic
-======
-
-.. autoclass:: sympy.physics.mechanics.essential.Dyadic
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/mechanics/api/functions.txt b/dev-py3k/_sources/modules/physics/mechanics/api/functions.txt
deleted file mode 100644
index f67ca1f057c..00000000000
--- a/dev-py3k/_sources/modules/physics/mechanics/api/functions.txt
+++ /dev/null
@@ -1,32 +0,0 @@
-========================================
-Related Essential Functions (Docstrings)
-========================================
-
-dynamicsymbols
-==============
-
-.. autofunction:: sympy.physics.mechanics.essential.dynamicsymbols
-
-
-dot
-===
-
-.. autofunction:: sympy.physics.mechanics.functions.dot
-
-
-cross
-=====
-
-.. autofunction:: sympy.physics.mechanics.functions.cross
-
-
-outer
-=====
-
-.. autofunction:: sympy.physics.mechanics.functions.outer
-
-
-express
-=======
-
-.. autofunction:: sympy.physics.mechanics.functions.express
diff --git a/dev-py3k/_sources/modules/physics/mechanics/api/kane.txt b/dev-py3k/_sources/modules/physics/mechanics/api/kane.txt
deleted file mode 100644
index 0d1424f245a..00000000000
--- a/dev-py3k/_sources/modules/physics/mechanics/api/kane.txt
+++ /dev/null
@@ -1,22 +0,0 @@
-====================================================================
-Kane's Method, Lagrange's Method & Associated Functions (Docstrings)
-====================================================================
-
-Kane
-====
-
-.. automodule:: sympy.physics.mechanics.kane
-   :members:
-
-
-partial_velocity
-================
-
-.. automodule:: sympy.physics.mechanics.functions
-   :members: partial_velocity
-
-LagrangesMethod
-===============
-
-.. automodule:: sympy.physics.mechanics.lagrange
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/mechanics/api/kinematics.txt b/dev-py3k/_sources/modules/physics/mechanics/api/kinematics.txt
deleted file mode 100644
index e3135261fd5..00000000000
--- a/dev-py3k/_sources/modules/physics/mechanics/api/kinematics.txt
+++ /dev/null
@@ -1,16 +0,0 @@
-=======================
-Kinematics (Docstrings)
-=======================
-
-Point
-=====
-
-.. automodule:: sympy.physics.mechanics.point
-   :members:
-
-
-kinematic_equations
-===================
-
-.. automodule:: sympy.physics.mechanics.functions
-   :members: kinematic_equations
diff --git a/dev-py3k/_sources/modules/physics/mechanics/api/part_bod.txt b/dev-py3k/_sources/modules/physics/mechanics/api/part_bod.txt
deleted file mode 100644
index 5fd0200876f..00000000000
--- a/dev-py3k/_sources/modules/physics/mechanics/api/part_bod.txt
+++ /dev/null
@@ -1,55 +0,0 @@
-=====================================================
-Masses, Inertias & Particles, RigidBodys (Docstrings)
-=====================================================
-
-Particle
-========
-
-.. automodule:: sympy.physics.mechanics.particle
-   :members:
-
-
-RigidBody
-=========
-
-.. automodule:: sympy.physics.mechanics.rigidbody
-   :members:
-
-
-inertia
-=======
-
-.. autofunction:: sympy.physics.mechanics.functions.inertia
-
-
-inertia_of_point_mass
-=====================
-
-.. autofunction:: sympy.physics.mechanics.functions.inertia_of_point_mass
-
-
-linear_momentum
-===============
-
-.. autofunction:: sympy.physics.mechanics.functions.linear_momentum
-
-
-angular_momentum
-================
-
-.. autofunction:: sympy.physics.mechanics.functions.angular_momentum
-
-kinetic_energy
-==============
-
-.. autofunction:: sympy.physics.mechanics.functions.kinetic_energy
-
-potential_energy
-================
-
-.. autofunction:: sympy.physics.mechanics.functions.potential_energy
-
-Lagrangian
-==========
-
-.. autofunction:: sympy.physics.mechanics.functions.Lagrangian
diff --git a/dev-py3k/_sources/modules/physics/mechanics/api/printing.txt b/dev-py3k/_sources/modules/physics/mechanics/api/printing.txt
deleted file mode 100644
index 99ba010e23e..00000000000
--- a/dev-py3k/_sources/modules/physics/mechanics/api/printing.txt
+++ /dev/null
@@ -1,26 +0,0 @@
-=====================
-Printing (Docstrings)
-=====================
-
-mechanics_printing
-==================
-
-.. autofunction:: sympy.physics.mechanics.functions.mechanics_printing
-
-
-mprint
-======
-
-.. autofunction:: sympy.physics.mechanics.functions.mprint
-
-
-mpprint
-=======
-
-.. autofunction:: sympy.physics.mechanics.functions.mpprint
-
-
-mlatex
-======
-
-.. autofunction:: sympy.physics.mechanics.functions.mlatex
diff --git a/dev-py3k/_sources/modules/physics/mechanics/examples.txt b/dev-py3k/_sources/modules/physics/mechanics/examples.txt
deleted file mode 100644
index 025a002d519..00000000000
--- a/dev-py3k/_sources/modules/physics/mechanics/examples.txt
+++ /dev/null
@@ -1,577 +0,0 @@
-==============================
-Examples for Physics.Mechanics
-==============================
-
-This module has been developed to aid in formulation of equations of motion;
-here, examples follow to help display how to use this module.
-
-The Rolling Disc
-================
-
-The first example is that of a rolling disc. The disc is assumed to be
-infinitely thin, in contact with the ground at only 1 point, and it is rolling
-without slip on the ground. See the image below.
-
-.. image:: ex_rd.*
-   :height: 550
-   :width: 550
-   :align: center
-
-Here the definition of the rolling disc's kinematics is formed from the contact
-point up, removing the need to introduce generalized speeds. Only 3
-configuration and three speed variables are need to describe this system, along
-with the disc's mass and radius, and the local gravity (note that mass will
-drop out). ::
-
-  >>> from sympy import symbols, sin, cos, tan
-  >>> from sympy.physics.mechanics import *
-  >>> q1, q2, q3, u1, u2, u3  = dynamicsymbols('q1 q2 q3 u1 u2 u3')
-  >>> q1d, q2d, q3d, u1d, u2d, u3d = dynamicsymbols('q1 q2 q3 u1 u2 u3', 1)
-  >>> r, m, g = symbols('r m g')
-  >>> mechanics_printing()
-
-The kinematics are formed by a series of simple rotations. Each simple rotation
-creates a new frame, and the next rotation is defined by the new frame's basis
-vectors. This example uses a 3-1-2 series of rotations, or Z, X, Y series of
-rotations. Angular velocity for this is defined using the second frame's basis
-(the lean frame); it is for this reason that we defined intermediate frames,
-rather than using a body-three orientation. ::
-
-  >>> N = ReferenceFrame('N')
-  >>> Y = N.orientnew('Y', 'Axis', [q1, N.z])
-  >>> L = Y.orientnew('L', 'Axis', [q2, Y.x])
-  >>> R = L.orientnew('R', 'Axis', [q3, L.y])
-  >>> R.set_ang_vel(N, u1 * L.x + u2 * L.y + u3 * L.z)
-  >>> R.set_ang_acc(N, R.ang_vel_in(N).dt(R) + (R.ang_vel_in(N) ^ R.ang_vel_in(N)))
-
-This is the translational kinematics. We create a point with no velocity
-in N; this is the contact point between the disc and ground. Next we form
-the position vector from the contact point to the disc's center of mass.
-Finally we form the velocity and acceleration of the disc. ::
-
-  >>> C = Point('C')
-  >>> C.set_vel(N, 0)
-  >>> Dmc = C.locatenew('Dmc', r * L.z)
-  >>> Dmc.v2pt_theory(C, N, R)
-  r*u2*L.x - r*u1*L.y
-  >>> Dmc.a2pt_theory(C, N, R)
-  (r*u1*u3 + r*u2')*L.x + (-r*(-u3*q3' + u1') + r*u2*u3)*L.y + (-r*u1**2 - r*u2**2)*L.z
-
-This is a simple way to form the inertia dyadic. The inertia of the disc does
-not change within the lean frame as the disc rolls; this will make for simpler
-equations in the end. ::
-
-  >>> I = inertia(L, m / 4 * r**2, m / 2 * r**2, m / 4 * r**2)
-  >>> mprint(I)
-  m*r**2/4*(L.x|L.x) + m*r**2/2*(L.y|L.y) + m*r**2/4*(L.z|L.z)
-
-Kinematic differential equations; how the generalized coordinate time
-derivatives relate to generalized speeds. Here these were computed by hand. ::
-
-  >>> kd = [q1d - u3/cos(q2), q2d - u1, q3d - u2 + u3 * tan(q2)]
-
-Creation of the force list; it is the gravitational force at the center of mass of
-the disc. Then we create the disc by assigning a Point to the center of mass
-attribute, a ReferenceFrame to the frame attribute, and mass and inertia. Then
-we form the body list. ::
-
-  >>> ForceList = [(Dmc, - m * g * Y.z)]
-  >>> BodyD = RigidBody('BodyD', Dmc, R, m, (I, Dmc))
-  >>> BodyList = [BodyD]
-
-Finally we form the equations of motion, using the same steps we did before.
-Specify inertial frame, supply generalized coordinates and speeds, supply
-kinematic differential equation dictionary, compute Fr from the force list and
-Fr* from the body list, compute the mass matrix and forcing terms, then solve
-for the u dots (time derivatives of the generalized speeds). ::
-
-  >>> KM = KanesMethod(N, q_ind=[q1, q2, q3], u_ind=[u1, u2, u3], kd_eqs=kd)
-  >>> (fr, frstar) = KM.kanes_equations(ForceList, BodyList)
-  >>> MM = KM.mass_matrix
-  >>> forcing = KM.forcing
-  >>> rhs = MM.inv() * forcing
-  >>> kdd = KM.kindiffdict()
-  >>> rhs = rhs.subs(kdd)
-  >>> rhs.simplify()
-  >>> mprint(rhs)
-  [(4*g*sin(q2)/5 + 2*r*u2*u3 - r*u3**2*tan(q2))/r]
-  [                                     -2*u1*u3/3]
-  [                        (-2*u2 + u3*tan(q2))*u1]
-
-This concludes the rolling disc example.
-
-The Rolling Disc, again
-=======================
-
-We will now revisit the rolling disc example, except this time we are bringing
-the non-contributing (constraint) forces into evidence. See [Kane1985]_ for a
-more thorough explanation of this. Here, we will turn on the automatic
-simplifcation done when doing vector operations. It makes the outputs nicer for
-small problems, but can cause larger vector operations to hang. ::
-
-  >>> from sympy.physics.mechanics import *
-  >>> mechanics_printing()
-  >>> Vector.simp = True
-  >>> q1, q2, q3, u1, u2, u3  = dynamicsymbols('q1 q2 q3 u1 u2 u3')
-  >>> q1d, q2d, q3d, u1d, u2d, u3d = dynamicsymbols('q1 q2 q3 u1 u2 u3', 1)
-  >>> r, m, g = symbols('r m g')
-
-These two lines introduce the extra quantities needed to find the constraint
-forces. ::
-
-  >>> u4, u5, u6, f1, f2, f3 = dynamicsymbols('u4 u5 u6 f1 f2 f3')
-
-Most of the main code is the same as before. ::
-
-  >>> N = ReferenceFrame('N')
-  >>> Y = N.orientnew('Y', 'Axis', [q1, N.z])
-  >>> L = Y.orientnew('L', 'Axis', [q2, Y.x])
-  >>> R = L.orientnew('R', 'Axis', [q3, L.y])
-  >>> R.set_ang_vel(N, u1 * L.x + u2 * L.y + u3 * L.z)
-  >>> R.set_ang_acc(N, R.ang_vel_in(N).dt(R) + (R.ang_vel_in(N) ^ R.ang_vel_in(N)))
-
-The definition of rolling without slip necessitates that the velocity of the
-contact point is zero; as part of bringing the constraint forces into evidence,
-we have to introduce speeds at this point, which will by definition always be
-zero. They are normal to the ground, along the path which the disc is rolling,
-and along the ground in an perpendicular direction. ::
-
-  >>> C = Point('C')
-  >>> C.set_vel(N, u4 * L.x + u5 * (Y.z ^ L.x) + u6 * Y.z)
-  >>> Dmc = C.locatenew('Dmc', r * L.z)
-  >>> vel = Dmc.v2pt_theory(C, N, R)
-  >>> acc = Dmc.a2pt_theory(C, N, R)
-  >>> I = inertia(L, m / 4 * r**2, m / 2 * r**2, m / 4 * r**2)
-  >>> kd = [q1d - u3/cos(q2), q2d - u1, q3d - u2 + u3 * tan(q2)]
-
-Just as we previously introduced three speeds as part of this process, we also
-introduce three forces; they are in the same direction as the speeds, and
-represent the constraint forces in those directions. ::
-
-  >>> ForceList = [(Dmc, - m * g * Y.z), (C, f1 * L.x + f2 * (Y.z ^ L.x) + f3 * Y.z)]
-  >>> BodyD = RigidBody('BodyD', Dmc, R, m, (I, Dmc))
-  >>> BodyList = [BodyD]
-
-  >>> KM = KanesMethod(N, q_ind=[q1, q2, q3], u_ind=[u1, u2, u3], kd_eqs=kd,
-  ...           u_auxiliary=[u4, u5, u6])
-  >>> (fr, frstar) = KM.kanes_equations(ForceList, BodyList)
-  >>> MM = KM.mass_matrix
-  >>> forcing = KM.forcing
-  >>> rhs = MM.inv() * forcing
-  >>> kdd = KM.kindiffdict()
-  >>> rhs = rhs.subs(kdd)
-  >>> rhs.simplify()
-  >>> mprint(rhs)
-  [(4*g*sin(q2)/5 + 2*r*u2*u3 - r*u3**2*tan(q2))/r]
-  [                                     -2*u1*u3/3]
-  [                        (-2*u2 + u3*tan(q2))*u1]
-  >>> from sympy import signsimp, factor_terms
-  >>> mprint(KM.auxiliary_eqs.applyfunc(lambda w: factor_terms(signsimp(w))))
-  [                                                   m*r*(u1*u3 + u2') - f1]
-  [      m*r*((u1**2 + u2**2)*sin(q2) + (u2*u3 + u3*q3' - u1')*cos(q2)) - f2]
-  [g*m - m*r*((u1**2 + u2**2)*cos(q2) - (u2*u3 + u3*q3' - u1')*sin(q2)) - f3]
-
-The Bicycle
-===========
-
-The bicycle is an interesting system in that it has multiple rigid bodies,
-non-holonomic constraints, and a holonomic constraint. The linearized equations
-of motion are presented in [Meijaard2007]_. This example will go through
-construction of the equations of motion in :mod:`mechanics`. ::
-
-  >>> from sympy import *
-  >>> from sympy.physics.mechanics import *
-  >>> print('Calculation of Linearized Bicycle \"A\" Matrix, '
-  ...       'with States: Roll, Steer, Roll Rate, Steer Rate')
-  Calculation of Linearized Bicycle "A" Matrix, with States: Roll, Steer, Roll Rate, Steer Rate
-
-
-Note that this code has been crudely ported from Autolev, which is the reason
-for some of the unusual naming conventions. It was purposefully as similar as
-possible in order to aid initial porting & debugging. We also turn off
-Vector.simp (turned on in the last example) to avoid hangups when doing
-computations in this problem. ::
-
-  >>> Vector.simp = False
-  >>> mechanics_printing()
-
-Declaration of Coordinates & Speeds:
-A Simple definitions for qdots: (qd = u) is used in this code.  Speeds are: yaw
-frame ang. rate, roll frame ang. rate, rear wheel frame ang.  rate (spinning
-motion), frame ang. rate (pitching motion), steering frame ang. rate, and front
-wheel ang. rate (spinning motion).  Wheel positions are ignorable coordinates,
-so they are not introduced. ::
-
-  >>> q1, q2, q4, q5 = dynamicsymbols('q1 q2 q4 q5')
-  >>> q1d, q2d, q4d, q5d = dynamicsymbols('q1 q2 q4 q5', 1)
-  >>> u1, u2, u3, u4, u5, u6 = dynamicsymbols('u1 u2 u3 u4 u5 u6')
-  >>> u1d, u2d, u3d, u4d, u5d, u6d = dynamicsymbols('u1 u2 u3 u4 u5 u6', 1)
-
-Declaration of System's Parameters:
-The below symbols should be fairly self-explanatory. ::
-
-  >>> WFrad, WRrad, htangle, forkoffset = symbols('WFrad WRrad htangle forkoffset')
-  >>> forklength, framelength, forkcg1 = symbols('forklength framelength forkcg1')
-  >>> forkcg3, framecg1, framecg3, Iwr11 = symbols('forkcg3 framecg1 framecg3 Iwr11')
-  >>> Iwr22, Iwf11, Iwf22, Iframe11 = symbols('Iwr22 Iwf11 Iwf22 Iframe11')
-  >>> Iframe22, Iframe33, Iframe31, Ifork11 = \
-  ...     symbols('Iframe22 Iframe33 Iframe31 Ifork11')
-  >>> Ifork22, Ifork33, Ifork31, g = symbols('Ifork22 Ifork33 Ifork31 g')
-  >>> mframe, mfork, mwf, mwr = symbols('mframe mfork mwf mwr')
-
-Set up reference frames for the system:
-N - inertial
-Y - yaw
-R - roll
-WR - rear wheel, rotation angle is ignorable coordinate so not oriented
-Frame - bicycle frame
-TempFrame - statically rotated frame for easier reference inertia definition
-Fork - bicycle fork
-TempFork - statically rotated frame for easier reference inertia definition
-WF - front wheel, again posses a ignorable coordinate ::
-
-  >>> N = ReferenceFrame('N')
-  >>> Y = N.orientnew('Y', 'Axis', [q1, N.z])
-  >>> R = Y.orientnew('R', 'Axis', [q2, Y.x])
-  >>> Frame = R.orientnew('Frame', 'Axis', [q4 + htangle, R.y])
-  >>> WR = ReferenceFrame('WR')
-  >>> TempFrame = Frame.orientnew('TempFrame', 'Axis', [-htangle, Frame.y])
-  >>> Fork = Frame.orientnew('Fork', 'Axis', [q5, Frame.x])
-  >>> TempFork = Fork.orientnew('TempFork', 'Axis', [-htangle, Fork.y])
-  >>> WF = ReferenceFrame('WF')
-
-
-Kinematics of the Bicycle:
-First block of code is forming the positions of the relevant points rear wheel
-contact -> rear wheel's center of mass -> frame's center of mass + frame/fork connection
--> fork's center of mass + front wheel's center of mass -> front wheel contact point. ::
-
-  >>> WR_cont = Point('WR_cont')
-  >>> WR_mc = WR_cont.locatenew('WR_mc', WRrad * R.z)
-  >>> Steer = WR_mc.locatenew('Steer', framelength * Frame.z)
-  >>> Frame_mc = WR_mc.locatenew('Frame_mc', -framecg1 * Frame.x + framecg3 * Frame.z)
-  >>> Fork_mc = Steer.locatenew('Fork_mc', -forkcg1 * Fork.x + forkcg3 * Fork.z)
-  >>> WF_mc = Steer.locatenew('WF_mc', forklength * Fork.x + forkoffset * Fork.z)
-  >>> WF_cont = WF_mc.locatenew('WF_cont', WFrad*(dot(Fork.y, Y.z)*Fork.y - \
-  ...                                             Y.z).normalize())
-
-Set the angular velocity of each frame:
-Angular accelerations end up being calculated automatically by differentiating
-the angular velocities when first needed. ::
-u1 is yaw rate
-u2 is roll rate
-u3 is rear wheel rate
-u4 is frame pitch rate
-u5 is fork steer rate
-u6 is front wheel rate ::
-
-  >>> Y.set_ang_vel(N, u1 * Y.z)
-  >>> R.set_ang_vel(Y, u2 * R.x)
-  >>> WR.set_ang_vel(Frame, u3 * Frame.y)
-  >>> Frame.set_ang_vel(R, u4 * Frame.y)
-  >>> Fork.set_ang_vel(Frame, u5 * Fork.x)
-  >>> WF.set_ang_vel(Fork, u6 * Fork.y)
-
-Form the velocities of the points, using the 2-point theorem.  Accelerations
-again are calculated automatically when first needed. ::
-
-  >>> WR_cont.set_vel(N, 0)
-  >>> WR_mc.v2pt_theory(WR_cont, N, WR)
-  WRrad*(u1*sin(q2) + u3 + u4)*R.x - WRrad*u2*R.y
-  >>> Steer.v2pt_theory(WR_mc, N, Frame)
-  WRrad*(u1*sin(q2) + u3 + u4)*R.x - WRrad*u2*R.y + framelength*(u1*sin(q2) + u4)*Frame.x - framelength*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4))*Frame.y
-  >>> Frame_mc.v2pt_theory(WR_mc, N, Frame)
-  WRrad*(u1*sin(q2) + u3 + u4)*R.x - WRrad*u2*R.y + framecg3*(u1*sin(q2) + u4)*Frame.x + (-framecg1*(u1*cos(htangle + q4)*cos(q2) + u2*sin(htangle + q4)) - framecg3*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4)))*Frame.y + framecg1*(u1*sin(q2) + u4)*Frame.z
-  >>> Fork_mc.v2pt_theory(Steer, N, Fork)
-  WRrad*(u1*sin(q2) + u3 + u4)*R.x - WRrad*u2*R.y + framelength*(u1*sin(q2) + u4)*Frame.x - framelength*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4))*Frame.y + forkcg3*((sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*u1 + u2*sin(htangle + q4)*sin(q5) + u4*cos(q5))*Fork.x + (-forkcg1*((-sin(q2)*sin(q5) + cos(htangle + q4)*cos(q2)*cos(q5))*u1 + u2*sin(htangle + q4)*cos(q5) - u4*sin(q5)) - forkcg3*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4) + u5))*Fork.y + forkcg1*((sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*u1 + u2*sin(htangle + q4)*sin(q5) + u4*cos(q5))*Fork.z
-  >>> WF_mc.v2pt_theory(Steer, N, Fork)
-  WRrad*(u1*sin(q2) + u3 + u4)*R.x - WRrad*u2*R.y + framelength*(u1*sin(q2) + u4)*Frame.x - framelength*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4))*Frame.y + forkoffset*((sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*u1 + u2*sin(htangle + q4)*sin(q5) + u4*cos(q5))*Fork.x + (forklength*((-sin(q2)*sin(q5) + cos(htangle + q4)*cos(q2)*cos(q5))*u1 + u2*sin(htangle + q4)*cos(q5) - u4*sin(q5)) - forkoffset*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4) + u5))*Fork.y - forklength*((sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*u1 + u2*sin(htangle + q4)*sin(q5) + u4*cos(q5))*Fork.z
-  >>> WF_cont.v2pt_theory(WF_mc, N, WF)
-  WRrad*(u1*sin(q2) + u3 + u4)*R.x - WRrad*u2*R.y + framelength*(u1*sin(q2) + u4)*Frame.x - framelength*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4))*Frame.y + (-WFrad*(sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*((-sin(q2)*sin(q5) + cos(htangle + q4)*cos(q2)*cos(q5))*u1 + u2*sin(htangle + q4)*cos(q5) - u4*sin(q5))/sqrt((-sin(q2)*cos(q5) - sin(q5)*cos(htangle + q4)*cos(q2))*(sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2)) + 1) + forkoffset*((sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*u1 + u2*sin(htangle + q4)*sin(q5) + u4*cos(q5)))*Fork.x + (forklength*((-sin(q2)*sin(q5) + cos(htangle + q4)*cos(q2)*cos(q5))*u1 + u2*sin(htangle + q4)*cos(q5) - u4*sin(q5)) - forkoffset*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4) + u5))*Fork.y + (WFrad*(sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4) + u5)/sqrt((-sin(q2)*cos(q5) - sin(q5)*cos(htangle + q4)*cos(q2))*(sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2)) + 1) - forklength*((sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*u1 + u2*sin(htangle + q4)*sin(q5) + u4*cos(q5)))*Fork.z - WFrad*((-sin(q2)*sin(q5)*cos(htangle + q4) + cos(q2)*cos(q5))*u6 + u4*cos(q2) + u5*sin(htangle + q4)*sin(q2))/sqrt((-sin(q2)*cos(q5) - sin(q5)*cos(htangle + q4)*cos(q2))*(sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2)) + 1)*Y.x + WFrad*(u2 + u5*cos(htangle + q4) + u6*sin(htangle + q4)*sin(q5))/sqrt((-sin(q2)*cos(q5) - sin(q5)*cos(htangle + q4)*cos(q2))*(sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2)) + 1)*Y.y
-
-
-Sets the inertias of each body. Uses the inertia frame to construct the inertia
-dyadics. Wheel inertias are only defined by principal moments of inertia, and
-are in fact constant in the frame and fork reference frames; it is for this
-reason that the orientations of the wheels does not need to be defined. The
-frame and fork inertias are defined in the 'Temp' frames which are fixed to the
-appropriate body frames; this is to allow easier input of the reference values
-of the benchmark paper. Note that due to slightly different orientations, the
-products of inertia need to have their signs flipped; this is done later when
-entering the numerical value. ::
-
-  >>> Frame_I = (inertia(TempFrame, Iframe11, Iframe22, Iframe33, 0, 0,
-  ...                                                   Iframe31), Frame_mc)
-  >>> Fork_I = (inertia(TempFork, Ifork11, Ifork22, Ifork33, 0, 0, Ifork31), Fork_mc)
-  >>> WR_I = (inertia(Frame, Iwr11, Iwr22, Iwr11), WR_mc)
-  >>> WF_I = (inertia(Fork, Iwf11, Iwf22, Iwf11), WF_mc)
-
-Declaration of the RigidBody containers. ::
-
-  >>> BodyFrame = RigidBody('BodyFrame', Frame_mc, Frame, mframe, Frame_I)
-  >>> BodyFork = RigidBody('BodyFork', Fork_mc, Fork, mfork, Fork_I)
-  >>> BodyWR = RigidBody('BodyWR', WR_mc, WR, mwr, WR_I)
-  >>> BodyWF = RigidBody('BodyWF', WF_mc, WF, mwf, WF_I)
-
-  >>> print('Before Forming the List of Nonholonomic Constraints.')
-  Before Forming the List of Nonholonomic Constraints.
-
-The kinematic differential equations; they are defined quite simply. Each entry
-in this list is equal to zero. ::
-
-  >>> kd = [q1d - u1, q2d - u2, q4d - u4, q5d - u5]
-
-The nonholonomic constraints are the velocity of the front wheel contact point
-dotted into the X, Y, and Z directions; the yaw frame is used as it is "closer"
-to the front wheel (1 less DCM connecting them). These constraints force the
-velocity of the front wheel contact point to be 0 in the inertial frame; the X
-and Y direction constraints enforce a "no-slip" condition, and the Z direction
-constraint forces the front wheel contact point to not move away from the
-ground frame, essentially replicating the holonomic constraint which does not
-allow the frame pitch to change in an invalid fashion. ::
-
-  >>> conlist_speed = [WF_cont.vel(N) & Y.x,
-  ...                  WF_cont.vel(N) & Y.y,
-  ...                  WF_cont.vel(N) & Y.z]
-
-The holonomic constraint is that the position from the rear wheel contact point
-to the front wheel contact point when dotted into the normal-to-ground plane
-direction must be zero; effectively that the front and rear wheel contact
-points are always touching the ground plane. This is actually not part of the
-dynamic equations, but instead is necessary for the linearization process. ::
-
-  >>> conlist_coord = [WF_cont.pos_from(WR_cont) & Y.z]
-
-The force list; each body has the appropriate gravitational force applied at
-its center of mass. ::
-
-  >>> FL = [(Frame_mc, -mframe * g * Y.z), (Fork_mc, -mfork * g * Y.z),
-  ...       (WF_mc, -mwf * g * Y.z), (WR_mc, -mwr * g * Y.z)]
-  >>> BL = [BodyFrame, BodyFork, BodyWR, BodyWF]
-
-The N frame is the inertial frame, coordinates are supplied in the order of
-independent, dependent coordinates. The kinematic differential equations are
-also entered here. Here the independent speeds are specified, followed by the
-dependent speeds, along with the non-holonomic constraints. The dependent
-coordinate is also provided, with the holonomic constraint. Again, this is only
-comes into play in the linearization process, but is necessary for the
-linearization to correctly work. ::
-
-  >>> KM = KanesMethod(N, q_ind=[q1, q2, q3],
-  ...           q_dependent=[q4], configuration_constraints=conlist_coord,
-  ...           u_ind=[u2, u3, u5],
-  ...           u_dependent=[u1, u4, u6], velocity_constraints=conlist_speed,
-  ...           kd_eqs=kd)
-  >>> print('Before Forming Generalized Active and Inertia Forces, Fr and Fr*')
-  Before Forming Generalized Active and Inertia Forces, Fr and Fr*
-  >>> (fr, frstar) = KM.kanes_equations(FL, BL)
-  >>> print('Base Equations of Motion Computed')
-  Base Equations of Motion Computed
-
-This is the start of entering in the numerical values from the benchmark paper
-to validate the eigenvalues of the linearized equations from this model to the
-reference eigenvalues. Look at the aforementioned paper for more information.
-Some of these are intermediate values, used to transform values from the paper
-into the coordinate systems used in this model. ::
-
-  >>> PaperRadRear  =  0.3
-  >>> PaperRadFront =  0.35
-  >>> HTA           =  evalf.N(pi/2-pi/10)
-  >>> TrailPaper    =  0.08
-  >>> rake          =  evalf.N(-(TrailPaper*sin(HTA)-(PaperRadFront*cos(HTA))))
-  >>> PaperWb       =  1.02
-  >>> PaperFrameCgX =  0.3
-  >>> PaperFrameCgZ =  0.9
-  >>> PaperForkCgX  =  0.9
-  >>> PaperForkCgZ  =  0.7
-  >>> FrameLength   =  evalf.N(PaperWb*sin(HTA) - (rake - \
-  ...                         (PaperRadFront - PaperRadRear)*cos(HTA)))
-  >>> FrameCGNorm   =  evalf.N((PaperFrameCgZ - PaperRadRear - \
-  ...                          (PaperFrameCgX/sin(HTA))*cos(HTA))*sin(HTA))
-  >>> FrameCGPar    =  evalf.N((PaperFrameCgX / sin(HTA) + \
-  ...                          (PaperFrameCgZ - PaperRadRear - \
-  ...                           PaperFrameCgX / sin(HTA) * cos(HTA)) * cos(HTA)))
-  >>> tempa         =  evalf.N((PaperForkCgZ - PaperRadFront))
-  >>> tempb         =  evalf.N((PaperWb-PaperForkCgX))
-  >>> tempc         =  evalf.N(sqrt(tempa**2 + tempb**2))
-  >>> PaperForkL    =  evalf.N((PaperWb*cos(HTA) - \
-  ...                          (PaperRadFront - PaperRadRear)*sin(HTA)))
-  >>> ForkCGNorm    =  evalf.N(rake + (tempc * sin(pi/2 - \
-  ...                          HTA - acos(tempa/tempc))))
-  >>> ForkCGPar     =  evalf.N(tempc * cos((pi/2 - HTA) - \
-  ...                          acos(tempa/tempc)) - PaperForkL)
-
-Here is the final assembly of the numerical values. The symbol 'v' is the
-forward speed of the bicycle (a concept which only makes sense in the upright,
-static equilibrium case?). These are in a dictionary which will later be
-substituted in. Again the sign on the *product* of inertia values is flipped
-here, due to different orientations of coordinate systems. ::
-
-  >>> v = Symbol('v')
-  >>> val_dict = {
-  ...       WFrad: PaperRadFront,
-  ...       WRrad: PaperRadRear,
-  ...       htangle: HTA,
-  ...       forkoffset: rake,
-  ...       forklength: PaperForkL,
-  ...       framelength: FrameLength,
-  ...       forkcg1: ForkCGPar,
-  ...       forkcg3: ForkCGNorm,
-  ...       framecg1: FrameCGNorm,
-  ...       framecg3: FrameCGPar,
-  ...       Iwr11: 0.0603,
-  ...       Iwr22: 0.12,
-  ...       Iwf11: 0.1405,
-  ...       Iwf22: 0.28,
-  ...       Ifork11: 0.05892,
-  ...       Ifork22: 0.06,
-  ...       Ifork33: 0.00708,
-  ...       Ifork31: 0.00756,
-  ...       Iframe11: 9.2,
-  ...       Iframe22: 11,
-  ...       Iframe33: 2.8,
-  ...       Iframe31: -2.4,
-  ...       mfork: 4,
-  ...       mframe: 85,
-  ...       mwf: 3,
-  ...       mwr: 2,
-  ...       g: 9.81,
-  ...       q1: 0,
-  ...       q2: 0,
-  ...       q4: 0,
-  ...       q5: 0,
-  ...       u1: 0,
-  ...       u2: 0,
-  ...       u3: v/PaperRadRear,
-  ...       u4: 0,
-  ...       u5: 0,
-  ...       u6: v/PaperRadFront}
-  >>> kdd = KM.kindiffdict()
-  >>> print('Before Linearization of the \"Forcing\" Term')
-  Before Linearization of the "Forcing" Term
-
-Linearizes the forcing vector; the equations are set up as MM udot = forcing,
-where MM is the mass matrix, udot is the vector representing the time
-derivatives of the generalized speeds, and forcing is a vector which contains
-both external forcing terms and internal forcing terms, such as centripetal or
-Coriolis forces.  This actually returns a matrix with as many rows as *total*
-coordinates and speeds, but only as many columns as independent coordinates and
-speeds. (Note that below this is commented out, as it takes a few minutes to
-run, which is not good when performing the doctests) ::
-
-  >>> # forcing_lin = KM.linearize()[0].subs(sub_dict)
-
-As mentioned above, the size of the linearized forcing terms is expanded to
-include both q's and u's, so the mass matrix must have this done as well.  This
-will likely be changed to be part of the linearized process, for future
-reference. ::
-
-  >>> MM_full = (KM._k_kqdot).row_join(zeros(4, 6)).col_join(
-  ...           (zeros(6, 4)).row_join(KM.mass_matrix))
-  >>> print('Before Substitution of Numerical Values')
-  Before Substitution of Numerical Values
-
-I think this is pretty self explanatory. It takes a really long time though.
-I've experimented with using evalf with substitution, this failed due to
-maximum recursion depth being exceeded; I also tried lambdifying this, and it
-is also not successful. (again commented out due to speed) ::
-
-  >>> # MM_full = MM_full.subs(val_dict)
-  >>> # forcing_lin = forcing_lin.subs(val_dict)
-  >>> # print('Before .evalf() call')
-
-  >>> # MM_full = MM_full.evalf()
-  >>> # forcing_lin = forcing_lin.evalf()
-
-Finally, we construct an "A" matrix for the form xdot = A x (x being the state
-vector, although in this case, the sizes are a little off). The following line
-extracts only the minimum entries required for eigenvalue analysis, which
-correspond to rows and columns for lean, steer, lean rate, and steer rate.
-(this is all commented out due to being dependent on the above code, which is
-also commented out)::
-
-  >>> # Amat = MM_full.inv() * forcing_lin
-  >>> # A = Amat.extract([1,2,4,6],[1,2,3,5])
-  >>> # print(A)
-  >>> # print('v = 1')
-  >>> # print(A.subs(v, 1).eigenvals())
-  >>> # print('v = 2')
-  >>> # print(A.subs(v, 2).eigenvals())
-  >>> # print('v = 3')
-  >>> # print(A.subs(v, 3).eigenvals())
-  >>> # print('v = 4')
-  >>> # print(A.subs(v, 4).eigenvals())
-  >>> # print('v = 5')
-  >>> # print(A.subs(v, 5).eigenvals())
-
-Upon running the above code yourself, enabling the commented out lines, compare
-the computed eigenvalues to those is the referenced paper. This concludes the
-bicycle example.
-
-The Rolling Disc Example using Lagranges Method
-===============================================
-
-Here the rolling disc is formed from the contact point up, removing the
-need to introduce generalized speeds. Only 3 configuration and 3
-speed variables are needed to describe this system, along with the
-disc's mass and radius, and the local gravity. ::
-
-  >>> from sympy import symbols, cos, sin
-  >>> from sympy.physics.mechanics import *
-  >>> mechanics_printing()
-  >>> q1, q2, q3 = dynamicsymbols('q1 q2 q3')
-  >>> q1d, q2d, q3d = dynamicsymbols('q1 q2 q3', 1)
-  >>> r, m, g = symbols('r m g')
-
-The kinematics are formed by a series of simple rotations. Each simple
-rotation creates a new frame, and the next rotation is defined by the new
-frame's basis vectors. This example uses a 3-1-2 series of rotations, or
-Z, X, Y series of rotations. Angular velocity for this is defined using
-the second frame's basis (the lean frame). ::
-
-  >>> N = ReferenceFrame('N')
-  >>> Y = N.orientnew('Y', 'Axis', [q1, N.z])
-  >>> L = Y.orientnew('L', 'Axis', [q2, Y.x])
-  >>> R = L.orientnew('R', 'Axis', [q3, L.y])
-
-This is the translational kinematics. We create a point with no velocity
-in N; this is the contact point between the disc and ground. Next we form
-the position vector from the contact point to the disc's center of mass.
-Finally we form the velocity and acceleration of the disc. ::
-
-  >>> C = Point('C')
-  >>> C.set_vel(N, 0)
-  >>> Dmc = C.locatenew('Dmc', r * L.z)
-  >>> Dmc.v2pt_theory(C, N, R)
-  r*(sin(q2)*q1' + q3')*L.x - r*q2'*L.y
-
-Forming the inertia dyadic. ::
-
-  >>> I = inertia(L, m / 4 * r**2, m / 2 * r**2, m / 4 * r**2)
-  >>> mprint(I)
-  m*r**2/4*(L.x|L.x) + m*r**2/2*(L.y|L.y) + m*r**2/4*(L.z|L.z)
-  >>> BodyD = RigidBody('BodyD', Dmc, R, m, (I, Dmc))
-
-We then set the potential energy and determine the Lagrangian of the rolling
-disc. ::
-
-  >>> BodyD.set_potential_energy(- m * g * r * cos(q2))
-  >>> Lag = Lagrangian(N, BodyD)
-
-Then the equations of motion are generated by initializing the
-``LagrangesMethod`` object. Finally we solve for the generalized
-accelerations(q double dots) with the ``rhs`` method. ::
-
-  >>> q = [q1, q2, q3]
-  >>> l = LagrangesMethod(Lag, q)
-  >>> l.form_lagranges_equations()
-  [5*m*r**2*(sin(q2)*q1' + q3')*cos(q2)*q2'/4 + 5*m*r**2*(sin(q2)*q1'' + cos(q2)*q1'*q2' + q3'')*sin(q2)/4 + m*r**2*sin(q2)*q3''/4 + m*r**2*cos(q2)**2*q1''/8 + m*r**2*cos(q2)*q2'*q3'/4 + (m*r**2*sin(q2)**2/4 + m*r**2*cos(q2)**2/8)*q1'']
-  [                                                                                                                                 g*m*r*sin(q2) - 5*m*r**2*(sin(q2)*q1' + q3')*cos(q2)*q1'/4 - m*r**2*cos(q2)*q1'*q3'/4 + 5*m*r**2*q2''/4]
-  [                                                            m*r**2*(sin(q2)*q1'' + cos(q2)*q1'*q2' + q3'')/4 + m*r**2*(2*sin(q2)*q1'' + 2*cos(q2)*q1'*q2' + 2*q3'')/2 + m*r**2*sin(q2)*q1''/4 + m*r**2*cos(q2)*q1'*q2'/4 + m*r**2*q3''/4]
-  >>> l.rhs()
-  [                                                                                                                                                                                                                             q1']
-  [                                                                                                                                                                                                                             q2']
-  [                                                                                                                                                                                                                             q3']
-  [(24*sin(q2)**2/(m*r**2*(5*sin(q2)**2 + 1)*cos(q2)**2) + 4/(m*r**2*(5*sin(q2)**2 + 1)))*(-5*m*r**2*(sin(q2)*q1' + q3')*cos(q2)*q2'/4 - 5*m*r**2*sin(q2)*cos(q2)*q1'*q2'/4 - m*r**2*cos(q2)*q2'*q3'/4) + 6*sin(q2)*q1'*q2'/cos(q2)]
-  [                                                                                                                           4*(-g*m*r*sin(q2) + 5*m*r**2*(sin(q2)*q1' + q3')*cos(q2)*q1'/4 + m*r**2*cos(q2)*q1'*q3'/4)/(5*m*r**2)]
-  [                                               -(5*sin(q2)**2 + 1)*q1'*q2'/cos(q2) - 4*(-5*m*r**2*(sin(q2)*q1' + q3')*cos(q2)*q2'/4 - 5*m*r**2*sin(q2)*cos(q2)*q1'*q2'/4 - m*r**2*cos(q2)*q2'*q3'/4)*sin(q2)/(m*r**2*cos(q2)**2)]
diff --git a/dev-py3k/_sources/modules/physics/mechanics/index.txt b/dev-py3k/_sources/modules/physics/mechanics/index.txt
deleted file mode 100644
index 60da9a54745..00000000000
--- a/dev-py3k/_sources/modules/physics/mechanics/index.txt
+++ /dev/null
@@ -1,50 +0,0 @@
-===================
-Classical Mechanics
-===================
-
-:Authors: Gilbert Gede, Luke Peterson, Angadh Nanjangud
-
-.. topic:: Abstract
-
-   In this documentation many components of the physics/mechanics module will
-   be discussed. :mod:`mechanics` has been written to allow for creation of
-   symbolic equations of motion for complicated multibody systems.
-
-Mechanics
-=========
-
-In physics, mechanics describes conditions of rest or motion; statics or
-dynamics. First, an idealized representation of a system is described. Next, we
-use physical laws to generate equations that define the system's behaviors.
-Then, we solve these equations, in multiple possible ways. Finally, we extract
-information from these equations and solutions. Mechanics is currently set up
-to create equations of motion, for dynamics. It is also limited to rigid bodies
-and particles.
-
-
-Guide to Mechanics
-==================
-
-.. toctree::
-    :maxdepth: 2
-
-    vectors.rst
-    kinematics.rst
-    masses.rst
-    kane.rst
-    examples.rst
-    advanced.rst
-    reference.rst
-
-Mechanics API
-=============
-
-.. toctree::
-    :maxdepth: 2
-
-    api/essential.rst
-    api/functions.rst
-    api/kinematics.rst
-    api/part_bod.rst
-    api/kane.rst
-    api/printing.rst
diff --git a/dev-py3k/_sources/modules/physics/mechanics/kane.txt b/dev-py3k/_sources/modules/physics/mechanics/kane.txt
deleted file mode 100644
index 467bdfb6936..00000000000
--- a/dev-py3k/_sources/modules/physics/mechanics/kane.txt
+++ /dev/null
@@ -1,302 +0,0 @@
-==================================
-Kane's Method in Physics/Mechanics
-==================================
-
-:mod:`mechanics` has been written for use with Kane's method of forming
-equations of motion [Kane1985]_. This document will describe Kane's Method
-as used in this module, but not how the equations are actually derived.
-
-Structure of Equations
-======================
-
-In :mod:`mechanics` we are assuming there are 5 basic sets of equations needed
-to describe a system. They are: holonomic constraints, non-holonomic
-constraints, kinematic differential equations, dynamic equations, and
-differentiated non-holonomic equations.
-
-.. math::
-  \mathbf{f_h}(q, t) &= 0\\
-  \mathbf{k_{nh}}(q, t) u + \mathbf{f_{nh}}(q, t) &= 0\\
-  \mathbf{k_{k\dot{q}}}(q, t) \dot{q} + \mathbf{k_{ku}}(q, t) u +
-  \mathbf{f_k}(q, t) &= 0\\
-  \mathbf{k_d}(q, t) \dot{u} + \mathbf{f_d}(q, \dot{q}, u, t) &= 0\\
-  \mathbf{k_{dnh}}(q, t) \dot{u} + \mathbf{f_{dnh}}(q, \dot{q}, u, t) &= 0\\
-
-In :mod:`mechanics` holonomic constraints are only used for the linearization
-process; it is assumed that they will be too complicated to solve for the
-dependent coordinate(s).  If you are able to easily solve a holonomic
-constraint, you should consider redefining your problem in terms of a smaller
-set of coordinates. Alternatively, the time-differentiated holonomic
-constraints can be supplied.
-
-Kane's method forms two expressions, :math:`F_r` and :math:`F_r^*`, whose sum
-is zero. In this module, these expressions are rearranged into the following
-form:
-
- :math:`\mathbf{M}(q, t) \dot{u} = \mathbf{f}(q, \dot{q}, u, t)`
-
-For a non-holonomic system with `o` total speeds and `m` motion constraints, we
-will get o - m equations. The mass-matrix/forcing equations are then augmented
-in the following fashion:
-
-.. math::
-  \mathbf{M}(q, t) &= \begin{bmatrix} \mathbf{k_d}(q, t) \\
-  \mathbf{k_{dnh}}(q, t) \end{bmatrix}\\
-  \mathbf{_{(forcing)}}(q, \dot{q}, u, t) &= \begin{bmatrix}
-  - \mathbf{f_d}(q, \dot{q}, u, t) \\ - \mathbf{f_{dnh}}(q, \dot{q}, u, t)
-  \end{bmatrix}\\
-
-
-Kane's Method in Physics/Mechanics
-==================================
-
-The formulation of the equations of motion in :mod:`mechanics` starts with
-creation of a ``KanesMethod`` object. Upon initialization of the
-``KanesMethod`` object, an inertial reference frame needs to be supplied. along
-with some basic system information, suchs as coordinates and speeds ::
-
-  >>> from sympy.physics.mechanics import *
-  >>> N = ReferenceFrame('N')
-  >>> q1, q2, u1, u2 = dynamicsymbols('q1 q2 u1 u2')
-  >>> q1d, q2d, u1d, u2d = dynamicsymbols('q1 q2 u1 u2', 1)
-  >>> KM = KanesMethod(N, [q1, q2], [u1, u2])
-
-It is also important to supply the order of coordinates and speeds properly if
-there are dependent coordinates and speeds. They must be supplied after
-independent coordinates and speeds or as a keyword argument; this is shown
-later. ::
-
-  >>> q1, q2, q3, q4 = dynamicsymbols('q1 q2 q3 q4')
-  >>> u1, u2, u3, u4 = dynamicsymbols('u1 u2 u3 u4')
-  >>> # Here we will assume q2 is dependent, and u2 and u3 are dependent
-  >>> # We need the constraint equations to enter them though
-  >>> KM = KanesMethod(N, [q1, q3, q4], [u1, u4])
-
-Additionally, if there are auxiliary speeds, they need to be identified here.
-See the examples for more information on this. In this example u4 is the
-auxiliary speed. ::
-
-  >>> KM = KanesMethod(N, [q1, q3, q4], [u1, u2, u3], u_auxiliary=[u4])
-
-Kinematic differential equations must also be supplied; there are to be
-provided as a list of expressions which are each equal to zero. A trivial
-example follows: ::
-
-  >>> kd = [q1d - u1, q2d - u2]
-
-Turning on ``mechanics_printing()`` makes the expressions significantly
-shorter and is recommended. Alternatively, the ``mprint`` and ``mpprint``
-commands can be used.
-
-If there are non-holonomic constraints, dependent speeds need to be specified
-(and so do dependent coordinates, but they only come into play when linearizing
-the system). The constraints need to be supplied in a list of expressions which
-are equal to zero, trivial motion and configuration constraints are shown
-below: ::
-
-  >>> N = ReferenceFrame('N')
-  >>> q1, q2, q3, q4 = dynamicsymbols('q1 q2 q3 q4')
-  >>> q1d, q2d, q3d, q4d = dynamicsymbols('q1 q2 q3 q4', 1)
-  >>> u1, u2, u3, u4 = dynamicsymbols('u1 u2 u3 u4')
-  >>> #Here we will assume q2 is dependent, and u2 and u3 are dependent
-  >>> speed_cons = [u2 - u1, u3 - u1 - u4]
-  >>> coord_cons = [q2 - q1]
-  >>> q_ind = [q1, q3, q4]
-  >>> q_dep = [q2]
-  >>> u_ind = [u1, u4]
-  >>> u_dep = [u2, u3]
-  >>> kd = [q1d - u1, q2d - u2, q3d - u3, q4d - u4]
-  >>> KM = KanesMethod(N, q_ind, u_ind, kd,
-  ...           q_dependent=q_dep,
-  ...           configuration_constraints=coord_cons,
-  ...           u_dependent=u_dep,
-  ...           velocity_constraints=speed_cons)
-
-A dictionary returning the solved :math:`\dot{q}`'s can also be solved for: ::
-
-  >>> mechanics_printing()
-  >>> KM.kindiffdict()
-  {q1': u1, q2': u2, q3': u3, q4': u4}
-
-The final step in forming the equations of motion is supplying a list of
-bodies and particles, and a list of 2-tuples of the form ``(Point, Vector)``
-or ``(ReferenceFrame, Vector)`` to represent applied forces and torques. ::
-
-  >>> N = ReferenceFrame('N')
-  >>> q, u = dynamicsymbols('q u')
-  >>> qd, ud = dynamicsymbols('q u', 1)
-  >>> P = Point('P')
-  >>> P.set_vel(N, u * N.x)
-  >>> Pa = Particle('Pa', P, 5)
-  >>> BL = [Pa]
-  >>> FL = [(P, 7 * N.x)]
-  >>> KM = KanesMethod(N, [q], [u], [qd - u])
-  >>> (fr, frstar) = KM.kanes_equations(FL, BL)
-  >>> KM.mass_matrix
-  [5]
-  >>> KM.forcing
-  [7]
-
-When there are motion constraints, the mass matrix is augmented by the
-:math:`k_{dnh}(q, t)` matrix, and the forcing vector by the :math:`f_{dnh}(q,
-\dot{q}, u, t)` vector.
-
-There are also the "full" mass matrix and "full" forcing vector terms, these
-include the kinematic differential equations; the mass matrix is of size (n +
-o) x (n + o), or square and the size of all coordinates and speeds. ::
-
-  >>> KM.mass_matrix_full
-  [1, 0]
-  [0, 5]
-  >>> KM.forcing_full
-  [u]
-  [7]
-
-The forcing vector can be linearized as well; its Jacobian is taken only with
-respect to the independent coordinates and speeds. The linearized forcing
-vector is of size (n + o) x (n - l + o - m), where l is the number of
-configuration constraints and m is the number of motion constraints. Two
-matrices are returned; the first is an "A" matrix, or the Jacobian with respect
-to the independent states, the second is a "B" matrix, or the Jacobian with
-respect to 'forces'; this can be an empty matrix if there are no 'forces'.
-Forces here are undefined functions of time (dynamic symbols); they are only
-allowed to be in the forcing vector and their derivatives are not allowed to be
-present. If dynamic symbols appear in the mass matrix or kinematic differential
-equations, an error with be raised. ::
-
-  >>> KM.linearize()[0]
-  [0, 1]
-  [0, 0]
-
-Exploration of the provided examples is encouraged in order to gain more
-understanding of the ``KanesMethod`` object.
-
-======================================
-Lagrange's Method in Physics/Mechanics
-======================================
-
-Structure of Equations
-======================
-
-In :mod:`mechanics` we are assuming there are 3 basic sets of equations needed
-to describe a system; the constraint equations, the time differentiated
-constraint equations and the dynamic equations.
-
-.. math::
-  \mathbf{m_{c}}(q, t) \dot{q} + \mathbf{f_{c}}(q, t) &= 0\\
-  \mathbf{m_{dc}}(\dot{q}, q, t) \ddot{q} + \mathbf{f_{dc}}(\dot{q}, q, t) &= 0\\
-  \mathbf{m_d}(\dot{q}, q, t) \ddot{q} + \mathbf{\Lambda_c}(q, t)
-  \lambda + \mathbf{f_d}(\dot{q}, q, t) &= 0\\
-
-In this module, the expressions formed by using Lagrange's equations of the
-second kind are rearranged into the following form:
-
- :math:`\mathbf{M}(q, t) x = \mathbf{f}(q, \dot{q}, t)`
-
-where in the case of a system without constraints:
-
- :math:`x = \ddot{q}`
-
-For a constrained system with `n` generalized speeds and `m` constraints, we
-will get n - m equations. The mass-matrix/forcing equations are then augmented
-in the following fashion:
-
-.. math::
-  x = \begin{bmatrix} \ddot{q} \\ \lambda \end{bmatrix} \\
-  \mathbf{M}(q, t) &= \begin{bmatrix} \mathbf{m_d}(q, t) &
-  \mathbf{\Lambda_c}(q, t) \end{bmatrix}\\
-  \mathbf{F}(\dot{q}, q, t) &= \begin{bmatrix} \mathbf{f_d}(q, \dot{q}, t)
-  \end{bmatrix}\\
-
-
-Lagrange's Method in Physics/Mechanics
-======================================
-
-The formulation of the equations of motion in :mod:`mechanics` using
-Lagrange's Method starts with the creation of generalized coordinates and a
-Lagrangian. The Lagrangian can either be created with the ``Lagrangian``
-function or can be a user supplied function. In this case we will supply the
-Lagrangian. ::
-
-  >>> from sympy.physics.mechanics import *
-  >>> q1, q2 = dynamicsymbols('q1 q2')
-  >>> q1d, q2d = dynamicsymbols('q1 q2', 1)
-  >>> L = q1d**2 + q2d**2
-
-To formulate the equations of motion we create a ``LagrangesMethod``
-object. The Lagrangian and generalized coordinates need to be supplied upon
-initialization. ::
-
-  >>> LM = LagrangesMethod(L, [q1, q2])
-
-With that the equations of motion can be formed. ::
-
-  >>> mechanics_printing()
-  >>> LM.form_lagranges_equations()
-  [2*q1'']
-  [2*q2'']
-
-It is possible to obtain the mass matrix and the forcing vector. ::
-
-  >>> LM.mass_matrix
-  [2, 0]
-  [0, 2]
-
-  >>> LM.forcing
-  [0]
-  [0]
-
-If there are any holonomic or non-holonomic constraints, they must be supplied
-as keyword arguments in a list of expressions which are equal to zero. It
-should be noted that :mod:`mechanics` requires that the holonomic constraint
-equations must be supplied as velocity level constraint equations i.e. the
-holonomic constraint equations must be supplied after they have been
-differentiated with respect to time. Modifying the example above, the equations
-of motion can then be generated: ::
-
-  >>> LM = LagrangesMethod(L, [q1, q2], coneqs = [q1d - q2d])
-
-When the equations of motion are generated in this case, the Lagrange
-multipliers are introduced; they are represented by ``lam1`` in this case. In
-general, there will be as many multipliers as there are constraint equations. ::
-
-  >>> LM.form_lagranges_equations()
-  [ lam1 + 2*q1'']
-  [-lam1 + 2*q2'']
-
-Also in the case of systems with constraints, the 'full' mass matrix is
-augmented by the :math:`k_{dc}(q, t)` matrix, and the forcing vector by the
-:math:`f_{dc}(q, \dot{q}, t)` vector. The 'full' mass matrix is of size
-(2n + o) x (2n + o), i.e. it's a square matrix. ::
-
-  >>> LM.mass_matrix_full
-  [1, 0, 0,  0,  0]
-  [0, 1, 0,  0,  0]
-  [0, 0, 2,  0, -1]
-  [0, 0, 0,  2,  1]
-  [0, 0, 1, -1,  0]
-  >>> LM.forcing_full
-  [q1']
-  [q2']
-  [  0]
-  [  0]
-  [  0]
-
-If there are any non-conservative forces or moments acting on the system,
-they must also be supplied as keyword arguments in a list of 2-tuples of the
-form ``(Point, Vector)`` or ``(ReferenceFrame, Vector)`` where the ``Vector``
-represents the non-conservative forces and torques. Along with this 2-tuple,
-the inertial frame must also be specified as a keyword argument. This is shown
-below by modifying the example above: ::
-
-  >>> N = ReferenceFrame('N')
-  >>> P = Point('P')
-  >>> P.set_vel(N, q1d * N.x)
-  >>> FL = [(P, 7 * N.x)]
-  >>> LM = LagrangesMethod(L, [q1, q2], forcelist = FL, frame = N)
-  >>> LM.form_lagranges_equations()
-  [2*q1'' - 7]
-  [    2*q2'']
-
-Exploration of the provided examples is encouraged in order to gain more
-understanding of the ``LagrangesMethod`` object.
diff --git a/dev-py3k/_sources/modules/physics/mechanics/kinematics.txt b/dev-py3k/_sources/modules/physics/mechanics/kinematics.txt
deleted file mode 100644
index ca8a91471d8..00000000000
--- a/dev-py3k/_sources/modules/physics/mechanics/kinematics.txt
+++ /dev/null
@@ -1,572 +0,0 @@
-=====================
-Mechanics: Kinematics
-=====================
-
-In mechanics, kinematics refers to the motion of bodies (or particles).
-It is the 'a' in 'f=ma'. This document will give some mathematical background
-to describing a system's kinematics as well as how to represent the kinematics
-in :mod:`mechanics`.
-
-Introduction to Kinematics
-==========================
-
-The first topic is rigid motion kinematics. A rigid body is an idealized
-representation of a physical object which has mass and rotational inertia.
-Rigid bodies are obviously not flexible. We can break down rigid body motion
-into translational motion, and rotational motion (when dealing with particles, we
-only have translational motion). Rotational motion can further be broken down
-into simple rotations and general rotations.
-
-Translation of a rigid body is defined as a motion where the orientation of the
-body does not change during the motion; or during the motion any line segment
-would be parallel to itself at the start of the motion.
-
-Simple rotations are rotations in which the orientation of the body may change,
-but there is always one line which remains parallel to itself at the start of
-the motion.
-
-General rotations are rotations which there is not always one line parallel to
-itself at the start of the motion.
-
-Angular Velocity
-----------------
-
-The angular velocity of a rigid body refers to the rate of change of its
-orientation. The angular velocity of a body is written down as:
-:math:`^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}}`, or the angular velocity of
-:math:`\mathbf{B}` in :math:`\mathbf{N}`, which is a vector. Note that here,
-the term rigid body was used, but reference frames can also have angular
-velocities. Further discussion of the distinction between a rigid body and a
-reference frame will occur later when describing the code representation.
-
-Angular velocity is defined as being positive in the direction which causes the
-orientation angles to increase (for simple rotations, or series of simple
-rotations).
-
-.. image:: kin_angvel1.*
-   :height: 350
-   :width: 250
-   :align: center
-
-The angular velocity vector represents the time derivative of the orientation.
-As a time derivative vector quantity, like those covered in the Vector &
-ReferenceFrame documentation, this quantity (angular velocity) needs to be
-defined in a reference frame. That is what the :math:`\mathbf{N}` is in the
-above definition of angular velocity; the frame in which the angular velocity
-is defined in.
-
-The angular velocity of :math:`\mathbf{B}` in :math:`\mathbf{N}` can also be
-defined by:
-
-.. math::
-  ^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}} =
-  (\frac{^{\mathbf{N}}d \mathbf{\hat{b}_y}}{dt}\cdot\mathbf{\hat{b}_z}
-  )\mathbf{\hat{b}_x} + (\frac{^{\mathbf{N}}d \mathbf{\hat{b}_z}}{dt}\cdot
-  \mathbf{\hat{b}_x})\mathbf{\hat{b}_y} + (\frac{^{\mathbf{N}}d
-  \mathbf{\hat{b}_x}}{dt}\cdot\mathbf{\hat{b}_y})\mathbf{\hat{b}_z}
-
-It is also common for a body's angular velocity to be written as:
-
-.. math::
-  ^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}} = w_x \mathbf{\hat{b}_x} +
-  w_y \mathbf{\hat{b}_y} + w_z \mathbf{\hat{b}_z}
-
-There are a few additional important points relating to angular velocity. The
-first is the addition theorem for angular velocities, a way of relating the
-angular velocities of multiple bodies and frames. The theorem follows:
-
-.. math::
-  ^{\mathbf{N}}\mathbf{\omega}^{\mathbf{D}} =
-  ^{\mathbf{N}}\mathbf{\omega}^{\mathbf{A}} +
-  ^{\mathbf{A}}\mathbf{\omega}^{\mathbf{B}} +
-  ^{\mathbf{B}}\mathbf{\omega}^{\mathbf{C}} +
-  ^{\mathbf{C}}\mathbf{\omega}^{\mathbf{D}}
-
-This is also shown in the following example:
-
-.. image:: kin_angvel2.*
-   :height: 300
-   :width: 450
-   :align: center
-
-.. math::
-  ^{\mathbf{N}}\mathbf{\omega}^{\mathbf{A}} &= 0\\
-  ^{\mathbf{A}}\mathbf{\omega}^{\mathbf{B}} &= \dot{q_1} \mathbf{\hat{a}_x}\\
-  ^{\mathbf{B}}\mathbf{\omega}^{\mathbf{C}} &= - \dot{q_2} \mathbf{\hat{b}_z}\\
-  ^{\mathbf{C}}\mathbf{\omega}^{\mathbf{D}} &= \dot{q_3} \mathbf{\hat{c}_y}\\
-  ^{\mathbf{N}}\mathbf{\omega}^{\mathbf{D}} &= \dot{q_1} \mathbf{\hat{a}_x}
-  - \dot{q_2} \mathbf{\hat{b}_z} + \dot{q_3} \mathbf{\hat{c}_y}\\
-
-Note the signs used in the angular velocity definitions, which are related to
-how the displacement angle is defined in this case.
-
-
-This theorem makes defining angular velocities of multibody systems much
-easier, as the angular velocity of a body in a chain needs to only be defined
-to the previous body in order to be fully defined (and the first body needs
-to be defined in the desired reference frame). The following figure shows an
-example of when using this theorem can make things easier.
-
-.. image:: kin_angvel3.*
-   :height: 250
-   :width: 400
-   :align: center
-
-Here we can easily write the angular velocity of the body
-:math:`\mathbf{D}` in the reference frame of the first body :math:`\mathbf{A}`:
-
-.. math::
-  ^\mathbf{A}\mathbf{\omega}^\mathbf{D} = w_1 \mathbf{\hat{p_1}} +
-  w_2 \mathbf{\hat{p_2}} + w_3 \mathbf{\hat{p_3}}\\
-
-It is very important to remember to only use this with angular velocities; you
-cannot use this theorem with the velocities of points.
-
-There is another theorem commonly used: the derivative theorem. It provides an
-alternative method (which can be easier) to calculate the time derivative of a
-vector in a reference frame:
-
-.. math::
-  \frac{^{\mathbf{N}} d \mathbf{v}}{dt} = \frac{^{\mathbf{B}} d \mathbf{v}}{dt}
-  + ^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}} \times \mathbf{v}
-
-The vector :math:`\mathbf{v}` can be any vector quantity: a position vector,
-a velocity vector, angular velocity vector, etc. Instead of taking the time
-derivative of the vector in :math:`\mathbf{N}`, we take it in
-:math:`\mathbf{B}`, where :math:`\mathbf{B}` can be any reference frame or
-body, usually one in which it is easy to take the derivative on
-:math:`\mathbf{v}` in (:math:`\mathbf{v}` is usually composed only of the basis
-vector set belonging to :math:`\mathbf{B}`). Then we add the cross product of
-the angular velocity of our newer frame,
-:math:`^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}}` and our vector quantity
-:math:`\mathbf{v}`. Again, you can choose any alternative frame for this.
-Examples follow:
-
-.. % need multiple examples here showing the derivative theorem
-
-
-Angular Acceleration
---------------------
-Angular acceleration refers to the time rate of change of the angular velocity
-vector. Just as the angular velocity vector is for a body and is specified in a
-frame, the angular acceleration vector is for a body and is specified in a
-frame: :math:`^{\mathbf{N}}\mathbf{\alpha}^{\mathbf{B}}`, or the angular
-acceleration of :math:`\mathbf{B}` in :math:`\mathbf{N}`, which is a vector.
-
-Calculating the angular acceleration is relatively straight forward:
-
-.. math::
-  ^{\mathbf{N}}\mathbf{\alpha}^{\mathbf{B}} =
-  \frac{^{\mathbf{N}} d ^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}}}{dt}
-
-Note that this can be calculated with the derivative theorem, and when the
-angular velocity is defined in a body fixed frame, becomes quite simple:
-
-.. math::
-
-  ^{\mathbf{N}}\mathbf{\alpha}^{\mathbf{B}} &=
-  \frac{^{\mathbf{N}} d ^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}}}{dt}\\
-
-  ^{\mathbf{N}}\mathbf{\alpha}^{\mathbf{B}} &=
-  \frac{^{\mathbf{B}} d ^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}}}{dt}
-  + ^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}} \times
-  ^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}}\\
-
-  \textrm{if } ^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}} &=
-  w_x \mathbf{\hat{b}_x} + w_y \mathbf{\hat{b}_y} + w_z \mathbf{\hat{b}_z}\\
-
-  \textrm{then } ^{\mathbf{N}}\mathbf{\alpha}^{\mathbf{B}} &=
-  \frac{^{\mathbf{B}} d ^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}}}{dt}
-  + \underbrace{^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}} \times
-  ^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}}}_{
-  \textrm{this is 0 by definition}}\\
-
-  ^{\mathbf{N}}\mathbf{\alpha}^{\mathbf{B}}&=\frac{d w_x}{dt}\mathbf{\hat{b}_x}
-  + \frac{d w_y}{dt}\mathbf{\hat{b}_y} + \frac{d w_z}{dt}\mathbf{\hat{b}_z}\\
-
-  ^{\mathbf{N}}\mathbf{\alpha}^{\mathbf{B}}&= \dot{w_x}\mathbf{\hat{b}_x} +
-  \dot{w_y}\mathbf{\hat{b}_y} + \dot{w_z}\mathbf{\hat{b}_z}\\
-
-Again, this is only for the case in which the angular velocity of the body is
-defined in body fixed components.
-
-
-
-Point Velocity & Acceleration
------------------------------
-
-Consider a point, :math:`P`: we can define some characteristics of the point.
-First, we can define a position vector from some other point to :math:`P`.
-Second, we can define the velocity vector of :math:`P` in a reference frame of
-our choice. Third, we can define the acceleration vector of :math:`P` in a
-reference frame of our choice.
-
-These three quantities are read as:
-
-.. math::
-  \mathbf{r}^{OP} \textrm{, the position vector from } O
-  \textrm{ to }P\\
-  ^{\mathbf{N}}\mathbf{v}^P \textrm{, the velocity of } P
-  \textrm{ in the reference frame } \mathbf{N}\\
-  ^{\mathbf{N}}\mathbf{a}^P \textrm{, the acceleration of } P
-  \textrm{ in the reference frame } \mathbf{N}\\
-
-Note that the position vector does not have a frame associated with it; this is
-because there is no time derivative involved, unlike the velocity and
-acceleration vectors.
-
-We can find these quantities for a simple example easily:
-
-.. image:: kin_1.*
-   :height: 300
-   :width: 300
-   :align: center
-
-.. math::
-  \textrm{Let's define: }
-  \mathbf{r}^{OP} &= q_x \mathbf{\hat{n}_x} + q_y \mathbf{\hat{n}_y}\\
-  ^{\mathbf{N}}\mathbf{v}^P &= \frac{^{\mathbf{N}} d \mathbf{r}^{OP}}{dt}\\
-  \textrm{then we can calculate: }
-  ^{\mathbf{N}}\mathbf{v}^P &= \dot{q}_x\mathbf{\hat{n}_x} +
-  \dot{q}_y\mathbf{\hat{n}_y}\\
-  \textrm{and :}
-  ^{\mathbf{N}}\mathbf{a}^P &= \frac{^{\mathbf{N}} d
-  ^{\mathbf{N}}\mathbf{v}^P}{dt}\\
-  ^{\mathbf{N}}\mathbf{a}^P &= \ddot{q}_x\mathbf{\hat{n}_x} +
-  \ddot{q}_y\mathbf{\hat{n}_y}\\
-
-It is critical to understand in the above example that the point :math:`O` is
-fixed in the reference frame :math:`\mathbf{N}`. There is no addition theorem
-for translational velocities; alternatives will be discussed later though.
-Also note that the position of every point might not
-always need to be defined to form the dynamic equations of motion.
-When you don't want to define the position vector of a point, you can start by
-just defining the velocity vector. For the above example:
-
-.. math::
-  \textrm{Let us instead define the velocity vector as: }
-  ^{\mathbf{N}}\mathbf{v}^P &= u_x \mathbf{\hat{n}_x} +
-  u_y \mathbf{\hat{n}_y}\\
-  \textrm{then acceleration can be written as: }
-  ^{\mathbf{N}}\mathbf{a}^P &= \dot{u}_x \mathbf{\hat{n}_x} +
-  \dot{u}_y \mathbf{\hat{n}_y}\\
-
-
-There will often be cases when the velocity of a point is desired and a related
-point's velocity is known. For the cases in which we have two points fixed on a
-rigid body, we use the 2-Point Theorem:
-
-.. image:: kin_2pt.*
-   :height: 300
-   :width: 300
-   :align: center
-
-Let's say we know the velocity of the point :math:`S` and the angular
-velocity of the body :math:`\mathbf{B}`, both defined in the reference frame
-:math:`\mathbf{N}`. We can calculate the velocity and acceleration
-of the point :math:`P` in :math:`\mathbf{N}` as follows:
-
-.. math::
-  ^{\mathbf{N}}\mathbf{v}^P &= ^\mathbf{N}\mathbf{v}^S +
-  ^\mathbf{N}\mathbf{\omega}^\mathbf{B} \times \mathbf{r}^{SP}\\
-  ^{\mathbf{N}}\mathbf{a}^P &= ^\mathbf{N}\mathbf{a}^S +
-  ^\mathbf{N}\mathbf{\alpha}^\mathbf{B} \times \mathbf{r}^{SP} +
-  ^\mathbf{N}\mathbf{\omega}^\mathbf{B} \times
-  (^\mathbf{N}\mathbf{\omega}^\mathbf{B} \times \mathbf{r}^{SP})\\
-
-When only one of the two points is fixed on a body, the 1 point theorem is used
-instead.
-
-.. image:: kin_1pt.*
-   :height: 400
-   :width: 400
-   :align: center
-
-Here, the velocity of point :math:`S` is known in the frame :math:`\mathbf{N}`,
-the angular velocity of :math:`\mathbf{B}` is known in :math:`\mathbf{N}`, and
-the velocity of the point :math:`P` is known in the frame associated with body
-:math:`\mathbf{B}`. We can then write the velocity and acceleration of
-:math:`P` in :math:`\mathbf{N}` as:
-
-.. math::
-  ^{\mathbf{N}}\mathbf{v}^P &= ^\mathbf{B}\mathbf{v}^P +
-  ^\mathbf{N}\mathbf{v}^S + ^\mathbf{N}\mathbf{\omega}^\mathbf{B} \times
-  \mathbf{r}^{SP}\\
-
-  ^{\mathbf{N}}\mathbf{a}^P &= ^\mathbf{B}\mathbf{a}^S +
-  ^\mathbf{N}\mathbf{a}^O + ^\mathbf{N}\mathbf{\alpha}^\mathbf{B}
-  \times \mathbf{r}^{SP} + ^\mathbf{N}\mathbf{\omega}^\mathbf{B} \times
-  (^\mathbf{N}\mathbf{\omega}^\mathbf{B} \times \mathbf{r}^{SP}) +
-  2 ^\mathbf{N}\mathbf{\omega}^\mathbf{B} \times ^\mathbf{B} \mathbf{v}^P \\
-
-
-Examples of applications of the 1 point and 2 point theorem follow.
-
-.. image:: kin_2.*
-   :height: 300
-   :width: 400
-   :align: center
-
-This example has a disc translating and rotating in a plane. We can easily
-define the angular velocity of the body :math:`\mathbf{B}` and velocity of the
-point :math:`O`:
-
-.. math::
-  ^\mathbf{N}\mathbf{\omega}^\mathbf{B} &= u_3 \mathbf{\hat{n}_z} = u_3
-  \mathbf{\hat{b}_z}\\
-  ^\mathbf{N}\mathbf{v}^O &= u_1 \mathbf{\hat{n}_x} + u_2 \mathbf{\hat{n}_y}\\
-
-and accelerations can be written as:
-
-.. math::
-  ^\mathbf{N}\mathbf{\alpha}^\mathbf{B} &= \dot{u_3} \mathbf{\hat{n}_z} =
-  \dot{u_3} \mathbf{\hat{b}_z}\\
-  ^\mathbf{N}\mathbf{a}^O &= \dot{u_1} \mathbf{\hat{n}_x} + \dot{u_2}
-  \mathbf{\hat{n}_y}\\
-
-We can use the 2 point theorem to calculate the velocity and acceleration of
-point :math:`P` now.
-
-.. math::
-  \mathbf{r}^{OP} &= R \mathbf{\hat{b}_x}\\
-  ^\mathbf{N}\mathbf{v}^P &= ^\mathbf{N}\mathbf{v}^O +
-  ^\mathbf{N}\mathbf{\omega}^\mathbf{B} \times \mathbf{r}^{OP}\\
-  ^\mathbf{N}\mathbf{v}^P &= u_1 \mathbf{\hat{n}_x} + u_2 \mathbf{\hat{n}_y}
-  + u_3 \mathbf{\hat{b}_z} \times R \mathbf{\hat{b}_x} = u_1
-  \mathbf{\hat{n}_x} + u_2 \mathbf{\hat{n}_y} + u_3 R \mathbf{\hat{b}_y}\\
-  ^{\mathbf{N}}\mathbf{a}^P &= ^\mathbf{N}\mathbf{a}^O +
-  ^\mathbf{N}\mathbf{\alpha}^\mathbf{B} \times \mathbf{r}^{OP} +
-  ^\mathbf{N}\mathbf{\omega}^\mathbf{B} \times
-  (^\mathbf{N}\mathbf{\omega}^\mathbf{B} \times \mathbf{r}^{OP})\\
-  ^{\mathbf{N}}\mathbf{a}^P &= \dot{u_1} \mathbf{\hat{n}_x} + \dot{u_2}
-  \mathbf{\hat{n}_y} + \dot{u_3}\mathbf{\hat{b}_z}\times R \mathbf{\hat{b}_x}
-  +u_3\mathbf{\hat{b}_z}\times(u_3\mathbf{\hat{b}_z}\times
-  R\mathbf{\hat{b}_x})\\
-  ^{\mathbf{N}}\mathbf{a}^P &= \dot{u_1} \mathbf{\hat{n}_x} + \dot{u_2}
-  \mathbf{\hat{n}_y} + R\dot{u_3}\mathbf{\hat{b}_y} - R u_3^2
-  \mathbf{\hat{b}_x}\\
-
-.. image:: kin_3.*
-   :height: 200
-   :width: 200
-   :align: center
-
-
-In this example we have a double pendulum. We can use the two point theorem
-twice here in order to find the velocity of points :math:`Q` and :math:`P`;
-point :math:`O`'s velocity is zero in :math:`\mathbf{N}`.
-
-.. math::
-  \mathbf{r}^{OQ} &= l \mathbf{\hat{b}_x}\\
-  \mathbf{r}^{QP} &= l \mathbf{\hat{c}_x}\\
-  ^\mathbf{N}\mathbf{\omega}^\mathbf{B} &= u_1 \mathbf{\hat{b}_z}\\
-  ^\mathbf{N}\mathbf{\omega}^\mathbf{C} &= u_2 \mathbf{\hat{c}_z}\\
-  ^\mathbf{N}\mathbf{v}^Q &= ^\mathbf{N}\mathbf{v}^O +
-  ^\mathbf{N}\mathbf{\omega}^\mathbf{B} \times \mathbf{r}^{OQ}\\
-  ^\mathbf{N}\mathbf{v}^Q &= u_1 l \mathbf{\hat{b}_y}\\
-  ^\mathbf{N}\mathbf{v}^P &= ^\mathbf{N}\mathbf{v}^Q +
-  ^\mathbf{N}\mathbf{\omega}^\mathbf{C} \times \mathbf{r}^{QP}\\
-  ^\mathbf{N}\mathbf{v}^Q &= u_1 l \mathbf{\hat{b}_y} +u_2 \mathbf{\hat{c}_z}
-  \times l \mathbf{\hat{c}_x}\\
-  ^\mathbf{N}\mathbf{v}^Q &= u_1 l\mathbf{\hat{b}_y}+u_2 l\mathbf{\hat{c}_y}\\
-
-.. image:: kin_4.*
-   :height: 400
-   :width: 300
-   :align: center
-
-In this example we have a particle moving on a ring; the ring is supported by a
-rod which can rotate about the :math:`\mathbf{\hat{n}_x}` axis. First we use
-the two point theorem to find the velocity of the center point of the ring,
-:math:`Q`, then use the 1 point theorem to find the velocity of the particle on
-the ring.
-
-.. math::
-  ^\mathbf{N}\mathbf{\omega}^\mathbf{C} &= u_1 \mathbf{\hat{n}_x}\\
-  \mathbf{r}^{OQ} &= -l \mathbf{\hat{c}_z}\\
-  ^\mathbf{N}\mathbf{v}^Q &= u_1 l \mathbf{\hat{c}_y}\\
-  \mathbf{r}^{QP} &= R(cos(q_2) \mathbf{\hat{c}_x}
-  + sin(q_2) \mathbf{\hat{c}_y} )\\
-  ^\mathbf{C}\mathbf{v}^P &= R u_2 (-sin(q_2) \mathbf{\hat{c}_x}
-  + cos(q_2) \mathbf{\hat{c}_y} )\\
-  ^\mathbf{N}\mathbf{v}^P &= ^\mathbf{C}\mathbf{v}^P +^\mathbf{N}\mathbf{v}^Q
-  + ^\mathbf{N}\mathbf{\omega}^\mathbf{C} \times \mathbf{r}^{QP}\\
-  ^\mathbf{N}\mathbf{v}^P &= R u_2 (-sin(q_2) \mathbf{\hat{c}_x}
-  + cos(q_2) \mathbf{\hat{c}_y} ) + u_1 l \mathbf{\hat{c}_y} +
-  u_1 \mathbf{\hat{c}_x} \times R(cos(q_2) \mathbf{\hat{c}_x}
-  + sin(q_2) \mathbf{\hat{c}_y}\\
-  ^\mathbf{N}\mathbf{v}^P &= - R u_2 sin(q_2) \mathbf{\hat{c}_x}
-  + (R u_2 cos(q_2)+u_1 l)\mathbf{\hat{c}_y} + R u_1 sin(q_2)
-  \mathbf{\hat{c}_z}\\
-
-A final topic in the description of velocities of points is that of rolling, or
-rather, rolling without slip. Two bodies are said to be rolling without slip if
-and only if the point of contact on each body has the same velocity in another
-frame. See the following figure:
-
-.. image:: kin_rolling.*
-   :height: 250
-   :width: 450
-   :align: center
-
-This is commonly used to form the velocity of a point on one object rolling on
-another fixed object, such as in the following example:
-
-.. % rolling disc kinematics here
-
-
-Kinematics in SymPy's Mechanics
-===============================
-
-It should be clear by now that the topic of kinematics here has been mostly
-describing the correct way to manipulate vectors into representing the
-velocities of points. Within :mod:`mechanics` there are convenient methods for
-storing these velocities associated with frames and points. We'll now revisit
-the above examples and show how to represent them in :mod:`sympy`.
-
-The topic of reference frame creation has already been covered. When a
-``ReferenceFrame`` is created though, it automatically calculates the angular
-velocity of the frame using the time derivative of the DCM and the angular
-velocity definition. ::
-
-  >>> from sympy import Symbol, sin, cos
-  >>> from sympy.physics.mechanics import *
-  >>> mechanics_printing()
-  >>> N = ReferenceFrame('N')
-  >>> q1 = dynamicsymbols('q1')
-  >>> A = N.orientnew('A', 'Axis', [q1, N.x])
-  >>> A.ang_vel_in(N)
-  q1'*N.x
-
-Note that the angular velocity can be defined in an alternate way: ::
-
-  >>> B = ReferenceFrame('B')
-  >>> u1 = dynamicsymbols('u1')
-  >>> B.set_ang_vel(N, u1 * B.y)
-  >>> B.ang_vel_in(N)
-  u1*B.y
-  >>> N.ang_vel_in(B)
-  - u1*B.y
-
-Both upon frame creation during ``orientnew`` and when calling ``set_ang_vel``,
-the angular velocity is set in both frames involved, as seen above.
-
-.. image:: kin_angvel2.*
-   :height: 300
-   :width: 450
-   :align: center
-
-Here we have multiple bodies with angular velocities defined relative to each
-other. This is coded as: ::
-
-  >>> N = ReferenceFrame('N')
-  >>> A = ReferenceFrame('A')
-  >>> B = ReferenceFrame('B')
-  >>> C = ReferenceFrame('C')
-  >>> D = ReferenceFrame('D')
-  >>> u1, u2, u3 = dynamicsymbols('u1 u2 u3')
-  >>> A.set_ang_vel(N, 0)
-  >>> B.set_ang_vel(A, u1 * A.x)
-  >>> C.set_ang_vel(B, -u2 * B.z)
-  >>> D.set_ang_vel(C, u3 * C.y)
-  >>> D.ang_vel_in(N)
-  u3*C.y - u2*B.z + u1*A.x
-
-In :mod:`mechanics` the shortest path between two frames is used when finding
-the angular velocity. That would mean if we went back and set: ::
-
-  >>> D.set_ang_vel(N, 0)
-  >>> D.ang_vel_in(N)
-  0
-
-The path that was just defined is what is used.
-This can cause problems though, as now the angular
-velocity definitions are inconsistent. It is recommended that you avoid
-doing this.
-
-.. % put some stuff to go with derivative theorem here
-
-Points are a translational analog to the rotational ``ReferenceFrame``.
-Creating a ``Point`` can be done in two ways, like ``ReferenceFrame``: ::
-
-  >>> O = Point('O')
-  >>> P = O.locatenew('P', 3 * N.x + N.y)
-  >>> P.pos_from(O)
-  3*N.x + N.y
-  >>> Q = Point('Q')
-  >>> Q.set_pos(P, N.z)
-  >>> Q.pos_from(P)
-  N.z
-  >>> Q.pos_from(O)
-  3*N.x + N.y + N.z
-
-Similar to ``ReferenceFrame``, the position vector between two points is found
-by the shortest path (number of intermediate points) between them. Unlike
-rotational motion, there is no addition theorem for the velocity of points. In
-order to have the velocity of a ``Point`` in a ``ReferenceFrame``, you have to
-set the value. ::
-
-  >>> O = Point('O')
-  >>> O.set_vel(N, u1*N.x)
-  >>> O.vel(N)
-  u1*N.x
-
-For both translational and rotational accelerations, the value is computed by
-taking the time derivative of the appropriate velocity, unless the user sets it
-otherwise.
-
-  >>> O.acc(N)
-  u1'*N.x
-  >>> O.set_acc(N, u2*u1*N.y)
-  >>> O.acc(N)
-  u1*u2*N.y
-
-
-Next is a description of the 2 point and 1 point theorems, as used in
-``sympy``.
-
-.. image:: kin_2.*
-   :height: 300
-   :width: 400
-   :align: center
-
-First is the translating, rotating disc. ::
-
-  >>> N = ReferenceFrame('N')
-  >>> u1, u2, u3 = dynamicsymbols('u1 u2 u3')
-  >>> R = Symbol('R')
-  >>> B = ReferenceFrame('B')
-  >>> O = Point('O')
-  >>> O.set_vel(N, u1 * N.x + u2 * N.y)
-  >>> P = O.locatenew('P', R * B.x)
-  >>> B.set_ang_vel(N, u3 * B.z)
-  >>> P.v2pt_theory(O, N, B)
-  u1*N.x + u2*N.y + R*u3*B.y
-  >>> P.a2pt_theory(O, N, B)
-  u1'*N.x + u2'*N.y - R*u3**2*B.x + R*u3'*B.y
-
-We will also cover implementation of the 1 point theorem.
-
-.. image:: kin_4.*
-   :height: 400
-   :width: 300
-   :align: center
-
-This is the particle moving on a ring, again. ::
-
-  >>> N = ReferenceFrame('N')
-  >>> u1, u2 = dynamicsymbols('u1 u2')
-  >>> q1, q2 = dynamicsymbols('q1 q2')
-  >>> l = Symbol('l')
-  >>> R = Symbol('R')
-  >>> C = N.orientnew('C', 'Axis', [q1, N.x])
-  >>> C.set_ang_vel(N, u1 * N.x)
-  >>> O = Point('O')
-  >>> O.set_vel(N, 0)
-  >>> Q = O.locatenew('Q', -l * C.z)
-  >>> P = Q.locatenew('P', R * (cos(q2) * C.x + sin(q2) * C.y))
-  >>> P.set_vel(C, R * u2 * (-sin(q2) * C.x + cos(q2) * C.y))
-  >>> Q.v2pt_theory(O, N, C)
-  l*u1*C.y
-  >>> P.v1pt_theory(Q, N, C)
-  - R*u2*sin(q2)*C.x + (R*u2*cos(q2) + l*u1)*C.y + R*u1*sin(q2)*C.z
diff --git a/dev-py3k/_sources/modules/physics/mechanics/masses.txt b/dev-py3k/_sources/modules/physics/mechanics/masses.txt
deleted file mode 100644
index 4e10111d692..00000000000
--- a/dev-py3k/_sources/modules/physics/mechanics/masses.txt
+++ /dev/null
@@ -1,395 +0,0 @@
-=======================================================
-Mechanics: Masses, Inertias, Particles and Rigid Bodies
-=======================================================
-
-This document will describe how to represent masses and inertias in
-:mod:`mechanics` and use of the ``RigidBody`` and ``Particle`` classes.
-
-It is assumed that the reader is familiar with the basics of these topics, such
-as finding the center of mass for a system of particles, how to manipulate an
-inertia tensor, and the definition of a particle and rigid body. Any advanced
-dynamics text can provide a reference for these details.
-
-Mass
-====
-
-The only requirement for a mass is that it needs to be a ``sympify``-able
-expression. Keep in mind that masses can be time varying.
-
-Particle
-========
-
-Particles are created with the class ``Particle`` in :mod:`mechanics`.
-A ``Particle`` object has an associated point and an associated mass which are
-the only two attributes of the object.::
-
-  >>> from sympy.physics.mechanics import Particle, Point
-  >>> from sympy import Symbol
-  >>> m = Symbol('m')
-  >>> po = Point('po')
-  >>> # create a particle container
-  >>> pa = Particle('pa', po, m)
-
-The associated point contains the position, velocity and acceleration of the
-particle. :mod:`mechanics` allows one to perform kinematic analysis of points
-separate from their association with masses.
-
-Inertia
-=======
-
-Dyadics are used to define the inertia of bodies within :mod:`mechanics`.
-Inertia dyadics can be defined explicitly but the ``inertia`` function is
-typically much more convenient for the user::
-
-  >>> from sympy.physics.mechanics import ReferenceFrame, inertia
-  >>> N = ReferenceFrame('N')
-
-  Supply a reference frame and the moments of inertia if the object
-  is symmetrical:
-
-  >>> inertia(N, 1, 2, 3)
-  (N.x|N.x) + 2*(N.y|N.y) + 3*(N.z|N.z)
-
-  Supply a reference frame along with the products and moments of inertia
-  for a general object:
-
-  >>> inertia(N, 1, 2, 3, 4, 5, 6)
-  (N.x|N.x) + 4*(N.x|N.y) + 6*(N.x|N.z) + 4*(N.y|N.x) + 2*(N.y|N.y) +
-              5*(N.y|N.z) + 6*(N.z|N.x) + 5*(N.z|N.y) + 3*(N.z|N.z)
-
-Notice that the ``inertia`` function returns a dyadic with each component
-represented as two unit vectors separated by a ``|``. Refer to the
-:ref:`Dyadic` section for more information about dyadics.
-
-Rigid Body
-==========
-
-Rigid bodies are created in a similar fashion as particles. The ``RigidBody``
-class generates objects with four attributes: mass, center of mass, a reference
-frame, and an inertia tuple::
-
-  >>> from sympy import Symbol
-  >>> from sympy.physics.mechanics import ReferenceFrame, Point, RigidBody
-  >>> from sympy.physics.mechanics import outer
-  >>> m = Symbol('m')
-  >>> A = ReferenceFrame('A')
-  >>> P = Point('P')
-  >>> I = outer(A.x, A.x)
-  >>> # create a rigid body
-  >>> B = RigidBody('B', P, A, m, (I, P))
-
-The mass is specified exactly as is in a particle. Similar to the
-``Particle``'s ``.point``, the ``RigidBody``'s center of mass, ``.masscenter``
-must be specified. The reference frame is stored in an analogous fashion and
-holds information about the body's orientation and angular velocity. Finally,
-the inertia for a rigid body needs to be specified about a point. In
-:mod:`mechanics`, you are allowed to specify any point for this. The most
-common is the center of mass, as shown in the above code. If a point is selected
-which is not the center of mass, ensure that the position between the point and
-the center of mass has been defined. The inertia is specified as a tuple of length
-two with the first entry being a ``Dyadic`` and the second entry being a
-``Point`` of which the inertia dyadic is defined about.
-
-.. _Dyadic:
-
-Dyadic
-======
-
-In :mod:`mechanics`, dyadics are used to represent inertia ([Kane1985]_,
-[WikiDyadics]_, [WikiDyadicProducts]_). A dyadic is a linear polynomial of
-component unit dyadics, similar to a vector being a linear polynomial of
-component unit vectors. A dyadic is the outer product between two vectors which
-returns a new quantity representing the juxtaposition of these two vectors. For
-example:
-
-.. math::
-  \mathbf{\hat{a}_x} \otimes \mathbf{\hat{a}_x} &= \mathbf{\hat{a}_x}
-  \mathbf{\hat{a}_x}\\
-  \mathbf{\hat{a}_x} \otimes \mathbf{\hat{a}_y} &= \mathbf{\hat{a}_x}
-  \mathbf{\hat{a}_y}\\
-
-Where :math:`\mathbf{\hat{a}_x}\mathbf{\hat{a}_x}` and
-`\mathbf{\hat{a}_x}\mathbf{\hat{a}_y}` are the outer products obtained by
-multiplying the left side as a column vector by the right side as a row vector.
-Note that the order is significant.
-
-Some additional properties of a dyadic are:
-
-.. math::
-  (x \mathbf{v}) \otimes \mathbf{w} &= \mathbf{v} \otimes (x \mathbf{w}) = x
-  (\mathbf{v} \otimes \mathbf{w})\\
-  \mathbf{v} \otimes (\mathbf{w} + \mathbf{u}) &= \mathbf{v} \otimes \mathbf{w}
-  + \mathbf{v} \otimes \mathbf{u}\\
-  (\mathbf{v} + \mathbf{w}) \otimes \mathbf{u} &= \mathbf{v} \otimes \mathbf{u}
-  + \mathbf{w} \otimes \mathbf{u}\\
-
-A vector in a reference frame can be represented as
-:math:`\begin{bmatrix}a\\b\\c\end{bmatrix}` or :math:`a \mathbf{\hat{i}} + b
-\mathbf{\hat{j}} + c \mathbf{\hat{k}}`. Similarly, a dyadic can be represented
-in tensor form:
-
-.. math::
-  \begin{bmatrix}
-  a_{11} & a_{12} & a_{13} \\
-  a_{21} & a_{22} & a_{23} \\
-  a_{31} & a_{32} & a_{33}
-  \end{bmatrix}\\
-
-or in dyadic form:
-
-.. math::
-  a_{11} \mathbf{\hat{a}_x}\mathbf{\hat{a}_x} +
-  a_{12} \mathbf{\hat{a}_x}\mathbf{\hat{a}_y} +
-  a_{13} \mathbf{\hat{a}_x}\mathbf{\hat{a}_z} +
-  a_{21} \mathbf{\hat{a}_y}\mathbf{\hat{a}_x} +
-  a_{22} \mathbf{\hat{a}_y}\mathbf{\hat{a}_y} +
-  a_{23} \mathbf{\hat{a}_y}\mathbf{\hat{a}_z} +
-  a_{31} \mathbf{\hat{a}_z}\mathbf{\hat{a}_x} +
-  a_{32} \mathbf{\hat{a}_z}\mathbf{\hat{a}_y} +
-  a_{33} \mathbf{\hat{a}_z}\mathbf{\hat{a}_z}\\
-
-Just as with vectors, the later representation makes it possible to keep track
-of which frames the dyadic is defined with respect to. Also, the two
-components of each term in the dyadic need not be in the same frame. The
-following is valid:
-
-.. math::
-  \mathbf{\hat{a}_x} \otimes \mathbf{\hat{b}_y} = \mathbf{\hat{a}_x}
-  \mathbf{\hat{b}_y}
-
-Dyadics can also be crossed and dotted with vectors; again, order matters:
-
-.. math::
-  \mathbf{\hat{a}_x}\mathbf{\hat{a}_x} \cdot \mathbf{\hat{a}_x} &=
-  \mathbf{\hat{a}_x}\\
-  \mathbf{\hat{a}_y}\mathbf{\hat{a}_x} \cdot \mathbf{\hat{a}_x} &=
-  \mathbf{\hat{a}_y}\\
-  \mathbf{\hat{a}_x}\mathbf{\hat{a}_y} \cdot \mathbf{\hat{a}_x} &= 0\\
-  \mathbf{\hat{a}_x} \cdot \mathbf{\hat{a}_x}\mathbf{\hat{a}_x} &=
-  \mathbf{\hat{a}_x}\\
-  \mathbf{\hat{a}_x} \cdot \mathbf{\hat{a}_x}\mathbf{\hat{a}_y} &=
-  \mathbf{\hat{a}_y}\\
-  \mathbf{\hat{a}_x} \cdot \mathbf{\hat{a}_y}\mathbf{\hat{a}_x} &= 0\\
-  \mathbf{\hat{a}_x} \times \mathbf{\hat{a}_y}\mathbf{\hat{a}_x} &=
-  \mathbf{\hat{a}_z}\mathbf{\hat{a}_x}\\
-  \mathbf{\hat{a}_x} \times \mathbf{\hat{a}_x}\mathbf{\hat{a}_x} &= 0\\
-  \mathbf{\hat{a}_y}\mathbf{\hat{a}_x} \times \mathbf{\hat{a}_z} &=
-  - \mathbf{\hat{a}_y}\mathbf{\hat{a}_y}\\
-
-One can also take the time derivative of dyadics or express them in different
-frames, just like with vectors.
-
-Linear Momentum
-===============
-
-The linear momentum of a particle P is defined as:
-
-.. math::
-  L_P = m\mathbf{v}
-
-where :math:`m` is the mass of the particle P and :math:`\mathbf{v}` is the
-velocity of the particle in the inertial frame.[Likins1973]_.
-
-Similarly the linear momentum of a rigid body is defined as:
-
-.. math::
-  L_B = m\mathbf{v^*}
-
-where :math:`m` is the mass of the rigid body, B, and :math:`\mathbf{v^*}` is
-the velocity of the mass center of B in the inertial frame.
-
-Angular Momentum
-================
-
-The angular momentum of a particle P about an arbitrary point O in an inertial
-frame N is defined as:
-
-.. math::
-  ^N \mathbf{H} ^ {P/O} = \mathbf{r} \times m\mathbf{v}
-
-where :math:`\mathbf{r}` is a position vector from point O to the particle of
-mass :math:`m` and :math:`\mathbf{v}` is the velocity of the particle in the
-inertial frame.
-
-Similarly the angular momentum of a rigid body B about a point O in an inertial
-frame N is defined as:
-
-.. math::
-  ^N \mathbf{H} ^ {B/O} = ^N \mathbf{H} ^ {B/B^*} + ^N \mathbf{H} ^ {B^*/O}
-
-where the angular momentum of the body about it's mass center is:
-
-.. math::
-  ^N \mathbf{H} ^ {B/B^*} = \mathbf{I^*} \cdot \omega
-
-and the angular momentum of the mass center about O is:
-
-.. math::
-  ^N \mathbf{H} ^ {B^*/O} = \mathbf{r^*} \times m \mathbf{v^*}
-
-where :math:`\mathbf{I^*}` is the central inertia dyadic of rigid body B,
-:math:`\omega` is the inertial angular velocity of B, :math:`\mathbf{r^*}` is a
-position vector from point O to the mass center of B, :math:`m` is the mass of
-B and :math:`\mathbf{v^*}` is the velocity of the mass center in the inertial
-frame.
-
-Using momenta functions in Mechanics
-====================================
-
-The following example shows how to use the momenta functions in
-:mod:`mechanics`.
-
-One begins by creating the requisite symbols to describe the system. Then
-the reference frame is created and the kinematics are done. ::
-
-  >> from sympy import symbols
-  >> from sympy.physics.mechanics import dynamicsymbols, ReferenceFrame
-  >> from sympy.physics.mechanics import RigidBody, Particle, Point, outer
-  >> from symp.physics.mechanics import linear_momentum, angular_momentum
-  >> m, M, l1 = symbols('m M l1')
-  >> q1d = dynamicsymbols('q1d')
-  >> N = ReferenceFrame('N')
-  >> O = Point('O')
-  >> O.set_vel(N, 0 * N.x)
-  >> Ac = O.locatenew('Ac', l1 * N.x)
-  >> P = Ac.locatenew('P', l1 * N.x)
-  >> a = ReferenceFrame('a')
-  >> a.set_ang_vel(N, q1d * N.z)
-  >> Ac.v2pt_theory(O, N, a)
-  >> P.v2pt_theory(O, N, a)
-
-Finally, the bodies that make up the system are created. In this case the
-system consists of a particle Pa and a RigidBody A. ::
-
-  >> Pa = Particle('Pa', P, m)
-  >> I = outer(N.z, N.z)
-  >> A = RigidBody('A', Ac, a, M, (I, Ac))
-
-Then one can either choose to evaluate the the momenta of individual components
-of the system or of the entire system itself. ::
-
-  >> linear_momentum(N,A)
-  M*l1*q1d*N.y
-  >> angular_momentum(O, N, Pa)
-  4*l1**2*m*q1d*N.z
-  >> linear_momentum(N, A, Pa)
-  (M*l1*q1d + 2*l1*m*q1d)*N.y
-  >> angular_momentum(O, N, A, Pa)
-  (4*l1**2*m*q1d + q1d)*N.z
-
-It should be noted that the user can determine either momenta in any frame
-in :mod:`mechanics` as the user is allowed to specify the reference frame when
-calling the function. In other words the user is not limited to determining
-just inertial linear and angular momenta. Please refer to the docstrings on
-each function to learn more about how each function works precisely.
-
-Kinetic Energy
-==============
-
-The kinetic energy of a particle P is defined as
-
-.. math::
-  T_P = \frac{1}{2} m \mathbf{v^2}
-
-where :math:`m` is the mass of the particle P and :math:`\mathbf{v}`
-is the velocity of the particle in the inertial frame.
-
-Similarly the kinetic energy of a rigid body B is defined as
-
-.. math::
-  T_B = T_t + T_r
-
-where the translational kinetic energy is given by:
-
-.. math::
-  T_t = \frac{1}{2} m \mathbf{v^*} \cdot \mathbf{v^*}
-
-and the rotational kinetic energy is given by:
-
-.. math::
-  T_r = \frac{1}{2} \omega \cdot \mathbf{I^*} \cdot \omega
-
-where :math:`m` is the mass of the rigid body, :math:`\mathbf{v^*}` is the
-velocity of the mass center in the inertial frame, :math:`\omega` is the
-inertial angular velocity of the body and :math:`\mathbf{I^*}` is the central
-inertia dyadic.
-
-Potential Energy
-================
-
-Potential energy is defined as the energy possessed by a body or system by
-virtue of its position or arrangement.
-
-Since there are a variety of definitions for potential energy, this is not
-discussed further here. One can learn more about this in any elementary text
-book on dynamics.
-
-Lagrangian
-==========
-
-The Lagrangian of a body or a system of bodies is defined as:
-
-.. math::
-   \mathcal{L} = T - V
-
-where :math:`T` and :math:`V` are the kinetic and potential energies
-respectively.
-
-Using energy functions in Mechanics
-===================================
-
-The following example shows how to use the energy functions in
-:mod:`mechanics`.
-
-As was discussed above in the momenta functions, one first creates the system
-by going through an identical procedure. ::
-
-  >> from sympy import symbols
-  >> from sympy.physics.mechanics import dynamicsymbols, ReferenceFrame, outer
-  >> from sympy.physics.mechanics import RigidBody, Particle, mechanics_printing
-  >> from symp.physics.mechanics import kinetic_energy, potential_energy, Point
-  >> mechanics_printing()
-  >> m, M, l1, g, h, H = symbols('m M l1 g h H')
-  >> omega = dynamicsymbols('omega')
-  >> N = ReferenceFrame('N')
-  >> O = Point('O')
-  >> O.set_vel(N, 0 * N.x)
-  >> Ac = O.locatenew('Ac', l1 * N.x)
-  >> P = Ac.locatenew('P', l1 * N.x)
-  >> a = ReferenceFrame('a')
-  >> a.set_ang_vel(N, omega * N.z)
-  >> Ac.v2pt_theory(O, N, a)
-  >> P.v2pt_theory(O, N, a)
-  >> Pa = Particle('Pa', P, m)
-  >> I = outer(N.z, N.z)
-  >> A = RigidBody('A', Ac, a, M, (I, Ac))
-
-The user can then determine the kinetic energy of any number of entities of the
-system: ::
-
-  >> kinetic_energy(N, Pa)
-  2*l1**2*m*q1d**2
-  >> kinetic_energy(N, Pa, A)
-  M*l1**2*q1d**2/2 + 2*l1**2*m*q1d**2 + q1d**2/2
-
-It should be noted that the user can determine either kinetic energy relative
-to any frame in :mod:`mechanics` as the user is allowed to specify the
-reference frame when calling the function. In other words the user is not
-limited to determining just inertial kinetic energy.
-
-For potential energies, the user must first specify the potential energy of
-every entity of the system using the :mod:`set_potential_energy` method. The
-potential energy of any number of entities comprising the system can then be
-determined: ::
-
-  >> Pa.set_potential_energy(m * g * h)
-  >> A.set_potential_energy(M * g * H)
-  >> potential_energy(A, Pa)
-  H*M*g + g*h*m
-
-One can also determine the Lagrangian for this system: ::
-
-  >> Lagrangian(Pa, A)
-  -H*M*g + M*l1**2*q1d**2/2 - g*h*m + 2*l1**2*m*q1d**2 + q1d**2/2
-
-Please refer to the docstrings to learn more about each function.
diff --git a/dev-py3k/_sources/modules/physics/mechanics/reference.txt b/dev-py3k/_sources/modules/physics/mechanics/reference.txt
deleted file mode 100644
index f7fa36fd397..00000000000
--- a/dev-py3k/_sources/modules/physics/mechanics/reference.txt
+++ /dev/null
@@ -1,18 +0,0 @@
-================================
-References for Physics/Mechanics
-================================
-
-.. [Kane1983] Kane, Thomas R., Peter W. Likins, and David A. Levinson.
-        Spacecraft Dynamics. New York: McGraw-Hill Book, 1983. Print.
-.. [Kane1985] Kane, Thomas R., and David A. Levinson. Dynamics, Theory and
-        Applications. New York: McGraw-Hill, 1985. Print.
-.. [Meijaard2007] Meijaard, J.P., Jim M. Papadopoulos, Andy Ruina, and A.L.
-        Schwab. "Linearized Dynamics Equations for the Balance and Steer of a Bicycle:
-        a Benchmark and Review." Proceedings of the Royal Society A: Mathematical,
-        Physical and Engineering Sciences 463.2084 (2007): 1955-982. Print.
-.. [WikiDyadics] "Dyadics." Wikipedia, the Free Encyclopedia. Web. 05 Aug.
-        2011. <http://en.wikipedia.org/wiki/Dyadics>.
-.. [WikiDyadicProducts] "Dyadic Product." Wikipedia, the Free Encyclopedia.
-        Web. 05 Aug. 2011. <http://en.wikipedia.org/wiki/Dyadic_product>.
-.. [Likins1973] Likins, Peter W. Elements of Engineering Mechanics.
-        McGraw-Hill, Inc. 1973. Print.
diff --git a/dev-py3k/_sources/modules/physics/mechanics/vectors.txt b/dev-py3k/_sources/modules/physics/mechanics/vectors.txt
deleted file mode 100644
index e4d3b2d2154..00000000000
--- a/dev-py3k/_sources/modules/physics/mechanics/vectors.txt
+++ /dev/null
@@ -1,767 +0,0 @@
-==================================
-Mechanics: Vector & ReferenceFrame
-==================================
-
-
-In :mod:`mechanics`, vectors and reference frames are the "building blocks" of
-dynamic systems. This document will describe these mathematically and describe
-how to use them with this module's code.
-
-Vector
-======
-
-A vector is a geometric object that has a magnitude (or length) and a
-direction.  Vectors in 3-space are often represented on paper as:
-
-.. image:: vec_rep.*
-   :height: 175
-   :width: 350
-   :align: center
-
-Vector Algebra
-==============
-
-Vector algebra is the first topic to be discussed.
-
-Two vectors are said to be equal if and only if (iff) they have the same same
-magnitude and orientation.
-
-Vector Operations
------------------
-Multiple algebraic operations can be done with vectors: addition between
-vectors, scalar multiplication, and vector multiplication.
-
-Vector addition as based on the parallelogram law.
-
-.. image:: vec_add.*
-   :height: 200
-   :width: 200
-   :align: center
-
-Vector addition is also commutative:
-
-.. math::
-  \mathbf{a} + \mathbf{b} &= \mathbf{b} + \mathbf{a} \\
-  (\mathbf{a} + \mathbf{b}) + \mathbf{c} &= \mathbf{a} + (\mathbf{b} +
-  \mathbf{c})
-
-Scalar multiplication is the product of a vector and a scalar; the result is a
-vector with the same orientation but whose magnitude is scaled by the scalar.
-Note that multiplication by -1 is equivalent to rotating the vector by 180
-degrees about an arbitrary axis in the plane perpendicular to the vector.
-
-.. image:: vec_mul.*
-   :height: 150
-   :width: 200
-   :align: center
-
-A unit vector is simply a vector whose magnitude is equal to 1.  Given any
-vector :math:`\mathbf{v}` we can define a unit vector as:
-
-.. math::
-  \mathbf{\hat{n}_v} = \frac{\mathbf{v}}{\Vert \mathbf{v} \Vert}
-
-Note that every vector can be written as the product of a scalar and unit
-vector.
-
-Three vector products are implemented in :mod:`mechanics`: the dot product, the
-cross product, and the outer product.
-
-The dot product operation maps two vectors to a scalar.  It is defined as:
-
-.. math::
-  \mathbf{a} \cdot \mathbf{b} = \Vert \mathbf{a} \Vert \Vert \mathbf{b}
-  \Vert \cos(\theta)\\
-
-where :math:`\theta` is the angle between :math:`\mathbf{a}` and
-:math:`\mathbf{b}`.
-
-The dot product of two unit vectors represent the magnitude of the common
-direction; for other vectors, it is the product of the magnitude of the common
-direction and the two vectors' magnitudes. The dot product of two perpendicular
-is zero. The figure below shows some examples:
-
-.. image:: vec_dot.*
-   :height: 250
-   :width: 450
-   :align: center
-
-The dot product is commutative:
-
-.. math::
-  \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}
-
-The cross product vector multiplication operation of two vectors returns a
-vector:
-
-.. math::
-  \mathbf{a} \times \mathbf{b} = \mathbf{c}
-
-The vector :math:`\mathbf{c}` has the following properties: it's orientation is
-perpendicular to both :math:`\mathbf{a}` and :math:`\mathbf{b}`, it's magnitude
-is defined as :math:`\Vert \mathbf{c} \Vert = \Vert \mathbf{a} \Vert \Vert
-\mathbf{b} \Vert \sin(\theta)` (where :math:`\theta` is the angle between
-:math:`\mathbf{a}` and :math:`\mathbf{b}`), and has a sense defined by using
-the right hand rule between :math:`\Vert \mathbf{a} \Vert \Vert \mathbf{b}
-\Vert`. The figure below shows this:
-
-.. image:: vec_cross.*
-   :height: 350
-   :width: 700
-   :align: center
-
-The cross product has the following properties:
-
-It is not commutative:
-
-.. math::
-  \mathbf{a} \times \mathbf{b} &\neq \mathbf{b} \times \mathbf{a} \\
-  \mathbf{a} \times \mathbf{b} &= - \mathbf{b} \times \mathbf{a}
-
-and not associative:
-
-.. math::
-  (\mathbf{a} \times \mathbf{b} ) \times \mathbf{c} \neq \mathbf{a} \times
-  (\mathbf{b} \times \mathbf{c})
-
-Two parallel vectors will have a zero cross product.
-
-The outer product between two vectors will not be not be discussed here, but
-instead in the inertia section (that is where it is used). Other useful vector
-properties and relationships are:
-
-.. math::
-  \alpha (\mathbf{a} + \mathbf{b}) &= \alpha \mathbf{a} + \alpha \mathbf{b}\\
-  \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) &= \mathbf{a} \cdot \mathbf{b} +
-  \mathbf{a} \cdot \mathbf{c}\\
-  \mathbf{a} \times (\mathbf{b} + \mathbf{c}) &= \mathbf{a} \times \mathbf{b} +
-  \mathbf{a} \times \mathbf{b}\\
-  (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} & \textrm{ gives the scalar
-  triple product.}\\
-  \mathbf{a} \times (\mathbf{b} \cdot \mathbf{c}) & \textrm{ does not work,
-  as you cannot cross a vector and a scalar.}\\
-  (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} &= \mathbf{a} \cdot
-  (\mathbf{b} \times \mathbf{c})\\
-  (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} &= (\mathbf{b} \times
-  \mathbf{c}) \cdot \mathbf{a} = (\mathbf{c} \times \mathbf{a}) \cdot
-  \mathbf{b}\\
-  (\mathbf{a} \times \mathbf{b}) \times \mathbf{c} &= \mathbf{b}(\mathbf{a}
-  \cdot \mathbf{c}) - \mathbf{a}(\mathbf{b} \cdot \mathbf{c})\\
-  \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) &= \mathbf{b}(\mathbf{a}
-  \cdot \mathbf{c}) - \mathbf{c}(\mathbf{a} \cdot \mathbf{b})\\
-
-Alternative Representation
---------------------------
-If we have three non-coplanar unit vectors
-:math:`\mathbf{\hat{n}_x},\mathbf{\hat{n}_y},\mathbf{\hat{n}_z}`,
-we can represent any vector
-:math:`\mathbf{a}` as :math:`\mathbf{a} = a_x \mathbf{\hat{n}_x} + a_y
-\mathbf{\hat{n}_y} + a_z \mathbf{\hat{n}_z}`. In this situation
-:math:`\mathbf{\hat{n}_x},\mathbf{\hat{n}_y},\mathbf{\hat{n}_z}`
-are referred to as a basis.  :math:`a_x, a_y, a_z`
-are called the measure numbers.
-Usually the unit vectors are mutually perpendicular, in which case we can refer
-to them as an orthonormal basis, and they are usually right-handed.
-
-To test equality between two vectors, now we can do the following. With
-vectors:
-
-.. math::
-  \mathbf{a} &= a_x \mathbf{\hat{n}_x} + a_y \mathbf{\hat{n}_y} + a_z
-  \mathbf{\hat{n}_z}\\
-  \mathbf{b} &= b_x \mathbf{\hat{n}_x} + b_y \mathbf{\hat{n}_y} + b_z
-  \mathbf{\hat{n}_z}\\
-
-We can claim equality if: :math:`a_x = b_x, a_y = b_y, a_z = b_z`.
-
-Vector addition is then represented, for the same two vectors, as:
-
-.. math::
-  \mathbf{a} + \mathbf{b} = (a_x + b_x)\mathbf{\hat{n}_x} + (a_y + b_y)
-  \mathbf{\hat{n}_y} + (a_z + b_z) \mathbf{\hat{n}_z}
-
-Multiplication operations are now defined as:
-
-.. math::
-  \alpha \mathbf{b} &= \alpha b_x \mathbf{\hat{n}_x} + \alpha b_y
-  \mathbf{\hat{n}_y} + \alpha b_z \mathbf{\hat{n}_z}\\
-  \mathbf{a} \cdot \mathbf{b} &= a_x b_x + a_y b_y + a_z b_z\\
-  \mathbf{a} \times \mathbf{b} &=
-  \textrm{det }\begin{bmatrix} \mathbf{\hat{n}_x} & \mathbf{\hat{n}_y} &
-  \mathbf{\hat{n}_z} \\ a_x & a_y & a_z \\ a_x & a_y & a_z \end{bmatrix}\\
-  (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} &=
-  \textrm{det }\begin{bmatrix} a_x & a_y & a_z \\ b_x & b_y & b_z \\ c_x & c_y
-  & c_z \end{bmatrix}\\
-
-To write a vector in a given basis, we can do the follow:
-
-.. math::
-  \mathbf{a} = (\mathbf{a}\cdot\mathbf{\hat{n}_x})\mathbf{\hat{n}_x} +
-  (\mathbf{a}\cdot\mathbf{\hat{n}_y})\mathbf{\hat{n}_y} +
-  (\mathbf{a}\cdot\mathbf{\hat{n}_z})\mathbf{\hat{n}_z}\\
-
-
-Examples
---------
-Some numeric examples of these operations follow:
-
-.. math::
-  \mathbf{a} &= \mathbf{\hat{n}_x} + 5 \mathbf{\hat{n}_y}\\
-  \mathbf{b} &= \mathbf{\hat{n}_y} + \alpha \mathbf{\hat{n}_z}\\
-  \mathbf{a} + \mathbf{b} &= \mathbf{\hat{n}_x} + 6 \mathbf{\hat{n}_y} + \alpha
-  \mathbf{\hat{n}_z}\\
-  \mathbf{a} \cdot \mathbf{b} &= 5\\
-  \mathbf{a} \cdot \mathbf{\hat{n}_y} &= 5\\
-  \mathbf{a} \cdot \mathbf{\hat{n}_z} &= 0\\
-  \mathbf{a} \times \mathbf{b} &= 5 \alpha \mathbf{\hat{n}_x} - \alpha
-  \mathbf{\hat{n}_y} + \mathbf{\hat{n}_z}\\
-  \mathbf{b} \times \mathbf{a} &= -5 \alpha \mathbf{\hat{n}_x} + \alpha
-  \mathbf{\hat{n}_y} - \mathbf{\hat{n}_z}\\
-
-
-Vector Calculus
-===============
-To deal with the calculus of vectors with moving object, we have to introduce
-the concept of a reference frame. A classic example is a train moving along its
-tracks, with you and a friend inside. If both you and your friend are sitting,
-the relative velocity between the two of you is zero. From an observer outside
-the train, you will both have velocity though.
-
-We will now apply more rigor to this definition. A reference frame is a virtual
-"platform" which we choose to observe vector quantities from. If we have a
-reference frame :math:`\mathbf{N}`, vector :math:`\mathbf{a}` is said to be
-fixed in the frame :math:`\mathbf{N}` if none of its properties ever change
-when observed from :math:`\mathbf{N}`. We will typically assign a fixed
-orthonormal basis vector set with each reference frame; :math:`\mathbf{N}` will
-have :math:`\mathbf{\hat{n}_x}, \mathbf{\hat{n}_y},\mathbf{\hat{n}_z}` as its
-basis vectors.
-
-Derivatives of Vectors
-----------------------
-
-A vector which is not fixed in a reference frame therefore has changing
-properties when observed from that frame. Calculus is the study of change, and
-in order to deal with the peculiarities of vectors fixed and not fixed in
-different reference frames, we need to be more explicit in our definitions.
-
-.. image:: vec_fix_notfix.*
-   :height: 300
-   :width: 450
-   :align: center
-
-In the above figure, we have vectors :math:`\mathbf{c,d,e,f}`. If one were to
-take the derivative of :math:`\mathbf{e}` with respect to :math:`\theta`:
-
-.. math::
-  \frac{d \mathbf{e}}{d \theta}
-
-it is not clear what the derivative is. If you are observing from frame
-:math:`\mathbf{A}`, it is clearly non-zero. If you are observing from frame
-:math:`\mathbf{B}`, the derivative is zero. We will therefore introduce the
-frame as part of the derivative notation:
-
-.. math::
-  \frac{^{\mathbf{A}} d \mathbf{e}}{d \theta} &\neq 0 \textrm{,
-  the derivative of } \mathbf{e} \textrm{ with respect to } \theta
-  \textrm{ in the reference frame } \mathbf{A}\\
-  \frac{^{\mathbf{B}} d \mathbf{e}}{d \theta} &= 0 \textrm{,
-   the derivative of } \mathbf{e} \textrm{ with respect to } \theta
-  \textrm{ in the reference frame } \mathbf{B}\\
-  \frac{^{\mathbf{A}} d \mathbf{c}}{d \theta} &= 0 \textrm{,
-   the derivative of } \mathbf{c} \textrm{ with respect to } \theta
-  \textrm{ in the reference frame } \mathbf{A}\\
-  \frac{^{\mathbf{B}} d \mathbf{c}}{d \theta} &\neq 0 \textrm{,
-   the derivative of } \mathbf{c} \textrm{ with respect to } \theta
-  \textrm{ in the reference frame } \mathbf{B}\\
-
-Here are some additional properties of derivatives of vectors in specific
-frames:
-
-.. math::
-  \frac{^{\mathbf{A}} d}{dt}(\mathbf{a} + \mathbf{b}) &= \frac{^{\mathbf{A}}
-  d\mathbf{a}}{dt} + \frac{^{\mathbf{A}} d\mathbf{b}}{dt}\\
-  \frac{^{\mathbf{A}} d}{dt}\gamma \mathbf{a} &= \frac{ d \gamma}{dt}\mathbf{a}
-  + \gamma\frac{^{\mathbf{A}} d\mathbf{a}}{dt}\\
-  \frac{^{\mathbf{A}} d}{dt}(\mathbf{a} \times \mathbf{b}) &=
-  \frac{^{\mathbf{A}} d\mathbf{a}}{dt} \times \mathbf{b} +
-  \mathbf{a} \times \frac{^{\mathbf{A}} d\mathbf{b}}{dt}\\
-
-Relating Sets of Basis Vectors
-------------------------------
-
-We need to now define the relationship between two different reference frames;
-or how to relate the basis vectors of one frame to another. We can do this
-using a direction cosine matrix (DCM). The direction cosine matrix relates
-the basis vectors of one frame to another, in the following fashion:
-
-.. math::
-  \begin{bmatrix}
-  \mathbf{\hat{a}_x} \\ \mathbf{\hat{a}_y} \\ \mathbf{\hat{a}_z} \\
-  \end{bmatrix}  =
-  \begin{bmatrix} ^{\mathbf{A}} \mathbf{C}^{\mathbf{B}} \end{bmatrix}
-  \begin{bmatrix}
-  \mathbf{\hat{b}_x} \\ \mathbf{\hat{b}_y} \\ \mathbf{\hat{b}_z} \\
-  \end{bmatrix}
-
-When two frames (say, :math:`\mathbf{A}` & :math:`\mathbf{B}`) are initially
-aligned, then one frame has all of its basis vectors rotated around an axis
-which is aligned with a basis vector, we say the frames are related by a simple
-rotation. The figure below shows this:
-
-.. image:: simp_rot.*
-   :height: 250
-   :width: 250
-   :align: center
-
-The above rotation is a simple rotation about the Z axis by an angle
-:math:`\theta`. Note that after the rotation, the basis vectors
-:math:`\mathbf{\hat{a}_z}` and :math:`\mathbf{\hat{b}_z}` are still aligned.
-
-This rotation can be characterized by the following direction cosine matrix:
-
-.. math::
-
-  ^{\mathbf{A}}\mathbf{C}^{\mathbf{B}} =
-  \begin{bmatrix}
-  \cos(\theta) & - \sin(\theta) & 0\\
-  \sin(\theta) & \cos(\theta) & 0\\
-  0 & 0 & 1\\
-  \end{bmatrix}
-
-Simple rotations about the X and Y axes are defined by:
-
-.. math::
-
-  \textrm{DCM for x-axis rotation: }
-  \begin{bmatrix}
-  1 & 0 & 0\\
-  0 & \cos(\theta) & -\sin(\theta)\\
-  0 & \sin(\theta) & \cos(\theta)
-  \end{bmatrix}
-
-  \textrm{DCM for y-axis rotation: }
-  \begin{bmatrix}
-  \cos(\theta) & 0 & \sin(\theta)\\
-  0 & 1 & 0\\
-  -\sin(\theta) & 0 & \cos(\theta)\\
-  \end{bmatrix}
-
-Rotation in the positive direction here will be defined by using the right-hand
-rule.
-
-The direction cosine matrix is also involved with the definition of the dot
-product between sets of basis vectors. If we have two reference frames with
-associated basis vectors, their direction cosine matrix can be defined as:
-
-.. math::
-
-  \begin{bmatrix}
-  C_{xx} & C_{xy} & C_{xz}\\
-  C_{yx} & C_{yy} & C_{yz}\\
-  C_{zx} & C_{zy} & C_{zz}\\
-  \end{bmatrix} =
-  \begin{bmatrix}
-  \mathbf{\hat{a}_x}\cdot\mathbf{\hat{b}_x} &
-  \mathbf{\hat{a}_x}\cdot\mathbf{\hat{b}_y} &
-  \mathbf{\hat{a}_x}\cdot\mathbf{\hat{b}_z}\\
-  \mathbf{\hat{a}_y}\cdot\mathbf{\hat{b}_x} &
-  \mathbf{\hat{a}_y}\cdot\mathbf{\hat{b}_y} &
-  \mathbf{\hat{a}_y}\cdot\mathbf{\hat{b}_z}\\
-  \mathbf{\hat{a}_z}\cdot\mathbf{\hat{b}_x} &
-  \mathbf{\hat{a}_z}\cdot\mathbf{\hat{b}_y} &
-  \mathbf{\hat{a}_z}\cdot\mathbf{\hat{b}_z}\\
-  \end{bmatrix}
-
-Additionally, the direction cosine matrix is orthogonal, in that:
-
-.. math::
-  ^{\mathbf{A}}\mathbf{C}^{\mathbf{B}} =
-  (^{\mathbf{B}}\mathbf{C}^{\mathbf{A}})^{-1}\\ =
-  (^{\mathbf{B}}\mathbf{C}^{\mathbf{A}})^T\\
-
-If we have reference frames :math:`\mathbf{A}` and :math:`\mathbf{B}`, which in
-this example have undergone a simple z-axis rotation by an amount
-:math:`\theta`, we will have two sets of basis vectors. We can then define two
-vectors: :math:`\mathbf{a} = \mathbf{\hat{a}_x} + \mathbf{\hat{a}_y} +
-\mathbf{\hat{a}_z}` and :math:`\mathbf{b} = \mathbf{\hat{b}_x} +
-\mathbf{\hat{b}_y} + \mathbf{\hat{b}_z}`. If we wish to express
-:math:`\mathbf{b}` in the :math:`\mathbf{A}` frame, we do the following:
-
-.. math::
-  \mathbf{b} &= \mathbf{\hat{b}_x} + \mathbf{\hat{b}_y} + \mathbf{\hat{b}_z}\\
-  \mathbf{b} &= \begin{bmatrix}\mathbf{\hat{a}_x}\cdot (\mathbf{\hat{b}_x} +
-  \mathbf{\hat{b}_y} + \mathbf{\hat{b}_z})\end{bmatrix} \mathbf{\hat{a}_x} +
-  \begin{bmatrix}\mathbf{\hat{a}_y}\cdot (\mathbf{\hat{b}_x} + \mathbf{\hat{b}_y}
-  + \mathbf{\hat{b}_z})\end{bmatrix} \mathbf{\hat{a}_y} +
-  \begin{bmatrix}\mathbf{\hat{a}_z}\cdot (\mathbf{\hat{b}_x} + \mathbf{\hat{b}_y}
-  + \mathbf{\hat{b}_z})\end{bmatrix} \mathbf{\hat{a}_z}\\ \mathbf{b} &=
-  (\cos(\theta) - \sin(\theta))\mathbf{\hat{a}_x} +
-  (\sin(\theta) + \cos(\theta))\mathbf{\hat{a}_y} + \mathbf{\hat{a}_z}
-
-And if we wish to express :math:`\mathbf{a}` in the :math:`\mathbf{B}`, we do:
-
-.. math::
-  \mathbf{a} &= \mathbf{\hat{a}_x} + \mathbf{\hat{a}_y} + \mathbf{\hat{a}_z}\\
-  \mathbf{a} &= \begin{bmatrix}\mathbf{\hat{b}_x}\cdot (\mathbf{\hat{a}_x} +
-  \mathbf{\hat{a}_y} + \mathbf{\hat{a}_z})\end{bmatrix} \mathbf{\hat{b}_x} +
-  \begin{bmatrix}\mathbf{\hat{b}_y}\cdot (\mathbf{\hat{a}_x} +
-  \mathbf{\hat{a}_y} + \mathbf{\hat{a}_z})\end{bmatrix} \mathbf{\hat{b}_y} +
-  \begin{bmatrix}\mathbf{\hat{b}_z}\cdot (\mathbf{\hat{a}_x} +
-  \mathbf{\hat{a}_y} + \mathbf{\hat{a}_z})\end{bmatrix} \mathbf{\hat{b}_z}\\
-  \mathbf{a} &= (\cos(\theta) + \sin(\theta))\mathbf{\hat{b}_x} +
-  (-\sin(\theta)+\cos(\theta))\mathbf{\hat{b}_y} + \mathbf{\hat{b}_z}
-
-
-Derivatives with Multiple Frames
---------------------------------
-
-If we have reference frames :math:`\mathbf{A}` and :math:`\mathbf{B}`
-we will have two sets of basis vectors. We can then define two
-vectors: :math:`\mathbf{a} = a_x\mathbf{\hat{a}_x} + a_y\mathbf{\hat{a}_y} +
-a_z\mathbf{\hat{a}_z}` and :math:`\mathbf{b} = b_x\mathbf{\hat{b}_x} +
-b_y\mathbf{\hat{b}_y} + b_z\mathbf{\hat{b}_z}`. If we want to take the
-derivative of :math:`\mathbf{b}` in the reference frame :math:`\mathbf{A}`, we
-must first express it in :math:`\mathbf{A}`, and the take the derivatives of
-the measure numbers:
-
-.. math::
-  \frac{^{\mathbf{A}} d\mathbf{b}}{dx} = \frac{d (\mathbf{b}\cdot
-  \mathbf{\hat{a}_x} )}{dx} \mathbf{\hat{a}_x} + \frac{d (\mathbf{b}\cdot
-  \mathbf{\hat{a}_y} )}{dx} \mathbf{\hat{a}_y} + \frac{d (\mathbf{b}\cdot
-  \mathbf{\hat{a}_z} )}{dx} \mathbf{\hat{a}_z} +
-
-
-Examples
---------
-
-An example of vector calculus:
-
-.. image:: vec_simp_der.*
-   :height: 500
-   :width: 350
-   :align: center
-
-In this example we have two bodies, each with an attached reference frame.
-We will say that :math:`\theta` and :math:`x` are functions of time.
-We wish to know the time derivative of vector :math:`\mathbf{c}` in both the
-:math:`\mathbf{A}` and :math:`\mathbf{B}` frames.
-
-First, we need to define :math:`\mathbf{c}`;
-:math:`\mathbf{c}=x\mathbf{\hat{b}_x}+l\mathbf{\hat{b}_y}`. This provides a
-definition in the :math:`\mathbf{B}` frame. We can now do the following:
-
-.. math::
-  \frac{^{\mathbf{B}} d \mathbf{c}}{dt} &= \frac{dx}{dt} \mathbf{\hat{b}_x} +
-  \frac{dl}{dt} \mathbf{\hat{b}_y}\\
-  &= \dot{x} \mathbf{\hat{b}_x}
-
-To take the derivative in the :math:`\mathbf{A}` frame, we have to first relate
-the two frames:
-
-.. math::
-  ^{\mathbf{A}} \mathbf{C} ^{\mathbf{B}} =
-  \begin{bmatrix}
-  \cos(\theta) & 0 & \sin(\theta)\\
-  0 & 1 & 0\\
-  -\sin(\theta) & 0 & \cos(\theta)\\
-  \end{bmatrix}
-
-Now we can do the following:
-
-.. math::
-  \frac{^{\mathbf{A}} d \mathbf{c}}{dt} &= \frac{d (\mathbf{c} \cdot
-  \mathbf{\hat{a}_x})}{dt} \mathbf{\hat{a}_x} + \frac{d (\mathbf{c} \cdot
-  \mathbf{\hat{a}_y})}{dt} \mathbf{\hat{a}_y} + \frac{d (\mathbf{c} \cdot
-  \mathbf{\hat{a}_z})}{dt} \mathbf{\hat{a}_z}\\
-  &= \frac{d (\cos(\theta) x)}{dt} \mathbf{\hat{a}_x} +
-  \frac{d (l)}{dt} \mathbf{\hat{a}_y} +
-  \frac{d (-\sin(\theta) x)}{dt} \mathbf{\hat{a}_z}\\
-  &= (-\dot{\theta}\sin(\theta)x + \cos(\theta)\dot{x}) \mathbf{\hat{a}_x} +
-  (\dot{\theta}\cos(\theta)x + \sin(\theta)\dot{x}) \mathbf{\hat{a}_z}
-
-Note that this is the time derivative of :math:`\mathbf{c}` in
-:math:`\mathbf{A}`, and is expressed in the :math:`\mathbf{A}` frame. We can
-express it in the :math:`\mathbf{B}` frame however, and the expression will
-still be valid:
-
-.. math::
-  \frac{^{\mathbf{A}} d \mathbf{c}}{dt} &= (-\dot{\theta}\sin(\theta)x +
-  \cos(\theta)\dot{x}) \mathbf{\hat{a}_x} + (\dot{\theta}\cos(\theta)x +
-  \sin(\theta)\dot{x}) \mathbf{\hat{a}_z}\\
-  &= \dot{x}\mathbf{\hat{b}_x} - \theta x \mathbf{\hat{b}_z}\\
-
-Note the difference in expression complexity between the two forms. They are
-equivalent, but one is much simpler. This is an extremely important concept, as
-defining vectors in the more complex forms can vastly slow down formulation of
-the equations of motion and increase their length, sometimes to a point where
-they cannot be shown on screen.
-
-Using Vectors and Reference Frames in SymPy's Mechanics
-=======================================================
-
-We have waited until after all of the relevant mathematical relationships have
-been defined for vectors and reference frames to introduce code. This is due to
-how vectors are formed. When starting any problem in :mod:`mechanics`, one of
-the first steps is defining a reference frame (remember to import
-sympy.physics.mechanics first)::
-
-  >>> from sympy.physics.mechanics import *
-  >>> N = ReferenceFrame('N')
-
-Now we have created a reference frame, :math:`\mathbf{N}`. To have access to
-any basis vectors, first a reference frame needs to be created. Now that we
-have made and object representing :math:`\mathbf{N}`, we can access its basis
-vectors::
-
-  >>> N.x
-  N.x
-  >>> N.y
-  N.y
-  >>> N.z
-  N.z
-
-Vector Algebra, in Mechanics
-----------------------------
-
-We can now do basic algebraic operations on these vectors.::
-
-  >>> N.x == N.x
-  True
-  >>> N.x == N.y
-  False
-  >>> N.x + N.y
-  N.x + N.y
-  >>> 2 * N.x + N.y
-  2*N.x + N.y
-
-Remember, don't add a scalar quantity to a vector (``N.x + 5``); this will
-raise an error. At this point, we'll use SymPy's Symbol in our vectors.
-Remember to refer to SymPy's Gotchas and Pitfalls when dealing with symbols.::
-
-  >>> from sympy import Symbol, symbols
-  >>> x = Symbol('x')
-  >>> x * N.x
-  x*N.x
-  >>> x*(N.x + N.y)
-  x*N.x + x*N.y
-
-In :mod:`mechanics` multiple interfaces to vector multiplication have been
-implemented, at the operator level, method level, and function level. The
-vector dot product can work as follows: ::
-
-  >>> N.x & N.x
-  1
-  >>> N.x & N.y
-  0
-  >>> N.x.dot(N.x)
-  1
-  >>> N.x.dot(N.y)
-  0
-  >>> dot(N.x, N.x)
-  1
-  >>> dot(N.x, N.y)
-  0
-
-The "official" interface is the function interface; this is what will be used
-in all examples. This is to avoid confusion with the attribute and methods
-being next to each other, and in the case of the operator operation priority.
-The operators used in :mod:`mechanics` for vector multiplication do not posses
-the correct order of operations; this can lead to errors. Care with parentheses
-is needed when using operators to represent vector multiplication.
-
-The cross product is the other vector multiplication which will be discussed
-here. It offers similar interfaces to the dot product, and comes with the same
-warnings. ::
-
-  >>> N.x ^ N.x
-  0
-  >>> N.x ^ N.y
-  N.z
-  >>> N.x.cross(N.x)
-  0
-  >>> N.x.cross(N.z)
-  - N.y
-  >>> cross(N.x, N.y)
-  N.z
-  >>> N.x ^ (N.y + N.z)
-  - N.y + N.z
-
-Two additional operations can be done with vectors: normalizing the vector to
-length 1, and getting its magnitude. These are done as follows:
-
-  >>> (N.x + N.y).normalize()
-  sqrt(2)/2*N.x + sqrt(2)/2*N.y
-  >>> (N.x + N.y).magnitude()
-  sqrt(2)
-
-Vector Calculus, in Mechanics
------------------------------
-
-We have already introduced our first reference frame. We can take the
-derivative in that frame right now, if we desire: ::
-
-  >>> (x * N.x + N.y).diff(x, N)
-  N.x
-
-SymPy has a ``diff`` function, but it does not currently work with
-:mod:`mechanics` Vectors, so please use ``Vector``'s ``diff`` method.  The
-reason for this is that when differentiating a ``Vector``, the frame of
-reference must be specified in addition to what you are taking the derivative
-with respect to; SymPy's ``diff`` function doesn't fit this mold.
-
-The more interesting case arise with multiple reference frames. If we introduce
-a second reference frame, :math:`\mathbf{A}`, we now have two frames. Note that
-at this point we can add components of :math:`\mathbf{N}` and
-:math:`\mathbf{A}` together, but cannot perform vector multiplication, as no
-relationship between the two frames has been defined. ::
-
-  >>> A = ReferenceFrame('A')
-  >>> A.x + N.x
-  A.x + N.x
-
-If we want to do vector multiplication, first we have to define and
-orientation. The ``orient`` method of ``ReferenceFrame`` provides that
-functionality. ::
-
-  >>> A.orient(N, 'Axis', [x, N.y])
-
-If we desire, we can view the DCM between these two frames at any time. This
-can be calculated with the ``dcm`` method. This code: ``N.dcm(A)`` gives the
-dcm :math:`^{\mathbf{N}} \mathbf{C} ^{\mathbf{A}}`.
-
-This orients the :math:`\mathbf{A}` frame relative to the :math:`\mathbf{N}`
-frame by a simple rotation around the Y axis, by an amount x. Other, more
-complicated rotation types include Body rotations, Space rotations,
-quaternions, and arbitrary axis rotations. Body and space rotations are
-equivalent to doing 3 simple rotations in a row, each about a basis vector in
-the new frame. An example follows: ::
-
-
-  >>> N = ReferenceFrame('N')
-  >>> Bp = ReferenceFrame('Bp')
-  >>> Bpp = ReferenceFrame('Bpp')
-  >>> B = ReferenceFrame('B')
-  >>> q1,q2,q3 = symbols('q1 q2 q3')
-  >>> Bpp.orient(N,'Axis', [q1, N.x])
-  >>> Bp.orient(Bpp,'Axis', [q2, Bpp.y])
-  >>> B.orient(Bp,'Axis', [q3, Bp.z])
-  >>> N.dcm(B)
-  [                          cos(q2)*cos(q3),                           -sin(q3)*cos(q2),          sin(q2)]
-  [sin(q1)*sin(q2)*cos(q3) + sin(q3)*cos(q1), -sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3), -sin(q1)*cos(q2)]
-  [sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3),  sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1),  cos(q1)*cos(q2)]
-  >>> B.orient(N,'Body',[q1,q2,q3],'XYZ')
-  >>> N.dcm(B)
-  [                          cos(q2)*cos(q3),                           -sin(q3)*cos(q2),          sin(q2)]
-  [sin(q1)*sin(q2)*cos(q3) + sin(q3)*cos(q1), -sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3), -sin(q1)*cos(q2)]
-  [sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3),  sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1),  cos(q1)*cos(q2)]
-
-Space orientations are similar to body orientation, but applied from the frame
-to body. Body and space rotations can involve either two or three axes: 'XYZ'
-works, as does 'YZX', 'ZXZ', 'YXY', etc. What is key is that each simple
-rotation is about a different axis than the previous one; 'ZZX' does not
-completely orient a set of basis vectors in 3 space.
-
-Sometimes it will be more convenient to create a new reference frame and orient
-relative to an existing one in one step. The ``orientnew`` method allows for
-this functionality, and essentially wraps the ``orient`` method. All of the
-things you can do in ``orient``, you can do in ``orientnew``. ::
-
-  >>> C = N.orientnew('C', 'Axis', [q1, N.x])
-
-Quaternions (or Euler Parameters) use 4 value to characterize the orientation
-of the frame. This and arbitrary axis rotations are described in the ``orient``
-and ``orientnew`` method help, or in the references [Kane1983]_.
-
-
-Finally, before starting multiframe calculus operations, we will introduce
-another :mod:`mechanics` tool: ``dynamicsymbols``. ``dynamicsymbols`` is
-a shortcut function to create undefined functions of time within SymPy. The
-derivative of such a 'dynamicsymbol' is shown below. ::
-
-  >>> from sympy import diff
-  >>> q1, q2, q3 = dynamicsymbols('q1 q2 q3')
-  >>> diff(q1, Symbol('t'))
-  Derivative(q1(t), t)
-
-The 'dynamicsymbol' printing is not very clear above; we will also introduce a
-few other tools here. We can use ``mprint`` instead of print for
-non-interactive sessions. ::
-
-  >>> q1
-  q1(t)
-  >>> q1d = diff(q1, Symbol('t'))
-  >>> mprint(q1)
-  q1
-  >>> mprint(q1d)
-  q1'
-
-For interactive sessions use ``mechanics_printing``. There also exist analogs
-for SymPy's ``pprint``, ``mpprint``, and ``latex``, ``mlatex``. ::
-
-  >>> mechanics_printing()
-  >>> q1
-  q1
-  >>> q1d
-  q1'
-
-A 'dynamicsymbol' should be used to represent any time varying quantity in
-:mod:`mechanics`, whether it is a coordinate, varying position, or force.  The
-primary use of a 'dynamicsymbol' is for speeds and coordinates (of which there
-will be more discussion in the Kinematics Section of the documentation).
-
-Now we will define the orientation of our new frames with a 'dynamicsymbol',
-and can take derivatives and time derivatives with ease. Some examples follow.
-::
-
-  >>> N = ReferenceFrame('N')
-  >>> B = N.orientnew('B', 'Axis', [q1, N.x])
-  >>> (B.y*q2 + B.z).diff(q2, N)
-  B.y
-  >>> (B.y*q2 + B.z).dt(N)
-  (-q1' + q2')*B.y + q2*q1'*B.z
-
-Note that the output vectors are kept in the same frames that they were
-provided in. This remains true for vectors with components made of basis
-vectors from multiple frames: ::
-
-  >>> (B.y*q2 + B.z + q2*N.x).diff(q2, N)
-  B.y + N.x
-
-
-How Vectors are Coded in Mechanics
-----------------------------------
-
-What follows is a short description of how vectors are defined by the code in
-:mod:`mechanics`. It is provided for those who want to learn more about how
-this part of :mod:`mechanics` works, and does not need to be read to use this
-module; don't read it unless you want to learn how this module was implemented.
-
-Every ``Vector``'s main information is stored in the ``args`` attribute, which
-stores the three measure numbers for each basis vector in a frame, for every
-relevant frame. A vector does not exist in code until a ``ReferenceFrame``
-is created. At this point, the ``x``, ``y``, and ``z`` attributes of the
-reference frame are immutable ``Vector``'s which have measure numbers of
-[1,0,0], [0,1,0], and [0,0,1] associated with that ``ReferenceFrame``. Once
-these vectors are accessible, new vectors can be created by doing algebraic
-operations with the basis vectors. A vector can have components from multiple
-frames though. That is why ``args`` is a list; it has as many elements in the
-list as there are unique ``ReferenceFrames`` in its components, i.e. if there
-are ``A`` and ``B`` frame basis vectors in our new vector, ``args`` is of
-length 2; if it has ``A``, ``B``, and ``C`` frame basis vector, ``args`` is of
-length three.
-
-Each element in the ``args`` list is a 2-tuple; the first element is a SymPy
-``Matrix`` (this is where the measure numbers for each set of basis vectors are
-stored) and the second element is a ``ReferenceFrame`` to associate those
-measure numbers with.
-
-``ReferenceFrame`` stores a few things. First, it stores the name you supply it
-on creation (``name`` attribute). It also stores the direction cosine matrices,
-defined upon creation with the ``orientnew`` method, or calling the ``orient``
-method after creation. The direction cosine matrices are represented by SymPy's
-``Matrix``, and are part of a dictionary where the keys are the
-``ReferenceFrame`` and the value the ``Matrix``; these are set
-bi-directionally; in that when you orient ``A`` to ``N`` you are setting ``A``'s
-orientation dictionary to include ``N`` and its ``Matrix``, but also you also
-are setting ``N``'s orientation dictionary to include ``A`` and its ``Matrix``
-(that DCM being the transpose of the other).
diff --git a/dev-py3k/_sources/modules/physics/paulialgebra.txt b/dev-py3k/_sources/modules/physics/paulialgebra.txt
deleted file mode 100644
index a6b916dd88e..00000000000
--- a/dev-py3k/_sources/modules/physics/paulialgebra.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-=============
-Pauli Algebra
-=============
-
-.. automodule:: sympy.physics.paulialgebra
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/qho_1d.txt b/dev-py3k/_sources/modules/physics/qho_1d.txt
deleted file mode 100644
index 8890e194a1a..00000000000
--- a/dev-py3k/_sources/modules/physics/qho_1d.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-==================================
-Quantum Harmonic Oscillator in 1-D
-==================================
-
-.. automodule:: sympy.physics.qho_1d
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/quantum/anticommutator.txt b/dev-py3k/_sources/modules/physics/quantum/anticommutator.txt
deleted file mode 100644
index a9d7c490d7d..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/anticommutator.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-==============
-Anticommutator
-==============
-
-.. automodule:: sympy.physics.quantum.anticommutator
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/quantum/cartesian.txt b/dev-py3k/_sources/modules/physics/quantum/cartesian.txt
deleted file mode 100644
index 9652226def7..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/cartesian.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-==============================
-Cartesian Operators and States
-==============================
-
-.. automodule:: sympy.physics.quantum.cartesian
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/quantum/cg.txt b/dev-py3k/_sources/modules/physics/quantum/cg.txt
deleted file mode 100644
index 375369902db..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/cg.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-===========================
-Clebsch-Gordan Coefficients
-===========================
-
-.. automodule:: sympy.physics.quantum.cg
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/quantum/circuitplot.txt b/dev-py3k/_sources/modules/physics/quantum/circuitplot.txt
deleted file mode 100644
index 6a821b1146c..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/circuitplot.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-============
-Circuit Plot
-============
-
-.. automodule:: sympy.physics.quantum.circuitplot
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/quantum/commutator.txt b/dev-py3k/_sources/modules/physics/quantum/commutator.txt
deleted file mode 100644
index 532195cc8cc..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/commutator.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-==========
-Commutator
-==========
-
-.. automodule:: sympy.physics.quantum.commutator
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/quantum/constants.txt b/dev-py3k/_sources/modules/physics/quantum/constants.txt
deleted file mode 100644
index 28258e8e2bd..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/constants.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-=========
-Constants
-=========
-
-.. automodule:: sympy.physics.quantum.constants
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/quantum/dagger.txt b/dev-py3k/_sources/modules/physics/quantum/dagger.txt
deleted file mode 100644
index 8aeac00dceb..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/dagger.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-======
-Dagger
-======
-
-.. automodule:: sympy.physics.quantum.dagger
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/quantum/gate.txt b/dev-py3k/_sources/modules/physics/quantum/gate.txt
deleted file mode 100644
index fcbb0a3d448..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/gate.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-=====
-Gates
-=====
-
-.. automodule:: sympy.physics.quantum.gate
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/quantum/grover.txt b/dev-py3k/_sources/modules/physics/quantum/grover.txt
deleted file mode 100644
index acea8fc038f..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/grover.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-==================
-Grover's Algorithm
-==================
-
-.. automodule:: sympy.physics.quantum.grover
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/quantum/hilbert.txt b/dev-py3k/_sources/modules/physics/quantum/hilbert.txt
deleted file mode 100644
index 00cc7408591..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/hilbert.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-=============
-Hilbert Space
-=============
-
-.. automodule:: sympy.physics.quantum.hilbert
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/quantum/index.txt b/dev-py3k/_sources/modules/physics/quantum/index.txt
deleted file mode 100644
index 3af790543ec..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/index.txt
+++ /dev/null
@@ -1,58 +0,0 @@
-=================
-Quantum Mechanics
-=================
-
-.. topic:: Abstract
-
-   Contains Docstrings of Physics-Quantum module
-
-Quantum Functions
-=================
-
-.. toctree::
-    :maxdepth: 3
-
-    anticommutator.rst
-    cg.rst
-    commutator.rst
-    constants.rst
-    dagger.rst
-    innerproduct.rst
-    tensorproduct.rst
-
-States and Operators
-====================
-
-.. toctree::
-    :maxdepth: 3
-
-    cartesian.rst
-    hilbert.rst
-    operator.rst
-    operatorset.rst
-    qapply.rst
-    represent.rst
-    spin.rst
-    state.rst
-
-Quantum Computation
-===================
-
-.. toctree::
-    :maxdepth: 3
-
-    circuitplot.rst
-    gate.rst
-    grover.rst
-    qft.rst
-    qubit.rst
-    shor.rst
-
-Analytic Solutions
-==================
-
-
-.. toctree::
-    :maxdepth: 3
-
-    piab.rst
diff --git a/dev-py3k/_sources/modules/physics/quantum/innerproduct.txt b/dev-py3k/_sources/modules/physics/quantum/innerproduct.txt
deleted file mode 100644
index 6fd1b23c2ec..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/innerproduct.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-=============
-Inner Product
-=============
-
-.. automodule:: sympy.physics.quantum.innerproduct
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/quantum/operator.txt b/dev-py3k/_sources/modules/physics/quantum/operator.txt
deleted file mode 100644
index 813a21cf7c2..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/operator.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-========
-Operator
-========
-
-.. automodule:: sympy.physics.quantum.operator
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/quantum/operatorset.txt b/dev-py3k/_sources/modules/physics/quantum/operatorset.txt
deleted file mode 100644
index e6c1457d630..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/operatorset.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-===============================
-Operator/State Helper Functions
-===============================
-
-.. automodule:: sympy.physics.quantum.operatorset
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/quantum/piab.txt b/dev-py3k/_sources/modules/physics/quantum/piab.txt
deleted file mode 100644
index 16d1c5d8251..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/piab.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-=================
-Particle in a Box
-=================
-
-.. automodule:: sympy.physics.quantum.piab
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/quantum/qapply.txt b/dev-py3k/_sources/modules/physics/quantum/qapply.txt
deleted file mode 100644
index 12df45aba7a..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/qapply.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-======
-Qapply
-======
-
-.. automodule:: sympy.physics.quantum.qapply
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/quantum/qft.txt b/dev-py3k/_sources/modules/physics/quantum/qft.txt
deleted file mode 100644
index 99d1a857535..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/qft.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-===
-QFT
-===
-
-.. automodule:: sympy.physics.quantum.qft
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/quantum/qubit.txt b/dev-py3k/_sources/modules/physics/quantum/qubit.txt
deleted file mode 100644
index 65c02630d18..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/qubit.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-=====
-Qubit
-=====
-
-.. automodule:: sympy.physics.quantum.qubit
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/quantum/represent.txt b/dev-py3k/_sources/modules/physics/quantum/represent.txt
deleted file mode 100644
index 1b4e4869c0e..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/represent.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-=========
-Represent
-=========
-
-.. automodule:: sympy.physics.quantum.represent
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/quantum/shor.txt b/dev-py3k/_sources/modules/physics/quantum/shor.txt
deleted file mode 100644
index 29210295a3f..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/shor.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-================
-Shor's Algorithm
-================
-
-.. automodule:: sympy.physics.quantum.shor
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/quantum/spin.txt b/dev-py3k/_sources/modules/physics/quantum/spin.txt
deleted file mode 100644
index 80bedaae041..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/spin.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-====
-Spin
-====
-
-.. automodule:: sympy.physics.quantum.spin
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/quantum/state.txt b/dev-py3k/_sources/modules/physics/quantum/state.txt
deleted file mode 100644
index 4d72c5443d7..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/state.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-=====
-State
-=====
-
-.. automodule:: sympy.physics.quantum.state
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/quantum/tensorproduct.txt b/dev-py3k/_sources/modules/physics/quantum/tensorproduct.txt
deleted file mode 100644
index 4bb2d99ce43..00000000000
--- a/dev-py3k/_sources/modules/physics/quantum/tensorproduct.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-==============
-Tensor Product
-==============
-
-.. automodule:: sympy.physics.quantum.tensorproduct
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/secondquant.txt b/dev-py3k/_sources/modules/physics/secondquant.txt
deleted file mode 100644
index d007b77482e..00000000000
--- a/dev-py3k/_sources/modules/physics/secondquant.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-===================
-Second Quantization
-===================
-
-.. automodule:: sympy.physics.secondquant
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/sho.txt b/dev-py3k/_sources/modules/physics/sho.txt
deleted file mode 100644
index 80564c250bf..00000000000
--- a/dev-py3k/_sources/modules/physics/sho.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-==================================
-Quantum Harmonic Oscillator in 3-D
-==================================
-
-.. automodule:: sympy.physics.sho
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/units.txt b/dev-py3k/_sources/modules/physics/units.txt
deleted file mode 100644
index 529068bb798..00000000000
--- a/dev-py3k/_sources/modules/physics/units.txt
+++ /dev/null
@@ -1,69 +0,0 @@
-.. _physics-units:
-
-=====
-Units
-=====
-
-Introduction
-============
-
-This module provides around 200 predefined units that are commonly used in the
-sciences.  Additionally, it provides the :class:`Unit` class which allows you
-to define your own units.
-
-Examples
-========
-
-All examples in this tutorial are computable, so one can just copy and
-paste them into a Python shell and do something useful with them. All
-computations were done using the following setup::
-
-    >>> from sympy.physics.units import *
-
-Dimensionless quantities
-------------------------
-
-Provides variables for commonly written dimensionless (unit-less) quantities.
-
-    >>> 2*ten
-    20
-    >>> 20*percent
-    1/5
-    >>> 300*kilo*20*percent
-    60000
-    >>> nano*deg
-    pi/180000000000
-
-Base units
-----------
-
-The SI base units are defined variable name that are commonly used in written
-and verbal communication.  The singular abbreviated versions are what is used
-for display purposes, but the plural non-abbreviated versions are defined in
-case it helps readability.
-
-    >>> 5*meters
-    5*m
-    >>> milli*kilogram
-    kg/1000
-    >>> gram
-    kg/1000
-
-Note that British Imperial and U.S. customary units are not included.
-We strongly urge the use of SI units; only Myanmar (Burma), Liberia, and the
-United States have not officially accepted the SI system.
-
-
-Derived units
--------------
-
-Common SI derived units.
-
-    >>> joule
-    kg*m**2/s**2
-
-Docstring
-=========
-
-.. automodule:: sympy.physics.units
-   :members:
diff --git a/dev-py3k/_sources/modules/physics/wigner.txt b/dev-py3k/_sources/modules/physics/wigner.txt
deleted file mode 100644
index 8b6d98ef4d4..00000000000
--- a/dev-py3k/_sources/modules/physics/wigner.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-==============
-Wigner Symbols
-==============
-
-.. automodule:: sympy.physics.wigner
-   :members:
diff --git a/dev-py3k/_sources/modules/plotting.txt b/dev-py3k/_sources/modules/plotting.txt
deleted file mode 100644
index 402a8fbe1df..00000000000
--- a/dev-py3k/_sources/modules/plotting.txt
+++ /dev/null
@@ -1,287 +0,0 @@
-Plotting Module
-===============
-
-.. module:: sympy.plotting.plot
-
-Introduction
-------------
-
-The plotting module allows you to plot 2-dimensional and 3-dimensional plots.
-Presently the plots are rendered using ``Matplotlib`` as its backend. It is
-also possible to plot 2-dimensional plots using a ``TextBackend`` if you don't
-have ``Matplotlib``.
-
-The plotting module has the following functions:
-
-* plot: Plots 2D line plots.
-* plot_parametric: Plots 2D parametric plots.
-* plot_implicit: Plots 2D implicit and region plots.
-* plot3d: Plots 3D plots of functions in two variables.
-* plot3d_parametric_line: Plots 3D line plots, defined by a parameter.
-* plot3d_parametric_surface: Plots 3D parametric surface plots.
-
-The above functions are only for convenience and ease of use. It is possible to
-plot any plot by passing the corresponding ``Series`` class to ``Plot`` as
-argument.
-
-Plot Class
-----------
-
-.. autoclass:: sympy.plotting.plot.Plot
-   :members:
-
-Plotting Function Reference
----------------------------
-
-.. autofunction:: plot
-
-.. autofunction:: plot_parametric
-
-.. autofunction:: plot3d
-
-.. autofunction:: plot3d_parametric_line
-
-.. autofunction:: plot3d_parametric_surface
-
-.. autofunction:: sympy.plotting.plot_implicit.plot_implicit
-
-Series Classes
---------------
-
-.. autoclass:: sympy.plotting.plot.BaseSeries
-   :members:
-
-.. autoclass:: sympy.plotting.plot.Line2DBaseSeries
-   :members:
-
-.. autoclass:: sympy.plotting.plot.LineOver1DRangeSeries
-   :members:
-
-.. autoclass:: sympy.plotting.plot.Parametric2DLineSeries
-   :members:
-
-.. autoclass:: sympy.plotting.plot.Line3DBaseSeries
-   :members:
-
-.. autoclass:: sympy.plotting.plot.Parametric3DLineSeries
-   :members:
-
-.. autoclass:: sympy.plotting.plot.SurfaceBaseSeries
-   :members:
-
-.. autoclass:: sympy.plotting.plot.SurfaceOver2DRangeSeries
-   :members:
-
-.. autoclass:: sympy.plotting.plot.ParametricSurfaceSeries
-   :members:
-
-.. autoclass:: sympy.plotting.plot_implicit.ImplicitSeries
-   :members:
-
-
-
-
-Pyglet Plotting Module
-======================
-
-.. module:: sympy.plotting.pygletplot
-
-This is the documentation for the old plotting module that uses pyglet.
-This module has some limitations and is not actively developed anymore.
-For an alternative you can look at the new plotting module.
-
-The pyglet plotting module can do nice 2D and 3D plots that can be
-controlled by console commands as well as keyboard and mouse, with
-the only dependency being ``pyglet``.
-
-Here is the simplest usage:
-
-    >>> from sympy import var, Plot
-    >>> var('x y z')
-    >>> Plot(x*y**3-y*x**3)
-
-To see lots of plotting examples, see ``examples/pyglet_plotting.py`` and try running
-it in interactive mode (python -i plotting.py)::
-
-    $ python -i examples/pyglet_plotting.py
-
-And type for instance ``example(7)`` or ``example(11)``.
-
-See also the `Plotting Module <https://github.com/sympy/sympy/wiki/Plotting-Module>`_
-wiki page for screenshots.
-
-
-Plot Window Controls
---------------------
-
-======================   ========
-Camera                   Keys
-======================   ========
-Sensitivity Modifier     SHIFT
-Zoom                     R and F, Page Up and Down, Numpad + and -
-Rotate View X,Y axis     Arrow Keys, A,S,D,W, Numpad 4,6,8,2
-Rotate View Z axis       Q and E, Numpad 7 and 9
-Rotate Ordinate Z axis   Z and C, Numpad 1 and 3
-View XY                  F1
-View XZ                  F2
-View YZ                  F3
-View Perspective         F4
-Reset                    X, Numpad 5
-======================   ========
-
-======================   ========
-Axes                     Keys
-======================   ========
-Toggle Visible           F5
-Toggle Colors            F6
-======================   ========
-
-======================   ========
-Window                   Keys
-======================   ========
-Close                    ESCAPE
-Screenshot               F8
-======================   ========
-
-The mouse can be used to rotate, zoom, and translate by dragging the left, middle, and right mouse buttons respectively.
-
-Coordinate Modes
-----------------
-
-Plot supports several curvilinear coordinate modes, and they are independent
-for each plotted function. You can specify a coordinate mode explicitly with
-the 'mode' named argument, but it can be automatically determined for cartesian
-or parametric plots, and therefore must only be specified for polar,
-cylindrical, and spherical modes.
-
-Specifically, Plot(function arguments) and Plot.__setitem__(i, function
-arguments) (accessed using array-index syntax on the Plot instance) will
-interpret your arguments as a cartesian plot if you provide one function and a
-parametric plot if you provide two or three functions. Similarly, the arguments
-will be interpreted as a curve is one variable is used, and a surface if two
-are used.
-
-Supported mode names by number of variables:
-
-* 1 (curves): parametric, cartesian, polar
-* 2 (surfaces): parametric, cartesian, cylindrical, spherical
-
-::
-
-    >>> Plot(1, 'mode=spherical; color=zfade4')
-
-Note that function parameters are given as option strings of the form
-"key1=value1; key2 = value2" (spaces are truncated). Keyword arguments given
-directly to plot apply to the plot itself.
-
-Specifying Intervals for Variables
-----------------------------------
-
-The basic format for variable intervals is [var, min, max, steps]. However, the
-syntax is quite flexible, and arguments not specified are taken from the
-defaults for the current coordinate mode:
-
-    >>> Plot(x**2) # implies [x,-5,5,100]
-    >>> Plot(x**2, [], []) # [x,-1,1,40], [y,-1,1,40]
-    >>> Plot(x**2-y**2, [100], [100]) # [x,-1,1,100], [y,-1,1,100]
-    >>> Plot(x**2, [x,-13,13,100])
-    >>> Plot(x**2, [-13,13]) # [x,-13,13,100]
-    >>> Plot(x**2, [x,-13,13]) # [x,-13,13,100]
-    >>> Plot(1*x, [], [x], 'mode=cylindrical') # [unbound_theta,0,2*Pi,40], [x,-1,1,20]
-
-Using the Interactive Interface
--------------------------------
-::
-
-    >>> p = Plot(visible=False)
-    >>> f = x**2
-    >>> p[1] = f
-    >>> p[2] = f.diff(x)
-    >>> p[3] = f.diff(x).diff(x)
-    >>> p
-    [1]: x**2, 'mode=cartesian'
-    [2]: 2*x, 'mode=cartesian'
-    [3]: 2, 'mode=cartesian'
-    >>> p.show()
-    >>> p.clear()
-    >>> p
-    <blank plot>
-    >>> p[1] =  x**2+y**2
-    >>> p[1].style = 'solid'
-    >>> p[2] = -x**2-y**2
-    >>> p[2].style = 'wireframe'
-    >>> p[1].color = z, (0.4,0.4,0.9), (0.9,0.4,0.4)
-    >>> p[1].style = 'both'
-    >>> p[2].style = 'both'
-    >>> p.close()
-
-Using Custom Color Functions
-----------------------------
-
-The following code plots a saddle and color it by the magnitude of its gradient:
-
-    >>> fz = x**2-y**2
-    >>> Fx, Fy, Fz = fz.diff(x), fz.diff(y), 0
-    >>> p[1] = fz, 'style=solid'
-    >>> p[1].color = (Fx**2 + Fy**2 + Fz**2)**(0.5)
-
-The coloring algorithm works like this:
-
-#. Evaluate the color function(s) across the curve or surface.
-#. Find the minimum and maximum value of each component.
-#. Scale each component to the color gradient.
-
-When not specified explicitly, the default color gradient is
-f(0.0)=(0.4,0.4,0.4) -> f(1.0)=(0.9,0.9,0.9). In our case, everything is
-gray-scale because we have applied the default color gradient uniformly for
-each color component. When defining a color scheme in this way, you might want
-to supply a color gradient as well:
-
-    >>> p[1].color = (Fx**2 + Fy**2 + Fz**2)**(0.5), (0.1,0.1,0.9), (0.9,0.1,0.1)
-
-Here's a color gradient with four steps:
-
-    >>> gradient = [ 0.0, (0.1,0.1,0.9), 0.3, (0.1,0.9,0.1),
-    ...              0.7, (0.9,0.9,0.1), 1.0, (1.0,0.0,0.0) ]
-    >>> p[1].color = (Fx**2 + Fy**2 + Fz**2)**(0.5), gradient
-
-The other way to specify a color scheme is to give a separate function for each
-component r, g, b. With this syntax, the default color scheme is defined:
-
-    >>> p[1].color = z,y,x, (0.4,0.4,0.4), (0.9,0.9,0.9)
-
-This maps z->red, y->green, and x->blue. In some cases, you might prefer to use
-the following alternative syntax:
-
-    >>> p[1].color = z,(0.4,0.9), y,(0.4,0.9), x,(0.4,0.9)
-
-You can still use multi-step gradients with three-function color schemes.
-
-Plotting Geometric Entities
----------------------------
-
-The plotting module is capable of plotting some 2D geometric entities like
-line, circle and ellipse. The following example plots a circle and a tangent
-line at a random point on the ellipse.
-::
-
-    In [1]: p = Plot(axes='label_axes=True')
-
-    In [2]: c = Circle(Point(0,0), 1)
-
-    In [3]: t = c.tangent_line(c.random_point())
-
-    In [4]: p[0] = c
-
-    In [5]: p[1] = t
-
-Plotting polygons (Polygon, RegularPolygon, Triangle) are not supported
-directly. However a polygon can be plotted through a loop as follows.
-::
-
-    In [6]: p = Plot(axes='label_axes=True')
-
-    In [7]: t = RegularPolygon(Point(0,0), 1, 5)
-
-    In [8]: for i in range(len(t.sides)):
-       ....:    p[i] = t.sides[i]
diff --git a/dev-py3k/_sources/modules/polys/agca.txt b/dev-py3k/_sources/modules/polys/agca.txt
deleted file mode 100644
index 6af960c3e6b..00000000000
--- a/dev-py3k/_sources/modules/polys/agca.txt
+++ /dev/null
@@ -1,308 +0,0 @@
-.. _polys-agca:
-
-========================================================
-AGCA - Algebraic Geometry and Commutative Algebra Module
-========================================================
-
-Introduction
-============
-
-  Algebraic geometry is a mixture of the ideas of two Mediterranean
-  cultures. It is the superposition of the Arab science of the lightening
-  calculation of the solutions of equations over the Greek art of position
-  and shape.
-  This tapestry was originally woven on European soil and is still being refined
-  under the influence of international fashion. Algebraic geometry studies the
-  delicate balance between the geometrically plausible and the algebraically
-  possible.  Whenever one side of this mathematical teeter-totter outweighs the
-  other, one immediately loses interest and runs off in search of a more exciting
-  amusement.
-
-    George R. Kempf
-    1944 -- 2002
-
-
-Algebraic Geometry refers to the study of geometric problems via algebraic
-methods (and sometimes vice versa). While this is a rather old topic,
-algebraic geometry as understood today is very much a 20th century
-development. Building on ideas of e.g. Riemann and Dedekind, it was realized
-that there is an intimate connection between properties of the set of
-solutions of a system of polynomial equations (called an algebraic variety)
-and the behavior of the set of polynomial functions on that variety
-(called the coordinate ring).
-
-As in many geometric disciplines, we can distinguish between local and global
-questions (and methods). Local investigations in algebraic geometry are
-essentially equivalent to the study of certain rings, their ideals and modules.
-This latter topic is also called commutative algebra. It is the basic local
-toolset of algebraic geometers, in much the same way that differential analysis
-is the local toolset of differential geometers.
-
-A good conceptual introduction to commutative algebra is [Atiyah69]_. An
-introduction more geared towards computations, and the work most of the
-algorithms in this module are based on, is [Greuel2008]_.
-
-This module aims to eventually allow expression and solution of both
-local and global geometric problems, both in the classical case over a field
-and in the more modern arithmetic cases. So far, however, there is no geometric
-functionality at all. Currently the module only provides tools for computational
-commutative algebra over fields.
-
-All code examples assume::
-
-    >>> from sympy import *
-    >>> x, y, z = symbols('x,y,z')
-    >>> init_printing(use_unicode=True, wrap_line=False, no_global=True)
-
-Reference
-=========
-
-In this section we document the usage of the AGCA module. For convenience of
-the reader, some definitions and examples/explanations are interspersed.
-
-Base Rings
-----------
-
-Almost all computations in commutative algebra are relative to a "base ring".
-(For example, when asking questions about an ideal, the base ring is the ring
-the ideal is a subset of.) In principle all polys "domains" can be used as base
-rings. However, useful functionality is only implemented for polynomial rings
-over fields, and various localizations and quotients thereof.
-
-As demonstrated in
-the examples below, the most convenient method to create objects you are
-interested in is to build them up from the ground field, and then use the
-various methods to create new objects from old. For example, in order to
-create the local ring of the nodal cubic `y^2 = x^3` at the origin, over
-`\mathbb{Q}`, you do::
-
-    >>> lr = QQ.poly_ring(x, y, order="ilex") / [y**2 - x**3]
-    >>> lr
-    ℚ[x, y, order=ilex]
-    ───────────────────
-        ╱   3    2╲
-        ╲- x  + y ╱
-
-Note how the python list notation can be used as a short cut to express ideals.
-You can use the ``convert`` method to return ordinary sympy objects into
-objects understood by the AGCA module (although in many cases this will be done
-automatically -- for example the list was automatically turned into an ideal,
-and in the process the symbols `x` and `y` were automatically converted into
-other representations). For example::
-
-    >>> X, Y = lr.convert(x), lr.convert(y) ; X
-        ╱   3    2╲
-    x + ╲- x  + y ╱
-
-    >>> x**3 == y**2
-    False
-
-    >>> X**3 == Y**2
-    True
-
-When no localisation is needed, a more mathematical notation can be
-used. For example, let us create the coordinate ring of three-dimensional
-affine space `\mathbb{A}^3`::
-
-    >>> ar = QQ[x, y, z] ; ar
-    ℚ[x, y, z]
-
-For more details, refer to the following class documentation. Note that
-the base rings, being domains, are the main point of overlap between the
-AGCA module and the rest of the polys module. All domains are documented
-in detail in the polys reference, so we show here only an abridged version,
-with the methods most pertinent to the AGCA module.
-
-.. currentmodule:: sympy.polys.domains
-.. autoclass:: Ring
-   :members: free_module, ideal, quotient_ring
-   :noindex:
-
-.. autofunction:: PolynomialRing
-   :noindex:
-
-.. autoclass:: QuotientRing
-   :noindex:
-
-Modules, Ideals and their Elementary Properties
------------------------------------------------
-
-Let `A` be a ring. An `A`-module is a set `M`, together with two binary
-operations `+: M \times M \to M` and `\times: R \times M \to M` called
-addition and scalar multiplication. These are required to satisfy certain
-axioms, which can be found in e.g. [Atiyah69]_. In this way modules are
-a direct generalisation of both vector spaces (`A` being a field) and abelian
-groups (`A = \mathbb{Z}`). A *submodule* of the `A`-module `M` is a subset
-`N \subset M`, such that the binary operations restrict to `N`, and `N` becomes
-an `A`-module with these operations.
-
-The ring `A` itself has a natural `A`-module structure where addition and
-multiplication in the module coincide with addition and multiplication in
-the ring. This `A`-module is also written as `A`. An `A`-submodule of `A`
-is called an *ideal* of `A`. Ideals come up very naturally in algebraic
-geometry. More general modules can be seen as a technically convenient "elbow
-room" beyond talking only about ideals.
-
-If `M`, `N` are `A`-modules,
-then there is a natural (componentwise) `A`-module structure on `M \times N`.
-Similarly there are `A`-module structures on cartesian products of more
-components. (For the categorically inclined:
-the cartesian product of finitely many `A`-modules, with this
-`A`-module structure, is the finite biproduct in the category of all
-`A`-modules. With infinitely many components, it is the direct product
-(but the infinite direct sum has to be constructed differently).) As usual,
-repeated product of the `A`-module `M` is denoted `M, M^2, M^3 \dots`, or
-`M^I` for arbitrary index sets `I`.
-
-An `A`-module `M` is called *free* if it is isomorphic to the `A`-module
-`A^I` for some (not necessarily finite) index set `I` (refer to the next
-section for a definition of isomorphism). The cardinality of `I` is called
-the *rank* of `M`; one may prove this is well-defined.
-In general, the AGCA module only works with free modules of finite rank, and
-other closely related modules. The easiest way to create modules is to use
-member methods of the objects they are made up from. For example, let us create
-a free module of rank 4 over the coordinate ring of `\mathbb{A}^2`
-we created above, together with a submodule::
-
-    >>> F = ar.free_module(4) ; F
-              4
-    ℚ[x, y, z]
-
-    >>> S = F.submodule([1, x, x**2, x**3], [0, 1, 0, y]) ; S
-    ╱⎡       2   3⎤              ╲
-    ╲⎣1, x, x , x ⎦, [0, 1, 0, y]╱
-
-Note how python lists can be used as a short-cut notation for module
-elements (vectors). As usual, the ``convert`` method can be used to convert
-sympy/python objects into the internal AGCA representation (see detailed
-reference below).
-
-Here is the detailed documentation of the classes for modules, free modules,
-and submodules:
-
-.. currentmodule:: sympy.polys.agca.modules
-
-.. autoclass:: Module
-   :members:
-
-.. autoclass:: FreeModule
-   :members:
-
-.. autoclass:: SubModule
-   :members:
-
-
-Ideals are created very similarly to modules. For example, let's verify
-that the nodal cubic is indeed singular at the origin::
-
-    >>> I = lr.ideal(x, y)
-    >>> I == lr.ideal(x)
-    False
-
-    >>> I == lr.ideal(y)
-    False
-
-We are using here the fact that a curve is non-singular at a point if and only
-if the maximal ideal of the local ring is principal, and that in this case at
-least one of `x` and `y` must be generators.
-
-This is the detailed documentation of the class ideal. Please note that most
-of the methods regarding properties of ideals (primality etc.) are not yet
-implemented.
-
-.. currentmodule:: sympy.polys.agca.ideals
-
-.. autoclass:: Ideal
-   :members:
-
-
-If `M` is an `A`-module and `N` is an `A`-submodule, we can define two elements
-`x` and `y` of `M` to be equivalent if `x - y \in N`. The set of equivalence
-classes is written `M/N`, and has a natural `A`-module structure. This is
-called the quotient module of `M` by `N`. If `K` is a submodule of `M`
-containing `N`, then `K/N` is in a natural way a submodule of `M/N`. Such a
-module is called a subquotient. Here is the documentation of quotient and
-subquotient modules:
-
-.. currentmodule:: sympy.polys.agca.modules
-
-.. autoclass:: QuotientModule
-   :members:
-
-.. autoclass:: SubQuotientModule
-   :members:
-
-Module Homomorphisms and Syzygies
----------------------------------
-
-Let `M` and `N` be `A`-modules. A mapping `f: M \to N` satisfying various
-obvious properties (see [Atiyah69]_) is called an `A`-module homomorphism.
-In this case `M` is called the *domain* and *N* the *codomain*. The
-set `\{x \in M | f(x) = 0\}` is called the *kernel* `ker(f)`, whereas the
-set `\{f(x) | x \in M\}` is called the *image* `im(f)`.
-The kernel is a submodule of `M`, the image is a submodule of `N`.
-The homomorphism `f` is injective if and only if `ker(f) = 0` and surjective
-if and only if `im(f) = N`.
-A bijective homomorphism is called an *isomorphism*. Equivalently, `ker(f) = 0`
-and `im(f) = N`. (A related notion, which currently has no special name in
-the AGCA module, is that of the *cokernel*, `coker(f) = N/im(f)`.)
-
-Suppose now `M` is an `A`-module. `M` is called *finitely generated* if there
-exists a surjective homomorphism `A^n \to M` for some `n`. If such a morphism
-`f` is chosen, the images of the standard basis of `A^n` are called the
-*generators* of `M`. The module `ker(f)` is called *syzygy module* with respect
-to the generators. A module is called *finitely presented* if it is finitely
-generated with a finitely generated syzygy module. The class of finitely
-presented modules is essentially the largest class we can hope to be able to
-meaningfully compute in.
-
-It is an important theorem that, for all the rings we are considering,
-all submodules of finitely generated modules are finitely generated, and hence
-finitely generated and finitely presented modules are the same.
-
-The notion of syzygies, while it may first seem rather abstract, is actually
-very computational. This is because there exist (fairly easy) algorithms for
-computing them, and more general questions (kernels, intersections, ...) are
-often reduced to syzygy computation.
-
-Let us say a few words about the definition of homomorphisms in the AGCA
-module. Suppose first that `f : M \to N` is an arbitrary morphism of
-`A`-modules. Then if `K` is a submodule of `M`, `f` naturally defines a new
-homomorphism `g: K \to N` (via `g(x) = f(x)`), called the *restriction* of
-`f` to `K`. If now `K` contained in the kernel of
-`f`, then moreover `f` defines in a natural homomorphism `g: M/K \to N`
-(same formula as above!), and we say that `f` *descends* to `M/K`.
-Similarly, if `L` is a submodule of `N`, there is a natural homomorphism
-`g: M \to N/L`, we say that `g` *factors* through `f`. Finally, if now `L`
-contains the image of `f`, then there is a natural homomorphism `g: M \to L`
-(defined, again, by the same formula), and we say `g` is obtained from `f`
-by restriction of codomain. Observe also that each of these four operations
-is reversible, in the sense that given `g`, one can always (non-uniquely)
-find `f` such that `g` is obtained from `f` in the above way.
-
-Note that all modules implemented in AGCA are obtained from free modules by
-taking a succession of submodules and quotients. Hence, in order to explain
-how to define a homomorphism between arbitrary modules, in light of the above,
-we need only explain how to define homomorphisms of free modules. But,
-essentially by the definition of free module, a homomorphism from a free module
-`A^n` to any module `M` is precisely the same as giving `n` elements of `M`
-(the images of the standard basis), and giving an element of a free module
-`A^m` is precisely the same as giving `m` elements of `A`. Hence a
-homomorphism of free modules `A^n \to A^m` can be specified via a matrix,
-entirely analogously to the case of vector spaces.
-
-The functions ``restrict_domain`` etc. of the class ``Homomorphism`` can be
-used
-to carry out the operations described above, and homomorphisms of free modules
-can in principle be instantiated by hand. Since these operations are so common,
-there is a convenience function ``homomorphism`` to define a homomorphism
-between arbitrary modules via the method outlined above. It is essentially
-the only way homomorphisms need ever be created by the user.
-
-.. currentmodule:: sympy.polys.agca.homomorphisms
-.. autofunction:: homomorphism
-
-Finally, here is the detailed reference of the actual homomorphism class:
-
-.. autoclass:: ModuleHomomorphism
-   :members:
diff --git a/dev-py3k/_sources/modules/polys/basics.txt b/dev-py3k/_sources/modules/polys/basics.txt
deleted file mode 100644
index 5c87e9a615b..00000000000
--- a/dev-py3k/_sources/modules/polys/basics.txt
+++ /dev/null
@@ -1,226 +0,0 @@
-.. _polys-basics:
-
-=================================
-Basic functionality of the module
-=================================
-
-Introduction
-============
-
-This tutorial tries to give an overview of the functionality concerning
-polynomials within SymPy. All code examples assume::
-
-    >>> from sympy import *
-    >>> x, y, z = symbols('x,y,z')
-    >>> init_printing(use_unicode=False, wrap_line=False, no_global=True)
-
-Basic functionality
-===================
-
-These functions provide different algorithms dealing with polynomials in the
-form of SymPy expression, like symbols, sums etc.
-
-Division
---------
-
-The function :func:`div` provides division of polynomials with remainder.
-That is, for polynomials ``f`` and ``g``, it computes ``q`` and ``r``, such
-that `f = g \cdot q + r` and `\deg(r) < q`. For polynomials in one variables
-with coefficients in a field, say, the rational numbers, ``q`` and ``r`` are
-uniquely defined this way::
-
-    >>> f = 5*x**2 + 10*x + 3
-    >>> g = 2*x + 2
-
-    >>> q, r = div(f, g, domain='QQ')
-    >>> q
-    5*x   5
-    --- + -
-     2    2
-    >>> r
-    -2
-    >>> (q*g + r).expand()
-       2
-    5*x  + 10*x + 3
-
-As you can see, ``q`` has a non-integer coefficient. If you want to do division
-only in the ring of polynomials with integer coefficients, you can specify an
-additional parameter::
-
-    >>> q, r = div(f, g, domain='ZZ')
-    >>> q
-    0
-    >>> r
-       2
-    5*x  + 10*x + 3
-
-But be warned, that this ring is no longer Euclidean and that the degree of the
-remainder doesn't need to be smaller than that of ``f``. Since 2 doesn't divide 5,
-`2 x` doesn't divide `5 x^2`, even if the degree is smaller. But::
-
-    >>> g = 5*x + 1
-
-    >>> q, r = div(f, g, domain='ZZ')
-    >>> q
-    x
-    >>> r
-    9*x + 3
-    >>> (q*g + r).expand()
-       2
-    5*x  + 10*x + 3
-
-This also works for polynomials with multiple variables::
-
-    >>> f = x*y + y*z
-    >>> g = 3*x + 3*z
-
-    >>> q, r = div(f, g, domain='QQ')
-    >>> q
-    y
-    -
-    3
-    >>> r
-    0
-
-In the last examples, all of the three variables ``x``, ``y`` and ``z`` are
-assumed to be variables of the polynomials. But if you have some unrelated
-constant as coefficient, you can specify the variables explicitly::
-
-    >>> a, b, c = symbols('a,b,c')
-    >>> f = a*x**2 + b*x + c
-    >>> g = 3*x + 2
-    >>> q, r = div(f, g, domain='QQ')
-    >>> q
-    a*x   2*a   b
-    --- - --- + -
-     3     9    3
-
-    >>> r
-    4*a   2*b
-    --- - --- + c
-     9     3
-
-GCD and LCM
------------
-
-With division, there is also the computation of the greatest common divisor and
-the least common multiple.
-
-When the polynomials have integer coefficients, the contents' gcd is also
-considered::
-
-    >>> f = (12*x + 12)*x
-    >>> g = 16*x**2
-    >>> gcd(f, g)
-    4*x
-
-But if the polynomials have rational coefficients, then the returned polynomial is
-monic::
-
-    >>> f = 3*x**2/2
-    >>> g = 9*x/4
-    >>> gcd(f, g)
-    x
-
-It also works with multiple variables. In this case, the variables are ordered
-alphabetically, be default, which has influence on the leading coefficient::
-
-    >>> f = x*y/2 + y**2
-    >>> g = 3*x + 6*y
-
-    >>> gcd(f, g)
-    x + 2*y
-
-The lcm is connected with the gcd and one can be computed using the other::
-
-    >>> f = x*y**2 + x**2*y
-    >>> g = x**2*y**2
-    >>> gcd(f, g)
-    x*y
-    >>> lcm(f, g)
-     3  2    2  3
-    x *y  + x *y
-    >>> (f*g).expand()
-     4  3    3  4
-    x *y  + x *y
-    >>> (gcd(f, g, x, y)*lcm(f, g, x, y)).expand()
-     4  3    3  4
-    x *y  + x *y
-
-Square-free factorization
--------------------------
-
-The square-free factorization of a univariate polynomial is the product of all
-factors (not necessarily irreducible) of degree 1, 2 etc.::
-
-    >>> f = 2*x**2 + 5*x**3 + 4*x**4 + x**5
-
-    >>> sqf_list(f)
-    (1, [(x + 2, 1), (x, 2), (x + 1, 2)])
-
-    >>> sqf(f)
-     2        2
-    x *(x + 1) *(x + 2)
-
-Factorization
--------------
-
-This function provides factorization of univariate and multivariate polynomials
-with rational coefficients::
-
-    >>> factor(x**4/2 + 5*x**3/12 - x**2/3)
-     2
-    x *(2*x - 1)*(3*x + 4)
-    ----------------------
-              12
-
-    >>> factor(x**2 + 4*x*y + 4*y**2)
-             2
-    (x + 2*y)
-
-Groebner bases
---------------
-
-Buchberger's algorithm is implemented, supporting various monomial orders::
-
-    >>> groebner([x**2 + 1, y**4*x + x**3], x, y, order='lex')
-                 /[ 2       4    ]                            \
-    GroebnerBasis\[x  + 1, y  - 1], x, y, domain=ZZ, order=lex/
-
-
-    >>> groebner([x**2 + 1, y**4*x + x**3, x*y*z**3], x, y, z, order='grevlex')
-                 /[ 4       3   2    ]                                   \
-    GroebnerBasis\[y  - 1, z , x  + 1], x, y, z, domain=ZZ, order=grevlex/
-
-Solving Equations
------------------
-
-We have (incomplete) methods to find the complex or even symbolic roots of
-polynomials and to solve some systems of polynomial equations::
-
-    >>> from sympy import roots, solve_poly_system
-
-    >>> solve(x**3 + 2*x + 3, x)
-               ____          ____
-         1   \/ 11 *I  1   \/ 11 *I
-    [-1, - - --------, - + --------]
-         2      2      2      2
-
-    >>> p = Symbol('p')
-    >>> q = Symbol('q')
-
-    >>> sorted(solve(x**2 + p*x + q, x))
-              __________           __________
-             /  2                 /  2
-       p   \/  p  - 4*q     p   \/  p  - 4*q
-    [- - + -------------, - - - -------------]
-       2         2          2         2
-
-    >>> solve_poly_system([y - x, x - 5], x, y)
-    [(5, 5)]
-
-    >>> solve_poly_system([y**2 - x**3 + 1, y*x], x, y)
-                                       ___                 ___
-                                 1   \/ 3 *I         1   \/ 3 *I
-    [(0, I), (0, -I), (1, 0), (- - + -------, 0), (- - - -------, 0)]
-                                 2      2            2      2
diff --git a/dev-py3k/_sources/modules/polys/index.txt b/dev-py3k/_sources/modules/polys/index.txt
deleted file mode 100644
index be212828af3..00000000000
--- a/dev-py3k/_sources/modules/polys/index.txt
+++ /dev/null
@@ -1,28 +0,0 @@
-.. _polys-docs:
-
-===============================
-Polynomials Manipulation Module
-===============================
-
-Computations with polynomials are at the core of computer algebra and
-having a fast and robust polynomials manipulation module is a key for
-building a powerful symbolic manipulation system. SymPy has a dedicated
-module :mod:`sympy.polys` for computing in polynomial algebras over
-various coefficient domains.
-
-There is a vast number of methods implemented, ranging from simple tools
-like polynomial division, to advanced concepts including Gröbner bases
-and multivariate factorization over algebraic number domains.
-
-Contents
-========
-
-.. toctree::
-    :maxdepth: 3
-
-    basics.rst
-    wester.rst
-    reference.rst
-    agca.rst
-    internals.rst
-    literature.rst
diff --git a/dev-py3k/_sources/modules/polys/internals.txt b/dev-py3k/_sources/modules/polys/internals.txt
deleted file mode 100644
index c18fe7c2caf..00000000000
--- a/dev-py3k/_sources/modules/polys/internals.txt
+++ /dev/null
@@ -1,551 +0,0 @@
-.. _polys-internals:
-
-===============================================
-Internals of the Polynomial Manipulation Module
-===============================================
-
-The implementation of the polynomials module is structured internally in
-"levels". There are four levels, called L0, L1, L2 and L3. The levels three
-and four contain the user-facing functionality and were described in the
-previous section. This section focuses on levels zero and one.
-
-Level zero provides core polynomial manipulation functionality with C-like,
-low-level interfaces. Level one wraps this low-level functionality into object
-oriented structures. These are *not* the classes seen by the user, but rather
-classes used internally throughout the polys module.
-
-There is one additional complication in the implementation. This comes from the
-fact that all polynomial manipulations are relative to a *ground domain*. For
-example, when factoring a polynomial like `x^{10} - 1`, one has to decide what
-ring the coefficients are supposed to belong to, or less trivially, what
-coefficients are allowed to appear in the factorization. This choice of
-coefficients is called a ground domain. Typical choices include the integers
-`\mathbb{Z}`, the rational numbers `\mathbb{Q}` or various related rings and
-fields. But it is perfectly legitimate (although in this case uninteresting)
-to factorize over polynomial rings such as `k[Y]`, where `k` is some fixed
-field.
-
-Thus the polynomial manipulation algorithms (both
-complicated ones like factoring, and simpler ones like addition or
-multiplication) have to rely on other code to manipulate the coefficients.
-In the polynomial manipulation module, such code is encapsulated in so-called
-"domains". A domain is basically a factory object: it takes various
-representations of data, and converts them into objects with unified interface.
-Every object created by a domain has to implement the arithmetic operations
-`+`, `-` and `\times`. Other operations are accessed through the domain, e.g.
-as in ``ZZ.quo(ZZ(4), ZZ(2))``.
-
-Note that there is some amount of *circularity*: the polynomial ring domains
-use the level one classes, the level one classes use the level zero functions,
-and level zero functions use domains. It is possible, in principle, but not in
-the current implementation, to work in rings like `k[X][Y]`. This would create
-even more layers. For this reason, working in the isomorphic ring `k[X, Y]`
-is preferred.
-
-Domains
-=======
-
-.. currentmodule:: sympy.polys.domains
-
-Here we document the various implemented ground domains. There are three
-types: abstract domains, concrete domains, and "implementation domains".
-Abstract domains cannot be (usefully) instantiated at all, and just collect
-together functionality shared by many other domains. Concrete domains are
-those meant to be instantiated and used in the polynomial manipulation
-algorithms. In some cases, there are various possible ways to implement the
-data type the domain provides. For example, depending on what libraries are
-available on the system, the integers are implemented either using the python
-built-in integers, or using gmpy. Note that various aliases are created
-automatically depending on the libraries available. As such e.g. ``ZZ`` always
-refers to the most efficient implementation of the integer ring available.
-
-Abstract Domains
-****************
-
-.. autoclass:: Domain
-   :members:
-
-.. autoclass:: Field
-   :members:
-
-.. autoclass:: Ring
-   :members:
-
-.. autoclass:: SimpleDomain
-   :members:
-
-.. autoclass:: CompositeDomain
-   :members:
-
-Concrete Domains
-****************
-
-.. autoclass:: FiniteField
-   :members:
-
-.. autoclass:: IntegerRing
-   :members:
-
-.. autoclass:: PolynomialRing
-   :members:
-
-.. autoclass:: RationalField
-   :members:
-
-.. autoclass:: AlgebraicField
-   :members:
-
-.. autoclass:: FractionField
-   :members:
-
-.. autoclass:: RealDomain
-   :members:
-
-.. autoclass:: ExpressionDomain
-   :members:
-
-Implementation Domains
-**********************
-
-.. autoclass:: PythonFiniteField
-.. autoclass:: SymPyFiniteField
-.. autoclass:: GMPYFiniteField
-
-.. autoclass:: PythonIntegerRing
-.. autoclass:: SymPyIntegerRing
-.. autoclass:: GMPYIntegerRing
-
-.. autoclass:: PythonRationalField
-.. autoclass:: SymPyRationalField
-.. autoclass:: GMPYRationalField
-
-.. autoclass:: MPmathRealDomain
-
-
-Level One
-=========
-
-.. currentmodule:: sympy.polys.polyclasses
-
-.. autoclass:: DMP
-   :members:
-
-.. autoclass:: DMF
-   :members:
-
-.. autoclass:: ANP
-   :members:
-
-Level Zero
-==========
-
-Level zero contains the bulk code of the polynomial manipulation module.
-
-Manipulation of dense, multivariate polynomials
-***********************************************
-
-These functions can be used to manipulate polynomials in `K[X_0, \dots, X_u]`.
-Functions for manipulating multivariate polynomials in the dense representation
-have the prefix ``dmp_``. Functions which only apply to univariate polynomials
-(i.e. `u = 0`)
-have the prefix ``dup__``. The ground domain `K` has to be passed explicitly.
-For many multivariate polynomial manipulation functions also the level `u`,
-i.e. the number of generators minus one, has to be passed.
-(Note that, in many cases, ``dup_`` versions of functions are available, which
-may be slightly more efficient.)
-
-**Basic manipulation:**
-
-.. currentmodule:: sympy.polys.densebasic
-
-.. autofunction:: dmp_LC
-.. autofunction:: dmp_TC
-.. autofunction:: dmp_ground_LC
-.. autofunction:: dmp_ground_TC
-.. autofunction:: dmp_true_LT
-.. autofunction:: dmp_degree
-.. autofunction:: dmp_degree_in
-.. autofunction:: dmp_degree_list
-.. autofunction:: dmp_strip
-.. autofunction:: dmp_validate
-.. autofunction:: dup_reverse
-.. autofunction:: dmp_copy
-.. autofunction:: dmp_to_tuple
-.. autofunction:: dmp_normal
-.. autofunction:: dmp_convert
-.. autofunction:: dmp_from_sympy
-.. autofunction:: dmp_nth
-.. autofunction:: dmp_ground_nth
-.. autofunction:: dmp_zero_p
-.. autofunction:: dmp_zero
-.. autofunction:: dmp_one_p
-.. autofunction:: dmp_one
-.. autofunction:: dmp_ground_p
-.. autofunction:: dmp_ground
-.. autofunction:: dmp_zeros
-.. autofunction:: dmp_grounds
-.. autofunction:: dmp_negative_p
-.. autofunction:: dmp_positive_p
-.. autofunction:: dmp_from_dict
-.. autofunction:: dmp_to_dict
-.. autofunction:: dmp_swap
-.. autofunction:: dmp_permute
-.. autofunction:: dmp_nest
-.. autofunction:: dmp_raise
-.. autofunction:: dmp_deflate
-.. autofunction:: dmp_multi_deflate
-.. autofunction:: dmp_inflate
-.. autofunction:: dmp_exclude
-.. autofunction:: dmp_include
-.. autofunction:: dmp_inject
-.. autofunction:: dmp_eject
-.. autofunction:: dmp_terms_gcd
-.. autofunction:: dmp_list_terms
-.. autofunction:: dmp_apply_pairs
-.. autofunction:: dmp_slice
-.. autofunction:: dup_random
-
-**Arithmetic operations:**
-
-.. currentmodule:: sympy.polys.densearith
-
-.. autofunction:: dmp_add_term
-.. autofunction:: dmp_sub_term
-.. autofunction:: dmp_mul_term
-.. autofunction:: dmp_add_ground
-.. autofunction:: dmp_sub_ground
-.. autofunction:: dmp_mul_ground
-.. autofunction:: dmp_quo_ground
-.. autofunction:: dmp_exquo_ground
-.. autofunction:: dup_lshift
-.. autofunction:: dup_rshift
-.. autofunction:: dmp_abs
-.. autofunction:: dmp_neg
-.. autofunction:: dmp_add
-.. autofunction:: dmp_sub
-.. autofunction:: dmp_add_mul
-.. autofunction:: dmp_sub_mul
-.. autofunction:: dmp_mul
-.. autofunction:: dmp_sqr
-.. autofunction:: dmp_pow
-.. autofunction:: dmp_pdiv
-.. autofunction:: dmp_prem
-.. autofunction:: dmp_pquo
-.. autofunction:: dmp_pexquo
-.. autofunction:: dmp_rr_div
-.. autofunction:: dmp_ff_div
-.. autofunction:: dmp_div
-.. autofunction:: dmp_rem
-.. autofunction:: dmp_quo
-.. autofunction:: dmp_exquo
-.. autofunction:: dmp_max_norm
-.. autofunction:: dmp_l1_norm
-.. autofunction:: dmp_expand
-
-**Further tools:**
-
-.. currentmodule:: sympy.polys.densetools
-
-.. autofunction:: dmp_integrate
-.. autofunction:: dmp_integrate_in
-.. autofunction:: dmp_diff
-.. autofunction:: dmp_diff_in
-.. autofunction:: dmp_eval
-.. autofunction:: dmp_eval_in
-.. autofunction:: dmp_eval_tail
-.. autofunction:: dmp_diff_eval_in
-.. autofunction:: dmp_trunc
-.. autofunction:: dmp_ground_trunc
-.. autofunction:: dup_monic
-.. autofunction:: dmp_ground_monic
-.. autofunction:: dup_content
-.. autofunction:: dmp_ground_content
-.. autofunction:: dup_primitive
-.. autofunction:: dmp_ground_primitive
-.. autofunction:: dup_extract
-.. autofunction:: dmp_ground_extract
-.. autofunction:: dup_real_imag
-.. autofunction:: dup_mirror
-.. autofunction:: dup_scale
-.. autofunction:: dup_shift
-.. autofunction:: dup_transform
-.. autofunction:: dmp_compose
-.. autofunction:: dup_decompose
-.. autofunction:: dmp_lift
-.. autofunction:: dup_sign_variations
-.. autofunction:: dmp_clear_denoms
-.. autofunction:: dmp_revert
-
-Manipulation of dense, univariate polynomials with finite field coefficients
-****************************************************************************
-.. currentmodule:: sympy.polys.galoistools
-
-Functions in this module carry the prefix ``gf_``, referring to the classical
-name "Galois Fields" for finite fields. Note that many polynomial
-factorization algorithms work by reduction to the finite field case, so having
-special implementations for this case is justified both by performance, and by
-the necessity of certain methods which do not even make sense over general
-fields.
-
-.. autofunction:: gf_crt
-.. autofunction:: gf_crt1
-.. autofunction:: gf_crt2
-.. autofunction:: gf_int
-.. autofunction:: gf_degree
-.. autofunction:: gf_LC
-.. autofunction:: gf_TC
-.. autofunction:: gf_strip
-.. autofunction:: gf_trunc
-.. autofunction:: gf_normal
-.. autofunction:: gf_convert
-.. autofunction:: gf_from_dict
-.. autofunction:: gf_to_dict
-.. autofunction:: gf_from_int_poly
-.. autofunction:: gf_to_int_poly
-.. autofunction:: gf_neg
-.. autofunction:: gf_add_ground
-.. autofunction:: gf_sub_ground
-.. autofunction:: gf_mul_ground
-.. autofunction:: gf_quo_ground
-.. autofunction:: gf_add
-.. autofunction:: gf_sub
-.. autofunction:: gf_mul
-.. autofunction:: gf_sqr
-.. autofunction:: gf_add_mul
-.. autofunction:: gf_sub_mul
-.. autofunction:: gf_expand
-.. autofunction:: gf_div
-.. autofunction:: gf_rem
-.. autofunction:: gf_quo
-.. autofunction:: gf_exquo
-.. autofunction:: gf_lshift
-.. autofunction:: gf_rshift
-.. autofunction:: gf_pow
-.. autofunction:: gf_pow_mod
-.. autofunction:: gf_gcd
-.. autofunction:: gf_lcm
-.. autofunction:: gf_cofactors
-.. autofunction:: gf_gcdex
-.. autofunction:: gf_monic
-.. autofunction:: gf_diff
-.. autofunction:: gf_eval
-.. autofunction:: gf_multi_eval
-.. autofunction:: gf_compose
-.. autofunction:: gf_compose_mod
-.. autofunction:: gf_trace_map
-.. autofunction:: gf_random
-.. autofunction:: gf_irreducible
-.. autofunction:: gf_irreducible_p
-.. autofunction:: gf_sqf_p
-.. autofunction:: gf_sqf_part
-.. autofunction:: gf_sqf_list
-.. autofunction:: gf_Qmatrix
-.. autofunction:: gf_Qbasis
-.. autofunction:: gf_berlekamp
-.. autofunction:: gf_zassenhaus
-.. autofunction:: gf_shoup
-.. autofunction:: gf_factor_sqf
-.. autofunction:: gf_factor
-.. autofunction:: gf_value
-.. autofunction:: gf_csolve
-
-Manipulation of sparse, distributed polynomials and vectors
-***********************************************************
-
-Dense representations quickly require infeasible amounts of storage and
-computation time if the number of variables increases. For this reason,
-there is code to manipulate polynomials in a *sparse* representation. These
-functions carry the ``sdp_`` prefix. As in the ``sdm_`` case, the ground domain
-`K` and level `u` often have to be passed explicitly. Furthermore, sparse
-representations are relative to a monomial order `O`, which also has to be
-passed.
-
-.. currentmodule:: sympy.polys.distributedpolys
-.. autofunction:: sdp_LC
-.. autofunction:: sdp_LM
-.. autofunction:: sdp_LT
-.. autofunction:: sdp_del_LT
-.. autofunction:: sdp_monoms
-.. autofunction:: sdp_sort
-.. autofunction:: sdp_strip
-.. autofunction:: sdp_normal
-.. autofunction:: sdp_from_dict
-.. autofunction:: sdp_to_dict
-.. autofunction:: sdp_indep_p
-.. autofunction:: sdp_one_p
-.. autofunction:: sdp_one
-.. autofunction:: sdp_term_p
-.. autofunction:: sdp_abs
-.. autofunction:: sdp_neg
-.. autofunction:: sdp_add_term
-.. autofunction:: sdp_sub_term
-.. autofunction:: sdp_mul_term
-.. autofunction:: sdp_add
-.. autofunction:: sdp_sub
-.. autofunction:: sdp_mul
-.. autofunction:: sdp_sqr
-.. autofunction:: sdp_pow
-.. autofunction:: sdp_monic
-.. autofunction:: sdp_content
-.. autofunction:: sdp_primitive
-.. autofunction:: sdp_div
-.. autofunction:: sdp_rem
-.. autofunction:: sdp_quo
-.. autofunction:: sdp_exquo
-.. autofunction:: sdp_lcm
-.. autofunction:: sdp_gcd
-
-In commutative algebra, one often studies not only polynomials, but also
-*modules* over polynomial rings. The polynomial manipulation module provides
-rudimentary low-level support for finitely generated free modules. This is
-mainly used for Groebner basis computations (see there), so manipulation
-functions are only provided to the extend needed. They carry the prefix
-``sdm_``. Note that in examples, the generators of the free module are called
-`f_1, f_2, \dots`.
-
-.. currentmodule:: sympy.polys.distributedmodules
-
-.. autofunction:: sdm_monomial_mul
-.. autofunction:: sdm_monomial_deg
-.. autofunction:: sdm_monomial_divides
-.. autofunction:: sdm_LC
-.. autofunction:: sdm_to_dict
-.. autofunction:: sdm_from_dict
-.. autofunction:: sdm_add
-.. autofunction:: sdm_LM
-.. autofunction:: sdm_LT
-.. autofunction:: sdm_mul_term
-.. autofunction:: sdm_zero
-.. autofunction:: sdm_deg
-.. autofunction:: sdm_from_vector
-.. autofunction:: sdm_to_vector
-
-Polynomial factorization algorithms
-***********************************
-
-Many variants of Euclid's algorithm:
-
-.. currentmodule:: sympy.polys.euclidtools
-
-.. autofunction:: dmp_half_gcdex
-.. autofunction:: dmp_gcdex
-.. autofunction:: dmp_invert
-.. autofunction:: dmp_euclidean_prs
-.. autofunction:: dmp_primitive_prs
-.. autofunction:: dmp_inner_subresultants
-.. autofunction:: dmp_subresultants
-.. autofunction:: dmp_prs_resultant
-.. autofunction:: dmp_zz_modular_resultant
-.. autofunction:: dmp_zz_collins_resultant
-.. autofunction:: dmp_qq_collins_resultant
-.. autofunction:: dmp_resultant
-.. autofunction:: dmp_discriminant
-.. autofunction:: dmp_rr_prs_gcd
-.. autofunction:: dmp_ff_prs_gcd
-.. autofunction:: dmp_zz_heu_gcd
-.. autofunction:: dmp_qq_heu_gcd
-.. autofunction:: dmp_inner_gcd
-.. autofunction:: dmp_gcd
-.. autofunction:: dmp_lcm
-.. autofunction:: dmp_content
-.. autofunction:: dmp_primitive
-.. autofunction:: dmp_cancel
-
-Polynomial factorization in characteristic zero:
-
-.. currentmodule:: sympy.polys.factortools
-
-.. autofunction:: dmp_trial_division
-.. autofunction:: dmp_zz_mignotte_bound
-.. autofunction:: dup_zz_hensel_step
-.. autofunction:: dup_zz_hensel_lift
-.. autofunction:: dup_zz_zassenhaus
-.. autofunction:: dup_zz_irreducible_p
-.. autofunction:: dup_cyclotomic_p
-.. autofunction:: dup_zz_cyclotomic_poly
-.. autofunction:: dup_zz_cyclotomic_factor
-.. autofunction:: dup_zz_factor_sqf
-.. autofunction:: dup_zz_factor
-.. autofunction:: dmp_zz_wang_non_divisors
-.. autofunction:: dmp_zz_wang_test_points
-.. autofunction:: dmp_zz_wang_lead_coeffs
-.. autofunction:: dmp_zz_diophantine
-.. autofunction:: dmp_zz_wang_hensel_lifting
-.. autofunction:: dmp_zz_wang
-.. autofunction:: dmp_zz_factor
-.. autofunction:: dmp_ext_factor
-.. autofunction:: dup_gf_factor
-.. autofunction:: dmp_factor_list
-.. autofunction:: dmp_factor_list_include
-.. autofunction:: dmp_irreducible_p
-
-Groebner basis algorithms
-*************************
-
-Groebner bases can be used to answer many problems in computational
-commutative algebra. Their computation in rather complicated, and very
-performance-sensitive. We present here various low-level implementations of
-Groebner basis computation algorithms; please see the previous section of the
-manual for usage.
-
-.. currentmodule:: sympy.polys.groebnertools
-
-.. autofunction:: sdp_groebner
-.. autofunction:: buchberger
-.. autofunction:: sdp_spoly
-.. autofunction:: f5b
-.. autofunction:: red_groebner
-.. autofunction:: is_groebner
-.. autofunction:: is_minimal
-.. autofunction:: is_reduced
-.. autofunction:: matrix_fglm
-
-Groebner basis algorithms for modules are also provided:
-
-.. currentmodule:: sympy.polys.distributedmodules
-
-.. autofunction:: sdm_spoly
-.. autofunction:: sdm_ecart
-.. autofunction:: sdm_nf_mora
-.. autofunction:: sdm_groebner
-
-Exceptions
-==========
-
-These are exceptions defined by the polynomials module.
-
-TODO sort and explain
-
-.. currentmodule:: sympy.polys.polyerrors
-
-.. autoclass:: BasePolynomialError
-
-.. autoclass:: ExactQuotientFailed
-.. autoclass:: OperationNotSupported
-.. autoclass:: HeuristicGCDFailed
-.. autoclass:: HomomorphismFailed
-.. autoclass:: IsomorphismFailed
-.. autoclass:: ExtraneousFactors
-.. autoclass:: EvaluationFailed
-.. autoclass:: RefinementFailed
-.. autoclass:: CoercionFailed
-.. autoclass:: NotInvertible
-.. autoclass:: NotReversible
-.. autoclass:: NotAlgebraic
-.. autoclass:: DomainError
-.. autoclass:: PolynomialError
-.. autoclass:: UnificationFailed
-.. autoclass:: GeneratorsNeeded
-.. autoclass:: ComputationFailed
-.. autoclass:: GeneratorsError
-.. autoclass:: UnivariatePolynomialError
-.. autoclass:: MultivariatePolynomialError
-.. autoclass:: PolificationFailed
-.. autoclass:: OptionError
-.. autoclass:: FlagError
-
-Undocumented
-============
-
-Many parts of the polys module are still undocumented, and even where there is
-documentation it is scarce. Please contribute!
diff --git a/dev-py3k/_sources/modules/polys/literature.txt b/dev-py3k/_sources/modules/polys/literature.txt
deleted file mode 100644
index 3774fdca1b1..00000000000
--- a/dev-py3k/_sources/modules/polys/literature.txt
+++ /dev/null
@@ -1,79 +0,0 @@
-.. _polys-literature:
-
-==========
-Literature
-==========
-
-The following is a non-comprehensive list of publications that were used as
-a theoretical foundation for implementing polynomials manipulation module.
-
-.. [Kozen89] D. Kozen, S. Landau, Polynomial decomposition algorithms,
-    Journal of Symbolic Computation 7 (1989), pp. 445-456
-
-.. [Liao95] Hsin-Chao Liao,  R. Fateman, Evaluation of the heuristic
-    polynomial GCD, International Symposium on Symbolic and Algebraic
-    Computation (ISSAC), ACM Press, Montreal, Quebec, Canada, 1995,
-    pp. 240--247
-
-.. [Gathen99] J. von zur Gathen, J. Gerhard, Modern Computer Algebra,
-    First Edition, Cambridge University Press, 1999
-
-.. [Weisstein09] Eric W. Weisstein, Cyclotomic Polynomial, From MathWorld - A
-    Wolfram Web Resource, http://mathworld.wolfram.com/CyclotomicPolynomial.html
-
-.. [Wang78] P. S. Wang, An Improved Multivariate Polynomial Factoring
-    Algorithm, Math. of Computation 32, 1978, pp. 1215--1231
-
-.. [Geddes92] K. Geddes, S. R. Czapor, G. Labahn, Algorithms for
-    Computer Algebra, Springer, 1992
-
-.. [Monagan93] Michael Monagan, In-place Arithmetic for Polynomials
-    over Z_n, Proceedings of DISCO '92, Springer-Verlag LNCS, 721,
-    1993, pp. 22--34
-
-.. [Kaltofen98] E. Kaltofen, V. Shoup, Subquadratic-time Factoring of
-    Polynomials over Finite Fields, Mathematics of Computation, Volume
-    67, Issue 223, 1998, pp. 1179--1197
-
-.. [Shoup95] V. Shoup, A New Polynomial Factorization Algorithm and
-    its Implementation, Journal of Symbolic Computation, Volume 20,
-    Issue 4, 1995, pp. 363--397
-
-.. [Gathen92] J. von zur Gathen, V. Shoup, Computing Frobenius Maps
-    and Factoring Polynomials, ACM Symposium on Theory of Computing,
-    1992, pp. 187--224
-
-.. [Shoup91] V. Shoup, A Fast Deterministic Algorithm for Factoring
-    Polynomials over Finite Fields of Small Characteristic, In Proceedings
-    of International Symposium on Symbolic and Algebraic Computation, 1991,
-    pp. 14--21
-
-.. [Cox97] D. Cox, J. Little, D. O'Shea, Ideals, Varieties and
-    Algorithms, Springer, Second Edition, 1997
-
-.. [Ajwa95] I.A. Ajwa, Z. Liu, P.S. Wang, Groebner Bases Algorithm,
-    http://citeseer.ist.psu.edu/ajwa95grbner.html, 1995
-
-.. [Bose03] N.K. Bose, B. Buchberger, J.P. Guiver, Multidimensional
-    Systems Theory and Applications, Springer, 2003
-
-.. [Giovini91] A. Giovini, T. Mora, "One sugar cube, please" or
-    Selection strategies in Buchberger algorithm, ISSAC '91, ACM
-
-.. [Bronstein93] M. Bronstein, B. Salvy, Full partial fraction
-    decomposition of rational functions, Proceedings ISSAC '93,
-    ACM Press, Kiev, Ukraine, 1993, pp. 157--160
-
-.. [Buchberger01] B. Buchberger, Groebner Bases: A Short Introduction for
-    Systems Theorists,  In: R. Moreno-Diaz,  B. Buchberger,
-    J. L. Freire, Proceedings of EUROCAST'01, February, 2001
-
-.. [Davenport88] J.H. Davenport, Y. Siret, E. Tournier, Computer Algebra
-    Systems and Algorithms for Algebraic Computation, Academic Press, London,
-    1988, pp. 124--128
-
-.. [Greuel2008] G.-M. Greuel, Gerhard Pfister, A Singular Introduction to
-    Commutative Algebra, Springer, 2008
-
-.. [Atiyah69] M.F. Atiyah, I.G. MacDonald, Introduction to Commutative Algebra,
-    Addison-Wesley, 1969
diff --git a/dev-py3k/_sources/modules/polys/reference.txt b/dev-py3k/_sources/modules/polys/reference.txt
deleted file mode 100644
index b3e5546b127..00000000000
--- a/dev-py3k/_sources/modules/polys/reference.txt
+++ /dev/null
@@ -1,171 +0,0 @@
-.. _polys-reference:
-
-=========================================
-Polynomials Manipulation Module Reference
-=========================================
-
-Basic polynomial manipulation functions
-=======================================
-
-.. currentmodule:: sympy.polys.polytools
-
-.. autofunction:: poly
-.. autofunction:: poly_from_expr
-.. autofunction:: parallel_poly_from_expr
-.. autofunction:: degree
-.. autofunction:: degree_list
-.. autofunction:: LC
-.. autofunction:: LM
-.. autofunction:: LT
-.. autofunction:: pdiv
-.. autofunction:: prem
-.. autofunction:: pquo
-.. autofunction:: pexquo
-.. autofunction:: div
-.. autofunction:: rem
-.. autofunction:: quo
-.. autofunction:: exquo
-.. autofunction:: half_gcdex
-.. autofunction:: gcdex
-.. autofunction:: invert
-.. autofunction:: subresultants
-.. autofunction:: resultant
-.. autofunction:: discriminant
-.. autofunction:: terms_gcd
-.. autofunction:: cofactors
-.. autofunction:: gcd
-.. autofunction:: gcd_list
-.. autofunction:: lcm
-.. autofunction:: lcm_list
-.. autofunction:: trunc
-.. autofunction:: monic
-.. autofunction:: content
-.. autofunction:: primitive
-.. autofunction:: compose
-.. autofunction:: decompose
-.. autofunction:: sturm
-.. autofunction:: gff_list
-.. autofunction:: gff
-.. autofunction:: sqf_norm
-.. autofunction:: sqf_part
-.. autofunction:: sqf_list
-.. autofunction:: sqf
-.. autofunction:: factor_list
-.. autofunction:: factor
-.. autofunction:: intervals
-.. autofunction:: refine_root
-.. autofunction:: count_roots
-.. autofunction:: real_roots
-.. autofunction:: nroots
-.. autofunction:: ground_roots
-.. autofunction:: nth_power_roots_poly
-.. autofunction:: cancel
-.. autofunction:: reduced
-.. autofunction:: groebner
-.. autofunction:: is_zero_dimensional
-
-.. autoclass:: Poly
-   :members:
-
-.. autoclass:: PurePoly
-   :members:
-
-.. autoclass:: GroebnerBasis
-   :members:
-
-Extra polynomial manipulation functions
-=======================================
-
-.. currentmodule:: sympy.polys.polyfuncs
-
-.. autofunction:: symmetrize
-.. autofunction:: horner
-.. autofunction:: interpolate
-.. autofunction:: viete
-
-Domain constructors
-===================
-
-.. currentmodule:: sympy.polys.constructor
-
-.. autofunction:: construct_domain
-
-Algebraic number fields
-=======================
-
-.. currentmodule:: sympy.polys.numberfields
-
-.. autofunction:: minimal_polynomial
-.. autofunction:: minpoly
-.. autofunction:: primitive_element
-.. autofunction:: field_isomorphism
-.. autofunction:: to_number_field
-.. autofunction:: isolate
-
-.. autoclass:: AlgebraicNumber
-   :members:
-
-Monomials encoded as tuples
-===========================
-
-.. currentmodule:: sympy.polys.monomialtools
-
-.. autoclass:: Monomial
-.. autofunction:: monomials
-.. autofunction:: monomial_count
-.. autoclass:: LexOrder
-.. autoclass:: GradedLexOrder
-.. autoclass:: ReversedGradedLexOrder
-
-Formal manipulation of roots of polynomials
-===========================================
-
-.. currentmodule:: sympy.polys.rootoftools
-
-.. autoclass:: RootOf
-.. autoclass:: RootSum
-
-Symbolic root-finding algorithms
-================================
-
-.. currentmodule:: sympy.polys.polyroots
-
-.. autofunction:: roots
-
-Special polynomials
-===================
-
-.. currentmodule:: sympy.polys.specialpolys
-
-.. autofunction:: swinnerton_dyer_poly
-.. autofunction:: interpolating_poly
-.. autofunction:: cyclotomic_poly
-.. autofunction:: symmetric_poly
-.. autofunction:: random_poly
-
-Orthogonal polynomials
-======================
-
-.. currentmodule:: sympy.polys.orthopolys
-
-.. autofunction:: chebyshevt_poly
-.. autofunction:: chebyshevu_poly
-.. autofunction:: gegenbauer_poly
-.. autofunction:: hermite_poly
-.. autofunction:: jacobi_poly
-.. autofunction:: legendre_poly
-.. autofunction:: laguerre_poly
-
-Manipulation of rational functions
-==================================
-
-.. currentmodule:: sympy.polys.rationaltools
-
-.. autofunction:: together
-
-Partial fraction decomposition
-==============================
-
-.. currentmodule:: sympy.polys.partfrac
-
-.. autofunction:: apart
diff --git a/dev-py3k/_sources/modules/polys/wester.txt b/dev-py3k/_sources/modules/polys/wester.txt
deleted file mode 100644
index 39a3b2d15b3..00000000000
--- a/dev-py3k/_sources/modules/polys/wester.txt
+++ /dev/null
@@ -1,451 +0,0 @@
-.. _polys-wester:
-
-==============================
-Examples from Wester's Article
-==============================
-
-Introduction
-============
-
-In this tutorial we present examples from Wester's article concerning
-comparison and critique of mathematical abilities of several computer
-algebra systems (see [Wester1999]_). All the examples are related to
-polynomial and algebraic computations and SymPy specific remarks were
-added to all of them.
-
-Examples
-========
-
-All examples in this tutorial are computable, so one can just copy and
-paste them into a Python shell and do something useful with them. All
-computations were done using the following setup::
-
-    >>> from sympy import *
-
-    >>> init_printing(use_unicode=True, wrap_line=False, no_global=True)
-
-    >>> var('x,y,z,s,c,n')
-    (x, y, z, s, c, n)
-
-Simple univariate polynomial factorization
-------------------------------------------
-
-To obtain a factorization of a polynomial use :func:`factor` function.
-By default :func:`factor` returns the result in unevaluated form, so the
-content of the input polynomial is left unexpanded, as in the following
-example::
-
-    >>> factor(6*x - 10)
-    2⋅(3⋅x - 5)
-
-To achieve the same effect in a more systematic way use :func:`primitive`
-function, which returns the content and the primitive part of the input
-polynomial::
-
-    >>> primitive(6*x - 10)
-    (2, 3⋅x - 5)
-
-.. note::
-
-    The content and the primitive part can be computed only over a ring. To
-    simplify coefficients of a polynomial over a field use :func:`monic`.
-
-Univariate GCD, resultant and factorization
--------------------------------------------
-
-Consider univariate polynomials ``f``, ``g`` and ``h`` over integers::
-
-    >>> f = 64*x**34 - 21*x**47 - 126*x**8 - 46*x**5 - 16*x**60 - 81
-    >>> g = 72*x**60 - 25*x**25 - 19*x**23 - 22*x**39 - 83*x**52 + 54*x**10 + 81
-    >>> h = 34*x**19 - 25*x**16 + 70*x**7 + 20*x**3 - 91*x - 86
-
-We can compute the greatest common divisor (GCD) of two polynomials using
-:func:`gcd` function::
-
-    >>> gcd(f, g)
-    1
-
-We see that ``f`` and ``g`` have no common factors. However, ``f*h`` and ``g*h``
-have an obvious factor ``h``::
-
-    >>> gcd(expand(f*h), expand(g*h)) - h
-    0
-
-The same can be verified using the resultant of univariate polynomials::
-
-    >>> resultant(expand(f*h), expand(g*h))
-    0
-
-Factorization of large univariate polynomials (of degree 120 in this case) over
-integers is also possible::
-
-    >>> factor(expand(f*g))
-     ⎛    60       47       34        8       5     ⎞ ⎛    60       52     39       25       23       10     ⎞
-    -⎝16⋅x   + 21⋅x   - 64⋅x   + 126⋅x  + 46⋅x  + 81⎠⋅⎝72⋅x   - 83⋅x - 22⋅x   - 25⋅x   - 19⋅x   + 54⋅x   + 81⎠
-
-Multivariate GCD and factorization
-----------------------------------
-
-What can be done in univariate case, can be also done for multivariate
-polynomials. Consider the following polynomials ``f``, ``g`` and ``h``
-in `\mathbb{Z}[x,y,z]`::
-
-    >>> f = 24*x*y**19*z**8 - 47*x**17*y**5*z**8 + 6*x**15*y**9*z**2 - 3*x**22 + 5
-    >>> g = 34*x**5*y**8*z**13 + 20*x**7*y**7*z**7 + 12*x**9*y**16*z**4 + 80*y**14*z
-    >>> h = 11*x**12*y**7*z**13 - 23*x**2*y**8*z**10 + 47*x**17*y**5*z**8
-
-As previously, we can verify that ``f`` and ``g`` have no common factors::
-
-    >>> gcd(f, g)
-    1
-
-However, ``f*h`` and ``g*h`` have an obvious factor ``h``::
-
-    >>> gcd(expand(f*h), expand(g*h)) - h
-    0
-
-Multivariate factorization of large polynomials is also possible::
-
-    >>> factor(expand(f*g))
-        7   ⎛   9  9  3       7  6       5    12       7⎞ ⎛   22       17  5  8      15  9  2         19  8    ⎞
-    -2⋅y ⋅z⋅⎝6⋅x ⋅y ⋅z  + 10⋅x ⋅z  + 17⋅x ⋅y⋅z   + 40⋅y ⎠⋅⎝3⋅x   + 47⋅x  ⋅y ⋅z  - 6⋅x  ⋅y ⋅z  - 24⋅x⋅y  ⋅z  - 5⎠
-
-Support for symbols in exponents
---------------------------------
-
-Polynomial manipulation functions provided by :mod:`sympy.polys` are mostly
-used with integer exponents. However, it's perfectly valid to compute with
-symbolic exponents, e.g.::
-
-    >>> gcd(2*x**(n + 4) - x**(n + 2), 4*x**(n + 1) + 3*x**n)
-     n
-    x
-
-Testing if polynomials have common zeros
-----------------------------------------
-
-To test if two polynomials have a root in common we can use :func:`resultant`
-function. The theory says that the resultant of two polynomials vanishes if
-there is a common zero of those polynomials. For example::
-
-    >>> resultant(3*x**4 + 3*x**3 + x**2 - x - 2, x**3 - 3*x**2 + x + 5)
-    0
-
-We can visualize this fact by factoring the polynomials::
-
-    >>> factor(3*x**4 + 3*x**3 + x**2 - x - 2)
-            ⎛   3        ⎞
-    (x + 1)⋅⎝3⋅x  + x - 2⎠
-
-    >>> factor(x**3 - 3*x**2 + x + 5)
-            ⎛ 2          ⎞
-    (x + 1)⋅⎝x  - 4⋅x + 5⎠
-
-In both cases we obtained the factor `x + 1` which tells us that the common
-root is `x = -1`.
-
-Normalizing simple rational functions
--------------------------------------
-
-To remove common factors from the numerator and the denominator of a rational
-function the elegant way, use :func:`cancel` function. For example::
-
-    >>> cancel((x**2 - 4)/(x**2 + 4*x + 4))
-    x - 2
-    ─────
-    x + 2
-
-Expanding expressions and factoring back
-----------------------------------------
-
-One can work easily we expressions in both expanded and factored forms.
-Consider a polynomial ``f`` in expanded form. We differentiate it and
-factor the result back::
-
-    >>> f = expand((x + 1)**20)
-
-    >>> g = diff(f, x)
-
-    >>> factor(g)
-              19
-    20⋅(x + 1)
-
-The same can be achieved in factored form::
-
-    >>> diff((x + 1)**20, x)
-              19
-    20⋅(x + 1)
-
-Factoring in terms of cyclotomic polynomials
---------------------------------------------
-
-SymPy can very efficiently decompose polynomials of the form `x^n \pm 1` in
-terms of cyclotomic polynomials::
-
-    >>> factor(x**15 - 1)
-            ⎛ 2        ⎞ ⎛ 4    3    2        ⎞ ⎛ 8    7    5    4    3       ⎞
-    (x - 1)⋅⎝x  + x + 1⎠⋅⎝x  + x  + x  + x + 1⎠⋅⎝x  - x  + x  - x  + x - x + 1⎠
-
-The original Wester`s example was `x^{100} - 1`, but was truncated for
-readability purpose. Note that this is not a big struggle for :func:`factor`
-to decompose polynomials of degree 1000 or greater.
-
-Univariate factoring over Gaussian numbers
-------------------------------------------
-
-Consider a univariate polynomial ``f`` with integer coefficients::
-
-    >>> f = 4*x**4 + 8*x**3 + 77*x**2 + 18*x + 153
-
-We want to obtain a factorization of ``f`` over Gaussian numbers. To do this
-we use :func:`factor` as previously, but this time we set ``gaussian`` keyword
-to ``True``::
-
-    >>> factor(f, gaussian=True)
-      ⎛    3⋅ⅈ⎞ ⎛    3⋅ⅈ⎞
-    4⋅⎜x - ───⎟⋅⎜x + ───⎟⋅(x + 1 - 4⋅ⅈ)⋅(x + 1 + 4⋅ⅈ)
-      ⎝     2 ⎠ ⎝     2 ⎠
-
-As the result we got a splitting factorization of ``f`` with monic factors
-(this is a general rule when computing in a field with SymPy). The ``gaussian``
-keyword is useful for improving code readability, however the same result can
-be computed using more general syntax::
-
-    >>> factor(f, extension=I)
-      ⎛    3⋅ⅈ⎞ ⎛    3⋅ⅈ⎞
-    4⋅⎜x - ───⎟⋅⎜x + ───⎟⋅(x + 1 - 4⋅ⅈ)⋅(x + 1 + 4⋅ⅈ)
-      ⎝     2 ⎠ ⎝     2 ⎠
-
-Computing with automatic field extensions
------------------------------------------
-
-Consider two univariate polynomials ``f`` and ``g``::
-
-    >>> f = x**3 + (sqrt(2) - 2)*x**2 - (2*sqrt(2) + 3)*x - 3*sqrt(2)
-    >>> g = x**2 - 2
-
-We would like to reduce degrees of the numerator and the denominator of a
-rational function ``f/g``. Do do this we employ :func:`cancel` function::
-
-    >>> cancel(f/g)
-     3      2     ___  2             ___         ___
-    x  - 2⋅x  + ╲╱ 2 ⋅x  - 3⋅x - 2⋅╲╱ 2 ⋅x - 3⋅╲╱ 2
-    ────────────────────────────────────────────────
-                          2
-                         x  - 2
-
-Unfortunately nothing interesting happened. This is because by default SymPy
-treats `\sqrt{2}` as a generator, obtaining a bivariate polynomial for the
-numerator. To make :func:`cancel` recognize algebraic properties of `\sqrt{2}`,
-one needs to use ``extension`` keyword::
-
-    >>> cancel(f/g, extension=True)
-     2
-    x  - 2⋅x - 3
-    ────────────
-           ___
-     x - ╲╱ 2
-
-Setting ``extension=True`` tells :func:`cancel` to find minimal algebraic
-number domain for the coefficients of ``f/g``. The automatically inferred
-domain is `\mathbb{Q}(\sqrt{2})`. If one doesn't want to rely on automatic
-inference, the same result can be obtained by setting the ``extension``
-keyword with an explicit algebraic number::
-
-    >>> cancel(f/g, extension=sqrt(2))
-     2
-    x  - 2⋅x - 3
-    ────────────
-           ___
-     x - ╲╱ 2
-
-Univariate factoring over various domains
------------------------------------------
-
-Consider a univariate polynomial ``f`` with integer coefficients::
-
-    >>> f = x**4 - 3*x**2 + 1
-
-With :mod:`sympy.polys` we can obtain factorizations of ``f`` over different
-domains, which includes:
-
-* rationals::
-
-    >>> factor(f)
-    ⎛ 2        ⎞ ⎛ 2        ⎞
-    ⎝x  - x - 1⎠⋅⎝x  + x - 1⎠
-
-* finite fields::
-
-    >>> factor(f, modulus=5)
-           2        2
-    (x - 2) ⋅(x + 2)
-
-* algebraic numbers::
-
-    >>> alg = AlgebraicNumber((sqrt(5) - 1)/2, alias='alpha')
-
-    >>> factor(f, extension=alg)
-    (x - 1 - α)⋅(x - α)⋅(x + α)⋅(x + 1 + α)
-
-Factoring polynomials into linear factors
------------------------------------------
-
-Currently SymPy can factor polynomials into irreducibles over various domains,
-which can result in a splitting factorization (into linear factors). However,
-there is currently no systematic way to infer a splitting field (algebraic
-number field) automatically. In future the following syntax will be
-implemented::
-
-    >>> factor(x**3 + x**2 - 7, split=True)
-    Traceback (most recent call last):
-    ...
-    NotImplementedError: 'split' option is not implemented yet
-
-Note this is different from ``extension=True``, because the later only tells how
-expression parsing should be done, not what should be the domain of computation.
-One can simulate the ``split`` keyword for several classes of polynomials using
-:func:`solve` function.
-
-Advanced factoring over finite fields
--------------------------------------
-
-Consider a univariate polynomial ``f`` with integer coefficients::
-
-    >>> f = x**11 + x + 1
-
-We can factor ``f`` over a large finite field `F_{65537}`::
-
-    >>> factor(f, modulus=65537)
-    ⎛ 2        ⎞ ⎛ 9    8    6    5    3    2    ⎞
-    ⎝x  + x + 1⎠⋅⎝x  - x  + x  - x  + x  - x  + 1⎠
-
-and expand the resulting factorization back::
-
-    >>> expand(_)
-     11
-    x   + x + 1
-
-obtaining polynomial ``f``. This was done using symmetric polynomial
-representation over finite fields The same thing can be done using
-non-symmetric representation::
-
-    >>> factor(f, modulus=65537, symmetric=False)
-    ⎛ 2        ⎞ ⎛ 9          8    6          5    3          2    ⎞
-    ⎝x  + x + 1⎠⋅⎝x  + 65536⋅x  + x  + 65536⋅x  + x  + 65536⋅x  + 1⎠
-
-As with symmetric representation we can expand the factorization
-to get the input polynomial back. This time, however, we need to
-truncate coefficients of the expanded polynomial modulo 65537::
-
-    >>> trunc(expand(_), 65537)
-     11
-    x   + x + 1
-
-Working with expressions as polynomials
----------------------------------------
-
-Consider a multivariate polynomial ``f`` in `\mathbb{Z}[x,y,z]`::
-
-    >>> f = expand((x - 2*y**2 + 3*z**3)**20)
-
-We want to compute factorization of ``f``. To do this we use ``factor`` as
-usually, however we note that the polynomial in consideration is already
-in expanded form, so we can tell the factorization routine to skip
-expanding ``f``::
-
-    >>> factor(f, expand=False)
-                     20
-    ⎛       2      3⎞
-    ⎝x - 2⋅y  + 3⋅z ⎠
-
-The default in :mod:`sympy.polys` is to expand all expressions given as
-arguments to polynomial manipulation functions and :class:`Poly` class.
-If we know that expanding is unnecessary, then by setting ``expand=False``
-we can save quite a lot of time for complicated inputs. This can be really
-important when computing with expressions like::
-
-    >>> g = expand((sin(x) - 2*cos(y)**2 + 3*tan(z)**3)**20)
-
-    >>> factor(g, expand=False)
-                                     20
-    ⎛               2           3   ⎞
-    ⎝-sin(x) + 2⋅cos (y) - 3⋅tan (z)⎠
-
-Computing reduced Gröbner bases
--------------------------------
-
-To compute a reduced Gröbner basis for a set of polynomials use
-:func:`groebner` function. The function accepts various monomial
-orderings, e.g.: ``lex``, ``grlex`` and ``grevlex``, or a user
-defined one, via ``order`` keyword. The ``lex`` ordering is the
-most interesting because it has elimination property, which means
-that if the system of polynomial equations to :func:`groebner` is
-zero-dimensional (has finite number of solutions) the last element
-of the basis is a univariate polynomial. Consider the following example::
-
-    >>> f = expand((1 - c**2)**5 * (1 - s**2)**5 * (c**2 + s**2)**10)
-
-    >>> groebner([f, c**2 + s**2 - 1])
-                 ⎛⎡ 2    2       20      18       16       14      12    10⎤                           ⎞
-    GroebnerBasis⎝⎣c  + s  - 1, c   - 5⋅c   + 10⋅c   - 10⋅c   + 5⋅c   - c  ⎦, s, c, domain=ℤ, order=lex⎠
-
-The result is an ordinary Python list, so we can easily apply a function to
-all its elements, for example we can factor those elements::
-
-    >>> list(map(factor, _))
-    ⎡ 2    2       10        5        5⎤
-    ⎣c  + s  - 1, c  ⋅(c - 1) ⋅(c + 1) ⎦
-
-From the above we can easily find all solutions of the system of polynomial
-equations. Or we can use :func:`solve` to achieve this in a more systematic
-way::
-
-    >>> solve([f, s**2 + c**2 - 1], c, s)
-    [(-1, 0), (0, -1), (0, 1), (1, 0)]
-
-Multivariate factoring over algebraic numbers
----------------------------------------------
-
-Computing with multivariate polynomials over various domains is as simple as
-in univariate case. For example consider the following factorization over
-`\mathbb{Q}(\sqrt{-3})`::
-
-    >>> factor(x**3 + y**3, extension=sqrt(-3))
-            ⎛      ⎛        ___  ⎞⎞ ⎛      ⎛        ___  ⎞⎞
-            ⎜      ⎜  1   ╲╱ 3 ⋅ⅈ⎟⎟ ⎜      ⎜  1   ╲╱ 3 ⋅ⅈ⎟⎟
-    (x + y)⋅⎜x + y⋅⎜- ─ - ───────⎟⎟⋅⎜x + y⋅⎜- ─ + ───────⎟⎟
-            ⎝      ⎝  2      2   ⎠⎠ ⎝      ⎝  2      2   ⎠⎠
-
-.. note:: Currently multivariate polynomials over finite fields aren't supported.
-
-Partial fraction decomposition
-------------------------------
-
-Consider a univariate rational function ``f`` with integer coefficients::
-
-    >>> f = (x**2 + 2*x + 3)/(x**3 + 4*x**2 + 5*x + 2)
-
-To decompose ``f`` into partial fractions use :func:`apart` function::
-
-    >>> apart(f)
-      3       2        2
-    ───── - ───── + ────────
-    x + 2   x + 1          2
-                    (x + 1)
-
-To return from partial fractions to the rational function use
-a composition of :func:`together` and :func:`cancel`::
-
-    >>> cancel(together(_))
-         2
-        x  + 2⋅x + 3
-    ───────────────────
-     3      2
-    x  + 4⋅x  + 5⋅x + 2
-
-Literature
-==========
-
-.. [Wester1999] Michael J. Wester, A Critique of the Mathematical Abilities of
-    CA Systems, 1999, `<http://www.math.unm.edu/~wester/cas/book/Wester.pdf>`_
diff --git a/dev-py3k/_sources/modules/printing.txt b/dev-py3k/_sources/modules/printing.txt
deleted file mode 100644
index 6317d575635..00000000000
--- a/dev-py3k/_sources/modules/printing.txt
+++ /dev/null
@@ -1,444 +0,0 @@
-Printing System
-===============
-
-See the :ref:`printing-tutorial` section in Tutorial for introduction into
-printing.
-
-This guide documents the printing system in SymPy and how it works
-internally.
-
-Printer Class
--------------
-
-.. automodule:: sympy.printing.printer
-
-The main class responsible for printing is ``Printer`` (see also its
-`source code
-<https://github.com/sympy/sympy/blob/master/sympy/printing/printer.py>`_):
-
-.. autoclass:: Printer
-    :members: doprint, _print, set_global_settings, order
-
-    .. autoattribute:: Printer.printmethod
-
-
-PrettyPrinter Class
--------------------
-
-Pretty printing subsystem is implemented in ``sympy.printing.pretty.pretty``
-by the ``PrettyPrinter`` class deriving from ``Printer``. It relies on
-modules ``sympy.printing.pretty.stringPict``, and
-``sympy.printing.pretty.pretty_symbology`` for rendering nice-looking
-formulas.
-
-The module ``stringPict`` provides a base class ``stringPict`` and a derived
-class ``prettyForm`` that ease the creation and manipulation of formulas
-that span across multiple lines.
-
-The module ``pretty_symbology`` provides primitives to construct 2D shapes
-(hline, vline, etc) together with a technique to use unicode automatically
-when possible.
-
-.. module:: sympy.printing.pretty.pretty
-
-.. autoclass:: PrettyPrinter
-   :members: _use_unicode, doprint
-
-   .. autoattribute:: PrettyPrinter.printmethod
-
-.. autofunction:: pretty
-.. autofunction:: pretty_print
-
-CCodePrinter
-------------
-
-.. module:: sympy.printing.ccode
-
-This class implements C code printing (i.e. it converts Python expressions
-to strings of C code).
-
-Usage::
-
-    >>> from sympy.printing import print_ccode
-    >>> from sympy.functions import sin, cos, Abs
-    >>> from sympy.abc import x
-    >>> print_ccode(sin(x)**2 + cos(x)**2)
-    pow(sin(x), 2) + pow(cos(x), 2)
-    >>> print_ccode(2*x + cos(x), assign_to="result")
-    result = 2*x + cos(x);
-    >>> print_ccode(Abs(x**2))
-    fabs(pow(x, 2))
-
-.. autodata:: sympy.printing.ccode.known_functions
-
-.. autoclass:: sympy.printing.ccode.CCodePrinter
-   :members:
-
-   .. autoattribute:: CCodePrinter.printmethod
-
-
-.. autofunction:: sympy.printing.ccode.ccode
-
-.. autofunction:: sympy.printing.ccode.print_ccode
-
-Fortran Printing
-----------------
-
-The fcode function translates a sympy expression into Fortran code. The main
-purpose is to take away the burden of manually translating long mathematical
-expressions. Therefore the resulting expression should also require no (or
-very little) manual tweaking to make it compilable. The optional arguments
-of fcode can be used to fine-tune the behavior of fcode in such a way that
-manual changes in the result are no longer needed.
-
-.. module:: sympy.printing.fcode
-.. autofunction:: fcode
-.. autofunction:: print_fcode
-.. autoclass:: FCodePrinter
-   :members:
-
-   .. autoattribute:: FCodePrinter.printmethod
-
-
-Two basic examples:
-
-    >>> from sympy import *
-    >>> x = symbols("x")
-    >>> fcode(sqrt(1-x**2))
-    '      sqrt(-x**2 + 1)'
-    >>> fcode((3 + 4*I)/(1 - conjugate(x)))
-    '      (cmplx(3,4))/(-conjg(x) + 1)'
-
-An example where line wrapping is required:
-
-    >>> expr = sqrt(1-x**2).series(x,n=20).removeO()
-    >>> print(fcode(expr))
-          -715*x**18/65536 - 429*x**16/32768 - 33*x**14/2048 - 21*x**12/1024
-         @ - 7*x**10/256 - 5*x**8/128 - x**6/16 - x**4/8 - x**2/2 + 1
-
-In case of line wrapping, it is handy to include the assignment so that lines
-are wrapped properly when the assignment part is added.
-
-    >>> print(fcode(expr, assign_to="var"))
-          var = -715*x**18/65536 - 429*x**16/32768 - 33*x**14/2048 - 21*x**
-         @ 12/1024 - 7*x**10/256 - 5*x**8/128 - x**6/16 - x**4/8 - x**2/2 +
-         @ 1
-
-For piecewise functions, the assign_to option is mandatory:
-
-    >>> print(fcode(Piecewise((x,x<1),(x**2,True)), assign_to="var"))
-          if (x < 1) then
-            var = x
-          else
-            var = x**2
-          end if
-
-Note that only top-level piecewise functions are supported due to the lack of
-a conditional operator in Fortran. Nested piecewise functions would require the
-introduction of temporary variables, which is a type of expression manipulation
-that goes beyond the scope of fcode.
-
-Loops are generated if there are Indexed objects in the expression.  This
-also requires use of the assign_to option.
-
-    >>> A, B = list(map(IndexedBase, ['A', 'B']))
-    >>> m = Symbol('m', integer=True)
-    >>> i = Idx('i', m)
-    >>> print(fcode(2*B[i], assign_to=A[i]))
-        do i = 1, m
-            A(i) = 2*B(i)
-        end do
-
-Repeated indices in an expression with Indexed objects are interpreted as
-summation. For instance, code for the trace of a matrix can be generated
-with
-
-    >>> print(fcode(A[i, i], assign_to=x))
-          x = 0
-          do i = 1, m
-              x = x + A(i, i)
-          end do
-
-By default, number symbols such as ``pi`` and ``E`` are detected and defined as
-Fortran parameters. The precision of the constants can be tuned with the
-precision argument. Parameter definitions are easily avoided using the ``N``
-function.
-
-    >>> print(fcode(x - pi**2 - E))
-          parameter (E = 2.71828182845905d0)
-          parameter (pi = 3.14159265358979d0)
-          x - pi**2 - E
-    >>> print(fcode(x - pi**2 - E, precision=25))
-          parameter (E = 2.718281828459045235360287d0)
-          parameter (pi = 3.141592653589793238462643d0)
-          x - pi**2 - E
-    >>> print(fcode(N(x - pi**2, 25)))
-          x - 9.869604401089358618834491d0
-
-When some functions are not part of the Fortran standard, it might be desirable
-to introduce the names of user-defined functions in the Fortran expression.
-
-    >>> print(fcode(1 - gamma(x)**2, user_functions={gamma: 'mygamma'}))
-          -mygamma(x)**2 + 1
-
-However, when the user_functions argument is not provided, fcode attempts to
-use a reasonable default and adds a comment to inform the user of the issue.
-
-    >>> print(fcode(1 - gamma(x)**2))
-    C     Not Fortran:
-    C     gamma(x)
-          -gamma(x)**2 + 1
-
-By default the output is human readable code, ready for copy and paste. With the
-option ``human=False``, the return value is suitable for post-processing with
-source code generators that write routines with multiple instructions. The
-return value is a three-tuple containing: (i) a set of number symbols that must
-be defined as 'Fortran parameters', (ii) a list functions that can not be
-translated in pure Fortran and (iii) a string of Fortran code. A few examples:
-
-    >>> fcode(1 - gamma(x)**2, human=False)
-    (set(), set([gamma(x)]), '      -gamma(x)**2 + 1')
-    >>> fcode(1 - sin(x)**2, human=False)
-    (set(), set(), '      -sin(x)**2 + 1')
-    >>> fcode(x - pi**2, human=False)
-    (set([(pi, '3.14159265358979d0')]), set(), '      x - pi**2')
-
-Gtk
----
-
-.. module:: sympy.printing.gtk
-
-You can print to a grkmathview widget using the function ``print_gtk``
-located in ``sympy.printing.gtk`` (it requires to have installed
-gtkmatmatview and libgtkmathview-bin in some systems).
-
-GtkMathView accepts MathML, so this rendering depends on the MathML
-representation of the expression.
-
-Usage::
-
-    from sympy import *
-    print_gtk(x**2 + 2*exp(x**3))
-
-.. autofunction:: print_gtk
-
-LambdaPrinter
--------------
-
-.. module:: sympy.printing.lambdarepr
-
-This classes implements printing to strings that can be used by the
-:py:func:`sympy.utilities.lambdify.lambdify` function.
-
-.. autoclass:: LambdaPrinter
-
-   .. autoattribute:: LambdaPrinter.printmethod
-
-
-.. autofunction:: lambdarepr
-
-LatexPrinter
-------------
-
-.. module:: sympy.printing.latex
-
-This class implements LaTeX printing. See ``sympy.printing.latex``.
-
-See also the extended LatexPrinter: :ref:`Extended LaTeXModule for SymPy <extended-latex>`
-
-.. autodata:: accepted_latex_functions
-
-.. autoclass:: LatexPrinter
-   :members:
-
-   .. autoattribute:: LatexPrinter.printmethod
-
-.. autofunction:: latex
-
-.. autofunction:: print_latex
-
-MathMLPrinter
--------------
-
-.. module:: sympy.printing.mathml
-
-This class is responsible for MathML printing. See ``sympy.printing.mathml``.
-
-More info on mathml content: http://www.w3.org/TR/MathML2/chapter4.html
-
-.. autoclass:: MathMLPrinter
-   :members:
-
-   .. autoattribute:: MathMLPrinter.printmethod
-
-.. autofunction:: mathml
-
-.. autofunction:: print_mathml
-
-PythonPrinter
--------------
-
-.. module:: sympy.printing.python
-
-This class implements Python printing. Usage::
-
-    >>> from sympy import print_python, sin
-    >>> from sympy.abc import x
-
-    >>> print_python(5*x**3 + sin(x))
-    x = Symbol('x')
-    e = 5*x**3 + sin(x)
-
-ReprPrinter
------------
-
-.. module:: sympy.printing.repr
-
-This printer generates executable code. This code satisfies the identity
-``eval(srepr(expr))=expr``.
-
-.. autoclass:: ReprPrinter
-   :members:
-
-   .. autoattribute:: ReprPrinter.printmethod
-
-.. autofunction:: srepr
-
-StrPrinter
-----------
-
-.. module:: sympy.printing.str
-
-This module generates readable representations of SymPy expressions.
-
-.. autoclass:: StrPrinter
-   :members: parenthesize, stringify, emptyPrinter
-
-   .. autoattribute:: StrPrinter.printmethod
-
-.. autofunction:: sstrrepr
-
-Tree Printing
--------------
-
-.. module:: sympy.printing.tree
-
-The functions in this module create a representation of an expression as a
-tree.
-
-.. autofunction:: pprint_nodes
-
-.. autofunction:: print_node
-
-.. autofunction:: tree
-
-.. autofunction:: print_tree
-
-Preview
--------
-
-A useful function is ``preview``:
-
-.. module:: sympy.printing.preview
-
-.. autofunction:: preview
-
-Implementation - Helper Classes/Functions
------------------------------------------
-
-.. module:: sympy.printing.conventions
-
-.. autofunction:: split_super_sub
-
-CodePrinter
-+++++++++++
-
-.. module:: sympy.printing.codeprinter
-
-This class is a base class for other classes that implement code-printing
-functionality, and additionally lists a number of functions that cannot be
-easily translated to C or Fortran.
-
-.. autoclass:: sympy.printing.codeprinter.CodePrinter
-
-   .. autoattribute:: CodePrinter.printmethod
-
-.. autoexception:: sympy.printing.codeprinter.AssignmentError
-
-Precedence
-++++++++++
-
-.. module:: sympy.printing.precedence
-
-.. autodata:: PRECEDENCE
-
-   Default precedence values for some basic types.
-
-.. autodata:: PRECEDENCE_VALUES
-
-   A dictionary assigning precedence values to certain classes. These values
-   are treated like they were inherited, so not every single class has to be
-   named here.
-
-.. autodata:: PRECEDENCE_FUNCTIONS
-
-   Sometimes it's not enough to assign a fixed precedence value to a
-   class. Then a function can be inserted in this dictionary that takes an
-   instance of this class as argument and returns the appropriate precedence
-   value.
-
-.. autofunction:: precedence
-
-Pretty-Printing Implementation Helpers
---------------------------------------
-
-.. module:: sympy.printing.pretty.pretty_symbology
-
-.. autofunction:: U
-.. autofunction:: pretty_use_unicode
-.. autofunction:: pretty_try_use_unicode
-.. autofunction:: xstr
-
-The following two functions return the Unicode version of the inputted Greek
-letter.
-
-.. autofunction:: g
-.. autofunction:: G
-.. autodata:: greek_letters
-.. autodata:: greek
-.. autodata:: digit_2txt
-.. autodata:: symb_2txt
-
-The following functions return the Unicode subscript/superscript version of
-the character.
-
-.. autodata:: sub
-.. autodata:: sup
-
-The following functions return Unicode vertical objects.
-
-.. autofunction:: xobj
-.. autofunction:: vobj
-.. autofunction:: hobj
-
-The following constants are for rendering roots and fractions.
-
-.. autodata:: root
-.. autofunction:: VF
-.. autodata:: frac
-
-The following constants/functions are for rendering atoms and symbols.
-
-.. autofunction:: xsym
-.. autodata:: atoms_table
-.. autofunction:: pretty_atom
-.. autofunction:: pretty_symbol
-.. autofunction:: annotated
-
-.. automodule:: sympy.printing.pretty.stringpict
-
-.. autoclass:: stringPict
-   :members:
-
-.. autoclass:: prettyForm
-   :members:
diff --git a/dev-py3k/_sources/modules/rewriting.txt b/dev-py3k/_sources/modules/rewriting.txt
deleted file mode 100644
index 266c4eff96e..00000000000
--- a/dev-py3k/_sources/modules/rewriting.txt
+++ /dev/null
@@ -1,92 +0,0 @@
-Term rewriting
-==============
-
-Term rewriting is a very general class of functionalities which are used to
-convert expressions of one type in terms of expressions of different kind. For
-example expanding, combining and converting expressions apply to term
-rewriting, and also simplification routines can be included here. Currently
-!SymPy has several functions and Basic built-in methods for performing various
-types of rewriting.
-
-Expanding
----------
-
-The simplest rewrite rule is expanding expressions into a _sparse_ form.
-Expanding has several flavors and include expanding complex valued expressions,
-arithmetic expand of products and powers but also expanding functions in terms
-of more general functions is possible. Below are listed all currently available
-expand rules.
-
-Expanding of arithmetic expressions involving products and powers:
-    >>> from sympy import *
-    >>> x, y, z = symbols('x,y,z')
-    >>> ((x + y)*(x - y)).expand(basic=True)
-    x**2 - y**2
-    >>> ((x + y + z)**2).expand(basic=True)
-    x**2 + 2*x*y + 2*x*z + y**2 + 2*y*z + z**2
-
-Arithmetic expand is done by default in ``expand()`` so the keyword ``basic`` can
-be omitted. However you can set ``basic=False`` to avoid this type of expand if
-you use rules described below. This give complete control on what is done with
-the expression.
-
-Another type of expand rule is expanding complex valued expressions and putting
-them into a normal form. For this ``complex`` keyword is used. Note that it will
-always perform arithmetic expand to obtain the desired normal form:
-
-    >>> (x + I*y).expand(complex=True)
-    re(x) + I*re(y) + I*im(x) - im(y)
-
-    >>> sin(x + I*y).expand(complex=True)
-    sin(re(x) - im(y))*cosh(re(y) + im(x)) + I*cos(re(x) - im(y))*sinh(re(y) + im(x))
-
-Note also that the same behavior can be obtained by using ``as_real_imag()``
-method. However it will return a tuple containing the real part in the first
-place and the imaginary part in the other. This can be also done in a two step
-process by using ``collect`` function:
-
-    >>> (x + I*y).as_real_imag()
-    (re(x) - im(y), re(y) + im(x))
-
-    >>> collect((x + I*y).expand(complex=True), I, evaluate=False)
-    {1: re(x) - im(y), I: re(y) + im(x)}
-
-There is also possibility for expanding expressions in terms of expressions of
-different kind. This is very general type of expanding and usually you would
-use ``rewrite()`` to do specific type of rewrite::
-
-    >>> GoldenRatio.expand(func=True)
-    1/2 + sqrt(5)/2
-
-Common Subexpression Detection and Collection
----------------------------------------------
-
-.. module:: sympy.simplify.cse_main
-
-Before evaluating a large expression, it is often useful to identify common
-subexpressions, collect them and evaluate them at once. This is implemented
-in the ``cse`` function. Examples::
-
-    >>> from sympy import cse, sqrt, sin, pprint
-    >>> from sympy.abc import x
-
-    >>> pprint(cse(sqrt(sin(x))), use_unicode=True)
-    ⎛    ⎡  ________⎤⎞
-    ⎝[], ⎣╲╱ sin(x) ⎦⎠
-
-    >>> pprint(cse(sqrt(sin(x)+5)*sqrt(sin(x)+4)), use_unicode=True)
-    ⎛                ⎡  ________   ________⎤⎞
-    ⎝[(x₀, sin(x))], ⎣╲╱ x₀ + 4 ⋅╲╱ x₀ + 5 ⎦⎠
-
-    >>> pprint(cse(sqrt(sin(x+1) + 5 + cos(y))*sqrt(sin(x+1) + 4 + cos(y))),
-    ...     use_unicode=True)
-    ⎛                             ⎡  ________   ________⎤⎞
-    ⎝[(x₀, sin(x + 1) + cos(y))], ⎣╲╱ x₀ + 4 ⋅╲╱ x₀ + 5 ⎦⎠
-
-    >>> pprint(cse((x-y)*(z-y) + sqrt((x-y)*(z-y))), use_unicode=True)
-    ⎛                          ⎡  ____     ⎤⎞
-    ⎝[(x₀, -(x - y)⋅(y - z))], ⎣╲╱ x₀  + x₀⎦⎠
-
-More information:
-
-.. autofunction:noindex: cse
diff --git a/dev-py3k/_sources/modules/series.txt b/dev-py3k/_sources/modules/series.txt
deleted file mode 100644
index 441172c8bd5..00000000000
--- a/dev-py3k/_sources/modules/series.txt
+++ /dev/null
@@ -1,188 +0,0 @@
-Series Expansions
-=================
-
-.. module:: sympy.series
-
-The series module implements series expansions as a function and many related
-functions.
-
-Limits
-------
-
-The main purpose of this module is the computation of limits.
-
-.. autofunction:: sympy.series.limits.limit
-
-.. autoclass:: sympy.series.limits.Limit
-   :members:
-
-As is explained above, the workhorse for limit computations is the
-function gruntz() which implements Gruntz' algorithm for computing limits.
-
-The Gruntz Algorithm
-^^^^^^^^^^^^^^^^^^^^
-
-This section explains the basics of the algorithm used for computing limits.
-Most of the time the limit() function should just work. However it is still
-useful to keep in mind how it is implemented in case something does not work
-as expected.
-
-First we define an ordering on functions. Suppose `f(x)` and `g(x)` are two
-real-valued functions such that `\lim_{x \to \infty} f(x) = \infty` and
-similarly `\lim_{x \to \infty} g(x) = \infty`. We shall say that `f(x)`
-*dominates*
-`g(x)`, written `f(x) \succ g(x)`, if for all `a, b \in \mathbb{R}_{>0}` we have
-`\lim_{x \to \infty} \frac{f(x)^a}{g(x)^b} = \infty`.
-We also say that `f(x)` and
-`g(x)` are *of the same comparability class* if neither `f(x) \succ g(x)` nor
-`g(x) \succ f(x)` and shall denote it as `f(x) \asymp g(x)`.
-
-Note that whenever `a, b \in \mathbb{R}_{>0}` then
-`a f(x)^b \asymp f(x)`, and we shall use this to extend the definition of
-`\succ` to all functions which tend to `0` or `\pm \infty` as `x \to \infty`.
-Thus we declare that `f(x) \asymp 1/f(x)` and `f(x) \asymp -f(x)`.
-
-It is easy to show the following examples:
-
-* `e^x \succ x^m`
-* `e^{x^2} \succ e^{mx}`
-* `e^{e^x} \succ e^{x^m}`
-* `x^m \asymp x^n`
-* `e^{x + \frac{1}{x}} \asymp e^{x + \log{x}} \asymp e^x`.
-
-From the above definition, it is possible to prove the following property:
-
-    Suppose `\omega`, `g_1, g_2, \dots` are functions of `x`,
-    `\lim_{x \to \infty} \omega = 0` and `\omega \succ g_i` for
-    all `i`. Let `c_1, c_2, \dots \in \mathbb{R}` with `c_1 < c_2 < \dots`.
-
-    Then `\lim_{x \to \infty} \sum_i g_i \omega^{c_i} = \lim_{x \to \infty} g_1 \omega^{c_1}`.
-
-For `g_1 = g` and `\omega` as above we also have the following easy result:
-
-    * `\lim_{x \to \infty} g \omega^c = 0` for `c > 0`
-    * `\lim_{x \to \infty} g \omega^c = \pm \infty` for `c < 0`,
-      where the sign is determined by the (eventual) sign of `g`
-    * `\lim_{x \to \infty} g \omega^0 = \lim_{x \to \infty} g`.
-
-
-Using these results yields the following strategy for computing
-`\lim_{x \to \infty} f(x)`:
-
-1. Find the set of *most rapidly varying subexpressions* (MRV set) of `f(x)`.
-   That is, from the set of all subexpressions of `f(x)`, find the elements that
-   are maximal under the relation `\succ`.
-2. Choose a function `\omega` that is in the same comparability class as
-   the elements in the MRV set, such that `\lim_{x \to \infty} \omega = 0`.
-3. Expand `f(x)` as a series in `\omega` in such a way that the antecedents of
-   the above theorem are satisfied.
-4. Apply the theorem and conclude the computation of
-   `\lim_{x \to \infty} f(x)`, possibly by recursively working on `g_1(x)`.
-
-
-Notes
-"""""
-
-This exposition glossed over several details. Many are described in the file
-gruntz.py, and all can be found in Gruntz' very readable thesis. The most
-important points that have not been explained are:
-
-1. Given f(x) and g(x), how do we determine if `f(x) \succ g(x)`,
-   `g(x) \succ f(x)` or `g(x) \asymp f(x)`?
-2. How do we find the MRV set of an expression?
-3. How do we compute series expansions?
-4. Why does the algorithm terminate?
-
-If you are interested, be sure to take a look at
-`Gruntz Thesis <http://www.cybertester.com/data/gruntz.pdf>`_.
-
-Reference
-"""""""""
-
-.. autofunction:: sympy.series.gruntz.gruntz
-
-.. autofunction:: sympy.series.gruntz.compare
-
-.. autofunction:: sympy.series.gruntz.rewrite
-
-.. autofunction:: sympy.series.gruntz.build_expression_tree
-
-.. autofunction:: sympy.series.gruntz.mrv_leadterm
-
-.. autofunction:: sympy.series.gruntz.calculate_series
-
-.. autofunction:: sympy.series.gruntz.limitinf
-
-.. autofunction:: sympy.series.gruntz.sign
-
-.. autofunction:: sympy.series.gruntz.mrv
-
-.. autofunction:: sympy.series.gruntz.mrv_max1
-
-.. autofunction:: sympy.series.gruntz.mrv_max3
-
-.. autoclass:: sympy.series.gruntz.SubsSet
-   :members:
-
-
-More Intuitive Series Expansion
--------------------------------
-
-This is achieved
-by creating a wrapper around Basic.series(). This allows for the use of
-series(x*cos(x),x), which is possibly more intuative than (x*cos(x)).series(x).
-
-Examples
-^^^^^^^^
-    >>> from sympy import Symbol, cos, series
-    >>> x = Symbol('x')
-    >>> series(cos(x),x)
-    1 - x**2/2 + x**4/24 + O(x**6)
-
-Reference
-^^^^^^^^^
-
-.. autofunction:: sympy.series.series.series
-
-Order Terms
------------
-
-This module also implements automatic keeping track of the order of your
-expansion.
-
-Examples
-^^^^^^^^
-     >>> from sympy import Symbol, Order
-     >>> x = Symbol('x')
-     >>> Order(x) + x**2
-     O(x)
-     >>> Order(x) + 1
-     1 + O(x)
-
-Reference
-^^^^^^^^^
-
-.. autoclass:: sympy.series.order.Order
-   :members:
-
-Series Acceleration
--------------------
-
-TODO
-
-Reference
-^^^^^^^^^
-
-.. autofunction:: sympy.series.acceleration.richardson
-
-.. autofunction:: sympy.series.acceleration.shanks
-
-Residues
---------
-
-TODO
-
-Reference
-^^^^^^^^^
-
-.. autofunction:: sympy.series.residues.residue
\ No newline at end of file
diff --git a/dev-py3k/_sources/modules/sets.txt b/dev-py3k/_sources/modules/sets.txt
deleted file mode 100644
index 7e3571db0e1..00000000000
--- a/dev-py3k/_sources/modules/sets.txt
+++ /dev/null
@@ -1,71 +0,0 @@
-Sets
-===========
-
-.. automodule:: sympy.core.sets
-
-Set
-^^^
-.. autoclass:: Set
-   :members:
-
-Elementary Sets
----------------
-
-Interval
-^^^^^^^^
-.. autoclass:: Interval
-   :members:
-
-FiniteSet
-^^^^^^^^^
-.. autoclass:: FiniteSet
-   :members:
-
-Compound Sets
--------------
-Union
-^^^^^
-.. autoclass:: Union
-   :members:
-
-Intersection
-^^^^^^^^^^^^
-.. autoclass:: Intersection
-   :members:
-
-ProductSet
-^^^^^^^^^^
-.. autoclass:: ProductSet
-   :members:
-
-Singleton Sets
---------------
-
-EmptySet
-^^^^^^^^
-.. autoclass:: EmptySet
-   :members:
-
-UniversalSet
-^^^^^^^^^^^^
-.. autoclass:: UniversalSet
-   :members:
-
-Special Sets
-------------
-.. automodule:: sympy.sets.fancysets
-
-Naturals
-^^^^^^^^
-.. autoclass:: Naturals
-   :members:
-
-Integers
-^^^^^^^^
-.. autoclass:: Integers
-   :members:
-
-TransformationSet
-^^^^^^^^^^^^^^^^^
-.. autoclass:: TransformationSet
-   :members:
diff --git a/dev-py3k/_sources/modules/simplify/hyperexpand.txt b/dev-py3k/_sources/modules/simplify/hyperexpand.txt
deleted file mode 100644
index 0a251df8541..00000000000
--- a/dev-py3k/_sources/modules/simplify/hyperexpand.txt
+++ /dev/null
@@ -1,634 +0,0 @@
-Details on the Hypergeometric Function Expansion Module
-#######################################################
-
-This page describes how the function :func:`hyperexpand` and related code
-work. For usage, see the documentation of the symplify module.
-
-Hypergeometric Function Expansion Algorithm
-*******************************************
-
-This section describes the algorithm used to expand hypergeometric functions.
-Most of it is based on the papers [Roach1996]_ and [Roach1997]_.
-
-Recall that the hypergeometric function is (initially) defined as
-
-.. math ::
-    {}_pF_q\left(\begin{matrix} a_1, \dots, a_p \\ b_1, \dots, b_q \end{matrix}
-                     \middle| z \right)
-        = \sum_{n=0}^\infty \frac{(a_1)_n \dots (a_p)_n}{(b_1)_n \dots (b_q)_n}
-                            \frac{z^n}{n!}.
-
-It turns out that there are certain differential operators that can change the
-`a_p` and `p_q` parameters by integers. If a sequence of such
-operators is known that converts the set of indices `a_r^0` and
-`b_s^0` into `a_p` and `b_q`, then we shall say the pair `a_p,
-b_q` is reachable from `a_r^0, b_s^0`. Our general strategy is thus as
-follows: given a set `a_p, b_q` of parameters, try to look up an origin
-`a_r^0, b_s^0` for which we know an expression, and then apply the
-sequence of differential operators to the known expression to find an
-expression for the Hypergeometric function we are interested in.
-
-Notation
-========
-
-In the following, the symbol `a` will always denote a numerator parameter
-and the symbol `b` will always denote a denominator parameter. The
-subscripts `p, q, r, s` denote vectors of that length, so e.g.
-`a_p` denotes a vector of `p` numerator parameters. The subscripts
-`i` and `j` denote "running indices", so they should usually be used in
-conjuction with a "for all `i`". E.g. `a_i < 4` for all `i`.
-Uppercase subscripts `I` and `J` denote a chosen, fixed index. So
-for example `a_I > 0` is true if the inequality holds for the one index
-`I` we are currently interested in.
-
-Incrementing and decrementing indices
-=====================================
-
-Suppose `a_i \ne 0`. Set `A(a_i) =
-\frac{z}{a_i}\frac{\mathrm{d}}{dz}+1`. It is then easy to show that
-`A(a_i) {}_p F_q\left({a_p \atop b_q} \middle| z \right) = {}_p F_q\left({a_p +
-e_i \atop b_q} \middle| z \right)`, where `e_i` is the i-th unit vector.
-Similarly for `b_j \ne 1` we set `B(b_j) = \frac{z}{b_j-1}
-\frac{\mathrm{d}}{dz}+1` and find `B(b_j) {}_p F_q\left({a_p \atop b_q}
-\middle| z \right) = {}_p F_q\left({a_p \atop b_q - e_i} \middle| z \right)`.
-Thus we can increment upper and decrement lower indices at will, as long as we
-don't go through zero. The `A(a_i)` and `B(b_j)` are called shift
-operators.
-
-It is also easy to show that `\frac{\mathrm{d}}{dz} {}_p F_q\left({a_p
-\atop b_q} \middle| z \right) = \frac{a_1 \dots a_p}{b_1 \dots b_q} {}_p
-F_q\left({a_p + 1 \atop b_q + 1} \middle| z \right)`, where `a_p + 1` is
-the vector `a_1 + 1, a_2 + 1, \dots` and similarly for `b_q + 1`.
-Combining this with the shift operators,  we arrive at one form of the
-Hypergeometric differential equation: `\left[ \frac{\mathrm{d}}{dz}
-\prod_{j=1}^q B(b_j) - \frac{a_1 \dots a_p}{(b_1-1) \dots (b_q-1)}
-\prod_{i=1}^p A(a_i) \right] {}_p F_q\left({a_p \atop b_q} \middle| z \right) =
-0`. This holds if all shift operators are defined, i.e. if no `a_i = 0`
-and no `b_j = 1`. Clearing denominators and multiplying through by z we
-arrive at the following equation: `\left[ z\frac{\mathrm{d}}{dz}
-\prod_{j=1}^q \left(z\frac{\mathrm{d}}{dz} + b_j-1 \right) - z \prod_{i=1}^p
-\left( z\frac{\mathrm{d}}{\mathrm{d}z} + a_i \right) \right] {}_p F_q\left({a_p
-\atop b_q} \middle| z\right) = 0`. Even though our derivation does not show it,
-it can be checked that this equation holds whenever the `{}_p F_q` is
-defined.
-
-Notice that, under suitable conditions on `a_I, b_J`, each of the
-operators `A(a_i)`, `B(b_j)` and
-`z\frac{\mathrm{d}}{\mathrm{d}z}` can be expressed in terms of `A(a_I)` or
-`B(b_J)`. Our next aim is to write the Hypergeometric differential
-equation as follows: `[X A(a_I) - r] {}_p F_q\left({a_p \atop b_q}
-\middle| z\right) = 0`, for some operator `X` and some constant `r`
-to be determined. If
-`r \ne 0`, then we can write this as `\frac{-1}{r} X {}_p F_q\left({a_p +
-e_I \atop b_q} \middle| z\right) = {}_p F_q\left({a_p \atop b_q} \middle|
-z\right)`, and so `\frac{-1}{r}X` undoes the shifting of `A(a_I)`,
-whence it will be called an inverse-shift operator.
-
-Now `A(a_I)` exists if `a_I \ne 0`, and then
-`z\frac{\mathrm{d}}{\mathrm{d}z} = a_I A(a_I) - a_I`. Observe also that all the
-operators `A(a_i)`, `B(b_j)` and
-`z\frac{\mathrm{d}}{\mathrm{d}z}` commute. We have `\prod_{i=1}^p \left(
-z\frac{\mathrm{d}}{\mathrm{d}z} + a_i \right) = \left(\prod_{i=1, i \ne I}^p
-\left( z\frac{\mathrm{d}}{\mathrm{d}z} + a_i \right)\right) a_I A(a_I)`, so
-this gives us the first half of `X`. The other half does not have such a
-nice expression. We find `z\frac{\mathrm{d}}{dz} \prod_{j=1}^q
-\left(z\frac{\mathrm{d}}{dz} + b_j-1 \right) = \left(a_I A(a_I) - a_I\right)
-\prod_{j=1}^q \left(a_I A(a_I) - a_I + b_j - 1\right)`. Since the first
-half had no constant term, we infer `r = -a_I\prod_{j=1}^q(b_j - 1
--a_I)`.
-
-This tells us under which conditions we can "un-shift" `A(a_I)`, namely
-when `a_I \ne 0` and `r \ne 0`. Substituting `a_I - 1` for
-`a_I` then tells us under what conditions we can decrement the index
-`a_I`. Doing a similar analysis for `B(a_J)`, we arrive at the
-following rules:
-
-* An index `a_I` can be decremented if `a_I \ne 1` and
-  `a_I \ne b_j` for all `b_j`.
-* An index `b_J` can be
-  incremented if `b_J \ne -1` and `b_J \ne a_i` for all
-  `a_i`.
-
-Combined with the conditions (stated above) for the existence of shift
-operators, we have thus established the rules of the game!
-
-
-Reduction of Order
-==================
-
-Notice that, quite trivially, if `a_I = b_J`, we have `{}_p
-F_q\left({a_p \atop b_q} \middle| z \right) = {}_{p-1} F_{q-1}\left({a_p^*
-\atop b_q^*} \middle| z \right)`, where `a_p^*` means `a_p` with
-`a_I` omitted, and similarly for `b_q^*`. We call this reduction of
-order.
-
-In fact, we can do even better. If `a_I - b_J \in \mathbb{Z}_{>0}`, then
-it is easy to see that `\frac{(a_I)_n}{(b_J)_n}` is actually a polynomial
-in `n`. It is also easy to see that
-`(z\frac{\mathrm{d}}{\mathrm{d}z})^k z^n = n^k z^n`. Combining these two remarks
-we find:
-
-  If `a_I - b_J \in \mathbb{Z}_{>0}`, then there exists a polynomial
-  `p(n) = p_0 + p_1 n + \dots` (of degree `a_I - b_J`) such
-  that `\frac{(a_I)_n}{(b_J)_n} = p(n)` and `{}_p F_q\left({a_p
-  \atop b_q} \middle| z \right) = \left(p_0 + p_1
-  z\frac{\mathrm{d}}{\mathrm{d}z} + p_2
-  \left(z\frac{\mathrm{d}}{\mathrm{d}z}\right)^2 + \dots \right) {}_{p-1}
-  F_{q-1}\left({a_p^* \atop b_q^*} \middle| z \right)`.
-
-Thus any set of parameters `a_p, b_q` is reachable from a set of
-parameters `c_r, d_s` where `c_i - d_j \in \mathbb{Z}` implies
-`c_i < d_j`. Such a set of parameters `c_r, d_s` is called
-suitable. Our database of known formulae should only contain suitable origins.
-The reasons are twofold: firstly, working from suitable origins is easier, and
-secondly, a formula for a non-suitable origin can be deduced from a lower order
-formula, and we should put this one into the database instead.
-
-Moving Around in the Parameter Space
-====================================
-
-It remains to investigate the following question: suppose `a_p, b_q` and
-`a_p^0, b_q^0` are both suitable, and also `a_i - a_i^0 \in
-\mathbb{Z}`, `b_j - b_j^0 \in \mathbb{Z}`. When is `a_p, b_q`
-reachable from `a_p^0, b_q^0`? It is clear that we can treat all
-parameters independently that are incongruent mod 1. So assume that `a_i`
-and `b_j` are congruent to `r` mod 1, for all `i` and
-`j`. The same then follows for `a_i^0` and `b_j^0`.
-
-If `r \ne 0`, then any such `a_p, b_q` is reachable from any
-`a_p^0, b_q^0`. To see this notice that there exist constants `c, c^0`,
-congruent mod 1, such that `a_i < c < b_j` for all `i` and
-`j`, and similarly `a_i^0 < c^0 < b_j^0`. If `n = c - c^0 > 0` then
-we first inverse-shift all the `b_j^0` `n` times up, and then
-similarly shift shift up all the `a_i^0` `n` times. If `n <
-0` then we first inverse-shift down the `a_i^0` and then shift down the
-`b_j^0`. This reduces to the case `c = c^0`. But evidently we can
-now shift or inverse-shift around the `a_i^0` arbitrarily so long as we
-keep them less than `c`, and similarly for the `b_j^0` so long as
-we keep them bigger than `c`. Thus `a_p, b_q` is reachable from
-`a_p^0, b_q^0`.
-
-If `r = 0` then the problem is slightly more involved. WLOG no parameter
-is zero. We now have one additional complication: no parameter can ever move
-through zero. Hence `a_p, b_q` is reachable from `a_p^0, b_q^0` if
-and only if the number of `a_i < 0` equals the number of `a_i^0 <
-0`, and similarly for the `b_i` and `b_i^0`. But in a suitable set
-of parameters, all `b_j > 0`! This is because the Hypergeometric function
-is undefined if one of the `b_j` is a non-positive integer and all
-`a_i` are smaller than the `b_j`. Hence the number of `b_j \le 0` is
-always zero.
-
-We can thus associate to every suitable set of parameters `a_p, b_q`,
-where no `a_i = 0`, the following invariants:
-
-    * For every `r \in [0, 1)` the number `\alpha_r` of parameters
-      `a_i \equiv r \pmod{1}`, and similarly the number `\beta_r`
-      of parameters `b_i \equiv r \pmod{1}`.
-    * The number `\gamma`
-      of integers `a_i` with `a_i < 0`.
-
-The above reasoning shows that `a_p, b_q` is reachable from `a_p^0,
-b_q^0` if and only if the invariants `\alpha_r, \beta_r, \gamma` all
-agree. Thus in particular "being reachable from" is a symmetric relation on
-suitable parameters without zeros.
-
-
-Applying the Operators
-======================
-
-If all goes well then for a given set of parameters we find an origin in our
-database for which we have a nice formula. We now have to apply (potentially)
-many differential operators to it. If we do this blindly then the result will
-be very messy. This is because with Hypergeometric type functions, the
-derivative is usually expressed as a sum of two contiguous functions. Hence if
-we compute `N` derivatives, then the answer will involve `2N`
-contiguous functions! This is clearly undesirable. In fact we know from the
-Hypergeometric differential equation that we need at most `\max(p, q+1)`
-contiguous functions to express all derivatives.
-
-Hence instead of differentiating blindly, we will work with a
-`\mathbb{C}(z)`-module basis: for an origin `a_r^0, b_s^0` we either store
-(for particularly pretty answers) or compute a set of `N` functions
-(typically `N = \max(r, s+1)`) with the property that the derivative of
-any of them is a `\mathbb{C}(z)`-linear combination of them. In formulae,
-we store a vector `B` of `N` functions, a matrix `M` and a
-vector `C` (the latter two with entries in `\mathbb{C}(z)`), with
-the following properties:
-
-* `{}_r F_s\left({a_r^0 \atop b_s^0} \middle| z \right) = C B`
-* `z\frac{\mathrm{d}}{\mathrm{d}z} B = M B`.
-
-Then we can compute as many derivatives as we want and we will always end up
-with `\mathbb{C}(z)`-linear combination of at most `N` special
-functions.
-
-As hinted above, `B`, `M` and `C` can either all be stored
-(for particularly pretty answers) or computed from a single `{}_p F_q`
-formula.
-
-
-Loose Ends
-==========
-
-This describes the bulk of the hypergeometric function algorithm. There a few
-further tricks, described in the hyperexpand.py source file. The extension to
-Meijer G-functions is also described there.
-
-Meijer G-Functions of Finite Confluence
-***************************************
-
-Slater's theorem essentially evaluates a `G`-function as a sum of residues.
-If all poles are simple, the resulting series can be recognised as
-hypergeometric series. Thus a `G`-function can be evaluated as a sum of
-Hypergeometric functions.
-
-If the poles are not simple, the resulting series are not hypergeometric. This
-is known as the "confluent" or "logarithmic" case (the latter because the
-resulting series tend to contain logarithms). The answer depends in a
-complicated way on the multiplicities of various poles, and there is no
-accepted notation for representing it (as far as I know).
-However if there are only finitely many
-multiple poles, we can evaluate the `G` function as a sum of hypergeometric
-functions, plus finitely many extra terms. I could not find any good reference
-for this, which is why I work it out here.
-
-Recall the general setup. We define
-
-.. math::
-    G(z) = \frac{1}{2\pi i} \int_L \frac{\prod_{j=1}^m \Gamma(b_j - s)
-      \prod_{j=1}^n \Gamma(1 - a_j + s)}{\prod_{j=m+1}^q \Gamma(1 - b_j + s)
-      \prod_{j=n+1}^p \Gamma(a_j - s)} z^s \mathrm{d}s,
-
-where `L` is a contour starting and ending at `+\infty`, enclosing all of the
-poles of `\Gamma(b_j - s)` for `j = 1, \dots, n` once in the negative
-direction, and no other poles. Also the integral is assumed absolutely
-convergent.
-
-In what follows, for any complex numbers `a, b`, we write `a \equiv b \pmod{1}` if
-and only if there exists an integer `k` such that `a - b = k`. Thus there are
-double poles iff `a_i \equiv a_j \pmod{1}` for some `i \ne j \le n`.
-
-We now assume that whenever `b_j \equiv a_i \pmod{1}` for `i \le m`, `j > n`
-then `b_j < a_i`. This means that no quotient of the relevant gamma functions
-is a polynomial, and can always be achieved by "reduction of order". Fix a
-complex number `c` such that `\{b_i | b_i \equiv c \pmod{1}, i \le  m\}` is
-not empty. Enumerate this set as `b, b+k_1, \dots, b+k_u`, with `k_i`
-non-negative integers. Enumerate similarly
-`\{a_j | a_j \equiv c \pmod{1}, j > n\}` as `b + l_1, \dots, b + l_v`.
-Then `l_i > k_j` for all `i, j`. For finite confluence, we need to assume
-`v \ge u` for all such `c`.
-
-Let `c_1, \dots, c_w` be distinct `\pmod{1}` and exhaust the congruence classes
-of the `b_i`. I claim
-
-.. math :: G(z) = -\sum_{j=1}^w (F_j(z) + R_j(z)),
-
-where `F_j(z)` is a hypergeometric function and `R_j(z)` is a finite sum, both
-to be specified later. Indeed corresponding to every `c_j` there is
-a sequence of poles, at mostly finitely many of them multiple poles. This is where
-the `j`-th term comes from.
-
-Hence fix again `c`, enumerate the relevant `b_i` as
-`b, b + k_1, \dots, b + k_u`. We will look at the `a_j` corresponding to
-`a + l_1, \dots, a + l_u`. The other `a_i` are not treated specially. The
-corresponding gamma functions have poles at (potentially) `s = b + r` for
-`r = 0, 1, \dots`. For `r \ge l_u`, pole of the integrand is simple. We thus set
-
-.. math :: R(z) = \sum_{r=0}^{l_u - 1} res_{s = r + b}.
-
-We finally need to investigate the other poles. Set `r = l_u + t`, `t \ge 0`.
-A computation shows
-
-.. math ::
-       \frac{\Gamma(k_i - l_u - t)}{\Gamma(l_i - l_u - t)}
-            = \frac{1}{(k_i - l_u - t)_{l_i - k_i}}
-            = \frac{(-1)^{\delta_i}}{(l_u - l_i + 1)_{\delta_i}}
-              \frac{(l_u - l_i + 1)_t}{(l_u - k_i + 1)_t},
-
-where `\delta_i = l_i - k_i`.
-
-Also
-
-.. math ::
-    \Gamma(b_j - l_u - b - t) =
-        \frac{\Gamma(b_j - l_u - b)}{(-1)^t(l_u + b + 1 - b_j)_t}, \\
-
-    \Gamma(1 - a_j + l_u + b + t) =
-        \Gamma(1 - a_j + l_u + b) (1 - a_j + l_u + b)_t
-
-and
-
-.. math ::
-    res_{s = b + l_u + t} \Gamma(b - s) = -\frac{(-1)^{l_u + t}}{(l_u + t)!}
-              = -\frac{(-1)^{l_u}}{l_u!} \frac{(-1)^t}{(l_u+1)_t}.
-
-Hence
-
-.. math ::
-    res_{s = b + l_u + t} =& -z^{b + l_u}
-       \frac{(-1)^{l_u}}{l_u!}
-       \prod_{i=1}^{u} \frac{(-1)^{\delta_i}}{(l_u - k_i + 1)_{\delta_i}}
-       \frac{\prod_{j=1}^n \Gamma(1 - a_j + l_u + b)
-             \prod_{j=1}^m \Gamma(b_j - l_u - b)^*}
-            {\prod_{j=n+1}^p \Gamma(a_j - l_u - b)^* \prod_{j=m+1}^q
-             \Gamma(1 - b_j + l_u + b)}
-       \\ &\times
-       z^t
-       \frac{(-1)^t}{(l_u+1)_t}
-       \prod_{i=1}^{u} \frac{(l_u - l_i + 1)_t}{(l_u - k_i + 1)_t}
-       \frac{\prod_{j=1}^n (1 - a_j + l_u + b)_t
-             \prod_{j=n+1}^p (-1)^t (l_u + b + 1 - a_j)_t^*}
-            {\prod_{j=1}^m (-1)^t (l_u + b + 1 - b_j)_t^*
-             \prod_{j=m+1}^q (1 - b_j + l_u + b)_t},
-
-where the `*` means to omit the terms we treated specially.
-
-We thus arrive at
-
-.. math ::
-    F(z) = C \times {}_{p+1}F_{q}\left(
-        \begin{matrix} 1, (1 + l_u - l_i), (1 + l_u + b - a_i)^* \\
-                       1 + l_u, (1 + l_u - k_i), (1 + l_u + b - b_i)^*
-        \end{matrix} \middle| (-1)^{p-m-n} z\right),
-
-where `C` designates the factor in the residue independent of `t`.
-(This result can also be written in slightly simpler form by converting
-all the `l_u` etc back to `a_* - b_*`, but doing so is going to require more
-notation still and is not helpful for computation.)
-
-Extending The Hypergeometric Tables
-***********************************
-
-Adding new formulae to the tables is straightforward. At the top of the file
-``sympy/simplify/hyperexpand.py``, there is a function called
-:func:`add_formulae`. Nested in it are defined two helpers,
-``add(ap, bq, res)`` and ``addb(ap, bq, B, C, M)``, as well as dummys
-``a``, ``b``, ``c``, and ``z``.
-
-The first step in adding a new formula is by using ``add(ap, bq, res)``. This
-declares ``hyper(ap, bq, z) == res``. Here ``ap`` and ``bq`` may use the
-dummys ``a``, ``b``, and ``c`` as free symbols. For example the well-known formula
-`\sum_0^\infty \frac{(-a)_n z^n}{n!} = (1-z)^a` is declared by the following
-line: ``add((-a, ), (), (1-z)**a)``.
-
-From the information provided, the matrices `B`, `C` and `M` will be computed,
-and the formula is now available when expanding hypergeometric functions.
-Next the test file ``sympy/simplify/tests/test_hyperexpand.py`` should be run,
-in particular the test :func:`test_formulae`. This will test the newly added
-formula numerically. If it fails, there is (presumably) a typo in what was
-entered.
-
-Since all newly-added formulae are probably relatively complicated, chances
-are that the automatically computed basis is rather suboptimal (there is no
-good way of testing this, other than observing very messy output). In this
-case the matrices `B`, `C` and `M` should be computed by hand. Then the helper
-``addb`` can be used to declare a hypergeometric formula with hand-computed
-basis.
-
-An example
-==========
-
-Because this explanation so far might be very theoretical and difficult to
-understand, we walk through an explicit example now. We take the Fresnel
-function `C(z)` which obeys the following hypergeometric representation:
-
-.. math ::
-    C(z) = z \cdot {}_{1}F_{2}\left.\left(
-        \begin{matrix} \frac{1}{4} \\
-                       \frac{1}{2}, \frac{5}{4}
-        \end{matrix} \right| -\frac{\pi^2 z^4}{16}\right) \,.
-
-First we try to add this formula to the lookup table by using the
-(simpler) function ``add(ap, bq, res)``. The first two arguments
-are simply the lists containing the parameter sets of `{}_{1}F_{2}`.
-The ``res`` argument is a little bit more complicated. We only know
-`C(z)` in terms of `{}_{1}F_{2}(\ldots | f(z))` with `f`
-a function of `z`, in our case
-
-.. math ::
-   f(z) = -\frac{\pi^2 z^4}{16} \,.
-
-What we need is a formula where the hypergeometric function has
-only `z` as argument `{}_{1}F_{2}(\ldots | z)`. We
-introduce the new complex symbol `w` and search for a function
-`g(w)` such that
-
-.. math ::
-   f(g(w)) = w
-
-holds. Then we can replace every `z` in `C(z)` by `g(w)`.
-In the case of our example the function `g` could look like
-
-.. math ::
-   g(w) = \frac{2}{\sqrt{\pi}} \exp\left(\frac{i \pi}{4}\right) w^{\frac{1}{4}} \,.
-
-We get these functions mainly by guessing and testing the result. Hence
-we proceed by computing `f(g(w))` (and simplifying naively)
-
-.. math ::
-   f(g(w)) &= -\frac{\pi^2 g(w)^4}{16} \\
-           &= -\frac{\pi^2 g\left(\frac{2}{\sqrt{\pi}} \exp\left(\frac{i \pi}{4}\right) w^{\frac{1}{4}}\right)^4}{16} \\
-           &= -\frac{\pi^2 \frac{2^4}{\sqrt{\pi}^4} \exp\left(\frac{i \pi}{4}\right)^4 {w^{\frac{1}{4}}}^4}{16} \\
-           &= -\exp\left(i \pi\right) w \\
-           &= w
-
-and indeed get back `w`. (In case of branched functions we have to be
-aware of branch cuts. In that case we take `w` to be a positive real
-number and check the formula. If what we have found works for positive
-`w`, then just replace :func:`exp` inside any branched function by
-:func:`exp\_polar` and what we get is right for `all` `w`.) Hence
-we can write the formula as
-
-.. math ::
-   C(g(w)) = g(w) \cdot {}_{1}F_{2}\left.\left(
-        \begin{matrix} \frac{1}{4} \\
-                       \frac{1}{2}, \frac{5}{4}
-        \end{matrix} \right| w\right) \,.
-
-and trivially
-
-.. math ::
-   {}_{1}F_{2}\left.\left(
-   \begin{matrix} \frac{1}{4} \\
-                  \frac{1}{2}, \frac{5}{4}
-   \end{matrix} \right| w\right)
-   = \frac{C(g(w))}{g(w)}
-   = \frac{C\left(\frac{2}{\sqrt{\pi}} \exp\left(\frac{i \pi}{4}\right) w^{\frac{1}{4}}\right)}
-          {\frac{2}{\sqrt{\pi}} \exp\left(\frac{i \pi}{4}\right) w^{\frac{1}{4}}}
-
-which is exactly what is needed for the third paramenter,
-``res``, in ``add``. Finally, the whole function call to add
-this rule to the table looks like::
-
-  add([S(1)/4],
-      [S(1)/2, S(5)/4],
-      fresnelc(exp(pi*I/4)*root(z,4)*2/sqrt(pi)) / (exp(pi*I/4)*root(z,4)*2/sqrt(pi))
-     )
-
-Using this rule we will find that it works but the results are not really nice
-in terms of simplicity and number of special function instances included.
-We can obtain much better results by adding the formula to the lookup table
-in another way. For this we use the (more complicated) function ``addb(ap, bq, B, C, M)``.
-The first two arguments are again the lists containing the parameter sets of
-`{}_{1}F_{2}`. The remaining three are the matrices mentioned earlier
-on this page.
-
-We know that the `n = \max{\left(p, q+1\right)}`-th derivative can be
-expressed as a linear combination of lower order derivatives. The matrix
-`B` contains the basis `\{B_0, B_1, \ldots\}` and is of shape
-`n \times 1`. The best way to get `B_i` is to take the first
-`n = \max(p, q+1)` derivatives of the expression for `{}_p F_q`
-and take out usefull pieces. In our case we find that
-`n = \max{\left(1, 2+1\right)} = 3`. For computing the derivatives,
-we have to use the operator `z\frac{\mathrm{d}}{\mathrm{d}z}`. The
-first basis element `B_0` is set to the expression for `{}_1 F_2`
-from above:
-
-.. math ::
-   B_0 = \frac{ \sqrt{\pi} \exp\left(-\frac{\mathbf{\imath}\pi}{4}\right)
-   C\left( \frac{2}{\sqrt{\pi}} \exp\left(\frac{\mathbf{\imath}\pi}{4}\right) z^{\frac{1}{4}}\right)}
-   {2 z^{\frac{1}{4}}}
-
-Next we compute `z\frac{\mathrm{d}}{\mathrm{d}z} B_0`. For this we can
-directly use SymPy!
-
-   >>> from sympy import Symbol, sqrt, exp, I, pi, fresnelc, root, diff, expand
-   >>> z = Symbol("z")
-   >>> B0 = sqrt(pi)*exp(-I*pi/4)*fresnelc(2*root(z,4)*exp(I*pi/4)/sqrt(pi))/\
-   ...          (2*root(z,4))
-   >>> z * diff(B0, z)
-   z*(cosh(2*sqrt(z))/(4*z) -
-   sqrt(pi)*exp(-I*pi/4)*fresnelc(2*z**(1/4)*exp(I*pi/4)/sqrt(pi))/(8*z**(5/4)))
-   >>> expand(_)
-   cosh(2*sqrt(z))/4 - sqrt(pi)*exp(-I*pi/4)*fresnelc(2*z**(1/4)*exp(I*pi/4)/sqrt(pi))/(8*z**(1/4))
-
-Formatting this result nicely we obtain
-
-.. math ::
-   B_1^\prime =
-   - \frac{1}{4} \frac{
-     \sqrt{\pi}
-     \exp\left(-\frac{\mathbf{\imath}\pi}{4}\right)
-     C\left( \frac{2}{\sqrt{\pi}} \exp\left(\frac{\mathbf{\imath}\pi}{4}\right) z^{\frac{1}{4}}\right)
-   }
-   {2 z^{\frac{1}{4}}}
-   + \frac{1}{4} \cosh{\left( 2 \sqrt{z} \right )}
-
-Computing the second derivative we find
-
-   >>> from sympy import (Symbol, cosh, sqrt, pi, exp, I, fresnelc, root,
-   ...                    diff, expand)
-   >>> z = Symbol("z")
-   >>> B1prime = cosh(2*sqrt(z))/4 - sqrt(pi)*exp(-I*pi/4)*\
-   ...           fresnelc(2*root(z,4)*exp(I*pi/4)/sqrt(pi))/(8*root(z,4))
-   >>> z * diff(B1prime, z)
-   z*(-cosh(2*sqrt(z))/(16*z) + sinh(2*sqrt(z))/(4*sqrt(z)) + sqrt(pi)*exp(-I*pi/4)*fresnelc(2*z**(1/4)*exp(I*pi/4)/sqrt(pi))/(32*z**(5/4)))
-   >>> expand(_)
-   sqrt(z)*sinh(2*sqrt(z))/4 - cosh(2*sqrt(z))/16 + sqrt(pi)*exp(-I*pi/4)*fresnelc(2*z**(1/4)*exp(I*pi/4)/sqrt(pi))/(32*z**(1/4))
-
-which can be printed as
-
-.. math ::
-   B_2^\prime =
-   \frac{1}{16} \frac{
-     \sqrt{\pi}
-     \exp\left(-\frac{\mathbf{\imath}\pi}{4}\right)
-     C\left( \frac{2}{\sqrt{\pi}} \exp\left(\frac{\mathbf{\imath}\pi}{4}\right) z^{\frac{1}{4}}\right)
-   }
-   {2 z^{\frac{1}{4}}}
-   - \frac{1}{16} \cosh{\left(2\sqrt{z}\right)}
-   + \frac{1}{4} \sinh{\left(2\sqrt{z}\right)} \sqrt{z}
-
-We see the common pattern and can collect the pieces. Hence it makes sense to
-choose `B_1` and `B_2` as follows
-
-.. math ::
-   B =
-   \left( \begin{matrix}
-     B_0 \\ B_1 \\ B_2
-   \end{matrix} \right)
-   =
-   \left( \begin{matrix}
-     \frac{
-       \sqrt{\pi}
-       \exp\left(-\frac{\mathbf{\imath}\pi}{4}\right)
-       C\left( \frac{2}{\sqrt{\pi}} \exp\left(\frac{\mathbf{\imath}\pi}{4}\right) z^{\frac{1}{4}}\right)
-     }{2 z^{\frac{1}{4}}} \\
-     \cosh\left(2\sqrt{z}\right) \\
-     \sinh\left(2\sqrt{z}\right) \sqrt{z}
-   \end{matrix} \right)
-
-(This is in contrast to the basis `B = \left(B_0, B_1^\prime, B_2^\prime\right)` that would
-have been computed automatically if we used just ``add(ap, bq, res)``.)
-
-Because it must hold that `{}_p F_q\left(\cdots \middle| z \right) = C B`
-the entries of `C` are obviously
-
-.. math ::
-   C =
-   \left( \begin{matrix}
-     1 \\ 0 \\ 0
-   \end{matrix} \right)
-
-Finally we have to compute the entries of the `3 \times 3` matrix `M`
-such that `z\frac{\mathrm{d}}{\mathrm{d}z} B = M B` holds. This is easy.
-We already computed the first part `z\frac{\mathrm{d}}{\mathrm{d}z} B_0`
-above. This gives us the first row of `M`. For the second row we have:
-
-   >>> from sympy import Symbol, cosh, sqrt, diff
-   >>> z = Symbol("z")
-   >>> B1 = cosh(2*sqrt(z))
-   >>> z * diff(B1, z)
-   sqrt(z)*sinh(2*sqrt(z))
-
-and for the third one
-
-   >>> from sympy import Symbol, sinh, sqrt, expand, diff
-   >>> z = Symbol("z")
-   >>> B2 = sinh(2*sqrt(z))*sqrt(z)
-   >>> expand(z * diff(B2, z))
-   sqrt(z)*sinh(2*sqrt(z))/2 + z*cosh(2*sqrt(z))
-
-Now we have computed the entries of this matrix to be
-
-.. math ::
-   M =
-   \left( \begin{matrix}
-     -\frac{1}{4} & \frac{1}{4} & 0 \\
-     0            & 0           & 1 \\
-     0            & z           & \frac{1}{2} \\
-   \end{matrix} \right)
-
-Note that the entries of `C` and `M` should typically be
-rational functions in `z`, with rational coefficients. This is all
-we need to do in order to add a new formula to the lookup table for
-``hyperexpand``.
-
-Implemented Hypergeometric Formulae
-***********************************
-
-A vital part of the algorithm is a relatively large table of hypergeometric
-function representations. The following automatically generated list contains
-all the representations implemented in SymPy (of course many more are
-derived from them). These formulae are mostly taken from [Luke1969]_ and
-[Prudnikov1990]_. They are all tested numerically.
-
-.. automodule:: sympy.simplify.hyperexpand_doc
-
-References
-**********
-
-.. [Roach1996] Kelly B. Roach.  Hypergeometric Function Representations.
-      In: Proceedings of the 1996 International Symposium on Symbolic and
-      Algebraic Computation, pages 301-308, New York, 1996. ACM.
-
-.. [Roach1997] Kelly B. Roach.  Meijer G Function Representations.
-      In: Proceedings of the 1997 International Symposium on Symbolic and
-      Algebraic Computation, pages 205-211, New York, 1997. ACM.
-
-.. [Luke1969] Luke, Y. L. (1969), The Special Functions and Their
-              Approximations, Volume 1.
-
-.. [Prudnikov1990] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev (1990).
-     Integrals and Series: More Special Functions, Vol. 3,
-     Gordon and Breach Science Publisher.
diff --git a/dev-py3k/_sources/modules/simplify/simplify.txt b/dev-py3k/_sources/modules/simplify/simplify.txt
deleted file mode 100644
index 8b038f659a7..00000000000
--- a/dev-py3k/_sources/modules/simplify/simplify.txt
+++ /dev/null
@@ -1,129 +0,0 @@
-Simplify
-********
-
-.. currentmodule:: sympy.simplify.simplify
-
-simplify
---------
-.. autofunction:: simplify
-
-collect
--------
-.. autofunction:: collect
-
-.. autofunction:: rcollect
-
-separate
---------
-.. autofunction:: separate
-
-separatevars
-------------
-
-.. autofunction:: separatevars
-
-radsimp
--------
-.. autofunction:: radsimp
-
-ratsimp
--------
-.. autofunction:: ratsimp
-
-fraction
---------
-.. autofunction:: fraction
-
-trigsimp
---------
-.. autofunction:: trigsimp
-
-besselsimp
-----------
-.. autofunction:: besselsimp
-
-powsimp
--------
-.. autofunction:: powsimp
-
-combsimp
---------
-.. autofunction:: combsimp
-
-hypersimp
----------
-.. autofunction:: hypersimp
-
-hypersimilar
-------------
-.. autofunction:: hypersimilar
-
-nsimplify
----------
-.. autofunction:: nsimplify
-
-collect_sqrt
-------------
-.. autofunction:: collect_sqrt
-
-collect_const
--------------
-.. autofunction:: collect_const
-
-posify
-------
-.. autofunction:: posify
-
-powdenest
----------
-.. autofunction:: powdenest
-
-logcombine
-----------
-.. autofunction:: logcombine
-
-
-Square Root Denest
-------------------
-.. module:: sympy.simplify.sqrtdenest
-
-sqrtdenest
-^^^^^^^^^^
-.. autofunction:: sqrtdenest
-
-Common Subexpresion Elimination
--------------------------------
-.. module:: sympy.simplify.cse_main
-
-cse
-^^^
-.. autofunction:: cse
-
-Hypergeometric Function Expansion
----------------------------------
-.. module:: sympy.simplify.hyperexpand
-
-hyperexpand
-^^^^^^^^^^^
-.. autofunction:: hyperexpand
-
-Traversal Tools
----------------
-.. module:: sympy.simplify.traversaltools
-
-use
-^^^
-.. autofunction:: use
-
-EPath Tools
------------
-.. module:: sympy.simplify.epathtools
-
-EPath class
-^^^^^^^^^^^
-.. autoclass:: EPath
-   :members:
-
-epath
-^^^^^
-.. autofunction:: epath
diff --git a/dev-py3k/_sources/modules/solvers/ode.txt b/dev-py3k/_sources/modules/solvers/ode.txt
deleted file mode 100644
index 44a5be8802e..00000000000
--- a/dev-py3k/_sources/modules/solvers/ode.txt
+++ /dev/null
@@ -1,107 +0,0 @@
-.. _ode-docs:
-
-ODE
-===
-
-.. module::sympy.solvers.ode
-
-User Functions
---------------
-These are functions that are imported into the global namespace with ``from sympy import *``.  They are intended for user use.
-
-:func:`dsolve`
-^^^^^^^^^^^^^^
-.. autofunction:: sympy.solvers.ode.dsolve
-
-:func:`classify_ode`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: sympy.solvers.ode.classify_ode
-
-:func:`ode_order`
-^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: sympy.solvers.ode.ode_order
-
-:func:`checkodesol`
-^^^^^^^^^^^^^^^^^^^
-.. autofunction:: sympy.solvers.ode.checkodesol
-
-:func:`homogeneous_order`
-^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: sympy.solvers.ode.homogeneous_order
-
-Hint Methods
-------------
-These functions are intended for internal use by :func:`dsolve` and others.  Nonetheless, they contain useful information in their docstrings on the various ODE solving methods.
-
-:obj:`preprocess`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: sympy.solvers.ode.preprocess
-
-:obj:`odesimp`
-^^^^^^^^^^^^^^
-.. autofunction:: sympy.solvers.ode.odesimp
-
-:obj:`constant_renumber`
-^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: sympy.solvers.ode.constant_renumber
-
-:obj:`constantsimp`
-^^^^^^^^^^^^^^^^^^^
-.. autofunction:: sympy.solvers.ode.constantsimp
-
-:obj:`sol_simplicity`
-^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: sympy.solvers.ode.ode_sol_simplicity
-
-:obj:`1st_exact`
-^^^^^^^^^^^^^^^^
-.. autofunction:: sympy.solvers.ode.ode_1st_exact
-
-:obj:`1st_homogeneous_coeff_best`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: sympy.solvers.ode.ode_1st_homogeneous_coeff_best
-
-:obj:`1st_homogeneous_coeff_subs_dep_div_indep`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: sympy.solvers.ode.ode_1st_homogeneous_coeff_subs_dep_div_indep
-
-:obj:`1st_homogeneous_coeff_subs_indep_div_dep`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: sympy.solvers.ode.ode_1st_homogeneous_coeff_subs_indep_div_dep
-
-:obj:`1st_linear`
-^^^^^^^^^^^^^^^^^
-.. autofunction:: sympy.solvers.ode.ode_1st_linear
-
-:obj:`Bernoulli`
-^^^^^^^^^^^^^^^^
-.. autofunction:: sympy.solvers.ode.ode_Bernoulli
-
-:obj:`Liouville`
-^^^^^^^^^^^^^^^^
-.. autofunction:: sympy.solvers.ode.ode_Liouville
-
-:obj:`Riccati_special_minus2`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: sympy.solvers.ode.ode_Riccati_special_minus2
-
-:obj:`nth_linear_constant_coeff_homogeneous`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: sympy.solvers.ode.ode_nth_linear_constant_coeff_homogeneous
-
-:obj:`nth_linear_constant_coeff_undetermined_coefficients`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: sympy.solvers.ode.ode_nth_linear_constant_coeff_undetermined_coefficients
-
-:obj:`nth_linear_constant_coeff_variation_of_parameters`
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-.. autofunction:: sympy.solvers.ode.ode_nth_linear_constant_coeff_variation_of_parameters
-
-:obj:`separable`
-^^^^^^^^^^^^^^^^
-.. autofunction:: sympy.solvers.ode.ode_separable
-
-Information on the ode module
------------------------------
-
-.. automodule:: sympy.solvers.ode
diff --git a/dev-py3k/_sources/modules/solvers/solvers.txt b/dev-py3k/_sources/modules/solvers/solvers.txt
deleted file mode 100644
index b7f3682b4e5..00000000000
--- a/dev-py3k/_sources/modules/solvers/solvers.txt
+++ /dev/null
@@ -1,71 +0,0 @@
-Solvers
-==========
-
-.. module:: sympy.solvers
-
-The *solvers* module in SymPy implements methods for solving equations.
-
-Algebraic equations
---------------------
-
-Use :func:`solve` to solve algebraic equations. We suppose all equations are equaled to 0,
-so solving x**2 == 1 translates into the following code::
-
-    >>> from sympy.solvers import solve
-    >>> from sympy import Symbol
-    >>> x = Symbol('x')
-    >>> solve(x**2 - 1, x)
-    [-1, 1]
-
-The first argument for :func:`solve` is an equation (equaled to zero) and the second argument
-is the symbol that we want to solve the equation for.
-
-.. autofunction:: sympy.solvers.solvers.solve
-
-.. autofunction:: sympy.solvers.solvers.solve_linear
-
-.. autofunction:: sympy.solvers.solvers.solve_linear_system
-
-.. autofunction:: sympy.solvers.solvers.solve_linear_system_LU
-
-.. autofunction:: sympy.solvers.solvers.solve_undetermined_coeffs
-
-.. autofunction:: sympy.solvers.solvers.tsolve
-
-.. autofunction:: sympy.solvers.solvers.nsolve
-
-.. autofunction:: sympy.solvers.solvers.check_assumptions
-
-.. autofunction:: sympy.solvers.solvers.checksol
-
-Ordinary Differential equations (ODEs)
---------------------------------------
-
-See :ref:`ode-docs`.
-
-Partial Differential Equations (PDEs)
--------------------------------------
-
-.. autofunction:: sympy.solvers.pde.pde_separate
-
-.. autofunction:: sympy.solvers.pde.pde_separate_add
-
-.. autofunction:: sympy.solvers.pde.pde_separate_mul
-
-Recurrence Equtions
--------------------
-
-.. autofunction:: sympy.solvers.recurr.rsolve
-
-.. autofunction:: sympy.solvers.recurr.rsolve_poly
-
-.. autofunction:: sympy.solvers.recurr.rsolve_ratio
-
-.. autofunction:: sympy.solvers.recurr.rsolve_hyper
-
-Systems of Polynomial Equations
--------------------------------
-
-.. autofunction:: sympy.solvers.polysys.solve_poly_system
-
-.. autofunction:: sympy.solvers.polysys.solve_triangulated
diff --git a/dev-py3k/_sources/modules/statistics.txt b/dev-py3k/_sources/modules/statistics.txt
deleted file mode 100644
index 43cfdf736e7..00000000000
--- a/dev-py3k/_sources/modules/statistics.txt
+++ /dev/null
@@ -1,136 +0,0 @@
-Statistics
-==========
-
-.. module:: sympy.statistics
-
-Note: This module has been deprecated. See the stats module.
-
-The *statistics* module in SymPy implements standard probability distributions
-and related tools. Its contents can be imported with the following statement::
-
-    >>> from sympy import *
-    >>> from sympy.statistics import *
-    >>> init_printing(use_unicode=False, wrap_line=False, no_global=True)
-
-Normal distributions
---------------------
-
-``Normal(mu, sigma)`` creates a normal distribution with mean value ``mu`` and
-standard deviation ``sigma``. The ``Normal`` class defines several
-useful methods and properties. Various properties can be accessed directly as
-follows:
-
-    >>> N = Normal(0, 1)
-    >>> N.mean
-    0
-    >>> N.median
-    0
-    >>> N.variance
-    1
-    >>> N.stddev
-    1
-
-You can generate random numbers from the desired distribution with the
-``random`` method:
-
-    >>> N = Normal(10, 5)
-    >>> N.random() #doctest: +SKIP
-    4.914375200829805834246144514
-    >>> N.random() #doctest: +SKIP
-    11.84331557474637897087177407
-    >>> N.random() #doctest: +SKIP
-    17.22474580071733640806996846
-    >>> N.random() #doctest: +SKIP
-    9.864643097429464546621602494
-
-The probability density function (pdf) and cumulative distribution function
-(cdf) of a distribution can be computed, either in symbolic form or for
-particular values:
-
-    >>> N = Normal(1, 1)
-    >>> x = Symbol('x')
-    >>> N.pdf(1)
-       ___
-     \/ 2
-    --------
-        ____
-    2*\/ pi
-    >>> N.pdf(3).evalf()
-    0.0539909665131880
-    >>> N.cdf(x)
-       /  ___        \
-       |\/ 2 *(x - 1)|
-    erf|-------------|
-       \      2      /   1
-    ------------------ + -
-            2            2
-    >>> N.cdf(-oo), N.cdf(1), N.cdf(oo)
-    (0, 1/2, 1)
-    >>> N.cdf(5).evalf()
-    0.999968328758167
-
-The method ``probability`` gives the total probability on a given interval (a
-convenient alternative syntax for cdf(b)-cdf(a)):
-
-    >>> N = Normal(0, 1)
-    >>> N.probability(-oo, 0)
-    1/2
-    >>> N.probability(-1, 1)
-       /  ___\
-       |\/ 2 |
-    erf|-----|
-       \  2  /
-    >>> N.probability(-1, 1).evalf()
-    0.682689492137086
-
-You can also generate a symmetric confidence interval from a given desired
-confidence level (given as a fraction 0-1). For the normal distribution, 68%,
-95% and 99.7% confidence levels respectively correspond to approximately 1, 2
-and 3 standard deviations:
-
-    >>> N = Normal(0, 1)
-    >>> N.confidence(0.68)
-    (-0.994457883209753, 0.994457883209753)
-    >>> N.confidence(0.95)
-    (-1.95996398454005, 1.95996398454005)
-    >>> N.confidence(0.997)
-    (-2.96773792534178, 2.96773792534178)
-
-Plug the interval back in to see that the value is correct:
-
-    >>> N.probability(*N.confidence(0.95)).evalf()
-    0.950000000000000
-
-Other distributions
--------------------
-
-Besides the normal distribution, uniform continuous distributions are also
-supported. ``Uniform(a, b)`` represents the distribution with uniform
-probability on the interval [a, b] and zero probability everywhere else. The
-``Uniform`` class supports the same methods as the ``Normal`` class.
-
-Additional distributions, including support for arbitrary user-defined distributions, are planned for the future.
-
-API Reference
--------------
-
-Sample
-^^^^^^
-
-.. autoclass:: sympy.statistics.distributions.Sample
-   :members:
-
-Continuous Probability Distributions
-^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
-
-.. autoclass:: sympy.statistics.distributions.ContinuousProbability
-   :members:
-
-.. autoclass:: sympy.statistics.distributions.Normal
-   :members:
-
-.. autoclass:: sympy.statistics.distributions.Uniform
-   :members:
-
-.. autoclass:: sympy.statistics.distributions.PDF
-   :members:
diff --git a/dev-py3k/_sources/modules/stats.txt b/dev-py3k/_sources/modules/stats.txt
deleted file mode 100644
index 7ea6045312b..00000000000
--- a/dev-py3k/_sources/modules/stats.txt
+++ /dev/null
@@ -1,170 +0,0 @@
-Stats
-===========
-
-.. automodule:: sympy.stats
-
-Random Variable Types
-^^^^^^^^^^^^^^^^^^^^^
-
-Finite Types
----------------
-.. autofunction:: DiscreteUniform
-.. autofunction:: Die
-.. autofunction:: Bernoulli
-.. autofunction:: Coin
-.. autofunction:: Binomial
-.. autofunction:: Hypergeometric
-.. autofunction:: FiniteRV
-
-Continuous Types
--------------------
-
-.. autofunction:: Arcsin
-.. autofunction:: Benini
-.. autofunction:: Beta
-.. autofunction:: BetaPrime
-.. autofunction:: Cauchy
-.. autofunction:: Chi
-.. autofunction:: ChiNoncentral
-.. autofunction:: ChiSquared
-.. autofunction:: Dagum
-.. autofunction:: Erlang
-.. autofunction:: Exponential
-.. autofunction:: FDistribution
-.. autofunction:: FisherZ
-.. autofunction:: Frechet
-.. autofunction:: Gamma
-.. autofunction:: GammaInverse
-.. autofunction:: Kumaraswamy
-.. autofunction:: Laplace
-.. autofunction:: Logistic
-.. autofunction:: LogNormal
-.. autofunction:: Maxwell
-.. autofunction:: Nakagami
-.. autofunction:: Normal
-.. autofunction:: Pareto
-.. autofunction:: QuadraticU
-.. autofunction:: RaisedCosine
-.. autofunction:: Rayleigh
-.. autofunction:: StudentT
-.. autofunction:: Triangular
-.. autofunction:: Uniform
-.. autofunction:: UniformSum
-.. autofunction:: VonMises
-.. autofunction:: Weibull
-.. autofunction:: WignerSemicircle
-.. autofunction:: ContinuousRV
-
-Interface
-^^^^^^^^^
-
-.. autofunction:: P
-.. autofunction:: E
-.. autofunction:: density
-.. autofunction:: given
-.. autofunction:: where
-.. autofunction:: variance
-.. autofunction:: std
-.. autofunction:: sample
-.. autofunction:: sample_iter
-
-Mechanics
-^^^^^^^^^
-.. module:: sympy.stats.rv
-
-SymPy Stats employs a relatively complex class hierarchy.
-
-RandomDomains are a mapping of variables to possible values. For example we
-might say that the symbol Symbol('x') can take on the values {1,2,3,4,5,6}.
-
-.. class:: RandomDomain
-
-A PSpace, or Probability Space, combines a RandomDomain with a density to
-provide probabilistic information. For example the above domain could be
-enhanced by a finite density {1:1/6, 2:1/6, 3:1/6, 4:1/6, 5:1/6, 6:1/6} to
-fully define the roll of a fair die named 'x'.
-
-.. class:: PSpace
-
-A RandomSymbol represents the PSpace's symbol 'x' inside of SymPy expressions.
-
-.. class:: RandomSymbol
-
-The RandomDomain and PSpace classes are almost never directly instantiated.
-Instead they are subclassed for a variety of situations.
-
-RandomDomains and PSpaces must be sufficiently general to represent domains and
-spaces of several variables with arbitrarily complex densities. This generality
-is often unnecessary. Instead we often build SingleDomains and SinglePSpaces to
-represent single, univariate events and processes such as a single die or a
-single normal variable.
-
-.. class:: SinglePSpace
-.. class:: SingleDomain
-
-
-Another common case is to collect together a set of such univariate random
-variables. A collection of independent SinglePSpaces or SingleDomains can be
-brought together to form a ProductDomain or ProductPSpace. These objects would
-be useful in representing three dice rolled together for example.
-
-.. class:: ProductDomain
-
-.. class:: ProductPSpace
-
-The Conditional adjective is added whenever we add a global condition to a
-RandomDomain or PSpace. A common example would be three independent dice where
-we know their sum to be greater than 12.
-
-.. class:: ConditionalDomain
-
-We specialize further into Finite and Continuous versions of these classes to
-represent finite (such as dice) and continuous (such as normals) random
-variables.
-
-.. module:: sympy.stats.frv
-.. class:: FiniteDomain
-.. class:: FinitePSpace
-
-.. module:: sympy.stats.crv
-.. class:: ContinuousDomain
-.. class:: ContinuousPSpace
-
-Additionally there are a few specialized classes that implement certain common
-random variable types. There is for example a DiePSpace that implements
-SingleFinitePSpace and a NormalPSpace that implements SingleContinuousPSpace.
-
-.. module:: sympy.stats.frv_types
-.. class:: DiePSpace
-
-.. module:: sympy.stats.crv_types
-.. class:: NormalPSpace
-
-RandomVariables can be extracted from these objects using the PSpace.values
-method.
-
-As previously mentioned SymPy Stats employs a relatively complex class
-structure. Inheritance is widely used in the implementation of end-level
-classes. This tactic was chosen to balance between the need to allow SymPy to
-represent arbitrarily defined random variables and optimizing for common cases.
-This complicates the code but is structured to only be important to those
-working on extending SymPy Stats to other random variable types.
-
-Users will not use this class structure. Instead these mechanics are exposed
-through variable creation functions Die, Coin, FiniteRV, Normal, Exponential,
-etc.... These build the appropriate SinglePSpaces and return the corresponding
-RandomVariable. Conditional and Product spaces are formed in the natural
-construction of SymPy expressions and the use of interface functions E, Given,
-Density, etc....
-
-
-.. function:: sympy.stats.Die
-.. function:: sympy.stats.Normal
-
-There are some additional functions that may be useful. They are largely used
-internally.
-
-
-.. autofunction:: sympy.stats.rv.random_symbols
-.. autofunction:: sympy.stats.rv.pspace
-.. autofunction:: sympy.stats.rv.rs_swap
diff --git a/dev-py3k/_sources/modules/tensor/index.txt b/dev-py3k/_sources/modules/tensor/index.txt
deleted file mode 100644
index 853f490aef7..00000000000
--- a/dev-py3k/_sources/modules/tensor/index.txt
+++ /dev/null
@@ -1,16 +0,0 @@
-.. _tensor_module:
-
-=============
-Tensor Module
-=============
-
-.. automodule:: sympy.tensor
-
-Contents
-========
-
-.. toctree::
-    :maxdepth: 3
-
-    indexed.rst
-    index_methods.rst
diff --git a/dev-py3k/_sources/modules/tensor/index_methods.txt b/dev-py3k/_sources/modules/tensor/index_methods.txt
deleted file mode 100644
index 1fbff8cb511..00000000000
--- a/dev-py3k/_sources/modules/tensor/index_methods.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-=======
-Methods
-=======
-
-.. automodule:: sympy.tensor.index_methods
-   :members:
diff --git a/dev-py3k/_sources/modules/tensor/indexed.txt b/dev-py3k/_sources/modules/tensor/indexed.txt
deleted file mode 100644
index d9ee71fff60..00000000000
--- a/dev-py3k/_sources/modules/tensor/indexed.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-===============
-Indexed Objects
-===============
-
-.. automodule:: sympy.tensor.indexed
-   :members:
diff --git a/dev-py3k/_sources/modules/utilities/autowrap.txt b/dev-py3k/_sources/modules/utilities/autowrap.txt
deleted file mode 100644
index eb8a699f26a..00000000000
--- a/dev-py3k/_sources/modules/utilities/autowrap.txt
+++ /dev/null
@@ -1,53 +0,0 @@
-===============
-Autowrap Module
-===============
-
-The autowrap module works very well in tandem with the Indexed classes of the
-:ref:`tensor_module`.  Here is a simple example that shows how to setup a binary
-routine that calculates a matrix-vector product.
-
->>> from sympy.utilities.autowrap import autowrap
->>> from sympy import symbols, IndexedBase, Idx, Eq
->>> A, x, y = list(map(IndexedBase, ['A', 'x', 'y']))
->>> m, n = symbols('m n', integer=True)
->>> i = Idx('i', m)
->>> j = Idx('j', n)
->>> instruction = Eq(y[i], A[i, j]*x[j]); instruction
-y[i] == A[i, j]*x[j]
-
-Because the code printers treat Indexed objects with repeated indices as a
-summation, the above equality instance will be translated to low-level code for
-a matrix vector product.  This is how you tell SymPy to generate the code,
-compile it and wrap it as a python function:
-
->>> matvec = autowrap(instruction)                 # doctest: +SKIP
-
-That's it.  Now let's test it with some numpy arrays.  The default wrapper
-backend is f2py.  The wrapper function it provides is set up to accept python
-lists, which it will silently convert to numpy arrays.  So we can test the
-matrix vector product like this:
-
->>> M = [[0, 1],
-...      [1, 0]]
->>> matvec(M, [2, 3])                              # doctest: +SKIP
-[ 3.  2.]
-
-Implementation details
-======================
-
-The autowrap module is implemented with a backend consisting of CodeWrapper
-objects.  The base class ``CodeWrapper`` takes care of details about module
-name, filenames and options.  It also contains the driver routine, which runs
-through all steps in the correct order, and also takes care of setting up and
-removing the temporary working directory.
-
-The actual compilation and wrapping is done by external resources, such as the
-system installed f2py command. The Cython backend runs a distutils setup script
-in a subprocess. Subclasses of CodeWrapper takes care of these
-backend-dependent details.
-
-API Reference
-=============
-
-.. automodule:: sympy.utilities.autowrap
-   :members:
diff --git a/dev-py3k/_sources/modules/utilities/codegen.txt b/dev-py3k/_sources/modules/utilities/codegen.txt
deleted file mode 100644
index 2897a9449b7..00000000000
--- a/dev-py3k/_sources/modules/utilities/codegen.txt
+++ /dev/null
@@ -1,50 +0,0 @@
-=======
-Codegen
-=======
-
-This module provides functionality to generate directly compilable code from
-SymPy expressions.  The ``codegen`` function is the user interface to the code
-generation functionality in SymPy.  Some details of the implementation is given
-below for advanced users that may want to use the framework directly.
-
-.. note:: The ``codegen`` callable is not in the sympy namespace automatically,
-   to use it you must first execute
-
-   >>> from sympy.utilities.codegen import codegen
-
-Impementation Details
-=====================
-
-Here we present the most important pieces of the internal structure, as
-advanced users may want to use it directly, for instance by subclassing a code
-generator for a specialized application.  **It is very likely that you would
-prefer to use the codegen() function documented above.**
-
-Basic assumptions:
-
-* A generic Routine data structure describes the routine that must be translated
-  into C/Fortran/... code. This data structure covers all features present in
-  one or more of the supported languages.
-
-* Descendants from the CodeGen class transform multiple Routine instances into
-  compilable code. Each derived class translates into a specific language.
-
-* In many cases, one wants a simple workflow. The friendly functions in the last
-  part are a simple api on top of the Routine/CodeGen stuff. They are easier to
-  use, but are less powerful.
-
-Routine
-=======
-
-The Routine class is a very important piece of the codegen module. Viewing the
-codegen utility as a translator of mathematical expressions into a set of
-statements in a programming language, the Routine instances are responsible for
-extracting and storing information about how the math can be encapsulated in a
-function call.  Thus, it is the Routine constructor that decides what arguments
-the routine will need and if there should be a return value.
-
-API Reference
-=============
-
-.. automodule:: sympy.utilities.codegen
-   :members:
diff --git a/dev-py3k/_sources/modules/utilities/cythonutils.txt b/dev-py3k/_sources/modules/utilities/cythonutils.txt
deleted file mode 100644
index d733adbffe8..00000000000
--- a/dev-py3k/_sources/modules/utilities/cythonutils.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-================
-Cython Utilities
-================
-
-.. automodule:: sympy.utilities.cythonutils
-   :members:
diff --git a/dev-py3k/_sources/modules/utilities/decorator.txt b/dev-py3k/_sources/modules/utilities/decorator.txt
deleted file mode 100644
index ee279faf4b6..00000000000
--- a/dev-py3k/_sources/modules/utilities/decorator.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-=========
-Decorator
-=========
-
-.. automodule:: sympy.utilities.decorator
-   :members:
diff --git a/dev-py3k/_sources/modules/utilities/index.txt b/dev-py3k/_sources/modules/utilities/index.txt
deleted file mode 100644
index 0b507bfc7c2..00000000000
--- a/dev-py3k/_sources/modules/utilities/index.txt
+++ /dev/null
@@ -1,29 +0,0 @@
-.. _utilities-docs:
-
-=========
-Utilities
-=========
-
-.. TODO: add compilef.rst and benchmarking.rst when they're fixed
-
-.. automodule:: sympy.utilities
-
-Contents:
-
-.. toctree::
-   :maxdepth: 2
-
-   autowrap.rst
-   codegen.rst
-   cythonutils.rst
-   decorator.rst
-   iterables.rst
-   lambdify.rst
-   memoization.rst
-   misc.rst
-   pkgdata.rst
-   pytest.rst
-   randtest.rst
-   runtests.rst
-   source.rst
-   timeutils.rst
diff --git a/dev-py3k/_sources/modules/utilities/iterables.txt b/dev-py3k/_sources/modules/utilities/iterables.txt
deleted file mode 100644
index f3376b148a5..00000000000
--- a/dev-py3k/_sources/modules/utilities/iterables.txt
+++ /dev/null
@@ -1,91 +0,0 @@
-=========
-Iterables
-=========
-
-cartes
-------
-
-Returns the cartesian product of sequences as a generator.
-
-Examples::
-    >>> from sympy.utilities.iterables import cartes
-    >>> list(cartes([1,2,3], 'ab'))
-    [(1, 'a'), (1, 'b'), (2, 'a'), (2, 'b'), (3, 'a'), (3, 'b')]
-
-
-
-variations
-----------
-
-variations(seq, n) Returns all the variations of the list of size n.
-
-Has an optional third argument. Must be a boolean value and makes the method
-return the variations with repetition if set to True, or the variations
-without repetition if set to False.
-
-Examples::
-    >>> from sympy.utilities.iterables import variations
-    >>> list(variations([1,2,3], 2))
-    [(1, 2), (1, 3), (2, 1), (2, 3), (3, 1), (3, 2)]
-    >>> list(variations([1,2,3], 2, True))
-    [(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)]
-
-
-partitions
-----------
-
-Although the combinatorics module contains Partition and IntegerPartition
-classes for investigation and manipulation of partitions, there are a few
-functions to generate partitions that can be used as low-level tools for
-routines:  ``partitions`` and ``multiset_partitions``. The former gives
-integer partitions, and the latter gives enumerated partitions of elements.
-There is also a routine ``kbins`` that will give a variety of permutations
-of partions.
-
-partitions::
-
-    >>> from sympy.utilities.iterables import partitions
-    >>> [p.copy() for s, p in partitions(7, m=2, size=True) if s == 2]
-    [{1: 1, 6: 1}, {2: 1, 5: 1}, {3: 1, 4: 1}]
-
-multiset_partitions::
-
-    >>> from sympy.utilities.iterables import multiset_partitions
-    >>> [p for p in multiset_partitions(3, 2)]
-    [[[0, 1], [2]], [[0, 2], [1]], [[0], [1, 2]]]
-    >>> [p for p in multiset_partitions([1, 1, 1, 2], 2)]
-    [[[1, 1, 1], [2]], [[1, 1, 2], [1]], [[1, 1], [1, 2]]]
-
-kbins::
-
-    >>> from sympy.utilities.iterables import kbins
-    >>> def show(k):
-    ...     rv = []
-    ...     for p in k:
-    ...         rv.append(','.join([''.join(j) for j in p]))
-    ...     return sorted(rv)
-    ...
-    >>> show(kbins("ABCD", 2))
-    ['A,BCD', 'AB,CD', 'ABC,D']
-    >>> show(kbins("ABC", 2))
-    ['A,BC', 'AB,C']
-    >>> show(kbins("ABC", 2, ordered=00))  # same as multiset_partitions
-    ['A,BC', 'AB,C', 'AC,B']
-    >>> show(kbins("ABC", 2, ordered=0o1))
-    ['A,BC', 'A,CB',
-     'B,AC', 'B,CA',
-     'C,AB', 'C,BA']
-    >>> show(kbins("ABC", 2, ordered=10))
-    ['A,BC', 'AB,C', 'AC,B',
-     'B,AC', 'BC,A',
-     'C,AB']
-    >>> show(kbins("ABC", 2, ordered=11))
-    ['A,BC', 'A,CB', 'AB,C', 'AC,B',
-     'B,AC', 'B,CA', 'BA,C', 'BC,A',
-     'C,AB', 'C,BA', 'CA,B', 'CB,A']
-
-Docstring
-=========
-
-.. automodule:: sympy.utilities.iterables
-   :members:
diff --git a/dev-py3k/_sources/modules/utilities/lambdify.txt b/dev-py3k/_sources/modules/utilities/lambdify.txt
deleted file mode 100644
index cf8466e0876..00000000000
--- a/dev-py3k/_sources/modules/utilities/lambdify.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-========
-Lambdify
-========
-
-.. automodule:: sympy.utilities.lambdify
-   :members:
diff --git a/dev-py3k/_sources/modules/utilities/memoization.txt b/dev-py3k/_sources/modules/utilities/memoization.txt
deleted file mode 100644
index c8137af1113..00000000000
--- a/dev-py3k/_sources/modules/utilities/memoization.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-===========
-Memoization
-===========
-
-.. automodule:: sympy.utilities.memoization
-   :members:
diff --git a/dev-py3k/_sources/modules/utilities/misc.txt b/dev-py3k/_sources/modules/utilities/misc.txt
deleted file mode 100644
index 8857c19ccdd..00000000000
--- a/dev-py3k/_sources/modules/utilities/misc.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-=============
-Miscellaneous
-=============
-
-.. automodule:: sympy.utilities.misc
-   :members:
diff --git a/dev-py3k/_sources/modules/utilities/pkgdata.txt b/dev-py3k/_sources/modules/utilities/pkgdata.txt
deleted file mode 100644
index 50a4a02a428..00000000000
--- a/dev-py3k/_sources/modules/utilities/pkgdata.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-=======
-PKGDATA
-=======
-
-.. automodule:: sympy.utilities.pkgdata
-   :members:
diff --git a/dev-py3k/_sources/modules/utilities/pytest.txt b/dev-py3k/_sources/modules/utilities/pytest.txt
deleted file mode 100644
index 2bf84507a7b..00000000000
--- a/dev-py3k/_sources/modules/utilities/pytest.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-======
-pytest
-======
-
-.. automodule:: sympy.utilities.pytest
-   :members:
diff --git a/dev-py3k/_sources/modules/utilities/randtest.txt b/dev-py3k/_sources/modules/utilities/randtest.txt
deleted file mode 100644
index a5e5e8f96df..00000000000
--- a/dev-py3k/_sources/modules/utilities/randtest.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-==================
-Randomised Testing
-==================
-
-.. automodule:: sympy.utilities.randtest
-   :members:
diff --git a/dev-py3k/_sources/modules/utilities/runtests.txt b/dev-py3k/_sources/modules/utilities/runtests.txt
deleted file mode 100644
index baa16deab3a..00000000000
--- a/dev-py3k/_sources/modules/utilities/runtests.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-=========
-Run Tests
-=========
-
-.. automodule:: sympy.utilities.runtests
-   :members:
diff --git a/dev-py3k/_sources/modules/utilities/source.txt b/dev-py3k/_sources/modules/utilities/source.txt
deleted file mode 100644
index 12dee2171e9..00000000000
--- a/dev-py3k/_sources/modules/utilities/source.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-======================
-Source Code Inspection
-======================
-
-.. automodule:: sympy.utilities.source
-   :members:
diff --git a/dev-py3k/_sources/modules/utilities/timeutils.txt b/dev-py3k/_sources/modules/utilities/timeutils.txt
deleted file mode 100644
index c6a5cdea0cf..00000000000
--- a/dev-py3k/_sources/modules/utilities/timeutils.txt
+++ /dev/null
@@ -1,6 +0,0 @@
-================
-Timing Utilities
-================
-
-.. automodule:: sympy.utilities.timeutils
-   :members:
diff --git a/dev-py3k/_sources/outreach.txt b/dev-py3k/_sources/outreach.txt
deleted file mode 100644
index 0d552892fbc..00000000000
--- a/dev-py3k/_sources/outreach.txt
+++ /dev/null
@@ -1,39 +0,0 @@
-SymPy Papers
-------------
-
-A list of papers which either use or mention SymPy, as well as presentations
-given about SymPy at conferences can be seen at `SymPy Papers
-<https://github.com/sympy/sympy/wiki/SymPy-Papers>`_.
-
-Planet SymPy
-------------
-
-We have a blog agregator at http://planet.sympy.org
-
-SymPy logos
------------
-
-SymPy has a collection of official logos, which can be accessed here:
-
-https://github.com/sympy/sympy/tree/master/doc/logo
-
-The license of all the logos is the same as SymPy: BSD. See the LICENSE file in
-the trunk for more information.
-
-Projects using SymPy
---------------------
-
-This is an (incomplete) list of projects that use SymPy. If you use SymPy in
-your project, please let us know on our mailinglist_, so that we can add your
-project here as well.
-
-* `SfePy <http://sfepy.org/>`_ (simple finite elements in Python)
-* `Quameon <http://quameon.sourceforge.net/>`_ (Quantum Monte Carlo in Python)
-
-.. _mailinglist:        http://groups.google.com/group/sympy
-
-Blogs, News, Magazines
-----------------------
-
-You can see a list of blogs/news/magazines mentioning SymPy (that we know of) on
-our `wiki page <https://github.com/sympy/sympy/wiki/SymPy-in-the-news>`_.
diff --git a/dev-py3k/_sources/python-comparisons.txt b/dev-py3k/_sources/python-comparisons.txt
deleted file mode 100644
index 4e82744b363..00000000000
--- a/dev-py3k/_sources/python-comparisons.txt
+++ /dev/null
@@ -1,388 +0,0 @@
-=======================================
-Development Tips: Comparisons in Python
-=======================================
-
-.. role:: input(strong)
-
-Introduction
-============
-
-When debugging comparisons and hashes in SymPy, it is necessary to understand
-when exactly Python calls each method.
-Unfortunately, the official Python documentation for this is
-not very detailed (see the docs for `rich comparison
-<http://docs.python.org/dev/reference/datamodel.html#object.__lt__>`_,
-`__cmp__() <http://docs.python.org/dev/reference/datamodel.html#object.__cmp__>`_
-and `__hash__()
-<http://docs.python.org/dev/reference/datamodel.html#object.__hash__>`_
-methods).
-
-We wrote this guide to fill in the missing gaps. After reading it, you should
-be able to understand which methods do (and do not) get called and the order in
-which they are called.
-
-Hashing
-=======
-
-Every Python class has a ``__hash__()`` method, the default
-implementation of which is::
-
-    def __hash__(self):
-        return id(self)
-
-You can reimplement it to return a different integer that you compute on your own.
-``hash(x)`` just calls ``x.__hash__()``. Python builtin classes usually redefine
-the ``__hash__()`` method. For example, an ``int`` has something like this::
-
-    def __hash__(self):
-        return int(self)
-
-and a ``list`` does something like this::
-
-    def __hash__(self):
-        raise TypeError("list objects are unhashable")
-
-The general
-idea about hashes is that if two objects have a different hash, they are not
-equal, but if they have the same hash, they *might* be equal. (This is usually
-called a "hash collision" and you need to use the methods described in the
-next section to determine if the objects really are equal).
-
-The only requirement from the Python side is
-that the hash value mustn't change after it is returned by the
-``__hash__()`` method.
-
-Please be aware that hashing is *platform-dependent*. This means that you can
-get different hashes for the same SymPy object on different platforms. This
-affects for instance sorting of sympy expressions. You can also get SymPy
-objects printed in different order.
-
-When developing, you have to be careful about this, especially when writing
-tests. It is possible that your test runs on a 32-bit platform, but not on
-64-bit. An example::
-
-    >> from sympy import *
-    >> x = Symbol('x')
-    >> r = rootfinding.roots_quartic(Poly(x**4 - 6*x**3 + 17*x**2 - 26*x + 20, x))
-    >> [i.evalf(2) for i in r]
-    [1.0 + 1.7*I, 2.0 - 1.0*I, 2.0 + I, 1.0 - 1.7*I]
-
-If you get this order of solutions, you are probably running 32-bit system.
-On a 64-bit system you would get the following::
-
-    >> [i.evalf(2) for i in r]
-    [1.0 - 1.7*I, 1.0 + 1.7*I, 2.0 + I, 2.0 - 1.0*I
-
-When you now write a test like this::
-
-    r = [i.evalf(2) for i in r]
-    assert r == [1.0 + 1.7*I, 2.0 - 1.0*I, 2.0 + I, 1.0 - 1.7*I]
-
-it will fail on a 64-bit platforms, even if it works for your 32-bit system. You can
-avoid this by using the ``sorted()`` or ``set()`` Python built-in::
-
-    r = [i.evalf(2) for i in r]
-    assert set(r) == set([1.0 + 1.7*I, 2.0 - 1.0*I, 2.0 + I, 1.0 - 1.7*I])
-
-This approach does not work for doctests since they always compare strings that would
-be printed after a prompt. In that case you could make your test print results using
-a combination of ``str()`` and ``sorted()``::
-
-    >> sorted([str(i.evalf(2)) for i in r])
-    ['1.0 + 1.7*I', '1.0 - 1.7*I', '2.0 + I', '2.0 - 1.0*I']
-
-or, if you don't want to show the values as strings, then sympify the results or the
-sorted list::
-
-    >> [S(s) for s in sorted([str(i.evalf(2)) for i in r])]
-    [1.0 + 1.7*I, 1.0 - 1.7*I, 2.0 + I, 2.0 - I]
-
-The printing of SymPy expressions might be also affected, so be careful
-with doctests. If you get the following on a 32-bit system::
-
-    >> print dsolve(f(x).diff(x, 2) + 2*f(x).diff(x) - f(x), f(x))
-    f(x) == C1*exp(-x + x*sqrt(2)) + C2*exp(-x - x*sqrt(2))
-
-you might get the following on a 64-bit platform::
-
-    >> print dsolve(f(x).diff(x, 2) + 2*f(x).diff(x) - f(x), f(x))
-    f(x) == C1*exp(-x - x*sqrt(2)) + C2*exp(-x + x*sqrt(2))
-
-Method Resolution
-=================
-
-Let ``a``, ``b`` and ``c`` be instances of any one of the Python classes.
-As can be easily checked by the `python script`_ at the end of this guide,
-if you write::
-
-    a == b
-
-Python calls the following -- in this order::
-
-    a.__eq__(b)
-    b.__eq__(a)
-    a.__cmp__(b)
-    b.__cmp__(a)
-    id(a) == id(b)
-
-If a particular method is not implemented (or a method
-returns ``NotImplemented`` [1]_) Python skips it
-and tries the next one until it succeeds (i.e. until the method returns something
-meaningful). The last line is a catch-all method that always succeeds.
-
-If you write::
-
-    a != b
-
-Python tries to call::
-
-    a.__ne__(b)
-    b.__ne__(a)
-    a.__cmp__(b)
-    b.__cmp__(a)
-    id(a) == id(b)
-
-If you write::
-
-    a < b
-
-Python tries to call::
-
-    a.__lt__(b)
-    b.__gt__(a)
-    a.__cmp__(b)
-    b.__cmp__(a)
-    id(a) < id(b)
-
-If you write::
-
-    a <= b
-
-Python tries to call::
-
-    a.__le__(b)
-    b.__ge__(a)
-    a.__cmp__(b)
-    b.__cmp__(a)
-    id(a) <= id(b)
-
-And similarly for ``a > b`` and ``a >= b``.
-
-If you write::
-
-    sorted([a, b, c])
-
-Python calls the same chain of methods as for the ``b < a`` and ``c < b``
-comparisons.
-
-If you write any of the following::
-
-    a in {d: 5}
-    a in set([d, d, d])
-    set([a, b]) == set([a, b])
-
-Python first compares hashes, e.g.::
-
-    a.__hash__()
-    d.__hash__()
-
-If ``hash(a) != hash(d)`` then the result of the statement ``a in {d: 5}`` is
-immediately ``False`` (remember how hashes work in general). If
-``hash(a) == hash(d)``) Python goes through the method resolution of the
-``==`` operator as shown above.
-
-General Notes and Caveats
-=========================
-
-In the method resolution for ``<``, ``<=``, ``==``, ``!=``, ``>=``, ``>`` and
-``sorted([a, b, c])`` operators the ``__hash__()`` method is *not* called, so
-in these cases it doesn't matter what it returns. The ``__hash__()`` method is
-only called for sets and dictionaries.
-
-In the official Python documentation you can read about `hashable and
-non-hashable <http://docs.python.org/dev/glossary.html#term-hashable>`_ objects.
-In reality, you don't have to think about it, you just follow the method
-resolution described here. E.g. if you try to use lists as dictionary keys, the
-list's ``__hash__()`` method will be called and it returns an exception.
-
-In SymPy, every instance of any subclass of ``Basic`` is
-immutable.  Technically this means, that it's behavior through all the methods
-above mustn't change once the instance is created. Especially, the hash value
-mustn't change (as already stated above) or else objects will get mixed up in
-dictionaries and wrong values will be returned for a given key, etc....
-
-.. _python script:
-
-Script To Verify This Guide
-============================
-
-The above method resolution can be verified using the following program::
-
-    class A(object):
-
-        def __init__(self, a, hash):
-            self.a = a
-            self._hash = hash
-
-        def __lt__(self, o):
-            print "%s.__lt__(%s)" % (self.a, o.a)
-            return NotImplemented
-
-        def __le__(self, o):
-            print "%s.__le__(%s)" % (self.a, o.a)
-            return NotImplemented
-
-        def __gt__(self, o):
-            print "%s.__gt__(%s)" % (self.a, o.a)
-            return NotImplemented
-
-        def __ge__(self, o):
-            print "%s.__ge__(%s)" % (self.a, o.a)
-            return NotImplemented
-
-        def __cmp__(self, o):
-            print "%s.__cmp__(%s)" % (self.a, o.a)
-            #return cmp(self._hash, o._hash)
-            return NotImplemented
-
-        def __eq__(self, o):
-            print "%s.__eq__(%s)" % (self.a, o.a)
-            return NotImplemented
-
-        def __ne__(self, o):
-            print "%s.__ne__(%s)" % (self.a, o.a)
-            return NotImplemented
-
-        def __hash__(self):
-            print "%s.__hash__()" % (self.a)
-            return self._hash
-
-    def show(s):
-        print "--- %s " % s + "-"*40
-        eval(s)
-
-    a = A("a", 1)
-    b = A("b", 2)
-    c = A("c", 3)
-    d = A("d", 1)
-
-    show("a == b")
-    show("a != b")
-    show("a < b")
-    show("a <= b")
-    show("a > b")
-    show("a >= b")
-    show("sorted([a, b, c])")
-    show("{d: 5}")
-    show("a in {d: 5}")
-    show("set([d, d, d])")
-    show("a in set([d, d, d])")
-    show("set([a, b])")
-
-    print "--- x = set([a, b]); y = set([a, b]); ---"
-    x = set([a, b])
-    y = set([a, b])
-    print "               x == y :"
-    x == y
-
-    print "--- x = set([a, b]); y = set([b, d]); ---"
-    x = set([a, b])
-    y = set([b, d])
-    print "               x == y :"
-    x == y
-
-
-and its output::
-
-    --- a == b ----------------------------------------
-    a.__eq__(b)
-    b.__eq__(a)
-    a.__cmp__(b)
-    b.__cmp__(a)
-    --- a != b ----------------------------------------
-    a.__ne__(b)
-    b.__ne__(a)
-    a.__cmp__(b)
-    b.__cmp__(a)
-    --- a < b ----------------------------------------
-    a.__lt__(b)
-    b.__gt__(a)
-    a.__cmp__(b)
-    b.__cmp__(a)
-    --- a <= b ----------------------------------------
-    a.__le__(b)
-    b.__ge__(a)
-    a.__cmp__(b)
-    b.__cmp__(a)
-    --- a > b ----------------------------------------
-    a.__gt__(b)
-    b.__lt__(a)
-    a.__cmp__(b)
-    b.__cmp__(a)
-    --- a >= b ----------------------------------------
-    a.__ge__(b)
-    b.__le__(a)
-    a.__cmp__(b)
-    b.__cmp__(a)
-    --- sorted([a, b, c]) ----------------------------------------
-    b.__lt__(a)
-    a.__gt__(b)
-    b.__cmp__(a)
-    a.__cmp__(b)
-    c.__lt__(b)
-    b.__gt__(c)
-    c.__cmp__(b)
-    b.__cmp__(c)
-    --- {d: 5} ----------------------------------------
-    d.__hash__()
-    --- a in {d: 5} ----------------------------------------
-    d.__hash__()
-    a.__hash__()
-    d.__eq__(a)
-    a.__eq__(d)
-    d.__cmp__(a)
-    a.__cmp__(d)
-    --- set([d, d, d]) ----------------------------------------
-    d.__hash__()
-    d.__hash__()
-    d.__hash__()
-    --- a in set([d, d, d]) ----------------------------------------
-    d.__hash__()
-    d.__hash__()
-    d.__hash__()
-    a.__hash__()
-    d.__eq__(a)
-    a.__eq__(d)
-    d.__cmp__(a)
-    a.__cmp__(d)
-    --- set([a, b]) ----------------------------------------
-    a.__hash__()
-    b.__hash__()
-    --- x = set([a, b]); y = set([a, b]); ---
-    a.__hash__()
-    b.__hash__()
-    a.__hash__()
-    b.__hash__()
-                   x == y :
-    --- x = set([a, b]); y = set([b, d]); ---
-    a.__hash__()
-    b.__hash__()
-    b.__hash__()
-    d.__hash__()
-                   x == y :
-    d.__eq__(a)
-    a.__eq__(d)
-    d.__cmp__(a)
-    a.__cmp__(d)
-
-----------
-
-.. [1] There is also the similar ``NotImplementedError`` exception, which one may
-       be tempted to raise to obtain the same effect as returning
-       ``NotImplemented``.
-
-       But these are **not** the same, and Python will completely ignore
-       ``NotImplementedError`` with respect to choosing appropriate comparison
-       method, and will just propagate this exception upwards, to the caller.
-
-       So ``return NotImplemented`` is not the same as ``raise NotImplementedError``.
diff --git a/dev-py3k/_sources/tutorial/tutorial.en.txt b/dev-py3k/_sources/tutorial/tutorial.en.txt
deleted file mode 100644
index e4ed9e50e77..00000000000
--- a/dev-py3k/_sources/tutorial/tutorial.en.txt
+++ /dev/null
@@ -1,1063 +0,0 @@
-.. _tutorial:
-
-========
-Tutorial
-========
-
-.. role:: input(strong)
-
-Introduction
-============
-
-SymPy is a Python library for symbolic mathematics. It aims to become a
-full-featured computer algebra system (CAS) while keeping the code as simple as
-possible in order to be comprehensible and easily extensible.  SymPy is written
-entirely in Python and does not require any external libraries.
-
-This tutorial gives an overview and introduction to SymPy.
-Read this to have an idea what SymPy can do for you (and how) and if you want
-to know more, read the :ref:`SymPy User's Guide <guide>`, the
-:ref:`SymPy Modules Reference <module-docs>`, or the
-`sources <https://github.com/sympy/sympy/>`_ directly.
-
-First Steps with SymPy
-======================
-
-The easiest way to download it is to go to
-http://code.google.com/p/sympy/ and
-download the latest tarball from the Featured Downloads:
-
-.. image:: ../figures/featured-downloads.png
-
-Unpack it:
-
-.. parsed-literal::
-
-    $ :input:`tar xzf sympy-0.5.12.tar.gz`
-
-and try it from a Python interpreter:
-
-.. parsed-literal::
-
-    $ :input:`cd sympy-0.5.12`
-    $ :input:`python`
-    Python 2.4.4 (#2, Jan  3 2008, 13:36:28)
-    [GCC 4.2.3 20071123 (prerelease) (Debian 4.2.2-4)] on linux2
-    Type "help", "copyright", "credits" or "license" for more information.
-    >>> from sympy import Symbol, cos
-    >>> x = Symbol("x")
-    >>> (1/cos(x)).series(x, 0, 10)
-    1 + x**2/2 + 5*x**4/24 + 61*x**6/720 + 277*x**8/8064 + O(x**10)
-
-You can use SymPy as shown above and this is indeed the recommended way if you
-use it in your program. You can also install it using ``./setup.py install`` as
-any other Python module, or just install a package in your favourite Linux
-distribution, e.g.:
-
-.. topic:: Installing SymPy in Debian
-
-  .. parsed-literal::
-
-    $ :input:`sudo apt-get install python-sympy`
-    Reading package lists... Done
-    Building dependency tree
-    Reading state information... Done
-    The following NEW packages will be installed:
-      python-sympy
-    0 upgraded, 1 newly installed, 0 to remove and 18 not upgraded.
-    Need to get 991kB of archives.
-    After this operation, 5976kB of additional disk space will be used.
-    Get:1 http://ftp.cz.debian.org unstable/main python-sympy 0.5.12-1 [991kB]
-    Fetched 991kB in 2s (361kB/s)
-    Selecting previously deselected package python-sympy.
-    (Reading database ... 232619 files and directories currently installed.)
-    Unpacking python-sympy (from .../python-sympy_0.5.12-1_all.deb) ...
-    Setting up python-sympy (0.5.12-1) ...
-
-
-For other means how to install SymPy, consult the wiki page
-`Download and Installation <https://github.com/sympy/sympy/wiki/Download-Installation>`_.
-
-
-isympy Console
---------------
-
-For experimenting with new features, or when figuring out how to do things, you
-can use our special wrapper around IPython called ``isympy`` (located in
-``bin/isympy`` if you are running from the source directory) which is just a
-standard Python shell that has already imported the relevant SymPy modules and
-defined the symbols x, y, z and some other things:
-
-.. parsed-literal::
-
-    $ :input:`cd sympy`
-    $ :input:`./bin/isympy`
-    IPython console for SymPy 0.7.2-git (Python 2.7.1) (ground types: gmpy)
-
-    These commands were executed:
-    >>>
-    >>> from sympy import *
-    >>> x, y, z, t = symbols('x y z t')
-    >>> k, m, n = symbols('k m n', integer=True)
-    >>> f, g, h = symbols('f g h', cls=Function)
-
-    Documentation can be found at http://www.sympy.org
-
-    In [1]: :input:`(1/cos(x)).series(x, 0, 10)`
-    Out[1]:
-         2      4       6        8
-        x    5*x    61*x    277*x     / 10\\
-    1 + -- + ---- + ----- + ------ + O\\x  /
-        2     24     720     8064
-
-.. note::
-
-    Commands entered by you are bold. Thus what we did in 3 lines in a regular
-    Python interpreter can be done in 1 line in isympy.
-
-
-Using SymPy as a calculator
----------------------------
-
-SymPy has three built-in numeric types: Float, Rational and Integer.
-
-The Rational class represents a rational number as a pair of two Integers:
-the numerator and the denominator. So Rational(1, 2) represents 1/2,
-Rational(5, 2) represents 5/2, and so on.
-
-::
-
-    >>> from sympy import Rational
-    >>> a = Rational(1, 2)
-
-    >>> a
-    1/2
-
-    >>> a*2
-    1
-
-    >>> Rational(2)**50/Rational(10)**50
-    1/88817841970012523233890533447265625
-
-
-Proceed with caution while working with Python int's and floating
-point numbers, especially in division, since you may create a
-Python number, not a SymPy number. A ratio of two Python ints may
-create a float -- the "true division" standard of Python 3
-and the default behavior of ``isympy`` which imports division
-from __future__:
-
-::
-
-    >>> 1/2 #doctest: +SKIP
-    0.5
-
-But in earlier Python versions where division has not been imported, a
-truncated int will result:
-
-::
-
-    >>> 1/2 #doctest: +SKIP
-    0
-
-In both cases, however, you are not dealing with a SymPy Number because
-Python created its own number. Most of the time you will probably be
-working with Rational numbers, so make sure to use Rational to get
-the SymPy result. One might find it convenient to equate ``R`` and
-Rational:
-
-::
-
-    >>> R = Rational
-    >>> R(1, 2)
-    1/2
-    >>> R(1)/2 # R(1) is a SymPy Integer and Integer/int gives a Rational
-    1/2
-
-We also have some special constants, like e and pi, that are treated as symbols
-(1 + pi won't evaluate to something numeric, rather it will remain as 1 + pi), and
-have arbitrary precision:
-
-::
-
-    >>> from sympy import pi, E
-    >>> pi**2
-    pi**2
-
-    >>> pi.evalf()
-    3.14159265358979
-
-    >>> (pi + E).evalf()
-    5.85987448204884
-
-as you see, evalf evaluates the expression to a floating-point number
-
-The symbol ``oo`` is used for a class defining mathematical infinity:
-
-::
-
-    >>> from sympy import oo
-    >>> oo > 99999
-    True
-    >>> oo + 1
-    oo
-
-Symbols
--------
-
-In contrast to other Computer Algebra Systems, in SymPy you have to declare
-symbolic variables explicitly:
-
-::
-
-    >>> from sympy import Symbol
-    >>> x = Symbol('x')
-    >>> y = Symbol('y')
-
-On the left is the normal Python variable which has been assigned to the
-SymPy Symbol class. Predefined symbols (including those for symbols with
-Greek names) are available for import from abc:
-
-    >>> from sympy.abc import x, theta
-
-Symbols can also be created with the ``symbols`` or ``var`` functions, the
-latter automatically adding the created symbols to the namespace, and both
-accepting a range notation:
-
-    >>> from sympy import symbols, var
-    >>> a, b, c = symbols('a,b,c')
-    >>> d, e, f = symbols('d:f')
-    >>> var('g:h')
-    (g, h)
-    >>> var('g:2')
-    (g0, g1)
-
-Instances of the Symbol class "play well together" and are the building blocks
-of expresions:
-
-::
-
-    >>> x + y + x - y
-    2*x
-
-    >>> (x + y)**2
-    (x + y)**2
-
-    >>> ((x + y)**2).expand()
-    x**2 + 2*x*y + y**2
-
-They can be substituted with other numbers, symbols or expressions using ``subs(old, new)``:
-
-::
-
-    >>> ((x + y)**2).subs(x, 1)
-    (y + 1)**2
-
-    >>> ((x + y)**2).subs(x, y)
-    4*y**2
-
-    >>> ((x + y)**2).subs(x, 1 - y)
-    1
-
-For the remainder of the tutorial, we assume that we have run:
-
-::
-
-    >>> from sympy import init_printing
-    >>> init_printing(use_unicode=False, wrap_line=False, no_global=True)
-
-This will make things look better when printed. See the :ref:`printing-tutorial`
-section below. If you have a unicode font installed, you can pass
-use_unicode=True for a slightly nicer output.
-
-Algebra
-=======
-
-For partial fraction decomposition, use ``apart(expr, x)``:
-
-::
-
-    >>> from sympy import apart
-    >>> from sympy.abc import x, y, z
-
-    >>> 1/( (x + 2)*(x + 1) )
-           1
-    ---------------
-    (x + 1)*(x + 2)
-
-    >>> apart(1/( (x + 2)*(x + 1) ), x)
-        1       1
-    - ----- + -----
-      x + 2   x + 1
-
-    >>> (x + 1)/(x - 1)
-    x + 1
-    -----
-    x - 1
-
-    >>> apart((x + 1)/(x - 1), x)
-          2
-    1 + -----
-        x - 1
-
-To combine things back together, use ``together(expr, x)``:
-
-::
-
-    >>> from sympy import together
-    >>> together(1/x + 1/y + 1/z)
-    x*y + x*z + y*z
-    ---------------
-         x*y*z
-
-    >>> together(apart((x + 1)/(x - 1), x), x)
-    x + 1
-    -----
-    x - 1
-
-    >>> together(apart(1/( (x + 2)*(x + 1) ), x), x)
-           1
-    ---------------
-    (x + 1)*(x + 2)
-
-
-.. index:: calculus
-
-Calculus
-========
-
-.. index:: limits
-
-Limits
-------
-
-Limits are easy to use in SymPy, they follow the syntax ``limit(function,
-variable, point)``, so to compute the limit of f(x) as x -> 0, you would issue
-``limit(f, x, 0)``:
-
-::
-
-   >>> from sympy import limit, Symbol, sin, oo
-   >>> x = Symbol("x")
-   >>> limit(sin(x)/x, x, 0)
-   1
-
-you can also calculate the limit at infinity:
-
-::
-
-   >>> limit(x, x, oo)
-   oo
-
-   >>> limit(1/x, x, oo)
-   0
-
-   >>> limit(x**x, x, 0)
-   1
-
-for some non-trivial examples on limits, you can read the test file
-`test_demidovich.py
-<https://github.com/sympy/sympy/blob/master/sympy/series/tests/test_demidovich.py>`_
-
-.. index:: differentiation, diff
-
-Differentiation
----------------
-
-You can differentiate any SymPy expression using ``diff(func, var)``. Examples:
-
-::
-
-    >>> from sympy import diff, Symbol, sin, tan
-    >>> x = Symbol('x')
-    >>> diff(sin(x), x)
-    cos(x)
-    >>> diff(sin(2*x), x)
-    2*cos(2*x)
-
-    >>> diff(tan(x), x)
-       2
-    tan (x) + 1
-
-You can check, that it is correct by:
-
-::
-
-    >>> from sympy import limit
-    >>> from sympy.abc import delta
-    >>> limit((tan(x + delta) - tan(x))/delta, delta, 0)
-       2
-    tan (x) + 1
-
-Higher derivatives can be calculated using the ``diff(func, var, n)`` method:
-
-::
-
-    >>> diff(sin(2*x), x, 1)
-    2*cos(2*x)
-
-    >>> diff(sin(2*x), x, 2)
-    -4*sin(2*x)
-
-    >>> diff(sin(2*x), x, 3)
-    -8*cos(2*x)
-
-
-.. index::
-    single: series expansion
-    single: expansion; series
-
-Series expansion
-----------------
-
-Use ``.series(var, point, order)``:
-
-::
-
-    >>> from sympy import Symbol, cos
-    >>> x = Symbol('x')
-    >>> cos(x).series(x, 0, 10)
-         2    4     6      8
-        x    x     x      x      / 10\
-    1 - -- + -- - --- + ----- + O\x  /
-        2    24   720   40320
-    >>> (1/cos(x)).series(x, 0, 10)
-         2      4       6        8
-        x    5*x    61*x    277*x     / 10\
-    1 + -- + ---- + ----- + ------ + O\x  /
-        2     24     720     8064
-
-Another simple example:
-
-::
-
-    >>> from sympy import Integral, pprint
-
-    >>> y = Symbol("y")
-    >>> e = 1/(x + y)
-    >>> s = e.series(x, 0, 5)
-
-    >>> print(s)
-    1/y - x/y**2 + x**2/y**3 - x**3/y**4 + x**4/y**5 + O(x**5)
-    >>> pprint(s)
-              2    3    4
-    1   x    x    x    x     / 5\
-    - - -- + -- - -- + -- + O\x /
-    y    2    3    4    5
-        y    y    y    y
-
-
-
-
-
-.. index:: summation
-
-Summation
----------
-
-Compute the summation of f with respect to the given summation variable over the given limits.
-
-summation(f, (i, a, b)) computes the sum of f with respect to i from a to b,
-i.e.,
-
-::
-
-                                b
-                              ____
-                              \   `
-    summation(f, (i, a, b)) =  )    f
-                              /___,
-                              i = a
-
-
-If it cannot compute the sum, it prints the corresponding summation formula.
-Repeated sums can be computed by introducing additional limits:
-
-::
-
-    >>> from sympy import summation, oo, symbols, log
-    >>> i, n, m = symbols('i n m', integer=True)
-
-    >>> summation(2*i - 1, (i, 1, n))
-     2
-    n
-    >>> summation(1/2**i, (i, 0, oo))
-    2
-    >>> summation(1/log(n)**n, (n, 2, oo))
-      oo
-     ___
-     \  `
-      \     -n
-      /   log (n)
-     /__,
-    n = 2
-    >>> summation(i, (i, 0, n), (n, 0, m))
-          3    2
-    m    m    m
-    -- + -- + -
-    6    2    3
-    >>> from sympy.abc import x
-    >>> from sympy import factorial
-    >>> summation(x**n/factorial(n), (n, 0, oo))
-     x
-    e
-
-
-.. index:: integration
-
-Integration
------------
-
-SymPy has support for indefinite and definite integration of transcendental
-elementary and special functions via ``integrate()`` facility, which uses
-powerful extended Risch-Norman algorithm and some heuristics and pattern
-matching:
-
-::
-
-    >>> from sympy import integrate, erf, exp, sin, log, oo, pi, sinh, symbols
-    >>> x, y = symbols('x,y')
-
-You can integrate elementary functions:
-
-::
-
-    >>> integrate(6*x**5, x)
-     6
-    x
-    >>> integrate(sin(x), x)
-    -cos(x)
-    >>> integrate(log(x), x)
-    x*log(x) - x
-    >>> integrate(2*x + sinh(x), x)
-     2
-    x  + cosh(x)
-
-Also special functions are handled easily:
-
-::
-
-    >>> integrate(exp(-x**2)*erf(x), x)
-      ____    2
-    \/ pi *erf (x)
-    --------------
-          4
-
-It is possible to compute definite integrals:
-
-::
-
-    >>> integrate(x**3, (x, -1, 1))
-    0
-    >>> integrate(sin(x), (x, 0, pi/2))
-    1
-    >>> integrate(cos(x), (x, -pi/2, pi/2))
-    2
-
-Also, improper integrals are supported as well:
-
-::
-
-    >>> integrate(exp(-x), (x, 0, oo))
-    1
-    >>> integrate(log(x), (x, 0, 1))
-    -1
-
-.. index::
-    single: complex numbers
-    single: expansion; complex
-
-Complex numbers
----------------
-
-Besides the imaginary unit, I, which is imaginary, symbols can be created with
-attributes (e.g. real, positive, complex, etc...) and this will affect how
-they behave:
-
-::
-
-    >>> from sympy import Symbol, exp, I
-    >>> x = Symbol("x") # a plain x with no attributes
-    >>> exp(I*x).expand()
-     I*x
-    e
-    >>> exp(I*x).expand(complex=True)
-       -im(x)               -im(x)
-    I*e      *sin(re(x)) + e      *cos(re(x))
-    >>> x = Symbol("x", real=True)
-    >>> exp(I*x).expand(complex=True)
-    I*sin(x) + cos(x)
-
-Functions
----------
-
-**trigonometric**
-
-::
-
-    >>> from sympy import asin, asinh, cos, sin, sinh, symbols, I
-    >>> x, y = symbols('x,y')
-
-    >>> sin(x + y).expand(trig=True)
-    sin(x)*cos(y) + sin(y)*cos(x)
-
-    >>> cos(x + y).expand(trig=True)
-    -sin(x)*sin(y) + cos(x)*cos(y)
-
-    >>> sin(I*x)
-    I*sinh(x)
-
-    >>> sinh(I*x)
-    I*sin(x)
-
-    >>> asinh(I)
-    I*pi
-    ----
-     2
-
-    >>> asinh(I*x)
-    I*asin(x)
-
-    >>> sin(x).series(x, 0, 10)
-         3     5     7       9
-        x     x     x       x       / 10\
-    x - -- + --- - ---- + ------ + O\x  /
-        6    120   5040   362880
-
-    >>> sinh(x).series(x, 0, 10)
-         3     5     7       9
-        x     x     x       x       / 10\
-    x + -- + --- + ---- + ------ + O\x  /
-        6    120   5040   362880
-
-    >>> asin(x).series(x, 0, 10)
-         3      5      7       9
-        x    3*x    5*x    35*x     / 10\
-    x + -- + ---- + ---- + ----- + O\x  /
-        6     40    112     1152
-
-    >>> asinh(x).series(x, 0, 10)
-         3      5      7       9
-        x    3*x    5*x    35*x     / 10\
-    x - -- + ---- - ---- + ----- + O\x  /
-        6     40    112     1152
-
-**spherical harmonics**
-
-::
-
-    >>> from sympy import Ylm
-    >>> from sympy.abc import theta, phi
-
-    >>> Ylm(1, 0, theta, phi)
-      ___
-    \/ 3 *cos(theta)
-    ----------------
-            ____
-        2*\/ pi
-
-    >>> Ylm(1, 1, theta, phi)
-       ___  I*phi
-    -\/ 6 *e     *sin(theta)
-    ------------------------
-                ____
-            4*\/ pi
-
-    >>> Ylm(2, 1, theta, phi)
-       ____  I*phi
-    -\/ 30 *e     *sin(theta)*cos(theta)
-    ------------------------------------
-                      ____
-                  4*\/ pi
-
-**factorials and gamma function**
-
-::
-
-    >>> from sympy import factorial, gamma, Symbol
-    >>> x = Symbol("x")
-    >>> n = Symbol("n", integer=True)
-
-    >>> factorial(x)
-    x!
-
-    >>> factorial(n)
-    n!
-
-    >>> gamma(x + 1).series(x, 0, 3) # i.e. factorial(x)
-                          /          2     2\
-                        2 |EulerGamma    pi |    / 3\
-    1 - EulerGamma*x + x *|----------- + ---| + O\x /
-                          \     2         12/
-
-**zeta function**
-
-::
-
-    >>> from sympy import zeta
-    >>> zeta(4, x)
-    zeta(4, x)
-
-    >>> zeta(4, 1)
-      4
-    pi
-    ---
-     90
-
-    >>> zeta(4, 2)
-           4
-         pi
-    -1 + ---
-          90
-
-    >>> zeta(4, 3)
-             4
-      17   pi
-    - -- + ---
-      16    90
-
-
-**polynomials**
-
-::
-
-    >>> from sympy import assoc_legendre, chebyshevt, legendre, hermite
-    >>> chebyshevt(2, x)
-       2
-    2*x  - 1
-
-    >>> chebyshevt(4, x)
-       4      2
-    8*x  - 8*x  + 1
-
-    >>> legendre(2, x)
-       2
-    3*x    1
-    ---- - -
-     2     2
-
-    >>> legendre(8, x)
-          8         6         4        2
-    6435*x    3003*x    3465*x    315*x     35
-    ------- - ------- + ------- - ------ + ---
-      128        32        64       32     128
-
-    >>> assoc_legendre(2, 1, x)
-            __________
-           /    2
-    -3*x*\/  - x  + 1
-
-    >>> assoc_legendre(2, 2, x)
-         2
-    - 3*x  + 3
-
-    >>> hermite(3, x)
-       3
-    8*x  - 12*x
-
-.. index:: equations; differential, diff, dsolve
-
-Differential Equations
-----------------------
-
-In ``isympy``:
-
-::
-
-    >>> from sympy import Function, Symbol, dsolve
-    >>> f = Function('f')
-    >>> x = Symbol('x')
-    >>> f(x).diff(x, x) + f(x)
-            2
-           d
-    f(x) + ---(f(x))
-             2
-           dx
-
-    >>> dsolve(f(x).diff(x, x) + f(x), f(x))
-    f(x) = C1*sin(x) + C2*cos(x)
-
-.. index:: equations; algebraic, solve
-
-Algebraic equations
--------------------
-
-In ``isympy``:
-
-::
-
-    >>> from sympy import solve, symbols
-    >>> x, y = symbols('x,y')
-    >>> solve(x**4 - 1, x)
-    [-1, 1, -I, I]
-
-    >>> solve([x + 5*y - 2, -3*x + 6*y - 15], [x, y])
-    {x: -3, y: 1}
-
-.. index:: linear algebra
-
-Linear Algebra
-==============
-
-.. index:: Matrix
-
-Matrices
---------
-
-Matrices are created as instances from the Matrix class:
-
-::
-
-    >>> from sympy import Matrix, Symbol
-    >>> Matrix([[1, 0], [0, 1]])
-    [1  0]
-    [    ]
-    [0  1]
-
-They can also contain symbols:
-
-::
-
-    >>> x = Symbol('x')
-    >>> y = Symbol('y')
-    >>> A = Matrix([[1, x], [y, 1]])
-    >>> A
-    [1  x]
-    [    ]
-    [y  1]
-
-    >>> A**2
-    [x*y + 1    2*x  ]
-    [                ]
-    [  2*y    x*y + 1]
-
-For more about Matrices, see the Linear Algebra tutorial.
-
-.. index:: pattern matching, match, Wild, WildFunction
-
-Pattern matching
-================
-
-Use the ``.match()`` method, along with the ``Wild`` class, to perform pattern
-matching on expressions. The method will return a dictionary with the required
-substitutions, as follows:
-
-::
-
-    >>> from sympy import Symbol, Wild
-    >>> x = Symbol('x')
-    >>> p = Wild('p')
-    >>> (5*x**2).match(p*x**2)
-    {p: 5}
-
-    >>> q = Wild('q')
-    >>> (x**2).match(p*x**q)
-    {p: 1, q: 2}
-
-If the match is unsuccessful, it returns ``None``:
-
-::
-
-    >>> print((x + 1).match(p**x))
-    None
-
-One can also use the exclude parameter of the ``Wild`` class to ensure that
-certain things do not show up in the result:
-
-::
-
-    >>> p = Wild('p', exclude=[1, x])
-    >>> print((x + 1).match(x + p)) # 1 is excluded
-    None
-    >>> print((x + 1).match(p + 1)) # x is excluded
-    None
-    >>> print((x + 1).match(x + 2 + p)) # -1 is not excluded
-    {p_: -1}
-
-.. _printing-tutorial:
-
-Printing
-========
-
-There are many ways to print expressions.
-
-**Standard**
-
-This is what ``str(expression)`` returns and it looks like this:
-
-    >>> from sympy import Integral
-    >>> from sympy.abc import x
-    >>> print(x**2)
-    x**2
-    >>> print(1/x)
-    1/x
-    >>> print(Integral(x**2, x))
-    Integral(x**2, x)
-
-**Pretty printing**
-
-Nice ascii-art printing is produced by the ``pprint`` function:
-
-    >>> from sympy import Integral, pprint
-    >>> from sympy.abc import x
-    >>> pprint(x**2)
-     2
-    x
-    >>> pprint(1/x)
-    1
-    -
-    x
-    >>> pprint(Integral(x**2, x))
-      /
-     |
-     |  2
-     | x  dx
-     |
-    /
-
-If you have a unicode font installed, the ``pprint`` function will use it by
-default. You can override this using the ``use_unicode`` option.:
-
-    >>> pprint(Integral(x**2, x), use_unicode=True)
-    ⌠
-    ⎮  2
-    ⎮ x  dx
-    ⌡
-
-See also the wiki `Pretty Printing
-<https://github.com/sympy/sympy/wiki/Pretty-Printing>`_ for more examples of a nice
-unicode printing.
-
-Tip: To make pretty printing the default in the Python interpreter, use:
-
-::
-
-    $ python
-    Python 2.5.2 (r252:60911, Jun 25 2008, 17:58:32)
-    [GCC 4.3.1] on linux2
-    Type "help", "copyright", "credits" or "license" for more information.
-    >>> from sympy import init_printing, var, Integral
-    >>> init_printing(use_unicode=False, wrap_line=False, no_global=True)
-    >>> var("x")
-    x
-    >>> x**3/3
-     3
-    x
-    --
-    3
-    >>> Integral(x**2, x) #doctest: +NORMALIZE_WHITESPACE
-      /
-     |
-     |  2
-     | x  dx
-     |
-    /
-
-**Python printing**
-
-::
-
-    >>> from sympy.printing.python import python
-    >>> from sympy import Integral
-    >>> from sympy.abc import x
-    >>> print(python(x**2))
-    x = Symbol('x')
-    e = x**2
-    >>> print(python(1/x))
-    x = Symbol('x')
-    e = 1/x
-    >>> print(python(Integral(x**2, x)))
-    x = Symbol('x')
-    e = Integral(x**2, x)
-
-
-**LaTeX printing**
-
-::
-
-    >>> from sympy import Integral, latex
-    >>> from sympy.abc import x
-    >>> latex(x**2)
-    x^{2}
-    >>> latex(x**2, mode='inline')
-    $x^{2}$
-    >>> latex(x**2, mode='equation')
-    \begin{equation}x^{2}\end{equation}
-    >>> latex(x**2, mode='equation*')
-    \begin{equation*}x^{2}\end{equation*}
-    >>> latex(1/x)
-    \frac{1}{x}
-    >>> latex(Integral(x**2, x))
-    \int x^{2}\, dx
-
-**MathML**
-
-::
-
-    >>> from sympy.printing.mathml import mathml
-    >>> from sympy import Integral, latex
-    >>> from sympy.abc import x
-    >>> print(mathml(x**2))
-    <apply><power/><ci>x</ci><cn>2</cn></apply>
-    >>> print(mathml(1/x))
-    <apply><power/><ci>x</ci><cn>-1</cn></apply>
-
-**Pyglet**
-
-::
-
-    >>> from sympy import Integral, preview
-    >>> from sympy.abc import x
-    >>> preview(Integral(x**2, x)) #doctest:+SKIP
-
-If pyglet is installed, a pyglet window will open containing the LaTeX
-rendered expression:
-
-.. image:: ../pics/pngview1.png
-
-Notes
------
-
-``isympy`` calls ``pprint`` automatically, so that's why you see pretty
-printing by default.
-
-Note that there is also a printing module available, ``sympy.printing``.  Other
-printing methods available through this module are:
-
-* ``pretty(expr)``, ``pretty_print(expr)``, ``pprint(expr)``: Return or print, respectively, a pretty representation of ``expr``. This is the same as the second level of representation described above.
-
-* ``latex(expr)``, ``print_latex(expr)``: Return or print, respectively, a `LaTeX <http://www.latex-project.org/>`_  representation of ``expr``
-
-* ``mathml(expr)``, ``print_mathml(expr)``: Return or print, respectively, a `MathML <http://www.w3.org/Math/>`_ representation of ``expr``.
-
-* ``print_gtk(expr)``: Print ``expr`` to `Gtkmathview <http://helm.cs.unibo.it/mml-widget/>`_, a GTK widget that displays MathML code. The `Gtkmathview <http://helm.cs.unibo.it/mml-widget/>`_ program is required.
-
-Further documentation
-=====================
-
-Now it's time to learn more about SymPy. Go through the
-:ref:`SymPy User's Guide <guide>` and
-:ref:`SymPy Modules Reference <module-docs>`.
-
-Be sure to also browse our public `wiki.sympy.org <http://wiki.sympy.org/>`_,
-that contains a lot of useful examples, tutorials, cookbooks that we and our
-users contributed, and feel free to edit it.
-
-.. only:: html or gettext
-
-    Translations
-    ------------
-
-    This tutorial is also available in other languages:
-
-.. only:: html
-
-        - `Български <tutorial.bg.html>`_
-        - `Česky <tutorial.cs.html>`_
-        - `Deutsch <tutorial.de.html>`_
-        - `English <tutorial.en.html>`_
-        - `Français <tutorial.fr.html>`_
-        - `Polski <tutorial.pl.html>`_
-        - `Русский <tutorial.ru.html>`_
-        - `Српски <tutorial.sr.html>`_
-        - `中文 <tutorial.zh.html>`_
diff --git a/dev-py3k/_sources/wiki.txt b/dev-py3k/_sources/wiki.txt
deleted file mode 100644
index d7c09ca61c7..00000000000
--- a/dev-py3k/_sources/wiki.txt
+++ /dev/null
@@ -1,12 +0,0 @@
-Wiki
-====
-
-We have a public wiki: http://wiki.sympy.org
-
-Anyone please feel free to contribute to it anything you find
-interesting/useful to other users.
-
-FAQ
----
-
-`FAQ <https://github.com/sympy/sympy/wiki/Faq>`_ is one of the very useful wiki pages.
diff --git a/dev-py3k/_static/SymPy-Favicon.ico b/dev-py3k/_static/SymPy-Favicon.ico
deleted file mode 100644
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deleted file mode 100644
index 7833719c136..00000000000
--- a/dev-py3k/_static/default.css
+++ /dev/null
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-/*
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- * Sphinx stylesheet -- default theme.
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- * :copyright: Copyright 2007-2011 by the Sphinx team, see AUTHORS.
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- */
-
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-    font-family: sans-serif;
-    font-size: 100%;
-    background-color: #11303d;
-    color: #000;
-    margin: 0;
-    padding: 0;
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-div.documentwrapper {
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-    margin: 0 0 0 230px;
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-    margin: 20px -20px 10px -20px;
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\ No newline at end of file
diff --git a/dev-py3k/_static/doctools.js b/dev-py3k/_static/doctools.js
deleted file mode 100644
index d4619fdfb10..00000000000
--- a/dev-py3k/_static/doctools.js
+++ /dev/null
@@ -1,247 +0,0 @@
-/*
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- * ~~~~~~~~~~~
- *
- * Sphinx JavaScript utilities for all documentation.
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diff --git a/dev-py3k/_static/searchtools.js b/dev-py3k/_static/searchtools.js
deleted file mode 100644
index cf88109558a..00000000000
--- a/dev-py3k/_static/searchtools.js
+++ /dev/null
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-/*
- * searchtools.js_t
- * ~~~~~~~~~~~~~~~~
- *
- * Sphinx JavaScript utilties for the full-text search.
- *
- * :copyright: Copyright 2007-2011 by the Sphinx team, see AUTHORS.
- * :license: BSD, see LICENSE for details.
- *
- */
-
-/**
- * helper function to return a node containing the
- * search summary for a given text. keywords is a list
- * of stemmed words, hlwords is the list of normal, unstemmed
- * words. the first one is used to find the occurance, the
- * latter for highlighting it.
- */
-
-jQuery.makeSearchSummary = function(text, keywords, hlwords) {
-  var textLower = text.toLowerCase();
-  var start = 0;
-  $.each(keywords, function() {
-    var i = textLower.indexOf(this.toLowerCase());
-    if (i > -1)
-      start = i;
-  });
-  start = Math.max(start - 120, 0);
-  var excerpt = ((start > 0) ? '...' : '') +
-  $.trim(text.substr(start, 240)) +
-  ((start + 240 - text.length) ? '...' : '');
-  var rv = $('<div class="context"></div>').text(excerpt);
-  $.each(hlwords, function() {
-    rv = rv.highlightText(this, 'highlighted');
-  });
-  return rv;
-}
-
-
-/**
- * Porter Stemmer
- */
-var Stemmer = function() {
-
-  var step2list = {
-    ational: 'ate',
-    tional: 'tion',
-    enci: 'ence',
-    anci: 'ance',
-    izer: 'ize',
-    bli: 'ble',
-    alli: 'al',
-    entli: 'ent',
-    eli: 'e',
-    ousli: 'ous',
-    ization: 'ize',
-    ation: 'ate',
-    ator: 'ate',
-    alism: 'al',
-    iveness: 'ive',
-    fulness: 'ful',
-    ousness: 'ous',
-    aliti: 'al',
-    iviti: 'ive',
-    biliti: 'ble',
-    logi: 'log'
-  };
-
-  var step3list = {
-    icate: 'ic',
-    ative: '',
-    alize: 'al',
-    iciti: 'ic',
-    ical: 'ic',
-    ful: '',
-    ness: ''
-  };
-
-  var c = "[^aeiou]";          // consonant
-  var v = "[aeiouy]";          // vowel
-  var C = c + "[^aeiouy]*";    // consonant sequence
-  var V = v + "[aeiou]*";      // vowel sequence
-
-  var mgr0 = "^(" + C + ")?" + V + C;                      // [C]VC... is m>0
-  var meq1 = "^(" + C + ")?" + V + C + "(" + V + ")?$";    // [C]VC[V] is m=1
-  var mgr1 = "^(" + C + ")?" + V + C + V + C;              // [C]VCVC... is m>1
-  var s_v   = "^(" + C + ")?" + v;                         // vowel in stem
-
-  this.stemWord = function (w) {
-    var stem;
-    var suffix;
-    var firstch;
-    var origword = w;
-
-    if (w.length < 3)
-      return w;
-
-    var re;
-    var re2;
-    var re3;
-    var re4;
-
-    firstch = w.substr(0,1);
-    if (firstch == "y")
-      w = firstch.toUpperCase() + w.substr(1);
-
-    // Step 1a
-    re = /^(.+?)(ss|i)es$/;
-    re2 = /^(.+?)([^s])s$/;
-
-    if (re.test(w))
-      w = w.replace(re,"$1$2");
-    else if (re2.test(w))
-      w = w.replace(re2,"$1$2");
-
-    // Step 1b
-    re = /^(.+?)eed$/;
-    re2 = /^(.+?)(ed|ing)$/;
-    if (re.test(w)) {
-      var fp = re.exec(w);
-      re = new RegExp(mgr0);
-      if (re.test(fp[1])) {
-        re = /.$/;
-        w = w.replace(re,"");
-      }
-    }
-    else if (re2.test(w)) {
-      var fp = re2.exec(w);
-      stem = fp[1];
-      re2 = new RegExp(s_v);
-      if (re2.test(stem)) {
-        w = stem;
-        re2 = /(at|bl|iz)$/;
-        re3 = new RegExp("([^aeiouylsz])\\1$");
-        re4 = new RegExp("^" + C + v + "[^aeiouwxy]$");
-        if (re2.test(w))
-          w = w + "e";
-        else if (re3.test(w)) {
-          re = /.$/;
-          w = w.replace(re,"");
-        }
-        else if (re4.test(w))
-          w = w + "e";
-      }
-    }
-
-    // Step 1c
-    re = /^(.+?)y$/;
-    if (re.test(w)) {
-      var fp = re.exec(w);
-      stem = fp[1];
-      re = new RegExp(s_v);
-      if (re.test(stem))
-        w = stem + "i";
-    }
-
-    // Step 2
-    re = /^(.+?)(ational|tional|enci|anci|izer|bli|alli|entli|eli|ousli|ization|ation|ator|alism|iveness|fulness|ousness|aliti|iviti|biliti|logi)$/;
-    if (re.test(w)) {
-      var fp = re.exec(w);
-      stem = fp[1];
-      suffix = fp[2];
-      re = new RegExp(mgr0);
-      if (re.test(stem))
-        w = stem + step2list[suffix];
-    }
-
-    // Step 3
-    re = /^(.+?)(icate|ative|alize|iciti|ical|ful|ness)$/;
-    if (re.test(w)) {
-      var fp = re.exec(w);
-      stem = fp[1];
-      suffix = fp[2];
-      re = new RegExp(mgr0);
-      if (re.test(stem))
-        w = stem + step3list[suffix];
-    }
-
-    // Step 4
-    re = /^(.+?)(al|ance|ence|er|ic|able|ible|ant|ement|ment|ent|ou|ism|ate|iti|ous|ive|ize)$/;
-    re2 = /^(.+?)(s|t)(ion)$/;
-    if (re.test(w)) {
-      var fp = re.exec(w);
-      stem = fp[1];
-      re = new RegExp(mgr1);
-      if (re.test(stem))
-        w = stem;
-    }
-    else if (re2.test(w)) {
-      var fp = re2.exec(w);
-      stem = fp[1] + fp[2];
-      re2 = new RegExp(mgr1);
-      if (re2.test(stem))
-        w = stem;
-    }
-
-    // Step 5
-    re = /^(.+?)e$/;
-    if (re.test(w)) {
-      var fp = re.exec(w);
-      stem = fp[1];
-      re = new RegExp(mgr1);
-      re2 = new RegExp(meq1);
-      re3 = new RegExp("^" + C + v + "[^aeiouwxy]$");
-      if (re.test(stem) || (re2.test(stem) && !(re3.test(stem))))
-        w = stem;
-    }
-    re = /ll$/;
-    re2 = new RegExp(mgr1);
-    if (re.test(w) && re2.test(w)) {
-      re = /.$/;
-      w = w.replace(re,"");
-    }
-
-    // and turn initial Y back to y
-    if (firstch == "y")
-      w = firstch.toLowerCase() + w.substr(1);
-    return w;
-  }
-}
-
-
-/**
- * Search Module
- */
-var Search = {
-
-  _index : null,
-  _queued_query : null,
-  _pulse_status : -1,
-
-  init : function() {
-      var params = $.getQueryParameters();
-      if (params.q) {
-          var query = params.q[0];
-          $('input[name="q"]')[0].value = query;
-          this.performSearch(query);
-      }
-  },
-
-  loadIndex : function(url) {
-    $.ajax({type: "GET", url: url, data: null, success: null,
-            dataType: "script", cache: true});
-  },
-
-  setIndex : function(index) {
-    var q;
-    this._index = index;
-    if ((q = this._queued_query) !== null) {
-      this._queued_query = null;
-      Search.query(q);
-    }
-  },
-
-  hasIndex : function() {
-      return this._index !== null;
-  },
-
-  deferQuery : function(query) {
-      this._queued_query = query;
-  },
-
-  stopPulse : function() {
-      this._pulse_status = 0;
-  },
-
-  startPulse : function() {
-    if (this._pulse_status >= 0)
-        return;
-    function pulse() {
-      Search._pulse_status = (Search._pulse_status + 1) % 4;
-      var dotString = '';
-      for (var i = 0; i < Search._pulse_status; i++)
-        dotString += '.';
-      Search.dots.text(dotString);
-      if (Search._pulse_status > -1)
-        window.setTimeout(pulse, 500);
-    };
-    pulse();
-  },
-
-  /**
-   * perform a search for something
-   */
-  performSearch : function(query) {
-    // create the required interface elements
-    this.out = $('#search-results');
-    this.title = $('<h2>' + _('Searching') + '</h2>').appendTo(this.out);
-    this.dots = $('<span></span>').appendTo(this.title);
-    this.status = $('<p style="display: none"></p>').appendTo(this.out);
-    this.output = $('<ul class="search"/>').appendTo(this.out);
-
-    $('#search-progress').text(_('Preparing search...'));
-    this.startPulse();
-
-    // index already loaded, the browser was quick!
-    if (this.hasIndex())
-      this.query(query);
-    else
-      this.deferQuery(query);
-  },
-
-  query : function(query) {
-    var stopwords = ["but","was","if","and","to","no","these","will","for","they","there","as","at","their","then","is","it","or","this","with","on","that","such","are","of","in","into","near","by","the","a","be","not"];
-
-    // Stem the searchterms and add them to the correct list
-    var stemmer = new Stemmer();
-    var searchterms = [];
-    var excluded = [];
-    var hlterms = [];
-    var tmp = query.split(/\s+/);
-    var objectterms = [];
-    for (var i = 0; i < tmp.length; i++) {
-      if (tmp[i] != "") {
-          objectterms.push(tmp[i].toLowerCase());
-      }
-
-      if ($u.indexOf(stopwords, tmp[i]) != -1 || tmp[i].match(/^\d+$/) ||
-          tmp[i] == "") {
-        // skip this "word"
-        continue;
-      }
-      // stem the word
-      var word = stemmer.stemWord(tmp[i]).toLowerCase();
-      // select the correct list
-      if (word[0] == '-') {
-        var toAppend = excluded;
-        word = word.substr(1);
-      }
-      else {
-        var toAppend = searchterms;
-        hlterms.push(tmp[i].toLowerCase());
-      }
-      // only add if not already in the list
-      if (!$.contains(toAppend, word))
-        toAppend.push(word);
-    };
-    var highlightstring = '?highlight=' + $.urlencode(hlterms.join(" "));
-
-    // console.debug('SEARCH: searching for:');
-    // console.info('required: ', searchterms);
-    // console.info('excluded: ', excluded);
-
-    // prepare search
-    var filenames = this._index.filenames;
-    var titles = this._index.titles;
-    var terms = this._index.terms;
-    var fileMap = {};
-    var files = null;
-    // different result priorities
-    var importantResults = [];
-    var objectResults = [];
-    var regularResults = [];
-    var unimportantResults = [];
-    $('#search-progress').empty();
-
-    // lookup as object
-    for (var i = 0; i < objectterms.length; i++) {
-      var others = [].concat(objectterms.slice(0,i),
-                             objectterms.slice(i+1, objectterms.length))
-      var results = this.performObjectSearch(objectterms[i], others);
-      // Assume first word is most likely to be the object,
-      // other words more likely to be in description.
-      // Therefore put matches for earlier words first.
-      // (Results are eventually used in reverse order).
-      objectResults = results[0].concat(objectResults);
-      importantResults = results[1].concat(importantResults);
-      unimportantResults = results[2].concat(unimportantResults);
-    }
-
-    // perform the search on the required terms
-    for (var i = 0; i < searchterms.length; i++) {
-      var word = searchterms[i];
-      // no match but word was a required one
-      if ((files = terms[word]) == null)
-        break;
-      if (files.length == undefined) {
-        files = [files];
-      }
-      // create the mapping
-      for (var j = 0; j < files.length; j++) {
-        var file = files[j];
-        if (file in fileMap)
-          fileMap[file].push(word);
-        else
-          fileMap[file] = [word];
-      }
-    }
-
-    // now check if the files don't contain excluded terms
-    for (var file in fileMap) {
-      var valid = true;
-
-      // check if all requirements are matched
-      if (fileMap[file].length != searchterms.length)
-        continue;
-
-      // ensure that none of the excluded terms is in the
-      // search result.
-      for (var i = 0; i < excluded.length; i++) {
-        if (terms[excluded[i]] == file ||
-            $.contains(terms[excluded[i]] || [], file)) {
-          valid = false;
-          break;
-        }
-      }
-
-      // if we have still a valid result we can add it
-      // to the result list
-      if (valid)
-        regularResults.push([filenames[file], titles[file], '', null]);
-    }
-
-    // delete unused variables in order to not waste
-    // memory until list is retrieved completely
-    delete filenames, titles, terms;
-
-    // now sort the regular results descending by title
-    regularResults.sort(function(a, b) {
-      var left = a[1].toLowerCase();
-      var right = b[1].toLowerCase();
-      return (left > right) ? -1 : ((left < right) ? 1 : 0);
-    });
-
-    // combine all results
-    var results = unimportantResults.concat(regularResults)
-      .concat(objectResults).concat(importantResults);
-
-    // print the results
-    var resultCount = results.length;
-    function displayNextItem() {
-      // results left, load the summary and display it
-      if (results.length) {
-        var item = results.pop();
-        var listItem = $('<li style="display:none"></li>');
-        if (DOCUMENTATION_OPTIONS.FILE_SUFFIX == '') {
-          // dirhtml builder
-          var dirname = item[0] + '/';
-          if (dirname.match(/\/index\/$/)) {
-            dirname = dirname.substring(0, dirname.length-6);
-          } else if (dirname == 'index/') {
-            dirname = '';
-          }
-          listItem.append($('<a/>').attr('href',
-            DOCUMENTATION_OPTIONS.URL_ROOT + dirname +
-            highlightstring + item[2]).html(item[1]));
-        } else {
-          // normal html builders
-          listItem.append($('<a/>').attr('href',
-            item[0] + DOCUMENTATION_OPTIONS.FILE_SUFFIX +
-            highlightstring + item[2]).html(item[1]));
-        }
-        if (item[3]) {
-          listItem.append($('<span> (' + item[3] + ')</span>'));
-          Search.output.append(listItem);
-          listItem.slideDown(5, function() {
-            displayNextItem();
-          });
-        } else if (DOCUMENTATION_OPTIONS.HAS_SOURCE) {
-          $.get(DOCUMENTATION_OPTIONS.URL_ROOT + '_sources/' +
-                item[0] + '.txt', function(data) {
-            if (data != '') {
-              listItem.append($.makeSearchSummary(data, searchterms, hlterms));
-              Search.output.append(listItem);
-            }
-            listItem.slideDown(5, function() {
-              displayNextItem();
-            });
-          }, "text");
-        } else {
-          // no source available, just display title
-          Search.output.append(listItem);
-          listItem.slideDown(5, function() {
-            displayNextItem();
-          });
-        }
-      }
-      // search finished, update title and status message
-      else {
-        Search.stopPulse();
-        Search.title.text(_('Search Results'));
-        if (!resultCount)
-          Search.status.text(_('Your search did not match any documents. Please make sure that all words are spelled correctly and that you\'ve selected enough categories.'));
-        else
-            Search.status.text(_('Search finished, found %s page(s) matching the search query.').replace('%s', resultCount));
-        Search.status.fadeIn(500);
-      }
-    }
-    displayNextItem();
-  },
-
-  performObjectSearch : function(object, otherterms) {
-    var filenames = this._index.filenames;
-    var objects = this._index.objects;
-    var objnames = this._index.objnames;
-    var titles = this._index.titles;
-
-    var importantResults = [];
-    var objectResults = [];
-    var unimportantResults = [];
-
-    for (var prefix in objects) {
-      for (var name in objects[prefix]) {
-        var fullname = (prefix ? prefix + '.' : '') + name;
-        if (fullname.toLowerCase().indexOf(object) > -1) {
-          var match = objects[prefix][name];
-          var objname = objnames[match[1]][2];
-          var title = titles[match[0]];
-          // If more than one term searched for, we require other words to be
-          // found in the name/title/description
-          if (otherterms.length > 0) {
-            var haystack = (prefix + ' ' + name + ' ' +
-                            objname + ' ' + title).toLowerCase();
-            var allfound = true;
-            for (var i = 0; i < otherterms.length; i++) {
-              if (haystack.indexOf(otherterms[i]) == -1) {
-                allfound = false;
-                break;
-              }
-            }
-            if (!allfound) {
-              continue;
-            }
-          }
-          var descr = objname + _(', in ') + title;
-          anchor = match[3];
-          if (anchor == '')
-            anchor = fullname;
-          else if (anchor == '-')
-            anchor = objnames[match[1]][1] + '-' + fullname;
-          result = [filenames[match[0]], fullname, '#'+anchor, descr];
-          switch (match[2]) {
-          case 1: objectResults.push(result); break;
-          case 0: importantResults.push(result); break;
-          case 2: unimportantResults.push(result); break;
-          }
-        }
-      }
-    }
-
-    // sort results descending
-    objectResults.sort(function(a, b) {
-      return (a[1] > b[1]) ? -1 : ((a[1] < b[1]) ? 1 : 0);
-    });
-
-    importantResults.sort(function(a, b) {
-      return (a[1] > b[1]) ? -1 : ((a[1] < b[1]) ? 1 : 0);
-    });
-
-    unimportantResults.sort(function(a, b) {
-      return (a[1] > b[1]) ? -1 : ((a[1] < b[1]) ? 1 : 0);
-    });
-
-    return [importantResults, objectResults, unimportantResults]
-  }
-}
-
-$(document).ready(function() {
-  Search.init();
-});
\ No newline at end of file
diff --git a/dev-py3k/_static/sidebar.js b/dev-py3k/_static/sidebar.js
deleted file mode 100644
index a45e1926add..00000000000
--- a/dev-py3k/_static/sidebar.js
+++ /dev/null
@@ -1,151 +0,0 @@
-/*
- * sidebar.js
- * ~~~~~~~~~~
- *
- * This script makes the Sphinx sidebar collapsible.
- *
- * .sphinxsidebar contains .sphinxsidebarwrapper.  This script adds
- * in .sphixsidebar, after .sphinxsidebarwrapper, the #sidebarbutton
- * used to collapse and expand the sidebar.
- *
- * When the sidebar is collapsed the .sphinxsidebarwrapper is hidden
- * and the width of the sidebar and the margin-left of the document
- * are decreased. When the sidebar is expanded the opposite happens.
- * This script saves a per-browser/per-session cookie used to
- * remember the position of the sidebar among the pages.
- * Once the browser is closed the cookie is deleted and the position
- * reset to the default (expanded).
- *
- * :copyright: Copyright 2007-2011 by the Sphinx team, see AUTHORS.
- * :license: BSD, see LICENSE for details.
- *
- */
-
-$(function() {
-  // global elements used by the functions.
-  // the 'sidebarbutton' element is defined as global after its
-  // creation, in the add_sidebar_button function
-  var bodywrapper = $('.bodywrapper');
-  var sidebar = $('.sphinxsidebar');
-  var sidebarwrapper = $('.sphinxsidebarwrapper');
-
-  // for some reason, the document has no sidebar; do not run into errors
-  if (!sidebar.length) return;
-
-  // original margin-left of the bodywrapper and width of the sidebar
-  // with the sidebar expanded
-  var bw_margin_expanded = bodywrapper.css('margin-left');
-  var ssb_width_expanded = sidebar.width();
-
-  // margin-left of the bodywrapper and width of the sidebar
-  // with the sidebar collapsed
-  var bw_margin_collapsed = '.8em';
-  var ssb_width_collapsed = '.8em';
-
-  // colors used by the current theme
-  var dark_color = $('.related').css('background-color');
-  var light_color = $('.document').css('background-color');
-
-  function sidebar_is_collapsed() {
-    return sidebarwrapper.is(':not(:visible)');
-  }
-
-  function toggle_sidebar() {
-    if (sidebar_is_collapsed())
-      expand_sidebar();
-    else
-      collapse_sidebar();
-  }
-
-  function collapse_sidebar() {
-    sidebarwrapper.hide();
-    sidebar.css('width', ssb_width_collapsed);
-    bodywrapper.css('margin-left', bw_margin_collapsed);
-    sidebarbutton.css({
-        'margin-left': '0',
-        'height': bodywrapper.height()
-    });
-    sidebarbutton.find('span').text('»');
-    sidebarbutton.attr('title', _('Expand sidebar'));
-    document.cookie = 'sidebar=collapsed';
-  }
-
-  function expand_sidebar() {
-    bodywrapper.css('margin-left', bw_margin_expanded);
-    sidebar.css('width', ssb_width_expanded);
-    sidebarwrapper.show();
-    sidebarbutton.css({
-        'margin-left': ssb_width_expanded-12,
-        'height': bodywrapper.height()
-    });
-    sidebarbutton.find('span').text('«');
-    sidebarbutton.attr('title', _('Collapse sidebar'));
-    document.cookie = 'sidebar=expanded';
-  }
-
-  function add_sidebar_button() {
-    sidebarwrapper.css({
-        'float': 'left',
-        'margin-right': '0',
-        'width': ssb_width_expanded - 28
-    });
-    // create the button
-    sidebar.append(
-        '<div id="sidebarbutton"><span>&laquo;</span></div>'
-    );
-    var sidebarbutton = $('#sidebarbutton');
-    light_color = sidebarbutton.css('background-color');
-    // find the height of the viewport to center the '<<' in the page
-    var viewport_height;
-    if (window.innerHeight)
- 	  viewport_height = window.innerHeight;
-    else
-	  viewport_height = $(window).height();
-    sidebarbutton.find('span').css({
-        'display': 'block',
-        'margin-top': (viewport_height - sidebar.position().top - 20) / 2
-    });
-
-    sidebarbutton.click(toggle_sidebar);
-    sidebarbutton.attr('title', _('Collapse sidebar'));
-    sidebarbutton.css({
-        'color': '#FFFFFF',
-        'border-left': '1px solid ' + dark_color,
-        'font-size': '1.2em',
-        'cursor': 'pointer',
-        'height': bodywrapper.height(),
-        'padding-top': '1px',
-        'margin-left': ssb_width_expanded - 12
-    });
-
-    sidebarbutton.hover(
-      function () {
-          $(this).css('background-color', dark_color);
-      },
-      function () {
-          $(this).css('background-color', light_color);
-      }
-    );
-  }
-
-  function set_position_from_cookie() {
-    if (!document.cookie)
-      return;
-    var items = document.cookie.split(';');
-    for(var k=0; k<items.length; k++) {
-      var key_val = items[k].split('=');
-      var key = key_val[0];
-      if (key == 'sidebar') {
-        var value = key_val[1];
-        if ((value == 'collapsed') && (!sidebar_is_collapsed()))
-          collapse_sidebar();
-        else if ((value == 'expanded') && (sidebar_is_collapsed()))
-          expand_sidebar();
-      }
-    }
-  }
-
-  add_sidebar_button();
-  var sidebarbutton = $('#sidebarbutton');
-  set_position_from_cookie();
-});
diff --git a/dev-py3k/_static/sympylogo.png b/dev-py3k/_static/sympylogo.png
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index 5d89914340f..00000000000
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-/*
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-      var i = by.substring(3);
-      comp = function(a, b) { return a[i] - b[i]; };
-    } else {
-      // Otherwise sort in descending order.
-      comp = function(a, b) { return b[by] - a[by]; };
-    }
-
-    // Reset link styles and format the selected sort option.
-    $('a.sel').attr('href', '#').removeClass('sel');
-    $('a.by' + by).removeAttr('href').addClass('sel');
-  }
-
-  /**
-   * Create a comp function. If the user has preferences stored in
-   * the sortBy cookie, use those, otherwise use the default.
-   */
-  function initComparator() {
-    by = 'rating'; // Default to sort by rating.
-    // If the sortBy cookie is set, use that instead.
-    if (document.cookie.length > 0) {
-      var start = document.cookie.indexOf('sortBy=');
-      if (start != -1) {
-        start = start + 7;
-        var end = document.cookie.indexOf(";", start);
-        if (end == -1) {
-          end = document.cookie.length;
-          by = unescape(document.cookie.substring(start, end));
-        }
-      }
-    }
-    setComparator();
-  }
-
-  /**
-   * Show a comment div.
-   */
-  function show(id) {
-    $('#ao' + id).hide();
-    $('#ah' + id).show();
-    var context = $.extend({id: id}, opts);
-    var popup = $(renderTemplate(popupTemplate, context)).hide();
-    popup.find('textarea[name="proposal"]').hide();
-    popup.find('a.by' + by).addClass('sel');
-    var form = popup.find('#cf' + id);
-    form.submit(function(event) {
-      event.preventDefault();
-      addComment(form);
-    });
-    $('#s' + id).after(popup);
-    popup.slideDown('fast', function() {
-      getComments(id);
-    });
-  }
-
-  /**
-   * Hide a comment div.
-   */
-  function hide(id) {
-    $('#ah' + id).hide();
-    $('#ao' + id).show();
-    var div = $('#sc' + id);
-    div.slideUp('fast', function() {
-      div.remove();
-    });
-  }
-
-  /**
-   * Perform an ajax request to get comments for a node
-   * and insert the comments into the comments tree.
-   */
-  function getComments(id) {
-    $.ajax({
-     type: 'GET',
-     url: opts.getCommentsURL,
-     data: {node: id},
-     success: function(data, textStatus, request) {
-       var ul = $('#cl' + id);
-       var speed = 100;
-       $('#cf' + id)
-         .find('textarea[name="proposal"]')
-         .data('source', data.source);
-
-       if (data.comments.length === 0) {
-         ul.html('<li>No comments yet.</li>');
-         ul.data('empty', true);
-       } else {
-         // If there are comments, sort them and put them in the list.
-         var comments = sortComments(data.comments);
-         speed = data.comments.length * 100;
-         appendComments(comments, ul);
-         ul.data('empty', false);
-       }
-       $('#cn' + id).slideUp(speed + 200);
-       ul.slideDown(speed);
-     },
-     error: function(request, textStatus, error) {
-       showError('Oops, there was a problem retrieving the comments.');
-     },
-     dataType: 'json'
-    });
-  }
-
-  /**
-   * Add a comment via ajax and insert the comment into the comment tree.
-   */
-  function addComment(form) {
-    var node_id = form.find('input[name="node"]').val();
-    var parent_id = form.find('input[name="parent"]').val();
-    var text = form.find('textarea[name="comment"]').val();
-    var proposal = form.find('textarea[name="proposal"]').val();
-
-    if (text == '') {
-      showError('Please enter a comment.');
-      return;
-    }
-
-    // Disable the form that is being submitted.
-    form.find('textarea,input').attr('disabled', 'disabled');
-
-    // Send the comment to the server.
-    $.ajax({
-      type: "POST",
-      url: opts.addCommentURL,
-      dataType: 'json',
-      data: {
-        node: node_id,
-        parent: parent_id,
-        text: text,
-        proposal: proposal
-      },
-      success: function(data, textStatus, error) {
-        // Reset the form.
-        if (node_id) {
-          hideProposeChange(node_id);
-        }
-        form.find('textarea')
-          .val('')
-          .add(form.find('input'))
-          .removeAttr('disabled');
-	var ul = $('#cl' + (node_id || parent_id));
-        if (ul.data('empty')) {
-          $(ul).empty();
-          ul.data('empty', false);
-        }
-        insertComment(data.comment);
-        var ao = $('#ao' + node_id);
-        ao.find('img').attr({'src': opts.commentBrightImage});
-        if (node_id) {
-          // if this was a "root" comment, remove the commenting box
-          // (the user can get it back by reopening the comment popup)
-          $('#ca' + node_id).slideUp();
-        }
-      },
-      error: function(request, textStatus, error) {
-        form.find('textarea,input').removeAttr('disabled');
-        showError('Oops, there was a problem adding the comment.');
-      }
-    });
-  }
-
-  /**
-   * Recursively append comments to the main comment list and children
-   * lists, creating the comment tree.
-   */
-  function appendComments(comments, ul) {
-    $.each(comments, function() {
-      var div = createCommentDiv(this);
-      ul.append($(document.createElement('li')).html(div));
-      appendComments(this.children, div.find('ul.comment-children'));
-      // To avoid stagnating data, don't store the comments children in data.
-      this.children = null;
-      div.data('comment', this);
-    });
-  }
-
-  /**
-   * After adding a new comment, it must be inserted in the correct
-   * location in the comment tree.
-   */
-  function insertComment(comment) {
-    var div = createCommentDiv(comment);
-
-    // To avoid stagnating data, don't store the comments children in data.
-    comment.children = null;
-    div.data('comment', comment);
-
-    var ul = $('#cl' + (comment.node || comment.parent));
-    var siblings = getChildren(ul);
-
-    var li = $(document.createElement('li'));
-    li.hide();
-
-    // Determine where in the parents children list to insert this comment.
-    for(i=0; i < siblings.length; i++) {
-      if (comp(comment, siblings[i]) <= 0) {
-        $('#cd' + siblings[i].id)
-          .parent()
-          .before(li.html(div));
-        li.slideDown('fast');
-        return;
-      }
-    }
-
-    // If we get here, this comment rates lower than all the others,
-    // or it is the only comment in the list.
-    ul.append(li.html(div));
-    li.slideDown('fast');
-  }
-
-  function acceptComment(id) {
-    $.ajax({
-      type: 'POST',
-      url: opts.acceptCommentURL,
-      data: {id: id},
-      success: function(data, textStatus, request) {
-        $('#cm' + id).fadeOut('fast');
-        $('#cd' + id).removeClass('moderate');
-      },
-      error: function(request, textStatus, error) {
-        showError('Oops, there was a problem accepting the comment.');
-      }
-    });
-  }
-
-  function deleteComment(id) {
-    $.ajax({
-      type: 'POST',
-      url: opts.deleteCommentURL,
-      data: {id: id},
-      success: function(data, textStatus, request) {
-        var div = $('#cd' + id);
-        if (data == 'delete') {
-          // Moderator mode: remove the comment and all children immediately
-          div.slideUp('fast', function() {
-            div.remove();
-          });
-          return;
-        }
-        // User mode: only mark the comment as deleted
-        div
-          .find('span.user-id:first')
-          .text('[deleted]').end()
-          .find('div.comment-text:first')
-          .text('[deleted]').end()
-          .find('#cm' + id + ', #dc' + id + ', #ac' + id + ', #rc' + id +
-                ', #sp' + id + ', #hp' + id + ', #cr' + id + ', #rl' + id)
-          .remove();
-        var comment = div.data('comment');
-        comment.username = '[deleted]';
-        comment.text = '[deleted]';
-        div.data('comment', comment);
-      },
-      error: function(request, textStatus, error) {
-        showError('Oops, there was a problem deleting the comment.');
-      }
-    });
-  }
-
-  function showProposal(id) {
-    $('#sp' + id).hide();
-    $('#hp' + id).show();
-    $('#pr' + id).slideDown('fast');
-  }
-
-  function hideProposal(id) {
-    $('#hp' + id).hide();
-    $('#sp' + id).show();
-    $('#pr' + id).slideUp('fast');
-  }
-
-  function showProposeChange(id) {
-    $('#pc' + id).hide();
-    $('#hc' + id).show();
-    var textarea = $('#pt' + id);
-    textarea.val(textarea.data('source'));
-    $.fn.autogrow.resize(textarea[0]);
-    textarea.slideDown('fast');
-  }
-
-  function hideProposeChange(id) {
-    $('#hc' + id).hide();
-    $('#pc' + id).show();
-    var textarea = $('#pt' + id);
-    textarea.val('').removeAttr('disabled');
-    textarea.slideUp('fast');
-  }
-
-  function toggleCommentMarkupBox(id) {
-    $('#mb' + id).toggle();
-  }
-
-  /** Handle when the user clicks on a sort by link. */
-  function handleReSort(link) {
-    var classes = link.attr('class').split(/\s+/);
-    for (var i=0; i<classes.length; i++) {
-      if (classes[i] != 'sort-option') {
-	by = classes[i].substring(2);
-      }
-    }
-    setComparator();
-    // Save/update the sortBy cookie.
-    var expiration = new Date();
-    expiration.setDate(expiration.getDate() + 365);
-    document.cookie= 'sortBy=' + escape(by) +
-                     ';expires=' + expiration.toUTCString();
-    $('ul.comment-ul').each(function(index, ul) {
-      var comments = getChildren($(ul), true);
-      comments = sortComments(comments);
-      appendComments(comments, $(ul).empty());
-    });
-  }
-
-  /**
-   * Function to process a vote when a user clicks an arrow.
-   */
-  function handleVote(link) {
-    if (!opts.voting) {
-      showError("You'll need to login to vote.");
-      return;
-    }
-
-    var id = link.attr('id');
-    if (!id) {
-      // Didn't click on one of the voting arrows.
-      return;
-    }
-    // If it is an unvote, the new vote value is 0,
-    // Otherwise it's 1 for an upvote, or -1 for a downvote.
-    var value = 0;
-    if (id.charAt(1) != 'u') {
-      value = id.charAt(0) == 'u' ? 1 : -1;
-    }
-    // The data to be sent to the server.
-    var d = {
-      comment_id: id.substring(2),
-      value: value
-    };
-
-    // Swap the vote and unvote links.
-    link.hide();
-    $('#' + id.charAt(0) + (id.charAt(1) == 'u' ? 'v' : 'u') + d.comment_id)
-      .show();
-
-    // The div the comment is displayed in.
-    var div = $('div#cd' + d.comment_id);
-    var data = div.data('comment');
-
-    // If this is not an unvote, and the other vote arrow has
-    // already been pressed, unpress it.
-    if ((d.value !== 0) && (data.vote === d.value * -1)) {
-      $('#' + (d.value == 1 ? 'd' : 'u') + 'u' + d.comment_id).hide();
-      $('#' + (d.value == 1 ? 'd' : 'u') + 'v' + d.comment_id).show();
-    }
-
-    // Update the comments rating in the local data.
-    data.rating += (data.vote === 0) ? d.value : (d.value - data.vote);
-    data.vote = d.value;
-    div.data('comment', data);
-
-    // Change the rating text.
-    div.find('.rating:first')
-      .text(data.rating + ' point' + (data.rating == 1 ? '' : 's'));
-
-    // Send the vote information to the server.
-    $.ajax({
-      type: "POST",
-      url: opts.processVoteURL,
-      data: d,
-      error: function(request, textStatus, error) {
-        showError('Oops, there was a problem casting that vote.');
-      }
-    });
-  }
-
-  /**
-   * Open a reply form used to reply to an existing comment.
-   */
-  function openReply(id) {
-    // Swap out the reply link for the hide link
-    $('#rl' + id).hide();
-    $('#cr' + id).show();
-
-    // Add the reply li to the children ul.
-    var div = $(renderTemplate(replyTemplate, {id: id})).hide();
-    $('#cl' + id)
-      .prepend(div)
-      // Setup the submit handler for the reply form.
-      .find('#rf' + id)
-      .submit(function(event) {
-        event.preventDefault();
-        addComment($('#rf' + id));
-        closeReply(id);
-      })
-      .find('input[type=button]')
-      .click(function() {
-        closeReply(id);
-      });
-    div.slideDown('fast', function() {
-      $('#rf' + id).find('textarea').focus();
-    });
-  }
-
-  /**
-   * Close the reply form opened with openReply.
-   */
-  function closeReply(id) {
-    // Remove the reply div from the DOM.
-    $('#rd' + id).slideUp('fast', function() {
-      $(this).remove();
-    });
-
-    // Swap out the hide link for the reply link
-    $('#cr' + id).hide();
-    $('#rl' + id).show();
-  }
-
-  /**
-   * Recursively sort a tree of comments using the comp comparator.
-   */
-  function sortComments(comments) {
-    comments.sort(comp);
-    $.each(comments, function() {
-      this.children = sortComments(this.children);
-    });
-    return comments;
-  }
-
-  /**
-   * Get the children comments from a ul. If recursive is true,
-   * recursively include childrens' children.
-   */
-  function getChildren(ul, recursive) {
-    var children = [];
-    ul.children().children("[id^='cd']")
-      .each(function() {
-        var comment = $(this).data('comment');
-        if (recursive)
-          comment.children = getChildren($(this).find('#cl' + comment.id), true);
-        children.push(comment);
-      });
-    return children;
-  }
-
-  /** Create a div to display a comment in. */
-  function createCommentDiv(comment) {
-    if (!comment.displayed && !opts.moderator) {
-      return $('<div class="moderate">Thank you!  Your comment will show up '
-               + 'once it is has been approved by a moderator.</div>');
-    }
-    // Prettify the comment rating.
-    comment.pretty_rating = comment.rating + ' point' +
-      (comment.rating == 1 ? '' : 's');
-    // Make a class (for displaying not yet moderated comments differently)
-    comment.css_class = comment.displayed ? '' : ' moderate';
-    // Create a div for this comment.
-    var context = $.extend({}, opts, comment);
-    var div = $(renderTemplate(commentTemplate, context));
-
-    // If the user has voted on this comment, highlight the correct arrow.
-    if (comment.vote) {
-      var direction = (comment.vote == 1) ? 'u' : 'd';
-      div.find('#' + direction + 'v' + comment.id).hide();
-      div.find('#' + direction + 'u' + comment.id).show();
-    }
-
-    if (opts.moderator || comment.text != '[deleted]') {
-      div.find('a.reply').show();
-      if (comment.proposal_diff)
-        div.find('#sp' + comment.id).show();
-      if (opts.moderator && !comment.displayed)
-        div.find('#cm' + comment.id).show();
-      if (opts.moderator || (opts.username == comment.username))
-        div.find('#dc' + comment.id).show();
-    }
-    return div;
-  }
-
-  /**
-   * A simple template renderer. Placeholders such as <%id%> are replaced
-   * by context['id'] with items being escaped. Placeholders such as <#id#>
-   * are not escaped.
-   */
-  function renderTemplate(template, context) {
-    var esc = $(document.createElement('div'));
-
-    function handle(ph, escape) {
-      var cur = context;
-      $.each(ph.split('.'), function() {
-        cur = cur[this];
-      });
-      return escape ? esc.text(cur || "").html() : cur;
-    }
-
-    return template.replace(/<([%#])([\w\.]*)\1>/g, function() {
-      return handle(arguments[2], arguments[1] == '%' ? true : false);
-    });
-  }
-
-  /** Flash an error message briefly. */
-  function showError(message) {
-    $(document.createElement('div')).attr({'class': 'popup-error'})
-      .append($(document.createElement('div'))
-               .attr({'class': 'error-message'}).text(message))
-      .appendTo('body')
-      .fadeIn("slow")
-      .delay(2000)
-      .fadeOut("slow");
-  }
-
-  /** Add a link the user uses to open the comments popup. */
-  $.fn.comment = function() {
-    return this.each(function() {
-      var id = $(this).attr('id').substring(1);
-      var count = COMMENT_METADATA[id];
-      var title = count + ' comment' + (count == 1 ? '' : 's');
-      var image = count > 0 ? opts.commentBrightImage : opts.commentImage;
-      var addcls = count == 0 ? ' nocomment' : '';
-      $(this)
-        .append(
-          $(document.createElement('a')).attr({
-            href: '#',
-            'class': 'sphinx-comment-open' + addcls,
-            id: 'ao' + id
-          })
-            .append($(document.createElement('img')).attr({
-              src: image,
-              alt: 'comment',
-              title: title
-            }))
-            .click(function(event) {
-              event.preventDefault();
-              show($(this).attr('id').substring(2));
-            })
-        )
-        .append(
-          $(document.createElement('a')).attr({
-            href: '#',
-            'class': 'sphinx-comment-close hidden',
-            id: 'ah' + id
-          })
-            .append($(document.createElement('img')).attr({
-              src: opts.closeCommentImage,
-              alt: 'close',
-              title: 'close'
-            }))
-            .click(function(event) {
-              event.preventDefault();
-              hide($(this).attr('id').substring(2));
-            })
-        );
-    });
-  };
-
-  var opts = {
-    processVoteURL: '/_process_vote',
-    addCommentURL: '/_add_comment',
-    getCommentsURL: '/_get_comments',
-    acceptCommentURL: '/_accept_comment',
-    deleteCommentURL: '/_delete_comment',
-    commentImage: '/static/_static/comment.png',
-    closeCommentImage: '/static/_static/comment-close.png',
-    loadingImage: '/static/_static/ajax-loader.gif',
-    commentBrightImage: '/static/_static/comment-bright.png',
-    upArrow: '/static/_static/up.png',
-    downArrow: '/static/_static/down.png',
-    upArrowPressed: '/static/_static/up-pressed.png',
-    downArrowPressed: '/static/_static/down-pressed.png',
-    voting: false,
-    moderator: false
-  };
-
-  if (typeof COMMENT_OPTIONS != "undefined") {
-    opts = jQuery.extend(opts, COMMENT_OPTIONS);
-  }
-
-  var popupTemplate = '\
-    <div class="sphinx-comments" id="sc<%id%>">\
-      <p class="sort-options">\
-        Sort by:\
-        <a href="#" class="sort-option byrating">best rated</a>\
-        <a href="#" class="sort-option byascage">newest</a>\
-        <a href="#" class="sort-option byage">oldest</a>\
-      </p>\
-      <div class="comment-header">Comments</div>\
-      <div class="comment-loading" id="cn<%id%>">\
-        loading comments... <img src="<%loadingImage%>" alt="" /></div>\
-      <ul id="cl<%id%>" class="comment-ul"></ul>\
-      <div id="ca<%id%>">\
-      <p class="add-a-comment">Add a comment\
-        (<a href="#" class="comment-markup" id="ab<%id%>">markup</a>):</p>\
-      <div class="comment-markup-box" id="mb<%id%>">\
-        reStructured text markup: <i>*emph*</i>, <b>**strong**</b>, \
-        <tt>``code``</tt>, \
-        code blocks: <tt>::</tt> and an indented block after blank line</div>\
-      <form method="post" id="cf<%id%>" class="comment-form" action="">\
-        <textarea name="comment" cols="80"></textarea>\
-        <p class="propose-button">\
-          <a href="#" id="pc<%id%>" class="show-propose-change">\
-            Propose a change &#9657;\
-          </a>\
-          <a href="#" id="hc<%id%>" class="hide-propose-change">\
-            Propose a change &#9663;\
-          </a>\
-        </p>\
-        <textarea name="proposal" id="pt<%id%>" cols="80"\
-                  spellcheck="false"></textarea>\
-        <input type="submit" value="Add comment" />\
-        <input type="hidden" name="node" value="<%id%>" />\
-        <input type="hidden" name="parent" value="" />\
-      </form>\
-      </div>\
-    </div>';
-
-  var commentTemplate = '\
-    <div id="cd<%id%>" class="sphinx-comment<%css_class%>">\
-      <div class="vote">\
-        <div class="arrow">\
-          <a href="#" id="uv<%id%>" class="vote" title="vote up">\
-            <img src="<%upArrow%>" />\
-          </a>\
-          <a href="#" id="uu<%id%>" class="un vote" title="vote up">\
-            <img src="<%upArrowPressed%>" />\
-          </a>\
-        </div>\
-        <div class="arrow">\
-          <a href="#" id="dv<%id%>" class="vote" title="vote down">\
-            <img src="<%downArrow%>" id="da<%id%>" />\
-          </a>\
-          <a href="#" id="du<%id%>" class="un vote" title="vote down">\
-            <img src="<%downArrowPressed%>" />\
-          </a>\
-        </div>\
-      </div>\
-      <div class="comment-content">\
-        <p class="tagline comment">\
-          <span class="user-id"><%username%></span>\
-          <span class="rating"><%pretty_rating%></span>\
-          <span class="delta"><%time.delta%></span>\
-        </p>\
-        <div class="comment-text comment"><#text#></div>\
-        <p class="comment-opts comment">\
-          <a href="#" class="reply hidden" id="rl<%id%>">reply &#9657;</a>\
-          <a href="#" class="close-reply" id="cr<%id%>">reply &#9663;</a>\
-          <a href="#" id="sp<%id%>" class="show-proposal">proposal &#9657;</a>\
-          <a href="#" id="hp<%id%>" class="hide-proposal">proposal &#9663;</a>\
-          <a href="#" id="dc<%id%>" class="delete-comment hidden">delete</a>\
-          <span id="cm<%id%>" class="moderation hidden">\
-            <a href="#" id="ac<%id%>" class="accept-comment">accept</a>\
-          </span>\
-        </p>\
-        <pre class="proposal" id="pr<%id%>">\
-<#proposal_diff#>\
-        </pre>\
-          <ul class="comment-children" id="cl<%id%>"></ul>\
-        </div>\
-        <div class="clearleft"></div>\
-      </div>\
-    </div>';
-
-  var replyTemplate = '\
-    <li>\
-      <div class="reply-div" id="rd<%id%>">\
-        <form id="rf<%id%>">\
-          <textarea name="comment" cols="80"></textarea>\
-          <input type="submit" value="Add reply" />\
-          <input type="button" value="Cancel" />\
-          <input type="hidden" name="parent" value="<%id%>" />\
-          <input type="hidden" name="node" value="" />\
-        </form>\
-      </div>\
-    </li>';
-
-  $(document).ready(function() {
-    init();
-  });
-})(jQuery);
-
-$(document).ready(function() {
-  // add comment anchors for all paragraphs that are commentable
-  $('.sphinx-has-comment').comment();
-
-  // highlight search words in search results
-  $("div.context").each(function() {
-    var params = $.getQueryParameters();
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-  <div class="section" id="about">
-<h1>About<a class="headerlink" href="#about" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="sympy-development-team">
-<h2>SymPy Development Team<a class="headerlink" href="#sympy-development-team" title="Permalink to this headline">¶</a></h2>
-<p>SymPy is a team project and it was developed by a lot of people.</p>
-<p>Here is a list of contributors together with what they do, (and in some cases links to their
-wiki pages), where they describe in
-more details what they do and what they are interested in (some people didn&#8217;t
-want to be mentioned here, so see our repository history for a full list).</p>
-<ol class="arabic simple">
-<li>Ondřej Čertík: started the project in 2006, on Jan 4, 2011 passed the project leadership to Aaron Meurer</li>
-<li>Fabian Pedregosa: everything, reviewing patches, releases, general advice (issues and mailinglist), GSoC 2009</li>
-<li>Jurjen N.E. Bos: pretty printing and other patches</li>
-<li>Mateusz Paprocki: GSoC 2007, concrete math module, integration module, new core integration, a lot of patches, general advice, new polynomial module, improvements to solvers, simplifications, patch review</li>
-<li>Marc-Etienne M.Leveille: matrix patch</li>
-<li>Brian Jorgensen: GSoC 2007, plotting module and related things, patches</li>
-<li>Jason Gedge: GSoC 2007, geometry module, a lot of patches and fixes, new core integration</li>
-<li>Robert Schwarz: GSoC 2007, polynomials module, patches</li>
-<li><a class="reference external" href="https://github.com/sympy/sympy/wiki/Pearu-Peterson">Pearu Peterson</a>: new core, sympycore project, general advice (issues and mailinglist)</li>
-<li><a class="reference external" href="https://github.com/sympy/sympy/wiki/Fredrik-Johansson">Fredrik Johansson</a>: mpmath project and its integration in SymPy, number theory, combinatorial functions, products &amp; summation, statistics, units, patches, documentation, general advice (issues and mailinglist)</li>
-<li>Chris Wu: GSoC 2007, linear algebra module</li>
-<li>Ulrich Hecht: pattern matching and other patches</li>
-<li>Goutham Lakshminarayan: number theory functions</li>
-<li>David Lawrence: GHOP, Mathematica parser, square root denesting</li>
-<li>Jaroslaw Tworek: GHOP, sympify AST implementation, sqrt() refactoring, maxima parser and other patches</li>
-<li>David Marek: GHOP, derivative evaluation patch, int(NumberSymbol) fix</li>
-<li>Bernhard R. Link: documentation patch</li>
-<li>Andrej Tokarčík: GHOP, python printer</li>
-<li>Or Dvory: GHOP, documentation</li>
-<li>Saroj Adhikari: bug fixes</li>
-<li>Pauli Virtanen: bug fix</li>
-<li>Robert Kern: bug fix, common subexpression elimination</li>
-<li>James Aspnes: bug fixes</li>
-<li>Nimish Telang: multivariate lambdas</li>
-<li>Abderrahim Kitouni: pretty printing + complex expansion bug fixes</li>
-<li>Pan Peng: ode solvers patch</li>
-<li>Friedrich Hagedorn: many bug fixes, refactorings and new features added</li>
-<li>Elrond der Elbenfuerst: pretty printing fix</li>
-<li>Rizgar Mella: BBP formula for pi calculating algorithm</li>
-<li>Felix Kaiser: documentation + whitespace testing patches</li>
-<li>Roberto Nobrega: several pretty printing patches</li>
-<li>David Roberts: latex printing patches</li>
-<li>Sebastian Krämer: implemented lambdify/numpy/mpmath cooperation, bug fixes, refactoring, lambdifying of matrices, large printing refactoring and bugfixes</li>
-<li>Vinzent Steinberg: docstring patches, a lot of bug fixes, nsolve (nonlinear equation systems solver), compiling functions to machine code, patches review</li>
-<li>Riccardo Gori: improvements and speedups to matrices, many bug fixes</li>
-<li>Case Van Horsen: implemented optional support for gmpy in mpmath</li>
-<li>Štěpán Roučka: a lot of bug fixes all over SymPy (matrix, simplification, limits, series, ...)</li>
-<li>Ali Raza Syed: pretty printing/isympy on windows fix</li>
-<li>Stefano Maggiolo: many bug fixes, polishings and improvements</li>
-<li>Robert Cimrman: matrix patches</li>
-<li>Bastian Weber: latex printing patches</li>
-<li>Sebastian Krause: match patches</li>
-<li>Sebastian Kreft: latex printing patches, Dirac delta function, other fixes</li>
-<li>Dan (coolg49964): documentation fixes</li>
-<li>Alan Bromborsky: geometric algebra modules</li>
-<li>Boris Timokhin: matrix fixes</li>
-<li>Robert (average.programmer): initial piecewise function patch</li>
-<li>Andy R. Terrel: piecewise function polishing and other patches</li>
-<li>Hubert Tsang: LaTeX printing fixes</li>
-<li>Konrad Meyer: policy patch</li>
-<li>Henrik Johansson: matrix zero finder patch</li>
-<li>Priit Laes: line integrals, cleanups in ODE, tests added</li>
-<li>Freddie Witherden: trigonometric simplifications fixes, LaTeX printer improvements</li>
-<li>Brian E. Granger: second quantization physics module</li>
-<li>Andrew Straw: lambdify() improvements</li>
-<li>Kaifeng Zhu: factorint() fix</li>
-<li>Ted Horst: Basic.__call__() fix, pretty printing fix</li>
-<li>Andrew Docherty: Fix to series</li>
-<li>Akshay Srinivasan: printing fix, improvements to integration</li>
-<li>Aaron Meurer: ODE solvers (GSoC 2009), The Risch Algorithm for integration (GSoC 2010), other fixes, project leader as of Jan 4, 2011</li>
-<li>Barry Wardell: implement series as function, several tests added</li>
-<li>Tomasz Buchert: ccode printing fixes, code quality concerning exceptions, documentation</li>
-<li>Vinay Kumar: polygamma tests</li>
-<li>Johann Cohen-Tanugi: commutative diff tests</li>
-<li>Jochen Voss: a test for the &#64;cachit decorator and several other fixes</li>
-<li>Luke Peterson: improve solve() to handle Derivatives</li>
-<li>Chris Smith: improvements to solvers, many bug fixes, documentation and test improvements</li>
-<li>Thomas Sidoti: MathML printer improvements</li>
-<li>Florian Mickler: reimplementation of convex_hull, several geometry module fixes</li>
-<li>Nicolas Pourcelot: Infinity comparison fixes, latex fixes</li>
-<li>Ben Goodrich: Matrix.jacobian() function improvements and fixes</li>
-<li>Toon Verstraelen: code generation module, latex printing improvements</li>
-<li>Ronan Lamy: test coverage script; limit, expansion and other fixes and improvements; cleanup</li>
-<li>James Abbatiello: fixes tests on windows</li>
-<li>Ryan Krauss: fixes could_extract_minus_sign(), latex fixes and improvements</li>
-<li>Bill Flynn: substitution fixes</li>
-<li>Kevin Goodsell: Fix to Integer/Rational</li>
-<li>Jorn Baayen: improvements to piecewise functions and latex printing, bug fixes</li>
-<li>Eh Tan: improve trigonometric simplification</li>
-<li>Renato Coutinho: derivative improvements</li>
-<li>Oscar Benjamin: latex printer fix, gcd bug fix</li>
-<li>Øyvind Jensen: implemented coupled cluster expansion and wick theorem, improvements to assumptions, bugfixes</li>
-<li>Julio Idichekop Filho: indentation fixes, docstring improvements</li>
-<li>Łukasz Pankowski: fix matrix multiplication with numpy scalars</li>
-<li>Chu-Ching Huang: fix 3d plotting example</li>
-<li>Fernando Perez: symarray() implemented</li>
-<li>Raffaele De Feo: fix non-commutative expansion</li>
-<li>Christian Muise: fixes to logic module</li>
-<li>Matt Curry: GSoC 2010 project (symbolic quantum mechanics)</li>
-<li>Kazuo Thow: cleanup pretty printing tests</li>
-<li>Christian Schubert: Fix to sympify()</li>
-<li>Jezreel Ng: fix hyperbolic functions rewrite</li>
-<li>James Pearson: Py3k related fixes</li>
-<li>Matthew Brett: fixes to lambdify</li>
-<li>Addison Cugini: GSoC 2010 project (quantum computing)</li>
-<li>Nicholas J.S. Kinar: improved documentation about &#8220;Immutability of Expressions&#8221;</li>
-<li>Harold Erbin: Geometry related work</li>
-<li>Thomas Dixon: fix a limit bug</li>
-<li>Cristóvão Sousa: implements _sage_ method for sign function.</li>
-<li>Andre de Fortier Smit: doctests in matrices improved</li>
-<li>Mark Dewing: Fixes to Integral/Sum, MathML work</li>
-<li>Alexey U. Gudchenko: various work in matrices</li>
-<li>Gary Kerr: fix examples, docs</li>
-<li>Sherjil Ozair: fixes to SparseMatrix</li>
-<li>Oleksandr Gituliar: fixes to Matrix</li>
-<li>Sean Vig: Quantum improvement</li>
-<li>Prafullkumar P. Tale: fixed plotting documentation</li>
-<li>Vladimir Perić: fix some Python 3 issues</li>
-<li>Tom Bachmann: fixes to Real</li>
-<li>Yuri Karadzhov: improvements to hyperbolic identities</li>
-<li>Vladimir Lagunov: improvements to the geometry module</li>
-<li>Matthew Rocklin: stats, sets, matrix expressions</li>
-<li>Saptarshi Mandal: Test for an integral</li>
-<li>Gilbert Gede: Tests for solvers</li>
-<li>Anatolii Koval: Infinite 1D box example</li>
-<li>Tomo Lazovich: fixes to quantum operators</li>
-<li>Pavel Fedotov: fix to is_number</li>
-<li>Kibeom Kim: fixes to cse function and preorder_traversal</li>
-<li>Gregory Ksionda: fixes to Integral instantiation</li>
-<li>Tomáš Bambas: prettier printing of examples</li>
-<li>Jeremias Yehdegho: fixes to the nthoery module</li>
-<li>Jack McCaffery: fixes to asin and acos</li>
-<li>Raymond Wong: Quantum work</li>
-<li>Luca Weihs: improvements to the geometry module</li>
-<li>Shai &#8216;Deshe&#8217; Wyborski: fix to numeric evaluation of hypergeometric sums</li>
-<li>Thomas Wiecki: Fix Sum.diff</li>
-<li>Óscar Nájera: better Laguerre polynomial generator</li>
-<li>Mario Pernici: faster algorithm for computing Groebner bases</li>
-<li>Benjamin McDonald: Fix bug in geometry</li>
-<li>Sam Magura: Improvements to Plot.saveimage</li>
-<li>Stefan Krastanov: Make Pyglet an external dependency</li>
-<li>Bradley Froehle: Fix shell command to generate modules in setup.py</li>
-<li>Min Ragan-Kelley: Fix isympy to work with IPython 0.11</li>
-<li>Nikhil Sarda: Fix to combinatorics/prufer</li>
-<li>Emma Hogan: Fixes to the documentation</li>
-<li>Jason Moore: Fixes to the mechanics module</li>
-<li>Julien Rioux: Fix behavior of some deprecated methods</li>
-<li>Roberto Colistete, Jr.: Add num_columns option to the pretty printer</li>
-<li>Raoul Bourquin: Implement Euler numbers</li>
-<li>Gert-Ludwig Ingold: Correct derivative of coth</li>
-<li>Srinivas Vasudevan: Implementation of Catalan numbers</li>
-<li>Miha Marolt: Add numpydoc extension to the Sphinx docs, fix ode_order</li>
-<li>Tim Lahey: Rotation matrices</li>
-<li>Luis Garcia: update examples in symarry</li>
-<li>Matt Rajca: Code quality fixes</li>
-<li>David Li: Documentation fixes</li>
-<li>David Ju: Increase test coverage, fixes to KroneckerDelta</li>
-<li>Alexandr Gudulin: Code quality fixes</li>
-<li>Bilal Akhtar: isympy man page</li>
-<li>Grzegorz Świrski: Fix to latex(), MacPorts portfile</li>
-<li>Matt Habel: SymPy-wide pyflakes editing</li>
-<li>Nikolay Lazarov: Translation of the tutorial to Bulgarian</li>
-<li>Nichita Utiu: Add pretty printing to Product()</li>
-<li>Tristan Hume: Fixes to test_lambdify.py</li>
-<li>Imran Ahmed Manzoor: Fixes to the test runner</li>
-<li>Steve Anton: pyflakes cleanup of various modules, documentation fixes for logic</li>
-<li>Sam Sleight: Fixes to geometry documentation</li>
-<li>tsmars15: Fixes to code quality</li>
-<li>Chancellor Arkantos: Fixes to the logo</li>
-<li>Stepan Simsa: Translation of the tutorial to Czech</li>
-<li>Tobias Lenz: Unicode pretty printer for Sum</li>
-<li>Siddhanathan Shanmugam: Documentation fixes for the Physics module</li>
-<li>Tiffany Zhu: Improved the latex() docstring</li>
-<li>Alexey Subach: Translation of the tutorial to Russian</li>
-<li>Joan Creus: Improvements to the test runner</li>
-<li>Geoffry Song: Improve code coverage</li>
-<li>Puneeth Chaganti: Fix for the tutorial</li>
-<li>Marcin Kostrzewa: SymPy Cheatsheet</li>
-<li>Jim Zhang: Fixes to AUTHORS and .mailmap</li>
-<li>Natalia Nawara: Fixes to Product()</li>
-<li>vishal: Update the Czech tutorial translation to use a .po file</li>
-<li>Shruti Mangipudi: added See Also documentation to Matrices</li>
-<li>Davy Mao: Documentation</li>
-<li>Swapnil Agarwal: added See Also documentation to ntheory, functions</li>
-<li>Kendhia: Add XFAIL tests</li>
-<li>jerryma1121: See Also documentation for geometry</li>
-<li>Joachim Durchholz: corrected spacing issues identified by PyDev</li>
-<li>Martin Povišer: fix problem with rationaltools</li>
-<li>Siddhant Jain: See Also documentation for functions/special</li>
-<li>Kevin Hunter: Improvements to the inequality classes</li>
-<li>Michael Mayorov: Improvement to series</li>
-<li>Nathan Alison: Additions to the stats module</li>
-<li>Christian Bühler: remove use of set_main() in GA</li>
-<li>Carsten Knoll: improvement to preview()</li>
-<li>M R Bharath: modified use of int_tested, improvement to Permutation</li>
-<li>Matthias Toews: File permissions</li>
-<li>Sergiu Ivanov: Fixes to zoo, SymPyDeprecationWarning</li>
-<li>Jorge E. Cardona: Cleanup in polys</li>
-<li>Sanket Agarwal: Rewrite coverage_doctest.py script</li>
-<li>Manoj Babu K.: Improve gamma function</li>
-<li>Sai Nikhil: Fix to Heavyside with complex arguments</li>
-<li>Aleksandar Makelov: Fixes regarding the dihedral group generator</li>
-<li>Raphael Michel: German translation of the tutorial</li>
-<li>Sachin Irukula: Changes to allow Dict sorting</li>
-<li>Ashwini Oruganti: Changes to Pow printing</li>
-<li>Andreas Kloeckner: Fix to cse()</li>
-<li>Prateek Papriwal: improve summation documentation</li>
-<li>Arpit Goyal: Improvements to Integral and Sum</li>
-<li>Angadh Nanjangud: in physics.mechanics, added a function and tests for the parallel axis theorem</li>
-<li>Comer Duncan: added dual, is_antisymmetric, and det_lu_decomposition to matrices.py</li>
-<li>Jens H. Nielsen: added sets to modules listing, update IPython printing extension</li>
-<li>Joseph Dougherty: modified whitespace cleaning to remove multiple newlines at eof</li>
-<li>marshall2389: Spelling correction</li>
-<li>Guru Devanla: Implemented quantum density operator</li>
-<li>George Waksman: Implemented JavaScript code printer and MathML printer</li>
-<li>Angus Griffith: Fix bug in rsolve</li>
-<li>Timothy Reluga: Rewrite trigonometric functions as rationals</li>
-<li>Brian Stephanik: Test for a bug in fcode</li>
-<li>Ljubiša Moćić: Serbian translation of the tutorial</li>
-<li>Piotr Korgul: Polish translation of the tutorial</li>
-<li>Rom le Clair: French translation of the tutorial</li>
-<li>Alexandr Popov: Fixes to Pauli algebra</li>
-<li>Saurabh Jha: Work on Kauers algorithm</li>
-<li>Tarun Gaba: Implemented some trigonometric integrals</li>
-<li>Takafumi Arakaki: Add info target to the doc Makefile</li>
-<li>Alexander Eberspächer: correct typo in aboutus.rst</li>
-<li>Sachin Joglekar: Simplification of logic expressions to SOP and POS forms</li>
-<li>Tyler Pirtle: Fix improperly formatted error message</li>
-<li>Vasily Povalyaev: Fix latex(Min)</li>
-<li>Colleen Lee: replace uses of fnan with S.NaN</li>
-<li>Niklas Thörne: Fix links in the docs</li>
-<li>Huijun Mai: Chinese translation of the tutorial</li>
-<li>Marek Šuppa: Improvements to symbols, tests</li>
-<li>Prasoon Shukla: Bug fixes</li>
-</ol>
-<p>Up-to-date list in the order of the first contribution is given in the <a class="reference external" href="https://github.com/sympy/sympy/blob/master/AUTHORS">AUTHORS</a> file.</p>
-<p>You can find a brief history of SymPy in the README.</p>
-</div>
-<div class="section" id="financial-and-infrastructure-support">
-<h2>Financial and Infrastructure Support<a class="headerlink" href="#financial-and-infrastructure-support" title="Permalink to this headline">¶</a></h2>
-<ul class="simple">
-<li><a class="reference external" href="http://www.google.com/corporate/">Google</a>: SymPy has received generous
-financial support from Google in various years through the <a class="reference external" href="http://www.google-melange.com/">Google Summer of
-Code</a> program by providing stipends:<ul>
-<li>in 2007 for 5 students (<a class="reference external" href="http://code.google.com/p/sympy/wiki/GSoC2007">GSoC 2007</a>)</li>
-<li>in 2008 for 1 student (<a class="reference external" href="http://code.google.com/p/sympy/wiki/GSoC2008">GSoC 2008</a>)</li>
-<li>in 2009 for 5 students (<a class="reference external" href="http://code.google.com/p/sympy/wiki/GSoC2010">GSoC 2009</a>)</li>
-<li>in 2010 for 5 students (<a class="reference external" href="http://code.google.com/p/sympy/wiki/GSoC2010">GSoC 2010</a>)</li>
-<li>in 2011 for 9 students (<a class="reference external" href="https://github.com/sympy/sympy/wiki/Gsoc-2011-report">GSoC 2011</a>)</li>
-<li>in 2012 for 6 students (<a class="reference external" href="https://github.com/sympy/sympy/wiki/Gsoc-2012-report">GSoC 2012</a>)</li>
-</ul>
-</li>
-<li><a class="reference external" href="http://www.python.org/psf/">Python Software Foundation (PSF)</a> has hosted
-various GSoC students over the years:<ul>
-<li>3 GSoC 2007 students (Brian, Robert and Jason)</li>
-<li>1 GSoC 2008 student (Fredrik)</li>
-<li>2 GSoC 2009 students (Freddie and Priit)</li>
-<li>4 GSoC 2010 students (Aaron, Christian, Matthew and Øyvind)</li>
-</ul>
-</li>
-<li><a class="reference external" href="http://www.pdx.edu/">Portland State University (PSU)</a> has hosted following
-GSoC students:<ul>
-<li>1 student (Chris) in 2007</li>
-<li>3 students (Aaron, Dale and Fabian) in 2009</li>
-<li>1 student (Addison) in 2010</li>
-</ul>
-</li>
-<li><a class="reference external" href="http://www.stsci.edu/portal/">The Space Telescope Science Institute</a>: STScI hosted 1 GSoC 2007 student (Mateusz)</li>
-<li>Several 13-17 year old pre-university students contributed as part of
-Google&#8217;s <a class="reference external" href="http://www.google-melange.com/gci/homepage/google/gci2011">Code-In</a> 2011.  (<a class="reference external" href="http://www.google-melange.com/gci/org/google/gci2011/sympy">GCI
-2011</a>)</li>
-<li><a class="reference external" href="http://www.simula.no/">Simula Research Laboratory</a>: supports Pearu Peterson work in SymPy/SymPy Core projects</li>
-<li><a class="reference external" href="https://github.com/about">GitHub</a> is providing us with development
-and collaboration tools</li>
-</ul>
-</div>
-<div class="section" id="license">
-<h2>License<a class="headerlink" href="#license" title="Permalink to this headline">¶</a></h2>
-<p>Unless stated otherwise, all files in the SymPy project, SymPy&#8217;s webpage (and
-wiki), all images and all documentation including this User&#8217;s Guide are licensed
-using the new BSD license:</p>
-<div class="highlight-python"><pre>Copyright (c) 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 SymPy Development Team
-
-All rights reserved.
-
-Redistribution and use in source and binary forms, with or without
-modification, are permitted provided that the following conditions are met:
-
-  a. Redistributions of source code must retain the above copyright notice,
-     this list of conditions and the following disclaimer.
-  b. Redistributions in binary form must reproduce the above copyright
-     notice, this list of conditions and the following disclaimer in the
-     documentation and/or other materials provided with the distribution.
-  c. Neither the name of SymPy nor the names of its contributors
-     may be used to endorse or promote products derived from this software
-     without specific prior written permission.
-
-
-THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS &quot;AS IS&quot;
-AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
-IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
-ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE FOR
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-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.ANP.to_dict">ANP.to_dict() (in module sympy.polys.polyclasses)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.ANP.to_list">ANP.to_list() (in module sympy.polys.polyclasses)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.ANP.to_sympy_dict">ANP.to_sympy_dict() (in module sympy.polys.polyclasses)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.ANP.to_sympy_list">ANP.to_sympy_list() (in module sympy.polys.polyclasses)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.ANP.to_tuple">ANP.to_tuple() (in module sympy.polys.polyclasses)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.ANP.unify">ANP.unify() (in module sympy.polys.polyclasses)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/anticommutator.html#sympy.physics.quantum.anticommutator.AntiCommutator">AntiCommutator (class in sympy.physics.quantum.anticommutator)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/anticommutator.html#sympy.physics.quantum.anticommutator.AntiCommutator.doit">AntiCommutator.doit() (in module sympy.physics.quantum.anticommutator)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.AntiSymmetricTensor">AntiSymmetricTensor (class in sympy.physics.secondquant)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.AntiSymmetricTensor.doit">AntiSymmetricTensor.doit() (in module sympy.physics.secondquant)</a>
-  </dt>
-
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-  <dt><a href="modules/parsing.html#sympy.parsing.sympy_tokenize.any">any() (in module sympy.parsing.sympy_tokenize)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.hyper.meijerg.aother">aother (sympy.functions.special.hyper.meijerg attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/functions/special.html#sympy.functions.special.hyper.hyper.ap">ap (sympy.functions.special.hyper.hyper attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/functions/special.html#sympy.functions.special.hyper.meijerg.ap">(sympy.functions.special.hyper.meijerg attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.partfrac.apart">apart() (in module sympy.polys.partfrac)</a>
-  </dt>
-
-  </dl></td>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/mpmath/functions/constants.html#mpmath.mp.apery">apery (mpmath.mp attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Ellipse.apoapsis">apoapsis (sympy.geometry.ellipse.Ellipse attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.RegularPolygon.apothem">apothem (sympy.geometry.polygon.RegularPolygon attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/functions/hypergeometric.html#mpmath.appellf1">appellf1() (in module mpmath)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/functions/hypergeometric.html#mpmath.appellf2">appellf2() (in module mpmath)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/functions/hypergeometric.html#mpmath.appellf3">appellf3() (in module mpmath)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/functions/hypergeometric.html#mpmath.appellf4">appellf4() (in module mpmath)</a>
-  </dt>
-
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-  <dt><a href="modules/assumptions/assume.html#sympy.assumptions.assume.AppliedPredicate">AppliedPredicate (class in sympy.assumptions.assume)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/quantum/grover.html#sympy.physics.quantum.grover.apply_grover">apply_grover() (in module sympy.physics.quantum.grover)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.apply_operators">apply_operators() (in module sympy.physics.secondquant)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/general.html#mpmath.arange">arange() (in module mpmath)</a>
-  </dt>
-
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-  <dt><a href="modules/stats.html#sympy.stats.Arcsin">Arcsin() (in module sympy.stats)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.util.are_similar">are_similar() (in module sympy.geometry.util)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Ellipse.area">area (sympy.geometry.ellipse.Ellipse attribute)</a>
-  </dt>
-
-      <dd><dl>
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Polygon.area">(sympy.geometry.polygon.Polygon attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.RegularPolygon.area">(sympy.geometry.polygon.RegularPolygon attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.complexes.arg">arg (class in sympy.functions.elementary.complexes)</a>
-  </dt>
-
-      <dd><dl>
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-  <dt><a href="modules/assumptions/assume.html#sympy.assumptions.assume.AppliedPredicate.arg">(sympy.assumptions.assume.AppliedPredicate attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/mpmath/general.html#mpmath.arg">arg() (in module mpmath)</a>
-  </dt>
-
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-  <dt><a href="modules/core.html#sympy.core.basic.Basic.args">args (sympy.core.basic.Basic attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.RegularPolygon.args">(sympy.geometry.polygon.RegularPolygon attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.args">(sympy.polys.polytools.Poly attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/tensor/indexed.html#sympy.tensor.indexed.IndexedBase.args">(sympy.tensor.indexed.IndexedBase attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/utilities/codegen.html#sympy.utilities.codegen.Argument">Argument (class in sympy.utilities.codegen)</a>
-  </dt>
-
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-  <dt><a href="modules/functions/special.html#sympy.functions.special.bessel.BesselBase.argument">argument (sympy.functions.special.bessel.BesselBase attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/functions/special.html#sympy.functions.special.hyper.hyper.argument">(sympy.functions.special.hyper.hyper attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/functions/special.html#sympy.functions.special.hyper.meijerg.argument">(sympy.functions.special.hyper.meijerg attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.array_form">array_form (sympy.combinatorics.permutations.Permutation attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/combinatorics/polyhedron.html#sympy.combinatorics.polyhedron.Polyhedron.array_form">(sympy.combinatorics.polyhedron.Polyhedron attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/categories.html#sympy.categories.diagram_drawing.ArrowStringDescription">ArrowStringDescription (class in sympy.categories.diagram_drawing)</a>
-  </dt>
-
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-  <dt><a href="modules/core.html#sympy.core.compatibility.as_int">as_int() (in module sympy.core.compatibility)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/functions/trigonometric.html#mpmath.asec">asec() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/hyperbolic.html#mpmath.asech">asech() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.trigonometric.asin">asin (class in sympy.functions.elementary.trigonometric)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/trigonometric.html#mpmath.asin">asin() (in module mpmath)</a>
-  </dt>
-
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-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.asinh">asinh (class in sympy.functions.elementary.hyperbolic)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/hyperbolic.html#mpmath.asinh">asinh() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/ask.html#sympy.assumptions.ask.ask">ask() (in module sympy.assumptions.ask)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/ask.html#sympy.assumptions.ask.ask_full_inference">ask_full_inference() (in module sympy.assumptions.ask)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskAlgebraicHandler">AskAlgebraicHandler (class in sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskAntiHermitianHandler">AskAntiHermitianHandler (class in sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskAntiHermitianHandler.Add">AskAntiHermitianHandler.Add() (in module sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskAntiHermitianHandler.Mul">AskAntiHermitianHandler.Mul() (in module sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskAntiHermitianHandler.Pow">AskAntiHermitianHandler.Pow() (in module sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/calculus.html#sympy.assumptions.handlers.calculus.AskBoundedHandler">AskBoundedHandler (class in sympy.assumptions.handlers.calculus)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/calculus.html#sympy.assumptions.handlers.calculus.AskBoundedHandler.Add">AskBoundedHandler.Add() (in module sympy.assumptions.handlers.calculus)</a>
-  </dt>
-
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-  <dt><a href="modules/assumptions/handlers/calculus.html#sympy.assumptions.handlers.calculus.AskBoundedHandler.Mul">AskBoundedHandler.Mul() (in module sympy.assumptions.handlers.calculus)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/calculus.html#sympy.assumptions.handlers.calculus.AskBoundedHandler.Pow">AskBoundedHandler.Pow() (in module sympy.assumptions.handlers.calculus)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/calculus.html#sympy.assumptions.handlers.calculus.AskBoundedHandler.Symbol">AskBoundedHandler.Symbol() (in module sympy.assumptions.handlers.calculus)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskComplexHandler">AskComplexHandler (class in sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskExtendedRealHandler">AskExtendedRealHandler (class in sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskHermitianHandler">AskHermitianHandler (class in sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskHermitianHandler.Add">AskHermitianHandler.Add() (in module sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskHermitianHandler.Mul">AskHermitianHandler.Mul() (in module sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskHermitianHandler.Pow">AskHermitianHandler.Pow() (in module sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskImaginaryHandler">AskImaginaryHandler (class in sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskImaginaryHandler.Add">AskImaginaryHandler.Add() (in module sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskImaginaryHandler.Mul">AskImaginaryHandler.Mul() (in module sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskImaginaryHandler.Pow">AskImaginaryHandler.Pow() (in module sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/calculus.html#sympy.assumptions.handlers.calculus.AskInfinitesimalHandler">AskInfinitesimalHandler (class in sympy.assumptions.handlers.calculus)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/calculus.html#sympy.assumptions.handlers.calculus.AskInfinitesimalHandler.Add">AskInfinitesimalHandler.Add() (in module sympy.assumptions.handlers.calculus)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/calculus.html#sympy.assumptions.handlers.calculus.AskInfinitesimalHandler.Mul">AskInfinitesimalHandler.Mul() (in module sympy.assumptions.handlers.calculus)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/calculus.html#sympy.assumptions.handlers.calculus.AskInfinitesimalHandler.Pow">AskInfinitesimalHandler.Pow() (in module sympy.assumptions.handlers.calculus)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskIntegerHandler">AskIntegerHandler (class in sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskIntegerHandler.Add">AskIntegerHandler.Add() (in module sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskIntegerHandler.Mul">AskIntegerHandler.Mul() (in module sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskIntegerHandler.Pow">AskIntegerHandler.Pow() (in module sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/order.html#sympy.assumptions.handlers.order.AskNegativeHandler">AskNegativeHandler (class in sympy.assumptions.handlers.order)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/order.html#sympy.assumptions.handlers.order.AskNegativeHandler.Add">AskNegativeHandler.Add() (in module sympy.assumptions.handlers.order)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/order.html#sympy.assumptions.handlers.order.AskNegativeHandler.Pow">AskNegativeHandler.Pow() (in module sympy.assumptions.handlers.order)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/order.html#sympy.assumptions.handlers.order.AskNonZeroHandler">AskNonZeroHandler (class in sympy.assumptions.handlers.order)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/ntheory.html#sympy.assumptions.handlers.ntheory.AskOddHandler">AskOddHandler (class in sympy.assumptions.handlers.ntheory)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/order.html#sympy.assumptions.handlers.order.AskPositiveHandler">AskPositiveHandler (class in sympy.assumptions.handlers.order)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/ntheory.html#sympy.assumptions.handlers.ntheory.AskPrimeHandler">AskPrimeHandler (class in sympy.assumptions.handlers.ntheory)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/ntheory.html#sympy.assumptions.handlers.ntheory.AskPrimeHandler.Pow">AskPrimeHandler.Pow() (in module sympy.assumptions.handlers.ntheory)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskRationalHandler">AskRationalHandler (class in sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskRationalHandler.Add">AskRationalHandler.Add() (in module sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskRationalHandler.Mul">AskRationalHandler.Mul() (in module sympy.assumptions.handlers.sets)</a>
-  </dt>
-
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-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskRationalHandler.Pow">AskRationalHandler.Pow() (in module sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskRealHandler">AskRealHandler (class in sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskRealHandler.Add">AskRealHandler.Add() (in module sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskRealHandler.Mul">AskRealHandler.Mul() (in module sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.AskRealHandler.Pow">AskRealHandler.Pow() (in module sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.codeprinter.AssignmentError">AssignmentError</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.polynomials.assoc_laguerre">assoc_laguerre (class in sympy.functions.special.polynomials)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.polynomials.assoc_legendre">assoc_legendre (class in sympy.functions.special.polynomials)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/memoization.html#sympy.utilities.memoization.assoc_recurrence_memo">assoc_recurrence_memo() (in module sympy.utilities.memoization)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.basic.Basic.assumptions0">assumptions0 (sympy.core.basic.Basic attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/assume.html#sympy.assumptions.assume.AssumptionsContext">AssumptionsContext (class in sympy.assumptions.assume)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/assume.html#sympy.assumptions.assume.AssumptionsContext.add">AssumptionsContext.add() (in module sympy.assumptions.assume)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.trigonometric.atan">atan (class in sympy.functions.elementary.trigonometric)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/trigonometric.html#mpmath.atan">atan() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.trigonometric.atan2">atan2 (class in sympy.functions.elementary.trigonometric)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/trigonometric.html#mpmath.atan2">atan2() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.atanh">atanh (class in sympy.functions.elementary.hyperbolic)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/hyperbolic.html#mpmath.atanh">atanh() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.basic.Atom">Atom (class in sympy.core.basic)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.expr.AtomicExpr">AtomicExpr (class in sympy.core.expr)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.pretty.pretty_symbology.atoms_table">atoms_table (in module sympy.printing.pretty.pretty_symbology)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/general.html#mpmath.autoprec">autoprec() (in module mpmath)</a>
-  </dt>
-
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-  <dt><a href="modules/utilities/autowrap.html#sympy.utilities.autowrap.autowrap">autowrap() (in module sympy.utilities.autowrap)</a>
-  </dt>
-
-  </dl></td>
-</tr></table>
-
-<h2 id="B">B</h2>
-<table style="width: 100%" class="indextable genindextable"><tr>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.B">B (in module sympy.physics.secondquant)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.RayTransferMatrix.B">(sympy.physics.gaussopt.RayTransferMatrix attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/mpmath/functions/zeta.html#mpmath.backlunds">backlunds() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/gamma.html#mpmath.barnesg">barnesg() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.base">base (sympy.combinatorics.perm_groups.PermutationGroup attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.exponential.exp.base">(sympy.functions.elementary.exponential.exp attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/tensor/indexed.html#sympy.tensor.indexed.Indexed.base">(sympy.tensor.indexed.Indexed attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/diffgeom.html#sympy.diffgeom.BaseCovarDerivativeOp">BaseCovarDerivativeOp (class in sympy.diffgeom)</a>
-  </dt>
-
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-  <dt><a href="modules/polys/internals.html#sympy.polys.polyerrors.BasePolynomialError">BasePolynomialError (class in sympy.polys.polyerrors)</a>
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-
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-  <dt><a href="modules/diffgeom.html#sympy.diffgeom.BaseScalarField">BaseScalarField (class in sympy.diffgeom)</a>
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-  <dt><a href="modules/plotting.html#sympy.plotting.plot.BaseSeries">BaseSeries (class in sympy.plotting.plot)</a>
-  </dt>
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-  <dt><a href="modules/diffgeom.html#sympy.diffgeom.BaseVectorField">BaseVectorField (class in sympy.diffgeom)</a>
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-  <dt><a href="modules/core.html#sympy.core.basic.Basic">Basic (class in sympy.core.basic)</a>
-  </dt>
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-  <dt><a href="modules/core.html#sympy.core.basic.Basic.as_content_primitive">Basic.as_content_primitive() (in module sympy.core.basic)</a>
-  </dt>
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-  <dt><a href="modules/core.html#sympy.core.basic.Basic.as_poly">Basic.as_poly() (in module sympy.core.basic)</a>
-  </dt>
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-  <dt><a href="modules/core.html#sympy.core.basic.Basic.atoms">Basic.atoms() (in module sympy.core.basic)</a>
-  </dt>
-
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-  <dt><a href="modules/core.html#sympy.core.basic.Basic.compare">Basic.compare() (in module sympy.core.basic)</a>
-  </dt>
-
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-  <dt><a href="modules/core.html#sympy.core.basic.Basic.compare_pretty">Basic.compare_pretty() (in module sympy.core.basic)</a>
-  </dt>
-
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-  <dt><a href="modules/core.html#sympy.core.basic.Basic.count">Basic.count() (in module sympy.core.basic)</a>
-  </dt>
-
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-  <dt><a href="modules/core.html#sympy.core.basic.Basic.count_ops">Basic.count_ops() (in module sympy.core.basic)</a>
-  </dt>
-
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-  <dt><a href="modules/core.html#sympy.core.basic.Basic.doit">Basic.doit() (in module sympy.core.basic)</a>
-  </dt>
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-  <dt><a href="modules/core.html#sympy.core.basic.Basic.dummy_eq">Basic.dummy_eq() (in module sympy.core.basic)</a>
-  </dt>
-
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-  <dt><a href="modules/core.html#sympy.core.basic.Basic.find">Basic.find() (in module sympy.core.basic)</a>
-  </dt>
-
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-  <dt><a href="modules/core.html#sympy.core.basic.Basic.has">Basic.has() (in module sympy.core.basic)</a>
-  </dt>
-
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-  <dt><a href="modules/core.html#sympy.core.basic.Basic.iter_basic_args">Basic.iter_basic_args() (in module sympy.core.basic)</a>
-  </dt>
-
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-  <dt><a href="modules/core.html#sympy.core.basic.Basic.match">Basic.match() (in module sympy.core.basic)</a>
-  </dt>
-
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-  <dt><a href="modules/core.html#sympy.core.basic.Basic.matches">Basic.matches() (in module sympy.core.basic)</a>
-  </dt>
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-  <dt><a href="modules/core.html#sympy.core.basic.Basic.replace">Basic.replace() (in module sympy.core.basic)</a>
-  </dt>
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-  <dt><a href="modules/core.html#sympy.core.basic.Basic.rewrite">Basic.rewrite() (in module sympy.core.basic)</a>
-  </dt>
-
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-  <dt><a href="modules/core.html#sympy.core.basic.Basic.sort_key">Basic.sort_key() (in module sympy.core.basic)</a>
-  </dt>
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-  <dt><a href="modules/core.html#sympy.core.basic.Basic.subs">Basic.subs() (in module sympy.core.basic)</a>
-  </dt>
-
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-  <dt><a href="modules/core.html#sympy.core.basic.Basic.xreplace">Basic.xreplace() (in module sympy.core.basic)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.basic_orbits">basic_orbits (sympy.combinatorics.perm_groups.PermutationGroup attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.basic_stabilizers">basic_stabilizers (sympy.combinatorics.perm_groups.PermutationGroup attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.basic_transversals">basic_transversals (sympy.combinatorics.perm_groups.PermutationGroup attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.BBra">BBra (in module sympy.physics.secondquant)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.Bd">Bd (in module sympy.physics.secondquant)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.BeamParameter">BeamParameter (class in sympy.physics.gaussopt)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.bei">bei() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/combinatorial.html#sympy.functions.combinatorial.numbers.bell">bell (class in sympy.functions.combinatorial.numbers)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/functions/numtheory.html#mpmath.bell">bell() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/stats.html#sympy.stats.Benini">Benini() (in module sympy.stats)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.ber">ber() (in module mpmath)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/functions/numtheory.html#mpmath.bernfrac">bernfrac() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/combinatorial.html#sympy.functions.combinatorial.numbers.bernoulli">bernoulli (class in sympy.functions.combinatorial.numbers)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/functions/numtheory.html#mpmath.bernoulli">bernoulli() (in module mpmath)</a>
-  </dt>
-
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-  <dt><a href="modules/stats.html#sympy.stats.Bernoulli">Bernoulli() (in module sympy.stats)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/functions/numtheory.html#mpmath.bernpoly">bernpoly() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.bessel.BesselBase">BesselBase (class in sympy.functions.special.bessel)</a>
-  </dt>
-
-  </dl></td>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.bessel.besseli">besseli (class in sympy.functions.special.bessel)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.besseli">besseli() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.bessel.besselj">besselj (class in sympy.functions.special.bessel)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.besselj">besselj() (in module mpmath)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.besseljzero">besseljzero() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.bessel.besselk">besselk (class in sympy.functions.special.bessel)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.besselk">besselk() (in module mpmath)</a>
-  </dt>
-
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-  <dt><a href="modules/simplify/simplify.html#sympy.simplify.simplify.besselsimp">besselsimp() (in module sympy.simplify.simplify)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.bessel.bessely">bessely (class in sympy.functions.special.bessel)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.bessely">bessely() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.besselyzero">besselyzero() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/gamma.html#mpmath.beta">beta() (in module mpmath)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/functions/special.html#sympy.functions.special.gamma_functions.beta">(in module sympy.functions.special.gamma_functions)</a>
-  </dt>
-
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-      
-  <dt><a href="modules/stats.html#sympy.stats.Beta">Beta() (in module sympy.stats)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/gamma.html#mpmath.betainc">betainc() (in module mpmath)</a>
-  </dt>
-
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-  <dt><a href="modules/stats.html#sympy.stats.BetaPrime">BetaPrime() (in module sympy.stats)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/hypergeometric.html#mpmath.bihyper">bihyper() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/graycode.html#sympy.combinatorics.graycode.bin_to_gray">bin_to_gray() (sympy.combinatorics.graycode method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/autowrap.html#sympy.utilities.autowrap.binary_function">binary_function() (in module sympy.utilities.autowrap)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.binary_partitions">binary_partitions() (in module sympy.utilities.iterables)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/combinatorial.html#sympy.functions.combinatorial.factorials.binomial">binomial (class in sympy.functions.combinatorial.factorials)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/gamma.html#mpmath.binomial">binomial() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/stats.html#sympy.stats.Binomial">Binomial() (in module sympy.stats)</a>
-  </dt>
-
-      
-  <dt><a href="modules/ntheory.html#sympy.ntheory.multinomial.binomial_coefficients">binomial_coefficients() (in module sympy.ntheory.multinomial)</a>
-  </dt>
-
-      
-  <dt><a href="modules/ntheory.html#sympy.ntheory.multinomial.binomial_coefficients_list">binomial_coefficients_list() (in module sympy.ntheory.multinomial)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/calculus/optimization.html#mpmath.calculus.optimization.Bisection">Bisection (class in mpmath.calculus.optimization)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.bitlist_from_subset">bitlist_from_subset() (sympy.combinatorics.subsets.Subset class method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.BKet">BKet (in module sympy.physics.secondquant)</a>
-  </dt>
-
-      
-  <dt><a href="modules/matrices/expressions.html#sympy.matrices.expressions.blockmatrix.block_collapse">block_collapse() (in module sympy.matrices.expressions.blockmatrix)</a>
-  </dt>
-
-      
-  <dt><a href="modules/matrices/expressions.html#sympy.matrices.expressions.blockmatrix.BlockDiagMatrix">BlockDiagMatrix (class in sympy.matrices.expressions.blockmatrix)</a>
-  </dt>
-
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-  <dt><a href="modules/matrices/expressions.html#sympy.matrices.expressions.blockmatrix.BlockMatrix">BlockMatrix (class in sympy.matrices.expressions.blockmatrix)</a>
-  </dt>
-
-      
-  <dt><a href="modules/matrices/expressions.html#sympy.matrices.expressions.blockmatrix.BlockMatrix.inverse">BlockMatrix.inverse() (in module sympy.matrices.expressions.blockmatrix)</a>
-  </dt>
-
-      
-  <dt><a href="modules/matrices/expressions.html#sympy.matrices.expressions.blockmatrix.BlockMatrix.transpose">BlockMatrix.transpose() (in module sympy.matrices.expressions.blockmatrix)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.hyper.meijerg.bm">bm (sympy.functions.special.hyper.meijerg attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.BosonicBasis">BosonicBasis (class in sympy.physics.secondquant)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.hyper.meijerg.bother">bother (sympy.functions.special.hyper.meijerg attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.hyper.hyper.bq">bq (sympy.functions.special.hyper.hyper attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/functions/special.html#sympy.functions.special.hyper.meijerg.bq">(sympy.functions.special.hyper.meijerg attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/physics/quantum/state.html#sympy.physics.quantum.state.Bra">Bra (class in sympy.physics.quantum.state)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/operator.html#sympy.physics.quantum.operator.OuterProduct.bra">bra (sympy.physics.quantum.operator.OuterProduct attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.InnerProduct.bra">(sympy.physics.secondquant.InnerProduct attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/physics/quantum/state.html#sympy.physics.quantum.state.BraBase">BraBase (class in sympy.physics.quantum.state)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.bracelets">bracelets() (in module sympy.utilities.iterables)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/tensor_can.html#sympy.combinatorics.tensor_can.bsgs_direct_product">bsgs_direct_product() (in module sympy.combinatorics.tensor_can)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.bsplines.bspline_basis">bspline_basis() (in module sympy.functions.special.bsplines)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.bsplines.bspline_basis_set">bspline_basis_set() (in module sympy.functions.special.bsplines)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.groebnertools.buchberger">buchberger() (in module sympy.polys.groebnertools)</a>
-  </dt>
-
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-  <dt><a href="modules/series.html#sympy.series.gruntz.build_expression_tree">build_expression_tree() (in module sympy.series.gruntz)</a>
-  </dt>
-
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-</tr></table>
-
-<h2 id="C">C</h2>
-<table style="width: 100%" class="indextable genindextable"><tr>
-  <td style="width: 33%" valign="top"><dl>
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-  <dt><a href="modules/core.html#sympy.core.basic.C">C (in module sympy.core.basic)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/matrices/immutablematrices.html#sympy.matrices.immutable.ImmutableMatrix.C">(sympy.matrices.immutable.ImmutableMatrix attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.C">(sympy.matrices.matrices.MatrixBase attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.RayTransferMatrix.C">(sympy.physics.gaussopt.RayTransferMatrix attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/core.html#sympy.core.cache.cacheit">cacheit() (in module sympy.core.cache)</a>
-  </dt>
-
-      
-  <dt><a href="modules/series.html#sympy.series.gruntz.calculate_series">calculate_series() (in module sympy.series.gruntz)</a>
-  </dt>
-
-      
-  <dt><a href="tutorial/tutorial.en.html#index-0">calculus</a>
-  </dt>
-
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-  <dt><a href="modules/galgebra/GA/GAsympy.html#cancel">cancel() (built-in function)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.cancel">(in module sympy.polys.polytools)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/combinatorics/tensor_can.html#sympy.combinatorics.tensor_can.canonicalize">canonicalize() (in module sympy.combinatorics.tensor_can)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.capture">capture() (in module sympy.utilities.iterables)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.cardinality">cardinality (sympy.combinatorics.permutations.Permutation attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.cardinality">(sympy.combinatorics.subsets.Subset attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.dense.casoratian">casoratian() (in module sympy.matrices.dense)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/combinatorial.html#sympy.functions.combinatorial.numbers.catalan">catalan (class in sympy.functions.combinatorial.numbers)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/mpmath/functions/constants.html#mpmath.mp.catalan">(mpmath.mp attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/categories.html#sympy.categories.Category">Category (class in sympy.categories)</a>
-  </dt>
-
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-  <dt><a href="modules/stats.html#sympy.stats.Cauchy">Cauchy() (in module sympy.stats)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/functions/powers.html#mpmath.cbrt">cbrt() (in module mpmath)</a>
-  </dt>
-
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-  <dt><a href="modules/printing.html#sympy.printing.ccode.ccode">ccode() (in module sympy.printing.ccode)</a>
-  </dt>
-
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-  <dt><a href="modules/utilities/codegen.html#sympy.utilities.codegen.CCodeGen">CCodeGen (class in sympy.utilities.codegen)</a>
-  </dt>
-
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-  <dt><a href="modules/utilities/codegen.html#sympy.utilities.codegen.CCodeGen.dump_c">CCodeGen.dump_c() (in module sympy.utilities.codegen)</a>
-  </dt>
-
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-  <dt><a href="modules/utilities/codegen.html#sympy.utilities.codegen.CCodeGen.dump_h">CCodeGen.dump_h() (in module sympy.utilities.codegen)</a>
-  </dt>
-
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-  <dt><a href="modules/utilities/codegen.html#sympy.utilities.codegen.CCodeGen.get_prototype">CCodeGen.get_prototype() (in module sympy.utilities.codegen)</a>
-  </dt>
-
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-  <dt><a href="modules/printing.html#sympy.printing.ccode.CCodePrinter">CCodePrinter (class in sympy.printing.ccode)</a>
-  </dt>
-
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-  <dt><a href="modules/printing.html#sympy.printing.ccode.CCodePrinter.doprint">CCodePrinter.doprint() (in module sympy.printing.ccode)</a>
-  </dt>
-
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-  <dt><a href="modules/printing.html#sympy.printing.ccode.CCodePrinter.indent_code">CCodePrinter.indent_code() (in module sympy.printing.ccode)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/general.html#mpmath.ceil">ceil() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.integers.ceiling">ceiling (class in sympy.functions.elementary.integers)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Ellipse.center">center (sympy.geometry.ellipse.Ellipse attribute)</a>
-  </dt>
-
-      <dd><dl>
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.RegularPolygon.center">(sympy.geometry.polygon.RegularPolygon attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Polygon.centroid">centroid (sympy.geometry.polygon.Polygon attribute)</a>
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-
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-  <dt><a href="modules/geometry.html#sympy.geometry.util.centroid">centroid() (in module sympy.geometry.util)</a>
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-
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-  <dt><a href="modules/physics/quantum/cg.html#sympy.physics.quantum.cg.CG">CG (class in sympy.physics.quantum.cg)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/quantum/cg.html#sympy.physics.quantum.cg.cg_simp">cg_simp() (in module sympy.physics.quantum.cg)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CGate">CGate (class in sympy.physics.quantum.gate)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CGate.decompose">CGate.decompose() (in module sympy.physics.quantum.gate)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CGate.eval_controls">CGate.eval_controls() (in module sympy.physics.quantum.gate)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/calculus/approximation.html#mpmath.chebyfit">chebyfit() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.polynomials.chebyshevt">chebyshevt (class in sympy.functions.special.polynomials)</a>
-  </dt>
-
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-  <dt><a href="modules/polys/reference.html#sympy.polys.orthopolys.chebyshevt_poly">chebyshevt_poly() (in module sympy.polys.orthopolys)</a>
-  </dt>
-
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-  <dt><a href="modules/functions/special.html#sympy.functions.special.polynomials.chebyshevt_root">chebyshevt_root (class in sympy.functions.special.polynomials)</a>
-  </dt>
-
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-  <dt><a href="modules/functions/special.html#sympy.functions.special.polynomials.chebyshevu">chebyshevu (class in sympy.functions.special.polynomials)</a>
-  </dt>
-
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-  <dt><a href="modules/polys/reference.html#sympy.polys.orthopolys.chebyshevu_poly">chebyshevu_poly() (in module sympy.polys.orthopolys)</a>
-  </dt>
-
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-  <dt><a href="modules/functions/special.html#sympy.functions.special.polynomials.chebyshevu_root">chebyshevu_root (class in sympy.functions.special.polynomials)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/functions/orthogonal.html#mpmath.chebyt">chebyt() (in module mpmath)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/functions/orthogonal.html#mpmath.chebyu">chebyu() (in module mpmath)</a>
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-
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-  <dt><a href="modules/solvers/solvers.html#sympy.solvers.solvers.check_assumptions">check_assumptions() (in module sympy.solvers.solvers)</a>
-  </dt>
-
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-  <dt><a href="modules/solvers/ode.html#sympy.solvers.ode.checkodesol">checkodesol() (in module sympy.solvers.ode)</a>
-  </dt>
-
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-  <dt><a href="modules/solvers/solvers.html#sympy.solvers.solvers.checksol">checksol() (in module sympy.solvers.solvers)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.error_functions.Chi">Chi (class in sympy.functions.special.error_functions)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/functions/expintegrals.html#mpmath.chi">chi() (in module mpmath)</a>
-  </dt>
-
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-  <dt><a href="modules/stats.html#sympy.stats.Chi">Chi() (in module sympy.stats)</a>
-  </dt>
-
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-  <dt><a href="modules/stats.html#sympy.stats.ChiNoncentral">ChiNoncentral() (in module sympy.stats)</a>
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-
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-  <dt><a href="modules/stats.html#sympy.stats.ChiSquared">ChiSquared() (in module sympy.stats)</a>
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-
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-  <dt><a href="modules/mpmath/matrices.html#mpmath.cholesky">cholesky() (in module mpmath)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/general.html#mpmath.chop">chop() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.error_functions.Ci">Ci (class in sympy.functions.special.error_functions)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/functions/expintegrals.html#mpmath.ci">ci() (in module mpmath)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Circle">Circle (class in sympy.geometry.ellipse)</a>
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-
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-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Circle.equation">Circle.equation() (in module sympy.geometry.ellipse)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Circle.intersection">Circle.intersection() (in module sympy.geometry.ellipse)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Circle.scale">Circle.scale() (in module sympy.geometry.ellipse)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.RegularPolygon.circumcenter">circumcenter (sympy.geometry.polygon.RegularPolygon attribute)</a>
-  </dt>
-
-      <dd><dl>
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Triangle.circumcenter">(sympy.geometry.polygon.Triangle attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.RegularPolygon.circumcircle">circumcircle (sympy.geometry.polygon.RegularPolygon attribute)</a>
-  </dt>
-
-      <dd><dl>
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Triangle.circumcircle">(sympy.geometry.polygon.Triangle attribute)</a>
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-
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-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Circle.circumference">circumference (sympy.geometry.ellipse.Circle attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Ellipse.circumference">(sympy.geometry.ellipse.Ellipse attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.RegularPolygon.circumradius">circumradius (sympy.geometry.polygon.RegularPolygon attribute)</a>
-  </dt>
-
-      <dd><dl>
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Triangle.circumradius">(sympy.geometry.polygon.Triangle attribute)</a>
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-
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-  <dt><a href="modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.CL">CL (sympy.matrices.sparse.SparseMatrix attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/core.html#sympy.core.add.Add.class_key">class_key() (sympy.core.add.Add class method)</a>
-  </dt>
-
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-  <dt><a href="modules/core.html#sympy.core.basic.Basic.class_key">(sympy.core.basic.Basic class method)</a>
-  </dt>
-
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-      
-  <dt><a href="modules/solvers/ode.html#sympy.solvers.ode.classify_ode">classify_ode() (in module sympy.solvers.ode)</a>
-  </dt>
-
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.classof">classof() (in module sympy.matrices.matrices)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/functions/zeta.html#mpmath.clcos">clcos() (in module mpmath)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/wigner.html#sympy.physics.wigner.clebsch_gordan">clebsch_gordan() (in module sympy.physics.wigner)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/zeta.html#mpmath.clsin">clsin() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/shor.html#sympy.physics.quantum.shor.CMod">CMod (class in sympy.physics.quantum.shor)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CNOT">CNOT (in module sympy.physics.quantum.gate)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CNotGate">CNotGate (class in sympy.physics.quantum.gate)</a>
-  </dt>
-
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-  <dt><a href="modules/utilities/codegen.html#sympy.utilities.codegen.CodeGen">CodeGen (class in sympy.utilities.codegen)</a>
-  </dt>
-
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-  <dt><a href="modules/utilities/codegen.html#sympy.utilities.codegen.codegen">codegen() (in module sympy.utilities.codegen)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/codegen.html#sympy.utilities.codegen.CodeGen.dump_code">CodeGen.dump_code() (in module sympy.utilities.codegen)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/codegen.html#sympy.utilities.codegen.CodeGen.write">CodeGen.write() (in module sympy.utilities.codegen)</a>
-  </dt>
-
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-  <dt><a href="modules/printing.html#sympy.printing.codeprinter.CodePrinter">CodePrinter (class in sympy.printing.codeprinter)</a>
-  </dt>
-
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-  <dt><a href="modules/utilities/autowrap.html#sympy.utilities.autowrap.CodeWrapper">CodeWrapper (class in sympy.utilities.autowrap)</a>
-  </dt>
-
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-  <dt><a href="modules/categories.html#sympy.categories.CompositeMorphism.codomain">codomain (sympy.categories.CompositeMorphism attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/categories.html#sympy.categories.Morphism.codomain">(sympy.categories.Morphism attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.LinearEntity.coefficients">coefficients (sympy.geometry.line.LinearEntity attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/polys/internals.html#sympy.polys.polyerrors.CoercionFailed">CoercionFailed (class in sympy.polys.polyerrors)</a>
-  </dt>
-
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.cofactors">cofactors() (in module sympy.polys.polytools)</a>
-  </dt>
-
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-  <dt><a href="modules/stats.html#sympy.stats.Coin">Coin() (in module sympy.stats)</a>
-  </dt>
-
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-  <dt><a href="modules/galgebra/GA/GAsympy.html#collect">collect() (built-in function)</a>
-  </dt>
-
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-  <dt><a href="modules/galgebra/GA.html#sympy.galgebra.GA.collect">(in module sympy.galgebra.GA)</a>
-  </dt>
-
-        
-  <dt><a href="modules/simplify/simplify.html#sympy.simplify.simplify.collect">(in module sympy.simplify.simplify)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/simplify/simplify.html#sympy.simplify.simplify.collect_const">collect_const() (in module sympy.simplify.simplify)</a>
-  </dt>
-
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-  <dt><a href="modules/simplify/simplify.html#sympy.simplify.simplify.collect_sqrt">collect_sqrt() (in module sympy.simplify.simplify)</a>
-  </dt>
-
-      
-  <dt><a href="modules/galgebra/GA.html#sympy.galgebra.GA.comb">comb() (in module sympy.galgebra.GA)</a>
-  </dt>
-
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-  <dt><a href="modules/simplify/simplify.html#sympy.simplify.simplify.combsimp">combsimp() (in module sympy.simplify.simplify)</a>
-  </dt>
-
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-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.common_prefix">common_prefix() (in module sympy.utilities.iterables)</a>
-  </dt>
-
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-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.common_suffix">common_suffix() (in module sympy.utilities.iterables)</a>
-  </dt>
-
-      
-  <dt><a href="modules/categories.html#sympy.categories.Category.commutative_diagrams">commutative_diagrams (sympy.categories.Category attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/commutator.html#sympy.physics.quantum.commutator.Commutator">Commutator (class in sympy.physics.quantum.commutator)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.Commutator">(class in sympy.physics.secondquant)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/physics/quantum/commutator.html#sympy.physics.quantum.commutator.Commutator.doit">Commutator.doit() (in module sympy.physics.quantum.commutator)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.Commutator.doit">(in module sympy.physics.secondquant)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/utilities/randtest.html#sympy.utilities.randtest.comp">comp() (in module sympy.utilities.randtest)</a>
-  </dt>
-
-      
-  <dt><a href="modules/series.html#sympy.series.gruntz.compare">compare() (in module sympy.series.gruntz)</a>
-  </dt>
-
-      
-  <dt><a href="modules/sets.html#sympy.core.sets.Set.complement">complement (sympy.core.sets.Set attribute)</a>
-  </dt>
-
-      
-  <dt><a href="tutorial/tutorial.en.html#index-6">complex numbers</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/hilbert.html#sympy.physics.quantum.hilbert.ComplexSpace">ComplexSpace (class in sympy.physics.quantum.hilbert)</a>
-  </dt>
-
-      
-  <dt><a href="modules/categories.html#sympy.categories.CompositeMorphism.components">components (sympy.categories.CompositeMorphism attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.compose">compose() (in module sympy.polys.polytools)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.CompositeDomain">CompositeDomain (class in sympy.polys.domains)</a>
-  </dt>
-
-  </dl></td>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.CompositeDomain.inject">CompositeDomain.inject() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/categories.html#sympy.categories.CompositeMorphism">CompositeMorphism (class in sympy.categories)</a>
-  </dt>
-
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-  <dt><a href="modules/categories.html#sympy.categories.CompositeMorphism.flatten">CompositeMorphism.flatten() (in module sympy.categories)</a>
-  </dt>
-
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-  <dt><a href="modules/polys/internals.html#sympy.polys.polyerrors.ComputationFailed">ComputationFailed (class in sympy.polys.polyerrors)</a>
-  </dt>
-
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-  <dt><a href="modules/assumptions/ask.html#sympy.assumptions.ask.compute_known_facts">compute_known_facts() (in module sympy.assumptions.ask)</a>
-  </dt>
-
-      
-  <dt><a href="modules/categories.html#sympy.categories.Diagram.conclusions">conclusions (sympy.categories.Diagram attribute)</a>
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-
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-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.piecewise.ExprCondPair.cond">cond (sympy.functions.elementary.piecewise.ExprCondPair attribute)</a>
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-
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-  <dt><a href="modules/stats.html#sympy.stats.rv.ConditionalDomain">ConditionalDomain (class in sympy.stats.rv)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/general.html#mpmath.conj">conj() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.complexes.conjugate">conjugate (class in sympy.functions.elementary.complexes)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/combinatorics/partitions.html#sympy.combinatorics.partitions.IntegerPartition.conjugate">(sympy.combinatorics.partitions.IntegerPartition attribute)</a>
-  </dt>
-
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-      
-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.conjugate_gauss_beams">conjugate_gauss_beams() (in module sympy.physics.gaussopt)</a>
-  </dt>
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-      
-  <dt><a href="modules/utilities/decorator.html#sympy.utilities.decorator.conserve_mpmath_dps">conserve_mpmath_dps() (in module sympy.utilities.decorator)</a>
-  </dt>
-
-      
-  <dt><a href="modules/solvers/ode.html#sympy.solvers.ode.constant_renumber">constant_renumber() (in module sympy.solvers.ode)</a>
-  </dt>
-
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-  <dt><a href="modules/solvers/ode.html#sympy.solvers.ode.constantsimp">constantsimp() (in module sympy.solvers.ode)</a>
-  </dt>
-
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-  <dt><a href="modules/polys/reference.html#sympy.polys.constructor.construct_domain">construct_domain() (in module sympy.polys.constructor)</a>
-  </dt>
-
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.content">content() (in module sympy.polys.polytools)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/quantum/shor.html#sympy.physics.quantum.shor.continued_fraction">continued_fraction() (in module sympy.physics.quantum.shor)</a>
-  </dt>
-
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-  <dt><a href="modules/stats.html#sympy.stats.crv.ContinuousDomain">ContinuousDomain (class in sympy.stats.crv)</a>
-  </dt>
-
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-  <dt><a href="modules/statistics.html#sympy.statistics.distributions.ContinuousProbability">ContinuousProbability (class in sympy.statistics.distributions)</a>
-  </dt>
-
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-  <dt><a href="modules/statistics.html#sympy.statistics.distributions.ContinuousProbability.probability">ContinuousProbability.probability() (in module sympy.statistics.distributions)</a>
-  </dt>
-
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-  <dt><a href="modules/statistics.html#sympy.statistics.distributions.ContinuousProbability.random">ContinuousProbability.random() (in module sympy.statistics.distributions)</a>
-  </dt>
-
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-  <dt><a href="modules/stats.html#sympy.stats.crv.ContinuousPSpace">ContinuousPSpace (class in sympy.stats.crv)</a>
-  </dt>
-
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-  <dt><a href="modules/stats.html#sympy.stats.ContinuousRV">ContinuousRV() (in module sympy.stats)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.contraction">contraction() (in module sympy.physics.secondquant)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CGate.controls">controls (sympy.physics.quantum.gate.CGate attribute)</a>
-  </dt>
-
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-        
-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CNotGate.controls">(sympy.physics.quantum.gate.CNotGate attribute)</a>
-  </dt>
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-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.hyper.hyper.convergence_statement">convergence_statement (sympy.functions.special.hyper.hyper attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/galgebra/GA/GAsympy.html#convert_from_blades">convert_from_blades() (built-in function)</a>
-  </dt>
-
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-  <dt><a href="modules/galgebra/GA/GAsympy.html#convert_to_blades">convert_to_blades() (built-in function)</a>
-  </dt>
-
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-  <dt><a href="modules/utilities/runtests.html#sympy.utilities.runtests.convert_to_native_paths">convert_to_native_paths() (in module sympy.utilities.runtests)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.util.convex_hull">convex_hull() (in module sympy.geometry.util)</a>
-  </dt>
-
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-  <dt><a href="modules/diffgeom.html#sympy.diffgeom.CoordSystem">CoordSystem (class in sympy.diffgeom)</a>
-  </dt>
-
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-  <dt><a href="modules/diffgeom.html#sympy.diffgeom.CoordSystem.base_oneform">CoordSystem.base_oneform() (in module sympy.diffgeom)</a>
-  </dt>
-
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-  <dt><a href="modules/diffgeom.html#sympy.diffgeom.CoordSystem.base_oneforms">CoordSystem.base_oneforms() (in module sympy.diffgeom)</a>
-  </dt>
-
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-  <dt><a href="modules/diffgeom.html#sympy.diffgeom.CoordSystem.base_vector">CoordSystem.base_vector() (in module sympy.diffgeom)</a>
-  </dt>
-
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-  <dt><a href="modules/diffgeom.html#sympy.diffgeom.CoordSystem.base_vectors">CoordSystem.base_vectors() (in module sympy.diffgeom)</a>
-  </dt>
-
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-  <dt><a href="modules/diffgeom.html#sympy.diffgeom.CoordSystem.connect_to">CoordSystem.connect_to() (in module sympy.diffgeom)</a>
-  </dt>
-
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-  <dt><a href="modules/diffgeom.html#sympy.diffgeom.CoordSystem.coord_function">CoordSystem.coord_function() (in module sympy.diffgeom)</a>
-  </dt>
-
-      
-  <dt><a href="modules/diffgeom.html#sympy.diffgeom.CoordSystem.coord_functions">CoordSystem.coord_functions() (in module sympy.diffgeom)</a>
-  </dt>
-
-      
-  <dt><a href="modules/diffgeom.html#sympy.diffgeom.CoordSystem.coord_tuple_transform_to">CoordSystem.coord_tuple_transform_to() (in module sympy.diffgeom)</a>
-  </dt>
-
-      
-  <dt><a href="modules/diffgeom.html#sympy.diffgeom.CoordSystem.jacobian">CoordSystem.jacobian() (in module sympy.diffgeom)</a>
-  </dt>
-
-      
-  <dt><a href="modules/diffgeom.html#sympy.diffgeom.CoordSystem.point">CoordSystem.point() (in module sympy.diffgeom)</a>
-  </dt>
-
-      
-  <dt><a href="modules/diffgeom.html#sympy.diffgeom.CoordSystem.point_to_coords">CoordSystem.point_to_coords() (in module sympy.diffgeom)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/polyhedron.html#sympy.combinatorics.polyhedron.Polyhedron.corners">corners (sympy.combinatorics.polyhedron.Polyhedron attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.trigonometric.cos">cos (class in sympy.functions.elementary.trigonometric)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/trigonometric.html#mpmath.cos">cos() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.cosh">cosh (class in sympy.functions.elementary.hyperbolic)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/hyperbolic.html#mpmath.cosh">cosh() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.cosh.inverse">cosh.inverse() (in module sympy.functions.elementary.hyperbolic)</a>
-  </dt>
-
-      
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.transforms.cosine_transform">cosine_transform() (in module sympy.integrals.transforms)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/matrices.html#mpmath.cosm">cosm() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/trigonometric.html#mpmath.cospi">cospi() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.trigonometric.cot">cot (class in sympy.functions.elementary.trigonometric)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/trigonometric.html#mpmath.cot">cot() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.trigonometric.cot.inverse">cot.inverse() (in module sympy.functions.elementary.trigonometric)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.coth">coth (class in sympy.functions.elementary.hyperbolic)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/hyperbolic.html#mpmath.coth">coth() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.coth.inverse">coth.inverse() (in module sympy.functions.elementary.hyperbolic)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.coulombc">coulombc() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.coulombf">coulombf() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.coulombg">coulombg() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.function.count_ops">count_ops() (in module sympy.core.function)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.count_roots">count_roots() (in module sympy.polys.polytools)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/spin.html#sympy.physics.quantum.spin.couple">couple() (in module sympy.physics.quantum.spin)</a>
-  </dt>
-
-      
-  <dt><a href="modules/diffgeom.html#sympy.diffgeom.CovarDerivativeOp">CovarDerivativeOp (class in sympy.diffgeom)</a>
-  </dt>
-
-      
-  <dt><a href="modules/galgebra/GA.html#sympy.galgebra.GA.cp">cp() (in module sympy.galgebra.GA)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/plotting.html#mpmath.cplot">cplot() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.CreateBoson">CreateBoson (class in sympy.physics.secondquant)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.CreateBoson.apply_operator">CreateBoson.apply_operator() (in module sympy.physics.secondquant)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.CreateFermion">CreateFermion (class in sympy.physics.secondquant)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.CreateFermion.apply_operator">CreateFermion.apply_operator() (in module sympy.physics.secondquant)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/mechanics/api/functions.html#sympy.physics.mechanics.functions.cross">cross() (in module sympy.physics.mechanics.functions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/ntheory.html#sympy.ntheory.modular.crt">crt() (in module sympy.ntheory.modular)</a>
-  </dt>
-
-      
-  <dt><a href="modules/ntheory.html#sympy.ntheory.modular.crt1">crt1() (in module sympy.ntheory.modular)</a>
-  </dt>
-
-      
-  <dt><a href="modules/ntheory.html#sympy.ntheory.modular.crt2">crt2() (in module sympy.ntheory.modular)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/trigonometric.html#mpmath.csc">csc() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/hyperbolic.html#mpmath.csch">csch() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/simplify/simplify.html#sympy.simplify.cse_main.cse">cse() (in module sympy.simplify.cse_main)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/graycode.html#sympy.combinatorics.graycode.GrayCode.current">current (sympy.combinatorics.graycode.GrayCode attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/geometry.html#sympy.geometry.curve.Curve">Curve (class in sympy.geometry.curve)</a>
-  </dt>
-
-      
-  <dt><a href="modules/geometry.html#sympy.geometry.curve.Curve.arbitrary_point">Curve.arbitrary_point() (in module sympy.geometry.curve)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.curve.Curve.plot_interval">Curve.plot_interval() (in module sympy.geometry.curve)</a>
-  </dt>
-
-      
-  <dt><a href="modules/geometry.html#sympy.geometry.curve.Curve.rotate">Curve.rotate() (in module sympy.geometry.curve)</a>
-  </dt>
-
-      
-  <dt><a href="modules/geometry.html#sympy.geometry.curve.Curve.scale">Curve.scale() (in module sympy.geometry.curve)</a>
-  </dt>
-
-      
-  <dt><a href="modules/geometry.html#sympy.geometry.curve.Curve.translate">Curve.translate() (in module sympy.geometry.curve)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.CurvedMirror">CurvedMirror (class in sympy.physics.gaussopt)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.CurvedRefraction">CurvedRefraction (class in sympy.physics.gaussopt)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Cycle">Cycle (class in sympy.combinatorics.permutations)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Cycle.list">Cycle.list() (in module sympy.combinatorics.permutations)</a>
-  </dt>
-
-      
-  <dt><a href="modules/ntheory.html#sympy.ntheory.generate.cycle_length">cycle_length() (in module sympy.ntheory.generate)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.cycle_structure">cycle_structure (sympy.combinatorics.permutations.Permutation attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.cycles">cycles (sympy.combinatorics.permutations.Permutation attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.generators.cyclic">cyclic() (sympy.combinatorics.generators method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.cyclic_form">cyclic_form (sympy.combinatorics.permutations.Permutation attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/combinatorics/polyhedron.html#sympy.combinatorics.polyhedron.Polyhedron.cyclic_form">(sympy.combinatorics.polyhedron.Polyhedron attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/combinatorics/named_groups.html#sympy.combinatorics.named_groups.CyclicGroup">CyclicGroup() (in module sympy.combinatorics.named_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/numtheory.html#mpmath.cyclotomic">cyclotomic() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.specialpolys.cyclotomic_poly">cyclotomic_poly() (in module sympy.polys.specialpolys)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/autowrap.html#sympy.utilities.autowrap.CythonCodeWrapper">CythonCodeWrapper (class in sympy.utilities.autowrap)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/autowrap.html#sympy.utilities.autowrap.CythonCodeWrapper.dump_pyx">CythonCodeWrapper.dump_pyx() (in module sympy.utilities.autowrap)</a>
-  </dt>
-
-  </dl></td>
-</tr></table>
-
-<h2 id="D">D</h2>
-<table style="width: 100%" class="indextable genindextable"><tr>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.D">D (sympy.matrices.matrices.MatrixBase attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.RayTransferMatrix.D">(sympy.physics.gaussopt.RayTransferMatrix attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/physics/quantum/spin.html#sympy.physics.quantum.spin.Rotation.d">d() (sympy.physics.quantum.spin.Rotation class method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/spin.html#sympy.physics.quantum.spin.Rotation.D">D() (sympy.physics.quantum.spin.Rotation class method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/dagger.html#sympy.physics.quantum.dagger.Dagger">Dagger (class in sympy.physics.quantum.dagger)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.Dagger">(class in sympy.physics.secondquant)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/stats.html#sympy.stats.Dagum">Dagum() (in module sympy.stats)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/codegen.html#sympy.utilities.codegen.DataType">DataType (class in sympy.utilities.codegen)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/misc.html#sympy.utilities.misc.debug">debug() (in module sympy.utilities.misc)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.decompose">decompose() (in module sympy.polys.polytools)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/constants.html#mpmath.mp.degree">degree (mpmath.mp attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.degree">(sympy.combinatorics.perm_groups.PermutationGroup attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.degree">degree() (in module sympy.polys.polytools)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.degree_list">degree_list() (in module sympy.polys.polytools)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/trigonometric.html#mpmath.degrees">degrees() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.hyper.meijerg.delta">delta (sympy.functions.special.hyper.meijerg attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.deltaintegrate">deltaintegrate() (sympy.integrals method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/stats.html#sympy.stats.density">density() (in module sympy.stats)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.function.Derivative">Derivative (class in sympy.core.function)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.function.Derivative.doit_numerically">Derivative.doit_numerically() (in module sympy.core.function)</a>
-  </dt>
-
-      
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.dense.diag">diag() (in module sympy.matrices.dense)</a>
-  </dt>
-
-      
-  <dt><a href="modules/galgebra/GA.html#sympy.galgebra.GA.diagpq">diagpq() (in module sympy.galgebra.GA)</a>
-  </dt>
-
-      
-  <dt><a href="modules/categories.html#sympy.categories.Diagram">Diagram (class in sympy.categories)</a>
-  </dt>
-
-      
-  <dt><a href="modules/categories.html#sympy.categories.Diagram.hom">Diagram.hom() (in module sympy.categories)</a>
-  </dt>
-
-      
-  <dt><a href="modules/categories.html#sympy.categories.Diagram.is_subdiagram">Diagram.is_subdiagram() (in module sympy.categories)</a>
-  </dt>
-
-      
-  <dt><a href="modules/categories.html#sympy.categories.Diagram.subdiagram_from_objects">Diagram.subdiagram_from_objects() (in module sympy.categories)</a>
-  </dt>
-
-      
-  <dt><a href="modules/categories.html#sympy.categories.diagram_drawing.DiagramGrid">DiagramGrid (class in sympy.categories.diagram_drawing)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.containers.Dict">Dict (class in sympy.core.containers)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.containers.Dict.get">Dict.get() (in module sympy.core.containers)</a>
-  </dt>
-
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-  <dt><a href="modules/core.html#sympy.core.containers.Dict.items">Dict.items() (in module sympy.core.containers)</a>
-  </dt>
-
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-  <dt><a href="modules/core.html#sympy.core.containers.Dict.keys">Dict.keys() (in module sympy.core.containers)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.containers.Dict.values">Dict.values() (in module sympy.core.containers)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.dict_merge">dict_merge() (in module sympy.utilities.iterables)</a>
-  </dt>
-
-      
-  <dt><a href="modules/stats.html#sympy.stats.Die">Die() (in module sympy.stats)</a>
-  </dt>
-
-      
-  <dt><a href="modules/stats.html#sympy.stats.frv_types.DiePSpace">DiePSpace (class in sympy.stats.frv_types)</a>
-  </dt>
-
-      
-  <dt><a href="tutorial/tutorial.en.html#index-2">diff</a>, <a href="tutorial/tutorial.en.html#index-7">[1]</a>
-  </dt>
-
-      
-  <dt><a href="modules/galgebra/GA/GAsympy.html#diff">diff() (built-in function)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/mpmath/calculus/differentiation.html#mpmath.diff">(in module mpmath)</a>
-  </dt>
-
-        
-  <dt><a href="modules/core.html#sympy.core.function.diff">(in module sympy.core.function)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/diffgeom.html#sympy.diffgeom.Differential">Differential (class in sympy.diffgeom)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/operator.html#sympy.physics.quantum.operator.DifferentialOperator">DifferentialOperator (class in sympy.physics.quantum.operator)</a>
-  </dt>
-
-      
-  <dt><a href="tutorial/tutorial.en.html#index-2">differentiation</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/calculus/differentiation.html#mpmath.differint">differint() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/calculus/differentiation.html#mpmath.diffs">diffs() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/calculus/differentiation.html#mpmath.diffs_exp">diffs_exp() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/calculus/differentiation.html#mpmath.diffs_prod">diffs_prod() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/gamma.html#mpmath.digamma">digamma() (in module mpmath)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/functions/special.html#sympy.functions.special.gamma_functions.digamma">(in module sympy.functions.special.gamma_functions)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/printing.html#sympy.printing.pretty.pretty_symbology.digit_2txt">digit_2txt (in module sympy.printing.pretty.pretty_symbology)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.generators.dihedral">dihedral() (sympy.combinatorics.generators method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/named_groups.html#sympy.combinatorics.named_groups.DihedralGroup">DihedralGroup() (in module sympy.combinatorics.named_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/hilbert.html#sympy.physics.quantum.hilbert.HilbertSpace.dimension">dimension (sympy.physics.quantum.hilbert.HilbertSpace attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.delta_functions.DiracDelta">DiracDelta (class in sympy.functions.special.delta_functions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.delta_functions.DiracDelta.is_simple">DiracDelta.is_simple() (in module sympy.functions.special.delta_functions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.delta_functions.DiracDelta.simplify">DiracDelta.simplify() (in module sympy.functions.special.delta_functions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/group_constructs.html#sympy.combinatorics.group_constructs.DirectProduct">DirectProduct() (in module sympy.combinatorics.group_constructs)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/zeta.html#mpmath.dirichlet">dirichlet() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.zeta_functions.dirichlet_eta">dirichlet_eta (class in sympy.functions.special.zeta_functions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/stats.html#sympy.stats.DiscreteUniform">DiscreteUniform() (in module sympy.stats)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.discriminant">discriminant() (in module sympy.polys.polytools)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.div">div() (in module sympy.polys.polytools)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.BeamParameter.divergence">divergence (sympy.physics.gaussopt.BeamParameter attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/ntheory.html#sympy.ntheory.factor_.divisor_count">divisor_count() (in module sympy.ntheory.factor_)</a>
-  </dt>
-
-      
-  <dt><a href="modules/ntheory.html#sympy.ntheory.factor_.divisors">divisors() (in module sympy.ntheory.factor_)</a>
-  </dt>
-
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-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.DMF">DMF (class in sympy.polys.polyclasses)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.factortools.dmp_zz_wang_non_divisors">dmp_zz_wang_non_divisors() (in module sympy.polys.factortools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.factortools.dmp_zz_wang_test_points">dmp_zz_wang_test_points() (in module sympy.polys.factortools)</a>
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-  <dt><a href="modules/utilities/runtests.html#sympy.utilities.runtests.doctest">doctest() (in module sympy.utilities.runtests)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain">Domain (class in sympy.polys.domains)</a>
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-  <dt><a href="modules/categories.html#sympy.categories.CompositeMorphism.domain">domain (sympy.categories.CompositeMorphism attribute)</a>
-  </dt>
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-  <dt><a href="modules/categories.html#sympy.categories.Morphism.domain">(sympy.categories.Morphism attribute)</a>
-  </dt>
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.domain">(sympy.polys.polytools.Poly attribute)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.abs">Domain.abs() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.add">Domain.add() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.algebraic_field">Domain.algebraic_field() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.characteristic">Domain.characteristic() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.cofactors">Domain.cofactors() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.convert">Domain.convert() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.denom">Domain.denom() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.div">Domain.div() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.evalf">Domain.evalf() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.exquo">Domain.exquo() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.frac_field">Domain.frac_field() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.from_AlgebraicField">Domain.from_AlgebraicField() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.from_ExpressionDomain">Domain.from_ExpressionDomain() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.from_FF_gmpy">Domain.from_FF_gmpy() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.from_FF_python">Domain.from_FF_python() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.from_FF_sympy">Domain.from_FF_sympy() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.from_FractionField">Domain.from_FractionField() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.from_GlobalPolynomialRing">Domain.from_GlobalPolynomialRing() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.from_QQ_gmpy">Domain.from_QQ_gmpy() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.from_QQ_python">Domain.from_QQ_python() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.from_QQ_sympy">Domain.from_QQ_sympy() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.from_RR_mpmath">Domain.from_RR_mpmath() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.from_RR_sympy">Domain.from_RR_sympy() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.from_sympy">Domain.from_sympy() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.from_ZZ_gmpy">Domain.from_ZZ_gmpy() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.from_ZZ_python">Domain.from_ZZ_python() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.from_ZZ_sympy">Domain.from_ZZ_sympy() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.gcd">Domain.gcd() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.gcdex">Domain.gcdex() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.get_exact">Domain.get_exact() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.get_field">Domain.get_field() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.get_ring">Domain.get_ring() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.half_gcdex">Domain.half_gcdex() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.inject">Domain.inject() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.invert">Domain.invert() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.is_negative">Domain.is_negative() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.is_nonnegative">Domain.is_nonnegative() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.is_nonpositive">Domain.is_nonpositive() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.is_positive">Domain.is_positive() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.log">Domain.log() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.map">Domain.map() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.mul">Domain.mul() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.n">Domain.n() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.neg">Domain.neg() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.poly_ring">Domain.poly_ring() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.pow">Domain.pow() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.quo">Domain.quo() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.rem">Domain.rem() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.revert">Domain.revert() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Domain.sub">Domain.sub() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.polyerrors.DomainError">DomainError (class in sympy.polys.polyerrors)</a>
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-  <dt><a href="modules/physics/mechanics/api/functions.html#sympy.physics.mechanics.functions.dot">dot() (in module sympy.physics.mechanics.functions)</a>
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-  <dt><a href="modules/combinatorics/tensor_can.html#sympy.combinatorics.tensor_can.double_coset_can_rep">double_coset_can_rep() (in module sympy.combinatorics.tensor_can)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.FreeModule.dtype">dtype (sympy.polys.agca.modules.FreeModule attribute)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.QuotientModule.dtype">(sympy.polys.agca.modules.QuotientModule attribute)</a>
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-  <dt><a href="modules/physics/quantum/state.html#sympy.physics.quantum.state.StateBase.dual">dual (sympy.physics.quantum.state.StateBase attribute)</a>
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-  <dt><a href="modules/physics/quantum/state.html#sympy.physics.quantum.state.StateBase.dual_class">dual_class() (sympy.physics.quantum.state.StateBase class method)</a>
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-  <dt><a href="modules/galgebra/GA.html#sympy.galgebra.GA.dualsort">dualsort() (in module sympy.galgebra.GA)</a>
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-  <dt><a href="modules/core.html#sympy.core.symbol.Dummy">Dummy (class in sympy.core.symbol)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.factortools.dup_cyclotomic_p">dup_cyclotomic_p() (in module sympy.polys.factortools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.densearith.dup_lshift">dup_lshift() (in module sympy.polys.densearith)</a>
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-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Dyadic.cross">Dyadic.cross() (in module sympy.physics.mechanics.essential)</a>
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-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Dyadic.doit">Dyadic.doit() (in module sympy.physics.mechanics.essential)</a>
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-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Dyadic.dot">Dyadic.dot() (in module sympy.physics.mechanics.essential)</a>
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-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Dyadic.dt">Dyadic.dt() (in module sympy.physics.mechanics.essential)</a>
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-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Dyadic.express">Dyadic.express() (in module sympy.physics.mechanics.essential)</a>
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-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Dyadic.simplify">Dyadic.simplify() (in module sympy.physics.mechanics.essential)</a>
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-  <dt><a href="modules/physics/mechanics/api/functions.html#sympy.physics.mechanics.essential.dynamicsymbols">dynamicsymbols() (in module sympy.physics.mechanics.essential)</a>
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-<h2 id="E">E</h2>
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-  <dt><a href="modules/stats.html#sympy.stats.E">E() (in module sympy.stats)</a>
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-  <dt><a href="modules/physics/qho_1d.html#sympy.physics.qho_1d.E_n">E_n() (in module sympy.physics.qho_1d)</a>
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-  <dt><a href="modules/physics/hydrogen.html#sympy.physics.hydrogen.E_nl">E_nl() (in module sympy.physics.hydrogen)</a>
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-  <dt><a href="modules/physics/hydrogen.html#sympy.physics.hydrogen.E_nl_dirac">E_nl_dirac() (in module sympy.physics.hydrogen)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Ellipse.eccentricity">eccentricity (sympy.geometry.ellipse.Ellipse attribute)</a>
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-  <dt><a href="modules/combinatorics/polyhedron.html#sympy.combinatorics.polyhedron.Polyhedron.edges">edges (sympy.combinatorics.polyhedron.Polyhedron attribute)</a>
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-  <dt><a href="modules/functions/special.html#sympy.functions.special.error_functions.Ei">Ei (class in sympy.functions.special.error_functions)</a>
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-  <dt><a href="modules/mpmath/functions/elliptic.html#mpmath.ellipe">ellipe() (in module mpmath)</a>
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-  <dt><a href="modules/mpmath/functions/elliptic.html#mpmath.ellipf">ellipf() (in module mpmath)</a>
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-  <dt><a href="modules/mpmath/functions/elliptic.html#mpmath.elliprf">elliprf() (in module mpmath)</a>
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-  <dt><a href="modules/mpmath/functions/elliptic.html#mpmath.elliprg">elliprg() (in module mpmath)</a>
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-  <dt><a href="modules/mpmath/functions/elliptic.html#mpmath.elliprj">elliprj() (in module mpmath)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Ellipse">Ellipse (class in sympy.geometry.ellipse)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Ellipse.encloses_point">Ellipse.encloses_point() (in module sympy.geometry.ellipse)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Ellipse.intersection">Ellipse.intersection() (in module sympy.geometry.ellipse)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Ellipse.is_tangent">Ellipse.is_tangent() (in module sympy.geometry.ellipse)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Ellipse.plot_interval">Ellipse.plot_interval() (in module sympy.geometry.ellipse)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Ellipse.random_point">Ellipse.random_point() (in module sympy.geometry.ellipse)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Ellipse.tangent_lines">Ellipse.tangent_lines() (in module sympy.geometry.ellipse)</a>
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-  <dt><a href="modules/mpmath/functions/expintegrals.html#mpmath.erf">erf() (in module mpmath)</a>
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-  <dt><a href="modules/stats.html#sympy.stats.Erlang">Erlang() (in module sympy.stats)</a>
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-  <dt><a href="modules/functions/special.html#sympy.functions.special.hyper.hyper.eta">eta (sympy.functions.special.hyper.hyper attribute)</a>
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-  <dt><a href="modules/functions/combinatorial.html#sympy.functions.combinatorial.numbers.euler">euler (class in sympy.functions.combinatorial.numbers)</a>
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-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.evaluate_deltas">evaluate_deltas() (in module sympy.physics.secondquant)</a>
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-  <dt><a href="modules/physics/paulialgebra.html#sympy.physics.paulialgebra.evaluate_pauli_product">evaluate_pauli_product() (in module sympy.physics.paulialgebra)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.polyerrors.EvaluationFailed">EvaluationFailed (class in sympy.polys.polyerrors)</a>
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-  <dt><a href="modules/galgebra/GA/GAsympy.html#even">even() (built-in function)</a>
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-  <dt><a href="modules/mpmath/functions/powers.html#mpmath.exp">exp() (in module mpmath)</a>
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-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.exponential.exp.taylor_term">exp.taylor_term() (in module sympy.functions.elementary.exponential)</a>
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-  <dt><a href="modules/core.html#sympy.core.function.expand_power_base">expand_power_base() (in module sympy.core.function)</a>
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-  <dt>
-    expansion
-  </dt>
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-  <dt><a href="modules/core.html#sympy.core.expr.Expr.as_coeff_mul">Expr.as_coeff_mul() (in module sympy.core.expr)</a>
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-  <dt><a href="modules/core.html#sympy.core.expr.Expr.is_polynomial">Expr.is_polynomial() (in module sympy.core.expr)</a>
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-  <dt><a href="modules/core.html#sympy.core.expr.Expr.is_rational_function">Expr.is_rational_function() (in module sympy.core.expr)</a>
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-  <dt><a href="modules/core.html#sympy.core.expr.Expr.limit">Expr.limit() (in module sympy.core.expr)</a>
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-  <dt><a href="modules/core.html#sympy.core.expr.Expr.nsimplify">Expr.nsimplify() (in module sympy.core.expr)</a>
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-  <dt><a href="modules/core.html#sympy.core.expr.Expr.powsimp">Expr.powsimp() (in module sympy.core.expr)</a>
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-  <dt><a href="modules/core.html#sympy.core.expr.Expr.radsimp">Expr.radsimp() (in module sympy.core.expr)</a>
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-  <dt><a href="modules/core.html#sympy.core.expr.Expr.ratsimp">Expr.ratsimp() (in module sympy.core.expr)</a>
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-  <dt><a href="modules/core.html#sympy.core.expr.Expr.refine">Expr.refine() (in module sympy.core.expr)</a>
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-  <dt><a href="modules/core.html#sympy.core.expr.Expr.removeO">Expr.removeO() (in module sympy.core.expr)</a>
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-  <dt><a href="modules/core.html#sympy.core.expr.Expr.round">Expr.round() (in module sympy.core.expr)</a>
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-  <dt><a href="modules/core.html#sympy.core.expr.Expr.separate">Expr.separate() (in module sympy.core.expr)</a>
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-  <dt><a href="modules/core.html#sympy.core.expr.Expr.trigsimp">Expr.trigsimp() (in module sympy.core.expr)</a>
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-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.piecewise.ExprCondPair">ExprCondPair (class in sympy.functions.elementary.piecewise)</a>
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-  <dt><a href="modules/physics/mechanics/api/functions.html#sympy.physics.mechanics.functions.express">express() (in module sympy.physics.mechanics.functions)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain">ExpressionDomain (class in sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain.denom">ExpressionDomain.denom() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain.Expression">ExpressionDomain.Expression (class in sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain.from_ExpressionDomain">ExpressionDomain.from_ExpressionDomain() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain.from_FractionField">ExpressionDomain.from_FractionField() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain.from_GlobalPolynomialRing">ExpressionDomain.from_GlobalPolynomialRing() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain.from_QQ_gmpy">ExpressionDomain.from_QQ_gmpy() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain.from_QQ_python">ExpressionDomain.from_QQ_python() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain.from_QQ_sympy">ExpressionDomain.from_QQ_sympy() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain.from_RR_mpmath">ExpressionDomain.from_RR_mpmath() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain.from_RR_sympy">ExpressionDomain.from_RR_sympy() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain.from_sympy">ExpressionDomain.from_sympy() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain.from_ZZ_gmpy">ExpressionDomain.from_ZZ_gmpy() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain.from_ZZ_python">ExpressionDomain.from_ZZ_python() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain.from_ZZ_sympy">ExpressionDomain.from_ZZ_sympy() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain.get_field">ExpressionDomain.get_field() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain.get_ring">ExpressionDomain.get_ring() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain.is_negative">ExpressionDomain.is_negative() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain.is_nonnegative">ExpressionDomain.is_nonnegative() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain.is_nonpositive">ExpressionDomain.is_nonpositive() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain.is_positive">ExpressionDomain.is_positive() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain.numer">ExpressionDomain.numer() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.ExpressionDomain.to_sympy">ExpressionDomain.to_sympy() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.exquo">exquo() (in module sympy.polys.polytools)</a>
-  </dt>
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.RegularPolygon.exterior_angle">exterior_angle (sympy.geometry.polygon.RegularPolygon attribute)</a>
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-  <dt><a href="modules/mpmath/general.html#mpmath.extradps">extradps() (in module mpmath)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.polyerrors.ExtraneousFactors">ExtraneousFactors (class in sympy.polys.polyerrors)</a>
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-  <dt><a href="modules/mpmath/general.html#mpmath.extraprec">extraprec() (in module mpmath)</a>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.dense.eye">eye() (in module sympy.matrices.dense)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.eye">(sympy.matrices.sparse.SparseMatrix class method)</a>
-  </dt>
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-  </dl></td>
-</tr></table>
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-  <td style="width: 33%" valign="top"><dl>
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-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.F">F (in module sympy.physics.secondquant)</a>
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-  <dt><a href="modules/utilities/autowrap.html#sympy.utilities.autowrap.F2PyCodeWrapper">F2PyCodeWrapper (class in sympy.utilities.autowrap)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.groebnertools.f5b">f5b() (in module sympy.polys.groebnertools)</a>
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-  <dt><a href="modules/mpmath/general.html#mpmath.fabs">fabs() (in module mpmath)</a>
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-  <dt><a href="modules/mpmath/functions/gamma.html#mpmath.fac2">fac2() (in module mpmath)</a>
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-  <dt><a href="modules/combinatorics/polyhedron.html#sympy.combinatorics.polyhedron.Polyhedron.faces">faces (sympy.combinatorics.polyhedron.Polyhedron attribute)</a>
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.factor">factor() (in module sympy.polys.polytools)</a>
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.factor_list">factor_list() (in module sympy.polys.polytools)</a>
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-  <dt><a href="modules/core.html#sympy.core.exprtools.factor_terms">factor_terms() (in module sympy.core.exprtools)</a>
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-  <dt><a href="modules/functions/combinatorial.html#sympy.functions.combinatorial.factorials.factorial">factorial (class in sympy.functions.combinatorial.factorials)</a>
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-  <dt><a href="modules/mpmath/functions/gamma.html#mpmath.factorial">factorial() (in module mpmath)</a>
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-  <dt><a href="modules/functions/combinatorial.html#sympy.functions.combinatorial.factorials.factorial2">factorial2 (class in sympy.functions.combinatorial.factorials)</a>
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-  <dt><a href="modules/ntheory.html#sympy.ntheory.factor_.factorint">factorint() (in module sympy.ntheory.factor_)</a>
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-  <dt><a href="modules/mpmath/general.html#mpmath.fadd">fadd() (in module mpmath)</a>
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-  <dt><a href="modules/functions/combinatorial.html#sympy.functions.combinatorial.factorials.FallingFactorial">FallingFactorial (class in sympy.functions.combinatorial.factorials)</a>
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-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.FBra">FBra (in module sympy.physics.secondquant)</a>
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-  <dt><a href="modules/printing.html#sympy.printing.fcode.fcode">fcode() (in module sympy.printing.fcode)</a>
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-  <dt><a href="modules/utilities/codegen.html#sympy.utilities.codegen.FCodeGen">FCodeGen (class in sympy.utilities.codegen)</a>
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-  <dt><a href="modules/utilities/codegen.html#sympy.utilities.codegen.FCodeGen.dump_f95">FCodeGen.dump_f95() (in module sympy.utilities.codegen)</a>
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-  <dt><a href="modules/utilities/codegen.html#sympy.utilities.codegen.FCodeGen.dump_h">FCodeGen.dump_h() (in module sympy.utilities.codegen)</a>
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-  <dt><a href="modules/utilities/codegen.html#sympy.utilities.codegen.FCodeGen.get_interface">FCodeGen.get_interface() (in module sympy.utilities.codegen)</a>
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-  <dt><a href="modules/printing.html#sympy.printing.fcode.FCodePrinter">FCodePrinter (class in sympy.printing.fcode)</a>
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-  <dt><a href="modules/printing.html#sympy.printing.fcode.FCodePrinter.doprint">FCodePrinter.doprint() (in module sympy.printing.fcode)</a>
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-  <dt><a href="modules/printing.html#sympy.printing.fcode.FCodePrinter.indent_code">FCodePrinter.indent_code() (in module sympy.printing.fcode)</a>
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-  <dt><a href="modules/galgebra/latex_ex/latex_ex.html#fct_format">fct_format() (built-in function)</a>
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-  </dt>
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-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.Fd">Fd (in module sympy.physics.secondquant)</a>
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-  <dt><a href="modules/stats.html#sympy.stats.FDistribution">FDistribution() (in module sympy.stats)</a>
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-  <dt><a href="modules/mpmath/general.html#mpmath.fdiv">fdiv() (in module mpmath)</a>
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-  <dt><a href="modules/mpmath/functions/gamma.html#mpmath.ff">ff() (in module mpmath)</a>
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-  <dt><a href="modules/functions/combinatorial.html#sympy.functions.combinatorial.numbers.fibonacci">fibonacci (class in sympy.functions.combinatorial.numbers)</a>
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-  <dt><a href="modules/mpmath/functions/numtheory.html#mpmath.fibonacci">fibonacci() (in module mpmath)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Field">Field (class in sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Field.div">Field.div() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Field.exquo">Field.exquo() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Field.gcd">Field.gcd() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Field.get_field">Field.get_field() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Field.get_ring">Field.get_ring() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Field.lcm">Field.lcm() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Field.quo">Field.quo() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Field.rem">Field.rem() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Field.revert">Field.revert() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/reference.html#sympy.polys.numberfields.field_isomorphism">field_isomorphism() (in module sympy.polys.numberfields)</a>
-  </dt>
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-  <dt><a href="modules/galgebra/latex_ex.html#sympy.galgebra.latex_ex.find_executable">find_executable() (in module sympy.galgebra.latex_ex)</a>
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-  <dt><a href="modules/physics/units.html#sympy.physics.units.find_unit">find_unit() (in module sympy.physics.units)</a>
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-  <dt><a href="modules/mpmath/identification.html#mpmath.findpoly">findpoly() (in module mpmath)</a>
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-  <dt><a href="modules/mpmath/calculus/optimization.html#mpmath.findroot">findroot() (in module mpmath)</a>
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-  <dt><a href="modules/stats.html#sympy.stats.frv.FiniteDomain">FiniteDomain (class in sympy.stats.frv)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FiniteField">FiniteField (class in sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FiniteField.characteristic">FiniteField.characteristic() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FiniteField.from_FF_gmpy">FiniteField.from_FF_gmpy() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FiniteField.from_FF_python">FiniteField.from_FF_python() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FiniteField.from_FF_sympy">FiniteField.from_FF_sympy() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FiniteField.from_QQ_gmpy">FiniteField.from_QQ_gmpy() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FiniteField.from_QQ_python">FiniteField.from_QQ_python() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FiniteField.from_QQ_sympy">FiniteField.from_QQ_sympy() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FiniteField.from_RR_mpmath">FiniteField.from_RR_mpmath() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FiniteField.from_RR_sympy">FiniteField.from_RR_sympy() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FiniteField.from_sympy">FiniteField.from_sympy() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FiniteField.from_ZZ_gmpy">FiniteField.from_ZZ_gmpy() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FiniteField.from_ZZ_python">FiniteField.from_ZZ_python() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FiniteField.from_ZZ_sympy">FiniteField.from_ZZ_sympy() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FiniteField.get_field">FiniteField.get_field() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FiniteField.to_sympy">FiniteField.to_sympy() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/stats.html#sympy.stats.frv.FinitePSpace">FinitePSpace (class in sympy.stats.frv)</a>
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-  <dt><a href="modules/stats.html#sympy.stats.FiniteRV">FiniteRV() (in module sympy.stats)</a>
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-  <dt><a href="modules/sets.html#sympy.core.sets.FiniteSet">FiniteSet (class in sympy.core.sets)</a>
-  </dt>
-
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-  <dt><a href="modules/sets.html#sympy.core.sets.FiniteSet.as_relational">FiniteSet.as_relational() (in module sympy.core.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/stats.html#sympy.stats.FisherZ">FisherZ() (in module sympy.stats)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.FixedBosonicBasis">FixedBosonicBasis (class in sympy.physics.secondquant)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.FixedBosonicBasis.index">FixedBosonicBasis.index() (in module sympy.physics.secondquant)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.FixedBosonicBasis.state">FixedBosonicBasis.state() (in module sympy.physics.secondquant)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.FKet">FKet (in module sympy.physics.secondquant)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyerrors.FlagError">FlagError (class in sympy.polys.polyerrors)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.FlatMirror">FlatMirror (class in sympy.physics.gaussopt)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.FlatRefraction">FlatRefraction (class in sympy.physics.gaussopt)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.flatten">flatten() (in module sympy.utilities.iterables)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/core.html#sympy.core.add.Add.flatten">(sympy.core.add.Add class method)</a>
-  </dt>
-
-        
-  <dt><a href="modules/core.html#sympy.core.mul.Mul.flatten">(sympy.core.mul.Mul class method)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/core.html#sympy.core.numbers.Float">Float (class in sympy.core.numbers)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.integers.floor">floor (class in sympy.functions.elementary.integers)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/general.html#mpmath.floor">floor() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/general.html#mpmath.fmod">fmod() (in module mpmath)</a>
-  </dt>
-
-  </dl></td>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/mpmath/general.html#mpmath.fmul">fmul() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/general.html#mpmath.fneg">fneg() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Ellipse.foci">foci (sympy.geometry.ellipse.Ellipse attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/hilbert.html#sympy.physics.quantum.hilbert.FockSpace">FockSpace (class in sympy.physics.quantum.hilbert)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.FockState">FockState (class in sympy.physics.secondquant)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.FockStateBosonBra">FockStateBosonBra (class in sympy.physics.secondquant)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.FockStateBosonKet">FockStateBosonKet (class in sympy.physics.secondquant)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.FockStateBra">FockStateBra (class in sympy.physics.secondquant)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.FockStateKet">FockStateKet (class in sympy.physics.secondquant)</a>
-  </dt>
-
-      
-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Ellipse.focus_distance">focus_distance (sympy.geometry.ellipse.Ellipse attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.lagrange.LagrangesMethod.forcing">forcing (sympy.physics.mechanics.lagrange.LagrangesMethod attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.lagrange.LagrangesMethod.forcing_full">forcing_full (sympy.physics.mechanics.lagrange.LagrangesMethod attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/galgebra/latex_ex/latex_ex.html#Format">Format() (built-in function)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/calculus/approximation.html#mpmath.fourier">fourier() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.transforms.fourier_transform">fourier_transform() (in module sympy.integrals.transforms)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/calculus/approximation.html#mpmath.fourierval">fourierval() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/general.html#mpmath.fprod">fprod() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.pretty.pretty_symbology.frac">frac (in module sympy.printing.pretty.pretty_symbology)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/general.html#mpmath.frac">frac() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/general.html#mpmath.fraction">fraction() (in module mpmath)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/simplify/simplify.html#sympy.simplify.simplify.fraction">(in module sympy.simplify.simplify)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField">FractionField (class in sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField.denom">FractionField.denom() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField.factorial">FractionField.factorial() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField.frac_field">FractionField.frac_field() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField.from_FractionField">FractionField.from_FractionField() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField.from_GlobalPolynomialRing">FractionField.from_GlobalPolynomialRing() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField.from_QQ_gmpy">FractionField.from_QQ_gmpy() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField.from_QQ_python">FractionField.from_QQ_python() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField.from_QQ_sympy">FractionField.from_QQ_sympy() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField.from_RR_mpmath">FractionField.from_RR_mpmath() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField.from_RR_sympy">FractionField.from_RR_sympy() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField.from_sympy">FractionField.from_sympy() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField.from_ZZ_gmpy">FractionField.from_ZZ_gmpy() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField.from_ZZ_python">FractionField.from_ZZ_python() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField.from_ZZ_sympy">FractionField.from_ZZ_sympy() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField.get_ring">FractionField.get_ring() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField.is_negative">FractionField.is_negative() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField.is_nonnegative">FractionField.is_nonnegative() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField.is_nonpositive">FractionField.is_nonpositive() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField.is_positive">FractionField.is_positive() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField.numer">FractionField.numer() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField.poly_ring">FractionField.poly_ring() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.FractionField.to_sympy">FractionField.to_sympy() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/stats.html#sympy.stats.Frechet">Frechet() (in module sympy.stats)</a>
-  </dt>
-
-      
-  <dt><a href="modules/concrete.html#sympy.concrete.products.Product.free_symbols">free_symbols (sympy.concrete.products.Product attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/concrete.html#sympy.concrete.summations.Sum.free_symbols">(sympy.concrete.summations.Sum attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/core.html#sympy.core.basic.Basic.free_symbols">(sympy.core.basic.Basic attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.piecewise.ExprCondPair.free_symbols">(sympy.functions.elementary.piecewise.ExprCondPair attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/geometry.html#sympy.geometry.curve.Curve.free_symbols">(sympy.geometry.curve.Curve attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.Integral.free_symbols">(sympy.integrals.Integral attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/physics/quantum/operator.html#sympy.physics.quantum.operator.DifferentialOperator.free_symbols">(sympy.physics.quantum.operator.DifferentialOperator attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.free_symbols">(sympy.polys.polytools.Poly attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.PurePoly.free_symbols">(sympy.polys.polytools.PurePoly attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.free_symbols_in_domain">free_symbols_in_domain (sympy.polys.polytools.Poly attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.FreeModule">FreeModule (class in sympy.polys.agca.modules)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.FreeModule.basis">FreeModule.basis() (in module sympy.polys.agca.modules)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.FreeModule.convert">FreeModule.convert() (in module sympy.polys.agca.modules)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.FreeModule.identity_hom">FreeModule.identity_hom() (in module sympy.polys.agca.modules)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.FreeModule.is_submodule">FreeModule.is_submodule() (in module sympy.polys.agca.modules)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.FreeModule.is_zero">FreeModule.is_zero() (in module sympy.polys.agca.modules)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.FreeModule.multiply_ideal">FreeModule.multiply_ideal() (in module sympy.polys.agca.modules)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.FreeModule.quotient_module">FreeModule.quotient_module() (in module sympy.polys.agca.modules)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.FreeSpace">FreeSpace (class in sympy.physics.gaussopt)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.error_functions.fresnelc">fresnelc (class in sympy.functions.special.error_functions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/expintegrals.html#mpmath.fresnelc">fresnelc() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.error_functions.FresnelIntegral">FresnelIntegral (class in sympy.functions.special.error_functions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.error_functions.fresnels">fresnels (class in sympy.functions.special.error_functions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/expintegrals.html#mpmath.fresnels">fresnels() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/general.html#mpmath.frexp">frexp() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.DMP.from_dict">from_dict() (sympy.polys.polyclasses.DMP class method)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.from_dict">(sympy.polys.polytools.Poly class method)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.from_expr">from_expr() (sympy.polys.polytools.Poly class method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.from_inversion_vector">from_inversion_vector() (sympy.combinatorics.permutations.Permutation class method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.DMP.from_list">from_list() (sympy.polys.polyclasses.DMP class method)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.from_list">(sympy.polys.polytools.Poly class method)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.from_poly">from_poly() (sympy.polys.polytools.Poly class method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/partitions.html#sympy.combinatorics.partitions.Partition.from_rgs">from_rgs() (sympy.combinatorics.partitions.Partition class method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.from_sequence">from_sequence() (sympy.combinatorics.permutations.Permutation class method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.DMP.from_sympy_list">from_sympy_list() (sympy.polys.polyclasses.DMP class method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.basic.Basic.fromiter">fromiter() (sympy.core.basic.Basic class method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/general.html#mpmath.fsub">fsub() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/general.html#mpmath.fsum">fsum() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.full_cyclic_form">full_cyclic_form (sympy.combinatorics.permutations.Permutation attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.basic.Basic.func">func (sympy.core.basic.Basic attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.function.Function">Function (class in sympy.core.function)</a>
-  </dt>
-
-      
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.Integral.function">function (sympy.integrals.Integral attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/quantum/operator.html#sympy.physics.quantum.operator.DifferentialOperator.function">(sympy.physics.quantum.operator.DifferentialOperator attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/core.html#sympy.core.function.Function.as_base_exp">Function.as_base_exp() (in module sympy.core.function)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.function.Function.fdiff">Function.fdiff() (in module sympy.core.function)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.function.FunctionClass">FunctionClass (class in sympy.core.function)</a>
-  </dt>
-
-      
-  <dt><a href="modules/matrices/expressions.html#sympy.matrices.expressions.FunctionMatrix">FunctionMatrix (class in sympy.matrices.expressions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/geometry.html#sympy.geometry.curve.Curve.functions">functions (sympy.geometry.curve.Curve attribute)</a>
-  </dt>
-
-  </dl></td>
-</tr></table>
-
-<h2 id="G">G</h2>
-<table style="width: 100%" class="indextable genindextable"><tr>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/printing.html#sympy.printing.pretty.pretty_symbology.g">g() (in module sympy.printing.pretty.pretty_symbology)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.pretty.pretty_symbology.G">G() (in module sympy.printing.pretty.pretty_symbology)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.gamma_functions.gamma">gamma (class in sympy.functions.special.gamma_functions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/gamma.html#mpmath.gamma">gamma() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/stats.html#sympy.stats.Gamma">Gamma() (in module sympy.stats)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/expintegrals.html#mpmath.gammainc">gammainc() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/stats.html#sympy.stats.GammaInverse">GammaInverse() (in module sympy.stats)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/gamma.html#mpmath.gammaprod">gammaprod() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.Gate">Gate (class in sympy.physics.quantum.gate)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CGate.gate">gate (sympy.physics.quantum.gate.CGate attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CNotGate.gate">(sympy.physics.quantum.gate.CNotGate attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.Gate.get_target_matrix">Gate.get_target_matrix() (in module sympy.physics.quantum.gate)</a>
-  </dt>
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-  <dt><a href="modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.GaussLegendre">GaussLegendre (class in mpmath.calculus.quadrature)</a>
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-  <dt><a href="modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.GaussLegendre.calc_nodes">GaussLegendre.calc_nodes() (in module mpmath.calculus.quadrature)</a>
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-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.generators">generators (sympy.combinatorics.perm_groups.PermutationGroup attribute)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.polyerrors.GeneratorsError">GeneratorsError (class in sympy.polys.polyerrors)</a>
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-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.geometric_conj_ab">geometric_conj_ab() (in module sympy.physics.gaussopt)</a>
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-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.geometric_conj_af">geometric_conj_af() (in module sympy.physics.gaussopt)</a>
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-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.geometric_conj_bf">geometric_conj_bf() (in module sympy.physics.gaussopt)</a>
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-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.GeometricRay">GeometricRay (class in sympy.physics.gaussopt)</a>
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-  <dt><a href="modules/utilities/codegen.html#sympy.utilities.codegen.get_default_datatype">get_default_datatype() (in module sympy.utilities.codegen)</a>
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-  <dt><a href="modules/tensor/index_methods.html#sympy.tensor.index_methods.get_indices">get_indices() (in module sympy.tensor.index_methods)</a>
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-  <dt><a href="modules/utilities/source.html#sympy.utilities.source.get_mod_func">get_mod_func() (in module sympy.utilities.source)</a>
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-  <dt><a href="modules/combinatorics/graycode.html#sympy.combinatorics.graycode.get_subset_from_bitstring">get_subset_from_bitstring() (sympy.combinatorics.graycode method)</a>
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-  <dt><a href="modules/combinatorics/tensor_can.html#sympy.combinatorics.tensor_can.get_symmetric_group_sgs">get_symmetric_group_sgs() (in module sympy.combinatorics.tensor_can)</a>
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-  <dt><a href="modules/utilities/runtests.html#sympy.utilities.runtests.get_sympy_dir">get_sympy_dir() (in module sympy.utilities.runtests)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_add">gf_add() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_add_ground">gf_add_ground() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_add_mul">gf_add_mul() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_berlekamp">gf_berlekamp() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_cofactors">gf_cofactors() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_compose">gf_compose() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_compose_mod">gf_compose_mod() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_crt1">gf_crt1() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_crt2">gf_crt2() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_csolve">gf_csolve() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_exquo">gf_exquo() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_factor_sqf">gf_factor_sqf() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_from_dict">gf_from_dict() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_gcd">gf_gcd() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_gcdex">gf_gcdex() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_irreducible_p">gf_irreducible_p() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_LC">gf_LC() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_lshift">gf_lshift() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_monic">gf_monic() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_mul_ground">gf_mul_ground() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_multi_eval">gf_multi_eval() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_neg">gf_neg() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_pow_mod">gf_pow_mod() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_Qbasis">gf_Qbasis() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_Qmatrix">gf_Qmatrix() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_quo">gf_quo() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_quo_ground">gf_quo_ground() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_random">gf_random() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_shoup">gf_shoup() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_sqf_list">gf_sqf_list() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_sqf_p">gf_sqf_p() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_sqf_part">gf_sqf_part() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_sqr">gf_sqr() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_strip">gf_strip() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_sub_ground">gf_sub_ground() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_sub_mul">gf_sub_mul() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.galoistools.gf_trace_map">gf_trace_map() (in module sympy.polys.galoistools)</a>
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.gff">gff() (in module sympy.polys.polytools)</a>
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-  <dt><a href="modules/concrete.html#sympy.concrete.gosper.gosper_normal">gosper_normal() (in module sympy.concrete.gosper)</a>
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-  <dt><a href="modules/concrete.html#sympy.concrete.gosper.gosper_sum">gosper_sum() (in module sympy.concrete.gosper)</a>
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-  <dt><a href="modules/concrete.html#sympy.concrete.gosper.gosper_term">gosper_term() (in module sympy.concrete.gosper)</a>
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-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.BeamParameter.gouy">gouy (sympy.physics.gaussopt.BeamParameter attribute)</a>
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-  <dt><a href="modules/galgebra/GA/GAsympy.html#grad">grad() (built-in function)</a>
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-  <dt><a href="modules/tensor/indexed.html#sympy.tensor.indexed.IndexedBase">IndexedBase (class in sympy.tensor.indexed)</a>
-  </dt>
-
-      
-  <dt><a href="modules/tensor/indexed.html#sympy.tensor.indexed.Indexed.indices">indices (sympy.tensor.indexed.Indexed attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.tensor_functions.KroneckerDelta.indices_contain_equal_information">indices_contain_equal_information (sympy.functions.special.tensor_functions.KroneckerDelta attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.KroneckerDelta.indices_contain_equal_information">(sympy.physics.secondquant.KroneckerDelta attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.functions.inertia">inertia() (in module sympy.physics.mechanics.functions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.functions.inertia_of_point_mass">inertia_of_point_mass() (in module sympy.physics.mechanics.functions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/sets.html#sympy.core.sets.Set.inf">inf (sympy.core.sets.Set attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/innerproduct.html#sympy.physics.quantum.innerproduct.InnerProduct">InnerProduct (class in sympy.physics.quantum.innerproduct)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.InnerProduct">(class in sympy.physics.secondquant)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.RegularPolygon.inradius">inradius (sympy.geometry.polygon.RegularPolygon attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Triangle.inradius">(sympy.geometry.polygon.Triangle attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/core.html#sympy.core.numbers.Integer">Integer (class in sympy.core.numbers)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.power.integer_nthroot">integer_nthroot() (in module sympy.core.power)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/partitions.html#sympy.combinatorics.partitions.IntegerPartition">IntegerPartition (class in sympy.combinatorics.partitions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/partitions.html#sympy.combinatorics.partitions.IntegerPartition.as_dict">IntegerPartition.as_dict() (in module sympy.combinatorics.partitions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/partitions.html#sympy.combinatorics.partitions.IntegerPartition.as_ferrers">IntegerPartition.as_ferrers() (in module sympy.combinatorics.partitions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/partitions.html#sympy.combinatorics.partitions.IntegerPartition.next_lex">IntegerPartition.next_lex() (in module sympy.combinatorics.partitions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/partitions.html#sympy.combinatorics.partitions.IntegerPartition.prev_lex">IntegerPartition.prev_lex() (in module sympy.combinatorics.partitions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.IntegerRing">IntegerRing (class in sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.IntegerRing.algebraic_field">IntegerRing.algebraic_field() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.IntegerRing.from_AlgebraicField">IntegerRing.from_AlgebraicField() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.IntegerRing.get_field">IntegerRing.get_field() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.IntegerRing.log">IntegerRing.log() (in module sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/sets.html#sympy.sets.fancysets.Integers">Integers (class in sympy.sets.fancysets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.Integral">Integral (class in sympy.integrals)</a>
-  </dt>
-
-      
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.Integral.as_dummy">Integral.as_dummy() (in module sympy.integrals)</a>
-  </dt>
-
-      
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.Integral.as_sum">Integral.as_sum() (in module sympy.integrals)</a>
-  </dt>
-
-      
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.Integral.doit">Integral.doit() (in module sympy.integrals)</a>
-  </dt>
-
-      
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.transforms.Integral.is_commutative">Integral.is_commutative (in module sympy.integrals.transforms)</a>
-  </dt>
-
-      
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.Integral.transform">Integral.transform() (in module sympy.integrals)</a>
-  </dt>
-
-      
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.integrate">integrate() (sympy.integrals method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/represent.html#sympy.physics.quantum.represent.integrate_result">integrate_result() (in module sympy.physics.quantum.represent)</a>
-  </dt>
-
-      
-  <dt><a href="tutorial/tutorial.en.html#index-5">integration</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.interactive_traversal">interactive_traversal() (in module sympy.utilities.iterables)</a>
-  </dt>
-
-      
-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.RegularPolygon.interior_angle">interior_angle (sympy.geometry.polygon.RegularPolygon attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polyfuncs.interpolate">interpolate() (in module sympy.polys.polyfuncs)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.specialpolys.interpolating_poly">interpolating_poly() (in module sympy.polys.specialpolys)</a>
-  </dt>
-
-      
-  <dt><a href="modules/sets.html#sympy.core.sets.Intersection">Intersection (class in sympy.core.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/geometry.html#sympy.geometry.util.intersection">intersection() (in module sympy.geometry.util)</a>
-  </dt>
-
-      
-  <dt><a href="modules/sets.html#sympy.core.sets.Intersection.as_relational">Intersection.as_relational() (in module sympy.core.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/sets.html#sympy.core.sets.Intersection.reduce">Intersection.reduce() (in module sympy.core.sets)</a>
-  </dt>
-
-  </dl></td>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/sets.html#sympy.core.sets.Interval">Interval (class in sympy.core.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/sets.html#sympy.core.sets.Interval.as_relational">Interval.as_relational() (in module sympy.core.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.intervals">intervals() (in module sympy.polys.polytools)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/qubit.html#sympy.physics.quantum.qubit.IntQubit">IntQubit (class in sympy.physics.quantum.qubit)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/qubit.html#sympy.physics.quantum.qubit.IntQubitBra">IntQubitBra (class in sympy.physics.quantum.qubit)</a>
-  </dt>
-
-      
-  <dt><a href="modules/matrices/expressions.html#sympy.matrices.expressions.Inverse">Inverse (class in sympy.matrices.expressions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.transforms.inverse_cosine_transform">inverse_cosine_transform() (in module sympy.integrals.transforms)</a>
-  </dt>
-
-      
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.transforms.inverse_fourier_transform">inverse_fourier_transform() (in module sympy.integrals.transforms)</a>
-  </dt>
-
-      
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.transforms.inverse_hankel_transform">inverse_hankel_transform() (in module sympy.integrals.transforms)</a>
-  </dt>
-
-      
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.transforms.inverse_laplace_transform">inverse_laplace_transform() (in module sympy.integrals.transforms)</a>
-  </dt>
-
-      
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.transforms.inverse_mellin_transform">inverse_mellin_transform() (in module sympy.integrals.transforms)</a>
-  </dt>
-
-      
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.transforms.inverse_sine_transform">inverse_sine_transform() (in module sympy.integrals.transforms)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.invert">invert() (in module sympy.polys.polytools)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/qft.html#sympy.physics.quantum.qft.IQFT">IQFT (class in sympy.physics.quantum.qft)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/qft.html#sympy.physics.quantum.qft.IQFT.decompose">IQFT.decompose() (in module sympy.physics.quantum.qft)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.is_abelian">is_abelian (sympy.combinatorics.perm_groups.PermutationGroup attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.tensor_functions.KroneckerDelta.is_above_fermi">is_above_fermi (sympy.functions.special.tensor_functions.KroneckerDelta attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.KroneckerDelta.is_above_fermi">(sympy.physics.secondquant.KroneckerDelta attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.numberfields.AlgebraicNumber.is_aliased">is_aliased (sympy.polys.numberfields.AlgebraicNumber attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.tensor_functions.KroneckerDelta.is_below_fermi">is_below_fermi (sympy.functions.special.tensor_functions.KroneckerDelta attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.KroneckerDelta.is_below_fermi">(sympy.physics.secondquant.KroneckerDelta attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/core.html#sympy.core.function.Function.is_commutative">is_commutative (sympy.core.function.Function attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/quantum/state.html#sympy.physics.quantum.state.Wavefunction.is_commutative">(sympy.physics.quantum.state.Wavefunction attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_cyclotomic">is_cyclotomic (sympy.polys.polyclasses.DMP attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.is_cyclotomic">(sympy.polys.polytools.Poly attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.is_Empty">is_Empty (sympy.combinatorics.permutations.Permutation attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.is_even">is_even (sympy.combinatorics.permutations.Permutation attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.groebnertools.is_groebner">is_groebner() (in module sympy.polys.groebnertools)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.ANP.is_ground">is_ground (sympy.polys.polyclasses.ANP attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_ground">(sympy.polys.polyclasses.DMP attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.is_ground">(sympy.polys.polytools.Poly attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_homogeneous">is_homogeneous (sympy.polys.polyclasses.DMP attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.is_homogeneous">(sympy.polys.polytools.Poly attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.is_Identity">is_Identity (sympy.combinatorics.permutations.Permutation attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.function.Lambda.is_identity">is_identity (sympy.core.function.Lambda attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_irreducible">is_irreducible (sympy.polys.polyclasses.DMP attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.is_irreducible">(sympy.polys.polytools.Poly attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/sets.html#sympy.core.sets.Interval.is_left_unbounded">is_left_unbounded (sympy.core.sets.Interval attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_linear">is_linear (sympy.polys.polyclasses.DMP attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.is_linear">(sympy.polys.polytools.Poly attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_lower">is_lower (sympy.matrices.matrices.MatrixBase attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_lower_hessenberg">is_lower_hessenberg (sympy.matrices.matrices.MatrixBase attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.groebnertools.is_minimal">is_minimal() (in module sympy.polys.groebnertools)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_monic">is_monic (sympy.polys.polyclasses.DMP attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.is_monic">(sympy.polys.polytools.Poly attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_monomial">is_monomial (sympy.polys.polyclasses.DMP attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.is_monomial">(sympy.polys.polytools.Poly attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.is_multivariate">is_multivariate (sympy.polys.polytools.Poly attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.is_nilpotent">is_nilpotent (sympy.combinatorics.perm_groups.PermutationGroup attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/state.html#sympy.physics.quantum.state.Wavefunction.is_normalized">is_normalized (sympy.physics.quantum.state.Wavefunction attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/concrete.html#sympy.concrete.products.Product.is_number">is_number (sympy.concrete.products.Product attribute)</a>
-  </dt>
-
-      <dd><dl>
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-  <dt><a href="modules/concrete.html#sympy.concrete.summations.Sum.is_number">(sympy.concrete.summations.Sum attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/core.html#sympy.core.basic.Basic.is_number">(sympy.core.basic.Basic attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/core.html#sympy.core.expr.Expr.is_number">(sympy.core.expr.Expr attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.Integral.is_number">(sympy.integrals.Integral attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.is_odd">is_odd (sympy.combinatorics.permutations.Permutation attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.ANP.is_one">is_one (sympy.polys.polyclasses.ANP attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.DMF.is_one">(sympy.polys.polyclasses.DMF attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_one">(sympy.polys.polyclasses.DMP attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.is_one">(sympy.polys.polytools.Poly attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.tensor_functions.KroneckerDelta.is_only_above_fermi">is_only_above_fermi (sympy.functions.special.tensor_functions.KroneckerDelta attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.KroneckerDelta.is_only_above_fermi">(sympy.physics.secondquant.KroneckerDelta attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.tensor_functions.KroneckerDelta.is_only_below_fermi">is_only_below_fermi (sympy.functions.special.tensor_functions.KroneckerDelta attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.KroneckerDelta.is_only_below_fermi">(sympy.physics.secondquant.KroneckerDelta attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.AnnihilateFermion.is_only_q_annihilator">is_only_q_annihilator (sympy.physics.secondquant.AnnihilateFermion attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.CreateFermion.is_only_q_annihilator">(sympy.physics.secondquant.CreateFermion attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.AnnihilateFermion.is_only_q_creator">is_only_q_creator (sympy.physics.secondquant.AnnihilateFermion attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.CreateFermion.is_only_q_creator">(sympy.physics.secondquant.CreateFermion attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_primitive">is_primitive (sympy.polys.polyclasses.DMP attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.is_primitive">(sympy.polys.polytools.Poly attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/ntheory.html#sympy.ntheory.residue_ntheory.is_primitive_root">is_primitive_root() (in module sympy.ntheory.residue_ntheory)</a>
-  </dt>
-
-      
-  <dt><a href="modules/galgebra/GA/GAsympy.html#is_pure">is_pure() (built-in function)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.AnnihilateFermion.is_q_annihilator">is_q_annihilator (sympy.physics.secondquant.AnnihilateFermion attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.CreateFermion.is_q_annihilator">(sympy.physics.secondquant.CreateFermion attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.AnnihilateFermion.is_q_creator">is_q_creator (sympy.physics.secondquant.AnnihilateFermion attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.CreateFermion.is_q_creator">(sympy.physics.secondquant.CreateFermion attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/ntheory.html#sympy.ntheory.residue_ntheory.is_quad_residue">is_quad_residue() (in module sympy.ntheory.residue_ntheory)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_quadratic">is_quadratic (sympy.polys.polyclasses.DMP attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.is_quadratic">(sympy.polys.polytools.Poly attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/galgebra/GA.html#sympy.galgebra.GA.is_quasi_unit_numpy_array">is_quasi_unit_numpy_array() (in module sympy.galgebra.GA)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.basic.Basic.is_Real">is_Real (sympy.core.basic.Basic attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.groebnertools.is_reduced">is_reduced() (in module sympy.polys.groebnertools)</a>
-  </dt>
-
-      
-  <dt><a href="modules/sets.html#sympy.core.sets.Interval.is_right_unbounded">is_right_unbounded (sympy.core.sets.Interval attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.compatibility.is_sequence">is_sequence() (in module sympy.core.compatibility)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.is_Singleton">is_Singleton (sympy.combinatorics.permutations.Permutation attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.is_solvable">is_solvable (sympy.combinatorics.perm_groups.PermutationGroup attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_sqf">is_sqf (sympy.polys.polyclasses.DMP attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.is_sqf">(sympy.polys.polytools.Poly attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_square">is_square (sympy.matrices.matrices.MatrixBase attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.is_trivial">is_trivial (sympy.combinatorics.perm_groups.PermutationGroup attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.is_univariate">is_univariate (sympy.polys.polytools.Poly attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_upper">is_upper (sympy.matrices.matrices.MatrixBase attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_upper_hessenberg">is_upper_hessenberg (sympy.matrices.matrices.MatrixBase attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/concrete.html#sympy.concrete.products.Product.is_zero">is_zero (sympy.concrete.products.Product attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/concrete.html#sympy.concrete.summations.Sum.is_zero">(sympy.concrete.summations.Sum attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.Integral.is_zero">(sympy.integrals.Integral attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_zero">(sympy.matrices.matrices.MatrixBase attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.ANP.is_zero">(sympy.polys.polyclasses.ANP attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.DMF.is_zero">(sympy.polys.polyclasses.DMF attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyclasses.DMP.is_zero">(sympy.polys.polyclasses.DMP attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.is_zero">(sympy.polys.polytools.Poly attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.GroebnerBasis.is_zero_dimensional">is_zero_dimensional (sympy.polys.polytools.GroebnerBasis attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.is_zero_dimensional">is_zero_dimensional() (in module sympy.polys.polytools)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/runtests.html#sympy.utilities.runtests.isgeneratorfunction">isgeneratorfunction() (in module sympy.utilities.runtests)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/general.html#mpmath.isinf">isinf() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/general.html#mpmath.isint">isint() (in module mpmath)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/galgebra/GA.html#sympy.galgebra.GA.isint">(in module sympy.galgebra.GA)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/mpmath/general.html#mpmath.isnan">isnan() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/general.html#mpmath.isnormal">isnormal() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.numberfields.isolate">isolate() (in module sympy.polys.numberfields)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyerrors.IsomorphismFailed">IsomorphismFailed (class in sympy.polys.polyerrors)</a>
-  </dt>
-
-      
-  <dt><a href="modules/ntheory.html#sympy.ntheory.primetest.isprime">isprime() (in module sympy.ntheory.primetest)</a>
-  </dt>
-
-      
-  <dt><a href="modules/galgebra/GA.html#sympy.galgebra.GA.israt">israt() (in module sympy.galgebra.GA)</a>
-  </dt>
-
-      
-  <dt><a href="modules/logic.html#sympy.logic.boolalg.ITE">ITE (class in sympy.logic.boolalg)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.compatibility.iterable">iterable() (in module sympy.core.compatibility)</a>
-  </dt>
-
-  </dl></td>
-</tr></table>
-
-<h2 id="J">J</h2>
-<table style="width: 100%" class="indextable genindextable"><tr>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.j0">j0() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.j1">j1() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.polynomials.jacobi">jacobi (class in sympy.functions.special.polynomials)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/orthogonal.html#mpmath.jacobi">jacobi() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.orthopolys.jacobi_poly">jacobi_poly() (in module sympy.polys.orthopolys)</a>
-  </dt>
-
-      
-  <dt><a href="modules/ntheory.html#sympy.ntheory.residue_ntheory.jacobi_symbol">jacobi_symbol() (in module sympy.ntheory.residue_ntheory)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.bessel.jn">jn (class in sympy.functions.special.bessel)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.bessel.jn_zeros">jn_zeros() (in module sympy.functions.special.bessel)</a>
-  </dt>
-
-      
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.dense.jordan_cell">jordan_cell() (in module sympy.matrices.dense)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.josephus">josephus() (sympy.combinatorics.permutations.Permutation class method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/elliptic.html#mpmath.jtheta">jtheta() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JxBra">JxBra (class in sympy.physics.quantum.spin)</a>
-  </dt>
-
-  </dl></td>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JxBraCoupled">JxBraCoupled (class in sympy.physics.quantum.spin)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JxKet">JxKet (class in sympy.physics.quantum.spin)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JxKetCoupled">JxKetCoupled (class in sympy.physics.quantum.spin)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JyBra">JyBra (class in sympy.physics.quantum.spin)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JyBraCoupled">JyBraCoupled (class in sympy.physics.quantum.spin)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JyKet">JyKet (class in sympy.physics.quantum.spin)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JyKetCoupled">JyKetCoupled (class in sympy.physics.quantum.spin)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JzBra">JzBra (class in sympy.physics.quantum.spin)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JzBraCoupled">JzBraCoupled (class in sympy.physics.quantum.spin)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JzKet">JzKet (class in sympy.physics.quantum.spin)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/spin.html#sympy.physics.quantum.spin.JzKetCoupled">JzKetCoupled (class in sympy.physics.quantum.spin)</a>
-  </dt>
-
-  </dl></td>
-</tr></table>
-
-<h2 id="K">K</h2>
-<table style="width: 100%" class="indextable genindextable"><tr>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.kane.KanesMethod">KanesMethod (class in sympy.physics.mechanics.kane)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.kane.KanesMethod.kanes_equations">KanesMethod.kanes_equations() (in module sympy.physics.mechanics.kane)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.kane.KanesMethod.kindiffdict">KanesMethod.kindiffdict() (in module sympy.physics.mechanics.kane)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.kane.KanesMethod.linearize">KanesMethod.linearize() (in module sympy.physics.mechanics.kane)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.kane.KanesMethod.rhs">KanesMethod.rhs() (in module sympy.physics.mechanics.kane)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.kbins">kbins() (in module sympy.utilities.iterables)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.kei">kei() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.ker">ker() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/state.html#sympy.physics.quantum.state.Ket">Ket (class in sympy.physics.quantum.state)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/operator.html#sympy.physics.quantum.operator.OuterProduct.ket">ket (sympy.physics.quantum.operator.OuterProduct attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.InnerProduct.ket">(sympy.physics.secondquant.InnerProduct attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/physics/quantum/state.html#sympy.physics.quantum.state.KetBase">KetBase (class in sympy.physics.quantum.state)</a>
-  </dt>
-
-  </dl></td>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/mpmath/functions/elliptic.html#mpmath.kfrom">kfrom() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/constants.html#mpmath.mp.khinchin">khinchin (mpmath.mp attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.tensor_functions.KroneckerDelta.killable_index">killable_index (sympy.functions.special.tensor_functions.KroneckerDelta attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.KroneckerDelta.killable_index">(sympy.physics.secondquant.KroneckerDelta attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/physics/mechanics/api/kinematics.html#sympy.physics.mechanics.functions.kinematic_equations">kinematic_equations() (in module sympy.physics.mechanics.functions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.functions.kinetic_energy">kinetic_energy() (in module sympy.physics.mechanics.functions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/elliptic.html#mpmath.kleinj">kleinj() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.ccode.known_functions">known_functions (in module sympy.printing.ccode)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.tensor_functions.KroneckerDelta">KroneckerDelta (class in sympy.functions.special.tensor_functions)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.KroneckerDelta">(class in sympy.physics.secondquant)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/combinatorics/subsets.html#sympy.combinatorics.subsets.ksubsets">ksubsets() (sympy.combinatorics.subsets method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/stats.html#sympy.stats.Kumaraswamy">Kumaraswamy() (in module sympy.stats)</a>
-  </dt>
-
-  </dl></td>
-</tr></table>
-
-<h2 id="L">L</h2>
-<table style="width: 100%" class="indextable genindextable"><tr>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/physics/quantum/hilbert.html#sympy.physics.quantum.hilbert.L2">L2 (class in sympy.physics.quantum.hilbert)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/state.html#sympy.physics.quantum.state.TimeDepState.label">label (sympy.physics.quantum.state.TimeDepState attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/tensor/indexed.html#sympy.tensor.indexed.Idx.label">(sympy.tensor.indexed.Idx attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/tensor/indexed.html#sympy.tensor.indexed.IndexedBase.label">(sympy.tensor.indexed.IndexedBase attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.lagrange.LagrangesMethod">LagrangesMethod (class in sympy.physics.mechanics.lagrange)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.lagrange.LagrangesMethod.form_lagranges_equations">LagrangesMethod.form_lagranges_equations() (in module sympy.physics.mechanics.lagrange)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.lagrange.LagrangesMethod.rhs">LagrangesMethod.rhs() (in module sympy.physics.mechanics.lagrange)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.functions.Lagrangian">Lagrangian() (in module sympy.physics.mechanics.functions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.polynomials.laguerre">laguerre (class in sympy.functions.special.polynomials)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/orthogonal.html#mpmath.laguerre">laguerre() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.orthopolys.laguerre_poly">laguerre_poly() (in module sympy.polys.orthopolys)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.function.Lambda">Lambda (class in sympy.core.function)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.lambdarepr.LambdaPrinter">LambdaPrinter (class in sympy.printing.lambdarepr)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.lambdarepr.lambdarepr">lambdarepr() (in module sympy.printing.lambdarepr)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/lambdify.html#sympy.utilities.lambdify.lambdastr">lambdastr() (in module sympy.utilities.lambdify)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/lambdify.html#sympy.utilities.lambdify.lambdify">lambdify() (in module sympy.utilities.lambdify)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.exponential.LambertW">LambertW (class in sympy.functions.elementary.exponential)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/powers.html#mpmath.lambertw">lambertw() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.exponential.LambertW.fdiff">LambertW.fdiff() (in module sympy.functions.elementary.exponential)</a>
-  </dt>
-
-      
-  <dt><a href="modules/stats.html#sympy.stats.Laplace">Laplace() (in module sympy.stats)</a>
-  </dt>
-
-      
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.transforms.laplace_transform">laplace_transform() (in module sympy.integrals.transforms)</a>
-  </dt>
-
-      
-  <dt><a href="modules/galgebra/latex_ex/latex_ex.html#LaTeX">LaTeX() (built-in function)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/galgebra/latex_ex.html#sympy.galgebra.latex_ex.LaTeX">(in module sympy.galgebra.latex_ex)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/printing.html#sympy.printing.latex.latex">latex() (in module sympy.printing.latex)</a>
-  </dt>
-
-      
-  <dt><a href="modules/galgebra/GA.html#sympy.galgebra.GA.LaTeX_lst">LaTeX_lst() (in module sympy.galgebra.GA)</a>
-  </dt>
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-      
-  <dt><a href="modules/galgebra/latex_ex.html#sympy.galgebra.latex_ex.LatexPrinter">LatexPrinter (class in sympy.galgebra.latex_ex)</a>
-  </dt>
-
-      <dd><dl>
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-  <dt><a href="modules/printing.html#sympy.printing.latex.LatexPrinter">(class in sympy.printing.latex)</a>
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-
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-      
-  <dt><a href="modules/galgebra/latex_ex.html#sympy.galgebra.latex_ex.LatexPrinter.latex_bases">LatexPrinter.latex_bases() (in module sympy.galgebra.latex_ex)</a>
-  </dt>
-
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.LC">LC() (in module sympy.polys.polytools)</a>
-  </dt>
-
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.lcm">lcm() (in module sympy.polys.polytools)</a>
-  </dt>
-
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.lcm_list">lcm_list() (in module sympy.polys.polytools)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/general.html#mpmath.ldexp">ldexp() (in module mpmath)</a>
-  </dt>
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-  <dt><a href="modules/core.html#sympy.core.relational.Le">Le (class in sympy.core.relational)</a>
-  </dt>
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-  <dt><a href="modules/sets.html#sympy.core.sets.Interval.left">left (sympy.core.sets.Interval attribute)</a>
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-  <dt><a href="modules/sets.html#sympy.core.sets.Interval.left_open">left_open (sympy.core.sets.Interval attribute)</a>
-  </dt>
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-  <dt><a href="modules/functions/special.html#sympy.functions.special.polynomials.legendre">legendre (class in sympy.functions.special.polynomials)</a>
-  </dt>
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-  <dt><a href="modules/mpmath/functions/orthogonal.html#mpmath.legendre">legendre() (in module mpmath)</a>
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-  <dt><a href="modules/polys/reference.html#sympy.polys.orthopolys.legendre_poly">legendre_poly() (in module sympy.polys.orthopolys)</a>
-  </dt>
-
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-  <dt><a href="modules/ntheory.html#sympy.ntheory.residue_ntheory.legendre_symbol">legendre_symbol() (in module sympy.ntheory.residue_ntheory)</a>
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-  <dt><a href="modules/mpmath/functions/orthogonal.html#mpmath.legenp">legenp() (in module mpmath)</a>
-  </dt>
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-  <dt><a href="modules/mpmath/functions/orthogonal.html#mpmath.legenq">legenq() (in module mpmath)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.LinearEntity.length">length (sympy.geometry.line.LinearEntity attribute)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.Segment.length">(sympy.geometry.line.Segment attribute)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.point.Point.length">(sympy.geometry.point.Point attribute)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.RegularPolygon.length">(sympy.geometry.polygon.RegularPolygon attribute)</a>
-  </dt>
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-  <dt><a href="modules/functions/special.html#sympy.functions.special.zeta_functions.lerchphi">lerchphi (class in sympy.functions.special.zeta_functions)</a>
-  </dt>
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-  <dt><a href="modules/mpmath/functions/zeta.html#mpmath.lerchphi">lerchphi() (in module mpmath)</a>
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-  <dt><a href="modules/core.html#sympy.core.relational.LessThan">LessThan (class in sympy.core.relational)</a>
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-  <dt><a href="modules/functions/special.html#sympy.functions.special.tensor_functions.LeviCivita">LeviCivita (class in sympy.functions.special.tensor_functions)</a>
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-  <dt><a href="modules/polys/reference.html#sympy.polys.monomialtools.LexOrder">LexOrder (class in sympy.polys.monomialtools)</a>
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-  <dt><a href="modules/mpmath/functions/expintegrals.html#mpmath.li">li() (in module mpmath)</a>
-  </dt>
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-  <dt><a href="modules/series.html#sympy.series.limits.Limit">Limit (class in sympy.series.limits)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/calculus/sums_limits.html#mpmath.limit">limit() (in module mpmath)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/series.html#sympy.series.limits.limit">(in module sympy.series.limits)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/series.html#sympy.series.limits.Limit.doit">Limit.doit() (in module sympy.series.limits)</a>
-  </dt>
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-  <dt><a href="modules/series.html#sympy.series.gruntz.limitinf">limitinf() (in module sympy.series.gruntz)</a>
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-
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-  <dt><a href="tutorial/tutorial.en.html#index-1">limits</a>
-  </dt>
-
-      <dd><dl>
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-  <dt><a href="modules/geometry.html#sympy.geometry.curve.Curve.limits">(sympy.geometry.curve.Curve attribute)</a>
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-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.Integral.limits">(sympy.integrals.Integral attribute)</a>
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-  <dt><a href="modules/physics/quantum/state.html#sympy.physics.quantum.state.Wavefunction.limits">(sympy.physics.quantum.state.Wavefunction attribute)</a>
-  </dt>
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-      
-  <dt><a href="modules/geometry.html#sympy.geometry.line.Line">Line (class in sympy.geometry.line)</a>
-  </dt>
-
-  </dl></td>
-  <td style="width: 33%" valign="top"><dl>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.Line.arbitrary_point">Line.arbitrary_point() (in module sympy.geometry.line)</a>
-  </dt>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.Line.contains">Line.contains() (in module sympy.geometry.line)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.Line.equation">Line.equation() (in module sympy.geometry.line)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.Line.plot_interval">Line.plot_interval() (in module sympy.geometry.line)</a>
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-  <dt><a href="modules/plotting.html#sympy.plotting.plot.Line2DBaseSeries">Line2DBaseSeries (class in sympy.plotting.plot)</a>
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-  <dt><a href="modules/plotting.html#sympy.plotting.plot.Line3DBaseSeries">Line3DBaseSeries (class in sympy.plotting.plot)</a>
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-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.line_integrate">line_integrate() (sympy.integrals method)</a>
-  </dt>
-
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-  <dt><a href="tutorial/tutorial.en.html#index-9">linear algebra</a>
-  </dt>
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-  <dt><a href="modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.functions.linear_momentum">linear_momentum() (in module sympy.physics.mechanics.functions)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.LinearEntity">LinearEntity (class in sympy.geometry.line)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.LinearEntity.angle_between">LinearEntity.angle_between() (in module sympy.geometry.line)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.LinearEntity.contains">LinearEntity.contains() (in module sympy.geometry.line)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.LinearEntity.intersection">LinearEntity.intersection() (in module sympy.geometry.line)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.LinearEntity.is_concurrent">LinearEntity.is_concurrent() (in module sympy.geometry.line)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.LinearEntity.is_parallel">LinearEntity.is_parallel() (in module sympy.geometry.line)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.LinearEntity.is_perpendicular">LinearEntity.is_perpendicular() (in module sympy.geometry.line)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.LinearEntity.is_similar">LinearEntity.is_similar() (in module sympy.geometry.line)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.LinearEntity.parallel_line">LinearEntity.parallel_line() (in module sympy.geometry.line)</a>
-  </dt>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.LinearEntity.perpendicular_line">LinearEntity.perpendicular_line() (in module sympy.geometry.line)</a>
-  </dt>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.LinearEntity.perpendicular_segment">LinearEntity.perpendicular_segment() (in module sympy.geometry.line)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.LinearEntity.projection">LinearEntity.projection() (in module sympy.geometry.line)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.LinearEntity.random_point">LinearEntity.random_point() (in module sympy.geometry.line)</a>
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-  <dt><a href="modules/plotting.html#sympy.plotting.plot.LineOver1DRangeSeries">LineOver1DRangeSeries (class in sympy.plotting.plot)</a>
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-  <dt><a href="modules/plotting.html#sympy.plotting.plot.LineOver1DRangeSeries.get_segments">LineOver1DRangeSeries.get_segments() (in module sympy.plotting.plot)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/general.html#mpmath.linspace">linspace() (in module mpmath)</a>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.dense.list2numpy">list2numpy() (in module sympy.matrices.dense)</a>
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.LM">LM() (in module sympy.polys.polytools)</a>
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-
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-  <dt><a href="modules/mpmath/functions/powers.html#mpmath.ln">ln() (in module mpmath)</a>
-  </dt>
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-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.exponential.log">log (class in sympy.functions.elementary.exponential)</a>
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-  <dt><a href="modules/mpmath/functions/powers.html#mpmath.log">log() (in module mpmath)</a>
-  </dt>
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-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.exponential.log.as_base_exp">log.as_base_exp() (in module sympy.functions.elementary.exponential)</a>
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-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.exponential.log.as_real_imag">log.as_real_imag() (in module sympy.functions.elementary.exponential)</a>
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-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.exponential.log.fdiff">log.fdiff() (in module sympy.functions.elementary.exponential)</a>
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-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.exponential.log.inverse">log.inverse() (in module sympy.functions.elementary.exponential)</a>
-  </dt>
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-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.exponential.log.taylor_term">log.taylor_term() (in module sympy.functions.elementary.exponential)</a>
-  </dt>
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-  <dt><a href="modules/mpmath/functions/powers.html#mpmath.log10">log10() (in module mpmath)</a>
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-  <dt><a href="modules/simplify/simplify.html#sympy.simplify.simplify.logcombine">logcombine() (in module sympy.simplify.simplify)</a>
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-  <dt><a href="modules/functions/special.html#sympy.functions.special.gamma_functions.loggamma">loggamma (class in sympy.functions.special.gamma_functions)</a>
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-  <dt><a href="modules/mpmath/functions/gamma.html#mpmath.loggamma">loggamma() (in module mpmath)</a>
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-  <dt><a href="modules/stats.html#sympy.stats.Logistic">Logistic() (in module sympy.stats)</a>
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-  <dt><a href="modules/mpmath/matrices.html#mpmath.logm">logm() (in module mpmath)</a>
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-  <dt><a href="modules/stats.html#sympy.stats.LogNormal">LogNormal() (in module sympy.stats)</a>
-  </dt>
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-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.lommels1">lommels1() (in module mpmath)</a>
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-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.lommels2">lommels2() (in module mpmath)</a>
-  </dt>
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-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.AntiSymmetricTensor.lower">lower (sympy.physics.secondquant.AntiSymmetricTensor attribute)</a>
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-  <dt><a href="modules/tensor/indexed.html#sympy.tensor.indexed.Idx.lower">(sympy.tensor.indexed.Idx attribute)</a>
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-
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-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.gamma_functions.lowergamma">lowergamma (class in sympy.functions.special.gamma_functions)</a>
-  </dt>
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-  <dt><a href="modules/core.html#sympy.core.relational.Lt">Lt (class in sympy.core.relational)</a>
-  </dt>
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.LT">LT() (in module sympy.polys.polytools)</a>
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-  <dt><a href="modules/functions/combinatorial.html#sympy.functions.combinatorial.numbers.lucas">lucas (class in sympy.functions.combinatorial.numbers)</a>
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-
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-</tr></table>
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-<h2 id="M">M</h2>
-<table style="width: 100%" class="indextable genindextable"><tr>
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-  <dt><a href="modules/mpmath/general.html#mpmath.mag">mag() (in module mpmath)</a>
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-  <dt><a href="modules/galgebra/GA.html#sympy.galgebra.GA.magnitude">magnitude() (in module sympy.galgebra.GA)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Ellipse.major">major (sympy.geometry.ellipse.Ellipse attribute)</a>
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-  <dt><a href="modules/galgebra/GA.html#sympy.galgebra.GA.make_null_array">make_null_array() (in module sympy.galgebra.GA)</a>
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-  <dt><a href="modules/galgebra/GA.html#sympy.galgebra.GA.make_scalars">make_scalars() (in module sympy.galgebra.GA)</a>
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-  <dt><a href="modules/mpmath/functions/numtheory.html#mpmath.mangoldt">mangoldt() (in module mpmath)</a>
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-  <dt><a href="modules/diffgeom.html#sympy.diffgeom.Manifold">Manifold (class in sympy.diffgeom)</a>
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-  <dt><a href="modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.particle.Particle.mass">mass (sympy.physics.mechanics.particle.Particle attribute)</a>
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-  <dt><a href="modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.lagrange.LagrangesMethod.mass_matrix">mass_matrix (sympy.physics.mechanics.lagrange.LagrangesMethod attribute)</a>
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-  <dt><a href="modules/matrices/expressions.html#sympy.matrices.expressions.MatAdd">MatAdd (class in sympy.matrices.expressions)</a>
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-  <dt><a href="tutorial/tutorial.en.html#index-11">match</a>
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-  <dt><a href="modules/parsing.html#sympy.parsing.mathematica.mathematica">mathematica() (in module sympy.parsing.mathematica)</a>
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-  <dt><a href="modules/printing.html#sympy.printing.mathml.mathml">mathml() (in module sympy.printing.mathml)</a>
-  </dt>
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-  <dt><a href="modules/printing.html#sympy.printing.mathml.MathMLPrinter">MathMLPrinter (class in sympy.printing.mathml)</a>
-  </dt>
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-  <dt><a href="modules/printing.html#sympy.printing.mathml.MathMLPrinter.doprint">MathMLPrinter.doprint() (in module sympy.printing.mathml)</a>
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-  <dt><a href="modules/printing.html#sympy.printing.mathml.MathMLPrinter.mathml_tag">MathMLPrinter.mathml_tag() (in module sympy.printing.mathml)</a>
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-  <dt><a href="modules/matrices/expressions.html#sympy.matrices.expressions.MatMul">MatMul (class in sympy.matrices.expressions)</a>
-  </dt>
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-  <dt><a href="modules/matrices/expressions.html#sympy.matrices.expressions.MatPow">MatPow (class in sympy.matrices.expressions)</a>
-  </dt>
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-  <dt><a href="tutorial/tutorial.en.html#index-10">Matrix</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.dense.matrix2numpy">matrix2numpy() (in module sympy.matrices.dense)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.groebnertools.matrix_fglm">matrix_fglm() (in module sympy.polys.groebnertools)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.dense.matrix_multiply_elementwise">matrix_multiply_elementwise() (in module sympy.matrices.dense)</a>
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-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.matrix_rep">matrix_rep() (in module sympy.physics.secondquant)</a>
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-  <dt><a href="modules/physics/quantum/qubit.html#sympy.physics.quantum.qubit.matrix_to_density">matrix_to_density() (in module sympy.physics.quantum.qubit)</a>
-  </dt>
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-  <dt><a href="modules/physics/quantum/qubit.html#sympy.physics.quantum.qubit.matrix_to_qubit">matrix_to_qubit() (in module sympy.physics.quantum.qubit)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase">MatrixBase (class in sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.add">MatrixBase.add() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.adjoint">MatrixBase.adjoint() (in module sympy.matrices.matrices)</a>
-  </dt>
-
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.adjugate">MatrixBase.adjugate() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.berkowitz">MatrixBase.berkowitz() (in module sympy.matrices.matrices)</a>
-  </dt>
-
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.berkowitz_charpoly">MatrixBase.berkowitz_charpoly() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.berkowitz_det">MatrixBase.berkowitz_det() (in module sympy.matrices.matrices)</a>
-  </dt>
-
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.berkowitz_eigenvals">MatrixBase.berkowitz_eigenvals() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.berkowitz_minors">MatrixBase.berkowitz_minors() (in module sympy.matrices.matrices)</a>
-  </dt>
-
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.charpoly">MatrixBase.charpoly() (in module sympy.matrices.matrices)</a>
-  </dt>
-
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.cholesky">MatrixBase.cholesky() (in module sympy.matrices.matrices)</a>
-  </dt>
-
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.cholesky_solve">MatrixBase.cholesky_solve() (in module sympy.matrices.matrices)</a>
-  </dt>
-
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.cofactor">MatrixBase.cofactor() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.cofactorMatrix">MatrixBase.cofactorMatrix() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.col_insert">MatrixBase.col_insert() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.col_join">MatrixBase.col_join() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.condition_number">MatrixBase.condition_number() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.cross">MatrixBase.cross() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.det">MatrixBase.det() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.det_bareis">MatrixBase.det_bareis() (in module sympy.matrices.matrices)</a>
-  </dt>
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-      
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.det_LU_decomposition">MatrixBase.det_LU_decomposition() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.diagonal_solve">MatrixBase.diagonal_solve() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.diagonalize">MatrixBase.diagonalize() (in module sympy.matrices.matrices)</a>
-  </dt>
-
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.diff">MatrixBase.diff() (in module sympy.matrices.matrices)</a>
-  </dt>
-
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.dot">MatrixBase.dot() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.dual">MatrixBase.dual() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.eigenvals">MatrixBase.eigenvals() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.eigenvects">MatrixBase.eigenvects() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.evalf">MatrixBase.evalf() (in module sympy.matrices.matrices)</a>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.exp">MatrixBase.exp() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.expand">MatrixBase.expand() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.extract">MatrixBase.extract() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.get_diag_blocks">MatrixBase.get_diag_blocks() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.has">MatrixBase.has() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.integrate">MatrixBase.integrate() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.inverse_ADJ">MatrixBase.inverse_ADJ() (in module sympy.matrices.matrices)</a>
-  </dt>
-
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.inverse_GE">MatrixBase.inverse_GE() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.inverse_LU">MatrixBase.inverse_LU() (in module sympy.matrices.matrices)</a>
-  </dt>
-
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_anti_symmetric">MatrixBase.is_anti_symmetric() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_diagonal">MatrixBase.is_diagonal() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_diagonalizable">MatrixBase.is_diagonalizable() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_nilpotent">MatrixBase.is_nilpotent() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_symbolic">MatrixBase.is_symbolic() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.is_symmetric">MatrixBase.is_symmetric() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.jacobian">MatrixBase.jacobian() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.jordan_cells">MatrixBase.jordan_cells() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.jordan_form">MatrixBase.jordan_form() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.key2bounds">MatrixBase.key2bounds() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.key2ij">MatrixBase.key2ij() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.LDLdecomposition">MatrixBase.LDLdecomposition() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.LDLsolve">MatrixBase.LDLsolve() (in module sympy.matrices.matrices)</a>
-  </dt>
-
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.limit">MatrixBase.limit() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.lower_triangular_solve">MatrixBase.lower_triangular_solve() (in module sympy.matrices.matrices)</a>
-  </dt>
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-      
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.LUdecomposition">MatrixBase.LUdecomposition() (in module sympy.matrices.matrices)</a>
-  </dt>
-
-      
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.LUdecomposition_Simple">MatrixBase.LUdecomposition_Simple() (in module sympy.matrices.matrices)</a>
-  </dt>
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-      
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.LUdecompositionFF">MatrixBase.LUdecompositionFF() (in module sympy.matrices.matrices)</a>
-  </dt>
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-      
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.LUsolve">MatrixBase.LUsolve() (in module sympy.matrices.matrices)</a>
-  </dt>
-
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.minorEntry">MatrixBase.minorEntry() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.minorMatrix">MatrixBase.minorMatrix() (in module sympy.matrices.matrices)</a>
-  </dt>
-
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.multiply">MatrixBase.multiply() (in module sympy.matrices.matrices)</a>
-  </dt>
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-      
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.multiply_elementwise">MatrixBase.multiply_elementwise() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.n">MatrixBase.n() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.norm">MatrixBase.norm() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.normalized">MatrixBase.normalized() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.nullspace">MatrixBase.nullspace() (in module sympy.matrices.matrices)</a>
-  </dt>
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-      
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.permuteBkwd">MatrixBase.permuteBkwd() (in module sympy.matrices.matrices)</a>
-  </dt>
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-      
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.permuteFwd">MatrixBase.permuteFwd() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.print_nonzero">MatrixBase.print_nonzero() (in module sympy.matrices.matrices)</a>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.project">MatrixBase.project() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.QRdecomposition">MatrixBase.QRdecomposition() (in module sympy.matrices.matrices)</a>
-  </dt>
-
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.QRsolve">MatrixBase.QRsolve() (in module sympy.matrices.matrices)</a>
-  </dt>
-
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.row_insert">MatrixBase.row_insert() (in module sympy.matrices.matrices)</a>
-  </dt>
-
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.row_join">MatrixBase.row_join() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.rref">MatrixBase.rref() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.simplify">MatrixBase.simplify() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.singular_values">MatrixBase.singular_values() (in module sympy.matrices.matrices)</a>
-  </dt>
-
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.solve">MatrixBase.solve() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.solve_least_squares">MatrixBase.solve_least_squares() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.submatrix">MatrixBase.submatrix() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.subs">MatrixBase.subs() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.upper_triangular_solve">MatrixBase.upper_triangular_solve() (in module sympy.matrices.matrices)</a>
-  </dt>
-
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.values">MatrixBase.values() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.vec">MatrixBase.vec() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.vech">MatrixBase.vech() (in module sympy.matrices.matrices)</a>
-  </dt>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixError">MatrixError (class in sympy.matrices.matrices)</a>
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-  <dt><a href="modules/matrices/expressions.html#sympy.matrices.expressions.MatrixExpr">MatrixExpr (class in sympy.matrices.expressions)</a>
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-  <dt><a href="modules/matrices/expressions.html#sympy.matrices.expressions.MatrixExpr.as_explicit">MatrixExpr.as_explicit() (in module sympy.matrices.expressions)</a>
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-  <dt><a href="modules/matrices/expressions.html#sympy.matrices.expressions.MatrixExpr.as_mutable">MatrixExpr.as_mutable() (in module sympy.matrices.expressions)</a>
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-  <dt><a href="modules/matrices/expressions.html#sympy.matrices.expressions.MatrixExpr.equals">MatrixExpr.equals() (in module sympy.matrices.expressions)</a>
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-  <dt><a href="modules/matrices/expressions.html#sympy.matrices.expressions.MatrixSymbol">MatrixSymbol (class in sympy.matrices.expressions)</a>
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-  <td style="width: 33%" valign="top"><dl>
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-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.miscellaneous.Max">Max (class in sympy.functions.elementary.miscellaneous)</a>
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-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.max_div">max_div (sympy.combinatorics.perm_groups.PermutationGroup attribute)</a>
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-  <dt><a href="modules/mpmath/general.html#mpmath.maxcalls">maxcalls() (in module mpmath)</a>
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-  <dt><a href="modules/stats.html#sympy.stats.Maxwell">Maxwell() (in module sympy.stats)</a>
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-  <dt><a href="modules/parsing.html#sympy.parsing.sympy_tokenize.maybe">maybe() (in module sympy.parsing.sympy_tokenize)</a>
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-  <dt><a href="modules/mpmath/calculus/optimization.html#mpmath.calculus.optimization.MDNewton">MDNewton (class in mpmath.calculus.optimization)</a>
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-  <dt><a href="modules/sets.html#sympy.core.sets.Set.measure">measure (sympy.core.sets.Set attribute)</a>
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-  <dt><a href="modules/physics/quantum/qubit.html#sympy.physics.quantum.qubit.measure_all">measure_all() (in module sympy.physics.quantum.qubit)</a>
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-  <dt><a href="modules/physics/quantum/qubit.html#sympy.physics.quantum.qubit.measure_all_oneshot">measure_all_oneshot() (in module sympy.physics.quantum.qubit)</a>
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-  <dt><a href="modules/physics/quantum/qubit.html#sympy.physics.quantum.qubit.measure_partial">measure_partial() (in module sympy.physics.quantum.qubit)</a>
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-  <dt><a href="modules/physics/quantum/qubit.html#sympy.physics.quantum.qubit.measure_partial_oneshot">measure_partial_oneshot() (in module sympy.physics.quantum.qubit)</a>
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-  <dt><a href="modules/physics/mechanics/api/printing.html#sympy.physics.mechanics.functions.mechanics_printing">mechanics_printing() (in module sympy.physics.mechanics.functions)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Triangle.medial">medial (sympy.geometry.polygon.Triangle attribute)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Triangle.medians">medians (sympy.geometry.polygon.Triangle attribute)</a>
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-  <dt><a href="modules/functions/special.html#sympy.functions.special.hyper.meijerg">meijerg (class in sympy.functions.special.hyper)</a>
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-  <dt><a href="modules/mpmath/functions/hypergeometric.html#mpmath.meijerg">meijerg() (in module mpmath)</a>
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-  <dt><a href="modules/functions/special.html#sympy.functions.special.hyper.meijerg.get_period">meijerg.get_period() (in module sympy.functions.special.hyper)</a>
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-  <dt><a href="modules/functions/special.html#sympy.functions.special.hyper.meijerg.integrand">meijerg.integrand() (in module sympy.functions.special.hyper)</a>
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-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.transforms.mellin_transform">mellin_transform() (in module sympy.integrals.transforms)</a>
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-  <dt><a href="modules/mpmath/general.html#mpmath.memoize">memoize() (in module mpmath)</a>
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-  <dt><a href="modules/mpmath/functions/constants.html#mpmath.mp.mertens">mertens (mpmath.mp attribute)</a>
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-  <dt><a href="modules/mpmath/functions/elliptic.html#mpmath.mfrom">mfrom() (in module mpmath)</a>
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-  <dt><a href="modules/physics/matrices.html#sympy.physics.matrices.mgamma">mgamma() (in module sympy.physics.matrices)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.Segment.midpoint">midpoint (sympy.geometry.line.Segment attribute)</a>
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-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.miscellaneous.Min">Min (class in sympy.functions.elementary.miscellaneous)</a>
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-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CGate.min_qubits">min_qubits (sympy.physics.quantum.gate.CGate attribute)</a>
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-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CNotGate.min_qubits">(sympy.physics.quantum.gate.CNotGate attribute)</a>
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-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.Gate.min_qubits">(sympy.physics.quantum.gate.Gate attribute)</a>
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-  <dt><a href="modules/polys/reference.html#sympy.polys.numberfields.minimal_polynomial">minimal_polynomial() (in module sympy.polys.numberfields)</a>
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-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.minlex">minlex() (in module sympy.utilities.iterables)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Ellipse.minor">minor (sympy.geometry.ellipse.Ellipse attribute)</a>
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-  <dt><a href="modules/polys/reference.html#sympy.polys.numberfields.minpoly">minpoly() (in module sympy.polys.numberfields)</a>
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-  <dt><a href="modules/physics/mechanics/api/printing.html#sympy.physics.mechanics.functions.mlatex">mlatex() (in module sympy.physics.mechanics.functions)</a>
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-  <dt><a href="modules/mpmath/calculus/optimization.html#mpmath.calculus.optimization.MNewton">MNewton (class in mpmath.calculus.optimization)</a>
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-  <dt><a href="modules/mpmath/matrices.html#mpmath.mnorm">mnorm() (in module mpmath)</a>
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-  <dt><a href="modules/core.html#sympy.core.mod.Mod">Mod (class in sympy.core.mod)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.Module">Module (class in sympy.polys.agca.modules)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.Module.contains">Module.contains() (in module sympy.polys.agca.modules)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.Module.convert">Module.convert() (in module sympy.polys.agca.modules)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.Module.identity_hom">Module.identity_hom() (in module sympy.polys.agca.modules)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.Module.is_submodule">Module.is_submodule() (in module sympy.polys.agca.modules)</a>
-  </dt>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.Module.is_zero">Module.is_zero() (in module sympy.polys.agca.modules)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.Module.quotient_module">Module.quotient_module() (in module sympy.polys.agca.modules)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.Module.submodule">Module.submodule() (in module sympy.polys.agca.modules)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.Module.subset">Module.subset() (in module sympy.polys.agca.modules)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.homomorphisms.ModuleHomomorphism">ModuleHomomorphism (class in sympy.polys.agca.homomorphisms)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.homomorphisms.ModuleHomomorphism.image">ModuleHomomorphism.image() (in module sympy.polys.agca.homomorphisms)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.homomorphisms.ModuleHomomorphism.is_injective">ModuleHomomorphism.is_injective() (in module sympy.polys.agca.homomorphisms)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.homomorphisms.ModuleHomomorphism.is_isomorphism">ModuleHomomorphism.is_isomorphism() (in module sympy.polys.agca.homomorphisms)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.homomorphisms.ModuleHomomorphism.is_surjective">ModuleHomomorphism.is_surjective() (in module sympy.polys.agca.homomorphisms)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.homomorphisms.ModuleHomomorphism.is_zero">ModuleHomomorphism.is_zero() (in module sympy.polys.agca.homomorphisms)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.homomorphisms.ModuleHomomorphism.quotient_codomain">ModuleHomomorphism.quotient_codomain() (in module sympy.polys.agca.homomorphisms)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.homomorphisms.ModuleHomomorphism.quotient_domain">ModuleHomomorphism.quotient_domain() (in module sympy.polys.agca.homomorphisms)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.homomorphisms.ModuleHomomorphism.restrict_codomain">ModuleHomomorphism.restrict_codomain() (in module sympy.polys.agca.homomorphisms)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.homomorphisms.ModuleHomomorphism.restrict_domain">ModuleHomomorphism.restrict_domain() (in module sympy.polys.agca.homomorphisms)</a>
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-  <dt><a href="modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.PxBra.momentum">momentum (sympy.physics.quantum.cartesian.PxBra attribute)</a>
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-  <dt><a href="modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.PxKet.momentum">(sympy.physics.quantum.cartesian.PxKet attribute)</a>
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.monic">monic() (in module sympy.polys.polytools)</a>
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-  <dt><a href="modules/mpmath/general.html#mpmath.monitor">monitor() (in module mpmath)</a>
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-  <dt><a href="modules/categories.html#sympy.categories.Morphism">Morphism (class in sympy.categories)</a>
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-  <dt><a href="modules/categories.html#sympy.categories.diagram_drawing.DiagramGrid.morphisms">morphisms (sympy.categories.diagram_drawing.DiagramGrid attribute)</a>
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-  <dt><a href="modules/mpmath/functions/elliptic.html#module-mpmath.functions.elliptic">mpmath.functions.elliptic (module)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.MPmathRealDomain">MPmathRealDomain (class in sympy.polys.domains)</a>
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-  <dt><a href="modules/physics/mechanics/api/printing.html#sympy.physics.mechanics.functions.mpprint">mpprint() (in module sympy.physics.mechanics.functions)</a>
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-  <dt><a href="modules/physics/mechanics/api/printing.html#sympy.physics.mechanics.functions.mprint">mprint() (in module sympy.physics.mechanics.functions)</a>
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-  <dt><a href="modules/ntheory.html#sympy.ntheory.primetest.mr">mr() (in module sympy.ntheory.primetest)</a>
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-  <dt><a href="modules/series.html#sympy.series.gruntz.mrv_max3">mrv_max3() (in module sympy.series.gruntz)</a>
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-  <dt><a href="modules/physics/matrices.html#sympy.physics.matrices.msigma">msigma() (in module sympy.physics.matrices)</a>
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-  <dt><a href="modules/core.html#sympy.core.mul.Mul">Mul (class in sympy.core.mul)</a>
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-  <dt><a href="modules/core.html#sympy.core.mul.Mul.as_coeff_Mul">Mul.as_coeff_Mul() (in module sympy.core.mul)</a>
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-  <dt><a href="modules/core.html#sympy.core.mul.Mul.as_content_primitive">Mul.as_content_primitive() (in module sympy.core.mul)</a>
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-  <dt><a href="modules/core.html#sympy.core.mul.Mul.as_ordered_factors">Mul.as_ordered_factors() (in module sympy.core.mul)</a>
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-  <dt><a href="modules/core.html#sympy.core.mul.Mul.as_two_terms">Mul.as_two_terms() (in module sympy.core.mul)</a>
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-  <dt><a href="modules/mpmath/calculus/optimization.html#mpmath.calculus.optimization.Muller">Muller (class in mpmath.calculus.optimization)</a>
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-  <dt><a href="modules/functions/combinatorial.html#sympy.functions.combinatorial.factorials.MultiFactorial">MultiFactorial (class in sympy.functions.combinatorial.factorials)</a>
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-  <dt><a href="modules/ntheory.html#sympy.ntheory.multinomial.multinomial_coefficients">multinomial_coefficients() (in module sympy.ntheory.multinomial)</a>
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-  <dt><a href="modules/ntheory.html#sympy.ntheory.multinomial.multinomial_coefficients_iterator">multinomial_coefficients_iterator() (in module sympy.ntheory.multinomial)</a>
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-  <dt><a href="modules/ntheory.html#sympy.ntheory.factor_.multiplicity">multiplicity() (in module sympy.ntheory.factor_)</a>
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-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.multiset_combinations">multiset_combinations() (in module sympy.utilities.iterables)</a>
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-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.multiset_partitions">multiset_partitions() (in module sympy.utilities.iterables)</a>
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-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.multiset_permutations">multiset_permutations() (in module sympy.utilities.iterables)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.polyerrors.MultivariatePolynomialError">MultivariatePolynomialError (class in sympy.polys.polyerrors)</a>
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-  <dt><a href="modules/matrices/dense.html#sympy.matrices.dense.MutableDenseMatrix">MutableDenseMatrix (class in sympy.matrices.dense)</a>
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-  <dt><a href="modules/matrices/dense.html#sympy.matrices.dense.MutableDenseMatrix.col_del">MutableDenseMatrix.col_del() (in module sympy.matrices.dense)</a>
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-  <dt><a href="modules/matrices/dense.html#sympy.matrices.dense.MutableDenseMatrix.col_op">MutableDenseMatrix.col_op() (in module sympy.matrices.dense)</a>
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-  <dt><a href="modules/matrices/dense.html#sympy.matrices.dense.MutableDenseMatrix.col_swap">MutableDenseMatrix.col_swap() (in module sympy.matrices.dense)</a>
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-  <dt><a href="modules/matrices/dense.html#sympy.matrices.dense.MutableDenseMatrix.copyin_list">MutableDenseMatrix.copyin_list() (in module sympy.matrices.dense)</a>
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-  <dt><a href="modules/matrices/dense.html#sympy.matrices.dense.MutableDenseMatrix.copyin_matrix">MutableDenseMatrix.copyin_matrix() (in module sympy.matrices.dense)</a>
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-  <dt><a href="modules/matrices/dense.html#sympy.matrices.dense.MutableDenseMatrix.fill">MutableDenseMatrix.fill() (in module sympy.matrices.dense)</a>
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-  <dt><a href="modules/matrices/dense.html#sympy.matrices.dense.MutableDenseMatrix.row_del">MutableDenseMatrix.row_del() (in module sympy.matrices.dense)</a>
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-  <dt><a href="modules/matrices/dense.html#sympy.matrices.dense.MutableDenseMatrix.row_op">MutableDenseMatrix.row_op() (in module sympy.matrices.dense)</a>
-  </dt>
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-  <dt><a href="modules/matrices/dense.html#sympy.matrices.dense.MutableDenseMatrix.row_swap">MutableDenseMatrix.row_swap() (in module sympy.matrices.dense)</a>
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-  <dt><a href="modules/matrices/dense.html#sympy.matrices.dense.MutableDenseMatrix.simplify">MutableDenseMatrix.simplify() (in module sympy.matrices.dense)</a>
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-  <dt><a href="modules/matrices/sparse.html#sympy.matrices.sparse.MutableSparseMatrix.col_del">MutableSparseMatrix.col_del() (in module sympy.matrices.sparse)</a>
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-  <dt><a href="modules/matrices/sparse.html#sympy.matrices.sparse.MutableSparseMatrix.col_join">MutableSparseMatrix.col_join() (in module sympy.matrices.sparse)</a>
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-  <dt><a href="modules/matrices/sparse.html#sympy.matrices.sparse.MutableSparseMatrix.col_op">MutableSparseMatrix.col_op() (in module sympy.matrices.sparse)</a>
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-  <dt><a href="modules/matrices/sparse.html#sympy.matrices.sparse.MutableSparseMatrix.col_swap">MutableSparseMatrix.col_swap() (in module sympy.matrices.sparse)</a>
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-  <dt><a href="modules/matrices/sparse.html#sympy.matrices.sparse.MutableSparseMatrix.fill">MutableSparseMatrix.fill() (in module sympy.matrices.sparse)</a>
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-  <dt><a href="modules/matrices/sparse.html#sympy.matrices.sparse.MutableSparseMatrix.row_del">MutableSparseMatrix.row_del() (in module sympy.matrices.sparse)</a>
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-  <dt><a href="modules/matrices/sparse.html#sympy.matrices.sparse.MutableSparseMatrix.row_join">MutableSparseMatrix.row_join() (in module sympy.matrices.sparse)</a>
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-  <dt><a href="modules/matrices/sparse.html#sympy.matrices.sparse.MutableSparseMatrix.row_op">MutableSparseMatrix.row_op() (in module sympy.matrices.sparse)</a>
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-  <dt><a href="modules/matrices/sparse.html#sympy.matrices.sparse.MutableSparseMatrix.row_swap">MutableSparseMatrix.row_swap() (in module sympy.matrices.sparse)</a>
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-  <dt><a href="modules/galgebra/GA/GAsympy.html#MV">MV (built-in class)</a>
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-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.NO.doit">NO.doit() (in module sympy.physics.secondquant)</a>
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-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.NO.get_subNO">NO.get_subNO() (in module sympy.physics.secondquant)</a>
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-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.NO.iter_q_annihilators">NO.iter_q_annihilators() (in module sympy.physics.secondquant)</a>
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-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.NO.iter_q_creators">NO.iter_q_creators() (in module sympy.physics.secondquant)</a>
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-  <dt><a href="modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.nodes">nodes (sympy.combinatorics.prufer.Prufer attribute)</a>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.NonSquareMatrixError">NonSquareMatrixError (class in sympy.matrices.matrices)</a>
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-  <dt><a href="modules/mpmath/matrices.html#mpmath.norm">norm() (in module mpmath)</a>
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-  <dt><a href="modules/statistics.html#sympy.statistics.distributions.Normal.confidence">Normal.confidence() (in module sympy.statistics.distributions)</a>
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-  <dt><a href="modules/galgebra/GA.html#sympy.galgebra.GA.normalize">normalize() (in module sympy.galgebra.GA)</a>
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-  <dt><a href="modules/stats.html#sympy.stats.crv_types.NormalPSpace">NormalPSpace (class in sympy.stats.crv_types)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.polyerrors.NotReversible">NotReversible (class in sympy.polys.polyerrors)</a>
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-  <dt><a href="modules/ntheory.html#sympy.ntheory.partitions_.npartitions">npartitions() (in module sympy.ntheory.partitions_)</a>
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-  <dt><a href="modules/mpmath/functions/expintegrals.html#mpmath.npdf">npdf() (in module mpmath)</a>
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-  <dt><a href="modules/mpmath/general.html#mpmath.nprint">nprint() (in module mpmath)</a>
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-  <dt><a href="modules/mpmath/calculus/sums_limits.html#mpmath.nprod">nprod() (in module mpmath)</a>
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-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CGate.nqubits">nqubits (sympy.physics.quantum.gate.CGate attribute)</a>
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-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.Gate.nqubits">(sympy.physics.quantum.gate.Gate attribute)</a>
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-
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.nroots">nroots() (in module sympy.polys.polytools)</a>
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-  <dt><a href="modules/simplify/simplify.html#sympy.simplify.simplify.nsimplify">nsimplify() (in module sympy.simplify.simplify)</a>
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-  <dt><a href="modules/solvers/solvers.html#sympy.solvers.solvers.nsolve">nsolve() (in module sympy.solvers.solvers)</a>
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-  <dt><a href="modules/mpmath/general.html#mpmath.nstr">nstr() (in module mpmath)</a>
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-  <dt><a href="modules/mpmath/calculus/sums_limits.html#mpmath.nsum">nsum() (in module mpmath)</a>
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.nth_power_roots_poly">nth_power_roots_poly() (in module sympy.polys.polytools)</a>
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-  <dt><a href="modules/functions/special.html#sympy.functions.special.hyper.meijerg.nu">nu (sympy.functions.special.hyper.meijerg attribute)</a>
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-  <dt><a href="modules/core.html#sympy.core.numbers.Number">Number (class in sympy.core.numbers)</a>
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-  <dt><a href="modules/core.html#sympy.core.numbers.Number.as_coeff_Add">Number.as_coeff_Add() (in module sympy.core.numbers)</a>
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-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.numbered_symbols">numbered_symbols() (in module sympy.utilities.iterables)</a>
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-  <dt><a href="modules/core.html#sympy.core.numbers.NumberSymbol">NumberSymbol (class in sympy.core.numbers)</a>
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-  <dt><a href="modules/core.html#sympy.core.numbers.NumberSymbol.approximation">NumberSymbol.approximation() (in module sympy.core.numbers)</a>
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-  <dt><a href="modules/galgebra/GA.html#sympy.galgebra.GA.numeric">numeric() (in module sympy.galgebra.GA)</a>
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-  <dt><a href="modules/galgebra/GA.html#sympy.galgebra.GA.NUMPAT">NUMPAT (in module sympy.galgebra.GA)</a>
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-  <dt><a href="modules/categories.html#sympy.categories.Diagram.objects">(sympy.categories.Diagram attribute)</a>
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-  <dt><a href="modules/galgebra/GA/GAsympy.html#odd">odd() (built-in function)</a>
-  </dt>
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-  <dt><a href="modules/solvers/ode.html#sympy.solvers.ode.ode_1st_homogeneous_coeff_best">ode_1st_homogeneous_coeff_best() (in module sympy.solvers.ode)</a>
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-  <dt><a href="modules/solvers/ode.html#sympy.solvers.ode.ode_1st_homogeneous_coeff_subs_dep_div_indep">ode_1st_homogeneous_coeff_subs_dep_div_indep() (in module sympy.solvers.ode)</a>
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-  <dt><a href="modules/solvers/ode.html#sympy.solvers.ode.ode_1st_homogeneous_coeff_subs_indep_div_dep">ode_1st_homogeneous_coeff_subs_indep_div_dep() (in module sympy.solvers.ode)</a>
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-  <dt><a href="modules/solvers/ode.html#sympy.solvers.ode.ode_1st_linear">ode_1st_linear() (in module sympy.solvers.ode)</a>
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-  <dt><a href="modules/solvers/ode.html#sympy.solvers.ode.ode_Bernoulli">ode_Bernoulli() (in module sympy.solvers.ode)</a>
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-  <dt><a href="modules/solvers/ode.html#sympy.solvers.ode.ode_Liouville">ode_Liouville() (in module sympy.solvers.ode)</a>
-  </dt>
-
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-  <dt><a href="modules/solvers/ode.html#sympy.solvers.ode.ode_nth_linear_constant_coeff_homogeneous">ode_nth_linear_constant_coeff_homogeneous() (in module sympy.solvers.ode)</a>
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-  <dt><a href="modules/solvers/ode.html#sympy.solvers.ode.ode_nth_linear_constant_coeff_undetermined_coefficients">ode_nth_linear_constant_coeff_undetermined_coefficients() (in module sympy.solvers.ode)</a>
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-
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-  <dt><a href="modules/solvers/ode.html#sympy.solvers.ode.ode_nth_linear_constant_coeff_variation_of_parameters">ode_nth_linear_constant_coeff_variation_of_parameters() (in module sympy.solvers.ode)</a>
-  </dt>
-
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-  <dt><a href="modules/solvers/ode.html#sympy.solvers.ode.ode_order">ode_order() (in module sympy.solvers.ode)</a>
-  </dt>
-
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-  <dt><a href="modules/solvers/ode.html#sympy.solvers.ode.ode_Riccati_special_minus2">ode_Riccati_special_minus2() (in module sympy.solvers.ode)</a>
-  </dt>
-
-      
-  <dt><a href="modules/solvers/ode.html#sympy.solvers.ode.ode_separable">ode_separable() (in module sympy.solvers.ode)</a>
-  </dt>
-
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-  <dt><a href="modules/solvers/ode.html#sympy.solvers.ode.ode_sol_simplicity">ode_sol_simplicity() (in module sympy.solvers.ode)</a>
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-  <dt><a href="modules/mpmath/calculus/odes.html#mpmath.odefun">odefun() (in module mpmath)</a>
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-
-  </dl></td>
-  <td style="width: 33%" valign="top"><dl>
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-  <dt><a href="modules/solvers/ode.html#sympy.solvers.ode.odesimp">odesimp() (in module sympy.solvers.ode)</a>
-  </dt>
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.one">one (sympy.polys.polytools.Poly attribute)</a>
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-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.OneQubitGate">OneQubitGate (class in sympy.physics.quantum.gate)</a>
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-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.dense.ones">ones() (in module sympy.matrices.dense)</a>
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-  <dt><a href="modules/core.html#sympy.core.numbers.oo">oo() (in module sympy.core.numbers)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.polyerrors.OperationNotSupported">OperationNotSupported (class in sympy.polys.polyerrors)</a>
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-  <dt><a href="modules/physics/quantum/operator.html#sympy.physics.quantum.operator.Operator">Operator (class in sympy.physics.quantum.operator)</a>
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-  <dt><a href="modules/physics/quantum/state.html#sympy.physics.quantum.state.StateBase.operators">operators (sympy.physics.quantum.state.StateBase attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/quantum/operatorset.html#sympy.physics.quantum.operatorset.operators_to_state">operators_to_state() (in module sympy.physics.quantum.operatorset)</a>
-  </dt>
-
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-  <dt><a href="modules/polys/internals.html#sympy.polys.polyerrors.OptionError">OptionError (class in sympy.polys.polyerrors)</a>
-  </dt>
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-  <dt><a href="modules/logic.html#sympy.logic.boolalg.Or">Or (class in sympy.logic.boolalg)</a>
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-  <dt><a href="modules/physics/quantum/grover.html#sympy.physics.quantum.grover.OracleGate">OracleGate (class in sympy.physics.quantum.grover)</a>
-  </dt>
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-  <dt><a href="modules/series.html#sympy.series.order.Order">Order (class in sympy.series.order)</a>
-  </dt>
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-  <dt><a href="modules/functions/special.html#sympy.functions.special.bessel.BesselBase.order">order (sympy.functions.special.bessel.BesselBase attribute)</a>
-  </dt>
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-  <dt><a href="modules/series.html#sympy.series.order.Order.contains">Order.contains() (in module sympy.series.order)</a>
-  </dt>
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Triangle.orthocenter">orthocenter (sympy.geometry.polygon.Triangle attribute)</a>
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-  <dt><a href="modules/physics/mechanics/api/functions.html#sympy.physics.mechanics.functions.outer">outer() (in module sympy.physics.mechanics.functions)</a>
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-  <dt><a href="modules/physics/quantum/operator.html#sympy.physics.quantum.operator.OuterProduct">OuterProduct (class in sympy.physics.quantum.operator)</a>
-  </dt>
-
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-</tr></table>
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-<table style="width: 100%" class="indextable genindextable"><tr>
-  <td style="width: 33%" valign="top"><dl>
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-  <dt><a href="modules/stats.html#sympy.stats.P">P() (in module sympy.stats)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.LinearEntity.p1">p1 (sympy.geometry.line.LinearEntity attribute)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.LinearEntity.p2">p2 (sympy.geometry.line.LinearEntity attribute)</a>
-  </dt>
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-  <dt><a href="modules/mpmath/calculus/approximation.html#mpmath.pade">pade() (in module mpmath)</a>
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.parallel_poly_from_expr">parallel_poly_from_expr() (in module sympy.polys.polytools)</a>
-  </dt>
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-  <dt><a href="modules/geometry.html#sympy.geometry.curve.Curve.parameter">parameter (sympy.geometry.curve.Curve attribute)</a>
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-  <dt><a href="modules/plotting.html#sympy.plotting.plot.Parametric2DLineSeries">Parametric2DLineSeries (class in sympy.plotting.plot)</a>
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-  <dt><a href="modules/plotting.html#sympy.plotting.plot.Parametric2DLineSeries.get_segments">Parametric2DLineSeries.get_segments() (in module sympy.plotting.plot)</a>
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-  <dt><a href="modules/plotting.html#sympy.plotting.plot.Parametric3DLineSeries">Parametric3DLineSeries (class in sympy.plotting.plot)</a>
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-  <dt><a href="modules/plotting.html#sympy.plotting.plot.ParametricSurfaceSeries">ParametricSurfaceSeries (class in sympy.plotting.plot)</a>
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-  <dt><a href="modules/stats.html#sympy.stats.Pareto">Pareto() (in module sympy.stats)</a>
-  </dt>
-
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-  <dt><a href="modules/parsing.html#sympy.parsing.sympy_parser.parse_expr">parse_expr() (in module sympy.parsing.sympy_parser)</a>
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-  <dt><a href="modules/parsing.html#sympy.parsing.maxima.parse_maxima">parse_maxima() (in module sympy.parsing.maxima)</a>
-  </dt>
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-  <dt><a href="modules/physics/mechanics/api/kane.html#sympy.physics.mechanics.functions.partial_velocity">partial_velocity() (in module sympy.physics.mechanics.functions)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.particle.Particle">Particle (class in sympy.physics.mechanics.particle)</a>
-  </dt>
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-  <dt><a href="modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.particle.Particle.angular_momentum">Particle.angular_momentum() (in module sympy.physics.mechanics.particle)</a>
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-  <dt><a href="modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.particle.Particle.get_mass">Particle.get_mass() (in module sympy.physics.mechanics.particle)</a>
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-  <dt><a href="modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.particle.Particle.get_point">Particle.get_point() (in module sympy.physics.mechanics.particle)</a>
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-  <dt><a href="modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.particle.Particle.kinetic_energy">Particle.kinetic_energy() (in module sympy.physics.mechanics.particle)</a>
-  </dt>
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-  <dt><a href="modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.particle.Particle.linear_momentum">Particle.linear_momentum() (in module sympy.physics.mechanics.particle)</a>
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-  <dt><a href="modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.particle.Particle.set_potential_energy">Particle.set_potential_energy() (in module sympy.physics.mechanics.particle)</a>
-  </dt>
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-  <dt><a href="modules/combinatorics/partitions.html#sympy.combinatorics.partitions.Partition">Partition (class in sympy.combinatorics.partitions)</a>
-  </dt>
-
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-  <dt><a href="modules/combinatorics/partitions.html#sympy.combinatorics.partitions.Partition.partition">partition (sympy.combinatorics.partitions.Partition attribute)</a>
-  </dt>
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-  <dt><a href="modules/combinatorics/partitions.html#sympy.combinatorics.partitions.Partition.sort_key">Partition.sort_key() (in module sympy.combinatorics.partitions)</a>
-  </dt>
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-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.partitions">partitions() (in module sympy.utilities.iterables)</a>
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-  <dt><a href="modules/physics/matrices.html#sympy.physics.matrices.pat_matrix">pat_matrix() (in module sympy.physics.matrices)</a>
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-  <dt><a href="modules/diffgeom.html#sympy.diffgeom.Patch">Patch (class in sympy.diffgeom)</a>
-  </dt>
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-  <dt><a href="tutorial/tutorial.en.html#index-11">pattern matching</a>
-  </dt>
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-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.pcfd">pcfd() (in module mpmath)</a>
-  </dt>
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-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.pcfu">pcfu() (in module mpmath)</a>
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-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.pcfv">pcfv() (in module mpmath)</a>
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-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.pcfw">pcfw() (in module mpmath)</a>
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-  <dt><a href="modules/solvers/solvers.html#sympy.solvers.pde.pde_separate">pde_separate() (in module sympy.solvers.pde)</a>
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-  <dt><a href="modules/solvers/solvers.html#sympy.solvers.pde.pde_separate_add">pde_separate_add() (in module sympy.solvers.pde)</a>
-  </dt>
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-  <dt><a href="modules/solvers/solvers.html#sympy.solvers.pde.pde_separate_mul">pde_separate_mul() (in module sympy.solvers.pde)</a>
-  </dt>
-
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-  <dt><a href="modules/statistics.html#sympy.statistics.distributions.PDF">PDF (class in sympy.statistics.distributions)</a>
-  </dt>
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-  <dt><a href="modules/statistics.html#sympy.statistics.distributions.PDF.cdf">PDF.cdf() (in module sympy.statistics.distributions)</a>
-  </dt>
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-  <dt><a href="modules/statistics.html#sympy.statistics.distributions.PDF.normalize">PDF.normalize() (in module sympy.statistics.distributions)</a>
-  </dt>
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-  <dt><a href="modules/statistics.html#sympy.statistics.distributions.PDF.transform">PDF.transform() (in module sympy.statistics.distributions)</a>
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-  <dt><a href="modules/galgebra/latex_ex/latex_ex.html#pdiff_format">pdiff_format() (built-in function)</a>
-  </dt>
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-  <dt><a href="modules/galgebra/latex_ex.html#sympy.galgebra.latex_ex.pdiff_format">(in module sympy.galgebra.latex_ex)</a>
-  </dt>
-
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-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.pdiv">pdiv() (in module sympy.polys.polytools)</a>
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-  <dt><a href="modules/mpmath/calculus/optimization.html#mpmath.calculus.optimization.Pegasus">Pegasus (class in mpmath.calculus.optimization)</a>
-  </dt>
-
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-  <dt><a href="modules/ntheory.html#sympy.ntheory.factor_.perfect_power">perfect_power() (in module sympy.ntheory.factor_)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Ellipse.periapsis">periapsis (sympy.geometry.ellipse.Ellipse attribute)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Polygon.perimeter">perimeter (sympy.geometry.polygon.Polygon attribute)</a>
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-
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-  <dt><a href="modules/physics/quantum/shor.html#sympy.physics.quantum.shor.period_find">period_find() (in module sympy.physics.quantum.shor)</a>
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-
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-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation">Permutation (class in sympy.combinatorics.permutations)</a>
-  </dt>
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-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.ascents">Permutation.ascents() (in module sympy.combinatorics.permutations)</a>
-  </dt>
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-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.atoms">Permutation.atoms() (in module sympy.combinatorics.permutations)</a>
-  </dt>
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-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.commutator">Permutation.commutator() (in module sympy.combinatorics.permutations)</a>
-  </dt>
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-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.commutes_with">Permutation.commutes_with() (in module sympy.combinatorics.permutations)</a>
-  </dt>
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-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.descents">Permutation.descents() (in module sympy.combinatorics.permutations)</a>
-  </dt>
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-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.get_adjacency_distance">Permutation.get_adjacency_distance() (in module sympy.combinatorics.permutations)</a>
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-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.get_adjacency_matrix">Permutation.get_adjacency_matrix() (in module sympy.combinatorics.permutations)</a>
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-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.get_positional_distance">Permutation.get_positional_distance() (in module sympy.combinatorics.permutations)</a>
-  </dt>
-
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-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.get_precedence_distance">Permutation.get_precedence_distance() (in module sympy.combinatorics.permutations)</a>
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-
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-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.get_precedence_matrix">Permutation.get_precedence_matrix() (in module sympy.combinatorics.permutations)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.index">Permutation.index() (in module sympy.combinatorics.permutations)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.inversion_vector">Permutation.inversion_vector() (in module sympy.combinatorics.permutations)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.inversions">Permutation.inversions() (in module sympy.combinatorics.permutations)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.length">Permutation.length() (in module sympy.combinatorics.permutations)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.list">Permutation.list() (in module sympy.combinatorics.permutations)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.max">Permutation.max() (in module sympy.combinatorics.permutations)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.min">Permutation.min() (in module sympy.combinatorics.permutations)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.mul_inv">Permutation.mul_inv() (in module sympy.combinatorics.permutations)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.next_lex">Permutation.next_lex() (in module sympy.combinatorics.permutations)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.next_nonlex">Permutation.next_nonlex() (in module sympy.combinatorics.permutations)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.next_trotterjohnson">Permutation.next_trotterjohnson() (in module sympy.combinatorics.permutations)</a>
-  </dt>
-
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-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.order">Permutation.order() (in module sympy.combinatorics.permutations)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.parity">Permutation.parity() (in module sympy.combinatorics.permutations)</a>
-  </dt>
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-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.rank">Permutation.rank() (in module sympy.combinatorics.permutations)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.rank_nonlex">Permutation.rank_nonlex() (in module sympy.combinatorics.permutations)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.rank_trotterjohnson">Permutation.rank_trotterjohnson() (in module sympy.combinatorics.permutations)</a>
-  </dt>
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-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.rmul">Permutation.rmul() (in module sympy.combinatorics.permutations)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.rmul_with_af">Permutation.rmul_with_af() (in module sympy.combinatorics.permutations)</a>
-  </dt>
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-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.runs">Permutation.runs() (in module sympy.combinatorics.permutations)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.signature">Permutation.signature() (in module sympy.combinatorics.permutations)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.support">Permutation.support() (in module sympy.combinatorics.permutations)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.transpositions">Permutation.transpositions() (in module sympy.combinatorics.permutations)</a>
-  </dt>
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-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup">PermutationGroup (class in sympy.combinatorics.perm_groups)</a>
-  </dt>
-
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-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.baseswap">PermutationGroup.baseswap() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
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-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.center">PermutationGroup.center() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
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-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.centralizer">PermutationGroup.centralizer() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
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-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.commutator">PermutationGroup.commutator() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
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-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.contains">PermutationGroup.contains() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
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-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.coset_factor">PermutationGroup.coset_factor() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
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-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.coset_rank">PermutationGroup.coset_rank() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
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-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.coset_unrank">PermutationGroup.coset_unrank() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.derived_series">PermutationGroup.derived_series() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
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-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.derived_subgroup">PermutationGroup.derived_subgroup() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
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-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.generate">PermutationGroup.generate() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.generate_dimino">PermutationGroup.generate_dimino() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.generate_schreier_sims">PermutationGroup.generate_schreier_sims() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.is_alt_sym">PermutationGroup.is_alt_sym() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.is_group">PermutationGroup.is_group() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.is_normal">PermutationGroup.is_normal() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.is_primitive">PermutationGroup.is_primitive() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.is_subgroup">PermutationGroup.is_subgroup() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.is_transitive">PermutationGroup.is_transitive() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.lower_central_series">PermutationGroup.lower_central_series() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.make_perm">PermutationGroup.make_perm() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.minimal_block">PermutationGroup.minimal_block() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.normal_closure">PermutationGroup.normal_closure() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
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-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.orbit">PermutationGroup.orbit() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.orbit_rep">PermutationGroup.orbit_rep() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.orbit_transversal">PermutationGroup.orbit_transversal() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.orbits">PermutationGroup.orbits() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.order">PermutationGroup.order() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.pointwise_stabilizer">PermutationGroup.pointwise_stabilizer() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.random">PermutationGroup.random() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.random_pr">PermutationGroup.random_pr() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.random_stab">PermutationGroup.random_stab() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims">PermutationGroup.schreier_sims() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims_incremental">PermutationGroup.schreier_sims_incremental() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims_random">PermutationGroup.schreier_sims_random() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.schreier_vector">PermutationGroup.schreier_vector() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.stabilizer">PermutationGroup.stabilizer() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.subgroup_search">PermutationGroup.subgroup_search() (in module sympy.combinatorics.perm_groups)</a>
-  </dt>
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-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.PermutationOperator">PermutationOperator (class in sympy.physics.secondquant)</a>
-  </dt>
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-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.PermutationOperator.get_permuted">PermutationOperator.get_permuted() (in module sympy.physics.secondquant)</a>
-  </dt>
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-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.pexquo">pexquo() (in module sympy.polys.polytools)</a>
-  </dt>
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-  <dt><a href="modules/combinatorics/polyhedron.html#sympy.combinatorics.polyhedron.Polyhedron.pgroup">pgroup (sympy.combinatorics.polyhedron.Polyhedron attribute)</a>
-  </dt>
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-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.Phase">Phase (in module sympy.physics.quantum.gate)</a>
-  </dt>
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-      
-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.PhaseGate">PhaseGate (class in sympy.physics.quantum.gate)</a>
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-  <dt><a href="modules/mpmath/functions/constants.html#mpmath.mp.phi">phi (mpmath.mp attribute)</a>
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-  <dt><a href="modules/mpmath/functions/constants.html#mpmath.mp.pi">pi (mpmath.mp attribute)</a>
-  </dt>
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-  <dt><a href="modules/core.html#sympy.core.numbers.pi">pi() (in module sympy.core.numbers)</a>
-  </dt>
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-  <dt><a href="modules/physics/quantum/piab.html#sympy.physics.quantum.piab.PIABBra">PIABBra (class in sympy.physics.quantum.piab)</a>
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-  <dt><a href="modules/physics/quantum/piab.html#sympy.physics.quantum.piab.PIABHamiltonian">PIABHamiltonian (class in sympy.physics.quantum.piab)</a>
-  </dt>
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-  <dt><a href="modules/physics/quantum/piab.html#sympy.physics.quantum.piab.PIABKet">PIABKet (class in sympy.physics.quantum.piab)</a>
-  </dt>
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-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.piecewise.Piecewise">Piecewise (class in sympy.functions.elementary.piecewise)</a>
-  </dt>
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-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.piecewise.Piecewise.doit">Piecewise.doit() (in module sympy.functions.elementary.piecewise)</a>
-  </dt>
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-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.piecewise.piecewise_fold">piecewise_fold() (in module sympy.functions.elementary.piecewise)</a>
-  </dt>
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-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.spherical_harmonics.Plmcos">Plmcos() (in module sympy.functions.special.spherical_harmonics)</a>
-  </dt>
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-  <dt><a href="modules/plotting.html#sympy.plotting.plot.Plot">Plot (class in sympy.plotting.plot)</a>
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-  <dt><a href="modules/mpmath/plotting.html#mpmath.plot">plot() (in module mpmath)</a>
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-      <dd><dl>
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-  <dt><a href="modules/plotting.html#sympy.plotting.plot.plot">(in module sympy.plotting.plot)</a>
-  </dt>
-
-      </dl></dd>
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-  <dt><a href="modules/plotting.html#sympy.plotting.plot.Plot.append">Plot.append() (in module sympy.plotting.plot)</a>
-  </dt>
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-  <dt><a href="modules/plotting.html#sympy.plotting.plot.Plot.extend">Plot.extend() (in module sympy.plotting.plot)</a>
-  </dt>
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-  <dt><a href="modules/plotting.html#sympy.plotting.plot.plot3d">plot3d() (in module sympy.plotting.plot)</a>
-  </dt>
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-  <dt><a href="modules/plotting.html#sympy.plotting.plot.plot3d_parametric_line">plot3d_parametric_line() (in module sympy.plotting.plot)</a>
-  </dt>
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-  <dt><a href="modules/plotting.html#sympy.plotting.plot.plot3d_parametric_surface">plot3d_parametric_surface() (in module sympy.plotting.plot)</a>
-  </dt>
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-  <dt><a href="modules/plotting.html#sympy.plotting.plot_implicit.plot_implicit">plot_implicit() (in module sympy.plotting.plot_implicit)</a>
-  </dt>
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-  <dt><a href="modules/plotting.html#sympy.plotting.plot.plot_parametric">plot_parametric() (in module sympy.plotting.plot)</a>
-  </dt>
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-  <dt><a href="modules/geometry.html#sympy.geometry.point.Point">Point (class in sympy.geometry.point)</a>
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-  <dt><a href="modules/physics/mechanics/api/kinematics.html#sympy.physics.mechanics.point.Point">(class in sympy.physics.mechanics.point)</a>
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-  <dt><a href="modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.particle.Particle.point">(sympy.physics.mechanics.particle.Particle attribute)</a>
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-  <dt><a href="modules/physics/mechanics/api/kinematics.html#sympy.physics.mechanics.point.Point.a1pt_theory">Point.a1pt_theory() (in module sympy.physics.mechanics.point)</a>
-  </dt>
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-  <dt><a href="modules/physics/mechanics/api/kinematics.html#sympy.physics.mechanics.point.Point.a2pt_theory">Point.a2pt_theory() (in module sympy.physics.mechanics.point)</a>
-  </dt>
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-  <dt><a href="modules/physics/mechanics/api/kinematics.html#sympy.physics.mechanics.point.Point.acc">Point.acc() (in module sympy.physics.mechanics.point)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.point.Point.distance">Point.distance() (in module sympy.geometry.point)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.point.Point.evalf">Point.evalf() (in module sympy.geometry.point)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.point.Point.intersection">Point.intersection() (in module sympy.geometry.point)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.point.Point.is_collinear">Point.is_collinear() (in module sympy.geometry.point)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.point.Point.is_concyclic">Point.is_concyclic() (in module sympy.geometry.point)</a>
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-  <dt><a href="modules/physics/mechanics/api/kinematics.html#sympy.physics.mechanics.point.Point.locatenew">Point.locatenew() (in module sympy.physics.mechanics.point)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.point.Point.midpoint">Point.midpoint() (in module sympy.geometry.point)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.point.Point.n">Point.n() (in module sympy.geometry.point)</a>
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-  <dt><a href="modules/physics/mechanics/api/kinematics.html#sympy.physics.mechanics.point.Point.pos_from">Point.pos_from() (in module sympy.physics.mechanics.point)</a>
-  </dt>
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-  <dt><a href="modules/geometry.html#sympy.geometry.point.Point.rotate">Point.rotate() (in module sympy.geometry.point)</a>
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-  </dt>
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-  <dt><a href="modules/physics/mechanics/api/kinematics.html#sympy.physics.mechanics.point.Point.set_pos">Point.set_pos() (in module sympy.physics.mechanics.point)</a>
-  </dt>
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-  <dt><a href="modules/physics/mechanics/api/kinematics.html#sympy.physics.mechanics.point.Point.set_vel">Point.set_vel() (in module sympy.physics.mechanics.point)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.point.Point.transform">Point.transform() (in module sympy.geometry.point)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Polygon.arbitrary_point">Polygon.arbitrary_point() (in module sympy.geometry.polygon)</a>
-  </dt>
-
-      
-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Polygon.distance">Polygon.distance() (in module sympy.geometry.polygon)</a>
-  </dt>
-
-      
-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Polygon.encloses_point">Polygon.encloses_point() (in module sympy.geometry.polygon)</a>
-  </dt>
-
-      
-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Polygon.intersection">Polygon.intersection() (in module sympy.geometry.polygon)</a>
-  </dt>
-
-      
-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Polygon.is_convex">Polygon.is_convex() (in module sympy.geometry.polygon)</a>
-  </dt>
-
-      
-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Polygon.plot_interval">Polygon.plot_interval() (in module sympy.geometry.polygon)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/polyhedron.html#sympy.combinatorics.polyhedron.Polyhedron">Polyhedron (class in sympy.combinatorics.polyhedron)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/polyhedron.html#sympy.combinatorics.polyhedron.Polyhedron.reset">Polyhedron.reset() (in module sympy.combinatorics.polyhedron)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/polyhedron.html#sympy.combinatorics.polyhedron.Polyhedron.rotate">Polyhedron.rotate() (in module sympy.combinatorics.polyhedron)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.zeta_functions.polylog">polylog (class in sympy.functions.special.zeta_functions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/zeta.html#mpmath.polylog">polylog() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyerrors.PolynomialError">PolynomialError (class in sympy.polys.polyerrors)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.PolynomialRing">PolynomialRing (class in sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/calculus/polynomials.html#mpmath.polyroots">polyroots() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/calculus/polynomials.html#mpmath.polyval">polyval() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/simplify/simplify.html#sympy.simplify.simplify.posify">posify() (in module sympy.simplify.simplify)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.XBra.position">position (sympy.physics.quantum.cartesian.XBra attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.XKet.position">(sympy.physics.quantum.cartesian.XKet attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.PositionState3D.position_x">position_x (sympy.physics.quantum.cartesian.PositionState3D attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.PositionState3D.position_y">position_y (sympy.physics.quantum.cartesian.PositionState3D attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.PositionState3D.position_z">position_z (sympy.physics.quantum.cartesian.PositionState3D attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.PositionBra3D">PositionBra3D (class in sympy.physics.quantum.cartesian)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.PositionKet3D">PositionKet3D (class in sympy.physics.quantum.cartesian)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.PositionState3D">PositionState3D (class in sympy.physics.quantum.cartesian)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.postfixes">postfixes() (in module sympy.utilities.iterables)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.postorder_traversal">postorder_traversal() (in module sympy.utilities.iterables)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.particle.Particle.potential_energy">potential_energy (sympy.physics.mechanics.particle.Particle attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.rigidbody.RigidBody.potential_energy">(sympy.physics.mechanics.rigidbody.RigidBody attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.functions.potential_energy">potential_energy() (in module sympy.physics.mechanics.functions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.power.Pow">Pow (class in sympy.core.power)</a>
-  </dt>
-
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-  <dt><a href="modules/core.html#sympy.core.power.Pow.as_base_exp">Pow.as_base_exp() (in module sympy.core.power)</a>
-  </dt>
-
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-  <dt><a href="modules/core.html#sympy.core.power.Pow.as_content_primitive">Pow.as_content_primitive() (in module sympy.core.power)</a>
-  </dt>
-
-      
-  <dt><a href="modules/simplify/simplify.html#sympy.simplify.simplify.powdenest">powdenest() (in module sympy.simplify.simplify)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/powers.html#mpmath.power">power() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/matrices.html#mpmath.powm">powm() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/powers.html#mpmath.powm1">powm1() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/simplify/simplify.html#sympy.simplify.simplify.powsimp">powsimp() (in module sympy.simplify.simplify)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.tree.pprint_nodes">pprint_nodes() (in module sympy.printing.tree)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.pquo">pquo() (in module sympy.polys.polytools)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.precedence.PRECEDENCE">PRECEDENCE (in module sympy.printing.precedence)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.precedence.precedence">precedence() (in module sympy.printing.precedence)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.precedence.PRECEDENCE_FUNCTIONS">PRECEDENCE_FUNCTIONS (in module sympy.printing.precedence)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.precedence.PRECEDENCE_VALUES">PRECEDENCE_VALUES (in module sympy.printing.precedence)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.evalf.PrecisionExhausted">PrecisionExhausted (class in sympy.core.evalf)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/assume.html#sympy.assumptions.assume.Predicate">Predicate (class in sympy.assumptions.assume)</a>
-  </dt>
-
-      
-  <dt><a href="modules/assumptions/assume.html#sympy.assumptions.assume.Predicate.eval">Predicate.eval() (in module sympy.assumptions.assume)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.tensor_functions.KroneckerDelta.preferred_index">preferred_index (sympy.functions.special.tensor_functions.KroneckerDelta attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.KroneckerDelta.preferred_index">(sympy.physics.secondquant.KroneckerDelta attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.prefixes">prefixes() (in module sympy.utilities.iterables)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.prem">prem() (in module sympy.polys.polytools)</a>
-  </dt>
-
-      
-  <dt><a href="modules/categories.html#sympy.categories.Diagram.premises">premises (sympy.categories.Diagram attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/solvers/ode.html#sympy.solvers.ode.preprocess">preprocess() (in module sympy.solvers.ode)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.pretty.pretty.pretty">pretty() (in module sympy.printing.pretty.pretty)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.pretty.pretty_symbology.pretty_atom">pretty_atom() (in module sympy.printing.pretty.pretty_symbology)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.pretty.pretty.pretty_print">pretty_print() (in module sympy.printing.pretty.pretty)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.pretty.pretty_symbology.pretty_symbol">pretty_symbol() (in module sympy.printing.pretty.pretty_symbology)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.pretty.pretty_symbology.pretty_try_use_unicode">pretty_try_use_unicode() (in module sympy.printing.pretty.pretty_symbology)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.pretty.pretty_symbology.pretty_use_unicode">pretty_use_unicode() (in module sympy.printing.pretty.pretty_symbology)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.pretty.stringpict.prettyForm">prettyForm (class in sympy.printing.pretty.stringpict)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.pretty.stringpict.prettyForm.apply">prettyForm.apply() (in module sympy.printing.pretty.stringpict)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.pretty.pretty.PrettyPrinter">PrettyPrinter (class in sympy.printing.pretty.pretty)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.preview.preview">preview() (in module sympy.printing.preview)</a>
-  </dt>
-
-      
-  <dt><a href="modules/categories.html#sympy.categories.diagram_drawing.preview_diagram">preview_diagram() (in module sympy.categories.diagram_drawing)</a>
-  </dt>
-
-      
-  <dt><a href="modules/ntheory.html#sympy.ntheory.generate.prevprime">prevprime() (in module sympy.ntheory.generate)</a>
-  </dt>
-
-      
-  <dt><a href="modules/ntheory.html#sympy.ntheory.generate.prime">prime() (in module sympy.ntheory.generate)</a>
-  </dt>
-
-      
-  <dt><a href="modules/ntheory.html#sympy.ntheory.factor_.primefactors">primefactors() (in module sympy.ntheory.factor_)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/numtheory.html#mpmath.primepi">primepi() (in module mpmath)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/ntheory.html#sympy.ntheory.generate.primepi">(in module sympy.ntheory.generate)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/mpmath/functions/numtheory.html#mpmath.primepi2">primepi2() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/ntheory.html#sympy.ntheory.generate.primerange">primerange() (in module sympy.ntheory.generate)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/zeta.html#mpmath.primezeta">primezeta() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.primitive">primitive() (in module sympy.polys.polytools)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.numberfields.primitive_element">primitive_element() (in module sympy.polys.numberfields)</a>
-  </dt>
-
-      
-  <dt><a href="modules/ntheory.html#sympy.ntheory.generate.primorial">primorial() (in module sympy.ntheory.generate)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.ccode.print_ccode">print_ccode() (in module sympy.printing.ccode)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.fcode.print_fcode">print_fcode() (in module sympy.printing.fcode)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.gtk.print_gtk">print_gtk() (in module sympy.printing.gtk)</a>
-  </dt>
-
-      
-  <dt><a href="modules/galgebra/latex_ex.html#sympy.galgebra.latex_ex.print_LaTeX">print_LaTeX() (in module sympy.galgebra.latex_ex)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.latex.print_latex">print_latex() (in module sympy.printing.latex)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.mathml.print_mathml">print_mathml() (in module sympy.printing.mathml)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.tree.print_node">print_node() (in module sympy.printing.tree)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.tree.print_tree">print_tree() (in module sympy.printing.tree)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.printer.Printer">Printer (class in sympy.printing.printer)</a>
-  </dt>
-
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-  <dt><a href="modules/printing.html#sympy.printing.printer.Printer._print">Printer._print() (in module sympy.printing.printer)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.printer.Printer.doprint">Printer.doprint() (in module sympy.printing.printer)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.ccode.CCodePrinter.printmethod">printmethod (sympy.printing.ccode.CCodePrinter attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/printing.html#sympy.printing.codeprinter.CodePrinter.printmethod">(sympy.printing.codeprinter.CodePrinter attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/printing.html#sympy.printing.fcode.FCodePrinter.printmethod">(sympy.printing.fcode.FCodePrinter attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/printing.html#sympy.printing.lambdarepr.LambdaPrinter.printmethod">(sympy.printing.lambdarepr.LambdaPrinter attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/printing.html#sympy.printing.latex.LatexPrinter.printmethod">(sympy.printing.latex.LatexPrinter attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/printing.html#sympy.printing.mathml.MathMLPrinter.printmethod">(sympy.printing.mathml.MathMLPrinter attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/printing.html#sympy.printing.pretty.pretty.PrettyPrinter.printmethod">(sympy.printing.pretty.pretty.PrettyPrinter attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/printing.html#sympy.printing.printer.Printer.printmethod">(sympy.printing.printer.Printer attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/printing.html#sympy.printing.repr.ReprPrinter.printmethod">(sympy.printing.repr.ReprPrinter attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/printing.html#sympy.printing.str.StrPrinter.printmethod">(sympy.printing.str.StrPrinter attribute)</a>
-  </dt>
-
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-      
-  <dt><a href="modules/parsing.html#sympy.parsing.sympy_tokenize.printtoken">printtoken() (in module sympy.parsing.sympy_tokenize)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.mul.prod">prod() (in module sympy.core.mul)</a>
-  </dt>
-
-      
-  <dt><a href="modules/concrete.html#sympy.concrete.products.Product">Product (class in sympy.concrete.products)</a>
-  </dt>
-
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-  <dt><a href="modules/concrete.html#sympy.concrete.products.product">product() (in module sympy.concrete.products)</a>
-  </dt>
-
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-  <dt><a href="modules/stats.html#sympy.stats.rv.ProductDomain">ProductDomain (class in sympy.stats.rv)</a>
-  </dt>
-
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-  <dt><a href="modules/stats.html#sympy.stats.rv.ProductPSpace">ProductPSpace (class in sympy.stats.rv)</a>
-  </dt>
-
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-  <dt><a href="modules/sets.html#sympy.core.sets.ProductSet">ProductSet (class in sympy.core.sets)</a>
-  </dt>
-
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-  <dt><a href="modules/galgebra/GA/GAsympy.html#project">project() (built-in function)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer">Prufer (class in sympy.combinatorics.prufer)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.edges">Prufer.edges() (in module sympy.combinatorics.prufer)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.next">Prufer.next() (in module sympy.combinatorics.prufer)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.prev">Prufer.prev() (in module sympy.combinatorics.prufer)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.prufer_rank">Prufer.prufer_rank() (in module sympy.combinatorics.prufer)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.to_prufer">Prufer.to_prufer() (in module sympy.combinatorics.prufer)</a>
-  </dt>
-
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-  <dt><a href="modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.to_tree">Prufer.to_tree() (in module sympy.combinatorics.prufer)</a>
-  </dt>
-
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-  <dt><a href="modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.prufer_repr">prufer_repr (sympy.combinatorics.prufer.Prufer attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/gamma.html#mpmath.psi">psi() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/qho_1d.html#sympy.physics.qho_1d.psi_n">psi_n() (in module sympy.physics.qho_1d)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/identification.html#mpmath.pslq">pslq() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/stats.html#sympy.stats.rv.PSpace">PSpace (class in sympy.stats.rv)</a>
-  </dt>
-
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-  <dt><a href="modules/stats.html#sympy.stats.rv.pspace">pspace() (in module sympy.stats.rv)</a>
-  </dt>
-
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.PurePoly">PurePoly (class in sympy.polys.polytools)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.PxBra">PxBra (class in sympy.physics.quantum.cartesian)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.PxKet">PxKet (class in sympy.physics.quantum.cartesian)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.PxOp">PxOp (class in sympy.physics.quantum.cartesian)</a>
-  </dt>
-
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-  <dt><a href="modules/utilities/runtests.html#sympy.utilities.runtests.PyTestReporter">PyTestReporter (class in sympy.utilities.runtests)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/runtests.html#sympy.utilities.runtests.PyTestReporter.write">PyTestReporter.write() (in module sympy.utilities.runtests)</a>
-  </dt>
-
-      
-  <dt>
-    Python Enhancement Proposals
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/core.html#index-0">PEP 335</a>, <a href="modules/core.html#index-1">[1]</a>, <a href="modules/core.html#index-2">[2]</a>, <a href="modules/core.html#index-3">[3]</a>
-  </dt>
-
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.PythonFiniteField">PythonFiniteField (class in sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.PythonIntegerRing">PythonIntegerRing (class in sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.PythonRationalField">PythonRationalField (class in sympy.polys.domains)</a>
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-
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-</tr></table>
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-<h2 id="Q">Q</h2>
-<table style="width: 100%" class="indextable genindextable"><tr>
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-  <dt><a href="modules/assumptions/ask.html#sympy.assumptions.ask.Q">Q (class in sympy.assumptions.ask)</a>
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-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.BeamParameter.q">q (sympy.physics.gaussopt.BeamParameter attribute)</a>
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-  <dt><a href="modules/physics/quantum/qapply.html#sympy.physics.quantum.qapply.qapply">qapply() (in module sympy.physics.quantum.qapply)</a>
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-  <dt><a href="modules/mpmath/functions/elliptic.html#mpmath.qbarfrom">qbarfrom() (in module mpmath)</a>
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-  <dt><a href="modules/mpmath/functions/qfunctions.html#mpmath.qfac">qfac() (in module mpmath)</a>
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-  <dt><a href="modules/mpmath/functions/elliptic.html#mpmath.qfrom">qfrom() (in module mpmath)</a>
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-  <dt><a href="modules/physics/quantum/qft.html#sympy.physics.quantum.qft.QFT">QFT (class in sympy.physics.quantum.qft)</a>
-  </dt>
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-  <dt><a href="modules/physics/quantum/qft.html#sympy.physics.quantum.qft.QFT.decompose">QFT.decompose() (in module sympy.physics.quantum.qft)</a>
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-  <dt><a href="modules/mpmath/functions/qfunctions.html#mpmath.qgamma">qgamma() (in module mpmath)</a>
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-  <dt><a href="modules/mpmath/functions/qfunctions.html#mpmath.qhyper">qhyper() (in module mpmath)</a>
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-  <dt><a href="modules/mpmath/functions/qfunctions.html#mpmath.qp">qp() (in module mpmath)</a>
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-  <dt><a href="modules/mpmath/calculus/integration.html#mpmath.quad">quad() (in module mpmath)</a>
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-  <dt><a href="modules/mpmath/calculus/integration.html#mpmath.quadosc">quadosc() (in module mpmath)</a>
-  </dt>
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-  <dt><a href="modules/stats.html#sympy.stats.QuadraticU">QuadraticU() (in module sympy.stats)</a>
-  </dt>
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-  <dt><a href="modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.QuadratureRule">QuadratureRule (class in mpmath.calculus.quadrature)</a>
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-  <dt><a href="modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.QuadratureRule.calc_nodes">QuadratureRule.calc_nodes() (in module mpmath.calculus.quadrature)</a>
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-  <dt><a href="modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.QuadratureRule.clear">QuadratureRule.clear() (in module mpmath.calculus.quadrature)</a>
-  </dt>
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-  <td style="width: 33%" valign="top"><dl>
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-  <dt><a href="modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.QuadratureRule.estimate_error">QuadratureRule.estimate_error() (in module mpmath.calculus.quadrature)</a>
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-  <dt><a href="modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.QuadratureRule.get_nodes">QuadratureRule.get_nodes() (in module mpmath.calculus.quadrature)</a>
-  </dt>
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-  <dt><a href="modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.QuadratureRule.guess_degree">QuadratureRule.guess_degree() (in module mpmath.calculus.quadrature)</a>
-  </dt>
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-  <dt><a href="modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.QuadratureRule.sum_next">QuadratureRule.sum_next() (in module mpmath.calculus.quadrature)</a>
-  </dt>
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-  <dt><a href="modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.QuadratureRule.summation">QuadratureRule.summation() (in module mpmath.calculus.quadrature)</a>
-  </dt>
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-  <dt><a href="modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.QuadratureRule.transform_nodes">QuadratureRule.transform_nodes() (in module mpmath.calculus.quadrature)</a>
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-  <dt><a href="modules/physics/quantum/qubit.html#sympy.physics.quantum.qubit.Qubit">Qubit (class in sympy.physics.quantum.qubit)</a>
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-  <dt><a href="modules/physics/quantum/qubit.html#sympy.physics.quantum.qubit.qubit_to_matrix">qubit_to_matrix() (in module sympy.physics.quantum.qubit)</a>
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-  <dt><a href="modules/physics/quantum/qubit.html#sympy.physics.quantum.qubit.QubitBra">QubitBra (class in sympy.physics.quantum.qubit)</a>
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-
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.quo">quo() (in module sympy.polys.polytools)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.QuotientModule">QuotientModule (class in sympy.polys.agca.modules)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.QuotientModule.convert">QuotientModule.convert() (in module sympy.polys.agca.modules)</a>
-  </dt>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.QuotientModule.identity_hom">QuotientModule.identity_hom() (in module sympy.polys.agca.modules)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.QuotientModule.is_submodule">QuotientModule.is_submodule() (in module sympy.polys.agca.modules)</a>
-  </dt>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.QuotientModule.is_zero">QuotientModule.is_zero() (in module sympy.polys.agca.modules)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.QuotientModule.quotient_hom">QuotientModule.quotient_hom() (in module sympy.polys.agca.modules)</a>
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-  <dt><a href="modules/polys/agca.html#sympy.polys.agca.modules.QuotientModule.submodule">QuotientModule.submodule() (in module sympy.polys.agca.modules)</a>
-  </dt>
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-</tr></table>
-
-<h2 id="R">R</h2>
-<table style="width: 100%" class="indextable genindextable"><tr>
-  <td style="width: 33%" valign="top"><dl>
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-  <dt><a href="modules/physics/hydrogen.html#sympy.physics.hydrogen.R_nl">R_nl() (in module sympy.physics.hydrogen)</a>
-  </dt>
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-      <dd><dl>
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-  <dt><a href="modules/physics/sho.html#sympy.physics.sho.R_nl">(in module sympy.physics.sho)</a>
-  </dt>
-
-      </dl></dd>
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-  <dt><a href="modules/physics/wigner.html#sympy.physics.wigner.racah">racah() (in module sympy.physics.wigner)</a>
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-  <dt><a href="modules/mpmath/functions/trigonometric.html#mpmath.radians">radians() (in module mpmath)</a>
-  </dt>
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-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Circle.radius">radius (sympy.geometry.ellipse.Circle attribute)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.RegularPolygon.radius">(sympy.geometry.polygon.RegularPolygon attribute)</a>
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-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.BeamParameter.radius">(sympy.physics.gaussopt.BeamParameter attribute)</a>
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-      </dl></dd>
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.hyper.hyper.radius_of_convergence">radius_of_convergence (sympy.functions.special.hyper.hyper attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/simplify/simplify.html#sympy.simplify.simplify.radsimp">radsimp() (in module sympy.simplify.simplify)</a>
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-      
-  <dt><a href="modules/stats.html#sympy.stats.RaisedCosine">RaisedCosine() (in module sympy.stats)</a>
-  </dt>
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-      
-  <dt><a href="modules/utilities/pytest.html#sympy.utilities.pytest.raises">raises() (in module sympy.utilities.pytest)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/general.html#mpmath.rand">rand() (in module mpmath)</a>
-  </dt>
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-      
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.dense.randMatrix">randMatrix() (in module sympy.matrices.dense)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.random">random() (sympy.combinatorics.permutations.Permutation class method)</a>
-  </dt>
-
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-  <dt><a href="modules/combinatorics/graycode.html#sympy.combinatorics.graycode.random_bitstring">random_bitstring() (sympy.combinatorics.graycode method)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.random_circuit">random_circuit() (in module sympy.physics.quantum.gate)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/randtest.html#sympy.utilities.randtest.random_complex_number">random_complex_number() (in module sympy.utilities.randtest)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/partitions.html#sympy.combinatorics.partitions.random_integer_partition">random_integer_partition() (in module sympy.combinatorics.partitions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.specialpolys.random_poly">random_poly() (in module sympy.polys.specialpolys)</a>
-  </dt>
-
-      
-  <dt><a href="modules/stats.html#sympy.stats.rv.random_symbols">random_symbols() (in module sympy.stats.rv)</a>
-  </dt>
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-  <dt><a href="modules/stats.html#sympy.stats.rv.RandomDomain">RandomDomain (class in sympy.stats.rv)</a>
-  </dt>
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-  <dt><a href="modules/stats.html#sympy.stats.rv.RandomSymbol">RandomSymbol (class in sympy.stats.rv)</a>
-  </dt>
-
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-  <dt><a href="modules/ntheory.html#sympy.ntheory.generate.randprime">randprime() (in module sympy.ntheory.generate)</a>
-  </dt>
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-  <dt><a href="modules/tensor/indexed.html#sympy.tensor.indexed.Indexed.ranges">ranges (sympy.tensor.indexed.Indexed attribute)</a>
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-  <dt><a href="modules/combinatorics/graycode.html#sympy.combinatorics.graycode.GrayCode.rank">rank (sympy.combinatorics.graycode.GrayCode attribute)</a>
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-  <dt><a href="modules/combinatorics/partitions.html#sympy.combinatorics.partitions.Partition.rank">(sympy.combinatorics.partitions.Partition attribute)</a>
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-  <dt><a href="modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.rank">(sympy.combinatorics.prufer.Prufer attribute)</a>
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-  <dt><a href="modules/tensor/indexed.html#sympy.tensor.indexed.Indexed.rank">(sympy.tensor.indexed.Indexed attribute)</a>
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-  <dt><a href="modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.rank_binary">rank_binary (sympy.combinatorics.subsets.Subset attribute)</a>
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-  <dt><a href="modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.rank_gray">rank_gray (sympy.combinatorics.subsets.Subset attribute)</a>
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-  <dt><a href="modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.rank_lexicographic">rank_lexicographic (sympy.combinatorics.subsets.Subset attribute)</a>
-  </dt>
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-      
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.ratint">ratint() (sympy.integrals method)</a>
-  </dt>
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-  <dt><a href="modules/core.html#sympy.core.numbers.Rational">Rational (class in sympy.core.numbers)</a>
-  </dt>
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-  <dt><a href="modules/core.html#sympy.core.numbers.Rational.as_content_primitive">Rational.as_content_primitive() (in module sympy.core.numbers)</a>
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-  <dt><a href="modules/core.html#sympy.core.numbers.Rational.factors">Rational.factors() (in module sympy.core.numbers)</a>
-  </dt>
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-  <dt><a href="modules/core.html#sympy.core.numbers.Rational.limit_denominator">Rational.limit_denominator() (in module sympy.core.numbers)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.RationalField">RationalField (class in sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.RationalField.algebraic_field">RationalField.algebraic_field() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.RationalField.from_AlgebraicField">RationalField.from_AlgebraicField() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.RationalField.get_ring">RationalField.get_ring() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/simplify/simplify.html#sympy.simplify.simplify.ratsimp">ratsimp() (in module sympy.simplify.simplify)</a>
-  </dt>
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-  <dt><a href="modules/utilities/misc.html#sympy.utilities.misc.rawlines">rawlines() (in module sympy.utilities.misc)</a>
-  </dt>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.Ray">Ray (class in sympy.geometry.line)</a>
-  </dt>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.Ray.arbitrary_point">Ray.arbitrary_point() (in module sympy.geometry.line)</a>
-  </dt>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.Ray.contains">Ray.contains() (in module sympy.geometry.line)</a>
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.Ray.plot_interval">Ray.plot_interval() (in module sympy.geometry.line)</a>
-  </dt>
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-  <dt><a href="modules/stats.html#sympy.stats.Rayleigh">Rayleigh() (in module sympy.stats)</a>
-  </dt>
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-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.rayleigh2waist">rayleigh2waist() (in module sympy.physics.gaussopt)</a>
-  </dt>
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-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.RayTransferMatrix">RayTransferMatrix (class in sympy.physics.gaussopt)</a>
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-  <dt><a href="modules/simplify/simplify.html#sympy.simplify.simplify.rcollect">rcollect() (in module sympy.simplify.simplify)</a>
-  </dt>
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-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.complexes.re">re (class in sympy.functions.elementary.complexes)</a>
-  </dt>
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-  <dt><a href="modules/mpmath/general.html#mpmath.re">re() (in module mpmath)</a>
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-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.complexes.re.as_real_imag">re.as_real_imag() (in module sympy.functions.elementary.complexes)</a>
-  </dt>
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-  <dt><a href="modules/core.html#sympy.core.numbers.Real">Real (class in sympy.core.numbers)</a>
-  </dt>
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.real_roots">real_roots() (in module sympy.polys.polytools)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.RealDomain">RealDomain (class in sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.RealDomain.as_integer_ratio">RealDomain.as_integer_ratio() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.RealDomain.div">RealDomain.div() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.RealDomain.exquo">RealDomain.exquo() (in module sympy.polys.domains)</a>
-  </dt>
-
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.RealDomain.from_sympy">RealDomain.from_sympy() (in module sympy.polys.domains)</a>
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-
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.RealDomain.gcd">RealDomain.gcd() (in module sympy.polys.domains)</a>
-  </dt>
-
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.RealDomain.get_exact">RealDomain.get_exact() (in module sympy.polys.domains)</a>
-  </dt>
-
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.RealDomain.get_field">RealDomain.get_field() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.RealDomain.get_ring">RealDomain.get_ring() (in module sympy.polys.domains)</a>
-  </dt>
-
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.RealDomain.lcm">RealDomain.lcm() (in module sympy.polys.domains)</a>
-  </dt>
-
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.RealDomain.limit_denom">RealDomain.limit_denom() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.RealDomain.quo">RealDomain.quo() (in module sympy.polys.domains)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.RealDomain.rem">RealDomain.rem() (in module sympy.polys.domains)</a>
-  </dt>
-
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.RealDomain.to_sympy">RealDomain.to_sympy() (in module sympy.polys.domains)</a>
-  </dt>
-
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-  <dt><a href="modules/core.html#sympy.core.numbers.RealNumber">RealNumber (in module sympy.core.numbers)</a>
-  </dt>
-
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-  <dt><a href="modules/galgebra/GA/GAsympy.html#reciprocal_frame">reciprocal_frame() (built-in function)</a>
-  </dt>
-
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-  <dt><a href="modules/galgebra/GA.html#sympy.galgebra.GA.reciprocal_frame">(in module sympy.galgebra.GA)</a>
-  </dt>
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-  <dt><a href="modules/mpmath/general.html#mpmath.rect">rect() (in module mpmath)</a>
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-  <dt><a href="modules/utilities/memoization.html#sympy.utilities.memoization.recurrence_memo">recurrence_memo() (in module sympy.utilities.memoization)</a>
-  </dt>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.groebnertools.red_groebner">red_groebner() (in module sympy.polys.groebnertools)</a>
-  </dt>
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-  <dt><a href="modules/galgebra/GA.html#sympy.galgebra.GA.reduce_base">reduce_base() (in module sympy.galgebra.GA)</a>
-  </dt>
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.reduced">reduced() (in module sympy.polys.polytools)</a>
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-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.ReferenceFrame">ReferenceFrame (class in sympy.physics.mechanics.essential)</a>
-  </dt>
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-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.ReferenceFrame.ang_acc_in">ReferenceFrame.ang_acc_in() (in module sympy.physics.mechanics.essential)</a>
-  </dt>
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-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.ReferenceFrame.ang_vel_in">ReferenceFrame.ang_vel_in() (in module sympy.physics.mechanics.essential)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.ReferenceFrame.dcm">ReferenceFrame.dcm() (in module sympy.physics.mechanics.essential)</a>
-  </dt>
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-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.ReferenceFrame.orient">ReferenceFrame.orient() (in module sympy.physics.mechanics.essential)</a>
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-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.ReferenceFrame.orientnew">ReferenceFrame.orientnew() (in module sympy.physics.mechanics.essential)</a>
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-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.ReferenceFrame.set_ang_acc">ReferenceFrame.set_ang_acc() (in module sympy.physics.mechanics.essential)</a>
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-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.ReferenceFrame.set_ang_vel">ReferenceFrame.set_ang_vel() (in module sympy.physics.mechanics.essential)</a>
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-  <dt><a href="modules/assumptions/refine.html#sympy.assumptions.refine.refine">refine() (in module sympy.assumptions.refine)</a>
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-  <dt><a href="modules/assumptions/refine.html#sympy.assumptions.refine.refine_abs">refine_abs() (in module sympy.assumptions.refine)</a>
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-  <dt><a href="modules/assumptions/refine.html#sympy.assumptions.refine.refine_Pow">refine_Pow() (in module sympy.assumptions.refine)</a>
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-  <td style="width: 33%" valign="top"><dl>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.polyerrors.RefinementFailed">RefinementFailed (class in sympy.polys.polyerrors)</a>
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-  <dt><a href="modules/assumptions/ask.html#sympy.assumptions.ask.register_handler">register_handler() (in module sympy.assumptions.ask)</a>
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-  <dt><a href="modules/core.html#sympy.core.relational.Rel">Rel (class in sympy.core.relational)</a>
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-  <dt><a href="modules/assumptions/ask.html#sympy.assumptions.ask.remove_handler">remove_handler() (in module sympy.assumptions.ask)</a>
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-  <dt><a href="modules/physics/quantum/represent.html#sympy.physics.quantum.represent.rep_expectation">rep_expectation() (in module sympy.physics.quantum.represent)</a>
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-  <dt><a href="modules/physics/quantum/represent.html#sympy.physics.quantum.represent.rep_innerproduct">rep_innerproduct() (in module sympy.physics.quantum.represent)</a>
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-  <dt><a href="modules/utilities/runtests.html#sympy.utilities.runtests.Reporter">Reporter (class in sympy.utilities.runtests)</a>
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-  <dt><a href="modules/physics/quantum/represent.html#sympy.physics.quantum.represent.represent">represent() (in module sympy.physics.quantum.represent)</a>
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-  <dt><a href="modules/printing.html#sympy.printing.repr.ReprPrinter">ReprPrinter (class in sympy.printing.repr)</a>
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-  <dt><a href="modules/printing.html#sympy.printing.repr.ReprPrinter.reprify">ReprPrinter.reprify() (in module sympy.printing.repr)</a>
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-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.reshape">reshape() (in module sympy.utilities.iterables)</a>
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-  <dt><a href="modules/series.html#sympy.series.residues.residue">residue() (in module sympy.series.residues)</a>
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-  <dt><a href="modules/utilities/codegen.html#sympy.utilities.codegen.Result">Result (class in sympy.utilities.codegen)</a>
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-  <dt><a href="modules/utilities/codegen.html#sympy.utilities.codegen.Routine.result_variables">result_variables (sympy.utilities.codegen.Routine attribute)</a>
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-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.resultant">resultant() (in module sympy.polys.polytools)</a>
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-  <dt><a href="modules/galgebra/GA/GAsympy.html#rev">rev() (built-in function)</a>
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-  <dt><a href="modules/mpmath/functions/gamma.html#mpmath.rf">rf() (in module mpmath)</a>
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-  <dt><a href="modules/combinatorics/partitions.html#sympy.combinatorics.partitions.Partition.RGS">RGS (sympy.combinatorics.partitions.Partition attribute)</a>
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-  <dt><a href="modules/combinatorics/partitions.html#sympy.combinatorics.partitions.RGS_generalized">RGS_generalized() (in module sympy.combinatorics.partitions)</a>
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-  <dt><a href="modules/combinatorics/partitions.html#sympy.combinatorics.partitions.RGS_rank">RGS_rank() (in module sympy.combinatorics.partitions)</a>
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-  <dt><a href="modules/combinatorics/partitions.html#sympy.combinatorics.partitions.RGS_unrank">RGS_unrank() (in module sympy.combinatorics.partitions)</a>
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-  <dt><a href="modules/mpmath/calculus/sums_limits.html#mpmath.richardson">richardson() (in module mpmath)</a>
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-  <dt><a href="modules/series.html#sympy.series.acceleration.richardson">(in module sympy.series.acceleration)</a>
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-  <dt><a href="modules/mpmath/calculus/optimization.html#mpmath.calculus.optimization.Ridder">Ridder (class in mpmath.calculus.optimization)</a>
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-  <dt><a href="modules/mpmath/functions/numtheory.html#mpmath.riemannr">riemannr() (in module mpmath)</a>
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-  <dt><a href="modules/sets.html#sympy.core.sets.Interval.right">right (sympy.core.sets.Interval attribute)</a>
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-  <dt><a href="modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.rigidbody.RigidBody">RigidBody (class in sympy.physics.mechanics.rigidbody)</a>
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-  <dt><a href="modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.rigidbody.RigidBody.angular_momentum">RigidBody.angular_momentum() (in module sympy.physics.mechanics.rigidbody)</a>
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-  <dt><a href="modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.rigidbody.RigidBody.kinetic_energy">RigidBody.kinetic_energy() (in module sympy.physics.mechanics.rigidbody)</a>
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-  <dt><a href="modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.rigidbody.RigidBody.linear_momentum">RigidBody.linear_momentum() (in module sympy.physics.mechanics.rigidbody)</a>
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-  <dt><a href="modules/physics/mechanics/api/part_bod.html#sympy.physics.mechanics.rigidbody.RigidBody.set_potential_energy">RigidBody.set_potential_energy() (in module sympy.physics.mechanics.rigidbody)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Ring">Ring (class in sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Ring.div">Ring.div() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Ring.exquo">Ring.exquo() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Ring.free_module">Ring.free_module() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Ring.get_ring">Ring.get_ring() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Ring.ideal">Ring.ideal() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Ring.numer">Ring.numer() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Ring.quo">Ring.quo() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Ring.quotient_ring">Ring.quotient_ring() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Ring.rem">Ring.rem() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.Ring.revert">Ring.revert() (in module sympy.polys.domains)</a>
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-  <dt><a href="modules/functions/combinatorial.html#sympy.functions.combinatorial.factorials.RisingFactorial">RisingFactorial (class in sympy.functions.combinatorial.factorials)</a>
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-  <dt><a href="modules/physics/quantum/qft.html#sympy.physics.quantum.qft.Rk">Rk (in module sympy.physics.quantum.qft)</a>
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-  <dt><a href="modules/physics/quantum/qft.html#sympy.physics.quantum.qft.RkGate">RkGate (class in sympy.physics.quantum.qft)</a>
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-  <dt><a href="modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.RL">RL (sympy.matrices.sparse.SparseMatrix attribute)</a>
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-  <dt><a href="modules/printing.html#sympy.printing.pretty.pretty_symbology.root">root (in module sympy.printing.pretty.pretty_symbology)</a>
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-  <dt><a href="modules/core.html#sympy.core.singleton.S">S (in module sympy.core.singleton)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.distributedpolys.sdp_indep_p">sdp_indep_p() (in module sympy.polys.distributedpolys)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.distributedpolys.sdp_mul_term">sdp_mul_term() (in module sympy.polys.distributedpolys)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.distributedpolys.sdp_one_p">sdp_one_p() (in module sympy.polys.distributedpolys)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.distributedpolys.sdp_quo">sdp_quo() (in module sympy.polys.distributedpolys)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.distributedpolys.sdp_strip">sdp_strip() (in module sympy.polys.distributedpolys)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.distributedpolys.sdp_sub">sdp_sub() (in module sympy.polys.distributedpolys)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.distributedpolys.sdp_sub_term">sdp_sub_term() (in module sympy.polys.distributedpolys)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.distributedpolys.sdp_term_p">sdp_term_p() (in module sympy.polys.distributedpolys)</a>
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-  <dt><a href="modules/polys/internals.html#sympy.polys.distributedpolys.sdp_to_dict">sdp_to_dict() (in module sympy.polys.distributedpolys)</a>
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-  <dt><a href="modules/physics/quantum/grover.html#sympy.physics.quantum.grover.OracleGate.search_function">search_function (sympy.physics.quantum.grover.OracleGate attribute)</a>
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-  <dt><a href="modules/simplify/simplify.html#sympy.simplify.simplify.separate">separate() (in module sympy.simplify.simplify)</a>
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-  <dt><a href="modules/simplify/simplify.html#sympy.simplify.simplify.separatevars">separatevars() (in module sympy.simplify.simplify)</a>
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-  <dt><a href="tutorial/tutorial.en.html#index-3">series expansion</a>
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-  <dt><a href="modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.next_lexicographic">Subset.next_lexicographic() (in module sympy.combinatorics.subsets)</a>
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-  <dt><a href="modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.prev_lexicographic">Subset.prev_lexicographic() (in module sympy.combinatorics.subsets)</a>
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-  <dt><a href="modules/printing.html#module-sympy.printing.pretty.stringpict">sympy.printing.pretty.stringpict (module)</a>
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-  <dt><a href="modules/printing.html#module-sympy.printing.preview">sympy.printing.preview (module)</a>
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-  <dt><a href="modules/printing.html#module-sympy.printing.printer">sympy.printing.printer (module)</a>
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-  <dt><a href="modules/rewriting.html#module-sympy.simplify.cse_main">sympy.simplify.cse_main (module)</a>, <a href="modules/simplify/simplify.html#module-sympy.simplify.cse_main">[1]</a>
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-  <dt><a href="modules/simplify/simplify.html#module-sympy.simplify.epathtools">sympy.simplify.epathtools (module)</a>
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-  <dt><a href="modules/simplify/simplify.html#module-sympy.simplify.hyperexpand">sympy.simplify.hyperexpand (module)</a>
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-  <dt><a href="modules/simplify/hyperexpand.html#module-sympy.simplify.hyperexpand_doc">sympy.simplify.hyperexpand_doc (module)</a>
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-  <dt><a href="gotchas.html#module-sympy.simplify.simplify">sympy.simplify.simplify (module)</a>
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-  <dt><a href="modules/simplify/simplify.html#module-sympy.simplify.sqrtdenest">sympy.simplify.sqrtdenest (module)</a>
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-  <dt><a href="modules/simplify/simplify.html#module-sympy.simplify.traversaltools">sympy.simplify.traversaltools (module)</a>
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-  <dt><a href="modules/solvers/ode.html#module-sympy.solvers.ode">sympy.solvers.ode (module)</a>
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-  <dt><a href="modules/statistics.html#module-sympy.statistics">sympy.statistics (module)</a>
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-  <dt><a href="modules/stats.html#module-sympy.stats">sympy.stats (module)</a>
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-  <dt><a href="modules/stats.html#sympy.stats.crv_types.sympy.stats.Die">sympy.stats.Die() (in module sympy.stats.crv_types)</a>
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-  <dt><a href="modules/stats.html#module-sympy.stats.frv">sympy.stats.frv (module)</a>
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-  <dt><a href="modules/stats.html#module-sympy.stats.frv_types">sympy.stats.frv_types (module)</a>
-  </dt>
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-  <dt><a href="modules/stats.html#sympy.stats.crv_types.sympy.stats.Normal">sympy.stats.Normal() (in module sympy.stats.crv_types)</a>
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-  <dt><a href="modules/stats.html#module-sympy.stats.rv">sympy.stats.rv (module)</a>
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-  <dt><a href="modules/tensor/index.html#module-sympy.tensor">sympy.tensor (module)</a>
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-  <dt><a href="modules/tensor/indexed.html#module-sympy.tensor.indexed">sympy.tensor.indexed (module)</a>
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-  <dt><a href="modules/utilities/autowrap.html#module-sympy.utilities.autowrap">sympy.utilities.autowrap (module)</a>
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-  <dt><a href="modules/utilities/codegen.html#module-sympy.utilities.codegen">sympy.utilities.codegen (module)</a>
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-  <dt><a href="modules/utilities/cythonutils.html#module-sympy.utilities.cythonutils">sympy.utilities.cythonutils (module)</a>
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-  <dt><a href="modules/utilities/decorator.html#module-sympy.utilities.decorator">sympy.utilities.decorator (module)</a>
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-  <dt><a href="modules/utilities/iterables.html#module-sympy.utilities.iterables">sympy.utilities.iterables (module)</a>
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-  <dt><a href="modules/utilities/lambdify.html#module-sympy.utilities.lambdify">sympy.utilities.lambdify (module)</a>
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-  <dt><a href="modules/utilities/memoization.html#module-sympy.utilities.memoization">sympy.utilities.memoization (module)</a>
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-  <dt><a href="modules/utilities/misc.html#module-sympy.utilities.misc">sympy.utilities.misc (module)</a>
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-  <dt><a href="modules/utilities/pkgdata.html#module-sympy.utilities.pkgdata">sympy.utilities.pkgdata (module)</a>
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-  <dt><a href="modules/utilities/pytest.html#module-sympy.utilities.pytest">sympy.utilities.pytest (module)</a>
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-  <dt><a href="modules/utilities/randtest.html#module-sympy.utilities.randtest">sympy.utilities.randtest (module)</a>
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-  <dt><a href="modules/utilities/runtests.html#module-sympy.utilities.runtests">sympy.utilities.runtests (module)</a>
-  </dt>
-
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-  <dt><a href="modules/utilities/source.html#module-sympy.utilities.source">sympy.utilities.source (module)</a>
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-      
-  <dt><a href="modules/utilities/timeutils.html#module-sympy.utilities.timeutils">sympy.utilities.timeutils (module)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/runtests.html#sympy.utilities.runtests.SymPyDocTestFinder">SymPyDocTestFinder (class in sympy.utilities.runtests)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/runtests.html#sympy.utilities.runtests.SymPyDocTestRunner">SymPyDocTestRunner (class in sympy.utilities.runtests)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/runtests.html#sympy.utilities.runtests.SymPyDocTestRunner.run">SymPyDocTestRunner.run() (in module sympy.utilities.runtests)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.SymPyFiniteField">SymPyFiniteField (class in sympy.polys.domains)</a>
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-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.SymPyIntegerRing">SymPyIntegerRing (class in sympy.polys.domains)</a>
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-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.domains.SymPyRationalField">SymPyRationalField (class in sympy.polys.domains)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/runtests.html#sympy.utilities.runtests.sympytestfile">sympytestfile() (in module sympy.utilities.runtests)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/runtests.html#sympy.utilities.runtests.SymPyTestResults">SymPyTestResults (in module sympy.utilities.runtests)</a>
-  </dt>
-
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-</tr></table>
-
-<h2 id="T">T</h2>
-<table style="width: 100%" class="indextable genindextable"><tr>
-  <td style="width: 33%" valign="top"><dl>
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-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.T">T (in module sympy.physics.quantum.gate)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/matrices/expressions.html#sympy.matrices.expressions.MatrixExpr.T">(sympy.matrices.expressions.MatrixExpr attribute)</a>
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-
-        
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.T">(sympy.matrices.matrices.MatrixBase attribute)</a>
-  </dt>
-
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-      
-  <dt><a href="modules/physics/quantum/shor.html#sympy.physics.quantum.shor.CMod.t">t (sympy.physics.quantum.shor.CMod attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.take">take() (in module sympy.utilities.iterables)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.trigonometric.tan">tan (class in sympy.functions.elementary.trigonometric)</a>
-  </dt>
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-  <dt><a href="modules/mpmath/functions/trigonometric.html#mpmath.tan">tan() (in module mpmath)</a>
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-      
-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.trigonometric.tan.inverse">tan.inverse() (in module sympy.functions.elementary.trigonometric)</a>
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-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.tanh">tanh (class in sympy.functions.elementary.hyperbolic)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/hyperbolic.html#mpmath.tanh">tanh() (in module mpmath)</a>
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-      
-  <dt><a href="modules/functions/elementary.html#sympy.functions.elementary.hyperbolic.tanh.inverse">tanh.inverse() (in module sympy.functions.elementary.hyperbolic)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.TanhSinh">TanhSinh (class in mpmath.calculus.quadrature)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.TanhSinh.calc_nodes">TanhSinh.calc_nodes() (in module mpmath.calculus.quadrature)</a>
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-
-      
-  <dt><a href="modules/mpmath/calculus/integration.html#mpmath.calculus.quadrature.TanhSinh.sum_next">TanhSinh.sum_next() (in module mpmath.calculus.quadrature)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CGate.targets">targets (sympy.physics.quantum.gate.CGate attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.CNotGate.targets">(sympy.physics.quantum.gate.CNotGate attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.Gate.targets">(sympy.physics.quantum.gate.Gate attribute)</a>
-  </dt>
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-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.UGate.targets">(sympy.physics.quantum.gate.UGate attribute)</a>
-  </dt>
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-        
-  <dt><a href="modules/physics/quantum/grover.html#sympy.physics.quantum.grover.OracleGate.targets">(sympy.physics.quantum.grover.OracleGate attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/mpmath/functions/elliptic.html#mpmath.taufrom">taufrom() (in module mpmath)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/calculus/approximation.html#mpmath.taylor">taylor() (in module mpmath)</a>
-  </dt>
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-  <dt><a href="modules/core.html#sympy.core.function.Function.taylor_term">taylor_term() (sympy.core.function.Function class method)</a>
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-
-      
-  <dt><a href="modules/physics/quantum/tensorproduct.html#sympy.physics.quantum.tensorproduct.tensor_product_simp">tensor_product_simp() (in module sympy.physics.quantum.tensorproduct)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/quantum/tensorproduct.html#sympy.physics.quantum.tensorproduct.TensorProduct">TensorProduct (class in sympy.physics.quantum.tensorproduct)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.terms_gcd">terms_gcd() (in module sympy.polys.polytools)</a>
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-
-      
-  <dt><a href="modules/utilities/runtests.html#sympy.utilities.runtests.test">test() (in module sympy.utilities.runtests)</a>
-  </dt>
-
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-  <dt><a href="modules/assumptions/handlers/sets.html#sympy.assumptions.handlers.sets.test_closed_group">test_closed_group() (in module sympy.assumptions.handlers.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/randtest.html#sympy.utilities.randtest.test_derivative_numerically">test_derivative_numerically() (in module sympy.utilities.randtest)</a>
-  </dt>
-
-      
-  <dt><a href="modules/galgebra/GA.html#sympy.galgebra.GA.test_int_flgs">test_int_flgs() (in module sympy.galgebra.GA)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/randtest.html#sympy.utilities.randtest.test_numerically">test_numerically() (in module sympy.utilities.randtest)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.TGate">TGate (class in sympy.physics.quantum.gate)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.ThinLens">ThinLens (class in sympy.physics.gaussopt)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/decorator.html#sympy.utilities.decorator.threaded">threaded() (in module sympy.utilities.decorator)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/decorator.html#sympy.utilities.decorator.threaded_factory">threaded_factory() (in module sympy.utilities.decorator)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/state.html#sympy.physics.quantum.state.TimeDepState.time">time (sympy.physics.quantum.state.TimeDepState attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/timeutils.html#sympy.utilities.timeutils.timed">timed() (in module sympy.utilities.timeutils)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/quantum/state.html#sympy.physics.quantum.state.TimeDepBra">TimeDepBra (class in sympy.physics.quantum.state)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/quantum/state.html#sympy.physics.quantum.state.TimeDepKet">TimeDepKet (class in sympy.physics.quantum.state)</a>
-  </dt>
-
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-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/physics/quantum/state.html#sympy.physics.quantum.state.TimeDepState">TimeDepState (class in sympy.physics.quantum.state)</a>
-  </dt>
-
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-  <dt><a href="modules/mpmath/general.html#mpmath.timing">timing() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/logic.html#sympy.logic.boolalg.to_cnf">to_cnf() (in module sympy.logic.boolalg)</a>
-  </dt>
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-  <dt><a href="modules/polys/reference.html#sympy.polys.numberfields.to_number_field">to_number_field() (in module sympy.polys.numberfields)</a>
-  </dt>
-
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-  <dt><a href="modules/polys/reference.html#sympy.polys.rationaltools.together">together() (in module sympy.polys.rationaltools)</a>
-  </dt>
-
-      
-  <dt><a href="modules/parsing.html#sympy.parsing.sympy_tokenize.TokenError">TokenError (class in sympy.parsing.sympy_tokenize)</a>
-  </dt>
-
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-  <dt><a href="modules/parsing.html#sympy.parsing.sympy_tokenize.tokenize">tokenize() (in module sympy.parsing.sympy_tokenize)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.topological_sort">topological_sort() (in module sympy.utilities.iterables)</a>
-  </dt>
-
-      
-  <dt><a href="modules/ntheory.html#sympy.ntheory.factor_.totient">totient() (in module sympy.ntheory.factor_)</a>
-  </dt>
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-  <dt><a href="modules/matrices/expressions.html#sympy.matrices.expressions.Trace">Trace (class in sympy.matrices.expressions)</a>
-  </dt>
-
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-  <dt><a href="modules/ntheory.html#sympy.ntheory.factor_.trailing">trailing() (in module sympy.ntheory.factor_)</a>
-  </dt>
-
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-  <dt><a href="modules/sets.html#sympy.sets.fancysets.TransformationSet">TransformationSet (class in sympy.sets.fancysets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.transitivity_degree">transitivity_degree (sympy.combinatorics.perm_groups.PermutationGroup attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/matrices/expressions.html#sympy.matrices.expressions.Transpose">Transpose (class in sympy.matrices.expressions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.tree.tree">tree() (in module sympy.printing.tree)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.tree_repr">tree_repr (sympy.combinatorics.prufer.Prufer attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Triangle">Triangle (class in sympy.geometry.polygon)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Triangle.bisectors">Triangle.bisectors() (in module sympy.geometry.polygon)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Triangle.is_equilateral">Triangle.is_equilateral() (in module sympy.geometry.polygon)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Triangle.is_isosceles">Triangle.is_isosceles() (in module sympy.geometry.polygon)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Triangle.is_right">Triangle.is_right() (in module sympy.geometry.polygon)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Triangle.is_scalene">Triangle.is_scalene() (in module sympy.geometry.polygon)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Triangle.is_similar">Triangle.is_similar() (in module sympy.geometry.polygon)</a>
-  </dt>
-
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-  <dt><a href="modules/stats.html#sympy.stats.Triangular">Triangular() (in module sympy.stats)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.gamma_functions.trigamma">trigamma() (in module sympy.functions.special.gamma_functions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.trigintegrate">trigintegrate() (sympy.integrals method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/galgebra/GA/GAsympy.html#trigsimp">trigsimp() (built-in function)</a>, <a href="modules/galgebra/GA/GAsympy.html#trigsimp">[1]</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/simplify/simplify.html#sympy.simplify.simplify.trigsimp">(in module sympy.simplify.simplify)</a>
-  </dt>
-
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-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.trunc">trunc() (in module sympy.polys.polytools)</a>
-  </dt>
-
-      
-  <dt><a href="modules/solvers/solvers.html#sympy.solvers.solvers.tsolve">tsolve() (in module sympy.solvers.solvers)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.containers.Tuple">Tuple (class in sympy.core.containers)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/constants.html#mpmath.mp.twinprime">twinprime (mpmath.mp attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.TwoQubitGate">TwoQubitGate (class in sympy.physics.quantum.gate)</a>
-  </dt>
-
-  </dl></td>
-</tr></table>
-
-<h2 id="U">U</h2>
-<table style="width: 100%" class="indextable genindextable"><tr>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/printing.html#sympy.printing.pretty.pretty_symbology.U">U() (in module sympy.printing.pretty.pretty_symbology)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/autowrap.html#sympy.utilities.autowrap.ufuncify">ufuncify() (in module sympy.utilities.autowrap)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.UGate">UGate (class in sympy.physics.quantum.gate)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.UGate.get_target_matrix">UGate.get_target_matrix() (in module sympy.physics.quantum.gate)</a>
-  </dt>
-
-      
-  <dt><a href="modules/galgebra/GA.html#sympy.galgebra.GA.unabs">unabs() (in module sympy.galgebra.GA)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/spin.html#sympy.physics.quantum.spin.uncouple">uncouple() (in module sympy.physics.quantum.spin)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.relational.Unequality">Unequality (class in sympy.core.relational)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.unflatten">unflatten() (in module sympy.utilities.iterables)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyerrors.UnificationFailed">UnificationFailed (class in sympy.polys.polyerrors)</a>
-  </dt>
-
-      
-  <dt><a href="modules/statistics.html#sympy.statistics.distributions.Uniform">Uniform (class in sympy.statistics.distributions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/stats.html#sympy.stats.Uniform">Uniform() (in module sympy.stats)</a>
-  </dt>
-
-      
-  <dt><a href="modules/statistics.html#sympy.statistics.distributions.Uniform.cdf">Uniform.cdf() (in module sympy.statistics.distributions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/statistics.html#sympy.statistics.distributions.Uniform.confidence">Uniform.confidence() (in module sympy.statistics.distributions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/statistics.html#sympy.statistics.distributions.Uniform.fit">Uniform.fit() (in module sympy.statistics.distributions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/statistics.html#sympy.statistics.distributions.Uniform.pdf">Uniform.pdf() (in module sympy.statistics.distributions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/stats.html#sympy.stats.UniformSum">UniformSum() (in module sympy.stats)</a>
-  </dt>
-
-      
-  <dt><a href="modules/sets.html#sympy.core.sets.Union">Union (class in sympy.core.sets)</a>
-  </dt>
-
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-  <dt><a href="modules/sets.html#sympy.core.sets.Union.as_relational">Union.as_relational() (in module sympy.core.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/sets.html#sympy.core.sets.Union.reduce">Union.reduce() (in module sympy.core.sets)</a>
-  </dt>
-
-  </dl></td>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.uniq">uniq() (in module sympy.utilities.iterables)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/units.html#sympy.physics.units.Unit">Unit (class in sympy.physics.units)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.unit">unit (sympy.polys.polytools.Poly attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/operator.html#sympy.physics.quantum.operator.UnitaryOperator">UnitaryOperator (class in sympy.physics.quantum.operator)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/powers.html#mpmath.unitroots">unitroots() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/internals.html#sympy.polys.polyerrors.UnivariatePolynomialError">UnivariatePolynomialError (class in sympy.polys.polyerrors)</a>
-  </dt>
-
-      
-  <dt><a href="modules/sets.html#sympy.core.sets.UniversalSet">UniversalSet (class in sympy.core.sets)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/graycode.html#sympy.combinatorics.graycode.GrayCode.unrank">unrank() (sympy.combinatorics.graycode.GrayCode class method)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/combinatorics/prufer.html#sympy.combinatorics.prufer.Prufer.unrank">(sympy.combinatorics.prufer.Prufer class method)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.unrank_binary">unrank_binary() (sympy.combinatorics.subsets.Subset class method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/subsets.html#sympy.combinatorics.subsets.Subset.unrank_gray">unrank_gray() (sympy.combinatorics.subsets.Subset class method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.unrank_lex">unrank_lex() (sympy.combinatorics.permutations.Permutation class method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.unrank_nonlex">unrank_nonlex() (sympy.combinatorics.permutations.Permutation class method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation.unrank_trotterjohnson">unrank_trotterjohnson() (sympy.combinatorics.permutations.Permutation class method)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.unrestricted_necklace">unrestricted_necklace() (in module sympy.utilities.iterables)</a>
-  </dt>
-
-      
-  <dt><a href="modules/parsing.html#sympy.parsing.sympy_tokenize.untokenize">untokenize() (in module sympy.parsing.sympy_tokenize)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.AntiSymmetricTensor.upper">upper (sympy.physics.secondquant.AntiSymmetricTensor attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/tensor/indexed.html#sympy.tensor.indexed.Idx.upper">(sympy.tensor.indexed.Idx attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.gamma_functions.uppergamma">uppergamma (class in sympy.functions.special.gamma_functions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/simplify/simplify.html#sympy.simplify.traversaltools.use">use() (in module sympy.simplify.traversaltools)</a>
-  </dt>
-
-  </dl></td>
-</tr></table>
-
-<h2 id="V">V</h2>
-<table style="width: 100%" class="indextable genindextable"><tr>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/core.html#sympy.core.symbol.var">var() (in module sympy.core.symbol)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.VarBosonicBasis">VarBosonicBasis (class in sympy.physics.secondquant)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.VarBosonicBasis.index">VarBosonicBasis.index() (in module sympy.physics.secondquant)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.VarBosonicBasis.state">VarBosonicBasis.state() (in module sympy.physics.secondquant)</a>
-  </dt>
-
-      
-  <dt><a href="modules/concrete.html#sympy.concrete.products.Product.variables">variables (sympy.concrete.products.Product attribute)</a>
-  </dt>
-
-      <dd><dl>
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-  <dt><a href="modules/concrete.html#sympy.concrete.summations.Sum.variables">(sympy.concrete.summations.Sum attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/core.html#sympy.core.function.Lambda.variables">(sympy.core.function.Lambda attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/core.html#sympy.core.function.Subs.variables">(sympy.core.function.Subs attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/integrals/integrals.html#sympy.integrals.Integral.variables">(sympy.integrals.Integral attribute)</a>
-  </dt>
-
-        
-  <dt><a href="modules/physics/quantum/operator.html#sympy.physics.quantum.operator.DifferentialOperator.variables">(sympy.physics.quantum.operator.DifferentialOperator attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/quantum/state.html#sympy.physics.quantum.state.Wavefunction.variables">(sympy.physics.quantum.state.Wavefunction attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/utilities/codegen.html#sympy.utilities.codegen.Routine.variables">(sympy.utilities.codegen.Routine attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/stats.html#sympy.stats.variance">variance() (in module sympy.stats)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/iterables.html#sympy.utilities.iterables.variations">variations() (in module sympy.utilities.iterables)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector">Vector (class in sympy.physics.mechanics.essential)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector.cross">Vector.cross() (in module sympy.physics.mechanics.essential)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector.diff">Vector.diff() (in module sympy.physics.mechanics.essential)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector.doit">Vector.doit() (in module sympy.physics.mechanics.essential)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector.dot">Vector.dot() (in module sympy.physics.mechanics.essential)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector.dt">Vector.dt() (in module sympy.physics.mechanics.essential)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector.express">Vector.express() (in module sympy.physics.mechanics.essential)</a>
-  </dt>
-
-  </dl></td>
-  <td style="width: 33%" valign="top"><dl>
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-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector.magnitude">Vector.magnitude() (in module sympy.physics.mechanics.essential)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector.normalize">Vector.normalize() (in module sympy.physics.mechanics.essential)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector.outer">Vector.outer() (in module sympy.physics.mechanics.essential)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector.simplify">Vector.simplify() (in module sympy.physics.mechanics.essential)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.Vector.subs">Vector.subs() (in module sympy.physics.mechanics.essential)</a>
-  </dt>
-
-      
-  <dt><a href="modules/galgebra/GA.html#sympy.galgebra.GA.vector_fct">vector_fct() (in module sympy.galgebra.GA)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.multidimensional.vectorize">vectorize (class in sympy.core.multidimensional)</a>
-  </dt>
-
-      
-  <dt><a href="modules/combinatorics/polyhedron.html#sympy.combinatorics.polyhedron.Polyhedron.vertices">vertices (sympy.combinatorics.polyhedron.Polyhedron attribute)</a>
-  </dt>
-
-      <dd><dl>
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Polygon.vertices">(sympy.geometry.polygon.Polygon attribute)</a>
-  </dt>
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.RegularPolygon.vertices">(sympy.geometry.polygon.RegularPolygon attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.polygon.Triangle.vertices">(sympy.geometry.polygon.Triangle attribute)</a>
-  </dt>
-
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-      
-  <dt><a href="modules/printing.html#sympy.printing.pretty.pretty_symbology.VF">VF() (in module sympy.printing.pretty.pretty_symbology)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polyfuncs.viete">viete() (in module sympy.polys.polyfuncs)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.pretty.pretty_symbology.vobj">vobj() (in module sympy.printing.pretty.pretty_symbology)</a>
-  </dt>
-
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-  <dt><a href="modules/stats.html#sympy.stats.VonMises">VonMises() (in module sympy.stats)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Circle.vradius">vradius (sympy.geometry.ellipse.Circle attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.ellipse.Ellipse.vradius">(sympy.geometry.ellipse.Ellipse attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.vstack">vstack() (sympy.matrices.matrices.MatrixBase class method)</a>
-  </dt>
-
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-</tr></table>
-
-<h2 id="W">W</h2>
-<table style="width: 100%" class="indextable genindextable"><tr>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.BeamParameter.w">w (sympy.physics.gaussopt.BeamParameter attribute)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.BeamParameter.w_0">w_0 (sympy.physics.gaussopt.BeamParameter attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.waist2rayleigh">waist2rayleigh() (in module sympy.physics.gaussopt)</a>
-  </dt>
-
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-  <dt><a href="modules/physics/gaussopt.html#sympy.physics.gaussopt.BeamParameter.waist_approximation_limit">waist_approximation_limit (sympy.physics.gaussopt.BeamParameter attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/state.html#sympy.physics.quantum.state.Wavefunction">Wavefunction (class in sympy.physics.quantum.state)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/state.html#sympy.physics.quantum.state.Wavefunction.normalize">Wavefunction.normalize() (in module sympy.physics.quantum.state)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/state.html#sympy.physics.quantum.state.Wavefunction.prob">Wavefunction.prob() (in module sympy.physics.quantum.state)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.webere">webere() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/stats.html#sympy.stats.Weibull">Weibull() (in module sympy.stats)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/grover.html#sympy.physics.quantum.grover.WGate">WGate (class in sympy.physics.quantum.grover)</a>
-  </dt>
-
-      
-  <dt><a href="modules/stats.html#sympy.stats.where">where() (in module sympy.stats)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.whitm">whitm() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/bessel.html#mpmath.whitw">whitw() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/secondquant.html#sympy.physics.secondquant.wicks">wicks() (in module sympy.physics.secondquant)</a>
-  </dt>
-
-  </dl></td>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/categories.html#sympy.categories.diagram_drawing.DiagramGrid.width">width (sympy.categories.diagram_drawing.DiagramGrid attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/cg.html#sympy.physics.quantum.cg.Wigner3j">Wigner3j (class in sympy.physics.quantum.cg)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/cg.html#sympy.physics.quantum.cg.Wigner6j">Wigner6j (class in sympy.physics.quantum.cg)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/cg.html#sympy.physics.quantum.cg.Wigner9j">Wigner9j (class in sympy.physics.quantum.cg)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/wigner.html#sympy.physics.wigner.wigner_3j">wigner_3j() (in module sympy.physics.wigner)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/wigner.html#sympy.physics.wigner.wigner_6j">wigner_6j() (in module sympy.physics.wigner)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/wigner.html#sympy.physics.wigner.wigner_9j">wigner_9j() (in module sympy.physics.wigner)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/spin.html#sympy.physics.quantum.spin.WignerD">WignerD (class in sympy.physics.quantum.spin)</a>
-  </dt>
-
-      
-  <dt><a href="modules/stats.html#sympy.stats.WignerSemicircle">WignerSemicircle() (in module sympy.stats)</a>
-  </dt>
-
-      
-  <dt><a href="tutorial/tutorial.en.html#index-11">Wild</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/core.html#sympy.core.symbol.Wild">(class in sympy.core.symbol)</a>
-  </dt>
-
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-      
-  <dt><a href="tutorial/tutorial.en.html#index-11">WildFunction</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/core.html#sympy.core.function.WildFunction">(class in sympy.core.function)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/mpmath/general.html#mpmath.workdps">workdps() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/general.html#mpmath.workprec">workprec() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.dense.wronskian">wronskian() (in module sympy.matrices.dense)</a>
-  </dt>
-
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-</tr></table>
-
-<h2 id="X">X</h2>
-<table style="width: 100%" class="indextable genindextable"><tr>
-  <td style="width: 33%" valign="top"><dl>
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-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.X">X (in module sympy.physics.quantum.gate)</a>
-  </dt>
-
-      
-  <dt><a href="modules/geometry.html#sympy.geometry.point.Point.x">x (sympy.geometry.point.Point attribute)</a>
-  </dt>
-
-      <dd><dl>
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-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.ReferenceFrame.x">(sympy.physics.mechanics.essential.ReferenceFrame attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.XBra">XBra (class in sympy.physics.quantum.cartesian)</a>
-  </dt>
-
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-  <dt><a href="modules/geometry.html#sympy.geometry.line.Ray.xdirection">xdirection (sympy.geometry.line.Ray attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/galgebra/latex_ex/latex_ex.html#xdvi">xdvi() (built-in function)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/galgebra/latex_ex.html#sympy.galgebra.latex_ex.xdvi">(in module sympy.galgebra.latex_ex)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.XGate">XGate (class in sympy.physics.quantum.gate)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.XKet">XKet (class in sympy.physics.quantum.cartesian)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.pretty.pretty_symbology.xobj">xobj() (in module sympy.printing.pretty.pretty_symbology)</a>
-  </dt>
-
-  </dl></td>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.XOp">XOp (class in sympy.physics.quantum.cartesian)</a>
-  </dt>
-
-      
-  <dt><a href="modules/logic.html#sympy.logic.boolalg.Xor">Xor (class in sympy.logic.boolalg)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.pretty.pretty_symbology.xstr">xstr() (in module sympy.printing.pretty.pretty_symbology)</a>
-  </dt>
-
-      
-  <dt><a href="modules/printing.html#sympy.printing.pretty.pretty_symbology.xsym">xsym() (in module sympy.printing.pretty.pretty_symbology)</a>
-  </dt>
-
-      
-  <dt><a href="modules/utilities/decorator.html#sympy.utilities.decorator.xthreaded">xthreaded() (in module sympy.utilities.decorator)</a>
-  </dt>
-
-      
-  <dt><a href="modules/categories.html#sympy.categories.diagram_drawing.xypic_draw_diagram">xypic_draw_diagram() (in module sympy.categories.diagram_drawing)</a>
-  </dt>
-
-      
-  <dt><a href="modules/categories.html#sympy.categories.diagram_drawing.XypicDiagramDrawer">XypicDiagramDrawer (class in sympy.categories.diagram_drawing)</a>
-  </dt>
-
-      
-  <dt><a href="modules/categories.html#sympy.categories.diagram_drawing.XypicDiagramDrawer.draw">XypicDiagramDrawer.draw() (in module sympy.categories.diagram_drawing)</a>
-  </dt>
-
-  </dl></td>
-</tr></table>
-
-<h2 id="Y">Y</h2>
-<table style="width: 100%" class="indextable genindextable"><tr>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.Y">Y (in module sympy.physics.quantum.gate)</a>
-  </dt>
-
-      
-  <dt><a href="modules/geometry.html#sympy.geometry.point.Point.y">y (sympy.geometry.point.Point attribute)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.ReferenceFrame.y">(sympy.physics.mechanics.essential.ReferenceFrame attribute)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/geometry.html#sympy.geometry.line.Ray.ydirection">ydirection (sympy.geometry.line.Ray attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.YGate">YGate (class in sympy.physics.quantum.gate)</a>
-  </dt>
-
-  </dl></td>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.spherical_harmonics.Ylm">Ylm() (in module sympy.functions.special.spherical_harmonics)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.spherical_harmonics.Ylm_c">Ylm_c() (in module sympy.functions.special.spherical_harmonics)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.bessel.yn">yn (class in sympy.functions.special.bessel)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.YOp">YOp (class in sympy.physics.quantum.cartesian)</a>
-  </dt>
-
-  </dl></td>
-</tr></table>
-
-<h2 id="Z">Z</h2>
-<table style="width: 100%" class="indextable genindextable"><tr>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.Z">Z (in module sympy.physics.quantum.gate)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/mechanics/api/essential.html#sympy.physics.mechanics.essential.ReferenceFrame.z">z (sympy.physics.mechanics.essential.ReferenceFrame attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/polys/reference.html#sympy.polys.polytools.Poly.zero">zero (sympy.polys.polytools.Poly attribute)</a>
-  </dt>
-
-      
-  <dt><a href="modules/matrices/expressions.html#sympy.matrices.expressions.ZeroMatrix">ZeroMatrix (class in sympy.matrices.expressions)</a>
-  </dt>
-
-      
-  <dt><a href="modules/matrices/matrices.html#sympy.matrices.dense.zeros">zeros() (in module sympy.matrices.dense)</a>
-  </dt>
-
-      <dd><dl>
-        
-  <dt><a href="modules/matrices/sparse.html#sympy.matrices.sparse.SparseMatrix.zeros">(sympy.matrices.sparse.SparseMatrix class method)</a>
-  </dt>
-
-      </dl></dd>
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.zeta_functions.zeta">zeta (class in sympy.functions.special.zeta_functions)</a>
-  </dt>
-
-  </dl></td>
-  <td style="width: 33%" valign="top"><dl>
-      
-  <dt><a href="modules/mpmath/functions/zeta.html#mpmath.zeta">zeta() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/mpmath/functions/zeta.html#mpmath.zetazero">zetazero() (in module mpmath)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/gate.html#sympy.physics.quantum.gate.ZGate">ZGate (class in sympy.physics.quantum.gate)</a>
-  </dt>
-
-      
-  <dt><a href="modules/functions/special.html#sympy.functions.special.spherical_harmonics.Zlm">Zlm() (in module sympy.functions.special.spherical_harmonics)</a>
-  </dt>
-
-      
-  <dt><a href="modules/core.html#sympy.core.numbers.zoo">zoo() (in module sympy.core.numbers)</a>
-  </dt>
-
-      
-  <dt><a href="modules/physics/quantum/cartesian.html#sympy.physics.quantum.cartesian.ZOp">ZOp (class in sympy.physics.quantum.cartesian)</a>
-  </dt>
-
-  </dl></td>
-</tr></table>
-
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deleted file mode 100644
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-            
-  <div class="section" id="gotchas-and-pitfalls">
-<span id="gotchas"></span><h1>Gotchas and Pitfalls<a class="headerlink" href="#gotchas-and-pitfalls" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="introduction">
-<h2>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline">¶</a></h2>
-<p>SymPy runs under the <a class="reference external" href="http://www.python.org/">Python Programming Language</a>, so there are some things that may behave
-differently than they do in other, independent computer algebra systems
-like Maple or Mathematica.  These are some of the gotchas and pitfalls
-that you may encounter when using SymPy.  See also the <a class="reference external" href="https://github.com/sympy/sympy/wiki/Faq">FAQ</a>, the <a class="reference internal" href="tutorial/tutorial.en.html#tutorial"><em>Tutorial</em></a>, the
-remainder of the SymPy Docs, and the <a class="reference external" href="http://docs.python.org/tutorial/">official Python Tutorial</a>.</p>
-<p>If you are already familiar with C or Java, you might also want to look
-this <a class="reference external" href="http://www.nerdparadise.com/tech/python/4minutecrashcourse/">4 minute Python tutorial</a>.</p>
-<p>Ignore <tt class="docutils literal"><span class="pre">#doctest:</span> <span class="pre">+SKIP</span></tt> in the examples.  That has to do with
-internal testing of the examples.</p>
-</div>
-<div class="section" id="equals-signs">
-<span id="id1"></span><h2>Equals Signs (=)<a class="headerlink" href="#equals-signs" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="single-equals-sign">
-<h3>Single Equals Sign<a class="headerlink" href="#single-equals-sign" title="Permalink to this headline">¶</a></h3>
-<p>The equals sign (<tt class="docutils literal"><span class="pre">=</span></tt>) is the assignment operator, not an equality.  If
-you want to do <span class="math">\(x = y\)</span>, use <tt class="docutils literal"><span class="pre">Eq(x,</span> <span class="pre">y)</span></tt> for equality.
-Alternatively, all expressions are assumed to equal zero, so you can
-just subtract one side and use <tt class="docutils literal"><span class="pre">x</span> <span class="pre">-</span> <span class="pre">y</span></tt>.</p>
-<p>The proper use of the equals sign is to assign expressions to variables.</p>
-<p>For example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">x</span> <span class="o">-</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="go">x - y</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="double-equals-signs">
-<h3>Double Equals Signs<a class="headerlink" href="#double-equals-signs" title="Permalink to this headline">¶</a></h3>
-<p>Double equals signs (<tt class="docutils literal"><span class="pre">==</span></tt>) are used to test equality.  However, this
-tests expressions exactly, not symbolically.  For example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">==</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">==</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>If you want to test for symbolic equality, one way is to subtract one
-expression from the other and run it through functions like
-<a class="reference internal" href="modules/galgebra/GA/GAsympy.html#expand" title="expand"><tt class="xref py py-func docutils literal"><span class="pre">expand()</span></tt></a>, <a class="reference internal" href="modules/galgebra/GA/GAsympy.html#simplify" title="simplify"><tt class="xref py py-func docutils literal"><span class="pre">simplify()</span></tt></a>, and <a class="reference internal" href="modules/galgebra/GA/GAsympy.html#trigsimp" title="trigsimp"><tt class="xref py py-func docutils literal"><span class="pre">trigsimp()</span></tt></a> and see if the
-equation reduces to 0.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">expand</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">-2*sin(x)*cos(x) + sin(2*x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">See also <a class="reference external" href="https://github.com/sympy/sympy/wiki/Faq">Why does SymPy say that two equal expressions are unequal?</a> in the FAQ.</p>
-</div>
-</div>
-</div>
-<div class="section" id="variables">
-<h2>Variables<a class="headerlink" href="#variables" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="variables-assignment-does-not-create-a-relation-between-expressions">
-<h3>Variables Assignment does not Create a Relation Between Expressions<a class="headerlink" href="#variables-assignment-does-not-create-a-relation-between-expressions" title="Permalink to this headline">¶</a></h3>
-<p>When you use <tt class="docutils literal"><span class="pre">=</span></tt> to do assignment, remember that in Python, as in most
-programming languages, the variable does not change if you change the
-value you assigned to it.  The equations you are typing use the values
-present at the time of creation to &#8220;fill in&#8221; values, just like regular
-Python definitions. They are not altered by changes made afterwards.
-Consider the following:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">)</span>  <span class="c"># Symbol, `a`, stored as variable &quot;a&quot;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">1</span>        <span class="c"># an expression involving `a` stored as variable &quot;b&quot;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-<span class="go">a + 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="mi">4</span>            <span class="c"># &quot;a&quot; now points to literal integer 4, not Symbol(&#39;a&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="go">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span>                <span class="c"># &quot;b&quot; is still pointing at the expression involving `a`</span>
-<span class="go">a + 1</span>
-</pre></div>
-</div>
-<p>Changing quantity <tt class="xref py py-obj docutils literal"><span class="pre">a</span></tt> does not change <tt class="xref py py-obj docutils literal"><span class="pre">b</span></tt>; you are not working
-with a set of simultaneous equations. It might be helpful to remember
-that the string that gets printed when you print a variable referring to
-a SymPy object is the string that was given to it when it was created;
-that string does not have to be the same as the variable that you assign
-it to.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">var</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">var</span><span class="p">(</span><span class="s">&#39;rate time short_life&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">r</span><span class="o">*</span><span class="n">t</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-<span class="go">rate*time</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span> <span class="o">=</span> <span class="mi">80</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>         <span class="c"># We haven&#39;t changed d, only r and t</span>
-<span class="go">rate*time</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">r</span><span class="o">*</span><span class="n">t</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>         <span class="c"># Now d is using the current values of r and t</span>
-<span class="go">160</span>
-</pre></div>
-</div>
-<p>If you need variables that have dependence on each other, you can define
-functions.  Use the <tt class="docutils literal"><span class="pre">def</span></tt> operator.  Indent the body of the function.
-See the Python docs for more information on defining functions.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">c</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">var</span><span class="p">(</span><span class="s">&#39;c d&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-<span class="go">c</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-<span class="go">d</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">ctimesd</span><span class="p">():</span>
-<span class="gp">... </span>    <span class="sd">&quot;&quot;&quot;</span>
-<span class="gp">... </span><span class="sd">    This function returns whatever c is times whatever d is.</span>
-<span class="gp">... </span><span class="sd">    &quot;&quot;&quot;</span>
-<span class="gp">... </span>    <span class="k">return</span> <span class="n">c</span><span class="o">*</span><span class="n">d</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ctimesd</span><span class="p">()</span>
-<span class="go">c*d</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ctimesd</span><span class="p">()</span>
-<span class="go">2*d</span>
-</pre></div>
-</div>
-<p>If you define a circular relationship, you will get a
-<tt class="xref py py-exc docutils literal"><span class="pre">RuntimeError</span></tt>.</p>
-<div class="highlight-python"><pre>&gt;&gt;&gt; def a():
-...     return b()
-...
-&gt;&gt;&gt; def b():
-...     return a()
-...
-&gt;&gt;&gt; a()
-Traceback (most recent call last):
-  File &quot;...&quot;, line ..., in ...
-    compileflags, 1) in test.globs
-  File &quot;&lt;...&gt;&quot;, line 1, in &lt;module&gt;
-    a()
-  File &quot;&lt;...&gt;&quot;, line 2, in a
-    return b()
-  File &quot;&lt;...&gt;&quot;, line 2, in b
-    return a()
-  File &quot;&lt;...&gt;&quot;, line 2, in a
-    return b()
-...
-RuntimeError: maximum recursion depth exceeded</pre>
-</div>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">See also <a class="reference external" href="https://github.com/sympy/sympy/wiki/Faq">Why doesn&#8217;t changing one variable change another that depends on it?</a> in the FAQ.</p>
-</div>
-</div>
-<div class="section" id="symbols">
-<span id="id2"></span><h3>Symbols<a class="headerlink" href="#symbols" title="Permalink to this headline">¶</a></h3>
-<p>Symbols are variables, and like all other variables, they need to be
-assigned before you can use them.  For example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">sympy</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span><span class="o">**</span><span class="mi">2</span>  <span class="c"># z is not defined yet </span>
-<span class="gt">Traceback (most recent call last):</span>
-  File <span class="nb">&quot;&lt;stdin&gt;&quot;</span>, line <span class="m">1</span>, in <span class="n">&lt;module&gt;</span>
-<span class="gr">NameError</span>: <span class="n">name &#39;z&#39; is not defined</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sympy</span><span class="o">.</span><span class="n">var</span><span class="p">(</span><span class="s">&#39;z&#39;</span><span class="p">)</span>  <span class="c"># This is the easiest way to define z as a standard symbol</span>
-<span class="go">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">z**2</span>
-</pre></div>
-</div>
-<p>If you use <strong class="command">isympy</strong>, it runs the following commands for you,
-giving you some default Symbols and Functions.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x y z t&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;k m n&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;f g h&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Function</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>You can also import common symbol names from <tt class="xref py py-mod docutils literal"><span class="pre">sympy.abc</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">w</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">w</span>
-<span class="go">w</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">sympy</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">dir</span><span class="p">(</span><span class="n">sympy</span><span class="o">.</span><span class="n">abc</span><span class="p">)</span>  
-<span class="go">[&#39;A&#39;, &#39;B&#39;, &#39;C&#39;, &#39;D&#39;, &#39;E&#39;, &#39;F&#39;, &#39;G&#39;, &#39;H&#39;, &#39;I&#39;, &#39;J&#39;, &#39;K&#39;, &#39;L&#39;, &#39;M&#39;, &#39;N&#39;, &#39;O&#39;,</span>
-<span class="go">&#39;P&#39;, &#39;Q&#39;, &#39;R&#39;, &#39;S&#39;, &#39;Symbol&#39;, &#39;T&#39;, &#39;U&#39;, &#39;V&#39;, &#39;W&#39;, &#39;X&#39;, &#39;Y&#39;, &#39;Z&#39;,</span>
-<span class="go">&#39;__builtins__&#39;, &#39;__doc__&#39;, &#39;__file__&#39;, &#39;__name__&#39;, &#39;__package__&#39;, &#39;_greek&#39;,</span>
-<span class="go">&#39;_latin&#39;, &#39;a&#39;, &#39;alpha&#39;, &#39;b&#39;, &#39;beta&#39;, &#39;c&#39;, &#39;chi&#39;, &#39;d&#39;, &#39;delta&#39;, &#39;e&#39;,</span>
-<span class="go">&#39;epsilon&#39;, &#39;eta&#39;, &#39;f&#39;, &#39;g&#39;, &#39;gamma&#39;, &#39;h&#39;, &#39;i&#39;, &#39;iota&#39;, &#39;j&#39;, &#39;k&#39;, &#39;kappa&#39;,</span>
-<span class="go">&#39;l&#39;, &#39;m&#39;, &#39;mu&#39;, &#39;n&#39;, &#39;nu&#39;, &#39;o&#39;, &#39;omega&#39;, &#39;omicron&#39;, &#39;p&#39;, &#39;phi&#39;, &#39;pi&#39;,</span>
-<span class="go">&#39;psi&#39;, &#39;q&#39;, &#39;r&#39;, &#39;rho&#39;, &#39;s&#39;, &#39;sigma&#39;, &#39;t&#39;, &#39;tau&#39;, &#39;theta&#39;, &#39;u&#39;, &#39;upsilon&#39;,</span>
-<span class="go">&#39;v&#39;, &#39;w&#39;, &#39;x&#39;, &#39;xi&#39;, &#39;y&#39;, &#39;z&#39;, &#39;zeta&#39;]</span>
-</pre></div>
-</div>
-<p>If you want control over the assumptions of the variables, use
-<tt class="xref py py-func docutils literal"><span class="pre">Symbol()</span></tt> and <tt class="xref py py-func docutils literal"><span class="pre">symbols()</span></tt>.  See <a class="reference internal" href="#keyword-arguments"><em>Keyword
-Arguments</em></a> below.</p>
-<p>Lastly, it is recommended that you not use <tt class="xref py py-obj docutils literal"><span class="pre">I</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">E</span></tt>, <a class="reference internal" href="modules/galgebra/GA/GAsympy.html#S" title="S"><tt class="xref py py-obj docutils literal"><span class="pre">S</span></tt></a>,
-<tt class="xref py py-obj docutils literal"><span class="pre">N</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">C</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">O</span></tt>, or <tt class="xref py py-obj docutils literal"><span class="pre">Q</span></tt> for variable or symbol names, as those
-are used for the imaginary unit (<span class="math">\(i\)</span>), the base of the natural
-logarithm (<span class="math">\(e\)</span>), the <tt class="xref py py-func docutils literal"><span class="pre">sympify()</span></tt> function (see <a class="reference internal" href="#symbolic-expressions"><em>Symbolic
-Expressions</em></a> below), numeric evaluation (<tt class="xref py py-func docutils literal"><span class="pre">N()</span></tt>
-is equivalent to <a class="reference internal" href="modules/evalf.html#evalf-label"><em>evalf()</em></a> ), the class registry (for
-things like <tt class="xref py py-func docutils literal"><span class="pre">C.cos()</span></tt>, to prevent cyclic imports in some code),
-the <a class="reference external" href="http://en.wikipedia.org/wiki/Big_O_notation">big O</a> order symbol
-(as in <span class="math">\(O(n\log{n})\)</span>), and the assumptions object that holds a list of
-supported ask keys (such as <tt class="xref py py-obj docutils literal"><span class="pre">Q.real</span></tt>), respectively.  You can use the
-mnemonic <tt class="docutils literal"><span class="pre">QCOSINE</span></tt> to remember what Symbols are defined by default in SymPy.
-Or better yet, always use lowercase letters for Symbol names.  Python will
-not prevent you from overriding default SymPy names or functions, so be
-careful.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>  <span class="c"># cos and pi are a built-in sympy names.</span>
-<span class="go">-1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span> <span class="o">=</span> <span class="mi">3</span>   <span class="c"># Notice that there is no warning for overriding pi.</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">cos(3)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">cos</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>  <span class="c"># No warning for overriding built-in functions either.</span>
-<span class="gp">... </span>    <span class="k">return</span> <span class="mi">5</span><span class="o">*</span><span class="n">x</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">cos</span>  <span class="c"># reimport to restore normal behavior</span>
-</pre></div>
-</div>
-<p>To get a full list of all default names in SymPy do:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">sympy</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">dir</span><span class="p">(</span><span class="n">sympy</span><span class="p">)</span>  
-<span class="go"># A big list of all default sympy names and functions follows.</span>
-<span class="go"># Ignore everything that starts and ends with __.</span>
-</pre></div>
-</div>
-<p>If you have <a class="reference external" href="http://ipython.scipy.org/moin/">iPython</a> installed and
-use <strong class="command">isympy</strong>, you can also press the TAB key to get a list of
-all built-in names and to autocomplete.  Also, see <a class="reference external" href="http://kogs-www.informatik.uni-hamburg.de/~meine/python_tricks">this page</a> for a
-trick for getting tab completion in the regular Python console.</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">See also <a class="reference external" href="https://github.com/sympy/sympy/wiki/Faq">What is the best way to create symbols?</a> in the FAQ.</p>
-</div>
-</div>
-</div>
-<div class="section" id="symbolic-expressions">
-<span id="id3"></span><h2>Symbolic Expressions<a class="headerlink" href="#symbolic-expressions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="python-numbers-vs-sympy-numbers">
-<span id="python-vs-sympy-numbers"></span><h3>Python numbers vs. SymPy Numbers<a class="headerlink" href="#python-numbers-vs-sympy-numbers" title="Permalink to this headline">¶</a></h3>
-<p>SymPy uses its own classes for integers, rational numbers, and floating
-point numbers instead of the default Python <tt class="xref py py-obj docutils literal"><span class="pre">int</span></tt> and <tt class="xref py py-obj docutils literal"><span class="pre">float</span></tt>
-types because it allows for more control.  But you have to be careful.
-If you type an expression that just has numbers in it, it will default
-to a Python expression.  Use the <tt class="xref py py-func docutils literal"><span class="pre">sympify()</span></tt> function, or just
-<a class="reference internal" href="modules/galgebra/GA/GAsympy.html#S" title="S"><tt class="xref py py-func docutils literal"><span class="pre">S()</span></tt></a>, to ensure that something is a SymPy expression.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mf">6.2</span>  <span class="c"># Python float. Notice the floating point accuracy problems. </span>
-<span class="go">6.2000000000000002</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="mf">6.2</span><span class="p">)</span>
-<span class="go">&lt;... &#39;float&#39;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="p">(</span><span class="mf">6.2</span><span class="p">)</span>  <span class="c"># SymPy Float has no such problems because of arbitrary precision.</span>
-<span class="go">6.20000000000000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mf">6.2</span><span class="p">))</span>
-<span class="go">&lt;class &#39;sympy.core.numbers.Float&#39;&gt;</span>
-</pre></div>
-</div>
-<p>If you include numbers in a SymPy expression, they will be sympified
-automatically, but there is one gotcha you should be aware of.  If you
-do <tt class="docutils literal"><span class="pre">&lt;number&gt;/&lt;number&gt;</span></tt> inside of a SymPy expression, Python will
-evaluate the two numbers before SymPy has a chance to get
-to them.  The solution is to <tt class="xref py py-func docutils literal"><span class="pre">sympify()</span></tt> one of the numbers, or use
-<tt class="xref py py-mod docutils literal"><span class="pre">Rational</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>  <span class="c"># evaluates to x**0 or x**0.5 </span>
-<span class="go">x**0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>  <span class="c"># sympyify one of the ints</span>
-<span class="go">sqrt(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">**</span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>  <span class="c"># use the Rational class</span>
-<span class="go">sqrt(x)</span>
-</pre></div>
-</div>
-<p>With a power of <tt class="docutils literal"><span class="pre">1/2</span></tt> you can also use <tt class="docutils literal"><span class="pre">sqrt</span></tt> shorthand:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">==</span> <span class="n">x</span><span class="o">**</span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>If the two integers are not directly separated by a division sign then
-you don&#8217;t have to worry about this problem:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">x**(2*x/3)</span>
-</pre></div>
-</div>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p>A common mistake is copying an expression that is printed and
-reusing it.  If the expression has a <tt class="xref py py-mod docutils literal"><span class="pre">Rational</span></tt> (i.e.,
-<tt class="docutils literal"><span class="pre">&lt;number&gt;/&lt;number&gt;</span></tt>) in it, you will not get the same result,
-obtaining the Python result for the division rather than a SymPy
-Rational.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">solve</span><span class="p">(</span><span class="mi">7</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span><span class="mi">22</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">[22/7]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">22</span><span class="o">/</span><span class="mi">7</span>  <span class="c"># If we just copy and paste we get int 3 or a float </span>
-<span class="go">3.142857142857143</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># One solution is to just assign the expression to a variable</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># if we need to use it again.</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="mi">7</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">22</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
-<span class="go">[22/7]</span>
-</pre></div>
-</div>
-<p>The other solution is to put quotes around the expression
-and run it through S() (i.e., sympify it):</p>
-<div class="last highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="p">(</span><span class="s">&quot;22/7&quot;</span><span class="p">)</span>
-<span class="go">22/7</span>
-</pre></div>
-</div>
-</div>
-<p>Also, if you do not use <strong class="command">isympy</strong>, you could use <tt class="docutils literal"><span class="pre">from</span>
-<span class="pre">__future__</span> <span class="pre">import</span> <span class="pre">division</span></tt> to prevent the <tt class="docutils literal"><span class="pre">/</span></tt> sign from performing
-<a class="reference external" href="http://en.wikipedia.org/wiki/Integer_division">integer division</a>.</p>
-<blockquote>
-<div><div class="highlight-python"><div class="highlight"><pre><span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span>   <span class="c"># With division imported it evaluates to a python float </span>
-<span class="go">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">//</span><span class="mi">2</span>  <span class="c"># You can still achieve integer division with //</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>But be careful: you will now receive floats where you might have desired
-a Rational:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>  
-<span class="go">x**0.5</span>
-</pre></div>
-</div>
-</div></blockquote>
-<p><tt class="xref py py-mod docutils literal"><span class="pre">Rational</span></tt> only works for number/number and is only meant for
-rational numbers.  If you want a fraction with symbols or expressions in
-it, just use <tt class="docutils literal"><span class="pre">/</span></tt>.  If you do number/expression or expression/number,
-then the number will automatically be converted into a SymPy Number.
-You only need to be careful with number/number.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">TypeError: invalid input</span>: <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span><span class="o">/</span><span class="n">x</span>
-<span class="go">2/x</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="evaluating-expressions-with-floats-and-rationals">
-<h3>Evaluating Expressions with Floats and Rationals<a class="headerlink" href="#evaluating-expressions-with-floats-and-rationals" title="Permalink to this headline">¶</a></h3>
-<p>SymPy keeps track of the precision of Floats. The default precision is
-15 digits. When expressions involving Floats are evaluated, the result
-will be expressed to 15 digits of precision but those digits (depending
-on the numbers involved with the calculation) may not all be significant.</p>
-<p>The first issue to keep in mind is how the Float is created: it is created
-with a value and a precision. The precision indicates how precise of a value
-to use when that Float (or an expression it appears in) is evaluated.</p>
-<p>The values can be given as strings, integers, floats, or Rationals.</p>
-<blockquote>
-<div><ul class="simple">
-<li>strings and integers are interpreted as exact</li>
-</ul>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">100.000000000000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="s">&#39;100&#39;</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">100.00</span>
-</pre></div>
-</div>
-<ul class="simple">
-<li>to have the precision match the number of digits, the null string
-can be used for the precision</li>
-</ul>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-<span class="go">100.</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="s">&#39;12.34&#39;</span><span class="p">)</span>
-<span class="go">12.3400000000000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="s">&#39;12.34&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-<span class="go">12.34</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="p">[</span><span class="n">Float</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="p">(</span><span class="s">&#39;0.25&#39;</span><span class="p">,</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">))]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="p">[</span><span class="n">s</span><span class="p">,</span> <span class="n">r</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-<span class="go">0.250</span>
-<span class="go">0.143</span>
-</pre></div>
-</div>
-</div></blockquote>
-<p>Next, notice that each of those values looks correct to 3 digits. But if we try
-to evaluate them to 20 digits, a difference will become apparent:</p>
-<blockquote>
-<div><p>The 0.25 (with precision of 3) represents a number that has a non-repeating
-binary decimal; 1/7 is repeating in binary and decimal &#8211; it cannot be
-represented accurately too far past those first 3 digits (the correct
-decimal is a repeating 142857):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="mi">20</span><span class="p">)</span>
-<span class="go">0.25000000000000000000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="mi">20</span><span class="p">)</span>
-<span class="go">0.14285278320312500000</span>
-</pre></div>
-</div>
-<p>It is important to realize that although a Float is being displayed in
-decimal at aritrary precision, it is actually stored in binary. Once the
-Float is created, its binary information is set at the given precision.
-The accuracy of that value cannot be subsequently changed; so 1/7, at a
-precision of 3 digits, can be padded with binary zeros, but these will
-not make it a more accurate value of 1/7.</p>
-</div></blockquote>
-<p>If inexact, low-precision numbers are involved in a calculation with
-with higher precision values, the evalf engine will increase the precision
-of the low precision values and inexact results will be obtained. This is
-feature of calculations with limited precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="s">&#39;0.1&#39;</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span> <span class="o">+</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;0.1&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">0.2000061035</span>
-</pre></div>
-</div>
-<p>Although the evalf engine tried to maintain 10 digits of precision (since
-that was the highest precision represented) the 3-digit precision used
-limits the accuracy to about 4 digits &#8211; not all the digits you see
-are significant. evalf doesn&#8217;t try to keep track of the number of
-significant digits.</p>
-<p>That very simple expression involving the addition of two numbers with
-different precisions will hopefully be instructive in helping you
-understand why more complicated expressions (like trig expressions that
-may not be simplified) will not evaluate to an exact zero even though,
-with the right simplification, they should be zero. Consider this
-unsimplified trig identity, multiplied by a big number:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">big</span> <span class="o">=</span> <span class="mi">12345678901234567890</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">big_trig_identity</span> <span class="o">=</span> <span class="n">big</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">big</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">big</span><span class="o">*</span><span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">abs</span><span class="p">(</span><span class="n">big_trig_identity</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">.</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span> <span class="o">&gt;</span> <span class="mi">1000</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>When the cos and sin terms were evaluated to 15 digits of precision and
-multiplied by the big number, they gave a large number that was only
-precise to 15 digits (approximately) and when the 20 digit big number
-was subtracted the result was not zero.</p>
-<p>There are three things that will help you obtain more precise numerical
-values for expressions:</p>
-<blockquote>
-<div><p>1) Pass the desired substitutions with the call to evaluate. By doing
-the subs first, the Float values can not be updated as necessary. By
-passing the desired substitutions with the call to evalf the ability
-to re-evaluate as necessary is gained and the results are impressively
-better:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">big_trig_identity</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">{</span><span class="n">x</span><span class="p">:</span> <span class="mf">0.1</span><span class="p">})</span>
-<span class="go">-0.e-91</span>
-</pre></div>
-</div>
-<p>2) Use Rationals, not Floats. During the evaluation process, the
-Rational can be computed to an arbitrary precision while the Float,
-once created &#8211; at a default of 15 digits &#8211; cannot. Compare the
-value of -1.4e+3 above with the nearly zero value obtained when
-replacing x with a Rational representing 1/10 &#8211; before the call
-to evaluate:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">big_trig_identity</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="s">&#39;1/10&#39;</span><span class="p">))</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.e-91</span>
-</pre></div>
-</div>
-<p>3) Try to simplify the expression. In this case, SymPy will recognize
-the trig identity and simplify it to zero so you don&#8217;t even have to
-evaluate it numerically:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">big_trig_identity</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</div></blockquote>
-</div>
-<div class="section" id="immutability-of-expressions">
-<span id="id4"></span><h3>Immutability of Expressions<a class="headerlink" href="#immutability-of-expressions" title="Permalink to this headline">¶</a></h3>
-<p>Expressions in SymPy are immutable, and cannot be modified by an in-place
-operation.  This means that a function will always return an object, and the
-original expression will not be modified. The following example snippet
-demonstrates how this works:</p>
-<div class="highlight-python"><pre>def main():
-    var(&#x27;x y a b&#x27;)
-    expr = 3*x + 4*y
-    print &#x27;original =&#x27;, expr
-    expr_modified = expr.subs({x: a, y: b})
-    print &#x27;modified =&#x27;, expr_modified
-
-if __name__ == &quot;__main__&quot;:
-    main()</pre>
-</div>
-<p>The output shows that the <a class="reference internal" href="modules/galgebra/GA/GAsympy.html#subs" title="subs"><tt class="xref py py-func docutils literal"><span class="pre">subs()</span></tt></a> function has replaced variable
-<tt class="xref py py-obj docutils literal"><span class="pre">x</span></tt> with variable <tt class="xref py py-obj docutils literal"><span class="pre">a</span></tt>, and variable <tt class="xref py py-obj docutils literal"><span class="pre">y</span></tt> with variable <tt class="xref py py-obj docutils literal"><span class="pre">b</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">original</span> <span class="o">=</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">y</span>
-<span class="n">modified</span> <span class="o">=</span> <span class="mi">3</span><span class="o">*</span><span class="n">a</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">b</span>
-</pre></div>
-</div>
-<p>The <a class="reference internal" href="modules/galgebra/GA/GAsympy.html#subs" title="subs"><tt class="xref py py-func docutils literal"><span class="pre">subs()</span></tt></a> function does not modify the original expression <tt class="xref py py-obj docutils literal"><span class="pre">expr</span></tt>.
-Rather, a modified copy of the expression is returned. This returned object
-is stored in the variable <tt class="xref py py-obj docutils literal"><span class="pre">expr_modified</span></tt>. Note that unlike C/C++ and
-other high-level languages, Python does not require you to declare a variable
-before it is used.</p>
-</div>
-<div class="section" id="mathematical-operators">
-<h3>Mathematical Operators<a class="headerlink" href="#mathematical-operators" title="Permalink to this headline">¶</a></h3>
-<p>SymPy uses the same default operators as Python.  Most of these, like
-<tt class="docutils literal"><span class="pre">*/+-</span></tt>, are standard.  Aside from integer division discussed in
-<a class="reference internal" href="#python-vs-sympy-numbers"><em>Python numbers vs. SymPy Numbers</em></a> above,
-you should also be aware that implied multiplication is not allowed. You
-need to use <tt class="docutils literal"><span class="pre">*</span></tt> whenever you wish to multiply something.  Also, to
-raise something to a power, use <tt class="docutils literal"><span class="pre">**</span></tt>, not <tt class="docutils literal"><span class="pre">^</span></tt> as many computer
-algebra systems use.  Parentheses <tt class="docutils literal"><span class="pre">()</span></tt> change operator precedence as
-you would normally expect.</p>
-<p>In <strong class="command">isympy</strong>, with the <strong class="command">ipython</strong> shell:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span><span class="n">x</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">SyntaxError</span>: <span class="n">invalid syntax</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span><span class="o">*</span><span class="n">x</span>
-<span class="go">2*x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">^</span><span class="mi">2</span>  <span class="c"># This is not power.  Use ** instead.</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">TypeError: unsupported operand type(s) for ^</span>: <span class="n">&#39;Add&#39; and &#39;int&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">(x + 1)**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="mi">3</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">    2*x</span>
-<span class="go">   x</span>
-<span class="go">- ----- + 3</span>
-<span class="go">  x + 1</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="inverse-trig-functions">
-<h3>Inverse Trig Functions<a class="headerlink" href="#inverse-trig-functions" title="Permalink to this headline">¶</a></h3>
-<p>SymPy uses different names for some functions than most computer algebra
-systems.  In particular, the inverse trig functions use the python names
-of <tt class="xref py py-func docutils literal"><span class="pre">asin()</span></tt>, <tt class="xref py py-func docutils literal"><span class="pre">acos()</span></tt> and so on instead of the usual <tt class="docutils literal"><span class="pre">arcsin</span></tt>
-and <tt class="docutils literal"><span class="pre">arccos</span></tt>.  Use the methods described in <a class="reference internal" href="#symbols"><em>Symbols</em></a>
-above to see the names of all SymPy functions.</p>
-</div>
-</div>
-<div class="section" id="special-symbols">
-<h2>Special Symbols<a class="headerlink" href="#special-symbols" title="Permalink to this headline">¶</a></h2>
-<p>The symbols <tt class="docutils literal"><span class="pre">[]</span></tt>, <tt class="docutils literal"><span class="pre">{}</span></tt>, <tt class="docutils literal"><span class="pre">=</span></tt>, and <tt class="docutils literal"><span class="pre">()</span></tt> have special meanings in
-Python, and thus in SymPy.  See the Python docs linked to above for
-additional information.</p>
-<div class="section" id="lists">
-<span id="id5"></span><h3>Lists<a class="headerlink" href="#lists" title="Permalink to this headline">¶</a></h3>
-<p>Square brackets <tt class="docutils literal"><span class="pre">[]</span></tt> denote a list.  A list is a container that holds
-any number of different objects.  A list can contain anything, including
-items of different types.  Lists are mutable, which means that you can
-change the elements of a list after it has been created.  You access the
-items of a list also using square brackets, placing them after the list
-or list variable.  Items are numbered using the space before the item.</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">List indexes begin at 0.</p>
-</div>
-<p>Example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>  <span class="c"># A simple list of two items</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
-<span class="go">[x, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>  <span class="c"># This is the first item</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">2</span>  <span class="c"># You can change values of lists after they have been created</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="go">[2, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">solve</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>  <span class="c"># Some functions return lists</span>
-<span class="go">[-1 + sqrt(2), -sqrt(2) - 1]</span>
-</pre></div>
-</div>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">See the Python docs for more information on lists and the square
-bracket notation for accessing elements of a list.</p>
-</div>
-</div>
-<div class="section" id="dictionaries">
-<h3>Dictionaries<a class="headerlink" href="#dictionaries" title="Permalink to this headline">¶</a></h3>
-<p>Curly brackets <tt class="docutils literal"><span class="pre">{}</span></tt> denote a dictionary, or a dict for short.  A
-dictionary is an unordered list of non-duplicate keys and values.  The
-syntax is <tt class="docutils literal"><span class="pre">{key:</span> <span class="pre">value}</span></tt>.  You can access values of keys using square
-bracket notation.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="p">{</span><span class="s">&#39;a&#39;</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;b&#39;</span><span class="p">:</span> <span class="mi">2</span><span class="p">}</span>  <span class="c"># A dictionary.</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span>
-<span class="go">{&#39;a&#39;: 1, &#39;b&#39;: 2}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">]</span>  <span class="c"># How to access items in a dict</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">roots</span><span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>  <span class="c"># Some functions return dicts</span>
-<span class="go">{1: 2, 2: 1}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># Some SymPy functions return dictionaries.  For example,</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># roots returns a dictionary of root:multiplicity items.</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">roots</span><span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="mi">5</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">3</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">{-3: 1, 5: 2}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># This means that the root -3 occurs once and the root 5 occurs twice.</span>
-</pre></div>
-</div>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">See the Python docs for more information on dictionaries.</p>
-</div>
-</div>
-<div class="section" id="tuples">
-<h3>Tuples<a class="headerlink" href="#tuples" title="Permalink to this headline">¶</a></h3>
-<p>Parentheses <tt class="docutils literal"><span class="pre">()</span></tt>, aside from changing operator precedence and their
-use in function calls, (like <tt class="docutils literal"><span class="pre">cos(x)</span></tt>), are also used for tuples.  A
-<tt class="docutils literal"><span class="pre">tuple</span></tt> is identical to a <a class="reference internal" href="#lists"><em>list</em></a>, except that it is not
-mutable.  That means that you can not change their values after they
-have been created.  In general, you will not need tuples in SymPy, but
-sometimes it can be more convenient to type parentheses instead of
-square brackets.</p>
-<blockquote>
-<div><div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>  <span class="c"># Tuples are like lists</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span>
-<span class="go">(1, 2, x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">4</span>  <span class="c"># Except you can not change them after they have been created</span>
-<span class="gt">Traceback (most recent call last):</span>
-  File <span class="nb">&quot;&lt;console&gt;&quot;</span>, line <span class="m">1</span>, in <span class="n">&lt;module&gt;</span>
-<span class="gr">TypeError</span>: <span class="n">&#39;tuple&#39; object does not support item assignment</span>
-</pre></div>
-</div>
-<p>Single element tuples, unlike lists, must have a comma in them:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="p">,)</span>
-<span class="go">(x,)</span>
-</pre></div>
-</div>
-<p>Without the comma, a single expression without a comma is not a tuple:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">x</span>
-</pre></div>
-</div>
-<p>integrate takes a sequence as the second argument if you want to integrate
-with limits (and a tuple or list will work):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">1/3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">1/3</span>
-</pre></div>
-</div>
-</div></blockquote>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">See the Python docs for more information on tuples.</p>
-</div>
-</div>
-<div class="section" id="keyword-arguments">
-<span id="id6"></span><h3>Keyword Arguments<a class="headerlink" href="#keyword-arguments" title="Permalink to this headline">¶</a></h3>
-<p>Aside from the usage described <a class="reference internal" href="#equals-signs"><em>above</em></a>, equals signs
-(<tt class="docutils literal"><span class="pre">=</span></tt>) are also used to give named arguments to functions.  Any
-function that has <tt class="docutils literal"><span class="pre">key=value</span></tt> in its parameters list (see below on how
-to find this out), then <tt class="docutils literal"><span class="pre">key</span></tt> is set to <tt class="docutils literal"><span class="pre">value</span></tt> by default.  You can
-change the value of the key by supplying your own value using the equals
-sign in the function call.  Also, functions that have <tt class="docutils literal"><span class="pre">**</span></tt> followed by
-a name in the parameters list (usually <tt class="docutils literal"><span class="pre">**kwargs</span></tt> or
-<tt class="docutils literal"><span class="pre">**assumptions</span></tt>) allow you to add any number of <tt class="docutils literal"><span class="pre">key=value</span></tt> pairs
-that you want, and they will all be evaluated according to the function.</p>
-<blockquote>
-<div><p>sqrt(x**2) doesn&#8217;t auto simplify to x because x is assumed to be
-complex by default, and, for example, sqrt((-1)**2) == sqrt(1) == 1 != -1:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">sqrt(x**2)</span>
-</pre></div>
-</div>
-<p>Giving assumptions to Symbols is an example of using the keyword argument:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>The square root will now simplify since it knows that x &gt;= 0:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x</span>
-</pre></div>
-</div>
-<p>powsimp has a default argument of combine=&#8217;all&#8217;:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">powsimp</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">m</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="n">n</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="n">m</span><span class="p">))</span>
-<span class="go">     m + n</span>
-<span class="go">(x*y)</span>
-</pre></div>
-</div>
-<p>Setting combine to the default value is the same as not setting it.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">powsimp</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">m</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="n">n</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="n">m</span><span class="p">,</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;all&#39;</span><span class="p">))</span>
-<span class="go">     m + n</span>
-<span class="go">(x*y)</span>
-</pre></div>
-</div>
-<p>The non-default options are &#8216;exp&#8217;, which combines exponents...</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">powsimp</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">m</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="n">n</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="n">m</span><span class="p">,</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;exp&#39;</span><span class="p">))</span>
-<span class="go"> m + n  m + n</span>
-<span class="go">x     *y</span>
-</pre></div>
-</div>
-<p>...and &#8216;base&#8217;, which combines bases.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">powsimp</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">m</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="n">n</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="n">m</span><span class="p">,</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;base&#39;</span><span class="p">))</span>
-<span class="go">     m      n</span>
-<span class="go">(x*y) *(x*y)</span>
-</pre></div>
-</div>
-</div></blockquote>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">See the Python docs for more information on function parameters.</p>
-</div>
-</div>
-</div>
-<div class="section" id="getting-help-from-within-sympy">
-<h2>Getting help from within SymPy<a class="headerlink" href="#getting-help-from-within-sympy" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="help">
-<h3>help()<a class="headerlink" href="#help" title="Permalink to this headline">¶</a></h3>
-<p>Although all docs are available at <a class="reference external" href="http://docs.sympy.org/">docs.sympy.org</a> or on the
-<a class="reference external" href="http://wiki.sympy.org/">SymPy Wiki</a>, you can also get info on functions from within the
-Python interpreter that runs SymPy.  The easiest way to do this is to do
-<tt class="docutils literal"><span class="pre">help(function)</span></tt>, or <tt class="docutils literal"><span class="pre">function?</span></tt> if you are using <strong class="command">ipython</strong>:</p>
-<div class="highlight-python"><pre>In [1]: help(powsimp)  # help() works everywhere
-
-In [2]: # But in ipython, you can also use ?, which is better because it
-In [3]: # it gives you more information
-In [4]: powsimp?</pre>
-</div>
-<p>These will give you the function parameters and docstring for
-<tt class="xref py py-func docutils literal"><span class="pre">powsimp()</span></tt>.  The output will look something like this:</p>
-<span class="target" id="module-sympy.simplify.simplify"></span></div>
-<div class="section" id="source">
-<h3>source()<a class="headerlink" href="#source" title="Permalink to this headline">¶</a></h3>
-<p>Another useful option is the <tt class="xref py py-func docutils literal"><span class="pre">source()</span></tt> function.  This will print
-the source code of a function, including any docstring that it may have.
-You can also do <tt class="docutils literal"><span class="pre">function??</span></tt> in <strong class="command">ipython</strong>.  For example,
-from SymPy 0.6.5:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">source</span><span class="p">(</span><span class="n">simplify</span><span class="p">)</span>  <span class="c"># simplify() is actually only 2 lines of code. </span>
-<span class="go">In file: ./sympy/simplify/simplify.py</span>
-<span class="go">def simplify(expr):</span>
-<span class="go">    &quot;&quot;&quot;Naively simplifies the given expression.</span>
-<span class="go">       ...</span>
-<span class="go">       Simplification is not a well defined term and the exact strategies</span>
-<span class="go">       this function tries can change in the future versions of SymPy. If</span>
-<span class="go">       your algorithm relies on &quot;simplification&quot; (whatever it is), try to</span>
-<span class="go">       determine what you need exactly  -  is it powsimp()? radsimp()?</span>
-<span class="go">       together()?, logcombine()?, or something else? And use this particular</span>
-<span class="go">       function directly, because those are well defined and thus your algorithm</span>
-<span class="go">       will be robust.</span>
-<span class="go">       ...</span>
-<span class="go">    &quot;&quot;&quot;</span>
-<span class="go">    expr = Poly.cancel(powsimp(expr))</span>
-<span class="go">    return powsimp(together(expr.expand()), combine=&#39;exp&#39;, deep=True)</span>
-</pre></div>
-</div>
-</div>
-</div>
-</div>
-
-
-          </div>
-        </div>
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-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="index.html">
-              <img class="logo" src="_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Gotchas and Pitfalls</a><ul>
-<li><a class="reference internal" href="#introduction">Introduction</a></li>
-<li><a class="reference internal" href="#equals-signs">Equals Signs (=)</a><ul>
-<li><a class="reference internal" href="#single-equals-sign">Single Equals Sign</a></li>
-<li><a class="reference internal" href="#double-equals-signs">Double Equals Signs</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#variables">Variables</a><ul>
-<li><a class="reference internal" href="#variables-assignment-does-not-create-a-relation-between-expressions">Variables Assignment does not Create a Relation Between Expressions</a></li>
-<li><a class="reference internal" href="#symbols">Symbols</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#symbolic-expressions">Symbolic Expressions</a><ul>
-<li><a class="reference internal" href="#python-numbers-vs-sympy-numbers">Python numbers vs. SymPy Numbers</a></li>
-<li><a class="reference internal" href="#evaluating-expressions-with-floats-and-rationals">Evaluating Expressions with Floats and Rationals</a></li>
-<li><a class="reference internal" href="#immutability-of-expressions">Immutability of Expressions</a></li>
-<li><a class="reference internal" href="#mathematical-operators">Mathematical Operators</a></li>
-<li><a class="reference internal" href="#inverse-trig-functions">Inverse Trig Functions</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#special-symbols">Special Symbols</a><ul>
-<li><a class="reference internal" href="#lists">Lists</a></li>
-<li><a class="reference internal" href="#dictionaries">Dictionaries</a></li>
-<li><a class="reference internal" href="#tuples">Tuples</a></li>
-<li><a class="reference internal" href="#keyword-arguments">Keyword Arguments</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#getting-help-from-within-sympy">Getting help from within SymPy</a><ul>
-<li><a class="reference internal" href="#help">help()</a></li>
-<li><a class="reference internal" href="#source">source()</a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="tutorial/tutorial.en.html"
-                        title="previous chapter">Tutorial</a></p>
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-  <div class="section" id="sympy-user-s-guide">
-<span id="guide"></span><h1>SymPy User&#8217;s Guide<a class="headerlink" href="#sympy-user-s-guide" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="introduction">
-<h2>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline">¶</a></h2>
-<p>If you are new to SymPy, start with the <a class="reference internal" href="tutorial/tutorial.en.html#tutorial"><em>Tutorial</em></a>. If you went
-through it, now
-it&#8217;s time to learn how SymPy works internally and this is what this guide is
-about. Once you grasp the idea behind SymPy, you will be able to use it
-effectively and also know how to extend it and fix it.
-You may also be just interested in <a class="reference internal" href="modules/index.html#module-docs"><em>SymPy Modules Reference</em></a>.</p>
-</div>
-<div class="section" id="learning-sympy">
-<h2>Learning SymPy<a class="headerlink" href="#learning-sympy" title="Permalink to this headline">¶</a></h2>
-<p>Everyone has different ways of understanding the code written by others.</p>
-<div class="section" id="ondrej-s-approach">
-<h3>Ondřej&#8217;s approach<a class="headerlink" href="#ondrej-s-approach" title="Permalink to this headline">¶</a></h3>
-<p>Let&#8217;s say I&#8217;d like to understand how <tt class="docutils literal"><span class="pre">x</span> <span class="pre">+</span> <span class="pre">y</span> <span class="pre">+</span> <span class="pre">x</span></tt> works and how it is possible
-that it gets simplified to <tt class="docutils literal"><span class="pre">2*x</span> <span class="pre">+</span> <span class="pre">y</span></tt>.</p>
-<p>I write a simple script, I usually call it <tt class="docutils literal"><span class="pre">t.py</span></tt> (I don&#8217;t remember anymore
-why I call it that way):</p>
-<div class="highlight-python"><pre>from sympy.abc import x, y
-
-e = x + y + x
-
-print e</pre>
-</div>
-<p>And I try if it works</p>
-<pre class="literal-block">
-$ <strong class="input">python t.py</strong>
-y + 2*x
-</pre>
-<p>Now I start <a class="reference external" href="http://winpdb.org/">winpdb</a> on it (if you&#8217;ve never used winpdb
-&#8211; it&#8217;s an excellent multiplatform debugger, works on Linux, Windows and Mac OS
-X):</p>
-<pre class="literal-block">
-$ <strong class="input">winpdb t.py</strong>
-y + 2*x
-</pre>
-<p>and a winpdb window will popup, I move to the next line using F6:</p>
-<img alt="_images/winpdb1.png" src="_images/winpdb1.png" />
-<p>Then I step into (F7) and after a little debugging I get for example:</p>
-<img alt="_images/winpdb2.png" src="_images/winpdb2.png" />
-<div class="admonition tip">
-<p class="first admonition-title">Tip</p>
-<p class="last">Make the winpdb window larger on your screen, it was just made smaller
-to fit in this guide.</p>
-</div>
-<p>I see values of all local variables in the left panel, so it&#8217;s very easy to see
-what&#8217;s happening. You can see, that the <tt class="docutils literal"><span class="pre">y</span> <span class="pre">+</span> <span class="pre">2*x</span></tt> is emerging in the <tt class="docutils literal"><span class="pre">obj</span></tt>
-variable. Observing that <tt class="docutils literal"><span class="pre">obj</span></tt> is constructed from <tt class="docutils literal"><span class="pre">c_part</span></tt> and <tt class="docutils literal"><span class="pre">nc_part</span></tt>
-and seeing what <tt class="docutils literal"><span class="pre">c_part</span></tt> contains (<tt class="docutils literal"><span class="pre">y</span></tt> and <tt class="docutils literal"><span class="pre">2*x</span></tt>). So looking at the line
-28 (the whole line is not visible on the screenshot, so here it is):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">c_part</span><span class="p">,</span> <span class="n">nc_part</span><span class="p">,</span> <span class="n">lambda_args</span><span class="p">,</span> <span class="n">order_symbols</span> <span class="o">=</span> <span class="n">cls</span><span class="o">.</span><span class="n">flatten</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">_sympify</span><span class="p">,</span> <span class="n">args</span><span class="p">))</span>
-</pre></div>
-</div>
-<p>you can see that the simplification happens in <tt class="docutils literal"><span class="pre">cls.flatten</span></tt>. Now you can set
-the breakpoint on the line 28, quit winpdb (it will remember the breakpoint),
-start it again, hit F5, this will stop at this breakpoint, hit F7, this will go
-into the function <tt class="docutils literal"><span class="pre">Add.flatten()</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="nd">@classmethod</span>
-<span class="k">def</span> <span class="nf">flatten</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">seq</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    Takes the sequence &quot;seq&quot; of nested Adds and returns a flatten list.</span>
-
-<span class="sd">    Returns: (commutative_part, noncommutative_part, lambda_args,</span>
-<span class="sd">        order_symbols)</span>
-
-<span class="sd">    Applies associativity, all terms are commutable with respect to</span>
-<span class="sd">    addition.</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="n">terms</span> <span class="o">=</span> <span class="p">{}</span>      <span class="c"># term -&gt; coeff</span>
-                    <span class="c"># e.g. x**2 -&gt; 5   for ... + 5*x**2 + ...</span>
-
-    <span class="n">coeff</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>  <span class="c"># standalone term</span>
-                    <span class="c"># e.g. 3 + ...</span>
-    <span class="n">lambda_args</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="n">order_factors</span> <span class="o">=</span> <span class="p">[]</span>
-    <span class="k">while</span> <span class="n">seq</span><span class="p">:</span>
-        <span class="n">o</span> <span class="o">=</span> <span class="n">seq</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>and then you can study how it works. I am going to stop here, this should be
-enough to get you going &#8211; with the above technique, I am able to understand
-almost any Python code.</p>
-</div>
-</div>
-<div class="section" id="sympy-s-architecture">
-<h2>SymPy&#8217;s Architecture<a class="headerlink" href="#sympy-s-architecture" title="Permalink to this headline">¶</a></h2>
-<p>We try to make the sources easily understandable, so you can look into the
-sources and read the doctests, it should be well documented and if you don&#8217;t
-understand something, ask on the <a class="reference external" href="http://groups.google.com/group/sympy">mailinglist</a>.</p>
-<p>You can find all the decisions archived in the <a class="reference external" href="http://code.google.com/p/sympy/issues/list">issues</a>, to see rationale for
-doing this and that.</p>
-<div class="section" id="basics">
-<h3>Basics<a class="headerlink" href="#basics" title="Permalink to this headline">¶</a></h3>
-<p>All symbolic things are implemented using subclasses of the <tt class="docutils literal"><span class="pre">Basic</span></tt> class.
-First, you need to create symbols using <tt class="docutils literal"><span class="pre">Symbol(&quot;x&quot;)</span></tt> or numbers using
-<tt class="docutils literal"><span class="pre">Integer(5)</span></tt> or <tt class="docutils literal"><span class="pre">Float(34.3)</span></tt>. Then you construct the expression using any
-class from SymPy.  For example <tt class="docutils literal"><span class="pre">Add(Symbol(&quot;a&quot;),</span> <span class="pre">Symbol(&quot;b&quot;))</span></tt> gives an
-instance of the <tt class="docutils literal"><span class="pre">Add</span></tt> class.  You can call all methods, which the particular
-class supports.</p>
-<p>For easier use, there is a syntactic sugar for expressions like:</p>
-<p><tt class="docutils literal"><span class="pre">cos(x)</span> <span class="pre">+</span> <span class="pre">1</span></tt> is equal to <tt class="docutils literal"><span class="pre">cos(x).__add__(1)</span></tt> is equal to
-<tt class="docutils literal"><span class="pre">Add(cos(x),</span> <span class="pre">Integer(1))</span></tt></p>
-<p>or</p>
-<p><tt class="docutils literal"><span class="pre">2/cos(x)</span></tt> is equal to <tt class="docutils literal"><span class="pre">cos(x).__rdiv__(2)</span></tt> is equal to
-<tt class="docutils literal"><span class="pre">Mul(Rational(2),</span> <span class="pre">Pow(cos(x),</span> <span class="pre">Rational(-1)))</span></tt>.</p>
-<p>So, you can write normal expressions using python arithmetics like this:</p>
-<div class="highlight-python"><pre>a = Symbol(&quot;a&quot;)
-b = Symbol(&quot;b&quot;)
-e = (a + b)**2
-print e</pre>
-</div>
-<p>but from the sympy point of view, we just need the classes <tt class="docutils literal"><span class="pre">Add</span></tt>, <tt class="docutils literal"><span class="pre">Mul</span></tt>,
-<tt class="docutils literal"><span class="pre">Pow</span></tt>, <tt class="docutils literal"><span class="pre">Rational</span></tt>, <tt class="docutils literal"><span class="pre">Integer</span></tt>.</p>
-</div>
-<div class="section" id="automatic-evaluation-to-canonical-form">
-<h3>Automatic evaluation to canonical form<a class="headerlink" href="#automatic-evaluation-to-canonical-form" title="Permalink to this headline">¶</a></h3>
-<p>For computation, all expressions need to be in a
-canonical form, this is done during the creation of the particular instance
-and only inexpensive operations are performed, necessary to put the expression
-in the
-canonical form.  So the canonical form doesn&#8217;t mean the simplest possible
-expression. The exact list of operations performed depend on the
-implementation.  Obviously, the definition of the canonical form is arbitrary,
-the only requirement is that all equivalent expressions must have the same
-canonical form.  We tried the conversion to a canonical (standard) form to be
-as fast as possible and also in a way so that the result is what you would
-write by hand - so for example <tt class="docutils literal"><span class="pre">b*a</span> <span class="pre">+</span> <span class="pre">-4</span> <span class="pre">+</span> <span class="pre">b</span> <span class="pre">+</span> <span class="pre">a*b</span> <span class="pre">+</span> <span class="pre">4</span> <span class="pre">+</span> <span class="pre">(a</span> <span class="pre">+</span> <span class="pre">b)**2</span></tt> becomes
-<tt class="docutils literal"><span class="pre">2*a*b</span> <span class="pre">+</span> <span class="pre">b</span> <span class="pre">+</span> <span class="pre">(a</span> <span class="pre">+</span> <span class="pre">b)**2</span></tt>.</p>
-<p>Whenever you construct an expression, for example <tt class="docutils literal"><span class="pre">Add(x,</span> <span class="pre">x)</span></tt>, the
-<tt class="docutils literal"><span class="pre">Add.__new__()</span></tt> is called and it determines what to return. In this case:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Add</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">Add</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span>
-<span class="go">2*x</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-<span class="go">&lt;class &#39;sympy.core.mul.Mul&#39;&gt;</span>
-</pre></div>
-</div>
-<p><tt class="docutils literal"><span class="pre">e</span></tt> is actually an instance of <tt class="docutils literal"><span class="pre">Mul(2,</span> <span class="pre">x)</span></tt>, because <tt class="docutils literal"><span class="pre">Add.__new__()</span></tt>
-returned <tt class="docutils literal"><span class="pre">Mul</span></tt>.</p>
-</div>
-<div class="section" id="comparisons">
-<h3>Comparisons<a class="headerlink" href="#comparisons" title="Permalink to this headline">¶</a></h3>
-<p>Expressions can be compared using a regular python syntax:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">==</span> <span class="n">y</span> <span class="o">+</span> <span class="n">x</span>
-<span class="go">True</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">==</span> <span class="n">y</span> <span class="o">-</span> <span class="n">x</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>We made the following decision in SymPy: <tt class="docutils literal"><span class="pre">a</span> <span class="pre">=</span> <span class="pre">Symbol(&quot;x&quot;)</span></tt> and another
-<tt class="docutils literal"><span class="pre">b</span> <span class="pre">=</span> <span class="pre">Symbol(&quot;x&quot;)</span></tt> (with the same string &#8220;x&#8221;) is the same thing, i.e <tt class="docutils literal"><span class="pre">a</span> <span class="pre">==</span> <span class="pre">b</span></tt> is
-<tt class="docutils literal"><span class="pre">True</span></tt>. We chose <tt class="docutils literal"><span class="pre">a</span> <span class="pre">==</span> <span class="pre">b</span></tt>, because it is more natural - <tt class="docutils literal"><span class="pre">exp(x)</span> <span class="pre">==</span> <span class="pre">exp(x)</span></tt> is
-also <tt class="docutils literal"><span class="pre">True</span></tt> for the same instance of <tt class="docutils literal"><span class="pre">x</span></tt> but different instances of <tt class="docutils literal"><span class="pre">exp</span></tt>,
-so we chose to have <tt class="docutils literal"><span class="pre">exp(x)</span> <span class="pre">==</span> <span class="pre">exp(x)</span></tt> even for different instances of <tt class="docutils literal"><span class="pre">x</span></tt>.</p>
-<p>Sometimes, you need to have a unique symbol, for example as a temporary one in
-some calculation, which is going to be substituted for something else at the
-end anyway. This is achieved using <tt class="docutils literal"><span class="pre">Dummy(&quot;x&quot;)</span></tt>. So, to sum it
-up:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Dummy</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span> <span class="o">==</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="go">True</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span> <span class="o">==</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="debugging">
-<h3>Debugging<a class="headerlink" href="#debugging" title="Permalink to this headline">¶</a></h3>
-<p>Starting with 0.6.4, you can turn on/off debug messages with the environment
-variable SYMPY_DEBUG, which is expected to have the values True or False. For
-example, to turn on debugging, you would issue:</p>
-<div class="highlight-python"><pre>[user@localhost]: SYMPY_DEBUG=True ./bin/isympy</pre>
-</div>
-</div>
-<div class="section" id="functionality">
-<h3>Functionality<a class="headerlink" href="#functionality" title="Permalink to this headline">¶</a></h3>
-<p>There are no given requirements on classes in the library. For example, if they
-don&#8217;t implement the <tt class="docutils literal"><span class="pre">fdiff()</span></tt> method and you construct an expression using
-such a class, then trying to use the <tt class="docutils literal"><span class="pre">Basic.series()</span></tt> method will raise an
-exception of not finding the <tt class="docutils literal"><span class="pre">fdiff()</span></tt> method in your class.  This &#8220;duck
-typing&#8221; has an advantage that you just implement the functionality which you
-need.</p>
-<p>You can define the <tt class="docutils literal"><span class="pre">cos</span></tt> class like this:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="k">class</span> <span class="nc">cos</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="k">pass</span>
-</pre></div>
-</div>
-<p>and use it like <tt class="docutils literal"><span class="pre">1</span> <span class="pre">+</span> <span class="pre">cos(x)</span></tt>, but if you don&#8217;t implement the <tt class="docutils literal"><span class="pre">fdiff()</span></tt> method,
-you will not be able to call <tt class="docutils literal"><span class="pre">(1</span> <span class="pre">+</span> <span class="pre">cos(x)).series()</span></tt>.</p>
-<p>The symbolic object is characterized (defined) by the things which it can do,
-so implementing more methods like <tt class="docutils literal"><span class="pre">fdiff()</span></tt>, <tt class="docutils literal"><span class="pre">subs()</span></tt> etc., you are creating
-a &#8220;shape&#8221; of the symbolic object. Useful things to implement in new classes are:
-<tt class="docutils literal"><span class="pre">hash()</span></tt> (to use the class in comparisons), <tt class="docutils literal"><span class="pre">fdiff()</span></tt> (to use it in series
-expansion), <tt class="docutils literal"><span class="pre">subs()</span></tt> (to use it in expressions, where some parts are being
-substituted) and <tt class="docutils literal"><span class="pre">series()</span></tt> (if the series cannot be computed using the
-general <tt class="docutils literal"><span class="pre">Basic.series()</span></tt> method). When you create a new class, don&#8217;t worry
-about this too much - just try to use it in your code and you will realize
-immediately which methods need to be implemented in each situation.</p>
-<p>All objects in sympy are immutable - in the sense that any operation just
-returns a new instance (it can return the same instance only if it didn&#8217;t
-change). This is a common mistake to change the current instance, like
-<tt class="docutils literal"><span class="pre">self.arg</span> <span class="pre">=</span> <span class="pre">self.arg</span> <span class="pre">+</span> <span class="pre">1</span></tt> (wrong!). Use <tt class="docutils literal"><span class="pre">arg</span> <span class="pre">=</span> <span class="pre">self.arg</span> <span class="pre">+</span> <span class="pre">1;</span> <span class="pre">return</span> <span class="pre">arg</span></tt> instead.
-The object is immutable in the
-sense of the symbolic expression it represents. It can modify itself to keep
-track of, for example, its hash. Or it can recalculate anything regarding the
-expression it contains. But the expression cannot be changed. So you can pass
-any instance to other objects, because you don&#8217;t have to worry that it will
-change, or that this would break anything.</p>
-</div>
-<div class="section" id="conclusion">
-<h3>Conclusion<a class="headerlink" href="#conclusion" title="Permalink to this headline">¶</a></h3>
-<p>Above are the main ideas behind SymPy that we try to obey. The rest
-depends on the current implementation and may possibly change in the future.
-The point of all of this is that the interdependencies inside SymPy should be
-kept to a minimum. If one wants to add new functionality to SymPy, all that is
-necessary is to create a subclass of <tt class="docutils literal"><span class="pre">Basic</span></tt> and implement what you want.</p>
-</div>
-<div class="section" id="functions">
-<h3>Functions<a class="headerlink" href="#functions" title="Permalink to this headline">¶</a></h3>
-<p>How to create a new function with one variable:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="k">class</span> <span class="nc">sign</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Basic</span><span class="o">.</span><span class="n">NaN</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Basic</span><span class="o">.</span><span class="n">Zero</span><span class="p">):</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_positive</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-        <span class="k">if</span> <span class="n">arg</span><span class="o">.</span><span class="n">is_negative</span><span class="p">:</span>
-            <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span>
-        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">Basic</span><span class="o">.</span><span class="n">Mul</span><span class="p">):</span>
-            <span class="n">coeff</span><span class="p">,</span> <span class="n">terms</span> <span class="o">=</span> <span class="n">arg</span><span class="o">.</span><span class="n">as_coeff_terms</span><span class="p">()</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">coeff</span><span class="p">,</span> <span class="n">Basic</span><span class="o">.</span><span class="n">One</span><span class="p">):</span>
-                <span class="k">return</span> <span class="n">cls</span><span class="p">(</span><span class="n">coeff</span><span class="p">)</span> <span class="o">*</span> <span class="n">cls</span><span class="p">(</span><span class="n">Basic</span><span class="o">.</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">terms</span><span class="p">))</span>
-
-    <span class="n">is_bounded</span> <span class="o">=</span> <span class="bp">True</span>
-
-    <span class="k">def</span> <span class="nf">_eval_conjugate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="bp">self</span>
-
-    <span class="k">def</span> <span class="nf">_eval_is_zero</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">return</span> <span class="nb">isinstance</span><span class="p">(</span><span class="bp">self</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">Basic</span><span class="o">.</span><span class="n">Zero</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>and that&#8217;s it. The <tt class="docutils literal"><span class="pre">_eval_*</span></tt> functions are called when something is needed.
-The <tt class="docutils literal"><span class="pre">eval</span></tt> is called when the class is about to be instantiated and it
-should return either some simplified instance of some other class or if the
-class should be unmodified, return <tt class="docutils literal"><span class="pre">None</span></tt> (see <tt class="docutils literal"><span class="pre">core/function.py</span></tt> in
-<tt class="docutils literal"><span class="pre">Function.__new__</span></tt> for implementation details). See also tests in
-<a class="reference external" href="https://github.com/sympy/sympy/blob/master/sympy/functions/elementary/tests/test_interface.py">sympy/functions/elementary/tests/test_interface.py</a> that test this interface. You can use them to create your own new functions.</p>
-<p>The applied function <tt class="docutils literal"><span class="pre">sign(x)</span></tt> is constructed using</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">sign</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>both inside and outside of SymPy. Unapplied functions <tt class="docutils literal"><span class="pre">sign</span></tt> is just the class
-itself:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">sign</span>
-</pre></div>
-</div>
-<p>both inside and outside of SymPy. This is the current structure of classes in
-SymPy:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="k">class</span> <span class="nc">BasicType</span><span class="p">(</span><span class="nb">type</span><span class="p">):</span>
-    <span class="k">pass</span>
-<span class="k">class</span> <span class="nc">MetaBasicMeths</span><span class="p">(</span><span class="n">BasicType</span><span class="p">):</span>
-    <span class="o">...</span>
-<span class="k">class</span> <span class="nc">BasicMeths</span><span class="p">(</span><span class="n">AssumeMeths</span><span class="p">):</span>
-    <span class="n">__metaclass__</span> <span class="o">=</span> <span class="n">MetaBasicMeths</span>
-    <span class="o">...</span>
-<span class="k">class</span> <span class="nc">Basic</span><span class="p">(</span><span class="n">BasicMeths</span><span class="p">):</span>
-    <span class="o">...</span>
-<span class="k">class</span> <span class="nc">FunctionClass</span><span class="p">(</span><span class="n">MetaBasicMeths</span><span class="p">):</span>
-    <span class="o">...</span>
-<span class="k">class</span> <span class="nc">Function</span><span class="p">(</span><span class="n">Basic</span><span class="p">,</span> <span class="n">RelMeths</span><span class="p">,</span> <span class="n">ArithMeths</span><span class="p">):</span>
-    <span class="n">__metaclass__</span> <span class="o">=</span> <span class="n">FunctionClass</span>
-    <span class="o">...</span>
-</pre></div>
-</div>
-<p>The exact names of the classes and the names of the methods and how they work
-can be changed in the future.</p>
-<p>This is how to create a function with two variables:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="k">class</span> <span class="nc">chebyshevt_root</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-
-    <span class="nd">@classmethod</span>
-    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-        <span class="k">if</span> <span class="ow">not</span> <span class="mi">0</span> <span class="o">&lt;=</span> <span class="n">k</span> <span class="o">&lt;</span> <span class="n">n</span><span class="p">:</span>
-            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&quot;must have 0 &lt;= k &lt; n&quot;</span><span class="p">)</span>
-        <span class="k">return</span> <span class="n">C</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Pi</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="p">))</span>
-</pre></div>
-</div>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">the first argument of a &#64;classmethod should be <tt class="docutils literal"><span class="pre">cls</span></tt> (i.e. not
-<tt class="docutils literal"><span class="pre">self</span></tt>).</p>
-</div>
-<p>Here it&#8217;s how to define a derivative of the function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">sympify</span><span class="p">,</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">my_function</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-<span class="gp">...</span>
-<span class="gp">... </span>    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span> <span class="o">=</span> <span class="mi">1</span><span class="p">):</span>
-<span class="gp">... </span>        <span class="k">return</span> <span class="n">cos</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-<span class="gp">...</span>
-<span class="gp">... </span>    <span class="nd">@classmethod</span>
-<span class="gp">... </span>    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-<span class="gp">... </span>        <span class="n">arg</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-<span class="gp">... </span>        <span class="k">if</span> <span class="n">arg</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-<span class="gp">... </span>            <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>So guess what this <tt class="docutils literal"><span class="pre">my_function</span></tt> is going to be? Well, it&#8217;s derivative is
-<tt class="docutils literal"><span class="pre">cos</span></tt> and the function value at 0 is 0, but let&#8217;s pretend we don&#8217;t know:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">my_function</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
-<span class="go">     3     5     7       9</span>
-<span class="go">    x     x     x       x       / 10\</span>
-<span class="go">x - -- + --- - ---- + ------ + O\x  /</span>
-<span class="go">    6    120   5040   362880</span>
-</pre></div>
-</div>
-<p>Looks familiar indeed:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
-<span class="go">     3     5     7       9</span>
-<span class="go">    x     x     x       x       / 10\</span>
-<span class="go">x - -- + --- - ---- + ------ + O\x  /</span>
-<span class="go">    6    120   5040   362880</span>
-</pre></div>
-</div>
-<p>Let&#8217;s try a more complicated example. Let&#8217;s define the derivative in terms of
-the function itself:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">what_am_i</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-<span class="gp">...</span>
-<span class="gp">... </span>    <span class="k">def</span> <span class="nf">fdiff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">argindex</span> <span class="o">=</span> <span class="mi">1</span><span class="p">):</span>
-<span class="gp">... </span>        <span class="k">return</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">what_am_i</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">**</span><span class="mi">2</span>
-<span class="gp">...</span>
-<span class="gp">... </span>    <span class="nd">@classmethod</span>
-<span class="gp">... </span>    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">arg</span><span class="p">):</span>
-<span class="gp">... </span>        <span class="n">arg</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">arg</span><span class="p">)</span>
-<span class="gp">... </span>        <span class="k">if</span> <span class="n">arg</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-<span class="gp">... </span>            <span class="k">return</span> <span class="n">sympify</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>So what is <tt class="docutils literal"><span class="pre">what_am_i</span></tt>?  Let&#8217;s try it:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">what_am_i</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
-<span class="go">     3      5       7       9</span>
-<span class="go">    x    2*x    17*x    62*x     / 10\</span>
-<span class="go">x - -- + ---- - ----- + ----- + O\x  /</span>
-<span class="go">    3     15     315     2835</span>
-</pre></div>
-</div>
-<p>Well, it&#8217;s <tt class="docutils literal"><span class="pre">tanh</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">tanh</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">tanh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
-<span class="go">     3      5       7       9</span>
-<span class="go">    x    2*x    17*x    62*x     / 10\</span>
-<span class="go">x - -- + ---- - ----- + ----- + O\x  /</span>
-<span class="go">    3     15     315     2835</span>
-</pre></div>
-</div>
-<p>The new functions we just defined are regular SymPy objects, you
-can use them all over SymPy, e.g.:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">limit</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">what_am_i</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="common-tasks">
-<h3>Common tasks<a class="headerlink" href="#common-tasks" title="Permalink to this headline">¶</a></h3>
-<p>Please use the same way as is shown below all across SymPy.</p>
-<p><strong>accessing parameters</strong>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sign</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">sign</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">args</span>
-<span class="go">(x**2,)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">x**2</span>
-
-<span class="go">Number arguments (in Adds and Muls) will always be the first argument;</span>
-<span class="go">other arguments might be in arbitrary order:</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="ow">in</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
-<span class="go">True</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">y</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">args</span>
-<span class="go">(y, z)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">y</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">args</span>
-<span class="go">(y*z,)</span>
-</pre></div>
-</div>
-<p>Never use internal methods or variables, prefixed with &#8220;<tt class="docutils literal"><span class="pre">_</span></tt>&#8221; (example: don&#8217;t
-use <tt class="docutils literal"><span class="pre">_args</span></tt>, use <tt class="docutils literal"><span class="pre">.args</span></tt> instead).</p>
-<p><strong>testing the structure of a SymPy expression</strong></p>
-<p>Applied functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sign</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">Function</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">sign</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">sign</span><span class="p">)</span>
-<span class="go">True</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-<span class="go">False</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">Function</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>So <tt class="docutils literal"><span class="pre">e</span></tt> is a <tt class="docutils literal"><span class="pre">sign(z)</span></tt> function, but not <tt class="docutils literal"><span class="pre">exp(z)</span></tt> function.</p>
-<p>Unapplied functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sign</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">FunctionClass</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">sign</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">exp</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">Add</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">isinstance</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">FunctionClass</span><span class="p">)</span>
-<span class="go">True</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">isinstance</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">FunctionClass</span><span class="p">)</span>
-<span class="go">True</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">isinstance</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">FunctionClass</span><span class="p">)</span>
-<span class="go">False</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="ow">is</span> <span class="n">Add</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>So <tt class="docutils literal"><span class="pre">e</span></tt> and <tt class="docutils literal"><span class="pre">f</span></tt> are functions, <tt class="docutils literal"><span class="pre">g</span></tt> is not a function.</p>
-</div>
-</div>
-<div class="section" id="contributing">
-<h2>Contributing<a class="headerlink" href="#contributing" title="Permalink to this headline">¶</a></h2>
-<p>We welcome every SymPy user to participate in it&#8217;s development. Don&#8217;t worry if
-you&#8217;ve never contributed to any open source project, we&#8217;ll help you learn
-anything necessary, just ask on our <a class="reference external" href="http://groups.google.com/group/sympy">mailinglist</a>.</p>
-<p>Don&#8217;t be afraid to ask anything and don&#8217;t worry that you are wasting our time
-if you are new to SymPy and ask questions that maybe most of the people know
-the answer to &#8211; you are not, because that&#8217;s exactly what the <a class="reference external" href="http://groups.google.com/group/sympy">mailinglist</a> is
-for and people answer your emails because they want to. Also we try hard to
-answer every email, so you&#8217;ll always get some feedback and pointers what to do
-next.</p>
-<div class="section" id="improving-the-code">
-<h3>Improving the code<a class="headerlink" href="#improving-the-code" title="Permalink to this headline">¶</a></h3>
-<p>Go to <a class="reference external" href="http://code.google.com/p/sympy/issues/list">issues</a> that are sorted by priority and simply find something that you
-would like to get fixed and fix it. If you find something odd, please report it
-into <a class="reference external" href="http://code.google.com/p/sympy/issues/list">issues</a> first before fixing it. Feel free to consult with us on the
-<a class="reference external" href="http://groups.google.com/group/sympy">mailinglist</a>.  Then send your patch either to the <a class="reference external" href="http://code.google.com/p/sympy/issues/list">issues</a> or the <a class="reference external" href="http://groups.google.com/group/sympy">mailinglist</a>.
-See the <span class="math">\(SymPy Development
-&lt;https://github.com/sympy/sympy/wiki/old-wiki-Sympy-Development&gt;\)</span> wiki page,
-but don&#8217;t worry about it too much if you find it too formal - simply get in touch
-with us on the <a class="reference external" href="http://groups.google.com/group/sympy">mailinglist</a> and we&#8217;ll help you get your patch accepted.</p>
-<p>Please read our excellent <a class="reference external" href="https://github.com/sympy/sympy/wiki/Development-workflow">SymPy Patches Tutorial</a> at our
-wiki for a guide on how to write patches to SymPy, how to work with Git,
-and how to make your life easier as you get started with SymPy.</p>
-</div>
-<div class="section" id="improving-the-docs">
-<h3>Improving the docs<a class="headerlink" href="#improving-the-docs" title="Permalink to this headline">¶</a></h3>
-<p>Please see <a class="reference internal" href="modules/index.html#module-docs"><em>the documentation</em></a> how to fix and improve
-SymPy&#8217;s documentation. All contribution is very welcome.</p>
-</div>
-</div>
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-  <h3><a href="index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">SymPy User&#8217;s Guide</a><ul>
-<li><a class="reference internal" href="#introduction">Introduction</a></li>
-<li><a class="reference internal" href="#learning-sympy">Learning SymPy</a><ul>
-<li><a class="reference internal" href="#ondrej-s-approach">Ondřej&#8217;s approach</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#sympy-s-architecture">SymPy&#8217;s Architecture</a><ul>
-<li><a class="reference internal" href="#basics">Basics</a></li>
-<li><a class="reference internal" href="#automatic-evaluation-to-canonical-form">Automatic evaluation to canonical form</a></li>
-<li><a class="reference internal" href="#comparisons">Comparisons</a></li>
-<li><a class="reference internal" href="#debugging">Debugging</a></li>
-<li><a class="reference internal" href="#functionality">Functionality</a></li>
-<li><a class="reference internal" href="#conclusion">Conclusion</a></li>
-<li><a class="reference internal" href="#functions">Functions</a></li>
-<li><a class="reference internal" href="#common-tasks">Common tasks</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#contributing">Contributing</a><ul>
-<li><a class="reference internal" href="#improving-the-code">Improving the code</a></li>
-<li><a class="reference internal" href="#improving-the-docs">Improving the docs</a></li>
-</ul>
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-<p><a class="reference external" href="http://sympy.org">SymPy</a> is a Python library for symbolic mathematics.
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-            
-  <div class="section" id="installation">
-<h1>Installation<a class="headerlink" href="#installation" title="Permalink to this headline">¶</a></h1>
-<p>The SymPy CAS can be installed on virtually any computer with Python 2.5 or
-above. SymPy does not require any special Python modules: let us know if you
-have any problems with SymPy on a standard Python install. The current
-recommended method of installation is directly from the source files.
-Alternatively, executables are available for Windows, and some Linux
-distributions have SymPy packages available.</p>
-<div class="section" id="source">
-<h2>Source<a class="headerlink" href="#source" title="Permalink to this headline">¶</a></h2>
-<p>SymPy currently recommends that users install directly from the source files.
-You will first have to download the source files via the archive. Download the
-latest release (tar.gz) from the <a class="reference external" href="https://code.google.com/p/sympy/downloads/list">downloads site</a> and open it with your
-operating system&#8217;s standard decompression utility.</p>
-<p>When downloading the archive, make sure to get the correct version (Python 2 or
-Python 3). If you&#8217;re not sure which one to use, you probably want the Python 2
-version. Note that you can install both if you want.</p>
-<p>After the download is complete, you should have a folder called &#8220;sympy&#8221;. From
-your favorite command line terminal, change directory into that folder and
-execute the following:</p>
-<div class="highlight-python"><pre>$ python setup.py install</pre>
-</div>
-<p>Alternatively, if you don&#8217;t want to install the package onto your computer, you
-may run SymPy with the &#8220;isympy&#8221; console (which automatically imports SymPy
-packages and defines common symbols) by executing within the &#8220;sympy&#8221; folder:</p>
-<div class="highlight-python"><pre>$ ./bin/isympy</pre>
-</div>
-<p>You may now run SymPy statements directly within the Python shell:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">t</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x y z t&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;k m n&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;f g h&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Function</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">x</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="git">
-<h2>Git<a class="headerlink" href="#git" title="Permalink to this headline">¶</a></h2>
-<p>If you are a developer or like to get the latest updates as they come, be sure
-to install from git. To download the repository, execute the following from the
-command line:</p>
-<div class="highlight-python"><pre>$ git clone git://github.com/sympy/sympy.git</pre>
-</div>
-<p>Then, execute either the setup.py or the bin/isympy scripts as demonstrated
-above.</p>
-<p>To update to the latest version, go into your repository and execute:</p>
-<div class="highlight-python"><pre>$ git pull origin master</pre>
-</div>
-<p>If you want to install SymPy, but still want to use the git version, you can run
-from your repository:</p>
-<div class="highlight-python"><pre>$ setupegg.py develop</pre>
-</div>
-<p>This will cause the installed version to always point to the version in the git
-directory.</p>
-<p>If you&#8217;re using the git repository with Python 3, you have to use the
-<tt class="docutils literal"><span class="pre">./bin/use2to3</span></tt> script to build the Python 3 version of SymPy. This will put
-everything in the py3k-sympy directory.</p>
-</div>
-<div class="section" id="other-methods">
-<h2>Other Methods<a class="headerlink" href="#other-methods" title="Permalink to this headline">¶</a></h2>
-<p>An installation executable is available for Windows users at the
-<a class="reference external" href="https://code.google.com/p/sympy/downloads/list">downloads site</a> (.exe). In addition, various Linux distributions have SymPy
-available as a package. Others are strongly encouraged to download from source
-(details above).</p>
-</div>
-<div class="section" id="run-sympy">
-<h2>Run SymPy<a class="headerlink" href="#run-sympy" title="Permalink to this headline">¶</a></h2>
-<p>After installation, it is best to verify that your freshly-installed SymPy
-works. To do this, start up Python and import the SymPy libraries:</p>
-<div class="highlight-python"><pre>$ python
-&gt;&gt;&gt; from sympy import *</pre>
-</div>
-<p>From here, execute some simple SymPy statements like the ones below:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">log(x)</span>
-</pre></div>
-</div>
-<p>For a starter guide on using SymPy effectively, refer to the <a class="reference internal" href="tutorial/tutorial.en.html#tutorial"><em>Tutorial</em></a>.</p>
-</div>
-<div class="section" id="questions">
-<h2>Questions<a class="headerlink" href="#questions" title="Permalink to this headline">¶</a></h2>
-<p>If you have a question about installation or SymPy in general, feel free to
-visit the IRC channel at irc.freenode.net, channel <a class="reference external" href="irc://irc.freenode.net/sympy">#sympy</a>. In addition,
-our <a class="reference external" href="http://groups.google.com/group/sympy">mailing list</a> is an excellent source of community support.</p>
-<p>If you think there&#8217;s a bug or you would like to request a feature, please open
-an <a class="reference external" href="http://code.google.com/p/sympy/issues/list">issue ticket</a>.</p>
-</div>
-</div>
-
-
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-              <img class="logo" src="_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Installation</a><ul>
-<li><a class="reference internal" href="#source">Source</a></li>
-<li><a class="reference internal" href="#git">Git</a></li>
-<li><a class="reference internal" href="#other-methods">Other Methods</a></li>
-<li><a class="reference internal" href="#run-sympy">Run SymPy</a></li>
-<li><a class="reference internal" href="#questions">Questions</a></li>
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-</li>
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-
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diff --git a/dev-py3k/modules/assumptions/ask.html b/dev-py3k/modules/assumptions/ask.html
deleted file mode 100644
index 14bf7616cc1..00000000000
--- a/dev-py3k/modules/assumptions/ask.html
+++ /dev/null
@@ -1,232 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
-
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-<html xmlns="http://www.w3.org/1999/xhtml">
-  <head>
-    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
-    
-    <title>Ask &mdash; SymPy 0.7.2-git documentation</title>
-    
-    <link rel="stylesheet" href="../../_static/default.css" type="text/css" />
-    <link rel="stylesheet" href="../../_static/pygments.css" type="text/css" />
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-    <link rel="up" title="Assumptions module" href="../assumptions/index.html" />
-    <link rel="next" title="Assume" href="../assumptions/assume.html" />
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-            
-  <div class="section" id="module-sympy.assumptions.ask">
-<span id="ask"></span><h1>Ask<a class="headerlink" href="#module-sympy.assumptions.ask" title="Permalink to this headline">¶</a></h1>
-<p>Module for querying SymPy objects about assumptions.</p>
-<dl class="class">
-<dt id="sympy.assumptions.ask.Q">
-<em class="property">class </em><tt class="descclassname">sympy.assumptions.ask.</tt><tt class="descname">Q</tt><a class="reference internal" href="../../_modules/sympy/assumptions/ask.html#Q"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.ask.Q" title="Permalink to this definition">¶</a></dt>
-<dd><p>Supported ask keys.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.ask.ask">
-<tt class="descclassname">sympy.assumptions.ask.</tt><tt class="descname">ask</tt><big>(</big><em>proposition</em>, <em>assumptions=True</em>, <em>context=AssumptionsContext()</em><big>)</big><a class="reference internal" href="../../_modules/sympy/assumptions/ask.html#ask"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.ask.ask" title="Permalink to this definition">¶</a></dt>
-<dd><p>Method for inferring properties about objects.</p>
-<p><strong>Syntax</strong></p>
-<blockquote>
-<div><ul>
-<li><p class="first">ask(proposition)</p>
-</li>
-<li><p class="first">ask(proposition, assumptions)</p>
-<blockquote>
-<div><p>where <tt class="docutils literal"><span class="pre">proposition</span></tt> is any boolean expression</p>
-</div></blockquote>
-</li>
-</ul>
-</div></blockquote>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">ask</span><span class="p">,</span> <span class="n">Q</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">rational</span><span class="p">(</span><span class="n">pi</span><span class="p">))</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">&amp;</span> <span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">prime</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">&amp;</span>  <span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<dl class="docutils">
-<dt><strong>Remarks</strong></dt>
-<dd><p class="first">Relations in assumptions are not implemented (yet), so the following
-will not give a meaningful result.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">is_true</span><span class="p">(</span><span class="n">x</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">))</span> 
-</pre></div>
-</div>
-<p class="last">It is however a work in progress.</p>
-</dd>
-</dl>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.ask.ask_full_inference">
-<tt class="descclassname">sympy.assumptions.ask.</tt><tt class="descname">ask_full_inference</tt><big>(</big><em>proposition</em>, <em>assumptions</em>, <em>known_facts_cnf</em><big>)</big><a class="reference internal" href="../../_modules/sympy/assumptions/ask.html#ask_full_inference"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.ask.ask_full_inference" title="Permalink to this definition">¶</a></dt>
-<dd><p>Method for inferring properties about objects.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.ask.compute_known_facts">
-<tt class="descclassname">sympy.assumptions.ask.</tt><tt class="descname">compute_known_facts</tt><big>(</big><em>known_facts</em>, <em>known_facts_keys</em><big>)</big><a class="reference internal" href="../../_modules/sympy/assumptions/ask.html#compute_known_facts"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.ask.compute_known_facts" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the various forms of knowledge compilation used by the
-assumptions system.</p>
-<p>This function is typically applied to the variables
-<tt class="docutils literal"><span class="pre">known_facts</span></tt> and <tt class="docutils literal"><span class="pre">known_facts_keys</span></tt> defined at the bottom of
-this file.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.ask.register_handler">
-<tt class="descclassname">sympy.assumptions.ask.</tt><tt class="descname">register_handler</tt><big>(</big><em>key</em>, <em>handler</em><big>)</big><a class="reference internal" href="../../_modules/sympy/assumptions/ask.html#register_handler"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.ask.register_handler" title="Permalink to this definition">¶</a></dt>
-<dd><p>Register a handler in the ask system. key must be a string and handler a
-class inheriting from AskHandler:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.assumptions</span> <span class="kn">import</span> <span class="n">register_handler</span><span class="p">,</span> <span class="n">ask</span><span class="p">,</span> <span class="n">Q</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.assumptions.handlers</span> <span class="kn">import</span> <span class="n">AskHandler</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">MersenneHandler</span><span class="p">(</span><span class="n">AskHandler</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="c"># Mersenne numbers are in the form 2**n + 1, n integer</span>
-<span class="gp">... </span>    <span class="nd">@staticmethod</span>
-<span class="gp">... </span>    <span class="k">def</span> <span class="nf">Integer</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-<span class="gp">... </span>        <span class="kn">import</span> <span class="nn">math</span>
-<span class="gp">... </span>        <span class="k">return</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">expr</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">register_handler</span><span class="p">(</span><span class="s">&#39;mersenne&#39;</span><span class="p">,</span> <span class="n">MersenneHandler</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">mersenne</span><span class="p">(</span><span class="mi">7</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.ask.remove_handler">
-<tt class="descclassname">sympy.assumptions.ask.</tt><tt class="descname">remove_handler</tt><big>(</big><em>key</em>, <em>handler</em><big>)</big><a class="reference internal" href="../../_modules/sympy/assumptions/ask.html#remove_handler"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.ask.remove_handler" title="Permalink to this definition">¶</a></dt>
-<dd><p>Removes a handler from the ask system. Same syntax as register_handler</p>
-</dd></dl>
-
-</div>
-
-
-          </div>
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-  <p class="topless"><a href="../assumptions/index.html"
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-            
-  <div class="section" id="module-sympy.assumptions.assume">
-<span id="assume"></span><h1>Assume<a class="headerlink" href="#module-sympy.assumptions.assume" title="Permalink to this headline">¶</a></h1>
-<dl class="class">
-<dt id="sympy.assumptions.assume.AppliedPredicate">
-<em class="property">class </em><tt class="descclassname">sympy.assumptions.assume.</tt><tt class="descname">AppliedPredicate</tt><a class="reference internal" href="../../_modules/sympy/assumptions/assume.html#AppliedPredicate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.assume.AppliedPredicate" title="Permalink to this definition">¶</a></dt>
-<dd><p>The class of expressions resulting from applying a Predicate.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Q</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">Q.integer(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">&lt;class &#39;sympy.assumptions.assume.AppliedPredicate&#39;&gt;</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.assumptions.assume.AppliedPredicate.arg">
-<tt class="descname">arg</tt><a class="reference internal" href="../../_modules/sympy/assumptions/assume.html#AppliedPredicate.arg"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.assume.AppliedPredicate.arg" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the expression used by this assumption.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Q</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">arg</span>
-<span class="go">x + 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.assumptions.assume.AssumptionsContext">
-<em class="property">class </em><tt class="descclassname">sympy.assumptions.assume.</tt><tt class="descname">AssumptionsContext</tt><a class="reference internal" href="../../_modules/sympy/assumptions/assume.html#AssumptionsContext"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.assume.AssumptionsContext" title="Permalink to this definition">¶</a></dt>
-<dd><p>Set representing assumptions.</p>
-<p>This is used to represent global assumptions, but you can also use this
-class to create your own local assumptions contexts. It is basically a thin
-wrapper to Python&#8217;s set, so see its documentation for advanced usage.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">global_assumptions</span><span class="p">,</span> <span class="n">AppliedPredicate</span><span class="p">,</span> <span class="n">Q</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">global_assumptions</span>
-<span class="go">AssumptionsContext()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">global_assumptions</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">global_assumptions</span>
-<span class="go">AssumptionsContext([Q.real(x)])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">global_assumptions</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">global_assumptions</span>
-<span class="go">AssumptionsContext()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">global_assumptions</span><span class="o">.</span><span class="n">clear</span><span class="p">()</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.assumptions.assume.AssumptionsContext.add">
-<tt class="descname">add</tt><big>(</big><em>self</em>, <em>*assumptions</em><big>)</big><a class="reference internal" href="../../_modules/sympy/assumptions/assume.html#AssumptionsContext.add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.assume.AssumptionsContext.add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Add an assumption.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.assumptions.assume.Predicate">
-<em class="property">class </em><tt class="descclassname">sympy.assumptions.assume.</tt><tt class="descname">Predicate</tt><a class="reference internal" href="../../_modules/sympy/assumptions/assume.html#Predicate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.assume.Predicate" title="Permalink to this definition">¶</a></dt>
-<dd><p>A predicate is a function that returns a boolean value.</p>
-<p>Predicates merely wrap their argument and remain unevaluated:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Q</span><span class="p">,</span> <span class="n">ask</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span><span class="o">.</span><span class="n">prime</span><span class="p">(</span><span class="mi">7</span><span class="p">)</span>
-<span class="go">Q.prime(7)</span>
-</pre></div>
-</div>
-<p>To obtain the truth value of an expression containing predicates, use
-the function <span class="math">\(ask\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">prime</span><span class="p">(</span><span class="mi">7</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>The tautological predicate <span class="math">\(Q.is_true\)</span> can be used to wrap other objects:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span><span class="o">.</span><span class="n">is_true</span><span class="p">(</span><span class="n">x</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">Q.is_true(x &gt; 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span><span class="o">.</span><span class="n">is_true</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">Q.is_true(1 &lt; x)</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.assumptions.assume.Predicate.eval">
-<tt class="descname">eval</tt><big>(</big><em>self</em>, <em>expr</em>, <em>assumptions=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/assumptions/assume.html#Predicate.eval"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.assume.Predicate.eval" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluate self(expr) under the given assumptions.</p>
-<p>This uses only direct resolution methods, not logical inference.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
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-            
-  <div class="section" id="module-sympy.assumptions.handlers.calculus">
-<span id="calculus"></span><h1>Calculus<a class="headerlink" href="#module-sympy.assumptions.handlers.calculus" title="Permalink to this headline">¶</a></h1>
-<p>This module contains query handlers responsible for calculus queries:
-infinitesimal, bounded, etc.</p>
-<dl class="class">
-<dt id="sympy.assumptions.handlers.calculus.AskBoundedHandler">
-<em class="property">class </em><tt class="descclassname">sympy.assumptions.handlers.calculus.</tt><tt class="descname">AskBoundedHandler</tt><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/calculus.html#AskBoundedHandler"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.calculus.AskBoundedHandler" title="Permalink to this definition">¶</a></dt>
-<dd><p>Handler for key &#8216;bounded&#8217;.</p>
-<p>Test that an expression is bounded respect to all its variables.</p>
-<p>Examples of usage:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Q</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.assumptions.handlers.calculus</span> <span class="kn">import</span> <span class="n">AskBoundedHandler</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">AskBoundedHandler</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="o">==</span> <span class="bp">None</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">bounded</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.assumptions.handlers.calculus.AskBoundedHandler.Add">
-<tt class="descname">Add</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/calculus.html#AskBoundedHandler.Add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.calculus.AskBoundedHandler.Add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if expr is bounded, False if not and None if unknown.</p>
-<blockquote>
-<div><p>Truth Table:</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="10%" />
-<col width="10%" />
-<col width="17%" />
-<col width="10%" />
-<col width="10%" />
-<col width="10%" />
-<col width="10%" />
-<col width="10%" />
-<col width="10%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td colspan="2">&nbsp;</td>
-<td>B</td>
-<td colspan="3">U</td>
-<td colspan="3">?</td>
-</tr>
-<tr class="row-even"><td colspan="2">&nbsp;</td>
-<td>&nbsp;</td>
-<td>&#8216;+&#8217;</td>
-<td>&#8216;-&#8216;</td>
-<td>&#8216;x&#8217;</td>
-<td>&#8216;+&#8217;</td>
-<td>&#8216;-&#8216;</td>
-<td>&#8216;x&#8217;</td>
-</tr>
-<tr class="row-odd"><td colspan="2">B</td>
-<td>B</td>
-<td colspan="3">U</td>
-<td colspan="3">?</td>
-</tr>
-<tr class="row-even"><td rowspan="3">U</td>
-<td>&#8216;+&#8217;</td>
-<td>&nbsp;</td>
-<td>U</td>
-<td>?</td>
-<td>?</td>
-<td>U</td>
-<td>?</td>
-<td>?</td>
-</tr>
-<tr class="row-odd"><td>&#8216;-&#8216;</td>
-<td>&nbsp;</td>
-<td>?</td>
-<td>U</td>
-<td>?</td>
-<td>?</td>
-<td>U</td>
-<td>?</td>
-</tr>
-<tr class="row-even"><td>&#8216;x&#8217;</td>
-<td>&nbsp;</td>
-<td colspan="3">?</td>
-<td colspan="3">?</td>
-</tr>
-<tr class="row-odd"><td colspan="2">?</td>
-<td>&nbsp;</td>
-<td colspan="3">&nbsp;</td>
-<td colspan="3">?</td>
-</tr>
-</tbody>
-</table>
-<blockquote>
-<div><ul class="simple">
-<li>&#8216;B&#8217; = Bounded</li>
-<li>&#8216;U&#8217; = Unbounded</li>
-<li>&#8216;?&#8217; = unknown boundedness</li>
-<li>&#8216;+&#8217; = positive sign</li>
-<li>&#8216;-&#8216; = negative sign</li>
-<li>&#8216;x&#8217; = sign unknown</li>
-</ul>
-</div></blockquote>
-</div></blockquote>
-<div class="line-block">
-<div class="line"><br /></div>
-</div>
-<blockquote>
-<div><blockquote>
-<div><ul class="simple">
-<li>All Bounded -&gt; True</li>
-<li>1 Unbounded and the rest Bounded -&gt; False</li>
-<li>&gt;1 Unbounded, all with same known sign -&gt; False</li>
-<li>Any Unknown and unknown sign -&gt; None</li>
-<li>Else -&gt; None</li>
-</ul>
-</div></blockquote>
-<p>When the signs are not the same you can have an undefined
-result as in oo - oo, hence &#8216;bounded&#8217; is also undefined.</p>
-</div></blockquote>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.handlers.calculus.AskBoundedHandler.Mul">
-<tt class="descname">Mul</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/calculus.html#AskBoundedHandler.Mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.calculus.AskBoundedHandler.Mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if expr is bounded, False if not and None if unknown.</p>
-<p>Truth Table:</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="19%" />
-<col width="19%" />
-<col width="19%" />
-<col width="19%" />
-<col width="25%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>&nbsp;</td>
-<td>B</td>
-<td>U</td>
-<td colspan="2">?</td>
-</tr>
-<tr class="row-even"><td>&nbsp;</td>
-<td>&nbsp;</td>
-<td>&nbsp;</td>
-<td>s</td>
-<td>/s</td>
-</tr>
-<tr class="row-odd"><td>B</td>
-<td>B</td>
-<td>U</td>
-<td colspan="2">?</td>
-</tr>
-<tr class="row-even"><td>U</td>
-<td>&nbsp;</td>
-<td>U</td>
-<td>U</td>
-<td>?</td>
-</tr>
-<tr class="row-odd"><td>?</td>
-<td>&nbsp;</td>
-<td>&nbsp;</td>
-<td colspan="2">?</td>
-</tr>
-</tbody>
-</table>
-<blockquote>
-<div><ul class="simple">
-<li>B = Bounded</li>
-<li>U = Unbounded</li>
-<li>? = unknown boundedness</li>
-<li>s = signed (hence nonzero)</li>
-<li>/s = not signed</li>
-</ul>
-</div></blockquote>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.handlers.calculus.AskBoundedHandler.Pow">
-<tt class="descname">Pow</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/calculus.html#AskBoundedHandler.Pow"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.calculus.AskBoundedHandler.Pow" title="Permalink to this definition">¶</a></dt>
-<dd><p>Unbounded ** NonZero -&gt; Unbounded
-Bounded ** Bounded -&gt; Bounded
-Abs()&lt;=1 ** Positive -&gt; Bounded
-Abs()&gt;=1 ** Negative -&gt; Bounded
-Otherwise unknown</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.handlers.calculus.AskBoundedHandler.Symbol">
-<tt class="descname">Symbol</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/calculus.html#AskBoundedHandler.Symbol"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.calculus.AskBoundedHandler.Symbol" title="Permalink to this definition">¶</a></dt>
-<dd><p>Handles Symbol.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Q</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.assumptions.handlers.calculus</span> <span class="kn">import</span> <span class="n">AskBoundedHandler</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">AskBoundedHandler</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="o">==</span> <span class="bp">None</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">Symbol</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">bounded</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.assumptions.handlers.calculus.AskInfinitesimalHandler">
-<em class="property">class </em><tt class="descclassname">sympy.assumptions.handlers.calculus.</tt><tt class="descname">AskInfinitesimalHandler</tt><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/calculus.html#AskInfinitesimalHandler"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.calculus.AskInfinitesimalHandler" title="Permalink to this definition">¶</a></dt>
-<dd><p>Handler for key &#8216;infinitesimal&#8217;
-Test that a given expression is equivalent to an infinitesimal
-number</p>
-<dl class="function">
-<dt id="sympy.assumptions.handlers.calculus.AskInfinitesimalHandler.Add">
-<tt class="descname">Add</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="headerlink" href="#sympy.assumptions.handlers.calculus.AskInfinitesimalHandler.Add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Infinitesimal*Bounded -&gt; Infinitesimal</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.handlers.calculus.AskInfinitesimalHandler.Mul">
-<tt class="descname">Mul</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/calculus.html#AskInfinitesimalHandler.Mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.calculus.AskInfinitesimalHandler.Mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Infinitesimal*Bounded -&gt; Infinitesimal</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.handlers.calculus.AskInfinitesimalHandler.Pow">
-<tt class="descname">Pow</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="headerlink" href="#sympy.assumptions.handlers.calculus.AskInfinitesimalHandler.Pow" title="Permalink to this definition">¶</a></dt>
-<dd><p>Infinitesimal*Bounded -&gt; Infinitesimal</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-
-
-          </div>
-        </div>
-      </div>
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-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../handlers/index.html"
-                        title="previous chapter">Handlers</a></p>
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-                        title="next chapter">nTheory</a></p>
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-  <div class="section" id="module-sympy.assumptions.handlers.ntheory">
-<span id="ntheory"></span><h1>nTheory<a class="headerlink" href="#module-sympy.assumptions.handlers.ntheory" title="Permalink to this headline">¶</a></h1>
-<p>Handlers for keys related to number theory: prime, even, odd, etc.</p>
-<dl class="class">
-<dt id="sympy.assumptions.handlers.ntheory.AskOddHandler">
-<em class="property">class </em><tt class="descclassname">sympy.assumptions.handlers.ntheory.</tt><tt class="descname">AskOddHandler</tt><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/ntheory.html#AskOddHandler"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.ntheory.AskOddHandler" title="Permalink to this definition">¶</a></dt>
-<dd><p>Handler for key &#8216;odd&#8217;
-Test that an expression represents an odd number</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.assumptions.handlers.ntheory.AskPrimeHandler">
-<em class="property">class </em><tt class="descclassname">sympy.assumptions.handlers.ntheory.</tt><tt class="descname">AskPrimeHandler</tt><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/ntheory.html#AskPrimeHandler"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.ntheory.AskPrimeHandler" title="Permalink to this definition">¶</a></dt>
-<dd><p>Handler for key &#8216;prime&#8217;
-Test that an expression represents a prime number. When the
-expression is a number the result, when True, is subject to
-the limitations of isprime() which is used to return the result.</p>
-<dl class="function">
-<dt id="sympy.assumptions.handlers.ntheory.AskPrimeHandler.Pow">
-<tt class="descname">Pow</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/ntheory.html#AskPrimeHandler.Pow"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.ntheory.AskPrimeHandler.Pow" title="Permalink to this definition">¶</a></dt>
-<dd><p>Integer**Integer     -&gt; !Prime</p>
-</dd></dl>
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-  <div class="section" id="module-sympy.assumptions.handlers.order">
-<span id="order"></span><h1>Order<a class="headerlink" href="#module-sympy.assumptions.handlers.order" title="Permalink to this headline">¶</a></h1>
-<p>AskHandlers related to order relations: positive, negative, etc.</p>
-<dl class="class">
-<dt id="sympy.assumptions.handlers.order.AskNegativeHandler">
-<em class="property">class </em><tt class="descclassname">sympy.assumptions.handlers.order.</tt><tt class="descname">AskNegativeHandler</tt><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/order.html#AskNegativeHandler"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.order.AskNegativeHandler" title="Permalink to this definition">¶</a></dt>
-<dd><p>This is called by ask() when key=&#8217;negative&#8217;</p>
-<p>Test that an expression is less (strict) than zero.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">ask</span><span class="p">,</span> <span class="n">Q</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">(</span><span class="n">pi</span><span class="o">+</span><span class="mi">1</span><span class="p">))</span> <span class="c"># this calls AskNegativeHandler.Add</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">(</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span> <span class="c"># this calls AskNegativeHandler.Pow</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.assumptions.handlers.order.AskNegativeHandler.Add">
-<tt class="descname">Add</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/order.html#AskNegativeHandler.Add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.order.AskNegativeHandler.Add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Positive + Positive -&gt; Positive,
-Negative + Negative -&gt; Negative</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.handlers.order.AskNegativeHandler.Pow">
-<tt class="descname">Pow</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/order.html#AskNegativeHandler.Pow"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.order.AskNegativeHandler.Pow" title="Permalink to this definition">¶</a></dt>
-<dd><p>Real ** Even -&gt; NonNegative
-Real ** Odd  -&gt; same_as_base
-NonNegative ** Positive -&gt; NonNegative</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.assumptions.handlers.order.AskNonZeroHandler">
-<em class="property">class </em><tt class="descclassname">sympy.assumptions.handlers.order.</tt><tt class="descname">AskNonZeroHandler</tt><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/order.html#AskNonZeroHandler"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.order.AskNonZeroHandler" title="Permalink to this definition">¶</a></dt>
-<dd><p>Handler for key &#8216;zero&#8217;
-Test that an expression is not identically zero</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.assumptions.handlers.order.AskPositiveHandler">
-<em class="property">class </em><tt class="descclassname">sympy.assumptions.handlers.order.</tt><tt class="descname">AskPositiveHandler</tt><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/order.html#AskPositiveHandler"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.order.AskPositiveHandler" title="Permalink to this definition">¶</a></dt>
-<dd><p>Handler for key &#8216;positive&#8217;
-Test that an expression is greater (strict) than zero</p>
-</dd></dl>
-
-</div>
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-            
-  <div class="section" id="module-sympy.assumptions.handlers.sets">
-<span id="sets"></span><h1>Sets<a class="headerlink" href="#module-sympy.assumptions.handlers.sets" title="Permalink to this headline">¶</a></h1>
-<p>Handlers for predicates related to set membership: integer, rational, etc.</p>
-<dl class="class">
-<dt id="sympy.assumptions.handlers.sets.AskAlgebraicHandler">
-<em class="property">class </em><tt class="descclassname">sympy.assumptions.handlers.sets.</tt><tt class="descname">AskAlgebraicHandler</tt><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskAlgebraicHandler"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskAlgebraicHandler" title="Permalink to this definition">¶</a></dt>
-<dd><p>Handler for Q.algebraic key.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.assumptions.handlers.sets.AskAntiHermitianHandler">
-<em class="property">class </em><tt class="descclassname">sympy.assumptions.handlers.sets.</tt><tt class="descname">AskAntiHermitianHandler</tt><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskAntiHermitianHandler"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskAntiHermitianHandler" title="Permalink to this definition">¶</a></dt>
-<dd><p>Handler for Q.antihermitian
-Test that an expression belongs to the field of anti-Hermitian operators,
-that is, operators in the form x*I, where x is Hermitian</p>
-<dl class="function">
-<dt id="sympy.assumptions.handlers.sets.AskAntiHermitianHandler.Add">
-<tt class="descname">Add</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskAntiHermitianHandler.Add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskAntiHermitianHandler.Add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Antihermitian + Antihermitian  -&gt; Antihermitian
-Antihermitian + !Antihermitian -&gt; !Antihermitian</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.handlers.sets.AskAntiHermitianHandler.Mul">
-<tt class="descname">Mul</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskAntiHermitianHandler.Mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskAntiHermitianHandler.Mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>As long as there is at most only one noncommutative term:
-Hermitian*Hermitian         -&gt; !Antihermitian
-Hermitian*Antihermitian     -&gt; Antihermitian
-Antihermitian*Antihermitian -&gt; !Antihermitian</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.handlers.sets.AskAntiHermitianHandler.Pow">
-<tt class="descname">Pow</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskAntiHermitianHandler.Pow"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskAntiHermitianHandler.Pow" title="Permalink to this definition">¶</a></dt>
-<dd><p>Hermitian**Integer  -&gt; !Antihermitian
-Antihermitian**Even -&gt; !Antihermitian
-Antihermitian**Odd  -&gt; Antihermitian</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.assumptions.handlers.sets.AskComplexHandler">
-<em class="property">class </em><tt class="descclassname">sympy.assumptions.handlers.sets.</tt><tt class="descname">AskComplexHandler</tt><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskComplexHandler"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskComplexHandler" title="Permalink to this definition">¶</a></dt>
-<dd><p>Handler for Q.complex
-Test that an expression belongs to the field of complex numbers</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.assumptions.handlers.sets.AskExtendedRealHandler">
-<em class="property">class </em><tt class="descclassname">sympy.assumptions.handlers.sets.</tt><tt class="descname">AskExtendedRealHandler</tt><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskExtendedRealHandler"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskExtendedRealHandler" title="Permalink to this definition">¶</a></dt>
-<dd><p>Handler for Q.extended_real
-Test that an expression belongs to the field of extended real numbers,
-that is real numbers union {Infinity, -Infinity}</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.assumptions.handlers.sets.AskHermitianHandler">
-<em class="property">class </em><tt class="descclassname">sympy.assumptions.handlers.sets.</tt><tt class="descname">AskHermitianHandler</tt><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskHermitianHandler"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskHermitianHandler" title="Permalink to this definition">¶</a></dt>
-<dd><p>Handler for Q.hermitian
-Test that an expression belongs to the field of Hermitian operators</p>
-<dl class="function">
-<dt id="sympy.assumptions.handlers.sets.AskHermitianHandler.Add">
-<tt class="descname">Add</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskHermitianHandler.Add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskHermitianHandler.Add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Hermitian + Hermitian  -&gt; Hermitian
-Hermitian + !Hermitian -&gt; !Hermitian</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.handlers.sets.AskHermitianHandler.Mul">
-<tt class="descname">Mul</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskHermitianHandler.Mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskHermitianHandler.Mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>As long as there is at most only one noncommutative term:
-Hermitian*Hermitian         -&gt; Hermitian
-Hermitian*Antihermitian     -&gt; !Hermitian
-Antihermitian*Antihermitian -&gt; Hermitian</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.handlers.sets.AskHermitianHandler.Pow">
-<tt class="descname">Pow</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskHermitianHandler.Pow"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskHermitianHandler.Pow" title="Permalink to this definition">¶</a></dt>
-<dd><p>Hermitian**Integer -&gt; Hermitian</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.assumptions.handlers.sets.AskImaginaryHandler">
-<em class="property">class </em><tt class="descclassname">sympy.assumptions.handlers.sets.</tt><tt class="descname">AskImaginaryHandler</tt><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskImaginaryHandler"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskImaginaryHandler" title="Permalink to this definition">¶</a></dt>
-<dd><p>Handler for Q.imaginary
-Test that an expression belongs to the field of imaginary numbers,
-that is, numbers in the form x*I, where x is real</p>
-<dl class="function">
-<dt id="sympy.assumptions.handlers.sets.AskImaginaryHandler.Add">
-<tt class="descname">Add</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskImaginaryHandler.Add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskImaginaryHandler.Add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Imaginary + Imaginary -&gt; Imaginary
-Imaginary + Complex   -&gt; ?
-Imaginary + Real      -&gt; !Imaginary</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.handlers.sets.AskImaginaryHandler.Mul">
-<tt class="descname">Mul</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskImaginaryHandler.Mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskImaginaryHandler.Mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Real*Imaginary      -&gt; Imaginary
-Imaginary*Imaginary -&gt; Real</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.handlers.sets.AskImaginaryHandler.Pow">
-<tt class="descname">Pow</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskImaginaryHandler.Pow"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskImaginaryHandler.Pow" title="Permalink to this definition">¶</a></dt>
-<dd><p>Imaginary**integer -&gt; Imaginary if integer % 2 == 1
-Imaginary**integer -&gt; real if integer % 2 == 0
-Imaginary**Imaginary    -&gt; ?
-Imaginary**Real         -&gt; ?</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.assumptions.handlers.sets.AskIntegerHandler">
-<em class="property">class </em><tt class="descclassname">sympy.assumptions.handlers.sets.</tt><tt class="descname">AskIntegerHandler</tt><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskIntegerHandler"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskIntegerHandler" title="Permalink to this definition">¶</a></dt>
-<dd><p>Handler for Q.integer
-Test that an expression belongs to the field of integer numbers</p>
-<dl class="function">
-<dt id="sympy.assumptions.handlers.sets.AskIntegerHandler.Add">
-<tt class="descname">Add</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskIntegerHandler.Add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskIntegerHandler.Add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Integer + Integer       -&gt; Integer
-Integer + !Integer      -&gt; !Integer
-!Integer + !Integer -&gt; ?</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.handlers.sets.AskIntegerHandler.Mul">
-<tt class="descname">Mul</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskIntegerHandler.Mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskIntegerHandler.Mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Integer*Integer      -&gt; Integer
-Integer*Irrational   -&gt; !Integer
-Odd/Even             -&gt; !Integer
-Integer*Rational     -&gt; ?</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.handlers.sets.AskIntegerHandler.Pow">
-<tt class="descname">Pow</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskIntegerHandler.Pow" title="Permalink to this definition">¶</a></dt>
-<dd><p>Integer + Integer       -&gt; Integer
-Integer + !Integer      -&gt; !Integer
-!Integer + !Integer -&gt; ?</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.assumptions.handlers.sets.AskRationalHandler">
-<em class="property">class </em><tt class="descclassname">sympy.assumptions.handlers.sets.</tt><tt class="descname">AskRationalHandler</tt><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskRationalHandler"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskRationalHandler" title="Permalink to this definition">¶</a></dt>
-<dd><p>Handler for Q.rational
-Test that an expression belongs to the field of rational numbers</p>
-<dl class="function">
-<dt id="sympy.assumptions.handlers.sets.AskRationalHandler.Add">
-<tt class="descname">Add</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskRationalHandler.Add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskRationalHandler.Add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Rational + Rational     -&gt; Rational
-Rational + !Rational    -&gt; !Rational
-!Rational + !Rational   -&gt; ?</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.handlers.sets.AskRationalHandler.Mul">
-<tt class="descname">Mul</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskRationalHandler.Mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Rational + Rational     -&gt; Rational
-Rational + !Rational    -&gt; !Rational
-!Rational + !Rational   -&gt; ?</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.handlers.sets.AskRationalHandler.Pow">
-<tt class="descname">Pow</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskRationalHandler.Pow"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskRationalHandler.Pow" title="Permalink to this definition">¶</a></dt>
-<dd><p>Rational ** Integer      -&gt; Rational
-Irrational ** Rational   -&gt; Irrational
-Rational ** Irrational   -&gt; ?</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.assumptions.handlers.sets.AskRealHandler">
-<em class="property">class </em><tt class="descclassname">sympy.assumptions.handlers.sets.</tt><tt class="descname">AskRealHandler</tt><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskRealHandler"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskRealHandler" title="Permalink to this definition">¶</a></dt>
-<dd><p>Handler for Q.real
-Test that an expression belongs to the field of real numbers</p>
-<dl class="function">
-<dt id="sympy.assumptions.handlers.sets.AskRealHandler.Add">
-<tt class="descname">Add</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskRealHandler.Add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskRealHandler.Add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Real + Real              -&gt; Real
-Real + (Complex &amp; !Real) -&gt; !Real</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.handlers.sets.AskRealHandler.Mul">
-<tt class="descname">Mul</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskRealHandler.Mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskRealHandler.Mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Real*Real               -&gt; Real
-Real*Imaginary          -&gt; !Real
-Imaginary*Imaginary     -&gt; Real</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.handlers.sets.AskRealHandler.Pow">
-<tt class="descname">Pow</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#AskRealHandler.Pow"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.AskRealHandler.Pow" title="Permalink to this definition">¶</a></dt>
-<dd><p>Real**Integer         -&gt; Real
-Positive**Real        -&gt; Real
-Real**(Integer/Even)  -&gt; Real if base is nonnegative
-Real**(Integer/Odd)   -&gt; Real
-Real**Imaginary       -&gt; ?
-Imaginary**Real       -&gt; ?
-Real**Real            -&gt; ?</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.handlers.sets.test_closed_group">
-<tt class="descclassname">sympy.assumptions.handlers.sets.</tt><tt class="descname">test_closed_group</tt><big>(</big><em>expr</em>, <em>assumptions</em>, <em>key</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/assumptions/handlers/sets.html#test_closed_group"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.handlers.sets.test_closed_group" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test for membership in a group with respect
-to the current operation</p>
-</dd></dl>
-
-</div>
-
-
-          </div>
-        </div>
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-            
-  <div class="section" id="module-sympy.assumptions">
-<span id="assumptions-module"></span><h1>Assumptions module<a class="headerlink" href="#module-sympy.assumptions" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="contents">
-<h2>Contents<a class="headerlink" href="#contents" title="Permalink to this headline">¶</a></h2>
-<div class="toctree-wrapper compound">
-<ul>
-<li class="toctree-l1"><a class="reference internal" href="../assumptions/ask.html">Ask</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../assumptions/assume.html">Assume</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../assumptions/refine.html">Refine</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../assumptions/handlers/index.html">Handlers</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../assumptions/handlers/index.html#contents">Contents</a><ul>
-<li class="toctree-l3"><a class="reference internal" href="../assumptions/handlers/calculus.html">Calculus</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../assumptions/handlers/ntheory.html">nTheory</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../assumptions/handlers/order.html">Order</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../assumptions/handlers/sets.html">Sets</a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-</div>
-<p>Queries are used to ask information about expressions. Main method for this
-is ask():</p>
-<dl class="function">
-<dt>
-<tt class="descclassname">sympy.assumptions.ask.</tt><tt class="descname">ask</tt><big>(</big><em>proposition</em>, <em>assumptions=True</em>, <em>context=AssumptionsContext()</em><big>)</big><a class="reference internal" href="../../_modules/sympy/assumptions/ask.html#ask"><span class="viewcode-link">[source]</span></a></dt>
-<dd><p>Method for inferring properties about objects.</p>
-<p><strong>Syntax</strong></p>
-<blockquote>
-<div><ul>
-<li><p class="first">ask(proposition)</p>
-</li>
-<li><p class="first">ask(proposition, assumptions)</p>
-<blockquote>
-<div><p>where <tt class="docutils literal"><span class="pre">proposition</span></tt> is any boolean expression</p>
-</div></blockquote>
-</li>
-</ul>
-</div></blockquote>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">ask</span><span class="p">,</span> <span class="n">Q</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">rational</span><span class="p">(</span><span class="n">pi</span><span class="p">))</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">&amp;</span> <span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">prime</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">&amp;</span>  <span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<dl class="docutils">
-<dt><strong>Remarks</strong></dt>
-<dd><p class="first">Relations in assumptions are not implemented (yet), so the following
-will not give a meaningful result.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">is_true</span><span class="p">(</span><span class="n">x</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">))</span> 
-</pre></div>
-</div>
-<p class="last">It is however a work in progress.</p>
-</dd>
-</dl>
-</dd></dl>
-
-</div>
-<div class="section" id="querying">
-<h2>Querying<a class="headerlink" href="#querying" title="Permalink to this headline">¶</a></h2>
-<p>ask&#8217;s optional second argument should be a boolean expression involving
-assumptions about objects in expr. Valid values include:</p>
-<blockquote>
-<div><ul class="simple">
-<li>Q.integer(x)</li>
-<li>Q.positive(x)</li>
-<li>Q.integer(x) &amp; Q.positive(x)</li>
-<li>etc.</li>
-</ul>
-</div></blockquote>
-<p>Q is a class in sympy.assumptions holding known predicates.</p>
-<p>See documentation for the logic module for a complete list of valid boolean
-expressions.</p>
-<p>You can also define global assumptions so you don&#8217;t have to pass that argument
-each time to function ask(). This is done by using the global_assumptions
-object from module sympy.assumptions. You can then clear global assumptions
-with global_assumptions.clear():</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">global_assumptions</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">global_assumptions</span><span class="o">.</span><span class="n">clear</span><span class="p">()</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="supported-predicates">
-<h2>Supported predicates<a class="headerlink" href="#supported-predicates" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="bounded">
-<h3>bounded<a class="headerlink" href="#bounded" title="Permalink to this headline">¶</a></h3>
-<p>Test that a function is bounded with respect to its variables. For example,
-sin(x) is a bounded functions, but exp(x) is not.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">bounded</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="o">~</span><span class="n">Q</span><span class="o">.</span><span class="n">bounded</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">bounded</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">bounded</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">bounded</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="o">~</span><span class="n">Q</span><span class="o">.</span><span class="n">bounded</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="commutative">
-<h3>commutative<a class="headerlink" href="#commutative" title="Permalink to this headline">¶</a></h3>
-<p>Test that objects are commutative. By default, symbols in SymPy are considered
-commutative except otherwise stated.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">commutative</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">commutative</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="o">~</span><span class="n">Q</span><span class="o">.</span><span class="n">commutative</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">commutative</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">),</span> <span class="o">~</span><span class="n">Q</span><span class="o">.</span><span class="n">commutative</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="complex">
-<h3>complex<a class="headerlink" href="#complex" title="Permalink to this headline">¶</a></h3>
-<p>Test that expression belongs to the field of complex numbers.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">complex</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">complex</span><span class="p">(</span><span class="n">I</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">complex</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">I</span><span class="o">*</span><span class="n">y</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">&amp;</span> <span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="even">
-<h3>even<a class="headerlink" href="#even" title="Permalink to this headline">¶</a></h3>
-<p>Test that expression represents an even number, that is, an number that
-can be written in the form 2*n, n integer.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="extended-real">
-<h3>extended_real<a class="headerlink" href="#extended-real" title="Permalink to this headline">¶</a></h3>
-<p>Test that an expression belongs to the field of extended real numbers, that is, real
-numbers union {Infinity, -Infinity}.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">extended_real</span><span class="p">(</span><span class="n">oo</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">extended_real</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">extended_real</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="imaginary">
-<h3>imaginary<a class="headerlink" href="#imaginary" title="Permalink to this headline">¶</a></h3>
-<dl class="docutils">
-<dt>Test that an expression belongs to the set of imaginary numbers, that is,</dt>
-<dd>it can be written as x*I, where x is real and I is the imaginary unit.</dd>
-</dl>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">imaginary</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">I</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">imaginary</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">I</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="infinitesimal">
-<h3>infinitesimal<a class="headerlink" href="#infinitesimal" title="Permalink to this headline">¶</a></h3>
-<p>Test that an expression is equivalent to an infinitesimal number.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">infinitesimal</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">oo</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">infinitesimal</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">infinitesimal</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">infinitesimal</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">infinitesimal</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">&amp;</span> <span class="n">Q</span><span class="o">.</span><span class="n">bounded</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="integer">
-<h3>integer<a class="headerlink" href="#integer" title="Permalink to this headline">¶</a></h3>
-<p>Test that an expression belongs to the set of integer numbers.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)))</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="mi">2</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="irrational">
-<h3>irrational<a class="headerlink" href="#irrational" title="Permalink to this headline">¶</a></h3>
-<p>Test that an expression represents an irrational number.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">irrational</span><span class="p">(</span><span class="n">pi</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">irrational</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">irrational</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)),</span> <span class="n">Q</span><span class="o">.</span><span class="n">rational</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="rational">
-<h3>rational<a class="headerlink" href="#rational" title="Permalink to this headline">¶</a></h3>
-<p>Test that an expression represents a rational number.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">rational</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">rational</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="mi">2</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">rational</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="n">y</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">&amp;</span> <span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="negative">
-<h3>negative<a class="headerlink" href="#negative" title="Permalink to this headline">¶</a></h3>
-<p>Test that an expression is less (strict) than zero.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">(</span><span class="mf">0.3</span><span class="p">))</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="section" id="remarks">
-<h4>Remarks<a class="headerlink" href="#remarks" title="Permalink to this headline">¶</a></h4>
-<p>negative numbers are defined as real numbers that are not zero nor positive, so
-complex numbers (with nontrivial imaginary coefficients) will return False
-for this predicate. The same applies to Q.positive.</p>
-</div>
-</div>
-<div class="section" id="positive">
-<h3>positive<a class="headerlink" href="#positive" title="Permalink to this headline">¶</a></h3>
-<p>Test that a given expression is greater (strict) than zero.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="mf">0.3</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="section" id="id1">
-<h4>Remarks<a class="headerlink" href="#id1" title="Permalink to this headline">¶</a></h4>
-<p>see Remarks for negative</p>
-</div>
-</div>
-<div class="section" id="prime">
-<h3>prime<a class="headerlink" href="#prime" title="Permalink to this headline">¶</a></h3>
-<p>Test that an expression represents a prime number.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">prime</span><span class="p">(</span><span class="mi">13</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>Remarks: Use sympy.ntheory.isprime to test numeric values efficiently.</p>
-</div>
-<div class="section" id="real">
-<h3>real<a class="headerlink" href="#real" title="Permalink to this headline">¶</a></h3>
-<p>Test that an expression belongs to the field of real numbers.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">&amp;</span> <span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="odd">
-<h3>odd<a class="headerlink" href="#odd" title="Permalink to this headline">¶</a></h3>
-<p>Test that an expression represents an odd number.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">odd</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">odd</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="nonzero">
-<h3>nonzero<a class="headerlink" href="#nonzero" title="Permalink to this headline">¶</a></h3>
-<p>Test that an expression is not zero.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">nonzero</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">|</span> <span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="section" id="design">
-<h2>Design<a class="headerlink" href="#design" title="Permalink to this headline">¶</a></h2>
-<p>Each time ask is called, the appropriate Handler for the current key is called. This is
-always a subclass of sympy.assumptions.AskHandler. It&#8217;s classmethods have the name&#8217;s of the classes
-it supports. For example, a (simplified) AskHandler for the ask &#8216;positive&#8217; would
-look like this:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="k">class</span> <span class="nc">AskPositiveHandler</span><span class="p">(</span><span class="n">CommonHandler</span><span class="p">):</span>
-
-    <span class="k">def</span> <span class="nf">Mul</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="c"># return True if all argument&#39;s in self.expr.args are positive</span>
-        <span class="o">...</span>
-
-    <span class="k">def</span> <span class="nf">Add</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-        <span class="k">for</span> <span class="n">arg</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">expr</span><span class="o">.</span><span class="n">args</span><span class="p">:</span>
-            <span class="k">if</span> <span class="ow">not</span> <span class="n">ask</span><span class="p">(</span><span class="n">arg</span><span class="p">,</span> <span class="n">positive</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">assumptions</span><span class="p">):</span>
-                <span class="k">break</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="c"># if all argument&#39;s are positive</span>
-            <span class="k">return</span> <span class="bp">True</span>
-    <span class="o">...</span>
-</pre></div>
-</div>
-<p>The .Mul() method is called when self.expr is an instance of Mul, the Add method
-would be called when self.expr is an instance of Add and so on.</p>
-</div>
-<div class="section" id="extensibility">
-<h2>Extensibility<a class="headerlink" href="#extensibility" title="Permalink to this headline">¶</a></h2>
-<dl class="docutils">
-<dt>You can define new queries or support new types by subclassing sympy.assumptions.AskHandler</dt>
-<dd>and registering that handler for a particular key by calling register_handler:</dd>
-</dl>
-<dl class="function">
-<dt>
-<tt class="descclassname">sympy.assumptions.ask.</tt><tt class="descname">register_handler</tt><big>(</big><em>key</em>, <em>handler</em><big>)</big><a class="reference internal" href="../../_modules/sympy/assumptions/ask.html#register_handler"><span class="viewcode-link">[source]</span></a></dt>
-<dd><p>Register a handler in the ask system. key must be a string and handler a
-class inheriting from AskHandler:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.assumptions</span> <span class="kn">import</span> <span class="n">register_handler</span><span class="p">,</span> <span class="n">ask</span><span class="p">,</span> <span class="n">Q</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.assumptions.handlers</span> <span class="kn">import</span> <span class="n">AskHandler</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">MersenneHandler</span><span class="p">(</span><span class="n">AskHandler</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="c"># Mersenne numbers are in the form 2**n + 1, n integer</span>
-<span class="gp">... </span>    <span class="nd">@staticmethod</span>
-<span class="gp">... </span>    <span class="k">def</span> <span class="nf">Integer</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-<span class="gp">... </span>        <span class="kn">import</span> <span class="nn">math</span>
-<span class="gp">... </span>        <span class="k">return</span> <span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">expr</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">register_handler</span><span class="p">(</span><span class="s">&#39;mersenne&#39;</span><span class="p">,</span> <span class="n">MersenneHandler</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">mersenne</span><span class="p">(</span><span class="mi">7</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<p>You can undo this operation by calling remove_handler.</p>
-<dl class="function">
-<dt>
-<tt class="descclassname">sympy.assumptions.ask.</tt><tt class="descname">remove_handler</tt><big>(</big><em>key</em>, <em>handler</em><big>)</big><a class="reference internal" href="../../_modules/sympy/assumptions/ask.html#remove_handler"><span class="viewcode-link">[source]</span></a></dt>
-<dd><p>Removes a handler from the ask system. Same syntax as register_handler</p>
-</dd></dl>
-
-<p>You can support new types <a class="footnote-reference" href="#id3" id="id2">[1]</a> by adding a handler to an existing key. In the
-following example, we will create a new type MyType and extend the key &#8216;prime&#8217;
-to accept this type (and return True)</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Basic</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.assumptions</span> <span class="kn">import</span> <span class="n">register_handler</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.assumptions.handlers</span> <span class="kn">import</span> <span class="n">AskHandler</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">MyType</span><span class="p">(</span><span class="n">Basic</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">pass</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">MyAskHandler</span><span class="p">(</span><span class="n">AskHandler</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="nd">@staticmethod</span>
-<span class="gp">... </span>    <span class="k">def</span> <span class="nf">MyType</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assumptions</span><span class="p">):</span>
-<span class="gp">... </span>        <span class="k">return</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">MyType</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">register_handler</span><span class="p">(</span><span class="s">&#39;prime&#39;</span><span class="p">,</span> <span class="n">MyAskHandler</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ask</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">prime</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="performance-improvements">
-<h2>Performance improvements<a class="headerlink" href="#performance-improvements" title="Permalink to this headline">¶</a></h2>
-<p>On queries that involve symbolic coefficients, logical inference is used. Work on
-improving satisfiable function (sympy.logic.inference.satisfiable) should result
-in notable speed improvements.</p>
-<p>Logic inference used in one ask could be used to speed up further queries, and
-current system does not take advantage of this. For example, a truth maintenance
-system (<a class="reference external" href="http://en.wikipedia.org/wiki/Truth_maintenance_system">http://en.wikipedia.org/wiki/Truth_maintenance_system</a>) could be implemented.</p>
-</div>
-<div class="section" id="misc">
-<h2>Misc<a class="headerlink" href="#misc" title="Permalink to this headline">¶</a></h2>
-<p>You can find more examples in the in the form of test under directory
-sympy/assumptions/tests/</p>
-<table class="docutils footnote" frame="void" id="id3" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id2">[1]</a></td><td>New type must inherit from Basic, otherwise an exception will be raised.
-This is a bug and should be fixed.</td></tr>
-</tbody>
-</table>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../index.html">
-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Assumptions module</a><ul>
-<li><a class="reference internal" href="#contents">Contents</a><ul>
-</ul>
-</li>
-<li><a class="reference internal" href="#querying">Querying</a></li>
-<li><a class="reference internal" href="#supported-predicates">Supported predicates</a><ul>
-<li><a class="reference internal" href="#bounded">bounded</a></li>
-<li><a class="reference internal" href="#commutative">commutative</a></li>
-<li><a class="reference internal" href="#complex">complex</a></li>
-<li><a class="reference internal" href="#even">even</a></li>
-<li><a class="reference internal" href="#extended-real">extended_real</a></li>
-<li><a class="reference internal" href="#imaginary">imaginary</a></li>
-<li><a class="reference internal" href="#infinitesimal">infinitesimal</a></li>
-<li><a class="reference internal" href="#integer">integer</a></li>
-<li><a class="reference internal" href="#irrational">irrational</a></li>
-<li><a class="reference internal" href="#rational">rational</a></li>
-<li><a class="reference internal" href="#negative">negative</a><ul>
-<li><a class="reference internal" href="#remarks">Remarks</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#positive">positive</a><ul>
-<li><a class="reference internal" href="#id1">Remarks</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#prime">prime</a></li>
-<li><a class="reference internal" href="#real">real</a></li>
-<li><a class="reference internal" href="#odd">odd</a></li>
-<li><a class="reference internal" href="#nonzero">nonzero</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#design">Design</a></li>
-<li><a class="reference internal" href="#extensibility">Extensibility</a></li>
-<li><a class="reference internal" href="#performance-improvements">Performance improvements</a></li>
-<li><a class="reference internal" href="#misc">Misc</a></li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../plotting.html"
-                        title="previous chapter">Plotting Module</a></p>
-  <h4>Next topic</h4>
-  <p class="topless"><a href="../assumptions/ask.html"
-                        title="next chapter">Ask</a></p>
-  <h3>This Page</h3>
-  <ul class="this-page-menu">
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-           rel="nofollow">Show Source</a></li>
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-            
-  <div class="section" id="module-sympy.assumptions.refine">
-<span id="refine"></span><h1>Refine<a class="headerlink" href="#module-sympy.assumptions.refine" title="Permalink to this headline">¶</a></h1>
-<dl class="function">
-<dt id="sympy.assumptions.refine.refine">
-<tt class="descclassname">sympy.assumptions.refine.</tt><tt class="descname">refine</tt><big>(</big><em>expr</em>, <em>assumptions=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/assumptions/refine.html#refine"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.refine.refine" title="Permalink to this definition">¶</a></dt>
-<dd><p>Simplify an expression using assumptions.</p>
-<p>Gives the form of expr that would be obtained if symbols
-in it were replaced by explicit numerical expressions satisfying
-the assumptions.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">refine</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">Q</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">refine</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">Abs(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">refine</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.refine.refine_Pow">
-<tt class="descclassname">sympy.assumptions.refine.</tt><tt class="descname">refine_Pow</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../_modules/sympy/assumptions/refine.html#refine_Pow"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.refine.refine_Pow" title="Permalink to this definition">¶</a></dt>
-<dd><p>Handler for instances of Pow.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Q</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.assumptions.refine</span> <span class="kn">import</span> <span class="n">refine_Pow</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">refine_Pow</span><span class="p">((</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">x</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">refine_Pow</span><span class="p">((</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">x</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">refine_Pow</span><span class="p">((</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">x</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">odd</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">-1</span>
-</pre></div>
-</div>
-<p>For powers of -1, even parts of the exponent can be simplified:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">refine_Pow</span><span class="p">((</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">even</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">(-1)**y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">refine_Pow</span><span class="p">((</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="o">+</span><span class="n">z</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">odd</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">&amp;</span> <span class="n">Q</span><span class="o">.</span><span class="n">odd</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="go">(-1)**y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">refine_Pow</span><span class="p">((</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="o">+</span><span class="mi">2</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">odd</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">(-1)**(y + 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">refine_Pow</span><span class="p">((</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mi">3</span><span class="p">),</span> <span class="bp">True</span><span class="p">)</span>
-<span class="go">(-1)**(x + 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.refine.refine_abs">
-<tt class="descclassname">sympy.assumptions.refine.</tt><tt class="descname">refine_abs</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../_modules/sympy/assumptions/refine.html#refine_abs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.refine.refine_abs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Handler for the absolute value.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Q</span><span class="p">,</span> <span class="n">refine</span><span class="p">,</span> <span class="n">Abs</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.assumptions.refine</span> <span class="kn">import</span> <span class="n">refine_abs</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">refine_abs</span><span class="p">(</span><span class="n">Abs</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">refine_abs</span><span class="p">(</span><span class="n">Abs</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">positive</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">refine_abs</span><span class="p">(</span><span class="n">Abs</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">negative</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">-x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.assumptions.refine.refine_exp">
-<tt class="descclassname">sympy.assumptions.refine.</tt><tt class="descname">refine_exp</tt><big>(</big><em>expr</em>, <em>assumptions</em><big>)</big><a class="reference internal" href="../../_modules/sympy/assumptions/refine.html#refine_exp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.assumptions.refine.refine_exp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Handler for exponential function.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Q</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.assumptions.refine</span> <span class="kn">import</span> <span class="n">refine_exp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">refine_exp</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">real</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">refine_exp</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">Q</span><span class="o">.</span><span class="n">integer</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-
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-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../assumptions/assume.html"
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-            
-  <div class="section" id="category-theory-module">
-<h1>Category Theory Module<a class="headerlink" href="#category-theory-module" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="introduction">
-<h2>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline">¶</a></h2>
-<p>The category theory module for SymPy will allow manipulating diagrams
-within a single category, including drawing them in TikZ and deciding
-whether they are commutative or not.</p>
-<p>The general reference work this module tries to follow is</p>
-<blockquote>
-<div><dl class="docutils">
-<dt>[JoyOfCats] J. Adamek, H. Herrlich. G. E. Strecker: Abstract and</dt>
-<dd>Concrete Categories. The Joy of Cats.</dd>
-</dl>
-</div></blockquote>
-<p>The latest version of this book should be available for free download
-from</p>
-<blockquote>
-<div>katmat.math.uni-bremen.de/acc/acc.pdf</div></blockquote>
-<p>The module is still in its pre-embryonic stage.</p>
-</div>
-<div class="section" id="module-sympy.categories">
-<span id="base-class-reference"></span><h2>Base Class Reference<a class="headerlink" href="#module-sympy.categories" title="Permalink to this headline">¶</a></h2>
-<p>This section lists the classes which implement some of the basic
-notions in category theory: objects, morphisms, categories, and
-diagrams.</p>
-<dl class="class">
-<dt id="sympy.categories.Object">
-<em class="property">class </em><tt class="descclassname">sympy.categories.</tt><tt class="descname">Object</tt><a class="headerlink" href="#sympy.categories.Object" title="Permalink to this definition">¶</a></dt>
-<dd><p>The base class for any kind of object in an abstract category.</p>
-<p>While technically any instance of <tt class="xref py py-class docutils literal"><span class="pre">Basic</span></tt> will do, this
-class is the recommended way to create abstract objects in
-abstract categories.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.categories.Morphism">
-<em class="property">class </em><tt class="descclassname">sympy.categories.</tt><tt class="descname">Morphism</tt><a class="headerlink" href="#sympy.categories.Morphism" title="Permalink to this definition">¶</a></dt>
-<dd><p>The base class for any morphism in an abstract category.</p>
-<p>In abstract categories, a morphism is an arrow between two
-category objects.  The object where the arrow starts is called the
-domain, while the object where the arrow ends is called the
-codomain.</p>
-<p>Two morphisms between the same pair of objects are considered to
-be the same morphisms.  To distinguish between morphisms between
-the same objects use <a class="reference internal" href="#sympy.categories.NamedMorphism" title="sympy.categories.NamedMorphism"><tt class="xref py py-class docutils literal"><span class="pre">NamedMorphism</span></tt></a>.</p>
-<p>It is prohibited to instantiate this class.  Use one of the
-derived classes instead.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.categories.IdentityMorphism" title="sympy.categories.IdentityMorphism"><tt class="xref py py-obj docutils literal"><span class="pre">IdentityMorphism</span></tt></a>, <a class="reference internal" href="#sympy.categories.NamedMorphism" title="sympy.categories.NamedMorphism"><tt class="xref py py-obj docutils literal"><span class="pre">NamedMorphism</span></tt></a>, <a class="reference internal" href="#sympy.categories.CompositeMorphism" title="sympy.categories.CompositeMorphism"><tt class="xref py py-obj docutils literal"><span class="pre">CompositeMorphism</span></tt></a></p>
-</div>
-<dl class="attribute">
-<dt id="sympy.categories.Morphism.codomain">
-<tt class="descname">codomain</tt><a class="headerlink" href="#sympy.categories.Morphism.codomain" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the codomain of the morphism.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">codomain</span>
-<span class="go">Object(&quot;B&quot;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.categories.Morphism.compose">
-<tt class="descname">compose</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="headerlink" href="#sympy.categories.Morphism.compose" title="Permalink to this definition">¶</a></dt>
-<dd><p>Composes self with the supplied morphism.</p>
-<p>The order of elements in the composition is the usual order,
-i.e., to construct <span class="math">\(g\circ f\)</span> use <tt class="docutils literal"><span class="pre">g.compose(f)</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">*</span> <span class="n">f</span>
-<span class="go">CompositeMorphism((NamedMorphism(Object(&quot;A&quot;), Object(&quot;B&quot;), &quot;f&quot;),</span>
-<span class="go">NamedMorphism(Object(&quot;B&quot;), Object(&quot;C&quot;), &quot;g&quot;)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">g</span> <span class="o">*</span> <span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">domain</span>
-<span class="go">Object(&quot;A&quot;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">g</span> <span class="o">*</span> <span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">codomain</span>
-<span class="go">Object(&quot;C&quot;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.categories.Morphism.domain">
-<tt class="descname">domain</tt><a class="headerlink" href="#sympy.categories.Morphism.domain" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the domain of the morphism.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">domain</span>
-<span class="go">Object(&quot;A&quot;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.categories.NamedMorphism">
-<em class="property">class </em><tt class="descclassname">sympy.categories.</tt><tt class="descname">NamedMorphism</tt><a class="headerlink" href="#sympy.categories.NamedMorphism" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents a morphism which has a name.</p>
-<p>Names are used to distinguish between morphisms which have the
-same domain and codomain: two named morphisms are equal if they
-have the same domains, codomains, and names.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.categories.Morphism" title="sympy.categories.Morphism"><tt class="xref py py-obj docutils literal"><span class="pre">Morphism</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span>
-<span class="go">NamedMorphism(Object(&quot;A&quot;), Object(&quot;B&quot;), &quot;f&quot;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">name</span>
-<span class="go">&#39;f&#39;</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.categories.NamedMorphism.name">
-<tt class="descname">name</tt><a class="headerlink" href="#sympy.categories.NamedMorphism.name" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the name of the morphism.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">name</span>
-<span class="go">&#39;f&#39;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.categories.CompositeMorphism">
-<em class="property">class </em><tt class="descclassname">sympy.categories.</tt><tt class="descname">CompositeMorphism</tt><a class="headerlink" href="#sympy.categories.CompositeMorphism" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents a morphism which is a composition of other morphisms.</p>
-<p>Two composite morphisms are equal if the morphisms they were
-obtained from (components) are the same and were listed in the
-same order.</p>
-<p>The arguments to the constructor for this class should be listed
-in diagram order: to obtain the composition <span class="math">\(g\circ f\)</span> from the
-instances of <a class="reference internal" href="#sympy.categories.Morphism" title="sympy.categories.Morphism"><tt class="xref py py-class docutils literal"><span class="pre">Morphism</span></tt></a> <tt class="docutils literal"><span class="pre">g</span></tt> and <tt class="docutils literal"><span class="pre">f</span></tt> use
-<tt class="docutils literal"><span class="pre">CompositeMorphism(f,</span> <span class="pre">g)</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span><span class="p">,</span> <span class="n">CompositeMorphism</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">*</span> <span class="n">f</span>
-<span class="go">CompositeMorphism((NamedMorphism(Object(&quot;A&quot;), Object(&quot;B&quot;), &quot;f&quot;),</span>
-<span class="go">NamedMorphism(Object(&quot;B&quot;), Object(&quot;C&quot;), &quot;g&quot;)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">CompositeMorphism</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span> <span class="o">==</span> <span class="n">g</span> <span class="o">*</span> <span class="n">f</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.categories.CompositeMorphism.codomain">
-<tt class="descname">codomain</tt><a class="headerlink" href="#sympy.categories.CompositeMorphism.codomain" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the codomain of this composite morphism.</p>
-<p>The codomain of the composite morphism is the codomain of its
-last component.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">g</span> <span class="o">*</span> <span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">codomain</span>
-<span class="go">Object(&quot;C&quot;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.categories.CompositeMorphism.components">
-<tt class="descname">components</tt><a class="headerlink" href="#sympy.categories.CompositeMorphism.components" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the components of this composite morphism.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">g</span> <span class="o">*</span> <span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">components</span>
-<span class="go">(NamedMorphism(Object(&quot;A&quot;), Object(&quot;B&quot;), &quot;f&quot;),</span>
-<span class="go">NamedMorphism(Object(&quot;B&quot;), Object(&quot;C&quot;), &quot;g&quot;))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.categories.CompositeMorphism.domain">
-<tt class="descname">domain</tt><a class="headerlink" href="#sympy.categories.CompositeMorphism.domain" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the domain of this composite morphism.</p>
-<p>The domain of the composite morphism is the domain of its
-first component.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">g</span> <span class="o">*</span> <span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">domain</span>
-<span class="go">Object(&quot;A&quot;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.categories.CompositeMorphism.flatten">
-<tt class="descname">flatten</tt><big>(</big><em>self</em>, <em>new_name</em><big>)</big><a class="headerlink" href="#sympy.categories.CompositeMorphism.flatten" title="Permalink to this definition">¶</a></dt>
-<dd><p>Forgets the composite structure of this morphism.</p>
-<p>If <tt class="docutils literal"><span class="pre">new_name</span></tt> is not empty, returns a <a class="reference internal" href="#sympy.categories.NamedMorphism" title="sympy.categories.NamedMorphism"><tt class="xref py py-class docutils literal"><span class="pre">NamedMorphism</span></tt></a>
-with the supplied name, otherwise returns a <a class="reference internal" href="#sympy.categories.Morphism" title="sympy.categories.Morphism"><tt class="xref py py-class docutils literal"><span class="pre">Morphism</span></tt></a>.
-In both cases the domain of the new morphism is the domain of
-this composite morphism and the codomain of the new morphism
-is the codomain of this composite morphism.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">g</span> <span class="o">*</span> <span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">flatten</span><span class="p">(</span><span class="s">&quot;h&quot;</span><span class="p">)</span>
-<span class="go">NamedMorphism(Object(&quot;A&quot;), Object(&quot;C&quot;), &quot;h&quot;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.categories.IdentityMorphism">
-<em class="property">class </em><tt class="descclassname">sympy.categories.</tt><tt class="descname">IdentityMorphism</tt><a class="headerlink" href="#sympy.categories.IdentityMorphism" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents an identity morphism.</p>
-<p>An identity morphism is a morphism with equal domain and codomain,
-which acts as an identity with respect to composition.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.categories.Morphism" title="sympy.categories.Morphism"><tt class="xref py py-obj docutils literal"><span class="pre">Morphism</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span><span class="p">,</span> <span class="n">IdentityMorphism</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">id_A</span> <span class="o">=</span> <span class="n">IdentityMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">id_B</span> <span class="o">=</span> <span class="n">IdentityMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">*</span> <span class="n">id_A</span> <span class="o">==</span> <span class="n">f</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">id_B</span> <span class="o">*</span> <span class="n">f</span> <span class="o">==</span> <span class="n">f</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.categories.Category">
-<em class="property">class </em><tt class="descclassname">sympy.categories.</tt><tt class="descname">Category</tt><a class="headerlink" href="#sympy.categories.Category" title="Permalink to this definition">¶</a></dt>
-<dd><p>An (abstract) category.</p>
-<p>A category [JoyOfCats] is a quadruple <span class="math">\(\mbox{K} = (O, \hom, id,
-\circ)\)</span> consisting of</p>
-<ul class="simple">
-<li>a (set-theoretical) class <span class="math">\(O\)</span>, whose members are called
-<span class="math">\(K\)</span>-objects,</li>
-<li>for each pair <span class="math">\((A, B)\)</span> of <span class="math">\(K\)</span>-objects, a set <span class="math">\(\hom(A, B)\)</span> whose
-members are called <span class="math">\(K\)</span>-morphisms from <span class="math">\(A\)</span> to <span class="math">\(B\)</span>,</li>
-<li>for a each <span class="math">\(K\)</span>-object <span class="math">\(A\)</span>, a morphism <span class="math">\(id:A\rightarrow A\)</span>,
-called the <span class="math">\(K\)</span>-identity of <span class="math">\(A\)</span>,</li>
-<li>a composition law <span class="math">\(\circ\)</span> associating with every <span class="math">\(K\)</span>-morphisms
-<span class="math">\(f:A\rightarrow B\)</span> and <span class="math">\(g:B\rightarrow C\)</span> a <span class="math">\(K\)</span>-morphism <span class="math">\(g\circ
-f:A\rightarrow C\)</span>, called the composite of <span class="math">\(f\)</span> and <span class="math">\(g\)</span>.</li>
-</ul>
-<p>Composition is associative, <span class="math">\(K\)</span>-identities are identities with
-respect to composition, and the sets <span class="math">\(\hom(A, B)\)</span> are pairwise
-disjoint.</p>
-<p>This class knows nothing about its objects and morphisms.
-Concrete cases of (abstract) categories should be implemented as
-classes derived from this one.</p>
-<p>Certain instances of <a class="reference internal" href="#sympy.categories.Diagram" title="sympy.categories.Diagram"><tt class="xref py py-class docutils literal"><span class="pre">Diagram</span></tt></a> can be asserted to be
-commutative in a <a class="reference internal" href="#sympy.categories.Category" title="sympy.categories.Category"><tt class="xref py py-class docutils literal"><span class="pre">Category</span></tt></a> by supplying the argument
-<tt class="docutils literal"><span class="pre">commutative_diagrams</span></tt> in the constructor.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.categories.Diagram" title="sympy.categories.Diagram"><tt class="xref py py-obj docutils literal"><span class="pre">Diagram</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span><span class="p">,</span> <span class="n">Diagram</span><span class="p">,</span> <span class="n">Category</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">FiniteSet</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">K</span> <span class="o">=</span> <span class="n">Category</span><span class="p">(</span><span class="s">&quot;K&quot;</span><span class="p">,</span> <span class="n">commutative_diagrams</span><span class="o">=</span><span class="p">[</span><span class="n">d</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">K</span><span class="o">.</span><span class="n">commutative_diagrams</span> <span class="o">==</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.categories.Category.commutative_diagrams">
-<tt class="descname">commutative_diagrams</tt><a class="headerlink" href="#sympy.categories.Category.commutative_diagrams" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the <tt class="xref py py-class docutils literal"><span class="pre">FiniteSet</span></tt> of diagrams which are known to
-be commutative in this category.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span><span class="p">,</span> <span class="n">Diagram</span><span class="p">,</span> <span class="n">Category</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">FiniteSet</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">K</span> <span class="o">=</span> <span class="n">Category</span><span class="p">(</span><span class="s">&quot;K&quot;</span><span class="p">,</span> <span class="n">commutative_diagrams</span><span class="o">=</span><span class="p">[</span><span class="n">d</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">K</span><span class="o">.</span><span class="n">commutative_diagrams</span> <span class="o">==</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.categories.Category.name">
-<tt class="descname">name</tt><a class="headerlink" href="#sympy.categories.Category.name" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the name of this category.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Category</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">K</span> <span class="o">=</span> <span class="n">Category</span><span class="p">(</span><span class="s">&quot;K&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">K</span><span class="o">.</span><span class="n">name</span>
-<span class="go">&#39;K&#39;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.categories.Category.objects">
-<tt class="descname">objects</tt><a class="headerlink" href="#sympy.categories.Category.objects" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the class of objects of this category.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">Category</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">FiniteSet</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">K</span> <span class="o">=</span> <span class="n">Category</span><span class="p">(</span><span class="s">&quot;K&quot;</span><span class="p">,</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">K</span><span class="o">.</span><span class="n">objects</span>
-<span class="go">Class({Object(&quot;A&quot;), Object(&quot;B&quot;)})</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.categories.Diagram">
-<em class="property">class </em><tt class="descclassname">sympy.categories.</tt><tt class="descname">Diagram</tt><a class="headerlink" href="#sympy.categories.Diagram" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents a diagram in a certain category.</p>
-<p>Informally, a diagram is a collection of objects of a category and
-certain morphisms between them.  A diagram is still a monoid with
-respect to morphism composition; i.e., identity morphisms, as well
-as all composites of morphisms included in the diagram belong to
-the diagram.  For a more formal approach to this notion see
-[Pare1970].</p>
-<p>The components of composite morphisms are also added to the
-diagram.  No properties are assigned to such morphisms by default.</p>
-<p>A commutative diagram is often accompanied by a statement of the
-following kind: &#8220;if such morphisms with such properties exist,
-then such morphisms which such properties exist and the diagram is
-commutative&#8221;.  To represent this, an instance of <a class="reference internal" href="#sympy.categories.Diagram" title="sympy.categories.Diagram"><tt class="xref py py-class docutils literal"><span class="pre">Diagram</span></tt></a>
-includes a collection of morphisms which are the premises and
-another collection of conclusions.  <tt class="docutils literal"><span class="pre">premises</span></tt> and
-<tt class="docutils literal"><span class="pre">conclusions</span></tt> associate morphisms belonging to the corresponding
-categories with the <tt class="xref py py-class docutils literal"><span class="pre">FiniteSet</span></tt>&#8216;s of their properties.</p>
-<p>The set of properties of a composite morphism is the intersection
-of the sets of properties of its components.  The domain and
-codomain of a conclusion morphism should be among the domains and
-codomains of the morphisms listed as the premises of a diagram.</p>
-<p>No checks are carried out of whether the supplied object and
-morphisms do belong to one and the same category.</p>
-<p class="rubric">References</p>
-<p>[Pare1970] B. Pareigis: Categories and functors.  Academic Press,
-1970.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span><span class="p">,</span> <span class="n">Diagram</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">FiniteSet</span><span class="p">,</span> <span class="n">pprint</span><span class="p">,</span> <span class="n">default_sort_key</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">premises_keys</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">premises</span><span class="o">.</span><span class="n">keys</span><span class="p">()),</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">premises_keys</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">[g*f:A--&gt;C, id:A--&gt;A, id:B--&gt;B, id:C--&gt;C, f:A--&gt;B, g:B--&gt;C]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">premises</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">{g*f:A--&gt;C: EmptySet(), id:A--&gt;A: EmptySet(), id:B--&gt;B: EmptySet(), id:C--&gt;C:</span>
-<span class="go">EmptySet(), f:A--&gt;B: EmptySet(), g:B--&gt;C: EmptySet()}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">],</span> <span class="p">{</span><span class="n">g</span> <span class="o">*</span> <span class="n">f</span><span class="p">:</span> <span class="s">&quot;unique&quot;</span><span class="p">})</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">conclusions</span><span class="p">)</span>
-<span class="go">{g*f:A--&gt;C: {unique}}</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.categories.Diagram.conclusions">
-<tt class="descname">conclusions</tt><a class="headerlink" href="#sympy.categories.Diagram.conclusions" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the conclusions of this diagram.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">IdentityMorphism</span><span class="p">,</span> <span class="n">Diagram</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">FiniteSet</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">IdentityMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">premises</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">*</span> <span class="n">f</span> <span class="ow">in</span> <span class="nb">list</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">premises</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">],</span> <span class="p">{</span><span class="n">g</span> <span class="o">*</span> <span class="n">f</span><span class="p">:</span> <span class="s">&quot;unique&quot;</span><span class="p">})</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="o">.</span><span class="n">conclusions</span><span class="p">[</span><span class="n">g</span> <span class="o">*</span> <span class="n">f</span><span class="p">]</span> <span class="o">==</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="s">&quot;unique&quot;</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.categories.Diagram.hom">
-<tt class="descname">hom</tt><big>(</big><em>self</em>, <em>A</em>, <em>B</em><big>)</big><a class="headerlink" href="#sympy.categories.Diagram.hom" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a 2-tuple of sets of morphisms between objects A and
-B: one set of morphisms listed as premises, and the other set
-of morphisms listed as conclusions.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.categories.Object" title="sympy.categories.Object"><tt class="xref py py-obj docutils literal"><span class="pre">Object</span></tt></a>, <a class="reference internal" href="#sympy.categories.Morphism" title="sympy.categories.Morphism"><tt class="xref py py-obj docutils literal"><span class="pre">Morphism</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span><span class="p">,</span> <span class="n">Diagram</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pretty</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">],</span> <span class="p">{</span><span class="n">g</span> <span class="o">*</span> <span class="n">f</span><span class="p">:</span> <span class="s">&quot;unique&quot;</span><span class="p">})</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">pretty</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">hom</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">C</span><span class="p">),</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">))</span>
-<span class="go">({g*f:A--&gt;C}, {g*f:A--&gt;C})</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.categories.Diagram.is_subdiagram">
-<tt class="descname">is_subdiagram</tt><big>(</big><em>self</em>, <em>diagram</em><big>)</big><a class="headerlink" href="#sympy.categories.Diagram.is_subdiagram" title="Permalink to this definition">¶</a></dt>
-<dd><p>Checks whether <tt class="docutils literal"><span class="pre">diagram</span></tt> is a subdiagram of <tt class="docutils literal"><span class="pre">self</span></tt>.
-Diagram <span class="math">\(D'\)</span> is a subdiagram of <span class="math">\(D\)</span> if all premises
-(conclusions) of <span class="math">\(D'\)</span> are contained in the premises
-(conclusions) of <span class="math">\(D\)</span>.  The morphisms contained
-both in <span class="math">\(D'\)</span> and <span class="math">\(D\)</span> should have the same properties for <span class="math">\(D'\)</span>
-to be a subdiagram of <span class="math">\(D\)</span>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span><span class="p">,</span> <span class="n">Diagram</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">],</span> <span class="p">{</span><span class="n">g</span> <span class="o">*</span> <span class="n">f</span><span class="p">:</span> <span class="s">&quot;unique&quot;</span><span class="p">})</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d1</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="o">.</span><span class="n">is_subdiagram</span><span class="p">(</span><span class="n">d1</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d1</span><span class="o">.</span><span class="n">is_subdiagram</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.categories.Diagram.objects">
-<tt class="descname">objects</tt><a class="headerlink" href="#sympy.categories.Diagram.objects" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the <tt class="xref py py-class docutils literal"><span class="pre">FiniteSet</span></tt> of objects that appear in this
-diagram.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span><span class="p">,</span> <span class="n">Diagram</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="o">.</span><span class="n">objects</span>
-<span class="go">{Object(&quot;A&quot;), Object(&quot;B&quot;), Object(&quot;C&quot;)}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.categories.Diagram.premises">
-<tt class="descname">premises</tt><a class="headerlink" href="#sympy.categories.Diagram.premises" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the premises of this diagram.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">IdentityMorphism</span><span class="p">,</span> <span class="n">Diagram</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pretty</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">id_A</span> <span class="o">=</span> <span class="n">IdentityMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">id_B</span> <span class="o">=</span> <span class="n">IdentityMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">pretty</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">premises</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">))</span>
-<span class="go">{id:A--&gt;A: EmptySet(), id:B--&gt;B: EmptySet(), f:A--&gt;B: EmptySet()}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.categories.Diagram.subdiagram_from_objects">
-<tt class="descname">subdiagram_from_objects</tt><big>(</big><em>self</em>, <em>objects</em><big>)</big><a class="headerlink" href="#sympy.categories.Diagram.subdiagram_from_objects" title="Permalink to this definition">¶</a></dt>
-<dd><p>If <tt class="docutils literal"><span class="pre">objects</span></tt> is a subset of the objects of <tt class="docutils literal"><span class="pre">self</span></tt>, returns
-a diagram which has as premises all those premises of <tt class="docutils literal"><span class="pre">self</span></tt>
-which have a domains and codomains in <tt class="docutils literal"><span class="pre">objects</span></tt>, likewise
-for conclusions.  Properties are preserved.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span><span class="p">,</span> <span class="n">Diagram</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">FiniteSet</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">],</span> <span class="p">{</span><span class="n">f</span><span class="p">:</span> <span class="s">&quot;unique&quot;</span><span class="p">,</span> <span class="n">g</span><span class="o">*</span><span class="n">f</span><span class="p">:</span> <span class="s">&quot;veryunique&quot;</span><span class="p">})</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d1</span> <span class="o">=</span> <span class="n">d</span><span class="o">.</span><span class="n">subdiagram_from_objects</span><span class="p">(</span><span class="n">FiniteSet</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d1</span> <span class="o">==</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">],</span> <span class="p">{</span><span class="n">f</span><span class="p">:</span> <span class="s">&quot;unique&quot;</span><span class="p">})</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.categories.diagram_drawing">
-<span id="diagram-drawing"></span><h2>Diagram Drawing<a class="headerlink" href="#module-sympy.categories.diagram_drawing" title="Permalink to this headline">¶</a></h2>
-<p>This section lists the classes which allow automatic drawing of
-diagrams.</p>
-<dl class="class">
-<dt id="sympy.categories.diagram_drawing.DiagramGrid">
-<em class="property">class </em><tt class="descclassname">sympy.categories.diagram_drawing.</tt><tt class="descname">DiagramGrid</tt><big>(</big><em>diagram</em>, <em>groups=None</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../_modules/sympy/categories/diagram_drawing.html#DiagramGrid"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.categories.diagram_drawing.DiagramGrid" title="Permalink to this definition">¶</a></dt>
-<dd><p>Constructs and holds the fitting of the diagram into a grid.</p>
-<p>The mission of this class is to analyse the structure of the
-supplied diagram and to place its objects on a grid such that,
-when the objects and the morphisms are actually drawn, the diagram
-would be &#8220;readable&#8221;, in the sense that there will not be many
-intersections of moprhisms.  This class does not perform any
-actual drawing.  It does strive nevertheless to offer sufficient
-metadata to draw a diagram.</p>
-<p>Consider the following simple diagram.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Diagram</span><span class="p">,</span> <span class="n">DiagramGrid</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diagram</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">])</span>
-</pre></div>
-</div>
-<p>The simplest way to have a diagram laid out is the following:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">grid</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="p">(</span><span class="n">diagram</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">grid</span><span class="o">.</span><span class="n">width</span><span class="p">,</span> <span class="n">grid</span><span class="o">.</span><span class="n">height</span><span class="p">)</span>
-<span class="go">(2, 2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">grid</span><span class="p">)</span>
-<span class="go">A  B</span>
-
-<span class="go">   C</span>
-</pre></div>
-</div>
-<p>Sometimes one sees the diagram as consisting of logical groups.
-One can advise <tt class="docutils literal"><span class="pre">DiagramGrid</span></tt> as to such groups by employing the
-<tt class="docutils literal"><span class="pre">groups</span></tt> keyword argument.</p>
-<p>Consider the following diagram:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;D&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="s">&quot;h&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;k&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diagram</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">k</span><span class="p">])</span>
-</pre></div>
-</div>
-<p>Lay it out with generic layout:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">grid</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="p">(</span><span class="n">diagram</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">grid</span><span class="p">)</span>
-<span class="go">A  B  D</span>
-
-<span class="go">   C</span>
-</pre></div>
-</div>
-<p>Now, we can group the objects <span class="math">\(A\)</span> and <span class="math">\(D\)</span> to have them near one
-another:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">grid</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="p">(</span><span class="n">diagram</span><span class="p">,</span> <span class="n">groups</span><span class="o">=</span><span class="p">[[</span><span class="n">A</span><span class="p">,</span> <span class="n">D</span><span class="p">],</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">grid</span><span class="p">)</span>
-<span class="go">B     C</span>
-
-<span class="go">A  D</span>
-</pre></div>
-</div>
-<p>Note how the positioning of the other objects changes.</p>
-<p>Further indications can be supplied to the constructor of
-<a class="reference internal" href="#sympy.categories.diagram_drawing.DiagramGrid" title="sympy.categories.diagram_drawing.DiagramGrid"><tt class="xref py py-class docutils literal"><span class="pre">DiagramGrid</span></tt></a> using keyword arguments.  The currently
-supported hints are explained in the following paragraphs.</p>
-<p><a class="reference internal" href="#sympy.categories.diagram_drawing.DiagramGrid" title="sympy.categories.diagram_drawing.DiagramGrid"><tt class="xref py py-class docutils literal"><span class="pre">DiagramGrid</span></tt></a> does not automatically guess which layout
-would suit the supplied diagram better.  Consider, for example,
-the following linear diagram:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">E</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;E&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="s">&quot;h&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="s">&quot;i&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diagram</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">i</span><span class="p">])</span>
-</pre></div>
-</div>
-<p>When laid out with the generic layout, it does not get to look
-linear:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">grid</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="p">(</span><span class="n">diagram</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">grid</span><span class="p">)</span>
-<span class="go">A  B</span>
-
-<span class="go">   C  D</span>
-
-<span class="go">      E</span>
-</pre></div>
-</div>
-<p>To get it laid out in a line, use <tt class="docutils literal"><span class="pre">layout=&quot;sequential&quot;</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">grid</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="p">(</span><span class="n">diagram</span><span class="p">,</span> <span class="n">layout</span><span class="o">=</span><span class="s">&quot;sequential&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">grid</span><span class="p">)</span>
-<span class="go">A  B  C  D  E</span>
-</pre></div>
-</div>
-<p>One may sometimes need to transpose the resulting layout.  While
-this can always be done by hand, <a class="reference internal" href="#sympy.categories.diagram_drawing.DiagramGrid" title="sympy.categories.diagram_drawing.DiagramGrid"><tt class="xref py py-class docutils literal"><span class="pre">DiagramGrid</span></tt></a> provides a
-hint for that purpose:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">grid</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="p">(</span><span class="n">diagram</span><span class="p">,</span> <span class="n">layout</span><span class="o">=</span><span class="s">&quot;sequential&quot;</span><span class="p">,</span> <span class="n">transpose</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">grid</span><span class="p">)</span>
-<span class="go">A</span>
-
-<span class="go">B</span>
-
-<span class="go">C</span>
-
-<span class="go">D</span>
-
-<span class="go">E</span>
-</pre></div>
-</div>
-<p>Separate hints can also be provided for each group.  For an
-example, refer to <tt class="docutils literal"><span class="pre">tests/test_drawing.py</span></tt>, and see the different
-ways in which the five lemma [FiveLemma] can be laid out.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">Diagram</span></tt></p>
-</div>
-<p class="rubric">References</p>
-<p>[FiveLemma] <a class="reference external" href="http://en.wikipedia.org/wiki/Five_lemma">http://en.wikipedia.org/wiki/Five_lemma</a></p>
-<dl class="attribute">
-<dt id="sympy.categories.diagram_drawing.DiagramGrid.height">
-<tt class="descname">height</tt><a class="reference internal" href="../_modules/sympy/categories/diagram_drawing.html#DiagramGrid.height"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.categories.diagram_drawing.DiagramGrid.height" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the number of rows in this diagram layout.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Diagram</span><span class="p">,</span> <span class="n">DiagramGrid</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diagram</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">grid</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="p">(</span><span class="n">diagram</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">grid</span><span class="o">.</span><span class="n">height</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.categories.diagram_drawing.DiagramGrid.morphisms">
-<tt class="descname">morphisms</tt><a class="reference internal" href="../_modules/sympy/categories/diagram_drawing.html#DiagramGrid.morphisms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.categories.diagram_drawing.DiagramGrid.morphisms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns those morphisms (and their properties) which are
-sufficiently meaningful to be drawn.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Diagram</span><span class="p">,</span> <span class="n">DiagramGrid</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diagram</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">grid</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="p">(</span><span class="n">diagram</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">grid</span><span class="o">.</span><span class="n">morphisms</span>
-<span class="go">{NamedMorphism(Object(&quot;A&quot;), Object(&quot;B&quot;), &quot;f&quot;): EmptySet(),</span>
-<span class="go">NamedMorphism(Object(&quot;B&quot;), Object(&quot;C&quot;), &quot;g&quot;): EmptySet()}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.categories.diagram_drawing.DiagramGrid.width">
-<tt class="descname">width</tt><a class="reference internal" href="../_modules/sympy/categories/diagram_drawing.html#DiagramGrid.width"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.categories.diagram_drawing.DiagramGrid.width" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the number of columns in this diagram layout.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Diagram</span><span class="p">,</span> <span class="n">DiagramGrid</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diagram</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">grid</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="p">(</span><span class="n">diagram</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">grid</span><span class="o">.</span><span class="n">width</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.categories.diagram_drawing.ArrowStringDescription">
-<em class="property">class </em><tt class="descclassname">sympy.categories.diagram_drawing.</tt><tt class="descname">ArrowStringDescription</tt><big>(</big><em>unit</em>, <em>curving</em>, <em>curving_amount</em>, <em>looping_start</em>, <em>looping_end</em>, <em>horizontal_direction</em>, <em>vertical_direction</em>, <em>label_position</em>, <em>label</em><big>)</big><a class="reference internal" href="../_modules/sympy/categories/diagram_drawing.html#ArrowStringDescription"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.categories.diagram_drawing.ArrowStringDescription" title="Permalink to this definition">¶</a></dt>
-<dd><p>Stores the information necessary for producing an Xy-pic
-description of an arrow.</p>
-<p>The principal goal of this class is to abstract away the string
-representation of an arrow and to also provide the functionality
-to produce the actual Xy-pic string.</p>
-<p><tt class="docutils literal"><span class="pre">unit</span></tt> sets the unit which will be used to specify the amount of
-curving and other distances.  <tt class="docutils literal"><span class="pre">horizontal_direction</span></tt> should be a
-string of <tt class="docutils literal"><span class="pre">&quot;r&quot;</span></tt> or <tt class="docutils literal"><span class="pre">&quot;l&quot;</span></tt> specifying the horizontal offset of the
-target cell of the arrow relatively to the current one.
-<tt class="docutils literal"><span class="pre">vertical_direction</span></tt> should  specify the vertical offset using a
-series of either <tt class="docutils literal"><span class="pre">&quot;d&quot;</span></tt> or <tt class="docutils literal"><span class="pre">&quot;u&quot;</span></tt>.  <tt class="docutils literal"><span class="pre">label_position</span></tt> should be
-either <tt class="docutils literal"><span class="pre">&quot;^&quot;</span></tt>, <tt class="docutils literal"><span class="pre">&quot;_&quot;</span></tt>,  or <tt class="docutils literal"><span class="pre">&quot;|&quot;</span></tt> to specify that the label should
-be positioned above the arrow, below the arrow or just over the arrow,
-in a break.  Note that the notions &#8220;above&#8221; and &#8220;below&#8221; are relative
-to arrow direction.  <tt class="docutils literal"><span class="pre">label</span></tt> stores the morphism label.</p>
-<p>This works as follows (disregard the yet unexplained arguments):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories.diagram_drawing</span> <span class="kn">import</span> <span class="n">ArrowStringDescription</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">astr</span> <span class="o">=</span> <span class="n">ArrowStringDescription</span><span class="p">(</span>
-<span class="gp">... </span><span class="n">unit</span><span class="o">=</span><span class="s">&quot;mm&quot;</span><span class="p">,</span> <span class="n">curving</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">curving_amount</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">looping_start</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">looping_end</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">horizontal_direction</span><span class="o">=</span><span class="s">&quot;d&quot;</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">vertical_direction</span><span class="o">=</span><span class="s">&quot;r&quot;</span><span class="p">,</span> <span class="n">label_position</span><span class="o">=</span><span class="s">&quot;_&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">astr</span><span class="p">))</span>
-<span class="go">\ar[dr]_{f}</span>
-</pre></div>
-</div>
-<p><tt class="docutils literal"><span class="pre">curving</span></tt> should be one of <tt class="docutils literal"><span class="pre">&quot;^&quot;</span></tt>, <tt class="docutils literal"><span class="pre">&quot;_&quot;</span></tt> to specify in which
-direction the arrow is going to curve. <tt class="docutils literal"><span class="pre">curving_amount</span></tt> is a number
-describing how many <tt class="docutils literal"><span class="pre">unit</span></tt>&#8216;s the morphism is going to curve:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">astr</span> <span class="o">=</span> <span class="n">ArrowStringDescription</span><span class="p">(</span>
-<span class="gp">... </span><span class="n">unit</span><span class="o">=</span><span class="s">&quot;mm&quot;</span><span class="p">,</span> <span class="n">curving</span><span class="o">=</span><span class="s">&quot;^&quot;</span><span class="p">,</span> <span class="n">curving_amount</span><span class="o">=</span><span class="mi">12</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">looping_start</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">looping_end</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">horizontal_direction</span><span class="o">=</span><span class="s">&quot;d&quot;</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">vertical_direction</span><span class="o">=</span><span class="s">&quot;r&quot;</span><span class="p">,</span> <span class="n">label_position</span><span class="o">=</span><span class="s">&quot;_&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">astr</span><span class="p">))</span>
-<span class="go">\ar@/^12mm/[dr]_{f}</span>
-</pre></div>
-</div>
-<p><tt class="docutils literal"><span class="pre">looping_start</span></tt> and <tt class="docutils literal"><span class="pre">looping_end</span></tt> are currently only used for
-loop morphisms, those which have the same domain and codomain.
-These two attributes should store a valid Xy-pic direction and
-specify, correspondingly, the direction the arrow gets out into
-and the direction the arrow gets back from:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">astr</span> <span class="o">=</span> <span class="n">ArrowStringDescription</span><span class="p">(</span>
-<span class="gp">... </span><span class="n">unit</span><span class="o">=</span><span class="s">&quot;mm&quot;</span><span class="p">,</span> <span class="n">curving</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">curving_amount</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">looping_start</span><span class="o">=</span><span class="s">&quot;u&quot;</span><span class="p">,</span> <span class="n">looping_end</span><span class="o">=</span><span class="s">&quot;l&quot;</span><span class="p">,</span> <span class="n">horizontal_direction</span><span class="o">=</span><span class="s">&quot;&quot;</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">vertical_direction</span><span class="o">=</span><span class="s">&quot;&quot;</span><span class="p">,</span> <span class="n">label_position</span><span class="o">=</span><span class="s">&quot;_&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">astr</span><span class="p">))</span>
-<span class="go">\ar@(u,l)[]_{f}</span>
-</pre></div>
-</div>
-<p><tt class="docutils literal"><span class="pre">label_displacement</span></tt> controls how far the arrow label is from
-the ends of the arrow.  For example, to position the arrow label
-near the arrow head, use &#8220;&gt;&#8221;:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">astr</span> <span class="o">=</span> <span class="n">ArrowStringDescription</span><span class="p">(</span>
-<span class="gp">... </span><span class="n">unit</span><span class="o">=</span><span class="s">&quot;mm&quot;</span><span class="p">,</span> <span class="n">curving</span><span class="o">=</span><span class="s">&quot;^&quot;</span><span class="p">,</span> <span class="n">curving_amount</span><span class="o">=</span><span class="mi">12</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">looping_start</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">looping_end</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">horizontal_direction</span><span class="o">=</span><span class="s">&quot;d&quot;</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">vertical_direction</span><span class="o">=</span><span class="s">&quot;r&quot;</span><span class="p">,</span> <span class="n">label_position</span><span class="o">=</span><span class="s">&quot;_&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">astr</span><span class="o">.</span><span class="n">label_displacement</span> <span class="o">=</span> <span class="s">&quot;&gt;&quot;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">astr</span><span class="p">))</span>
-<span class="go">\ar@/^12mm/[dr]_&gt;{f}</span>
-</pre></div>
-</div>
-<p>Finally, <tt class="docutils literal"><span class="pre">arrow_style</span></tt> is used to specify the arrow style.  To
-get a dashed arrow, for example, use &#8220;{&#8211;&gt;}&#8221; as arrow style:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">astr</span> <span class="o">=</span> <span class="n">ArrowStringDescription</span><span class="p">(</span>
-<span class="gp">... </span><span class="n">unit</span><span class="o">=</span><span class="s">&quot;mm&quot;</span><span class="p">,</span> <span class="n">curving</span><span class="o">=</span><span class="s">&quot;^&quot;</span><span class="p">,</span> <span class="n">curving_amount</span><span class="o">=</span><span class="mi">12</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">looping_start</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">looping_end</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span> <span class="n">horizontal_direction</span><span class="o">=</span><span class="s">&quot;d&quot;</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">vertical_direction</span><span class="o">=</span><span class="s">&quot;r&quot;</span><span class="p">,</span> <span class="n">label_position</span><span class="o">=</span><span class="s">&quot;_&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">astr</span><span class="o">.</span><span class="n">arrow_style</span> <span class="o">=</span> <span class="s">&quot;{--&gt;}&quot;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">astr</span><span class="p">))</span>
-<span class="go">\ar@/^12mm/@{--&gt;}[dr]_{f}</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.categories.diagram_drawing.XypicDiagramDrawer" title="sympy.categories.diagram_drawing.XypicDiagramDrawer"><tt class="xref py py-obj docutils literal"><span class="pre">XypicDiagramDrawer</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>Instances of <a class="reference internal" href="#sympy.categories.diagram_drawing.ArrowStringDescription" title="sympy.categories.diagram_drawing.ArrowStringDescription"><tt class="xref py py-class docutils literal"><span class="pre">ArrowStringDescription</span></tt></a> will be constructed
-by <a class="reference internal" href="#sympy.categories.diagram_drawing.XypicDiagramDrawer" title="sympy.categories.diagram_drawing.XypicDiagramDrawer"><tt class="xref py py-class docutils literal"><span class="pre">XypicDiagramDrawer</span></tt></a> and provided for further use in
-formatters.  The user is not expected to construct instances of
-<a class="reference internal" href="#sympy.categories.diagram_drawing.ArrowStringDescription" title="sympy.categories.diagram_drawing.ArrowStringDescription"><tt class="xref py py-class docutils literal"><span class="pre">ArrowStringDescription</span></tt></a> themselves.</p>
-<p>To be able to properly utilise this class, the reader is encouraged
-to checkout the Xy-pic user guide, available at [Xypic].</p>
-<p class="rubric">References</p>
-<p>[Xypic] <a class="reference external" href="http://www.tug.org/applications/Xy-pic/">http://www.tug.org/applications/Xy-pic/</a></p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.categories.diagram_drawing.XypicDiagramDrawer">
-<em class="property">class </em><tt class="descclassname">sympy.categories.diagram_drawing.</tt><tt class="descname">XypicDiagramDrawer</tt><a class="reference internal" href="../_modules/sympy/categories/diagram_drawing.html#XypicDiagramDrawer"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.categories.diagram_drawing.XypicDiagramDrawer" title="Permalink to this definition">¶</a></dt>
-<dd><p>Given a <tt class="xref py py-class docutils literal"><span class="pre">Diagram</span></tt> and the corresponding
-<a class="reference internal" href="#sympy.categories.diagram_drawing.DiagramGrid" title="sympy.categories.diagram_drawing.DiagramGrid"><tt class="xref py py-class docutils literal"><span class="pre">DiagramGrid</span></tt></a>, produces the Xy-pic representation of the
-diagram.</p>
-<p>The most important method in this class is <tt class="docutils literal"><span class="pre">draw</span></tt>.  Consider the
-following triangle diagram:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span><span class="p">,</span> <span class="n">Diagram</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">DiagramGrid</span><span class="p">,</span> <span class="n">XypicDiagramDrawer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diagram</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">],</span> <span class="p">{</span><span class="n">g</span> <span class="o">*</span> <span class="n">f</span><span class="p">:</span> <span class="s">&quot;unique&quot;</span><span class="p">})</span>
-</pre></div>
-</div>
-<p>To draw this diagram, its objects need to be laid out with a
-<a class="reference internal" href="#sympy.categories.diagram_drawing.DiagramGrid" title="sympy.categories.diagram_drawing.DiagramGrid"><tt class="xref py py-class docutils literal"><span class="pre">DiagramGrid</span></tt></a>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">grid</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="p">(</span><span class="n">diagram</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Finally, the drawing:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">drawer</span> <span class="o">=</span> <span class="n">XypicDiagramDrawer</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">drawer</span><span class="o">.</span><span class="n">draw</span><span class="p">(</span><span class="n">diagram</span><span class="p">,</span> <span class="n">grid</span><span class="p">))</span>
-<span class="go">\xymatrix{</span>
-<span class="go">A \ar[d]_{g\circ f} \ar[r]^{f} &amp; B \ar[ld]^{g} \\</span>
-<span class="go">C &amp;</span>
-<span class="go">}</span>
-</pre></div>
-</div>
-<p>For further details see the docstring of this method.</p>
-<p>To control the appearance of the arrows, formatters are used.  The
-dictionary <tt class="docutils literal"><span class="pre">arrow_formatters</span></tt> maps morphisms to formatter
-functions.  A formatter is accepts an
-<a class="reference internal" href="#sympy.categories.diagram_drawing.ArrowStringDescription" title="sympy.categories.diagram_drawing.ArrowStringDescription"><tt class="xref py py-class docutils literal"><span class="pre">ArrowStringDescription</span></tt></a> and is allowed to modify any of
-the arrow properties exposed thereby.  For example, to have all
-morphisms with the property <tt class="docutils literal"><span class="pre">unique</span></tt> appear as dashed arrows,
-and to have their names prepended with <span class="math">\(\exists !\)</span>, the following
-should be done:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">formatter</span><span class="p">(</span><span class="n">astr</span><span class="p">):</span>
-<span class="gp">... </span>  <span class="n">astr</span><span class="o">.</span><span class="n">label</span> <span class="o">=</span> <span class="s">&quot;\exists !&quot;</span> <span class="o">+</span> <span class="n">astr</span><span class="o">.</span><span class="n">label</span>
-<span class="gp">... </span>  <span class="n">astr</span><span class="o">.</span><span class="n">arrow_style</span> <span class="o">=</span> <span class="s">&quot;{--&gt;}&quot;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">drawer</span><span class="o">.</span><span class="n">arrow_formatters</span><span class="p">[</span><span class="s">&quot;unique&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">formatter</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">drawer</span><span class="o">.</span><span class="n">draw</span><span class="p">(</span><span class="n">diagram</span><span class="p">,</span> <span class="n">grid</span><span class="p">))</span>
-<span class="go">\xymatrix{</span>
-<span class="go">A \ar@{--&gt;}[d]_{\exists !g\circ f} \ar[r]^{f} &amp; B \ar[ld]^{g} \\</span>
-<span class="go">C &amp;</span>
-<span class="go">}</span>
-</pre></div>
-</div>
-<p>To modify the appearance of all arrows in the diagram, set
-<tt class="docutils literal"><span class="pre">default_arrow_formatter</span></tt>.  For example, to place all morphism
-labels a little bit farther from the arrow head so that they look
-more centred, do as follows:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">default_formatter</span><span class="p">(</span><span class="n">astr</span><span class="p">):</span>
-<span class="gp">... </span>  <span class="n">astr</span><span class="o">.</span><span class="n">label_displacement</span> <span class="o">=</span> <span class="s">&quot;(0.45)&quot;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">drawer</span><span class="o">.</span><span class="n">default_arrow_formatter</span> <span class="o">=</span> <span class="n">default_formatter</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">drawer</span><span class="o">.</span><span class="n">draw</span><span class="p">(</span><span class="n">diagram</span><span class="p">,</span> <span class="n">grid</span><span class="p">))</span>
-<span class="go">\xymatrix{</span>
-<span class="go">A \ar@{--&gt;}[d]_(0.45){\exists !g\circ f} \ar[r]^(0.45){f} &amp; B \ar[ld]^(0.45){g} \\</span>
-<span class="go">C &amp;</span>
-<span class="go">}</span>
-</pre></div>
-</div>
-<p>In some diagrams some morphisms are drawn as curved arrows.
-Consider the following diagram:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;D&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;E&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="s">&quot;h&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;k&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diagram</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">k</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">grid</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="p">(</span><span class="n">diagram</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">drawer</span> <span class="o">=</span> <span class="n">XypicDiagramDrawer</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">drawer</span><span class="o">.</span><span class="n">draw</span><span class="p">(</span><span class="n">diagram</span><span class="p">,</span> <span class="n">grid</span><span class="p">))</span>
-<span class="go">\xymatrix{</span>
-<span class="go">A \ar[r]_{f} &amp; B \ar[d]^{g} &amp; D \ar[l]^{k} \ar@/_3mm/[ll]_{h} \\</span>
-<span class="go">&amp; C &amp;</span>
-<span class="go">}</span>
-</pre></div>
-</div>
-<p>To control how far the morphisms are curved by default, one can
-use the <tt class="docutils literal"><span class="pre">unit</span></tt> and <tt class="docutils literal"><span class="pre">default_curving_amount</span></tt> attributes:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">drawer</span><span class="o">.</span><span class="n">unit</span> <span class="o">=</span> <span class="s">&quot;cm&quot;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">drawer</span><span class="o">.</span><span class="n">default_curving_amount</span> <span class="o">=</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">drawer</span><span class="o">.</span><span class="n">draw</span><span class="p">(</span><span class="n">diagram</span><span class="p">,</span> <span class="n">grid</span><span class="p">))</span>
-<span class="go">\xymatrix{</span>
-<span class="go">A \ar[r]_{f} &amp; B \ar[d]^{g} &amp; D \ar[l]^{k} \ar@/_1cm/[ll]_{h} \\</span>
-<span class="go">&amp; C &amp;</span>
-<span class="go">}</span>
-</pre></div>
-</div>
-<p>In some diagrams, there are multiple curved morphisms between the
-same two objects.  To control by how much the curving changes
-between two such successive morphisms, use
-<tt class="docutils literal"><span class="pre">default_curving_step</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">drawer</span><span class="o">.</span><span class="n">default_curving_step</span> <span class="o">=</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h1</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">D</span><span class="p">,</span> <span class="s">&quot;h1&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diagram</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">h1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">grid</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="p">(</span><span class="n">diagram</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">drawer</span><span class="o">.</span><span class="n">draw</span><span class="p">(</span><span class="n">diagram</span><span class="p">,</span> <span class="n">grid</span><span class="p">))</span>
-<span class="go">\xymatrix{</span>
-<span class="go">A \ar[r]_{f} \ar@/^1cm/[rr]^{h_{1}} &amp; B \ar[d]^{g} &amp; D \ar[l]^{k} \ar@/_2cm/[ll]_{h} \\</span>
-<span class="go">&amp; C &amp;</span>
-<span class="go">}</span>
-</pre></div>
-</div>
-<p>The default value of <tt class="docutils literal"><span class="pre">default_curving_step</span></tt> is 4 units.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.categories.diagram_drawing.XypicDiagramDrawer.draw" title="sympy.categories.diagram_drawing.XypicDiagramDrawer.draw"><tt class="xref py py-obj docutils literal"><span class="pre">draw</span></tt></a>, <a class="reference internal" href="#sympy.categories.diagram_drawing.ArrowStringDescription" title="sympy.categories.diagram_drawing.ArrowStringDescription"><tt class="xref py py-obj docutils literal"><span class="pre">ArrowStringDescription</span></tt></a></p>
-</div>
-<dl class="function">
-<dt id="sympy.categories.diagram_drawing.XypicDiagramDrawer.draw">
-<tt class="descname">draw</tt><big>(</big><em>self</em>, <em>diagram</em>, <em>grid</em>, <em>masked=None</em>, <em>diagram_format=''</em><big>)</big><a class="reference internal" href="../_modules/sympy/categories/diagram_drawing.html#XypicDiagramDrawer.draw"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.categories.diagram_drawing.XypicDiagramDrawer.draw" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the Xy-pic representation of <tt class="docutils literal"><span class="pre">diagram</span></tt> laid out in
-<tt class="docutils literal"><span class="pre">grid</span></tt>.</p>
-<p>Consider the following simple triangle diagram.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span><span class="p">,</span> <span class="n">Diagram</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">DiagramGrid</span><span class="p">,</span> <span class="n">XypicDiagramDrawer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diagram</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">],</span> <span class="p">{</span><span class="n">g</span> <span class="o">*</span> <span class="n">f</span><span class="p">:</span> <span class="s">&quot;unique&quot;</span><span class="p">})</span>
-</pre></div>
-</div>
-<p>To draw this diagram, its objects need to be laid out with a
-<a class="reference internal" href="#sympy.categories.diagram_drawing.DiagramGrid" title="sympy.categories.diagram_drawing.DiagramGrid"><tt class="xref py py-class docutils literal"><span class="pre">DiagramGrid</span></tt></a>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">grid</span> <span class="o">=</span> <span class="n">DiagramGrid</span><span class="p">(</span><span class="n">diagram</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Finally, the drawing:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">drawer</span> <span class="o">=</span> <span class="n">XypicDiagramDrawer</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">drawer</span><span class="o">.</span><span class="n">draw</span><span class="p">(</span><span class="n">diagram</span><span class="p">,</span> <span class="n">grid</span><span class="p">))</span>
-<span class="go">\xymatrix{</span>
-<span class="go">A \ar[d]_{g\circ f} \ar[r]^{f} &amp; B \ar[ld]^{g} \\</span>
-<span class="go">C &amp;</span>
-<span class="go">}</span>
-</pre></div>
-</div>
-<p>The argument <tt class="docutils literal"><span class="pre">masked</span></tt> can be used to skip morphisms in the
-presentation of the diagram:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">drawer</span><span class="o">.</span><span class="n">draw</span><span class="p">(</span><span class="n">diagram</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">masked</span><span class="o">=</span><span class="p">[</span><span class="n">g</span> <span class="o">*</span> <span class="n">f</span><span class="p">]))</span>
-<span class="go">\xymatrix{</span>
-<span class="go">A \ar[r]^{f} &amp; B \ar[ld]^{g} \\</span>
-<span class="go">C &amp;</span>
-<span class="go">}</span>
-</pre></div>
-</div>
-<p>Finally, the <tt class="docutils literal"><span class="pre">diagram_format</span></tt> argument can be used to
-specify the format string of the diagram.  For example, to
-increase the spacing by 1 cm, proceeding as follows:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">drawer</span><span class="o">.</span><span class="n">draw</span><span class="p">(</span><span class="n">diagram</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">diagram_format</span><span class="o">=</span><span class="s">&quot;@+1cm&quot;</span><span class="p">))</span>
-<span class="go">\xymatrix@+1cm{</span>
-<span class="go">A \ar[d]_{g\circ f} \ar[r]^{f} &amp; B \ar[ld]^{g} \\</span>
-<span class="go">C &amp;</span>
-<span class="go">}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.categories.diagram_drawing.xypic_draw_diagram">
-<tt class="descclassname">sympy.categories.diagram_drawing.</tt><tt class="descname">xypic_draw_diagram</tt><big>(</big><em>diagram</em>, <em>masked=None</em>, <em>diagram_format=''</em>, <em>groups=None</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../_modules/sympy/categories/diagram_drawing.html#xypic_draw_diagram"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.categories.diagram_drawing.xypic_draw_diagram" title="Permalink to this definition">¶</a></dt>
-<dd><p>Provides a shortcut combining <a class="reference internal" href="#sympy.categories.diagram_drawing.DiagramGrid" title="sympy.categories.diagram_drawing.DiagramGrid"><tt class="xref py py-class docutils literal"><span class="pre">DiagramGrid</span></tt></a> and
-<a class="reference internal" href="#sympy.categories.diagram_drawing.XypicDiagramDrawer" title="sympy.categories.diagram_drawing.XypicDiagramDrawer"><tt class="xref py py-class docutils literal"><span class="pre">XypicDiagramDrawer</span></tt></a>.  Returns an Xy-pic presentation of
-<tt class="docutils literal"><span class="pre">diagram</span></tt>.  The argument <tt class="docutils literal"><span class="pre">masked</span></tt> is a list of morphisms which
-will be not be drawn.  The argument <tt class="docutils literal"><span class="pre">diagram_format</span></tt> is the
-format string inserted after &#8220;xymatrix&#8221;.  <tt class="docutils literal"><span class="pre">groups</span></tt> should be a
-set of logical groups.  The <tt class="docutils literal"><span class="pre">hints</span></tt> will be passed directly to
-the constructor of <a class="reference internal" href="#sympy.categories.diagram_drawing.DiagramGrid" title="sympy.categories.diagram_drawing.DiagramGrid"><tt class="xref py py-class docutils literal"><span class="pre">DiagramGrid</span></tt></a>.</p>
-<p>For more information about the arguments, see the docstrings of
-<a class="reference internal" href="#sympy.categories.diagram_drawing.DiagramGrid" title="sympy.categories.diagram_drawing.DiagramGrid"><tt class="xref py py-class docutils literal"><span class="pre">DiagramGrid</span></tt></a> and <tt class="docutils literal"><span class="pre">XypicDiagramDrawer.draw</span></tt>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.categories.diagram_drawing.XypicDiagramDrawer" title="sympy.categories.diagram_drawing.XypicDiagramDrawer"><tt class="xref py py-obj docutils literal"><span class="pre">XypicDiagramDrawer</span></tt></a>, <a class="reference internal" href="#sympy.categories.diagram_drawing.DiagramGrid" title="sympy.categories.diagram_drawing.DiagramGrid"><tt class="xref py py-obj docutils literal"><span class="pre">DiagramGrid</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span><span class="p">,</span> <span class="n">Diagram</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">xypic_draw_diagram</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diagram</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">],</span> <span class="p">{</span><span class="n">g</span> <span class="o">*</span> <span class="n">f</span><span class="p">:</span> <span class="s">&quot;unique&quot;</span><span class="p">})</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">xypic_draw_diagram</span><span class="p">(</span><span class="n">diagram</span><span class="p">))</span>
-<span class="go">\xymatrix{</span>
-<span class="go">A \ar[d]_{g\circ f} \ar[r]^{f} &amp; B \ar[ld]^{g} \\</span>
-<span class="go">C &amp;</span>
-<span class="go">}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.categories.diagram_drawing.preview_diagram">
-<tt class="descclassname">sympy.categories.diagram_drawing.</tt><tt class="descname">preview_diagram</tt><big>(</big><em>diagram</em>, <em>masked=None</em>, <em>diagram_format=''</em>, <em>groups=None</em>, <em>output='png'</em>, <em>viewer=None</em>, <em>euler=True</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../_modules/sympy/categories/diagram_drawing.html#preview_diagram"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.categories.diagram_drawing.preview_diagram" title="Permalink to this definition">¶</a></dt>
-<dd><p>Combines the functionality of <tt class="docutils literal"><span class="pre">xypic_draw_diagram</span></tt> and
-<tt class="docutils literal"><span class="pre">sympy.printing.preview</span></tt>.  The arguments <tt class="docutils literal"><span class="pre">masked</span></tt>,
-<tt class="docutils literal"><span class="pre">diagram_format</span></tt>, <tt class="docutils literal"><span class="pre">groups</span></tt>, and <tt class="docutils literal"><span class="pre">hints</span></tt> are passed to
-<tt class="docutils literal"><span class="pre">xypic_draw_diagram</span></tt>, while <tt class="docutils literal"><span class="pre">output</span></tt>, <tt class="docutils literal"><span class="pre">viewer,</span> <span class="pre">and</span> <span class="pre">``euler</span></tt>
-are passed to <tt class="docutils literal"><span class="pre">preview</span></tt>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">xypic_diagram_drawer</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">Object</span><span class="p">,</span> <span class="n">NamedMorphism</span><span class="p">,</span> <span class="n">Diagram</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.categories</span> <span class="kn">import</span> <span class="n">preview_diagram</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;A&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;B&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Object</span><span class="p">(</span><span class="s">&quot;C&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="s">&quot;f&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">NamedMorphism</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="s">&quot;g&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">Diagram</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">],</span> <span class="p">{</span><span class="n">g</span> <span class="o">*</span> <span class="n">f</span><span class="p">:</span> <span class="s">&quot;unique&quot;</span><span class="p">})</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">preview_diagram</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>  
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../index.html">
-              <img class="logo" src="../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Category Theory Module</a><ul>
-<li><a class="reference internal" href="#introduction">Introduction</a></li>
-<li><a class="reference internal" href="#module-sympy.categories">Base Class Reference</a></li>
-<li><a class="reference internal" href="#module-sympy.categories.diagram_drawing">Diagram Drawing</a></li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="physics/quantum/piab.html"
-                        title="previous chapter">Particle in a Box</a></p>
-  <h4>Next topic</h4>
-  <p class="topless"><a href="diffgeom.html"
-                        title="next chapter">Differential Geometry Module</a></p>
-  <h3>This Page</h3>
-  <ul class="this-page-menu">
-    <li><a href="../_sources/modules/categories.txt"
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-  <div class="section" id="module-sympy.combinatorics.graycode">
-<span id="gray-code"></span><span id="combinatorics-graycode"></span><h1>Gray Code<a class="headerlink" href="#module-sympy.combinatorics.graycode" title="Permalink to this headline">¶</a></h1>
-<dl class="class">
-<dt id="sympy.combinatorics.graycode.GrayCode">
-<em class="property">class </em><tt class="descclassname">sympy.combinatorics.graycode.</tt><tt class="descname">GrayCode</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/graycode.html#GrayCode"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.graycode.GrayCode" title="Permalink to this definition">¶</a></dt>
-<dd><p>A Gray code is essentially a Hamiltonian walk on
-a n-dimensional cube with edge length of one.
-The vertices of the cube are represented by vectors
-whose values are binary. The Hamilton walk visits
-each vertex exactly once. The Gray code for a 3d
-cube is [&#8216;000&#8217;,&#8216;100&#8217;,&#8216;110&#8217;,&#8216;010&#8217;,&#8216;011&#8217;,&#8216;111&#8217;,&#8216;101&#8217;,
-&#8216;001&#8217;].</p>
-<p>A Gray code solves the problem of sequentially
-generating all possible subsets of n objects in such
-a way that each subset is obtained from the previous
-one by either deleting or adding a single object.
-In the above example, 1 indicates that the object is
-present, and 0 indicates that its absent.</p>
-<p>Gray codes have applications in statistics as well when
-we want to compute various statistics related to subsets
-in an efficient manner.</p>
-<p>References:
-[1] Nijenhuis,A. and Wilf,H.S.(1978).
-Combinatorial Algorithms. Academic Press.
-[2] Knuth, D. (2011). The Art of Computer Programming, Vol 4
-Addison Wesley</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.graycode</span> <span class="kn">import</span> <span class="n">GrayCode</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">GrayCode</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">generate_gray</span><span class="p">())</span>
-<span class="go">[&#39;000&#39;, &#39;001&#39;, &#39;011&#39;, &#39;010&#39;, &#39;110&#39;, &#39;111&#39;, &#39;101&#39;, &#39;100&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">GrayCode</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">generate_gray</span><span class="p">())</span>
-<span class="go">[&#39;0000&#39;, &#39;0001&#39;, &#39;0011&#39;, &#39;0010&#39;, &#39;0110&#39;, &#39;0111&#39;, &#39;0101&#39;, &#39;0100&#39;,     &#39;1100&#39;, &#39;1101&#39;, &#39;1111&#39;, &#39;1110&#39;, &#39;1010&#39;, &#39;1011&#39;, &#39;1001&#39;, &#39;1000&#39;]</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.combinatorics.graycode.GrayCode.current">
-<tt class="descname">current</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/graycode.html#GrayCode.current"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.graycode.GrayCode.current" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the currently referenced Gray code as a bit string.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.graycode</span> <span class="kn">import</span> <span class="n">GrayCode</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">GrayCode</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">start</span><span class="o">=</span><span class="s">&#39;100&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">current</span>
-<span class="go">&#39;100&#39;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.graycode.GrayCode.generate_gray">
-<tt class="descname">generate_gray</tt><big>(</big><em>self</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/graycode.html#GrayCode.generate_gray"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.graycode.GrayCode.generate_gray" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates the sequence of bit vectors of a Gray Code.</p>
-<p>[1] Knuth, D. (2011). The Art of Computer Programming,
-Vol 4, Addison Wesley</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.graycode.GrayCode.skip" title="sympy.combinatorics.graycode.GrayCode.skip"><tt class="xref py py-obj docutils literal"><span class="pre">skip</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.graycode</span> <span class="kn">import</span> <span class="n">GrayCode</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">GrayCode</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">generate_gray</span><span class="p">())</span>
-<span class="go">[&#39;000&#39;, &#39;001&#39;, &#39;011&#39;, &#39;010&#39;, &#39;110&#39;, &#39;111&#39;, &#39;101&#39;, &#39;100&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">generate_gray</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="s">&#39;011&#39;</span><span class="p">))</span>
-<span class="go">[&#39;011&#39;, &#39;010&#39;, &#39;110&#39;, &#39;111&#39;, &#39;101&#39;, &#39;100&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">generate_gray</span><span class="p">(</span><span class="n">rank</span><span class="o">=</span><span class="mi">4</span><span class="p">))</span>
-<span class="go">[&#39;110&#39;, &#39;111&#39;, &#39;101&#39;, &#39;100&#39;]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.graycode.GrayCode.n">
-<tt class="descname">n</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/graycode.html#GrayCode.n"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.graycode.GrayCode.n" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the dimension of the Gray code.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.graycode</span> <span class="kn">import</span> <span class="n">GrayCode</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">GrayCode</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">n</span>
-<span class="go">5</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.graycode.GrayCode.next">
-<tt class="descname">next</tt><big>(</big><em>self</em>, <em>delta=1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/graycode.html#GrayCode.next"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.graycode.GrayCode.next" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the Gray code a distance <tt class="docutils literal"><span class="pre">delta</span></tt> (default = 1) from the
-current value in canonical order.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.graycode</span> <span class="kn">import</span> <span class="n">GrayCode</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">GrayCode</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">start</span><span class="o">=</span><span class="s">&#39;110&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">next</span><span class="p">()</span><span class="o">.</span><span class="n">current</span>
-<span class="go">&#39;111&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">next</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">current</span>
-<span class="go">&#39;010&#39;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.graycode.GrayCode.rank">
-<tt class="descname">rank</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/graycode.html#GrayCode.rank"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.graycode.GrayCode.rank" title="Permalink to this definition">¶</a></dt>
-<dd><p>Ranks the Gray code.</p>
-<p>A ranking algorithm determines the position (or rank)
-of a combinatorial object among all the objects w.r.t.
-a given order. For example, the 4 bit binary reflected
-Gray code (BRGC) &#8216;0101&#8217; has a rank of 6 as it appears in
-the 6th position in the canonical ordering of the family
-of 4 bit Gray codes.</p>
-<p>References:
-[1] <a class="reference external" href="http://www-stat.stanford.edu/~susan/courses/s208/node12.html">http://www-stat.stanford.edu/~susan/courses/s208/node12.html</a></p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.graycode.GrayCode.unrank" title="sympy.combinatorics.graycode.GrayCode.unrank"><tt class="xref py py-obj docutils literal"><span class="pre">unrank</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.graycode</span> <span class="kn">import</span> <span class="n">GrayCode</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">GrayCode</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">generate_gray</span><span class="p">())</span>
-<span class="go">[&#39;000&#39;, &#39;001&#39;, &#39;011&#39;, &#39;010&#39;, &#39;110&#39;, &#39;111&#39;, &#39;101&#39;, &#39;100&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">GrayCode</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">start</span><span class="o">=</span><span class="s">&#39;100&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">rank</span>
-<span class="go">7</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">GrayCode</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">rank</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span><span class="o">.</span><span class="n">current</span>
-<span class="go">&#39;100&#39;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.graycode.GrayCode.selections">
-<tt class="descname">selections</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/graycode.html#GrayCode.selections"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.graycode.GrayCode.selections" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the number of bit vectors in the Gray code.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.graycode</span> <span class="kn">import</span> <span class="n">GrayCode</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">GrayCode</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">selections</span>
-<span class="go">8</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.graycode.GrayCode.skip">
-<tt class="descname">skip</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/graycode.html#GrayCode.skip"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.graycode.GrayCode.skip" title="Permalink to this definition">¶</a></dt>
-<dd><p>Skips the bit generation.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.graycode.GrayCode.generate_gray" title="sympy.combinatorics.graycode.GrayCode.generate_gray"><tt class="xref py py-obj docutils literal"><span class="pre">generate_gray</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.graycode</span> <span class="kn">import</span> <span class="n">GrayCode</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">GrayCode</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">a</span><span class="o">.</span><span class="n">generate_gray</span><span class="p">():</span>
-<span class="gp">... </span>    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="s">&#39;010&#39;</span><span class="p">:</span>
-<span class="gp">... </span>        <span class="n">a</span><span class="o">.</span><span class="n">skip</span><span class="p">()</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="go">000</span>
-<span class="go">001</span>
-<span class="go">011</span>
-<span class="go">010</span>
-<span class="go">111</span>
-<span class="go">101</span>
-<span class="go">100</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.combinatorics.graycode.GrayCode.unrank">
-<em class="property">classmethod </em><tt class="descname">unrank</tt><big>(</big><em>n</em>, <em>rank</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/graycode.html#GrayCode.unrank"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.graycode.GrayCode.unrank" title="Permalink to this definition">¶</a></dt>
-<dd><p>Unranks an n-bit sized Gray code of rank k. This method exists
-so that a derivative GrayCode class can define its own code of
-a given rank.</p>
-<p>The string here is generated in reverse order to allow for tail-call
-optimization.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.graycode.GrayCode.rank" title="sympy.combinatorics.graycode.GrayCode.rank"><tt class="xref py py-obj docutils literal"><span class="pre">rank</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.graycode</span> <span class="kn">import</span> <span class="n">GrayCode</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">GrayCode</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="n">rank</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">current</span>
-<span class="go">&#39;00010&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">GrayCode</span><span class="o">.</span><span class="n">unrank</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">&#39;00010&#39;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="method">
-<dt id="sympy.combinatorics.graycode.random_bitstring">
-<tt class="descclassname">graycode.</tt><tt class="descname">random_bitstring</tt><big>(</big><em>n</em><big>)</big><a class="headerlink" href="#sympy.combinatorics.graycode.random_bitstring" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates a random bitlist of length n.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.graycode</span> <span class="kn">import</span> <span class="n">random_bitstring</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">random_bitstring</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> 
-<span class="go">100</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="method">
-<dt id="sympy.combinatorics.graycode.gray_to_bin">
-<tt class="descclassname">graycode.</tt><tt class="descname">gray_to_bin</tt><big>(</big><em>bin_list</em><big>)</big><a class="headerlink" href="#sympy.combinatorics.graycode.gray_to_bin" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert from Gray coding to binary coding.</p>
-<p>We assume big endian encoding.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.graycode.bin_to_gray" title="sympy.combinatorics.graycode.bin_to_gray"><tt class="xref py py-obj docutils literal"><span class="pre">bin_to_gray</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.graycode</span> <span class="kn">import</span> <span class="n">gray_to_bin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gray_to_bin</span><span class="p">(</span><span class="s">&#39;100&#39;</span><span class="p">)</span>
-<span class="go">&#39;111&#39;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="method">
-<dt id="sympy.combinatorics.graycode.bin_to_gray">
-<tt class="descclassname">graycode.</tt><tt class="descname">bin_to_gray</tt><big>(</big><em>bin_list</em><big>)</big><a class="headerlink" href="#sympy.combinatorics.graycode.bin_to_gray" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert from binary coding to gray coding.</p>
-<p>We assume big endian encoding.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.graycode.gray_to_bin" title="sympy.combinatorics.graycode.gray_to_bin"><tt class="xref py py-obj docutils literal"><span class="pre">gray_to_bin</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.graycode</span> <span class="kn">import</span> <span class="n">bin_to_gray</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bin_to_gray</span><span class="p">(</span><span class="s">&#39;111&#39;</span><span class="p">)</span>
-<span class="go">&#39;100&#39;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="method">
-<dt id="sympy.combinatorics.graycode.get_subset_from_bitstring">
-<tt class="descclassname">graycode.</tt><tt class="descname">get_subset_from_bitstring</tt><big>(</big><em>super_set</em>, <em>bitstring</em><big>)</big><a class="headerlink" href="#sympy.combinatorics.graycode.get_subset_from_bitstring" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gets the subset defined by the bitstring.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.graycode.graycode_subsets" title="sympy.combinatorics.graycode.graycode_subsets"><tt class="xref py py-obj docutils literal"><span class="pre">graycode_subsets</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.graycode</span> <span class="kn">import</span> <span class="n">get_subset_from_bitstring</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">get_subset_from_bitstring</span><span class="p">([</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">],</span> <span class="s">&#39;0011&#39;</span><span class="p">)</span>
-<span class="go">[&#39;c&#39;, &#39;d&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">get_subset_from_bitstring</span><span class="p">([</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">],</span> <span class="s">&#39;1100&#39;</span><span class="p">)</span>
-<span class="go">[&#39;c&#39;, &#39;a&#39;]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="method">
-<dt id="sympy.combinatorics.graycode.graycode_subsets">
-<tt class="descclassname">graycode.</tt><tt class="descname">graycode_subsets</tt><big>(</big><em>gray_code_set</em><big>)</big><a class="headerlink" href="#sympy.combinatorics.graycode.graycode_subsets" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates the subsets as enumerated by a Gray code.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.graycode.get_subset_from_bitstring" title="sympy.combinatorics.graycode.get_subset_from_bitstring"><tt class="xref py py-obj docutils literal"><span class="pre">get_subset_from_bitstring</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.graycode</span> <span class="kn">import</span> <span class="n">graycode_subsets</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">graycode_subsets</span><span class="p">([</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">]))</span>
-<span class="go">[[], [&#39;c&#39;], [&#39;b&#39;, &#39;c&#39;], [&#39;b&#39;], [&#39;a&#39;, &#39;b&#39;], [&#39;a&#39;, &#39;b&#39;, &#39;c&#39;],     [&#39;a&#39;, &#39;c&#39;], [&#39;a&#39;]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">graycode_subsets</span><span class="p">([</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">]))</span>
-<span class="go">[[], [&#39;c&#39;], [&#39;c&#39;, &#39;c&#39;], [&#39;c&#39;], [&#39;b&#39;, &#39;c&#39;], [&#39;b&#39;, &#39;c&#39;, &#39;c&#39;],     [&#39;b&#39;, &#39;c&#39;], [&#39;b&#39;], [&#39;a&#39;, &#39;b&#39;], [&#39;a&#39;, &#39;b&#39;, &#39;c&#39;], [&#39;a&#39;, &#39;b&#39;, &#39;c&#39;, &#39;c&#39;],     [&#39;a&#39;, &#39;b&#39;, &#39;c&#39;], [&#39;a&#39;, &#39;c&#39;], [&#39;a&#39;, &#39;c&#39;, &#39;c&#39;], [&#39;a&#39;, &#39;c&#39;], [&#39;a&#39;]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
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-  <div class="section" id="module-sympy.combinatorics.group_constructs">
-<span id="group-constructors"></span><span id="combinatorics-group-constructs"></span><h1>Group constructors<a class="headerlink" href="#module-sympy.combinatorics.group_constructs" title="Permalink to this headline">¶</a></h1>
-<dl class="function">
-<dt id="sympy.combinatorics.group_constructs.DirectProduct">
-<tt class="descclassname">sympy.combinatorics.group_constructs.</tt><tt class="descname">DirectProduct</tt><big>(</big><em>*groups</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/group_constructs.html#DirectProduct"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.group_constructs.DirectProduct" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the direct product of several groups as a permutation group.</p>
-<p>This is implemented much like the __mul__ procedure for taking the direct
-product of two permutation groups, but the idea of shifting the
-generators is realized in the case of an arbitrary number of groups.
-A call to DirectProduct(G1, G2, ..., Gn) is generally expected to be faster
-than a call to G1*G2*...*Gn (and thus the need for this algorithm).</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">__mul__</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.group_constructs</span> <span class="kn">import</span> <span class="n">DirectProduct</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">CyclicGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">CyclicGroup</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">DirectProduct</span><span class="p">(</span><span class="n">C</span><span class="p">,</span><span class="n">C</span><span class="p">,</span><span class="n">C</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">order</span><span class="p">()</span>
-<span class="go">64</span>
-</pre></div>
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-  <div class="section" id="module-sympy.combinatorics.named_groups">
-<span id="named-groups"></span><span id="combinatorics-named-groups"></span><h1>Named Groups<a class="headerlink" href="#module-sympy.combinatorics.named_groups" title="Permalink to this headline">¶</a></h1>
-<dl class="function">
-<dt id="sympy.combinatorics.named_groups.SymmetricGroup">
-<tt class="descclassname">sympy.combinatorics.named_groups.</tt><tt class="descname">SymmetricGroup</tt><big>(</big><em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/named_groups.html#SymmetricGroup"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.named_groups.SymmetricGroup" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates the symmetric group on <tt class="docutils literal"><span class="pre">n</span></tt> elements as a permutation group.</p>
-<p>The generators taken are the <tt class="docutils literal"><span class="pre">n</span></tt>-cycle
-<tt class="docutils literal"><span class="pre">(0</span> <span class="pre">1</span> <span class="pre">2</span> <span class="pre">...</span> <span class="pre">n-1)</span></tt> and the transposition <tt class="docutils literal"><span class="pre">(0</span> <span class="pre">1)</span></tt> (in cycle notation).
-(See [1]). After the group is generated, some of its basic properties
-are set.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.named_groups.CyclicGroup" title="sympy.combinatorics.named_groups.CyclicGroup"><tt class="xref py py-obj docutils literal"><span class="pre">CyclicGroup</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.named_groups.DihedralGroup" title="sympy.combinatorics.named_groups.DihedralGroup"><tt class="xref py py-obj docutils literal"><span class="pre">DihedralGroup</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.named_groups.AlternatingGroup" title="sympy.combinatorics.named_groups.AlternatingGroup"><tt class="xref py py-obj docutils literal"><span class="pre">AlternatingGroup</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Symmetric_group#Generators_and_relations">http://en.wikipedia.org/wiki/Symmetric_group#Generators_and_relations</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">SymmetricGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">SymmetricGroup</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">is_group</span><span class="p">()</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">order</span><span class="p">()</span>
-<span class="go">24</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">generate_schreier_sims</span><span class="p">(</span><span class="n">af</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 1, 2, 0], [0, 2, 3, 1],</span>
-<span class="go">[1, 3, 0, 2], [2, 0, 1, 3], [3, 2, 0, 1], [0, 3, 1, 2], [1, 0, 2, 3],</span>
-<span class="go">[2, 1, 3, 0], [3, 0, 1, 2], [0, 1, 3, 2], [1, 2, 0, 3], [2, 3, 1, 0],</span>
-<span class="go">[3, 1, 0, 2], [0, 2, 1, 3], [1, 3, 2, 0], [2, 0, 3, 1], [3, 2, 1, 0],</span>
-<span class="go">[0, 3, 2, 1], [1, 0, 3, 2], [2, 1, 0, 3], [3, 0, 2, 1]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.named_groups.CyclicGroup">
-<tt class="descclassname">sympy.combinatorics.named_groups.</tt><tt class="descname">CyclicGroup</tt><big>(</big><em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/named_groups.html#CyclicGroup"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.named_groups.CyclicGroup" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates the cyclic group of order <tt class="docutils literal"><span class="pre">n</span></tt> as a permutation group.</p>
-<p>The generator taken is the <tt class="docutils literal"><span class="pre">n</span></tt>-cycle <tt class="docutils literal"><span class="pre">(0</span> <span class="pre">1</span> <span class="pre">2</span> <span class="pre">...</span> <span class="pre">n-1)</span></tt>
-(in cycle notation). After the group is generated, some of its basic
-properties are set.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.named_groups.SymmetricGroup" title="sympy.combinatorics.named_groups.SymmetricGroup"><tt class="xref py py-obj docutils literal"><span class="pre">SymmetricGroup</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.named_groups.DihedralGroup" title="sympy.combinatorics.named_groups.DihedralGroup"><tt class="xref py py-obj docutils literal"><span class="pre">DihedralGroup</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.named_groups.AlternatingGroup" title="sympy.combinatorics.named_groups.AlternatingGroup"><tt class="xref py py-obj docutils literal"><span class="pre">AlternatingGroup</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">CyclicGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">CyclicGroup</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">is_group</span><span class="p">()</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">order</span><span class="p">()</span>
-<span class="go">6</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">generate_schreier_sims</span><span class="p">(</span><span class="n">af</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1],</span>
-<span class="go">[3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.named_groups.DihedralGroup">
-<tt class="descclassname">sympy.combinatorics.named_groups.</tt><tt class="descname">DihedralGroup</tt><big>(</big><em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/named_groups.html#DihedralGroup"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.named_groups.DihedralGroup" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates the dihedral group <span class="math">\(D_n\)</span> as a permutation group.</p>
-<p>The dihedral group <span class="math">\(D_n\)</span> is the group of symmetries of the regular
-<tt class="docutils literal"><span class="pre">n</span></tt>-gon. The generators taken are the <tt class="docutils literal"><span class="pre">n</span></tt>-cycle <tt class="docutils literal"><span class="pre">a</span> <span class="pre">=</span> <span class="pre">(0</span> <span class="pre">1</span> <span class="pre">2</span> <span class="pre">...</span> <span class="pre">n-1)</span></tt>
-(a rotation of the <tt class="docutils literal"><span class="pre">n</span></tt>-gon) and <tt class="docutils literal"><span class="pre">b</span> <span class="pre">=</span> <span class="pre">(0</span> <span class="pre">n-1)(1</span> <span class="pre">n-2)...</span></tt>
-(a reflection of the <tt class="docutils literal"><span class="pre">n</span></tt>-gon) in cycle rotation. It is easy to see that
-these satisfy <tt class="docutils literal"><span class="pre">a**n</span> <span class="pre">=</span> <span class="pre">b**2</span> <span class="pre">=</span> <span class="pre">1</span></tt> and <tt class="docutils literal"><span class="pre">bab</span> <span class="pre">=</span> <span class="pre">~a</span></tt> so they indeed generate
-<span class="math">\(D_n\)</span> (See [1]). After the group is generated, some of its basic properties
-are set.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.named_groups.SymmetricGroup" title="sympy.combinatorics.named_groups.SymmetricGroup"><tt class="xref py py-obj docutils literal"><span class="pre">SymmetricGroup</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.named_groups.CyclicGroup" title="sympy.combinatorics.named_groups.CyclicGroup"><tt class="xref py py-obj docutils literal"><span class="pre">CyclicGroup</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.named_groups.AlternatingGroup" title="sympy.combinatorics.named_groups.AlternatingGroup"><tt class="xref py py-obj docutils literal"><span class="pre">AlternatingGroup</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Dihedral_group">http://en.wikipedia.org/wiki/Dihedral_group</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">DihedralGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">DihedralGroup</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">is_group</span><span class="p">()</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">generate_dimino</span><span class="p">())</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">perm</span><span class="o">.</span><span class="n">cyclic_form</span> <span class="k">for</span> <span class="n">perm</span> <span class="ow">in</span> <span class="n">a</span><span class="p">]</span>
-<span class="go">[[], [[0, 1, 2, 3, 4]], [[0, 2, 4, 1, 3]],</span>
-<span class="go">[[0, 3, 1, 4, 2]], [[0, 4, 3, 2, 1]], [[0, 4], [1, 3]],</span>
-<span class="go">[[1, 4], [2, 3]], [[0, 1], [2, 4]], [[0, 2], [3, 4]],</span>
-<span class="go">[[0, 3], [1, 2]]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.named_groups.AlternatingGroup">
-<tt class="descclassname">sympy.combinatorics.named_groups.</tt><tt class="descname">AlternatingGroup</tt><big>(</big><em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/named_groups.html#AlternatingGroup"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.named_groups.AlternatingGroup" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates the alternating group on <tt class="docutils literal"><span class="pre">n</span></tt> elements as a permutation group.</p>
-<p>For <tt class="docutils literal"><span class="pre">n</span> <span class="pre">&gt;</span> <span class="pre">2</span></tt>, the generators taken are <tt class="docutils literal"><span class="pre">(0</span> <span class="pre">1</span> <span class="pre">2),</span> <span class="pre">(0</span> <span class="pre">1</span> <span class="pre">2</span> <span class="pre">...</span> <span class="pre">n-1)</span></tt> for
-<tt class="docutils literal"><span class="pre">n</span></tt> odd
-and <tt class="docutils literal"><span class="pre">(0</span> <span class="pre">1</span> <span class="pre">2),</span> <span class="pre">(1</span> <span class="pre">2</span> <span class="pre">...</span> <span class="pre">n-1)</span></tt> for <tt class="docutils literal"><span class="pre">n</span></tt> even (See [1], p.31, ex.6.9.).
-After the group is generated, some of its basic properties are set.
-The cases <tt class="docutils literal"><span class="pre">n</span> <span class="pre">=</span> <span class="pre">1,</span> <span class="pre">2</span></tt> are handled separately.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.named_groups.SymmetricGroup" title="sympy.combinatorics.named_groups.SymmetricGroup"><tt class="xref py py-obj docutils literal"><span class="pre">SymmetricGroup</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.named_groups.CyclicGroup" title="sympy.combinatorics.named_groups.CyclicGroup"><tt class="xref py py-obj docutils literal"><span class="pre">CyclicGroup</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.named_groups.DihedralGroup" title="sympy.combinatorics.named_groups.DihedralGroup"><tt class="xref py py-obj docutils literal"><span class="pre">DihedralGroup</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<p>[1] Armstrong, M. &#8220;Groups and Symmetry&#8221;</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">AlternatingGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">AlternatingGroup</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">is_group</span><span class="p">()</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">generate_dimino</span><span class="p">())</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="go">12</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">all</span><span class="p">(</span><span class="n">perm</span><span class="o">.</span><span class="n">is_even</span> <span class="k">for</span> <span class="n">perm</span> <span class="ow">in</span> <span class="n">a</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.named_groups.AbelianGroup">
-<tt class="descclassname">sympy.combinatorics.named_groups.</tt><tt class="descname">AbelianGroup</tt><big>(</big><em>*cyclic_orders</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/named_groups.html#AbelianGroup"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.named_groups.AbelianGroup" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the direct product of cyclic groups with the given orders.</p>
-<p>According to the structure theorem for finite abelian groups ([1]),
-every finite abelian group can be written as the direct product of finitely
-many cyclic groups.
-[1] <a class="reference external" href="http://groupprops.subwiki.org/wiki/Structure_theorem_for_finitely_generated_abelian_groups">http://groupprops.subwiki.org/wiki/Structure_theorem_for_finitely_generated_abelian_groups</a></p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">DirectProduct</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">AbelianGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">AbelianGroup</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">PermutationGroup([</span>
-<span class="go">        Permutation(6)(0, 1, 2),</span>
-<span class="go">        Permutation(3, 4, 5, 6)])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">.</span><span class="n">is_group</span><span class="p">()</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
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-  <div class="section" id="module-sympy.combinatorics.partitions">
-<span id="partitions"></span><span id="combinatorics-partitions"></span><h1>Partitions<a class="headerlink" href="#module-sympy.combinatorics.partitions" title="Permalink to this headline">¶</a></h1>
-<dl class="class">
-<dt id="sympy.combinatorics.partitions.Partition">
-<em class="property">class </em><tt class="descclassname">sympy.combinatorics.partitions.</tt><tt class="descname">Partition</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/partitions.html#Partition"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.partitions.Partition" title="Permalink to this definition">¶</a></dt>
-<dd><p>This class represents an abstract partition.</p>
-<p>A partition is a set of disjoint sets whose union equals a given set.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="../utilities/iterables.html#sympy.utilities.iterables.partitions" title="sympy.utilities.iterables.partitions"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.utilities.iterables.partitions</span></tt></a>, <a class="reference internal" href="../utilities/iterables.html#sympy.utilities.iterables.multiset_partitions" title="sympy.utilities.iterables.multiset_partitions"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.utilities.iterables.multiset_partitions</span></tt></a></p>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="60%" />
-<col width="40%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>is_EmptySet</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>is_Intersection</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>is_UniversalSet</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="attribute">
-<dt id="sympy.combinatorics.partitions.Partition.RGS">
-<tt class="descname">RGS</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/partitions.html#Partition.RGS"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.partitions.Partition.RGS" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the &#8220;restricted growth string&#8221; of the partition.</p>
-<p>The RGS is returned as a list of indices, L, where L[i] indicates
-the block in which element i appears. For example, in a partition
-of 3 elements (a, b, c) into 2 blocks ([c], [a, b]) the RGS is
-[1, 1, 0]: &#8220;a&#8221; is in block 1, &#8220;b&#8221; is in block 1 and &#8220;c&#8221; is in block 0.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.partitions</span> <span class="kn">import</span> <span class="n">Partition</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Partition</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">members</span>
-<span class="go">(1, 2, 3, 4, 5)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">RGS</span>
-<span class="go">(0, 0, 1, 2, 2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">+</span> <span class="mi">1</span>
-<span class="go">{{1, 2}, {3}, {4}, {5}}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">.</span><span class="n">RGS</span>
-<span class="go">(0, 0, 1, 2, 3)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.combinatorics.partitions.Partition.from_rgs">
-<em class="property">classmethod </em><tt class="descname">from_rgs</tt><big>(</big><em>rgs</em>, <em>elements</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/partitions.html#Partition.from_rgs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.partitions.Partition.from_rgs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Creates a set partition from a restricted growth string.</p>
-<p>The indices given in rgs are assumed to be the index
-of the element as given in elements <em>as provided</em> (the
-elements are not sorted by this routine). Block numbering
-starts from 0. If any block was not referenced in <tt class="docutils literal"><span class="pre">rgs</span></tt>
-an error will be raised.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.partitions</span> <span class="kn">import</span> <span class="n">Partition</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Partition</span><span class="o">.</span><span class="n">from_rgs</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="nb">list</span><span class="p">(</span><span class="s">&#39;abcde&#39;</span><span class="p">))</span>
-<span class="go">{{c}, {a, d}, {b, e}}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Partition</span><span class="o">.</span><span class="n">from_rgs</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="nb">list</span><span class="p">(</span><span class="s">&#39;cbead&#39;</span><span class="p">))</span>
-<span class="go">{{e}, {a, c}, {b, d}}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Partition</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Partition</span><span class="o">.</span><span class="n">from_rgs</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">RGS</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">members</span><span class="p">)</span>
-<span class="go">{{1, 4}, {2}, {3, 5}}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.partitions.Partition.partition">
-<tt class="descname">partition</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/partitions.html#Partition.partition"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.partitions.Partition.partition" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return partition as a sorted list of lists.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.partitions</span> <span class="kn">import</span> <span class="n">Partition</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Partition</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span><span class="o">.</span><span class="n">partition</span>
-<span class="go">[[1], [2, 3]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.partitions.Partition.rank">
-<tt class="descname">rank</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/partitions.html#Partition.rank"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.partitions.Partition.rank" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gets the rank of a partition.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.partitions</span> <span class="kn">import</span> <span class="n">Partition</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Partition</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">rank</span>
-<span class="go">13</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.partitions.Partition.sort_key">
-<tt class="descname">sort_key</tt><big>(</big><em>self</em>, <em>order=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/partitions.html#Partition.sort_key"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.partitions.Partition.sort_key" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a canonical key that can be used for sorting.</p>
-<p>Ordering is based on the size and sorted elements of the partition
-and ties are broken with the rank.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">default_sort_key</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.partitions</span> <span class="kn">import</span> <span class="n">Partition</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Partition</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Partition</span><span class="p">([[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">Partition</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">Partition</span><span class="p">([</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">))])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l</span> <span class="o">=</span> <span class="p">[</span><span class="n">d</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">c</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">);</span> <span class="n">l</span>
-<span class="go">[{{1, 2}}, {{1}, {2}}, {{1, x}}, {{3, 4}}, {{0, 1, 2, 3}}]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.combinatorics.partitions.IntegerPartition">
-<em class="property">class </em><tt class="descclassname">sympy.combinatorics.partitions.</tt><tt class="descname">IntegerPartition</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/partitions.html#IntegerPartition"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.partitions.IntegerPartition" title="Permalink to this definition">¶</a></dt>
-<dd><p>This class represents an integer partition.</p>
-<p>In number theory and combinatorics, a partition of a positive integer,
-<tt class="docutils literal"><span class="pre">n</span></tt>, also called an integer partition, is a way of writing <tt class="docutils literal"><span class="pre">n</span></tt> as a
-list of positive integers that sum to n. Two partitions that differ only
-in the order of summands are considered to be the same partition; if order
-matters then the partitions are referred to as compositions. For example,
-4 has five partitions: [4], [3, 1], [2, 2], [2, 1, 1], and [1, 1, 1, 1];
-the compositions [1, 2, 1] and [1, 1, 2] are the same as partition
-[2, 1, 1].</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p><a class="reference internal" href="../utilities/iterables.html#sympy.utilities.iterables.partitions" title="sympy.utilities.iterables.partitions"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.utilities.iterables.partitions</span></tt></a>, <a class="reference internal" href="../utilities/iterables.html#sympy.utilities.iterables.multiset_partitions" title="sympy.utilities.iterables.multiset_partitions"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.utilities.iterables.multiset_partitions</span></tt></a></p>
-<dl class="last docutils">
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">Reference</span></tt></dt>
-<dd><a class="reference external" href="http://en.wikipedia.org/wiki/Partition_%28number_theory%29">http://en.wikipedia.org/wiki/Partition_%28number_theory%29</a></dd>
-</dl>
-</div>
-<dl class="function">
-<dt id="sympy.combinatorics.partitions.IntegerPartition.as_dict">
-<tt class="descname">as_dict</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/partitions.html#IntegerPartition.as_dict"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.partitions.IntegerPartition.as_dict" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the partition as a dictionary whose keys are the
-partition integers and the values are the multiplicity of that
-integer.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.partitions</span> <span class="kn">import</span> <span class="n">IntegerPartition</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">IntegerPartition</span><span class="p">([</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="mi">3</span> <span class="o">+</span> <span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="mi">3</span><span class="p">]</span><span class="o">*</span><span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">as_dict</span><span class="p">()</span>
-<span class="go">{1: 3, 2: 1, 3: 4}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.partitions.IntegerPartition.as_ferrers">
-<tt class="descname">as_ferrers</tt><big>(</big><em>self</em>, <em>char='#'</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/partitions.html#IntegerPartition.as_ferrers"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.partitions.IntegerPartition.as_ferrers" title="Permalink to this definition">¶</a></dt>
-<dd><p>Prints the ferrer diagram of a partition.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.partitions</span> <span class="kn">import</span> <span class="n">IntegerPartition</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">IntegerPartition</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span><span class="o">.</span><span class="n">as_ferrers</span><span class="p">())</span>
-<span class="go">#####</span>
-<span class="go">#</span>
-<span class="go">#</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.partitions.IntegerPartition.conjugate">
-<tt class="descname">conjugate</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/partitions.html#IntegerPartition.conjugate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.partitions.IntegerPartition.conjugate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the conjugate partition of itself.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.partitions</span> <span class="kn">import</span> <span class="n">IntegerPartition</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">IntegerPartition</span><span class="p">([</span><span class="mi">6</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">conjugate</span>
-<span class="go">[5, 4, 3, 1, 1, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.partitions.IntegerPartition.next_lex">
-<tt class="descname">next_lex</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/partitions.html#IntegerPartition.next_lex"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.partitions.IntegerPartition.next_lex" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the next partition of the integer, n, in lexical order,
-wrapping around to [n] if the partition is [1, ..., 1].</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.partitions</span> <span class="kn">import</span> <span class="n">IntegerPartition</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">IntegerPartition</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">next_lex</span><span class="p">())</span>
-<span class="go">[4]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">partition</span> <span class="o">&lt;</span> <span class="n">p</span><span class="o">.</span><span class="n">next_lex</span><span class="p">()</span><span class="o">.</span><span class="n">partition</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.partitions.IntegerPartition.prev_lex">
-<tt class="descname">prev_lex</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/partitions.html#IntegerPartition.prev_lex"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.partitions.IntegerPartition.prev_lex" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the previous partition of the integer, n, in lexical order,
-wrapping around to [1, ..., 1] if the partition is [n].</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.partitions</span> <span class="kn">import</span> <span class="n">IntegerPartition</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">IntegerPartition</span><span class="p">([</span><span class="mi">4</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">prev_lex</span><span class="p">())</span>
-<span class="go">[3, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">partition</span> <span class="o">&gt;</span> <span class="n">p</span><span class="o">.</span><span class="n">prev_lex</span><span class="p">()</span><span class="o">.</span><span class="n">partition</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.partitions.random_integer_partition">
-<tt class="descclassname">sympy.combinatorics.partitions.</tt><tt class="descname">random_integer_partition</tt><big>(</big><em>n</em>, <em>seed=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/partitions.html#random_integer_partition"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.partitions.random_integer_partition" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates a random integer partition summing to <tt class="docutils literal"><span class="pre">n</span></tt> as a list
-of reverse-sorted integers.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.partitions</span> <span class="kn">import</span> <span class="n">random_integer_partition</span>
-</pre></div>
-</div>
-<p>For the following, a seed is given so a known value can be shown; in
-practice, the seed would not be given.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">random_integer_partition</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">85</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">[85, 12, 2, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">random_integer_partition</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">[5, 3, 1, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">random_integer_partition</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">[1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.partitions.RGS_generalized">
-<tt class="descclassname">sympy.combinatorics.partitions.</tt><tt class="descname">RGS_generalized</tt><big>(</big><em>m</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/partitions.html#RGS_generalized"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.partitions.RGS_generalized" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the m + 1 generalized unrestricted growth strings
-and returns them as rows in matrix.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.partitions</span> <span class="kn">import</span> <span class="n">RGS_generalized</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">RGS_generalized</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span>
-<span class="go">[  1,   1,   1,  1,  1, 1, 1]</span>
-<span class="go">[  1,   2,   3,  4,  5, 6, 0]</span>
-<span class="go">[  2,   5,  10, 17, 26, 0, 0]</span>
-<span class="go">[  5,  15,  37, 77,  0, 0, 0]</span>
-<span class="go">[ 15,  52, 151,  0,  0, 0, 0]</span>
-<span class="go">[ 52, 203,   0,  0,  0, 0, 0]</span>
-<span class="go">[203,   0,   0,  0,  0, 0, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.partitions.RGS_enum">
-<tt class="descclassname">sympy.combinatorics.partitions.</tt><tt class="descname">RGS_enum</tt><big>(</big><em>m</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/partitions.html#RGS_enum"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.partitions.RGS_enum" title="Permalink to this definition">¶</a></dt>
-<dd><p>RGS_enum computes the total number of restricted growth strings
-possible for a superset of size m.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.partitions</span> <span class="kn">import</span> <span class="n">RGS_enum</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.partitions</span> <span class="kn">import</span> <span class="n">Partition</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">RGS_enum</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">RGS_enum</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">52</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">RGS_enum</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span>
-<span class="go">203</span>
-</pre></div>
-</div>
-<p>We can check that the enumeration is correct by actually generating
-the partitions. Here, the 15 partitions of 4 items are generated:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Partition</span><span class="p">([</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">))])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">20</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">s</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="gp">... </span>    <span class="n">a</span> <span class="o">+=</span> <span class="mi">1</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">)</span> <span class="o">==</span> <span class="mi">15</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.partitions.RGS_unrank">
-<tt class="descclassname">sympy.combinatorics.partitions.</tt><tt class="descname">RGS_unrank</tt><big>(</big><em>rank</em>, <em>m</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/partitions.html#RGS_unrank"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.partitions.RGS_unrank" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the unranked restricted growth string for a given
-superset size.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.partitions</span> <span class="kn">import</span> <span class="n">RGS_unrank</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">RGS_unrank</span><span class="p">(</span><span class="mi">14</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">[0, 1, 2, 3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">RGS_unrank</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">[0, 0, 0, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.partitions.RGS_rank">
-<tt class="descclassname">sympy.combinatorics.partitions.</tt><tt class="descname">RGS_rank</tt><big>(</big><em>rgs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/partitions.html#RGS_rank"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.partitions.RGS_rank" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the rank of a restricted growth string.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.partitions</span> <span class="kn">import</span> <span class="n">RGS_rank</span><span class="p">,</span> <span class="n">RGS_unrank</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">RGS_rank</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="go">42</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">RGS_rank</span><span class="p">(</span><span class="n">RGS_unrank</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">7</span><span class="p">))</span>
-<span class="go">4</span>
-</pre></div>
-</div>
-</dd></dl>
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-  <div class="section" id="module-sympy.combinatorics.perm_groups">
-<span id="permutation-groups"></span><span id="combinatorics-perm-groups"></span><h1>Permutation Groups<a class="headerlink" href="#module-sympy.combinatorics.perm_groups" title="Permalink to this headline">¶</a></h1>
-<dl class="class">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup">
-<em class="property">class </em><tt class="descclassname">sympy.combinatorics.perm_groups.</tt><tt class="descname">PermutationGroup</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup" title="Permalink to this definition">¶</a></dt>
-<dd><p>The class defining a Permutation group.</p>
-<p>PermutationGroup([p1, p2, ..., pn]) returns the permutation group
-generated by the list of permutations. This group can be supplied
-to Polyhedron if one desires to decorate the elements to which the
-indices of the permutation refer.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="../combinatorics/polyhedron.html#sympy.combinatorics.polyhedron.Polyhedron" title="sympy.combinatorics.polyhedron.Polyhedron"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.combinatorics.polyhedron.Polyhedron</span></tt></a>, <a class="reference internal" href="../combinatorics/permutations.html#sympy.combinatorics.permutations.Permutation" title="sympy.combinatorics.permutations.Permutation"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.combinatorics.permutations.Permutation</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<p>[1] Holt, D., Eick, B., O&#8217;Brien, E.
-&#8220;Handbook of Computational Group Theory&#8221;</p>
-<p>[2] Seress, A.
-&#8220;Permutation Group Algorithms&#8221;</p>
-<p>[3] <a class="reference external" href="http://en.wikipedia.org/wiki/Schreier_vector">http://en.wikipedia.org/wiki/Schreier_vector</a></p>
-<p>[4] <a class="reference external" href="http://en.wikipedia.org/wiki/Nielsen_transformation">http://en.wikipedia.org/wiki/Nielsen_transformation</a>
-#Product_replacement_algorithm</p>
-<p>[5] Frank Celler, Charles R.Leedham-Green, Scott H.Murray,
-Alice C.Niemeyer, and E.A.O&#8217;Brien. &#8220;Generating Random
-Elements of a Finite Group&#8221;</p>
-<p>[6] <a class="reference external" href="http://en.wikipedia.org/wiki/Block_%28permutation_group_theory%29">http://en.wikipedia.org/wiki/Block_%28permutation_group_theory%29</a></p>
-<p>[7] <a class="reference external" href="http://www.algorithmist.com/index.php/Union_Find">http://www.algorithmist.com/index.php/Union_Find</a></p>
-<p>[8] <a class="reference external" href="http://en.wikipedia.org/wiki/Multiply_transitive_group#Multiply_transitive_groups">http://en.wikipedia.org/wiki/Multiply_transitive_group#Multiply_transitive_groups</a></p>
-<p>[9] <a class="reference external" href="http://en.wikipedia.org/wiki/Center_%28group_theory%29">http://en.wikipedia.org/wiki/Center_%28group_theory%29</a></p>
-<p>[10] <a class="reference external" href="http://en.wikipedia.org/wiki/Centralizer_and_normalizer">http://en.wikipedia.org/wiki/Centralizer_and_normalizer</a></p>
-<p>[11] <a class="reference external" href="http://groupprops.subwiki.org/wiki/Derived_subgroup">http://groupprops.subwiki.org/wiki/Derived_subgroup</a></p>
-<p>[12] <a class="reference external" href="http://en.wikipedia.org/wiki/Nilpotent_group">http://en.wikipedia.org/wiki/Nilpotent_group</a></p>
-<p>[13] <a class="reference external" href="http://www.math.colostate.edu/~hulpke/CGT/cgtnotes.pdf">http://www.math.colostate.edu/~hulpke/CGT/cgtnotes.pdf</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Cycle</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.polyhedron</span> <span class="kn">import</span> <span class="n">Polyhedron</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-</pre></div>
-</div>
-<p>The permutations corresponding to motion of the front, right and
-bottom face of a 2x2 Rubik&#8217;s cube are defined:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">19</span><span class="p">,</span> <span class="mi">21</span><span class="p">,</span> <span class="mi">8</span><span class="p">)(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">17</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">10</span><span class="p">)(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">21</span><span class="p">,</span> <span class="mi">14</span><span class="p">)(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">23</span><span class="p">,</span> <span class="mi">12</span><span class="p">)(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">9</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">18</span><span class="p">,</span> <span class="mi">14</span><span class="p">,</span> <span class="mi">10</span><span class="p">)(</span><span class="mi">7</span><span class="p">,</span> <span class="mi">19</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">11</span><span class="p">)(</span><span class="mi">20</span><span class="p">,</span> <span class="mi">22</span><span class="p">,</span> <span class="mi">23</span><span class="p">,</span> <span class="mi">21</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>These are passed as permutations to PermutationGroup:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="n">D</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">order</span><span class="p">()</span>
-<span class="go">3674160</span>
-</pre></div>
-</div>
-<p>The group can be supplied to a Polyhedron in order to track the
-objects being moved. An example involving the 2x2 Rubik&#8217;s cube is
-given there, but here is a simple demonstration:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">Polyhedron</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="s">&#39;ABC&#39;</span><span class="p">),</span> <span class="n">pgroup</span><span class="o">=</span><span class="n">G</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">corners</span>
-<span class="go">(A, B, C)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">rotate</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="c"># apply permutation 0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">corners</span>
-<span class="go">(A, C, B)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">reset</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">corners</span>
-<span class="go">(A, B, C)</span>
-</pre></div>
-</div>
-<p>Or one can make a permutation as a product of selected permutations
-and apply them to an iterable directly:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">P10</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">make_perm</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P10</span><span class="p">(</span><span class="s">&#39;ABC&#39;</span><span class="p">)</span>
-<span class="go">[&#39;C&#39;, &#39;A&#39;, &#39;B&#39;]</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.base">
-<tt class="descname">base</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.base"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.base" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a base from the Schreier-Sims algorithm.</p>
-<p>For a permutation group <tt class="docutils literal"><span class="pre">G</span></tt>, a base is a sequence of points
-<tt class="docutils literal"><span class="pre">B</span> <span class="pre">=</span> <span class="pre">(b_1,</span> <span class="pre">b_2,</span> <span class="pre">...,</span> <span class="pre">b_k)</span></tt> such that no element of <tt class="docutils literal"><span class="pre">G</span></tt> apart
-from the identity fixes all the points in <tt class="docutils literal"><span class="pre">B</span></tt>. The concepts of
-a base and strong generating set and their applications are
-discussed in depth in [1], pp. 87-89 and [2], pp. 55-57.</p>
-<p>An alternative way to think of <tt class="docutils literal"><span class="pre">B</span></tt> is that it gives the
-indices of the stabilizer cosets that contain more than the
-identity permutation.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.strong_gens" title="sympy.combinatorics.perm_groups.PermutationGroup.strong_gens"><tt class="xref py py-obj docutils literal"><span class="pre">strong_gens</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.basic_transversals" title="sympy.combinatorics.perm_groups.PermutationGroup.basic_transversals"><tt class="xref py py-obj docutils literal"><span class="pre">basic_transversals</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.basic_orbits" title="sympy.combinatorics.perm_groups.PermutationGroup.basic_orbits"><tt class="xref py py-obj docutils literal"><span class="pre">basic_orbits</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.basic_stabilizers" title="sympy.combinatorics.perm_groups.PermutationGroup.basic_stabilizers"><tt class="xref py py-obj docutils literal"><span class="pre">basic_stabilizers</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span><span class="p">,</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">Permutation</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">base</span>
-<span class="go">[0, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.baseswap">
-<tt class="descname">baseswap</tt><big>(</big><em>self</em>, <em>base</em>, <em>strong_gens</em>, <em>pos</em>, <em>randomized=False</em>, <em>transversals=None</em>, <em>basic_orbits=None</em>, <em>strong_gens_distr=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.baseswap"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.baseswap" title="Permalink to this definition">¶</a></dt>
-<dd><p>Swap two consecutive base points in base and strong generating set.</p>
-<p>If a base for a group <tt class="docutils literal"><span class="pre">G</span></tt> is given by <tt class="docutils literal"><span class="pre">(b_1,</span> <span class="pre">b_2,</span> <span class="pre">...,</span> <span class="pre">b_k)</span></tt>, this
-function returns a base <tt class="docutils literal"><span class="pre">(b_1,</span> <span class="pre">b_2,</span> <span class="pre">...,</span> <span class="pre">b_{i+1},</span> <span class="pre">b_i,</span> <span class="pre">...,</span> <span class="pre">b_k)</span></tt>,
-where <tt class="docutils literal"><span class="pre">i</span></tt> is given by <tt class="docutils literal"><span class="pre">pos</span></tt>, and a strong generating set relative
-to that base. The original base and strong generating set are not
-modified.</p>
-<p>The randomized version (default) is of Las Vegas type.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>base, strong_gens</strong> :</p>
-<blockquote>
-<div><p>The base and strong generating set.</p>
-</div></blockquote>
-<p><strong>pos</strong> :</p>
-<blockquote>
-<div><p>The position at which swapping is performed.</p>
-</div></blockquote>
-<p><strong>randomized</strong> :</p>
-<blockquote>
-<div><p>A switch between randomized and deterministic version.</p>
-</div></blockquote>
-<p><strong>transversals</strong> :</p>
-<blockquote>
-<div><p>The transversals for the basic orbits, if known.</p>
-</div></blockquote>
-<p><strong>basic_orbits</strong> :</p>
-<blockquote>
-<div><p>The basic orbits, if known.</p>
-</div></blockquote>
-<p><strong>strong_gens_distr</strong> :</p>
-<blockquote>
-<div><p>The strong generators distributed by basic stabilizers, if known.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>(base, strong_gens)</strong> :</p>
-<blockquote class="last">
-<div><p><tt class="docutils literal"><span class="pre">base</span></tt> is the new base, and <tt class="docutils literal"><span class="pre">strong_gens</span></tt> is a generating set
-relative to it.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims" title="sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims"><tt class="xref py py-obj docutils literal"><span class="pre">schreier_sims</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>The deterministic version of the algorithm is discussed in
-[1], pp. 102-103; the randomized version is discussed in [1], p.103, and
-[2], p.98. It is of Las Vegas type.
-Notice that [1] contains a mistake in the pseudocode and
-discussion of BASESWAP: on line 3 of the pseudocode,
-<tt class="docutils literal"><span class="pre">|\beta_{i+1}^{\left\langle</span> <span class="pre">T\right\rangle}|</span></tt> should be replaced by
-<tt class="docutils literal"><span class="pre">|\beta_{i}^{\left\langle</span> <span class="pre">T\right\rangle}|</span></tt>, and the same for the
-discussion of the algorithm.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">SymmetricGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.testutil</span> <span class="kn">import</span> <span class="n">_verify_bsgs</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">SymmetricGroup</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">.</span><span class="n">schreier_sims</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">.</span><span class="n">base</span>
-<span class="go">[0, 1, 2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">base</span><span class="p">,</span> <span class="n">gens</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">baseswap</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">strong_gens</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">randomized</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">base</span><span class="p">,</span> <span class="n">gens</span>
-<span class="go">([0, 2, 1],</span>
-<span class="go">[Permutation(0, 1, 2, 3), Permutation(3)(0, 1), Permutation(1, 3, 2),</span>
-<span class="go"> Permutation(2, 3), Permutation(1, 3)])</span>
-</pre></div>
-</div>
-<p>check that base, gens is a BSGS</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">S1</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="n">gens</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_verify_bsgs</span><span class="p">(</span><span class="n">S1</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="n">gens</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.basic_orbits">
-<tt class="descname">basic_orbits</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.basic_orbits"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.basic_orbits" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the basic orbits relative to a base and strong generating set.</p>
-<p>If <tt class="docutils literal"><span class="pre">(b_1,</span> <span class="pre">b_2,</span> <span class="pre">...,</span> <span class="pre">b_k)</span></tt> is a base for a group <tt class="docutils literal"><span class="pre">G</span></tt>, and
-<tt class="docutils literal"><span class="pre">G^{(i)}</span> <span class="pre">=</span> <span class="pre">G_{b_1,</span> <span class="pre">b_2,</span> <span class="pre">...,</span> <span class="pre">b_{i-1}}</span></tt> is the <tt class="docutils literal"><span class="pre">i</span></tt>-th basic stabilizer
-(so that <tt class="docutils literal"><span class="pre">G^{(1)}</span> <span class="pre">=</span> <span class="pre">G</span></tt>), the <tt class="docutils literal"><span class="pre">i</span></tt>-th basic orbit relative to this base
-is the orbit of <tt class="docutils literal"><span class="pre">b_i</span></tt> under <tt class="docutils literal"><span class="pre">G^{(i)}</span></tt>. See [1], pp. 87-89 for more
-information.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.base" title="sympy.combinatorics.perm_groups.PermutationGroup.base"><tt class="xref py py-obj docutils literal"><span class="pre">base</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.strong_gens" title="sympy.combinatorics.perm_groups.PermutationGroup.strong_gens"><tt class="xref py py-obj docutils literal"><span class="pre">strong_gens</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.basic_transversals" title="sympy.combinatorics.perm_groups.PermutationGroup.basic_transversals"><tt class="xref py py-obj docutils literal"><span class="pre">basic_transversals</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.basic_stabilizers" title="sympy.combinatorics.perm_groups.PermutationGroup.basic_stabilizers"><tt class="xref py py-obj docutils literal"><span class="pre">basic_stabilizers</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">SymmetricGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">SymmetricGroup</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">.</span><span class="n">basic_orbits</span>
-<span class="go">[[0, 1, 2, 3], [1, 2, 3], [2, 3]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.basic_stabilizers">
-<tt class="descname">basic_stabilizers</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.basic_stabilizers"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.basic_stabilizers" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a chain of stabilizers relative to a base and strong generating
-set.</p>
-<p>The <tt class="docutils literal"><span class="pre">i</span></tt>-th basic stabilizer <tt class="docutils literal"><span class="pre">G^{(i)}</span></tt> relative to a base
-<tt class="docutils literal"><span class="pre">(b_1,</span> <span class="pre">b_2,</span> <span class="pre">...,</span> <span class="pre">b_k)</span></tt> is <tt class="docutils literal"><span class="pre">G_{b_1,</span> <span class="pre">b_2,</span> <span class="pre">...,</span> <span class="pre">b_{i-1}}</span></tt>. For more
-information, see [1], pp. 87-89.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.base" title="sympy.combinatorics.perm_groups.PermutationGroup.base"><tt class="xref py py-obj docutils literal"><span class="pre">base</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.strong_gens" title="sympy.combinatorics.perm_groups.PermutationGroup.strong_gens"><tt class="xref py py-obj docutils literal"><span class="pre">strong_gens</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.basic_orbits" title="sympy.combinatorics.perm_groups.PermutationGroup.basic_orbits"><tt class="xref py py-obj docutils literal"><span class="pre">basic_orbits</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.basic_transversals" title="sympy.combinatorics.perm_groups.PermutationGroup.basic_transversals"><tt class="xref py py-obj docutils literal"><span class="pre">basic_transversals</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">AlternatingGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">AlternatingGroup</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">schreier_sims</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">base</span>
-<span class="go">[0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">A</span><span class="o">.</span><span class="n">basic_stabilizers</span><span class="p">:</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="go">PermutationGroup([</span>
-<span class="go">    Permutation(3)(0, 1, 2),</span>
-<span class="go">    Permutation(1, 2, 3)])</span>
-<span class="go">PermutationGroup([</span>
-<span class="go">    Permutation(1, 2, 3)])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.basic_transversals">
-<tt class="descname">basic_transversals</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.basic_transversals"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.basic_transversals" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return basic transversals relative to a base and strong generating set.</p>
-<p>The basic transversals are transversals of the basic orbits. They
-are provided as a list of dictionaries, each dictionary having
-keys - the elements of one of the basic orbits, and values - the
-corresponding transversal elements. See [1], pp. 87-89 for more
-information.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.strong_gens" title="sympy.combinatorics.perm_groups.PermutationGroup.strong_gens"><tt class="xref py py-obj docutils literal"><span class="pre">strong_gens</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.base" title="sympy.combinatorics.perm_groups.PermutationGroup.base"><tt class="xref py py-obj docutils literal"><span class="pre">base</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.basic_orbits" title="sympy.combinatorics.perm_groups.PermutationGroup.basic_orbits"><tt class="xref py py-obj docutils literal"><span class="pre">basic_orbits</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.basic_stabilizers" title="sympy.combinatorics.perm_groups.PermutationGroup.basic_stabilizers"><tt class="xref py py-obj docutils literal"><span class="pre">basic_stabilizers</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">AlternatingGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">AlternatingGroup</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">basic_transversals</span>
-<span class="go">[{0: Permutation(3),</span>
-<span class="go">  1: Permutation(3)(0, 1, 2),</span>
-<span class="go">  2: Permutation(3)(0, 2, 1),</span>
-<span class="go">  3: Permutation(0, 3, 1)},</span>
-<span class="go"> {1: Permutation(3),</span>
-<span class="go">  2: Permutation(1, 2, 3),</span>
-<span class="go">  3: Permutation(1, 3, 2)}]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.center">
-<tt class="descname">center</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.center"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.center" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the center of a permutation group.</p>
-<p>The center for a group <tt class="docutils literal"><span class="pre">G</span></tt> is defined as
-<tt class="docutils literal"><span class="pre">Z(G)</span> <span class="pre">=</span> <span class="pre">\{z\in</span> <span class="pre">G</span> <span class="pre">|</span> <span class="pre">\forall</span> <span class="pre">g\in</span> <span class="pre">G,</span> <span class="pre">zg</span> <span class="pre">=</span> <span class="pre">gz</span> <span class="pre">\}</span></tt>,
-the set of elements of <tt class="docutils literal"><span class="pre">G</span></tt> that commute with all elements of <tt class="docutils literal"><span class="pre">G</span></tt>.
-It is equal to the centralizer of <tt class="docutils literal"><span class="pre">G</span></tt> inside <tt class="docutils literal"><span class="pre">G</span></tt>, and is naturally a
-subgroup of <tt class="docutils literal"><span class="pre">G</span></tt> ([9]).</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.centralizer" title="sympy.combinatorics.perm_groups.PermutationGroup.centralizer"><tt class="xref py py-obj docutils literal"><span class="pre">centralizer</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>This is a naive implementation that is a straightforward application
-of <tt class="docutils literal"><span class="pre">.centralizer()</span></tt></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">DihedralGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">DihedralGroup</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">D</span><span class="o">.</span><span class="n">center</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">order</span><span class="p">()</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.centralizer">
-<tt class="descname">centralizer</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.centralizer"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.centralizer" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the centralizer of a group/set/element.</p>
-<p>The centralizer of a set of permutations <tt class="docutils literal"><span class="pre">S</span></tt> inside
-a group <tt class="docutils literal"><span class="pre">G</span></tt> is the set of elements of <tt class="docutils literal"><span class="pre">G</span></tt> that commute with all
-elements of <tt class="docutils literal"><span class="pre">S</span></tt>:</p>
-<div class="highlight-python"><pre>``C_G(S) = \{ g \in G | gs = sg \forall s \in S\}`` ([10])</pre>
-</div>
-<p>Usually, <tt class="docutils literal"><span class="pre">S</span></tt> is a subset of <tt class="docutils literal"><span class="pre">G</span></tt>, but if <tt class="docutils literal"><span class="pre">G</span></tt> is a proper subgroup of
-the full symmetric group, we allow for <tt class="docutils literal"><span class="pre">S</span></tt> to have elements outside
-<tt class="docutils literal"><span class="pre">G</span></tt>.</p>
-<p>It is naturally a subgroup of <tt class="docutils literal"><span class="pre">G</span></tt>; the centralizer of a permutation
-group is equal to the centralizer of any set of generators for that
-group, since any element commuting with the generators commutes with
-any product of the  generators.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>other</strong> :</p>
-<blockquote class="last">
-<div><p>a permutation group/list of permutations/single permutation</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.subgroup_search" title="sympy.combinatorics.perm_groups.PermutationGroup.subgroup_search"><tt class="xref py py-obj docutils literal"><span class="pre">subgroup_search</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>The implementation is an application of <tt class="docutils literal"><span class="pre">.subgroup_search()</span></tt> with
-tests using a specific base for the group <tt class="docutils literal"><span class="pre">G</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="p">(</span><span class="n">SymmetricGroup</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">CyclicGroup</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">SymmetricGroup</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">CyclicGroup</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">H</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">centralizer</span><span class="p">(</span><span class="n">C</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">H</span><span class="o">.</span><span class="n">is_subgroup</span><span class="p">(</span><span class="n">C</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.commutator">
-<tt class="descname">commutator</tt><big>(</big><em>self</em>, <em>G</em>, <em>H</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.commutator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.commutator" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the commutator of two subgroups.</p>
-<p>For a permutation group <tt class="docutils literal"><span class="pre">K</span></tt> and subgroups <tt class="docutils literal"><span class="pre">G</span></tt>, <tt class="docutils literal"><span class="pre">H</span></tt>, the
-commutator of <tt class="docutils literal"><span class="pre">G</span></tt> and <tt class="docutils literal"><span class="pre">H</span></tt> is defined as the group generated
-by all the commutators <tt class="docutils literal"><span class="pre">[g,</span> <span class="pre">h]</span> <span class="pre">=</span> <span class="pre">hgh^{-1}g^{-1}</span></tt> for <tt class="docutils literal"><span class="pre">g</span></tt> in <tt class="docutils literal"><span class="pre">G</span></tt> and
-<tt class="docutils literal"><span class="pre">h</span></tt> in <tt class="docutils literal"><span class="pre">H</span></tt>. It is naturally a subgroup of <tt class="docutils literal"><span class="pre">K</span></tt> ([1], p.27).</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.derived_subgroup" title="sympy.combinatorics.perm_groups.PermutationGroup.derived_subgroup"><tt class="xref py py-obj docutils literal"><span class="pre">derived_subgroup</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>The commutator of two subgroups <tt class="docutils literal"><span class="pre">H,</span> <span class="pre">G</span></tt> is equal to the normal closure
-of the commutators of all the generators, i.e. <tt class="docutils literal"><span class="pre">hgh^{-1}g^{-1}</span></tt> for <tt class="docutils literal"><span class="pre">h</span></tt>
-a generator of <tt class="docutils literal"><span class="pre">H</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt> a generator of <tt class="docutils literal"><span class="pre">G</span></tt> ([1], p.28)</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="p">(</span><span class="n">SymmetricGroup</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">AlternatingGroup</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">SymmetricGroup</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">AlternatingGroup</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">commutator</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">A</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">is_subgroup</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.contains">
-<tt class="descname">contains</tt><big>(</big><em>self</em>, <em>g</em>, <em>strict=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.contains"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.contains" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test if permutation <tt class="docutils literal"><span class="pre">g</span></tt> belong to self, <tt class="docutils literal"><span class="pre">G</span></tt>.</p>
-<p>If <tt class="docutils literal"><span class="pre">g</span></tt> is an element of <tt class="docutils literal"><span class="pre">G</span></tt> it can be written as a product
-of factors drawn from the cosets of <tt class="docutils literal"><span class="pre">G</span></tt>&#8216;s stabilizers. To see
-if <tt class="docutils literal"><span class="pre">g</span></tt> is one of the actual generators defining the group use
-<tt class="docutils literal"><span class="pre">G.has(g)</span></tt>.</p>
-<p>If <tt class="docutils literal"><span class="pre">strict</span></tt> is not True, <tt class="docutils literal"><span class="pre">g</span></tt> will be resized, if necessary,
-to match the size of permutations in <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.coset_factor" title="sympy.combinatorics.perm_groups.PermutationGroup.coset_factor"><tt class="xref py py-obj docutils literal"><span class="pre">coset_factor</span></tt></a>, <tt class="xref py py-obj docutils literal"><span class="pre">has</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">in</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">G</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="c"># trivial check</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elem</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]],</span> <span class="n">size</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">elem</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">Permutation</span><span class="p">(</span><span class="mi">4</span><span class="p">)(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>If strict is False, a permutation will be resized, if
-necessary:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">H</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="n">Permutation</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">H</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">Permutation</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">H</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">Permutation</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span> <span class="n">strict</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>To test if a given permutation is present in the group:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">elem</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">generators</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">elem</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.coset_factor">
-<tt class="descname">coset_factor</tt><big>(</big><em>self</em>, <em>g</em>, <em>factor_index=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.coset_factor"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.coset_factor" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return <tt class="docutils literal"><span class="pre">G</span></tt>&#8216;s (self&#8217;s) coset factorization of <tt class="docutils literal"><span class="pre">g</span></tt></p>
-<p>If <tt class="docutils literal"><span class="pre">g</span></tt> is an element of <tt class="docutils literal"><span class="pre">G</span></tt> then it can be written as the product
-of permutations drawn from the Schreier-Sims coset decomposition,</p>
-<p>The permutations returned in <tt class="docutils literal"><span class="pre">f</span></tt> are those for which
-the product gives <tt class="docutils literal"><span class="pre">g</span></tt>: <tt class="docutils literal"><span class="pre">g</span> <span class="pre">=</span> <span class="pre">f[n]*...f[1]*f[0]</span></tt> where <tt class="docutils literal"><span class="pre">n</span> <span class="pre">=</span> <span class="pre">len(B)</span></tt>
-and <tt class="docutils literal"><span class="pre">B</span> <span class="pre">=</span> <span class="pre">G.base</span></tt>. f[i] is one of the permutations in
-<tt class="docutils literal"><span class="pre">self._basic_orbits[i]</span></tt>.</p>
-<p>If factor_index==True,
-returns a tuple <tt class="docutils literal"><span class="pre">[b[0],..,b[n]]</span></tt>, where <tt class="docutils literal"><span class="pre">b[i]</span></tt>
-belongs to <tt class="docutils literal"><span class="pre">self._basic_orbits[i]</span></tt></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span><span class="p">,</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">4</span><span class="p">)(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">])</span>
-</pre></div>
-</div>
-<p>Define g:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">7</span><span class="p">)(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Confirm that it is an element of G:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>Thus, it can be written as a product of factors (up to
-3) drawn from u. See below that a factor from u1 and u2
-and the Identity permutation have been used:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">coset_factor</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">g</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f1</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">coset_factor</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="bp">True</span><span class="p">);</span> <span class="n">f1</span>
-<span class="go">[0, 4, 4]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tr</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">basic_transversals</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">tr</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">f1</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>If g is not an element of G then [] is returned:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">coset_factor</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-<span class="go">[]</span>
-</pre></div>
-</div>
-<p>see util._strip</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.coset_rank">
-<tt class="descname">coset_rank</tt><big>(</big><em>self</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.coset_rank"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.coset_rank" title="Permalink to this definition">¶</a></dt>
-<dd><p>rank using Schreier-Sims representation</p>
-<p>The coset rank of <tt class="docutils literal"><span class="pre">g</span></tt> is the ordering number in which
-it appears in the lexicographic listing according to the
-coset decomposition</p>
-<p>The ordering is the same as in G.generate(method=&#8217;coset&#8217;).
-If <tt class="docutils literal"><span class="pre">g</span></tt> does not belong to the group it returns None.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.coset_factor" title="sympy.combinatorics.perm_groups.PermutationGroup.coset_factor"><tt class="xref py py-obj docutils literal"><span class="pre">coset_factor</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">4</span><span class="p">)(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">7</span><span class="p">)(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">coset_rank</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-<span class="go">16</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">coset_unrank</span><span class="p">(</span><span class="mi">16</span><span class="p">)</span>
-<span class="go">Permutation(7)(2, 4)(3, 5)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.coset_unrank">
-<tt class="descname">coset_unrank</tt><big>(</big><em>self</em>, <em>rank</em>, <em>af=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.coset_unrank"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.coset_unrank" title="Permalink to this definition">¶</a></dt>
-<dd><p>unrank using Schreier-Sims representation</p>
-<p>coset_unrank is the inverse operation of coset_rank
-if 0 &lt;= rank &lt; order; otherwise it returns None.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.degree">
-<tt class="descname">degree</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.degree"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.degree" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the size of the permutations in the group.</p>
-<p>The number of permutations comprising the group is given by
-len(group); the number of permutations that can be generated
-by the group is given by group.order().</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.order" title="sympy.combinatorics.perm_groups.PermutationGroup.order"><tt class="xref py py-obj docutils literal"><span class="pre">order</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">a</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">degree</span>
-<span class="go">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">order</span><span class="p">()</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">generate</span><span class="p">())</span>
-<span class="go">[Permutation(2), Permutation(2)(0, 1)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.derived_series">
-<tt class="descname">derived_series</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.derived_series"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.derived_series" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the derived series for the group.</p>
-<p>The derived series for a group <tt class="docutils literal"><span class="pre">G</span></tt> is defined as
-<tt class="docutils literal"><span class="pre">G</span> <span class="pre">=</span> <span class="pre">G_0</span> <span class="pre">&gt;</span> <span class="pre">G_1</span> <span class="pre">&gt;</span> <span class="pre">G_2</span> <span class="pre">&gt;</span> <span class="pre">\ldots</span></tt> where <tt class="docutils literal"><span class="pre">G_i</span> <span class="pre">=</span> <span class="pre">[G_{i-1},</span> <span class="pre">G_{i-1}]</span></tt>,
-i.e. <tt class="docutils literal"><span class="pre">G_i</span></tt> is the derived subgroup of <tt class="docutils literal"><span class="pre">G_{i-1}</span></tt>, for
-<tt class="docutils literal"><span class="pre">i\in\mathbb{N}</span></tt>. When we have <tt class="docutils literal"><span class="pre">G_k</span> <span class="pre">=</span> <span class="pre">G_{k-1}</span></tt> for some
-<tt class="docutils literal"><span class="pre">k\in\mathbb{N}</span></tt>, the series terminates.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>A list of permutation groups containing the members of the derived</strong> :</p>
-<p class="last"><strong>series in the order ``G = G_0, G_1, G_2, ldots``.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.derived_subgroup" title="sympy.combinatorics.perm_groups.PermutationGroup.derived_subgroup"><tt class="xref py py-obj docutils literal"><span class="pre">derived_subgroup</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="p">(</span><span class="n">SymmetricGroup</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">AlternatingGroup</span><span class="p">,</span> <span class="n">DihedralGroup</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">AlternatingGroup</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">len</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">derived_series</span><span class="p">())</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">SymmetricGroup</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">len</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">derived_series</span><span class="p">())</span>
-<span class="go">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">.</span><span class="n">derived_series</span><span class="p">()[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">is_subgroup</span><span class="p">(</span><span class="n">AlternatingGroup</span><span class="p">(</span><span class="mi">4</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">.</span><span class="n">derived_series</span><span class="p">()[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">is_subgroup</span><span class="p">(</span><span class="n">DihedralGroup</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.derived_subgroup">
-<tt class="descname">derived_subgroup</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.derived_subgroup"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.derived_subgroup" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the derived subgroup.</p>
-<p>The derived subgroup, or commutator subgroup is the subgroup generated
-by all commutators <tt class="docutils literal"><span class="pre">[g,</span> <span class="pre">h]</span> <span class="pre">=</span> <span class="pre">hgh^{-1}g^{-1}</span></tt> for <tt class="docutils literal"><span class="pre">g,</span> <span class="pre">h\in</span> <span class="pre">G</span></tt> ; it is
-equal to the normal closure of the set of commutators of the generators
-([1], p.28, [11]).</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.derived_series" title="sympy.combinatorics.perm_groups.PermutationGroup.derived_series"><tt class="xref py py-obj docutils literal"><span class="pre">derived_series</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">derived_subgroup</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">generate</span><span class="p">(</span><span class="n">af</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">[[0, 1, 2, 3, 4], [0, 1, 3, 4, 2], [0, 1, 4, 2, 3]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.generate">
-<tt class="descname">generate</tt><big>(</big><em>self</em>, <em>method='coset'</em>, <em>af=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.generate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.generate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return iterator to generate the elements of the group</p>
-<p>Iteration is done with one of these methods:</p>
-<div class="highlight-python"><pre>method=&#x27;coset&#x27;  using the Schreier-Sims coset representation
-method=&#x27;dimino&#x27; using the Dimino method</pre>
-</div>
-<p>If af = True it yields the array form of the permutations</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.polyhedron</span> <span class="kn">import</span> <span class="n">tetrahedron</span>
-</pre></div>
-</div>
-<p>The permutation group given in the tetrahedron object is not
-true groups:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">tetrahedron</span><span class="o">.</span><span class="n">pgroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">is_group</span><span class="p">()</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>But the group generated by the permutations in the tetrahedron
-pgroup &#8211; even the first two &#8211; is a proper group:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">H</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="n">G</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">G</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">J</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">H</span><span class="o">.</span><span class="n">generate</span><span class="p">()));</span> <span class="n">J</span>
-<span class="go">PermutationGroup([</span>
-<span class="go">    Permutation(0, 1)(2, 3),</span>
-<span class="go">    Permutation(3),</span>
-<span class="go">    Permutation(1, 2, 3),</span>
-<span class="go">    Permutation(1, 3, 2),</span>
-<span class="go">    Permutation(0, 3, 1),</span>
-<span class="go">    Permutation(0, 2, 3),</span>
-<span class="go">    Permutation(0, 3)(1, 2),</span>
-<span class="go">    Permutation(0, 1, 3),</span>
-<span class="go">    Permutation(3)(0, 2, 1),</span>
-<span class="go">    Permutation(0, 3, 2),</span>
-<span class="go">    Permutation(3)(0, 1, 2),</span>
-<span class="go">    Permutation(0, 2)(1, 3)])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">.</span><span class="n">is_group</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.generate_dimino">
-<tt class="descname">generate_dimino</tt><big>(</big><em>self</em>, <em>af=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.generate_dimino"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.generate_dimino" title="Permalink to this definition">¶</a></dt>
-<dd><p>Yield group elements using Dimino&#8217;s algorithm</p>
-<p>If af == True it yields the array form of the permutations</p>
-<p class="rubric">References</p>
-<p>[1] The Implementation of Various Algorithms for Permutation Groups in
-the Computer Algebra System: AXIOM, N.J. Doye, M.Sc. Thesis</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">generate_dimino</span><span class="p">(</span><span class="n">af</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">[[0, 1, 2, 3], [0, 2, 1, 3], [0, 2, 3, 1],</span>
-<span class="go"> [0, 1, 3, 2], [0, 3, 2, 1], [0, 3, 1, 2]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.generate_schreier_sims">
-<tt class="descname">generate_schreier_sims</tt><big>(</big><em>self</em>, <em>af=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.generate_schreier_sims"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.generate_schreier_sims" title="Permalink to this definition">¶</a></dt>
-<dd><p>Yield group elements using the Schreier-Sims representation
-in coset_rank order</p>
-<p>If af = True it yields the array form of the permutations</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">generate_schreier_sims</span><span class="p">(</span><span class="n">af</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">[[0, 1, 2, 3], [0, 2, 1, 3], [0, 3, 2, 1],</span>
-<span class="go"> [0, 1, 3, 2], [0, 2, 3, 1], [0, 3, 1, 2]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.generators">
-<tt class="descname">generators</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.generators"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.generators" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the generators of the group.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">generators</span>
-<span class="go">[Permutation(1, 2), Permutation(2)(0, 1)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.is_abelian">
-<tt class="descname">is_abelian</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.is_abelian"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.is_abelian" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test if the group is Abelian.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">is_abelian</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">a</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">is_abelian</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.is_alt_sym">
-<tt class="descname">is_alt_sym</tt><big>(</big><em>self</em>, <em>eps=0.05</em>, <em>_random_prec=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.is_alt_sym"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.is_alt_sym" title="Permalink to this definition">¶</a></dt>
-<dd><p>Monte Carlo test for the symmetric/alternating group for degrees
-&gt;= 8.</p>
-<p>More specifically, it is one-sided Monte Carlo with the
-answer True (i.e., G is symmetric/alternating) guaranteed to be
-correct, and the answer False being incorrect with probability eps.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">_check_cycles_alt_sym</span></tt></p>
-</div>
-<p class="rubric">Notes</p>
-<p>The algorithm itself uses some nontrivial results from group theory and
-number theory:
-1) If a transitive group <tt class="docutils literal"><span class="pre">G</span></tt> of degree <tt class="docutils literal"><span class="pre">n</span></tt> contains an element
-with a cycle of length <tt class="docutils literal"><span class="pre">n/2</span> <span class="pre">&lt;</span> <span class="pre">p</span> <span class="pre">&lt;</span> <span class="pre">n-2</span></tt> for <tt class="docutils literal"><span class="pre">p</span></tt> a prime, <tt class="docutils literal"><span class="pre">G</span></tt> is the
-symmetric or alternating group ([1], pp. 81-82)
-2) The proportion of elements in the symmetric/alternating group having
-the property described in 1) is approximately <tt class="docutils literal"><span class="pre">\log(2)/\log(n)</span></tt>
-([1], p.82; [2], pp. 226-227).
-The helper function <tt class="docutils literal"><span class="pre">_check_cycles_alt_sym</span></tt> is used to
-go over the cycles in a permutation and look for ones satisfying 1).</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">DihedralGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">DihedralGroup</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span><span class="o">.</span><span class="n">is_alt_sym</span><span class="p">()</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.is_group">
-<tt class="descname">is_group</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.is_group"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.is_group" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if the group meets three criteria: identity is present,
-the inverse of every element is also an element, and the product of
-any two elements is also an element. If any of the tests fail, False
-is returned.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.polyhedron</span> <span class="kn">import</span> <span class="n">tetrahedron</span>
-</pre></div>
-</div>
-<p>The permutation group given in the tetrahedron object is not
-a true group:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">tetrahedron</span><span class="o">.</span><span class="n">pgroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">is_group</span><span class="p">()</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>But the group generated by the permutations in the tetrahedron
-pgroup is a proper group:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">H</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">generate</span><span class="p">()))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">H</span><span class="o">.</span><span class="n">is_group</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>The identity permutation is present:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">H</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Permutation</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>The product of any two elements from the group is also in the group:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">TableForm</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">H</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="p">[]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">g</span><span class="p">:</span>
-<span class="gp">... </span>    <span class="n">m</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="n">g</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">i</span><span class="o">*</span><span class="n">H</span><span class="p">)</span> <span class="k">for</span> <span class="n">H</span> <span class="ow">in</span> <span class="n">g</span><span class="p">])</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">TableForm</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">headings</span><span class="o">=</span><span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)),</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">))],</span> <span class="n">wipe_zeros</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">   | 0  1  2  3  4  5  6  7  8  9  10 11</span>
-<span class="go">----------------------------------------</span>
-<span class="go"> 0 | 11 0  8  10 6  2  7  4  5  3  9  1</span>
-<span class="go"> 1 | 0  1  2  3  4  5  6  7  8  9  10 11</span>
-<span class="go"> 2 | 6  2  7  4  5  3  9  1  11 0  8  10</span>
-<span class="go"> 3 | 5  3  9  1  11 0  8  10 6  2  7  4</span>
-<span class="go"> 4 | 3  4  0  2  10 6  11 8  9  7  1  5</span>
-<span class="go"> 5 | 4  5  6  7  8  9  10 11 0  1  2  3</span>
-<span class="go"> 6 | 10 6  11 8  9  7  1  5  3  4  0  2</span>
-<span class="go"> 7 | 9  7  1  5  3  4  0  2  10 6  11 8</span>
-<span class="go"> 8 | 7  8  4  6  2  10 3  0  1  11 5  9</span>
-<span class="go"> 9 | 8  9  10 11 0  1  2  3  4  5  6  7</span>
-<span class="go">10 | 2  10 3  0  1  11 5  9  7  8  4  6</span>
-<span class="go">11 | 1  11 5  9  7  8  4  6  2  10 3  0</span>
-<span class="go">&gt;&gt;&gt;</span>
-<span class="go">The entries in the table give the element in the group corresponding</span>
-<span class="go">to the product of a given column element and row element:</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">g</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span><span class="o">*</span><span class="n">g</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="n">g</span><span class="p">[</span><span class="mi">9</span><span class="p">]</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>The inverse of every element is also in the group:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">TableForm</span><span class="p">([[</span><span class="n">g</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="o">~</span><span class="n">gi</span><span class="p">)</span> <span class="k">for</span> <span class="n">gi</span> <span class="ow">in</span> <span class="n">g</span><span class="p">]],</span> <span class="n">headings</span><span class="o">=</span><span class="p">[[],</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">))],</span>
-<span class="gp">... </span>   <span class="n">wipe_zeros</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">0  1 2 3 4  5 6 7 8 9 10 11</span>
-<span class="go">---------------------------</span>
-<span class="go">11 1 7 3 10 9 6 2 8 5 4  0</span>
-</pre></div>
-</div>
-<p>So we see that g[1] and g[3] are equivalent to their inverse while
-g[7] == ~g[2].</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.is_nilpotent">
-<tt class="descname">is_nilpotent</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.is_nilpotent"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.is_nilpotent" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test if the group is nilpotent.</p>
-<p>A group <tt class="docutils literal"><span class="pre">G</span></tt> is nilpotent if it has a central series of finite length.
-Alternatively, <tt class="docutils literal"><span class="pre">G</span></tt> is nilpotent if its lower central series terminates
-with the trivial group. Every nilpotent group is also solvable
-([1], p.29, [12]).</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.lower_central_series" title="sympy.combinatorics.perm_groups.PermutationGroup.lower_central_series"><tt class="xref py py-obj docutils literal"><span class="pre">lower_central_series</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.is_solvable" title="sympy.combinatorics.perm_groups.PermutationGroup.is_solvable"><tt class="xref py py-obj docutils literal"><span class="pre">is_solvable</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="p">(</span><span class="n">SymmetricGroup</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">CyclicGroup</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">CyclicGroup</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span><span class="o">.</span><span class="n">is_nilpotent</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">SymmetricGroup</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">.</span><span class="n">is_nilpotent</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.is_normal">
-<tt class="descname">is_normal</tt><big>(</big><em>self</em>, <em>gr</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.is_normal"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.is_normal" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test if G=self is a normal subgroup of gr.</p>
-<p>G is normal in gr if
-for each g2 in G, g1 in gr, g = g1*g2*g1**-1 belongs to G
-It is sufficient to check this for each g1 in gr.generator and
-g2 g2 in G.generator</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G1</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G1</span><span class="o">.</span><span class="n">is_normal</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.is_primitive">
-<tt class="descname">is_primitive</tt><big>(</big><em>self</em>, <em>randomized=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.is_primitive"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.is_primitive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test if a group is primitive.</p>
-<p>A permutation group <tt class="docutils literal"><span class="pre">G</span></tt> acting on a set <tt class="docutils literal"><span class="pre">S</span></tt> is called primitive if
-<tt class="docutils literal"><span class="pre">S</span></tt> contains no nontrivial block under the action of <tt class="docutils literal"><span class="pre">G</span></tt>
-(a block is nontrivial if its cardinality is more than <tt class="docutils literal"><span class="pre">1</span></tt>).</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.minimal_block" title="sympy.combinatorics.perm_groups.PermutationGroup.minimal_block"><tt class="xref py py-obj docutils literal"><span class="pre">minimal_block</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.random_stab" title="sympy.combinatorics.perm_groups.PermutationGroup.random_stab"><tt class="xref py py-obj docutils literal"><span class="pre">random_stab</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>The algorithm is described in [1], p.83, and uses the function
-minimal_block to search for blocks of the form <tt class="docutils literal"><span class="pre">\{0,</span> <span class="pre">k\}</span></tt> for <tt class="docutils literal"><span class="pre">k</span></tt>
-ranging over representatives for the orbits of <tt class="docutils literal"><span class="pre">G_0</span></tt>, the stabilizer of
-<tt class="docutils literal"><span class="pre">0</span></tt>. This algorithm has complexity <tt class="docutils literal"><span class="pre">O(n^2)</span></tt> where <tt class="docutils literal"><span class="pre">n</span></tt> is the degree
-of the group, and will perform badly if <tt class="docutils literal"><span class="pre">G_0</span></tt> is small.</p>
-<p>There are two implementations offered: one finds <tt class="docutils literal"><span class="pre">G_0</span></tt>
-deterministically using the function <tt class="docutils literal"><span class="pre">stabilizer</span></tt>, and the other
-(default) produces random elements of <tt class="docutils literal"><span class="pre">G_0</span></tt> using <tt class="docutils literal"><span class="pre">random_stab</span></tt>,
-hoping that they generate a subgroup of <tt class="docutils literal"><span class="pre">G_0</span></tt> with not too many more
-orbits than G_0 (this is suggested in [1], p.83). Behavior is changed
-by the <tt class="docutils literal"><span class="pre">randomized</span></tt> flag.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">DihedralGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">DihedralGroup</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span><span class="o">.</span><span class="n">is_primitive</span><span class="p">()</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.is_solvable">
-<tt class="descname">is_solvable</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.is_solvable"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.is_solvable" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test if the group is solvable.</p>
-<p><tt class="docutils literal"><span class="pre">G</span></tt> is solvable if its derived series terminates with the trivial
-group ([1], p.29).</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.is_nilpotent" title="sympy.combinatorics.perm_groups.PermutationGroup.is_nilpotent"><tt class="xref py py-obj docutils literal"><span class="pre">is_nilpotent</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.derived_series" title="sympy.combinatorics.perm_groups.PermutationGroup.derived_series"><tt class="xref py py-obj docutils literal"><span class="pre">derived_series</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">SymmetricGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">SymmetricGroup</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">.</span><span class="n">is_solvable</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.is_subgroup">
-<tt class="descname">is_subgroup</tt><big>(</big><em>self</em>, <em>G</em>, <em>strict=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.is_subgroup"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.is_subgroup" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if all elements of self belong to G.</p>
-<p>If <tt class="docutils literal"><span class="pre">strict</span></tt> is False then if <tt class="docutils literal"><span class="pre">self</span></tt>&#8216;s degree is smaller
-than <tt class="docutils literal"><span class="pre">G</span></tt>&#8216;s, the elements will be resized to have the same degree.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span><span class="p">,</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="p">(</span><span class="n">SymmetricGroup</span><span class="p">,</span>
-<span class="gp">... </span>   <span class="n">CyclicGroup</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Testing is strict by default: the degree of each group must be the
-same:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G1</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">Permutation</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G2</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">Permutation</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G3</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">p</span><span class="p">,</span> <span class="n">p</span><span class="o">**</span><span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">assert</span> <span class="n">G1</span><span class="o">.</span><span class="n">order</span><span class="p">()</span> <span class="o">==</span> <span class="n">G2</span><span class="o">.</span><span class="n">order</span><span class="p">()</span> <span class="o">==</span> <span class="n">G3</span><span class="o">.</span><span class="n">order</span><span class="p">()</span> <span class="o">==</span> <span class="mi">6</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G1</span><span class="o">.</span><span class="n">is_subgroup</span><span class="p">(</span><span class="n">G2</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G1</span><span class="o">.</span><span class="n">is_subgroup</span><span class="p">(</span><span class="n">G3</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G3</span><span class="o">.</span><span class="n">is_subgroup</span><span class="p">(</span><span class="n">PermutationGroup</span><span class="p">(</span><span class="n">G3</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G3</span><span class="o">.</span><span class="n">is_subgroup</span><span class="p">(</span><span class="n">PermutationGroup</span><span class="p">(</span><span class="n">G3</span><span class="p">[</span><span class="mi">0</span><span class="p">]))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>To ignore the size, set <tt class="docutils literal"><span class="pre">strict</span></tt> to False:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">S3</span> <span class="o">=</span> <span class="n">SymmetricGroup</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S5</span> <span class="o">=</span> <span class="n">SymmetricGroup</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S3</span><span class="o">.</span><span class="n">is_subgroup</span><span class="p">(</span><span class="n">S5</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C7</span> <span class="o">=</span> <span class="n">CyclicGroup</span><span class="p">(</span><span class="mi">7</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">S5</span><span class="o">*</span><span class="n">C7</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S5</span><span class="o">.</span><span class="n">is_subgroup</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C7</span><span class="o">.</span><span class="n">is_subgroup</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.is_transitive">
-<tt class="descname">is_transitive</tt><big>(</big><em>self</em>, <em>strict=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.is_transitive"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.is_transitive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test if the group is transitive.</p>
-<p>A group is transitive if it has a single orbit.</p>
-<p>If <tt class="docutils literal"><span class="pre">strict</span></tt> is False the group is transitive if it has
-a single orbit of length different from 1.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G1</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G1</span><span class="o">.</span><span class="n">is_transitive</span><span class="p">()</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G1</span><span class="o">.</span><span class="n">is_transitive</span><span class="p">(</span><span class="n">strict</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G2</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">c</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G2</span><span class="o">.</span><span class="n">is_transitive</span><span class="p">()</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G3</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">d</span><span class="p">,</span> <span class="n">e</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G3</span><span class="o">.</span><span class="n">is_transitive</span><span class="p">()</span> <span class="ow">or</span> <span class="n">G3</span><span class="o">.</span><span class="n">is_transitive</span><span class="p">(</span><span class="n">strict</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.is_trivial">
-<tt class="descname">is_trivial</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.is_trivial"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.is_trivial" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test if the group is the trivial group.</p>
-<p>This is true if the group contains only the identity permutation.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">is_trivial</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.lower_central_series">
-<tt class="descname">lower_central_series</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.lower_central_series"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.lower_central_series" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the lower central series for the group.</p>
-<p>The lower central series for a group <tt class="docutils literal"><span class="pre">G</span></tt> is the series
-<tt class="docutils literal"><span class="pre">G</span> <span class="pre">=</span> <span class="pre">G_0</span> <span class="pre">&gt;</span> <span class="pre">G_1</span> <span class="pre">&gt;</span> <span class="pre">G_2</span> <span class="pre">&gt;</span> <span class="pre">\ldots</span></tt> where
-<tt class="docutils literal"><span class="pre">G_k</span> <span class="pre">=</span> <span class="pre">[G,</span> <span class="pre">G_{k-1}]</span></tt>, i.e. every term after the first is equal to the
-commutator of <tt class="docutils literal"><span class="pre">G</span></tt> and the previous term in <tt class="docutils literal"><span class="pre">G1</span></tt> ([1], p.29).</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>A list of permutation groups in the order</strong> :</p>
-<p class="last"><strong>``G = G_0, G_1, G_2, ldots``</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.commutator" title="sympy.combinatorics.perm_groups.PermutationGroup.commutator"><tt class="xref py py-obj docutils literal"><span class="pre">commutator</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.derived_series" title="sympy.combinatorics.perm_groups.PermutationGroup.derived_series"><tt class="xref py py-obj docutils literal"><span class="pre">derived_series</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="p">(</span><span class="n">AlternatingGroup</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">DihedralGroup</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">AlternatingGroup</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">len</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">lower_central_series</span><span class="p">())</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">lower_central_series</span><span class="p">()[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">is_subgroup</span><span class="p">(</span><span class="n">DihedralGroup</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.make_perm">
-<tt class="descname">make_perm</tt><big>(</big><em>self</em>, <em>n</em>, <em>seed=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.make_perm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.make_perm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Multiply <tt class="docutils literal"><span class="pre">n</span></tt> randomly selected permutations from
-pgroup together, starting with the identity
-permutation. If <tt class="docutils literal"><span class="pre">n</span></tt> is a list of integers, those
-integers will be used to select the permutations and they
-will be applied in L to R order: make_perm((A, B, C)) will
-give CBA(I) where I is the identity permutation.</p>
-<p><tt class="docutils literal"><span class="pre">seed</span></tt> is used to set the seed for the random selection
-of permutations from pgroup. If this is a list of integers,
-the corresponding permutations from pgroup will be selected
-in the order give. This is mainly used for testing purposes.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.random" title="sympy.combinatorics.perm_groups.PermutationGroup.random"><tt class="xref py py-obj docutils literal"><span class="pre">random</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="p">[</span><span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">]),</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">make_perm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-<span class="go">Permutation(0, 1)(2, 3)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">make_perm</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="go">Permutation(0, 2, 3, 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">make_perm</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="go">Permutation(0, 2, 3, 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.max_div">
-<tt class="descname">max_div</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.max_div"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.max_div" title="Permalink to this definition">¶</a></dt>
-<dd><p>Maximum proper divisor of the degree of a permutation group.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.minimal_block" title="sympy.combinatorics.perm_groups.PermutationGroup.minimal_block"><tt class="xref py py-obj docutils literal"><span class="pre">minimal_block</span></tt></a>, <tt class="xref py py-obj docutils literal"><span class="pre">_union_find_merge</span></tt></p>
-</div>
-<p class="rubric">Notes</p>
-<p>Obviously, this is the degree divided by its minimal proper divisor
-(larger than <tt class="docutils literal"><span class="pre">1</span></tt>, if one exists). As it is guaranteed to be prime,
-the <tt class="docutils literal"><span class="pre">sieve</span></tt> from <tt class="docutils literal"><span class="pre">sympy.ntheory</span></tt> is used.
-This function is also used as an optimization tool for the functions
-<tt class="docutils literal"><span class="pre">minimal_block</span></tt> and <tt class="docutils literal"><span class="pre">_union_find_merge</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">])])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">max_div</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.minimal_block">
-<tt class="descname">minimal_block</tt><big>(</big><em>self</em>, <em>points</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.minimal_block"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.minimal_block" title="Permalink to this definition">¶</a></dt>
-<dd><p>For a transitive group, finds the block system generated by
-<tt class="docutils literal"><span class="pre">points</span></tt>.</p>
-<p>If a group <tt class="docutils literal"><span class="pre">G</span></tt> acts on a set <tt class="docutils literal"><span class="pre">S</span></tt>, a nonempty subset <tt class="docutils literal"><span class="pre">B</span></tt> of <tt class="docutils literal"><span class="pre">S</span></tt>
-is called a block under the action of <tt class="docutils literal"><span class="pre">G</span></tt> if for all <tt class="docutils literal"><span class="pre">g</span></tt> in <tt class="docutils literal"><span class="pre">G</span></tt>
-we have <tt class="docutils literal"><span class="pre">gB</span> <span class="pre">=</span> <span class="pre">B</span></tt> (<tt class="docutils literal"><span class="pre">g</span></tt> fixes <tt class="docutils literal"><span class="pre">B</span></tt>) or <tt class="docutils literal"><span class="pre">gB</span></tt> and <tt class="docutils literal"><span class="pre">B</span></tt> have no
-common points (<tt class="docutils literal"><span class="pre">g</span></tt> moves <tt class="docutils literal"><span class="pre">B</span></tt> entirely). ([1], p.23; [6]).</p>
-<p>The distinct translates <tt class="docutils literal"><span class="pre">gB</span></tt> of a block <tt class="docutils literal"><span class="pre">B</span></tt> for <tt class="docutils literal"><span class="pre">g</span></tt> in <tt class="docutils literal"><span class="pre">G</span></tt>
-partition the set <tt class="docutils literal"><span class="pre">S</span></tt> and this set of translates is known as a block
-system. Moreover, we obviously have that all blocks in the partition
-have the same size, hence the block size divides <tt class="docutils literal"><span class="pre">|S|</span></tt> ([1], p.23).
-A <tt class="docutils literal"><span class="pre">G</span></tt>-congruence is an equivalence relation <tt class="docutils literal"><span class="pre">~</span></tt> on the set <tt class="docutils literal"><span class="pre">S</span></tt>
-such that <tt class="docutils literal"><span class="pre">a</span> <span class="pre">~</span> <span class="pre">b</span></tt> implies <tt class="docutils literal"><span class="pre">g(a)</span> <span class="pre">~</span> <span class="pre">g(b)</span></tt> for all <tt class="docutils literal"><span class="pre">g</span></tt> in <tt class="docutils literal"><span class="pre">G</span></tt>.
-For a transitive group, the equivalence classes of a <tt class="docutils literal"><span class="pre">G</span></tt>-congruence
-and the blocks of a block system are the same thing ([1], p.23).</p>
-<p>The algorithm below checks the group for transitivity, and then finds
-the <tt class="docutils literal"><span class="pre">G</span></tt>-congruence generated by the pairs <tt class="docutils literal"><span class="pre">(p_0,</span> <span class="pre">p_1),</span> <span class="pre">(p_0,</span> <span class="pre">p_2),</span>
-<span class="pre">...,</span> <span class="pre">(p_0,p_{k-1})</span></tt> which is the same as finding the maximal block
-system (i.e., the one with minimum block size) such that
-<tt class="docutils literal"><span class="pre">p_0,</span> <span class="pre">...,</span> <span class="pre">p_{k-1}</span></tt> are in the same block ([1], p.83).</p>
-<p>It is an implementation of Atkinson&#8217;s algorithm, as suggested in [1],
-and manipulates an equivalence relation on the set <tt class="docutils literal"><span class="pre">S</span></tt> using a
-union-find data structure. The running time is just above
-<tt class="docutils literal"><span class="pre">O(|points||S|)</span></tt>. ([1], pp. 83-87; [7]).</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">_union_find_rep</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">_union_find_merge</span></tt>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.is_transitive" title="sympy.combinatorics.perm_groups.PermutationGroup.is_transitive"><tt class="xref py py-obj docutils literal"><span class="pre">is_transitive</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.is_primitive" title="sympy.combinatorics.perm_groups.PermutationGroup.is_primitive"><tt class="xref py py-obj docutils literal"><span class="pre">is_primitive</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">DihedralGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">DihedralGroup</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span><span class="o">.</span><span class="n">minimal_block</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
-<span class="go">[0, 6, 2, 8, 4, 0, 6, 2, 8, 4]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span><span class="o">.</span><span class="n">minimal_block</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.normal_closure">
-<tt class="descname">normal_closure</tt><big>(</big><em>self</em>, <em>other</em>, <em>k=10</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.normal_closure"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.normal_closure" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the normal closure of a subgroup/set of permutations.</p>
-<p>If <tt class="docutils literal"><span class="pre">S</span></tt> is a subset of a group <tt class="docutils literal"><span class="pre">G</span></tt>, the normal closure of <tt class="docutils literal"><span class="pre">A</span></tt> in <tt class="docutils literal"><span class="pre">G</span></tt>
-is defined as the intersection of all normal subgroups of <tt class="docutils literal"><span class="pre">G</span></tt> that
-contain <tt class="docutils literal"><span class="pre">A</span></tt> ([1], p.14). Alternatively, it is the group generated by
-the conjugates <tt class="docutils literal"><span class="pre">x^{-1}yx</span></tt> for <tt class="docutils literal"><span class="pre">x</span></tt> a generator of <tt class="docutils literal"><span class="pre">G</span></tt> and <tt class="docutils literal"><span class="pre">y</span></tt> a
-generator of the subgroup <tt class="docutils literal"><span class="pre">\left\langle</span> <span class="pre">S\right\rangle</span></tt> generated by
-<tt class="docutils literal"><span class="pre">S</span></tt> (for some chosen generating set for <tt class="docutils literal"><span class="pre">\left\langle</span> <span class="pre">S\right\rangle</span></tt>)
-([1], p.73).</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>other</strong> :</p>
-<blockquote>
-<div><p>a subgroup/list of permutations/single permutation</p>
-</div></blockquote>
-<p><strong>k</strong> :</p>
-<blockquote class="last">
-<div><p>an implementation-specific parameter that determines the number
-of conjugates that are adjoined to <tt class="docutils literal"><span class="pre">other</span></tt> at once</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.commutator" title="sympy.combinatorics.perm_groups.PermutationGroup.commutator"><tt class="xref py py-obj docutils literal"><span class="pre">commutator</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.derived_subgroup" title="sympy.combinatorics.perm_groups.PermutationGroup.derived_subgroup"><tt class="xref py py-obj docutils literal"><span class="pre">derived_subgroup</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.random_pr" title="sympy.combinatorics.perm_groups.PermutationGroup.random_pr"><tt class="xref py py-obj docutils literal"><span class="pre">random_pr</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>The algorithm is described in [1], pp. 73-74; it makes use of the
-generation of random elements for permutation groups by the product
-replacement algorithm.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="p">(</span><span class="n">SymmetricGroup</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">CyclicGroup</span><span class="p">,</span> <span class="n">AlternatingGroup</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">SymmetricGroup</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">CyclicGroup</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">normal_closure</span><span class="p">(</span><span class="n">C</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">order</span><span class="p">()</span>
-<span class="go">60</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">is_subgroup</span><span class="p">(</span><span class="n">AlternatingGroup</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.orbit">
-<tt class="descname">orbit</tt><big>(</big><em>self</em>, <em>alpha</em>, <em>action='tuples'</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.orbit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.orbit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the orbit of alpha <tt class="docutils literal"><span class="pre">\{g(\alpha)</span> <span class="pre">|</span> <span class="pre">g</span> <span class="pre">\in</span> <span class="pre">G\}</span></tt> as a set.</p>
-<p>The time complexity of the algorithm used here is <tt class="docutils literal"><span class="pre">O(|Orb|*r)</span></tt> where
-<tt class="docutils literal"><span class="pre">|Orb|</span></tt> is the size of the orbit and <tt class="docutils literal"><span class="pre">r</span></tt> is the number of generators of
-the group. For a more detailed analysis, see [1], p.78, [2], pp. 19-21.
-Here alpha can be a single point, or a list of points.</p>
-<p>If alpha is a single point, the ordinary orbit is computed.
-if alpha is a list of points, there are three available options:</p>
-<p>&#8216;union&#8217; - computes the union of the orbits of the points in the list
-&#8216;tuples&#8217; - computes the orbit of the list interpreted as an ordered
-tuple under the group action ( i.e., g((1,2,3)) = (g(1), g(2), g(3)) )
-&#8216;sets&#8217; - computes the orbit of the list interpreted as a sets</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.orbit_transversal" title="sympy.combinatorics.perm_groups.PermutationGroup.orbit_transversal"><tt class="xref py py-obj docutils literal"><span class="pre">orbit_transversal</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">a</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">orbit</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">set([0, 1, 2])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">orbit</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="s">&#39;union&#39;</span><span class="p">)</span>
-<span class="go">set([0, 1, 2, 3, 4, 5, 6])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.orbit_rep">
-<tt class="descname">orbit_rep</tt><big>(</big><em>self</em>, <em>alpha</em>, <em>beta</em>, <em>schreier_vector=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.orbit_rep"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.orbit_rep" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a group element which sends <tt class="docutils literal"><span class="pre">alpha</span></tt> to <tt class="docutils literal"><span class="pre">beta</span></tt>.</p>
-<p>If <tt class="docutils literal"><span class="pre">beta</span></tt> is not in the orbit of <tt class="docutils literal"><span class="pre">alpha</span></tt>, the function returns
-<tt class="docutils literal"><span class="pre">False</span></tt>. This implementation makes use of the schreier vector.
-For a proof of correctness, see [1], p.80</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.schreier_vector" title="sympy.combinatorics.perm_groups.PermutationGroup.schreier_vector"><tt class="xref py py-obj docutils literal"><span class="pre">schreier_vector</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">AlternatingGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">AlternatingGroup</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">orbit_rep</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">Permutation(0, 4, 1, 2, 3)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.orbit_transversal">
-<tt class="descname">orbit_transversal</tt><big>(</big><em>self</em>, <em>alpha</em>, <em>pairs=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.orbit_transversal"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.orbit_transversal" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes a transversal for the orbit of <tt class="docutils literal"><span class="pre">alpha</span></tt> as a set.</p>
-<p>For a permutation group <tt class="docutils literal"><span class="pre">G</span></tt>, a transversal for the orbit
-<tt class="docutils literal"><span class="pre">Orb</span> <span class="pre">=</span> <span class="pre">\{g(\alpha)</span> <span class="pre">|</span> <span class="pre">g</span> <span class="pre">\in</span> <span class="pre">G\}</span></tt> is a set
-<tt class="docutils literal"><span class="pre">\{g_\beta</span> <span class="pre">|</span> <span class="pre">g_\beta(\alpha)</span> <span class="pre">=</span> <span class="pre">\beta\}</span></tt> for <tt class="docutils literal"><span class="pre">\beta</span> <span class="pre">\in</span> <span class="pre">Orb</span></tt>.
-Note that there may be more than one possible transversal.
-If <tt class="docutils literal"><span class="pre">pairs</span></tt> is set to <tt class="docutils literal"><span class="pre">True</span></tt>, it returns the list of pairs
-<tt class="docutils literal"><span class="pre">(\beta,</span> <span class="pre">g_\beta)</span></tt>. For a proof of correctness, see [1], p.79</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.orbit" title="sympy.combinatorics.perm_groups.PermutationGroup.orbit"><tt class="xref py py-obj docutils literal"><span class="pre">orbit</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">DihedralGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">DihedralGroup</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">orbit_transversal</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">[Permutation(5),</span>
-<span class="go"> Permutation(0, 1, 2, 3, 4, 5),</span>
-<span class="go"> Permutation(0, 5)(1, 4)(2, 3),</span>
-<span class="go"> Permutation(0, 2, 4)(1, 3, 5),</span>
-<span class="go"> Permutation(5)(0, 4)(1, 3),</span>
-<span class="go"> Permutation(0, 3)(1, 4)(2, 5)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.orbits">
-<tt class="descname">orbits</tt><big>(</big><em>self</em>, <em>rep=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.orbits"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.orbits" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the orbits of self, ordered according to lowest element
-in each orbit.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">5</span><span class="p">)(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)(</span><span class="mi">4</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">5</span><span class="p">)(</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">)(</span><span class="mi">2</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">orbits</span><span class="p">()</span>
-<span class="go">[set([0, 2, 3, 4, 6]), set([1, 5])]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.order">
-<tt class="descname">order</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.order"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.order" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the order of the group: the number of permutations that
-can be generated from elements of the group.</p>
-<p>The number of permutations comprising the group is given by
-len(group); the length of each permutation in the group is
-given by group.size.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.degree" title="sympy.combinatorics.perm_groups.PermutationGroup.degree"><tt class="xref py py-obj docutils literal"><span class="pre">degree</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">a</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">degree</span>
-<span class="go">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">order</span><span class="p">()</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">generate</span><span class="p">())</span>
-<span class="go">[Permutation(2), Permutation(2)(0, 1)]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">order</span><span class="p">()</span>
-<span class="go">6</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.pointwise_stabilizer">
-<tt class="descname">pointwise_stabilizer</tt><big>(</big><em>self</em>, <em>points</em>, <em>incremental=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.pointwise_stabilizer"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.pointwise_stabilizer" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the pointwise stabilizer for a set of points.</p>
-<p>For a permutation group <tt class="docutils literal"><span class="pre">G</span></tt> and a set of points
-<tt class="docutils literal"><span class="pre">\{p_1,</span> <span class="pre">p_2,\ldots,</span> <span class="pre">p_k\}</span></tt>, the pointwise stabilizer of
-<tt class="docutils literal"><span class="pre">p_1,</span> <span class="pre">p_2,</span> <span class="pre">\ldots,</span> <span class="pre">p_k</span></tt> is defined as
-<tt class="docutils literal"><span class="pre">G_{p_1,\ldots,</span> <span class="pre">p_k}</span> <span class="pre">=</span>
-<span class="pre">\{g\in</span> <span class="pre">G</span> <span class="pre">|</span> <span class="pre">g(p_i)</span> <span class="pre">=</span> <span class="pre">p_i</span> <span class="pre">\forall</span> <span class="pre">i\in\{1,</span> <span class="pre">2,\ldots,k\}\}</span> <span class="pre">([1],p20).</span>
-<span class="pre">It</span> <span class="pre">is</span> <span class="pre">a</span> <span class="pre">subgroup</span> <span class="pre">of</span> <span class="pre">``G</span></tt>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.stabilizer" title="sympy.combinatorics.perm_groups.PermutationGroup.stabilizer"><tt class="xref py py-obj docutils literal"><span class="pre">stabilizer</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims_incremental" title="sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims_incremental"><tt class="xref py py-obj docutils literal"><span class="pre">schreier_sims_incremental</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>When incremental == True,
-rather than the obvious implementation using successive calls to
-.stabilizer(), this uses the incremental Schreier-Sims algorithm
-to obtain a base with starting segment - the given points.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">SymmetricGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">SymmetricGroup</span><span class="p">(</span><span class="mi">7</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Stab</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">pointwise_stabilizer</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Stab</span><span class="o">.</span><span class="n">is_subgroup</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">stabilizer</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">stabilizer</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">stabilizer</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.random">
-<tt class="descname">random</tt><big>(</big><em>self</em>, <em>af=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.random"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.random" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a random group element</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.random_pr">
-<tt class="descname">random_pr</tt><big>(</big><em>self</em>, <em>gen_count=11</em>, <em>iterations=50</em>, <em>_random_prec=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.random_pr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.random_pr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a random group element using product replacement.</p>
-<p>For the details of the product replacement algorithm, see
-<tt class="docutils literal"><span class="pre">_random_pr_init</span></tt> In <tt class="docutils literal"><span class="pre">random_pr</span></tt> the actual &#8216;product replacement&#8217;
-is performed. Notice that if the attribute <tt class="docutils literal"><span class="pre">_random_gens</span></tt>
-is empty, it needs to be initialized by <tt class="docutils literal"><span class="pre">_random_pr_init</span></tt>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">_random_pr_init</span></tt></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.random_stab">
-<tt class="descname">random_stab</tt><big>(</big><em>self</em>, <em>alpha</em>, <em>schreier_vector=None</em>, <em>_random_prec=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.random_stab"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.random_stab" title="Permalink to this definition">¶</a></dt>
-<dd><p>Random element from the stabilizer of <tt class="docutils literal"><span class="pre">alpha</span></tt>.</p>
-<p>The schreier vector for <tt class="docutils literal"><span class="pre">alpha</span></tt> is an optional argument used
-for speeding up repeated calls. The algorithm is described in [1], p.81</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.random_pr" title="sympy.combinatorics.perm_groups.PermutationGroup.random_pr"><tt class="xref py py-obj docutils literal"><span class="pre">random_pr</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.orbit_rep" title="sympy.combinatorics.perm_groups.PermutationGroup.orbit_rep"><tt class="xref py py-obj docutils literal"><span class="pre">orbit_rep</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims">
-<tt class="descname">schreier_sims</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.schreier_sims"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims" title="Permalink to this definition">¶</a></dt>
-<dd><p>Schreier-Sims algorithm.</p>
-<p>It computes the generators of the chain of stabilizers
-G &gt; G_{b_1} &gt; .. &gt; G_{b1,..,b_r} &gt; 1
-in which G_{b_1,..,b_i} stabilizes b_1,..,b_i,
-and the corresponding <tt class="docutils literal"><span class="pre">s</span></tt> cosets.
-An element of the group can be written as the product
-h_1*..*h_s.</p>
-<p>We use the incremental Schreier-Sims algorithm.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">schreier_sims</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">basic_transversals</span>
-<span class="go">[{0: Permutation(2)(0, 1), 1: Permutation(2), 2: Permutation(1, 2)},</span>
-<span class="go"> {0: Permutation(2), 2: Permutation(0, 2)}]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims_incremental">
-<tt class="descname">schreier_sims_incremental</tt><big>(</big><em>self</em>, <em>base=None</em>, <em>gens=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.schreier_sims_incremental"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims_incremental" title="Permalink to this definition">¶</a></dt>
-<dd><p>Extend a sequence of points and generating set to a base and strong
-generating set.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>base</strong> :</p>
-<blockquote>
-<div><p>The sequence of points to be extended to a base. Optional
-parameter with default value <tt class="docutils literal"><span class="pre">[]</span></tt>.</p>
-</div></blockquote>
-<p><strong>gens</strong> :</p>
-<blockquote>
-<div><p>The generating set to be extended to a strong generating set
-relative to the base obtained. Optional parameter with default
-value <tt class="docutils literal"><span class="pre">self.generators</span></tt>.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>(base, strong_gens)</strong> :</p>
-<blockquote class="last">
-<div><p><tt class="docutils literal"><span class="pre">base</span></tt> is the base obtained, and <tt class="docutils literal"><span class="pre">strong_gens</span></tt> is the strong
-generating set relative to it. The original parameters <tt class="docutils literal"><span class="pre">base</span></tt>,
-<tt class="docutils literal"><span class="pre">gens</span></tt> remain unchanged.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims" title="sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims"><tt class="xref py py-obj docutils literal"><span class="pre">schreier_sims</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims_random" title="sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims_random"><tt class="xref py py-obj docutils literal"><span class="pre">schreier_sims_random</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>This version of the Schreier-Sims algorithm runs in polynomial time.
-There are certain assumptions in the implementation - if the trivial
-group is provided, <tt class="docutils literal"><span class="pre">base</span></tt> and <tt class="docutils literal"><span class="pre">gens</span></tt> are returned immediately,
-as any sequence of points is a base for the trivial group. If the
-identity is present in the generators <tt class="docutils literal"><span class="pre">gens</span></tt>, it is removed as
-it is a redundant generator.
-The implementation is described in [1], pp. 90-93.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">AlternatingGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.testutil</span> <span class="kn">import</span> <span class="n">_verify_bsgs</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">AlternatingGroup</span><span class="p">(</span><span class="mi">7</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">base</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">seq</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">schreier_sims_incremental</span><span class="p">(</span><span class="n">base</span><span class="o">=</span><span class="n">seq</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_verify_bsgs</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">base</span><span class="p">[:</span><span class="mi">2</span><span class="p">]</span>
-<span class="go">[2, 3]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims_random">
-<tt class="descname">schreier_sims_random</tt><big>(</big><em>self</em>, <em>base=None</em>, <em>gens=None</em>, <em>consec_succ=10</em>, <em>_random_prec=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.schreier_sims_random"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims_random" title="Permalink to this definition">¶</a></dt>
-<dd><p>Randomized Schreier-Sims algorithm.</p>
-<p>The randomized Schreier-Sims algorithm takes the sequence <tt class="docutils literal"><span class="pre">base</span></tt>
-and the generating set <tt class="docutils literal"><span class="pre">gens</span></tt>, and extends <tt class="docutils literal"><span class="pre">base</span></tt> to a base, and
-<tt class="docutils literal"><span class="pre">gens</span></tt> to a strong generating set relative to that base with
-probability of a wrong answer at most <tt class="docutils literal"><span class="pre">2^{-consec\_succ}</span></tt>,
-provided the random generators are sufficiently random.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>base</strong> :</p>
-<blockquote>
-<div><p>The sequence to be extended to a base.</p>
-</div></blockquote>
-<p><strong>gens</strong> :</p>
-<blockquote>
-<div><p>The generating set to be extended to a strong generating set.</p>
-</div></blockquote>
-<p><strong>consec_succ</strong> :</p>
-<blockquote>
-<div><p>The parameter defining the probability of a wrong answer.</p>
-</div></blockquote>
-<p><strong>_random_prec</strong> :</p>
-<blockquote>
-<div><p>An internal parameter used for testing purposes.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>(base, strong_gens)</strong> :</p>
-<blockquote class="last">
-<div><p><tt class="docutils literal"><span class="pre">base</span></tt> is the base and <tt class="docutils literal"><span class="pre">strong_gens</span></tt> is the strong generating
-set relative to it.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims" title="sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims"><tt class="xref py py-obj docutils literal"><span class="pre">schreier_sims</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>The algorithm is described in detail in [1], pp. 97-98. It extends
-the orbits <tt class="docutils literal"><span class="pre">orbs</span></tt> and the permutation groups <tt class="docutils literal"><span class="pre">stabs</span></tt> to
-basic orbits and basic stabilizers for the base and strong generating
-set produced in the end.
-The idea of the extension process
-is to &#8220;sift&#8221; random group elements through the stabilizer chain
-and amend the stabilizers/orbits along the way when a sift
-is not successful.
-The helper function <tt class="docutils literal"><span class="pre">_strip</span></tt> is used to attempt
-to decompose a random group element according to the current
-state of the stabilizer chain and report whether the element was
-fully decomposed (successful sift) or not (unsuccessful sift). In
-the latter case, the level at which the sift failed is reported and
-used to amend <tt class="docutils literal"><span class="pre">stabs</span></tt>, <tt class="docutils literal"><span class="pre">base</span></tt>, <tt class="docutils literal"><span class="pre">gens</span></tt> and <tt class="docutils literal"><span class="pre">orbs</span></tt> accordingly.
-The halting condition is for <tt class="docutils literal"><span class="pre">consec_succ</span></tt> consecutive successful
-sifts to pass. This makes sure that the current <tt class="docutils literal"><span class="pre">base</span></tt> and <tt class="docutils literal"><span class="pre">gens</span></tt>
-form a BSGS with probability at least <tt class="docutils literal"><span class="pre">1</span> <span class="pre">-</span> <span class="pre">1/\text{consec\_succ}</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.testutil</span> <span class="kn">import</span> <span class="n">_verify_bsgs</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">SymmetricGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">SymmetricGroup</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">schreier_sims_random</span><span class="p">(</span><span class="n">consec_succ</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_verify_bsgs</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span><span class="p">)</span> 
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.schreier_vector">
-<tt class="descname">schreier_vector</tt><big>(</big><em>self</em>, <em>alpha</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.schreier_vector"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.schreier_vector" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the schreier vector for <tt class="docutils literal"><span class="pre">alpha</span></tt>.</p>
-<p>The Schreier vector efficiently stores information
-about the orbit of <tt class="docutils literal"><span class="pre">alpha</span></tt>. It can later be used to quickly obtain
-elements of the group that send <tt class="docutils literal"><span class="pre">alpha</span></tt> to a particular element
-in the orbit. Notice that the Schreier vector depends on the order
-in which the group generators are listed. For a definition, see [3].
-Since list indices start from zero, we adopt the convention to use
-&#8220;None&#8221; instead of 0 to signify that an element doesn&#8217;t belong
-to the orbit.
-For the algorithm and its correctness, see [2], pp.78-80.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.orbit" title="sympy.combinatorics.perm_groups.PermutationGroup.orbit"><tt class="xref py py-obj docutils literal"><span class="pre">orbit</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">schreier_vector</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">[-1, None, 0, 1, None, 1, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.stabilizer">
-<tt class="descname">stabilizer</tt><big>(</big><em>self</em>, <em>alpha</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.stabilizer"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.stabilizer" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the stabilizer subgroup of <tt class="docutils literal"><span class="pre">alpha</span></tt>.</p>
-<p>The stabilizer of <tt class="docutils literal"><span class="pre">\alpha</span></tt> is the group <tt class="docutils literal"><span class="pre">G_\alpha</span> <span class="pre">=</span>
-<span class="pre">\{g</span> <span class="pre">\in</span> <span class="pre">G</span> <span class="pre">|</span> <span class="pre">g(\alpha)</span> <span class="pre">=</span> <span class="pre">\alpha\}</span></tt>.
-For a proof of correctness, see [1], p.79.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.orbit" title="sympy.combinatorics.perm_groups.PermutationGroup.orbit"><tt class="xref py py-obj docutils literal"><span class="pre">orbit</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">DihedralGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">DihedralGroup</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">stabilizer</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">PermutationGroup([</span>
-<span class="go">    Permutation(5)(0, 4)(1, 3),</span>
-<span class="go">    Permutation(5)])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.strong_gens">
-<tt class="descname">strong_gens</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.strong_gens"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.strong_gens" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a strong generating set from the Schreier-Sims algorithm.</p>
-<p>A generating set <tt class="docutils literal"><span class="pre">S</span> <span class="pre">=</span> <span class="pre">\{g_1,</span> <span class="pre">g_2,</span> <span class="pre">...,</span> <span class="pre">g_t\}</span></tt> for a permutation group
-<tt class="docutils literal"><span class="pre">G</span></tt> is a strong generating set relative to the sequence of points
-(referred to as a &#8220;base&#8221;) <tt class="docutils literal"><span class="pre">(b_1,</span> <span class="pre">b_2,</span> <span class="pre">...,</span> <span class="pre">b_k)</span></tt> if, for
-<tt class="docutils literal"><span class="pre">1</span> <span class="pre">\leq</span> <span class="pre">i</span> <span class="pre">\leq</span> <span class="pre">k</span></tt> we have that the intersection of the pointwise
-stabilizer <tt class="docutils literal"><span class="pre">G^{(i+1)}</span> <span class="pre">:=</span> <span class="pre">G_{b_1,</span> <span class="pre">b_2,</span> <span class="pre">...,</span> <span class="pre">b_i}</span></tt> with <tt class="docutils literal"><span class="pre">S</span></tt> generates
-the pointwise stabilizer <tt class="docutils literal"><span class="pre">G^{(i+1)}</span></tt>. The concepts of a base and
-strong generating set and their applications are discussed in depth
-in [1], pp. 87-89 and [2], pp. 55-57.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.base" title="sympy.combinatorics.perm_groups.PermutationGroup.base"><tt class="xref py py-obj docutils literal"><span class="pre">base</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.basic_transversals" title="sympy.combinatorics.perm_groups.PermutationGroup.basic_transversals"><tt class="xref py py-obj docutils literal"><span class="pre">basic_transversals</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.basic_orbits" title="sympy.combinatorics.perm_groups.PermutationGroup.basic_orbits"><tt class="xref py py-obj docutils literal"><span class="pre">basic_orbits</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.basic_stabilizers" title="sympy.combinatorics.perm_groups.PermutationGroup.basic_stabilizers"><tt class="xref py py-obj docutils literal"><span class="pre">basic_stabilizers</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">DihedralGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">DihedralGroup</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span><span class="o">.</span><span class="n">strong_gens</span>
-<span class="go">[Permutation(0, 1, 2, 3), Permutation(0, 3)(1, 2), Permutation(1, 3)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span><span class="o">.</span><span class="n">base</span>
-<span class="go">[0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.subgroup_search">
-<tt class="descname">subgroup_search</tt><big>(</big><em>self</em>, <em>prop</em>, <em>base=None</em>, <em>strong_gens=None</em>, <em>tests=None</em>, <em>init_subgroup=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.subgroup_search"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.subgroup_search" title="Permalink to this definition">¶</a></dt>
-<dd><p>Find the subgroup of all elements satisfying the property <tt class="docutils literal"><span class="pre">prop</span></tt>.</p>
-<p>This is done by a depth-first search with respect to base images that
-uses several tests to prune the search tree.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>prop</strong> :</p>
-<blockquote>
-<div><p>The property to be used. Has to be callable on group elements
-and always return <tt class="docutils literal"><span class="pre">True</span></tt> or <tt class="docutils literal"><span class="pre">False</span></tt>. It is assumed that
-all group elements satisfying <tt class="docutils literal"><span class="pre">prop</span></tt> indeed form a subgroup.</p>
-</div></blockquote>
-<p><strong>base</strong> :</p>
-<blockquote>
-<div><p>A base for the supergroup.</p>
-</div></blockquote>
-<p><strong>strong_gens</strong> :</p>
-<blockquote>
-<div><p>A strong generating set for the supergroup.</p>
-</div></blockquote>
-<p><strong>tests</strong> :</p>
-<blockquote>
-<div><p>A list of callables of length equal to the length of <tt class="docutils literal"><span class="pre">base</span></tt>.
-These are used to rule out group elements by partial base images,
-so that <tt class="docutils literal"><span class="pre">tests[l](g)</span></tt> returns False if the element <tt class="docutils literal"><span class="pre">g</span></tt> is known
-not to satisfy prop base on where g sends the first <tt class="docutils literal"><span class="pre">l</span> <span class="pre">+</span> <span class="pre">1</span></tt> base
-points.</p>
-</div></blockquote>
-<p><strong>init_subgroup</strong> :</p>
-<blockquote>
-<div><p>if a subgroup of the sought group is
-known in advance, it can be passed to the function as this
-parameter.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>res</strong> :</p>
-<blockquote class="last">
-<div><p>The subgroup of all elements satisfying <tt class="docutils literal"><span class="pre">prop</span></tt>. The generating
-set for this group is guaranteed to be a strong generating set
-relative to the base <tt class="docutils literal"><span class="pre">base</span></tt>.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Notes</p>
-<p>This function is extremely lenghty and complicated and will require
-some careful attention. The implementation is described in
-[1], pp. 114-117, and the comments for the code here follow the lines
-of the pseudocode in the book for clarity.</p>
-<p>The complexity is exponential in general, since the search process by
-itself visits all members of the supergroup. However, there are a lot
-of tests which are used to prune the search tree, and users can define
-their own tests via the <tt class="docutils literal"><span class="pre">tests</span></tt> parameter, so in practice, and for
-some computations, it&#8217;s not terrible.</p>
-<p>A crucial part in the procedure is the frequent base change performed
-(this is line 11 in the pseudocode) in order to obtain a new basic
-stabilizer. The book mentiones that this can be done by using
-<tt class="docutils literal"><span class="pre">.baseswap(...)</span></tt>, however the current imlementation uses a more
-straightforward way to find the next basic stabilizer - calling the
-function <tt class="docutils literal"><span class="pre">.stabilizer(...)</span></tt> on the previous basic stabilizer.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="p">(</span><span class="n">SymmetricGroup</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">AlternatingGroup</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.testutil</span> <span class="kn">import</span> <span class="n">_verify_bsgs</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">SymmetricGroup</span><span class="p">(</span><span class="mi">7</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">prop_even</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">is_even</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">schreier_sims_incremental</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">subgroup_search</span><span class="p">(</span><span class="n">prop_even</span><span class="p">,</span> <span class="n">base</span><span class="o">=</span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span><span class="o">=</span><span class="n">strong_gens</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">is_subgroup</span><span class="p">(</span><span class="n">AlternatingGroup</span><span class="p">(</span><span class="mi">7</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_verify_bsgs</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="n">G</span><span class="o">.</span><span class="n">generators</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.perm_groups.PermutationGroup.transitivity_degree">
-<tt class="descname">transitivity_degree</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/perm_groups.html#PermutationGroup.transitivity_degree"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.perm_groups.PermutationGroup.transitivity_degree" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the degree of transitivity of the group.</p>
-<p>A permutation group <tt class="docutils literal"><span class="pre">G</span></tt> acting on <tt class="docutils literal"><span class="pre">\Omega</span> <span class="pre">=</span> <span class="pre">\{0,</span> <span class="pre">1,</span> <span class="pre">...,</span> <span class="pre">n-1\}</span></tt> is
-<tt class="docutils literal"><span class="pre">k</span></tt>-fold transitive, if, for any k points
-<tt class="docutils literal"><span class="pre">(a_1,</span> <span class="pre">a_2,</span> <span class="pre">...,</span> <span class="pre">a_k)\in\Omega</span></tt> and any k points
-<tt class="docutils literal"><span class="pre">(b_1,</span> <span class="pre">b_2,</span> <span class="pre">...,</span> <span class="pre">b_k)\in\Omega</span></tt> there exists <tt class="docutils literal"><span class="pre">g\in</span> <span class="pre">G</span></tt> such that
-<tt class="docutils literal"><span class="pre">g(a_1)=b_1,</span> <span class="pre">g(a_2)=b_2,</span> <span class="pre">...,</span> <span class="pre">g(a_k)=b_k</span></tt>
-The degree of transitivity of <tt class="docutils literal"><span class="pre">G</span></tt> is the maximum <tt class="docutils literal"><span class="pre">k</span></tt> such that
-<tt class="docutils literal"><span class="pre">G</span></tt> is <tt class="docutils literal"><span class="pre">k</span></tt>-fold transitive. ([8])</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.is_transitive" title="sympy.combinatorics.perm_groups.PermutationGroup.is_transitive"><tt class="xref py py-obj docutils literal"><span class="pre">is_transitive</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.perm_groups.PermutationGroup.orbit" title="sympy.combinatorics.perm_groups.PermutationGroup.orbit"><tt class="xref py py-obj docutils literal"><span class="pre">orbit</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">transitivity_degree</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
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-            
-  <div class="section" id="module-sympy.combinatorics.permutations">
-<span id="permutations"></span><span id="combinatorics-permutations"></span><h1>Permutations<a class="headerlink" href="#module-sympy.combinatorics.permutations" title="Permalink to this headline">¶</a></h1>
-<dl class="class">
-<dt id="sympy.combinatorics.permutations.Permutation">
-<em class="property">class </em><tt class="descclassname">sympy.combinatorics.permutations.</tt><tt class="descname">Permutation</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation" title="Permalink to this definition">¶</a></dt>
-<dd><p>A permutation, alternatively known as an &#8216;arrangement number&#8217; or &#8216;ordering&#8217;
-is an arrangement of the elements of an ordered list into a one-to-one
-mapping with itself. The permutation of a given arrangement is given by
-indicating the positions of the elements after re-arrangment <a class="reference internal" href="#r5">[R5]</a>. For
-example, if one started with elements [x, y, a, b] (in that order) and
-they were reordered as [x, y, b, a] then the permutation would be
-[0, 1, 3, 2]. Notice that (in SymPy) the first element is always referred
-to as 0 and the permutation uses the indices of the elements in the
-original ordering, not the elements (a, b, etc...) themselves.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">False</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Cycle" title="sympy.combinatorics.permutations.Cycle"><tt class="xref py py-obj docutils literal"><span class="pre">Cycle</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r4" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id2">[R4]</a></td><td>Skiena, S. &#8216;Permutations.&#8217; 1.1 in Implementing Discrete Mathematics
-Combinatorics and Graph Theory with Mathematica.  Reading, MA:
-Addison-Wesley, pp. 3-16, 1990.</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r5" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[R5]</td><td><em>(<a class="fn-backref" href="#id1">1</a>, <a class="fn-backref" href="#id3">2</a>)</em> Knuth, D. E. The Art of Computer Programming, Vol. 4: Combinatorial
-Algorithms, 1st ed. Reading, MA: Addison-Wesley, 2011.</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r6" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id4">[R6]</a></td><td>Wendy Myrvold and Frank Ruskey. 2001. Ranking and unranking
-permutations in linear time. Inf. Process. Lett. 79, 6 (September 2001),
-281-284. DOI=10.1016/S0020-0190(01)00141-7</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r7" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id5">[R7]</a></td><td>D. L. Kreher, D. R. Stinson &#8216;Combinatorial Algorithms&#8217;
-CRC Press, 1999</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r8" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id6">[R8]</a></td><td>Graham, R. L.; Knuth, D. E.; and Patashnik, O.
-Concrete Mathematics: A Foundation for Computer Science, 2nd ed.
-Reading, MA: Addison-Wesley, 1994.</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r9" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[R9]</td><td><em>(<a class="fn-backref" href="#id7">1</a>, <a class="fn-backref" href="#id9">2</a>)</em> <a class="reference external" href="http://en.wikipedia.org/wiki/Permutation#Product_and_inverse">http://en.wikipedia.org/wiki/Permutation#Product_and_inverse</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r10" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id8">[R10]</a></td><td><a class="reference external" href="http://en.wikipedia.org/wiki/Lehmer_code">http://en.wikipedia.org/wiki/Lehmer_code</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Permutations Notation</p>
-<p>Permutations are commonly represented in disjoint cycle or array forms.</p>
-<p class="rubric">Array Notation And 2-line Form</p>
-<p>In the 2-line form, the elements and their final positions are shown
-as a matrix with 2 rows:</p>
-<p>[0    1    2     ... n-1]
-[p(0) p(1) p(2)  ... p(n-1)]</p>
-<p>Since the first line is always range(n), where n is the size of p,
-it is sufficient to represent the permutation by the second line,
-referred to as the &#8220;array form&#8221; of the permutation. This is entered
-in brackets as the argument to the Permutation class:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]);</span> <span class="n">p</span>
-<span class="go">Permutation([0, 2, 1])</span>
-</pre></div>
-</div>
-<p>Given i in range(p.size), the permutation maps i to i^p</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">i</span><span class="o">^</span><span class="n">p</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">size</span><span class="p">)]</span>
-<span class="go">[0, 2, 1]</span>
-</pre></div>
-</div>
-<p>The composite of two permutations p*q means first apply p, then q, so
-i^(p*q) = (i^p)^q which is i^p^q according to Python precedence rules:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">i</span><span class="o">^</span><span class="n">p</span><span class="o">^</span><span class="n">q</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">)]</span>
-<span class="go">[2, 0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">i</span><span class="o">^</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">q</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">)]</span>
-<span class="go">[2, 0, 1]</span>
-</pre></div>
-</div>
-<p>One can use also the notation p(i) = i^p, but then the composition
-rule is (p*q)(i) = q(p(i)), not p(q(i)):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">[(</span><span class="n">p</span><span class="o">*</span><span class="n">q</span><span class="p">)(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">size</span><span class="p">)]</span>
-<span class="go">[2, 0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">q</span><span class="p">(</span><span class="n">p</span><span class="p">(</span><span class="n">i</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">size</span><span class="p">)]</span>
-<span class="go">[2, 0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">p</span><span class="p">(</span><span class="n">q</span><span class="p">(</span><span class="n">i</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">size</span><span class="p">)]</span>
-<span class="go">[1, 2, 0]</span>
-</pre></div>
-</div>
-<p class="rubric">Disjoint Cycle Notation</p>
-<p>In disjoint cycle notation, only the elements that have shifted are
-indicated. In the above case, the 2 and 1 switched places. This can
-be entered in two ways:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="o">==</span> <span class="n">Permutation</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span> <span class="o">==</span> <span class="n">p</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>Only the relative ordering of elements in a cycle matter:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span> <span class="o">==</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>The disjoint cycle notation is convenient when representing permutations
-that have several cycles in them:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span> <span class="o">==</span> <span class="n">Permutation</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">]])</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>It also provides some economy in entry when computing products of
-permutations that are written in disjoint cycle notation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">Permutation([0, 3, 2, 1])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span> <span class="o">==</span> <span class="n">Permutation</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span><span class="o">*</span><span class="n">Permutation</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span><span class="o">*</span><span class="n">Permutation</span><span class="p">([[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>Entering a singleton in a permutation is a way to indicate the size of the
-permutation. The <tt class="docutils literal"><span class="pre">size</span></tt> keyword can also be used.</p>
-<p>Array-form entry:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">9</span><span class="p">]])</span>
-<span class="go">Permutation([0, 2, 1], size=10)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]],</span> <span class="n">size</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">Permutation([0, 2, 1], size=10)</span>
-</pre></div>
-</div>
-<p>Cyclic-form entry:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">Permutation([0, 2, 1], size=10)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">(</span><span class="mi">9</span><span class="p">)(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">Permutation([0, 2, 1], size=10)</span>
-</pre></div>
-</div>
-<p>Caution: no singleton containing an element larger than the largest
-in any previous cycle can be entered. This is an important difference
-in how Permutation and Cycle handle the __call__ syntax. A singleton
-argument at the start of a Permutation performs instantiation of the
-Permutation and is permitted:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">Permutation([], size=6)</span>
-</pre></div>
-</div>
-<p>A singleton entered after instantiation is a call to the permutation
-&#8211; a function call &#8211; and if the argument is out of range it will
-trigger an error. For this reason, it is better to start the cycle
-with the singleton:</p>
-<p>The following fails because there is is no element 3:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">IndexError</span>: <span class="n">list index out of range</span>
-</pre></div>
-</div>
-<p>This is ok: only the call to an out of range singleton is prohibited;
-otherwise the permutation autosizes:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">(</span><span class="mi">3</span><span class="p">)(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">Permutation([0, 2, 1, 3])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="o">==</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p class="rubric">Equality Testing</p>
-<p>The array forms must be the same in order for permutations to be equal:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span> <span class="o">==</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p class="rubric">Identity Permutation</p>
-<p>The identity permutation is a permutation in which no element is out of
-place. It can be entered in a variety of ways. All the following create
-an identity permutation of size 4:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">all</span><span class="p">(</span><span class="n">p</span> <span class="o">==</span> <span class="n">I</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="p">[</span>
-<span class="gp">... </span><span class="n">Permutation</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span>
-<span class="gp">... </span><span class="n">Permutation</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">))),</span>
-<span class="gp">... </span><span class="n">Permutation</span><span class="p">([],</span> <span class="n">size</span><span class="o">=</span><span class="mi">4</span><span class="p">),</span>
-<span class="gp">... </span><span class="n">Permutation</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="mi">4</span><span class="p">)])</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>Watch out for entering the range <em>inside</em> a set of brackets (which is
-cycle notation):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">==</span> <span class="n">Permutation</span><span class="p">([</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">))])</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p class="rubric">Permutation Printing</p>
-<p>There are a few things to note about how Permutations are printed.</p>
-<p>1) If you prefer one form (array or cycle) over another, you can set that
-with the print_cyclic flag.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">Permutation([0, 2, 1, 4, 5, 3])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">_</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span>
-<span class="go">Permutation(1, 2)(3, 4, 5)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">False</span>
-</pre></div>
-</div>
-<p>2) Regardless of the setting, a list of elements in the array for cyclic
-form can be obtained and either of those can be copied and supplied as
-the argument to Permutation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">array_form</span>
-<span class="go">[0, 2, 1, 4, 5, 3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">cyclic_form</span>
-<span class="go">[[1, 2], [3, 4, 5]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">(</span><span class="n">_</span><span class="p">)</span> <span class="o">==</span> <span class="n">p</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>3) Printing is economical in that as little as possible is printed while
-retaining all information about the size of the permutation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="go">Permutation([1, 0, 2, 3])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="n">size</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
-<span class="go">Permutation([1, 0], size=20)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="n">size</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
-<span class="go">Permutation([1, 0, 2, 4, 3], size=20)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span>
-<span class="go">Permutation(3)(0, 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">False</span>
-</pre></div>
-</div>
-<p>The 2 was not printed but it is still there as can be seen with the
-array_form and size methods:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">array_form</span>
-<span class="go">[1, 0, 2, 3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">size</span>
-<span class="go">4</span>
-</pre></div>
-</div>
-<p class="rubric">Short Introduction To Other Methods</p>
-<p>The permutation can act as a bijective function, telling what element is
-located at a given position</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">5</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="o">.</span><span class="n">array_form</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="c"># the hard way</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="c"># the easy way</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">dict</span><span class="p">([(</span><span class="n">i</span><span class="p">,</span> <span class="n">q</span><span class="p">(</span><span class="n">i</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">size</span><span class="p">)])</span> <span class="c"># showing the bijection</span>
-<span class="go">{0: 5, 1: 2, 2: 3, 3: 4, 4: 1, 5: 0}</span>
-</pre></div>
-</div>
-<p>The full cyclic form (including singletons) can be obtained:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">full_cyclic_form</span>
-<span class="go">[[0, 1], [2], [3]]</span>
-</pre></div>
-</div>
-<p>Any permutation can be factored into transpositions of pairs of elements:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">]])</span><span class="o">.</span><span class="n">transpositions</span><span class="p">()</span>
-<span class="go">[(1, 2), (3, 5), (3, 4)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">rmul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">Permutation</span><span class="p">([</span><span class="n">ti</span><span class="p">],</span> <span class="n">size</span><span class="o">=</span><span class="mi">6</span><span class="p">)</span> <span class="k">for</span> <span class="n">ti</span> <span class="ow">in</span> <span class="n">_</span><span class="p">])</span><span class="o">.</span><span class="n">cyclic_form</span>
-<span class="go">[[1, 2], [3, 4, 5]]</span>
-</pre></div>
-</div>
-<p>The number of permutations on a set of n elements is given by n! and is
-called the cardinality.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">size</span>
-<span class="go">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">cardinality</span>
-<span class="go">24</span>
-</pre></div>
-</div>
-<p>A given permutation has a rank among all the possible permutations of the
-same elements, but what that rank is depends on how the permutations are
-enumerated. (There are a number of different methods of doing so.) The
-lexicographic rank is given by the rank method and this rank is used to
-increment a partion with addition/subtraction:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">rank</span><span class="p">()</span>
-<span class="go">6</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">+</span> <span class="mi">1</span>
-<span class="go">Permutation([1, 0, 3, 2])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">next_lex</span><span class="p">()</span>
-<span class="go">Permutation([1, 0, 3, 2])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">.</span><span class="n">rank</span><span class="p">()</span>
-<span class="go">7</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">unrank_lex</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">size</span><span class="p">,</span> <span class="n">rank</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
-<span class="go">Permutation([1, 0, 3, 2])</span>
-</pre></div>
-</div>
-<p>The product of two permutations p and q is defined as their composition as
-functions, (p*q)(i) = q(p(i)) <a class="reference internal" href="#r9">[R9]</a>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">q</span><span class="o">*</span><span class="n">p</span><span class="p">)</span>
-<span class="go">[2, 3, 0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">q</span><span class="p">)</span>
-<span class="go">[3, 2, 1, 0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">q</span><span class="p">(</span><span class="n">p</span><span class="p">(</span><span class="n">i</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">size</span><span class="p">)]</span>
-<span class="go">[3, 2, 1, 0]</span>
-</pre></div>
-</div>
-<p>The permutation can be &#8216;applied&#8217; to any list-like object, not only
-Permutations:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">([</span><span class="s">&#39;zero&#39;</span><span class="p">,</span> <span class="s">&#39;one&#39;</span><span class="p">,</span> <span class="s">&#39;four&#39;</span><span class="p">,</span> <span class="s">&#39;two&#39;</span><span class="p">])</span>
-<span class="go"> [&#39;one&#39;, &#39;zero&#39;, &#39;four&#39;, &#39;two&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">(</span><span class="s">&#39;zo42&#39;</span><span class="p">)</span>
-<span class="go">[&#39;o&#39;, &#39;z&#39;, &#39;4&#39;, &#39;2&#39;]</span>
-</pre></div>
-</div>
-<p>If you have a list of arbitrary elements, the corresponding permutation
-can be found with the from_sequence method:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">from_sequence</span><span class="p">(</span><span class="s">&#39;SymPy&#39;</span><span class="p">)</span>
-<span class="go">Permutation([1, 3, 2, 0, 4])</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.combinatorics.permutations.Permutation.array_form">
-<tt class="descname">array_form</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.array_form"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.array_form" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a copy of the attribute _array_form
-Examples
-========</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([[</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">array_form</span>
-<span class="go">[2, 3, 0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([[</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">]])</span><span class="o">.</span><span class="n">array_form</span>
-<span class="go">[3, 2, 0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">array_form</span>
-<span class="go">[2, 0, 3, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">]])</span><span class="o">.</span><span class="n">array_form</span>
-<span class="go">[0, 2, 1, 3, 5, 4]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.ascents">
-<tt class="descname">ascents</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.ascents"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.ascents" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the positions of ascents in a permutation, ie, the location
-where p[i] &lt; p[i+1]</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.descents" title="sympy.combinatorics.permutations.Permutation.descents"><tt class="xref py py-obj docutils literal"><span class="pre">descents</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.inversions" title="sympy.combinatorics.permutations.Permutation.inversions"><tt class="xref py py-obj docutils literal"><span class="pre">inversions</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.min" title="sympy.combinatorics.permutations.Permutation.min"><tt class="xref py py-obj docutils literal"><span class="pre">min</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.max" title="sympy.combinatorics.permutations.Permutation.max"><tt class="xref py py-obj docutils literal"><span class="pre">max</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">ascents</span><span class="p">()</span>
-<span class="go">[1, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.atoms">
-<tt class="descname">atoms</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.atoms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.atoms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns all the elements of a permutation</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span><span class="o">.</span><span class="n">atoms</span><span class="p">()</span>
-<span class="go">set([0, 1, 2, 3, 4, 5])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">]])</span><span class="o">.</span><span class="n">atoms</span><span class="p">()</span>
-<span class="go">set([0, 1, 2, 3, 4, 5])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.permutations.Permutation.cardinality">
-<tt class="descname">cardinality</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.cardinality"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.cardinality" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the number of all possible permutations.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.length" title="sympy.combinatorics.permutations.Permutation.length"><tt class="xref py py-obj docutils literal"><span class="pre">length</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.order" title="sympy.combinatorics.permutations.Permutation.order"><tt class="xref py py-obj docutils literal"><span class="pre">order</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.rank" title="sympy.combinatorics.permutations.Permutation.rank"><tt class="xref py py-obj docutils literal"><span class="pre">rank</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.size" title="sympy.combinatorics.permutations.Permutation.size"><tt class="xref py py-obj docutils literal"><span class="pre">size</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">cardinality</span>
-<span class="go">24</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.commutator">
-<tt class="descname">commutator</tt><big>(</big><em>self</em>, <em>x</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.commutator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.commutator" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the commutator of self and x: <tt class="docutils literal"><span class="pre">~x*~self*x*self</span></tt></p>
-<p>If f and g are part of a group, G, then the commutator of f and g
-is the group identity iff f and g commute, i.e. fg == gf.</p>
-<p class="rubric">References</p>
-<p><a class="reference external" href="http://en.wikipedia.org/wiki/Commutator">http://en.wikipedia.org/wiki/Commutator</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">commutator</span><span class="p">(</span><span class="n">x</span><span class="p">);</span> <span class="n">c</span>
-<span class="go">Permutation([2, 1, 3, 0])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">==</span> <span class="o">~</span><span class="n">x</span><span class="o">*~</span><span class="n">p</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">p</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="p">[</span><span class="n">I</span> <span class="o">+</span> <span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">6</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="p">)):</span>
-<span class="gp">... </span>    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="p">)):</span>
-<span class="gp">... </span>        <span class="n">c</span> <span class="o">=</span> <span class="n">p</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">commutator</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-<span class="gp">... </span>        <span class="k">if</span> <span class="n">p</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="n">p</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-<span class="gp">... </span>            <span class="k">assert</span> <span class="n">c</span> <span class="o">==</span> <span class="n">I</span>
-<span class="gp">... </span>        <span class="k">else</span><span class="p">:</span>
-<span class="gp">... </span>            <span class="k">assert</span> <span class="n">c</span> <span class="o">!=</span> <span class="n">I</span>
-<span class="gp">...</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.commutes_with">
-<tt class="descname">commutes_with</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.commutes_with"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.commutes_with" title="Permalink to this definition">¶</a></dt>
-<dd><p>Checks if the elements are commuting.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">commutes_with</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">commutes_with</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.permutations.Permutation.cycle_structure">
-<tt class="descname">cycle_structure</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.cycle_structure"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.cycle_structure" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the cycle structure of the permutation as a dictionary
-indicating the multiplicity of each cycle length.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">cycle_structure</span>
-<span class="go">{1: 4}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span><span class="o">.</span><span class="n">cycle_structure</span>
-<span class="go">{2: 2, 3: 1}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.permutations.Permutation.cycles">
-<tt class="descname">cycles</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.cycles"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.cycles" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the number of cycles contained in the permutation
-(including singletons).</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span><span class="o">.</span><span class="n">cycles</span>
-<span class="go">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span><span class="o">.</span><span class="n">full_cyclic_form</span>
-<span class="go">[[0], [1], [2]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">cycles</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.permutations.Permutation.cyclic_form">
-<tt class="descname">cyclic_form</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.cyclic_form"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.cyclic_form" title="Permalink to this definition">¶</a></dt>
-<dd><p>This is used to convert to the cyclic notation
-from the canonical notation. Singletons are omitted.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.array_form" title="sympy.combinatorics.permutations.Permutation.array_form"><tt class="xref py py-obj docutils literal"><span class="pre">array_form</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.full_cyclic_form" title="sympy.combinatorics.permutations.Permutation.full_cyclic_form"><tt class="xref py py-obj docutils literal"><span class="pre">full_cyclic_form</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">cyclic_form</span>
-<span class="go">[[1, 3, 2]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span><span class="o">.</span><span class="n">cyclic_form</span>
-<span class="go">[[0, 1], [3, 4]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.descents">
-<tt class="descname">descents</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.descents"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.descents" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the positions of descents in a permutation, ie, the location
-where p[i] &gt; p[i+1]</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.ascents" title="sympy.combinatorics.permutations.Permutation.ascents"><tt class="xref py py-obj docutils literal"><span class="pre">ascents</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.inversions" title="sympy.combinatorics.permutations.Permutation.inversions"><tt class="xref py py-obj docutils literal"><span class="pre">inversions</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.min" title="sympy.combinatorics.permutations.Permutation.min"><tt class="xref py py-obj docutils literal"><span class="pre">min</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.max" title="sympy.combinatorics.permutations.Permutation.max"><tt class="xref py py-obj docutils literal"><span class="pre">max</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">descents</span><span class="p">()</span>
-<span class="go">[0, 3]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.combinatorics.permutations.Permutation.from_inversion_vector">
-<em class="property">classmethod </em><tt class="descname">from_inversion_vector</tt><big>(</big><em>inversion</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.from_inversion_vector"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.from_inversion_vector" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculates the permutation from the inversion vector.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">from_inversion_vector</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="go">Permutation([3, 2, 1, 0, 4, 5])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.combinatorics.permutations.Permutation.from_sequence">
-<em class="property">classmethod </em><tt class="descname">from_sequence</tt><big>(</big><em>i</em>, <em>key=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.from_sequence"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.from_sequence" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the permutation needed to obtain <tt class="docutils literal"><span class="pre">i</span></tt> from the sorted
-elements of <tt class="docutils literal"><span class="pre">i</span></tt>. If custom sorting is desired, a key can be given.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">from_sequence</span><span class="p">(</span><span class="s">&#39;SymPy&#39;</span><span class="p">)</span>
-<span class="go">Permutation(4)(0, 1, 3)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="s">&quot;SymPy&quot;</span><span class="p">))</span>
-<span class="go">[&#39;S&#39;, &#39;y&#39;, &#39;m&#39;, &#39;P&#39;, &#39;y&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">from_sequence</span><span class="p">(</span><span class="s">&#39;SymPy&#39;</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">lower</span><span class="p">())</span>
-<span class="go">Permutation(4)(0, 2)(1, 3)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.permutations.Permutation.full_cyclic_form">
-<tt class="descname">full_cyclic_form</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.full_cyclic_form"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.full_cyclic_form" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return permutation in cyclic form including singletons.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">full_cyclic_form</span>
-<span class="go">[[0], [1, 2]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.get_adjacency_distance">
-<tt class="descname">get_adjacency_distance</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.get_adjacency_distance"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.get_adjacency_distance" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the adjacency distance between two permutations.</p>
-<p>This metric counts the number of times a pair i,j of jobs is
-adjacent in both p and p&#8217;. If n_adj is this quantity then
-the adjacency distance is n - n_adj - 1 [1]</p>
-<p>[1] Reeves, Colin R. Landscapes, Operators and Heuristic search, Annals
-of Operational Research, 86, pp 473-490. (1999)</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.get_precedence_matrix" title="sympy.combinatorics.permutations.Permutation.get_precedence_matrix"><tt class="xref py py-obj docutils literal"><span class="pre">get_precedence_matrix</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.get_precedence_distance" title="sympy.combinatorics.permutations.Permutation.get_precedence_distance"><tt class="xref py py-obj docutils literal"><span class="pre">get_precedence_distance</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.get_adjacency_matrix" title="sympy.combinatorics.permutations.Permutation.get_adjacency_matrix"><tt class="xref py py-obj docutils literal"><span class="pre">get_adjacency_matrix</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Permutation</span><span class="o">.</span><span class="n">josephus</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">get_adjacency_distance</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-<span class="go">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">get_adjacency_distance</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-<span class="go">4</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.get_adjacency_matrix">
-<tt class="descname">get_adjacency_matrix</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.get_adjacency_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.get_adjacency_matrix" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the adjacency matrix of a permutation.</p>
-<p>If job i is adjacent to job j in a permutation p
-then we set m[i, j] = 1 where m is the adjacency
-matrix of p.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.get_precedence_matrix" title="sympy.combinatorics.permutations.Permutation.get_precedence_matrix"><tt class="xref py py-obj docutils literal"><span class="pre">get_precedence_matrix</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.get_precedence_distance" title="sympy.combinatorics.permutations.Permutation.get_precedence_distance"><tt class="xref py py-obj docutils literal"><span class="pre">get_precedence_distance</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.get_adjacency_distance" title="sympy.combinatorics.permutations.Permutation.get_adjacency_distance"><tt class="xref py py-obj docutils literal"><span class="pre">get_adjacency_distance</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="o">.</span><span class="n">josephus</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">get_adjacency_matrix</span><span class="p">()</span>
-<span class="go">[0, 0, 0, 0, 0, 0]</span>
-<span class="go">[0, 0, 0, 0, 1, 0]</span>
-<span class="go">[0, 0, 0, 0, 0, 1]</span>
-<span class="go">[0, 1, 0, 0, 0, 0]</span>
-<span class="go">[1, 0, 0, 0, 0, 0]</span>
-<span class="go">[0, 0, 0, 1, 0, 0]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="o">.</span><span class="n">get_adjacency_matrix</span><span class="p">()</span>
-<span class="go">[0, 1, 0, 0]</span>
-<span class="go">[0, 0, 1, 0]</span>
-<span class="go">[0, 0, 0, 1]</span>
-<span class="go">[0, 0, 0, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.get_positional_distance">
-<tt class="descname">get_positional_distance</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.get_positional_distance"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.get_positional_distance" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the positional distance between two permutations.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.get_precedence_distance" title="sympy.combinatorics.permutations.Permutation.get_precedence_distance"><tt class="xref py py-obj docutils literal"><span class="pre">get_precedence_distance</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.get_adjacency_distance" title="sympy.combinatorics.permutations.Permutation.get_adjacency_distance"><tt class="xref py py-obj docutils literal"><span class="pre">get_adjacency_distance</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Permutation</span><span class="o">.</span><span class="n">josephus</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">get_positional_distance</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-<span class="go">12</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">get_positional_distance</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-<span class="go">12</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.get_precedence_distance">
-<tt class="descname">get_precedence_distance</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.get_precedence_distance"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.get_precedence_distance" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the precedence distance between two permutations.</p>
-<p>Suppose p and p&#8217; represent n jobs. The precedence metric
-counts the number of times a job j is prededed by job i
-in both p and p&#8217;. This metric is commutative.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.get_precedence_matrix" title="sympy.combinatorics.permutations.Permutation.get_precedence_matrix"><tt class="xref py py-obj docutils literal"><span class="pre">get_precedence_matrix</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.get_adjacency_matrix" title="sympy.combinatorics.permutations.Permutation.get_adjacency_matrix"><tt class="xref py py-obj docutils literal"><span class="pre">get_adjacency_matrix</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.get_adjacency_distance" title="sympy.combinatorics.permutations.Permutation.get_adjacency_distance"><tt class="xref py py-obj docutils literal"><span class="pre">get_adjacency_distance</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">get_precedence_distance</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-<span class="go">7</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="o">.</span><span class="n">get_precedence_distance</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-<span class="go">7</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.get_precedence_matrix">
-<tt class="descname">get_precedence_matrix</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.get_precedence_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.get_precedence_matrix" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gets the precedence matrix. This is used for computing the
-distance between two permutations.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.get_precedence_distance" title="sympy.combinatorics.permutations.Permutation.get_precedence_distance"><tt class="xref py py-obj docutils literal"><span class="pre">get_precedence_distance</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.get_adjacency_matrix" title="sympy.combinatorics.permutations.Permutation.get_adjacency_matrix"><tt class="xref py py-obj docutils literal"><span class="pre">get_adjacency_matrix</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.get_adjacency_distance" title="sympy.combinatorics.permutations.Permutation.get_adjacency_distance"><tt class="xref py py-obj docutils literal"><span class="pre">get_adjacency_distance</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="o">.</span><span class="n">josephus</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span>
-<span class="go">Permutation([2, 5, 3, 1, 4, 0])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">get_precedence_matrix</span><span class="p">()</span>
-<span class="go">[0, 0, 0, 0, 0, 0]</span>
-<span class="go">[1, 0, 0, 0, 1, 0]</span>
-<span class="go">[1, 1, 0, 1, 1, 1]</span>
-<span class="go">[1, 1, 0, 0, 1, 0]</span>
-<span class="go">[1, 0, 0, 0, 0, 0]</span>
-<span class="go">[1, 1, 0, 1, 1, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.index">
-<tt class="descname">index</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.index"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.index" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the index of a permutation.</p>
-<p>The index of a permutation is the sum of all subscripts j such
-that p[j] is greater than p[j+1].</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">index</span><span class="p">()</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.inversion_vector">
-<tt class="descname">inversion_vector</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.inversion_vector"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.inversion_vector" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the inversion vector of the permutation.</p>
-<p>The inversion vector consists of elements whose value
-indicates the number of elements in the permutation
-that are lesser than it and lie on its right hand side.</p>
-<p>The inversion vector is the same as the Lehmer encoding of a
-permutation.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.from_inversion_vector" title="sympy.combinatorics.permutations.Permutation.from_inversion_vector"><tt class="xref py py-obj docutils literal"><span class="pre">from_inversion_vector</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">inversion_vector</span><span class="p">()</span>
-<span class="go">[4, 7, 0, 5, 0, 2, 1, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">inversion_vector</span><span class="p">()</span>
-<span class="go">[3, 2, 1]</span>
-</pre></div>
-</div>
-<p>The inversion vector increases lexicographically with the rank
-of the permutation, the -ith element cycling through 0..i.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">while</span> <span class="n">p</span><span class="p">:</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">inversion_vector</span><span class="p">(),</span> <span class="n">p</span><span class="o">.</span><span class="n">rank</span><span class="p">())</span>
-<span class="gp">... </span>    <span class="n">p</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">next_lex</span><span class="p">()</span>
-<span class="gp">...</span>
-<span class="go">Permutation([0, 1, 2]) [0, 0] 0</span>
-<span class="go">Permutation([0, 2, 1]) [0, 1] 1</span>
-<span class="go">Permutation([1, 0, 2]) [1, 0] 2</span>
-<span class="go">Permutation([1, 2, 0]) [1, 1] 3</span>
-<span class="go">Permutation([2, 0, 1]) [2, 0] 4</span>
-<span class="go">Permutation([2, 1, 0]) [2, 1] 5</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.inversions">
-<tt class="descname">inversions</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.inversions"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.inversions" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the number of inversions of a permutation.</p>
-<p>An inversion is where i &gt; j but p[i] &lt; p[j].</p>
-<p>For small length of p, it iterates over all i and j
-values and calculates the number of inversions.
-For large length of p, it uses a variation of merge
-sort to calculate the number of inversions.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.descents" title="sympy.combinatorics.permutations.Permutation.descents"><tt class="xref py py-obj docutils literal"><span class="pre">descents</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.ascents" title="sympy.combinatorics.permutations.Permutation.ascents"><tt class="xref py py-obj docutils literal"><span class="pre">ascents</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.min" title="sympy.combinatorics.permutations.Permutation.min"><tt class="xref py py-obj docutils literal"><span class="pre">min</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.max" title="sympy.combinatorics.permutations.Permutation.max"><tt class="xref py py-obj docutils literal"><span class="pre">max</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://www.cp.eng.chula.ac.th/~piak/teaching/algo/algo2008/count-inv.htm">http://www.cp.eng.chula.ac.th/~piak/teaching/algo/algo2008/count-inv.htm</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">inversions</span><span class="p">()</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">inversions</span><span class="p">()</span>
-<span class="go">6</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.permutations.Permutation.is_Empty">
-<tt class="descname">is_Empty</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.is_Empty"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.is_Empty" title="Permalink to this definition">¶</a></dt>
-<dd><p>Checks to see if the permutation is a set with zero elements</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.is_Singleton" title="sympy.combinatorics.permutations.Permutation.is_Singleton"><tt class="xref py py-obj docutils literal"><span class="pre">is_Singleton</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([])</span><span class="o">.</span><span class="n">is_Empty</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">is_Empty</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.permutations.Permutation.is_Identity">
-<tt class="descname">is_Identity</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.is_Identity"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.is_Identity" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if the Permutation is an identity permutation.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.order" title="sympy.combinatorics.permutations.Permutation.order"><tt class="xref py py-obj docutils literal"><span class="pre">order</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">is_Identity</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([[</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">is_Identity</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">is_Identity</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">is_Identity</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.permutations.Permutation.is_Singleton">
-<tt class="descname">is_Singleton</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.is_Singleton"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.is_Singleton" title="Permalink to this definition">¶</a></dt>
-<dd><p>Checks to see if the permutation contains only one number and is
-thus the only possible permutation of this set of numbers</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.is_Empty" title="sympy.combinatorics.permutations.Permutation.is_Empty"><tt class="xref py py-obj docutils literal"><span class="pre">is_Empty</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">is_Singleton</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">is_Singleton</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.permutations.Permutation.is_even">
-<tt class="descname">is_even</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.is_even"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.is_even" title="Permalink to this definition">¶</a></dt>
-<dd><p>Checks if a permutation is even.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.is_odd" title="sympy.combinatorics.permutations.Permutation.is_odd"><tt class="xref py py-obj docutils literal"><span class="pre">is_odd</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">is_even</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">is_even</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.permutations.Permutation.is_odd">
-<tt class="descname">is_odd</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.is_odd"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.is_odd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Checks if a permutation is odd.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.is_even" title="sympy.combinatorics.permutations.Permutation.is_even"><tt class="xref py py-obj docutils literal"><span class="pre">is_even</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">is_odd</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">is_odd</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.combinatorics.permutations.Permutation.josephus">
-<em class="property">classmethod </em><tt class="descname">josephus</tt><big>(</big><em>m</em>, <em>n</em>, <em>s=1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.josephus"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.josephus" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return as a permutation the shuffling of range(n) using the Josephus
-scheme in which every m-th item is selected until all have been chosen.
-The returned permutation has elements listed by the order in which they
-were selected.</p>
-<p>The parameter <tt class="docutils literal"><span class="pre">s</span></tt> stops the selection process when there are <tt class="docutils literal"><span class="pre">s</span></tt>
-items remaining and these are selected by countinuing the selection,
-counting by 1 rather than by <tt class="docutils literal"><span class="pre">m</span></tt>.</p>
-<p>Consider selecting every 3rd item from 6 until only 2 remain:</p>
-<div class="highlight-python"><pre>choices    chosen
-========   ======
-  012345
-  01 345   2
-  01 34    25
-  01  4    253
-  0   4    2531
-  0        25314
-           253140</pre>
-</div>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Flavius_Josephus">http://en.wikipedia.org/wiki/Flavius_Josephus</a></li>
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Josephus_problem">http://en.wikipedia.org/wiki/Josephus_problem</a></li>
-<li><a class="reference external" href="http://www.wou.edu/~burtonl/josephus.html">http://www.wou.edu/~burtonl/josephus.html</a></li>
-</ol>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">josephus</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">array_form</span>
-<span class="go">[2, 5, 3, 1, 4, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.length">
-<tt class="descname">length</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.length"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.length" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the number of integers moved by a permutation.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.min" title="sympy.combinatorics.permutations.Permutation.min"><tt class="xref py py-obj docutils literal"><span class="pre">min</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.max" title="sympy.combinatorics.permutations.Permutation.max"><tt class="xref py py-obj docutils literal"><span class="pre">max</span></tt></a>, <tt class="xref py py-obj docutils literal"><span class="pre">suppport</span></tt>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.cardinality" title="sympy.combinatorics.permutations.Permutation.cardinality"><tt class="xref py py-obj docutils literal"><span class="pre">cardinality</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.order" title="sympy.combinatorics.permutations.Permutation.order"><tt class="xref py py-obj docutils literal"><span class="pre">order</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.rank" title="sympy.combinatorics.permutations.Permutation.rank"><tt class="xref py py-obj docutils literal"><span class="pre">rank</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.size" title="sympy.combinatorics.permutations.Permutation.size"><tt class="xref py py-obj docutils literal"><span class="pre">size</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">length</span><span class="p">()</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span><span class="o">.</span><span class="n">length</span><span class="p">()</span>
-<span class="go">4</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.list">
-<tt class="descname">list</tt><big>(</big><em>self</em>, <em>size=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the permutation as an explicit list, possibly
-trimming unmoved elements if size is less than the maximum
-element in the permutation; if this is desired, setting
-<tt class="docutils literal"><span class="pre">size=-1</span></tt> will guarantee such trimming.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">list</span><span class="p">()</span>
-<span class="go">[0, 1, 3, 2, 5, 4]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">list</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">[0, 1, 3, 2, 5, 4, 6, 7, 8, 9]</span>
-</pre></div>
-</div>
-<p>Passing a length too small will trim trailing, unchanged elements
-in the permutation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">list</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">[0, 2, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">list</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">[]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.max">
-<tt class="descname">max</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.max"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.max" title="Permalink to this definition">¶</a></dt>
-<dd><p>The maximum element moved by the permutation.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.min" title="sympy.combinatorics.permutations.Permutation.min"><tt class="xref py py-obj docutils literal"><span class="pre">min</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.descents" title="sympy.combinatorics.permutations.Permutation.descents"><tt class="xref py py-obj docutils literal"><span class="pre">descents</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.ascents" title="sympy.combinatorics.permutations.Permutation.ascents"><tt class="xref py py-obj docutils literal"><span class="pre">ascents</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.inversions" title="sympy.combinatorics.permutations.Permutation.inversions"><tt class="xref py py-obj docutils literal"><span class="pre">inversions</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">max</span><span class="p">()</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.min">
-<tt class="descname">min</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.min"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.min" title="Permalink to this definition">¶</a></dt>
-<dd><p>The minimum element moved by the permutation.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.max" title="sympy.combinatorics.permutations.Permutation.max"><tt class="xref py py-obj docutils literal"><span class="pre">max</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.descents" title="sympy.combinatorics.permutations.Permutation.descents"><tt class="xref py py-obj docutils literal"><span class="pre">descents</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.ascents" title="sympy.combinatorics.permutations.Permutation.ascents"><tt class="xref py py-obj docutils literal"><span class="pre">ascents</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.inversions" title="sympy.combinatorics.permutations.Permutation.inversions"><tt class="xref py py-obj docutils literal"><span class="pre">inversions</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">min</span><span class="p">()</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.mul_inv">
-<tt class="descname">mul_inv</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.mul_inv"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.mul_inv" title="Permalink to this definition">¶</a></dt>
-<dd><p>other*~self, self and other have _array_form</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.next_lex">
-<tt class="descname">next_lex</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.next_lex"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.next_lex" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the next permutation in lexicographical order.
-If self is the last permutation in lexicographical order
-it returns None.
-See [4] section 2.4.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.rank" title="sympy.combinatorics.permutations.Permutation.rank"><tt class="xref py py-obj docutils literal"><span class="pre">rank</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.unrank_lex" title="sympy.combinatorics.permutations.Permutation.unrank_lex"><tt class="xref py py-obj docutils literal"><span class="pre">unrank_lex</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]);</span> <span class="n">p</span><span class="o">.</span><span class="n">rank</span><span class="p">()</span>
-<span class="go">17</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">next_lex</span><span class="p">();</span> <span class="n">p</span><span class="o">.</span><span class="n">rank</span><span class="p">()</span>
-<span class="go">18</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.next_nonlex">
-<tt class="descname">next_nonlex</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.next_nonlex"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.next_nonlex" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the next permutation in nonlex order [3].
-If self is the last permutation in this order it returns None.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.rank_nonlex" title="sympy.combinatorics.permutations.Permutation.rank_nonlex"><tt class="xref py py-obj docutils literal"><span class="pre">rank_nonlex</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.unrank_nonlex" title="sympy.combinatorics.permutations.Permutation.unrank_nonlex"><tt class="xref py py-obj docutils literal"><span class="pre">unrank_nonlex</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">]);</span> <span class="n">p</span><span class="o">.</span><span class="n">rank_nonlex</span><span class="p">()</span>
-<span class="go">5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">next_nonlex</span><span class="p">();</span> <span class="n">p</span>
-<span class="go">Permutation([3, 0, 1, 2])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">rank_nonlex</span><span class="p">()</span>
-<span class="go">6</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.next_trotterjohnson">
-<tt class="descname">next_trotterjohnson</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.next_trotterjohnson"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.next_trotterjohnson" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the next permutation in Trotter-Johnson order.
-If self is the last permutation it returns None.
-See [4] section 2.4.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.rank_trotterjohnson" title="sympy.combinatorics.permutations.Permutation.rank_trotterjohnson"><tt class="xref py py-obj docutils literal"><span class="pre">rank_trotterjohnson</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.unrank_trotterjohnson" title="sympy.combinatorics.permutations.Permutation.unrank_trotterjohnson"><tt class="xref py py-obj docutils literal"><span class="pre">unrank_trotterjohnson</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">rank_trotterjohnson</span><span class="p">()</span>
-<span class="go">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">next_trotterjohnson</span><span class="p">();</span> <span class="n">p</span>
-<span class="go">Permutation([0, 3, 2, 1])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">rank_trotterjohnson</span><span class="p">()</span>
-<span class="go">5</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.order">
-<tt class="descname">order</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.order"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.order" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the order of a permutation.</p>
-<p>When the permutation is raised to the power of its
-order it equals the identity permutation.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">identity</span></tt>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.cardinality" title="sympy.combinatorics.permutations.Permutation.cardinality"><tt class="xref py py-obj docutils literal"><span class="pre">cardinality</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.length" title="sympy.combinatorics.permutations.Permutation.length"><tt class="xref py py-obj docutils literal"><span class="pre">length</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.rank" title="sympy.combinatorics.permutations.Permutation.rank"><tt class="xref py py-obj docutils literal"><span class="pre">rank</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.size" title="sympy.combinatorics.permutations.Permutation.size"><tt class="xref py py-obj docutils literal"><span class="pre">size</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">order</span><span class="p">()</span>
-<span class="go">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">p</span><span class="o">**</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">order</span><span class="p">()))</span>
-<span class="go">Permutation([], size=6)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.parity">
-<tt class="descname">parity</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.parity"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.parity" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the parity of a permutation.</p>
-<p>The parity of a permutation reflects the parity of the
-number of inversions in the permutation, i.e., the
-number of pairs of x and y such that <tt class="docutils literal"><span class="pre">x</span> <span class="pre">&gt;</span> <span class="pre">y</span></tt> but <tt class="docutils literal"><span class="pre">p[x]</span> <span class="pre">&lt;</span> <span class="pre">p[y]</span></tt>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">_af_parity</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">parity</span><span class="p">()</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">parity</span><span class="p">()</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.combinatorics.permutations.Permutation.random">
-<em class="property">classmethod </em><tt class="descname">random</tt><big>(</big><em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.random"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.random" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates a random permutation of length <tt class="docutils literal"><span class="pre">n</span></tt>.</p>
-<p>Uses the underlying Python psuedo-random number generator.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="ow">in</span> <span class="p">(</span><span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]),</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.rank">
-<tt class="descname">rank</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.rank"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.rank" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the lexicographic rank of the permutation.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.next_lex" title="sympy.combinatorics.permutations.Permutation.next_lex"><tt class="xref py py-obj docutils literal"><span class="pre">next_lex</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.unrank_lex" title="sympy.combinatorics.permutations.Permutation.unrank_lex"><tt class="xref py py-obj docutils literal"><span class="pre">unrank_lex</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.cardinality" title="sympy.combinatorics.permutations.Permutation.cardinality"><tt class="xref py py-obj docutils literal"><span class="pre">cardinality</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.length" title="sympy.combinatorics.permutations.Permutation.length"><tt class="xref py py-obj docutils literal"><span class="pre">length</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.order" title="sympy.combinatorics.permutations.Permutation.order"><tt class="xref py py-obj docutils literal"><span class="pre">order</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.size" title="sympy.combinatorics.permutations.Permutation.size"><tt class="xref py py-obj docutils literal"><span class="pre">size</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">rank</span><span class="p">()</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">rank</span><span class="p">()</span>
-<span class="go">23</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.rank_nonlex">
-<tt class="descname">rank_nonlex</tt><big>(</big><em>self</em>, <em>inv_perm=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.rank_nonlex"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.rank_nonlex" title="Permalink to this definition">¶</a></dt>
-<dd><p>This is a linear time ranking algorithm that does not
-enforce lexicographic order [3].</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.next_nonlex" title="sympy.combinatorics.permutations.Permutation.next_nonlex"><tt class="xref py py-obj docutils literal"><span class="pre">next_nonlex</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.unrank_nonlex" title="sympy.combinatorics.permutations.Permutation.unrank_nonlex"><tt class="xref py py-obj docutils literal"><span class="pre">unrank_nonlex</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">rank_nonlex</span><span class="p">()</span>
-<span class="go">23</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.rank_trotterjohnson">
-<tt class="descname">rank_trotterjohnson</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.rank_trotterjohnson"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.rank_trotterjohnson" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the Trotter Johnson rank, which we get from the minimal
-change algorithm. See [4] section 2.4.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.unrank_trotterjohnson" title="sympy.combinatorics.permutations.Permutation.unrank_trotterjohnson"><tt class="xref py py-obj docutils literal"><span class="pre">unrank_trotterjohnson</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.next_trotterjohnson" title="sympy.combinatorics.permutations.Permutation.next_trotterjohnson"><tt class="xref py py-obj docutils literal"><span class="pre">next_trotterjohnson</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">rank_trotterjohnson</span><span class="p">()</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">rank_trotterjohnson</span><span class="p">()</span>
-<span class="go">7</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.rmul">
-<tt class="descname">rmul</tt><big>(</big><em>*args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.rmul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.rmul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return product of Permutations [a, b, c, ...] as the Permutation whose
-ith value is a(b(c(i))).</p>
-<p>a, b, c, ... can be Permutation objects or tuples.</p>
-<p class="rubric">Notes</p>
-<p>All items in the sequence will be parsed by Permutation as
-necessary as long as the first item is a Permutation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">rmul</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span> <span class="o">==</span> <span class="n">Permutation</span><span class="o">.</span><span class="n">rmul</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>The reverse order of arguments will raise a TypeError.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">_af_rmul</span><span class="p">,</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">False</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="n">a</span><span class="p">);</span> <span class="n">b</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">Permutation</span><span class="o">.</span><span class="n">rmul</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-<span class="go">[1, 2, 0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">a</span><span class="p">(</span><span class="n">b</span><span class="p">(</span><span class="n">i</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">)]</span>
-<span class="go">[1, 2, 0]</span>
-</pre></div>
-</div>
-<p>This handles the operands in reverse order compared to the <tt class="docutils literal"><span class="pre">*</span></tt> operator:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="n">a</span><span class="p">);</span> <span class="n">b</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">b</span><span class="p">)</span>
-<span class="go">[2, 0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">b</span><span class="p">(</span><span class="n">a</span><span class="p">(</span><span class="n">i</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">)]</span>
-<span class="go">[2, 0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.rmul_with_af">
-<tt class="descname">rmul_with_af</tt><big>(</big><em>*args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.rmul_with_af"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.rmul_with_af" title="Permalink to this definition">¶</a></dt>
-<dd><p>same as rmul, but the elements of args are Permutation objects
-which have _array_form</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.runs">
-<tt class="descname">runs</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.runs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.runs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the runs of a permutation.</p>
-<p>An ascending sequence in a permutation is called a run [5].</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">8</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">runs</span><span class="p">()</span>
-<span class="go">[[2, 5, 7], [3, 6], [0, 1, 4, 8]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="o">.</span><span class="n">runs</span><span class="p">()</span>
-<span class="go">[[1, 3], [2], [0]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.signature">
-<tt class="descname">signature</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.signature"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.signature" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the signature of the permutation needed to place the
-elements of the permutation in canonical order.</p>
-<p>The signature is calculated as (-1)^&lt;number of inversions&gt;</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.inversions" title="sympy.combinatorics.permutations.Permutation.inversions"><tt class="xref py py-obj docutils literal"><span class="pre">inversions</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">inversions</span><span class="p">()</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">signature</span><span class="p">()</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="o">.</span><span class="n">inversions</span><span class="p">()</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="o">.</span><span class="n">signature</span><span class="p">()</span>
-<span class="go">-1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.permutations.Permutation.size">
-<tt class="descname">size</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.size"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.size" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the number of elements in the permutation.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.cardinality" title="sympy.combinatorics.permutations.Permutation.cardinality"><tt class="xref py py-obj docutils literal"><span class="pre">cardinality</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.length" title="sympy.combinatorics.permutations.Permutation.length"><tt class="xref py py-obj docutils literal"><span class="pre">length</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.order" title="sympy.combinatorics.permutations.Permutation.order"><tt class="xref py py-obj docutils literal"><span class="pre">order</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.rank" title="sympy.combinatorics.permutations.Permutation.rank"><tt class="xref py py-obj docutils literal"><span class="pre">rank</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="p">([[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span><span class="o">.</span><span class="n">size</span>
-<span class="go">4</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.support">
-<tt class="descname">support</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.support"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.support" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the elements in permutation, P, for which P[i] != i.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">array_form</span>
-<span class="go">[1, 0, 3, 2, 4]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">support</span><span class="p">()</span>
-<span class="go">[0, 1, 2, 3]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Permutation.transpositions">
-<tt class="descname">transpositions</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.transpositions"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.transpositions" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the permutation decomposed into a list of transpositions.</p>
-<p>It is always possible to express a permutation as the product of
-transpositions, see [1]</p>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Transposition_%28mathematics%29#Properties">http://en.wikipedia.org/wiki/Transposition_%28mathematics%29#Properties</a></li>
-</ol>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">transpositions</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span>
-<span class="go">[(0, 7), (0, 6), (0, 5), (0, 4), (1, 3), (1, 2)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">t</span><span class="p">))</span>
-<span class="go">(0, 7)(0, 6)(0, 5)(0, 4)(1, 3)(1, 2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">rmul</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="n">Permutation</span><span class="p">([</span><span class="n">ti</span><span class="p">],</span> <span class="n">size</span><span class="o">=</span><span class="n">p</span><span class="o">.</span><span class="n">size</span><span class="p">)</span> <span class="k">for</span> <span class="n">ti</span> <span class="ow">in</span> <span class="n">t</span><span class="p">])</span> <span class="o">==</span> <span class="n">p</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.combinatorics.permutations.Permutation.unrank_lex">
-<em class="property">classmethod </em><tt class="descname">unrank_lex</tt><big>(</big><em>size</em>, <em>rank</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.unrank_lex"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.unrank_lex" title="Permalink to this definition">¶</a></dt>
-<dd><p>Lexicographic permutation unranking.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.rank" title="sympy.combinatorics.permutations.Permutation.rank"><tt class="xref py py-obj docutils literal"><span class="pre">rank</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.next_lex" title="sympy.combinatorics.permutations.Permutation.next_lex"><tt class="xref py py-obj docutils literal"><span class="pre">next_lex</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="o">.</span><span class="n">unrank_lex</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">rank</span><span class="p">()</span>
-<span class="go">10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
-<span class="go">Permutation([0, 2, 4, 1, 3])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.combinatorics.permutations.Permutation.unrank_nonlex">
-<em class="property">classmethod </em><tt class="descname">unrank_nonlex</tt><big>(</big><em>n</em>, <em>r</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.unrank_nonlex"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.unrank_nonlex" title="Permalink to this definition">¶</a></dt>
-<dd><p>This is a linear time unranking algorithm that does not
-respect lexicographic order [3].</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.next_nonlex" title="sympy.combinatorics.permutations.Permutation.next_nonlex"><tt class="xref py py-obj docutils literal"><span class="pre">next_nonlex</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.rank_nonlex" title="sympy.combinatorics.permutations.Permutation.rank_nonlex"><tt class="xref py py-obj docutils literal"><span class="pre">rank_nonlex</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">unrank_nonlex</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">Permutation([2, 0, 3, 1])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">unrank_nonlex</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">Permutation([0, 1, 2, 3])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.combinatorics.permutations.Permutation.unrank_trotterjohnson">
-<em class="property">classmethod </em><tt class="descname">unrank_trotterjohnson</tt><big>(</big><em>size</em>, <em>rank</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Permutation.unrank_trotterjohnson"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Permutation.unrank_trotterjohnson" title="Permalink to this definition">¶</a></dt>
-<dd><p>Trotter Johnson permutation unranking. See [4] section 2.4.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.rank_trotterjohnson" title="sympy.combinatorics.permutations.Permutation.rank_trotterjohnson"><tt class="xref py py-obj docutils literal"><span class="pre">rank_trotterjohnson</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.permutations.Permutation.next_trotterjohnson" title="sympy.combinatorics.permutations.Permutation.next_trotterjohnson"><tt class="xref py py-obj docutils literal"><span class="pre">next_trotterjohnson</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">unrank_trotterjohnson</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">Permutation([0, 3, 1, 2, 4])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.combinatorics.permutations.Cycle">
-<em class="property">class </em><tt class="descclassname">sympy.combinatorics.permutations.</tt><tt class="descname">Cycle</tt><big>(</big><em>*args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Cycle"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Cycle" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wrapper around dict which provides the functionality of a disjoint cycle.</p>
-<p>A cycle shows the rule to use to move subsets of elements to obtain
-a permutation. The Cycle class is more flexible that Permutation in
-that 1) all elements need not be present in order to investigate how
-multiple cycles act in sequence and 2) it can contain singletons:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Perm</span><span class="p">,</span> <span class="n">Cycle</span>
-</pre></div>
-</div>
-<p>A Cycle will automatically parse a cycle given as a tuple on the rhs:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Cycle</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">Cycle(1, 3, 2)</span>
-</pre></div>
-</div>
-<p>The identity cycle, Cycle(), can be used to start a product:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Cycle</span><span class="p">()(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">Cycle(1, 3, 2)</span>
-</pre></div>
-</div>
-<p>The array form of a Cycle can be obtained by calling the list
-method (or passing it to the list function) and all elements from
-0 will be shown:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Cycle</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">list</span><span class="p">()</span>
-<span class="go">[0, 2, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="go">[0, 2, 1]</span>
-</pre></div>
-</div>
-<p>If a larger (or smaller) range is desired use the list method and
-provide the desired size &#8211; but the Cycle cannot be truncated to
-a size smaller than the largest element that is out of place:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Cycle</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">)(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">)(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">list</span><span class="p">()</span>
-<span class="go">[0, 2, 1, 3, 4]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">list</span><span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">size</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">[0, 2, 1, 3, 4, 5]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">list</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">[0, 2, 1]</span>
-</pre></div>
-</div>
-<p>Singletons are not shown when printing with one exception: the largest
-element is always shown &#8211; as a singleton if necessary:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Cycle</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">10</span><span class="p">)(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">Cycle(1, 5, 4, 10)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Cycle</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)(</span><span class="mi">4</span><span class="p">)(</span><span class="mi">5</span><span class="p">)(</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">Cycle(1, 2)(10)</span>
-</pre></div>
-</div>
-<p>The array form can be used to instantiate a Permutation so other
-properties of the permutation can be investigated:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Perm</span><span class="p">(</span><span class="n">Cycle</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)(</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">list</span><span class="p">())</span><span class="o">.</span><span class="n">transpositions</span><span class="p">()</span>
-<span class="go">[(1, 2), (3, 4)]</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.permutations.Permutation" title="sympy.combinatorics.permutations.Permutation"><tt class="xref py py-obj docutils literal"><span class="pre">Permutation</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>The underlying structure of the Cycle is a dictionary and although
-the __iter__ method has been redefiend to give the array form of the
-cycle, the underlying dictionary items are still available with the
-such methods as items():</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">Cycle</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">items</span><span class="p">())</span>
-<span class="go">[(1, 2), (2, 1)]</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.combinatorics.permutations.Cycle.list">
-<tt class="descname">list</tt><big>(</big><em>self</em>, <em>size=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/permutations.html#Cycle.list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.permutations.Cycle.list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the cycles as an explicit list starting from 0 up
-to the greater of the largest value in the cycles and size.</p>
-<p>Truncation of trailing unmoved items will occur when size
-is less than the maximum element in the cycle; if this is
-desired, setting <tt class="docutils literal"><span class="pre">size=-1</span></tt> will guarantee such trimming.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Cycle</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Cycle</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">list</span><span class="p">()</span>
-<span class="go">[0, 1, 3, 2, 5, 4]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">list</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">[0, 1, 3, 2, 5, 4, 6, 7, 8, 9]</span>
-</pre></div>
-</div>
-<p>Passing a length too small will trim trailing, unchanged elements
-in the permutation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Cycle</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">list</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">[0, 2, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<div class="section" id="module-sympy.combinatorics.generators">
-<span id="generators"></span><span id="combinatorics-generators"></span><h2>Generators<a class="headerlink" href="#module-sympy.combinatorics.generators" title="Permalink to this headline">¶</a></h2>
-<dl class="method">
-<dt id="sympy.combinatorics.generators.symmetric">
-<tt class="descclassname">generators.</tt><tt class="descname">symmetric</tt><big>(</big><em>n</em><big>)</big><a class="headerlink" href="#sympy.combinatorics.generators.symmetric" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates the symmetric group of order n, Sn.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.generators</span> <span class="kn">import</span> <span class="n">symmetric</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">symmetric</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">[Permutation(2), Permutation(1, 2), Permutation(2)(0, 1),</span>
-<span class="go"> Permutation(0, 1, 2), Permutation(0, 2, 1), Permutation(0, 2)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="method">
-<dt id="sympy.combinatorics.generators.cyclic">
-<tt class="descclassname">generators.</tt><tt class="descname">cyclic</tt><big>(</big><em>n</em><big>)</big><a class="headerlink" href="#sympy.combinatorics.generators.cyclic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates the cyclic group of order n, Cn.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.generators.dihedral" title="sympy.combinatorics.generators.dihedral"><tt class="xref py py-obj docutils literal"><span class="pre">dihedral</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.generators</span> <span class="kn">import</span> <span class="n">cyclic</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">cyclic</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
-<span class="go">[Permutation(4), Permutation(0, 1, 2, 3, 4), Permutation(0, 2, 4, 1, 3),</span>
-<span class="go"> Permutation(0, 3, 1, 4, 2), Permutation(0, 4, 3, 2, 1)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="method">
-<dt id="sympy.combinatorics.generators.alternating">
-<tt class="descclassname">generators.</tt><tt class="descname">alternating</tt><big>(</big><em>n</em><big>)</big><a class="headerlink" href="#sympy.combinatorics.generators.alternating" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates the alternating group of order n, An.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.generators</span> <span class="kn">import</span> <span class="n">alternating</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">alternating</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">[Permutation(2), Permutation(0, 1, 2), Permutation(0, 2, 1)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="method">
-<dt id="sympy.combinatorics.generators.dihedral">
-<tt class="descclassname">generators.</tt><tt class="descname">dihedral</tt><big>(</big><em>n</em><big>)</big><a class="headerlink" href="#sympy.combinatorics.generators.dihedral" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates the dihedral group of order 2n, Dn.</p>
-<p>The result is given as a subgroup of Sn, except for the special cases n=1
-(the group S2) and n=2 (the Klein 4-group) where that&#8217;s not possible
-and embeddings in S2 and S4 respectively are given.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.generators.cyclic" title="sympy.combinatorics.generators.cyclic"><tt class="xref py py-obj docutils literal"><span class="pre">cyclic</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.generators</span> <span class="kn">import</span> <span class="n">dihedral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">dihedral</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">[Permutation(2), Permutation(0, 2), Permutation(0, 1, 2),</span>
-<span class="go"> Permutation(1, 2), Permutation(0, 2, 1), Permutation(2)(0, 1)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-
-
-          </div>
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-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Permutations</a><ul>
-<li><a class="reference internal" href="#module-sympy.combinatorics.generators">Generators</a></li>
-</ul>
-</li>
-</ul>
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-  <div class="section" id="module-sympy.combinatorics.polyhedron">
-<span id="polyhedron"></span><span id="combinatorics-polyhedron"></span><h1>Polyhedron<a class="headerlink" href="#module-sympy.combinatorics.polyhedron" title="Permalink to this headline">¶</a></h1>
-<dl class="class">
-<dt id="sympy.combinatorics.polyhedron.Polyhedron">
-<em class="property">class </em><tt class="descclassname">sympy.combinatorics.polyhedron.</tt><tt class="descname">Polyhedron</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/polyhedron.html#Polyhedron"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.polyhedron.Polyhedron" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents the polyhedral symmetry group (PSG).</p>
-<p>The PSG is one of the symmetry groups of the Platonic solids.
-There are three polyhedral groups: the tetrahedral group
-of order 12, the octahedral group of order 24, and the
-icosahedral group of order 60.</p>
-<p>All doctests have been given in the docstring of the
-constructor of the object.</p>
-<p class="rubric">References</p>
-<p><a class="reference external" href="http://mathworld.wolfram.com/PolyhedralGroup.html">http://mathworld.wolfram.com/PolyhedralGroup.html</a></p>
-<dl class="attribute">
-<dt id="sympy.combinatorics.polyhedron.Polyhedron.array_form">
-<tt class="descname">array_form</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/polyhedron.html#Polyhedron.array_form"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.polyhedron.Polyhedron.array_form" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the indices of the corners.</p>
-<p>The indices are given relative to the original position of corners.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.polyhedron.Polyhedron.corners" title="sympy.combinatorics.polyhedron.Polyhedron.corners"><tt class="xref py py-obj docutils literal"><span class="pre">corners</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.polyhedron.Polyhedron.cyclic_form" title="sympy.combinatorics.polyhedron.Polyhedron.cyclic_form"><tt class="xref py py-obj docutils literal"><span class="pre">cyclic_form</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span><span class="p">,</span> <span class="n">Cycle</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.polyhedron</span> <span class="kn">import</span> <span class="n">tetrahedron</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tetrahedron</span><span class="o">.</span><span class="n">array_form</span>
-<span class="go">[0, 1, 2, 3]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">tetrahedron</span><span class="o">.</span><span class="n">rotate</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tetrahedron</span><span class="o">.</span><span class="n">array_form</span>
-<span class="go">[0, 2, 3, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tetrahedron</span><span class="o">.</span><span class="n">pgroup</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">array_form</span>
-<span class="go">[0, 2, 3, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.polyhedron.Polyhedron.corners">
-<tt class="descname">corners</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/polyhedron.html#Polyhedron.corners"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.polyhedron.Polyhedron.corners" title="Permalink to this definition">¶</a></dt>
-<dd><p>Get the corners of the Polyhedron.</p>
-<p>The method <tt class="docutils literal"><span class="pre">vertices</span></tt> is an alias for <tt class="docutils literal"><span class="pre">corners</span></tt>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.polyhedron.Polyhedron.array_form" title="sympy.combinatorics.polyhedron.Polyhedron.array_form"><tt class="xref py py-obj docutils literal"><span class="pre">array_form</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.polyhedron.Polyhedron.cyclic_form" title="sympy.combinatorics.polyhedron.Polyhedron.cyclic_form"><tt class="xref py py-obj docutils literal"><span class="pre">cyclic_form</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Polyhedron</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Polyhedron</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="s">&#39;abcd&#39;</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">corners</span> <span class="o">==</span> <span class="n">p</span><span class="o">.</span><span class="n">vertices</span> <span class="o">==</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.polyhedron.Polyhedron.cyclic_form">
-<tt class="descname">cyclic_form</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/polyhedron.html#Polyhedron.cyclic_form"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.polyhedron.Polyhedron.cyclic_form" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the indices of the corners in cyclic notation.</p>
-<p>The indices are given relative to the original position of corners.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.polyhedron.Polyhedron.corners" title="sympy.combinatorics.polyhedron.Polyhedron.corners"><tt class="xref py py-obj docutils literal"><span class="pre">corners</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.polyhedron.Polyhedron.array_form" title="sympy.combinatorics.polyhedron.Polyhedron.array_form"><tt class="xref py py-obj docutils literal"><span class="pre">array_form</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.polyhedron.Polyhedron.edges">
-<tt class="descname">edges</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/polyhedron.html#Polyhedron.edges"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.polyhedron.Polyhedron.edges" title="Permalink to this definition">¶</a></dt>
-<dd><p>Given the faces of the polyhedra we can get the edges.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Polyhedron</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">corners</span> <span class="o">=</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">faces</span> <span class="o">=</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Polyhedron</span><span class="p">(</span><span class="n">corners</span><span class="p">,</span> <span class="n">faces</span><span class="p">)</span><span class="o">.</span><span class="n">edges</span>
-<span class="go">{(0, 1), (0, 2), (1, 2)}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.polyhedron.Polyhedron.faces">
-<tt class="descname">faces</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/polyhedron.html#Polyhedron.faces"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.polyhedron.Polyhedron.faces" title="Permalink to this definition">¶</a></dt>
-<dd><p>Get the faces of the Polyhedron.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.polyhedron.Polyhedron.pgroup">
-<tt class="descname">pgroup</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/polyhedron.html#Polyhedron.pgroup"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.polyhedron.Polyhedron.pgroup" title="Permalink to this definition">¶</a></dt>
-<dd><p>Get the permutations of the Polyhedron.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.polyhedron.Polyhedron.reset">
-<tt class="descname">reset</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/polyhedron.html#Polyhedron.reset"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.polyhedron.Polyhedron.reset" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return corners to their original positions.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.polyhedron</span> <span class="kn">import</span> <span class="n">tetrahedron</span> <span class="k">as</span> <span class="n">T</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">T</span><span class="o">.</span><span class="n">corners</span>
-<span class="go">(0, 1, 2, 3)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">T</span><span class="o">.</span><span class="n">rotate</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">T</span><span class="o">.</span><span class="n">corners</span>
-<span class="go">(0, 2, 3, 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">T</span><span class="o">.</span><span class="n">reset</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">T</span><span class="o">.</span><span class="n">corners</span>
-<span class="go">(0, 1, 2, 3)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.polyhedron.Polyhedron.rotate">
-<tt class="descname">rotate</tt><big>(</big><em>self</em>, <em>perm</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/polyhedron.html#Polyhedron.rotate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.polyhedron.Polyhedron.rotate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Apply a permutation to the polyhedron <em>in place</em>. The permutation
-may be given as a Permutation instance or an integer indicating
-which permutation from pgroup of the Polyhedron should be
-applied.</p>
-<p>This is an operation that is analogous to rotation about
-an axis by a fixed increment.</p>
-<p class="rubric">Notes</p>
-<p>When a Permutation is applied, no check is done to see if that
-is a valid permutation for the Polyhedron. For example, a cube
-could be given a permutation which effectively swaps only 2
-vertices. A valid permutation (that rotates the object in a
-physical way) will be obtained if one only uses
-permutations from the <tt class="docutils literal"><span class="pre">pgroup</span></tt> of the Polyhedron. On the other
-hand, allowing arbitrary rotations (applications of permutations)
-gives a way to follow named elements rather than indices since
-Polyhedron allows vertices to be named while Permutation works
-only with indices.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Polyhedron</span><span class="p">,</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.polyhedron</span> <span class="kn">import</span> <span class="n">cube</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cube</span><span class="o">.</span><span class="n">corners</span>
-<span class="go">(0, 1, 2, 3, 4, 5, 6, 7)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cube</span><span class="o">.</span><span class="n">rotate</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cube</span><span class="o">.</span><span class="n">corners</span>
-<span class="go">(1, 2, 3, 0, 5, 6, 7, 4)</span>
-</pre></div>
-</div>
-<p>A non-physical &#8220;rotation&#8221; that is not prohibited by this method:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">cube</span><span class="o">.</span><span class="n">reset</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cube</span><span class="o">.</span><span class="n">rotate</span><span class="p">(</span><span class="n">Permutation</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">]],</span> <span class="n">size</span><span class="o">=</span><span class="mi">8</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cube</span><span class="o">.</span><span class="n">corners</span>
-<span class="go">(0, 2, 1, 3, 4, 5, 6, 7)</span>
-</pre></div>
-</div>
-<p>Polyhedron can be used to follow elements of set that are
-identified by letters instead of integers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">shadow</span> <span class="o">=</span> <span class="n">h5</span> <span class="o">=</span> <span class="n">Polyhedron</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="s">&#39;abcde&#39;</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h5</span><span class="o">.</span><span class="n">rotate</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h5</span><span class="o">.</span><span class="n">corners</span>
-<span class="go">(d, a, b, c, e)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span> <span class="o">==</span> <span class="n">shadow</span><span class="o">.</span><span class="n">corners</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">copy</span> <span class="o">=</span> <span class="n">h5</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h5</span><span class="o">.</span><span class="n">rotate</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h5</span><span class="o">.</span><span class="n">corners</span> <span class="o">==</span> <span class="n">copy</span><span class="o">.</span><span class="n">corners</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.polyhedron.Polyhedron.size">
-<tt class="descname">size</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/polyhedron.html#Polyhedron.size"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.polyhedron.Polyhedron.size" title="Permalink to this definition">¶</a></dt>
-<dd><p>Get the number of corners of the Polyhedron.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.polyhedron.Polyhedron.vertices">
-<tt class="descname">vertices</tt><a class="headerlink" href="#sympy.combinatorics.polyhedron.Polyhedron.vertices" title="Permalink to this definition">¶</a></dt>
-<dd><p>Get the corners of the Polyhedron.</p>
-<p>The method <tt class="docutils literal"><span class="pre">vertices</span></tt> is an alias for <tt class="docutils literal"><span class="pre">corners</span></tt>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.polyhedron.Polyhedron.array_form" title="sympy.combinatorics.polyhedron.Polyhedron.array_form"><tt class="xref py py-obj docutils literal"><span class="pre">array_form</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.polyhedron.Polyhedron.cyclic_form" title="sympy.combinatorics.polyhedron.Polyhedron.cyclic_form"><tt class="xref py py-obj docutils literal"><span class="pre">cyclic_form</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Polyhedron</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Polyhedron</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="s">&#39;abcd&#39;</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">corners</span> <span class="o">==</span> <span class="n">p</span><span class="o">.</span><span class="n">vertices</span> <span class="o">==</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
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-  <div class="section" id="module-sympy.combinatorics.prufer">
-<span id="prufer-sequences"></span><span id="combinatorics-prufer"></span><h1>Prufer Sequences<a class="headerlink" href="#module-sympy.combinatorics.prufer" title="Permalink to this headline">¶</a></h1>
-<dl class="class">
-<dt id="sympy.combinatorics.prufer.Prufer">
-<em class="property">class </em><tt class="descclassname">sympy.combinatorics.prufer.</tt><tt class="descname">Prufer</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/prufer.html#Prufer"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.prufer.Prufer" title="Permalink to this definition">¶</a></dt>
-<dd><p>The Prufer correspondence is an algorithm that describes the
-bijection between labeled trees and the Prufer code. A Prufer
-code of a labeled tree is unique up to isomorphism and has
-a length of n - 2.</p>
-<p>Prufer sequences were first used by Heinz Prufer to give a
-proof of Cayley&#8217;s formula.</p>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r11" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id1">[R11]</a></td><td><a class="reference external" href="http://mathworld.wolfram.com/LabeledTree.html">http://mathworld.wolfram.com/LabeledTree.html</a></td></tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.combinatorics.prufer.Prufer.edges">
-<tt class="descname">edges</tt><big>(</big><em>*runs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/prufer.html#Prufer.edges"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.prufer.Prufer.edges" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a list of edges and the number of nodes from the given runs
-that connect nodes in an integer-labelled tree.</p>
-<p>All node numbers will be shifted so that the minimum node is 0. It is
-not a problem if edges are repeated in the runs; only unique edges are
-returned. There is no assumption made about what the range of the node
-labels should be, but all nodes from the smallest through the largest
-must be present.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.prufer</span> <span class="kn">import</span> <span class="n">Prufer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Prufer</span><span class="o">.</span><span class="n">edges</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span> <span class="c"># a T</span>
-<span class="go">([[0, 1], [3, 4], [1, 2], [1, 3]], 5)</span>
-</pre></div>
-</div>
-<p>Duplicate edges are removed:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Prufer</span><span class="o">.</span><span class="n">edges</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">6</span><span class="p">])</span> <span class="c"># a K</span>
-<span class="go">([[0, 1], [1, 2], [4, 6], [4, 5], [1, 4], [2, 3]], 7)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.prufer.Prufer.next">
-<tt class="descname">next</tt><big>(</big><em>self</em>, <em>delta=1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/prufer.html#Prufer.next"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.prufer.Prufer.next" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates the Prufer sequence that is delta beyond the current one.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.prufer_rank" title="sympy.combinatorics.prufer.Prufer.prufer_rank"><tt class="xref py py-obj docutils literal"><span class="pre">prufer_rank</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.rank" title="sympy.combinatorics.prufer.Prufer.rank"><tt class="xref py py-obj docutils literal"><span class="pre">rank</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.prev" title="sympy.combinatorics.prufer.Prufer.prev"><tt class="xref py py-obj docutils literal"><span class="pre">prev</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.size" title="sympy.combinatorics.prufer.Prufer.size"><tt class="xref py py-obj docutils literal"><span class="pre">size</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.prufer</span> <span class="kn">import</span> <span class="n">Prufer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Prufer</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">next</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="c"># == a.next()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">tree_repr</span>
-<span class="go">[[0, 2], [0, 1], [1, 3]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">rank</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.prufer.Prufer.nodes">
-<tt class="descname">nodes</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/prufer.html#Prufer.nodes"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.prufer.Prufer.nodes" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the number of nodes in the tree.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.prufer</span> <span class="kn">import</span> <span class="n">Prufer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Prufer</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">]])</span><span class="o">.</span><span class="n">nodes</span>
-<span class="go">6</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Prufer</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">nodes</span>
-<span class="go">5</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.prufer.Prufer.prev">
-<tt class="descname">prev</tt><big>(</big><em>self</em>, <em>delta=1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/prufer.html#Prufer.prev"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.prufer.Prufer.prev" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates the Prufer sequence that is -delta before the current one.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.prufer_rank" title="sympy.combinatorics.prufer.Prufer.prufer_rank"><tt class="xref py py-obj docutils literal"><span class="pre">prufer_rank</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.rank" title="sympy.combinatorics.prufer.Prufer.rank"><tt class="xref py py-obj docutils literal"><span class="pre">rank</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.next" title="sympy.combinatorics.prufer.Prufer.next"><tt class="xref py py-obj docutils literal"><span class="pre">next</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.size" title="sympy.combinatorics.prufer.Prufer.size"><tt class="xref py py-obj docutils literal"><span class="pre">size</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.prufer</span> <span class="kn">import</span> <span class="n">Prufer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Prufer</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">rank</span>
-<span class="go">36</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">prev</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span>
-<span class="go">Prufer([1, 2, 0])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">rank</span>
-<span class="go">35</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.prufer.Prufer.prufer_rank">
-<tt class="descname">prufer_rank</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/prufer.html#Prufer.prufer_rank"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.prufer.Prufer.prufer_rank" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the rank of a Prufer sequence.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.rank" title="sympy.combinatorics.prufer.Prufer.rank"><tt class="xref py py-obj docutils literal"><span class="pre">rank</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.next" title="sympy.combinatorics.prufer.Prufer.next"><tt class="xref py py-obj docutils literal"><span class="pre">next</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.prev" title="sympy.combinatorics.prufer.Prufer.prev"><tt class="xref py py-obj docutils literal"><span class="pre">prev</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.size" title="sympy.combinatorics.prufer.Prufer.size"><tt class="xref py py-obj docutils literal"><span class="pre">size</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.prufer</span> <span class="kn">import</span> <span class="n">Prufer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Prufer</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">prufer_rank</span><span class="p">()</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.prufer.Prufer.prufer_repr">
-<tt class="descname">prufer_repr</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/prufer.html#Prufer.prufer_repr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.prufer.Prufer.prufer_repr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns Prufer sequence for the Prufer object.</p>
-<p>This sequence is found by removing the highest numbered vertex,
-recording the node it was attached to, and continuuing until only
-two verices remain. The Prufer sequence is the list of recorded nodes.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.to_prufer" title="sympy.combinatorics.prufer.Prufer.to_prufer"><tt class="xref py py-obj docutils literal"><span class="pre">to_prufer</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.prufer</span> <span class="kn">import</span> <span class="n">Prufer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Prufer</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">]])</span><span class="o">.</span><span class="n">prufer_repr</span>
-<span class="go">[3, 3, 3, 4]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Prufer</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">prufer_repr</span>
-<span class="go">[1, 0, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.prufer.Prufer.rank">
-<tt class="descname">rank</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/prufer.html#Prufer.rank"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.prufer.Prufer.rank" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the rank of the Prufer sequence.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.prufer_rank" title="sympy.combinatorics.prufer.Prufer.prufer_rank"><tt class="xref py py-obj docutils literal"><span class="pre">prufer_rank</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.next" title="sympy.combinatorics.prufer.Prufer.next"><tt class="xref py py-obj docutils literal"><span class="pre">next</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.prev" title="sympy.combinatorics.prufer.Prufer.prev"><tt class="xref py py-obj docutils literal"><span class="pre">prev</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.size" title="sympy.combinatorics.prufer.Prufer.size"><tt class="xref py py-obj docutils literal"><span class="pre">size</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.prufer</span> <span class="kn">import</span> <span class="n">Prufer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Prufer</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">rank</span>
-<span class="go">778</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">next</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">rank</span>
-<span class="go">779</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">prev</span><span class="p">()</span><span class="o">.</span><span class="n">rank</span>
-<span class="go">777</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.prufer.Prufer.size">
-<tt class="descname">size</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/prufer.html#Prufer.size"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.prufer.Prufer.size" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the number of possible trees of this Prufer object.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.prufer_rank" title="sympy.combinatorics.prufer.Prufer.prufer_rank"><tt class="xref py py-obj docutils literal"><span class="pre">prufer_rank</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.rank" title="sympy.combinatorics.prufer.Prufer.rank"><tt class="xref py py-obj docutils literal"><span class="pre">rank</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.next" title="sympy.combinatorics.prufer.Prufer.next"><tt class="xref py py-obj docutils literal"><span class="pre">next</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.prev" title="sympy.combinatorics.prufer.Prufer.prev"><tt class="xref py py-obj docutils literal"><span class="pre">prev</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.prufer</span> <span class="kn">import</span> <span class="n">Prufer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Prufer</span><span class="p">([</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">size</span> <span class="o">==</span> <span class="n">Prufer</span><span class="p">([</span><span class="mi">6</span><span class="p">]</span><span class="o">*</span><span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">size</span> <span class="o">==</span> <span class="mi">1296</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.prufer.Prufer.to_prufer">
-<tt class="descname">to_prufer</tt><big>(</big><em>tree</em>, <em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/prufer.html#Prufer.to_prufer"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.prufer.Prufer.to_prufer" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the Prufer sequence for a tree given as a list of edges where
-<tt class="docutils literal"><span class="pre">n</span></tt> is the number of nodes in the tree.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.prufer_repr" title="sympy.combinatorics.prufer.Prufer.prufer_repr"><tt class="xref py py-obj docutils literal"><span class="pre">prufer_repr</span></tt></a></dt>
-<dd>returns Prufer sequence of a Prufer object.</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.prufer</span> <span class="kn">import</span> <span class="n">Prufer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Prufer</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">prufer_repr</span>
-<span class="go">[0, 0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Prufer</span><span class="o">.</span><span class="n">to_prufer</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">]],</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">[0, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.prufer.Prufer.to_tree">
-<tt class="descname">to_tree</tt><big>(</big><em>prufer</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/prufer.html#Prufer.to_tree"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.prufer.Prufer.to_tree" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the tree (as a list of edges) of the given Prufer sequence.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.tree_repr" title="sympy.combinatorics.prufer.Prufer.tree_repr"><tt class="xref py py-obj docutils literal"><span class="pre">tree_repr</span></tt></a></dt>
-<dd>returns tree representation of a Prufer object.</dd>
-</dl>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://hamberg.no/erlend/2010/11/06/prufer-sequence/">http://hamberg.no/erlend/2010/11/06/prufer-sequence/</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.prufer</span> <span class="kn">import</span> <span class="n">Prufer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Prufer</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">tree_repr</span>
-<span class="go">[[0, 1], [0, 2], [2, 3]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Prufer</span><span class="o">.</span><span class="n">to_tree</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="go">[[0, 1], [0, 2], [2, 3]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.prufer.Prufer.tree_repr">
-<tt class="descname">tree_repr</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/prufer.html#Prufer.tree_repr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.prufer.Prufer.tree_repr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the tree representation of the Prufer object.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.prufer.Prufer.to_tree" title="sympy.combinatorics.prufer.Prufer.to_tree"><tt class="xref py py-obj docutils literal"><span class="pre">to_tree</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.prufer</span> <span class="kn">import</span> <span class="n">Prufer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Prufer</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">]])</span><span class="o">.</span><span class="n">tree_repr</span>
-<span class="go">[[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Prufer</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">tree_repr</span>
-<span class="go">[[1, 2], [0, 1], [0, 3], [0, 4]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.combinatorics.prufer.Prufer.unrank">
-<em class="property">classmethod </em><tt class="descname">unrank</tt><big>(</big><em>rank</em>, <em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/prufer.html#Prufer.unrank"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.prufer.Prufer.unrank" title="Permalink to this definition">¶</a></dt>
-<dd><p>Finds the unranked Prufer sequence.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.prufer</span> <span class="kn">import</span> <span class="n">Prufer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Prufer</span><span class="o">.</span><span class="n">unrank</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">Prufer([0, 0])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-
-
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-  <div class="section" id="module-sympy.combinatorics.subsets">
-<span id="subsets"></span><span id="combinatorics-subsets"></span><h1>Subsets<a class="headerlink" href="#module-sympy.combinatorics.subsets" title="Permalink to this headline">¶</a></h1>
-<dl class="class">
-<dt id="sympy.combinatorics.subsets.Subset">
-<em class="property">class </em><tt class="descclassname">sympy.combinatorics.subsets.</tt><tt class="descname">Subset</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/subsets.html#Subset"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.subsets.Subset" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents a basic subset object.</p>
-<p>We generate subsets using essentially two techniques,
-binary enumeration and lexicographic enumeration.
-The Subset class takes two arguments, the first one
-describes the initial subset to consider and the second
-describes the superset.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.subsets</span> <span class="kn">import</span> <span class="n">Subset</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Subset</span><span class="p">([</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">],</span> <span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">next_binary</span><span class="p">()</span><span class="o">.</span><span class="n">subset</span>
-<span class="go">[&#39;b&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">prev_binary</span><span class="p">()</span><span class="o">.</span><span class="n">subset</span>
-<span class="go">[&#39;c&#39;]</span>
-</pre></div>
-</div>
-<dl class="classmethod">
-<dt id="sympy.combinatorics.subsets.Subset.bitlist_from_subset">
-<em class="property">classmethod </em><tt class="descname">bitlist_from_subset</tt><big>(</big><em>subset</em>, <em>superset</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/subsets.html#Subset.bitlist_from_subset"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.subsets.Subset.bitlist_from_subset" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gets the bitlist corresponding to a subset.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.subsets.Subset.subset_from_bitlist" title="sympy.combinatorics.subsets.Subset.subset_from_bitlist"><tt class="xref py py-obj docutils literal"><span class="pre">subset_from_bitlist</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.subsets</span> <span class="kn">import</span> <span class="n">Subset</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Subset</span><span class="o">.</span><span class="n">bitlist_from_subset</span><span class="p">([</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">],</span> <span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">])</span>
-<span class="go">&#39;0011&#39;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.subsets.Subset.cardinality">
-<tt class="descname">cardinality</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/subsets.html#Subset.cardinality"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.subsets.Subset.cardinality" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the number of all possible subsets.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.subsets.Subset.subset" title="sympy.combinatorics.subsets.Subset.subset"><tt class="xref py py-obj docutils literal"><span class="pre">subset</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.superset" title="sympy.combinatorics.subsets.Subset.superset"><tt class="xref py py-obj docutils literal"><span class="pre">superset</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.size" title="sympy.combinatorics.subsets.Subset.size"><tt class="xref py py-obj docutils literal"><span class="pre">size</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.superset_size" title="sympy.combinatorics.subsets.Subset.superset_size"><tt class="xref py py-obj docutils literal"><span class="pre">superset_size</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.subsets</span> <span class="kn">import</span> <span class="n">Subset</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Subset</span><span class="p">([</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">],</span> <span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">cardinality</span>
-<span class="go">16</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.subsets.Subset.iterate_binary">
-<tt class="descname">iterate_binary</tt><big>(</big><em>self</em>, <em>k</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/subsets.html#Subset.iterate_binary"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.subsets.Subset.iterate_binary" title="Permalink to this definition">¶</a></dt>
-<dd><p>This is a helper function. It iterates over the
-binary subsets by k steps. This variable can be
-both positive or negative.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.subsets.Subset.next_binary" title="sympy.combinatorics.subsets.Subset.next_binary"><tt class="xref py py-obj docutils literal"><span class="pre">next_binary</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.prev_binary" title="sympy.combinatorics.subsets.Subset.prev_binary"><tt class="xref py py-obj docutils literal"><span class="pre">prev_binary</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.subsets</span> <span class="kn">import</span> <span class="n">Subset</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Subset</span><span class="p">([</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">],</span> <span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">iterate_binary</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subset</span>
-<span class="go">[&#39;d&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Subset</span><span class="p">([</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">],</span> <span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">iterate_binary</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subset</span>
-<span class="go">[]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.subsets.Subset.iterate_graycode">
-<tt class="descname">iterate_graycode</tt><big>(</big><em>self</em>, <em>k</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/subsets.html#Subset.iterate_graycode"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.subsets.Subset.iterate_graycode" title="Permalink to this definition">¶</a></dt>
-<dd><p>Helper function used for prev_gray and next_gray.
-It performs k step overs to get the respective Gray codes.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.subsets.Subset.next_gray" title="sympy.combinatorics.subsets.Subset.next_gray"><tt class="xref py py-obj docutils literal"><span class="pre">next_gray</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.prev_gray" title="sympy.combinatorics.subsets.Subset.prev_gray"><tt class="xref py py-obj docutils literal"><span class="pre">prev_gray</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.subsets</span> <span class="kn">import</span> <span class="n">Subset</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Subset</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">iterate_graycode</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">subset</span>
-<span class="go">[1, 4]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">iterate_graycode</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subset</span>
-<span class="go">[1, 2, 4]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.subsets.Subset.next_binary">
-<tt class="descname">next_binary</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/subsets.html#Subset.next_binary"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.subsets.Subset.next_binary" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates the next binary ordered subset.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.subsets.Subset.prev_binary" title="sympy.combinatorics.subsets.Subset.prev_binary"><tt class="xref py py-obj docutils literal"><span class="pre">prev_binary</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.iterate_binary" title="sympy.combinatorics.subsets.Subset.iterate_binary"><tt class="xref py py-obj docutils literal"><span class="pre">iterate_binary</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.subsets</span> <span class="kn">import</span> <span class="n">Subset</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Subset</span><span class="p">([</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">],</span> <span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">next_binary</span><span class="p">()</span><span class="o">.</span><span class="n">subset</span>
-<span class="go">[&#39;b&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Subset</span><span class="p">([</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">],</span> <span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">next_binary</span><span class="p">()</span><span class="o">.</span><span class="n">subset</span>
-<span class="go">[]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.subsets.Subset.next_gray">
-<tt class="descname">next_gray</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/subsets.html#Subset.next_gray"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.subsets.Subset.next_gray" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates the next Gray code ordered subset.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.subsets.Subset.iterate_graycode" title="sympy.combinatorics.subsets.Subset.iterate_graycode"><tt class="xref py py-obj docutils literal"><span class="pre">iterate_graycode</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.prev_gray" title="sympy.combinatorics.subsets.Subset.prev_gray"><tt class="xref py py-obj docutils literal"><span class="pre">prev_gray</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.subsets</span> <span class="kn">import</span> <span class="n">Subset</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Subset</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">next_gray</span><span class="p">()</span><span class="o">.</span><span class="n">subset</span>
-<span class="go">[1, 3]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.subsets.Subset.next_lexicographic">
-<tt class="descname">next_lexicographic</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/subsets.html#Subset.next_lexicographic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.subsets.Subset.next_lexicographic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates the next lexicographically ordered subset.</p>
-<p>NOT IMPLEMENTED</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.subsets.Subset.prev_binary">
-<tt class="descname">prev_binary</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/subsets.html#Subset.prev_binary"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.subsets.Subset.prev_binary" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates the previous binary ordered subset.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.subsets.Subset.next_binary" title="sympy.combinatorics.subsets.Subset.next_binary"><tt class="xref py py-obj docutils literal"><span class="pre">next_binary</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.iterate_binary" title="sympy.combinatorics.subsets.Subset.iterate_binary"><tt class="xref py py-obj docutils literal"><span class="pre">iterate_binary</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.subsets</span> <span class="kn">import</span> <span class="n">Subset</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Subset</span><span class="p">([],</span> <span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">prev_binary</span><span class="p">()</span><span class="o">.</span><span class="n">subset</span>
-<span class="go">[&#39;a&#39;, &#39;b&#39;, &#39;c&#39;, &#39;d&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Subset</span><span class="p">([</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">],</span> <span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">prev_binary</span><span class="p">()</span><span class="o">.</span><span class="n">subset</span>
-<span class="go">[&#39;c&#39;]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.subsets.Subset.prev_gray">
-<tt class="descname">prev_gray</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/subsets.html#Subset.prev_gray"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.subsets.Subset.prev_gray" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates the previous Gray code ordered subset.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.subsets.Subset.iterate_graycode" title="sympy.combinatorics.subsets.Subset.iterate_graycode"><tt class="xref py py-obj docutils literal"><span class="pre">iterate_graycode</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.next_gray" title="sympy.combinatorics.subsets.Subset.next_gray"><tt class="xref py py-obj docutils literal"><span class="pre">next_gray</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.subsets</span> <span class="kn">import</span> <span class="n">Subset</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Subset</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">prev_gray</span><span class="p">()</span><span class="o">.</span><span class="n">subset</span>
-<span class="go">[2, 3, 4, 5]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.subsets.Subset.prev_lexicographic">
-<tt class="descname">prev_lexicographic</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/subsets.html#Subset.prev_lexicographic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.subsets.Subset.prev_lexicographic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates the previous lexicographically ordered subset.</p>
-<p>NOT YET IMPLEMENTED</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.subsets.Subset.rank_binary">
-<tt class="descname">rank_binary</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/subsets.html#Subset.rank_binary"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.subsets.Subset.rank_binary" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the binary ordered rank.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.subsets.Subset.iterate_binary" title="sympy.combinatorics.subsets.Subset.iterate_binary"><tt class="xref py py-obj docutils literal"><span class="pre">iterate_binary</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.unrank_binary" title="sympy.combinatorics.subsets.Subset.unrank_binary"><tt class="xref py py-obj docutils literal"><span class="pre">unrank_binary</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.subsets</span> <span class="kn">import</span> <span class="n">Subset</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Subset</span><span class="p">([],</span> <span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">rank_binary</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Subset</span><span class="p">([</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">],</span> <span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">rank_binary</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.subsets.Subset.rank_gray">
-<tt class="descname">rank_gray</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/subsets.html#Subset.rank_gray"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.subsets.Subset.rank_gray" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the Gray code ranking of the subset.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.subsets.Subset.iterate_graycode" title="sympy.combinatorics.subsets.Subset.iterate_graycode"><tt class="xref py py-obj docutils literal"><span class="pre">iterate_graycode</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.unrank_gray" title="sympy.combinatorics.subsets.Subset.unrank_gray"><tt class="xref py py-obj docutils literal"><span class="pre">unrank_gray</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.subsets</span> <span class="kn">import</span> <span class="n">Subset</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Subset</span><span class="p">([</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">],</span> <span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">rank_gray</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Subset</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">rank_gray</span>
-<span class="go">27</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.subsets.Subset.rank_lexicographic">
-<tt class="descname">rank_lexicographic</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/subsets.html#Subset.rank_lexicographic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.subsets.Subset.rank_lexicographic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the lexicographic ranking of the subset.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.subsets</span> <span class="kn">import</span> <span class="n">Subset</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Subset</span><span class="p">([</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">],</span> <span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">rank_lexicographic</span>
-<span class="go">14</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Subset</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">rank_lexicographic</span>
-<span class="go">43</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.subsets.Subset.size">
-<tt class="descname">size</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/subsets.html#Subset.size"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.subsets.Subset.size" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gets the size of the subset.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.subsets.Subset.subset" title="sympy.combinatorics.subsets.Subset.subset"><tt class="xref py py-obj docutils literal"><span class="pre">subset</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.superset" title="sympy.combinatorics.subsets.Subset.superset"><tt class="xref py py-obj docutils literal"><span class="pre">superset</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.superset_size" title="sympy.combinatorics.subsets.Subset.superset_size"><tt class="xref py py-obj docutils literal"><span class="pre">superset_size</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.cardinality" title="sympy.combinatorics.subsets.Subset.cardinality"><tt class="xref py py-obj docutils literal"><span class="pre">cardinality</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.subsets</span> <span class="kn">import</span> <span class="n">Subset</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Subset</span><span class="p">([</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">],</span> <span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">size</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.subsets.Subset.subset">
-<tt class="descname">subset</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/subsets.html#Subset.subset"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.subsets.Subset.subset" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gets the subset represented by the current instance.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.subsets.Subset.superset" title="sympy.combinatorics.subsets.Subset.superset"><tt class="xref py py-obj docutils literal"><span class="pre">superset</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.size" title="sympy.combinatorics.subsets.Subset.size"><tt class="xref py py-obj docutils literal"><span class="pre">size</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.superset_size" title="sympy.combinatorics.subsets.Subset.superset_size"><tt class="xref py py-obj docutils literal"><span class="pre">superset_size</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.cardinality" title="sympy.combinatorics.subsets.Subset.cardinality"><tt class="xref py py-obj docutils literal"><span class="pre">cardinality</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.subsets</span> <span class="kn">import</span> <span class="n">Subset</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Subset</span><span class="p">([</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">],</span> <span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">subset</span>
-<span class="go">[&#39;c&#39;, &#39;d&#39;]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.combinatorics.subsets.Subset.subset_from_bitlist">
-<em class="property">classmethod </em><tt class="descname">subset_from_bitlist</tt><big>(</big><em>super_set</em>, <em>bitlist</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/subsets.html#Subset.subset_from_bitlist"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.subsets.Subset.subset_from_bitlist" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gets the subset defined by the bitlist.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.subsets.Subset.bitlist_from_subset" title="sympy.combinatorics.subsets.Subset.bitlist_from_subset"><tt class="xref py py-obj docutils literal"><span class="pre">bitlist_from_subset</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.subsets</span> <span class="kn">import</span> <span class="n">Subset</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Subset</span><span class="o">.</span><span class="n">subset_from_bitlist</span><span class="p">([</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">],</span> <span class="s">&#39;0011&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">subset</span>
-<span class="go">[&#39;c&#39;, &#39;d&#39;]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.combinatorics.subsets.Subset.subset_indices">
-<em class="property">classmethod </em><tt class="descname">subset_indices</tt><big>(</big><em>subset</em>, <em>superset</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/subsets.html#Subset.subset_indices"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.subsets.Subset.subset_indices" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return indices of subset in superset in a list; the list is empty
-if all elements of subset are not in superset.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Subset</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">superset</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Subset</span><span class="o">.</span><span class="n">subset_indices</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">superset</span><span class="p">)</span>
-<span class="go">[1, 2, 0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Subset</span><span class="o">.</span><span class="n">subset_indices</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="n">superset</span><span class="p">)</span>
-<span class="go">[]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Subset</span><span class="o">.</span><span class="n">subset_indices</span><span class="p">([],</span> <span class="n">superset</span><span class="p">)</span>
-<span class="go">[]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.subsets.Subset.superset">
-<tt class="descname">superset</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/subsets.html#Subset.superset"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.subsets.Subset.superset" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gets the superset of the subset.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.subsets.Subset.subset" title="sympy.combinatorics.subsets.Subset.subset"><tt class="xref py py-obj docutils literal"><span class="pre">subset</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.size" title="sympy.combinatorics.subsets.Subset.size"><tt class="xref py py-obj docutils literal"><span class="pre">size</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.superset_size" title="sympy.combinatorics.subsets.Subset.superset_size"><tt class="xref py py-obj docutils literal"><span class="pre">superset_size</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.cardinality" title="sympy.combinatorics.subsets.Subset.cardinality"><tt class="xref py py-obj docutils literal"><span class="pre">cardinality</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.subsets</span> <span class="kn">import</span> <span class="n">Subset</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Subset</span><span class="p">([</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">],</span> <span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">superset</span>
-<span class="go">[&#39;a&#39;, &#39;b&#39;, &#39;c&#39;, &#39;d&#39;]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.combinatorics.subsets.Subset.superset_size">
-<tt class="descname">superset_size</tt><a class="reference internal" href="../../_modules/sympy/combinatorics/subsets.html#Subset.superset_size"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.subsets.Subset.superset_size" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the size of the superset.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.subsets.Subset.subset" title="sympy.combinatorics.subsets.Subset.subset"><tt class="xref py py-obj docutils literal"><span class="pre">subset</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.superset" title="sympy.combinatorics.subsets.Subset.superset"><tt class="xref py py-obj docutils literal"><span class="pre">superset</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.size" title="sympy.combinatorics.subsets.Subset.size"><tt class="xref py py-obj docutils literal"><span class="pre">size</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.cardinality" title="sympy.combinatorics.subsets.Subset.cardinality"><tt class="xref py py-obj docutils literal"><span class="pre">cardinality</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.subsets</span> <span class="kn">import</span> <span class="n">Subset</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Subset</span><span class="p">([</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">],</span> <span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">superset_size</span>
-<span class="go">4</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.combinatorics.subsets.Subset.unrank_binary">
-<em class="property">classmethod </em><tt class="descname">unrank_binary</tt><big>(</big><em>rank</em>, <em>superset</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/subsets.html#Subset.unrank_binary"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.subsets.Subset.unrank_binary" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gets the binary ordered subset of the specified rank.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.subsets.Subset.iterate_binary" title="sympy.combinatorics.subsets.Subset.iterate_binary"><tt class="xref py py-obj docutils literal"><span class="pre">iterate_binary</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.rank_binary" title="sympy.combinatorics.subsets.Subset.rank_binary"><tt class="xref py py-obj docutils literal"><span class="pre">rank_binary</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.subsets</span> <span class="kn">import</span> <span class="n">Subset</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Subset</span><span class="o">.</span><span class="n">unrank_binary</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">,</span><span class="s">&#39;d&#39;</span><span class="p">])</span><span class="o">.</span><span class="n">subset</span>
-<span class="go">[&#39;b&#39;]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.combinatorics.subsets.Subset.unrank_gray">
-<em class="property">classmethod </em><tt class="descname">unrank_gray</tt><big>(</big><em>rank</em>, <em>superset</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/subsets.html#Subset.unrank_gray"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.subsets.Subset.unrank_gray" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gets the Gray code ordered subset of the specified rank.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.subsets.Subset.iterate_graycode" title="sympy.combinatorics.subsets.Subset.iterate_graycode"><tt class="xref py py-obj docutils literal"><span class="pre">iterate_graycode</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.subsets.Subset.rank_gray" title="sympy.combinatorics.subsets.Subset.rank_gray"><tt class="xref py py-obj docutils literal"><span class="pre">rank_gray</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.subsets</span> <span class="kn">import</span> <span class="n">Subset</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Subset</span><span class="o">.</span><span class="n">unrank_gray</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">])</span><span class="o">.</span><span class="n">subset</span>
-<span class="go">[&#39;a&#39;, &#39;b&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Subset</span><span class="o">.</span><span class="n">unrank_gray</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">,</span><span class="s">&#39;b&#39;</span><span class="p">,</span><span class="s">&#39;c&#39;</span><span class="p">])</span><span class="o">.</span><span class="n">subset</span>
-<span class="go">[]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="method">
-<dt id="sympy.combinatorics.subsets.ksubsets">
-<tt class="descclassname">subsets.</tt><tt class="descname">ksubsets</tt><big>(</big><em>superset</em>, <em>k</em><big>)</big><a class="headerlink" href="#sympy.combinatorics.subsets.ksubsets" title="Permalink to this definition">¶</a></dt>
-<dd><p>Finds the subsets of size k in lexicographic order.</p>
-<p>This uses the itertools generator.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">class</span></tt></dt>
-<dd>Subset</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.subsets</span> <span class="kn">import</span> <span class="n">ksubsets</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">ksubsets</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">],</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">[(1, 2), (1, 3), (2, 3)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">ksubsets</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">],</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">[(1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4),     (2, 5), (3, 4), (3, 5), (4, 5)]</span>
-</pre></div>
-</div>
-</dd></dl>
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-  <div class="section" id="module-sympy.combinatorics.tensor_can">
-<span id="tensor-canonicalization"></span><span id="combinatorics-tensor-can"></span><h1>Tensor Canonicalization<a class="headerlink" href="#module-sympy.combinatorics.tensor_can" title="Permalink to this headline">¶</a></h1>
-<dl class="function">
-<dt id="sympy.combinatorics.tensor_can.canonicalize">
-<tt class="descclassname">sympy.combinatorics.tensor_can.</tt><tt class="descname">canonicalize</tt><big>(</big><em>g</em>, <em>dummies</em>, <em>msym</em>, <em>*v</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/tensor_can.html#canonicalize"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.tensor_can.canonicalize" title="Permalink to this definition">¶</a></dt>
-<dd><p>canonicalize tensor formed by tensors</p>
-<blockquote>
-<div><p>g     permutation representing the tensor</p>
-<dl class="docutils">
-<dt>dummies list representing the dummy indices</dt>
-<dd>it can be a list of dummy indices of the same type
-or a list of lists of dummy indices, one list for each
-type of index;
-the dummy indices must come after the free indices,
-and put in order contravariant, covariant
-[d0, -d0, d1,-d1,...]</dd>
-<dt>msym  symmetry of the metric(s)</dt>
-<dd>it can be an integer or a list;
-in the first case it is the symmetry of the dummy index metric;
-in the second case it is the list of the symmetries of the
-index metric for each type</dd>
-<dt>v     is a list of (base_i, gens_i, n_i, sym_i) for tensors of type <span class="math">\(i\)</span></dt>
-<dd><p class="first">base_i, gens_i BSGS for tensors of this type.</p>
-<p>The BSGS should have minimal base under lexicographic ordering;
-if not, an attempt is made do get the minimal BSGS;
-in case of failure,
-canonicalize_naive is used, which is much slower.</p>
-<p>n_i     number ot tensors of type <span class="math">\(i\)</span>.</p>
-<p class="last">sym_i   symmetry under exchange of component tensors of type <span class="math">\(i\)</span>.</p>
-</dd>
-<dt>Both for msym and sym_i the cases are</dt>
-<dd><ul class="first last simple">
-<li>None  no symmetry</li>
-<li>0     commuting</li>
-<li>1     anticommuting</li>
-</ul>
-</dd>
-</dl>
-</div></blockquote>
-<p>Return 0 if the tensor is zero, else return the array form of
-the permutation representing the canonical form of the tensor.</p>
-<p class="rubric">Examples</p>
-<p>one type of index with commuting metric;</p>
-<p><span class="math">\(A_{a b}\)</span> and <span class="math">\(B_{a b}\)</span> antisymmetric and commuting</p>
-<p><span class="math">\(T = A_{d0 d1} * B^{d0}{}_{d2} * B^{d2 d1}\)</span></p>
-<p><span class="math">\(ord = [d0,-d0,d1,-d1,d2,-d2]\)</span> order of the indices</p>
-<p>g = [1,3,0,5,4,2,6,7]</p>
-<p>T_c = 0</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.tensor_can</span> <span class="kn">import</span> <span class="n">get_symmetric_group_sgs</span><span class="p">,</span> <span class="n">canonicalize</span><span class="p">,</span> <span class="n">bsgs_direct_product</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">base2a</span><span class="p">,</span> <span class="n">gens2a</span> <span class="o">=</span> <span class="n">get_symmetric_group_sgs</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t0</span> <span class="o">=</span> <span class="p">(</span><span class="n">base2a</span><span class="p">,</span> <span class="n">gens2a</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t1</span> <span class="o">=</span> <span class="p">(</span><span class="n">base2a</span><span class="p">,</span> <span class="n">gens2a</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">canonicalize</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">6</span><span class="p">)),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">t0</span><span class="p">,</span> <span class="n">t1</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>same as above, but with <span class="math">\(B_{a b}\)</span> anticommuting</p>
-<p><span class="math">\(T_c = -A^{d0 d1} * B_{d0}{}^{d2} * B_{d1 d2}\)</span></p>
-<p><span class="math">\(can = [0,2,1,4,3,5,7,6]\)</span></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">t1</span> <span class="o">=</span> <span class="p">(</span><span class="n">base2a</span><span class="p">,</span> <span class="n">gens2a</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">canonicalize</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">6</span><span class="p">)),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">t0</span><span class="p">,</span> <span class="n">t1</span><span class="p">)</span>
-<span class="go">[0, 2, 1, 4, 3, 5, 7, 6]</span>
-</pre></div>
-</div>
-<p>two types of indices <span class="math">\([a,b,c,d,e,f]\)</span> and <span class="math">\([m,n]\)</span>, in this order,
-both with commuting metric</p>
-<p><span class="math">\(f^{a b c}\)</span> antisymmetric, commuting</p>
-<p><span class="math">\(A_{m a}\)</span> no symmetry, commuting</p>
-<p><span class="math">\(T = f^c{}_{d a} * f^f{}_{e b} * A_m{}^d * A^{m b} * A_n{}^a * A^{n e}\)</span></p>
-<p>ord = [c,f,a,-a,b,-b,d,-d,e,-e,m,-m,n,-n]</p>
-<p>g = [0,7,3, 1,9,5, 11,6, 10,4, 13,2, 12,8, 14,15]</p>
-<p>The canonical tensor is
-<span class="math">\(T_c = -f^{c a b} * f^{f d e} * A^m{}_a * A_{m d} * A^n{}_b * A_{n e}\)</span></p>
-<p>can = [0,2,4, 1,6,8, 10,3, 11,7, 12,5, 13,9, 15,14]</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">base_f</span><span class="p">,</span> <span class="n">gens_f</span> <span class="o">=</span> <span class="n">get_symmetric_group_sgs</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">base1</span><span class="p">,</span> <span class="n">gens1</span> <span class="o">=</span> <span class="n">get_symmetric_group_sgs</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">base_A</span><span class="p">,</span> <span class="n">gens_A</span> <span class="o">=</span> <span class="n">bsgs_direct_product</span><span class="p">(</span><span class="n">base1</span><span class="p">,</span> <span class="n">gens1</span><span class="p">,</span> <span class="n">base1</span><span class="p">,</span> <span class="n">gens1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t0</span> <span class="o">=</span> <span class="p">(</span><span class="n">base_f</span><span class="p">,</span> <span class="n">gens_f</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t1</span> <span class="o">=</span> <span class="p">(</span><span class="n">base_A</span><span class="p">,</span> <span class="n">gens_A</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dummies</span> <span class="o">=</span> <span class="p">[</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">10</span><span class="p">)),</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">14</span><span class="p">))]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">11</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">13</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">12</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">14</span><span class="p">,</span><span class="mi">15</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">canonicalize</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">dummies</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">t0</span><span class="p">,</span> <span class="n">t1</span><span class="p">)</span>
-<span class="go">[0, 2, 4, 1, 6, 8, 10, 3, 11, 7, 12, 5, 13, 9, 15, 14]</span>
-</pre></div>
-</div>
-<p class="rubric">Algorithm</p>
-<p>First one uses canonical_free to get the minimum tensor under
-lexicographic order, using only the slot symmetries.
-If the component tensors have not minimal BSGS, it is attempted
-to find it; if the attempt fails canonicalize_naive
-is used instead.</p>
-<p>Compute the residual slot symmetry keeping fixed the free indices
-using tensor_gens(base, gens, list_free_indices, sym).</p>
-<p>Reduce the problem eliminating the free indices.</p>
-<p>Then use double_coset_can_rep and lift back the result reintroducing
-the free indices.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.tensor_can.double_coset_can_rep">
-<tt class="descclassname">sympy.combinatorics.tensor_can.</tt><tt class="descname">double_coset_can_rep</tt><big>(</big><em>dummies</em>, <em>sym</em>, <em>b_S</em>, <em>sgens</em>, <em>S_transversals</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/tensor_can.html#double_coset_can_rep"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.tensor_can.double_coset_can_rep" title="Permalink to this definition">¶</a></dt>
-<dd><p>Butler-Portugal algorithm for tensor canonicalization with dummy indices</p>
-<blockquote>
-<div><dl class="docutils">
-<dt>dummies</dt>
-<dd>list of lists of dummy indices,
-one list for each type of index;
-the dummy indices are put in order contravariant, covariant
-[d0, -d0, d1,-d1,...].</dd>
-<dt>sym</dt>
-<dd>list of the symmetries of the index metric for each type.</dd>
-<dt>possible symmetries of the metrics</dt>
-<dd><ul class="first last simple">
-<li>0     symmetric</li>
-<li>1     antisymmetric</li>
-<li>None  no symmetry</li>
-</ul>
-</dd>
-<dt>b_S</dt>
-<dd>base of a minimal slot symmetry BSGS.</dd>
-<dt>sgens</dt>
-<dd>generators of the slot symmetry BSGS.</dd>
-<dt>S_transversals</dt>
-<dd>transversals for the slot BSGS.</dd>
-<dt>g</dt>
-<dd>permutation representing the tensor.</dd>
-</dl>
-</div></blockquote>
-<p>Return 0 if the tensor is zero, else return the array form of
-the permutation representing the canonical form of the tensor.</p>
-<p>A tensor with dummy indices can be represented in a number
-of equivalent ways which typically grows exponentially with
-the number of indices. To be able to establish if two tensors
-with many indices are equal becomes computationally very slow
-in absence of an efficient algorithm.</p>
-<p>The Butler-Portugal algorithm [3] is an efficient algorithm to
-put tensors in canonical form, solving the above problem.</p>
-<p>Portugal observed that a tensor can be represented by a permutation,
-and that the class of tensors equivalent to it under slot and dummy
-symmetries is equivalent to the double coset <span class="math">\(D*g*S\)</span>
-(Note: in this documentation we use the conventions for multiplication
-of permutations p, q with (p*q)(i) = p[q[i]] which is opposite
-to the one used in the Permutation class)</p>
-<p>Using the algorithm by Butler to find a representative of the
-double coset one can find a canonical form for the tensor.</p>
-<p>To see this correspondence,
-let <span class="math">\(g\)</span> be a permutation in array form; a tensor with indices <span class="math">\(ind\)</span>
-(the indices including both the contravariant and the covariant ones)
-can be written as</p>
-<p><span class="math">\(t = T(ind[g[0],..., ind[g[n-1]])\)</span>,</p>
-<p>where <span class="math">\(n= len(ind)\)</span>;
-<span class="math">\(g\)</span> has size <span class="math">\(n + 2\)</span>, the last two indices for the sign of the tensor
-(trick introduced in [4]).</p>
-<p>A slot symmetry transformation <span class="math">\(s\)</span> is a permutation acting on the slots
-<span class="math">\(t -&gt; T(ind[(g*s)[0]],..., ind[(g*s)[n-1]])\)</span></p>
-<p>A dummy symmetry transformation acts on <span class="math">\(ind\)</span>
-<span class="math">\(t -&gt; T(ind[(d*g)[0]],..., ind[(d*g)[n-1]])\)</span></p>
-<p>Being interested only in the transformations of the tensor under
-these symmetries, one can represent the tensor by <span class="math">\(g\)</span>, which transforms
-as</p>
-<p><span class="math">\(g -&gt; d*g*s\)</span>, so it belongs to the coset <span class="math">\(D*g*S\)</span>.</p>
-<p>Let us explain the conventions by an example.</p>
-<dl class="docutils">
-<dt>Given a tensor <span class="math">\(T^{d3 d2 d1}{}_{d1 d2 d3}\)</span> with the slot symmetries</dt>
-<dd><p class="first"><span class="math">\(T^{a0 a1 a2 a3 a4 a5} = -T^{a2 a1 a0 a3 a4 a5}\)</span></p>
-<p class="last"><span class="math">\(T^{a0 a1 a2 a3 a4 a5} = -T^{a4 a1 a2 a3 a0 a5}\)</span></p>
-</dd>
-</dl>
-<p>and symmetric metric, find the tensor equivalent to it which
-is the lowest under the ordering of indices:
-lexicographic ordering <span class="math">\(d1, d2, d3\)</span> then and contravariant index
-before covariant index; that is the canonical form of the tensor.</p>
-<p>The canonical form is <span class="math">\(-T^{d1 d2 d3}{}_{d1 d2 d3}\)</span>
-obtained using <span class="math">\(T^{a0 a1 a2 a3 a4 a5} = -T^{a2 a1 a0 a3 a4 a5}\)</span>.</p>
-<p>To convert this problem in the input for this function,
-use the following labelling of the index names
-(- for covariant for short) <span class="math">\(d1, -d1, d2, -d2, d3, -d3\)</span></p>
-<p><span class="math">\(T^{d3 d2 d1}{}_{d1 d2 d3}\)</span> corresponds to <span class="math">\(g = [4,2,0,1,3,5,6,7]\)</span>
-where the last two indices are for the sign</p>
-<p><span class="math">\(sgens = [Permutation(0,2)(6,7), Permutation(0,4)(6,7)]\)</span></p>
-<p>sgens[0] is the slot symmetry <span class="math">\(-(0,2)\)</span>
-<span class="math">\(T^{a0 a1 a2 a3 a4 a5} = -T^{a2 a1 a0 a3 a4 a5}\)</span></p>
-<p>sgens[1] is the slot symmetry <span class="math">\(-(0,4)\)</span>
-<span class="math">\(T^{a0 a1 a2 a3 a4 a5} = -T^{a4 a1 a2 a3 a0 a5}\)</span></p>
-<p>The dummy symmetry group D is generated by the strong base generators
-<span class="math">\([(0,1),(2,3),(4,5),(0,1)(2,3),(2,3)(4,5)]\)</span></p>
-<p>The dummy symmetry acts from the left
-<span class="math">\(d = [1,0,2,3,4,5,6,7]\)</span>  exchange <span class="math">\(d1 -&gt; -d1\)</span>
-<span class="math">\(T^{d3 d2 d1}{}_{d1 d2 d3} == T^{d3 d2}{}_{d1}{}^{d1}{}_{d2 d3}\)</span></p>
-<p><span class="math">\(g=[4,2,0,1,3,5,6,7]  -&gt; [4,2,1,0,3,5,6,7] = _af_rmul(d, g)\)</span>
-which differs from <span class="math">\(_af_rmul(g, d)\)</span>.</p>
-<p>The slot symmetry acts from the right
-<span class="math">\(s = [2,1,0,3,4,5,7,6]\)</span>  exchanges slots 0 and 2 and changes sign
-<span class="math">\(T^{d3 d2 d1}{}_{d1 d2 d3} == -T^{d1 d2 d3}{}_{d1 d2 d3}\)</span></p>
-<p><span class="math">\(g=[4,2,0,1,3,5,6,7]  -&gt; [0,2,4,1,3,5,7,6] = _af_rmul(g, s)\)</span></p>
-<p>Example in which the tensor is zero, same slot symmetries as above:
-<span class="math">\(T^{d3}{}_{d1,d2}{}^{d1}{}_{d3}{}^{d2}\)</span></p>
-<p><span class="math">\(= -T^{d3}{}_{d1,d3}{}^{d1}{}_{d2}{}^{d2}\)</span>   under slot symmetry <span class="math">\(-(2,4)\)</span>;</p>
-<p><span class="math">\(= T_{d3 d1}{}^{d3}{}^{d1}{}_{d2}{}^{d2}\)</span>    under slot symmetry <span class="math">\(-(0,2)\)</span>;</p>
-<p><span class="math">\(= T^{d3}{}_{d1 d3}{}^{d1}{}_{d2}{}^{d2}\)</span>    symmetric metric;</p>
-<p><span class="math">\(= 0\)</span>  since two of these lines have tensors differ only for the sign.</p>
-<p>The double coset D*g*S consists of permutations <span class="math">\(h = d*g*s\)</span> corresponding
-to equivalent tensors; if there are two <span class="math">\(h\)</span> which are the same apart
-from the sign, return zero; otherwise
-choose as representative the tensor with indices
-ordered lexicographically according to <span class="math">\([d1, -d1, d2, -d2, d3, -d3]\)</span>
-that is <span class="math">\(rep = min(D*g*S) = min([d*g*s for d in D for s in S])\)</span></p>
-<p>The indices are fixed one by one; first choose the lowest index
-for slot 0, then the lowest remaining index for slot 1, etc.
-Doing this one obtains a chain of stabilizers</p>
-<p><span class="math">\(S -&gt; S_{b0} -&gt; S_{b0,b1} -&gt; ...\)</span> and
-<span class="math">\(D -&gt; D_{p0} -&gt; D_{p0,p1} -&gt; ...\)</span></p>
-<p>where <span class="math">\([b0, b1, ...] = range(b)\)</span> is a base of the symmetric group;
-the strong base <span class="math">\(b_S\)</span> of S is an ordered sublist of it;
-therefore it is sufficient to compute once the
-strong base generators of S using the Schreier-Sims algorithm;
-the stabilizers of the strong base generators are the
-strong base generators of the stabilizer subgroup.</p>
-<p><span class="math">\(dbase = [p0,p1,...]\)</span> is not in general in lexicographic order,
-so that one must recompute the strong base generators each time;
-however this is trivial, there is no need to use the Schreier-Sims
-algorithm for D.</p>
-<p>The algorithm keeps a TAB of elements <span class="math">\((s_i, d_i, h_i)\)</span>
-where <span class="math">\(h_i = d_i*g*s_i\)</span> satisfying <span class="math">\(h_i[j] = p_j\)</span> for <span class="math">\(0 &lt;= j &lt; i\)</span>
-starting from <span class="math">\(s_0 = id, d_0 = id, h_0 = g\)</span>.</p>
-<p>The equations <span class="math">\(h_0[0] = p_0, h_1[1] = p_1,...\)</span> are solved in this order,
-choosing each time the lowest possible value of p_i</p>
-<p>For <span class="math">\(j &lt; i\)</span>
-<span class="math">\(d_i*g*s_i*S_{b_0,...,b_{i-1}}*b_j = D_{p_0,...,p_{i-1}}*p_j\)</span>
-so that for dx in <span class="math">\(D_{p_0,...,p_{i-1}}\)</span> and sx in
-<span class="math">\(S_{base[0],...,base[i-1]}\)</span> one has <span class="math">\(dx*d_i*g*s_i*sx*b_j = p_j\)</span></p>
-<p>Search for dx, sx such that this equation holds for <span class="math">\(j = i\)</span>;
-it can be written as <span class="math">\(s_i*sx*b_j = J, dx*d_i*g*J = p_j\)</span>
-<span class="math">\(sx*b_j = s_i**-1*J; sx = trace(s_i**-1, S_{b_0,...,b_{i-1}})\)</span>
-<span class="math">\(dx**-1*p_j = d_i*g*J; dx = trace(d_i*g*J, D_{p_0,...,p_{i-1}})\)</span></p>
-<p><span class="math">\(s_{i+1} = s_i*trace(s_i**-1*J, S_{b_0,...,b_{i-1}})\)</span>
-<span class="math">\(d_{i+1} = trace(d_i*g*J, D_{p_0,...,p_{i-1}})**-1*d_i\)</span>
-<span class="math">\(h_{i+1}*b_i = d_{i+1}*g*s_{i+1}*b_i = p_i\)</span></p>
-<p><span class="math">\(h_n*b_j = p_j\)</span> for all j, so that <span class="math">\(h_n\)</span> is the solution.</p>
-<p>Add the found <span class="math">\((s, d, h)\)</span> to TAB1.</p>
-<p>At the end of the iteration sort TAB1 with respect to the <span class="math">\(h\)</span>;
-if there are two consecutive <span class="math">\(h\)</span> in TAB1 which differ only for the
-sign, the tensor is zero, so return 0;
-if there are two consecutive <span class="math">\(h\)</span> which are equal, keep only one.</p>
-<p>Then stabilize the slot generators under <span class="math">\(i\)</span> and the dummy generators
-under <span class="math">\(p_i\)</span>.</p>
-<p>Assign <span class="math">\(TAB = TAB1\)</span> at the end of the iteration step.</p>
-<p>At the end <span class="math">\(TAB\)</span> contains a unique <span class="math">\((s, d, h)\)</span>, since all the slots
-of the tensor <span class="math">\(h\)</span> have been fixed to have the minimum value according
-to the symmetries. The algorithm returns <span class="math">\(h\)</span>.</p>
-<p>It is important that the slot BSGS has lexicographic minimal base,
-otherwise there is an <span class="math">\(i\)</span> which does not belong to the slot base
-for which <span class="math">\(p_i\)</span> is fixed by the dummy symmetry only, while <span class="math">\(i\)</span>
-is not invariant from the slot stabilizer, so <span class="math">\(p_i\)</span> is not in
-general the minimal value.</p>
-<dl class="docutils">
-<dt>This algorithm differs slightly from the original algorithm [3]:</dt>
-<dd>the canonical form is minimal lexicographically, and
-the BSGS has minimal base under lexicographic order.
-Equal tensors <span class="math">\(h\)</span> are eliminated from TAB.</dd>
-</dl>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.tensor_can</span> <span class="kn">import</span> <span class="n">double_coset_can_rep</span><span class="p">,</span> <span class="n">get_transversals</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gens</span> <span class="o">=</span> <span class="p">[</span><span class="n">Permutation</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="p">[[</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">6</span><span class="p">]]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">base</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">transversals</span> <span class="o">=</span> <span class="n">get_transversals</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">gens</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">double_coset_can_rep</span><span class="p">([</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">6</span><span class="p">))],</span> <span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">base</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">transversals</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-<span class="go">[0, 1, 2, 3, 4, 5, 7, 6]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">double_coset_can_rep</span><span class="p">([</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">6</span><span class="p">))],</span> <span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">base</span><span class="p">,</span> <span class="n">gens</span><span class="p">,</span> <span class="n">transversals</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.tensor_can.get_symmetric_group_sgs">
-<tt class="descclassname">sympy.combinatorics.tensor_can.</tt><tt class="descname">get_symmetric_group_sgs</tt><big>(</big><em>n</em>, <em>sym=0</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/tensor_can.html#get_symmetric_group_sgs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.tensor_can.get_symmetric_group_sgs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return base, gens of the minimal BSGS for (anti)symmetric group
-with n elements</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.tensor_can</span> <span class="kn">import</span> <span class="n">get_symmetric_group_sgs</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">get_symmetric_group_sgs</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">([0, 1], [Permutation(4)(0, 1), Permutation(4)(1, 2)])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.tensor_can.bsgs_direct_product">
-<tt class="descclassname">sympy.combinatorics.tensor_can.</tt><tt class="descname">bsgs_direct_product</tt><big>(</big><em>base1</em>, <em>gens1</em>, <em>base2</em>, <em>gens2</em>, <em>signed=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/tensor_can.html#bsgs_direct_product"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.tensor_can.bsgs_direct_product" title="Permalink to this definition">¶</a></dt>
-<dd><p>direct product of two BSGS</p>
-<p>base1    base of the first BSGS.</p>
-<p>gens1    strong generating sequence of the first BSGS.</p>
-<p>base2, gens2   similarly for the second BSGS.</p>
-<p>signed   flag for signed permutations.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.tensor_can</span> <span class="kn">import</span> <span class="p">(</span><span class="n">get_symmetric_group_sgs</span><span class="p">,</span> <span class="n">bsgs_direct_product</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">base1</span><span class="p">,</span> <span class="n">gens1</span> <span class="o">=</span> <span class="n">get_symmetric_group_sgs</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">base2</span><span class="p">,</span> <span class="n">gens2</span> <span class="o">=</span> <span class="n">get_symmetric_group_sgs</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bsgs_direct_product</span><span class="p">(</span><span class="n">base1</span><span class="p">,</span> <span class="n">gens1</span><span class="p">,</span> <span class="n">base2</span><span class="p">,</span> <span class="n">gens2</span><span class="p">)</span>
-<span class="go">([1], [Permutation(4)(1, 2)])</span>
-</pre></div>
-</div>
-</dd></dl>
-
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-  <div class="section" id="module-sympy.combinatorics.testutil">
-<span id="test-utilities"></span><span id="combinatorics-testutil"></span><h1>Test Utilities<a class="headerlink" href="#module-sympy.combinatorics.testutil" title="Permalink to this headline">¶</a></h1>
-<dl class="function">
-<dt id="sympy.combinatorics.testutil._cmp_perm_lists">
-<tt class="descclassname">sympy.combinatorics.testutil.</tt><tt class="descname">_cmp_perm_lists</tt><big>(</big><em>first</em>, <em>second</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/testutil.html#_cmp_perm_lists"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.testutil._cmp_perm_lists" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compare two lists of permutations as sets.</p>
-<p>This is used for testing purposes. Since the array form of a
-permutation is currently a list, Permutation is not hashable
-and cannot be put into a set.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.testutil</span> <span class="kn">import</span> <span class="n">_cmp_perm_lists</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ls1</span> <span class="o">=</span> <span class="p">[</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ls2</span> <span class="o">=</span> <span class="p">[</span><span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">a</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_cmp_perm_lists</span><span class="p">(</span><span class="n">ls1</span><span class="p">,</span> <span class="n">ls2</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.testutil._naive_list_centralizer">
-<tt class="descclassname">sympy.combinatorics.testutil.</tt><tt class="descname">_naive_list_centralizer</tt><big>(</big><em>self</em>, <em>other</em>, <em>af=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/testutil.html#_naive_list_centralizer"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.testutil._naive_list_centralizer" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.testutil._verify_bsgs">
-<tt class="descclassname">sympy.combinatorics.testutil.</tt><tt class="descname">_verify_bsgs</tt><big>(</big><em>group</em>, <em>base</em>, <em>gens</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/testutil.html#_verify_bsgs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.testutil._verify_bsgs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Verify the correctness of a base and strong generating set.</p>
-<p>This is a naive implementation using the definition of a base and a strong
-generating set relative to it. There are other procedures for
-verifying a base and strong generating set, but this one will
-serve for more robust testing.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="../combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims" title="sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">AlternatingGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.testutil</span> <span class="kn">import</span> <span class="n">_verify_bsgs</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">AlternatingGroup</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">schreier_sims</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_verify_bsgs</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">A</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">A</span><span class="o">.</span><span class="n">strong_gens</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.testutil._verify_centralizer">
-<tt class="descclassname">sympy.combinatorics.testutil.</tt><tt class="descname">_verify_centralizer</tt><big>(</big><em>group</em>, <em>arg</em>, <em>centr=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/testutil.html#_verify_centralizer"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.testutil._verify_centralizer" title="Permalink to this definition">¶</a></dt>
-<dd><p>Verify the centralizer of a group/set/element inside another group.</p>
-<p>This is used for testing <tt class="docutils literal"><span class="pre">.centralizer()</span></tt> from
-<tt class="docutils literal"><span class="pre">sympy.combinatorics.perm_groups</span></tt></p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.testutil._naive_list_centralizer" title="sympy.combinatorics.testutil._naive_list_centralizer"><tt class="xref py py-obj docutils literal"><span class="pre">_naive_list_centralizer</span></tt></a>, <a class="reference internal" href="../combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.centralizer" title="sympy.combinatorics.perm_groups.PermutationGroup.centralizer"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.combinatorics.perm_groups.PermutationGroup.centralizer</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.testutil._cmp_perm_lists" title="sympy.combinatorics.testutil._cmp_perm_lists"><tt class="xref py py-obj docutils literal"><span class="pre">_cmp_perm_lists</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="p">(</span><span class="n">SymmetricGroup</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">AlternatingGroup</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.testutil</span> <span class="kn">import</span> <span class="n">_verify_centralizer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">SymmetricGroup</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">AlternatingGroup</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">centr</span> <span class="o">=</span> <span class="n">PermutationGroup</span><span class="p">([</span><span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">])])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_verify_centralizer</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">centr</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.testutil._verify_normal_closure">
-<tt class="descclassname">sympy.combinatorics.testutil.</tt><tt class="descname">_verify_normal_closure</tt><big>(</big><em>group</em>, <em>arg</em>, <em>closure=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/testutil.html#_verify_normal_closure"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.testutil._verify_normal_closure" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
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-  <div class="section" id="module-sympy.combinatorics.util">
-<span id="utilities"></span><span id="combinatorics-util"></span><h1>Utilities<a class="headerlink" href="#module-sympy.combinatorics.util" title="Permalink to this headline">¶</a></h1>
-<dl class="function">
-<dt id="sympy.combinatorics.util._base_ordering">
-<tt class="descclassname">sympy.combinatorics.util.</tt><tt class="descname">_base_ordering</tt><big>(</big><em>base</em>, <em>degree</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/util.html#_base_ordering"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.util._base_ordering" title="Permalink to this definition">¶</a></dt>
-<dd><p>Order <span class="math">\(\{0, 1, ..., n-1\}\)</span> so that base points come first and in order.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>``base`` - the base</strong> :</p>
-<p><strong>``degree`` - the degree of the associated permutation group</strong> :</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>A list ``base_ordering`` such that ``base_ordering[point]`` is the</strong> :</p>
-<p><strong>number of ``point`` in the ordering.</strong> :</p>
-<p><strong>Examples</strong> :</p>
-<p><strong>========</strong> :</p>
-<p><strong>&gt;&gt;&gt; from sympy.combinatorics.named_groups import SymmetricGroup</strong> :</p>
-<p><strong>&gt;&gt;&gt; from sympy.combinatorics.util import _base_ordering</strong> :</p>
-<p><strong>&gt;&gt;&gt; S = SymmetricGroup(4)</strong> :</p>
-<p><strong>&gt;&gt;&gt; S.schreier_sims()</strong> :</p>
-<p><strong>&gt;&gt;&gt; _base_ordering(S.base, S.degree)</strong> :</p>
-<p class="last"><strong>[0, 1, 2, 3]</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Notes</p>
-<p>This is used in backtrack searches, when we define a relation <span class="math">\(&lt;&lt;\)</span> on
-the underlying set for a permutation group of degree <span class="math">\(n\)</span>,
-<span class="math">\(\{0, 1, ..., n-1\}\)</span>, so that if <span class="math">\((b_1, b_2, ..., b_k)\)</span> is a base we
-have <span class="math">\(b_i &lt;&lt; b_j\)</span> whenever <span class="math">\(i&lt;j\)</span> and <span class="math">\(b_i &lt;&lt; a\)</span> for all
-<span class="math">\(i\in\{1,2, ..., k\}\)</span> and <span class="math">\(a\)</span> is not in the base. The idea is developed
-and applied to backtracking algorithms in [1], pp.108-132. The points
-that are not in the base are taken in increasing order.</p>
-<p class="rubric">References</p>
-<p>[1] Holt, D., Eick, B., O&#8217;Brien, E.
-&#8220;Handbook of computational group theory&#8221;</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.util._check_cycles_alt_sym">
-<tt class="descclassname">sympy.combinatorics.util.</tt><tt class="descname">_check_cycles_alt_sym</tt><big>(</big><em>perm</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/util.html#_check_cycles_alt_sym"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.util._check_cycles_alt_sym" title="Permalink to this definition">¶</a></dt>
-<dd><p>Checks for cycles of prime length p with n/2 &lt; p &lt; n-2.</p>
-<p>Here <span class="math">\(n\)</span> is the degree of the permutation. This is a helper function for
-the function is_alt_sym from sympy.combinatorics.perm_groups.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="../combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.is_alt_sym" title="sympy.combinatorics.perm_groups.PermutationGroup.is_alt_sym"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.combinatorics.perm_groups.PermutationGroup.is_alt_sym</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.util</span> <span class="kn">import</span> <span class="n">_check_cycles_alt_sym</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">10</span><span class="p">],</span> <span class="p">[</span><span class="mi">11</span><span class="p">,</span> <span class="mi">12</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_check_cycles_alt_sym</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">10</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_check_cycles_alt_sym</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.util._distribute_gens_by_base">
-<tt class="descclassname">sympy.combinatorics.util.</tt><tt class="descname">_distribute_gens_by_base</tt><big>(</big><em>base</em>, <em>gens</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/util.html#_distribute_gens_by_base"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.util._distribute_gens_by_base" title="Permalink to this definition">¶</a></dt>
-<dd><p>Distribute the group elements <tt class="docutils literal"><span class="pre">gens</span></tt> by membership in basic stabilizers.</p>
-<p>Notice that for a base <span class="math">\((b_1, b_2, ..., b_k)\)</span>, the basic stabilizers
-are defined as <span class="math">\(G^{(i)} = G_{b_1, ..., b_{i-1}}\)</span> for
-<span class="math">\(i \in\{1, 2, ..., k\}\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>``base`` - a sequence of points in `{0, 1, ..., n-1}`</strong> :</p>
-<p><strong>``gens`` - a list of elements of a permutation group of degree `n`.</strong> :</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>List of length `k`, where `k` is</strong> :</p>
-<p><strong>the length of ``base``. The `i`-th entry contains those elements in</strong> :</p>
-<p><strong>``gens`` which fix the first `i` elements of ``base`` (so that the</strong> :</p>
-<p><strong>`0`-th entry is equal to ``gens`` itself). If no element fixes the first</strong> :</p>
-<p><strong>`i` elements of ``base``, the `i`-th element is set to a list containing</strong> :</p>
-<p class="last"><strong>the identity element.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.util._strong_gens_from_distr" title="sympy.combinatorics.util._strong_gens_from_distr"><tt class="xref py py-obj docutils literal"><span class="pre">_strong_gens_from_distr</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.util._orbits_transversals_from_bsgs" title="sympy.combinatorics.util._orbits_transversals_from_bsgs"><tt class="xref py py-obj docutils literal"><span class="pre">_orbits_transversals_from_bsgs</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.util._handle_precomputed_bsgs" title="sympy.combinatorics.util._handle_precomputed_bsgs"><tt class="xref py py-obj docutils literal"><span class="pre">_handle_precomputed_bsgs</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">DihedralGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.util</span> <span class="kn">import</span> <span class="n">_distribute_gens_by_base</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">DihedralGroup</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span><span class="o">.</span><span class="n">schreier_sims</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span><span class="o">.</span><span class="n">strong_gens</span>
-<span class="go">[Permutation(0, 1, 2), Permutation(0, 2), Permutation(1, 2)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span><span class="o">.</span><span class="n">base</span>
-<span class="go">[0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_distribute_gens_by_base</span><span class="p">(</span><span class="n">D</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">D</span><span class="o">.</span><span class="n">strong_gens</span><span class="p">)</span>
-<span class="go">[[Permutation(0, 1, 2), Permutation(0, 2), Permutation(1, 2)],</span>
-<span class="go"> [Permutation(1, 2)]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.util._handle_precomputed_bsgs">
-<tt class="descclassname">sympy.combinatorics.util.</tt><tt class="descname">_handle_precomputed_bsgs</tt><big>(</big><em>base</em>, <em>strong_gens</em>, <em>transversals=None</em>, <em>basic_orbits=None</em>, <em>strong_gens_distr=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/util.html#_handle_precomputed_bsgs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.util._handle_precomputed_bsgs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculate BSGS-related structures from those present.</p>
-<p>The base and strong generating set must be provided; if any of the
-transversals, basic orbits or distributed strong generators are not
-provided, they will be calculated from the base and strong generating set.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>``base`` - the base</strong> :</p>
-<p><strong>``strong_gens`` - the strong generators</strong> :</p>
-<p><strong>``transversals`` - basic transversals</strong> :</p>
-<p><strong>``basic_orbits`` - basic orbits</strong> :</p>
-<p><strong>``strong_gens_distr`` - strong generators distributed by membership in basic</strong> :</p>
-<p><strong>stabilizers</strong> :</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>``(transversals, basic_orbits, strong_gens_distr)`` where ``transversals``</strong> :</p>
-<p><strong>are the basic transversals, ``basic_orbits`` are the basic orbits, and</strong> :</p>
-<p><strong>``strong_gens_distr`` are the strong generators distributed by membership</strong> :</p>
-<p class="last"><strong>in basic stabilizers.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.util._orbits_transversals_from_bsgs" title="sympy.combinatorics.util._orbits_transversals_from_bsgs"><tt class="xref py py-obj docutils literal"><span class="pre">_orbits_transversals_from_bsgs</span></tt></a>, <tt class="xref py py-obj docutils literal"><span class="pre">distribute_gens_by_base</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">DihedralGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.util</span> <span class="kn">import</span> <span class="n">_handle_precomputed_bsgs</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">DihedralGroup</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span><span class="o">.</span><span class="n">schreier_sims</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_handle_precomputed_bsgs</span><span class="p">(</span><span class="n">D</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">D</span><span class="o">.</span><span class="n">strong_gens</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">basic_orbits</span><span class="o">=</span><span class="n">D</span><span class="o">.</span><span class="n">basic_orbits</span><span class="p">)</span>
-<span class="go">([{0: Permutation(2), 1: Permutation(0, 1, 2), 2: Permutation(0, 2)},</span>
-<span class="go">{1: Permutation(2), 2: Permutation(1, 2)}],</span>
-<span class="go">[[0, 1, 2], [1, 2]], [[Permutation(0, 1, 2),</span>
-<span class="go">                       Permutation(0, 2),</span>
-<span class="go">                       Permutation(1, 2)],</span>
-<span class="go">                      [Permutation(1, 2)]])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.util._orbits_transversals_from_bsgs">
-<tt class="descclassname">sympy.combinatorics.util.</tt><tt class="descname">_orbits_transversals_from_bsgs</tt><big>(</big><em>base</em>, <em>strong_gens_distr</em>, <em>transversals_only=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/util.html#_orbits_transversals_from_bsgs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.util._orbits_transversals_from_bsgs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute basic orbits and transversals from a base and strong generating set.</p>
-<p>The generators are provided as distributed across the basic stabilizers.
-If the optional argument <tt class="docutils literal"><span class="pre">transversals_only</span></tt> is set to True, only the
-transversals are returned.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>``base`` - the base</strong> :</p>
-<p><strong>``strong_gens_distr`` - strong generators distributed by membership in basic</strong> :</p>
-<p><strong>stabilizers</strong> :</p>
-<p><strong>``transversals_only`` - a flag swithing between returning only the</strong> :</p>
-<p class="last"><strong>transversals/ both orbits and transversals</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.util._distribute_gens_by_base" title="sympy.combinatorics.util._distribute_gens_by_base"><tt class="xref py py-obj docutils literal"><span class="pre">_distribute_gens_by_base</span></tt></a>, <a class="reference internal" href="#sympy.combinatorics.util._handle_precomputed_bsgs" title="sympy.combinatorics.util._handle_precomputed_bsgs"><tt class="xref py py-obj docutils literal"><span class="pre">_handle_precomputed_bsgs</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">SymmetricGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.util</span> <span class="kn">import</span> <span class="n">_orbits_transversals_from_bsgs</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.util</span> <span class="kn">import</span> <span class="p">(</span><span class="n">_orbits_transversals_from_bsgs</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">_distribute_gens_by_base</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">SymmetricGroup</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">.</span><span class="n">schreier_sims</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">strong_gens_distr</span> <span class="o">=</span> <span class="n">_distribute_gens_by_base</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">strong_gens</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_orbits_transversals_from_bsgs</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens_distr</span><span class="p">)</span>
-<span class="go">([[0, 1, 2], [1, 2]],</span>
-<span class="go">[{0: Permutation(2), 1: Permutation(0, 1, 2), 2: Permutation(0, 2, 1)},</span>
-<span class="go">{1: Permutation(2), 2: Permutation(1, 2)}])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.util._remove_gens">
-<tt class="descclassname">sympy.combinatorics.util.</tt><tt class="descname">_remove_gens</tt><big>(</big><em>base</em>, <em>strong_gens</em>, <em>basic_orbits=None</em>, <em>strong_gens_distr=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/util.html#_remove_gens"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.util._remove_gens" title="Permalink to this definition">¶</a></dt>
-<dd><p>Remove redundant generators from a strong generating set.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>``base`` - a base</strong> :</p>
-<p><strong>``strong_gens`` - a strong generating set relative to ``base``</strong> :</p>
-<p><strong>``basic_orbits`` - basic orbits</strong> :</p>
-<p><strong>``strong_gens_distr`` - strong generators distributed by membership in basic</strong> :</p>
-<p><strong>stabilizers</strong> :</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>A strong generating set with respect to ``base`` which is a subset of</strong> :</p>
-<p class="last"><strong>``strong_gens``.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Notes</p>
-<p>This procedure is outlined in [1],p.95.</p>
-<p class="rubric">References</p>
-<p>[1] Holt, D., Eick, B., O&#8217;Brien, E.
-&#8220;Handbook of computational group theory&#8221;</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">SymmetricGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.perm_groups</span> <span class="kn">import</span> <span class="n">PermutationGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.util</span> <span class="kn">import</span> <span class="n">_remove_gens</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.testutil</span> <span class="kn">import</span> <span class="n">_verify_bsgs</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">SymmetricGroup</span><span class="p">(</span><span class="mi">15</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">schreier_sims_incremental</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">len</span><span class="p">(</span><span class="n">strong_gens</span><span class="p">)</span>
-<span class="go">26</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">new_gens</span> <span class="o">=</span> <span class="n">_remove_gens</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">strong_gens</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">len</span><span class="p">(</span><span class="n">new_gens</span><span class="p">)</span>
-<span class="go">14</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_verify_bsgs</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">base</span><span class="p">,</span> <span class="n">new_gens</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.util._strip">
-<tt class="descclassname">sympy.combinatorics.util.</tt><tt class="descname">_strip</tt><big>(</big><em>g</em>, <em>base</em>, <em>orbits</em>, <em>transversals</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/util.html#_strip"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.util._strip" title="Permalink to this definition">¶</a></dt>
-<dd><p>Attempt to decompose a permutation using a (possibly partial) BSGS
-structure.</p>
-<p>This is done by treating the sequence <tt class="docutils literal"><span class="pre">base</span></tt> as an actual base, and
-the orbits <tt class="docutils literal"><span class="pre">orbits</span></tt> and transversals <tt class="docutils literal"><span class="pre">transversals</span></tt> as basic orbits and
-transversals relative to it.</p>
-<p>This process is called &#8220;sifting&#8221;. A sift is unsuccessful when a certain
-orbit element is not found or when after the sift the decomposition
-doesn&#8217;t end with the identity element.</p>
-<p>The argument <tt class="docutils literal"><span class="pre">transversals</span></tt> is a list of dictionaries that provides
-transversal elements for the orbits <tt class="docutils literal"><span class="pre">orbits</span></tt>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>``g`` - permutation to be decomposed</strong> :</p>
-<p><strong>``base`` - sequence of points</strong> :</p>
-<p><strong>``orbits`` - a list in which the ``i``-th entry is an orbit of ``base[i]``</strong> :</p>
-<p><strong>under some subgroup of the pointwise stabilizer of `</strong> :</p>
-<p><strong>`base[0], base[1], ..., base[i - 1]``. The groups themselves are implicit</strong> :</p>
-<p><strong>in this function since the only infromation we need is encoded in the orbits</strong> :</p>
-<p><strong>and transversals</strong> :</p>
-<p><strong>``transversals`` - a list of orbit transversals associated with the orbits</strong> :</p>
-<p class="last"><strong>``orbits``.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="../combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims" title="sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims</span></tt></a>, <a class="reference internal" href="../combinatorics/perm_groups.html#sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims_random" title="sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims_random"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims_random</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>The algorithm is described in [1],pp.89-90. The reason for returning
-both the current state of the element being decomposed and the level
-at which the sifting ends is that they provide important information for
-the randomized version of the Schreier-Sims algorithm.</p>
-<p class="rubric">References</p>
-<p>[1] Holt, D., Eick, B., O&#8217;Brien, E.
-&#8220;Handbook of computational group theory&#8221;</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">SymmetricGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.permutations</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.util</span> <span class="kn">import</span> <span class="n">_strip</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">SymmetricGroup</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">.</span><span class="n">schreier_sims</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">Permutation</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_strip</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">basic_orbits</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">basic_transversals</span><span class="p">)</span>
-<span class="go">(Permutation(4), 5)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.combinatorics.util._strong_gens_from_distr">
-<tt class="descclassname">sympy.combinatorics.util.</tt><tt class="descname">_strong_gens_from_distr</tt><big>(</big><em>strong_gens_distr</em><big>)</big><a class="reference internal" href="../../_modules/sympy/combinatorics/util.html#_strong_gens_from_distr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.combinatorics.util._strong_gens_from_distr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Retrieve strong generating set from generators of basic stabilizers.</p>
-<p>This is just the union of the generators of the first and second basic
-stabilizers.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>``strong_gens_distr`` - strong generators distributed by membership in basic</strong> :</p>
-<p class="last"><strong>stabilizers</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.combinatorics.util._distribute_gens_by_base" title="sympy.combinatorics.util._distribute_gens_by_base"><tt class="xref py py-obj docutils literal"><span class="pre">_distribute_gens_by_base</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics</span> <span class="kn">import</span> <span class="n">Permutation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Permutation</span><span class="o">.</span><span class="n">print_cyclic</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">SymmetricGroup</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.util</span> <span class="kn">import</span> <span class="p">(</span><span class="n">_strong_gens_from_distr</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">_distribute_gens_by_base</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">SymmetricGroup</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">.</span><span class="n">schreier_sims</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">.</span><span class="n">strong_gens</span>
-<span class="go">[Permutation(0, 1, 2), Permutation(2)(0, 1), Permutation(1, 2)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">strong_gens_distr</span> <span class="o">=</span> <span class="n">_distribute_gens_by_base</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">strong_gens</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_strong_gens_from_distr</span><span class="p">(</span><span class="n">strong_gens_distr</span><span class="p">)</span>
-<span class="go">[Permutation(0, 1, 2), Permutation(2)(0, 1), Permutation(1, 2)]</span>
-</pre></div>
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-</dd></dl>
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-            
-  <div class="section" id="concrete-mathematics">
-<h1>Concrete Mathematics<a class="headerlink" href="#concrete-mathematics" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="hypergeometric-terms">
-<h2>Hypergeometric terms<a class="headerlink" href="#hypergeometric-terms" title="Permalink to this headline">¶</a></h2>
-<p>The center stage, in recurrence solving and summations, play hypergeometric
-terms. Formally these are sequences annihilated by first order linear
-recurrence operators. In simple words if we are given term <span class="math">\(a(n)\)</span> then it is
-hypergeometric if its consecutive term ratio is a rational function in <span class="math">\(n\)</span>.</p>
-<p>To check if a sequence is of this type you can use the <tt class="docutils literal"><span class="pre">is_hypergeometric</span></tt>
-method which is available in Basic class. Here is simple example involving a
-polynomial:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;n,k&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">n</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">is_hypergeometric</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>Of course polynomials are hypergeometric but are there any more complicated
-sequences of this type? Here are some trivial examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">is_hypergeometric</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">binomial</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span><span class="o">.</span><span class="n">is_hypergeometric</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rf</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span><span class="o">.</span><span class="n">is_hypergeometric</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ff</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span><span class="o">.</span><span class="n">is_hypergeometric</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">is_hypergeometric</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">is_hypergeometric</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>We see that all species used in summations and other parts of concrete
-mathematics are hypergeometric. Note also that binomial coefficients and both
-rising and falling factorials are hypergeometric in both their arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">binomial</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span><span class="o">.</span><span class="n">is_hypergeometric</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rf</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span><span class="o">.</span><span class="n">is_hypergeometric</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ff</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span><span class="o">.</span><span class="n">is_hypergeometric</span><span class="p">(</span><span class="n">k</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>To say more, all previously shown examples are valid for integer linear
-arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">is_hypergeometric</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">binomial</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span><span class="o">.</span><span class="n">is_hypergeometric</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rf</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">k</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">is_hypergeometric</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ff</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">is_hypergeometric</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="mi">5</span><span class="o">*</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">is_hypergeometric</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mi">7</span><span class="p">))</span><span class="o">.</span><span class="n">is_hypergeometric</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>However nonlinear arguments make those sequences fail to be hypergeometric:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">is_hypergeometric</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">is_hypergeometric</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>If not only the knowledge of being hypergeometric or not is needed, you can use
-<tt class="docutils literal"><span class="pre">hypersimp()</span></tt> function. It will try to simplify combinatorial expression and
-if the term given is hypergeometric it will return a quotient of polynomials of
-minimal degree. Otherwise is will return <span class="math">\(None\)</span> to say that sequence is not
-hypergeometric:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">hypersimp</span><span class="p">(</span><span class="n">factorial</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="p">),</span> <span class="n">n</span><span class="p">)</span>
-<span class="go">4*n**2 + 6*n + 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hypersimp</span><span class="p">(</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="n">n</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="concrete-class-reference">
-<h2>Concrete Class Reference<a class="headerlink" href="#concrete-class-reference" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.concrete.summations.Sum">
-<em class="property">class </em><tt class="descclassname">sympy.concrete.summations.</tt><tt class="descname">Sum</tt><a class="reference internal" href="../_modules/sympy/concrete/summations.html#Sum"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.concrete.summations.Sum" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents unevaluated summation.</p>
-<dl class="function">
-<dt id="sympy.concrete.summations.Sum.euler_maclaurin">
-<tt class="descname">euler_maclaurin</tt><big>(</big><em>self</em>, <em>m=0</em>, <em>n=0</em>, <em>eps=0</em>, <em>eval_integral=True</em><big>)</big><a class="reference internal" href="../_modules/sympy/concrete/summations.html#Sum.euler_maclaurin"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.concrete.summations.Sum.euler_maclaurin" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return an Euler-Maclaurin approximation of self, where m is the
-number of leading terms to sum directly and n is the number of
-terms in the tail.</p>
-<p>With m = n = 0, this is simply the corresponding integral
-plus a first-order endpoint correction.</p>
-<p>Returns (s, e) where s is the Euler-Maclaurin approximation
-and e is the estimated error (taken to be the magnitude of
-the first omitted term in the tail):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">k</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Sum</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">1.28333333333333</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">Sum</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span><span class="o">.</span><span class="n">euler_maclaurin</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span>
-<span class="go">-log(2) + 7/20 + log(5)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sstr</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">sstr</span><span class="p">((</span><span class="n">s</span><span class="o">.</span><span class="n">evalf</span><span class="p">(),</span> <span class="n">e</span><span class="o">.</span><span class="n">evalf</span><span class="p">()),</span> <span class="n">full_prec</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">(1.26629073187415, 0.0175000000000000)</span>
-</pre></div>
-</div>
-<p>The endpoints may be symbolic:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">Sum</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span><span class="o">.</span><span class="n">euler_maclaurin</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span>
-<span class="go">-log(a) + log(b) + 1/(2*b) + 1/(2*a)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span>
-<span class="go">Abs(-1/(12*b**2) + 1/(12*a**2))</span>
-</pre></div>
-</div>
-<p>If the function is a polynomial of degree at most 2n+1, the
-Euler-Maclaurin formula becomes exact (and e = 0 is returned):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span><span class="o">.</span><span class="n">euler_maclaurin</span><span class="p">()</span>
-<span class="go">(b**2/2 + b/2 - 1, 0)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-<span class="go">b**2/2 + b/2 - 1</span>
-</pre></div>
-</div>
-<p>With a nonzero eps specified, the summation is ended
-as soon as the remainder term is less than the epsilon.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.concrete.summations.Sum.free_symbols">
-<tt class="descname">free_symbols</tt><a class="reference internal" href="../_modules/sympy/concrete/summations.html#Sum.free_symbols"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.concrete.summations.Sum.free_symbols" title="Permalink to this definition">¶</a></dt>
-<dd><p>This method returns the symbols that will exist when the
-summation is evaluated. This is useful if one is trying to
-determine whether a sum depends on a certain symbol or not.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Sum</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">free_symbols</span>
-<span class="go">set([y])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.concrete.summations.Sum.is_number">
-<tt class="descname">is_number</tt><a class="reference internal" href="../_modules/sympy/concrete/summations.html#Sum.is_number"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.concrete.summations.Sum.is_number" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if the Sum will result in a number, else False.</p>
-<p>sympy considers anything that will result in a number to have
-is_number == True.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">log</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>Sums are a special case since they contain symbols that can
-be replaced with numbers. Whether the integral can be done or not is
-another issue. But answering whether the final result is a number is
-not difficult.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Sum</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.concrete.summations.Sum.is_zero">
-<tt class="descname">is_zero</tt><a class="reference internal" href="../_modules/sympy/concrete/summations.html#Sum.is_zero"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.concrete.summations.Sum.is_zero" title="Permalink to this definition">¶</a></dt>
-<dd><p>A Sum is only zero if its function is zero or if all terms
-cancel out. This only answers whether the summand zero.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.concrete.summations.Sum.variables">
-<tt class="descname">variables</tt><a class="reference internal" href="../_modules/sympy/concrete/summations.html#Sum.variables"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.concrete.summations.Sum.variables" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a list of the summation variables</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Sum</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">i</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span><span class="o">.</span><span class="n">variables</span>
-<span class="go">[i]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.concrete.products.Product">
-<em class="property">class </em><tt class="descclassname">sympy.concrete.products.</tt><tt class="descname">Product</tt><a class="reference internal" href="../_modules/sympy/concrete/products.html#Product"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.concrete.products.Product" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents unevaluated product.</p>
-<dl class="attribute">
-<dt id="sympy.concrete.products.Product.free_symbols">
-<tt class="descname">free_symbols</tt><a class="reference internal" href="../_modules/sympy/concrete/products.html#Product.free_symbols"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.concrete.products.Product.free_symbols" title="Permalink to this definition">¶</a></dt>
-<dd><p>This method returns the symbols that will affect the value of
-the Product when evaluated. This is useful if one is trying to
-determine whether a product depends on a certain symbol or not.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Product</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Product</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">free_symbols</span>
-<span class="go">set([y])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.concrete.products.Product.is_number">
-<tt class="descname">is_number</tt><a class="reference internal" href="../_modules/sympy/concrete/products.html#Product.is_number"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.concrete.products.Product.is_number" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if the Product will result in a number, else False.</p>
-<p>sympy considers anything that will result in a number to have
-is_number == True.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">log</span><span class="p">,</span> <span class="n">Product</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Product</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Product</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Product</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Product</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.concrete.products.Product.is_zero">
-<tt class="descname">is_zero</tt><a class="reference internal" href="../_modules/sympy/concrete/products.html#Product.is_zero"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.concrete.products.Product.is_zero" title="Permalink to this definition">¶</a></dt>
-<dd><p>A Product is zero only if its term is zero.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.concrete.products.Product.variables">
-<tt class="descname">variables</tt><a class="reference internal" href="../_modules/sympy/concrete/products.html#Product.variables"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.concrete.products.Product.variables" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a list of the product variables</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Product</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">i</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Product</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span><span class="o">.</span><span class="n">variables</span>
-<span class="go">[i]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="concrete-functions-reference">
-<h2>Concrete Functions Reference<a class="headerlink" href="#concrete-functions-reference" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.concrete.summations.summation">
-<tt class="descclassname">sympy.concrete.summations.</tt><tt class="descname">summation</tt><big>(</big><em>f</em>, <em>*symbols</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../_modules/sympy/concrete/summations.html#summation"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.concrete.summations.summation" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the summation of f with respect to symbols.</p>
-<p>The notation for symbols is similar to the notation used in Integral.
-summation(f, (i, a, b)) computes the sum of f with respect to i from a to b,
-i.e.,</p>
-<div class="highlight-python"><pre>                            b
-                          ____
-                          \   `
-summation(f, (i, a, b)) =  )    f
-                          /___,
-                          i = a</pre>
-</div>
-<p>If it cannot compute the sum, it returns an unevaluated Sum object.
-Repeated sums can be computed by introducing additional symbols tuples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">summation</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">log</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i n m&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-<span class="go">n**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="o">**</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">Sum(log(n)**(-n), (n, 2, oo))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">),</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">m</span><span class="p">))</span>
-<span class="go">m**3/6 + m**2/2 + m/3</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">exp(x)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.concrete.products.product">
-<tt class="descclassname">sympy.concrete.products.</tt><tt class="descname">product</tt><big>(</big><em>*args</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../_modules/sympy/concrete/products.html#product"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.concrete.products.product" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the product.</p>
-<p>The notation for symbols is similiar to the notation used in Sum or
-Integral. product(f, (i, a, b)) computes the product of f with
-respect to i from a to b, i.e.,</p>
-<div class="highlight-python"><pre>                             b
-                           _____
-product(f(n), (i, a, b)) = |   | f(n)
-                           |   |
-                           i = a</pre>
-</div>
-<p>If it cannot compute the product, it returns an unevaluated Product object.
-Repeated products can be computed by introducing additional symbols tuples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">product</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i n m k&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">product</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">k</span><span class="p">))</span>
-<span class="go">k!</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">product</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">k</span><span class="p">))</span>
-<span class="go">m**k</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">product</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">k</span><span class="p">),</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-<span class="go">Product(k!, (k, 1, n))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.concrete.gosper.gosper_normal">
-<tt class="descclassname">sympy.concrete.gosper.</tt><tt class="descname">gosper_normal</tt><big>(</big><em>f</em>, <em>g</em>, <em>n</em>, <em>polys=True</em><big>)</big><a class="reference internal" href="../_modules/sympy/concrete/gosper.html#gosper_normal"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.concrete.gosper.gosper_normal" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the Gosper&#8217;s normal form of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p>Given relatively prime univariate polynomials <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>,
-rewrite their quotient to a normal form defined as follows:</p>
-<div class="math">
-\[\frac{f(n)}{g(n)} = Z \cdot \frac{A(n) C(n+1)}{B(n) C(n)}\]</div>
-<p>where <tt class="docutils literal"><span class="pre">Z</span></tt> is an arbitrary constant and <tt class="docutils literal"><span class="pre">A</span></tt>, <tt class="docutils literal"><span class="pre">B</span></tt>, <tt class="docutils literal"><span class="pre">C</span></tt> are
-monic polynomials in <tt class="docutils literal"><span class="pre">n</span></tt> with the following properties:</p>
-<ol class="arabic simple">
-<li><span class="math">\(\gcd(A(n), B(n+h)) = 1 \forall h \in \mathbb{N}\)</span></li>
-<li><span class="math">\(\gcd(B(n), C(n+1)) = 1\)</span></li>
-<li><span class="math">\(\gcd(A(n), C(n)) = 1\)</span></li>
-</ol>
-<p>This normal form, or rational factorization in other words, is a
-crucial step in Gosper&#8217;s algorithm and in solving of difference
-equations. It can be also used to decide if two hypergeometric
-terms are similar or not.</p>
-<p>This procedure will return a tuple containing elements of this
-factorization in the form <tt class="docutils literal"><span class="pre">(Z*A,</span> <span class="pre">B,</span> <span class="pre">C)</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.concrete.gosper</span> <span class="kn">import</span> <span class="n">gosper_normal</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">n</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gosper_normal</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">n</span><span class="o">+</span><span class="mi">5</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">+</span><span class="mi">3</span><span class="p">),</span> <span class="n">n</span><span class="p">,</span> <span class="n">polys</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">(1/4, n + 3/2, n + 1/4)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.concrete.gosper.gosper_term">
-<tt class="descclassname">sympy.concrete.gosper.</tt><tt class="descname">gosper_term</tt><big>(</big><em>f</em>, <em>n</em><big>)</big><a class="reference internal" href="../_modules/sympy/concrete/gosper.html#gosper_term"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.concrete.gosper.gosper_term" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute Gosper&#8217;s hypergeometric term for <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p>Suppose <tt class="docutils literal"><span class="pre">f</span></tt> is a hypergeometric term such that:</p>
-<div class="math">
-\[s_n = \sum_{k=0}^{n-1} f_k\]</div>
-<p>and <span class="math">\(f_k\)</span> doesn&#8217;t depend on <span class="math">\(n\)</span>. Returns a hypergeometric
-term <span class="math">\(g_n\)</span> such that <span class="math">\(g_{n+1} - g_n = f_n\)</span>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.concrete.gosper</span> <span class="kn">import</span> <span class="n">gosper_term</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">factorial</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">n</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gosper_term</span><span class="p">((</span><span class="mi">4</span><span class="o">*</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">n</span><span class="p">)</span>
-<span class="go">(-n - 1/2)/(n + 1/4)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.concrete.gosper.gosper_sum">
-<tt class="descclassname">sympy.concrete.gosper.</tt><tt class="descname">gosper_sum</tt><big>(</big><em>f</em>, <em>k</em><big>)</big><a class="reference internal" href="../_modules/sympy/concrete/gosper.html#gosper_sum"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.concrete.gosper.gosper_sum" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gosper&#8217;s hypergeometric summation algorithm.</p>
-<p>Given a hypergeometric term <tt class="docutils literal"><span class="pre">f</span></tt> such that:</p>
-<div class="math">
-\[s_n = \sum_{k=0}^{n-1} f_k\]</div>
-<p>and <span class="math">\(f(n)\)</span> doesn&#8217;t depend on <span class="math">\(n\)</span>, returns <span class="math">\(g_{n} - g(0)\)</span> where
-<span class="math">\(g_{n+1} - g_n = f_n\)</span>, or <tt class="docutils literal"><span class="pre">None</span></tt> if <span class="math">\(s_n\)</span> can not be expressed
-in closed form as a sum of hypergeometric terms.</p>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r12" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id1">[R12]</a></td><td>Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger, A = B,
-AK Peters, Ltd., Wellesley, MA, USA, 1997, pp. 73&#8211;100</td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.concrete.gosper</span> <span class="kn">import</span> <span class="n">gosper_sum</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">factorial</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">i</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">factorial</span><span class="p">(</span><span class="n">k</span><span class="p">)</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gosper_sum</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-<span class="go">(-n! + 2*(2*n + 1)!)/(2*n + 1)!</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="o">==</span> <span class="nb">sum</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gosper_sum</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-<span class="go">(-60*n! + (2*n + 1)!)/(60*(2*n + 1)!)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span> <span class="o">==</span> <span class="nb">sum</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-
-
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-            </a></p>
-  <h3><a href="../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Concrete Mathematics</a><ul>
-<li><a class="reference internal" href="#hypergeometric-terms">Hypergeometric terms</a></li>
-<li><a class="reference internal" href="#concrete-class-reference">Concrete Class Reference</a></li>
-<li><a class="reference internal" href="#concrete-functions-reference">Concrete Functions Reference</a></li>
-</ul>
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-  <div class="section" id="sympy-core">
-<h1>SymPy Core<a class="headerlink" href="#sympy-core" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="module-sympy.core.sympify">
-<span id="sympify"></span><h2>sympify<a class="headerlink" href="#module-sympy.core.sympify" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="id1">
-<h3>sympify<a class="headerlink" href="#id1" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.sympify.sympify">
-<tt class="descclassname">sympy.core.sympify.</tt><tt class="descname">sympify</tt><big>(</big><em>a</em>, <em>locals=None</em>, <em>convert_xor=True</em>, <em>strict=False</em>, <em>rational=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/sympify.html#sympify"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sympify.sympify" title="Permalink to this definition">¶</a></dt>
-<dd><p>Converts an arbitrary expression to a type that can be used inside sympy.</p>
-<p>For example, it will convert python ints into instance of sympy.Rational,
-floats into instances of sympy.Float, etc. It is also able to coerce symbolic
-expressions which inherit from Basic. This can be useful in cooperation
-with SAGE.</p>
-<dl class="docutils">
-<dt>It currently accepts as arguments:</dt>
-<dd><ul class="first last simple">
-<li>any object defined in sympy (except matrices [TODO])</li>
-<li>standard numeric python types: int, long, float, Decimal</li>
-<li>strings (like &#8220;0.09&#8221; or &#8220;2e-19&#8221;)</li>
-<li>booleans, including <tt class="docutils literal"><span class="pre">None</span></tt> (will leave them unchanged)</li>
-<li>lists, sets or tuples containing any of the above</li>
-</ul>
-</dd>
-</dl>
-<p>If the argument is already a type that sympy understands, it will do
-nothing but return that value. This can be used at the beginning of a
-function to ensure you are working with the correct type.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sympify</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sympify</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">is_integer</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sympify</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">is_real</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sympify</span><span class="p">(</span><span class="mf">2.0</span><span class="p">)</span><span class="o">.</span><span class="n">is_real</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sympify</span><span class="p">(</span><span class="s">&quot;2.0&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">is_real</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sympify</span><span class="p">(</span><span class="s">&quot;2e-45&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">is_real</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>If the expression could not be converted, a SympifyError is raised.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sympify</span><span class="p">(</span><span class="s">&quot;x***2&quot;</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">SympifyError: SympifyError</span>: <span class="n">&quot;could not parse u&#39;x***2&#39;&quot;</span>
-</pre></div>
-</div>
-<p class="rubric">Locals</p>
-<p>The sympification happens with access to everything that is loaded
-by <tt class="docutils literal"><span class="pre">from</span> <span class="pre">sympy</span> <span class="pre">import</span> <span class="pre">*</span></tt>; anything used in a string that is not
-defined by that import will be converted to a symbol. In the following,
-the <tt class="docutils literal"><span class="pre">bitcout</span></tt> function is treated as a symbol and the <tt class="docutils literal"><span class="pre">O</span></tt> is
-interpreted as the Order object (used with series) and it raises
-an error when used improperly:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="s">&#39;bitcount(42)&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sympify</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="go">bitcount(42)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sympify</span><span class="p">(</span><span class="s">&quot;O(x)&quot;</span><span class="p">)</span>
-<span class="go">O(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sympify</span><span class="p">(</span><span class="s">&quot;O + 1&quot;</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">TypeError</span>: <span class="n">unbound method...</span>
-</pre></div>
-</div>
-<p>In order to have <tt class="docutils literal"><span class="pre">bitcount</span></tt> be recognized it can be imported into a
-namespace dictionary and passed as locals:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ns</span> <span class="o">=</span> <span class="p">{}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">exec</span><span class="p">(</span><span class="s">&#39;from sympy.core.evalf import bitcount&#39;</span><span class="p">,</span> <span class="n">ns</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sympify</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="nb">locals</span><span class="o">=</span><span class="n">ns</span><span class="p">)</span>
-<span class="go">6</span>
-</pre></div>
-</div>
-<p>In order to have the <tt class="docutils literal"><span class="pre">O</span></tt> interpreted as a Symbol, identify it as such
-in the namespace dictionary. This can be done in a variety of ways; all
-three of the following are possibilities:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ns</span><span class="p">[</span><span class="s">&quot;O&quot;</span><span class="p">]</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;O&quot;</span><span class="p">)</span>  <span class="c"># method 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">exec</span><span class="p">(</span><span class="s">&#39;from sympy.abc import O&#39;</span><span class="p">,</span> <span class="n">ns</span><span class="p">)</span>  <span class="c"># method 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ns</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">O</span><span class="o">=</span><span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;O&quot;</span><span class="p">)))</span>  <span class="c"># method 3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sympify</span><span class="p">(</span><span class="s">&quot;O + 1&quot;</span><span class="p">,</span> <span class="nb">locals</span><span class="o">=</span><span class="n">ns</span><span class="p">)</span>
-<span class="go">O + 1</span>
-</pre></div>
-</div>
-<p>If you want <em>all</em> single-letter and Greek-letter variables to be symbols
-then you can use the clashing-symbols dictionaries that have been defined
-there as private variables: _clash1 (single-letter variables), _clash2
-(the multi-letter Greek names) or _clash (both single and multi-letter
-names that are defined in abc).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">_clash1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_clash1</span>
-<span class="go">{&#39;C&#39;: C, &#39;E&#39;: E, &#39;I&#39;: I, &#39;N&#39;: N, &#39;O&#39;: O, &#39;Q&#39;: Q, &#39;S&#39;: S}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sympify</span><span class="p">(</span><span class="s">&#39;C &amp; Q&#39;</span><span class="p">,</span> <span class="n">_clash1</span><span class="p">)</span>
-<span class="go">And(C, Q)</span>
-</pre></div>
-</div>
-<p class="rubric">Strict</p>
-<p>If the option <tt class="docutils literal"><span class="pre">strict</span></tt> is set to <tt class="docutils literal"><span class="pre">True</span></tt>, only the types for which an
-explicit conversion has been defined are converted. In the other
-cases, a SympifyError is raised.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sympify</span><span class="p">(</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sympify</span><span class="p">(</span><span class="bp">True</span><span class="p">,</span> <span class="n">strict</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">SympifyError: SympifyError</span>: <span class="n">True</span>
-</pre></div>
-</div>
-<p class="rubric">Extending</p>
-<p>To extend <tt class="docutils literal"><span class="pre">sympify</span></tt> to convert custom objects (not derived from <tt class="docutils literal"><span class="pre">Basic</span></tt>),
-just define a <tt class="docutils literal"><span class="pre">_sympy_</span></tt> method to your class. You can do that even to
-classes that you do not own by subclassing or adding the method at runtime.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">MyList1</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-<span class="gp">... </span>        <span class="k">yield</span> <span class="mi">1</span>
-<span class="gp">... </span>        <span class="k">yield</span> <span class="mi">2</span>
-<span class="gp">... </span>        <span class="k">raise</span> <span class="ne">StopIteration</span>
-<span class="gp">... </span>    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span> <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="p">)[</span><span class="n">i</span><span class="p">]</span>
-<span class="gp">... </span>    <span class="k">def</span> <span class="nf">_sympy_</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span> <span class="k">return</span> <span class="n">Matrix</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sympify</span><span class="p">(</span><span class="n">MyList1</span><span class="p">())</span>
-<span class="go">[1]</span>
-<span class="go">[2]</span>
-</pre></div>
-</div>
-<p>If you do not have control over the class definition you could also use the
-<tt class="docutils literal"><span class="pre">converter</span></tt> global dictionary. The key is the class and the value is a
-function that takes a single argument and returns the desired SymPy
-object, e.g. <tt class="docutils literal"><span class="pre">converter[MyList]</span> <span class="pre">=</span> <span class="pre">lambda</span> <span class="pre">x:</span> <span class="pre">Matrix(x)</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">MyList2</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>   <span class="c"># XXX Do not do this if you control the class!</span>
-<span class="gp">... </span>    <span class="k">def</span> <span class="nf">__iter__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>  <span class="c">#     Use _sympy_!</span>
-<span class="gp">... </span>        <span class="k">yield</span> <span class="mi">1</span>
-<span class="gp">... </span>        <span class="k">yield</span> <span class="mi">2</span>
-<span class="gp">... </span>        <span class="k">raise</span> <span class="ne">StopIteration</span>
-<span class="gp">... </span>    <span class="k">def</span> <span class="nf">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span> <span class="k">return</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="p">)[</span><span class="n">i</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core.sympify</span> <span class="kn">import</span> <span class="n">converter</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">converter</span><span class="p">[</span><span class="n">MyList2</span><span class="p">]</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sympify</span><span class="p">(</span><span class="n">MyList2</span><span class="p">())</span>
-<span class="go">[1]</span>
-<span class="go">[2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.core.cache">
-<span id="cache"></span><h2>cache<a class="headerlink" href="#module-sympy.core.cache" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="cacheit">
-<h3>cacheit<a class="headerlink" href="#cacheit" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.cache.cacheit">
-<tt class="descclassname">sympy.core.cache.</tt><tt class="descname">cacheit</tt><big>(</big><em>func</em><big>)</big><a class="headerlink" href="#sympy.core.cache.cacheit" title="Permalink to this definition">¶</a></dt>
-<dd><p>caching decorator.</p>
-<p>important: the result of cached function must be <em>immutable</em></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core.cache</span> <span class="kn">import</span> <span class="n">cacheit</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nd">@cacheit</span>
-<span class="gp">... </span><span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">):</span>
-<span class="gp">... </span>   <span class="k">return</span> <span class="n">a</span><span class="o">+</span><span class="n">b</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nd">@cacheit</span>
-<span class="gp">... </span><span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">):</span>
-<span class="gp">... </span>   <span class="k">return</span> <span class="p">[</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">]</span> <span class="c"># &lt;-- WRONG, returns mutable object</span>
-</pre></div>
-</div>
-<p>to force cacheit to check returned results mutability and consistency,
-set environment variable SYMPY_USE_CACHE to &#8216;debug&#8217;</p>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.core.basic">
-<span id="basic"></span><h2>basic<a class="headerlink" href="#module-sympy.core.basic" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="id2">
-<h3>Basic<a class="headerlink" href="#id2" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.basic.Basic">
-<em class="property">class </em><tt class="descclassname">sympy.core.basic.</tt><tt class="descname">Basic</tt><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Base class for all objects in sympy.</p>
-<p>Conventions:</p>
-<ol class="arabic">
-<li><p class="first">Always use <tt class="docutils literal"><span class="pre">.args</span></tt>, when accessing parameters of some instance:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">cot</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">cot</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">args</span>
-<span class="go">(x,)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">cot</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">args</span>
-<span class="go">(x, y)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-<span class="go">y</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">Never use internal methods or variables (the ones prefixed with <tt class="docutils literal"><span class="pre">_</span></tt>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">cot</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">_args</span>    <span class="c"># do not use this, use cot(x).args instead</span>
-<span class="go">(x,)</span>
-</pre></div>
-</div>
-</li>
-</ol>
-<dl class="attribute">
-<dt id="sympy.core.basic.Basic.args">
-<tt class="descname">args</tt><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.args"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.args" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a tuple of arguments of &#8216;self&#8217;.</p>
-<p class="rubric">Notes</p>
-<p>Never use self._args, always use self.args.
-Only use _args in __new__ when creating a new function.
-Don&#8217;t override .args() from Basic (so that it&#8217;s easy to
-change the interface in the future if needed).</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">cot</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">cot</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">args</span>
-<span class="go">(x,)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">cot</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">args</span>
-<span class="go">(x, y)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-<span class="go">y</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.basic.Basic.as_content_primitive">
-<tt class="descname">as_content_primitive</tt><big>(</big><em>self</em>, <em>radical=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.as_content_primitive"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.as_content_primitive" title="Permalink to this definition">¶</a></dt>
-<dd><p>A stub to allow Basic args (like Tuple) to be skipped when computing
-the content and primitive components of an expression.</p>
-<p>See docstring of Expr.as_content_primitive</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.basic.Basic.as_poly">
-<tt class="descname">as_poly</tt><big>(</big><em>self</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.as_poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.as_poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Converts <tt class="docutils literal"><span class="pre">self</span></tt> to a polynomial or returns <tt class="docutils literal"><span class="pre">None</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">as_poly</span><span class="p">())</span>
-<span class="go">Poly(x**2 + x*y, x, y, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">as_poly</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span>
-<span class="go">Poly(x**2 + x*y, x, y, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">sin</span><span class="p">(</span><span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">as_poly</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span>
-<span class="go">None</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.basic.Basic.assumptions0">
-<tt class="descname">assumptions0</tt><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.assumptions0"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.assumptions0" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return object <span class="math">\(type\)</span> assumptions.</p>
-<p>For example:</p>
-<blockquote>
-<div>Symbol(&#8216;x&#8217;, real=True)
-Symbol(&#8216;x&#8217;, integer=True)</div></blockquote>
-<p>are different objects. In other words, besides Python type (Symbol in
-this case), the initial assumptions are also forming their typeinfo.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">assumptions0</span>
-<span class="go">{&#39;commutative&#39;: True}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">assumptions0</span>
-<span class="go">{&#39;commutative&#39;: True, &#39;complex&#39;: True, &#39;hermitian&#39;: True,</span>
-<span class="go">&#39;imaginary&#39;: False, &#39;negative&#39;: False, &#39;nonnegative&#39;: True,</span>
-<span class="go">&#39;nonpositive&#39;: False, &#39;nonzero&#39;: True, &#39;positive&#39;: True, &#39;real&#39;: True,</span>
-<span class="go">&#39;zero&#39;: False}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.basic.Basic.atoms">
-<tt class="descname">atoms</tt><big>(</big><em>self</em>, <em>*types</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.atoms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.atoms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the atoms that form the current object.</p>
-<p>By default, only objects that are truly atomic and can&#8217;t
-be divided into smaller pieces are returned: symbols, numbers,
-and number symbols like I and pi. It is possible to request
-atoms of any type, however, as demonstrated below.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Number</span><span class="p">,</span> <span class="n">NumberSymbol</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Symbol</span><span class="p">)</span>
-<span class="go">set([x, y])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Number</span><span class="p">)</span>
-<span class="go">set([1, 2])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Number</span><span class="p">,</span> <span class="n">NumberSymbol</span><span class="p">)</span>
-<span class="go">set([1, 2, pi])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Number</span><span class="p">,</span> <span class="n">NumberSymbol</span><span class="p">,</span> <span class="n">I</span><span class="p">)</span>
-<span class="go">set([1, 2, I, pi])</span>
-</pre></div>
-</div>
-<p>Note that I (imaginary unit) and zoo (complex infinity) are special
-types of number symbols and are not part of the NumberSymbol class.</p>
-<p>The type can be given implicitly, too:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="c"># x is a Symbol</span>
-<span class="go">set([x, y])</span>
-</pre></div>
-</div>
-<p>Be careful to check your assumptions when using the implicit option
-since <tt class="docutils literal"><span class="pre">S(1).is_Integer</span> <span class="pre">=</span> <span class="pre">True</span></tt> but <tt class="docutils literal"><span class="pre">type(S(1))</span></tt> is <tt class="docutils literal"><span class="pre">One</span></tt>, a special type
-of sympy atom, while <tt class="docutils literal"><span class="pre">type(S(2))</span></tt> is type <tt class="docutils literal"><span class="pre">Integer</span></tt> and will find all
-integers in an expression:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span>
-<span class="go">set([1])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">set([1, 2])</span>
-</pre></div>
-</div>
-<p>Finally, arguments to atoms() can select more than atomic atoms: any
-sympy type (loaded in core/__init__.py) can be listed as an argument
-and those types of &#8220;atoms&#8221; as found in scanning the arguments of the
-expression recursively:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Mul</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">AppliedUndef</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Function</span><span class="p">)</span>
-<span class="go">set([f(x), sin(y + I*pi)])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">AppliedUndef</span><span class="p">)</span>
-<span class="go">set([f(x)])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Mul</span><span class="p">)</span>
-<span class="go">set([I*pi, 2*sin(y + I*pi)])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.core.basic.Basic.class_key">
-<em class="property">classmethod </em><tt class="descname">class_key</tt><big>(</big><big>)</big><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.class_key"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.class_key" title="Permalink to this definition">¶</a></dt>
-<dd><p>Nice order of classes.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.basic.Basic.compare">
-<tt class="descname">compare</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.compare"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.compare" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return -1, 0, 1 if the object is smaller, equal, or greater than other.</p>
-<p>Not in the mathematical sense. If the object is of a different type
-from the &#8220;other&#8221; then their classes are ordered according to
-the sorted_classes list.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">compare</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">-1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">compare</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span><span class="o">.</span><span class="n">compare</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.basic.Basic.compare_pretty">
-<tt class="descname">compare_pretty</tt><big>(</big><em>*args</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.compare_pretty"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.compare_pretty" title="Permalink to this definition">¶</a></dt>
-<dd><p>Is a &gt; b in the sense of ordering in printing?</p>
-<p>THIS FUNCTION IS DEPRECATED.  Use <tt class="docutils literal"><span class="pre">default_sort_key</span></tt> instead.</p>
-<div class="highlight-python"><pre>yes ..... return 1
-no ...... return -1
-equal ... return 0</pre>
-</div>
-<p>Strategy:</p>
-<p>It uses Basic.compare as a fallback, but improves it in many cases,
-like <tt class="docutils literal"><span class="pre">x**3</span></tt>, <tt class="docutils literal"><span class="pre">x**4</span></tt>, <tt class="docutils literal"><span class="pre">O(x**3)</span></tt> etc. In those simple cases, it just parses the
-expression and returns the &#8220;sane&#8221; ordering such as:</p>
-<div class="highlight-python"><pre>1 &lt; x &lt; x**2 &lt; x**3 &lt; O(x**4) etc.</pre>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">Number</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Basic</span><span class="o">.</span><span class="n">_compare_pretty</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">-1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Basic</span><span class="o">.</span><span class="n">_compare_pretty</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Basic</span><span class="o">.</span><span class="n">_compare_pretty</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Basic</span><span class="o">.</span><span class="n">_compare_pretty</span><span class="p">(</span><span class="n">Number</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">Number</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Basic</span><span class="o">.</span><span class="n">_compare_pretty</span><span class="p">(</span><span class="n">Number</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">Number</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.basic.Basic.count">
-<tt class="descname">count</tt><big>(</big><em>self</em>, <em>query</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.count"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.count" title="Permalink to this definition">¶</a></dt>
-<dd><p>Count the number of matching subexpressions.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.basic.Basic.count_ops">
-<tt class="descname">count_ops</tt><big>(</big><em>self</em>, <em>visual=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.count_ops"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.count_ops" title="Permalink to this definition">¶</a></dt>
-<dd><p>wrapper for count_ops that returns the operation count.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.basic.Basic.doit">
-<tt class="descname">doit</tt><big>(</big><em>self</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.doit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.doit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluate objects that are not evaluated by default like limits,
-integrals, sums and products. All objects of this kind will be
-evaluated recursively, unless some species were excluded via &#8216;hints&#8217;
-or unless the &#8216;deep&#8217; hint was set to &#8216;False&#8217;.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span><span class="o">*</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">2*Integral(x, x)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-<span class="go">x**2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="n">deep</span> <span class="o">=</span> <span class="bp">False</span><span class="p">)</span>
-<span class="go">2*Integral(x, x)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.basic.Basic.dummy_eq">
-<tt class="descname">dummy_eq</tt><big>(</big><em>self</em>, <em>other</em>, <em>symbol=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.dummy_eq"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.dummy_eq" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compare two expressions and handle dummy symbols.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Dummy</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">u</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&#39;u&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">u</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">dummy_eq</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">u</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">u</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">dummy_eq</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">u</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">dummy_eq</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.basic.Basic.find">
-<tt class="descname">find</tt><big>(</big><em>self</em>, <em>query</em>, <em>group=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.find"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.find" title="Permalink to this definition">¶</a></dt>
-<dd><p>Find all subexpressions matching a query.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.basic.Basic.free_symbols">
-<tt class="descname">free_symbols</tt><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.free_symbols"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.free_symbols" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return from the atoms of self those which are free symbols.</p>
-<p>For most expressions, all symbols are free symbols. For some classes
-this is not true. e.g. Integrals use Symbols for the dummy variables
-which are bound variables, so Integral has a method to return all symbols
-except those. Derivative keeps track of symbols with respect to which it
-will perform a derivative; those are bound variables, too, so it has
-its own symbols method.</p>
-<p>Any other method that uses bound variables should implement a symbols
-method.</p>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.core.basic.Basic.fromiter">
-<em class="property">classmethod </em><tt class="descname">fromiter</tt><big>(</big><em>args</em>, <em>**assumptions</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.fromiter"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.fromiter" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a new object from an iterable.</p>
-<p>This is a convenience function that allows one to create objects from
-any iterable, without having to convert to a list or tuple first.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Tuple</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Tuple</span><span class="o">.</span><span class="n">fromiter</span><span class="p">(</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
-<span class="go">(0, 1, 2, 3, 4)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.basic.Basic.func">
-<tt class="descname">func</tt><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.func"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.func" title="Permalink to this definition">¶</a></dt>
-<dd><p>The top-level function in an expression.</p>
-<p>The following should hold for all objects:</p>
-<div class="highlight-python"><pre>&gt;&gt; x == x.func(*x.args)</pre>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">func</span>
-<span class="go">&lt;class &#39;sympy.core.mul.Mul&#39;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">args</span>
-<span class="go">(2, x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">a</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-<span class="go">2*x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">==</span> <span class="n">a</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="n">a</span><span class="o">.</span><span class="n">args</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.basic.Basic.has">
-<tt class="descname">has</tt><big>(</big><em>*args</em>, <em>**kw_args</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.has"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.has" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test whether any subexpression matches any of the patterns.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>Note that <tt class="docutils literal"><span class="pre">expr.has(*patterns)</span></tt> is exactly equivalent to
-<tt class="docutils literal"><span class="pre">any(expr.has(p)</span> <span class="pre">for</span> <span class="pre">p</span> <span class="pre">in</span> <span class="pre">patterns)</span></tt>. In particular, <tt class="docutils literal"><span class="pre">False</span></tt> is
-returned when the list of patterns is empty.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">has</span><span class="p">()</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.basic.Basic.is_Real">
-<tt class="descname">is_Real</tt><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.is_Real"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.is_Real" title="Permalink to this definition">¶</a></dt>
-<dd><p>Deprecated alias for <tt class="docutils literal"><span class="pre">is_Float</span></tt></p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.basic.Basic.is_number">
-<tt class="descname">is_number</tt><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.is_number"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.is_number" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if &#8216;self&#8217; is a number.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">log</span><span class="p">,</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="n">Integral</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="n">Integral</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.basic.Basic.iter_basic_args">
-<tt class="descname">iter_basic_args</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.iter_basic_args"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.iter_basic_args" title="Permalink to this definition">¶</a></dt>
-<dd><p>Iterates arguments of <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">iter_basic_args</span><span class="p">()</span>
-<span class="go">&lt;...iterator object at 0x...&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">iter_basic_args</span><span class="p">())</span>
-<span class="go">[2, x]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.basic.Basic.match">
-<tt class="descname">match</tt><big>(</big><em>self</em>, <em>pattern</em>, <em>old=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.match"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.match" title="Permalink to this definition">¶</a></dt>
-<dd><p>Pattern matching.</p>
-<p>Wild symbols match all.</p>
-<p>Return <tt class="docutils literal"><span class="pre">None</span></tt> when expression (self) does not match
-with pattern. Otherwise return a dictionary such that:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">pattern</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">pattern</span><span class="p">))</span> <span class="o">==</span> <span class="bp">self</span>
-</pre></div>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">Wild</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&quot;p&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&quot;q&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&quot;r&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">**</span><span class="n">p</span><span class="p">)</span>
-<span class="go">{p_: x + y}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">**</span><span class="n">q</span><span class="p">)</span>
-<span class="go">{p_: x + y, q_: x + y}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">q</span><span class="o">**</span><span class="n">r</span><span class="p">)</span>
-<span class="go">{p_: 4, q_: x, r_: 2}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">q</span><span class="o">**</span><span class="n">r</span><span class="p">)</span><span class="o">.</span><span class="n">xreplace</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">q</span><span class="o">**</span><span class="n">r</span><span class="p">))</span>
-<span class="go">4*x**2</span>
-</pre></div>
-</div>
-<p>The <tt class="docutils literal"><span class="pre">old</span></tt> flag will give the old-style pattern matching where
-expressions and patterns are essentially solved to give the
-match. Both of the following give None unless <tt class="docutils literal"><span class="pre">old=True</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span> <span class="o">-</span> <span class="n">x</span><span class="p">,</span> <span class="n">old</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">{p_: 2*x - 2}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">/</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">x</span><span class="p">,</span> <span class="n">old</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">{p_: 2/x**2}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.basic.Basic.matches">
-<tt class="descname">matches</tt><big>(</big><em>self</em>, <em>expr</em>, <em>repl_dict={}</em>, <em>old=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.matches"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.matches" title="Permalink to this definition">¶</a></dt>
-<dd><p>Helper method for match() that looks for a match between wild symbols
-in self and expressions in expr.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">Wild</span><span class="p">,</span> <span class="n">Integer</span><span class="p">,</span> <span class="n">Basic</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a b c&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Basic</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">matches</span><span class="p">(</span><span class="n">Basic</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">))</span> <span class="ow">is</span> <span class="bp">None</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Basic</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">matches</span><span class="p">(</span><span class="n">Basic</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">b</span> <span class="o">+</span> <span class="n">c</span><span class="p">))</span>
-<span class="go">{x_: b + c}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.basic.Basic.replace">
-<tt class="descname">replace</tt><big>(</big><em>self</em>, <em>query</em>, <em>value</em>, <em>map=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.replace"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.replace" title="Permalink to this definition">¶</a></dt>
-<dd><p>Replace matching subexpressions of <tt class="docutils literal"><span class="pre">self</span></tt> with <tt class="docutils literal"><span class="pre">value</span></tt>.</p>
-<p>If <tt class="docutils literal"><span class="pre">map</span> <span class="pre">=</span> <span class="pre">True</span></tt> then also return the mapping {old: new} where <tt class="docutils literal"><span class="pre">old</span></tt>
-was a sub-expression found with query and <tt class="docutils literal"><span class="pre">new</span></tt> is the replacement
-value for it.</p>
-<p>Traverses an expression tree and performs replacement of matching
-subexpressions from the bottom to the top of the tree. The list of
-possible combinations of queries and replacement values is listed
-below:</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="galgebra/GA/GAsympy.html#subs" title="subs"><tt class="xref py py-obj docutils literal"><span class="pre">subs</span></tt></a></dt>
-<dd>substitution of subexpressions as defined by the objects themselves.</dd>
-<dt><a class="reference internal" href="#sympy.core.basic.Basic.xreplace" title="sympy.core.basic.Basic.xreplace"><tt class="xref py py-obj docutils literal"><span class="pre">xreplace</span></tt></a></dt>
-<dd>exact node replacement in expr tree; also capable of using matching rules</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<p>Initial setup</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">log</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">tan</span><span class="p">,</span> <span class="n">Wild</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">log</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="o">+</span> <span class="n">tan</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-</pre></div>
-</div>
-<dl class="docutils">
-<dt>1.1. type -&gt; type</dt>
-<dd><p class="first">obj.replace(sin, tan)</p>
-<div class="last highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">)</span>
-<span class="go">log(cos(x)) + tan(cos(x**2))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="nb">map</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(cos(x), {sin(x): cos(x)})</span>
-</pre></div>
-</div>
-</dd>
-<dt>1.2. type -&gt; func</dt>
-<dd><p class="first">obj.replace(sin, lambda arg: ...)</p>
-<div class="last highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">arg</span><span class="p">:</span> <span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">arg</span><span class="p">))</span>
-<span class="go">log(sin(2*x)) + tan(sin(2*x**2))</span>
-</pre></div>
-</div>
-</dd>
-<dt>2.1. expr -&gt; expr</dt>
-<dd><p class="first">obj.replace(sin(a), tan(a))</p>
-<div class="last highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">tan</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
-<span class="go">log(tan(x)) + tan(tan(x**2))</span>
-</pre></div>
-</div>
-</dd>
-<dt>2.2. expr -&gt; func</dt>
-<dd><p class="first">obj.replace(sin(a), lambda a: ...)</p>
-<div class="last highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">cos</span><span class="p">(</span><span class="n">a</span><span class="p">))</span>
-<span class="go">log(cos(x)) + tan(cos(x**2))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="k">lambda</span> <span class="n">a</span><span class="p">:</span> <span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">))</span>
-<span class="go">log(sin(2*x)) + tan(sin(2*x**2))</span>
-</pre></div>
-</div>
-</dd>
-<dt>3.1. func -&gt; func</dt>
-<dd><p class="first">obj.replace(lambda expr: ..., lambda expr: ...)</p>
-<div class="last highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="k">lambda</span> <span class="n">expr</span><span class="p">:</span> <span class="n">expr</span><span class="o">.</span><span class="n">is_Number</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">expr</span><span class="p">:</span> <span class="n">expr</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">4*sin(x**9)</span>
-</pre></div>
-</div>
-</dd>
-</dl>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.basic.Basic.rewrite">
-<tt class="descname">rewrite</tt><big>(</big><em>self</em>, <em>*args</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.rewrite"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.rewrite" title="Permalink to this definition">¶</a></dt>
-<dd><p>Rewrite functions in terms of other functions.</p>
-<p>Rewrites expression containing applications of functions
-of one kind in terms of functions of different kind. For
-example you can rewrite trigonometric functions as complex
-exponentials or combinatorial functions as gamma function.</p>
-<p>As a pattern this function accepts a list of functions to
-to rewrite (instances of DefinedFunction class). As rule
-you can use string or a destination function instance (in
-this case rewrite() will use the str() function).</p>
-<p>There is also possibility to pass hints on how to rewrite
-the given expressions. For now there is only one such hint
-defined called &#8216;deep&#8217;. When &#8216;deep&#8217; is set to False it will
-forbid functions to rewrite their contents.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<p>Unspecified pattern:
-&gt;&gt;&gt; sin(x).rewrite(exp)
--I*(exp(I*x) - exp(-I*x))/2</p>
-<p>Pattern as a single function:
-&gt;&gt;&gt; sin(x).rewrite(sin, exp)
--I*(exp(I*x) - exp(-I*x))/2</p>
-<p>Pattern as a list of functions:
-&gt;&gt;&gt; sin(x).rewrite([sin, ], exp)
--I*(exp(I*x) - exp(-I*x))/2</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.basic.Basic.sort_key">
-<tt class="descname">sort_key</tt><big>(</big><em>*args</em>, <em>**kw_args</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.sort_key"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.sort_key" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a sort key.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Basic</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">sorted</span><span class="p">([</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="o">-</span><span class="n">I</span><span class="p">],</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">sort_key</span><span class="p">())</span>
-<span class="go">[1/2, -I, I]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="p">(</span><span class="s">&quot;[x, 1/x, 1/x**2, x**2, x**(1/2), x**(1/4), x**(3/2)]&quot;</span><span class="p">)</span>
-<span class="go">[x, 1/x, x**(-2), x**2, sqrt(x), x**(1/4), x**(3/2)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">sorted</span><span class="p">(</span><span class="n">_</span><span class="p">,</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">sort_key</span><span class="p">())</span>
-<span class="go">[x**(-2), 1/x, x**(1/4), sqrt(x), x, x**(3/2), x**2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.basic.Basic.subs">
-<tt class="descname">subs</tt><big>(</big><em>self</em>, <em>*args</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.subs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.subs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Substitutes old for new in an expression after sympifying args.</p>
-<dl class="docutils">
-<dt><span class="math">\(args\)</span> is either:</dt>
-<dd><ul class="first last">
-<li><p class="first">two arguments, e.g. foo.subs(old, new)</p>
-</li>
-<li><dl class="first docutils">
-<dt>one iterable argument, e.g. foo.subs(iterable). The iterable may be</dt>
-<dd><dl class="first last docutils">
-<dt>o an iterable container with (old, new) pairs. In this case the</dt>
-<dd><p class="first last">replacements are processed in the order given with successive
-patterns possibly affecting replacements already made.</p>
-</dd>
-<dt>o a dict or set whose key/value items correspond to old/new pairs.</dt>
-<dd><p class="first last">In this case the old/new pairs will be sorted by op count and in
-case of a tie, by number of args and the default_sort_key. The
-resulting sorted list is then processed as an iterable container
-(see previous).</p>
-</dd>
-</dl>
-</dd>
-</dl>
-</li>
-</ul>
-</dd>
-</dl>
-<p>If the keyword <tt class="docutils literal"><span class="pre">simultaneous</span></tt> is True, the subexpressions will not be
-evaluated until all the substitutions have been made.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.core.basic.Basic.replace" title="sympy.core.basic.Basic.replace"><tt class="xref py py-obj docutils literal"><span class="pre">replace</span></tt></a></dt>
-<dd>replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements</dd>
-<dt><a class="reference internal" href="#sympy.core.basic.Basic.xreplace" title="sympy.core.basic.Basic.xreplace"><tt class="xref py py-obj docutils literal"><span class="pre">xreplace</span></tt></a></dt>
-<dd>exact node replacement in expr tree; also capable of using matching rules</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">exp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">pi</span><span class="p">)</span>
-<span class="go">pi*y + 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">({</span><span class="n">x</span><span class="p">:</span><span class="n">pi</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span><span class="mi">2</span><span class="p">})</span>
-<span class="go">1 + 2*pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">([(</span><span class="n">x</span><span class="p">,</span> <span class="n">pi</span><span class="p">),</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="mi">2</span><span class="p">)])</span>
-<span class="go">1 + 2*pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">reps</span> <span class="o">=</span> <span class="p">[(</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">reps</span><span class="p">)</span>
-<span class="go">6</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">reps</span><span class="p">))</span>
-<span class="go">x**2 + 2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">y**2 + y</span>
-</pre></div>
-</div>
-<p>To replace only the x**2 but not the x**4, use xreplace:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">xreplace</span><span class="p">({</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">:</span> <span class="n">y</span><span class="p">})</span>
-<span class="go">x**4 + y</span>
-</pre></div>
-</div>
-<p>To delay evaluation until all substitutions have been made,
-set the keyword <tt class="docutils literal"><span class="pre">simultaneous</span></tt> to True:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">([(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="mi">0</span><span class="p">)])</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">([(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="mi">0</span><span class="p">)],</span> <span class="n">simultaneous</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">nan</span>
-</pre></div>
-</div>
-<p>This has the added feature of not allowing subsequent substitutions
-to affect those already made:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">/</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">({</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">:</span> <span class="n">y</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">})</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">/</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">({</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">:</span> <span class="n">y</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">},</span> <span class="n">simultaneous</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">y/(x + y)</span>
-</pre></div>
-</div>
-<p>In order to obtain a canonical result, unordered iterables are
-sorted by count_op length, number of arguments and by the
-default_sort_key to break any ties. All other iterables are left
-unsorted.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">exp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">e</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)),</span> <span class="n">a</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">b</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">c</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E</span> <span class="o">=</span> <span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">e</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expr</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">))</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expr</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="nb">dict</span><span class="p">([</span><span class="n">A</span><span class="p">,</span><span class="n">B</span><span class="p">,</span><span class="n">C</span><span class="p">,</span><span class="n">D</span><span class="p">,</span><span class="n">E</span><span class="p">]))</span>
-<span class="go">a*c*sin(d*e) + b</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.basic.Basic.xreplace">
-<tt class="descname">xreplace</tt><big>(</big><em>self</em>, <em>rule</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/basic.html#Basic.xreplace"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Basic.xreplace" title="Permalink to this definition">¶</a></dt>
-<dd><p>Replace occurrences of objects within the expression.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>rule</strong> : dict-like</p>
-<blockquote>
-<div><p>Expresses a replacement rule</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>xreplace</strong> : the result of the replacement</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.core.basic.Basic.replace" title="sympy.core.basic.Basic.replace"><tt class="xref py py-obj docutils literal"><span class="pre">replace</span></tt></a></dt>
-<dd>replacement capable of doing wildcard-like matching, parsing of match, and conditional replacements</dd>
-<dt><a class="reference internal" href="galgebra/GA/GAsympy.html#subs" title="subs"><tt class="xref py py-obj docutils literal"><span class="pre">subs</span></tt></a></dt>
-<dd>substitution of subexpressions as defined by the objects themselves.</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">exp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x y z&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">xreplace</span><span class="p">({</span><span class="n">x</span><span class="p">:</span> <span class="n">pi</span><span class="p">})</span>
-<span class="go">pi*y + 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">xreplace</span><span class="p">({</span><span class="n">x</span><span class="p">:</span><span class="n">pi</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span><span class="mi">2</span><span class="p">})</span>
-<span class="go">1 + 2*pi</span>
-</pre></div>
-</div>
-<p>Replacements occur only if an entire node in the expression tree is
-matched:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">xreplace</span><span class="p">({</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">:</span> <span class="n">pi</span><span class="p">})</span>
-<span class="go">z + pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">xreplace</span><span class="p">({</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">:</span> <span class="n">pi</span><span class="p">})</span>
-<span class="go">x*y*z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">xreplace</span><span class="p">({</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">:</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">:</span> <span class="n">z</span><span class="p">})</span>
-<span class="go">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">xreplace</span><span class="p">({</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">:</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">:</span> <span class="n">z</span><span class="p">})</span>
-<span class="go">4*z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">xreplace</span><span class="p">({</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">:</span> <span class="mi">2</span><span class="p">})</span>
-<span class="go">x + y + 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">xreplace</span><span class="p">({</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">:</span> <span class="n">y</span><span class="p">})</span>
-<span class="go">x + exp(y) + 2</span>
-</pre></div>
-</div>
-<p>xreplace doesn&#8217;t differentiate between free and bound symbols. In the
-following, subs(x, y) would not change x since it is a bound symbol,
-but xreplace does:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">xreplace</span><span class="p">({</span><span class="n">x</span><span class="p">:</span> <span class="n">y</span><span class="p">})</span>
-<span class="go">Integral(y, (y, 1, 2*y))</span>
-</pre></div>
-</div>
-<p>Trying to replace x with an expression raises an error:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">xreplace</span><span class="p">({</span><span class="n">x</span><span class="p">:</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="p">})</span> 
-<span class="go">ValueError: Invalid limits given: ((2*y, 1, 4*y),)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="atom">
-<h3>Atom<a class="headerlink" href="#atom" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.basic.Atom">
-<em class="property">class </em><tt class="descclassname">sympy.core.basic.</tt><tt class="descname">Atom</tt><a class="reference internal" href="../_modules/sympy/core/basic.html#Atom"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.basic.Atom" title="Permalink to this definition">¶</a></dt>
-<dd><p>A parent class for atomic things. An atom is an expression with no subexpressions.</p>
-<p class="rubric">Examples</p>
-<p>Symbol, Number, Rational, Integer, ...
-But not: Add, Mul, Pow, ...</p>
-</dd></dl>
-
-</div>
-<div class="section" id="c">
-<h3>C<a class="headerlink" href="#c" title="Permalink to this headline">¶</a></h3>
-<dl class="attribute">
-<dt id="sympy.core.basic.C">
-<tt class="descclassname">sympy.core.basic.</tt><tt class="descname">C</tt><a class="headerlink" href="#sympy.core.basic.C" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.core.singleton">
-<span id="singleton"></span><h2>singleton<a class="headerlink" href="#module-sympy.core.singleton" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="s">
-<h3>S<a class="headerlink" href="#s" title="Permalink to this headline">¶</a></h3>
-<dl class="attribute">
-<dt id="sympy.core.singleton.S">
-<tt class="descclassname">sympy.core.singleton.</tt><tt class="descname">S</tt><a class="headerlink" href="#sympy.core.singleton.S" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.core.expr">
-<span id="expr"></span><h2>expr<a class="headerlink" href="#module-sympy.core.expr" title="Permalink to this headline">¶</a></h2>
-</div>
-<div class="section" id="id3">
-<h2>Expr<a class="headerlink" href="#id3" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.core.expr.Expr">
-<em class="property">class </em><tt class="descclassname">sympy.core.expr.</tt><tt class="descname">Expr</tt><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr" title="Permalink to this definition">¶</a></dt>
-<dd><dl class="function">
-<dt id="sympy.core.expr.Expr.apart">
-<tt class="descname">apart</tt><big>(</big><em>self</em>, <em>x=None</em>, <em>**args</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.apart"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.apart" title="Permalink to this definition">¶</a></dt>
-<dd><p>See the apart function in sympy.polys</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.args_cnc">
-<tt class="descname">args_cnc</tt><big>(</big><em>self</em>, <em>cset=False</em>, <em>warn=True</em>, <em>split_1=True</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.args_cnc"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.args_cnc" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return [commutative factors, non-commutative factors] of self.</p>
-<p>self is treated as a Mul and the ordering of the factors is maintained.
-If <tt class="docutils literal"><span class="pre">cset</span></tt> is True the commutative factors will be returned in a set.
-If there were repeated factors (as may happen with an unevaluated Mul)
-then an error will be raised unless it is explicitly supressed by
-setting <tt class="docutils literal"><span class="pre">warn</span></tt> to False.</p>
-<p>Note: -1 is always separated from a Number unless split_1 is False.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="p">,</span> <span class="n">B</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;A B&#39;</span><span class="p">,</span> <span class="n">commutative</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-<span class="go">[[-1, 2, x, y], []]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="o">-</span><span class="mf">2.5</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-<span class="go">[[-1, 2.5, x], []]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">A</span><span class="o">*</span><span class="n">B</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-<span class="go">[[-1, 2, x, y], [A, B]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">A</span><span class="o">*</span><span class="n">B</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">(</span><span class="n">split_1</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">[[-2, x, y], [A, B]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">(</span><span class="n">cset</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">[set([-1, 2, x, y]), []]</span>
-</pre></div>
-</div>
-<p>The arg is always treated as a Mul:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="o">-</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="n">A</span><span class="p">)</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span>
-<span class="go">[[], [x - 2 + A]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="o">-</span><span class="n">oo</span><span class="p">)</span><span class="o">.</span><span class="n">args_cnc</span><span class="p">()</span> <span class="c"># -oo is a singleton</span>
-<span class="go">[[-1, oo], []]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.as_coeff_Add">
-<tt class="descname">as_coeff_Add</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.as_coeff_Add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.as_coeff_Add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Efficiently extract the coefficient of a summation.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.as_coeff_Mul">
-<tt class="descname">as_coeff_Mul</tt><big>(</big><em>self</em>, <em>rational=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.as_coeff_Mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.as_coeff_Mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Efficiently extract the coefficient of a product.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.as_coeff_add">
-<tt class="descname">as_coeff_add</tt><big>(</big><em>self</em>, <em>*deps</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.as_coeff_add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.as_coeff_add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the tuple (c, args) where self is written as an Add, <tt class="docutils literal"><span class="pre">a</span></tt>.</p>
-<p>c should be a Rational added to any terms of the Add that are
-independent of deps.</p>
-<p>args should be a tuple of all other terms of <tt class="docutils literal"><span class="pre">a</span></tt>; args is empty
-if self is a Number or if self is independent of deps (when given).</p>
-<p>This should be used when you don&#8217;t know if self is an Add or not but
-you want to treat self as an Add or if you want to process the
-individual arguments of the tail of self as an Add.</p>
-<ul class="simple">
-<li>if you know self is an Add and want only the head, use self.args[0];</li>
-<li>if you don&#8217;t want to process the arguments of the tail but need the
-tail then use self.as_two_terms() which gives the head and tail.</li>
-<li>if you want to split self into an independent and dependent parts
-use <tt class="docutils literal"><span class="pre">self.as_independent(*deps)</span></tt></li>
-</ul>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span><span class="o">.</span><span class="n">as_coeff_add</span><span class="p">()</span>
-<span class="go">(3, ())</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">as_coeff_add</span><span class="p">()</span>
-<span class="go">(3, (x,))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">as_coeff_add</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(y + 3, (x,))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">as_coeff_add</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(y + 3, ())</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.as_coeff_exponent">
-<tt class="descname">as_coeff_exponent</tt><big>(</big><em>self</em>, <em>x</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.as_coeff_exponent"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.as_coeff_exponent" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">c*x**e</span> <span class="pre">-&gt;</span> <span class="pre">c,e</span></tt> where x can be any symbolic expression.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.as_coeff_factors">
-<tt class="descname">as_coeff_factors</tt><big>(</big><em>self</em>, <em>*deps</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.as_coeff_factors"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.as_coeff_factors" title="Permalink to this definition">¶</a></dt>
-<dd><p>This method is deprecated.  Use .as_coeff_add() instead.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.as_coeff_mul">
-<tt class="descname">as_coeff_mul</tt><big>(</big><em>self</em>, <em>*deps</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.as_coeff_mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.as_coeff_mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the tuple (c, args) where self is written as a Mul, <tt class="docutils literal"><span class="pre">m</span></tt>.</p>
-<p>c should be a Rational multiplied by any terms of the Mul that are
-independent of deps.</p>
-<p>args should be a tuple of all other terms of m; args is empty
-if self is a Number or if self is independent of deps (when given).</p>
-<p>This should be used when you don&#8217;t know if self is a Mul or not but
-you want to treat self as a Mul or if you want to process the
-individual arguments of the tail of self as a Mul.</p>
-<ul class="simple">
-<li>if you know self is a Mul and want only the head, use self.args[0];</li>
-<li>if you don&#8217;t want to process the arguments of the tail but need the
-tail then use self.as_two_terms() which gives the head and tail;</li>
-<li>if you want to split self into an independent and dependent parts
-use <tt class="docutils literal"><span class="pre">self.as_independent(*deps)</span></tt></li>
-</ul>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">()</span>
-<span class="go">(3, ())</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">()</span>
-<span class="go">(3, (x, y))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(3*y, (x,))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">as_coeff_mul</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(3*y, ())</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.as_coeff_terms">
-<tt class="descname">as_coeff_terms</tt><big>(</big><em>self</em>, <em>*deps</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.as_coeff_terms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.as_coeff_terms" title="Permalink to this definition">¶</a></dt>
-<dd><p>This method is deprecated. Use .as_coeff_mul() instead.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.as_coefficient">
-<tt class="descname">as_coefficient</tt><big>(</big><em>self</em>, <em>expr</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.as_coefficient"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.as_coefficient" title="Permalink to this definition">¶</a></dt>
-<dd><p>Extracts symbolic coefficient at the given expression. In
-other words, this functions separates &#8216;self&#8217; into the product
-of &#8216;expr&#8217; and &#8216;expr&#8217;-free coefficient. If such separation
-is not possible it will return None.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.core.expr.Expr.coeff" title="sympy.core.expr.Expr.coeff"><tt class="xref py py-obj docutils literal"><span class="pre">coeff</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">E</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">E</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">E</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">E</span><span class="p">)</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">E</span><span class="p">)</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">E</span><span class="p">)</span><span class="o">*</span><span class="n">E</span><span class="p">)</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">E</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">E</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">E</span><span class="p">)</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">E</span><span class="p">)</span>
-<span class="go">x + 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">E</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">E</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">E</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">E</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="p">)</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">as_coefficient</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="p">)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.as_coefficients_dict">
-<tt class="descname">as_coefficients_dict</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.as_coefficients_dict"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.as_coefficients_dict" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a dictionary mapping terms to their Rational coefficient.
-Since the dictionary is a defaultdict, inquiries about terms which
-were not present will return a coefficient of 0. If an expression is
-not an Add it is considered to have a single term.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">as_coefficients_dict</span><span class="p">()</span>
-<span class="go">{1: 4, x: 3, a*x: 1}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="p">[</span><span class="n">a</span><span class="p">]</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">as_coefficients_dict</span><span class="p">()</span>
-<span class="go">{a*x: 3}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.as_content_primitive">
-<tt class="descname">as_content_primitive</tt><big>(</big><em>self</em>, <em>radical=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.as_content_primitive"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.as_content_primitive" title="Permalink to this definition">¶</a></dt>
-<dd><p>This method should recursively remove a Rational from all arguments
-and return that (content) and the new self (primitive). The content
-should always be positive and <tt class="docutils literal"><span class="pre">Mul(*foo.as_content_primitive())</span> <span class="pre">==</span> <span class="pre">foo</span></tt>.
-The primitive need no be in canonical form and should try to preserve
-the underlying structure if possible (i.e. expand_mul should not be
-applied to self).</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="o">*</span><span class="p">(</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>The as_content_primitive function is recursive and retains structure:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-<span class="go">(2, x + 3*y*(y + 1) + 1)</span>
-</pre></div>
-</div>
-<p>Integer powers will have Rationals extracted from the base:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">6</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-<span class="go">(4, (3*x + 1)**2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">6</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-<span class="go">(1, (2*(3*x + 1))**(2*y))</span>
-</pre></div>
-</div>
-<p>Terms may end up joining once their as_content_primitives are added:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="mi">5</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">y</span><span class="p">))</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="p">)))</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-<span class="go">(11, x*(y + 1))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="mi">3</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">y</span><span class="p">))</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="p">)))</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-<span class="go">(9, x*(y + 1))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="mi">3</span><span class="o">*</span><span class="p">(</span><span class="n">z</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">y</span><span class="p">))</span> <span class="o">+</span> <span class="mf">2.0</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="p">)))</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-<span class="go">(1, 6.0*x*(y + 1) + 3*z*(y + 1))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="mi">5</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">y</span><span class="p">))</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-<span class="go">(121, x**2*(y + 1)**2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="mi">5</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">y</span><span class="p">))</span> <span class="o">+</span> <span class="mf">2.0</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-<span class="go">(1, 121.0*x**2*(y + 1)**2)</span>
-</pre></div>
-</div>
-<p>Radical content can also be factored out of the primitive:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">10</span><span class="p">))</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">(</span><span class="n">radical</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(2, sqrt(2)*(1 + 2*sqrt(5)))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.as_expr">
-<tt class="descname">as_expr</tt><big>(</big><em>self</em>, <em>*gens</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.as_expr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.as_expr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a polynomial to a SymPy expression.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">as_poly</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-<span class="go">x**2 + x*y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-<span class="go">sin(x)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.as_independent">
-<tt class="descname">as_independent</tt><big>(</big><em>self</em>, <em>*deps</em>, <em>**hint</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.as_independent"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.as_independent" title="Permalink to this definition">¶</a></dt>
-<dd><p>A mostly naive separation of a Mul or Add into arguments that are not
-are dependent on deps. To obtain as complete a separation of variables
-as possible, use a separation method first, e.g.:</p>
-<ul class="simple">
-<li>separatevars() to change Mul, Add and Pow (including exp) into Mul</li>
-<li>.expand(mul=True) to change Add or Mul into Add</li>
-<li>.expand(log=True) to change log expr into an Add</li>
-</ul>
-<p>The only non-naive thing that is done here is to respect noncommutative
-ordering of variables.</p>
-<p>The returned tuple (i, d) has the following interpretation:</p>
-<ul class="simple">
-<li>i will has no variable that appears in deps</li>
-<li>d will be 1 or else have terms that contain variables that are in deps</li>
-<li>if self is an Add then self = i + d</li>
-<li>if self is a Mul then self = i*d</li>
-<li>if self is anything else, either tuple (self, S.One) or (S.One, self)
-is returned.</li>
-</ul>
-<p>To force the expression to be treated as an Add, use the hint as_Add=True</p>
-<p class="rubric">Examples</p>
-<p>&#8211; self is an Add</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">exp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(0, x*y + x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">(x, x*y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(y + z, 2*x*sin(x) + x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">(z, 2*x*sin(x) + x + y)</span>
-</pre></div>
-</div>
-<p>&#8211; self is a Mul</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(cos(y), x*sin(x))</span>
-</pre></div>
-</div>
-<p>non-commutative terms cannot always be separated out when self is a Mul</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">,</span> <span class="n">n3</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;n1 n2 n3&#39;</span><span class="p">,</span> <span class="n">commutative</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">n1</span> <span class="o">+</span> <span class="n">n1</span><span class="o">*</span><span class="n">n2</span><span class="p">)</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">n2</span><span class="p">)</span>
-<span class="go">(n1, n1*n2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">n2</span><span class="o">*</span><span class="n">n1</span> <span class="o">+</span> <span class="n">n1</span><span class="o">*</span><span class="n">n2</span><span class="p">)</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">n2</span><span class="p">)</span>
-<span class="go">(0, n1*n2 + n2*n1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">n1</span><span class="o">*</span><span class="n">n2</span><span class="o">*</span><span class="n">n3</span><span class="p">)</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">n1</span><span class="p">)</span>
-<span class="go">(1, n1*n2*n3)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">n1</span><span class="o">*</span><span class="n">n2</span><span class="o">*</span><span class="n">n3</span><span class="p">)</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">n2</span><span class="p">)</span>
-<span class="go">(n1, n2*n3)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span><span class="o">-</span><span class="n">n1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(1, (x - y)*(x - n1))</span>
-</pre></div>
-</div>
-<p>&#8211; self is anything else:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(1, sin(x))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">(sin(x), 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(1, exp(x + y))</span>
-</pre></div>
-</div>
-<p>&#8211; force self to be treated as an Add:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">as_Add</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(0, 3*x)</span>
-</pre></div>
-</div>
-<p>&#8211; force self to be treated as a Mul:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">as_Add</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">(1, x + 3)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="o">+</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">as_Add</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">(1, x - 3)</span>
-</pre></div>
-</div>
-<p>Note how the below differs from the above in making the
-constant on the dep term positive.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">y</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="o">+</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(y, x - 3)</span>
-</pre></div>
-</div>
-<dl class="docutils">
-<dt>&#8211; use .as_independent() for true independence testing instead</dt>
-<dd>of .has(). The former considers only symbols in the free
-symbols while the latter considers all symbols</dd>
-</dl>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="ow">in</span> <span class="n">I</span><span class="o">.</span><span class="n">free_symbols</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">==</span> <span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">I</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">==</span> <span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>Note: when trying to get independent terms, a separation method
-might need to be used first. In this case, it is important to keep
-track of what you send to this routine so you know how to interpret
-the returned values</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">separatevars</span><span class="p">,</span> <span class="n">log</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">separatevars</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(exp(y), exp(x))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">(x, x*y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">separatevars</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">(x, y + 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">(x, y + 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">mul</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">(x, x*y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a b&#39;</span><span class="p">,</span><span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">b</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">log</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span><span class="o">.</span><span class="n">as_independent</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-<span class="go">(log(a), log(b))</span>
-</pre></div>
-</div>
-<dl class="docutils">
-<dt>See also: .separatevars(), .expand(log=True),</dt>
-<dd>.as_two_terms(), .as_coeff_add(), .as_coeff_mul()</dd>
-</dl>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.as_leading_term">
-<tt class="descname">as_leading_term</tt><big>(</big><em>*args</em>, <em>**kw_args</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.as_leading_term"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.as_leading_term" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the leading (nonzero) term of the series expansion of self.</p>
-<p>The _eval_as_leading_term routines are used to do this, and they must
-always return a non-zero value.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">as_leading_term</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">x**(-2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.as_numer_denom">
-<tt class="descname">as_numer_denom</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.as_numer_denom"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.as_numer_denom" title="Permalink to this definition">¶</a></dt>
-<dd><p>expression -&gt; a/b -&gt; a, b</p>
-<p>This is just a stub that should be defined by
-an object&#8217;s class methods to get anything else.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">normal</span></tt></dt>
-<dd>return a/b instead of a, b</dd>
-</dl>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.as_ordered_factors">
-<tt class="descname">as_ordered_factors</tt><big>(</big><em>self</em>, <em>order=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.as_ordered_factors"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.as_ordered_factors" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return list of ordered factors (if Mul) else [self].</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.as_ordered_terms">
-<tt class="descname">as_ordered_terms</tt><big>(</big><em>self</em>, <em>order=None</em>, <em>data=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.as_ordered_terms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.as_ordered_terms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Transform an expression to an ordered list of terms.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">as_ordered_terms</span><span class="p">()</span>
-<span class="go">[sin(x)**2*cos(x), sin(x)**2, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.as_powers_dict">
-<tt class="descname">as_powers_dict</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.as_powers_dict"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.as_powers_dict" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return self as a dictionary of factors with each factor being
-treated as a power. The keys are the bases of the factors and the
-values, the corresponding exponents. The resulting dictionary should
-be used with caution if the expression is a Mul and contains non-
-commutative factors since the order that they appeared will be lost in
-the dictionary.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.as_real_imag">
-<tt class="descname">as_real_imag</tt><big>(</big><em>self</em>, <em>deep=True</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.as_real_imag"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.as_real_imag" title="Permalink to this definition">¶</a></dt>
-<dd><p>Performs complex expansion on &#8216;self&#8217; and returns a tuple
-containing collected both real and imaginary parts. This
-method can&#8217;t be confused with re() and im() functions,
-which does not perform complex expansion at evaluation.</p>
-<p>However it is possible to expand both re() and im()
-functions and get exactly the same results as with
-a single call to this function.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">I</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="o">*</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-<span class="go">(x, y)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">z</span><span class="p">,</span> <span class="n">w</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">z</span> <span class="o">+</span> <span class="n">w</span><span class="o">*</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-<span class="go">(re(z) - im(w), re(w) + im(z))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.as_terms">
-<tt class="descname">as_terms</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.as_terms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.as_terms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Transform an expression to a list of terms.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.cancel">
-<tt class="descname">cancel</tt><big>(</big><em>self</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.cancel"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.cancel" title="Permalink to this definition">¶</a></dt>
-<dd><p>See the cancel function in sympy.polys</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.coeff">
-<tt class="descname">coeff</tt><big>(</big><em>self</em>, <em>x</em>, <em>n=1</em>, <em>right=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.coeff"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.coeff" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the coefficient from the term containing &#8220;x**n&#8221; or None if
-there is no such term. If <tt class="docutils literal"><span class="pre">n</span></tt> is zero then all terms independent of
-x will be returned.</p>
-<p>When x is noncommutative, the coeff to the left (default) or right of x
-can be returned. The keyword &#8216;right&#8217; is ignored when x is commutative.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.core.expr.Expr.as_coeff_Add" title="sympy.core.expr.Expr.as_coeff_Add"><tt class="xref py py-obj docutils literal"><span class="pre">as_coeff_Add</span></tt></a></dt>
-<dd>a method to separate the additive constant from an expression</dd>
-<dt><a class="reference internal" href="#sympy.core.expr.Expr.as_coeff_Mul" title="sympy.core.expr.Expr.as_coeff_Mul"><tt class="xref py py-obj docutils literal"><span class="pre">as_coeff_Mul</span></tt></a></dt>
-<dd>a method to separate the multiplicative constant from an expression</dd>
-<dt><a class="reference internal" href="#sympy.core.expr.Expr.as_independent" title="sympy.core.expr.Expr.as_independent"><tt class="xref py py-obj docutils literal"><span class="pre">as_independent</span></tt></a></dt>
-<dd>a method to separate x dependent terms/factors from others</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-</pre></div>
-</div>
-<p>You can select terms that have an explicit negative in front of them:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="o">-</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">2*y</span>
-</pre></div>
-</div>
-<p>You can select terms with no Rational coefficient:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>You can select terms independent of x by making n=0; in this case
-expr.as_independent(x)[0] is returned (and 0 will be returned instead
-of None):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span> <span class="o">=</span> <span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span> <span class="o">+</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span>
-<span class="go">x**3 + 3*x**2 + 3*x + 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">eq</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">)))]</span>
-<span class="go">[1, 3, 3, 2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span> <span class="o">-=</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">eq</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">)))]</span>
-<span class="go">[1, 3, 3, 0]</span>
-</pre></div>
-</div>
-<p>You can select terms that have a numerical term in front of them:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="o">-</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">-y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">x</span>
-</pre></div>
-</div>
-<p>The matching is exact:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">z</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">coeff</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">z</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>In addition, no factoring is done, so 1 + z*(1 + y) is not obtained
-from the following:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">z</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>If such factoring is desired, factor_terms can be used first:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factor_terms</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factor_terms</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">z</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">z*(y + 1) + 1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">o</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;n m o&#39;</span><span class="p">,</span> <span class="n">commutative</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">m</span> <span class="o">+</span> <span class="n">m</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">m</span><span class="p">)</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="c"># = (1 + m)*n*m</span>
-<span class="go">1 + m</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">m</span> <span class="o">+</span> <span class="n">m</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">m</span><span class="p">)</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">right</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="c"># = (1 + m)*n*m</span>
-<span class="go">m</span>
-</pre></div>
-</div>
-<p>If there is more than one possible coefficient 0 is returned:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">m</span> <span class="o">+</span> <span class="n">m</span><span class="o">*</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>If there is only one possible coefficient, it is returned:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">m</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">m</span><span class="o">*</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">m</span><span class="o">*</span><span class="n">n</span><span class="p">)</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">m</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">m</span><span class="o">*</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">coeff</span><span class="p">(</span><span class="n">m</span><span class="o">*</span><span class="n">n</span><span class="p">,</span> <span class="n">right</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.collect">
-<tt class="descname">collect</tt><big>(</big><em>self</em>, <em>syms</em>, <em>func=None</em>, <em>evaluate=True</em>, <em>exact=False</em>, <em>distribute_order_term=True</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.collect"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.collect" title="Permalink to this definition">¶</a></dt>
-<dd><p>See the collect function in sympy.simplify</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.combsimp">
-<tt class="descname">combsimp</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.combsimp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.combsimp" title="Permalink to this definition">¶</a></dt>
-<dd><p>See the combsimp function in sympy.simplify</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.compute_leading_term">
-<tt class="descname">compute_leading_term</tt><big>(</big><em>self</em>, <em>x</em>, <em>skip_abs=False</em>, <em>logx=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.compute_leading_term"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.compute_leading_term" title="Permalink to this definition">¶</a></dt>
-<dd><p>as_leading_term is only allowed for results of .series()
-This is a wrapper to compute a series first.
-If skip_abs is true, the absolute term is assumed to be zero.
-(This is necessary because sometimes it cannot be simplified
-to zero without a lot of work, but is still known to be zero.
-See log._eval_nseries for an example.)
-If skip_log is true, log(x) is treated as an independent symbol.
-(This is needed for the gruntz algorithm.)</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.could_extract_minus_sign">
-<tt class="descname">could_extract_minus_sign</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.could_extract_minus_sign"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.could_extract_minus_sign" title="Permalink to this definition">¶</a></dt>
-<dd><p>Canonical way to choose an element in the set {e, -e} where
-e is any expression. If the canonical element is e, we have
-e.could_extract_minus_sign() == True, else
-e.could_extract_minus_sign() == False.</p>
-<p>For any expression, the set <tt class="docutils literal"><span class="pre">{e.could_extract_minus_sign(),</span>
-<span class="pre">(-e).could_extract_minus_sign()}</span></tt> must be <tt class="docutils literal"><span class="pre">{True,</span> <span class="pre">False}</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">()</span> <span class="o">!=</span> <span class="p">(</span><span class="n">y</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">could_extract_minus_sign</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.count_ops">
-<tt class="descname">count_ops</tt><big>(</big><em>self</em>, <em>visual=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.count_ops"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.count_ops" title="Permalink to this definition">¶</a></dt>
-<dd><p>wrapper for count_ops that returns the operation count.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.equals">
-<tt class="descname">equals</tt><big>(</big><em>self</em>, <em>other</em>, <em>failing_expression=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.equals"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.equals" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if self == other, False if it doesn&#8217;t, or None. If
-failing_expression is True then the expression which did not simplify
-to a 0 will be returned instead of None.</p>
-<p>If <tt class="docutils literal"><span class="pre">self</span></tt> is a Number (or complex number) that is not zero, then
-the result is False.</p>
-<p>If <tt class="docutils literal"><span class="pre">self</span></tt> is a number and has not evaluated to zero, evalf will be
-used to test whether the expression evaluates to zero. If it does so
-and the result has significance (i.e. the precision is either -1, for
-a Rational result, or is greater than 1) then the evalf value will be
-used to return True or False.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.expand">
-<tt class="descname">expand</tt><big>(</big><em>*args</em>, <em>**kw_args</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.expand"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.expand" title="Permalink to this definition">¶</a></dt>
-<dd><p>Expand an expression using hints.</p>
-<p>See the docstring of the expand() function in sympy.core.function for
-more information.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.extract_additively">
-<tt class="descname">extract_additively</tt><big>(</big><em>self</em>, <em>c</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.extract_additively"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.extract_additively" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return self - c if it&#8217;s possible to subtract c from self and
-make all matching coefficients move towards zero, else return None.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.core.expr.Expr.extract_multiplicatively" title="sympy.core.expr.Expr.extract_multiplicatively"><tt class="xref py py-obj docutils literal"><span class="pre">extract_multiplicatively</span></tt></a>, <a class="reference internal" href="#sympy.core.expr.Expr.coeff" title="sympy.core.expr.Expr.coeff"><tt class="xref py py-obj docutils literal"><span class="pre">coeff</span></tt></a>, <a class="reference internal" href="#sympy.core.expr.Expr.as_coefficient" title="sympy.core.expr.Expr.as_coefficient"><tt class="xref py py-obj docutils literal"><span class="pre">as_coefficient</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">extract_additively</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">x + 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">extract_additively</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">extract_additively</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">y</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">extract_additively</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">extract_additively</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">(x + 1)*(x + 2*y) + 3</span>
-</pre></div>
-</div>
-<p>Sometimes auto-expansion will return a less simplified result
-than desired; gcd_terms might be used in such cases:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">gcd_terms</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">extract_additively</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">4*x*(y + 1) + x*(4*y + 3) - x*(4*y + 4) + y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gcd_terms</span><span class="p">(</span><span class="n">_</span><span class="p">)</span>
-<span class="go">x*(4*y + 3) + y</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.extract_branch_factor">
-<tt class="descname">extract_branch_factor</tt><big>(</big><em>self</em>, <em>allow_half=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.extract_branch_factor"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.extract_branch_factor" title="Permalink to this definition">¶</a></dt>
-<dd><p>Try to write self as <tt class="docutils literal"><span class="pre">exp_polar(2*pi*I*n)*z</span></tt> in a nice way.
-Return (z, n).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exp_polar</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp_polar</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">.</span><span class="n">extract_branch_factor</span><span class="p">()</span>
-<span class="go">(exp_polar(I*pi), 0)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp_polar</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">.</span><span class="n">extract_branch_factor</span><span class="p">()</span>
-<span class="go">(1, 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp_polar</span><span class="p">(</span><span class="o">-</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">extract_branch_factor</span><span class="p">()</span>
-<span class="go">(exp_polar(I*pi), -1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp_polar</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">extract_branch_factor</span><span class="p">()</span>
-<span class="go">(exp_polar(x + I*pi), 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">y</span><span class="o">*</span><span class="n">exp_polar</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="p">)</span><span class="o">*</span><span class="n">exp_polar</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">extract_branch_factor</span><span class="p">()</span>
-<span class="go">(y*exp_polar(2*pi*x), -1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp_polar</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">extract_branch_factor</span><span class="p">()</span>
-<span class="go">(exp_polar(-I*pi/2), 0)</span>
-</pre></div>
-</div>
-<p>If allow_half is True, also extract exp_polar(I*pi):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">exp_polar</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">.</span><span class="n">extract_branch_factor</span><span class="p">(</span><span class="n">allow_half</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(1, 1/2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp_polar</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">.</span><span class="n">extract_branch_factor</span><span class="p">(</span><span class="n">allow_half</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(1, 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp_polar</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">.</span><span class="n">extract_branch_factor</span><span class="p">(</span><span class="n">allow_half</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(1, 3/2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp_polar</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">.</span><span class="n">extract_branch_factor</span><span class="p">(</span><span class="n">allow_half</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(1, -1/2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.extract_multiplicatively">
-<tt class="descname">extract_multiplicatively</tt><big>(</big><em>self</em>, <em>c</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.extract_multiplicatively"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.extract_multiplicatively" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return None if it&#8217;s not possible to make self in the form
-c * something in a nice way, i.e. preserving the properties
-of arguments of self.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">Rational</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">x*y**2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">*</span> <span class="n">y</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">extract_multiplicatively</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">x/6</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.factor">
-<tt class="descname">factor</tt><big>(</big><em>self</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.factor"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.factor" title="Permalink to this definition">¶</a></dt>
-<dd><p>See the factor() function in sympy.polys.polytools</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.getO">
-<tt class="descname">getO</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.getO"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.getO" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the additive O(..) symbol if there is one, else None.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.getn">
-<tt class="descname">getn</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.getn"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.getn" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the order of the expression.</p>
-<p>The order is determined either from the O(...) term. If there
-is no O(...) term, it returns None.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">O</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="n">O</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">getn</span><span class="p">()</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">getn</span><span class="p">()</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.integrate">
-<tt class="descname">integrate</tt><big>(</big><em>self</em>, <em>*args</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.integrate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.integrate" title="Permalink to this definition">¶</a></dt>
-<dd><p>See the integrate function in sympy.integrals</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.invert">
-<tt class="descname">invert</tt><big>(</big><em>self</em>, <em>g</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.invert"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.invert" title="Permalink to this definition">¶</a></dt>
-<dd><p>See the invert function in sympy.polys</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.is_constant">
-<tt class="descname">is_constant</tt><big>(</big><em>self</em>, <em>*wrt</em>, <em>**flags</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.is_constant"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.is_constant" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if self is constant, False if not, or None if
-the constancy could not be determined conclusively.</p>
-<p>If an expression has no free symbols then it is a constant. If
-there are free symbols it is possible that the expression is a
-constant, perhaps (but not necessarily) zero. To test such
-expressions, two strategies are tried:</p>
-<p>1) numerical evaluation at two random points. If two such evaluations
-give two different values and the values have a precision greater than
-1 then self is not constant. If the evaluations agree or could not be
-obtained with any precision, no decision is made. The numerical testing
-is done only if <tt class="docutils literal"><span class="pre">wrt</span></tt> is different than the free symbols.</p>
-<p>2) differentiation with respect to variables in &#8216;wrt&#8217; (or all free
-symbols if omitted) to see if the expression is constant or not. This
-will not always lead to an expression that is zero even though an
-expression is constant (see added test in test_expr.py). If
-all derivatives are zero then self is constant with respect to the
-given symbols.</p>
-<p>If neither evaluation nor differentiation can prove the expression is
-constant, None is returned unless two numerical values happened to be
-the same and the flag <tt class="docutils literal"><span class="pre">failing_number</span></tt> is True &#8211; in that case the
-numerical value will be returned.</p>
-<p>If flag simplify=False is passed, self will not be simplified;
-the default is True since self should be simplified before testing.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">Sum</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">is_constant</span><span class="p">()</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">is_constant</span><span class="p">()</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span><span class="o">.</span><span class="n">is_constant</span><span class="p">()</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span><span class="o">.</span><span class="n">is_constant</span><span class="p">()</span>  
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span><span class="o">.</span><span class="n">is_constant</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span><span class="o">.</span><span class="n">is_constant</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> 
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span><span class="o">.</span><span class="n">is_constant</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span> <span class="o">=</span> <span class="n">a</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">a</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span><span class="o">.</span><span class="n">is_constant</span><span class="p">()</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span><span class="o">.</span><span class="n">subs</span><span class="p">({</span><span class="n">x</span><span class="p">:</span><span class="n">pi</span><span class="p">,</span> <span class="n">a</span><span class="p">:</span><span class="mi">2</span><span class="p">})</span> <span class="o">==</span> <span class="n">eq</span><span class="o">.</span><span class="n">subs</span><span class="p">({</span><span class="n">x</span><span class="p">:</span><span class="n">pi</span><span class="p">,</span> <span class="n">a</span><span class="p">:</span><span class="mi">3</span><span class="p">})</span> <span class="o">==</span> <span class="mi">0</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">0</span><span class="o">**</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_constant</span><span class="p">()</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">is_constant</span><span class="p">()</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_constant</span><span class="p">()</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">one</span> <span class="o">=</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">one</span><span class="o">.</span><span class="n">is_constant</span><span class="p">()</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">one</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">is_constant</span><span class="p">()</span> <span class="c"># could be 0 or 1</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.expr.Expr.is_number">
-<tt class="descname">is_number</tt><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.is_number"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.is_number" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if &#8216;self&#8217; is a number.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">log</span><span class="p">,</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="n">Integral</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="n">Integral</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.is_polynomial">
-<tt class="descname">is_polynomial</tt><big>(</big><em>self</em>, <em>*syms</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.is_polynomial"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.is_polynomial" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if self is a polynomial in syms and False otherwise.</p>
-<p>This checks if self is an exact polynomial in syms.  This function
-returns False for expressions that are &#8220;polynomials&#8221; with symbolic
-exponents.  Thus, you should be able to apply polynomial algorithms to
-expressions for which this returns True, and Poly(expr, *syms) should
-work only if and only if expr.is_polynomial(*syms) returns True. The
-polynomial does not have to be in expanded form.  If no symbols are
-given, all free symbols in the expression will be used.</p>
-<p>This is not part of the assumptions system.  You cannot do
-Symbol(&#8216;z&#8217;, polynomial=True).</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">is_polynomial</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">is_polynomial</span><span class="p">()</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">is_polynomial</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">,</span> <span class="n">nonnegative</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">is_polynomial</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>This function does not attempt any nontrivial simplifications that may
-result in an expression that does not appear to be a polynomial to
-become one.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">factor</span><span class="p">,</span> <span class="n">cancel</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">is_polynomial</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="go">y + 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">is_polynomial</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="p">(</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">is_polynomial</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cancel</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-<span class="go">y + 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cancel</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">.</span><span class="n">is_polynomial</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>See also .is_rational_function()</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.is_rational_function">
-<tt class="descname">is_rational_function</tt><big>(</big><em>self</em>, <em>*syms</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.is_rational_function"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.is_rational_function" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test whether function is a ratio of two polynomials in the given
-symbols, syms. When syms is not given, all free symbols will be used.
-The rational function does not have to be in expanded or in any kind of
-canonical form.</p>
-<p>This function returns False for expressions that are &#8220;rational
-functions&#8221; with symbolic exponents.  Thus, you should be able to call
-.as_numer_denom() and apply polynomial algorithms to the result for
-expressions for which this returns True.</p>
-<p>This is not part of the assumptions system.  You cannot do
-Symbol(&#8216;z&#8217;, rational_function=True).</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">is_rational_function</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">is_rational_function</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="n">sin</span><span class="p">(</span><span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">is_rational_function</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">is_rational_function</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>This function does not attempt any nontrivial simplifications that may
-result in an expression that does not appear to be a rational function
-to become one.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">factor</span><span class="p">,</span> <span class="n">cancel</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">is_rational_function</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="go">(y + 1)/y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">is_rational_function</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>See also is_rational_function().</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.leadterm">
-<tt class="descname">leadterm</tt><big>(</big><em>self</em>, <em>x</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.leadterm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.leadterm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the leading term a*x**b as a tuple (a, b).</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">x</span><span class="o">+</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">leadterm</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(1, 0)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">x</span><span class="o">+</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">leadterm</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(1, -2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.limit">
-<tt class="descname">limit</tt><big>(</big><em>self</em>, <em>x</em>, <em>xlim</em>, <em>dir='+'</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.limit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.limit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute limit x-&gt;xlim.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.lseries">
-<tt class="descname">lseries</tt><big>(</big><em>self</em>, <em>x=None</em>, <em>x0=0</em>, <em>dir='+'</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.lseries"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.lseries" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wrapper for series yielding an iterator of the terms of the series.</p>
-<p>Note: an infinite series will yield an infinite iterator. The following,
-for exaxmple, will never terminate. It will just keep printing terms
-of the sin(x) series:</p>
-<div class="highlight-python"><pre>for term in sin(x).lseries(x):
-    print term</pre>
-</div>
-<p>The advantage of lseries() over nseries() is that many times you are
-just interested in the next term in the series (i.e. the first term for
-example), but you don&#8217;t know how many you should ask for in nseries()
-using the &#8220;n&#8221; parameter.</p>
-<p>See also nseries().</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.nseries">
-<tt class="descname">nseries</tt><big>(</big><em>self</em>, <em>x=None</em>, <em>x0=0</em>, <em>n=6</em>, <em>dir='+'</em>, <em>logx=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.nseries"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.nseries" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wrapper to _eval_nseries if assumptions allow, else to series.</p>
-<p>If x is given, x0 is 0, dir=&#8217;+&#8217;, and self has x, then _eval_nseries is
-called. This calculates &#8220;n&#8221; terms in the innermost expressions and
-then builds up the final series just by &#8220;cross-multiplying&#8221; everything
-out.</p>
-<p>Advantage &#8211; it&#8217;s fast, because we don&#8217;t have to determine how many
-terms we need to calculate in advance.</p>
-<p>Disadvantage &#8211; you may end up with less terms than you may have
-expected, but the O(x**n) term appended will always be correct and
-so the result, though perhaps shorter, will also be correct.</p>
-<p>If any of those assumptions is not met, this is treated like a
-wrapper to series which will try harder to return the correct
-number of terms.</p>
-<p>See also lseries().</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.nsimplify">
-<tt class="descname">nsimplify</tt><big>(</big><em>self</em>, <em>constants=</em><span class="optional">[</span><span class="optional">]</span>, <em>tolerance=None</em>, <em>full=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.nsimplify"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.nsimplify" title="Permalink to this definition">¶</a></dt>
-<dd><p>See the nsimplify function in sympy.simplify</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.powsimp">
-<tt class="descname">powsimp</tt><big>(</big><em>self</em>, <em>deep=False</em>, <em>combine='all'</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.powsimp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.powsimp" title="Permalink to this definition">¶</a></dt>
-<dd><p>See the powsimp function in sympy.simplify</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.primitive">
-<tt class="descname">primitive</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.primitive"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.primitive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the positive Rational that can be extracted non-recursively
-from every term of self (i.e., self is treated like an Add). This is
-like the as_coeff_Mul() method but primitive always extracts a positive
-Rational (never a negative or a Float).</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">primitive</span><span class="p">()</span>
-<span class="go">(3, (x + 1)**2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="p">(</span><span class="mi">6</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">);</span> <span class="n">a</span><span class="o">.</span><span class="n">primitive</span><span class="p">()</span>
-<span class="go">(2, 3*x + 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">3</span><span class="p">);</span> <span class="n">b</span><span class="o">.</span><span class="n">primitive</span><span class="p">()</span>
-<span class="go">(1/2, x + 6)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">b</span><span class="p">)</span><span class="o">.</span><span class="n">primitive</span><span class="p">()</span> <span class="o">==</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="o">*</span><span class="n">b</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.radsimp">
-<tt class="descname">radsimp</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.radsimp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.radsimp" title="Permalink to this definition">¶</a></dt>
-<dd><p>See the radsimp function in sympy.simplify</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.ratsimp">
-<tt class="descname">ratsimp</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.ratsimp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.ratsimp" title="Permalink to this definition">¶</a></dt>
-<dd><p>See the ratsimp function in sympy.simplify</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.refine">
-<tt class="descname">refine</tt><big>(</big><em>self</em>, <em>assumption=True</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.refine"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.refine" title="Permalink to this definition">¶</a></dt>
-<dd><p>See the refine function in sympy.assumptions</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.removeO">
-<tt class="descname">removeO</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.removeO"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.removeO" title="Permalink to this definition">¶</a></dt>
-<dd><p>Removes the additive O(..) symbol if there is one</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.round">
-<tt class="descname">round</tt><big>(</big><em>self</em>, <em>p=0</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.round"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.round" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return x rounded to the given decimal place.</p>
-<p>If a complex number would results, apply round to the real
-and imaginary components of the number.</p>
-<p class="rubric">Notes</p>
-<p>Do not confuse the Python builtin function, round, with the
-SymPy method of the same name. The former always returns a float
-(or raises an error if applied to a complex value) while the
-latter returns either a Number or a complex number:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">isinstance</span><span class="p">(</span><span class="nb">round</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">123</span><span class="p">),</span> <span class="o">-</span><span class="mi">2</span><span class="p">),</span> <span class="n">Number</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">isinstance</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">123</span><span class="p">)</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">),</span> <span class="n">Number</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">isinstance</span><span class="p">((</span><span class="mi">3</span><span class="o">*</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">round</span><span class="p">(),</span> <span class="n">Mul</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">isinstance</span><span class="p">((</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">round</span><span class="p">(),</span> <span class="n">Add</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">Add</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Number</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="p">(</span><span class="mf">10.5</span><span class="p">)</span><span class="o">.</span><span class="n">round</span><span class="p">()</span>
-<span class="go">11.</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">.</span><span class="n">round</span><span class="p">()</span>
-<span class="go">3.</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">3.14</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span> <span class="o">+</span> <span class="n">E</span><span class="o">*</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">round</span><span class="p">()</span>              
-<span class="go">6. + 3.*I</span>
-</pre></div>
-</div>
-<p>The round method has a chopping effect:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span> <span class="o">+</span> <span class="n">I</span><span class="o">/</span><span class="mi">10</span><span class="p">)</span><span class="o">.</span><span class="n">round</span><span class="p">()</span>
-<span class="go">6.</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">10</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">round</span><span class="p">()</span>             
-<span class="go">2.*I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">10</span> <span class="o">+</span> <span class="n">E</span><span class="o">*</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.31 + 2.72*I</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.separate">
-<tt class="descname">separate</tt><big>(</big><em>self</em>, <em>deep=False</em>, <em>force=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.separate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.separate" title="Permalink to this definition">¶</a></dt>
-<dd><p>See the separate function in sympy.simplify</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.series">
-<tt class="descname">series</tt><big>(</big><em>self</em>, <em>x=None</em>, <em>x0=0</em>, <em>n=6</em>, <em>dir='+'</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.series"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.series" title="Permalink to this definition">¶</a></dt>
-<dd><p>Series expansion of &#8220;self&#8221; around <tt class="docutils literal"><span class="pre">x</span> <span class="pre">=</span> <span class="pre">x0</span></tt> yielding either terms of
-the series one by one (the lazy series given when n=None), else
-all the terms at once when n != None.</p>
-<p>Note: when n != None, if an O() term is returned then the x in the
-in it and the entire expression represents x - x0, the displacement
-from x0. (If there is no O() term then the series was exact and x has
-it&#8217;s normal meaning.) This is currently necessary since sympy&#8217;s O()
-can only represent terms at x0=0. So instead of:</p>
-<div class="highlight-python"><pre>cos(x).series(x0=1, n=2) --&gt; (1 - x)*sin(1) + cos(1) + O((x - 1)**2)</pre>
-</div>
-<p>which graphically looks like this:</p>
-<div class="highlight-python"><pre>   |
-  .|.         . .
- . | \      .     .
----+----------------------
-   |   . .          . .
-   |                  x=0</pre>
-</div>
-<p>the following is returned instead:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="o">-</span><span class="n">x</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">cos</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">O</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>whose graph is this:</p>
-<div class="highlight-python"><pre>   \ |
-  . .|        . .
- .   \      .     .
------+\------------------.
-     | . .          . .
-     |                  x=0</pre>
-</div>
-<p>which is identical to <tt class="docutils literal"><span class="pre">cos(x</span> <span class="pre">+</span> <span class="pre">1).series(n=2)</span></tt>.</p>
-<dl class="docutils">
-<dt>Usage:</dt>
-<dd><p class="first">Returns the series expansion of &#8220;self&#8221; around the point <tt class="docutils literal"><span class="pre">x</span> <span class="pre">=</span> <span class="pre">x0</span></tt>
-with respect to <tt class="docutils literal"><span class="pre">x</span></tt> up to O(x**n) (default n is 6).</p>
-<p>If <tt class="docutils literal"><span class="pre">x=None</span></tt> and <tt class="docutils literal"><span class="pre">self</span></tt> is univariate, the univariate symbol will
-be supplied, otherwise an error will be raised.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">cos</span><span class="p">,</span> <span class="n">exp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">()</span>
-<span class="go">1 - x**2/2 + x**4/24 + O(x**6)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">n</span><span class="o">=</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">1 - x**2/2 + O(x**4)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">exp</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">cos(x + 1) - y*sin(x + 1) + O(y**2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">cos(exp(y)) - x*sin(exp(y)) + O(x**2)</span>
-</pre></div>
-</div>
-<p>If <tt class="docutils literal"><span class="pre">n=None</span></tt> then an iterator of the series terms will be returned.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">term</span><span class="o">=</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">n</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="nb">next</span><span class="p">(</span><span class="n">term</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">)]</span>
-<span class="go">[1, -x**2/2]</span>
-</pre></div>
-</div>
-<p>For <tt class="docutils literal"><span class="pre">dir=+</span></tt> (default) the series is calculated from the right and
-for <tt class="docutils literal"><span class="pre">dir=-</span></tt> the series from the left. For smooth functions this
-flag will not alter the results.</p>
-<div class="last highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="nb">dir</span><span class="o">=</span><span class="s">&quot;+&quot;</span><span class="p">)</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="nb">dir</span><span class="o">=</span><span class="s">&quot;-&quot;</span><span class="p">)</span>
-<span class="go">-x</span>
-</pre></div>
-</div>
-</dd>
-</dl>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.simplify">
-<tt class="descname">simplify</tt><big>(</big><em>self</em>, <em>ratio=1.7</em>, <em>measure=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.simplify"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.simplify" title="Permalink to this definition">¶</a></dt>
-<dd><p>See the simplify function in sympy.simplify</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.together">
-<tt class="descname">together</tt><big>(</big><em>self</em>, <em>*args</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.together"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.together" title="Permalink to this definition">¶</a></dt>
-<dd><p>See the together function in sympy.polys</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.expr.Expr.trigsimp">
-<tt class="descname">trigsimp</tt><big>(</big><em>self</em>, <em>deep=False</em>, <em>recursive=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/expr.html#Expr.trigsimp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.Expr.trigsimp" title="Permalink to this definition">¶</a></dt>
-<dd><p>See the trigsimp function in sympy.simplify</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="atomicexpr">
-<h2>AtomicExpr<a class="headerlink" href="#atomicexpr" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.core.expr.AtomicExpr">
-<em class="property">class </em><tt class="descclassname">sympy.core.expr.</tt><tt class="descname">AtomicExpr</tt><a class="reference internal" href="../_modules/sympy/core/expr.html#AtomicExpr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.expr.AtomicExpr" title="Permalink to this definition">¶</a></dt>
-<dd><p>A parent class for object which are both atoms and Exprs.</p>
-<p>For example: Symbol, Number, Rational, Integer, ...
-But not: Add, Mul, Pow, ...</p>
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.core.symbol">
-<span id="symbol"></span><h2>symbol<a class="headerlink" href="#module-sympy.core.symbol" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="id4">
-<h3>Symbol<a class="headerlink" href="#id4" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.symbol.Symbol">
-<em class="property">class </em><tt class="descclassname">sympy.core.symbol.</tt><tt class="descname">Symbol</tt><a class="reference internal" href="../_modules/sympy/core/symbol.html#Symbol"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.symbol.Symbol" title="Permalink to this definition">¶</a></dt>
-<dd><dl class="docutils">
-<dt>Assumptions:</dt>
-<dd>commutative = True</dd>
-</dl>
-<p>You can override the default assumptions in the constructor:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="p">,</span><span class="n">B</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;A,B&#39;</span><span class="p">,</span> <span class="n">commutative</span> <span class="o">=</span> <span class="bp">False</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">bool</span><span class="p">(</span><span class="n">A</span><span class="o">*</span><span class="n">B</span> <span class="o">!=</span> <span class="n">B</span><span class="o">*</span><span class="n">A</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">bool</span><span class="p">(</span><span class="n">A</span><span class="o">*</span><span class="n">B</span><span class="o">*</span><span class="mi">2</span> <span class="o">==</span> <span class="mi">2</span><span class="o">*</span><span class="n">A</span><span class="o">*</span><span class="n">B</span><span class="p">)</span> <span class="o">==</span> <span class="bp">True</span> <span class="c"># multiplication by scalars is commutative</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="wild">
-<h3>Wild<a class="headerlink" href="#wild" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.symbol.Wild">
-<em class="property">class </em><tt class="descclassname">sympy.core.symbol.</tt><tt class="descname">Wild</tt><a class="reference internal" href="../_modules/sympy/core/symbol.html#Wild"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.symbol.Wild" title="Permalink to this definition">¶</a></dt>
-<dd><p>A Wild symbol matches anything.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Wild</span><span class="p">,</span> <span class="n">WildFunction</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;b&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="go">{a_: b_}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="go">{a_: x}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="go">{a_: pi}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="go">{a_: x**2}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="go">{a_: cos(x)}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">WildFunction</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="go">{a_: A_}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="dummy">
-<h3>Dummy<a class="headerlink" href="#dummy" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.symbol.Dummy">
-<em class="property">class </em><tt class="descclassname">sympy.core.symbol.</tt><tt class="descname">Dummy</tt><a class="reference internal" href="../_modules/sympy/core/symbol.html#Dummy"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.symbol.Dummy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Dummy symbols are each unique, identified by an internal count index:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Dummy</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">bool</span><span class="p">(</span><span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span> <span class="o">==</span> <span class="n">Dummy</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">))</span> <span class="o">==</span> <span class="bp">True</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>If a name is not supplied then a string value of the count index will be
-used. This is useful when a temporary variable is needed and the name
-of the variable used in the expression is not important.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Dummy</span><span class="o">.</span><span class="n">_count</span> <span class="o">=</span> <span class="mi">0</span> <span class="c"># /!\ this should generally not be changed; it is being</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dummy</span><span class="p">()</span>          <span class="c"># used here to make sure that the doctest passes.</span>
-<span class="go">_0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="symbols">
-<h3>symbols<a class="headerlink" href="#symbols" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.symbol.symbols">
-<tt class="descclassname">sympy.core.symbol.</tt><tt class="descname">symbols</tt><big>(</big><em>names</em>, <em>**args</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/symbol.html#symbols"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.symbol.symbols" title="Permalink to this definition">¶</a></dt>
-<dd><p>Transform strings into instances of <a class="reference internal" href="#sympy.core.symbol.Symbol" title="sympy.core.symbol.Symbol"><tt class="xref py py-class docutils literal"><span class="pre">Symbol</span></tt></a> class.</p>
-<p><a class="reference internal" href="#sympy.core.symbol.symbols" title="sympy.core.symbol.symbols"><tt class="xref py py-func docutils literal"><span class="pre">symbols()</span></tt></a> function returns a sequence of symbols with names taken
-from <tt class="docutils literal"><span class="pre">names</span></tt> argument, which can be a comma or whitespace delimited
-string, or a sequence of strings:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">Function</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y,z&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a b c&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>The type of output is dependent on the properties of input arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,&#39;</span><span class="p">)</span>
-<span class="go">(x,)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-<span class="go">(x, y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">symbols</span><span class="p">((</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="s">&#39;b&#39;</span><span class="p">,</span> <span class="s">&#39;c&#39;</span><span class="p">))</span>
-<span class="go">(a, b, c)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">symbols</span><span class="p">([</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="s">&#39;b&#39;</span><span class="p">,</span> <span class="s">&#39;c&#39;</span><span class="p">])</span>
-<span class="go">[a, b, c]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">symbols</span><span class="p">(</span><span class="nb">set</span><span class="p">([</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="s">&#39;b&#39;</span><span class="p">,</span> <span class="s">&#39;c&#39;</span><span class="p">]))</span>
-<span class="go">set([a, b, c])</span>
-</pre></div>
-</div>
-<p>If an iterable container is needed for a single symbol, set the <tt class="docutils literal"><span class="pre">seq</span></tt>
-argument to <tt class="docutils literal"><span class="pre">True</span></tt> or terminate the symbol name with a comma:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="n">seq</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(x,)</span>
-</pre></div>
-</div>
-<p>To reduce typing, range syntax is supported to create indexed symbols:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x:10&#39;</span><span class="p">)</span>
-<span class="go">(x0, x1, x2, x3, x4, x5, x6, x7, x8, x9)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x5:10&#39;</span><span class="p">)</span>
-<span class="go">(x5, x6, x7, x8, x9)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x5:10,y:5&#39;</span><span class="p">)</span>
-<span class="go">(x5, x6, x7, x8, x9, y0, y1, y2, y3, y4)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">symbols</span><span class="p">((</span><span class="s">&#39;x5:10&#39;</span><span class="p">,</span> <span class="s">&#39;y:5&#39;</span><span class="p">))</span>
-<span class="go">((x5, x6, x7, x8, x9), (y0, y1, y2, y3, y4))</span>
-</pre></div>
-</div>
-<p>To reduce typing even more, lexicographic range syntax is supported:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x:z&#39;</span><span class="p">)</span>
-<span class="go">(x, y, z)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a:d,x:z&#39;</span><span class="p">)</span>
-<span class="go">(a, b, c, d, x, y, z)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">symbols</span><span class="p">((</span><span class="s">&#39;a:d&#39;</span><span class="p">,</span> <span class="s">&#39;x:z&#39;</span><span class="p">))</span>
-<span class="go">((a, b, c, d), (x, y, z))</span>
-</pre></div>
-</div>
-<p>All newly created symbols have assumptions set accordingly to <tt class="docutils literal"><span class="pre">args</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">is_integer</span>
-<span class="go">True</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y,z&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span> <span class="n">y</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span> <span class="n">z</span><span class="o">.</span><span class="n">is_real</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>Despite its name, <a class="reference internal" href="#sympy.core.symbol.symbols" title="sympy.core.symbol.symbols"><tt class="xref py py-func docutils literal"><span class="pre">symbols()</span></tt></a> can create symbol&#8211;like objects of
-other type, for example instances of Function or Wild classes. To
-achieve this, set <tt class="docutils literal"><span class="pre">cls</span></tt> keyword argument to the desired type:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">symbols</span><span class="p">(</span><span class="s">&#39;f,g,h&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Function</span><span class="p">)</span>
-<span class="go">(f, g, h)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="n">_</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-<span class="go">&lt;class &#39;sympy.core.function.UndefinedFunction&#39;&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="var">
-<h3>var<a class="headerlink" href="#var" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.symbol.var">
-<tt class="descclassname">sympy.core.symbol.</tt><tt class="descname">var</tt><big>(</big><em>names</em>, <em>**args</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/symbol.html#var"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.symbol.var" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create symbols and inject them into the global namespace.</p>
-<p>This calls <a class="reference internal" href="#sympy.core.symbol.symbols" title="sympy.core.symbol.symbols"><tt class="xref py py-func docutils literal"><span class="pre">symbols()</span></tt></a> with the same arguments and puts the results
-into the <em>global</em> namespace. It&#8217;s recommended not to use <a class="reference internal" href="#sympy.core.symbol.var" title="sympy.core.symbol.var"><tt class="xref py py-func docutils literal"><span class="pre">var()</span></tt></a> in
-library code, where <a class="reference internal" href="#sympy.core.symbol.symbols" title="sympy.core.symbol.symbols"><tt class="xref py py-func docutils literal"><span class="pre">symbols()</span></tt></a> has to be used:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">var</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span>
-<span class="go">x</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;a,ab,abc&#39;</span><span class="p">)</span>
-<span class="go">(a, ab, abc)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">abc</span>
-<span class="go">abc</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(x, y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">is_real</span> <span class="ow">and</span> <span class="n">y</span><span class="o">.</span><span class="n">is_real</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>See <tt class="xref py py-func docutils literal"><span class="pre">symbol()</span></tt> documentation for more details on what kinds of
-arguments can be passed to <a class="reference internal" href="#sympy.core.symbol.var" title="sympy.core.symbol.var"><tt class="xref py py-func docutils literal"><span class="pre">var()</span></tt></a>.</p>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.core.numbers">
-<span id="numbers"></span><h2>numbers<a class="headerlink" href="#module-sympy.core.numbers" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="number">
-<h3>Number<a class="headerlink" href="#number" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.numbers.Number">
-<em class="property">class </em><tt class="descclassname">sympy.core.numbers.</tt><tt class="descname">Number</tt><a class="reference internal" href="../_modules/sympy/core/numbers.html#Number"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.numbers.Number" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents any kind of number in sympy.</p>
-<p>Floating point numbers are represented by the Float class.
-Integer numbers (of any size), together with rational numbers (again,
-there is no limit on their size) are represented by the Rational class.</p>
-<p>If you want to represent, for example, <tt class="docutils literal"><span class="pre">1+sqrt(2)</span></tt>, then you need to do:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.core.numbers.Number.as_coeff_Add">
-<tt class="descname">as_coeff_Add</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/numbers.html#Number.as_coeff_Add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.numbers.Number.as_coeff_Add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Efficiently extract the coefficient of a summation.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.numbers.Number.as_coeff_Mul">
-<tt class="descname">as_coeff_Mul</tt><big>(</big><em>self</em>, <em>rational=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/numbers.html#Number.as_coeff_Mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.numbers.Number.as_coeff_Mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Efficiently extract the coefficient of a product.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.numbers.Number.cofactors">
-<tt class="descname">cofactors</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/numbers.html#Number.cofactors"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.numbers.Number.cofactors" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute GCD and cofactors of <span class="math">\(self\)</span> and <span class="math">\(other\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.numbers.Number.gcd">
-<tt class="descname">gcd</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/numbers.html#Number.gcd"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.numbers.Number.gcd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute GCD of <span class="math">\(self\)</span> and <span class="math">\(other\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.numbers.Number.lcm">
-<tt class="descname">lcm</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/numbers.html#Number.lcm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.numbers.Number.lcm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute LCM of <span class="math">\(self\)</span> and <span class="math">\(other\)</span>.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="float">
-<h3>Float<a class="headerlink" href="#float" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.numbers.Float">
-<em class="property">class </em><tt class="descclassname">sympy.core.numbers.</tt><tt class="descname">Float</tt><a class="reference internal" href="../_modules/sympy/core/numbers.html#Float"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.numbers.Float" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents a floating point number. It is capable of representing
-arbitrary-precision floating-point numbers.</p>
-<p class="rubric">Notes</p>
-<p>Floats are inexact by their nature unless their value is a binary-exact
-value.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">approx</span><span class="p">,</span> <span class="n">exact</span> <span class="o">=</span> <span class="n">Float</span><span class="p">(</span><span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Float</span><span class="p">(</span><span class="o">.</span><span class="mi">125</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>For calculation purposes, evalf needs to be able to change the precision
-but this will not increase the accuracy of the inexact value. The
-following is the most accurate 5-digit approximation of a value of 0.1
-that had only 1 digit of precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">approx</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">0.099609</span>
-</pre></div>
-</div>
-<p>By contrast, 0.125 is exact in binary (as it is in base 10) and so it
-can be passed to Float or evalf to obtain an arbitrary precision with
-matching accuracy:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="n">exact</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">0.12500</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exact</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">20</span><span class="p">)</span>
-<span class="go">0.12500000000000000000</span>
-</pre></div>
-</div>
-<p>Trying to make a high-precision Float from a float is not disallowed,
-but one must keep in mind that the <em>underlying float</em> (not the apparent
-decimal value) is being obtained with high precision. For example, 0.3
-does not have a finite binary representation. The closest rational is
-the fraction 5404319552844595/2**54. So if you try to obtain a Float of
-0.3 to 20 digits of precision you will not see the same thing as 0.3
-followed by 19 zeros:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="mf">0.3</span><span class="p">,</span> <span class="mi">20</span><span class="p">)</span>
-<span class="go">0.29999999999999998890</span>
-</pre></div>
-</div>
-<p>If you want a 20-digit value of the decimal 0.3 (not the floating point
-approximation of 0.3) you should send the 0.3 as a string. The underlying
-representation is still binary but a higher precision than Python&#8217;s float
-is used:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="s">&#39;0.3&#39;</span><span class="p">,</span> <span class="mi">20</span><span class="p">)</span>
-<span class="go">0.30000000000000000000</span>
-</pre></div>
-</div>
-<p>Although you can increase the precision of an existing Float using Float
-it will not increase the accuracy &#8211; the underlying value is not changed:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">show</span><span class="p">(</span><span class="n">f</span><span class="p">):</span> <span class="c"># binary rep of Float</span>
-<span class="gp">... </span>    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Pow</span>
-<span class="gp">... </span>    <span class="n">s</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">_mpf_</span>
-<span class="gp">... </span>    <span class="n">v</span> <span class="o">=</span> <span class="n">Mul</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">Pow</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">),</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> at prec=</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">f</span><span class="o">.</span><span class="n">_prec</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Float</span><span class="p">(</span><span class="s">&#39;0.3&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">show</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-<span class="go">4915/2**14 at prec=13</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">show</span><span class="p">(</span><span class="n">Float</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mi">20</span><span class="p">))</span> <span class="c"># higher prec, not higher accuracy</span>
-<span class="go">4915/2**14 at prec=70</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">show</span><span class="p">(</span><span class="n">Float</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span> <span class="c"># lower prec</span>
-<span class="go">307/2**10 at prec=10</span>
-</pre></div>
-</div>
-<p>The same thing happens when evalf is used on a Float:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">show</span><span class="p">(</span><span class="n">t</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">20</span><span class="p">))</span>
-<span class="go">4915/2**14 at prec=70</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">show</span><span class="p">(</span><span class="n">t</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">307/2**10 at prec=10</span>
-</pre></div>
-</div>
-<p>Finally, Floats can be instantiated with an mpf tuple (n, c, p) to
-produce the number (-1)**n*c*2**p:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="o">*</span><span class="n">c</span><span class="o">*</span><span class="mi">2</span><span class="o">**</span><span class="n">p</span>
-<span class="go">-5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">-5.00000000000000</span>
-</pre></div>
-</div>
-<p>An actual mpf tuple also contains the number of bits in c as the last
-element of the tuple, but this is not needed for instantiation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">.</span><span class="n">_mpf_</span>
-<span class="go">(1, 5, 0, 3)</span>
-</pre></div>
-</div>
-<p>That last value can be used to create the corresponding Float, however,
-by passing the tuple and prec=&#8217;&#8217; to Float to copy the precision from the
-mpf tuple:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="n">pi</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span><span class="o">.</span><span class="n">_mpf_</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-<span class="go">3.1416</span>
-</pre></div>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Float</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="mf">3.5</span><span class="p">)</span>
-<span class="go">3.50000000000000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">3.00000000000000</span>
-</pre></div>
-</div>
-<p>Floats can be created from a string representations of Python floats
-to force ints to Float or to enter high-precision (&gt; 15 significant
-digits) values:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="s">&#39;.0010&#39;</span><span class="p">)</span>
-<span class="go">0.00100000000000000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="s">&#39;1e-3&#39;</span><span class="p">)</span>
-<span class="go">0.00100000000000000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="s">&#39;1e-3&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">0.00100</span>
-</pre></div>
-</div>
-<p>Float can automatically count significant figures if a null string
-is sent for the precision; space are also allowed in the string. (Auto-
-counting is only allowed for strings, ints and longs).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="s">&#39;123 456 789 . 123 456&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-<span class="go">123456789.123456</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="s">&#39;12e-3&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-<span class="go">0.012</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-<span class="go">3.</span>
-</pre></div>
-</div>
-<p>If a number is written in scientific notation, only the digits before the
-exponent are considered significant if a decimal appears, otherwise the
-&#8220;e&#8221; signifies only how to move the decimal:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="s">&#39;60.e2&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>  <span class="c"># 2 digits significant</span>
-<span class="go">6.0e+3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="s">&#39;60e2&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>  <span class="c"># 4 digits significant</span>
-<span class="go">6000.</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="s">&#39;600e-2&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>  <span class="c"># 3 digits significant</span>
-<span class="go">6.00</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="rational">
-<h3>Rational<a class="headerlink" href="#rational" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.numbers.Rational">
-<em class="property">class </em><tt class="descclassname">sympy.core.numbers.</tt><tt class="descname">Rational</tt><a class="reference internal" href="../_modules/sympy/core/numbers.html#Rational"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.numbers.Rational" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents integers and rational numbers (p/q) of any size.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">sympify</span></tt>, <a class="reference internal" href="simplify/simplify.html#sympy.simplify.simplify.nsimplify" title="sympy.simplify.simplify.nsimplify"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.simplify.simplify.nsimplify</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Rational</span><span class="p">,</span> <span class="n">nsimplify</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">1/2</span>
-</pre></div>
-</div>
-<p>Rational is unprejudiced in accepting input. If a float is passed, the
-underlying value of the binary representation will be returned:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="o">.</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="o">.</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">3602879701896397/18014398509481984</span>
-</pre></div>
-</div>
-<p>If the simpler representation of the float is desired then consider
-limiting the denominator to the desired value or convert the float to
-a string (which is roughly equivalent to limiting the denominator to
-10**12):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="o">.</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">1/5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="o">.</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">limit_denominator</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">12</span><span class="p">)</span>
-<span class="go">1/5</span>
-</pre></div>
-</div>
-<p>An arbitrarily precise Rational is obtained when a string literal is
-passed:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="s">&quot;1.23&quot;</span><span class="p">)</span>
-<span class="go">123/100</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="s">&#39;1e-2&#39;</span><span class="p">)</span>
-<span class="go">1/100</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="s">&quot;.1&quot;</span><span class="p">)</span>
-<span class="go">1/10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="s">&#39;1e-2/3.2&#39;</span><span class="p">)</span>
-<span class="go">1/320</span>
-</pre></div>
-</div>
-<p>The conversion of other types of strings can be handled by
-the sympify() function, and conversion of floats to expressions
-or simple fractions can be handled with nsimplify:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="p">(</span><span class="s">&#39;.[3]&#39;</span><span class="p">)</span>  <span class="c"># repeating digits in brackets</span>
-<span class="go">1/3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="p">(</span><span class="s">&#39;3**2/10&#39;</span><span class="p">)</span>  <span class="c"># general expressions</span>
-<span class="go">9/10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsimplify</span><span class="p">(</span><span class="o">.</span><span class="mi">3</span><span class="p">)</span>  <span class="c"># numbers that have a simple form</span>
-<span class="go">3/10</span>
-</pre></div>
-</div>
-<p>But if the input does not reduce to a literal Rational, an error will
-be raised:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">TypeError: invalid input</span>: <span class="n">pi</span>
-</pre></div>
-</div>
-<p class="rubric">Low-level</p>
-<p>Access numerator and denominator as .p and .q:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">r</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span>
-<span class="go">3/4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="o">.</span><span class="n">p</span>
-<span class="go">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="o">.</span><span class="n">q</span>
-<span class="go">4</span>
-</pre></div>
-</div>
-<p>Note that p and q return integers (not sympy Integers) so some care
-is needed when using them in expressions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="o">.</span><span class="n">p</span><span class="o">//</span><span class="n">r</span><span class="o">.</span><span class="n">q</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.core.numbers.Rational.as_content_primitive">
-<tt class="descname">as_content_primitive</tt><big>(</big><em>self</em>, <em>radical=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/numbers.html#Rational.as_content_primitive"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.numbers.Rational.as_content_primitive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the tuple (R, self/R) where R is the positive Rational
-extracted from self.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-<span class="go">(3/2, -1)</span>
-</pre></div>
-</div>
-<p>See docstring of Expr.as_content_primitive for more examples.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.numbers.Rational.factors">
-<tt class="descname">factors</tt><big>(</big><em>self</em>, <em>limit=None</em>, <em>use_trial=True</em>, <em>use_rho=False</em>, <em>use_pm1=False</em>, <em>verbose=False</em>, <em>visual=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/numbers.html#Rational.factors"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.numbers.Rational.factors" title="Permalink to this definition">¶</a></dt>
-<dd><p>A wrapper to factorint which return factors of self that are
-smaller than limit (or cheap to compute). Special methods of
-factoring are disabled by default so that only trial division is used.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.numbers.Rational.limit_denominator">
-<tt class="descname">limit_denominator</tt><big>(</big><em>self</em>, <em>max_denominator=1000000</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/numbers.html#Rational.limit_denominator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.numbers.Rational.limit_denominator" title="Permalink to this definition">¶</a></dt>
-<dd><p>Closest Rational to self with denominator at most max_denominator.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="s">&#39;3.141592653589793&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">limit_denominator</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">22/7</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="s">&#39;3.141592653589793&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">limit_denominator</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">311/99</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="integer">
-<h3>Integer<a class="headerlink" href="#integer" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.numbers.Integer">
-<em class="property">class </em><tt class="descclassname">sympy.core.numbers.</tt><tt class="descname">Integer</tt><a class="reference internal" href="../_modules/sympy/core/numbers.html#Integer"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.numbers.Integer" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="numbersymbol">
-<h3>NumberSymbol<a class="headerlink" href="#numbersymbol" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.numbers.NumberSymbol">
-<em class="property">class </em><tt class="descclassname">sympy.core.numbers.</tt><tt class="descname">NumberSymbol</tt><a class="reference internal" href="../_modules/sympy/core/numbers.html#NumberSymbol"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.numbers.NumberSymbol" title="Permalink to this definition">¶</a></dt>
-<dd><dl class="function">
-<dt id="sympy.core.numbers.NumberSymbol.approximation">
-<tt class="descname">approximation</tt><big>(</big><em>self</em>, <em>number_cls</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/numbers.html#NumberSymbol.approximation"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.numbers.NumberSymbol.approximation" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return an interval with number_cls endpoints
-that contains the value of NumberSymbol.
-If not implemented, then return None.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="realnumber">
-<h3>RealNumber<a class="headerlink" href="#realnumber" title="Permalink to this headline">¶</a></h3>
-<dl class="attribute">
-<dt id="sympy.core.numbers.RealNumber">
-<tt class="descclassname">sympy.core.numbers.</tt><tt class="descname">RealNumber</tt><a class="headerlink" href="#sympy.core.numbers.RealNumber" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <a class="reference internal" href="#sympy.core.numbers.Float" title="sympy.core.numbers.Float"><tt class="xref py py-class docutils literal"><span class="pre">Float</span></tt></a></p>
-</dd></dl>
-
-</div>
-<div class="section" id="real">
-<h3>Real<a class="headerlink" href="#real" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.numbers.Real">
-<em class="property">class </em><tt class="descclassname">sympy.core.numbers.</tt><tt class="descname">Real</tt><a class="reference internal" href="../_modules/sympy/core/numbers.html#Real"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.numbers.Real" title="Permalink to this definition">¶</a></dt>
-<dd><p>Deprecated alias for the Float constructor.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="igcd">
-<h3>igcd<a class="headerlink" href="#igcd" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.numbers.igcd">
-<tt class="descclassname">sympy.core.numbers.</tt><tt class="descname">igcd</tt><big>(</big><em>a</em>, <em>b</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/numbers.html#igcd"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.numbers.igcd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes positive, integer greatest common divisor of two numbers.</p>
-<p>The algorithm is based on the well known Euclid&#8217;s algorithm. To
-improve speed, igcd() has its own caching mechanism implemented.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="ilcm">
-<h3>ilcm<a class="headerlink" href="#ilcm" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.numbers.ilcm">
-<tt class="descclassname">sympy.core.numbers.</tt><tt class="descname">ilcm</tt><big>(</big><em>a</em>, <em>b</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/numbers.html#ilcm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.numbers.ilcm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes integer least common multiple of two numbers.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="seterr">
-<h3>seterr<a class="headerlink" href="#seterr" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.numbers.seterr">
-<tt class="descclassname">sympy.core.numbers.</tt><tt class="descname">seterr</tt><big>(</big><em>divide=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/numbers.html#seterr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.numbers.seterr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Should sympy raise an exception on 0/0 or return a nan?</p>
-<p>divide == True .... raise an exception
-divide == False ... return nan</p>
-</dd></dl>
-
-</div>
-<div class="section" id="e">
-<h3>E<a class="headerlink" href="#e" title="Permalink to this headline">¶</a></h3>
-<dl class="attribute">
-<dt id="sympy.core.numbers.E">
-<tt class="descclassname">sympy.core.numbers.</tt><tt class="descname">E</tt><a class="headerlink" href="#sympy.core.numbers.E" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="i">
-<h3>I<a class="headerlink" href="#i" title="Permalink to this headline">¶</a></h3>
-<dl class="attribute">
-<dt id="sympy.core.numbers.I">
-<tt class="descclassname">sympy.core.numbers.</tt><tt class="descname">I</tt><a class="headerlink" href="#sympy.core.numbers.I" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="nan">
-<h3>nan<a class="headerlink" href="#nan" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.numbers.nan">
-<tt class="descclassname">sympy.core.numbers.</tt><tt class="descname">nan</tt><big>(</big><em>cls</em><big>)</big><a class="headerlink" href="#sympy.core.numbers.nan" title="Permalink to this definition">¶</a></dt>
-<dd><p>Not a Number.</p>
-<p>This represents the corresponding data type to floating point nan, which
-is defined in the IEEE 754 floating point standard, and corresponds to the
-Python <tt class="docutils literal"><span class="pre">float('nan')</span></tt>.</p>
-<p>NaN serves as a place holder for numeric values that are indeterminate,
-but not infinite.  Most operations on nan, produce another nan.  Most
-indeterminate forms, such as <tt class="docutils literal"><span class="pre">0/0</span></tt> or <tt class="docutils literal"><span class="pre">oo</span> <span class="pre">-</span> <span class="pre">oo`</span> <span class="pre">produce</span> <span class="pre">nan.</span>&nbsp; <span class="pre">Three</span>
-<span class="pre">exceptions</span> <span class="pre">are</span> <span class="pre">``0**0</span></tt>, <tt class="docutils literal"><span class="pre">1**oo</span></tt>, and <tt class="docutils literal"><span class="pre">oo**0</span></tt>, which all produce <tt class="docutils literal"><span class="pre">1</span></tt>
-(this is consistent with Python&#8217;s float).</p>
-<p>NaN is a singleton, and can be accessed by <tt class="docutils literal"><span class="pre">S.NaN</span></tt>, or can be imported
-as <tt class="docutils literal"><span class="pre">nan</span></tt>.</p>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/NaN">http://en.wikipedia.org/wiki/NaN</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">nan</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nan</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">NaN</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">oo</span> <span class="o">-</span> <span class="n">oo</span>
-<span class="go">nan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nan</span> <span class="o">+</span> <span class="mi">1</span>
-<span class="go">nan</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="oo">
-<h3>oo<a class="headerlink" href="#oo" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.numbers.oo">
-<tt class="descclassname">sympy.core.numbers.</tt><tt class="descname">oo</tt><big>(</big><em>cls</em><big>)</big><a class="headerlink" href="#sympy.core.numbers.oo" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="pi">
-<h3>pi<a class="headerlink" href="#pi" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.numbers.pi">
-<tt class="descclassname">sympy.core.numbers.</tt><tt class="descname">pi</tt><big>(</big><em>cls</em><big>)</big><a class="headerlink" href="#sympy.core.numbers.pi" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="zoo">
-<h3>zoo<a class="headerlink" href="#zoo" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.numbers.zoo">
-<tt class="descclassname">sympy.core.numbers.</tt><tt class="descname">zoo</tt><big>(</big><em>cls</em><big>)</big><a class="headerlink" href="#sympy.core.numbers.zoo" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.core.power">
-<span id="power"></span><h2>power<a class="headerlink" href="#module-sympy.core.power" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="pow">
-<h3>Pow<a class="headerlink" href="#pow" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.power.Pow">
-<em class="property">class </em><tt class="descclassname">sympy.core.power.</tt><tt class="descname">Pow</tt><a class="reference internal" href="../_modules/sympy/core/power.html#Pow"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.power.Pow" title="Permalink to this definition">¶</a></dt>
-<dd><dl class="function">
-<dt id="sympy.core.power.Pow.as_base_exp">
-<tt class="descname">as_base_exp</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/power.html#Pow.as_base_exp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.power.Pow.as_base_exp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return base and exp of self.</p>
-<p>If base is 1/Integer, then return Integer, -exp. If this extra
-processing is not needed, the base and exp properties will
-give the raw arguments</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Pow</span><span class="p">,</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Pow</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()</span>
-<span class="go">(2, -2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">args</span>
-<span class="go">(1/2, 2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.power.Pow.as_content_primitive">
-<tt class="descname">as_content_primitive</tt><big>(</big><em>self</em>, <em>radical=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/power.html#Pow.as_content_primitive"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.power.Pow.as_content_primitive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the tuple (R, self/R) where R is the positive Rational
-extracted from self.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="mi">4</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-<span class="go">(2, sqrt(1 + sqrt(2)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-<span class="go">(1, sqrt(3)*sqrt(1 + sqrt(2)))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expand_power_base</span><span class="p">,</span> <span class="n">powsimp</span><span class="p">,</span> <span class="n">Mul</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-<span class="go">(4, (x + 1)**2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">4</span><span class="o">**</span><span class="p">((</span><span class="mi">1</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-<span class="go">(2, 4**(y/2))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span><span class="o">**</span><span class="p">((</span><span class="mi">1</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-<span class="go">(1, 3**((y + 1)/2))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span><span class="o">**</span><span class="p">((</span><span class="mi">5</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-<span class="go">(9, 3**((y + 1)/2))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span> <span class="o">=</span> <span class="mi">3</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powsimp</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span> <span class="o">==</span> <span class="n">eq</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-<span class="go">(9, 3**(2*x))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powsimp</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">_</span><span class="p">))</span>
-<span class="go">9**(x + 1)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">expand_power_base</span><span class="p">(</span><span class="n">eq</span><span class="p">);</span> <span class="n">s</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">,</span> <span class="n">s</span>
-<span class="go">(False, (2*x + 2)**y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-<span class="go">(1, (2*(x + 1))**y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">expand_power_base</span><span class="p">(</span><span class="n">_</span><span class="p">[</span><span class="mi">1</span><span class="p">]);</span> <span class="n">s</span><span class="o">.</span><span class="n">is_Mul</span><span class="p">,</span> <span class="n">s</span>
-<span class="go">(True, 2**y*(x + 1)**y)</span>
-</pre></div>
-</div>
-<p>See docstring of Expr.as_content_primitive for more examples.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="integer-nthroot">
-<h3>integer_nthroot<a class="headerlink" href="#integer-nthroot" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.power.integer_nthroot">
-<tt class="descclassname">sympy.core.power.</tt><tt class="descname">integer_nthroot</tt><big>(</big><em>y</em>, <em>n</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/power.html#integer_nthroot"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.power.integer_nthroot" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a tuple containing x = floor(y**(1/n))
-and a boolean indicating whether the result is exact (that is,
-whether x**n == y).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">integer_nthroot</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integer_nthroot</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(4, True)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integer_nthroot</span><span class="p">(</span><span class="mi">26</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(5, False)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.core.mul">
-<span id="mul"></span><h2>mul<a class="headerlink" href="#module-sympy.core.mul" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="id5">
-<h3>Mul<a class="headerlink" href="#id5" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.mul.Mul">
-<em class="property">class </em><tt class="descclassname">sympy.core.mul.</tt><tt class="descname">Mul</tt><a class="reference internal" href="../_modules/sympy/core/mul.html#Mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.mul.Mul" title="Permalink to this definition">¶</a></dt>
-<dd><dl class="function">
-<dt id="sympy.core.mul.Mul.as_coeff_Mul">
-<tt class="descname">as_coeff_Mul</tt><big>(</big><em>self</em>, <em>rational=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/mul.html#Mul.as_coeff_Mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.mul.Mul.as_coeff_Mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Efficiently extract the coefficient of a product.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.mul.Mul.as_content_primitive">
-<tt class="descname">as_content_primitive</tt><big>(</big><em>self</em>, <em>radical=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/mul.html#Mul.as_content_primitive"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.mul.Mul.as_content_primitive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the tuple (R, self/R) where R is the positive Rational
-extracted from self.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)))</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-<span class="go">(6, -sqrt(2)*(-sqrt(2) + 1))</span>
-</pre></div>
-</div>
-<p>See docstring of Expr.as_content_primitive for more examples.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.mul.Mul.as_ordered_factors">
-<tt class="descname">as_ordered_factors</tt><big>(</big><em>self</em>, <em>order=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/mul.html#Mul.as_ordered_factors"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.mul.Mul.as_ordered_factors" title="Permalink to this definition">¶</a></dt>
-<dd><p>Transform an expression into an ordered list of factors.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">as_ordered_factors</span><span class="p">()</span>
-<span class="go">[2, x, y, sin(x), cos(x)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.mul.Mul.as_two_terms">
-<tt class="descname">as_two_terms</tt><big>(</big><em>*args</em>, <em>**kw_args</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/mul.html#Mul.as_two_terms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.mul.Mul.as_two_terms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return head and tail of self.</p>
-<p>This is the most efficient way to get the head and tail of an
-expression.</p>
-<ul class="simple">
-<li>if you want only the head, use self.args[0];</li>
-<li>if you want to process the arguments of the tail then use
-self.as_coef_mul() which gives the head and a tuple containing
-the arguments of the tail when treated as a Mul.</li>
-<li>if you want the coefficient when self is treated as an Add
-then use self.as_coeff_add()[0]</li>
-</ul>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">as_two_terms</span><span class="p">()</span>
-<span class="go">(3, x*y)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.core.mul.Mul.flatten">
-<em class="property">classmethod </em><tt class="descname">flatten</tt><big>(</big><em>seq</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/mul.html#Mul.flatten"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.mul.Mul.flatten" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return commutative, noncommutative and order arguments by
-combining related terms.</p>
-<p class="rubric">Notes</p>
-<ul>
-<li><p class="first">In an expression like <tt class="docutils literal"><span class="pre">a*b*c</span></tt>, python process this through sympy
-as <tt class="docutils literal"><span class="pre">Mul(Mul(a,</span> <span class="pre">b),</span> <span class="pre">c)</span></tt>. This can have undesirable consequences.</p>
-<ul class="simple">
-<li>Sometimes terms are not combined as one would like:
-{c.f. <a class="reference external" href="http://code.google.com/p/sympy/issues/detail?id=1497">http://code.google.com/p/sympy/issues/detail?id=1497</a>}</li>
-</ul>
-<blockquote>
-<div><div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="c"># this is the 2-arg Mul behavior</span>
-<span class="go">2*x + 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="mi">2</span>
-<span class="go">2*y*(x + 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">y</span> <span class="c"># 2-arg result will be obtained first</span>
-<span class="go">y*(2*x + 2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Mul</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span> <span class="c"># all 3 args simultaneously processed</span>
-<span class="go">2*y*(x + 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span><span class="o">*</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">y</span><span class="p">)</span> <span class="c"># parentheses can control this behavior</span>
-<span class="go">2*y*(x + 1)</span>
-</pre></div>
-</div>
-<p>Powers with compound bases may not find a single base to
-combine with unless all arguments are processed at once.
-Post-processing may be necessary in such cases.
-{c.f. <a class="reference external" href="http://code.google.com/p/sympy/issues/detail?id=2629">http://code.google.com/p/sympy/issues/detail?id=2629</a>}</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">**</span><span class="mi">3</span>
-<span class="go">(x*sqrt(y))**(3/2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Mul</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">a</span><span class="p">)</span>
-<span class="go">(x*sqrt(y))**(3/2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">a</span>
-<span class="go">x*sqrt(y)*sqrt(x*sqrt(y))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">base</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">base</span><span class="p">)</span>
-<span class="go">(x*sqrt(y))**(3/2)</span>
-</pre></div>
-</div>
-</div></blockquote>
-<ul>
-<li><p class="first">If more than two terms are being multiplied then all the
-previous terms will be re-processed for each new argument.
-So if each of <tt class="docutils literal"><span class="pre">a</span></tt>, <tt class="docutils literal"><span class="pre">b</span></tt> and <tt class="docutils literal"><span class="pre">c</span></tt> were <a class="reference internal" href="#sympy.core.mul.Mul" title="sympy.core.mul.Mul"><tt class="xref py py-class docutils literal"><span class="pre">Mul</span></tt></a>
-expression, then <tt class="docutils literal"><span class="pre">a*b*c</span></tt> (or building up the product
-with <tt class="docutils literal"><span class="pre">*=</span></tt>) will process all the arguments of <tt class="docutils literal"><span class="pre">a</span></tt> and
-<tt class="docutils literal"><span class="pre">b</span></tt> twice: once when <tt class="docutils literal"><span class="pre">a*b</span></tt> is computed and again when
-<tt class="docutils literal"><span class="pre">c</span></tt> is multiplied.</p>
-<p>Using <tt class="docutils literal"><span class="pre">Mul(a,</span> <span class="pre">b,</span> <span class="pre">c)</span></tt> will process all arguments once.</p>
-</li>
-</ul>
-</li>
-<li><p class="first">The results of Mul are cached according to arguments, so flatten
-will only be called once for <tt class="docutils literal"><span class="pre">Mul(a,</span> <span class="pre">b,</span> <span class="pre">c)</span></tt>. If you can
-structure a calculation so the arguments are most likely to be
-repeats then this can save time in computing the answer. For
-example, say you had a Mul, M, that you wished to divide by <tt class="docutils literal"><span class="pre">d[i]</span></tt>
-and multiply by <tt class="docutils literal"><span class="pre">n[i]</span></tt> and you suspect there are many repeats
-in <tt class="docutils literal"><span class="pre">n</span></tt>. It would be better to compute <tt class="docutils literal"><span class="pre">M*n[i]/d[i]</span></tt> rather
-than <tt class="docutils literal"><span class="pre">M/d[i]*n[i]</span></tt> since every time n[i] is a repeat, the
-product, <tt class="docutils literal"><span class="pre">M*n[i]</span></tt> will be returned without flattening &#8211; the
-cached value will be returned. If you divide by the <tt class="docutils literal"><span class="pre">d[i]</span></tt>
-first (and those are more unique than the <tt class="docutils literal"><span class="pre">n[i]</span></tt>) then that will
-create a new Mul, <tt class="docutils literal"><span class="pre">M/d[i]</span></tt> the args of which will be traversed
-again when it is multiplied by <tt class="docutils literal"><span class="pre">n[i]</span></tt>.</p>
-<p>{c.f. <a class="reference external" href="http://code.google.com/p/sympy/issues/detail?id=2607">http://code.google.com/p/sympy/issues/detail?id=2607</a>}</p>
-<p>This consideration is moot if the cache is turned off.</p>
-</li>
-</ul>
-<p class="rubric">Nb</p>
-<p>The validity of the above notes depends on the implementation
-details of Mul and flatten which may change at any time. Therefore,
-you should only consider them when your code is highly performance
-sensitive.</p>
-<p>Removal of 1 from the sequence is already handled by AssocOp.__new__.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="prod">
-<h3>prod<a class="headerlink" href="#prod" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.mul.prod">
-<tt class="descclassname">sympy.core.mul.</tt><tt class="descname">prod</tt><big>(</big><em>a</em>, <em>start=1</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/mul.html#prod"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.mul.prod" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return product of elements of a. Start with int 1 so if only
-ints are included then an int result is returned.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">prod</span><span class="p">,</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">prod</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">)))</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="n">_</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">int</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">prod</span><span class="p">([</span><span class="n">S</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="mi">3</span><span class="p">])</span>
-<span class="go">6</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">.</span><span class="n">is_Integer</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>You can start the product at something other than 1:
-&gt;&gt;&gt; prod([1, 2], 3)
-6</p>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.core.add">
-<span id="add"></span><h2>add<a class="headerlink" href="#module-sympy.core.add" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="id6">
-<h3>Add<a class="headerlink" href="#id6" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.add.Add">
-<em class="property">class </em><tt class="descclassname">sympy.core.add.</tt><tt class="descname">Add</tt><a class="reference internal" href="../_modules/sympy/core/add.html#Add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.add.Add" title="Permalink to this definition">¶</a></dt>
-<dd><dl class="function">
-<dt id="sympy.core.add.Add.as_coeff_Add">
-<tt class="descname">as_coeff_Add</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/add.html#Add.as_coeff_Add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.add.Add.as_coeff_Add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Efficiently extract the coefficient of a summation.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.add.Add.as_coeff_add">
-<tt class="descname">as_coeff_add</tt><big>(</big><em>*args</em>, <em>**kw_args</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/add.html#Add.as_coeff_add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.add.Add.as_coeff_add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a tuple (coeff, args) where self is treated as an Add and coeff
-is the Number term and args is a tuple of all other terms.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">7</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">as_coeff_add</span><span class="p">()</span>
-<span class="go">(7, (3*x,))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">7</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">as_coeff_add</span><span class="p">()</span>
-<span class="go">(0, (7*x,))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.add.Add.as_coefficients_dict">
-<tt class="descname">as_coefficients_dict</tt><big>(</big><em>a</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/add.html#Add.as_coefficients_dict"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.add.Add.as_coefficients_dict" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a dictionary mapping terms to their Rational coefficient.
-Since the dictionary is a defaultdict, inquiries about terms which
-were not present will return a coefficient of 0. If an expression is
-not an Add it is considered to have a single term.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">as_coefficients_dict</span><span class="p">()</span>
-<span class="go">{1: 4, x: 3, a*x: 1}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="p">[</span><span class="n">a</span><span class="p">]</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">as_coefficients_dict</span><span class="p">()</span>
-<span class="go">{a*x: 3}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.add.Add.as_content_primitive">
-<tt class="descname">as_content_primitive</tt><big>(</big><em>self</em>, <em>radical=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/add.html#Add.as_content_primitive"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.add.Add.as_content_primitive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the tuple (R, self/R) where R is the positive Rational
-extracted from self. If radical is True (default is False) then
-common radicals will be removed and included as a factor of the
-primitive expression.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-<span class="go">(3, 1 + sqrt(2))</span>
-</pre></div>
-</div>
-<p>Radical content can also be factored out of the primitive:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">10</span><span class="p">))</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">(</span><span class="n">radical</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(2, sqrt(2)*(1 + 2*sqrt(5)))</span>
-</pre></div>
-</div>
-<p>See docstring of Expr.as_content_primitive for more examples.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.add.Add.as_real_imag">
-<tt class="descname">as_real_imag</tt><big>(</big><em>self</em>, <em>deep=True</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/add.html#Add.as_real_imag"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.add.Add.as_real_imag" title="Permalink to this definition">¶</a></dt>
-<dd><p>returns a tuple represeting a complex numbers</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">7</span> <span class="o">+</span> <span class="mi">9</span><span class="o">*</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-<span class="go">(7, 9)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.add.Add.as_two_terms">
-<tt class="descname">as_two_terms</tt><big>(</big><em>*args</em>, <em>**kw_args</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/add.html#Add.as_two_terms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.add.Add.as_two_terms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return head and tail of self.</p>
-<p>This is the most efficient way to get the head and tail of an
-expression.</p>
-<ul class="simple">
-<li>if you want only the head, use self.args[0];</li>
-<li>if you want to process the arguments of the tail then use
-self.as_coef_add() which gives the head and a tuple containing
-the arguments of the tail when treated as an Add.</li>
-<li>if you want the coefficient when self is treated as a Mul
-then use self.as_coeff_mul()[0]</li>
-</ul>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">as_two_terms</span><span class="p">()</span>
-<span class="go">(3, x*y)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.core.add.Add.class_key">
-<em class="property">classmethod </em><tt class="descname">class_key</tt><big>(</big><big>)</big><a class="reference internal" href="../_modules/sympy/core/add.html#Add.class_key"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.add.Add.class_key" title="Permalink to this definition">¶</a></dt>
-<dd><p>Nice order of classes</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.add.Add.extract_leading_order">
-<tt class="descname">extract_leading_order</tt><big>(</big><em>*args</em>, <em>**kw_args</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/add.html#Add.extract_leading_order"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.add.Add.extract_leading_order" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the leading term and it&#8217;s order.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="o">**</span><span class="mi">5</span><span class="p">)</span><span class="o">.</span><span class="n">extract_leading_order</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">((x**(-5), O(x**(-5))),)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">extract_leading_order</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">((1, O(1)),)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">extract_leading_order</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">((x, O(x)),)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.core.add.Add.flatten">
-<em class="property">classmethod </em><tt class="descname">flatten</tt><big>(</big><em>seq</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/add.html#Add.flatten"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.add.Add.flatten" title="Permalink to this definition">¶</a></dt>
-<dd><p>Takes the sequence &#8220;seq&#8221; of nested Adds and returns a flatten list.</p>
-<p>Returns: (commutative_part, noncommutative_part, order_symbols)</p>
-<p>Applies associativity, all terms are commutable with respect to
-addition.</p>
-<p>NB: the removal of 0 is already handled by AssocOp.__new__</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.core.mul.Mul.flatten" title="sympy.core.mul.Mul.flatten"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.core.mul.Mul.flatten</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.add.Add.primitive">
-<tt class="descname">primitive</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/add.html#Add.primitive"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.add.Add.primitive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return <tt class="docutils literal"><span class="pre">(R,</span> <span class="pre">self/R)</span></tt> where <tt class="docutils literal"><span class="pre">R`</span></tt> is the Rational GCD of <tt class="docutils literal"><span class="pre">self`</span></tt>.</p>
-<p><tt class="docutils literal"><span class="pre">R</span></tt> is collected only from the leading coefficient of each term.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">primitive</span><span class="p">()</span>
-<span class="go">(2, x + 2*y)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">/</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">y</span><span class="o">/</span><span class="mi">9</span><span class="p">)</span><span class="o">.</span><span class="n">primitive</span><span class="p">()</span>
-<span class="go">(2/9, 3*x + 2*y)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">/</span><span class="mi">3</span> <span class="o">+</span> <span class="mf">4.2</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">primitive</span><span class="p">()</span>
-<span class="go">(1/3, 2*x + 12.6*y)</span>
-</pre></div>
-</div>
-<p>No subprocessing of term factors is performed:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">primitive</span><span class="p">()</span>
-<span class="go">(1, x*(2*x + 2) + 2)</span>
-</pre></div>
-</div>
-<p>Recursive subprocessing can be done with the as_content_primitive()
-method:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-<span class="go">(2, x*(x + 1) + 1)</span>
-</pre></div>
-</div>
-<p>See also: primitive() function in polytools.py</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.core.mod">
-<span id="mod"></span><h2>mod<a class="headerlink" href="#module-sympy.core.mod" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="id7">
-<h3>Mod<a class="headerlink" href="#id7" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.mod.Mod">
-<em class="property">class </em><tt class="descclassname">sympy.core.mod.</tt><tt class="descname">Mod</tt><a class="reference internal" href="../_modules/sympy/core/mod.html#Mod"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.mod.Mod" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents a modulo operation on symbolic expressions.</p>
-<p>Receives two arguments, dividend p and divisor q.</p>
-<p>The convention used is the same as python&#8217;s: the remainder always has the
-same sign as the divisor.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">%</span> <span class="n">y</span>
-<span class="go">Mod(x**2, y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">.</span><span class="n">subs</span><span class="p">({</span><span class="n">x</span><span class="p">:</span> <span class="mi">5</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="mi">6</span><span class="p">})</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.core.relational">
-<span id="relational"></span><h2>relational<a class="headerlink" href="#module-sympy.core.relational" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="rel">
-<h3>Rel<a class="headerlink" href="#rel" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.relational.Rel">
-<em class="property">class </em><tt class="descclassname">sympy.core.relational.</tt><tt class="descname">Rel</tt><a class="reference internal" href="../_modules/sympy/core/relational.html#Rel"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.relational.Rel" title="Permalink to this definition">¶</a></dt>
-<dd><p>A handy wrapper around the Relational class.
-Rel(a,b, op)</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Rel</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rel</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="o">+</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="s">&#39;==&#39;</span><span class="p">)</span>
-<span class="go">y == x**2 + x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="eq">
-<h3>Eq<a class="headerlink" href="#eq" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.relational.Eq">
-<em class="property">class </em><tt class="descclassname">sympy.core.relational.</tt><tt class="descname">Eq</tt><a class="reference internal" href="../_modules/sympy/core/relational.html#Eq"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.relational.Eq" title="Permalink to this definition">¶</a></dt>
-<dd><p>A handy wrapper around the Relational class.
-Eq(a,b)</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Eq</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Eq</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="o">+</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">y == x**2 + x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="ne">
-<h3>Ne<a class="headerlink" href="#ne" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.relational.Ne">
-<em class="property">class </em><tt class="descclassname">sympy.core.relational.</tt><tt class="descname">Ne</tt><a class="reference internal" href="../_modules/sympy/core/relational.html#Ne"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.relational.Ne" title="Permalink to this definition">¶</a></dt>
-<dd><p>A handy wrapper around the Relational class.
-Ne(a,b)</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Ne</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ne</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="o">+</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">y != x**2 + x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="lt">
-<h3>Lt<a class="headerlink" href="#lt" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.relational.Lt">
-<em class="property">class </em><tt class="descclassname">sympy.core.relational.</tt><tt class="descname">Lt</tt><a class="reference internal" href="../_modules/sympy/core/relational.html#Lt"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.relational.Lt" title="Permalink to this definition">¶</a></dt>
-<dd><p>A handy wrapper around the Relational class.
-Lt(a,b)</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Lt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Lt</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="o">+</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">y &lt; x**2 + x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="le">
-<h3>Le<a class="headerlink" href="#le" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.relational.Le">
-<em class="property">class </em><tt class="descclassname">sympy.core.relational.</tt><tt class="descname">Le</tt><a class="reference internal" href="../_modules/sympy/core/relational.html#Le"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.relational.Le" title="Permalink to this definition">¶</a></dt>
-<dd><p>A handy wrapper around the Relational class.
-Le(a,b)</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Le</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Le</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="o">+</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">y &lt;= x**2 + x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="gt">
-<h3>Gt<a class="headerlink" href="#gt" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.relational.Gt">
-<em class="property">class </em><tt class="descclassname">sympy.core.relational.</tt><tt class="descname">Gt</tt><a class="reference internal" href="../_modules/sympy/core/relational.html#Gt"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.relational.Gt" title="Permalink to this definition">¶</a></dt>
-<dd><p>A handy wrapper around the Relational class.
-Gt(a,b)</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Gt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Gt</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">y &gt; x**2 + x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="ge">
-<h3>Ge<a class="headerlink" href="#ge" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.relational.Ge">
-<em class="property">class </em><tt class="descclassname">sympy.core.relational.</tt><tt class="descname">Ge</tt><a class="reference internal" href="../_modules/sympy/core/relational.html#Ge"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.relational.Ge" title="Permalink to this definition">¶</a></dt>
-<dd><p>A handy wrapper around the Relational class.
-Ge(a,b)</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Ge</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ge</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">y &gt;= x**2 + x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="equality">
-<h3>Equality<a class="headerlink" href="#equality" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.relational.Equality">
-<em class="property">class </em><tt class="descclassname">sympy.core.relational.</tt><tt class="descname">Equality</tt><a class="reference internal" href="../_modules/sympy/core/relational.html#Equality"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.relational.Equality" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="greaterthan">
-<h3>GreaterThan<a class="headerlink" href="#greaterthan" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.relational.GreaterThan">
-<em class="property">class </em><tt class="descclassname">sympy.core.relational.</tt><tt class="descname">GreaterThan</tt><a class="reference internal" href="../_modules/sympy/core/relational.html#GreaterThan"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.relational.GreaterThan" title="Permalink to this definition">¶</a></dt>
-<dd><p>Class representations of inequalities.</p>
-<p>The <tt class="docutils literal"><span class="pre">*Than</span></tt> classes represent inequal relationships, where the left-hand
-side is generally bigger or smaller than the right-hand side.  For example,
-the GreaterThan class represents an inequal relationship where the
-left-hand side is at least as big as the right side, if not bigger.  In
-mathematical notation:</p>
-<p>lhs &gt;= rhs</p>
-<p>In total, there are four <tt class="docutils literal"><span class="pre">*Than</span></tt> classes, to represent the four
-inequalities:</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="68%" />
-<col width="32%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head">Class Name</th>
-<th class="head">Symbol</th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-even"><td>GreaterThan</td>
-<td>(&gt;=)</td>
-</tr>
-<tr class="row-odd"><td>LessThan</td>
-<td>(&lt;=)</td>
-</tr>
-<tr class="row-even"><td>StrictGreaterThan</td>
-<td>(&gt;)</td>
-</tr>
-<tr class="row-odd"><td>StrictLessThan</td>
-<td>(&lt;)</td>
-</tr>
-</tbody>
-</table>
-<p>All classes take two arguments, lhs and rhs.</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="62%" />
-<col width="38%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head">Signature Example</th>
-<th class="head">Math equivalent</th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-even"><td>GreaterThan(lhs, rhs)</td>
-<td>lhs &gt;= rhs</td>
-</tr>
-<tr class="row-odd"><td>LessThan(lhs, rhs)</td>
-<td>lhs &lt;= rhs</td>
-</tr>
-<tr class="row-even"><td>StrictGreaterThan(lhs, rhs)</td>
-<td>lhs &gt;  rhs</td>
-</tr>
-<tr class="row-odd"><td>StrictLessThan(lhs, rhs)</td>
-<td>lhs &lt;  rhs</td>
-</tr>
-</tbody>
-</table>
-<p>In addition to the normal .lhs and .rhs of Relations, <tt class="docutils literal"><span class="pre">*Than</span></tt> inequality
-objects also have the .lts and .gts properties, which represent the &#8220;less
-than side&#8221; and &#8220;greater than side&#8221; of the operator.  Use of .lts and .gts
-in an algorithm rather than .lhs and .rhs as an assumption of inequality
-direction will make more explicit the intent of a certain section of code,
-and will make it similarly more robust to client code changes:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">GreaterThan</span><span class="p">,</span> <span class="n">StrictGreaterThan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">LessThan</span><span class="p">,</span>    <span class="n">StrictLessThan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">And</span><span class="p">,</span> <span class="n">Ge</span><span class="p">,</span> <span class="n">Gt</span><span class="p">,</span> <span class="n">Le</span><span class="p">,</span> <span class="n">Lt</span><span class="p">,</span> <span class="n">Rel</span><span class="p">,</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Relational</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">GreaterThan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span>
-<span class="go">x &gt;= 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> &gt;= </span><span class="si">%s</span><span class="s"> is the same as </span><span class="si">%s</span><span class="s"> &lt;= </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">gts</span><span class="p">,</span> <span class="n">e</span><span class="o">.</span><span class="n">lts</span><span class="p">,</span> <span class="n">e</span><span class="o">.</span><span class="n">lts</span><span class="p">,</span> <span class="n">e</span><span class="o">.</span><span class="n">gts</span><span class="p">)</span>
-<span class="go">&#39;x &gt;= 1 is the same as 1 &lt;= x&#39;</span>
-</pre></div>
-</div>
-<p class="rubric">Notes</p>
-<p>There are a couple of &#8220;gotchas&#8221; when using Python&#8217;s operators.</p>
-<p>The first enters the mix when comparing against a literal number as the lhs
-argument.  Due to the order that Python decides to parse a statement, it may
-not immediately find two objects comparable.  For example, to evaluate the
-statement (1 &lt; x), Python will first recognize the number 1 as a native
-number, and then that x is <em>not</em> a native number.  At this point, because a
-native Python number does not know how to compare itself with a SymPy object
-Python will try the reflective operation, (x &gt; 1).  Unfortunately, there is
-no way available to SymPy to recognize this has happened, so the statement
-(1 &lt; x) will turn silently into (x &gt; 1).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&gt;</span>  <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e2</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&gt;=</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e3</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&lt;</span>  <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e4</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&lt;=</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e5</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">&gt;</span>  <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e6</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">&gt;=</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e7</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">&lt;</span>  <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e8</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">&lt;=</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s">     </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span><span class="o">*</span><span class="mi">4</span> <span class="o">%</span> <span class="p">(</span><span class="n">e1</span><span class="p">,</span> <span class="n">e2</span><span class="p">,</span> <span class="n">e3</span><span class="p">,</span> <span class="n">e4</span><span class="p">,</span> <span class="n">e5</span><span class="p">,</span> <span class="n">e6</span><span class="p">,</span> <span class="n">e7</span><span class="p">,</span> <span class="n">e8</span><span class="p">))</span>
-<span class="go">x &gt; 1     x &gt;= 1</span>
-<span class="go">x &lt; 1     x &lt;= 1</span>
-<span class="go">x &lt; 1     x &lt;= 1</span>
-<span class="go">x &gt; 1     x &gt;= 1</span>
-</pre></div>
-</div>
-<p>If the order of the statement is important (for visual output to the
-console, perhaps), one can work around this annoyance in a couple ways: (1)
-&#8220;sympify&#8221; the literal before comparison, (2) use one of the wrappers, or (3)
-use the less succinct methods described above:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">&gt;</span>  <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e2</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e3</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">&lt;</span>  <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e4</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e5</span> <span class="o">=</span> <span class="n">Gt</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e6</span> <span class="o">=</span> <span class="n">Ge</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e7</span> <span class="o">=</span> <span class="n">Lt</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e8</span> <span class="o">=</span> <span class="n">Le</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s">     </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span><span class="o">*</span><span class="mi">4</span> <span class="o">%</span> <span class="p">(</span><span class="n">e1</span><span class="p">,</span> <span class="n">e2</span><span class="p">,</span> <span class="n">e3</span><span class="p">,</span> <span class="n">e4</span><span class="p">,</span> <span class="n">e5</span><span class="p">,</span> <span class="n">e6</span><span class="p">,</span> <span class="n">e7</span><span class="p">,</span> <span class="n">e8</span><span class="p">))</span>
-<span class="go">1 &gt; x     1 &gt;= x</span>
-<span class="go">1 &lt; x     1 &lt;= x</span>
-<span class="go">1 &gt; x     1 &gt;= x</span>
-<span class="go">1 &lt; x     1 &lt;= x</span>
-</pre></div>
-</div>
-<p>The other gotcha is with chained inequalities.  Occasionally, one may be
-tempted to write statements like:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&lt;</span> <span class="n">y</span> <span class="o">&lt;</span> <span class="n">z</span>  <span class="c"># silent error!  Where did ``x`` go?</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> 
-<span class="go">y &lt; z</span>
-</pre></div>
-</div>
-<p>Due to an implementation detail or decision of Python <a class="reference internal" href="#r13">[R13]</a>, there is no way
-for SymPy to reliably create that as a chained inequality.  To create a
-chained inequality, the only method currently available is to make use of
-And:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">And</span><span class="p">(</span><span class="n">x</span> <span class="o">&lt;</span> <span class="n">y</span><span class="p">,</span> <span class="n">y</span> <span class="o">&lt;</span> <span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span> <span class="n">e</span> <span class="p">)</span>
-<span class="go">And</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span>
-<span class="go">And(x &lt; y, y &lt; z)</span>
-</pre></div>
-</div>
-<p>Note that this is different than chaining an equality directly via use of
-parenthesis (this is currently an open bug in SymPy <a class="reference internal" href="#r14">[R14]</a>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">&lt;</span> <span class="n">y</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span> <span class="n">e</span> <span class="p">)</span>
-<span class="go">&lt;class &#39;sympy.core.relational.StrictLessThan&#39;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span>
-<span class="go">(x &lt; y) &lt; z</span>
-</pre></div>
-</div>
-<p>Any code that explicitly relies on this latter functionality will not be
-robust as this behaviour is completely wrong and will be corrected at some
-point.  For the time being (circa Jan 2012), use And to create chained
-inequalities.</p>
-<table class="docutils citation" frame="void" id="r13" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id8">[R13]</a></td><td><p class="first">This implementation detail is that Python provides no reliable
-method to determine that a chained inequality is being built.  Chained
-comparison operators are evaluated pairwise, using &#8220;and&#8221; logic (see
-<a class="reference external" href="http://docs.python.org/reference/expressions.html#notin">http://docs.python.org/reference/expressions.html#notin</a>).  This is done
-in an efficient way, so that each object being compared is only
-evaluated once and the comparison can short-circuit.  For example, <tt class="docutils literal"><span class="pre">1</span>
-<span class="pre">&gt;</span> <span class="pre">2</span> <span class="pre">&gt;</span> <span class="pre">3</span></tt> is evaluated by Python as <tt class="docutils literal"><span class="pre">(1</span> <span class="pre">&gt;</span> <span class="pre">2)</span> <span class="pre">and</span> <span class="pre">(2</span> <span class="pre">&gt;</span> <span class="pre">3)</span></tt>.  The
-<tt class="docutils literal"><span class="pre">and</span></tt> operator coerces each side into a bool, returning the object
-itself when it short-circuits.  Currently, the bool of the &#8211;Than
-operators will give True or False arbitrarily.  Thus, if we were to
-compute <tt class="docutils literal"><span class="pre">x</span> <span class="pre">&gt;</span> <span class="pre">y</span> <span class="pre">&gt;</span> <span class="pre">z</span></tt>, with <tt class="docutils literal"><span class="pre">x</span></tt>, <tt class="docutils literal"><span class="pre">y</span></tt>, and <tt class="docutils literal"><span class="pre">z</span></tt> being Symbols,
-Python converts the statement (roughly) into these steps:</p>
-<blockquote>
-<div><ol class="arabic simple">
-<li>x &gt; y &gt; z</li>
-<li>(x &gt; y) and (y &gt; z)</li>
-<li>(GreaterThanObject) and (y &gt; z)</li>
-<li>(GreaterThanObject.__nonzero__()) and (y &gt; z)</li>
-<li>(True) and (y &gt; z)</li>
-<li>(y &gt; z)</li>
-<li>LessThanObject</li>
-</ol>
-</div></blockquote>
-<p>Because of the &#8220;and&#8221; added at step 2, the statement gets turned into a
-weak ternary statement.  If the first object evalutes __nonzero__ as
-True, then the second object, (y &gt; z) is returned.  If the first object
-evaluates __nonzero__ as False (step 5), then (x &gt; y) is returned.</p>
-<blockquote class="last">
-<div>In Python, there is no way to override the <tt class="docutils literal"><span class="pre">and</span></tt> operator, or to
-control how it short circuits, so it is impossible to make something
-like <tt class="docutils literal"><span class="pre">x</span> <span class="pre">&gt;</span> <span class="pre">y</span> <span class="pre">&gt;</span> <span class="pre">z</span></tt> work.  There is an open PEP to change this,
-<span class="target" id="index-0"></span><a class="pep reference external" href="http://www.python.org/dev/peps/pep-0335"><strong>PEP 335</strong></a>, but until that is implemented, this cannot be made to work.</div></blockquote>
-</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r14" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id9">[R14]</a></td><td><p class="first">For more information, see these two bug reports:</p>
-<p>&#8220;Separate boolean and symbolic relationals&#8221;
-<a class="reference external" href="http://code.google.com/p/sympy/issues/detail?id=1887">Issue 1887</a></p>
-<p class="last">&#8220;It right 0 &lt; x &lt; 1 ?&#8221;
-<a class="reference external" href="http://code.google.com/p/sympy/issues/detail?id=2960">Issue 2960</a></p>
-</td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>One generally does not instantiate these classes directly, but uses various
-convenience methods:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ge</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span> <span class="p">)</span>      <span class="c"># Ge is a convenience wrapper</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">e1</span><span class="p">)</span>
-<span class="go">x &gt;= 2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rels</span> <span class="o">=</span> <span class="n">Ge</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span> <span class="p">),</span> <span class="n">Gt</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span> <span class="p">),</span> <span class="n">Le</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span> <span class="p">),</span> <span class="n">Lt</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span> <span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">rels</span><span class="p">)</span>
-<span class="go">x &gt;= 2</span>
-<span class="go">x &gt; 2</span>
-<span class="go">x &lt;= 2</span>
-<span class="go">x &lt; 2</span>
-</pre></div>
-</div>
-<p>Another option is to use the Python inequality operators (&gt;=, &gt;, &lt;=, &lt;)
-directly.  Their main advantage over the Ge, Gt, Le, and Lt counterparts, is
-that one can write a more &#8220;mathematical looking&#8221; statement rather than
-littering the math with oddball function calls.  However there are certain
-(minor) caveats of which to be aware (search for &#8216;gotcha&#8217;, below).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e2</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&gt;=</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">e2</span><span class="p">)</span>
-<span class="go">x &gt;= 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&quot;e1: </span><span class="si">%s</span><span class="s">,    e2: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">e1</span><span class="p">,</span> <span class="n">e2</span><span class="p">))</span>
-<span class="go">e1: x &gt;= 2,    e2: x &gt;= 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">==</span> <span class="n">e2</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>However, it is also perfectly valid to instantiate a <tt class="docutils literal"><span class="pre">*Than</span></tt> class less
-succinctly and less conveniently:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rels</span> <span class="o">=</span> <span class="n">Rel</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&gt;=&#39;</span><span class="p">),</span> <span class="n">Relational</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&gt;=&#39;</span><span class="p">),</span> <span class="n">GreaterThan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">rels</span><span class="p">)</span>
-<span class="go">x &gt;= 1</span>
-<span class="go">x &gt;= 1</span>
-<span class="go">x &gt;= 1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rels</span> <span class="o">=</span> <span class="n">Rel</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&gt;&#39;</span><span class="p">),</span> <span class="n">Relational</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&gt;&#39;</span><span class="p">),</span> <span class="n">StrictGreaterThan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">rels</span><span class="p">)</span>
-<span class="go">x &gt; 1</span>
-<span class="go">x &gt; 1</span>
-<span class="go">x &gt; 1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rels</span> <span class="o">=</span> <span class="n">Rel</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&lt;=&#39;</span><span class="p">),</span> <span class="n">Relational</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&lt;=&#39;</span><span class="p">),</span> <span class="n">LessThan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">rels</span><span class="p">)</span>
-<span class="go">x &lt;= 1</span>
-<span class="go">x &lt;= 1</span>
-<span class="go">x &lt;= 1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rels</span> <span class="o">=</span> <span class="n">Rel</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&lt;&#39;</span><span class="p">),</span> <span class="n">Relational</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&lt;&#39;</span><span class="p">),</span> <span class="n">StrictLessThan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">rels</span><span class="p">)</span>
-<span class="go">x &lt; 1</span>
-<span class="go">x &lt; 1</span>
-<span class="go">x &lt; 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="lessthan">
-<h3>LessThan<a class="headerlink" href="#lessthan" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.relational.LessThan">
-<em class="property">class </em><tt class="descclassname">sympy.core.relational.</tt><tt class="descname">LessThan</tt><a class="reference internal" href="../_modules/sympy/core/relational.html#LessThan"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.relational.LessThan" title="Permalink to this definition">¶</a></dt>
-<dd><p>Class representations of inequalities.</p>
-<p>The <tt class="docutils literal"><span class="pre">*Than</span></tt> classes represent inequal relationships, where the left-hand
-side is generally bigger or smaller than the right-hand side.  For example,
-the GreaterThan class represents an inequal relationship where the
-left-hand side is at least as big as the right side, if not bigger.  In
-mathematical notation:</p>
-<p>lhs &gt;= rhs</p>
-<p>In total, there are four <tt class="docutils literal"><span class="pre">*Than</span></tt> classes, to represent the four
-inequalities:</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="68%" />
-<col width="32%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head">Class Name</th>
-<th class="head">Symbol</th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-even"><td>GreaterThan</td>
-<td>(&gt;=)</td>
-</tr>
-<tr class="row-odd"><td>LessThan</td>
-<td>(&lt;=)</td>
-</tr>
-<tr class="row-even"><td>StrictGreaterThan</td>
-<td>(&gt;)</td>
-</tr>
-<tr class="row-odd"><td>StrictLessThan</td>
-<td>(&lt;)</td>
-</tr>
-</tbody>
-</table>
-<p>All classes take two arguments, lhs and rhs.</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="62%" />
-<col width="38%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head">Signature Example</th>
-<th class="head">Math equivalent</th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-even"><td>GreaterThan(lhs, rhs)</td>
-<td>lhs &gt;= rhs</td>
-</tr>
-<tr class="row-odd"><td>LessThan(lhs, rhs)</td>
-<td>lhs &lt;= rhs</td>
-</tr>
-<tr class="row-even"><td>StrictGreaterThan(lhs, rhs)</td>
-<td>lhs &gt;  rhs</td>
-</tr>
-<tr class="row-odd"><td>StrictLessThan(lhs, rhs)</td>
-<td>lhs &lt;  rhs</td>
-</tr>
-</tbody>
-</table>
-<p>In addition to the normal .lhs and .rhs of Relations, <tt class="docutils literal"><span class="pre">*Than</span></tt> inequality
-objects also have the .lts and .gts properties, which represent the &#8220;less
-than side&#8221; and &#8220;greater than side&#8221; of the operator.  Use of .lts and .gts
-in an algorithm rather than .lhs and .rhs as an assumption of inequality
-direction will make more explicit the intent of a certain section of code,
-and will make it similarly more robust to client code changes:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">GreaterThan</span><span class="p">,</span> <span class="n">StrictGreaterThan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">LessThan</span><span class="p">,</span>    <span class="n">StrictLessThan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">And</span><span class="p">,</span> <span class="n">Ge</span><span class="p">,</span> <span class="n">Gt</span><span class="p">,</span> <span class="n">Le</span><span class="p">,</span> <span class="n">Lt</span><span class="p">,</span> <span class="n">Rel</span><span class="p">,</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Relational</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">GreaterThan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span>
-<span class="go">x &gt;= 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> &gt;= </span><span class="si">%s</span><span class="s"> is the same as </span><span class="si">%s</span><span class="s"> &lt;= </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">gts</span><span class="p">,</span> <span class="n">e</span><span class="o">.</span><span class="n">lts</span><span class="p">,</span> <span class="n">e</span><span class="o">.</span><span class="n">lts</span><span class="p">,</span> <span class="n">e</span><span class="o">.</span><span class="n">gts</span><span class="p">)</span>
-<span class="go">&#39;x &gt;= 1 is the same as 1 &lt;= x&#39;</span>
-</pre></div>
-</div>
-<p class="rubric">Notes</p>
-<p>There are a couple of &#8220;gotchas&#8221; when using Python&#8217;s operators.</p>
-<p>The first enters the mix when comparing against a literal number as the lhs
-argument.  Due to the order that Python decides to parse a statement, it may
-not immediately find two objects comparable.  For example, to evaluate the
-statement (1 &lt; x), Python will first recognize the number 1 as a native
-number, and then that x is <em>not</em> a native number.  At this point, because a
-native Python number does not know how to compare itself with a SymPy object
-Python will try the reflective operation, (x &gt; 1).  Unfortunately, there is
-no way available to SymPy to recognize this has happened, so the statement
-(1 &lt; x) will turn silently into (x &gt; 1).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&gt;</span>  <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e2</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&gt;=</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e3</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&lt;</span>  <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e4</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&lt;=</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e5</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">&gt;</span>  <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e6</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">&gt;=</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e7</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">&lt;</span>  <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e8</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">&lt;=</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s">     </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span><span class="o">*</span><span class="mi">4</span> <span class="o">%</span> <span class="p">(</span><span class="n">e1</span><span class="p">,</span> <span class="n">e2</span><span class="p">,</span> <span class="n">e3</span><span class="p">,</span> <span class="n">e4</span><span class="p">,</span> <span class="n">e5</span><span class="p">,</span> <span class="n">e6</span><span class="p">,</span> <span class="n">e7</span><span class="p">,</span> <span class="n">e8</span><span class="p">))</span>
-<span class="go">x &gt; 1     x &gt;= 1</span>
-<span class="go">x &lt; 1     x &lt;= 1</span>
-<span class="go">x &lt; 1     x &lt;= 1</span>
-<span class="go">x &gt; 1     x &gt;= 1</span>
-</pre></div>
-</div>
-<p>If the order of the statement is important (for visual output to the
-console, perhaps), one can work around this annoyance in a couple ways: (1)
-&#8220;sympify&#8221; the literal before comparison, (2) use one of the wrappers, or (3)
-use the less succinct methods described above:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">&gt;</span>  <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e2</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e3</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">&lt;</span>  <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e4</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e5</span> <span class="o">=</span> <span class="n">Gt</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e6</span> <span class="o">=</span> <span class="n">Ge</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e7</span> <span class="o">=</span> <span class="n">Lt</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e8</span> <span class="o">=</span> <span class="n">Le</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s">     </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span><span class="o">*</span><span class="mi">4</span> <span class="o">%</span> <span class="p">(</span><span class="n">e1</span><span class="p">,</span> <span class="n">e2</span><span class="p">,</span> <span class="n">e3</span><span class="p">,</span> <span class="n">e4</span><span class="p">,</span> <span class="n">e5</span><span class="p">,</span> <span class="n">e6</span><span class="p">,</span> <span class="n">e7</span><span class="p">,</span> <span class="n">e8</span><span class="p">))</span>
-<span class="go">1 &gt; x     1 &gt;= x</span>
-<span class="go">1 &lt; x     1 &lt;= x</span>
-<span class="go">1 &gt; x     1 &gt;= x</span>
-<span class="go">1 &lt; x     1 &lt;= x</span>
-</pre></div>
-</div>
-<p>The other gotcha is with chained inequalities.  Occasionally, one may be
-tempted to write statements like:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&lt;</span> <span class="n">y</span> <span class="o">&lt;</span> <span class="n">z</span>  <span class="c"># silent error!  Where did ``x`` go?</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> 
-<span class="go">y &lt; z</span>
-</pre></div>
-</div>
-<p>Due to an implementation detail or decision of Python <a class="reference internal" href="#r15">[R15]</a>, there is no way
-for SymPy to reliably create that as a chained inequality.  To create a
-chained inequality, the only method currently available is to make use of
-And:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">And</span><span class="p">(</span><span class="n">x</span> <span class="o">&lt;</span> <span class="n">y</span><span class="p">,</span> <span class="n">y</span> <span class="o">&lt;</span> <span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span> <span class="n">e</span> <span class="p">)</span>
-<span class="go">And</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span>
-<span class="go">And(x &lt; y, y &lt; z)</span>
-</pre></div>
-</div>
-<p>Note that this is different than chaining an equality directly via use of
-parenthesis (this is currently an open bug in SymPy <a class="reference internal" href="#r16">[R16]</a>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">&lt;</span> <span class="n">y</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span> <span class="n">e</span> <span class="p">)</span>
-<span class="go">&lt;class &#39;sympy.core.relational.StrictLessThan&#39;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span>
-<span class="go">(x &lt; y) &lt; z</span>
-</pre></div>
-</div>
-<p>Any code that explicitly relies on this latter functionality will not be
-robust as this behaviour is completely wrong and will be corrected at some
-point.  For the time being (circa Jan 2012), use And to create chained
-inequalities.</p>
-<table class="docutils citation" frame="void" id="r15" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id10">[R15]</a></td><td><p class="first">This implementation detail is that Python provides no reliable
-method to determine that a chained inequality is being built.  Chained
-comparison operators are evaluated pairwise, using &#8220;and&#8221; logic (see
-<a class="reference external" href="http://docs.python.org/reference/expressions.html#notin">http://docs.python.org/reference/expressions.html#notin</a>).  This is done
-in an efficient way, so that each object being compared is only
-evaluated once and the comparison can short-circuit.  For example, <tt class="docutils literal"><span class="pre">1</span>
-<span class="pre">&gt;</span> <span class="pre">2</span> <span class="pre">&gt;</span> <span class="pre">3</span></tt> is evaluated by Python as <tt class="docutils literal"><span class="pre">(1</span> <span class="pre">&gt;</span> <span class="pre">2)</span> <span class="pre">and</span> <span class="pre">(2</span> <span class="pre">&gt;</span> <span class="pre">3)</span></tt>.  The
-<tt class="docutils literal"><span class="pre">and</span></tt> operator coerces each side into a bool, returning the object
-itself when it short-circuits.  Currently, the bool of the &#8211;Than
-operators will give True or False arbitrarily.  Thus, if we were to
-compute <tt class="docutils literal"><span class="pre">x</span> <span class="pre">&gt;</span> <span class="pre">y</span> <span class="pre">&gt;</span> <span class="pre">z</span></tt>, with <tt class="docutils literal"><span class="pre">x</span></tt>, <tt class="docutils literal"><span class="pre">y</span></tt>, and <tt class="docutils literal"><span class="pre">z</span></tt> being Symbols,
-Python converts the statement (roughly) into these steps:</p>
-<blockquote>
-<div><ol class="arabic simple">
-<li>x &gt; y &gt; z</li>
-<li>(x &gt; y) and (y &gt; z)</li>
-<li>(GreaterThanObject) and (y &gt; z)</li>
-<li>(GreaterThanObject.__nonzero__()) and (y &gt; z)</li>
-<li>(True) and (y &gt; z)</li>
-<li>(y &gt; z)</li>
-<li>LessThanObject</li>
-</ol>
-</div></blockquote>
-<p>Because of the &#8220;and&#8221; added at step 2, the statement gets turned into a
-weak ternary statement.  If the first object evalutes __nonzero__ as
-True, then the second object, (y &gt; z) is returned.  If the first object
-evaluates __nonzero__ as False (step 5), then (x &gt; y) is returned.</p>
-<blockquote class="last">
-<div>In Python, there is no way to override the <tt class="docutils literal"><span class="pre">and</span></tt> operator, or to
-control how it short circuits, so it is impossible to make something
-like <tt class="docutils literal"><span class="pre">x</span> <span class="pre">&gt;</span> <span class="pre">y</span> <span class="pre">&gt;</span> <span class="pre">z</span></tt> work.  There is an open PEP to change this,
-<span class="target" id="index-1"></span><a class="pep reference external" href="http://www.python.org/dev/peps/pep-0335"><strong>PEP 335</strong></a>, but until that is implemented, this cannot be made to work.</div></blockquote>
-</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r16" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id11">[R16]</a></td><td><p class="first">For more information, see these two bug reports:</p>
-<p>&#8220;Separate boolean and symbolic relationals&#8221;
-<a class="reference external" href="http://code.google.com/p/sympy/issues/detail?id=1887">Issue 1887</a></p>
-<p class="last">&#8220;It right 0 &lt; x &lt; 1 ?&#8221;
-<a class="reference external" href="http://code.google.com/p/sympy/issues/detail?id=2960">Issue 2960</a></p>
-</td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>One generally does not instantiate these classes directly, but uses various
-convenience methods:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ge</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span> <span class="p">)</span>      <span class="c"># Ge is a convenience wrapper</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">e1</span><span class="p">)</span>
-<span class="go">x &gt;= 2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rels</span> <span class="o">=</span> <span class="n">Ge</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span> <span class="p">),</span> <span class="n">Gt</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span> <span class="p">),</span> <span class="n">Le</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span> <span class="p">),</span> <span class="n">Lt</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span> <span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">rels</span><span class="p">)</span>
-<span class="go">x &gt;= 2</span>
-<span class="go">x &gt; 2</span>
-<span class="go">x &lt;= 2</span>
-<span class="go">x &lt; 2</span>
-</pre></div>
-</div>
-<p>Another option is to use the Python inequality operators (&gt;=, &gt;, &lt;=, &lt;)
-directly.  Their main advantage over the Ge, Gt, Le, and Lt counterparts, is
-that one can write a more &#8220;mathematical looking&#8221; statement rather than
-littering the math with oddball function calls.  However there are certain
-(minor) caveats of which to be aware (search for &#8216;gotcha&#8217;, below).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e2</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&gt;=</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">e2</span><span class="p">)</span>
-<span class="go">x &gt;= 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&quot;e1: </span><span class="si">%s</span><span class="s">,    e2: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">e1</span><span class="p">,</span> <span class="n">e2</span><span class="p">))</span>
-<span class="go">e1: x &gt;= 2,    e2: x &gt;= 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">==</span> <span class="n">e2</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>However, it is also perfectly valid to instantiate a <tt class="docutils literal"><span class="pre">*Than</span></tt> class less
-succinctly and less conveniently:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rels</span> <span class="o">=</span> <span class="n">Rel</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&gt;=&#39;</span><span class="p">),</span> <span class="n">Relational</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&gt;=&#39;</span><span class="p">),</span> <span class="n">GreaterThan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">rels</span><span class="p">)</span>
-<span class="go">x &gt;= 1</span>
-<span class="go">x &gt;= 1</span>
-<span class="go">x &gt;= 1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rels</span> <span class="o">=</span> <span class="n">Rel</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&gt;&#39;</span><span class="p">),</span> <span class="n">Relational</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&gt;&#39;</span><span class="p">),</span> <span class="n">StrictGreaterThan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">rels</span><span class="p">)</span>
-<span class="go">x &gt; 1</span>
-<span class="go">x &gt; 1</span>
-<span class="go">x &gt; 1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rels</span> <span class="o">=</span> <span class="n">Rel</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&lt;=&#39;</span><span class="p">),</span> <span class="n">Relational</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&lt;=&#39;</span><span class="p">),</span> <span class="n">LessThan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">rels</span><span class="p">)</span>
-<span class="go">x &lt;= 1</span>
-<span class="go">x &lt;= 1</span>
-<span class="go">x &lt;= 1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rels</span> <span class="o">=</span> <span class="n">Rel</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&lt;&#39;</span><span class="p">),</span> <span class="n">Relational</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&lt;&#39;</span><span class="p">),</span> <span class="n">StrictLessThan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">rels</span><span class="p">)</span>
-<span class="go">x &lt; 1</span>
-<span class="go">x &lt; 1</span>
-<span class="go">x &lt; 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="unequality">
-<h3>Unequality<a class="headerlink" href="#unequality" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.relational.Unequality">
-<em class="property">class </em><tt class="descclassname">sympy.core.relational.</tt><tt class="descname">Unequality</tt><a class="reference internal" href="../_modules/sympy/core/relational.html#Unequality"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.relational.Unequality" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="strictgreaterthan">
-<h3>StrictGreaterThan<a class="headerlink" href="#strictgreaterthan" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.relational.StrictGreaterThan">
-<em class="property">class </em><tt class="descclassname">sympy.core.relational.</tt><tt class="descname">StrictGreaterThan</tt><a class="reference internal" href="../_modules/sympy/core/relational.html#StrictGreaterThan"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.relational.StrictGreaterThan" title="Permalink to this definition">¶</a></dt>
-<dd><p>Class representations of inequalities.</p>
-<p>The <tt class="docutils literal"><span class="pre">*Than</span></tt> classes represent inequal relationships, where the left-hand
-side is generally bigger or smaller than the right-hand side.  For example,
-the GreaterThan class represents an inequal relationship where the
-left-hand side is at least as big as the right side, if not bigger.  In
-mathematical notation:</p>
-<p>lhs &gt;= rhs</p>
-<p>In total, there are four <tt class="docutils literal"><span class="pre">*Than</span></tt> classes, to represent the four
-inequalities:</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="68%" />
-<col width="32%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head">Class Name</th>
-<th class="head">Symbol</th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-even"><td>GreaterThan</td>
-<td>(&gt;=)</td>
-</tr>
-<tr class="row-odd"><td>LessThan</td>
-<td>(&lt;=)</td>
-</tr>
-<tr class="row-even"><td>StrictGreaterThan</td>
-<td>(&gt;)</td>
-</tr>
-<tr class="row-odd"><td>StrictLessThan</td>
-<td>(&lt;)</td>
-</tr>
-</tbody>
-</table>
-<p>All classes take two arguments, lhs and rhs.</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="62%" />
-<col width="38%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head">Signature Example</th>
-<th class="head">Math equivalent</th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-even"><td>GreaterThan(lhs, rhs)</td>
-<td>lhs &gt;= rhs</td>
-</tr>
-<tr class="row-odd"><td>LessThan(lhs, rhs)</td>
-<td>lhs &lt;= rhs</td>
-</tr>
-<tr class="row-even"><td>StrictGreaterThan(lhs, rhs)</td>
-<td>lhs &gt;  rhs</td>
-</tr>
-<tr class="row-odd"><td>StrictLessThan(lhs, rhs)</td>
-<td>lhs &lt;  rhs</td>
-</tr>
-</tbody>
-</table>
-<p>In addition to the normal .lhs and .rhs of Relations, <tt class="docutils literal"><span class="pre">*Than</span></tt> inequality
-objects also have the .lts and .gts properties, which represent the &#8220;less
-than side&#8221; and &#8220;greater than side&#8221; of the operator.  Use of .lts and .gts
-in an algorithm rather than .lhs and .rhs as an assumption of inequality
-direction will make more explicit the intent of a certain section of code,
-and will make it similarly more robust to client code changes:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">GreaterThan</span><span class="p">,</span> <span class="n">StrictGreaterThan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">LessThan</span><span class="p">,</span>    <span class="n">StrictLessThan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">And</span><span class="p">,</span> <span class="n">Ge</span><span class="p">,</span> <span class="n">Gt</span><span class="p">,</span> <span class="n">Le</span><span class="p">,</span> <span class="n">Lt</span><span class="p">,</span> <span class="n">Rel</span><span class="p">,</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Relational</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">GreaterThan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span>
-<span class="go">x &gt;= 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> &gt;= </span><span class="si">%s</span><span class="s"> is the same as </span><span class="si">%s</span><span class="s"> &lt;= </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">gts</span><span class="p">,</span> <span class="n">e</span><span class="o">.</span><span class="n">lts</span><span class="p">,</span> <span class="n">e</span><span class="o">.</span><span class="n">lts</span><span class="p">,</span> <span class="n">e</span><span class="o">.</span><span class="n">gts</span><span class="p">)</span>
-<span class="go">&#39;x &gt;= 1 is the same as 1 &lt;= x&#39;</span>
-</pre></div>
-</div>
-<p class="rubric">Notes</p>
-<p>There are a couple of &#8220;gotchas&#8221; when using Python&#8217;s operators.</p>
-<p>The first enters the mix when comparing against a literal number as the lhs
-argument.  Due to the order that Python decides to parse a statement, it may
-not immediately find two objects comparable.  For example, to evaluate the
-statement (1 &lt; x), Python will first recognize the number 1 as a native
-number, and then that x is <em>not</em> a native number.  At this point, because a
-native Python number does not know how to compare itself with a SymPy object
-Python will try the reflective operation, (x &gt; 1).  Unfortunately, there is
-no way available to SymPy to recognize this has happened, so the statement
-(1 &lt; x) will turn silently into (x &gt; 1).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&gt;</span>  <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e2</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&gt;=</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e3</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&lt;</span>  <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e4</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&lt;=</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e5</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">&gt;</span>  <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e6</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">&gt;=</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e7</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">&lt;</span>  <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e8</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">&lt;=</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s">     </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span><span class="o">*</span><span class="mi">4</span> <span class="o">%</span> <span class="p">(</span><span class="n">e1</span><span class="p">,</span> <span class="n">e2</span><span class="p">,</span> <span class="n">e3</span><span class="p">,</span> <span class="n">e4</span><span class="p">,</span> <span class="n">e5</span><span class="p">,</span> <span class="n">e6</span><span class="p">,</span> <span class="n">e7</span><span class="p">,</span> <span class="n">e8</span><span class="p">))</span>
-<span class="go">x &gt; 1     x &gt;= 1</span>
-<span class="go">x &lt; 1     x &lt;= 1</span>
-<span class="go">x &lt; 1     x &lt;= 1</span>
-<span class="go">x &gt; 1     x &gt;= 1</span>
-</pre></div>
-</div>
-<p>If the order of the statement is important (for visual output to the
-console, perhaps), one can work around this annoyance in a couple ways: (1)
-&#8220;sympify&#8221; the literal before comparison, (2) use one of the wrappers, or (3)
-use the less succinct methods described above:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">&gt;</span>  <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e2</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e3</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">&lt;</span>  <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e4</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e5</span> <span class="o">=</span> <span class="n">Gt</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e6</span> <span class="o">=</span> <span class="n">Ge</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e7</span> <span class="o">=</span> <span class="n">Lt</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e8</span> <span class="o">=</span> <span class="n">Le</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s">     </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span><span class="o">*</span><span class="mi">4</span> <span class="o">%</span> <span class="p">(</span><span class="n">e1</span><span class="p">,</span> <span class="n">e2</span><span class="p">,</span> <span class="n">e3</span><span class="p">,</span> <span class="n">e4</span><span class="p">,</span> <span class="n">e5</span><span class="p">,</span> <span class="n">e6</span><span class="p">,</span> <span class="n">e7</span><span class="p">,</span> <span class="n">e8</span><span class="p">))</span>
-<span class="go">1 &gt; x     1 &gt;= x</span>
-<span class="go">1 &lt; x     1 &lt;= x</span>
-<span class="go">1 &gt; x     1 &gt;= x</span>
-<span class="go">1 &lt; x     1 &lt;= x</span>
-</pre></div>
-</div>
-<p>The other gotcha is with chained inequalities.  Occasionally, one may be
-tempted to write statements like:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&lt;</span> <span class="n">y</span> <span class="o">&lt;</span> <span class="n">z</span>  <span class="c"># silent error!  Where did ``x`` go?</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> 
-<span class="go">y &lt; z</span>
-</pre></div>
-</div>
-<p>Due to an implementation detail or decision of Python <a class="reference internal" href="#r17">[R17]</a>, there is no way
-for SymPy to reliably create that as a chained inequality.  To create a
-chained inequality, the only method currently available is to make use of
-And:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">And</span><span class="p">(</span><span class="n">x</span> <span class="o">&lt;</span> <span class="n">y</span><span class="p">,</span> <span class="n">y</span> <span class="o">&lt;</span> <span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span> <span class="n">e</span> <span class="p">)</span>
-<span class="go">And</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span>
-<span class="go">And(x &lt; y, y &lt; z)</span>
-</pre></div>
-</div>
-<p>Note that this is different than chaining an equality directly via use of
-parenthesis (this is currently an open bug in SymPy <a class="reference internal" href="#r18">[R18]</a>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">&lt;</span> <span class="n">y</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span> <span class="n">e</span> <span class="p">)</span>
-<span class="go">&lt;class &#39;sympy.core.relational.StrictLessThan&#39;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span>
-<span class="go">(x &lt; y) &lt; z</span>
-</pre></div>
-</div>
-<p>Any code that explicitly relies on this latter functionality will not be
-robust as this behaviour is completely wrong and will be corrected at some
-point.  For the time being (circa Jan 2012), use And to create chained
-inequalities.</p>
-<table class="docutils citation" frame="void" id="r17" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id14">[R17]</a></td><td><p class="first">This implementation detail is that Python provides no reliable
-method to determine that a chained inequality is being built.  Chained
-comparison operators are evaluated pairwise, using &#8220;and&#8221; logic (see
-<a class="reference external" href="http://docs.python.org/reference/expressions.html#notin">http://docs.python.org/reference/expressions.html#notin</a>).  This is done
-in an efficient way, so that each object being compared is only
-evaluated once and the comparison can short-circuit.  For example, <tt class="docutils literal"><span class="pre">1</span>
-<span class="pre">&gt;</span> <span class="pre">2</span> <span class="pre">&gt;</span> <span class="pre">3</span></tt> is evaluated by Python as <tt class="docutils literal"><span class="pre">(1</span> <span class="pre">&gt;</span> <span class="pre">2)</span> <span class="pre">and</span> <span class="pre">(2</span> <span class="pre">&gt;</span> <span class="pre">3)</span></tt>.  The
-<tt class="docutils literal"><span class="pre">and</span></tt> operator coerces each side into a bool, returning the object
-itself when it short-circuits.  Currently, the bool of the &#8211;Than
-operators will give True or False arbitrarily.  Thus, if we were to
-compute <tt class="docutils literal"><span class="pre">x</span> <span class="pre">&gt;</span> <span class="pre">y</span> <span class="pre">&gt;</span> <span class="pre">z</span></tt>, with <tt class="docutils literal"><span class="pre">x</span></tt>, <tt class="docutils literal"><span class="pre">y</span></tt>, and <tt class="docutils literal"><span class="pre">z</span></tt> being Symbols,
-Python converts the statement (roughly) into these steps:</p>
-<blockquote>
-<div><ol class="arabic simple">
-<li>x &gt; y &gt; z</li>
-<li>(x &gt; y) and (y &gt; z)</li>
-<li>(GreaterThanObject) and (y &gt; z)</li>
-<li>(GreaterThanObject.__nonzero__()) and (y &gt; z)</li>
-<li>(True) and (y &gt; z)</li>
-<li>(y &gt; z)</li>
-<li>LessThanObject</li>
-</ol>
-</div></blockquote>
-<p>Because of the &#8220;and&#8221; added at step 2, the statement gets turned into a
-weak ternary statement.  If the first object evalutes __nonzero__ as
-True, then the second object, (y &gt; z) is returned.  If the first object
-evaluates __nonzero__ as False (step 5), then (x &gt; y) is returned.</p>
-<blockquote class="last">
-<div>In Python, there is no way to override the <tt class="docutils literal"><span class="pre">and</span></tt> operator, or to
-control how it short circuits, so it is impossible to make something
-like <tt class="docutils literal"><span class="pre">x</span> <span class="pre">&gt;</span> <span class="pre">y</span> <span class="pre">&gt;</span> <span class="pre">z</span></tt> work.  There is an open PEP to change this,
-<span class="target" id="index-2"></span><a class="pep reference external" href="http://www.python.org/dev/peps/pep-0335"><strong>PEP 335</strong></a>, but until that is implemented, this cannot be made to work.</div></blockquote>
-</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r18" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id15">[R18]</a></td><td><p class="first">For more information, see these two bug reports:</p>
-<p>&#8220;Separate boolean and symbolic relationals&#8221;
-<a class="reference external" href="http://code.google.com/p/sympy/issues/detail?id=1887">Issue 1887</a></p>
-<p class="last">&#8220;It right 0 &lt; x &lt; 1 ?&#8221;
-<a class="reference external" href="http://code.google.com/p/sympy/issues/detail?id=2960">Issue 2960</a></p>
-</td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>One generally does not instantiate these classes directly, but uses various
-convenience methods:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ge</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span> <span class="p">)</span>      <span class="c"># Ge is a convenience wrapper</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">e1</span><span class="p">)</span>
-<span class="go">x &gt;= 2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rels</span> <span class="o">=</span> <span class="n">Ge</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span> <span class="p">),</span> <span class="n">Gt</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span> <span class="p">),</span> <span class="n">Le</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span> <span class="p">),</span> <span class="n">Lt</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span> <span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">rels</span><span class="p">)</span>
-<span class="go">x &gt;= 2</span>
-<span class="go">x &gt; 2</span>
-<span class="go">x &lt;= 2</span>
-<span class="go">x &lt; 2</span>
-</pre></div>
-</div>
-<p>Another option is to use the Python inequality operators (&gt;=, &gt;, &lt;=, &lt;)
-directly.  Their main advantage over the Ge, Gt, Le, and Lt counterparts, is
-that one can write a more &#8220;mathematical looking&#8221; statement rather than
-littering the math with oddball function calls.  However there are certain
-(minor) caveats of which to be aware (search for &#8216;gotcha&#8217;, below).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e2</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&gt;=</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">e2</span><span class="p">)</span>
-<span class="go">x &gt;= 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&quot;e1: </span><span class="si">%s</span><span class="s">,    e2: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">e1</span><span class="p">,</span> <span class="n">e2</span><span class="p">))</span>
-<span class="go">e1: x &gt;= 2,    e2: x &gt;= 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">==</span> <span class="n">e2</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>However, it is also perfectly valid to instantiate a <tt class="docutils literal"><span class="pre">*Than</span></tt> class less
-succinctly and less conveniently:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rels</span> <span class="o">=</span> <span class="n">Rel</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&gt;=&#39;</span><span class="p">),</span> <span class="n">Relational</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&gt;=&#39;</span><span class="p">),</span> <span class="n">GreaterThan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">rels</span><span class="p">)</span>
-<span class="go">x &gt;= 1</span>
-<span class="go">x &gt;= 1</span>
-<span class="go">x &gt;= 1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rels</span> <span class="o">=</span> <span class="n">Rel</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&gt;&#39;</span><span class="p">),</span> <span class="n">Relational</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&gt;&#39;</span><span class="p">),</span> <span class="n">StrictGreaterThan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">rels</span><span class="p">)</span>
-<span class="go">x &gt; 1</span>
-<span class="go">x &gt; 1</span>
-<span class="go">x &gt; 1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rels</span> <span class="o">=</span> <span class="n">Rel</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&lt;=&#39;</span><span class="p">),</span> <span class="n">Relational</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&lt;=&#39;</span><span class="p">),</span> <span class="n">LessThan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">rels</span><span class="p">)</span>
-<span class="go">x &lt;= 1</span>
-<span class="go">x &lt;= 1</span>
-<span class="go">x &lt;= 1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rels</span> <span class="o">=</span> <span class="n">Rel</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&lt;&#39;</span><span class="p">),</span> <span class="n">Relational</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&lt;&#39;</span><span class="p">),</span> <span class="n">StrictLessThan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">rels</span><span class="p">)</span>
-<span class="go">x &lt; 1</span>
-<span class="go">x &lt; 1</span>
-<span class="go">x &lt; 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="strictlessthan">
-<h3>StrictLessThan<a class="headerlink" href="#strictlessthan" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.relational.StrictLessThan">
-<em class="property">class </em><tt class="descclassname">sympy.core.relational.</tt><tt class="descname">StrictLessThan</tt><a class="reference internal" href="../_modules/sympy/core/relational.html#StrictLessThan"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.relational.StrictLessThan" title="Permalink to this definition">¶</a></dt>
-<dd><p>Class representations of inequalities.</p>
-<p>The <tt class="docutils literal"><span class="pre">*Than</span></tt> classes represent inequal relationships, where the left-hand
-side is generally bigger or smaller than the right-hand side.  For example,
-the GreaterThan class represents an inequal relationship where the
-left-hand side is at least as big as the right side, if not bigger.  In
-mathematical notation:</p>
-<p>lhs &gt;= rhs</p>
-<p>In total, there are four <tt class="docutils literal"><span class="pre">*Than</span></tt> classes, to represent the four
-inequalities:</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="68%" />
-<col width="32%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head">Class Name</th>
-<th class="head">Symbol</th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-even"><td>GreaterThan</td>
-<td>(&gt;=)</td>
-</tr>
-<tr class="row-odd"><td>LessThan</td>
-<td>(&lt;=)</td>
-</tr>
-<tr class="row-even"><td>StrictGreaterThan</td>
-<td>(&gt;)</td>
-</tr>
-<tr class="row-odd"><td>StrictLessThan</td>
-<td>(&lt;)</td>
-</tr>
-</tbody>
-</table>
-<p>All classes take two arguments, lhs and rhs.</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="62%" />
-<col width="38%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head">Signature Example</th>
-<th class="head">Math equivalent</th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-even"><td>GreaterThan(lhs, rhs)</td>
-<td>lhs &gt;= rhs</td>
-</tr>
-<tr class="row-odd"><td>LessThan(lhs, rhs)</td>
-<td>lhs &lt;= rhs</td>
-</tr>
-<tr class="row-even"><td>StrictGreaterThan(lhs, rhs)</td>
-<td>lhs &gt;  rhs</td>
-</tr>
-<tr class="row-odd"><td>StrictLessThan(lhs, rhs)</td>
-<td>lhs &lt;  rhs</td>
-</tr>
-</tbody>
-</table>
-<p>In addition to the normal .lhs and .rhs of Relations, <tt class="docutils literal"><span class="pre">*Than</span></tt> inequality
-objects also have the .lts and .gts properties, which represent the &#8220;less
-than side&#8221; and &#8220;greater than side&#8221; of the operator.  Use of .lts and .gts
-in an algorithm rather than .lhs and .rhs as an assumption of inequality
-direction will make more explicit the intent of a certain section of code,
-and will make it similarly more robust to client code changes:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">GreaterThan</span><span class="p">,</span> <span class="n">StrictGreaterThan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">LessThan</span><span class="p">,</span>    <span class="n">StrictLessThan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">And</span><span class="p">,</span> <span class="n">Ge</span><span class="p">,</span> <span class="n">Gt</span><span class="p">,</span> <span class="n">Le</span><span class="p">,</span> <span class="n">Lt</span><span class="p">,</span> <span class="n">Rel</span><span class="p">,</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core.relational</span> <span class="kn">import</span> <span class="n">Relational</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">GreaterThan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span>
-<span class="go">x &gt;= 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="s">&#39;</span><span class="si">%s</span><span class="s"> &gt;= </span><span class="si">%s</span><span class="s"> is the same as </span><span class="si">%s</span><span class="s"> &lt;= </span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">gts</span><span class="p">,</span> <span class="n">e</span><span class="o">.</span><span class="n">lts</span><span class="p">,</span> <span class="n">e</span><span class="o">.</span><span class="n">lts</span><span class="p">,</span> <span class="n">e</span><span class="o">.</span><span class="n">gts</span><span class="p">)</span>
-<span class="go">&#39;x &gt;= 1 is the same as 1 &lt;= x&#39;</span>
-</pre></div>
-</div>
-<p class="rubric">Notes</p>
-<p>There are a couple of &#8220;gotchas&#8221; when using Python&#8217;s operators.</p>
-<p>The first enters the mix when comparing against a literal number as the lhs
-argument.  Due to the order that Python decides to parse a statement, it may
-not immediately find two objects comparable.  For example, to evaluate the
-statement (1 &lt; x), Python will first recognize the number 1 as a native
-number, and then that x is <em>not</em> a native number.  At this point, because a
-native Python number does not know how to compare itself with a SymPy object
-Python will try the reflective operation, (x &gt; 1).  Unfortunately, there is
-no way available to SymPy to recognize this has happened, so the statement
-(1 &lt; x) will turn silently into (x &gt; 1).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&gt;</span>  <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e2</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&gt;=</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e3</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&lt;</span>  <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e4</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&lt;=</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e5</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">&gt;</span>  <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e6</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">&gt;=</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e7</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">&lt;</span>  <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e8</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">&lt;=</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s">     </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span><span class="o">*</span><span class="mi">4</span> <span class="o">%</span> <span class="p">(</span><span class="n">e1</span><span class="p">,</span> <span class="n">e2</span><span class="p">,</span> <span class="n">e3</span><span class="p">,</span> <span class="n">e4</span><span class="p">,</span> <span class="n">e5</span><span class="p">,</span> <span class="n">e6</span><span class="p">,</span> <span class="n">e7</span><span class="p">,</span> <span class="n">e8</span><span class="p">))</span>
-<span class="go">x &gt; 1     x &gt;= 1</span>
-<span class="go">x &lt; 1     x &lt;= 1</span>
-<span class="go">x &lt; 1     x &lt;= 1</span>
-<span class="go">x &gt; 1     x &gt;= 1</span>
-</pre></div>
-</div>
-<p>If the order of the statement is important (for visual output to the
-console, perhaps), one can work around this annoyance in a couple ways: (1)
-&#8220;sympify&#8221; the literal before comparison, (2) use one of the wrappers, or (3)
-use the less succinct methods described above:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">&gt;</span>  <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e2</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e3</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">&lt;</span>  <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e4</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e5</span> <span class="o">=</span> <span class="n">Gt</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e6</span> <span class="o">=</span> <span class="n">Ge</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e7</span> <span class="o">=</span> <span class="n">Lt</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e8</span> <span class="o">=</span> <span class="n">Le</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s">     </span><span class="si">%s</span><span class="se">\n</span><span class="s">&quot;</span><span class="o">*</span><span class="mi">4</span> <span class="o">%</span> <span class="p">(</span><span class="n">e1</span><span class="p">,</span> <span class="n">e2</span><span class="p">,</span> <span class="n">e3</span><span class="p">,</span> <span class="n">e4</span><span class="p">,</span> <span class="n">e5</span><span class="p">,</span> <span class="n">e6</span><span class="p">,</span> <span class="n">e7</span><span class="p">,</span> <span class="n">e8</span><span class="p">))</span>
-<span class="go">1 &gt; x     1 &gt;= x</span>
-<span class="go">1 &lt; x     1 &lt;= x</span>
-<span class="go">1 &gt; x     1 &gt;= x</span>
-<span class="go">1 &lt; x     1 &lt;= x</span>
-</pre></div>
-</div>
-<p>The other gotcha is with chained inequalities.  Occasionally, one may be
-tempted to write statements like:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&lt;</span> <span class="n">y</span> <span class="o">&lt;</span> <span class="n">z</span>  <span class="c"># silent error!  Where did ``x`` go?</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> 
-<span class="go">y &lt; z</span>
-</pre></div>
-</div>
-<p>Due to an implementation detail or decision of Python <a class="reference internal" href="#r19">[R19]</a>, there is no way
-for SymPy to reliably create that as a chained inequality.  To create a
-chained inequality, the only method currently available is to make use of
-And:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">And</span><span class="p">(</span><span class="n">x</span> <span class="o">&lt;</span> <span class="n">y</span><span class="p">,</span> <span class="n">y</span> <span class="o">&lt;</span> <span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span> <span class="n">e</span> <span class="p">)</span>
-<span class="go">And</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span>
-<span class="go">And(x &lt; y, y &lt; z)</span>
-</pre></div>
-</div>
-<p>Note that this is different than chaining an equality directly via use of
-parenthesis (this is currently an open bug in SymPy <a class="reference internal" href="#r20">[R20]</a>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">&lt;</span> <span class="n">y</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span> <span class="n">e</span> <span class="p">)</span>
-<span class="go">&lt;class &#39;sympy.core.relational.StrictLessThan&#39;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span>
-<span class="go">(x &lt; y) &lt; z</span>
-</pre></div>
-</div>
-<p>Any code that explicitly relies on this latter functionality will not be
-robust as this behaviour is completely wrong and will be corrected at some
-point.  For the time being (circa Jan 2012), use And to create chained
-inequalities.</p>
-<table class="docutils citation" frame="void" id="r19" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id18">[R19]</a></td><td><p class="first">This implementation detail is that Python provides no reliable
-method to determine that a chained inequality is being built.  Chained
-comparison operators are evaluated pairwise, using &#8220;and&#8221; logic (see
-<a class="reference external" href="http://docs.python.org/reference/expressions.html#notin">http://docs.python.org/reference/expressions.html#notin</a>).  This is done
-in an efficient way, so that each object being compared is only
-evaluated once and the comparison can short-circuit.  For example, <tt class="docutils literal"><span class="pre">1</span>
-<span class="pre">&gt;</span> <span class="pre">2</span> <span class="pre">&gt;</span> <span class="pre">3</span></tt> is evaluated by Python as <tt class="docutils literal"><span class="pre">(1</span> <span class="pre">&gt;</span> <span class="pre">2)</span> <span class="pre">and</span> <span class="pre">(2</span> <span class="pre">&gt;</span> <span class="pre">3)</span></tt>.  The
-<tt class="docutils literal"><span class="pre">and</span></tt> operator coerces each side into a bool, returning the object
-itself when it short-circuits.  Currently, the bool of the &#8211;Than
-operators will give True or False arbitrarily.  Thus, if we were to
-compute <tt class="docutils literal"><span class="pre">x</span> <span class="pre">&gt;</span> <span class="pre">y</span> <span class="pre">&gt;</span> <span class="pre">z</span></tt>, with <tt class="docutils literal"><span class="pre">x</span></tt>, <tt class="docutils literal"><span class="pre">y</span></tt>, and <tt class="docutils literal"><span class="pre">z</span></tt> being Symbols,
-Python converts the statement (roughly) into these steps:</p>
-<blockquote>
-<div><ol class="arabic simple">
-<li>x &gt; y &gt; z</li>
-<li>(x &gt; y) and (y &gt; z)</li>
-<li>(GreaterThanObject) and (y &gt; z)</li>
-<li>(GreaterThanObject.__nonzero__()) and (y &gt; z)</li>
-<li>(True) and (y &gt; z)</li>
-<li>(y &gt; z)</li>
-<li>LessThanObject</li>
-</ol>
-</div></blockquote>
-<p>Because of the &#8220;and&#8221; added at step 2, the statement gets turned into a
-weak ternary statement.  If the first object evalutes __nonzero__ as
-True, then the second object, (y &gt; z) is returned.  If the first object
-evaluates __nonzero__ as False (step 5), then (x &gt; y) is returned.</p>
-<blockquote class="last">
-<div>In Python, there is no way to override the <tt class="docutils literal"><span class="pre">and</span></tt> operator, or to
-control how it short circuits, so it is impossible to make something
-like <tt class="docutils literal"><span class="pre">x</span> <span class="pre">&gt;</span> <span class="pre">y</span> <span class="pre">&gt;</span> <span class="pre">z</span></tt> work.  There is an open PEP to change this,
-<span class="target" id="index-3"></span><a class="pep reference external" href="http://www.python.org/dev/peps/pep-0335"><strong>PEP 335</strong></a>, but until that is implemented, this cannot be made to work.</div></blockquote>
-</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r20" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id19">[R20]</a></td><td><p class="first">For more information, see these two bug reports:</p>
-<p>&#8220;Separate boolean and symbolic relationals&#8221;
-<a class="reference external" href="http://code.google.com/p/sympy/issues/detail?id=1887">Issue 1887</a></p>
-<p class="last">&#8220;It right 0 &lt; x &lt; 1 ?&#8221;
-<a class="reference external" href="http://code.google.com/p/sympy/issues/detail?id=2960">Issue 2960</a></p>
-</td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>One generally does not instantiate these classes directly, but uses various
-convenience methods:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ge</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span> <span class="p">)</span>      <span class="c"># Ge is a convenience wrapper</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">e1</span><span class="p">)</span>
-<span class="go">x &gt;= 2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rels</span> <span class="o">=</span> <span class="n">Ge</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span> <span class="p">),</span> <span class="n">Gt</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span> <span class="p">),</span> <span class="n">Le</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span> <span class="p">),</span> <span class="n">Lt</span><span class="p">(</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span> <span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">rels</span><span class="p">)</span>
-<span class="go">x &gt;= 2</span>
-<span class="go">x &gt; 2</span>
-<span class="go">x &lt;= 2</span>
-<span class="go">x &lt; 2</span>
-</pre></div>
-</div>
-<p>Another option is to use the Python inequality operators (&gt;=, &gt;, &lt;=, &lt;)
-directly.  Their main advantage over the Ge, Gt, Le, and Lt counterparts, is
-that one can write a more &#8220;mathematical looking&#8221; statement rather than
-littering the math with oddball function calls.  However there are certain
-(minor) caveats of which to be aware (search for &#8216;gotcha&#8217;, below).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e2</span> <span class="o">=</span> <span class="n">x</span> <span class="o">&gt;=</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">e2</span><span class="p">)</span>
-<span class="go">x &gt;= 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&quot;e1: </span><span class="si">%s</span><span class="s">,    e2: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">e1</span><span class="p">,</span> <span class="n">e2</span><span class="p">))</span>
-<span class="go">e1: x &gt;= 2,    e2: x &gt;= 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">==</span> <span class="n">e2</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>However, it is also perfectly valid to instantiate a <tt class="docutils literal"><span class="pre">*Than</span></tt> class less
-succinctly and less conveniently:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rels</span> <span class="o">=</span> <span class="n">Rel</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&gt;=&#39;</span><span class="p">),</span> <span class="n">Relational</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&gt;=&#39;</span><span class="p">),</span> <span class="n">GreaterThan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">rels</span><span class="p">)</span>
-<span class="go">x &gt;= 1</span>
-<span class="go">x &gt;= 1</span>
-<span class="go">x &gt;= 1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rels</span> <span class="o">=</span> <span class="n">Rel</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&gt;&#39;</span><span class="p">),</span> <span class="n">Relational</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&gt;&#39;</span><span class="p">),</span> <span class="n">StrictGreaterThan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">rels</span><span class="p">)</span>
-<span class="go">x &gt; 1</span>
-<span class="go">x &gt; 1</span>
-<span class="go">x &gt; 1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rels</span> <span class="o">=</span> <span class="n">Rel</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&lt;=&#39;</span><span class="p">),</span> <span class="n">Relational</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&lt;=&#39;</span><span class="p">),</span> <span class="n">LessThan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">rels</span><span class="p">)</span>
-<span class="go">x &lt;= 1</span>
-<span class="go">x &lt;= 1</span>
-<span class="go">x &lt;= 1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rels</span> <span class="o">=</span> <span class="n">Rel</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&lt;&#39;</span><span class="p">),</span> <span class="n">Relational</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="s">&#39;&lt;&#39;</span><span class="p">),</span> <span class="n">StrictLessThan</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="se">\n</span><span class="si">%s</span><span class="s">&#39;</span> <span class="o">%</span> <span class="n">rels</span><span class="p">)</span>
-<span class="go">x &lt; 1</span>
-<span class="go">x &lt; 1</span>
-<span class="go">x &lt; 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.core.multidimensional">
-<span id="multidimensional"></span><h2>multidimensional<a class="headerlink" href="#module-sympy.core.multidimensional" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="vectorize">
-<h3>vectorize<a class="headerlink" href="#vectorize" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.multidimensional.vectorize">
-<em class="property">class </em><tt class="descclassname">sympy.core.multidimensional.</tt><tt class="descname">vectorize</tt><big>(</big><em>*mdargs</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/multidimensional.html#vectorize"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.multidimensional.vectorize" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generalizes a function taking scalars to accept multidimensional arguments.</p>
-<p>For example:</p>
-<div class="highlight-python"><pre>(1) @vectorize(0)
-    def sin(x):
-        ....
-
-    sin([1, x, y])
-    --&gt; [sin(1), sin(x), sin(y)]
-
-(2) @vectorize(0,1)
-    def diff(f(y), y)
-        ....
-
-    diff([f(x,y,z),g(x,y,z),h(x,y,z)], [x,y,z])
-    --&gt; [[d/dx f, d/dy f, d/dz f], [d/dx g, d/dy g, d/dz g], [d/dx h, d/dy h, d/dz h]]</pre>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.core.function">
-<span id="function"></span><h2>function<a class="headerlink" href="#module-sympy.core.function" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="lambda">
-<h3>Lambda<a class="headerlink" href="#lambda" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.function.Lambda">
-<em class="property">class </em><tt class="descclassname">sympy.core.function.</tt><tt class="descname">Lambda</tt><a class="reference internal" href="../_modules/sympy/core/function.html#Lambda"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.Lambda" title="Permalink to this definition">¶</a></dt>
-<dd><p>Lambda(x, expr) represents a lambda function similar to Python&#8217;s
-&#8216;lambda x: expr&#8217;. A function of several variables is written as
-Lambda((x, y, ...), expr).</p>
-<p>A simple example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Lambda</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Lambda</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">16</span>
-</pre></div>
-</div>
-<p>For multivariate functions, use:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">t</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f2</span> <span class="o">=</span> <span class="n">Lambda</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">t</span><span class="p">),</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="n">z</span> <span class="o">+</span> <span class="n">t</span><span class="o">**</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f2</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">73</span>
-</pre></div>
-</div>
-<p>A handy shortcut for lots of arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Lambda</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="o">*</span><span class="n">p</span><span class="p">)</span>
-<span class="go">x + y*z</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.core.function.Lambda.expr">
-<tt class="descname">expr</tt><a class="reference internal" href="../_modules/sympy/core/function.html#Lambda.expr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.Lambda.expr" title="Permalink to this definition">¶</a></dt>
-<dd><p>The return value of the function</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.function.Lambda.is_identity">
-<tt class="descname">is_identity</tt><a class="reference internal" href="../_modules/sympy/core/function.html#Lambda.is_identity"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.Lambda.is_identity" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return <tt class="docutils literal"><span class="pre">True</span></tt> if this <tt class="docutils literal"><span class="pre">Lambda</span></tt> is an identity function.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.function.Lambda.nargs">
-<tt class="descname">nargs</tt><a class="reference internal" href="../_modules/sympy/core/function.html#Lambda.nargs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.Lambda.nargs" title="Permalink to this definition">¶</a></dt>
-<dd><p>The number of arguments that this function takes</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.function.Lambda.variables">
-<tt class="descname">variables</tt><a class="reference internal" href="../_modules/sympy/core/function.html#Lambda.variables"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.Lambda.variables" title="Permalink to this definition">¶</a></dt>
-<dd><p>The variables used in the internal representation of the function</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="wildfunction">
-<h3>WildFunction<a class="headerlink" href="#wildfunction" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.function.WildFunction">
-<em class="property">class </em><tt class="descclassname">sympy.core.function.</tt><tt class="descname">WildFunction</tt><a class="reference internal" href="../_modules/sympy/core/function.html#WildFunction"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.WildFunction" title="Permalink to this definition">¶</a></dt>
-<dd><p>A WildFunction function matches any function (with its arguments).</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Wild</span><span class="p">,</span> <span class="n">WildFunction</span><span class="p">,</span> <span class="n">Function</span><span class="p">,</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">WildFunction</span><span class="p">(</span><span class="s">&#39;F&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">F</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">F</span><span class="p">)</span>
-<span class="go">{F_: F_}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">F</span><span class="p">)</span>
-<span class="go">{F_: f(x)}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">F</span><span class="p">)</span>
-<span class="go">{F_: cos(x)}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">F</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>To match functions with more than 1 arguments, set <tt class="docutils literal"><span class="pre">nargs</span></tt> to the
-desired value:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="o">.</span><span class="n">nargs</span> <span class="o">=</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">F</span><span class="p">)</span>
-<span class="go">{F_: f(x, y)}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="derivative">
-<h3>Derivative<a class="headerlink" href="#derivative" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.function.Derivative">
-<em class="property">class </em><tt class="descclassname">sympy.core.function.</tt><tt class="descname">Derivative</tt><a class="reference internal" href="../_modules/sympy/core/function.html#Derivative"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.Derivative" title="Permalink to this definition">¶</a></dt>
-<dd><p>Carries out differentiation of the given expression with respect to symbols.</p>
-<p>expr must define ._eval_derivative(symbol) method that returns
-the differentiation result. This function only needs to consider the
-non-trivial case where expr contains symbol and it should call the diff()
-method internally (not _eval_derivative); Derivative should be the only
-one to call _eval_derivative.</p>
-<p>Ordering of variables:</p>
-<p>If evaluate is set to True and the expression can not be evaluated, the
-list of differentiation symbols will be sorted, that is, the expression is
-assumed to have continuous derivatives up to the order asked. This sorting
-assumes that derivatives wrt Symbols commute, derivatives wrt non-Symbols
-commute, but Symbol and non-Symbol derivatives don&#8217;t commute with each
-other.</p>
-<p>Derivative wrt non-Symbols:</p>
-<p>This class also allows derivatives wrt non-Symbols that have _diff_wrt
-set to True, such as Function and Derivative. When a derivative wrt a non-
-Symbol is attempted, the non-Symbol is temporarily converted to a Symbol
-while the differentiation is performed.</p>
-<p>Note that this may seem strange, that Derivative allows things like
-f(g(x)).diff(g(x)), or even f(cos(x)).diff(cos(x)).  The motivation for
-allowing this syntax is to make it easier to work with variational calculus
-(i.e., the Euler-Lagrange method).  The best way to understand this is that
-the action of derivative with respect to a non-Symbol is defined by the
-above description:  the object is substituted for a Symbol and the
-derivative is taken with respect to that.  This action is only allowed for
-objects for which this can be done unambiguously, for example Function and
-Derivative objects.  Note that this leads to what may appear to be
-mathematically inconsistent results.  For example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>This appears wrong because in fact 2*cos(x) and 2*sqrt(1 - sin(x)**2) are
-identically equal.  However this is the wrong way to think of this.  Think
-of it instead as if we have something like this:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">c</span><span class="p">,</span> <span class="n">s</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">F</span><span class="p">(</span><span class="n">u</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">u</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">G</span><span class="p">(</span><span class="n">u</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">u</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">2*cos(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">2*sqrt(-sin(x)**2 + 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="p">(</span><span class="n">c</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="p">(</span><span class="n">c</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="p">(</span><span class="n">s</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>Here, the Symbols c and s act just like the functions cos(x) and sin(x),
-respectively. Think of 2*cos(x) as f(c).subs(c, cos(x)) (or f(c) <em>at</em>
-c = cos(x)) and 2*sqrt(1 - sin(x)**2) as g(s).subs(s, sin(x)) (or g(s) <em>at</em>
-s = sin(x)), where f(u) == 2*u and g(u) == 2*sqrt(1 - u**2).  Here, we
-define the function first and evaluate it at the function, but we can
-actually unambiguously do this in reverse in SymPy, because
-expr.subs(Function, Symbol) is well-defined:  just structurally replace the
-function everywhere it appears in the expression.</p>
-<p>This is actually the same notational convenience used in the Euler-Lagrange
-method when one says F(t, f(t), f&#8217;(t)).diff(f(t)).  What is actually meant
-is that the expression in question is represented by some F(t, u, v) at
-u = f(t) and v = f&#8217;(t), and F(t, f(t), f&#8217;(t)).diff(f(t)) simply means
-F(t, u, v).diff(u) at u = f(t).</p>
-<p>We do not allow to take derivative with respect to expressions where this
-is not so well defined.  For example, we do not allow expr.diff(x*y)
-because there are multiple ways of structurally defining where x*y appears
-in an expression, some of which may surprise the reader (for example, a
-very strict definition would have that (x*y*z).diff(x*y) == 0).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">ValueError: Can&#39;t differentiate wrt the variable</span>: <span class="n">x*y, 1</span>
-</pre></div>
-</div>
-<p>Note that this definition also fits in nicely with the definition of the
-chain rule.  Note how the chain rule in SymPy is defined using unevaluated
-Subs objects:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">Function</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;f g&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Function</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">g</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">2*Derivative(g(x), x)*Subs(Derivative(f(_xi_1), _xi_1),</span>
-<span class="go">                                      (_xi_1,), (2*g(x),))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">g</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">Derivative(g(x), x)*Subs(Derivative(f(_xi_1), _xi_1),</span>
-<span class="go">                                    (_xi_1,), (g(x),))</span>
-</pre></div>
-</div>
-<p>Finally, note that, to be consistent with variational calculus, and to
-ensure that the definition of substituting a Function for a Symbol in an
-expression is well-defined, derivatives of functions are assumed to not be
-related to the function.  In other words, we have:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">diff</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>The same is actually true for derivatives of different orders:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>Note, any class can allow derivatives to be taken with respect to itself.
-See the docstring of Expr._diff_wrt.</p>
-<p class="rubric">Examples</p>
-<p>Some basic examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Derivative</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Function</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Derivative</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">2*x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Derivative</span><span class="p">(</span><span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">Derivative(f(x, y), x, y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">Derivative(f(x), x, x, x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">Derivative(f(x, y), x, y)</span>
-</pre></div>
-</div>
-<p>Now some derivatives wrt functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">2*f(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">g</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="n">x</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">Derivative(g(x), x)*Subs(Derivative(f(_xi_1), _xi_1),</span>
-<span class="go">                                    (_xi_1,), (g(x),))</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.core.function.Derivative.doit_numerically">
-<tt class="descname">doit_numerically</tt><big>(</big><em>a</em>, <em>b</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/function.html#Derivative.doit_numerically"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.Derivative.doit_numerically" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluate the derivative at z numerically.</p>
-<p>When we can represent derivatives at a point, this should be folded
-into the normal evalf. For now, we need a special method.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="diff">
-<h3>diff<a class="headerlink" href="#diff" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.function.diff">
-<tt class="descclassname">sympy.core.function.</tt><tt class="descname">diff</tt><big>(</big><em>f</em>, <em>*symbols</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/function.html#diff"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.diff" title="Permalink to this definition">¶</a></dt>
-<dd><p>Differentiate f with respect to symbols.</p>
-<p>This is just a wrapper to unify .diff() and the Derivative class; its
-interface is similar to that of integrate().  You can use the same
-shortcuts for multiple variables as with Derivative.  For example,
-diff(f(x), x, x, x) and diff(f(x), x, 3) both return the third derivative
-of f(x).</p>
-<p>You can pass evaluate=False to get an unevaluated Derivative class.  Note
-that if there are 0 symbols (such as diff(f(x), x, 0), then the result will
-be the function (the zeroth derivative), even if evaluate=False.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.core.function.Derivative" title="sympy.core.function.Derivative"><tt class="xref py py-obj docutils literal"><span class="pre">Derivative</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<p><a class="reference external" href="http://reference.wolfram.com/legacy/v5_2/Built-inFunctions/AlgebraicComputation/Calculus/D.html">http://reference.wolfram.com/legacy/v5_2/Built-inFunctions/AlgebraicComputation/Calculus/D.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">Function</span><span class="p">,</span> <span class="n">diff</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">cos(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">Derivative(f(x), x, x, x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">Derivative(f(x), x, x, x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">y</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">sin(x)*cos(y)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">))</span>
-<span class="go">&lt;class &#39;sympy.core.function.Derivative&#39;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">))</span>
-<span class="go">sin</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">cos(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">))</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">ValueError</span>: <span class="n">specify differentiation variables to differentiate sin(x*y)</span>
-</pre></div>
-</div>
-<p>Note that <tt class="docutils literal"><span class="pre">diff(sin(x))</span></tt> syntax is meant only for convenience
-in interactive sessions and should be avoided in library code.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="functionclass">
-<h3>FunctionClass<a class="headerlink" href="#functionclass" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.function.FunctionClass">
-<em class="property">class </em><tt class="descclassname">sympy.core.function.</tt><tt class="descname">FunctionClass</tt><big>(</big><em>*args</em>, <em>**kws</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/function.html#FunctionClass"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.FunctionClass" title="Permalink to this definition">¶</a></dt>
-<dd><p>Base class for function classes. FunctionClass is a subclass of type.</p>
-<p>Use Function(&#8216;&lt;function name&gt;&#8217; [ , signature ]) to create
-undefined function classes.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="id22">
-<h3>Function<a class="headerlink" href="#id22" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.function.Function">
-<em class="property">class </em><tt class="descclassname">sympy.core.function.</tt><tt class="descname">Function</tt><a class="reference internal" href="../_modules/sympy/core/function.html#Function"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.Function" title="Permalink to this definition">¶</a></dt>
-<dd><p>Base class for applied mathematical functions.</p>
-<p>It also serves as a constructor for undefined function classes.</p>
-<p class="rubric">Examples</p>
-<p>First example shows how to use Function as a constructor for undefined
-function classes:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">)(</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span>
-<span class="go">f</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">f(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span>
-<span class="go">g(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">Derivative(f(x), x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">Derivative(g(x), x)</span>
-</pre></div>
-</div>
-<p>In the following example Function is used as a base class for
-<tt class="docutils literal"><span class="pre">my_func</span></tt> that represents a mathematical function <em>my_func</em>. Suppose
-that it is well known, that <em>my_func(0)</em> is <em>1</em> and <em>my_func</em> at infinity
-goes to <em>0</em>, so we want those two simplifications to occur automatically.
-Suppose also that <em>my_func(x)</em> is real exactly when <em>x</em> is real. Here is
-an implementation that honours those requirements:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">my_func</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-<span class="gp">...</span>
-<span class="gp">... </span>    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-<span class="gp">...</span>
-<span class="gp">... </span>    <span class="nd">@classmethod</span>
-<span class="gp">... </span>    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-<span class="gp">... </span>        <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-<span class="gp">... </span>            <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-<span class="gp">... </span>                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-<span class="gp">... </span>            <span class="k">elif</span> <span class="n">x</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-<span class="gp">... </span>                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-<span class="gp">...</span>
-<span class="gp">... </span>    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-<span class="gp">... </span>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">my_func</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="o">+</span> <span class="n">sin</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">my_func</span><span class="p">(</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">my_func</span><span class="p">(</span><span class="mf">3.54</span><span class="p">)</span><span class="o">.</span><span class="n">n</span><span class="p">()</span> <span class="c"># Not yet implemented for my_func.</span>
-<span class="go">my_func(3.54)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">my_func</span><span class="p">(</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">is_real</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>In order for <tt class="docutils literal"><span class="pre">my_func</span></tt> to become useful, several other methods would
-need to be implemented. See source code of some of the already
-implemented functions for more complete examples.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.core.function.Function.as_base_exp">
-<tt class="descname">as_base_exp</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/function.html#Function.as_base_exp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.Function.as_base_exp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the method as the 2-tuple (base, exponent).</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.function.Function.fdiff">
-<tt class="descname">fdiff</tt><big>(</big><em>self</em>, <em>argindex=1</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/function.html#Function.fdiff"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.Function.fdiff" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the first derivative of the function.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.function.Function.is_commutative">
-<tt class="descname">is_commutative</tt><a class="reference internal" href="../_modules/sympy/core/function.html#Function.is_commutative"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.Function.is_commutative" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns whether the functon is commutative.</p>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.core.function.Function.taylor_term">
-<em class="property">classmethod </em><tt class="descname">taylor_term</tt><big>(</big><em>n</em>, <em>x</em>, <em>*previous_terms</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/function.html#Function.taylor_term"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.Function.taylor_term" title="Permalink to this definition">¶</a></dt>
-<dd><p>General method for the taylor term.</p>
-<p>This method is slow, because it differentiates n-times. Subclasses can
-redefine it to make it faster by using the &#8220;previous_terms&#8221;.</p>
-</dd></dl>
-
-</dd></dl>
-
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p>Not all functions are the same</p>
-<p>SymPy defines many functions (like <tt class="docutils literal"><span class="pre">cos</span></tt> and <tt class="docutils literal"><span class="pre">factorial</span></tt>). It also
-allows the user to create generic functions which act as argument
-holders. Such functions are created just like symbols:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">f(2) + f(x)</span>
-</pre></div>
-</div>
-<p>If you want to see which functions appear in an expression you can use
-the atoms method:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Function</span><span class="p">)</span>
-<span class="go">set([f(x), cos(x)])</span>
-</pre></div>
-</div>
-<p>If you just want the function you defined, not SymPy functions, the
-thing to search for is AppliedUndef:</p>
-<div class="last highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">AppliedUndef</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">AppliedUndef</span><span class="p">)</span>
-<span class="go">set([f(x)])</span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="section" id="subs">
-<h3>Subs<a class="headerlink" href="#subs" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.function.Subs">
-<em class="property">class </em><tt class="descclassname">sympy.core.function.</tt><tt class="descname">Subs</tt><a class="reference internal" href="../_modules/sympy/core/function.html#Subs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.Subs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents unevaluated substitutions of an expression.</p>
-<p><tt class="docutils literal"><span class="pre">Subs(expr,</span> <span class="pre">x,</span> <span class="pre">x0)</span></tt> receives 3 arguments: an expression, a variable or
-list of distinct variables and a point or list of evaluation points
-corresponding to those variables.</p>
-<p><tt class="docutils literal"><span class="pre">Subs</span></tt> objects are generally useful to represent unevaluated derivatives
-calculated at a point.</p>
-<p>The variables may be expressions, but they are subjected to the limitations
-of subs(), so it is usually a good practice to use only symbols for
-variables, since in that case there can be no ambiguity.</p>
-<p>There&#8217;s no automatic expansion - use the method .doit() to effect all
-possible substitutions of the object and also of objects inside the
-expression.</p>
-<p>When evaluating derivatives at a point that is not a symbol, a Subs object
-is returned. One is also able to calculate derivatives of Subs objects - in
-this case the expression is always expanded (for the unevaluated form, use
-Derivative()).</p>
-<p>A simple example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Subs</span><span class="p">,</span> <span class="n">Function</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">Subs</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">Subs(Derivative(f(x), x), (x,), (0,))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">sin</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-<span class="go">cos(y)</span>
-</pre></div>
-</div>
-<p>An example with several variables:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Subs</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">y</span><span class="p">)</span> <span class="o">+</span> <span class="n">z</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">Subs(z + f(x)*sin(y), (x, y), (0, 1))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-<span class="go">z + f(0)*sin(1)</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.core.function.Subs.expr">
-<tt class="descname">expr</tt><a class="reference internal" href="../_modules/sympy/core/function.html#Subs.expr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.Subs.expr" title="Permalink to this definition">¶</a></dt>
-<dd><p>The expression on which the substitution operates</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.function.Subs.point">
-<tt class="descname">point</tt><a class="reference internal" href="../_modules/sympy/core/function.html#Subs.point"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.Subs.point" title="Permalink to this definition">¶</a></dt>
-<dd><p>The values for which the variables are to be substituted</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.function.Subs.variables">
-<tt class="descname">variables</tt><a class="reference internal" href="../_modules/sympy/core/function.html#Subs.variables"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.Subs.variables" title="Permalink to this definition">¶</a></dt>
-<dd><p>The variables to be evaluated</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="expand">
-<h3>expand<a class="headerlink" href="#expand" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.function.expand">
-<tt class="descclassname">sympy.core.function.</tt><tt class="descname">expand</tt><big>(</big><em>e</em>, <em>deep=True</em>, <em>modulus=None</em>, <em>power_base=True</em>, <em>power_exp=True</em>, <em>mul=True</em>, <em>log=True</em>, <em>multinomial=True</em>, <em>basic=True</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/function.html#expand"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.expand" title="Permalink to this definition">¶</a></dt>
-<dd><p>Expand an expression using methods given as hints.</p>
-<p>Hints evaluated unless explicitly set to False are:  <tt class="docutils literal"><span class="pre">basic</span></tt>, <tt class="docutils literal"><span class="pre">log</span></tt>,
-<tt class="docutils literal"><span class="pre">multinomial</span></tt>, <tt class="docutils literal"><span class="pre">mul</span></tt>, <tt class="docutils literal"><span class="pre">power_base</span></tt>, and <tt class="docutils literal"><span class="pre">power_exp</span></tt> The following
-hints are supported but not applied unless set to True:  <tt class="docutils literal"><span class="pre">complex</span></tt>,
-<tt class="docutils literal"><span class="pre">func</span></tt>, and <tt class="docutils literal"><span class="pre">trig</span></tt>.  In addition, the following meta-hints are
-supported by some or all of the other hints:  <tt class="docutils literal"><span class="pre">frac</span></tt>, <tt class="docutils literal"><span class="pre">numer</span></tt>,
-<tt class="docutils literal"><span class="pre">denom</span></tt>, <tt class="docutils literal"><span class="pre">modulus</span></tt>, and <tt class="docutils literal"><span class="pre">force</span></tt>.  <tt class="docutils literal"><span class="pre">deep</span></tt> is supported by all
-hints.  Additionally, subclasses of Expr may define their own hints or
-meta-hints.</p>
-<p>The <tt class="docutils literal"><span class="pre">basic</span></tt> hint is used for any special rewriting of an object that
-should be done automatically (along with the other hints like <tt class="docutils literal"><span class="pre">mul</span></tt>)
-when expand is called. This is a catch-all hint to handle any sort of
-expansion that may not be described by the existing hint names. To use
-this hint an object should override the <tt class="docutils literal"><span class="pre">_eval_expand_basic</span></tt> method.
-Objects may also define their own expand methods, which are not run by
-default.  See the API section below.</p>
-<p>If <tt class="docutils literal"><span class="pre">deep</span></tt> is set to <tt class="docutils literal"><span class="pre">True</span></tt> (the default), things like arguments of
-functions are recursively expanded.  Use <tt class="docutils literal"><span class="pre">deep=False</span></tt> to only expand on
-the top level.</p>
-<p>If the <tt class="docutils literal"><span class="pre">force</span></tt> hint is used, assumptions about variables will be ignored
-in making the expansion.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.core.function.expand_log" title="sympy.core.function.expand_log"><tt class="xref py py-obj docutils literal"><span class="pre">expand_log</span></tt></a>, <a class="reference internal" href="#sympy.core.function.expand_mul" title="sympy.core.function.expand_mul"><tt class="xref py py-obj docutils literal"><span class="pre">expand_mul</span></tt></a>, <a class="reference internal" href="#sympy.core.function.expand_multinomial" title="sympy.core.function.expand_multinomial"><tt class="xref py py-obj docutils literal"><span class="pre">expand_multinomial</span></tt></a>, <a class="reference internal" href="#sympy.core.function.expand_complex" title="sympy.core.function.expand_complex"><tt class="xref py py-obj docutils literal"><span class="pre">expand_complex</span></tt></a>, <a class="reference internal" href="#sympy.core.function.expand_trig" title="sympy.core.function.expand_trig"><tt class="xref py py-obj docutils literal"><span class="pre">expand_trig</span></tt></a>, <a class="reference internal" href="#sympy.core.function.expand_power_base" title="sympy.core.function.expand_power_base"><tt class="xref py py-obj docutils literal"><span class="pre">expand_power_base</span></tt></a>, <a class="reference internal" href="#sympy.core.function.expand_power_exp" title="sympy.core.function.expand_power_exp"><tt class="xref py py-obj docutils literal"><span class="pre">expand_power_exp</span></tt></a>, <a class="reference internal" href="#sympy.core.function.expand_func" title="sympy.core.function.expand_func"><tt class="xref py py-obj docutils literal"><span class="pre">expand_func</span></tt></a>, <tt class="xref py py-obj docutils literal"><span class="pre">hyperexpand</span></tt></p>
-</div>
-<p class="rubric">Notes</p>
-<ul>
-<li><p class="first">You can shut off unwanted methods:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go">x*exp(x)*exp(y) + y*exp(x)*exp(y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">power_exp</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">x*exp(x + y) + y*exp(x + y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">mul</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">(x + y)*exp(x)*exp(y)</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">Use deep=False to only expand on the top level:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go">exp(x)*exp(exp(x)*exp(y))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">exp(x)*exp(exp(x + y))</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">Hints are applied in an arbitrary, but consistent order (in the current
-implementation, they are applied in alphabetical order, except
-multinomial comes before mul, but this may change).  Because of this,
-some hints may prevent expansion by other hints if they are applied
-first. For example, <tt class="docutils literal"><span class="pre">mul</span></tt> may distribute multiplications and prevent
-<tt class="docutils literal"><span class="pre">log</span></tt> and <tt class="docutils literal"><span class="pre">power_base</span></tt> from expanding them. Also, if <tt class="docutils literal"><span class="pre">mul</span></tt> is
-applied before <tt class="docutils literal"><span class="pre">multinomial`,</span> <span class="pre">the</span> <span class="pre">expression</span> <span class="pre">might</span> <span class="pre">not</span> <span class="pre">be</span> <span class="pre">fully</span>
-<span class="pre">distributed.</span> <span class="pre">The</span> <span class="pre">solution</span> <span class="pre">is</span> <span class="pre">to</span> <span class="pre">use</span> <span class="pre">the</span> <span class="pre">various</span> <span class="pre">``expand_hint</span></tt> helper
-functions or to use <tt class="docutils literal"><span class="pre">hint=False</span></tt> to this function to finely control
-which hints are applied. Here are some examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expand_log</span><span class="p">,</span> <span class="n">expand</span><span class="p">,</span> <span class="n">expand_mul</span><span class="p">,</span> <span class="n">expand_power_base</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y,z&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="n">z</span><span class="p">)))</span>
-<span class="go">log(x) + log(y + z)</span>
-</pre></div>
-</div>
-<p>Here, we see that <tt class="docutils literal"><span class="pre">log</span></tt> was applied before <tt class="docutils literal"><span class="pre">mul</span></tt>.  To get the log
-expanded form, either of the following will work:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expand_log</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="n">z</span><span class="p">)))</span>
-<span class="go">log(x) + log(y + z)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="n">z</span><span class="p">)),</span> <span class="n">mul</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">log(x) + log(y + z)</span>
-</pre></div>
-</div>
-<p>A similar thing can happen with the <tt class="docutils literal"><span class="pre">power_base</span></tt> hint:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expand</span><span class="p">((</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="n">z</span><span class="p">))</span><span class="o">**</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(x*y + x*z)**x</span>
-</pre></div>
-</div>
-<p>To get the <tt class="docutils literal"><span class="pre">power_base</span></tt> expanded form, either of the following will
-work:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expand</span><span class="p">((</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="n">z</span><span class="p">))</span><span class="o">**</span><span class="n">x</span><span class="p">,</span> <span class="n">mul</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">x**x*(y + z)**x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_power_base</span><span class="p">((</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="n">z</span><span class="p">))</span><span class="o">**</span><span class="n">x</span><span class="p">)</span>
-<span class="go">x**x*(y + z)**x</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">y</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">y + y**2/x</span>
-</pre></div>
-</div>
-<p>The parts of a rational expression can be targeted:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expand</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">y</span><span class="o">/</span><span class="n">x</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">frac</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(x*y + y**2)/(x**2 + x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">y</span><span class="o">/</span><span class="n">x</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">numer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(x*y + y**2)/(x*(x + 1))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">y</span><span class="o">/</span><span class="n">x</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">denom</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">y*(x + y)/(x**2 + x)</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">The <tt class="docutils literal"><span class="pre">modulus</span></tt> meta-hint can be used to reduce the coefficients of an
-expression post-expansion:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expand</span><span class="p">((</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">9*x**2 + 6*x + 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand</span><span class="p">((</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">modulus</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">4*x**2 + x + 1</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">Either <tt class="docutils literal"><span class="pre">expand()</span></tt> the function or <tt class="docutils literal"><span class="pre">.expand()</span></tt> the method can be
-used.  Both are equivalent:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expand</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x**2 + 2*x + 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go">x**2 + 2*x + 1</span>
-</pre></div>
-</div>
-</li>
-</ul>
-<p class="rubric">Hints</p>
-<p>These hints are run by default</p>
-<p class="rubric">Mul</p>
-<p>Distributes multiplication over addition:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">cos</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">y</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">z</span><span class="p">))</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">mul</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">x*y + y*z</span>
-</pre></div>
-</div>
-<p class="rubric">Multinomial</p>
-<p>Expand (x + y + ...)**n where n is a positive integer.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">multinomial</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">x**2 + 2*x*y + 2*x*z + y**2 + 2*y*z + z**2</span>
-</pre></div>
-</div>
-<p class="rubric">Power_exp</p>
-<p>Expand addition in exponents into multiplied bases.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">power_exp</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">exp(x)*exp(y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">power_exp</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">2**x*2**y</span>
-</pre></div>
-</div>
-<p class="rubric">Power_base</p>
-<p>Split powers of multiplied bases.</p>
-<p>This only happens by default if assumptions allow, or if the
-<tt class="docutils literal"><span class="pre">force</span></tt> meta-hint is used:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">power_base</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(x*y)**z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">power_base</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">x**z*y**z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">power_base</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">2**z*y**z</span>
-</pre></div>
-</div>
-<p>Note that in some cases where this expansion always holds, SymPy performs
-it automatically:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">x**2*y**2</span>
-</pre></div>
-</div>
-<p class="rubric">Log</p>
-<p>Pull out power of an argument as a coefficient and split logs products
-into sums of logs.</p>
-<p>Note that these only work if the arguments of the log function have the
-proper assumptions&#8211;the arguments must be positive and the exponents must
-be real&#8211;or else the <tt class="docutils literal"><span class="pre">force</span></tt> hint must be True:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">log</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">log</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">log(x**2*y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">log</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">2*log(x) + log(y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">log</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">2*log(x) + log(y)</span>
-</pre></div>
-</div>
-<p class="rubric">Basic</p>
-<p>This hint is intended primarily as a way for custom subclasses to enable
-expansion by default.</p>
-<p>These hints are not run by default:</p>
-<p class="rubric">Complex</p>
-<p>Split an expression into real and imaginary parts.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">re(x) + re(y) + I*im(x) + I*im(y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">-I*sin(re(x))*sinh(im(x)) + cos(re(x))*cosh(im(x))</span>
-</pre></div>
-</div>
-<p>Note that this is just a wrapper around <tt class="docutils literal"><span class="pre">as_real_imag()</span></tt>.  Most objects
-that wish to redefine <tt class="docutils literal"><span class="pre">_eval_expand_complex()</span></tt> should consider
-redefining <tt class="docutils literal"><span class="pre">as_real_imag()</span></tt> instead.</p>
-<p class="rubric">Func</p>
-<p>Expand other functions.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">gamma</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">func</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">x*gamma(x)</span>
-</pre></div>
-</div>
-<p class="rubric">Trig</p>
-<p>Do trigonometric expansions.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">-sin(x)*sin(y) + cos(x)*cos(y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">2*sin(x)*cos(x)</span>
-</pre></div>
-</div>
-<p>Note that the forms of <tt class="docutils literal"><span class="pre">sin(n*x)</span></tt> and <tt class="docutils literal"><span class="pre">cos(n*x)</span></tt> in terms of <tt class="docutils literal"><span class="pre">sin(x)</span></tt>
-and <tt class="docutils literal"><span class="pre">cos(x)</span></tt> are not unique, due to the identity <span class="math">\(\sin^2(x) + \cos^2(x)
-= 1\)</span>.  The current implementation uses the form obtained from Chebyshev
-polynomials, but this may change.  See <a class="reference external" href="http://mathworld.wolfram.com/Multiple-AngleFormulas.html">this MathWorld article</a> for more
-information.</p>
-<p class="rubric">Api</p>
-<p>Objects can define their own expand hints by defining
-<tt class="docutils literal"><span class="pre">_eval_expand_hint()</span></tt>.  The function should take the form:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="k">def</span> <span class="nf">_eval_expand_hint</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-    <span class="c"># Only apply the method to the top-level expression</span>
-    <span class="o">...</span>
-</pre></div>
-</div>
-<p>See also the example below.  Objects should define <tt class="docutils literal"><span class="pre">_eval_expand_hint()</span></tt>
-methods only if <tt class="docutils literal"><span class="pre">hint</span></tt> applies to that specific object.  The generic
-<tt class="docutils literal"><span class="pre">_eval_expand_hint()</span></tt> method defined in Expr will handle the no-op case.</p>
-<p>Each hint should be responsible for expanding that hint only.
-Furthermore, the expansion should be applied to the top-level expression
-only.  <tt class="docutils literal"><span class="pre">expand()</span></tt> takes care of the recursion that happens when
-<tt class="docutils literal"><span class="pre">deep=True</span></tt>.</p>
-<p>You should only call <tt class="docutils literal"><span class="pre">_eval_expand_hint()</span></tt> methods directly if you are
-100% sure that the object has the method, as otherwise you are liable to
-get unexpected <tt class="docutils literal"><span class="pre">AttributeError``s.</span>&nbsp; <span class="pre">Note,</span> <span class="pre">again,</span> <span class="pre">that</span> <span class="pre">you</span> <span class="pre">do</span> <span class="pre">not</span> <span class="pre">need</span> <span class="pre">to</span>
-<span class="pre">recursively</span> <span class="pre">apply</span> <span class="pre">the</span> <span class="pre">hint</span> <span class="pre">to</span> <span class="pre">args</span> <span class="pre">of</span> <span class="pre">your</span> <span class="pre">object:</span> <span class="pre">this</span> <span class="pre">is</span> <span class="pre">handled</span>
-<span class="pre">automatically</span> <span class="pre">by</span> <span class="pre">``expand()</span></tt>.  <tt class="docutils literal"><span class="pre">_eval_expand_hint()</span></tt> should
-generally not be used at all outside of an <tt class="docutils literal"><span class="pre">_eval_expand_hint()</span></tt> method.
-If you want to apply a specific expansion from within another method, use
-the public <tt class="docutils literal"><span class="pre">expand()</span></tt> function, method, or <tt class="docutils literal"><span class="pre">expand_hint()</span></tt> functions.</p>
-<p>In order for expand to work, objects must be rebuildable by their args,
-i.e., <tt class="docutils literal"><span class="pre">obj.func(*obj.args)</span> <span class="pre">==</span> <span class="pre">obj</span></tt> must hold.</p>
-<p>Expand methods are passed <tt class="docutils literal"><span class="pre">**hints</span></tt> so that expand hints may use
-&#8216;metahints&#8217;&#8211;hints that control how different expand methods are applied.
-For example, the <tt class="docutils literal"><span class="pre">force=True</span></tt> hint described above that causes
-<tt class="docutils literal"><span class="pre">expand(log=True)</span></tt> to ignore assumptions is such a metahint.  The
-<tt class="docutils literal"><span class="pre">deep</span></tt> meta-hint is handled exclusively by <tt class="docutils literal"><span class="pre">expand()</span></tt> and is not
-passed to <tt class="docutils literal"><span class="pre">_eval_expand_hint()</span></tt> methods.</p>
-<p>Note that expansion hints should generally be methods that perform some
-kind of &#8216;expansion&#8217;.  For hints that simply rewrite an expression, use the
-.rewrite() API.</p>
-<p class="rubric">Example</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Expr</span><span class="p">,</span> <span class="n">sympify</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">MyClass</span><span class="p">(</span><span class="n">Expr</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">def</span> <span class="nf">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">):</span>
-<span class="gp">... </span>        <span class="n">args</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">args</span><span class="p">)</span>
-<span class="gp">... </span>        <span class="k">return</span> <span class="n">Expr</span><span class="o">.</span><span class="n">__new__</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="o">*</span><span class="n">args</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="gp">... </span>    <span class="k">def</span> <span class="nf">_eval_expand_double</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">hints</span><span class="p">):</span>
-<span class="gp">... </span>        <span class="sd">&#39;&#39;&#39;</span>
-<span class="gp">... </span><span class="sd">        Doubles the args of MyClass.</span>
-<span class="gp">...</span><span class="sd"></span>
-<span class="gp">... </span><span class="sd">        If there more than four args, doubling is not performed,</span>
-<span class="gp">... </span><span class="sd">        unless force=True is also used (False by default).</span>
-<span class="gp">... </span><span class="sd">        &#39;&#39;&#39;</span>
-<span class="gp">... </span>        <span class="n">force</span> <span class="o">=</span> <span class="n">hints</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="s">&#39;force&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-<span class="gp">... </span>        <span class="k">if</span> <span class="ow">not</span> <span class="n">force</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">4</span><span class="p">:</span>
-<span class="gp">... </span>            <span class="k">return</span> <span class="bp">self</span>
-<span class="gp">... </span>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="o">*</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">args</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">MyClass</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">MyClass</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
-<span class="go">MyClass(1, 2, MyClass(3, 4))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">double</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">MyClass(1, 2, MyClass(3, 4, 3, 4), 1, 2, MyClass(3, 4, 3, 4))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">double</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">MyClass(1, 2, MyClass(3, 4), 1, 2, MyClass(3, 4))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">MyClass</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">double</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">MyClass(1, 2, 3, 4, 5)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">double</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">MyClass(1, 2, 3, 4, 5, 1, 2, 3, 4, 5)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="poleerror">
-<h3>PoleError<a class="headerlink" href="#poleerror" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.function.PoleError">
-<em class="property">class </em><tt class="descclassname">sympy.core.function.</tt><tt class="descname">PoleError</tt><a class="reference internal" href="../_modules/sympy/core/function.html#PoleError"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.PoleError" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="count-ops">
-<h3>count_ops<a class="headerlink" href="#count-ops" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.function.count_ops">
-<tt class="descclassname">sympy.core.function.</tt><tt class="descname">count_ops</tt><big>(</big><em>expr</em>, <em>visual=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/function.html#count_ops"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.count_ops" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a representation (integer or expression) of the operations in expr.</p>
-<p>If <tt class="docutils literal"><span class="pre">visual</span></tt> is <tt class="docutils literal"><span class="pre">False</span></tt> (default) then the sum of the coefficients of the
-visual expression will be returned.</p>
-<p>If <tt class="docutils literal"><span class="pre">visual</span></tt> is <tt class="docutils literal"><span class="pre">True</span></tt> then the number of each type of operation is shown
-with the core class types (or their virtual equivalent) multiplied by the
-number of times they occur.</p>
-<p>If expr is an iterable, the sum of the op counts of the
-items will be returned.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">count_ops</span>
-</pre></div>
-</div>
-<p>Although there isn&#8217;t a SUB object, minus signs are interpreted as
-either negations or subtractions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">count_ops</span><span class="p">(</span><span class="n">visual</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">SUB</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">count_ops</span><span class="p">(</span><span class="n">visual</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">NEG</span>
-</pre></div>
-</div>
-<p>Here, there are two Adds and a Pow:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">count_ops</span><span class="p">(</span><span class="n">visual</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">2*ADD + POW</span>
-</pre></div>
-</div>
-<p>In the following, an Add, Mul, Pow and two functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">count_ops</span><span class="p">(</span><span class="n">visual</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">ADD + MUL + POW + 2*SIN</span>
-</pre></div>
-</div>
-<p>for a total of 5:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">count_ops</span><span class="p">(</span><span class="n">visual</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">5</span>
-</pre></div>
-</div>
-<p>Note that &#8220;what you type&#8221; is not always what you get. The expression
-1/x/y is translated by sympy into 1/(x*y) so it gives a DIV and MUL rather
-than two DIVs:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="o">/</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">count_ops</span><span class="p">(</span><span class="n">visual</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">DIV + MUL</span>
-</pre></div>
-</div>
-<p>The visual option can be used to demonstrate the difference in
-operations for expressions in different forms. Here, the Horner
-representation is compared with the expanded form of a polynomial:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span><span class="o">=</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">3</span> <span class="o">+</span> <span class="n">x</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">count_ops</span><span class="p">(</span><span class="n">eq</span><span class="o">.</span><span class="n">expand</span><span class="p">(),</span> <span class="n">visual</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="o">-</span> <span class="n">count_ops</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">-MUL + 3*POW</span>
-</pre></div>
-</div>
-<p>The count_ops function also handles iterables:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">count_ops</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">True</span><span class="p">,</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">],</span> <span class="n">visual</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">count_ops</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">True</span><span class="p">,</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">],</span> <span class="n">visual</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">ADD + SIN</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">count_ops</span><span class="p">({</span><span class="n">x</span><span class="p">:</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">:</span> <span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">},</span> <span class="n">visual</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">2*ADD + SIN</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="expand-mul">
-<h3>expand_mul<a class="headerlink" href="#expand-mul" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.function.expand_mul">
-<tt class="descclassname">sympy.core.function.</tt><tt class="descname">expand_mul</tt><big>(</big><em>expr</em>, <em>deep=True</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/function.html#expand_mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.expand_mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wrapper around expand that only uses the mul hint.  See the expand
-docstring for more information.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">expand_mul</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">log</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_mul</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">x*exp(x + y)*log(x*y**2) + y*exp(x + y)*log(x*y**2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="expand-log">
-<h3>expand_log<a class="headerlink" href="#expand-log" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.function.expand_log">
-<tt class="descclassname">sympy.core.function.</tt><tt class="descname">expand_log</tt><big>(</big><em>expr</em>, <em>deep=True</em>, <em>force=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/function.html#expand_log"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.expand_log" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wrapper around expand that only uses the log hint.  See the expand
-docstring for more information.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">expand_log</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">log</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_log</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">(x + y)*(log(x) + 2*log(y))*exp(x + y)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="expand-func">
-<h3>expand_func<a class="headerlink" href="#expand-func" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.function.expand_func">
-<tt class="descclassname">sympy.core.function.</tt><tt class="descname">expand_func</tt><big>(</big><em>expr</em>, <em>deep=True</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/function.html#expand_func"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.expand_func" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wrapper around expand that only uses the func hint.  See the expand
-docstring for more information.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expand_func</span><span class="p">,</span> <span class="n">gamma</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_func</span><span class="p">(</span><span class="n">gamma</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">x*(x + 1)*gamma(x)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="expand-trig">
-<h3>expand_trig<a class="headerlink" href="#expand-trig" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.function.expand_trig">
-<tt class="descclassname">sympy.core.function.</tt><tt class="descname">expand_trig</tt><big>(</big><em>expr</em>, <em>deep=True</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/function.html#expand_trig"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.expand_trig" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wrapper around expand that only uses the trig hint.  See the expand
-docstring for more information.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expand_trig</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_trig</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="p">))</span>
-<span class="go">(x + y)*(sin(x)*cos(y) + sin(y)*cos(x))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="expand-complex">
-<h3>expand_complex<a class="headerlink" href="#expand-complex" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.function.expand_complex">
-<tt class="descclassname">sympy.core.function.</tt><tt class="descname">expand_complex</tt><big>(</big><em>expr</em>, <em>deep=True</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/function.html#expand_complex"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.expand_complex" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wrapper around expand that only uses the complex hint.  See the expand
-docstring for more information.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">Expr.as_real_imag</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expand_complex</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_complex</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="go">I*exp(re(z))*sin(im(z)) + exp(re(z))*cos(im(z))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_complex</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">I</span><span class="p">))</span>
-<span class="go">sqrt(2)/2 + sqrt(2)*I/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="expand-multinomial">
-<h3>expand_multinomial<a class="headerlink" href="#expand-multinomial" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.function.expand_multinomial">
-<tt class="descclassname">sympy.core.function.</tt><tt class="descname">expand_multinomial</tt><big>(</big><em>expr</em>, <em>deep=True</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/function.html#expand_multinomial"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.expand_multinomial" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wrapper around expand that only uses the multinomial hint.  See the expand
-docstring for more information.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">expand_multinomial</span><span class="p">,</span> <span class="n">exp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x y&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_multinomial</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x**2 + 2*x*exp(x + 1) + exp(2*x + 2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="expand-power-exp">
-<h3>expand_power_exp<a class="headerlink" href="#expand-power-exp" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.function.expand_power_exp">
-<tt class="descclassname">sympy.core.function.</tt><tt class="descname">expand_power_exp</tt><big>(</big><em>expr</em>, <em>deep=True</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/function.html#expand_power_exp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.expand_power_exp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wrapper around expand that only uses the power_exp hint.</p>
-<p>See the expand docstring for more information.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expand_power_exp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_power_exp</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">x**2*x**y</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="expand-power-base">
-<h3>expand_power_base<a class="headerlink" href="#expand-power-base" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.function.expand_power_base">
-<tt class="descclassname">sympy.core.function.</tt><tt class="descname">expand_power_base</tt><big>(</big><em>expr</em>, <em>deep=True</em>, <em>force=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/function.html#expand_power_base"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.expand_power_base" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wrapper around expand that only uses the power_base hint.</p>
-<p>See the expand docstring for more information.</p>
-<p>A wrapper to expand(power_base=True) which separates a power with a base
-that is a Mul into a product of powers, without performing any other
-expansions, provided that assumptions about the power&#8217;s base and exponent
-allow.</p>
-<p>deep=False (default is True) will only apply to the top-level expression.</p>
-<p>force=True (default is False) will cause the expansion to ignore
-assumptions about the base and exponent. When False, the expansion will
-only happen if the base is non-negative or the exponent is an integer.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expand_power_base</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">exp</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">x**2*y**2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="n">y</span>
-<span class="go">(2*x)**y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_power_base</span><span class="p">(</span><span class="n">_</span><span class="p">)</span>
-<span class="go">2**y*x**y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expand_power_base</span><span class="p">((</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="n">z</span><span class="p">)</span>
-<span class="go">(x*y)**z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_power_base</span><span class="p">((</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="n">z</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">x**z*y**z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_power_base</span><span class="p">(</span><span class="n">sin</span><span class="p">((</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="n">z</span><span class="p">),</span> <span class="n">deep</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">sin((x*y)**z)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_power_base</span><span class="p">(</span><span class="n">sin</span><span class="p">((</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="n">z</span><span class="p">),</span> <span class="n">force</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">sin(x**z*y**z)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expand_power_base</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">**</span><span class="n">y</span> <span class="o">+</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">**</span><span class="n">y</span><span class="p">)</span>
-<span class="go">2**y*sin(x)**y + 2**y*cos(x)**y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expand_power_base</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">y</span><span class="p">))</span><span class="o">**</span><span class="n">x</span><span class="p">)</span>
-<span class="go">2**x*exp(y)**x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expand_power_base</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">**</span><span class="n">y</span><span class="p">)</span>
-<span class="go">2**y*cos(x)**y</span>
-</pre></div>
-</div>
-<p>Notice that sums are left untouched. If this is not the desired behavior,
-apply full <tt class="docutils literal"><span class="pre">expand()</span></tt> to the expression:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expand_power_base</span><span class="p">(((</span><span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">z**2*(x + y)**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(((</span><span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go">x**2*z**2 + 2*x*y*z**2 + y**2*z**2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expand_power_base</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">z</span><span class="p">))</span>
-<span class="go">2**(z + 1)*y**(z + 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">z</span><span class="p">))</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go">2*2**z*y*y**z</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="nfloat">
-<h3>nfloat<a class="headerlink" href="#nfloat" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.function.nfloat">
-<tt class="descclassname">sympy.core.function.</tt><tt class="descname">nfloat</tt><big>(</big><em>expr</em>, <em>n=15</em>, <em>exponent=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/function.html#nfloat"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.function.nfloat" title="Permalink to this definition">¶</a></dt>
-<dd><p>Make all Rationals in expr Floats except those in exponents
-(unless the exponents flag is set to True).</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core.function</span> <span class="kn">import</span> <span class="n">nfloat</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">cos</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nfloat</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">+</span> <span class="n">x</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>
-<span class="go">x**4 + 0.5*x + sqrt(y) + 1.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nfloat</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">y</span><span class="p">),</span> <span class="n">exponent</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">x**4.0 + y**0.5</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.core.evalf">
-<span id="evalf"></span><h2>evalf<a class="headerlink" href="#module-sympy.core.evalf" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="precisionexhausted">
-<h3>PrecisionExhausted<a class="headerlink" href="#precisionexhausted" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.evalf.PrecisionExhausted">
-<em class="property">class </em><tt class="descclassname">sympy.core.evalf.</tt><tt class="descname">PrecisionExhausted</tt><a class="reference internal" href="../_modules/sympy/core/evalf.html#PrecisionExhausted"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.evalf.PrecisionExhausted" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="n">
-<h3>N<a class="headerlink" href="#n" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.evalf.N">
-<em class="property">class </em><tt class="descclassname">sympy.core.evalf.</tt><tt class="descname">N</tt><a class="reference internal" href="../_modules/sympy/core/evalf.html#N"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.evalf.N" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calls x.evalf(n, **options).</p>
-<p>Both .n() and N() are equivalent to .evalf(); use the one that you like better.
-See also the docstring of .evalf() for information on the options.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Sum</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">N</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">k</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="o">**</span><span class="n">k</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">Sum(k**(-k), (k, 1, oo))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">(</span><span class="n">_</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">1.291</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.core.containers">
-<span id="containers"></span><h2>containers<a class="headerlink" href="#module-sympy.core.containers" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="tuple">
-<h3>Tuple<a class="headerlink" href="#tuple" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.containers.Tuple">
-<em class="property">class </em><tt class="descclassname">sympy.core.containers.</tt><tt class="descname">Tuple</tt><a class="reference internal" href="../_modules/sympy/core/containers.html#Tuple"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.containers.Tuple" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wrapper around the builtin tuple object</p>
-<p>The Tuple is a subclass of Basic, so that it works well in the
-SymPy framework.  The wrapped tuple is available as self.args, but
-you can also access elements or slices with [:] syntax.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core.containers</span> <span class="kn">import</span> <span class="n">Tuple</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a b c d&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Tuple</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">)[</span><span class="mi">1</span><span class="p">:]</span>
-<span class="go">(b, c)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Tuple</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
-<span class="go">(d, b, c)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="dict">
-<h3>Dict<a class="headerlink" href="#dict" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.core.containers.Dict">
-<em class="property">class </em><tt class="descclassname">sympy.core.containers.</tt><tt class="descname">Dict</tt><a class="reference internal" href="../_modules/sympy/core/containers.html#Dict"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.containers.Dict" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wrapper around the builtin dict object</p>
-<p>The Dict is a subclass of Basic, so that it works well in the
-SymPy framework.  Because it is immutable, it may be included
-in sets, but its values must all be given at instantiation and
-cannot be changed afterwards.  Otherwise it behaves identically
-to the Python dict.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core.containers</span> <span class="kn">import</span> <span class="n">Dict</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">Dict</span><span class="p">({</span><span class="mi">1</span><span class="p">:</span> <span class="s">&#39;one&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">:</span> <span class="s">&#39;two&#39;</span><span class="p">})</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">key</span> <span class="ow">in</span> <span class="n">D</span><span class="p">:</span>
-<span class="gp">... </span>   <span class="k">if</span> <span class="n">key</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-<span class="gp">... </span>       <span class="k">print</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">D</span><span class="p">[</span><span class="n">key</span><span class="p">])</span>
-<span class="go">1 one</span>
-</pre></div>
-</div>
-<p>The args are sympified so the 1 and 2 are Integers and the values
-are Symbols. Queries automatically sympify args so the following work:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span> <span class="ow">in</span> <span class="n">D</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="s">&#39;one&#39;</span><span class="p">)</span> <span class="c"># searches keys and values</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="s">&#39;one&#39;</span> <span class="ow">in</span> <span class="n">D</span> <span class="c"># not in the keys</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-<span class="go">one</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.core.containers.Dict.get">
-<tt class="descname">get</tt><big>(</big><em>k</em><span class="optional">[</span>, <em>d</em><span class="optional">]</span><big>)</big> &rarr; D[k] if k in D, else d.  d defaults to None.<a class="reference internal" href="../_modules/sympy/core/containers.html#Dict.get"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.containers.Dict.get" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.containers.Dict.items">
-<tt class="descname">items</tt><big>(</big><big>)</big> &rarr; list of D's (key, value) pairs, as 2-tuples<a class="reference internal" href="../_modules/sympy/core/containers.html#Dict.items"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.containers.Dict.items" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.containers.Dict.keys">
-<tt class="descname">keys</tt><big>(</big><big>)</big> &rarr; list of D's keys<a class="reference internal" href="../_modules/sympy/core/containers.html#Dict.keys"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.containers.Dict.keys" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.containers.Dict.values">
-<tt class="descname">values</tt><big>(</big><big>)</big> &rarr; list of D's values<a class="reference internal" href="../_modules/sympy/core/containers.html#Dict.values"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.containers.Dict.values" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.core.compatibility">
-<span id="compatibility"></span><h2>compatibility<a class="headerlink" href="#module-sympy.core.compatibility" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="iterable">
-<h3>iterable<a class="headerlink" href="#iterable" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.compatibility.iterable">
-<tt class="descclassname">sympy.core.compatibility.</tt><tt class="descname">iterable</tt><big>(</big><em>i</em>, <em>exclude=(&lt;class 'str'&gt;</em>, <em>&lt;class 'dict'&gt;)</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/compatibility.html#iterable"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.compatibility.iterable" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a boolean indicating whether <tt class="docutils literal"><span class="pre">i</span></tt> is SymPy iterable.</p>
-<p>When SymPy is working with iterables, it is almost always assuming
-that the iterable is not a string or a mapping, so those are excluded
-by default. If you want a pure Python definition, make exclude=None. To
-exclude multiple items, pass them as a tuple.</p>
-<p>See also: is_sequence</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">iterable</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Tuple</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">things</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">1</span><span class="p">],</span> <span class="p">(</span><span class="mi">1</span><span class="p">,),</span> <span class="nb">set</span><span class="p">([</span><span class="mi">1</span><span class="p">]),</span> <span class="n">Tuple</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="n">j</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]),</span> <span class="p">{</span><span class="mi">1</span><span class="p">:</span><span class="mi">2</span><span class="p">},</span> <span class="s">&#39;1&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">things</span><span class="p">:</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="n">iterable</span><span class="p">(</span><span class="n">i</span><span class="p">),</span> <span class="nb">type</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>
-<span class="go">True &lt;... &#39;list&#39;&gt;</span>
-<span class="go">True &lt;... &#39;tuple&#39;&gt;</span>
-<span class="go">True &lt;... &#39;set&#39;&gt;</span>
-<span class="go">True &lt;class &#39;sympy.core.containers.Tuple&#39;&gt;</span>
-<span class="go">True &lt;... &#39;generator&#39;&gt;</span>
-<span class="go">False &lt;... &#39;dict&#39;&gt;</span>
-<span class="go">False &lt;... &#39;str&#39;&gt;</span>
-<span class="go">False &lt;... &#39;int&#39;&gt;</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">iterable</span><span class="p">({},</span> <span class="n">exclude</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iterable</span><span class="p">({},</span> <span class="n">exclude</span><span class="o">=</span><span class="nb">str</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iterable</span><span class="p">(</span><span class="s">&quot;no&quot;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="nb">str</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="is-sequence">
-<h3>is_sequence<a class="headerlink" href="#is-sequence" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.compatibility.is_sequence">
-<tt class="descclassname">sympy.core.compatibility.</tt><tt class="descname">is_sequence</tt><big>(</big><em>i</em>, <em>include=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/compatibility.html#is_sequence"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.compatibility.is_sequence" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a boolean indicating whether <tt class="docutils literal"><span class="pre">i</span></tt> is a sequence in the SymPy
-sense. If anything that fails the test below should be included as
-being a sequence for your application, set &#8216;include&#8217; to that object&#8217;s
-type; multiple types should be passed as a tuple of types.</p>
-<p>Note: although generators can generate a sequence, they often need special
-handling to make sure their elements are captured before the generator is
-exhausted, so these are not included by default in the definition of a
-sequence.</p>
-<p>See also: iterable</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">is_sequence</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">types</span> <span class="kn">import</span> <span class="n">GeneratorType</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">is_sequence</span><span class="p">([])</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">is_sequence</span><span class="p">(</span><span class="nb">set</span><span class="p">())</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">is_sequence</span><span class="p">(</span><span class="s">&#39;abc&#39;</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">is_sequence</span><span class="p">(</span><span class="s">&#39;abc&#39;</span><span class="p">,</span> <span class="n">include</span><span class="o">=</span><span class="nb">str</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">generator</span> <span class="o">=</span> <span class="p">(</span><span class="n">c</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="s">&#39;abc&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">is_sequence</span><span class="p">(</span><span class="n">generator</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">is_sequence</span><span class="p">(</span><span class="n">generator</span><span class="p">,</span> <span class="n">include</span><span class="o">=</span><span class="p">(</span><span class="nb">str</span><span class="p">,</span> <span class="n">GeneratorType</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="set-intersection">
-<h3>set_intersection<a class="headerlink" href="#set-intersection" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.compatibility.set_intersection">
-<tt class="descclassname">sympy.core.compatibility.</tt><tt class="descname">set_intersection</tt><big>(</big><em>*sets</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/compatibility.html#set_intersection"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.compatibility.set_intersection" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the intersection of all the given sets.</p>
-<p>As of Python 2.6 you can write <tt class="docutils literal"><span class="pre">set.intersection(*sets)</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">set_intersection</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">set_intersection</span><span class="p">(</span><span class="nb">set</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]),</span> <span class="nb">set</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]))</span>
-<span class="go">set([2])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">set_intersection</span><span class="p">()</span>
-<span class="go">set()</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="set-union">
-<h3>set_union<a class="headerlink" href="#set-union" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.compatibility.set_union">
-<tt class="descclassname">sympy.core.compatibility.</tt><tt class="descname">set_union</tt><big>(</big><em>*sets</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/compatibility.html#set_union"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.compatibility.set_union" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the union of all the given sets.</p>
-<p>As of Python 2.6 you can write <tt class="docutils literal"><span class="pre">set.union(*sets)</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">set_union</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">set_union</span><span class="p">(</span><span class="nb">set</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]),</span> <span class="nb">set</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]))</span>
-<span class="go">set([1, 2, 3])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">set_union</span><span class="p">()</span>
-<span class="go">set()</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="as-int">
-<h3>as_int<a class="headerlink" href="#as-int" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.compatibility.as_int">
-<tt class="descclassname">sympy.core.compatibility.</tt><tt class="descname">as_int</tt><big>(</big><em>n</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/compatibility.html#as_int"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.compatibility.as_int" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert the argument to a builtin integer.</p>
-<p>The return value is guaranteed to be equal to the input. ValueError is
-raised if the input has a non-integral value.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">as_int</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mf">3.0</span>
-<span class="go">3.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">as_int</span><span class="p">(</span><span class="mf">3.0</span><span class="p">)</span> <span class="c"># convert to int and test for equality</span>
-<span class="go">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">int</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">10</span><span class="p">))</span>
-<span class="go">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">as_int</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">10</span><span class="p">))</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">ValueError</span>: <span class="n">... is not an integer</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.core.exprtools">
-<span id="exprtools"></span><h2>exprtools<a class="headerlink" href="#module-sympy.core.exprtools" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="gcd-terms">
-<h3>gcd_terms<a class="headerlink" href="#gcd-terms" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.exprtools.gcd_terms">
-<tt class="descclassname">sympy.core.exprtools.</tt><tt class="descname">gcd_terms</tt><big>(</big><em>terms</em>, <em>isprimitive=False</em>, <em>clear=True</em>, <em>fraction=True</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/exprtools.html#gcd_terms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.exprtools.gcd_terms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the GCD of <tt class="docutils literal"><span class="pre">terms</span></tt> and put them together.</p>
-<p><tt class="docutils literal"><span class="pre">terms</span></tt> can be an expression or a non-Basic sequence of expressions
-which will be handled as though they are terms from a sum.</p>
-<p>If <tt class="docutils literal"><span class="pre">isprimitive</span></tt> is True the _gcd_terms will not run the primitive
-method on the terms.</p>
-<p><tt class="docutils literal"><span class="pre">clear</span></tt> controls the removal of integers from the denominator of an Add
-expression. When True (default), all numerical denominator will be cleared;
-when False the denominators will be cleared only if all terms had numerical
-denominators other than 1.</p>
-<p><tt class="docutils literal"><span class="pre">fraction</span></tt>, when True (default), will put the expression over a common
-denominator.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.core.exprtools.factor_terms" title="sympy.core.exprtools.factor_terms"><tt class="xref py py-obj docutils literal"><span class="pre">factor_terms</span></tt></a>, <a class="reference internal" href="polys/reference.html#sympy.polys.polytools.terms_gcd" title="sympy.polys.polytools.terms_gcd"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.polytools.terms_gcd</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">gcd_terms</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gcd_terms</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">y*(x + 1)*(x + y + 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gcd_terms</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">(x + 2)/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gcd_terms</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">clear</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">x/2 + 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gcd_terms</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">clear</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">(x + y)/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gcd_terms</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(x**2 + 2)/(2*x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gcd_terms</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">fraction</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">(x + 2/x)/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gcd_terms</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">fraction</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">clear</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">x/2 + 1/x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gcd_terms</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="mi">2</span><span class="o">/</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="o">/</span><span class="n">y</span><span class="p">)</span>
-<span class="go">(x**2 + 2)/(2*x*y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gcd_terms</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="mi">2</span><span class="o">/</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="o">/</span><span class="n">y</span><span class="p">,</span> <span class="n">fraction</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">clear</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">(x + 2/x)/(2*y)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="factor-terms">
-<h3>factor_terms<a class="headerlink" href="#factor-terms" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.core.exprtools.factor_terms">
-<tt class="descclassname">sympy.core.exprtools.</tt><tt class="descname">factor_terms</tt><big>(</big><em>expr</em>, <em>radical=False</em>, <em>clear=False</em>, <em>fraction=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/exprtools.html#factor_terms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.exprtools.factor_terms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Remove common factors from terms in all arguments without
-changing the underlying structure of the expr. No expansion or
-simplification (and no processing of non-commutatives) is performed.</p>
-<p>If radical=True then a radical common to all terms will be factored
-out of any Add sub-expressions of the expr.</p>
-<p>If clear=False (default) then coefficients will not be separated
-from a single Add if they can be distributed to leave one or more
-terms with integer coefficients.</p>
-<p>If fraction=True (default is False) then a common denominator will be
-constructed for the expression.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.core.exprtools.gcd_terms" title="sympy.core.exprtools.gcd_terms"><tt class="xref py py-obj docutils literal"><span class="pre">gcd_terms</span></tt></a>, <a class="reference internal" href="polys/reference.html#sympy.polys.polytools.terms_gcd" title="sympy.polys.polytools.terms_gcd"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.polytools.terms_gcd</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factor_terms</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">primitive</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factor_terms</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">x*(8*(2*y + 1)**3 + 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">commutative</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factor_terms</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">A</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">A</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">*</span><span class="n">A</span><span class="p">)</span>
-<span class="go">x*(y*A + 2*A)</span>
-</pre></div>
-</div>
-<p>When <tt class="docutils literal"><span class="pre">clear</span></tt> is False, a rational will only be factored out of an
-Add expression if all terms of the Add have coefficients that are
-fractions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor_terms</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">clear</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">x/2 + 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factor_terms</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">clear</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(x + 2)/2</span>
-</pre></div>
-</div>
-<p>This only applies when there is a single Add that the coefficient
-multiplies:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor_terms</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="p">,</span> <span class="n">clear</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">y*(x + 2)/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factor_terms</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="p">,</span> <span class="n">clear</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span> <span class="o">==</span> <span class="n">_</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../index.html">
-              <img class="logo" src="../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">SymPy Core</a><ul>
-<li><a class="reference internal" href="#module-sympy.core.sympify">sympify</a><ul>
-<li><a class="reference internal" href="#id1">sympify</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.core.cache">cache</a><ul>
-<li><a class="reference internal" href="#cacheit">cacheit</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.core.basic">basic</a><ul>
-<li><a class="reference internal" href="#id2">Basic</a></li>
-<li><a class="reference internal" href="#atom">Atom</a></li>
-<li><a class="reference internal" href="#c">C</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.core.singleton">singleton</a><ul>
-<li><a class="reference internal" href="#s">S</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.core.expr">expr</a></li>
-<li><a class="reference internal" href="#id3">Expr</a></li>
-<li><a class="reference internal" href="#atomicexpr">AtomicExpr</a></li>
-<li><a class="reference internal" href="#module-sympy.core.symbol">symbol</a><ul>
-<li><a class="reference internal" href="#id4">Symbol</a></li>
-<li><a class="reference internal" href="#wild">Wild</a></li>
-<li><a class="reference internal" href="#dummy">Dummy</a></li>
-<li><a class="reference internal" href="#symbols">symbols</a></li>
-<li><a class="reference internal" href="#var">var</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.core.numbers">numbers</a><ul>
-<li><a class="reference internal" href="#number">Number</a></li>
-<li><a class="reference internal" href="#float">Float</a></li>
-<li><a class="reference internal" href="#rational">Rational</a></li>
-<li><a class="reference internal" href="#integer">Integer</a></li>
-<li><a class="reference internal" href="#numbersymbol">NumberSymbol</a></li>
-<li><a class="reference internal" href="#realnumber">RealNumber</a></li>
-<li><a class="reference internal" href="#real">Real</a></li>
-<li><a class="reference internal" href="#igcd">igcd</a></li>
-<li><a class="reference internal" href="#ilcm">ilcm</a></li>
-<li><a class="reference internal" href="#seterr">seterr</a></li>
-<li><a class="reference internal" href="#e">E</a></li>
-<li><a class="reference internal" href="#i">I</a></li>
-<li><a class="reference internal" href="#nan">nan</a></li>
-<li><a class="reference internal" href="#oo">oo</a></li>
-<li><a class="reference internal" href="#pi">pi</a></li>
-<li><a class="reference internal" href="#zoo">zoo</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.core.power">power</a><ul>
-<li><a class="reference internal" href="#pow">Pow</a></li>
-<li><a class="reference internal" href="#integer-nthroot">integer_nthroot</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.core.mul">mul</a><ul>
-<li><a class="reference internal" href="#id5">Mul</a></li>
-<li><a class="reference internal" href="#prod">prod</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.core.add">add</a><ul>
-<li><a class="reference internal" href="#id6">Add</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.core.mod">mod</a><ul>
-<li><a class="reference internal" href="#id7">Mod</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.core.relational">relational</a><ul>
-<li><a class="reference internal" href="#rel">Rel</a></li>
-<li><a class="reference internal" href="#eq">Eq</a></li>
-<li><a class="reference internal" href="#ne">Ne</a></li>
-<li><a class="reference internal" href="#lt">Lt</a></li>
-<li><a class="reference internal" href="#le">Le</a></li>
-<li><a class="reference internal" href="#gt">Gt</a></li>
-<li><a class="reference internal" href="#ge">Ge</a></li>
-<li><a class="reference internal" href="#equality">Equality</a></li>
-<li><a class="reference internal" href="#greaterthan">GreaterThan</a></li>
-<li><a class="reference internal" href="#lessthan">LessThan</a></li>
-<li><a class="reference internal" href="#unequality">Unequality</a></li>
-<li><a class="reference internal" href="#strictgreaterthan">StrictGreaterThan</a></li>
-<li><a class="reference internal" href="#strictlessthan">StrictLessThan</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.core.multidimensional">multidimensional</a><ul>
-<li><a class="reference internal" href="#vectorize">vectorize</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.core.function">function</a><ul>
-<li><a class="reference internal" href="#lambda">Lambda</a></li>
-<li><a class="reference internal" href="#wildfunction">WildFunction</a></li>
-<li><a class="reference internal" href="#derivative">Derivative</a></li>
-<li><a class="reference internal" href="#diff">diff</a></li>
-<li><a class="reference internal" href="#functionclass">FunctionClass</a></li>
-<li><a class="reference internal" href="#id22">Function</a></li>
-<li><a class="reference internal" href="#subs">Subs</a></li>
-<li><a class="reference internal" href="#expand">expand</a></li>
-<li><a class="reference internal" href="#poleerror">PoleError</a></li>
-<li><a class="reference internal" href="#count-ops">count_ops</a></li>
-<li><a class="reference internal" href="#expand-mul">expand_mul</a></li>
-<li><a class="reference internal" href="#expand-log">expand_log</a></li>
-<li><a class="reference internal" href="#expand-func">expand_func</a></li>
-<li><a class="reference internal" href="#expand-trig">expand_trig</a></li>
-<li><a class="reference internal" href="#expand-complex">expand_complex</a></li>
-<li><a class="reference internal" href="#expand-multinomial">expand_multinomial</a></li>
-<li><a class="reference internal" href="#expand-power-exp">expand_power_exp</a></li>
-<li><a class="reference internal" href="#expand-power-base">expand_power_base</a></li>
-<li><a class="reference internal" href="#nfloat">nfloat</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.core.evalf">evalf</a><ul>
-<li><a class="reference internal" href="#precisionexhausted">PrecisionExhausted</a></li>
-<li><a class="reference internal" href="#n">N</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.core.containers">containers</a><ul>
-<li><a class="reference internal" href="#tuple">Tuple</a></li>
-<li><a class="reference internal" href="#dict">Dict</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.core.compatibility">compatibility</a><ul>
-<li><a class="reference internal" href="#iterable">iterable</a></li>
-<li><a class="reference internal" href="#is-sequence">is_sequence</a></li>
-<li><a class="reference internal" href="#set-intersection">set_intersection</a></li>
-<li><a class="reference internal" href="#set-union">set_union</a></li>
-<li><a class="reference internal" href="#as-int">as_int</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.core.exprtools">exprtools</a><ul>
-<li><a class="reference internal" href="#gcd-terms">gcd_terms</a></li>
-<li><a class="reference internal" href="#factor-terms">factor_terms</a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
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-            
-  <div class="section" id="module-sympy.diffgeom">
-<span id="differential-geometry-module"></span><h1>Differential Geometry Module<a class="headerlink" href="#module-sympy.diffgeom" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="introduction">
-<h2>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline">¶</a></h2>
-</div>
-<div class="section" id="base-class-reference">
-<h2>Base Class Reference<a class="headerlink" href="#base-class-reference" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.diffgeom.Manifold">
-<em class="property">class </em><tt class="descclassname">sympy.diffgeom.</tt><tt class="descname">Manifold</tt><big>(</big><em>name</em>, <em>dim</em><big>)</big><a class="headerlink" href="#sympy.diffgeom.Manifold" title="Permalink to this definition">¶</a></dt>
-<dd><p>Object representing a mathematical manifold.</p>
-<p>The only role that this object plays is to keep a list of all patches
-defined on the manifold. It does not provide any means to study the
-topological characteristics of the manifold that it represents.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.diffgeom.Patch">
-<em class="property">class </em><tt class="descclassname">sympy.diffgeom.</tt><tt class="descname">Patch</tt><big>(</big><em>name</em>, <em>manifold</em><big>)</big><a class="headerlink" href="#sympy.diffgeom.Patch" title="Permalink to this definition">¶</a></dt>
-<dd><p>Object representing a patch on a manifold.</p>
-<p>On a manifold one can have many patches that do not always include the
-whole manifold. On these patches coordinate charts can be defined that
-permit the parametrization of any point on the patch in terms of a tuple
-of real numbers (the coordinates).</p>
-<p>This object serves as a container/parent for all coordinate system charts
-that can be defined on the patch it represents.</p>
-<p class="rubric">Examples:</p>
-<p>Define a Manifold and a Patch on that Manifold:
-&gt;&gt;&gt; from sympy.diffgeom import Manifold, Patch
-&gt;&gt;&gt; m = Manifold(&#8216;M&#8217;, 3)
-&gt;&gt;&gt; p = Patch(&#8216;P&#8217;, m)
-&gt;&gt;&gt; p in m.patches
-True</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.diffgeom.CoordSystem">
-<em class="property">class </em><tt class="descclassname">sympy.diffgeom.</tt><tt class="descname">CoordSystem</tt><big>(</big><em>name</em>, <em>patch</em>, <em>names=None</em><big>)</big><a class="headerlink" href="#sympy.diffgeom.CoordSystem" title="Permalink to this definition">¶</a></dt>
-<dd><p>Contains all coordinate transformation logic.</p>
-<p class="rubric">Examples</p>
-<p>Define a Manifold and a Patch, and then define two coord systems on that
-patch:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.diffgeom</span> <span class="kn">import</span> <span class="n">Manifold</span><span class="p">,</span> <span class="n">Patch</span><span class="p">,</span> <span class="n">CoordSystem</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="p">,</span> <span class="n">theta</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;r, theta&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Manifold</span><span class="p">(</span><span class="s">&#39;M&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">patch</span> <span class="o">=</span> <span class="n">Patch</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rect</span> <span class="o">=</span> <span class="n">CoordSystem</span><span class="p">(</span><span class="s">&#39;rect&#39;</span><span class="p">,</span> <span class="n">patch</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polar</span> <span class="o">=</span> <span class="n">CoordSystem</span><span class="p">(</span><span class="s">&#39;polar&#39;</span><span class="p">,</span> <span class="n">patch</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rect</span> <span class="ow">in</span> <span class="n">patch</span><span class="o">.</span><span class="n">coord_systems</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>Connect the coordinate systems. An inverse transformation is automatically
-found by <tt class="docutils literal"><span class="pre">solve</span></tt> when possible:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">polar</span><span class="o">.</span><span class="n">connect_to</span><span class="p">(</span><span class="n">rect</span><span class="p">,</span> <span class="p">[</span><span class="n">r</span><span class="p">,</span> <span class="n">theta</span><span class="p">],</span> <span class="p">[</span><span class="n">r</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="p">),</span> <span class="n">r</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polar</span><span class="o">.</span><span class="n">coord_tuple_transform_to</span><span class="p">(</span><span class="n">rect</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="go">[0]</span>
-<span class="go">[0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polar</span><span class="o">.</span><span class="n">coord_tuple_transform_to</span><span class="p">(</span><span class="n">rect</span><span class="p">,</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">])</span>
-<span class="go">[0]</span>
-<span class="go">[2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rect</span><span class="o">.</span><span class="n">coord_tuple_transform_to</span><span class="p">(</span><span class="n">polar</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">[sqrt(2)]</span>
-<span class="go">[   pi/4]</span>
-</pre></div>
-</div>
-<p>Calculate the jacobian of the polar to cartesian transformation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">polar</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">rect</span><span class="p">,</span> <span class="p">[</span><span class="n">r</span><span class="p">,</span> <span class="n">theta</span><span class="p">])</span>
-<span class="go">[cos(theta), -r*sin(theta)]</span>
-<span class="go">[sin(theta),  r*cos(theta)]</span>
-</pre></div>
-</div>
-<p>Define a point using coordinates in one of the coordinate systems:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">polar</span><span class="o">.</span><span class="n">point</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rect</span><span class="o">.</span><span class="n">point_to_coords</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-<span class="go">[-sqrt(2)/2]</span>
-<span class="go">[ sqrt(2)/2]</span>
-</pre></div>
-</div>
-<p>Define a basis scalar field (i.e. a coordinate function), that takes a
-point and returns its coordinates. It is an instance of <tt class="docutils literal"><span class="pre">BaseScalarField</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rect</span><span class="o">.</span><span class="n">coord_function</span><span class="p">(</span><span class="mi">0</span><span class="p">)(</span><span class="n">p</span><span class="p">)</span>
-<span class="go">-sqrt(2)/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rect</span><span class="o">.</span><span class="n">coord_function</span><span class="p">(</span><span class="mi">1</span><span class="p">)(</span><span class="n">p</span><span class="p">)</span>
-<span class="go">sqrt(2)/2</span>
-</pre></div>
-</div>
-<p>Define a basis vector field (i.e. a unit vector field along the coordinate
-line). Vectors are also differential operators on scalar fields. It is an
-instance of <tt class="docutils literal"><span class="pre">BaseVectorField</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">v_x</span> <span class="o">=</span> <span class="n">rect</span><span class="o">.</span><span class="n">base_vector</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">rect</span><span class="o">.</span><span class="n">coord_function</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v_x</span><span class="p">(</span><span class="n">x</span><span class="p">)(</span><span class="n">p</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v_x</span><span class="p">(</span><span class="n">v_x</span><span class="p">(</span><span class="n">x</span><span class="p">))(</span><span class="n">p</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>Define a basis oneform field:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dx</span> <span class="o">=</span> <span class="n">rect</span><span class="o">.</span><span class="n">base_oneform</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dx</span><span class="p">(</span><span class="n">v_x</span><span class="p">)(</span><span class="n">p</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>If you provide a list of names the fields will print nicely:
-- without provided names:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">v_x</span><span class="p">,</span> <span class="n">dx</span>
-<span class="go">(rect_0, e_rect_0, drect_0)</span>
-</pre></div>
-</div>
-<ul class="simple">
-<li>with provided names</li>
-</ul>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rect</span> <span class="o">=</span> <span class="n">CoordSystem</span><span class="p">(</span><span class="s">&#39;rect&#39;</span><span class="p">,</span> <span class="n">patch</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="s">&#39;y&#39;</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rect</span><span class="o">.</span><span class="n">coord_function</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">rect</span><span class="o">.</span><span class="n">base_vector</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">rect</span><span class="o">.</span><span class="n">base_oneform</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">(x, e_x, dx)</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.diffgeom.CoordSystem.base_oneform">
-<tt class="descname">base_oneform</tt><big>(</big><em>self</em>, <em>coord_index</em><big>)</big><a class="headerlink" href="#sympy.diffgeom.CoordSystem.base_oneform" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a basis 1-form field.</p>
-<p>The basis one-form field for this coordinate system. It is also an
-operator on vector fields.</p>
-<p>See the docstring of <tt class="docutils literal"><span class="pre">CoordSystem</span></tt> for examples.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.diffgeom.CoordSystem.base_oneforms">
-<tt class="descname">base_oneforms</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.diffgeom.CoordSystem.base_oneforms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a list of all base oneforms.</p>
-<p>For more details see the <tt class="docutils literal"><span class="pre">base_oneform</span></tt> method of this class.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.diffgeom.CoordSystem.base_vector">
-<tt class="descname">base_vector</tt><big>(</big><em>self</em>, <em>coord_index</em><big>)</big><a class="headerlink" href="#sympy.diffgeom.CoordSystem.base_vector" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a basis vector field.</p>
-<p>The basis vector field for this coordinate system. It is also an
-operator on scalar fields.</p>
-<p>See the docstring of <tt class="docutils literal"><span class="pre">CoordSystem</span></tt> for examples.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.diffgeom.CoordSystem.base_vectors">
-<tt class="descname">base_vectors</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.diffgeom.CoordSystem.base_vectors" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a list of all base vectors.</p>
-<p>For more details see the <tt class="docutils literal"><span class="pre">base_vector</span></tt> method of this class.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.diffgeom.CoordSystem.connect_to">
-<tt class="descname">connect_to</tt><big>(</big><em>self</em>, <em>to_sys</em>, <em>from_coords</em>, <em>to_exprs</em>, <em>inverse=True</em>, <em>fill_in_gaps=False</em><big>)</big><a class="headerlink" href="#sympy.diffgeom.CoordSystem.connect_to" title="Permalink to this definition">¶</a></dt>
-<dd><p>Register the transformation used to switch to another coordinate system.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>to_sys</strong> :</p>
-<blockquote>
-<div><p>another instance of <tt class="docutils literal"><span class="pre">CoordSystem</span></tt></p>
-</div></blockquote>
-<p><strong>from_coords</strong> :</p>
-<blockquote>
-<div><p>list of symbols in terms of which <tt class="docutils literal"><span class="pre">to_exprs</span></tt> is given</p>
-</div></blockquote>
-<p><strong>to_exprs</strong> :</p>
-<blockquote>
-<div><p>list of the expressions of the new coordinate tuple</p>
-</div></blockquote>
-<p><strong>inverse</strong> :</p>
-<blockquote>
-<div><p>try to deduce and register the inverse transformation</p>
-</div></blockquote>
-<p><strong>fill_in_gaps</strong> :</p>
-<blockquote class="last">
-<div><p>try to deduce other transformation that are made
-possible by composing the present transformation with other already
-registered transformation</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.diffgeom.CoordSystem.coord_function">
-<tt class="descname">coord_function</tt><big>(</big><em>self</em>, <em>coord_index</em><big>)</big><a class="headerlink" href="#sympy.diffgeom.CoordSystem.coord_function" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a <tt class="docutils literal"><span class="pre">BaseScalarField</span></tt> that takes a point and returns one of the coords.</p>
-<p>Takes a point and returns its coordinate in this coordinate system.</p>
-<p>See the docstring of <tt class="docutils literal"><span class="pre">CoordSystem</span></tt> for examples.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.diffgeom.CoordSystem.coord_functions">
-<tt class="descname">coord_functions</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.diffgeom.CoordSystem.coord_functions" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a list of all coordinate functions.</p>
-<p>For more details see the <tt class="docutils literal"><span class="pre">coord_function</span></tt> method of this class.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.diffgeom.CoordSystem.coord_tuple_transform_to">
-<tt class="descname">coord_tuple_transform_to</tt><big>(</big><em>self</em>, <em>to_sys</em>, <em>coords</em><big>)</big><a class="headerlink" href="#sympy.diffgeom.CoordSystem.coord_tuple_transform_to" title="Permalink to this definition">¶</a></dt>
-<dd><p>Transform <tt class="docutils literal"><span class="pre">coords</span></tt> to coord system <tt class="docutils literal"><span class="pre">to_sys</span></tt>.</p>
-<p>See the docstring of <tt class="docutils literal"><span class="pre">CoordSystem</span></tt> for examples.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.diffgeom.CoordSystem.jacobian">
-<tt class="descname">jacobian</tt><big>(</big><em>self</em>, <em>to_sys</em>, <em>coords</em><big>)</big><a class="headerlink" href="#sympy.diffgeom.CoordSystem.jacobian" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the jacobian matrix of a transformation.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.diffgeom.CoordSystem.point">
-<tt class="descname">point</tt><big>(</big><em>self</em>, <em>coords</em><big>)</big><a class="headerlink" href="#sympy.diffgeom.CoordSystem.point" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a <tt class="docutils literal"><span class="pre">Point</span></tt> with coordinates given in this coord system.</p>
-<p>See the docstring of <tt class="docutils literal"><span class="pre">CoordSystem</span></tt> for examples.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.diffgeom.CoordSystem.point_to_coords">
-<tt class="descname">point_to_coords</tt><big>(</big><em>self</em>, <em>point</em><big>)</big><a class="headerlink" href="#sympy.diffgeom.CoordSystem.point_to_coords" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculate the coordinates of a point in this coord system.</p>
-<p>See the docstring of <tt class="docutils literal"><span class="pre">CoordSystem</span></tt> for examples.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.diffgeom.BaseScalarField">
-<em class="property">class </em><tt class="descclassname">sympy.diffgeom.</tt><tt class="descname">BaseScalarField</tt><big>(</big><em>coord_sys</em>, <em>index</em><big>)</big><a class="headerlink" href="#sympy.diffgeom.BaseScalarField" title="Permalink to this definition">¶</a></dt>
-<dd><p>Base Scalar Field over a Manifold for a given Coordinate System.</p>
-<p>A scalar field takes a point as an argument and returns a scalar.</p>
-<p>A base scalar field of a coordinate system takes a point and returns one of
-the coordinates of that point in the coordinate system in question.</p>
-<p>To define a scalar field you need to choose the coordinate system and the
-index of the coordinate.</p>
-<p>The use of the scalar field after its definition is independent of the
-coordinate system in which it was defined, however due to limitations in
-the simplification routines you may arrive at more complicated
-expression if you use unappropriate coordinate systems.</p>
-<p>You can build complicated scalar fields by just building up SymPy
-expressions containing <tt class="docutils literal"><span class="pre">BaseScalarField</span></tt> instances.</p>
-<p class="rubric">Examples</p>
-<p>Define boilerplate Manifold, Patch and coordinate systems:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">Function</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.diffgeom</span> <span class="kn">import</span> <span class="p">(</span>
-<span class="gp">... </span>       <span class="n">Manifold</span><span class="p">,</span> <span class="n">Patch</span><span class="p">,</span> <span class="n">CoordSystem</span><span class="p">,</span> <span class="n">Point</span><span class="p">,</span> <span class="n">BaseScalarField</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r0</span><span class="p">,</span> <span class="n">theta0</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;r0, theta0&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Manifold</span><span class="p">(</span><span class="s">&#39;M&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Patch</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rect</span> <span class="o">=</span> <span class="n">CoordSystem</span><span class="p">(</span><span class="s">&#39;rect&#39;</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polar</span> <span class="o">=</span> <span class="n">CoordSystem</span><span class="p">(</span><span class="s">&#39;polar&#39;</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polar</span><span class="o">.</span><span class="n">connect_to</span><span class="p">(</span><span class="n">rect</span><span class="p">,</span> <span class="p">[</span><span class="n">r0</span><span class="p">,</span> <span class="n">theta0</span><span class="p">],</span> <span class="p">[</span><span class="n">r0</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">theta0</span><span class="p">),</span> <span class="n">r0</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">theta0</span><span class="p">)])</span>
-</pre></div>
-</div>
-<p>Point to be used as an argument for the filed:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">point</span> <span class="o">=</span> <span class="n">polar</span><span class="o">.</span><span class="n">point</span><span class="p">([</span><span class="n">r0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-</pre></div>
-</div>
-<p>Examples of fields:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fx</span> <span class="o">=</span> <span class="n">BaseScalarField</span><span class="p">(</span><span class="n">rect</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fy</span> <span class="o">=</span> <span class="n">BaseScalarField</span><span class="p">(</span><span class="n">rect</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">fx</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">fy</span><span class="o">**</span><span class="mi">2</span><span class="p">)(</span><span class="n">point</span><span class="p">)</span>
-<span class="go">r0**2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ftheta</span> <span class="o">=</span> <span class="n">BaseScalarField</span><span class="p">(</span><span class="n">polar</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fg</span> <span class="o">=</span> <span class="n">g</span><span class="p">(</span><span class="n">ftheta</span><span class="o">-</span><span class="n">pi</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fg</span><span class="p">(</span><span class="n">point</span><span class="p">)</span>
-<span class="go">g(-pi)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.diffgeom.BaseVectorField">
-<em class="property">class </em><tt class="descclassname">sympy.diffgeom.</tt><tt class="descname">BaseVectorField</tt><big>(</big><em>coord_sys</em>, <em>index</em><big>)</big><a class="headerlink" href="#sympy.diffgeom.BaseVectorField" title="Permalink to this definition">¶</a></dt>
-<dd><p>Vector Field over a Manifold.</p>
-<p>A vector field is an operator taking a scalar field and returning a
-directional derivative (which is also a scalar field).</p>
-<p>A base vector field is the same type of operator, however the derivation is
-specifically done wrt a chosen coordinate.</p>
-<p>To define a base vector field you need to choose the coordinate system and
-the index of the coordinate.</p>
-<p>The use of the vector field after its definition is independent of the
-coordinate system in which it was defined, however due to limitations in
-the simplification routines you may arrive at more complicated
-expression if you use unappropriate coordinate systems.</p>
-<p class="rubric">Examples</p>
-<p>Use the predefined R2 manifold, setup some boilerplate.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">Function</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.diffgeom.rn</span> <span class="kn">import</span> <span class="n">R2</span><span class="p">,</span> <span class="n">R2_p</span><span class="p">,</span> <span class="n">R2_r</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.diffgeom</span> <span class="kn">import</span> <span class="n">BaseVectorField</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x0</span><span class="p">,</span> <span class="n">y0</span><span class="p">,</span> <span class="n">r0</span><span class="p">,</span> <span class="n">theta0</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x0, y0, r0, theta0&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Points to be used as arguments for the field:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">point_p</span> <span class="o">=</span> <span class="n">R2_p</span><span class="o">.</span><span class="n">point</span><span class="p">([</span><span class="n">r0</span><span class="p">,</span> <span class="n">theta0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">point_r</span> <span class="o">=</span> <span class="n">R2_r</span><span class="o">.</span><span class="n">point</span><span class="p">([</span><span class="n">x0</span><span class="p">,</span> <span class="n">y0</span><span class="p">])</span>
-</pre></div>
-</div>
-<p>Scalar field to operate on:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s_field</span> <span class="o">=</span> <span class="n">g</span><span class="p">(</span><span class="n">R2</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">R2</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s_field</span><span class="p">(</span><span class="n">point_r</span><span class="p">)</span>
-<span class="go">g(x0, y0)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s_field</span><span class="p">(</span><span class="n">point_p</span><span class="p">)</span>
-<span class="go">g(r0*cos(theta0), r0*sin(theta0))</span>
-</pre></div>
-</div>
-<p>Vector field:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">v</span> <span class="o">=</span> <span class="n">BaseVectorField</span><span class="p">(</span><span class="n">R2_r</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">v</span><span class="p">(</span><span class="n">s_field</span><span class="p">))</span>
-<span class="go">/  d              \|</span>
-<span class="go">|-----(g(x, xi_2))||</span>
-<span class="go">\dxi_2            /|xi_2=y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">v</span><span class="p">(</span><span class="n">s_field</span><span class="p">)(</span><span class="n">point_r</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">())</span>
-<span class="go"> d</span>
-<span class="go">---(g(x0, y0))</span>
-<span class="go">dy0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">v</span><span class="p">(</span><span class="n">s_field</span><span class="p">)(</span><span class="n">point_p</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">())</span>
-<span class="go">/  d                           \|</span>
-<span class="go">|-----(g(r0*cos(theta0), xi_2))||</span>
-<span class="go">\dxi_2                         /|xi_2=r0*sin(theta0)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.diffgeom.Differential">
-<em class="property">class </em><tt class="descclassname">sympy.diffgeom.</tt><tt class="descname">Differential</tt><big>(</big><em>form_field</em><big>)</big><a class="headerlink" href="#sympy.diffgeom.Differential" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the differential (exterior derivative) of a form field.</p>
-<p>The differential of a form (i.e. the exterior derivative) has a complicated
-definition in the general case.</p>
-<p>The differential <span class="math">\(df\)</span> of the 0-form <span class="math">\(f\)</span> is defined for any vector field <span class="math">\(v\)</span>
-as <span class="math">\(df(v) = v(f)\)</span>.</p>
-<p class="rubric">Examples</p>
-<p>Use the predefined R2 manifold, setup some boilerplate.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.diffgeom.rn</span> <span class="kn">import</span> <span class="n">R2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.diffgeom</span> <span class="kn">import</span> <span class="n">Differential</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pprint</span>
-</pre></div>
-</div>
-<p>Scalar field (0-forms):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s_field</span> <span class="o">=</span> <span class="n">g</span><span class="p">(</span><span class="n">R2</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">R2</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Vector fields:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e_x</span><span class="p">,</span> <span class="n">e_y</span><span class="p">,</span> <span class="o">=</span> <span class="n">R2</span><span class="o">.</span><span class="n">e_x</span><span class="p">,</span> <span class="n">R2</span><span class="o">.</span><span class="n">e_y</span>
-</pre></div>
-</div>
-<p>Differentials:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dg</span> <span class="o">=</span> <span class="n">Differential</span><span class="p">(</span><span class="n">s_field</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dg</span>
-<span class="go">d(g(x, y))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">dg</span><span class="p">(</span><span class="n">e_x</span><span class="p">))</span>
-<span class="go">/  d              \|</span>
-<span class="go">|-----(g(xi_1, y))||</span>
-<span class="go">\dxi_1            /|xi_1=x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">dg</span><span class="p">(</span><span class="n">e_y</span><span class="p">))</span>
-<span class="go">/  d              \|</span>
-<span class="go">|-----(g(x, xi_2))||</span>
-<span class="go">\dxi_2            /|xi_2=y</span>
-</pre></div>
-</div>
-<p>Applying the exterior derivative operator twice always results in:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Differential</span><span class="p">(</span><span class="n">dg</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.diffgeom.BaseCovarDerivativeOp">
-<em class="property">class </em><tt class="descclassname">sympy.diffgeom.</tt><tt class="descname">BaseCovarDerivativeOp</tt><big>(</big><em>coord_sys</em>, <em>index</em>, <em>christoffel</em><big>)</big><a class="headerlink" href="#sympy.diffgeom.BaseCovarDerivativeOp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Covariant derivative operator wrt a base vector.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.diffgeom.rn</span> <span class="kn">import</span> <span class="n">R2</span><span class="p">,</span> <span class="n">R2_r</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.diffgeom</span> <span class="kn">import</span> <span class="n">BaseCovarDerivativeOp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.diffgeom</span> <span class="kn">import</span> <span class="n">metric_to_Christoffel_2nd</span><span class="p">,</span> <span class="n">TensorProduct</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">TP</span> <span class="o">=</span> <span class="n">TensorProduct</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ch</span> <span class="o">=</span> <span class="n">metric_to_Christoffel_2nd</span><span class="p">(</span><span class="n">TP</span><span class="p">(</span><span class="n">R2</span><span class="o">.</span><span class="n">dx</span><span class="p">,</span> <span class="n">R2</span><span class="o">.</span><span class="n">dx</span><span class="p">)</span> <span class="o">+</span> <span class="n">TP</span><span class="p">(</span><span class="n">R2</span><span class="o">.</span><span class="n">dy</span><span class="p">,</span> <span class="n">R2</span><span class="o">.</span><span class="n">dy</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ch</span>
-<span class="go">(((0, 0), (0, 0)), ((0, 0), (0, 0)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cvd</span> <span class="o">=</span> <span class="n">BaseCovarDerivativeOp</span><span class="p">(</span><span class="n">R2_r</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">ch</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cvd</span><span class="p">(</span><span class="n">R2</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cvd</span><span class="p">(</span><span class="n">R2</span><span class="o">.</span><span class="n">x</span><span class="o">*</span><span class="n">R2</span><span class="o">.</span><span class="n">e_x</span><span class="p">)</span>
-<span class="go">e_x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.diffgeom.CovarDerivativeOp">
-<em class="property">class </em><tt class="descclassname">sympy.diffgeom.</tt><tt class="descname">CovarDerivativeOp</tt><big>(</big><em>wrt</em>, <em>christoffel</em><big>)</big><a class="headerlink" href="#sympy.diffgeom.CovarDerivativeOp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Covariant derivative operator.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.diffgeom.rn</span> <span class="kn">import</span> <span class="n">R2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.diffgeom</span> <span class="kn">import</span> <span class="n">CovarDerivativeOp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.diffgeom</span> <span class="kn">import</span> <span class="n">metric_to_Christoffel_2nd</span><span class="p">,</span> <span class="n">TensorProduct</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">TP</span> <span class="o">=</span> <span class="n">TensorProduct</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ch</span> <span class="o">=</span> <span class="n">metric_to_Christoffel_2nd</span><span class="p">(</span><span class="n">TP</span><span class="p">(</span><span class="n">R2</span><span class="o">.</span><span class="n">dx</span><span class="p">,</span> <span class="n">R2</span><span class="o">.</span><span class="n">dx</span><span class="p">)</span> <span class="o">+</span> <span class="n">TP</span><span class="p">(</span><span class="n">R2</span><span class="o">.</span><span class="n">dy</span><span class="p">,</span> <span class="n">R2</span><span class="o">.</span><span class="n">dy</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ch</span>
-<span class="go">(((0, 0), (0, 0)), ((0, 0), (0, 0)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cvd</span> <span class="o">=</span> <span class="n">CovarDerivativeOp</span><span class="p">(</span><span class="n">R2</span><span class="o">.</span><span class="n">x</span><span class="o">*</span><span class="n">R2</span><span class="o">.</span><span class="n">e_x</span><span class="p">,</span> <span class="n">ch</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cvd</span><span class="p">(</span><span class="n">R2</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cvd</span><span class="p">(</span><span class="n">R2</span><span class="o">.</span><span class="n">x</span><span class="o">*</span><span class="n">R2</span><span class="o">.</span><span class="n">e_x</span><span class="p">)</span>
-<span class="go">x*e_x</span>
-</pre></div>
-</div>
-</dd></dl>
-
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-  <h3><a href="../index.html">Table Of Contents</a></h3>
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-<li><a class="reference internal" href="#">Differential Geometry Module</a><ul>
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-            
-  <div class="section" id="numerical-evaluation">
-<span id="evalf-label"></span><h1>Numerical evaluation<a class="headerlink" href="#numerical-evaluation" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="basics">
-<h2>Basics<a class="headerlink" href="#basics" title="Permalink to this headline">¶</a></h2>
-<p>Exact SymPy expressions can be converted to floating-point approximations
-(decimal numbers) using either the <tt class="docutils literal"><span class="pre">.evalf()</span></tt> method or the <tt class="docutils literal"><span class="pre">N()</span></tt> function.
-<tt class="docutils literal"><span class="pre">N(expr,</span> <span class="pre">&lt;args&gt;)</span></tt> is equivalent to <tt class="docutils literal"><span class="pre">sympify(expr).evalf(&lt;args&gt;)</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">4.44288293815837</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">4.44288293815837</span>
-</pre></div>
-</div>
-<p>By default, numerical evaluation is performed to an accuracy of 15 decimal
-digits. You can optionally pass a desired accuracy (which should be a positive
-integer) as an argument to <tt class="docutils literal"><span class="pre">evalf</span></tt> or <tt class="docutils literal"><span class="pre">N</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">pi</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">4.4429</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">pi</span><span class="p">,</span> <span class="mi">50</span><span class="p">)</span>
-<span class="go">4.4428829381583662470158809900606936986146216893757</span>
-</pre></div>
-</div>
-<p>Complex numbers are supported:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">pi</span> <span class="o">+</span> <span class="n">I</span><span class="p">),</span> <span class="mi">20</span><span class="p">)</span>
-<span class="go">0.28902548222223624241 - 0.091999668350375232456*I</span>
-</pre></div>
-</div>
-<p>If the expression contains symbols or for some other reason cannot be evaluated
-numerically, calling <tt class="docutils literal"><span class="pre">.evalf()</span></tt> or <tt class="docutils literal"><span class="pre">N()</span></tt> returns the original expression, or
-in some cases a partially evaluated expression. For example, when the
-expression is a polynomial in expanded form, the coefficients are evaluated:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">3.14159265358979*x**2 + 0.333333333333333*x</span>
-</pre></div>
-</div>
-<p>You can also use the standard Python functions <tt class="docutils literal"><span class="pre">float()</span></tt>, <tt class="docutils literal"><span class="pre">complex()</span></tt> to
-convert SymPy expressions to regular Python numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">float</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span> 
-<span class="go">3.1415926535...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">complex</span><span class="p">(</span><span class="n">pi</span><span class="o">+</span><span class="n">E</span><span class="o">*</span><span class="n">I</span><span class="p">)</span> 
-<span class="go">(3.1415926535...+2.7182818284...j)</span>
-</pre></div>
-</div>
-<p>If these functions are used, failure to evaluate the expression to an explicit
-number (for example if the expression contains symbols) will raise an exception.</p>
-<p>There is essentially no upper precision limit. The following command, for
-example, computes the first 100,000 digits of π/e:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="n">E</span><span class="p">,</span> <span class="mi">100000</span><span class="p">)</span> 
-<span class="gp">...</span>
-</pre></div>
-</div>
-<p>This shows digits 999,951 through 1,000,000 of pi:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">str</span><span class="p">(</span><span class="n">N</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="mi">10</span><span class="o">**</span><span class="mi">6</span><span class="p">))[</span><span class="o">-</span><span class="mi">50</span><span class="p">:]</span> 
-<span class="go">&#39;95678796130331164628399634646042209010610577945815&#39;</span>
-</pre></div>
-</div>
-<p>High-precision calculations can be slow. It is recommended (but entirely
-optional) to install gmpy (<a class="reference external" href="http://code.google.com/p/gmpy/">http://code.google.com/p/gmpy/</a>), which will
-significantly speed up computations such as the one above.</p>
-</div>
-<div class="section" id="floating-point-numbers">
-<h2>Floating-point numbers<a class="headerlink" href="#floating-point-numbers" title="Permalink to this headline">¶</a></h2>
-<p>Floating-point numbers in SymPy are instances of the class <tt class="docutils literal"><span class="pre">Float</span></tt>. A <tt class="docutils literal"><span class="pre">Float</span></tt>
-can be created with a custom precision as second argument:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="mf">0.1</span><span class="p">)</span>
-<span class="go">0.100000000000000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">0.1000000000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="mf">0.125</span><span class="p">,</span> <span class="mi">30</span><span class="p">)</span>
-<span class="go">0.125000000000000000000000000000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span> <span class="mi">30</span><span class="p">)</span>
-<span class="go">0.100000000000000005551115123126</span>
-</pre></div>
-</div>
-<p>As the last example shows, some Python floats are only accurate to about 15
-digits as inputs, while others (those that have a denominator that is a
-power of 2, like .125 = 1/4) are exact. To create a <tt class="docutils literal"><span class="pre">Float</span></tt> from a
-high-precision decimal number, it is better to pass a string, <tt class="docutils literal"><span class="pre">Rational</span></tt>,
-or <tt class="docutils literal"><span class="pre">evalf</span></tt> a <tt class="docutils literal"><span class="pre">Rational</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="s">&#39;0.1&#39;</span><span class="p">,</span> <span class="mi">30</span><span class="p">)</span>
-<span class="go">0.100000000000000000000000000000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">),</span> <span class="mi">30</span><span class="p">)</span>
-<span class="go">0.100000000000000000000000000000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">30</span><span class="p">)</span>
-<span class="go">0.100000000000000000000000000000</span>
-</pre></div>
-</div>
-<p>The precision of a number determines 1) the precision to use when performing
-arithmetic with the number, and 2) the number of digits to display when printing
-the number. When two numbers with different precision are used together in an
-arithmetic operation, the higher of the precisions is used for the result. The
-product of 0.1 +/- 0.001 and 3.1415 +/- 0.0001 has an uncertainty of about 0.003
-and yet 5 digits of precision are shown.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span><span class="o">*</span><span class="n">Float</span><span class="p">(</span><span class="mf">3.1415</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">0.31417</span>
-</pre></div>
-</div>
-<p>So the displayed precision should not be used as a model of error propagation or
-significance arithmetic; rather, this scheme is employed to ensure stability of
-numerical algorithms.</p>
-<p><tt class="docutils literal"><span class="pre">N</span></tt> and <tt class="docutils literal"><span class="pre">evalf</span></tt> can be used to change the precision of existing
-floating-point numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">(</span><span class="mf">3.5</span><span class="p">)</span>
-<span class="go">3.50000000000000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">(</span><span class="mf">3.5</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">3.5000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">(</span><span class="mf">3.5</span><span class="p">,</span> <span class="mi">30</span><span class="p">)</span>
-<span class="go">3.50000000000000000000000000000</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="accuracy-and-error-handling">
-<h2>Accuracy and error handling<a class="headerlink" href="#accuracy-and-error-handling" title="Permalink to this headline">¶</a></h2>
-<p>When the input to <tt class="docutils literal"><span class="pre">N</span></tt> or <tt class="docutils literal"><span class="pre">evalf</span></tt> is a complicated expression, numerical
-error propagation becomes a concern. As an example, consider the 100&#8217;th
-Fibonacci number and the excellent (but not exact) approximation <span class="math">\(\varphi^{100} / \sqrt{5}\)</span>
-where <span class="math">\(\varphi\)</span> is the golden ratio. With ordinary floating-point arithmetic,
-subtracting these numbers from each other erroneously results in a complete
-cancellation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">GoldenRatio</span><span class="o">**</span><span class="mi">1000</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">fibonacci</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">float</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> 
-<span class="go">4.34665576869...e+208</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">float</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> 
-<span class="go">4.34665576869...e+208</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">float</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">-</span> <span class="nb">float</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p><tt class="docutils literal"><span class="pre">N</span></tt> and <tt class="docutils literal"><span class="pre">evalf</span></tt> keep track of errors and automatically increase the
-precision used internally in order to obtain a correct result:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">(</span><span class="n">fibonacci</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span> <span class="o">-</span> <span class="n">GoldenRatio</span><span class="o">**</span><span class="mi">100</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
-<span class="go">-5.64613129282185e-22</span>
-</pre></div>
-</div>
-<p>Unfortunately, numerical evaluation cannot tell an expression that is exactly
-zero apart from one that is merely very small. The working precision is
-therefore capped, by default to around 100 digits. If we try with the 1000&#8217;th
-Fibonacci number, the following happens:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">(</span><span class="n">fibonacci</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="n">GoldenRatio</span><span class="p">)</span><span class="o">**</span><span class="mi">1000</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
-<span class="go">0.e+85</span>
-</pre></div>
-</div>
-<p>The lack of digits in the returned number indicates that <tt class="docutils literal"><span class="pre">N</span></tt> failed to achieve
-full accuracy. The result indicates that the magnitude of the expression is something less than 10^84, but that is not a particularly good answer. To force a higher working precision, the <tt class="docutils literal"><span class="pre">maxn</span></tt> keyword argument can be used:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">(</span><span class="n">fibonacci</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="n">GoldenRatio</span><span class="p">)</span><span class="o">**</span><span class="mi">1000</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">maxn</span><span class="o">=</span><span class="mi">500</span><span class="p">)</span>
-<span class="go">-4.60123853010113e-210</span>
-</pre></div>
-</div>
-<p>Normally, <tt class="docutils literal"><span class="pre">maxn</span></tt> can be set very high (thousands of digits), but be aware that
-this may cause significant slowdown in extreme cases. Alternatively, the
-<tt class="docutils literal"><span class="pre">strict=True</span></tt> option can be set to force an exception instead of silently
-returning a value with less than the requested accuracy:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">(</span><span class="n">fibonacci</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="n">GoldenRatio</span><span class="p">)</span><span class="o">**</span><span class="mi">1000</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">strict</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">PrecisionExhausted</span>: <span class="n">Failed to distinguish the expression:</span>
-
-<span class="go">-sqrt(5)*GoldenRatio**1000/5 + 43466557686937456435688527675040625802564660517371780402481729089536555417949051890403879840079255169295922593080322634775209689623239873322471161642996440906533187938298969649928516003704476137795166849228875</span>
-
-<span class="go">from zero. Try simplifying the input, using chop=True, or providing a higher maxn for evalf</span>
-</pre></div>
-</div>
-<p>If we add a term so that the Fibonacci approximation becomes exact (the full
-form of Binet&#8217;s formula), we get an expression that is exactly zero, but <tt class="docutils literal"><span class="pre">N</span></tt>
-does not know this:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">fibonacci</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="n">GoldenRatio</span><span class="o">**</span><span class="mi">100</span> <span class="o">-</span> <span class="p">(</span><span class="n">GoldenRatio</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">100</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-<span class="go">0.e-104</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">maxn</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
-<span class="go">0.e-1336</span>
-</pre></div>
-</div>
-<p>In situations where such cancellations are known to occur, the <tt class="docutils literal"><span class="pre">chop</span></tt> options
-is useful. This basically replaces very small numbers in the real or
-imaginary portions of a number with exact zeros:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">chop</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">(</span><span class="mi">3</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">f</span><span class="p">,</span> <span class="n">chop</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">3.00000000000000</span>
-</pre></div>
-</div>
-<p>In situations where you wish to remove meaningless digits, re-evaluation or
-the use of the <tt class="docutils literal"><span class="pre">round</span></tt> method are useful:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Float</span><span class="p">(</span><span class="s">&#39;.1&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span><span class="o">*</span><span class="n">Float</span><span class="p">(</span><span class="s">&#39;.12345&#39;</span><span class="p">,</span> <span class="s">&#39;&#39;</span><span class="p">)</span>
-<span class="go">0.012297</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ans</span> <span class="o">=</span> <span class="n">_</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">(</span><span class="n">ans</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">0.01</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ans</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.01</span>
-</pre></div>
-</div>
-<p>If you are dealing with a numeric expression that contains no floats, it
-can be evaluated to arbitrary precision. To round the result relative to
-a given decimal, the round method is useful:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">v</span> <span class="o">=</span> <span class="mi">10</span><span class="o">*</span><span class="n">pi</span> <span class="o">+</span> <span class="n">cos</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-<span class="go">31.9562288417661</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">31.956</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="sums-and-integrals">
-<h2>Sums and integrals<a class="headerlink" href="#sums-and-integrals" title="Permalink to this headline">¶</a></h2>
-<p>Sums (in particular, infinite series) and integrals can be used like regular
-closed-form expressions, and support arbitrary-precision evaluation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;n x&#39;</span><span class="p">)</span>
-<span class="go">(n, x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">n</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">1.29128599706266</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">1.29128599706266</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">n</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">50</span><span class="p">)</span>
-<span class="go">1.2912859970626635404072825905956005414986193682745</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">50</span><span class="p">)</span>
-<span class="go">1.2912859970626635404072825905956005414986193682745</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">oo</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">30</span><span class="p">)</span>
-<span class="go">3.14159265358979323846264338328</span>
-</pre></div>
-</div>
-<p>By default, the tanh-sinh quadrature algorithm is used to evaluate integrals.
-This algorithm is very efficient and robust for smooth integrands (and even
-integrals with endpoint singularities), but may struggle with integrals that
-are highly oscillatory or have mid-interval discontinuities. In many cases,
-<tt class="docutils literal"><span class="pre">evalf</span></tt>/<tt class="docutils literal"><span class="pre">N</span></tt> will correctly estimate the error. With the following integral,
-the result is accurate but only good to four digits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">2.346</span>
-</pre></div>
-</div>
-<p>It is better to split this integral into two pieces:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="p">))</span> <span class="o">+</span> <span class="n">Integral</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="mi">4</span><span class="p">)))</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">2.34635637913639</span>
-</pre></div>
-</div>
-<p>A similar example is the following oscillatory integral:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">maxn</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
-<span class="go">0.5</span>
-</pre></div>
-</div>
-<p>It can be dealt with much more efficiently by telling <tt class="docutils literal"><span class="pre">evalf</span></tt> or <tt class="docutils literal"><span class="pre">N</span></tt> to
-use an oscillatory quadrature algorithm:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="n">quad</span><span class="o">=</span><span class="s">&#39;osc&#39;</span><span class="p">)</span>
-<span class="go">0.504067061906928</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">20</span><span class="p">,</span> <span class="n">quad</span><span class="o">=</span><span class="s">&#39;osc&#39;</span><span class="p">)</span>
-<span class="go">0.50406706190692837199</span>
-</pre></div>
-</div>
-<p>Oscillatory quadrature requires an integrand containing a factor cos(ax+b) or
-sin(ax+b). Note that many other oscillatory integrals can be transformed to
-this form with a change of variables:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">init_printing</span><span class="p">(</span><span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">wrap_line</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">no_global</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">intgrl</span> <span class="o">=</span> <span class="n">Integral</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">intgrl</span>
-<span class="go"> oo</span>
-<span class="go">  /</span>
-<span class="go"> |</span>
-<span class="go"> |  sin(x)</span>
-<span class="go"> |  ------ dx</span>
-<span class="go"> |     2</span>
-<span class="go"> |    x</span>
-<span class="go"> |</span>
-<span class="go">/</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">(</span><span class="n">intgrl</span><span class="p">,</span> <span class="n">quad</span><span class="o">=</span><span class="s">&#39;osc&#39;</span><span class="p">)</span>
-<span class="go">0.504067061906928</span>
-</pre></div>
-</div>
-<p>Infinite series use direct summation if the series converges quickly enough.
-Otherwise, extrapolation methods (generally the Euler-Maclaurin formula but
-also Richardson extrapolation) are used to speed up convergence. This allows
-high-precision evaluation of slowly convergent series:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;k&#39;</span><span class="p">)</span>
-<span class="go">k</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">1.64493406684823</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">1.64493406684823</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="o">-</span><span class="n">log</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="p">),</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">0.577215664901533</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="o">-</span><span class="n">log</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="p">),</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">50</span><span class="p">)</span>
-<span class="go">0.57721566490153286060651209008240243104215933593992</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">EulerGamma</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">50</span><span class="p">)</span>
-<span class="go">0.57721566490153286060651209008240243104215933593992</span>
-</pre></div>
-</div>
-<p>The Euler-Maclaurin formula is also used for finite series, allowing them to
-be approximated quickly without evaluating all terms:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Sum</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">10000000</span><span class="p">,</span> <span class="mi">20000000</span><span class="p">))</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">0.693147255559946</span>
-</pre></div>
-</div>
-<p>Note that <tt class="docutils literal"><span class="pre">evalf</span></tt> makes some assumptions that are not always optimal. For
-fine-tuned control over numerical summation, it might be worthwhile to manually
-use the method <tt class="docutils literal"><span class="pre">Sum.euler_maclaurin</span></tt>.</p>
-<p>Special optimizations are used for rational hypergeometric series (where the
-term is a product of polynomials, powers, factorials, binomial coefficients and
-the like). <tt class="docutils literal"><span class="pre">N</span></tt>/<tt class="docutils literal"><span class="pre">evalf</span></tt> sum series of this type very rapidly to high
-precision. For example, this Ramanujan formula for pi can be summed to 10,000
-digits in a fraction of a second with a simple command:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">factorial</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="mi">9801</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">8</span><span class="p">)</span><span class="o">/</span><span class="n">Sum</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1103</span><span class="o">+</span><span class="mi">26390</span><span class="o">*</span><span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">f</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">**</span><span class="mi">4</span><span class="o">/</span><span class="mi">396</span><span class="o">**</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">n</span><span class="p">),</span>
-<span class="gp">... </span>                        <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span> 
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="mi">10000</span><span class="p">)</span> 
-<span class="go">3.141592653589793238462643383279502884197169399375105820974944592307816406286208</span>
-<span class="go">99862803482534211706798214808651328230664709384460955058223172535940812848111745</span>
-<span class="go">02841027019385211055596446229489549303819644288109756659334461284756482337867831</span>
-<span class="gp">...</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="numerical-simplification">
-<h2>Numerical simplification<a class="headerlink" href="#numerical-simplification" title="Permalink to this headline">¶</a></h2>
-<p>The function <tt class="docutils literal"><span class="pre">nsimplify</span></tt> attempts to find a formula that is numerically equal
-to the given input. This feature can be used to guess an exact formula for an
-approximate floating-point input, or to guess a simpler formula for a
-complicated symbolic input. The algorithm used by <tt class="docutils literal"><span class="pre">nsimplify</span></tt> is capable of
-identifying simple fractions, simple algebraic expressions, linear combinations
-of given constants, and certain elementary functional transformations of any of
-the preceding.</p>
-<p>Optionally, <tt class="docutils literal"><span class="pre">nsimplify</span></tt> can be passed a list of constants to include (e.g. pi)
-and a minimum numerical tolerance. Here are some elementary examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nsimplify</span><span class="p">(</span><span class="mf">0.1</span><span class="p">)</span>
-<span class="go">1/10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsimplify</span><span class="p">(</span><span class="mf">6.28</span><span class="p">,</span> <span class="p">[</span><span class="n">pi</span><span class="p">],</span> <span class="n">tolerance</span><span class="o">=</span><span class="mf">0.01</span><span class="p">)</span>
-<span class="go">2*pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsimplify</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="n">tolerance</span><span class="o">=</span><span class="mf">0.01</span><span class="p">)</span>
-<span class="go">22/7</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsimplify</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="n">tolerance</span><span class="o">=</span><span class="mf">0.001</span><span class="p">)</span>
-<span class="go">355</span>
-<span class="go">---</span>
-<span class="go">113</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsimplify</span><span class="p">(</span><span class="mf">0.33333</span><span class="p">,</span> <span class="n">tolerance</span><span class="o">=</span><span class="mf">1e-4</span><span class="p">)</span>
-<span class="go">1/3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsimplify</span><span class="p">(</span><span class="mf">2.0</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mf">3.</span><span class="p">),</span> <span class="n">tolerance</span><span class="o">=</span><span class="mf">0.001</span><span class="p">)</span>
-<span class="go">635</span>
-<span class="go">---</span>
-<span class="go">504</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsimplify</span><span class="p">(</span><span class="mf">2.0</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mf">3.</span><span class="p">),</span> <span class="n">tolerance</span><span class="o">=</span><span class="mf">0.001</span><span class="p">,</span> <span class="n">full</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">3 ___</span>
-<span class="go">\/ 2</span>
-</pre></div>
-</div>
-<p>Here are several more advanced examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nsimplify</span><span class="p">(</span><span class="n">Float</span><span class="p">(</span><span class="s">&#39;0.130198866629986772369127970337&#39;</span><span class="p">,</span><span class="mi">30</span><span class="p">),</span> <span class="p">[</span><span class="n">pi</span><span class="p">,</span> <span class="n">E</span><span class="p">])</span>
-<span class="go">    1</span>
-<span class="go">----------</span>
-<span class="go">5*pi</span>
-<span class="go">---- + 2*E</span>
-<span class="go"> 7</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsimplify</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">atan</span><span class="p">(</span><span class="s">&#39;1/3&#39;</span><span class="p">)))</span>
-<span class="go">    ____</span>
-<span class="go">3*\/ 10</span>
-<span class="go">--------</span>
-<span class="go">   10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsimplify</span><span class="p">(</span><span class="mi">4</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">)),</span> <span class="p">[</span><span class="n">GoldenRatio</span><span class="p">])</span>
-<span class="go">-2 + 2*GoldenRatio</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsimplify</span><span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">atan</span><span class="p">(</span><span class="s">&#39;1/4&#39;</span><span class="p">)</span><span class="o">*</span><span class="n">I</span><span class="p">))</span>
-<span class="go">49   8*I</span>
-<span class="go">-- + ---</span>
-<span class="go">17    17</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsimplify</span><span class="p">((</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">/</span><span class="mi">5</span><span class="p">)</span><span class="o">+</span><span class="mi">1</span><span class="p">)))</span>
-<span class="go">           ___________</span>
-<span class="go">          /   ___</span>
-<span class="go">1        /  \/ 5    1</span>
-<span class="go">- - I*  /   ----- + -</span>
-<span class="go">2     \/      10    4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsimplify</span><span class="p">(</span><span class="n">I</span><span class="o">**</span><span class="n">I</span><span class="p">,</span> <span class="p">[</span><span class="n">pi</span><span class="p">])</span>
-<span class="go"> -pi</span>
-<span class="go"> ---</span>
-<span class="go">  2</span>
-<span class="go">e</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsimplify</span><span class="p">(</span><span class="n">Sum</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">n</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">oo</span><span class="p">)),</span> <span class="p">[</span><span class="n">pi</span><span class="p">])</span>
-<span class="go">  2</span>
-<span class="go">pi</span>
-<span class="go">---</span>
-<span class="go"> 6</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsimplify</span><span class="p">(</span><span class="n">gamma</span><span class="p">(</span><span class="s">&#39;1/4&#39;</span><span class="p">)</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="s">&#39;3/4&#39;</span><span class="p">),</span> <span class="p">[</span><span class="n">pi</span><span class="p">])</span>
-<span class="go">  ___</span>
-<span class="go">\/ 2 *pi</span>
-</pre></div>
-</div>
-</div>
-</div>
-
-
-          </div>
-        </div>
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-              <img class="logo" src="../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Numerical evaluation</a><ul>
-<li><a class="reference internal" href="#basics">Basics</a></li>
-<li><a class="reference internal" href="#floating-point-numbers">Floating-point numbers</a></li>
-<li><a class="reference internal" href="#accuracy-and-error-handling">Accuracy and error handling</a></li>
-<li><a class="reference internal" href="#sums-and-integrals">Sums and integrals</a></li>
-<li><a class="reference internal" href="#numerical-simplification">Numerical simplification</a></li>
-</ul>
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-  <div class="section" id="combinatorial">
-<h1>Combinatorial<a class="headerlink" href="#combinatorial" title="Permalink to this headline">¶</a></h1>
-<p>This module implements various combinatorial functions.</p>
-<div class="section" id="bell">
-<h2>bell<a class="headerlink" href="#bell" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.combinatorial.numbers.bell">
-<em class="property">class </em><tt class="descclassname">sympy.functions.combinatorial.numbers.</tt><tt class="descname">bell</tt><a class="reference internal" href="../../_modules/sympy/functions/combinatorial/numbers.html#bell"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.combinatorial.numbers.bell" title="Permalink to this definition">¶</a></dt>
-<dd><p>Bell numbers / Bell polynomials</p>
-<p>The Bell numbers satisfy <span class="math">\(B_0 = 1\)</span> and</p>
-<div class="math">
-\[B_n = \sum_{k=0}^{n-1} \binom{n-1}{k} B_k.\]</div>
-<p>They are also given by:</p>
-<div class="math">
-\[B_n = \frac{1}{e} \sum_{k=0}^{\infty} \frac{k^n}{k!}.\]</div>
-<p>The Bell polynomials are given by <span class="math">\(B_0(x) = 1\)</span> and</p>
-<div class="math">
-\[B_n(x) = x \sum_{k=1}^{n-1} \binom{n-1}{k-1} B_{k-1}(x).\]</div>
-<p>The second kind of Bell polynomials (are sometimes called &#8220;partial&#8221; Bell
-polynomials or incomplete Bell polynomials) are defined as</p>
-<div class="math">
-\[B_{n,k}(x_1, x_2,\dotsc x_{n-k+1}) =
-\sum_{j_1+j_2+j_2+\dotsb=k \atop j_1+2j_2+3j_2+\dotsb=n}
-    \frac{n!}{j_1!j_2!\dotsb j_{n-k+1}!}
-    \left(\frac{x_1}{1!} \right)^{j_1}
-    \left(\frac{x_2}{2!} \right)^{j_2} \dotsb
-    \left(\frac{x_{n-k+1}}{(n-k+1)!} \right) ^{j_{n-k+1}}.\]</div>
-<ul class="simple">
-<li>bell(n) gives the <span class="math">\(n^{th}\)</span> Bell number, <span class="math">\(B_n\)</span>.</li>
-<li>bell(n, x) gives the <span class="math">\(n^{th}\)</span> Bell polynomial, <span class="math">\(B_n(x)\)</span>.</li>
-<li>bell(n, k, (x1, x2, ...)) gives Bell polynomials of the second kind,
-<span class="math">\(B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1})\)</span>.</li>
-</ul>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">euler</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">bernoulli</span></tt></p>
-</div>
-<p class="rubric">Notes</p>
-<p>Not to be confused with Bernoulli numbers and Bernoulli polynomials,
-which use the same notation.</p>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Bell_number">http://en.wikipedia.org/wiki/Bell_number</a></li>
-<li><a class="reference external" href="http://mathworld.wolfram.com/BellNumber.html">http://mathworld.wolfram.com/BellNumber.html</a></li>
-<li><a class="reference external" href="http://mathworld.wolfram.com/BellPolynomial.html">http://mathworld.wolfram.com/BellPolynomial.html</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">bell</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">symbols</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">bell</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">11</span><span class="p">)]</span>
-<span class="go">[1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bell</span><span class="p">(</span><span class="mi">30</span><span class="p">)</span>
-<span class="go">846749014511809332450147</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bell</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">))</span>
-<span class="go">t**4 + 6*t**3 + 7*t**2 + t</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bell</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x:6&#39;</span><span class="p">)[</span><span class="mi">1</span><span class="p">:])</span>
-<span class="go">6*x1*x5 + 15*x2*x4 + 10*x3**2</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</div>
-<div class="section" id="bernoulli">
-<h2>bernoulli<a class="headerlink" href="#bernoulli" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.combinatorial.numbers.bernoulli">
-<em class="property">class </em><tt class="descclassname">sympy.functions.combinatorial.numbers.</tt><tt class="descname">bernoulli</tt><a class="reference internal" href="../../_modules/sympy/functions/combinatorial/numbers.html#bernoulli"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.combinatorial.numbers.bernoulli" title="Permalink to this definition">¶</a></dt>
-<dd><p>Bernoulli numbers / Bernoulli polynomials</p>
-<p>The Bernoulli numbers are a sequence of rational numbers
-defined by B_0 = 1 and the recursive relation (n &gt; 0):</p>
-<div class="highlight-python"><pre>      n
-     ___
-    \      / n + 1 \
-0 =  )     |       | * B .
-    /___   \   k   /    k
-    k = 0</pre>
-</div>
-<p>They are also commonly defined by their exponential generating
-function, which is x/(exp(x) - 1). For odd indices &gt; 1, the
-Bernoulli numbers are zero.</p>
-<p>The Bernoulli polynomials satisfy the analogous formula:</p>
-<div class="highlight-python"><pre>          n
-         ___
-        \      / n \         n-k
-B (x) =  )     |   | * B  * x   .
- n      /___   \ k /    k
-        k = 0</pre>
-</div>
-<p>Bernoulli numbers and Bernoulli polynomials are related as
-B_n(0) = B_n.</p>
-<p>We compute Bernoulli numbers using Ramanujan&#8217;s formula:</p>
-<div class="highlight-python"><pre>                         / n + 3 \
-B   =  (A(n) - S(n))  /  |       |
- n                       \   n   /</pre>
-</div>
-<p>where A(n) = (n+3)/3 when n = 0 or 2 (mod 6), A(n) = -(n+3)/6
-when n = 4 (mod 6), and:</p>
-<div class="highlight-python"><pre>       [n/6]
-        ___
-       \      /  n + 3  \
-S(n) =  )     |         | * B
-       /___   \ n - 6*k /    n-6*k
-       k = 1</pre>
-</div>
-<p>This formula is similar to the sum given in the definition, but
-cuts 2/3 of the terms. For Bernoulli polynomials, we use the
-formula in the definition.</p>
-<ul class="simple">
-<li>bernoulli(n) gives the nth Bernoulli number, B_n</li>
-<li>bernoulli(n, x) gives the nth Bernoulli polynomial in x, B_n(x)</li>
-</ul>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">euler</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">bell</span></tt></p>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Bernoulli_number">http://en.wikipedia.org/wiki/Bernoulli_number</a></li>
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Bernoulli_polynomial">http://en.wikipedia.org/wiki/Bernoulli_polynomial</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">bernoulli</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">bernoulli</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">11</span><span class="p">)]</span>
-<span class="go">[1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bernoulli</span><span class="p">(</span><span class="mi">1000001</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</div>
-<div class="section" id="binomial">
-<h2>binomial<a class="headerlink" href="#binomial" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.combinatorial.factorials.binomial">
-<em class="property">class </em><tt class="descclassname">sympy.functions.combinatorial.factorials.</tt><tt class="descname">binomial</tt><a class="reference internal" href="../../_modules/sympy/functions/combinatorial/factorials.html#binomial"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.combinatorial.factorials.binomial" title="Permalink to this definition">¶</a></dt>
-<dd><p>Implementation of the binomial coefficient. It can be defined
-in two ways depending on its desired interpretation:</p>
-<blockquote>
-<div>C(n,k) = n!/(k!(n-k)!)   or   C(n, k) = ff(n, k)/k!</div></blockquote>
-<p>First, in a strict combinatorial sense it defines the
-number of ways we can choose &#8216;k&#8217; elements from a set of
-&#8216;n&#8217; elements. In this case both arguments are nonnegative
-integers and binomial is computed using an efficient
-algorithm based on prime factorization.</p>
-<p>The other definition is generalization for arbitrary &#8216;n&#8217;,
-however &#8216;k&#8217; must also be nonnegative. This case is very
-useful when evaluating summations.</p>
-<p>For the sake of convenience for negative &#8216;k&#8217; this function
-will return zero no matter what valued is the other argument.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Rational</span><span class="p">,</span> <span class="n">binomial</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">binomial</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span>
-<span class="go">6435</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">binomial</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">[</span> <span class="n">binomial</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">)]</span>
-<span class="go">[1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span> <span class="n">binomial</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">)]</span>
-<span class="go">[1, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span> <span class="n">binomial</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">)]</span>
-<span class="go">[1, 2, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span> <span class="n">binomial</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">)]</span>
-<span class="go">[1, 3, 3, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span> <span class="n">binomial</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">)]</span>
-<span class="go">[1, 4, 6, 4, 1]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">binomial</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">4</span><span class="p">),</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">-5/128</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">binomial</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">n*(n - 2)*(n - 1)/6</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="catalan">
-<h2>catalan<a class="headerlink" href="#catalan" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.combinatorial.numbers.catalan">
-<em class="property">class </em><tt class="descclassname">sympy.functions.combinatorial.numbers.</tt><tt class="descname">catalan</tt><a class="reference internal" href="../../_modules/sympy/functions/combinatorial/numbers.html#catalan"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.combinatorial.numbers.catalan" title="Permalink to this definition">¶</a></dt>
-<dd><p>Catalan numbers</p>
-<p>The n-th catalan number is given by:</p>
-<div class="highlight-python"><pre>       1   / 2*n \
-C  = ----- |     |
- n   n + 1 \  n  /</pre>
-</div>
-<ul class="simple">
-<li>catalan(n) gives the n-th Catalan number, C_n</li>
-</ul>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.functions.combinatorial.factorials.binomial" title="sympy.functions.combinatorial.factorials.binomial"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.functions.combinatorial.factorials.binomial</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Catalan_number">http://en.wikipedia.org/wiki/Catalan_number</a></li>
-<li><a class="reference external" href="http://mathworld.wolfram.com/CatalanNumber.html">http://mathworld.wolfram.com/CatalanNumber.html</a></li>
-<li><a class="reference external" href="http://geometer.org/mathcircles/catalan.pdf">http://geometer.org/mathcircles/catalan.pdf</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="p">(</span><span class="n">Symbol</span><span class="p">,</span> <span class="n">binomial</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">hyper</span><span class="p">,</span> <span class="n">polygamma</span><span class="p">,</span>
-<span class="gp">... </span>            <span class="n">catalan</span><span class="p">,</span> <span class="n">diff</span><span class="p">,</span> <span class="n">combsimp</span><span class="p">,</span> <span class="n">Rational</span><span class="p">,</span> <span class="n">I</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">[</span> <span class="n">catalan</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">10</span><span class="p">)</span> <span class="p">]</span>
-<span class="go">[1, 2, 5, 14, 42, 132, 429, 1430, 4862]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">catalan</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">catalan(n)</span>
-</pre></div>
-</div>
-<p>Catalan numbers can be transformed into several other, identical
-expressions involving other mathematical functions</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">catalan</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">binomial</span><span class="p">)</span>
-<span class="go">binomial(2*n, n)/(n + 1)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">catalan</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">gamma</span><span class="p">)</span>
-<span class="go">4**n*gamma(n + 1/2)/(sqrt(pi)*gamma(n + 2))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">catalan</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">hyper</span><span class="p">)</span>
-<span class="go">hyper((-n + 1, -n), (2,), 1)</span>
-</pre></div>
-</div>
-<p>For some non-integer values of n we can get closed form
-expressions by rewriting in terms of gamma functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">catalan</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">gamma</span><span class="p">)</span>
-<span class="go">8/(3*pi)</span>
-</pre></div>
-</div>
-<p>We can differentiate the Catalan numbers C(n) interpreted as a
-continuous real funtion in n:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">catalan</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">n</span><span class="p">)</span>
-<span class="go">(polygamma(0, n + 1/2) - polygamma(0, n + 2) + 2*log(2))*catalan(n)</span>
-</pre></div>
-</div>
-<p>As a more advanced example consider the following ratio
-between consecutive numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">combsimp</span><span class="p">((</span><span class="n">catalan</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">catalan</span><span class="p">(</span><span class="n">n</span><span class="p">))</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">binomial</span><span class="p">))</span>
-<span class="go">2*(2*n + 1)/(n + 2)</span>
-</pre></div>
-</div>
-<p>The Catalan numbers can be generalized to complex numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">catalan</span><span class="p">(</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">gamma</span><span class="p">)</span>
-<span class="go">4**I*gamma(1/2 + I)/(sqrt(pi)*gamma(2 + I))</span>
-</pre></div>
-</div>
-<p>and evaluated with arbitrary precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">catalan</span><span class="p">(</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">20</span><span class="p">)</span>
-<span class="go">0.39764993382373624267 - 0.020884341620842555705*I</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</div>
-<div class="section" id="euler">
-<h2>euler<a class="headerlink" href="#euler" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.combinatorial.numbers.euler">
-<em class="property">class </em><tt class="descclassname">sympy.functions.combinatorial.numbers.</tt><tt class="descname">euler</tt><a class="reference internal" href="../../_modules/sympy/functions/combinatorial/numbers.html#euler"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.combinatorial.numbers.euler" title="Permalink to this definition">¶</a></dt>
-<dd><p>Euler numbers</p>
-<p>The euler numbers are given by:</p>
-<div class="highlight-python"><pre>        2*n+1   k
-         ___   ___            j          2*n+1
-        \     \     / k \ (-1)  * (k-2*j)
-E   = I  )     )    |   | --------------------
- 2n     /___  /___  \ j /      k    k
-        k = 1 j = 0           2  * I  * k
-
-E     = 0
- 2n+1</pre>
-</div>
-<ul class="simple">
-<li>euler(n) gives the n-th Euler number, E_n</li>
-</ul>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">bernoulli</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">bell</span></tt></p>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Euler_numbers">http://en.wikipedia.org/wiki/Euler_numbers</a></li>
-<li><a class="reference external" href="http://mathworld.wolfram.com/EulerNumber.html">http://mathworld.wolfram.com/EulerNumber.html</a></li>
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Alternating_permutation">http://en.wikipedia.org/wiki/Alternating_permutation</a></li>
-<li><a class="reference external" href="http://mathworld.wolfram.com/AlternatingPermutation.html">http://mathworld.wolfram.com/AlternatingPermutation.html</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">euler</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">euler</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">)]</span>
-<span class="go">[1, 0, -1, 0, 5, 0, -61, 0, 1385, 0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">euler</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="p">)</span>
-<span class="go">euler(3*n)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="factorial">
-<h2>factorial<a class="headerlink" href="#factorial" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.combinatorial.factorials.factorial">
-<em class="property">class </em><tt class="descclassname">sympy.functions.combinatorial.factorials.</tt><tt class="descname">factorial</tt><a class="reference internal" href="../../_modules/sympy/functions/combinatorial/factorials.html#factorial"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.combinatorial.factorials.factorial" title="Permalink to this definition">¶</a></dt>
-<dd><p>Implementation of factorial function over nonnegative integers.
-For the sake of convenience and simplicity of procedures using
-this function it is defined for negative integers and returns
-zero in this case.</p>
-<p>The factorial is very important in combinatorics where it gives
-the number of ways in which &#8216;n&#8217; objects can be permuted. It also
-arises in calculus, probability, number theory etc.</p>
-<p>There is strict relation of factorial with gamma function. In
-fact n! = gamma(n+1) for nonnegative integers. Rewrite of this
-kind is very useful in case of combinatorial simplification.</p>
-<p>Computation of the factorial is done using two algorithms. For
-small arguments naive product is evaluated. However for bigger
-input algorithm Prime-Swing is used. It is the fastest algorithm
-known and computes n! via prime factorization of special class
-of numbers, called here the &#8216;Swing Numbers&#8217;.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">factorial2</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">RisingFactorial</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">FallingFactorial</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">factorial</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="mi">7</span><span class="p">)</span>
-<span class="go">5040</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">n!</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="p">)</span>
-<span class="go">(2*n)!</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="factorial2-double-factorial">
-<h2>factorial2 / double factorial<a class="headerlink" href="#factorial2-double-factorial" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.combinatorial.factorials.factorial2">
-<em class="property">class </em><tt class="descclassname">sympy.functions.combinatorial.factorials.</tt><tt class="descname">factorial2</tt><a class="reference internal" href="../../_modules/sympy/functions/combinatorial/factorials.html#factorial2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.combinatorial.factorials.factorial2" title="Permalink to this definition">¶</a></dt>
-<dd><p>The double factorial n!!, not to be confused with (n!)!</p>
-<p>The double factorial is defined for integers &gt;= -1 as:</p>
-<div class="highlight-python"><pre>       ,
-      |  n*(n - 2)*(n - 4)* ... * 1    for n odd
-n!! = {  n*(n - 2)*(n - 4)* ... * 2    for n even
-      |  1                             for n = 0, -1
-       `</pre>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">factorial</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">RisingFactorial</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">FallingFactorial</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial2</span><span class="p">,</span> <span class="n">var</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">)</span>
-<span class="go">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorial2</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">(n + 1)!!</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorial2</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorial2</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="fallingfactorial">
-<h2>FallingFactorial<a class="headerlink" href="#fallingfactorial" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.combinatorial.factorials.FallingFactorial">
-<em class="property">class </em><tt class="descclassname">sympy.functions.combinatorial.factorials.</tt><tt class="descname">FallingFactorial</tt><a class="reference internal" href="../../_modules/sympy/functions/combinatorial/factorials.html#FallingFactorial"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.combinatorial.factorials.FallingFactorial" title="Permalink to this definition">¶</a></dt>
-<dd><p>Falling factorial (related to rising factorial) is a double valued
-function arising in concrete mathematics, hypergeometric functions
-and series expansions. It is defined by</p>
-<blockquote>
-<div>ff(x, k) = x * (x-1) * ... * (x - k+1)</div></blockquote>
-<p>where &#8216;x&#8217; can be arbitrary expression and &#8216;k&#8217; is an integer. For
-more information check &#8220;Concrete mathematics&#8221; by Graham, pp. 66
-or visit <a class="reference external" href="http://mathworld.wolfram.com/FallingFactorial.html">http://mathworld.wolfram.com/FallingFactorial.html</a> page.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">ff</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ff</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">120</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span> <span class="o">==</span> <span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">3</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">factorial</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">factorial2</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">RisingFactorial</span></tt></p>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="fibonacci">
-<h2>fibonacci<a class="headerlink" href="#fibonacci" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.combinatorial.numbers.fibonacci">
-<em class="property">class </em><tt class="descclassname">sympy.functions.combinatorial.numbers.</tt><tt class="descname">fibonacci</tt><a class="reference internal" href="../../_modules/sympy/functions/combinatorial/numbers.html#fibonacci"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.combinatorial.numbers.fibonacci" title="Permalink to this definition">¶</a></dt>
-<dd><p>Fibonacci numbers / Fibonacci polynomials</p>
-<p>The Fibonacci numbers are the integer sequence defined by the
-initial terms F_0 = 0, F_1 = 1 and the two-term recurrence
-relation F_n = F_{n-1} + F_{n-2}.</p>
-<p>The Fibonacci polynomials are defined by F_1(x) = 1,
-F_2(x) = x, and F_n(x) = x*F_{n-1}(x) + F_{n-2}(x) for n &gt; 2.
-For all positive integers n, F_n(1) = F_n.</p>
-<ul class="simple">
-<li>fibonacci(n) gives the nth Fibonacci number, F_n</li>
-<li>fibonacci(n, x) gives the nth Fibonacci polynomial in x, F_n(x)</li>
-</ul>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">lucas</span></tt></p>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Fibonacci_number">http://en.wikipedia.org/wiki/Fibonacci_number</a></li>
-<li><a class="reference external" href="http://mathworld.wolfram.com/FibonacciNumber.html">http://mathworld.wolfram.com/FibonacciNumber.html</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">fibonacci</span><span class="p">,</span> <span class="n">Symbol</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">fibonacci</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">11</span><span class="p">)]</span>
-<span class="go">[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fibonacci</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">))</span>
-<span class="go">t**4 + 3*t**2 + 1</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</div>
-<div class="section" id="harmonic">
-<h2>harmonic<a class="headerlink" href="#harmonic" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.combinatorial.numbers.harmonic">
-<em class="property">class </em><tt class="descclassname">sympy.functions.combinatorial.numbers.</tt><tt class="descname">harmonic</tt><a class="reference internal" href="../../_modules/sympy/functions/combinatorial/numbers.html#harmonic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.combinatorial.numbers.harmonic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Harmonic numbers</p>
-<p>The nth harmonic number is given by 1 + 1/2 + 1/3 + ... + 1/n.</p>
-<p>More generally:</p>
-<div class="highlight-python"><pre>         n
-        ___
-       \       -m
-H    =  )     k   .
- n,m   /___
-       k = 1</pre>
-</div>
-<p>As n -&gt; oo, H_{n,m} -&gt; zeta(m) (the Riemann zeta function)</p>
-<ul class="simple">
-<li>harmonic(n) gives the nth harmonic number, H_n</li>
-<li>harmonic(n, m) gives the nth generalized harmonic number
-of order m, H_{n,m}, where harmonic(n) == harmonic(n, 1)</li>
-</ul>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Harmonic_number">http://en.wikipedia.org/wiki/Harmonic_number</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">harmonic</span><span class="p">,</span> <span class="n">oo</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">harmonic</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">6</span><span class="p">)]</span>
-<span class="go">[0, 1, 3/2, 11/6, 25/12, 137/60]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">harmonic</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">6</span><span class="p">)]</span>
-<span class="go">[0, 1, 5/4, 49/36, 205/144, 5269/3600]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">harmonic</span><span class="p">(</span><span class="n">oo</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">pi**2/6</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="lucas">
-<h2>lucas<a class="headerlink" href="#lucas" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.combinatorial.numbers.lucas">
-<em class="property">class </em><tt class="descclassname">sympy.functions.combinatorial.numbers.</tt><tt class="descname">lucas</tt><a class="reference internal" href="../../_modules/sympy/functions/combinatorial/numbers.html#lucas"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.combinatorial.numbers.lucas" title="Permalink to this definition">¶</a></dt>
-<dd><p>Lucas numbers</p>
-<p>Lucas numbers satisfy a recurrence relation similar to that of
-the Fibonacci sequence, in which each term is the sum of the
-preceding two. They are generated by choosing the initial
-values L_0 = 2 and L_1 = 1.</p>
-<ul class="simple">
-<li>lucas(n) gives the nth Lucas number</li>
-</ul>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">fibonacci</span></tt></p>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Lucas_number">http://en.wikipedia.org/wiki/Lucas_number</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">lucas</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">lucas</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">11</span><span class="p">)]</span>
-<span class="go">[2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123]</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</div>
-<div class="section" id="multifactorial">
-<h2>MultiFactorial<a class="headerlink" href="#multifactorial" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.combinatorial.factorials.MultiFactorial">
-<em class="property">class </em><tt class="descclassname">sympy.functions.combinatorial.factorials.</tt><tt class="descname">MultiFactorial</tt><a class="reference internal" href="../../_modules/sympy/functions/combinatorial/factorials.html#MultiFactorial"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.combinatorial.factorials.MultiFactorial" title="Permalink to this definition">¶</a></dt>
-<dd><p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</div>
-<div class="section" id="risingfactorial">
-<h2>RisingFactorial<a class="headerlink" href="#risingfactorial" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.combinatorial.factorials.RisingFactorial">
-<em class="property">class </em><tt class="descclassname">sympy.functions.combinatorial.factorials.</tt><tt class="descname">RisingFactorial</tt><a class="reference internal" href="../../_modules/sympy/functions/combinatorial/factorials.html#RisingFactorial"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.combinatorial.factorials.RisingFactorial" title="Permalink to this definition">¶</a></dt>
-<dd><p>Rising factorial (also called Pochhammer symbol) is a double valued
-function arising in concrete mathematics, hypergeometric functions
-and series expansions. It is defined by:</p>
-<blockquote>
-<div>rf(x, k) = x * (x+1) * ... * (x + k-1)</div></blockquote>
-<p>where &#8216;x&#8217; can be arbitrary expression and &#8216;k&#8217; is an integer. For
-more information check &#8220;Concrete mathematics&#8221; by Graham, pp. 66
-or visit <a class="reference external" href="http://mathworld.wolfram.com/RisingFactorial.html">http://mathworld.wolfram.com/RisingFactorial.html</a> page.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">factorial</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">factorial2</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">FallingFactorial</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">rf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rf</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rf</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">120</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rf</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span> <span class="o">==</span> <span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">3</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">4</span> <span class="o">+</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../index.html">
-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Combinatorial</a><ul>
-<li><a class="reference internal" href="#bell">bell</a></li>
-<li><a class="reference internal" href="#bernoulli">bernoulli</a></li>
-<li><a class="reference internal" href="#binomial">binomial</a></li>
-<li><a class="reference internal" href="#catalan">catalan</a></li>
-<li><a class="reference internal" href="#euler">euler</a></li>
-<li><a class="reference internal" href="#factorial">factorial</a></li>
-<li><a class="reference internal" href="#factorial2-double-factorial">factorial2 / double factorial</a></li>
-<li><a class="reference internal" href="#fallingfactorial">FallingFactorial</a></li>
-<li><a class="reference internal" href="#fibonacci">fibonacci</a></li>
-<li><a class="reference internal" href="#harmonic">harmonic</a></li>
-<li><a class="reference internal" href="#lucas">lucas</a></li>
-<li><a class="reference internal" href="#multifactorial">MultiFactorial</a></li>
-<li><a class="reference internal" href="#risingfactorial">RisingFactorial</a></li>
-</ul>
-</li>
-</ul>
-
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-            
-  <div class="section" id="elementary">
-<h1>Elementary<a class="headerlink" href="#elementary" title="Permalink to this headline">¶</a></h1>
-<p>This module implements elementary functions, as well as functions like Abs,
-Max, etc.</p>
-<div class="section" id="abs">
-<h2>Abs<a class="headerlink" href="#abs" title="Permalink to this headline">¶</a></h2>
-<p>Returns the absolute value of the argument.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">Abs</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Abs</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<dl class="class">
-<dt id="sympy.functions.elementary.complexes.Abs">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.complexes.</tt><tt class="descname">Abs</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/complexes.html#Abs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.complexes.Abs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the absolute value of the argument.</p>
-<p>This is an extension of the built-in function abs() to accept symbolic
-values.  If you pass a SymPy expression to the built-in abs(), it will
-pass it automatically to Abs().</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">sign</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">conjugate</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Abs</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Abs</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Abs</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">)</span>
-<span class="go">Abs(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Abs</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">abs</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">)</span> <span class="c"># The Python built-in</span>
-<span class="go">Abs(x)</span>
-</pre></div>
-</div>
-<p>Note that the Python built-in will return either an Expr or int depending on
-the argument:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span>
-<span class="go">&lt;... &#39;int&#39;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">NegativeOne</span><span class="p">))</span>
-<span class="go">&lt;class &#39;sympy.core.numbers.One&#39;&gt;</span>
-</pre></div>
-</div>
-<p>Abs will always return a sympy object.</p>
-<dl class="function">
-<dt id="sympy.functions.elementary.complexes.Abs.fdiff">
-<tt class="descname">fdiff</tt><big>(</big><em>self</em>, <em>argindex=1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/complexes.html#Abs.fdiff"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.complexes.Abs.fdiff" title="Permalink to this definition">¶</a></dt>
-<dd><p>Get the first derivative of the argument to Abs().</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">Abs</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Abs</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">fdiff</span><span class="p">()</span>
-<span class="go">sign(x)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="acos">
-<h2>acos<a class="headerlink" href="#acos" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.trigonometric.acos">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.trigonometric.</tt><tt class="descname">acos</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/trigonometric.html#acos"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.trigonometric.acos" title="Permalink to this definition">¶</a></dt>
-<dd><div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">asin</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">atan</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">cos</span></tt></p>
-</div>
-<p class="rubric">Notes</p>
-<ul class="simple">
-<li>acos(x) will evaluate automatically in the cases
-oo, -oo, 0, 1, -1</li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">acos</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">acos</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">acos</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">pi/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">acos</span><span class="p">(</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">oo*I</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="acosh">
-<h2>acosh<a class="headerlink" href="#acosh" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.hyperbolic.acosh">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.hyperbolic.</tt><tt class="descname">acosh</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/hyperbolic.html#acosh"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.hyperbolic.acosh" title="Permalink to this definition">¶</a></dt>
-<dd><p>The inverse hyperbolic cosine function.</p>
-<ul class="simple">
-<li>acosh(x) -&gt; Returns the inverse hyperbolic cosine of x</li>
-</ul>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">asinh</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">atanh</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">cosh</span></tt></p>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="acot">
-<h2>acot<a class="headerlink" href="#acot" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.trigonometric.acot">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.trigonometric.</tt><tt class="descname">acot</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/trigonometric.html#acot"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.trigonometric.acot" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="acoth">
-<h2>acoth<a class="headerlink" href="#acoth" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.hyperbolic.acoth">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.hyperbolic.</tt><tt class="descname">acoth</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/hyperbolic.html#acoth"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.hyperbolic.acoth" title="Permalink to this definition">¶</a></dt>
-<dd><p>The inverse hyperbolic cotangent function.</p>
-<ul class="simple">
-<li>acoth(x) -&gt; Returns the inverse hyperbolic cotangent of x</li>
-</ul>
-</dd></dl>
-
-</div>
-<div class="section" id="arg">
-<h2>arg<a class="headerlink" href="#arg" title="Permalink to this headline">¶</a></h2>
-<p>Returns the argument (in radians) of a complex number. For a real
-number, the argument is always 0.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">arg</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">I</span><span class="p">,</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">arg</span><span class="p">(</span><span class="mf">2.0</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">arg</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
-<span class="go">pi/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">arg</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">pi/4</span>
-</pre></div>
-</div>
-<dl class="class">
-<dt id="sympy.functions.elementary.complexes.arg">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.complexes.</tt><tt class="descname">arg</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/complexes.html#arg"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.complexes.arg" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the argument (in radians) of a complex number</p>
-</dd></dl>
-
-</div>
-<div class="section" id="asin">
-<h2>asin<a class="headerlink" href="#asin" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.trigonometric.asin">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.trigonometric.</tt><tt class="descname">asin</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/trigonometric.html#asin"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.trigonometric.asin" title="Permalink to this definition">¶</a></dt>
-<dd><div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">acos</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">atan</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">sin</span></tt></p>
-</div>
-<p class="rubric">Notes</p>
-<ul class="simple">
-<li>asin(x) will evaluate automatically in the cases
-oo, -oo, 0, 1, -1</li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">asin</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asin</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">pi/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asin</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-pi/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="asinh">
-<h2>asinh<a class="headerlink" href="#asinh" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.hyperbolic.asinh">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.hyperbolic.</tt><tt class="descname">asinh</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/hyperbolic.html#asinh"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.hyperbolic.asinh" title="Permalink to this definition">¶</a></dt>
-<dd><p>The inverse hyperbolic sine function.</p>
-<ul class="simple">
-<li>asinh(x) -&gt; Returns the inverse hyperbolic sine of x</li>
-</ul>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">acosh</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">atanh</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">sinh</span></tt></p>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="atan">
-<h2>atan<a class="headerlink" href="#atan" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.trigonometric.atan">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.trigonometric.</tt><tt class="descname">atan</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/trigonometric.html#atan"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.trigonometric.atan" title="Permalink to this definition">¶</a></dt>
-<dd><div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">acos</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">asin</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">tan</span></tt></p>
-</div>
-<p class="rubric">Notes</p>
-<ul class="simple">
-<li>atan(x) will evaluate automatically in the cases
-oo, -oo, 0, 1, -1</li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">atan</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">atan</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">atan</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">pi/4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">atan</span><span class="p">(</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">pi/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="atan2">
-<h2>atan2<a class="headerlink" href="#atan2" title="Permalink to this headline">¶</a></h2>
-<p>This function is like <span class="math">\(atan\)</span>, but considers the sign of both arguments in
-order to correctly determine the quadrant of its result.</p>
-<dl class="class">
-<dt id="sympy.functions.elementary.trigonometric.atan2">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.trigonometric.</tt><tt class="descname">atan2</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/trigonometric.html#atan2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.trigonometric.atan2" title="Permalink to this definition">¶</a></dt>
-<dd><p>atan2(y,x) -&gt; Returns the atan(y/x) taking two arguments y and x.
-Signs of both y and x are considered to determine the appropriate
-quadrant of atan(y/x). The range is (-pi, pi].</p>
-</dd></dl>
-
-</div>
-<div class="section" id="atanh">
-<h2>atanh<a class="headerlink" href="#atanh" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.hyperbolic.atanh">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.hyperbolic.</tt><tt class="descname">atanh</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/hyperbolic.html#atanh"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.hyperbolic.atanh" title="Permalink to this definition">¶</a></dt>
-<dd><p>The inverse hyperbolic tangent function.</p>
-<ul class="simple">
-<li>atanh(x) -&gt; Returns the inverse hyperbolic tangent of x</li>
-</ul>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">asinh</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">acosh</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">tanh</span></tt></p>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="ceiling">
-<h2>ceiling<a class="headerlink" href="#ceiling" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.integers.ceiling">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.integers.</tt><tt class="descname">ceiling</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/integers.html#ceiling"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.integers.ceiling" title="Permalink to this definition">¶</a></dt>
-<dd><p>Ceiling is a univariate function which returns the smallest integer
-value not less than its argument. Ceiling function is generalized
-in this implementation to complex numbers.</p>
-<p>More information can be found in &#8220;Concrete mathematics&#8221; by Graham,
-pp. 87 or visit <a class="reference external" href="http://mathworld.wolfram.com/CeilingFunction.html">http://mathworld.wolfram.com/CeilingFunction.html</a>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">ceiling</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">Float</span><span class="p">,</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ceiling</span><span class="p">(</span><span class="mi">17</span><span class="p">)</span>
-<span class="go">17</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ceiling</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="mi">23</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
-<span class="go">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ceiling</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">E</span><span class="p">)</span>
-<span class="go">6</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ceiling</span><span class="p">(</span><span class="o">-</span><span class="n">Float</span><span class="p">(</span><span class="mf">0.567</span><span class="p">))</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ceiling</span><span class="p">(</span><span class="n">I</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">I</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">floor</span></tt></p>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="conjugate">
-<h2>conjugate<a class="headerlink" href="#conjugate" title="Permalink to this headline">¶</a></h2>
-<p>Returns the <a class="reference external" href="http://en.wikipedia.org/wiki/Complex_conjugation">complex conjugate</a>
-of an argument. In mathematics, the complex conjugate of a complex number is given
-by changing the sign of the imaginary part. Thus, the conjugate of the complex number</p>
-<blockquote>
-<div><span class="math">\(a + ib\)</span></div></blockquote>
-<p>(where a and b are real numbers) is</p>
-<blockquote>
-<div><span class="math">\(a - ib\)</span></div></blockquote>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">conjugate</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">conjugate</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">conjugate</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
-<span class="go">-I</span>
-</pre></div>
-</div>
-<dl class="class">
-<dt id="sympy.functions.elementary.complexes.conjugate">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.complexes.</tt><tt class="descname">conjugate</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/complexes.html#conjugate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.complexes.conjugate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Changes the sign of the imaginary part of a complex number.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">sign</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">Abs</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">conjugate</span><span class="p">,</span> <span class="n">I</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">conjugate</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">I</span><span class="p">)</span>
-<span class="go">1 - I</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="cos">
-<h2>cos<a class="headerlink" href="#cos" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.trigonometric.cos">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.trigonometric.</tt><tt class="descname">cos</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/trigonometric.html#cos"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.trigonometric.cos" title="Permalink to this definition">¶</a></dt>
-<dd><p>The cosine function.</p>
-<ul class="simple">
-<li>cos(x) -&gt; Returns the cosine of x (measured in radians)</li>
-</ul>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">sin</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">tan</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">acos</span></tt></p>
-</div>
-<p class="rubric">Notes</p>
-<ul class="simple">
-<li>cos(x) will evaluate automatically in the case x
-is a multiple of pi, pi/2, pi/3, pi/4 and pi/6.</li>
-</ul>
-<p class="rubric">References</p>
-<p>U{Definitions in trigonometry&lt;<a class="reference external" href="http://planetmath.org/encyclopedia/DefinitionsInTrigonometry.html">http://planetmath.org/encyclopedia/DefinitionsInTrigonometry.html</a>&gt;}</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">cos</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">-2*x*sin(x**2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">-1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">-1/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="cosh">
-<h2>cosh<a class="headerlink" href="#cosh" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.hyperbolic.cosh">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.hyperbolic.</tt><tt class="descname">cosh</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/hyperbolic.html#cosh"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.hyperbolic.cosh" title="Permalink to this definition">¶</a></dt>
-<dd><p>The hyperbolic cosine function, <span class="math">\(\frac{exp(x) + exp(-x)}{2}\)</span>.</p>
-<ul class="simple">
-<li>cosh(x) -&gt; Returns the hyperbolic cosine of x</li>
-</ul>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">sinh</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">tanh</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">acosh</span></tt></p>
-</div>
-<dl class="function">
-<dt id="sympy.functions.elementary.hyperbolic.cosh.inverse">
-<tt class="descname">inverse</tt><big>(</big><em>self</em>, <em>argindex=1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/hyperbolic.html#cosh.inverse"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.hyperbolic.cosh.inverse" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the inverse of this function.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="cot">
-<h2>cot<a class="headerlink" href="#cot" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.trigonometric.cot">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.trigonometric.</tt><tt class="descname">cot</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/trigonometric.html#cot"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.trigonometric.cot" title="Permalink to this definition">¶</a></dt>
-<dd><dl class="function">
-<dt id="sympy.functions.elementary.trigonometric.cot.inverse">
-<tt class="descname">inverse</tt><big>(</big><em>self</em>, <em>argindex=1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/trigonometric.html#cot.inverse"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.trigonometric.cot.inverse" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the inverse of this function.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="coth">
-<h2>coth<a class="headerlink" href="#coth" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.hyperbolic.coth">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.hyperbolic.</tt><tt class="descname">coth</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/hyperbolic.html#coth"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.hyperbolic.coth" title="Permalink to this definition">¶</a></dt>
-<dd><p>The hyperbolic tangent function, <span class="math">\(\frac{cosh(x)}{sinh(x)}\)</span>.</p>
-<ul class="simple">
-<li>coth(x) -&gt; Returns the hyperbolic cotangent of x</li>
-</ul>
-<dl class="function">
-<dt id="sympy.functions.elementary.hyperbolic.coth.inverse">
-<tt class="descname">inverse</tt><big>(</big><em>self</em>, <em>argindex=1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/hyperbolic.html#coth.inverse"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.hyperbolic.coth.inverse" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the inverse of this function.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="exp">
-<h2>exp<a class="headerlink" href="#exp" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.exponential.exp">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.exponential.</tt><tt class="descname">exp</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/exponential.html#exp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.exponential.exp" title="Permalink to this definition">¶</a></dt>
-<dd><p>The exponential function, <span class="math">\(e^x\)</span>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">log</span></tt></p>
-</div>
-<dl class="function">
-<dt id="sympy.functions.elementary.exponential.exp.as_real_imag">
-<tt class="descname">as_real_imag</tt><big>(</big><em>self</em>, <em>deep=True</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/exponential.html#exp.as_real_imag"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.exponential.exp.as_real_imag" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns this function as a 2-tuple representing a complex number.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.functions.elementary.complexes.re" title="sympy.functions.elementary.complexes.re"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.functions.elementary.complexes.re</span></tt></a>, <tt class="xref py py-obj docutils literal"><span class="pre">sympy.functions.elementary.complexes.im</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">exp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-<span class="go">(exp(re(x))*cos(im(x)), exp(re(x))*sin(im(x)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-<span class="go">(E, 0)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-<span class="go">(cos(1), sin(1))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-<span class="go">(E*cos(1), E*sin(1))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.elementary.exponential.exp.base">
-<tt class="descname">base</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/exponential.html#exp.base"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.exponential.exp.base" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the base of the exponential function.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.functions.elementary.exponential.exp.fdiff">
-<tt class="descname">fdiff</tt><big>(</big><em>self</em>, <em>argindex=1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/exponential.html#exp.fdiff"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.exponential.exp.fdiff" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the first derivative of this function.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.functions.elementary.exponential.exp.taylor_term">
-<tt class="descname">taylor_term</tt><big>(</big><em>*args</em>, <em>**kw_args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/exponential.html#exp.taylor_term"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.exponential.exp.taylor_term" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculates the next term in the Taylor series expansion.</p>
-</dd></dl>
-
-</dd></dl>
-
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last">classes <a class="reference internal" href="#sympy.functions.elementary.exponential.log" title="sympy.functions.elementary.exponential.log"><tt class="xref py py-class docutils literal"><span class="pre">sympy.functions.elementary.exponential.log</span></tt></a></p>
-</div>
-</div>
-<div class="section" id="exprcondpair">
-<h2>ExprCondPair<a class="headerlink" href="#exprcondpair" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.piecewise.ExprCondPair">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.piecewise.</tt><tt class="descname">ExprCondPair</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/piecewise.html#ExprCondPair"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.piecewise.ExprCondPair" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents an expression, condition pair.</p>
-<dl class="attribute">
-<dt id="sympy.functions.elementary.piecewise.ExprCondPair.cond">
-<tt class="descname">cond</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/piecewise.html#ExprCondPair.cond"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.piecewise.ExprCondPair.cond" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the condition of this pair.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.elementary.piecewise.ExprCondPair.expr">
-<tt class="descname">expr</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/piecewise.html#ExprCondPair.expr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.piecewise.ExprCondPair.expr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the expression of this pair.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.elementary.piecewise.ExprCondPair.free_symbols">
-<tt class="descname">free_symbols</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/piecewise.html#ExprCondPair.free_symbols"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.piecewise.ExprCondPair.free_symbols" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the free symbols of this pair.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="floor">
-<h2>floor<a class="headerlink" href="#floor" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.integers.floor">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.integers.</tt><tt class="descname">floor</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/integers.html#floor"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.integers.floor" title="Permalink to this definition">¶</a></dt>
-<dd><p>Floor is a univariate function which returns the largest integer
-value not greater than its argument. However this implementation
-generalizes floor to complex numbers.</p>
-<p>More information can be found in &#8220;Concrete mathematics&#8221; by Graham,
-pp. 87 or visit <a class="reference external" href="http://mathworld.wolfram.com/FloorFunction.html">http://mathworld.wolfram.com/FloorFunction.html</a>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">floor</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">Float</span><span class="p">,</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">floor</span><span class="p">(</span><span class="mi">17</span><span class="p">)</span>
-<span class="go">17</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">floor</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="mi">23</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">floor</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">E</span><span class="p">)</span>
-<span class="go">5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">floor</span><span class="p">(</span><span class="o">-</span><span class="n">Float</span><span class="p">(</span><span class="mf">0.567</span><span class="p">))</span>
-<span class="go">-1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">floor</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">-I</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">ceiling</span></tt></p>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="hyperbolicfunction">
-<h2>HyperbolicFunction<a class="headerlink" href="#hyperbolicfunction" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.hyperbolic.HyperbolicFunction">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.hyperbolic.</tt><tt class="descname">HyperbolicFunction</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/hyperbolic.html#HyperbolicFunction"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.hyperbolic.HyperbolicFunction" title="Permalink to this definition">¶</a></dt>
-<dd><p>Base class for hyperbolic functions.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</div>
-<div class="section" id="identityfunction">
-<h2>IdentityFunction<a class="headerlink" href="#identityfunction" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.miscellaneous.IdentityFunction">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.miscellaneous.</tt><tt class="descname">IdentityFunction</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/miscellaneous.html#IdentityFunction"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.miscellaneous.IdentityFunction" title="Permalink to this definition">¶</a></dt>
-<dd><p>The identity function</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Id</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Id</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="im">
-<h2>im<a class="headerlink" href="#im" title="Permalink to this headline">¶</a></h2>
-<p>Returns the imaginary part of an expression.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">im</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">im</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">I</span><span class="p">)</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.functions.elementary.complexes.re" title="sympy.functions.elementary.complexes.re"><tt class="xref py py-class docutils literal"><span class="pre">sympy.functions.elementary.complexes.re</span></tt></a></p>
-</div>
-</div>
-<div class="section" id="lambertw">
-<h2>LambertW<a class="headerlink" href="#lambertw" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.exponential.LambertW">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.exponential.</tt><tt class="descname">LambertW</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/exponential.html#LambertW"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.exponential.LambertW" title="Permalink to this definition">¶</a></dt>
-<dd><p>Lambert W function, defined as the inverse function of
-x*exp(x). This function represents the principal branch
-of this inverse function, which like the natural logarithm
-is multivalued.</p>
-<p>For more information, see:
-<a class="reference external" href="http://en.wikipedia.org/wiki/Lambert_W_function">http://en.wikipedia.org/wiki/Lambert_W_function</a></p>
-<dl class="function">
-<dt id="sympy.functions.elementary.exponential.LambertW.fdiff">
-<tt class="descname">fdiff</tt><big>(</big><em>self</em>, <em>argindex=1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/exponential.html#LambertW.fdiff"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.exponential.LambertW.fdiff" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the first derivative of this function.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="log">
-<h2>log<a class="headerlink" href="#log" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.exponential.log">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.exponential.</tt><tt class="descname">log</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/exponential.html#log"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.exponential.log" title="Permalink to this definition">¶</a></dt>
-<dd><p>The logarithmic function <span class="math">\(ln(x)\)</span> or <span class="math">\(log(x)\)</span>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">exp</span></tt></p>
-</div>
-<dl class="function">
-<dt id="sympy.functions.elementary.exponential.log.as_base_exp">
-<tt class="descname">as_base_exp</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/exponential.html#log.as_base_exp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.exponential.log.as_base_exp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns this function in the form (base, exponent).</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.functions.elementary.exponential.log.as_real_imag">
-<tt class="descname">as_real_imag</tt><big>(</big><em>self</em>, <em>deep=True</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/exponential.html#log.as_real_imag"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.exponential.log.as_real_imag" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns this function as a complex coordinate.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">log</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-<span class="go">(log(Abs(x)), arg(x))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-<span class="go">(0, pi/2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-<span class="go">(log(sqrt(2)), pi/4)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-<span class="go">(log(Abs(x)), arg(I*x))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.functions.elementary.exponential.log.fdiff">
-<tt class="descname">fdiff</tt><big>(</big><em>self</em>, <em>argindex=1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/exponential.html#log.fdiff"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.exponential.log.fdiff" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the first derivative of the function.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.functions.elementary.exponential.log.inverse">
-<tt class="descname">inverse</tt><big>(</big><em>self</em>, <em>argindex=1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/exponential.html#log.inverse"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.exponential.log.inverse" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the inverse function, log(x) (or ln(x)).</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.functions.elementary.exponential.log.taylor_term">
-<tt class="descname">taylor_term</tt><big>(</big><em>*args</em>, <em>**kw_args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/exponential.html#log.taylor_term"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.exponential.log.taylor_term" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the next term in the Taylor series expansion of log(1+x).</p>
-</dd></dl>
-
-</dd></dl>
-
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last">classes <a class="reference internal" href="#sympy.functions.elementary.exponential.exp" title="sympy.functions.elementary.exponential.exp"><tt class="xref py py-class docutils literal"><span class="pre">sympy.functions.elementary.exponential.exp</span></tt></a></p>
-</div>
-</div>
-<div class="section" id="min">
-<h2>Min<a class="headerlink" href="#min" title="Permalink to this headline">¶</a></h2>
-<p>Returns the minimum of two (comparable) expressions.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">Min</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Min</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Min</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">Min(1, x)</span>
-</pre></div>
-</div>
-<p>It is named Min and not min to avoid conflicts with the built-in function min.</p>
-<dl class="class">
-<dt id="sympy.functions.elementary.miscellaneous.Min">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.miscellaneous.</tt><tt class="descname">Min</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/miscellaneous.html#Min"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.miscellaneous.Min" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return, if possible, the minimum value of the list.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">Max</span></tt></dt>
-<dd>find maximum values</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Min</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">,</span> <span class="n">negative</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Min</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">)</span>                  
-<span class="go">Min(x, -2)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Min</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">-2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Min</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">-3</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Min</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>                   
-<span class="go">Min(x, y)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Min</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="o">-</span><span class="mi">7</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>     
-<span class="go">Min(n, -7)</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</div>
-<div class="section" id="max">
-<h2>Max<a class="headerlink" href="#max" title="Permalink to this headline">¶</a></h2>
-<p>Returns the maximum of two (comparable) expressions</p>
-<p>It is named Max and not max to avoid conflicts with the built-in function max.</p>
-<dl class="class">
-<dt id="sympy.functions.elementary.miscellaneous.Max">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.miscellaneous.</tt><tt class="descname">Max</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/miscellaneous.html#Max"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.miscellaneous.Max" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return, if possible, the maximum value of the list.</p>
-<p>When number of arguments is equal one, then
-return this argument.</p>
-<p>When number of arguments is equal two, then
-return, if possible, the value from (a, b) that is &gt;= the other.</p>
-<p>In common case, when the length of list greater than 2, the task
-is more complicated. Return only the arguments, which are greater
-than others, if it is possible to determine directional relation.</p>
-<p>If is not possible to determine such a relation, return a partially
-evaluated result.</p>
-<p>Assumptions are used to make the decision too.</p>
-<p>Also, only comparable arguments are permitted.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">Min</span></tt></dt>
-<dd>find minimum values</dd>
-</dl>
-</div>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r21" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[R21]</td><td><em>(<a class="fn-backref" href="#id1">1</a>, <a class="fn-backref" href="#id3">2</a>)</em> <a class="reference external" href="http://en.wikipedia.org/wiki/Directed_complete_partial_order">http://en.wikipedia.org/wiki/Directed_complete_partial_order</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r22" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id2">[R22]</a></td><td><a class="reference external" href="http://en.wikipedia.org/wiki/Lattice_%28order%29">http://en.wikipedia.org/wiki/Lattice_%28order%29</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Max</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">,</span> <span class="n">negative</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Max</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">)</span>                  
-<span class="go">Max(x, -2)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Max</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Max</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">p</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Max</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>                   
-<span class="go">Max(x, y)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Max</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span> <span class="o">==</span> <span class="n">Max</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Max</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">Max</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">))</span>           
-<span class="go">Max(x, y, z)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Max</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="o">-</span><span class="n">oo</span><span class="p">)</span>        
-<span class="go">Max(8, p)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Max</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">oo</span>
-</pre></div>
-</div>
-<p>Algorithm</p>
-<p>The task can be considered as searching of supremums in the
-directed complete partial orders <a class="reference internal" href="#r21">[R21]</a>.</p>
-<p>The source values are sequentially allocated by the isolated subsets
-in which supremums are searched and result as Max arguments.</p>
-<p>If the resulted supremum is single, then it is returned.</p>
-<p>The isolated subsets are the sets of values which are only the comparable
-with each other in the current set. E.g. natural numbers are comparable with
-each other, but not comparable with the <span class="math">\(x\)</span> symbol. Another example: the
-symbol <span class="math">\(x\)</span> with negative assumption is comparable with a natural number.</p>
-<p>Also there are &#8220;least&#8221; elements, which are comparable with all others,
-and have a zero property (maximum or minimum for all elements). E.g. <span class="math">\(oo\)</span>.
-In case of it the allocation operation is terminated and only this value is
-returned.</p>
-<dl class="docutils">
-<dt>Assumption:</dt>
-<dd><ul class="first last simple">
-<li>if A &gt; B &gt; C then A &gt; C</li>
-<li>if A==B then B can be removed</li>
-</ul>
-</dd>
-</dl>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</div>
-<div class="section" id="piecewise">
-<h2>Piecewise<a class="headerlink" href="#piecewise" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.piecewise.Piecewise">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.piecewise.</tt><tt class="descname">Piecewise</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/piecewise.html#Piecewise"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.piecewise.Piecewise" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents a piecewise function.</p>
-<p>Usage:</p>
-<blockquote>
-<div><dl class="docutils">
-<dt>Piecewise( (expr,cond), (expr,cond), ... )</dt>
-<dd><ul class="first last simple">
-<li>Each argument is a 2-tuple defining a expression and condition</li>
-<li>The conds are evaluated in turn returning the first that is True.
-If any of the evaluated conds are not determined explicitly False,
-e.g. x &lt; 1, the function is returned in symbolic form.</li>
-<li>If the function is evaluated at a place where all conditions are False,
-a ValueError exception will be raised.</li>
-<li>Pairs where the cond is explicitly False, will be removed.</li>
-</ul>
-</dd>
-</dl>
-</div></blockquote>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">piecewise_fold</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Piecewise</span><span class="p">,</span> <span class="n">log</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Piecewise</span><span class="p">(</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="o">&lt;-</span><span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="o">&lt;=</span><span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="bp">True</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">log(5)</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.functions.elementary.piecewise.Piecewise.doit">
-<tt class="descname">doit</tt><big>(</big><em>self</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/piecewise.html#Piecewise.doit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.piecewise.Piecewise.doit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluate this piecewise function.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.functions.elementary.piecewise.piecewise_fold">
-<tt class="descclassname">sympy.functions.elementary.piecewise.</tt><tt class="descname">piecewise_fold</tt><big>(</big><em>expr</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/piecewise.html#piecewise_fold"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.piecewise.piecewise_fold" title="Permalink to this definition">¶</a></dt>
-<dd><p>Takes an expression containing a piecewise function and returns the
-expression in piecewise form.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">Piecewise</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Piecewise</span><span class="p">,</span> <span class="n">piecewise_fold</span><span class="p">,</span> <span class="n">sympify</span> <span class="k">as</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Piecewise</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span> <span class="o">&lt;</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="n">x</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">piecewise_fold</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">p</span><span class="p">)</span>
-<span class="go">Piecewise((x**2, x &lt; 1), (x, 1 &lt;= x))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="re">
-<h2>re<a class="headerlink" href="#re" title="Permalink to this headline">¶</a></h2>
-<p>Return the real part of an expression.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">re</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">re</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">I</span><span class="p">)</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-<dl class="class">
-<dt id="sympy.functions.elementary.complexes.re">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.complexes.</tt><tt class="descname">re</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/complexes.html#re"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.complexes.re" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns real part of expression. This function performs only
-elementary analysis and so it will fail to decompose properly
-more complicated expressions. If completely simplified result
-is needed then use Basic.as_real_imag() or perform complex
-expansion on instance of this function.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">re</span><span class="p">,</span> <span class="n">im</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">E</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">re</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">E</span><span class="p">)</span>
-<span class="go">2*E</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">re</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">I</span> <span class="o">+</span> <span class="mi">17</span><span class="p">)</span>
-<span class="go">17</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">re</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">I</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">re</span><span class="p">(</span><span class="n">im</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">I</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">im</span></tt></p>
-</div>
-<dl class="function">
-<dt id="sympy.functions.elementary.complexes.re.as_real_imag">
-<tt class="descname">as_real_imag</tt><big>(</big><em>self</em>, <em>deep=True</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/complexes.html#re.as_real_imag"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.complexes.re.as_real_imag" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the real number with a zero complex part.</p>
-</dd></dl>
-
-</dd></dl>
-
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-class docutils literal"><span class="pre">sympy.functions.elementary.complexes.im</span></tt></p>
-</div>
-</div>
-<div class="section" id="root">
-<h2>root<a class="headerlink" href="#root" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.functions.elementary.miscellaneous.root">
-<tt class="descclassname">sympy.functions.elementary.miscellaneous.</tt><tt class="descname">root</tt><big>(</big><em>arg</em>, <em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/miscellaneous.html#root"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.miscellaneous.root" title="Permalink to this definition">¶</a></dt>
-<dd><p>The n-th root function (a shortcut for arg**(1/n))</p>
-<p>root(x, n) -&gt; Returns the principal n-th root of x.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="../polys/reference.html#sympy.polys.rootoftools.RootOf" title="sympy.polys.rootoftools.RootOf"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.rootoftools.RootOf</span></tt></a>, <a class="reference internal" href="../core.html#sympy.core.power.integer_nthroot" title="sympy.core.power.integer_nthroot"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.core.power.integer_nthroot</span></tt></a>, <tt class="xref py py-obj docutils literal"><span class="pre">sqrt</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">real_root</span></tt></p>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Square_root">http://en.wikipedia.org/wiki/Square_root</a></li>
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/real_root">http://en.wikipedia.org/wiki/real_root</a></li>
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Root_of_unity">http://en.wikipedia.org/wiki/Root_of_unity</a></li>
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Principal_value">http://en.wikipedia.org/wiki/Principal_value</a></li>
-<li><a class="reference external" href="http://mathworld.wolfram.com/CubeRoot.html">http://mathworld.wolfram.com/CubeRoot.html</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">root</span><span class="p">,</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">root</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">sqrt(x)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">root</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">x**(1/3)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">root</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-<span class="go">x**(1/n)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">root</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">Rational</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
-<span class="go">x**(-3/2)</span>
-</pre></div>
-</div>
-<p>To get all n n-th roots you can use the RootOf function.
-The following examples show the roots of unity for n
-equal 2, 3 and 4:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">RootOf</span><span class="p">,</span> <span class="n">I</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">[</span> <span class="n">RootOf</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span> <span class="p">]</span>
-<span class="go">[-1, 1]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">[</span> <span class="n">RootOf</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="p">]</span>
-<span class="go">[1, -1/2 - sqrt(3)*I/2, -1/2 + sqrt(3)*I/2]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">[</span> <span class="n">RootOf</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span> <span class="p">]</span>
-<span class="go">[-1, 1, -I, I]</span>
-</pre></div>
-</div>
-<p>SymPy, like other symbolic algebra systems, returns the
-complex root of negative numbers. This is the principal
-root and differs from the text-book result that one might
-be expecting. For example, the cube root of -8 does not
-come back as -2:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">root</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">2*(-1)**(1/3)</span>
-</pre></div>
-</div>
-<p>The real_root function can be used to either make such a result
-real or simply return the real root in the first place:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">real_root</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">real_root</span><span class="p">(</span><span class="n">_</span><span class="p">)</span>
-<span class="go">-2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">real_root</span><span class="p">(</span><span class="o">-</span><span class="mi">32</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">-2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="roundfunction">
-<h2>RoundFunction<a class="headerlink" href="#roundfunction" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.integers.RoundFunction">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.integers.</tt><tt class="descname">RoundFunction</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/integers.html#RoundFunction"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.integers.RoundFunction" title="Permalink to this definition">¶</a></dt>
-<dd><p>The base class for rounding functions.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="sin">
-<h2>sin<a class="headerlink" href="#sin" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.trigonometric.sin">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.trigonometric.</tt><tt class="descname">sin</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/trigonometric.html#sin"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.trigonometric.sin" title="Permalink to this definition">¶</a></dt>
-<dd><p>The sine function.</p>
-<ul class="simple">
-<li>sin(x) -&gt; Returns the sine of x (measured in radians)</li>
-</ul>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">cos</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">tan</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">asin</span></tt></p>
-</div>
-<p class="rubric">Notes</p>
-<ul class="simple">
-<li>sin(x) will evaluate automatically in the case x
-is a multiple of pi, pi/2, pi/3, pi/4 and pi/6.</li>
-</ul>
-<p class="rubric">References</p>
-<p>U{Definitions in trigonometry&lt;<a class="reference external" href="http://planetmath.org/encyclopedia/DefinitionsInTrigonometry.html">http://planetmath.org/encyclopedia/DefinitionsInTrigonometry.html</a>&gt;}</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">2*x*cos(x**2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">6</span><span class="p">)</span>
-<span class="go">1/2</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.functions.elementary.trigonometric.sin.inverse">
-<tt class="descname">inverse</tt><big>(</big><em>self</em>, <em>argindex=1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/trigonometric.html#sin.inverse"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.trigonometric.sin.inverse" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the inverse of this function.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="sinh">
-<h2>sinh<a class="headerlink" href="#sinh" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.hyperbolic.sinh">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.hyperbolic.</tt><tt class="descname">sinh</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/hyperbolic.html#sinh"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.hyperbolic.sinh" title="Permalink to this definition">¶</a></dt>
-<dd><p>The hyperbolic sine function, <span class="math">\(\frac{exp(x) - exp(-x)}{2}\)</span>.</p>
-<ul class="simple">
-<li>sinh(x) -&gt; Returns the hyperbolic sine of x</li>
-</ul>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">cosh</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">tanh</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">asinh</span></tt></p>
-</div>
-<dl class="function">
-<dt id="sympy.functions.elementary.hyperbolic.sinh.as_real_imag">
-<tt class="descname">as_real_imag</tt><big>(</big><em>self</em>, <em>deep=True</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/hyperbolic.html#sinh.as_real_imag"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.hyperbolic.sinh.as_real_imag" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns this function as a complex coordinate.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.functions.elementary.hyperbolic.sinh.fdiff">
-<tt class="descname">fdiff</tt><big>(</big><em>self</em>, <em>argindex=1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/hyperbolic.html#sinh.fdiff"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.hyperbolic.sinh.fdiff" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the first derivative of this function.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.functions.elementary.hyperbolic.sinh.inverse">
-<tt class="descname">inverse</tt><big>(</big><em>self</em>, <em>argindex=1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/hyperbolic.html#sinh.inverse"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.hyperbolic.sinh.inverse" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the inverse of this function.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.functions.elementary.hyperbolic.sinh.taylor_term">
-<tt class="descname">taylor_term</tt><big>(</big><em>*args</em>, <em>**kw_args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/hyperbolic.html#sinh.taylor_term"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.hyperbolic.sinh.taylor_term" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the next term in the Taylor series expansion.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="sqrt">
-<h2>sqrt<a class="headerlink" href="#sqrt" title="Permalink to this headline">¶</a></h2>
-<p>Returns the square root of an expression. It is equivalent to raise to Rational(1,2)</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span><span class="o">**</span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<dl class="class">
-<dt id="sympy.functions.elementary.miscellaneous.sqrt">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.miscellaneous.</tt><tt class="descname">sqrt</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/miscellaneous.html#sqrt"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.miscellaneous.sqrt" title="Permalink to this definition">¶</a></dt>
-<dd><p>The square root function</p>
-<p>sqrt(x) -&gt; Returns the principal square root of x.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="../polys/reference.html#sympy.polys.rootoftools.RootOf" title="sympy.polys.rootoftools.RootOf"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.rootoftools.RootOf</span></tt></a>, <tt class="xref py py-obj docutils literal"><span class="pre">root</span></tt></p>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Square_root">http://en.wikipedia.org/wiki/Square_root</a></li>
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Principal_value">http://en.wikipedia.org/wiki/Principal_value</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">sqrt(x)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">x</span>
-</pre></div>
-</div>
-<p>Note that sqrt(x**2) does not simplify to x.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">sqrt(x**2)</span>
-</pre></div>
-</div>
-<p>This is because the two are not equal to each other in general.
-For example, consider x == -1:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-1</span>
-</pre></div>
-</div>
-<p>This is because sqrt computes the principal square root, so the square may
-put the argument in a different branch.  This identity does hold if x is
-positive:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">y</span>
-</pre></div>
-</div>
-<p>You can force this simplification by using the powdenest() function with
-the force option set to True:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">powdenest</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">sqrt(x**2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powdenest</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="n">force</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">x</span>
-</pre></div>
-</div>
-<p>To get both branches of the square root you can use the RootOf function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">RootOf</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">[</span> <span class="n">RootOf</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span> <span class="p">]</span>
-<span class="go">[-sqrt(3), sqrt(3)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="sign">
-<h2>sign<a class="headerlink" href="#sign" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.complexes.sign">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.complexes.</tt><tt class="descname">sign</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/complexes.html#sign"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.complexes.sign" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the sign of an expression, that is:</p>
-<ul class="simple">
-<li>1 if expression is positive</li>
-<li>0 if expression is equal to zero</li>
-<li>-1 if expression is negative</li>
-</ul>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">Abs</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">conjugate</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">sign</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sign</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sign</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="tan">
-<h2>tan<a class="headerlink" href="#tan" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.trigonometric.tan">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.trigonometric.</tt><tt class="descname">tan</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/trigonometric.html#tan"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.trigonometric.tan" title="Permalink to this definition">¶</a></dt>
-<dd><div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">sin</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">cos</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">atan</span></tt></p>
-</div>
-<p class="rubric">Notes</p>
-<ul class="simple">
-<li>tan(x) will evaluate automatically in the case x is a
-multiple of pi.</li>
-</ul>
-<p class="rubric">References</p>
-<p>U{Definitions in trigonometry&lt;<a class="reference external" href="http://planetmath.org/encyclopedia/DefinitionsInTrigonometry.html">http://planetmath.org/encyclopedia/DefinitionsInTrigonometry.html</a>&gt;}</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">tan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">2*x*(tan(x**2)**2 + 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tan</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.functions.elementary.trigonometric.tan.inverse">
-<tt class="descname">inverse</tt><big>(</big><em>self</em>, <em>argindex=1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/trigonometric.html#tan.inverse"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.trigonometric.tan.inverse" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the inverse of this function.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="tanh">
-<h2>tanh<a class="headerlink" href="#tanh" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.elementary.hyperbolic.tanh">
-<em class="property">class </em><tt class="descclassname">sympy.functions.elementary.hyperbolic.</tt><tt class="descname">tanh</tt><a class="reference internal" href="../../_modules/sympy/functions/elementary/hyperbolic.html#tanh"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.hyperbolic.tanh" title="Permalink to this definition">¶</a></dt>
-<dd><p>The hyperbolic tangent function, <span class="math">\(\frac{sinh(x)}{cosh(x)}\)</span>.</p>
-<ul class="simple">
-<li>tanh(x) -&gt; Returns the hyperbolic tangent of x</li>
-</ul>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">sinh</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">cosh</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">atanh</span></tt></p>
-</div>
-<dl class="function">
-<dt id="sympy.functions.elementary.hyperbolic.tanh.inverse">
-<tt class="descname">inverse</tt><big>(</big><em>self</em>, <em>argindex=1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/elementary/hyperbolic.html#tanh.inverse"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.elementary.hyperbolic.tanh.inverse" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the inverse of this function.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
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-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../index.html">
-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Elementary</a><ul>
-<li><a class="reference internal" href="#abs">Abs</a></li>
-<li><a class="reference internal" href="#acos">acos</a></li>
-<li><a class="reference internal" href="#acosh">acosh</a></li>
-<li><a class="reference internal" href="#acot">acot</a></li>
-<li><a class="reference internal" href="#acoth">acoth</a></li>
-<li><a class="reference internal" href="#arg">arg</a></li>
-<li><a class="reference internal" href="#asin">asin</a></li>
-<li><a class="reference internal" href="#asinh">asinh</a></li>
-<li><a class="reference internal" href="#atan">atan</a></li>
-<li><a class="reference internal" href="#atan2">atan2</a></li>
-<li><a class="reference internal" href="#atanh">atanh</a></li>
-<li><a class="reference internal" href="#ceiling">ceiling</a></li>
-<li><a class="reference internal" href="#conjugate">conjugate</a></li>
-<li><a class="reference internal" href="#cos">cos</a></li>
-<li><a class="reference internal" href="#cosh">cosh</a></li>
-<li><a class="reference internal" href="#cot">cot</a></li>
-<li><a class="reference internal" href="#coth">coth</a></li>
-<li><a class="reference internal" href="#exp">exp</a></li>
-<li><a class="reference internal" href="#exprcondpair">ExprCondPair</a></li>
-<li><a class="reference internal" href="#floor">floor</a></li>
-<li><a class="reference internal" href="#hyperbolicfunction">HyperbolicFunction</a></li>
-<li><a class="reference internal" href="#identityfunction">IdentityFunction</a></li>
-<li><a class="reference internal" href="#im">im</a></li>
-<li><a class="reference internal" href="#lambertw">LambertW</a></li>
-<li><a class="reference internal" href="#log">log</a></li>
-<li><a class="reference internal" href="#min">Min</a></li>
-<li><a class="reference internal" href="#max">Max</a></li>
-<li><a class="reference internal" href="#piecewise">Piecewise</a></li>
-<li><a class="reference internal" href="#re">re</a></li>
-<li><a class="reference internal" href="#root">root</a></li>
-<li><a class="reference internal" href="#roundfunction">RoundFunction</a></li>
-<li><a class="reference internal" href="#sin">sin</a></li>
-<li><a class="reference internal" href="#sinh">sinh</a></li>
-<li><a class="reference internal" href="#sqrt">sqrt</a></li>
-<li><a class="reference internal" href="#sign">sign</a></li>
-<li><a class="reference internal" href="#tan">tan</a></li>
-<li><a class="reference internal" href="#tanh">tanh</a></li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../functions/index.html"
-                        title="previous chapter">Functions Module</a></p>
-  <h4>Next topic</h4>
-  <p class="topless"><a href="../functions/combinatorial.html"
-                        title="next chapter">Combinatorial</a></p>
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-  <div class="section" id="module-sympy.functions">
-<span id="functions-module"></span><h1>Functions Module<a class="headerlink" href="#module-sympy.functions" title="Permalink to this headline">¶</a></h1>
-<p>All functions support the methods documented below, inherited from
-<a class="reference internal" href="../core.html#sympy.core.function.Function" title="sympy.core.function.Function"><tt class="xref py py-class docutils literal"><span class="pre">sympy.core.function.Function</span></tt></a>.</p>
-<dl class="class">
-<dt>
-<em class="property">class </em><tt class="descclassname">sympy.core.function.</tt><tt class="descname">Function</tt><a class="reference internal" href="../../_modules/sympy/core/function.html#Function"><span class="viewcode-link">[source]</span></a></dt>
-<dd><p>Base class for applied mathematical functions.</p>
-<p>It also serves as a constructor for undefined function classes.</p>
-<p class="rubric">Examples</p>
-<p>First example shows how to use Function as a constructor for undefined
-function classes:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">)(</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span>
-<span class="go">f</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">f(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span>
-<span class="go">g(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">Derivative(f(x), x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">Derivative(g(x), x)</span>
-</pre></div>
-</div>
-<p>In the following example Function is used as a base class for
-<tt class="docutils literal"><span class="pre">my_func</span></tt> that represents a mathematical function <em>my_func</em>. Suppose
-that it is well known, that <em>my_func(0)</em> is <em>1</em> and <em>my_func</em> at infinity
-goes to <em>0</em>, so we want those two simplifications to occur automatically.
-Suppose also that <em>my_func(x)</em> is real exactly when <em>x</em> is real. Here is
-an implementation that honours those requirements:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">my_func</span><span class="p">(</span><span class="n">Function</span><span class="p">):</span>
-<span class="gp">...</span>
-<span class="gp">... </span>    <span class="n">nargs</span> <span class="o">=</span> <span class="mi">1</span>
-<span class="gp">...</span>
-<span class="gp">... </span>    <span class="nd">@classmethod</span>
-<span class="gp">... </span>    <span class="k">def</span> <span class="nf">eval</span><span class="p">(</span><span class="n">cls</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
-<span class="gp">... </span>        <span class="k">if</span> <span class="n">x</span><span class="o">.</span><span class="n">is_Number</span><span class="p">:</span>
-<span class="gp">... </span>            <span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span><span class="p">:</span>
-<span class="gp">... </span>                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span>
-<span class="gp">... </span>            <span class="k">elif</span> <span class="n">x</span> <span class="ow">is</span> <span class="n">S</span><span class="o">.</span><span class="n">Infinity</span><span class="p">:</span>
-<span class="gp">... </span>                <span class="k">return</span> <span class="n">S</span><span class="o">.</span><span class="n">Zero</span>
-<span class="gp">...</span>
-<span class="gp">... </span>    <span class="k">def</span> <span class="nf">_eval_is_real</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-<span class="gp">... </span>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">is_real</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">S</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">my_func</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="o">+</span> <span class="n">sin</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">my_func</span><span class="p">(</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">my_func</span><span class="p">(</span><span class="mf">3.54</span><span class="p">)</span><span class="o">.</span><span class="n">n</span><span class="p">()</span> <span class="c"># Not yet implemented for my_func.</span>
-<span class="go">my_func(3.54)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">my_func</span><span class="p">(</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">is_real</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>In order for <tt class="docutils literal"><span class="pre">my_func</span></tt> to become useful, several other methods would
-need to be implemented. See source code of some of the already
-implemented functions for more complete examples.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt>
-<tt class="descclassname">Function.</tt><tt class="descname">as_base_exp</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/core/function.html#Function.as_base_exp"><span class="viewcode-link">[source]</span></a></dt>
-<dd><p>Returns the method as the 2-tuple (base, exponent).</p>
-</dd></dl>
-
-<dl class="function">
-<dt>
-<tt class="descclassname">Function.</tt><tt class="descname">fdiff</tt><big>(</big><em>self</em>, <em>argindex=1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/core/function.html#Function.fdiff"><span class="viewcode-link">[source]</span></a></dt>
-<dd><p>Returns the first derivative of the function.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt>
-<tt class="descclassname">Function.</tt><tt class="descname">is_commutative</tt><a class="reference internal" href="../../_modules/sympy/core/function.html#Function.is_commutative"><span class="viewcode-link">[source]</span></a></dt>
-<dd><p>Returns whether the functon is commutative.</p>
-</dd></dl>
-
-<dl class="classmethod">
-<dt>
-<em class="property">classmethod </em><tt class="descclassname">Function.</tt><tt class="descname">taylor_term</tt><big>(</big><em>n</em>, <em>x</em>, <em>*previous_terms</em><big>)</big><a class="reference internal" href="../../_modules/sympy/core/function.html#Function.taylor_term"><span class="viewcode-link">[source]</span></a></dt>
-<dd><p>General method for the taylor term.</p>
-<p>This method is slow, because it differentiates n-times. Subclasses can
-redefine it to make it faster by using the &#8220;previous_terms&#8221;.</p>
-</dd></dl>
-
-</dd></dl>
-
-<div class="section" id="contents">
-<h2>Contents<a class="headerlink" href="#contents" title="Permalink to this headline">¶</a></h2>
-<div class="toctree-wrapper compound">
-<ul>
-<li class="toctree-l1"><a class="reference internal" href="../functions/elementary.html">Elementary</a><ul>
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-</ul>
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-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../functions/special.html">Special</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../functions/special.html#diracdelta">DiracDelta</a></li>
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-          <li><a href="../functions/index.html" accesskey="U">Functions Module</a> &raquo;</li> 
-      </ul>
-    </div>  
-
-    <div class="document">
-      <div class="documentwrapper">
-        <div class="bodywrapper">
-          <div class="body">
-            
-  <div class="section" id="special">
-<h1>Special<a class="headerlink" href="#special" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="diracdelta">
-<h2>DiracDelta<a class="headerlink" href="#diracdelta" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.special.delta_functions.DiracDelta">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.delta_functions.</tt><tt class="descname">DiracDelta</tt><a class="reference internal" href="../../_modules/sympy/functions/special/delta_functions.html#DiracDelta"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.delta_functions.DiracDelta" title="Permalink to this definition">¶</a></dt>
-<dd><p>The DiracDelta function and its derivatives.</p>
-<p>DiracDelta function has the following properties:</p>
-<ol class="arabic simple">
-<li><tt class="docutils literal"><span class="pre">diff(Heaviside(x),x)</span> <span class="pre">=</span> <span class="pre">DiracDelta(x)</span></tt></li>
-<li><tt class="docutils literal"><span class="pre">integrate(DiracDelta(x-a)*f(x),(x,-oo,oo))</span> <span class="pre">=</span> <span class="pre">f(a)</span></tt> and
-<tt class="docutils literal"><span class="pre">integrate(DiracDelta(x-a)*f(x),(x,a-e,a+e))</span> <span class="pre">=</span> <span class="pre">f(a)</span></tt></li>
-<li><tt class="docutils literal"><span class="pre">DiracDelta(x)</span> <span class="pre">=</span> <span class="pre">0</span></tt> for all <tt class="docutils literal"><span class="pre">x</span> <span class="pre">!=</span> <span class="pre">0</span></tt></li>
-<li><tt class="docutils literal"><span class="pre">DiracDelta(g(x))</span> <span class="pre">=</span> <span class="pre">Sum_i(DiracDelta(x-x_i)/abs(g'(x_i)))</span></tt>
-Where <tt class="docutils literal"><span class="pre">x_i</span></tt>-s are the roots of <tt class="docutils literal"><span class="pre">g</span></tt></li>
-</ol>
-<p>Derivatives of <tt class="docutils literal"><span class="pre">k</span></tt>-th order of DiracDelta have the following property:</p>
-<ol class="arabic simple" start="5">
-<li><tt class="docutils literal"><span class="pre">DiracDelta(x,k)</span> <span class="pre">=</span> <span class="pre">0</span></tt>, for all <tt class="docutils literal"><span class="pre">x</span> <span class="pre">!=</span> <span class="pre">0</span></tt></li>
-</ol>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">Heaviside</span></tt>, <a class="reference internal" href="../galgebra/GA/GAsympy.html#simplify" title="simplify"><tt class="xref py py-obj docutils literal"><span class="pre">simplify</span></tt></a>, <tt class="xref py py-obj docutils literal"><span class="pre">is_simple</span></tt>, <a class="reference internal" href="#sympy.functions.special.tensor_functions.KroneckerDelta" title="sympy.functions.special.tensor_functions.KroneckerDelta"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.functions.special.tensor_functions.KroneckerDelta</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<p><a class="reference external" href="http://mathworld.wolfram.com/DeltaFunction.html">http://mathworld.wolfram.com/DeltaFunction.html</a></p>
-<dl class="function">
-<dt id="sympy.functions.special.delta_functions.DiracDelta.is_simple">
-<tt class="descname">is_simple</tt><big>(</big><em>self</em>, <em>x</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/special/delta_functions.html#DiracDelta.is_simple"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.delta_functions.DiracDelta.is_simple" title="Permalink to this definition">¶</a></dt>
-<dd><p>Tells whether the argument(args[0]) of DiracDelta is a linear
-expression in x.</p>
-<p>x can be:</p>
-<ul class="simple">
-<li>a symbol</li>
-</ul>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="../galgebra/GA/GAsympy.html#simplify" title="simplify"><tt class="xref py py-obj docutils literal"><span class="pre">simplify</span></tt></a>, <tt class="xref py py-obj docutils literal"><span class="pre">Directdelta</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">DiracDelta</span><span class="p">,</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">DiracDelta</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">is_simple</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">DiracDelta</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">is_simple</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">DiracDelta</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">x</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">is_simple</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">DiracDelta</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">is_simple</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.functions.special.delta_functions.DiracDelta.simplify">
-<tt class="descname">simplify</tt><big>(</big><em>self</em>, <em>x</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/special/delta_functions.html#DiracDelta.simplify"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.delta_functions.DiracDelta.simplify" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute a simplified representation of the function using
-property number 4.</p>
-<p>x can be:</p>
-<ul class="simple">
-<li>a symbol</li>
-</ul>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">is_simple</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">Directdelta</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">DiracDelta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">DiracDelta</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">DiracDelta(x)/Abs(y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">DiracDelta</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">DiracDelta(y)/Abs(x)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">DiracDelta</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">simplify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">DiracDelta(x - 1)/3 + DiracDelta(x + 2)/3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="heaviside">
-<h2>Heaviside<a class="headerlink" href="#heaviside" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.special.delta_functions.Heaviside">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.delta_functions.</tt><tt class="descname">Heaviside</tt><a class="reference internal" href="../../_modules/sympy/functions/special/delta_functions.html#Heaviside"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.delta_functions.Heaviside" title="Permalink to this definition">¶</a></dt>
-<dd><p>Heaviside Piecewise function</p>
-<p>Heaviside function has the following properties <a class="footnote-reference" href="#id2" id="id1">[*]</a>:</p>
-<ol class="arabic">
-<li><dl class="first docutils">
-<dt><tt class="docutils literal"><span class="pre">diff(Heaviside(x),x)</span> <span class="pre">=</span> <span class="pre">DiracDelta(x)</span></tt></dt>
-<dd><p class="first last"><tt class="docutils literal"><span class="pre">(</span> <span class="pre">0,</span> <span class="pre">if</span> <span class="pre">x</span> <span class="pre">&lt;</span> <span class="pre">0</span></tt></p>
-</dd>
-</dl>
-</li>
-<li><dl class="first docutils">
-<dt><tt class="docutils literal"><span class="pre">Heaviside(x)</span> <span class="pre">=</span> <span class="pre">&lt;</span> <span class="pre">(</span> <span class="pre">1/2</span> <span class="pre">if</span> <span class="pre">x==0</span> <span class="pre">[*]</span></tt></dt>
-<dd><p class="first last"><tt class="docutils literal"><span class="pre">(</span> <span class="pre">1,</span> <span class="pre">if</span> <span class="pre">x</span> <span class="pre">&gt;</span> <span class="pre">0</span></tt></p>
-</dd>
-</dl>
-</li>
-</ol>
-<table class="docutils footnote" frame="void" id="id2" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id1">[*]</a></td><td>Regarding to the value at 0, Mathematica defines <tt class="docutils literal"><span class="pre">H(0)</span> <span class="pre">=</span> <span class="pre">1</span></tt>,
-but Maple uses <tt class="docutils literal"><span class="pre">H(0)</span> <span class="pre">=</span> <span class="pre">undefined</span></tt></td></tr>
-</tbody>
-</table>
-<p>I think is better to have H(0) = 1/2, due to the following:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">integrate</span><span class="p">(</span><span class="n">DiracDelta</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span> <span class="o">=</span> <span class="n">Heaviside</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="n">integrate</span><span class="p">(</span><span class="n">DiracDelta</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">oo</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span> <span class="o">=</span> <span class="mi">1</span>
-</pre></div>
-</div>
-<p>and since DiracDelta is a symmetric function,
-<tt class="docutils literal"><span class="pre">integrate(DiracDelta(x),</span> <span class="pre">(x,</span> <span class="pre">0,</span> <span class="pre">oo))</span></tt> should be 1/2 (which is what
-Maple returns).</p>
-<p>If we take <tt class="docutils literal"><span class="pre">Heaviside(0)</span> <span class="pre">=</span> <span class="pre">1/2</span></tt>, we would have
-<tt class="docutils literal"><span class="pre">integrate(DiracDelta(x),</span> <span class="pre">(x,</span> <span class="pre">0,</span> <span class="pre">oo))</span> <span class="pre">=</span> <span class="pre">``</span>
-<span class="pre">``Heaviside(oo)</span> <span class="pre">-</span> <span class="pre">Heaviside(0)</span> <span class="pre">=</span> <span class="pre">1</span> <span class="pre">-</span> <span class="pre">1/2</span> <span class="pre">=</span> <span class="pre">1/2</span></tt>
-and
-<tt class="docutils literal"><span class="pre">integrate(DiracDelta(x),</span> <span class="pre">(x,</span> <span class="pre">-oo,</span> <span class="pre">0))</span> <span class="pre">=</span> <span class="pre">``</span>
-<span class="pre">``Heaviside(0)</span> <span class="pre">-</span> <span class="pre">Heaviside(-oo)</span> <span class="pre">=</span> <span class="pre">1/2</span> <span class="pre">-</span> <span class="pre">0</span> <span class="pre">=</span> <span class="pre">1/2</span></tt></p>
-<p>If we consider, instead <tt class="docutils literal"><span class="pre">Heaviside(0)</span> <span class="pre">=</span> <span class="pre">1</span></tt>, we would have
-<tt class="docutils literal"><span class="pre">integrate(DiracDelta(x),</span> <span class="pre">(x,</span> <span class="pre">0,</span> <span class="pre">oo))</span> <span class="pre">=</span> <span class="pre">Heaviside(oo)</span> <span class="pre">-</span> <span class="pre">Heaviside(0)</span> <span class="pre">=</span> <span class="pre">0</span></tt>
-and
-<tt class="docutils literal"><span class="pre">integrate(DiracDelta(x),</span> <span class="pre">(x,</span> <span class="pre">-oo,</span> <span class="pre">0))</span> <span class="pre">=</span> <span class="pre">Heaviside(0)</span> <span class="pre">-</span> <span class="pre">Heaviside(-oo)</span> <span class="pre">=</span> <span class="pre">1</span></tt></p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">DiracDelta</span></tt></p>
-</div>
-<p class="rubric">References</p>
-<p><a class="reference external" href="http://mathworld.wolfram.com/HeavisideStepFunction.html">http://mathworld.wolfram.com/HeavisideStepFunction.html</a></p>
-</dd></dl>
-
-</div>
-<div class="section" id="beta">
-<h2>beta<a class="headerlink" href="#beta" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.functions.special.gamma_functions.beta">
-<tt class="descclassname">sympy.functions.special.gamma_functions.</tt><tt class="descname">beta</tt><big>(</big><em>x</em>, <em>y</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/special/gamma_functions.html#beta"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.gamma_functions.beta" title="Permalink to this definition">¶</a></dt>
-<dd><p>Euler Beta function</p>
-<p><tt class="docutils literal"><span class="pre">beta(x,</span> <span class="pre">y)</span> <span class="pre">==</span> <span class="pre">gamma(x)*gamma(y)</span> <span class="pre">/</span> <span class="pre">gamma(x+y)</span></tt></p>
-</dd></dl>
-
-</div>
-<div class="section" id="erf">
-<h2>erf<a class="headerlink" href="#erf" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.special.error_functions.erf">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.error_functions.</tt><tt class="descname">erf</tt><a class="reference internal" href="../../_modules/sympy/functions/special/error_functions.html#erf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.error_functions.erf" title="Permalink to this definition">¶</a></dt>
-<dd><p>The Gauss error function.</p>
-<p>This function is defined as:</p>
-<p><span class="math">\(\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \, \mathrm{d}x\)</span></p>
-<p>Or, in ASCII:</p>
-<div class="highlight-python"><pre>        x
-    /
-   |
-   |     2
-   |   -t
-2* |  e    dt
-   |
-  /
-  0
--------------
-      ____
-    \/ pi</pre>
-</div>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r23" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id3">[R23]</a></td><td><a class="reference external" href="http://en.wikipedia.org/wiki/Error_function">http://en.wikipedia.org/wiki/Error_function</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r24" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id4">[R24]</a></td><td><a class="reference external" href="http://dlmf.nist.gov/7">http://dlmf.nist.gov/7</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r25" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id5">[R25]</a></td><td><a class="reference external" href="http://mathworld.wolfram.com/Erf.html">http://mathworld.wolfram.com/Erf.html</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r26" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id6">[R26]</a></td><td><a class="reference external" href="http://functions.wolfram.com/GammaBetaErf/Erf">http://functions.wolfram.com/GammaBetaErf/Erf</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">I</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">erf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">z</span>
-</pre></div>
-</div>
-<p>Several special values are known:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="o">-</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">-1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">oo*I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">-oo*I</span>
-</pre></div>
-</div>
-<p>In general one can pull out factors of -1 and I from the argument:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">)</span>
-<span class="go">-erf(z)</span>
-</pre></div>
-</div>
-<p>The error function obeys the mirror symmetry:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">conjugate</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">conjugate</span><span class="p">(</span><span class="n">erf</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="go">erf(conjugate(z))</span>
-</pre></div>
-</div>
-<p>Differentiation with respect to z is supported:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">diff</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">erf</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">2*exp(-z**2)/sqrt(pi)</span>
-</pre></div>
-</div>
-<p>We can numerically evaluate the error function to arbitrary precision
-on the whole complex plane:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">30</span><span class="p">)</span>
-<span class="go">0.999999984582742099719981147840</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="o">*</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">30</span><span class="p">)</span>
-<span class="go">-1296959.73071763923152794095062*I</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="gamma-and-related-functions">
-<h2>Gamma and Related Functions<a class="headerlink" href="#gamma-and-related-functions" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.special.gamma_functions.gamma">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.gamma_functions.</tt><tt class="descname">gamma</tt><a class="reference internal" href="../../_modules/sympy/functions/special/gamma_functions.html#gamma"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.gamma_functions.gamma" title="Permalink to this definition">¶</a></dt>
-<dd><p>The gamma function returns a function which passes through the integral
-values of the factorial function, i.e. though defined in the complex plane,
-when n is an integer, <span class="math">\(gamma(n) = (n - 1)!\)</span></p>
-<dl class="docutils">
-<dt>Reference:</dt>
-<dd><a class="reference external" href="http://en.wikipedia.org/wiki/Gamma_function">http://en.wikipedia.org/wiki/Gamma_function</a></dd>
-</dl>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.gamma_functions.loggamma">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.gamma_functions.</tt><tt class="descname">loggamma</tt><a class="reference internal" href="../../_modules/sympy/functions/special/gamma_functions.html#loggamma"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.gamma_functions.loggamma" title="Permalink to this definition">¶</a></dt>
-<dd><p>The loggamma function is <span class="math">\(ln(gamma(x))\)</span>.</p>
-<p class="rubric">References</p>
-<p><a class="reference external" href="http://mathworld.wolfram.com/LogGammaFunction.html">http://mathworld.wolfram.com/LogGammaFunction.html</a></p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.gamma_functions.polygamma">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.gamma_functions.</tt><tt class="descname">polygamma</tt><a class="reference internal" href="../../_modules/sympy/functions/special/gamma_functions.html#polygamma"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.gamma_functions.polygamma" title="Permalink to this definition">¶</a></dt>
-<dd><p>The function <span class="math">\(polygamma(n, z)\)</span> returns <span class="math">\(log(gamma(z)).diff(n + 1)\)</span></p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p><tt class="xref py py-obj docutils literal"><span class="pre">Reference</span></tt></p>
-<dl class="last docutils">
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">http</span></tt></dt>
-<dd>//en.wikipedia.org/wiki/Polygamma_function</dd>
-</dl>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.functions.special.gamma_functions.digamma">
-<tt class="descclassname">sympy.functions.special.gamma_functions.</tt><tt class="descname">digamma</tt><big>(</big><em>x</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/special/gamma_functions.html#digamma"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.gamma_functions.digamma" title="Permalink to this definition">¶</a></dt>
-<dd><p>The digamma function is the logarithmic derivative of the gamma function.</p>
-<p>In this case, <span class="math">\(digamma(x) = polygamma(0, x)\)</span>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">gamma</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">trigamma</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">polygamma</span></tt></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.functions.special.gamma_functions.trigamma">
-<tt class="descclassname">sympy.functions.special.gamma_functions.</tt><tt class="descname">trigamma</tt><big>(</big><em>x</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/special/gamma_functions.html#trigamma"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.gamma_functions.trigamma" title="Permalink to this definition">¶</a></dt>
-<dd><p>The trigamma function is the second of the polygamma functions.</p>
-<p>In this case, <span class="math">\(trigamma(x) = polygamma(1, x)\)</span>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">gamma</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">digamma</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">polygamma</span></tt></p>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.gamma_functions.uppergamma">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.gamma_functions.</tt><tt class="descname">uppergamma</tt><a class="reference internal" href="../../_modules/sympy/functions/special/gamma_functions.html#uppergamma"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.gamma_functions.uppergamma" title="Permalink to this definition">¶</a></dt>
-<dd><p>Upper incomplete gamma function</p>
-<p>It can be defined as the meromorphic continuation of</p>
-<div class="math">
-\[\Gamma(s, x) = \int_x^\infty t^{s-1} e^{-t} \mathrm{d}t
-             = \Gamma(s) - \gamma(s, x).\]</div>
-<p>This can be shown to be the same as</p>
-<div class="math">
-\[\Gamma(s, x) = \Gamma(s)
-        - \frac{x^s}{s} {}_1F_1\left({s \atop s+1} \middle| -x\right),\]</div>
-<p>where <span class="math">\({}_1F_1\)</span> is the (confluent) hypergeometric function.</p>
-<p>The upper incomplete gamma function is also essentially equivalent to the
-generalized exponential integral.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">gamma</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">lowergamma</span></tt>, <a class="reference internal" href="#sympy.functions.special.hyper.hyper" title="sympy.functions.special.hyper.hyper"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.functions.special.hyper.hyper</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li>Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6, Section 5,
-Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical
-Tables</li>
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Incomplete_gamma_function">http://en.wikipedia.org/wiki/Incomplete_gamma_function</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">uppergamma</span><span class="p">,</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">s</span><span class="p">,</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">uppergamma</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">uppergamma(s, x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">uppergamma</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x**2*exp(-x) + 2*x*exp(-x) + 2*exp(-x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">uppergamma</span><span class="p">(</span><span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">-2*sqrt(pi)*(-erf(sqrt(x)) + 1) + 2*exp(-x)/sqrt(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">uppergamma</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">expint(3, x)/x**2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.gamma_functions.lowergamma">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.gamma_functions.</tt><tt class="descname">lowergamma</tt><a class="reference internal" href="../../_modules/sympy/functions/special/gamma_functions.html#lowergamma"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.gamma_functions.lowergamma" title="Permalink to this definition">¶</a></dt>
-<dd><p>The lower incomplete gamma function.</p>
-<p>It can be defined as the meromorphic continuation of</p>
-<div class="math">
-\[\gamma(s, x) = \int_0^x t^{s-1} e^{-t} \mathrm{d}t.\]</div>
-<p>This can be shown to be the same as</p>
-<div class="math">
-\[\gamma(s, x) = \frac{x^s}{s} {}_1F_1\left({s \atop s+1} \middle| -x\right),\]</div>
-<p>where <span class="math">\({}_1F_1\)</span> is the (confluent) hypergeometric function.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">gamma</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">uppergamma</span></tt>, <a class="reference internal" href="#sympy.functions.special.hyper.hyper" title="sympy.functions.special.hyper.hyper"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.functions.special.hyper.hyper</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li>Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6, Section 5,
-Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical
-Tables</li>
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Incomplete_gamma_function">http://en.wikipedia.org/wiki/Incomplete_gamma_function</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">lowergamma</span><span class="p">,</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">s</span><span class="p">,</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lowergamma</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">lowergamma(s, x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lowergamma</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">-x**2*exp(-x) - 2*x*exp(-x) + 2 - 2*exp(-x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lowergamma</span><span class="p">(</span><span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">-2*sqrt(pi)*erf(sqrt(x)) - 2*exp(-x)/sqrt(x)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.functions.special.error_functions">
-<span id="special-cases-of-the-incomplete-gamma-functions"></span><h2>Special Cases of the Incomplete Gamma Functions<a class="headerlink" href="#module-sympy.functions.special.error_functions" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt>
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.error_functions.</tt><tt class="descname">erf</tt><a class="reference internal" href="../../_modules/sympy/functions/special/error_functions.html#erf"><span class="viewcode-link">[source]</span></a></dt>
-<dd><p>The Gauss error function.</p>
-<p>This function is defined as:</p>
-<p><span class="math">\(\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \, \mathrm{d}x\)</span></p>
-<p>Or, in ASCII:</p>
-<div class="highlight-python"><pre>        x
-    /
-   |
-   |     2
-   |   -t
-2* |  e    dt
-   |
-  /
-  0
--------------
-      ____
-    \/ pi</pre>
-</div>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r27" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id7">[R27]</a></td><td><a class="reference external" href="http://en.wikipedia.org/wiki/Error_function">http://en.wikipedia.org/wiki/Error_function</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r28" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id8">[R28]</a></td><td><a class="reference external" href="http://dlmf.nist.gov/7">http://dlmf.nist.gov/7</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r29" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id9">[R29]</a></td><td><a class="reference external" href="http://mathworld.wolfram.com/Erf.html">http://mathworld.wolfram.com/Erf.html</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r30" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id10">[R30]</a></td><td><a class="reference external" href="http://functions.wolfram.com/GammaBetaErf/Erf">http://functions.wolfram.com/GammaBetaErf/Erf</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">I</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">erf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">z</span>
-</pre></div>
-</div>
-<p>Several special values are known:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="o">-</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">-1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">oo*I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">-oo*I</span>
-</pre></div>
-</div>
-<p>In general one can pull out factors of -1 and I from the argument:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">)</span>
-<span class="go">-erf(z)</span>
-</pre></div>
-</div>
-<p>The error function obeys the mirror symmetry:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">conjugate</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">conjugate</span><span class="p">(</span><span class="n">erf</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="go">erf(conjugate(z))</span>
-</pre></div>
-</div>
-<p>Differentiation with respect to z is supported:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">diff</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">erf</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">2*exp(-z**2)/sqrt(pi)</span>
-</pre></div>
-</div>
-<p>We can numerically evaluate the error function to arbitrary precision
-on the whole complex plane:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">30</span><span class="p">)</span>
-<span class="go">0.999999984582742099719981147840</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="o">*</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">30</span><span class="p">)</span>
-<span class="go">-1296959.73071763923152794095062*I</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.error_functions.Ei">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.error_functions.</tt><tt class="descname">Ei</tt><a class="reference internal" href="../../_modules/sympy/functions/special/error_functions.html#Ei"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.error_functions.Ei" title="Permalink to this definition">¶</a></dt>
-<dd><p>The classical exponential integral.</p>
-<p>For the use in SymPy, this function is defined as</p>
-<div class="math">
-\[\operatorname{Ei}(x) = \sum_{n=1}^\infty \frac{x^n}{n\, n!}
-+ \log(x) + \gamma,\]</div>
-<p>where <span class="math">\(\gamma\)</span> is the Euler-Mascheroni constant.</p>
-<p>If <span class="math">\(x\)</span> is a polar number, this defines an analytic function on the
-riemann surface of the logarithm. Otherwise this defines an analytic
-function in the cut plane <span class="math">\(\mathbb{C} \setminus (-\infty, 0]\)</span>.</p>
-<p><strong>Background</strong></p>
-<p>The name &#8216;exponential integral&#8217; comes from the following statement:</p>
-<div class="math">
-\[\operatorname{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} \mathrm{d}t\]</div>
-<p>If the integral is interpreted as a Cauchy principal value, this statement
-holds for <span class="math">\(x &gt; 0\)</span> and <span class="math">\(\operatorname{Ei}(x)\)</span> as defined above.</p>
-<p>Note that we carefully avoided defining <span class="math">\(\operatorname{Ei}(x)\)</span> for
-negative real x. This is because above integral formula does not hold for
-any polar lift of such <span class="math">\(x\)</span>, indeed all branches of
-<span class="math">\(\operatorname{Ei}(x)\)</span> above the negative reals are imaginary.</p>
-<p>However, the following statement holds for all <span class="math">\(x \in \mathbb{R}^*\)</span>:</p>
-<div class="math">
-\[\int_{-\infty}^x \frac{e^t}{t} \mathrm{d}t =
-\frac{\operatorname{Ei}\left(|x|e^{i \arg(x)}\right) +
-      \operatorname{Ei}\left(|x|e^{- i \arg(x)}\right)}{2},\]</div>
-<p>where the integral is again understood to be a principal value if
-<span class="math">\(x &gt; 0\)</span>, and <span class="math">\(|x|e^{i \arg(x)}\)</span>,
-<span class="math">\(|x|e^{- i \arg(x)}\)</span> denote two conjugate polar lifts of <span class="math">\(x\)</span>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.functions.special.error_functions.expint" title="sympy.functions.special.error_functions.expint"><tt class="xref py py-obj docutils literal"><span class="pre">expint</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.gamma_functions.uppergamma" title="sympy.functions.special.gamma_functions.uppergamma"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.functions.special.gamma_functions.uppergamma</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li>Abramowitz &amp; Stegun, section 5: <a class="reference external" href="http://www.math.sfu.ca/~cbm/aands/page_228.htm">http://www.math.sfu.ca/~cbm/aands/page_228.htm</a></li>
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Exponential_integral">http://en.wikipedia.org/wiki/Exponential_integral</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Ei</span><span class="p">,</span> <span class="n">polar_lift</span><span class="p">,</span> <span class="n">exp_polar</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<p>The exponential integral in SymPy is strictly undefined for negative values
-of the argument. For convenience, exponential integrals with negative
-arguments are immediately converted into an expression that agrees with
-the classical integral definition:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Ei</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-I*pi + Ei(exp_polar(I*pi))</span>
-</pre></div>
-</div>
-<p>This yields a real value:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Ei</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="n">chop</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">-0.219383934395520</span>
-</pre></div>
-</div>
-<p>On the other hand the analytic continuation is not real:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Ei</span><span class="p">(</span><span class="n">polar_lift</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="n">chop</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">-0.21938393439552 + 3.14159265358979*I</span>
-</pre></div>
-</div>
-<p>The exponential integral has a logarithmic branch point at the origin:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Ei</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">exp_polar</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span>
-<span class="go">Ei(x) + 2*I*pi</span>
-</pre></div>
-</div>
-<p>Differentiation is supported:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Ei</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">exp(x)/x</span>
-</pre></div>
-</div>
-<p>The exponential integral is related to many other special functions.
-For example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">uppergamma</span><span class="p">,</span> <span class="n">expint</span><span class="p">,</span> <span class="n">Shi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ei</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">expint</span><span class="p">)</span>
-<span class="go">-expint(1, x*exp_polar(I*pi)) - I*pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ei</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">Shi</span><span class="p">)</span>
-<span class="go">Chi(x) + Shi(x)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.error_functions.expint">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.error_functions.</tt><tt class="descname">expint</tt><a class="reference internal" href="../../_modules/sympy/functions/special/error_functions.html#expint"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.error_functions.expint" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generalized exponential integral.</p>
-<p>This function is defined as</p>
-<div class="math">
-\[\operatorname{E}_\nu(z) = z^{\nu - 1} \Gamma(1 - \nu, z),\]</div>
-<p>where <span class="math">\(\Gamma(1 - \nu, z)\)</span> is the upper incomplete gamma function
-(<tt class="docutils literal"><span class="pre">uppergamma</span></tt>).</p>
-<p>Hence for <span class="math">\(z\)</span> with positive real part we have</p>
-<div class="math">
-\[\operatorname{E}_\nu(z)
-=   \int_1^\infty \frac{e^{-zt}}{z^\nu} \mathrm{d}t,\]</div>
-<p>which explains the name.</p>
-<p>The representation as an incomplete gamma function provides an analytic
-continuation for <span class="math">\(\operatorname{E}_\nu(z)\)</span>. If <span class="math">\(\nu\)</span> is a
-non-positive integer the exponential integral is thus an unbranched
-function of <span class="math">\(z\)</span>, otherwise there is a branch point at the origin.
-Refer to the incomplete gamma function documentation for details of the
-branching behavior.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="docutils">
-<dt><a class="reference internal" href="#sympy.functions.special.error_functions.E1" title="sympy.functions.special.error_functions.E1"><tt class="xref py py-obj docutils literal"><span class="pre">E1</span></tt></a></dt>
-<dd>The classical case, returns expint(1, z).</dd>
-<dt><a class="reference internal" href="#sympy.functions.special.error_functions.Ei" title="sympy.functions.special.error_functions.Ei"><tt class="xref py py-obj docutils literal"><span class="pre">Ei</span></tt></a></dt>
-<dd>Another related function called exponential integral.</dd>
-</dl>
-<p class="last"><a class="reference internal" href="#sympy.functions.special.gamma_functions.uppergamma" title="sympy.functions.special.gamma_functions.uppergamma"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.functions.special.gamma_functions.uppergamma</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://dlmf.nist.gov/8.19">http://dlmf.nist.gov/8.19</a></li>
-<li><a class="reference external" href="http://functions.wolfram.com/GammaBetaErf/ExpIntegralE/">http://functions.wolfram.com/GammaBetaErf/ExpIntegralE/</a></li>
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Exponential_integral">http://en.wikipedia.org/wiki/Exponential_integral</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expint</span><span class="p">,</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">nu</span><span class="p">,</span> <span class="n">z</span>
-</pre></div>
-</div>
-<p>Differentiation is supported. Differentiation with respect to z explains
-further the name: for integral orders, the exponential integral is an
-iterated integral of the exponential function.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expint</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">-expint(nu - 1, z)</span>
-</pre></div>
-</div>
-<p>Differentiation with respect to nu has no classical expression:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expint</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">nu</span><span class="p">)</span>
-<span class="go">-z**(nu - 1)*meijerg(((), (1, 1)), ((0, 0, -nu + 1), ()), z)</span>
-</pre></div>
-</div>
-<p>At non-postive integer orders, the exponential integral reduces to the
-exponential function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">exp(-z)/z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expint</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">exp(-z)/z + exp(-z)/z**2</span>
-</pre></div>
-</div>
-<p>At half-integers it reduces to error functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expint</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">-sqrt(pi)*erf(sqrt(z))/sqrt(z) + sqrt(pi)/sqrt(z)</span>
-</pre></div>
-</div>
-<p>At positive integer orders it can be rewritten in terms of exponentials
-and expint(1, z). Use expand_func() to do this:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expand_func</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_func</span><span class="p">(</span><span class="n">expint</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="n">z</span><span class="p">))</span>
-<span class="go">z**4*expint(1, z)/24 + (-z**3 + z**2 - 2*z + 6)*exp(-z)/24</span>
-</pre></div>
-</div>
-<p>The generalised exponential integral is essentially equivalent to the
-incomplete gamma function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">uppergamma</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expint</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">uppergamma</span><span class="p">)</span>
-<span class="go">z**(nu - 1)*uppergamma(-nu + 1, z)</span>
-</pre></div>
-</div>
-<p>As such it is branched at the origin:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exp_polar</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expint</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">z</span><span class="o">*</span><span class="n">exp_polar</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="p">))</span>
-<span class="go">I*pi*z**3/3 + expint(4, z)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expint</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="o">*</span><span class="n">exp_polar</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="p">))</span>
-<span class="go">z**(nu - 1)*(exp(2*I*pi*nu) - 1)*gamma(-nu + 1) + expint(nu, z)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.functions.special.error_functions.E1">
-<tt class="descclassname">sympy.functions.special.error_functions.</tt><tt class="descname">E1</tt><big>(</big><em>z</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/special/error_functions.html#E1"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.error_functions.E1" title="Permalink to this definition">¶</a></dt>
-<dd><p>Classical case of the generalized exponential integral.</p>
-<p>This is equivalent to <tt class="docutils literal"><span class="pre">expint(1,</span> <span class="pre">z)</span></tt>.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.error_functions.Si">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.error_functions.</tt><tt class="descname">Si</tt><a class="reference internal" href="../../_modules/sympy/functions/special/error_functions.html#Si"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.error_functions.Si" title="Permalink to this definition">¶</a></dt>
-<dd><p>Sine integral.</p>
-<p>This function is defined by</p>
-<div class="math">
-\[\operatorname{Si}(z) = \int_0^z \frac{\sin{t}}{t} \mathrm{d}t.\]</div>
-<p>It is an entire function.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.functions.special.error_functions.Ci" title="sympy.functions.special.error_functions.Ci"><tt class="xref py py-obj docutils literal"><span class="pre">Ci</span></tt></a></dt>
-<dd>Cosine integral.</dd>
-<dt><a class="reference internal" href="#sympy.functions.special.error_functions.Shi" title="sympy.functions.special.error_functions.Shi"><tt class="xref py py-obj docutils literal"><span class="pre">Shi</span></tt></a></dt>
-<dd>Sinh integral.</dd>
-<dt><a class="reference internal" href="#sympy.functions.special.error_functions.Chi" title="sympy.functions.special.error_functions.Chi"><tt class="xref py py-obj docutils literal"><span class="pre">Chi</span></tt></a></dt>
-<dd>Cosh integral.</dd>
-<dt><a class="reference internal" href="#sympy.functions.special.error_functions.expint" title="sympy.functions.special.error_functions.expint"><tt class="xref py py-obj docutils literal"><span class="pre">expint</span></tt></a></dt>
-<dd>The generalised exponential integral.</dd>
-</dl>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Trigonometric_integral">http://en.wikipedia.org/wiki/Trigonometric_integral</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Si</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">z</span>
-</pre></div>
-</div>
-<p>The sine integral is an antiderivative of sin(z)/z:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Si</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">sin(z)/z</span>
-</pre></div>
-</div>
-<p>It is unbranched:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exp_polar</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Si</span><span class="p">(</span><span class="n">z</span><span class="o">*</span><span class="n">exp_polar</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span>
-<span class="go">Si(z)</span>
-</pre></div>
-</div>
-<p>Sine integral behaves much like ordinary sine under multiplication by I:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Si</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
-<span class="go">I*Shi(z)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Si</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">)</span>
-<span class="go">-Si(z)</span>
-</pre></div>
-</div>
-<p>It can also be expressed in terms of exponential integrals, but beware
-that the latter is branched:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Si</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">expint</span><span class="p">)</span>
-<span class="go">-I*(-expint(1, z*exp_polar(-I*pi/2))/2 +</span>
-<span class="go">     expint(1, z*exp_polar(I*pi/2))/2) + pi/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.error_functions.Ci">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.error_functions.</tt><tt class="descname">Ci</tt><a class="reference internal" href="../../_modules/sympy/functions/special/error_functions.html#Ci"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.error_functions.Ci" title="Permalink to this definition">¶</a></dt>
-<dd><p>Cosine integral.</p>
-<p>This function is defined for positive <span class="math">\(x\)</span> by</p>
-<div class="math">
-\[\operatorname{Ci}(x) = \gamma + \log{x}
-              + \int_0^x \frac{\cos{t} - 1}{t} \mathrm{d}t
-= -\int_x^\infty \frac{\cos{t}}{t} \mathrm{d}t,\]</div>
-<p>where <span class="math">\(\gamma\)</span> is the Euler-Mascheroni constant.</p>
-<p>We have</p>
-<div class="math">
-\[\operatorname{Ci}(z) =
--\frac{\operatorname{E}_1\left(e^{i\pi/2} z\right)
-       + \operatorname{E}_1\left(e^{-i \pi/2} z\right)}{2}\]</div>
-<p>which holds for all polar <span class="math">\(z\)</span> and thus provides an analytic
-continuation to the Riemann surface of the logarithm.</p>
-<p>The formula also holds as stated
-for <span class="math">\(z \in \mathbb{C}\)</span> with <span class="math">\(Re(z) &gt; 0\)</span>.
-By lifting to the principal branch we obtain an analytic function on the
-cut complex plane.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.functions.special.error_functions.Si" title="sympy.functions.special.error_functions.Si"><tt class="xref py py-obj docutils literal"><span class="pre">Si</span></tt></a></dt>
-<dd>Sine integral.</dd>
-<dt><a class="reference internal" href="#sympy.functions.special.error_functions.Shi" title="sympy.functions.special.error_functions.Shi"><tt class="xref py py-obj docutils literal"><span class="pre">Shi</span></tt></a></dt>
-<dd>Sinh integral.</dd>
-<dt><a class="reference internal" href="#sympy.functions.special.error_functions.Chi" title="sympy.functions.special.error_functions.Chi"><tt class="xref py py-obj docutils literal"><span class="pre">Chi</span></tt></a></dt>
-<dd>Cosh integral.</dd>
-<dt><a class="reference internal" href="#sympy.functions.special.error_functions.expint" title="sympy.functions.special.error_functions.expint"><tt class="xref py py-obj docutils literal"><span class="pre">expint</span></tt></a></dt>
-<dd>The generalised exponential integral.</dd>
-</dl>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Trigonometric_integral">http://en.wikipedia.org/wiki/Trigonometric_integral</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Ci</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">z</span>
-</pre></div>
-</div>
-<p>The cosine integral is a primitive of cos(z)/z:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Ci</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">cos(z)/z</span>
-</pre></div>
-</div>
-<p>It has a logarithmic branch point at the origin:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exp_polar</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ci</span><span class="p">(</span><span class="n">z</span><span class="o">*</span><span class="n">exp_polar</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span>
-<span class="go">Ci(z) + 2*I*pi</span>
-</pre></div>
-</div>
-<p>Cosine integral behaves somewhat like ordinary cos under multiplication by I:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">polar_lift</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ci</span><span class="p">(</span><span class="n">polar_lift</span><span class="p">(</span><span class="n">I</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
-<span class="go">Chi(z) + I*pi/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ci</span><span class="p">(</span><span class="n">polar_lift</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
-<span class="go">Ci(z) + I*pi</span>
-</pre></div>
-</div>
-<p>It can also be expressed in terms of exponential integrals:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ci</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">expint</span><span class="p">)</span>
-<span class="go">-expint(1, z*exp_polar(-I*pi/2))/2 - expint(1, z*exp_polar(I*pi/2))/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.error_functions.Shi">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.error_functions.</tt><tt class="descname">Shi</tt><a class="reference internal" href="../../_modules/sympy/functions/special/error_functions.html#Shi"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.error_functions.Shi" title="Permalink to this definition">¶</a></dt>
-<dd><p>Sinh integral.</p>
-<p>This function is defined by</p>
-<div class="math">
-\[\operatorname{Shi}(z) = \int_0^z \frac{\sinh{t}}{t} \mathrm{d}t.\]</div>
-<p>It is an entire function.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.functions.special.error_functions.Si" title="sympy.functions.special.error_functions.Si"><tt class="xref py py-obj docutils literal"><span class="pre">Si</span></tt></a></dt>
-<dd>Sine integral.</dd>
-<dt><a class="reference internal" href="#sympy.functions.special.error_functions.Ci" title="sympy.functions.special.error_functions.Ci"><tt class="xref py py-obj docutils literal"><span class="pre">Ci</span></tt></a></dt>
-<dd>Cosine integral.</dd>
-<dt><a class="reference internal" href="#sympy.functions.special.error_functions.Chi" title="sympy.functions.special.error_functions.Chi"><tt class="xref py py-obj docutils literal"><span class="pre">Chi</span></tt></a></dt>
-<dd>Cosh integral.</dd>
-<dt><a class="reference internal" href="#sympy.functions.special.error_functions.expint" title="sympy.functions.special.error_functions.expint"><tt class="xref py py-obj docutils literal"><span class="pre">expint</span></tt></a></dt>
-<dd>The generalised exponential integral.</dd>
-</dl>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Trigonometric_integral">http://en.wikipedia.org/wiki/Trigonometric_integral</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Shi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">z</span>
-</pre></div>
-</div>
-<p>The Sinh integral is a primitive of sinh(z)/z:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Shi</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">sinh(z)/z</span>
-</pre></div>
-</div>
-<p>It is unbranched:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exp_polar</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Shi</span><span class="p">(</span><span class="n">z</span><span class="o">*</span><span class="n">exp_polar</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span>
-<span class="go">Shi(z)</span>
-</pre></div>
-</div>
-<p>Sinh integral behaves much like ordinary sinh under multiplication by I:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Shi</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
-<span class="go">I*Si(z)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Shi</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">)</span>
-<span class="go">-Shi(z)</span>
-</pre></div>
-</div>
-<p>It can also be expressed in terms of exponential integrals, but beware
-that the latter is branched:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Shi</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">expint</span><span class="p">)</span>
-<span class="go">expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.error_functions.Chi">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.error_functions.</tt><tt class="descname">Chi</tt><a class="reference internal" href="../../_modules/sympy/functions/special/error_functions.html#Chi"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.error_functions.Chi" title="Permalink to this definition">¶</a></dt>
-<dd><p>Cosh integral.</p>
-<p>This function is defined for positive <span class="math">\(x\)</span> by</p>
-<div class="math">
-\[\operatorname{Chi}(x) = \gamma + \log{x}
-+ \int_0^x \frac{\cosh{t} - 1}{t} \mathrm{d}t,\]</div>
-<p>where <span class="math">\(\gamma\)</span> is the Euler-Mascheroni constant.</p>
-<p>We have</p>
-<div class="math">
-\[\operatorname{Chi}(z) = \operatorname{Ci}\left(e^{i \pi/2}z\right)
-- i\frac{\pi}{2},\]</div>
-<p>which holds for all polar <span class="math">\(z\)</span> and thus provides an analytic
-continuation to the Riemann surface of the logarithm.
-By lifting to the principal branch we obtain an analytic function on the
-cut complex plane.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.functions.special.error_functions.Si" title="sympy.functions.special.error_functions.Si"><tt class="xref py py-obj docutils literal"><span class="pre">Si</span></tt></a></dt>
-<dd>Sine integral.</dd>
-<dt><a class="reference internal" href="#sympy.functions.special.error_functions.Ci" title="sympy.functions.special.error_functions.Ci"><tt class="xref py py-obj docutils literal"><span class="pre">Ci</span></tt></a></dt>
-<dd>Cosine integral.</dd>
-<dt><a class="reference internal" href="#sympy.functions.special.error_functions.Shi" title="sympy.functions.special.error_functions.Shi"><tt class="xref py py-obj docutils literal"><span class="pre">Shi</span></tt></a></dt>
-<dd>Sinh integral.</dd>
-<dt><a class="reference internal" href="#sympy.functions.special.error_functions.expint" title="sympy.functions.special.error_functions.expint"><tt class="xref py py-obj docutils literal"><span class="pre">expint</span></tt></a></dt>
-<dd>The generalised exponential integral.</dd>
-</dl>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Trigonometric_integral">http://en.wikipedia.org/wiki/Trigonometric_integral</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Chi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">z</span>
-</pre></div>
-</div>
-<p>The cosh integral is a primitive of cosh(z)/z:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Chi</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">cosh(z)/z</span>
-</pre></div>
-</div>
-<p>It has a logarithmic branch point at the origin:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exp_polar</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Chi</span><span class="p">(</span><span class="n">z</span><span class="o">*</span><span class="n">exp_polar</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span>
-<span class="go">Chi(z) + 2*I*pi</span>
-</pre></div>
-</div>
-<p>Cosh integral behaves somewhat like ordinary cosh under multiplication by I:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">polar_lift</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Chi</span><span class="p">(</span><span class="n">polar_lift</span><span class="p">(</span><span class="n">I</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
-<span class="go">Ci(z) + I*pi/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Chi</span><span class="p">(</span><span class="n">polar_lift</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
-<span class="go">Chi(z) + I*pi</span>
-</pre></div>
-</div>
-<p>It can also be expressed in terms of exponential integrals:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Chi</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">expint</span><span class="p">)</span>
-<span class="go">-expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.error_functions.FresnelIntegral">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.error_functions.</tt><tt class="descname">FresnelIntegral</tt><a class="reference internal" href="../../_modules/sympy/functions/special/error_functions.html#FresnelIntegral"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.error_functions.FresnelIntegral" title="Permalink to this definition">¶</a></dt>
-<dd><p>Base class for the Fresnel integrals.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.error_functions.fresnels">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.error_functions.</tt><tt class="descname">fresnels</tt><a class="reference internal" href="../../_modules/sympy/functions/special/error_functions.html#fresnels"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.error_functions.fresnels" title="Permalink to this definition">¶</a></dt>
-<dd><p>Fresnel integral S.</p>
-<p>This function is defined by</p>
-<div class="math">
-\[\operatorname{S}(z) = \int_0^z \sin{\frac{\pi}{2} t^2} \mathrm{d}t.\]</div>
-<p>It is an entire function.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.functions.special.error_functions.fresnelc" title="sympy.functions.special.error_functions.fresnelc"><tt class="xref py py-obj docutils literal"><span class="pre">fresnelc</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r31" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id11">[R31]</a></td><td><a class="reference external" href="http://en.wikipedia.org/wiki/Fresnel_integral">http://en.wikipedia.org/wiki/Fresnel_integral</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r32" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id12">[R32]</a></td><td><a class="reference external" href="http://dlmf.nist.gov/7">http://dlmf.nist.gov/7</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r33" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id13">[R33]</a></td><td><a class="reference external" href="http://mathworld.wolfram.com/FresnelIntegrals.html">http://mathworld.wolfram.com/FresnelIntegrals.html</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r34" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id14">[R34]</a></td><td><a class="reference external" href="http://functions.wolfram.com/GammaBetaErf/FresnelS">http://functions.wolfram.com/GammaBetaErf/FresnelS</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">I</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">fresnels</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">z</span>
-</pre></div>
-</div>
-<p>Several special values are known:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fresnels</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fresnels</span><span class="p">(</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fresnels</span><span class="p">(</span><span class="o">-</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">-1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fresnels</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">-I/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fresnels</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">I/2</span>
-</pre></div>
-</div>
-<p>In general one can pull out factors of -1 and I from the argument:
-&gt;&gt;&gt; fresnels(-z)
--fresnels(z)</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fresnels</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
-<span class="go">-I*fresnels(z)</span>
-</pre></div>
-</div>
-<p>The Fresnel S integral obeys the mirror symmetry:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">conjugate</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">conjugate</span><span class="p">(</span><span class="n">fresnels</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="go">fresnels(conjugate(z))</span>
-</pre></div>
-</div>
-<p>Differentiation with respect to z is supported:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">diff</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">fresnels</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">sin(pi*z**2/2)</span>
-</pre></div>
-</div>
-<p>Defining the Fresnel functions via an integral</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">integrate</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">expand_func</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">2</span><span class="p">),</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">3*fresnels(z)*gamma(3/4)/(4*gamma(7/4))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_func</span><span class="p">(</span><span class="n">integrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">2</span><span class="p">),</span> <span class="n">z</span><span class="p">))</span>
-<span class="go">fresnels(z)</span>
-</pre></div>
-</div>
-<p>We can numerically evaluate the Fresnel integral to arbitrary precision
-on the whole complex plane:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fresnels</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">30</span><span class="p">)</span>
-<span class="go">0.343415678363698242195300815958</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fresnels</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">30</span><span class="p">)</span>
-<span class="go">0.343415678363698242195300815958*I</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.error_functions.fresnelc">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.error_functions.</tt><tt class="descname">fresnelc</tt><a class="reference internal" href="../../_modules/sympy/functions/special/error_functions.html#fresnelc"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.error_functions.fresnelc" title="Permalink to this definition">¶</a></dt>
-<dd><p>Fresnel integral C.</p>
-<p>This function is defined by</p>
-<div class="math">
-\[\operatorname{C}(z) = \int_0^z \cos{\frac{\pi}{2} t^2} \mathrm{d}t.\]</div>
-<p>It is an entire function.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.functions.special.error_functions.fresnels" title="sympy.functions.special.error_functions.fresnels"><tt class="xref py py-obj docutils literal"><span class="pre">fresnels</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r35" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id15">[R35]</a></td><td><a class="reference external" href="http://en.wikipedia.org/wiki/Fresnel_integral">http://en.wikipedia.org/wiki/Fresnel_integral</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r36" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id16">[R36]</a></td><td><a class="reference external" href="http://dlmf.nist.gov/7">http://dlmf.nist.gov/7</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r37" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id17">[R37]</a></td><td><a class="reference external" href="http://mathworld.wolfram.com/FresnelIntegrals.html">http://mathworld.wolfram.com/FresnelIntegrals.html</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r38" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id18">[R38]</a></td><td><a class="reference external" href="http://functions.wolfram.com/GammaBetaErf/FresnelC">http://functions.wolfram.com/GammaBetaErf/FresnelC</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">I</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">fresnelc</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">z</span>
-</pre></div>
-</div>
-<p>Several special values are known:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fresnelc</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fresnelc</span><span class="p">(</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fresnelc</span><span class="p">(</span><span class="o">-</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">-1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fresnelc</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">I/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fresnelc</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">-I/2</span>
-</pre></div>
-</div>
-<p>In general one can pull out factors of -1 and I from the argument:
-&gt;&gt;&gt; fresnelc(-z)
--fresnelc(z)</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fresnelc</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
-<span class="go">I*fresnelc(z)</span>
-</pre></div>
-</div>
-<p>The Fresnel C integral obeys the mirror symmetry:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">conjugate</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">conjugate</span><span class="p">(</span><span class="n">fresnelc</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="go">fresnelc(conjugate(z))</span>
-</pre></div>
-</div>
-<p>Differentiation with respect to z is supported:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">diff</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">fresnelc</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">cos(pi*z**2/2)</span>
-</pre></div>
-</div>
-<p>Defining the Fresnel functions via an integral</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">integrate</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">expand_func</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">2</span><span class="p">),</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">fresnelc(z)*gamma(1/4)/(4*gamma(5/4))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_func</span><span class="p">(</span><span class="n">integrate</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">2</span><span class="p">),</span> <span class="n">z</span><span class="p">))</span>
-<span class="go">fresnelc(z)</span>
-</pre></div>
-</div>
-<p>We can numerically evaluate the Fresnel integral to arbitrary precision
-on the whole complex plane:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fresnelc</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">30</span><span class="p">)</span>
-<span class="go">0.488253406075340754500223503357</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fresnelc</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">30</span><span class="p">)</span>
-<span class="go">-0.488253406075340754500223503357*I</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="bessel-type-functions">
-<h2>Bessel Type Functions<a class="headerlink" href="#bessel-type-functions" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.special.bessel.BesselBase">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.bessel.</tt><tt class="descname">BesselBase</tt><a class="reference internal" href="../../_modules/sympy/functions/special/bessel.html#BesselBase"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.bessel.BesselBase" title="Permalink to this definition">¶</a></dt>
-<dd><p>Abstract base class for bessel-type functions.</p>
-<p>This class is meant to reduce code duplication.
-All bessel type functions can 1) be differentiated, and the derivatives
-expressed in terms of similar functions and 2) be rewritten in terms
-of other bessel-type functions.</p>
-<p>Here &#8220;bessel-type functions&#8221; are assumed to have one complex parameter.</p>
-<p>To use this base class, define class attributes <tt class="docutils literal"><span class="pre">_a</span></tt> and <tt class="docutils literal"><span class="pre">_b</span></tt> such that
-<tt class="docutils literal"><span class="pre">2*F_n'</span> <span class="pre">=</span> <span class="pre">-_a*F_{n+1}</span> <span class="pre">b*F_{n-1}</span></tt>.</p>
-<dl class="attribute">
-<dt id="sympy.functions.special.bessel.BesselBase.argument">
-<tt class="descname">argument</tt><a class="reference internal" href="../../_modules/sympy/functions/special/bessel.html#BesselBase.argument"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.bessel.BesselBase.argument" title="Permalink to this definition">¶</a></dt>
-<dd><p>The argument of the bessel-type function.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.special.bessel.BesselBase.order">
-<tt class="descname">order</tt><a class="reference internal" href="../../_modules/sympy/functions/special/bessel.html#BesselBase.order"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.bessel.BesselBase.order" title="Permalink to this definition">¶</a></dt>
-<dd><p>The order of the bessel-type function.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.bessel.besselj">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.bessel.</tt><tt class="descname">besselj</tt><a class="reference internal" href="../../_modules/sympy/functions/special/bessel.html#besselj"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.bessel.besselj" title="Permalink to this definition">¶</a></dt>
-<dd><p>Bessel function of the first kind.</p>
-<p>The Bessel J function of order <span class="math">\(\nu\)</span> is defined to be the function
-satisfying Bessel&#8217;s differential equation</p>
-<div class="math">
-\[z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2}
-+ z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu^2) w = 0,\]</div>
-<p>with Laurent expansion</p>
-<div class="math">
-\[J_\nu(z) = z^\nu \left(\frac{1}{\Gamma(\nu + 1) 2^\nu} + O(z^2) \right),\]</div>
-<p>if <span class="math">\(\nu\)</span> is not a negative integer. If <span class="math">\(\nu=-n \in \mathbb{Z}_{&lt;0}\)</span>
-<em>is</em> a negative integer, then the definition is</p>
-<div class="math">
-\[J_{-n}(z) = (-1)^n J_n(z).\]</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">bessely</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">besseli</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">besselk</span></tt></p>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li>Abramowitz, Milton; Stegun, Irene A., eds. (1965), &#8220;Chapter 9&#8221;,
-Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical
-Tables</li>
-<li>Luke, Y. L. (1969), The Special Functions and Their Approximations,
-Volume 1</li>
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Bessel_function">http://en.wikipedia.org/wiki/Bessel_function</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<p>Create a bessel function object:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">besselj</span><span class="p">,</span> <span class="n">jn</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">besselj</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Differentiate it:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">besselj(n - 1, z)/2 - besselj(n + 1, z)/2</span>
-</pre></div>
-</div>
-<p>Rewrite in terms of spherical bessel functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">jn</span><span class="p">)</span>
-<span class="go">sqrt(2)*sqrt(z)*jn(n - 1/2, z)/sqrt(pi)</span>
-</pre></div>
-</div>
-<p>Access the parameter and argument:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">order</span>
-<span class="go">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">argument</span>
-<span class="go">z</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.bessel.bessely">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.bessel.</tt><tt class="descname">bessely</tt><a class="reference internal" href="../../_modules/sympy/functions/special/bessel.html#bessely"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.bessel.bessely" title="Permalink to this definition">¶</a></dt>
-<dd><p>Bessel function of the second kind.</p>
-<p>The Bessel Y function of order <span class="math">\(\nu\)</span> is defined as</p>
-<div class="math">
-\[Y_\nu(z) = \lim_{\mu \to \nu} \frac{J_\mu(z) \cos(\pi \mu)
-                                    - J_{-\mu}(z)}{\sin(\pi \mu)},\]</div>
-<p>where <span class="math">\(J_\mu(z)\)</span> is the Bessel function of the first kind.</p>
-<p>It is a solution to Bessel&#8217;s equation, and linearly independent from
-<span class="math">\(J_\nu\)</span>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">besselj</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">besseli</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">besselk</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">bessely</span><span class="p">,</span> <span class="n">yn</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">bessely</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">bessely(n - 1, z)/2 - bessely(n + 1, z)/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">yn</span><span class="p">)</span>
-<span class="go">sqrt(2)*sqrt(z)*yn(n - 1/2, z)/sqrt(pi)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.bessel.besseli">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.bessel.</tt><tt class="descname">besseli</tt><a class="reference internal" href="../../_modules/sympy/functions/special/bessel.html#besseli"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.bessel.besseli" title="Permalink to this definition">¶</a></dt>
-<dd><p>Modified Bessel function of the first kind.</p>
-<p>The Bessel I function is a solution to the modified Bessel equation</p>
-<div class="math">
-\[z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2}
-+ z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 + \nu^2)^2 w = 0.\]</div>
-<p>It can be defined as</p>
-<div class="math">
-\[I_\nu(z) = i^{-\nu} J_\nu(iz),\]</div>
-<p>where <span class="math">\(J_\mu(z)\)</span> is the Bessel function of the first kind.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">besselj</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">bessely</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">besselk</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">besseli</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besseli</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">besseli(n - 1, z)/2 + besseli(n + 1, z)/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.bessel.besselk">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.bessel.</tt><tt class="descname">besselk</tt><a class="reference internal" href="../../_modules/sympy/functions/special/bessel.html#besselk"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.bessel.besselk" title="Permalink to this definition">¶</a></dt>
-<dd><p>Modified Bessel function of the second kind.</p>
-<p>The Bessel K function of order <span class="math">\(\nu\)</span> is defined as</p>
-<div class="math">
-\[K_\nu(z) = \lim_{\mu \to \nu} \frac{\pi}{2}
-           \frac{I_{-\mu}(z) -I_\mu(z)}{\sin(\pi \mu)},\]</div>
-<p>where <span class="math">\(I_\mu(z)\)</span> is the modified Bessel function of the first kind.</p>
-<p>It is a solution of the modified Bessel equation, and linearly independent
-from <span class="math">\(Y_\nu\)</span>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">besselj</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">besseli</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">bessely</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">besselk</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselk</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">-besselk(n - 1, z)/2 - besselk(n + 1, z)/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.bessel.hankel1">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.bessel.</tt><tt class="descname">hankel1</tt><a class="reference internal" href="../../_modules/sympy/functions/special/bessel.html#hankel1"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.bessel.hankel1" title="Permalink to this definition">¶</a></dt>
-<dd><p>Hankel function of the first kind.</p>
-<p>This function is defined as</p>
-<div class="math">
-\[H_\nu^{(1)} = J_\nu(z) + iY_\nu(z),\]</div>
-<p>where <span class="math">\(J_\nu(z)\)</span> is the Bessel function of the first kind, and
-<span class="math">\(Y_\nu(z)\)</span> is the Bessel function of the second kind.</p>
-<p>It is a solution to Bessel&#8217;s equation.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">hankel2</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">besselj</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">bessely</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">hankel1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hankel1</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">hankel1(n - 1, z)/2 - hankel1(n + 1, z)/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.bessel.hankel2">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.bessel.</tt><tt class="descname">hankel2</tt><a class="reference internal" href="../../_modules/sympy/functions/special/bessel.html#hankel2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.bessel.hankel2" title="Permalink to this definition">¶</a></dt>
-<dd><p>Hankel function of the second kind.</p>
-<p>This function is defined as</p>
-<div class="math">
-\[H_\nu^{(2)} = J_\nu(z) - iY_\nu(z),\]</div>
-<p>where <span class="math">\(J_\nu(z)\)</span> is the Bessel function of the first kind, and
-<span class="math">\(Y_\nu(z)\)</span> is the Bessel function of the second kind.</p>
-<p>It is a solution to Bessel&#8217;s equation, and linearly independent from
-<span class="math">\(H_\nu^{(1)}\)</span>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">hankel1</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">besselj</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">bessely</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">hankel2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hankel2</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">hankel2(n - 1, z)/2 - hankel2(n + 1, z)/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.bessel.jn">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.bessel.</tt><tt class="descname">jn</tt><a class="reference internal" href="../../_modules/sympy/functions/special/bessel.html#jn"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.bessel.jn" title="Permalink to this definition">¶</a></dt>
-<dd><p>Spherical Bessel function of the first kind.</p>
-<p>This function is a solution to the spherical bessel equation</p>
-<div class="math">
-\[z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2}
-  + 2z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu(\nu + 1)) w = 0.\]</div>
-<p>It can be defined as</p>
-<div class="math">
-\[j_\nu(z) = \sqrt{\frac{\pi}{2z}} J_{\nu + \frac{1}{2}}(z),\]</div>
-<p>where <span class="math">\(J_\nu(z)\)</span> is the Bessel function of the first kind.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">besselj</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">bessely</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">besselk</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">yn</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">jn</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">expand_func</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;z&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">jn</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">func</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">sin(z)/z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jn</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">func</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="o">==</span> <span class="n">sin</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">cos</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="n">z</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_func</span><span class="p">(</span><span class="n">jn</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">z</span><span class="p">))</span>
-<span class="go">(-6/z**2 + 15/z**4)*sin(z) + (1/z - 15/z**3)*cos(z)</span>
-</pre></div>
-</div>
-<p>The spherical Bessel functions of integral order
-are calculated using the formula:</p>
-<div class="math">
-\[j_n(z) = f_n(z) \sin{z} + (-1)^{n+1} f_{-n-1}(z) \cos{z},\]</div>
-<p>where the coefficients <span class="math">\(f_n(z)\)</span> are available as
-<tt class="xref py py-func docutils literal"><span class="pre">polys.orthopolys.spherical_bessel_fn()</span></tt>.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.bessel.yn">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.bessel.</tt><tt class="descname">yn</tt><a class="reference internal" href="../../_modules/sympy/functions/special/bessel.html#yn"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.bessel.yn" title="Permalink to this definition">¶</a></dt>
-<dd><p>Spherical Bessel function of the second kind.</p>
-<p>This function is another solution to the spherical bessel equation, and
-linearly independent from <span class="math">\(j_n\)</span>. It can be defined as</p>
-<div class="math">
-\[j_\nu(z) = \sqrt{\frac{\pi}{2z}} Y_{\nu + \frac{1}{2}}(z),\]</div>
-<p>where <span class="math">\(Y_\nu(z)\)</span> is the Bessel function of the second kind.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">besselj</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">bessely</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">besselk</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">jn</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">yn</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">expand_func</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;z&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">expand_func</span><span class="p">(</span><span class="n">yn</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">z</span><span class="p">)))</span>
-<span class="go">-cos(z)/z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_func</span><span class="p">(</span><span class="n">yn</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">))</span> <span class="o">==</span> <span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="n">z</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>For integral orders <span class="math">\(n\)</span>, <span class="math">\(y_n\)</span> is calculated using the formula:</p>
-<div class="math">
-\[y_n(z) = (-1)^{n+1} j_{-n-1}(z)\]</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.functions.special.bessel.jn_zeros">
-<tt class="descclassname">sympy.functions.special.bessel.</tt><tt class="descname">jn_zeros</tt><big>(</big><em>n</em>, <em>k</em>, <em>method='sympy'</em>, <em>dps=15</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/special/bessel.html#jn_zeros"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.bessel.jn_zeros" title="Permalink to this definition">¶</a></dt>
-<dd><p>Zeros of the spherical Bessel function of the first kind.</p>
-<p>This returns an array of zeros of jn up to the k-th zero.</p>
-<ul class="simple">
-<li>method = &#8220;sympy&#8221;: uses mpmath besseljzero</li>
-<li>method = &#8220;scipy&#8221;: uses the SciPy&#8217;s sph_jn and newton to find all
-roots, which is faster than computing the zeros using a general
-numerical solver, but it requires SciPy and only works with low
-precision floating point numbers.  [the function used with
-method=&#8221;sympy&#8221; is a recent addition to mpmath, before that a general
-solver was used]</li>
-</ul>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">jn</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">yn</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">besselj</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">besselk</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">bessely</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">jn_zeros</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jn_zeros</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="n">dps</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">[5.7635, 9.095, 12.323, 15.515]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="b-splines">
-<h2>B-Splines<a class="headerlink" href="#b-splines" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.functions.special.bsplines.bspline_basis">
-<tt class="descclassname">sympy.functions.special.bsplines.</tt><tt class="descname">bspline_basis</tt><big>(</big><em>d</em>, <em>knots</em>, <em>n</em>, <em>x</em>, <em>close=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/special/bsplines.html#bspline_basis"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.bsplines.bspline_basis" title="Permalink to this definition">¶</a></dt>
-<dd><p>The n-th B-spline at x of degree d with knots.</p>
-<p>B-Splines are piecewise polynomials of degree d [1].  They are defined on
-a set of knots, which is a sequence of integers or floats.</p>
-<p>The 0th degree splines have a value of one on a single interval:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">bspline_basis</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="mi">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">knots</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bspline_basis</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">knots</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">Piecewise((1, And(x &lt;= 1, x &gt;= 0)), (0, True))</span>
-</pre></div>
-</div>
-<p>For a given (d, knots) there are len(knots)-d-1 B-splines defined, that
-are indexed by n (starting at 0).</p>
-<p>Here is an example of a cubic B-spline:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">bspline_basis</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">)),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">Piecewise((x**3/6, And(x &lt; 1, x &gt;= 0)),</span>
-<span class="go">          (-x**3/2 + 2*x**2 - 2*x + 2/3, And(x &lt; 2, x &gt;= 1)),</span>
-<span class="go">          (x**3/2 - 4*x**2 + 10*x - 22/3, And(x &lt; 3, x &gt;= 2)),</span>
-<span class="go">          (-x**3/6 + 2*x**2 - 8*x + 32/3, And(x &lt;= 4, x &gt;= 3)),</span>
-<span class="go">          (0, True))</span>
-</pre></div>
-</div>
-<p>By repeating knot points, you can introduce discontinuities in the
-B-splines and their derivatives:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">knots</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bspline_basis</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">knots</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">Piecewise((-x/2 + 1, And(x &lt;= 2, x &gt;= 0)), (0, True))</span>
-</pre></div>
-</div>
-<p>It is quite time consuming to construct and evaluate B-splines. If you
-need to evaluate a B-splines many times, it is best to lambdify them
-first:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">lambdify</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">knots</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b0</span> <span class="o">=</span> <span class="n">bspline_basis</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">knots</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">lambdify</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">b0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">bsplines_basis_set</span></tt></p>
-</div>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/B-spline">http://en.wikipedia.org/wiki/B-spline</a></p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.functions.special.bsplines.bspline_basis_set">
-<tt class="descclassname">sympy.functions.special.bsplines.</tt><tt class="descname">bspline_basis_set</tt><big>(</big><em>d</em>, <em>knots</em>, <em>x</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/special/bsplines.html#bspline_basis_set"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.bsplines.bspline_basis_set" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the len(knots)-d-1 B-splines at x of degree d with knots.</p>
-<p>This function returns a list of Piecewise polynomials that are the
-len(knots)-d-1 B-splines of degree d for the given knots. This function
-calls bspline_basis(d, knots, n, x) for different values of n.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">bsplines_basis</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">bspline_basis_set</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">knots</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">splines</span> <span class="o">=</span> <span class="n">bspline_basis_set</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">knots</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">splines</span>
-<span class="go">[Piecewise((x**2/2, And(x &lt; 1, x &gt;= 0)),</span>
-<span class="go">           (-x**2 + 3*x - 3/2, And(x &lt; 2, x &gt;= 1)),</span>
-<span class="go">           (x**2/2 - 3*x + 9/2, And(x &lt;= 3, x &gt;= 2)),</span>
-<span class="go">           (0, True)),</span>
-<span class="go"> Piecewise((x**2/2 - x + 1/2, And(x &lt; 2, x &gt;= 1)),</span>
-<span class="go">           (-x**2 + 5*x - 11/2, And(x &lt; 3, x &gt;= 2)),</span>
-<span class="go">           (x**2/2 - 4*x + 8, And(x &lt;= 4, x &gt;= 3)),</span>
-<span class="go">           (0, True))]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.functions.special.zeta_functions">
-<span id="riemann-zeta-and-related-functions"></span><h2>Riemann Zeta and Related Functions<a class="headerlink" href="#module-sympy.functions.special.zeta_functions" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.special.zeta_functions.zeta">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.zeta_functions.</tt><tt class="descname">zeta</tt><a class="reference internal" href="../../_modules/sympy/functions/special/zeta_functions.html#zeta"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.zeta_functions.zeta" title="Permalink to this definition">¶</a></dt>
-<dd><p>Hurwitz zeta function (or Riemann zeta function).</p>
-<p>For <span class="math">\(Re(a) &gt; 0\)</span> and <span class="math">\(Re(s) &gt; 1\)</span>, this function is defined as</p>
-<div class="math">
-\[\zeta(s, a) = \sum_{n=0}^\infty \frac{1}{(n + a)^s},\]</div>
-<p>where the standard choice of argument for <span class="math">\(n + a\)</span> is used. For fixed
-<span class="math">\(a\)</span> with <span class="math">\(Re(a) &gt; 0\)</span> the Hurwitz zeta function admits a
-meromorphic continuation to all of <span class="math">\(\mathbb{C}\)</span>, it is an unbranched
-function with a simple pole at <span class="math">\(s = 1\)</span>.</p>
-<p>Analytic continuation to other <span class="math">\(a\)</span> is possible under some circumstances,
-but this is not typically done.</p>
-<p>The Hurwitz zeta function is a special case of the Lerch transcendent:</p>
-<div class="math">
-\[\zeta(s, a) = \Phi(1, s, a).\]</div>
-<p>This formula defines an analytic continuation for all possible values of
-<span class="math">\(s\)</span> and <span class="math">\(a\)</span> (also <span class="math">\(Re(a) &lt; 0\)</span>), see the documentation of
-<a class="reference internal" href="#sympy.functions.special.zeta_functions.lerchphi" title="sympy.functions.special.zeta_functions.lerchphi"><tt class="xref py py-class docutils literal"><span class="pre">lerchphi</span></tt></a> for a description of the branching behavior.</p>
-<p>If no value is passed for <span class="math">\(a\)</span>, by this function assumes a default value
-of <span class="math">\(a = 1\)</span>, yielding the Riemann zeta function.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.functions.special.zeta_functions.dirichlet_eta" title="sympy.functions.special.zeta_functions.dirichlet_eta"><tt class="xref py py-obj docutils literal"><span class="pre">dirichlet_eta</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.zeta_functions.lerchphi" title="sympy.functions.special.zeta_functions.lerchphi"><tt class="xref py py-obj docutils literal"><span class="pre">lerchphi</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.zeta_functions.polylog" title="sympy.functions.special.zeta_functions.polylog"><tt class="xref py py-obj docutils literal"><span class="pre">polylog</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://dlmf.nist.gov/25.11">http://dlmf.nist.gov/25.11</a></li>
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Hurwitz_zeta_function">http://en.wikipedia.org/wiki/Hurwitz_zeta_function</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<p>For <span class="math">\(a = 1\)</span> the Hurwitz zeta function reduces to the famous Riemann
-zeta function:</p>
-<div class="math">
-\[\zeta(s, 1) = \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.\]</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">zeta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">s</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">zeta(s)</span>
-</pre></div>
-</div>
-<p>The Riemann zeta function can also be expressed using the Dirichlet eta
-function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">dirichlet_eta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="n">s</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">dirichlet_eta</span><span class="p">)</span>
-<span class="go">dirichlet_eta(s)/(-2**(-s + 1) + 1)</span>
-</pre></div>
-</div>
-<p>The Riemann zeta function at positive even integer and negative odd integer
-values is related to the Bernoulli numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">pi**2/6</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">pi**4/90</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-1/12</span>
-</pre></div>
-</div>
-<p>The specific formulae are:</p>
-<div class="math">
-\[\zeta(2n) = (-1)^{n+1} \frac{B_{2n} (2\pi)^{2n}}{2(2n)!}\]</div>
-<div class="math">
-\[\zeta(-n) = -\frac{B_{n+1}}{n+1}\]</div>
-<p>At negative even integers the Riemann zeta function is zero:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>No closed-form expressions are known at positive odd integers, but
-numerical evaluation is possible:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">n</span><span class="p">()</span>
-<span class="go">1.20205690315959</span>
-</pre></div>
-</div>
-<p>The derivative of <span class="math">\(\zeta(s, a)\)</span> with respect to <span class="math">\(a\)</span> is easily
-computed:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">a</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="go">-s*zeta(s + 1, a)</span>
-</pre></div>
-</div>
-<p>However the derivative with respect to <span class="math">\(s\)</span> has no useful closed form
-expression:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="go">Derivative(zeta(s, a), s)</span>
-</pre></div>
-</div>
-<p>The Hurwitz zeta function can be expressed in terms of the Lerch transcendent,
-<tt class="xref py py-class docutils literal"><span class="pre">sympy.functions.special.lerchphi</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">lerchphi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">lerchphi</span><span class="p">)</span>
-<span class="go">lerchphi(1, s, a)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.zeta_functions.dirichlet_eta">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.zeta_functions.</tt><tt class="descname">dirichlet_eta</tt><a class="reference internal" href="../../_modules/sympy/functions/special/zeta_functions.html#dirichlet_eta"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.zeta_functions.dirichlet_eta" title="Permalink to this definition">¶</a></dt>
-<dd><p>Dirichlet eta function.</p>
-<p>For <span class="math">\(Re(s) &gt; 0\)</span>, this function is defined as</p>
-<div class="math">
-\[\eta(s) = \sum_{n=1}^\infty \frac{(-1)^n}{n^s}.\]</div>
-<p>It admits a unique analytic continuation to all of <span class="math">\(\mathbb{C}\)</span>.
-It is an entire, unbranched function.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.functions.special.zeta_functions.zeta" title="sympy.functions.special.zeta_functions.zeta"><tt class="xref py py-obj docutils literal"><span class="pre">zeta</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Dirichlet_eta_function">http://en.wikipedia.org/wiki/Dirichlet_eta_function</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<p>The Dirichlet eta function is closely related to the Riemann zeta function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">dirichlet_eta</span><span class="p">,</span> <span class="n">zeta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">s</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dirichlet_eta</span><span class="p">(</span><span class="n">s</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">zeta</span><span class="p">)</span>
-<span class="go">(-2**(-s + 1) + 1)*zeta(s)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.zeta_functions.polylog">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.zeta_functions.</tt><tt class="descname">polylog</tt><a class="reference internal" href="../../_modules/sympy/functions/special/zeta_functions.html#polylog"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.zeta_functions.polylog" title="Permalink to this definition">¶</a></dt>
-<dd><p>Polylogarithm function.</p>
-<p>For <span class="math">\(|z| &lt; 1\)</span> and <span class="math">\(s \in \mathbb{C}\)</span>, the polylogarithm is
-defined by</p>
-<div class="math">
-\[\operatorname{Li}_s(z) = \sum_{n=1}^\infty \frac{z^n}{n^s},\]</div>
-<p>where the standard branch of the argument is used for <span class="math">\(n\)</span>. It admits
-an analytic continuation which is branched at <span class="math">\(z=1\)</span> (notably not on the
-sheet of initial definition), <span class="math">\(z=0\)</span> and <span class="math">\(z=\infty\)</span>.</p>
-<p>The name polylogarithm comes from the fact that for <span class="math">\(s=1\)</span>, the
-polylogarithm is related to the ordinary logarithm (see examples), and that</p>
-<div class="math">
-\[\operatorname{Li}_{s+1}(z) =
-\int_0^z \frac{\operatorname{Li}_s(t)}{t} \mathrm{d}t.\]</div>
-<p>The polylogarithm is a special case of the Lerch transcendent:</p>
-<div class="math">
-\[\operatorname{Li}_{s}(z) = z \Phi(z, s, 1)\]</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.functions.special.zeta_functions.zeta" title="sympy.functions.special.zeta_functions.zeta"><tt class="xref py py-obj docutils literal"><span class="pre">zeta</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.zeta_functions.lerchphi" title="sympy.functions.special.zeta_functions.lerchphi"><tt class="xref py py-obj docutils literal"><span class="pre">lerchphi</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<p>For <span class="math">\(z \in \{0, 1, -1\}\)</span>, the polylogarithm is automatically expressed
-using other functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">polylog</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">s</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">zeta(s)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">dirichlet_eta(s)</span>
-</pre></div>
-</div>
-<p>If <span class="math">\(s\)</span> is a negative integer, <span class="math">\(0\)</span> or <span class="math">\(1\)</span>, the
-polylogarithm can be expressed using elementary functions. This can be
-done using expand_func():</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expand_func</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_func</span><span class="p">(</span><span class="n">polylog</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">))</span>
-<span class="go">-log(z*exp_polar(-I*pi) + 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_func</span><span class="p">(</span><span class="n">polylog</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">z</span><span class="p">))</span>
-<span class="go">z/(-z + 1)</span>
-</pre></div>
-</div>
-<p>The derivative with respect to <span class="math">\(z\)</span> can be computed in closed form:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">polylog(s - 1, z)/z</span>
-</pre></div>
-</div>
-<p>The polylogarithm can be expressed in terms of the lerch transcendent:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">lerchphi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="n">lerchphi</span><span class="p">)</span>
-<span class="go">z*lerchphi(z, s, 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.zeta_functions.lerchphi">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.zeta_functions.</tt><tt class="descname">lerchphi</tt><a class="reference internal" href="../../_modules/sympy/functions/special/zeta_functions.html#lerchphi"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.zeta_functions.lerchphi" title="Permalink to this definition">¶</a></dt>
-<dd><p>Lerch transcendent (Lerch phi function).</p>
-<p>For <span class="math">\(Re(a) &gt; 0\)</span>, <span class="math">\(|z| &lt; 1\)</span> and <span class="math">\(s \in \mathbb{C}\)</span>, the
-Lerch transcendent is defined as</p>
-<div class="math">
-\[\Phi(z, s, a) = \sum_{n=0}^\infty \frac{z^n}{(n + a)^s},\]</div>
-<p>where the standard branch of the argument is used for <span class="math">\(n + a\)</span>,
-and by analytic continuation for other values of the parameters.</p>
-<p>A commonly used related function is the Lerch zeta function, defined by</p>
-<div class="math">
-\[L(q, s, a) = \Phi(e^{2\pi i q}, s, a).\]</div>
-<p><strong>Analytic Continuation and Branching Behavior</strong></p>
-<p>It can be shown that</p>
-<div class="math">
-\[\Phi(z, s, a) = z\Phi(z, s, a+1) + a^{-s}.\]</div>
-<p>This provides the analytic continuation to <span class="math">\(Re(a) \le 0\)</span>.</p>
-<p>Assume now <span class="math">\(Re(a) &gt; 0\)</span>. The integral representation</p>
-<div class="math">
-\[\Phi_0(z, s, a) = \int_0^\infty \frac{t^{s-1} e^{-at}}{1 - ze^{-t}}
-\frac{\mathrm{d}t}{\Gamma(s)}\]</div>
-<p>provides an analytic continuation to <span class="math">\(\mathbb{C} - [1, \infty)\)</span>.
-Finally, for <span class="math">\(x \in (1, \infty)\)</span> we find</p>
-<div class="math">
-\[\lim_{\epsilon \to 0^+} \Phi_0(x + i\epsilon, s, a)
--\lim_{\epsilon \to 0^+} \Phi_0(x - i\epsilon, s, a)
-= \frac{2\pi i \log^{s-1}{x}}{x^a \Gamma(s)},\]</div>
-<p>using the standard branch for both <span class="math">\(\log{x}\)</span> and
-<span class="math">\(\log{\log{x}}\)</span> (a branch of <span class="math">\(\log{\log{x}}\)</span> is needed to
-evaluate <span class="math">\(\log{x}^{s-1}\)</span>).
-This concludes the analytic continuation. The Lerch transcendent is thus
-branched at <span class="math">\(z \in \{0, 1, \infty\}\)</span> and
-<span class="math">\(a \in \mathbb{Z}_{\le 0}\)</span>. For fixed <span class="math">\(z, a\)</span> outside these
-branch points, it is an entire function of <span class="math">\(s\)</span>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.functions.special.zeta_functions.polylog" title="sympy.functions.special.zeta_functions.polylog"><tt class="xref py py-obj docutils literal"><span class="pre">polylog</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.zeta_functions.zeta" title="sympy.functions.special.zeta_functions.zeta"><tt class="xref py py-obj docutils literal"><span class="pre">zeta</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li>Bateman, H.; Erdelyi, A. (1953), Higher Transcendental Functions,
-Vol. I, New York: McGraw-Hill. Section 1.11.</li>
-<li><a class="reference external" href="http://dlmf.nist.gov/25.14">http://dlmf.nist.gov/25.14</a></li>
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Lerch_transcendent">http://en.wikipedia.org/wiki/Lerch_transcendent</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<p>The Lerch transcendent is a fairly general function, for this reason it does
-not automatically evaluate to simpler functions. Use expand_func() to
-achieve this.</p>
-<p>If <span class="math">\(z=1\)</span>, the Lerch transcendent reduces to the Hurwitz zeta function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">lerchphi</span><span class="p">,</span> <span class="n">expand_func</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">z</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_func</span><span class="p">(</span><span class="n">lerchphi</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">))</span>
-<span class="go">zeta(s, a)</span>
-</pre></div>
-</div>
-<p>More generally, if <span class="math">\(z\)</span> is a root of unity, the Lerch transcendent
-reduces to a sum of Hurwitz zeta functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expand_func</span><span class="p">(</span><span class="n">lerchphi</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">))</span>
-<span class="go">2**(-s)*zeta(s, a/2) - 2**(-s)*zeta(s, a/2 + 1/2)</span>
-</pre></div>
-</div>
-<p>If <span class="math">\(a=1\)</span>, the Lerch transcendent reduces to the polylogarithm:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expand_func</span><span class="p">(</span><span class="n">lerchphi</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">polylog(s, z)/z</span>
-</pre></div>
-</div>
-<p>More generally, if <span class="math">\(a\)</span> is rational, the Lerch transcendent reduces
-to a sum of polylogarithms:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_func</span><span class="p">(</span><span class="n">lerchphi</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) -</span>
-<span class="go">            polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_func</span><span class="p">(</span><span class="n">lerchphi</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">-2**s/z + 2**(s - 1)*(polylog(s, sqrt(z))/sqrt(z) -</span>
-<span class="go">                      polylog(s, sqrt(z)*exp_polar(I*pi))/sqrt(z))/z</span>
-</pre></div>
-</div>
-<p>The derivatives with respect to <span class="math">\(z\)</span> and <span class="math">\(a\)</span> can be computed in
-closed form:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">lerchphi</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">(-a*lerchphi(z, s, a) + lerchphi(z, s - 1, a))/z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lerchphi</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="go">-s*lerchphi(z, s + 1, a)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="hypergeometric-functions">
-<h2>Hypergeometric Functions<a class="headerlink" href="#hypergeometric-functions" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.functions.special.hyper.hyper">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.hyper.</tt><tt class="descname">hyper</tt><a class="reference internal" href="../../_modules/sympy/functions/special/hyper.html#hyper"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.hyper.hyper" title="Permalink to this definition">¶</a></dt>
-<dd><p>The (generalized) hypergeometric function is defined by a series where
-the ratios of successive terms are a rational function of the summation
-index. When convergent, it is continued analytically to the largest
-possible domain.</p>
-<p>The hypergeometric function depends on two vectors of parameters, called
-the numerator parameters <span class="math">\(a_p\)</span>, and the denominator parameters
-<span class="math">\(b_q\)</span>. It also has an argument <span class="math">\(z\)</span>. The series definition is</p>
-<div class="math">
-\[\begin{split}{}_pF_q\left(\begin{matrix} a_1, \dots, a_p \\ b_1, \dots, b_q \end{matrix}
-             \middle| z \right)
-= \sum_{n=0}^\infty \frac{(a_1)_n \dots (a_p)_n}{(b_1)_n \dots (b_q)_n}
-                    \frac{z^n}{n!},\end{split}\]</div>
-<p>where <span class="math">\((a)_n = (a)(a+1)\dots(a+n-1)\)</span> denotes the rising factorial.</p>
-<p>If one of the <span class="math">\(b_q\)</span> is a non-positive integer then the series is
-undefined unless one of the <span class="math">\(a_p\)</span> is a larger (i.e. smaller in
-magnitude) non-positive integer. If none of the <span class="math">\(b_q\)</span> is a
-non-positive integer and one of the <span class="math">\(a_p\)</span> is a non-positive
-integer, then the series reduces to a polynomial. To simplify the
-following discussion, we assume that none of the <span class="math">\(a_p\)</span> or
-<span class="math">\(b_q\)</span> is a non-positive integer. For more details, see the
-references.</p>
-<p>The series converges for all <span class="math">\(z\)</span> if <span class="math">\(p \le q\)</span>, and thus
-defines an entire single-valued function in this case. If <span class="math">\(p =
-q+1\)</span> the series converges for <span class="math">\(|z| &lt; 1\)</span>, and can be continued
-analytically into a half-plane. If <span class="math">\(p &gt; q+1\)</span> the series is
-divergent for all <span class="math">\(z\)</span>.</p>
-<p>Note: The hypergeometric function constructor currently does <em>not</em> check
-if the parameters actually yield a well-defined function.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="../simplify/simplify.html#module-sympy.simplify.hyperexpand" title="sympy.simplify.hyperexpand"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.simplify.hyperexpand</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.gamma_functions.gamma" title="sympy.functions.special.gamma_functions.gamma"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.functions.special.gamma_functions.gamma</span></tt></a>, <tt class="xref py py-obj docutils literal"><span class="pre">meijerg</span></tt></p>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li>Luke, Y. L. (1969), The Special Functions and Their Approximations,
-Volume 1</li>
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Generalized_hypergeometric_function">http://en.wikipedia.org/wiki/Generalized_hypergeometric_function</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<p>The parameters <span class="math">\(a_p\)</span> and <span class="math">\(b_q\)</span> can be passed as arbitrary
-iterables, for example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">hyper</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">a</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">hyper((1, 2, 3), (3, 4), x)</span>
-</pre></div>
-</div>
-<p>There is also pretty printing (it looks better using unicode):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">hyper</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="n">x</span><span class="p">),</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">  _</span>
-<span class="go"> |_  /1, 2, 3 |  \</span>
-<span class="go"> |   |        | x|</span>
-<span class="go">3  2 \  3, 4  |  /</span>
-</pre></div>
-</div>
-<p>The parameters must always be iterables, even if they are vectors of
-length one or zero:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="p">),</span> <span class="p">[],</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">hyper((1,), (), x)</span>
-</pre></div>
-</div>
-<p>But of course they may be variables (but if they depend on x then you
-should not expect much implemented functionality):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">),</span> <span class="p">(</span><span class="n">n</span><span class="o">**</span><span class="mi">2</span><span class="p">,),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">hyper((n, a), (n**2,), x)</span>
-</pre></div>
-</div>
-<p>The hypergeometric function generalizes many named special functions.
-The function hyperexpand() tries to express a hypergeometric function
-using named special functions.
-For example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">hyperexpand</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyperexpand</span><span class="p">(</span><span class="n">hyper</span><span class="p">([],</span> <span class="p">[],</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">exp(x)</span>
-</pre></div>
-</div>
-<p>You can also use expand_func:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expand_func</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_func</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">hyper</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="o">-</span><span class="n">x</span><span class="p">))</span>
-<span class="go">log(x + 1)</span>
-</pre></div>
-</div>
-<p>More examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyperexpand</span><span class="p">(</span><span class="n">hyper</span><span class="p">([],</span> <span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">],</span> <span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">4</span><span class="p">))</span>
-<span class="go">cos(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyperexpand</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">hyper</span><span class="p">([</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">],</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">asin(x)</span>
-</pre></div>
-</div>
-<p>We can also sometimes hyperexpand parametric functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">a</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyperexpand</span><span class="p">(</span><span class="n">hyper</span><span class="p">([</span><span class="o">-</span><span class="n">a</span><span class="p">],</span> <span class="p">[],</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">(-x + 1)**a</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.functions.special.hyper.hyper.ap">
-<tt class="descname">ap</tt><a class="reference internal" href="../../_modules/sympy/functions/special/hyper.html#hyper.ap"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.hyper.hyper.ap" title="Permalink to this definition">¶</a></dt>
-<dd><p>Numerator parameters of the hypergeometric function.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.special.hyper.hyper.argument">
-<tt class="descname">argument</tt><a class="reference internal" href="../../_modules/sympy/functions/special/hyper.html#hyper.argument"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.hyper.hyper.argument" title="Permalink to this definition">¶</a></dt>
-<dd><p>Argument of the hypergeometric function.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.special.hyper.hyper.bq">
-<tt class="descname">bq</tt><a class="reference internal" href="../../_modules/sympy/functions/special/hyper.html#hyper.bq"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.hyper.hyper.bq" title="Permalink to this definition">¶</a></dt>
-<dd><p>Denominator parameters of the hypergeometric function.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.special.hyper.hyper.convergence_statement">
-<tt class="descname">convergence_statement</tt><a class="reference internal" href="../../_modules/sympy/functions/special/hyper.html#hyper.convergence_statement"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.hyper.hyper.convergence_statement" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a condition on z under which the series converges.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.special.hyper.hyper.eta">
-<tt class="descname">eta</tt><a class="reference internal" href="../../_modules/sympy/functions/special/hyper.html#hyper.eta"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.hyper.hyper.eta" title="Permalink to this definition">¶</a></dt>
-<dd><p>A quantity related to the convergence of the series.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.special.hyper.hyper.radius_of_convergence">
-<tt class="descname">radius_of_convergence</tt><a class="reference internal" href="../../_modules/sympy/functions/special/hyper.html#hyper.radius_of_convergence"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.hyper.hyper.radius_of_convergence" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the radius of convergence of the defining series.</p>
-<p>Note that even if this is not oo, the function may still be evaluated
-outside of the radius of convergence by analytic continuation. But if
-this is zero, then the function is not actually defined anywhere else.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">hyper</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">radius_of_convergence</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="p">[</span><span class="mi">4</span><span class="p">],</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">radius_of_convergence</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">radius_of_convergence</span>
-<span class="go">oo</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.hyper.meijerg">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.hyper.</tt><tt class="descname">meijerg</tt><a class="reference internal" href="../../_modules/sympy/functions/special/hyper.html#meijerg"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.hyper.meijerg" title="Permalink to this definition">¶</a></dt>
-<dd><p>The Meijer G-function is defined by a Mellin-Barnes type integral that
-resembles an inverse Mellin transform. It generalizes the hypergeometric
-functions.</p>
-<p>The Meijer G-function depends on four sets of parameters. There are
-&#8220;<em>numerator parameters</em>&#8221;
-<span class="math">\(a_1, \dots, a_n\)</span> and <span class="math">\(a_{n+1}, \dots, a_p\)</span>, and there are
-&#8220;<em>denominator parameters</em>&#8221;
-<span class="math">\(b_1, \dots, b_m\)</span> and <span class="math">\(b_{m+1}, \dots, b_q\)</span>.
-Confusingly, it is traditionally denoted as follows (note the position
-of <span class="math">\(m\)</span>, <span class="math">\(n\)</span>, <span class="math">\(p\)</span>, <span class="math">\(q\)</span>, and how they relate to the lengths of the four
-parameter vectors):</p>
-<div class="math">
-\[\begin{split}G_{p,q}^{m,n} \left(\begin{matrix}a_1, \dots, a_n &amp; a_{n+1}, \dots, a_p \\
-                                b_1, \dots, b_m &amp; b_{m+1}, \dots, b_q
-                  \end{matrix} \middle| z \right).\end{split}\]</div>
-<p>However, in sympy the four parameter vectors are always available
-separately (see examples), so that there is no need to keep track of the
-decorating sub- and super-scripts on the G symbol.</p>
-<p>The G function is defined as the following integral:</p>
-<div class="math">
-\[\frac{1}{2 \pi i} \int_L \frac{\prod_{j=1}^m \Gamma(b_j - s)
-\prod_{j=1}^n \Gamma(1 - a_j + s)}{\prod_{j=m+1}^q \Gamma(1- b_j +s)
-\prod_{j=n+1}^p \Gamma(a_j - s)} z^s \mathrm{d}s,\]</div>
-<p>where <span class="math">\(\Gamma(z)\)</span> is the gamma function. There are three possible
-contours which we will not describe in detail here (see the references).
-If the integral converges along more than one of them the definitions
-agree. The contours all separate the poles of <span class="math">\(\Gamma(1-a_j+s)\)</span>
-from the poles of <span class="math">\(\Gamma(b_k-s)\)</span>, so in particular the G function
-is undefined if <span class="math">\(a_j - b_k \in \mathbb{Z}_{&gt;0}\)</span> for some
-<span class="math">\(j \le n\)</span> and <span class="math">\(k \le m\)</span>.</p>
-<p>The conditions under which one of the contours yields a convergent integral
-are complicated and we do not state them here, see the references.</p>
-<p>Note: Currently the Meijer G-function constructor does <em>not</em> check any
-convergence conditions.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">hyper</span></tt>, <a class="reference internal" href="../simplify/simplify.html#module-sympy.simplify.hyperexpand" title="sympy.simplify.hyperexpand"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.simplify.hyperexpand</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li>Luke, Y. L. (1969), The Special Functions and Their Approximations,
-Volume 1</li>
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Meijer_G-function">http://en.wikipedia.org/wiki/Meijer_G-function</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<p>You can pass the parameters either as four separate vectors:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">meijerg</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">a</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core.containers</span> <span class="kn">import</span> <span class="n">Tuple</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">meijerg</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="p">(</span><span class="mi">5</span><span class="p">,),</span> <span class="p">[],</span> <span class="n">x</span><span class="p">),</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go"> __1, 2 /1, 2  a, 4 |  \</span>
-<span class="go">/__     |           | x|</span>
-<span class="go">\_|4, 1 \ 5         |  /</span>
-</pre></div>
-</div>
-<p>or as two nested vectors:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">meijerg</span><span class="p">([(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)],</span> <span class="p">([</span><span class="mi">5</span><span class="p">],</span> <span class="n">Tuple</span><span class="p">()),</span> <span class="n">x</span><span class="p">),</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go"> __1, 2 /1, 2  3, 4 |  \</span>
-<span class="go">/__     |           | x|</span>
-<span class="go">\_|4, 1 \ 5         |  /</span>
-</pre></div>
-</div>
-<p>As with the hypergeometric function, the parameters may be passed as
-arbitrary iterables. Vectors of length zero and one also have to be
-passed as iterables. The parameters need not be constants, but if they
-depend on the argument then not much implemented functionality should be
-expected.</p>
-<p>All the subvectors of parameters are available:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">meijerg</span><span class="p">([</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go"> __1, 1 /1  2 |  \</span>
-<span class="go">/__     |     | x|</span>
-<span class="go">\_|2, 2 \3  4 |  /</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span><span class="o">.</span><span class="n">an</span>
-<span class="go">(1,)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span><span class="o">.</span><span class="n">ap</span>
-<span class="go">(1, 2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span><span class="o">.</span><span class="n">aother</span>
-<span class="go">(2,)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span><span class="o">.</span><span class="n">bm</span>
-<span class="go">(3,)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span><span class="o">.</span><span class="n">bq</span>
-<span class="go">(3, 4)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span><span class="o">.</span><span class="n">bother</span>
-<span class="go">(4,)</span>
-</pre></div>
-</div>
-<p>The Meijer G-function generalizes the hypergeometric functions.
-In some cases it can be expressed in terms of hypergeometric functions,
-using Slater&#8217;s theorem. For example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">hyperexpand</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyperexpand</span><span class="p">(</span><span class="n">meijerg</span><span class="p">([</span><span class="n">a</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="n">c</span><span class="p">],</span> <span class="p">[</span><span class="n">b</span><span class="p">],</span> <span class="n">x</span><span class="p">),</span> <span class="n">allow_hyper</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">x**c*gamma(-a + c + 1)*hyper((-a + c + 1,),</span>
-<span class="go">                             (-b + c + 1,), -x)/gamma(-b + c + 1)</span>
-</pre></div>
-</div>
-<p>Thus the Meijer G-function also subsumes many named functions as special
-cases. You can use expand_func or hyperexpand to (try to) rewrite a
-Meijer G-function in terms of named special functions. For example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">expand_func</span><span class="p">,</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand_func</span><span class="p">(</span><span class="n">meijerg</span><span class="p">([[],[]],</span> <span class="p">[[</span><span class="mi">0</span><span class="p">],[]],</span> <span class="o">-</span><span class="n">x</span><span class="p">))</span>
-<span class="go">exp(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyperexpand</span><span class="p">(</span><span class="n">meijerg</span><span class="p">([[],[]],</span> <span class="p">[[</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">],[</span><span class="mi">0</span><span class="p">]],</span> <span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">sin(x)/sqrt(pi)</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.functions.special.hyper.meijerg.an">
-<tt class="descname">an</tt><a class="reference internal" href="../../_modules/sympy/functions/special/hyper.html#meijerg.an"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.hyper.meijerg.an" title="Permalink to this definition">¶</a></dt>
-<dd><p>First set of numerator parameters.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.special.hyper.meijerg.aother">
-<tt class="descname">aother</tt><a class="reference internal" href="../../_modules/sympy/functions/special/hyper.html#meijerg.aother"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.hyper.meijerg.aother" title="Permalink to this definition">¶</a></dt>
-<dd><p>Second set of numerator parameters.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.special.hyper.meijerg.ap">
-<tt class="descname">ap</tt><a class="reference internal" href="../../_modules/sympy/functions/special/hyper.html#meijerg.ap"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.hyper.meijerg.ap" title="Permalink to this definition">¶</a></dt>
-<dd><p>Combined numerator parameters.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.special.hyper.meijerg.argument">
-<tt class="descname">argument</tt><a class="reference internal" href="../../_modules/sympy/functions/special/hyper.html#meijerg.argument"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.hyper.meijerg.argument" title="Permalink to this definition">¶</a></dt>
-<dd><p>Argument of the Meijer G-function.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.special.hyper.meijerg.bm">
-<tt class="descname">bm</tt><a class="reference internal" href="../../_modules/sympy/functions/special/hyper.html#meijerg.bm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.hyper.meijerg.bm" title="Permalink to this definition">¶</a></dt>
-<dd><p>First set of denominator parameters.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.special.hyper.meijerg.bother">
-<tt class="descname">bother</tt><a class="reference internal" href="../../_modules/sympy/functions/special/hyper.html#meijerg.bother"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.hyper.meijerg.bother" title="Permalink to this definition">¶</a></dt>
-<dd><p>Second set of denominator parameters.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.special.hyper.meijerg.bq">
-<tt class="descname">bq</tt><a class="reference internal" href="../../_modules/sympy/functions/special/hyper.html#meijerg.bq"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.hyper.meijerg.bq" title="Permalink to this definition">¶</a></dt>
-<dd><p>Combined denominator parameters.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.special.hyper.meijerg.delta">
-<tt class="descname">delta</tt><a class="reference internal" href="../../_modules/sympy/functions/special/hyper.html#meijerg.delta"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.hyper.meijerg.delta" title="Permalink to this definition">¶</a></dt>
-<dd><p>A quantity related to the convergence region of the integral,
-c.f. references.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.functions.special.hyper.meijerg.get_period">
-<tt class="descname">get_period</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/special/hyper.html#meijerg.get_period"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.hyper.meijerg.get_period" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a number P such that G(x*exp(I*P)) == G(x).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions.special.hyper</span> <span class="kn">import</span> <span class="n">meijerg</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">S</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">meijerg</span><span class="p">([</span><span class="mi">1</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">get_period</span><span class="p">()</span>
-<span class="go">2*pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">meijerg</span><span class="p">([</span><span class="n">pi</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">get_period</span><span class="p">()</span>
-<span class="go">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">meijerg</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="p">[],</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">get_period</span><span class="p">()</span>
-<span class="go">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">meijerg</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">get_period</span><span class="p">()</span>
-<span class="go">12*pi</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.functions.special.hyper.meijerg.integrand">
-<tt class="descname">integrand</tt><big>(</big><em>self</em>, <em>s</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/special/hyper.html#meijerg.integrand"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.hyper.meijerg.integrand" title="Permalink to this definition">¶</a></dt>
-<dd><p>Get the defining integrand D(s).</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.special.hyper.meijerg.nu">
-<tt class="descname">nu</tt><a class="reference internal" href="../../_modules/sympy/functions/special/hyper.html#meijerg.nu"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.hyper.meijerg.nu" title="Permalink to this definition">¶</a></dt>
-<dd><p>A quantity related to the convergence region of the integral,
-c.f. references.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.functions.special.polynomials">
-<span id="orthogonal-polynomials"></span><h2>Orthogonal Polynomials<a class="headerlink" href="#module-sympy.functions.special.polynomials" title="Permalink to this headline">¶</a></h2>
-<p>This module mainly implements special orthogonal polynomials.</p>
-<p>See also functions.combinatorial.numbers which contains some
-combinatorial polynomials.</p>
-<div class="section" id="jacobi-polynomials">
-<h3>Jacobi Polynomials<a class="headerlink" href="#jacobi-polynomials" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.functions.special.polynomials.jacobi">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.polynomials.</tt><tt class="descname">jacobi</tt><a class="reference internal" href="../../_modules/sympy/functions/special/polynomials.html#jacobi"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.polynomials.jacobi" title="Permalink to this definition">¶</a></dt>
-<dd><p>Jacobi polynomial <span class="math">\(P_n^{\left(\alpha, \beta\right)}(x)\)</span></p>
-<p>jacobi(n, alpha, beta, x) gives the nth Jacobi polynomial
-in x, <span class="math">\(P_n^{\left(\alpha, \beta\right)}(x)\)</span>.</p>
-<p>The Jacobi polynomials are orthogonal on <span class="math">\([-1, 1]\)</span> with respect
-to the weight <span class="math">\(\left(1-x\right)^\alpha \left(1+x\right)^\beta\)</span>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.functions.special.polynomials.gegenbauer" title="sympy.functions.special.polynomials.gegenbauer"><tt class="xref py py-obj docutils literal"><span class="pre">gegenbauer</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevt_root" title="sympy.functions.special.polynomials.chebyshevt_root"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevt_root</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevu" title="sympy.functions.special.polynomials.chebyshevu"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevu</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevu_root" title="sympy.functions.special.polynomials.chebyshevu_root"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevu_root</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.legendre" title="sympy.functions.special.polynomials.legendre"><tt class="xref py py-obj docutils literal"><span class="pre">legendre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.assoc_legendre" title="sympy.functions.special.polynomials.assoc_legendre"><tt class="xref py py-obj docutils literal"><span class="pre">assoc_legendre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.hermite" title="sympy.functions.special.polynomials.hermite"><tt class="xref py py-obj docutils literal"><span class="pre">hermite</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.laguerre" title="sympy.functions.special.polynomials.laguerre"><tt class="xref py py-obj docutils literal"><span class="pre">laguerre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.assoc_laguerre" title="sympy.functions.special.polynomials.assoc_laguerre"><tt class="xref py py-obj docutils literal"><span class="pre">assoc_laguerre</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.jacobi_poly" title="sympy.polys.orthopolys.jacobi_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.jacobi_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.gegenbauer_poly" title="sympy.polys.orthopolys.gegenbauer_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.gegenbauer_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.chebyshevt_poly" title="sympy.polys.orthopolys.chebyshevt_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.chebyshevt_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.chebyshevu_poly" title="sympy.polys.orthopolys.chebyshevu_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.chebyshevu_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.hermite_poly" title="sympy.polys.orthopolys.hermite_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.hermite_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.legendre_poly" title="sympy.polys.orthopolys.legendre_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.legendre_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.laguerre_poly" title="sympy.polys.orthopolys.laguerre_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.laguerre_poly</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r39" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id19">[R39]</a></td><td><a class="reference external" href="http://en.wikipedia.org/wiki/Jacobi_polynomials">http://en.wikipedia.org/wiki/Jacobi_polynomials</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r40" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id20">[R40]</a></td><td><a class="reference external" href="http://mathworld.wolfram.com/JacobiPolynomial.html">http://mathworld.wolfram.com/JacobiPolynomial.html</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r41" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id21">[R41]</a></td><td><a class="reference external" href="http://functions.wolfram.com/Polynomials/JacobiP/">http://functions.wolfram.com/Polynomials/JacobiP/</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">jacobi</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">conjugate</span><span class="p">,</span> <span class="n">diff</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">n</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">jacobi</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jacobi</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">a/2 - b/2 + x*(a/2 + b/2 + 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jacobi</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>   
-<span class="go">(a**2/8 - a*b/4 - a/8 + b**2/8 - b/8 + x**2*(a**2/8 + a*b/4 + 7*a/8 +</span>
-<span class="go">b**2/8 + 7*b/8 + 3/2) + x*(a**2/4 + 3*a/4 - b**2/4 - 3*b/4) - 1/2)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">jacobi</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">jacobi(n, a, b, x)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">jacobi</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">RisingFactorial(a + 1, n)*gegenbauer(n,</span>
-<span class="go">    a + 1/2, x)/RisingFactorial(2*a + 1, n)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">jacobi</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">legendre(n, x)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">jacobi</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">RisingFactorial(3/2, n)*chebyshevu(n, x)/(n + 1)!</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">jacobi</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">RisingFactorial(1/2, n)*chebyshevt(n, x)/n!</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">jacobi</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="o">-</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(-1)**n*jacobi(n, b, a, x)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">jacobi</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">2**(-n)*gamma(a + n + 1)*hyper((-b - n, -n), (a + 1,), -1)/(n!*gamma(a + 1))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jacobi</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">RisingFactorial(a + 1, n)/n!</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">conjugate</span><span class="p">(</span><span class="n">jacobi</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">jacobi(n, conjugate(a), conjugate(b), conjugate(x))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">jacobi</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">(a/2 + b/2 + n/2 + 1/2)*jacobi(n - 1, a + 1, b + 1, x)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="gegenbauer-polynomials">
-<h3>Gegenbauer Polynomials<a class="headerlink" href="#gegenbauer-polynomials" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.functions.special.polynomials.gegenbauer">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.polynomials.</tt><tt class="descname">gegenbauer</tt><a class="reference internal" href="../../_modules/sympy/functions/special/polynomials.html#gegenbauer"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.polynomials.gegenbauer" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gegenbauer polynomial <span class="math">\(C_n^{\left(\alpha\right)}(x)\)</span></p>
-<p>gegenbauer(n, alpha, x) gives the nth Gegenbauer polynomial
-in x, <span class="math">\(C_n^{\left(\alpha\right)}(x)\)</span>.</p>
-<p>The Gegenbauer polynomials are orthogonal on <span class="math">\([-1, 1]\)</span> with
-respect to the weight <span class="math">\(\left(1-x^2\right)^{\alpha-\frac{1}{2}}\)</span>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.functions.special.polynomials.jacobi" title="sympy.functions.special.polynomials.jacobi"><tt class="xref py py-obj docutils literal"><span class="pre">jacobi</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevt_root" title="sympy.functions.special.polynomials.chebyshevt_root"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevt_root</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevu" title="sympy.functions.special.polynomials.chebyshevu"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevu</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevu_root" title="sympy.functions.special.polynomials.chebyshevu_root"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevu_root</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.legendre" title="sympy.functions.special.polynomials.legendre"><tt class="xref py py-obj docutils literal"><span class="pre">legendre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.assoc_legendre" title="sympy.functions.special.polynomials.assoc_legendre"><tt class="xref py py-obj docutils literal"><span class="pre">assoc_legendre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.hermite" title="sympy.functions.special.polynomials.hermite"><tt class="xref py py-obj docutils literal"><span class="pre">hermite</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.laguerre" title="sympy.functions.special.polynomials.laguerre"><tt class="xref py py-obj docutils literal"><span class="pre">laguerre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.assoc_laguerre" title="sympy.functions.special.polynomials.assoc_laguerre"><tt class="xref py py-obj docutils literal"><span class="pre">assoc_laguerre</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.jacobi_poly" title="sympy.polys.orthopolys.jacobi_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.jacobi_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.gegenbauer_poly" title="sympy.polys.orthopolys.gegenbauer_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.gegenbauer_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.chebyshevt_poly" title="sympy.polys.orthopolys.chebyshevt_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.chebyshevt_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.chebyshevu_poly" title="sympy.polys.orthopolys.chebyshevu_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.chebyshevu_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.hermite_poly" title="sympy.polys.orthopolys.hermite_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.hermite_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.legendre_poly" title="sympy.polys.orthopolys.legendre_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.legendre_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.laguerre_poly" title="sympy.polys.orthopolys.laguerre_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.laguerre_poly</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r42" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id22">[R42]</a></td><td><a class="reference external" href="http://en.wikipedia.org/wiki/Gegenbauer_polynomials">http://en.wikipedia.org/wiki/Gegenbauer_polynomials</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r43" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id23">[R43]</a></td><td><a class="reference external" href="http://mathworld.wolfram.com/GegenbauerPolynomial.html">http://mathworld.wolfram.com/GegenbauerPolynomial.html</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r44" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id24">[R44]</a></td><td><a class="reference external" href="http://functions.wolfram.com/Polynomials/GegenbauerC3/">http://functions.wolfram.com/Polynomials/GegenbauerC3/</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">gegenbauer</span><span class="p">,</span> <span class="n">conjugate</span><span class="p">,</span> <span class="n">diff</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">n</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gegenbauer</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gegenbauer</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">2*a*x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gegenbauer</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">-a + x**2*(2*a**2 + 2*a)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gegenbauer</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x**3*(4*a**3/3 + 4*a**2 + 8*a/3) + x*(-2*a**2 - 2*a)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gegenbauer</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">gegenbauer(n, a, x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gegenbauer</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="o">-</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(-1)**n*gegenbauer(n, a, x)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gegenbauer</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">2**n*sqrt(pi)*gamma(a + n/2)/(gamma(a)*gamma(-n/2 + 1/2)*gamma(n + 1))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gegenbauer</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">gamma(2*a + n)/(gamma(2*a)*gamma(n + 1))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">conjugate</span><span class="p">(</span><span class="n">gegenbauer</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">gegenbauer(n, conjugate(a), conjugate(x))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">gegenbauer</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">2*a*gegenbauer(n - 1, a + 1, x)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="chebyshev-polynomials">
-<h3>Chebyshev Polynomials<a class="headerlink" href="#chebyshev-polynomials" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.functions.special.polynomials.chebyshevt">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.polynomials.</tt><tt class="descname">chebyshevt</tt><a class="reference internal" href="../../_modules/sympy/functions/special/polynomials.html#chebyshevt"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.polynomials.chebyshevt" title="Permalink to this definition">¶</a></dt>
-<dd><p>Chebyshev polynomial of the first kind, <span class="math">\(T_n(x)\)</span></p>
-<p>chebyshevt(n, x) gives the nth Chebyshev polynomial (of the first
-kind) in x, <span class="math">\(T_n(x)\)</span>.</p>
-<p>The Chebyshev polynomials of the first kind are orthogonal on
-<span class="math">\([-1, 1]\)</span> with respect to the weight <span class="math">\(\frac{1}{\sqrt{1-x^2}}\)</span>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.functions.special.polynomials.jacobi" title="sympy.functions.special.polynomials.jacobi"><tt class="xref py py-obj docutils literal"><span class="pre">jacobi</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.gegenbauer" title="sympy.functions.special.polynomials.gegenbauer"><tt class="xref py py-obj docutils literal"><span class="pre">gegenbauer</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevt_root" title="sympy.functions.special.polynomials.chebyshevt_root"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevt_root</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevu" title="sympy.functions.special.polynomials.chebyshevu"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevu</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevu_root" title="sympy.functions.special.polynomials.chebyshevu_root"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevu_root</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.legendre" title="sympy.functions.special.polynomials.legendre"><tt class="xref py py-obj docutils literal"><span class="pre">legendre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.assoc_legendre" title="sympy.functions.special.polynomials.assoc_legendre"><tt class="xref py py-obj docutils literal"><span class="pre">assoc_legendre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.hermite" title="sympy.functions.special.polynomials.hermite"><tt class="xref py py-obj docutils literal"><span class="pre">hermite</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.laguerre" title="sympy.functions.special.polynomials.laguerre"><tt class="xref py py-obj docutils literal"><span class="pre">laguerre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.assoc_laguerre" title="sympy.functions.special.polynomials.assoc_laguerre"><tt class="xref py py-obj docutils literal"><span class="pre">assoc_laguerre</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.jacobi_poly" title="sympy.polys.orthopolys.jacobi_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.jacobi_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.gegenbauer_poly" title="sympy.polys.orthopolys.gegenbauer_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.gegenbauer_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.chebyshevt_poly" title="sympy.polys.orthopolys.chebyshevt_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.chebyshevt_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.chebyshevu_poly" title="sympy.polys.orthopolys.chebyshevu_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.chebyshevu_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.hermite_poly" title="sympy.polys.orthopolys.hermite_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.hermite_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.legendre_poly" title="sympy.polys.orthopolys.legendre_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.legendre_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.laguerre_poly" title="sympy.polys.orthopolys.laguerre_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.laguerre_poly</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r45" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id25">[R45]</a></td><td><a class="reference external" href="http://en.wikipedia.org/wiki/Chebyshev_polynomial">http://en.wikipedia.org/wiki/Chebyshev_polynomial</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r46" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id26">[R46]</a></td><td><a class="reference external" href="http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html">http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r47" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id27">[R47]</a></td><td><a class="reference external" href="http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html">http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r48" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id28">[R48]</a></td><td><a class="reference external" href="http://functions.wolfram.com/Polynomials/ChebyshevT/">http://functions.wolfram.com/Polynomials/ChebyshevT/</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r49" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id29">[R49]</a></td><td><a class="reference external" href="http://functions.wolfram.com/Polynomials/ChebyshevU/">http://functions.wolfram.com/Polynomials/ChebyshevU/</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">chebyshevt</span><span class="p">,</span> <span class="n">chebyshevu</span><span class="p">,</span> <span class="n">diff</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">n</span><span class="p">,</span><span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">2*x**2 - 1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">chebyshevt(n, x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(-1)**n*chebyshevt(n, x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">chebyshevt(n, x)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">cos(pi*n/2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(-1)**n</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">chebyshevt</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">n*chebyshevu(n - 1, x)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.polynomials.chebyshevu">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.polynomials.</tt><tt class="descname">chebyshevu</tt><a class="reference internal" href="../../_modules/sympy/functions/special/polynomials.html#chebyshevu"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.polynomials.chebyshevu" title="Permalink to this definition">¶</a></dt>
-<dd><p>Chebyshev polynomial of the second kind, <span class="math">\(U_n(x)\)</span></p>
-<p>chebyshevu(n, x) gives the nth Chebyshev polynomial of the second
-kind in x, <span class="math">\(U_n(x)\)</span>.</p>
-<p>The Chebyshev polynomials of the second kind are orthogonal on
-<span class="math">\([-1, 1]\)</span> with respect to the weight <span class="math">\(\sqrt{1-x^2}\)</span>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.functions.special.polynomials.jacobi" title="sympy.functions.special.polynomials.jacobi"><tt class="xref py py-obj docutils literal"><span class="pre">jacobi</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.gegenbauer" title="sympy.functions.special.polynomials.gegenbauer"><tt class="xref py py-obj docutils literal"><span class="pre">gegenbauer</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevt" title="sympy.functions.special.polynomials.chebyshevt"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevt</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevt_root" title="sympy.functions.special.polynomials.chebyshevt_root"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevt_root</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevu_root" title="sympy.functions.special.polynomials.chebyshevu_root"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevu_root</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.legendre" title="sympy.functions.special.polynomials.legendre"><tt class="xref py py-obj docutils literal"><span class="pre">legendre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.assoc_legendre" title="sympy.functions.special.polynomials.assoc_legendre"><tt class="xref py py-obj docutils literal"><span class="pre">assoc_legendre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.hermite" title="sympy.functions.special.polynomials.hermite"><tt class="xref py py-obj docutils literal"><span class="pre">hermite</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.laguerre" title="sympy.functions.special.polynomials.laguerre"><tt class="xref py py-obj docutils literal"><span class="pre">laguerre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.assoc_laguerre" title="sympy.functions.special.polynomials.assoc_laguerre"><tt class="xref py py-obj docutils literal"><span class="pre">assoc_laguerre</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.jacobi_poly" title="sympy.polys.orthopolys.jacobi_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.jacobi_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.gegenbauer_poly" title="sympy.polys.orthopolys.gegenbauer_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.gegenbauer_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.chebyshevt_poly" title="sympy.polys.orthopolys.chebyshevt_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.chebyshevt_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.chebyshevu_poly" title="sympy.polys.orthopolys.chebyshevu_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.chebyshevu_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.hermite_poly" title="sympy.polys.orthopolys.hermite_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.hermite_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.legendre_poly" title="sympy.polys.orthopolys.legendre_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.legendre_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.laguerre_poly" title="sympy.polys.orthopolys.laguerre_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.laguerre_poly</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r50" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id30">[R50]</a></td><td><a class="reference external" href="http://en.wikipedia.org/wiki/Chebyshev_polynomial">http://en.wikipedia.org/wiki/Chebyshev_polynomial</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r51" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id31">[R51]</a></td><td><a class="reference external" href="http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html">http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r52" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id32">[R52]</a></td><td><a class="reference external" href="http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html">http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r53" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id33">[R53]</a></td><td><a class="reference external" href="http://functions.wolfram.com/Polynomials/ChebyshevT/">http://functions.wolfram.com/Polynomials/ChebyshevT/</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r54" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id34">[R54]</a></td><td><a class="reference external" href="http://functions.wolfram.com/Polynomials/ChebyshevU/">http://functions.wolfram.com/Polynomials/ChebyshevU/</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">chebyshevt</span><span class="p">,</span> <span class="n">chebyshevu</span><span class="p">,</span> <span class="n">diff</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">n</span><span class="p">,</span><span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevu</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevu</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">2*x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevu</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">4*x**2 - 1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevu</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">chebyshevu(n, x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevu</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(-1)**n*chebyshevu(n, x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevu</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">-chebyshevu(n - 2, x)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevu</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">cos(pi*n/2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevu</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">n + 1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">chebyshevu</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">(-x*chebyshevu(n, x) + (n + 1)*chebyshevt(n + 1, x))/(x**2 - 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.polynomials.chebyshevt_root">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.polynomials.</tt><tt class="descname">chebyshevt_root</tt><a class="reference internal" href="../../_modules/sympy/functions/special/polynomials.html#chebyshevt_root"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.polynomials.chebyshevt_root" title="Permalink to this definition">¶</a></dt>
-<dd><p>chebyshev_root(n, k) returns the kth root (indexed from zero) of
-the nth Chebyshev polynomial of the first kind; that is, if
-0 &lt;= k &lt; n, chebyshevt(n, chebyshevt_root(n, k)) == 0.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.functions.special.polynomials.jacobi" title="sympy.functions.special.polynomials.jacobi"><tt class="xref py py-obj docutils literal"><span class="pre">jacobi</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.gegenbauer" title="sympy.functions.special.polynomials.gegenbauer"><tt class="xref py py-obj docutils literal"><span class="pre">gegenbauer</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevt" title="sympy.functions.special.polynomials.chebyshevt"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevt</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevu" title="sympy.functions.special.polynomials.chebyshevu"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevu</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevu_root" title="sympy.functions.special.polynomials.chebyshevu_root"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevu_root</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.legendre" title="sympy.functions.special.polynomials.legendre"><tt class="xref py py-obj docutils literal"><span class="pre">legendre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.assoc_legendre" title="sympy.functions.special.polynomials.assoc_legendre"><tt class="xref py py-obj docutils literal"><span class="pre">assoc_legendre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.hermite" title="sympy.functions.special.polynomials.hermite"><tt class="xref py py-obj docutils literal"><span class="pre">hermite</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.laguerre" title="sympy.functions.special.polynomials.laguerre"><tt class="xref py py-obj docutils literal"><span class="pre">laguerre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.assoc_laguerre" title="sympy.functions.special.polynomials.assoc_laguerre"><tt class="xref py py-obj docutils literal"><span class="pre">assoc_laguerre</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.jacobi_poly" title="sympy.polys.orthopolys.jacobi_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.jacobi_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.gegenbauer_poly" title="sympy.polys.orthopolys.gegenbauer_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.gegenbauer_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.chebyshevt_poly" title="sympy.polys.orthopolys.chebyshevt_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.chebyshevt_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.chebyshevu_poly" title="sympy.polys.orthopolys.chebyshevu_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.chebyshevu_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.hermite_poly" title="sympy.polys.orthopolys.hermite_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.hermite_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.legendre_poly" title="sympy.polys.orthopolys.legendre_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.legendre_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.laguerre_poly" title="sympy.polys.orthopolys.laguerre_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.laguerre_poly</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">chebyshevt</span><span class="p">,</span> <span class="n">chebyshevt_root</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt_root</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">-sqrt(3)/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">chebyshevt_root</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.polynomials.chebyshevu_root">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.polynomials.</tt><tt class="descname">chebyshevu_root</tt><a class="reference internal" href="../../_modules/sympy/functions/special/polynomials.html#chebyshevu_root"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.polynomials.chebyshevu_root" title="Permalink to this definition">¶</a></dt>
-<dd><p>chebyshevu_root(n, k) returns the kth root (indexed from zero) of the
-nth Chebyshev polynomial of the second kind; that is, if 0 &lt;= k &lt; n,
-chebyshevu(n, chebyshevu_root(n, k)) == 0.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevt" title="sympy.functions.special.polynomials.chebyshevt"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevt</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevt_root" title="sympy.functions.special.polynomials.chebyshevt_root"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevt_root</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevu" title="sympy.functions.special.polynomials.chebyshevu"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevu</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.legendre" title="sympy.functions.special.polynomials.legendre"><tt class="xref py py-obj docutils literal"><span class="pre">legendre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.assoc_legendre" title="sympy.functions.special.polynomials.assoc_legendre"><tt class="xref py py-obj docutils literal"><span class="pre">assoc_legendre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.hermite" title="sympy.functions.special.polynomials.hermite"><tt class="xref py py-obj docutils literal"><span class="pre">hermite</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.laguerre" title="sympy.functions.special.polynomials.laguerre"><tt class="xref py py-obj docutils literal"><span class="pre">laguerre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.assoc_laguerre" title="sympy.functions.special.polynomials.assoc_laguerre"><tt class="xref py py-obj docutils literal"><span class="pre">assoc_laguerre</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.jacobi_poly" title="sympy.polys.orthopolys.jacobi_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.jacobi_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.gegenbauer_poly" title="sympy.polys.orthopolys.gegenbauer_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.gegenbauer_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.chebyshevt_poly" title="sympy.polys.orthopolys.chebyshevt_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.chebyshevt_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.chebyshevu_poly" title="sympy.polys.orthopolys.chebyshevu_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.chebyshevu_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.hermite_poly" title="sympy.polys.orthopolys.hermite_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.hermite_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.legendre_poly" title="sympy.polys.orthopolys.legendre_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.legendre_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.laguerre_poly" title="sympy.polys.orthopolys.laguerre_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.laguerre_poly</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">chebyshevu</span><span class="p">,</span> <span class="n">chebyshevu_root</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevu_root</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">-sqrt(2)/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevu</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">chebyshevu_root</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="legendre-polynomials">
-<h3>Legendre Polynomials<a class="headerlink" href="#legendre-polynomials" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.functions.special.polynomials.legendre">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.polynomials.</tt><tt class="descname">legendre</tt><a class="reference internal" href="../../_modules/sympy/functions/special/polynomials.html#legendre"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.polynomials.legendre" title="Permalink to this definition">¶</a></dt>
-<dd><p>legendre(n, x) gives the nth Legendre polynomial of x, <span class="math">\(P_n(x)\)</span></p>
-<p>The Legendre polynomials are orthogonal on [-1, 1] with respect to
-the constant weight 1. They satisfy <span class="math">\(P_n(1) = 1\)</span> for all n; further,
-<span class="math">\(P_n\)</span> is odd for odd n and even for even n.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.functions.special.polynomials.jacobi" title="sympy.functions.special.polynomials.jacobi"><tt class="xref py py-obj docutils literal"><span class="pre">jacobi</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.gegenbauer" title="sympy.functions.special.polynomials.gegenbauer"><tt class="xref py py-obj docutils literal"><span class="pre">gegenbauer</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevt" title="sympy.functions.special.polynomials.chebyshevt"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevt</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevt_root" title="sympy.functions.special.polynomials.chebyshevt_root"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevt_root</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevu" title="sympy.functions.special.polynomials.chebyshevu"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevu</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevu_root" title="sympy.functions.special.polynomials.chebyshevu_root"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevu_root</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.assoc_legendre" title="sympy.functions.special.polynomials.assoc_legendre"><tt class="xref py py-obj docutils literal"><span class="pre">assoc_legendre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.hermite" title="sympy.functions.special.polynomials.hermite"><tt class="xref py py-obj docutils literal"><span class="pre">hermite</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.laguerre" title="sympy.functions.special.polynomials.laguerre"><tt class="xref py py-obj docutils literal"><span class="pre">laguerre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.assoc_laguerre" title="sympy.functions.special.polynomials.assoc_laguerre"><tt class="xref py py-obj docutils literal"><span class="pre">assoc_laguerre</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.jacobi_poly" title="sympy.polys.orthopolys.jacobi_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.jacobi_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.gegenbauer_poly" title="sympy.polys.orthopolys.gegenbauer_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.gegenbauer_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.chebyshevt_poly" title="sympy.polys.orthopolys.chebyshevt_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.chebyshevt_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.chebyshevu_poly" title="sympy.polys.orthopolys.chebyshevu_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.chebyshevu_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.hermite_poly" title="sympy.polys.orthopolys.hermite_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.hermite_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.legendre_poly" title="sympy.polys.orthopolys.legendre_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.legendre_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.laguerre_poly" title="sympy.polys.orthopolys.laguerre_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.laguerre_poly</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r55" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id35">[R55]</a></td><td><a class="reference external" href="http://en.wikipedia.org/wiki/Legendre_polynomial">http://en.wikipedia.org/wiki/Legendre_polynomial</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r56" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id36">[R56]</a></td><td><a class="reference external" href="http://mathworld.wolfram.com/LegendrePolynomial.html">http://mathworld.wolfram.com/LegendrePolynomial.html</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r57" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id37">[R57]</a></td><td><a class="reference external" href="http://functions.wolfram.com/Polynomials/LegendreP/">http://functions.wolfram.com/Polynomials/LegendreP/</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r58" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id38">[R58]</a></td><td><a class="reference external" href="http://functions.wolfram.com/Polynomials/LegendreP2/">http://functions.wolfram.com/Polynomials/LegendreP2/</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">legendre</span><span class="p">,</span> <span class="n">diff</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">3*x**2/2 - 1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">legendre(n, x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">n*(x*legendre(n, x) - legendre(n - 1, x))/(x**2 - 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.polynomials.assoc_legendre">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.polynomials.</tt><tt class="descname">assoc_legendre</tt><a class="reference internal" href="../../_modules/sympy/functions/special/polynomials.html#assoc_legendre"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.polynomials.assoc_legendre" title="Permalink to this definition">¶</a></dt>
-<dd><p>assoc_legendre(n,m, x) gives <span class="math">\(P_n^m(x)\)</span>, where n and m are
-the degree and order or an expression which is related to the nth
-order Legendre polynomial, <span class="math">\(P_n(x)\)</span> in the following manner:</p>
-<div class="math">
-\[P_n^m(x) = (-1)^m (1 - x^2)^{\frac{m}{2}}
-           \frac{\mathrm{d}^m P_n(x)}{\mathrm{d} x^m}\]</div>
-<p>Associated Legendre polynomial are orthogonal on [-1, 1] with:</p>
-<ul class="simple">
-<li>weight = 1            for the same m, and different n.</li>
-<li>weight = 1/(1-x**2)   for the same n, and different m.</li>
-</ul>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.functions.special.polynomials.jacobi" title="sympy.functions.special.polynomials.jacobi"><tt class="xref py py-obj docutils literal"><span class="pre">jacobi</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.gegenbauer" title="sympy.functions.special.polynomials.gegenbauer"><tt class="xref py py-obj docutils literal"><span class="pre">gegenbauer</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevt" title="sympy.functions.special.polynomials.chebyshevt"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevt</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevt_root" title="sympy.functions.special.polynomials.chebyshevt_root"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevt_root</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevu" title="sympy.functions.special.polynomials.chebyshevu"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevu</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevu_root" title="sympy.functions.special.polynomials.chebyshevu_root"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevu_root</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.legendre" title="sympy.functions.special.polynomials.legendre"><tt class="xref py py-obj docutils literal"><span class="pre">legendre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.hermite" title="sympy.functions.special.polynomials.hermite"><tt class="xref py py-obj docutils literal"><span class="pre">hermite</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.laguerre" title="sympy.functions.special.polynomials.laguerre"><tt class="xref py py-obj docutils literal"><span class="pre">laguerre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.assoc_laguerre" title="sympy.functions.special.polynomials.assoc_laguerre"><tt class="xref py py-obj docutils literal"><span class="pre">assoc_laguerre</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.jacobi_poly" title="sympy.polys.orthopolys.jacobi_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.jacobi_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.gegenbauer_poly" title="sympy.polys.orthopolys.gegenbauer_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.gegenbauer_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.chebyshevt_poly" title="sympy.polys.orthopolys.chebyshevt_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.chebyshevt_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.chebyshevu_poly" title="sympy.polys.orthopolys.chebyshevu_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.chebyshevu_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.hermite_poly" title="sympy.polys.orthopolys.hermite_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.hermite_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.legendre_poly" title="sympy.polys.orthopolys.legendre_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.legendre_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.laguerre_poly" title="sympy.polys.orthopolys.laguerre_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.laguerre_poly</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r59" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id39">[R59]</a></td><td><a class="reference external" href="http://en.wikipedia.org/wiki/Associated_Legendre_polynomials">http://en.wikipedia.org/wiki/Associated_Legendre_polynomials</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r60" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id40">[R60]</a></td><td><a class="reference external" href="http://mathworld.wolfram.com/LegendrePolynomial.html">http://mathworld.wolfram.com/LegendrePolynomial.html</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r61" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id41">[R61]</a></td><td><a class="reference external" href="http://functions.wolfram.com/Polynomials/LegendreP/">http://functions.wolfram.com/Polynomials/LegendreP/</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r62" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id42">[R62]</a></td><td><a class="reference external" href="http://functions.wolfram.com/Polynomials/LegendreP2/">http://functions.wolfram.com/Polynomials/LegendreP2/</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">assoc_legendre</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_legendre</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_legendre</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_legendre</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">-sqrt(-x**2 + 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">m</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="go">assoc_legendre(n, m, x)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="hermite-polynomials">
-<h3>Hermite Polynomials<a class="headerlink" href="#hermite-polynomials" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.functions.special.polynomials.hermite">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.polynomials.</tt><tt class="descname">hermite</tt><a class="reference internal" href="../../_modules/sympy/functions/special/polynomials.html#hermite"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.polynomials.hermite" title="Permalink to this definition">¶</a></dt>
-<dd><p>hermite(n, x) gives the nth Hermite polynomial in x, <span class="math">\(H_n(x)\)</span></p>
-<p>The Hermite polynomials are orthogonal on <span class="math">\((-\infty, \infty)\)</span>
-with respect to the weight <span class="math">\(\exp\left(-\frac{x^2}{2}\right)\)</span>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.functions.special.polynomials.jacobi" title="sympy.functions.special.polynomials.jacobi"><tt class="xref py py-obj docutils literal"><span class="pre">jacobi</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.gegenbauer" title="sympy.functions.special.polynomials.gegenbauer"><tt class="xref py py-obj docutils literal"><span class="pre">gegenbauer</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevt" title="sympy.functions.special.polynomials.chebyshevt"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevt</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevt_root" title="sympy.functions.special.polynomials.chebyshevt_root"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevt_root</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevu" title="sympy.functions.special.polynomials.chebyshevu"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevu</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevu_root" title="sympy.functions.special.polynomials.chebyshevu_root"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevu_root</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.legendre" title="sympy.functions.special.polynomials.legendre"><tt class="xref py py-obj docutils literal"><span class="pre">legendre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.assoc_legendre" title="sympy.functions.special.polynomials.assoc_legendre"><tt class="xref py py-obj docutils literal"><span class="pre">assoc_legendre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.laguerre" title="sympy.functions.special.polynomials.laguerre"><tt class="xref py py-obj docutils literal"><span class="pre">laguerre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.assoc_laguerre" title="sympy.functions.special.polynomials.assoc_laguerre"><tt class="xref py py-obj docutils literal"><span class="pre">assoc_laguerre</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.jacobi_poly" title="sympy.polys.orthopolys.jacobi_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.jacobi_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.gegenbauer_poly" title="sympy.polys.orthopolys.gegenbauer_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.gegenbauer_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.chebyshevt_poly" title="sympy.polys.orthopolys.chebyshevt_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.chebyshevt_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.chebyshevu_poly" title="sympy.polys.orthopolys.chebyshevu_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.chebyshevu_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.hermite_poly" title="sympy.polys.orthopolys.hermite_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.hermite_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.legendre_poly" title="sympy.polys.orthopolys.legendre_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.legendre_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.laguerre_poly" title="sympy.polys.orthopolys.laguerre_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.laguerre_poly</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r63" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id43">[R63]</a></td><td><a class="reference external" href="http://en.wikipedia.org/wiki/Hermite_polynomial">http://en.wikipedia.org/wiki/Hermite_polynomial</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r64" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id44">[R64]</a></td><td><a class="reference external" href="http://mathworld.wolfram.com/HermitePolynomial.html">http://mathworld.wolfram.com/HermitePolynomial.html</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r65" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id45">[R65]</a></td><td><a class="reference external" href="http://functions.wolfram.com/Polynomials/HermiteH/">http://functions.wolfram.com/Polynomials/HermiteH/</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">hermite</span><span class="p">,</span> <span class="n">diff</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hermite</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hermite</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">2*x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hermite</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">4*x**2 - 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hermite</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">hermite(n, x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">hermite</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">2*n*hermite(n - 1, x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">hermite</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">2*n*hermite(n - 1, x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hermite</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(-1)**n*hermite(n, x)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="laguerre-polynomials">
-<h3>Laguerre Polynomials<a class="headerlink" href="#laguerre-polynomials" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.functions.special.polynomials.laguerre">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.polynomials.</tt><tt class="descname">laguerre</tt><a class="reference internal" href="../../_modules/sympy/functions/special/polynomials.html#laguerre"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.polynomials.laguerre" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the nth Laguerre polynomial in x, <span class="math">\(L_n(x)\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>n</strong> : int</p>
-<blockquote class="last">
-<div><p>Degree of Laguerre polynomial. Must be <tt class="docutils literal"><span class="pre">n</span> <span class="pre">&gt;=</span> <span class="pre">0</span></tt>.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.functions.special.polynomials.jacobi" title="sympy.functions.special.polynomials.jacobi"><tt class="xref py py-obj docutils literal"><span class="pre">jacobi</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.gegenbauer" title="sympy.functions.special.polynomials.gegenbauer"><tt class="xref py py-obj docutils literal"><span class="pre">gegenbauer</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevt" title="sympy.functions.special.polynomials.chebyshevt"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevt</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevt_root" title="sympy.functions.special.polynomials.chebyshevt_root"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevt_root</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevu" title="sympy.functions.special.polynomials.chebyshevu"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevu</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevu_root" title="sympy.functions.special.polynomials.chebyshevu_root"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevu_root</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.legendre" title="sympy.functions.special.polynomials.legendre"><tt class="xref py py-obj docutils literal"><span class="pre">legendre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.assoc_legendre" title="sympy.functions.special.polynomials.assoc_legendre"><tt class="xref py py-obj docutils literal"><span class="pre">assoc_legendre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.hermite" title="sympy.functions.special.polynomials.hermite"><tt class="xref py py-obj docutils literal"><span class="pre">hermite</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.assoc_laguerre" title="sympy.functions.special.polynomials.assoc_laguerre"><tt class="xref py py-obj docutils literal"><span class="pre">assoc_laguerre</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.jacobi_poly" title="sympy.polys.orthopolys.jacobi_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.jacobi_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.gegenbauer_poly" title="sympy.polys.orthopolys.gegenbauer_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.gegenbauer_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.chebyshevt_poly" title="sympy.polys.orthopolys.chebyshevt_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.chebyshevt_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.chebyshevu_poly" title="sympy.polys.orthopolys.chebyshevu_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.chebyshevu_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.hermite_poly" title="sympy.polys.orthopolys.hermite_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.hermite_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.legendre_poly" title="sympy.polys.orthopolys.legendre_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.legendre_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.laguerre_poly" title="sympy.polys.orthopolys.laguerre_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.laguerre_poly</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r66" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id46">[R66]</a></td><td><a class="reference external" href="http://en.wikipedia.org/wiki/Laguerre_polynomial">http://en.wikipedia.org/wiki/Laguerre_polynomial</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r67" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id47">[R67]</a></td><td><a class="reference external" href="http://mathworld.wolfram.com/LaguerrePolynomial.html">http://mathworld.wolfram.com/LaguerrePolynomial.html</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r68" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id48">[R68]</a></td><td><a class="reference external" href="http://functions.wolfram.com/Polynomials/LaguerreL/">http://functions.wolfram.com/Polynomials/LaguerreL/</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r69" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id49">[R69]</a></td><td><a class="reference external" href="http://functions.wolfram.com/Polynomials/LaguerreL3/">http://functions.wolfram.com/Polynomials/LaguerreL3/</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">laguerre</span><span class="p">,</span> <span class="n">diff</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">laguerre</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">laguerre</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">-x + 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">laguerre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x**2/2 - 2*x + 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">laguerre</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">-x**3/6 + 3*x**2/2 - 3*x + 1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">laguerre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">laguerre(n, x)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">laguerre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">-assoc_laguerre(n - 1, 1, x)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.polynomials.assoc_laguerre">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.polynomials.</tt><tt class="descname">assoc_laguerre</tt><a class="reference internal" href="../../_modules/sympy/functions/special/polynomials.html#assoc_laguerre"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.polynomials.assoc_laguerre" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the nth generalized Laguerre polynomial in x, <span class="math">\(L_n(x)\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>n</strong> : int</p>
-<blockquote>
-<div><p>Degree of Laguerre polynomial. Must be <tt class="docutils literal"><span class="pre">n</span> <span class="pre">&gt;=</span> <span class="pre">0</span></tt>.</p>
-</div></blockquote>
-<p><strong>alpha</strong> : Expr</p>
-<blockquote class="last">
-<div><p>Arbitrary expression. For <tt class="docutils literal"><span class="pre">alpha=0</span></tt> regular Laguerre
-polynomials will be generated.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.functions.special.polynomials.jacobi" title="sympy.functions.special.polynomials.jacobi"><tt class="xref py py-obj docutils literal"><span class="pre">jacobi</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.gegenbauer" title="sympy.functions.special.polynomials.gegenbauer"><tt class="xref py py-obj docutils literal"><span class="pre">gegenbauer</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevt" title="sympy.functions.special.polynomials.chebyshevt"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevt</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevt_root" title="sympy.functions.special.polynomials.chebyshevt_root"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevt_root</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevu" title="sympy.functions.special.polynomials.chebyshevu"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevu</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.chebyshevu_root" title="sympy.functions.special.polynomials.chebyshevu_root"><tt class="xref py py-obj docutils literal"><span class="pre">chebyshevu_root</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.legendre" title="sympy.functions.special.polynomials.legendre"><tt class="xref py py-obj docutils literal"><span class="pre">legendre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.assoc_legendre" title="sympy.functions.special.polynomials.assoc_legendre"><tt class="xref py py-obj docutils literal"><span class="pre">assoc_legendre</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.hermite" title="sympy.functions.special.polynomials.hermite"><tt class="xref py py-obj docutils literal"><span class="pre">hermite</span></tt></a>, <a class="reference internal" href="#sympy.functions.special.polynomials.laguerre" title="sympy.functions.special.polynomials.laguerre"><tt class="xref py py-obj docutils literal"><span class="pre">laguerre</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.jacobi_poly" title="sympy.polys.orthopolys.jacobi_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.jacobi_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.gegenbauer_poly" title="sympy.polys.orthopolys.gegenbauer_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.gegenbauer_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.chebyshevt_poly" title="sympy.polys.orthopolys.chebyshevt_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.chebyshevt_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.chebyshevu_poly" title="sympy.polys.orthopolys.chebyshevu_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.chebyshevu_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.hermite_poly" title="sympy.polys.orthopolys.hermite_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.hermite_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.legendre_poly" title="sympy.polys.orthopolys.legendre_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.legendre_poly</span></tt></a>, <a class="reference internal" href="../polys/reference.html#sympy.polys.orthopolys.laguerre_poly" title="sympy.polys.orthopolys.laguerre_poly"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.orthopolys.laguerre_poly</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r70" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id50">[R70]</a></td><td><a class="reference external" href="http://en.wikipedia.org/wiki/Laguerre_polynomial#Assoc_laguerre_polynomials">http://en.wikipedia.org/wiki/Laguerre_polynomial#Assoc_laguerre_polynomials</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r71" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id51">[R71]</a></td><td><a class="reference external" href="http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html">http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r72" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id52">[R72]</a></td><td><a class="reference external" href="http://functions.wolfram.com/Polynomials/LaguerreL/">http://functions.wolfram.com/Polynomials/LaguerreL/</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r73" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id53">[R73]</a></td><td><a class="reference external" href="http://functions.wolfram.com/Polynomials/LaguerreL3/">http://functions.wolfram.com/Polynomials/LaguerreL3/</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">laguerre</span><span class="p">,</span> <span class="n">assoc_laguerre</span><span class="p">,</span> <span class="n">diff</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">a</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_laguerre</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_laguerre</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">a - x + 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_laguerre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">a**2/2 + 3*a/2 + x**2/2 + x*(-a - 2) + 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_laguerre</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">a**3/6 + a**2 + 11*a/6 - x**3/6 + x**2*(a/2 + 3/2) +</span>
-<span class="go">    x*(-a**2/2 - 5*a/2 - 3) + 1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_laguerre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">binomial(a + n, a)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_laguerre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">assoc_laguerre(n, a, x)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_laguerre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">laguerre(n, x)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">assoc_laguerre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">-assoc_laguerre(n - 1, a + 1, x)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">assoc_laguerre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">a</span><span class="p">)</span>
-<span class="go">Sum(assoc_laguerre(_k, a, x)/(-a + n), (_k, 0, n - 1))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="spherical-harmonics">
-<h2>Spherical Harmonics<a class="headerlink" href="#spherical-harmonics" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.functions.special.spherical_harmonics.Plmcos">
-<tt class="descclassname">sympy.functions.special.spherical_harmonics.</tt><tt class="descname">Plmcos</tt><big>(</big><em>l</em>, <em>m</em>, <em>th</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/special/spherical_harmonics.html#Plmcos"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.spherical_harmonics.Plmcos" title="Permalink to this definition">¶</a></dt>
-<dd><p>Plm(cos(th)).</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.functions.special.spherical_harmonics.Ylm">
-<tt class="descclassname">sympy.functions.special.spherical_harmonics.</tt><tt class="descname">Ylm</tt><big>(</big><em>l</em>, <em>m</em>, <em>theta</em>, <em>phi</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/special/spherical_harmonics.html#Ylm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.spherical_harmonics.Ylm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Spherical harmonics Ylm.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">Ylm</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">theta</span><span class="p">,</span> <span class="n">phi</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&quot;theta phi&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">1/(2*sqrt(pi))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">sqrt(6)*exp(-I*phi)*sin(theta)/(4*sqrt(pi))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">sqrt(3)*cos(theta)/(2*sqrt(pi))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.functions.special.spherical_harmonics.Ylm_c">
-<tt class="descclassname">sympy.functions.special.spherical_harmonics.</tt><tt class="descname">Ylm_c</tt><big>(</big><em>l</em>, <em>m</em>, <em>theta</em>, <em>phi</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/special/spherical_harmonics.html#Ylm_c"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.spherical_harmonics.Ylm_c" title="Permalink to this definition">¶</a></dt>
-<dd><p>Conjugate spherical harmonics.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.functions.special.spherical_harmonics.Zlm">
-<tt class="descclassname">sympy.functions.special.spherical_harmonics.</tt><tt class="descname">Zlm</tt><big>(</big><em>l</em>, <em>m</em>, <em>th</em>, <em>ph</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/special/spherical_harmonics.html#Zlm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.spherical_harmonics.Zlm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Real spherical harmonics.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="tensor-functions">
-<h2>Tensor Functions<a class="headerlink" href="#tensor-functions" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.functions.special.tensor_functions.Eijk">
-<tt class="descclassname">sympy.functions.special.tensor_functions.</tt><tt class="descname">Eijk</tt><big>(</big><em>*args</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/special/tensor_functions.html#Eijk"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.tensor_functions.Eijk" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represent the Levi-Civita symbol.</p>
-<p>This is just compatibility wrapper to LeviCivita().</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">LeviCivita</span></tt></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.functions.special.tensor_functions.eval_levicivita">
-<tt class="descclassname">sympy.functions.special.tensor_functions.</tt><tt class="descname">eval_levicivita</tt><big>(</big><em>*args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/special/tensor_functions.html#eval_levicivita"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.tensor_functions.eval_levicivita" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluate Levi-Civita symbol.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.tensor_functions.LeviCivita">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.tensor_functions.</tt><tt class="descname">LeviCivita</tt><a class="reference internal" href="../../_modules/sympy/functions/special/tensor_functions.html#LeviCivita"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.tensor_functions.LeviCivita" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represent the Levi-Civita symbol.</p>
-<p>For even permutations of indices it returns 1, for odd permutations -1, and
-for everything else (a repeated index) it returns 0.</p>
-<p>Thus it represents an alternating pseudotensor.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">Eijk</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">LeviCivita</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">k</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">LeviCivita</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">LeviCivita</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">-1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">LeviCivita</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">LeviCivita</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-<span class="go">LeviCivita(i, j, k)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">LeviCivita</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.functions.special.tensor_functions.KroneckerDelta">
-<em class="property">class </em><tt class="descclassname">sympy.functions.special.tensor_functions.</tt><tt class="descname">KroneckerDelta</tt><a class="reference internal" href="../../_modules/sympy/functions/special/tensor_functions.html#KroneckerDelta"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.tensor_functions.KroneckerDelta" title="Permalink to this definition">¶</a></dt>
-<dd><p>The discrete, or Kronecker, delta function.</p>
-<p>A function that takes in two integers i and j. It returns 0 if i and j are
-not equal or it returns 1 if i and j are equal.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>i</strong> : Number, Symbol</p>
-<blockquote>
-<div><p>The first index of the delta function.</p>
-</div></blockquote>
-<p><strong>j</strong> : Number, Symbol</p>
-<blockquote class="last">
-<div><p>The second index of the delta function.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">eval</span></tt>, <a class="reference internal" href="#sympy.functions.special.delta_functions.DiracDelta" title="sympy.functions.special.delta_functions.DiracDelta"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.functions.special.delta_functions.DiracDelta</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<p><a class="reference external" href="http://en.wikipedia.org/wiki/Kronecker_delta">http://en.wikipedia.org/wiki/Kronecker_delta</a></p>
-<p class="rubric">Examples</p>
-<p>A simple example with integer indices:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions.special.tensor_functions</span> <span class="kn">import</span> <span class="n">KroneckerDelta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>Symbolic indices:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">k</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
-<span class="go">KroneckerDelta(i, j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span>
-<span class="go">KroneckerDelta(i, i + k + 1)</span>
-</pre></div>
-</div>
-<dl class="classmethod">
-<dt id="sympy.functions.special.tensor_functions.KroneckerDelta.eval">
-<em class="property">classmethod </em><tt class="descname">eval</tt><big>(</big><em>i</em>, <em>j</em><big>)</big><a class="reference internal" href="../../_modules/sympy/functions/special/tensor_functions.html#KroneckerDelta.eval"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.tensor_functions.KroneckerDelta.eval" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the discrete delta function.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions.special.tensor_functions</span> <span class="kn">import</span> <span class="n">KroneckerDelta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">k</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
-<span class="go">KroneckerDelta(i, j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span>
-<span class="go">KroneckerDelta(i, i + k + 1)</span>
-</pre></div>
-</div>
-<p># indirect doctest</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.special.tensor_functions.KroneckerDelta.indices_contain_equal_information">
-<tt class="descname">indices_contain_equal_information</tt><a class="reference internal" href="../../_modules/sympy/functions/special/tensor_functions.html#KroneckerDelta.indices_contain_equal_information"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.tensor_functions.KroneckerDelta.indices_contain_equal_information" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if indices are either both above or below fermi.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions.special.tensor_functions</span> <span class="kn">import</span> <span class="n">KroneckerDelta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span><span class="o">.</span><span class="n">indices_contain_equal_information</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">indices_contain_equal_information</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">indices_contain_equal_information</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.special.tensor_functions.KroneckerDelta.is_above_fermi">
-<tt class="descname">is_above_fermi</tt><a class="reference internal" href="../../_modules/sympy/functions/special/tensor_functions.html#KroneckerDelta.is_above_fermi"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.tensor_functions.KroneckerDelta.is_above_fermi" title="Permalink to this definition">¶</a></dt>
-<dd><p>True if Delta can be non-zero above fermi</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">is_below_fermi</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">is_only_below_fermi</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">is_only_above_fermi</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions.special.tensor_functions</span> <span class="kn">import</span> <span class="n">KroneckerDelta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">is_above_fermi</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">is_above_fermi</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span><span class="o">.</span><span class="n">is_above_fermi</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.special.tensor_functions.KroneckerDelta.is_below_fermi">
-<tt class="descname">is_below_fermi</tt><a class="reference internal" href="../../_modules/sympy/functions/special/tensor_functions.html#KroneckerDelta.is_below_fermi"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.tensor_functions.KroneckerDelta.is_below_fermi" title="Permalink to this definition">¶</a></dt>
-<dd><p>True if Delta can be non-zero below fermi</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">is_above_fermi</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">is_only_above_fermi</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">is_only_below_fermi</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions.special.tensor_functions</span> <span class="kn">import</span> <span class="n">KroneckerDelta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">is_below_fermi</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">is_below_fermi</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span><span class="o">.</span><span class="n">is_below_fermi</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.special.tensor_functions.KroneckerDelta.is_only_above_fermi">
-<tt class="descname">is_only_above_fermi</tt><a class="reference internal" href="../../_modules/sympy/functions/special/tensor_functions.html#KroneckerDelta.is_only_above_fermi"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.tensor_functions.KroneckerDelta.is_only_above_fermi" title="Permalink to this definition">¶</a></dt>
-<dd><p>True if Delta is restricted to above fermi</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">is_above_fermi</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">is_below_fermi</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">is_only_below_fermi</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions.special.tensor_functions</span> <span class="kn">import</span> <span class="n">KroneckerDelta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_above_fermi</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_above_fermi</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_above_fermi</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.special.tensor_functions.KroneckerDelta.is_only_below_fermi">
-<tt class="descname">is_only_below_fermi</tt><a class="reference internal" href="../../_modules/sympy/functions/special/tensor_functions.html#KroneckerDelta.is_only_below_fermi"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.tensor_functions.KroneckerDelta.is_only_below_fermi" title="Permalink to this definition">¶</a></dt>
-<dd><p>True if Delta is restricted to below fermi</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">is_above_fermi</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">is_below_fermi</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">is_only_above_fermi</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions.special.tensor_functions</span> <span class="kn">import</span> <span class="n">KroneckerDelta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_below_fermi</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_below_fermi</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_below_fermi</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.special.tensor_functions.KroneckerDelta.killable_index">
-<tt class="descname">killable_index</tt><a class="reference internal" href="../../_modules/sympy/functions/special/tensor_functions.html#KroneckerDelta.killable_index"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.tensor_functions.KroneckerDelta.killable_index" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the index which is preferred to substitute in the final
-expression.</p>
-<p>The index to substitute is the index with less information regarding
-fermi level.  If indices contain same information, &#8216;a&#8217; is preferred
-before &#8216;b&#8217;.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">preferred_index</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions.special.tensor_functions</span> <span class="kn">import</span> <span class="n">KroneckerDelta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">j</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;j&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">killable_index</span>
-<span class="go">p</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">killable_index</span>
-<span class="go">p</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span><span class="o">.</span><span class="n">killable_index</span>
-<span class="go">j</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.functions.special.tensor_functions.KroneckerDelta.preferred_index">
-<tt class="descname">preferred_index</tt><a class="reference internal" href="../../_modules/sympy/functions/special/tensor_functions.html#KroneckerDelta.preferred_index"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.functions.special.tensor_functions.KroneckerDelta.preferred_index" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the index which is preferred to keep in the final expression.</p>
-<p>The preferred index is the index with more information regarding fermi
-level.  If indices contain same information, &#8216;a&#8217; is preferred before
-&#8216;b&#8217;.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">killable_index</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions.special.tensor_functions</span> <span class="kn">import</span> <span class="n">KroneckerDelta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">j</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;j&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">preferred_index</span>
-<span class="go">i</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">preferred_index</span>
-<span class="go">a</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span><span class="o">.</span><span class="n">preferred_index</span>
-<span class="go">i</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-</div>
-
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-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Special</a><ul>
-<li><a class="reference internal" href="#diracdelta">DiracDelta</a></li>
-<li><a class="reference internal" href="#heaviside">Heaviside</a></li>
-<li><a class="reference internal" href="#beta">beta</a></li>
-<li><a class="reference internal" href="#erf">erf</a></li>
-<li><a class="reference internal" href="#gamma-and-related-functions">Gamma and Related Functions</a></li>
-<li><a class="reference internal" href="#module-sympy.functions.special.error_functions">Special Cases of the Incomplete Gamma Functions</a></li>
-<li><a class="reference internal" href="#bessel-type-functions">Bessel Type Functions</a></li>
-<li><a class="reference internal" href="#b-splines">B-Splines</a></li>
-<li><a class="reference internal" href="#module-sympy.functions.special.zeta_functions">Riemann Zeta and Related Functions</a></li>
-<li><a class="reference internal" href="#hypergeometric-functions">Hypergeometric Functions</a></li>
-<li><a class="reference internal" href="#module-sympy.functions.special.polynomials">Orthogonal Polynomials</a><ul>
-<li><a class="reference internal" href="#jacobi-polynomials">Jacobi Polynomials</a></li>
-<li><a class="reference internal" href="#gegenbauer-polynomials">Gegenbauer Polynomials</a></li>
-<li><a class="reference internal" href="#chebyshev-polynomials">Chebyshev Polynomials</a></li>
-<li><a class="reference internal" href="#legendre-polynomials">Legendre Polynomials</a></li>
-<li><a class="reference internal" href="#hermite-polynomials">Hermite Polynomials</a></li>
-<li><a class="reference internal" href="#laguerre-polynomials">Laguerre Polynomials</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#spherical-harmonics">Spherical Harmonics</a></li>
-<li><a class="reference internal" href="#tensor-functions">Tensor Functions</a></li>
-</ul>
-</li>
-</ul>
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-            
-  <div class="section" id="module-sympy.galgebra.GA">
-<span id="galgebra"></span><h1>GAlgebra<a class="headerlink" href="#module-sympy.galgebra.GA" title="Permalink to this headline">¶</a></h1>
-<p>The module symbolicGA implements symbolic Geometric Algebra in python.
-The relevant references for this module are:</p>
-<blockquote>
-<div><ol class="arabic simple">
-<li>&#8220;Geometric Algebra for Physicists&#8221; by C. Doran and A. Lazenby,
-Cambridge University Press, 2003.</li>
-<li>&#8220;Geometric Algebra for Computer Science&#8221; by Leo Dorst,
-Daniel Fontijne, and Stephen Mann, Morgan Kaufmann Publishers, 2007.</li>
-<li>SymPy Tutorial, <a class="reference external" href="http://docs.sympy.org/">http://docs.sympy.org/</a></li>
-</ol>
-</div></blockquote>
-<dl class="function">
-<dt id="sympy.galgebra.GA.LaTeX_lst">
-<tt class="descclassname">sympy.galgebra.GA.</tt><tt class="descname">LaTeX_lst</tt><big>(</big><em>lst</em>, <em>title=''</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/GA.html#LaTeX_lst"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.GA.LaTeX_lst" title="Permalink to this definition">¶</a></dt>
-<dd><p>Output a list in LaTeX format.</p>
-</dd></dl>
-
-<dl class="data">
-<dt id="sympy.galgebra.GA.NUMPAT">
-<tt class="descclassname">sympy.galgebra.GA.</tt><tt class="descname">NUMPAT</tt><em class="property"> = &lt;_sre.SRE_Pattern object at 0x10f4fbb70&gt;</em><a class="headerlink" href="#sympy.galgebra.GA.NUMPAT" title="Permalink to this definition">¶</a></dt>
-<dd><p>Re pattern for rational number</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.GA.collect">
-<tt class="descclassname">sympy.galgebra.GA.</tt><tt class="descname">collect</tt><big>(</big><em>expr</em>, <em>lst</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/GA.html#collect"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.GA.collect" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wrapper for sympy.collect.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.GA.comb">
-<tt class="descclassname">sympy.galgebra.GA.</tt><tt class="descname">comb</tt><big>(</big><em>N</em>, <em>P</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/GA.html#comb"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.GA.comb" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculates the combinations of the integers [0,N-1] taken P at a time.
-The output is a list of lists of integers where the inner lists are
-the different combinations. Each combination is sorted in ascending
-order.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.GA.cp">
-<tt class="descclassname">sympy.galgebra.GA.</tt><tt class="descname">cp</tt><big>(</big><em>A</em>, <em>B</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/GA.html#cp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.GA.cp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculates the commutator product (A*B-B*A)/2 for
-the objects A and B.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.GA.diagpq">
-<tt class="descclassname">sympy.galgebra.GA.</tt><tt class="descname">diagpq</tt><big>(</big><em>p</em>, <em>q=0</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/GA.html#diagpq"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.GA.diagpq" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return string equivalent metric tensor for signature (p, q).</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.GA.dualsort">
-<tt class="descclassname">sympy.galgebra.GA.</tt><tt class="descname">dualsort</tt><big>(</big><em>lst1</em>, <em>lst2</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/GA.html#dualsort"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.GA.dualsort" title="Permalink to this definition">¶</a></dt>
-<dd><p>Inplace dual sort of lst1 and lst2 keyed on sorted lst1.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.GA.is_quasi_unit_numpy_array">
-<tt class="descclassname">sympy.galgebra.GA.</tt><tt class="descname">is_quasi_unit_numpy_array</tt><big>(</big><em>array</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/GA.html#is_quasi_unit_numpy_array"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.GA.is_quasi_unit_numpy_array" title="Permalink to this definition">¶</a></dt>
-<dd><p>Determine if a array is square and diagonal with
-entries of +1 or -1.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.GA.isint">
-<tt class="descclassname">sympy.galgebra.GA.</tt><tt class="descname">isint</tt><big>(</big><em>a</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/GA.html#isint"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.GA.isint" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test for integer.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.GA.israt">
-<tt class="descclassname">sympy.galgebra.GA.</tt><tt class="descname">israt</tt><big>(</big><em>numstr</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/GA.html#israt"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.GA.israt" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test if string represents a rational number.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.GA.magnitude">
-<tt class="descclassname">sympy.galgebra.GA.</tt><tt class="descname">magnitude</tt><big>(</big><em>vector</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/GA.html#magnitude"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.GA.magnitude" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculate magnitude of vector containing trig expressions
-and simplify.  This is a hack because of way he sign of
-magsq is determined and because of the way that absolute
-values are removed.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.GA.make_null_array">
-<tt class="descclassname">sympy.galgebra.GA.</tt><tt class="descname">make_null_array</tt><big>(</big><em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/GA.html#make_null_array"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.GA.make_null_array" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return list of n empty lists.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.GA.make_scalars">
-<tt class="descclassname">sympy.galgebra.GA.</tt><tt class="descname">make_scalars</tt><big>(</big><em>symnamelst</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/GA.html#make_scalars"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.GA.make_scalars" title="Permalink to this definition">¶</a></dt>
-<dd><p>make_scalars takes a string of symbol names separated by
-blanks and converts them to MV scalars and returns a list
-of the symbols.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.GA.normalize">
-<tt class="descclassname">sympy.galgebra.GA.</tt><tt class="descname">normalize</tt><big>(</big><em>elst</em>, <em>nname_lst</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/GA.html#normalize"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.GA.normalize" title="Permalink to this definition">¶</a></dt>
-<dd><p>Normalize a list of vectors and rename the normalized vectors. &#8216;elist&#8217; is the list
-(or array) of vectors to be normalized and nname_lst is a list of the names for the
-normalized vectors.  The function returns the numpy arrays enlst and mags containing
-the normalized vectors (enlst) and the magnitudes of the original vectors (mags).</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.GA.numeric">
-<tt class="descclassname">sympy.galgebra.GA.</tt><tt class="descname">numeric</tt><big>(</big><em>num_str</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/GA.html#numeric"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.GA.numeric" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns rational numbers compatible with symbols.
-Input is a string representing a fraction or integer
-or a simple integer.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.GA.reciprocal_frame">
-<tt class="descclassname">sympy.galgebra.GA.</tt><tt class="descname">reciprocal_frame</tt><big>(</big><em>vlst</em>, <em>names=''</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/GA.html#reciprocal_frame"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.GA.reciprocal_frame" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculate reciprocal frame of list (vlst) of vectors.  If desired name each
-vector in list of reciprocal vectors with names in space delimited string
-(names).</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.GA.reduce_base">
-<tt class="descclassname">sympy.galgebra.GA.</tt><tt class="descname">reduce_base</tt><big>(</big><em>k</em>, <em>base</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/GA.html#reduce_base"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.GA.reduce_base" title="Permalink to this definition">¶</a></dt>
-<dd><p>If base is a list of sorted integers [i_1,...,i_R] then reduce_base
-sorts the list [k,i_1,...,i_R] and calculates whether an odd or even
-number of permutations is required to sort the list. The sorted list
-is returned and +1 for even permutations or -1 for odd permutations.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.GA.set_names">
-<tt class="descclassname">sympy.galgebra.GA.</tt><tt class="descname">set_names</tt><big>(</big><em>var_lst</em>, <em>var_str</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/GA.html#set_names"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.GA.set_names" title="Permalink to this definition">¶</a></dt>
-<dd><p>Set the names of a list of multivectors (var_lst) for a space delimited
-string (var_str) containing the names.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.GA.sub_base">
-<tt class="descclassname">sympy.galgebra.GA.</tt><tt class="descname">sub_base</tt><big>(</big><em>k</em>, <em>base</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/GA.html#sub_base"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.GA.sub_base" title="Permalink to this definition">¶</a></dt>
-<dd><p>If base is a list of sorted integers [i_1,...,i_R] then sub_base returns
-a list with the k^th element removed. Note that k=0 removes the first
-element.  There is no test to see if k is in the range of the list.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.GA.test_int_flgs">
-<tt class="descclassname">sympy.galgebra.GA.</tt><tt class="descname">test_int_flgs</tt><big>(</big><em>lst</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/GA.html#test_int_flgs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.GA.test_int_flgs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test if all elements in list are 0.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.GA.unabs">
-<tt class="descclassname">sympy.galgebra.GA.</tt><tt class="descname">unabs</tt><big>(</big><em>x</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/GA.html#unabs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.GA.unabs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Remove absolute values from expressions so a = sqrt(a**2).
-This is a hack.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.GA.vector_fct">
-<tt class="descclassname">sympy.galgebra.GA.</tt><tt class="descname">vector_fct</tt><big>(</big><em>Fstr</em>, <em>x</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/GA.html#vector_fct"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.GA.vector_fct" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a list of functions of arguments x.  One function is
-created for each variable in x.  Fstr is a string that is
-the base name of each function while each function in the
-list is given the name Fstr+&#8217;__&#8217;+str(x[ix]) so that if
-Fstr = &#8216;f&#8217; and str(x[1]) = &#8216;theta&#8217; then the LaTeX output
-of the second element in the output list would be &#8216;f^{theta}&#8217;.</p>
-</dd></dl>
-
-</div>
-
-
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-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../galgebra/index.html"
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-            
-  <div class="section" id="geometric-algebra-module-for-sympy">
-<h1>Geometric Algebra Module for SymPy<a class="headerlink" href="#geometric-algebra-module-for-sympy" title="Permalink to this headline">¶</a></h1>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Author:</th><td class="field-body">Alan Bromborsky</td>
-</tr>
-</tbody>
-</table>
-<div class="topic">
-<p class="topic-title first">Abstract</p>
-<p>This document describes the implementation of a geometric algebra module in
-python that utilizes the <a class="reference internal" href="../../matrices/immutablematrices.html#module-sympy" title="sympy"><tt class="xref py py-mod docutils literal"><span class="pre">sympy</span></tt></a> symbolic algebra library.  The python
-module <tt class="xref py py-mod docutils literal"><span class="pre">GA</span></tt> has been developed for coordinate free calculations using
-the operations (geometric, outer, and inner products etc.) of geometric algebra.
-The operations can be defined using a completely arbitrary metric defined
-by the inner products of a set of arbitrary vectors or the metric can be
-restricted to enforce orthogonality and signature constraints on the set of
-vectors.  In addition the module includes the geometric, outer (curl) and inner
-(div) derivatives and the ability to define a curvilinear coordinate system.
-The module requires the numpy and the sympy modules.</p>
-</div>
-<div class="section" id="what-is-geometric-algebra">
-<h2>What is Geometric Algebra?<a class="headerlink" href="#what-is-geometric-algebra" title="Permalink to this headline">¶</a></h2>
-<p>Geometric algebra is the Clifford algebra of a real finite dimensional vector
-space or the algebra that results when a real finite dimensional vector space
-is extended with a product of vectors (geometric product) that is associative,
-left and right distributive, and yields a real number for the square (geometric
-product) of any vector [Hestenes,Lasenby].  The elements of the geometric
-algebra are called multivectors and consist of the linear combination of
-scalars, vectros, and the geometric product of two or more vectors. The
-additional axioms for the geometric algebra are that for any vectors <span class="math">\(a\)</span>,
-<span class="math">\(b\)</span>, and <span class="math">\(c\)</span> in the base vector space:</p>
-<div class="math">
-\[\begin{equation*}
-\begin{array}{c}
-a \left( bc \right) = \left( ab \right) c \\
-a \left( b+c \right) = ab+ac \\
-\left( a + b \right) c = ac+bc \\
-aa = a^{2} \in \Re
-\end{array}
-\end{equation*}\]</div><p>By induction these also apply to any multivectors.</p>
-<p>Several software packages for numerical geometric algebra calculations are
-available from Doran-Lazenby group and the Dorst group. Symbolic packages for
-Clifford algebra using orthongonal bases such as
-<span class="math">\(e_{i}e_{j}+e_{j}e_{i} = 2\eta_{ij}\)</span>, where <span class="math">\(\eta_{ij}\)</span> is a numeric
-array are available in Maple and Mathematica. The symbolic algebra module,
-<tt class="xref py py-mod docutils literal"><span class="pre">GA</span></tt>, developed for python does not depend on an orthogonal basis
-representation, but rather is generated from a set of <span class="math">\(n\)</span> arbitrary
-symbolic vectors,  <span class="math">\(a_{1},a_{2},\dots,a_{n}\)</span> and a symbolic metric
-tensor <span class="math">\(g_{ij} = a_{i}\cdot a_{j}\)</span>.</p>
-<p>In order not to reinvent the wheel all scalar symbolic algebra is handled by the
-python module  <a class="reference internal" href="../../matrices/immutablematrices.html#module-sympy" title="sympy"><tt class="xref py py-mod docutils literal"><span class="pre">sympy</span></tt></a>.</p>
-<p>The basic geometic algebra operations will be implemented in python by defining
-a multivector class, MV, and overloading the python operators in Table
-<a class="reference internal" href="#table1"><em>1</em></a> where <tt class="docutils literal"><span class="pre">A</span></tt> and <tt class="docutils literal"><span class="pre">B</span></tt>  are any two multivectors (In the case of
-<tt class="docutils literal"><span class="pre">+</span></tt>, <tt class="docutils literal"><span class="pre">-</span></tt>, <tt class="docutils literal"><span class="pre">*</span></tt>, <tt class="docutils literal"><span class="pre">^</span></tt>, <tt class="docutils literal"><span class="pre">|</span></tt>, <tt class="docutils literal"><span class="pre">&lt;&lt;</span></tt>, and <tt class="docutils literal"><span class="pre">&gt;&gt;</span></tt> the operation is also defined if <tt class="docutils literal"><span class="pre">A</span></tt> or
-<tt class="docutils literal"><span class="pre">B</span></tt> is a sympy symbol or a sympy real number).</p>
-<table border="1" class="docutils" id="table1">
-<colgroup>
-<col width="20%" />
-<col width="80%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head">Operation</th>
-<th class="head">Result</th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-even"><td><tt class="docutils literal"><span class="pre">A+B</span></tt></td>
-<td>Sum of multivectors</td>
-</tr>
-<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">A-B</span></tt></td>
-<td>Difference of multivectors</td>
-</tr>
-<tr class="row-even"><td><tt class="docutils literal"><span class="pre">A*B</span></tt></td>
-<td>Geometric product</td>
-</tr>
-<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">A^B</span></tt></td>
-<td>Outer product of multivectors</td>
-</tr>
-<tr class="row-even"><td><tt class="docutils literal"><span class="pre">A|B</span></tt></td>
-<td>Inner product of multivectors</td>
-</tr>
-<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">A&lt;B</span></tt> or <tt class="docutils literal"><span class="pre">A&lt;&lt;B</span></tt></td>
-<td>Left contraction of multivectors</td>
-</tr>
-<tr class="row-even"><td><tt class="docutils literal"><span class="pre">A&gt;B</span></tt> or <tt class="docutils literal"><span class="pre">A&gt;&gt;B</span></tt></td>
-<td>Right contraction of multivectors</td>
-</tr>
-</tbody>
-</table>
-<p>Table <a class="reference internal" href="#table1"><em>1</em></a>. Multivector operations for symbolicGA</p>
-<p>The option to use <tt class="docutils literal"><span class="pre">&lt;</span></tt> or <tt class="docutils literal"><span class="pre">&lt;&lt;</span></tt> for left contraction and <tt class="docutils literal"><span class="pre">&gt;</span></tt> or <tt class="docutils literal"><span class="pre">&gt;&gt;</span></tt> is given since the <tt class="docutils literal"><span class="pre">&lt;</span></tt> and
-<tt class="docutils literal"><span class="pre">&gt;</span></tt> operators do not have r-forms (there are no <tt class="xref py py-func docutils literal"><span class="pre">__rlt__()</span></tt> and <tt class="xref py py-func docutils literal"><span class="pre">__rgt__()</span></tt> functions to overload) while
-<tt class="docutils literal"><span class="pre">&lt;&lt;</span></tt> and <tt class="docutils literal"><span class="pre">&gt;&gt;</span></tt> do have r-forms so that <tt class="docutils literal"><span class="pre">x</span> <span class="pre">&lt;&lt;</span> <span class="pre">A</span></tt> and <tt class="docutils literal"><span class="pre">x</span> <span class="pre">&gt;&gt;</span> <span class="pre">A</span></tt> are allowed where <tt class="docutils literal"><span class="pre">x</span></tt> is a scalar (symbol or integer)
-and <tt class="docutils literal"><span class="pre">A</span></tt> is a multivector. With <tt class="docutils literal"><span class="pre">&lt;</span></tt> and <tt class="docutils literal"><span class="pre">&gt;</span></tt> we can only have mixed modes (scalars and multivectors) if the
-multivector is the first operand.</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">Except for <tt class="docutils literal"><span class="pre">&lt;</span></tt> and <tt class="docutils literal"><span class="pre">&gt;</span></tt> all the multivector operators have r-forms so that as long as one of the
-operands, left or right, is a multivector the other can be a multivector or a scalar (sympy symbol or integer).</p>
-</div>
-<div class="admonition warning">
-<p class="first admonition-title">Warning</p>
-<p class="last">Note that the operator order precedence is determined by python and is not
-neccessarily that used by geometric algebra. It is <strong>absolutely essential</strong> to
-use parenthesis in multivector
-expressions containing <tt class="docutils literal"><span class="pre">^</span></tt>, <tt class="docutils literal"><span class="pre">|</span></tt>, <tt class="docutils literal"><span class="pre">&lt;</span></tt>, <tt class="docutils literal"><span class="pre">&gt;</span></tt>, <tt class="docutils literal"><span class="pre">&lt;&lt;</span></tt> and/or <tt class="docutils literal"><span class="pre">&gt;&gt;</span></tt>.  As an example let
-<tt class="docutils literal"><span class="pre">A</span></tt> and <tt class="docutils literal"><span class="pre">B</span></tt> be any two multivectors. Then <tt class="docutils literal"><span class="pre">A</span> <span class="pre">+</span> <span class="pre">A*B</span> <span class="pre">=</span> <span class="pre">A</span> <span class="pre">+(A*B)</span></tt>, but
-<tt class="docutils literal"><span class="pre">A+A^B</span> <span class="pre">=</span> <span class="pre">(2*A)^B</span></tt> since in python the <tt class="docutils literal"><span class="pre">^</span></tt> operator has a lower precedence
-than the &#8216;+&#8217; operator.  In geometric algebra the outer and inner products and
-the left and right contractions have a higher precedence than the geometric
-product and the geometric product has a higher precedence than addition and
-subtraction.  In python the <tt class="docutils literal"><span class="pre">^</span></tt>, <tt class="docutils literal"><span class="pre">|</span></tt>, <tt class="docutils literal"><span class="pre">&lt;</span></tt>, <tt class="docutils literal"><span class="pre">&gt;</span></tt>, <tt class="docutils literal"><span class="pre">&lt;&lt;</span></tt> and <tt class="docutils literal"><span class="pre">&gt;&gt;</span></tt> all have a lower
-precedence than <tt class="docutils literal"><span class="pre">+</span></tt> and <tt class="docutils literal"><span class="pre">-</span></tt> while <tt class="docutils literal"><span class="pre">*</span></tt> has a higher precedence than
-<tt class="docutils literal"><span class="pre">+</span></tt> and <tt class="docutils literal"><span class="pre">-</span></tt>.</p>
-</div>
-</div>
-<div class="section" id="vector-basis-and-metric">
-<span id="vbm"></span><h2>Vector Basis and Metric<a class="headerlink" href="#vector-basis-and-metric" title="Permalink to this headline">¶</a></h2>
-<p>The two structures that define the <a class="reference internal" href="#MV" title="MV"><tt class="xref py py-class docutils literal"><span class="pre">MV</span></tt></a> (multivector) class are the
-symbolic basis vectors and the symbolic metric.  The symbolic basis
-vectors are input as a string with the symbol name separated by spaces.  For
-example if we are calculating the geometric algebra of a system with three
-vectors that we wish to denote as <tt class="docutils literal"><span class="pre">a0</span></tt>, <tt class="docutils literal"><span class="pre">a1</span></tt>, and <tt class="docutils literal"><span class="pre">a2</span></tt> we would define the
-string variable:</p>
-<p><tt class="docutils literal"><span class="pre">basis</span> <span class="pre">=</span> <span class="pre">'a0</span> <span class="pre">a1</span> <span class="pre">a2'</span></tt></p>
-<p>that would be input into the multivector setup function.  The next step would be
-to define the symbolic metric for the geometric algebra of the basis we
-have defined. The default metric is the most general and is the matrix of
-the following symbols</p>
-<div class="math" id="equation-1">
-<span id="eq1"></span>\[\begin{equation*}
-g = \left[
-\begin{array}{ccc}
-  a0**2 & (a0.a1) & (a0.a2) \\
-  (a0.a1) & a1**2 & (a1.a2) \\
-  (a0.a2) & (a1.a2) & a2**2 \\
-\end{array}
-\right]
-\end{equation*}\]</div><p>where each of the <span class="math">\(g_{ij}\)</span> is a symbol representing all of the dot
-products of the basis vectors. Note that the symbols are named so that
-<span class="math">\(g_{ij} = g_{ji}\)</span> since for the symbol function
-<span class="math">\((a0.a1) \ne (a1.a0)\)</span>.</p>
-<p>Note that the strings shown in equation <a class="reference internal" href="#eq1"><em>1</em></a> are only used when the values
-of <span class="math">\(g_{ij}\)</span> are output (printed).   In the <tt class="xref py py-mod docutils literal"><span class="pre">GA</span></tt> module (library)
-the <span class="math">\(g_{ij}\)</span> symbols are stored in a static member list of the multivector
-class <a class="reference internal" href="#MV" title="MV"><tt class="xref py py-class docutils literal"><span class="pre">MV</span></tt></a> as the double list <tt class="docutils literal"><span class="pre">MV.metric</span></tt> (<span class="math">\(g_{ij}\)</span> = <tt class="docutils literal"><span class="pre">MV.metric[i][j]</span></tt>).</p>
-<p>The default definition of <span class="math">\(g\)</span> can be overwritten by specifying a string
-that will define <span class="math">\(g\)</span>. As an example consider a symbolic representation
-for conformal geometry. Define for a basis</p>
-<p><tt class="docutils literal"><span class="pre">basis</span> <span class="pre">=</span> <span class="pre">'a0</span> <span class="pre">a1</span> <span class="pre">a2</span> <span class="pre">n</span> <span class="pre">nbar'</span></tt></p>
-<p>and for a metric</p>
-<p><tt class="docutils literal"><span class="pre">metric</span> <span class="pre">=</span> <span class="pre">'#</span> <span class="pre">#</span> <span class="pre">#</span> <span class="pre">0</span> <span class="pre">0,</span> <span class="pre">#</span> <span class="pre">#</span> <span class="pre">#</span> <span class="pre">0</span> <span class="pre">0,</span> <span class="pre">#</span> <span class="pre">#</span> <span class="pre">#</span> <span class="pre">0</span> <span class="pre">0,</span> <span class="pre">0</span> <span class="pre">0</span> <span class="pre">0</span> <span class="pre">0</span> <span class="pre">2,</span> <span class="pre">0</span> <span class="pre">0</span> <span class="pre">0</span> <span class="pre">2</span> <span class="pre">0'</span></tt></p>
-<p>then calling <span class="math">\(MV.setup(basis,metric)\)</span> would initialize</p>
-<div class="math">
-\[\begin{equation*}
-g = \left[
-\begin{array}{ccccc}
-  a0**2 & (a0.a1)  & (a0.a2) & 0 & 0\\
-  (a0.a1) & a1**2  & (a1.a2) & 0 & 0\\
-  (a0.a2) & (a1.a2) & a2**2 & 0 & 0 \\
-  0 & 0 & 0 & 0 & 2 \\
-  0 & 0 & 0 & 2 & 0
-\end{array}
-\right]
-\end{equation*}\]</div><p>Here we have specified that <span class="math">\(n\)</span> and <span class="math">\(nbar\)</span> are orthonal to all the
-<span class="math">\(a\)</span>&#8216;s, <span class="math">\(n**2 = nbar**2 = 0\)</span>, and <span class="math">\((n.nbar) = 2\)</span>. Using
-<span class="math">\(\#\)</span> in the metric definition string just tells the program to use the
-default symbol for that value.</p>
-<p>When <tt class="docutils literal"><span class="pre">MV.setup</span></tt> is called multivector representations of the basis local to
-the program are instantiated.  For our first example that means that the
-symbolic vectors named <tt class="docutils literal"><span class="pre">a0</span></tt>, <tt class="docutils literal"><span class="pre">a1</span></tt>, and <tt class="docutils literal"><span class="pre">a2</span></tt> are created and made available
-to the programmer for future calculations.</p>
-<p>In addition to the basis vectors the <span class="math">\(g_{ij}\)</span> are also made available to
-the programer with the following convention. If <tt class="docutils literal"><span class="pre">a0</span></tt> and <tt class="docutils literal"><span class="pre">a1</span></tt> are basis
-vectors, then their dot products are denoted by  <tt class="docutils literal"><span class="pre">a0sq</span></tt>, <tt class="docutils literal"><span class="pre">a2sq</span></tt>, and
-<tt class="docutils literal"><span class="pre">a0dota1</span></tt> for use as python program varibles. If you print  <tt class="docutils literal"><span class="pre">a0sq</span></tt> the
-output would be <tt class="docutils literal"><span class="pre">a0**2</span></tt> and the output for <tt class="docutils literal"><span class="pre">a0dota1</span></tt> would be <tt class="docutils literal"><span class="pre">(a0.a1)</span></tt> as
-shown in equation <a class="reference internal" href="#eq1"><em>1</em></a>.  If the default value are overridden the new
-values are output by print.  For examle if <span class="math">\(g_{00} = 0\)</span> then &#8220;<tt class="docutils literal"><span class="pre">print</span>
-<span class="pre">a0sq</span></tt>&#8221; would output &#8220;0.&#8221;</p>
-<p>More generally, if <tt class="docutils literal"><span class="pre">metric</span></tt> is not a string, but a list of lists or a two
-dimension numpy array, it is assumed that each element of <tt class="docutils literal"><span class="pre">metric</span></tt> is symbolic
-variable so that the <span class="math">\(g_{ij}\)</span> could be defined as symbolic functions as
-well as variables. For example instead of letting <span class="math">\(g_{01} = (a0.a1)\)</span> we
-could have  <span class="math">\(g_{01} = cos(theta)\)</span> where we use a symbolic <span class="math">\(\cos\)</span>
-function.</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p>Additionally <tt class="docutils literal"><span class="pre">MV.setup</span></tt> has an option for an othogonal basis where the signature
-of the metric space is defined by a string.  For example if the signature of the
-vector space is <span class="math">\((1,1,1)\)</span> (Euclidian 3-space) set</p>
-<p><tt class="docutils literal"><span class="pre">metric</span> <span class="pre">=</span> <span class="pre">'[1,1,1]'</span></tt></p>
-<p>Likewise if the signature is that of spacetime, <span class="math">\((1,-1,-1,-1)\)</span> then define</p>
-<p class="last"><tt class="docutils literal"><span class="pre">metric</span> <span class="pre">=</span> <span class="pre">'[1,-1,-1,-1]'</span></tt>.</p>
-</div>
-</div>
-<div class="section" id="representation-and-reduction-of-multivector-bases">
-<h2>Representation and Reduction of Multivector Bases<a class="headerlink" href="#representation-and-reduction-of-multivector-bases" title="Permalink to this headline">¶</a></h2>
-<p>In our symbolic geometric algebra we assume that all multivectors of interest to
-us can be obtained from the symbolic basis vectors we have input, via the
-different operations available to geometric algebra. The first problem we have
-is representing the general multivector in terms of the basis vectors.  To
-do this we form the ordered geometric products of the basis vectors and develop
-an internal representation of these products in terms of python classes.  The
-ordered geometric products are all multivectors of the form
-<span class="math">\(a_{i_{1}}a_{i_{2}}\dots a_{i_{r}}\)</span> where <span class="math">\(i_{1}&lt;i_{2}&lt;\dots &lt;i_{r}\)</span>
-and <span class="math">\(r \le n\)</span>. We call these multivectors bases and represent them
-internally with the list of integers
-<span class="math">\(\left[ i_{1},i_{2},\dots,i_{r}\right]\)</span>. The bases are labeled, for the
-purpose of output display, with strings that are concatenations of the strings
-representing the basis vectors.  So that in our example  <tt class="docutils literal"><span class="pre">[1,2]</span></tt> would be
-labeled with the string <tt class="docutils literal"><span class="pre">'a1a2'</span></tt> and represents the geometric product
-<tt class="docutils literal"><span class="pre">a1*a2</span></tt>. Thus the list <tt class="docutils literal"><span class="pre">[0,1,2]</span></tt> represents <tt class="docutils literal"><span class="pre">a0*a1*a2</span></tt>.For our example the
-complete set of bases and labels are shown in Table <a class="reference internal" href="#table2"><em>2</em></a></p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">The empty list, <tt class="docutils literal"><span class="pre">[]</span></tt>, represents the scalar 1.</p>
-</div>
-<div class="highlight-python" id="table2"><div class="highlight"><pre><span class="n">MV</span><span class="o">.</span><span class="n">basislabel</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;1&#39;</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;a0&#39;</span><span class="p">,</span> <span class="s">&#39;a1&#39;</span><span class="p">,</span> <span class="s">&#39;a2&#39;</span><span class="p">],</span> <span class="p">[</span><span class="s">&#39;a0a1&#39;</span><span class="p">,</span> <span class="s">&#39;a0a2&#39;</span><span class="p">,</span> <span class="s">&#39;a1a2&#39;</span><span class="p">],</span>
-                 <span class="p">[</span><span class="s">&#39;a0a1a2&#39;</span><span class="p">]]</span>
-<span class="n">MV</span><span class="o">.</span><span class="n">basis</span>      <span class="o">=</span> <span class="p">[[],</span> <span class="p">[[</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]],</span> <span class="p">[[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]],</span>
-                 <span class="p">[[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]]]</span>
-</pre></div>
-</div>
-<p>Table <a class="reference internal" href="#table2"><em>2</em></a>. Multivector basis labels and internal basis representation.</p>
-<p>Since there are <span class="math">\(2^{n}\)</span> bases and the number of bases with equal list
-lengths is the same as for the grade decomposition of a dimension <span class="math">\(n\)</span>
-geometric algebra we will call the collections of bases of equal length
-<tt class="docutils literal"><span class="pre">psuedogrades</span></tt>.</p>
-<p>The critical operation in setting up the geometric algebra module is reducing
-the geomertric product of any two bases to a linear combination of bases so that
-we can calculate a multiplication table for the bases.  First we represent the
-product as the concatenation of two base lists.  For example <tt class="docutils literal"><span class="pre">a1a2*a0a1</span></tt> is
-represented by the list <tt class="docutils literal"><span class="pre">[1,2]+[0,1]</span> <span class="pre">=</span> <span class="pre">[1,2,0,1]</span></tt>. The representation of the
-product is reduced via two operations, contraction and revision. The state of
-the reduction is saved in two lists of equal length.  The first list contains
-symbolic scale factors (symbol or numeric types) for the corresponding interger
-list representing the product of bases.  If we wish to reduce
-<span class="math">\(\left[ i_{1},\dots,i_{r}\right]\)</span> the starting point is the coefficient list
-<span class="math">\(C = \left[ 1 \right]\)</span> and the bases list
-<span class="math">\(B = \left[ \left[ i_{1},\dots,i_{r}\right] \right]\)</span>.  We now operate on each
-element of the lists as follows:</p>
-<ul class="simple">
-<li><tt class="docutils literal"><span class="pre">contraction</span></tt>: Consider a basis list <span class="math">\(B\)</span> with element
-<span class="math">\(B[j] = \left[ i_{1},\dots,i_{l},i_{l+1},\dots,i_{s}\right]\)</span> where
-<span class="math">\(i_{l} = i_{l+1}\)</span>. Then the product of the <span class="math">\(l\)</span> and <span class="math">\(l+1\)</span> terms
-result in a scalar and <span class="math">\(B[j]\)</span> is replaced by the new list representation
-<span class="math">\(\left[ i_{1},\dots,i_{l-1},i_{l+2},\dots,i_{r}\right]\)</span> which is of psuedo
-grade <span class="math">\(r-2\)</span> and <span class="math">\(C[j]\)</span> is replaced by the symbol
-<span class="math">\(g_{i_{l}i_{l}}C[j]\)</span>.</li>
-<li><tt class="docutils literal"><span class="pre">revision</span></tt>: Consider a basis list <span class="math">\(B\)</span> with element
-<span class="math">\(B[j] = \left[ i_{1},\dots,i_{l},i_{l+1},\dots,i_{s}\right]\)</span> where
-<span class="math">\(i_{l} &gt; i_{l+1}\)</span>. Then the <span class="math">\(l\)</span> and <span class="math">\(l+1\)</span> elements must be
-reversed to be put in normal order, but we have <span class="math">\(a_{i_{l}}a_{i_{l+1}} = 2g_{i_{l}i_{l+1}}-a_{i_{l+1}}a_{i_{l}}\)</span>
-(From the geometric algebra definition of the dot product of two vectors). Thus
-we append the list representing the reduced element,
-<span class="math">\(\left[ i_{1},\dots,i_{l-1},i_{l+2},\dots,i_{s}\right]\)</span>, to the pseudo bases
-list, <span class="math">\(B\)</span>, and append <span class="math">\(2g_{i{l}i_{l+1}}C[j]\)</span> to the coefficients
-list, then we replace <span class="math">\(B[j]\)</span> with
-<span class="math">\(\left[ i_{1},\dots,i_{l+1},i_{l},\dots,i_{s}\right]\)</span>  and <span class="math">\(C[j]\)</span> with
-<span class="math">\(-C[j]\)</span>. Both lists are increased by one element if
-<span class="math">\(g_{i_{l}i_{l+1}} \ne 0\)</span>.</li>
-</ul>
-<p>These processes are repeated untill every basis list in <span class="math">\(B\)</span> is in normal
-(ascending) order with no repeated elements.  Then the coefficents of equivalent
-bases are summed and the bases sorted according to psuedograde and ascending
-order.  We now have a way of calculating the geometric product of any two bases
-as a symbolic linear combination of all the bases with the coefficients
-determined by <span class="math">\(g\)</span>. The base multiplication table for our simple example of
-three vectors is given by (the coefficient of each psuedo base is enclosed with
-{} for clarity):</p>
-<div class="highlight-python"><pre>(1)(1) = 1
-(1)(a0) = a0
-(1)(a1) = a1
-(1)(a2) = a2
-(1)(a0a1) = a0a1
-(1)(a0a2) = a0a2
-(1)(a1a2) = a1a2
-(1)(a0a1a2) = a0a1a2
-
-(a0)(1) = a0
-(a0)(a0) = {a0**2}1
-(a0)(a1) = a0a1
-(a0)(a2) = a0a2
-(a0)(a0a1) = {a0**2}a1
-(a0)(a0a2) = {a0**2}a2
-(a0)(a1a2) = a0a1a2
-(a0)(a0a1a2) = {a0**2}a1a2
-
-(a1)(1) = a1
-(a1)(a0) = {2*(a0.a1)}1-a0a1
-(a1)(a1) = {a1**2}1
-(a1)(a2) = a1a2
-(a1)(a0a1) = {-a1**2}a0+{2*(a0.a1)}a1
-(a1)(a0a2) = {2*(a0.a1)}a2-a0a1a2
-(a1)(a1a2) = {a1**2}a2
-(a1)(a0a1a2) = {-a1**2}a0a2+{2*(a0.a1)}a1a2
-
-(a2)(1) = a2
-(a2)(a0) = {2*(a0.a2)}1-a0a2
-(a2)(a1) = {2*(a1.a2)}1-a1a2
-(a2)(a2) = {a2**2}1
-(a2)(a0a1) = {-2*(a1.a2)}a0+{2*(a0.a2)}a1+a0a1a2
-(a2)(a0a2) = {-a2**2}a0+{2*(a0.a2)}a2
-(a2)(a1a2) = {-a2**2}a1+{2*(a1.a2)}a2
-(a2)(a0a1a2) = {a2**2}a0a1+{-2*(a1.a2)}a0a2+{2*(a0.a2)}a1a2
-
-(a0a1)(1) = a0a1
-(a0a1)(a0) = {2*(a0.a1)}a0+{-a0**2}a1
-(a0a1)(a1) = {a1**2}a0
-(a0a1)(a2) = a0a1a2
-(a0a1)(a0a1) = {-a0**2*a1**2}1+{2*(a0.a1)}a0a1
-(a0a1)(a0a2) = {2*(a0.a1)}a0a2+{-a0**2}a1a2
-(a0a1)(a1a2) = {a1**2}a0a2
-(a0a1)(a0a1a2) = {-a0**2*a1**2}a2+{2*(a0.a1)}a0a1a2
-
-(a0a2)(1) = a0a2
-(a0a2)(a0) = {2*(a0.a2)}a0+{-a0**2}a2
-(a0a2)(a1) = {2*(a1.a2)}a0-a0a1a2
-(a0a2)(a2) = {a2**2}a0
-(a0a2)(a0a1) = {-2*a0**2*(a1.a2)}1+{2*(a0.a2)}a0a1+{a0**2}a1a2
-(a0a2)(a0a2) = {-a0**2*a2**2}1+{2*(a0.a2)}a0a2
-(a0a2)(a1a2) = {-a2**2}a0a1+{2*(a1.a2)}a0a2
-(a0a2)(a0a1a2) = {a0**2*a2**2}a1+{-2*a0**2*(a1.a2)}a2+{2*(a0.a2)}a0a1a2
-
-(a1a2)(1) = a1a2
-(a1a2)(a0) = {2*(a0.a2)}a1+{-2*(a0.a1)}a2+a0a1a2
-(a1a2)(a1) = {2*(a1.a2)}a1+{-a1**2}a2
-(a1a2)(a2) = {a2**2}a1
-(a1a2)(a0a1) = {2*a1**2*(a0.a2)-4*(a0.a1)*(a1.a2)}1+{2*(a1.a2)}a0a1+{-a1**2}a0a2
-              +{2*(a0.a1)}a1a2
-(a1a2)(a0a2) = {-2*a2**2*(a0.a1)}1+{a2**2}a0a1+{2*(a0.a2)}a1a2
-(a1a2)(a1a2) = {-a1**2*a2**2}1+{2*(a1.a2)}a1a2
-(a1a2)(a0a1a2) = {-a1**2*a2**2}a0+{2*a2**2*(a0.a1)}a1+{2*a1**2*(a0.a2)
-                -4*(a0.a1)*(a1.a2)}a2+{2*(a1.a2)}a0a1a2
-
-(a0a1a2)(1) = a0a1a2
-(a0a1a2)(a0) = {2*(a0.a2)}a0a1+{-2*(a0.a1)}a0a2+{a0**2}a1a2
-(a0a1a2)(a1) = {2*(a1.a2)}a0a1+{-a1**2}a0a2
-(a0a1a2)(a2) = {a2**2}a0a1
-(a0a1a2)(a0a1) = {2*a1**2*(a0.a2)-4*(a0.a1)*(a1.a2)}a0+{2*a0**2*(a1.a2)}a1
-                +{-a0**2*a1**2}a2+{2*(a0.a1)}a0a1a2
-(a0a1a2)(a0a2) = {-2*a2**2*(a0.a1)}a0+{a0**2*a2**2}a1+{2*(a0.a2)}a0a1a2
-(a0a1a2)(a1a2) = {-a1**2*a2**2}a0+{2*(a1.a2)}a0a1a2
-(a0a1a2)(a0a1a2) = {-a0**2*a1**2*a2**2}1+{2*a2**2*(a0.a1)}a0a1+{2*a1**2*(a0.a2)
-                  -4*(a0.a1)*(a1.a2)}a0a2+{2*a0**2*(a1.a2)}a1a2</pre>
-</div>
-</div>
-<div class="section" id="base-representation-of-multivectors">
-<h2>Base Representation of Multivectors<a class="headerlink" href="#base-representation-of-multivectors" title="Permalink to this headline">¶</a></h2>
-<p>In terms of the bases defined an arbitrary multivector can be represented as a
-list of arrays  (we use the numpy python module to implement arrays).   If we
-have <span class="math">\(n\)</span> basis vectors we initialize the list <tt class="docutils literal"><span class="pre">self.mv</span> <span class="pre">=</span> <span class="pre">[0,0,...,0]</span></tt>
-with <span class="math">\(n+1\)</span> integers all zero.  Each zero is a placeholder for an array of
-python objects (in this case the objects will be sympy symbol objects). If
-<tt class="docutils literal"><span class="pre">self.mv[r]</span> <span class="pre">=</span> <span class="pre">numpy.array([list</span> <span class="pre">of</span> <span class="pre">symbol</span> <span class="pre">objects])</span></tt> each entry in the
-<tt class="docutils literal"><span class="pre">numpy.array</span></tt> will be a coefficient of the corresponding psuedo base.
-<tt class="docutils literal"><span class="pre">self.mv[r]</span> <span class="pre">=</span> <span class="pre">0</span></tt> indicates that the coefficients of every base of psuedo grade
-<span class="math">\(r\)</span> are 0.  The length of the array <tt class="docutils literal"><span class="pre">self.mv[r]</span></tt> is <span class="math">\(n \choose r\)</span>
-the binomial coefficient. For example the psuedo basis vector <tt class="docutils literal"><span class="pre">a1</span></tt> would be
-represented as a multivector by the list:</p>
-<p><tt class="docutils literal"><span class="pre">a1.mv</span> <span class="pre">=</span> <span class="pre">[0,numpy.array([numeric(0),numeric(1),numeric(0)]),0,0]</span></tt></p>
-<p>and <tt class="docutils literal"><span class="pre">a0a1a2</span></tt> by:</p>
-<p><tt class="docutils literal"><span class="pre">a0a1a2.mv</span> <span class="pre">=</span> <span class="pre">[0,0,0,numpy.array([numeric(1)])]</span></tt></p>
-<p>The array is stuffed with sympy numeric objects instead of python integers so
-that we can perform symbolically manipulate sympy expressions that consist of
-scalar algebraic symbols and exact rational numbers which sympy can also
-represent.</p>
-<p>The <tt class="docutils literal"><span class="pre">numpy.array</span></tt> is used because operations of addition, substraction, and
-multiplication by an object are defined for the array if they are defined for
-the objects making up the array, which they are by sympy.  We call this
-representation a base type because the <tt class="docutils literal"><span class="pre">r</span></tt> index is not a grade index since
-the bases we are using are not blades. In a blade representation the structure
-would be identical, but the bases would be replaced by blades and
-<tt class="docutils literal"><span class="pre">self.mv[r]</span></tt> would represent the <tt class="docutils literal"><span class="pre">r</span></tt> grade components of the multivector.
-The first use of the base representation is to store the results of the
-multiplication tabel for the bases in the class variable <tt class="docutils literal"><span class="pre">MV.mtabel</span></tt>.  This
-variable is a group of nested lists so that the geometric product of the
-<tt class="docutils literal"><span class="pre">igrade</span></tt> and  <tt class="docutils literal"><span class="pre">ibase</span></tt> with the <tt class="docutils literal"><span class="pre">jgrade</span></tt> and <tt class="docutils literal"><span class="pre">jbase</span></tt> is
-<tt class="docutils literal"><span class="pre">MV.mtabel[igrade][ibase][jgrade][jbase]</span></tt>.  We can then use this table to
-calculate the geometric product of any two multivectors.</p>
-</div>
-<div class="section" id="blade-representation-of-multivectors">
-<h2>Blade Representation of Multivectors<a class="headerlink" href="#blade-representation-of-multivectors" title="Permalink to this headline">¶</a></h2>
-<p>Since we can now calculate the symbolic geometric product of any two
-multivectors we can also calculate the blades corresponding to the product of
-the symbolic basis vectors using the formula</p>
-<div class="math">
-\[\begin{equation*}
-  A_{r}\wedge b = \frac{1}{2}\left( A_{r}b-\left( -1 \right)^{r}bA_{r} \right),
-\end{equation*}\]</div><p>where <span class="math">\(A_{r}\)</span> is a multivector of grade <span class="math">\(r\)</span> and <span class="math">\(b\)</span> is a
-vector.  For our example basis the result is shown in Table <a class="reference internal" href="#table3"><em>3</em></a>.</p>
-<div class="highlight-python" id="table3"><pre>1 = 1
-a0 = a0
-a1 = a1
-a2 = a2
-a0^a1 = {-(a0.a1)}1+a0a1
-a0^a2 = {-(a0.a2)}1+a0a2
-a1^a2 = {-(a1.a2)}1+a1a2
-a0^a1^a2 = {-(a1.a2)}a0+{(a0.a2)}a1+{-(a0.a1)}a2+a0a1a2</pre>
-</div>
-<p>Table <a class="reference internal" href="#table3"><em>3</em></a>. Bases blades in terms of bases.</p>
-<p>The important thing to notice about Table <a class="reference internal" href="#table3"><em>3</em></a> is that it is a
-triagonal (lower triangular) system of equations so that using a simple back
-substitution algorithym we can solve for the psuedo bases in terms of the blades
-giving Table <a class="reference internal" href="#table4"><em>4</em></a>.</p>
-<div class="highlight-python" id="table4"><pre>1 = 1
-a0 = a0
-a1 = a1
-a2 = a2
-a0a1 = {(a0.a1)}1+a0^a1
-a0a2 = {(a0.a2)}1+a0^a2
-a1a2 = {(a1.a2)}1+a1^a2
-a0a1a2 = {(a1.a2)}a0+{-(a0.a2)}a1+{(a0.a1)}a2+a0^a1^a2</pre>
-</div>
-<p>Table <a class="reference internal" href="#table4"><em>4</em></a>. Bases in terms of basis blades.</p>
-<p>Using Table <a class="reference internal" href="#table4"><em>4</em></a> and simple substitution we can convert from a base
-multivector representation to a blade representation.  Likewise, using Table
-<a class="reference internal" href="#table3"><em>3</em></a> we can convert from blades to bases.</p>
-<p>Using the blade representation it becomes simple to program functions that will
-calculate the grade projection, reverse, even, and odd multivector functions.</p>
-<p>Note that in the multivector class <tt class="docutils literal"><span class="pre">MV</span></tt> there is a class variable for each
-instantiation, <tt class="docutils literal"><span class="pre">self.bladeflg</span></tt>, that is set to zero for a base representation
-and 1 for a blade representation.  One needs to keep track of which
-representation is in use since various multivector operations require conversion
-from one representation to the other.</p>
-<div class="admonition warning">
-<p class="first admonition-title">Warning</p>
-<p class="last">When the geometric product of two multivectors is calculated the module looks to
-see if either multivector is in blade representation.  If either is the result of
-the geometric product is converted to a blade representation.  One result of this
-is that if either of the multivectors is a simple vector (which is automatically a
-blade) the result will be in a blade representation.  If <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt> are vectors
-then the result <tt class="docutils literal"><span class="pre">a*b</span></tt> will be <tt class="docutils literal"><span class="pre">(a.b)+a^b</span></tt> or simply <tt class="docutils literal"><span class="pre">a^b</span></tt> if <tt class="docutils literal"><span class="pre">(a.b)</span> <span class="pre">=</span> <span class="pre">0</span></tt>.</p>
-</div>
-</div>
-<div class="section" id="outer-and-inner-products-left-and-right-contractions">
-<h2>Outer and Inner Products, Left and Right Contractions<a class="headerlink" href="#outer-and-inner-products-left-and-right-contractions" title="Permalink to this headline">¶</a></h2>
-<p>In geometric algebra any general multivector <span class="math">\(A\)</span> can be decomposed into
-pure grade multivectors (a linear combination of blades of all the same order)
-so that in a <span class="math">\(n\)</span>-dimensional vector space</p>
-<div class="math">
-\[\begin{equation*}
-A = \sum_{r = 0}^{n}A_{r}
-\end{equation*}\]</div><p>The geometric product of two pure grade multivectors <span class="math">\(A_{r}\)</span> and
-<span class="math">\(B_{s}\)</span> has the form</p>
-<div class="math">
-\[\begin{equation*}
-A_{r}B_{s} = \left\langle A_{r}B_{s} \right\rangle_{\left| r-s \right|} + \left\langle A_{r}B_{s} \right\rangle_{\left| r-s \right|+2} + \cdots + \left\langle A_{r}B_{s} \right\rangle_{r+s}
-\end{equation*}\]</div><p>where <span class="math">\(\left\langle  \right\rangle_{t}\)</span> projects the <span class="math">\(t\)</span> grade components of the
-multivector argument.  The inner and outer products of <span class="math">\(A_{r}\)</span> and
-<span class="math">\(B_{s}\)</span> are then defined to be</p>
-<div class="math">
-\[\begin{equation*}
-A_{r}\cdot B_{s} = \left\langle A_{r}B_{s} \right\rangle_{\left| r-s \right|}
-\end{equation*}\]</div><div class="math">
-\[\begin{equation*}
-A_{r}\wedge B_{s} = \left\langle A_{r}B_{s} \right\rangle_{r+s}
-\end{equation*}\]</div><p>and</p>
-<div class="math">
-\[\begin{equation*}
-A\cdot B = \sum_{r,s}A_{r}\cdot B_{s}
-\end{equation*}\]</div><div class="math">
-\[\begin{equation*}
-A\wedge B = \sum_{r,s}A_{r}\wedge B_{s}
-\end{equation*}\]</div><p>Likewise the right (<span class="math">\(\lfloor\)</span>) and left (<span class="math">\(\rfloor\)</span>) contractions are defined as</p>
-<div class="math">
-\[\begin{equation*}
-A_{r}\lfloor B_{s} = \left \{ \begin{array}{cc}
-  \left\langle A_{r}B_{s} \right\rangle_{r-s} &  r \ge s \\
-                   0                          &  r < s \end{array} \right \}
-\end{equation*}\]</div><div class="math">
-\[\begin{equation*}
-A_{r}\rfloor B_{s} = \left \{ \begin{array}{cc}
-   \left\langle A_{r}B_{s} \right\rangle_{s-r} &  s \ge r \\
-                    0                          &  s < r \end{array} \right \}
-\end{equation*}\]</div><p>and</p>
-<div class="math">
-\[\begin{equation*}
-A\lfloor B = \sum_{r,s}A_{r}\lfloor B_{s}
-\end{equation*}\]</div><div class="math">
-\[\begin{equation*}
-A\rfloor B = \sum_{r,s}A_{r}\rfloor B_{s}
-\end{equation*}\]</div><p>The <tt class="docutils literal"><span class="pre">MV</span></tt> class function for the outer product of the multivectors <tt class="docutils literal"><span class="pre">mv1</span></tt> and
-<tt class="docutils literal"><span class="pre">mv2</span></tt> is</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="nd">@staticmethod</span>
-<span class="k">def</span> <span class="nf">outer_product</span><span class="p">(</span><span class="n">mv1</span><span class="p">,</span><span class="n">mv2</span><span class="p">):</span>
-     <span class="n">product</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-     <span class="n">product</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="mi">1</span>
-     <span class="n">mv1</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-     <span class="n">mv2</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-     <span class="k">for</span> <span class="n">igrade1</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-         <span class="k">if</span> <span class="ow">not</span> <span class="n">isint</span><span class="p">(</span><span class="n">mv1</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade1</span><span class="p">]):</span>
-             <span class="n">pg1</span> <span class="o">=</span> <span class="n">mv1</span><span class="o">.</span><span class="n">project</span><span class="p">(</span><span class="n">igrade1</span><span class="p">)</span>
-             <span class="k">for</span> <span class="n">igrade2</span> <span class="ow">in</span> <span class="n">MV</span><span class="o">.</span><span class="n">n1rg</span><span class="p">:</span>
-                 <span class="n">igrade</span> <span class="o">=</span> <span class="n">igrade1</span><span class="o">+</span><span class="n">igrade2</span>
-                 <span class="k">if</span> <span class="n">igrade</span> <span class="o">&lt;=</span> <span class="n">MV</span><span class="o">.</span><span class="n">n</span><span class="p">:</span>
-                     <span class="k">if</span> <span class="ow">not</span> <span class="n">isint</span><span class="p">(</span><span class="n">mv2</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade2</span><span class="p">]):</span>
-                         <span class="n">pg2</span> <span class="o">=</span> <span class="n">mv2</span><span class="o">.</span><span class="n">project</span><span class="p">(</span><span class="n">igrade2</span><span class="p">)</span>
-                         <span class="n">pg1pg2</span> <span class="o">=</span> <span class="n">pg1</span><span class="o">*</span><span class="n">pg2</span>
-                         <span class="n">product</span><span class="o">.</span><span class="n">add_in_place</span><span class="p">(</span><span class="n">pg1pg2</span><span class="o">.</span><span class="n">project</span><span class="p">(</span><span class="n">igrade</span><span class="p">))</span>
-     <span class="k">return</span><span class="p">(</span><span class="n">product</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>The steps for calculating the outer product are:</p>
-<ol class="arabic simple">
-<li>Convert <tt class="docutils literal"><span class="pre">mv1</span></tt> and <tt class="docutils literal"><span class="pre">mv2</span></tt> to blade representation if they are not already
-in that form.</li>
-<li>Project and loop through each grade <tt class="docutils literal"><span class="pre">mv1.mv[i1]</span></tt> and <tt class="docutils literal"><span class="pre">mv2.mv[i2]</span></tt>.</li>
-<li>Calculate the geometric product <tt class="docutils literal"><span class="pre">pg1*pg2</span></tt>.</li>
-<li>Project the <tt class="docutils literal"><span class="pre">i1+i2</span></tt> grade from <tt class="docutils literal"><span class="pre">pg1*pg2</span></tt>.</li>
-<li>Accumulate the results for each pair of grades in the input multivectors.</li>
-</ol>
-<div class="admonition warning">
-<p class="first admonition-title">Warning</p>
-<p class="last">In the  <tt class="docutils literal"><span class="pre">MV</span></tt> class we have overloaded the <tt class="docutils literal"><span class="pre">^</span></tt> operator to represent the outer
-product so that instead of calling the outer product function we can write <tt class="docutils literal"><span class="pre">mv1^</span> <span class="pre">mv2</span></tt>.
-Due to the precedence rules for python it is <strong>absolutely essential</strong> to enclose outer products
-in parenthesis.</p>
-</div>
-<p>For the inner product of the multivectors <tt class="docutils literal"><span class="pre">mv1</span></tt> and <tt class="docutils literal"><span class="pre">mv2</span></tt> the <tt class="docutils literal"><span class="pre">MV</span></tt> class
-function is</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="nd">@staticmethod</span>
-<span class="k">def</span> <span class="nf">inner_product</span><span class="p">(</span><span class="n">mv1</span><span class="p">,</span><span class="n">mv2</span><span class="p">,</span><span class="n">mode</span><span class="o">=</span><span class="s">&#39;s&#39;</span><span class="p">):</span>
-    <span class="sd">&quot;&quot;&quot;</span>
-<span class="sd">    MV.inner_product(mv1,mv2) calculates the inner</span>
-
-<span class="sd">    mode = &#39;s&#39; - symmetic (Doran &amp; Lasenby)</span>
-<span class="sd">    mode = &#39;l&#39; - left contraction (Dorst)</span>
-<span class="sd">    mode = &#39;r&#39; - right contraction (Dorst)</span>
-<span class="sd">    &quot;&quot;&quot;</span>
-    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv1</span><span class="p">,</span><span class="n">MV</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv2</span><span class="p">,</span><span class="n">MV</span><span class="p">):</span>
-        <span class="n">product</span> <span class="o">=</span> <span class="n">MV</span><span class="p">()</span>
-        <span class="n">product</span><span class="o">.</span><span class="n">bladeflg</span> <span class="o">=</span> <span class="mi">1</span>
-        <span class="n">mv1</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-        <span class="n">mv2</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-        <span class="k">for</span> <span class="n">igrade1</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n1</span><span class="p">):</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv1</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade1</span><span class="p">],</span><span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                <span class="n">pg1</span> <span class="o">=</span> <span class="n">mv1</span><span class="o">.</span><span class="n">project</span><span class="p">(</span><span class="n">igrade1</span><span class="p">)</span>
-                <span class="k">for</span> <span class="n">igrade2</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">MV</span><span class="o">.</span><span class="n">n1</span><span class="p">):</span>
-                    <span class="n">igrade</span> <span class="o">=</span> <span class="n">igrade1</span><span class="o">-</span><span class="n">igrade2</span>
-                    <span class="k">if</span> <span class="n">mode</span> <span class="o">==</span> <span class="s">&#39;s&#39;</span><span class="p">:</span>
-                        <span class="n">igrade</span> <span class="o">=</span> <span class="n">igrade</span><span class="o">.</span><span class="n">__abs__</span><span class="p">()</span>
-                    <span class="k">else</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="n">mode</span> <span class="o">==</span> <span class="s">&#39;l&#39;</span><span class="p">:</span>
-                            <span class="n">igrade</span> <span class="o">=</span> <span class="o">-</span><span class="n">igrade</span>
-                    <span class="k">if</span> <span class="n">igrade</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-                        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv2</span><span class="o">.</span><span class="n">mv</span><span class="p">[</span><span class="n">igrade2</span><span class="p">],</span><span class="n">numpy</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
-                            <span class="n">pg2</span> <span class="o">=</span> <span class="n">mv2</span><span class="o">.</span><span class="n">project</span><span class="p">(</span><span class="n">igrade2</span><span class="p">)</span>
-                            <span class="n">pg1pg2</span> <span class="o">=</span> <span class="n">pg1</span><span class="o">*</span><span class="n">pg2</span>
-                            <span class="n">product</span><span class="o">.</span><span class="n">add_in_place</span><span class="p">(</span><span class="n">pg1pg2</span><span class="o">.</span><span class="n">project</span><span class="p">(</span><span class="n">igrade</span><span class="p">))</span>
-        <span class="k">return</span><span class="p">(</span><span class="n">product</span><span class="p">)</span>
-    <span class="k">else</span><span class="p">:</span>
-        <span class="k">if</span> <span class="n">mode</span> <span class="o">==</span> <span class="s">&#39;s&#39;</span><span class="p">:</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv1</span><span class="p">,</span><span class="n">MV</span><span class="p">):</span>
-                <span class="n">product</span> <span class="o">=</span> <span class="n">mv1</span><span class="o">.</span><span class="n">scalar_mul</span><span class="p">(</span><span class="n">mv2</span><span class="p">)</span>
-            <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">mv2</span><span class="p">,</span><span class="n">MV</span><span class="p">):</span>
-                <span class="n">product</span> <span class="o">=</span> <span class="n">mv2</span><span class="o">.</span><span class="n">scalar_mul</span><span class="p">(</span><span class="n">mv1</span><span class="p">)</span>
-        <span class="k">else</span><span class="p">:</span>
-            <span class="n">product</span> <span class="o">=</span> <span class="bp">None</span>
-    <span class="k">return</span><span class="p">(</span><span class="n">product</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>The inner product is calculated the same way as the outer product except that in
-step 4, <tt class="docutils literal"><span class="pre">i1+i2</span></tt> is replaced by <tt class="docutils literal"><span class="pre">abs(i1-i2)</span></tt> or <tt class="docutils literal"><span class="pre">i1-i2</span></tt> for the right contraction or
-<tt class="docutils literal"><span class="pre">i2-i1</span></tt> for the left contraction.  If <tt class="docutils literal"><span class="pre">i1-i2</span></tt> is less than zero there is no contribution
-to the right contraction.  If <tt class="docutils literal"><span class="pre">i2-i1</span></tt> is less than zero there is no contribution to the
-left contraction.</p>
-<div class="admonition warning">
-<p class="first admonition-title">Warning</p>
-<p class="last">In the <tt class="docutils literal"><span class="pre">MV</span></tt> class we have overloaded the <tt class="docutils literal"><span class="pre">|</span></tt> operator for the inner product,
-<tt class="docutils literal"><span class="pre">&gt;</span></tt> operator for the right contraction, and <tt class="docutils literal"><span class="pre">&lt;</span></tt> operator for the left contraction.
-Instead of calling the inner product function we can write <tt class="docutils literal"><span class="pre">mv1|mv2</span></tt>, <tt class="docutils literal"><span class="pre">mv1&gt;mv2</span></tt>, or
-<tt class="docutils literal"><span class="pre">mv1&lt;mv2</span></tt> respectively for the inner product, right contraction, or left contraction.
-Again, due to the precedence rules for python it is <strong>absolutely essential</strong> to enclose inner
-products and/or contractions in parenthesis.</p>
-</div>
-</div>
-<div class="section" id="reverse-of-multivector">
-<span id="reverse"></span><h2>Reverse of Multivector<a class="headerlink" href="#reverse-of-multivector" title="Permalink to this headline">¶</a></h2>
-<p>If <span class="math">\(A\)</span> is the geometric product of <span class="math">\(r\)</span> vectors</p>
-<div class="math">
-\[\begin{equation*}
-  A = a_{1}\dots a_{r}
-\end{equation*}\]</div><p>then the reverse of <span class="math">\(A\)</span> designated <span class="math">\(A^{\dagger}\)</span> is defined by</p>
-<div class="math">
-\[\begin{equation*}
-  A^{\dagger} \equiv a_{r}\dots a_{1}.
-\end{equation*}\]</div><p>The reverse is simply the product with the order of terms reversed.  The reverse
-of a sum of products is defined as the sum of the reverses so that for a general
-multivector A we have</p>
-<div class="math">
-\[\begin{equation*}
-  A^{\dagger} = \sum_{i=0}^{N} \left\langle A \right\rangle_{i}^{\dagger}
-\end{equation*}\]</div><p>but</p>
-<div class="math" id="equation-2">
-<span id="eq14"></span>\[\begin{equation*}
-  \left\langle A \right\rangle_{i}^{\dagger} = \left( -1\right)^{\frac{i\left( i-1\right)}{2}}\left\langle A \right\rangle_{i}
-\end{equation*}\]</div><p>which is proved by expanding the blade bases in terms of orthogonal vectors and
-showing that equation <a class="reference internal" href="#eq14"><em>2</em></a> holds for the geometric product of orthogonal
-vectors.</p>
-<p>The reverse is important in the theory of rotations in <span class="math">\(n\)</span>-dimensions.  If
-<span class="math">\(R\)</span> is the product of an even number of vectors and <span class="math">\(RR^{\dagger} = 1\)</span>
-then <span class="math">\(RaR^{\dagger}\)</span> is a composition of rotations of the vector <span class="math">\(a\)</span>.
-If <span class="math">\(R\)</span> is the product of two vectors then the plane that <span class="math">\(R\)</span> defines
-is the plane of the rotation.  That is to say that <span class="math">\(RaR^{\dagger}\)</span> rotates the
-component of <span class="math">\(a\)</span> that is projected into the plane defined by <span class="math">\(a\)</span> and
-<span class="math">\(b\)</span> where <span class="math">\(R=ab\)</span>.  <span class="math">\(R\)</span> may be written
-<span class="math">\(R = e^{\frac{\theta}{2}U}\)</span>, where <span class="math">\(\theta\)</span> is the angle of rotation
-and <span class="math">\(u\)</span> is a unit blade <span class="math">\(\left( u^{2} = \pm 1\right)\)</span> that defines the
-plane of rotation.</p>
-</div>
-<div class="section" id="reciprocal-frames">
-<span id="recframe"></span><h2>Reciprocal Frames<a class="headerlink" href="#reciprocal-frames" title="Permalink to this headline">¶</a></h2>
-<p>If we have <span class="math">\(M\)</span> linearly independent vectors (a frame),
-<span class="math">\(a_{1},\dots,a_{M}\)</span>, then the reciprocal frame is
-<span class="math">\(a^{1},\dots,a^{M}\)</span> where <span class="math">\(a_{i}\cdot a^{j} = \delta_{i}^{j}\)</span>,
-<span class="math">\(\delta_{i}^{j}\)</span> is the Kronecker delta (zero if <span class="math">\(i \ne j\)</span> and one
-if <span class="math">\(i = j\)</span>). The reciprocal frame is constructed as follows:</p>
-<div class="math">
-\[\begin{equation*}
-  E_{M} = a_{1}\wedge\dots\wedge a_{M}
-\end{equation*}\]</div><div class="math">
-\[\begin{equation*}
-  E_{M}^{-1} = \displaystyle\frac{E_{M}}{E_{M}^{2}}
-\end{equation*}\]</div><p>Then</p>
-<div class="math">
-\[\begin{equation*}
-  a^{i} = \left( -1\right)^{i-1}\left( a_{1}\wedge\dots\wedge \breve{a}_{i} \wedge\dots\wedge a_{M}\right) E_{M}^{-1}
-\end{equation*}\]</div><p>where <span class="math">\(\breve{a}_{i}\)</span> indicates that <span class="math">\(a_{i}\)</span> is to be deleted from
-the product.  In the standard notation if a vector is denoted with a subscript
-the reciprocal vector is denoted with a superscript. The multivector setup
-function <tt class="docutils literal"><span class="pre">MV.setup(basis,metric,rframe)</span></tt> has the argument <tt class="docutils literal"><span class="pre">rframe</span></tt> with a
-default value of <tt class="docutils literal"><span class="pre">False</span></tt>.  If it is set to <tt class="docutils literal"><span class="pre">True</span></tt> the reciprocal frame of
-the basis vectors is calculated. Additionaly there is the function
-<tt class="docutils literal"><span class="pre">reciprocal_frame(vlst,names='')</span></tt> external to the <tt class="docutils literal"><span class="pre">MV</span></tt> class that will
-calculate the reciprocal frame of a list, <tt class="docutils literal"><span class="pre">vlst</span></tt>, of vectors.  If the argument
-<tt class="docutils literal"><span class="pre">names</span></tt> is set to a space delimited string of names for the vectors the
-reciprocal vectors will be given these names.</p>
-</div>
-<div class="section" id="geometric-derivative">
-<span id="deriv"></span><h2>Geometric Derivative<a class="headerlink" href="#geometric-derivative" title="Permalink to this headline">¶</a></h2>
-<p>If <span class="math">\(F\)</span> is a multivector field that is a function of a vector
-<span class="math">\(x = x^{i}\boldsymbol{\gamma}_{i}\)</span> (we are using the summation convention that
-pairs of subscripts and superscripts are summed over the dimension of the vector
-space) then the geometric derivative <span class="math">\(\nabla F\)</span> is given by (in this
-section the summation convention is used):</p>
-<div class="math">
-\[\begin{equation*}
-  \nabla F = \boldsymbol{\gamma}^{i}\displaystyle\frac{\partial F}{\partial x^{i}}
-\end{equation*}\]</div><p>If <span class="math">\(F_{R}\)</span> is a grade-<span class="math">\(R\)</span> multivector and
-<span class="math">\(F_{R} = F_{R}^{i_{1}\dots i_{R}}\boldsymbol{\gamma}_{i_{1}}\wedge\dots\wedge \boldsymbol{\gamma}_{i_{R}}\)</span>
-then</p>
-<div class="math">
-\[\begin{equation*}
-  \nabla F_{R} = \displaystyle\frac{\partial F_{R}^{i_{1}\dots i_{R}}}{\partial x^{j}}\boldsymbol{\gamma}^{j}\left(\boldsymbol{\gamma}_{i_{1}}\wedge
-                 \dots\wedge \boldsymbol{\gamma}_{i_{R}} \right)
-\end{equation*}\]</div><p>Note that
-<span class="math">\(\boldsymbol{\gamma}^{j}\left(\boldsymbol{\gamma}_{i_{1}}\wedge\dots\wedge \boldsymbol{\gamma}_{i_{R}} \right)\)</span>
-can only contain grades <span class="math">\(R-1\)</span> and <span class="math">\(R+1\)</span> so that <span class="math">\(\nabla F_{R}\)</span>
-also can only contain those grades. For a grade-<span class="math">\(R\)</span> multivector
-<span class="math">\(F_{R}\)</span> the inner (div) and outer (curl) derivatives are defined as</p>
-<div class="math">
-\[\begin{equation*}
-\nabla\cdot F_{R} = \left < \nabla F_{R}\right >_{R-1}
-\end{equation*}\]</div><p>and</p>
-<div class="math">
-\[\begin{equation*}
-\nabla\wedge F_{R} = \left < \nabla F_{R}\right >_{R+1}
-\end{equation*}\]</div><p>For a general multivector function <span class="math">\(F\)</span> the inner and outer derivatives are
-just the sum of the inner and outer dervatives of each grade of the multivector
-function.</p>
-<p>Curvilinear coordinates are derived from a vector function
-<span class="math">\(x(\boldsymbol{\theta})\)</span> where
-<span class="math">\(\boldsymbol{\theta} = \left(\theta_{1},\dots,\theta_{N}\right)\)</span> where the number of
-coordinates is equal to the dimension of the vector space.  In the case of
-3-dimensional spherical coordinates <span class="math">\(\boldsymbol{\theta} = \left( r,\theta,\phi \right)\)</span>
-and the coordinate generating function <span class="math">\(x(\boldsymbol{\theta})\)</span> is</p>
-<div class="math">
-\[\begin{equation*}
-x =  r \cos\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+ r \sin\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{y}}}+ r \cos\left({\theta}\right){\boldsymbol{{\gamma}_{z}}}
-\end{equation*}\]</div><p>A coordinate frame is derived from <span class="math">\(x\)</span> by
-<span class="math">\(\boldsymbol{e}_{i} = \displaystyle\frac{\partial x}{\partial \theta^{i}}\)</span>.  The following show the frame for
-spherical coordinates.</p>
-<div class="math">
-\[\begin{equation*}
-\boldsymbol{e}_{r} = \cos\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+\sin\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{y}}}+\cos\left({\theta}\right){\boldsymbol{{\gamma}_{z}}}
-\end{equation*}\]</div><div class="math">
-\[\begin{equation*}
-\boldsymbol{e}_{{\theta}} = \cos\left({\phi}\right) \cos\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+r \cos\left({\theta}\right) \sin\left({\phi}\right){\boldsymbol{{\gamma}_{y}}} - r \sin\left({\theta}\right){\boldsymbol{{\gamma}_{z}}}
-\end{equation*}\]</div><div class="math">
-\[\begin{equation*}
-\boldsymbol{e}_{{\phi}} = - r \sin\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+r \cos\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{y}}}
-\end{equation*}\]</div><p>The coordinate frame generated in this manner is not necessarily normalized so
-define a normalized frame by</p>
-<div class="math">
-\[\begin{equation*}
-\boldsymbol{\hat{e}}_{i} = \displaystyle\frac{\boldsymbol{e}_{i}}{\sqrt{\left| \boldsymbol{e}_{i}^{2} \right|}} = \displaystyle\frac{\boldsymbol{e}_{i}}{\left| \boldsymbol{e}_{i} \right|}
-\end{equation*}\]</div><p>This works for all <span class="math">\(\boldsymbol{e}_{i}^{2} \neq 0\)</span> since we have defined
-<span class="math">\(\left| \boldsymbol{e}_{i} \right| = \sqrt{\left| \boldsymbol{e}_{i}^{2} \right|}\)</span>.   For spherical
-coordinates the normalized frame vectors are</p>
-<div class="math">
-\[\begin{equation*}
-\boldsymbol{\hat{e}}_{r} = \cos\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+\sin\left({\phi}\right) \sin\left({\theta}\right){\boldsymbol{{\gamma}_{y}}}+\cos\left({\theta}\right){\boldsymbol{{\gamma}_{z}}}
-\end{equation*}\]</div><div class="math">
-\[\begin{equation*}
-\boldsymbol{\hat{e}}_{{\theta}} = \cos\left({\phi}\right) \cos\left({\theta}\right){\boldsymbol{{\gamma}_{x}}}+\cos\left({\theta}\right) \sin\left({\phi}\right){\boldsymbol{{\gamma}_{y}}}- \sin\left({\theta}\right){\boldsymbol{{\gamma}_{z}}}
-\end{equation*}\]</div><div class="math">
-\[\begin{equation*}
-\boldsymbol{\hat{e}}_{{\phi}} = - \sin\left({\phi}\right){\boldsymbol{{\gamma}_{x}}}+\cos\left({\phi}\right){\boldsymbol{{\gamma}_{y}}}
-\end{equation*}\]</div><p>The geometric derivative in curvilinear coordinates is given by</p>
-<div class="math">
-\[\begin{align*}
-  \nabla F_{R} & = \boldsymbol{\gamma}^{i}\displaystyle\frac{\partial }{\partial x^{i}}\left( F_{R}^{i_{1}\dots i_{R}}
-                   \boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right)  \\
-               & = \boldsymbol{e^{j}}\displaystyle\frac{\partial }{\partial \theta^{j}}\left( F_{R}^{i_{1}\dots i_{R}}
-                   \boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right)  \\
-               & = \left(\displaystyle\frac{\partial }{\partial \theta^{j}} F_{R}^{i_{1}\dots i_{R}}\right)
-                   \boldsymbol{e^{j}}\left(\boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right)+
-                   F_{R}^{i_{1}\dots i_{R}}\boldsymbol{e^{j}}
-                   \displaystyle\frac{\partial }{\partial \theta^{j}}\left(\boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right) \\
-               & = \left(\displaystyle\frac{\partial }{\partial \theta^{j}} F_{R}^{i_{1}\dots i_{R}}\right)
-                   \boldsymbol{e^{j}}\left(\boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right)+
-                   F_{R}^{i_{1}\dots i_{R}}C\left\{ \boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right\}
-\end{align*}\]</div><p>where</p>
-<div class="math">
-\[\begin{equation*}
-C\left\{ \boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right\} = \boldsymbol{e^{j}}\displaystyle\frac{\partial }{\partial \theta^{j}}
-                                                                                             \left(\boldsymbol{\hat{e}}_{i_{1}}\wedge\dots\wedge\boldsymbol{\hat{e}}_{i_{R}}\right)
-\end{equation*}\]</div><p>are the connection multivectors for the curvilinear coordinate system. For a
-spherical coordinate system they are</p>
-<div class="math">
-\[\begin{equation*}
-C\left\{\boldsymbol{\hat{e}}_{r}\right\} =  \frac{2}{r}
-\end{equation*}\]</div><div class="math">
-\[\begin{equation*}
-C\left\{\boldsymbol{\hat{e}}_{\theta}\right\} = \frac{\cos\left({\theta}\right)}{r \sin\left({\theta}\right)}
-                                              + \frac{1}{r}\boldsymbol{\hat{e}}_{r}\wedge\boldsymbol{\hat{e}}_{\theta}
-\end{equation*}\]</div><div class="math">
-\[\begin{equation*}
-C\left\{\boldsymbol{\hat{e}}_{\phi}\right\} = \frac{1}{r}{\boldsymbol{\boldsymbol{\hat{e}}_{r}}}\wedge\boldsymbol{\hat{e}}_{{\phi}}+ \frac{\cos\left({\theta}\right)}{r \sin\left({\theta}\right)}\boldsymbol{\hat{e}}_{{\theta}}\wedge\boldsymbol{\hat{e}}_{{\phi}}
-\end{equation*}\]</div><div class="math">
-\[\begin{equation*}
-C\left\{\hat{e}_{r}\wedge\hat{e}_{\theta}\right\} = - \frac{\cos\left({\theta}\right)}{r \sin\left({\theta}\right)}
-                                                      \boldsymbol{\hat{e}}_{r}+\frac{1}{r}\boldsymbol{\hat{e}}_{{\theta}}
-\end{equation*}\]</div><div class="math">
-\[\begin{equation*}
-C\left\{\boldsymbol{\hat{e}}_{r}\wedge\boldsymbol{\hat{e}}_{\phi}\right\} = \frac{1}{r}\boldsymbol{\hat{e}}_{{\phi}}
-                                                                          - \frac{\cos\left({\theta}\right)}{r \sin\left({\theta}\right)}\boldsymbol{\hat{e}}_{r}\wedge\boldsymbol{\hat{e}}_{{\theta}}\wedge\boldsymbol{\hat{e}}_{{\phi}}
-\end{equation*}\]</div><div class="math">
-\[\begin{equation*}
-C\left\{\boldsymbol{\hat{e}}_{\theta}\wedge\boldsymbol{\hat{e}}_{\phi}\right\} = \frac{2}{r}\boldsymbol{\hat{e}}_{r}\wedge
-                                                                                 \boldsymbol{\hat{e}}_{\theta}\wedge\boldsymbol{\hat{e}}_{\phi}
-\end{equation*}\]</div><div class="math">
-\[\begin{equation*}
-C\left\{\boldsymbol{\hat{e}}_r\wedge\boldsymbol{\hat{e}}_{\theta}\wedge\boldsymbol{\hat{e}}_{\phi}\right\} = 0
-\end{equation*}\]</div></div>
-<div class="section" id="module-components">
-<h2>Module Components<a class="headerlink" href="#module-components" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="initializing-multivector-class">
-<h3>Initializing Multivector Class<a class="headerlink" href="#initializing-multivector-class" title="Permalink to this headline">¶</a></h3>
-<p>The multivector class is initialized with:</p>
-<dl class="function">
-<dt id="MV.setup">
-<tt class="descclassname">MV.</tt><tt class="descname">setup</tt><big>(</big><em>basis</em>, <em>metric=''</em>, <em>rframe=False</em>, <em>coords=None</em>, <em>debug=False</em>, <em>offset=0</em><big>)</big><a class="headerlink" href="#MV.setup" title="Permalink to this definition">¶</a></dt>
-<dd><p>The <tt class="docutils literal"><span class="pre">basis</span></tt> and <tt class="docutils literal"><span class="pre">metric</span></tt> parameters were described in section <a class="reference internal" href="#vbm"><em>Vector Basis and Metric</em></a>. If
-<tt class="docutils literal"><span class="pre">rframe=True</span></tt> the reciprocal frame of the symbolic bases vectors is calculated.
-If <tt class="docutils literal"><span class="pre">debug=True</span></tt> the data structure required to initialize the <a class="reference internal" href="#MV" title="MV"><tt class="xref py py-class docutils literal"><span class="pre">MV</span></tt></a> class
-are printer out. <tt class="docutils literal"><span class="pre">coords</span></tt> is a list of <a class="reference internal" href="../../matrices/immutablematrices.html#module-sympy" title="sympy"><tt class="xref py py-class docutils literal"><span class="pre">sympy</span></tt></a> symbols equal in length to
-the number of basis vectors.  These symbols are used as the arguments of a
-multivector field as a function of position and for calculating the derivatives
-of a multivector field (if <tt class="docutils literal"><span class="pre">coords</span></tt> is defined then <tt class="docutils literal"><span class="pre">rframe</span></tt> is automatically
-set equal to <tt class="docutils literal"><span class="pre">True</span></tt>). <tt class="docutils literal"><span class="pre">offset</span></tt> is an integer that is added to the multivector
-coefficient index. For example if one wishes to start labeling vector coefficient
-indexes at one instead of zero then set <tt class="docutils literal"><span class="pre">offset=1</span></tt>.  Additionally, <a class="reference internal" href="#MV.setup" title="MV.setup"><tt class="xref py py-func docutils literal"><span class="pre">MV.setup()</span></tt></a>
-calculates the pseudo scalar, <span class="math">\(I\)</span> and its inverse, <span class="math">\(I^{-1}\)</span> and makes
-them available to the programmer as <tt class="docutils literal"><span class="pre">MV.I</span></tt> and <tt class="docutils literal"><span class="pre">MV.Iinv</span></tt>.</p>
-</dd></dl>
-
-<p>After <a class="reference internal" href="#MV.setup" title="MV.setup"><tt class="xref py py-func docutils literal"><span class="pre">MV.setup()</span></tt></a> is run one can reinialize the <a class="reference internal" href="#MV" title="MV"><tt class="xref py py-class docutils literal"><span class="pre">MV</span></tt></a> class with
-curvilinear coordinates using:</p>
-<dl class="function">
-<dt id="MV.rebase">
-<tt class="descclassname">MV.</tt><tt class="descname">rebase</tt><big>(</big><em>x</em>, <em>coords</em>, <em>base_name</em>, <em>debug=False</em>, <em>debug_level=0</em><big>)</big><a class="headerlink" href="#MV.rebase" title="Permalink to this definition">¶</a></dt>
-<dd><p>A typical usage of <tt class="docutils literal"><span class="pre">MV.rebase</span></tt> for generating spherical curvilinear coordinate
-is:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">metric</span> <span class="o">=</span> <span class="s">&#39;1 0 0,0 1 0,0 0 1&#39;</span>
-<span class="n">gamma_x</span><span class="p">,</span><span class="n">gamma_y</span><span class="p">,</span><span class="n">gamma_z</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">setup</span><span class="p">(</span><span class="s">&#39;gamma_x gamma_y gamma_z&#39;</span><span class="p">,</span><span class="n">metric</span><span class="p">,</span><span class="bp">True</span><span class="p">)</span>
-<span class="n">coords</span> <span class="o">=</span> <span class="n">r</span><span class="p">,</span><span class="n">theta</span><span class="p">,</span><span class="n">phi</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">symbols</span><span class="p">(</span><span class="s">&#39;r theta phi&#39;</span><span class="p">)</span>
-<span class="n">x</span> <span class="o">=</span> <span class="n">r</span><span class="o">*</span><span class="p">(</span><span class="n">sympy</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span><span class="o">*</span><span class="n">gamma_z</span><span class="o">+</span><span class="n">sympy</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span><span class="o">*</span>\
-    <span class="p">(</span><span class="n">sympy</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">phi</span><span class="p">)</span><span class="o">*</span><span class="n">gamma_x</span><span class="o">+</span><span class="n">sympy</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">phi</span><span class="p">)</span><span class="o">*</span><span class="n">gamma_y</span><span class="p">))</span>
-<span class="n">x</span><span class="o">.</span><span class="n">set_name</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="n">MV</span><span class="o">.</span><span class="n">rebase</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">coords</span><span class="p">,</span><span class="s">&#39;e&#39;</span><span class="p">,</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>The input parameters for <tt class="docutils literal"><span class="pre">MV.rebase</span></tt> are</p>
-</dd></dl>
-
-<ul>
-<li><p class="first"><tt class="docutils literal"><span class="pre">x</span></tt>: Vector function of coordinates (derivatives define curvilinear basis)</p>
-</li>
-<li><p class="first"><tt class="docutils literal"><span class="pre">coords</span></tt>: List of sympy symbols for curvilinear coordinates</p>
-</li>
-<li><p class="first"><tt class="docutils literal"><span class="pre">debug</span></tt>: If <tt class="docutils literal"><span class="pre">True</span></tt> printout (LaTeX) all quantities required for derivative calculation</p>
-</li>
-<li><p class="first"><tt class="docutils literal"><span class="pre">debug_level</span></tt>: Set to 0,1,2, or 3 to stop curvilinear calculation before all quatities are
-calculated.  This is done when debugging new curvilinear coordinate systems since simplification
-of expressions is not sufficiently automated to insure success of process of any coordinate system
-defined by vector function <tt class="docutils literal"><span class="pre">x</span></tt></p>
-<p>To date <tt class="docutils literal"><span class="pre">MV.rebase</span></tt> works for cylindrical and spherical coordinate systems in
-any number of dimensions  (until the execution time becomes too long).  To make
-it work for these systems required creating some hacks for expression
-simplification since both trigsimp and simplify were not general enough to
-perform the required simplification.</p>
-</li>
-</ul>
-<dl class="function">
-<dt id="MV.set_str_format">
-<tt class="descclassname">MV.</tt><tt class="descname">set_str_format</tt><big>(</big><em>str_mode=0</em><big>)</big><a class="headerlink" href="#MV.set_str_format" title="Permalink to this definition">¶</a></dt>
-<dd><p>If <tt class="docutils literal"><span class="pre">str_mode=0</span></tt> the string representation of the multivector contains no
-newline characters (prints on one line).   If <tt class="docutils literal"><span class="pre">str_mode=1</span></tt> the string
-representation of the multivector places a newline after each grade of the
-multivector  (prints one grade per line).  If <tt class="docutils literal"><span class="pre">str_mode=2</span></tt> the string
-representation of the multivector places a newline after each base of the
-multivector  (prints one base per line). In both cases bases with zero
-coefficients are not printed.</p>
-<blockquote>
-<div><div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p>This function directly affects the way multivectors are printed with the
-print command since it interacts with the <tt class="xref py py-func docutils literal"><span class="pre">__str__()</span></tt> function for the
-multivector class which is used by the <tt class="docutils literal"><span class="pre">print</span></tt> command.</p>
-<table border="1" class="last docutils">
-<colgroup>
-<col width="17%" />
-<col width="83%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head"><tt class="docutils literal"><span class="pre">str_mode</span></tt></th>
-<th class="head">Effect on print</th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-even"><td>0</td>
-<td>One multivector per line</td>
-</tr>
-<tr class="row-odd"><td>1</td>
-<td>One grade per line</td>
-</tr>
-<tr class="row-even"><td>2</td>
-<td>One base per line</td>
-</tr>
-</tbody>
-</table>
-</div>
-</div></blockquote>
-</dd></dl>
-
-</div>
-<div class="section" id="instantiating-a-multivector">
-<h3>Instantiating a Multivector<a class="headerlink" href="#instantiating-a-multivector" title="Permalink to this headline">¶</a></h3>
-<p>Now that grades and bases have been described we can show all the ways that a
-multivector can be instantiated. As an example assume that the multivector space
-is initialized with <tt class="docutils literal"><span class="pre">e1,e2,e3</span> <span class="pre">=</span> <span class="pre">MV.setup('e1</span> <span class="pre">e2</span> <span class="pre">e3')</span></tt>.</p>
-<p>So that multivectors
-could be instantiated with statements such as (<tt class="docutils literal"><span class="pre">a1</span></tt>, <tt class="docutils literal"><span class="pre">a2</span></tt>, and <tt class="docutils literal"><span class="pre">a3</span></tt> are
-<tt class="docutils literal"><span class="pre">sympy</span></tt> symbols):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">x</span> <span class="o">=</span> <span class="n">a1</span><span class="o">*</span><span class="n">e1</span><span class="o">+</span><span class="n">a2</span><span class="o">*</span><span class="n">e2</span><span class="o">+</span><span class="n">a3</span><span class="o">*</span><span class="n">e3</span>
-<span class="n">y</span> <span class="o">=</span> <span class="n">x</span><span class="o">*</span><span class="n">e1</span><span class="o">*</span><span class="n">e2</span>
-<span class="n">z</span> <span class="o">=</span> <span class="n">x</span><span class="o">|</span><span class="n">y</span>
-<span class="n">w</span> <span class="o">=</span> <span class="n">x</span><span class="o">^</span><span class="n">y</span>
-</pre></div>
-</div>
-<p>or with the multivector class constructor:</p>
-<dl class="class">
-<dt id="MV">
-<em class="property">class </em><tt class="descname">MV</tt><big>(</big><em>value=''</em>, <em>mvtype=''</em>, <em>mvname=''</em>, <em>fct=False</em><big>)</big><a class="headerlink" href="#MV" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">mvname</span></tt> is a string that defines the name of the multivector for output
-purposes. <tt class="docutils literal"><span class="pre">value</span></tt> and  <tt class="docutils literal"><span class="pre">type</span></tt> are defined by the following table and <tt class="docutils literal"><span class="pre">fct</span></tt> is a
-switch that will convert the symbolic coefficients of a multivector to functions
-if coordinate variables have been defined when <a class="reference internal" href="#MV.setup" title="MV.setup"><tt class="xref py py-func docutils literal"><span class="pre">MV.setup()</span></tt></a> is called:</p>
-<blockquote>
-<div><table border="1" class="docutils">
-<colgroup>
-<col width="14%" />
-<col width="43%" />
-<col width="43%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head"><tt class="docutils literal"><span class="pre">mvtype</span></tt></th>
-<th class="head"><tt class="docutils literal"><span class="pre">value</span></tt></th>
-<th class="head">Result</th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-even"><td>default</td>
-<td>default</td>
-<td>Zero multivector</td>
-</tr>
-<tr class="row-odd"><td>&#8216;basisvector&#8217;</td>
-<td>int i</td>
-<td><span class="math">\(\mbox{i}^{th}\)</span> basis vector</td>
-</tr>
-<tr class="row-even"><td>&#8216;basisbivector&#8217;</td>
-<td>int i</td>
-<td><span class="math">\(\mbox{i}^{th}\)</span> basis bivector</td>
-</tr>
-<tr class="row-odd"><td>&#8216;scalar&#8217;</td>
-<td>symbol x</td>
-<td>Symbolic scalar of value x</td>
-</tr>
-<tr class="row-even"><td>&nbsp;</td>
-<td>string s</td>
-<td>Symbolic scalar of value Symbol(s)</td>
-</tr>
-<tr class="row-odd"><td>&nbsp;</td>
-<td>int i</td>
-<td>SymPy integer of value i</td>
-</tr>
-<tr class="row-even"><td>&#8216;grade&#8217;</td>
-<td>[int r, 1-D symbol array A]</td>
-<td>X.grade(r) = A</td>
-</tr>
-<tr class="row-odd"><td>&nbsp;</td>
-<td>[int r, string s]</td>
-<td>Symbolic grade r multivector</td>
-</tr>
-<tr class="row-even"><td>&#8216;vector&#8217;</td>
-<td>1-D symbol array A</td>
-<td>X.grade(1) = A</td>
-</tr>
-<tr class="row-odd"><td>&nbsp;</td>
-<td>string s</td>
-<td>Symbolic vector</td>
-</tr>
-<tr class="row-even"><td>&#8216;grade2&#8217;</td>
-<td>1-D symbol array A</td>
-<td>X.grade(2) = A</td>
-</tr>
-<tr class="row-odd"><td>&#8216;pseudo&#8217;</td>
-<td>symbol x</td>
-<td>X.grade(n) = x</td>
-</tr>
-<tr class="row-even"><td>&#8216;spinor&#8217;</td>
-<td>string s</td>
-<td>Symbolic even multivector</td>
-</tr>
-<tr class="row-odd"><td>&nbsp;</td>
-<td>&nbsp;</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>default</td>
-<td>string s</td>
-<td>Symbolic general multivector</td>
-</tr>
-</tbody>
-</table>
-</div></blockquote>
-<p>If the <tt class="docutils literal"><span class="pre">value</span></tt> argument has the option of being a string s then a general symbolic
-multivector will be constructed constrained by the value of <tt class="docutils literal"><span class="pre">mvtype</span></tt>.  The string
-s will be the base name of the multivector symbolic coefficients.  If <tt class="docutils literal"><span class="pre">coords</span></tt> is
-not defined in <a class="reference internal" href="#MV.setup" title="MV.setup"><tt class="xref py py-func docutils literal"><span class="pre">MV.setup()</span></tt></a> the indices of the multivector bases are appended to
-the base name with a double underscore (superscript notation).  If <tt class="docutils literal"><span class="pre">coords</span></tt> is defined
-the coordinate names will replace the indices in the coefficient names.  For example if
-the base string is <em>A</em> and the coordinates <em>(x,y,z)</em> then the
-coefficients of a spinor in 3d space would be <em>A</em>, <em>A__xy</em>, <em>A__xz</em>, and <em>A__yz</em>.  If
-the <tt class="xref py py-mod docutils literal"><span class="pre">latex_ex</span></tt> is used to print the multivector the coefficients would print as
-<span class="math">\(A\)</span>, <span class="math">\(A^{xy}\)</span>, <span class="math">\(A^{xz}\)</span>, and <span class="math">\(A^{yz}\)</span>.</p>
-<p>If the <tt class="docutils literal"><span class="pre">fct</span></tt> argrument of <a class="reference internal" href="#MV" title="MV"><tt class="xref py py-func docutils literal"><span class="pre">MV()</span></tt></a> is set to <tt class="docutils literal"><span class="pre">True</span></tt> and the <tt class="docutils literal"><span class="pre">coords</span></tt> argument in
-<a class="reference internal" href="#MV.setup" title="MV.setup"><tt class="xref py py-func docutils literal"><span class="pre">MV.setup()</span></tt></a> is defined the symbolic coefficients of the multivector are functions
-of the coordinates.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="basic-multivector-class-functions">
-<h3>Basic Multivector Class Functions<a class="headerlink" href="#basic-multivector-class-functions" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="__call__">
-<tt class="descname">__call__</tt><big>(</big><em>self</em>, <em>igrade=0</em>, <em>ibase=0</em><big>)</big><a class="headerlink" href="#__call__" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">__call__</span></tt> returns the the <tt class="docutils literal"><span class="pre">igrade</span></tt>, <tt class="docutils literal"><span class="pre">ibase</span></tt> coefficient of the
-multivector.  The defaults return the scalar component of the multivector.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="convert_to_blades">
-<tt class="descname">convert_to_blades</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#convert_to_blades" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert multivector from the base representation to the blade representation.
-If multivector is already in blade representation nothing is done.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="convert_from_blades">
-<tt class="descname">convert_from_blades</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#convert_from_blades" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert multivector from the blade representation to the base representation.
-If multivector is already in base representation nothing is done.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="project">
-<tt class="descname">project</tt><big>(</big><em>self</em>, <em>r</em><big>)</big><a class="headerlink" href="#project" title="Permalink to this definition">¶</a></dt>
-<dd><p>If r is a integer return the grade-<span class="math">\(r\)</span> components of the multivector. If
-r is a multivector return the grades of the multivector that correspond to the
-non-zero grades of r. For example if one is projecting a general multivector and
-r is a spinor, <tt class="docutils literal"><span class="pre">A.project(r)</span></tt> will return only the even grades of the multivector
-A since a spinor only has even grades that are non-zero.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="even">
-<tt class="descname">even</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#even" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the even grade components of the multivector.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="odd">
-<tt class="descname">odd</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#odd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the odd grade components of the multivector.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="rev">
-<tt class="descname">rev</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#rev" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the reverse of the multivector.  See section <a class="reference internal" href="#reverse"><em>Reverse of Multivector</em></a>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="is_pure">
-<tt class="descname">is_pure</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#is_pure" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return true if multivector is pure grade (all grades but one are zero).</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="diff">
-<tt class="descname">diff</tt><big>(</big><em>self</em>, <em>x</em><big>)</big><a class="headerlink" href="#diff" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the partial derivative of the multivector function with respect to
-variable <span class="math">\(x\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="grad">
-<tt class="descname">grad</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#grad" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the geometric derivative of the multivector function.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="grad_ext">
-<tt class="descname">grad_ext</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#grad_ext" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the outer (curl) derivative of the multivector function. Equivalent to
-<tt class="xref py py-func docutils literal"><span class="pre">curl()</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="grad_int">
-<tt class="descname">grad_int</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#grad_int" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the inner (div) derivative of the multivector function. Equivalent to
-<tt class="xref py py-func docutils literal"><span class="pre">div()</span></tt>.</p>
-</dd></dl>
-
-<div class="admonition warning">
-<p class="first admonition-title">Warning</p>
-<p class="last">If <em>A</em> is a vector field in three dimensions <span class="math">\(\nabla\cdot {\bf A}\)</span> = A.grad_int() = A.div(), but
-<span class="math">\(\nabla\times {\bf A}\)</span> = -MV.I*A.grad_ext() = -MV.I*A.curl(). Note that grad_int() lowers the grade
-of all blades by one grade and grad_ext() raises the grade of all blades by one.</p>
-</div>
-<dl class="function">
-<dt id="set_coef">
-<tt class="descname">set_coef</tt><big>(</big><em>self</em>, <em>grade</em>, <em>base</em>, <em>value</em><big>)</big><a class="headerlink" href="#set_coef" title="Permalink to this definition">¶</a></dt>
-<dd><p>Set the multivector coefficient of index <tt class="docutils literal"><span class="pre">(grade,base)</span></tt> to <tt class="docutils literal"><span class="pre">value</span></tt>.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="sympy-functions-applied-inplace-to-multivector-coefficients">
-<h3>SymPy Functions Applied Inplace to Multivector Coefficients<a class="headerlink" href="#sympy-functions-applied-inplace-to-multivector-coefficients" title="Permalink to this headline">¶</a></h3>
-<p>All the following fuctions belong to the <a class="reference internal" href="#MV" title="MV"><tt class="xref py py-class docutils literal"><span class="pre">MV</span></tt></a> class and apply the
-corresponding <a class="reference internal" href="../../matrices/immutablematrices.html#module-sympy" title="sympy"><tt class="xref py py-mod docutils literal"><span class="pre">sympy</span></tt></a> function to each component of a multivector.  All
-these functions perform the operations inplace (<tt class="docutils literal"><span class="pre">None</span></tt> is returned)  on each
-coefficient.  For example if you wished to simplify all the components of the
-multivector <tt class="docutils literal"><span class="pre">A</span></tt> you would invoke <tt class="docutils literal"><span class="pre">A.simplify()</span></tt>.  The argument list for each
-function is the same as for the corresponding <a class="reference internal" href="../../matrices/immutablematrices.html#module-sympy" title="sympy"><tt class="xref py py-mod docutils literal"><span class="pre">sympy</span></tt></a> function.  The only
-function that differs in its action from the <a class="reference internal" href="../../matrices/immutablematrices.html#module-sympy" title="sympy"><tt class="xref py py-mod docutils literal"><span class="pre">sympy</span></tt></a> version is
-<a class="reference internal" href="#trigsimp" title="trigsimp"><tt class="xref py py-func docutils literal"><span class="pre">trigsimp()</span></tt></a> which is applied using the recursive option.</p>
-<dl class="function">
-<dt id="collect">
-<tt class="descname">collect</tt><big>(</big><em>self</em>, <em>lst</em><big>)</big><a class="headerlink" href="#collect" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="sqrfree">
-<tt class="descname">sqrfree</tt><big>(</big><em>self</em>, <em>lst</em><big>)</big><a class="headerlink" href="#sqrfree" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="subs">
-<tt class="descname">subs</tt><big>(</big><em>self</em>, <em>*args</em><big>)</big><a class="headerlink" href="#subs" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="simplify">
-<tt class="descname">simplify</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#simplify" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="trigsimp">
-<tt class="descname">trigsimp</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#trigsimp" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="cancel">
-<tt class="descname">cancel</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#cancel" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="expand">
-<tt class="descname">expand</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#expand" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="helper-functions">
-<h3>Helper Functions<a class="headerlink" href="#helper-functions" title="Permalink to this headline">¶</a></h3>
-<p>These are functions in <tt class="xref py py-mod docutils literal"><span class="pre">GA</span></tt>, but not in the multivector (<a class="reference internal" href="#MV" title="MV"><tt class="xref py py-class docutils literal"><span class="pre">MV</span></tt></a>)
-class.</p>
-<dl class="function">
-<dt id="set_names">
-<tt class="descname">set_names</tt><big>(</big><em>var_lst</em>, <em>var_str</em><big>)</big><a class="headerlink" href="#set_names" title="Permalink to this definition">¶</a></dt>
-<dd><p><a class="reference internal" href="#set_names" title="set_names"><tt class="xref py py-func docutils literal"><span class="pre">set_names()</span></tt></a> allows one to name a list, <tt class="docutils literal"><span class="pre">var_lst</span></tt>, of multivectors enmass.
-The names are in <tt class="docutils literal"><span class="pre">var_str</span></tt>, a blank separated string of names.  An error is
-generated if the number of name is not equal to the length of <tt class="docutils literal"><span class="pre">var_lst</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="reciprocal_frame">
-<tt class="descname">reciprocal_frame</tt><big>(</big><em>vlst</em>, <em>names=''</em><big>)</big><a class="headerlink" href="#reciprocal_frame" title="Permalink to this definition">¶</a></dt>
-<dd><p><a class="reference internal" href="#reciprocal_frame" title="reciprocal_frame"><tt class="xref py py-func docutils literal"><span class="pre">reciprocal_frame()</span></tt></a> implements the proceedure described in section
-<a class="reference internal" href="#recframe"><em>Reciprocal Frames</em></a>.  <tt class="docutils literal"><span class="pre">vlst</span></tt> is a list of independent vectors that you wish the
-reciprocal frame calculated for. <tt class="docutils literal"><span class="pre">names</span></tt> is a blank separated string of names
-for the reciprocal vectors if names are required by you application.  The
-function returns a list containing the reciprocal vectors.</p>
-</dd></dl>
-
-<dl class="function">
-<dt>
-<tt class="descname">trigsimp</tt><big>(</big><em>f</em><big>)</big></dt>
-<dd><p>In general <tt class="xref py py-func docutils literal"><span class="pre">sympy.trigsimp()</span></tt> will not catch all the trigonometric
-simplifications in an <a class="reference internal" href="../../matrices/immutablematrices.html#module-sympy" title="sympy"><tt class="xref py py-mod docutils literal"><span class="pre">sympy</span></tt></a> expression, but by using the recursive
-option, more of them are caught. So <a class="reference internal" href="#trigsimp" title="trigsimp"><tt class="xref py py-func docutils literal"><span class="pre">trigsimp()</span></tt></a> uses this option by
-default.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="S">
-<tt class="descname">S</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#S" title="Permalink to this definition">¶</a></dt>
-<dd><p><a class="reference internal" href="#S" title="S"><tt class="xref py py-func docutils literal"><span class="pre">S()</span></tt></a> instanciates a scaler multivector of value <tt class="docutils literal"><span class="pre">x</span></tt>, where <tt class="docutils literal"><span class="pre">x</span></tt> can be a
-<a class="reference internal" href="../../matrices/immutablematrices.html#module-sympy" title="sympy"><tt class="xref py py-mod docutils literal"><span class="pre">sympy</span></tt></a> variable or integer.  This is just a shorthand method for
-constructing scalar multivectors and can be used when there is any ambiguity
-in a multivector expression as to whether a symbol or constant should be
-treated as a scalar multivector or not.</p>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="examples">
-<h2>Examples<a class="headerlink" href="#examples" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="algebra">
-<h3>Algebra<a class="headerlink" href="#algebra" title="Permalink to this headline">¶</a></h3>
-<p>The following examples of geometric algebra (not calculus) are all in the file
-<strong class="program">testsymbolicGA.py</strong> which is included in the sympy distribution
-examples under the galbebra directory.  The section of code in the program for
-each example is show with the respective output following the code section.</p>
-<div class="section" id="example-header">
-<h4>Example Header<a class="headerlink" href="#example-header" title="Permalink to this headline">¶</a></h4>
-<p>This is the header of <strong class="program">testsymbolicGA.py</strong> that allows access to the
-required modules and also allow variables and certain multivectors to be
-broadcast from the <tt class="xref py py-mod docutils literal"><span class="pre">GA</span></tt> module to the main program.</p>
-<div class="highlight-python"><pre>import os
-import sys
-import sympy
-from sympy.galgebra.GA import MV, ZERO, ONE, HALF
-from sympy import collect, symbols
-
-
-def F(x, n, nbar):
-        &quot;&quot;&quot;
-        Conformal Mapping Function
-        &quot;&quot;&quot;
-        Fx = HALF*((x*x)*n + 2*x - nbar)
-        return(Fx)
-
-if __name__ == &#x27;__main__&#x27;:
-</pre>
-</div>
-</div>
-<div class="section" id="basic-geometric-algebra-operations">
-<h4>Basic Geometric Algebra Operations<a class="headerlink" href="#basic-geometric-algebra-operations" title="Permalink to this headline">¶</a></h4>
-<p>Example of basic geometric algebra operation of geometric, outer, and inner
-products.</p>
-<div class="highlight-python"><pre>        a, b, c, d, e = MV.setup(&#x27;a b c d e&#x27;)
-        MV.set_str_format(1)
-
-        print &#x27;e|(a^b) =&#x27;, e | (a ^ b)
-        print &#x27;e|(a^b^c) =&#x27;, e | (a ^ b ^ c)
-        print &#x27;a*(b^c)-b*(a^c)+c*(a^b) =&#x27;, a*(b ^ c) - b*(a ^ c) + c*(a ^ b)
-        print &#x27;e|(a^b^c^d) =&#x27;, e | (a ^ b ^ c ^ d)
-        print -d*(a ^ b ^ c) + c*(a ^ b ^ d) - b*(a ^ c ^ d) + a*(b ^ c ^ d)
-
-        print (a ^ b) | (c ^ d)
-
-e | (a ^ b) = {-(b.e)}a
-+{(a.e)}b
-
-e | (a ^ b ^ c) = {(c.e)}a ^ b
-+{-(b.e)}a ^ c
-+{(a.e)}b ^ c
-
-a*(b ^ c) - b*(a ^ c) + c*(a ^ b) = {3}a ^ b ^ c
-
-e | (a ^ b ^ c ^ d) = {-(d.e)}a ^ b ^ c
-+{(c.e)}a ^ b ^ d
-+{-(b.e)}a ^ c ^ d
-+{(a.e)}b ^ c ^ d
-
-{4}a ^ b ^ c ^ d
-
-{(a.d)*(b.c) - (a.c)*(b.d)}1
-</pre>
-</div>
-</div>
-<div class="section" id="examples-of-conformal-geometry">
-<h4>Examples of Conformal Geometry<a class="headerlink" href="#examples-of-conformal-geometry" title="Permalink to this headline">¶</a></h4>
-<p>Examples of comformal geometry [Lasenby,Chapter 10]. The examples show that
-basic geometric entities (lines, circles, planes, and spheres) in three
-dimensions can be represented by blades in a five dimensional (conformal) space.</p>
-<div class="highlight-python"><pre>        print &#x27;\nExample: Conformal representations of circles, lines, spheres, and planes&#x27;
-
-        metric = &#x27;1 0 0 0 0,0 1 0 0 0,0 0 1 0 0,0 0 0 0 2,0 0 0 2 0&#x27;
-
-        e0, e1, e2, n, nbar = MV.setup(&#x27;e0 e1 e2 n nbar&#x27;, metric, debug=0)
-        MV.set_str_format(1)
-        e = n + nbar
-        #conformal representation of points
-
-        A = F(e0, n, nbar)     # point a = (1,0,0)  A = F(a)
-        B = F(e1, n, nbar)     # point b = (0,1,0)  B = F(b)
-        C = F(-1*e0, n, nbar)  # point c = (-1,0,0) C = F(c)
-        D = F(e2, n, nbar)     # point d = (0,0,1)  D = F(d)
-        x0, x1, x2 = sympy.symbols(&#x27;x0 x1 x2&#x27;)
-        X = F(MV([x0, x1, x2], &#x27;vector&#x27;), n, nbar)
-
-        print &#x27;a = e0, b = e1, c = -e0, and d = e2&#x27;
-        print &#x27;A = F(a) = 1/2*(a*a*n+2*a-nbar), etc.&#x27;
-        print &#x27;Circle through a, b, and c&#x27;
-        print &#x27;Circle: A^B^C^X = 0 =&#x27;, (A ^ B ^ C ^ X)
-        print &#x27;Line through a and b&#x27;
-        print &#x27;Line  : A^B^n^X = 0 =&#x27;, (A ^ B ^ n ^ X)
-        print &#x27;Sphere through a, b, c, and d&#x27;
-        print &#x27;Sphere: A^B^C^D^X = 0 =&#x27;, (A ^ B ^ C ^ D ^ X)
-        print &#x27;Plane through a, b, and d&#x27;
-        print &#x27;Plane : A^B^n^D^X = 0 =&#x27;, (A ^ B ^ n ^ D ^ X)
-
-Example:
-        Conformal representations of circles, lines, spheres, and planes
-a = e0, b = e1, c = -e0, and d = e2
-A = F(a) = 1/2*(a*a*n + 2*a - nbar), etc.
-Circle through a, b, and c
-Circle:
-        A ^ B ^ C ^ X = 0 = {-x2}e0 ^ e1 ^ e2 ^ n
-+{x2}e0 ^ e1 ^ e2 ^ nbar
-+{-1/2 + 1/2*x0**2 + 1/2*x1**2 + 1/2*x2**2}e0 ^ e1 ^ n ^ nbar
-
-Line through a and b
-Line:
-        A ^ B ^ n ^ X = 0 = {-x2}e0 ^ e1 ^ e2 ^ n
-+{-1/2 + x0/2 + x1/2}e0 ^ e1 ^ n ^ nbar
-+{x2/2}e0 ^ e2 ^ n ^ nbar
-+{-x2/2}e1 ^ e2 ^ n ^ nbar
-
-Sphere through a, b, c, and d
-Sphere:
-        A ^ B ^ C ^ D ^ X = 0 = {1/2 - 1/2*x0**2 - 1/2*x1**2 - 1/2*x2**2}e0 ^ e1 ^ e2 ^ n ^ nbar
-
-Plane through a, b, and d
-Plane:
-        A ^ B ^ n ^ D ^ X = 0 = {1/2 - x0/2 - x1/2 - x2/2}e0 ^ e1 ^ e2 ^ n ^ nbar
-</pre>
-</div>
-</div>
-<div class="section" id="calculation-of-reciprocal-frame">
-<h4>Calculation of Reciprocal Frame<a class="headerlink" href="#calculation-of-reciprocal-frame" title="Permalink to this headline">¶</a></h4>
-<p>Example shows the calculation of the reciprocal frame for three arbitrary
-vectors and verifies that the calculated reciprocal vectors have the correct
-properties.</p>
-<div class="highlight-python"><pre>        e1, e2, e3 = MV.setup(&#x27;e1 e2 e3&#x27;)
-
-        print &#x27;Example: Reciprocal Frames e1, e2, and e3 unit vectors.\n\n&#x27;
-
-        E = e1 ^ e2 ^ e3
-        Esq = (E*E)()
-        print &#x27;E =&#x27;, E
-        print &#x27;E^2 =&#x27;, Esq
-        Esq_inv = 1/Esq
-
-        E1 = (e2 ^ e3)*E
-        E2 = (-1)*(e1 ^ e3)*E
-        E3 = (e1 ^ e2)*E
-
-        print &#x27;E1 = (e2^e3)*E =&#x27;, E1
-        print &#x27;E2 =-(e1^e3)*E =&#x27;, E2
-        print &#x27;E3 = (e1^e2)*E =&#x27;, E3
-
-        w = (E1 | e2)
-        w.collect(MV.g)
-        w = w().expand()
-        print &#x27;E1|e2 =&#x27;, w
-
-        w = (E1 | e3)
-        w.collect(MV.g)
-        w = w().expand()
-        print &#x27;E1|e3 =&#x27;, w
-
-        w = (E2 | e1)
-        w.collect(MV.g)
-        w = w().expand()
-        print &#x27;E2|e1 =&#x27;, w
-
-        w = (E2 | e3)
-        w.collect(MV.g)
-        w = w().expand()
-        print &#x27;E2|e3 =&#x27;, w
-
-        w = (E3 | e1)
-        w.collect(MV.g)
-        w = w().expand()
-        print &#x27;E3|e1 =&#x27;, w
-
-        w = (E3 | e2)
-        w.collect(MV.g)
-        w = w().expand()
-        print &#x27;E3|e2 =&#x27;, w
-
-        w = (E1 | e1)
-        w = w().expand()
-        Esq = Esq.expand()
-        print &#x27;(E1|e1)/E^2 =&#x27;, w/Esq
-
-        w = (E2 | e2)
-        w = w().expand()
-        print &#x27;(E2|e2)/E^2 =&#x27;, w/Esq
-
-        w = (E3 | e3)
-        w = w().expand()
-        print &#x27;(E3|e3)/E^2 =&#x27;, w/Esq
-
-Example:
-        Reciprocal Frames e1, e2, and e3 unit vectors.
-
-
-E = e1 ^ e2 ^ e3
-E ^ 2 = -1 - 2*(e1.e2)*(e1.e3)*(e2.e3) + (e1.e2)**2 + (e1.e3)**2 + (e2.e3)**2
-E1 = (e2 ^ e3)*E = {-1 + (e2.e3)**2}e1 + {(e1.e2) - (e1.e3)*(e2.e3)}e2 + {(e1.e3) - (e1.e2)*(e2.e3)}e3
-E2 = -(e1 ^ e3)*E = {(e1.e2) - (e1.e3)*(e2.e3)}e1 + {-1 + (e1.e3)**2}e2 + {(e2.e3) - (e1.e2)*(e1.e3)}e3
-E3 = (e1 ^ e2)*E = {(e1.e3) - (e1.e2)*(e2.e3)}e1 + {(e2.e3) - (e1.e2)*(e1.e3)}e2 + {-1 + (e1.e2)**2}e3
-E1 | e2 = 0
-E1 | e3 = 0
-E2 | e1 = 0
-E2 | e3 = 0
-E3 | e1 = 0
-E3 | e2 = 0
-(E1 | e1)/E ^ 2 = 1
-(E2 | e2)/E ^ 2 = 1
-(E3 | e3)/E ^ 2 = 1
-</pre>
-</div>
-</div>
-<div class="section" id="hyperbolic-geometry">
-<h4>Hyperbolic Geometry<a class="headerlink" href="#hyperbolic-geometry" title="Permalink to this headline">¶</a></h4>
-<p>Examples of calculation of distance in hyperbolic geometry [Lasenby,pp373-375].
-This is a good example of the utility of not restricting the basis vector to be
-orthogonal. Note that most of the calculation is simplifying a scalar
-expression.</p>
-<div class="highlight-python"><pre>    print &#x27;Example: non-euclidian distance calculation&#x27;
-
-    metric = &#x27;0 # #,# 0 #,# # 1&#x27;
-    X, Y, e = MV.setup(&#x27;X Y e&#x27;, metric)
-    XdotY = sympy.Symbol(&#x27;(X.Y)&#x27;)
-    Xdote = sympy.Symbol(&#x27;(X.e)&#x27;)
-    Ydote = sympy.Symbol(&#x27;(Y.e)&#x27;)
-    MV.set_str_format(1)
-    L = X ^ Y ^ e
-    B = L*e
-    Bsq = (B*B)()
-    print &#x27;L = X^Y^e is a non-euclidian line&#x27;
-    print &#x27;B = L*e =&#x27;, B
-    BeBr = B*e*B.rev()
-    print &#x27;B*e*B.rev() =&#x27;, BeBr
-    print &#x27;B^2 =&#x27;, Bsq
-    print &#x27;L^2 =&#x27;, (L*L)()
-    s, c, Binv, M, S, C, alpha = symbols(&#x27;s c Binv M S C alpha&#x27;)
-    Bhat = Binv*B  # Normalize translation generator
-    R = c + s*Bhat  # Rotor R = exp(alpha*Bhat/2)
-    print &#x27;s = sinh(alpha/2) and c = cosh(alpha/2)&#x27;
-    print &#x27;R = exp(alpha*B/(2*|B|)) =&#x27;, R
-    Z = R*X*R.rev()
-    Z.expand()
-    Z.collect([Binv, s, c, XdotY])
-    print &#x27;R*X*R.rev() =&#x27;, Z
-    W = Z | Y
-    W.expand()
-    W.collect([s*Binv])
-    print &#x27;(R*X*rev(R)).Y =&#x27;, W
-    M = 1/Bsq
-    W.subs(Binv**2, M)
-    W.simplify()
-    Bmag = sympy.sqrt(XdotY**2 - 2*XdotY*Xdote*Ydote)
-    W.collect([Binv*c*s, XdotY])
-
-    W.subs(2*XdotY**2 - 4*XdotY*Xdote*Ydote, 2/(Binv**2))
-    W.subs(2*c*s, S)
-    W.subs(c**2, (C + 1)/2)
-    W.subs(s**2, (C - 1)/2)
-    W.simplify()
-    W.subs(1/Binv, Bmag)
-    W = W().expand()
-    print &#x27;(R*X*R.rev()).Y =&#x27;, W
-    nl = &#x27;\n&#x27;
-
-    Wd = collect(W, [C, S], exact=True, evaluate=False)
-    print &#x27;Wd =&#x27;, Wd
-    Wd_1 = Wd[ONE]
-    Wd_C = Wd[C]
-    Wd_S = Wd[S]
-    print &#x27;|B| =&#x27;, Bmag
-    Wd_1 = Wd_1.subs(Bmag, 1/Binv)
-    Wd_C = Wd_C.subs(Bmag, 1/Binv)
-    Wd_S = Wd_S.subs(Bmag, 1/Binv)
-    print &#x27;Wd[ONE] =&#x27;, Wd_1
-    print &#x27;Wd[C] =&#x27;, Wd_C
-    print &#x27;Wd[S] =&#x27;, Wd_S
-
-    lhs = Wd_1 + Wd_C*C
-    rhs = -Wd_S*S
-    lhs = lhs**2
-    rhs = rhs**2
-    W = (lhs - rhs).expand()
-    W = (W.subs(1/Binv**2, Bmag**2)).expand()
-    print &#x27;W =&#x27;, W
-    W = (W.subs(S**2, C**2 - 1)).expand()
-    print &#x27;W =&#x27;, W
-    W = collect(W, [C, C**2], evaluate=False)
-    print &#x27;W =&#x27;, W
-
-    a = W[C**2]
-    b = W[C]
-    c = W[ONE]
-
-    print &#x27;a =&#x27;, a
-    print &#x27;b =&#x27;, b
-    print &#x27;c =&#x27;, c
-
-    D = (b**2 - 4*a*c).expand()
-    print &#x27;Setting to 0 and solving for C gives:&#x27;
-    print &#x27;Descriminant D = b^2-4*a*c =&#x27;, D
-    C = (-b/(2*a)).expand()
-    print &#x27;C = cosh(alpha) = -b/(2*a) =&#x27;, C
-
-Example:
-    non - euclidian distance calculation
-L = X ^ Y ^ e is a non - euclidian line
-B = L*e = X ^ Y
-+{-(Y.e)}X ^ e
-+{(X.e)}Y ^ e
-
-B*e*B.rev() = {2*(X.Y)*(X.e)*(Y.e) - (X.Y)**2}e
-
-B ^ 2 = -2*(X.Y)*(X.e)*(Y.e) + (X.Y)**2
-L ^ 2 = -2*(X.Y)*(X.e)*(Y.e) + (X.Y)**2
-s = sinh(alpha/2) and c = cosh(alpha/2)
-R = exp(alpha*B/(2* | B | )) = {c}1
-+{Binv*s}X ^ Y
-+{-(Y.e)*Binv*s}X ^ e
-+{(X.e)*Binv*s}Y ^ e
-
-R*X*R.rev() = {Binv*(2*(X.Y)*c*s - 2*(X.e)*(Y.e)*c*s) + Binv**2*((X.Y)**2*s**2
-               - 2*(X.Y)*(X.e)*(Y.e)*s**2) + c**2}X
-    +{2*Binv*c*s*(X.e)**2}Y
-    +{Binv**2*(-2*(X.e)*(X.Y)**2*s**2 + 4*(X.Y)*(Y.e)*(X.e)**2*s**2)
-      - 2*(X.Y)*(X.e)*Binv*c*s}e
-
-(R*X*rev(R)).Y = {Binv*s*(-4*(X.Y)*(X.e)*(Y.e)*c + 2*c*(X.Y)**2)
-                 + Binv**2*s**2*(-4*(X.e)*(Y.e)*(X.Y)**2 +
-                 4*(X.Y)*(X.e)**2*(Y.e)**2 + (X.Y)**3) + (X.Y)*c**2}1
-
-(R*X*R.rev()).Y = S*(-2*(X.Y)*(X.e)*(Y.e) + (X.Y)**2)**(1/2)
-    + (X.Y)*Binv*C*(-2*(X.Y)*(X.e)*(Y.e) +
-    (X.Y)**2)**(1/2) + (X.e)*(Y.e)*Binv*(-2*(X.Y)*(X.e)*(Y.e)
-    + (X.Y)**2)**(1/2) -
-    (X.e)*(Y.e)*Binv*C*(-2*(X.Y)*(X.e)*(Y.e) + (X.Y)**2)**(1/2)
-Wd = {1: (X.e)*(Y.e)*Binv*(-2*(X.Y)*(X.e)*(Y.e) + (X.Y)**2)**(1/2),
-      S: (-2*(X.Y)*(X.e)*(Y.e) + (X.Y)**2)**(1/2),
-      C: (X.Y)*Binv*(-2*(X.Y)*(X.e)*(Y.e) + (X.Y)**2)**(1/2) -
-         (X.e)*(Y.e)*Binv*(-2*(X.Y)*(X.e)*(Y.e) + (X.Y)**2)**(1/2)}
-|B | = (-2*(X.Y)*(X.e)*(Y.e) + (X.Y)**2)**(1/2)
-
-Wd[ONE] = (X.e)*(Y.e)
-Wd[C] = (X.Y) - (X.e)*(Y.e)
-Wd[S] = 1/Binv
-
-W = 2*(X.Y)*(X.e)*(Y.e)*C + (X.Y)**2*C**2 + (X.e)**2*(Y.e)**2
-    - (X.Y)**2*S**2 + (X.e)**2*(Y.e)**2*C**2 - 2*C*(X.e)**2*(Y.e)**2
-    - 2*(X.Y)*(X.e)*(Y.e)*C**2 + 2*(X.Y)*(X.e)*(Y.e)*S**2
-W = -2*(X.Y)*(X.e)*(Y.e) + 2*(X.Y)*(X.e)*(Y.e)*C + (X.Y)**2
-    + (X.e)**2*(Y.e)**2 + (X.e)**2*(Y.e)**2*C**2 -
-        2*C*(X.e)**2*(Y.e)**2
-W = {1: -2*(X.Y)*(X.e)*(Y.e) + (X.Y)**2 + (X.e)**2*(Y.e)**2,
-     C**2: (X.e)**2*(Y.e)**2,
-     C: 2*(X.Y)*(X.e)*(Y.e) - 2*(X.e)**2*(Y.e)**2}
-
-a = (X.e)**2*(Y.e)**2
-b = 2*(X.Y)*(X.e)*(Y.e) - 2*(X.e)**2*(Y.e)**2
-c = -2*(X.Y)*(X.e)*(Y.e) + (X.Y)**2 + (X.e)**2*(Y.e)**2
-
-Setting to 0 and solving for C gives:
-Descriminant D = b ^ 2 - 4*a*c = 0
-C = cosh(alpha) = -b/(2*a) = 1 - (X.Y)/((X.e)*(Y.e))
-</pre>
-</div>
-</div>
-</div>
-<div class="section" id="calculus">
-<h3>Calculus<a class="headerlink" href="#calculus" title="Permalink to this headline">¶</a></h3>
-<p>The calculus examples all use the extened LaTeXoutput module, <tt class="docutils literal"><span class="pre">latex_ex</span></tt>, for
-clarity.</p>
-<div class="section" id="maxwell-s-equations">
-<h4>Maxwell&#8217;s Equations<a class="headerlink" href="#maxwell-s-equations" title="Permalink to this headline">¶</a></h4>
-<p>In geometric calculus the equivalent of the electromagnetic tensor is
-<span class="math">\(F = E\gamma_{0}+IB\gamma_{0}\)</span> where a spacetime vector is given by
-<span class="math">\(x = x^{0}\gamma_{0}+x^{1}\gamma_{1}+x^{2}\gamma_{2}+x^{3}\gamma_{3}\)</span>
-where <span class="math">\(x^{0} = ct\)</span>, the pseudoscalar
-<span class="math">\(I = \gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}\)</span>, <span class="math">\(E\)</span> and <span class="math">\(B\)</span>
-are four vectors where the time component is zero and the spatial components
-equal to the electric and magnetic field components.  Then Maxwell&#8217;s equations
-can be all written as <span class="math">\(\nabla F = J\)</span> with <span class="math">\(J\)</span> the four current.
-This example shows that this equations generates all of Maxwell&#8217;s equations
-correctly  (in our units:math:<span class="math">\(c=1\)</span>) [Lasenby,pp229-231].</p>
-<p><tt class="docutils literal"><span class="pre">Begin</span> <span class="pre">Program</span> <span class="pre">Maxwell.py</span></tt></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="kn">from</span> <span class="nn">sympy.galgebra.GA</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="kn">from</span> <span class="nn">sympy.galgebra.latex_ex</span> <span class="kn">import</span> <span class="o">*</span>
-
-<span class="k">if</span> <span class="n">__name__</span> <span class="o">==</span> <span class="s">&#39;__main__&#39;</span><span class="p">:</span>
-
-    <span class="n">metric</span> <span class="o">=</span> <span class="s">&#39;1  0  0  0,&#39;</span> \
-             <span class="s">&#39;0 -1  0  0,&#39;</span> \
-             <span class="s">&#39;0  0 -1  0,&#39;</span> \
-             <span class="s">&#39;0  0  0 -1&#39;</span>
-
-    <span class="nb">vars</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;t x y z&#39;</span><span class="p">)</span>
-    <span class="n">gamma_t</span><span class="p">,</span> <span class="n">gamma_x</span><span class="p">,</span> <span class="n">gamma_y</span><span class="p">,</span> <span class="n">gamma_z</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">setup</span><span class="p">(</span><span class="s">&#39;gamma_t gamma_x gamma_y gamma_z&#39;</span><span class="p">,</span> <span class="n">metric</span><span class="p">,</span> <span class="bp">True</span><span class="p">,</span> <span class="nb">vars</span><span class="p">)</span>
-    <span class="n">LatexPrinter</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-    <span class="n">I</span> <span class="o">=</span> <span class="n">MV</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="s">&#39;pseudo&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;$I$ Pseudo-Scalar&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;I =&#39;</span><span class="p">,</span> <span class="n">I</span><span class="p">)</span>
-    <span class="n">B</span> <span class="o">=</span> <span class="n">MV</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="s">&#39;vector&#39;</span><span class="p">,</span> <span class="n">fct</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">E</span> <span class="o">=</span> <span class="n">MV</span><span class="p">(</span><span class="s">&#39;E&#39;</span><span class="p">,</span> <span class="s">&#39;vector&#39;</span><span class="p">,</span> <span class="n">fct</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">B</span><span class="o">.</span><span class="n">set_coef</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-    <span class="n">E</span><span class="o">.</span><span class="n">set_coef</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-    <span class="n">B</span> <span class="o">*=</span> <span class="n">gamma_t</span>
-    <span class="n">E</span> <span class="o">*=</span> <span class="n">gamma_t</span>
-    <span class="n">J</span> <span class="o">=</span> <span class="n">MV</span><span class="p">(</span><span class="s">&#39;J&#39;</span><span class="p">,</span> <span class="s">&#39;vector&#39;</span><span class="p">,</span> <span class="n">fct</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">F</span> <span class="o">=</span> <span class="n">E</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">B</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39; &#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;$B$ Magnetic Field Bi-Vector&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;B = Bvec gamma_0 =&#39;</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;$F$ Electric Field Bi-Vector&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;E = Evec gamma_0 =&#39;</span><span class="p">,</span> <span class="n">E</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;$E+IB$ Electo-Magnetic Field Bi-Vector&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;F = E+IB =&#39;</span><span class="p">,</span> <span class="n">F</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;$J$ Four Current&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;J =&#39;</span><span class="p">,</span> <span class="n">J</span><span class="p">)</span>
-    <span class="n">gradF</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">grad</span><span class="p">()</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;Geometric Derivative of Electo-Magnetic Field Bi-Vector&#39;</span><span class="p">)</span>
-    <span class="n">MV_format</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\\</span><span class="s">nabla F =&#39;</span><span class="p">,</span> <span class="n">gradF</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;All Maxwell Equations are&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\\</span><span class="s">nabla F = J&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;Div $E$ and Curl $H$ Equations&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;&lt;</span><span class="se">\\</span><span class="s">nabla F&gt;_1 -J =&#39;</span><span class="p">,</span> <span class="n">gradF</span><span class="o">.</span><span class="n">project</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">J</span><span class="p">,</span> <span class="s">&#39; = 0&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;Curl $E$ and Div $B$ equations&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;&lt;</span><span class="se">\\</span><span class="s">nabla F&gt;_3 =&#39;</span><span class="p">,</span> <span class="n">gradF</span><span class="o">.</span><span class="n">project</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span> <span class="s">&#39; = 0&#39;</span><span class="p">)</span>
-    <span class="n">xdvi</span><span class="p">(</span><span class="n">filename</span><span class="o">=</span><span class="s">&#39;Maxwell.tex&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p><tt class="docutils literal"><span class="pre">End</span> <span class="pre">Program</span> <span class="pre">Maxwell.py</span></tt></p>
-<p><tt class="docutils literal"><span class="pre">Begin</span> <span class="pre">Program</span> <span class="pre">Output</span></tt></p>
-<p><span class="math">\(I\)</span> Pseudo-Scalar</p>
-<div class="math">
-\[\begin{equation*}
-I = {\gamma}_{t}{\gamma}_{x}{\gamma}_{y}{\gamma}_{z}
-\end{equation*}\]</div><p><span class="math">\(B\)</span> Magnetic Field Bi-Vector</p>
-<div class="math">
-\[\begin{equation*}
-B = - {B^{x}}{\gamma}_{t}{\gamma}_{x}- {B^{y}}{\gamma}_{t}{\gamma}_{y}- {B^{z}}{\gamma}_{t}{\gamma}_{z}
-\end{equation*}\]</div><p><span class="math">\(F\)</span> Electric Field Bi-Vector</p>
-<div class="math">
-\[\begin{equation*}
-E = - {E^{x}}{\gamma}_{t}{\gamma}_{x}- {E^{y}}{\gamma}_{t}{\gamma}_{y}- {E^{z}}{\gamma}_{t}{\gamma}_{z}
-\end{equation*}\]</div><p><span class="math">\(E+IB\)</span> Electo-Magnetic Field Bi-Vector</p>
-<div class="math">
-\[\begin{equation*}
-F = - {E^{x}}{\gamma}_{t}{\gamma}_{x}- {E^{y}}{\gamma}_{t}{\gamma}_{y}- {B^{z}}{\gamma}_{x}{\gamma}_{y}- {E^{z}}{\gamma}_{t}{\gamma}_{z}+ {B^{y}}{\gamma}_{x}{\gamma}_{z}- {B^{x}}{\gamma}_{y}{\gamma}_{z}
-\end{equation*}\]</div><p><span class="math">\(J\)</span> Four Current</p>
-<div class="math">
-\[\begin{equation*}
-J =  {J^{t}}{\gamma}_{t}+ {J^{x}}{\gamma}_{x}+ {J^{y}}{\gamma}_{y}+ {J^{z}}{\gamma}_{z}
-\end{equation*}\]</div><p>Geometric Derivative of Electo-Magnetic Field Bi-Vector</p>
-<div class="math">
-\[\begin{align*}
-\nabla F & = \left( \partial_{z} {E^{z}} + \partial_{y} {E^{y}} + \partial_{x} {E^{x}}\right){\gamma}_{t} \\
-         & + \left(-\partial_{t} {E^{x}} + \partial_{y} {B^{z}} - \partial_{z} {B^{y}}\right){\gamma}_{x} \\
-         & + \left( \partial_{z} {B^{x}} - \partial_{t} {E^{y}} - \partial_{x} {B^{z}}\right){\gamma}_{y} \\
-         & + \left(-\partial_{y} {B^{x}} - \partial_{t} {E^{z}} + \partial_{x} {B^{y}}\right){\gamma}_{z} \\
-         & + \left(-\partial_{x} {E^{y}} - \partial_{t} {B^{z}} + \partial_{y} {E^{x}}\right){\gamma}_{t}{\gamma}_{x}{\gamma}_{y} \\
-         & + \left(-\partial_{x} {E^{z}} + \partial_{t} {B^{y}} + \partial_{z} {E^{x}}\right){\gamma}_{t}{\gamma}_{x}{\gamma}_{z} \\
-         & + \left(-\partial_{t} {B^{x}} - \partial_{y} {E^{z}} + \partial_{z} {E^{y}}\right){\gamma}_{t}{\gamma}_{y}{\gamma}_{z} \\
-         & + \left( \partial_{y} {B^{y}} + \partial_{z} {B^{z}} + \partial_{x} {B^{x}}\right){\gamma}_{x}{\gamma}_{y}{\gamma}_{z}
-\end{align*}\]</div><p>All Maxwell Equations are</p>
-<div class="math">
-\[\begin{equation*}
-\nabla F = J
-\end{equation*}\]</div><p>Div <span class="math">\(E\)</span> and Curl <span class="math">\(H\)</span> Equations</p>
-<div class="math">
-\[\begin{align*}
-<\nabla F>_1 -J & = \left(-{J^{t}} + \partial_{z} {E^{z}} + \partial_{y} {E^{y}} + \partial_{x} {E^{x}}\right){\gamma}_{t} \\
-                & + \left(-{J^{x}} - \partial_{t} {E^{x}} + \partial_{y} {B^{z}} - \partial_{z} {B^{y}}\right){\gamma}_{x} \\
-                & + \left( \partial_{z} {B^{x}} - \partial_{t} {E^{y}} -{J^{y}} - \partial_{x} {B^{z}}\right){\gamma}_{y} \\
-                & + \left(-\partial_{y} {B^{x}} - \partial_{t} {E^{z}} -{J^{z}} + \partial_{x} {B^{y}}\right){\gamma}_{z}
-\end{align*}\]</div><div class="math">
-\[\begin{equation*}
-  = 0
-\end{equation*}\]</div><p>Curl <span class="math">\(E\)</span> and Div <span class="math">\(B\)</span> equations</p>
-<div class="math">
-\[\begin{align*}
-<\nabla F>_3 & = \left(-\partial_{x} {E^{y}} - \partial_{t} {B^{z}} + \partial_{y} {E^{x}}\right){\gamma}_{t}\wedge {\gamma}_{x}\wedge {\gamma}_{y} \\
-             & + \left(-\partial_{x} {E^{z}} + \partial_{t} {B^{y}} + \partial_{z} {E^{x}}\right){\gamma}_{t}\wedge {\gamma}_{x}\wedge {\gamma}_{z} \\
-             & + \left(-\partial_{t} {B^{x}} - \partial_{y} {E^{z}} + \partial_{z} {E^{y}}\right){\gamma}_{t}\wedge {\gamma}_{y}\wedge {\gamma}_{z} \\
-             & + \left( \partial_{y} {B^{y}} + \partial_{z} {B^{z}} + \partial_{x} {B^{x}}\right){\gamma}_{x}\wedge {\gamma}_{y}\wedge {\gamma}_{z}
-\end{align*}\]</div><div class="math">
-\[\begin{equation*}
-  = 0
-\end{equation*}\]</div><p><tt class="docutils literal"><span class="pre">End</span> <span class="pre">Program</span> <span class="pre">Output</span></tt></p>
-</div>
-<div class="section" id="dirac-s-equation">
-<h4>Dirac&#8217;s Equation<a class="headerlink" href="#dirac-s-equation" title="Permalink to this headline">¶</a></h4>
-<p>The geometric algebra/calculus allows one to formulate the Dirac equation in
-real terms (no <span class="math">\(\sqrt{-1}\)</span>).  Spinors  <span class="math">\(\left(\Psi\right)\)</span> are even
-multivectors in space time (Minkowski space with signature (1,-1,-1,-1)) and the
-Dirac equation becomes
-<span class="math">\(\nabla \Psi I \sigma_{z}-eA\Psi = m\Psi\gamma_{t}\)</span>.  All the terms in the
-real equation are explined in Doran and Lasenby [Lasenby,pp281-283].</p>
-<p><tt class="docutils literal"><span class="pre">Begin</span> <span class="pre">Program</span> <span class="pre">Dirac.py</span></tt></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c">#!/usr/local/bin/python</span>
-<span class="c">#Dirac.py</span>
-
-<span class="kn">from</span> <span class="nn">sympy.galgebra.GA</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="kn">from</span> <span class="nn">sympy.galgebra.latex_ex</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-
-<span class="k">if</span> <span class="n">__name__</span> <span class="o">==</span> <span class="s">&#39;__main__&#39;</span><span class="p">:</span>
-
-    <span class="n">metric</span> <span class="o">=</span> <span class="s">&#39;1  0  0  0,&#39;</span> \
-             <span class="s">&#39;0 -1  0  0,&#39;</span> \
-             <span class="s">&#39;0  0 -1  0,&#39;</span> \
-             <span class="s">&#39;0  0  0 -1&#39;</span>
-
-    <span class="nb">vars</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;t x y z&#39;</span><span class="p">)</span>
-    <span class="n">gamma_t</span><span class="p">,</span> <span class="n">gamma_x</span><span class="p">,</span> <span class="n">gamma_y</span><span class="p">,</span> <span class="n">gamma_z</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">setup</span><span class="p">(</span><span class="s">&#39;gamma_t gamma_x gamma_y gamma_z&#39;</span><span class="p">,</span> <span class="n">metric</span><span class="p">,</span> <span class="bp">True</span><span class="p">,</span> <span class="nb">vars</span><span class="p">)</span>
-
-    <span class="n">m</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;m e&#39;</span><span class="p">)</span>
-    <span class="n">Format</span><span class="p">(</span><span class="s">&#39;1 1 1 1&#39;</span><span class="p">)</span>
-    <span class="n">I</span> <span class="o">=</span> <span class="n">MV</span><span class="p">(</span><span class="n">ONE</span><span class="p">,</span> <span class="s">&#39;pseudo&#39;</span><span class="p">)</span>
-    <span class="n">nvars</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="nb">vars</span><span class="p">)</span>
-    <span class="n">psi</span> <span class="o">=</span> <span class="n">MV</span><span class="p">(</span><span class="s">&#39;psi&#39;</span><span class="p">,</span> <span class="s">&#39;spinor&#39;</span><span class="p">,</span> <span class="n">fct</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">A</span> <span class="o">=</span> <span class="n">MV</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="s">&#39;vector&#39;</span><span class="p">,</span> <span class="n">fct</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">sig_x</span> <span class="o">=</span> <span class="n">gamma_x</span><span class="o">*</span><span class="n">gamma_t</span>
-    <span class="n">sig_y</span> <span class="o">=</span> <span class="n">gamma_y</span><span class="o">*</span><span class="n">gamma_t</span>
-    <span class="n">sig_z</span> <span class="o">=</span> <span class="n">gamma_z</span><span class="o">*</span><span class="n">gamma_t</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;$A$ is 4-vector potential&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">r&#39;$\bm{\psi}$ is 8-component real spinor (even multi-vector)&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="n">psi</span><span class="p">)</span>
-    <span class="n">dirac_eq</span> <span class="o">=</span> <span class="n">psi</span><span class="o">.</span><span class="n">grad</span><span class="p">()</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">sig_z</span> <span class="o">-</span> <span class="n">e</span><span class="o">*</span><span class="n">A</span><span class="o">*</span><span class="n">psi</span> <span class="o">-</span> <span class="n">m</span><span class="o">*</span><span class="n">psi</span><span class="o">*</span><span class="n">gamma_t</span>
-    <span class="n">dirac_eq</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;Dirac equation in terms of real geometric algebra/calculus &#39;</span> \
-          <span class="s">r&#39;$\lp\nabla \bm{\psi} I \sigma_{z}-eA\bm{\psi} = m\bm{\psi}\gamma_{t}\rp$&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;Spin measured with respect to $z$ axis&#39;</span><span class="p">)</span>
-    <span class="n">Format</span><span class="p">(</span><span class="s">&#39;mv=3&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">r&#39;\nabla \bm{\psi} I \sigma_{z}-eA\bm{\psi}-m\bm{\psi}\gamma_{t} = &#39;</span><span class="p">,</span> <span class="n">dirac_eq</span><span class="p">,</span> <span class="s">&#39; = 0&#39;</span><span class="p">)</span>
-    <span class="n">xdvi</span><span class="p">(</span><span class="n">filename</span><span class="o">=</span><span class="s">&#39;Dirac.tex&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p><tt class="docutils literal"><span class="pre">End</span> <span class="pre">Program</span> <span class="pre">Dirac.py</span></tt></p>
-<p><tt class="docutils literal"><span class="pre">Begin</span> <span class="pre">Program</span> <span class="pre">Output</span></tt></p>
-<p><span class="math">\(A\)</span> is 4-vector potential</p>
-<div class="math">
-\[\begin{equation*}
-A =  {A^{t}}{\gamma}_{t}+ {A^{x}}{\gamma}_{x}+ {A^{y}}{\gamma}_{y}+ {A^{z}}{\gamma}_{z}
-\end{equation*}\]</div><p><span class="math">\(\boldsymbol{\psi}\)</span> is 8-component real spinor (even multi-vector)</p>
-<div class="math">
-\[\begin{equation*}
-{}\boldsymbol{{\psi}} = {{\psi}}+ {{\psi}^{tx}}{\gamma}_{t}{\gamma}_{x}+ {{\psi}^{ty}}{\gamma}_{t}{\gamma}_{y}+ {{\psi}^{xy}}{\gamma}_{x}{\gamma}_{y}+ {{\psi}^{tz}}{\gamma}_{t}{\gamma}_{z}+ {{\psi}^{xz}}{\gamma}_{x}{\gamma}_{z}+ {{\psi}^{yz}}{\gamma}_{y}{\gamma}_{z}+ {{\psi}^{txyz}}{\gamma}_{t}{\gamma}_{x}{\gamma}_{y}{\gamma}_{z}
-\end{equation*}\]</div><p>Dirac equation in terms of real geometric algebra/calculus
-<span class="math">\(\left(\nabla \boldsymbol{\psi} I \sigma_{z}-eA\boldsymbol{\psi} = m\boldsymbol{\psi}\gamma_{t}\right)\)</span>
-Spin measured with respect to <span class="math">\(z\)</span> axis</p>
-<div class="math">
-\[\begin{align*}
-\nabla \boldsymbol{\psi} I \sigma_{z}-eA\boldsymbol{\psi}-m\boldsymbol{\psi}\gamma_{t}
-  & = \left(-e {A^{y}} {{\psi}^{ty}} - m {{\psi}} - \partial_{z} {{\psi}^{txyz}} - \partial_{y} {{\psi}^{tx}} - e {A^{z}} {{\psi}^{tz}} - e {A^{x}} {{\psi}^{tx}} + \partial_{x} {{\psi}^{ty}} - e {A^{t}} {{\psi}} + \partial_{t} {{\psi}^{xy}}\right){\gamma}_{t} \\
-  & + \left(-e {A^{y}} {{\psi}^{xy}} + \partial_{z} {{\psi}^{yz}} - e {A^{z}} {{\psi}^{xz}} - e {A^{t}} {{\psi}^{tx}} + m {{\psi}^{tx}} - \partial_{x} {{\psi}^{xy}} + \partial_{y} {{\psi}} - e {A^{x}} {{\psi}} - \partial_{t} {{\psi}^{ty}}\right){\gamma}_{x} \\
-  & + \left(-e {A^{y}} {{\psi}} + e {A^{x}} {{\psi}^{xy}} - e {A^{t}} {{\psi}^{ty}} - \partial_{x} {{\psi}} + \partial_{t} {{\psi}^{tx}} - \partial_{z} {{\psi}^{xz}} - e {A^{z}} {{\psi}^{yz}} + m {{\psi}^{ty}} - \partial_{y} {{\psi}^{xy}}\right){\gamma}_{y} \\
-  & + \left(-\partial_{z} {{\psi}^{xy}} + e {A^{x}} {{\psi}^{xz}} - e {A^{t}} {{\psi}^{tz}} - \partial_{x} {{\psi}^{yz}} + \partial_{y} {{\psi}^{xz}} + e {A^{y}} {{\psi}^{yz}} + \partial_{t} {{\psi}^{txyz}} + m {{\psi}^{tz}} - e {A^{z}} {{\psi}}\right){\gamma}_{z} \\
-  & + \left( \partial_{y} {{\psi}^{ty}} - e {A^{y}} {{\psi}^{tx}} + e {A^{x}} {{\psi}^{ty}} + \partial_{z} {{\psi}^{tz}} - e {A^{z}} {{\psi}^{txyz}} + \partial_{x} {{\psi}^{tx}} - e {A^{t}} {{\psi}^{xy}} - \partial_{t} {{\psi}} - m {{\psi}^{xy}}\right){\gamma}_{t}{\gamma}_{x}{\gamma}_{y} \\
-  & + \left(-\partial_{y} {{\psi}^{tz}} + \partial_{x} {{\psi}^{txyz}} - e {A^{t}} {{\psi}^{xz}} + e {A^{x}} {{\psi}^{tz}} + e {A^{y}} {{\psi}^{txyz}} + \partial_{z} {{\psi}^{ty}} - m {{\psi}^{xz}} - e {A^{z}} {{\psi}^{tx}} - \partial_{t} {{\psi}^{yz}}\right){\gamma}_{t}{\gamma}_{x}{\gamma}_{z} \\
-  & + \left(-e {A^{t}} {{\psi}^{yz}} - \partial_{z} {{\psi}^{tx}} + \partial_{x} {{\psi}^{tz}} - m {{\psi}^{yz}} + \partial_{y} {{\psi}^{txyz}} + \partial_{t} {{\psi}^{xz}} - e {A^{z}} {{\psi}^{ty}} - e {A^{x}} {{\psi}^{txyz}} + e {A^{y}} {{\psi}^{tz}}\right){\gamma}_{t}{\gamma}_{y}{\gamma}_{z} \\
-  & + \left(\partial_{z} {{\psi}} + e {A^{y}} {{\psi}^{xz}} + m {{\psi}^{txyz}} -\partial_{t} {{\psi}^{tz}} -\partial_{x} {{\psi}^{xz}} -e {A^{x}} {{\psi}^{yz}} -e {A^{t}} {{\psi}^{txyz}} -\partial_{y} {{\psi}^{yz}} -e {A^{z}} {{\psi}^{xy}}\right){\gamma}_{x}{\gamma}_{y}{\gamma}_{z}
-\end{align*}\]</div><div class="math">
-\[\begin{equation*}
-  = 0
-\end{equation*}\]</div><p><tt class="docutils literal"><span class="pre">End</span> <span class="pre">Program</span> <span class="pre">Output</span></tt></p>
-</div>
-<div class="section" id="spherical-coordinates">
-<h4>Spherical Coordinates<a class="headerlink" href="#spherical-coordinates" title="Permalink to this headline">¶</a></h4>
-<p>Curvilinear coodinates are implemented as shown in section <a class="reference internal" href="#deriv"><em>Geometric Derivative</em></a>.  The
-gradient of a scalar function and the divergence and curl of a vector function
-(<span class="math">\(-I\left(\nabla\wedge A\right)\)</span> is the curl in three dimensions in the notation of
-geometric algebra) to demonstrate the formulas derived in section <a class="reference internal" href="#deriv"><em>Geometric Derivative</em></a>.</p>
-<p><tt class="docutils literal"><span class="pre">Begin</span> <span class="pre">Program</span> <span class="pre">coords.py</span></tt></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c">#!/usrlocal/bin/python</span>
-<span class="c">#EandM.py</span>
-
-<span class="kn">from</span> <span class="nn">sympy.galgebra.GA</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="kn">from</span> <span class="nn">sympy.galgebra.latex_ex</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-
-<span class="kn">import</span> <span class="nn">sympy</span>
-<span class="kn">import</span> <span class="nn">numpy</span>
-<span class="kn">import</span> <span class="nn">sys</span>
-
-<span class="k">if</span> <span class="n">__name__</span> <span class="o">==</span> <span class="s">&#39;__main__&#39;</span><span class="p">:</span>
-    <span class="n">metric</span> <span class="o">=</span> <span class="s">&#39;1 0 0,&#39;</span> <span class="o">+</span>\
-             <span class="s">&#39;0 1 0,&#39;</span> <span class="o">+</span>\
-             <span class="s">&#39;0 0 1&#39;</span>
-
-    <span class="n">gamma_x</span><span class="p">,</span> <span class="n">gamma_y</span><span class="p">,</span> <span class="n">gamma_z</span> <span class="o">=</span> <span class="n">MV</span><span class="o">.</span><span class="n">setup</span><span class="p">(</span><span class="s">&#39;gamma_x gamma_y gamma_z&#39;</span><span class="p">,</span> <span class="n">metric</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-    <span class="n">Format</span><span class="p">(</span><span class="s">&#39;1 1 1 1&#39;</span><span class="p">)</span>
-
-    <span class="n">coords</span> <span class="o">=</span> <span class="n">r</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;r theta phi&#39;</span><span class="p">)</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">r</span><span class="o">*</span><span class="p">(</span><span class="n">sympy</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span><span class="o">*</span><span class="n">gamma_z</span> <span class="o">+</span> <span class="n">sympy</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span><span class="o">*</span>
-        <span class="p">(</span><span class="n">sympy</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">phi</span><span class="p">)</span><span class="o">*</span><span class="n">gamma_x</span> <span class="o">+</span> <span class="n">sympy</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">phi</span><span class="p">)</span><span class="o">*</span><span class="n">gamma_y</span><span class="p">))</span>
-    <span class="n">x</span><span class="o">.</span><span class="n">set_name</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-
-    <span class="n">MV</span><span class="o">.</span><span class="n">rebase</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">coords</span><span class="p">,</span> <span class="s">&#39;e&#39;</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-
-    <span class="c">#psi = MV.scalar_fct(&#39;psi&#39;)</span>
-    <span class="n">psi</span> <span class="o">=</span> <span class="n">MV</span><span class="p">(</span><span class="s">&#39;psi&#39;</span><span class="p">,</span> <span class="s">&#39;scalar&#39;</span><span class="p">,</span> <span class="n">fct</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="c">#psi.name = &#39;psi&#39;</span>
-    <span class="n">dpsi</span> <span class="o">=</span> <span class="n">psi</span><span class="o">.</span><span class="n">grad</span><span class="p">()</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;Gradient of Scalar Function $</span><span class="se">\\</span><span class="s">psi$&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\\</span><span class="s">nabla</span><span class="se">\\</span><span class="s">psi =&#39;</span><span class="p">,</span> <span class="n">dpsi</span><span class="p">)</span>
-
-    <span class="c">#A = MV.vector_fct(&#39;A&#39;)</span>
-    <span class="n">A</span> <span class="o">=</span> <span class="n">MV</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="s">&#39;vector&#39;</span><span class="p">,</span> <span class="n">fct</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="c">#A.name = &#39;A&#39;</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;Div and Curl of Vector Function $A$&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-
-    <span class="n">gradA</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">grad</span><span class="p">()</span>
-    <span class="n">I</span> <span class="o">=</span> <span class="n">MV</span><span class="p">(</span><span class="n">ONE</span><span class="p">,</span> <span class="s">&#39;pseudo&#39;</span><span class="p">)</span>
-    <span class="n">divA</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">grad_int</span><span class="p">()</span>
-    <span class="n">curlA</span> <span class="o">=</span> <span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">A</span><span class="o">.</span><span class="n">grad_ext</span><span class="p">()</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\\</span><span class="s">nabla </span><span class="se">\\</span><span class="s">cdot A =&#39;</span><span class="p">,</span> <span class="n">divA</span><span class="p">)</span>
-    <span class="n">Format</span><span class="p">(</span><span class="s">&#39;mv=3&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;-I</span><span class="se">\\</span><span class="s">lp</span><span class="se">\\</span><span class="s">nabla </span><span class="se">\\</span><span class="s">W A</span><span class="se">\\</span><span class="s">rp =&#39;</span><span class="p">,</span> <span class="n">curlA</span><span class="p">)</span>
-
-    <span class="n">xdvi</span><span class="p">(</span><span class="n">filename</span><span class="o">=</span><span class="s">&#39;coords.tex&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p><tt class="docutils literal"><span class="pre">End</span> <span class="pre">Program</span> <span class="pre">coords.py</span></tt></p>
-<p><tt class="docutils literal"><span class="pre">Begin</span> <span class="pre">Program</span> <span class="pre">Output</span></tt></p>
-<p>Gradient of Scalar Function <span class="math">\(\psi\)</span></p>
-<div class="math">
-\[\begin{equation*}
-\nabla\psi =  \partial_{r} {{\psi}}{}\boldsymbol{e}_{r}+\frac{1 \partial_{{\theta}} {{\psi}}}{r}{}\boldsymbol{e}_{{\theta}}+\frac{1 \partial_{{\phi}} {{\psi}}}{r \operatorname{sin}\left({\theta}\right)}{}\boldsymbol{e}_{{\phi}}
-\end{equation*}\]</div><p>Div and Curl of Vector Function <span class="math">\(A\)</span></p>
-<div class="math">
-\[\begin{equation*}
-A =  {A^{r}}{}\boldsymbol{e}_{r}+ {A^{{\theta}}}{}\boldsymbol{e}_{{\theta}}+ {A^{{\phi}}}{}\boldsymbol{e}_{{\phi}}
-\end{equation*}\]</div><div class="math">
-\[\begin{equation*}
-\nabla \cdot A =  \left(\frac{{A^{{\theta}}} \operatorname{cos}\left({\theta}\right)}{r \operatorname{sin}\left({\theta}\right)} + 2 \frac{{A^{r}}}{r} + \partial_{r} {A^{r}} + \frac{\partial_{{\theta}} {A^{{\theta}}}}{r} + \frac{\partial_{{\phi}} {A^{{\phi}}}}{r \operatorname{sin}\left({\theta}\right)}\right)
-\end{equation*}\]</div><div class="math">
-\[\begin{align*}
--I\left(\nabla \wedge A\right) & = \left(\frac{\partial_{{\theta}} {A^{{\phi}}}}{r} + \frac{{A^{{\phi}}} \operatorname{cos}\left({\theta}\right)}{r \operatorname{sin}\left({\theta}\right)} -\frac{\partial_{{\phi}} {A^{{\theta}}}}{r \operatorname{sin}\left({\theta}\right)}\right){}\boldsymbol{e}_{r} \\
-                               & - \left(\frac{{A^{{\phi}}}}{r} + \partial_{r} {A^{{\phi}}} -\frac{\partial_{{\phi}} {A^{r}}}{r \operatorname{sin}\left({\theta}\right)}\right){}\boldsymbol{e}_{{\theta}} \\
-                               & + \left(\frac{{A^{{\theta}}}}{r} + \partial_{r} {A^{{\theta}}} -\frac{\partial_{{\theta}} {A^{r}}}{r}\right){}\boldsymbol{e}_{{\phi}}
-\end{align*}\]</div><p><tt class="docutils literal"><span class="pre">End</span> <span class="pre">Program</span> <span class="pre">Output</span></tt></p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference external" href="http://geocalc.clas.asu.edu/html/CA_to_GC.html">Hestenes</a></dt>
-<dd><tt class="docutils literal"><span class="pre">Clifford</span> <span class="pre">Algebra</span> <span class="pre">to</span> <span class="pre">Geometric</span> <span class="pre">Calculus</span></tt> by D.Hestenes and G. Sobczyk, Kluwer
-Academic Publishers, 1984.</dd>
-<dt><a class="reference external" href="http://www.mrao.cam.ac.uk/~cjld1/pages/book.htm">Lasenby</a></dt>
-<dd><tt class="docutils literal"><span class="pre">Geometric</span> <span class="pre">Algebra</span> <span class="pre">for</span> <span class="pre">Physicists</span></tt> by C. Doran and A. Lasenby, Cambridge
-University Press, 2003.</dd>
-</dl>
-</div>
-</div>
-</div>
-</div>
-</div>
-
-
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-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Geometric Algebra Module for SymPy</a><ul>
-<li><a class="reference internal" href="#what-is-geometric-algebra">What is Geometric Algebra?</a></li>
-<li><a class="reference internal" href="#vector-basis-and-metric">Vector Basis and Metric</a></li>
-<li><a class="reference internal" href="#representation-and-reduction-of-multivector-bases">Representation and Reduction of Multivector Bases</a></li>
-<li><a class="reference internal" href="#base-representation-of-multivectors">Base Representation of Multivectors</a></li>
-<li><a class="reference internal" href="#blade-representation-of-multivectors">Blade Representation of Multivectors</a></li>
-<li><a class="reference internal" href="#outer-and-inner-products-left-and-right-contractions">Outer and Inner Products, Left and Right Contractions</a></li>
-<li><a class="reference internal" href="#reverse-of-multivector">Reverse of Multivector</a></li>
-<li><a class="reference internal" href="#reciprocal-frames">Reciprocal Frames</a></li>
-<li><a class="reference internal" href="#geometric-derivative">Geometric Derivative</a></li>
-<li><a class="reference internal" href="#module-components">Module Components</a><ul>
-<li><a class="reference internal" href="#initializing-multivector-class">Initializing Multivector Class</a></li>
-<li><a class="reference internal" href="#instantiating-a-multivector">Instantiating a Multivector</a></li>
-<li><a class="reference internal" href="#basic-multivector-class-functions">Basic Multivector Class Functions</a></li>
-<li><a class="reference internal" href="#sympy-functions-applied-inplace-to-multivector-coefficients">SymPy Functions Applied Inplace to Multivector Coefficients</a></li>
-<li><a class="reference internal" href="#helper-functions">Helper Functions</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#examples">Examples</a><ul>
-<li><a class="reference internal" href="#algebra">Algebra</a><ul>
-<li><a class="reference internal" href="#example-header">Example Header</a></li>
-<li><a class="reference internal" href="#basic-geometric-algebra-operations">Basic Geometric Algebra Operations</a></li>
-<li><a class="reference internal" href="#examples-of-conformal-geometry">Examples of Conformal Geometry</a></li>
-<li><a class="reference internal" href="#calculation-of-reciprocal-frame">Calculation of Reciprocal Frame</a></li>
-<li><a class="reference internal" href="#hyperbolic-geometry">Hyperbolic Geometry</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#calculus">Calculus</a><ul>
-<li><a class="reference internal" href="#maxwell-s-equations">Maxwell&#8217;s Equations</a></li>
-<li><a class="reference internal" href="#dirac-s-equation">Dirac&#8217;s Equation</a></li>
-<li><a class="reference internal" href="#spherical-coordinates">Spherical Coordinates</a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-</li>
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-  <div class="section" id="geometric-algebra-module-docstring">
-<h1>Geometric Algebra Module Docstring<a class="headerlink" href="#geometric-algebra-module-docstring" title="Permalink to this headline">¶</a></h1>
-<p>Documentation for Geometric Algebra module</p>
-<p>Contents:</p>
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-  <div class="section" id="module-sympy.galgebra.latex_ex">
-<span id="latex-ex"></span><h1>Latex_ex<a class="headerlink" href="#module-sympy.galgebra.latex_ex" title="Permalink to this headline">¶</a></h1>
-<dl class="function">
-<dt id="sympy.galgebra.latex_ex.LaTeX">
-<tt class="descclassname">sympy.galgebra.latex_ex.</tt><tt class="descname">LaTeX</tt><big>(</big><em>expr</em>, <em>inline=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/latex_ex.html#LaTeX"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.latex_ex.LaTeX" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert the given expression to LaTeX representation.</p>
-<p>You can specify how the generated code will be delimited.
-If the &#8216;inline&#8217; keyword is set then inline LaTeX $ $ will
-be used. Otherwise the resulting code will be enclosed in
-&#8216;equation*&#8217; environment (remember to import &#8216;amsmath&#8217;).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">tau</span><span class="p">,</span> <span class="n">mu</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">tau</span><span class="p">)</span><span class="o">**</span><span class="n">Rational</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">&#39;$8 \\sqrt{2} \\sqrt[7]{\\tau}$&#39;</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">mu</span><span class="p">)</span><span class="o">**</span><span class="n">Rational</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="n">inline</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">&#39;\\begin{equation*}8 \\sqrt{2} \\sqrt[7]{\\mu}\\end{equation*}&#39;</span>
-</pre></div>
-</div>
-<p>Besides all Basic based expressions, you can recursively
-convert Python containers (lists, tuples and dicts) and
-also SymPy matrices:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">([</span><span class="mi">2</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">])</span>
-<span class="go">&#39;$\\begin{bmatrix}\\frac{2}{x}, &amp; y\\end{bmatrix}$&#39;</span>
-</pre></div>
-</div>
-<p>The extended latex printer will also append the output to a
-string (LatexPrinter.body) that will be processed by xdvi()
-for immediate display one xdvi() is called.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.galgebra.latex_ex.LatexPrinter">
-<em class="property">class </em><tt class="descclassname">sympy.galgebra.latex_ex.</tt><tt class="descname">LatexPrinter</tt><big>(</big><em>inline=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/latex_ex.html#LatexPrinter"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.latex_ex.LatexPrinter" title="Permalink to this definition">¶</a></dt>
-<dd><p>A printer class which converts an expression into its LaTeX equivalent.
-This class extends the LatexPrinter class currently in sympy in the
-following ways:</p>
-<blockquote>
-<div><ol class="arabic simple">
-<li>Variable and function names can now encode multiple Greek symbols,
-number, Greek, and roman super and subscripts and accents plus bold
-math in an alphanumeric ASCII string consisting of <tt class="docutils literal"><span class="pre">[A-Za-z0-9_]</span></tt>
-symbols<ol class="loweralpha">
-<li>Accents and bold math are implemented in reverse notation. For
-example if you wished the LaTeX output to be <tt class="docutils literal"><span class="pre">\bm{\hat{\sigma}}</span></tt>
-you would give the variable the name sigmahatbm.</li>
-<li>Subscripts are denoted by a single underscore and superscripts
-by a double underscore so that <tt class="docutils literal"><span class="pre">A_{\rho\beta}^{25}</span></tt> would be
-input as A_rhobeta__25.</li>
-</ol>
-</li>
-<li>Some standard function names have been improved such as asin is now
-denoted by sin^{-1} and log by ln.</li>
-<li>Several LaTeX formats for multivectors are available:<ol class="loweralpha">
-<li>Print multivector on one line</li>
-<li>Print each grade of multivector on one line</li>
-<li>Print each base of multivector on one line</li>
-</ol>
-</li>
-<li>A LaTeX output for numpy arrays containing sympy expressions is
-implemented for up to a three dimensional array.</li>
-<li>LaTeX formatting for raw LaTeX, eqnarray, and array is available
-in simple output strings.<ol class="loweralpha">
-<li>The delimiter for raw LaTeX input is &#8216;%&#8217;.  The raw input starts
-on the line where &#8216;%&#8217; is first encountered and continues until
-the next line where &#8216;%&#8217; is encountered. It does not matter where
-&#8216;%&#8217; is in the line.</li>
-<li>The delimiter for eqnarray input is &#8216;&#64;&#8217;. The rules are the same
-as for raw input except that &#8216;=&#8217; in the first line is replaced
-be &#8216;&amp;=&amp;&#8217; and &#8216;begin{eqnarray*}&#8217; is added before the first line
-and &#8216;end{eqnarray*}&#8217; to after the last line in the group of
-lines.</li>
-<li>The delimiter for array input is &#8216;#&#8217;. The rules are the same
-as for raw input except that &#8216;begin{equation*}&#8217; is added before
-the first line and &#8216;end{equation*}&#8217; to after the last line in
-the group of lines.</li>
-</ol>
-</li>
-<li>Additional formats for partial derivatives:<ol class="loweralpha">
-<li>Same as sympy latex module</li>
-<li>Use subscript notation with partial symbol to indicate which
-variable the differentiation is with respect to.  Symbol is of
-form partial_{differentiation variable}</li>
-</ol>
-</li>
-</ol>
-</div></blockquote>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="52%" />
-<col width="48%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>printmethod</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.galgebra.latex_ex.LatexPrinter.latex_bases">
-<tt class="descname">latex_bases</tt><big>(</big><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/latex_ex.html#LatexPrinter.latex_bases"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.latex_ex.LatexPrinter.latex_bases" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generate LaTeX strings for multivector bases</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.latex_ex.MV_format">
-<tt class="descclassname">sympy.galgebra.latex_ex.</tt><tt class="descname">MV_format</tt><big>(</big><em>mv_fmt</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/latex_ex.html#MV_format"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.latex_ex.MV_format" title="Permalink to this definition">¶</a></dt>
-<dd><p>0 or 1 - Print multivector on one line</p>
-<p>2      - Print each multivector grade on one line</p>
-<p>3      - Print each multivector base on one line</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.latex_ex.fct_format">
-<tt class="descclassname">sympy.galgebra.latex_ex.</tt><tt class="descname">fct_format</tt><big>(</big><em>fct_fmt</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/latex_ex.html#fct_format"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.latex_ex.fct_format" title="Permalink to this definition">¶</a></dt>
-<dd><p>0 - Default sympy latex format</p>
-<dl class="docutils">
-<dt>1 - Do not print arguments of arbitrary functions.</dt>
-<dd>Use symbol font for arbitrary functions.
-Use enhanced symbol naming for arbitrary functions.
-Use new names for standard functions (acos -&gt; Cos^{-1})</dd>
-</dl>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.latex_ex.find_executable">
-<tt class="descclassname">sympy.galgebra.latex_ex.</tt><tt class="descname">find_executable</tt><big>(</big><em>executable</em>, <em>path=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/latex_ex.html#find_executable"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.latex_ex.find_executable" title="Permalink to this definition">¶</a></dt>
-<dd><p>Try to find &#8216;executable&#8217; in the directories listed in &#8216;path&#8217; (a
-string listing directories separated by &#8216;os.pathsep&#8217;; defaults to
-os.environ[&#8216;PATH&#8217;]).  Returns the complete filename or None if not
-found</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.latex_ex.pdiff_format">
-<tt class="descclassname">sympy.galgebra.latex_ex.</tt><tt class="descname">pdiff_format</tt><big>(</big><em>pdiff_fmt</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/latex_ex.html#pdiff_format"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.latex_ex.pdiff_format" title="Permalink to this definition">¶</a></dt>
-<dd><p>0 - Use default sympy partial derivative format</p>
-<p>1 - Contracted derivative format (no fraction symbols)</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.latex_ex.print_LaTeX">
-<tt class="descclassname">sympy.galgebra.latex_ex.</tt><tt class="descname">print_LaTeX</tt><big>(</big><em>expr</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/latex_ex.html#print_LaTeX"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.latex_ex.print_LaTeX" title="Permalink to this definition">¶</a></dt>
-<dd><p>Prints LaTeX representation of the given expression.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.latex_ex.str_format">
-<tt class="descclassname">sympy.galgebra.latex_ex.</tt><tt class="descname">str_format</tt><big>(</big><em>str_fmt</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/latex_ex.html#str_format"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.latex_ex.str_format" title="Permalink to this definition">¶</a></dt>
-<dd><p>0 - Use default sympy format</p>
-<dl class="docutils">
-<dt>1 - Use extended symbol format including multiple Greek letters in</dt>
-<dd>basic symbol (symbol preceding sub and superscripts)and in
-sub and superscripts of basic symbol and accents in basic symbol</dd>
-</dl>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.latex_ex.sym_format">
-<tt class="descclassname">sympy.galgebra.latex_ex.</tt><tt class="descname">sym_format</tt><big>(</big><em>sym_fmt</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/latex_ex.html#sym_format"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.latex_ex.sym_format" title="Permalink to this definition">¶</a></dt>
-<dd><p>0 - Use default sympy format</p>
-<dl class="docutils">
-<dt>1 - Use extended symbol format including multiple Greek letters in</dt>
-<dd>basic symbol (symbol preceding sub and superscripts)and in
-sub and superscripts of basic symbol and accents in basic symbol</dd>
-</dl>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.galgebra.latex_ex.xdvi">
-<tt class="descclassname">sympy.galgebra.latex_ex.</tt><tt class="descname">xdvi</tt><big>(</big><em>filename='tmplatex.tex'</em>, <em>debug=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/galgebra/latex_ex.html#xdvi"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.galgebra.latex_ex.xdvi" title="Permalink to this definition">¶</a></dt>
-<dd><p>Post processes LaTeX output (see comments below), adds preamble and
-postscript, generates tex file, inputs file to latex, displays resulting
-dvi file with xdvi or yap.</p>
-</dd></dl>
-
-</div>
-
-
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-        </div>
-      </div>
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-        <div class="sphinxsidebarwrapper">
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-            
-  <div class="section" id="extended-latexmodule-for-sympy">
-<span id="extended-latex"></span><h1>Extended LaTeXModule for SymPy<a class="headerlink" href="#extended-latexmodule-for-sympy" title="Permalink to this headline">¶</a></h1>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Author:</th><td class="field-body">Alan Bromborsky</td>
-</tr>
-</tbody>
-</table>
-<div class="topic">
-<p class="topic-title first">Abstract</p>
-<p>This document describes the extension of the latex module for  <a class="reference internal" href="../../matrices/immutablematrices.html#module-sympy" title="sympy"><tt class="xref py py-mod docutils literal"><span class="pre">sympy</span></tt></a>. The
-python module <tt class="xref py py-mod docutils literal"><span class="pre">latex_ex</span></tt> extends the capabilities of the current
-<tt class="xref py py-mod docutils literal"><span class="pre">latex</span></tt> (while preserving the current capabilities) to geometric algebra
-multivectors, <tt class="xref py py-mod docutils literal"><span class="pre">numpy</span></tt> array&#8217;s, and extends the ascii formatting of greek
-symbols, accents, and subscripts and superscripts.  Additionally the module is
-configured to use the print command to generate a LaTeX output file and display
-it using xdvi in Linux and yap in Windows.  To get LaTeX displayed text latex and
-xdvi must be installed on a Linux system and MikTex on a Windows system.</p>
-</div>
-<div class="section" id="extended-symbol-coding">
-<h2>Extended Symbol Coding<a class="headerlink" href="#extended-symbol-coding" title="Permalink to this headline">¶</a></h2>
-<p>One of the main extensions in <tt class="xref py py-mod docutils literal"><span class="pre">latex_ex</span></tt> is the ability to encode complex
-symbols (multiple greek letters with accents and superscripts and subscripts) is
-ascii strings containing only letters, numbers, and underscores.  These
-restrictions allow <a class="reference internal" href="../../matrices/immutablematrices.html#module-sympy" title="sympy"><tt class="xref py py-mod docutils literal"><span class="pre">sympy</span></tt></a> variable names to represent complex symbols. For
-example if we use the function <tt class="xref py py-func docutils literal"><span class="pre">symbols()</span></tt> as
-follows:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">xalpha</span><span class="p">,</span><span class="n">Gammavec__1_rho</span><span class="p">,</span><span class="n">delta__j_k</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;xalpha Gammavec__1_rho delta__j_k&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p><tt class="docutils literal"><span class="pre">symbols</span></tt> creates the three <a class="reference internal" href="../../matrices/immutablematrices.html#module-sympy" title="sympy"><tt class="xref py py-mod docutils literal"><span class="pre">sympy</span></tt></a> symbols <em>xalpha</em>,
-<em>Gammavec__1_rho</em>, and <em>delta__j_k</em>.  If these symbols are printed with the
-<tt class="xref py py-mod docutils literal"><span class="pre">latex_ex</span></tt> modules the results are</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="50%" />
-<col width="50%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head">Ascii String</th>
-<th class="head">LaTeX Output</th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-even"><td><tt class="docutils literal"><span class="pre">xalpha</span></tt></td>
-<td><span class="math">\(x\alpha\)</span></td>
-</tr>
-<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">Gammavec__1_rho</span></tt></td>
-<td><span class="math">\(\vec{\Gamma}^{1}_{\rho}\)</span></td>
-</tr>
-<tr class="row-even"><td><tt class="docutils literal"><span class="pre">delta__j_k</span></tt></td>
-<td><span class="math">\(\delta^{j}_{k}\)</span></td>
-</tr>
-</tbody>
-</table>
-<p>A single underscore denotes a subscript and a double undscore a superscript.</p>
-<p>In addition to all normal LaTeX accents boldmath is supported so that
-<tt class="docutils literal"><span class="pre">omegaomegabm</span></tt><span class="math">\(\rightarrow \Omega\boldsymbol{\Omega}\)</span> so an accent (or boldmath)
-only applies to the character (characters in the case of the string representing
-a greek letter) immediately preceeding the accent command.</p>
-</div>
-<div class="section" id="how-latexprinter-works">
-<h2>How LatexPrinter Works<a class="headerlink" href="#how-latexprinter-works" title="Permalink to this headline">¶</a></h2>
-<p>The actual <tt class="xref py py-class docutils literal"><span class="pre">LatexPrinter</span></tt> class is hidden with the helper functions
-<a class="reference internal" href="#Format" title="Format"><tt class="xref py py-func docutils literal"><span class="pre">Format()</span></tt></a>,  <a class="reference internal" href="#LaTeX" title="LaTeX"><tt class="xref py py-func docutils literal"><span class="pre">LaTeX()</span></tt></a>, and <a class="reference internal" href="#xdvi" title="xdvi"><tt class="xref py py-func docutils literal"><span class="pre">xdvi()</span></tt></a>.  <tt class="xref py py-class docutils literal"><span class="pre">LatexPrinter</span></tt> is
-setup when <a class="reference internal" href="#Format" title="Format"><tt class="xref py py-func docutils literal"><span class="pre">Format()</span></tt></a> is called.  In addition to setting format switches in
-<tt class="xref py py-class docutils literal"><span class="pre">LatexPrinter</span></tt>, <a class="reference internal" href="#Format" title="Format"><tt class="xref py py-func docutils literal"><span class="pre">Format()</span></tt></a> does two other critical tasks. Firstly,
-the <a class="reference internal" href="../../matrices/immutablematrices.html#module-sympy" title="sympy"><tt class="xref py py-mod docutils literal"><span class="pre">sympy</span></tt></a> function <tt class="xref py py-func docutils literal"><span class="pre">Basic.__str__()</span></tt> is redirected to the
-<tt class="xref py py-class docutils literal"><span class="pre">LatexPrinter</span></tt> helper function <a class="reference internal" href="#LaTeX" title="LaTeX"><tt class="xref py py-func docutils literal"><span class="pre">LaTeX()</span></tt></a>.  If nothing more that this
-were done the python print command would output LaTeX code.  Secondly,
-<em>sys.stdout</em> is redirected to a file until <a class="reference internal" href="#xdvi" title="xdvi"><tt class="xref py py-func docutils literal"><span class="pre">xdvi()</span></tt></a> is called.  This file
-is then compiled with the <strong class="program">latex</strong> program (if present) and the dvi
-output file is displayed with the <strong class="program">xdvi</strong> program (if present).  Thus
-for <tt class="xref py py-class docutils literal"><span class="pre">LatexPrinter</span></tt> to display the output in a window both
-<strong class="program">latex</strong> and <strong class="program">xdvi</strong> must be installed on your system and in the
-execution path.</p>
-<p>One problem for <tt class="xref py py-class docutils literal"><span class="pre">LatexPrinter</span></tt> is determining when equation mode should be
-use in the output formatting.  To allow for printing output not in equation mode the
-program attemps to determine from the output string context when to use equation mode.
-The current test is to use equation mode if the string contains an =, _, ^, or \.
-This is not a foolproof method.  The safest thing to do if you wish to print an object, <em>X</em>,
-in math mode is to use <em>print &#8216;X =&#8217;,X</em> so the = sign triggers math mode.</p>
-</div>
-<div class="section" id="latexprinter-functions">
-<h2>LatexPrinter Functions<a class="headerlink" href="#latexprinter-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="latexprinter-class-functions-for-extending-latexprinter-class">
-<h3>LatexPrinter Class Functions for Extending LatexPrinter Class<a class="headerlink" href="#latexprinter-class-functions-for-extending-latexprinter-class" title="Permalink to this headline">¶</a></h3>
-<p>The <tt class="xref py py-class docutils literal"><span class="pre">LatexPrinter</span></tt> class functions are not called directly, but rather
-are called when <tt class="xref py py-func docutils literal"><span class="pre">print()</span></tt>, <a class="reference internal" href="#LaTeX" title="LaTeX"><tt class="xref py py-func docutils literal"><span class="pre">LaTeX()</span></tt></a>, or <tt class="xref py py-func docutils literal"><span class="pre">str()</span></tt> are called.  The
-two new functions for extending the <tt class="xref py py-mod docutils literal"><span class="pre">latex</span></tt> module are
-<a class="reference internal" href="#_print_ndarray" title="_print_ndarray"><tt class="xref py py-func docutils literal"><span class="pre">_print_ndarray()</span></tt></a> and <a class="reference internal" href="#_print_MV" title="_print_MV"><tt class="xref py py-func docutils literal"><span class="pre">_print_MV()</span></tt></a>.  Some other functions in
-<tt class="xref py py-class docutils literal"><span class="pre">LatexPrinter</span></tt> have been modified to increase their utility.</p>
-<dl class="function">
-<dt id="_print_ndarray">
-<tt class="descname">_print_ndarray</tt><big>(</big><em>self</em>, <em>expr</em><big>)</big><a class="headerlink" href="#_print_ndarray" title="Permalink to this definition">¶</a></dt>
-<dd><p><a class="reference internal" href="#_print_ndarray" title="_print_ndarray"><tt class="xref py py-func docutils literal"><span class="pre">_print_ndarray()</span></tt></a> returns a latex formatted string for the <em>expr</em> equal to
-a <tt class="xref py py-class docutils literal"><span class="pre">numpy</span></tt> array with elements that can be <a class="reference internal" href="../../matrices/immutablematrices.html#module-sympy" title="sympy"><tt class="xref py py-class docutils literal"><span class="pre">sympy</span></tt></a> expressions.
-The <tt class="xref py py-class docutils literal"><span class="pre">numpy</span></tt> array&#8217;s can have up to three dimensions.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="_print_MV">
-<tt class="descname">_print_MV</tt><big>(</big><em>self</em>, <em>expr</em><big>)</big><a class="headerlink" href="#_print_MV" title="Permalink to this definition">¶</a></dt>
-<dd><p><a class="reference internal" href="#_print_MV" title="_print_MV"><tt class="xref py py-func docutils literal"><span class="pre">_print_MV()</span></tt></a> returns a latex formatted string for the <em>expr</em> equal to a
-<tt class="xref py py-class docutils literal"><span class="pre">GA</span></tt> multivector.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="str_basic">
-<tt class="descname">str_basic</tt><big>(</big><em>in_str</em><big>)</big><a class="headerlink" href="#str_basic" title="Permalink to this definition">¶</a></dt>
-<dd><p><a class="reference internal" href="#str_basic" title="str_basic"><tt class="xref py py-func docutils literal"><span class="pre">str_basic()</span></tt></a> returns a string without the latex formatting provided by
-<tt class="xref py py-class docutils literal"><span class="pre">LatexPrinter</span></tt>.  This is needed since <tt class="xref py py-class docutils literal"><span class="pre">LatexPrinter</span></tt> takes over
-the <tt class="xref py py-func docutils literal"><span class="pre">str()</span></tt> fuction and there are instances when the unformatted string is
-needed such as during the automatic generation of multivector coefficients and
-the reduction of multivector coefficients for printing.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="helper-functions-for-extending-latexprinter-class">
-<h3>Helper Functions for Extending LatexPrinter Class<a class="headerlink" href="#helper-functions-for-extending-latexprinter-class" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="Format">
-<tt class="descname">Format</tt><big>(</big><em>fmt='1 1 1 1'</em><big>)</big><a class="headerlink" href="#Format" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">Format</span></tt> iniailizes <tt class="docutils literal"><span class="pre">LatexPrinter</span></tt> and set the format for <tt class="docutils literal"><span class="pre">sympy</span></tt> symbols,
-functions, and derivatives and for <tt class="docutils literal"><span class="pre">GA</span></tt> multivectors. The switch are encoded
-in the text string argument of <tt class="docutils literal"><span class="pre">Format</span></tt> as follows.  It is assumed that the
-text string <tt class="docutils literal"><span class="pre">fmt</span></tt> always contains four integers separated by blanks.</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="8%" />
-<col width="12%" />
-<col width="80%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head">Position</th>
-<th class="head">Switch</th>
-<th class="head">Values</th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-even"><td><span class="math">\(1^{st}\)</span></td>
-<td>symbol</td>
-<td>0: Use symbol encoding in <tt class="docutils literal"><span class="pre">latex.py</span></tt></td>
-</tr>
-<tr class="row-odd"><td>&nbsp;</td>
-<td>&nbsp;</td>
-<td>1: Use extended symbol encoding in <tt class="docutils literal"><span class="pre">latex_ex.py</span></tt></td>
-</tr>
-<tr class="row-even"><td><span class="math">\(2^{nd}\)</span></td>
-<td>function</td>
-<td>0: Use symbol encoding in  <tt class="docutils literal"><span class="pre">latex.py</span></tt>. Print functions args, use <tt class="docutils literal"><span class="pre">\operator{</span> <span class="pre">}</span></tt> format.</td>
-</tr>
-<tr class="row-odd"><td>&nbsp;</td>
-<td>&nbsp;</td>
-<td>1: Do not print function args. Do not use <tt class="docutils literal"><span class="pre">\operator{}</span></tt> format. Suppress printing of function arguments.</td>
-</tr>
-<tr class="row-even"><td><span class="math">\(3^{rd}\)</span></td>
-<td>partial derivative</td>
-<td>0: Use partial derivative format in <tt class="docutils literal"><span class="pre">latex.py</span></tt>.</td>
-</tr>
-<tr class="row-odd"><td>&nbsp;</td>
-<td>&nbsp;</td>
-<td>1: Use format <span class="math">\(\partial_{x}\)</span> instead of <span class="math">\(\partial/\partial x\)</span>.</td>
-</tr>
-<tr class="row-even"><td><span class="math">\(4^{th}\)</span></td>
-<td>multivector</td>
-<td>1: Print entire multivector on one line.</td>
-</tr>
-<tr class="row-odd"><td>&nbsp;</td>
-<td>&nbsp;</td>
-<td>2: Print each grade of multivector on one line.</td>
-</tr>
-<tr class="row-even"><td>&nbsp;</td>
-<td>&nbsp;</td>
-<td>3: Print each base of multivector on one line.</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="function">
-<dt id="LaTeX">
-<tt class="descname">LaTeX</tt><big>(</big><em>expr</em>, <em>inline=True</em><big>)</big><a class="headerlink" href="#LaTeX" title="Permalink to this definition">¶</a></dt>
-<dd><p><a class="reference internal" href="#LaTeX" title="LaTeX"><tt class="xref py py-func docutils literal"><span class="pre">LaTeX()</span></tt></a> returns the latex formatted string for the <a class="reference internal" href="../../matrices/immutablematrices.html#module-sympy" title="sympy"><tt class="xref py py-class docutils literal"><span class="pre">sympy</span></tt></a>,
-<tt class="xref py py-class docutils literal"><span class="pre">GA</span></tt>, or <tt class="xref py py-class docutils literal"><span class="pre">numpy</span></tt>  expression <em>expr</em>.  This is needed since
-<tt class="xref py py-class docutils literal"><span class="pre">numpy</span></tt> cannot be subclassed and hence cannot be used with the
-<tt class="xref py py-class docutils literal"><span class="pre">LatexPrinter</span></tt> modified <tt class="xref py py-func docutils literal"><span class="pre">print()</span></tt> command. Thus is <em>A</em> is a
-<tt class="xref py py-class docutils literal"><span class="pre">numpy</span></tt> array containing <a class="reference internal" href="../../matrices/immutablematrices.html#module-sympy" title="sympy"><tt class="xref py py-class docutils literal"><span class="pre">sympy</span></tt></a> expressions one cannot simply
-code</p>
-<div class="highlight-python"><pre>print A</pre>
-</div>
-<p>but rather must use</p>
-<div class="highlight-python"><pre>print LaTeX(A)</pre>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="xdvi">
-<tt class="descname">xdvi</tt><big>(</big><em>filename='tmplatex.tex'</em>, <em>debug=False</em><big>)</big><a class="headerlink" href="#xdvi" title="Permalink to this definition">¶</a></dt>
-<dd><p><a class="reference internal" href="#xdvi" title="xdvi"><tt class="xref py py-func docutils literal"><span class="pre">xdvi()</span></tt></a> postprocesses the output of the print statements and generates the
-latex file with name <em>filename</em>.  If the <strong class="program">latex</strong> and <strong class="program">xdvi</strong>
-programs are present on the system they are invoked to display the latex file in
-a window.  If <em>debug=True</em> the associated output of <strong class="program">latex</strong> is sent to
-<em>stdout</em>, otherwise it is sent to  <em>/dev/null</em> for linux and <em>NUL</em> for Windows.
-If <tt class="xref py py-class docutils literal"><span class="pre">LatexPrinter</span></tt> has not been initialized <a class="reference internal" href="#xdvi" title="xdvi"><tt class="xref py py-func docutils literal"><span class="pre">xdvi()</span></tt></a> does nothing.  After the
-.dvi file is generated it is displayed with <strong class="program">xdvi</strong> for linux (if latex and xdvi
-are installed ) and <strong class="program">yap</strong> for Windows (if MikTex is installed).</p>
-</dd></dl>
-
-<p>The functions <a class="reference internal" href="#sym_format" title="sym_format"><tt class="xref py py-func docutils literal"><span class="pre">sym_format()</span></tt></a>, <a class="reference internal" href="#fct_format" title="fct_format"><tt class="xref py py-func docutils literal"><span class="pre">fct_format()</span></tt></a>, <a class="reference internal" href="#pdiff_format" title="pdiff_format"><tt class="xref py py-func docutils literal"><span class="pre">pdiff_format()</span></tt></a>, and
-<a class="reference internal" href="#MV_format" title="MV_format"><tt class="xref py py-func docutils literal"><span class="pre">MV_format()</span></tt></a> allow one to change various formatting aspects of the
-<tt class="xref py py-class docutils literal"><span class="pre">LatexPrinter</span></tt>.  They do not initialize the class and if they are called
-with the class not initialized they have no effect.  These functions and the
-function <a class="reference internal" href="#xdvi" title="xdvi"><tt class="xref py py-func docutils literal"><span class="pre">xdvi()</span></tt></a> are designed so that if the <tt class="xref py py-class docutils literal"><span class="pre">LatexPrinter</span></tt> class is
-not initialized the program output is as if the <tt class="xref py py-class docutils literal"><span class="pre">LatexPrinter</span></tt> class is
-not used. Thus all one needs to do to get simple ascii output (possibly for
-program debugging) is to comment out the one function call that initializes the
-<tt class="xref py py-class docutils literal"><span class="pre">LatexPrinter</span></tt> class.  All other <tt class="xref py py-mod docutils literal"><span class="pre">latex_ex</span></tt> function calls can
-remain in the program and have no effect on program output.</p>
-<dl class="function">
-<dt id="sym_format">
-<tt class="descname">sym_format</tt><big>(</big><em>sym_fmt</em><big>)</big><a class="headerlink" href="#sym_format" title="Permalink to this definition">¶</a></dt>
-<dd><p><a class="reference internal" href="#sym_format" title="sym_format"><tt class="xref py py-func docutils literal"><span class="pre">sym_format()</span></tt></a> allows one to change the latex format options for
-<a class="reference internal" href="../../matrices/immutablematrices.html#module-sympy" title="sympy"><tt class="xref py py-class docutils literal"><span class="pre">sympy</span></tt></a> symbol output independent of other format switches (see
-<span class="math">\(1^{st}\)</span> switch in Table I).</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="fct_format">
-<tt class="descname">fct_format</tt><big>(</big><em>fct_fmt</em><big>)</big><a class="headerlink" href="#fct_format" title="Permalink to this definition">¶</a></dt>
-<dd><p><a class="reference internal" href="#fct_format" title="fct_format"><tt class="xref py py-func docutils literal"><span class="pre">fct_format()</span></tt></a> allows one to change the latex format options for
-<a class="reference internal" href="../../matrices/immutablematrices.html#module-sympy" title="sympy"><tt class="xref py py-class docutils literal"><span class="pre">sympy</span></tt></a> function output independent of other format switches (see
-<span class="math">\(2^{nd}\)</span> switch in Table I).</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="pdiff_format">
-<tt class="descname">pdiff_format</tt><big>(</big><em>pdiff_fmt</em><big>)</big><a class="headerlink" href="#pdiff_format" title="Permalink to this definition">¶</a></dt>
-<dd><p><a class="reference internal" href="#pdiff_format" title="pdiff_format"><tt class="xref py py-func docutils literal"><span class="pre">pdiff_format()</span></tt></a> allows one to change the latex format options for
-<a class="reference internal" href="../../matrices/immutablematrices.html#module-sympy" title="sympy"><tt class="xref py py-class docutils literal"><span class="pre">sympy</span></tt></a> partial derivative output independent of other format switches
-(see <span class="math">\(3^{rd}\)</span> switch in Table I).</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="MV_format">
-<tt class="descname">MV_format</tt><big>(</big><em>mv_fmt</em><big>)</big><a class="headerlink" href="#MV_format" title="Permalink to this definition">¶</a></dt>
-<dd><p><a class="reference internal" href="#MV_format" title="MV_format"><tt class="xref py py-func docutils literal"><span class="pre">MV_format()</span></tt></a> allows one to change the latex format options for
-<a class="reference internal" href="../../matrices/immutablematrices.html#module-sympy" title="sympy"><tt class="xref py py-class docutils literal"><span class="pre">sympy</span></tt></a> partial derivative output independent of other format switches
-(see <span class="math">\(3^{rd}\)</span> switch in Table I).</p>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="examples">
-<h2>Examples<a class="headerlink" href="#examples" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="latexdemo-py-a-simple-example">
-<h3><strong class="program">latexdemo.py</strong> a simple example<a class="headerlink" href="#latexdemo-py-a-simple-example" title="Permalink to this headline">¶</a></h3>
-<p><strong class="program">latexdemo.py</strong> example of using <tt class="xref py py-mod docutils literal"><span class="pre">latex_ex</span></tt> with <a class="reference internal" href="../../matrices/immutablematrices.html#module-sympy" title="sympy"><tt class="xref py py-mod docutils literal"><span class="pre">sympy</span></tt></a></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="kn">import</span> <span class="nn">sys</span>
-<span class="kn">import</span> <span class="nn">sympy</span>
-<span class="kn">import</span> <span class="nn">sympy.galgebra.latex_ex</span> <span class="kn">as</span> <span class="nn">tex</span>
-
-<span class="k">if</span> <span class="n">__name__</span> <span class="o">==</span> <span class="s">&#39;__main__&#39;</span><span class="p">:</span>
-
-    <span class="n">tex</span><span class="o">.</span><span class="n">Format</span><span class="p">()</span>
-    <span class="n">xbm</span><span class="p">,</span> <span class="n">alpha_1</span><span class="p">,</span> <span class="n">delta__nugamma_r</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">symbols</span><span class="p">(</span><span class="s">&#39;xbm alpha_1 delta__nugamma_r&#39;</span><span class="p">)</span>
-
-    <span class="n">x</span> <span class="o">=</span> <span class="n">alpha_1</span><span class="o">*</span><span class="n">xbm</span><span class="o">/</span><span class="n">delta__nugamma_r</span>
-
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;x =&#39;</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-    <span class="n">tex</span><span class="o">.</span><span class="n">xdvi</span><span class="p">()</span>
-</pre></div>
-</div>
-<p>Start of Program Output</p>
-<div class="math">
-\[\begin{equation*}
-x = \frac{{\alpha}_{1} {}\boldsymbol{x}}{{\delta}^{{\nu}{\gamma}}_{r}}
-\end{equation*}\]</div><p>End of Program Output</p>
-<p>The program <strong class="program">latexdemo.py</strong> demonstrates the extended symbol naming
-conventions in <tt class="xref py py-mod docutils literal"><span class="pre">latex_ex</span></tt>.   the statment <tt class="docutils literal"><span class="pre">Format()</span></tt> starts the
-<tt class="xref py py-class docutils literal"><span class="pre">LatexPrinter</span></tt> driver with default formatting. Note that on the right
-hand side of the output that <em>xbm</em> gives <span class="math">\(\boldsymbol{x}\)</span>, <em>alpha_1</em> gives
-<span class="math">\(\alpha_{1}\)</span> and  <em>delta__nugamma_r</em> gives <span class="math">\(\delta^{\nu\gamma}_{r}\)</span>.
-Also the fraction is printed correctly.  The statment <tt class="docutils literal"><span class="pre">print</span> <span class="pre">'x</span> <span class="pre">=',x</span></tt> sends
-the string <tt class="docutils literal"><span class="pre">'x</span> <span class="pre">=</span> <span class="pre">'+str(x)</span></tt> to the output processor (<a class="reference internal" href="#xdvi" title="xdvi"><tt class="xref py py-func docutils literal"><span class="pre">xdvi()</span></tt></a>).  Because
-the string contains an <span class="math">\(=\)</span> sign the processor treats the string as an
-LaTeX equation (unnumbered).  If <tt class="docutils literal"><span class="pre">'x</span> <span class="pre">='</span></tt>  was not in the print statment a
-LaTeX error would be generated.  In the case of a <tt class="xref py py-class docutils literal"><span class="pre">GA</span></tt> multivector one
-does not need the <tt class="docutils literal"><span class="pre">'x</span> <span class="pre">='</span></tt> if the multivector has been given a name.</p>
-</div>
-<div class="section" id="maxwell-py-a-multivector-example">
-<h3><strong class="program">Maxwell.py</strong> a multivector example<a class="headerlink" href="#maxwell-py-a-multivector-example" title="Permalink to this headline">¶</a></h3>
-<p><strong class="program">Maxwell.py</strong> example of using <tt class="xref py py-mod docutils literal"><span class="pre">latex_ex</span></tt> with <tt class="xref py py-mod docutils literal"><span class="pre">GA</span></tt></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="kn">import</span> <span class="nn">sys</span>
-<span class="kn">import</span> <span class="nn">sympy</span>
-<span class="kn">import</span> <span class="nn">sympy.galgebra.GAsympy</span> <span class="kn">as</span> <span class="nn">GA</span>
-<span class="kn">import</span> <span class="nn">sympy.galgebra.latex_ex</span> <span class="kn">as</span> <span class="nn">tex</span>
-
-<span class="k">if</span> <span class="n">__name__</span> <span class="o">==</span> <span class="s">&#39;__main__&#39;</span><span class="p">:</span>
-
-    <span class="n">metric</span> <span class="o">=</span> <span class="s">&#39;1  0  0  0,&#39;</span> \
-             <span class="s">&#39;0 -1  0  0,&#39;</span> \
-             <span class="s">&#39;0  0 -1  0,&#39;</span> \
-             <span class="s">&#39;0  0  0 -1&#39;</span>
-
-    <span class="nb">vars</span> <span class="o">=</span> <span class="n">sympy</span><span class="o">.</span><span class="n">symbols</span><span class="p">(</span><span class="s">&#39;t x y z&#39;</span><span class="p">)</span>
-    <span class="n">gamma_t</span><span class="p">,</span> <span class="n">gamma_x</span><span class="p">,</span> <span class="n">gamma_y</span><span class="p">,</span> <span class="n">gamma_z</span> <span class="o">=</span> <span class="n">GA</span><span class="o">.</span><span class="n">MV</span><span class="o">.</span><span class="n">setup</span><span class="p">(</span><span class="s">&#39;gamma_t gamma_x gamma_y gamma_z&#39;</span><span class="p">,</span> <span class="n">metric</span><span class="p">,</span> <span class="bp">True</span><span class="p">,</span> <span class="nb">vars</span><span class="p">)</span>
-    <span class="n">tex</span><span class="o">.</span><span class="n">Format</span><span class="p">()</span>
-    <span class="n">I</span> <span class="o">=</span> <span class="n">GA</span><span class="o">.</span><span class="n">MV</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="s">&#39;pseudo&#39;</span><span class="p">)</span>
-    <span class="n">I</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;$I$ Pseudo-Scalar&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;I =&#39;</span><span class="p">,</span> <span class="n">I</span><span class="p">)</span>
-    <span class="n">B</span> <span class="o">=</span> <span class="n">GA</span><span class="o">.</span><span class="n">MV</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="s">&#39;vector&#39;</span><span class="p">,</span> <span class="n">fct</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">E</span> <span class="o">=</span> <span class="n">GA</span><span class="o">.</span><span class="n">MV</span><span class="p">(</span><span class="s">&#39;E&#39;</span><span class="p">,</span> <span class="s">&#39;vector&#39;</span><span class="p">,</span> <span class="n">fct</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="n">B</span><span class="o">.</span><span class="n">set_coef</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-    <span class="n">E</span><span class="o">.</span><span class="n">set_coef</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-    <span class="n">B</span> <span class="o">*=</span> <span class="n">gamma_t</span>
-    <span class="n">E</span> <span class="o">*=</span> <span class="n">gamma_t</span>
-    <span class="n">B</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-    <span class="n">E</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-    <span class="n">J</span> <span class="o">=</span> <span class="n">GA</span><span class="o">.</span><span class="n">MV</span><span class="p">(</span><span class="s">&#39;J&#39;</span><span class="p">,</span> <span class="s">&#39;vector&#39;</span><span class="p">,</span> <span class="n">fct</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;$B$ Magnetic Field Bi-Vector&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;B = Bvec gamma_0 =&#39;</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;$E$ Electric Field Bi-Vector&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;E = Evec gamma_0 =&#39;</span><span class="p">,</span> <span class="n">E</span><span class="p">)</span>
-    <span class="n">F</span> <span class="o">=</span> <span class="n">E</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">B</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;$E+IB$ Electo-Magnetic Field Bi-Vector&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;F = E+IB =&#39;</span><span class="p">,</span> <span class="n">F</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;$J$ Four Current&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;J =&#39;</span><span class="p">,</span> <span class="n">J</span><span class="p">)</span>
-    <span class="n">gradF</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">grad</span><span class="p">()</span>
-    <span class="n">gradF</span><span class="o">.</span><span class="n">convert_to_blades</span><span class="p">()</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;Geometric Derivative of Electo-Magnetic Field Bi-Vector&#39;</span><span class="p">)</span>
-    <span class="n">tex</span><span class="o">.</span><span class="n">MV_format</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\\</span><span class="s">nabla F =&#39;</span><span class="p">,</span> <span class="n">gradF</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;All Maxwell Equations are&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;</span><span class="se">\\</span><span class="s">nabla F = J&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;Div $E$ and Curl $H$ Equations&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;&lt;</span><span class="se">\\</span><span class="s">nabla F&gt;_1 -J =&#39;</span><span class="p">,</span> <span class="n">gradF</span><span class="o">.</span><span class="n">project</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">J</span><span class="p">,</span> <span class="s">&#39; = 0&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;Curl $E$ and Div $B$ equations&#39;</span><span class="p">)</span>
-    <span class="k">print</span><span class="p">(</span><span class="s">&#39;&lt;</span><span class="se">\\</span><span class="s">nabla F&gt;_3 =&#39;</span><span class="p">,</span> <span class="n">gradF</span><span class="o">.</span><span class="n">project</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span> <span class="s">&#39; = 0&#39;</span><span class="p">)</span>
-    <span class="n">tex</span><span class="o">.</span><span class="n">xdvi</span><span class="p">(</span><span class="n">filename</span><span class="o">=</span><span class="s">&#39;Maxwell.tex&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Start of Program Output</p>
-<p><span class="math">\(I\)</span> Pseudo-Scalar</p>
-<div class="math">
-\[\begin{equation*}
-I = {\gamma}_{t}{\gamma}_{x}{\gamma}_{y}{\gamma}_{z}
-\end{equation*}\]</div><p><span class="math">\(B\)</span> Magnetic Field Bi-Vector</p>
-<div class="math">
-\[\begin{equation*}
-B = - {B^{x}}{\gamma}_{t}{\gamma}_{x}- {B^{y}}{\gamma}_{t}{\gamma}_{y}- {B^{z}}{\gamma}_{t}{\gamma}_{z}
-\end{equation*}\]</div><p><span class="math">\(F\)</span> Electric Field Bi-Vector</p>
-<div class="math">
-\[\begin{equation*}
-E = - {E^{x}}{\gamma}_{t}{\gamma}_{x}- {E^{y}}{\gamma}_{t}{\gamma}_{y}- {E^{z}}{\gamma}_{t}{\gamma}_{z}
-\end{equation*}\]</div><p><span class="math">\(E+IB\)</span> Electo-Magnetic Field Bi-Vector</p>
-<div class="math">
-\[\begin{equation*}
-F = - {E^{x}}{\gamma}_{t}{\gamma}_{x}- {E^{y}}{\gamma}_{t}{\gamma}_{y}- {B^{z}}{\gamma}_{x}{\gamma}_{y}- {E^{z}}{\gamma}_{t}{\gamma}_{z}+ {B^{y}}{\gamma}_{x}{\gamma}_{z}- {B^{x}}{\gamma}_{y}{\gamma}_{z}
-\end{equation*}\]</div><p><span class="math">\(J\)</span> Four Current</p>
-<div class="math">
-\[\begin{equation*}
-J =  {J^{t}}{\gamma}_{t}+ {J^{x}}{\gamma}_{x}+ {J^{y}}{\gamma}_{y}+ {J^{z}}{\gamma}_{z}
-\end{equation*}\]</div><p>Geometric Derivative of Electo-Magnetic Field Bi-Vector</p>
-<div class="math">
-\[\begin{align*}
-\nabla F & =   \left(\partial_{z} {E^{z}} + \partial_{y} {E^{y}} + \partial_{x} {E^{x}}\right){\gamma}_{t} \\ & + \left(-\partial_{t} {E^{x}} + \partial_{y} {B^{z}} -\partial_{z} {B^{y}}\right){\gamma}_{x} \\ & + \left(\partial_{z} {B^{x}} -\partial_{t} {E^{y}} -\partial_{x} {B^{z}}\right){\gamma}_{y} \\ & + \left(-\partial_{y} {B^{x}} -\partial_{t} {E^{z}} + \partial_{x} {B^{y}}\right){\gamma}_{z} \\ & + \left(-\partial_{x} {E^{y}} -\partial_{t} {B^{z}} + \partial_{y} {E^{x}}\right){\gamma}_{t}{\gamma}_{x}{\gamma}_{y} \\ & + \left(-\partial_{x} {E^{z}} + \partial_{t} {B^{y}} + \partial_{z} {E^{x}}\right){\gamma}_{t}{\gamma}_{x}{\gamma}_{z} \\ & + \left(-\partial_{t} {B^{x}} -\partial_{y} {E^{z}} + \partial_{z} {E^{y}}\right){\gamma}_{t}{\gamma}_{y}{\gamma}_{z} \\ & + \left(\partial_{y} {B^{y}} + \partial_{z} {B^{z}} + \partial_{x} {B^{x}}\right){\gamma}_{x}{\gamma}_{y}{\gamma}_{z}\end{align*}\]</div><p>All Maxwell Equations are</p>
-<div class="math">
-\[\begin{equation*}
-\nabla F = J
-\end{equation*}\]</div><p>Div <span class="math">\(E\)</span> and Curl <span class="math">\(H\)</span> Equations</p>
-<div class="math">
-\[\begin{align*}
-<\nabla F>_1 -J & =   \left(-{J^{t}} + \partial_{z} {E^{z}} + \partial_{y} {E^{y}} + \partial_{x} {E^{x}}\right){\gamma}_{t} \\ & + \left(-{J^{x}} -\partial_{t} {E^{x}} + \partial_{y} {B^{z}} -\partial_{z} {B^{y}}\right){\gamma}_{x} \\ & + \left(\partial_{z} {B^{x}} -\partial_{t} {E^{y}} -{J^{y}} -\partial_{x} {B^{z}}\right){\gamma}_{y} \\ & + \left(-\partial_{y} {B^{x}} -\partial_{t} {E^{z}} -{J^{z}} + \partial_{x} {B^{y}}\right){\gamma}_{z}\end{align*}\]</div><div class="math">
-\[\begin{equation*}
-  = 0
-\end{equation*}\]</div><p>Curl <span class="math">\(E\)</span> and Div <span class="math">\(B\)</span> equations</p>
-<div class="math">
-\[\begin{align*}
-<\nabla F>_3 & =   \left( -\partial_{x} {E^{y}} -\partial_{t} {B^{z}} + \partial_{y} {E^{x}}\right){\gamma}_{t}\wedge {\gamma}_{x}\wedge {\gamma}_{y} \\ & + \left( -\partial_{x} {E^{z}} + \partial_{t} {B^{y}} + \partial_{z} {E^{x}}\right){\gamma}_{t}\wedge {\gamma}_{x}\wedge {\gamma}_{z} \\ & + \left( -\partial_{t} {B^{x}} -\partial_{y} {E^{z}} + \partial_{z} {E^{y}}\right){\gamma}_{t}\wedge {\gamma}_{y}\wedge {\gamma}_{z} \\ & + \left( \partial_{y} {B^{y}} + \partial_{z} {B^{z}} + \partial_{x} {B^{x}}\right){\gamma}_{x}\wedge {\gamma}_{y}\wedge {\gamma}_{z}\end{align*}\]</div><div class="math">
-\[\begin{equation*}
-  = 0
-\end{equation*}\]</div><p>End of Program Output</p>
-<p>The program <strong class="program">Maxwell.py</strong> demonstrates the use of the
-<tt class="xref py py-class docutils literal"><span class="pre">LatexPrinter</span></tt> class with the <tt class="xref py py-mod docutils literal"><span class="pre">GA</span></tt> module multivector class,
-<a class="reference internal" href="../../galgebra/GA/GAsympy.html#MV" title="MV"><tt class="xref py py-class docutils literal"><span class="pre">MV</span></tt></a>.  The <a class="reference internal" href="#Format" title="Format"><tt class="xref py py-func docutils literal"><span class="pre">Format()</span></tt></a> call initializes <tt class="xref py py-class docutils literal"><span class="pre">LatexPrinter</span></tt>.  The
-only other explicit <tt class="xref py py-mod docutils literal"><span class="pre">latex_x</span></tt>  module formatting statement used is
-<tt class="docutils literal"><span class="pre">MV_format(3)</span></tt>.  This statment changes the multivector latex format so that
-instead of printing the entire multivector on one line, which would run off the
-page, each multivector base and its coefficient are printed on individual lines
-using the latex align environment.  Another option used is that the printing of
-function arguments is suppressed since <span class="math">\(E\)</span>, <span class="math">\(B\)</span>, <span class="math">\(J\)</span>, and
-<span class="math">\(F\)</span> are multivector fields and printing out the argument,
-<span class="math">\((t,x,y,z)\)</span>, for every field component would greatly lengthen the output
-and make it more difficult to format in a pleasing way.</p>
-</div>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Extended LaTeXModule for SymPy</a><ul>
-<li><a class="reference internal" href="#extended-symbol-coding">Extended Symbol Coding</a></li>
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-  <div class="section" id="module-sympy.geometry">
-<span id="geometry-module"></span><h1>Geometry Module<a class="headerlink" href="#module-sympy.geometry" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="introduction">
-<h2>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline">¶</a></h2>
-<p>The geometry module for SymPy allows one to create two-dimensional geometrical
-entities, such as lines and circles, and query information about these
-entities. This could include asking the area of an ellipse, checking for
-collinearity of a set of points, or finding the intersection between two lines.
-The primary use case of the module involves entities with numerical values, but
-it is possible to also use symbolic representations.</p>
-</div>
-<div class="section" id="available-entities">
-<h2>Available Entities<a class="headerlink" href="#available-entities" title="Permalink to this headline">¶</a></h2>
-<p>The following entities are currently available in the geometry module:</p>
-<ul class="simple">
-<li>Point</li>
-<li>Line, Ray, Segment</li>
-<li>Ellipse, Circle</li>
-<li>Polygon, RegularPolygon, Triangle</li>
-</ul>
-<p>Most of the work one will do will be through the properties and methods of
-these entities, but several global methods exist for one&#8217;s usage:</p>
-<ul class="simple">
-<li>intersection(entity1, entity2)</li>
-<li>are_similar(entity1, entity2)</li>
-<li>convex_hull(points)</li>
-</ul>
-<p>For a full API listing and an explanation of the methods and their return
-values please see the list of classes at the end of this document.</p>
-</div>
-<div class="section" id="example-usage">
-<h2>Example Usage<a class="headerlink" href="#example-usage" title="Permalink to this headline">¶</a></h2>
-<p>The following Python session gives one an idea of how to work with some of the
-geometry module.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zp</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Point</span><span class="o">.</span><span class="n">is_collinear</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Point</span><span class="o">.</span><span class="n">is_collinear</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">zp</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">zp</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">area</span>
-<span class="go">1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">medians</span><span class="p">[</span><span class="n">x</span><span class="p">]</span>
-<span class="go">Segment(Point(0, 0), Point(1, 1/2))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Segment</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">Segment(Point(0, 0), Point(1, 1/2))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">t</span><span class="o">.</span><span class="n">medians</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">intersection</span><span class="p">(</span><span class="n">m</span><span class="p">[</span><span class="n">x</span><span class="p">],</span> <span class="n">m</span><span class="p">[</span><span class="n">y</span><span class="p">],</span> <span class="n">m</span><span class="p">[</span><span class="n">zp</span><span class="p">])</span>
-<span class="go">[Point(2/3, 1/3)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">Circle</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span><span class="o">.</span><span class="n">is_tangent</span><span class="p">(</span><span class="n">l</span><span class="p">)</span> <span class="c"># is l tangent to c?</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span><span class="o">.</span><span class="n">is_tangent</span><span class="p">(</span><span class="n">l</span><span class="p">)</span> <span class="c"># is l tangent to c?</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">intersection</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">l</span><span class="p">)</span>
-<span class="go">[Point(-5*sqrt(2)/2, -5*sqrt(2)/2), Point(5*sqrt(2)/2, 5*sqrt(2)/2)]</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="intersection-of-medians">
-<h2>Intersection of medians<a class="headerlink" href="#intersection-of-medians" title="Permalink to this headline">¶</a></h2>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Triangle</span><span class="p">,</span> <span class="n">intersection</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&quot;a,b&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">area</span>
-<span class="go">a*b/2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">medians</span><span class="p">[</span><span class="n">x</span><span class="p">]</span>
-<span class="go">Segment(Point(0, 0), Point(3*a/2, b/2))</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">intersection</span><span class="p">(</span><span class="n">t</span><span class="o">.</span><span class="n">medians</span><span class="p">[</span><span class="n">x</span><span class="p">],</span> <span class="n">t</span><span class="o">.</span><span class="n">medians</span><span class="p">[</span><span class="n">y</span><span class="p">],</span> <span class="n">t</span><span class="o">.</span><span class="n">medians</span><span class="p">[</span><span class="n">z</span><span class="p">])</span>
-<span class="go">[Point(a, b/3)]</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="an-in-depth-example-pappus-theorem">
-<h2>An in-depth example: Pappus&#8217; Theorem<a class="headerlink" href="#an-in-depth-example-pappus-theorem" title="Permalink to this headline">¶</a></h2>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l2</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">subs_point</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">val</span><span class="p">):</span>
-<span class="gp">... </span>   <span class="sd">&quot;&quot;&quot;Take an arbitrary point and make it a fixed point.&quot;&quot;&quot;</span>
-<span class="gp">... </span>   <span class="n">t</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">... </span>   <span class="n">ap</span> <span class="o">=</span> <span class="n">l</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">()</span>
-<span class="gp">... </span>   <span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="n">ap</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">val</span><span class="p">),</span> <span class="n">ap</span><span class="o">.</span><span class="n">y</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">val</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p11</span> <span class="o">=</span> <span class="n">subs_point</span><span class="p">(</span><span class="n">l1</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p12</span> <span class="o">=</span> <span class="n">subs_point</span><span class="p">(</span><span class="n">l1</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p13</span> <span class="o">=</span> <span class="n">subs_point</span><span class="p">(</span><span class="n">l1</span><span class="p">,</span> <span class="mi">11</span><span class="p">)</span>
-<span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p21</span> <span class="o">=</span> <span class="n">subs_point</span><span class="p">(</span><span class="n">l2</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p22</span> <span class="o">=</span> <span class="n">subs_point</span><span class="p">(</span><span class="n">l2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p23</span> <span class="o">=</span> <span class="n">subs_point</span><span class="p">(</span><span class="n">l2</span><span class="p">,</span> <span class="mi">13</span><span class="p">)</span>
-<span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ll1</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p11</span><span class="p">,</span> <span class="n">p22</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ll2</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p11</span><span class="p">,</span> <span class="n">p23</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ll3</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p12</span><span class="p">,</span> <span class="n">p21</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ll4</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p12</span><span class="p">,</span> <span class="n">p23</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ll5</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p13</span><span class="p">,</span> <span class="n">p21</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ll6</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p13</span><span class="p">,</span> <span class="n">p22</span><span class="p">)</span>
-<span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pp1</span> <span class="o">=</span> <span class="n">intersection</span><span class="p">(</span><span class="n">ll1</span><span class="p">,</span> <span class="n">ll3</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pp2</span> <span class="o">=</span> <span class="n">intersection</span><span class="p">(</span><span class="n">ll2</span><span class="p">,</span> <span class="n">ll5</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pp3</span> <span class="o">=</span> <span class="n">intersection</span><span class="p">(</span><span class="n">ll4</span><span class="p">,</span> <span class="n">ll6</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Point</span><span class="o">.</span><span class="n">is_collinear</span><span class="p">(</span><span class="n">pp1</span><span class="p">,</span> <span class="n">pp2</span><span class="p">,</span> <span class="n">pp3</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="miscellaneous-notes">
-<h2>Miscellaneous Notes<a class="headerlink" href="#miscellaneous-notes" title="Permalink to this headline">¶</a></h2>
-<ul class="simple">
-<li>The area property of Polygon and Triangle may return a positive or
-negative value, depending on whether or not the points are oriented
-counter-clockwise or clockwise, respectively. If you always want a
-positive value be sure to use the <tt class="docutils literal"><span class="pre">abs</span></tt> function.</li>
-<li>Although Polygon can refer to any type of polygon, the code has been
-written for simple polygons. Hence, expect potential problems if dealing
-with complex polygons (overlapping sides).</li>
-<li>Since !SymPy is still in its infancy some things may not simplify
-properly and hence some things that should return True (e.g.,
-Point.is_collinear) may not actually do so. Similarly, attempting to find
-the intersection of entities that do intersect may result in an empty
-result.</li>
-</ul>
-</div>
-<div class="section" id="future-work">
-<h2>Future Work<a class="headerlink" href="#future-work" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="truth-setting-expressions">
-<h3>Truth Setting Expressions<a class="headerlink" href="#truth-setting-expressions" title="Permalink to this headline">¶</a></h3>
-<p>When one deals with symbolic entities, it often happens that an assertion
-cannot be guaranteed. For example, consider the following code:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Symbol</span><span class="p">,</span> <span class="s">&#39;xyz&#39;</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span><span class="n">p2</span><span class="p">,</span><span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Point</span><span class="o">.</span><span class="n">is_collinear</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>Even though the result is currently <tt class="docutils literal"><span class="pre">False</span></tt>, this is not <em>always</em> true. If the
-quantity <span class="math">\(z - y - 2*y*z + 2*y**2 == 0\)</span> then the points will be collinear. It
-would be really nice to inform the user of this because such a quantity may be
-useful to a user for further calculation and, at the very least, being nice to
-know. This could be potentially done by returning an object (e.g.,
-GeometryResult) that the user could use. This actually would not involve an
-extensive amount of work.</p>
-</div>
-<div class="section" id="three-dimensions-and-beyond">
-<h3>Three Dimensions and Beyond<a class="headerlink" href="#three-dimensions-and-beyond" title="Permalink to this headline">¶</a></h3>
-<p>Currently there are no plans for extending the module to three dimensions, but
-it certainly would be a good addition. This would probably involve a fair
-amount of work since many of the algorithms used are specific to two
-dimensions.</p>
-</div>
-<div class="section" id="geometry-visualization">
-<h3>Geometry Visualization<a class="headerlink" href="#geometry-visualization" title="Permalink to this headline">¶</a></h3>
-<p>The plotting module is capable of plotting geometric entities. See
-<a class="reference external" href="https://github.com/sympy/sympy/wiki/Plotting-Module">Plotting Geometric Entities</a> in
-the plotting module entry.</p>
-</div>
-</div>
-<div class="section" id="api-reference">
-<h2>API Reference<a class="headerlink" href="#api-reference" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="module-sympy.geometry.entity">
-<span id="entities"></span><h3>Entities<a class="headerlink" href="#module-sympy.geometry.entity" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.geometry.entity.GeometryEntity">
-<em class="property">class </em><tt class="descclassname">sympy.geometry.entity.</tt><tt class="descname">GeometryEntity</tt><a class="reference internal" href="../_modules/sympy/geometry/entity.html#GeometryEntity"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.entity.GeometryEntity" title="Permalink to this definition">¶</a></dt>
-<dd><p>The base class for all geometrical entities.</p>
-<p>This class doesn&#8217;t represent any particular geometric entity, it only
-provides the implementation of some methods common to all subclasses.</p>
-<dl class="function">
-<dt id="sympy.geometry.entity.GeometryEntity.encloses">
-<tt class="descname">encloses</tt><big>(</big><em>self</em>, <em>o</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/entity.html#GeometryEntity.encloses"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.entity.GeometryEntity.encloses" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if o is inside (not on or outside) the boundaries of self.</p>
-<p>The object will be decomposed into Points and individual Entities need
-only define an encloses_point method for their class.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.encloses_point" title="sympy.geometry.ellipse.Ellipse.encloses_point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.ellipse.Ellipse.encloses_point</span></tt></a>, <a class="reference internal" href="#sympy.geometry.polygon.Polygon.encloses_point" title="sympy.geometry.polygon.Polygon.encloses_point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.polygon.Polygon.encloses_point</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">RegularPolygon</span><span class="p">,</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Polygon</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span>  <span class="o">=</span> <span class="n">Polygon</span><span class="p">(</span><span class="o">*</span><span class="n">RegularPolygon</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">vertices</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t2</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">(</span><span class="o">*</span><span class="n">RegularPolygon</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">vertices</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t2</span><span class="o">.</span><span class="n">encloses</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">encloses</span><span class="p">(</span><span class="n">t2</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.entity.GeometryEntity.intersection">
-<tt class="descname">intersection</tt><big>(</big><em>self</em>, <em>o</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/entity.html#GeometryEntity.intersection"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.entity.GeometryEntity.intersection" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a list of all of the intersections of self with o.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.util.intersection" title="sympy.geometry.util.intersection"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.util.intersection</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>An entity is not required to implement this method.</p>
-<p>If two different types of entities can intersect, the item with
-higher index in ordering_of_classes should implement
-intersections with anything having a lower index.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.entity.GeometryEntity.is_similar">
-<tt class="descname">is_similar</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/entity.html#GeometryEntity.is_similar"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.entity.GeometryEntity.is_similar" title="Permalink to this definition">¶</a></dt>
-<dd><p>Is this geometrical entity similar to another geometrical entity?</p>
-<p>Two entities are similar if a uniform scaling (enlarging or
-shrinking) of one of the entities will allow one to obtain the other.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.entity.GeometryEntity.scale" title="sympy.geometry.entity.GeometryEntity.scale"><tt class="xref py py-obj docutils literal"><span class="pre">scale</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>This method is not intended to be used directly but rather
-through the <span class="math">\(are_similar\)</span> function found in util.py.
-An entity is not required to implement this method.
-If two different types of entities can be similar, it is only
-required that one of them be able to determine this.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.entity.GeometryEntity.rotate">
-<tt class="descname">rotate</tt><big>(</big><em>self</em>, <em>angle</em>, <em>pt=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/entity.html#GeometryEntity.rotate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.entity.GeometryEntity.rotate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Rotate <tt class="docutils literal"><span class="pre">angle</span></tt> radians counterclockwise about Point <tt class="docutils literal"><span class="pre">pt</span></tt>.</p>
-<p>The default pt is the origin, Point(0, 0)</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.entity.GeometryEntity.scale" title="sympy.geometry.entity.GeometryEntity.scale"><tt class="xref py py-obj docutils literal"><span class="pre">scale</span></tt></a>, <a class="reference internal" href="#sympy.geometry.entity.GeometryEntity.translate" title="sympy.geometry.entity.GeometryEntity.translate"><tt class="xref py py-obj docutils literal"><span class="pre">translate</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">RegularPolygon</span><span class="p">,</span> <span class="n">Polygon</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">(</span><span class="o">*</span><span class="n">RegularPolygon</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">vertices</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="c"># vertex on x axis</span>
-<span class="go">Triangle(Point(1, 0), Point(-1/2, sqrt(3)/2), Point(-1/2, -sqrt(3)/2))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">rotate</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="c"># vertex on y axis now</span>
-<span class="go">Triangle(Point(0, 1), Point(-sqrt(3)/2, -1/2), Point(sqrt(3)/2, -1/2))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.entity.GeometryEntity.scale">
-<tt class="descname">scale</tt><big>(</big><em>self</em>, <em>x=1</em>, <em>y=1</em>, <em>pt=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/entity.html#GeometryEntity.scale"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.entity.GeometryEntity.scale" title="Permalink to this definition">¶</a></dt>
-<dd><p>Scale the object by multiplying the x,y-coordinates by x and y.</p>
-<p>If pt is given, the scaling is done relative to that point; the
-object is shifted by -pt, scaled, and shifted by pt.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.entity.GeometryEntity.rotate" title="sympy.geometry.entity.GeometryEntity.rotate"><tt class="xref py py-obj docutils literal"><span class="pre">rotate</span></tt></a>, <a class="reference internal" href="#sympy.geometry.entity.GeometryEntity.translate" title="sympy.geometry.entity.GeometryEntity.translate"><tt class="xref py py-obj docutils literal"><span class="pre">translate</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">RegularPolygon</span><span class="p">,</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Polygon</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">(</span><span class="o">*</span><span class="n">RegularPolygon</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">vertices</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span>
-<span class="go">Triangle(Point(1, 0), Point(-1/2, sqrt(3)/2), Point(-1/2, -sqrt(3)/2))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">scale</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">Triangle(Point(2, 0), Point(-1, sqrt(3)/2), Point(-1, -sqrt(3)/2))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">scale</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">Triangle(Point(2, 0), Point(-1, sqrt(3)), Point(-1, -sqrt(3)))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.entity.GeometryEntity.translate">
-<tt class="descname">translate</tt><big>(</big><em>self</em>, <em>x=0</em>, <em>y=0</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/entity.html#GeometryEntity.translate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.entity.GeometryEntity.translate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Shift the object by adding to the x,y-coordinates the values x and y.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.entity.GeometryEntity.rotate" title="sympy.geometry.entity.GeometryEntity.rotate"><tt class="xref py py-obj docutils literal"><span class="pre">rotate</span></tt></a>, <a class="reference internal" href="#sympy.geometry.entity.GeometryEntity.scale" title="sympy.geometry.entity.GeometryEntity.scale"><tt class="xref py py-obj docutils literal"><span class="pre">scale</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">RegularPolygon</span><span class="p">,</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Polygon</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">(</span><span class="o">*</span><span class="n">RegularPolygon</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">vertices</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span>
-<span class="go">Triangle(Point(1, 0), Point(-1/2, sqrt(3)/2), Point(-1/2, -sqrt(3)/2))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">Triangle(Point(3, 0), Point(3/2, sqrt(3)/2), Point(3/2, -sqrt(3)/2))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">Triangle(Point(3, 2), Point(3/2, sqrt(3)/2 + 2),</span>
-<span class="go">    Point(3/2, -sqrt(3)/2 + 2))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.geometry.util">
-<span id="utils"></span><h3>Utils<a class="headerlink" href="#module-sympy.geometry.util" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.geometry.util.intersection">
-<tt class="descclassname">sympy.geometry.util.</tt><tt class="descname">intersection</tt><big>(</big><em>*entities</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/util.html#intersection"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.util.intersection" title="Permalink to this definition">¶</a></dt>
-<dd><p>The intersection of a collection of GeometryEntity instances.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>entities</strong> : sequence of GeometryEntity</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>intersection</strong> : list of GeometryEntity</p>
-</td>
-</tr>
-<tr class="field-odd field"><th class="field-name">Raises :</th><td class="field-body"><p class="first"><strong>NotImplementedError</strong> :</p>
-<blockquote class="last">
-<div><p>When unable to calculate intersection.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.entity.GeometryEntity.intersection" title="sympy.geometry.entity.GeometryEntity.intersection"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.entity.GeometryEntity.intersection</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>The intersection of any geometrical entity with itself should return
-a list with one item: the entity in question.
-An intersection requires two or more entities. If only a single
-entity is given then the function will return an empty list.
-It is possible for <span class="math">\(intersection\)</span> to miss intersections that one
-knows exists because the required quantities were not fully
-simplified internally.
-Reals should be converted to Rationals, e.g. Rational(str(real_num))
-or else failures due to floating point issues may result.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Line</span><span class="p">,</span> <span class="n">Circle</span><span class="p">,</span> <span class="n">intersection</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="p">,</span> <span class="n">l2</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">),</span> <span class="n">Line</span><span class="p">(</span><span class="n">p3</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">Circle</span><span class="p">(</span><span class="n">p2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">intersection</span><span class="p">(</span><span class="n">l1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="go">[Point(1, 1)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">intersection</span><span class="p">(</span><span class="n">l1</span><span class="p">,</span> <span class="n">l2</span><span class="p">)</span>
-<span class="go">[Point(1, 1)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">intersection</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="go">[]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">intersection</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">[Point(1, 0)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">intersection</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">l2</span><span class="p">)</span>
-<span class="go">[Point(-sqrt(5)/5 + 1, 2*sqrt(5)/5 + 1),</span>
-<span class="go"> Point(sqrt(5)/5 + 1, -2*sqrt(5)/5 + 1)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.util.convex_hull">
-<tt class="descclassname">sympy.geometry.util.</tt><tt class="descname">convex_hull</tt><big>(</big><em>*args</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/util.html#convex_hull"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.util.convex_hull" title="Permalink to this definition">¶</a></dt>
-<dd><p>The convex hull surrounding the Points contained in the list of entities.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>args</strong> : a collection of Points, Segments and/or Polygons</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>convex_hull</strong> : Polygon</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a>, <a class="reference internal" href="#sympy.geometry.polygon.Polygon" title="sympy.geometry.polygon.Polygon"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.polygon.Polygon</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>This can only be performed on a set of non-symbolic points.</p>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Graham_scan">http://en.wikipedia.org/wiki/Graham_scan</a></p>
-<p>[2] Andrew&#8217;s Monotone Chain Algorithm
-(A.M. Andrew,
-&#8220;Another Efficient Algorithm for Convex Hulls in Two Dimensions&#8221;, 1979)
-<a class="reference external" href="http://softsurfer.com/Archive/algorithm_0109/algorithm_0109.htm">http://softsurfer.com/Archive/algorithm_0109/algorithm_0109.htm</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">convex_hull</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">points</span> <span class="o">=</span> <span class="p">[(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">15</span><span class="p">,</span><span class="mi">4</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">convex_hull</span><span class="p">(</span><span class="o">*</span><span class="n">points</span><span class="p">)</span>
-<span class="go">Polygon(Point(-5, 2), Point(1, 1), Point(3, 1), Point(15, 4))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.util.are_similar">
-<tt class="descclassname">sympy.geometry.util.</tt><tt class="descname">are_similar</tt><big>(</big><em>e1</em>, <em>e2</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/util.html#are_similar"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.util.are_similar" title="Permalink to this definition">¶</a></dt>
-<dd><p>Are two geometrical entities similar.</p>
-<p>Can one geometrical entity be uniformly scaled to the other?</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>e1</strong> : GeometryEntity</p>
-<p><strong>e2</strong> : GeometryEntity</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>are_similar</strong> : boolean</p>
-</td>
-</tr>
-<tr class="field-odd field"><th class="field-name">Raises :</th><td class="field-body"><p class="first"><strong>GeometryError</strong> :</p>
-<blockquote class="last">
-<div><p>When <span class="math">\(e1\)</span> and <span class="math">\(e2\)</span> cannot be compared.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.entity.GeometryEntity.is_similar" title="sympy.geometry.entity.GeometryEntity.is_similar"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.entity.GeometryEntity.is_similar</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>If the two objects are equal then they are similar.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Circle</span><span class="p">,</span> <span class="n">Triangle</span><span class="p">,</span> <span class="n">are_similar</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span><span class="p">,</span> <span class="n">c2</span> <span class="o">=</span> <span class="n">Circle</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">4</span><span class="p">),</span> <span class="n">Circle</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t1</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t2</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t3</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">are_similar</span><span class="p">(</span><span class="n">t1</span><span class="p">,</span> <span class="n">t2</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">are_similar</span><span class="p">(</span><span class="n">t1</span><span class="p">,</span> <span class="n">t3</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.util.centroid">
-<tt class="descclassname">sympy.geometry.util.</tt><tt class="descname">centroid</tt><big>(</big><em>*args</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/util.html#centroid"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.util.centroid" title="Permalink to this definition">¶</a></dt>
-<dd><p>Find the centroid (center of mass) of the collection containing only Points,
-Segments or Polygons. The centroid is the weighted average of the individual centroid
-where the weights are the lengths (of segments) or areas (of polygons).
-Overlapping regions will add to the weight of that region.</p>
-<p>If there are no objects (or a mixture of objects) then None is returned.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a>, <a class="reference internal" href="#sympy.geometry.line.Segment" title="sympy.geometry.line.Segment"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.line.Segment</span></tt></a>, <a class="reference internal" href="#sympy.geometry.polygon.Polygon" title="sympy.geometry.polygon.Polygon"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.polygon.Polygon</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Segment</span><span class="p">,</span> <span class="n">Polygon</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry.util</span> <span class="kn">import</span> <span class="n">centroid</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">20</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">centroid</span><span class="p">,</span> <span class="n">q</span><span class="o">.</span><span class="n">centroid</span>
-<span class="go">(Point(20/3, 10/3), Point(20/3, 70/3))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">centroid</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
-<span class="go">Point(20/3, 40/3)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">)),</span> <span class="n">Segment</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">centroid</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
-<span class="go">Point(1, -sqrt(2) + 2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">centroid</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">Point(1, 0)</span>
-</pre></div>
-</div>
-<p>Stacking 3 polygons on top of each other effectively triples the
-weight of that polygon:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">centroid</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
-<span class="go">Point(3/2, 1/2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">centroid</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span> <span class="c"># centroid x-coord shifts left</span>
-<span class="go">Point(11/10, 1/2)</span>
-</pre></div>
-</div>
-<p>Stacking the squares vertically above and below p has the same
-effect:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">centroid</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">p</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">q</span><span class="p">)</span>
-<span class="go">Point(11/10, 1/2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.geometry.point">
-<span id="points"></span><h3>Points<a class="headerlink" href="#module-sympy.geometry.point" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.geometry.point.Point">
-<em class="property">class </em><tt class="descclassname">sympy.geometry.point.</tt><tt class="descname">Point</tt><a class="reference internal" href="../_modules/sympy/geometry/point.html#Point"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.point.Point" title="Permalink to this definition">¶</a></dt>
-<dd><p>A point in a 2-dimensional Euclidean space.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>coords</strong> : sequence of 2 coordinate values.</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Raises :</th><td class="field-body"><p class="first"><strong>NotImplementedError</strong> :</p>
-<blockquote>
-<div><p>When trying to create a point with more than two dimensions.
-When <span class="math">\(intersection\)</span> is called with object other than a Point.</p>
-</div></blockquote>
-<p><strong>TypeError</strong> :</p>
-<blockquote class="last">
-<div><p>When trying to add or subtract points with different dimensions.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.geometry.line.Segment" title="sympy.geometry.line.Segment"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.line.Segment</span></tt></a></dt>
-<dd>Connects two Points</dd>
-</dl>
-</div>
-<p class="rubric">Notes</p>
-<p>Currently only 2-dimensional points are supported.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">Point(1, 2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Point</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="go">Point(1, 2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">Point(0, x)</span>
-</pre></div>
-</div>
-<p>Floats are automatically converted to Rational unless the
-evaluate flag is False:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Point</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">)</span>
-<span class="go">Point(1/2, 1/4)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Point</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">Point(0.5, 0.25)</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="38%" />
-<col width="63%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>x</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>y</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>length</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.geometry.point.Point.distance">
-<tt class="descname">distance</tt><big>(</big><em>self</em>, <em>p</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/point.html#Point.distance"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.point.Point.distance" title="Permalink to this definition">¶</a></dt>
-<dd><p>The Euclidean distance from self to point p.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>p</strong> : Point</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>distance</strong> : number or symbolic expression.</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.line.Segment.length" title="sympy.geometry.line.Segment.length"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.line.Segment.length</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">p2</span><span class="p">)</span>
-<span class="go">5</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p3</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">sqrt(x**2 + y**2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.point.Point.dot">
-<tt class="descname">dot</tt><big>(</big><em>self</em>, <em>p2</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/point.html#Point.dot"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.point.Point.dot" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return dot product of self with another Point.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.point.Point.evalf">
-<tt class="descname">evalf</tt><big>(</big><em>self</em>, <em>prec=None</em>, <em>**options</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/point.html#Point.evalf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.point.Point.evalf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluate the coordinates of the point.</p>
-<p>This method will, where possible, create and return a new Point
-where the coordinates are evaluated as floating point numbers to
-the precision indicated (default=15).</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>point</strong> : Point</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span>
-<span class="go">Point(1/2, 3/2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">Point(0.5, 1.5)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.point.Point.intersection">
-<tt class="descname">intersection</tt><big>(</big><em>self</em>, <em>o</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/point.html#Point.intersection"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.point.Point.intersection" title="Permalink to this definition">¶</a></dt>
-<dd><p>The intersection between this point and another point.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>other</strong> : Point</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>intersection</strong> : list of Points</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Notes</p>
-<p>The return value will either be an empty list if there is no
-intersection, otherwise it will contain this point.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">p2</span><span class="p">)</span>
-<span class="go">[]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">p3</span><span class="p">)</span>
-<span class="go">[Point(0, 0)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.point.Point.is_collinear">
-<tt class="descname">is_collinear</tt><big>(</big><em>*points</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/point.html#Point.is_collinear"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.point.Point.is_collinear" title="Permalink to this definition">¶</a></dt>
-<dd><p>Is a sequence of points collinear?</p>
-<p>Test whether or not a set of points are collinear. Returns True if
-the set of points are collinear, or False otherwise.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>points</strong> : sequence of Point</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>is_collinear</strong> : boolean</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.line.Line" title="sympy.geometry.line.Line"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.line.Line</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>Slope is preserved everywhere on a line, so the slope between
-any two points on the line should be the same. Take the first
-two points, p1 and p2, and create a translated point v1
-with p1 as the origin. Now for every other point we create
-a translated point, vi with p1 also as the origin. Note that
-these translations preserve slope since everything is
-consistently translated to a new origin of p1. Since slope
-is preserved then we have the following equality:</p>
-<blockquote>
-<div><ul class="simple">
-<li>v1_slope = vi_slope</li>
-<li>v1.y/v1.x = vi.y/vi.x (due to translation)</li>
-<li>v1.y*vi.x = vi.y*v1.x</li>
-<li>v1.y*vi.x - vi.y*v1.x = 0           (*)</li>
-</ul>
-</div></blockquote>
-<p>Hence, if we have a vi such that the equality in (*) is False
-then the points are not collinear. We do this test for every
-point in the list, and if all pass then they are collinear.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p3</span><span class="p">,</span> <span class="n">p4</span><span class="p">,</span> <span class="n">p5</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Point</span><span class="o">.</span><span class="n">is_collinear</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Point</span><span class="o">.</span><span class="n">is_collinear</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p5</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.point.Point.is_concyclic">
-<tt class="descname">is_concyclic</tt><big>(</big><em>*points</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/point.html#Point.is_concyclic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.point.Point.is_concyclic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Is a sequence of points concyclic?</p>
-<p>Test whether or not a sequence of points are concyclic (i.e., they lie
-on a circle).</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>points</strong> : sequence of Points</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>is_concyclic</strong> : boolean</p>
-<blockquote class="last">
-<div><p>True if points are concyclic, False otherwise.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.ellipse.Circle" title="sympy.geometry.ellipse.Circle"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.ellipse.Circle</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>No points are not considered to be concyclic. One or two points
-are definitely concyclic and three points are conyclic iff they
-are not collinear.</p>
-<p>For more than three points, create a circle from the first three
-points. If the circle cannot be created (i.e., they are collinear)
-then all of the points cannot be concyclic. If the circle is created
-successfully then simply check the remaining points for containment
-in the circle.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p3</span><span class="p">,</span> <span class="n">p4</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Point</span><span class="o">.</span><span class="n">is_concyclic</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Point</span><span class="o">.</span><span class="n">is_concyclic</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.point.Point.length">
-<tt class="descname">length</tt><a class="reference internal" href="../_modules/sympy/geometry/point.html#Point.length"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.point.Point.length" title="Permalink to this definition">¶</a></dt>
-<dd><p>Treating a Point as a Line, this returns 0 for the length of a Point.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">length</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.point.Point.midpoint">
-<tt class="descname">midpoint</tt><big>(</big><em>self</em>, <em>p</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/point.html#Point.midpoint"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.point.Point.midpoint" title="Permalink to this definition">¶</a></dt>
-<dd><p>The midpoint between self and point p.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>p</strong> : Point</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>midpoint</strong> : Point</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.line.Segment.midpoint" title="sympy.geometry.line.Segment.midpoint"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.line.Segment.midpoint</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">13</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="o">.</span><span class="n">midpoint</span><span class="p">(</span><span class="n">p2</span><span class="p">)</span>
-<span class="go">Point(7, 3)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.point.Point.n">
-<tt class="descname">n</tt><big>(</big><em>self</em>, <em>prec=None</em>, <em>**options</em><big>)</big><a class="headerlink" href="#sympy.geometry.point.Point.n" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluate the coordinates of the point.</p>
-<p>This method will, where possible, create and return a new Point
-where the coordinates are evaluated as floating point numbers to
-the precision indicated (default=15).</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>point</strong> : Point</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span>
-<span class="go">Point(1/2, 3/2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">Point(0.5, 1.5)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.point.Point.rotate">
-<tt class="descname">rotate</tt><big>(</big><em>self</em>, <em>angle</em>, <em>pt=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/point.html#Point.rotate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.point.Point.rotate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Rotate <tt class="docutils literal"><span class="pre">angle</span></tt> radians counterclockwise about Point <tt class="docutils literal"><span class="pre">pt</span></tt>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point.rotate" title="sympy.geometry.point.Point.rotate"><tt class="xref py py-obj docutils literal"><span class="pre">rotate</span></tt></a>, <a class="reference internal" href="#sympy.geometry.point.Point.scale" title="sympy.geometry.point.Point.scale"><tt class="xref py py-obj docutils literal"><span class="pre">scale</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">rotate</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">Point(0, 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">rotate</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">Point(2, -1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.point.Point.scale">
-<tt class="descname">scale</tt><big>(</big><em>self</em>, <em>x=1</em>, <em>y=1</em>, <em>pt=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/point.html#Point.scale"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.point.Point.scale" title="Permalink to this definition">¶</a></dt>
-<dd><p>Scale the coordinates of the Point by multiplying by
-<tt class="docutils literal"><span class="pre">x</span></tt> and <tt class="docutils literal"><span class="pre">y</span></tt> after subtracting <tt class="docutils literal"><span class="pre">pt</span></tt> &#8211; default is (0, 0) &#8211;
-and then adding <tt class="docutils literal"><span class="pre">pt</span></tt> back again (i.e. <tt class="docutils literal"><span class="pre">pt</span></tt> is the point of
-reference for the scaling).</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point.rotate" title="sympy.geometry.point.Point.rotate"><tt class="xref py py-obj docutils literal"><span class="pre">rotate</span></tt></a>, <a class="reference internal" href="#sympy.geometry.point.Point.translate" title="sympy.geometry.point.Point.translate"><tt class="xref py py-obj docutils literal"><span class="pre">translate</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">scale</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">Point(2, 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">scale</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">Point(2, 2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.point.Point.transform">
-<tt class="descname">transform</tt><big>(</big><em>self</em>, <em>matrix</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/point.html#Point.transform"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.point.Point.transform" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the point after applying the transformation described
-by the 3x3 Matrix, <tt class="docutils literal"><span class="pre">matrix</span></tt>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">geometry.entity.rotate</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">geometry.entity.scale</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">geometry.entity.translate</span></tt></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.point.Point.translate">
-<tt class="descname">translate</tt><big>(</big><em>self</em>, <em>x=0</em>, <em>y=0</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/point.html#Point.translate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.point.Point.translate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Shift the Point by adding x and y to the coordinates of the Point.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point.rotate" title="sympy.geometry.point.Point.rotate"><tt class="xref py py-obj docutils literal"><span class="pre">rotate</span></tt></a>, <a class="reference internal" href="#sympy.geometry.point.Point.scale" title="sympy.geometry.point.Point.scale"><tt class="xref py py-obj docutils literal"><span class="pre">scale</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">Point(2, 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">Point(2, 3)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">+</span> <span class="n">Point</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">Point(2, 3)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.point.Point.x">
-<tt class="descname">x</tt><a class="reference internal" href="../_modules/sympy/geometry/point.html#Point.x"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.point.Point.x" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the X coordinate of the Point.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">x</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.point.Point.y">
-<tt class="descname">y</tt><a class="reference internal" href="../_modules/sympy/geometry/point.html#Point.y"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.point.Point.y" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the Y coordinate of the Point.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">y</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.geometry.line">
-<span id="lines"></span><h3>Lines<a class="headerlink" href="#module-sympy.geometry.line" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.geometry.line.LinearEntity">
-<em class="property">class </em><tt class="descclassname">sympy.geometry.line.</tt><tt class="descname">LinearEntity</tt><a class="reference internal" href="../_modules/sympy/geometry/line.html#LinearEntity"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.LinearEntity" title="Permalink to this definition">¶</a></dt>
-<dd><p>An abstract base class for all linear entities (line, ray and segment)
-in a 2-dimensional Euclidean space.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.entity.GeometryEntity" title="sympy.geometry.entity.GeometryEntity"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.entity.GeometryEntity</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>This is an abstract class and is not meant to be instantiated.
-Subclasses should implement the following methods:</p>
-<blockquote>
-<div><ul class="simple">
-<li>__eq__</li>
-<li>contains</li>
-</ul>
-</div></blockquote>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="55%" />
-<col width="45%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>p1</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>p2</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>coefficients</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>slope</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>points</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.geometry.line.LinearEntity.angle_between">
-<tt class="descname">angle_between</tt><big>(</big><em>l1</em>, <em>l2</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#LinearEntity.angle_between"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.LinearEntity.angle_between" title="Permalink to this definition">¶</a></dt>
-<dd><p>The angle formed between the two linear entities.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>l1</strong> : LinearEntity</p>
-<p><strong>l2</strong> : LinearEntity</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>angle</strong> : angle in radians</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.line.LinearEntity.is_perpendicular" title="sympy.geometry.line.LinearEntity.is_perpendicular"><tt class="xref py py-obj docutils literal"><span class="pre">is_perpendicular</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>From the dot product of vectors v1 and v2 it is known that:</p>
-<blockquote>
-<div><tt class="docutils literal"><span class="pre">dot(v1,</span> <span class="pre">v2)</span> <span class="pre">=</span> <span class="pre">|v1|*|v2|*cos(A)</span></tt></div></blockquote>
-<p>where A is the angle formed between the two vectors. We can
-get the directional vectors of the two lines and readily
-find the angle between the two using the above formula.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Line</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="p">,</span> <span class="n">l2</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">),</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="o">.</span><span class="n">angle_between</span><span class="p">(</span><span class="n">l2</span><span class="p">)</span>
-<span class="go">pi/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.line.LinearEntity.coefficients">
-<tt class="descname">coefficients</tt><a class="reference internal" href="../_modules/sympy/geometry/line.html#LinearEntity.coefficients"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.LinearEntity.coefficients" title="Permalink to this definition">¶</a></dt>
-<dd><p>The coefficients (<span class="math">\(a\)</span>, <span class="math">\(b\)</span>, <span class="math">\(c\)</span>) for the linear equation <span class="math">\(ax + by + c = 0\)</span>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.line.Line.equation" title="sympy.geometry.line.Line.equation"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.line.Line.equation</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Line</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l</span><span class="o">.</span><span class="n">coefficients</span>
-<span class="go">(-3, 5, 0)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l2</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l2</span><span class="o">.</span><span class="n">coefficients</span>
-<span class="go">(-y, x, 0)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.line.LinearEntity.contains">
-<tt class="descname">contains</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#LinearEntity.contains"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.LinearEntity.contains" title="Permalink to this definition">¶</a></dt>
-<dd><p>Subclasses should implement this method and should return
-True if other is on the boundaries of self;
-False if not on the boundaries of self;
-None if a determination cannot be made.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.line.LinearEntity.intersection">
-<tt class="descname">intersection</tt><big>(</big><em>self</em>, <em>o</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#LinearEntity.intersection"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.LinearEntity.intersection" title="Permalink to this definition">¶</a></dt>
-<dd><p>The intersection with another geometrical entity.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>o</strong> : Point or LinearEntity</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>intersection</strong> : list of geometrical entities</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Line</span><span class="p">,</span> <span class="n">Segment</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">p3</span><span class="p">)</span>
-<span class="go">[Point(7, 7)]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p4</span><span class="p">,</span> <span class="n">p5</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l2</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p4</span><span class="p">,</span> <span class="n">p5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">l2</span><span class="p">)</span>
-<span class="go">[Point(15/8, 15/8)]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p6</span><span class="p">,</span> <span class="n">p7</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s1</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="n">p6</span><span class="p">,</span> <span class="n">p7</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">s1</span><span class="p">)</span>
-<span class="go">[]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.line.LinearEntity.is_concurrent">
-<tt class="descname">is_concurrent</tt><big>(</big><em>*lines</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#LinearEntity.is_concurrent"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.LinearEntity.is_concurrent" title="Permalink to this definition">¶</a></dt>
-<dd><p>Is a sequence of linear entities concurrent?</p>
-<p>Two or more linear entities are concurrent if they all
-intersect at a single point.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>lines</strong> : a sequence of linear entities.</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>True</strong> : if the set of linear entities are concurrent,</p>
-<p class="last"><strong>False</strong> : otherwise.</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.util.intersection" title="sympy.geometry.util.intersection"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.util.intersection</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>Simply take the first two lines and find their intersection.
-If there is no intersection, then the first two lines were
-parallel and had no intersection so concurrency is impossible
-amongst the whole set. Otherwise, check to see if the
-intersection point of the first two lines is a member on
-the rest of the lines. If so, the lines are concurrent.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Line</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p3</span><span class="p">,</span> <span class="n">p4</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="p">,</span> <span class="n">l2</span><span class="p">,</span> <span class="n">l3</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">),</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p3</span><span class="p">),</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="o">.</span><span class="n">is_concurrent</span><span class="p">(</span><span class="n">l2</span><span class="p">,</span> <span class="n">l3</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">l4</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l4</span><span class="o">.</span><span class="n">is_concurrent</span><span class="p">(</span><span class="n">l2</span><span class="p">,</span> <span class="n">l3</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.line.LinearEntity.is_parallel">
-<tt class="descname">is_parallel</tt><big>(</big><em>l1</em>, <em>l2</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#LinearEntity.is_parallel"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.LinearEntity.is_parallel" title="Permalink to this definition">¶</a></dt>
-<dd><p>Are two linear entities parallel?</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>l1</strong> : LinearEntity</p>
-<p><strong>l2</strong> : LinearEntity</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>True</strong> : if l1 and l2 are parallel,</p>
-<p class="last"><strong>False</strong> : otherwise.</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.line.LinearEntity.coefficients" title="sympy.geometry.line.LinearEntity.coefficients"><tt class="xref py py-obj docutils literal"><span class="pre">coefficients</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Line</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p3</span><span class="p">,</span> <span class="n">p4</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="p">,</span> <span class="n">l2</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">),</span> <span class="n">Line</span><span class="p">(</span><span class="n">p3</span><span class="p">,</span> <span class="n">p4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Line</span><span class="o">.</span><span class="n">is_parallel</span><span class="p">(</span><span class="n">l1</span><span class="p">,</span> <span class="n">l2</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p5</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l3</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p3</span><span class="p">,</span> <span class="n">p5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Line</span><span class="o">.</span><span class="n">is_parallel</span><span class="p">(</span><span class="n">l1</span><span class="p">,</span> <span class="n">l3</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.line.LinearEntity.is_perpendicular">
-<tt class="descname">is_perpendicular</tt><big>(</big><em>l1</em>, <em>l2</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#LinearEntity.is_perpendicular"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.LinearEntity.is_perpendicular" title="Permalink to this definition">¶</a></dt>
-<dd><p>Are two linear entities perpendicular?</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>l1</strong> : LinearEntity</p>
-<p><strong>l2</strong> : LinearEntity</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>True</strong> : if l1 and l2 are perpendicular,</p>
-<p class="last"><strong>False</strong> : otherwise.</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.line.LinearEntity.coefficients" title="sympy.geometry.line.LinearEntity.coefficients"><tt class="xref py py-obj docutils literal"><span class="pre">coefficients</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Line</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="p">,</span> <span class="n">l2</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">),</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="o">.</span><span class="n">is_perpendicular</span><span class="p">(</span><span class="n">l2</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p4</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l3</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="o">.</span><span class="n">is_perpendicular</span><span class="p">(</span><span class="n">l3</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.line.LinearEntity.is_similar">
-<tt class="descname">is_similar</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#LinearEntity.is_similar"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.LinearEntity.is_similar" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if self and other are contained in the same line.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Line</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l2</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="o">.</span><span class="n">is_similar</span><span class="p">(</span><span class="n">l2</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.line.LinearEntity.length">
-<tt class="descname">length</tt><a class="reference internal" href="../_modules/sympy/geometry/line.html#LinearEntity.length"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.LinearEntity.length" title="Permalink to this definition">¶</a></dt>
-<dd><p>The length of the line.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Line</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="o">.</span><span class="n">length</span>
-<span class="go">oo</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.line.LinearEntity.p1">
-<tt class="descname">p1</tt><a class="reference internal" href="../_modules/sympy/geometry/line.html#LinearEntity.p1"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.LinearEntity.p1" title="Permalink to this definition">¶</a></dt>
-<dd><p>The first defining point of a linear entity.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Line</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l</span><span class="o">.</span><span class="n">p1</span>
-<span class="go">Point(0, 0)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.line.LinearEntity.p2">
-<tt class="descname">p2</tt><a class="reference internal" href="../_modules/sympy/geometry/line.html#LinearEntity.p2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.LinearEntity.p2" title="Permalink to this definition">¶</a></dt>
-<dd><p>The second defining point of a linear entity.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Line</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l</span><span class="o">.</span><span class="n">p2</span>
-<span class="go">Point(5, 3)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.line.LinearEntity.parallel_line">
-<tt class="descname">parallel_line</tt><big>(</big><em>self</em>, <em>p</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#LinearEntity.parallel_line"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.LinearEntity.parallel_line" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a new Line parallel to this linear entity which passes
-through the point <span class="math">\(p\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>p</strong> : Point</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>line</strong> : Line</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.line.LinearEntity.is_parallel" title="sympy.geometry.line.LinearEntity.is_parallel"><tt class="xref py py-obj docutils literal"><span class="pre">is_parallel</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Line</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l2</span> <span class="o">=</span> <span class="n">l1</span><span class="o">.</span><span class="n">parallel_line</span><span class="p">(</span><span class="n">p3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p3</span> <span class="ow">in</span> <span class="n">l2</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="o">.</span><span class="n">is_parallel</span><span class="p">(</span><span class="n">l2</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.line.LinearEntity.perpendicular_line">
-<tt class="descname">perpendicular_line</tt><big>(</big><em>self</em>, <em>p</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#LinearEntity.perpendicular_line"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.LinearEntity.perpendicular_line" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a new Line perpendicular to this linear entity which passes
-through the point <span class="math">\(p\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>p</strong> : Point</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>line</strong> : Line</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.line.LinearEntity.is_perpendicular" title="sympy.geometry.line.LinearEntity.is_perpendicular"><tt class="xref py py-obj docutils literal"><span class="pre">is_perpendicular</span></tt></a>, <a class="reference internal" href="#sympy.geometry.line.LinearEntity.perpendicular_segment" title="sympy.geometry.line.LinearEntity.perpendicular_segment"><tt class="xref py py-obj docutils literal"><span class="pre">perpendicular_segment</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Line</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l2</span> <span class="o">=</span> <span class="n">l1</span><span class="o">.</span><span class="n">perpendicular_line</span><span class="p">(</span><span class="n">p3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p3</span> <span class="ow">in</span> <span class="n">l2</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="o">.</span><span class="n">is_perpendicular</span><span class="p">(</span><span class="n">l2</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.line.LinearEntity.perpendicular_segment">
-<tt class="descname">perpendicular_segment</tt><big>(</big><em>self</em>, <em>p</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#LinearEntity.perpendicular_segment"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.LinearEntity.perpendicular_segment" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a perpendicular line segment from <span class="math">\(p\)</span> to this line.</p>
-<p>The enpoints of the segment are <tt class="docutils literal"><span class="pre">p</span></tt> and the closest point in
-the line containing self. (If self is not a line, the point might
-not be in self.)</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>p</strong> : Point</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>segment</strong> : Segment</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.line.LinearEntity.perpendicular_line" title="sympy.geometry.line.LinearEntity.perpendicular_line"><tt class="xref py py-obj docutils literal"><span class="pre">perpendicular_line</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>Returns <span class="math">\(p\)</span> itself if <span class="math">\(p\)</span> is on this linear entity.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Line</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s1</span> <span class="o">=</span> <span class="n">l1</span><span class="o">.</span><span class="n">perpendicular_segment</span><span class="p">(</span><span class="n">p3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="o">.</span><span class="n">is_perpendicular</span><span class="p">(</span><span class="n">s1</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p3</span> <span class="ow">in</span> <span class="n">s1</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="o">.</span><span class="n">perpendicular_segment</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">Segment(Point(2, 2), Point(4, 0))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.line.LinearEntity.points">
-<tt class="descname">points</tt><a class="reference internal" href="../_modules/sympy/geometry/line.html#LinearEntity.points"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.LinearEntity.points" title="Permalink to this definition">¶</a></dt>
-<dd><p>The two points used to define this linear entity.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>points</strong> : tuple of Points</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Line</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">11</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="o">.</span><span class="n">points</span>
-<span class="go">(Point(0, 0), Point(5, 11))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.line.LinearEntity.projection">
-<tt class="descname">projection</tt><big>(</big><em>self</em>, <em>o</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#LinearEntity.projection"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.LinearEntity.projection" title="Permalink to this definition">¶</a></dt>
-<dd><p>Project a point, line, ray, or segment onto this linear entity.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>other</strong> : Point or LinearEntity (Line, Ray, Segment)</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>projection</strong> : Point or LinearEntity (Line, Ray, Segment)</p>
-<blockquote>
-<div><p>The return type matches the type of the parameter <tt class="docutils literal"><span class="pre">other</span></tt>.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-odd field"><th class="field-name">Raises :</th><td class="field-body"><p class="first"><strong>GeometryError</strong> :</p>
-<blockquote class="last">
-<div><p>When method is unable to perform projection.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a>, <a class="reference internal" href="#sympy.geometry.line.LinearEntity.perpendicular_line" title="sympy.geometry.line.LinearEntity.perpendicular_line"><tt class="xref py py-obj docutils literal"><span class="pre">perpendicular_line</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>A projection involves taking the two points that define
-the linear entity and projecting those points onto a
-Line and then reforming the linear entity using these
-projections.
-A point P is projected onto a line L by finding the point
-on L that is closest to P. This is done by creating a
-perpendicular line through P and L and finding its
-intersection with L.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Line</span><span class="p">,</span> <span class="n">Segment</span><span class="p">,</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="o">.</span><span class="n">projection</span><span class="p">(</span><span class="n">p3</span><span class="p">)</span>
-<span class="go">Point(1/4, 1/4)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p4</span><span class="p">,</span> <span class="n">p5</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s1</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="n">p4</span><span class="p">,</span> <span class="n">p5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="o">.</span><span class="n">projection</span><span class="p">(</span><span class="n">s1</span><span class="p">)</span>
-<span class="go">Segment(Point(5, 5), Point(13/2, 13/2))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.line.LinearEntity.random_point">
-<tt class="descname">random_point</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#LinearEntity.random_point"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.LinearEntity.random_point" title="Permalink to this definition">¶</a></dt>
-<dd><p>A random point on a LinearEntity.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>point</strong> : Point</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Line</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p3</span> <span class="o">=</span> <span class="n">l1</span><span class="o">.</span><span class="n">random_point</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># random point - don&#39;t know its coords in advance</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p3</span> 
-<span class="go">Point(...)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># point should belong to the line</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p3</span> <span class="ow">in</span> <span class="n">l1</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.line.LinearEntity.slope">
-<tt class="descname">slope</tt><a class="reference internal" href="../_modules/sympy/geometry/line.html#LinearEntity.slope"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.LinearEntity.slope" title="Permalink to this definition">¶</a></dt>
-<dd><p>The slope of this linear entity, or infinity if vertical.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>slope</strong> : number or sympy expression</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.line.LinearEntity.coefficients" title="sympy.geometry.line.LinearEntity.coefficients"><tt class="xref py py-obj docutils literal"><span class="pre">coefficients</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Line</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="o">.</span><span class="n">slope</span>
-<span class="go">5/3</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l2</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l2</span><span class="o">.</span><span class="n">slope</span>
-<span class="go">oo</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.geometry.line.Line">
-<em class="property">class </em><tt class="descclassname">sympy.geometry.line.</tt><tt class="descname">Line</tt><a class="reference internal" href="../_modules/sympy/geometry/line.html#Line"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.Line" title="Permalink to this definition">¶</a></dt>
-<dd><p>An infinite line in space.</p>
-<p>A line is declared with two distinct points or a point and slope
-as defined using keyword <span class="math">\(slope\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>p1</strong> : Point</p>
-<p><strong>pt</strong> : Point</p>
-<p class="last"><strong>slope</strong> : sympy expression</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>At the moment only lines in a 2D space can be declared, because
-Points can be defined only for 2D spaces.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">sympy</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">L</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Line</span><span class="p">,</span> <span class="n">Segment</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span>
-<span class="go">Line(Point(2, 3), Point(3, 5))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span><span class="o">.</span><span class="n">points</span>
-<span class="go">(Point(2, 3), Point(3, 5))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span><span class="o">.</span><span class="n">equation</span><span class="p">()</span>
-<span class="go">-2*x + y + 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span><span class="o">.</span><span class="n">coefficients</span>
-<span class="go">(-2, 1, 1)</span>
-</pre></div>
-</div>
-<p>Instantiate with keyword <tt class="docutils literal"><span class="pre">slope</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Line</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">slope</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">Line(Point(0, 0), Point(1, 0))</span>
-</pre></div>
-</div>
-<p>Instantiate with another linear object</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Line</span><span class="p">(</span><span class="n">s</span><span class="p">)</span><span class="o">.</span><span class="n">equation</span><span class="p">()</span>
-<span class="go">x</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.geometry.line.Line.arbitrary_point">
-<tt class="descname">arbitrary_point</tt><big>(</big><em>self</em>, <em>parameter='t'</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#Line.arbitrary_point"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.Line.arbitrary_point" title="Permalink to this definition">¶</a></dt>
-<dd><p>A parameterized point on the Line.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>parameter</strong> : str, optional</p>
-<blockquote>
-<div><p>The name of the parameter which will be used for the parametric
-point. The default value is &#8216;t&#8217;.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>point</strong> : Point</p>
-</td>
-</tr>
-<tr class="field-odd field"><th class="field-name">Raises :</th><td class="field-body"><p class="first"><strong>ValueError</strong> :</p>
-<blockquote class="last">
-<div><p>When <tt class="docutils literal"><span class="pre">parameter</span></tt> already appears in the Line&#8217;s definition.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Line</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">()</span>
-<span class="go">Point(4*t + 1, 3*t)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.line.Line.contains">
-<tt class="descname">contains</tt><big>(</big><em>self</em>, <em>o</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#Line.contains"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.Line.contains" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if o is on this Line, or False otherwise.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.line.Line.equation">
-<tt class="descname">equation</tt><big>(</big><em>self</em>, <em>x='x'</em>, <em>y='y'</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#Line.equation"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.Line.equation" title="Permalink to this definition">¶</a></dt>
-<dd><p>The equation of the line: ax + by + c.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>x</strong> : str, optional</p>
-<blockquote>
-<div><p>The name to use for the x-axis, default value is &#8216;x&#8217;.</p>
-</div></blockquote>
-<p><strong>y</strong> : str, optional</p>
-<blockquote>
-<div><p>The name to use for the y-axis, default value is &#8216;y&#8217;.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>equation</strong> : sympy expression</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.line.LinearEntity.coefficients" title="sympy.geometry.line.LinearEntity.coefficients"><tt class="xref py py-obj docutils literal"><span class="pre">LinearEntity.coefficients</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Line</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="o">.</span><span class="n">equation</span><span class="p">()</span>
-<span class="go">-3*x + 4*y + 3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.line.Line.plot_interval">
-<tt class="descname">plot_interval</tt><big>(</big><em>self</em>, <em>parameter='t'</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#Line.plot_interval"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.Line.plot_interval" title="Permalink to this definition">¶</a></dt>
-<dd><p>The plot interval for the default geometric plot of line.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>parameter</strong> : str, optional</p>
-<blockquote>
-<div><p>Default value is &#8216;t&#8217;.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>plot_interval</strong> : list (plot interval)</p>
-<blockquote class="last">
-<div><p>[parameter, lower_bound, upper_bound]</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Line</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span><span class="o">.</span><span class="n">plot_interval</span><span class="p">()</span>
-<span class="go">[t, -5, 5]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.geometry.line.Ray">
-<em class="property">class </em><tt class="descclassname">sympy.geometry.line.</tt><tt class="descname">Ray</tt><a class="reference internal" href="../_modules/sympy/geometry/line.html#Ray"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.Ray" title="Permalink to this definition">¶</a></dt>
-<dd><p>A Ray is a semi-line in the space with a source point and a direction.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>p1</strong> : Point</p>
-<blockquote>
-<div><p>The source of the Ray</p>
-</div></blockquote>
-<p><strong>p2</strong> : Point or radian value</p>
-<blockquote class="last">
-<div><p>This point determines the direction in which the Ray propagates.
-If given as an angle it is interpreted in radians with the positive
-direction being ccw.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a>, <a class="reference internal" href="#sympy.geometry.line.Line" title="sympy.geometry.line.Line"><tt class="xref py py-obj docutils literal"><span class="pre">Line</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>At the moment only rays in a 2D space can be declared, because
-Points can be defined only for 2D spaces.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">sympy</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">r</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Ray</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span> <span class="o">=</span> <span class="n">Ray</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span> <span class="o">=</span> <span class="n">Ray</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span>
-<span class="go">Ray(Point(2, 3), Point(3, 5))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="o">.</span><span class="n">points</span>
-<span class="go">(Point(2, 3), Point(3, 5))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="o">.</span><span class="n">source</span>
-<span class="go">Point(2, 3)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="o">.</span><span class="n">xdirection</span>
-<span class="go">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="o">.</span><span class="n">ydirection</span>
-<span class="go">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="o">.</span><span class="n">slope</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ray</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">angle</span><span class="o">=</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">slope</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="50%" />
-<col width="50%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>source</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>xdirection</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>ydirection</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.geometry.line.Ray.arbitrary_point">
-<tt class="descname">arbitrary_point</tt><big>(</big><em>self</em>, <em>parameter='t'</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#Ray.arbitrary_point"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.Ray.arbitrary_point" title="Permalink to this definition">¶</a></dt>
-<dd><p>A parameterized point on the Ray.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>parameter</strong> : str, optional</p>
-<blockquote>
-<div><p>The name of the parameter which will be used for the parametric
-point. The default value is &#8216;t&#8217;.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>point</strong> : Point</p>
-</td>
-</tr>
-<tr class="field-odd field"><th class="field-name">Raises :</th><td class="field-body"><p class="first"><strong>ValueError</strong> :</p>
-<blockquote class="last">
-<div><p>When <tt class="docutils literal"><span class="pre">parameter</span></tt> already appears in the Ray&#8217;s definition.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Ray</span><span class="p">,</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Segment</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">solve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">t</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span> <span class="o">=</span> <span class="n">Ray</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">r</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>The parameter <span class="math">\(t\)</span> used in the arbitrary point maps 0 to the
-origin of the ray and 1 to the end of the ray at infinity
-(which will show up as NaN).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">p</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">(Point(0, 0), Point(oo, oo))</span>
-</pre></div>
-</div>
-<p>The unit that <span class="math">\(t\)</span> moves you is based on the spacing of the
-points used to define the ray.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span> <span class="c"># one unit</span>
-<span class="go">Point(2, 3)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span><span class="p">))</span> <span class="c"># two units out</span>
-<span class="go">Point(4, 6)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="o">/</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">))</span> <span class="c"># half a unit out</span>
-<span class="go">Point(1, 3/2)</span>
-</pre></div>
-</div>
-<p>If you want to be located a distance of 1 from the origin of the
-ray, what value of <span class="math">\(t\)</span> is needed?</p>
-<ol class="loweralpha">
-<li><p class="first">Find the unit length and pick <span class="math">\(t\)</span> accordingly.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">u</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="n">r</span><span class="o">.</span><span class="n">p1</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">))</span><span class="o">.</span><span class="n">length</span> <span class="c"># S.Half = 1/(1 + 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">want</span> <span class="o">=</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t_need</span> <span class="o">=</span> <span class="n">want</span><span class="o">/</span><span class="n">u</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p_want</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">t_need</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">t_need</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">Segment</span><span class="p">(</span><span class="n">r</span><span class="o">.</span><span class="n">p1</span><span class="p">,</span> <span class="n">p_want</span><span class="p">)</span><span class="o">.</span><span class="n">length</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">Find the <span class="math">\(t\)</span> that makes the length from origin to <span class="math">\(p\)</span> equal to 1.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">l</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="n">r</span><span class="o">.</span><span class="n">p1</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">length</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t_need</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">l</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">want</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span> <span class="c"># use the square to remove abs() if it is there</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t_need</span> <span class="o">=</span> <span class="p">[</span><span class="n">w</span> <span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">t_need</span> <span class="k">if</span> <span class="n">w</span><span class="o">.</span><span class="n">n</span><span class="p">()</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="c"># take positive t</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p_want</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">t_need</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">Segment</span><span class="p">(</span><span class="n">r</span><span class="o">.</span><span class="n">p1</span><span class="p">,</span> <span class="n">p_want</span><span class="p">)</span><span class="o">.</span><span class="n">length</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</li>
-</ol>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.line.Ray.contains">
-<tt class="descname">contains</tt><big>(</big><em>self</em>, <em>o</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#Ray.contains"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.Ray.contains" title="Permalink to this definition">¶</a></dt>
-<dd><p>Is other GeometryEntity contained in this Ray?</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.line.Ray.plot_interval">
-<tt class="descname">plot_interval</tt><big>(</big><em>self</em>, <em>parameter='t'</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#Ray.plot_interval"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.Ray.plot_interval" title="Permalink to this definition">¶</a></dt>
-<dd><p>The plot interval for the default geometric plot of the Ray.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>parameter</strong> : str, optional</p>
-<blockquote>
-<div><p>Default value is &#8216;t&#8217;.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>plot_interval</strong> : list</p>
-<blockquote class="last">
-<div><p>[parameter, lower_bound, upper_bound]</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ray</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span> <span class="o">=</span> <span class="n">Ray</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">angle</span><span class="o">=</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="o">.</span><span class="n">plot_interval</span><span class="p">()</span>
-<span class="go">[t, 0, 5*sqrt(2)/(1 + 5*sqrt(2))]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.line.Ray.source">
-<tt class="descname">source</tt><a class="reference internal" href="../_modules/sympy/geometry/line.html#Ray.source"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.Ray.source" title="Permalink to this definition">¶</a></dt>
-<dd><p>The point from which the ray emanates.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ray</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r1</span> <span class="o">=</span> <span class="n">Ray</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r1</span><span class="o">.</span><span class="n">source</span>
-<span class="go">Point(0, 0)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.line.Ray.xdirection">
-<tt class="descname">xdirection</tt><a class="reference internal" href="../_modules/sympy/geometry/line.html#Ray.xdirection"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.Ray.xdirection" title="Permalink to this definition">¶</a></dt>
-<dd><p>The x direction of the ray.</p>
-<p>Positive infinity if the ray points in the positive x direction,
-negative infinity if the ray points in the negative x direction,
-or 0 if the ray is vertical.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.line.Ray.ydirection" title="sympy.geometry.line.Ray.ydirection"><tt class="xref py py-obj docutils literal"><span class="pre">ydirection</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ray</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r1</span><span class="p">,</span> <span class="n">r2</span> <span class="o">=</span> <span class="n">Ray</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">),</span> <span class="n">Ray</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r1</span><span class="o">.</span><span class="n">xdirection</span>
-<span class="go">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r2</span><span class="o">.</span><span class="n">xdirection</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.line.Ray.ydirection">
-<tt class="descname">ydirection</tt><a class="reference internal" href="../_modules/sympy/geometry/line.html#Ray.ydirection"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.Ray.ydirection" title="Permalink to this definition">¶</a></dt>
-<dd><p>The y direction of the ray.</p>
-<p>Positive infinity if the ray points in the positive y direction,
-negative infinity if the ray points in the negative y direction,
-or 0 if the ray is horizontal.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.line.Ray.xdirection" title="sympy.geometry.line.Ray.xdirection"><tt class="xref py py-obj docutils literal"><span class="pre">xdirection</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ray</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r1</span><span class="p">,</span> <span class="n">r2</span> <span class="o">=</span> <span class="n">Ray</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">),</span> <span class="n">Ray</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r1</span><span class="o">.</span><span class="n">ydirection</span>
-<span class="go">-oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r2</span><span class="o">.</span><span class="n">ydirection</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.geometry.line.Segment">
-<em class="property">class </em><tt class="descclassname">sympy.geometry.line.</tt><tt class="descname">Segment</tt><a class="reference internal" href="../_modules/sympy/geometry/line.html#Segment"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.Segment" title="Permalink to this definition">¶</a></dt>
-<dd><p>An undirected line segment in space.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>p1</strong> : Point</p>
-<p class="last"><strong>p2</strong> : Point</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a>, <a class="reference internal" href="#sympy.geometry.line.Line" title="sympy.geometry.line.Line"><tt class="xref py py-obj docutils literal"><span class="pre">Line</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>At the moment only segments in a 2D space can be declared, because
-Points can be defined only for 2D spaces.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">sympy</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">s</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Segment</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Segment</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span> <span class="c"># tuples are interpreted as pts</span>
-<span class="go">Segment(Point(1, 0), Point(1, 1))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span>
-<span class="go">Segment(Point(1, 1), Point(4, 3))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="o">.</span><span class="n">points</span>
-<span class="go">(Point(1, 1), Point(4, 3))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="o">.</span><span class="n">slope</span>
-<span class="go">2/3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="o">.</span><span class="n">length</span>
-<span class="go">sqrt(13)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="o">.</span><span class="n">midpoint</span>
-<span class="go">Point(5/2, 2)</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="18%" />
-<col width="59%" />
-<col width="23%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>length</td>
-<td>number or sympy expression</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>midpoint</td>
-<td>Point</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.geometry.line.Segment.arbitrary_point">
-<tt class="descname">arbitrary_point</tt><big>(</big><em>self</em>, <em>parameter='t'</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#Segment.arbitrary_point"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.Segment.arbitrary_point" title="Permalink to this definition">¶</a></dt>
-<dd><p>A parameterized point on the Segment.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>parameter</strong> : str, optional</p>
-<blockquote>
-<div><p>The name of the parameter which will be used for the parametric
-point. The default value is &#8216;t&#8217;.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>point</strong> : Point</p>
-</td>
-</tr>
-<tr class="field-odd field"><th class="field-name">Raises :</th><td class="field-body"><p class="first"><strong>ValueError</strong> :</p>
-<blockquote class="last">
-<div><p>When <tt class="docutils literal"><span class="pre">parameter</span></tt> already appears in the Segment&#8217;s definition.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Segment</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s1</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s1</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">()</span>
-<span class="go">Point(4*t + 1, 3*t)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.line.Segment.contains">
-<tt class="descname">contains</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#Segment.contains"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.Segment.contains" title="Permalink to this definition">¶</a></dt>
-<dd><p>Is the other GeometryEntity contained within this Segment?</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Segment</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s2</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="n">p2</span><span class="p">,</span> <span class="n">p1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">s2</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.line.Segment.distance">
-<tt class="descname">distance</tt><big>(</big><em>self</em>, <em>o</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#Segment.distance"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.Segment.distance" title="Permalink to this definition">¶</a></dt>
-<dd><p>Finds the shortest distance between a line segment and a point.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Raises :</th><td class="field-body"><strong>NotImplementedError is raised if o is not a Point</strong> :</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Segment</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">15</span><span class="p">))</span>
-<span class="go">sqrt(170)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.line.Segment.length">
-<tt class="descname">length</tt><a class="reference internal" href="../_modules/sympy/geometry/line.html#Segment.length"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.Segment.length" title="Permalink to this definition">¶</a></dt>
-<dd><p>The length of the line segment.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point.distance" title="sympy.geometry.point.Point.distance"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point.distance</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Segment</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s1</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s1</span><span class="o">.</span><span class="n">length</span>
-<span class="go">5</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.line.Segment.midpoint">
-<tt class="descname">midpoint</tt><a class="reference internal" href="../_modules/sympy/geometry/line.html#Segment.midpoint"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.Segment.midpoint" title="Permalink to this definition">¶</a></dt>
-<dd><p>The midpoint of the line segment.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point.midpoint" title="sympy.geometry.point.Point.midpoint"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point.midpoint</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Segment</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s1</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s1</span><span class="o">.</span><span class="n">midpoint</span>
-<span class="go">Point(2, 3/2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.line.Segment.perpendicular_bisector">
-<tt class="descname">perpendicular_bisector</tt><big>(</big><em>self</em>, <em>p=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#Segment.perpendicular_bisector"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.Segment.perpendicular_bisector" title="Permalink to this definition">¶</a></dt>
-<dd><p>The perpendicular bisector of this segment.</p>
-<p>If no point is specified or the point specified is not on the
-bisector then the bisector is returned as a Line. Otherwise a
-Segment is returned that joins the point specified and the
-intersection of the bisector and the segment.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>p</strong> : Point</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>bisector</strong> : Line or Segment</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.line.LinearEntity.perpendicular_segment" title="sympy.geometry.line.LinearEntity.perpendicular_segment"><tt class="xref py py-obj docutils literal"><span class="pre">LinearEntity.perpendicular_segment</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Segment</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s1</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s1</span><span class="o">.</span><span class="n">perpendicular_bisector</span><span class="p">()</span>
-<span class="go">Line(Point(3, 3), Point(9, -3))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">s1</span><span class="o">.</span><span class="n">perpendicular_bisector</span><span class="p">(</span><span class="n">p3</span><span class="p">)</span>
-<span class="go">Segment(Point(3, 3), Point(5, 1))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.line.Segment.plot_interval">
-<tt class="descname">plot_interval</tt><big>(</big><em>self</em>, <em>parameter='t'</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/line.html#Segment.plot_interval"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.line.Segment.plot_interval" title="Permalink to this definition">¶</a></dt>
-<dd><p>The plot interval for the default geometric plot of the Segment.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>parameter</strong> : str, optional</p>
-<blockquote>
-<div><p>Default value is &#8216;t&#8217;.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>plot_interval</strong> : list</p>
-<blockquote class="last">
-<div><p>[parameter, lower_bound, upper_bound]</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Segment</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s1</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s1</span><span class="o">.</span><span class="n">plot_interval</span><span class="p">()</span>
-<span class="go">[t, 0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.geometry.curve">
-<span id="curves"></span><h3>Curves<a class="headerlink" href="#module-sympy.geometry.curve" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.geometry.curve.Curve">
-<em class="property">class </em><tt class="descclassname">sympy.geometry.curve.</tt><tt class="descname">Curve</tt><a class="reference internal" href="../_modules/sympy/geometry/curve.html#Curve"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.curve.Curve" title="Permalink to this definition">¶</a></dt>
-<dd><p>A curve in space.</p>
-<p>A curve is defined by parametric functions for the coordinates, a
-parameter and the lower and upper bounds for the parameter value.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>function</strong> : list of functions</p>
-<p><strong>limits</strong> : 3-tuple</p>
-<blockquote>
-<div><p>Function parameter and lower and upper bounds.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Raises :</th><td class="field-body"><p class="first"><strong>ValueError</strong> :</p>
-<blockquote class="last">
-<div><p>When <span class="math">\(functions\)</span> are specified incorrectly.
-When <span class="math">\(limits\)</span> are specified incorrectly.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="core.html#sympy.core.function.Function" title="sympy.core.function.Function"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.core.function.Function</span></tt></a>, <a class="reference internal" href="polys/reference.html#sympy.polys.polyfuncs.interpolate" title="sympy.polys.polyfuncs.interpolate"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.polyfuncs.interpolate</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">interpolate</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">t</span><span class="p">,</span> <span class="n">a</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Curve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Curve</span><span class="p">((</span><span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="n">cos</span><span class="p">(</span><span class="n">t</span><span class="p">)),</span> <span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span><span class="o">.</span><span class="n">functions</span>
-<span class="go">(sin(t), cos(t))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span><span class="o">.</span><span class="n">limits</span>
-<span class="go">(t, 0, 2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span><span class="o">.</span><span class="n">parameter</span>
-<span class="go">t</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Curve</span><span class="p">((</span><span class="n">t</span><span class="p">,</span> <span class="n">interpolate</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">16</span><span class="p">],</span> <span class="n">t</span><span class="p">)),</span> <span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">));</span> <span class="n">C</span>
-<span class="go">Curve((t, t**2), (t, 0, 1))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">Point(4, 16)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="go">Point(a, a**2)</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="47%" />
-<col width="53%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>functions</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>parameter</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>limits</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.geometry.curve.Curve.arbitrary_point">
-<tt class="descname">arbitrary_point</tt><big>(</big><em>self</em>, <em>parameter='t'</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/curve.html#Curve.arbitrary_point"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.curve.Curve.arbitrary_point" title="Permalink to this definition">¶</a></dt>
-<dd><p>A parameterized point on the curve.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>parameter</strong> : str or Symbol, optional</p>
-<blockquote>
-<div><p>Default value is &#8216;t&#8217;;
-the Curve&#8217;s parameter is selected with None or self.parameter
-otherwise the provided symbol is used.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>arbitrary_point</strong> : Point</p>
-</td>
-</tr>
-<tr class="field-odd field"><th class="field-name">Raises :</th><td class="field-body"><p class="first"><strong>ValueError</strong> :</p>
-<blockquote class="last">
-<div><p>When <span class="math">\(parameter\)</span> already appears in the functions.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">s</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Curve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Curve</span><span class="p">([</span><span class="mi">2</span><span class="o">*</span><span class="n">s</span><span class="p">,</span> <span class="n">s</span><span class="o">**</span><span class="mi">2</span><span class="p">],</span> <span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">()</span>
-<span class="go">Point(2*t, t**2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">parameter</span><span class="p">)</span>
-<span class="go">Point(2*s, s**2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">(</span><span class="bp">None</span><span class="p">)</span>
-<span class="go">Point(2*s, s**2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">(</span><span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">))</span>
-<span class="go">Point(2*a, a**2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.curve.Curve.free_symbols">
-<tt class="descname">free_symbols</tt><a class="reference internal" href="../_modules/sympy/geometry/curve.html#Curve.free_symbols"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.curve.Curve.free_symbols" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a set of symbols other than the bound symbols used to
-parametrically define the Curve.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">t</span><span class="p">,</span> <span class="n">a</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Curve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Curve</span><span class="p">((</span><span class="n">t</span><span class="p">,</span> <span class="n">t</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">free_symbols</span>
-<span class="go">set()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Curve</span><span class="p">((</span><span class="n">t</span><span class="p">,</span> <span class="n">t</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">free_symbols</span>
-<span class="go">set([a])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.curve.Curve.functions">
-<tt class="descname">functions</tt><a class="reference internal" href="../_modules/sympy/geometry/curve.html#Curve.functions"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.curve.Curve.functions" title="Permalink to this definition">¶</a></dt>
-<dd><p>The functions specifying the curve.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>functions</strong> : list of parameterized coordinate functions.</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.curve.Curve.parameter" title="sympy.geometry.curve.Curve.parameter"><tt class="xref py py-obj docutils literal"><span class="pre">parameter</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">t</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Curve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Curve</span><span class="p">((</span><span class="n">t</span><span class="p">,</span> <span class="n">t</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span><span class="o">.</span><span class="n">functions</span>
-<span class="go">(t, t**2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.curve.Curve.limits">
-<tt class="descname">limits</tt><a class="reference internal" href="../_modules/sympy/geometry/curve.html#Curve.limits"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.curve.Curve.limits" title="Permalink to this definition">¶</a></dt>
-<dd><p>The limits for the curve.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>limits</strong> : tuple</p>
-<blockquote class="last">
-<div><p>Contains parameter and lower and upper limits.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.curve.Curve.plot_interval" title="sympy.geometry.curve.Curve.plot_interval"><tt class="xref py py-obj docutils literal"><span class="pre">plot_interval</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">t</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Curve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Curve</span><span class="p">([</span><span class="n">t</span><span class="p">,</span> <span class="n">t</span><span class="o">**</span><span class="mi">3</span><span class="p">],</span> <span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span><span class="o">.</span><span class="n">limits</span>
-<span class="go">(t, -2, 2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.curve.Curve.parameter">
-<tt class="descname">parameter</tt><a class="reference internal" href="../_modules/sympy/geometry/curve.html#Curve.parameter"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.curve.Curve.parameter" title="Permalink to this definition">¶</a></dt>
-<dd><p>The curve function variable.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>parameter</strong> : SymPy symbol</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.curve.Curve.functions" title="sympy.geometry.curve.Curve.functions"><tt class="xref py py-obj docutils literal"><span class="pre">functions</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">t</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Curve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Curve</span><span class="p">([</span><span class="n">t</span><span class="p">,</span> <span class="n">t</span><span class="o">**</span><span class="mi">2</span><span class="p">],</span> <span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span><span class="o">.</span><span class="n">parameter</span>
-<span class="go">t</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.curve.Curve.plot_interval">
-<tt class="descname">plot_interval</tt><big>(</big><em>self</em>, <em>parameter='t'</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/curve.html#Curve.plot_interval"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.curve.Curve.plot_interval" title="Permalink to this definition">¶</a></dt>
-<dd><p>The plot interval for the default geometric plot of the curve.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>parameter</strong> : str or Symbol, optional</p>
-<blockquote>
-<div><p>Default value is &#8216;t&#8217;;
-otherwise the provided symbol is used.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>plot_interval</strong> : list (plot interval)</p>
-<blockquote class="last">
-<div><p>[parameter, lower_bound, upper_bound]</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.geometry.curve.Curve.limits" title="sympy.geometry.curve.Curve.limits"><tt class="xref py py-obj docutils literal"><span class="pre">limits</span></tt></a></dt>
-<dd>Returns limits of the parameter interval</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Curve</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">s</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Curve</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">plot_interval</span><span class="p">()</span>
-<span class="go">[t, 1, 2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Curve</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">plot_interval</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="go">[s, 1, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.curve.Curve.rotate">
-<tt class="descname">rotate</tt><big>(</big><em>self</em>, <em>angle=0</em>, <em>pt=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/curve.html#Curve.rotate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.curve.Curve.rotate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Rotate <tt class="docutils literal"><span class="pre">angle</span></tt> radians counterclockwise about Point <tt class="docutils literal"><span class="pre">pt</span></tt>.</p>
-<p>The default pt is the origin, Point(0, 0).</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry.curve</span> <span class="kn">import</span> <span class="n">Curve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Curve</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">rotate</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">Curve((-x, x), (x, 0, 1))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.curve.Curve.scale">
-<tt class="descname">scale</tt><big>(</big><em>self</em>, <em>x=1</em>, <em>y=1</em>, <em>pt=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/curve.html#Curve.scale"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.curve.Curve.scale" title="Permalink to this definition">¶</a></dt>
-<dd><p>Override GeometryEntity.scale since Curve is not made up of Points.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry.curve</span> <span class="kn">import</span> <span class="n">Curve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Curve</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">scale</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">Curve((2*x, x), (x, 0, 1))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.curve.Curve.translate">
-<tt class="descname">translate</tt><big>(</big><em>self</em>, <em>x=0</em>, <em>y=0</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/curve.html#Curve.translate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.curve.Curve.translate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Translate the Curve by (x, y).</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry.curve</span> <span class="kn">import</span> <span class="n">Curve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Curve</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">translate</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">Curve((x + 1, x + 2), (x, 0, 1))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.geometry.ellipse">
-<span id="ellipses"></span><h3>Ellipses<a class="headerlink" href="#module-sympy.geometry.ellipse" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.geometry.ellipse.Ellipse">
-<em class="property">class </em><tt class="descclassname">sympy.geometry.ellipse.</tt><tt class="descname">Ellipse</tt><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse" title="Permalink to this definition">¶</a></dt>
-<dd><p>An elliptical GeometryEntity.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>center</strong> : Point, optional</p>
-<blockquote>
-<div><p>Default value is Point(0, 0)</p>
-</div></blockquote>
-<p><strong>hradius</strong> : number or SymPy expression, optional</p>
-<p><strong>vradius</strong> : number or SymPy expression, optional</p>
-<p><strong>eccentricity</strong> : number or SymPy expression, optional</p>
-<blockquote>
-<div><p>Two of <span class="math">\(hradius\)</span>, <span class="math">\(vradius\)</span> and <span class="math">\(eccentricity\)</span> must be supplied to
-create an Ellipse. The third is derived from the two supplied.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Raises :</th><td class="field-body"><p class="first"><strong>GeometryError</strong> :</p>
-<blockquote>
-<div><p>When <span class="math">\(hradius\)</span>, <span class="math">\(vradius\)</span> and <span class="math">\(eccentricity\)</span> are incorrectly supplied
-as parameters.</p>
-</div></blockquote>
-<p><strong>TypeError</strong> :</p>
-<blockquote class="last">
-<div><p>When <span class="math">\(center\)</span> is not a Point.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.ellipse.Circle" title="sympy.geometry.ellipse.Circle"><tt class="xref py py-obj docutils literal"><span class="pre">Circle</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>Constructed from a center and two radii, the first being the horizontal
-radius (along the x-axis) and the second being the vertical radius (along
-the y-axis).</p>
-<p>When symbolic value for hradius and vradius are used, any calculation that
-refers to the foci or the major or minor axis will assume that the ellipse
-has its major radius on the x-axis. If this is not true then a manual
-rotation is necessary.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Ellipse</span><span class="p">,</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="o">.</span><span class="n">hradius</span><span class="p">,</span> <span class="n">e1</span><span class="o">.</span><span class="n">vradius</span>
-<span class="go">(5, 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e2</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">hradius</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">eccentricity</span><span class="o">=</span><span class="n">Rational</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e2</span>
-<span class="go">Ellipse(Point(3, 1), 3, 9/5)</span>
-</pre></div>
-</div>
-<p>Plotting:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Circle</span><span class="p">,</span> <span class="n">Plot</span><span class="p">,</span> <span class="n">Segment</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span> <span class="o">=</span> <span class="n">Circle</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Plot</span><span class="p">(</span><span class="n">c1</span><span class="p">)</span>                                
-<span class="go">[0]: cos(t), sin(t), &#39;mode=parametric&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Plot</span><span class="p">()</span>                              
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">c1</span>                               
-<span class="gp">&gt;&gt;&gt; </span><span class="n">radius</span> <span class="o">=</span> <span class="n">Segment</span><span class="p">(</span><span class="n">c1</span><span class="o">.</span><span class="n">center</span><span class="p">,</span> <span class="n">c1</span><span class="o">.</span><span class="n">random_point</span><span class="p">())</span>  
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">radius</span>                           
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span>                                       
-<span class="go">[0]: cos(t), sin(t), &#39;mode=parametric&#39;</span>
-<span class="go">[1]: t*cos(1.546086215036205357975518382),</span>
-<span class="go">t*sin(1.546086215036205357975518382), &#39;mode=parametric&#39;</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="58%" />
-<col width="42%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>center</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>hradius</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>vradius</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>area</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>circumference</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>eccentricity</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>periapsis</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>apoapsis</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>focus_distance</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>foci</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="attribute">
-<dt id="sympy.geometry.ellipse.Ellipse.apoapsis">
-<tt class="descname">apoapsis</tt><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse.apoapsis"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse.apoapsis" title="Permalink to this definition">¶</a></dt>
-<dd><p>The apoapsis of the ellipse.</p>
-<p>The greatest distance between the focus and the contour.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>apoapsis</strong> : number</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.periapsis" title="sympy.geometry.ellipse.Ellipse.periapsis"><tt class="xref py py-obj docutils literal"><span class="pre">periapsis</span></tt></a></dt>
-<dd>Returns shortest distance between foci and contour</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ellipse</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="o">.</span><span class="n">apoapsis</span>
-<span class="go">2*sqrt(2) + 3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.ellipse.Ellipse.arbitrary_point">
-<tt class="descname">arbitrary_point</tt><big>(</big><em>self</em>, <em>parameter='t'</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse.arbitrary_point"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse.arbitrary_point" title="Permalink to this definition">¶</a></dt>
-<dd><p>A parameterized point on the ellipse.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>parameter</strong> : str, optional</p>
-<blockquote>
-<div><p>Default value is &#8216;t&#8217;.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>arbitrary_point</strong> : Point</p>
-</td>
-</tr>
-<tr class="field-odd field"><th class="field-name">Raises :</th><td class="field-body"><p class="first"><strong>ValueError</strong> :</p>
-<blockquote class="last">
-<div><p>When <span class="math">\(parameter\)</span> already appears in the functions.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ellipse</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">()</span>
-<span class="go">Point(3*cos(t), 2*sin(t))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.ellipse.Ellipse.area">
-<tt class="descname">area</tt><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse.area"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse.area" title="Permalink to this definition">¶</a></dt>
-<dd><p>The area of the ellipse.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>area</strong> : number</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ellipse</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="o">.</span><span class="n">area</span>
-<span class="go">3*pi</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.ellipse.Ellipse.center">
-<tt class="descname">center</tt><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse.center"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse.center" title="Permalink to this definition">¶</a></dt>
-<dd><p>The center of the ellipse.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>center</strong> : number</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ellipse</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="o">.</span><span class="n">center</span>
-<span class="go">Point(0, 0)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.ellipse.Ellipse.circumference">
-<tt class="descname">circumference</tt><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse.circumference"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse.circumference" title="Permalink to this definition">¶</a></dt>
-<dd><p>The circumference of the ellipse.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ellipse</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="o">.</span><span class="n">circumference</span>
-<span class="go">12*Integral(sqrt((-8*_x**2/9 + 1)/(-_x**2 + 1)), (_x, 0, 1))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.ellipse.Ellipse.eccentricity">
-<tt class="descname">eccentricity</tt><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse.eccentricity"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse.eccentricity" title="Permalink to this definition">¶</a></dt>
-<dd><p>The eccentricity of the ellipse.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>eccentricity</strong> : number</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ellipse</span><span class="p">,</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="o">.</span><span class="n">eccentricity</span>
-<span class="go">sqrt(7)/3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.ellipse.Ellipse.encloses_point">
-<tt class="descname">encloses_point</tt><big>(</big><em>self</em>, <em>p</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse.encloses_point"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse.encloses_point" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if p is enclosed by (is inside of) self.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>p</strong> : Point</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>encloses_point</strong> : True, False or None</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>Being on the border of self is considered False.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Ellipse</span><span class="p">,</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">t</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">encloses_point</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">encloses_point</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">))</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">encloses_point</span><span class="p">((</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.ellipse.Ellipse.equation">
-<tt class="descname">equation</tt><big>(</big><em>self</em>, <em>x='x'</em>, <em>y='y'</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse.equation"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse.equation" title="Permalink to this definition">¶</a></dt>
-<dd><p>The equation of the ellipse.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>x</strong> : str, optional</p>
-<blockquote>
-<div><p>Label for the x-axis. Default value is &#8216;x&#8217;.</p>
-</div></blockquote>
-<p><strong>y</strong> : str, optional</p>
-<blockquote>
-<div><p>Label for the y-axis. Default value is &#8216;y&#8217;.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>equation</strong> : sympy expression</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.arbitrary_point" title="sympy.geometry.ellipse.Ellipse.arbitrary_point"><tt class="xref py py-obj docutils literal"><span class="pre">arbitrary_point</span></tt></a></dt>
-<dd>Returns parameterized point on ellipse</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ellipse</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="o">.</span><span class="n">equation</span><span class="p">()</span>
-<span class="go">y**2/4 + (x/3 - 1/3)**2 - 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.ellipse.Ellipse.foci">
-<tt class="descname">foci</tt><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse.foci"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse.foci" title="Permalink to this definition">¶</a></dt>
-<dd><p>The foci of the ellipse.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Raises :</th><td class="field-body"><p class="first"><strong>ValueError</strong> :</p>
-<blockquote class="last">
-<div><p>When the major and minor axis cannot be determined.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.focus_distance" title="sympy.geometry.ellipse.Ellipse.focus_distance"><tt class="xref py py-obj docutils literal"><span class="pre">focus_distance</span></tt></a></dt>
-<dd>Returns the distance between focus and center</dd>
-</dl>
-</div>
-<p class="rubric">Notes</p>
-<p>The foci can only be calculated if the major/minor axes are known.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ellipse</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="o">.</span><span class="n">foci</span>
-<span class="go">(Point(-2*sqrt(2), 0), Point(2*sqrt(2), 0))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.ellipse.Ellipse.focus_distance">
-<tt class="descname">focus_distance</tt><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse.focus_distance"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse.focus_distance" title="Permalink to this definition">¶</a></dt>
-<dd><p>The focale distance of the ellipse.</p>
-<p>The distance between the center and one focus.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>focus_distance</strong> : number</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.foci" title="sympy.geometry.ellipse.Ellipse.foci"><tt class="xref py py-obj docutils literal"><span class="pre">foci</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ellipse</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="o">.</span><span class="n">focus_distance</span>
-<span class="go">2*sqrt(2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.ellipse.Ellipse.hradius">
-<tt class="descname">hradius</tt><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse.hradius"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse.hradius" title="Permalink to this definition">¶</a></dt>
-<dd><p>The horizontal radius of the ellipse.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>hradius</strong> : number</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.vradius" title="sympy.geometry.ellipse.Ellipse.vradius"><tt class="xref py py-obj docutils literal"><span class="pre">vradius</span></tt></a>, <a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.major" title="sympy.geometry.ellipse.Ellipse.major"><tt class="xref py py-obj docutils literal"><span class="pre">major</span></tt></a>, <a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.minor" title="sympy.geometry.ellipse.Ellipse.minor"><tt class="xref py py-obj docutils literal"><span class="pre">minor</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ellipse</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="o">.</span><span class="n">hradius</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.ellipse.Ellipse.intersection">
-<tt class="descname">intersection</tt><big>(</big><em>self</em>, <em>o</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse.intersection"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse.intersection" title="Permalink to this definition">¶</a></dt>
-<dd><p>The intersection of this ellipse and another geometrical entity
-<span class="math">\(o\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>o</strong> : GeometryEntity</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>intersection</strong> : list of GeometryEntity objects</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.entity.GeometryEntity" title="sympy.geometry.entity.GeometryEntity"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.entity.GeometryEntity</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>Currently supports intersections with Point, Line, Segment, Ray,
-Circle and Ellipse types.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Ellipse</span><span class="p">,</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Line</span><span class="p">,</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">[]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">[Point(5, 0)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">Line</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)))</span>
-<span class="go">[Point(0, -7), Point(0, 7)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">Line</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">)))</span>
-<span class="go">[Point(5, 0)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">Line</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">1</span><span class="p">)))</span>
-<span class="go">[]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">Ellipse</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
-<span class="go">[Point(0, -3*sqrt(15)/4), Point(0, 3*sqrt(15)/4)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">Ellipse</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
-<span class="go">[Point(2, -3*sqrt(7)/4), Point(2, 3*sqrt(7)/4)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">Ellipse</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">100500</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
-<span class="go">[]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">Ellipse</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
-<span class="go">[Point(-363/175, -48*sqrt(111)/175), Point(-363/175, 48*sqrt(111)/175),</span>
-<span class="go">Point(3, 0)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">Ellipse</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
-<span class="go">[Point(-17/5, -12/5), Point(-17/5, 12/5), Point(7/5, -12/5),</span>
-<span class="go">Point(7/5, 12/5)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.ellipse.Ellipse.is_tangent">
-<tt class="descname">is_tangent</tt><big>(</big><em>self</em>, <em>o</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse.is_tangent"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse.is_tangent" title="Permalink to this definition">¶</a></dt>
-<dd><p>Is <span class="math">\(o\)</span> tangent to the ellipse?</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>o</strong> : GeometryEntity</p>
-<blockquote>
-<div><p>An Ellipse, LinearEntity or Polygon</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>is_tangent: boolean</strong> :</p>
-<blockquote>
-<div><p>True if o is tangent to the ellipse, False otherwise.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-odd field"><th class="field-name">Raises :</th><td class="field-body"><p class="first"><strong>NotImplementedError</strong> :</p>
-<blockquote class="last">
-<div><p>When the wrong type of argument is supplied.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.tangent_lines" title="sympy.geometry.ellipse.Ellipse.tangent_lines"><tt class="xref py py-obj docutils literal"><span class="pre">tangent_lines</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ellipse</span><span class="p">,</span> <span class="n">Line</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p0</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">p0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span> <span class="o">=</span> <span class="n">Line</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="o">.</span><span class="n">is_tangent</span><span class="p">(</span><span class="n">l1</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.ellipse.Ellipse.major">
-<tt class="descname">major</tt><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse.major"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse.major" title="Permalink to this definition">¶</a></dt>
-<dd><p>Longer axis of the ellipse (if it can be determined) else hradius.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>major</strong> : number or expression</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.hradius" title="sympy.geometry.ellipse.Ellipse.hradius"><tt class="xref py py-obj docutils literal"><span class="pre">hradius</span></tt></a>, <a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.vradius" title="sympy.geometry.ellipse.Ellipse.vradius"><tt class="xref py py-obj docutils literal"><span class="pre">vradius</span></tt></a>, <a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.minor" title="sympy.geometry.ellipse.Ellipse.minor"><tt class="xref py py-obj docutils literal"><span class="pre">minor</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ellipse</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="o">.</span><span class="n">major</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;b&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ellipse</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span><span class="o">.</span><span class="n">major</span>
-<span class="go">a</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ellipse</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">major</span>
-<span class="go">b</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;m&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ellipse</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">M</span><span class="p">)</span><span class="o">.</span><span class="n">major</span>
-<span class="go">m + 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.ellipse.Ellipse.minor">
-<tt class="descname">minor</tt><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse.minor"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse.minor" title="Permalink to this definition">¶</a></dt>
-<dd><p>Shorter axis of the ellipse (if it can be determined) else vradius.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>minor</strong> : number or expression</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.hradius" title="sympy.geometry.ellipse.Ellipse.hradius"><tt class="xref py py-obj docutils literal"><span class="pre">hradius</span></tt></a>, <a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.vradius" title="sympy.geometry.ellipse.Ellipse.vradius"><tt class="xref py py-obj docutils literal"><span class="pre">vradius</span></tt></a>, <a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.major" title="sympy.geometry.ellipse.Ellipse.major"><tt class="xref py py-obj docutils literal"><span class="pre">major</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ellipse</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="o">.</span><span class="n">minor</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;b&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ellipse</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span><span class="o">.</span><span class="n">minor</span>
-<span class="go">b</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ellipse</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">minor</span>
-<span class="go">a</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;m&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ellipse</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">M</span><span class="p">)</span><span class="o">.</span><span class="n">minor</span>
-<span class="go">m</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.ellipse.Ellipse.periapsis">
-<tt class="descname">periapsis</tt><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse.periapsis"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse.periapsis" title="Permalink to this definition">¶</a></dt>
-<dd><p>The periapsis of the ellipse.</p>
-<p>The shortest distance between the focus and the contour.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>periapsis</strong> : number</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.apoapsis" title="sympy.geometry.ellipse.Ellipse.apoapsis"><tt class="xref py py-obj docutils literal"><span class="pre">apoapsis</span></tt></a></dt>
-<dd>Returns greatest distance between focus and contour</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ellipse</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="o">.</span><span class="n">periapsis</span>
-<span class="go">-2*sqrt(2) + 3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.ellipse.Ellipse.plot_interval">
-<tt class="descname">plot_interval</tt><big>(</big><em>self</em>, <em>parameter='t'</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse.plot_interval"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse.plot_interval" title="Permalink to this definition">¶</a></dt>
-<dd><p>The plot interval for the default geometric plot of the Ellipse.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>parameter</strong> : str, optional</p>
-<blockquote>
-<div><p>Default value is &#8216;t&#8217;.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>plot_interval</strong> : list</p>
-<blockquote class="last">
-<div><p>[parameter, lower_bound, upper_bound]</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ellipse</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="o">.</span><span class="n">plot_interval</span><span class="p">()</span>
-<span class="go">[t, -pi, pi]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.ellipse.Ellipse.random_point">
-<tt class="descname">random_point</tt><big>(</big><em>self</em>, <em>seed=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse.random_point"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse.random_point" title="Permalink to this definition">¶</a></dt>
-<dd><p>A random point on the ellipse.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>point</strong> : Point</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.arbitrary_point" title="sympy.geometry.ellipse.Ellipse.arbitrary_point"><tt class="xref py py-obj docutils literal"><span class="pre">arbitrary_point</span></tt></a></dt>
-<dd>Returns parameterized point on ellipse</dd>
-</dl>
-</div>
-<p class="rubric">Notes</p>
-<p>An arbitrary_point with a random value of t substituted into it may
-not test as being on the ellipse because the expression tested that
-a point is on the ellipse doesn&#8217;t simplify to zero and doesn&#8217;t evaluate
-exactly to zero:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">t</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-<span class="go">Point(3*cos(t), 2*sin(t))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p2</span> <span class="o">=</span> <span class="n">_</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p2</span> <span class="ow">in</span> <span class="n">e1</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>Note that arbitrary_point routine does not take this approach. A value for
-cos(t) and sin(t) (not t) is substituted into the arbitrary point. There is
-a small chance that this will give a point that will not test as being
-in the ellipse, so the process is repeated (up to 10 times) until a
-valid point is obtained.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ellipse</span><span class="p">,</span> <span class="n">Segment</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="o">.</span><span class="n">random_point</span><span class="p">()</span> <span class="c"># gives some random point</span>
-<span class="go">Point(...)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">e1</span><span class="o">.</span><span class="n">random_point</span><span class="p">(</span><span class="n">seed</span><span class="o">=</span><span class="mi">0</span><span class="p">);</span> <span class="n">p1</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">Point(2.1, 1.4)</span>
-</pre></div>
-</div>
-<p>The random_point method assures that the point will test as being
-in the ellipse:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="ow">in</span> <span class="n">e1</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.ellipse.Ellipse.rotate">
-<tt class="descname">rotate</tt><big>(</big><em>self</em>, <em>angle=0</em>, <em>pt=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse.rotate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse.rotate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Rotate <tt class="docutils literal"><span class="pre">angle</span></tt> radians counterclockwise about Point <tt class="docutils literal"><span class="pre">pt</span></tt>.</p>
-<p>Note: since the general ellipse is not supported, the axes of
-the ellipse will not be rotated. Only the center is rotated to
-a new position.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Ellipse</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ellipse</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">rotate</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">Ellipse(Point(0, 1), 2, 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.ellipse.Ellipse.scale">
-<tt class="descname">scale</tt><big>(</big><em>self</em>, <em>x=1</em>, <em>y=1</em>, <em>pt=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse.scale"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse.scale" title="Permalink to this definition">¶</a></dt>
-<dd><p>Override GeometryEntity.scale since it is the major and minor
-axes which must be scaled and they are not GeometryEntities.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Ellipse</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ellipse</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">scale</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">Circle(Point(0, 0), 4)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ellipse</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">scale</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">Ellipse(Point(0, 0), 4, 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.ellipse.Ellipse.tangent_lines">
-<tt class="descname">tangent_lines</tt><big>(</big><em>self</em>, <em>p</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse.tangent_lines"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse.tangent_lines" title="Permalink to this definition">¶</a></dt>
-<dd><p>Tangent lines between <span class="math">\(p\)</span> and the ellipse.</p>
-<p>If <span class="math">\(p\)</span> is on the ellipse, returns the tangent line through point <span class="math">\(p\)</span>.
-Otherwise, returns the tangent line(s) from <span class="math">\(p\)</span> to the ellipse, or
-None if no tangent line is possible (e.g., <span class="math">\(p\)</span> inside ellipse).</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>p</strong> : Point</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>tangent_lines</strong> : list with 1 or 2 Lines</p>
-</td>
-</tr>
-<tr class="field-odd field"><th class="field-name">Raises :</th><td class="field-body"><p class="first"><strong>NotImplementedError</strong> :</p>
-<blockquote class="last">
-<div><p>Can only find tangent lines for a point, <span class="math">\(p\)</span>, on the ellipse.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a>, <a class="reference internal" href="#sympy.geometry.line.Line" title="sympy.geometry.line.Line"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.line.Line</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ellipse</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="o">.</span><span class="n">tangent_lines</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">[Line(Point(3, 0), Point(3, -12))]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="c"># This will plot an ellipse together with a tangent line.</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ellipse</span><span class="p">,</span> <span class="n">Plot</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">tangent_lines</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">random_point</span><span class="p">())</span> 
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Plot</span><span class="p">()</span> 
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">e</span> 
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">t</span> 
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.ellipse.Ellipse.vradius">
-<tt class="descname">vradius</tt><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Ellipse.vradius"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Ellipse.vradius" title="Permalink to this definition">¶</a></dt>
-<dd><p>The vertical radius of the ellipse.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>vradius</strong> : number</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.hradius" title="sympy.geometry.ellipse.Ellipse.hradius"><tt class="xref py py-obj docutils literal"><span class="pre">hradius</span></tt></a>, <a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.major" title="sympy.geometry.ellipse.Ellipse.major"><tt class="xref py py-obj docutils literal"><span class="pre">major</span></tt></a>, <a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.minor" title="sympy.geometry.ellipse.Ellipse.minor"><tt class="xref py py-obj docutils literal"><span class="pre">minor</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Ellipse</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span> <span class="o">=</span> <span class="n">Ellipse</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="o">.</span><span class="n">vradius</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.geometry.ellipse.Circle">
-<em class="property">class </em><tt class="descclassname">sympy.geometry.ellipse.</tt><tt class="descname">Circle</tt><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Circle"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Circle" title="Permalink to this definition">¶</a></dt>
-<dd><p>A circle in space.</p>
-<p>Constructed simply from a center and a radius, or from three
-non-collinear points.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>center</strong> : Point</p>
-<p><strong>radius</strong> : number or sympy expression</p>
-<p><strong>points</strong> : sequence of three Points</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Raises :</th><td class="field-body"><p class="first"><strong>GeometryError</strong> :</p>
-<blockquote class="last">
-<div><p>When trying to construct circle from three collinear points.
-When trying to construct circle from incorrect parameters.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.ellipse.Ellipse" title="sympy.geometry.ellipse.Ellipse"><tt class="xref py py-obj docutils literal"><span class="pre">Ellipse</span></tt></a>, <a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Circle</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># a circle constructed from a center and radius</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span> <span class="o">=</span> <span class="n">Circle</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span><span class="o">.</span><span class="n">hradius</span><span class="p">,</span> <span class="n">c1</span><span class="o">.</span><span class="n">vradius</span><span class="p">,</span> <span class="n">c1</span><span class="o">.</span><span class="n">radius</span>
-<span class="go">(5, 5, 5)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="c"># a circle costructed from three points</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c2</span> <span class="o">=</span> <span class="n">Circle</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c2</span><span class="o">.</span><span class="n">hradius</span><span class="p">,</span> <span class="n">c2</span><span class="o">.</span><span class="n">vradius</span><span class="p">,</span> <span class="n">c2</span><span class="o">.</span><span class="n">radius</span><span class="p">,</span> <span class="n">c2</span><span class="o">.</span><span class="n">center</span>
-<span class="go">(sqrt(2)/2, sqrt(2)/2, sqrt(2)/2, Point(1/2, 1/2))</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="85%" />
-<col width="15%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>radius (synonymous with hradius, vradius, major and minor)</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>circumference</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>equation</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="attribute">
-<dt id="sympy.geometry.ellipse.Circle.circumference">
-<tt class="descname">circumference</tt><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Circle.circumference"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Circle.circumference" title="Permalink to this definition">¶</a></dt>
-<dd><p>The circumference of the circle.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>circumference</strong> : number or SymPy expression</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Circle</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span> <span class="o">=</span> <span class="n">Circle</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span><span class="o">.</span><span class="n">circumference</span>
-<span class="go">12*pi</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.ellipse.Circle.equation">
-<tt class="descname">equation</tt><big>(</big><em>self</em>, <em>x='x'</em>, <em>y='y'</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Circle.equation"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Circle.equation" title="Permalink to this definition">¶</a></dt>
-<dd><p>The equation of the circle.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>x</strong> : str or Symbol, optional</p>
-<blockquote>
-<div><p>Default value is &#8216;x&#8217;.</p>
-</div></blockquote>
-<p><strong>y</strong> : str or Symbol, optional</p>
-<blockquote>
-<div><p>Default value is &#8216;y&#8217;.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>equation</strong> : SymPy expression</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Circle</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span> <span class="o">=</span> <span class="n">Circle</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span><span class="o">.</span><span class="n">equation</span><span class="p">()</span>
-<span class="go">x**2 + y**2 - 25</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.ellipse.Circle.intersection">
-<tt class="descname">intersection</tt><big>(</big><em>self</em>, <em>o</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Circle.intersection"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Circle.intersection" title="Permalink to this definition">¶</a></dt>
-<dd><p>The intersection of this circle with another geometrical entity.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>o</strong> : GeometryEntity</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>intersection</strong> : list of GeometryEntities</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Circle</span><span class="p">,</span> <span class="n">Line</span><span class="p">,</span> <span class="n">Ray</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p4</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span> <span class="o">=</span> <span class="n">Circle</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">p2</span><span class="p">)</span>
-<span class="go">[]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">p4</span><span class="p">)</span>
-<span class="go">[Point(5, 0)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">Ray</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">))</span>
-<span class="go">[Point(5*sqrt(2)/2, 5*sqrt(2)/2)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">Line</span><span class="p">(</span><span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">))</span>
-<span class="go">[]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.ellipse.Circle.radius">
-<tt class="descname">radius</tt><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Circle.radius"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Circle.radius" title="Permalink to this definition">¶</a></dt>
-<dd><p>The radius of the circle.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>radius</strong> : number or sympy expression</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.major" title="sympy.geometry.ellipse.Ellipse.major"><tt class="xref py py-obj docutils literal"><span class="pre">Ellipse.major</span></tt></a>, <a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.minor" title="sympy.geometry.ellipse.Ellipse.minor"><tt class="xref py py-obj docutils literal"><span class="pre">Ellipse.minor</span></tt></a>, <a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.hradius" title="sympy.geometry.ellipse.Ellipse.hradius"><tt class="xref py py-obj docutils literal"><span class="pre">Ellipse.hradius</span></tt></a>, <a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.vradius" title="sympy.geometry.ellipse.Ellipse.vradius"><tt class="xref py py-obj docutils literal"><span class="pre">Ellipse.vradius</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Circle</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span> <span class="o">=</span> <span class="n">Circle</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span><span class="o">.</span><span class="n">radius</span>
-<span class="go">6</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.ellipse.Circle.scale">
-<tt class="descname">scale</tt><big>(</big><em>self</em>, <em>x=1</em>, <em>y=1</em>, <em>pt=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Circle.scale"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Circle.scale" title="Permalink to this definition">¶</a></dt>
-<dd><p>Override GeometryEntity.scale since the radius
-is not a GeometryEntity.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Circle</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Circle</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">scale</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">Circle(Point(0, 0), 2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Circle</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">scale</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">Ellipse(Point(0, 0), 2, 4)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.ellipse.Circle.vradius">
-<tt class="descname">vradius</tt><a class="reference internal" href="../_modules/sympy/geometry/ellipse.html#Circle.vradius"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.ellipse.Circle.vradius" title="Permalink to this definition">¶</a></dt>
-<dd><p>This Ellipse property is an alias for the Circle&#8217;s radius.</p>
-<p>Whereas hradius, major and minor can use Ellipse&#8217;s conventions,
-the vradius does not exist for a circle.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Circle</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span> <span class="o">=</span> <span class="n">Circle</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span><span class="o">.</span><span class="n">vradius</span>
-<span class="go">6</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.geometry.polygon">
-<span id="polygons"></span><h3>Polygons<a class="headerlink" href="#module-sympy.geometry.polygon" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.geometry.polygon.Polygon">
-<em class="property">class </em><tt class="descclassname">sympy.geometry.polygon.</tt><tt class="descname">Polygon</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Polygon"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Polygon" title="Permalink to this definition">¶</a></dt>
-<dd><p>A two-dimensional polygon.</p>
-<p>A simple polygon in space. Can be constructed from a sequence of points
-or from a center, radius, number of sides and rotation angle.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>vertices</strong> : sequence of Points</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Raises :</th><td class="field-body"><p class="first"><strong>GeometryError</strong> :</p>
-<blockquote class="last">
-<div><p>If all parameters are not Points.</p>
-<p>If the Polygon has intersecting sides.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a>, <a class="reference internal" href="#sympy.geometry.line.Segment" title="sympy.geometry.line.Segment"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.line.Segment</span></tt></a>, <a class="reference internal" href="#sympy.geometry.polygon.Triangle" title="sympy.geometry.polygon.Triangle"><tt class="xref py py-obj docutils literal"><span class="pre">Triangle</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>Polygons are treated as closed paths rather than 2D areas so
-some calculations can be be negative or positive (e.g., area)
-based on the orientation of the points.</p>
-<p>Any consecutive identical points are reduced to a single point
-and any points collinear and between two points will be removed
-unless they are needed to define an explicit intersection (see examples).</p>
-<p>A Triangle, Segment or Point will be returned when there are 3 or
-fewer points provided.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Polygon</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span><span class="p">,</span> <span class="n">p5</span> <span class="o">=</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Polygon</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span><span class="p">)</span>
-<span class="go">Polygon(Point(0, 0), Point(1, 0), Point(5, 1), Point(0, 1))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Polygon</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">)</span>
-<span class="go">Segment(Point(0, 0), Point(1, 0))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Polygon</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p5</span><span class="p">)</span>
-<span class="go">Segment(Point(0, 0), Point(3, 0))</span>
-</pre></div>
-</div>
-<p>While the sides of a polygon are not allowed to cross implicitly, they
-can do so explicitly. For example, a polygon shaped like a Z with the top
-left connecting to the bottom right of the Z must have the point in the
-middle of the Z explicitly given:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mid</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Polygon</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">mid</span><span class="p">,</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">mid</span><span class="p">)</span><span class="o">.</span><span class="n">area</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Polygon</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">mid</span><span class="p">,</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">mid</span><span class="p">)</span><span class="o">.</span><span class="n">area</span>
-<span class="go">-2</span>
-</pre></div>
-</div>
-<p>When the the keyword <span class="math">\(n\)</span> is used to define the number of sides of the
-Polygon then a RegularPolygon is created and the other arguments are
-interpreted as center, radius and rotation. The unrotated RegularPolygon
-will always have a vertex at Point(r, 0) where <span class="math">\(r\)</span> is the radius of the
-circle that circumscribes the RegularPolygon. Its method <span class="math">\(spin\)</span> can be
-used to increment that angle.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span>
-<span class="go">RegularPolygon(Point(0, 0), 1, 3, 0)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">vertices</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">Point(1, 0)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">Point(0, 0)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">spin</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">vertices</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">Point(0, 1)</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="47%" />
-<col width="53%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>area</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>angles</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>perimeter</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>vertices</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>centroid</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>sides</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.Polygon.angles">
-<tt class="descname">angles</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Polygon.angles"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Polygon.angles" title="Permalink to this definition">¶</a></dt>
-<dd><p>The internal angle at each vertex.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>angles</strong> : dict</p>
-<blockquote class="last">
-<div><p>A dictionary where each key is a vertex and each value is the
-internal angle at that vertex. The vertices are represented as
-Points.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a>, <a class="reference internal" href="#sympy.geometry.line.LinearEntity.angle_between" title="sympy.geometry.line.LinearEntity.angle_between"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.line.LinearEntity.angle_between</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Polygon</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Point</span><span class="p">,</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly</span><span class="o">.</span><span class="n">angles</span><span class="p">[</span><span class="n">p1</span><span class="p">]</span>
-<span class="go">pi/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly</span><span class="o">.</span><span class="n">angles</span><span class="p">[</span><span class="n">p2</span><span class="p">]</span>
-<span class="go">acos(-4*sqrt(17)/17)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.polygon.Polygon.arbitrary_point">
-<tt class="descname">arbitrary_point</tt><big>(</big><em>self</em>, <em>parameter='t'</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Polygon.arbitrary_point"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Polygon.arbitrary_point" title="Permalink to this definition">¶</a></dt>
-<dd><p>A parameterized point on the polygon.</p>
-<p>The parameter, varying from 0 to 1, assigns points to the position on
-the perimeter that is that fraction of the total perimeter. So the
-point evaluated at t=1/2 would return the point from the first vertex
-that is 1/2 way around the polygon.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>parameter</strong> : str, optional</p>
-<blockquote>
-<div><p>Default value is &#8216;t&#8217;.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>arbitrary_point</strong> : Point</p>
-</td>
-</tr>
-<tr class="field-odd field"><th class="field-name">Raises :</th><td class="field-body"><p class="first"><strong>ValueError</strong> :</p>
-<blockquote class="last">
-<div><p>When <span class="math">\(parameter\)</span> already appears in the Polygon&#8217;s definition.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Polygon</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tri</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">tri</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">perimeter</span> <span class="o">=</span> <span class="n">tri</span><span class="o">.</span><span class="n">perimeter</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s1</span><span class="p">,</span> <span class="n">s2</span> <span class="o">=</span> <span class="p">[</span><span class="n">s</span><span class="o">.</span><span class="n">length</span> <span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">tri</span><span class="o">.</span><span class="n">sides</span><span class="p">[:</span><span class="mi">2</span><span class="p">]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="p">(</span><span class="n">s1</span> <span class="o">+</span> <span class="n">s2</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">perimeter</span><span class="p">)</span>
-<span class="go">Point(1, 1/2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.Polygon.area">
-<tt class="descname">area</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Polygon.area"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Polygon.area" title="Permalink to this definition">¶</a></dt>
-<dd><p>The area of the polygon.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.area" title="sympy.geometry.ellipse.Ellipse.area"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.ellipse.Ellipse.area</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>The area calculation can be positive or negative based on the
-orientation of the points.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Polygon</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Point</span><span class="p">,</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly</span><span class="o">.</span><span class="n">area</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.Polygon.centroid">
-<tt class="descname">centroid</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Polygon.centroid"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Polygon.centroid" title="Permalink to this definition">¶</a></dt>
-<dd><p>The centroid of the polygon.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>centroid</strong> : Point</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a>, <a class="reference internal" href="#sympy.geometry.util.centroid" title="sympy.geometry.util.centroid"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.util.centroid</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Polygon</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Point</span><span class="p">,</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly</span><span class="o">.</span><span class="n">centroid</span>
-<span class="go">Point(31/18, 11/18)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.polygon.Polygon.distance">
-<tt class="descname">distance</tt><big>(</big><em>self</em>, <em>o</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Polygon.distance"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Polygon.distance" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the shortest distance between self and o.</p>
-<p>If o is a point, then self does not need to be convex.
-If o is another polygon self and o must be complex.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Polygon</span><span class="p">,</span> <span class="n">RegularPolygon</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Point</span><span class="p">,</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="mi">5</span><span class="p">)]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">(</span><span class="o">*</span><span class="n">RegularPolygon</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">vertices</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly</span><span class="o">.</span><span class="n">distance</span><span class="p">(</span><span class="n">p2</span><span class="p">)</span>
-<span class="go">sqrt(61)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.polygon.Polygon.encloses_point">
-<tt class="descname">encloses_point</tt><big>(</big><em>self</em>, <em>p</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Polygon.encloses_point"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Polygon.encloses_point" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if p is enclosed by (is inside of) self.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>p</strong> : Point</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>encloses_point</strong> : True, False or None</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a>, <a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.encloses_point" title="sympy.geometry.ellipse.Ellipse.encloses_point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.ellipse.Ellipse.encloses_point</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>Being on the border of self is considered False.</p>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://www.ariel.com.au/a/python-point-int-poly.html">http://www.ariel.com.au/a/python-point-int-poly.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Polygon</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">t</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">encloses_point</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">encloses_point</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">encloses_point</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.polygon.Polygon.intersection">
-<tt class="descname">intersection</tt><big>(</big><em>self</em>, <em>o</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Polygon.intersection"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Polygon.intersection" title="Permalink to this definition">¶</a></dt>
-<dd><p>The intersection of two polygons.</p>
-<p>The intersection may be empty and can contain individual Points and
-complete Line Segments.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>other: Polygon</strong> :</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>intersection</strong> : list</p>
-<blockquote class="last">
-<div><p>The list of Segments and Points</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a>, <a class="reference internal" href="#sympy.geometry.line.Segment" title="sympy.geometry.line.Segment"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.line.Segment</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Polygon</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Point</span><span class="p">,</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly1</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p5</span><span class="p">,</span> <span class="n">p6</span><span class="p">,</span> <span class="n">p7</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Point</span><span class="p">,</span> <span class="p">[(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">)]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly2</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">(</span><span class="n">p5</span><span class="p">,</span> <span class="n">p6</span><span class="p">,</span> <span class="n">p7</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly1</span><span class="o">.</span><span class="n">intersection</span><span class="p">(</span><span class="n">poly2</span><span class="p">)</span>
-<span class="go">[Point(2/3, 0), Point(9/5, 1/5), Point(7/3, 1), Point(1/3, 1)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.polygon.Polygon.is_convex">
-<tt class="descname">is_convex</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Polygon.is_convex"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Polygon.is_convex" title="Permalink to this definition">¶</a></dt>
-<dd><p>Is the polygon convex?</p>
-<p>A polygon is convex if all its interior angles are less than 180
-degrees.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>is_convex</strong> : boolean</p>
-<blockquote class="last">
-<div><p>True if this polygon is convex, False otherwise.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.util.convex_hull" title="sympy.geometry.util.convex_hull"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.util.convex_hull</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Polygon</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Point</span><span class="p">,</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly</span><span class="o">.</span><span class="n">is_convex</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.Polygon.perimeter">
-<tt class="descname">perimeter</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Polygon.perimeter"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Polygon.perimeter" title="Permalink to this definition">¶</a></dt>
-<dd><p>The perimeter of the polygon.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>perimeter</strong> : number or Basic instance</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.line.Segment.length" title="sympy.geometry.line.Segment.length"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.line.Segment.length</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Polygon</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Point</span><span class="p">,</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly</span><span class="o">.</span><span class="n">perimeter</span>
-<span class="go">sqrt(17) + 7</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.polygon.Polygon.plot_interval">
-<tt class="descname">plot_interval</tt><big>(</big><em>self</em>, <em>parameter='t'</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Polygon.plot_interval"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Polygon.plot_interval" title="Permalink to this definition">¶</a></dt>
-<dd><p>The plot interval for the default geometric plot of the polygon.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>parameter</strong> : str, optional</p>
-<blockquote>
-<div><p>Default value is &#8216;t&#8217;.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>plot_interval</strong> : list (plot interval)</p>
-<blockquote class="last">
-<div><p>[parameter, lower_bound, upper_bound]</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Polygon</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">plot_interval</span><span class="p">()</span>
-<span class="go">[t, 0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.Polygon.sides">
-<tt class="descname">sides</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Polygon.sides"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Polygon.sides" title="Permalink to this definition">¶</a></dt>
-<dd><p>The line segments that form the sides of the polygon.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>sides</strong> : list of sides</p>
-<blockquote class="last">
-<div><p>Each side is a Segment.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a>, <a class="reference internal" href="#sympy.geometry.line.Segment" title="sympy.geometry.line.Segment"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.line.Segment</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>The Segments that represent the sides are an undirected
-line segment so cannot be used to tell the orientation of
-the polygon.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Polygon</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Point</span><span class="p">,</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly</span><span class="o">.</span><span class="n">sides</span>
-<span class="go">[Segment(Point(0, 0), Point(1, 0)),</span>
-<span class="go">Segment(Point(1, 0), Point(5, 1)),</span>
-<span class="go">Segment(Point(0, 1), Point(5, 1)), Segment(Point(0, 0), Point(0, 1))]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.Polygon.vertices">
-<tt class="descname">vertices</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Polygon.vertices"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Polygon.vertices" title="Permalink to this definition">¶</a></dt>
-<dd><p>The vertices of the polygon.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>vertices</strong> : tuple of Points</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>When iterating over the vertices, it is more efficient to index self
-rather than to request the vertices and index them. Only use the
-vertices when you want to process all of them at once. This is even
-more important with RegularPolygons that calculate each vertex.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Polygon</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Point</span><span class="p">,</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">,</span> <span class="n">p4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly</span><span class="o">.</span><span class="n">vertices</span>
-<span class="go">(Point(0, 0), Point(1, 0), Point(5, 1), Point(0, 1))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">Point(0, 0)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.geometry.polygon.RegularPolygon">
-<em class="property">class </em><tt class="descclassname">sympy.geometry.polygon.</tt><tt class="descname">RegularPolygon</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#RegularPolygon"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.RegularPolygon" title="Permalink to this definition">¶</a></dt>
-<dd><p>A regular polygon.</p>
-<p>Such a polygon has all internal angles equal and all sides the same length.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>center</strong> : Point</p>
-<p><strong>radius</strong> : number or Basic instance</p>
-<blockquote>
-<div><p>The distance from the center to a vertex</p>
-</div></blockquote>
-<p><strong>n</strong> : int</p>
-<blockquote>
-<div><p>The number of sides</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Raises :</th><td class="field-body"><p class="first"><strong>GeometryError</strong> :</p>
-<blockquote class="last">
-<div><p>If the <span class="math">\(center\)</span> is not a Point, or the <span class="math">\(radius\)</span> is not a number or Basic
-instance, or the number of sides, <span class="math">\(n\)</span>, is less than three.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a>, <a class="reference internal" href="#sympy.geometry.polygon.Polygon" title="sympy.geometry.polygon.Polygon"><tt class="xref py py-obj docutils literal"><span class="pre">Polygon</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>A RegularPolygon can be instantiated with Polygon with the kwarg n.</p>
-<p>Regular polygons are instantiated with a center, radius, number of sides
-and a rotation angle. Whereas the arguments of a Polygon are vertices, the
-vertices of the RegularPolygon must be obtained with the vertices method.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">RegularPolygon</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span> <span class="o">=</span> <span class="n">RegularPolygon</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span>
-<span class="go">RegularPolygon(Point(0, 0), 5, 3, 0)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="o">.</span><span class="n">vertices</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">Point(5, 0)</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="58%" />
-<col width="42%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>vertices</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>center</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>radius</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>rotation</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>apothem</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>interior_angle</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>exterior_angle</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>circumcircle</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>incircle</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>angles</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.RegularPolygon.angles">
-<tt class="descname">angles</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#RegularPolygon.angles"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.RegularPolygon.angles" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a dictionary with keys, the vertices of the Polygon,
-and values, the interior angle at each vertex.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">RegularPolygon</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span> <span class="o">=</span> <span class="n">RegularPolygon</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="o">.</span><span class="n">angles</span>
-<span class="go">{Point(-5/2, -5*sqrt(3)/2): pi/3,</span>
-<span class="go"> Point(-5/2, 5*sqrt(3)/2): pi/3,</span>
-<span class="go"> Point(5, 0): pi/3}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.RegularPolygon.apothem">
-<tt class="descname">apothem</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#RegularPolygon.apothem"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.RegularPolygon.apothem" title="Permalink to this definition">¶</a></dt>
-<dd><p>The inradius of the RegularPolygon.</p>
-<p>The apothem/inradius is the radius of the inscribed circle.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>apothem</strong> : number or instance of Basic</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.line.Segment.length" title="sympy.geometry.line.Segment.length"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.line.Segment.length</span></tt></a>, <a class="reference internal" href="#sympy.geometry.ellipse.Circle.radius" title="sympy.geometry.ellipse.Circle.radius"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.ellipse.Circle.radius</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">RegularPolygon</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">radius</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;r&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rp</span> <span class="o">=</span> <span class="n">RegularPolygon</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">radius</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rp</span><span class="o">.</span><span class="n">apothem</span>
-<span class="go">sqrt(2)*r/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.RegularPolygon.area">
-<tt class="descname">area</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#RegularPolygon.area"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.RegularPolygon.area" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the area.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">RegularPolygon</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">square</span> <span class="o">=</span> <span class="n">RegularPolygon</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">square</span><span class="o">.</span><span class="n">area</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span> <span class="o">==</span> <span class="n">square</span><span class="o">.</span><span class="n">length</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.RegularPolygon.args">
-<tt class="descname">args</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#RegularPolygon.args"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.RegularPolygon.args" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the center point, the radius,
-the number of sides, and the orientation angle.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">RegularPolygon</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span> <span class="o">=</span> <span class="n">RegularPolygon</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="o">.</span><span class="n">args</span>
-<span class="go">(Point(0, 0), 5, 3, 0)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.RegularPolygon.center">
-<tt class="descname">center</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#RegularPolygon.center"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.RegularPolygon.center" title="Permalink to this definition">¶</a></dt>
-<dd><p>The center of the RegularPolygon</p>
-<p>This is also the center of the circumscribing circle.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>center</strong> : Point</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a>, <a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.center" title="sympy.geometry.ellipse.Ellipse.center"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.ellipse.Ellipse.center</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">RegularPolygon</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rp</span> <span class="o">=</span> <span class="n">RegularPolygon</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rp</span><span class="o">.</span><span class="n">center</span>
-<span class="go">Point(0, 0)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.RegularPolygon.circumcenter">
-<tt class="descname">circumcenter</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#RegularPolygon.circumcenter"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.RegularPolygon.circumcenter" title="Permalink to this definition">¶</a></dt>
-<dd><p>Alias for center.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">RegularPolygon</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rp</span> <span class="o">=</span> <span class="n">RegularPolygon</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rp</span><span class="o">.</span><span class="n">circumcenter</span>
-<span class="go">Point(0, 0)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.RegularPolygon.circumcircle">
-<tt class="descname">circumcircle</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#RegularPolygon.circumcircle"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.RegularPolygon.circumcircle" title="Permalink to this definition">¶</a></dt>
-<dd><p>The circumcircle of the RegularPolygon.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>circumcircle</strong> : Circle</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.polygon.RegularPolygon.circumcenter" title="sympy.geometry.polygon.RegularPolygon.circumcenter"><tt class="xref py py-obj docutils literal"><span class="pre">circumcenter</span></tt></a>, <a class="reference internal" href="#sympy.geometry.ellipse.Circle" title="sympy.geometry.ellipse.Circle"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.ellipse.Circle</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">RegularPolygon</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rp</span> <span class="o">=</span> <span class="n">RegularPolygon</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rp</span><span class="o">.</span><span class="n">circumcircle</span>
-<span class="go">Circle(Point(0, 0), 4)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.RegularPolygon.circumradius">
-<tt class="descname">circumradius</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#RegularPolygon.circumradius"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.RegularPolygon.circumradius" title="Permalink to this definition">¶</a></dt>
-<dd><p>Alias for radius.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">RegularPolygon</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">radius</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;r&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rp</span> <span class="o">=</span> <span class="n">RegularPolygon</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">radius</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rp</span><span class="o">.</span><span class="n">circumradius</span>
-<span class="go">r</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.polygon.RegularPolygon.encloses_point">
-<tt class="descname">encloses_point</tt><big>(</big><em>self</em>, <em>p</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#RegularPolygon.encloses_point"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.RegularPolygon.encloses_point" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if p is enclosed by (is inside of) self.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>p</strong> : Point</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>encloses_point</strong> : True, False or None</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.ellipse.Ellipse.encloses_point" title="sympy.geometry.ellipse.Ellipse.encloses_point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.ellipse.Ellipse.encloses_point</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>Being on the border of self is considered False.</p>
-<p>The general Polygon.encloses_point method is called only if
-a point is not within or beyond the incircle or circumcircle,
-respectively.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">RegularPolygon</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">RegularPolygon</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">encloses_point</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="p">,</span> <span class="n">R</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">inradius</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">circumradius</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">encloses_point</span><span class="p">(</span><span class="n">Point</span><span class="p">((</span><span class="n">r</span> <span class="o">+</span> <span class="n">R</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">encloses_point</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="n">R</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">R</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="p">(</span><span class="n">R</span> <span class="o">-</span> <span class="n">r</span><span class="p">)</span><span class="o">/</span><span class="mi">10</span><span class="p">))</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">encloses_point</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">arbitrary_point</span><span class="p">()</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">))</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">encloses_point</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.RegularPolygon.exterior_angle">
-<tt class="descname">exterior_angle</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#RegularPolygon.exterior_angle"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.RegularPolygon.exterior_angle" title="Permalink to this definition">¶</a></dt>
-<dd><p>Measure of the exterior angles.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>exterior_angle</strong> : number</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.line.LinearEntity.angle_between" title="sympy.geometry.line.LinearEntity.angle_between"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.line.LinearEntity.angle_between</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">RegularPolygon</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rp</span> <span class="o">=</span> <span class="n">RegularPolygon</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rp</span><span class="o">.</span><span class="n">exterior_angle</span>
-<span class="go">pi/4</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.RegularPolygon.incircle">
-<tt class="descname">incircle</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#RegularPolygon.incircle"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.RegularPolygon.incircle" title="Permalink to this definition">¶</a></dt>
-<dd><p>The incircle of the RegularPolygon.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>incircle</strong> : Circle</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.polygon.RegularPolygon.inradius" title="sympy.geometry.polygon.RegularPolygon.inradius"><tt class="xref py py-obj docutils literal"><span class="pre">inradius</span></tt></a>, <a class="reference internal" href="#sympy.geometry.ellipse.Circle" title="sympy.geometry.ellipse.Circle"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.ellipse.Circle</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">RegularPolygon</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rp</span> <span class="o">=</span> <span class="n">RegularPolygon</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rp</span><span class="o">.</span><span class="n">incircle</span>
-<span class="go">Circle(Point(0, 0), 4*cos(pi/7))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.RegularPolygon.inradius">
-<tt class="descname">inradius</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#RegularPolygon.inradius"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.RegularPolygon.inradius" title="Permalink to this definition">¶</a></dt>
-<dd><p>Alias for apothem.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">RegularPolygon</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">radius</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;r&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rp</span> <span class="o">=</span> <span class="n">RegularPolygon</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">radius</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rp</span><span class="o">.</span><span class="n">inradius</span>
-<span class="go">sqrt(2)*r/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.RegularPolygon.interior_angle">
-<tt class="descname">interior_angle</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#RegularPolygon.interior_angle"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.RegularPolygon.interior_angle" title="Permalink to this definition">¶</a></dt>
-<dd><p>Measure of the interior angles.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>interior_angle</strong> : number</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.line.LinearEntity.angle_between" title="sympy.geometry.line.LinearEntity.angle_between"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.line.LinearEntity.angle_between</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">RegularPolygon</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rp</span> <span class="o">=</span> <span class="n">RegularPolygon</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rp</span><span class="o">.</span><span class="n">interior_angle</span>
-<span class="go">3*pi/4</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.RegularPolygon.length">
-<tt class="descname">length</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#RegularPolygon.length"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.RegularPolygon.length" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the length of the sides.</p>
-<p>The half-length of the side and the apothem form two legs
-of a right triangle whose hypotenuse is the radius of the
-regular polygon.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">RegularPolygon</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">square_in_unit_circle</span> <span class="o">=</span> <span class="n">RegularPolygon</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="o">.</span><span class="n">length</span>
-<span class="go">sqrt(2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">((</span><span class="n">_</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">s</span><span class="o">.</span><span class="n">apothem</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">==</span> <span class="n">s</span><span class="o">.</span><span class="n">radius</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.RegularPolygon.radius">
-<tt class="descname">radius</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#RegularPolygon.radius"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.RegularPolygon.radius" title="Permalink to this definition">¶</a></dt>
-<dd><p>Radius of the RegularPolygon</p>
-<p>This is also the radius of the circumscribing circle.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>radius</strong> : number or instance of Basic</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.line.Segment.length" title="sympy.geometry.line.Segment.length"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.line.Segment.length</span></tt></a>, <a class="reference internal" href="#sympy.geometry.ellipse.Circle.radius" title="sympy.geometry.ellipse.Circle.radius"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.ellipse.Circle.radius</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">RegularPolygon</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">radius</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;r&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rp</span> <span class="o">=</span> <span class="n">RegularPolygon</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">radius</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rp</span><span class="o">.</span><span class="n">radius</span>
-<span class="go">r</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.polygon.RegularPolygon.rotate">
-<tt class="descname">rotate</tt><big>(</big><em>self</em>, <em>angle</em>, <em>pt=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#RegularPolygon.rotate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.RegularPolygon.rotate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Override GeometryEntity.rotate to first rotate the RegularPolygon
-about its center.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">RegularPolygon</span><span class="p">,</span> <span class="n">Polygon</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">RegularPolygon</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">vertices</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="c"># vertex on x-axis</span>
-<span class="go">Point(2, 0)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">rotate</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">vertices</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="c"># vertex on y axis now</span>
-<span class="go">Point(0, 2)</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p><a class="reference internal" href="#sympy.geometry.polygon.RegularPolygon.rotation" title="sympy.geometry.polygon.RegularPolygon.rotation"><tt class="xref py py-obj docutils literal"><span class="pre">rotation</span></tt></a></p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.geometry.polygon.RegularPolygon.spin" title="sympy.geometry.polygon.RegularPolygon.spin"><tt class="xref py py-obj docutils literal"><span class="pre">spin</span></tt></a></dt>
-<dd>Rotates a RegularPolygon in place</dd>
-</dl>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.RegularPolygon.rotation">
-<tt class="descname">rotation</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#RegularPolygon.rotation"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.RegularPolygon.rotation" title="Permalink to this definition">¶</a></dt>
-<dd><p>CCW angle by which the RegularPolygon is rotated</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>rotation</strong> : number or instance of Basic</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">RegularPolygon</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">RegularPolygon</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="n">pi</span><span class="p">)</span><span class="o">.</span><span class="n">rotation</span>
-<span class="go">pi</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.polygon.RegularPolygon.scale">
-<tt class="descname">scale</tt><big>(</big><em>self</em>, <em>x=1</em>, <em>y=1</em>, <em>pt=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#RegularPolygon.scale"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.RegularPolygon.scale" title="Permalink to this definition">¶</a></dt>
-<dd><p>Override GeometryEntity.scale since it is the radius that must be
-scaled (if x == y) or else a new Polygon must be returned.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">RegularPolygon</span>
-</pre></div>
-</div>
-<p>Symmetric scaling returns a RegularPolygon:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">RegularPolygon</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">scale</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">RegularPolygon(Point(0, 0), 2, 4, 0)</span>
-</pre></div>
-</div>
-<p>Asymmetric scaling returns a kite as a Polygon:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">RegularPolygon</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">scale</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">Polygon(Point(2, 0), Point(0, 1), Point(-2, 0), Point(0, -1))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.polygon.RegularPolygon.spin">
-<tt class="descname">spin</tt><big>(</big><em>self</em>, <em>angle</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#RegularPolygon.spin"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.RegularPolygon.spin" title="Permalink to this definition">¶</a></dt>
-<dd><p>Increment <em>in place</em> the virtual Polygon&#8217;s rotation by ccw angle.</p>
-<p>See also: rotate method which moves the center.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Polygon</span><span class="p">,</span> <span class="n">Point</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span> <span class="o">=</span> <span class="n">Polygon</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="o">.</span><span class="n">vertices</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">Point(1, 0)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="o">.</span><span class="n">spin</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="o">.</span><span class="n">vertices</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">Point(sqrt(3)/2, 1/2)</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p><a class="reference internal" href="#sympy.geometry.polygon.RegularPolygon.rotation" title="sympy.geometry.polygon.RegularPolygon.rotation"><tt class="xref py py-obj docutils literal"><span class="pre">rotation</span></tt></a></p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.geometry.polygon.RegularPolygon.rotate" title="sympy.geometry.polygon.RegularPolygon.rotate"><tt class="xref py py-obj docutils literal"><span class="pre">rotate</span></tt></a></dt>
-<dd>Creates a copy of the RegularPolygon rotated about a Point</dd>
-</dl>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.RegularPolygon.vertices">
-<tt class="descname">vertices</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#RegularPolygon.vertices"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.RegularPolygon.vertices" title="Permalink to this definition">¶</a></dt>
-<dd><p>The vertices of the RegularPolygon.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>vertices</strong> : list</p>
-<blockquote class="last">
-<div><p>Each vertex is a Point.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">RegularPolygon</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rp</span> <span class="o">=</span> <span class="n">RegularPolygon</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rp</span><span class="o">.</span><span class="n">vertices</span>
-<span class="go">[Point(5, 0), Point(0, 5), Point(-5, 0), Point(0, -5)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.geometry.polygon.Triangle">
-<em class="property">class </em><tt class="descclassname">sympy.geometry.polygon.</tt><tt class="descname">Triangle</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Triangle"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Triangle" title="Permalink to this definition">¶</a></dt>
-<dd><p>A polygon with three vertices and three sides.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>points</strong> : sequence of Points</p>
-<p><strong>keyword: asa, sas, or sss to specify sides/angles of the triangle</strong> :</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Raises :</th><td class="field-body"><p class="first"><strong>GeometryError</strong> :</p>
-<blockquote class="last">
-<div><p>If the number of vertices is not equal to three, or one of the vertices
-is not a Point, or a valid keyword is not given.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a>, <a class="reference internal" href="#sympy.geometry.polygon.Polygon" title="sympy.geometry.polygon.Polygon"><tt class="xref py py-obj docutils literal"><span class="pre">Polygon</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Triangle</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Triangle</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
-<span class="go">Triangle(Point(0, 0), Point(4, 0), Point(4, 3))</span>
-</pre></div>
-</div>
-<p>Keywords sss, sas, or asa can be used to give the desired
-side lengths (in order) and interior angles (in degrees) that
-define the triangle:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Triangle</span><span class="p">(</span><span class="n">sss</span><span class="o">=</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
-<span class="go">Triangle(Point(0, 0), Point(3, 0), Point(3, 4))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Triangle</span><span class="p">(</span><span class="n">asa</span><span class="o">=</span><span class="p">(</span><span class="mi">30</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">30</span><span class="p">))</span>
-<span class="go">Triangle(Point(0, 0), Point(1, 0), Point(1/2, sqrt(3)/6))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Triangle</span><span class="p">(</span><span class="n">sas</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">45</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">Triangle(Point(0, 0), Point(2, 0), Point(sqrt(2)/2, sqrt(2)/2))</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="55%" />
-<col width="45%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>vertices</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>altitudes</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>orthocenter</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>circumcenter</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>circumradius</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>circumcircle</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>inradius</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>incircle</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>medians</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>medial</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.Triangle.altitudes">
-<tt class="descname">altitudes</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Triangle.altitudes"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Triangle.altitudes" title="Permalink to this definition">¶</a></dt>
-<dd><p>The altitudes of the triangle.</p>
-<p>An altitude of a triangle is a segment through a vertex,
-perpendicular to the opposite side, with length being the
-height of the vertex measured from the line containing the side.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>altitudes</strong> : dict</p>
-<blockquote class="last">
-<div><p>The dictionary consists of keys which are vertices and values
-which are Segments.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a>, <a class="reference internal" href="#sympy.geometry.line.Segment.length" title="sympy.geometry.line.Segment.length"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.line.Segment.length</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Triangle</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">altitudes</span><span class="p">[</span><span class="n">p1</span><span class="p">]</span>
-<span class="go">Segment(Point(0, 0), Point(1/2, 1/2))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.polygon.Triangle.bisectors">
-<tt class="descname">bisectors</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Triangle.bisectors"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Triangle.bisectors" title="Permalink to this definition">¶</a></dt>
-<dd><p>The angle bisectors of the triangle.</p>
-<p>An angle bisector of a triangle is a straight line through a vertex
-which cuts the corresponding angle in half.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>bisectors</strong> : dict</p>
-<blockquote class="last">
-<div><p>Each key is a vertex (Point) and each value is the corresponding
-bisector (Segment).</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a>, <a class="reference internal" href="#sympy.geometry.line.Segment" title="sympy.geometry.line.Segment"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.line.Segment</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Triangle</span><span class="p">,</span> <span class="n">Segment</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">bisectors</span><span class="p">()[</span><span class="n">p2</span><span class="p">]</span> <span class="o">==</span> <span class="n">Segment</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.Triangle.circumcenter">
-<tt class="descname">circumcenter</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Triangle.circumcenter"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Triangle.circumcenter" title="Permalink to this definition">¶</a></dt>
-<dd><p>The circumcenter of the triangle</p>
-<p>The circumcenter is the center of the circumcircle.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>circumcenter</strong> : Point</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Triangle</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">circumcenter</span>
-<span class="go">Point(1/2, 1/2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.Triangle.circumcircle">
-<tt class="descname">circumcircle</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Triangle.circumcircle"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Triangle.circumcircle" title="Permalink to this definition">¶</a></dt>
-<dd><p>The circle which passes through the three vertices of the triangle.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>circumcircle</strong> : Circle</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.ellipse.Circle" title="sympy.geometry.ellipse.Circle"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.ellipse.Circle</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Triangle</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">circumcircle</span>
-<span class="go">Circle(Point(1/2, 1/2), sqrt(2)/2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.Triangle.circumradius">
-<tt class="descname">circumradius</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Triangle.circumradius"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Triangle.circumradius" title="Permalink to this definition">¶</a></dt>
-<dd><p>The radius of the circumcircle of the triangle.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>circumradius</strong> : number of Basic instance</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.ellipse.Circle.radius" title="sympy.geometry.ellipse.Circle.radius"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.ellipse.Circle.radius</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Triangle</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">circumradius</span>
-<span class="go">sqrt(a**2/4 + 1/4)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.Triangle.incenter">
-<tt class="descname">incenter</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Triangle.incenter"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Triangle.incenter" title="Permalink to this definition">¶</a></dt>
-<dd><p>The center of the incircle.</p>
-<p>The incircle is the circle which lies inside the triangle and touches
-all three sides.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>incenter</strong> : Point</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.polygon.Triangle.incircle" title="sympy.geometry.polygon.Triangle.incircle"><tt class="xref py py-obj docutils literal"><span class="pre">incircle</span></tt></a>, <a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Triangle</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">incenter</span>
-<span class="go">Point(-sqrt(2)/2 + 1, -sqrt(2)/2 + 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.Triangle.incircle">
-<tt class="descname">incircle</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Triangle.incircle"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Triangle.incircle" title="Permalink to this definition">¶</a></dt>
-<dd><p>The incircle of the triangle.</p>
-<p>The incircle is the circle which lies inside the triangle and touches
-all three sides.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>incircle</strong> : Circle</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.ellipse.Circle" title="sympy.geometry.ellipse.Circle"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.ellipse.Circle</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Triangle</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">incircle</span>
-<span class="go">Circle(Point(-sqrt(2) + 2, -sqrt(2) + 2), -sqrt(2) + 2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.Triangle.inradius">
-<tt class="descname">inradius</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Triangle.inradius"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Triangle.inradius" title="Permalink to this definition">¶</a></dt>
-<dd><p>The radius of the incircle.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>inradius</strong> : number of Basic instance</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.polygon.Triangle.incircle" title="sympy.geometry.polygon.Triangle.incircle"><tt class="xref py py-obj docutils literal"><span class="pre">incircle</span></tt></a>, <a class="reference internal" href="#sympy.geometry.ellipse.Circle.radius" title="sympy.geometry.ellipse.Circle.radius"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.ellipse.Circle.radius</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Triangle</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">inradius</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.polygon.Triangle.is_equilateral">
-<tt class="descname">is_equilateral</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Triangle.is_equilateral"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Triangle.is_equilateral" title="Permalink to this definition">¶</a></dt>
-<dd><p>Are all the sides the same length?</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>is_equilateral</strong> : boolean</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.entity.GeometryEntity.is_similar" title="sympy.geometry.entity.GeometryEntity.is_similar"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.entity.GeometryEntity.is_similar</span></tt></a>, <a class="reference internal" href="#sympy.geometry.polygon.RegularPolygon" title="sympy.geometry.polygon.RegularPolygon"><tt class="xref py py-obj docutils literal"><span class="pre">RegularPolygon</span></tt></a>, <a class="reference internal" href="#sympy.geometry.polygon.Triangle.is_isosceles" title="sympy.geometry.polygon.Triangle.is_isosceles"><tt class="xref py py-obj docutils literal"><span class="pre">is_isosceles</span></tt></a>, <a class="reference internal" href="#sympy.geometry.polygon.Triangle.is_right" title="sympy.geometry.polygon.Triangle.is_right"><tt class="xref py py-obj docutils literal"><span class="pre">is_right</span></tt></a>, <a class="reference internal" href="#sympy.geometry.polygon.Triangle.is_scalene" title="sympy.geometry.polygon.Triangle.is_scalene"><tt class="xref py py-obj docutils literal"><span class="pre">is_scalene</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Triangle</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t1</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t1</span><span class="o">.</span><span class="n">is_equilateral</span><span class="p">()</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t2</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t2</span><span class="o">.</span><span class="n">is_equilateral</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.polygon.Triangle.is_isosceles">
-<tt class="descname">is_isosceles</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Triangle.is_isosceles"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Triangle.is_isosceles" title="Permalink to this definition">¶</a></dt>
-<dd><p>Are two or more of the sides the same length?</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>is_isosceles</strong> : boolean</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.polygon.Triangle.is_equilateral" title="sympy.geometry.polygon.Triangle.is_equilateral"><tt class="xref py py-obj docutils literal"><span class="pre">is_equilateral</span></tt></a>, <a class="reference internal" href="#sympy.geometry.polygon.Triangle.is_right" title="sympy.geometry.polygon.Triangle.is_right"><tt class="xref py py-obj docutils literal"><span class="pre">is_right</span></tt></a>, <a class="reference internal" href="#sympy.geometry.polygon.Triangle.is_scalene" title="sympy.geometry.polygon.Triangle.is_scalene"><tt class="xref py py-obj docutils literal"><span class="pre">is_scalene</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Triangle</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t1</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t1</span><span class="o">.</span><span class="n">is_isosceles</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.polygon.Triangle.is_right">
-<tt class="descname">is_right</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Triangle.is_right"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Triangle.is_right" title="Permalink to this definition">¶</a></dt>
-<dd><p>Is the triangle right-angled.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>is_right</strong> : boolean</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.line.LinearEntity.is_perpendicular" title="sympy.geometry.line.LinearEntity.is_perpendicular"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.line.LinearEntity.is_perpendicular</span></tt></a>, <a class="reference internal" href="#sympy.geometry.polygon.Triangle.is_equilateral" title="sympy.geometry.polygon.Triangle.is_equilateral"><tt class="xref py py-obj docutils literal"><span class="pre">is_equilateral</span></tt></a>, <a class="reference internal" href="#sympy.geometry.polygon.Triangle.is_isosceles" title="sympy.geometry.polygon.Triangle.is_isosceles"><tt class="xref py py-obj docutils literal"><span class="pre">is_isosceles</span></tt></a>, <a class="reference internal" href="#sympy.geometry.polygon.Triangle.is_scalene" title="sympy.geometry.polygon.Triangle.is_scalene"><tt class="xref py py-obj docutils literal"><span class="pre">is_scalene</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Triangle</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t1</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t1</span><span class="o">.</span><span class="n">is_right</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.polygon.Triangle.is_scalene">
-<tt class="descname">is_scalene</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Triangle.is_scalene"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Triangle.is_scalene" title="Permalink to this definition">¶</a></dt>
-<dd><p>Are all the sides of the triangle of different lengths?</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>is_scalene</strong> : boolean</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.polygon.Triangle.is_equilateral" title="sympy.geometry.polygon.Triangle.is_equilateral"><tt class="xref py py-obj docutils literal"><span class="pre">is_equilateral</span></tt></a>, <a class="reference internal" href="#sympy.geometry.polygon.Triangle.is_isosceles" title="sympy.geometry.polygon.Triangle.is_isosceles"><tt class="xref py py-obj docutils literal"><span class="pre">is_isosceles</span></tt></a>, <a class="reference internal" href="#sympy.geometry.polygon.Triangle.is_right" title="sympy.geometry.polygon.Triangle.is_right"><tt class="xref py py-obj docutils literal"><span class="pre">is_right</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Triangle</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t1</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t1</span><span class="o">.</span><span class="n">is_scalene</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.geometry.polygon.Triangle.is_similar">
-<tt class="descname">is_similar</tt><big>(</big><em>t1</em>, <em>t2</em><big>)</big><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Triangle.is_similar"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Triangle.is_similar" title="Permalink to this definition">¶</a></dt>
-<dd><p>Is another triangle similar to this one.</p>
-<p>Two triangles are similar if one can be uniformly scaled to the other.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>other: Triangle</strong> :</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>is_similar</strong> : boolean</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.entity.GeometryEntity.is_similar" title="sympy.geometry.entity.GeometryEntity.is_similar"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.entity.GeometryEntity.is_similar</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Triangle</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t1</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t2</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t1</span><span class="o">.</span><span class="n">is_similar</span><span class="p">(</span><span class="n">t2</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">t2</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t1</span><span class="o">.</span><span class="n">is_similar</span><span class="p">(</span><span class="n">t2</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.Triangle.medial">
-<tt class="descname">medial</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Triangle.medial"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Triangle.medial" title="Permalink to this definition">¶</a></dt>
-<dd><p>The medial triangle of the triangle.</p>
-<p>The triangle which is formed from the midpoints of the three sides.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>medial</strong> : Triangle</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.line.Segment.midpoint" title="sympy.geometry.line.Segment.midpoint"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.line.Segment.midpoint</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Triangle</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">medial</span>
-<span class="go">Triangle(Point(1/2, 0), Point(1/2, 1/2), Point(0, 1/2))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.Triangle.medians">
-<tt class="descname">medians</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Triangle.medians"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Triangle.medians" title="Permalink to this definition">¶</a></dt>
-<dd><p>The medians of the triangle.</p>
-<p>A median of a triangle is a straight line through a vertex and the
-midpoint of the opposite side, and divides the triangle into two
-equal areas.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>medians</strong> : dict</p>
-<blockquote class="last">
-<div><p>Each key is a vertex (Point) and each value is the median (Segment)
-at that point.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point.midpoint" title="sympy.geometry.point.Point.midpoint"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point.midpoint</span></tt></a>, <a class="reference internal" href="#sympy.geometry.line.Segment.midpoint" title="sympy.geometry.line.Segment.midpoint"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.line.Segment.midpoint</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Triangle</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">medians</span><span class="p">[</span><span class="n">p1</span><span class="p">]</span>
-<span class="go">Segment(Point(0, 0), Point(1/2, 1/2))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.Triangle.orthocenter">
-<tt class="descname">orthocenter</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Triangle.orthocenter"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Triangle.orthocenter" title="Permalink to this definition">¶</a></dt>
-<dd><p>The orthocenter of the triangle.</p>
-<p>The orthocenter is the intersection of the altitudes of a triangle.
-It may lie inside, outside or on the triangle.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><strong>orthocenter</strong> : Point</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Triangle</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">p3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">orthocenter</span>
-<span class="go">Point(0, 0)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.geometry.polygon.Triangle.vertices">
-<tt class="descname">vertices</tt><a class="reference internal" href="../_modules/sympy/geometry/polygon.html#Triangle.vertices"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.geometry.polygon.Triangle.vertices" title="Permalink to this definition">¶</a></dt>
-<dd><p>The triangle&#8217;s vertices</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>vertices</strong> : tuple</p>
-<blockquote class="last">
-<div><p>Each element in the tuple is a Point</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.geometry.point.Point" title="sympy.geometry.point.Point"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.geometry.point.Point</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.geometry</span> <span class="kn">import</span> <span class="n">Triangle</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Triangle</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">Point</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span><span class="o">.</span><span class="n">vertices</span>
-<span class="go">(Point(0, 0), Point(4, 0), Point(4, 3))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../index.html">
-              <img class="logo" src="../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Geometry Module</a><ul>
-<li><a class="reference internal" href="#introduction">Introduction</a></li>
-<li><a class="reference internal" href="#available-entities">Available Entities</a></li>
-<li><a class="reference internal" href="#example-usage">Example Usage</a></li>
-<li><a class="reference internal" href="#intersection-of-medians">Intersection of medians</a></li>
-<li><a class="reference internal" href="#an-in-depth-example-pappus-theorem">An in-depth example: Pappus&#8217; Theorem</a></li>
-<li><a class="reference internal" href="#miscellaneous-notes">Miscellaneous Notes</a></li>
-<li><a class="reference internal" href="#future-work">Future Work</a><ul>
-<li><a class="reference internal" href="#truth-setting-expressions">Truth Setting Expressions</a></li>
-<li><a class="reference internal" href="#three-dimensions-and-beyond">Three Dimensions and Beyond</a></li>
-<li><a class="reference internal" href="#geometry-visualization">Geometry Visualization</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#api-reference">API Reference</a><ul>
-<li><a class="reference internal" href="#module-sympy.geometry.entity">Entities</a></li>
-<li><a class="reference internal" href="#module-sympy.geometry.util">Utils</a></li>
-<li><a class="reference internal" href="#module-sympy.geometry.point">Points</a></li>
-<li><a class="reference internal" href="#module-sympy.geometry.line">Lines</a></li>
-<li><a class="reference internal" href="#module-sympy.geometry.curve">Curves</a></li>
-<li><a class="reference internal" href="#module-sympy.geometry.ellipse">Ellipses</a></li>
-<li><a class="reference internal" href="#module-sympy.geometry.polygon">Polygons</a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
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-                        title="previous chapter">Special</a></p>
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-                        title="next chapter">Geometric Algebra Module Docstring</a></p>
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-  <div class="section" id="sympy-modules-reference">
-<span id="module-docs"></span><h1>SymPy Modules Reference<a class="headerlink" href="#sympy-modules-reference" title="Permalink to this headline">¶</a></h1>
-<p>Because every feature of SymPy must have a test case, when you are not sure how
-to use something, just look into the <tt class="docutils literal"><span class="pre">tests/</span></tt> directories, find that feature
-and read the tests for it, that will tell you everything you need to know.</p>
-<p>Most of the things are already documented though in this document, that is
-automatically generated using SymPy&#8217;s docstrings.</p>
-<p>Click the  &#8220;modules&#8221; (<a class="reference internal" href="../py-modindex.html"><em>Module Index</em></a>) link in the top right corner to easily
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-</ul>
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-<li class="toctree-l1"><a class="reference internal" href="geometry.html">Geometry Module</a><ul>
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-<li class="toctree-l2"><a class="reference internal" href="geometry.html#an-in-depth-example-pappus-theorem">An in-depth example: Pappus&#8217; Theorem</a></li>
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-<li class="toctree-l1"><a class="reference internal" href="galgebra/index.html">Geometric Algebra Module Docstring</a><ul>
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-<li class="toctree-l2"><a class="reference internal" href="galgebra/GA/GAsympy.html#what-is-geometric-algebra">What is Geometric Algebra?</a></li>
-<li class="toctree-l2"><a class="reference internal" href="galgebra/GA/GAsympy.html#vector-basis-and-metric">Vector Basis and Metric</a></li>
-<li class="toctree-l2"><a class="reference internal" href="galgebra/GA/GAsympy.html#representation-and-reduction-of-multivector-bases">Representation and Reduction of Multivector Bases</a></li>
-<li class="toctree-l2"><a class="reference internal" href="galgebra/GA/GAsympy.html#base-representation-of-multivectors">Base Representation of Multivectors</a></li>
-<li class="toctree-l2"><a class="reference internal" href="galgebra/GA/GAsympy.html#blade-representation-of-multivectors">Blade Representation of Multivectors</a></li>
-<li class="toctree-l2"><a class="reference internal" href="galgebra/GA/GAsympy.html#outer-and-inner-products-left-and-right-contractions">Outer and Inner Products, Left and Right Contractions</a></li>
-<li class="toctree-l2"><a class="reference internal" href="galgebra/GA/GAsympy.html#reverse-of-multivector">Reverse of Multivector</a></li>
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-<li class="toctree-l2"><a class="reference internal" href="galgebra/GA/GAsympy.html#geometric-derivative">Geometric Derivative</a></li>
-<li class="toctree-l2"><a class="reference internal" href="galgebra/GA/GAsympy.html#module-components">Module Components</a></li>
-<li class="toctree-l2"><a class="reference internal" href="galgebra/GA/GAsympy.html#examples">Examples</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="galgebra/latex_ex/latex_ex.html">Extended LaTeXModule for SymPy</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="galgebra/latex_ex/latex_ex.html#extended-symbol-coding">Extended Symbol Coding</a></li>
-<li class="toctree-l2"><a class="reference internal" href="galgebra/latex_ex/latex_ex.html#how-latexprinter-works">How LatexPrinter Works</a></li>
-<li class="toctree-l2"><a class="reference internal" href="galgebra/latex_ex/latex_ex.html#latexprinter-functions">LatexPrinter Functions</a></li>
-<li class="toctree-l2"><a class="reference internal" href="galgebra/latex_ex/latex_ex.html#examples">Examples</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="integrals/integrals.html">Integrals</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="integrals/integrals.html#examples">Examples</a></li>
-<li class="toctree-l2"><a class="reference internal" href="integrals/integrals.html#module-sympy.integrals.transforms">Integral Transforms</a></li>
-<li class="toctree-l2"><a class="reference internal" href="integrals/integrals.html#internals">Internals</a></li>
-<li class="toctree-l2"><a class="reference internal" href="integrals/integrals.html#api-reference">API reference</a></li>
-<li class="toctree-l2"><a class="reference internal" href="integrals/integrals.html#todo-and-bugs">TODO and Bugs</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="logic.html">Logic Module</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="logic.html#introduction">Introduction</a></li>
-<li class="toctree-l2"><a class="reference internal" href="logic.html#forming-logical-expressions">Forming logical expressions</a></li>
-<li class="toctree-l2"><a class="reference internal" href="logic.html#boolean-functions">Boolean functions</a></li>
-<li class="toctree-l2"><a class="reference internal" href="logic.html#inference">Inference</a></li>
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-<li class="toctree-l1"><a class="reference internal" href="matrices/index.html">Matrices</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="matrices/matrices.html">Matrices (linear algebra)</a></li>
-<li class="toctree-l2"><a class="reference internal" href="matrices/dense.html">Dense Matrices</a></li>
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-<li class="toctree-l2"><a class="reference internal" href="matrices/expressions.html">Matrix Expressions</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="mpmath/index.html">Welcome to mpmath&#8217;s documentation!</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="mpmath/index.html#introduction">Introduction</a></li>
-<li class="toctree-l2"><a class="reference internal" href="mpmath/index.html#basic-features">Basic features</a></li>
-<li class="toctree-l2"><a class="reference internal" href="mpmath/index.html#advanced-mathematics">Advanced mathematics</a></li>
-<li class="toctree-l2"><a class="reference internal" href="mpmath/index.html#end-matter">End matter</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="polys/index.html">Polynomials Manipulation Module</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="polys/index.html#contents">Contents</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="printing.html">Printing System</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="printing.html#module-sympy.printing.printer">Printer Class</a></li>
-<li class="toctree-l2"><a class="reference internal" href="printing.html#prettyprinter-class">PrettyPrinter Class</a></li>
-<li class="toctree-l2"><a class="reference internal" href="printing.html#module-sympy.printing.ccode">CCodePrinter</a></li>
-<li class="toctree-l2"><a class="reference internal" href="printing.html#fortran-printing">Fortran Printing</a></li>
-<li class="toctree-l2"><a class="reference internal" href="printing.html#module-sympy.printing.gtk">Gtk</a></li>
-<li class="toctree-l2"><a class="reference internal" href="printing.html#module-sympy.printing.lambdarepr">LambdaPrinter</a></li>
-<li class="toctree-l2"><a class="reference internal" href="printing.html#module-sympy.printing.latex">LatexPrinter</a></li>
-<li class="toctree-l2"><a class="reference internal" href="printing.html#module-sympy.printing.mathml">MathMLPrinter</a></li>
-<li class="toctree-l2"><a class="reference internal" href="printing.html#module-sympy.printing.python">PythonPrinter</a></li>
-<li class="toctree-l2"><a class="reference internal" href="printing.html#module-sympy.printing.repr">ReprPrinter</a></li>
-<li class="toctree-l2"><a class="reference internal" href="printing.html#module-sympy.printing.str">StrPrinter</a></li>
-<li class="toctree-l2"><a class="reference internal" href="printing.html#module-sympy.printing.tree">Tree Printing</a></li>
-<li class="toctree-l2"><a class="reference internal" href="printing.html#preview">Preview</a></li>
-<li class="toctree-l2"><a class="reference internal" href="printing.html#module-sympy.printing.conventions">Implementation - Helper Classes/Functions</a></li>
-<li class="toctree-l2"><a class="reference internal" href="printing.html#module-sympy.printing.pretty.pretty_symbology">Pretty-Printing Implementation Helpers</a></li>
-</ul>
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-<li class="toctree-l1"><a class="reference internal" href="plotting.html">Plotting Module</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="plotting.html#introduction">Introduction</a></li>
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-<li class="toctree-l2"><a class="reference internal" href="plotting.html#plotting-function-reference">Plotting Function Reference</a></li>
-<li class="toctree-l2"><a class="reference internal" href="plotting.html#series-classes">Series Classes</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="plotting.html#module-sympy.plotting.pygletplot">Pyglet Plotting Module</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="plotting.html#plot-window-controls">Plot Window Controls</a></li>
-<li class="toctree-l2"><a class="reference internal" href="plotting.html#coordinate-modes">Coordinate Modes</a></li>
-<li class="toctree-l2"><a class="reference internal" href="plotting.html#specifying-intervals-for-variables">Specifying Intervals for Variables</a></li>
-<li class="toctree-l2"><a class="reference internal" href="plotting.html#using-the-interactive-interface">Using the Interactive Interface</a></li>
-<li class="toctree-l2"><a class="reference internal" href="plotting.html#using-custom-color-functions">Using Custom Color Functions</a></li>
-<li class="toctree-l2"><a class="reference internal" href="plotting.html#plotting-geometric-entities">Plotting Geometric Entities</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="assumptions/index.html">Assumptions module</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="assumptions/index.html#contents">Contents</a></li>
-<li class="toctree-l2"><a class="reference internal" href="assumptions/index.html#querying">Querying</a></li>
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-  <div class="section" id="computing-integrals-using-meijer-g-functions">
-<h1>Computing Integrals using Meijer G-Functions<a class="headerlink" href="#computing-integrals-using-meijer-g-functions" title="Permalink to this headline">¶</a></h1>
-<p>This text aims do describe in some detail the steps (and subtleties) involved
-in using Meijer G-functions for computing definite and indefinite integrals.
-We shall ignore proofs completely.</p>
-<div class="section" id="overview">
-<h2>Overview<a class="headerlink" href="#overview" title="Permalink to this headline">¶</a></h2>
-<p>The algorithm to compute <span class="math">\(\int f(x) \mathrm{d}x\)</span> or
-<span class="math">\(\int_0^\infty f(x) \mathrm{d}x\)</span> generally consists of three steps:</p>
-<ol class="arabic simple">
-<li>Rewrite the integrand using Meijer G-functions (one or sometimes two).</li>
-<li>Apply an integration theorem, to get the answer (usually expressed as another
-G-function).</li>
-<li>Expand the result in named special functions.</li>
-</ol>
-<p>Step (3) is implemented in the function hyperexpand (q.v.). Steps (1) and (2)
-are described below. Morever, G-functions are usually branched. Thus our treatment
-of branched functions is described first.</p>
-<p>Some other integrals (e.g. <span class="math">\(\int_{-\infty}^\infty\)</span>) can also be computed by first
-recasting them into one of the above forms. There is a lot of choice involved
-here, and the algorithm is heuristic at best.</p>
-</div>
-<div class="section" id="polar-numbers-and-branched-functions">
-<h2>Polar Numbers and Branched Functions<a class="headerlink" href="#polar-numbers-and-branched-functions" title="Permalink to this headline">¶</a></h2>
-<p>Both Meijer G-Functions and Hypergeometric functions are typically branched
-(possible branchpoints being <span class="math">\(0\)</span>, <span class="math">\(\pm 1\)</span>, <span class="math">\(\infty\)</span>). This is not very important
-when e.g. expanding a single hypergeometric function into named special functions,
-since sorting out the branches can be left to the human user. However this
-algorithm manipulates and transforms G-functions, and to do this correctly it needs
-at least some crude understanding of the branchings involved.</p>
-<p>To begin, we consider the set
-<span class="math">\(\mathcal{S} = \{(r, \theta) : r &gt; 0, \theta \in \mathbb{R}\}\)</span>. We have a map
-<span class="math">\(p: \mathcal{S}: \rightarrow \mathbb{C}-\{0\}, (r, \theta) \mapsto r e^{i \theta}\)</span>.
-Decreeing this to be a local biholomorphism gives <span class="math">\(\mathcal{S}\)</span> both a topology
-and a complex structure. This Riemann Surface is usually referred to as the
-Riemann Surface of the logarithm, for the following reason:
-We can define maps
-<span class="math">\(Exp: \mathbb{C} \rightarrow \mathcal{S}, Exp(x + i y) = (exp(x), y)\)</span> and
-<span class="math">\(Log: \mathcal{S} \rightarrow \mathbb{C}, (e^x, y) \mapsto x + iy\)</span>.
-These can be shown to be both holomorphic, and are indeed mutual inverses.</p>
-<p>We also sometimes formally attach a point &#8220;zero&#8221; to <span class="math">\(\mathcal{S}\)</span> and denote the
-resulting object <span class="math">\(\mathcal{S}_0\)</span>. Notably there is no complex structure
-defined near <span class="math">\(0\)</span>. A fundamental system of neighbourhoods is given by
-<span class="math">\(\{Exp(z) : Re(z) &lt; k\}\)</span>, this at least defines a topology. Elements of
-<span class="math">\(\mathcal{S}_0\)</span> shall be called polar numbers.
-We further define functions
-<span class="math">\(Arg: \mathcal{S} \rightarrow \mathbb{R}, (r, \theta) \mapsto \theta\)</span> and
-<span class="math">\(|.|: \mathcal{S}_0 \rightarrow \mathbb{R}_{&gt;0}, (r, \theta) \mapsto r\)</span>.
-These have evident meaning and are both continuous everywhere.</p>
-<p>Using these maps many operations can be extended from <span class="math">\(\mathbb{C}\)</span> to
-<span class="math">\(\mathcal{S}\)</span>. We define <span class="math">\(Exp(a) Exp(b) = Exp(a + b)\)</span> for <span class="math">\(a, b \in \mathbb{C}\)</span>,
-also for <span class="math">\(a \in \mathcal{S}\)</span> and <span class="math">\(b \in \mathbb{C}\)</span> we define
-<span class="math">\(a^b = Exp(b Log(a))\)</span>.
-It can be checked easily that using these definitions, many algebraic properties
-holding for positive reals (e.g. <span class="math">\((ab)^c = a^c b^c\)</span>) which hold in <span class="math">\(\mathbb{C}\)</span>
-only for some numbers (because of branch cuts) hold indeed for all polar numbers.</p>
-<p>As one peculiarity it should be mentioned that addition of polar numbers is not
-usually defined. However, formal sums of polar numbers can be used to express
-branching behaviour. For example, consider the functions <span class="math">\(F(z) = \sqrt{1 + z}\)</span>
-and <span class="math">\(G(a, b) = \sqrt{a + b}\)</span>, where <span class="math">\(a, b, z\)</span> are polar numbers.
-The general rule is that functions of a single polar variable are defined in
-such a way that they are continuous on circles, and agree with the usual
-definition for positive reals. Thus if <span class="math">\(S(z)\)</span> denotes the standard branch of
-the square root function on <span class="math">\(\mathbb{C}\)</span>, we are forced to define</p>
-<div class="math">
-\[\begin{split}F(z) = \begin{cases}
- S(p(z)) &amp;: |z| &lt; 1 \\
- S(p(z)) &amp;: -\pi &lt; Arg(z) + 4\pi n \le \pi \text{ for some } n \in \mathbb{Z} \\
- -S(p(z)) &amp;: \text{else}
-\end{cases}.\end{split}\]</div>
-<p>(We are omitting <span class="math">\(|z| = 1\)</span> here, this does not matter for integration.)
-Finally we define <span class="math">\(G(a, b) = \sqrt{a}F(b/a)\)</span>.</p>
-</div>
-<div class="section" id="representing-branched-functions-on-the-argand-plane">
-<h2>Representing Branched Functions on the Argand Plane<a class="headerlink" href="#representing-branched-functions-on-the-argand-plane" title="Permalink to this headline">¶</a></h2>
-<p>Suppose <span class="math">\(f: \mathcal{S} \to \mathbb{C}\)</span> is a holomorphic function. We wish to
-define a function <span class="math">\(F\)</span> on (part of) the complex numbers <span class="math">\(\mathbb{C}\)</span> that
-represents <span class="math">\(f\)</span> as closely as possible. This process is knows as &#8220;introducing
-branch cuts&#8221;. In our situation, there is actually a canonical way of doing this
-(which is adhered to in all of SymPy), as follows: Introduce the &#8220;cut complex
-plane&#8221;
-<span class="math">\(C = \mathbb{C} \setminus \mathbb{R}_{\le 0}\)</span>. Define a function
-<span class="math">\(l: C \to \mathcal{S}\)</span> via <span class="math">\(re^{i\theta} \mapsto r Exp(i\theta)\)</span>. Here <span class="math">\(r &gt; 0\)</span>
-and <span class="math">\(-\pi &lt; \theta \le \pi\)</span>. Then <span class="math">\(l\)</span> is holomorphic, and we define
-<span class="math">\(G = f \circ l\)</span>. This called &#8220;lifting to the principal branch&#8221; throughout the
-SymPy documentation.</p>
-</div>
-<div class="section" id="table-lookups-and-inverse-mellin-transforms">
-<h2>Table Lookups and Inverse Mellin Transforms<a class="headerlink" href="#table-lookups-and-inverse-mellin-transforms" title="Permalink to this headline">¶</a></h2>
-<p>Suppose we are given an integrand <span class="math">\(f(x)\)</span> and are trying to rewrite it as a
-single G-function. To do this, we first split <span class="math">\(f(x)\)</span> into the form <span class="math">\(x^s g(x)\)</span>
-(where <span class="math">\(g(x)\)</span> is supposed to be simpler than <span class="math">\(f(x)\)</span>). This is because multiplicative
-powers can be absorbed into the G-function later. This splitting is done by
-<tt class="docutils literal"><span class="pre">_split_mul(f,</span> <span class="pre">x)</span></tt>. Then we assemble a tuple of functions that occur in
-<span class="math">\(f\)</span> (e.g. if <span class="math">\(f(x) = e^x \cos{x}\)</span>, we would assemble the tuple <span class="math">\((\cos, \exp)\)</span>).
-This is done by the function <tt class="docutils literal"><span class="pre">_mytype(f,</span> <span class="pre">x)</span></tt>. Next we index a lookup table
-(created using <tt class="docutils literal"><span class="pre">_create_lookup_table()</span></tt>) with this tuple. This (hopefully)
-yields a list of Meijer G-function formulae involving these functions, we then
-pattern-match all of them. If one fits, we were successful, otherwise not and we
-have to try something else.</p>
-<p>Suppose now we want to rewrite as a product of two G-functions. To do this,
-we (try to) find all inequivalent ways of splitting <span class="math">\(f(x)\)</span> into a product
-<span class="math">\(f_1(x) f_2(x)\)</span>.
-We could try these splittings in any order, but it is often a good idea to
-minimise (a) the number of powers occuring in <span class="math">\(f_i(x)\)</span> and (b) the number of
-different functions occuring in <span class="math">\(f_i(x)\)</span>. Thus given e.g.
-<span class="math">\(f(x) = \sin{x}\, e^{x} \sin{2x}\)</span> we should try <span class="math">\(f_1(x) = \sin{x}\, \sin{2x}\)</span>,
-<span class="math">\(f_2(x) = e^{x}\)</span> first.
-All of this is done by the function <tt class="docutils literal"><span class="pre">_mul_as_two_parts(f)</span></tt>.</p>
-<p>Finally, we can try a recursive Mellin transform technique. Since the Meijer
-G-function is defined essentially as a certain inverse mellin transform,
-if we want to write a function <span class="math">\(f(x)\)</span> as a G-function, we can compute its mellin
-transform <span class="math">\(F(s)\)</span>. If <span class="math">\(F(s)\)</span> is in the right form, the G-function expression
-can be read off. This technique generalises many standard rewritings, e.g.
-<span class="math">\(e^{ax} e^{bx} = e^{(a + b) x}\)</span>.</p>
-<p>One twist is that some functions don&#8217;t have mellin transforms, even though they
-can be written as G-functions. This is true for example for <span class="math">\(f(x) = e^x \sin{x}\)</span>
-(the function grows too rapidly to have a mellin transform). However if the function
-is recognised to be analytic, then we can try to compute the mellin-transform of
-<span class="math">\(f(ax)\)</span> for a parameter <span class="math">\(a\)</span>, and deduce the G-function expression by analytic
-continuation. (Checking for analyticity is easy. Since we can only deal with a
-certain subset of functions anyway, we only have to filter out those which are
-not analyitc.)</p>
-<p>The function <tt class="docutils literal"><span class="pre">_rewrite_single</span></tt> does the table lookup and recursive mellin
-transform. The functions <tt class="docutils literal"><span class="pre">_rewrite1</span></tt> and <tt class="docutils literal"><span class="pre">_rewrite2</span></tt> respectively use
-above-mentioned helpers and <tt class="docutils literal"><span class="pre">_rewrite_single</span></tt> to rewrite their argument as
-respectively one or two G-functions.</p>
-</div>
-<div class="section" id="applying-the-integral-theorems">
-<h2>Applying the Integral Theorems<a class="headerlink" href="#applying-the-integral-theorems" title="Permalink to this headline">¶</a></h2>
-<p>If the integrand has been recast into G-functions, evaluating the integral is
-relatively easy. We first do some substitutions to reduce e.g. the exponent
-of the argument of the G-function to unity (see <tt class="docutils literal"><span class="pre">_rewrite_saxena_1</span></tt> and
-<tt class="docutils literal"><span class="pre">_rewrite_saxena</span></tt>, respectively, for one or two G-functions). Next we go through
-a list of conditions under which the integral theorem applies. It can fail for
-basically two reasons: either the integral does not exist, or the manipulations
-in deriving the theorem may not be allowed (for more details, see this <a class="reference internal" href="#blogpost">[BlogPost]</a>).</p>
-<p>Sometimes this can be remedied by reducing the argument of the G-functions
-involved. For example it is clear that the G-function representing <span class="math">\(e^z\)</span>
-is satisfies <span class="math">\(G(Exp(2 \pi i)z) = G(z)\)</span> for all <span class="math">\(z \in \mathcal{S}\)</span>. The function
-<tt class="docutils literal"><span class="pre">meijerg.get_period()</span></tt> can be used to discover this, and the function
-<tt class="docutils literal"><span class="pre">principal_branch(z,</span> <span class="pre">period)</span></tt> in <tt class="docutils literal"><span class="pre">functions/elementary/complexes.py</span></tt> can
-be used to exploit the information. This is done transparently by the
-integration code.</p>
-<table class="docutils citation" frame="void" id="blogpost" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id1">[BlogPost]</a></td><td><a class="reference external" href="http://nessgrh.wordpress.com/2011/07/07/tricky-branch-cuts/">http://nessgrh.wordpress.com/2011/07/07/tricky-branch-cuts/</a></td></tr>
-</tbody>
-</table>
-</div>
-</div>
-<div class="section" id="the-g-function-integration-theorems">
-<h1>The G-Function Integration Theorems<a class="headerlink" href="#the-g-function-integration-theorems" title="Permalink to this headline">¶</a></h1>
-<p>This section intends to display in detail the definite integration theorems
-used in the code. The following two formulae go back to Meijer (In fact he
-proved more general formulae; indeed in the literature formulae are usually
-staded in more general form. However it is very easy to deduce the general
-formulae from the ones we give here. It seemed best to keep the theorems as
-simple as possible, since they are very complicated anyway.):</p>
-<ol class="arabic">
-<li><div class="first math">
-\[\begin{split}\int_0^\infty
-G_{p, q}^{m, n} \left.\left(\begin{matrix} a_1, \dots, a_p \\
-                                           b_1, \dots, b_q \end{matrix}
-        \right| \eta x \right) \mathrm{d}x =
- \frac{\prod_{j=1}^m \Gamma(b_j + 1) \prod_{j=1}^n \Gamma(-a_j)}{\eta
-       \prod_{j=m+1}^q \Gamma(-b_j) \prod_{j=n+1}^p \Gamma(a_j + 1)}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}\int_0^\infty
-G_{u, v}^{s, t} \left.\left(\begin{matrix} c_1, \dots, c_u \\
-                                           d_1, \dots, d_v \end{matrix}
-        \right| \sigma x \right)
-G_{p, q}^{m, n} \left.\left(\begin{matrix} a_1, \dots, a_p \\
-                                           b_1, \dots, b_q \end{matrix}
-        \right| \omega x \right)
-\mathrm{d}x =
-G_{v+p, u+q}^{m+t, n+s} \left.\left(
-      \begin{matrix} a_1, \dots, a_n, -d_1, \dots, -d_v, a_{n+1}, \dots, a_p \\
-                     b_1, \dots, b_m, -c_1, \dots, -c_u, b_{m+1}, \dots, b_q
-      \end{matrix}
-        \right| \frac{\omega}{\sigma} \right)\end{split}\]</div>
-</li>
-</ol>
-<p>The more interesting question is under what conditions these formulae are
-valid. Below we detail the conditions implemented in SymPy. They are an
-amalgamation of conditions found in <a class="reference internal" href="../simplify/hyperexpand.html#prudnikov1990">[Prudnikov1990]</a> and <a class="reference internal" href="../simplify/hyperexpand.html#luke1969">[Luke1969]</a>; please
-let us know if you find any errors.</p>
-<div class="section" id="conditions-of-convergence-for-integral-1">
-<h2>Conditions of Convergence for Integral (1)<a class="headerlink" href="#conditions-of-convergence-for-integral-1" title="Permalink to this headline">¶</a></h2>
-<p>We can without loss of generality assume <span class="math">\(p \le q\)</span>, since the G-functions
-of indices <span class="math">\(m, n, p, q\)</span> and of indices <span class="math">\(n, m, q, p\)</span> can be related easily
-(see e.g. <a class="reference internal" href="../simplify/hyperexpand.html#luke1969">[Luke1969]</a>, section 5.3). We introduce the following notation:</p>
-<div class="math">
-\[\begin{split}\xi = m + n - p \\
-\delta = m + n - \frac{p + q}{2}\end{split}\]</div>
-<div class="math">
-\[\begin{split}C_3: -\Re(b_j) &lt; 1 \text{ for } j=1, \dots, m \\
-0 &lt; -\Re(a_j) \text{ for } j=1, \dots, n\end{split}\]</div>
-<div class="math">
-\[\begin{split}C_3^*: -\Re(b_j) &lt; 1 \text{ for } j=1, \dots, q \\
-0 &lt; -\Re(a_j) \text{ for } j=1, \dots, p\end{split}\]</div>
-<div class="math">
-\[\begin{split}C_4: -\Re(\delta) + \frac{q + 1 - p}{2} &gt; q - p\end{split}\]</div>
-<p>The convergence conditions will be detailed in several &#8220;cases&#8221;, numbered one
-to five. For later use it will be helpful to separate conditions &#8220;at infinity&#8221;
-from conditions &#8220;at zero&#8221;. By conditions &#8220;at infinity&#8221; we mean conditions that
-only depend on the behaviour of the integrand for large, positive values
-of <span class="math">\(x\)</span>, whereas by conditions &#8220;at zero&#8221; we mean conditions that only depend on
-the behaviour of the integrand on <span class="math">\((0, \epsilon)\)</span> for any <span class="math">\(\epsilon &gt; 0\)</span>.
-Since all our conditions are specified in terms of parameters of the
-G-functions, this distinction is not immediately visible. They are, however, of
-very distinct character mathematically; the conditions at infinity being in
-particular much harder to control.</p>
-<p>In order for the integral theorem to be valid, conditions
-<span class="math">\(n\)</span> &#8220;at zero&#8221; and &#8220;at infinity&#8221; both have to be fulfilled, for some <span class="math">\(n\)</span>.</p>
-<p>These are the conditions &#8220;at infinity&#8221;:</p>
-<ol class="arabic">
-<li><div class="first math">
-\[\begin{split}\delta &gt; 0 \wedge |\arg(\eta)| &lt; \delta \pi \wedge (A \vee B \vee C),\end{split}\]</div>
-<p>where</p>
-<div class="math">
-\[\begin{split}A = 1 \le n \wedge p &lt; q \wedge 1 \le m\end{split}\]</div>
-<div class="math">
-\[B = 1 \le p \wedge 1 \le m \wedge q = p+1 \wedge
-            \neg (n = 0 \wedge m = p + 1 )\]</div>
-<div class="math">
-\[C = 1 \le n \wedge q = p \wedge |\arg(\eta)| \ne (\delta - 2k)\pi
-       \text{ for } k = 0, 1, \dots
-         \left\lceil \frac{\delta}{2} \right\rceil.\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}n = 0 \wedge p + 1 \le m \wedge |\arg(\eta)| &lt; \delta \pi\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}(p &lt; q \wedge 1 \le m \wedge \delta &gt; 0 \wedge |\arg(\eta)| = \delta \pi)
-\vee (p \le q - 2 \wedge \delta = 0 \wedge \arg(\eta) = 0)\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}p = q \wedge \delta = 0 \wedge \arg(\eta) = 0 \wedge \eta \ne 0
-\wedge \Re\left(\sum_{j=1}^p b_j - a_j \right) &lt; 0\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}\delta &gt; 0 \wedge |\arg(\eta)| &lt; \delta \pi\end{split}\]</div>
-</li>
-</ol>
-<p>And these are the conditions &#8220;at zero&#8221;:</p>
-<ol class="arabic">
-<li><div class="first math">
-\[\eta \ne 0 \wedge C_3\]</div>
-</li>
-<li><div class="first math">
-\[C_3\]</div>
-</li>
-<li><div class="first math">
-\[C_3 \wedge C_4\]</div>
-</li>
-<li><div class="first math">
-\[C_3\]</div>
-</li>
-<li><div class="first math">
-\[C_3\]</div>
-</li>
-</ol>
-</div>
-<div class="section" id="conditions-of-convergence-for-integral-2">
-<h2>Conditions of Convergence for Integral (2)<a class="headerlink" href="#conditions-of-convergence-for-integral-2" title="Permalink to this headline">¶</a></h2>
-<p>We introduce the following notation:</p>
-<div class="math">
-\[b^* = s + t - \frac{u + v}{2}\]</div>
-<div class="math">
-\[c^* = m + n - \frac{p + q}{2}\]</div>
-<div class="math">
-\[\rho = \sum_{j=1}^v d_j - \sum_{j=1}^u c_j + \frac{u - v}{2} + 1\]</div>
-<div class="math">
-\[\mu = \sum_{j=1}^q b_j - \sum_{j=1}^p a_j + \frac{p - q}{2} + 1\]</div>
-<div class="math">
-\[\phi = q - p - \frac{u - v}{2} + 1\]</div>
-<div class="math">
-\[\eta = 1 - (v - u) - \mu - \rho\]</div>
-<div class="math">
-\[\psi = \frac{\pi(q - m - n) + |\arg(\omega)|}{q - p}\]</div>
-<div class="math">
-\[\theta = \frac{\pi(v - s - t) + |\arg(\sigma)|)}{v - u}\]</div>
-<div class="math">
-\[\lambda_c = (q - p)|\omega|^{1/(q - p)} \cos{\psi}
-+ (v - u)|\sigma|^{1/(v - u)} \cos{\theta}\]</div>
-<div class="math">
-\[\lambda_{s0}(c_1, c_2) = c_1 (q - p)|\omega|^{1/(q - p)} \sin{\psi}
-+ c_2 (v - u)|\sigma|^{1/(v - u)} \sin{\theta}\]</div>
-<div class="math">
-\[\begin{split}\lambda_s =
-\begin{cases} \operatorname{\lambda_{s0}}\left(-1,-1\right) \operatorname{\lambda_{s0}}\left(1,1\right) &amp; \text{for}\: \arg(\omega) = 0 \wedge \arg(\sigma) = 0 \\\operatorname{\lambda_{s0}}\left(\operatorname{sign}\left(\operatorname{\arg}\left(\omega\right)\right),-1\right) \operatorname{\lambda_{s0}}\left(\operatorname{sign}\left(\operatorname{\arg}\left(\omega\right)\right),1\right) &amp; \text{for}\: \arg(\omega) \ne 0 \wedge \arg(\sigma) = 0 \\\operatorname{\lambda_{s0}}\left(-1,\operatorname{sign}\left(\operatorname{\arg}\left(\sigma\right)\right)\right) \operatorname{\lambda_{s0}}\left(1,\operatorname{sign}\left(\operatorname{\arg}\left(\sigma\right)\right)\right) &amp; \text{for}\: \arg(\omega) = 0 \wedge \arg(\sigma) \ne 0) \\\operatorname{\lambda_{s0}}\left(\operatorname{sign}\left(\operatorname{\arg}\left(\omega\right)\right),\operatorname{sign}\left(\operatorname{\arg}\left(\sigma\right)\right)\right) &amp; \text{otherwise} \end{cases}\end{split}\]</div>
-<div class="math">
-\[z_0 = \frac{\omega}{\sigma} e^{-i\pi (b^* + c^*)}\]</div>
-<div class="math">
-\[z_1 = \frac{\sigma}{\omega} e^{-i\pi (b^* + c^*)}\]</div>
-<p>The following conditions will be helpful:</p>
-<div class="math">
-\[\begin{split}C_1: (a_i - b_j \notin \mathbb{Z}_{&gt;0} \text{ for } i = 1, \dots, n, j = 1, \dots, m) \\
-\wedge
-(c_i - d_j \notin \mathbb{Z}_{&gt;0} \text{ for } i = 1, \dots, t, j = 1, \dots, s)\end{split}\]</div>
-<div class="math">
-\[\begin{split}C_2:
-\Re(1 + b_i + d_j) &gt; 0 \text{ for } i = 1, \dots, m, j = 1, \dots, s\end{split}\]</div>
-<div class="math">
-\[\begin{split}C_3:
-\Re(a_i + c_j) &lt; 1 \text{ for } i = 1, \dots, n, j = 1, \dots, t\end{split}\]</div>
-<div class="math">
-\[\begin{split}C_4:
-(p - q)\Re(c_i) - \Re(\mu) &gt; -\frac{3}{2} \text{ for } i=1, \dots, t\end{split}\]</div>
-<div class="math">
-\[\begin{split}C_5:
-(p - q)\Re(1 + d_i) - \Re(\mu) &gt; -\frac{3}{2} \text{ for } i=1, \dots, s\end{split}\]</div>
-<div class="math">
-\[\begin{split}C_6:
-(u - v)\Re(a_i) - \Re(\rho) &gt; -\frac{3}{2} \text{ for } i=1, \dots, n\end{split}\]</div>
-<div class="math">
-\[\begin{split}C_7:
-(u - v)\Re(1 + b_i) - \Re(\rho) &gt; -\frac{3}{2} \text{ for } i=1, \dots, m\end{split}\]</div>
-<div class="math">
-\[\begin{split}C_8:
-0 &lt; \lvert{\phi}\rvert + 2 \Re\left(\left(\mu -1\right) \left(- u + v\right) + \left(- p + q\right) \left(\rho -1\right) + \left(- p + q\right) \left(- u + v\right)\right)\end{split}\]</div>
-<div class="math">
-\[\begin{split}C_9:
-0 &lt; \lvert{\phi}\rvert - 2 \Re\left(\left(\mu -1\right) \left(- u + v\right) + \left(- p + q\right) \left(\rho -1\right) + \left(- p + q\right) \left(- u + v\right)\right)\end{split}\]</div>
-<div class="math">
-\[\begin{split}C_{10}:
-\lvert{\operatorname{arg}\left(\sigma\right)}\rvert &lt; \pi b^{*}\end{split}\]</div>
-<div class="math">
-\[C_{11}:
-\lvert{\operatorname{arg}\left(\sigma\right)}\rvert = \pi b^{*}\]</div>
-<div class="math">
-\[\begin{split}C_{12}:
-|\arg(\omega)| &lt; c^*\pi\end{split}\]</div>
-<div class="math">
-\[C_{13}:
-|\arg(\omega)| = c^*\pi\]</div>
-<div class="math">
-\[\begin{split}C_{14}^1:
-\left(z_0 \ne 1 \wedge |\arg(1 - z_0)| &lt; \pi \right) \vee
-\left(z_0 = 1 \wedge \Re(\mu + \rho - u + v) &lt; 1 \right)\end{split}\]</div>
-<div class="math">
-\[\begin{split}C_{14}^2:
-\left(z_1 \ne 1 \wedge |\arg(1 - z_1)| &lt; \pi \right) \vee
-\left(z_1 = 1 \wedge \Re(\mu + \rho - p + q) &lt; 1 \right)\end{split}\]</div>
-<div class="math">
-\[C_{14}:
-\phi = 0 \wedge b^* + c^* \le 1 \wedge (C_{14}^1 \vee C_{14}^2)\]</div>
-<div class="math">
-\[\begin{split}C_{15}:
-\lambda_c &gt; 0 \vee (\lambda_c = 0 \wedge \lambda_s \ne 0 \wedge \Re(\eta) &gt; -1)
-              \vee (\lambda_c = 0 \wedge \lambda_s = 0 \wedge \Re(\eta) &gt; 0)\end{split}\]</div>
-<div class="math">
-\[C_{16}: \int_0^\infty G_{u, v}^{s, t}(\sigma x) \mathrm{d} x
-\text{ converges at infinity }\]</div>
-<div class="math">
-\[C_{17}: \int_0^\infty G_{p, q}^{m, n}(\omega x) \mathrm{d} x
-\text{ converges at infinity }\]</div>
-<p>Note that <span class="math">\(C_{16}\)</span> and <span class="math">\(C_{17}\)</span> are the reason we split the convergence conditions for
-integral (1).</p>
-<p>With this notation established, the implemented convergence conditions can be enumerated
-as follows:</p>
-<ol class="arabic">
-<li><div class="first math">
-\[\begin{split}m n s t \neq 0 \wedge 0 &lt; b^{*} \wedge 0 &lt; c^{*} \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{10} \wedge C_{12}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}u = v \wedge b^{*} = 0 \wedge 0 &lt; c^{*} \wedge 0 &lt; \sigma \wedge \Re{\rho} &lt; 1 \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{12}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}p = q \wedge u = v \wedge b^{*} = 0 \wedge c^{*} = 0 \wedge 0 &lt; \sigma \wedge 0 &lt; \omega \wedge \Re{\mu} &lt; 1 \wedge \Re{\rho} &lt; 1 \wedge \sigma \neq \omega \wedge C_{1} \wedge C_{2} \wedge C_{3}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}p = q \wedge u = v \wedge b^{*} = 0 \wedge c^{*} = 0 \wedge 0 &lt; \sigma \wedge 0 &lt; \omega \wedge \Re\left(\mu + \rho\right) &lt; 1 \wedge \omega \neq \sigma \wedge C_{1} \wedge C_{2} \wedge C_{3}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}p = q \wedge u = v \wedge b^{*} = 0 \wedge c^{*} = 0 \wedge 0 &lt; \sigma \wedge 0 &lt; \omega \wedge \Re\left(\mu + \rho\right) &lt; 1 \wedge \omega \neq \sigma \wedge C_{1} \wedge C_{2} \wedge C_{3}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}q &lt; p \wedge 0 &lt; s \wedge 0 &lt; b^{*} \wedge 0 \leq c^{*} \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{5} \wedge C_{10} \wedge C_{13}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}p &lt; q \wedge 0 &lt; t \wedge 0 &lt; b^{*} \wedge 0 \leq c^{*} \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{4} \wedge C_{10} \wedge C_{13}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}v &lt; u \wedge 0 &lt; m \wedge 0 &lt; c^{*} \wedge 0 \leq b^{*} \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{7} \wedge C_{11} \wedge C_{12}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}u &lt; v \wedge 0 &lt; n \wedge 0 &lt; c^{*} \wedge 0 \leq b^{*} \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{6} \wedge C_{11} \wedge C_{12}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}q &lt; p \wedge u = v \wedge b^{*} = 0 \wedge 0 \leq c^{*} \wedge 0 &lt; \sigma \wedge \Re{\rho} &lt; 1 \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{5} \wedge C_{13}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}p &lt; q \wedge u = v \wedge b^{*} = 0 \wedge 0 \leq c^{*} \wedge 0 &lt; \sigma \wedge \Re{\rho} &lt; 1 \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{4} \wedge C_{13}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}p = q \wedge v &lt; u \wedge 0 \leq b^{*} \wedge c^{*} = 0 \wedge 0 &lt; \omega \wedge \Re{\mu} &lt; 1 \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{7} \wedge C_{11}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}p = q \wedge u &lt; v \wedge 0 \leq b^{*} \wedge c^{*} = 0 \wedge 0 &lt; \omega \wedge \Re{\mu} &lt; 1 \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{6} \wedge C_{11}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}p &lt; q \wedge v &lt; u \wedge 0 \leq b^{*} \wedge 0 \leq c^{*} \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{4} \wedge C_{7} \wedge C_{11} \wedge C_{13}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}q &lt; p \wedge u &lt; v \wedge 0 \leq b^{*} \wedge 0 \leq c^{*} \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{5} \wedge C_{6} \wedge C_{11} \wedge C_{13}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}q &lt; p \wedge v &lt; u \wedge 0 \leq b^{*} \wedge 0 \leq c^{*} \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{5} \wedge C_{7} \wedge C_{8} \wedge C_{11} \wedge C_{13} \wedge C_{14}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}p &lt; q \wedge u &lt; v \wedge 0 \leq b^{*} \wedge 0 \leq c^{*} \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{4} \wedge C_{6} \wedge C_{9} \wedge C_{11} \wedge C_{13} \wedge C_{14}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}t = 0 \wedge 0 &lt; s \wedge 0 &lt; b^{*} \wedge 0 &lt; \phi \wedge C_{1} \wedge C_{2} \wedge C_{10}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}s = 0 \wedge 0 &lt; t \wedge 0 &lt; b^{*} \wedge \phi &lt; 0 \wedge C_{1} \wedge C_{3} \wedge C_{10}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}n = 0 \wedge 0 &lt; m \wedge 0 &lt; c^{*} \wedge \phi &lt; 0 \wedge C_{1} \wedge C_{2} \wedge C_{12}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}m = 0 \wedge 0 &lt; n \wedge 0 &lt; c^{*} \wedge 0 &lt; \phi \wedge C_{1} \wedge C_{3} \wedge C_{12}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}s t = 0 \wedge 0 &lt; b^{*} \wedge 0 &lt; c^{*} \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{10} \wedge C_{12}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}m n = 0 \wedge 0 &lt; b^{*} \wedge 0 &lt; c^{*} \wedge C_{1} \wedge C_{2} \wedge C_{3} \wedge C_{10} \wedge C_{12}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}p &lt; m + n \wedge t = 0 \wedge \phi = 0 \wedge 0 &lt; s \wedge 0 &lt; b^{*} \wedge c^{*} &lt; 0 \wedge \lvert{\operatorname{arg}\left(\omega\right)}\rvert &lt; \pi \left(m + n - p + 1\right) \wedge C_{1} \wedge C_{2} \wedge C_{10} \wedge C_{14} \wedge C_{15}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}q &lt; m + n \wedge s = 0 \wedge \phi = 0 \wedge 0 &lt; t \wedge 0 &lt; b^{*} \wedge c^{*} &lt; 0 \wedge \lvert{\operatorname{arg}\left(\omega\right)}\rvert &lt; \pi \left(m + n - q + 1\right) \wedge C_{1} \wedge C_{3} \wedge C_{10} \wedge C_{14} \wedge C_{15}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}p = q -1 \wedge t = 0 \wedge \phi = 0 \wedge 0 &lt; s \wedge 0 &lt; b^{*} \wedge 0 \leq c^{*} \wedge \pi c^{*} &lt; \lvert{\operatorname{arg}\left(\omega\right)}\rvert \wedge C_{1} \wedge C_{2} \wedge C_{10} \wedge C_{14} \wedge C_{15}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}p = q + 1 \wedge s = 0 \wedge \phi = 0 \wedge 0 &lt; t \wedge 0 &lt; b^{*} \wedge 0 \leq c^{*} \wedge \pi c^{*} &lt; \lvert{\operatorname{arg}\left(\omega\right)}\rvert \wedge C_{1} \wedge C_{3} \wedge C_{10} \wedge C_{14} \wedge C_{15}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}p &lt; q -1 \wedge t = 0 \wedge \phi = 0 \wedge 0 &lt; s \wedge 0 &lt; b^{*} \wedge 0 \leq c^{*} \wedge \pi c^{*} &lt; \lvert{\operatorname{arg}\left(\omega\right)}\rvert \wedge \lvert{\operatorname{arg}\left(\omega\right)}\rvert &lt; \pi \left(m + n - p + 1\right) \wedge C_{1} \wedge C_{2} \wedge C_{10} \wedge C_{14} \wedge C_{15}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}q + 1 &lt; p \wedge s = 0 \wedge \phi = 0 \wedge 0 &lt; t \wedge 0 &lt; b^{*} \wedge 0 \leq c^{*} \wedge \pi c^{*} &lt; \lvert{\operatorname{arg}\left(\omega\right)}\rvert \wedge \lvert{\operatorname{arg}\left(\omega\right)}\rvert &lt; \pi \left(m + n - q + 1 \right) \wedge C_{1} \wedge C_{3} \wedge C_{10} \wedge C_{14} \wedge C_{15}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}n = 0 \wedge \phi = 0 \wedge 0 &lt; s + t \wedge 0 &lt; m \wedge 0 &lt; c^{*} \wedge b^{*} &lt; 0 \wedge \lvert{\operatorname{arg}\left(\sigma\right)}\rvert &lt; \pi \left(s + t - u + 1\right) \wedge C_{1} \wedge C_{2} \wedge C_{12} \wedge C_{14} \wedge C_{15}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}m = 0 \wedge \phi = 0 \wedge v &lt; s + t \wedge 0 &lt; n \wedge 0 &lt; c^{*} \wedge b^{*} &lt; 0 \wedge \lvert{\operatorname{arg}\left(\sigma\right)}\rvert &lt; \pi \left(s + t - v + 1\right) \wedge C_{1} \wedge C_{3} \wedge C_{12} \wedge C_{14} \wedge C_{15}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}n = 0 \wedge \phi = 0 \wedge u = v -1 \wedge 0 &lt; m \wedge 0 &lt; c^{*} \wedge 0 \leq b^{*} \wedge \pi b^{*} &lt; \lvert{\operatorname{arg}\left(\sigma\right)}\rvert \wedge \lvert{\operatorname{arg}\left(\sigma\right)}\rvert &lt; \pi \left(b^{*} + 1\right) \wedge C_{1} \wedge C_{2} \wedge C_{12} \wedge C_{14} \wedge C_{15}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}m = 0 \wedge \phi = 0 \wedge u = v + 1 \wedge 0 &lt; n \wedge 0 &lt; c^{*} \wedge 0 \leq b^{*} \wedge \pi b^{*} &lt; \lvert{\operatorname{arg}\left(\sigma\right)}\rvert \wedge \lvert{\operatorname{arg}\left(\sigma\right)}\rvert &lt; \pi \left(b^{*} + 1\right) \wedge C_{1} \wedge C_{3} \wedge C_{12} \wedge C_{14} \wedge C_{15}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}n = 0 \wedge \phi = 0 \wedge u &lt; v -1 \wedge 0 &lt; m \wedge 0 &lt; c^{*} \wedge 0 \leq b^{*} \wedge \pi b^{*} &lt; \lvert{\operatorname{arg}\left(\sigma\right)}\rvert \wedge \lvert{\operatorname{arg}\left(\sigma\right)}\rvert &lt; \pi \left(s + t - u + 1\right) \wedge C_{1} \wedge C_{2} \wedge C_{12} \wedge C_{14} \wedge C_{15}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}m = 0 \wedge \phi = 0 \wedge v + 1 &lt; u \wedge 0 &lt; n \wedge 0 &lt; c^{*} \wedge 0 \leq b^{*} \wedge \pi b^{*} &lt; \lvert{\operatorname{arg}\left(\sigma\right)}\rvert \wedge \lvert{\operatorname{arg}\left(\sigma\right)}\rvert &lt; \pi \left(s + t - v + 1 \right) \wedge C_{1} \wedge C_{3} \wedge C_{12} \wedge C_{14} \wedge C_{15}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}C_{17} \wedge t = 0 \wedge u &lt; s \wedge 0 &lt; b^{*} \wedge C_{10} \wedge C_{1} \wedge C_{2} \wedge C_{3}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}C_{17} \wedge s = 0 \wedge v &lt; t \wedge 0 &lt; b^{*} \wedge C_{10} \wedge C_{1} \wedge C_{2} \wedge C_{3}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}C_{16} \wedge n = 0 \wedge p &lt; m \wedge 0 &lt; c^{*} \wedge C_{12} \wedge C_{1} \wedge C_{2} \wedge C_{3}\end{split}\]</div>
-</li>
-<li><div class="first math">
-\[\begin{split}C_{16} \wedge m = 0 \wedge q &lt; n \wedge 0 &lt; c^{*} \wedge C_{12} \wedge C_{1} \wedge C_{2} \wedge C_{3}\end{split}\]</div>
-</li>
-</ol>
-</div>
-</div>
-<div class="section" id="the-inverse-laplace-transform-of-a-g-function">
-<h1>The Inverse Laplace Transform of a G-function<a class="headerlink" href="#the-inverse-laplace-transform-of-a-g-function" title="Permalink to this headline">¶</a></h1>
-<p>The inverse laplace transform of a Meijer G-function can be expressed as
-another G-function. This is a fairly versatile method for computing this
-transform. However, I could not find the details in the literature, so I work
-them out here. In <a class="reference internal" href="../simplify/hyperexpand.html#luke1969">[Luke1969]</a>, section 5.6.3, there is a formula for the inverse
-Laplace transform of a G-function of argument <span class="math">\(bz\)</span>, and convergence conditions
-are also given. However, we need a formula for argument <span class="math">\(bz^a\)</span> for rational <span class="math">\(a\)</span>.</p>
-<p>We are asked to compute</p>
-<div class="math">
-\[f(t) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} e^{zt} G(bz^a) \mathrm{d}z,\]</div>
-<p>for positive real <span class="math">\(t\)</span>. Three questions arise:</p>
-<ol class="arabic simple">
-<li>When does this integral converge?</li>
-<li>How can we compute the integral?</li>
-<li>When is our computation valid?</li>
-</ol>
-<div class="section" id="how-to-compute-the-integral">
-<h2>How to compute the integral<a class="headerlink" href="#how-to-compute-the-integral" title="Permalink to this headline">¶</a></h2>
-<p>We shall work formally for now. Denote by <span class="math">\(\Delta(s)\)</span> the product of gamma
-functions appearing in the definition of <span class="math">\(G\)</span>, so that</p>
-<div class="math">
-\[G(z) = \frac{1}{2\pi i} \int_L \Delta(s) z^s \mathrm{d}s.\]</div>
-<p>Thus</p>
-<div class="math">
-\[f(t) = \frac{1}{(2\pi i)^2} \int_{c - i\infty}^{c + i\infty} \int_L
-              e^{zt} \Delta(s) b^s z^{as} \mathrm{d}s \mathrm{d}z.\]</div>
-<p>We interchange the order of integration to get</p>
-<div class="math">
-\[f(t) = \frac{1}{2\pi i} \int_L b^s \Delta(s)
-      \int_{c-i\infty}^{c+i\infty} e^{zt} z^{as} \frac{\mathrm{d}z}{2\pi i}
-            \mathrm{d}s.\]</div>
-<p>The inner integral is easily seen to be
-<span class="math">\(\frac{1}{\Gamma(-as)} \frac{1}{t^{1+as}}\)</span>. (Using Cauchy&#8217;s theorem and Jordan&#8217;s
-lemma deform the contour to run from <span class="math">\(-\infty\)</span> to <span class="math">\(-\infty\)</span>, encircling <span class="math">\(0\)</span> once
-in the negative sense. For <span class="math">\(as\)</span> real and greater than one,
-this contour can be pushed onto
-the negative real axis and the integral is recognised as a product of a sine and
-a gamma function. The formula is then proved using the functional equation of the
-gamma function, and extended to the entire domain of convergence of the original
-integral by appealing to analytic continuation.)
-Hence we find</p>
-<div class="math">
-\[f(t) = \frac{1}{t} \frac{1}{2\pi i} \int_L \Delta(s) \frac{1}{\Gamma(-as)}
-              \left(\frac{b}{t^a}\right)^s \mathrm{d}s,\]</div>
-<p>which is a so-called Fox H function (of argument <span class="math">\(\frac{b}{t^a}\)</span>). For rational
-<span class="math">\(a\)</span>, this can be expressed as a Meijer G-function using the gamma function
-multiplication theorem.</p>
-</div>
-<div class="section" id="when-this-computation-is-valid">
-<h2>When this computation is valid<a class="headerlink" href="#when-this-computation-is-valid" title="Permalink to this headline">¶</a></h2>
-<p>There are a number of obstacles in this computation. Interchange of integrals
-is only valid if all integrals involved are absolutely convergent. In
-particular the inner integral has to converge. Also, for our identification of
-the final integral as a Fox H / Meijer G-function to be correct, the poles of
-the newly obtained gamma function must be separated properly.</p>
-<p>It is easy to check that the inner integal converges absolutely for
-<span class="math">\(Re(as) &lt; -1\)</span>. Thus the contour <span class="math">\(L\)</span> has to run left of the line <span class="math">\(Re(as) = -1\)</span>.
-Under this condition, the poles of the newly-introduced gamma function are
-separated properly.</p>
-<p>It remains to observe that the Meijer G-function is an analytic, unbranched
-function of its parameters, and of the coefficient <span class="math">\(b\)</span>. Hence so is <span class="math">\(f(t)\)</span>.
-Thus the final computation remains valid as long as the initial integral
-converges, and if there exists a changed set of parameters where the computation
-is valid. If we assume w.l.o.g. that <span class="math">\(a &gt; 0\)</span>, then the latter condition is
-fulfilled if <span class="math">\(G\)</span> converges along contours (2) or (3) of <a class="reference internal" href="../simplify/hyperexpand.html#luke1969">[Luke1969]</a>,
-section 5.2, i.e. either <span class="math">\(\delta &gt;= \frac{a}{2}\)</span> or <span class="math">\(p \ge 1, p \ge q\)</span>.</p>
-</div>
-<div class="section" id="when-the-integral-exists">
-<h2>When the integral exists<a class="headerlink" href="#when-the-integral-exists" title="Permalink to this headline">¶</a></h2>
-<p>Using <a class="reference internal" href="../simplify/hyperexpand.html#luke1969">[Luke1969]</a>, section 5.10, for any given meijer G-function we can find a
-dominant term of the form <span class="math">\(z^a e^{bz^c}\)</span> (although this expression might not be
-the best possible, because of cancellation).</p>
-<p>We must thus investigate</p>
-<div class="math">
-\[\lim_{T \to \infty} \int_{c-iT}^{c+iT}
-e^{zt} z^a e^{bz^c} \mathrm{d}z.\]</div>
-<p>(This principal value integral is the exact statement used in the Laplace
-inversion theorem.) We write <span class="math">\(z = c + i \tau\)</span>. Then
-<span class="math">\(arg(z) \to \pm \frac{\pi}{2}\)</span>, and so <span class="math">\(e^{zt} \sim e^{it \tau}\)</span> (where <span class="math">\(\sim\)</span>
-shall always mean &#8220;asymptotically equivalent up to a positive real
-multiplicative constant&#8221;). Also
-<span class="math">\(z^{x + iy} \sim |\tau|^x e^{i y \log{|\tau|}} e^{\pm x i \frac{\pi}{2}}.\)</span></p>
-<p>Set <span class="math">\(\omega_{\pm} = b e^{\pm i Re(c) \frac{\pi}{2}}\)</span>. We have three cases:</p>
-<ol class="arabic simple">
-<li><span class="math">\(b=0\)</span> or <span class="math">\(Re(c) \le 0\)</span>.
-In this case the integral converges if <span class="math">\(Re(a) \le -1\)</span>.</li>
-<li><span class="math">\(b \ne 0\)</span>, <span class="math">\(Im(c) = 0\)</span>, <span class="math">\(Re(c) &gt; 0\)</span>.
-In this case the integral converges if <span class="math">\(Re(\omega_{\pm}) &lt; 0\)</span>.</li>
-<li><span class="math">\(b \ne 0\)</span>, <span class="math">\(Im(c) = 0\)</span>, <span class="math">\(Re(c) &gt; 0\)</span>, <span class="math">\(Re(\omega_{\pm}) \le 0\)</span>, and at least
-one of <span class="math">\(Re(\omega_{\pm}) = 0\)</span>.
-Here the same condition as in (1) applies.</li>
-</ol>
-</div>
-</div>
-<div class="section" id="implemented-g-function-formulae">
-<h1>Implemented G-Function Formulae<a class="headerlink" href="#implemented-g-function-formulae" title="Permalink to this headline">¶</a></h1>
-<p>An important part of the algorithm is a table expressing various functions
-as Meijer G-functions. This is essentially a table of Mellin Transforms in
-disguise. The following automatically generated table shows the formulae
-currently implemented in SymPy. An entry &#8220;generated&#8221; means that the
-corresponding G-function has a variable number of parameters.
-This table is intended to shrink in future, when the algorithm&#8217;s capabilities
-of deriving new formulae improve. Of course it has to grow whenever a new class
-of special functions is to be dealt with.</p>
-<span class="target" id="module-sympy.integrals.meijerint_doc"></span><p>Elementary functions:</p>
-<div class="math">
-\[\begin{split}a = a {G_{1, 1}^{1, 0}\left(\begin{matrix}  &amp; 1 \\0 &amp;  \end{matrix} \middle| {z} \right)} + a {G_{1, 1}^{0, 1}\left(\begin{matrix} 1 &amp;  \\ &amp; 0 \end{matrix} \middle| {z} \right)}\end{split}\]</div>
-<div class="math">
-\[\begin{split}\left(z^{q} p + b\right)^{- a} = \frac{b^{- a} {G_{1, 1}^{1, 1}\left(\begin{matrix} - a + 1 &amp;  \\0 &amp;  \end{matrix} \middle| {\frac{z^{q} p}{b}} \right)}}{\Gamma\left(a\right)}\end{split}\]</div>
-<div class="math">
-\[\begin{split}\frac{- b^{a} + \left(z^{q} p\right)^{a}}{z^{q} p - b} = \frac{b^{a - 1} \sin{\left (\pi a \right )} {G_{2, 2}^{2, 2}\left(\begin{matrix} 0, a &amp;  \\0, a &amp;  \end{matrix} \middle| {\frac{z^{q} p}{b}} \right)}}{\pi}\end{split}\]</div>
-<div class="math">
-\[\begin{split}\left(a + \sqrt{z^{q} p + a^{2}}\right)^{b} = - \frac{a^{b} b {G_{2, 2}^{1, 2}\left(\begin{matrix} \frac{1}{2} b + \frac{1}{2}, \frac{1}{2} b + 1 &amp;  \\0 &amp; b \end{matrix} \middle| {\frac{z^{q} p}{a^{2}}} \right)}}{2 \sqrt{\pi}}\end{split}\]</div>
-<div class="math">
-\[\begin{split}\left(- a + \sqrt{z^{q} p + a^{2}}\right)^{b} = \frac{a^{b} b {G_{2, 2}^{1, 2}\left(\begin{matrix} \frac{1}{2} b + \frac{1}{2}, \frac{1}{2} b + 1 &amp;  \\b &amp; 0 \end{matrix} \middle| {\frac{z^{q} p}{a^{2}}} \right)}}{2 \sqrt{\pi}}\end{split}\]</div>
-<div class="math">
-\[\begin{split}\frac{\left(a + \sqrt{z^{q} p + a^{2}}\right)^{b}}{\sqrt{z^{q} p + a^{2}}} = \frac{a^{b - 1} {G_{2, 2}^{1, 2}\left(\begin{matrix} \frac{1}{2} b + \frac{1}{2}, \frac{1}{2} b &amp;  \\0 &amp; b \end{matrix} \middle| {\frac{z^{q} p}{a^{2}}} \right)}}{\sqrt{\pi}}\end{split}\]</div>
-<div class="math">
-\[\begin{split}\frac{\left(- a + \sqrt{z^{q} p + a^{2}}\right)^{b}}{\sqrt{z^{q} p + a^{2}}} = \frac{a^{b - 1} {G_{2, 2}^{1, 2}\left(\begin{matrix} \frac{1}{2} b + \frac{1}{2}, \frac{1}{2} b &amp;  \\b &amp; 0 \end{matrix} \middle| {\frac{z^{q} p}{a^{2}}} \right)}}{\sqrt{\pi}}\end{split}\]</div>
-<div class="math">
-\[\begin{split}\left(z^{\frac{1}{2} q} \sqrt{p} + \sqrt{z^{q} p + a}\right)^{b} = - \frac{a^{\frac{1}{2} b} b {G_{2, 2}^{2, 1}\left(\begin{matrix} \frac{1}{2} b + 1 &amp; - \frac{1}{2} b + 1 \\0, \frac{1}{2} &amp;  \end{matrix} \middle| {\frac{z^{q} p}{a}} \right)}}{2 \sqrt{\pi}}\end{split}\]</div>
-<div class="math">
-\[\begin{split}\left(- z^{\frac{1}{2} q} \sqrt{p} + \sqrt{z^{q} p + a}\right)^{b} = \frac{a^{\frac{1}{2} b} b {G_{2, 2}^{2, 1}\left(\begin{matrix} - \frac{1}{2} b + 1 &amp; \frac{1}{2} b + 1 \\0, \frac{1}{2} &amp;  \end{matrix} \middle| {\frac{z^{q} p}{a}} \right)}}{2 \sqrt{\pi}}\end{split}\]</div>
-<div class="math">
-\[\begin{split}\frac{\left(z^{\frac{1}{2} q} \sqrt{p} + \sqrt{z^{q} p + a}\right)^{b}}{\sqrt{z^{q} p + a}} = \frac{a^{\frac{1}{2} b - \frac{1}{2}} {G_{2, 2}^{2, 1}\left(\begin{matrix} \frac{1}{2} b + \frac{1}{2} &amp; - \frac{1}{2} b + \frac{1}{2} \\0, \frac{1}{2} &amp;  \end{matrix} \middle| {\frac{z^{q} p}{a}} \right)}}{\sqrt{\pi}}\end{split}\]</div>
-<div class="math">
-\[\begin{split}\frac{\left(- z^{\frac{1}{2} q} \sqrt{p} + \sqrt{z^{q} p + a}\right)^{b}}{\sqrt{z^{q} p + a}} = \frac{a^{\frac{1}{2} b - \frac{1}{2}} {G_{2, 2}^{2, 1}\left(\begin{matrix} - \frac{1}{2} b + \frac{1}{2} &amp; \frac{1}{2} b + \frac{1}{2} \\0, \frac{1}{2} &amp;  \end{matrix} \middle| {\frac{z^{q} p}{a}} \right)}}{\sqrt{\pi}}\end{split}\]</div>
-<p>Functions involving <span class="math">\(\lvert{- z^{q} p + b}\rvert\)</span>:</p>
-<div class="math">
-\[\begin{split}\lvert{- z^{q} p + b}\rvert^{- a} = \frac{\pi \lvert{b}\rvert^{- a} {G_{2, 2}^{1, 1}\left(\begin{matrix} - a + 1 &amp; - \frac{1}{2} a + \frac{1}{2} \\0 &amp; - \frac{1}{2} a + \frac{1}{2} \end{matrix} \middle| {\frac{z^{q} p}{b}} \right)}}{\cos{\left (\frac{1}{2} \pi a \right )} \Gamma\left(a\right)},\text{ if } \Re{a} &lt; 1\end{split}\]</div>
-<p>Functions involving <span class="math">\(\operatorname{Chi}{\left (z^{q} p \right )}\)</span>:</p>
-<div class="math">
-\[\begin{split}\operatorname{Chi}{\left (z^{q} p \right )} = - \frac{1}{2} \pi^{\frac{3}{2}} {G_{2, 4}^{2, 0}\left(\begin{matrix}  &amp; \frac{1}{2}, 1 \\0, 0 &amp; \frac{1}{2}, \frac{1}{2} \end{matrix} \middle| {\frac{1}{4} z^{2 q} p^{2}} \right)}\end{split}\]</div>
-<p>Functions involving <span class="math">\(\operatorname{Ci}{\left (z^{q} p \right )}\)</span>:</p>
-<div class="math">
-\[\begin{split}\operatorname{Ci}{\left (z^{q} p \right )} = - \frac{1}{2} \sqrt{\pi} {G_{1, 3}^{2, 0}\left(\begin{matrix}  &amp; 1 \\0, 0 &amp; \frac{1}{2} \end{matrix} \middle| {\frac{1}{4} z^{2 q} p^{2}} \right)}\end{split}\]</div>
-<p>Functions involving <span class="math">\(\operatorname{Ei}{\left (z^{q} p \right )}\)</span>:</p>
-<div class="math">
-\[\begin{split}\operatorname{Ei}{\left (z^{q} p \right )} = - \mathbf{\imath} \pi {G_{1, 1}^{1, 0}\left(\begin{matrix}  &amp; 1 \\0 &amp;  \end{matrix} \middle| {z} \right)} - {G_{1, 2}^{2, 0}\left(\begin{matrix}  &amp; 1 \\0, 0 &amp;  \end{matrix} \middle| {z^{q} p e^{\mathbf{\imath} \pi}} \right)} - \mathbf{\imath} \pi {G_{1, 1}^{0, 1}\left(\begin{matrix} 1 &amp;  \\ &amp; 0 \end{matrix} \middle| {z} \right)}\end{split}\]</div>
-<p>Functions involving <span class="math">\(\theta\left(z^{q} p - b\right)\)</span>:</p>
-<div class="math">
-\[\begin{split}\left(z^{q} p - b\right)^{a - 1} \theta\left(z^{q} p - b\right) = b^{a - 1} \Gamma\left(a\right) {G_{1, 1}^{0, 1}\left(\begin{matrix} a &amp;  \\ &amp; 0 \end{matrix} \middle| {\frac{z^{q} p}{b}} \right)},\text{ if } b &gt; 0\end{split}\]</div>
-<div class="math">
-\[\begin{split}\left(- z^{q} p + b\right)^{a - 1} \theta\left(- z^{q} p + b\right) = b^{a - 1} \Gamma\left(a\right) {G_{1, 1}^{1, 0}\left(\begin{matrix}  &amp; a \\0 &amp;  \end{matrix} \middle| {\frac{z^{q} p}{b}} \right)},\text{ if } b &gt; 0\end{split}\]</div>
-<div class="math">
-\[\begin{split}\left(z^{q} p - b\right)^{a - 1} \theta\left(z - \left(\frac{b}{p}\right)^{\frac{1}{q}}\right) = b^{a - 1} \Gamma\left(a\right) {G_{1, 1}^{0, 1}\left(\begin{matrix} a &amp;  \\ &amp; 0 \end{matrix} \middle| {\frac{z^{q} p}{b}} \right)},\text{ if } b &gt; 0\end{split}\]</div>
-<div class="math">
-\[\begin{split}\left(- z^{q} p + b\right)^{a - 1} \theta\left(- z + \left(\frac{b}{p}\right)^{\frac{1}{q}}\right) = b^{a - 1} \Gamma\left(a\right) {G_{1, 1}^{1, 0}\left(\begin{matrix}  &amp; a \\0 &amp;  \end{matrix} \middle| {\frac{z^{q} p}{b}} \right)},\text{ if } b &gt; 0\end{split}\]</div>
-<p>Functions involving <span class="math">\(\theta\left(- z^{q} p + 1\right)\)</span>, <span class="math">\(\log{\left (z^{q} p \right )}\)</span>:</p>
-<div class="math">
-\[\log^{n}{\left (z^{q} p \right )} \theta\left(- z^{q} p + 1\right) = \text{generated}\]</div>
-<div class="math">
-\[\log^{n}{\left (z^{q} p \right )} \theta\left(z^{q} p - 1\right) = \text{generated}\]</div>
-<p>Functions involving <span class="math">\(\operatorname{Shi}{\left (z^{q} p \right )}\)</span>:</p>
-<div class="math">
-\[\begin{split}\operatorname{Shi}{\left (z^{q} p \right )} = \frac{1}{4} z^{q} \sqrt{\pi} p {G_{1, 3}^{1, 1}\left(\begin{matrix} \frac{1}{2} &amp;  \\0 &amp; - \frac{1}{2}, - \frac{1}{2} \end{matrix} \middle| {\frac{1}{4} z^{2 q} p^{2} e^{\mathbf{\imath} \pi}} \right)}\end{split}\]</div>
-<p>Functions involving <span class="math">\(\operatorname{Si}{\left (z^{q} p \right )}\)</span>:</p>
-<div class="math">
-\[\begin{split}\operatorname{Si}{\left (z^{q} p \right )} = \frac{1}{2} \sqrt{\pi} {G_{1, 3}^{1, 1}\left(\begin{matrix} 1 &amp;  \\\frac{1}{2} &amp; 0, 0 \end{matrix} \middle| {\frac{1}{4} z^{2 q} p^{2}} \right)}\end{split}\]</div>
-<p>Functions involving <span class="math">\(I_{a}\left(z^{q} p\right)\)</span>:</p>
-<div class="math">
-\[\begin{split}I_{a}\left(z^{q} p\right) = \pi {G_{1, 3}^{1, 0}\left(\begin{matrix}  &amp; \frac{1}{2} a + \frac{1}{2} \\\frac{1}{2} a &amp; - \frac{1}{2} a, \frac{1}{2} a + \frac{1}{2} \end{matrix} \middle| {\frac{1}{4} z^{2 q} p^{2}} \right)}\end{split}\]</div>
-<p>Functions involving <span class="math">\(J_{a}\left(z^{q} p\right)\)</span>:</p>
-<div class="math">
-\[\begin{split}J_{a}\left(z^{q} p\right) = {G_{0, 2}^{1, 0}\left(\begin{matrix}  &amp;  \\\frac{1}{2} a &amp; - \frac{1}{2} a \end{matrix} \middle| {\frac{1}{4} z^{2 q} p^{2}} \right)}\end{split}\]</div>
-<p>Functions involving <span class="math">\(K_{a}\left(z^{q} p\right)\)</span>:</p>
-<div class="math">
-\[\begin{split}K_{a}\left(z^{q} p\right) = \frac{1}{2} {G_{0, 2}^{2, 0}\left(\begin{matrix}  &amp;  \\\frac{1}{2} a, - \frac{1}{2} a &amp;  \end{matrix} \middle| {\frac{1}{4} z^{2 q} p^{2}} \right)}\end{split}\]</div>
-<p>Functions involving <span class="math">\(Y_{a}\left(z^{q} p\right)\)</span>:</p>
-<div class="math">
-\[\begin{split}Y_{a}\left(z^{q} p\right) = {G_{1, 3}^{2, 0}\left(\begin{matrix}  &amp; - \frac{1}{2} a - \frac{1}{2} \\\frac{1}{2} a, - \frac{1}{2} a &amp; - \frac{1}{2} a - \frac{1}{2} \end{matrix} \middle| {\frac{1}{4} z^{2 q} p^{2}} \right)}\end{split}\]</div>
-<p>Functions involving <span class="math">\(\cos{\left (z^{q} p \right )}\)</span>:</p>
-<div class="math">
-\[\begin{split}\cos{\left (z^{q} p \right )} = \sqrt{\pi} {G_{0, 2}^{1, 0}\left(\begin{matrix}  &amp;  \\0 &amp; \frac{1}{2} \end{matrix} \middle| {\frac{1}{4} z^{2 q} p^{2}} \right)}\end{split}\]</div>
-<p>Functions involving <span class="math">\(\cosh{\left (z^{q} p \right )}\)</span>:</p>
-<div class="math">
-\[\begin{split}\cosh{\left (z^{q} p \right )} = \pi^{\frac{3}{2}} {G_{1, 3}^{1, 0}\left(\begin{matrix}  &amp; \frac{1}{2} \\0 &amp; \frac{1}{2}, \frac{1}{2} \end{matrix} \middle| {\frac{1}{4} z^{2 q} p^{2}} \right)}\end{split}\]</div>
-<p>Functions involving <span class="math">\(\operatorname{erf}{\left (z^{q} p \right )}\)</span>:</p>
-<div class="math">
-\[\begin{split}\operatorname{erf}{\left (z^{q} p \right )} = \frac{{G_{1, 2}^{1, 1}\left(\begin{matrix} 1 &amp;  \\\frac{1}{2} &amp; 0 \end{matrix} \middle| {z^{2 q} p^{2}} \right)}}{\sqrt{\pi}}\end{split}\]</div>
-<p>Functions involving <span class="math">\(e^{z^{q} p e^{\mathbf{\imath} \pi}}\)</span>:</p>
-<div class="math">
-\[\begin{split}e^{z^{q} p e^{\mathbf{\imath} \pi}} = {G_{0, 1}^{1, 0}\left(\begin{matrix}  &amp;  \\0 &amp;  \end{matrix} \middle| {z^{q} p} \right)}\end{split}\]</div>
-<p>Functions involving <span class="math">\(\operatorname{E}_{a}\left(z^{q} p\right)\)</span>:</p>
-<div class="math">
-\[\begin{split}\operatorname{E}_{a}\left(z^{q} p\right) = {G_{1, 2}^{2, 0}\left(\begin{matrix}  &amp; a \\a - 1, 0 &amp;  \end{matrix} \middle| {z^{q} p} \right)}\end{split}\]</div>
-<p>Functions involving <span class="math">\(C\left(z^{q} p\right)\)</span>:</p>
-<div class="math">
-\[\begin{split}C\left(z^{q} p\right) = \frac{1}{2} {G_{1, 3}^{1, 1}\left(\begin{matrix} 1 &amp;  \\\frac{1}{4} &amp; 0, \frac{3}{4} \end{matrix} \middle| {\frac{1}{16} z^{4 q} \pi^{2} p^{4}} \right)}\end{split}\]</div>
-<p>Functions involving <span class="math">\(S\left(z^{q} p\right)\)</span>:</p>
-<div class="math">
-\[\begin{split}S\left(z^{q} p\right) = \frac{1}{2} {G_{1, 3}^{1, 1}\left(\begin{matrix} 1 &amp;  \\\frac{3}{4} &amp; 0, \frac{1}{4} \end{matrix} \middle| {\frac{1}{16} z^{4 q} \pi^{2} p^{4}} \right)}\end{split}\]</div>
-<p>Functions involving <span class="math">\(\log{\left (z^{q} p \right )}\)</span>:</p>
-<div class="math">
-\[\log^{n}{\left (z^{q} p \right )} = \text{generated}\]</div>
-<div class="math">
-\[\begin{split}\log{\left (z^{q} p + a \right )} = \log{\left (a \right )} {G_{1, 1}^{1, 0}\left(\begin{matrix}  &amp; 1 \\0 &amp;  \end{matrix} \middle| {z} \right)} + \log{\left (a \right )} {G_{1, 1}^{0, 1}\left(\begin{matrix} 1 &amp;  \\ &amp; 0 \end{matrix} \middle| {z} \right)} + {G_{2, 2}^{1, 2}\left(\begin{matrix} 1, 1 &amp;  \\1 &amp; 0 \end{matrix} \middle| {\frac{z^{q} p}{a}} \right)}\end{split}\]</div>
-<div class="math">
-\[\begin{split}\log{\left (\lvert{z^{q} p - a}\rvert \right )} = \log{\left (\lvert{a}\rvert \right )} {G_{1, 1}^{1, 0}\left(\begin{matrix}  &amp; 1 \\0 &amp;  \end{matrix} \middle| {z} \right)} + \log{\left (\lvert{a}\rvert \right )} {G_{1, 1}^{0, 1}\left(\begin{matrix} 1 &amp;  \\ &amp; 0 \end{matrix} \middle| {z} \right)} + \pi {G_{3, 3}^{1, 2}\left(\begin{matrix} 1, 1 &amp; \frac{1}{2} \\1 &amp; 0, \frac{1}{2} \end{matrix} \middle| {\frac{z^{q} p}{a}} \right)}\end{split}\]</div>
-<p>Functions involving <span class="math">\(\sin{\left (z^{q} p \right )}\)</span>:</p>
-<div class="math">
-\[\begin{split}\sin{\left (z^{q} p \right )} = \sqrt{\pi} {G_{0, 2}^{1, 0}\left(\begin{matrix}  &amp;  \\\frac{1}{2} &amp; 0 \end{matrix} \middle| {\frac{1}{4} z^{2 q} p^{2}} \right)}\end{split}\]</div>
-<p>Functions involving <span class="math">\(\sinh{\left (z^{q} p \right )}\)</span>:</p>
-<div class="math">
-\[\begin{split}\sinh{\left (z^{q} p \right )} = \pi^{\frac{3}{2}} {G_{1, 3}^{1, 0}\left(\begin{matrix}  &amp; 1 \\\frac{1}{2} &amp; 1, 0 \end{matrix} \middle| {\frac{1}{4} z^{2 q} p^{2}} \right)}\end{split}\]</div>
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-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Computing Integrals using Meijer G-Functions</a><ul>
-<li><a class="reference internal" href="#overview">Overview</a></li>
-<li><a class="reference internal" href="#polar-numbers-and-branched-functions">Polar Numbers and Branched Functions</a></li>
-<li><a class="reference internal" href="#representing-branched-functions-on-the-argand-plane">Representing Branched Functions on the Argand Plane</a></li>
-<li><a class="reference internal" href="#table-lookups-and-inverse-mellin-transforms">Table Lookups and Inverse Mellin Transforms</a></li>
-<li><a class="reference internal" href="#applying-the-integral-theorems">Applying the Integral Theorems</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#the-g-function-integration-theorems">The G-Function Integration Theorems</a><ul>
-<li><a class="reference internal" href="#conditions-of-convergence-for-integral-1">Conditions of Convergence for Integral (1)</a></li>
-<li><a class="reference internal" href="#conditions-of-convergence-for-integral-2">Conditions of Convergence for Integral (2)</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#the-inverse-laplace-transform-of-a-g-function">The Inverse Laplace Transform of a G-function</a><ul>
-<li><a class="reference internal" href="#how-to-compute-the-integral">How to compute the integral</a></li>
-<li><a class="reference internal" href="#when-this-computation-is-valid">When this computation is valid</a></li>
-<li><a class="reference internal" href="#when-the-integral-exists">When the integral exists</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#implemented-g-function-formulae">Implemented G-Function Formulae</a></li>
-</ul>
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-  <div class="section" id="module-sympy.integrals">
-<span id="integrals"></span><h1>Integrals<a class="headerlink" href="#module-sympy.integrals" title="Permalink to this headline">¶</a></h1>
-<p>The <em>integrals</em> module in SymPy implements methods to calculate definite and indefinite integrals of expressions.</p>
-<p>Principal method in this module is integrate()</p>
-<blockquote>
-<div><ul class="simple">
-<li>integrate(f, x) returns the indefinite integral <span class="math">\(\int f\,dx\)</span></li>
-<li>integrate(f, (x, a, b)) returns the definite integral <span class="math">\(\int_{a}^{b} f\,dx\)</span></li>
-</ul>
-</div></blockquote>
-<div class="section" id="examples">
-<h2>Examples<a class="headerlink" href="#examples" title="Permalink to this headline">¶</a></h2>
-<p>SymPy can integrate a vast array of functions. It can integrate polynomial functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">init_printing</span><span class="p">(</span><span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">wrap_line</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">no_global</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go"> 3    2</span>
-<span class="go">x    x</span>
-<span class="go">-- + -- + x</span>
-<span class="go">3    2</span>
-</pre></div>
-</div>
-<p>Rational functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">               1</span>
-<span class="go">log(x + 1) + -----</span>
-<span class="go">             x + 1</span>
-</pre></div>
-</div>
-<p>Exponential-polynomial functions. Multiplicative combinations of polynomials and the functions exp, cos and sin can be integrated by hand using repeated integration by parts, which is an extremely tedious process. Happily, SymPy will deal with these integrals.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go"> 2  x           2  x                         x           x</span>
-<span class="go">x *e *sin(x)   x *e *cos(x)      x          e *sin(x)   e *cos(x)</span>
-<span class="go">------------ + ------------ - x*e *sin(x) + --------- - ---------</span>
-<span class="go">     2              2                           2           2</span>
-</pre></div>
-</div>
-<p>even a few nonelementary integrals (in particular, some integrals involving the error function) can be evaluated:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">erf</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">  ____    2</span>
-<span class="go">\/ pi *erf (x)</span>
-<span class="go">--------------</span>
-<span class="go">      4</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="module-sympy.integrals.transforms">
-<span id="integral-transforms"></span><h2>Integral Transforms<a class="headerlink" href="#module-sympy.integrals.transforms" title="Permalink to this headline">¶</a></h2>
-<p>SymPy has special support for definite integrals, and integral transforms.</p>
-<dl class="function">
-<dt id="sympy.integrals.transforms.mellin_transform">
-<tt class="descclassname">sympy.integrals.transforms.</tt><tt class="descname">mellin_transform</tt><big>(</big><em>f</em>, <em>x</em>, <em>s</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/integrals/transforms.html#mellin_transform"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.integrals.transforms.mellin_transform" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the Mellin transform <span class="math">\(F(s)\)</span> of <span class="math">\(f(x)\)</span>,</p>
-<div class="math">
-\[F(s) = \int_0^\infty x^{s-1} f(x) \mathrm{d}x.\]</div>
-<dl class="docutils">
-<dt>For all &#8220;sensible&#8221; functions, this converges absolutely in a strip</dt>
-<dd><span class="math">\(a &lt; Re(s) &lt; b\)</span>.</dd>
-</dl>
-<p>The Mellin transform is related via change of variables to the Fourier
-transform, and also to the (bilateral) Laplace transform.</p>
-<p>This function returns (F, (a, b), cond)
-where <span class="math">\(F\)</span> is the Mellin transform of <span class="math">\(f\)</span>, <span class="math">\((a, b)\)</span> is the fundamental strip
-(as above), and cond are auxiliary convergence conditions.</p>
-<p>If the integral cannot be computed in closed form, this function returns
-an unevaluated MellinTransform object.</p>
-<p>For a description of possible hints, refer to the docstring of
-<tt class="xref py py-func docutils literal"><span class="pre">sympy.integrals.transforms.IntegralTransform.doit()</span></tt>. If <tt class="docutils literal"><span class="pre">noconds=False</span></tt>,
-then only <span class="math">\(F\)</span> will be returned (i.e. not <tt class="docutils literal"><span class="pre">cond</span></tt>, and also not the strip
-<tt class="docutils literal"><span class="pre">(a,</span> <span class="pre">b)</span></tt>).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.integrals.transforms</span> <span class="kn">import</span> <span class="n">mellin_transform</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">s</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mellin_transform</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span>
-<span class="go">(gamma(s), (0, oo), True)</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.integrals.transforms.inverse_mellin_transform" title="sympy.integrals.transforms.inverse_mellin_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_mellin_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.laplace_transform" title="sympy.integrals.transforms.laplace_transform"><tt class="xref py py-obj docutils literal"><span class="pre">laplace_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.fourier_transform" title="sympy.integrals.transforms.fourier_transform"><tt class="xref py py-obj docutils literal"><span class="pre">fourier_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.hankel_transform" title="sympy.integrals.transforms.hankel_transform"><tt class="xref py py-obj docutils literal"><span class="pre">hankel_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_hankel_transform" title="sympy.integrals.transforms.inverse_hankel_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_hankel_transform</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.integrals.transforms.inverse_mellin_transform">
-<tt class="descclassname">sympy.integrals.transforms.</tt><tt class="descname">inverse_mellin_transform</tt><big>(</big><em>F</em>, <em>s</em>, <em>x</em>, <em>strip</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/integrals/transforms.html#inverse_mellin_transform"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.integrals.transforms.inverse_mellin_transform" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the inverse Mellin transform of <span class="math">\(F(s)\)</span> over the fundamental
-strip given by <tt class="docutils literal"><span class="pre">strip=(a,</span> <span class="pre">b)</span></tt>.</p>
-<p>This can be defined as</p>
-<div class="math">
-\[f(x) = \int_{c - i\infty}^{c + i\infty} x^{-s} F(s) \mathrm{d}s,\]</div>
-<p>for any <span class="math">\(c\)</span> in the fundamental strip. Under certain regularity
-conditions on <span class="math">\(F\)</span> and/or <span class="math">\(f\)</span>,
-this recovers <span class="math">\(f\)</span> from its Mellin transform <span class="math">\(F\)</span>
-(and vice versa), for positive real <span class="math">\(x\)</span>.</p>
-<p>One of <span class="math">\(a\)</span> or <span class="math">\(b\)</span> may be passed as None; a suitable <span class="math">\(c\)</span> will be
-inferred.</p>
-<p>If the integral cannot be computed in closed form, this function returns
-an unevaluated InverseMellinTransform object.</p>
-<p>Note that this function will assume x to be positive and real, regardless
-of the sympy assumptions!</p>
-<p>For a description of possible hints, refer to the docstring of
-<tt class="xref py py-func docutils literal"><span class="pre">sympy.integrals.transforms.IntegralTransform.doit()</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.integrals.transforms</span> <span class="kn">import</span> <span class="n">inverse_mellin_transform</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">oo</span><span class="p">,</span> <span class="n">gamma</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">s</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">inverse_mellin_transform</span><span class="p">(</span><span class="n">gamma</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">exp(-x)</span>
-</pre></div>
-</div>
-<p>The fundamental strip matters:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">s</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">inverse_mellin_transform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="o">-</span><span class="n">oo</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">))</span>
-<span class="go">x*(1 - 1/x**2)*Heaviside(x - 1)/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">inverse_mellin_transform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">-x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">inverse_mellin_transform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">(-x**2/2 + 1/2)*Heaviside(-x + 1)/x</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.integrals.transforms.mellin_transform" title="sympy.integrals.transforms.mellin_transform"><tt class="xref py py-obj docutils literal"><span class="pre">mellin_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.hankel_transform" title="sympy.integrals.transforms.hankel_transform"><tt class="xref py py-obj docutils literal"><span class="pre">hankel_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_hankel_transform" title="sympy.integrals.transforms.inverse_hankel_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_hankel_transform</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.integrals.transforms.laplace_transform">
-<tt class="descclassname">sympy.integrals.transforms.</tt><tt class="descname">laplace_transform</tt><big>(</big><em>f</em>, <em>t</em>, <em>s</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/integrals/transforms.html#laplace_transform"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.integrals.transforms.laplace_transform" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the Laplace Transform <span class="math">\(F(s)\)</span> of <span class="math">\(f(t)\)</span>,</p>
-<div class="math">
-\[F(s) = \int_0^\infty e^{-st} f(t) \mathrm{d}t.\]</div>
-<p>For all &#8220;sensible&#8221; functions, this converges absolutely in a
-half plane  <span class="math">\(a &lt; Re(s)\)</span>.</p>
-<p>This function returns (F, a, cond)
-where <span class="math">\(F\)</span> is the Laplace transform of <span class="math">\(f\)</span>, <span class="math">\(Re(s) &gt; a\)</span> is the half-plane
-of convergence, and cond are auxiliary convergence conditions.</p>
-<p>If the integral cannot be computed in closed form, this function returns
-an unevaluated LaplaceTransform object.</p>
-<p>For a description of possible hints, refer to the docstring of
-<tt class="xref py py-func docutils literal"><span class="pre">sympy.integrals.transforms.IntegralTransform.doit()</span></tt>. If <tt class="docutils literal"><span class="pre">noconds=True</span></tt>,
-only <span class="math">\(F\)</span> will be returned (i.e. not <tt class="docutils literal"><span class="pre">cond</span></tt>, and also not the plane <tt class="docutils literal"><span class="pre">a</span></tt>).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.integrals</span> <span class="kn">import</span> <span class="n">laplace_transform</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">t</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">a</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">laplace_transform</span><span class="p">(</span><span class="n">t</span><span class="o">**</span><span class="n">a</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span>
-<span class="go">(s**(-a - 1)*gamma(a + 1), 0, -re(a) &lt; 1)</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.integrals.transforms.inverse_laplace_transform" title="sympy.integrals.transforms.inverse_laplace_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_laplace_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.mellin_transform" title="sympy.integrals.transforms.mellin_transform"><tt class="xref py py-obj docutils literal"><span class="pre">mellin_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.fourier_transform" title="sympy.integrals.transforms.fourier_transform"><tt class="xref py py-obj docutils literal"><span class="pre">fourier_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.hankel_transform" title="sympy.integrals.transforms.hankel_transform"><tt class="xref py py-obj docutils literal"><span class="pre">hankel_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_hankel_transform" title="sympy.integrals.transforms.inverse_hankel_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_hankel_transform</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.integrals.transforms.inverse_laplace_transform">
-<tt class="descclassname">sympy.integrals.transforms.</tt><tt class="descname">inverse_laplace_transform</tt><big>(</big><em>F</em>, <em>s</em>, <em>t</em>, <em>plane=None</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/integrals/transforms.html#inverse_laplace_transform"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.integrals.transforms.inverse_laplace_transform" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the inverse Laplace transform of <span class="math">\(F(s)\)</span>, defined as</p>
-<div class="math">
-\[f(t) = \int_{c-i\infty}^{c+i\infty} e^{st} F(s) \mathrm{d}s,\]</div>
-<p>for <span class="math">\(c\)</span> so large that <span class="math">\(F(s)\)</span> has no singularites in the
-half-plane <span class="math">\(Re(s) &gt; c-\epsilon\)</span>.</p>
-<p>The plane can be specified by
-argument <tt class="docutils literal"><span class="pre">plane</span></tt>, but will be inferred if passed as None.</p>
-<p>Under certain regularity conditions, this recovers <span class="math">\(f(t)\)</span> from its
-Laplace Transform <span class="math">\(F(s)\)</span>, for non-negative <span class="math">\(t\)</span>, and vice
-versa.</p>
-<p>If the integral cannot be computed in closed form, this function returns
-an unevaluated InverseLaplaceTransform object.</p>
-<p>Note that this function will always assume <span class="math">\(t\)</span> to be real,
-regardless of the sympy assumption on <span class="math">\(t\)</span>.</p>
-<p>For a description of possible hints, refer to the docstring of
-<tt class="xref py py-func docutils literal"><span class="pre">sympy.integrals.transforms.IntegralTransform.doit()</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.integrals.transforms</span> <span class="kn">import</span> <span class="n">inverse_laplace_transform</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exp</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">inverse_laplace_transform</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="o">*</span><span class="n">s</span><span class="p">)</span><span class="o">/</span><span class="n">s</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
-<span class="go">Heaviside(-a + t)</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.integrals.transforms.laplace_transform" title="sympy.integrals.transforms.laplace_transform"><tt class="xref py py-obj docutils literal"><span class="pre">laplace_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.hankel_transform" title="sympy.integrals.transforms.hankel_transform"><tt class="xref py py-obj docutils literal"><span class="pre">hankel_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_hankel_transform" title="sympy.integrals.transforms.inverse_hankel_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_hankel_transform</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.integrals.transforms.fourier_transform">
-<tt class="descclassname">sympy.integrals.transforms.</tt><tt class="descname">fourier_transform</tt><big>(</big><em>f</em>, <em>x</em>, <em>k</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/integrals/transforms.html#fourier_transform"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.integrals.transforms.fourier_transform" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the unitary, ordinary-frequency Fourier transform of <span class="math">\(f\)</span>, defined
-as</p>
-<div class="math">
-\[F(k) = \int_{-\infty}^\infty f(x) e^{-2\pi i x k} \mathrm{d} x.\]</div>
-<p>If the transform cannot be computed in closed form, this
-function returns an unevaluated FourierTransform object.</p>
-<p>For other Fourier transform conventions, see the function
-<tt class="xref py py-func docutils literal"><span class="pre">sympy.integrals.transforms._fourier_transform()</span></tt>.</p>
-<p>For a description of possible hints, refer to the docstring of
-<tt class="xref py py-func docutils literal"><span class="pre">sympy.integrals.transforms.IntegralTransform.doit()</span></tt>.
-Note that for this transform, by default <tt class="docutils literal"><span class="pre">noconds=True</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">fourier_transform</span><span class="p">,</span> <span class="n">exp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fourier_transform</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-<span class="go">sqrt(pi)*exp(-pi**2*k**2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fourier_transform</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">noconds</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">(sqrt(pi)*exp(-pi**2*k**2), True)</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.integrals.transforms.inverse_fourier_transform" title="sympy.integrals.transforms.inverse_fourier_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_fourier_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.sine_transform" title="sympy.integrals.transforms.sine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">sine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_sine_transform" title="sympy.integrals.transforms.inverse_sine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_sine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.cosine_transform" title="sympy.integrals.transforms.cosine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">cosine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_cosine_transform" title="sympy.integrals.transforms.inverse_cosine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_cosine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.hankel_transform" title="sympy.integrals.transforms.hankel_transform"><tt class="xref py py-obj docutils literal"><span class="pre">hankel_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_hankel_transform" title="sympy.integrals.transforms.inverse_hankel_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_hankel_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.mellin_transform" title="sympy.integrals.transforms.mellin_transform"><tt class="xref py py-obj docutils literal"><span class="pre">mellin_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.laplace_transform" title="sympy.integrals.transforms.laplace_transform"><tt class="xref py py-obj docutils literal"><span class="pre">laplace_transform</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.integrals.transforms.inverse_fourier_transform">
-<tt class="descclassname">sympy.integrals.transforms.</tt><tt class="descname">inverse_fourier_transform</tt><big>(</big><em>F</em>, <em>k</em>, <em>x</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/integrals/transforms.html#inverse_fourier_transform"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.integrals.transforms.inverse_fourier_transform" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the unitary, ordinary-frequency inverse Fourier transform of <span class="math">\(F\)</span>,
-defined as</p>
-<div class="math">
-\[f(x) = \int_{-\infty}^\infty F(k) e^{2\pi i x k} \mathrm{d} k.\]</div>
-<p>If the transform cannot be computed in closed form, this
-function returns an unevaluated InverseFourierTransform object.</p>
-<p>For other Fourier transform conventions, see the function
-<tt class="xref py py-func docutils literal"><span class="pre">sympy.integrals.transforms._fourier_transform()</span></tt>.</p>
-<p>For a description of possible hints, refer to the docstring of
-<tt class="xref py py-func docutils literal"><span class="pre">sympy.integrals.transforms.IntegralTransform.doit()</span></tt>.
-Note that for this transform, by default <tt class="docutils literal"><span class="pre">noconds=True</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">inverse_fourier_transform</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">inverse_fourier_transform</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">k</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="n">k</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">exp(-x**2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">inverse_fourier_transform</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">k</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="n">k</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">noconds</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">(exp(-x**2), True)</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.integrals.transforms.fourier_transform" title="sympy.integrals.transforms.fourier_transform"><tt class="xref py py-obj docutils literal"><span class="pre">fourier_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.sine_transform" title="sympy.integrals.transforms.sine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">sine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_sine_transform" title="sympy.integrals.transforms.inverse_sine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_sine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.cosine_transform" title="sympy.integrals.transforms.cosine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">cosine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_cosine_transform" title="sympy.integrals.transforms.inverse_cosine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_cosine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.hankel_transform" title="sympy.integrals.transforms.hankel_transform"><tt class="xref py py-obj docutils literal"><span class="pre">hankel_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_hankel_transform" title="sympy.integrals.transforms.inverse_hankel_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_hankel_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.mellin_transform" title="sympy.integrals.transforms.mellin_transform"><tt class="xref py py-obj docutils literal"><span class="pre">mellin_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.laplace_transform" title="sympy.integrals.transforms.laplace_transform"><tt class="xref py py-obj docutils literal"><span class="pre">laplace_transform</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.integrals.transforms.sine_transform">
-<tt class="descclassname">sympy.integrals.transforms.</tt><tt class="descname">sine_transform</tt><big>(</big><em>f</em>, <em>x</em>, <em>k</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/integrals/transforms.html#sine_transform"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.integrals.transforms.sine_transform" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the unitary, ordinary-frequency sine transform of <span class="math">\(f\)</span>, defined
-as</p>
-<div class="math">
-\[F(k) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty f(x) \sin(2\pi x k) \mathrm{d} x.\]</div>
-<p>If the transform cannot be computed in closed form, this
-function returns an unevaluated SineTransform object.</p>
-<p>For a description of possible hints, refer to the docstring of
-<tt class="xref py py-func docutils literal"><span class="pre">sympy.integrals.transforms.IntegralTransform.doit()</span></tt>.
-Note that for this transform, by default <tt class="docutils literal"><span class="pre">noconds=True</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sine_transform</span><span class="p">,</span> <span class="n">exp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">a</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sine_transform</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-<span class="go">sqrt(2)*k*exp(-k**2/(4*a))/(4*a**(3/2))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sine_transform</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-<span class="go">2**(-a + 1/2)*k**(a - 1)*gamma(-a/2 + 1)/gamma(a/2 + 1/2)</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.integrals.transforms.fourier_transform" title="sympy.integrals.transforms.fourier_transform"><tt class="xref py py-obj docutils literal"><span class="pre">fourier_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_fourier_transform" title="sympy.integrals.transforms.inverse_fourier_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_fourier_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_sine_transform" title="sympy.integrals.transforms.inverse_sine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_sine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.cosine_transform" title="sympy.integrals.transforms.cosine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">cosine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_cosine_transform" title="sympy.integrals.transforms.inverse_cosine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_cosine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.hankel_transform" title="sympy.integrals.transforms.hankel_transform"><tt class="xref py py-obj docutils literal"><span class="pre">hankel_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_hankel_transform" title="sympy.integrals.transforms.inverse_hankel_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_hankel_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.mellin_transform" title="sympy.integrals.transforms.mellin_transform"><tt class="xref py py-obj docutils literal"><span class="pre">mellin_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.laplace_transform" title="sympy.integrals.transforms.laplace_transform"><tt class="xref py py-obj docutils literal"><span class="pre">laplace_transform</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.integrals.transforms.inverse_sine_transform">
-<tt class="descclassname">sympy.integrals.transforms.</tt><tt class="descname">inverse_sine_transform</tt><big>(</big><em>F</em>, <em>k</em>, <em>x</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/integrals/transforms.html#inverse_sine_transform"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.integrals.transforms.inverse_sine_transform" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the unitary, ordinary-frequency inverse sine transform of <span class="math">\(F\)</span>,
-defined as</p>
-<div class="math">
-\[f(x) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty F(k) \sin(2\pi x k) \mathrm{d} k.\]</div>
-<p>If the transform cannot be computed in closed form, this
-function returns an unevaluated InverseSineTransform object.</p>
-<p>For a description of possible hints, refer to the docstring of
-<tt class="xref py py-func docutils literal"><span class="pre">sympy.integrals.transforms.IntegralTransform.doit()</span></tt>.
-Note that for this transform, by default <tt class="docutils literal"><span class="pre">noconds=True</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">inverse_sine_transform</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">a</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">inverse_sine_transform</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="p">((</span><span class="mi">1</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">k</span><span class="o">**</span><span class="p">(</span><span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span>
-<span class="gp">... </span>    <span class="n">gamma</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">gamma</span><span class="p">((</span><span class="n">a</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">),</span> <span class="n">k</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x**(-a)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">inverse_sine_transform</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">k</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">k</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">a</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">**</span><span class="mi">3</span><span class="p">),</span> <span class="n">k</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x*exp(-a*x**2)</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.integrals.transforms.fourier_transform" title="sympy.integrals.transforms.fourier_transform"><tt class="xref py py-obj docutils literal"><span class="pre">fourier_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_fourier_transform" title="sympy.integrals.transforms.inverse_fourier_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_fourier_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.sine_transform" title="sympy.integrals.transforms.sine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">sine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.cosine_transform" title="sympy.integrals.transforms.cosine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">cosine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_cosine_transform" title="sympy.integrals.transforms.inverse_cosine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_cosine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.hankel_transform" title="sympy.integrals.transforms.hankel_transform"><tt class="xref py py-obj docutils literal"><span class="pre">hankel_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_hankel_transform" title="sympy.integrals.transforms.inverse_hankel_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_hankel_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.mellin_transform" title="sympy.integrals.transforms.mellin_transform"><tt class="xref py py-obj docutils literal"><span class="pre">mellin_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.laplace_transform" title="sympy.integrals.transforms.laplace_transform"><tt class="xref py py-obj docutils literal"><span class="pre">laplace_transform</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.integrals.transforms.cosine_transform">
-<tt class="descclassname">sympy.integrals.transforms.</tt><tt class="descname">cosine_transform</tt><big>(</big><em>f</em>, <em>x</em>, <em>k</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/integrals/transforms.html#cosine_transform"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.integrals.transforms.cosine_transform" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the unitary, ordinary-frequency cosine transform of <span class="math">\(f\)</span>, defined
-as</p>
-<div class="math">
-\[F(k) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty f(x) \cos(2\pi x k) \mathrm{d} x.\]</div>
-<p>If the transform cannot be computed in closed form, this
-function returns an unevaluated CosineTransform object.</p>
-<p>For a description of possible hints, refer to the docstring of
-<tt class="xref py py-func docutils literal"><span class="pre">sympy.integrals.transforms.IntegralTransform.doit()</span></tt>.
-Note that for this transform, by default <tt class="docutils literal"><span class="pre">noconds=True</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">cosine_transform</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">a</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cosine_transform</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-<span class="go">sqrt(2)*a/(sqrt(pi)*(a**2 + k**2))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cosine_transform</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-<span class="go">a*(-sinh(a**2/(2*k)) + cosh(a**2/(2*k)))/(2*k**(3/2))</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.integrals.transforms.fourier_transform" title="sympy.integrals.transforms.fourier_transform"><tt class="xref py py-obj docutils literal"><span class="pre">fourier_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_fourier_transform" title="sympy.integrals.transforms.inverse_fourier_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_fourier_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.sine_transform" title="sympy.integrals.transforms.sine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">sine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_sine_transform" title="sympy.integrals.transforms.inverse_sine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_sine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_cosine_transform" title="sympy.integrals.transforms.inverse_cosine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_cosine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.hankel_transform" title="sympy.integrals.transforms.hankel_transform"><tt class="xref py py-obj docutils literal"><span class="pre">hankel_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_hankel_transform" title="sympy.integrals.transforms.inverse_hankel_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_hankel_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.mellin_transform" title="sympy.integrals.transforms.mellin_transform"><tt class="xref py py-obj docutils literal"><span class="pre">mellin_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.laplace_transform" title="sympy.integrals.transforms.laplace_transform"><tt class="xref py py-obj docutils literal"><span class="pre">laplace_transform</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.integrals.transforms.inverse_cosine_transform">
-<tt class="descclassname">sympy.integrals.transforms.</tt><tt class="descname">inverse_cosine_transform</tt><big>(</big><em>F</em>, <em>k</em>, <em>x</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/integrals/transforms.html#inverse_cosine_transform"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.integrals.transforms.inverse_cosine_transform" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the unitary, ordinary-frequency inverse cosine transform of <span class="math">\(F\)</span>,
-defined as</p>
-<div class="math">
-\[f(x) = \sqrt{\frac{2}{\pi}} \int_{0}^\infty F(k) \cos(2\pi x k) \mathrm{d} k.\]</div>
-<p>If the transform cannot be computed in closed form, this
-function returns an unevaluated InverseCosineTransform object.</p>
-<p>For a description of possible hints, refer to the docstring of
-<tt class="xref py py-func docutils literal"><span class="pre">sympy.integrals.transforms.IntegralTransform.doit()</span></tt>.
-Note that for this transform, by default <tt class="docutils literal"><span class="pre">noconds=True</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">inverse_cosine_transform</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">a</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">inverse_cosine_transform</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">a</span><span class="o">/</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">a</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">k</span><span class="o">**</span><span class="mi">2</span><span class="p">)),</span> <span class="n">k</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">-sinh(a*x) + cosh(a*x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">inverse_cosine_transform</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">k</span><span class="p">),</span> <span class="n">k</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">1/sqrt(x)</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.integrals.transforms.fourier_transform" title="sympy.integrals.transforms.fourier_transform"><tt class="xref py py-obj docutils literal"><span class="pre">fourier_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_fourier_transform" title="sympy.integrals.transforms.inverse_fourier_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_fourier_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.sine_transform" title="sympy.integrals.transforms.sine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">sine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_sine_transform" title="sympy.integrals.transforms.inverse_sine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_sine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.cosine_transform" title="sympy.integrals.transforms.cosine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">cosine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.hankel_transform" title="sympy.integrals.transforms.hankel_transform"><tt class="xref py py-obj docutils literal"><span class="pre">hankel_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_hankel_transform" title="sympy.integrals.transforms.inverse_hankel_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_hankel_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.mellin_transform" title="sympy.integrals.transforms.mellin_transform"><tt class="xref py py-obj docutils literal"><span class="pre">mellin_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.laplace_transform" title="sympy.integrals.transforms.laplace_transform"><tt class="xref py py-obj docutils literal"><span class="pre">laplace_transform</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.integrals.transforms.hankel_transform">
-<tt class="descclassname">sympy.integrals.transforms.</tt><tt class="descname">hankel_transform</tt><big>(</big><em>f</em>, <em>r</em>, <em>k</em>, <em>nu</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/integrals/transforms.html#hankel_transform"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.integrals.transforms.hankel_transform" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the Hankel transform of <span class="math">\(f\)</span>, defined as</p>
-<div class="math">
-\[F_\nu(k) = \int_{0}^\infty f(r) J_\nu(k r) r \mathrm{d} r.\]</div>
-<p>If the transform cannot be computed in closed form, this
-function returns an unevaluated <tt class="xref py py-class docutils literal"><span class="pre">HankelTransform</span></tt> object.</p>
-<p>For a description of possible hints, refer to the docstring of
-<tt class="xref py py-func docutils literal"><span class="pre">sympy.integrals.transforms.IntegralTransform.doit()</span></tt>.
-Note that for this transform, by default <tt class="docutils literal"><span class="pre">noconds=True</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">hankel_transform</span><span class="p">,</span> <span class="n">inverse_hankel_transform</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">cosh</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">r</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="n">a</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ht</span> <span class="o">=</span> <span class="n">hankel_transform</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">r</span><span class="o">**</span><span class="n">m</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">nu</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ht</span>
-<span class="go">2**(-m + 1)*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">inverse_hankel_transform</span><span class="p">(</span><span class="n">ht</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">nu</span><span class="p">)</span>
-<span class="go">r**(-m)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ht</span> <span class="o">=</span> <span class="n">hankel_transform</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="o">*</span><span class="n">r</span><span class="p">),</span> <span class="n">r</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ht</span>
-<span class="go">a/(k**3*(a**2/k**2 + 1)**(3/2))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">inverse_hankel_transform</span><span class="p">(</span><span class="n">ht</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">-sinh(a*r) + cosh(a*r)</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.integrals.transforms.fourier_transform" title="sympy.integrals.transforms.fourier_transform"><tt class="xref py py-obj docutils literal"><span class="pre">fourier_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_fourier_transform" title="sympy.integrals.transforms.inverse_fourier_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_fourier_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.sine_transform" title="sympy.integrals.transforms.sine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">sine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_sine_transform" title="sympy.integrals.transforms.inverse_sine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_sine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.cosine_transform" title="sympy.integrals.transforms.cosine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">cosine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_cosine_transform" title="sympy.integrals.transforms.inverse_cosine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_cosine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_hankel_transform" title="sympy.integrals.transforms.inverse_hankel_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_hankel_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.mellin_transform" title="sympy.integrals.transforms.mellin_transform"><tt class="xref py py-obj docutils literal"><span class="pre">mellin_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.laplace_transform" title="sympy.integrals.transforms.laplace_transform"><tt class="xref py py-obj docutils literal"><span class="pre">laplace_transform</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.integrals.transforms.inverse_hankel_transform">
-<tt class="descclassname">sympy.integrals.transforms.</tt><tt class="descname">inverse_hankel_transform</tt><big>(</big><em>F</em>, <em>k</em>, <em>r</em>, <em>nu</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/integrals/transforms.html#inverse_hankel_transform"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.integrals.transforms.inverse_hankel_transform" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the inverse Hankel transform of <span class="math">\(F\)</span> defined as</p>
-<div class="math">
-\[f(r) = \int_{0}^\infty F_\nu(k) J_\nu(k r) k \mathrm{d} k.\]</div>
-<p>If the transform cannot be computed in closed form, this
-function returns an unevaluated <tt class="xref py py-class docutils literal"><span class="pre">InverseHankelTransform</span></tt> object.</p>
-<p>For a description of possible hints, refer to the docstring of
-<tt class="xref py py-func docutils literal"><span class="pre">sympy.integrals.transforms.IntegralTransform.doit()</span></tt>.
-Note that for this transform, by default <tt class="docutils literal"><span class="pre">noconds=True</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">hankel_transform</span><span class="p">,</span> <span class="n">inverse_hankel_transform</span><span class="p">,</span> <span class="n">gamma</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">cosh</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">r</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="n">a</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ht</span> <span class="o">=</span> <span class="n">hankel_transform</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">r</span><span class="o">**</span><span class="n">m</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">nu</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ht</span>
-<span class="go">2**(-m + 1)*k**(m - 2)*gamma(-m/2 + nu/2 + 1)/gamma(m/2 + nu/2)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">inverse_hankel_transform</span><span class="p">(</span><span class="n">ht</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">nu</span><span class="p">)</span>
-<span class="go">r**(-m)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ht</span> <span class="o">=</span> <span class="n">hankel_transform</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="o">*</span><span class="n">r</span><span class="p">),</span> <span class="n">r</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ht</span>
-<span class="go">a/(k**3*(a**2/k**2 + 1)**(3/2))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">inverse_hankel_transform</span><span class="p">(</span><span class="n">ht</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">-sinh(a*r) + cosh(a*r)</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.integrals.transforms.fourier_transform" title="sympy.integrals.transforms.fourier_transform"><tt class="xref py py-obj docutils literal"><span class="pre">fourier_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_fourier_transform" title="sympy.integrals.transforms.inverse_fourier_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_fourier_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.sine_transform" title="sympy.integrals.transforms.sine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">sine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_sine_transform" title="sympy.integrals.transforms.inverse_sine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_sine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.cosine_transform" title="sympy.integrals.transforms.cosine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">cosine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.inverse_cosine_transform" title="sympy.integrals.transforms.inverse_cosine_transform"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_cosine_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.hankel_transform" title="sympy.integrals.transforms.hankel_transform"><tt class="xref py py-obj docutils literal"><span class="pre">hankel_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.mellin_transform" title="sympy.integrals.transforms.mellin_transform"><tt class="xref py py-obj docutils literal"><span class="pre">mellin_transform</span></tt></a>, <a class="reference internal" href="#sympy.integrals.transforms.laplace_transform" title="sympy.integrals.transforms.laplace_transform"><tt class="xref py py-obj docutils literal"><span class="pre">laplace_transform</span></tt></a></p>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="internals">
-<h2>Internals<a class="headerlink" href="#internals" title="Permalink to this headline">¶</a></h2>
-<p>There is a general method for calculating antiderivatives of elementary functions, called the Risch algorithm. The Risch algorithm is a decision procedure that can determine whether an elementary solution exists, and in that case calculate it. It can be extended to handle many nonelementary functions in addition to the elementary ones.</p>
-<p>SymPy currently uses a simplified version of the Risch algorithm, called the Risch-Norman algorithm. This algorithm is much faster, but may fail to find an antiderivative, although it is still very powerful. SymPy also uses pattern matching and heuristics to speed up evaluation of some types of integrals, e.g. polynomials.</p>
-<p>For non-elementary definite integrals, sympy uses so-called Meijer G-functions.
-Details are described here:</p>
-<div class="toctree-wrapper compound">
-<ul>
-<li class="toctree-l1"><a class="reference internal" href="../integrals/g-functions.html">Computing Integrals using Meijer G-Functions</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../integrals/g-functions.html#the-g-function-integration-theorems">The G-Function Integration Theorems</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../integrals/g-functions.html#the-inverse-laplace-transform-of-a-g-function">The Inverse Laplace Transform of a G-function</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../integrals/g-functions.html#implemented-g-function-formulae">Implemented G-Function Formulae</a></li>
-</ul>
-</div>
-</div>
-<div class="section" id="api-reference">
-<h2>API reference<a class="headerlink" href="#api-reference" title="Permalink to this headline">¶</a></h2>
-<dl class="method">
-<dt id="sympy.integrals.integrate">
-<tt class="descclassname">integrals.</tt><tt class="descname">integrate</tt><big>(</big><em>f</em>, <em>var</em>, <em>...</em><big>)</big><a class="headerlink" href="#sympy.integrals.integrate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute definite or indefinite integral of one or more variables
-using Risch-Norman algorithm and table lookup. This procedure is
-able to handle elementary algebraic and transcendental functions
-and also a huge class of special functions, including Airy,
-Bessel, Whittaker and Lambert.</p>
-<p>var can be:</p>
-<ul>
-<li><p class="first">a symbol                   &#8211; indefinite integration</p>
-</li>
-<li><dl class="first docutils">
-<dt>a tuple (symbol, a)        &#8211; indefinite integration with result</dt>
-<dd><p class="first last">given with <span class="math">\(a\)</span> replacing <span class="math">\(symbol\)</span></p>
-</dd>
-</dl>
-</li>
-<li><p class="first">a tuple (symbol, a, b)     &#8211; definite integration</p>
-</li>
-</ul>
-<p>Several variables can be specified, in which case the result is
-multiple integration. (If var is omitted and the integrand is
-univariate, the indefinite integral in that variable will be performed.)</p>
-<p>Indefinite integrals are returned without terms that are independent
-of the integration variables. (see examples)</p>
-<p>Definite improper integrals often entail delicate convergence
-conditions. Pass conds=&#8217;piecewise&#8217;, &#8216;separate&#8217; or &#8216;none&#8217; to have
-these returned, respectively, as a Piecewise function, as a separate
-result (i.e. result will be a tuple), or not at all (default is
-&#8216;piecewise&#8217;).</p>
-<p><strong>Strategy</strong></p>
-<p>SymPy uses various approaches to definite integration. One method is to
-find an antiderivative for the integrand, and then use the fundamental
-theorem of calculus. Various functions are implemented to integrate
-polynomial, rational and trigonometric functions, and integrands
-containing DiracDelta terms.</p>
-<p>SymPy also implements the part of the Risch algorithm, which is a decision
-procedure for integrating elementary functions, i.e., the algorithm can
-either find an elementary antiderivative, or prove that one does not
-exist.  There is also a (very successful, albeit somewhat slow) general
-implementation of the heuristic Risch algorithm.  This algorithm will
-eventually be phased out as more of the full Risch algorithm is
-implemented. See the docstring of Integral._eval_integral() for more
-details on computing the antiderivative using algebraic methods.</p>
-<p>The option risch=True can be used to use only the (full) Risch algorithm.
-This is useful if you want to know if an elementary function has an
-elementary antiderivative.  If the indefinite Integral returned by this
-function is an instance of NonElementaryIntegral, that means that the
-Risch algorithm has proven that integral to be non-elementary.  Note that
-by default, additional methods (such as the Meijer G method outlined
-below) are tried on these integrals, as they may be expressible in terms
-of special functions, so if you only care about elementary answers, use
-risch=True.  Also note that an unevaluated Integral returned by this
-function is not necessarily a NonElementaryIntegral, even with risch=True,
-as it may just be an indication that the particular part of the Risch
-algorithm needed to integrate that function is not yet implemented.</p>
-<p>Another family of strategies comes from re-writing the integrand in
-terms of so-called Meijer G-functions. Indefinite integrals of a
-single G-function can always be computed, and the definite integral
-of a product of two G-functions can be computed from zero to
-infinity. Various strategies are implemented to rewrite integrands
-as G-functions, and use this information to compute integrals (see
-the <tt class="docutils literal"><span class="pre">meijerint</span></tt> module).</p>
-<p>In general, the algebraic methods work best for computing
-antiderivatives of (possibly complicated) combinations of elementary
-functions. The G-function methods work best for computing definite
-integrals from zero to infinity of moderately complicated
-combinations of special functions, or indefinite integrals of very
-simple combinations of special functions.</p>
-<p>The strategy employed by the integration code is as follows:</p>
-<ul class="simple">
-<li>If computing a definite integral, and both limits are real,
-and at least one limit is +- oo, try the G-function method of
-definite integration first.</li>
-<li>Try to find an antiderivative, using all available methods, ordered
-by performance (that is try fastest method first, slowest last; in
-particular polynomial integration is tried first, meijer
-g-functions second to last, and heuristic risch last).</li>
-<li>If still not successful, try G-functions irrespective of the
-limits.</li>
-</ul>
-<p>The option meijerg=True, False, None can be used to, respectively:
-always use G-function methods and no others, never use G-function
-methods, or use all available methods (in order as described above).
-It defaults to None.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">Integral</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">Integral.doit</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">integrate</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x**2*y/2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x*log(x) - x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="p">))</span>
-<span class="go">a*log(a) - a + 1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">x**2/2</span>
-</pre></div>
-</div>
-<p>Terms that are independent of x are dropped by indefinite integration:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">2*(x + 1)**(3/2)/3 - 2/3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">2*(x + 1)**(3/2)/3</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">ValueError</span>: <span class="n">specify integration variables to integrate x*y</span>
-</pre></div>
-</div>
-<p>Note that <tt class="docutils literal"><span class="pre">integrate(x)</span></tt> syntax is meant only for convenience
-in interactive sessions and should be avoided in library code.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">a</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span> <span class="c"># same as conds=&#39;piecewise&#39;</span>
-<span class="go">Piecewise((gamma(a + 1), -re(a) &lt; 1),</span>
-<span class="go">    (Integral(x**a*exp(-x), (x, 0, oo)), True))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">a</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">),</span> <span class="n">conds</span><span class="o">=</span><span class="s">&#39;none&#39;</span><span class="p">)</span>
-<span class="go">gamma(a + 1)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">a</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">),</span> <span class="n">conds</span><span class="o">=</span><span class="s">&#39;separate&#39;</span><span class="p">)</span>
-<span class="go">(gamma(a + 1), -re(a) &lt; 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="method">
-<dt id="sympy.integrals.line_integrate">
-<tt class="descclassname">integrals.</tt><tt class="descname">line_integrate</tt><big>(</big><em>field</em>, <em>Curve</em>, <em>variables</em><big>)</big><a class="headerlink" href="#sympy.integrals.line_integrate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the line integral.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">integrate</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">Integral</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Curve</span><span class="p">,</span> <span class="n">line_integrate</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">ln</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">t</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Curve</span><span class="p">([</span><span class="n">E</span><span class="o">**</span><span class="n">t</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">E</span><span class="o">**</span><span class="n">t</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">ln</span><span class="p">(</span><span class="mi">2</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">line_integrate</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">])</span>
-<span class="go">3*sqrt(2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="method">
-<dt id="sympy.integrals.deltaintegrate">
-<tt class="descclassname">integrals.</tt><tt class="descname">deltaintegrate</tt><big>(</big><em>f</em>, <em>x</em><big>)</big><a class="headerlink" href="#sympy.integrals.deltaintegrate" title="Permalink to this definition">¶</a></dt>
-<dd><p>The idea for integration is the following:</p>
-<ul>
-<li><p class="first">If we are dealing with a DiracDelta expression, i.e. DiracDelta(g(x)),
-we try to simplify it.</p>
-<p>If we could simplify it, then we integrate the resulting expression.
-We already know we can integrate a simplified expression, because only
-simple DiracDelta expressions are involved.</p>
-<p>If we couldn&#8217;t simplify it, there are two cases:</p>
-<ol class="arabic simple">
-<li>The expression is a simple expression: we return the integral,
-taking care if we are dealing with a Derivative or with a proper
-DiracDelta.</li>
-<li>The expression is not simple (i.e. DiracDelta(cos(x))): we can do
-nothing at all.</li>
-</ol>
-</li>
-<li><p class="first">If the node is a multiplication node having a DiracDelta term:</p>
-<p>First we expand it.</p>
-<p>If the expansion did work, the we try to integrate the expansion.</p>
-<p>If not, we try to extract a simple DiracDelta term, then we have two
-cases:</p>
-<ol class="arabic simple">
-<li>We have a simple DiracDelta term, so we return the integral.</li>
-<li>We didn&#8217;t have a simple term, but we do have an expression with
-simplified DiracDelta terms, so we integrate this expression.</li>
-</ol>
-</li>
-</ul>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="../functions/special.html#sympy.functions.special.delta_functions.DiracDelta" title="sympy.functions.special.delta_functions.DiracDelta"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.functions.special.delta_functions.DiracDelta</span></tt></a>, <tt class="xref py py-obj docutils literal"><span class="pre">sympy.integrals.integrals.Integral</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.integrals.deltafunctions</span> <span class="kn">import</span> <span class="n">deltaintegrate</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">DiracDelta</span><span class="p">,</span> <span class="n">Heaviside</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">deltaintegrate</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">DiracDelta</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">sin(1)*cos(1)*Heaviside(x - 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">deltaintegrate</span><span class="p">(</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">DiracDelta</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">DiracDelta</span><span class="p">(</span><span class="n">y</span> <span class="o">-</span> <span class="n">z</span><span class="p">),</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">z**2*DiracDelta(x - z)*Heaviside(y - z)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="method">
-<dt id="sympy.integrals.ratint">
-<tt class="descclassname">integrals.</tt><tt class="descname">ratint</tt><big>(</big><em>f</em>, <em>x</em>, <em>**flags</em><big>)</big><a class="headerlink" href="#sympy.integrals.ratint" title="Permalink to this definition">¶</a></dt>
-<dd><p>Performs indefinite integration of rational functions.</p>
-<p>Given a field <span class="math">\(K\)</span> and a rational function <span class="math">\(f = p/q\)</span>,
-where <span class="math">\(p\)</span> and <span class="math">\(q\)</span> are polynomials in <span class="math">\(K[x]\)</span>,
-returns a function <span class="math">\(g\)</span> such that <span class="math">\(f = g'\)</span>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.integrals.rationaltools</span> <span class="kn">import</span> <span class="n">ratint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ratint</span><span class="p">(</span><span class="mi">36</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">5</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">(12*x + 6)/(x**2 - 1) + 4*log(x - 2) - 4*log(x + 1)</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.integrals.integrals.Integral.doit</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">ratint_logpart</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">ratint_ratpart</span></tt></p>
-</div>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="bro05" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id1">[Bro05]</a></td><td>M. Bronstein, Symbolic Integration I: Transcendental
-Functions, Second Edition, Springer-Verlag, 2005, pp. 35-70</td></tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="method">
-<dt id="sympy.integrals.heurisch">
-<tt class="descclassname">integrals.</tt><tt class="descname">heurisch</tt><big>(</big><em>f</em>, <em>x</em>, <em>rewrite=False</em>, <em>hints=None</em>, <em>mappings=None</em>, <em>retries=3</em><big>)</big><a class="headerlink" href="#sympy.integrals.heurisch" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute indefinite integral using heuristic Risch algorithm.</p>
-<p>This is a heuristic approach to indefinite integration in finite
-terms using the extended heuristic (parallel) Risch algorithm, based
-on Manuel Bronstein&#8217;s &#8220;Poor Man&#8217;s Integrator&#8221;.</p>
-<p>The algorithm supports various classes of functions including
-transcendental elementary or special functions like Airy,
-Bessel, Whittaker and Lambert.</p>
-<p>Note that this algorithm is not a decision procedure. If it isn&#8217;t
-able to compute the antiderivative for a given function, then this is
-not a proof that such a functions does not exist.  One should use
-recursive Risch algorithm in such case.  It&#8217;s an open question if
-this algorithm can be made a full decision procedure.</p>
-<p>This is an internal integrator procedure. You should use toplevel
-&#8216;integrate&#8217; function in most cases,  as this procedure needs some
-preprocessing steps and otherwise may fail.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.integrals.integrals.Integral.doit</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">sympy.integrals.integrals.Integral</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">components</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">tan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.integrals.heurisch</span> <span class="kn">import</span> <span class="n">heurisch</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">heurisch</span><span class="p">(</span><span class="n">y</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">y*log(tan(x)**2 + 1)/2</span>
-</pre></div>
-</div>
-<p>See Manuel Bronstein&#8217;s &#8220;Poor Man&#8217;s Integrator&#8221;:</p>
-<p>[1] <a class="reference external" href="http://www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/index.html">http://www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/index.html</a></p>
-<p>For more information on the implemented algorithm refer to:</p>
-<dl class="docutils">
-<dt>[2] K. Geddes, L. Stefanus, On the Risch-Norman Integration</dt>
-<dd>Method and its Implementation in Maple, Proceedings of
-ISSAC&#8216;89, ACM Press, 212-217.</dd>
-<dt>[3] J. H. Davenport, On the Parallel Risch Algorithm (I),</dt>
-<dd>Proceedings of EUROCAM&#8216;82, LNCS 144, Springer, 144-157.</dd>
-<dt>[4] J. H. Davenport, On the Parallel Risch Algorithm (III):</dt>
-<dd>Use of Tangents, SIGSAM Bulletin 16 (1982), 3-6.</dd>
-<dt>[5] J. H. Davenport, B. M. Trager, On the Parallel Risch</dt>
-<dd>Algorithm (II), ACM Transactions on Mathematical
-Software 11 (1985), 356-362.</dd>
-</dl>
-<p class="rubric">Specification</p>
-<p>heurisch(f, x, rewrite=False, hints=None)</p>
-<blockquote>
-<div><dl class="docutils">
-<dt>where</dt>
-<dd><p class="first">f : expression
-x : symbol</p>
-<p>rewrite -&gt; force rewrite &#8216;f&#8217; in terms of &#8216;tan&#8217; and &#8216;tanh&#8217;
-hints   -&gt; a list of functions that may appear in anti-derivate</p>
-<blockquote class="last">
-<div><ul class="simple">
-<li>hints = None          &#8211;&gt; no suggestions at all</li>
-<li>hints = [ ]           &#8211;&gt; try to figure out</li>
-<li>hints = [f1, ..., fn] &#8211;&gt; we know better</li>
-</ul>
-</div></blockquote>
-</dd>
-</dl>
-</div></blockquote>
-</dd></dl>
-
-<dl class="method">
-<dt id="sympy.integrals.trigintegrate">
-<tt class="descclassname">integrals.</tt><tt class="descname">trigintegrate</tt><big>(</big><em>f</em>, <em>x</em><big>)</big><a class="headerlink" href="#sympy.integrals.trigintegrate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Integrate f = Mul(trig) over x</p>
-<blockquote>
-<div><div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">tan</span><span class="p">,</span> <span class="n">sec</span><span class="p">,</span> <span class="n">csc</span><span class="p">,</span> <span class="n">cot</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.integrals.trigonometry</span> <span class="kn">import</span> <span class="n">trigintegrate</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">trigintegrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">sin(x)**2/2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">trigintegrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x/2 - sin(x)*cos(x)/2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">trigintegrate</span><span class="p">(</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sec</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">1/cos(x)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">trigintegrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">-log(sin(x) - 1)/2 + log(sin(x) + 1)/2 - sin(x)</span>
-</pre></div>
-</div>
-<p><a class="reference external" href="http://en.wikibooks.org/wiki/Calculus/Integration_techniques">http://en.wikibooks.org/wiki/Calculus/Integration_techniques</a></p>
-</div></blockquote>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.integrals.integrals.Integral.doit</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">sympy.integrals.integrals.Integral</span></tt></p>
-</div>
-</dd></dl>
-
-<p>Class Integral represents an unevaluated integral and has some methods that help in the integration of an expression.</p>
-<dl class="class">
-<dt id="sympy.integrals.Integral">
-<em class="property">class </em><tt class="descclassname">sympy.integrals.</tt><tt class="descname">Integral</tt><a class="headerlink" href="#sympy.integrals.Integral" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents unevaluated integral.</p>
-<dl class="data">
-<dt id="sympy.integrals.transforms.Integral.is_commutative">
-<tt class="descname">is_commutative</tt><a class="headerlink" href="#sympy.integrals.transforms.Integral.is_commutative" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns whether all the free symbols in the integral are commutative.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.integrals.Integral.as_dummy">
-<tt class="descname">as_dummy</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.integrals.Integral.as_dummy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Replace instances of the integration variables with their dummy
-counterparts to make clear what are dummy variables and what
-are real-world symbols in an Integral.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">as_dummy</span><span class="p">()</span>
-<span class="go">Integral(_x, (_x, x, _y), (_y, x, y))</span>
-</pre></div>
-</div>
-<p>The &#8220;integral at&#8221; limit that has a length of 1 is not treated as
-though the integration symbol is a dummy, but the explicit form
-of length 2 does treat the integration variable as a dummy.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">as_dummy</span><span class="p">()</span>
-<span class="go">Integral(x, x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">as_dummy</span><span class="p">()</span>
-<span class="go">Integral(_x, (_x, x))</span>
-</pre></div>
-</div>
-<p>If there were no dummies in the original expression, then the
-output of this function will show which symbols cannot be
-changed by subs(), those with an underscore prefix.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">variables</span></tt></dt>
-<dd>Lists the integration variables</dd>
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">transform</span></tt></dt>
-<dd>Perform mapping on the integration variable</dd>
-</dl>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.integrals.Integral.as_sum">
-<tt class="descname">as_sum</tt><big>(</big><em>self</em>, <em>n</em>, <em>method='midpoint'</em><big>)</big><a class="headerlink" href="#sympy.integrals.Integral.as_sum" title="Permalink to this definition">¶</a></dt>
-<dd><p>Approximates the integral by a sum.</p>
-<p>method ... one of: left, right, midpoint</p>
-<p>This is basically just the rectangle method [1], the only difference is
-where the function value is taken in each interval.</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Rectangle_method">http://en.wikipedia.org/wiki/Rectangle_method</a></p>
-<p><strong>method = midpoint</strong>:</p>
-<p>Uses the n-order midpoint rule to evaluate the integral.</p>
-<p>Midpoint rule uses rectangles approximation for the given area (e.g.
-definite integral) of the function with heights equal to the point on
-the curve exactly in the middle of each interval (thus midpoint
-method). See [1] for more information.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">Integral.doit</span></tt></dt>
-<dd>Perform the integration using any hints</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">Integral</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="o">+</span><span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span>
-<span class="go">Integral(sqrt(x**3 + 1), (x, 2, 10))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">as_sum</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">&quot;left&quot;</span><span class="p">)</span>
-<span class="go">6 + 2*sqrt(65) + 2*sqrt(217) + 6*sqrt(57)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">as_sum</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">&quot;left&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">n</span><span class="p">()</span>
-<span class="go">96.8853618335341</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">n</span><span class="p">()</span>
-<span class="go">124.616199194723</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.integrals.Integral.doit">
-<tt class="descname">doit</tt><big>(</big><em>self</em>, <em>**hints</em><big>)</big><a class="headerlink" href="#sympy.integrals.Integral.doit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Perform the integration using any hints given.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p><tt class="xref py py-obj docutils literal"><span class="pre">sympy.integrals.trigonometry.trigintegrate</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">sympy.integrals.risch.heurisch</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">sympy.integrals.rationaltools.ratint</span></tt></p>
-<dl class="last docutils">
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">as_sum</span></tt></dt>
-<dd>Approximate the integral using a sum</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">i</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-<span class="go">x**3/log(x) - x/log(x)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.integrals.Integral.free_symbols">
-<tt class="descname">free_symbols</tt><a class="headerlink" href="#sympy.integrals.Integral.free_symbols" title="Permalink to this definition">¶</a></dt>
-<dd><p>This method returns the symbols that will exist when the
-integral is evaluated. This is useful if one is trying to
-determine whether an integral depends on a certain
-symbol or not.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">function</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">limits</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">variables</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">free_symbols</span>
-<span class="go">set([y])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.integrals.Integral.function">
-<tt class="descname">function</tt><a class="headerlink" href="#sympy.integrals.Integral.function" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the function to be integrated.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">limits</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">variables</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">free_symbols</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,))</span><span class="o">.</span><span class="n">function</span>
-<span class="go">x**2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.integrals.Integral.is_number">
-<tt class="descname">is_number</tt><a class="headerlink" href="#sympy.integrals.Integral.is_number" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if the Integral will result in a number, else False.</p>
-<p>sympy considers anything that will result in a number to have
-is_number == True.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">log</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>Integrals are a special case since they contain symbols that can
-be replaced with numbers. Whether the integral can be done or not is
-another issue. But answering whether the final result is a number is
-not difficult.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">is_number</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">is_zero</span></tt></p>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.integrals.Integral.is_zero">
-<tt class="descname">is_zero</tt><a class="headerlink" href="#sympy.integrals.Integral.is_zero" title="Permalink to this definition">¶</a></dt>
-<dd><p>Since Integral doesn&#8217;t autosimplify it it useful to see if
-it would simplify to zero or not in a trivial manner, i.e. when
-the function is 0 or two limits of a definite integral are the same.</p>
-<p>This is a very naive and quick test, not intended to check for special
-patterns like Integral(sin(m*x)*cos(n*x), (x, 0, 2*pi)) == 0.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">is_number</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">is_zero</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">))</span><span class="o">.</span><span class="n">is_zero</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">is_zero</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.integrals.Integral.limits">
-<tt class="descname">limits</tt><a class="headerlink" href="#sympy.integrals.Integral.limits" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the limits of integration.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">function</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">variables</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">free_symbols</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">i</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span><span class="o">.</span><span class="n">limits</span>
-<span class="go">((i, 1, 3),)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.integrals.Integral.transform">
-<tt class="descname">transform</tt><big>(</big><em>self</em>, <em>x</em>, <em>u</em>, <em>inverse=False</em><big>)</big><a class="headerlink" href="#sympy.integrals.Integral.transform" title="Permalink to this definition">¶</a></dt>
-<dd><p>Performs a change of variables from <span class="math">\(x\)</span> to <span class="math">\(u\)</span> using the relationship
-given by <span class="math">\(x\)</span> and <span class="math">\(u\)</span> which will define the transformations <span class="math">\(f\)</span> and <span class="math">\(F\)</span>
-(which are inverses of each other) as follows:</p>
-<ol class="arabic simple">
-<li>If <span class="math">\(x\)</span> is a Symbol (which is a variable of integration) then <span class="math">\(u\)</span>
-will be interpreted as some function, f(u), with inverse F(u).
-This, in effect, just makes the substitution of x with f(x).</li>
-<li>If <span class="math">\(u\)</span> is a Symbol then <span class="math">\(x\)</span> will be interpreted as some function,
-F(x), with inverse f(u). This is commonly referred to as
-u-substitution.</li>
-</ol>
-<p>The <span class="math">\(inverse\)</span> option will reverse <span class="math">\(x\)</span> and <span class="math">\(u\)</span>. It is a deprecated option
-since <span class="math">\(x\)</span> and <span class="math">\(u\)</span> can just be passed in reverse order.</p>
-<p>Once f and F have been identified, the transformation is made as
-follows:</p>
-<div class="math">
-\[\int_a^b x \mathrm{d}x \rightarrow \int_{F(a)}^{F(b)} f(x)
-\frac{\mathrm{d}}{\mathrm{d}x}\]</div>
-<p>where <span class="math">\(F(x)\)</span> is the inverse of <span class="math">\(f(x)\)</span> and the limits and integrand have
-been corrected so as to retain the same value after integration.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">variables</span></tt></dt>
-<dd>Lists the integration variables</dd>
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">as_dummy</span></tt></dt>
-<dd>Replace integration variables with dummy ones</dd>
-</dl>
-</div>
-<p class="rubric">Notes</p>
-<p>The mappings, F(x) or f(u), must lead to a unique integral. Linear
-or rational linear expression, <span class="math">\(2*x\)</span>, <span class="math">\(1/x\)</span> and <span class="math">\(sqrt(x)\)</span>, will
-always work; quadratic expressions like <span class="math">\(x**2 - 1\)</span> are acceptable
-as long as the resulting integrand does not depend on the sign of
-the solutions (see examples).</p>
-<p>The integral will be returned unchanged if <span class="math">\(x\)</span> is not a variable of
-integration.</p>
-<p><span class="math">\(x\)</span> must be (or contain) only one of of the integration variables. If
-<span class="math">\(u\)</span> has more than one free symbol then it should be sent as a tuple
-(<span class="math">\(u\)</span>, <span class="math">\(uvar\)</span>) where <span class="math">\(uvar\)</span> identifies which variable is replacing
-the integration variable.
-XXX can it contain another integration variable?</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sqrt</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-</pre></div>
-</div>
-<p>transform can change the variable of integration</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-<span class="go">Integral(u*cos(u**2 - 1), (u, 0, 1))</span>
-</pre></div>
-</div>
-<p>transform can perform u-substitution as long as a unique
-integrand is obtained:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-<span class="go">Integral(cos(u)/2, (u, -1, 0))</span>
-</pre></div>
-</div>
-<p>This attempt fails because x = +/-sqrt(u + 1) and the
-sign does not cancel out of the integrand:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">ValueError</span>:
-<span class="go">The mapping between F(x) and f(u) did not give a unique integrand.</span>
-</pre></div>
-</div>
-<p>transform can do a substitution. Here, the previous
-result is transformed back into the original expression
-using &#8220;u-substitution&#8221;:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ui</span> <span class="o">=</span> <span class="n">_</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span> <span class="o">==</span> <span class="n">i</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>We can accomplish the same with a regular substitution:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ui</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="n">i</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>If the <span class="math">\(x\)</span> does not contain a symbol of integration then
-the integral will be returned unchanged. Integral <span class="math">\(i\)</span> does
-not have an integration variable <span class="math">\(a\)</span> so no change is made:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">==</span> <span class="n">i</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>When <span class="math">\(u\)</span> has more than one free symbol the symbol that is
-replacing <span class="math">\(x\)</span> must be identified by passing <span class="math">\(u\)</span> as a tuple:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="n">a</span><span class="p">,</span> <span class="n">u</span><span class="p">))</span>
-<span class="go">Integral(a + u, (u, -a, -a + 1))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="n">a</span><span class="p">,</span> <span class="n">a</span><span class="p">))</span>
-<span class="go">Integral(a + u, (a, -u, -u + 1))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.integrals.Integral.variables">
-<tt class="descname">variables</tt><a class="headerlink" href="#sympy.integrals.Integral.variables" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a list of the integration variables.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p><tt class="xref py py-obj docutils literal"><span class="pre">function</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">limits</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">free_symbols</span></tt></p>
-<dl class="last docutils">
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">as_dummy</span></tt></dt>
-<dd>Replace integration variables with dummy ones</dd>
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">transform</span></tt></dt>
-<dd>Perform mapping on the integration variable</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">i</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span><span class="o">.</span><span class="n">variables</span>
-<span class="go">[i]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="todo-and-bugs">
-<h2>TODO and Bugs<a class="headerlink" href="#todo-and-bugs" title="Permalink to this headline">¶</a></h2>
-<p>There are still lots of functions that sympy does not know how to integrate. For bugs related to this module, see <a class="reference external" href="http://code.google.com/p/sympy/issues/list?q=label:Integration">http://code.google.com/p/sympy/issues/list?q=label:Integration</a></p>
-</div>
-</div>
-
-
-          </div>
-        </div>
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-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../index.html">
-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Integrals</a><ul>
-<li><a class="reference internal" href="#examples">Examples</a></li>
-<li><a class="reference internal" href="#module-sympy.integrals.transforms">Integral Transforms</a></li>
-<li><a class="reference internal" href="#internals">Internals</a><ul>
-</ul>
-</li>
-<li><a class="reference internal" href="#api-reference">API reference</a></li>
-<li><a class="reference internal" href="#todo-and-bugs">TODO and Bugs</a></li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../galgebra/latex_ex/latex_ex.html"
-                        title="previous chapter">Extended LaTeXModule for SymPy</a></p>
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-  <div class="section" id="module-sympy.logic">
-<span id="logic-module"></span><h1>Logic Module<a class="headerlink" href="#module-sympy.logic" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="introduction">
-<h2>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline">¶</a></h2>
-<p>The logic module for SymPy allows to form and manipulate logic expressions
-using symbolic and boolean values.</p>
-</div>
-<div class="section" id="forming-logical-expressions">
-<h2>Forming logical expressions<a class="headerlink" href="#forming-logical-expressions" title="Permalink to this headline">¶</a></h2>
-<p>You can build boolean expressions with the standard python operators <tt class="docutils literal"><span class="pre">&amp;</span></tt>
-(<tt class="xref py py-class docutils literal"><span class="pre">And</span></tt>), <tt class="docutils literal"><span class="pre">|</span></tt> (<tt class="xref py py-class docutils literal"><span class="pre">Or</span></tt>), <tt class="docutils literal"><span class="pre">~</span></tt> (<tt class="xref py py-class docutils literal"><span class="pre">Not</span></tt>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">|</span> <span class="p">(</span><span class="n">x</span> <span class="o">&amp;</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">Or(And(x, y), y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">|</span> <span class="n">y</span>
-<span class="go">Or(x, y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">~</span><span class="n">x</span>
-<span class="go">Not(x)</span>
-</pre></div>
-</div>
-<p>You can also form implications with <tt class="docutils literal"><span class="pre">&gt;&gt;</span></tt> and <tt class="docutils literal"><span class="pre">&lt;&lt;</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">&gt;&gt;</span> <span class="n">y</span>
-<span class="go">Implies(x, y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">&lt;&lt;</span> <span class="n">y</span>
-<span class="go">Implies(y, x)</span>
-</pre></div>
-</div>
-<p>Like most types in SymPy, Boolean expressions inherit from <tt class="xref py py-class docutils literal"><span class="pre">Basic</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">y</span> <span class="o">&amp;</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">({</span><span class="n">x</span><span class="p">:</span> <span class="bp">True</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="bp">True</span><span class="p">})</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">|</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">atoms</span><span class="p">()</span>
-<span class="go">set([x, y])</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="boolean-functions">
-<h2>Boolean functions<a class="headerlink" href="#boolean-functions" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.logic.boolalg.to_cnf">
-<tt class="descclassname">sympy.logic.boolalg.</tt><tt class="descname">to_cnf</tt><big>(</big><em>expr</em><big>)</big><a class="reference internal" href="../_modules/sympy/logic/boolalg.html#to_cnf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.logic.boolalg.to_cnf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a propositional logical sentence s to conjunctive normal form.
-That is, of the form ((A | ~B | ...) &amp; (B | C | ...) &amp; ...)</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.logic.boolalg</span> <span class="kn">import</span> <span class="n">to_cnf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">D</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">to_cnf</span><span class="p">(</span><span class="o">~</span><span class="p">(</span><span class="n">A</span> <span class="o">|</span> <span class="n">B</span><span class="p">)</span> <span class="o">|</span> <span class="n">D</span><span class="p">)</span>
-<span class="go">And(Or(D, Not(A)), Or(D, Not(B)))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.logic.boolalg.And">
-<em class="property">class </em><tt class="descclassname">sympy.logic.boolalg.</tt><tt class="descname">And</tt><a class="reference internal" href="../_modules/sympy/logic/boolalg.html#And"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.logic.boolalg.And" title="Permalink to this definition">¶</a></dt>
-<dd><p>Logical AND function.</p>
-<p>It evaluates its arguments in order, giving False immediately
-if any of them are False, and True if they are all True.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">&amp;</span> <span class="n">y</span>
-<span class="go">And(x, y)</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.logic.boolalg.Or">
-<em class="property">class </em><tt class="descclassname">sympy.logic.boolalg.</tt><tt class="descname">Or</tt><a class="reference internal" href="../_modules/sympy/logic/boolalg.html#Or"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.logic.boolalg.Or" title="Permalink to this definition">¶</a></dt>
-<dd><p>Logical OR function</p>
-<p>It evaluates its arguments in order, giving True immediately
-if any of them are True, and False if they are all False.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.logic.boolalg.Not">
-<em class="property">class </em><tt class="descclassname">sympy.logic.boolalg.</tt><tt class="descname">Not</tt><a class="reference internal" href="../_modules/sympy/logic/boolalg.html#Not"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.logic.boolalg.Not" title="Permalink to this definition">¶</a></dt>
-<dd><p>Logical Not function (negation)</p>
-<p>Note: De Morgan rules applied automatically</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.logic.boolalg.Xor">
-<em class="property">class </em><tt class="descclassname">sympy.logic.boolalg.</tt><tt class="descname">Xor</tt><a class="reference internal" href="../_modules/sympy/logic/boolalg.html#Xor"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.logic.boolalg.Xor" title="Permalink to this definition">¶</a></dt>
-<dd><p>Logical XOR (exclusive OR) function.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.logic.boolalg.Nand">
-<em class="property">class </em><tt class="descclassname">sympy.logic.boolalg.</tt><tt class="descname">Nand</tt><a class="reference internal" href="../_modules/sympy/logic/boolalg.html#Nand"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.logic.boolalg.Nand" title="Permalink to this definition">¶</a></dt>
-<dd><p>Logical NAND function.</p>
-<p>It evaluates its arguments in order, giving True immediately if any
-of them are False, and False if they are all True.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.logic.boolalg.Nor">
-<em class="property">class </em><tt class="descclassname">sympy.logic.boolalg.</tt><tt class="descname">Nor</tt><a class="reference internal" href="../_modules/sympy/logic/boolalg.html#Nor"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.logic.boolalg.Nor" title="Permalink to this definition">¶</a></dt>
-<dd><p>Logical NOR function.</p>
-<p>It evaluates its arguments in order, giving False immediately if any
-of them are True, and True if they are all False.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.logic.boolalg.Implies">
-<em class="property">class </em><tt class="descclassname">sympy.logic.boolalg.</tt><tt class="descname">Implies</tt><a class="reference internal" href="../_modules/sympy/logic/boolalg.html#Implies"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.logic.boolalg.Implies" title="Permalink to this definition">¶</a></dt>
-<dd><p>Logical implication.</p>
-<p>A implies B is equivalent to !A v B</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.logic.boolalg.Equivalent">
-<em class="property">class </em><tt class="descclassname">sympy.logic.boolalg.</tt><tt class="descname">Equivalent</tt><a class="reference internal" href="../_modules/sympy/logic/boolalg.html#Equivalent"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.logic.boolalg.Equivalent" title="Permalink to this definition">¶</a></dt>
-<dd><p>Equivalence relation.</p>
-<p>Equivalent(A, B) is True iff A and B are both True or both False</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.logic.boolalg.ITE">
-<em class="property">class </em><tt class="descclassname">sympy.logic.boolalg.</tt><tt class="descname">ITE</tt><a class="reference internal" href="../_modules/sympy/logic/boolalg.html#ITE"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.logic.boolalg.ITE" title="Permalink to this definition">¶</a></dt>
-<dd><p>If then else clause.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</div>
-<div class="section" id="inference">
-<h2>Inference<a class="headerlink" href="#inference" title="Permalink to this headline">¶</a></h2>
-<p>This module implements some inference routines in propositional logic.</p>
-<p>The function satisfiable will test that a given boolean expression is satisfiable,
-that is, you can assign values to the variables to make the sentence True.</p>
-<p>For example, the expression x &amp; ~x is not satisfiable, since there are no values
-for x that make this sentence True. On the other hand, (x | y) &amp; (x | ~y) &amp; (~x | y)
-is satisfiable with both x and y being True.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.logic.inference</span> <span class="kn">import</span> <span class="n">satisfiable</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">satisfiable</span><span class="p">(</span><span class="n">x</span> <span class="o">&amp;</span> <span class="o">~</span><span class="n">x</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">satisfiable</span><span class="p">((</span><span class="n">x</span> <span class="o">|</span> <span class="n">y</span><span class="p">)</span> <span class="o">&amp;</span> <span class="p">(</span><span class="n">x</span> <span class="o">|</span> <span class="o">~</span><span class="n">y</span><span class="p">)</span> <span class="o">&amp;</span> <span class="p">(</span><span class="o">~</span><span class="n">x</span> <span class="o">|</span> <span class="n">y</span><span class="p">))</span>
-<span class="go">{x: True, y: True}</span>
-</pre></div>
-</div>
-<p>As you see, when a sentence is satisfiable, it returns a model that makes that
-sentence True. If it is not satisfiable it will return False</p>
-<dl class="function">
-<dt id="sympy.logic.inference.satisfiable">
-<tt class="descclassname">sympy.logic.inference.</tt><tt class="descname">satisfiable</tt><big>(</big><em>expr</em>, <em>algorithm='dpll2'</em><big>)</big><a class="reference internal" href="../_modules/sympy/logic/inference.html#satisfiable"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.logic.inference.satisfiable" title="Permalink to this definition">¶</a></dt>
-<dd><p>Check satisfiability of a propositional sentence.
-Returns a model when it succeeds</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">A</span><span class="p">,</span> <span class="n">B</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.logic.inference</span> <span class="kn">import</span> <span class="n">satisfiable</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">satisfiable</span><span class="p">(</span><span class="n">A</span> <span class="o">&amp;</span> <span class="o">~</span><span class="n">B</span><span class="p">)</span>
-<span class="go">{A: True, B: False}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">satisfiable</span><span class="p">(</span><span class="n">A</span> <span class="o">&amp;</span> <span class="o">~</span><span class="n">A</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../index.html">
-              <img class="logo" src="../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Logic Module</a><ul>
-<li><a class="reference internal" href="#introduction">Introduction</a></li>
-<li><a class="reference internal" href="#forming-logical-expressions">Forming logical expressions</a></li>
-<li><a class="reference internal" href="#boolean-functions">Boolean functions</a></li>
-<li><a class="reference internal" href="#inference">Inference</a></li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="integrals/g-functions.html"
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-            
-  <div class="section" id="dense-matrices">
-<h1>Dense Matrices<a class="headerlink" href="#dense-matrices" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="matrix-class-reference">
-<h2>Matrix Class Reference<a class="headerlink" href="#matrix-class-reference" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.matrices.dense.MutableDenseMatrix">
-<em class="property">class </em><tt class="descclassname">sympy.matrices.dense.</tt><tt class="descname">MutableDenseMatrix</tt><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#MutableDenseMatrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.MutableDenseMatrix" title="Permalink to this definition">¶</a></dt>
-<dd><dl class="function">
-<dt id="sympy.matrices.dense.MutableDenseMatrix.col_del">
-<tt class="descname">col_del</tt><big>(</big><em>self</em>, <em>i</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#MutableDenseMatrix.col_del"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.MutableDenseMatrix.col_del" title="Permalink to this definition">¶</a></dt>
-<dd><p>Delete the given column.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">col</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">row_del</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">col_del</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span>
-<span class="go">[1, 0]</span>
-<span class="go">[0, 0]</span>
-<span class="go">[0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.MutableDenseMatrix.col_op">
-<tt class="descname">col_op</tt><big>(</big><em>self</em>, <em>j</em>, <em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#MutableDenseMatrix.col_op"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.MutableDenseMatrix.col_op" title="Permalink to this definition">¶</a></dt>
-<dd><p>In-place operation on col j using two-arg functor whose args are
-interpreted as (self[i, j], i).</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">col</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">row_op</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">col_op</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">v</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="n">v</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">]);</span> <span class="n">M</span>
-<span class="go">[1, 2, 0]</span>
-<span class="go">[0, 1, 0]</span>
-<span class="go">[0, 0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.MutableDenseMatrix.col_swap">
-<tt class="descname">col_swap</tt><big>(</big><em>self</em>, <em>i</em>, <em>j</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#MutableDenseMatrix.col_swap"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.MutableDenseMatrix.col_swap" title="Permalink to this definition">¶</a></dt>
-<dd><p>Swap the two given columns of the matrix in-place.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">col</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">row_swap</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span>
-<span class="go">[1, 0]</span>
-<span class="go">[1, 0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">col_swap</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span>
-<span class="go">[0, 1]</span>
-<span class="go">[0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.MutableDenseMatrix.copyin_list">
-<tt class="descname">copyin_list</tt><big>(</big><em>self</em>, <em>key</em>, <em>value</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#MutableDenseMatrix.copyin_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.MutableDenseMatrix.copyin_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Copy in elements from a list.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>key</strong> : slice</p>
-<blockquote>
-<div><p>The section of this matrix to replace.</p>
-</div></blockquote>
-<p><strong>value</strong> : iterable</p>
-<blockquote class="last">
-<div><p>The iterable to copy values from.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">copyin_matrix</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span><span class="p">[:</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span> <span class="c"># col</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span>
-<span class="go">[1, 0, 0]</span>
-<span class="go">[2, 1, 0]</span>
-<span class="go">[0, 0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span>
-<span class="go">[1, 0, 0]</span>
-<span class="go">[3, 4, 0]</span>
-<span class="go">[0, 0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.MutableDenseMatrix.copyin_matrix">
-<tt class="descname">copyin_matrix</tt><big>(</big><em>self</em>, <em>key</em>, <em>value</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#MutableDenseMatrix.copyin_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.MutableDenseMatrix.copyin_matrix" title="Permalink to this definition">¶</a></dt>
-<dd><p>Copy in values from a matrix into the given bounds.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>key</strong> : slice</p>
-<blockquote>
-<div><p>The section of this matrix to replace.</p>
-</div></blockquote>
-<p><strong>value</strong> : Matrix</p>
-<blockquote class="last">
-<div><p>The matrix to copy values from.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">copyin_list</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span><span class="p">[:</span><span class="mi">3</span><span class="p">,</span> <span class="p">:</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">M</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span>
-<span class="go">[0, 1, 0]</span>
-<span class="go">[2, 3, 0]</span>
-<span class="go">[4, 5, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">M</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span>
-<span class="go">[0, 0, 1]</span>
-<span class="go">[2, 2, 3]</span>
-<span class="go">[4, 4, 5]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.MutableDenseMatrix.fill">
-<tt class="descname">fill</tt><big>(</big><em>self</em>, <em>value</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#MutableDenseMatrix.fill"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.MutableDenseMatrix.fill" title="Permalink to this definition">¶</a></dt>
-<dd><p>Fill the matrix with the scalar value.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">zeros</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">ones</span></tt></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.MutableDenseMatrix.row_del">
-<tt class="descname">row_del</tt><big>(</big><em>self</em>, <em>i</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#MutableDenseMatrix.row_del"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.MutableDenseMatrix.row_del" title="Permalink to this definition">¶</a></dt>
-<dd><p>Delete the given row.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">row</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">col_del</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">row_del</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span>
-<span class="go">[1, 0, 0]</span>
-<span class="go">[0, 0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.MutableDenseMatrix.row_op">
-<tt class="descname">row_op</tt><big>(</big><em>self</em>, <em>i</em>, <em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#MutableDenseMatrix.row_op"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.MutableDenseMatrix.row_op" title="Permalink to this definition">¶</a></dt>
-<dd><p>In-place operation on row i using two-arg functor whose args are
-interpreted as (self[i, j], j).</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">row</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">col_op</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">row_op</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">v</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="n">v</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">j</span><span class="p">]);</span> <span class="n">M</span>
-<span class="go">[1, 0, 0]</span>
-<span class="go">[2, 1, 0]</span>
-<span class="go">[0, 0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.MutableDenseMatrix.row_swap">
-<tt class="descname">row_swap</tt><big>(</big><em>self</em>, <em>i</em>, <em>j</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#MutableDenseMatrix.row_swap"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.MutableDenseMatrix.row_swap" title="Permalink to this definition">¶</a></dt>
-<dd><p>Swap the two given rows of the matrix in-place.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">row</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">col_swap</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span>
-<span class="go">[0, 1]</span>
-<span class="go">[1, 0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">row_swap</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span>
-<span class="go">[1, 0]</span>
-<span class="go">[0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.MutableDenseMatrix.simplify">
-<tt class="descname">simplify</tt><big>(</big><em>self</em>, <em>ratio=1.7</em>, <em>measure=&lt;function count_ops at 0x10c198b00&gt;</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#MutableDenseMatrix.simplify"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.MutableDenseMatrix.simplify" title="Permalink to this definition">¶</a></dt>
-<dd><p>Applies simplify to the elements of a matrix in place.</p>
-<p>This is a shortcut for M.applyfunc(lambda x: simplify(x, ratio, measure))</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="../simplify/simplify.html#sympy.simplify.simplify.simplify" title="sympy.simplify.simplify.simplify"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.simplify.simplify.simplify</span></tt></a></p>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="immutablematrix-class-reference">
-<h2>ImmutableMatrix Class Reference<a class="headerlink" href="#immutablematrix-class-reference" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt>
-<em class="property">class </em><tt class="descclassname">sympy.matrices.immutable.</tt><tt class="descname">ImmutableMatrix</tt><a class="reference internal" href="../../_modules/sympy/matrices/immutable.html#ImmutableMatrix"><span class="viewcode-link">[source]</span></a></dt>
-<dd><p>Create an immutable version of a matrix.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">ImmutableMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ImmutableMatrix</span><span class="p">(</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">[1, 0, 0]</span>
-<span class="go">[0, 1, 0]</span>
-<span class="go">[0, 0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">42</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">TypeError</span>: <span class="n">Cannot set values of ImmutableDenseMatrix</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt>
-<tt class="descclassname">ImmutableMatrix.</tt><tt class="descname">C</tt></dt>
-<dd><p>By-element conjugation.</p>
-</dd></dl>
-
-<dl class="function">
-<dt>
-<tt class="descclassname">ImmutableMatrix.</tt><tt class="descname">adjoint</tt><big>(</big><em>self</em><big>)</big></dt>
-<dd><p>Conjugate transpose or Hermitian conjugation.</p>
-</dd></dl>
-
-<dl class="function">
-<dt>
-<tt class="descclassname">ImmutableMatrix.</tt><tt class="descname">as_mutable</tt><big>(</big><em>self</em><big>)</big></dt>
-<dd><p>Returns a mutable version of this matrix</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">ImmutableMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">ImmutableMatrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">as_mutable</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">5</span> <span class="c"># Can set values in Y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span>
-<span class="go">[1, 2]</span>
-<span class="go">[3, 5]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt>
-<tt class="descclassname">ImmutableMatrix.</tt><tt class="descname">equals</tt><big>(</big><em>self</em>, <em>other</em>, <em>failing_expression=False</em><big>)</big></dt>
-<dd><p>Applies <tt class="docutils literal"><span class="pre">equals</span></tt> to corresponding elements of the matrices,
-trying to prove that the elements are equivalent, returning True
-if they are, False if any pair is not, and None (or the first
-failing expression if failing_expression is True) if it cannot
-be decided if the expressions are equivalent or not. This is, in
-general, an expensive operation.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.core.expr.equals</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">==</span> <span class="n">B</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span> <span class="o">==</span> <span class="n">B</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
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-            <p class="logo"><a href="../../index.html">
-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Dense Matrices</a><ul>
-<li><a class="reference internal" href="#matrix-class-reference">Matrix Class Reference</a></li>
-<li><a class="reference internal" href="#immutablematrix-class-reference">ImmutableMatrix Class Reference</a></li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../matrices/matrices.html"
-                        title="previous chapter">Matrices (linear algebra)</a></p>
-  <h4>Next topic</h4>
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-                        title="next chapter">Sparse Matrices</a></p>
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-  <div class="section" id="module-sympy.matrices.expressions">
-<span id="matrix-expressions"></span><h1>Matrix Expressions<a class="headerlink" href="#module-sympy.matrices.expressions" title="Permalink to this headline">¶</a></h1>
-<p>The Matrix expression module allows users to write down statements like</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">MatrixSymbol</span><span class="p">,</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;Y&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span><span class="o">*</span><span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">I</span><span class="o">*</span><span class="n">Y</span>
-<span class="go">X^-1*X&#39;^-1*Y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">[X_00, X_01, X_02]</span>
-<span class="go">[X_10, X_11, X_12]</span>
-<span class="go">[X_20, X_21, X_22]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">X</span><span class="o">*</span><span class="n">Y</span><span class="p">)[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
-<span class="go">X_10*Y_02 + X_11*Y_12 + X_12*Y_22</span>
-</pre></div>
-</div>
-<p>where <tt class="docutils literal"><span class="pre">X</span></tt> and <tt class="docutils literal"><span class="pre">Y</span></tt> are <a class="reference internal" href="#sympy.matrices.expressions.MatrixSymbol" title="sympy.matrices.expressions.MatrixSymbol"><tt class="xref py py-class docutils literal"><span class="pre">MatrixSymbol</span></tt></a>&#8216;s rather than scalar symbols.</p>
-<div class="section" id="matrix-expressions-core-reference">
-<h2>Matrix Expressions Core Reference<a class="headerlink" href="#matrix-expressions-core-reference" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.matrices.expressions.MatrixExpr">
-<em class="property">class </em><tt class="descclassname">sympy.matrices.expressions.</tt><tt class="descname">MatrixExpr</tt><a class="headerlink" href="#sympy.matrices.expressions.MatrixExpr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Matrix Expression Class
-Matrix Expressions subclass SymPy Expr&#8217;s so that
-MatAdd inherits from Add
-MatMul inherits from Mul
-MatPow inherits from Pow</p>
-<p>They use _op_priority to gain control with binary operations (+, <em>, -, *</em>)
-are used</p>
-<p>They implement operations specific to Matrix Algebra.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="52%" />
-<col width="48%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>is_Identity</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="attribute">
-<dt id="sympy.matrices.expressions.MatrixExpr.T">
-<tt class="descname">T</tt><a class="headerlink" href="#sympy.matrices.expressions.MatrixExpr.T" title="Permalink to this definition">¶</a></dt>
-<dd><p>Matrix transposition.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.expressions.MatrixExpr.as_explicit">
-<tt class="descname">as_explicit</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.matrices.expressions.MatrixExpr.as_explicit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a dense Matrix with elements represented explicitly</p>
-<p>Returns an object of type ImmutableMatrix.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.matrices.expressions.MatrixExpr.as_mutable" title="sympy.matrices.expressions.MatrixExpr.as_mutable"><tt class="xref py py-obj docutils literal"><span class="pre">as_mutable</span></tt></a></dt>
-<dd>returns mutable Matrix type</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Identity</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">Identity</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span>
-<span class="go">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span><span class="o">.</span><span class="n">as_explicit</span><span class="p">()</span>
-<span class="go">[1, 0, 0]</span>
-<span class="go">[0, 1, 0]</span>
-<span class="go">[0, 0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.expressions.MatrixExpr.as_mutable">
-<tt class="descname">as_mutable</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.matrices.expressions.MatrixExpr.as_mutable" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a dense, mutable matrix with elements represented explicitly</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.matrices.expressions.MatrixExpr.as_explicit" title="sympy.matrices.expressions.MatrixExpr.as_explicit"><tt class="xref py py-obj docutils literal"><span class="pre">as_explicit</span></tt></a></dt>
-<dd>returns ImmutableMatrix</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Identity</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">Identity</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span>
-<span class="go">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span><span class="o">.</span><span class="n">shape</span>
-<span class="go">(3, 3)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span><span class="o">.</span><span class="n">as_mutable</span><span class="p">()</span>
-<span class="go">[1, 0, 0]</span>
-<span class="go">[0, 1, 0]</span>
-<span class="go">[0, 0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.expressions.MatrixExpr.equals">
-<tt class="descname">equals</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="headerlink" href="#sympy.matrices.expressions.MatrixExpr.equals" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test elementwise equality between matrices, potentially of different
-types</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Identity</span><span class="p">,</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Identity</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.matrices.expressions.MatrixSymbol">
-<em class="property">class </em><tt class="descclassname">sympy.matrices.expressions.</tt><tt class="descname">MatrixSymbol</tt><a class="headerlink" href="#sympy.matrices.expressions.MatrixSymbol" title="Permalink to this definition">¶</a></dt>
-<dd><p>Symbolic representation of a Matrix object</p>
-<p>Creates a SymPy Symbol to represent a Matrix. This matrix has a shape and
-can be included in Matrix Expressions</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">MatrixSymbol</span><span class="p">,</span> <span class="n">Identity</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="c"># A 3 by 4 Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="c"># A 4 by 3 Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">shape</span>
-<span class="go">(3, 4)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span><span class="o">*</span><span class="n">A</span><span class="o">*</span><span class="n">B</span> <span class="o">+</span> <span class="n">Identity</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">2*A*B + I</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="52%" />
-<col width="48%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>is_Identity</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.matrices.expressions.MatAdd">
-<em class="property">class </em><tt class="descclassname">sympy.matrices.expressions.</tt><tt class="descname">MatAdd</tt><a class="headerlink" href="#sympy.matrices.expressions.MatAdd" title="Permalink to this definition">¶</a></dt>
-<dd><p>A Sum of Matrix Expressions</p>
-<p>MatAdd inherits from and operates like SymPy Add</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">MatAdd</span><span class="p">,</span> <span class="n">MatrixSymbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;C&#39;</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">MatAdd</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">)</span>
-<span class="go">A + B + C</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="52%" />
-<col width="48%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>is_Identity</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.matrices.expressions.MatMul">
-<em class="property">class </em><tt class="descclassname">sympy.matrices.expressions.</tt><tt class="descname">MatMul</tt><a class="headerlink" href="#sympy.matrices.expressions.MatMul" title="Permalink to this definition">¶</a></dt>
-<dd><p>A product of matrix expressions</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">MatMul</span><span class="p">,</span> <span class="n">MatrixSymbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;C&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">MatMul</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">)</span>
-<span class="go">A*B*C</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="52%" />
-<col width="48%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>is_Identity</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.matrices.expressions.MatPow">
-<em class="property">class </em><tt class="descclassname">sympy.matrices.expressions.</tt><tt class="descname">MatPow</tt><a class="headerlink" href="#sympy.matrices.expressions.MatPow" title="Permalink to this definition">¶</a></dt>
-<dd><p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="52%" />
-<col width="48%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>is_Identity</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.matrices.expressions.Inverse">
-<em class="property">class </em><tt class="descclassname">sympy.matrices.expressions.</tt><tt class="descname">Inverse</tt><a class="headerlink" href="#sympy.matrices.expressions.Inverse" title="Permalink to this definition">¶</a></dt>
-<dd><p>The multiplicative inverse of a matrix expression</p>
-<p>This is a symbolic object that simply stores its argument without
-evaluating it. To actually compute the inverse, use the <tt class="docutils literal"><span class="pre">.inverse()</span></tt>
-method of matrices.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">MatrixSymbol</span><span class="p">,</span> <span class="n">Inverse</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Inverse</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-<span class="go">A^-1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">inverse</span><span class="p">()</span> <span class="o">==</span> <span class="n">Inverse</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">A</span><span class="o">*</span><span class="n">B</span><span class="p">)</span><span class="o">.</span><span class="n">inverse</span><span class="p">()</span>
-<span class="go">B^-1*A^-1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Inverse</span><span class="p">(</span><span class="n">A</span><span class="o">*</span><span class="n">B</span><span class="p">)</span>
-<span class="go">(A*B)^-1</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="52%" />
-<col width="48%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>is_Identity</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.matrices.expressions.Transpose">
-<em class="property">class </em><tt class="descclassname">sympy.matrices.expressions.</tt><tt class="descname">Transpose</tt><a class="headerlink" href="#sympy.matrices.expressions.Transpose" title="Permalink to this definition">¶</a></dt>
-<dd><p>The transpose of a matrix expression.</p>
-<p>This is a symbolic object that simply stores its argument without
-evaluating it. To actually compute the transpose, use the <tt class="docutils literal"><span class="pre">transpose()</span></tt>
-function, or the <tt class="docutils literal"><span class="pre">.T</span></tt> attribute of matrices.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">MatrixSymbol</span><span class="p">,</span> <span class="n">Transpose</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">transpose</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Transpose</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-<span class="go">A&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">T</span> <span class="o">==</span> <span class="n">transpose</span><span class="p">(</span><span class="n">A</span><span class="p">)</span> <span class="o">==</span> <span class="n">Transpose</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Transpose</span><span class="p">(</span><span class="n">A</span><span class="o">*</span><span class="n">B</span><span class="p">)</span>
-<span class="go">(A*B)&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">transpose</span><span class="p">(</span><span class="n">A</span><span class="o">*</span><span class="n">B</span><span class="p">)</span>
-<span class="go">B&#39;*A&#39;</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="52%" />
-<col width="48%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>is_Identity</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.matrices.expressions.Trace">
-<em class="property">class </em><tt class="descclassname">sympy.matrices.expressions.</tt><tt class="descname">Trace</tt><a class="headerlink" href="#sympy.matrices.expressions.Trace" title="Permalink to this definition">¶</a></dt>
-<dd><p>Matrix Trace</p>
-<p>Represents the trace of a matrix expression.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">MatrixSymbol</span><span class="p">,</span> <span class="n">Trace</span><span class="p">,</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Trace</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-<span class="go">Trace(A)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Trace</span><span class="p">(</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.matrices.expressions.FunctionMatrix">
-<em class="property">class </em><tt class="descclassname">sympy.matrices.expressions.</tt><tt class="descname">FunctionMatrix</tt><a class="headerlink" href="#sympy.matrices.expressions.FunctionMatrix" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents a Matrix using a function (Lambda)</p>
-<p>This class is an alternative to SparseMatrix</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">FunctionMatrix</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">Lambda</span><span class="p">,</span> <span class="n">MatMul</span><span class="p">,</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i,j&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">FunctionMatrix</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">Lambda</span><span class="p">((</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">),</span> <span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">[0, 1, 2]</span>
-<span class="go">[1, 2, 3]</span>
-<span class="go">[2, 3, 4]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span> <span class="o">=</span> <span class="n">FunctionMatrix</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span> <span class="mi">1000</span><span class="p">,</span> <span class="n">Lambda</span><span class="p">((</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">),</span> <span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">isinstance</span><span class="p">(</span><span class="n">Y</span><span class="o">*</span><span class="n">Y</span><span class="p">,</span> <span class="n">MatMul</span><span class="p">)</span> <span class="c"># this is an expression object</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">Y</span><span class="o">**</span><span class="mi">2</span><span class="p">)[</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">]</span> <span class="c"># So this is evaluated lazily</span>
-<span class="go">342923500</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="52%" />
-<col width="48%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>is_Identity</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.matrices.expressions.Identity">
-<em class="property">class </em><tt class="descclassname">sympy.matrices.expressions.</tt><tt class="descname">Identity</tt><a class="headerlink" href="#sympy.matrices.expressions.Identity" title="Permalink to this definition">¶</a></dt>
-<dd><p>The Matrix Identity I - multiplicative identity</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Identity</span><span class="p">,</span> <span class="n">MatrixSymbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">Identity</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span><span class="o">*</span><span class="n">A</span>
-<span class="go">A</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.matrices.expressions.ZeroMatrix">
-<em class="property">class </em><tt class="descclassname">sympy.matrices.expressions.</tt><tt class="descname">ZeroMatrix</tt><a class="headerlink" href="#sympy.matrices.expressions.ZeroMatrix" title="Permalink to this definition">¶</a></dt>
-<dd><p>The Matrix Zero 0 - additive identity</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">MatrixSymbol</span><span class="p">,</span> <span class="n">ZeroMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Z</span> <span class="o">=</span> <span class="n">ZeroMatrix</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">+</span><span class="n">Z</span>
-<span class="go">A</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Z</span><span class="o">*</span><span class="n">A</span><span class="o">.</span><span class="n">T</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="52%" />
-<col width="48%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>is_Identity</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</div>
-<div class="section" id="block-matrices">
-<h2>Block Matrices<a class="headerlink" href="#block-matrices" title="Permalink to this headline">¶</a></h2>
-<p>Block matrices allow you to construct larger matrices out of smaller
-sub-blocks. They can work with <a class="reference internal" href="#sympy.matrices.expressions.MatrixExpr" title="sympy.matrices.expressions.MatrixExpr"><tt class="xref py py-class docutils literal"><span class="pre">MatrixExpr</span></tt></a> or
-<tt class="xref py py-class docutils literal"><span class="pre">ImmutableMatrix</span></tt> objects.</p>
-<span class="target" id="module-sympy.matrices.expressions.blockmatrix"></span><dl class="class">
-<dt id="sympy.matrices.expressions.blockmatrix.BlockMatrix">
-<em class="property">class </em><tt class="descclassname">sympy.matrices.expressions.blockmatrix.</tt><tt class="descname">BlockMatrix</tt><a class="reference internal" href="../../_modules/sympy/matrices/expressions/blockmatrix.html#BlockMatrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.expressions.blockmatrix.BlockMatrix" title="Permalink to this definition">¶</a></dt>
-<dd><p>A BlockMatrix is a Matrix composed of other smaller, submatrices</p>
-<p>The submatrices are stored in a SymPy Matrix object but accessed as part of
-a Matrix Expression</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="p">(</span><span class="n">MatrixSymbol</span><span class="p">,</span> <span class="n">BlockMatrix</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span>
-<span class="gp">... </span>    <span class="n">Identity</span><span class="p">,</span> <span class="n">ZeroMatrix</span><span class="p">,</span> <span class="n">block_collapse</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span><span class="n">m</span><span class="p">,</span><span class="n">l</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;n m l&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;Y&#39;</span><span class="p">,</span> <span class="n">m</span> <span class="p">,</span><span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Z</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;Z&#39;</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">BlockMatrix</span><span class="p">([[</span><span class="n">X</span><span class="p">,</span> <span class="n">Z</span><span class="p">],</span> <span class="p">[</span><span class="n">ZeroMatrix</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="n">n</span><span class="p">),</span> <span class="n">Y</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
-<span class="go">[X, Z]</span>
-<span class="go">[0, Y]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">BlockMatrix</span><span class="p">([[</span><span class="n">Identity</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">Z</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">C</span><span class="p">)</span>
-<span class="go">[I, Z]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">block_collapse</span><span class="p">(</span><span class="n">C</span><span class="o">*</span><span class="n">B</span><span class="p">))</span>
-<span class="go">[X, Z + Z*Y]</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.matrices.expressions.blockmatrix.BlockMatrix.inverse">
-<tt class="descname">inverse</tt><big>(</big><em>self</em>, <em>expand=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/expressions/blockmatrix.html#BlockMatrix.inverse"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.expressions.blockmatrix.BlockMatrix.inverse" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return inverse of matrix.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">MatrixSymbol</span><span class="p">,</span> <span class="n">BlockMatrix</span><span class="p">,</span> <span class="n">ZeroMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">l</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">BlockMatrix</span><span class="p">([[</span><span class="n">X</span><span class="p">]])</span><span class="o">.</span><span class="n">inverse</span><span class="p">()</span>
-<span class="go">[X^-1]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;Y&#39;</span><span class="p">,</span> <span class="n">m</span> <span class="p">,</span><span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Z</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;Z&#39;</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">BlockMatrix</span><span class="p">([[</span><span class="n">X</span><span class="p">,</span> <span class="n">Z</span><span class="p">],</span> <span class="p">[</span><span class="n">ZeroMatrix</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="n">n</span><span class="p">),</span> <span class="n">Y</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span>
-<span class="go">[X, Z]</span>
-<span class="go">[0, Y]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">inverse</span><span class="p">(</span><span class="n">expand</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">[X^-1, (-1)*X^-1*Z*Y^-1]</span>
-<span class="go">[   0,             Y^-1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.expressions.blockmatrix.BlockMatrix.transpose">
-<tt class="descname">transpose</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/expressions/blockmatrix.html#BlockMatrix.transpose"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.expressions.blockmatrix.BlockMatrix.transpose" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return transpose of matrix.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">MatrixSymbol</span><span class="p">,</span> <span class="n">BlockMatrix</span><span class="p">,</span> <span class="n">ZeroMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">l</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;Y&#39;</span><span class="p">,</span> <span class="n">m</span> <span class="p">,</span><span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Z</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;Z&#39;</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">BlockMatrix</span><span class="p">([[</span><span class="n">X</span><span class="p">,</span> <span class="n">Z</span><span class="p">],</span> <span class="p">[</span><span class="n">ZeroMatrix</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="n">n</span><span class="p">),</span> <span class="n">Y</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span>
-<span class="go">[X&#39;,  0]</span>
-<span class="go">[Z&#39;, Y&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span>
-<span class="go">[X, Z]</span>
-<span class="go">[0, Y]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.matrices.expressions.blockmatrix.BlockDiagMatrix">
-<em class="property">class </em><tt class="descclassname">sympy.matrices.expressions.blockmatrix.</tt><tt class="descname">BlockDiagMatrix</tt><a class="reference internal" href="../../_modules/sympy/matrices/expressions/blockmatrix.html#BlockDiagMatrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.expressions.blockmatrix.BlockDiagMatrix" title="Permalink to this definition">¶</a></dt>
-<dd><p>A BlockDiagMatrix is a BlockMatrix with matrices only along the diagonal</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">MatrixSymbol</span><span class="p">,</span> <span class="n">BlockDiagMatrix</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">Identity</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span><span class="n">m</span><span class="p">,</span><span class="n">l</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;n m l&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;Y&#39;</span><span class="p">,</span> <span class="n">m</span> <span class="p">,</span><span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">BlockDiagMatrix</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
-<span class="go">[X, 0]</span>
-<span class="go">[0, Y]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.expressions.blockmatrix.block_collapse">
-<tt class="descclassname">sympy.matrices.expressions.blockmatrix.</tt><tt class="descname">block_collapse</tt><big>(</big><em>expr</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/expressions/blockmatrix.html#block_collapse"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.expressions.blockmatrix.block_collapse" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates a block matrix expression</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">MatrixSymbol</span><span class="p">,</span> <span class="n">BlockMatrix</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span>                           <span class="n">Identity</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">ZeroMatrix</span><span class="p">,</span> <span class="n">block_collapse</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span><span class="n">m</span><span class="p">,</span><span class="n">l</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;n m l&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;Y&#39;</span><span class="p">,</span> <span class="n">m</span> <span class="p">,</span><span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Z</span> <span class="o">=</span> <span class="n">MatrixSymbol</span><span class="p">(</span><span class="s">&#39;Z&#39;</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">BlockMatrix</span><span class="p">([[</span><span class="n">X</span><span class="p">,</span> <span class="n">Z</span><span class="p">],</span> <span class="p">[</span><span class="n">ZeroMatrix</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">),</span> <span class="n">Y</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
-<span class="go">[X, Z]</span>
-<span class="go">[0, Y]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">BlockMatrix</span><span class="p">([[</span><span class="n">Identity</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">Z</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">C</span><span class="p">)</span>
-<span class="go">[I, Z]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">block_collapse</span><span class="p">(</span><span class="n">C</span><span class="o">*</span><span class="n">B</span><span class="p">))</span>
-<span class="go">[X, Z + Z*Y]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-
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-        </div>
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-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Matrix Expressions</a><ul>
-<li><a class="reference internal" href="#matrix-expressions-core-reference">Matrix Expressions Core Reference</a></li>
-<li><a class="reference internal" href="#block-matrices">Block Matrices</a></li>
-</ul>
-</li>
-</ul>
-
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-  <p class="topless"><a href="../matrices/immutablematrices.html"
-                        title="previous chapter">Immutable Matrices</a></p>
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-            
-  <div class="section" id="module-sympy">
-<span id="immutable-matrices"></span><h1>Immutable Matrices<a class="headerlink" href="#module-sympy" title="Permalink to this headline">¶</a></h1>
-<p>The standard <tt class="xref py py-class docutils literal"><span class="pre">Matrix</span></tt> class in SymPy is mutable. This is important for
-performance reasons but means that standard matrices can not interact well with
-the rest of SymPy. This is because the <tt class="xref py py-class docutils literal"><span class="pre">Basic</span></tt> object, from which most
-SymPy classes inherit, is immutable.</p>
-<p>The mission of the <tt class="xref py py-class docutils literal"><span class="pre">ImmutableMatrix</span></tt> class is to bridge the tension
-between performance/mutability and safety/immutability. Immutable matrices can
-do almost everything that normal matrices can do but they inherit from
-<tt class="xref py py-class docutils literal"><span class="pre">Basic</span></tt> and can thus interact more naturally with the rest of SymPy.
-<tt class="xref py py-class docutils literal"><span class="pre">ImmutableMatrix</span></tt> also inherits from <tt class="xref py py-class docutils literal"><span class="pre">MatrixExpr</span></tt>, allowing it to
-interact freely with SymPy&#8217;s Matrix Expression module.</p>
-<p>You can turn any Matrix-like object into an <tt class="xref py py-class docutils literal"><span class="pre">ImmutableMatrix</span></tt> by calling
-the constructor</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">ImmutableMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">IM</span> <span class="o">=</span> <span class="n">ImmutableMatrix</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">IM</span>
-<span class="go">[1, 2, 3]</span>
-<span class="go">[4, 0, 6]</span>
-<span class="go">[7, 8, 9]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">IM</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">5</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">TypeError</span>: <span class="n">Can not set values in Immutable Matrix. Use Matrix instead.</span>
-</pre></div>
-</div>
-<div class="section" id="module-sympy.matrices.immutable">
-<span id="immutablematrix-class-reference"></span><h2>ImmutableMatrix Class Reference<a class="headerlink" href="#module-sympy.matrices.immutable" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.matrices.immutable.ImmutableMatrix">
-<em class="property">class </em><tt class="descclassname">sympy.matrices.immutable.</tt><tt class="descname">ImmutableMatrix</tt><a class="reference internal" href="../../_modules/sympy/matrices/immutable.html#ImmutableMatrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.immutable.ImmutableMatrix" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create an immutable version of a matrix.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">ImmutableMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ImmutableMatrix</span><span class="p">(</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">[1, 0, 0]</span>
-<span class="go">[0, 1, 0]</span>
-<span class="go">[0, 0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">42</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">TypeError</span>: <span class="n">Cannot set values of ImmutableDenseMatrix</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.matrices.immutable.ImmutableMatrix.C">
-<tt class="descname">C</tt><a class="headerlink" href="#sympy.matrices.immutable.ImmutableMatrix.C" title="Permalink to this definition">¶</a></dt>
-<dd><p>By-element conjugation.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.immutable.ImmutableMatrix.adjoint">
-<tt class="descname">adjoint</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.matrices.immutable.ImmutableMatrix.adjoint" title="Permalink to this definition">¶</a></dt>
-<dd><p>Conjugate transpose or Hermitian conjugation.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.immutable.ImmutableMatrix.as_mutable">
-<tt class="descname">as_mutable</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.matrices.immutable.ImmutableMatrix.as_mutable" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a mutable version of this matrix</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">ImmutableMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">ImmutableMatrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">as_mutable</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">5</span> <span class="c"># Can set values in Y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span>
-<span class="go">[1, 2]</span>
-<span class="go">[3, 5]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.immutable.ImmutableMatrix.equals">
-<tt class="descname">equals</tt><big>(</big><em>self</em>, <em>other</em>, <em>failing_expression=False</em><big>)</big><a class="headerlink" href="#sympy.matrices.immutable.ImmutableMatrix.equals" title="Permalink to this definition">¶</a></dt>
-<dd><p>Applies <tt class="docutils literal"><span class="pre">equals</span></tt> to corresponding elements of the matrices,
-trying to prove that the elements are equivalent, returning True
-if they are, False if any pair is not, and None (or the first
-failing expression if failing_expression is True) if it cannot
-be decided if the expressions are equivalent or not. This is, in
-general, an expensive operation.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.core.expr.equals</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">==</span> <span class="n">B</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span> <span class="o">==</span> <span class="n">B</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">equals</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-</div>
-
-
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-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../index.html">
-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Immutable Matrices</a><ul>
-<li><a class="reference internal" href="#module-sympy.matrices.immutable">ImmutableMatrix Class Reference</a></li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../matrices/sparse.html"
-                        title="previous chapter">Sparse Matrices</a></p>
-  <h4>Next topic</h4>
-  <p class="topless"><a href="../matrices/expressions.html"
-                        title="next chapter">Matrix Expressions</a></p>
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-<span id="matrices"></span><span id="matrices-docs"></span><h1>Matrices<a class="headerlink" href="#module-sympy.matrices" title="Permalink to this headline">¶</a></h1>
-<p>A module that handles matrices.</p>
-<p>Includes functions for fast creating matrices like zero, one/eye, random
-matrix, etc.</p>
-<p>Contents:</p>
-<div class="toctree-wrapper compound">
-<ul>
-<li class="toctree-l1"><a class="reference internal" href="../matrices/matrices.html">Matrices (linear algebra)</a><ul>
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-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../matrices/sparse.html">Sparse Matrices</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../matrices/sparse.html#sparsematrix-class-reference">SparseMatrix Class Reference</a></li>
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-</ul>
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-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../matrices/expressions.html">Matrix Expressions</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../matrices/expressions.html#matrix-expressions-core-reference">Matrix Expressions Core Reference</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../matrices/expressions.html#block-matrices">Block Matrices</a></li>
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-  <div class="section" id="module-sympy.matrices.matrices">
-<span id="matrices-linear-algebra"></span><h1>Matrices (linear algebra)<a class="headerlink" href="#module-sympy.matrices.matrices" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="creating-matrices">
-<h2>Creating Matrices<a class="headerlink" href="#creating-matrices" title="Permalink to this headline">¶</a></h2>
-<p>The linear algebra module is designed to be as simple as possible. First, we
-import and declare our first Matrix object:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.interactive.printing</span> <span class="kn">import</span> <span class="n">init_printing</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">init_printing</span><span class="p">(</span><span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">wrap_line</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">no_global</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">eye</span><span class="p">,</span> <span class="n">zeros</span><span class="p">,</span> <span class="n">ones</span><span class="p">,</span> <span class="n">diag</span><span class="p">,</span> <span class="n">GramSchmidt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]]);</span> <span class="n">M</span>
-<span class="go">[1  0  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  0  0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">([</span><span class="n">M</span><span class="p">,</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)])</span>
-<span class="go">[1  0  0 ]</span>
-<span class="go">[        ]</span>
-<span class="go">[0  0  0 ]</span>
-<span class="go">[        ]</span>
-<span class="go">[0  0  -1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span>
-<span class="go">[1 2 3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="go">[1]</span>
-<span class="go">[ ]</span>
-<span class="go">[2]</span>
-<span class="go">[ ]</span>
-<span class="go">[3]</span>
-</pre></div>
-</div>
-<p>In addition to creating a matrix from a list of appropriately-sized lists
-and/or matrices, SymPy also supports more advanced methods of matrix creation
-including a single list of values and dimension inputs:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">])</span>
-<span class="go">[1  2  3]</span>
-<span class="go">[       ]</span>
-<span class="go">[4  5  6]</span>
-</pre></div>
-</div>
-<p>More interesting (and useful), is the ability to use a 2-variable function
-(or lambda) to create a matrix. Here we create an indicator function which
-is 1 on the diagonal and then use it to make the identity matrix:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
-<span class="gp">... </span>        <span class="k">return</span> <span class="mi">1</span>
-<span class="gp">... </span>    <span class="k">else</span><span class="p">:</span>
-<span class="gp">... </span>        <span class="k">return</span> <span class="mi">0</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
-<span class="go">[1  0  0  0]</span>
-<span class="go">[          ]</span>
-<span class="go">[0  1  0  0]</span>
-<span class="go">[          ]</span>
-<span class="go">[0  0  1  0]</span>
-<span class="go">[          ]</span>
-<span class="go">[0  0  0  1]</span>
-</pre></div>
-</div>
-<p>Finally let&#8217;s use lambda to create a 1-line matrix with 1&#8217;s in the even
-permutation entries:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">:</span> <span class="mi">1</span> <span class="o">-</span> <span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="n">j</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">[1  0  1  0]</span>
-<span class="go">[          ]</span>
-<span class="go">[0  1  0  1]</span>
-<span class="go">[          ]</span>
-<span class="go">[1  0  1  0]</span>
-</pre></div>
-</div>
-<p>There are also a couple of special constructors for quick matrix construction:
-<tt class="docutils literal"><span class="pre">eye</span></tt> is the identity matrix, <tt class="docutils literal"><span class="pre">zeros</span></tt> and <tt class="docutils literal"><span class="pre">ones</span></tt> for matrices of all
-zeros and ones, respectively, and <tt class="docutils literal"><span class="pre">diag</span></tt> to put matrices or elements along
-the diagonal:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">eye</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">[1  0  0  0]</span>
-<span class="go">[          ]</span>
-<span class="go">[0  1  0  0]</span>
-<span class="go">[          ]</span>
-<span class="go">[0  0  1  0]</span>
-<span class="go">[          ]</span>
-<span class="go">[0  0  0  1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeros</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">[0  0]</span>
-<span class="go">[    ]</span>
-<span class="go">[0  0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeros</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">[0  0  0  0  0]</span>
-<span class="go">[             ]</span>
-<span class="go">[0  0  0  0  0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ones</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">[1  1  1]</span>
-<span class="go">[       ]</span>
-<span class="go">[1  1  1]</span>
-<span class="go">[       ]</span>
-<span class="go">[1  1  1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ones</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">[1  1  1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diag</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">]]))</span>
-<span class="go">[1  0  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  1  2]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  3  4]</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="basic-manipulation">
-<h2>Basic Manipulation<a class="headerlink" href="#basic-manipulation" title="Permalink to this headline">¶</a></h2>
-<p>While learning to work with matrices, let&#8217;s choose one where the entries are
-readily identifiable. One useful thing to know is that while matrices are
-2-dimensional, the storage is not and so it is allowable - though one should be
-careful - to access the entries as if they were a 1-d list.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">[</span><span class="mi">4</span><span class="p">]</span>
-<span class="go">5</span>
-</pre></div>
-</div>
-<p>Now, the more standard entry access is a pair of indices which will always
-return the value at the corresponding row and column of the matrix:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
-<span class="go">6</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
-<span class="go">5</span>
-</pre></div>
-</div>
-<p>Since this is Python we&#8217;re also able to slice submatrices; slices always
-give a matrix in return, even if the dimension is 1 x 1:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">:</span><span class="mi">2</span><span class="p">]</span>
-<span class="go">[1  2]</span>
-<span class="go">[    ]</span>
-<span class="go">[4  5]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">[</span><span class="mi">2</span><span class="p">:</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
-<span class="go">[]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">[:,</span> <span class="mi">2</span><span class="p">]</span>
-<span class="go">[3]</span>
-<span class="go">[ ]</span>
-<span class="go">[6]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">[:</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
-<span class="go">[3]</span>
-</pre></div>
-</div>
-<p>In the 2nd example above notice that the slice 2:2 gives an empty range. Note
-also (in keeping with 0-based indexing of Python) the first row/column is 0.</p>
-<p>You cannot access rows or columns that are not present unless they
-are in a slice:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">[:,</span> <span class="mi">10</span><span class="p">]</span> <span class="c"># the 10-th column (not there)</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">IndexError: Index out of range</span>: <span class="n">a[[0, 10]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">[:,</span> <span class="mi">10</span><span class="p">:</span><span class="mi">11</span><span class="p">]</span> <span class="c"># the 10-th column (if there)</span>
-<span class="go">[]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">[:,</span> <span class="p">:</span><span class="mi">10</span><span class="p">]</span> <span class="c"># all columns up to the 10-th</span>
-<span class="go">[1  2  3]</span>
-<span class="go">[       ]</span>
-<span class="go">[4  5  6]</span>
-</pre></div>
-</div>
-<p>Slicing an empty matrix works as long as you use a slice for the coordinate
-that has no size:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="p">[])[:,</span> <span class="mi">1</span><span class="p">]</span>
-<span class="go">[]</span>
-</pre></div>
-</div>
-<p>Slicing gives a copy of what is sliced, so modifications of one object
-do not affect the other:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">M2</span> <span class="o">=</span> <span class="n">M</span><span class="p">[:,</span> <span class="p">:]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M2</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">100</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="mi">100</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>Notice that changing M2 didn&#8217;t change M. Since we can slice, we can also assign
-entries:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">],[</span><span class="mi">9</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="mi">11</span><span class="p">,</span><span class="mi">12</span><span class="p">],[</span><span class="mi">13</span><span class="p">,</span><span class="mi">14</span><span class="p">,</span><span class="mi">15</span><span class="p">,</span><span class="mi">16</span><span class="p">]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span>
-<span class="go">[1   2   3   4 ]</span>
-<span class="go">[              ]</span>
-<span class="go">[5   6   7   8 ]</span>
-<span class="go">[              ]</span>
-<span class="go">[9   10  11  12]</span>
-<span class="go">[              ]</span>
-<span class="go">[13  14  15  16]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span>
-<span class="go">[1   2   3   0 ]</span>
-<span class="go">[              ]</span>
-<span class="go">[5   6   7   8 ]</span>
-<span class="go">[              ]</span>
-<span class="go">[9   10  0   12]</span>
-<span class="go">[              ]</span>
-<span class="go">[13  14  15  16]</span>
-</pre></div>
-</div>
-<p>as well as assign slices:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">],[</span><span class="mi">9</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="mi">11</span><span class="p">,</span><span class="mi">12</span><span class="p">],[</span><span class="mi">13</span><span class="p">,</span><span class="mi">14</span><span class="p">,</span><span class="mi">15</span><span class="p">,</span><span class="mi">16</span><span class="p">]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">[</span><span class="mi">2</span><span class="p">:,</span><span class="mi">2</span><span class="p">:]</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="k">lambda</span> <span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">:</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span>
-<span class="go">[1   2   3  4]</span>
-<span class="go">[            ]</span>
-<span class="go">[5   6   7  8]</span>
-<span class="go">[            ]</span>
-<span class="go">[9   10  0  0]</span>
-<span class="go">[            ]</span>
-<span class="go">[13  14  0  0]</span>
-</pre></div>
-</div>
-<p>All the standard arithmetic operations are supported:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">],[</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">],[</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">9</span><span class="p">]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">-</span> <span class="n">M</span>
-<span class="go">[0  0  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  0  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  0  0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">+</span> <span class="n">M</span>
-<span class="go">[2   4   6 ]</span>
-<span class="go">[          ]</span>
-<span class="go">[8   10  12]</span>
-<span class="go">[          ]</span>
-<span class="go">[14  16  18]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">*</span> <span class="n">M</span>
-<span class="go">[30   36   42 ]</span>
-<span class="go">[             ]</span>
-<span class="go">[66   81   96 ]</span>
-<span class="go">[             ]</span>
-<span class="go">[102  126  150]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M2</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,[</span><span class="mi">1</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">*</span><span class="n">M2</span>
-<span class="go">[11]</span>
-<span class="go">[  ]</span>
-<span class="go">[29]</span>
-<span class="go">[  ]</span>
-<span class="go">[47]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">[30   36   42 ]</span>
-<span class="go">[             ]</span>
-<span class="go">[66   81   96 ]</span>
-<span class="go">[             ]</span>
-<span class="go">[102  126  150]</span>
-</pre></div>
-</div>
-<p>As well as some useful vector operations:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">row_del</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span>
-<span class="go">[4  5  6]</span>
-<span class="go">[       ]</span>
-<span class="go">[7  8  9]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">col_del</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span>
-<span class="go">[4  6]</span>
-<span class="go">[    ]</span>
-<span class="go">[7  9]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v1</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v2</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v3</span> <span class="o">=</span> <span class="n">v1</span><span class="o">.</span><span class="n">cross</span><span class="p">(</span><span class="n">v2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v1</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">v2</span><span class="p">)</span>
-<span class="go">32</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v2</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">v3</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v1</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">v3</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>Recall that the row_del() and col_del() operations don&#8217;t return a value - they
-simply change the matrix object. We can also &#8216;&#8217;glue&#8217;&#8217; together matrices of the
-appropriate size:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">M1</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M2</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M1</span><span class="o">.</span><span class="n">row_join</span><span class="p">(</span><span class="n">M2</span><span class="p">)</span>
-<span class="go">[1  0  0  0  0  0  0]</span>
-<span class="go">[                   ]</span>
-<span class="go">[0  1  0  0  0  0  0]</span>
-<span class="go">[                   ]</span>
-<span class="go">[0  0  1  0  0  0  0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M3</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M1</span><span class="o">.</span><span class="n">col_join</span><span class="p">(</span><span class="n">M3</span><span class="p">)</span>
-<span class="go">[1  0  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  1  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  0  1]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  0  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  0  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  0  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  0  0]</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="operations-on-entries">
-<h2>Operations on entries<a class="headerlink" href="#operations-on-entries" title="Permalink to this headline">¶</a></h2>
-<p>We are not restricted to having multiplication between two matrices:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span><span class="o">*</span><span class="n">M</span>
-<span class="go">[2  0  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  2  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  0  2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">3</span><span class="o">*</span><span class="n">M</span>
-<span class="go">[3  0  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  3  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  0  3]</span>
-</pre></div>
-</div>
-<p>but we can also apply functions to our matrix entries using applyfunc(). Here we&#8217;ll declare a function that double any input number. Then we apply it to the 3x3 identity matrix:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">applyfunc</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-<span class="go">[2  0  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  2  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  0  2]</span>
-</pre></div>
-</div>
-<p>One more useful matrix-wide entry application function is the substitution function. Let&#8217;s declare a matrix with symbolic entries then substitute a value. Remember we can substitute anything - even another symbol!:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">*</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span>
-<span class="go">[x  0  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  x  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  0  x]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">[4  0  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  4  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  0  4]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">[y  0  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  y  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  0  y]</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="linear-algebra">
-<h2>Linear algebra<a class="headerlink" href="#linear-algebra" title="Permalink to this headline">¶</a></h2>
-<p>Now that we have the basics out of the way, let&#8217;s see what we can do with the
-actual matrices. Of course, one of the first things that comes to mind is the
-determinant:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">((</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">det</span><span class="p">()</span>
-<span class="go">-28</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M2</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M2</span><span class="o">.</span><span class="n">det</span><span class="p">()</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M3</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">((</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M3</span><span class="o">.</span><span class="n">det</span><span class="p">()</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>Another common operation is the inevers: In SymPy, this is computed by Gaussian
-elimination by default (for dense matrices) but we can specify it be done by LU
-decomposition as well:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">M2</span><span class="o">.</span><span class="n">inv</span><span class="p">()</span>
-<span class="go">[1  0  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  1  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  0  1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M2</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">method</span><span class="o">=</span><span class="s">&quot;LU&quot;</span><span class="p">)</span>
-<span class="go">[1  0  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  1  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  0  1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">method</span><span class="o">=</span><span class="s">&quot;LU&quot;</span><span class="p">)</span>
-<span class="go">[-3/14  1/14  1/2 ]</span>
-<span class="go">[                 ]</span>
-<span class="go">[-1/28  5/28  -1/4]</span>
-<span class="go">[                 ]</span>
-<span class="go">[ 3/7   -1/7   0  ]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">*</span> <span class="n">M</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">method</span><span class="o">=</span><span class="s">&quot;LU&quot;</span><span class="p">)</span>
-<span class="go">[1  0  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  1  0]</span>
-<span class="go">[       ]</span>
-<span class="go">[0  0  1]</span>
-</pre></div>
-</div>
-<p>We can perform a QR factorization which is handy for solving systems:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">],[</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span><span class="p">,</span> <span class="n">R</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">QRdecomposition</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span>
-<span class="go">[  ___     ___     ___]</span>
-<span class="go">[\/ 6   -\/ 3   -\/ 2 ]</span>
-<span class="go">[-----  ------  ------]</span>
-<span class="go">[  6      3       2   ]</span>
-<span class="go">[                     ]</span>
-<span class="go">[  ___     ___    ___ ]</span>
-<span class="go">[\/ 6   -\/ 3   \/ 2  ]</span>
-<span class="go">[-----  ------  ----- ]</span>
-<span class="go">[  6      3       2   ]</span>
-<span class="go">[                     ]</span>
-<span class="go">[  ___    ___         ]</span>
-<span class="go">[\/ 6   \/ 3          ]</span>
-<span class="go">[-----  -----     0   ]</span>
-<span class="go">[  3      3           ]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span>
-<span class="go">[           ___         ]</span>
-<span class="go">[  ___  4*\/ 6       ___]</span>
-<span class="go">[\/ 6   -------  2*\/ 6 ]</span>
-<span class="go">[          3            ]</span>
-<span class="go">[                       ]</span>
-<span class="go">[          ___          ]</span>
-<span class="go">[        \/ 3           ]</span>
-<span class="go">[  0     -----      0   ]</span>
-<span class="go">[          3            ]</span>
-<span class="go">[                       ]</span>
-<span class="go">[                   ___ ]</span>
-<span class="go">[  0       0      \/ 2  ]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span><span class="o">*</span><span class="n">R</span>
-<span class="go">[1  1  1]</span>
-<span class="go">[       ]</span>
-<span class="go">[1  1  3]</span>
-<span class="go">[       ]</span>
-<span class="go">[2  3  4]</span>
-</pre></div>
-</div>
-<p>In addition to the solvers in the solver.py file, we can solve the system Ax=b
-by passing the b vector to the matrix A&#8217;s LUsolve function. Here we&#8217;ll cheat a
-little choose A and x then multiply to get b. Then we can solve for x and check
-that it&#8217;s correct:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">6</span><span class="p">]</span> <span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,[</span><span class="mi">3</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">A</span><span class="o">*</span><span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">soln</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">LUsolve</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">soln</span>
-<span class="go">[3]</span>
-<span class="go">[ ]</span>
-<span class="go">[7]</span>
-<span class="go">[ ]</span>
-<span class="go">[5]</span>
-</pre></div>
-</div>
-<p>There&#8217;s also a nice Gram-Schmidt orthogonalizer which will take a set of
-vectors and orthogonalize then with respect to another another. There is an
-optional argument which specifies whether or not the output should also be
-normalized, it defaults to False. Let&#8217;s take some vectors and orthogonalize
-them - one normalized and one not:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">L</span> <span class="o">=</span> <span class="p">[</span><span class="n">Matrix</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">]),</span> <span class="n">Matrix</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">2</span><span class="p">]),</span> <span class="n">Matrix</span><span class="p">([</span><span class="mi">8</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">6</span><span class="p">])]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">out1</span> <span class="o">=</span> <span class="n">GramSchmidt</span><span class="p">(</span><span class="n">L</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">out2</span> <span class="o">=</span> <span class="n">GramSchmidt</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Let&#8217;s take a look at the vectors:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">out1</span><span class="p">:</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="go">[2]</span>
-<span class="go">[3]</span>
-<span class="go">[5]</span>
-<span class="go">[ 23/19]</span>
-<span class="go">[ 63/19]</span>
-<span class="go">[-47/19]</span>
-<span class="go">[ 1692/353]</span>
-<span class="go">[-1551/706]</span>
-<span class="go">[ -423/706]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">out2</span><span class="p">:</span>
-<span class="gp">... </span>     <span class="k">print</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="go">[  sqrt(38)/19]</span>
-<span class="go">[3*sqrt(38)/38]</span>
-<span class="go">[5*sqrt(38)/38]</span>
-<span class="go">[ 23*sqrt(6707)/6707]</span>
-<span class="go">[ 63*sqrt(6707)/6707]</span>
-<span class="go">[-47*sqrt(6707)/6707]</span>
-<span class="go">[ 12*sqrt(706)/353]</span>
-<span class="go">[-11*sqrt(706)/706]</span>
-<span class="go">[ -3*sqrt(706)/706]</span>
-</pre></div>
-</div>
-<p>We can spot-check their orthogonality with dot() and their normality with
-norm():</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">out1</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">out1</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">out1</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">out1</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">out1</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">out1</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">out2</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">norm</span><span class="p">()</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">out2</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">norm</span><span class="p">()</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">out2</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">norm</span><span class="p">()</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>So there is quite a bit that can be done with the module including eigenvalues,
-eigenvectors, nullspace calculation, cofactor expansion tools, and so on. From
-here one might want to look over the matrices.py file for all functionality.</p>
-</div>
-<div class="section" id="matrixbase-class-reference">
-<h2>MatrixBase Class Reference<a class="headerlink" href="#matrixbase-class-reference" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.matrices.matrices.MatrixBase">
-<em class="property">class </em><tt class="descclassname">sympy.matrices.matrices.</tt><tt class="descname">MatrixBase</tt><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase" title="Permalink to this definition">¶</a></dt>
-<dd><p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="52%" />
-<col width="48%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>is_Identity</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="attribute">
-<dt id="sympy.matrices.matrices.MatrixBase.C">
-<tt class="descname">C</tt><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.C" title="Permalink to this definition">¶</a></dt>
-<dd><p>By-element conjugation.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.matrices.matrices.MatrixBase.D">
-<tt class="descname">D</tt><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.D"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.D" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return Dirac conjugate (if self.rows == 4).</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">conjugate</span></tt></dt>
-<dd>By-element conjugation</dd>
-<dt><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.H" title="sympy.matrices.matrices.MatrixBase.H"><tt class="xref py py-obj docutils literal"><span class="pre">H</span></tt></a></dt>
-<dd>Hermite conjugation</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">I</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">D</span>
-<span class="go">[0, 1 - I, -2, -3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="p">(</span><span class="n">eye</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">eye</span><span class="p">(</span><span class="mi">4</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">D</span>
-<span class="go">[1 - I,     0,      0,      0]</span>
-<span class="go">[    0, 1 - I,      0,      0]</span>
-<span class="go">[    0,     0, -1 + I,      0]</span>
-<span class="go">[    2,     0,      0, -1 + I]</span>
-</pre></div>
-</div>
-<p>If the matrix does not have 4 rows an AttributeError will be raised
-because this property is only defined for matrices with 4 rows.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">(</span><span class="n">eye</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">.</span><span class="n">D</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">AttributeError</span>: <span class="n">Matrix has no attribute D.</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.matrices.matrices.MatrixBase.H">
-<tt class="descname">H</tt><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.H"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.H" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return Hermite conjugate.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">conjugate</span></tt></dt>
-<dd>By-element conjugation</dd>
-<dt><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.D" title="sympy.matrices.matrices.MatrixBase.D"><tt class="xref py py-obj docutils literal"><span class="pre">D</span></tt></a></dt>
-<dd>Dirac conjugation</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">I</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[    0]</span>
-<span class="go">[1 + I]</span>
-<span class="go">[    2]</span>
-<span class="go">[    3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">H</span>
-<span class="go">[0, 1 - I, 2, 3]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.LDLdecomposition">
-<tt class="descname">LDLdecomposition</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.LDLdecomposition"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.LDLdecomposition" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the LDL Decomposition (L, D) of matrix A,
-such that L * D * L.T == A
-This method eliminates the use of square root.
-Further this ensures that all the diagonal entries of L are 1.
-A must be a square, symmetric, positive-definite
-and non-singular matrix.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.cholesky" title="sympy.matrices.matrices.MatrixBase.cholesky"><tt class="xref py py-obj docutils literal"><span class="pre">cholesky</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LUdecomposition" title="sympy.matrices.matrices.MatrixBase.LUdecomposition"><tt class="xref py py-obj docutils literal"><span class="pre">LUdecomposition</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.QRdecomposition" title="sympy.matrices.matrices.MatrixBase.QRdecomposition"><tt class="xref py py-obj docutils literal"><span class="pre">QRdecomposition</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(((</span><span class="mi">25</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">),</span> <span class="p">(</span><span class="mi">15</span><span class="p">,</span> <span class="mi">18</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">11</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span><span class="p">,</span> <span class="n">D</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">LDLdecomposition</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span>
-<span class="go">[   1,   0, 0]</span>
-<span class="go">[ 3/5,   1, 0]</span>
-<span class="go">[-1/5, 1/3, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span>
-<span class="go">[25, 0, 0]</span>
-<span class="go">[ 0, 9, 0]</span>
-<span class="go">[ 0, 0, 9]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span> <span class="o">*</span> <span class="n">D</span> <span class="o">*</span> <span class="n">L</span><span class="o">.</span><span class="n">T</span> <span class="o">*</span> <span class="n">A</span><span class="o">.</span><span class="n">inv</span><span class="p">()</span> <span class="o">==</span> <span class="n">eye</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">rows</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.LDLsolve">
-<tt class="descname">LDLsolve</tt><big>(</big><em>self</em>, <em>rhs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.LDLsolve"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.LDLsolve" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solves Ax = B using LDL decomposition,
-for a general square and non-singular matrix.</p>
-<p>For a non-square matrix with rows &gt; cols,
-the least squares solution is returned.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LDLdecomposition" title="sympy.matrices.matrices.MatrixBase.LDLdecomposition"><tt class="xref py py-obj docutils literal"><span class="pre">LDLdecomposition</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.lower_triangular_solve" title="sympy.matrices.matrices.MatrixBase.lower_triangular_solve"><tt class="xref py py-obj docutils literal"><span class="pre">lower_triangular_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.upper_triangular_solve" title="sympy.matrices.matrices.MatrixBase.upper_triangular_solve"><tt class="xref py py-obj docutils literal"><span class="pre">upper_triangular_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.cholesky_solve" title="sympy.matrices.matrices.MatrixBase.cholesky_solve"><tt class="xref py py-obj docutils literal"><span class="pre">cholesky_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.diagonal_solve" title="sympy.matrices.matrices.MatrixBase.diagonal_solve"><tt class="xref py py-obj docutils literal"><span class="pre">diagonal_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LUsolve" title="sympy.matrices.matrices.MatrixBase.LUsolve"><tt class="xref py py-obj docutils literal"><span class="pre">LUsolve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.QRsolve" title="sympy.matrices.matrices.MatrixBase.QRsolve"><tt class="xref py py-obj docutils literal"><span class="pre">QRsolve</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">LDLsolve</span><span class="p">(</span><span class="n">B</span><span class="p">)</span> <span class="o">==</span> <span class="n">B</span><span class="o">/</span><span class="mi">2</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.LUdecomposition">
-<tt class="descname">LUdecomposition</tt><big>(</big><em>self</em>, <em>iszerofunc=&lt;function _iszero at 0x10c6e2cb0&gt;</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.LUdecomposition"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.LUdecomposition" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the decomposition LU and the row swaps p.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.cholesky" title="sympy.matrices.matrices.MatrixBase.cholesky"><tt class="xref py py-obj docutils literal"><span class="pre">cholesky</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LDLdecomposition" title="sympy.matrices.matrices.MatrixBase.LDLdecomposition"><tt class="xref py py-obj docutils literal"><span class="pre">LDLdecomposition</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.QRdecomposition" title="sympy.matrices.matrices.MatrixBase.QRdecomposition"><tt class="xref py py-obj docutils literal"><span class="pre">QRdecomposition</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LUdecomposition_Simple" title="sympy.matrices.matrices.MatrixBase.LUdecomposition_Simple"><tt class="xref py py-obj docutils literal"><span class="pre">LUdecomposition_Simple</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LUdecompositionFF" title="sympy.matrices.matrices.MatrixBase.LUdecompositionFF"><tt class="xref py py-obj docutils literal"><span class="pre">LUdecompositionFF</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LUsolve" title="sympy.matrices.matrices.MatrixBase.LUsolve"><tt class="xref py py-obj docutils literal"><span class="pre">LUsolve</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">6</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span><span class="p">,</span> <span class="n">U</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">LUdecomposition</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span>
-<span class="go">[  1, 0]</span>
-<span class="go">[3/2, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">U</span>
-<span class="go">[4,    3]</span>
-<span class="go">[0, -3/2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.LUdecompositionFF">
-<tt class="descname">LUdecompositionFF</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.LUdecompositionFF"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.LUdecompositionFF" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute a fraction-free LU decomposition.</p>
-<p>Returns 4 matrices P, L, D, U such that PA = L D**-1 U.
-If the elements of the matrix belong to some integral domain I, then all
-elements of L, D and U are guaranteed to belong to I.</p>
-<dl class="docutils">
-<dt><strong>Reference</strong></dt>
-<dd><ul class="first last simple">
-<li>W. Zhou &amp; D.J. Jeffrey, &#8220;Fraction-free matrix factors: new forms
-for LU and QR factors&#8221;. Frontiers in Computer Science in China,
-Vol 2, no. 1, pp. 67-80, 2008.</li>
-</ul>
-</dd>
-</dl>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LUdecomposition" title="sympy.matrices.matrices.MatrixBase.LUdecomposition"><tt class="xref py py-obj docutils literal"><span class="pre">LUdecomposition</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LUdecomposition_Simple" title="sympy.matrices.matrices.MatrixBase.LUdecomposition_Simple"><tt class="xref py py-obj docutils literal"><span class="pre">LUdecomposition_Simple</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LUsolve" title="sympy.matrices.matrices.MatrixBase.LUsolve"><tt class="xref py py-obj docutils literal"><span class="pre">LUsolve</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.LUdecomposition_Simple">
-<tt class="descname">LUdecomposition_Simple</tt><big>(</big><em>self</em>, <em>iszerofunc=&lt;function _iszero at 0x10c6e2cb0&gt;</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.LUdecomposition_Simple"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.LUdecomposition_Simple" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns A comprised of L, U (L&#8217;s diag entries are 1) and
-p which is the list of the row swaps (in order).</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LUdecomposition" title="sympy.matrices.matrices.MatrixBase.LUdecomposition"><tt class="xref py py-obj docutils literal"><span class="pre">LUdecomposition</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LUdecompositionFF" title="sympy.matrices.matrices.MatrixBase.LUdecompositionFF"><tt class="xref py py-obj docutils literal"><span class="pre">LUdecompositionFF</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LUsolve" title="sympy.matrices.matrices.MatrixBase.LUsolve"><tt class="xref py py-obj docutils literal"><span class="pre">LUsolve</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.LUsolve">
-<tt class="descname">LUsolve</tt><big>(</big><em>self</em>, <em>rhs</em>, <em>iszerofunc=&lt;function _iszero at 0x10c6e2cb0&gt;</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.LUsolve"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.LUsolve" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solve the linear system Ax = rhs for x where A = self.</p>
-<p>This is for symbolic matrices, for real or complex ones use
-sympy.mpmath.lu_solve or sympy.mpmath.qr_solve.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.lower_triangular_solve" title="sympy.matrices.matrices.MatrixBase.lower_triangular_solve"><tt class="xref py py-obj docutils literal"><span class="pre">lower_triangular_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.upper_triangular_solve" title="sympy.matrices.matrices.MatrixBase.upper_triangular_solve"><tt class="xref py py-obj docutils literal"><span class="pre">upper_triangular_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.cholesky_solve" title="sympy.matrices.matrices.MatrixBase.cholesky_solve"><tt class="xref py py-obj docutils literal"><span class="pre">cholesky_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.diagonal_solve" title="sympy.matrices.matrices.MatrixBase.diagonal_solve"><tt class="xref py py-obj docutils literal"><span class="pre">diagonal_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LDLsolve" title="sympy.matrices.matrices.MatrixBase.LDLsolve"><tt class="xref py py-obj docutils literal"><span class="pre">LDLsolve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.QRsolve" title="sympy.matrices.matrices.MatrixBase.QRsolve"><tt class="xref py py-obj docutils literal"><span class="pre">QRsolve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LUdecomposition" title="sympy.matrices.matrices.MatrixBase.LUdecomposition"><tt class="xref py py-obj docutils literal"><span class="pre">LUdecomposition</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.QRdecomposition">
-<tt class="descname">QRdecomposition</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.QRdecomposition"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.QRdecomposition" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return Q, R where A = Q*R, Q is orthogonal and R is upper triangular.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.cholesky" title="sympy.matrices.matrices.MatrixBase.cholesky"><tt class="xref py py-obj docutils literal"><span class="pre">cholesky</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LDLdecomposition" title="sympy.matrices.matrices.MatrixBase.LDLdecomposition"><tt class="xref py py-obj docutils literal"><span class="pre">LDLdecomposition</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LUdecomposition" title="sympy.matrices.matrices.MatrixBase.LUdecomposition"><tt class="xref py py-obj docutils literal"><span class="pre">LUdecomposition</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.QRsolve" title="sympy.matrices.matrices.MatrixBase.QRsolve"><tt class="xref py py-obj docutils literal"><span class="pre">QRsolve</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<p>This is the example from wikipedia:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">12</span><span class="p">,</span> <span class="o">-</span><span class="mi">51</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">6</span><span class="p">,</span> <span class="mi">167</span><span class="p">,</span> <span class="o">-</span><span class="mi">68</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">24</span><span class="p">,</span> <span class="o">-</span><span class="mi">41</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span><span class="p">,</span> <span class="n">R</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">QRdecomposition</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span>
-<span class="go">[ 6/7, -69/175, -58/175]</span>
-<span class="go">[ 3/7, 158/175,   6/175]</span>
-<span class="go">[-2/7,    6/35,  -33/35]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span>
-<span class="go">[14,  21, -14]</span>
-<span class="go">[ 0, 175, -70]</span>
-<span class="go">[ 0,   0,  35]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">==</span> <span class="n">Q</span><span class="o">*</span><span class="n">R</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>QR factorization of an identity matrix:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span><span class="p">,</span> <span class="n">R</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">QRdecomposition</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span>
-<span class="go">[1, 0, 0]</span>
-<span class="go">[0, 1, 0]</span>
-<span class="go">[0, 0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span>
-<span class="go">[1, 0, 0]</span>
-<span class="go">[0, 1, 0]</span>
-<span class="go">[0, 0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.QRsolve">
-<tt class="descname">QRsolve</tt><big>(</big><em>self</em>, <em>b</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.QRsolve"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.QRsolve" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solve the linear system &#8216;Ax = b&#8217;.</p>
-<p>&#8216;self&#8217; is the matrix &#8216;A&#8217;, the method argument is the vector
-&#8216;b&#8217;.  The method returns the solution vector &#8216;x&#8217;.  If &#8216;b&#8217; is a
-matrix, the system is solved for each column of &#8216;b&#8217; and the
-return value is a matrix of the same shape as &#8216;b&#8217;.</p>
-<p>This method is slower (approximately by a factor of 2) but
-more stable for floating-point arithmetic than the LUsolve method.
-However, LUsolve usually uses an exact arithmetic, so you don&#8217;t need
-to use QRsolve.</p>
-<p>This is mainly for educational purposes and symbolic matrices, for real
-(or complex) matrices use sympy.mpmath.qr_solve.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.lower_triangular_solve" title="sympy.matrices.matrices.MatrixBase.lower_triangular_solve"><tt class="xref py py-obj docutils literal"><span class="pre">lower_triangular_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.upper_triangular_solve" title="sympy.matrices.matrices.MatrixBase.upper_triangular_solve"><tt class="xref py py-obj docutils literal"><span class="pre">upper_triangular_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.cholesky_solve" title="sympy.matrices.matrices.MatrixBase.cholesky_solve"><tt class="xref py py-obj docutils literal"><span class="pre">cholesky_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.diagonal_solve" title="sympy.matrices.matrices.MatrixBase.diagonal_solve"><tt class="xref py py-obj docutils literal"><span class="pre">diagonal_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LDLsolve" title="sympy.matrices.matrices.MatrixBase.LDLsolve"><tt class="xref py py-obj docutils literal"><span class="pre">LDLsolve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LUsolve" title="sympy.matrices.matrices.MatrixBase.LUsolve"><tt class="xref py py-obj docutils literal"><span class="pre">LUsolve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.QRdecomposition" title="sympy.matrices.matrices.MatrixBase.QRdecomposition"><tt class="xref py py-obj docutils literal"><span class="pre">QRdecomposition</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.matrices.matrices.MatrixBase.T">
-<tt class="descname">T</tt><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.T" title="Permalink to this definition">¶</a></dt>
-<dd><p>Matrix transposition.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.add">
-<tt class="descname">add</tt><big>(</big><em>self</em>, <em>b</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return self + b</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.adjoint">
-<tt class="descname">adjoint</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.adjoint"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.adjoint" title="Permalink to this definition">¶</a></dt>
-<dd><p>Conjugate transpose or Hermitian conjugation.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.adjugate">
-<tt class="descname">adjugate</tt><big>(</big><em>self</em>, <em>method='berkowitz'</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.adjugate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.adjugate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the adjugate matrix.</p>
-<p>Adjugate matrix is the transpose of the cofactor matrix.</p>
-<p><a class="reference external" href="http://en.wikipedia.org/wiki/Adjugate">http://en.wikipedia.org/wiki/Adjugate</a></p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.cofactorMatrix" title="sympy.matrices.matrices.MatrixBase.cofactorMatrix"><tt class="xref py py-obj docutils literal"><span class="pre">cofactorMatrix</span></tt></a>, <tt class="xref py py-obj docutils literal"><span class="pre">transpose</span></tt>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.berkowitz" title="sympy.matrices.matrices.MatrixBase.berkowitz"><tt class="xref py py-obj docutils literal"><span class="pre">berkowitz</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.berkowitz">
-<tt class="descname">berkowitz</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.berkowitz"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.berkowitz" title="Permalink to this definition">¶</a></dt>
-<dd><p>The Berkowitz algorithm.</p>
-<blockquote>
-<div><p>Given N x N matrix with symbolic content, compute efficiently
-coefficients of characteristic polynomials of &#8216;self&#8217; and all
-its square sub-matrices composed by removing both i-th row
-and column, without division in the ground domain.</p>
-<p>This method is particularly useful for computing determinant,
-principal minors and characteristic polynomial, when &#8216;self&#8217;
-has complicated coefficients e.g. polynomials. Semi-direct
-usage of this algorithm is also important in computing
-efficiently sub-resultant PRS.</p>
-<p>Assuming that M is a square matrix of dimension N x N and
-I is N x N identity matrix,  then the following following
-definition of characteristic polynomial is begin used:</p>
-<blockquote>
-<div>charpoly(M) = det(t*I - M)</div></blockquote>
-<p>As a consequence, all polynomials generated by Berkowitz
-algorithm are monic.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">x</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">berkowitz</span><span class="p">()</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="c"># 1 x 1 M&#39;s sub-matrix</span>
-<span class="go">(1, -x)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="c"># 2 x 2 M&#39;s sub-matrix</span>
-<span class="go">(1, -x, -y)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">r</span> <span class="c"># 3 x 3 M&#39;s sub-matrix</span>
-<span class="go">(1, -2*x, x**2 - y*z - y, x*y - z**2)</span>
-</pre></div>
-</div>
-<p>For more information on the implemented algorithm refer to:</p>
-<dl class="docutils">
-<dt>[1] S.J. Berkowitz, On computing the determinant in small</dt>
-<dd>parallel time using a small number of processors, ACM,
-Information Processing Letters 18, 1984, pp. 147-150</dd>
-<dt>[2] M. Keber, Division-Free computation of sub-resultants</dt>
-<dd>using Bezout matrices, Tech. Report MPI-I-2006-1-006,
-Saarbrucken, 2006</dd>
-</dl>
-</div></blockquote>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.berkowitz_det" title="sympy.matrices.matrices.MatrixBase.berkowitz_det"><tt class="xref py py-obj docutils literal"><span class="pre">berkowitz_det</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.berkowitz_minors" title="sympy.matrices.matrices.MatrixBase.berkowitz_minors"><tt class="xref py py-obj docutils literal"><span class="pre">berkowitz_minors</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.berkowitz_charpoly" title="sympy.matrices.matrices.MatrixBase.berkowitz_charpoly"><tt class="xref py py-obj docutils literal"><span class="pre">berkowitz_charpoly</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.berkowitz_eigenvals" title="sympy.matrices.matrices.MatrixBase.berkowitz_eigenvals"><tt class="xref py py-obj docutils literal"><span class="pre">berkowitz_eigenvals</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.berkowitz_charpoly">
-<tt class="descname">berkowitz_charpoly</tt><big>(</big><em>self</em>, <em>x=_lambda</em>, <em>simplify=&lt;function simplify at 0x10c635c20&gt;</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.berkowitz_charpoly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.berkowitz_charpoly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes characteristic polynomial minors using Berkowitz method.</p>
-<p>A PurePoly is returned so using different variables for <tt class="docutils literal"><span class="pre">x</span></tt> does
-not affect the comparison or the polynomials:</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.berkowitz" title="sympy.matrices.matrices.MatrixBase.berkowitz"><tt class="xref py py-obj docutils literal"><span class="pre">berkowitz</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">berkowitz_charpoly</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">==</span> <span class="n">A</span><span class="o">.</span><span class="n">berkowitz_charpoly</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>Specifying <tt class="docutils literal"><span class="pre">x</span></tt> is optional; a Dummy with name <tt class="docutils literal"><span class="pre">lambda</span></tt> is used by
-default (which looks good when pretty-printed in unicode):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">berkowitz_charpoly</span><span class="p">()</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-<span class="go">_lambda**2 - _lambda - 6</span>
-</pre></div>
-</div>
-<p>No test is done to see that <tt class="docutils literal"><span class="pre">x</span></tt> doesn&#8217;t clash with an existing
-symbol, so using the default (<tt class="docutils literal"><span class="pre">lambda</span></tt>) or your own Dummy symbol is
-the safest option:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">charpoly</span><span class="p">()</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-<span class="go">_lambda**2 - _lambda - 2*x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">charpoly</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-<span class="go">x**2 - 3*x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.berkowitz_det">
-<tt class="descname">berkowitz_det</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.berkowitz_det"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.berkowitz_det" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes determinant using Berkowitz method.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.det" title="sympy.matrices.matrices.MatrixBase.det"><tt class="xref py py-obj docutils literal"><span class="pre">det</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.berkowitz" title="sympy.matrices.matrices.MatrixBase.berkowitz"><tt class="xref py py-obj docutils literal"><span class="pre">berkowitz</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.berkowitz_eigenvals">
-<tt class="descname">berkowitz_eigenvals</tt><big>(</big><em>self</em>, <em>**flags</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.berkowitz_eigenvals"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.berkowitz_eigenvals" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes eigenvalues of a Matrix using Berkowitz method.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.berkowitz" title="sympy.matrices.matrices.MatrixBase.berkowitz"><tt class="xref py py-obj docutils literal"><span class="pre">berkowitz</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.berkowitz_minors">
-<tt class="descname">berkowitz_minors</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.berkowitz_minors"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.berkowitz_minors" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes principal minors using Berkowitz method.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.berkowitz" title="sympy.matrices.matrices.MatrixBase.berkowitz"><tt class="xref py py-obj docutils literal"><span class="pre">berkowitz</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.charpoly">
-<tt class="descname">charpoly</tt><big>(</big><em>self</em>, <em>x=_lambda</em>, <em>simplify=&lt;function simplify at 0x10c635c20&gt;</em><big>)</big><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.charpoly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes characteristic polynomial minors using Berkowitz method.</p>
-<p>A PurePoly is returned so using different variables for <tt class="docutils literal"><span class="pre">x</span></tt> does
-not affect the comparison or the polynomials:</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.berkowitz" title="sympy.matrices.matrices.MatrixBase.berkowitz"><tt class="xref py py-obj docutils literal"><span class="pre">berkowitz</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">berkowitz_charpoly</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">==</span> <span class="n">A</span><span class="o">.</span><span class="n">berkowitz_charpoly</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>Specifying <tt class="docutils literal"><span class="pre">x</span></tt> is optional; a Dummy with name <tt class="docutils literal"><span class="pre">lambda</span></tt> is used by
-default (which looks good when pretty-printed in unicode):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">berkowitz_charpoly</span><span class="p">()</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-<span class="go">_lambda**2 - _lambda - 6</span>
-</pre></div>
-</div>
-<p>No test is done to see that <tt class="docutils literal"><span class="pre">x</span></tt> doesn&#8217;t clash with an existing
-symbol, so using the default (<tt class="docutils literal"><span class="pre">lambda</span></tt>) or your own Dummy symbol is
-the safest option:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">charpoly</span><span class="p">()</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-<span class="go">_lambda**2 - _lambda - 2*x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">charpoly</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-<span class="go">x**2 - 3*x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.cholesky">
-<tt class="descname">cholesky</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.cholesky"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.cholesky" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the Cholesky decomposition L of a matrix A
-such that L * L.T = A</p>
-<p>A must be a square, symmetric, positive-definite
-and non-singular matrix.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LDLdecomposition" title="sympy.matrices.matrices.MatrixBase.LDLdecomposition"><tt class="xref py py-obj docutils literal"><span class="pre">LDLdecomposition</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LUdecomposition" title="sympy.matrices.matrices.MatrixBase.LUdecomposition"><tt class="xref py py-obj docutils literal"><span class="pre">LUdecomposition</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.QRdecomposition" title="sympy.matrices.matrices.MatrixBase.QRdecomposition"><tt class="xref py py-obj docutils literal"><span class="pre">QRdecomposition</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(((</span><span class="mi">25</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">),</span> <span class="p">(</span><span class="mi">15</span><span class="p">,</span> <span class="mi">18</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">11</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">cholesky</span><span class="p">()</span>
-<span class="go">[ 5, 0, 0]</span>
-<span class="go">[ 3, 3, 0]</span>
-<span class="go">[-1, 1, 3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">cholesky</span><span class="p">()</span> <span class="o">*</span> <span class="n">A</span><span class="o">.</span><span class="n">cholesky</span><span class="p">()</span><span class="o">.</span><span class="n">T</span>
-<span class="go">[25, 15, -5]</span>
-<span class="go">[15, 18,  0]</span>
-<span class="go">[-5,  0, 11]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.cholesky_solve">
-<tt class="descname">cholesky_solve</tt><big>(</big><em>self</em>, <em>rhs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.cholesky_solve"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.cholesky_solve" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solves Ax = B using Cholesky decomposition,
-for a general square non-singular matrix.
-For a non-square matrix with rows &gt; cols,
-the least squares solution is returned.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.lower_triangular_solve" title="sympy.matrices.matrices.MatrixBase.lower_triangular_solve"><tt class="xref py py-obj docutils literal"><span class="pre">lower_triangular_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.upper_triangular_solve" title="sympy.matrices.matrices.MatrixBase.upper_triangular_solve"><tt class="xref py py-obj docutils literal"><span class="pre">upper_triangular_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.diagonal_solve" title="sympy.matrices.matrices.MatrixBase.diagonal_solve"><tt class="xref py py-obj docutils literal"><span class="pre">diagonal_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LDLsolve" title="sympy.matrices.matrices.MatrixBase.LDLsolve"><tt class="xref py py-obj docutils literal"><span class="pre">LDLsolve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LUsolve" title="sympy.matrices.matrices.MatrixBase.LUsolve"><tt class="xref py py-obj docutils literal"><span class="pre">LUsolve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.QRsolve" title="sympy.matrices.matrices.MatrixBase.QRsolve"><tt class="xref py py-obj docutils literal"><span class="pre">QRsolve</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.cofactor">
-<tt class="descname">cofactor</tt><big>(</big><em>self</em>, <em>i</em>, <em>j</em>, <em>method='berkowitz'</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.cofactor"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.cofactor" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculate the cofactor of an element.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.cofactorMatrix" title="sympy.matrices.matrices.MatrixBase.cofactorMatrix"><tt class="xref py py-obj docutils literal"><span class="pre">cofactorMatrix</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.minorEntry" title="sympy.matrices.matrices.MatrixBase.minorEntry"><tt class="xref py py-obj docutils literal"><span class="pre">minorEntry</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.minorMatrix" title="sympy.matrices.matrices.MatrixBase.minorMatrix"><tt class="xref py py-obj docutils literal"><span class="pre">minorMatrix</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.cofactorMatrix">
-<tt class="descname">cofactorMatrix</tt><big>(</big><em>self</em>, <em>method='berkowitz'</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.cofactorMatrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.cofactorMatrix" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a matrix containing the cofactor of each element.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.cofactor" title="sympy.matrices.matrices.MatrixBase.cofactor"><tt class="xref py py-obj docutils literal"><span class="pre">cofactor</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.minorEntry" title="sympy.matrices.matrices.MatrixBase.minorEntry"><tt class="xref py py-obj docutils literal"><span class="pre">minorEntry</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.minorMatrix" title="sympy.matrices.matrices.MatrixBase.minorMatrix"><tt class="xref py py-obj docutils literal"><span class="pre">minorMatrix</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.adjugate" title="sympy.matrices.matrices.MatrixBase.adjugate"><tt class="xref py py-obj docutils literal"><span class="pre">adjugate</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.col_insert">
-<tt class="descname">col_insert</tt><big>(</big><em>self</em>, <em>pos</em>, <em>mti</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.col_insert"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.col_insert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Insert one or more columns at the given column position.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">col</span></tt>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.row_insert" title="sympy.matrices.matrices.MatrixBase.row_insert"><tt class="xref py py-obj docutils literal"><span class="pre">row_insert</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">zeros</span><span class="p">,</span> <span class="n">ones</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">V</span> <span class="o">=</span> <span class="n">ones</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">col_insert</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">V</span><span class="p">)</span>
-<span class="go">[0, 1, 0, 0]</span>
-<span class="go">[0, 1, 0, 0]</span>
-<span class="go">[0, 1, 0, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.col_join">
-<tt class="descname">col_join</tt><big>(</big><em>self</em>, <em>bott</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.col_join"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.col_join" title="Permalink to this definition">¶</a></dt>
-<dd><p>Concatenates two matrices along self&#8217;s last and bott&#8217;s first row</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">col</span></tt>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.row_join" title="sympy.matrices.matrices.MatrixBase.row_join"><tt class="xref py py-obj docutils literal"><span class="pre">row_join</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">zeros</span><span class="p">,</span> <span class="n">ones</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">V</span> <span class="o">=</span> <span class="n">ones</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">col_join</span><span class="p">(</span><span class="n">V</span><span class="p">)</span>
-<span class="go">[0, 0, 0]</span>
-<span class="go">[0, 0, 0]</span>
-<span class="go">[0, 0, 0]</span>
-<span class="go">[1, 1, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.condition_number">
-<tt class="descname">condition_number</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.condition_number"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.condition_number" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the condition number of a matrix.</p>
-<p>This is the maximum singular value divided by the minimum singular value</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.singular_values" title="sympy.matrices.matrices.MatrixBase.singular_values"><tt class="xref py py-obj docutils literal"><span class="pre">singular_values</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="o">/</span><span class="mi">10</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">condition_number</span><span class="p">()</span>
-<span class="go">100</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.cross">
-<tt class="descname">cross</tt><big>(</big><em>self</em>, <em>b</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.cross"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.cross" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculate the cross product of <tt class="docutils literal"><span class="pre">self</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.dot" title="sympy.matrices.matrices.MatrixBase.dot"><tt class="xref py py-obj docutils literal"><span class="pre">dot</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.multiply" title="sympy.matrices.matrices.MatrixBase.multiply"><tt class="xref py py-obj docutils literal"><span class="pre">multiply</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.multiply_elementwise" title="sympy.matrices.matrices.MatrixBase.multiply_elementwise"><tt class="xref py py-obj docutils literal"><span class="pre">multiply_elementwise</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.det">
-<tt class="descname">det</tt><big>(</big><em>self</em>, <em>method='bareis'</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.det"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.det" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the matrix determinant using the method &#8220;method&#8221;.</p>
-<dl class="docutils">
-<dt>Possible values for &#8220;method&#8221;:</dt>
-<dd>bareis ... det_bareis
-berkowitz ... berkowitz_det
-lu_decomposition ... det_LU</dd>
-</dl>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.det_bareis" title="sympy.matrices.matrices.MatrixBase.det_bareis"><tt class="xref py py-obj docutils literal"><span class="pre">det_bareis</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.berkowitz_det" title="sympy.matrices.matrices.MatrixBase.berkowitz_det"><tt class="xref py py-obj docutils literal"><span class="pre">berkowitz_det</span></tt></a>, <tt class="xref py py-obj docutils literal"><span class="pre">det_LU</span></tt></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.det_LU_decomposition">
-<tt class="descname">det_LU_decomposition</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.det_LU_decomposition"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.det_LU_decomposition" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute matrix determinant using LU decomposition</p>
-<p>Note that this method fails if the LU decomposition itself
-fails. In particular, if the matrix has no inverse this method
-will fail.</p>
-<p>TODO: Implement algorithm for sparse matrices (SFF),
-<a class="reference external" href="http://www.eecis.udel.edu/~saunders/papers/sffge/it5.ps">http://www.eecis.udel.edu/~saunders/papers/sffge/it5.ps</a>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.det" title="sympy.matrices.matrices.MatrixBase.det"><tt class="xref py py-obj docutils literal"><span class="pre">det</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.det_bareis" title="sympy.matrices.matrices.MatrixBase.det_bareis"><tt class="xref py py-obj docutils literal"><span class="pre">det_bareis</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.berkowitz_det" title="sympy.matrices.matrices.MatrixBase.berkowitz_det"><tt class="xref py py-obj docutils literal"><span class="pre">berkowitz_det</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.det_bareis">
-<tt class="descname">det_bareis</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.det_bareis"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.det_bareis" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute matrix determinant using Bareis&#8217; fraction-free
-algorithm which is an extension of the well known Gaussian
-elimination method. This approach is best suited for dense
-symbolic matrices and will result in a determinant with
-minimal number of fractions. It means that less term
-rewriting is needed on resulting formulae.</p>
-<p>TODO: Implement algorithm for sparse matrices (SFF),
-<a class="reference external" href="http://www.eecis.udel.edu/~saunders/papers/sffge/it5.ps">http://www.eecis.udel.edu/~saunders/papers/sffge/it5.ps</a>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.det" title="sympy.matrices.matrices.MatrixBase.det"><tt class="xref py py-obj docutils literal"><span class="pre">det</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.berkowitz_det" title="sympy.matrices.matrices.MatrixBase.berkowitz_det"><tt class="xref py py-obj docutils literal"><span class="pre">berkowitz_det</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.diagonal_solve">
-<tt class="descname">diagonal_solve</tt><big>(</big><em>self</em>, <em>rhs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.diagonal_solve"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.diagonal_solve" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solves Ax = B efficiently, where A is a diagonal Matrix,
-with non-zero diagonal entries.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.lower_triangular_solve" title="sympy.matrices.matrices.MatrixBase.lower_triangular_solve"><tt class="xref py py-obj docutils literal"><span class="pre">lower_triangular_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.upper_triangular_solve" title="sympy.matrices.matrices.MatrixBase.upper_triangular_solve"><tt class="xref py py-obj docutils literal"><span class="pre">upper_triangular_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.cholesky_solve" title="sympy.matrices.matrices.MatrixBase.cholesky_solve"><tt class="xref py py-obj docutils literal"><span class="pre">cholesky_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LDLsolve" title="sympy.matrices.matrices.MatrixBase.LDLsolve"><tt class="xref py py-obj docutils literal"><span class="pre">LDLsolve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LUsolve" title="sympy.matrices.matrices.MatrixBase.LUsolve"><tt class="xref py py-obj docutils literal"><span class="pre">LUsolve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.QRsolve" title="sympy.matrices.matrices.MatrixBase.QRsolve"><tt class="xref py py-obj docutils literal"><span class="pre">QRsolve</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">diagonal_solve</span><span class="p">(</span><span class="n">B</span><span class="p">)</span> <span class="o">==</span> <span class="n">B</span><span class="o">/</span><span class="mi">2</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.diagonalize">
-<tt class="descname">diagonalize</tt><big>(</big><em>self</em>, <em>reals_only=False</em>, <em>sort=False</em>, <em>normalize=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.diagonalize"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.diagonalize" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return diagonalized matrix D and transformation P such as</p>
-<blockquote>
-<div>D = P^-1 * M * P</div></blockquote>
-<p>where M is current matrix.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.is_diagonal" title="sympy.matrices.matrices.MatrixBase.is_diagonal"><tt class="xref py py-obj docutils literal"><span class="pre">is_diagonal</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.is_diagonalizable" title="sympy.matrices.matrices.MatrixBase.is_diagonalizable"><tt class="xref py py-obj docutils literal"><span class="pre">is_diagonalizable</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[1,  2, 0]</span>
-<span class="go">[0,  3, 0]</span>
-<span class="go">[2, -4, 2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">D</span><span class="p">)</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">diagonalize</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span>
-<span class="go">[1, 0, 0]</span>
-<span class="go">[0, 2, 0]</span>
-<span class="go">[0, 0, 3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span>
-<span class="go">[-1, 0, -1]</span>
-<span class="go">[ 0, 0, -1]</span>
-<span class="go">[ 2, 1,  2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">inv</span><span class="p">()</span> <span class="o">*</span> <span class="n">m</span> <span class="o">*</span> <span class="n">P</span>
-<span class="go">[1, 0, 0]</span>
-<span class="go">[0, 2, 0]</span>
-<span class="go">[0, 0, 3]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.diff">
-<tt class="descname">diff</tt><big>(</big><em>self</em>, <em>*args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.diff"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.diff" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculate the derivative of each element in the matrix.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.integrate" title="sympy.matrices.matrices.MatrixBase.integrate"><tt class="xref py py-obj docutils literal"><span class="pre">integrate</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.limit" title="sympy.matrices.matrices.MatrixBase.limit"><tt class="xref py py-obj docutils literal"><span class="pre">limit</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">[1, 0]</span>
-<span class="go">[0, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.dot">
-<tt class="descname">dot</tt><big>(</big><em>self</em>, <em>b</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.dot"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.dot" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the dot product of Matrix self and b relaxing the condition
-of compatible dimensions: if either the number of rows or columns are
-the same as the length of b then the dot product is returned. If self
-is a row or column vector, a scalar is returned. Otherwise, a list
-of results is returned (and in that case the number of columns in self
-must match the length of b).</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.cross" title="sympy.matrices.matrices.MatrixBase.cross"><tt class="xref py py-obj docutils literal"><span class="pre">cross</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.multiply" title="sympy.matrices.matrices.MatrixBase.multiply"><tt class="xref py py-obj docutils literal"><span class="pre">multiply</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.multiply_elementwise" title="sympy.matrices.matrices.MatrixBase.multiply_elementwise"><tt class="xref py py-obj docutils literal"><span class="pre">multiply_elementwise</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">row</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-<span class="go">6</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">col</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-<span class="go">12</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-<span class="go">[6, 15, 24]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.dual">
-<tt class="descname">dual</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.dual"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.dual" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the dual of a matrix, which is:</p>
-<p><span class="math">\((1/2)*levicivita(i, j, k, l)*M(k, l)\)</span> summed over indices <span class="math">\(k\)</span> and <span class="math">\(l\)</span></p>
-<p>Since the levicivita method is anti_symmetric for any pairwise
-exchange of indices, the dual of a symmetric matrix is the zero
-matrix. Strictly speaking the dual defined here assumes that the
-&#8216;matrix&#8217; <span class="math">\(M\)</span> is a contravariant anti_symmetric second rank tensor,
-so that the dual is a covariant second rank tensor.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.eigenvals">
-<tt class="descname">eigenvals</tt><big>(</big><em>self</em>, <em>**flags</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.eigenvals"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.eigenvals" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return eigen values using the berkowitz_eigenvals routine.</p>
-<p>Since the roots routine doesn&#8217;t always work well with Floats,
-they will be replaced with Rationals before calling that
-routine. If this is not desired, set flag <tt class="docutils literal"><span class="pre">rational</span></tt> to False.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.eigenvects">
-<tt class="descname">eigenvects</tt><big>(</big><em>self</em>, <em>**flags</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.eigenvects"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.eigenvects" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return list of triples (eigenval, multiplicity, basis).</p>
-<dl class="docutils">
-<dt>The flag <tt class="docutils literal"><span class="pre">simplify</span></tt> has two effects:</dt>
-<dd>1) if bool(simplify) is True, as_content_primitive()
-will be used to tidy up normalization artifacts;
-2) if nullspace needs simplification to compute the
-basis, the simplify flag will be passed on to the
-nullspace routine which will interpret it there.</dd>
-</dl>
-<p>If the matrix contains any Floats, they will be changed to Rationals
-for computation purposes, but the answers will be returned after being
-evaluated with evalf. If it is desired to removed small imaginary
-portions during the evalf step, pass a value for the <tt class="docutils literal"><span class="pre">chop</span></tt> flag.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.evalf">
-<tt class="descname">evalf</tt><big>(</big><em>self</em>, <em>prec=None</em>, <em>**options</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.evalf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.evalf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Apply evalf() to each element of self.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.exp">
-<tt class="descname">exp</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.exp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.exp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the exponentiation of a square matrix.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.expand">
-<tt class="descname">expand</tt><big>(</big><em>self</em>, <em>deep=True</em>, <em>modulus=None</em>, <em>power_base=True</em>, <em>power_exp=True</em>, <em>mul=True</em>, <em>log=True</em>, <em>multinomial=True</em>, <em>basic=True</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.expand"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.expand" title="Permalink to this definition">¶</a></dt>
-<dd><p>Apply core.function.expand to each entry of the matrix.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="p">[</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mi">1</span><span class="p">)])</span>
-<span class="go">[x*(x + 1)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go">[x**2 + x]</span>
-<span class="go">&gt;&gt;&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.extract">
-<tt class="descname">extract</tt><big>(</big><em>self</em>, <em>rowsList</em>, <em>colsList</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.extract"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.extract" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a submatrix by specifying a list of rows and columns.
-Negative indices can be given. All indices must be in the range
--n &lt;= i &lt; n where n is the number of rows or columns.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.submatrix" title="sympy.matrices.matrices.MatrixBase.submatrix"><tt class="xref py py-obj docutils literal"><span class="pre">submatrix</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">12</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[0,  1,  2]</span>
-<span class="go">[3,  4,  5]</span>
-<span class="go">[6,  7,  8]</span>
-<span class="go">[9, 10, 11]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">extract</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">[0,  1]</span>
-<span class="go">[3,  4]</span>
-<span class="go">[9, 10]</span>
-</pre></div>
-</div>
-<p>Rows or columns can be repeated:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">extract</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">[2]</span>
-<span class="go">[2]</span>
-<span class="go">[5]</span>
-</pre></div>
-</div>
-<p>Every other row can be taken by using range to provide the indices:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">extract</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">m</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="mi">2</span><span class="p">)),</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">[2]</span>
-<span class="go">[8]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.get_diag_blocks">
-<tt class="descname">get_diag_blocks</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.get_diag_blocks"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.get_diag_blocks" title="Permalink to this definition">¶</a></dt>
-<dd><p>Obtains the square sub-matrices on the main diagonal of a square matrix.</p>
-<p>Useful for inverting symbolic matrices or solving systems of
-linear equations which may be decoupled by having a block diagonal
-structure.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="o">*</span><span class="n">z</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a1</span><span class="p">,</span> <span class="n">a2</span><span class="p">,</span> <span class="n">a3</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">get_diag_blocks</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a1</span>
-<span class="go">[1,    3]</span>
-<span class="go">[y, z**2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a2</span>
-<span class="go">[x]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a3</span>
-<span class="go">[0]</span>
-<span class="go">&gt;&gt;&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.has">
-<tt class="descname">has</tt><big>(</big><em>self</em>, <em>*patterns</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.has"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.has" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test whether any subexpression matches any of the patterns.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">Float</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(((</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="mf">0.2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Float</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.matrices.matrices.MatrixBase.hstack">
-<em class="property">classmethod </em><tt class="descname">hstack</tt><big>(</big><em>*args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.hstack"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.hstack" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a matrix formed by joining args horizontally (i.e.
-by repeated application of row_join).</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="o">.</span><span class="n">hstack</span><span class="p">(</span><span class="n">eye</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="mi">2</span><span class="o">*</span><span class="n">eye</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">[1, 0, 2, 0]</span>
-<span class="go">[0, 1, 0, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.integrate">
-<tt class="descname">integrate</tt><big>(</big><em>self</em>, <em>*args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.integrate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.integrate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Integrate each element of the matrix.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.limit" title="sympy.matrices.matrices.MatrixBase.limit"><tt class="xref py py-obj docutils literal"><span class="pre">limit</span></tt></a>, <a class="reference internal" href="../galgebra/GA/GAsympy.html#diff" title="diff"><tt class="xref py py-obj docutils literal"><span class="pre">diff</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">integrate</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="p">))</span>
-<span class="go">[x**2/2, x*y]</span>
-<span class="go">[     x,   0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">integrate</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">[2, 2*y]</span>
-<span class="go">[2,   0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.inverse_ADJ">
-<tt class="descname">inverse_ADJ</tt><big>(</big><em>self</em>, <em>iszerofunc=&lt;function _iszero at 0x10c6e2cb0&gt;</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.inverse_ADJ"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.inverse_ADJ" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculates the inverse using the adjugate matrix and a determinant.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">inv</span></tt>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.inverse_LU" title="sympy.matrices.matrices.MatrixBase.inverse_LU"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_LU</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.inverse_GE" title="sympy.matrices.matrices.MatrixBase.inverse_GE"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_GE</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.inverse_GE">
-<tt class="descname">inverse_GE</tt><big>(</big><em>self</em>, <em>iszerofunc=&lt;function _iszero at 0x10c6e2cb0&gt;</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.inverse_GE"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.inverse_GE" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculates the inverse using Gaussian elimination.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">inv</span></tt>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.inverse_LU" title="sympy.matrices.matrices.MatrixBase.inverse_LU"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_LU</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.inverse_ADJ" title="sympy.matrices.matrices.MatrixBase.inverse_ADJ"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_ADJ</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.inverse_LU">
-<tt class="descname">inverse_LU</tt><big>(</big><em>self</em>, <em>iszerofunc=&lt;function _iszero at 0x10c6e2cb0&gt;</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.inverse_LU"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.inverse_LU" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculates the inverse using LU decomposition.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">inv</span></tt>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.inverse_GE" title="sympy.matrices.matrices.MatrixBase.inverse_GE"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_GE</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.inverse_ADJ" title="sympy.matrices.matrices.MatrixBase.inverse_ADJ"><tt class="xref py py-obj docutils literal"><span class="pre">inverse_ADJ</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.is_anti_symmetric">
-<tt class="descname">is_anti_symmetric</tt><big>(</big><em>self</em>, <em>simplify=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.is_anti_symmetric"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.is_anti_symmetric" title="Permalink to this definition">¶</a></dt>
-<dd><p>Check if matrix M is an antisymmetric matrix,
-that is, M is a square matrix with all M[i, j] == -M[j, i].</p>
-<p>When <tt class="docutils literal"><span class="pre">simplify=True</span></tt> (default), the sum M[i, j] + M[j, i] is
-simplified before testing to see if it is zero. By default,
-the SymPy simplify function is used. To use a custom function
-set simplify to a function that accepts a single argument which
-returns a simplified expression. To skip simplification, set
-simplify to False but note that although this will be faster,
-it may induce false negatives.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[ 0, 1]</span>
-<span class="go">[-1, 0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_anti_symmetric</span><span class="p">()</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">y</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[ 0, 0, x]</span>
-<span class="go">[-y, 0, 0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_anti_symmetric</span><span class="p">()</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span>
-<span class="gp">... </span>                  <span class="o">-</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">,</span>
-<span class="gp">... </span>                  <span class="o">-</span><span class="n">y</span><span class="p">,</span> <span class="o">-</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-</pre></div>
-</div>
-<p>Simplification of matrix elements is done by default so even
-though two elements which should be equal and opposite wouldn&#8217;t
-pass an equality test, the matrix is still reported as
-anti-symmetric:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="o">-</span><span class="n">m</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_anti_symmetric</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>If &#8216;simplify=False&#8217; is used for the case when a Matrix is already
-simplified, this will speed things up. Here, we see that without
-simplification the matrix does not appear anti-symmetric:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_anti_symmetric</span><span class="p">(</span><span class="n">simplify</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>But if the matrix were already expanded, then it would appear
-anti-symmetric and simplification in the is_anti_symmetric routine
-is not needed:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_anti_symmetric</span><span class="p">(</span><span class="n">simplify</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.is_diagonal">
-<tt class="descname">is_diagonal</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.is_diagonal"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.is_diagonal" title="Permalink to this definition">¶</a></dt>
-<dd><p>Check if matrix is diagonal,
-that is matrix in which the entries outside the main diagonal are all zero.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.is_lower" title="sympy.matrices.matrices.MatrixBase.is_lower"><tt class="xref py py-obj docutils literal"><span class="pre">is_lower</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.is_upper" title="sympy.matrices.matrices.MatrixBase.is_upper"><tt class="xref py py-obj docutils literal"><span class="pre">is_upper</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.is_diagonalizable" title="sympy.matrices.matrices.MatrixBase.is_diagonalizable"><tt class="xref py py-obj docutils literal"><span class="pre">is_diagonalizable</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.diagonalize" title="sympy.matrices.matrices.MatrixBase.diagonalize"><tt class="xref py py-obj docutils literal"><span class="pre">diagonalize</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">diag</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[1, 0]</span>
-<span class="go">[0, 2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_diagonal</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[1, 1]</span>
-<span class="go">[0, 2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_diagonal</span><span class="p">()</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">diag</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[1, 0, 0]</span>
-<span class="go">[0, 2, 0]</span>
-<span class="go">[0, 0, 3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_diagonal</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.is_diagonalizable">
-<tt class="descname">is_diagonalizable</tt><big>(</big><em>self</em>, <em>reals_only=False</em>, <em>clear_subproducts=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.is_diagonalizable"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.is_diagonalizable" title="Permalink to this definition">¶</a></dt>
-<dd><p>Check if matrix is diagonalizable.</p>
-<p>If reals_only==True then check that diagonalized matrix consists of the only not complex values.</p>
-<p>Some subproducts could be used further in other methods to avoid double calculations,
-By default (if clear_subproducts==True) they will be deleted.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.is_diagonal" title="sympy.matrices.matrices.MatrixBase.is_diagonal"><tt class="xref py py-obj docutils literal"><span class="pre">is_diagonal</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.diagonalize" title="sympy.matrices.matrices.MatrixBase.diagonalize"><tt class="xref py py-obj docutils literal"><span class="pre">diagonalize</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[1,  2, 0]</span>
-<span class="go">[0,  3, 0]</span>
-<span class="go">[2, -4, 2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_diagonalizable</span><span class="p">()</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[0, 1]</span>
-<span class="go">[0, 0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_diagonalizable</span><span class="p">()</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[ 0, 1]</span>
-<span class="go">[-1, 0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_diagonalizable</span><span class="p">()</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_diagonalizable</span><span class="p">(</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.matrices.matrices.MatrixBase.is_lower">
-<tt class="descname">is_lower</tt><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.is_lower"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.is_lower" title="Permalink to this definition">¶</a></dt>
-<dd><p>Check if matrix is a lower triangular matrix. True can be returned
-even if the matrix is not square.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.is_upper" title="sympy.matrices.matrices.MatrixBase.is_upper"><tt class="xref py py-obj docutils literal"><span class="pre">is_upper</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.is_diagonal" title="sympy.matrices.matrices.MatrixBase.is_diagonal"><tt class="xref py py-obj docutils literal"><span class="pre">is_diagonal</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.is_lower_hessenberg" title="sympy.matrices.matrices.MatrixBase.is_lower_hessenberg"><tt class="xref py py-obj docutils literal"><span class="pre">is_lower_hessenberg</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[1, 0]</span>
-<span class="go">[0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_lower</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span> <span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[0, 0, 0]</span>
-<span class="go">[2, 0, 0]</span>
-<span class="go">[1, 4, 0]</span>
-<span class="go">[6, 6, 5]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_lower</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="p">,</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[x**2 + y, x + y**2]</span>
-<span class="go">[       0,    x + y]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_lower</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.matrices.matrices.MatrixBase.is_lower_hessenberg">
-<tt class="descname">is_lower_hessenberg</tt><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.is_lower_hessenberg"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.is_lower_hessenberg" title="Permalink to this definition">¶</a></dt>
-<dd><p>Checks if the matrix is in the lower-Hessenberg form.</p>
-<p>The lower hessenberg matrix has zero entries
-above the first superdiagonal.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.is_upper_hessenberg" title="sympy.matrices.matrices.MatrixBase.is_upper_hessenberg"><tt class="xref py py-obj docutils literal"><span class="pre">is_upper_hessenberg</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.is_lower" title="sympy.matrices.matrices.MatrixBase.is_lower"><tt class="xref py py-obj docutils literal"><span class="pre">is_lower</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">],</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
-<span class="go">[1, 2, 0, 0]</span>
-<span class="go">[5, 2, 3, 0]</span>
-<span class="go">[3, 4, 3, 7]</span>
-<span class="go">[5, 6, 1, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">is_lower_hessenberg</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.is_nilpotent">
-<tt class="descname">is_nilpotent</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.is_nilpotent"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.is_nilpotent" title="Permalink to this definition">¶</a></dt>
-<dd><p>Checks if a matrix is nilpotent.</p>
-<p>A matrix B is nilpotent if for some integer k, B**k is
-a zero matrix.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">is_nilpotent</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">is_nilpotent</span><span class="p">()</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.matrices.matrices.MatrixBase.is_square">
-<tt class="descname">is_square</tt><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.is_square"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.is_square" title="Permalink to this definition">¶</a></dt>
-<dd><p>Checks if a matrix is square.</p>
-<p>A matrix is square if the number of rows equals the number of columns.
-The empty matrix is square by definition, since the number of rows and
-the number of columns are both zero.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">is_square</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">is_square</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span><span class="o">.</span><span class="n">is_square</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.is_symbolic">
-<tt class="descname">is_symbolic</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.is_symbolic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.is_symbolic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Checks if any elements contain Symbols.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">is_symbolic</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.is_symmetric">
-<tt class="descname">is_symmetric</tt><big>(</big><em>self</em>, <em>simplify=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.is_symmetric"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.is_symmetric" title="Permalink to this definition">¶</a></dt>
-<dd><p>Check if matrix is symmetric matrix,
-that is square matrix and is equal to its transpose.</p>
-<p>By default, simplifications occur before testing symmetry.
-They can be skipped using &#8216;simplify=False&#8217;; while speeding things a bit,
-this may however induce false negatives.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[0, 1]</span>
-<span class="go">[1, 2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_symmetric</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[0, 1]</span>
-<span class="go">[2, 0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_symmetric</span><span class="p">()</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[0, 0, 0]</span>
-<span class="go">[0, 0, 0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_symmetric</span><span class="p">()</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[         1, x**2 + 2*x + 1, y]</span>
-<span class="go">[(x + 1)**2,              2, 0]</span>
-<span class="go">[         y,              0, 3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_symmetric</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>If the matrix is already simplified, you may speed-up is_symmetric()
-test by using &#8216;simplify=False&#8217;.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_symmetric</span><span class="p">(</span><span class="n">simplify</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m1</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m1</span><span class="o">.</span><span class="n">is_symmetric</span><span class="p">(</span><span class="n">simplify</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.matrices.matrices.MatrixBase.is_upper">
-<tt class="descname">is_upper</tt><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.is_upper"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.is_upper" title="Permalink to this definition">¶</a></dt>
-<dd><p>Check if matrix is an upper triangular matrix. True can be returned
-even if the matrix is not square.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.is_lower" title="sympy.matrices.matrices.MatrixBase.is_lower"><tt class="xref py py-obj docutils literal"><span class="pre">is_lower</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.is_diagonal" title="sympy.matrices.matrices.MatrixBase.is_diagonal"><tt class="xref py py-obj docutils literal"><span class="pre">is_diagonal</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.is_upper_hessenberg" title="sympy.matrices.matrices.MatrixBase.is_upper_hessenberg"><tt class="xref py py-obj docutils literal"><span class="pre">is_upper_hessenberg</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[1, 0]</span>
-<span class="go">[0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_upper</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span> <span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[5, 1, 9]</span>
-<span class="go">[0, 4, 6]</span>
-<span class="go">[0, 0, 5]</span>
-<span class="go">[0, 0, 0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_upper</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[4, 2, 5]</span>
-<span class="go">[6, 1, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">is_upper</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.matrices.matrices.MatrixBase.is_upper_hessenberg">
-<tt class="descname">is_upper_hessenberg</tt><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.is_upper_hessenberg"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.is_upper_hessenberg" title="Permalink to this definition">¶</a></dt>
-<dd><p>Checks if the matrix is the upper-Hessenberg form.</p>
-<p>The upper hessenberg matrix has zero entries
-below the first subdiagonal.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.is_lower_hessenberg" title="sympy.matrices.matrices.MatrixBase.is_lower_hessenberg"><tt class="xref py py-obj docutils literal"><span class="pre">is_lower_hessenberg</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.is_upper" title="sympy.matrices.matrices.MatrixBase.is_upper"><tt class="xref py py-obj docutils literal"><span class="pre">is_upper</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
-<span class="go">[1, 4, 2, 3]</span>
-<span class="go">[3, 4, 1, 7]</span>
-<span class="go">[0, 2, 3, 4]</span>
-<span class="go">[0, 0, 1, 3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">is_upper_hessenberg</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.matrices.matrices.MatrixBase.is_zero">
-<tt class="descname">is_zero</tt><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.is_zero"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.is_zero" title="Permalink to this definition">¶</a></dt>
-<dd><p>Checks if a matrix is a zero matrix.</p>
-<p>A matrix is zero if every element is zero.  A matrix need not be square
-to be considered zero.  The empty matrix is zero by the principle of
-vacuous truth.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">zeros</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">is_zero</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">is_zero</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span><span class="o">.</span><span class="n">is_zero</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="o">.</span><span class="n">is_zero</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.jacobian">
-<tt class="descname">jacobian</tt><big>(</big><em>self</em>, <em>X</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.jacobian"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.jacobian" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculates the Jacobian matrix (derivative of a vectorial function).</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>self</strong> : vector of expressions representing functions f_i(x_1, ..., x_n).</p>
-<p><strong>X</strong> : set of x_i&#8217;s in order, it can be a list or a Matrix</p>
-<p><strong>Both self and X can be a row or a column matrix in any order</strong> :</p>
-<p class="last"><strong>(i.e., jacobian() should always work).</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">hessian</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">wronskian</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">rho</span><span class="p">,</span> <span class="n">phi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span><span class="n">rho</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">phi</span><span class="p">),</span> <span class="n">rho</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">phi</span><span class="p">),</span> <span class="n">rho</span><span class="o">**</span><span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span><span class="n">rho</span><span class="p">,</span> <span class="n">phi</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span>
-<span class="go">[cos(phi), -rho*sin(phi)]</span>
-<span class="go">[sin(phi),  rho*cos(phi)]</span>
-<span class="go">[   2*rho,             0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span><span class="n">rho</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">phi</span><span class="p">),</span> <span class="n">rho</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">phi</span><span class="p">)])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span>
-<span class="go">[cos(phi), -rho*sin(phi)]</span>
-<span class="go">[sin(phi),  rho*cos(phi)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.jordan_cells">
-<tt class="descname">jordan_cells</tt><big>(</big><em>self</em>, <em>calc_transformation=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.jordan_cells"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.jordan_cells" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a list of Jordan cells of current matrix.
-This list shape Jordan matrix J.</p>
-<p>If calc_transformation is specified as False, then transformation P such that</p>
-<blockquote>
-<div>J = P^-1 * M * P</div></blockquote>
-<p>will not be calculated.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.jordan_form" title="sympy.matrices.matrices.MatrixBase.jordan_form"><tt class="xref py py-obj docutils literal"><span class="pre">jordan_form</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>Calculation of transformation P is not implemented yet.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="p">[</span>
-<span class="gp">... </span> <span class="mi">6</span><span class="p">,</span>  <span class="mi">5</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span>
-<span class="gp">... </span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span>  <span class="mi">3</span><span class="p">,</span>  <span class="mi">3</span><span class="p">,</span>
-<span class="gp">... </span> <span class="mi">2</span><span class="p">,</span>  <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span>
-<span class="gp">... </span><span class="o">-</span><span class="mi">1</span><span class="p">,</span>  <span class="mi">1</span><span class="p">,</span>  <span class="mi">5</span><span class="p">,</span>  <span class="mi">5</span><span class="p">])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">Jcells</span><span class="p">)</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">jordan_cells</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Jcells</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">[2, 1]</span>
-<span class="go">[0, 2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Jcells</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-<span class="go">[2, 1]</span>
-<span class="go">[0, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.jordan_form">
-<tt class="descname">jordan_form</tt><big>(</big><em>self</em>, <em>calc_transformation=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.jordan_form"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.jordan_form" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return Jordan form J of current matrix.</p>
-<p>If calc_transformation is specified as False, then transformation P such that</p>
-<blockquote>
-<div>J = P^-1 * M * P</div></blockquote>
-<p>will not be calculated.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.jordan_cells" title="sympy.matrices.matrices.MatrixBase.jordan_cells"><tt class="xref py py-obj docutils literal"><span class="pre">jordan_cells</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>Calculation of transformation P is not implemented yet.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span>
-<span class="gp">... </span>       <span class="p">[</span> <span class="mi">6</span><span class="p">,</span>  <span class="mi">5</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">],</span>
-<span class="gp">... </span>       <span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span>  <span class="mi">3</span><span class="p">,</span>  <span class="mi">3</span><span class="p">],</span>
-<span class="gp">... </span>       <span class="p">[</span> <span class="mi">2</span><span class="p">,</span>  <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">],</span>
-<span class="gp">... </span>       <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span>  <span class="mi">1</span><span class="p">,</span>  <span class="mi">5</span><span class="p">,</span>  <span class="mi">5</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">J</span><span class="p">)</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">jordan_form</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">J</span>
-<span class="go">[2, 1, 0, 0]</span>
-<span class="go">[0, 2, 0, 0]</span>
-<span class="go">[0, 0, 2, 1]</span>
-<span class="go">[0, 0, 0, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.key2bounds">
-<tt class="descname">key2bounds</tt><big>(</big><em>self</em>, <em>keys</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.key2bounds"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.key2bounds" title="Permalink to this definition">¶</a></dt>
-<dd><p>Converts a key with potentially mixed types of keys (integer and slice)
-into a tuple of ranges and raises an error if any index is out of self&#8217;s
-range.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.key2ij" title="sympy.matrices.matrices.MatrixBase.key2ij"><tt class="xref py py-obj docutils literal"><span class="pre">key2ij</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.key2ij">
-<tt class="descname">key2ij</tt><big>(</big><em>self</em>, <em>key</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.key2ij"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.key2ij" title="Permalink to this definition">¶</a></dt>
-<dd><p>Converts key into canonical form, converting integers or indexable
-items into valid integers for self&#8217;s range or returning slices
-unchanged.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.key2bounds" title="sympy.matrices.matrices.MatrixBase.key2bounds"><tt class="xref py py-obj docutils literal"><span class="pre">key2bounds</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.limit">
-<tt class="descname">limit</tt><big>(</big><em>self</em>, <em>*args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.limit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.limit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculate the limit of each element in the matrix.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.integrate" title="sympy.matrices.matrices.MatrixBase.integrate"><tt class="xref py py-obj docutils literal"><span class="pre">integrate</span></tt></a>, <a class="reference internal" href="../galgebra/GA/GAsympy.html#diff" title="diff"><tt class="xref py py-obj docutils literal"><span class="pre">diff</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">[2, y]</span>
-<span class="go">[1, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.lower_triangular_solve">
-<tt class="descname">lower_triangular_solve</tt><big>(</big><em>self</em>, <em>rhs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.lower_triangular_solve"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.lower_triangular_solve" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solves Ax = B, where A is a lower triangular matrix.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.upper_triangular_solve" title="sympy.matrices.matrices.MatrixBase.upper_triangular_solve"><tt class="xref py py-obj docutils literal"><span class="pre">upper_triangular_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.cholesky_solve" title="sympy.matrices.matrices.MatrixBase.cholesky_solve"><tt class="xref py py-obj docutils literal"><span class="pre">cholesky_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.diagonal_solve" title="sympy.matrices.matrices.MatrixBase.diagonal_solve"><tt class="xref py py-obj docutils literal"><span class="pre">diagonal_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LDLsolve" title="sympy.matrices.matrices.MatrixBase.LDLsolve"><tt class="xref py py-obj docutils literal"><span class="pre">LDLsolve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LUsolve" title="sympy.matrices.matrices.MatrixBase.LUsolve"><tt class="xref py py-obj docutils literal"><span class="pre">LUsolve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.QRsolve" title="sympy.matrices.matrices.MatrixBase.QRsolve"><tt class="xref py py-obj docutils literal"><span class="pre">QRsolve</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.minorEntry">
-<tt class="descname">minorEntry</tt><big>(</big><em>self</em>, <em>i</em>, <em>j</em>, <em>method='berkowitz'</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.minorEntry"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.minorEntry" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculate the minor of an element.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.minorMatrix" title="sympy.matrices.matrices.MatrixBase.minorMatrix"><tt class="xref py py-obj docutils literal"><span class="pre">minorMatrix</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.cofactor" title="sympy.matrices.matrices.MatrixBase.cofactor"><tt class="xref py py-obj docutils literal"><span class="pre">cofactor</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.cofactorMatrix" title="sympy.matrices.matrices.MatrixBase.cofactorMatrix"><tt class="xref py py-obj docutils literal"><span class="pre">cofactorMatrix</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.minorMatrix">
-<tt class="descname">minorMatrix</tt><big>(</big><em>self</em>, <em>i</em>, <em>j</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.minorMatrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.minorMatrix" title="Permalink to this definition">¶</a></dt>
-<dd><p>Creates the minor matrix of a given element.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.minorEntry" title="sympy.matrices.matrices.MatrixBase.minorEntry"><tt class="xref py py-obj docutils literal"><span class="pre">minorEntry</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.cofactor" title="sympy.matrices.matrices.MatrixBase.cofactor"><tt class="xref py py-obj docutils literal"><span class="pre">cofactor</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.cofactorMatrix" title="sympy.matrices.matrices.MatrixBase.cofactorMatrix"><tt class="xref py py-obj docutils literal"><span class="pre">cofactorMatrix</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.multiply">
-<tt class="descname">multiply</tt><big>(</big><em>self</em>, <em>b</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.multiply"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.multiply" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns self*b</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.dot" title="sympy.matrices.matrices.MatrixBase.dot"><tt class="xref py py-obj docutils literal"><span class="pre">dot</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.cross" title="sympy.matrices.matrices.MatrixBase.cross"><tt class="xref py py-obj docutils literal"><span class="pre">cross</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.multiply_elementwise" title="sympy.matrices.matrices.MatrixBase.multiply_elementwise"><tt class="xref py py-obj docutils literal"><span class="pre">multiply_elementwise</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.multiply_elementwise">
-<tt class="descname">multiply_elementwise</tt><big>(</big><em>self</em>, <em>b</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.multiply_elementwise"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.multiply_elementwise" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the Hadamard product (elementwise product) of A and B</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.cross" title="sympy.matrices.matrices.MatrixBase.cross"><tt class="xref py py-obj docutils literal"><span class="pre">cross</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.dot" title="sympy.matrices.matrices.MatrixBase.dot"><tt class="xref py py-obj docutils literal"><span class="pre">dot</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.multiply" title="sympy.matrices.matrices.MatrixBase.multiply"><tt class="xref py py-obj docutils literal"><span class="pre">multiply</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">100</span><span class="p">],</span> <span class="p">[</span><span class="mi">100</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">multiply_elementwise</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
-<span class="go">[  0, 10, 200]</span>
-<span class="go">[300, 40,   5]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.n">
-<tt class="descname">n</tt><big>(</big><em>self</em>, <em>prec=None</em>, <em>**options</em><big>)</big><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.n" title="Permalink to this definition">¶</a></dt>
-<dd><p>Apply evalf() to each element of self.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.norm">
-<tt class="descname">norm</tt><big>(</big><em>self</em>, <em>ord=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.norm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.norm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the Norm of a Matrix or Vector.
-In the simplest case this is the geometric size of the vector
-Other norms can be specified by the ord parameter</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="8%" />
-<col width="47%" />
-<col width="44%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head">ord</th>
-<th class="head">norm for matrices</th>
-<th class="head">norm for vectors</th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-even"><td>None</td>
-<td>Frobenius norm</td>
-<td>2-norm</td>
-</tr>
-<tr class="row-odd"><td>&#8216;fro&#8217;</td>
-<td>Frobenius norm</td>
-<td><ul class="first last simple">
-<li>does not exist</li>
-</ul>
-</td>
-</tr>
-<tr class="row-even"><td>inf</td>
-<td>&#8211;</td>
-<td>max(abs(x))</td>
-</tr>
-<tr class="row-odd"><td>-inf</td>
-<td>&#8211;</td>
-<td>min(abs(x))</td>
-</tr>
-<tr class="row-even"><td>1</td>
-<td>&#8211;</td>
-<td>as below</td>
-</tr>
-<tr class="row-odd"><td>-1</td>
-<td>&#8211;</td>
-<td>as below</td>
-</tr>
-<tr class="row-even"><td>2</td>
-<td>2-norm (largest sing. value)</td>
-<td>as below</td>
-</tr>
-<tr class="row-odd"><td>-2</td>
-<td>smallest singular value</td>
-<td>as below</td>
-</tr>
-<tr class="row-even"><td>other</td>
-<td><ul class="first last simple">
-<li>does not exist</li>
-</ul>
-</td>
-<td>sum(abs(x)**ord)**(1./ord)</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.normalized" title="sympy.matrices.matrices.MatrixBase.normalized"><tt class="xref py py-obj docutils literal"><span class="pre">normalized</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">trigsimp</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">trigsimp</span><span class="p">(</span> <span class="n">v</span><span class="o">.</span><span class="n">norm</span><span class="p">()</span> <span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">(sin(x)**10 + cos(x)**10)**(1/10)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="c"># Spectral norm (max of |Ax|/|x| under 2-vector-norm)</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span> <span class="c"># Inverse spectral norm (smallest singular value)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">norm</span><span class="p">()</span> <span class="c"># Frobenius Norm</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">])</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="o">-</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.normalized">
-<tt class="descname">normalized</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.normalized"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.normalized" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the normalized version of <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.norm" title="sympy.matrices.matrices.MatrixBase.norm"><tt class="xref py py-obj docutils literal"><span class="pre">norm</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.nullspace">
-<tt class="descname">nullspace</tt><big>(</big><em>self</em>, <em>simplified=False</em>, <em>simplify=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.nullspace"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.nullspace" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns list of vectors (Matrix objects) that span nullspace of self</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.permuteBkwd">
-<tt class="descname">permuteBkwd</tt><big>(</big><em>self</em>, <em>perm</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.permuteBkwd"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.permuteBkwd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Permute the rows of the matrix with the given permutation in reverse.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.permuteFwd" title="sympy.matrices.matrices.MatrixBase.permuteFwd"><tt class="xref py py-obj docutils literal"><span class="pre">permuteFwd</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">permuteBkwd</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
-<span class="go">[0, 1, 0]</span>
-<span class="go">[0, 0, 1]</span>
-<span class="go">[1, 0, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.permuteFwd">
-<tt class="descname">permuteFwd</tt><big>(</big><em>self</em>, <em>perm</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.permuteFwd"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.permuteFwd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Permute the rows of the matrix with the given permutation.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.permuteBkwd" title="sympy.matrices.matrices.MatrixBase.permuteBkwd"><tt class="xref py py-obj docutils literal"><span class="pre">permuteBkwd</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">permuteFwd</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
-<span class="go">[0, 0, 1]</span>
-<span class="go">[1, 0, 0]</span>
-<span class="go">[0, 1, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.print_nonzero">
-<tt class="descname">print_nonzero</tt><big>(</big><em>self</em>, <em>symb='X'</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.print_nonzero"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.print_nonzero" title="Permalink to this definition">¶</a></dt>
-<dd><p>Shows location of non-zero entries for fast shape lookup.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="n">i</span><span class="o">*</span><span class="mi">3</span><span class="o">+</span><span class="n">j</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[0, 1, 2]</span>
-<span class="go">[3, 4, 5]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">print_nonzero</span><span class="p">()</span>
-<span class="go">[ XX]</span>
-<span class="go">[XXX]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">print_nonzero</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="go">[x   ]</span>
-<span class="go">[ x  ]</span>
-<span class="go">[  x ]</span>
-<span class="go">[   x]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.project">
-<tt class="descname">project</tt><big>(</big><em>self</em>, <em>v</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.project"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.project" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the projection of <tt class="docutils literal"><span class="pre">self</span></tt> onto the line containing <tt class="docutils literal"><span class="pre">v</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">V</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">V</span><span class="o">.</span><span class="n">project</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">[sqrt(3)/2, 0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">V</span><span class="o">.</span><span class="n">project</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">)</span>
-<span class="go">[sqrt(3)/2, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.row_insert">
-<tt class="descname">row_insert</tt><big>(</big><em>self</em>, <em>pos</em>, <em>mti</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.row_insert"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.row_insert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Insert one or more rows at the given row position.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">row</span></tt>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.col_insert" title="sympy.matrices.matrices.MatrixBase.col_insert"><tt class="xref py py-obj docutils literal"><span class="pre">col_insert</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">zeros</span><span class="p">,</span> <span class="n">ones</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">V</span> <span class="o">=</span> <span class="n">ones</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">row_insert</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">V</span><span class="p">)</span>
-<span class="go">[0, 0, 0]</span>
-<span class="go">[1, 1, 1]</span>
-<span class="go">[0, 0, 0]</span>
-<span class="go">[0, 0, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.row_join">
-<tt class="descname">row_join</tt><big>(</big><em>self</em>, <em>rhs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.row_join"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.row_join" title="Permalink to this definition">¶</a></dt>
-<dd><p>Concatenates two matrices along self&#8217;s last and rhs&#8217;s first column</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">row</span></tt>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.col_join" title="sympy.matrices.matrices.MatrixBase.col_join"><tt class="xref py py-obj docutils literal"><span class="pre">col_join</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">zeros</span><span class="p">,</span> <span class="n">ones</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">V</span> <span class="o">=</span> <span class="n">ones</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">row_join</span><span class="p">(</span><span class="n">V</span><span class="p">)</span>
-<span class="go">[0, 0, 0, 1]</span>
-<span class="go">[0, 0, 0, 1]</span>
-<span class="go">[0, 0, 0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.rref">
-<tt class="descname">rref</tt><big>(</big><em>self</em>, <em>simplified=False</em>, <em>iszerofunc=&lt;function _iszero at 0x10c6e2cb0&gt;</em>, <em>simplify=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.rref"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.rref" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return reduced row-echelon form of matrix and indices of pivot vars.</p>
-<p>To simplify elements before finding nonzero pivots set simplify=True
-(to use the default SymPy simplify function) or pass a custom
-simplify function.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span> <span class="o">-</span> <span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">rref</span><span class="p">()</span>
-<span class="go">([1, 0]</span>
-<span class="go">[0, 1], [0, 1])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.matrices.matrices.MatrixBase.shape">
-<tt class="descname">shape</tt><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.shape"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.shape" title="Permalink to this definition">¶</a></dt>
-<dd><p>The shape (dimensions) of the matrix as the 2-tuple (rows, cols).</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">zeros</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">shape</span>
-<span class="go">(2, 3)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">rows</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">cols</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.simplify">
-<tt class="descname">simplify</tt><big>(</big><em>self</em>, <em>ratio=1.7</em>, <em>measure=&lt;function count_ops at 0x10c198b00&gt;</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.simplify"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.simplify" title="Permalink to this definition">¶</a></dt>
-<dd><p>Apply simplify to each element of the matrix.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">SparseMatrix</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="p">[</span><span class="n">x</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">])</span>
-<span class="go">[x*sin(y)**2 + x*cos(y)**2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-<span class="go">[x]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.singular_values">
-<tt class="descname">singular_values</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.singular_values"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.singular_values" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the singular values of a Matrix</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.condition_number" title="sympy.matrices.matrices.MatrixBase.condition_number"><tt class="xref py py-obj docutils literal"><span class="pre">condition_number</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">singular_values</span><span class="p">()</span>
-<span class="go">[sqrt(x**2 + 1), 1, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.solve">
-<tt class="descname">solve</tt><big>(</big><em>self</em>, <em>rhs</em>, <em>method='GE'</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.solve"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.solve" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return solution to self*soln = rhs using given inversion method.</p>
-<p>For a list of possible inversion methods, see the .inv() docstring.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.solve_least_squares">
-<tt class="descname">solve_least_squares</tt><big>(</big><em>self</em>, <em>rhs</em>, <em>method='CH'</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.solve_least_squares"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.solve_least_squares" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the least-square fit to the data.</p>
-<p>By default the cholesky_solve routine is used (method=&#8217;CH&#8217;); other
-methods of matrix inversion can be used. To find out which are
-available, see the docstring of the .inv() method.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">ones</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">row_join</span><span class="p">(</span><span class="n">B</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span>
-<span class="go">[1, 2]</span>
-<span class="go">[2, 3]</span>
-<span class="go">[3, 4]</span>
-</pre></div>
-</div>
-<p>If each line of S represent coefficients of Ax + By
-and x and y are [2, 3] then S*xy is:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">r</span> <span class="o">=</span> <span class="n">S</span><span class="o">*</span><span class="n">Matrix</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]);</span> <span class="n">r</span>
-<span class="go">[ 8]</span>
-<span class="go">[13]</span>
-<span class="go">[18]</span>
-</pre></div>
-</div>
-<p>But let&#8217;s add 1 to the middle value and then solve for the
-least-squares value of xy:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">xy</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">solve_least_squares</span><span class="p">(</span><span class="n">Matrix</span><span class="p">([</span><span class="mi">8</span><span class="p">,</span> <span class="mi">14</span><span class="p">,</span> <span class="mi">18</span><span class="p">]));</span> <span class="n">xy</span>
-<span class="go">[ 5/3]</span>
-<span class="go">[10/3]</span>
-</pre></div>
-</div>
-<p>The error is given by S*xy - r:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">*</span><span class="n">xy</span> <span class="o">-</span> <span class="n">r</span>
-<span class="go">[1/3]</span>
-<span class="go">[1/3]</span>
-<span class="go">[1/3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">.</span><span class="n">norm</span><span class="p">()</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.58</span>
-</pre></div>
-</div>
-<p>If a different xy is used, the norm will be higher:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">xy</span> <span class="o">+=</span> <span class="n">ones</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">S</span><span class="o">*</span><span class="n">xy</span> <span class="o">-</span> <span class="n">r</span><span class="p">)</span><span class="o">.</span><span class="n">norm</span><span class="p">()</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">1.5</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.submatrix">
-<tt class="descname">submatrix</tt><big>(</big><em>self</em>, <em>keys</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.submatrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.submatrix" title="Permalink to this definition">¶</a></dt>
-<dd><p>Get a slice/submatrix of the matrix using the given slice.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.extract" title="sympy.matrices.matrices.MatrixBase.extract"><tt class="xref py py-obj docutils literal"><span class="pre">extract</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="n">i</span><span class="o">+</span><span class="n">j</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[0, 1, 2, 3]</span>
-<span class="go">[1, 2, 3, 4]</span>
-<span class="go">[2, 3, 4, 5]</span>
-<span class="go">[3, 4, 5, 6]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="p">[:</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
-<span class="go">[1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="p">[:</span><span class="mi">2</span><span class="p">,</span> <span class="p">:</span><span class="mi">1</span><span class="p">]</span>
-<span class="go">[0]</span>
-<span class="go">[1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="p">[</span><span class="mi">2</span><span class="p">:</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">:</span><span class="mi">4</span><span class="p">]</span>
-<span class="go">[4, 5]</span>
-<span class="go">[5, 6]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.subs">
-<tt class="descname">subs</tt><big>(</big><em>self</em>, <em>*args</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.subs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.subs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a new matrix with subs applied to each entry.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span><span class="p">,</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">SparseMatrix</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="p">[</span><span class="n">x</span><span class="p">])</span>
-<span class="go">[x]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">[y]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">(</span><span class="n">_</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">[x]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.upper_triangular_solve">
-<tt class="descname">upper_triangular_solve</tt><big>(</big><em>self</em>, <em>rhs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.upper_triangular_solve"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.upper_triangular_solve" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solves Ax = B, where A is an upper triangular matrix.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.lower_triangular_solve" title="sympy.matrices.matrices.MatrixBase.lower_triangular_solve"><tt class="xref py py-obj docutils literal"><span class="pre">lower_triangular_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.cholesky_solve" title="sympy.matrices.matrices.MatrixBase.cholesky_solve"><tt class="xref py py-obj docutils literal"><span class="pre">cholesky_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.diagonal_solve" title="sympy.matrices.matrices.MatrixBase.diagonal_solve"><tt class="xref py py-obj docutils literal"><span class="pre">diagonal_solve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LDLsolve" title="sympy.matrices.matrices.MatrixBase.LDLsolve"><tt class="xref py py-obj docutils literal"><span class="pre">LDLsolve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.LUsolve" title="sympy.matrices.matrices.MatrixBase.LUsolve"><tt class="xref py py-obj docutils literal"><span class="pre">LUsolve</span></tt></a>, <a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.QRsolve" title="sympy.matrices.matrices.MatrixBase.QRsolve"><tt class="xref py py-obj docutils literal"><span class="pre">QRsolve</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.values">
-<tt class="descname">values</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.values"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.values" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return non-zero values of self.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.vec">
-<tt class="descname">vec</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.vec"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.vec" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the Matrix converted into a one column matrix by stacking columns</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.vech" title="sympy.matrices.matrices.MatrixBase.vech"><tt class="xref py py-obj docutils literal"><span class="pre">vech</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">=</span><span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[1, 3]</span>
-<span class="go">[2, 4]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">vec</span><span class="p">()</span>
-<span class="go">[1]</span>
-<span class="go">[2]</span>
-<span class="go">[3]</span>
-<span class="go">[4]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.MatrixBase.vech">
-<tt class="descname">vech</tt><big>(</big><em>self</em>, <em>diagonal=True</em>, <em>check_symmetry=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.vech"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.vech" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the unique elements of a symmetric Matrix as a one column matrix
-by stacking the elements in the lower triangle.</p>
-<p>Arguments:
-diagonal &#8211; include the diagonal cells of self or not
-check_symmetry &#8211; checks symmetry of self but not completely reliably</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.matrices.MatrixBase.vec" title="sympy.matrices.matrices.MatrixBase.vec"><tt class="xref py py-obj docutils literal"><span class="pre">vec</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">=</span><span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[1, 2]</span>
-<span class="go">[2, 3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">vech</span><span class="p">()</span>
-<span class="go">[1]</span>
-<span class="go">[2]</span>
-<span class="go">[3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">vech</span><span class="p">(</span><span class="n">diagonal</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">[2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.matrices.matrices.MatrixBase.vstack">
-<em class="property">classmethod </em><tt class="descname">vstack</tt><big>(</big><em>*args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixBase.vstack"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixBase.vstack" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a matrix formed by joining args vertically (i.e.
-by repeated application of col_join).</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="o">.</span><span class="n">vstack</span><span class="p">(</span><span class="n">eye</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="mi">2</span><span class="o">*</span><span class="n">eye</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">[1, 0]</span>
-<span class="go">[0, 1]</span>
-<span class="go">[2, 0]</span>
-<span class="go">[0, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="matrix-exceptions-reference">
-<h2>Matrix Exceptions Reference<a class="headerlink" href="#matrix-exceptions-reference" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.matrices.matrices.MatrixError">
-<em class="property">class </em><tt class="descclassname">sympy.matrices.matrices.</tt><tt class="descname">MatrixError</tt><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#MatrixError"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.MatrixError" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.matrices.matrices.ShapeError">
-<em class="property">class </em><tt class="descclassname">sympy.matrices.matrices.</tt><tt class="descname">ShapeError</tt><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#ShapeError"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.ShapeError" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wrong matrix shape</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.matrices.matrices.NonSquareMatrixError">
-<em class="property">class </em><tt class="descclassname">sympy.matrices.matrices.</tt><tt class="descname">NonSquareMatrixError</tt><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#NonSquareMatrixError"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.NonSquareMatrixError" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="matrix-functions-reference">
-<h2>Matrix Functions Reference<a class="headerlink" href="#matrix-functions-reference" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.matrices.matrices.classof">
-<tt class="descclassname">sympy.matrices.matrices.</tt><tt class="descname">classof</tt><big>(</big><em>A</em>, <em>B</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#classof"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.classof" title="Permalink to this definition">¶</a></dt>
-<dd><p>Get the type of the result when combining matrices of different types.</p>
-<p>Currently the strategy is that immutability is contagious.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">ImmutableMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices.matrices</span> <span class="kn">import</span> <span class="n">classof</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span> <span class="c"># a Mutable Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">IM</span> <span class="o">=</span> <span class="n">ImmutableMatrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">classof</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">IM</span><span class="p">)</span>
-<span class="go">&lt;class &#39;sympy.matrices.immutable.ImmutableMatrix&#39;&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.matrix_multiply_elementwise">
-<tt class="descclassname">sympy.matrices.dense.</tt><tt class="descname">matrix_multiply_elementwise</tt><big>(</big><em>A</em>, <em>B</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#matrix_multiply_elementwise"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.matrix_multiply_elementwise" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the Hadamard product (elementwise product) of A and B</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">matrix_multiply_elementwise</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">100</span><span class="p">],</span> <span class="p">[</span><span class="mi">100</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">matrix_multiply_elementwise</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-<span class="go">[  0, 10, 200]</span>
-<span class="go">[300, 40,   5]</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">__mul__</span></tt></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.zeros">
-<tt class="descclassname">sympy.matrices.dense.</tt><tt class="descname">zeros</tt><big>(</big><em>r</em>, <em>c=None</em>, <em>cls=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#zeros"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.zeros" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a matrix of zeros with <tt class="docutils literal"><span class="pre">r</span></tt> rows and <tt class="docutils literal"><span class="pre">c</span></tt> columns;
-if <tt class="docutils literal"><span class="pre">c</span></tt> is omitted a square matrix will be returned.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">ones</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">eye</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">diag</span></tt></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.ones">
-<tt class="descclassname">sympy.matrices.dense.</tt><tt class="descname">ones</tt><big>(</big><em>r</em>, <em>c=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#ones"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.ones" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a matrix of ones with <tt class="docutils literal"><span class="pre">r</span></tt> rows and <tt class="docutils literal"><span class="pre">c</span></tt> columns;
-if <tt class="docutils literal"><span class="pre">c</span></tt> is omitted a square matrix will be returned.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">zeros</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">eye</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">diag</span></tt></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.eye">
-<tt class="descclassname">sympy.matrices.dense.</tt><tt class="descname">eye</tt><big>(</big><em>n</em>, <em>cls=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#eye"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.eye" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create square identity matrix n x n</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">diag</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">zeros</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">ones</span></tt></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.diag">
-<tt class="descclassname">sympy.matrices.dense.</tt><tt class="descname">diag</tt><big>(</big><em>*values</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#diag"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.diag" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a sparse, diagonal matrix from a list of diagonal values.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">eye</span></tt></p>
-</div>
-<p class="rubric">Notes</p>
-<p>When arguments are matrices they are fitted in resultant matrix.</p>
-<p>The returned matrix is a mutable, dense matrix. To make it a different
-type, send the desired class for keyword <tt class="docutils literal"><span class="pre">cls</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">diag</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">ones</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diag</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">[1, 0, 0]</span>
-<span class="go">[0, 2, 0]</span>
-<span class="go">[0, 0, 3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diag</span><span class="p">(</span><span class="o">*</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="go">[1, 0, 0]</span>
-<span class="go">[0, 2, 0]</span>
-<span class="go">[0, 0, 3]</span>
-</pre></div>
-</div>
-<p>The diagonal elements can be matrices; diagonal filling will
-continue on the diagonal from the last element of the matrix:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diag</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
-<span class="go">[x, 0, 0, 0, 0, 0]</span>
-<span class="go">[y, 0, 0, 0, 0, 0]</span>
-<span class="go">[z, 0, 0, 0, 0, 0]</span>
-<span class="go">[0, 7, 0, 0, 0, 0]</span>
-<span class="go">[0, 0, 1, 2, 0, 0]</span>
-<span class="go">[0, 0, 3, 4, 0, 0]</span>
-<span class="go">[0, 0, 0, 0, 5, 6]</span>
-</pre></div>
-</div>
-<p>When diagonal elements are lists, they will be treated as arguments
-to Matrix:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diag</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">[1, 0]</span>
-<span class="go">[2, 0]</span>
-<span class="go">[3, 0]</span>
-<span class="go">[0, 4]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diag</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]],</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">[1, 2, 3, 0]</span>
-<span class="go">[0, 0, 0, 4]</span>
-</pre></div>
-</div>
-<p>A given band off the diagonal can be made by padding with a
-vertical or horizontal &#8220;kerning&#8221; vector:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">hpad</span> <span class="o">=</span> <span class="n">ones</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">vpad</span> <span class="o">=</span> <span class="n">ones</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diag</span><span class="p">(</span><span class="n">vpad</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">hpad</span><span class="p">)</span> <span class="o">+</span> <span class="n">diag</span><span class="p">(</span><span class="n">hpad</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="n">vpad</span><span class="p">)</span>
-<span class="go">[0, 0, 4, 0, 0]</span>
-<span class="go">[0, 0, 0, 5, 0]</span>
-<span class="go">[1, 0, 0, 0, 6]</span>
-<span class="go">[0, 2, 0, 0, 0]</span>
-<span class="go">[0, 0, 3, 0, 0]</span>
-</pre></div>
-</div>
-<p>The type is mutable by default but can be made immutable by setting
-the <tt class="docutils literal"><span class="pre">mutable</span></tt> flag to False:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="n">diag</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span>
-<span class="go">&lt;class &#39;sympy.matrices.dense.MutableDenseMatrix&#39;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">ImmutableMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="n">diag</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">ImmutableMatrix</span><span class="p">))</span>
-<span class="go">&lt;class &#39;sympy.matrices.immutable.ImmutableMatrix&#39;&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.jordan_cell">
-<tt class="descclassname">sympy.matrices.dense.</tt><tt class="descname">jordan_cell</tt><big>(</big><em>eigenval</em>, <em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#jordan_cell"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.jordan_cell" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create matrix of Jordan cell kind:</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">jordan_cell</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jordan_cell</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">[x, 1, 0, 0]</span>
-<span class="go">[0, x, 1, 0]</span>
-<span class="go">[0, 0, x, 1]</span>
-<span class="go">[0, 0, 0, x]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.hessian">
-<tt class="descclassname">sympy.matrices.dense.</tt><tt class="descname">hessian</tt><big>(</big><em>f</em>, <em>varlist</em>, <em>constraints=</em><span class="optional">[</span><span class="optional">]</span><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#hessian"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.hessian" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute Hessian matrix for a function f wrt parameters in varlist
-which may be given as a sequence or a row/column vector. A list of
-constraints may optionally be given.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.matrices.mutable.Matrix.jacobian</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">wronskian</span></tt></p>
-</div>
-<p class="rubric">References</p>
-<p><a class="reference external" href="http://en.wikipedia.org/wiki/Hessian_matrix">http://en.wikipedia.org/wiki/Hessian_matrix</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">hessian</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g1</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">)(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g2</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">hessian</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="p">[</span><span class="n">g1</span><span class="p">,</span> <span class="n">g2</span><span class="p">]))</span>
-<span class="go">[                   d               d            ]</span>
-<span class="go">[     0        0    --(g(x, y))     --(g(x, y))  ]</span>
-<span class="go">[                   dx              dy           ]</span>
-<span class="go">[                                                ]</span>
-<span class="go">[     0        0        2*x              3       ]</span>
-<span class="go">[                                                ]</span>
-<span class="go">[                     2               2          ]</span>
-<span class="go">[d                   d               d           ]</span>
-<span class="go">[--(g(x, y))  2*x   ---(f(x, y))   -----(f(x, y))]</span>
-<span class="go">[dx                   2            dy dx         ]</span>
-<span class="go">[                   dx                           ]</span>
-<span class="go">[                                                ]</span>
-<span class="go">[                     2               2          ]</span>
-<span class="go">[d                   d               d           ]</span>
-<span class="go">[--(g(x, y))   3   -----(f(x, y))   ---(f(x, y)) ]</span>
-<span class="go">[dy                dy dx              2          ]</span>
-<span class="go">[                                   dy           ]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.GramSchmidt">
-<tt class="descclassname">sympy.matrices.dense.</tt><tt class="descname">GramSchmidt</tt><big>(</big><em>vlist</em>, <em>orthog=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#GramSchmidt"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.GramSchmidt" title="Permalink to this definition">¶</a></dt>
-<dd><p>Apply the Gram-Schmidt process to a set of vectors.</p>
-<p>see: <a class="reference external" href="http://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process">http://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process</a></p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.wronskian">
-<tt class="descclassname">sympy.matrices.dense.</tt><tt class="descname">wronskian</tt><big>(</big><em>functions</em>, <em>var</em>, <em>method='bareis'</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#wronskian"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.wronskian" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute Wronskian for [] of functions</p>
-<div class="highlight-python"><pre>                 | f1       f2        ...   fn      |
-                 | f1&#x27;      f2&#x27;       ...   fn&#x27;     |
-                 |  .        .        .      .      |
-W(f1, ..., fn) = |  .        .         .     .      |
-                 |  .        .          .    .      |
-                 |  (n)      (n)            (n)     |
-                 | D   (f1) D   (f2)  ...  D   (fn) |</pre>
-</div>
-<p>see: <a class="reference external" href="http://en.wikipedia.org/wiki/Wronskian">http://en.wikipedia.org/wiki/Wronskian</a></p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.matrices.mutable.Matrix.jacobian</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">hessian</span></tt></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.casoratian">
-<tt class="descclassname">sympy.matrices.dense.</tt><tt class="descname">casoratian</tt><big>(</big><em>seqs</em>, <em>n</em>, <em>zero=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#casoratian"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.casoratian" title="Permalink to this definition">¶</a></dt>
-<dd><p>Given linear difference operator L of order &#8216;k&#8217; and homogeneous
-equation Ly = 0 we want to compute kernel of L, which is a set
-of &#8216;k&#8217; sequences: a(n), b(n), ... z(n).</p>
-<p>Solutions of L are linearly independent iff their Casoratian,
-denoted as C(a, b, ..., z), do not vanish for n = 0.</p>
-<p>Casoratian is defined by k x k determinant:</p>
-<div class="highlight-python"><pre>+  a(n)     b(n)     . . . z(n)     +
-|  a(n+1)   b(n+1)   . . . z(n+1)   |
-|    .         .     .        .     |
-|    .         .       .      .     |
-|    .         .         .    .     |
-+  a(n+k-1) b(n+k-1) . . . z(n+k-1) +</pre>
-</div>
-<p>It proves very useful in rsolve_hyper() where it is applied
-to a generating set of a recurrence to factor out linearly
-dependent solutions and return a basis:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">casoratian</span><span class="p">,</span> <span class="n">factorial</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Exponential and factorial are linearly independent:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">casoratian</span><span class="p">([</span><span class="mi">2</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)],</span> <span class="n">n</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">0</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.randMatrix">
-<tt class="descclassname">sympy.matrices.dense.</tt><tt class="descname">randMatrix</tt><big>(</big><em>r</em>, <em>c=None</em>, <em>min=0</em>, <em>max=99</em>, <em>seed=None</em>, <em>symmetric=False</em>, <em>percent=100</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#randMatrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.randMatrix" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create random matrix with dimensions <tt class="docutils literal"><span class="pre">r</span></tt> x <tt class="docutils literal"><span class="pre">c</span></tt>. If <tt class="docutils literal"><span class="pre">c</span></tt> is omitted
-the matrix will be square. If <tt class="docutils literal"><span class="pre">symmetric</span></tt> is True the matrix must be
-square. If <tt class="docutils literal"><span class="pre">percent</span></tt> is less than 100 then only approximately the given
-percentage of elements will be non-zero.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">randMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">randMatrix</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> 
-<span class="go">[25, 45, 27]</span>
-<span class="go">[44, 54,  9]</span>
-<span class="go">[23, 96, 46]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">randMatrix</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> 
-<span class="go">[87, 29]</span>
-<span class="go">[23, 37]</span>
-<span class="go">[90, 26]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">randMatrix</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> 
-<span class="go">[0, 2, 0]</span>
-<span class="go">[2, 0, 1]</span>
-<span class="go">[0, 0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">randMatrix</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">symmetric</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> 
-<span class="go">[85, 26, 29]</span>
-<span class="go">[26, 71, 43]</span>
-<span class="go">[29, 43, 57]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">randMatrix</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">randMatrix</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">==</span> <span class="n">B</span> 
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">==</span> <span class="n">randMatrix</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">randMatrix</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">symmetric</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">percent</span><span class="o">=</span><span class="mi">50</span><span class="p">)</span> 
-<span class="go">[0, 68, 43]</span>
-<span class="go">[0, 68,  0]</span>
-<span class="go">[0, 91, 34]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="numpy-utility-functions-reference">
-<h2>Numpy Utility Functions Reference<a class="headerlink" href="#numpy-utility-functions-reference" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.matrices.dense.list2numpy">
-<tt class="descclassname">sympy.matrices.dense.</tt><tt class="descname">list2numpy</tt><big>(</big><em>l</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#list2numpy"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.list2numpy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Converts python list of SymPy expressions to a NumPy array.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">matrix2numpy</span></tt></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.matrix2numpy">
-<tt class="descclassname">sympy.matrices.dense.</tt><tt class="descname">matrix2numpy</tt><big>(</big><em>m</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#matrix2numpy"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.matrix2numpy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Converts SymPy&#8217;s matrix to a NumPy array.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">list2numpy</span></tt></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.symarray">
-<tt class="descclassname">sympy.matrices.dense.</tt><tt class="descname">symarray</tt><big>(</big><em>prefix</em>, <em>shape</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#symarray"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.symarray" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a numpy ndarray of symbols (as an object array).</p>
-<p>The created symbols are named <tt class="docutils literal"><span class="pre">prefix_i1_i2_</span></tt>...  You should thus provide a
-non-empty prefix if you want your symbols to be unique for different output
-arrays, as SymPy symbols with identical names are the same object.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>prefix</strong> : string</p>
-<blockquote>
-<div><p>A prefix prepended to the name of every symbol.</p>
-</div></blockquote>
-<p><strong>shape</strong> : int or tuple</p>
-<blockquote class="last">
-<div><p>Shape of the created array.  If an int, the array is one-dimensional; for
-more than one dimension the shape must be a tuple.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>These doctests require numpy.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symarray</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">symarray</span><span class="p">(</span><span class="s">&#39;&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> 
-<span class="go">[_0, _1, _2]</span>
-</pre></div>
-</div>
-<p>If you want multiple symarrays to contain distinct symbols, you <em>must</em>
-provide unique prefixes:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">symarray</span><span class="p">(</span><span class="s">&#39;&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> 
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">symarray</span><span class="p">(</span><span class="s">&#39;&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> 
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">is</span> <span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> 
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">symarray</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> 
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">symarray</span><span class="p">(</span><span class="s">&#39;b&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> 
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">is</span> <span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> 
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>Creating symarrays with a prefix:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">symarray</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> 
-<span class="go">[a_0, a_1, a_2]</span>
-</pre></div>
-</div>
-<p>For more than one dimension, the shape must be given as a tuple:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">symarray</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span> 
-<span class="go">[[a_0_0, a_0_1, a_0_2],</span>
-<span class="go"> [a_1_0, a_1_1, a_1_2]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">symarray</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span> 
-<span class="go">[[[a_0_0_0, a_0_0_1],</span>
-<span class="go">  [a_0_1_0, a_0_1_1],</span>
-<span class="go">  [a_0_2_0, a_0_2_1]],</span>
-
-<span class="go"> [[a_1_0_0, a_1_0_1],</span>
-<span class="go">  [a_1_1_0, a_1_1_1],</span>
-<span class="go">  [a_1_2_0, a_1_2_1]]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.rot_axis1">
-<tt class="descclassname">sympy.matrices.dense.</tt><tt class="descname">rot_axis1</tt><big>(</big><em>theta</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#rot_axis1"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.rot_axis1" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a rotation matrix for a rotation of theta (in radians) about
-the 1-axis.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">rot_axis2</span></tt></dt>
-<dd>Returns a rotation matrix for a rotation of theta (in radians) about the 2-axis</dd>
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">rot_axis3</span></tt></dt>
-<dd>Returns a rotation matrix for a rotation of theta (in radians) about the 3-axis</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">rot_axis1</span>
-</pre></div>
-</div>
-<p>A rotation of pi/3 (60 degrees):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">theta</span> <span class="o">=</span> <span class="n">pi</span><span class="o">/</span><span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rot_axis1</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
-<span class="go">[1,          0,         0]</span>
-<span class="go">[0,        1/2, sqrt(3)/2]</span>
-<span class="go">[0, -sqrt(3)/2,       1/2]</span>
-</pre></div>
-</div>
-<p>If we rotate by pi/2 (90 degrees):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rot_axis1</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">[1,  0, 0]</span>
-<span class="go">[0,  0, 1]</span>
-<span class="go">[0, -1, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.rot_axis2">
-<tt class="descclassname">sympy.matrices.dense.</tt><tt class="descname">rot_axis2</tt><big>(</big><em>theta</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#rot_axis2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.rot_axis2" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a rotation matrix for a rotation of theta (in radians) about
-the 2-axis.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">rot_axis1</span></tt></dt>
-<dd>Returns a rotation matrix for a rotation of theta (in radians) about the 1-axis</dd>
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">rot_axis3</span></tt></dt>
-<dd>Returns a rotation matrix for a rotation of theta (in radians) about the 3-axis</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">rot_axis2</span>
-</pre></div>
-</div>
-<p>A rotation of pi/3 (60 degrees):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">theta</span> <span class="o">=</span> <span class="n">pi</span><span class="o">/</span><span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rot_axis2</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
-<span class="go">[      1/2, 0, -sqrt(3)/2]</span>
-<span class="go">[        0, 1,          0]</span>
-<span class="go">[sqrt(3)/2, 0,        1/2]</span>
-</pre></div>
-</div>
-<p>If we rotate by pi/2 (90 degrees):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rot_axis2</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">[0, 0, -1]</span>
-<span class="go">[0, 1,  0]</span>
-<span class="go">[1, 0,  0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.dense.rot_axis3">
-<tt class="descclassname">sympy.matrices.dense.</tt><tt class="descname">rot_axis3</tt><big>(</big><em>theta</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/dense.html#rot_axis3"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.dense.rot_axis3" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a rotation matrix for a rotation of theta (in radians) about
-the 3-axis.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">rot_axis1</span></tt></dt>
-<dd>Returns a rotation matrix for a rotation of theta (in radians) about the 1-axis</dd>
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">rot_axis2</span></tt></dt>
-<dd>Returns a rotation matrix for a rotation of theta (in radians) about the 2-axis</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">rot_axis3</span>
-</pre></div>
-</div>
-<p>A rotation of pi/3 (60 degrees):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">theta</span> <span class="o">=</span> <span class="n">pi</span><span class="o">/</span><span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rot_axis3</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
-<span class="go">[       1/2, sqrt(3)/2, 0]</span>
-<span class="go">[-sqrt(3)/2,       1/2, 0]</span>
-<span class="go">[         0,         0, 1]</span>
-</pre></div>
-</div>
-<p>If we rotate by pi/2 (90 degrees):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rot_axis3</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">[ 0, 1, 0]</span>
-<span class="go">[-1, 0, 0]</span>
-<span class="go">[ 0, 0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.matrices.a2idx">
-<tt class="descclassname">sympy.matrices.matrices.</tt><tt class="descname">a2idx</tt><big>(</big><em>j</em>, <em>n=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/matrices.html#a2idx"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.matrices.a2idx" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return integer after making positive and validating against n.</p>
-</dd></dl>
-
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../index.html">
-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Matrices (linear algebra)</a><ul>
-<li><a class="reference internal" href="#creating-matrices">Creating Matrices</a></li>
-<li><a class="reference internal" href="#basic-manipulation">Basic Manipulation</a></li>
-<li><a class="reference internal" href="#operations-on-entries">Operations on entries</a></li>
-<li><a class="reference internal" href="#linear-algebra">Linear algebra</a></li>
-<li><a class="reference internal" href="#matrixbase-class-reference">MatrixBase Class Reference</a></li>
-<li><a class="reference internal" href="#matrix-exceptions-reference">Matrix Exceptions Reference</a></li>
-<li><a class="reference internal" href="#matrix-functions-reference">Matrix Functions Reference</a></li>
-<li><a class="reference internal" href="#numpy-utility-functions-reference">Numpy Utility Functions Reference</a></li>
-</ul>
-</li>
-</ul>
-
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-            
-  <div class="section" id="module-sympy.matrices.sparse">
-<span id="sparse-matrices"></span><h1>Sparse Matrices<a class="headerlink" href="#module-sympy.matrices.sparse" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="sparsematrix-class-reference">
-<h2>SparseMatrix Class Reference<a class="headerlink" href="#sparsematrix-class-reference" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.matrices.sparse.SparseMatrix">
-<em class="property">class </em><tt class="descclassname">sympy.matrices.sparse.</tt><tt class="descname">SparseMatrix</tt><big>(</big><em>*args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix" title="Permalink to this definition">¶</a></dt>
-<dd><p>A sparse matrix (a matrix with a large number of zero elements).</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.matrices.dense.Matrix</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">SparseMatrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">)))</span>
-<span class="go">[0, 1]</span>
-<span class="go">[2, 3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">SparseMatrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">{(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">):</span> <span class="mi">2</span><span class="p">})</span>
-<span class="go">[0, 0]</span>
-<span class="go">[0, 2]</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.matrices.sparse.SparseMatrix.CL">
-<tt class="descname">CL</tt><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.CL" title="Permalink to this definition">¶</a></dt>
-<dd><p>Alternate faster representation</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.SparseMatrix.LDLdecomposition">
-<tt class="descname">LDLdecomposition</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix.LDLdecomposition"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.LDLdecomposition" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the LDL Decomposition (matrices <tt class="docutils literal"><span class="pre">L</span></tt> and <tt class="docutils literal"><span class="pre">D</span></tt>) of matrix
-<tt class="docutils literal"><span class="pre">A</span></tt>, such that <tt class="docutils literal"><span class="pre">L</span> <span class="pre">*</span> <span class="pre">D</span> <span class="pre">*</span> <span class="pre">L.T</span> <span class="pre">==</span> <span class="pre">A</span></tt>. <tt class="docutils literal"><span class="pre">A</span></tt> must be a square,
-symmetric, positive-definite and non-singular.</p>
-<p>This method eliminates the use of square root and ensures that all
-the diagonal entries of L are 1.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="p">(((</span><span class="mi">25</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">),</span> <span class="p">(</span><span class="mi">15</span><span class="p">,</span> <span class="mi">18</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">11</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span><span class="p">,</span> <span class="n">D</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">LDLdecomposition</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span>
-<span class="go">[   1,   0, 0]</span>
-<span class="go">[ 3/5,   1, 0]</span>
-<span class="go">[-1/5, 1/3, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span>
-<span class="go">[25, 0, 0]</span>
-<span class="go">[ 0, 9, 0]</span>
-<span class="go">[ 0, 0, 9]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span> <span class="o">*</span> <span class="n">D</span> <span class="o">*</span> <span class="n">L</span><span class="o">.</span><span class="n">T</span> <span class="o">==</span> <span class="n">A</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.matrices.sparse.SparseMatrix.RL">
-<tt class="descname">RL</tt><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.RL" title="Permalink to this definition">¶</a></dt>
-<dd><p>Alternate faster representation</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.SparseMatrix.add">
-<tt class="descname">add</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix.add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Add two sparse matrices with dictionary representation.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.sparse.SparseMatrix.multiply" title="sympy.matrices.sparse.SparseMatrix.multiply"><tt class="xref py py-obj docutils literal"><span class="pre">multiply</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span><span class="p">,</span> <span class="n">eye</span><span class="p">,</span> <span class="n">ones</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">SparseMatrix</span><span class="p">(</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">SparseMatrix</span><span class="p">(</span><span class="n">ones</span><span class="p">(</span><span class="mi">3</span><span class="p">)))</span>
-<span class="go">[2, 1, 1]</span>
-<span class="go">[1, 2, 1]</span>
-<span class="go">[1, 1, 2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">SparseMatrix</span><span class="p">(</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="o">-</span><span class="n">SparseMatrix</span><span class="p">(</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)))</span>
-<span class="go">[0, 0, 0]</span>
-<span class="go">[0, 0, 0]</span>
-<span class="go">[0, 0, 0]</span>
-</pre></div>
-</div>
-<p>Only the non-zero elements are stored, so the resulting dictionary
-that is used to represent the sparse matrix is empty:
-&gt;&gt;&gt; _._smat
-{}</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.SparseMatrix.applyfunc">
-<tt class="descname">applyfunc</tt><big>(</big><em>self</em>, <em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix.applyfunc"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.applyfunc" title="Permalink to this definition">¶</a></dt>
-<dd><p>Apply a function to each element of the matrix.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="n">i</span><span class="o">*</span><span class="mi">2</span><span class="o">+</span><span class="n">j</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[0, 1]</span>
-<span class="go">[2, 3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">applyfunc</span><span class="p">(</span><span class="k">lambda</span> <span class="n">i</span><span class="p">:</span> <span class="mi">2</span><span class="o">*</span><span class="n">i</span><span class="p">)</span>
-<span class="go">[0, 2]</span>
-<span class="go">[4, 6]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.SparseMatrix.as_immutable">
-<tt class="descname">as_immutable</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix.as_immutable"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.as_immutable" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns an Immutable version of this Matrix.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.SparseMatrix.as_mutable">
-<tt class="descname">as_mutable</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix.as_mutable"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.as_mutable" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a mutable version of this matrix.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">ImmutableMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">ImmutableMatrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">as_mutable</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">5</span> <span class="c"># Can set values in Y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span>
-<span class="go">[1, 2]</span>
-<span class="go">[3, 5]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.SparseMatrix.cholesky">
-<tt class="descname">cholesky</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix.cholesky"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.cholesky" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the Cholesky decomposition L of a matrix A
-such that L * L.T = A</p>
-<p>A must be a square, symmetric, positive-definite
-and non-singular matrix</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="p">(((</span><span class="mi">25</span><span class="p">,</span><span class="mi">15</span><span class="p">,</span><span class="o">-</span><span class="mi">5</span><span class="p">),(</span><span class="mi">15</span><span class="p">,</span><span class="mi">18</span><span class="p">,</span><span class="mi">0</span><span class="p">),(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">11</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">cholesky</span><span class="p">()</span>
-<span class="go">[ 5, 0, 0]</span>
-<span class="go">[ 3, 3, 0]</span>
-<span class="go">[-1, 1, 3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">cholesky</span><span class="p">()</span> <span class="o">*</span> <span class="n">A</span><span class="o">.</span><span class="n">cholesky</span><span class="p">()</span><span class="o">.</span><span class="n">T</span> <span class="o">==</span> <span class="n">A</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.SparseMatrix.col">
-<tt class="descname">col</tt><big>(</big><em>self</em>, <em>j</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix.col"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.col" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns column j from self as a column vector.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.sparse.SparseMatrix.row" title="sympy.matrices.sparse.SparseMatrix.row"><tt class="xref py py-obj docutils literal"><span class="pre">row</span></tt></a>, <a class="reference internal" href="#sympy.matrices.sparse.SparseMatrix.col_list" title="sympy.matrices.sparse.SparseMatrix.col_list"><tt class="xref py py-obj docutils literal"><span class="pre">col_list</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="p">(((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">col</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">[1]</span>
-<span class="go">[3]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.SparseMatrix.col_list">
-<tt class="descname">col_list</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix.col_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.col_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a column-sorted list of non-zero elements of the matrix.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">col_op</span></tt>, <a class="reference internal" href="#sympy.matrices.sparse.SparseMatrix.row_list" title="sympy.matrices.sparse.SparseMatrix.row_list"><tt class="xref py py-obj docutils literal"><span class="pre">row_list</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">=</span><span class="n">SparseMatrix</span><span class="p">(((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
-<span class="go">[1, 2]</span>
-<span class="go">[3, 4]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">CL</span>
-<span class="go">[(0, 0, 1), (1, 0, 3), (0, 1, 2), (1, 1, 4)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.matrices.sparse.SparseMatrix.eye">
-<em class="property">classmethod </em><tt class="descname">eye</tt><big>(</big><em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix.eye"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.eye" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return an n x n identity matrix.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.SparseMatrix.has">
-<tt class="descname">has</tt><big>(</big><em>self</em>, <em>*patterns</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix.has"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.has" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test whether any subexpression matches any of the patterns.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">SparseMatrix</span><span class="p">,</span> <span class="n">Float</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="p">(((</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="mf">0.2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">has</span><span class="p">(</span><span class="n">Float</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.SparseMatrix.is_symmetric">
-<tt class="descname">is_symmetric</tt><big>(</big><em>self</em>, <em>simplify=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix.is_symmetric"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.is_symmetric" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if self is symmetric.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span><span class="p">,</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="p">(</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">is_symmetric</span><span class="p">()</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">is_symmetric</span><span class="p">()</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.SparseMatrix.liupc">
-<tt class="descname">liupc</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix.liupc"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.liupc" title="Permalink to this definition">¶</a></dt>
-<dd><p>Liu&#8217;s algorithm, for pre-determination of the Elimination Tree of
-the given matrix, used in row-based symbolic Cholesky factorization.</p>
-<p class="rubric">References</p>
-<p>Symbolic Sparse Cholesky Factorization using Elimination Trees,
-Jeroen Van Grondelle (1999)
-<a class="reference external" href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.39.7582">http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.39.7582</a>,
-downloaded from <a class="reference external" href="http://tinyurl.com/9o2jsxj">http://tinyurl.com/9o2jsxj</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="p">([</span>
-<span class="gp">... </span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span>
-<span class="gp">... </span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
-<span class="gp">... </span><span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</span>
-<span class="gp">... </span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">.</span><span class="n">liupc</span><span class="p">()</span>
-<span class="go">([[0], [], [0], [1, 2]], [4, 3, 4, 4])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.SparseMatrix.multiply">
-<tt class="descname">multiply</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix.multiply"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.multiply" title="Permalink to this definition">¶</a></dt>
-<dd><p>Fast multiplication exploiting the sparsity of the matrix.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.sparse.SparseMatrix.add" title="sympy.matrices.sparse.SparseMatrix.add"><tt class="xref py py-obj docutils literal"><span class="pre">add</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span><span class="p">,</span> <span class="n">ones</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="p">,</span> <span class="n">B</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="p">(</span><span class="n">ones</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)),</span> <span class="n">SparseMatrix</span><span class="p">(</span><span class="n">ones</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">multiply</span><span class="p">(</span><span class="n">B</span><span class="p">)</span> <span class="o">==</span> <span class="mi">3</span><span class="o">*</span><span class="n">ones</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.SparseMatrix.reshape">
-<tt class="descname">reshape</tt><big>(</big><em>self</em>, <em>rows</em>, <em>cols</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix.reshape"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.reshape" title="Permalink to this definition">¶</a></dt>
-<dd><p>Reshape matrix while retaining original size.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">8</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">[0, 1, 2, 3]</span>
-<span class="go">[4, 5, 6, 7]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.SparseMatrix.row">
-<tt class="descname">row</tt><big>(</big><em>self</em>, <em>i</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix.row"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.row" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns column i from self as a row vector.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.sparse.SparseMatrix.col" title="sympy.matrices.sparse.SparseMatrix.col"><tt class="xref py py-obj docutils literal"><span class="pre">col</span></tt></a>, <a class="reference internal" href="#sympy.matrices.sparse.SparseMatrix.row_list" title="sympy.matrices.sparse.SparseMatrix.row_list"><tt class="xref py py-obj docutils literal"><span class="pre">row_list</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="p">(((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">row</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">[1, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.SparseMatrix.row_list">
-<tt class="descname">row_list</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix.row_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.row_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a row-sorted list of non-zero elements of the matrix.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">row_op</span></tt>, <a class="reference internal" href="#sympy.matrices.sparse.SparseMatrix.col_list" title="sympy.matrices.sparse.SparseMatrix.col_list"><tt class="xref py py-obj docutils literal"><span class="pre">col_list</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="p">(((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
-<span class="go">[1, 2]</span>
-<span class="go">[3, 4]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">RL</span>
-<span class="go">[(0, 0, 1), (0, 1, 2), (1, 0, 3), (1, 1, 4)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.SparseMatrix.row_structure_symbolic_cholesky">
-<tt class="descname">row_structure_symbolic_cholesky</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix.row_structure_symbolic_cholesky"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.row_structure_symbolic_cholesky" title="Permalink to this definition">¶</a></dt>
-<dd><p>Symbolic cholesky factorization, for pre-determination of the
-non-zero structure of the Cholesky factororization.</p>
-<p class="rubric">References</p>
-<p>Symbolic Sparse Cholesky Factorization using Elimination Trees,
-Jeroen Van Grondelle (1999)
-<a class="reference external" href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.39.7582">http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.39.7582</a>,
-downloaded from <a class="reference external" href="http://tinyurl.com/9o2jsxj">http://tinyurl.com/9o2jsxj</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="p">([</span>
-<span class="gp">... </span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span>
-<span class="gp">... </span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
-<span class="gp">... </span><span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</span>
-<span class="gp">... </span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">.</span><span class="n">row_structure_symbolic_cholesky</span><span class="p">()</span>
-<span class="go">[[0], [], [0], [1, 2]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.SparseMatrix.scalar_multiply">
-<tt class="descname">scalar_multiply</tt><big>(</big><em>self</em>, <em>scalar</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix.scalar_multiply"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.scalar_multiply" title="Permalink to this definition">¶</a></dt>
-<dd><p>Scalar element-wise multiplication</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.SparseMatrix.solve">
-<tt class="descname">solve</tt><big>(</big><em>self</em>, <em>rhs</em>, <em>method='LDL'</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix.solve"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.solve" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return solution to self*soln = rhs using given inversion method.</p>
-<p>For a list of possible inversion methods, see the .inv() docstring.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.SparseMatrix.solve_least_squares">
-<tt class="descname">solve_least_squares</tt><big>(</big><em>self</em>, <em>rhs</em>, <em>method='LDL'</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix.solve_least_squares"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.solve_least_squares" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the least-square fit to the data.</p>
-<p>By default the cholesky_solve routine is used (method=&#8217;CH&#8217;); other
-methods of matrix inversion can be used. To find out which are
-available, see the docstring of the .inv() method.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">ones</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">row_join</span><span class="p">(</span><span class="n">B</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span>
-<span class="go">[1, 2]</span>
-<span class="go">[2, 3]</span>
-<span class="go">[3, 4]</span>
-</pre></div>
-</div>
-<p>If each line of S represent coefficients of Ax + By
-and x and y are [2, 3] then S*xy is:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">r</span> <span class="o">=</span> <span class="n">S</span><span class="o">*</span><span class="n">Matrix</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]);</span> <span class="n">r</span>
-<span class="go">[ 8]</span>
-<span class="go">[13]</span>
-<span class="go">[18]</span>
-</pre></div>
-</div>
-<p>But let&#8217;s add 1 to the middle value and then solve for the
-least-squares value of xy:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">xy</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">solve_least_squares</span><span class="p">(</span><span class="n">Matrix</span><span class="p">([</span><span class="mi">8</span><span class="p">,</span> <span class="mi">14</span><span class="p">,</span> <span class="mi">18</span><span class="p">]));</span> <span class="n">xy</span>
-<span class="go">[ 5/3]</span>
-<span class="go">[10/3]</span>
-</pre></div>
-</div>
-<p>The error is given by S*xy - r:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">*</span><span class="n">xy</span> <span class="o">-</span> <span class="n">r</span>
-<span class="go">[1/3]</span>
-<span class="go">[1/3]</span>
-<span class="go">[1/3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">.</span><span class="n">norm</span><span class="p">()</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.58</span>
-</pre></div>
-</div>
-<p>If a different xy is used, the norm will be higher:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">xy</span> <span class="o">+=</span> <span class="n">ones</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">S</span><span class="o">*</span><span class="n">xy</span> <span class="o">-</span> <span class="n">r</span><span class="p">)</span><span class="o">.</span><span class="n">norm</span><span class="p">()</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">1.5</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.SparseMatrix.tolist">
-<tt class="descname">tolist</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix.tolist"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.tolist" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert this sparse matrix into a list of nested Python lists.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span><span class="p">,</span> <span class="n">ones</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="p">(((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">tolist</span><span class="p">()</span>
-<span class="go">[[1, 2], [3, 4]]</span>
-</pre></div>
-</div>
-<p>When there are no rows then it will not be possible to tell how
-many columns were in the original matrix:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">SparseMatrix</span><span class="p">(</span><span class="n">ones</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span><span class="o">.</span><span class="n">tolist</span><span class="p">()</span>
-<span class="go">[]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.matrices.sparse.SparseMatrix.zeros">
-<em class="property">classmethod </em><tt class="descname">zeros</tt><big>(</big><em>r</em>, <em>c=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#SparseMatrix.zeros"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.SparseMatrix.zeros" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return an r x c matrix of zeros, square if c is omitted.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.matrices.sparse.MutableSparseMatrix">
-<em class="property">class </em><tt class="descclassname">sympy.matrices.sparse.</tt><tt class="descname">MutableSparseMatrix</tt><big>(</big><em>*args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#MutableSparseMatrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.MutableSparseMatrix" title="Permalink to this definition">¶</a></dt>
-<dd><dl class="function">
-<dt id="sympy.matrices.sparse.MutableSparseMatrix.col_del">
-<tt class="descname">col_del</tt><big>(</big><em>self</em>, <em>k</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#MutableSparseMatrix.col_del"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.MutableSparseMatrix.col_del" title="Permalink to this definition">¶</a></dt>
-<dd><p>Delete the given column of the matrix.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.sparse.MutableSparseMatrix.row_del" title="sympy.matrices.sparse.MutableSparseMatrix.row_del"><tt class="xref py py-obj docutils literal"><span class="pre">row_del</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span>
-<span class="go">[0, 0]</span>
-<span class="go">[0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">col_del</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span>
-<span class="go">[0]</span>
-<span class="go">[1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.MutableSparseMatrix.col_join">
-<tt class="descname">col_join</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#MutableSparseMatrix.col_join"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.MutableSparseMatrix.col_join" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns B augmented beneath A (row-wise joining):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="p">[</span><span class="n">A</span><span class="p">]</span>
-<span class="p">[</span><span class="n">B</span><span class="p">]</span>
-</pre></div>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">SparseMatrix</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">ones</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="p">(</span><span class="n">ones</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span>
-<span class="go">[1, 1, 1]</span>
-<span class="go">[1, 1, 1]</span>
-<span class="go">[1, 1, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span>
-<span class="go">[1, 0, 0]</span>
-<span class="go">[0, 1, 0]</span>
-<span class="go">[0, 0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">col_join</span><span class="p">(</span><span class="n">B</span><span class="p">);</span> <span class="n">C</span>
-<span class="go">[1, 1, 1]</span>
-<span class="go">[1, 1, 1]</span>
-<span class="go">[1, 1, 1]</span>
-<span class="go">[1, 0, 0]</span>
-<span class="go">[0, 1, 0]</span>
-<span class="go">[0, 0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">==</span> <span class="n">A</span><span class="o">.</span><span class="n">col_join</span><span class="p">(</span><span class="n">Matrix</span><span class="p">(</span><span class="n">B</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>Joining along columns is the same as appending rows at the end
-of the matrix:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">==</span> <span class="n">A</span><span class="o">.</span><span class="n">row_insert</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">rows</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">(</span><span class="n">B</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.MutableSparseMatrix.col_op">
-<tt class="descname">col_op</tt><big>(</big><em>self</em>, <em>j</em>, <em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#MutableSparseMatrix.col_op"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.MutableSparseMatrix.col_op" title="Permalink to this definition">¶</a></dt>
-<dd><p>In-place operation on col j using two-arg functor whose args are
-interpreted as (self[i, j], i) for i in range(self.rows).</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">*</span><span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">col_op</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">v</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="n">v</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">]);</span> <span class="n">M</span>
-<span class="go">[ 2, 4, 0]</span>
-<span class="go">[-1, 0, 0]</span>
-<span class="go">[ 0, 0, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.MutableSparseMatrix.col_swap">
-<tt class="descname">col_swap</tt><big>(</big><em>self</em>, <em>i</em>, <em>j</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#MutableSparseMatrix.col_swap"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.MutableSparseMatrix.col_swap" title="Permalink to this definition">¶</a></dt>
-<dd><p>Swap, in place, columns i and j.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span> <span class="n">S</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">.</span><span class="n">col_swap</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">);</span> <span class="n">S</span>
-<span class="go">[0, 1, 0]</span>
-<span class="go">[1, 0, 0]</span>
-<span class="go">[2, 0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.MutableSparseMatrix.fill">
-<tt class="descname">fill</tt><big>(</big><em>self</em>, <em>value</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#MutableSparseMatrix.fill"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.MutableSparseMatrix.fill" title="Permalink to this definition">¶</a></dt>
-<dd><p>Fill self with the given value.</p>
-<p class="rubric">Notes</p>
-<p>Unless many values are going to be deleted (i.e. set to zero)
-this will create a matrix that is slower than a dense matrix in
-operations.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span> <span class="n">M</span>
-<span class="go">[0, 0, 0]</span>
-<span class="go">[0, 0, 0]</span>
-<span class="go">[0, 0, 0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">fill</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span> <span class="n">M</span>
-<span class="go">[1, 1, 1]</span>
-<span class="go">[1, 1, 1]</span>
-<span class="go">[1, 1, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.MutableSparseMatrix.row_del">
-<tt class="descname">row_del</tt><big>(</big><em>self</em>, <em>k</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#MutableSparseMatrix.row_del"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.MutableSparseMatrix.row_del" title="Permalink to this definition">¶</a></dt>
-<dd><p>Delete the given row of the matrix.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.matrices.sparse.MutableSparseMatrix.col_del" title="sympy.matrices.sparse.MutableSparseMatrix.col_del"><tt class="xref py py-obj docutils literal"><span class="pre">col_del</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span>
-<span class="go">[0, 0]</span>
-<span class="go">[0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">row_del</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span>
-<span class="go">[0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.MutableSparseMatrix.row_join">
-<tt class="descname">row_join</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#MutableSparseMatrix.row_join"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.MutableSparseMatrix.row_join" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns B appended after A (column-wise augmenting):</p>
-<div class="highlight-python"><pre>[A B]</pre>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">SparseMatrix</span><span class="p">,</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="p">(((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span>
-<span class="go">[1, 0, 1]</span>
-<span class="go">[0, 1, 0]</span>
-<span class="go">[1, 1, 0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="p">(((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span>
-<span class="go">[1, 0, 0]</span>
-<span class="go">[0, 1, 0]</span>
-<span class="go">[0, 0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">row_join</span><span class="p">(</span><span class="n">B</span><span class="p">);</span> <span class="n">C</span>
-<span class="go">[1, 0, 1, 1, 0, 0]</span>
-<span class="go">[0, 1, 0, 0, 1, 0]</span>
-<span class="go">[1, 1, 0, 0, 0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">==</span> <span class="n">A</span><span class="o">.</span><span class="n">row_join</span><span class="p">(</span><span class="n">Matrix</span><span class="p">(</span><span class="n">B</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>Joining at row ends is the same as appending columns at the end
-of the matrix:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">==</span> <span class="n">A</span><span class="o">.</span><span class="n">col_insert</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">cols</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.MutableSparseMatrix.row_op">
-<tt class="descname">row_op</tt><big>(</big><em>self</em>, <em>i</em>, <em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#MutableSparseMatrix.row_op"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.MutableSparseMatrix.row_op" title="Permalink to this definition">¶</a></dt>
-<dd><p>In-place operation on row i using two-arg functor whose args are
-interpreted as (self[i, j], j) for j in range(self.cols).</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">*</span><span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">row_op</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">v</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="n">v</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">M</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">j</span><span class="p">]);</span> <span class="n">M</span>
-<span class="go">[2, -1, 0]</span>
-<span class="go">[4,  0, 0]</span>
-<span class="go">[0,  0, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.matrices.sparse.MutableSparseMatrix.row_swap">
-<tt class="descname">row_swap</tt><big>(</big><em>self</em>, <em>i</em>, <em>j</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/sparse.html#MutableSparseMatrix.row_swap"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.sparse.MutableSparseMatrix.row_swap" title="Permalink to this definition">¶</a></dt>
-<dd><p>Swap, in place, columns i and j.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">SparseMatrix</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span> <span class="n">S</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">.</span><span class="n">row_swap</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">);</span> <span class="n">S</span>
-<span class="go">[0, 1, 0]</span>
-<span class="go">[1, 0, 0]</span>
-<span class="go">[0, 2, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="immutablesparsematrix-class-reference">
-<h2>ImmutableSparseMatrix Class Reference<a class="headerlink" href="#immutablesparsematrix-class-reference" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.matrices.immutable.ImmutableSparseMatrix">
-<em class="property">class </em><tt class="descclassname">sympy.matrices.immutable.</tt><tt class="descname">ImmutableSparseMatrix</tt><big>(</big><em>*args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/matrices/immutable.html#ImmutableSparseMatrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.matrices.immutable.ImmutableSparseMatrix" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create an immutable version of a sparse matrix.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">eye</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices.immutable</span> <span class="kn">import</span> <span class="n">ImmutableSparseMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ImmutableSparseMatrix</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="p">{})</span>
-<span class="go">[0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ImmutableSparseMatrix</span><span class="p">(</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">[1, 0, 0]</span>
-<span class="go">[0, 1, 0]</span>
-<span class="go">[0, 0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">42</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">TypeError</span>: <span class="n">Cannot set values of ImmutableSparseMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">.</span><span class="n">shape</span>
-<span class="go">(3, 3)</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.matrices.immutable.ImmutableSparseMatrix.subs">
-<tt class="descname">subs</tt><big>(</big><em>self</em>, <em>*args</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#sympy.matrices.immutable.ImmutableSparseMatrix.subs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a new matrix with subs applied to each entry.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.matrices</span> <span class="kn">import</span> <span class="n">SparseMatrix</span><span class="p">,</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">SparseMatrix</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="p">[</span><span class="n">x</span><span class="p">])</span>
-<span class="go">[x]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">[y]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">(</span><span class="n">_</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">[x]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-</div>
-
-
-          </div>
-        </div>
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-            <p class="logo"><a href="../../index.html">
-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Sparse Matrices</a><ul>
-<li><a class="reference internal" href="#sparsematrix-class-reference">SparseMatrix Class Reference</a></li>
-<li><a class="reference internal" href="#immutablesparsematrix-class-reference">ImmutableSparseMatrix Class Reference</a></li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../matrices/dense.html"
-                        title="previous chapter">Dense Matrices</a></p>
-  <h4>Next topic</h4>
-  <p class="topless"><a href="../matrices/immutablematrices.html"
-                        title="next chapter">Immutable Matrices</a></p>
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-            
-  <div class="section" id="basic-usage">
-<h1>Basic usage<a class="headerlink" href="#basic-usage" title="Permalink to this headline">¶</a></h1>
-<p>In interactive code examples that follow, it will be assumed that
-all items in the <tt class="docutils literal"><span class="pre">mpmath</span></tt> namespace have been imported:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-</pre></div>
-</div>
-<p>Importing everything can be convenient, especially when using mpmath interactively, but be
-careful when mixing mpmath with other libraries! To avoid inadvertently overriding
-other functions or objects, explicitly import only the needed objects, or use
-the <tt class="docutils literal"><span class="pre">mpmath.</span></tt> or <tt class="docutils literal"><span class="pre">mp.</span></tt> namespaces:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span>
-<span class="n">sin</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">cos</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-
-<span class="kn">import</span> <span class="nn">mpmath</span>
-<span class="n">mpmath</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">mpmath</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-
-<span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="n">mp</span>    <span class="c"># mp context object -- to be explained</span>
-<span class="n">mp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">mp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="section" id="number-types">
-<h2>Number types<a class="headerlink" href="#number-types" title="Permalink to this headline">¶</a></h2>
-<p>Mpmath provides the following numerical types:</p>
-<blockquote>
-<div><table border="1" class="docutils">
-<colgroup>
-<col width="43%" />
-<col width="57%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head">Class</th>
-<th class="head">Description</th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-even"><td><tt class="docutils literal"><span class="pre">mpf</span></tt></td>
-<td>Real float</td>
-</tr>
-<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">mpc</span></tt></td>
-<td>Complex float</td>
-</tr>
-<tr class="row-even"><td><tt class="docutils literal"><span class="pre">matrix</span></tt></td>
-<td>Matrix</td>
-</tr>
-</tbody>
-</table>
-</div></blockquote>
-<p>The following section will provide a very short introduction to the types <tt class="docutils literal"><span class="pre">mpf</span></tt> and <tt class="docutils literal"><span class="pre">mpc</span></tt>. Intervals and matrices are described further in the documentation chapters on interval arithmetic and matrices / linear algebra.</p>
-<p>The <tt class="docutils literal"><span class="pre">mpf</span></tt> type is analogous to Python&#8217;s built-in <tt class="docutils literal"><span class="pre">float</span></tt>. It holds a real number or one of the special values <tt class="docutils literal"><span class="pre">inf</span></tt> (positive infinity), <tt class="docutils literal"><span class="pre">-inf</span></tt> (negative infinity) and <tt class="docutils literal"><span class="pre">nan</span></tt> (not-a-number, indicating an indeterminate result). You can create <tt class="docutils literal"><span class="pre">mpf</span></tt> instances from strings, integers, floats, and other <tt class="docutils literal"><span class="pre">mpf</span></tt> instances:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mpf</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">mpf(&#39;4.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpf</span><span class="p">(</span><span class="mf">2.5</span><span class="p">)</span>
-<span class="go">mpf(&#39;2.5&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpf</span><span class="p">(</span><span class="s">&quot;1.25e6&quot;</span><span class="p">)</span>
-<span class="go">mpf(&#39;1250000.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpf</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">mpf(&#39;2.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpf</span><span class="p">(</span><span class="s">&quot;inf&quot;</span><span class="p">)</span>
-<span class="go">mpf(&#39;+inf&#39;)</span>
-</pre></div>
-</div>
-<p>The <tt class="docutils literal"><span class="pre">mpc</span></tt> type represents a complex number in rectangular form as a pair of <tt class="docutils literal"><span class="pre">mpf</span></tt> instances. It can be constructed from a Python <tt class="docutils literal"><span class="pre">complex</span></tt>, a real number, or a pair of real numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mpc</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">mpc(real=&#39;2.0&#39;, imag=&#39;3.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpc</span><span class="p">(</span><span class="nb">complex</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span><span class="o">.</span><span class="n">imag</span>
-<span class="go">mpf(&#39;3.0&#39;)</span>
-</pre></div>
-</div>
-<p>You can mix <tt class="docutils literal"><span class="pre">mpf</span></tt> and <tt class="docutils literal"><span class="pre">mpc</span></tt> instances with each other and with Python numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mpf</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">mpf</span><span class="p">(</span><span class="s">&#39;2.5&#39;</span><span class="p">)</span> <span class="o">+</span> <span class="mf">1.0</span>
-<span class="go">mpf(&#39;9.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>      <span class="c"># Set precision (see below)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpc</span><span class="p">(</span><span class="mi">1j</span><span class="p">)</span><span class="o">**</span><span class="mf">0.5</span>
-<span class="go">mpc(real=&#39;0.70710678118654757&#39;, imag=&#39;0.70710678118654757&#39;)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="setting-the-precision">
-<h2>Setting the precision<a class="headerlink" href="#setting-the-precision" title="Permalink to this headline">¶</a></h2>
-<p>Mpmath uses a global working precision; it does not keep track of the precision or accuracy of individual numbers. Performing an arithmetic operation or calling <tt class="docutils literal"><span class="pre">mpf()</span></tt> rounds the result to the current working precision. The working precision is controlled by a context object called <tt class="docutils literal"><span class="pre">mp</span></tt>, which has the following default state:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">mp</span>
-<span class="go">Mpmath settings:</span>
-<span class="go">  mp.prec = 53                [default: 53]</span>
-<span class="go">  mp.dps = 15                 [default: 15]</span>
-<span class="go">  mp.trap_complex = False     [default: False]</span>
-</pre></div>
-</div>
-<p>The term <strong>prec</strong> denotes the binary precision (measured in bits) while <strong>dps</strong> (short for <em>decimal places</em>) is the decimal precision. Binary and decimal precision are related roughly according to the formula <tt class="docutils literal"><span class="pre">prec</span> <span class="pre">=</span> <span class="pre">3.33*dps</span></tt>. For example, it takes a precision of roughly 333 bits to hold an approximation of pi that is accurate to 100 decimal places (actually slightly more than 333 bits is used).</p>
-<p>Changing either precision property of the <tt class="docutils literal"><span class="pre">mp</span></tt> object automatically updates the other; usually you just want to change the <tt class="docutils literal"><span class="pre">dps</span></tt> value:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">100</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span>
-<span class="go">100</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">prec</span>
-<span class="go">336</span>
-</pre></div>
-</div>
-<p>When the precision has been set, all <tt class="docutils literal"><span class="pre">mpf</span></tt> operations are carried out at that precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="mi">6</span>
-<span class="go">mpf(&#39;0.16666666666666666666666666666666666666666666666666656&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">**</span> <span class="n">mpf</span><span class="p">(</span><span class="s">&#39;0.5&#39;</span><span class="p">)</span>
-<span class="go">mpf(&#39;1.414213562373095048801688713&#39;)</span>
-</pre></div>
-</div>
-<p>The precision of complex arithmetic is also controlled by the <tt class="docutils literal"><span class="pre">mp</span></tt> object:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpc</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="mi">3</span>
-<span class="go">mpc(real=&#39;0.3333333333321&#39;, imag=&#39;0.6666666666642&#39;)</span>
-</pre></div>
-</div>
-<p>There is no restriction on the magnitude of numbers. An <tt class="docutils literal"><span class="pre">mpf</span></tt> can for example hold an approximation of a large Mersenne prime:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">32582657</span> <span class="o">-</span> <span class="mi">1</span>
-<span class="go">1.24575026015369e+9808357</span>
-</pre></div>
-</div>
-<p>Or why not 1 googolplex:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span> <span class="o">**</span> <span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">100</span><span class="p">)</span>  
-<span class="go">1.0e+100000000000000000000000000000000000000000000000000...</span>
-</pre></div>
-</div>
-<p>The (binary) exponent is stored exactly and is independent of the precision.</p>
-<div class="section" id="temporarily-changing-the-precision">
-<h3>Temporarily changing the precision<a class="headerlink" href="#temporarily-changing-the-precision" title="Permalink to this headline">¶</a></h3>
-<p>It is often useful to change the precision during only part of a calculation. A way to temporarily increase the precision and then restore it is as follows:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">prec</span> <span class="o">+=</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># do_something()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">prec</span> <span class="o">-=</span> <span class="mi">2</span>
-</pre></div>
-</div>
-<p>In Python 2.5, the <tt class="docutils literal"><span class="pre">with</span></tt> statement along with the mpmath functions <tt class="docutils literal"><span class="pre">workprec</span></tt>, <tt class="docutils literal"><span class="pre">workdps</span></tt>, <tt class="docutils literal"><span class="pre">extraprec</span></tt> and <tt class="docutils literal"><span class="pre">extradps</span></tt> can be used to temporarily change precision in a more safe manner:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">__future__</span> <span class="kn">import</span> <span class="n">with_statement</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">with</span> <span class="n">workdps</span><span class="p">(</span><span class="mi">20</span><span class="p">):</span>  
-<span class="gp">... </span>    <span class="k">print</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">7</span>
-<span class="gp">... </span>    <span class="k">with</span> <span class="n">extradps</span><span class="p">(</span><span class="mi">10</span><span class="p">):</span>
-<span class="gp">... </span>        <span class="k">print</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">7</span>
-<span class="gp">...</span>
-<span class="go">0.14285714285714285714</span>
-<span class="go">0.142857142857142857142857142857</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span>
-<span class="go">15</span>
-</pre></div>
-</div>
-<p>The <tt class="docutils literal"><span class="pre">with</span></tt> statement ensures that the precision gets reset when exiting the block, even in the case that an exception is raised. (The effect of the <tt class="docutils literal"><span class="pre">with</span></tt> statement can be emulated in Python 2.4 by using a <tt class="docutils literal"><span class="pre">try/finally</span></tt> block.)</p>
-<p>The <tt class="docutils literal"><span class="pre">workprec</span></tt> family of functions can also be used as function decorators:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nd">@workdps</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span>
-<span class="gp">... </span><span class="k">def</span> <span class="nf">f</span><span class="p">():</span>
-<span class="gp">... </span>    <span class="k">return</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">()</span>
-<span class="go">mpf(&#39;0.33333331346511841&#39;)</span>
-</pre></div>
-</div>
-<p>Some functions accept the <tt class="docutils literal"><span class="pre">prec</span></tt> and <tt class="docutils literal"><span class="pre">dps</span></tt> keyword arguments and this will override the global working precision. Note that this will not affect the precision at which the result is printed, so to get all digits, you must either use increase precision afterward when printing or use <tt class="docutils literal"><span class="pre">nstr</span></tt>/<tt class="docutils literal"><span class="pre">nprint</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">2.71828182845905</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">dps</span><span class="o">=</span><span class="mi">50</span><span class="p">)</span>      <span class="c"># Extra digits won&#39;t be printed</span>
-<span class="go">2.71828182845905</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">dps</span><span class="o">=</span><span class="mi">50</span><span class="p">),</span> <span class="mi">50</span><span class="p">)</span>
-<span class="go">2.7182818284590452353602874713526624977572470937</span>
-</pre></div>
-</div>
-<p>Finally, instead of using the global context object <tt class="docutils literal"><span class="pre">mp</span></tt>, you can create custom contexts and work with methods of those instances instead of global functions. The working precision will be local to each context object:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp2</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">clone</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp2</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">20</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">mp</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="mi">3</span>
-<span class="go">0.3333333333</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">mp2</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="mi">3</span>
-<span class="go">0.33333333333333333333</span>
-</pre></div>
-</div>
-<p><strong>Note</strong>: the ability to create multiple contexts is a new feature that is only partially implemented. Not all mpmath functions are yet available as context-local methods. In the present version, you are likely to encounter bugs if you try mixing different contexts.</p>
-</div>
-</div>
-<div class="section" id="providing-correct-input">
-<h2>Providing correct input<a class="headerlink" href="#providing-correct-input" title="Permalink to this headline">¶</a></h2>
-<p>Note that when creating a new <tt class="docutils literal"><span class="pre">mpf</span></tt>, the value will at most be as accurate as the input. <em>Be careful when mixing mpmath numbers with Python floats</em>. When working at high precision, fractional <tt class="docutils literal"><span class="pre">mpf</span></tt> values should be created from strings or integers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">30</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpf</span><span class="p">(</span><span class="mf">10.9</span><span class="p">)</span>   <span class="c"># bad</span>
-<span class="go">mpf(&#39;10.9000000000000003552713678800501&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpf</span><span class="p">(</span><span class="s">&#39;10.9&#39;</span><span class="p">)</span>  <span class="c"># good</span>
-<span class="go">mpf(&#39;10.8999999999999999999999999999997&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpf</span><span class="p">(</span><span class="mi">109</span><span class="p">)</span> <span class="o">/</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>   <span class="c"># also good</span>
-<span class="go">mpf(&#39;10.8999999999999999999999999999997&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-</pre></div>
-</div>
-<p>(Binary fractions such as 0.5, 1.5, 0.75, 0.125, etc, are generally safe as input, however, since those can be represented exactly by Python floats.)</p>
-</div>
-<div class="section" id="printing">
-<h2>Printing<a class="headerlink" href="#printing" title="Permalink to this headline">¶</a></h2>
-<p>By default, the <tt class="docutils literal"><span class="pre">repr()</span></tt> of a number includes its type signature. This way <tt class="docutils literal"><span class="pre">eval</span></tt> can be used to recreate a number from its string representation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">eval</span><span class="p">(</span><span class="nb">repr</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="mf">2.5</span><span class="p">)))</span>
-<span class="go">mpf(&#39;2.5&#39;)</span>
-</pre></div>
-</div>
-<p>Prettier output can be obtained by using <tt class="docutils literal"><span class="pre">str()</span></tt> or <tt class="docutils literal"><span class="pre">print</span></tt>, which hide the <tt class="docutils literal"><span class="pre">mpf</span></tt> and <tt class="docutils literal"><span class="pre">mpc</span></tt> signatures and also suppress rounding artifacts in the last few digits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mpf</span><span class="p">(</span><span class="s">&quot;3.14159&quot;</span><span class="p">)</span>
-<span class="go">mpf(&#39;3.1415899999999999&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">mpf</span><span class="p">(</span><span class="s">&quot;3.14159&quot;</span><span class="p">)</span>
-<span class="go">3.14159</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">mpc</span><span class="p">(</span><span class="mi">1j</span><span class="p">)</span><span class="o">**</span><span class="mf">0.5</span>
-<span class="go">(0.707106781186548 + 0.707106781186548j)</span>
-</pre></div>
-</div>
-<p>Setting the <tt class="docutils literal"><span class="pre">mp.pretty</span></tt> option will use the <tt class="docutils literal"><span class="pre">str()</span></tt>-style output for <tt class="docutils literal"><span class="pre">repr()</span></tt> as well:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpf</span><span class="p">(</span><span class="mf">0.6</span><span class="p">)</span>
-<span class="go">0.6</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpf</span><span class="p">(</span><span class="mf">0.6</span><span class="p">)</span>
-<span class="go">mpf(&#39;0.59999999999999998&#39;)</span>
-</pre></div>
-</div>
-<p>The number of digits with which numbers are printed by default is determined by the working precision. To specify the number of digits to show without changing the working precision, use <a class="reference internal" href="../mpmath/general.html#mpmath.nstr" title="mpmath.nstr"><tt class="xref py py-func docutils literal"><span class="pre">mpmath.nstr()</span></tt></a> and <a class="reference internal" href="../mpmath/general.html#mpmath.nprint" title="mpmath.nprint"><tt class="xref py py-func docutils literal"><span class="pre">mpmath.nprint()</span></tt></a>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="mi">6</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
-<span class="go">mpf(&#39;0.16666666666666666&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nstr</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span>
-<span class="go">&#39;0.16666667&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span>
-<span class="go">0.16666667</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nstr</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">50</span><span class="p">)</span>
-<span class="go">&#39;0.16666666666666665741480812812369549646973609924316&#39;</span>
-</pre></div>
-</div>
-</div>
-</div>
-
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-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Basic usage</a><ul>
-<li><a class="reference internal" href="#number-types">Number types</a></li>
-<li><a class="reference internal" href="#setting-the-precision">Setting the precision</a><ul>
-<li><a class="reference internal" href="#temporarily-changing-the-precision">Temporarily changing the precision</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#providing-correct-input">Providing correct input</a></li>
-<li><a class="reference internal" href="#printing">Printing</a></li>
-</ul>
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-</ul>
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-            
-  <div class="section" id="function-approximation">
-<h1>Function approximation<a class="headerlink" href="#function-approximation" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="taylor-series-taylor">
-<h2>Taylor series (<tt class="docutils literal"><span class="pre">taylor</span></tt>)<a class="headerlink" href="#taylor-series-taylor" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="mpmath.taylor">
-<tt class="descclassname">mpmath.</tt><tt class="descname">taylor</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>x</em>, <em>n</em>, <em>**options</em><big>)</big><a class="headerlink" href="#mpmath.taylor" title="Permalink to this definition">¶</a></dt>
-<dd><p>Produces a degree-<span class="math">\(n\)</span> Taylor polynomial around the point <span class="math">\(x\)</span> of the
-given function <span class="math">\(f\)</span>. The coefficients are returned as a list.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)))</span>
-<span class="go">[0.0, 1.0, 0.0, -0.166667, 0.0, 0.00833333]</span>
-</pre></div>
-</div>
-<p>The coefficients are computed using high-order numerical
-differentiation. The function must be possible to evaluate
-to arbitrary precision. See <a class="reference internal" href="../calculus/differentiation.html#mpmath.diff" title="mpmath.diff"><tt class="xref py py-func docutils literal"><span class="pre">diff()</span></tt></a> for additional details
-and supported keyword options.</p>
-<p>Note that to evaluate the Taylor polynomial as an approximation
-of <span class="math">\(f\)</span>, e.g. with <a class="reference internal" href="../calculus/polynomials.html#mpmath.polyval" title="mpmath.polyval"><tt class="xref py py-func docutils literal"><span class="pre">polyval()</span></tt></a>, the coefficients must be reversed,
-and the point of the Taylor expansion must be subtracted from
-the argument:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">taylor</span><span class="p">(</span><span class="n">exp</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polyval</span><span class="p">(</span><span class="n">p</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="mf">2.5</span> <span class="o">-</span> <span class="mf">2.0</span><span class="p">)</span>
-<span class="go">12.1824939606092</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="mf">2.5</span><span class="p">)</span>
-<span class="go">12.1824939607035</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="pade-approximation-pade">
-<h2>Pade approximation (<tt class="docutils literal"><span class="pre">pade</span></tt>)<a class="headerlink" href="#pade-approximation-pade" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="mpmath.pade">
-<tt class="descclassname">mpmath.</tt><tt class="descname">pade</tt><big>(</big><em>ctx</em>, <em>a</em>, <em>L</em>, <em>M</em><big>)</big><a class="headerlink" href="#mpmath.pade" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes a Pade approximation of degree <span class="math">\((L, M)\)</span> to a function.
-Given at least <span class="math">\(L+M+1\)</span> Taylor coefficients <span class="math">\(a\)</span> approximating
-a function <span class="math">\(A(x)\)</span>, <a class="reference internal" href="#mpmath.pade" title="mpmath.pade"><tt class="xref py py-func docutils literal"><span class="pre">pade()</span></tt></a> returns coefficients of
-polynomials <span class="math">\(P, Q\)</span> satisfying</p>
-<div class="math">
-\[P = \sum_{k=0}^L p_k x^k\]\[Q = \sum_{k=0}^M q_k x^k\]\[Q_0 = 1\]\[A(x) Q(x) = P(x) + O(x^{L+M+1})\]</div>
-<p><span class="math">\(P(x)/Q(x)\)</span> can provide a good approximation to an analytic function
-beyond the radius of convergence of its Taylor series (example
-from G.A. Baker &#8216;Essentials of Pade Approximants&#8217; Academic Press,
-Ch.1A):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">one</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">return</span> <span class="n">sqrt</span><span class="p">((</span><span class="n">one</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">one</span> <span class="o">+</span> <span class="n">x</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">taylor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">pade</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="mi">10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polyval</span><span class="p">(</span><span class="n">p</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">polyval</span><span class="p">(</span><span class="n">q</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">1.38169105566806</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1.38169855941551</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="chebyshev-approximation-chebyfit">
-<h2>Chebyshev approximation (<tt class="docutils literal"><span class="pre">chebyfit</span></tt>)<a class="headerlink" href="#chebyshev-approximation-chebyfit" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="mpmath.chebyfit">
-<tt class="descclassname">mpmath.</tt><tt class="descname">chebyfit</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>interval</em>, <em>N</em>, <em>error=False</em><big>)</big><a class="headerlink" href="#mpmath.chebyfit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes a polynomial of degree <span class="math">\(N-1\)</span> that approximates the
-given function <span class="math">\(f\)</span> on the interval <span class="math">\([a, b]\)</span>. With <tt class="docutils literal"><span class="pre">error=True</span></tt>,
-<a class="reference internal" href="#mpmath.chebyfit" title="mpmath.chebyfit"><tt class="xref py py-func docutils literal"><span class="pre">chebyfit()</span></tt></a> also returns an accurate estimate of the
-maximum absolute error; that is, the maximum value of
-<span class="math">\(|f(x) - P(x)|\)</span> for <span class="math">\(x \in [a, b]\)</span>.</p>
-<p><a class="reference internal" href="#mpmath.chebyfit" title="mpmath.chebyfit"><tt class="xref py py-func docutils literal"><span class="pre">chebyfit()</span></tt></a> uses the Chebyshev approximation formula,
-which gives a nearly optimal solution: that is, the maximum
-error of the approximating polynomial is very close to
-the smallest possible for any polynomial of the same degree.</p>
-<p>Chebyshev approximation is very useful if one needs repeated
-evaluation of an expensive function, such as function defined
-implicitly by an integral or a differential equation. (For
-example, it could be used to turn a slow mpmath function
-into a fast machine-precision version of the same.)</p>
-<p><strong>Examples</strong></p>
-<p>Here we use <a class="reference internal" href="#mpmath.chebyfit" title="mpmath.chebyfit"><tt class="xref py py-func docutils literal"><span class="pre">chebyfit()</span></tt></a> to generate a low-degree approximation
-of <span class="math">\(f(x) = \cos(x)\)</span>, valid on the interval <span class="math">\([1, 2]\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly</span><span class="p">,</span> <span class="n">err</span> <span class="o">=</span> <span class="n">chebyfit</span><span class="p">(</span><span class="n">cos</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="mi">5</span><span class="p">,</span> <span class="n">error</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">poly</span><span class="p">)</span>
-<span class="go">[0.00291682, 0.146166, -0.732491, 0.174141, 0.949553]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">err</span><span class="p">,</span> <span class="mi">12</span><span class="p">)</span>
-<span class="go">1.61351758081e-5</span>
-</pre></div>
-</div>
-<p>The polynomial can be evaluated using <tt class="docutils literal"><span class="pre">polyval</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">polyval</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="mf">1.6</span><span class="p">),</span> <span class="mi">12</span><span class="p">)</span>
-<span class="go">-0.0291858904138</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="mf">1.6</span><span class="p">),</span> <span class="mi">12</span><span class="p">)</span>
-<span class="go">-0.0291995223013</span>
-</pre></div>
-</div>
-<p>Sampling the true error at 1000 points shows that the error
-estimate generated by <tt class="docutils literal"><span class="pre">chebyfit</span></tt> is remarkably good:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">error</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="nb">abs</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">polyval</span><span class="p">(</span><span class="n">poly</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="nb">max</span><span class="p">([</span><span class="n">error</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">n</span><span class="o">/</span><span class="mf">1000.</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1000</span><span class="p">)]),</span> <span class="mi">12</span><span class="p">)</span>
-<span class="go">1.61349954245e-5</span>
-</pre></div>
-</div>
-<p><strong>Choice of degree</strong></p>
-<p>The degree <span class="math">\(N\)</span> can be set arbitrarily high, to obtain an
-arbitrarily good approximation. As a rule of thumb, an
-<span class="math">\(N\)</span>-term Chebyshev approximation is good to <span class="math">\(N/(b-a)\)</span> decimal
-places on a unit interval (although this depends on how
-well-behaved <span class="math">\(f\)</span> is). The cost grows accordingly: <tt class="docutils literal"><span class="pre">chebyfit</span></tt>
-evaluates the function <span class="math">\((N^2)/2\)</span> times to compute the
-coefficients and an additional <span class="math">\(N\)</span> times to estimate the error.</p>
-<p><strong>Possible issues</strong></p>
-<p>One should be careful to use a sufficiently high working
-precision both when calling <tt class="docutils literal"><span class="pre">chebyfit</span></tt> and when evaluating
-the resulting polynomial, as the polynomial is sometimes
-ill-conditioned. It is for example difficult to reach
-15-digit accuracy when evaluating the polynomial using
-machine precision floats, no matter the theoretical
-accuracy of the polynomial. (The option to return the
-coefficients in Chebyshev form should be made available
-in the future.)</p>
-<p>It is important to note the Chebyshev approximation works
-poorly if <span class="math">\(f\)</span> is not smooth. A function containing singularities,
-rapid oscillation, etc can be approximated more effectively by
-multiplying it by a weight function that cancels out the
-nonsmooth features, or by dividing the interval into several
-segments.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="fourier-series-fourier-fourierval">
-<h2>Fourier series (<tt class="docutils literal"><span class="pre">fourier</span></tt>, <tt class="docutils literal"><span class="pre">fourierval</span></tt>)<a class="headerlink" href="#fourier-series-fourier-fourierval" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="mpmath.fourier">
-<tt class="descclassname">mpmath.</tt><tt class="descname">fourier</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>interval</em>, <em>N</em><big>)</big><a class="headerlink" href="#mpmath.fourier" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the Fourier series of degree <span class="math">\(N\)</span> of the given function
-on the interval <span class="math">\([a, b]\)</span>. More precisely, <a class="reference internal" href="#mpmath.fourier" title="mpmath.fourier"><tt class="xref py py-func docutils literal"><span class="pre">fourier()</span></tt></a> returns
-two lists <span class="math">\((c, s)\)</span> of coefficients (the cosine series and sine
-series, respectively), such that</p>
-<div class="math">
-\[f(x) \sim \sum_{k=0}^N
-    c_k \cos(k m) + s_k \sin(k m)\]</div>
-<p>where <span class="math">\(m = 2 \pi / (b-a)\)</span>.</p>
-<p>Note that many texts define the first coefficient as <span class="math">\(2 c_0\)</span> instead
-of <span class="math">\(c_0\)</span>. The easiest way to evaluate the computed series correctly
-is to pass it to <a class="reference internal" href="#mpmath.fourierval" title="mpmath.fourierval"><tt class="xref py py-func docutils literal"><span class="pre">fourierval()</span></tt></a>.</p>
-<p><strong>Examples</strong></p>
-<p>The function <span class="math">\(f(x) = x\)</span> has a simple Fourier series on the standard
-interval <span class="math">\([-\pi, \pi]\)</span>. The cosine coefficients are all zero (because
-the function has odd symmetry), and the sine coefficients are
-rational numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span><span class="p">,</span> <span class="n">s</span> <span class="o">=</span> <span class="n">fourier</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="n">pi</span><span class="p">,</span> <span class="n">pi</span><span class="p">],</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-<span class="go">[0.0, 0.0, 0.0, 0.0, 0.0, 0.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="go">[0.0, 2.0, -1.0, 0.666667, -0.5, 0.4]</span>
-</pre></div>
-</div>
-<p>This computes a Fourier series of a nonsymmetric function on
-a nonstandard interval:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cs</span> <span class="o">=</span> <span class="n">fourier</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">cs</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-<span class="go">[0.583333, 1.12479, -1.27552, 0.904708, -0.441296]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">cs</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">[0.0, -2.6255, 0.580905, 0.219974, -0.540057]</span>
-</pre></div>
-</div>
-<p>It is instructive to plot a function along with its truncated
-Fourier series:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">plot</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">fourierval</span><span class="p">(</span><span class="n">cs</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">x</span><span class="p">)],</span> <span class="n">I</span><span class="p">)</span> 
-</pre></div>
-</div>
-<p>Fourier series generally converge slowly (and may not converge
-pointwise). For example, if <span class="math">\(f(x) = \cosh(x)\)</span>, a 10-term Fourier
-series gives an <span class="math">\(L^2\)</span> error corresponding to 2-digit accuracy:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cs</span> <span class="o">=</span> <span class="n">fourier</span><span class="p">(</span><span class="n">cosh</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="mi">9</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="p">(</span><span class="n">cosh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">fourierval</span><span class="p">(</span><span class="n">cs</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">quad</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">I</span><span class="p">)))</span>
-<span class="go">0.00467963</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.fourier" title="mpmath.fourier"><tt class="xref py py-func docutils literal"><span class="pre">fourier()</span></tt></a> uses numerical quadrature. For nonsmooth functions,
-the accuracy (and speed) can be improved by including all singular
-points in the interval specification:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">fourier</span><span class="p">(</span><span class="nb">abs</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">([0.5000441648], [0.0])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">fourier</span><span class="p">(</span><span class="nb">abs</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="mi">0</span><span class="p">),</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">([0.5], [0.0])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="mpmath.fourierval">
-<tt class="descclassname">mpmath.</tt><tt class="descname">fourierval</tt><big>(</big><em>ctx</em>, <em>series</em>, <em>interval</em>, <em>x</em><big>)</big><a class="headerlink" href="#mpmath.fourierval" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates a Fourier series (in the format computed by
-by <a class="reference internal" href="#mpmath.fourier" title="mpmath.fourier"><tt class="xref py py-func docutils literal"><span class="pre">fourier()</span></tt></a> for the given interval) at the point <span class="math">\(x\)</span>.</p>
-<p>The series should be a pair <span class="math">\((c, s)\)</span> where <span class="math">\(c\)</span> is the
-cosine series and <span class="math">\(s\)</span> is the sine series. The two lists
-need not have the same length.</p>
-</dd></dl>
-
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Function approximation</a><ul>
-<li><a class="reference internal" href="#taylor-series-taylor">Taylor series (<tt class="docutils literal"><span class="pre">taylor</span></tt>)</a></li>
-<li><a class="reference internal" href="#pade-approximation-pade">Pade approximation (<tt class="docutils literal"><span class="pre">pade</span></tt>)</a></li>
-<li><a class="reference internal" href="#chebyshev-approximation-chebyfit">Chebyshev approximation (<tt class="docutils literal"><span class="pre">chebyfit</span></tt>)</a></li>
-<li><a class="reference internal" href="#fourier-series-fourier-fourierval">Fourier series (<tt class="docutils literal"><span class="pre">fourier</span></tt>, <tt class="docutils literal"><span class="pre">fourierval</span></tt>)</a></li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../calculus/odes.html"
-                        title="previous chapter">Ordinary differential equations</a></p>
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-                        title="next chapter">Matrices</a></p>
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-            
-  <div class="section" id="differentiation">
-<h1>Differentiation<a class="headerlink" href="#differentiation" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="numerical-derivatives-diff-diffs">
-<h2>Numerical derivatives (<tt class="docutils literal"><span class="pre">diff</span></tt>, <tt class="docutils literal"><span class="pre">diffs</span></tt>)<a class="headerlink" href="#numerical-derivatives-diff-diffs" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="mpmath.diff">
-<tt class="descclassname">mpmath.</tt><tt class="descname">diff</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>x</em>, <em>n=1</em>, <em>**options</em><big>)</big><a class="headerlink" href="#mpmath.diff" title="Permalink to this definition">¶</a></dt>
-<dd><p>Numerically computes the derivative of <span class="math">\(f\)</span>, <span class="math">\(f'(x)\)</span>, or generally for
-an integer <span class="math">\(n \ge 0\)</span>, the <span class="math">\(n\)</span>-th derivative <span class="math">\(f^{(n)}(x)\)</span>.
-A few basic examples are:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">)</span>
-<span class="go">3.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">2.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">([</span><span class="n">diff</span><span class="p">(</span><span class="n">exp</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">)])</span>   <span class="c"># exp&#39;(x) = exp(x)</span>
-<span class="go">[20.0855, 20.0855, 20.0855, 20.0855, 20.0855]</span>
-</pre></div>
-</div>
-<p>Even more generally, given a tuple of arguments <span class="math">\((x_1, \ldots, x_k)\)</span>
-and order <span class="math">\((n_1, \ldots, n_k)\)</span>, the partial derivative
-<span class="math">\(f^{(n_1,\ldots,n_k)}(x_1,\ldots,x_k)\)</span> is evaluated. For example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">:</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="mf">0.25</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span>
-<span class="go">2.75</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">:</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="mf">0.25</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span>
-<span class="go">3.0</span>
-</pre></div>
-</div>
-<p><strong>Options</strong></p>
-<p>The following optional keyword arguments are recognized:</p>
-<dl class="docutils">
-<dt><tt class="docutils literal"><span class="pre">method</span></tt></dt>
-<dd>Supported methods are <tt class="docutils literal"><span class="pre">'step'</span></tt> or <tt class="docutils literal"><span class="pre">'quad'</span></tt>: derivatives may be
-computed using either a finite difference with a small step
-size <span class="math">\(h\)</span> (default), or numerical quadrature.</dd>
-<dt><tt class="docutils literal"><span class="pre">direction</span></tt></dt>
-<dd>Direction of finite difference: can be -1 for a left
-difference, 0 for a central difference (default), or +1
-for a right difference; more generally can be any complex number.</dd>
-<dt><tt class="docutils literal"><span class="pre">addprec</span></tt></dt>
-<dd>Extra precision for <span class="math">\(h\)</span> used to account for the function&#8217;s
-sensitivity to perturbations (default = 10).</dd>
-<dt><tt class="docutils literal"><span class="pre">relative</span></tt></dt>
-<dd>Choose <span class="math">\(h\)</span> relative to the magnitude of <span class="math">\(x\)</span>, rather than an
-absolute value; useful for large or tiny <span class="math">\(x\)</span> (default = False).</dd>
-<dt><tt class="docutils literal"><span class="pre">h</span></tt></dt>
-<dd>As an alternative to <tt class="docutils literal"><span class="pre">addprec</span></tt> and <tt class="docutils literal"><span class="pre">relative</span></tt>, manually
-select the step size <span class="math">\(h\)</span>.</dd>
-<dt><tt class="docutils literal"><span class="pre">singular</span></tt></dt>
-<dd>If True, evaluation exactly at the point <span class="math">\(x\)</span> is avoided; this is
-useful for differentiating functions with removable singularities.
-Default = False.</dd>
-<dt><tt class="docutils literal"><span class="pre">radius</span></tt></dt>
-<dd>Radius of integration contour (with <tt class="docutils literal"><span class="pre">method</span> <span class="pre">=</span> <span class="pre">'quad'</span></tt>).
-Default = 0.25. A larger radius typically is faster and more
-accurate, but it must be chosen so that <span class="math">\(f\)</span> has no
-singularities within the radius from the evaluation point.</dd>
-</dl>
-<p>A finite difference requires <span class="math">\(n+1\)</span> function evaluations and must be
-performed at <span class="math">\((n+1)\)</span> times the target precision. Accordingly, <span class="math">\(f\)</span> must
-support fast evaluation at high precision.</p>
-<p>With integration, a larger number of function evaluations is
-required, but not much extra precision is required. For high order
-derivatives, this method may thus be faster if f is very expensive to
-evaluate at high precision.</p>
-<p><strong>Further examples</strong></p>
-<p>The direction option is useful for computing left- or right-sided
-derivatives of nonsmooth functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="nb">abs</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">direction</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="nb">abs</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">direction</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="nb">abs</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">direction</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-1.0</span>
-</pre></div>
-</div>
-<p>More generally, if the direction is nonzero, a right difference
-is computed where the step size is multiplied by sign(direction).
-For example, with direction=+j, the derivative from the positive
-imaginary direction will be computed:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="nb">abs</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">direction</span><span class="o">=</span><span class="n">j</span><span class="p">)</span>
-<span class="go">(0.0 - 1.0j)</span>
-</pre></div>
-</div>
-<p>With integration, the result may have a small imaginary part
-even even if the result is purely real:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sqrt</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;quad&#39;</span><span class="p">)</span>    
-<span class="go">(0.5 - 4.59...e-26j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">_</span><span class="p">)</span>
-<span class="go">0.5</span>
-</pre></div>
-</div>
-<p>Adding precision to obtain an accurate value:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">cos</span><span class="p">,</span> <span class="mf">1e-30</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">cos</span><span class="p">,</span> <span class="mf">1e-30</span><span class="p">,</span> <span class="n">h</span><span class="o">=</span><span class="mf">0.0001</span><span class="p">)</span>
-<span class="go">-9.99999998328279e-31</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">cos</span><span class="p">,</span> <span class="mf">1e-30</span><span class="p">,</span> <span class="n">addprec</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">-1.0e-30</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="mpmath.diffs">
-<tt class="descclassname">mpmath.</tt><tt class="descname">diffs</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>x</em>, <em>n=None</em>, <em>**options</em><big>)</big><a class="headerlink" href="#mpmath.diffs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a generator that yields the sequence of derivatives</p>
-<div class="math">
-\[f(x), f'(x), f''(x), \ldots, f^{(k)}(x), \ldots\]</div>
-<p>With <tt class="docutils literal"><span class="pre">method='step'</span></tt>, <a class="reference internal" href="#mpmath.diffs" title="mpmath.diffs"><tt class="xref py py-func docutils literal"><span class="pre">diffs()</span></tt></a> uses only <span class="math">\(O(k)\)</span>
-function evaluations to generate the first <span class="math">\(k\)</span> derivatives,
-rather than the roughly <span class="math">\(O(k^2)\)</span> evaluations
-required if one calls <a class="reference internal" href="#mpmath.diff" title="mpmath.diff"><tt class="xref py py-func docutils literal"><span class="pre">diff()</span></tt></a> <span class="math">\(k\)</span> separate times.</p>
-<p>With <span class="math">\(n &lt; \infty\)</span>, the generator stops as soon as the
-<span class="math">\(n\)</span>-th derivative has been generated. If the exact number of
-needed derivatives is known in advance, this is further
-slightly more efficient.</p>
-<p>Options are the same as for <a class="reference internal" href="#mpmath.diff" title="mpmath.diff"><tt class="xref py py-func docutils literal"><span class="pre">diff()</span></tt></a>.</p>
-<p><strong>Examples</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">diffs</span><span class="p">(</span><span class="n">cos</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">)))</span>
-<span class="go">[0.540302, -0.841471, -0.540302, 0.841471, 0.540302, -0.841471]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">6</span><span class="p">),</span> <span class="n">diffs</span><span class="p">(</span><span class="n">cos</span><span class="p">,</span> <span class="mi">1</span><span class="p">)):</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">d</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">0 0.54030230586814</span>
-<span class="go">1 -0.841470984807897</span>
-<span class="go">2 -0.54030230586814</span>
-<span class="go">3 0.841470984807897</span>
-<span class="go">4 0.54030230586814</span>
-<span class="go">5 -0.841470984807897</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="composition-of-derivatives-diffs-prod-diffs-exp">
-<h2>Composition of derivatives (<tt class="docutils literal"><span class="pre">diffs_prod</span></tt>, <tt class="docutils literal"><span class="pre">diffs_exp</span></tt>)<a class="headerlink" href="#composition-of-derivatives-diffs-prod-diffs-exp" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="mpmath.diffs_prod">
-<tt class="descclassname">mpmath.</tt><tt class="descname">diffs_prod</tt><big>(</big><em>ctx</em>, <em>factors</em><big>)</big><a class="headerlink" href="#mpmath.diffs_prod" title="Permalink to this definition">¶</a></dt>
-<dd><p>Given a list of <span class="math">\(N\)</span> iterables or generators yielding
-<span class="math">\(f_k(x), f'_k(x), f''_k(x), \ldots\)</span> for <span class="math">\(k = 1, \ldots, N\)</span>,
-generate <span class="math">\(g(x), g'(x), g''(x), \ldots\)</span> where
-<span class="math">\(g(x) = f_1(x) f_2(x) \cdots f_N(x)\)</span>.</p>
-<p>At high precision and for large orders, this is typically more efficient
-than numerical differentiation if the derivatives of each <span class="math">\(f_k(x)\)</span>
-admit direct computation.</p>
-<p>Note: This function does not increase the working precision internally,
-so guard digits may have to be added externally for full accuracy.</p>
-<p><strong>Examples</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span> <span class="o">=</span> <span class="n">diffs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">diffs_prod</span><span class="p">([</span><span class="n">diffs</span><span class="p">(</span><span class="n">exp</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span> <span class="n">diffs</span><span class="p">(</span><span class="n">cos</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span> <span class="n">diffs</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span><span class="mi">1</span><span class="p">)])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">next</span><span class="p">(</span><span class="n">u</span><span class="p">);</span> <span class="nb">next</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-<span class="go">1.23586333600241</span>
-<span class="go">1.23586333600241</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">next</span><span class="p">(</span><span class="n">u</span><span class="p">);</span> <span class="nb">next</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-<span class="go">0.104658952245596</span>
-<span class="go">0.104658952245596</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">next</span><span class="p">(</span><span class="n">u</span><span class="p">);</span> <span class="nb">next</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-<span class="go">-5.96999877552086</span>
-<span class="go">-5.96999877552086</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">next</span><span class="p">(</span><span class="n">u</span><span class="p">);</span> <span class="nb">next</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-<span class="go">-12.4632923122697</span>
-<span class="go">-12.4632923122697</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="mpmath.diffs_exp">
-<tt class="descclassname">mpmath.</tt><tt class="descname">diffs_exp</tt><big>(</big><em>ctx</em>, <em>fdiffs</em><big>)</big><a class="headerlink" href="#mpmath.diffs_exp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Given an iterable or generator yielding <span class="math">\(f(x), f'(x), f''(x), \ldots\)</span>
-generate <span class="math">\(g(x), g'(x), g''(x), \ldots\)</span> where <span class="math">\(g(x) = \exp(f(x))\)</span>.</p>
-<p>At high precision and for large orders, this is typically more efficient
-than numerical differentiation if the derivatives of <span class="math">\(f(x)\)</span>
-admit direct computation.</p>
-<p>Note: This function does not increase the working precision internally,
-so guard digits may have to be added externally for full accuracy.</p>
-<p><strong>Examples</strong></p>
-<p>The derivatives of the gamma function can be computed using
-logarithmic differentiation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">diffs_loggamma</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">yield</span> <span class="n">loggamma</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">... </span>    <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-<span class="gp">... </span>    <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-<span class="gp">... </span>        <span class="k">yield</span> <span class="n">psi</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">... </span>        <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span> <span class="o">=</span> <span class="n">diffs_exp</span><span class="p">(</span><span class="n">diffs_loggamma</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v</span> <span class="o">=</span> <span class="n">diffs</span><span class="p">(</span><span class="n">gamma</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">next</span><span class="p">(</span><span class="n">u</span><span class="p">);</span> <span class="nb">next</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-<span class="go">2.0</span>
-<span class="go">2.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">next</span><span class="p">(</span><span class="n">u</span><span class="p">);</span> <span class="nb">next</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-<span class="go">1.84556867019693</span>
-<span class="go">1.84556867019693</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">next</span><span class="p">(</span><span class="n">u</span><span class="p">);</span> <span class="nb">next</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-<span class="go">2.49292999190269</span>
-<span class="go">2.49292999190269</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">next</span><span class="p">(</span><span class="n">u</span><span class="p">);</span> <span class="nb">next</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-<span class="go">3.44996501352367</span>
-<span class="go">3.44996501352367</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="fractional-derivatives-differintegration-differint">
-<h2>Fractional derivatives / differintegration (<tt class="docutils literal"><span class="pre">differint</span></tt>)<a class="headerlink" href="#fractional-derivatives-differintegration-differint" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="mpmath.differint">
-<tt class="descclassname">mpmath.</tt><tt class="descname">differint</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>x</em>, <em>n=1</em>, <em>x0=0</em><big>)</big><a class="headerlink" href="#mpmath.differint" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculates the Riemann-Liouville differintegral, or fractional
-derivative, defined by</p>
-<div class="math">
-\[\,_{x_0}{\mathbb{D}}^n_xf(x) \frac{1}{\Gamma(m-n)} \frac{d^m}{dx^m}
-\int_{x_0}^{x}(x-t)^{m-n-1}f(t)dt\]</div>
-<p>where <span class="math">\(f\)</span> is a given (presumably well-behaved) function,
-<span class="math">\(x\)</span> is the evaluation point, <span class="math">\(n\)</span> is the order, and <span class="math">\(x_0\)</span> is
-the reference point of integration (<span class="math">\(m\)</span> is an arbitrary
-parameter selected automatically).</p>
-<p>With <span class="math">\(n = 1\)</span>, this is just the standard derivative <span class="math">\(f'(x)\)</span>; with <span class="math">\(n = 2\)</span>,
-the second derivative <span class="math">\(f''(x)\)</span>, etc. With <span class="math">\(n = -1\)</span>, it gives
-<span class="math">\(\int_{x_0}^x f(t) dt\)</span>, with <span class="math">\(n = -2\)</span>
-it gives <span class="math">\(\int_{x_0}^x \left( \int_{x_0}^t f(u) du \right) dt\)</span>, etc.</p>
-<p>As <span class="math">\(n\)</span> is permitted to be any number, this operator generalizes
-iterated differentiation and iterated integration to a single
-operator with a continuous order parameter.</p>
-<p><strong>Examples</strong></p>
-<p>There is an exact formula for the fractional derivative of a
-monomial <span class="math">\(x^p\)</span>, which may be used as a reference. For example,
-the following gives a half-derivative (order 0.5):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span> <span class="n">p</span> <span class="o">=</span> <span class="mi">2</span><span class="p">;</span> <span class="n">n</span> <span class="o">=</span> <span class="mf">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">differint</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">t</span><span class="o">**</span><span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-<span class="go">7.81764019044672</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="n">p</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">gamma</span><span class="p">(</span><span class="n">p</span><span class="o">-</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">p</span><span class="o">-</span><span class="n">n</span><span class="p">)</span>
-<span class="go">7.81764019044672</span>
-</pre></div>
-</div>
-<p>Another useful test function is the exponential function, whose
-integration / differentiation formula easy generalizes
-to arbitrary order. Here we first compute a third derivative,
-and then a triply nested integral. (The reference point <span class="math">\(x_0\)</span>
-is set to <span class="math">\(-\infty\)</span> to avoid nonzero endpoint terms.):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">differint</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="o">-</span><span class="mf">1.5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">0.278538406900792</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">pi</span><span class="o">*-</span><span class="mf">1.5</span><span class="p">)</span> <span class="o">*</span> <span class="n">pi</span><span class="o">**</span><span class="mi">3</span>
-<span class="go">0.278538406900792</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">differint</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="mf">3.5</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">1922.50563031149</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="mf">3.5</span><span class="p">)</span> <span class="o">/</span> <span class="n">pi</span><span class="o">**</span><span class="mi">3</span>
-<span class="go">1922.50563031149</span>
-</pre></div>
-</div>
-<p>However, for noninteger <span class="math">\(n\)</span>, the differentiation formula for the
-exponential function must be modified to give the same result as the
-Riemann-Liouville differintegral:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">3.5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="mi">1</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">j</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">differint</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="n">c</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-<span class="go">(-123295.005390743 + 140955.117867654j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">)</span> <span class="o">*</span> <span class="n">exp</span><span class="p">(</span><span class="n">c</span><span class="p">)</span><span class="o">**</span><span class="n">x</span> <span class="o">*</span> <span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">c</span><span class="p">)</span><span class="o">**</span><span class="n">n</span> <span class="o">*</span> <span class="n">gammainc</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="o">*</span><span class="n">c</span><span class="p">)</span> <span class="o">/</span> <span class="n">gamma</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">)</span>
-<span class="go">(-123295.005390743 + 140955.117867654j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
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-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Differentiation</a><ul>
-<li><a class="reference internal" href="#numerical-derivatives-diff-diffs">Numerical derivatives (<tt class="docutils literal"><span class="pre">diff</span></tt>, <tt class="docutils literal"><span class="pre">diffs</span></tt>)</a></li>
-<li><a class="reference internal" href="#composition-of-derivatives-diffs-prod-diffs-exp">Composition of derivatives (<tt class="docutils literal"><span class="pre">diffs_prod</span></tt>, <tt class="docutils literal"><span class="pre">diffs_exp</span></tt>)</a></li>
-<li><a class="reference internal" href="#fractional-derivatives-differintegration-differint">Fractional derivatives / differintegration (<tt class="docutils literal"><span class="pre">differint</span></tt>)</a></li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../calculus/sums_limits.html"
-                        title="previous chapter">Sums, products, limits and extrapolation</a></p>
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-                        title="next chapter">Numerical integration (quadrature)</a></p>
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-  <div class="section" id="numerical-integration-quadrature">
-<h1>Numerical integration (quadrature)<a class="headerlink" href="#numerical-integration-quadrature" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="standard-quadrature-quad">
-<h2>Standard quadrature (<tt class="docutils literal"><span class="pre">quad</span></tt>)<a class="headerlink" href="#standard-quadrature-quad" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="mpmath.quad">
-<tt class="descclassname">mpmath.</tt><tt class="descname">quad</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>*points</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.quad" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes a single, double or triple integral over a given
-1D interval, 2D rectangle, or 3D cuboid. A basic example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="p">])</span>
-<span class="go">2.0</span>
-</pre></div>
-</div>
-<p>A basic 2D integral:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="p">])</span>
-<span class="go">4.0</span>
-</pre></div>
-</div>
-<p><strong>Interval format</strong></p>
-<p>The integration range for each dimension may be specified
-using a list or tuple. Arguments are interpreted as follows:</p>
-<p><tt class="docutils literal"><span class="pre">quad(f,</span> <span class="pre">[x1,</span> <span class="pre">x2])</span></tt> &#8211; calculates
-<span class="math">\(\int_{x_1}^{x_2} f(x) \, dx\)</span></p>
-<p><tt class="docutils literal"><span class="pre">quad(f,</span> <span class="pre">[x1,</span> <span class="pre">x2],</span> <span class="pre">[y1,</span> <span class="pre">y2])</span></tt> &#8211; calculates
-<span class="math">\(\int_{x_1}^{x_2} \int_{y_1}^{y_2} f(x,y) \, dy \, dx\)</span></p>
-<p><tt class="docutils literal"><span class="pre">quad(f,</span> <span class="pre">[x1,</span> <span class="pre">x2],</span> <span class="pre">[y1,</span> <span class="pre">y2],</span> <span class="pre">[z1,</span> <span class="pre">z2])</span></tt> &#8211; calculates
-<span class="math">\(\int_{x_1}^{x_2} \int_{y_1}^{y_2} \int_{z_1}^{z_2} f(x,y,z)
-\, dz \, dy \, dx\)</span></p>
-<p>Endpoints may be finite or infinite. An interval descriptor
-may also contain more than two points. In this
-case, the integration is split into subintervals, between
-each pair of consecutive points. This is useful for
-dealing with mid-interval discontinuities, or integrating
-over large intervals where the function is irregular or
-oscillates.</p>
-<p><strong>Options</strong></p>
-<p><a class="reference internal" href="#mpmath.quad" title="mpmath.quad"><tt class="xref py py-func docutils literal"><span class="pre">quad()</span></tt></a> recognizes the following keyword arguments:</p>
-<dl class="docutils">
-<dt><em>method</em></dt>
-<dd>Chooses integration algorithm (described below).</dd>
-<dt><em>error</em></dt>
-<dd>If set to true, <a class="reference internal" href="#mpmath.quad" title="mpmath.quad"><tt class="xref py py-func docutils literal"><span class="pre">quad()</span></tt></a> returns <span class="math">\((v, e)\)</span> where <span class="math">\(v\)</span> is the
-integral and <span class="math">\(e\)</span> is the estimated error.</dd>
-<dt><em>maxdegree</em></dt>
-<dd>Maximum degree of the quadrature rule to try before
-quitting.</dd>
-<dt><em>verbose</em></dt>
-<dd>Print details about progress.</dd>
-</dl>
-<p><strong>Algorithms</strong></p>
-<p>Mpmath presently implements two integration algorithms: tanh-sinh
-quadrature and Gauss-Legendre quadrature. These can be selected
-using <em>method=&#8217;tanh-sinh&#8217;</em> or <em>method=&#8217;gauss-legendre&#8217;</em> or by
-passing the classes <em>method=TanhSinh</em>, <em>method=GaussLegendre</em>.
-The functions <tt class="xref py py-func docutils literal"><span class="pre">quadts()</span></tt> and <tt class="xref py py-func docutils literal"><span class="pre">quadgl()</span></tt> are also available
-as shortcuts.</p>
-<p>Both algorithms have the property that doubling the number of
-evaluation points roughly doubles the accuracy, so both are ideal
-for high precision quadrature (hundreds or thousands of digits).</p>
-<p>At high precision, computing the nodes and weights for the
-integration can be expensive (more expensive than computing the
-function values). To make repeated integrations fast, nodes
-are automatically cached.</p>
-<p>The advantages of the tanh-sinh algorithm are that it tends to
-handle endpoint singularities well, and that the nodes are cheap
-to compute on the first run. For these reasons, it is used by
-<a class="reference internal" href="#mpmath.quad" title="mpmath.quad"><tt class="xref py py-func docutils literal"><span class="pre">quad()</span></tt></a> as the default algorithm.</p>
-<p>Gauss-Legendre quadrature often requires fewer function
-evaluations, and is therefore often faster for repeated use, but
-the algorithm does not handle endpoint singularities as well and
-the nodes are more expensive to compute. Gauss-Legendre quadrature
-can be a better choice if the integrand is smooth and repeated
-integrations are required (e.g. for multiple integrals).</p>
-<p>See the documentation for <tt class="xref py py-class docutils literal"><span class="pre">TanhSinh</span></tt> and
-<tt class="xref py py-class docutils literal"><span class="pre">GaussLegendre</span></tt> for additional details.</p>
-<p><strong>Examples of 1D integrals</strong></p>
-<p>Intervals may be infinite or half-infinite. The following two
-examples evaluate the limits of the inverse tangent function
-(<span class="math">\(\int 1/(1+x^2) = \tan^{-1} x\)</span>), and the Gaussian integral
-<span class="math">\(\int_{\infty}^{\infty} \exp(-x^2)\,dx = \sqrt{\pi}\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="mi">1</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">3.14159265358979</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">3.14159265358979</span>
-</pre></div>
-</div>
-<p>Integrals can typically be resolved to high precision.
-The following computes 50 digits of <span class="math">\(\pi\)</span> by integrating the
-area of the half-circle defined by <span class="math">\(x^2 + y^2 \le 1\)</span>,
-<span class="math">\(-1 \le x \le 1\)</span>, <span class="math">\(y \ge 0\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span><span class="o">*</span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">3.1415926535897932384626433832795028841971693993751</span>
-</pre></div>
-</div>
-<p>One can just as well compute 1000 digits (output truncated):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">1000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span><span class="o">*</span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>  
-<span class="go">3.141592653589793238462643383279502884...216420198</span>
-</pre></div>
-</div>
-<p>Complex integrals are supported. The following computes
-a residue at <span class="math">\(z = 0\)</span> by integrating counterclockwise along the
-diamond-shaped path from <span class="math">\(1\)</span> to <span class="math">\(+i\)</span> to <span class="math">\(-1\)</span> to <span class="math">\(-i\)</span> to <span class="math">\(1\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">j</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="n">j</span><span class="p">,</span><span class="mi">1</span><span class="p">]))</span>
-<span class="go">(0.0 + 6.28318530717959j)</span>
-</pre></div>
-</div>
-<p><strong>Examples of 2D and 3D integrals</strong></p>
-<p>Here are several nice examples of analytically solvable
-2D integrals (taken from MathWorld [1]) that can be evaluated
-to high precision fairly rapidly by <a class="reference internal" href="#mpmath.quad" title="mpmath.quad"><tt class="xref py py-func docutils literal"><span class="pre">quad()</span></tt></a>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">30</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">((</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">0.577215664901532860606512090082</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">+</span><span class="n">euler</span>
-<span class="go">0.577215664901532860606512090082</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">3.17343648530607134219175646705</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">4</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span>
-<span class="go">3.17343648530607134219175646705</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">1.23370055013616982735431137498</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span> <span class="mi">8</span>
-<span class="go">1.23370055013616982735431137498</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">1.64493406684822643647241516665</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span> <span class="mi">6</span>
-<span class="go">1.64493406684822643647241516665</span>
-</pre></div>
-</div>
-<p>Multiple integrals may be done over infinite ranges:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">-</span><span class="n">y</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">inf</span><span class="p">]))</span>
-<span class="go">0.367879441171442</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">e</span><span class="p">)</span>
-<span class="go">0.367879441171442</span>
-</pre></div>
-</div>
-<p>For nonrectangular areas, one can call <a class="reference internal" href="#mpmath.quad" title="mpmath.quad"><tt class="xref py py-func docutils literal"><span class="pre">quad()</span></tt></a> recursively.
-For example, we can replicate the earlier example of calculating
-<span class="math">\(\pi\)</span> by integrating over the unit-circle, and actually use double
-quadrature to actually measure the area circle:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">y</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">3.14159265358979</span>
-</pre></div>
-</div>
-<p>Here is a simple triple integral:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span><span class="p">:</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;gauss-legendre&#39;</span><span class="p">)</span>
-<span class="go">0.101366277027041</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">-</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">/</span><span class="mi">4</span>
-<span class="go">0.101366277027041</span>
-</pre></div>
-</div>
-<p><strong>Singularities</strong></p>
-<p>Both tanh-sinh and Gauss-Legendre quadrature are designed to
-integrate smooth (infinitely differentiable) functions. Neither
-algorithm copes well with mid-interval singularities (such as
-mid-interval discontinuities in <span class="math">\(f(x)\)</span> or <span class="math">\(f'(x)\)</span>).
-The best solution is to split the integral into parts:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="nb">abs</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">])</span>   <span class="c"># Bad</span>
-<span class="go">3.99900894176779</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="nb">abs</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">])</span>  <span class="c"># Good</span>
-<span class="go">4.0</span>
-</pre></div>
-</div>
-<p>The tanh-sinh rule often works well for integrands having a
-singularity at one or both endpoints:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">log</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;tanh-sinh&#39;</span><span class="p">)</span>  <span class="c"># Good</span>
-<span class="go">-1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">log</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;gauss-legendre&#39;</span><span class="p">)</span>  <span class="c"># Bad</span>
-<span class="go">-0.999932197413801</span>
-</pre></div>
-</div>
-<p>However, the result may still be inaccurate for some functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;tanh-sinh&#39;</span><span class="p">)</span>
-<span class="go">1.99999999946942</span>
-</pre></div>
-</div>
-<p>This problem is not due to the quadrature rule per se, but to
-numerical amplification of errors in the nodes. The problem can be
-circumvented by temporarily increasing the precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">30</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;tanh-sinh&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">+</span><span class="n">a</span>
-<span class="go">2.0</span>
-</pre></div>
-</div>
-<p><strong>Highly variable functions</strong></p>
-<p>For functions that are smooth (in the sense of being infinitely
-differentiable) but contain sharp mid-interval peaks or many
-&#8220;bumps&#8221;, <a class="reference internal" href="#mpmath.quad" title="mpmath.quad"><tt class="xref py py-func docutils literal"><span class="pre">quad()</span></tt></a> may fail to provide full accuracy. For
-example, with default settings, <a class="reference internal" href="#mpmath.quad" title="mpmath.quad"><tt class="xref py py-func docutils literal"><span class="pre">quad()</span></tt></a> is able to integrate
-<span class="math">\(\sin(x)\)</span> accurately over an interval of length 100 but not over
-length 1000:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">100</span><span class="p">]);</span> <span class="mi">1</span><span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>   <span class="c"># Good</span>
-<span class="go">0.137681127712316</span>
-<span class="go">0.137681127712316</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1000</span><span class="p">]);</span> <span class="mi">1</span><span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span>   <span class="c"># Bad</span>
-<span class="go">-37.8587612408485</span>
-<span class="go">0.437620923709297</span>
-</pre></div>
-</div>
-<p>One solution is to break the integration into 10 intervals of
-length 100:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1000</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>   <span class="c"># Good</span>
-<span class="go">0.437620923709297</span>
-</pre></div>
-</div>
-<p>Another is to increase the degree of the quadrature:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1000</span><span class="p">],</span> <span class="n">maxdegree</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>   <span class="c"># Also good</span>
-<span class="go">0.437620923709297</span>
-</pre></div>
-</div>
-<p>Whether splitting the interval or increasing the degree is
-more efficient differs from case to case. Another example is the
-function <span class="math">\(1/(1+x^2)\)</span>, which has a sharp peak centered around
-<span class="math">\(x = 0\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">100</span><span class="p">,</span> <span class="mi">100</span><span class="p">])</span>   <span class="c"># Bad</span>
-<span class="go">3.64804647105268</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">100</span><span class="p">,</span> <span class="mi">100</span><span class="p">],</span> <span class="n">maxdegree</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>   <span class="c"># Good</span>
-<span class="go">3.12159332021646</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">100</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">100</span><span class="p">])</span>   <span class="c"># Also good</span>
-<span class="go">3.12159332021646</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference external" href="http://mathworld.wolfram.com/DoubleIntegral.html">http://mathworld.wolfram.com/DoubleIntegral.html</a></li>
-</ol>
-</dd></dl>
-
-</div>
-<div class="section" id="oscillatory-quadrature-quadosc">
-<h2>Oscillatory quadrature (<tt class="docutils literal"><span class="pre">quadosc</span></tt>)<a class="headerlink" href="#oscillatory-quadrature-quadosc" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="mpmath.quadosc">
-<tt class="descclassname">mpmath.</tt><tt class="descname">quadosc</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>interval</em>, <em>omega=None</em>, <em>period=None</em>, <em>zeros=None</em><big>)</big><a class="headerlink" href="#mpmath.quadosc" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculates</p>
-<div class="math">
-\[I = \int_a^b f(x) dx\]</div>
-<p>where at least one of <span class="math">\(a\)</span> and <span class="math">\(b\)</span> is infinite and where
-<span class="math">\(f(x) = g(x) \cos(\omega x  + \phi)\)</span> for some slowly
-decreasing function <span class="math">\(g(x)\)</span>. With proper input, <a class="reference internal" href="#mpmath.quadosc" title="mpmath.quadosc"><tt class="xref py py-func docutils literal"><span class="pre">quadosc()</span></tt></a>
-can also handle oscillatory integrals where the oscillation
-rate is different from a pure sine or cosine wave.</p>
-<p>In the standard case when <span class="math">\(|a| &lt; \infty, b = \infty\)</span>,
-<a class="reference internal" href="#mpmath.quadosc" title="mpmath.quadosc"><tt class="xref py py-func docutils literal"><span class="pre">quadosc()</span></tt></a> works by evaluating the infinite series</p>
-<div class="math">
-\[I = \int_a^{x_1} f(x) dx +
-\sum_{k=1}^{\infty} \int_{x_k}^{x_{k+1}} f(x) dx\]</div>
-<p>where <span class="math">\(x_k\)</span> are consecutive zeros (alternatively
-some other periodic reference point) of <span class="math">\(f(x)\)</span>.
-Accordingly, <a class="reference internal" href="#mpmath.quadosc" title="mpmath.quadosc"><tt class="xref py py-func docutils literal"><span class="pre">quadosc()</span></tt></a> requires information about the
-zeros of <span class="math">\(f(x)\)</span>. For a periodic function, you can specify
-the zeros by either providing the angular frequency <span class="math">\(\omega\)</span>
-(<em>omega</em>) or the <em>period</em> <span class="math">\(2 \pi/\omega\)</span>. In general, you can
-specify the <span class="math">\(n\)</span>-th zero by providing the <em>zeros</em> arguments.
-Below is an example of each:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">sin</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quadosc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="n">omega</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.37833007080198</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quadosc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="n">period</span><span class="o">=</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.37833007080198</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quadosc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="n">zeros</span><span class="o">=</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="n">pi</span><span class="o">*</span><span class="n">n</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.37833007080198</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">ei</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">)</span><span class="o">-</span><span class="n">exp</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">*</span><span class="n">ei</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span>  <span class="c"># Computed by Mathematica</span>
-<span class="go">0.37833007080198</span>
-</pre></div>
-</div>
-<p>Note that <em>zeros</em> was specified to multiply <span class="math">\(n\)</span> by the
-<em>half-period</em>, not the full period. In theory, it does not matter
-whether each partial integral is done over a half period or a full
-period. However, if done over half-periods, the infinite series
-passed to <a class="reference internal" href="../calculus/sums_limits.html#mpmath.nsum" title="mpmath.nsum"><tt class="xref py py-func docutils literal"><span class="pre">nsum()</span></tt></a> becomes an <em>alternating series</em> and this
-typically makes the extrapolation much more efficient.</p>
-<p>Here is an example of an integration over the entire real line,
-and a half-infinite integration starting at <span class="math">\(-\infty\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">quadosc</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span> <span class="n">omega</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1.15572734979092</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">/</span><span class="n">e</span>
-<span class="go">1.15572734979092</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quadosc</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">period</span><span class="o">=</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">-0.0844109505595739</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">+</span><span class="n">si</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">-</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span>
-<span class="go">-0.0844109505595738</span>
-</pre></div>
-</div>
-<p>Of course, the integrand may contain a complex exponential just as
-well as a real sine or cosine:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">quadosc</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">j</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="n">omega</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">(0.156410688228254 + 0.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">/</span><span class="n">e</span><span class="o">**</span><span class="mi">3</span>
-<span class="go">0.156410688228254</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quadosc</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">j</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="n">x</span><span class="o">+</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="n">omega</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">(0.00317486988463794 - 0.0447701735209082j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">7</span><span class="p">)</span><span class="o">/</span><span class="n">exp</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="p">(</span><span class="n">j</span><span class="o">+</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">7</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(0.00317486988463794 - 0.0447701735209082j)</span>
-</pre></div>
-</div>
-<p><strong>Non-periodic functions</strong></p>
-<p>If <span class="math">\(f(x) = g(x) h(x)\)</span> for some function <span class="math">\(h(x)\)</span> that is not
-strictly periodic, <em>omega</em> or <em>period</em> might not work, and it might
-be necessary to use <em>zeros</em>.</p>
-<p>A notable exception can be made for Bessel functions which, though not
-periodic, are &#8220;asymptotically periodic&#8221; in a sufficiently strong sense
-that the sum extrapolation will work out:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">quadosc</span><span class="p">(</span><span class="n">j0</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span> <span class="n">period</span><span class="o">=</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quadosc</span><span class="p">(</span><span class="n">j1</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span> <span class="n">period</span><span class="o">=</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">1.0</span>
-</pre></div>
-</div>
-<p>More properly, one should provide the exact Bessel function zeros:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">j0zero</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="n">findroot</span><span class="p">(</span><span class="n">j0</span><span class="p">,</span> <span class="n">pi</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mf">0.25</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quadosc</span><span class="p">(</span><span class="n">j0</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span> <span class="n">zeros</span><span class="o">=</span><span class="n">j0zero</span><span class="p">)</span>
-<span class="go">1.0</span>
-</pre></div>
-</div>
-<p>For an example where <em>zeros</em> becomes necessary, consider the
-complete Fresnel integrals</p>
-<div class="math">
-\[\int_0^{\infty} \cos x^2\,dx = \int_0^{\infty} \sin x^2\,dx
-= \sqrt{\frac{\pi}{8}}.\]</div>
-<p>Although the integrands do not decrease in magnitude as
-<span class="math">\(x \to \infty\)</span>, the integrals are convergent since the oscillation
-rate increases (causing consecutive periods to asymptotically
-cancel out). These integrals are virtually impossible to calculate
-to any kind of accuracy using standard quadrature rules. However,
-if one provides the correct asymptotic distribution of zeros
-(<span class="math">\(x_n \sim \sqrt{n}\)</span>), <a class="reference internal" href="#mpmath.quadosc" title="mpmath.quadosc"><tt class="xref py py-func docutils literal"><span class="pre">quadosc()</span></tt></a> works:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">30</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quadosc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="n">zeros</span><span class="o">=</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">n</span><span class="p">))</span>
-<span class="go">0.626657068657750125603941321203</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quadosc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="n">zeros</span><span class="o">=</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">n</span><span class="p">))</span>
-<span class="go">0.626657068657750125603941321203</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">8</span><span class="p">)</span>
-<span class="go">0.626657068657750125603941321203</span>
-</pre></div>
-</div>
-<p>(Interestingly, these integrals can still be evaluated if one
-places some other constant than <span class="math">\(\pi\)</span> in the square root sign.)</p>
-<p>In general, if <span class="math">\(f(x) \sim g(x) \cos(h(x))\)</span>, the zeros follow
-the inverse-function distribution <span class="math">\(h^{-1}(x)\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">sin</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quadosc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="n">zeros</span><span class="o">=</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="n">log</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-<span class="go">-0.25024394235267</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="o">-</span><span class="n">si</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-<span class="go">-0.250243942352671</span>
-</pre></div>
-</div>
-<p><strong>Non-alternating functions</strong></p>
-<p>If the integrand oscillates around a positive value, without
-alternating signs, the extrapolation might fail. A simple trick
-that sometimes works is to multiply or divide the frequency by 2:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quadosc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="n">omega</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>  <span class="c"># Bad</span>
-<span class="go">1.28642190869861</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quadosc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="n">omega</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>  <span class="c"># Perfect</span>
-<span class="go">1.28652953559617</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">+</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">+</span><span class="n">ci</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">+</span><span class="n">sin</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span><span class="o">/</span><span class="mi">6</span>
-<span class="go">1.28652953559617</span>
-</pre></div>
-</div>
-<p><strong>Fast decay</strong></p>
-<p><a class="reference internal" href="#mpmath.quadosc" title="mpmath.quadosc"><tt class="xref py py-func docutils literal"><span class="pre">quadosc()</span></tt></a> is primarily useful for slowly decaying
-integrands. If the integrand decreases exponentially or faster,
-<a class="reference internal" href="#mpmath.quad" title="mpmath.quad"><tt class="xref py py-func docutils literal"><span class="pre">quad()</span></tt></a> will likely handle it without trouble (and generally be
-much faster than <a class="reference internal" href="#mpmath.quadosc" title="mpmath.quadosc"><tt class="xref py py-func docutils literal"><span class="pre">quadosc()</span></tt></a>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">quadosc</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span> <span class="n">omega</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">0.5</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="quadrature-rules">
-<h2>Quadrature rules<a class="headerlink" href="#quadrature-rules" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="mpmath.calculus.quadrature.QuadratureRule">
-<em class="property">class </em><tt class="descclassname">mpmath.calculus.quadrature.</tt><tt class="descname">QuadratureRule</tt><big>(</big><em>ctx</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/quadrature.html#QuadratureRule"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.quadrature.QuadratureRule" title="Permalink to this definition">¶</a></dt>
-<dd><p>Quadrature rules are implemented using this class, in order to
-simplify the code and provide a common infrastructure
-for tasks such as error estimation and node caching.</p>
-<p>You can implement a custom quadrature rule by subclassing
-<tt class="xref py py-class docutils literal"><span class="pre">QuadratureRule</span></tt> and implementing the appropriate
-methods. The subclass can then be used by <a class="reference internal" href="#mpmath.quad" title="mpmath.quad"><tt class="xref py py-func docutils literal"><span class="pre">quad()</span></tt></a> by
-passing it as the <em>method</em> argument.</p>
-<p><tt class="xref py py-class docutils literal"><span class="pre">QuadratureRule</span></tt> instances are supposed to be singletons.
-<tt class="xref py py-class docutils literal"><span class="pre">QuadratureRule</span></tt> therefore implements instance caching
-in <tt class="xref py py-func docutils literal"><span class="pre">__new__()</span></tt>.</p>
-<dl class="function">
-<dt id="mpmath.calculus.quadrature.QuadratureRule.calc_nodes">
-<tt class="descname">calc_nodes</tt><big>(</big><em>self</em>, <em>degree</em>, <em>prec</em>, <em>verbose=False</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/quadrature.html#QuadratureRule.calc_nodes"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.quadrature.QuadratureRule.calc_nodes" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute nodes for the standard interval <span class="math">\([-1, 1]\)</span>. Subclasses
-should probably implement only this method, and use
-<tt class="xref py py-func docutils literal"><span class="pre">get_nodes()</span></tt> method to retrieve the nodes.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="mpmath.calculus.quadrature.QuadratureRule.clear">
-<tt class="descname">clear</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/quadrature.html#QuadratureRule.clear"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.quadrature.QuadratureRule.clear" title="Permalink to this definition">¶</a></dt>
-<dd><p>Delete cached node data.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="mpmath.calculus.quadrature.QuadratureRule.estimate_error">
-<tt class="descname">estimate_error</tt><big>(</big><em>self</em>, <em>results</em>, <em>prec</em>, <em>epsilon</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/quadrature.html#QuadratureRule.estimate_error"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.quadrature.QuadratureRule.estimate_error" title="Permalink to this definition">¶</a></dt>
-<dd><p>Given results from integrations <span class="math">\([I_1, I_2, \ldots, I_k]\)</span> done
-with a quadrature of rule of degree <span class="math">\(1, 2, \ldots, k\)</span>, estimate
-the error of <span class="math">\(I_k\)</span>.</p>
-<p>For <span class="math">\(k = 2\)</span>, we estimate  <span class="math">\(|I_{\infty}-I_2|\)</span> as <span class="math">\(|I_2-I_1|\)</span>.</p>
-<p>For <span class="math">\(k &gt; 2\)</span>, we extrapolate <span class="math">\(|I_{\infty}-I_k| \approx |I_{k+1}-I_k|\)</span>
-from <span class="math">\(|I_k-I_{k-1}|\)</span> and <span class="math">\(|I_k-I_{k-2}|\)</span> under the assumption
-that each degree increment roughly doubles the accuracy of
-the quadrature rule (this is true for both <tt class="xref py py-class docutils literal"><span class="pre">TanhSinh</span></tt>
-and <tt class="xref py py-class docutils literal"><span class="pre">GaussLegendre</span></tt>). The extrapolation formula is given
-by Borwein, Bailey &amp; Girgensohn. Although not very conservative,
-this method seems to be very robust in practice.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="mpmath.calculus.quadrature.QuadratureRule.get_nodes">
-<tt class="descname">get_nodes</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em>, <em>degree</em>, <em>prec</em>, <em>verbose=False</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/quadrature.html#QuadratureRule.get_nodes"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.quadrature.QuadratureRule.get_nodes" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return nodes for given interval, degree and precision. The
-nodes are retrieved from a cache if already computed;
-otherwise they are computed by calling <tt class="xref py py-func docutils literal"><span class="pre">calc_nodes()</span></tt>
-and are then cached.</p>
-<p>Subclasses should probably not implement this method,
-but just implement <tt class="xref py py-func docutils literal"><span class="pre">calc_nodes()</span></tt> for the actual
-node computation.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="mpmath.calculus.quadrature.QuadratureRule.guess_degree">
-<tt class="descname">guess_degree</tt><big>(</big><em>self</em>, <em>prec</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/quadrature.html#QuadratureRule.guess_degree"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.quadrature.QuadratureRule.guess_degree" title="Permalink to this definition">¶</a></dt>
-<dd><p>Given a desired precision <span class="math">\(p\)</span> in bits, estimate the degree <span class="math">\(m\)</span>
-of the quadrature required to accomplish full accuracy for
-typical integrals. By default, <a class="reference internal" href="#mpmath.quad" title="mpmath.quad"><tt class="xref py py-func docutils literal"><span class="pre">quad()</span></tt></a> will perform up
-to <span class="math">\(m\)</span> iterations. The value of <span class="math">\(m\)</span> should be a slight
-overestimate, so that &#8220;slightly bad&#8221; integrals can be dealt
-with automatically using a few extra iterations. On the
-other hand, it should not be too big, so <a class="reference internal" href="#mpmath.quad" title="mpmath.quad"><tt class="xref py py-func docutils literal"><span class="pre">quad()</span></tt></a> can
-quit within a reasonable amount of time when it is given
-an &#8220;unsolvable&#8221; integral.</p>
-<p>The default formula used by <tt class="xref py py-func docutils literal"><span class="pre">guess_degree()</span></tt> is tuned
-for both <tt class="xref py py-class docutils literal"><span class="pre">TanhSinh</span></tt> and <tt class="xref py py-class docutils literal"><span class="pre">GaussLegendre</span></tt>.
-The output is roughly as follows:</p>
-<blockquote>
-<div><table border="1" class="docutils">
-<colgroup>
-<col width="50%" />
-<col width="50%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head"><span class="math">\(p\)</span></th>
-<th class="head"><span class="math">\(m\)</span></th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-even"><td>50</td>
-<td>6</td>
-</tr>
-<tr class="row-odd"><td>100</td>
-<td>7</td>
-</tr>
-<tr class="row-even"><td>500</td>
-<td>10</td>
-</tr>
-<tr class="row-odd"><td>3000</td>
-<td>12</td>
-</tr>
-</tbody>
-</table>
-</div></blockquote>
-<p>This formula is based purely on a limited amount of
-experimentation and will sometimes be wrong.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="mpmath.calculus.quadrature.QuadratureRule.sum_next">
-<tt class="descname">sum_next</tt><big>(</big><em>self</em>, <em>f</em>, <em>nodes</em>, <em>degree</em>, <em>prec</em>, <em>previous</em>, <em>verbose=False</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/quadrature.html#QuadratureRule.sum_next"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.quadrature.QuadratureRule.sum_next" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the step sum <span class="math">\(\sum w_k f(x_k)\)</span> where the <em>nodes</em> list
-contains the <span class="math">\((w_k, x_k)\)</span> pairs.</p>
-<p><tt class="xref py py-func docutils literal"><span class="pre">summation()</span></tt> will supply the list <em>results</em> of
-values computed by <tt class="xref py py-func docutils literal"><span class="pre">sum_next()</span></tt> at previous degrees, in
-case the quadrature rule is able to reuse them.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="mpmath.calculus.quadrature.QuadratureRule.summation">
-<tt class="descname">summation</tt><big>(</big><em>self</em>, <em>f</em>, <em>points</em>, <em>prec</em>, <em>epsilon</em>, <em>max_degree</em>, <em>verbose=False</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/quadrature.html#QuadratureRule.summation"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.quadrature.QuadratureRule.summation" title="Permalink to this definition">¶</a></dt>
-<dd><p>Main integration function. Computes the 1D integral over
-the interval specified by <em>points</em>. For each subinterval,
-performs quadrature of degree from 1 up to <em>max_degree</em>
-until <tt class="xref py py-func docutils literal"><span class="pre">estimate_error()</span></tt> signals convergence.</p>
-<p><tt class="xref py py-func docutils literal"><span class="pre">summation()</span></tt> transforms each subintegration to
-the standard interval and then calls <tt class="xref py py-func docutils literal"><span class="pre">sum_next()</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="mpmath.calculus.quadrature.QuadratureRule.transform_nodes">
-<tt class="descname">transform_nodes</tt><big>(</big><em>self</em>, <em>nodes</em>, <em>a</em>, <em>b</em>, <em>verbose=False</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/quadrature.html#QuadratureRule.transform_nodes"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.quadrature.QuadratureRule.transform_nodes" title="Permalink to this definition">¶</a></dt>
-<dd><p>Rescale standardized nodes (for <span class="math">\([-1, 1]\)</span>) to a general
-interval <span class="math">\([a, b]\)</span>. For a finite interval, a simple linear
-change of variables is used. Otherwise, the following
-transformations are used:</p>
-<div class="math">
-\[[a, \infty] : t = \frac{1}{x} + (a-1)\]\[[-\infty, b] : t = (b+1) - \frac{1}{x}\]\[[-\infty, \infty] : t = \frac{x}{\sqrt{1-x^2}}\]</div>
-</dd></dl>
-
-</dd></dl>
-
-<div class="section" id="tanh-sinh-rule">
-<h3>Tanh-sinh rule<a class="headerlink" href="#tanh-sinh-rule" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="mpmath.calculus.quadrature.TanhSinh">
-<em class="property">class </em><tt class="descclassname">mpmath.calculus.quadrature.</tt><tt class="descname">TanhSinh</tt><big>(</big><em>ctx</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/quadrature.html#TanhSinh"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.quadrature.TanhSinh" title="Permalink to this definition">¶</a></dt>
-<dd><p>This class implements &#8220;tanh-sinh&#8221; or &#8220;doubly exponential&#8221;
-quadrature. This quadrature rule is based on the Euler-Maclaurin
-integral formula. By performing a change of variables involving
-nested exponentials / hyperbolic functions (hence the name), the
-derivatives at the endpoints vanish rapidly. Since the error term
-in the Euler-Maclaurin formula depends on the derivatives at the
-endpoints, a simple step sum becomes extremely accurate. In
-practice, this means that doubling the number of evaluation
-points roughly doubles the number of accurate digits.</p>
-<dl class="docutils">
-<dt>Comparison to Gauss-Legendre:</dt>
-<dd><ul class="first last simple">
-<li>Initial computation of nodes is usually faster</li>
-<li>Handles endpoint singularities better</li>
-<li>Handles infinite integration intervals better</li>
-<li>Is slower for smooth integrands once nodes have been computed</li>
-</ul>
-</dd>
-</dl>
-<p>The implementation of the tanh-sinh algorithm is based on the
-description given in Borwein, Bailey &amp; Girgensohn, &#8220;Experimentation
-in Mathematics - Computational Paths to Discovery&#8221;, A K Peters,
-2003, pages 312-313. In the present implementation, a few
-improvements have been made:</p>
-<blockquote>
-<div><ul class="simple">
-<li>A more efficient scheme is used to compute nodes (exploiting
-recurrence for the exponential function)</li>
-<li>The nodes are computed successively instead of all at once</li>
-</ul>
-</div></blockquote>
-<p>Various documents describing the algorithm are available online, e.g.:</p>
-<blockquote>
-<div><ul class="simple">
-<li><a class="reference external" href="http://crd.lbl.gov/~dhbailey/dhbpapers/dhb-tanh-sinh.pdf">http://crd.lbl.gov/~dhbailey/dhbpapers/dhb-tanh-sinh.pdf</a></li>
-<li><a class="reference external" href="http://users.cs.dal.ca/~jborwein/tanh-sinh.pdf">http://users.cs.dal.ca/~jborwein/tanh-sinh.pdf</a></li>
-</ul>
-</div></blockquote>
-<dl class="function">
-<dt id="mpmath.calculus.quadrature.TanhSinh.calc_nodes">
-<tt class="descname">calc_nodes</tt><big>(</big><em>self</em>, <em>degree</em>, <em>prec</em>, <em>verbose=False</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/quadrature.html#TanhSinh.calc_nodes"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.quadrature.TanhSinh.calc_nodes" title="Permalink to this definition">¶</a></dt>
-<dd><p>The abscissas and weights for tanh-sinh quadrature of degree
-<span class="math">\(m\)</span> are given by</p>
-<div class="math">
-\[x_k = \tanh(\pi/2 \sinh(t_k))\]\[w_k = \pi/2 \cosh(t_k) / \cosh(\pi/2 \sinh(t_k))^2\]</div>
-<p>where <span class="math">\(t_k = t_0 + hk\)</span> for a step length <span class="math">\(h \sim 2^{-m}\)</span>. The
-list of nodes is actually infinite, but the weights die off so
-rapidly that only a few are needed.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="mpmath.calculus.quadrature.TanhSinh.sum_next">
-<tt class="descname">sum_next</tt><big>(</big><em>self</em>, <em>f</em>, <em>nodes</em>, <em>degree</em>, <em>prec</em>, <em>previous</em>, <em>verbose=False</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/quadrature.html#TanhSinh.sum_next"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.quadrature.TanhSinh.sum_next" title="Permalink to this definition">¶</a></dt>
-<dd><p>Step sum for tanh-sinh quadrature of degree <span class="math">\(m\)</span>. We exploit the
-fact that half of the abscissas at degree <span class="math">\(m\)</span> are precisely the
-abscissas from degree <span class="math">\(m-1\)</span>. Thus reusing the result from
-the previous level allows a 2x speedup.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="gauss-legendre-rule">
-<h3>Gauss-Legendre rule<a class="headerlink" href="#gauss-legendre-rule" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="mpmath.calculus.quadrature.GaussLegendre">
-<em class="property">class </em><tt class="descclassname">mpmath.calculus.quadrature.</tt><tt class="descname">GaussLegendre</tt><big>(</big><em>ctx</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/quadrature.html#GaussLegendre"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.quadrature.GaussLegendre" title="Permalink to this definition">¶</a></dt>
-<dd><p>This class implements Gauss-Legendre quadrature, which is
-exceptionally efficient for polynomials and polynomial-like (i.e.
-very smooth) integrands.</p>
-<p>The abscissas and weights are given by roots and values of
-Legendre polynomials, which are the orthogonal polynomials
-on <span class="math">\([-1, 1]\)</span> with respect to the unit weight
-(see <a class="reference internal" href="../../mpmath/functions/orthogonal.html#mpmath.legendre" title="mpmath.legendre"><tt class="xref py py-func docutils literal"><span class="pre">legendre()</span></tt></a>).</p>
-<p>In this implementation, we take the &#8220;degree&#8221; <span class="math">\(m\)</span> of the quadrature
-to denote a Gauss-Legendre rule of degree <span class="math">\(3 \cdot 2^m\)</span> (following
-Borwein, Bailey &amp; Girgensohn). This way we get quadratic, rather
-than linear, convergence as the degree is incremented.</p>
-<dl class="docutils">
-<dt>Comparison to tanh-sinh quadrature:</dt>
-<dd><ul class="first last simple">
-<li>Is faster for smooth integrands once nodes have been computed</li>
-<li>Initial computation of nodes is usually slower</li>
-<li>Handles endpoint singularities worse</li>
-<li>Handles infinite integration intervals worse</li>
-</ul>
-</dd>
-</dl>
-<dl class="function">
-<dt id="mpmath.calculus.quadrature.GaussLegendre.calc_nodes">
-<tt class="descname">calc_nodes</tt><big>(</big><em>self</em>, <em>degree</em>, <em>prec</em>, <em>verbose=False</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/quadrature.html#GaussLegendre.calc_nodes"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.quadrature.GaussLegendre.calc_nodes" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculates the abscissas and weights for Gauss-Legendre
-quadrature of degree of given degree (actually <span class="math">\(3 \cdot 2^m\)</span>).</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Numerical integration (quadrature)</a><ul>
-<li><a class="reference internal" href="#standard-quadrature-quad">Standard quadrature (<tt class="docutils literal"><span class="pre">quad</span></tt>)</a></li>
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-  <div class="section" id="ordinary-differential-equations">
-<h1>Ordinary differential equations<a class="headerlink" href="#ordinary-differential-equations" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="solving-the-ode-initial-value-problem-odefun">
-<h2>Solving the ODE initial value problem (<tt class="docutils literal"><span class="pre">odefun</span></tt>)<a class="headerlink" href="#solving-the-ode-initial-value-problem-odefun" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="mpmath.odefun">
-<tt class="descclassname">mpmath.</tt><tt class="descname">odefun</tt><big>(</big><em>ctx</em>, <em>F</em>, <em>x0</em>, <em>y0</em>, <em>tol=None</em>, <em>degree=None</em>, <em>method='taylor'</em>, <em>verbose=False</em><big>)</big><a class="headerlink" href="#mpmath.odefun" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a function <span class="math">\(y(x) = [y_0(x), y_1(x), \ldots, y_n(x)]\)</span>
-that is a numerical solution of the <span class="math">\(n+1\)</span>-dimensional first-order
-ordinary differential equation (ODE) system</p>
-<div class="math">
-\[y_0'(x) = F_0(x, [y_0(x), y_1(x), \ldots, y_n(x)])\]\[y_1'(x) = F_1(x, [y_0(x), y_1(x), \ldots, y_n(x)])\]\[\vdots\]\[y_n'(x) = F_n(x, [y_0(x), y_1(x), \ldots, y_n(x)])\]</div>
-<p>The derivatives are specified by the vector-valued function
-<em>F</em> that evaluates
-<span class="math">\([y_0', \ldots, y_n'] = F(x, [y_0, \ldots, y_n])\)</span>.
-The initial point <span class="math">\(x_0\)</span> is specified by the scalar argument <em>x0</em>,
-and the initial value <span class="math">\(y(x_0) =  [y_0(x_0), \ldots, y_n(x_0)]\)</span> is
-specified by the vector argument <em>y0</em>.</p>
-<p>For convenience, if the system is one-dimensional, you may optionally
-provide just a scalar value for <em>y0</em>. In this case, <em>F</em> should accept
-a scalar <em>y</em> argument and return a scalar. The solution function
-<em>y</em> will return scalar values instead of length-1 vectors.</p>
-<p>Evaluation of the solution function <span class="math">\(y(x)\)</span> is permitted
-for any <span class="math">\(x \ge x_0\)</span>.</p>
-<p>A high-order ODE can be solved by transforming it into first-order
-vector form. This transformation is described in standard texts
-on ODEs. Examples will also be given below.</p>
-<p><strong>Options, speed and accuracy</strong></p>
-<p>By default, <a class="reference internal" href="#mpmath.odefun" title="mpmath.odefun"><tt class="xref py py-func docutils literal"><span class="pre">odefun()</span></tt></a> uses a high-order Taylor series
-method. For reasonably well-behaved problems, the solution will
-be fully accurate to within the working precision. Note that
-<em>F</em> must be possible to evaluate to very high precision
-for the generation of Taylor series to work.</p>
-<p>To get a faster but less accurate solution, you can set a large
-value for <em>tol</em> (which defaults roughly to <em>eps</em>). If you just
-want to plot the solution or perform a basic simulation,
-<em>tol = 0.01</em> is likely sufficient.</p>
-<p>The <em>degree</em> argument controls the degree of the solver (with
-<em>method=&#8217;taylor&#8217;</em>, this is the degree of the Taylor series
-expansion). A higher degree means that a longer step can be taken
-before a new local solution must be generated from <em>F</em>,
-meaning that fewer steps are required to get from <span class="math">\(x_0\)</span> to a given
-<span class="math">\(x_1\)</span>. On the other hand, a higher degree also means that each
-local solution becomes more expensive (i.e., more evaluations of
-<em>F</em> are required per step, and at higher precision).</p>
-<p>The optimal setting therefore involves a tradeoff. Generally,
-decreasing the <em>degree</em> for Taylor series is likely to give faster
-solution at low precision, while increasing is likely to be better
-at higher precision.</p>
-<p>The function
-object returned by <a class="reference internal" href="#mpmath.odefun" title="mpmath.odefun"><tt class="xref py py-func docutils literal"><span class="pre">odefun()</span></tt></a> caches the solutions at all step
-points and uses polynomial interpolation between step points.
-Therefore, once <span class="math">\(y(x_1)\)</span> has been evaluated for some <span class="math">\(x_1\)</span>,
-<span class="math">\(y(x)\)</span> can be evaluated very quickly for any <span class="math">\(x_0 \le x \le x_1\)</span>.
-and continuing the evaluation up to <span class="math">\(x_2 &gt; x_1\)</span> is also fast.</p>
-<p><strong>Examples of first-order ODEs</strong></p>
-<p>We will solve the standard test problem <span class="math">\(y'(x) = y(x), y(0) = 1\)</span>
-which has explicit solution <span class="math">\(y(x) = \exp(x)\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">odefun</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">y</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">((</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
-<span class="gp">...</span>
-<span class="go">(1.0, 1.0)</span>
-<span class="go">(2.71828182845905, 2.71828182845905)</span>
-<span class="go">(12.1824939607035, 12.1824939607035)</span>
-</pre></div>
-</div>
-<p>The solution with high precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">odefun</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">y</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">2.7182818284590452353602874713526624977572470937</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">2.7182818284590452353602874713526624977572470937</span>
-</pre></div>
-</div>
-<p>Using the more general vectorized form, the test problem
-can be input as (note that <em>f</em> returns a 1-element vector):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">odefun</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="p">[</span><span class="n">y</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span> <span class="mi">0</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">[2.71828182845905]</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.odefun" title="mpmath.odefun"><tt class="xref py py-func docutils literal"><span class="pre">odefun()</span></tt></a> can solve nonlinear ODEs, which are generally
-impossible (and at best difficult) to solve analytically. As
-an example of a nonlinear ODE, we will solve <span class="math">\(y'(x) = x \sin(y(x))\)</span>
-for <span class="math">\(y(0) = \pi/2\)</span>. An exact solution happens to be known
-for this problem, and is given by
-<span class="math">\(y(x) = 2 \tan^{-1}\left(\exp\left(x^2/2\right)\right)\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">odefun</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">x</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">y</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">10</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">((</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="mi">2</span><span class="o">*</span><span class="n">atan</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">2</span><span class="p">))))</span>
-<span class="gp">...</span>
-<span class="go">(2.87255666284091, 2.87255666284091)</span>
-<span class="go">(3.14158520028345, 3.14158520028345)</span>
-<span class="go">(3.14159265358979, 3.14159265358979)</span>
-</pre></div>
-</div>
-<p>If <span class="math">\(F\)</span> is independent of <span class="math">\(y\)</span>, an ODE can be solved using direct
-integration. We can therefore obtain a reference solution with
-<a class="reference internal" href="../calculus/integration.html#mpmath.quad" title="mpmath.quad"><tt class="xref py py-func docutils literal"><span class="pre">quad()</span></tt></a>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">odefun</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">pi</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">0.72128263801696</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="n">pi</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">])</span>
-<span class="go">0.72128263801696</span>
-</pre></div>
-</div>
-<p><strong>Examples of second-order ODEs</strong></p>
-<p>We will solve the harmonic oscillator equation <span class="math">\(y''(x) + y(x) = 0\)</span>.
-To do this, we introduce the helper functions <span class="math">\(y_0 = y, y_1 = y_0'\)</span>
-whereby the original equation can be written as <span class="math">\(y_1' + y_0' = 0\)</span>. Put
-together, we get the first-order, two-dimensional vector ODE</p>
-<div class="math">
-\[\begin{split}\begin{cases}
-y_0' = y_1 \\
-y_1' = -y_0
-\end{cases}\end{split}\]</div>
-<p>To get a well-defined IVP, we need two initial values. With
-<span class="math">\(y(0) = y_0(0) = 1\)</span> and <span class="math">\(-y'(0) = y_1(0) = 0\)</span>, the problem will of
-course be solved by <span class="math">\(y(x) = y_0(x) = \cos(x)\)</span> and
-<span class="math">\(-y'(x) = y_1(x) = \sin(x)\)</span>. We check this:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">odefun</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="p">[</span><span class="o">-</span><span class="n">y</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">y</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span> <span class="mi">0</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mi">10</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="n">nprint</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="mi">15</span><span class="p">)</span>
-<span class="gp">... </span>    <span class="n">nprint</span><span class="p">([</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)],</span> <span class="mi">15</span><span class="p">)</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&quot;---&quot;</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="go">[1.0, 0.0]</span>
-<span class="go">[1.0, 0.0]</span>
-<span class="go">---</span>
-<span class="go">[0.54030230586814, 0.841470984807897]</span>
-<span class="go">[0.54030230586814, 0.841470984807897]</span>
-<span class="go">---</span>
-<span class="go">[-0.801143615546934, 0.598472144103957]</span>
-<span class="go">[-0.801143615546934, 0.598472144103957]</span>
-<span class="go">---</span>
-<span class="go">[-0.839071529076452, -0.54402111088937]</span>
-<span class="go">[-0.839071529076452, -0.54402111088937]</span>
-<span class="go">---</span>
-</pre></div>
-</div>
-<p>Note that we get both the sine and the cosine solutions
-simultaneously.</p>
-<p><strong>TODO</strong></p>
-<ul class="simple">
-<li>Better automatic choice of degree and step size</li>
-<li>Make determination of Taylor series convergence radius
-more robust</li>
-<li>Allow solution for <span class="math">\(x &lt; x_0\)</span></li>
-<li>Allow solution for complex <span class="math">\(x\)</span></li>
-<li>Test for difficult (ill-conditioned) problems</li>
-<li>Implement Runge-Kutta and other algorithms</li>
-</ul>
-</dd></dl>
-
-</div>
-</div>
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-  <ul>
-<li><a class="reference internal" href="#">Ordinary differential equations</a><ul>
-<li><a class="reference internal" href="#solving-the-ode-initial-value-problem-odefun">Solving the ODE initial value problem (<tt class="docutils literal"><span class="pre">odefun</span></tt>)</a></li>
-</ul>
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-  <div class="section" id="root-finding-and-optimization">
-<h1>Root-finding and optimization<a class="headerlink" href="#root-finding-and-optimization" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="root-finding-findroot">
-<h2>Root-finding (<tt class="docutils literal"><span class="pre">findroot</span></tt>)<a class="headerlink" href="#root-finding-findroot" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="mpmath.findroot">
-<tt class="descclassname">mpmath.</tt><tt class="descname">findroot</tt><big>(</big><em>f</em>, <em>x0</em>, <em>solver=Secant</em>, <em>tol=None</em>, <em>verbose=False</em>, <em>verify=True</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.findroot" title="Permalink to this definition">¶</a></dt>
-<dd><p>Find a solution to <span class="math">\(f(x) = 0\)</span>, using <em>x0</em> as starting point or
-interval for <em>x</em>.</p>
-<p>Multidimensional overdetermined systems are supported.
-You can specify them using a function or a list of functions.</p>
-<p>If the found root does not satisfy <span class="math">\(|f(x)^2 &lt; \mathrm{tol}|\)</span>,
-an exception is raised (this can be disabled with <em>verify=False</em>).</p>
-<p><strong>Arguments</strong></p>
-<dl class="docutils">
-<dt><em>f</em></dt>
-<dd>one dimensional function</dd>
-<dt><em>x0</em></dt>
-<dd>starting point, several starting points or interval (depends on solver)</dd>
-<dt><em>tol</em></dt>
-<dd>the returned solution has an error smaller than this</dd>
-<dt><em>verbose</em></dt>
-<dd>print additional information for each iteration if true</dd>
-<dt><em>verify</em></dt>
-<dd>verify the solution and raise a ValueError if <span class="math">\(|f(x) &gt; \mathrm{tol}|\)</span></dd>
-<dt><em>solver</em></dt>
-<dd>a generator for <em>f</em> and <em>x0</em> returning approximative solution and error</dd>
-<dt><em>maxsteps</em></dt>
-<dd>after how many steps the solver will cancel</dd>
-<dt><em>df</em></dt>
-<dd>first derivative of <em>f</em> (used by some solvers)</dd>
-<dt><em>d2f</em></dt>
-<dd>second derivative of <em>f</em> (used by some solvers)</dd>
-<dt><em>multidimensional</em></dt>
-<dd>force multidimensional solving</dd>
-<dt><em>J</em></dt>
-<dd>Jacobian matrix of <em>f</em> (used by multidimensional solvers)</dd>
-<dt><em>norm</em></dt>
-<dd>used vector norm (used by multidimensional solvers)</dd>
-</dl>
-<p>solver has to be callable with <tt class="docutils literal"><span class="pre">(f,</span> <span class="pre">x0,</span> <span class="pre">**kwargs)</span></tt> and return an generator
-yielding pairs of approximative solution and estimated error (which is
-expected to be positive).
-You can use the following string aliases:
-&#8216;secant&#8217;, &#8216;mnewton&#8217;, &#8216;halley&#8217;, &#8216;muller&#8217;, &#8216;illinois&#8217;, &#8216;pegasus&#8217;, &#8216;anderson&#8217;,
-&#8216;ridder&#8217;, &#8216;anewton&#8217;, &#8216;bisect&#8217;</p>
-<p>See mpmath.optimization for their documentation.</p>
-<p><strong>Examples</strong></p>
-<p>The function <a class="reference internal" href="#mpmath.findroot" title="mpmath.findroot"><tt class="xref py py-func docutils literal"><span class="pre">findroot()</span></tt></a> locates a root of a given function using the
-secant method by default. A simple example use of the secant method is to
-compute <span class="math">\(\pi\)</span> as the root of <span class="math">\(\sin x\)</span> closest to <span class="math">\(x_0 = 3\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">30</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">3.14159265358979323846264338328</span>
-</pre></div>
-</div>
-<p>The secant method can be used to find complex roots of analytic functions,
-although it must in that case generally be given a nonreal starting value
-(or else it will never leave the real line):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
-<span class="go">(0.226698825758202 + 1.46771150871022j)</span>
-</pre></div>
-</div>
-<p>A nice application is to compute nontrivial roots of the Riemann zeta
-function with many digits (good initial values are needed for convergence):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">30</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">zeta</span><span class="p">,</span> <span class="mf">0.5</span><span class="o">+</span><span class="mi">14j</span><span class="p">)</span>
-<span class="go">(0.5 + 14.1347251417346937904572519836j)</span>
-</pre></div>
-</div>
-<p>The secant method can also be used as an optimization algorithm, by passing
-it a derivative of a function. The following example locates the positive
-minimum of the gamma function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">20</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">diff</span><span class="p">(</span><span class="n">gamma</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">1.4616321449683623413</span>
-</pre></div>
-</div>
-<p>Finally, a useful application is to compute inverse functions, such as the
-Lambert W function which is the inverse of <span class="math">\(w e^w\)</span>, given the first
-term of the solution&#8217;s asymptotic expansion as the initial value. In basic
-cases, this gives identical results to mpmath&#8217;s built-in <tt class="docutils literal"><span class="pre">lambertw</span></tt>
-function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">lambert</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">return</span> <span class="n">findroot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">w</span><span class="p">:</span> <span class="n">w</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">w</span><span class="p">)</span> <span class="o">-</span> <span class="n">x</span><span class="p">,</span> <span class="n">log</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">x</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lambert</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span> <span class="n">lambertw</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.567143290409784</span>
-<span class="go">0.567143290409784</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lambert</span><span class="p">(</span><span class="mi">1000</span><span class="p">);</span> <span class="n">lambert</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span>
-<span class="go">5.2496028524016</span>
-<span class="go">5.2496028524016</span>
-</pre></div>
-</div>
-<p>Multidimensional functions are also supported:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="p">[</span><span class="k">lambda</span> <span class="n">x1</span><span class="p">,</span> <span class="n">x2</span><span class="p">:</span> <span class="n">x1</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x2</span><span class="p">,</span>
-<span class="gp">... </span>     <span class="k">lambda</span> <span class="n">x1</span><span class="p">,</span> <span class="n">x2</span><span class="p">:</span> <span class="mi">5</span><span class="o">*</span><span class="n">x1</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">x1</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x2</span> <span class="o">-</span> <span class="mi">3</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">[-0.618033988749895]</span>
-<span class="go">[-0.381966011250105]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
-<span class="go">[ 1.61803398874989]</span>
-<span class="go">[-2.61803398874989]</span>
-</pre></div>
-</div>
-<p>You can verify this by solving the system manually.</p>
-<p>Please note that the following (more general) syntax also works:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span> <span class="n">x2</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">return</span> <span class="n">x1</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x2</span><span class="p">,</span> <span class="mi">5</span><span class="o">*</span><span class="n">x1</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">x1</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x2</span> <span class="o">-</span> <span class="mi">3</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">[-0.618033988749895]</span>
-<span class="go">[-0.381966011250105]</span>
-</pre></div>
-</div>
-<p><strong>Multiple roots</strong></p>
-<p>For multiple roots all methods of the Newtonian family (including secant)
-converge slowly. Consider this example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">99</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mf">0.9</span><span class="p">,</span> <span class="n">verify</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">0.918073542444929</span>
-</pre></div>
-</div>
-<p>Even for a very close starting point the secant method converges very
-slowly. Use <tt class="docutils literal"><span class="pre">verbose=True</span></tt> to illustrate this.</p>
-<p>It is possible to modify Newton&#8217;s method to make it converge regardless of
-the root&#8217;s multiplicity:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="n">solver</span><span class="o">=</span><span class="s">&#39;mnewton&#39;</span><span class="p">)</span>
-<span class="go">1.0</span>
-</pre></div>
-</div>
-<p>This variant uses the first and second derivative of the function, which is
-not very efficient.</p>
-<p>Alternatively you can use an experimental Newtonian solver that keeps track
-of the speed of convergence and accelerates it using Steffensen&#8217;s method if
-necessary:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="n">solver</span><span class="o">=</span><span class="s">&#39;anewton&#39;</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">x:     -9.88888888888888888889</span>
-<span class="go">error: 0.111111111111111111111</span>
-<span class="go">converging slowly</span>
-<span class="go">x:     -9.77890011223344556678</span>
-<span class="go">error: 0.10998877665544332211</span>
-<span class="go">converging slowly</span>
-<span class="go">x:     -9.67002233332199662166</span>
-<span class="go">error: 0.108877778911448945119</span>
-<span class="go">converging slowly</span>
-<span class="go">accelerating convergence</span>
-<span class="go">x:     -9.5622443299551077669</span>
-<span class="go">error: 0.107778003366888854764</span>
-<span class="go">converging slowly</span>
-<span class="go">x:     0.99999999999999999214</span>
-<span class="go">error: 10.562244329955107759</span>
-<span class="go">x:     1.0</span>
-<span class="go">error: 7.8598304758094664213e-18</span>
-<span class="go">1.0</span>
-</pre></div>
-</div>
-<p><strong>Complex roots</strong></p>
-<p>For complex roots it&#8217;s recommended to use Muller&#8217;s method as it converges
-even for real starting points very fast:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">solver</span><span class="o">=</span><span class="s">&#39;muller&#39;</span><span class="p">)</span>
-<span class="go">(0.727136084491197 + 0.934099289460529j)</span>
-</pre></div>
-</div>
-<p><strong>Intersection methods</strong></p>
-<p>When you need to find a root in a known interval, it&#8217;s highly recommended to
-use an intersection-based solver like <tt class="docutils literal"><span class="pre">'anderson'</span></tt> or <tt class="docutils literal"><span class="pre">'ridder'</span></tt>.
-Usually they converge faster and more reliable. They have however problems
-with multiple roots and usually need a sign change to find a root:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">solver</span><span class="o">=</span><span class="s">&#39;anderson&#39;</span><span class="p">)</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>Be careful with symmetric functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">solver</span><span class="o">=</span><span class="s">&#39;anderson&#39;</span><span class="p">)</span> 
-<span class="gt">Traceback (most recent call last):</span>
-  <span class="c">...</span>
-<span class="gr">ZeroDivisionError</span>
-</pre></div>
-</div>
-<p>It fails even for better starting points, because there is no sign change:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">.</span><span class="mi">5</span><span class="p">),</span> <span class="n">solver</span><span class="o">=</span><span class="s">&#39;anderson&#39;</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-  <span class="c">...</span>
-<span class="gr">ValueError</span>: <span class="n">Could not find root within given tolerance. (1 &gt; 2.1684e-19)</span>
-<span class="go">Try another starting point or tweak arguments.</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<div class="section" id="solvers">
-<h3>Solvers<a class="headerlink" href="#solvers" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="mpmath.calculus.optimization.Secant">
-<em class="property">class </em><tt class="descclassname">mpmath.calculus.optimization.</tt><tt class="descname">Secant</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>x0</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/optimization.html#Secant"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.optimization.Secant" title="Permalink to this definition">¶</a></dt>
-<dd><p>1d-solver generating pairs of approximative root and error.</p>
-<p>Needs starting points x0 and x1 close to the root.
-x1 defaults to x0 + 0.25.</p>
-<p>Pro:</p>
-<ul class="simple">
-<li>converges fast</li>
-</ul>
-<p>Contra:</p>
-<ul class="simple">
-<li>converges slowly for multiple roots</li>
-</ul>
-</dd></dl>
-
-<dl class="class">
-<dt id="mpmath.calculus.optimization.Newton">
-<em class="property">class </em><tt class="descclassname">mpmath.calculus.optimization.</tt><tt class="descname">Newton</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>x0</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/optimization.html#Newton"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.optimization.Newton" title="Permalink to this definition">¶</a></dt>
-<dd><p>1d-solver generating pairs of approximative root and error.</p>
-<p>Needs starting points x0 close to the root.</p>
-<p>Pro:</p>
-<ul class="simple">
-<li>converges fast</li>
-<li>sometimes more robust than secant with bad second starting point</li>
-</ul>
-<p>Contra:</p>
-<ul class="simple">
-<li>converges slowly for multiple roots</li>
-<li>needs first derivative</li>
-<li>2 function evaluations per iteration</li>
-</ul>
-</dd></dl>
-
-<dl class="class">
-<dt id="mpmath.calculus.optimization.MNewton">
-<em class="property">class </em><tt class="descclassname">mpmath.calculus.optimization.</tt><tt class="descname">MNewton</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>x0</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/optimization.html#MNewton"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.optimization.MNewton" title="Permalink to this definition">¶</a></dt>
-<dd><p>1d-solver generating pairs of approximative root and error.</p>
-<p>Needs starting point x0 close to the root.
-Uses modified Newton&#8217;s method that converges fast regardless of the
-multiplicity of the root.</p>
-<p>Pro:</p>
-<ul class="simple">
-<li>converges fast for multiple roots</li>
-</ul>
-<p>Contra:</p>
-<ul class="simple">
-<li>needs first and second derivative of f</li>
-<li>3 function evaluations per iteration</li>
-</ul>
-</dd></dl>
-
-<dl class="class">
-<dt id="mpmath.calculus.optimization.Halley">
-<em class="property">class </em><tt class="descclassname">mpmath.calculus.optimization.</tt><tt class="descname">Halley</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>x0</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/optimization.html#Halley"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.optimization.Halley" title="Permalink to this definition">¶</a></dt>
-<dd><p>1d-solver generating pairs of approximative root and error.</p>
-<p>Needs a starting point x0 close to the root.
-Uses Halley&#8217;s method with cubic convergence rate.</p>
-<p>Pro:</p>
-<ul class="simple">
-<li>converges even faster the Newton&#8217;s method</li>
-<li>useful when computing with <em>many</em> digits</li>
-</ul>
-<p>Contra:</p>
-<ul class="simple">
-<li>needs first and second derivative of f</li>
-<li>3 function evaluations per iteration</li>
-<li>converges slowly for multiple roots</li>
-</ul>
-</dd></dl>
-
-<dl class="class">
-<dt id="mpmath.calculus.optimization.Muller">
-<em class="property">class </em><tt class="descclassname">mpmath.calculus.optimization.</tt><tt class="descname">Muller</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>x0</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/optimization.html#Muller"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.optimization.Muller" title="Permalink to this definition">¶</a></dt>
-<dd><p>1d-solver generating pairs of approximative root and error.</p>
-<p>Needs starting points x0, x1 and x2 close to the root.
-x1 defaults to x0 + 0.25; x2 to x1 + 0.25.
-Uses Muller&#8217;s method that converges towards complex roots.</p>
-<p>Pro:</p>
-<ul class="simple">
-<li>converges fast (somewhat faster than secant)</li>
-<li>can find complex roots</li>
-</ul>
-<p>Contra:</p>
-<ul class="simple">
-<li>converges slowly for multiple roots</li>
-<li>may have complex values for real starting points and real roots</li>
-</ul>
-<p><a class="reference external" href="http://en.wikipedia.org/wiki/Muller's_method">http://en.wikipedia.org/wiki/Muller&#8217;s_method</a></p>
-</dd></dl>
-
-<dl class="class">
-<dt id="mpmath.calculus.optimization.Bisection">
-<em class="property">class </em><tt class="descclassname">mpmath.calculus.optimization.</tt><tt class="descname">Bisection</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>x0</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/optimization.html#Bisection"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.optimization.Bisection" title="Permalink to this definition">¶</a></dt>
-<dd><p>1d-solver generating pairs of approximative root and error.</p>
-<p>Uses bisection method to find a root of f in [a, b].
-Might fail for multiple roots (needs sign change).</p>
-<p>Pro:</p>
-<ul class="simple">
-<li>robust and reliable</li>
-</ul>
-<p>Contra:</p>
-<ul class="simple">
-<li>converges slowly</li>
-<li>needs sign change</li>
-</ul>
-</dd></dl>
-
-<dl class="class">
-<dt id="mpmath.calculus.optimization.Illinois">
-<em class="property">class </em><tt class="descclassname">mpmath.calculus.optimization.</tt><tt class="descname">Illinois</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>x0</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/optimization.html#Illinois"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.optimization.Illinois" title="Permalink to this definition">¶</a></dt>
-<dd><p>1d-solver generating pairs of approximative root and error.</p>
-<p>Uses Illinois method or similar to find a root of f in [a, b].
-Might fail for multiple roots (needs sign change).
-Combines bisect with secant (improved regula falsi).</p>
-<p>The only difference between the methods is the scaling factor m, which is
-used to ensure convergence (you can choose one using the &#8216;method&#8217; keyword):</p>
-<dl class="docutils">
-<dt>Illinois method (&#8216;illinois&#8217;):</dt>
-<dd>m = 0.5</dd>
-<dt>Pegasus method (&#8216;pegasus&#8217;):</dt>
-<dd>m = fb/(fb + fz)</dd>
-<dt>Anderson-Bjoerk method (&#8216;anderson&#8217;):</dt>
-<dd>m = 1 - fz/fb if positive else 0.5</dd>
-</dl>
-<p>Pro:</p>
-<ul class="simple">
-<li>converges very fast</li>
-</ul>
-<p>Contra:</p>
-<ul class="simple">
-<li>has problems with multiple roots</li>
-<li>needs sign change</li>
-</ul>
-</dd></dl>
-
-<dl class="class">
-<dt id="mpmath.calculus.optimization.Pegasus">
-<em class="property">class </em><tt class="descclassname">mpmath.calculus.optimization.</tt><tt class="descname">Pegasus</tt><a class="reference internal" href="../../../_modules/mpmath/calculus/optimization.html#Pegasus"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.optimization.Pegasus" title="Permalink to this definition">¶</a></dt>
-<dd><p>1d-solver generating pairs of approximative root and error.</p>
-<p>Uses Pegasus method to find a root of f in [a, b].
-Wrapper for illinois to use method=&#8217;pegasus&#8217;.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="mpmath.calculus.optimization.Anderson">
-<em class="property">class </em><tt class="descclassname">mpmath.calculus.optimization.</tt><tt class="descname">Anderson</tt><a class="reference internal" href="../../../_modules/mpmath/calculus/optimization.html#Anderson"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.optimization.Anderson" title="Permalink to this definition">¶</a></dt>
-<dd><p>1d-solver generating pairs of approximative root and error.</p>
-<p>Uses Anderson-Bjoerk method to find a root of f in [a, b].
-Wrapper for illinois to use method=&#8217;pegasus&#8217;.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="mpmath.calculus.optimization.Ridder">
-<em class="property">class </em><tt class="descclassname">mpmath.calculus.optimization.</tt><tt class="descname">Ridder</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>x0</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/optimization.html#Ridder"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.optimization.Ridder" title="Permalink to this definition">¶</a></dt>
-<dd><p>1d-solver generating pairs of approximative root and error.</p>
-<p>Ridders&#8217; method to find a root of f in [a, b].
-Is told to perform as well as Brent&#8217;s method while being simpler.</p>
-<p>Pro:</p>
-<ul class="simple">
-<li>very fast</li>
-<li>simpler than Brent&#8217;s method</li>
-</ul>
-<p>Contra:</p>
-<ul class="simple">
-<li>two function evaluations per step</li>
-<li>has problems with multiple roots</li>
-<li>needs sign change</li>
-</ul>
-<p><a class="reference external" href="http://en.wikipedia.org/wiki/Ridders'_method">http://en.wikipedia.org/wiki/Ridders&#8217;_method</a></p>
-</dd></dl>
-
-<dl class="class">
-<dt id="mpmath.calculus.optimization.ANewton">
-<em class="property">class </em><tt class="descclassname">mpmath.calculus.optimization.</tt><tt class="descname">ANewton</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>x0</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/optimization.html#ANewton"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.optimization.ANewton" title="Permalink to this definition">¶</a></dt>
-<dd><p>EXPERIMENTAL 1d-solver generating pairs of approximative root and error.</p>
-<p>Uses Newton&#8217;s method modified to use Steffensens method when convergence is
-slow. (I.e. for multiple roots.)</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="mpmath.calculus.optimization.MDNewton">
-<em class="property">class </em><tt class="descclassname">mpmath.calculus.optimization.</tt><tt class="descname">MDNewton</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>x0</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../../_modules/mpmath/calculus/optimization.html#MDNewton"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#mpmath.calculus.optimization.MDNewton" title="Permalink to this definition">¶</a></dt>
-<dd><p>Find the root of a vector function numerically using Newton&#8217;s method.</p>
-<p>f is a vector function representing a nonlinear equation system.</p>
-<p>x0 is the starting point close to the root.</p>
-<p>J is a function returning the Jacobian matrix for a point.</p>
-<p>Supports overdetermined systems.</p>
-<p>Use the &#8216;norm&#8217; keyword to specify which norm to use. Defaults to max-norm.
-The function to calculate the Jacobian matrix can be given using the
-keyword &#8216;J&#8217;. Otherwise it will be calculated numerically.</p>
-<p>Please note that this method converges only locally. Especially for high-
-dimensional systems it is not trivial to find a good starting point being
-close enough to the root.</p>
-<p>It is recommended to use a faster, low-precision solver from SciPy [1] or
-OpenOpt [2] to get an initial guess. Afterwards you can use this method for
-root-polishing to any precision.</p>
-<p>[1] <a class="reference external" href="http://scipy.org">http://scipy.org</a></p>
-<p>[2] <a class="reference external" href="http://openopt.org">http://openopt.org</a></p>
-</dd></dl>
-
-</div>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Root-finding and optimization</a><ul>
-<li><a class="reference internal" href="#root-finding-findroot">Root-finding (<tt class="docutils literal"><span class="pre">findroot</span></tt>)</a><ul>
-<li><a class="reference internal" href="#solvers">Solvers</a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../calculus/polynomials.html"
-                        title="previous chapter">Polynomials</a></p>
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-  <div class="section" id="polynomials">
-<h1>Polynomials<a class="headerlink" href="#polynomials" title="Permalink to this headline">¶</a></h1>
-<p>See also <tt class="xref py py-func docutils literal"><span class="pre">taylor()</span></tt> and <tt class="xref py py-func docutils literal"><span class="pre">chebyfit()</span></tt> for
-approximation of functions by polynomials.</p>
-<div class="section" id="polynomial-evaluation-polyval">
-<h2>Polynomial evaluation (<tt class="docutils literal"><span class="pre">polyval</span></tt>)<a class="headerlink" href="#polynomial-evaluation-polyval" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="mpmath.polyval">
-<tt class="descclassname">mpmath.</tt><tt class="descname">polyval</tt><big>(</big><em>ctx</em>, <em>coeffs</em>, <em>x</em>, <em>derivative=False</em><big>)</big><a class="headerlink" href="#mpmath.polyval" title="Permalink to this definition">¶</a></dt>
-<dd><p>Given coefficients <span class="math">\([c_n, \ldots, c_2, c_1, c_0]\)</span> and a number <span class="math">\(x\)</span>,
-<a class="reference internal" href="#mpmath.polyval" title="mpmath.polyval"><tt class="xref py py-func docutils literal"><span class="pre">polyval()</span></tt></a> evaluates the polynomial</p>
-<div class="math">
-\[P(x) = c_n x^n + \ldots + c_2 x^2 + c_1 x + c_0.\]</div>
-<p>If <em>derivative=True</em> is set, <a class="reference internal" href="#mpmath.polyval" title="mpmath.polyval"><tt class="xref py py-func docutils literal"><span class="pre">polyval()</span></tt></a> simultaneously
-evaluates <span class="math">\(P(x)\)</span> with the derivative, <span class="math">\(P'(x)\)</span>, and returns the
-tuple <span class="math">\((P(x), P'(x))\)</span>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polyval</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="mf">0.5</span><span class="p">)</span>
-<span class="go">2.75</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polyval</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="mf">0.5</span><span class="p">,</span> <span class="n">derivative</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(2.75, 3.0)</span>
-</pre></div>
-</div>
-<p>The coefficients and the evaluation point may be any combination
-of real or complex numbers.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="polynomial-roots-polyroots">
-<h2>Polynomial roots (<tt class="docutils literal"><span class="pre">polyroots</span></tt>)<a class="headerlink" href="#polynomial-roots-polyroots" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="mpmath.polyroots">
-<tt class="descclassname">mpmath.</tt><tt class="descname">polyroots</tt><big>(</big><em>ctx</em>, <em>coeffs</em>, <em>maxsteps=50</em>, <em>cleanup=True</em>, <em>extraprec=10</em>, <em>error=False</em><big>)</big><a class="headerlink" href="#mpmath.polyroots" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes all roots (real or complex) of a given polynomial. The roots are
-returned as a sorted list, where real roots appear first followed by
-complex conjugate roots as adjacent elements. The polynomial should be
-given as a list of coefficients, in the format used by <a class="reference internal" href="#mpmath.polyval" title="mpmath.polyval"><tt class="xref py py-func docutils literal"><span class="pre">polyval()</span></tt></a>.
-The leading coefficient must be nonzero.</p>
-<p>With <em>error=True</em>, <a class="reference internal" href="#mpmath.polyroots" title="mpmath.polyroots"><tt class="xref py py-func docutils literal"><span class="pre">polyroots()</span></tt></a> returns a tuple <em>(roots, err)</em> where
-<em>err</em> is an estimate of the maximum error among the computed roots.</p>
-<p><strong>Examples</strong></p>
-<p>Finding the three real roots of <span class="math">\(x^3 - x^2 - 14x + 24\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">polyroots</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">14</span><span class="p">,</span><span class="mi">24</span><span class="p">]),</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">[-4.0, 2.0, 3.0]</span>
-</pre></div>
-</div>
-<p>Finding the two complex conjugate roots of <span class="math">\(4x^2 + 3x + 2\)</span>, with an
-error estimate:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">roots</span><span class="p">,</span> <span class="n">err</span> <span class="o">=</span> <span class="n">polyroots</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span> <span class="n">error</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">roots</span><span class="p">:</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="go">(-0.375 + 0.59947894041409j)</span>
-<span class="go">(-0.375 - 0.59947894041409j)</span>
-<span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">err</span>
-<span class="go">2.22044604925031e-16</span>
-<span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polyval</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span> <span class="n">roots</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-<span class="go">(2.22044604925031e-16 + 0.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polyval</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span> <span class="n">roots</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">(2.22044604925031e-16 + 0.0j)</span>
-</pre></div>
-</div>
-<p>The following example computes all the 5th roots of unity; that is,
-the roots of <span class="math">\(x^5 - 1\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">20</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">polyroots</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">]):</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="go">1.0</span>
-<span class="go">(-0.8090169943749474241 + 0.58778525229247312917j)</span>
-<span class="go">(-0.8090169943749474241 - 0.58778525229247312917j)</span>
-<span class="go">(0.3090169943749474241 + 0.95105651629515357212j)</span>
-<span class="go">(0.3090169943749474241 - 0.95105651629515357212j)</span>
-</pre></div>
-</div>
-<p><strong>Precision and conditioning</strong></p>
-<p>Provided there are no repeated roots, <a class="reference internal" href="#mpmath.polyroots" title="mpmath.polyroots"><tt class="xref py py-func docutils literal"><span class="pre">polyroots()</span></tt></a> can typically
-compute all roots of an arbitrary polynomial to high precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">60</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">polyroots</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]):</span>
-<span class="gp">... </span>    <span class="k">print</span> <span class="n">r</span>
-<span class="gp">...</span>
-<span class="go">-3.14626436994197234232913506571557044551247712918732870123249</span>
-<span class="go">-0.317837245195782244725757617296174288373133378433432554879127</span>
-<span class="go">0.317837245195782244725757617296174288373133378433432554879127</span>
-<span class="go">3.14626436994197234232913506571557044551247712918732870123249</span>
-<span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">3.14626436994197234232913506571557044551247712918732870123249</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.317837245195782244725757617296174288373133378433432554879127</span>
-</pre></div>
-</div>
-<p><strong>Algorithm</strong></p>
-<p><a class="reference internal" href="#mpmath.polyroots" title="mpmath.polyroots"><tt class="xref py py-func docutils literal"><span class="pre">polyroots()</span></tt></a> implements the Durand-Kerner method [1], which
-uses complex arithmetic to locate all roots simultaneously.
-The Durand-Kerner method can be viewed as approximately performing
-simultaneous Newton iteration for all the roots. In particular,
-the convergence to simple roots is quadratic, just like Newton&#8217;s
-method.</p>
-<p>Although all roots are internally calculated using complex arithmetic,
-any root found to have an imaginary part smaller than the estimated
-numerical error is truncated to a real number. Real roots are placed
-first in the returned list, sorted by value. The remaining complex
-roots are sorted by real their parts so that conjugate roots end up
-next to each other.</p>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Durand-Kerner_method">http://en.wikipedia.org/wiki/Durand-Kerner_method</a></li>
-</ol>
-</dd></dl>
-
-</div>
-</div>
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-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Polynomials</a><ul>
-<li><a class="reference internal" href="#polynomial-evaluation-polyval">Polynomial evaluation (<tt class="docutils literal"><span class="pre">polyval</span></tt>)</a></li>
-<li><a class="reference internal" href="#polynomial-roots-polyroots">Polynomial roots (<tt class="docutils literal"><span class="pre">polyroots</span></tt>)</a></li>
-</ul>
-</li>
-</ul>
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-  <div class="section" id="sums-products-limits-and-extrapolation">
-<h1>Sums, products, limits and extrapolation<a class="headerlink" href="#sums-products-limits-and-extrapolation" title="Permalink to this headline">¶</a></h1>
-<p>The functions listed here permit approximation of infinite
-sums, products, and other sequence limits.
-Use <a class="reference internal" href="../../mpmath/general.html#mpmath.fsum" title="mpmath.fsum"><tt class="xref py py-func docutils literal"><span class="pre">mpmath.fsum()</span></tt></a> and <a class="reference internal" href="../../mpmath/general.html#mpmath.fprod" title="mpmath.fprod"><tt class="xref py py-func docutils literal"><span class="pre">mpmath.fprod()</span></tt></a>
-for summation and multiplication of finite sequences.</p>
-<div class="section" id="summation">
-<h2>Summation<a class="headerlink" href="#summation" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="nsum">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">nsum()</span></tt><a class="headerlink" href="#nsum" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.nsum">
-<tt class="descclassname">mpmath.</tt><tt class="descname">nsum</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>*intervals</em>, <em>**options</em><big>)</big><a class="headerlink" href="#mpmath.nsum" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the sum</p>
-<div class="math">
-\[S = \sum_{k=a}^b f(k)\]</div>
-<p>where <span class="math">\((a, b)\)</span> = <em>interval</em>, and where <span class="math">\(a = -\infty\)</span> and/or
-<span class="math">\(b = \infty\)</span> are allowed, or more generally</p>
-<div class="math">
-\[S = \sum_{k_1=a_1}^{b_1} \cdots
-\sum_{k_n=a_n}^{b_n} f(k_1,\ldots,k_n)\]</div>
-<p>if multiple intervals are given.</p>
-<p>Two examples of infinite series that can be summed by <a class="reference internal" href="#mpmath.nsum" title="mpmath.nsum"><tt class="xref py py-func docutils literal"><span class="pre">nsum()</span></tt></a>,
-where the first converges rapidly and the second converges slowly,
-are:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">2.71828182845905</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">n</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">1.64493406684823</span>
-</pre></div>
-</div>
-<p>When appropriate, <a class="reference internal" href="#mpmath.nsum" title="mpmath.nsum"><tt class="xref py py-func docutils literal"><span class="pre">nsum()</span></tt></a> applies convergence acceleration to
-accurately estimate the sums of slowly convergent series. If the series is
-finite, <a class="reference internal" href="#mpmath.nsum" title="mpmath.nsum"><tt class="xref py py-func docutils literal"><span class="pre">nsum()</span></tt></a> currently does not attempt to perform any
-extrapolation, and simply calls <a class="reference internal" href="../../mpmath/general.html#mpmath.fsum" title="mpmath.fsum"><tt class="xref py py-func docutils literal"><span class="pre">fsum()</span></tt></a>.</p>
-<p>Multidimensional infinite series are reduced to a single-dimensional
-series over expanding hypercubes; if both infinite and finite dimensions
-are present, the finite ranges are moved innermost. For more advanced
-control over the summation order, use nested calls to <a class="reference internal" href="#mpmath.nsum" title="mpmath.nsum"><tt class="xref py py-func docutils literal"><span class="pre">nsum()</span></tt></a>,
-or manually rewrite the sum as a single-dimensional series.</p>
-<p><strong>Options</strong></p>
-<dl class="docutils">
-<dt><em>tol</em></dt>
-<dd>Desired maximum final error. Defaults roughly to the
-epsilon of the working precision.</dd>
-<dt><em>method</em></dt>
-<dd>Which summation algorithm to use (described below).
-Default: <tt class="docutils literal"><span class="pre">'richardson+shanks'</span></tt>.</dd>
-<dt><em>maxterms</em></dt>
-<dd>Cancel after at most this many terms. Default: 10*dps.</dd>
-<dt><em>steps</em></dt>
-<dd>An iterable giving the number of terms to add between
-each extrapolation attempt. The default sequence is
-[10, 20, 30, 40, ...]. For example, if you know that
-approximately 100 terms will be required, efficiency might be
-improved by setting this to [100, 10]. Then the first
-extrapolation will be performed after 100 terms, the second
-after 110, etc.</dd>
-<dt><em>verbose</em></dt>
-<dd>Print details about progress.</dd>
-<dt><em>ignore</em></dt>
-<dd>If enabled, any term that raises <tt class="docutils literal"><span class="pre">ArithmeticError</span></tt>
-or <tt class="docutils literal"><span class="pre">ValueError</span></tt> (e.g. through division by zero) is replaced
-by a zero. This is convenient for lattice sums with
-a singular term near the origin.</dd>
-</dl>
-<p><strong>Methods</strong></p>
-<p>Unfortunately, an algorithm that can efficiently sum any infinite
-series does not exist. <a class="reference internal" href="#mpmath.nsum" title="mpmath.nsum"><tt class="xref py py-func docutils literal"><span class="pre">nsum()</span></tt></a> implements several different
-algorithms that each work well in different cases. The <em>method</em>
-keyword argument selects a method.</p>
-<p>The default method is <tt class="docutils literal"><span class="pre">'r+s'</span></tt>, i.e. both Richardson extrapolation
-and Shanks transformation is attempted. A slower method that
-handles more cases is <tt class="docutils literal"><span class="pre">'r+s+e'</span></tt>. For very high precision
-summation, or if the summation needs to be fast (for example if
-multiple sums need to be evaluated), it is a good idea to
-investigate which one method works best and only use that.</p>
-<dl class="docutils">
-<dt><tt class="docutils literal"><span class="pre">'richardson'</span></tt> / <tt class="docutils literal"><span class="pre">'r'</span></tt>:</dt>
-<dd>Uses Richardson extrapolation. Provides useful extrapolation
-when <span class="math">\(f(k) \sim P(k)/Q(k)\)</span> or when <span class="math">\(f(k) \sim (-1)^k P(k)/Q(k)\)</span>
-for polynomials <span class="math">\(P\)</span> and <span class="math">\(Q\)</span>. See <a class="reference internal" href="#mpmath.richardson" title="mpmath.richardson"><tt class="xref py py-func docutils literal"><span class="pre">richardson()</span></tt></a> for
-additional information.</dd>
-<dt><tt class="docutils literal"><span class="pre">'shanks'</span></tt> / <tt class="docutils literal"><span class="pre">'s'</span></tt>:</dt>
-<dd>Uses Shanks transformation. Typically provides useful
-extrapolation when <span class="math">\(f(k) \sim c^k\)</span> or when successive terms
-alternate signs. Is able to sum some divergent series.
-See <a class="reference internal" href="#mpmath.shanks" title="mpmath.shanks"><tt class="xref py py-func docutils literal"><span class="pre">shanks()</span></tt></a> for additional information.</dd>
-<dt><tt class="docutils literal"><span class="pre">'euler-maclaurin'</span></tt> / <tt class="docutils literal"><span class="pre">'e'</span></tt>:</dt>
-<dd>Uses the Euler-Maclaurin summation formula to approximate
-the remainder sum by an integral. This requires high-order
-numerical derivatives and numerical integration. The advantage
-of this algorithm is that it works regardless of the
-decay rate of <span class="math">\(f\)</span>, as long as <span class="math">\(f\)</span> is sufficiently smooth.
-See <a class="reference internal" href="#mpmath.sumem" title="mpmath.sumem"><tt class="xref py py-func docutils literal"><span class="pre">sumem()</span></tt></a> for additional information.</dd>
-<dt><tt class="docutils literal"><span class="pre">'direct'</span></tt> / <tt class="docutils literal"><span class="pre">'d'</span></tt>:</dt>
-<dd>Does not perform any extrapolation. This can be used
-(and should only be used for) rapidly convergent series.
-The summation automatically stops when the terms
-decrease below the target tolerance.</dd>
-</dl>
-<p><strong>Basic examples</strong></p>
-<p>A finite sum:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">])</span>
-<span class="go">2.45</span>
-</pre></div>
-</div>
-<p>Summation of a series going to negative infinity and a doubly
-infinite series:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">1.64493406684823</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">k</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">3.15334809493716</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.nsum" title="mpmath.nsum"><tt class="xref py py-func docutils literal"><span class="pre">nsum()</span></tt></a> handles sums of complex numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="mf">0.5</span><span class="o">+</span><span class="mf">0.25j</span><span class="p">)</span><span class="o">**</span><span class="n">k</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">(1.6 + 0.8j)</span>
-</pre></div>
-</div>
-<p>The following sum converges very rapidly, so it is most
-efficient to sum it by disabling convergence acceleration:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">1000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="o">-</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">k</span> <span class="o">*</span> <span class="n">k</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span> <span class="n">fac</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span>
-<span class="gp">... </span>    <span class="n">method</span><span class="o">=</span><span class="s">&#39;direct&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">+</span><span class="n">sin</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span><span class="o">/</span><span class="mi">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">abs</span><span class="p">(</span><span class="n">a</span><span class="o">-</span><span class="n">b</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">mpf</span><span class="p">(</span><span class="s">&#39;1e-998&#39;</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p><strong>Examples with Richardson extrapolation</strong></p>
-<p>Richardson extrapolation works well for sums over rational
-functions, as well as their alternating counterparts:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">k</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span>
-<span class="gp">... </span>    <span class="n">method</span><span class="o">=</span><span class="s">&#39;richardson&#39;</span><span class="p">)</span>
-<span class="go">1.2020569031595942853997381615114499907649862923405</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">1.2020569031595942853997381615114499907649862923405</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">n</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">n</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span>
-<span class="gp">... </span>    <span class="n">method</span><span class="o">=</span><span class="s">&#39;richardson&#39;</span><span class="p">)</span>
-<span class="go">2.9348022005446793094172454999380755676568497036204</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">2</span><span class="o">-</span><span class="mi">2</span>
-<span class="go">2.9348022005446793094172454999380755676568497036204</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">k</span> <span class="o">/</span> <span class="n">k</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span>
-<span class="gp">... </span>    <span class="n">method</span><span class="o">=</span><span class="s">&#39;richardson&#39;</span><span class="p">)</span>
-<span class="go">-0.90154267736969571404980362113358749307373971925537</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">zeta</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span>
-<span class="go">-0.90154267736969571404980362113358749307373971925538</span>
-</pre></div>
-</div>
-<p><strong>Examples with Shanks transformation</strong></p>
-<p>The Shanks transformation works well for geometric series
-and typically provides excellent acceleration for Taylor
-series near the border of their disk of convergence.
-Here we apply it to a series for <span class="math">\(\log(2)\)</span>, which can be
-seen as the Taylor series for <span class="math">\(\log(1+x)\)</span> with <span class="math">\(x = 1\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="o">-</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">k</span><span class="o">/</span><span class="n">k</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span>
-<span class="gp">... </span>    <span class="n">method</span><span class="o">=</span><span class="s">&#39;shanks&#39;</span><span class="p">)</span>
-<span class="go">0.69314718055994530941723212145817656807550013436025</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.69314718055994530941723212145817656807550013436025</span>
-</pre></div>
-</div>
-<p>Here we apply it to a slowly convergent geometric series:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="n">mpf</span><span class="p">(</span><span class="s">&#39;0.995&#39;</span><span class="p">)</span><span class="o">**</span><span class="n">k</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span>
-<span class="gp">... </span>    <span class="n">method</span><span class="o">=</span><span class="s">&#39;shanks&#39;</span><span class="p">)</span>
-<span class="go">200.0</span>
-</pre></div>
-</div>
-<p>Finally, Shanks&#8217; method works very well for alternating series
-where <span class="math">\(f(k) = (-1)^k g(k)\)</span>, and often does so regardless of
-the exact decay rate of <span class="math">\(g(k)\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="n">k</span><span class="o">**</span><span class="mf">1.5</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span>
-<span class="gp">... </span>    <span class="n">method</span><span class="o">=</span><span class="s">&#39;shanks&#39;</span><span class="p">)</span>
-<span class="go">0.765147024625408</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">*</span><span class="n">zeta</span><span class="p">(</span><span class="mf">1.5</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-<span class="go">0.765147024625408</span>
-</pre></div>
-</div>
-<p>The following slowly convergent alternating series has no known
-closed-form value. Evaluating the sum a second time at higher
-precision indicates that the value is probably correct:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">k</span> <span class="o">/</span> <span class="n">log</span><span class="p">(</span><span class="n">k</span><span class="p">),</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span>
-<span class="gp">... </span>    <span class="n">method</span><span class="o">=</span><span class="s">&#39;shanks&#39;</span><span class="p">)</span>
-<span class="go">0.924299897222939</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">30</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">k</span> <span class="o">/</span> <span class="n">log</span><span class="p">(</span><span class="n">k</span><span class="p">),</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span>
-<span class="gp">... </span>    <span class="n">method</span><span class="o">=</span><span class="s">&#39;shanks&#39;</span><span class="p">)</span>
-<span class="go">0.92429989722293885595957018136</span>
-</pre></div>
-</div>
-<p><strong>Examples with Euler-Maclaurin summation</strong></p>
-<p>The sum in the following example has the wrong rate of convergence
-for either Richardson or Shanks to be effective.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="n">log</span><span class="p">(</span><span class="n">k</span><span class="p">)</span><span class="o">/</span><span class="n">k</span><span class="o">**</span><span class="mf">2.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;euler-maclaurin&#39;</span><span class="p">)</span>
-<span class="go">0.38734195032621</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="n">diff</span><span class="p">(</span><span class="n">zeta</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">)</span>
-<span class="go">0.38734195032621</span>
-</pre></div>
-</div>
-<p>Increasing <tt class="docutils literal"><span class="pre">steps</span></tt> improves speed at higher precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;euler-maclaurin&#39;</span><span class="p">,</span> <span class="n">steps</span><span class="o">=</span><span class="p">[</span><span class="mi">250</span><span class="p">])</span>
-<span class="go">0.38734195032620997271199237593105101319948228874688</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="n">diff</span><span class="p">(</span><span class="n">zeta</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">)</span>
-<span class="go">0.38734195032620997271199237593105101319948228874688</span>
-</pre></div>
-</div>
-<p><strong>Divergent series</strong></p>
-<p>The Shanks transformation is able to sum some <em>divergent</em>
-series. In particular, it is often able to sum Taylor series
-beyond their radius of convergence (this is due to a relation
-between the Shanks transformation and Pade approximations;
-see <a class="reference internal" href="../calculus/approximation.html#mpmath.pade" title="mpmath.pade"><tt class="xref py py-func docutils literal"><span class="pre">pade()</span></tt></a> for an alternative way to evaluate divergent
-Taylor series).</p>
-<p>Here we apply it to <span class="math">\(\log(1+x)\)</span> far outside the region of
-convergence:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="o">-</span><span class="p">(</span><span class="o">-</span><span class="mi">9</span><span class="p">)</span><span class="o">**</span><span class="n">k</span><span class="o">/</span><span class="n">k</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span>
-<span class="gp">... </span>    <span class="n">method</span><span class="o">=</span><span class="s">&#39;shanks&#39;</span><span class="p">)</span>
-<span class="go">2.3025850929940456840179914546843642076011014886288</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">2.3025850929940456840179914546843642076011014886288</span>
-</pre></div>
-</div>
-<p>A particular type of divergent series that can be summed
-using the Shanks transformation is geometric series.
-The result is the same as using the closed-form formula
-for an infinite geometric series:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span> <span class="mi">8</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-<span class="gp">... </span>        <span class="k">continue</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">n</span><span class="p">),</span>
-<span class="gp">... </span>        <span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="n">n</span><span class="o">**</span><span class="n">k</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;shanks&#39;</span><span class="p">)))</span>
-<span class="gp">...</span>
-<span class="go">-8.0 0.111111111111111 0.111111111111111</span>
-<span class="go">-7.0 0.125 0.125</span>
-<span class="go">-6.0 0.142857142857143 0.142857142857143</span>
-<span class="go">-5.0 0.166666666666667 0.166666666666667</span>
-<span class="go">-4.0 0.2 0.2</span>
-<span class="go">-3.0 0.25 0.25</span>
-<span class="go">-2.0 0.333333333333333 0.333333333333333</span>
-<span class="go">-1.0 0.5 0.5</span>
-<span class="go">0.0 1.0 1.0</span>
-<span class="go">2.0 -1.0 -1.0</span>
-<span class="go">3.0 -0.5 -0.5</span>
-<span class="go">4.0 -0.333333333333333 -0.333333333333333</span>
-<span class="go">5.0 -0.25 -0.25</span>
-<span class="go">6.0 -0.2 -0.2</span>
-<span class="go">7.0 -0.166666666666667 -0.166666666666667</span>
-</pre></div>
-</div>
-<p><strong>Multidimensional sums</strong></p>
-<p>Any combination of finite and infinite ranges is allowed for the
-summation indices:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">:</span> <span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="p">,</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
-<span class="go">28.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">:</span> <span class="n">x</span><span class="o">/</span><span class="mi">2</span><span class="o">**</span><span class="n">y</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">6.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">:</span> <span class="n">y</span><span class="o">/</span><span class="mi">2</span><span class="o">**</span><span class="n">x</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-<span class="go">6.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span><span class="p">:</span> <span class="n">z</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">x</span><span class="o">*</span><span class="mi">2</span><span class="o">**</span><span class="n">y</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
-<span class="go">7.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span><span class="p">:</span> <span class="n">y</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">x</span><span class="o">*</span><span class="mi">2</span><span class="o">**</span><span class="n">z</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">7.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span><span class="p">:</span> <span class="n">x</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">z</span><span class="o">*</span><span class="mi">2</span><span class="o">**</span><span class="n">y</span><span class="p">),</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">7.0</span>
-</pre></div>
-</div>
-<p>Some nice examples of double series with analytic solutions or
-reductions to single-dimensional series (see [1]):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="n">m</span><span class="o">*</span><span class="n">n</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">1.60669515241529</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">1.60669515241529</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">:</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="n">j</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">i</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">j</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">0.278070510848213</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">*</span><span class="p">(</span><span class="n">pi</span><span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">ln2</span><span class="p">)</span><span class="o">/</span><span class="mi">12</span>
-<span class="go">0.278070510848213</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">:</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="n">j</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="n">j</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">0.129319852864168</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">altzeta</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="n">altzeta</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.129319852864168</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">:</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="n">j</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="n">j</span><span class="p">)</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">0.0790756439455825</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">altzeta</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="n">altzeta</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.0790756439455825</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">m</span><span class="p">,</span><span class="n">n</span><span class="p">:</span> <span class="n">m</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">/</span><span class="p">(</span><span class="mi">3</span><span class="o">**</span><span class="n">m</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="mi">3</span><span class="o">**</span><span class="n">m</span><span class="o">+</span><span class="n">m</span><span class="o">*</span><span class="mi">3</span><span class="o">**</span><span class="n">n</span><span class="p">)),</span>
-<span class="gp">... </span>    <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">0.28125</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpf</span><span class="p">(</span><span class="mi">9</span><span class="p">)</span><span class="o">/</span><span class="mi">32</span>
-<span class="go">0.28125</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">:</span> <span class="n">fac</span><span class="p">(</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">fac</span><span class="p">(</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="n">j</span><span class="p">),</span>
-<span class="gp">... </span>    <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="n">workprec</span><span class="o">=</span><span class="mi">400</span><span class="p">)</span>
-<span class="go">1.64493406684823</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">1.64493406684823</span>
-</pre></div>
-</div>
-<p>A hard example of a multidimensional sum is the Madelung constant
-in three dimensions (see [2]). The defining sum converges very
-slowly and only conditionally, so <a class="reference internal" href="#mpmath.nsum" title="mpmath.nsum"><tt class="xref py py-func docutils literal"><span class="pre">nsum()</span></tt></a> is lucky to
-obtain an accurate value through convergence acceleration. The
-second evaluation below uses a much more efficient, rapidly
-convergent 2D sum:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span><span class="p">:</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="o">+</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="o">*</span><span class="n">y</span><span class="o">+</span><span class="n">z</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="mf">0.5</span><span class="p">,</span>
-<span class="gp">... </span>    <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="n">ignore</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">-1.74756459463318</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">:</span> <span class="o">-</span><span class="mi">12</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">sech</span><span class="p">(</span><span class="mf">0.5</span><span class="o">*</span><span class="n">pi</span> <span class="o">*</span> \
-<span class="gp">... </span>    <span class="n">sqrt</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">-1.74756459463318</span>
-</pre></div>
-</div>
-<p>Another example of a lattice sum in 2D:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">:</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span>
-<span class="gp">... </span>    <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="n">ignore</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">-2.1775860903036</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="n">pi</span><span class="o">*</span><span class="n">ln2</span>
-<span class="go">-2.1775860903036</span>
-</pre></div>
-</div>
-<p>An example of an Eisenstein series:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">m</span><span class="p">,</span><span class="n">n</span><span class="p">:</span> <span class="p">(</span><span class="n">m</span><span class="o">+</span><span class="n">n</span><span class="o">*</span><span class="mi">1j</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span>
-<span class="gp">... </span>    <span class="n">ignore</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(3.1512120021539 + 0.0j)</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../../mpmath/references.html#weisstein">[Weisstein]</a> <a class="reference external" href="http://mathworld.wolfram.com/DoubleSeries.html">http://mathworld.wolfram.com/DoubleSeries.html</a>,</li>
-<li><a class="reference internal" href="../../mpmath/references.html#weisstein">[Weisstein]</a> <a class="reference external" href="http://mathworld.wolfram.com/MadelungConstants.html">http://mathworld.wolfram.com/MadelungConstants.html</a></li>
-</ol>
-</dd></dl>
-
-</div>
-<div class="section" id="sumem">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">sumem()</span></tt><a class="headerlink" href="#sumem" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.sumem">
-<tt class="descclassname">mpmath.</tt><tt class="descname">sumem</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>interval</em>, <em>tol=None</em>, <em>reject=10</em>, <em>integral=None</em>, <em>adiffs=None</em>, <em>bdiffs=None</em>, <em>verbose=False</em>, <em>error=False</em>, <em>_fast_abort=False</em><big>)</big><a class="headerlink" href="#mpmath.sumem" title="Permalink to this definition">¶</a></dt>
-<dd><p>Uses the Euler-Maclaurin formula to compute an approximation accurate
-to within <tt class="docutils literal"><span class="pre">tol</span></tt> (which defaults to the present epsilon) of the sum</p>
-<div class="math">
-\[S = \sum_{k=a}^b f(k)\]</div>
-<p>where <span class="math">\((a,b)\)</span> are given by <tt class="docutils literal"><span class="pre">interval</span></tt> and <span class="math">\(a\)</span> or <span class="math">\(b\)</span> may be
-infinite. The approximation is</p>
-<div class="math">
-\[S \sim \int_a^b f(x) \,dx + \frac{f(a)+f(b)}{2} +
-\sum_{k=1}^{\infty} \frac{B_{2k}}{(2k)!}
-\left(f^{(2k-1)}(b)-f^{(2k-1)}(a)\right).\]</div>
-<p>The last sum in the Euler-Maclaurin formula is not generally
-convergent (a notable exception is if <span class="math">\(f\)</span> is a polynomial, in
-which case Euler-Maclaurin actually gives an exact result).</p>
-<p>The summation is stopped as soon as the quotient between two
-consecutive terms falls below <em>reject</em>. That is, by default
-(<em>reject</em> = 10), the summation is continued as long as each
-term adds at least one decimal.</p>
-<p>Although not convergent, convergence to a given tolerance can
-often be &#8220;forced&#8221; if <span class="math">\(b = \infty\)</span> by summing up to <span class="math">\(a+N\)</span> and then
-applying the Euler-Maclaurin formula to the sum over the range
-<span class="math">\((a+N+1, \ldots, \infty)\)</span>. This procedure is implemented by
-<a class="reference internal" href="#mpmath.nsum" title="mpmath.nsum"><tt class="xref py py-func docutils literal"><span class="pre">nsum()</span></tt></a>.</p>
-<p>By default numerical quadrature and differentiation is used.
-If the symbolic values of the integral and endpoint derivatives
-are known, it is more efficient to pass the value of the
-integral explicitly as <tt class="docutils literal"><span class="pre">integral</span></tt> and the derivatives
-explicitly as <tt class="docutils literal"><span class="pre">adiffs</span></tt> and <tt class="docutils literal"><span class="pre">bdiffs</span></tt>. The derivatives
-should be given as iterables that yield
-<span class="math">\(f(a), f'(a), f''(a), \ldots\)</span> (and the equivalent for <span class="math">\(b\)</span>).</p>
-<p><strong>Examples</strong></p>
-<p>Summation of an infinite series, with automatic and symbolic
-integral and derivative values (the second should be much faster):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sumem</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">n</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">32</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">0.03174336652030209012658168043874142714132886413417</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">32</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">adiffs</span><span class="o">=</span><span class="p">((</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="o">*</span><span class="n">fac</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="mi">32</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">-</span><span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">999</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sumem</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">n</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">32</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span> <span class="n">integral</span><span class="o">=</span><span class="n">I</span><span class="p">,</span> <span class="n">adiffs</span><span class="o">=</span><span class="n">D</span><span class="p">)</span>
-<span class="go">0.03174336652030209012658168043874142714132886413417</span>
-</pre></div>
-</div>
-<p>An exact evaluation of a finite polynomial sum:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sumem</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="n">n</span><span class="o">**</span><span class="mi">5</span><span class="o">-</span><span class="mi">12</span><span class="o">*</span><span class="n">n</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">n</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">100000</span><span class="p">,</span> <span class="mi">200000</span><span class="p">])</span>
-<span class="go">10500155000624963999742499550000.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">sum</span><span class="p">(</span><span class="n">n</span><span class="o">**</span><span class="mi">5</span><span class="o">-</span><span class="mi">12</span><span class="o">*</span><span class="n">n</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">n</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="mi">100000</span><span class="p">,</span> <span class="mi">200001</span><span class="p">)))</span>
-<span class="go">10500155000624963999742499550000</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="sumap">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">sumap()</span></tt><a class="headerlink" href="#sumap" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.sumap">
-<tt class="descclassname">mpmath.</tt><tt class="descname">sumap</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>interval</em>, <em>integral=None</em>, <em>error=False</em><big>)</big><a class="headerlink" href="#mpmath.sumap" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates an infinite series of an analytic summand <em>f</em> using the
-Abel-Plana formula</p>
-<div class="math">
-\[\sum_{k=0}^{\infty} f(k) = \int_0^{\infty} f(t) dt + \frac{1}{2} f(0) +
-    i \int_0^{\infty} \frac{f(it)-f(-it)}{e^{2\pi t}-1} dt.\]</div>
-<p>Unlike the Euler-Maclaurin formula (see <a class="reference internal" href="#mpmath.sumem" title="mpmath.sumem"><tt class="xref py py-func docutils literal"><span class="pre">sumem()</span></tt></a>),
-the Abel-Plana formula does not require derivatives. However,
-it only works when <span class="math">\(|f(it)-f(-it)|\)</span> does not
-increase too rapidly with <span class="math">\(t\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>The Abel-Plana formula is particularly useful when the summand
-decreases like a power of <span class="math">\(k\)</span>; for example when the sum is a pure
-zeta function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sumap</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="o">**</span><span class="mf">2.5</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">1.34148725725091717975677</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mf">2.5</span><span class="p">)</span>
-<span class="go">1.34148725725091717975677</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sumap</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1j</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mf">2.5</span><span class="o">+</span><span class="mf">2.5j</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">(-3.385361068546473342286084 - 0.7432082105196321803869551j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mf">2.5</span><span class="o">+</span><span class="mf">2.5j</span><span class="p">,</span> <span class="mi">1</span><span class="o">+</span><span class="mi">1j</span><span class="p">)</span>
-<span class="go">(-3.385361068546473342286084 - 0.7432082105196321803869551j)</span>
-</pre></div>
-</div>
-<p>If the series is alternating, numerical quadrature along the real
-line is likely to give poor results, so it is better to evaluate
-the first term symbolically whenever possible:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="o">=</span><span class="mi">3</span><span class="p">;</span> <span class="n">z</span><span class="o">=-</span><span class="mf">0.75</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">expint</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="o">-</span><span class="n">log</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">sumap</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="n">z</span><span class="o">**</span><span class="n">k</span> <span class="o">/</span> <span class="n">k</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="n">integral</span><span class="o">=</span><span class="n">I</span><span class="p">))</span>
-<span class="go">-0.6917036036904594510141448</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">-0.6917036036904594510141448</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="products">
-<h2>Products<a class="headerlink" href="#products" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="nprod">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">nprod()</span></tt><a class="headerlink" href="#nprod" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.nprod">
-<tt class="descclassname">mpmath.</tt><tt class="descname">nprod</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>interval</em>, <em>nsum=False</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.nprod" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the product</p>
-<div class="math">
-\[P = \prod_{k=a}^b f(k)\]</div>
-<p>where <span class="math">\((a, b)\)</span> = <em>interval</em>, and where <span class="math">\(a = -\infty\)</span> and/or
-<span class="math">\(b = \infty\)</span> are allowed.</p>
-<p>By default, <a class="reference internal" href="#mpmath.nprod" title="mpmath.nprod"><tt class="xref py py-func docutils literal"><span class="pre">nprod()</span></tt></a> uses the same extrapolation methods as
-<a class="reference internal" href="#mpmath.nsum" title="mpmath.nsum"><tt class="xref py py-func docutils literal"><span class="pre">nsum()</span></tt></a>, except applied to the partial products rather than
-partial sums, and the same keyword options as for <a class="reference internal" href="#mpmath.nsum" title="mpmath.nsum"><tt class="xref py py-func docutils literal"><span class="pre">nsum()</span></tt></a> are
-supported. If <tt class="docutils literal"><span class="pre">nsum=True</span></tt>, the product is instead computed via
-<a class="reference internal" href="#mpmath.nsum" title="mpmath.nsum"><tt class="xref py py-func docutils literal"><span class="pre">nsum()</span></tt></a> as</p>
-<div class="math">
-\[P = \exp\left( \sum_{k=a}^b \log(f(k)) \right).\]</div>
-<p>This is slower, but can sometimes yield better results. It is
-also required (and used automatically) when Euler-Maclaurin
-summation is requested.</p>
-<p><strong>Examples</strong></p>
-<p>A simple finite product:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprod</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="n">k</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
-<span class="go">24.0</span>
-</pre></div>
-</div>
-<p>A large number of infinite products have known exact values,
-and can therefore be used as a reference. Most of the following
-examples are taken from MathWorld [1].</p>
-<p>A few infinite products with simple values are:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span><span class="o">*</span><span class="n">nprod</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">k</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">k</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">3.141592653589793238462643</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprod</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mi">2</span><span class="o">/</span><span class="n">k</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">2.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprod</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="n">k</span><span class="o">**</span><span class="mi">3</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">k</span><span class="o">**</span><span class="mi">3</span><span class="o">+</span><span class="mi">1</span><span class="p">),</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">0.6666666666666666666666667</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprod</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">0.5</span>
-</pre></div>
-</div>
-<p>Next, several more infinite products with more complicated
-values:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprod</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">inf</span><span class="p">]);</span> <span class="n">exp</span><span class="p">(</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">6</span><span class="p">)</span>
-<span class="go">5.180668317897115748416626</span>
-<span class="go">5.180668317897115748416626</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprod</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="n">k</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">k</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="mi">1</span><span class="p">),</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="n">inf</span><span class="p">]);</span> <span class="n">pi</span><span class="o">*</span><span class="n">csch</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">0.2720290549821331629502366</span>
-<span class="go">0.2720290549821331629502366</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprod</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="n">k</span><span class="o">**</span><span class="mi">4</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">k</span><span class="o">**</span><span class="mi">4</span><span class="o">+</span><span class="mi">1</span><span class="p">),</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">0.8480540493529003921296502</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">*</span><span class="n">sinh</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">cosh</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span>
-<span class="go">0.8480540493529003921296502</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprod</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mi">2</span><span class="o">/</span><span class="n">k</span><span class="o">+</span><span class="mi">3</span><span class="o">/</span><span class="n">k</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">1.848936182858244485224927</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">3</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">cosh</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">csch</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">/</span><span class="n">pi</span>
-<span class="go">1.848936182858244485224927</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprod</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="o">**</span><span class="mi">4</span><span class="p">),</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="n">inf</span><span class="p">]);</span> <span class="n">sinh</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">0.9190194775937444301739244</span>
-<span class="go">0.9190194775937444301739244</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprod</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="o">**</span><span class="mi">6</span><span class="p">),</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">0.9826842777421925183244759</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">cosh</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)))</span><span class="o">/</span><span class="p">(</span><span class="mi">12</span><span class="o">*</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.9826842777421925183244759</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprod</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="n">inf</span><span class="p">]);</span> <span class="n">sinh</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">1.838038955187488860347849</span>
-<span class="go">1.838038955187488860347849</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprod</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mi">1</span><span class="o">/</span><span class="n">n</span><span class="p">)</span><span class="o">**</span><span class="n">n</span> <span class="o">*</span> <span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">1.447255926890365298959138</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">euler</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">1.447255926890365298959138</span>
-</pre></div>
-</div>
-<p>The following two products are equivalent and can be evaluated in
-terms of a Jacobi theta function. Pi can be replaced by any value
-(as long as convergence is preserved):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprod</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">pi</span><span class="o">**-</span><span class="n">k</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">pi</span><span class="o">**-</span><span class="n">k</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">0.3838451207481672404778686</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprod</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="n">tanh</span><span class="p">(</span><span class="n">k</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">0.3838451207481672404778686</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jtheta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="o">/</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">0.3838451207481672404778686</span>
-</pre></div>
-</div>
-<p>This product does not have a known closed form value:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprod</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="o">**</span><span class="n">k</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">0.2887880950866024212788997</span>
-</pre></div>
-</div>
-<p>A product taken from <span class="math">\(-\infty\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprod</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="mi">1</span><span class="o">-</span><span class="n">k</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">])</span>
-<span class="go">0.8093965973662901095786805</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cosh</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">0.8093965973662901095786805</span>
-</pre></div>
-</div>
-<p>A doubly infinite product:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprod</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">k</span><span class="o">**</span><span class="mi">2</span><span class="p">)),</span> <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">23.41432688231864337420035</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="n">tanh</span><span class="p">(</span><span class="n">pi</span><span class="p">))</span>
-<span class="go">23.41432688231864337420035</span>
-</pre></div>
-</div>
-<p>A product requiring the use of Euler-Maclaurin summation to compute
-an accurate value:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprod</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="o">**</span><span class="mf">2.5</span><span class="p">),</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;e&#39;</span><span class="p">)</span>
-<span class="go">0.696155111336231052898125</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../../mpmath/references.html#weisstein">[Weisstein]</a> <a class="reference external" href="http://mathworld.wolfram.com/InfiniteProduct.html">http://mathworld.wolfram.com/InfiniteProduct.html</a></li>
-</ol>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="limits-limit">
-<h2>Limits (<tt class="docutils literal"><span class="pre">limit</span></tt>)<a class="headerlink" href="#limits-limit" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="limit">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">limit()</span></tt><a class="headerlink" href="#limit" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.limit">
-<tt class="descclassname">mpmath.</tt><tt class="descname">limit</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>x</em>, <em>direction=1</em>, <em>exp=False</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.limit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes an estimate of the limit</p>
-<div class="math">
-\[\lim_{t \to x} f(t)\]</div>
-<p>where <span class="math">\(x\)</span> may be finite or infinite.</p>
-<p>For finite <span class="math">\(x\)</span>, <a class="reference internal" href="#mpmath.limit" title="mpmath.limit"><tt class="xref py py-func docutils literal"><span class="pre">limit()</span></tt></a> evaluates <span class="math">\(f(x + d/n)\)</span> for
-consecutive integer values of <span class="math">\(n\)</span>, where the approach direction
-<span class="math">\(d\)</span> may be specified using the <em>direction</em> keyword argument.
-For infinite <span class="math">\(x\)</span>, <a class="reference internal" href="#mpmath.limit" title="mpmath.limit"><tt class="xref py py-func docutils literal"><span class="pre">limit()</span></tt></a> evaluates values of
-<span class="math">\(f(\mathrm{sign}(x) \cdot n)\)</span>.</p>
-<p>If the approach to the limit is not sufficiently fast to give
-an accurate estimate directly, <a class="reference internal" href="#mpmath.limit" title="mpmath.limit"><tt class="xref py py-func docutils literal"><span class="pre">limit()</span></tt></a> attempts to find
-the limit using Richardson extrapolation or the Shanks
-transformation. You can select between these methods using
-the <em>method</em> keyword (see documentation of <a class="reference internal" href="#mpmath.nsum" title="mpmath.nsum"><tt class="xref py py-func docutils literal"><span class="pre">nsum()</span></tt></a> for
-more information).</p>
-<p><strong>Options</strong></p>
-<p>The following options are available with essentially the
-same meaning as for <a class="reference internal" href="#mpmath.nsum" title="mpmath.nsum"><tt class="xref py py-func docutils literal"><span class="pre">nsum()</span></tt></a>: <em>tol</em>, <em>method</em>, <em>maxterms</em>,
-<em>steps</em>, <em>verbose</em>.</p>
-<p>If the option <em>exp=True</em> is set, <span class="math">\(f\)</span> will be
-sampled at exponentially spaced points <span class="math">\(n = 2^1, 2^2, 2^3, \ldots\)</span>
-instead of the linearly spaced points <span class="math">\(n = 1, 2, 3, \ldots\)</span>.
-This can sometimes improve the rate of convergence so that
-<a class="reference internal" href="#mpmath.limit" title="mpmath.limit"><tt class="xref py py-func docutils literal"><span class="pre">limit()</span></tt></a> may return a more accurate answer (and faster).
-However, do note that this can only be used if <span class="math">\(f\)</span>
-supports fast and accurate evaluation for arguments that
-are extremely close to the limit point (or if infinite,
-very large arguments).</p>
-<p><strong>Examples</strong></p>
-<p>A basic evaluation of a removable singularity:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">30</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">/</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">0.166666666666666666666666666667</span>
-</pre></div>
-</div>
-<p>Computing the exponential function using its limit definition:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mi">3</span><span class="o">/</span><span class="n">n</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="n">inf</span><span class="p">)</span>
-<span class="go">20.0855369231876677409285296546</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">20.0855369231876677409285296546</span>
-</pre></div>
-</div>
-<p>A limit for <span class="math">\(\pi\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="mi">2</span><span class="o">**</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">fac</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">**</span><span class="mi">4</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">inf</span><span class="p">)</span>
-<span class="go">3.14159265358979323846264338328</span>
-</pre></div>
-</div>
-<p>Calculating the coefficient in Stirling&#8217;s formula:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="n">fac</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="n">e</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="p">),</span> <span class="n">inf</span><span class="p">)</span>
-<span class="go">2.50662827463100050241576528481</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">2.50662827463100050241576528481</span>
-</pre></div>
-</div>
-<p>Evaluating Euler&#8217;s constant <span class="math">\(\gamma\)</span> using the limit representation</p>
-<div class="math">
-\[\gamma = \lim_{n \rightarrow \infty } \left[ \left(
-\sum_{k=1}^n \frac{1}{k} \right) - \log(n) \right]\]</div>
-<p>(which converges notoriously slowly):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="nb">sum</span><span class="p">([</span><span class="n">mpf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">k</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="nb">int</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">+</span><span class="mi">1</span><span class="p">)])</span> <span class="o">-</span> <span class="n">log</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">inf</span><span class="p">)</span>
-<span class="go">0.577215664901532860606512090082</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">+</span><span class="n">euler</span>
-<span class="go">0.577215664901532860606512090082</span>
-</pre></div>
-</div>
-<p>With default settings, the following limit converges too slowly
-to be evaluated accurately. Changing to exponential sampling
-however gives a perfect result:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="o">+</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span><span class="o">+</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">inf</span><span class="p">)</span>
-<span class="go">0.992831158558330281129249686491</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">inf</span><span class="p">,</span> <span class="n">exp</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">1.0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="extrapolation">
-<h2>Extrapolation<a class="headerlink" href="#extrapolation" title="Permalink to this headline">¶</a></h2>
-<p>The following functions provide a direct interface to
-extrapolation algorithms. <tt class="xref py py-func docutils literal"><span class="pre">nsum()</span></tt> and <tt class="xref py py-func docutils literal"><span class="pre">limit()</span></tt> essentially
-work by calling the following functions with an increasing
-number of terms until the extrapolated limit is accurate enough.</p>
-<p>The following functions may be useful to call directly if the
-precise number of terms needed to achieve a desired accuracy is
-known in advance, or if one wishes to study the convergence
-properties of the algorithms.</p>
-<div class="section" id="richardson">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">richardson()</span></tt><a class="headerlink" href="#richardson" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.richardson">
-<tt class="descclassname">mpmath.</tt><tt class="descname">richardson</tt><big>(</big><em>ctx</em>, <em>seq</em><big>)</big><a class="headerlink" href="#mpmath.richardson" title="Permalink to this definition">¶</a></dt>
-<dd><p>Given a list <tt class="docutils literal"><span class="pre">seq</span></tt> of the first <span class="math">\(N\)</span> elements of a slowly convergent
-infinite sequence, <a class="reference internal" href="#mpmath.richardson" title="mpmath.richardson"><tt class="xref py py-func docutils literal"><span class="pre">richardson()</span></tt></a> computes the <span class="math">\(N\)</span>-term
-Richardson extrapolate for the limit.</p>
-<p><a class="reference internal" href="#mpmath.richardson" title="mpmath.richardson"><tt class="xref py py-func docutils literal"><span class="pre">richardson()</span></tt></a> returns <span class="math">\((v, c)\)</span> where <span class="math">\(v\)</span> is the estimated
-limit and <span class="math">\(c\)</span> is the magnitude of the largest weight used during the
-computation. The weight provides an estimate of the precision
-lost to cancellation. Due to cancellation effects, the sequence must
-be typically be computed at a much higher precision than the target
-accuracy of the extrapolation.</p>
-<p><strong>Applicability and issues</strong></p>
-<p>The <span class="math">\(N\)</span>-step Richardson extrapolation algorithm used by
-<a class="reference internal" href="#mpmath.richardson" title="mpmath.richardson"><tt class="xref py py-func docutils literal"><span class="pre">richardson()</span></tt></a> is described in [1].</p>
-<p>Richardson extrapolation only works for a specific type of sequence,
-namely one converging like partial sums of
-<span class="math">\(P(1)/Q(1) + P(2)/Q(2) + \ldots\)</span> where <span class="math">\(P\)</span> and <span class="math">\(Q\)</span> are polynomials.
-When the sequence does not convergence at such a rate
-<a class="reference internal" href="#mpmath.richardson" title="mpmath.richardson"><tt class="xref py py-func docutils literal"><span class="pre">richardson()</span></tt></a> generally produces garbage.</p>
-<p>Richardson extrapolation has the advantage of being fast: the <span class="math">\(N\)</span>-term
-extrapolate requires only <span class="math">\(O(N)\)</span> arithmetic operations, and usually
-produces an estimate that is accurate to <span class="math">\(O(N)\)</span> digits. Contrast with
-the Shanks transformation (see <a class="reference internal" href="#mpmath.shanks" title="mpmath.shanks"><tt class="xref py py-func docutils literal"><span class="pre">shanks()</span></tt></a>), which requires
-<span class="math">\(O(N^2)\)</span> operations.</p>
-<p><a class="reference internal" href="#mpmath.richardson" title="mpmath.richardson"><tt class="xref py py-func docutils literal"><span class="pre">richardson()</span></tt></a> is unable to produce an estimate for the
-approximation error. One way to estimate the error is to perform
-two extrapolations with slightly different <span class="math">\(N\)</span> and comparing the
-results.</p>
-<p>Richardson extrapolation does not work for oscillating sequences.
-As a simple workaround, <a class="reference internal" href="#mpmath.richardson" title="mpmath.richardson"><tt class="xref py py-func docutils literal"><span class="pre">richardson()</span></tt></a> detects if the last
-three elements do not differ monotonically, and in that case
-applies extrapolation only to the even-index elements.</p>
-<p><strong>Examples</strong></p>
-<p>Applying Richardson extrapolation to the Leibniz series for <span class="math">\(\pi\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">30</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="p">[</span><span class="mi">4</span><span class="o">*</span><span class="nb">sum</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="p">))</span>
-<span class="gp">... </span>    <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">30</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">richardson</span><span class="p">(</span><span class="n">S</span><span class="p">[:</span><span class="mi">10</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v</span>
-<span class="go">3.2126984126984126984126984127</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">([</span><span class="n">v</span><span class="o">-</span><span class="n">pi</span><span class="p">,</span> <span class="n">c</span><span class="p">])</span>
-<span class="go">[0.0711058, 2.0]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">richardson</span><span class="p">(</span><span class="n">S</span><span class="p">[:</span><span class="mi">30</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v</span>
-<span class="go">3.14159265468624052829954206226</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">([</span><span class="n">v</span><span class="o">-</span><span class="n">pi</span><span class="p">,</span> <span class="n">c</span><span class="p">])</span>
-<span class="go">[1.09645e-9, 20833.3]</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../../mpmath/references.html#benderorszag">[BenderOrszag]</a> pp. 375-376</li>
-</ol>
-</dd></dl>
-
-</div>
-<div class="section" id="shanks">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">shanks()</span></tt><a class="headerlink" href="#shanks" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.shanks">
-<tt class="descclassname">mpmath.</tt><tt class="descname">shanks</tt><big>(</big><em>ctx</em>, <em>seq</em>, <em>table=None</em>, <em>randomized=False</em><big>)</big><a class="headerlink" href="#mpmath.shanks" title="Permalink to this definition">¶</a></dt>
-<dd><p>Given a list <tt class="docutils literal"><span class="pre">seq</span></tt> of the first <span class="math">\(N\)</span> elements of a slowly
-convergent infinite sequence <span class="math">\((A_k)\)</span>, <a class="reference internal" href="#mpmath.shanks" title="mpmath.shanks"><tt class="xref py py-func docutils literal"><span class="pre">shanks()</span></tt></a> computes the iterated
-Shanks transformation <span class="math">\(S(A), S(S(A)), \ldots, S^{N/2}(A)\)</span>. The Shanks
-transformation often provides strong convergence acceleration,
-especially if the sequence is oscillating.</p>
-<p>The iterated Shanks transformation is computed using the Wynn
-epsilon algorithm (see [1]). <a class="reference internal" href="#mpmath.shanks" title="mpmath.shanks"><tt class="xref py py-func docutils literal"><span class="pre">shanks()</span></tt></a> returns the full
-epsilon table generated by Wynn&#8217;s algorithm, which can be read
-off as follows:</p>
-<ul class="simple">
-<li>The table is a list of lists forming a lower triangular matrix,
-where higher row and column indices correspond to more accurate
-values.</li>
-<li>The columns with even index hold dummy entries (required for the
-computation) and the columns with odd index hold the actual
-extrapolates.</li>
-<li>The last element in the last row is typically the most
-accurate estimate of the limit.</li>
-<li>The difference to the third last element in the last row
-provides an estimate of the approximation error.</li>
-<li>The magnitude of the second last element provides an estimate
-of the numerical accuracy lost to cancellation.</li>
-</ul>
-<p>For convenience, so the extrapolation is stopped at an odd index
-so that <tt class="docutils literal"><span class="pre">shanks(seq)[-1][-1]</span></tt> always gives an estimate of the
-limit.</p>
-<p>Optionally, an existing table can be passed to <a class="reference internal" href="#mpmath.shanks" title="mpmath.shanks"><tt class="xref py py-func docutils literal"><span class="pre">shanks()</span></tt></a>.
-This can be used to efficiently extend a previous computation after
-new elements have been appended to the sequence. The table will
-then be updated in-place.</p>
-<p><strong>The Shanks transformation</strong></p>
-<p>The Shanks transformation is defined as follows (see [2]): given
-the input sequence <span class="math">\((A_0, A_1, \ldots)\)</span>, the transformed sequence is
-given by</p>
-<div class="math">
-\[S(A_k) = \frac{A_{k+1}A_{k-1}-A_k^2}{A_{k+1}+A_{k-1}-2 A_k}\]</div>
-<p>The Shanks transformation gives the exact limit <span class="math">\(A_{\infty}\)</span> in a
-single step if <span class="math">\(A_k = A + a q^k\)</span>. Note in particular that it
-extrapolates the exact sum of a geometric series in a single step.</p>
-<p>Applying the Shanks transformation once often improves convergence
-substantially for an arbitrary sequence, but the optimal effect is
-obtained by applying it iteratively:
-<span class="math">\(S(S(A_k)), S(S(S(A_k))), \ldots\)</span>.</p>
-<p>Wynn&#8217;s epsilon algorithm provides an efficient way to generate
-the table of iterated Shanks transformations. It reduces the
-computation of each element to essentially a single division, at
-the cost of requiring dummy elements in the table. See [1] for
-details.</p>
-<p><strong>Precision issues</strong></p>
-<p>Due to cancellation effects, the sequence must be typically be
-computed at a much higher precision than the target accuracy
-of the extrapolation.</p>
-<p>If the Shanks transformation converges to the exact limit (such
-as if the sequence is a geometric series), then a division by
-zero occurs. By default, <a class="reference internal" href="#mpmath.shanks" title="mpmath.shanks"><tt class="xref py py-func docutils literal"><span class="pre">shanks()</span></tt></a> handles this case by
-terminating the iteration and returning the table it has
-generated so far. With <em>randomized=True</em>, it will instead
-replace the zero by a pseudorandom number close to zero.
-(TODO: find a better solution to this problem.)</p>
-<p><strong>Examples</strong></p>
-<p>We illustrate by applying Shanks transformation to the Leibniz
-series for <span class="math">\(\pi\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="p">[</span><span class="mi">4</span><span class="o">*</span><span class="nb">sum</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="p">))</span>
-<span class="gp">... </span>    <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">30</span><span class="p">)]</span>
-<span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">T</span> <span class="o">=</span> <span class="n">shanks</span><span class="p">(</span><span class="n">S</span><span class="p">[:</span><span class="mi">7</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="n">T</span><span class="p">:</span>
-<span class="gp">... </span>    <span class="n">nprint</span><span class="p">(</span><span class="n">row</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="go">[-0.75]</span>
-<span class="go">[1.25, 3.16667]</span>
-<span class="go">[-1.75, 3.13333, -28.75]</span>
-<span class="go">[2.25, 3.14524, 82.25, 3.14234]</span>
-<span class="go">[-2.75, 3.13968, -177.75, 3.14139, -969.937]</span>
-<span class="go">[3.25, 3.14271, 327.25, 3.14166, 3515.06, 3.14161]</span>
-</pre></div>
-</div>
-<p>The extrapolated accuracy is about 4 digits, and about 4 digits
-may have been lost due to cancellation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">L</span> <span class="o">=</span> <span class="n">T</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">([</span><span class="nb">abs</span><span class="p">(</span><span class="n">L</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">pi</span><span class="p">),</span> <span class="nb">abs</span><span class="p">(</span><span class="n">L</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">L</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]),</span> <span class="nb">abs</span><span class="p">(</span><span class="n">L</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">])])</span>
-<span class="go">[2.22532e-5, 4.78309e-5, 3515.06]</span>
-</pre></div>
-</div>
-<p>Now we extend the computation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">T</span> <span class="o">=</span> <span class="n">shanks</span><span class="p">(</span><span class="n">S</span><span class="p">[:</span><span class="mi">25</span><span class="p">],</span> <span class="n">T</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span> <span class="o">=</span> <span class="n">T</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">([</span><span class="nb">abs</span><span class="p">(</span><span class="n">L</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">pi</span><span class="p">),</span> <span class="nb">abs</span><span class="p">(</span><span class="n">L</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">L</span><span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]),</span> <span class="nb">abs</span><span class="p">(</span><span class="n">L</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">])])</span>
-<span class="go">[3.75527e-19, 1.48478e-19, 2.96014e+17]</span>
-</pre></div>
-</div>
-<p>The value for pi is now accurate to 18 digits. About 18 digits may
-also have been lost to cancellation.</p>
-<p>Here is an example with a geometric series, where the convergence
-is immediate (the sum is exactly 1):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="n">shanks</span><span class="p">([</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.75</span><span class="p">,</span> <span class="mf">0.875</span><span class="p">,</span> <span class="mf">0.9375</span><span class="p">,</span> <span class="mf">0.96875</span><span class="p">]):</span>
-<span class="gp">... </span>    <span class="n">nprint</span><span class="p">(</span><span class="n">row</span><span class="p">)</span>
-<span class="go">[4.0]</span>
-<span class="go">[8.0, 1.0]</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../../mpmath/references.html#gravesmorris">[GravesMorris]</a></li>
-<li><a class="reference internal" href="../../mpmath/references.html#benderorszag">[BenderOrszag]</a> pp. 368-375</li>
-</ol>
-</dd></dl>
-
-</div>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
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-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Sums, products, limits and extrapolation</a><ul>
-<li><a class="reference internal" href="#summation">Summation</a><ul>
-<li><a class="reference internal" href="#nsum"><tt class="docutils literal"><span class="pre">nsum()</span></tt></a></li>
-<li><a class="reference internal" href="#sumem"><tt class="docutils literal"><span class="pre">sumem()</span></tt></a></li>
-<li><a class="reference internal" href="#sumap"><tt class="docutils literal"><span class="pre">sumap()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#products">Products</a><ul>
-<li><a class="reference internal" href="#nprod"><tt class="docutils literal"><span class="pre">nprod()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#limits-limit">Limits (<tt class="docutils literal"><span class="pre">limit</span></tt>)</a><ul>
-<li><a class="reference internal" href="#limit"><tt class="docutils literal"><span class="pre">limit()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#extrapolation">Extrapolation</a><ul>
-<li><a class="reference internal" href="#richardson"><tt class="docutils literal"><span class="pre">richardson()</span></tt></a></li>
-<li><a class="reference internal" href="#shanks"><tt class="docutils literal"><span class="pre">shanks()</span></tt></a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../calculus/optimization.html"
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-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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-            
-  <div class="section" id="contexts">
-<h1>Contexts<a class="headerlink" href="#contexts" title="Permalink to this headline">¶</a></h1>
-<p>High-level code in mpmath is implemented as methods on a &#8220;context object&#8221;. The context implements arithmetic, type conversions and other fundamental operations. The context also holds settings such as precision, and stores cache data. A few different contexts (with a mostly compatible interface) are provided so that the high-level algorithms can be used with different implementations of the underlying arithmetic, allowing different features and speed-accuracy tradeoffs. Currently, mpmath provides the following contexts:</p>
-<blockquote>
-<div><ul class="simple">
-<li>Arbitrary-precision arithmetic (<tt class="docutils literal"><span class="pre">mp</span></tt>)</li>
-<li>A faster Cython-based version of <tt class="docutils literal"><span class="pre">mp</span></tt> (used by default in Sage, and currently only available there)</li>
-<li>Arbitrary-precision interval arithmetic (<tt class="docutils literal"><span class="pre">iv</span></tt>)</li>
-<li>Double-precision arithmetic using Python&#8217;s builtin <tt class="docutils literal"><span class="pre">float</span></tt> and <tt class="docutils literal"><span class="pre">complex</span></tt> types (<tt class="docutils literal"><span class="pre">fp</span></tt>)</li>
-</ul>
-</div></blockquote>
-<p>Most global functions in the global mpmath namespace are actually methods of the <tt class="docutils literal"><span class="pre">mp</span></tt>
-context. This fact is usually transparent to the user, but sometimes shows up in the
-form of an initial parameter called &#8220;ctx&#8221; visible in the help for the function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">mpmath</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">help</span><span class="p">(</span><span class="n">mpmath</span><span class="o">.</span><span class="n">fsum</span><span class="p">)</span>   
-<span class="go">Help on method fsum in module mpmath.ctx_mp_python:</span>
-
-<span class="go">fsum(ctx, terms, absolute=False, squared=False) method of mpmath.ctx_mp.MPContext instance</span>
-<span class="go">    Calculates a sum containing a finite number of terms (for infinite</span>
-<span class="go">    series, see :func:`~mpmath.nsum`). The terms will be converted to</span>
-<span class="gp">...</span>
-</pre></div>
-</div>
-<p>The following operations are equivalent:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpmath</span><span class="o">.</span><span class="n">fsum</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-<span class="go">mpf(&#39;6.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">fsum</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-<span class="go">mpf(&#39;6.0&#39;)</span>
-</pre></div>
-</div>
-<p>The corresponding operation using the <tt class="docutils literal"><span class="pre">fp</span></tt> context:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mpmath</span><span class="o">.</span><span class="n">fp</span><span class="o">.</span><span class="n">fsum</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-<span class="go">6.0</span>
-</pre></div>
-</div>
-<div class="section" id="common-interface">
-<h2>Common interface<a class="headerlink" href="#common-interface" title="Permalink to this headline">¶</a></h2>
-<p><tt class="docutils literal"><span class="pre">ctx.mpf</span></tt> creates a real number:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="n">mp</span><span class="p">,</span> <span class="n">fp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">mpf(&#39;3.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fp</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">3.0</span>
-</pre></div>
-</div>
-<p><tt class="docutils literal"><span class="pre">ctx.mpc</span></tt> creates a complex number:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">mpc</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">mpc(real=&#39;2.0&#39;, imag=&#39;3.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fp</span><span class="o">.</span><span class="n">mpc</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">(2+3j)</span>
-</pre></div>
-</div>
-<p><tt class="docutils literal"><span class="pre">ctx.matrix</span></tt> creates a matrix:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]])</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;1.0&#39;, &#39;0.0&#39;],</span>
-<span class="go"> [&#39;0.0&#39;, &#39;1.0&#39;]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">mpf(&#39;1.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fp</span><span class="o">.</span><span class="n">matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]])</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;1.0&#39;, &#39;0.0&#39;],</span>
-<span class="go"> [&#39;0.0&#39;, &#39;1.0&#39;]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">1.0</span>
-</pre></div>
-</div>
-<p><tt class="docutils literal"><span class="pre">ctx.prec</span></tt> holds the current precision (in bits):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">prec</span>
-<span class="go">53</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fp</span><span class="o">.</span><span class="n">prec</span>
-<span class="go">53</span>
-</pre></div>
-</div>
-<p><tt class="docutils literal"><span class="pre">ctx.dps</span></tt> holds the current precision (in digits):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span>
-<span class="go">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fp</span><span class="o">.</span><span class="n">dps</span>
-<span class="go">15</span>
-</pre></div>
-</div>
-<p><tt class="docutils literal"><span class="pre">ctx.pretty</span></tt> controls whether objects should be pretty-printed automatically by <tt class="xref py py-func docutils literal"><span class="pre">repr()</span></tt>. Pretty-printing for <tt class="docutils literal"><span class="pre">mp</span></tt> numbers is disabled by default so that they can clearly be distinguished from Python numbers and so that <tt class="docutils literal"><span class="pre">eval(repr(x))</span> <span class="pre">==</span> <span class="pre">x</span></tt> works:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">mpf(&#39;3.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpf</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">mpf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">eval</span><span class="p">(</span><span class="nb">repr</span><span class="p">(</span><span class="n">mp</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span><span class="mi">3</span><span class="p">)))</span>
-<span class="go">mpf(&#39;3.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">3.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fp</span><span class="o">.</span><span class="n">matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]])</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;1.0&#39;, &#39;0.0&#39;],</span>
-<span class="go"> [&#39;0.0&#39;, &#39;1.0&#39;]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fp</span><span class="o">.</span><span class="n">matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]])</span>
-<span class="go">[1.0  0.0]</span>
-<span class="go">[0.0  1.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="arbitrary-precision-floating-point-mp">
-<h2>Arbitrary-precision floating-point (<tt class="docutils literal"><span class="pre">mp</span></tt>)<a class="headerlink" href="#arbitrary-precision-floating-point-mp" title="Permalink to this headline">¶</a></h2>
-<p>The <tt class="docutils literal"><span class="pre">mp</span></tt> context is what most users probably want to use most of the time, as it supports the most functions, is most well-tested, and is implemented with a high level of optimization. Nearly all examples in this documentation use <tt class="docutils literal"><span class="pre">mp</span></tt> functions.</p>
-<p>See <a class="reference internal" href="../mpmath/basics.html"><em>Basic usage</em></a> for a description of basic usage.</p>
-</div>
-<div class="section" id="arbitrary-precision-interval-arithmetic-iv">
-<h2>Arbitrary-precision interval arithmetic (<tt class="docutils literal"><span class="pre">iv</span></tt>)<a class="headerlink" href="#arbitrary-precision-interval-arithmetic-iv" title="Permalink to this headline">¶</a></h2>
-<p>The <tt class="docutils literal"><span class="pre">iv.mpf</span></tt> type represents a closed interval <span class="math">\([a,b]\)</span>; that is, the set <span class="math">\(\{x : a \le x \le b\}\)</span>, where <span class="math">\(a\)</span> and <span class="math">\(b\)</span> are arbitrary-precision floating-point values, possibly <span class="math">\(\pm \infty\)</span>. The <tt class="docutils literal"><span class="pre">iv.mpc</span></tt> type represents a rectangular complex interval <span class="math">\([a,b] + [c,d]i\)</span>; that is, the set <span class="math">\(\{z = x+iy : a \le x \le b \land c \le y \le d\}\)</span>.</p>
-<p>Interval arithmetic provides rigorous error tracking. If <span class="math">\(f\)</span> is a mathematical function and <span class="math">\(\hat f\)</span> is its interval arithmetic version, then the basic guarantee of interval arithmetic is that <span class="math">\(f(v) \subseteq \hat f(v)\)</span> for any input interval <span class="math">\(v\)</span>. Put differently, if an interval represents the known uncertainty for a fixed number, any sequence of interval operations will produce an interval that contains what would be the result of applying the same sequence of operations to the exact number. The principal drawbacks of interval arithmetic are speed (<tt class="docutils literal"><span class="pre">iv</span></tt> arithmetic is typically at least two times slower than <tt class="docutils literal"><span class="pre">mp</span></tt> arithmetic) and that it sometimes provides far too pessimistic bounds.</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">The support for interval arithmetic in mpmath is still experimental, and many functions
-do not yet properly support intervals. Please use this feature with caution.</p>
-</div>
-<p>Intervals can be created from single numbers (treated as zero-width intervals) or pairs of endpoint numbers. Strings are treated as exact decimal numbers. Note that a Python float like <tt class="docutils literal"><span class="pre">0.1</span></tt> generally does not represent the same number as its literal; use <tt class="docutils literal"><span class="pre">'0.1'</span></tt> instead:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="n">iv</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">iv</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">mpi(&#39;3.0&#39;, &#39;3.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">[3.0, 3.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-<span class="go">[2.0, 3.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span><span class="mf">0.1</span><span class="p">)</span>   <span class="c"># probably not intended</span>
-<span class="go">[0.10000000000000000555, 0.10000000000000000555]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span><span class="s">&#39;0.1&#39;</span><span class="p">)</span>   <span class="c"># good, gives a containing interval</span>
-<span class="go">[0.099999999999999991673, 0.10000000000000000555]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="s">&#39;0.1&#39;</span><span class="p">,</span> <span class="s">&#39;0.2&#39;</span><span class="p">])</span>
-<span class="go">[0.099999999999999991673, 0.2000000000000000111]</span>
-</pre></div>
-</div>
-<p>The fact that <tt class="docutils literal"><span class="pre">'0.1'</span></tt> results in an interval of nonzero width indicates that 1/10 cannot be represented using binary floating-point numbers at this precision level (in fact, it cannot be represented exactly at any precision).</p>
-<p>Intervals may be infinite or half-infinite:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="s">&#39;inf&#39;</span><span class="p">])</span>
-<span class="go">[0.0, 0.5]</span>
-</pre></div>
-</div>
-<p>The equality testing operators <tt class="docutils literal"><span class="pre">==</span></tt> and <tt class="docutils literal"><span class="pre">!=</span></tt> check whether their operands are identical as intervals; that is, have the same endpoints. The ordering operators <tt class="docutils literal"><span class="pre">&lt;</span> <span class="pre">&lt;=</span> <span class="pre">&gt;</span> <span class="pre">&gt;=</span></tt> permit inequality testing using triple-valued logic: a guaranteed inequality returns <tt class="docutils literal"><span class="pre">True</span></tt> or <tt class="docutils literal"><span class="pre">False</span></tt> while an indeterminate inequality returns <tt class="docutils literal"><span class="pre">None</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span> <span class="o">==</span> <span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span> <span class="o">!=</span> <span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span> <span class="o">&lt;=</span> <span class="mi">2</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span> <span class="o">&gt;</span> <span class="mi">0</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span> <span class="o">&lt;</span> <span class="mi">1</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span> <span class="o">&lt;</span> <span class="mi">2</span>    <span class="c"># returns None</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span> <span class="o">&lt;</span> <span class="mi">2</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span> <span class="o">&lt;=</span> <span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span> <span class="o">&lt;</span> <span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>  <span class="c"># returns None</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span> <span class="o">&lt;</span> <span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>The <tt class="docutils literal"><span class="pre">in</span></tt> operator tests whether a number or interval is contained in another interval:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span> <span class="ow">in</span> <span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">10</span><span class="p">])</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">3</span> <span class="ow">in</span> <span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="s">&#39;-inf&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>Intervals have the properties <tt class="docutils literal"><span class="pre">.a</span></tt>, <tt class="docutils literal"><span class="pre">.b</span></tt> (endpoints), <tt class="docutils literal"><span class="pre">.mid</span></tt>, and <tt class="docutils literal"><span class="pre">.delta</span></tt> (width):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">a</span>
-<span class="go">[2.0, 2.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">b</span>
-<span class="go">[5.0, 5.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">mid</span>
-<span class="go">[3.5, 3.5]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">delta</span>
-<span class="go">[3.0, 3.0]</span>
-</pre></div>
-</div>
-<p>Some transcendental functions are supported:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">1.5</span><span class="p">])</span> <span class="o">**</span> <span class="n">iv</span><span class="o">.</span><span class="n">mpf</span><span class="p">([</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">])</span>
-<span class="go">[0.35355339059327373086, 1.837117307087383633]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">[1.0, 1.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">exp</span><span class="p">([</span><span class="s">&#39;-inf&#39;</span><span class="p">,</span><span class="s">&#39;inf&#39;</span><span class="p">])</span>
-<span class="go">[0.0, +inf]</span>
-<span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">exp</span><span class="p">([</span><span class="s">&#39;-inf&#39;</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span>
-<span class="go">[0.0, 1.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">exp</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="s">&#39;inf&#39;</span><span class="p">])</span>
-<span class="go">[1.0, +inf]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">exp</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">[1.0, 2.7182818284590455349]</span>
-<span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">[0.0, 0.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">log</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">[-inf, 0.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">log</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="s">&#39;inf&#39;</span><span class="p">])</span>
-<span class="go">[-inf, +inf]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">[0.69314718055994528623, 0.69314718055994539725]</span>
-<span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">sin</span><span class="p">([</span><span class="mi">100</span><span class="p">,</span><span class="s">&#39;inf&#39;</span><span class="p">])</span>
-<span class="go">[-1.0, 1.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">cos</span><span class="p">([</span><span class="s">&#39;-0.1&#39;</span><span class="p">,</span><span class="s">&#39;0.1&#39;</span><span class="p">])</span>
-<span class="go">[0.99500416527802570954, 1.0]</span>
-</pre></div>
-</div>
-<p>Interval arithmetic is useful for proving inequalities involving irrational numbers.
-Naive use of <tt class="docutils literal"><span class="pre">mp</span></tt> arithmetic may result in wrong conclusions, such as the following:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">mp</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">mp</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">163</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">mp</span><span class="o">.</span><span class="n">mpf</span><span class="p">(</span><span class="mi">640320</span><span class="o">**</span><span class="mi">3</span><span class="o">+</span><span class="mi">744</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">x</span>
-<span class="go">262537412640768744.0000001</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">y</span>
-<span class="go">262537412640768744.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">&gt;</span> <span class="n">y</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>But the correct result is <span class="math">\(e^{\pi \sqrt{163}} &lt; 262537412640768744\)</span>, as can be
-seen by increasing the precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">mp</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">mp</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">mp</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">163</span><span class="p">))</span>
-<span class="go">262537412640768743.99999999999925007259719818568888</span>
-</pre></div>
-</div>
-<p>With interval arithmetic, the comparison returns <tt class="docutils literal"><span class="pre">None</span></tt> until the precision
-is large enough for <span class="math">\(x-y\)</span> to have a definite sign:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">iv</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">iv</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">163</span><span class="p">))</span> <span class="o">&gt;</span> <span class="p">(</span><span class="mi">640320</span><span class="o">**</span><span class="mi">3</span><span class="o">+</span><span class="mi">744</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">30</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">iv</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">iv</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">163</span><span class="p">))</span> <span class="o">&gt;</span> <span class="p">(</span><span class="mi">640320</span><span class="o">**</span><span class="mi">3</span><span class="o">+</span><span class="mi">744</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">60</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">iv</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">iv</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">163</span><span class="p">))</span> <span class="o">&gt;</span> <span class="p">(</span><span class="mi">640320</span><span class="o">**</span><span class="mi">3</span><span class="o">+</span><span class="mi">744</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="fast-low-precision-arithmetic-fp">
-<h2>Fast low-precision arithmetic (<tt class="docutils literal"><span class="pre">fp</span></tt>)<a class="headerlink" href="#fast-low-precision-arithmetic-fp" title="Permalink to this headline">¶</a></h2>
-<p>Although mpmath is generally designed for arbitrary-precision arithmetic, many of the high-level algorithms work perfectly well with ordinary Python <tt class="docutils literal"><span class="pre">float</span></tt> and <tt class="docutils literal"><span class="pre">complex</span></tt> numbers, which use hardware double precision (on most systems, this corresponds to 53 bits of precision). Whereas the global functions (which are methods of the <tt class="docutils literal"><span class="pre">mp</span></tt> object) always convert inputs to mpmath numbers, the <tt class="docutils literal"><span class="pre">fp</span></tt> object instead converts them to <tt class="docutils literal"><span class="pre">float</span></tt> or <tt class="docutils literal"><span class="pre">complex</span></tt>, and in some cases employs basic functions optimized for double precision. When large amounts of function evaluations (numerical integration, plotting, etc) are required, and when <tt class="docutils literal"><span class="pre">fp</span></tt> arithmetic provides sufficient accuracy, this can give a significant speedup over <tt class="docutils literal"><span class="pre">mp</span></tt> arithmetic.</p>
-<p>To take advantage of this feature, simply use the <tt class="docutils literal"><span class="pre">fp</span></tt> prefix, i.e. write <tt class="docutils literal"><span class="pre">fp.func</span></tt> instead of <tt class="docutils literal"><span class="pre">func</span></tt> or <tt class="docutils literal"><span class="pre">mp.func</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">u</span> <span class="o">=</span> <span class="n">fp</span><span class="o">.</span><span class="n">erfc</span><span class="p">(</span><span class="mf">2.5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">u</span>
-<span class="go">0.000406952017445</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
-<span class="go">&lt;type &#39;float&#39;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">mp</span><span class="o">.</span><span class="n">erfc</span><span class="p">(</span><span class="mf">2.5</span><span class="p">)</span>
-<span class="go">0.000406952017444959</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fp</span><span class="o">.</span><span class="n">matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">],[</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">]])</span> <span class="o">**</span> <span class="mi">2</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;7.0&#39;, &#39;10.0&#39;],</span>
-<span class="go"> [&#39;15.0&#39;, &#39;22.0&#39;]])</span>
-<span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="n">_</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span>
-<span class="go">&lt;type &#39;float&#39;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">fp</span><span class="o">.</span><span class="n">quad</span><span class="p">(</span><span class="n">fp</span><span class="o">.</span><span class="n">sin</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">fp</span><span class="o">.</span><span class="n">pi</span><span class="p">])</span>    <span class="c"># numerical integration</span>
-<span class="go">2.0</span>
-</pre></div>
-</div>
-<p>The <tt class="docutils literal"><span class="pre">fp</span></tt> context wraps Python&#8217;s <tt class="docutils literal"><span class="pre">math</span></tt> and <tt class="docutils literal"><span class="pre">cmath</span></tt> modules for elementary functions. It supports both real and complex numbers and automatically generates complex results for real inputs (<tt class="docutils literal"><span class="pre">math</span></tt> raises an exception):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fp</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">2.2360679774997898</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fp</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">2.2360679774997898j</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">-0.54402111088936977</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fp</span><span class="o">.</span><span class="n">power</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">)</span>
-<span class="go">(0.70710678118654757+0.70710678118654746j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">**</span> <span class="mf">0.25</span>
-<span class="gt">Traceback (most recent call last):</span>
-  <span class="c">...</span>
-<span class="gr">ValueError</span>: <span class="n">negative number cannot be raised to a fractional power</span>
-</pre></div>
-</div>
-<p>The <tt class="docutils literal"><span class="pre">prec</span></tt> and <tt class="docutils literal"><span class="pre">dps</span></tt> attributes can be changed (for interface compatibility with the <tt class="docutils literal"><span class="pre">mp</span></tt> context) but this has no effect:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fp</span><span class="o">.</span><span class="n">prec</span>
-<span class="go">53</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fp</span><span class="o">.</span><span class="n">dps</span>
-<span class="go">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fp</span><span class="o">.</span><span class="n">prec</span> <span class="o">=</span> <span class="mi">80</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fp</span><span class="o">.</span><span class="n">prec</span>
-<span class="go">53</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fp</span><span class="o">.</span><span class="n">dps</span>
-<span class="go">15</span>
-</pre></div>
-</div>
-<p>Due to intermediate rounding and cancellation errors, results computed with <tt class="docutils literal"><span class="pre">fp</span></tt> arithmetic may be much less accurate than those computed with <tt class="docutils literal"><span class="pre">mp</span></tt> using an equivalent precision (<tt class="docutils literal"><span class="pre">mp.prec</span> <span class="pre">=</span> <span class="pre">53</span></tt>), since the latter often uses increased internal precision. The accuracy is highly problem-dependent: for some functions, <tt class="docutils literal"><span class="pre">fp</span></tt> almost always gives 14-15 correct digits; for others, results can be accurate to only 2-3 digits or even completely wrong. The recommended use for <tt class="docutils literal"><span class="pre">fp</span></tt> is therefore to speed up large-scale computations where accuracy can be verified in advance on a subset of the input set, or where results can be verified afterwards.</p>
-</div>
-</div>
-
-
-          </div>
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-            <p class="logo"><a href="../../index.html">
-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Contexts</a><ul>
-<li><a class="reference internal" href="#common-interface">Common interface</a></li>
-<li><a class="reference internal" href="#arbitrary-precision-floating-point-mp">Arbitrary-precision floating-point (<tt class="docutils literal"><span class="pre">mp</span></tt>)</a></li>
-<li><a class="reference internal" href="#arbitrary-precision-interval-arithmetic-iv">Arbitrary-precision interval arithmetic (<tt class="docutils literal"><span class="pre">iv</span></tt>)</a></li>
-<li><a class="reference internal" href="#fast-low-precision-arithmetic-fp">Fast low-precision arithmetic (<tt class="docutils literal"><span class="pre">fp</span></tt>)</a></li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
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diff --git a/dev-py3k/modules/mpmath/functions/bessel.html b/dev-py3k/modules/mpmath/functions/bessel.html
deleted file mode 100644
index 60999249f30..00000000000
--- a/dev-py3k/modules/mpmath/functions/bessel.html
+++ /dev/null
@@ -1,2607 +0,0 @@
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-  <div class="section" id="bessel-functions-and-related-functions">
-<h1>Bessel functions and related functions<a class="headerlink" href="#bessel-functions-and-related-functions" title="Permalink to this headline">¶</a></h1>
-<p>The functions in this section arise as solutions to various differential equations in physics, typically describing wavelike oscillatory behavior or a combination of oscillation and exponential decay or growth. Mathematically, they are special cases of the confluent hypergeometric functions <span class="math">\(\,_0F_1\)</span>, <span class="math">\(\,_1F_1\)</span> and <span class="math">\(\,_1F_2\)</span> (see <a class="reference internal" href="../functions/hypergeometric.html"><em>Hypergeometric functions</em></a>).</p>
-<div class="section" id="bessel-functions">
-<h2>Bessel functions<a class="headerlink" href="#bessel-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="besselj">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">besselj()</span></tt><a class="headerlink" href="#besselj" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.besselj">
-<tt class="descclassname">mpmath.</tt><tt class="descname">besselj</tt><big>(</big><em>n</em>, <em>x</em>, <em>derivative=0</em><big>)</big><a class="headerlink" href="#mpmath.besselj" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">besselj(n,</span> <span class="pre">x,</span> <span class="pre">derivative=0)</span></tt> gives the Bessel function of the first kind
-<span class="math">\(J_n(x)\)</span>. Bessel functions of the first kind are defined as
-solutions of the differential equation</p>
-<div class="math">
-\[x^2 y'' + x y' + (x^2 - n^2) y = 0\]</div>
-<p>which appears, among other things, when solving the radial
-part of Laplace&#8217;s equation in cylindrical coordinates. This
-equation has two solutions for given <span class="math">\(n\)</span>, where the
-<span class="math">\(J_n\)</span>-function is the solution that is nonsingular at <span class="math">\(x = 0\)</span>.
-For positive integer <span class="math">\(n\)</span>, <span class="math">\(J_n(x)\)</span> behaves roughly like a sine
-(odd <span class="math">\(n\)</span>) or cosine (even <span class="math">\(n\)</span>) multiplied by a magnitude factor
-that decays slowly as <span class="math">\(x \to \pm\infty\)</span>.</p>
-<p>Generally, <span class="math">\(J_n\)</span> is a special case of the hypergeometric
-function <span class="math">\(\,_0F_1\)</span>:</p>
-<div class="math">
-\[J_n(x) = \frac{x^n}{2^n \Gamma(n+1)}
-         \,_0F_1\left(n+1,-\frac{x^2}{4}\right)\]</div>
-<p>With <em>derivative</em> = <span class="math">\(m \ne 0\)</span>, the <span class="math">\(m\)</span>-th derivative</p>
-<div class="math">
-\[\frac{d^m}{dx^m} J_n(x)\]</div>
-<p>is computed.</p>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Bessel function J_n(x) on the real line for n=0,1,2,3</span>
-<span class="n">j0</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">besselj</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">j1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">besselj</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">j2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">besselj</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">j3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">besselj</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">j0</span><span class="p">,</span><span class="n">j1</span><span class="p">,</span><span class="n">j2</span><span class="p">,</span><span class="n">j3</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">14</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/besselj.png" src="../../../_images/besselj.png" />
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Bessel function J_n(z) in the complex plane</span>
-<span class="n">cplot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">besselj</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">z</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span> <span class="n">points</span><span class="o">=</span><span class="mi">50000</span><span class="p">)</span>
-</pre></div>
-</div>
-<img alt="../../../_images/besselj_c.png" src="../../../_images/besselj_c.png" />
-<p><strong>Examples</strong></p>
-<p>Evaluation is supported for arbitrary arguments, and at
-arbitrary precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselj</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
-<span class="go">-0.024777229528606</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselj</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mf">0.75</span><span class="p">)</span>
-<span class="go">0.000801070086542314</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselj</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1000j</span><span class="p">)</span>
-<span class="go">(-2.48071721019185e+432 + 6.41567059811949e-437j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselj</span><span class="p">(</span><span class="mf">0.75j</span><span class="p">,</span> <span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(-2.778118364828153309919653 - 1.5863603889018621585533j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselj</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">pi</span><span class="p">)</span>
-<span class="go">0.28461534317975275734531059968613140570981118184947</span>
-</pre></div>
-</div>
-<p>Arguments may be large:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselj</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10000</span><span class="p">)</span>
-<span class="go">-0.007096160353388801477265164</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselj</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">0.000002175591750246891726859055</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselj</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">10</span><span class="o">**</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">7.337048736538615712436929e-51</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselj</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">10</span><span class="o">**</span><span class="mi">5</span><span class="o">*</span><span class="n">j</span><span class="p">)</span>
-<span class="go">(-3.540725411970948860173735e+43426 + 4.4949812409615803110051e-43433j)</span>
-</pre></div>
-</div>
-<p>The Bessel functions of the first kind satisfy simple
-symmetries around <span class="math">\(x = 0\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">([</span><span class="n">besselj</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">)])</span>
-<span class="go">[1.0, 0.0, 0.0, 0.0, 0.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">([</span><span class="n">besselj</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">pi</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">)])</span>
-<span class="go">[-0.304242, 0.284615, 0.485434, 0.333458, 0.151425]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">([</span><span class="n">besselj</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="o">-</span><span class="n">pi</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">)])</span>
-<span class="go">[-0.304242, -0.284615, 0.485434, -0.333458, 0.151425]</span>
-</pre></div>
-</div>
-<p>Roots of Bessel functions are often used:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">([</span><span class="n">findroot</span><span class="p">(</span><span class="n">j0</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">14</span><span class="p">]])</span>
-<span class="go">[2.40483, 5.52008, 8.65373, 11.7915, 14.9309]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">([</span><span class="n">findroot</span><span class="p">(</span><span class="n">j1</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">13</span><span class="p">,</span> <span class="mi">16</span><span class="p">]])</span>
-<span class="go">[3.83171, 7.01559, 10.1735, 13.3237, 16.4706]</span>
-</pre></div>
-</div>
-<p>The roots are not periodic, but the distance between successive
-roots asymptotically approaches <span class="math">\(2 \pi\)</span>. Bessel functions of
-the first kind have the following normalization:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">quadosc</span><span class="p">(</span><span class="n">j0</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span> <span class="n">period</span><span class="o">=</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quadosc</span><span class="p">(</span><span class="n">j1</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span> <span class="n">period</span><span class="o">=</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">1.0</span>
-</pre></div>
-</div>
-<p>For <span class="math">\(n = 1/2\)</span> or <span class="math">\(n = -1/2\)</span>, the Bessel function reduces to a
-trigonometric function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="mi">10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselj</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">))</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(-0.13726373575505, -0.13726373575505)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselj</span><span class="p">(</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">))</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(-0.211708866331398, -0.211708866331398)</span>
-</pre></div>
-</div>
-<p>Derivatives of any order can be computed (negative orders
-correspond to integration):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselj</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">7.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">-0.1352484275797055051822405</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">besselj</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="mf">7.5</span><span class="p">)</span>
-<span class="go">-0.1352484275797055051822405</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselj</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">7.5</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">-0.1377811164763244890135677</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">besselj</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="mf">7.5</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">-0.1377811164763244890135677</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselj</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mf">7.5</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">besselj</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mf">3.5</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-0.1241343240399987693521378</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">j0</span><span class="p">,</span> <span class="p">[</span><span class="mf">3.5</span><span class="p">,</span> <span class="mf">7.5</span><span class="p">])</span>
-<span class="go">-0.1241343240399987693521378</span>
-</pre></div>
-</div>
-<p>Differentiation with a noninteger order gives the fractional derivative
-in the sense of the Riemann-Liouville differintegral, as computed by
-<a class="reference internal" href="../../mpmath/calculus/differentiation.html#mpmath.differint" title="mpmath.differint"><tt class="xref py py-func docutils literal"><span class="pre">differint()</span></tt></a>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselj</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">,</span> <span class="mf">0.75</span><span class="p">)</span>
-<span class="go">-0.385977722939384</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">differint</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">besselj</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="mf">3.5</span><span class="p">,</span> <span class="mf">0.75</span><span class="p">)</span>
-<span class="go">-0.385977722939384</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="mpmath.j0">
-<tt class="descclassname">mpmath.</tt><tt class="descname">j0</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.j0" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the Bessel function <span class="math">\(J_0(x)\)</span>. See <a class="reference internal" href="#mpmath.besselj" title="mpmath.besselj"><tt class="xref py py-func docutils literal"><span class="pre">besselj()</span></tt></a>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="mpmath.j1">
-<tt class="descclassname">mpmath.</tt><tt class="descname">j1</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.j1" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the Bessel function <span class="math">\(J_1(x)\)</span>.  See <a class="reference internal" href="#mpmath.besselj" title="mpmath.besselj"><tt class="xref py py-func docutils literal"><span class="pre">besselj()</span></tt></a>.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="bessely">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">bessely()</span></tt><a class="headerlink" href="#bessely" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.bessely">
-<tt class="descclassname">mpmath.</tt><tt class="descname">bessely</tt><big>(</big><em>n</em>, <em>x</em>, <em>derivative=0</em><big>)</big><a class="headerlink" href="#mpmath.bessely" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">bessely(n,</span> <span class="pre">x,</span> <span class="pre">derivative=0)</span></tt> gives the Bessel function of the second kind,</p>
-<div class="math">
-\[Y_n(x) = \frac{J_n(x) \cos(\pi n) - J_{-n}(x)}{\sin(\pi n)}.\]</div>
-<p>For <span class="math">\(n\)</span> an integer, this formula should be understood as a
-limit. With <em>derivative</em> = <span class="math">\(m \ne 0\)</span>, the <span class="math">\(m\)</span>-th derivative</p>
-<div class="math">
-\[\frac{d^m}{dx^m} Y_n(x)\]</div>
-<p>is computed.</p>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Bessel function of 2nd kind Y_n(x) on the real line for n=0,1,2,3</span>
-<span class="n">y0</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">bessely</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">y1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">bessely</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">y2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">bessely</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">y3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">bessely</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">y0</span><span class="p">,</span><span class="n">y1</span><span class="p">,</span><span class="n">y2</span><span class="p">,</span><span class="n">y3</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">10</span><span class="p">],[</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/bessely.png" src="../../../_images/bessely.png" />
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Bessel function of 2nd kind Y_n(z) in the complex plane</span>
-<span class="n">cplot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">bessely</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">z</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span> <span class="n">points</span><span class="o">=</span><span class="mi">50000</span><span class="p">)</span>
-</pre></div>
-</div>
-<img alt="../../../_images/bessely_c.png" src="../../../_images/bessely_c.png" />
-<p><strong>Examples</strong></p>
-<p>Some values of <span class="math">\(Y_n(x)\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bessely</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> <span class="n">bessely</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> <span class="n">bessely</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">(-inf, -inf, -inf)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bessely</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">pi</span><span class="p">)</span>
-<span class="go">0.3588729167767189594679827</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bessely</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(9.242861436961450520325216 - 3.085042824915332562522402j)</span>
-</pre></div>
-</div>
-<p>Arguments may be large:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">bessely</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10000</span><span class="p">)</span>
-<span class="go">0.00364780555898660588668872</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bessely</span><span class="p">(</span><span class="mf">2.5</span><span class="p">,</span> <span class="mi">10</span><span class="o">**</span><span class="mi">50</span><span class="p">)</span>
-<span class="go">-4.8952500412050989295774e-26</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bessely</span><span class="p">(</span><span class="mf">2.5</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="o">**</span><span class="mi">50</span><span class="p">)</span>
-<span class="go">(0.0 + 4.8952500412050989295774e-26j)</span>
-</pre></div>
-</div>
-<p>Derivatives and antiderivatives of any order can be computed:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">bessely</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">0.3842618820422660066089231</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">bessely</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="mf">3.5</span><span class="p">)</span>
-<span class="go">0.3842618820422660066089231</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bessely</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">-0.2066598304156764337900417</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">bessely</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="mf">3.5</span><span class="p">)</span>
-<span class="go">-0.2066598304156764337900417</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">bessely</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">-208173867409.5547350101511</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bessely</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">-208173867409.5547350101511</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bessely</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">100.5</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
-<span class="go">0.02668487547301372334849043</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">bessely</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-<span class="go">-1.377046859093181969213262</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bessely</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">bessely</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-1.377046859093181969213262</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="besseli">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">besseli()</span></tt><a class="headerlink" href="#besseli" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.besseli">
-<tt class="descclassname">mpmath.</tt><tt class="descname">besseli</tt><big>(</big><em>n</em>, <em>x</em>, <em>derivative=0</em><big>)</big><a class="headerlink" href="#mpmath.besseli" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">besseli(n,</span> <span class="pre">x,</span> <span class="pre">derivative=0)</span></tt> gives the modified Bessel function of the
-first kind,</p>
-<div class="math">
-\[I_n(x) = i^{-n} J_n(ix).\]</div>
-<p>With <em>derivative</em> = <span class="math">\(m \ne 0\)</span>, the <span class="math">\(m\)</span>-th derivative</p>
-<div class="math">
-\[\frac{d^m}{dx^m} I_n(x)\]</div>
-<p>is computed.</p>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Modified Bessel function I_n(x) on the real line for n=0,1,2,3</span>
-<span class="n">i0</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">besseli</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">i1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">besseli</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">i2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">besseli</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">i3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">besseli</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">i0</span><span class="p">,</span><span class="n">i1</span><span class="p">,</span><span class="n">i2</span><span class="p">,</span><span class="n">i3</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">5</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/besseli.png" src="../../../_images/besseli.png" />
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Modified Bessel function I_n(z) in the complex plane</span>
-<span class="n">cplot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">besseli</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">z</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span> <span class="n">points</span><span class="o">=</span><span class="mi">50000</span><span class="p">)</span>
-</pre></div>
-</div>
-<img alt="../../../_images/besseli_c.png" src="../../../_images/besseli_c.png" />
-<p><strong>Examples</strong></p>
-<p>Some values of <span class="math">\(I_n(x)\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besseli</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besseli</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besseli</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1.266065877752008335598245</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besseli</span><span class="p">(</span><span class="mf">3.5</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(-0.2904369752642538144289025 - 0.4469098397654815837307006j)</span>
-</pre></div>
-</div>
-<p>Arguments may be large:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">besseli</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
-<span class="go">2.480717210191852440616782e+432</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besseli</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">4.299602851624027900335391e+4342944813</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besseli</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">6000</span><span class="o">+</span><span class="mi">10000j</span><span class="p">)</span>
-<span class="go">(-2.114650753239580827144204e+2603 + 4.385040221241629041351886e+2602j)</span>
-</pre></div>
-</div>
-<p>For integers <span class="math">\(n\)</span>, the following integral representation holds:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="mf">2.3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">t</span><span class="p">))</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">t</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">pi</span><span class="p">])</span><span class="o">/</span><span class="n">pi</span>
-<span class="go">0.349223221159309</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besseli</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="go">0.349223221159309</span>
-</pre></div>
-</div>
-<p>Derivatives and antiderivatives of any order can be computed:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besseli</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">7.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">195.8229038931399062565883</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">besseli</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="mf">7.5</span><span class="p">)</span>
-<span class="go">195.8229038931399062565883</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besseli</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">7.5</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">153.3296508971734525525176</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">besseli</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="mf">7.5</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">153.3296508971734525525176</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besseli</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mf">7.5</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">besseli</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mf">3.5</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">202.5043900051930141956876</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">besseli</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="p">[</span><span class="mf">3.5</span><span class="p">,</span> <span class="mf">7.5</span><span class="p">])</span>
-<span class="go">202.5043900051930141956876</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="besselk">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">besselk()</span></tt><a class="headerlink" href="#besselk" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.besselk">
-<tt class="descclassname">mpmath.</tt><tt class="descname">besselk</tt><big>(</big><em>n</em>, <em>x</em><big>)</big><a class="headerlink" href="#mpmath.besselk" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">besselk(n,</span> <span class="pre">x)</span></tt> gives the modified Bessel function of the
-second kind,</p>
-<div class="math">
-\[K_n(x) = \frac{\pi}{2} \frac{I_{-n}(x)-I_{n}(x)}{\sin(\pi n)}\]</div>
-<p>For <span class="math">\(n\)</span> an integer, this formula should be understood as a
-limit.</p>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Modified Bessel function of 2nd kind K_n(x) on the real line for n=0,1,2,3</span>
-<span class="n">k0</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">besselk</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">k1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">besselk</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">k2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">besselk</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">k3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">besselk</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">k0</span><span class="p">,</span><span class="n">k1</span><span class="p">,</span><span class="n">k2</span><span class="p">,</span><span class="n">k3</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">8</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/besselk.png" src="../../../_images/besselk.png" />
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Modified Bessel function of 2nd kind K_n(z) in the complex plane</span>
-<span class="n">cplot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">besselk</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">z</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span> <span class="n">points</span><span class="o">=</span><span class="mi">50000</span><span class="p">)</span>
-</pre></div>
-</div>
-<img alt="../../../_images/besselk_c.png" src="../../../_images/besselk_c.png" />
-<p><strong>Examples</strong></p>
-<p>Evaluation is supported for arbitrary complex arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselk</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.4210244382407083333356274</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselk</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(0.4210244382407083333356274 - 3.97746326050642263725661j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselk</span><span class="p">(</span><span class="mf">3.5</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(-0.02090732889633760668464128 + 0.2464022641351420167819697j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselk</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
-<span class="go">(0.9615816021726349402626083 + 0.1918250181801757416908224j)</span>
-</pre></div>
-</div>
-<p>Arguments may be large:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">besselk</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
-<span class="go">4.656628229175902018939005e-45</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselk</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="o">**</span><span class="mi">6</span><span class="p">)</span>
-<span class="go">4.131967049321725588398296e-434298</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselk</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="o">**</span><span class="mi">6</span><span class="o">*</span><span class="n">j</span><span class="p">)</span>
-<span class="go">(0.001140348428252385844876706 - 0.0005200017201681152909000961j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselk</span><span class="p">(</span><span class="mf">4.5</span><span class="p">,</span> <span class="n">fmul</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">50</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">(1.561034538142413947789221e-26 + 1.243554598118700063281496e-25j)</span>
-</pre></div>
-</div>
-<p>The point <span class="math">\(x = 0\)</span> is a singularity (logarithmic if <span class="math">\(n = 0\)</span>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">besselk</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselk</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="n">besselk</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="s">&#39;1e-1000&#39;</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">4.8e+4001</span>
-<span class="go">8.0e+3000</span>
-<span class="go">2.0e+2000</span>
-<span class="go">1.0e+1000</span>
-<span class="go">2302.701024509704096466802</span>
-<span class="go">1.0e+1000</span>
-<span class="go">2.0e+2000</span>
-<span class="go">8.0e+3000</span>
-<span class="go">4.8e+4001</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="bessel-function-zeros">
-<h2>Bessel function zeros<a class="headerlink" href="#bessel-function-zeros" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="besseljzero">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">besseljzero()</span></tt><a class="headerlink" href="#besseljzero" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.besseljzero">
-<tt class="descclassname">mpmath.</tt><tt class="descname">besseljzero</tt><big>(</big><em>v</em>, <em>m</em>, <em>derivative=0</em><big>)</big><a class="headerlink" href="#mpmath.besseljzero" title="Permalink to this definition">¶</a></dt>
-<dd><p>For a real order <span class="math">\(\nu \ge 0\)</span> and a positive integer <span class="math">\(m\)</span>, returns
-<span class="math">\(j_{\nu,m}\)</span>, the <span class="math">\(m\)</span>-th positive zero of the Bessel function of the
-first kind <span class="math">\(J_{\nu}(z)\)</span> (see <a class="reference internal" href="#mpmath.besselj" title="mpmath.besselj"><tt class="xref py py-func docutils literal"><span class="pre">besselj()</span></tt></a>). Alternatively,
-with <em>derivative=1</em>, gives the first nonnegative simple zero
-<span class="math">\(j'_{\nu,m}\)</span> of <span class="math">\(J'_{\nu}(z)\)</span>.</p>
-<p>The indexing convention is that used by Abramowitz &amp; Stegun
-and the DLMF. Note the special case <span class="math">\(j'_{0,1} = 0\)</span>, while all other
-zeros are positive. In effect, only simple zeros are counted
-(all zeros of Bessel functions are simple except possibly <span class="math">\(z = 0\)</span>)
-and <span class="math">\(j_{\nu,m}\)</span> becomes a monotonic function of both <span class="math">\(\nu\)</span>
-and <span class="math">\(m\)</span>.</p>
-<p>The zeros are interlaced according to the inequalities</p>
-<div class="math">
-\[\begin{split}j'_{\nu,k} &lt; j_{\nu,k} &lt; j'_{\nu,k+1}\end{split}\]\[\begin{split}j_{\nu,1} &lt; j_{\nu+1,2} &lt; j_{\nu,2} &lt; j_{\nu+1,2} &lt; j_{\nu,3} &lt; \cdots\end{split}\]</div>
-<p><strong>Examples</strong></p>
-<p>Initial zeros of the Bessel functions <span class="math">\(J_0(z), J_1(z), J_2(z)\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besseljzero</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">besseljzero</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">);</span> <span class="n">besseljzero</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">2.404825557695772768621632</span>
-<span class="go">5.520078110286310649596604</span>
-<span class="go">8.653727912911012216954199</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besseljzero</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">besseljzero</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">);</span> <span class="n">besseljzero</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">3.831705970207512315614436</span>
-<span class="go">7.01558666981561875353705</span>
-<span class="go">10.17346813506272207718571</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besseljzero</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">besseljzero</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">);</span> <span class="n">besseljzero</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">5.135622301840682556301402</span>
-<span class="go">8.417244140399864857783614</span>
-<span class="go">11.61984117214905942709415</span>
-</pre></div>
-</div>
-<p>Initial zeros of <span class="math">\(J'_0(z), J'_1(z), J'_2(z)\)</span>:</p>
-<div class="highlight-python"><pre>0.0
-3.831705970207512315614436
-7.01558666981561875353705
-&gt;&gt;&gt; besseljzero(1,1,1); besseljzero(1,2,1); besseljzero(1,3,1)
-1.84118378134065930264363
-5.331442773525032636884016
-8.536316366346285834358961
-&gt;&gt;&gt; besseljzero(2,1,1); besseljzero(2,2,1); besseljzero(2,3,1)
-3.054236928227140322755932
-6.706133194158459146634394
-9.969467823087595793179143</pre>
-</div>
-<p>Zeros with large index:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">besseljzero</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">100</span><span class="p">);</span> <span class="n">besseljzero</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1000</span><span class="p">);</span> <span class="n">besseljzero</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">10000</span><span class="p">)</span>
-<span class="go">313.3742660775278447196902</span>
-<span class="go">3140.807295225078628895545</span>
-<span class="go">31415.14114171350798533666</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besseljzero</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">100</span><span class="p">);</span> <span class="n">besseljzero</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">1000</span><span class="p">);</span> <span class="n">besseljzero</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">10000</span><span class="p">)</span>
-<span class="go">321.1893195676003157339222</span>
-<span class="go">3148.657306813047523500494</span>
-<span class="go">31422.9947255486291798943</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besseljzero</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">100</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">besseljzero</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1000</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">besseljzero</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">10000</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">311.8018681873704508125112</span>
-<span class="go">3139.236339643802482833973</span>
-<span class="go">31413.57032947022399485808</span>
-</pre></div>
-</div>
-<p>Zeros of functions with large order:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">besseljzero</span><span class="p">(</span><span class="mi">50</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">57.11689916011917411936228</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besseljzero</span><span class="p">(</span><span class="mi">50</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">62.80769876483536093435393</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besseljzero</span><span class="p">(</span><span class="mi">50</span><span class="p">,</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">388.6936600656058834640981</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besseljzero</span><span class="p">(</span><span class="mi">50</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">52.99764038731665010944037</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besseljzero</span><span class="p">(</span><span class="mi">50</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">60.02631933279942589882363</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besseljzero</span><span class="p">(</span><span class="mi">50</span><span class="p">,</span><span class="mi">100</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">387.1083151608726181086283</span>
-</pre></div>
-</div>
-<p>Zeros of functions with fractional order:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">besseljzero</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">besseljzero</span><span class="p">(</span><span class="mf">1.5</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">besseljzero</span><span class="p">(</span><span class="mf">2.25</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">3.141592653589793238462643</span>
-<span class="go">4.493409457909064175307881</span>
-<span class="go">15.15657692957458622921634</span>
-</pre></div>
-</div>
-<p>Both <span class="math">\(J_{\nu}(z)\)</span> and <span class="math">\(J'_{\nu}(z)\)</span> can be expressed as infinite
-products over their zeros:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">v</span><span class="p">,</span><span class="n">z</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="n">v</span><span class="o">/</span><span class="n">gamma</span><span class="p">(</span><span class="n">v</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span> <span class="o">*</span> \
-<span class="gp">... </span>    <span class="n">nprod</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="mi">1</span><span class="o">-</span><span class="p">(</span><span class="n">z</span><span class="o">/</span><span class="n">besseljzero</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">k</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="gp">...</span>
-<span class="go">0.1149034849319004804696469</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselj</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.1149034849319004804696469</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">v</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="o">/</span><span class="n">gamma</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="o">*</span> \
-<span class="gp">... </span>    <span class="n">nprod</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="mi">1</span><span class="o">-</span><span class="p">(</span><span class="n">z</span><span class="o">/</span><span class="n">besseljzero</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">k</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="gp">...</span>
-<span class="go">0.2102436158811325550203884</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselj</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.2102436158811325550203884</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="besselyzero">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">besselyzero()</span></tt><a class="headerlink" href="#besselyzero" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.besselyzero">
-<tt class="descclassname">mpmath.</tt><tt class="descname">besselyzero</tt><big>(</big><em>v</em>, <em>m</em>, <em>derivative=0</em><big>)</big><a class="headerlink" href="#mpmath.besselyzero" title="Permalink to this definition">¶</a></dt>
-<dd><p>For a real order <span class="math">\(\nu \ge 0\)</span> and a positive integer <span class="math">\(m\)</span>, returns
-<span class="math">\(y_{\nu,m}\)</span>, the <span class="math">\(m\)</span>-th positive zero of the Bessel function of the
-second kind <span class="math">\(Y_{\nu}(z)\)</span> (see <a class="reference internal" href="#mpmath.bessely" title="mpmath.bessely"><tt class="xref py py-func docutils literal"><span class="pre">bessely()</span></tt></a>). Alternatively,
-with <em>derivative=1</em>, gives the first positive zero <span class="math">\(y'_{\nu,m}\)</span> of
-<span class="math">\(Y'_{\nu}(z)\)</span>.</p>
-<p>The zeros are interlaced according to the inequalities</p>
-<div class="math">
-\[\begin{split}y_{\nu,k} &lt; y'_{\nu,k} &lt; y_{\nu,k+1}\end{split}\]\[\begin{split}y_{\nu,1} &lt; y_{\nu+1,2} &lt; y_{\nu,2} &lt; y_{\nu+1,2} &lt; y_{\nu,3} &lt; \cdots\end{split}\]</div>
-<p><strong>Examples</strong></p>
-<p>Initial zeros of the Bessel functions <span class="math">\(Y_0(z), Y_1(z), Y_2(z)\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselyzero</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">besselyzero</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">);</span> <span class="n">besselyzero</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.8935769662791675215848871</span>
-<span class="go">3.957678419314857868375677</span>
-<span class="go">7.086051060301772697623625</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselyzero</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">besselyzero</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">);</span> <span class="n">besselyzero</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">2.197141326031017035149034</span>
-<span class="go">5.429681040794135132772005</span>
-<span class="go">8.596005868331168926429606</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselyzero</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">besselyzero</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">);</span> <span class="n">besselyzero</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">3.384241767149593472701426</span>
-<span class="go">6.793807513268267538291167</span>
-<span class="go">10.02347797936003797850539</span>
-</pre></div>
-</div>
-<p>Initial zeros of <span class="math">\(Y'_0(z), Y'_1(z), Y'_2(z)\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">besselyzero</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">besselyzero</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">besselyzero</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">2.197141326031017035149034</span>
-<span class="go">5.429681040794135132772005</span>
-<span class="go">8.596005868331168926429606</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselyzero</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">besselyzero</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">besselyzero</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">3.683022856585177699898967</span>
-<span class="go">6.941499953654175655751944</span>
-<span class="go">10.12340465543661307978775</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselyzero</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">besselyzero</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">besselyzero</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">5.002582931446063945200176</span>
-<span class="go">8.350724701413079526349714</span>
-<span class="go">11.57419546521764654624265</span>
-</pre></div>
-</div>
-<p>Zeros with large index:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">besselyzero</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">100</span><span class="p">);</span> <span class="n">besselyzero</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1000</span><span class="p">);</span> <span class="n">besselyzero</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">10000</span><span class="p">)</span>
-<span class="go">311.8034717601871549333419</span>
-<span class="go">3139.236498918198006794026</span>
-<span class="go">31413.57034538691205229188</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselyzero</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">100</span><span class="p">);</span> <span class="n">besselyzero</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">1000</span><span class="p">);</span> <span class="n">besselyzero</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">10000</span><span class="p">)</span>
-<span class="go">319.6183338562782156235062</span>
-<span class="go">3147.086508524556404473186</span>
-<span class="go">31421.42392920214673402828</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselyzero</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">100</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">besselyzero</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1000</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">besselyzero</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">10000</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">313.3726705426359345050449</span>
-<span class="go">3140.807136030340213610065</span>
-<span class="go">31415.14112579761578220175</span>
-</pre></div>
-</div>
-<p>Zeros of functions with large order:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">besselyzero</span><span class="p">(</span><span class="mi">50</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">53.50285882040036394680237</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselyzero</span><span class="p">(</span><span class="mi">50</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">60.11244442774058114686022</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselyzero</span><span class="p">(</span><span class="mi">50</span><span class="p">,</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">387.1096509824943957706835</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselyzero</span><span class="p">(</span><span class="mi">50</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">56.96290427516751320063605</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselyzero</span><span class="p">(</span><span class="mi">50</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">62.74888166945933944036623</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselyzero</span><span class="p">(</span><span class="mi">50</span><span class="p">,</span><span class="mi">100</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">388.6923300548309258355475</span>
-</pre></div>
-</div>
-<p>Zeros of functions with fractional order:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">besselyzero</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">besselyzero</span><span class="p">(</span><span class="mf">1.5</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">besselyzero</span><span class="p">(</span><span class="mf">2.25</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">1.570796326794896619231322</span>
-<span class="go">2.798386045783887136720249</span>
-<span class="go">13.56721208770735123376018</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="hankel-functions">
-<h2>Hankel functions<a class="headerlink" href="#hankel-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="hankel1">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">hankel1()</span></tt><a class="headerlink" href="#hankel1" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.hankel1">
-<tt class="descclassname">mpmath.</tt><tt class="descname">hankel1</tt><big>(</big><em>n</em>, <em>x</em><big>)</big><a class="headerlink" href="#mpmath.hankel1" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">hankel1(n,x)</span></tt> computes the Hankel function of the first kind,
-which is the complex combination of Bessel functions given by</p>
-<div class="math">
-\[H_n^{(1)}(x) = J_n(x) + i Y_n(x).\]</div>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Hankel function H1_n(x) on the real line for n=0,1,2,3</span>
-<span class="n">h0</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">hankel1</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">h1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">hankel1</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">h2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">hankel1</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">h3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">hankel1</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">h0</span><span class="p">,</span><span class="n">h1</span><span class="p">,</span><span class="n">h2</span><span class="p">,</span><span class="n">h3</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">6</span><span class="p">],[</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/hankel1.png" src="../../../_images/hankel1.png" />
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Hankel function H1_n(z) in the complex plane</span>
-<span class="n">cplot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">hankel1</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">z</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span> <span class="n">points</span><span class="o">=</span><span class="mi">50000</span><span class="p">)</span>
-</pre></div>
-</div>
-<img alt="../../../_images/hankel1_c.png" src="../../../_images/hankel1_c.png" />
-<p><strong>Examples</strong></p>
-<p>The Hankel function is generally complex-valued:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hankel1</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">pi</span><span class="p">)</span>
-<span class="go">(0.4854339326315091097054957 - 0.0999007139290278787734903j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hankel1</span><span class="p">(</span><span class="mf">3.5</span><span class="p">,</span> <span class="n">pi</span><span class="p">)</span>
-<span class="go">(0.2340002029630507922628888 - 0.6419643823412927142424049j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="hankel2">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">hankel2()</span></tt><a class="headerlink" href="#hankel2" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.hankel2">
-<tt class="descclassname">mpmath.</tt><tt class="descname">hankel2</tt><big>(</big><em>n</em>, <em>x</em><big>)</big><a class="headerlink" href="#mpmath.hankel2" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">hankel2(n,x)</span></tt> computes the Hankel function of the second kind,
-which is the complex combination of Bessel functions given by</p>
-<div class="math">
-\[H_n^{(2)}(x) = J_n(x) - i Y_n(x).\]</div>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Hankel function H2_n(x) on the real line for n=0,1,2,3</span>
-<span class="n">h0</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">hankel2</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">h1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">hankel2</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">h2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">hankel2</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">h3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">hankel2</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">h0</span><span class="p">,</span><span class="n">h1</span><span class="p">,</span><span class="n">h2</span><span class="p">,</span><span class="n">h3</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">6</span><span class="p">],[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/hankel2.png" src="../../../_images/hankel2.png" />
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Hankel function H2_n(z) in the complex plane</span>
-<span class="n">cplot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">hankel2</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">z</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span> <span class="n">points</span><span class="o">=</span><span class="mi">50000</span><span class="p">)</span>
-</pre></div>
-</div>
-<img alt="../../../_images/hankel2_c.png" src="../../../_images/hankel2_c.png" />
-<p><strong>Examples</strong></p>
-<p>The Hankel function is generally complex-valued:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hankel2</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">pi</span><span class="p">)</span>
-<span class="go">(0.4854339326315091097054957 + 0.0999007139290278787734903j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hankel2</span><span class="p">(</span><span class="mf">3.5</span><span class="p">,</span> <span class="n">pi</span><span class="p">)</span>
-<span class="go">(0.2340002029630507922628888 + 0.6419643823412927142424049j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="kelvin-functions">
-<h2>Kelvin functions<a class="headerlink" href="#kelvin-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="ber">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">ber()</span></tt><a class="headerlink" href="#ber" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.ber">
-<tt class="descclassname">mpmath.</tt><tt class="descname">ber</tt><big>(</big><em>ctx</em>, <em>n</em>, <em>z</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.ber" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the Kelvin function ber, which for real arguments gives the real part
-of the Bessel J function of a rotated argument</p>
-<div class="math">
-\[J_n\left(x e^{3\pi i/4}\right) = \mathrm{ber}_n(x) + i \mathrm{bei}_n(x).\]</div>
-<p>The imaginary part is given by <a class="reference internal" href="#mpmath.bei" title="mpmath.bei"><tt class="xref py py-func docutils literal"><span class="pre">bei()</span></tt></a>.</p>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Kelvin functions ber_n(x) and bei_n(x) on the real line for n=0,2</span>
-<span class="n">f0</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">ber</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">bei</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">ber</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">bei</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">f0</span><span class="p">,</span><span class="n">f1</span><span class="p">,</span><span class="n">f2</span><span class="p">,</span><span class="n">f3</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">10</span><span class="p">],[</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/ber.png" src="../../../_images/ber.png" />
-<p><strong>Examples</strong></p>
-<p>Verifying the defining relation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="mf">3.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ber</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1.442338852571888752631129</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bei</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="go">-0.948359035324558320217678</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselj</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="o">*</span><span class="n">root</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">(1.442338852571888752631129 - 0.948359035324558320217678j)</span>
-</pre></div>
-</div>
-<p>The ber and bei functions are also defined by analytic continuation
-for complex arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ber</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(4.675445984756614424069563 - 15.84901771719130765656316j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bei</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(15.83886679193707699364398 + 4.684053288183046528703611j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="bei">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">bei()</span></tt><a class="headerlink" href="#bei" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.bei">
-<tt class="descclassname">mpmath.</tt><tt class="descname">bei</tt><big>(</big><em>ctx</em>, <em>n</em>, <em>z</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.bei" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the Kelvin function bei, which for real arguments gives the
-imaginary part of the Bessel J function of a rotated argument.
-See <a class="reference internal" href="#mpmath.ber" title="mpmath.ber"><tt class="xref py py-func docutils literal"><span class="pre">ber()</span></tt></a>.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="ker">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">ker()</span></tt><a class="headerlink" href="#ker" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.ker">
-<tt class="descclassname">mpmath.</tt><tt class="descname">ker</tt><big>(</big><em>ctx</em>, <em>n</em>, <em>z</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.ker" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the Kelvin function ker, which for real arguments gives the real part
-of the (rescaled) Bessel K function of a rotated argument</p>
-<div class="math">
-\[e^{-\pi i/2} K_n\left(x e^{3\pi i/4}\right) = \mathrm{ker}_n(x) + i \mathrm{kei}_n(x).\]</div>
-<p>The imaginary part is given by <a class="reference internal" href="#mpmath.kei" title="mpmath.kei"><tt class="xref py py-func docutils literal"><span class="pre">kei()</span></tt></a>.</p>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Kelvin functions ker_n(x) and kei_n(x) on the real line for n=0,2</span>
-<span class="n">f0</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">ker</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">kei</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">ker</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">kei</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">f0</span><span class="p">,</span><span class="n">f1</span><span class="p">,</span><span class="n">f2</span><span class="p">,</span><span class="n">f3</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">5</span><span class="p">],[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/ker.png" src="../../../_images/ker.png" />
-<p><strong>Examples</strong></p>
-<p>Verifying the defining relation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span> <span class="n">x</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="mf">4.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ker</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="go">0.02542895201906369640249801</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">kei</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="go">-0.02074960467222823237055351</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">j</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">besselk</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="o">*</span><span class="n">root</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span>
-<span class="go">(0.02542895201906369640249801 - 0.02074960467222823237055351j)</span>
-</pre></div>
-</div>
-<p>The ker and kei functions are also defined by analytic continuation
-for complex arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ker</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(1.586084268115490421090533 - 2.939717517906339193598719j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">kei</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(-2.940403256319453402690132 - 1.585621643835618941044855j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="kei">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">kei()</span></tt><a class="headerlink" href="#kei" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.kei">
-<tt class="descclassname">mpmath.</tt><tt class="descname">kei</tt><big>(</big><em>ctx</em>, <em>n</em>, <em>z</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.kei" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the Kelvin function kei, which for real arguments gives the
-imaginary part of the (rescaled) Bessel K function of a rotated argument.
-See <a class="reference internal" href="#mpmath.ker" title="mpmath.ker"><tt class="xref py py-func docutils literal"><span class="pre">ker()</span></tt></a>.</p>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="struve-functions">
-<h2>Struve functions<a class="headerlink" href="#struve-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="struveh">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">struveh()</span></tt><a class="headerlink" href="#struveh" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.struveh">
-<tt class="descclassname">mpmath.</tt><tt class="descname">struveh</tt><big>(</big><em>ctx</em>, <em>n</em>, <em>z</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.struveh" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the Struve function</p>
-<div class="math">
-\[\,\mathbf{H}_n(z) =
-\sum_{k=0}^\infty \frac{(-1)^k}{\Gamma(k+\frac{3}{2})
-    \Gamma(k+n+\frac{3}{2})} {\left({\frac{z}{2}}\right)}^{2k+n+1}\]</div>
-<p>which is a solution to the Struve differential equation</p>
-<div class="math">
-\[z^2 f''(z) + z f'(z) + (z^2-n^2) f(z) = \frac{2 z^{n+1}}{\pi (2n-1)!!}.\]</div>
-<p><strong>Examples</strong></p>
-<p>Evaluation for arbitrary real and complex arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">struveh</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">)</span>
-<span class="go">0.3608207733778295024977797</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">struveh</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">-0.255212719726956768034732</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">struveh</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mf">100.5</span><span class="p">)</span>
-<span class="go">0.5819566816797362287502246</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">struveh</span><span class="p">(</span><span class="mf">2.5</span><span class="p">,</span> <span class="mi">10000000000000</span><span class="p">)</span>
-<span class="go">3153915652525200060.308937</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">struveh</span><span class="p">(</span><span class="mf">2.5</span><span class="p">,</span> <span class="o">-</span><span class="mi">10000000000000</span><span class="p">)</span>
-<span class="go">(0.0 - 3153915652525200060.308937j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">struveh</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mi">1000000</span><span class="o">+</span><span class="mi">4000000j</span><span class="p">)</span>
-<span class="go">(-3.066421087689197632388731e+1737173 - 1.596619701076529803290973e+1737173j)</span>
-</pre></div>
-</div>
-<p>A Struve function of half-integer order is elementary; for example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">struveh</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">0.9167076867564138178671595</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="p">))</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">cos</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="go">0.9167076867564138178671595</span>
-</pre></div>
-</div>
-<p>Numerically verifying the differential equation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">4.5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">struveh</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lhs</span> <span class="o">=</span> <span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">z</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="n">n</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rhs</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">fac2</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lhs</span>
-<span class="go">17.40359302709875496632744</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rhs</span>
-<span class="go">17.40359302709875496632744</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="struvel">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">struvel()</span></tt><a class="headerlink" href="#struvel" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.struvel">
-<tt class="descclassname">mpmath.</tt><tt class="descname">struvel</tt><big>(</big><em>ctx</em>, <em>n</em>, <em>z</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.struvel" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the modified Struve function</p>
-<div class="math">
-\[\,\mathbf{L}_n(z) = -i e^{-n\pi i/2} \mathbf{H}_n(i z)\]</div>
-<p>which solves to the modified Struve differential equation</p>
-<div class="math">
-\[z^2 f''(z) + z f'(z) - (z^2+n^2) f(z) = \frac{2 z^{n+1}}{\pi (2n-1)!!}.\]</div>
-<p><strong>Examples</strong></p>
-<p>Evaluation for arbitrary real and complex arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">struvel</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">)</span>
-<span class="go">7.180846515103737996249972</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">struvel</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">2670.994904980850550721511</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">struvel</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mf">100.5</span><span class="p">)</span>
-<span class="go">1.757089288053346261497686e+42</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">struvel</span><span class="p">(</span><span class="mf">2.5</span><span class="p">,</span> <span class="mi">10000000000000</span><span class="p">)</span>
-<span class="go">4.160893281017115450519948e+4342944819025</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">struvel</span><span class="p">(</span><span class="mf">2.5</span><span class="p">,</span> <span class="o">-</span><span class="mi">10000000000000</span><span class="p">)</span>
-<span class="go">(0.0 - 4.160893281017115450519948e+4342944819025j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">struvel</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mi">700j</span><span class="p">)</span>
-<span class="go">(-0.1721150049480079451246076 + 0.1240770953126831093464055j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">struvel</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mi">1000000</span><span class="o">+</span><span class="mi">4000000j</span><span class="p">)</span>
-<span class="go">(-2.973341637511505389128708e+434290 - 5.164633059729968297147448e+434290j)</span>
-</pre></div>
-</div>
-<p>Numerically verifying the differential equation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">3.5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">struvel</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lhs</span> <span class="o">=</span> <span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">z</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">n</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rhs</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">fac2</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lhs</span>
-<span class="go">6.368850306060678353018165</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rhs</span>
-<span class="go">6.368850306060678353018165</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="anger-weber-functions">
-<h2>Anger-Weber functions<a class="headerlink" href="#anger-weber-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="angerj">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">angerj()</span></tt><a class="headerlink" href="#angerj" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.angerj">
-<tt class="descclassname">mpmath.</tt><tt class="descname">angerj</tt><big>(</big><em>ctx</em>, <em>v</em>, <em>z</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.angerj" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the Anger function</p>
-<div class="math">
-\[\mathbf{J}_{\nu}(z) = \frac{1}{\pi}
-    \int_0^{\pi} \cos(\nu t - z \sin t) dt\]</div>
-<p>which is an entire function of both the parameter <span class="math">\(\nu\)</span> and
-the argument <span class="math">\(z\)</span>. It solves the inhomogeneous Bessel differential
-equation</p>
-<div class="math">
-\[f''(z) + \frac{1}{z}f'(z) + \left(1-\frac{\nu^2}{z^2}\right) f(z)
-    = \frac{(z-\nu)}{\pi z^2} \sin(\pi \nu).\]</div>
-<p><strong>Examples</strong></p>
-<p>Evaluation for real and complex parameter and argument:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">angerj</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.4860912605858910769078311</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">angerj</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">5j</span><span class="p">)</span>
-<span class="go">(-5033.358320403384472395612 + 585.8011892476145118551756j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">angerj</span><span class="p">(</span><span class="mf">3.25</span><span class="p">,</span> <span class="mf">1e6j</span><span class="p">)</span>
-<span class="go">(4.630743639715893346570743e+434290 - 1.117960409887505906848456e+434291j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">angerj</span><span class="p">(</span><span class="o">-</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">1e6</span><span class="p">)</span>
-<span class="go">0.0002795719747073879393087011</span>
-</pre></div>
-</div>
-<p>The Anger function coincides with the Bessel J-function when <span class="math">\(\nu\)</span>
-is an integer:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">angerj</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">);</span> <span class="n">besselj</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.3390589585259364589255146</span>
-<span class="go">0.3390589585259364589255146</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">angerj</span><span class="p">(</span><span class="mf">1.5</span><span class="p">,</span><span class="mi">3</span><span class="p">);</span> <span class="n">besselj</span><span class="p">(</span><span class="mf">1.5</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.4088969848691080859328847</span>
-<span class="go">0.4777182150870917715515015</span>
-</pre></div>
-</div>
-<p>Verifying the differential equation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">v</span><span class="p">,</span><span class="n">z</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">2.25</span><span class="p">),</span> <span class="mf">0.75</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">angerj</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="n">z</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="p">(</span><span class="n">v</span><span class="o">/</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">-0.6002108774380707130367995</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">z</span><span class="o">-</span><span class="n">v</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">sinpi</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-<span class="go">-0.6002108774380707130367995</span>
-</pre></div>
-</div>
-<p>Verifying the integral representation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">angerj</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.1145380759919333180900501</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">cos</span><span class="p">(</span><span class="n">v</span><span class="o">*</span><span class="n">t</span><span class="o">-</span><span class="n">z</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">))</span><span class="o">/</span><span class="n">pi</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">pi</span><span class="p">])</span>
-<span class="go">0.1145380759919333180900501</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../../mpmath/references.html#dlmf">[DLMF]</a> section 11.10: Anger-Weber Functions</li>
-</ol>
-</dd></dl>
-
-</div>
-<div class="section" id="webere">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">webere()</span></tt><a class="headerlink" href="#webere" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.webere">
-<tt class="descclassname">mpmath.</tt><tt class="descname">webere</tt><big>(</big><em>ctx</em>, <em>v</em>, <em>z</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.webere" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the Weber function</p>
-<div class="math">
-\[\mathbf{E}_{\nu}(z) = \frac{1}{\pi}
-    \int_0^{\pi} \sin(\nu t - z \sin t) dt\]</div>
-<p>which is an entire function of both the parameter <span class="math">\(\nu\)</span> and
-the argument <span class="math">\(z\)</span>. It solves the inhomogeneous Bessel differential
-equation</p>
-<div class="math">
-\[f''(z) + \frac{1}{z}f'(z) + \left(1-\frac{\nu^2}{z^2}\right) f(z)
-    = -\frac{1}{\pi z^2} (z+\nu+(z-\nu)\cos(\pi \nu)).\]</div>
-<p><strong>Examples</strong></p>
-<p>Evaluation for real and complex parameter and argument:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">webere</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">-0.1057668973099018425662646</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">webere</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">5j</span><span class="p">)</span>
-<span class="go">(-585.8081418209852019290498 - 5033.314488899926921597203j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">webere</span><span class="p">(</span><span class="mf">3.25</span><span class="p">,</span> <span class="mf">1e6j</span><span class="p">)</span>
-<span class="go">(-1.117960409887505906848456e+434291 - 4.630743639715893346570743e+434290j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">webere</span><span class="p">(</span><span class="mf">3.25</span><span class="p">,</span> <span class="mf">1e6</span><span class="p">)</span>
-<span class="go">-0.00002812518265894315604914453</span>
-</pre></div>
-</div>
-<p>Up to addition of a rational function of <span class="math">\(z\)</span>, the Weber function coincides
-with the Struve H-function when <span class="math">\(\nu\)</span> is an integer:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">webere</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">);</span> <span class="mi">2</span><span class="o">/</span><span class="n">pi</span><span class="o">-</span><span class="n">struveh</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">-0.3834897968188690177372881</span>
-<span class="go">-0.3834897968188690177372881</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">webere</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">3</span><span class="p">);</span> <span class="mi">26</span><span class="o">/</span><span class="p">(</span><span class="mi">35</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">-</span><span class="n">struveh</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.2009680659308154011878075</span>
-<span class="go">0.2009680659308154011878075</span>
-</pre></div>
-</div>
-<p>Verifying the differential equation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">v</span><span class="p">,</span><span class="n">z</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">2.25</span><span class="p">),</span> <span class="mf">0.75</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">webere</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="n">z</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="p">(</span><span class="n">v</span><span class="o">/</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">-1.097441848875479535164627</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="p">(</span><span class="n">z</span><span class="o">+</span><span class="n">v</span><span class="o">+</span><span class="p">(</span><span class="n">z</span><span class="o">-</span><span class="n">v</span><span class="p">)</span><span class="o">*</span><span class="n">cospi</span><span class="p">(</span><span class="n">v</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">-1.097441848875479535164627</span>
-</pre></div>
-</div>
-<p>Verifying the integral representation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">webere</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.1486507351534283744485421</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">sin</span><span class="p">(</span><span class="n">v</span><span class="o">*</span><span class="n">t</span><span class="o">-</span><span class="n">z</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">))</span><span class="o">/</span><span class="n">pi</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">pi</span><span class="p">])</span>
-<span class="go">0.1486507351534283744485421</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../../mpmath/references.html#dlmf">[DLMF]</a> section 11.10: Anger-Weber Functions</li>
-</ol>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="lommel-functions">
-<h2>Lommel functions<a class="headerlink" href="#lommel-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="lommels1">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">lommels1()</span></tt><a class="headerlink" href="#lommels1" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.lommels1">
-<tt class="descclassname">mpmath.</tt><tt class="descname">lommels1</tt><big>(</big><em>ctx</em>, <em>u</em>, <em>v</em>, <em>z</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.lommels1" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the Lommel function <span class="math">\(s_{\mu,\nu}\)</span> or <span class="math">\(s^{(1)}_{\mu,\nu}\)</span></p>
-<div class="math">
-\[s_{\mu,\nu}(z) = \frac{z^{\mu+1}}{(\mu-\nu+1)(\mu+\nu+1)}
-    \,_1F_2\left(1; \frac{\mu-\nu+3}{2}, \frac{\mu+\nu+3}{2};
-    -\frac{z^2}{4} \right)\]</div>
-<p>which solves the inhomogeneous Bessel equation</p>
-<div class="math">
-\[z^2 f''(z) + z f'(z) + (z^2-\nu^2) f(z) = z^{\mu+1}.\]</div>
-<p>A second solution is given by <a class="reference internal" href="#mpmath.lommels2" title="mpmath.lommels2"><tt class="xref py py-func docutils literal"><span class="pre">lommels2()</span></tt></a>.</p>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Lommel function s_(u,v)(x) on the real line for a few different u,v</span>
-<span class="n">f1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">lommels1</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mf">2.5</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">lommels1</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">lommels1</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f4</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">lommels1</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">f1</span><span class="p">,</span><span class="n">f2</span><span class="p">,</span><span class="n">f3</span><span class="p">,</span><span class="n">f4</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">20</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/lommels1.png" src="../../../_images/lommels1.png" />
-<p><strong>Examples</strong></p>
-<p>An integral representation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">,</span><span class="n">z</span> <span class="o">=</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mf">0.125</span><span class="p">,</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">0.75</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lommels1</span><span class="p">(</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.4276243877565150372999126</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">bessely</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">t</span><span class="o">**</span><span class="n">u</span><span class="o">*</span><span class="n">besselj</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">t</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">z</span><span class="p">])</span> <span class="o">-</span> \
-<span class="gp">... </span> <span class="n">besselj</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">t</span><span class="o">**</span><span class="n">u</span><span class="o">*</span><span class="n">bessely</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">t</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">z</span><span class="p">]))</span><span class="o">*</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.4276243877565150372999126</span>
-</pre></div>
-</div>
-<p>A special value:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">lommels1</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">v</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.5461221367746048054932553</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="n">v</span><span class="o">+</span><span class="mf">0.5</span><span class="p">)</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">power</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">v</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">struveh</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.5461221367746048054932553</span>
-</pre></div>
-</div>
-<p>Verifying the differential equation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">lommels1</span><span class="p">(</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">z</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="n">v</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.6979536443265746992059141</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">u</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.6979536443265746992059141</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../../mpmath/references.html#gradshteynryzhik">[GradshteynRyzhik]</a></li>
-<li><a class="reference internal" href="../../mpmath/references.html#weisstein">[Weisstein]</a> <a class="reference external" href="http://mathworld.wolfram.com/LommelFunction.html">http://mathworld.wolfram.com/LommelFunction.html</a></li>
-</ol>
-</dd></dl>
-
-</div>
-<div class="section" id="lommels2">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">lommels2()</span></tt><a class="headerlink" href="#lommels2" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.lommels2">
-<tt class="descclassname">mpmath.</tt><tt class="descname">lommels2</tt><big>(</big><em>ctx</em>, <em>u</em>, <em>v</em>, <em>z</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.lommels2" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the second Lommel function <span class="math">\(S_{\mu,\nu}\)</span> or <span class="math">\(s^{(2)}_{\mu,\nu}\)</span></p>
-<div class="math">
-\[S_{\mu,\nu}(z) = s_{\mu,\nu}(z) + 2^{\mu-1}
-    \Gamma\left(\tfrac{1}{2}(\mu-\nu+1)\right)
-    \Gamma\left(\tfrac{1}{2}(\mu+\nu+1)\right) \times\]\[    \left[\sin(\tfrac{1}{2}(\mu-\nu)\pi) J_{\nu}(z) -
-          \cos(\tfrac{1}{2}(\mu-\nu)\pi) Y_{\nu}(z)
-    \right]\]</div>
-<p>which solves the same differential equation as
-<a class="reference internal" href="#mpmath.lommels1" title="mpmath.lommels1"><tt class="xref py py-func docutils literal"><span class="pre">lommels1()</span></tt></a>.</p>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Lommel function S_(u,v)(x) on the real line for a few different u,v</span>
-<span class="n">f1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">lommels2</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mf">2.5</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">lommels2</span><span class="p">(</span><span class="mf">1.5</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">lommels2</span><span class="p">(</span><span class="mf">2.5</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f4</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">lommels2</span><span class="p">(</span><span class="mf">3.5</span><span class="p">,</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">f1</span><span class="p">,</span><span class="n">f2</span><span class="p">,</span><span class="n">f3</span><span class="p">,</span><span class="n">f4</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/lommels2.png" src="../../../_images/lommels2.png" />
-<p><strong>Examples</strong></p>
-<p>For large <span class="math">\(|z|\)</span>, <span class="math">\(S_{\mu,\nu} \sim z^{\mu-1}\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lommels2</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">30000</span><span class="p">)</span>
-<span class="go">1.968299831601008419949804e+40</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">power</span><span class="p">(</span><span class="mi">30000</span><span class="p">,</span><span class="mi">9</span><span class="p">)</span>
-<span class="go">1.9683e+40</span>
-</pre></div>
-</div>
-<p>A special value:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">,</span><span class="n">z</span> <span class="o">=</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.125</span><span class="p">,</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">0.75</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lommels2</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">v</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.9589683199624672099969765</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">struveh</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">z</span><span class="p">)</span><span class="o">-</span><span class="n">bessely</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">z</span><span class="p">))</span><span class="o">*</span><span class="n">power</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">v</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="n">v</span><span class="o">+</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="go">0.9589683199624672099969765</span>
-</pre></div>
-</div>
-<p>Verifying the differential equation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">lommels2</span><span class="p">(</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">z</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="n">v</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.6495190528383289850727924</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">u</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.6495190528383289850727924</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../../mpmath/references.html#gradshteynryzhik">[GradshteynRyzhik]</a></li>
-<li><a class="reference internal" href="../../mpmath/references.html#weisstein">[Weisstein]</a> <a class="reference external" href="http://mathworld.wolfram.com/LommelFunction.html">http://mathworld.wolfram.com/LommelFunction.html</a></li>
-</ol>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="airy-and-scorer-functions">
-<h2>Airy and Scorer functions<a class="headerlink" href="#airy-and-scorer-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="airyai">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">airyai()</span></tt><a class="headerlink" href="#airyai" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.airyai">
-<tt class="descclassname">mpmath.</tt><tt class="descname">airyai</tt><big>(</big><em>z</em>, <em>derivative=0</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.airyai" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the Airy function <span class="math">\(\operatorname{Ai}(z)\)</span>, which is
-the solution of the Airy differential equation <span class="math">\(f''(z) - z f(z) = 0\)</span>
-with initial conditions</p>
-<div class="math">
-\[\operatorname{Ai}(0) =
-    \frac{1}{3^{2/3}\Gamma\left(\frac{2}{3}\right)}\]\[\operatorname{Ai}'(0) =
-    -\frac{1}{3^{1/3}\Gamma\left(\frac{1}{3}\right)}.\]</div>
-<p>Other common ways of defining the Ai-function include
-integrals such as</p>
-<div class="math">
-\[\operatorname{Ai}(x) = \frac{1}{\pi}
-    \int_0^{\infty} \cos\left(\frac{1}{3}t^3+xt\right) dt
-    \qquad x \in \mathbb{R}\]\[\operatorname{Ai}(z) = \frac{\sqrt{3}}{2\pi}
-    \int_0^{\infty}
-    \exp\left(-\frac{t^3}{3}-\frac{z^3}{3t^3}\right) dt.\]</div>
-<p>The Ai-function is an entire function with a turning point,
-behaving roughly like a slowly decaying sine wave for <span class="math">\(z &lt; 0\)</span> and
-like a rapidly decreasing exponential for <span class="math">\(z &gt; 0\)</span>.
-A second solution of the Airy differential equation
-is given by <span class="math">\(\operatorname{Bi}(z)\)</span> (see <a class="reference internal" href="#mpmath.airybi" title="mpmath.airybi"><tt class="xref py py-func docutils literal"><span class="pre">airybi()</span></tt></a>).</p>
-<p>Optionally, with <em>derivative=alpha</em>, <tt class="xref py py-func docutils literal"><span class="pre">airyai()</span></tt> can compute the
-<span class="math">\(\alpha\)</span>-th order fractional derivative with respect to <span class="math">\(z\)</span>.
-For <span class="math">\(\alpha = n = 1,2,3,\ldots\)</span> this gives the derivative
-<span class="math">\(\operatorname{Ai}^{(n)}(z)\)</span>, and for <span class="math">\(\alpha = -n = -1,-2,-3,\ldots\)</span>
-this gives the <span class="math">\(n\)</span>-fold iterated integral</p>
-<div class="math">
-\[f_0(z) = \operatorname{Ai}(z)\]\[f_n(z) = \int_0^z f_{n-1}(t) dt.\]</div>
-<p>The Ai-function has infinitely many zeros, all located along the
-negative half of the real axis. They can be computed with
-<a class="reference internal" href="#mpmath.airyaizero" title="mpmath.airyaizero"><tt class="xref py py-func docutils literal"><span class="pre">airyaizero()</span></tt></a>.</p>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Airy function Ai(x), Ai&#39;(x) and int_0^x Ai(t) dt on the real line</span>
-<span class="n">f</span> <span class="o">=</span> <span class="n">airyai</span>
-<span class="n">f_diff</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">airyai</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">derivative</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="n">f_int</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">airyai</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">derivative</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">f_diff</span><span class="p">,</span> <span class="n">f_int</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/ai.png" src="../../../_images/ai.png" />
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Airy function Ai(z) in the complex plane</span>
-<span class="n">cplot</span><span class="p">(</span><span class="n">airyai</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span> <span class="n">points</span><span class="o">=</span><span class="mi">50000</span><span class="p">)</span>
-</pre></div>
-</div>
-<img alt="../../../_images/ai_c.png" src="../../../_images/ai_c.png" />
-<p><strong>Basic examples</strong></p>
-<p>Limits and values include:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">power</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="s">&#39;2/3&#39;</span><span class="p">)</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="s">&#39;2/3&#39;</span><span class="p">))</span>
-<span class="go">0.3550280538878172392600632</span>
-<span class="go">0.3550280538878172392600632</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.1352924163128814155241474</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.5355608832923521187995166</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="n">inf</span><span class="p">);</span> <span class="n">airyai</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>Evaluation is supported for large magnitudes of the argument:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="o">-</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">0.1767533932395528780908311</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">2.634482152088184489550553e-291</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="mi">50</span><span class="o">+</span><span class="mi">50j</span><span class="p">)</span>
-<span class="go">(-5.31790195707456404099817e-68 - 1.163588003770709748720107e-67j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="o">-</span><span class="mi">50</span><span class="o">+</span><span class="mi">50j</span><span class="p">)</span>
-<span class="go">(1.041242537363167632587245e+158 + 3.347525544923600321838281e+157j)</span>
-</pre></div>
-</div>
-<p>Huge arguments are also fine:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">1.162235978298741779953693e-289529654602171</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">0.0001736206448152818510510181</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">w</span> <span class="o">=</span> <span class="n">airyai</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">w</span><span class="o">.</span><span class="n">real</span>
-<span class="go">5.711508683721355528322567e-186339621747698</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">w</span><span class="o">.</span><span class="n">imag</span>
-<span class="go">1.867245506962312577848166e-186339621747697</span>
-</pre></div>
-</div>
-<p>The first root of the Ai-function is:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">airyai</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">-2.338107410459767038489197</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyaizero</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-2.338107410459767038489197</span>
-</pre></div>
-</div>
-<p><strong>Properties and relations</strong></p>
-<p>Verifying the Airy differential equation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="p">[</span><span class="o">-</span><span class="mf">3.4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mi">1</span><span class="o">+</span><span class="mi">2j</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="n">chop</span><span class="p">(</span><span class="n">airyai</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="n">z</span><span class="o">*</span><span class="n">airyai</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>The first few terms of the Taylor series expansion around <span class="math">\(z = 0\)</span>
-(every third term is zero):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">airyai</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
-<span class="go">[0.355028, -0.258819, 0.0, 0.0591713, -0.0215683, 0.0]</span>
-</pre></div>
-</div>
-<p>The Airy functions satisfy the Wronskian relation
-<span class="math">\(\operatorname{Ai}(z) \operatorname{Bi}'(z) -
-\operatorname{Ai}'(z) \operatorname{Bi}(z) = 1/\pi\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="o">-</span><span class="mf">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">airybi</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">airyai</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">airybi</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.3183098861837906715377675</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="n">pi</span>
-<span class="go">0.3183098861837906715377675</span>
-</pre></div>
-</div>
-<p>The Airy functions can be expressed in terms of Bessel
-functions of order <span class="math">\(\pm 1/3\)</span>. For <span class="math">\(\Re[z] \le 0\)</span>, we have:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="o">-</span><span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">-0.3788142936776580743472439</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">power</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">,</span><span class="s">&#39;3/2&#39;</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">besselj</span><span class="p">(</span><span class="s">&#39;1/3&#39;</span><span class="p">,</span><span class="n">y</span><span class="p">)</span> <span class="o">+</span> <span class="n">besselj</span><span class="p">(</span><span class="s">&#39;-1/3&#39;</span><span class="p">,</span><span class="n">y</span><span class="p">)))</span><span class="o">/</span><span class="mi">3</span>
-<span class="go">-0.3788142936776580743472439</span>
-</pre></div>
-</div>
-<p><strong>Derivatives and integrals</strong></p>
-<p>Derivatives of the Ai-function (directly and using <a class="reference internal" href="../../mpmath/calculus/differentiation.html#mpmath.diff" title="mpmath.diff"><tt class="xref py py-func docutils literal"><span class="pre">diff()</span></tt></a>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">diff</span><span class="p">(</span><span class="n">airyai</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.3145837692165988136507873</span>
-<span class="go">0.3145837692165988136507873</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">);</span> <span class="n">diff</span><span class="p">(</span><span class="n">airyai</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">1.136442881032974223041732</span>
-<span class="go">1.136442881032974223041732</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">diff</span><span class="p">(</span><span class="n">airyai</span><span class="p">,</span><span class="mi">1000</span><span class="p">)</span>
-<span class="go">-2.943133917910336090459748e-9156</span>
-<span class="go">-2.943133917910336090459748e-9156</span>
-</pre></div>
-</div>
-<p>Several derivatives at <span class="math">\(z = 0\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">);</span> <span class="n">airyai</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">airyai</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.3550280538878172392600632</span>
-<span class="go">-0.2588194037928067984051836</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">);</span> <span class="n">airyai</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">4</span><span class="p">);</span> <span class="n">airyai</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">0.3550280538878172392600632</span>
-<span class="go">-0.5176388075856135968103671</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">15</span><span class="p">);</span> <span class="n">airyai</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">16</span><span class="p">);</span> <span class="n">airyai</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">17</span><span class="p">)</span>
-<span class="go">1292.30211615165475090663</span>
-<span class="go">-3188.655054727379756351861</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>The integral of the Ai-function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">);</span> <span class="n">quad</span><span class="p">(</span><span class="n">airyai</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-<span class="go">0.3299203760070217725002701</span>
-<span class="go">0.3299203760070217725002701</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">);</span> <span class="n">quad</span><span class="p">(</span><span class="n">airyai</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">10</span><span class="p">])</span>
-<span class="go">-0.765698403134212917425148</span>
-<span class="go">-0.765698403134212917425148</span>
-</pre></div>
-</div>
-<p>Integrals of high or fractional order:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="mf">0.5</span><span class="p">);</span> <span class="n">differint</span><span class="p">(</span><span class="n">airyai</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">(0.0 + 0.2453596101351438273844725j)</span>
-<span class="go">(0.0 + 0.2453596101351438273844725j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="mi">4</span><span class="p">);</span> <span class="n">differint</span><span class="p">(</span><span class="n">airyai</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.2939176441636809580339365</span>
-<span class="go">0.2939176441636809580339365</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">);</span> <span class="n">airyai</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">);</span> <span class="n">airyai</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>Integrals of the Ai-function can be evaluated at limit points:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="o">-</span><span class="mi">1000000</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">);</span> <span class="n">airyai</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-0.6666843728311539978751512</span>
-<span class="go">-0.6666666666666666666666667</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">);</span> <span class="n">airyai</span><span class="p">(</span><span class="o">+</span><span class="n">inf</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.3333333332991690159427932</span>
-<span class="go">0.3333333333333333333333333</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="o">+</span><span class="n">inf</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">);</span> <span class="n">airyai</span><span class="p">(</span><span class="o">+</span><span class="n">inf</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="o">-</span><span class="mi">1000000</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">);</span> <span class="n">airyai</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">666666.4078472650651209742</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyai</span><span class="p">(</span><span class="o">-</span><span class="mi">1000000</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">);</span> <span class="n">airyai</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">-333333074513.7520264995733</span>
-<span class="go">-inf</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../../mpmath/references.html#dlmf">[DLMF]</a> Chapter 9: Airy and Related Functions</li>
-<li><a class="reference internal" href="../../mpmath/references.html#wolframfunctions">[WolframFunctions]</a> section: Bessel-Type Functions</li>
-</ol>
-</dd></dl>
-
-</div>
-<div class="section" id="airybi">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">airybi()</span></tt><a class="headerlink" href="#airybi" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.airybi">
-<tt class="descclassname">mpmath.</tt><tt class="descname">airybi</tt><big>(</big><em>z</em>, <em>derivative=0</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.airybi" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the Airy function <span class="math">\(\operatorname{Bi}(z)\)</span>, which is
-the solution of the Airy differential equation <span class="math">\(f''(z) - z f(z) = 0\)</span>
-with initial conditions</p>
-<div class="math">
-\[\operatorname{Bi}(0) =
-    \frac{1}{3^{1/6}\Gamma\left(\frac{2}{3}\right)}\]\[\operatorname{Bi}'(0) =
-    \frac{3^{1/6}}{\Gamma\left(\frac{1}{3}\right)}.\]</div>
-<p>Like the Ai-function (see <a class="reference internal" href="#mpmath.airyai" title="mpmath.airyai"><tt class="xref py py-func docutils literal"><span class="pre">airyai()</span></tt></a>), the Bi-function
-is oscillatory for <span class="math">\(z &lt; 0\)</span>, but it grows rather than decreases
-for <span class="math">\(z &gt; 0\)</span>.</p>
-<p>Optionally, as for <a class="reference internal" href="#mpmath.airyai" title="mpmath.airyai"><tt class="xref py py-func docutils literal"><span class="pre">airyai()</span></tt></a>, derivatives, integrals
-and fractional derivatives can be computed with the <em>derivative</em>
-parameter.</p>
-<p>The Bi-function has infinitely many zeros along the negative
-half-axis, as well as complex zeros, which can all be computed
-with <a class="reference internal" href="#mpmath.airybizero" title="mpmath.airybizero"><tt class="xref py py-func docutils literal"><span class="pre">airybizero()</span></tt></a>.</p>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Airy function Bi(x), Bi&#39;(x) and int_0^x Bi(t) dt on the real line</span>
-<span class="n">f</span> <span class="o">=</span> <span class="n">airybi</span>
-<span class="n">f_diff</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">airybi</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">derivative</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="n">f_int</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">airybi</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">derivative</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">f_diff</span><span class="p">,</span> <span class="n">f_int</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/bi.png" src="../../../_images/bi.png" />
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Airy function Bi(z) in the complex plane</span>
-<span class="n">cplot</span><span class="p">(</span><span class="n">airybi</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span> <span class="n">points</span><span class="o">=</span><span class="mi">50000</span><span class="p">)</span>
-</pre></div>
-</div>
-<img alt="../../../_images/bi_c.png" src="../../../_images/bi_c.png" />
-<p><strong>Basic examples</strong></p>
-<p>Limits and values include:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">power</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="s">&#39;1/6&#39;</span><span class="p">)</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="s">&#39;2/3&#39;</span><span class="p">))</span>
-<span class="go">0.6149266274460007351509224</span>
-<span class="go">0.6149266274460007351509224</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1.207423594952871259436379</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.10399738949694461188869</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="n">inf</span><span class="p">);</span> <span class="n">airybi</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>Evaluation is supported for large magnitudes of the argument:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="o">-</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">0.02427388768016013160566747</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">6.041223996670201399005265e+288</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="mi">50</span><span class="o">+</span><span class="mi">50j</span><span class="p">)</span>
-<span class="go">(-5.322076267321435669290334e+63 + 1.478450291165243789749427e+65j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="o">-</span><span class="mi">50</span><span class="o">+</span><span class="mi">50j</span><span class="p">)</span>
-<span class="go">(-3.347525544923600321838281e+157 + 1.041242537363167632587245e+158j)</span>
-</pre></div>
-</div>
-<p>Huge arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">1.369385787943539818688433e+289529654602165</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">0.001775656141692932747610973</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">w</span> <span class="o">=</span> <span class="n">airybi</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">w</span><span class="o">.</span><span class="n">real</span>
-<span class="go">-6.559955931096196875845858e+186339621747689</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">w</span><span class="o">.</span><span class="n">imag</span>
-<span class="go">-6.822462726981357180929024e+186339621747690</span>
-</pre></div>
-</div>
-<p>The first real root of the Bi-function is:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">airybi</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">);</span> <span class="n">airybizero</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-1.17371322270912792491998</span>
-<span class="go">-1.17371322270912792491998</span>
-</pre></div>
-</div>
-<p><strong>Properties and relations</strong></p>
-<p>Verifying the Airy differential equation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="p">[</span><span class="o">-</span><span class="mf">3.4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mi">1</span><span class="o">+</span><span class="mi">2j</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="n">chop</span><span class="p">(</span><span class="n">airybi</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="n">z</span><span class="o">*</span><span class="n">airybi</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>The first few terms of the Taylor series expansion around <span class="math">\(z = 0\)</span>
-(every third term is zero):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">airybi</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
-<span class="go">[0.614927, 0.448288, 0.0, 0.102488, 0.0373574, 0.0]</span>
-</pre></div>
-</div>
-<p>The Airy functions can be expressed in terms of Bessel
-functions of order <span class="math">\(\pm 1/3\)</span>. For <span class="math">\(\Re[z] \le 0\)</span>, we have:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="o">-</span><span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">-0.1982896263749265432206449</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">power</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">,</span><span class="s">&#39;3/2&#39;</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="o">-</span><span class="n">mpf</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">besselj</span><span class="p">(</span><span class="s">&#39;-1/3&#39;</span><span class="p">,</span><span class="n">p</span><span class="p">)</span> <span class="o">-</span> <span class="n">besselj</span><span class="p">(</span><span class="s">&#39;1/3&#39;</span><span class="p">,</span><span class="n">p</span><span class="p">))</span>
-<span class="go">-0.1982896263749265432206449</span>
-</pre></div>
-</div>
-<p><strong>Derivatives and integrals</strong></p>
-<p>Derivatives of the Bi-function (directly and using <a class="reference internal" href="../../mpmath/calculus/differentiation.html#mpmath.diff" title="mpmath.diff"><tt class="xref py py-func docutils literal"><span class="pre">diff()</span></tt></a>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">diff</span><span class="p">(</span><span class="n">airybi</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">-0.675611222685258537668032</span>
-<span class="go">-0.675611222685258537668032</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">);</span> <span class="n">diff</span><span class="p">(</span><span class="n">airybi</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.5948688791247796296619346</span>
-<span class="go">0.5948688791247796296619346</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">diff</span><span class="p">(</span><span class="n">airybi</span><span class="p">,</span><span class="mi">1000</span><span class="p">)</span>
-<span class="go">1.710055114624614989262335e+9156</span>
-<span class="go">1.710055114624614989262335e+9156</span>
-</pre></div>
-</div>
-<p>Several derivatives at <span class="math">\(z = 0\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">);</span> <span class="n">airybi</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">airybi</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.6149266274460007351509224</span>
-<span class="go">0.4482883573538263579148237</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">);</span> <span class="n">airybi</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">4</span><span class="p">);</span> <span class="n">airybi</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">0.6149266274460007351509224</span>
-<span class="go">0.8965767147076527158296474</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">15</span><span class="p">);</span> <span class="n">airybi</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">16</span><span class="p">);</span> <span class="n">airybi</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">17</span><span class="p">)</span>
-<span class="go">2238.332923903442675949357</span>
-<span class="go">5522.912562599140729510628</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>The integral of the Bi-function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">);</span> <span class="n">quad</span><span class="p">(</span><span class="n">airybi</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-<span class="go">10.06200303130620056316655</span>
-<span class="go">10.06200303130620056316655</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">);</span> <span class="n">quad</span><span class="p">(</span><span class="n">airybi</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">10</span><span class="p">])</span>
-<span class="go">-0.01504042480614002045135483</span>
-<span class="go">-0.01504042480614002045135483</span>
-</pre></div>
-</div>
-<p>Integrals of high or fractional order:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="mf">0.5</span><span class="p">);</span> <span class="n">differint</span><span class="p">(</span><span class="n">airybi</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">(0.0 + 0.5019859055341699223453257j)</span>
-<span class="go">(0.0 + 0.5019859055341699223453257j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="mi">4</span><span class="p">);</span> <span class="n">differint</span><span class="p">(</span><span class="n">airybi</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.2809314599922447252139092</span>
-<span class="go">0.2809314599922447252139092</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">);</span> <span class="n">airybi</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">);</span> <span class="n">airybi</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>Integrals of the Bi-function can be evaluated at limit points:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="o">-</span><span class="mi">1000000</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">);</span> <span class="n">airybi</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.000002191261128063434047966873</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">);</span> <span class="n">airybi</span><span class="p">(</span><span class="o">+</span><span class="n">inf</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">147809803.1074067161675853</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="o">+</span><span class="n">inf</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">);</span> <span class="n">airybi</span><span class="p">(</span><span class="o">+</span><span class="n">inf</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="o">-</span><span class="mi">1000000</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">);</span> <span class="n">airybi</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.4482883750599908479851085</span>
-<span class="go">0.4482883573538263579148237</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="s">&#39;2/3&#39;</span><span class="p">)</span><span class="o">*</span><span class="n">power</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="s">&#39;2/3&#39;</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">0.4482883573538263579148237</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="o">-</span><span class="mi">100000</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">);</span> <span class="n">airybi</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">-44828.52827206932872493133</span>
-<span class="go">-inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="o">-</span><span class="mi">100000</span><span class="p">,</span><span class="o">-</span><span class="mi">4</span><span class="p">);</span> <span class="n">airybi</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span><span class="o">-</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">2241411040.437759489540248</span>
-<span class="go">+inf</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="airyaizero">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">airyaizero()</span></tt><a class="headerlink" href="#airyaizero" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.airyaizero">
-<tt class="descclassname">mpmath.</tt><tt class="descname">airyaizero</tt><big>(</big><em>k</em>, <em>derivative=0</em><big>)</big><a class="headerlink" href="#mpmath.airyaizero" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the <span class="math">\(k\)</span>-th zero of the Airy Ai-function,
-i.e. the <span class="math">\(k\)</span>-th number <span class="math">\(a_k\)</span> ordered by magnitude for which
-<span class="math">\(\operatorname{Ai}(a_k) = 0\)</span>.</p>
-<p>Optionally, with <em>derivative=1</em>, the corresponding
-zero <span class="math">\(a'_k\)</span> of the derivative function, i.e.
-<span class="math">\(\operatorname{Ai}'(a'_k) = 0\)</span>, is computed.</p>
-<p><strong>Examples</strong></p>
-<p>Some values of <span class="math">\(a_k\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyaizero</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-2.338107410459767038489197</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyaizero</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">-4.087949444130970616636989</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyaizero</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">-5.520559828095551059129856</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyaizero</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span>
-<span class="go">-281.0315196125215528353364</span>
-</pre></div>
-</div>
-<p>Some values of <span class="math">\(a'_k\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">airyaizero</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-1.018792971647471089017325</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyaizero</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-3.248197582179836537875424</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyaizero</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-4.820099211178735639400616</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airyaizero</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-280.9378080358935070607097</span>
-</pre></div>
-</div>
-<p>Verification:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">airyai</span><span class="p">(</span><span class="n">airyaizero</span><span class="p">(</span><span class="mi">1</span><span class="p">)))</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">airyai</span><span class="p">(</span><span class="n">airyaizero</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span><span class="mi">1</span><span class="p">))</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="airybizero">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">airybizero()</span></tt><a class="headerlink" href="#airybizero" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.airybizero">
-<tt class="descclassname">mpmath.</tt><tt class="descname">airybizero</tt><big>(</big><em>k</em>, <em>derivative=0</em>, <em>complex=0</em><big>)</big><a class="headerlink" href="#mpmath.airybizero" title="Permalink to this definition">¶</a></dt>
-<dd><p>With <em>complex=False</em>, gives the <span class="math">\(k\)</span>-th real zero of the Airy Bi-function,
-i.e. the <span class="math">\(k\)</span>-th number <span class="math">\(b_k\)</span> ordered by magnitude for which
-<span class="math">\(\operatorname{Bi}(b_k) = 0\)</span>.</p>
-<p>With <em>complex=True</em>, gives the <span class="math">\(k\)</span>-th complex zero in the upper
-half plane <span class="math">\(\beta_k\)</span>. Also the conjugate <span class="math">\(\overline{\beta_k}\)</span>
-is a zero.</p>
-<p>Optionally, with <em>derivative=1</em>, the corresponding
-zero <span class="math">\(b'_k\)</span> or <span class="math">\(\beta'_k\)</span> of the derivative function, i.e.
-<span class="math">\(\operatorname{Bi}'(b'_k) = 0\)</span> or <span class="math">\(\operatorname{Bi}'(\beta'_k) = 0\)</span>,
-is computed.</p>
-<p><strong>Examples</strong></p>
-<p>Some values of <span class="math">\(b_k\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybizero</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-1.17371322270912792491998</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybizero</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">-3.271093302836352715680228</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybizero</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">-4.830737841662015932667709</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybizero</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span>
-<span class="go">-280.9378112034152401578834</span>
-</pre></div>
-</div>
-<p>Some values of <span class="math">\(b_k\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">airybizero</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-2.294439682614123246622459</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybizero</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-4.073155089071828215552369</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybizero</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-5.512395729663599496259593</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybizero</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-281.0315164471118527161362</span>
-</pre></div>
-</div>
-<p>Some values of <span class="math">\(\beta_k\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">airybizero</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(0.9775448867316206859469927 + 2.141290706038744575749139j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybizero</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(1.896775013895336346627217 + 3.627291764358919410440499j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybizero</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(2.633157739354946595708019 + 4.855468179979844983174628j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybizero</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(140.4978560578493018899793 + 243.3907724215792121244867j)</span>
-</pre></div>
-</div>
-<p>Some values of <span class="math">\(\beta'_k\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">airybizero</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(0.2149470745374305676088329 + 1.100600143302797880647194j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybizero</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(1.458168309223507392028211 + 2.912249367458445419235083j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybizero</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(2.273760763013482299792362 + 4.254528549217097862167015j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybizero</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(140.4509972835270559730423 + 243.3096175398562811896208j)</span>
-</pre></div>
-</div>
-<p>Verification:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">airybi</span><span class="p">(</span><span class="n">airybizero</span><span class="p">(</span><span class="mi">1</span><span class="p">)))</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">airybi</span><span class="p">(</span><span class="n">airybizero</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span><span class="mi">1</span><span class="p">))</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span> <span class="o">=</span> <span class="n">airybizero</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">airybi</span><span class="p">(</span><span class="n">u</span><span class="p">))</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">airybi</span><span class="p">(</span><span class="n">conj</span><span class="p">(</span><span class="n">u</span><span class="p">)))</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>The complex zeros (in the upper and lower half-planes respectively)
-asymptotically approach the rays <span class="math">\(z = R \exp(\pm i \pi /3)\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">arg</span><span class="p">(</span><span class="n">airybizero</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">1.142532510286334022305364</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">arg</span><span class="p">(</span><span class="n">airybizero</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">1.047271114786212061583917</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">arg</span><span class="p">(</span><span class="n">airybizero</span><span class="p">(</span><span class="mi">1000000</span><span class="p">,</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">1.047197624741816183341355</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span>
-<span class="go">1.047197551196597746154214</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="scorergi">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">scorergi()</span></tt><a class="headerlink" href="#scorergi" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.scorergi">
-<tt class="descclassname">mpmath.</tt><tt class="descname">scorergi</tt><big>(</big><em>z</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.scorergi" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the Scorer function</p>
-<div class="math">
-\[\operatorname{Gi}(z) =
-\operatorname{Ai}(z) \int_0^z \operatorname{Bi}(t) dt +
-\operatorname{Bi}(z) \int_z^{\infty} \operatorname{Ai}(t) dt\]</div>
-<p>which gives a particular solution to the inhomogeneous Airy
-differential equation <span class="math">\(f''(z) - z f(z) = 1/\pi\)</span>. Another
-particular solution is given by the Scorer Hi-function
-(<a class="reference internal" href="#mpmath.scorerhi" title="mpmath.scorerhi"><tt class="xref py py-func docutils literal"><span class="pre">scorerhi()</span></tt></a>). The two functions are related as
-<span class="math">\(\operatorname{Gi}(z) + \operatorname{Hi}(z) = \operatorname{Bi}(z)\)</span>.</p>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Scorer function Gi(x) and Gi&#39;(x) on the real line</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">scorergi</span><span class="p">,</span> <span class="n">diffun</span><span class="p">(</span><span class="n">scorergi</span><span class="p">)],</span> <span class="p">[</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/gi.png" src="../../../_images/gi.png" />
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Scorer function Gi(z) in the complex plane</span>
-<span class="n">cplot</span><span class="p">(</span><span class="n">scorergi</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span> <span class="n">points</span><span class="o">=</span><span class="mi">50000</span><span class="p">)</span>
-</pre></div>
-</div>
-<img alt="../../../_images/gi_c.png" src="../../../_images/gi_c.png" />
-<p><strong>Examples</strong></p>
-<p>Some values and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorergi</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">power</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="s">&#39;7/6&#39;</span><span class="p">)</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="s">&#39;2/3&#39;</span><span class="p">))</span>
-<span class="go">0.2049755424820002450503075</span>
-<span class="go">0.2049755424820002450503075</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">scorergi</span><span class="p">,</span> <span class="mi">0</span><span class="p">);</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">power</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="s">&#39;5/6&#39;</span><span class="p">)</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="s">&#39;1/3&#39;</span><span class="p">))</span>
-<span class="go">0.1494294524512754526382746</span>
-<span class="go">0.1494294524512754526382746</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorergi</span><span class="p">(</span><span class="o">+</span><span class="n">inf</span><span class="p">);</span> <span class="n">scorergi</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorergi</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.2352184398104379375986902</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorergi</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-0.1166722172960152826494198</span>
-</pre></div>
-</div>
-<p>Evaluation for large arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">scorergi</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">0.03189600510067958798062034</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorergi</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">0.003183105228162961476590531</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorergi</span><span class="p">(</span><span class="mi">1000000</span><span class="p">)</span>
-<span class="go">0.0000003183098861837906721743873</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="mi">1000000</span><span class="p">)</span>
-<span class="go">0.0000003183098861837906715377675</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorergi</span><span class="p">(</span><span class="o">-</span><span class="mi">1000</span><span class="p">)</span>
-<span class="go">-0.08358288400262780392338014</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorergi</span><span class="p">(</span><span class="o">-</span><span class="mi">100000</span><span class="p">)</span>
-<span class="go">0.02886866118619660226809581</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorergi</span><span class="p">(</span><span class="mi">50</span><span class="o">+</span><span class="mi">10j</span><span class="p">)</span>
-<span class="go">(0.0061214102799778578790984 - 0.001224335676457532180747917j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorergi</span><span class="p">(</span><span class="o">-</span><span class="mi">50</span><span class="o">-</span><span class="mi">10j</span><span class="p">)</span>
-<span class="go">(5.236047850352252236372551e+29 - 3.08254224233701381482228e+29j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorergi</span><span class="p">(</span><span class="mi">100000j</span><span class="p">)</span>
-<span class="go">(-8.806659285336231052679025e+6474077 + 8.684731303500835514850962e+6474077j)</span>
-</pre></div>
-</div>
-<p>Verifying the connection between Gi and Hi:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mf">0.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorergi</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="n">scorerhi</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.7287469039362150078694543</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">airybi</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.7287469039362150078694543</span>
-</pre></div>
-</div>
-<p>Verifying the differential equation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="p">[</span><span class="o">-</span><span class="mf">3.4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mi">1</span><span class="o">+</span><span class="mi">2j</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="n">chop</span><span class="p">(</span><span class="n">diff</span><span class="p">(</span><span class="n">scorergi</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="n">z</span><span class="o">*</span><span class="n">scorergi</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">-0.3183098861837906715377675</span>
-<span class="go">-0.3183098861837906715377675</span>
-<span class="go">-0.3183098861837906715377675</span>
-<span class="go">-0.3183098861837906715377675</span>
-</pre></div>
-</div>
-<p>Verifying the integral representation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mf">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorergi</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.2447210432765581976910539</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ai</span><span class="p">,</span><span class="n">Bi</span> <span class="o">=</span> <span class="n">airyai</span><span class="p">,</span><span class="n">airybi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Bi</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">Ai</span><span class="p">(</span><span class="n">inf</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">-</span><span class="n">Ai</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span> <span class="o">+</span> <span class="n">Ai</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">Bi</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">-</span><span class="n">Bi</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span>
-<span class="go">0.2447210432765581976910539</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../../mpmath/references.html#dlmf">[DLMF]</a> section 9.12: Scorer Functions</li>
-</ol>
-</dd></dl>
-
-</div>
-<div class="section" id="scorerhi">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">scorerhi()</span></tt><a class="headerlink" href="#scorerhi" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.scorerhi">
-<tt class="descclassname">mpmath.</tt><tt class="descname">scorerhi</tt><big>(</big><em>z</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.scorerhi" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the second Scorer function</p>
-<div class="math">
-\[\operatorname{Hi}(z) =
-\operatorname{Bi}(z) \int_{-\infty}^z \operatorname{Ai}(t) dt -
-\operatorname{Ai}(z) \int_{-\infty}^z \operatorname{Bi}(t) dt\]</div>
-<p>which gives a particular solution to the inhomogeneous Airy
-differential equation <span class="math">\(f''(z) - z f(z) = 1/\pi\)</span>. See also
-<a class="reference internal" href="#mpmath.scorergi" title="mpmath.scorergi"><tt class="xref py py-func docutils literal"><span class="pre">scorergi()</span></tt></a>.</p>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Scorer function Hi(x) and Hi&#39;(x) on the real line</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">scorerhi</span><span class="p">,</span> <span class="n">diffun</span><span class="p">(</span><span class="n">scorerhi</span><span class="p">)],</span> <span class="p">[</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/hi.png" src="../../../_images/hi.png" />
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Scorer function Hi(z) in the complex plane</span>
-<span class="n">cplot</span><span class="p">(</span><span class="n">scorerhi</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">],</span> <span class="n">points</span><span class="o">=</span><span class="mi">50000</span><span class="p">)</span>
-</pre></div>
-</div>
-<img alt="../../../_images/hi_c.png" src="../../../_images/hi_c.png" />
-<p><strong>Examples</strong></p>
-<p>Some values and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorerhi</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span> <span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="n">power</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="s">&#39;7/6&#39;</span><span class="p">)</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="s">&#39;2/3&#39;</span><span class="p">))</span>
-<span class="go">0.4099510849640004901006149</span>
-<span class="go">0.4099510849640004901006149</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">scorerhi</span><span class="p">,</span><span class="mi">0</span><span class="p">);</span> <span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="n">power</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="s">&#39;5/6&#39;</span><span class="p">)</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="s">&#39;1/3&#39;</span><span class="p">))</span>
-<span class="go">0.2988589049025509052765491</span>
-<span class="go">0.2988589049025509052765491</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorerhi</span><span class="p">(</span><span class="o">+</span><span class="n">inf</span><span class="p">);</span> <span class="n">scorerhi</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorerhi</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.9722051551424333218376886</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorerhi</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.2206696067929598945381098</span>
-</pre></div>
-</div>
-<p>Evaluation for large arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">scorerhi</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">455641153.5163291358991077</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorerhi</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">6.041223996670201399005265e+288</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorerhi</span><span class="p">(</span><span class="mi">1000000</span><span class="p">)</span>
-<span class="go">7.138269638197858094311122e+289529652</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorerhi</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">0.0317685352825022727415011</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorerhi</span><span class="p">(</span><span class="o">-</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">0.003183092495767499864680483</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorerhi</span><span class="p">(</span><span class="mi">100j</span><span class="p">)</span>
-<span class="go">(-6.366197716545672122983857e-9 + 0.003183098861710582761688475j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorerhi</span><span class="p">(</span><span class="mi">50</span><span class="o">+</span><span class="mi">50j</span><span class="p">)</span>
-<span class="go">(-5.322076267321435669290334e+63 + 1.478450291165243789749427e+65j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorerhi</span><span class="p">(</span><span class="o">-</span><span class="mi">1000</span><span class="o">-</span><span class="mi">1000j</span><span class="p">)</span>
-<span class="go">(0.0001591549432510502796565538 - 0.000159154943091895334973109j)</span>
-</pre></div>
-</div>
-<p>Verifying the differential equation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="p">[</span><span class="o">-</span><span class="mf">3.4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="o">+</span><span class="mi">2j</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="n">chop</span><span class="p">(</span><span class="n">diff</span><span class="p">(</span><span class="n">scorerhi</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="n">z</span><span class="o">*</span><span class="n">scorerhi</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">0.3183098861837906715377675</span>
-<span class="go">0.3183098861837906715377675</span>
-<span class="go">0.3183098861837906715377675</span>
-<span class="go">0.3183098861837906715377675</span>
-</pre></div>
-</div>
-<p>Verifying the integral representation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mf">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">scorerhi</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.6095559998265972956089949</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ai</span><span class="p">,</span><span class="n">Bi</span> <span class="o">=</span> <span class="n">airyai</span><span class="p">,</span><span class="n">airybi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Bi</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">Ai</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">-</span><span class="n">Ai</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span> <span class="o">-</span> <span class="n">Ai</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">Bi</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">-</span><span class="n">Bi</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span>
-<span class="go">0.6095559998265972956089949</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="coulomb-wave-functions">
-<h2>Coulomb wave functions<a class="headerlink" href="#coulomb-wave-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="coulombf">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">coulombf()</span></tt><a class="headerlink" href="#coulombf" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.coulombf">
-<tt class="descclassname">mpmath.</tt><tt class="descname">coulombf</tt><big>(</big><em>l</em>, <em>eta</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.coulombf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculates the regular Coulomb wave function</p>
-<div class="math">
-\[F_l(\eta,z) = C_l(\eta) z^{l+1} e^{-iz} \,_1F_1(l+1-i\eta, 2l+2, 2iz)\]</div>
-<p>where the normalization constant <span class="math">\(C_l(\eta)\)</span> is as calculated by
-<a class="reference internal" href="#mpmath.coulombc" title="mpmath.coulombc"><tt class="xref py py-func docutils literal"><span class="pre">coulombc()</span></tt></a>. This function solves the differential equation</p>
-<div class="math">
-\[f''(z) + \left(1-\frac{2\eta}{z}-\frac{l(l+1)}{z^2}\right) f(z) = 0.\]</div>
-<p>A second linearly independent solution is given by the irregular
-Coulomb wave function <span class="math">\(G_l(\eta,z)\)</span> (see <a class="reference internal" href="#mpmath.coulombg" title="mpmath.coulombg"><tt class="xref py py-func docutils literal"><span class="pre">coulombg()</span></tt></a>)
-and thus the general solution is
-<span class="math">\(f(z) = C_1 F_l(\eta,z) + C_2 G_l(\eta,z)\)</span> for arbitrary
-constants <span class="math">\(C_1\)</span>, <span class="math">\(C_2\)</span>.
-Physically, the Coulomb wave functions give the radial solution
-to the Schrodinger equation for a point particle in a <span class="math">\(1/z\)</span> potential; <span class="math">\(z\)</span> is
-then the radius and <span class="math">\(l\)</span>, <span class="math">\(\eta\)</span> are quantum numbers.</p>
-<p>The Coulomb wave functions with real parameters are defined
-in Abramowitz &amp; Stegun, section 14. However, all parameters are permitted
-to be complex in this implementation (see references).</p>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Regular Coulomb wave functions -- equivalent to figure 14.3 in A&amp;S</span>
-<span class="n">F1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">coulombf</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">F2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">coulombf</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">F3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">coulombf</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">F4</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">coulombf</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">F5</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">coulombf</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">F1</span><span class="p">,</span><span class="n">F2</span><span class="p">,</span><span class="n">F3</span><span class="p">,</span><span class="n">F4</span><span class="p">,</span><span class="n">F5</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">25</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mf">1.2</span><span class="p">,</span><span class="mf">1.6</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/coulombf.png" src="../../../_images/coulombf.png" />
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Regular Coulomb wave function in the complex plane</span>
-<span class="n">cplot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">coulombf</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">z</span><span class="p">),</span> <span class="n">points</span><span class="o">=</span><span class="mi">50000</span><span class="p">)</span>
-</pre></div>
-</div>
-<img alt="../../../_images/coulombf_c.png" src="../../../_images/coulombf_c.png" />
-<p><strong>Examples</strong></p>
-<p>Evaluation is supported for arbitrary magnitudes of <span class="math">\(z\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">coulombf</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">)</span>
-<span class="go">0.4080998961088761187426445</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">coulombf</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">)</span>
-<span class="go">0.7103040849492536747533465</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">coulombf</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="s">&#39;1e-10&#39;</span><span class="p">)</span>
-<span class="go">4.143324917492256448770769e-33</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">coulombf</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
-<span class="go">0.4482623140325567050716179</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">coulombf</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">-0.066804196437694360046619</span>
-</pre></div>
-</div>
-<p>Verifying the differential equation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">l</span><span class="p">,</span> <span class="n">eta</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">2.75</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="p">,</span> <span class="n">B</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">A</span><span class="o">*</span><span class="n">coulombf</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="n">eta</span><span class="p">,</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="n">B</span><span class="o">*</span><span class="n">coulombg</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="n">eta</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">eta</span><span class="o">/</span><span class="n">z</span> <span class="o">-</span> <span class="n">l</span><span class="o">*</span><span class="p">(</span><span class="n">l</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>A Wronskian relation satisfied by the Coulomb wave functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">l</span> <span class="o">=</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eta</span> <span class="o">=</span> <span class="mf">1.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">coulombf</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="n">eta</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">coulombg</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="n">eta</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="p">[</span><span class="mf">3.5</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="n">chop</span><span class="p">(</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">G</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">-</span> <span class="n">F</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">G</span><span class="p">,</span><span class="n">z</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">1.0</span>
-<span class="go">1.0</span>
-<span class="go">1.0</span>
-</pre></div>
-</div>
-<p>Another Wronskian relation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">coulombf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">coulombg</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="p">[</span><span class="mf">3.5</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="n">chop</span><span class="p">(</span><span class="n">F</span><span class="p">(</span><span class="n">l</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="n">eta</span><span class="p">,</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">G</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="n">eta</span><span class="p">,</span><span class="n">z</span><span class="p">)</span><span class="o">-</span><span class="n">F</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="n">eta</span><span class="p">,</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">G</span><span class="p">(</span><span class="n">l</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="n">eta</span><span class="p">,</span><span class="n">z</span><span class="p">)</span> <span class="o">-</span> <span class="n">l</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">l</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">eta</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>An integral identity connecting the regular and irregular wave functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">l</span><span class="p">,</span> <span class="n">eta</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="mi">4</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mi">2</span><span class="o">-</span><span class="n">j</span><span class="p">,</span> <span class="mi">5</span><span class="o">+</span><span class="mi">2j</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">coulombf</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="n">eta</span><span class="p">,</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="n">j</span><span class="o">*</span><span class="n">coulombg</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="n">eta</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">(0.7997977752284033239714479 + 0.9294486669502295512503127j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">t</span><span class="p">)</span><span class="o">*</span><span class="n">t</span><span class="o">**</span><span class="p">(</span><span class="n">l</span><span class="o">-</span><span class="n">j</span><span class="o">*</span><span class="n">eta</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">t</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">j</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">l</span><span class="o">+</span><span class="n">j</span><span class="o">*</span><span class="n">eta</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">j</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">j</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">l</span><span class="p">)</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">l</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">coulombc</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="n">eta</span><span class="p">)</span><span class="o">*</span><span class="n">quad</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">(0.7997977752284033239714479 + 0.9294486669502295512503127j)</span>
-</pre></div>
-</div>
-<p>Some test case with complex parameters, taken from Michel [2]:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">coulombf</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mf">0.1j</span><span class="p">,</span> <span class="mi">50</span><span class="o">+</span><span class="mi">50j</span><span class="p">,</span> <span class="mf">100.156</span><span class="p">)</span>
-<span class="go">(-1.02107292320897e+15 - 2.83675545731519e+15j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">coulombg</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mf">0.1j</span><span class="p">,</span> <span class="mi">50</span><span class="o">+</span><span class="mi">50j</span><span class="p">,</span> <span class="mf">100.156</span><span class="p">)</span>
-<span class="go">(2.83675545731519e+15 - 1.02107292320897e+15j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">coulombf</span><span class="p">(</span><span class="mf">1e-5j</span><span class="p">,</span> <span class="mi">10</span><span class="o">+</span><span class="mf">1e-5j</span><span class="p">,</span> <span class="mf">0.1</span><span class="o">+</span><span class="mf">1e-6j</span><span class="p">)</span>
-<span class="go">(4.30566371247811e-14 - 9.03347835361657e-19j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">coulombg</span><span class="p">(</span><span class="mf">1e-5j</span><span class="p">,</span> <span class="mi">10</span><span class="o">+</span><span class="mf">1e-5j</span><span class="p">,</span> <span class="mf">0.1</span><span class="o">+</span><span class="mf">1e-6j</span><span class="p">)</span>
-<span class="go">(778709182061.134 + 18418936.2660553j)</span>
-</pre></div>
-</div>
-<p>The following reproduces a table in Abramowitz &amp; Stegun, at twice
-the precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eta</span> <span class="o">=</span> <span class="mi">2</span><span class="p">;</span> <span class="n">z</span> <span class="o">=</span> <span class="mi">5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">coulombf</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="n">eta</span><span class="p">,</span><span class="n">z</span><span class="p">),</span>
-<span class="gp">... </span>        <span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">coulombf</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="n">eta</span><span class="p">,</span><span class="n">z</span><span class="p">),</span> <span class="n">z</span><span class="p">)))</span>
-<span class="gp">...</span>
-<span class="go">5 0.09079533488 0.1042553261</span>
-<span class="go">4 0.2148205331 0.2029591779</span>
-<span class="go">3 0.4313159311 0.320534053</span>
-<span class="go">2 0.7212774133 0.3952408216</span>
-<span class="go">1 0.9935056752 0.3708676452</span>
-<span class="go">0 1.143337392 0.2937960375</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li>I.J. Thompson &amp; A.R. Barnett, &#8220;Coulomb and Bessel Functions of Complex
-Arguments and Order&#8221;, J. Comp. Phys., vol 64, no. 2, June 1986.</li>
-<li>N. Michel, &#8220;Precise Coulomb wave functions for a wide range of
-complex <span class="math">\(l\)</span>, <span class="math">\(\eta\)</span> and <span class="math">\(z\)</span>&#8221;, <a class="reference external" href="http://arxiv.org/abs/physics/0702051v1">http://arxiv.org/abs/physics/0702051v1</a></li>
-</ol>
-</dd></dl>
-
-</div>
-<div class="section" id="coulombg">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">coulombg()</span></tt><a class="headerlink" href="#coulombg" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.coulombg">
-<tt class="descclassname">mpmath.</tt><tt class="descname">coulombg</tt><big>(</big><em>l</em>, <em>eta</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.coulombg" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculates the irregular Coulomb wave function</p>
-<div class="math">
-\[G_l(\eta,z) = \frac{F_l(\eta,z) \cos(\chi) - F_{-l-1}(\eta,z)}{\sin(\chi)}\]</div>
-<p>where <span class="math">\(\chi = \sigma_l - \sigma_{-l-1} - (l+1/2) \pi\)</span>
-and <span class="math">\(\sigma_l(\eta) = (\ln \Gamma(1+l+i\eta)-\ln \Gamma(1+l-i\eta))/(2i)\)</span>.</p>
-<p>See <a class="reference internal" href="#mpmath.coulombf" title="mpmath.coulombf"><tt class="xref py py-func docutils literal"><span class="pre">coulombf()</span></tt></a> for additional information.</p>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Irregular Coulomb wave functions -- equivalent to figure 14.5 in A&amp;S</span>
-<span class="n">F1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">coulombg</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">F2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">coulombg</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">F3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">coulombg</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">F4</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">coulombg</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">F5</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">coulombg</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">F1</span><span class="p">,</span><span class="n">F2</span><span class="p">,</span><span class="n">F3</span><span class="p">,</span><span class="n">F4</span><span class="p">,</span><span class="n">F5</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">30</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/coulombg.png" src="../../../_images/coulombg.png" />
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Irregular Coulomb wave function in the complex plane</span>
-<span class="n">cplot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">coulombg</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">z</span><span class="p">),</span> <span class="n">points</span><span class="o">=</span><span class="mi">50000</span><span class="p">)</span>
-</pre></div>
-</div>
-<img alt="../../../_images/coulombg_c.png" src="../../../_images/coulombg_c.png" />
-<p><strong>Examples</strong></p>
-<p>Evaluation is supported for arbitrary magnitudes of <span class="math">\(z\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">coulombg</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">)</span>
-<span class="go">1.380011900612186346255524</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">coulombg</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">)</span>
-<span class="go">1.919153700722748795245926</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">coulombg</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="s">&#39;1e-10&#39;</span><span class="p">)</span>
-<span class="go">201126715824.7329115106793</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">coulombg</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
-<span class="go">0.1802071520691149410425512</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">coulombg</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">0.652103020061678070929794</span>
-</pre></div>
-</div>
-<p>The following reproduces a table in Abramowitz &amp; Stegun,
-at twice the precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eta</span> <span class="o">=</span> <span class="mi">2</span><span class="p">;</span> <span class="n">z</span> <span class="o">=</span> <span class="mi">5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">l</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">coulombg</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="n">eta</span><span class="p">,</span><span class="n">z</span><span class="p">),</span>
-<span class="gp">... </span>        <span class="o">-</span><span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">coulombg</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="n">eta</span><span class="p">,</span><span class="n">z</span><span class="p">),</span> <span class="n">z</span><span class="p">)))</span>
-<span class="gp">...</span>
-<span class="go">1 1.08148276 0.6028279961</span>
-<span class="go">2 1.496877075 0.5661803178</span>
-<span class="go">3 2.048694714 0.7959909551</span>
-<span class="go">4 3.09408669 1.731802374</span>
-<span class="go">5 5.629840456 4.549343289</span>
-</pre></div>
-</div>
-<p>Evaluation close to the singularity at <span class="math">\(z = 0\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">coulombg</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">3088184933.67358</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">coulombg</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="s">&#39;1e-10&#39;</span><span class="p">)</span>
-<span class="go">5554866000719.8</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">coulombg</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="s">&#39;1e-100&#39;</span><span class="p">)</span>
-<span class="go">5554866221524.1</span>
-</pre></div>
-</div>
-<p>Evaluation with a half-integer value for <span class="math">\(l\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">coulombg</span><span class="p">(</span><span class="mf">1.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">0.852320038297334</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="coulombc">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">coulombc()</span></tt><a class="headerlink" href="#coulombc" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.coulombc">
-<tt class="descclassname">mpmath.</tt><tt class="descname">coulombc</tt><big>(</big><em>l</em>, <em>eta</em><big>)</big><a class="headerlink" href="#mpmath.coulombc" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the normalizing Gamow constant for Coulomb wave functions,</p>
-<div class="math">
-\[C_l(\eta) = 2^l \exp\left(-\pi \eta/2 + [\ln \Gamma(1+l+i\eta) +
-    \ln \Gamma(1+l-i\eta)]/2 - \ln \Gamma(2l+2)\right),\]</div>
-<p>where the log gamma function with continuous imaginary part
-away from the negative half axis (see <a class="reference internal" href="../functions/gamma.html#mpmath.loggamma" title="mpmath.loggamma"><tt class="xref py py-func docutils literal"><span class="pre">loggamma()</span></tt></a>) is implied.</p>
-<p>This function is used internally for the calculation of
-Coulomb wave functions, and automatically cached to make multiple
-evaluations with fixed <span class="math">\(l\)</span>, <span class="math">\(\eta\)</span> fast.</p>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="confluent-u-and-whittaker-functions">
-<h2>Confluent U and Whittaker functions<a class="headerlink" href="#confluent-u-and-whittaker-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="hyperu">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">hyperu()</span></tt><a class="headerlink" href="#hyperu" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.hyperu">
-<tt class="descclassname">mpmath.</tt><tt class="descname">hyperu</tt><big>(</big><em>a</em>, <em>b</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.hyperu" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the Tricomi confluent hypergeometric function <span class="math">\(U\)</span>, also known as
-the Kummer or confluent hypergeometric function of the second kind. This
-function gives a second linearly independent solution to the confluent
-hypergeometric differential equation (the first is provided by <span class="math">\(\,_1F_1\)</span>  &#8211;
-see <a class="reference internal" href="../functions/hypergeometric.html#mpmath.hyp1f1" title="mpmath.hyp1f1"><tt class="xref py py-func docutils literal"><span class="pre">hyp1f1()</span></tt></a>).</p>
-<p><strong>Examples</strong></p>
-<p>Evaluation for arbitrary complex arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyperu</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">0.0625</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyperu</span><span class="p">(</span><span class="mf">0.25</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
-<span class="go">0.1779949416140579573763523</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyperu</span><span class="p">(</span><span class="mf">0.25</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="o">-</span><span class="mi">1000</span><span class="p">)</span>
-<span class="go">(0.1256256609322773150118907 - 0.1256256609322773150118907j)</span>
-</pre></div>
-</div>
-<p>The <span class="math">\(U\)</span> function may be singular at <span class="math">\(z = 0\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">hyperu</span><span class="p">(</span><span class="mf">1.5</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyperu</span><span class="p">(</span><span class="mf">1.5</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">0.1719434921288400112603671</span>
-</pre></div>
-</div>
-<p>Verifying the differential equation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">hyperu</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="p">[</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="n">chop</span><span class="p">(</span><span class="n">z</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">)</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>An integral representation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">z</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">,</span> <span class="mf">4.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyperu</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.06674960718150520648014567</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="o">*</span><span class="n">t</span><span class="p">)</span><span class="o">*</span><span class="n">t</span><span class="o">**</span><span class="p">(</span><span class="n">a</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">t</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="o">-</span><span class="mi">1</span><span class="p">),[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span> <span class="o">/</span> <span class="n">gamma</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="go">0.06674960718150520648014567</span>
-</pre></div>
-</div>
-<p>[1] <a class="reference external" href="http://people.math.sfu.ca/~cbm/aands/page_504.htm">http://people.math.sfu.ca/~cbm/aands/page_504.htm</a></p>
-</dd></dl>
-
-</div>
-<div class="section" id="whitm">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">whitm()</span></tt><a class="headerlink" href="#whitm" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.whitm">
-<tt class="descclassname">mpmath.</tt><tt class="descname">whitm</tt><big>(</big><em>k</em>, <em>m</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.whitm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the Whittaker function <span class="math">\(M(k,m,z)\)</span>, which gives a solution
-to the Whittaker differential equation</p>
-<div class="math">
-\[\frac{d^2f}{dz^2} + \left(-\frac{1}{4}+\frac{k}{z}+
-  \frac{(\frac{1}{4}-m^2)}{z^2}\right) f = 0.\]</div>
-<p>A second solution is given by <a class="reference internal" href="#mpmath.whitw" title="mpmath.whitw"><tt class="xref py py-func docutils literal"><span class="pre">whitw()</span></tt></a>.</p>
-<p>The Whittaker functions are defined in Abramowitz &amp; Stegun, section 13.1.
-They are alternate forms of the confluent hypergeometric functions
-<span class="math">\(\,_1F_1\)</span> and <span class="math">\(U\)</span>:</p>
-<div class="math">
-\[M(k,m,z) = e^{-\frac{1}{2}z} z^{\frac{1}{2}+m}
-    \,_1F_1(\tfrac{1}{2}+m-k, 1+2m, z)\]\[W(k,m,z) = e^{-\frac{1}{2}z} z^{\frac{1}{2}+m}
-    U(\tfrac{1}{2}+m-k, 1+2m, z).\]</div>
-<p><strong>Examples</strong></p>
-<p>Evaluation for arbitrary real and complex arguments is supported:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">whitm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">0.7302596799460411820509668</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">whitm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(0.0 - 1.417977827655098025684246j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">whitm</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(3.245477713363581112736478 - 0.822879187542699127327782j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">whitm</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">100000</span><span class="p">)</span>
-<span class="go">4.303985255686378497193063e+21707</span>
-</pre></div>
-</div>
-<p>Evaluation at zero:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">whitm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">);</span> <span class="n">whitm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span><span class="mi">0</span><span class="p">);</span> <span class="n">whitm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="go">nan</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>We can verify that <a class="reference internal" href="#mpmath.whitm" title="mpmath.whitm"><tt class="xref py py-func docutils literal"><span class="pre">whitm()</span></tt></a> numerically satisfies the
-differential equation for arbitrarily chosen values:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">0.25</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">1.5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">whitm</span><span class="p">(</span><span class="n">k</span><span class="p">,</span><span class="n">m</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="o">+</span><span class="mi">2j</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="n">chop</span><span class="p">(</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="o">-</span><span class="mf">0.25</span> <span class="o">+</span> <span class="n">k</span><span class="o">/</span><span class="n">z</span> <span class="o">+</span> <span class="p">(</span><span class="mf">0.25</span><span class="o">-</span><span class="n">m</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>An integral involving both <a class="reference internal" href="#mpmath.whitm" title="mpmath.whitm"><tt class="xref py py-func docutils literal"><span class="pre">whitm()</span></tt></a> and <a class="reference internal" href="#mpmath.whitw" title="mpmath.whitw"><tt class="xref py py-func docutils literal"><span class="pre">whitw()</span></tt></a>,
-verifying evaluation along the real axis:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">whitm</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">whitw</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">3.438869842576800225207341</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">128</span><span class="o">/</span><span class="p">(</span><span class="mi">21</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">))</span>
-<span class="go">3.438869842576800225207341</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="whitw">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">whitw()</span></tt><a class="headerlink" href="#whitw" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.whitw">
-<tt class="descclassname">mpmath.</tt><tt class="descname">whitw</tt><big>(</big><em>k</em>, <em>m</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.whitw" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the Whittaker function <span class="math">\(W(k,m,z)\)</span>, which gives a second
-solution to the Whittaker differential equation. (See <a class="reference internal" href="#mpmath.whitm" title="mpmath.whitm"><tt class="xref py py-func docutils literal"><span class="pre">whitm()</span></tt></a>.)</p>
-<p><strong>Examples</strong></p>
-<p>Evaluation for arbitrary real and complex arguments is supported:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">whitw</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">1.19532063107581155661012</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">whitw</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(-0.9424875979222187313924639 - 0.2607738054097702293308689j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">whitw</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(0.1782899315111033879430369 - 0.01609578360403649340169406j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">whitw</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">100000</span><span class="p">)</span>
-<span class="go">1.887705114889527446891274e-21705</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">whitw</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
-<span class="go">1.905250692824046162462058e-24</span>
-</pre></div>
-</div>
-<p>Evaluation at zero:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="n">whitw</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="go">+inf</span>
-<span class="go">nan</span>
-<span class="go">0.0</span>
-<span class="go">nan</span>
-<span class="go">+inf</span>
-</pre></div>
-</div>
-<p>We can verify that <a class="reference internal" href="#mpmath.whitw" title="mpmath.whitw"><tt class="xref py py-func docutils literal"><span class="pre">whitw()</span></tt></a> numerically satisfies the
-differential equation for arbitrarily chosen values:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">0.25</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">1.5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">whitw</span><span class="p">(</span><span class="n">k</span><span class="p">,</span><span class="n">m</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="o">+</span><span class="mi">2j</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="n">chop</span><span class="p">(</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="o">-</span><span class="mf">0.25</span> <span class="o">+</span> <span class="n">k</span><span class="o">/</span><span class="n">z</span> <span class="o">+</span> <span class="p">(</span><span class="mf">0.25</span><span class="o">-</span><span class="n">m</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="parabolic-cylinder-functions">
-<h2>Parabolic cylinder functions<a class="headerlink" href="#parabolic-cylinder-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="pcfd">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">pcfd()</span></tt><a class="headerlink" href="#pcfd" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.pcfd">
-<tt class="descclassname">mpmath.</tt><tt class="descname">pcfd</tt><big>(</big><em>n</em>, <em>z</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.pcfd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the parabolic cylinder function in Whittaker&#8217;s notation
-<span class="math">\(D_n(z) = U(-n-1/2, z)\)</span> (see <a class="reference internal" href="#mpmath.pcfu" title="mpmath.pcfu"><tt class="xref py py-func docutils literal"><span class="pre">pcfu()</span></tt></a>).
-It solves the differential equation</p>
-<div class="math">
-\[y'' + \left(n + \frac{1}{2} - \frac{1}{4} z^2\right) y = 0.\]</div>
-<p>and can be represented in terms of Hermite polynomials
-(see <a class="reference internal" href="../functions/orthogonal.html#mpmath.hermite" title="mpmath.hermite"><tt class="xref py py-func docutils literal"><span class="pre">hermite()</span></tt></a>) as</p>
-<div class="math">
-\[D_n(z) = 2^{-n/2} e^{-z^2/4} H_n\left(\frac{z}{\sqrt{2}}\right).\]</div>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Parabolic cylinder function D_n(x) on the real line for n=0,1,2,3,4</span>
-<span class="n">d0</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">pcfd</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">d1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">pcfd</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">d2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">pcfd</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">d3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">pcfd</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">d4</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">pcfd</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">d0</span><span class="p">,</span><span class="n">d1</span><span class="p">,</span><span class="n">d2</span><span class="p">,</span><span class="n">d3</span><span class="p">,</span><span class="n">d4</span><span class="p">],[</span><span class="o">-</span><span class="mi">7</span><span class="p">,</span><span class="mi">7</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/pcfd.png" src="../../../_images/pcfd.png" />
-<p><strong>Examples</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pcfd</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">);</span> <span class="n">pcfd</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">);</span> <span class="n">pcfd</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">);</span> <span class="n">pcfd</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="go">0.0</span>
-<span class="go">-1.0</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pcfd</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="mi">0</span><span class="p">);</span> <span class="n">pcfd</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">3.0</span>
-<span class="go">0.6266570686577501256039413</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pcfd</span><span class="p">(</span><span class="s">&#39;1/2&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(-5.363331161232920734849056 - 3.858877821790010714163487j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pcfd</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">1.374906442631438038871515e-9</span>
-</pre></div>
-</div>
-<p>Verifying the differential equation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">2.5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">pcfd</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mf">1.75</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">diff</span><span class="p">(</span><span class="n">y</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mf">0.5</span><span class="o">-</span><span class="mf">0.25</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">y</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>Rational Taylor series expansion when <span class="math">\(n\)</span> is an integer:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">pcfd</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="n">z</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
-<span class="go">[0.0, 15.0, 0.0, -13.75, 0.0, 3.96875, 0.0, -0.6015625]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="pcfu">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">pcfu()</span></tt><a class="headerlink" href="#pcfu" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.pcfu">
-<tt class="descclassname">mpmath.</tt><tt class="descname">pcfu</tt><big>(</big><em>a</em>, <em>z</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.pcfu" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the parabolic cylinder function <span class="math">\(U(a,z)\)</span>, which may be
-defined for <span class="math">\(\Re(z) &gt; 0\)</span> in terms of the confluent
-U-function (see <a class="reference internal" href="#mpmath.hyperu" title="mpmath.hyperu"><tt class="xref py py-func docutils literal"><span class="pre">hyperu()</span></tt></a>) by</p>
-<div class="math">
-\[U(a,z) = 2^{-\frac{1}{4}-\frac{a}{2}} e^{-\frac{1}{4} z^2}
-    U\left(\frac{a}{2}+\frac{1}{4},
-    \frac{1}{2}, \frac{1}{2}z^2\right)\]</div>
-<p>or, for arbitrary <span class="math">\(z\)</span>,</p>
-<div class="math">
-\[e^{-\frac{1}{4}z^2} U(a,z) =
-    U(a,0) \,_1F_1\left(-\tfrac{a}{2}+\tfrac{1}{4};
-    \tfrac{1}{2}; -\tfrac{1}{2}z^2\right) +
-    U'(a,0) z \,_1F_1\left(-\tfrac{a}{2}+\tfrac{3}{4};
-    \tfrac{3}{2}; -\tfrac{1}{2}z^2\right).\]</div>
-<p><strong>Examples</strong></p>
-<p>Connection to other functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pcfu</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.03210358129311151450551963</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="n">erfc</span><span class="p">(</span><span class="n">z</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">0.03210358129311151450551963</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pcfu</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span><span class="o">-</span><span class="n">z</span><span class="p">)</span>
-<span class="go">23.75012332835297233711255</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="n">erfc</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">23.75012332835297233711255</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pcfu</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span><span class="o">-</span><span class="n">z</span><span class="p">)</span>
-<span class="go">23.75012332835297233711255</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="n">erfc</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">23.75012332835297233711255</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="pcfv">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">pcfv()</span></tt><a class="headerlink" href="#pcfv" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.pcfv">
-<tt class="descclassname">mpmath.</tt><tt class="descname">pcfv</tt><big>(</big><em>a</em>, <em>z</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.pcfv" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the parabolic cylinder function <span class="math">\(V(a,z)\)</span>, which can be
-represented in terms of <a class="reference internal" href="#mpmath.pcfu" title="mpmath.pcfu"><tt class="xref py py-func docutils literal"><span class="pre">pcfu()</span></tt></a> as</p>
-<div class="math">
-\[V(a,z) = \frac{\Gamma(a+\tfrac{1}{2}) (U(a,-z)-\sin(\pi a) U(a,z)}{\pi}.\]</div>
-<p><strong>Examples</strong></p>
-<p>Wronskian relation between <span class="math">\(U\)</span> and <span class="math">\(V\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pcfu</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">pcfv</span><span class="p">,(</span><span class="n">a</span><span class="p">,</span><span class="n">z</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span><span class="o">-</span><span class="n">diff</span><span class="p">(</span><span class="n">pcfu</span><span class="p">,(</span><span class="n">a</span><span class="p">,</span><span class="n">z</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span><span class="o">*</span><span class="n">pcfv</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.7978845608028653558798921</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">/</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">0.7978845608028653558798921</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pcfu</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">pcfv</span><span class="p">,(</span><span class="n">a</span><span class="p">,</span><span class="n">z</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span><span class="o">-</span><span class="n">diff</span><span class="p">(</span><span class="n">pcfu</span><span class="p">,(</span><span class="n">a</span><span class="p">,</span><span class="n">z</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span><span class="o">*</span><span class="n">pcfv</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.7978845608028653558798921</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="mf">0.25</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pcfu</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">pcfv</span><span class="p">,(</span><span class="n">a</span><span class="p">,</span><span class="n">z</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span><span class="o">-</span><span class="n">diff</span><span class="p">(</span><span class="n">pcfu</span><span class="p">,(</span><span class="n">a</span><span class="p">,</span><span class="n">z</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span><span class="o">*</span><span class="n">pcfv</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.7978845608028653558798921</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="mi">2</span><span class="o">+</span><span class="mi">1j</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">3j</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">pcfu</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">pcfv</span><span class="p">,(</span><span class="n">a</span><span class="p">,</span><span class="n">z</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span><span class="o">-</span><span class="n">diff</span><span class="p">(</span><span class="n">pcfu</span><span class="p">,(</span><span class="n">a</span><span class="p">,</span><span class="n">z</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span><span class="o">*</span><span class="n">pcfv</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">z</span><span class="p">))</span>
-<span class="go">0.7978845608028653558798921</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="pcfw">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">pcfw()</span></tt><a class="headerlink" href="#pcfw" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.pcfw">
-<tt class="descclassname">mpmath.</tt><tt class="descname">pcfw</tt><big>(</big><em>a</em>, <em>z</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.pcfw" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the parabolic cylinder function <span class="math">\(W(a,z)\)</span> defined in (DLMF 12.14).</p>
-<p><strong>Examples</strong></p>
-<p>Value at the origin:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">0.25</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pcfw</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.9722833245718180765617104</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">power</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="mf">0.75</span><span class="p">)</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">gamma</span><span class="p">(</span><span class="mf">0.25</span><span class="o">+</span><span class="mf">0.5j</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="n">gamma</span><span class="p">(</span><span class="mf">0.75</span><span class="o">+</span><span class="mf">0.5j</span><span class="o">*</span><span class="n">a</span><span class="p">)))</span>
-<span class="go">0.9722833245718180765617104</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">pcfw</span><span class="p">,(</span><span class="n">a</span><span class="p">,</span><span class="mi">0</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span>
-<span class="go">-0.5142533944210078966003624</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="n">power</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="mf">0.25</span><span class="p">)</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">gamma</span><span class="p">(</span><span class="mf">0.75</span><span class="o">+</span><span class="mf">0.5j</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="n">gamma</span><span class="p">(</span><span class="mf">0.25</span><span class="o">+</span><span class="mf">0.5j</span><span class="o">*</span><span class="n">a</span><span class="p">)))</span>
-<span class="go">-0.5142533944210078966003624</span>
-</pre></div>
-</div>
-</dd></dl>
-
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-</div>
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-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Bessel functions and related functions</a><ul>
-<li><a class="reference internal" href="#bessel-functions">Bessel functions</a><ul>
-<li><a class="reference internal" href="#besselj"><tt class="docutils literal"><span class="pre">besselj()</span></tt></a></li>
-<li><a class="reference internal" href="#bessely"><tt class="docutils literal"><span class="pre">bessely()</span></tt></a></li>
-<li><a class="reference internal" href="#besseli"><tt class="docutils literal"><span class="pre">besseli()</span></tt></a></li>
-<li><a class="reference internal" href="#besselk"><tt class="docutils literal"><span class="pre">besselk()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#bessel-function-zeros">Bessel function zeros</a><ul>
-<li><a class="reference internal" href="#besseljzero"><tt class="docutils literal"><span class="pre">besseljzero()</span></tt></a></li>
-<li><a class="reference internal" href="#besselyzero"><tt class="docutils literal"><span class="pre">besselyzero()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#hankel-functions">Hankel functions</a><ul>
-<li><a class="reference internal" href="#hankel1"><tt class="docutils literal"><span class="pre">hankel1()</span></tt></a></li>
-<li><a class="reference internal" href="#hankel2"><tt class="docutils literal"><span class="pre">hankel2()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#kelvin-functions">Kelvin functions</a><ul>
-<li><a class="reference internal" href="#ber"><tt class="docutils literal"><span class="pre">ber()</span></tt></a></li>
-<li><a class="reference internal" href="#bei"><tt class="docutils literal"><span class="pre">bei()</span></tt></a></li>
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-<li><a class="reference internal" href="#kei"><tt class="docutils literal"><span class="pre">kei()</span></tt></a></li>
-</ul>
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-<li><a class="reference internal" href="#struve-functions">Struve functions</a><ul>
-<li><a class="reference internal" href="#struveh"><tt class="docutils literal"><span class="pre">struveh()</span></tt></a></li>
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-<li><a class="reference internal" href="#webere"><tt class="docutils literal"><span class="pre">webere()</span></tt></a></li>
-</ul>
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-<li><a class="reference internal" href="#lommels1"><tt class="docutils literal"><span class="pre">lommels1()</span></tt></a></li>
-<li><a class="reference internal" href="#lommels2"><tt class="docutils literal"><span class="pre">lommels2()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#airy-and-scorer-functions">Airy and Scorer functions</a><ul>
-<li><a class="reference internal" href="#airyai"><tt class="docutils literal"><span class="pre">airyai()</span></tt></a></li>
-<li><a class="reference internal" href="#airybi"><tt class="docutils literal"><span class="pre">airybi()</span></tt></a></li>
-<li><a class="reference internal" href="#airyaizero"><tt class="docutils literal"><span class="pre">airyaizero()</span></tt></a></li>
-<li><a class="reference internal" href="#airybizero"><tt class="docutils literal"><span class="pre">airybizero()</span></tt></a></li>
-<li><a class="reference internal" href="#scorergi"><tt class="docutils literal"><span class="pre">scorergi()</span></tt></a></li>
-<li><a class="reference internal" href="#scorerhi"><tt class="docutils literal"><span class="pre">scorerhi()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#coulomb-wave-functions">Coulomb wave functions</a><ul>
-<li><a class="reference internal" href="#coulombf"><tt class="docutils literal"><span class="pre">coulombf()</span></tt></a></li>
-<li><a class="reference internal" href="#coulombg"><tt class="docutils literal"><span class="pre">coulombg()</span></tt></a></li>
-<li><a class="reference internal" href="#coulombc"><tt class="docutils literal"><span class="pre">coulombc()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#confluent-u-and-whittaker-functions">Confluent U and Whittaker functions</a><ul>
-<li><a class="reference internal" href="#hyperu"><tt class="docutils literal"><span class="pre">hyperu()</span></tt></a></li>
-<li><a class="reference internal" href="#whitm"><tt class="docutils literal"><span class="pre">whitm()</span></tt></a></li>
-<li><a class="reference internal" href="#whitw"><tt class="docutils literal"><span class="pre">whitw()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#parabolic-cylinder-functions">Parabolic cylinder functions</a><ul>
-<li><a class="reference internal" href="#pcfd"><tt class="docutils literal"><span class="pre">pcfd()</span></tt></a></li>
-<li><a class="reference internal" href="#pcfu"><tt class="docutils literal"><span class="pre">pcfu()</span></tt></a></li>
-<li><a class="reference internal" href="#pcfv"><tt class="docutils literal"><span class="pre">pcfv()</span></tt></a></li>
-<li><a class="reference internal" href="#pcfw"><tt class="docutils literal"><span class="pre">pcfw()</span></tt></a></li>
-</ul>
-</li>
-</ul>
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-</ul>
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-  <div class="section" id="mathematical-constants">
-<h1>Mathematical constants<a class="headerlink" href="#mathematical-constants" title="Permalink to this headline">¶</a></h1>
-<p>Mpmath supports arbitrary-precision computation of various common (and less common) mathematical constants. These constants are implemented as lazy objects that can evaluate to any precision. Whenever the objects are used as function arguments or as operands in arithmetic operations, they automagically evaluate to the current working precision. A lazy number can be converted to a regular <tt class="docutils literal"><span class="pre">mpf</span></tt> using the unary <tt class="docutils literal"><span class="pre">+</span></tt> operator, or by calling it as a function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span>
-<span class="go">&lt;pi: 3.14159~&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span>
-<span class="go">mpf(&#39;6.2831853071795862&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">+</span><span class="n">pi</span>
-<span class="go">mpf(&#39;3.1415926535897931&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="p">()</span>
-<span class="go">mpf(&#39;3.1415926535897931&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">40</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span>
-<span class="go">&lt;pi: 3.14159~&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span>
-<span class="go">mpf(&#39;6.283185307179586476925286766559005768394338&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">+</span><span class="n">pi</span>
-<span class="go">mpf(&#39;3.141592653589793238462643383279502884197169&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="p">()</span>
-<span class="go">mpf(&#39;3.141592653589793238462643383279502884197169&#39;)</span>
-</pre></div>
-</div>
-<div class="section" id="exact-constants">
-<h2>Exact constants<a class="headerlink" href="#exact-constants" title="Permalink to this headline">¶</a></h2>
-<p>The predefined objects <tt class="xref py py-data docutils literal"><span class="pre">j</span></tt> (imaginary unit), <tt class="xref py py-data docutils literal"><span class="pre">inf</span></tt> (positive infinity) and <tt class="xref py py-data docutils literal"><span class="pre">nan</span></tt> (not-a-number) are shortcuts to <tt class="xref py py-class docutils literal"><span class="pre">mpc</span></tt> and <tt class="xref py py-class docutils literal"><span class="pre">mpf</span></tt> instances with these fixed values.</p>
-</div>
-<div class="section" id="pi-pi">
-<h2>Pi (<tt class="docutils literal"><span class="pre">pi</span></tt>)<a class="headerlink" href="#pi-pi" title="Permalink to this headline">¶</a></h2>
-<dl class="attribute">
-<dt id="mpmath.mp.pi">
-<tt class="descclassname">mp.</tt><tt class="descname">pi</tt><em class="property"> = &lt;pi: 3.14159~&gt;</em><a class="headerlink" href="#mpmath.mp.pi" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="degree-degree">
-<h2>Degree (<tt class="docutils literal"><span class="pre">degree</span></tt>)<a class="headerlink" href="#degree-degree" title="Permalink to this headline">¶</a></h2>
-<dl class="attribute">
-<dt id="mpmath.mp.degree">
-<tt class="descclassname">mp.</tt><tt class="descname">degree</tt><em class="property"> = &lt;1 deg = pi / 180: 0.0174533~&gt;</em><a class="headerlink" href="#mpmath.mp.degree" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="base-of-the-natural-logarithm-e">
-<h2>Base of the natural logarithm (<tt class="docutils literal"><span class="pre">e</span></tt>)<a class="headerlink" href="#base-of-the-natural-logarithm-e" title="Permalink to this headline">¶</a></h2>
-<dl class="attribute">
-<dt id="mpmath.mp.e">
-<tt class="descclassname">mp.</tt><tt class="descname">e</tt><em class="property"> = &lt;e = exp(1): 2.71828~&gt;</em><a class="headerlink" href="#mpmath.mp.e" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="golden-ratio-phi">
-<h2>Golden ratio (<tt class="docutils literal"><span class="pre">phi</span></tt>)<a class="headerlink" href="#golden-ratio-phi" title="Permalink to this headline">¶</a></h2>
-<dl class="attribute">
-<dt id="mpmath.mp.phi">
-<tt class="descclassname">mp.</tt><tt class="descname">phi</tt><em class="property"> = &lt;Golden ratio phi: 1.61803~&gt;</em><a class="headerlink" href="#mpmath.mp.phi" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="euler-s-constant-euler">
-<h2>Euler&#8217;s constant (<tt class="docutils literal"><span class="pre">euler</span></tt>)<a class="headerlink" href="#euler-s-constant-euler" title="Permalink to this headline">¶</a></h2>
-<dl class="attribute">
-<dt id="mpmath.mp.euler">
-<tt class="descclassname">mp.</tt><tt class="descname">euler</tt><em class="property"> = &lt;Euler's constant: 0.577216~&gt;</em><a class="headerlink" href="#mpmath.mp.euler" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="catalan-s-constant-catalan">
-<h2>Catalan&#8217;s constant (<tt class="docutils literal"><span class="pre">catalan</span></tt>)<a class="headerlink" href="#catalan-s-constant-catalan" title="Permalink to this headline">¶</a></h2>
-<dl class="attribute">
-<dt id="mpmath.mp.catalan">
-<tt class="descclassname">mp.</tt><tt class="descname">catalan</tt><em class="property"> = &lt;Catalan's constant: 0.915966~&gt;</em><a class="headerlink" href="#mpmath.mp.catalan" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="apery-s-constant-apery">
-<h2>Apery&#8217;s constant (<tt class="docutils literal"><span class="pre">apery</span></tt>)<a class="headerlink" href="#apery-s-constant-apery" title="Permalink to this headline">¶</a></h2>
-<dl class="attribute">
-<dt id="mpmath.mp.apery">
-<tt class="descclassname">mp.</tt><tt class="descname">apery</tt><em class="property"> = &lt;Apery's constant: 1.20206~&gt;</em><a class="headerlink" href="#mpmath.mp.apery" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="khinchin-s-constant-khinchin">
-<h2>Khinchin&#8217;s constant (<tt class="docutils literal"><span class="pre">khinchin</span></tt>)<a class="headerlink" href="#khinchin-s-constant-khinchin" title="Permalink to this headline">¶</a></h2>
-<dl class="attribute">
-<dt id="mpmath.mp.khinchin">
-<tt class="descclassname">mp.</tt><tt class="descname">khinchin</tt><em class="property"> = &lt;Khinchin's constant: 2.68545~&gt;</em><a class="headerlink" href="#mpmath.mp.khinchin" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="glaisher-s-constant-glaisher">
-<h2>Glaisher&#8217;s constant (<tt class="docutils literal"><span class="pre">glaisher</span></tt>)<a class="headerlink" href="#glaisher-s-constant-glaisher" title="Permalink to this headline">¶</a></h2>
-<dl class="attribute">
-<dt id="mpmath.mp.glaisher">
-<tt class="descclassname">mp.</tt><tt class="descname">glaisher</tt><em class="property"> = &lt;Glaisher's constant: 1.28243~&gt;</em><a class="headerlink" href="#mpmath.mp.glaisher" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="mertens-constant-mertens">
-<h2>Mertens constant (<tt class="docutils literal"><span class="pre">mertens</span></tt>)<a class="headerlink" href="#mertens-constant-mertens" title="Permalink to this headline">¶</a></h2>
-<dl class="attribute">
-<dt id="mpmath.mp.mertens">
-<tt class="descclassname">mp.</tt><tt class="descname">mertens</tt><em class="property"> = &lt;Mertens' constant: 0.261497~&gt;</em><a class="headerlink" href="#mpmath.mp.mertens" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="twin-prime-constant-twinprime">
-<h2>Twin prime constant (<tt class="docutils literal"><span class="pre">twinprime</span></tt>)<a class="headerlink" href="#twin-prime-constant-twinprime" title="Permalink to this headline">¶</a></h2>
-<dl class="attribute">
-<dt id="mpmath.mp.twinprime">
-<tt class="descclassname">mp.</tt><tt class="descname">twinprime</tt><em class="property"> = &lt;Twin prime constant: 0.660162~&gt;</em><a class="headerlink" href="#mpmath.mp.twinprime" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Mathematical constants</a><ul>
-<li><a class="reference internal" href="#exact-constants">Exact constants</a></li>
-<li><a class="reference internal" href="#pi-pi">Pi (<tt class="docutils literal"><span class="pre">pi</span></tt>)</a></li>
-<li><a class="reference internal" href="#degree-degree">Degree (<tt class="docutils literal"><span class="pre">degree</span></tt>)</a></li>
-<li><a class="reference internal" href="#base-of-the-natural-logarithm-e">Base of the natural logarithm (<tt class="docutils literal"><span class="pre">e</span></tt>)</a></li>
-<li><a class="reference internal" href="#golden-ratio-phi">Golden ratio (<tt class="docutils literal"><span class="pre">phi</span></tt>)</a></li>
-<li><a class="reference internal" href="#euler-s-constant-euler">Euler&#8217;s constant (<tt class="docutils literal"><span class="pre">euler</span></tt>)</a></li>
-<li><a class="reference internal" href="#catalan-s-constant-catalan">Catalan&#8217;s constant (<tt class="docutils literal"><span class="pre">catalan</span></tt>)</a></li>
-<li><a class="reference internal" href="#apery-s-constant-apery">Apery&#8217;s constant (<tt class="docutils literal"><span class="pre">apery</span></tt>)</a></li>
-<li><a class="reference internal" href="#khinchin-s-constant-khinchin">Khinchin&#8217;s constant (<tt class="docutils literal"><span class="pre">khinchin</span></tt>)</a></li>
-<li><a class="reference internal" href="#glaisher-s-constant-glaisher">Glaisher&#8217;s constant (<tt class="docutils literal"><span class="pre">glaisher</span></tt>)</a></li>
-<li><a class="reference internal" href="#mertens-constant-mertens">Mertens constant (<tt class="docutils literal"><span class="pre">mertens</span></tt>)</a></li>
-<li><a class="reference internal" href="#twin-prime-constant-twinprime">Twin prime constant (<tt class="docutils literal"><span class="pre">twinprime</span></tt>)</a></li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../functions/index.html"
-                        title="previous chapter">Mathematical functions</a></p>
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-  <div class="section" id="module-mpmath.functions.elliptic">
-<span id="elliptic-functions"></span><h1>Elliptic functions<a class="headerlink" href="#module-mpmath.functions.elliptic" title="Permalink to this headline">¶</a></h1>
-<p>Elliptic functions historically comprise the elliptic integrals
-and their inverses, and originate from the problem of computing the
-arc length of an ellipse. From a more modern point of view,
-an elliptic function is defined as a doubly periodic function, i.e.
-a function which satisfies</p>
-<div class="math">
-\[f(z + 2 \omega_1) = f(z + 2 \omega_2) = f(z)\]</div>
-<p>for some half-periods <span class="math">\(\omega_1, \omega_2\)</span> with
-<span class="math">\(\mathrm{Im}[\omega_1 / \omega_2] &gt; 0\)</span>. The canonical elliptic
-functions are the Jacobi elliptic functions. More broadly, this section
-includes  quasi-doubly periodic functions (such as the Jacobi theta
-functions) and other functions useful in the study of elliptic functions.</p>
-<p>Many different conventions for the arguments of
-elliptic functions are in use. It is even standard to use
-different parameterizations for different functions in the same
-text or software (and mpmath is no exception).
-The usual parameters are the elliptic nome <span class="math">\(q\)</span>, which usually
-must satisfy <span class="math">\(|q| &lt; 1\)</span>; the elliptic parameter <span class="math">\(m\)</span> (an arbitrary
-complex number); the elliptic modulus <span class="math">\(k\)</span> (an arbitrary complex
-number); and the half-period ratio <span class="math">\(\tau\)</span>, which usually must
-satisfy <span class="math">\(\mathrm{Im}[\tau] &gt; 0\)</span>.
-These quantities can be expressed in terms of each other
-using the following relations:</p>
-<div class="math">
-\[m = k^2\]</div>
-<div class="math">
-\[\tau = -i \frac{K(1-m)}{K(m)}\]</div>
-<div class="math">
-\[q = e^{i \pi \tau}\]</div>
-<div class="math">
-\[k = \frac{\vartheta_2^4(q)}{\vartheta_3^4(q)}\]</div>
-<p>In addition, an alternative definition is used for the nome in
-number theory, which we here denote by q-bar:</p>
-<div class="math">
-\[\bar{q} = q^2 = e^{2 i \pi \tau}\]</div>
-<p>For convenience, mpmath provides functions to convert
-between the various parameters (<a class="reference internal" href="#mpmath.qfrom" title="mpmath.qfrom"><tt class="xref py py-func docutils literal"><span class="pre">qfrom()</span></tt></a>, <a class="reference internal" href="#mpmath.mfrom" title="mpmath.mfrom"><tt class="xref py py-func docutils literal"><span class="pre">mfrom()</span></tt></a>,
-<a class="reference internal" href="#mpmath.kfrom" title="mpmath.kfrom"><tt class="xref py py-func docutils literal"><span class="pre">kfrom()</span></tt></a>, <a class="reference internal" href="#mpmath.taufrom" title="mpmath.taufrom"><tt class="xref py py-func docutils literal"><span class="pre">taufrom()</span></tt></a>, <a class="reference internal" href="#mpmath.qbarfrom" title="mpmath.qbarfrom"><tt class="xref py py-func docutils literal"><span class="pre">qbarfrom()</span></tt></a>).</p>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../../mpmath/references.html#abramowitzstegun">[AbramowitzStegun]</a></li>
-<li><a class="reference internal" href="../../mpmath/references.html#whittakerwatson">[WhittakerWatson]</a></li>
-</ol>
-<div class="section" id="elliptic-arguments">
-<h2>Elliptic arguments<a class="headerlink" href="#elliptic-arguments" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="qfrom">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">qfrom()</span></tt><a class="headerlink" href="#qfrom" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.qfrom">
-<tt class="descclassname">mpmath.</tt><tt class="descname">qfrom</tt><big>(</big><em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.qfrom" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the elliptic nome <span class="math">\(q\)</span>, given any of <span class="math">\(q, m, k, \tau, \bar{q}\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qfrom</span><span class="p">(</span><span class="n">q</span><span class="o">=</span><span class="mf">0.25</span><span class="p">)</span>
-<span class="go">0.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qfrom</span><span class="p">(</span><span class="n">m</span><span class="o">=</span><span class="n">mfrom</span><span class="p">(</span><span class="n">q</span><span class="o">=</span><span class="mf">0.25</span><span class="p">))</span>
-<span class="go">0.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qfrom</span><span class="p">(</span><span class="n">k</span><span class="o">=</span><span class="n">kfrom</span><span class="p">(</span><span class="n">q</span><span class="o">=</span><span class="mf">0.25</span><span class="p">))</span>
-<span class="go">0.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qfrom</span><span class="p">(</span><span class="n">tau</span><span class="o">=</span><span class="n">taufrom</span><span class="p">(</span><span class="n">q</span><span class="o">=</span><span class="mf">0.25</span><span class="p">))</span>
-<span class="go">(0.25 + 0.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qfrom</span><span class="p">(</span><span class="n">qbar</span><span class="o">=</span><span class="n">qbarfrom</span><span class="p">(</span><span class="n">q</span><span class="o">=</span><span class="mf">0.25</span><span class="p">))</span>
-<span class="go">0.25</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="qbarfrom">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">qbarfrom()</span></tt><a class="headerlink" href="#qbarfrom" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.qbarfrom">
-<tt class="descclassname">mpmath.</tt><tt class="descname">qbarfrom</tt><big>(</big><em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.qbarfrom" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the number-theoretic nome <span class="math">\(\bar q\)</span>, given any of
-<span class="math">\(q, m, k, \tau, \bar{q}\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qbarfrom</span><span class="p">(</span><span class="n">qbar</span><span class="o">=</span><span class="mf">0.25</span><span class="p">)</span>
-<span class="go">0.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qbarfrom</span><span class="p">(</span><span class="n">q</span><span class="o">=</span><span class="n">qfrom</span><span class="p">(</span><span class="n">qbar</span><span class="o">=</span><span class="mf">0.25</span><span class="p">))</span>
-<span class="go">0.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qbarfrom</span><span class="p">(</span><span class="n">m</span><span class="o">=</span><span class="n">extraprec</span><span class="p">(</span><span class="mi">20</span><span class="p">)(</span><span class="n">mfrom</span><span class="p">)(</span><span class="n">qbar</span><span class="o">=</span><span class="mf">0.25</span><span class="p">))</span>  <span class="c"># ill-conditioned</span>
-<span class="go">0.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qbarfrom</span><span class="p">(</span><span class="n">k</span><span class="o">=</span><span class="n">extraprec</span><span class="p">(</span><span class="mi">20</span><span class="p">)(</span><span class="n">kfrom</span><span class="p">)(</span><span class="n">qbar</span><span class="o">=</span><span class="mf">0.25</span><span class="p">))</span>  <span class="c"># ill-conditioned</span>
-<span class="go">0.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qbarfrom</span><span class="p">(</span><span class="n">tau</span><span class="o">=</span><span class="n">taufrom</span><span class="p">(</span><span class="n">qbar</span><span class="o">=</span><span class="mf">0.25</span><span class="p">))</span>
-<span class="go">(0.25 + 0.0j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="mfrom">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">mfrom()</span></tt><a class="headerlink" href="#mfrom" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.mfrom">
-<tt class="descclassname">mpmath.</tt><tt class="descname">mfrom</tt><big>(</big><em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.mfrom" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the elliptic parameter <span class="math">\(m\)</span>, given any of
-<span class="math">\(q, m, k, \tau, \bar{q}\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mfrom</span><span class="p">(</span><span class="n">m</span><span class="o">=</span><span class="mf">0.25</span><span class="p">)</span>
-<span class="go">0.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mfrom</span><span class="p">(</span><span class="n">q</span><span class="o">=</span><span class="n">qfrom</span><span class="p">(</span><span class="n">m</span><span class="o">=</span><span class="mf">0.25</span><span class="p">))</span>
-<span class="go">0.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mfrom</span><span class="p">(</span><span class="n">k</span><span class="o">=</span><span class="n">kfrom</span><span class="p">(</span><span class="n">m</span><span class="o">=</span><span class="mf">0.25</span><span class="p">))</span>
-<span class="go">0.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mfrom</span><span class="p">(</span><span class="n">tau</span><span class="o">=</span><span class="n">taufrom</span><span class="p">(</span><span class="n">m</span><span class="o">=</span><span class="mf">0.25</span><span class="p">))</span>
-<span class="go">(0.25 + 0.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mfrom</span><span class="p">(</span><span class="n">qbar</span><span class="o">=</span><span class="n">qbarfrom</span><span class="p">(</span><span class="n">m</span><span class="o">=</span><span class="mf">0.25</span><span class="p">))</span>
-<span class="go">0.25</span>
-</pre></div>
-</div>
-<p>As <span class="math">\(q \to 1\)</span> and <span class="math">\(q \to -1\)</span>, <span class="math">\(m\)</span> rapidly approaches
-<span class="math">\(1\)</span> and <span class="math">\(-\infty\)</span> respectively:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mfrom</span><span class="p">(</span><span class="n">q</span><span class="o">=</span><span class="mf">0.75</span><span class="p">)</span>
-<span class="go">0.9999999999999798332943533</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mfrom</span><span class="p">(</span><span class="n">q</span><span class="o">=-</span><span class="mf">0.75</span><span class="p">)</span>
-<span class="go">-49586681013729.32611558353</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mfrom</span><span class="p">(</span><span class="n">q</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mfrom</span><span class="p">(</span><span class="n">q</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-inf</span>
-</pre></div>
-</div>
-<p>The inverse nome as a function of <span class="math">\(q\)</span> has an integer
-Taylor series expansion:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">q</span><span class="p">:</span> <span class="n">mfrom</span><span class="p">(</span><span class="n">q</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
-<span class="go">[0.0, 16.0, -128.0, 704.0, -3072.0, 11488.0, -38400.0, 117632.0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="kfrom">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">kfrom()</span></tt><a class="headerlink" href="#kfrom" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.kfrom">
-<tt class="descclassname">mpmath.</tt><tt class="descname">kfrom</tt><big>(</big><em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.kfrom" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the elliptic modulus <span class="math">\(k\)</span>, given any of
-<span class="math">\(q, m, k, \tau, \bar{q}\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">kfrom</span><span class="p">(</span><span class="n">k</span><span class="o">=</span><span class="mf">0.25</span><span class="p">)</span>
-<span class="go">0.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">kfrom</span><span class="p">(</span><span class="n">m</span><span class="o">=</span><span class="n">mfrom</span><span class="p">(</span><span class="n">k</span><span class="o">=</span><span class="mf">0.25</span><span class="p">))</span>
-<span class="go">0.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">kfrom</span><span class="p">(</span><span class="n">q</span><span class="o">=</span><span class="n">qfrom</span><span class="p">(</span><span class="n">k</span><span class="o">=</span><span class="mf">0.25</span><span class="p">))</span>
-<span class="go">0.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">kfrom</span><span class="p">(</span><span class="n">tau</span><span class="o">=</span><span class="n">taufrom</span><span class="p">(</span><span class="n">k</span><span class="o">=</span><span class="mf">0.25</span><span class="p">))</span>
-<span class="go">(0.25 + 0.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">kfrom</span><span class="p">(</span><span class="n">qbar</span><span class="o">=</span><span class="n">qbarfrom</span><span class="p">(</span><span class="n">k</span><span class="o">=</span><span class="mf">0.25</span><span class="p">))</span>
-<span class="go">0.25</span>
-</pre></div>
-</div>
-<p>As <span class="math">\(q \to 1\)</span> and <span class="math">\(q \to -1\)</span>, <span class="math">\(k\)</span> rapidly approaches
-<span class="math">\(1\)</span> and <span class="math">\(i \infty\)</span> respectively:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">kfrom</span><span class="p">(</span><span class="n">q</span><span class="o">=</span><span class="mf">0.75</span><span class="p">)</span>
-<span class="go">0.9999999999999899166471767</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">kfrom</span><span class="p">(</span><span class="n">q</span><span class="o">=-</span><span class="mf">0.75</span><span class="p">)</span>
-<span class="go">(0.0 + 7041781.096692038332790615j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">kfrom</span><span class="p">(</span><span class="n">q</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">kfrom</span><span class="p">(</span><span class="n">q</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(0.0 + +infj)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="taufrom">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">taufrom()</span></tt><a class="headerlink" href="#taufrom" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.taufrom">
-<tt class="descclassname">mpmath.</tt><tt class="descname">taufrom</tt><big>(</big><em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.taufrom" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the elliptic half-period ratio <span class="math">\(\tau\)</span>, given any of
-<span class="math">\(q, m, k, \tau, \bar{q}\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">taufrom</span><span class="p">(</span><span class="n">tau</span><span class="o">=</span><span class="mf">0.5j</span><span class="p">)</span>
-<span class="go">(0.0 + 0.5j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">taufrom</span><span class="p">(</span><span class="n">q</span><span class="o">=</span><span class="n">qfrom</span><span class="p">(</span><span class="n">tau</span><span class="o">=</span><span class="mf">0.5j</span><span class="p">))</span>
-<span class="go">(0.0 + 0.5j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">taufrom</span><span class="p">(</span><span class="n">m</span><span class="o">=</span><span class="n">mfrom</span><span class="p">(</span><span class="n">tau</span><span class="o">=</span><span class="mf">0.5j</span><span class="p">))</span>
-<span class="go">(0.0 + 0.5j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">taufrom</span><span class="p">(</span><span class="n">k</span><span class="o">=</span><span class="n">kfrom</span><span class="p">(</span><span class="n">tau</span><span class="o">=</span><span class="mf">0.5j</span><span class="p">))</span>
-<span class="go">(0.0 + 0.5j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">taufrom</span><span class="p">(</span><span class="n">qbar</span><span class="o">=</span><span class="n">qbarfrom</span><span class="p">(</span><span class="n">tau</span><span class="o">=</span><span class="mf">0.5j</span><span class="p">))</span>
-<span class="go">(0.0 + 0.5j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="legendre-elliptic-integrals">
-<h2>Legendre elliptic integrals<a class="headerlink" href="#legendre-elliptic-integrals" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="ellipk">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">ellipk()</span></tt><a class="headerlink" href="#ellipk" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.ellipk">
-<tt class="descclassname">mpmath.</tt><tt class="descname">ellipk</tt><big>(</big><em>m</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.ellipk" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the complete elliptic integral of the first kind,
-<span class="math">\(K(m)\)</span>, defined by</p>
-<div class="math">
-\[K(m) = \int_0^{\pi/2} \frac{dt}{\sqrt{1-m \sin^2 t}} \, = \,
-\frac{\pi}{2} \,_2F_1\left(\frac{1}{2}, \frac{1}{2}, 1, m\right).\]</div>
-<p>Note that the argument is the parameter <span class="math">\(m = k^2\)</span>,
-not the modulus <span class="math">\(k\)</span> which is sometimes used.</p>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Complete elliptic integrals K(m) and E(m)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">ellipk</span><span class="p">,</span> <span class="n">ellipe</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">],</span> <span class="n">points</span><span class="o">=</span><span class="mi">600</span><span class="p">)</span>
-</pre></div>
-</div>
-<img alt="../../../_images/ellipk.png" src="../../../_images/ellipk.png" />
-<p><strong>Examples</strong></p>
-<p>Values and limits include:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipk</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">1.570796326794896619231322</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipk</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">(0.0 + 0.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipk</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipk</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipk</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1.31102877714605990523242</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipk</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(1.31102877714605990523242 - 1.31102877714605990523242j)</span>
-</pre></div>
-</div>
-<p>Verifying the defining integral and hypergeometric
-representation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ellipk</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="go">1.85407467730137191843385</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="mf">0.5</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">**-</span><span class="mf">0.5</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">])</span>
-<span class="go">1.85407467730137191843385</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="o">*</span><span class="n">hyp2f1</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="go">1.85407467730137191843385</span>
-</pre></div>
-</div>
-<p>Evaluation is supported for arbitrary complex <span class="math">\(m\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ellipk</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(0.9111955638049650086562171 + 0.6313342832413452438845091j)</span>
-</pre></div>
-</div>
-<p>A definite integral:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">ellipk</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">2.0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="ellipf">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">ellipf()</span></tt><a class="headerlink" href="#ellipf" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.ellipf">
-<tt class="descclassname">mpmath.</tt><tt class="descname">ellipf</tt><big>(</big><em>phi</em>, <em>m</em><big>)</big><a class="headerlink" href="#mpmath.ellipf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the Legendre incomplete elliptic integral of the first kind</p>
-<blockquote>
-<div><div class="math">
-\[F(\phi,m) = \int_0^{\phi} \frac{dt}{\sqrt{1-m \sin^2 t}}\]</div>
-</div></blockquote>
-<p>or equivalently</p>
-<div class="math">
-\[F(\phi,m) = \int_0^{\sin z}
-\frac{dt}{\left(\sqrt{1-t^2}\right)\left(\sqrt{1-mt^2}\right)}.\]</div>
-<p>The function reduces to a complete elliptic integral of the first kind
-(see <a class="reference internal" href="#mpmath.ellipk" title="mpmath.ellipk"><tt class="xref py py-func docutils literal"><span class="pre">ellipk()</span></tt></a>) when <span class="math">\(\phi = \frac{\pi}{2}\)</span>; that is,</p>
-<div class="math">
-\[F\left(\frac{\pi}{2}, m\right) = K(m).\]</div>
-<p>In the defining integral, it is assumed that the principal branch
-of the square root is taken and that the path of integration avoids
-crossing any branch cuts. Outside <span class="math">\(-\pi/2 \le \Re(z) \le \pi/2\)</span>,
-the function extends quasi-periodically as</p>
-<div class="math">
-\[F(\phi + n \pi, m) = 2 n K(m) + F(\phi,m), n \in \mathbb{Z}.\]</div>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Elliptic integral F(z,m) for some different m</span>
-<span class="n">f1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">ellipf</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="n">f2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">ellipf</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="o">-</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="n">f3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">ellipf</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="n">f4</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">ellipf</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="n">f5</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">ellipf</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">f1</span><span class="p">,</span><span class="n">f2</span><span class="p">,</span><span class="n">f3</span><span class="p">,</span><span class="n">f4</span><span class="p">,</span><span class="n">f5</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">pi</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/ellipf.png" src="../../../_images/ellipf.png" />
-<p><strong>Examples</strong></p>
-<p>Basic values and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipf</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipf</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipf</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">);</span> <span class="n">ellipf</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="go">(2.0 + 3.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipf</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">log</span><span class="p">(</span><span class="n">sec</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">+</span><span class="n">tan</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span>
-<span class="go">1.226191170883517070813061</span>
-<span class="go">1.226191170883517070813061</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipf</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.5</span><span class="p">);</span> <span class="n">ellipk</span><span class="p">(</span><span class="o">-</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="go">1.415737208425956198892166</span>
-<span class="go">1.415737208425956198892166</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipf</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="o">+</span><span class="n">eps</span><span class="p">,</span> <span class="mi">1</span><span class="p">);</span> <span class="n">ellipf</span><span class="p">(</span><span class="o">-</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="o">-</span><span class="n">eps</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipf</span><span class="p">(</span><span class="mf">1.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">3.340677542798311003320813</span>
-</pre></div>
-</div>
-<p>Comparing with numerical integration:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span><span class="p">,</span><span class="n">m</span> <span class="o">=</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipf</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="n">m</span><span class="p">)</span>
-<span class="go">0.5287219202206327872978255</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">m</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="mf">0.5</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">z</span><span class="p">])</span>
-<span class="go">0.5287219202206327872978255</span>
-</pre></div>
-</div>
-<p>The arguments may be complex numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ellipf</span><span class="p">(</span><span class="mi">3j</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
-<span class="go">(0.0 + 1.713602407841590234804143j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipf</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">,</span> <span class="mi">5</span><span class="o">-</span><span class="mi">6j</span><span class="p">)</span>
-<span class="go">(1.269131241950351323305741 - 0.3561052815014558335412538j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span><span class="p">,</span><span class="n">m</span> <span class="o">=</span> <span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span> <span class="mf">1.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="mi">1011</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipf</span><span class="p">(</span><span class="n">z</span><span class="o">+</span><span class="n">pi</span><span class="o">*</span><span class="n">k</span><span class="p">,</span><span class="n">m</span><span class="p">);</span> <span class="n">ellipf</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="n">m</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="o">*</span><span class="n">ellipk</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-<span class="go">(4086.184383622179764082821 - 3003.003538923749396546871j)</span>
-<span class="go">(4086.184383622179764082821 - 3003.003538923749396546871j)</span>
-</pre></div>
-</div>
-<p>For <span class="math">\(|\Re(z)| &lt; \pi/2\)</span>, the function can be expressed as a
-hypergeometric series of two variables
-(see <a class="reference internal" href="../functions/hypergeometric.html#mpmath.appellf1" title="mpmath.appellf1"><tt class="xref py py-func docutils literal"><span class="pre">appellf1()</span></tt></a>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span><span class="p">,</span><span class="n">m</span> <span class="o">=</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipf</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="n">m</span><span class="p">)</span>
-<span class="go">0.5050887275786480788831083</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">appellf1</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">1.5</span><span class="p">,</span><span class="n">sin</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span><span class="n">m</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.5050887275786480788831083</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="ellipe">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">ellipe()</span></tt><a class="headerlink" href="#ellipe" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.ellipe">
-<tt class="descclassname">mpmath.</tt><tt class="descname">ellipe</tt><big>(</big><em>*args</em><big>)</big><a class="headerlink" href="#mpmath.ellipe" title="Permalink to this definition">¶</a></dt>
-<dd><p>Called with a single argument <span class="math">\(m\)</span>, evaluates the Legendre complete
-elliptic integral of the second kind, <span class="math">\(E(m)\)</span>, defined by</p>
-<blockquote>
-<div><div class="math">
-\[E(m) = \int_0^{\pi/2} \sqrt{1-m \sin^2 t} \, dt \,=\,
-\frac{\pi}{2}
-\,_2F_1\left(\frac{1}{2}, -\frac{1}{2}, 1, m\right).\]</div>
-</div></blockquote>
-<p>Called with two arguments <span class="math">\(\phi, m\)</span>, evaluates the incomplete elliptic
-integral of the second kind</p>
-<blockquote>
-<div><div class="math">
-\[E(\phi,m) = \int_0^{\phi} \sqrt{1-m \sin^2 t} \, dt =
-            \int_0^{\sin z}
-            \frac{\sqrt{1-mt^2}}{\sqrt{1-t^2}} \, dt.\]</div>
-</div></blockquote>
-<p>The incomplete integral reduces to a complete integral when
-<span class="math">\(\phi = \frac{\pi}{2}\)</span>; that is,</p>
-<div class="math">
-\[E\left(\frac{\pi}{2}, m\right) = E(m).\]</div>
-<p>In the defining integral, it is assumed that the principal branch
-of the square root is taken and that the path of integration avoids
-crossing any branch cuts. Outside <span class="math">\(-\pi/2 \le \Re(z) \le \pi/2\)</span>,
-the function extends quasi-periodically as</p>
-<div class="math">
-\[E(\phi + n \pi, m) = 2 n E(m) + F(\phi,m), n \in \mathbb{Z}.\]</div>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Elliptic integral E(z,m) for some different m</span>
-<span class="n">f1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">ellipe</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
-<span class="n">f2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">ellipe</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="n">f3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">ellipe</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="n">f4</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">ellipe</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="n">f5</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">ellipe</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">f1</span><span class="p">,</span><span class="n">f2</span><span class="p">,</span><span class="n">f3</span><span class="p">,</span><span class="n">f4</span><span class="p">,</span><span class="n">f5</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">pi</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/ellipe.png" src="../../../_images/ellipe.png" />
-<p><strong>Examples for the complete integral</strong></p>
-<p>Basic values and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">1.570796326794896619231322</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1.910098894513856008952381</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(0.5990701173677961037199612 + 0.5990701173677961037199612j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">(0.0 + +infj)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">+inf</span>
-</pre></div>
-</div>
-<p>Verifying the defining integral and hypergeometric
-representation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="go">1.350643881047675502520175</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="mf">0.5</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">])</span>
-<span class="go">1.350643881047675502520175</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="o">*</span><span class="n">hyp2f1</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="go">1.350643881047675502520175</span>
-</pre></div>
-</div>
-<p>Evaluation is supported for arbitrary complex <span class="math">\(m\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="mf">0.5</span><span class="o">+</span><span class="mf">0.25j</span><span class="p">)</span>
-<span class="go">(1.360868682163129682716687 - 0.1238733442561786843557315j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(1.499553520933346954333612 - 1.577879007912758274533309j)</span>
-</pre></div>
-</div>
-<p>A definite integral:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">ellipe</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">1.333333333333333333333333</span>
-</pre></div>
-</div>
-<p><strong>Examples for the incomplete integral</strong></p>
-<p>Basic values and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">(2.0 + 3.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">sin</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.8414709848078965066525023</span>
-<span class="go">0.8414709848078965066525023</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.5</span><span class="p">);</span> <span class="n">ellipe</span><span class="p">(</span><span class="o">-</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="go">1.751771275694817862026502</span>
-<span class="go">1.751771275694817862026502</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">);</span> <span class="n">ellipe</span><span class="p">(</span><span class="o">-</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="go">-1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="mf">1.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">0.9974949866040544309417234</span>
-</pre></div>
-</div>
-<p>Comparing with numerical integration:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span><span class="p">,</span><span class="n">m</span> <span class="o">=</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="n">m</span><span class="p">)</span>
-<span class="go">0.4740152182652628394264449</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">m</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">z</span><span class="p">])</span>
-<span class="go">0.4740152182652628394264449</span>
-</pre></div>
-</div>
-<p>The arguments may be complex numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="mi">3j</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
-<span class="go">(0.0 + 7.551991234890371873502105j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">,</span> <span class="mi">5</span><span class="o">-</span><span class="mi">6j</span><span class="p">)</span>
-<span class="go">(24.15299022574220502424466 + 75.2503670480325997418156j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="mi">35</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span><span class="p">,</span><span class="n">m</span> <span class="o">=</span> <span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span> <span class="mf">1.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="n">z</span><span class="o">+</span><span class="n">pi</span><span class="o">*</span><span class="n">k</span><span class="p">,</span><span class="n">m</span><span class="p">);</span> <span class="n">ellipe</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="n">m</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="o">*</span><span class="n">ellipe</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-<span class="go">(48.30138799412005235090766 + 17.47255216721987688224357j)</span>
-<span class="go">(48.30138799412005235090766 + 17.47255216721987688224357j)</span>
-</pre></div>
-</div>
-<p>For <span class="math">\(|\Re(z)| &lt; \pi/2\)</span>, the function can be expressed as a
-hypergeometric series of two variables
-(see <a class="reference internal" href="../functions/hypergeometric.html#mpmath.appellf1" title="mpmath.appellf1"><tt class="xref py py-func docutils literal"><span class="pre">appellf1()</span></tt></a>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span><span class="p">,</span><span class="n">m</span> <span class="o">=</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="n">m</span><span class="p">)</span>
-<span class="go">0.4950017030164151928870375</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">appellf1</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">1.5</span><span class="p">,</span><span class="n">sin</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span><span class="n">m</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.4950017030164151928870376</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="ellippi">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">ellippi()</span></tt><a class="headerlink" href="#ellippi" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.ellippi">
-<tt class="descclassname">mpmath.</tt><tt class="descname">ellippi</tt><big>(</big><em>*args</em><big>)</big><a class="headerlink" href="#mpmath.ellippi" title="Permalink to this definition">¶</a></dt>
-<dd><p>Called with three arguments <span class="math">\(n, \phi, m\)</span>, evaluates the Legendre
-incomplete elliptic integral of the third kind</p>
-<div class="math">
-\[\Pi(n; \phi, m) = \int_0^{\phi}
-    \frac{dt}{(1-n \sin^2 t) \sqrt{1-m \sin^2 t}} =
-    \int_0^{\sin \phi}
-    \frac{dt}{(1-nt^2) \sqrt{1-t^2} \sqrt{1-mt^2}}.\]</div>
-<p>Called with two arguments <span class="math">\(n, m\)</span>, evaluates the complete
-elliptic integral of the third kind
-<span class="math">\(\Pi(n,m) = \Pi(n; \frac{\pi}{2},m)\)</span>.</p>
-<p>In the defining integral, it is assumed that the principal branch
-of the square root is taken and that the path of integration avoids
-crossing any branch cuts. Outside <span class="math">\(-\pi/2 \le \Re(z) \le \pi/2\)</span>,
-the function extends quasi-periodically as</p>
-<div class="math">
-\[\Pi(n,\phi+k\pi,m) = 2k\Pi(n,m) + \Pi(n,\phi,m), k \in \mathbb{Z}.\]</div>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Elliptic integral Pi(n,z,m) for some different n, m</span>
-<span class="n">f1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">ellippi</span><span class="p">(</span><span class="mf">0.9</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mf">0.9</span><span class="p">)</span>
-<span class="n">f2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">ellippi</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="n">f3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">ellippi</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="o">-</span><span class="mf">0.9</span><span class="p">)</span>
-<span class="n">f4</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">ellippi</span><span class="p">(</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="n">f5</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">ellippi</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">f1</span><span class="p">,</span><span class="n">f2</span><span class="p">,</span><span class="n">f3</span><span class="p">,</span><span class="n">f4</span><span class="p">,</span><span class="n">f5</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">pi</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/ellippi.png" src="../../../_images/ellippi.png" />
-<p><strong>Examples for the complete integral</strong></p>
-<p>Some basic values and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellippi</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">5</span><span class="p">);</span> <span class="n">ellipk</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">0.9555039270640439337379334</span>
-<span class="go">0.9555039270640439337379334</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellippi</span><span class="p">(</span><span class="n">inf</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellippi</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">abs</span><span class="p">(</span><span class="n">ellippi</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">5</span><span class="p">))</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">abs</span><span class="p">(</span><span class="n">ellippi</span><span class="p">(</span><span class="mf">0.25</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span>
-<span class="go">+inf</span>
-</pre></div>
-</div>
-<p>Evaluation in terms of simpler functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ellippi</span><span class="p">(</span><span class="mf">0.25</span><span class="p">,</span><span class="mf">0.25</span><span class="p">);</span> <span class="n">ellipe</span><span class="p">(</span><span class="mf">0.25</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="mf">0.25</span><span class="p">)</span>
-<span class="go">1.956616279119236207279727</span>
-<span class="go">1.956616279119236207279727</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellippi</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">0</span><span class="p">);</span> <span class="n">pi</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">(0.0 - 1.11072073453959156175397j)</span>
-<span class="go">(0.0 - 1.11072073453959156175397j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellippi</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">0</span><span class="p">);</span> <span class="n">pi</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">4</span><span class="p">))</span>
-<span class="go">0.7853981633974483096156609</span>
-<span class="go">0.7853981633974483096156609</span>
-</pre></div>
-</div>
-<p><strong>Examples for the incomplete integral</strong></p>
-<p>Basic values and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ellippi</span><span class="p">(</span><span class="mf">0.25</span><span class="p">,</span><span class="o">-</span><span class="mf">0.5</span><span class="p">);</span> <span class="n">ellippi</span><span class="p">(</span><span class="mf">0.25</span><span class="p">,</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="go">1.622944760954741603710555</span>
-<span class="go">1.622944760954741603710555</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellippi</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellippi</span><span class="p">(</span><span class="n">inf</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellippi</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mf">0.25</span><span class="p">,</span><span class="mf">0.5</span><span class="p">);</span> <span class="n">ellipf</span><span class="p">(</span><span class="mf">0.25</span><span class="p">,</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="go">0.2513040086544925794134591</span>
-<span class="go">0.2513040086544925794134591</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellippi</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">sec</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">+</span><span class="n">tan</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span><span class="o">+</span><span class="n">sec</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span>
-<span class="go">2.054332933256248668692452</span>
-<span class="go">2.054332933256248668692452</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellippi</span><span class="p">(</span><span class="mf">0.25</span><span class="p">,</span> <span class="mi">53</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mf">0.75</span><span class="p">);</span> <span class="mi">53</span><span class="o">*</span><span class="n">ellippi</span><span class="p">(</span><span class="mf">0.25</span><span class="p">,</span><span class="mf">0.75</span><span class="p">)</span>
-<span class="go">135.240868757890840755058</span>
-<span class="go">135.240868757890840755058</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellippi</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span><span class="mf">0.5</span><span class="p">);</span> <span class="mi">2</span><span class="o">*</span><span class="n">ellipe</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span><span class="mf">0.5</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.9190227391656969903987269</span>
-<span class="go">0.9190227391656969903987269</span>
-</pre></div>
-</div>
-<p>Complex arguments are supported:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ellippi</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mi">5</span><span class="o">+</span><span class="mi">6j</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">,</span> <span class="o">-</span><span class="mi">7</span><span class="o">-</span><span class="mi">8j</span><span class="p">)</span>
-<span class="go">(-0.3612856620076747660410167 + 0.5217735339984807829755815j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="carlson-symmetric-elliptic-integrals">
-<h2>Carlson symmetric elliptic integrals<a class="headerlink" href="#carlson-symmetric-elliptic-integrals" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="elliprf">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">elliprf()</span></tt><a class="headerlink" href="#elliprf" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.elliprf">
-<tt class="descclassname">mpmath.</tt><tt class="descname">elliprf</tt><big>(</big><em>x</em>, <em>y</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.elliprf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the Carlson symmetric elliptic integral of the first kind</p>
-<div class="math">
-\[R_F(x,y,z) = \frac{1}{2}
-    \int_0^{\infty} \frac{dt}{\sqrt{(t+x)(t+y)(t+z)}}\]</div>
-<p>which is defined for <span class="math">\(x,y,z \notin (-\infty,0)\)</span>, and with
-at most one of <span class="math">\(x,y,z\)</span> being zero.</p>
-<p>For real <span class="math">\(x,y,z \ge 0\)</span>, the principal square root is taken in the integrand.
-For complex <span class="math">\(x,y,z\)</span>, the principal square root is taken as <span class="math">\(t \to \infty\)</span>
-and as <span class="math">\(t \to 0\)</span> non-principal branches are chosen as necessary so as to
-make the integrand continuous.</p>
-<p><strong>Examples</strong></p>
-<p>Some basic values and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprf</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span>
-<span class="go">1.570796326794896619231322</span>
-<span class="go">1.570796326794896619231322</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprf</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprf</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprf</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprf</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">);</span> <span class="n">elliprf</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">elliprf</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">);</span> <span class="n">elliprf</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="go">+inf</span>
-<span class="go">+inf</span>
-<span class="go">+inf</span>
-</pre></div>
-</div>
-<p>Representing complete elliptic integrals in terms of <span class="math">\(R_F\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">0.75</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipk</span><span class="p">(</span><span class="n">m</span><span class="p">);</span> <span class="n">elliprf</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="o">-</span><span class="n">m</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">2.156515647499643235438675</span>
-<span class="go">2.156515647499643235438675</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipe</span><span class="p">(</span><span class="n">m</span><span class="p">);</span> <span class="n">elliprf</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="o">-</span><span class="n">m</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span><span class="o">-</span><span class="n">m</span><span class="o">*</span><span class="n">elliprd</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="o">-</span><span class="n">m</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span>
-<span class="go">1.211056027568459524803563</span>
-<span class="go">1.211056027568459524803563</span>
-</pre></div>
-</div>
-<p>Some symmetries and argument transformations:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprf</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span><span class="p">);</span> <span class="n">elliprf</span><span class="p">(</span><span class="n">y</span><span class="p">,</span><span class="n">x</span><span class="p">,</span><span class="n">z</span><span class="p">);</span> <span class="n">elliprf</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="go">0.5840828416771517066928492</span>
-<span class="go">0.5840828416771517066928492</span>
-<span class="go">0.5840828416771517066928492</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">100000</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprf</span><span class="p">(</span><span class="n">k</span><span class="o">*</span><span class="n">x</span><span class="p">,</span><span class="n">k</span><span class="o">*</span><span class="n">y</span><span class="p">,</span><span class="n">k</span><span class="o">*</span><span class="n">z</span><span class="p">);</span> <span class="n">k</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="mf">0.5</span><span class="p">)</span> <span class="o">*</span> <span class="n">elliprf</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.001847032121923321253219284</span>
-<span class="go">0.001847032121923321253219284</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">y</span><span class="o">*</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprf</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span><span class="p">);</span> <span class="mi">2</span><span class="o">*</span><span class="n">elliprf</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">l</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="n">l</span><span class="p">,</span><span class="n">z</span><span class="o">+</span><span class="n">l</span><span class="p">)</span>
-<span class="go">0.5840828416771517066928492</span>
-<span class="go">0.5840828416771517066928492</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprf</span><span class="p">((</span><span class="n">x</span><span class="o">+</span><span class="n">l</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">,(</span><span class="n">y</span><span class="o">+</span><span class="n">l</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">,(</span><span class="n">z</span><span class="o">+</span><span class="n">l</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">0.5840828416771517066928492</span>
-</pre></div>
-</div>
-<p>Comparing with numerical integration:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprf</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.5840828416771517066928492</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="mf">0.5</span><span class="o">*</span><span class="p">((</span><span class="n">t</span><span class="o">+</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">t</span><span class="o">+</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">t</span><span class="o">+</span><span class="n">z</span><span class="p">))</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">extradps</span><span class="p">(</span><span class="mi">25</span><span class="p">)(</span><span class="n">quad</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">0.5840828416771517066928492</span>
-</pre></div>
-</div>
-<p>With the following arguments, the square root in the integrand becomes
-discontinuous at <span class="math">\(t = 1/2\)</span> if the principal branch is used. To obtain
-the right value, <span class="math">\(-\sqrt{r}\)</span> must be taken instead of <span class="math">\(\sqrt{r}\)</span>
-on <span class="math">\(t \in (0, 1/2)\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span> <span class="o">=</span> <span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="n">j</span><span class="p">,</span><span class="mi">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprf</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">(0.7961258658423391329305694 - 1.213856669836495986430094j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="n">q</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mf">0.5</span><span class="p">])</span> <span class="o">+</span> <span class="n">q</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.5</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">(0.7961258658423391329305694 - 1.213856669836495986430094j)</span>
-</pre></div>
-</div>
-<p>The so-called <em>first lemniscate constant</em>, a transcendental number:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">elliprf</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">1.31102877714605990523242</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">extradps</span><span class="p">(</span><span class="mi">25</span><span class="p">)(</span><span class="n">quad</span><span class="p">)(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">t</span><span class="o">**</span><span class="mi">4</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">1.31102877714605990523242</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="s">&#39;1/4&#39;</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span>
-<span class="go">1.31102877714605990523242</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../../mpmath/references.html#carlson">[Carlson]</a></li>
-<li><a class="reference internal" href="../../mpmath/references.html#dlmf">[DLMF]</a> Chapter 19. Elliptic Integrals</li>
-</ol>
-</dd></dl>
-
-</div>
-<div class="section" id="elliprc">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">elliprc()</span></tt><a class="headerlink" href="#elliprc" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.elliprc">
-<tt class="descclassname">mpmath.</tt><tt class="descname">elliprc</tt><big>(</big><em>x</em>, <em>y</em>, <em>pv=True</em><big>)</big><a class="headerlink" href="#mpmath.elliprc" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the degenerate Carlson symmetric elliptic integral
-of the first kind</p>
-<div class="math">
-\[R_C(x,y) = R_F(x,y,y) =
-    \frac{1}{2} \int_0^{\infty} \frac{dt}{(t+y) \sqrt{(t+x)}}.\]</div>
-<p>If <span class="math">\(y \in (-\infty,0)\)</span>, either a value defined by continuity,
-or with <em>pv=True</em> the Cauchy principal value, can be computed.</p>
-<p>If <span class="math">\(x \ge 0, y &gt; 0\)</span>, the value can be expressed in terms of
-elementary functions as</p>
-<div class="math">
-\[\begin{split}R_C(x,y) =
-\begin{cases}
-  \dfrac{1}{\sqrt{y-x}}
-    \cos^{-1}\left(\sqrt{\dfrac{x}{y}}\right),   &amp; x &lt; y \\
-  \dfrac{1}{\sqrt{y}},                          &amp; x = y \\
-  \dfrac{1}{\sqrt{x-y}}
-    \cosh^{-1}\left(\sqrt{\dfrac{x}{y}}\right),  &amp; x &gt; y \\
-\end{cases}.\end{split}\]</div>
-<p><strong>Examples</strong></p>
-<p>Some special values and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprc</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="mi">4</span><span class="p">;</span> <span class="n">elliprc</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="mi">2</span><span class="p">;</span> <span class="o">+</span><span class="n">pi</span>
-<span class="go">3.141592653589793238462643</span>
-<span class="go">3.141592653589793238462643</span>
-<span class="go">3.141592653589793238462643</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprc</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprc</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">0.2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprc</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">);</span> <span class="n">elliprc</span><span class="p">(</span><span class="n">inf</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">elliprc</span><span class="p">(</span><span class="n">inf</span><span class="p">,</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>Comparing with the elementary closed-form solution:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">elliprc</span><span class="p">(</span><span class="s">&#39;1/3&#39;</span><span class="p">,</span> <span class="s">&#39;1/5&#39;</span><span class="p">);</span> <span class="n">sqrt</span><span class="p">(</span><span class="mf">7.5</span><span class="p">)</span><span class="o">*</span><span class="n">acosh</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="s">&#39;5/3&#39;</span><span class="p">))</span>
-<span class="go">2.041630778983498390751238</span>
-<span class="go">2.041630778983498390751238</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprc</span><span class="p">(</span><span class="s">&#39;1/5&#39;</span><span class="p">,</span> <span class="s">&#39;1/3&#39;</span><span class="p">);</span> <span class="n">sqrt</span><span class="p">(</span><span class="mf">7.5</span><span class="p">)</span><span class="o">*</span><span class="n">acos</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="s">&#39;3/5&#39;</span><span class="p">))</span>
-<span class="go">1.875180765206547065111085</span>
-<span class="go">1.875180765206547065111085</span>
-</pre></div>
-</div>
-<p>Comparing with numerical integration:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">extradps</span><span class="p">(</span><span class="mi">25</span><span class="p">)(</span><span class="n">quad</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprc</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="n">pv</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">0.3333969101113672670749334</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprc</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="n">pv</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">(0.3333969101113672670749334 + 0.7024814731040726393156375j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mf">0.5</span><span class="o">*</span><span class="n">q</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">t</span><span class="o">+</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">t</span><span class="o">-</span><span class="mi">3</span><span class="p">)),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="o">-</span><span class="n">j</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">(0.3333969101113672670749334 + 0.7024814731040726393156375j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="elliprj">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">elliprj()</span></tt><a class="headerlink" href="#elliprj" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.elliprj">
-<tt class="descclassname">mpmath.</tt><tt class="descname">elliprj</tt><big>(</big><em>x</em>, <em>y</em>, <em>z</em>, <em>p</em><big>)</big><a class="headerlink" href="#mpmath.elliprj" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the Carlson symmetric elliptic integral of the third kind</p>
-<div class="math">
-\[R_J(x,y,z,p) = \frac{3}{2}
-    \int_0^{\infty} \frac{dt}{(t+p)\sqrt{(t+x)(t+y)(t+z)}}.\]</div>
-<p>Like <a class="reference internal" href="#mpmath.elliprf" title="mpmath.elliprf"><tt class="xref py py-func docutils literal"><span class="pre">elliprf()</span></tt></a>, the branch of the square root in the integrand
-is defined so as to be continuous along the path of integration for
-complex values of the arguments.</p>
-<p><strong>Examples</strong></p>
-<p>Some values and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprj</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprj</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">);</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">0.3535533905932737622004222</span>
-<span class="go">0.3535533905932737622004222</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprj</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">1.067937989667395702268688</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">3</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="s">&#39;5/4&#39;</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="n">gamma</span><span class="p">(</span><span class="s">&#39;1/4&#39;</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span>
-<span class="go">1.067937989667395702268688</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprj</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">);</span> <span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">/</span><span class="mi">4</span>
-<span class="go">1.380226776765915172432054</span>
-<span class="go">1.380226776765915172432054</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprj</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">);</span> <span class="n">elliprj</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">);</span> <span class="n">elliprj</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="go">+inf</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprj</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">);</span> <span class="n">elliprj</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">elliprj</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mi">1</span><span class="o">-</span><span class="n">j</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">0.8505007163686739432927844</span>
-</pre></div>
-</div>
-<p>Scale transformation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="n">p</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">100000</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprj</span><span class="p">(</span><span class="n">k</span><span class="o">*</span><span class="n">x</span><span class="p">,</span><span class="n">k</span><span class="o">*</span><span class="n">y</span><span class="p">,</span><span class="n">k</span><span class="o">*</span><span class="n">z</span><span class="p">,</span><span class="n">k</span><span class="o">*</span><span class="n">p</span><span class="p">);</span> <span class="n">k</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="mf">1.5</span><span class="p">)</span><span class="o">*</span><span class="n">elliprj</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
-<span class="go">4.521291677592745527851168e-9</span>
-<span class="go">4.521291677592745527851168e-9</span>
-</pre></div>
-</div>
-<p>Comparing with numerical integration:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">elliprj</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">0.2398480997495677621758617</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="p">((</span><span class="n">t</span><span class="o">+</span><span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="n">sqrt</span><span class="p">((</span><span class="n">t</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">t</span><span class="o">+</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">t</span><span class="o">+</span><span class="mi">3</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mf">1.5</span><span class="o">*</span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">0.2398480997495677621758617</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprj</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="o">+</span><span class="mi">1j</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="o">-</span><span class="mi">2j</span><span class="p">)</span>
-<span class="go">(0.216888906014633498739952 + 0.04081912627366673332369512j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="p">((</span><span class="n">t</span><span class="o">+</span><span class="mi">4</span><span class="o">-</span><span class="mi">2j</span><span class="p">)</span><span class="o">*</span><span class="n">sqrt</span><span class="p">((</span><span class="n">t</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">t</span><span class="o">+</span><span class="mi">2</span><span class="o">+</span><span class="mi">1j</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">t</span><span class="o">+</span><span class="mi">3</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mf">1.5</span><span class="o">*</span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">(0.216888906014633498739952 + 0.04081912627366673332369511j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="elliprd">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">elliprd()</span></tt><a class="headerlink" href="#elliprd" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.elliprd">
-<tt class="descclassname">mpmath.</tt><tt class="descname">elliprd</tt><big>(</big><em>x</em>, <em>y</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.elliprd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the degenerate Carlson symmetric elliptic integral
-of the third kind or Carlson elliptic integral of the
-second kind <span class="math">\(R_D(x,y,z) = R_J(x,y,z,z)\)</span>.</p>
-<p>See <a class="reference internal" href="#mpmath.elliprj" title="mpmath.elliprj"><tt class="xref py py-func docutils literal"><span class="pre">elliprj()</span></tt></a> for additional information.</p>
-<p><strong>Examples</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprd</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.2904602810289906442326534</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprj</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.2904602810289906442326534</span>
-</pre></div>
-</div>
-<p>The so-called <em>second lemniscate constant</em>, a transcendental number:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">elliprd</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span>
-<span class="go">0.5990701173677961037199612</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">extradps</span><span class="p">(</span><span class="mi">25</span><span class="p">)(</span><span class="n">quad</span><span class="p">)(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">t</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">t</span><span class="o">**</span><span class="mi">4</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">0.5990701173677961037199612</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="s">&#39;3/4&#39;</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">0.5990701173677961037199612</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="elliprg">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">elliprg()</span></tt><a class="headerlink" href="#elliprg" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.elliprg">
-<tt class="descclassname">mpmath.</tt><tt class="descname">elliprg</tt><big>(</big><em>x</em>, <em>y</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.elliprg" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the Carlson completely symmetric elliptic integral
-of the second kind</p>
-<div class="math">
-\[R_G(x,y,z) = \frac{1}{4} \int_0^{\infty}
-    \frac{t}{\sqrt{(t+x)(t+y)(t+z)}}
-    \left( \frac{x}{t+x} + \frac{y}{t+y} + \frac{z}{t+z}\right) dt.\]</div>
-<p><strong>Examples</strong></p>
-<p>Evaluation for real and complex arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprg</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="mi">4</span><span class="p">;</span> <span class="o">+</span><span class="n">pi</span>
-<span class="go">3.141592653589793238462643</span>
-<span class="go">3.141592653589793238462643</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">elliprg</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.6753219405238377512600874</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">elliprg</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mi">1</span><span class="o">-</span><span class="n">j</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">1.172431327676416604532822</span>
-</pre></div>
-</div>
-<p>A double integral that can be evaluated in terms of <span class="math">\(R_G\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">u</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">st</span> <span class="o">=</span> <span class="n">fp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">);</span> <span class="n">ct</span> <span class="o">=</span> <span class="n">fp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-<span class="gp">... </span>    <span class="n">su</span> <span class="o">=</span> <span class="n">fp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">u</span><span class="p">);</span> <span class="n">cu</span> <span class="o">=</span> <span class="n">fp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
-<span class="gp">... </span>    <span class="k">return</span> <span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="n">st</span><span class="o">*</span><span class="n">cu</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">*</span><span class="p">(</span><span class="n">st</span><span class="o">*</span><span class="n">su</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">z</span><span class="o">*</span><span class="n">ct</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mf">0.5</span> <span class="o">*</span> <span class="n">st</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="n">fp</span><span class="o">.</span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">fp</span><span class="o">.</span><span class="n">pi</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="o">*</span><span class="n">fp</span><span class="o">.</span><span class="n">pi</span><span class="p">])</span><span class="o">/</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">fp</span><span class="o">.</span><span class="n">pi</span><span class="p">)),</span> <span class="mi">13</span><span class="p">)</span>
-<span class="go">1.725503028069</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">elliprg</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span><span class="p">),</span> <span class="mi">13</span><span class="p">)</span>
-<span class="go">1.725503028069</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="jacobi-theta-functions">
-<h2>Jacobi theta functions<a class="headerlink" href="#jacobi-theta-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="jtheta">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">jtheta()</span></tt><a class="headerlink" href="#jtheta" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.jtheta">
-<tt class="descclassname">mpmath.</tt><tt class="descname">jtheta</tt><big>(</big><em>n</em>, <em>z</em>, <em>q</em>, <em>derivative=0</em><big>)</big><a class="headerlink" href="#mpmath.jtheta" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the Jacobi theta function <span class="math">\(\vartheta_n(z, q)\)</span>, where
-<span class="math">\(n = 1, 2, 3, 4\)</span>, defined by the infinite series:</p>
-<div class="math">
-\[\vartheta_1(z,q) = 2 q^{1/4} \sum_{n=0}^{\infty}
-  (-1)^n q^{n^2+n\,} \sin((2n+1)z)\]\[\vartheta_2(z,q) = 2 q^{1/4} \sum_{n=0}^{\infty}
-  q^{n^{2\,} + n} \cos((2n+1)z)\]\[\vartheta_3(z,q) = 1 + 2 \sum_{n=1}^{\infty}
-  q^{n^2\,} \cos(2 n z)\]\[\vartheta_4(z,q) = 1 + 2 \sum_{n=1}^{\infty}
-  (-q)^{n^2\,} \cos(2 n z)\]</div>
-<p>The theta functions are functions of two variables:</p>
-<ul class="simple">
-<li><span class="math">\(z\)</span> is the <em>argument</em>, an arbitrary real or complex number</li>
-<li><span class="math">\(q\)</span> is the <em>nome</em>, which must be a real or complex number
-in the unit disk (i.e. <span class="math">\(|q| &lt; 1\)</span>). For <span class="math">\(|q| \ll 1\)</span>, the
-series converge very quickly, so the Jacobi theta functions
-can efficiently be evaluated to high precision.</li>
-</ul>
-<p>The compact notations <span class="math">\(\vartheta_n(q) = \vartheta_n(0,q)\)</span>
-and <span class="math">\(\vartheta_n = \vartheta_n(0,q)\)</span> are also frequently
-encountered. Finally, Jacobi theta functions are frequently
-considered as functions of the half-period ratio <span class="math">\(\tau\)</span>
-and then usually denoted by <span class="math">\(\vartheta_n(z|\tau)\)</span>.</p>
-<p>Optionally, <tt class="docutils literal"><span class="pre">jtheta(n,</span> <span class="pre">z,</span> <span class="pre">q,</span> <span class="pre">derivative=d)</span></tt> with <span class="math">\(d &gt; 0\)</span> computes
-a <span class="math">\(d\)</span>-th derivative with respect to <span class="math">\(z\)</span>.</p>
-<p><strong>Examples and basic properties</strong></p>
-<p>Considered as functions of <span class="math">\(z\)</span>, the Jacobi theta functions may be
-viewed as generalizations of the ordinary trigonometric functions
-cos and sin. They are periodic functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jtheta</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="s">&#39;0.2&#39;</span><span class="p">)</span>
-<span class="go">0.2945120798627300045053104</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jtheta</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mf">0.25</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">,</span> <span class="s">&#39;0.2&#39;</span><span class="p">)</span>
-<span class="go">0.2945120798627300045053104</span>
-</pre></div>
-</div>
-<p>Indeed, the series defining the theta functions are essentially
-trigonometric Fourier series. The coefficients can be retrieved
-using <a class="reference internal" href="../../mpmath/calculus/approximation.html#mpmath.fourier" title="mpmath.fourier"><tt class="xref py py-func docutils literal"><span class="pre">fourier()</span></tt></a>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">fourier</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">jtheta</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="n">pi</span><span class="p">,</span> <span class="n">pi</span><span class="p">],</span> <span class="mi">4</span><span class="p">))</span>
-<span class="go">([0.0, 1.68179, 0.0, 0.420448, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0])</span>
-</pre></div>
-</div>
-<p>The Jacobi theta functions are also so-called quasiperiodic
-functions of <span class="math">\(z\)</span> and <span class="math">\(\tau\)</span>, meaning that for fixed <span class="math">\(\tau\)</span>,
-<span class="math">\(\vartheta_n(z, q)\)</span> and <span class="math">\(\vartheta_n(z+\pi \tau, q)\)</span> are the same
-except for an exponential factor:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tau</span> <span class="o">=</span> <span class="mi">3</span><span class="o">*</span><span class="n">j</span><span class="o">/</span><span class="mi">10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">j</span><span class="o">*</span><span class="n">tau</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mi">10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jtheta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">z</span><span class="o">+</span><span class="n">tau</span><span class="o">*</span><span class="n">pi</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
-<span class="go">(-0.682420280786034687520568 + 1.526683999721399103332021j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">j</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="n">q</span> <span class="o">*</span> <span class="n">jtheta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
-<span class="go">(-0.682420280786034687520568 + 1.526683999721399103332021j)</span>
-</pre></div>
-</div>
-<p>The Jacobi theta functions satisfy a huge number of other
-functional equations, such as the following identity (valid for
-any <span class="math">\(q\)</span>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jtheta</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">q</span><span class="p">)</span><span class="o">**</span><span class="mi">4</span>
-<span class="go">6.823744089352763305137427</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jtheta</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">q</span><span class="p">)</span><span class="o">**</span><span class="mi">4</span> <span class="o">+</span> <span class="n">jtheta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">q</span><span class="p">)</span><span class="o">**</span><span class="mi">4</span>
-<span class="go">6.823744089352763305137427</span>
-</pre></div>
-</div>
-<p>Extensive listings of identities satisfied by the Jacobi theta
-functions can be found in standard reference works.</p>
-<p>The Jacobi theta functions are related to the gamma function
-for special arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">jtheta</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">pi</span><span class="p">))</span>
-<span class="go">1.086434811213308014575316</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mf">4.</span><span class="p">)</span> <span class="o">/</span> <span class="n">gamma</span><span class="p">(</span><span class="mi">3</span><span class="o">/</span><span class="mf">4.</span><span class="p">)</span>
-<span class="go">1.086434811213308014575316</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.jtheta" title="mpmath.jtheta"><tt class="xref py py-func docutils literal"><span class="pre">jtheta()</span></tt></a> supports arbitrary precision evaluation and complex
-arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jtheta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="mf">0.5</span><span class="p">)</span>
-<span class="go">2.0549510717571539127004115835148878097035750653737</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jtheta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="o">+</span><span class="mi">2j</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">)</span><span class="o">/</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">(7.180331760146805926356634 - 1.634292858119162417301683j)</span>
-</pre></div>
-</div>
-<p>Evaluation of derivatives:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jtheta</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mi">1</span><span class="p">);</span> <span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">jtheta</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">),</span> <span class="mi">7</span><span class="p">)</span>
-<span class="go">1.209857192844475388637236</span>
-<span class="go">1.209857192844475388637236</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jtheta</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mi">2</span><span class="p">);</span> <span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">jtheta</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">),</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">-0.2598718791650217206533052</span>
-<span class="go">-0.2598718791650217206533052</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jtheta</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mi">1</span><span class="p">);</span> <span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">jtheta</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">),</span> <span class="mi">7</span><span class="p">)</span>
-<span class="go">-1.150231437070259644461474</span>
-<span class="go">-1.150231437070259644461474</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jtheta</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mi">2</span><span class="p">);</span> <span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">jtheta</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">),</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">-0.6226636990043777445898114</span>
-<span class="go">-0.6226636990043777445898114</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jtheta</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mi">1</span><span class="p">);</span> <span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">jtheta</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">),</span> <span class="mi">7</span><span class="p">)</span>
-<span class="go">-0.9990312046096634316587882</span>
-<span class="go">-0.9990312046096634316587882</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jtheta</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mi">2</span><span class="p">);</span> <span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">jtheta</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">),</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">-0.1530388693066334936151174</span>
-<span class="go">-0.1530388693066334936151174</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jtheta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mi">1</span><span class="p">);</span> <span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">jtheta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">),</span> <span class="mi">7</span><span class="p">)</span>
-<span class="go">0.9820995967262793943571139</span>
-<span class="go">0.9820995967262793943571139</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jtheta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mi">2</span><span class="p">);</span> <span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">jtheta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">),</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">0.3936902850291437081667755</span>
-<span class="go">0.3936902850291437081667755</span>
-</pre></div>
-</div>
-<p><strong>Possible issues</strong></p>
-<p>For <span class="math">\(|q| \ge 1\)</span> or <span class="math">\(\Im(\tau) \le 0\)</span>, <a class="reference internal" href="#mpmath.jtheta" title="mpmath.jtheta"><tt class="xref py py-func docutils literal"><span class="pre">jtheta()</span></tt></a> raises
-<tt class="docutils literal"><span class="pre">ValueError</span></tt>. This exception is also raised for <span class="math">\(|q|\)</span> extremely
-close to 1 (or equivalently <span class="math">\(\tau\)</span> very close to 0), since the
-series would converge too slowly:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">jtheta</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mf">0.99999999</span> <span class="o">*</span> <span class="n">exp</span><span class="p">(</span><span class="mf">0.5</span><span class="o">*</span><span class="n">j</span><span class="p">))</span>
-<span class="gt">Traceback (most recent call last):</span>
-  <span class="c">...</span>
-<span class="gr">ValueError</span>: <span class="n">abs(q) &gt; THETA_Q_LIM = 1.000000</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="jacobi-elliptic-functions">
-<h2>Jacobi elliptic functions<a class="headerlink" href="#jacobi-elliptic-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="ellipfun">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">ellipfun()</span></tt><a class="headerlink" href="#ellipfun" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.ellipfun">
-<tt class="descclassname">mpmath.</tt><tt class="descname">ellipfun</tt><big>(</big><em>kind</em>, <em>u=None</em>, <em>m=None</em>, <em>q=None</em>, <em>k=None</em>, <em>tau=None</em><big>)</big><a class="headerlink" href="#mpmath.ellipfun" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes any of the Jacobi elliptic functions, defined
-in terms of Jacobi theta functions as</p>
-<div class="math">
-\[\mathrm{sn}(u,m) = \frac{\vartheta_3(0,q)}{\vartheta_2(0,q)}
-    \frac{\vartheta_1(t,q)}{\vartheta_4(t,q)}\]\[\mathrm{cn}(u,m) = \frac{\vartheta_4(0,q)}{\vartheta_2(0,q)}
-    \frac{\vartheta_2(t,q)}{\vartheta_4(t,q)}\]\[\mathrm{dn}(u,m) = \frac{\vartheta_4(0,q)}{\vartheta_3(0,q)}
-    \frac{\vartheta_3(t,q)}{\vartheta_4(t,q)},\]</div>
-<p>or more generally computes a ratio of two such functions. Here
-<span class="math">\(t = u/\vartheta_3(0,q)^2\)</span>, and <span class="math">\(q = q(m)\)</span> denotes the nome (see
-<tt class="xref py py-func docutils literal"><span class="pre">nome()</span></tt>). Optionally, you can specify the nome directly
-instead of <span class="math">\(m\)</span> by passing <tt class="docutils literal"><span class="pre">q=&lt;value&gt;</span></tt>, or you can directly
-specify the elliptic parameter <span class="math">\(k\)</span> with <tt class="docutils literal"><span class="pre">k=&lt;value&gt;</span></tt>.</p>
-<p>The first argument should be a two-character string specifying the
-function using any combination of <tt class="docutils literal"><span class="pre">'s'</span></tt>, <tt class="docutils literal"><span class="pre">'c'</span></tt>, <tt class="docutils literal"><span class="pre">'d'</span></tt>, <tt class="docutils literal"><span class="pre">'n'</span></tt>. These
-letters respectively denote the basic functions
-<span class="math">\(\mathrm{sn}(u,m)\)</span>, <span class="math">\(\mathrm{cn}(u,m)\)</span>, <span class="math">\(\mathrm{dn}(u,m)\)</span>, and <span class="math">\(1\)</span>.
-The identifier specifies the ratio of two such functions.
-For example, <tt class="docutils literal"><span class="pre">'ns'</span></tt> identifies the function</p>
-<div class="math">
-\[\mathrm{ns}(u,m) = \frac{1}{\mathrm{sn}(u,m)}\]</div>
-<p>and <tt class="docutils literal"><span class="pre">'cd'</span></tt> identifies the function</p>
-<div class="math">
-\[\mathrm{cd}(u,m) = \frac{\mathrm{cn}(u,m)}{\mathrm{dn}(u,m)}.\]</div>
-<p>If called with only the first argument, a function object
-evaluating the chosen function for given arguments is returned.</p>
-<p><strong>Examples</strong></p>
-<p>Basic evaluation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipfun</span><span class="p">(</span><span class="s">&#39;cd&#39;</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
-<span class="go">-0.9891101840595543931308394</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ellipfun</span><span class="p">(</span><span class="s">&#39;cd&#39;</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">,</span> <span class="n">q</span><span class="o">=</span><span class="mf">0.25</span><span class="p">)</span>
-<span class="go">0.07111979240214668158441418</span>
-</pre></div>
-</div>
-<p>The sn-function is doubly periodic in the complex plane with periods
-<span class="math">\(4 K(m)\)</span> and <span class="math">\(2 i K(1-m)\)</span> (see <a class="reference internal" href="#mpmath.ellipk" title="mpmath.ellipk"><tt class="xref py py-func docutils literal"><span class="pre">ellipk()</span></tt></a>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sn</span> <span class="o">=</span> <span class="n">ellipfun</span><span class="p">(</span><span class="s">&#39;sn&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">)</span>
-<span class="go">0.9628981775982774425751399</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sn</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">4</span><span class="o">*</span><span class="n">ellipk</span><span class="p">(</span><span class="mf">0.25</span><span class="p">),</span> <span class="mf">0.25</span><span class="p">)</span>
-<span class="go">0.9628981775982774425751399</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">sn</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">j</span><span class="o">*</span><span class="n">ellipk</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="mf">0.25</span><span class="p">),</span> <span class="mf">0.25</span><span class="p">))</span>
-<span class="go">0.9628981775982774425751399</span>
-</pre></div>
-</div>
-<p>The cn-function is doubly periodic with periods <span class="math">\(4 K(m)\)</span> and <span class="math">\(4 i K(1-m)\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">cn</span> <span class="o">=</span> <span class="n">ellipfun</span><span class="p">(</span><span class="s">&#39;cn&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">)</span>
-<span class="go">-0.2698649654510865792581416</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cn</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">4</span><span class="o">*</span><span class="n">ellipk</span><span class="p">(</span><span class="mf">0.25</span><span class="p">),</span> <span class="mf">0.25</span><span class="p">)</span>
-<span class="go">-0.2698649654510865792581416</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">cn</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">4</span><span class="o">*</span><span class="n">j</span><span class="o">*</span><span class="n">ellipk</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="mf">0.25</span><span class="p">),</span> <span class="mf">0.25</span><span class="p">))</span>
-<span class="go">-0.2698649654510865792581416</span>
-</pre></div>
-</div>
-<p>The dn-function is doubly periodic with periods <span class="math">\(2 K(m)\)</span> and <span class="math">\(4 i K(1-m)\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dn</span> <span class="o">=</span> <span class="n">ellipfun</span><span class="p">(</span><span class="s">&#39;dn&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">)</span>
-<span class="go">0.8764740583123262286931578</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dn</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">ellipk</span><span class="p">(</span><span class="mf">0.25</span><span class="p">),</span> <span class="mf">0.25</span><span class="p">)</span>
-<span class="go">0.8764740583123262286931578</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">dn</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">4</span><span class="o">*</span><span class="n">j</span><span class="o">*</span><span class="n">ellipk</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="mf">0.25</span><span class="p">),</span> <span class="mf">0.25</span><span class="p">))</span>
-<span class="go">0.8764740583123262286931578</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="modular-functions">
-<h2>Modular functions<a class="headerlink" href="#modular-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="kleinj">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">kleinj()</span></tt><a class="headerlink" href="#kleinj" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.kleinj">
-<tt class="descclassname">mpmath.</tt><tt class="descname">kleinj</tt><big>(</big><em>tau=None</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.kleinj" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the Klein j-invariant, which is a modular function defined for
-<span class="math">\(\tau\)</span> in the upper half-plane as</p>
-<div class="math">
-\[J(\tau) = \frac{g_2^3(\tau)}{g_2^3(\tau) - 27 g_3^2(\tau)}\]</div>
-<p>where <span class="math">\(g_2\)</span> and <span class="math">\(g_3\)</span> are the modular invariants of the Weierstrass
-elliptic function,</p>
-<div class="math">
-\[g_2(\tau) = 60 \sum_{(m,n) \in \mathbb{Z}^2 \setminus (0,0)} (m \tau+n)^{-4}\]\[g_3(\tau) = 140 \sum_{(m,n) \in \mathbb{Z}^2 \setminus (0,0)} (m \tau+n)^{-6}.\]</div>
-<p>An alternative, common notation is that of the j-function
-<span class="math">\(j(\tau) = 1728 J(\tau)\)</span>.</p>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Klein J-function as function of the number-theoretic nome</span>
-<span class="n">fp</span><span class="o">.</span><span class="n">cplot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">q</span><span class="p">:</span> <span class="n">fp</span><span class="o">.</span><span class="n">kleinj</span><span class="p">(</span><span class="n">qbar</span><span class="o">=</span><span class="n">q</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="n">points</span><span class="o">=</span><span class="mi">50000</span><span class="p">)</span>
-</pre></div>
-</div>
-<img alt="../../../_images/kleinj.png" src="../../../_images/kleinj.png" />
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Klein J-function as function of the half-period ratio</span>
-<span class="n">fp</span><span class="o">.</span><span class="n">cplot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">fp</span><span class="o">.</span><span class="n">kleinj</span><span class="p">(</span><span class="n">tau</span><span class="o">=</span><span class="n">t</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mf">1.5</span><span class="p">],</span> <span class="n">points</span><span class="o">=</span><span class="mi">50000</span><span class="p">)</span>
-</pre></div>
-</div>
-<img alt="../../../_images/kleinj2.png" src="../../../_images/kleinj2.png" />
-<p><strong>Examples</strong></p>
-<p>Verifying the functional equation <span class="math">\(J(\tau) = J(\tau+1) = J(-\tau^{-1})\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tau</span> <span class="o">=</span> <span class="mf">0.625</span><span class="o">+</span><span class="mf">0.75</span><span class="o">*</span><span class="n">j</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tau</span> <span class="o">=</span> <span class="mf">0.625</span><span class="o">+</span><span class="mf">0.75</span><span class="o">*</span><span class="n">j</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">kleinj</span><span class="p">(</span><span class="n">tau</span><span class="p">)</span>
-<span class="go">(-0.1507492166511182267125242 + 0.07595948379084571927228948j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">kleinj</span><span class="p">(</span><span class="n">tau</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(-0.1507492166511182267125242 + 0.07595948379084571927228948j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">kleinj</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">tau</span><span class="p">)</span>
-<span class="go">(-0.1507492166511182267125242 + 0.07595948379084571927228946j)</span>
-</pre></div>
-</div>
-<p>The j-function has a famous Laurent series expansion in terms of the nome
-<span class="math">\(\bar{q}\)</span>, <span class="math">\(j(\tau) = \bar{q}^{-1} + 744 + 196884\bar{q} + \ldots\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">q</span><span class="p">:</span> <span class="mi">1728</span><span class="o">*</span><span class="n">q</span><span class="o">*</span><span class="n">kleinj</span><span class="p">(</span><span class="n">qbar</span><span class="o">=</span><span class="n">q</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">singular</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">[1.0, 744.0, 196884.0, 21493760.0, 864299970.0, 20245856256.0]</span>
-</pre></div>
-</div>
-<p>The j-function admits exact evaluation at special algebraic points
-related to the Heegner numbers 1, 2, 3, 7, 11, 19, 43, 67, 163:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nd">@extraprec</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="gp">... </span><span class="k">def</span> <span class="nf">h</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">v</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">sqrt</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">j</span><span class="p">)</span>
-<span class="gp">... </span>    <span class="k">if</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">2</span><span class="p">:</span>
-<span class="gp">... </span>        <span class="n">v</span> <span class="o">*=</span> <span class="mf">0.5</span>
-<span class="gp">... </span>    <span class="k">return</span> <span class="n">v</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">11</span><span class="p">,</span><span class="mi">19</span><span class="p">,</span><span class="mi">43</span><span class="p">,</span><span class="mi">67</span><span class="p">,</span><span class="mi">163</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="n">n</span><span class="p">,</span> <span class="n">chop</span><span class="p">(</span><span class="mi">1728</span><span class="o">*</span><span class="n">kleinj</span><span class="p">(</span><span class="n">h</span><span class="p">(</span><span class="n">n</span><span class="p">)))</span>
-<span class="gp">...</span>
-<span class="go">(1, 1728.0)</span>
-<span class="go">(2, 8000.0)</span>
-<span class="go">(3, 0.0)</span>
-<span class="go">(7, -3375.0)</span>
-<span class="go">(11, -32768.0)</span>
-<span class="go">(19, -884736.0)</span>
-<span class="go">(43, -884736000.0)</span>
-<span class="go">(67, -147197952000.0)</span>
-<span class="go">(163, -262537412640768000.0)</span>
-</pre></div>
-</div>
-<p>Also at other special points, the j-function assumes explicit
-algebraic values, e.g.:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="mi">1728</span><span class="o">*</span><span class="n">kleinj</span><span class="p">(</span><span class="n">j</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">)))</span>
-<span class="go">1264538.909475140509320227</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="n">cbrt</span><span class="p">(</span><span class="n">_</span><span class="p">))</span>      <span class="c"># note: not simplified</span>
-<span class="go">&#39;((100+sqrt(13520))/2)&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">50</span><span class="o">+</span><span class="mi">26</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span><span class="o">**</span><span class="mi">3</span>
-<span class="go">1264538.909475140509320227</span>
-</pre></div>
-</div>
-</dd></dl>
-
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-<li><a class="reference internal" href="#">Elliptic functions</a><ul>
-<li><a class="reference internal" href="#elliptic-arguments">Elliptic arguments</a><ul>
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-  <div class="section" id="exponential-integrals-and-error-functions">
-<h1>Exponential integrals and error functions<a class="headerlink" href="#exponential-integrals-and-error-functions" title="Permalink to this headline">¶</a></h1>
-<p>Exponential integrals give closed-form solutions to a large class of commonly occurring transcendental integrals that cannot be evaluated using elementary functions. Integrals of this type include those with an integrand of the form <span class="math">\(t^a e^{t}\)</span> or <span class="math">\(e^{-x^2}\)</span>, the latter giving rise to the Gaussian (or normal) probability distribution.</p>
-<p>The most general function in this section is the incomplete gamma function, to which all others can be reduced. The incomplete gamma function, in turn, can be expressed using hypergeometric functions (see <a class="reference internal" href="../functions/hypergeometric.html"><em>Hypergeometric functions</em></a>).</p>
-<div class="section" id="incomplete-gamma-functions">
-<h2>Incomplete gamma functions<a class="headerlink" href="#incomplete-gamma-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="gammainc">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">gammainc()</span></tt><a class="headerlink" href="#gammainc" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.gammainc">
-<tt class="descclassname">mpmath.</tt><tt class="descname">gammainc</tt><big>(</big><em>z</em>, <em>a=0</em>, <em>b=inf</em>, <em>regularized=False</em><big>)</big><a class="headerlink" href="#mpmath.gammainc" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">gammainc(z,</span> <span class="pre">a=0,</span> <span class="pre">b=inf)</span></tt> computes the (generalized) incomplete
-gamma function with integration limits <span class="math">\([a, b]\)</span>:</p>
-<div class="math">
-\[\Gamma(z,a,b) = \int_a^b t^{z-1} e^{-t} \, dt\]</div>
-<p>The generalized incomplete gamma function reduces to the
-following special cases when one or both endpoints are fixed:</p>
-<ul class="simple">
-<li><span class="math">\(\Gamma(z,0,\infty)\)</span> is the standard (&#8220;complete&#8221;)
-gamma function, <span class="math">\(\Gamma(z)\)</span> (available directly
-as the mpmath function <a class="reference internal" href="../functions/gamma.html#mpmath.gamma" title="mpmath.gamma"><tt class="xref py py-func docutils literal"><span class="pre">gamma()</span></tt></a>)</li>
-<li><span class="math">\(\Gamma(z,a,\infty)\)</span> is the &#8220;upper&#8221; incomplete gamma
-function, <span class="math">\(\Gamma(z,a)\)</span></li>
-<li><span class="math">\(\Gamma(z,0,b)\)</span> is the &#8220;lower&#8221; incomplete gamma
-function, <span class="math">\(\gamma(z,b)\)</span>.</li>
-</ul>
-<p>Of course, we have
-<span class="math">\(\Gamma(z,0,x) + \Gamma(z,x,\infty) = \Gamma(z)\)</span>
-for all <span class="math">\(z\)</span> and <span class="math">\(x\)</span>.</p>
-<p>Note however that some authors reverse the order of the
-arguments when defining the lower and upper incomplete
-gamma function, so one should be careful to get the correct
-definition.</p>
-<p>If also given the keyword argument <tt class="docutils literal"><span class="pre">regularized=True</span></tt>,
-<a class="reference internal" href="#mpmath.gammainc" title="mpmath.gammainc"><tt class="xref py py-func docutils literal"><span class="pre">gammainc()</span></tt></a> computes the &#8220;regularized&#8221; incomplete gamma
-function</p>
-<div class="math">
-\[P(z,a,b) = \frac{\Gamma(z,a,b)}{\Gamma(z)}.\]</div>
-<p><strong>Examples</strong></p>
-<p>We can compare with numerical quadrature to verify that
-<a class="reference internal" href="#mpmath.gammainc" title="mpmath.gammainc"><tt class="xref py py-func docutils literal"><span class="pre">gammainc()</span></tt></a> computes the integral in the definition:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gammainc</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">(0.00977212668627705160602312 - 0.0770637306312989892451977j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">t</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">t</span><span class="p">),</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">10</span><span class="p">])</span>
-<span class="go">(0.00977212668627705160602312 - 0.0770637306312989892451977j)</span>
-</pre></div>
-</div>
-<p>Argument symmetries follow directly from the integral definition:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gammainc</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span> <span class="o">+</span> <span class="n">gammainc</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gammainc</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">gammainc</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">);</span> <span class="n">gammainc</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">1.523793388892911312363331</span>
-<span class="go">1.523793388892911312363331</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">gammainc</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">3.0</span>
-</pre></div>
-</div>
-<p>Evaluation for arbitrarily large arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gammainc</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
-<span class="go">4.083660630910611272288592e-26</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gammainc</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10000000000000000</span><span class="p">)</span>
-<span class="go">5.290402449901174752972486e-4342944819032375</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gammainc</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">,</span> <span class="mi">1000000</span><span class="o">+</span><span class="mi">1000000j</span><span class="p">)</span>
-<span class="go">(-1.257913707524362408877881e-434284 + 2.556691003883483531962095e-434284j)</span>
-</pre></div>
-</div>
-<p>Evaluation of a generalized incomplete gamma function automatically chooses
-the representation that gives a more accurate result, depending on which
-parameter is larger:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gammainc</span><span class="p">(</span><span class="mi">10000000</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="n">gammainc</span><span class="p">(</span><span class="mi">10000000</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>   <span class="c"># Bad</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gammainc</span><span class="p">(</span><span class="mi">10000000</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>   <span class="c"># Good</span>
-<span class="go">1.755146243738946045873491e+4771204</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gammainc</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">100000001</span><span class="p">)</span> <span class="o">-</span> <span class="n">gammainc</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">100000000</span><span class="p">)</span>   <span class="c"># Bad</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gammainc</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">100000000</span><span class="p">,</span> <span class="mi">100000001</span><span class="p">)</span>   <span class="c"># Good</span>
-<span class="go">4.078258353474186729184421e-43429441</span>
-</pre></div>
-</div>
-<p>The incomplete gamma functions satisfy simple recurrence
-relations:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span><span class="p">,</span> <span class="n">a</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">3.5</span><span class="p">),</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gammainc</span><span class="p">(</span><span class="n">z</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">a</span><span class="p">);</span> <span class="n">z</span><span class="o">*</span><span class="n">gammainc</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="n">a</span><span class="p">)</span> <span class="o">+</span> <span class="n">a</span><span class="o">**</span><span class="n">z</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="p">)</span>
-<span class="go">10.60130296933533459267329</span>
-<span class="go">10.60130296933533459267329</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gammainc</span><span class="p">(</span><span class="n">z</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">a</span><span class="p">);</span> <span class="n">z</span><span class="o">*</span><span class="n">gammainc</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">a</span><span class="p">)</span> <span class="o">-</span> <span class="n">a</span><span class="o">**</span><span class="n">z</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="p">)</span>
-<span class="go">1.030425427232114336470932</span>
-<span class="go">1.030425427232114336470932</span>
-</pre></div>
-</div>
-<p>Evaluation at integers and poles:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gammainc</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">(-0.2214577048967798566234192 + 0.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gammainc</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">+inf</span>
-</pre></div>
-</div>
-<p>If <span class="math">\(z\)</span> is an integer, the recurrence reduces the incomplete gamma
-function to <span class="math">\(P(a) \exp(-a) + Q(b) \exp(-b)\)</span> where <span class="math">\(P\)</span> and
-<span class="math">\(Q\)</span> are polynomials:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gammainc</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">);</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.1353352832366126918939995</span>
-<span class="go">0.1353352832366126918939995</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="n">gammainc</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="s">&#39;exp(-1)&#39;</span><span class="p">,</span> <span class="s">&#39;exp(-2)&#39;</span><span class="p">])</span>
-<span class="go">&#39;(326*exp(-1) + (-872)*exp(-2))&#39;</span>
-</pre></div>
-</div>
-<p>The incomplete gamma functions reduce to functions such as
-the exponential integral Ei and the error function for special
-arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gammainc</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">);</span> <span class="o">-</span><span class="n">ei</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">0.00377935240984890647887486</span>
-<span class="go">0.00377935240984890647887486</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gammainc</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">);</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">erf</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">1.691806732945198336509541</span>
-<span class="go">1.691806732945198336509541</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="exponential-integrals">
-<h2>Exponential integrals<a class="headerlink" href="#exponential-integrals" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="ei">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">ei()</span></tt><a class="headerlink" href="#ei" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.ei">
-<tt class="descclassname">mpmath.</tt><tt class="descname">ei</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.ei" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the exponential integral or Ei-function, <span class="math">\(\mathrm{Ei}(x)\)</span>.
-The exponential integral is defined as</p>
-<div class="math">
-\[\mathrm{Ei}(x) = \int_{-\infty\,}^x \frac{e^t}{t} \, dt.\]</div>
-<p>When the integration range includes <span class="math">\(t = 0\)</span>, the exponential
-integral is interpreted as providing the Cauchy principal value.</p>
-<p>For real <span class="math">\(x\)</span>, the Ei-function behaves roughly like
-<span class="math">\(\mathrm{Ei}(x) \approx \exp(x) + \log(|x|)\)</span>.</p>
-<p>The Ei-function is related to the more general family of exponential
-integral functions denoted by <span class="math">\(E_n\)</span>, which are available as <a class="reference internal" href="#mpmath.expint" title="mpmath.expint"><tt class="xref py py-func docutils literal"><span class="pre">expint()</span></tt></a>.</p>
-<p><strong>Basic examples</strong></p>
-<p>Some basic values and limits are:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ei</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">-inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ei</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1.89511781635594</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ei</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ei</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>For <span class="math">\(x &lt; 0\)</span>, the defining integral can be evaluated
-numerically as a reference:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ei</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">-0.00377935240984891</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">/</span><span class="n">t</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="p">])</span>
-<span class="go">-0.00377935240984891</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.ei" title="mpmath.ei"><tt class="xref py py-func docutils literal"><span class="pre">ei()</span></tt></a> supports complex arguments and arbitrary
-precision evaluation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ei</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">10.928374389331410348638445906907535171566338835056</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ei</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(-4.154091651642689822535359 + 4.294418620024357476985535j)</span>
-</pre></div>
-</div>
-<p><strong>Related functions</strong></p>
-<p>The exponential integral is closely related to the logarithmic
-integral. See <a class="reference internal" href="#mpmath.li" title="mpmath.li"><tt class="xref py py-func docutils literal"><span class="pre">li()</span></tt></a> for additional information.</p>
-<p>The exponential integral is related to the hyperbolic
-and trigonometric integrals (see <a class="reference internal" href="#mpmath.chi" title="mpmath.chi"><tt class="xref py py-func docutils literal"><span class="pre">chi()</span></tt></a>, <a class="reference internal" href="#mpmath.shi" title="mpmath.shi"><tt class="xref py py-func docutils literal"><span class="pre">shi()</span></tt></a>,
-<a class="reference internal" href="#mpmath.ci" title="mpmath.ci"><tt class="xref py py-func docutils literal"><span class="pre">ci()</span></tt></a>, <a class="reference internal" href="#mpmath.si" title="mpmath.si"><tt class="xref py py-func docutils literal"><span class="pre">si()</span></tt></a>) similarly to how the ordinary
-exponential function is related to the hyperbolic and
-trigonometric functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ei</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">9.93383257062542</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chi</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">+</span> <span class="n">shi</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">9.93383257062542</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">ci</span><span class="p">(</span><span class="mi">3j</span><span class="p">)</span> <span class="o">-</span> <span class="n">j</span><span class="o">*</span><span class="n">si</span><span class="p">(</span><span class="mi">3j</span><span class="p">)</span> <span class="o">-</span> <span class="n">pi</span><span class="o">*</span><span class="n">j</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">9.93383257062542</span>
-</pre></div>
-</div>
-<p>Beware that logarithmic corrections, as in the last example
-above, are required to obtain the correct branch in general.
-For details, see [1].</p>
-<p>The exponential integral is also a special case of the
-hypergeometric function <span class="math">\(\,_2F_2\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mf">0.6</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span><span class="o">*</span><span class="n">hyper</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">ln</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">-</span><span class="n">ln</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="n">euler</span>
-<span class="go">0.769881289937359</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ei</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.769881289937359</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li>Relations between Ei and other functions:
-<a class="reference external" href="http://functions.wolfram.com/GammaBetaErf/ExpIntegralEi/27/01/">http://functions.wolfram.com/GammaBetaErf/ExpIntegralEi/27/01/</a></li>
-<li>Abramowitz &amp; Stegun, section 5:
-<a class="reference external" href="http://www.math.sfu.ca/~cbm/aands/page_228.htm">http://www.math.sfu.ca/~cbm/aands/page_228.htm</a></li>
-<li>Asymptotic expansion for Ei:
-<a class="reference external" href="http://mathworld.wolfram.com/En-Function.html">http://mathworld.wolfram.com/En-Function.html</a></li>
-</ol>
-</dd></dl>
-
-</div>
-<div class="section" id="e1">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">e1()</span></tt><a class="headerlink" href="#e1" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.e1">
-<tt class="descclassname">mpmath.</tt><tt class="descname">e1</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.e1" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the exponential integral <span class="math">\(\mathrm{E}_1(z)\)</span>, given by</p>
-<div class="math">
-\[\mathrm{E}_1(z) = \int_z^{\infty} \frac{e^{-t}}{t} dt.\]</div>
-<p>This is equivalent to <a class="reference internal" href="#mpmath.expint" title="mpmath.expint"><tt class="xref py py-func docutils literal"><span class="pre">expint()</span></tt></a> with <span class="math">\(n = 1\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>Two ways to evaluate this function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="p">(</span><span class="mf">6.25</span><span class="p">)</span>
-<span class="go">0.0002704758872637179088496194</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expint</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mf">6.25</span><span class="p">)</span>
-<span class="go">0.0002704758872637179088496194</span>
-</pre></div>
-</div>
-<p>The E1-function is essentially the same as the Ei-function (<a class="reference internal" href="#mpmath.ei" title="mpmath.ei"><tt class="xref py py-func docutils literal"><span class="pre">ei()</span></tt></a>)
-with negated argument, except for an imaginary branch cut term:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="p">(</span><span class="mf">2.5</span><span class="p">)</span>
-<span class="go">0.02491491787026973549562801</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="n">ei</span><span class="p">(</span><span class="o">-</span><span class="mf">2.5</span><span class="p">)</span>
-<span class="go">0.02491491787026973549562801</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e1</span><span class="p">(</span><span class="o">-</span><span class="mf">2.5</span><span class="p">)</span>
-<span class="go">(-7.073765894578600711923552 - 3.141592653589793238462643j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="n">ei</span><span class="p">(</span><span class="mf">2.5</span><span class="p">)</span>
-<span class="go">-7.073765894578600711923552</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="expint">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">expint()</span></tt><a class="headerlink" href="#expint" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.expint">
-<tt class="descclassname">mpmath.</tt><tt class="descname">expint</tt><big>(</big><em>*args</em><big>)</big><a class="headerlink" href="#mpmath.expint" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="xref py py-func docutils literal"><span class="pre">expint(n,z)()</span></tt> gives the generalized exponential integral
-or En-function,</p>
-<div class="math">
-\[\mathrm{E}_n(z) = \int_1^{\infty} \frac{e^{-zt}}{t^n} dt,\]</div>
-<p>where <span class="math">\(n\)</span> and <span class="math">\(z\)</span> may both be complex numbers. The case with <span class="math">\(n = 1\)</span> is
-also given by <a class="reference internal" href="#mpmath.e1" title="mpmath.e1"><tt class="xref py py-func docutils literal"><span class="pre">e1()</span></tt></a>.</p>
-<p><strong>Examples</strong></p>
-<p>Evaluation at real and complex arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expint</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mf">6.25</span><span class="p">)</span>
-<span class="go">0.0002704758872637179088496194</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expint</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(0.00299658467335472929656159 + 0.06100816202125885450319632j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expint</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span> <span class="mi">4</span><span class="o">-</span><span class="mi">5j</span><span class="p">)</span>
-<span class="go">(0.001803529474663565056945248 - 0.002235061547756185403349091j)</span>
-</pre></div>
-</div>
-<p>At negative integer values of <span class="math">\(n\)</span>, <span class="math">\(E_n(z)\)</span> reduces to a
-rational-exponential function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">n</span><span class="p">,</span> <span class="n">z</span><span class="p">:</span> <span class="n">fac</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="nb">sum</span><span class="p">(</span><span class="n">z</span><span class="o">**</span><span class="n">k</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">k</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">n</span><span class="o">+</span><span class="mi">2</span><span class="p">))</span><span class="o">/</span>\
-<span class="gp">... </span>    <span class="n">exp</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expint</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">584.2604820613019908668219</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">584.2604820613019908668219</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="mi">5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expint</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">115366.5762594725451811138</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">115366.5762594725451811138</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="logarithmic-integral">
-<h2>Logarithmic integral<a class="headerlink" href="#logarithmic-integral" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="li">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">li()</span></tt><a class="headerlink" href="#li" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.li">
-<tt class="descclassname">mpmath.</tt><tt class="descname">li</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.li" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the logarithmic integral or li-function
-<span class="math">\(\mathrm{li}(x)\)</span>, defined by</p>
-<div class="math">
-\[\mathrm{li}(x) = \int_0^x \frac{1}{\log t} \, dt\]</div>
-<p>The logarithmic integral has a singularity at <span class="math">\(x = 1\)</span>.</p>
-<p>Alternatively, <tt class="docutils literal"><span class="pre">li(x,</span> <span class="pre">offset=True)</span></tt> computes the offset
-logarithmic integral (used in number theory)</p>
-<div class="math">
-\[\mathrm{Li}(x) = \int_2^x \frac{1}{\log t} \, dt.\]</div>
-<p>These two functions are related via the simple identity
-<span class="math">\(\mathrm{Li}(x) = \mathrm{li}(x) - \mathrm{li}(2)\)</span>.</p>
-<p>The logarithmic integral should also not be confused with
-the polylogarithm (also denoted by Li), which is implemented
-as <a class="reference internal" href="../functions/zeta.html#mpmath.polylog" title="mpmath.polylog"><tt class="xref py py-func docutils literal"><span class="pre">polylog()</span></tt></a>.</p>
-<p><strong>Examples</strong></p>
-<p>Some basic values and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">30</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">li</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">li</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">li</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">li</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">1.04516378011749278484458888919</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">li</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">1.45136923488338105028396848589</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">li</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">li</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">offset</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">li</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">offset</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">-inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">li</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">offset</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">-1.04516378011749278484458888919</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">li</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="n">offset</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">5.12043572466980515267839286347</span>
-</pre></div>
-</div>
-<p>The logarithmic integral can be evaluated for arbitrary
-complex arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">20</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">li</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(3.1343755504645775265 + 2.6769247817778742392j)</span>
-</pre></div>
-</div>
-<p>The logarithmic integral is related to the exponential integral:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ei</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">2.1635885946671919729</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">li</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">2.1635885946671919729</span>
-</pre></div>
-</div>
-<p>The logarithmic integral grows like <span class="math">\(O(x/\log(x))\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="mi">10</span><span class="o">**</span><span class="mi">100</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">4.34294481903252e+97</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">li</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">4.3619719871407e+97</span>
-</pre></div>
-</div>
-<p>The prime number theorem states that the number of primes less
-than <span class="math">\(x\)</span> is asymptotic to <span class="math">\(\mathrm{Li}(x)\)</span> (equivalently
-<span class="math">\(\mathrm{li}(x)\)</span>). For example, it is known that there are
-exactly 1,925,320,391,606,803,968,923 prime numbers less than
-<span class="math">\(10^{23}\)</span> [1]. The logarithmic integral provides a very
-accurate estimate:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">li</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">23</span><span class="p">,</span> <span class="n">offset</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">1.92532039161405e+21</span>
-</pre></div>
-</div>
-<p>A definite integral is:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">li</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">-0.693147180559945</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="n">ln</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">-0.693147180559945</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference external" href="http://mathworld.wolfram.com/PrimeCountingFunction.html">http://mathworld.wolfram.com/PrimeCountingFunction.html</a></li>
-<li><a class="reference external" href="http://mathworld.wolfram.com/LogarithmicIntegral.html">http://mathworld.wolfram.com/LogarithmicIntegral.html</a></li>
-</ol>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="trigonometric-integrals">
-<h2>Trigonometric integrals<a class="headerlink" href="#trigonometric-integrals" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="ci">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">ci()</span></tt><a class="headerlink" href="#ci" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.ci">
-<tt class="descclassname">mpmath.</tt><tt class="descname">ci</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.ci" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the cosine integral,</p>
-<div class="math">
-\[\mathrm{Ci}(x) = -\int_x^{\infty} \frac{\cos t}{t}\,dt
-= \gamma + \log x + \int_0^x \frac{\cos t - 1}{t}\,dt\]</div>
-<p><strong>Examples</strong></p>
-<p>Some values and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ci</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">-inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ci</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.3374039229009681346626462</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ci</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">0.07366791204642548599010096</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ci</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ci</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">(0.0 + 3.141592653589793238462643j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ci</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(1.408292501520849518759125 - 2.983617742029605093121118j)</span>
-</pre></div>
-</div>
-<p>The cosine integral behaves roughly like the sinc function
-(see <a class="reference internal" href="../functions/trigonometric.html#mpmath.sinc" title="mpmath.sinc"><tt class="xref py py-func docutils literal"><span class="pre">sinc()</span></tt></a>) for large real <span class="math">\(x\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ci</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">-4.875060251748226537857298e-11</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinc</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">-4.875060250875106915277943e-11</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">limit</span><span class="p">(</span><span class="n">ci</span><span class="p">,</span> <span class="n">inf</span><span class="p">))</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>It has infinitely many roots on the positive real axis:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">ci</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">0.6165054856207162337971104</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">ci</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">3.384180422551186426397851</span>
-</pre></div>
-</div>
-<p>Evaluation is supported for <span class="math">\(z\)</span> anywhere in the complex plane:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ci</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">6</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">))</span>
-<span class="go">(4.449410587611035724984376e+434287 + 9.75744874290013526417059e+434287j)</span>
-</pre></div>
-</div>
-<p>We can evaluate the defining integral as a reference:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="n">quadosc</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">cos</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">/</span><span class="n">t</span><span class="p">,</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span> <span class="n">omega</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-0.190029749656644</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ci</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">-0.190029749656644</span>
-</pre></div>
-</div>
-<p>Some infinite series can be evaluated using the
-cosine integral:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">k</span><span class="o">/</span><span class="p">(</span><span class="n">fac</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">)),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">-0.239811742000565</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ci</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">euler</span>
-<span class="go">-0.239811742000565</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="si">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">si()</span></tt><a class="headerlink" href="#si" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.si">
-<tt class="descclassname">mpmath.</tt><tt class="descname">si</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.si" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the sine integral,</p>
-<div class="math">
-\[\mathrm{Si}(x) = \int_0^x \frac{\sin t}{t}\,dt.\]</div>
-<p>The sine integral is thus the antiderivative of the sinc
-function (see <a class="reference internal" href="../functions/trigonometric.html#mpmath.sinc" title="mpmath.sinc"><tt class="xref py py-func docutils literal"><span class="pre">sinc()</span></tt></a>).</p>
-<p><strong>Examples</strong></p>
-<p>Some values and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">si</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">si</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.9460830703671830149413533</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">si</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-0.9460830703671830149413533</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">si</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">1.851937051982466170361053</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">si</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">1.570796326794896619231322</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">si</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">-1.570796326794896619231322</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">si</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(4.547513889562289219853204 + 1.399196580646054789459839j)</span>
-</pre></div>
-</div>
-<p>The sine integral approaches <span class="math">\(\pi/2\)</span> for large real <span class="math">\(x\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">si</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">1.570796326707584656968511</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span>
-<span class="go">1.570796326794896619231322</span>
-</pre></div>
-</div>
-<p>Evaluation is supported for <span class="math">\(z\)</span> anywhere in the complex plane:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">si</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">6</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">))</span>
-<span class="go">(-9.75744874290013526417059e+434287 + 4.449410587611035724984376e+434287j)</span>
-</pre></div>
-</div>
-<p>We can evaluate the defining integral as a reference:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">sinc</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
-<span class="go">1.54993124494467</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">si</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">1.54993124494467</span>
-</pre></div>
-</div>
-<p>Some infinite series can be evaluated using the
-sine integral:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">k</span><span class="o">/</span><span class="p">(</span><span class="n">fac</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">0.946083070367183</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">si</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.946083070367183</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="hyperbolic-integrals">
-<h2>Hyperbolic integrals<a class="headerlink" href="#hyperbolic-integrals" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="chi">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">chi()</span></tt><a class="headerlink" href="#chi" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.chi">
-<tt class="descclassname">mpmath.</tt><tt class="descname">chi</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.chi" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the hyperbolic cosine integral, defined
-in analogy with the cosine integral (see <a class="reference internal" href="#mpmath.ci" title="mpmath.ci"><tt class="xref py py-func docutils literal"><span class="pre">ci()</span></tt></a>) as</p>
-<div class="math">
-\[\mathrm{Chi}(x) = -\int_x^{\infty} \frac{\cosh t}{t}\,dt
-= \gamma + \log x + \int_0^x \frac{\cosh t - 1}{t}\,dt\]</div>
-<p>Some values and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chi</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">-inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chi</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.8378669409802082408946786</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chi</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">chi</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
-<span class="go">0.5238225713898644064509583</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chi</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(-0.1683628683277204662429321 + 2.625115880451325002151688j)</span>
-</pre></div>
-</div>
-<p>Evaluation is supported for <span class="math">\(z\)</span> anywhere in the complex plane:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">chi</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">6</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">))</span>
-<span class="go">(4.449410587611035724984376e+434287 - 9.75744874290013526417059e+434287j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="shi">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">shi()</span></tt><a class="headerlink" href="#shi" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.shi">
-<tt class="descclassname">mpmath.</tt><tt class="descname">shi</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.shi" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the hyperbolic sine integral, defined
-in analogy with the sine integral (see <a class="reference internal" href="#mpmath.si" title="mpmath.si"><tt class="xref py py-func docutils literal"><span class="pre">si()</span></tt></a>) as</p>
-<div class="math">
-\[\mathrm{Shi}(x) = \int_0^x \frac{\sinh t}{t}\,dt.\]</div>
-<p>Some values and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">shi</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">shi</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1.057250875375728514571842</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">shi</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-1.057250875375728514571842</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">shi</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">shi</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(-0.1931890762719198291678095 + 2.645432555362369624818525j)</span>
-</pre></div>
-</div>
-<p>Evaluation is supported for <span class="math">\(z\)</span> anywhere in the complex plane:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">shi</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">6</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">))</span>
-<span class="go">(4.449410587611035724984376e+434287 - 9.75744874290013526417059e+434287j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="error-functions">
-<h2>Error functions<a class="headerlink" href="#error-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="erf">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">erf()</span></tt><a class="headerlink" href="#erf" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.erf">
-<tt class="descclassname">mpmath.</tt><tt class="descname">erf</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.erf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the error function, <span class="math">\(\mathrm{erf}(x)\)</span>. The error
-function is the normalized antiderivative of the Gaussian function
-<span class="math">\(\exp(-t^2)\)</span>. More precisely,</p>
-<div class="math">
-\[\mathrm{erf}(x) = \frac{2}{\sqrt \pi} \int_0^x \exp(-t^2) \,dt\]</div>
-<p><strong>Basic examples</strong></p>
-<p>Simple values and limits include:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.842700792949715</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-0.842700792949715</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">-1.0</span>
-</pre></div>
-</div>
-<p>For large real <span class="math">\(x\)</span>, <span class="math">\(\mathrm{erf}(x)\)</span> approaches 1 very
-rapidly:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.999977909503001</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">0.999999999998463</span>
-</pre></div>
-</div>
-<p>The error function is an odd function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">erf</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)))</span>
-<span class="go">[0.0, 1.12838, 0.0, -0.376126, 0.0, 0.112838]</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.erf" title="mpmath.erf"><tt class="xref py py-func docutils literal"><span class="pre">erf()</span></tt></a> implements arbitrary-precision evaluation and
-supports complex numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="go">0.52049987781304653768274665389196452873645157575796</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">)</span>
-<span class="go">(1.316151281697947644880271 + 0.1904534692378346862841089j)</span>
-</pre></div>
-</div>
-<p>Evaluation is supported for large arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="s">&#39;1e1000&#39;</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="s">&#39;-1e1000&#39;</span><span class="p">)</span>
-<span class="go">-1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="s">&#39;1e-1000&#39;</span><span class="p">)</span>
-<span class="go">1.128379167095512573896159e-1000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="s">&#39;1e7j&#39;</span><span class="p">)</span>
-<span class="go">(0.0 + 8.593897639029319267398803e+43429448190317j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="s">&#39;1e7+1e7j&#39;</span><span class="p">)</span>
-<span class="go">(0.9999999858172446172631323 + 3.728805278735270407053139e-8j)</span>
-</pre></div>
-</div>
-<p><strong>Related functions</strong></p>
-<p>See also <a class="reference internal" href="#mpmath.erfc" title="mpmath.erfc"><tt class="xref py py-func docutils literal"><span class="pre">erfc()</span></tt></a>, which is more accurate for large <span class="math">\(x\)</span>,
-and <a class="reference internal" href="#mpmath.erfi" title="mpmath.erfi"><tt class="xref py py-func docutils literal"><span class="pre">erfi()</span></tt></a> which gives the antiderivative of
-<span class="math">\(\exp(t^2)\)</span>.</p>
-<p>The Fresnel integrals <a class="reference internal" href="#mpmath.fresnels" title="mpmath.fresnels"><tt class="xref py py-func docutils literal"><span class="pre">fresnels()</span></tt></a> and <a class="reference internal" href="#mpmath.fresnelc" title="mpmath.fresnelc"><tt class="xref py py-func docutils literal"><span class="pre">fresnelc()</span></tt></a>
-are also related to the error function.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="erfc">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">erfc()</span></tt><a class="headerlink" href="#erfc" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.erfc">
-<tt class="descclassname">mpmath.</tt><tt class="descname">erfc</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.erfc" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the complementary error function,
-<span class="math">\(\mathrm{erfc}(x) = 1-\mathrm{erf}(x)\)</span>.
-This function avoids cancellation that occurs when naively
-computing the complementary error function as <tt class="docutils literal"><span class="pre">1-erf(x)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span> <span class="o">-</span> <span class="n">erf</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erfc</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">2.08848758376254e-45</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.erfc" title="mpmath.erfc"><tt class="xref py py-func docutils literal"><span class="pre">erfc()</span></tt></a> works accurately even for ludicrously large
-arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">erfc</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">4.3504398860243e-43429448190325182776</span>
-</pre></div>
-</div>
-<p>Complex arguments are supported:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">erfc</span><span class="p">(</span><span class="mi">500</span><span class="o">+</span><span class="mi">50j</span><span class="p">)</span>
-<span class="go">(1.19739830969552e-107492 + 1.46072418957528e-107491j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="erfi">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">erfi()</span></tt><a class="headerlink" href="#erfi" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.erfi">
-<tt class="descclassname">mpmath.</tt><tt class="descname">erfi</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.erfi" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the imaginary error function, <span class="math">\(\mathrm{erfi}(x)\)</span>.
-The imaginary error function is defined in analogy with the
-error function, but with a positive sign in the integrand:</p>
-<div class="math">
-\[\mathrm{erfi}(x) = \frac{2}{\sqrt \pi} \int_0^x \exp(t^2) \,dt\]</div>
-<p>Whereas the error function rapidly converges to 1 as <span class="math">\(x\)</span> grows,
-the imaginary error function rapidly diverges to infinity.
-The functions are related as
-<span class="math">\(\mathrm{erfi}(x) = -i\,\mathrm{erf}(ix)\)</span> for all complex
-numbers <span class="math">\(x\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>Basic values and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erfi</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erfi</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1.65042575879754</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erfi</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-1.65042575879754</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erfi</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erfi</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">-inf</span>
-</pre></div>
-</div>
-<p>Note the symmetry between erf and erfi:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">erfi</span><span class="p">(</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(0.0 + 0.999977909503001j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.999977909503001</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mi">2j</span><span class="p">)</span>
-<span class="go">(-0.536643565778565 - 5.04914370344703j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erfi</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">1j</span><span class="p">)</span>
-<span class="go">(-5.04914370344703 - 0.536643565778565j)</span>
-</pre></div>
-</div>
-<p>Large arguments are supported:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">erfi</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span>
-<span class="go">1.71130938718796e+434291</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erfi</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">7.3167287567024e+43429448190325182754</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erfi</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">-7.3167287567024e+43429448190325182754</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erfi</span><span class="p">(</span><span class="mi">1000</span><span class="o">-</span><span class="mi">500j</span><span class="p">)</span>
-<span class="go">(2.49895233563961e+325717 + 2.6846779342253e+325717j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erfi</span><span class="p">(</span><span class="mi">100000j</span><span class="p">)</span>
-<span class="go">(0.0 + 1.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erfi</span><span class="p">(</span><span class="o">-</span><span class="mi">100000j</span><span class="p">)</span>
-<span class="go">(0.0 - 1.0j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="erfinv">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">erfinv()</span></tt><a class="headerlink" href="#erfinv" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.erfinv">
-<tt class="descclassname">mpmath.</tt><tt class="descname">erfinv</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.erfinv" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the inverse error function, satisfying</p>
-<div class="math">
-\[\mathrm{erf}(\mathrm{erfinv}(x)) =
-\mathrm{erfinv}(\mathrm{erf}(x)) = x.\]</div>
-<p>This function is defined only for <span class="math">\(-1 \le x \le 1\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>Special values include:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erfinv</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erfinv</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erfinv</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-inf</span>
-</pre></div>
-</div>
-<p>The domain is limited to the standard interval:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">erfinv</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-  <span class="c">...</span>
-<span class="gr">ValueError</span>: <span class="n">erfinv(x) is defined only for -1 &lt;= x &lt;= 1</span>
-</pre></div>
-</div>
-<p>It is simple to check that <a class="reference internal" href="#mpmath.erfinv" title="mpmath.erfinv"><tt class="xref py py-func docutils literal"><span class="pre">erfinv()</span></tt></a> computes inverse values of
-<a class="reference internal" href="#mpmath.erf" title="mpmath.erf"><tt class="xref py py-func docutils literal"><span class="pre">erf()</span></tt></a> as promised:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="n">erfinv</span><span class="p">(</span><span class="mf">0.75</span><span class="p">))</span>
-<span class="go">0.75</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erf</span><span class="p">(</span><span class="n">erfinv</span><span class="p">(</span><span class="o">-</span><span class="mf">0.995</span><span class="p">))</span>
-<span class="go">-0.995</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.erfinv" title="mpmath.erfinv"><tt class="xref py py-func docutils literal"><span class="pre">erfinv()</span></tt></a> supports arbitrary-precision evaluation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">erf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span>
-<span class="go">0.99532226501895273416206925636725292861089179704006</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">erfinv</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">2.0</span>
-</pre></div>
-</div>
-<p>A definite integral involving the inverse error function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">erfinv</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">0.564189583547756</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">0.564189583547756</span>
-</pre></div>
-</div>
-<p>The inverse error function can be used to generate random numbers
-with a Gaussian distribution (although this is a relatively
-inefficient algorithm):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">([</span><span class="n">erfinv</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">rand</span><span class="p">()</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">6</span><span class="p">)])</span> 
-<span class="go">[-0.586747, 1.10233, -0.376796, 0.926037, -0.708142, -0.732012]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="the-normal-distribution">
-<h2>The normal distribution<a class="headerlink" href="#the-normal-distribution" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="npdf">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">npdf()</span></tt><a class="headerlink" href="#npdf" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.npdf">
-<tt class="descclassname">mpmath.</tt><tt class="descname">npdf</tt><big>(</big><em>x</em>, <em>mu=0</em>, <em>sigma=1</em><big>)</big><a class="headerlink" href="#mpmath.npdf" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">npdf(x,</span> <span class="pre">mu=0,</span> <span class="pre">sigma=1)</span></tt> evaluates the probability density
-function of a normal distribution with mean value <span class="math">\(\mu\)</span>
-and variance <span class="math">\(\sigma^2\)</span>.</p>
-<p>Elementary properties of the probability distribution can
-be verified using numerical integration:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">npdf</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">npdf</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">npdf</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">0.5</span>
-</pre></div>
-</div>
-<p>See also <a class="reference internal" href="#mpmath.ncdf" title="mpmath.ncdf"><tt class="xref py py-func docutils literal"><span class="pre">ncdf()</span></tt></a>, which gives the cumulative
-distribution.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="ncdf">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">ncdf()</span></tt><a class="headerlink" href="#ncdf" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.ncdf">
-<tt class="descclassname">mpmath.</tt><tt class="descname">ncdf</tt><big>(</big><em>x</em>, <em>mu=0</em>, <em>sigma=1</em><big>)</big><a class="headerlink" href="#mpmath.ncdf" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">ncdf(x,</span> <span class="pre">mu=0,</span> <span class="pre">sigma=1)</span></tt> evaluates the cumulative distribution
-function of a normal distribution with mean value <span class="math">\(\mu\)</span>
-and variance <span class="math">\(\sigma^2\)</span>.</p>
-<p>See also <a class="reference internal" href="#mpmath.npdf" title="mpmath.npdf"><tt class="xref py py-func docutils literal"><span class="pre">npdf()</span></tt></a>, which gives the probability density.</p>
-<p>Elementary properties include:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ncdf</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="n">mu</span><span class="o">=</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ncdf</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ncdf</span><span class="p">(</span><span class="o">+</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">1.0</span>
-</pre></div>
-</div>
-<p>The cumulative distribution is the integral of the density
-function having identical mu and sigma:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">ncdf</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">0.053990966513188</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">npdf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.053990966513188</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">ncdf</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">),</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">0.107981933026376</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">npdf</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
-<span class="go">0.107981933026376</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="fresnel-integrals">
-<h2>Fresnel integrals<a class="headerlink" href="#fresnel-integrals" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="fresnels">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">fresnels()</span></tt><a class="headerlink" href="#fresnels" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.fresnels">
-<tt class="descclassname">mpmath.</tt><tt class="descname">fresnels</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.fresnels" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the Fresnel sine integral</p>
-<div class="math">
-\[S(x) = \int_0^x \sin\left(\frac{\pi t^2}{2}\right) \,dt\]</div>
-<p>Note that some sources define this function
-without the normalization factor <span class="math">\(\pi/2\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>Some basic values and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fresnels</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fresnels</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fresnels</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">-0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fresnels</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.4382591473903547660767567</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fresnels</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mi">2j</span><span class="p">)</span>
-<span class="go">(36.72546488399143842838788 + 15.58775110440458732748279j)</span>
-</pre></div>
-</div>
-<p>Comparing with the definition:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fresnels</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.4963129989673750360976123</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">t</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-<span class="go">0.4963129989673750360976123</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="fresnelc">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">fresnelc()</span></tt><a class="headerlink" href="#fresnelc" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.fresnelc">
-<tt class="descclassname">mpmath.</tt><tt class="descname">fresnelc</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.fresnelc" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the Fresnel cosine integral</p>
-<div class="math">
-\[C(x) = \int_0^x \cos\left(\frac{\pi t^2}{2}\right) \,dt\]</div>
-<p>Note that some sources define this function
-without the normalization factor <span class="math">\(\pi/2\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>Some basic values and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fresnelc</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fresnelc</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fresnelc</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">-0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fresnelc</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.7798934003768228294742064</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fresnelc</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mi">2j</span><span class="p">)</span>
-<span class="go">(16.08787137412548041729489 - 36.22568799288165021578758j)</span>
-</pre></div>
-</div>
-<p>Comparing with the definition:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fresnelc</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.6057207892976856295561611</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">t</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">])</span>
-<span class="go">0.6057207892976856295561611</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-</div>
-
-
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-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Exponential integrals and error functions</a><ul>
-<li><a class="reference internal" href="#incomplete-gamma-functions">Incomplete gamma functions</a><ul>
-<li><a class="reference internal" href="#gammainc"><tt class="docutils literal"><span class="pre">gammainc()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#exponential-integrals">Exponential integrals</a><ul>
-<li><a class="reference internal" href="#ei"><tt class="docutils literal"><span class="pre">ei()</span></tt></a></li>
-<li><a class="reference internal" href="#e1"><tt class="docutils literal"><span class="pre">e1()</span></tt></a></li>
-<li><a class="reference internal" href="#expint"><tt class="docutils literal"><span class="pre">expint()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#logarithmic-integral">Logarithmic integral</a><ul>
-<li><a class="reference internal" href="#li"><tt class="docutils literal"><span class="pre">li()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#trigonometric-integrals">Trigonometric integrals</a><ul>
-<li><a class="reference internal" href="#ci"><tt class="docutils literal"><span class="pre">ci()</span></tt></a></li>
-<li><a class="reference internal" href="#si"><tt class="docutils literal"><span class="pre">si()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#hyperbolic-integrals">Hyperbolic integrals</a><ul>
-<li><a class="reference internal" href="#chi"><tt class="docutils literal"><span class="pre">chi()</span></tt></a></li>
-<li><a class="reference internal" href="#shi"><tt class="docutils literal"><span class="pre">shi()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#error-functions">Error functions</a><ul>
-<li><a class="reference internal" href="#erf"><tt class="docutils literal"><span class="pre">erf()</span></tt></a></li>
-<li><a class="reference internal" href="#erfc"><tt class="docutils literal"><span class="pre">erfc()</span></tt></a></li>
-<li><a class="reference internal" href="#erfi"><tt class="docutils literal"><span class="pre">erfi()</span></tt></a></li>
-<li><a class="reference internal" href="#erfinv"><tt class="docutils literal"><span class="pre">erfinv()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#the-normal-distribution">The normal distribution</a><ul>
-<li><a class="reference internal" href="#npdf"><tt class="docutils literal"><span class="pre">npdf()</span></tt></a></li>
-<li><a class="reference internal" href="#ncdf"><tt class="docutils literal"><span class="pre">ncdf()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#fresnel-integrals">Fresnel integrals</a><ul>
-<li><a class="reference internal" href="#fresnels"><tt class="docutils literal"><span class="pre">fresnels()</span></tt></a></li>
-<li><a class="reference internal" href="#fresnelc"><tt class="docutils literal"><span class="pre">fresnelc()</span></tt></a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../functions/gamma.html"
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-  <div class="section" id="factorials-and-gamma-functions">
-<h1>Factorials and gamma functions<a class="headerlink" href="#factorials-and-gamma-functions" title="Permalink to this headline">¶</a></h1>
-<p>Factorials and factorial-like sums and products are basic tools of combinatorics and number theory. Much like the exponential function is fundamental to differential equations and analysis in general, the factorial function (and its extension to complex numbers, the gamma function) is fundamental to difference equations and functional equations.</p>
-<p>A large selection of factorial-like functions is implemented in mpmath. All functions support complex arguments, and arguments may be arbitrarily large. Results are numerical approximations, so to compute <em>exact</em> values a high enough precision must be set manually:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fac</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">9.33262154439442e+157</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="nb">int</span><span class="p">(</span><span class="n">_</span><span class="p">)</span>    <span class="c"># most digits are wrong</span>
-<span class="go">93326215443944150965646704795953882578400970373184098831012889540582227238570431</span>
-<span class="go">295066113089288327277825849664006524270554535976289719382852181865895959724032</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">160</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fac</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">93326215443944152681699238856266700490715968264381621468592963895217599993229915</span>
-<span class="go">608941463976156518286253697920827223758251185210916864000000000000000000000000.0</span>
-</pre></div>
-</div>
-<p>The gamma and polygamma functions are closely related to <a class="reference internal" href="../functions/zeta.html"><em>Zeta functions, L-series and polylogarithms</em></a>. See also <a class="reference internal" href="../functions/qfunctions.html"><em>q-functions</em></a> for q-analogs of factorial-like functions.</p>
-<div class="section" id="factorials">
-<h2>Factorials<a class="headerlink" href="#factorials" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="factorial-fac">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">factorial()</span></tt>/<tt class="xref py py-func docutils literal"><span class="pre">fac()</span></tt><a class="headerlink" href="#factorial-fac" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.factorial">
-<tt class="descclassname">mpmath.</tt><tt class="descname">factorial</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.factorial" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the factorial, <span class="math">\(x!\)</span>. For integers <span class="math">\(n \ge 0\)</span>, we have
-<span class="math">\(n! = 1 \cdot 2 \cdots (n-1) \cdot n\)</span> and more generally the factorial
-is defined for real or complex <span class="math">\(x\)</span> by <span class="math">\(x! = \Gamma(x+1)\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>Basic values and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">6</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">fac</span><span class="p">(</span><span class="n">k</span><span class="p">)))</span>
-<span class="gp">...</span>
-<span class="go">0 1.0</span>
-<span class="go">1 1.0</span>
-<span class="go">2 2.0</span>
-<span class="go">3 6.0</span>
-<span class="go">4 24.0</span>
-<span class="go">5 120.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fac</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fac</span><span class="p">(</span><span class="mf">0.5</span><span class="p">),</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-<span class="go">(0.886226925452758, 0.886226925452758)</span>
-</pre></div>
-</div>
-<p>For large positive <span class="math">\(x\)</span>, <span class="math">\(x!\)</span> can be approximated by
-Stirling&#8217;s formula:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="mi">10</span><span class="o">**</span><span class="mi">10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fac</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">2.32579620567308e+95657055186</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="n">e</span><span class="p">)</span><span class="o">**</span><span class="n">x</span>
-<span class="go">2.32579597597705e+95657055186</span>
-</pre></div>
-</div>
-<p><tt class="xref py py-func docutils literal"><span class="pre">fac()</span></tt> supports evaluation for astronomically large values:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fac</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">30</span><span class="p">)</span>
-<span class="go">6.22311232304258e+29565705518096748172348871081098</span>
-</pre></div>
-</div>
-<p>Reciprocal factorials appear in the Taylor series of the
-exponential function (among many other contexts):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">k</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">inf</span><span class="p">]),</span> <span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(2.71828182845905, 2.71828182845905)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="n">pi</span><span class="o">**</span><span class="n">k</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">k</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">inf</span><span class="p">]),</span> <span class="n">exp</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">(23.1406926327793, 23.1406926327793)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="fac2">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">fac2()</span></tt><a class="headerlink" href="#fac2" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.fac2">
-<tt class="descclassname">mpmath.</tt><tt class="descname">fac2</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.fac2" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the double factorial <span class="math">\(x!!\)</span>, defined for integers
-<span class="math">\(x &gt; 0\)</span> by</p>
-<div class="math">
-\[\begin{split}x!! = \begin{cases}
-    1 \cdot 3 \cdots (x-2) \cdot x &amp; x \;\mathrm{odd} \\
-    2 \cdot 4 \cdots (x-2) \cdot x &amp; x \;\mathrm{even}
-\end{cases}\end{split}\]</div>
-<p>and more generally by [1]</p>
-<div class="math">
-\[x!! = 2^{x/2} \left(\frac{\pi}{2}\right)^{(\cos(\pi x)-1)/4}
-      \Gamma\left(\frac{x}{2}+1\right).\]</div>
-<p><strong>Examples</strong></p>
-<p>The integer sequence of double factorials begins:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">([</span><span class="n">fac2</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">)])</span>
-<span class="go">[1.0, 1.0, 2.0, 3.0, 8.0, 15.0, 48.0, 105.0, 384.0, 945.0]</span>
-</pre></div>
-</div>
-<p>For large <span class="math">\(x\)</span>, double factorials follow a Stirling-like asymptotic
-approximation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">10000</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fac2</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">5.97272691416282e+17830</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="p">((</span><span class="n">x</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">5.97262736954392e+17830</span>
-</pre></div>
-</div>
-<p>The recurrence formula <span class="math">\(x!! = x (x-2)!!\)</span> can be reversed to
-define the double factorial of negative odd integers (but
-not negative even integers):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fac2</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">fac2</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">),</span> <span class="n">fac2</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">),</span> <span class="n">fac2</span><span class="p">(</span><span class="o">-</span><span class="mi">7</span><span class="p">)</span>
-<span class="go">(1.0, -1.0, 0.333333333333333, -0.0666666666666667)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fac2</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-  <span class="c">...</span>
-<span class="gr">ValueError</span>: <span class="n">gamma function pole</span>
-</pre></div>
-</div>
-<p>With the exception of the poles at negative even integers,
-<a class="reference internal" href="#mpmath.fac2" title="mpmath.fac2"><tt class="xref py py-func docutils literal"><span class="pre">fac2()</span></tt></a> supports evaluation for arbitrary complex arguments.
-The recurrence formula is valid generally:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fac2</span><span class="p">(</span><span class="n">pi</span><span class="o">+</span><span class="mi">2j</span><span class="p">)</span>
-<span class="go">(-1.3697207890154e-12 + 3.93665300979176e-12j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">pi</span><span class="o">+</span><span class="mi">2j</span><span class="p">)</span><span class="o">*</span><span class="n">fac2</span><span class="p">(</span><span class="n">pi</span><span class="o">-</span><span class="mi">2</span><span class="o">+</span><span class="mi">2j</span><span class="p">)</span>
-<span class="go">(-1.3697207890154e-12 + 3.93665300979176e-12j)</span>
-</pre></div>
-</div>
-<p>Double factorials should not be confused with nested factorials,
-which are immensely larger:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fac</span><span class="p">(</span><span class="n">fac</span><span class="p">(</span><span class="mi">20</span><span class="p">))</span>
-<span class="go">5.13805976125208e+43675043585825292774</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fac2</span><span class="p">(</span><span class="mi">20</span><span class="p">)</span>
-<span class="go">3715891200.0</span>
-</pre></div>
-</div>
-<p>Double factorials appear, among other things, in series expansions
-of Gaussian functions and the error function. Infinite series
-include:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">fac2</span><span class="p">(</span><span class="n">k</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">3.05940740534258</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="n">e</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">erf</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">3.05940740534258</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="mi">2</span><span class="o">**</span><span class="n">k</span><span class="o">/</span><span class="n">fac2</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">4.06015693855741</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">*</span> <span class="n">erf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">4.06015693855741</span>
-</pre></div>
-</div>
-<p>A beautiful Ramanujan sum:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">k</span><span class="o">*</span><span class="p">(</span><span class="n">fac2</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">fac2</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="p">))</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">0.90917279454693</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">gamma</span><span class="p">(</span><span class="s">&#39;9/8&#39;</span><span class="p">)</span><span class="o">/</span><span class="n">gamma</span><span class="p">(</span><span class="s">&#39;5/4&#39;</span><span class="p">)</span><span class="o">/</span><span class="n">gamma</span><span class="p">(</span><span class="s">&#39;7/8&#39;</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">0.90917279454693</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference external" href="http://functions.wolfram.com/GammaBetaErf/Factorial2/27/01/0002/">http://functions.wolfram.com/GammaBetaErf/Factorial2/27/01/0002/</a></li>
-<li><a class="reference external" href="http://mathworld.wolfram.com/DoubleFactorial.html">http://mathworld.wolfram.com/DoubleFactorial.html</a></li>
-</ol>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="binomial-coefficients">
-<h2>Binomial coefficients<a class="headerlink" href="#binomial-coefficients" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="binomial">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">binomial()</span></tt><a class="headerlink" href="#binomial" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.binomial">
-<tt class="descclassname">mpmath.</tt><tt class="descname">binomial</tt><big>(</big><em>n</em>, <em>k</em><big>)</big><a class="headerlink" href="#mpmath.binomial" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the binomial coefficient</p>
-<div class="math">
-\[{n \choose k} = \frac{n!}{k!(n-k)!}.\]</div>
-<p>The binomial coefficient gives the number of ways that <span class="math">\(k\)</span> items
-can be chosen from a set of <span class="math">\(n\)</span> items. More generally, the binomial
-coefficient is a well-defined function of arbitrary real or
-complex <span class="math">\(n\)</span> and <span class="math">\(k\)</span>, via the gamma function.</p>
-<p><strong>Examples</strong></p>
-<p>Generate Pascal&#8217;s triangle:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">nprint</span><span class="p">([</span><span class="n">binomial</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)])</span>
-<span class="gp">...</span>
-<span class="go">[1.0]</span>
-<span class="go">[1.0, 1.0]</span>
-<span class="go">[1.0, 2.0, 1.0]</span>
-<span class="go">[1.0, 3.0, 3.0, 1.0]</span>
-<span class="go">[1.0, 4.0, 6.0, 4.0, 1.0]</span>
-</pre></div>
-</div>
-<p>There is 1 way to select 0 items from the empty set, and 0 ways to
-select 1 item from the empty set:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">binomial</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">binomial</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.binomial" title="mpmath.binomial"><tt class="xref py py-func docutils literal"><span class="pre">binomial()</span></tt></a> supports large arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">binomial</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">20</span><span class="p">,</span> <span class="mi">10</span><span class="o">**</span><span class="mi">20</span><span class="o">-</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">8.33333333333333e+97</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">binomial</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">20</span><span class="p">,</span> <span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">2.60784095465201e+104342944813</span>
-</pre></div>
-</div>
-<p>Nonintegral binomial coefficients find use in series
-expansions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mf">0.25</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
-<span class="go">[1.0, 0.25, -0.09375, 0.0546875, -0.0375977]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">([</span><span class="n">binomial</span><span class="p">(</span><span class="mf">0.25</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">)])</span>
-<span class="go">[1.0, 0.25, -0.09375, 0.0546875, -0.0375977]</span>
-</pre></div>
-</div>
-<p>An integral representation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">j</span><span class="o">*</span><span class="n">k</span><span class="o">*</span><span class="n">t</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">exp</span><span class="p">(</span><span class="n">j</span><span class="o">*</span><span class="n">t</span><span class="p">))</span><span class="o">**</span><span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="n">pi</span><span class="p">,</span><span class="n">pi</span><span class="p">])</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span>
-<span class="go">10.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">binomial</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">k</span><span class="p">)</span>
-<span class="go">10.0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="gamma-function">
-<h2>Gamma function<a class="headerlink" href="#gamma-function" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="gamma">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">gamma()</span></tt><a class="headerlink" href="#gamma" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.gamma">
-<tt class="descclassname">mpmath.</tt><tt class="descname">gamma</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.gamma" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the gamma function, <span class="math">\(\Gamma(x)\)</span>. The gamma function is a
-shifted version of the ordinary factorial, satisfying
-<span class="math">\(\Gamma(n) = (n-1)!\)</span> for integers <span class="math">\(n &gt; 0\)</span>. More generally, it
-is defined by</p>
-<div class="math">
-\[\Gamma(x) = \int_0^{\infty} t^{x-1} e^{-t}\, dt\]</div>
-<p>for any real or complex <span class="math">\(x\)</span> with <span class="math">\(\Re(x) &gt; 0\)</span> and for <span class="math">\(\Re(x) &lt; 0\)</span>
-by analytic continuation.</p>
-<p><strong>Examples</strong></p>
-<p>Basic values and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">gamma</span><span class="p">(</span><span class="n">k</span><span class="p">)))</span>
-<span class="gp">...</span>
-<span class="go">1 1.0</span>
-<span class="go">2 1.0</span>
-<span class="go">3 2.0</span>
-<span class="go">4 6.0</span>
-<span class="go">5 24.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-  <span class="c">...</span>
-<span class="gr">ValueError</span>: <span class="n">gamma function pole</span>
-</pre></div>
-</div>
-<p>The gamma function of a half-integer is a rational multiple of
-<span class="math">\(\sqrt{\pi}\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="mf">0.5</span><span class="p">),</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">(1.77245385090552, 1.77245385090552)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="mf">1.5</span><span class="p">),</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-<span class="go">(0.886226925452758, 0.886226925452758)</span>
-</pre></div>
-</div>
-<p>We can check the integral definition:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="mf">3.5</span><span class="p">)</span>
-<span class="go">3.32335097044784</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">t</span><span class="o">**</span><span class="mf">2.5</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">t</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">3.32335097044784</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.gamma" title="mpmath.gamma"><tt class="xref py py-func docutils literal"><span class="pre">gamma()</span></tt></a> supports arbitrary-precision evaluation and
-complex arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">0.91510229697308632046045539308226554038315280564184</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="mi">2j</span><span class="p">)</span>
-<span class="go">(0.009902440080927490985955066 - 0.07595200133501806872408048j)</span>
-</pre></div>
-</div>
-<p>Arguments can also be large. Note that the gamma function grows
-very quickly:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">20</span><span class="p">)</span>
-<span class="go">1.9328495143101e+1956570551809674817225</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="rgamma">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">rgamma()</span></tt><a class="headerlink" href="#rgamma" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.rgamma">
-<tt class="descclassname">mpmath.</tt><tt class="descname">rgamma</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.rgamma" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the reciprocal of the gamma function, <span class="math">\(1/\Gamma(z)\)</span>. This
-function evaluates to zero at the poles
-of the gamma function, <span class="math">\(z = 0, -1, -2, \ldots\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>Basic examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rgamma</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rgamma</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">0.1666666666666666666666667</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rgamma</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span> <span class="n">rgamma</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rgamma</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span>
-<span class="go">2.485168143266784862783596e-2565</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rgamma</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>A definite integral that can be evaluated in terms of elementary
-integrals:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">rgamma</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">2.807770242028519365221501</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">+</span> <span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">t</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">log</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">2.807770242028519365221501</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="gammaprod">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">gammaprod()</span></tt><a class="headerlink" href="#gammaprod" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.gammaprod">
-<tt class="descclassname">mpmath.</tt><tt class="descname">gammaprod</tt><big>(</big><em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#mpmath.gammaprod" title="Permalink to this definition">¶</a></dt>
-<dd><p>Given iterables <span class="math">\(a\)</span> and <span class="math">\(b\)</span>, <tt class="docutils literal"><span class="pre">gammaprod(a,</span> <span class="pre">b)</span></tt> computes the
-product / quotient of gamma functions:</p>
-<div class="math">
-\[\frac{\Gamma(a_0) \Gamma(a_1) \cdots \Gamma(a_p)}
-     {\Gamma(b_0) \Gamma(b_1) \cdots \Gamma(b_q)}\]</div>
-<p>Unlike direct calls to <a class="reference internal" href="#mpmath.gamma" title="mpmath.gamma"><tt class="xref py py-func docutils literal"><span class="pre">gamma()</span></tt></a>, <a class="reference internal" href="#mpmath.gammaprod" title="mpmath.gammaprod"><tt class="xref py py-func docutils literal"><span class="pre">gammaprod()</span></tt></a> considers
-the entire product as a limit and evaluates this limit properly if
-any of the numerator or denominator arguments are nonpositive
-integers such that poles of the gamma function are encountered.
-That is, <a class="reference internal" href="#mpmath.gammaprod" title="mpmath.gammaprod"><tt class="xref py py-func docutils literal"><span class="pre">gammaprod()</span></tt></a> evaluates</p>
-<div class="math">
-\[\lim_{\epsilon \to 0}
-\frac{\Gamma(a_0+\epsilon) \Gamma(a_1+\epsilon) \cdots
-    \Gamma(a_p+\epsilon)}
-     {\Gamma(b_0+\epsilon) \Gamma(b_1+\epsilon) \cdots
-    \Gamma(b_q+\epsilon)}\]</div>
-<p>In particular:</p>
-<ul class="simple">
-<li>If there are equally many poles in the numerator and the
-denominator, the limit is a rational number times the remaining,
-regular part of the product.</li>
-<li>If there are more poles in the numerator, <a class="reference internal" href="#mpmath.gammaprod" title="mpmath.gammaprod"><tt class="xref py py-func docutils literal"><span class="pre">gammaprod()</span></tt></a>
-returns <tt class="docutils literal"><span class="pre">+inf</span></tt>.</li>
-<li>If there are more poles in the denominator, <a class="reference internal" href="#mpmath.gammaprod" title="mpmath.gammaprod"><tt class="xref py py-func docutils literal"><span class="pre">gammaprod()</span></tt></a>
-returns 0.</li>
-</ul>
-<p><strong>Examples</strong></p>
-<p>The reciprocal gamma function <span class="math">\(1/\Gamma(x)\)</span> evaluated at <span class="math">\(x = 0\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gammaprod</span><span class="p">([],</span> <span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>A limit:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gammaprod</span><span class="p">([</span><span class="o">-</span><span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">])</span>
-<span class="go">-0.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">gamma</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">gamma</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="n">direction</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-0.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">gamma</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">gamma</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="n">direction</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-0.25</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="loggamma">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">loggamma()</span></tt><a class="headerlink" href="#loggamma" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.loggamma">
-<tt class="descclassname">mpmath.</tt><tt class="descname">loggamma</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.loggamma" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the principal branch of the log-gamma function,
-<span class="math">\(\ln \Gamma(z)\)</span>. Unlike <span class="math">\(\ln(\Gamma(z))\)</span>, which has infinitely many
-complex branch cuts, the principal log-gamma function only has a single
-branch cut along the negative half-axis. The principal branch
-continuously matches the asymptotic Stirling expansion</p>
-<div class="math">
-\[\ln \Gamma(z) \sim \frac{\ln(2 \pi)}{2} +
-    \left(z-\frac{1}{2}\right) \ln(z) - z + O(z^{-1}).\]</div>
-<p>The real parts of both functions agree, but their imaginary
-parts generally differ by <span class="math">\(2 n \pi\)</span> for some <span class="math">\(n \in \mathbb{Z}\)</span>.
-They coincide for <span class="math">\(z \in \mathbb{R}, z &gt; 0\)</span>.</p>
-<p>Computationally, it is advantageous to use <a class="reference internal" href="#mpmath.loggamma" title="mpmath.loggamma"><tt class="xref py py-func docutils literal"><span class="pre">loggamma()</span></tt></a>
-instead of <a class="reference internal" href="#mpmath.gamma" title="mpmath.gamma"><tt class="xref py py-func docutils literal"><span class="pre">gamma()</span></tt></a> for extremely large arguments.</p>
-<p><strong>Examples</strong></p>
-<p>Comparing with <span class="math">\(\ln(\Gamma(z))\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">loggamma</span><span class="p">(</span><span class="s">&#39;13.2&#39;</span><span class="p">);</span> <span class="n">log</span><span class="p">(</span><span class="n">gamma</span><span class="p">(</span><span class="s">&#39;13.2&#39;</span><span class="p">))</span>
-<span class="go">20.49400419456603678498394</span>
-<span class="go">20.49400419456603678498394</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">loggamma</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(-1.756626784603784110530604 + 4.742664438034657928194889j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="n">gamma</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">))</span>
-<span class="go">(-1.756626784603784110530604 - 1.540520869144928548730397j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="n">gamma</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">))</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">j</span>
-<span class="go">(-1.756626784603784110530604 + 4.742664438034657928194889j)</span>
-</pre></div>
-</div>
-<p>Note the imaginary parts for negative arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">loggamma</span><span class="p">(</span><span class="o">-</span><span class="mf">0.5</span><span class="p">);</span> <span class="n">loggamma</span><span class="p">(</span><span class="o">-</span><span class="mf">1.5</span><span class="p">);</span> <span class="n">loggamma</span><span class="p">(</span><span class="o">-</span><span class="mf">2.5</span><span class="p">)</span>
-<span class="go">(1.265512123484645396488946 - 3.141592653589793238462643j)</span>
-<span class="go">(0.8600470153764810145109327 - 6.283185307179586476925287j)</span>
-<span class="go">(-0.05624371649767405067259453 - 9.42477796076937971538793j)</span>
-</pre></div>
-</div>
-<p>Some special values:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">loggamma</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span> <span class="n">loggamma</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">loggamma</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span> <span class="o">+</span><span class="n">ln2</span>
-<span class="go">0.6931471805599453094172321</span>
-<span class="go">0.6931471805599453094172321</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">loggamma</span><span class="p">(</span><span class="mf">3.5</span><span class="p">);</span> <span class="n">log</span><span class="p">(</span><span class="mi">15</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">/</span><span class="mi">8</span><span class="p">)</span>
-<span class="go">1.200973602347074224816022</span>
-<span class="go">1.200973602347074224816022</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">loggamma</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">+inf</span>
-</pre></div>
-</div>
-<p>Huge arguments are permitted:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">loggamma</span><span class="p">(</span><span class="s">&#39;1e30&#39;</span><span class="p">)</span>
-<span class="go">6.807755278982137052053974e+31</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">loggamma</span><span class="p">(</span><span class="s">&#39;1e300&#39;</span><span class="p">)</span>
-<span class="go">6.897755278982137052053974e+302</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">loggamma</span><span class="p">(</span><span class="s">&#39;1e3000&#39;</span><span class="p">)</span>
-<span class="go">6.906755278982137052053974e+3003</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">loggamma</span><span class="p">(</span><span class="s">&#39;1e100000000000000000000&#39;</span><span class="p">)</span>
-<span class="go">2.302585092994045684007991e+100000000000000000020</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">loggamma</span><span class="p">(</span><span class="s">&#39;1e30j&#39;</span><span class="p">)</span>
-<span class="go">(-1.570796326794896619231322e+30 + 6.807755278982137052053974e+31j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">loggamma</span><span class="p">(</span><span class="s">&#39;1e300j&#39;</span><span class="p">)</span>
-<span class="go">(-1.570796326794896619231322e+300 + 6.897755278982137052053974e+302j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">loggamma</span><span class="p">(</span><span class="s">&#39;1e3000j&#39;</span><span class="p">)</span>
-<span class="go">(-1.570796326794896619231322e+3000 + 6.906755278982137052053974e+3003j)</span>
-</pre></div>
-</div>
-<p>The log-gamma function can be integrated analytically
-on any interval of unit length:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mi">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">loggamma</span><span class="p">,</span> <span class="p">[</span><span class="n">z</span><span class="p">,</span><span class="n">z</span><span class="o">+</span><span class="mi">1</span><span class="p">]);</span> <span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-<span class="go">0.9189385332046727417803297</span>
-<span class="go">0.9189385332046727417803297</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mi">3</span><span class="o">+</span><span class="mi">4j</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">loggamma</span><span class="p">,</span> <span class="p">[</span><span class="n">z</span><span class="p">,</span><span class="n">z</span><span class="o">+</span><span class="mi">1</span><span class="p">]);</span> <span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">z</span> <span class="o">+</span> <span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-<span class="go">(-0.9619286014994750641314421 + 5.219637303741238195688575j)</span>
-<span class="go">(-0.9619286014994750641314421 + 5.219637303741238195688575j)</span>
-</pre></div>
-</div>
-<p>The derivatives of the log-gamma function are given by the
-polygamma function (<a class="reference internal" href="#mpmath.psi" title="mpmath.psi"><tt class="xref py py-func docutils literal"><span class="pre">psi()</span></tt></a>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">loggamma</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="o">+</span><span class="mi">3j</span><span class="p">);</span> <span class="n">psi</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(1.688493531222971393607153 + 2.554898911356806978892748j)</span>
-<span class="go">(1.688493531222971393607153 + 2.554898911356806978892748j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">loggamma</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span> <span class="mi">2</span><span class="p">);</span> <span class="n">psi</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(-0.1539414829219882371561038 - 0.1020485197430267719746479j)</span>
-<span class="go">(-0.1539414829219882371561038 - 0.1020485197430267719746479j)</span>
-</pre></div>
-</div>
-<p>The log-gamma function satisfies an additive form of the
-recurrence relation for the ordinary gamma function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mi">2</span><span class="o">+</span><span class="mi">3j</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">loggamma</span><span class="p">(</span><span class="n">z</span><span class="p">);</span> <span class="n">loggamma</span><span class="p">(</span><span class="n">z</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">log</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">(-2.092851753092733349564189 + 2.302396543466867626153708j)</span>
-<span class="go">(-2.092851753092733349564189 + 2.302396543466867626153708j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="rising-and-falling-factorials">
-<h2>Rising and falling factorials<a class="headerlink" href="#rising-and-falling-factorials" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="rf">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">rf()</span></tt><a class="headerlink" href="#rf" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.rf">
-<tt class="descclassname">mpmath.</tt><tt class="descname">rf</tt><big>(</big><em>x</em>, <em>n</em><big>)</big><a class="headerlink" href="#mpmath.rf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the rising factorial or Pochhammer symbol,</p>
-<div class="math">
-\[x^{(n)} = x (x+1) \cdots (x+n-1) = \frac{\Gamma(x+n)}{\Gamma(x)}\]</div>
-<p>where the rightmost expression is valid for nonintegral <span class="math">\(n\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>For integral <span class="math">\(n\)</span>, the rising factorial is a polynomial:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">nprint</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">rf</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">n</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">[1.0]</span>
-<span class="go">[0.0, 1.0]</span>
-<span class="go">[0.0, 1.0, 1.0]</span>
-<span class="go">[0.0, 2.0, 3.0, 1.0]</span>
-<span class="go">[0.0, 6.0, 11.0, 6.0, 1.0]</span>
-</pre></div>
-</div>
-<p>Evaluation is supported for arbitrary arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rf</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span> <span class="mf">5.5</span><span class="p">)</span>
-<span class="go">(-7202.03920483347 - 3777.58810701527j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="ff">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">ff()</span></tt><a class="headerlink" href="#ff" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.ff">
-<tt class="descclassname">mpmath.</tt><tt class="descname">ff</tt><big>(</big><em>x</em>, <em>n</em><big>)</big><a class="headerlink" href="#mpmath.ff" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the falling factorial,</p>
-<div class="math">
-\[(x)_n = x (x-1) \cdots (x-n+1) = \frac{\Gamma(x+1)}{\Gamma(x-n+1)}\]</div>
-<p>where the rightmost expression is valid for nonintegral <span class="math">\(n\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>For integral <span class="math">\(n\)</span>, the falling factorial is a polynomial:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">nprint</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">ff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">n</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">[1.0]</span>
-<span class="go">[0.0, 1.0]</span>
-<span class="go">[0.0, -1.0, 1.0]</span>
-<span class="go">[0.0, 2.0, -3.0, 1.0]</span>
-<span class="go">[0.0, -6.0, 11.0, -6.0, 1.0]</span>
-</pre></div>
-</div>
-<p>Evaluation is supported for arbitrary arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ff</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span> <span class="mf">5.5</span><span class="p">)</span>
-<span class="go">(-720.41085888203 + 316.101124983878j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="beta-function">
-<h2>Beta function<a class="headerlink" href="#beta-function" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="beta">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">beta()</span></tt><a class="headerlink" href="#beta" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.beta">
-<tt class="descclassname">mpmath.</tt><tt class="descname">beta</tt><big>(</big><em>x</em>, <em>y</em><big>)</big><a class="headerlink" href="#mpmath.beta" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the beta function,
-<span class="math">\(B(x,y) = \Gamma(x) \Gamma(y) / \Gamma(x+y)\)</span>.
-The beta function is also commonly defined by the integral
-representation</p>
-<div class="math">
-\[B(x,y) = \int_0^1 t^{x-1} (1-t)^{y-1} \, dt\]</div>
-<p><strong>Examples</strong></p>
-<p>For integer and half-integer arguments where all three gamma
-functions are finite, the beta function becomes either rational
-number or a rational multiple of <span class="math">\(\pi\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">beta</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">0.0333333333333333</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">beta</span><span class="p">(</span><span class="mf">1.5</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">0.266666666666667</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">16</span><span class="o">*</span><span class="n">beta</span><span class="p">(</span><span class="mf">2.5</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">)</span>
-<span class="go">3.14159265358979</span>
-</pre></div>
-</div>
-<p>Where appropriate, <a class="reference internal" href="#mpmath.beta" title="mpmath.beta"><tt class="xref py py-func docutils literal"><span class="pre">beta()</span></tt></a> evaluates limits. A pole
-of the beta function is taken to result in <tt class="docutils literal"><span class="pre">+inf</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">beta</span><span class="p">(</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">beta</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">-0.333333333333333</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">beta</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">beta</span><span class="p">(</span><span class="n">inf</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">beta</span><span class="p">(</span><span class="n">inf</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">nan</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.beta" title="mpmath.beta"><tt class="xref py py-func docutils literal"><span class="pre">beta()</span></tt></a> supports complex numbers and arbitrary precision
-evaluation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">beta</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="n">j</span><span class="p">)</span>
-<span class="go">(0.4 - 0.2j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">beta</span><span class="p">(</span><span class="n">j</span><span class="p">,</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="go">(1.079424249270925780135675 - 1.410032405664160838288752j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">beta</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span>
-<span class="go">0.037890298781212201348153837138927165984170287886464</span>
-</pre></div>
-</div>
-<p>Various integrals can be computed by means of the
-beta function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">t</span><span class="o">**</span><span class="mf">2.5</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">t</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">0.0230880230880231</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">beta</span><span class="p">(</span><span class="mf">3.5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">0.0230880230880231</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">**</span><span class="mi">4</span> <span class="o">*</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">t</span><span class="p">)),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">])</span>
-<span class="go">0.319504062596158</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">beta</span><span class="p">(</span><span class="mf">2.5</span><span class="p">,</span> <span class="mf">0.75</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-<span class="go">0.319504062596158</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="betainc">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">betainc()</span></tt><a class="headerlink" href="#betainc" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.betainc">
-<tt class="descclassname">mpmath.</tt><tt class="descname">betainc</tt><big>(</big><em>a</em>, <em>b</em>, <em>x1=0</em>, <em>x2=1</em>, <em>regularized=False</em><big>)</big><a class="headerlink" href="#mpmath.betainc" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">betainc(a,</span> <span class="pre">b,</span> <span class="pre">x1=0,</span> <span class="pre">x2=1,</span> <span class="pre">regularized=False)</span></tt> gives the generalized
-incomplete beta function,</p>
-<div class="math">
-\[I_{x_1}^{x_2}(a,b) = \int_{x_1}^{x_2} t^{a-1} (1-t)^{b-1} dt.\]</div>
-<p>When <span class="math">\(x_1 = 0, x_2 = 1\)</span>, this reduces to the ordinary (complete)
-beta function <span class="math">\(B(a,b)\)</span>; see <a class="reference internal" href="#mpmath.beta" title="mpmath.beta"><tt class="xref py py-func docutils literal"><span class="pre">beta()</span></tt></a>.</p>
-<p>With the keyword argument <tt class="docutils literal"><span class="pre">regularized=True</span></tt>, <a class="reference internal" href="#mpmath.betainc" title="mpmath.betainc"><tt class="xref py py-func docutils literal"><span class="pre">betainc()</span></tt></a>
-computes the regularized incomplete beta function
-<span class="math">\(I_{x_1}^{x_2}(a,b) / B(a,b)\)</span>. This is the cumulative distribution of the
-beta distribution with parameters <span class="math">\(a\)</span>, <span class="math">\(b\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>Verifying that <a class="reference internal" href="#mpmath.betainc" title="mpmath.betainc"><tt class="xref py py-func docutils literal"><span class="pre">betainc()</span></tt></a> computes the integral in the
-definition:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">b</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">betainc</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-<span class="go">-4010.4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">t</span><span class="o">**</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">t</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">y</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="p">[</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">])</span>
-<span class="go">-4010.4</span>
-</pre></div>
-</div>
-<p>The arguments may be arbitrary complex numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">betainc</span><span class="p">(</span><span class="mf">0.75</span><span class="p">,</span> <span class="mi">1</span><span class="o">-</span><span class="mi">4j</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(0.2241657956955709603655887 + 0.3619619242700451992411724j)</span>
-</pre></div>
-</div>
-<p>With regularization:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">betainc</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="n">regularized</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">0.4375</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">betainc</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">regularized</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>   <span class="c"># Complete</span>
-<span class="go">1.0</span>
-</pre></div>
-</div>
-<p>The beta integral satisfies some simple argument transformation
-symmetries:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">betainc</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">),</span> <span class="o">-</span><span class="n">betainc</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">4</span><span class="p">),</span> <span class="n">betainc</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">1</span><span class="o">-</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">(56.0833333333333, 56.0833333333333, 56.0833333333333)</span>
-</pre></div>
-</div>
-<p>The beta integral can often be evaluated analytically. For integer and
-rational arguments, the incomplete beta function typically reduces to a
-simple algebraic-logarithmic expression:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">betainc</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)))</span>
-<span class="go">&#39;-(log((9/8)))&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="n">betainc</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
-<span class="go">&#39;(673/12)&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="n">betainc</span><span class="p">(</span><span class="mf">1.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">&#39;((-12+sqrt(1152))/18)&#39;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="super-and-hyperfactorials">
-<h2>Super- and hyperfactorials<a class="headerlink" href="#super-and-hyperfactorials" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="superfac">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">superfac()</span></tt><a class="headerlink" href="#superfac" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.superfac">
-<tt class="descclassname">mpmath.</tt><tt class="descname">superfac</tt><big>(</big><em>z</em><big>)</big><a class="headerlink" href="#mpmath.superfac" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the superfactorial, defined as the product of
-consecutive factorials</p>
-<div class="math">
-\[\mathrm{sf}(n) = \prod_{k=1}^n k!\]</div>
-<p>For general complex <span class="math">\(z\)</span>, <span class="math">\(\mathrm{sf}(z)\)</span> is defined
-in terms of the Barnes G-function (see <a class="reference internal" href="#mpmath.barnesg" title="mpmath.barnesg"><tt class="xref py py-func docutils literal"><span class="pre">barnesg()</span></tt></a>).</p>
-<p><strong>Examples</strong></p>
-<p>The first few superfactorials are (OEIS A000178):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">superfac</span><span class="p">(</span><span class="n">n</span><span class="p">)))</span>
-<span class="gp">...</span>
-<span class="go">0 1.0</span>
-<span class="go">1 1.0</span>
-<span class="go">2 2.0</span>
-<span class="go">3 12.0</span>
-<span class="go">4 288.0</span>
-<span class="go">5 34560.0</span>
-<span class="go">6 24883200.0</span>
-<span class="go">7 125411328000.0</span>
-<span class="go">8 5.05658474496e+15</span>
-<span class="go">9 1.83493347225108e+21</span>
-</pre></div>
-</div>
-<p>Superfactorials grow very rapidly:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">superfac</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span>
-<span class="go">3.24570818422368e+1177245</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">superfac</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">2.61398543581249e+467427913956904067453</span>
-</pre></div>
-</div>
-<p>Evaluation is supported for arbitrary arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">superfac</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">17.20051550121297985285333</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">superfac</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(-0.005915485633199789627466468 + 0.008156449464604044948738263j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">superfac</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">0.2645072034016070205673056</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference external" href="http://www.research.att.com/~njas/sequences/A000178">http://www.research.att.com/~njas/sequences/A000178</a></li>
-</ol>
-</dd></dl>
-
-</div>
-<div class="section" id="hyperfac">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">hyperfac()</span></tt><a class="headerlink" href="#hyperfac" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.hyperfac">
-<tt class="descclassname">mpmath.</tt><tt class="descname">hyperfac</tt><big>(</big><em>z</em><big>)</big><a class="headerlink" href="#mpmath.hyperfac" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the hyperfactorial, defined for integers as the product</p>
-<div class="math">
-\[H(n) = \prod_{k=1}^n k^k.\]</div>
-<p>The hyperfactorial satisfies the recurrence formula <span class="math">\(H(z) = z^z H(z-1)\)</span>.
-It can be defined more generally in terms of the Barnes G-function (see
-<a class="reference internal" href="#mpmath.barnesg" title="mpmath.barnesg"><tt class="xref py py-func docutils literal"><span class="pre">barnesg()</span></tt></a>) and the gamma function by the formula</p>
-<div class="math">
-\[H(z) = \frac{\Gamma(z+1)^z}{G(z)}.\]</div>
-<p>The extension to complex numbers can also be done via
-the integral representation</p>
-<div class="math">
-\[H(z) = (2\pi)^{-z/2} \exp \left[
-    {z+1 \choose 2} + \int_0^z \log(t!)\,dt
-    \right].\]</div>
-<p><strong>Examples</strong></p>
-<p>The rapidly-growing sequence of hyperfactorials begins
-(OEIS A002109):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">hyperfac</span><span class="p">(</span><span class="n">n</span><span class="p">)))</span>
-<span class="gp">...</span>
-<span class="go">0 1.0</span>
-<span class="go">1 1.0</span>
-<span class="go">2 4.0</span>
-<span class="go">3 108.0</span>
-<span class="go">4 27648.0</span>
-<span class="go">5 86400000.0</span>
-<span class="go">6 4031078400000.0</span>
-<span class="go">7 3.3197663987712e+18</span>
-<span class="go">8 5.56964379417266e+25</span>
-<span class="go">9 2.15779412229419e+34</span>
-</pre></div>
-</div>
-<p>Some even larger hyperfactorials are:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">hyperfac</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span>
-<span class="go">5.46458120882585e+1392926</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyperfac</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">4.60408207642219e+489142638002418704309</span>
-</pre></div>
-</div>
-<p>The hyperfactorial can be evaluated for arbitrary arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">hyperfac</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="go">0.880449235173423</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">hyperfac</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">0.581061466795327</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyperfac</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">205.211134637462</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyperfac</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="o">+</span><span class="mi">1j</span><span class="p">)</span>
-<span class="go">(3.01144471378225e+46 - 2.45285242480185e+46j)</span>
-</pre></div>
-</div>
-<p>The recurrence property of the hyperfactorial holds
-generally:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mi">3</span><span class="o">-</span><span class="mi">4</span><span class="o">*</span><span class="n">j</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyperfac</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">(-4.49795891462086e-7 - 6.33262283196162e-7j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span><span class="o">**</span><span class="n">z</span> <span class="o">*</span> <span class="n">hyperfac</span><span class="p">(</span><span class="n">z</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(-4.49795891462086e-7 - 6.33262283196162e-7j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="o">-</span><span class="mf">0.6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">z</span><span class="o">**</span><span class="n">z</span> <span class="o">*</span> <span class="n">hyperfac</span><span class="p">(</span><span class="n">z</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span>
-<span class="go">1.28170142849352</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyperfac</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">1.28170142849352</span>
-</pre></div>
-</div>
-<p>The hyperfactorial may also be computed using the integral
-definition:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mf">2.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyperfac</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">15.9842119922237</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">binomial</span><span class="p">(</span><span class="n">z</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span>
-<span class="gp">... </span>    <span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">loggamma</span><span class="p">(</span><span class="n">t</span><span class="o">+</span><span class="mi">1</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">z</span><span class="p">]))</span>
-<span class="go">15.9842119922237</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.hyperfac" title="mpmath.hyperfac"><tt class="xref py py-func docutils literal"><span class="pre">hyperfac()</span></tt></a> supports arbitrary-precision evaluation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyperfac</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">215779412229418562091680268288000000000000000.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyperfac</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">0.89404818005227001975423476035729076375705084390942</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference external" href="http://www.research.att.com/~njas/sequences/A002109">http://www.research.att.com/~njas/sequences/A002109</a></li>
-<li><a class="reference external" href="http://mathworld.wolfram.com/Hyperfactorial.html">http://mathworld.wolfram.com/Hyperfactorial.html</a></li>
-</ol>
-</dd></dl>
-
-</div>
-<div class="section" id="barnesg">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">barnesg()</span></tt><a class="headerlink" href="#barnesg" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.barnesg">
-<tt class="descclassname">mpmath.</tt><tt class="descname">barnesg</tt><big>(</big><em>z</em><big>)</big><a class="headerlink" href="#mpmath.barnesg" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the Barnes G-function, which generalizes the
-superfactorial (<a class="reference internal" href="#mpmath.superfac" title="mpmath.superfac"><tt class="xref py py-func docutils literal"><span class="pre">superfac()</span></tt></a>) and by extension also the
-hyperfactorial (<a class="reference internal" href="#mpmath.hyperfac" title="mpmath.hyperfac"><tt class="xref py py-func docutils literal"><span class="pre">hyperfac()</span></tt></a>) to the complex numbers
-in an analogous way to how the gamma function generalizes
-the ordinary factorial.</p>
-<p>The Barnes G-function may be defined in terms of a Weierstrass
-product:</p>
-<div class="math">
-\[G(z+1) = (2\pi)^{z/2} e^{-[z(z+1)+\gamma z^2]/2}
-\prod_{n=1}^\infty
-\left[\left(1+\frac{z}{n}\right)^ne^{-z+z^2/(2n)}\right]\]</div>
-<p>For positive integers <span class="math">\(n\)</span>, we have have relation to superfactorials
-<span class="math">\(G(n) = \mathrm{sf}(n-2) = 0! \cdot 1! \cdots (n-2)!\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>Some elementary values and limits of the Barnes G-function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">barnesg</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">barnesg</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">barnesg</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">(1.0, 1.0, 1.0)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">barnesg</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">2.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">barnesg</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">12.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">barnesg</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span>
-<span class="go">288.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">barnesg</span><span class="p">(</span><span class="mi">7</span><span class="p">)</span>
-<span class="go">34560.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">barnesg</span><span class="p">(</span><span class="mi">8</span><span class="p">)</span>
-<span class="go">24883200.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">barnesg</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">barnesg</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">barnesg</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">barnesg</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(0.0, 0.0, 0.0)</span>
-</pre></div>
-</div>
-<p>Closed-form values are known for some rational arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">barnesg</span><span class="p">(</span><span class="s">&#39;1/2&#39;</span><span class="p">)</span>
-<span class="go">0.603244281209446</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="mf">0.25</span><span class="o">+</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">12</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">/</span><span class="n">glaisher</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.603244281209446</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">barnesg</span><span class="p">(</span><span class="s">&#39;1/4&#39;</span><span class="p">)</span>
-<span class="go">0.29375596533861</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nthroot</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="s">&#39;3/8&#39;</span><span class="p">)</span><span class="o">/</span><span class="n">exp</span><span class="p">(</span><span class="n">catalan</span><span class="o">/</span><span class="n">pi</span><span class="p">)</span><span class="o">/</span>
-<span class="gp">... </span>     <span class="n">gamma</span><span class="p">(</span><span class="mf">0.25</span><span class="p">)</span><span class="o">**</span><span class="mi">3</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">glaisher</span><span class="p">)</span><span class="o">**</span><span class="mi">9</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">0.29375596533861</span>
-</pre></div>
-</div>
-<p>The Barnes G-function satisfies the functional equation
-<span class="math">\(G(z+1) = \Gamma(z) G(z)\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">barnesg</span><span class="p">(</span><span class="n">z</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">2.39292119327948</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">barnesg</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">2.39292119327948</span>
-</pre></div>
-</div>
-<p>The asymptotic growth rate of the Barnes G-function is related to
-the Glaisher-Kinkelin constant:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="n">barnesg</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">n</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">2</span><span class="o">-</span><span class="n">mpf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">12</span><span class="p">)</span><span class="o">*</span>
-<span class="gp">... </span>    <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">n</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">4</span><span class="p">)),</span> <span class="n">inf</span><span class="p">)</span>
-<span class="go">0.847536694177301</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="s">&#39;1/12&#39;</span><span class="p">)</span><span class="o">/</span><span class="n">glaisher</span>
-<span class="go">0.847536694177301</span>
-</pre></div>
-</div>
-<p>The Barnes G-function can be differentiated in closed form:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">barnesg</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">0.264507203401607</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">barnesg</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="p">((</span><span class="n">z</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">psi</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">z</span><span class="p">)</span><span class="o">-</span><span class="n">z</span><span class="o">+</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.264507203401607</span>
-</pre></div>
-</div>
-<p>Evaluation is supported for arbitrary arguments and at arbitrary
-precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">barnesg</span><span class="p">(</span><span class="mf">6.5</span><span class="p">)</span>
-<span class="go">2548.7457695685</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">barnesg</span><span class="p">(</span><span class="o">-</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">0.00535976768353037</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">barnesg</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(-0.000676375932234244 - 4.42236140124728e-5j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">barnesg</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">0.81305501090451340843586085064413533788206204124732</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">barnesg</span><span class="p">(</span><span class="mi">10j</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="o">.</span><span class="n">real</span>
-<span class="go">0.000000000021852360840356557241543036724799812371995850552234</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="o">.</span><span class="n">imag</span>
-<span class="go">-0.00000000000070035335320062304849020654215545839053210041457588</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">barnesg</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">3.10361006263698e+6626</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">barnesg</span><span class="p">(</span><span class="o">-</span><span class="mi">101</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">barnesg</span><span class="p">(</span><span class="o">-</span><span class="mf">10.5</span><span class="p">)</span>
-<span class="go">5.94463017605008e+25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">barnesg</span><span class="p">(</span><span class="o">-</span><span class="mf">10000.5</span><span class="p">)</span>
-<span class="go">-6.14322868174828e+167480422</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">barnesg</span><span class="p">(</span><span class="mi">1000j</span><span class="p">)</span>
-<span class="go">(5.21133054865546e-1173597 + 4.27461836811016e-1173597j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">barnesg</span><span class="p">(</span><span class="o">-</span><span class="mi">1000</span><span class="o">+</span><span class="mi">1000j</span><span class="p">)</span>
-<span class="go">(2.43114569750291e+1026623 + 2.24851410674842e+1026623j)</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li>Whittaker &amp; Watson, <em>A Course of Modern Analysis</em>,
-Cambridge University Press, 4th edition (1927), p.264</li>
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Barnes_G-function">http://en.wikipedia.org/wiki/Barnes_G-function</a></li>
-<li><a class="reference external" href="http://mathworld.wolfram.com/BarnesG-Function.html">http://mathworld.wolfram.com/BarnesG-Function.html</a></li>
-</ol>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="polygamma-functions-and-harmonic-numbers">
-<h2>Polygamma functions and harmonic numbers<a class="headerlink" href="#polygamma-functions-and-harmonic-numbers" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="psi-digamma">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">psi()</span></tt>/<tt class="xref py py-func docutils literal"><span class="pre">digamma()</span></tt><a class="headerlink" href="#psi-digamma" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.psi">
-<tt class="descclassname">mpmath.</tt><tt class="descname">psi</tt><big>(</big><em>m</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.psi" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="mpmath.digamma">
-<tt class="descclassname">mpmath.</tt><tt class="descname">digamma</tt><big>(</big><em>z</em><big>)</big><a class="headerlink" href="#mpmath.digamma" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the &lt;no doc&gt; of x</p>
-</dd></dl>
-
-</div>
-<div class="section" id="harmonic">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">harmonic()</span></tt><a class="headerlink" href="#harmonic" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.harmonic">
-<tt class="descclassname">mpmath.</tt><tt class="descname">harmonic</tt><big>(</big><em>z</em><big>)</big><a class="headerlink" href="#mpmath.harmonic" title="Permalink to this definition">¶</a></dt>
-<dd><p>If <span class="math">\(n\)</span> is an integer, <tt class="docutils literal"><span class="pre">harmonic(n)</span></tt> gives a floating-point
-approximation of the <span class="math">\(n\)</span>-th harmonic number <span class="math">\(H(n)\)</span>, defined as</p>
-<div class="math">
-\[H(n) = 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n}\]</div>
-<p>The first few harmonic numbers are:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">8</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">harmonic</span><span class="p">(</span><span class="n">n</span><span class="p">)))</span>
-<span class="gp">...</span>
-<span class="go">0 0.0</span>
-<span class="go">1 1.0</span>
-<span class="go">2 1.5</span>
-<span class="go">3 1.83333333333333</span>
-<span class="go">4 2.08333333333333</span>
-<span class="go">5 2.28333333333333</span>
-<span class="go">6 2.45</span>
-<span class="go">7 2.59285714285714</span>
-</pre></div>
-</div>
-<p>The infinite harmonic series <span class="math">\(1 + 1/2 + 1/3 + \ldots\)</span> diverges:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">harmonic</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">+inf</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.harmonic" title="mpmath.harmonic"><tt class="xref py py-func docutils literal"><span class="pre">harmonic()</span></tt></a> is evaluated using the digamma function rather
-than by summing the harmonic series term by term. It can therefore
-be computed quickly for arbitrarily large <span class="math">\(n\)</span>, and even for
-nonintegral arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">harmonic</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">230.835724964306</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">harmonic</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="go">0.613705638880109</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">harmonic</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(2.24757548223494 + 0.850502209186044j)</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.harmonic" title="mpmath.harmonic"><tt class="xref py py-func docutils literal"><span class="pre">harmonic()</span></tt></a> supports arbitrary precision evaluation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">harmonic</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span>
-<span class="go">3.0198773448773448773448773448773448773448773448773</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">harmonic</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">1.8727388590273302654363491032336134987519132374152</span>
-</pre></div>
-</div>
-<p>The harmonic series diverges, but at a glacial pace. It is possible
-to calculate the exact number of terms required before the sum
-exceeds a given amount, say 100:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v</span> <span class="o">=</span> <span class="mi">10</span><span class="o">**</span><span class="n">findroot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">harmonic</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="mi">100</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v</span>
-<span class="go">15092688622113788323693563264538101449859496.864101</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">ceil</span><span class="p">(</span><span class="n">v</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-<span class="go">15092688622113788323693563264538101449859497</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">harmonic</span><span class="p">(</span><span class="n">v</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">99.999999999999999999999999999999999999999999942747</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">harmonic</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
-<span class="go">100.000000000000000000000000000000000000000000009</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Factorials and gamma functions</a><ul>
-<li><a class="reference internal" href="#factorials">Factorials</a><ul>
-<li><a class="reference internal" href="#factorial-fac"><tt class="docutils literal"><span class="pre">factorial()</span></tt>/<tt class="docutils literal"><span class="pre">fac()</span></tt></a></li>
-<li><a class="reference internal" href="#fac2"><tt class="docutils literal"><span class="pre">fac2()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#binomial-coefficients">Binomial coefficients</a><ul>
-<li><a class="reference internal" href="#binomial"><tt class="docutils literal"><span class="pre">binomial()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#gamma-function">Gamma function</a><ul>
-<li><a class="reference internal" href="#gamma"><tt class="docutils literal"><span class="pre">gamma()</span></tt></a></li>
-<li><a class="reference internal" href="#rgamma"><tt class="docutils literal"><span class="pre">rgamma()</span></tt></a></li>
-<li><a class="reference internal" href="#gammaprod"><tt class="docutils literal"><span class="pre">gammaprod()</span></tt></a></li>
-<li><a class="reference internal" href="#loggamma"><tt class="docutils literal"><span class="pre">loggamma()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#rising-and-falling-factorials">Rising and falling factorials</a><ul>
-<li><a class="reference internal" href="#rf"><tt class="docutils literal"><span class="pre">rf()</span></tt></a></li>
-<li><a class="reference internal" href="#ff"><tt class="docutils literal"><span class="pre">ff()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#beta-function">Beta function</a><ul>
-<li><a class="reference internal" href="#beta"><tt class="docutils literal"><span class="pre">beta()</span></tt></a></li>
-<li><a class="reference internal" href="#betainc"><tt class="docutils literal"><span class="pre">betainc()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#super-and-hyperfactorials">Super- and hyperfactorials</a><ul>
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-            
-  <div class="section" id="hyperbolic-functions">
-<h1>Hyperbolic functions<a class="headerlink" href="#hyperbolic-functions" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="id1">
-<h2>Hyperbolic functions<a class="headerlink" href="#id1" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="cosh">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">cosh()</span></tt><a class="headerlink" href="#cosh" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.cosh">
-<tt class="descclassname">mpmath.</tt><tt class="descname">cosh</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.cosh" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the hyperbolic cosine of <span class="math">\(x\)</span>,
-<span class="math">\(\cosh(x) = (e^x + e^{-x})/2\)</span>. Values and limits include:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cosh</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cosh</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1.543080634815243778477906</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cosh</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">),</span> <span class="n">cosh</span><span class="p">(</span><span class="o">+</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">(+inf, +inf)</span>
-</pre></div>
-</div>
-<p>The hyperbolic cosine is an even, convex function with
-a global minimum at <span class="math">\(x = 0\)</span>, having a Maclaurin series
-that starts:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">cosh</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)))</span>
-<span class="go">[1.0, 0.0, 0.5, 0.0, 0.0416667, 0.0]</span>
-</pre></div>
-</div>
-<p>Generalized to complex numbers, the hyperbolic cosine is
-equivalent to a cosine with the argument rotated
-in the imaginary direction, or <span class="math">\(\cosh x = \cos ix\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">cosh</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(-3.724545504915322565473971 + 0.5118225699873846088344638j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="mi">3</span><span class="o">-</span><span class="mi">2j</span><span class="p">)</span>
-<span class="go">(-3.724545504915322565473971 + 0.5118225699873846088344638j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="sinh">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">sinh()</span></tt><a class="headerlink" href="#sinh" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.sinh">
-<tt class="descclassname">mpmath.</tt><tt class="descname">sinh</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.sinh" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the hyperbolic sine of <span class="math">\(x\)</span>,
-<span class="math">\(\sinh(x) = (e^x - e^{-x})/2\)</span>. Values and limits include:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinh</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinh</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1.175201193643801456882382</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinh</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">),</span> <span class="n">sinh</span><span class="p">(</span><span class="o">+</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">(-inf, +inf)</span>
-</pre></div>
-</div>
-<p>The hyperbolic sine is an odd function, with a Maclaurin
-series that starts:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">sinh</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)))</span>
-<span class="go">[0.0, 1.0, 0.0, 0.166667, 0.0, 0.00833333]</span>
-</pre></div>
-</div>
-<p>Generalized to complex numbers, the hyperbolic sine is
-essentially a sine with a rotation <span class="math">\(i\)</span> applied to
-the argument; more precisely, <span class="math">\(\sinh x = -i \sin ix\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sinh</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(-3.590564589985779952012565 + 0.5309210862485198052670401j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">j</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="mi">3</span><span class="o">-</span><span class="mi">2j</span><span class="p">)</span>
-<span class="go">(-3.590564589985779952012565 + 0.5309210862485198052670401j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="tanh">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">tanh()</span></tt><a class="headerlink" href="#tanh" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.tanh">
-<tt class="descclassname">mpmath.</tt><tt class="descname">tanh</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.tanh" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the hyperbolic tangent of <span class="math">\(x\)</span>,
-<span class="math">\(\tanh(x) = \sinh(x)/\cosh(x)\)</span>. Values and limits include:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tanh</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tanh</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.7615941559557648881194583</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tanh</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">),</span> <span class="n">tanh</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">(-1.0, 1.0)</span>
-</pre></div>
-</div>
-<p>The hyperbolic tangent is an odd, sigmoidal function, similar
-to the inverse tangent and error function. Its Maclaurin
-series is:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">tanh</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)))</span>
-<span class="go">[0.0, 1.0, 0.0, -0.333333, 0.0, 0.133333]</span>
-</pre></div>
-</div>
-<p>Generalized to complex numbers, the hyperbolic tangent is
-essentially a tangent with a rotation <span class="math">\(i\)</span> applied to
-the argument; more precisely, <span class="math">\(\tanh x = -i \tan ix\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">tanh</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(0.9653858790221331242784803 - 0.009884375038322493720314034j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">j</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="mi">3</span><span class="o">-</span><span class="mi">2j</span><span class="p">)</span>
-<span class="go">(0.9653858790221331242784803 - 0.009884375038322493720314034j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="sech">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">sech()</span></tt><a class="headerlink" href="#sech" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.sech">
-<tt class="descclassname">mpmath.</tt><tt class="descname">sech</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.sech" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the hyperbolic secant of <span class="math">\(x\)</span>,
-<span class="math">\(\mathrm{sech}(x) = \frac{1}{\cosh(x)}\)</span>.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="csch">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">csch()</span></tt><a class="headerlink" href="#csch" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.csch">
-<tt class="descclassname">mpmath.</tt><tt class="descname">csch</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.csch" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the hyperbolic cosecant of <span class="math">\(x\)</span>,
-<span class="math">\(\mathrm{csch}(x) = \frac{1}{\sinh(x)}\)</span>.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="coth">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">coth()</span></tt><a class="headerlink" href="#coth" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.coth">
-<tt class="descclassname">mpmath.</tt><tt class="descname">coth</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.coth" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the hyperbolic cotangent of <span class="math">\(x\)</span>,
-<span class="math">\(\mathrm{coth}(x) = \frac{\cosh(x)}{\sinh(x)}\)</span>.</p>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="inverse-hyperbolic-functions">
-<h2>Inverse hyperbolic functions<a class="headerlink" href="#inverse-hyperbolic-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="acosh">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">acosh()</span></tt><a class="headerlink" href="#acosh" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.acosh">
-<tt class="descclassname">mpmath.</tt><tt class="descname">acosh</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.acosh" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the inverse hyperbolic cosine of <span class="math">\(x\)</span>,
-<span class="math">\(\mathrm{cosh}^{-1}(x) = \log(x+\sqrt{x+1}\sqrt{x-1})\)</span>.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="asinh">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">asinh()</span></tt><a class="headerlink" href="#asinh" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.asinh">
-<tt class="descclassname">mpmath.</tt><tt class="descname">asinh</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.asinh" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the inverse hyperbolic sine of <span class="math">\(x\)</span>,
-<span class="math">\(\mathrm{sinh}^{-1}(x) = \log(x+\sqrt{1+x^2})\)</span>.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="atanh">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">atanh()</span></tt><a class="headerlink" href="#atanh" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.atanh">
-<tt class="descclassname">mpmath.</tt><tt class="descname">atanh</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.atanh" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the inverse hyperbolic tangent of <span class="math">\(x\)</span>,
-<span class="math">\(\mathrm{tanh}^{-1}(x) = \frac{1}{2}\left(\log(1+x)-\log(1-x)\right)\)</span>.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="asech">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">asech()</span></tt><a class="headerlink" href="#asech" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.asech">
-<tt class="descclassname">mpmath.</tt><tt class="descname">asech</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.asech" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the inverse hyperbolic secant of <span class="math">\(x\)</span>,
-<span class="math">\(\mathrm{sech}^{-1}(x) = \cosh^{-1}(1/x)\)</span>.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="acsch">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">acsch()</span></tt><a class="headerlink" href="#acsch" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.acsch">
-<tt class="descclassname">mpmath.</tt><tt class="descname">acsch</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.acsch" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the inverse hyperbolic cosecant of <span class="math">\(x\)</span>,
-<span class="math">\(\mathrm{csch}^{-1}(x) = \sinh^{-1}(1/x)\)</span>.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="acoth">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">acoth()</span></tt><a class="headerlink" href="#acoth" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.acoth">
-<tt class="descclassname">mpmath.</tt><tt class="descname">acoth</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.acoth" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the inverse hyperbolic cotangent of <span class="math">\(x\)</span>,
-<span class="math">\(\mathrm{coth}^{-1}(x) = \tanh^{-1}(1/x)\)</span>.</p>
-</dd></dl>
-
-</div>
-</div>
-</div>
-
-
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-            </a></p>
-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Hyperbolic functions</a><ul>
-<li><a class="reference internal" href="#id1">Hyperbolic functions</a><ul>
-<li><a class="reference internal" href="#cosh"><tt class="docutils literal"><span class="pre">cosh()</span></tt></a></li>
-<li><a class="reference internal" href="#sinh"><tt class="docutils literal"><span class="pre">sinh()</span></tt></a></li>
-<li><a class="reference internal" href="#tanh"><tt class="docutils literal"><span class="pre">tanh()</span></tt></a></li>
-<li><a class="reference internal" href="#sech"><tt class="docutils literal"><span class="pre">sech()</span></tt></a></li>
-<li><a class="reference internal" href="#csch"><tt class="docutils literal"><span class="pre">csch()</span></tt></a></li>
-<li><a class="reference internal" href="#coth"><tt class="docutils literal"><span class="pre">coth()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#inverse-hyperbolic-functions">Inverse hyperbolic functions</a><ul>
-<li><a class="reference internal" href="#acosh"><tt class="docutils literal"><span class="pre">acosh()</span></tt></a></li>
-<li><a class="reference internal" href="#asinh"><tt class="docutils literal"><span class="pre">asinh()</span></tt></a></li>
-<li><a class="reference internal" href="#atanh"><tt class="docutils literal"><span class="pre">atanh()</span></tt></a></li>
-<li><a class="reference internal" href="#asech"><tt class="docutils literal"><span class="pre">asech()</span></tt></a></li>
-<li><a class="reference internal" href="#acsch"><tt class="docutils literal"><span class="pre">acsch()</span></tt></a></li>
-<li><a class="reference internal" href="#acoth"><tt class="docutils literal"><span class="pre">acoth()</span></tt></a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-
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-            
-  <div class="section" id="hypergeometric-functions">
-<h1>Hypergeometric functions<a class="headerlink" href="#hypergeometric-functions" title="Permalink to this headline">¶</a></h1>
-<p>The functions listed in <a class="reference internal" href="../functions/expintegrals.html"><em>Exponential integrals and error functions</em></a>, <a class="reference internal" href="../functions/bessel.html"><em>Bessel functions and related functions</em></a> and
-<a class="reference internal" href="../functions/orthogonal.html"><em>Orthogonal polynomials</em></a>, and many other functions as well, are merely
-particular instances of the generalized hypergeometric function <span class="math">\(\,_pF_q\)</span>.
-The functions listed in the following section enable efficient
-direct evaluation of the underlying hypergeometric series, as
-well as linear combinations, limits with respect to parameters,
-and analytic continuations thereof. Extensions to twodimensional
-series are also provided. See also the basic or q-analog of
-the hypergeometric series in <a class="reference internal" href="../functions/qfunctions.html"><em>q-functions</em></a>.</p>
-<p>For convenience, most of the hypergeometric series of low order are
-provided as standalone functions. They can equivalently be evaluated using
-<a class="reference internal" href="#mpmath.hyper" title="mpmath.hyper"><tt class="xref py py-func docutils literal"><span class="pre">hyper()</span></tt></a>. As will be demonstrated in the respective docstrings,
-all the <tt class="docutils literal"><span class="pre">hyp#f#</span></tt> functions implement analytic continuations and/or asymptotic
-expansions with respect to the argument <span class="math">\(z\)</span>, thereby permitting evaluation
-for <span class="math">\(z\)</span> anywhere in the complex plane. Functions of higher degree can be
-computed via <a class="reference internal" href="#mpmath.hyper" title="mpmath.hyper"><tt class="xref py py-func docutils literal"><span class="pre">hyper()</span></tt></a>, but generally only in rapidly convergent
-instances.</p>
-<p>Most hypergeometric and hypergeometric-derived functions accept optional
-keyword arguments to specify options for <tt class="xref py py-func docutils literal"><span class="pre">hypercomb()</span></tt> or
-<tt class="xref py py-func docutils literal"><span class="pre">hyper()</span></tt>. Some useful options are <em>maxprec</em>, <em>maxterms</em>,
-<em>zeroprec</em>, <em>accurate_small</em>, <em>hmag</em>, <em>force_series</em>,
-<em>asymp_tol</em> and <em>eliminate</em>. These options give control over what to
-do in case of slow convergence, extreme loss of accuracy or
-evaluation at zeros (these two cases cannot generally be
-distinguished from each other automatically),
-and singular parameter combinations.</p>
-<div class="section" id="common-hypergeometric-series">
-<h2>Common hypergeometric series<a class="headerlink" href="#common-hypergeometric-series" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="hyp0f1">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">hyp0f1()</span></tt><a class="headerlink" href="#hyp0f1" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.hyp0f1">
-<tt class="descclassname">mpmath.</tt><tt class="descname">hyp0f1</tt><big>(</big><em>a</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.hyp0f1" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the hypergeometric function <span class="math">\(\,_0F_1\)</span>, sometimes known as the
-confluent limit function, defined as</p>
-<div class="math">
-\[\,_0F_1(a,z) = \sum_{k=0}^{\infty} \frac{1}{(a)_k} \frac{z^k}{k!}.\]</div>
-<p>This function satisfies the differential equation <span class="math">\(z f''(z) + a f'(z) = f(z)\)</span>,
-and is related to the Bessel function of the first kind (see <a class="reference internal" href="../functions/bessel.html#mpmath.besselj" title="mpmath.besselj"><tt class="xref py py-func docutils literal"><span class="pre">besselj()</span></tt></a>).</p>
-<p><tt class="docutils literal"><span class="pre">hyp0f1(a,z)</span></tt> is equivalent to <tt class="docutils literal"><span class="pre">hyper([],[a],z)</span></tt>; see documentation for
-<a class="reference internal" href="#mpmath.hyper" title="mpmath.hyper"><tt class="xref py py-func docutils literal"><span class="pre">hyper()</span></tt></a> for more information.</p>
-<p><strong>Examples</strong></p>
-<p>Evaluation for arbitrary arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp0f1</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">)</span>
-<span class="go">1.130318207984970054415392</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp0f1</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="mi">1234567</span><span class="p">)</span>
-<span class="go">6.27287187546220705604627e+964</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp0f1</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">,</span> <span class="mi">1000000j</span><span class="p">)</span>
-<span class="go">(3.905169561300910030267132e+606 + 3.807708544441684513934213e+606j)</span>
-</pre></div>
-</div>
-<p>Evaluation is supported for arbitrarily large values of <span class="math">\(z\)</span>,
-using asymptotic expansions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">hyp0f1</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="o">**</span><span class="mi">50</span><span class="p">)</span>
-<span class="go">2.131705322874965310390701e+8685889638065036553022565</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp0f1</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="o">**</span><span class="mi">50</span><span class="p">)</span>
-<span class="go">1.115945364792025420300208e-13</span>
-</pre></div>
-</div>
-<p>Verifying the differential equation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="mf">2.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">hyp0f1</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="n">chop</span><span class="p">(</span><span class="n">z</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">)</span> <span class="o">-</span> <span class="n">f</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="hyp1f1">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">hyp1f1()</span></tt><a class="headerlink" href="#hyp1f1" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.hyp1f1">
-<tt class="descclassname">mpmath.</tt><tt class="descname">hyp1f1</tt><big>(</big><em>a</em>, <em>b</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.hyp1f1" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the confluent hypergeometric function of the first kind,</p>
-<div class="math">
-\[\,_1F_1(a,b,z) = \sum_{k=0}^{\infty} \frac{(a)_k}{(b)_k} \frac{z^k}{k!},\]</div>
-<p>also known as Kummer&#8217;s function and sometimes denoted by <span class="math">\(M(a,b,z)\)</span>. This
-function gives one solution to the confluent (Kummer&#8217;s) differential equation</p>
-<div class="math">
-\[z f''(z) + (b-z) f'(z) - af(z) = 0.\]</div>
-<p>A second solution is given by the <span class="math">\(U\)</span> function; see <a class="reference internal" href="../functions/bessel.html#mpmath.hyperu" title="mpmath.hyperu"><tt class="xref py py-func docutils literal"><span class="pre">hyperu()</span></tt></a>.
-Solutions are also given in an alternate form by the Whittaker
-functions (<a class="reference internal" href="../functions/bessel.html#mpmath.whitm" title="mpmath.whitm"><tt class="xref py py-func docutils literal"><span class="pre">whitm()</span></tt></a>, <a class="reference internal" href="../functions/bessel.html#mpmath.whitw" title="mpmath.whitw"><tt class="xref py py-func docutils literal"><span class="pre">whitw()</span></tt></a>).</p>
-<p><tt class="docutils literal"><span class="pre">hyp1f1(a,b,z)</span></tt> is equivalent
-to <tt class="docutils literal"><span class="pre">hyper([a],[b],z)</span></tt>; see documentation for <a class="reference internal" href="#mpmath.hyper" title="mpmath.hyper"><tt class="xref py py-func docutils literal"><span class="pre">hyper()</span></tt></a> for more
-information.</p>
-<p><strong>Examples</strong></p>
-<p>Evaluation for real and complex values of the argument <span class="math">\(z\)</span>, with
-fixed parameters <span class="math">\(a = 2, b = -1/3\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp1f1</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="mf">3.25</span><span class="p">)</span>
-<span class="go">-2815.956856924817275640248</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp1f1</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="o">-</span><span class="mf">3.25</span><span class="p">)</span>
-<span class="go">-1.145036502407444445553107</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp1f1</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="mi">1000</span><span class="p">)</span>
-<span class="go">-8.021799872770764149793693e+441</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp1f1</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="o">-</span><span class="mi">1000</span><span class="p">)</span>
-<span class="go">0.000003131987633006813594535331</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp1f1</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="mi">100</span><span class="o">+</span><span class="mi">100j</span><span class="p">)</span>
-<span class="go">(-3.189190365227034385898282e+48 - 1.106169926814270418999315e+49j)</span>
-</pre></div>
-</div>
-<p>Parameters may be complex:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">hyp1f1</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mi">10j</span><span class="p">)</span>
-<span class="go">(261.8977905181045142673351 + 160.8930312845682213562172j)</span>
-</pre></div>
-</div>
-<p>Arbitrarily large values of <span class="math">\(z\)</span> are supported:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">hyp1f1</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">10</span><span class="o">**</span><span class="mi">20</span><span class="p">)</span>
-<span class="go">3.890569218254486878220752e+43429448190325182745</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp1f1</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="o">**</span><span class="mi">20</span><span class="p">)</span>
-<span class="go">6.0e-60</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp1f1</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">10</span><span class="o">**</span><span class="mi">20</span><span class="o">*</span><span class="n">j</span><span class="p">)</span>
-<span class="go">(-1.935753855797342532571597e-20 - 2.291911213325184901239155e-20j)</span>
-</pre></div>
-</div>
-<p>Verifying the differential equation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">hyp1f1</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="n">chop</span><span class="p">(</span><span class="n">z</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">)</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>An integral representation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mf">1.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp1f1</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">2.269381460919952778587441</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="n">z</span><span class="o">*</span><span class="n">t</span><span class="p">)</span><span class="o">*</span><span class="n">t</span><span class="o">**</span><span class="p">(</span><span class="n">a</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">t</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gammaprod</span><span class="p">([</span><span class="n">b</span><span class="p">],[</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="p">])</span><span class="o">*</span><span class="n">quad</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">2.269381460919952778587441</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="hyp1f2">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">hyp1f2()</span></tt><a class="headerlink" href="#hyp1f2" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.hyp1f2">
-<tt class="descclassname">mpmath.</tt><tt class="descname">hyp1f2</tt><big>(</big><em>a1</em>, <em>b1</em>, <em>b2</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.hyp1f2" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the hypergeometric function <span class="math">\(\,_1F_2(a_1,a_2;b_1,b_2; z)\)</span>.
-The call <tt class="docutils literal"><span class="pre">hyp1f2(a1,b1,b2,z)</span></tt> is equivalent to
-<tt class="docutils literal"><span class="pre">hyper([a1],[b1,b2],z)</span></tt>.</p>
-<p>Evaluation works for complex and arbitrarily large arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="mf">1.5</span><span class="p">,</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="mf">2.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp1f2</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="mi">10</span><span class="o">**</span><span class="mi">20</span><span class="p">)</span>
-<span class="go">-1.159388148811981535941434e+8685889639</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp1f2</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="o">**</span><span class="mi">20</span><span class="p">)</span>
-<span class="go">-12.60262607892655945795907</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp1f2</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="mi">10</span><span class="o">**</span><span class="mi">20</span><span class="o">*</span><span class="n">j</span><span class="p">)</span>
-<span class="go">(4.237220401382240876065501e+6141851464 - 2.950930337531768015892987e+6141851464j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp1f2</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span> <span class="o">-</span><span class="mi">2j</span><span class="p">,</span> <span class="mf">0.5j</span><span class="p">,</span> <span class="mi">10</span><span class="o">-</span><span class="mi">20j</span><span class="p">)</span>
-<span class="go">(135881.9905586966432662004 - 86681.95885418079535738828j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="hyp2f0">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">hyp2f0()</span></tt><a class="headerlink" href="#hyp2f0" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.hyp2f0">
-<tt class="descclassname">mpmath.</tt><tt class="descname">hyp2f0</tt><big>(</big><em>a</em>, <em>b</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.hyp2f0" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the hypergeometric function <span class="math">\(\,_2F_0\)</span>, defined formally by the
-series</p>
-<div class="math">
-\[\,_2F_0(a,b;;z) = \sum_{n=0}^{\infty} (a)_n (b)_n \frac{z^n}{n!}.\]</div>
-<p>This series usually does not converge. For small enough <span class="math">\(z\)</span>, it can be viewed
-as an asymptotic series that may be summed directly with an appropriate
-truncation. When this is not the case, <a class="reference internal" href="#mpmath.hyp2f0" title="mpmath.hyp2f0"><tt class="xref py py-func docutils literal"><span class="pre">hyp2f0()</span></tt></a> gives a regularized sum,
-or equivalently, it uses a representation in terms of the
-hypergeometric U function [1]. The series also converges when either <span class="math">\(a\)</span> or <span class="math">\(b\)</span>
-is a nonpositive integer, as it then terminates into a polynomial
-after <span class="math">\(-a\)</span> or <span class="math">\(-b\)</span> terms.</p>
-<p><strong>Examples</strong></p>
-<p>Evaluation is supported for arbitrary complex arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f0</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="mf">1.25</span><span class="p">,</span> <span class="o">-</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">0.07095851870980052763312791</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f0</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="mf">1.25</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
-<span class="go">(-0.03254379032170590665041131 + 0.07269254613282301012735797j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f0</span><span class="p">(</span><span class="o">-</span><span class="mf">0.75</span><span class="p">,</span> <span class="mi">1</span><span class="o">-</span><span class="n">j</span><span class="p">,</span> <span class="mi">4j</span><span class="p">)</span>
-<span class="go">(-0.3579987031082732264862155 - 3.052951783922142735255881j)</span>
-</pre></div>
-</div>
-<p>Even with real arguments, the regularized value of 2F0 is often complex-valued,
-but the imaginary part decreases exponentially as <span class="math">\(z \to 0\)</span>. In the following
-example, the first call uses complex evaluation while the second has a small
-enough <span class="math">\(z\)</span> to evaluate using the direct series and thus the returned value
-is strictly real (this should be taken to indicate that the imaginary
-part is less than <tt class="docutils literal"><span class="pre">eps</span></tt>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f0</span><span class="p">(</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">)</span>
-<span class="go">(1.04166637647907 + 8.34584913683906e-8j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f0</span><span class="p">(</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.0005</span><span class="p">)</span>
-<span class="go">1.00037535207621</span>
-</pre></div>
-</div>
-<p>The imaginary part can be retrieved by increasing the working precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">80</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">hyp2f0</span><span class="p">(</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.009</span><span class="p">)</span><span class="o">.</span><span class="n">imag</span><span class="p">)</span>
-<span class="go">1.23828e-46</span>
-</pre></div>
-</div>
-<p>In the polynomial case (the series terminating), 2F0 can evaluate exactly:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f0</span><span class="p">(</span><span class="o">-</span><span class="mi">6</span><span class="p">,</span><span class="o">-</span><span class="mi">6</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">291793.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="n">hyp2f0</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mf">0.25</span><span class="p">))</span>
-<span class="go">&#39;(5/8)&#39;</span>
-</pre></div>
-</div>
-<p>The coefficients of the polynomials can be recovered using Taylor expansion:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">hyp2f0</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
-<span class="go">[1.0, -1.5, 2.25, -1.875, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">hyp2f0</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
-<span class="go">[1.0, -2.0, 4.5, -7.5, 6.5625, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]</span>
-</pre></div>
-</div>
-<p>[1] <a class="reference external" href="http://people.math.sfu.ca/~cbm/aands/page_504.htm">http://people.math.sfu.ca/~cbm/aands/page_504.htm</a></p>
-</dd></dl>
-
-</div>
-<div class="section" id="hyp2f1">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">hyp2f1()</span></tt><a class="headerlink" href="#hyp2f1" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.hyp2f1">
-<tt class="descclassname">mpmath.</tt><tt class="descname">hyp2f1</tt><big>(</big><em>a</em>, <em>b</em>, <em>c</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.hyp2f1" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the Gauss hypergeometric function <span class="math">\(\,_2F_1\)</span> (often simply referred to as
-<em>the</em> hypergeometric function), defined for <span class="math">\(|z| &lt; 1\)</span> as</p>
-<div class="math">
-\[\,_2F_1(a,b,c,z) = \sum_{k=0}^{\infty}
-    \frac{(a)_k (b)_k}{(c)_k} \frac{z^k}{k!}.\]</div>
-<p>and for <span class="math">\(|z| \ge 1\)</span> by analytic continuation, with a branch cut on <span class="math">\((1, \infty)\)</span>
-when necessary.</p>
-<p>Special cases of this function include many of the orthogonal polynomials as
-well as the incomplete beta function and other functions. Properties of the
-Gauss hypergeometric function are documented comprehensively in many references,
-for example Abramowitz &amp; Stegun, section 15.</p>
-<p>The implementation supports the analytic continuation as well as evaluation
-close to the unit circle where <span class="math">\(|z| \approx 1\)</span>. The syntax <tt class="docutils literal"><span class="pre">hyp2f1(a,b,c,z)</span></tt>
-is equivalent to <tt class="docutils literal"><span class="pre">hyper([a,b],[c],z)</span></tt>.</p>
-<p><strong>Examples</strong></p>
-<p>Evaluation with <span class="math">\(z\)</span> inside, outside and on the unit circle, for
-fixed parameters:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f1</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="mi">4</span><span class="p">,</span> <span class="mf">0.75</span><span class="p">)</span>
-<span class="go">1.303703703703703703703704</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f1</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="mi">4</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.75</span><span class="p">)</span>
-<span class="go">0.7431290566046919177853916</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f1</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="mi">4</span><span class="p">,</span> <span class="mf">1.75</span><span class="p">)</span>
-<span class="go">(1.418075801749271137026239 - 1.114976146679907015775102j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f1</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">1.6</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f1</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="mi">4</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.8235498012182875315037882</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f1</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="mi">4</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
-<span class="go">(0.9144026291433065674259078 + 0.2050415770437884900574923j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f1</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="n">j</span><span class="p">)</span>
-<span class="go">(0.9274013540258103029011549 + 0.7455257875808100868984496j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f1</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="mi">4</span><span class="p">,</span> <span class="mf">0.25j</span><span class="p">)</span>
-<span class="go">(0.9931169055799728251931672 + 0.06154836525312066938147793j)</span>
-</pre></div>
-</div>
-<p>Evaluation with complex parameter values:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f1</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mf">0.75</span><span class="p">,</span> <span class="mi">10j</span><span class="p">,</span> <span class="mi">1</span><span class="o">+</span><span class="mi">5j</span><span class="p">)</span>
-<span class="go">(0.8834833319713479923389638 + 0.7053886880648105068343509j)</span>
-</pre></div>
-</div>
-<p>Evaluation with <span class="math">\(z = 1\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f1</span><span class="p">(</span><span class="o">-</span><span class="mf">2.5</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f1</span><span class="p">(</span><span class="o">-</span><span class="mf">2.5</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">0.06926406926406926406926407</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f1</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">+inf</span>
-</pre></div>
-</div>
-<p>Evaluation for huge arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f1</span><span class="p">((</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="mf">1.75</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="s">&#39;1e100&#39;</span><span class="p">)</span>
-<span class="go">(7.883714220959876246415651e+32 + 1.365499358305579597618785e+33j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f1</span><span class="p">((</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="mf">1.75</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="s">&#39;1e1000000&#39;</span><span class="p">)</span>
-<span class="go">(7.883714220959876246415651e+333332 + 1.365499358305579597618785e+333333j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f1</span><span class="p">((</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="mf">1.75</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="s">&#39;1e1000000j&#39;</span><span class="p">)</span>
-<span class="go">(1.365499358305579597618785e+333333 - 7.883714220959876246415651e+333332j)</span>
-</pre></div>
-</div>
-<p>An integral representation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">c</span><span class="p">,</span><span class="n">z</span> <span class="o">=</span> <span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mf">0.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">t</span><span class="o">**</span><span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">t</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">c</span><span class="o">-</span><span class="n">b</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">t</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gammaprod</span><span class="p">([</span><span class="n">c</span><span class="p">],[</span><span class="n">b</span><span class="p">,</span><span class="n">c</span><span class="o">-</span><span class="n">b</span><span class="p">])</span> <span class="o">*</span> <span class="n">quad</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">0.9480458814362824478852618</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f1</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">c</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.9480458814362824478852618</span>
-</pre></div>
-</div>
-<p>Verifying the hypergeometric differential equation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">hyp2f1</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">c</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">z</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">c</span><span class="o">-</span><span class="p">(</span><span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">)</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="n">b</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="hyp2f2">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">hyp2f2()</span></tt><a class="headerlink" href="#hyp2f2" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.hyp2f2">
-<tt class="descclassname">mpmath.</tt><tt class="descname">hyp2f2</tt><big>(</big><em>a1</em>, <em>a2</em>, <em>b1</em>, <em>b2</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.hyp2f2" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the hypergeometric function <span class="math">\(\,_2F_2(a_1,a_2;b_1,b_2; z)\)</span>.
-The call <tt class="docutils literal"><span class="pre">hyp2f2(a1,a2,b1,b2,z)</span></tt> is equivalent to
-<tt class="docutils literal"><span class="pre">hyper([a1,a2],[b1,b2],z)</span></tt>.</p>
-<p>Evaluation works for complex and arbitrarily large arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="mf">1.5</span><span class="p">,</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="mf">2.25</span><span class="p">,</span> <span class="mi">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f2</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="mi">10</span><span class="o">**</span><span class="mi">20</span><span class="p">)</span>
-<span class="go">-5.275758229007902299823821e+43429448190325182663</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f2</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="o">**</span><span class="mi">20</span><span class="p">)</span>
-<span class="go">2561445.079983207701073448</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f2</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="mi">10</span><span class="o">**</span><span class="mi">20</span><span class="o">*</span><span class="n">j</span><span class="p">)</span>
-<span class="go">(2218276.509664121194836667 - 1280722.539991603850462856j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f2</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span> <span class="o">-</span><span class="mi">2j</span><span class="p">,</span> <span class="mf">0.5j</span><span class="p">,</span> <span class="mi">4j</span><span class="p">,</span> <span class="mi">10</span><span class="o">-</span><span class="mi">20j</span><span class="p">)</span>
-<span class="go">(80500.68321405666957342788 - 20346.82752982813540993502j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="hyp2f3">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">hyp2f3()</span></tt><a class="headerlink" href="#hyp2f3" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.hyp2f3">
-<tt class="descclassname">mpmath.</tt><tt class="descname">hyp2f3</tt><big>(</big><em>a1</em>, <em>a2</em>, <em>b1</em>, <em>b2</em>, <em>b3</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.hyp2f3" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the hypergeometric function <span class="math">\(\,_2F_3(a_1,a_2;b_1,b_2,b_3; z)\)</span>.
-The call <tt class="docutils literal"><span class="pre">hyp2f3(a1,a2,b1,b2,b3,z)</span></tt> is equivalent to
-<tt class="docutils literal"><span class="pre">hyper([a1,a2],[b1,b2,b3],z)</span></tt>.</p>
-<p>Evaluation works for arbitrarily large arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a1</span><span class="p">,</span><span class="n">a2</span><span class="p">,</span><span class="n">b1</span><span class="p">,</span><span class="n">b2</span><span class="p">,</span><span class="n">b3</span> <span class="o">=</span> <span class="mf">1.5</span><span class="p">,</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="mf">2.25</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f3</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span><span class="n">a2</span><span class="p">,</span><span class="n">b1</span><span class="p">,</span><span class="n">b2</span><span class="p">,</span><span class="n">b3</span><span class="p">,</span><span class="mi">10</span><span class="o">**</span><span class="mi">20</span><span class="p">)</span>
-<span class="go">-4.169178177065714963568963e+8685889590</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f3</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span><span class="n">a2</span><span class="p">,</span><span class="n">b1</span><span class="p">,</span><span class="n">b2</span><span class="p">,</span><span class="n">b3</span><span class="p">,</span><span class="o">-</span><span class="mi">10</span><span class="o">**</span><span class="mi">20</span><span class="p">)</span>
-<span class="go">7064472.587757755088178629</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f3</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span><span class="n">a2</span><span class="p">,</span><span class="n">b1</span><span class="p">,</span><span class="n">b2</span><span class="p">,</span><span class="n">b3</span><span class="p">,</span><span class="mi">10</span><span class="o">**</span><span class="mi">20</span><span class="o">*</span><span class="n">j</span><span class="p">)</span>
-<span class="go">(-5.163368465314934589818543e+6141851415 + 1.783578125755972803440364e+6141851416j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f3</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span> <span class="o">-</span><span class="mi">2j</span><span class="p">,</span> <span class="mf">0.5j</span><span class="p">,</span> <span class="mi">4j</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="o">-</span><span class="n">j</span><span class="p">,</span> <span class="mi">10</span><span class="o">-</span><span class="mi">20j</span><span class="p">)</span>
-<span class="go">(-2280.938956687033150740228 + 13620.97336609573659199632j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f3</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span> <span class="o">-</span><span class="mi">2j</span><span class="p">,</span> <span class="mf">0.5j</span><span class="p">,</span> <span class="mi">4j</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="o">-</span><span class="n">j</span><span class="p">,</span> <span class="mi">10000000</span><span class="o">-</span><span class="mi">20000000j</span><span class="p">)</span>
-<span class="go">(4.849835186175096516193e+3504 - 3.365981529122220091353633e+3504j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="hyp3f2">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">hyp3f2()</span></tt><a class="headerlink" href="#hyp3f2" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.hyp3f2">
-<tt class="descclassname">mpmath.</tt><tt class="descname">hyp3f2</tt><big>(</big><em>a1</em>, <em>a2</em>, <em>a3</em>, <em>b1</em>, <em>b2</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.hyp3f2" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the generalized hypergeometric function <span class="math">\(\,_3F_2\)</span>, defined for <span class="math">\(|z| &lt; 1\)</span>
-as</p>
-<div class="math">
-\[\,_3F_2(a_1,a_2,a_3,b_1,b_2,z) = \sum_{k=0}^{\infty}
-    \frac{(a_1)_k (a_2)_k (a_3)_k}{(b_1)_k (b_2)_k} \frac{z^k}{k!}.\]</div>
-<p>and for <span class="math">\(|z| \ge 1\)</span> by analytic continuation. The analytic structure of this
-function is similar to that of <span class="math">\(\,_2F_1\)</span>, generally with a singularity at
-<span class="math">\(z = 1\)</span> and a branch cut on <span class="math">\((1, \infty)\)</span>.</p>
-<p>Evaluation is supported inside, on, and outside
-the circle of convergence <span class="math">\(|z| = 1\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp3f2</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mf">0.25</span><span class="p">)</span>
-<span class="go">1.083533123380934241548707</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp3f2</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="o">+</span><span class="mi">2j</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="o">-</span><span class="mi">10</span><span class="o">+</span><span class="mi">10j</span><span class="p">)</span>
-<span class="go">(0.1574651066006004632914361 - 0.03194209021885226400892963j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp3f2</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="o">-</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">0.3071141169208772603266489</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp3f2</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">(-0.4857045320523947050581423 - 0.5988311440454888436888028j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp3f2</span><span class="p">(</span><span class="mf">0.25</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mf">1.5</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1.157370995096772047567631</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">8</span><span class="o">-</span><span class="n">pi</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">ln2</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span>
-<span class="go">1.157370995096772047567631</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp3f2</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span><span class="mf">0.5j</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">2j</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(1.74518490615029486475959 + 0.1454701525056682297614029j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp3f2</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span><span class="mf">0.5j</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">2j</span><span class="p">,</span><span class="n">sqrt</span><span class="p">(</span><span class="n">j</span><span class="p">))</span>
-<span class="go">(0.9829816481834277511138055 - 0.4059040020276937085081127j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp3f2</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1.41</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp3f2</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">2.12</span>
-</pre></div>
-</div>
-<p>Evaluation very close to the unit circle:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">hyp3f2</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="s">&#39;1.0001&#39;</span><span class="p">)</span>
-<span class="go">(1.564877796743282766872279 - 3.76821518787438186031973e-11j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp3f2</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="s">&#39;1+0.0001j&#39;</span><span class="p">)</span>
-<span class="go">(1.564747153061671573212831 + 0.0001305757570366084557648482j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp3f2</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="s">&#39;0.9999&#39;</span><span class="p">)</span>
-<span class="go">1.564616644881686134983664</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp3f2</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="s">&#39;-0.9999&#39;</span><span class="p">)</span>
-<span class="go">0.7823896253461678060196207</span>
-</pre></div>
-</div>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">Evaluation for <span class="math">\(|z-1|\)</span> small can currently be inaccurate or slow
-for some parameter combinations.</p>
-</div>
-<p>For various parameter combinations, <span class="math">\(\,_3F_2\)</span> admits representation in terms
-of hypergeometric functions of lower degree, or in terms of
-simpler functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">z</span> <span class="ow">in</span> <span class="p">[(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mi">1</span><span class="p">)]:</span>
-<span class="gp">... </span>    <span class="n">hyp2f1</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="o">+</span><span class="mf">0.5</span><span class="p">,</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="gp">... </span>    <span class="n">hyp3f2</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">,</span><span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="p">,</span><span class="mi">2</span><span class="o">*</span><span class="n">b</span><span class="p">,</span><span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="o">+</span><span class="mf">0.5</span><span class="p">,</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">b</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="go">0.4246104461966439006086308</span>
-<span class="go">0.4246104461966439006086308</span>
-<span class="go">7.111111111111111111111111</span>
-<span class="go">7.111111111111111111111111</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mi">2</span><span class="o">+</span><span class="mi">3j</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp3f2</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mf">1.5</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">(0.7621440939243342419729144 + 0.4249117735058037649915723j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">4</span><span class="o">*</span><span class="p">(</span><span class="n">pi</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">ellipe</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
-<span class="go">(0.7621440939243342419729144 + 0.4249117735058037649915723j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="generalized-hypergeometric-functions">
-<h2>Generalized hypergeometric functions<a class="headerlink" href="#generalized-hypergeometric-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="hyper">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">hyper()</span></tt><a class="headerlink" href="#hyper" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.hyper">
-<tt class="descclassname">mpmath.</tt><tt class="descname">hyper</tt><big>(</big><em>a_s</em>, <em>b_s</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.hyper" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the generalized hypergeometric function</p>
-<div class="math">
-\[\,_pF_q(a_1,\ldots,a_p; b_1,\ldots,b_q; z) =
-\sum_{n=0}^\infty \frac{(a_1)_n (a_2)_n \ldots (a_p)_n}
-   {(b_1)_n(b_2)_n\ldots(b_q)_n} \frac{z^n}{n!}\]</div>
-<p>where <span class="math">\((x)_n\)</span> denotes the rising factorial (see <a class="reference internal" href="../functions/gamma.html#mpmath.rf" title="mpmath.rf"><tt class="xref py py-func docutils literal"><span class="pre">rf()</span></tt></a>).</p>
-<p>The parameters lists <tt class="docutils literal"><span class="pre">a_s</span></tt> and <tt class="docutils literal"><span class="pre">b_s</span></tt> may contain integers,
-real numbers, complex numbers, as well as exact fractions given in
-the form of tuples <span class="math">\((p, q)\)</span>. <a class="reference internal" href="#mpmath.hyper" title="mpmath.hyper"><tt class="xref py py-func docutils literal"><span class="pre">hyper()</span></tt></a> is optimized to handle
-integers and fractions more efficiently than arbitrary
-floating-point parameters (since rational parameters are by
-far the most common).</p>
-<p><strong>Examples</strong></p>
-<p>Verifying that <a class="reference internal" href="#mpmath.hyper" title="mpmath.hyper"><tt class="xref py py-func docutils literal"><span class="pre">hyper()</span></tt></a> gives the sum in the definition, by
-comparison with <a class="reference internal" href="../../mpmath/calculus/sums_limits.html#mpmath.nsum" title="mpmath.nsum"><tt class="xref py py-func docutils literal"><span class="pre">nsum()</span></tt></a>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">c</span><span class="p">,</span><span class="n">d</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="mf">0.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">],[</span><span class="n">c</span><span class="p">,</span><span class="n">d</span><span class="p">],</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1.078903941164934876086237</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fn</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="n">rf</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">rf</span><span class="p">(</span><span class="n">b</span><span class="p">,</span><span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">rf</span><span class="p">(</span><span class="n">c</span><span class="p">,</span><span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">rf</span><span class="p">(</span><span class="n">d</span><span class="p">,</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="n">fn</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">1.078903941164934876086237</span>
-</pre></div>
-</div>
-<p>The parameters can be any combination of integers, fractions,
-floats and complex numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="n">pi</span><span class="p">,</span> <span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">,</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="mf">0.2j</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">],[</span><span class="n">c</span><span class="p">,</span><span class="n">d</span><span class="p">,</span><span class="n">e</span><span class="p">],</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(0.9923571616434024810831887 - 0.005753848733883879742993122j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fn</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="n">rf</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">rf</span><span class="p">(</span><span class="n">b</span><span class="p">,</span><span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">rf</span><span class="p">(</span><span class="n">c</span><span class="p">,</span><span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">rf</span><span class="p">(</span><span class="n">d</span><span class="p">,</span><span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">rf</span><span class="p">(</span><span class="n">e</span><span class="p">,</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="n">fn</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">(0.9923571616434024810831887 - 0.005753848733883879742993122j)</span>
-</pre></div>
-</div>
-<p>The <span class="math">\(\,_0F_0\)</span> and <span class="math">\(\,_1F_0\)</span> series are just elementary functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="o">+</span><span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([],[],</span><span class="n">z</span><span class="p">)</span>
-<span class="go">23.14069263277926900572909</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">23.14069263277926900572909</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="n">a</span><span class="p">],[],</span><span class="n">z</span><span class="p">)</span>
-<span class="go">(-0.09069132879922920160334114 + 0.3283224323946162083579656j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="p">)</span>
-<span class="go">(-0.09069132879922920160334114 + 0.3283224323946162083579656j)</span>
-</pre></div>
-</div>
-<p>If any <span class="math">\(a_k\)</span> coefficient is a nonpositive integer, the series terminates
-into a finite polynomial:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">],[</span><span class="mi">2</span><span class="p">,</span><span class="mi">5</span><span class="p">],</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.7904761904761904761904762</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="n">_</span><span class="p">)</span>
-<span class="go">&#39;(83/105)&#39;</span>
-</pre></div>
-</div>
-<p>If any <span class="math">\(b_k\)</span> is a nonpositive integer, the function is undefined (unless the
-series terminates before the division by zero occurs):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">],[</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="mi">5</span><span class="p">],</span><span class="mi">1</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-  <span class="c">...</span>
-<span class="gr">ZeroDivisionError</span>: <span class="n">pole in hypergeometric series</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">],[</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="mi">5</span><span class="p">],</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1.1</span>
-</pre></div>
-</div>
-<p>Except for polynomial cases, the radius of convergence <span class="math">\(R\)</span> of the hypergeometric
-series is either <span class="math">\(R = \infty\)</span> (if <span class="math">\(p \le q\)</span>), <span class="math">\(R = 1\)</span> (if <span class="math">\(p = q+1\)</span>), or
-<span class="math">\(R = 0\)</span> (if <span class="math">\(p &gt; q+1\)</span>).</p>
-<p>The analytic continuations of the functions with <span class="math">\(p = q+1\)</span>, i.e. <span class="math">\(\,_2F_1\)</span>,
-<span class="math">\(\,_3F_2\)</span>,  <span class="math">\(\,_4F_3\)</span>, etc, are all implemented and therefore these functions
-can be evaluated for <span class="math">\(|z| \ge 1\)</span>. The shortcuts <a class="reference internal" href="#mpmath.hyp2f1" title="mpmath.hyp2f1"><tt class="xref py py-func docutils literal"><span class="pre">hyp2f1()</span></tt></a>, <a class="reference internal" href="#mpmath.hyp3f2" title="mpmath.hyp3f2"><tt class="xref py py-func docutils literal"><span class="pre">hyp3f2()</span></tt></a>
-are available to handle the most common cases (see their documentation),
-but functions of higher degree are also supported via <a class="reference internal" href="#mpmath.hyper" title="mpmath.hyper"><tt class="xref py py-func docutils literal"><span class="pre">hyper()</span></tt></a>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">],</span> <span class="mi">1</span><span class="p">)</span>   <span class="c"># 4F3 at finite-valued branch point</span>
-<span class="go">1.141783505526870731311423</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">],</span> <span class="mi">1</span><span class="p">)</span>   <span class="c"># 4F3 at pole</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">],</span> <span class="p">[</span><span class="mi">6</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">9</span><span class="p">],</span> <span class="mi">10</span><span class="p">)</span>    <span class="c"># 5F4</span>
-<span class="go">(1.543998916527972259717257 - 0.5876309929580408028816365j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="mi">11</span><span class="p">],</span> <span class="mi">1j</span><span class="p">)</span>   <span class="c"># 6F5</span>
-<span class="go">(0.9996565821853579063502466 + 0.0129721075905630604445669j)</span>
-</pre></div>
-</div>
-<p>Near <span class="math">\(z = 1\)</span> with noninteger parameters:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="s">&#39;1/3&#39;</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="s">&#39;3/2&#39;</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="s">&#39;1/5&#39;</span><span class="p">,</span><span class="s">&#39;11/6&#39;</span><span class="p">,</span><span class="s">&#39;41/8&#39;</span><span class="p">],</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">2.219433352235586121250027</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="s">&#39;1/3&#39;</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="s">&#39;3/2&#39;</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="s">&#39;1/5&#39;</span><span class="p">,</span><span class="s">&#39;11/6&#39;</span><span class="p">,</span><span class="s">&#39;5/4&#39;</span><span class="p">],</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eps1</span> <span class="o">=</span> <span class="n">extradps</span><span class="p">(</span><span class="mi">6</span><span class="p">)(</span><span class="k">lambda</span><span class="p">:</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">mpf</span><span class="p">(</span><span class="s">&#39;1e-6&#39;</span><span class="p">))()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="s">&#39;1/3&#39;</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="s">&#39;3/2&#39;</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="s">&#39;1/5&#39;</span><span class="p">,</span><span class="s">&#39;11/6&#39;</span><span class="p">,</span><span class="s">&#39;5/4&#39;</span><span class="p">],</span> <span class="n">eps1</span><span class="p">)</span>
-<span class="go">2923978034.412973409330956</span>
-</pre></div>
-</div>
-<p>Please note that, as currently implemented, evaluation of <span class="math">\(\,_pF_{p-1}\)</span>
-with <span class="math">\(p \ge 3\)</span> may be slow or inaccurate when <span class="math">\(|z-1|\)</span> is small,
-for some parameter values.</p>
-<p>When <span class="math">\(p &gt; q+1\)</span>, <tt class="docutils literal"><span class="pre">hyper</span></tt> computes the (iterated) Borel sum of the divergent
-series. For <span class="math">\(\,_2F_0\)</span> the Borel sum has an analytic solution and can be
-computed efficiently (see <a class="reference internal" href="#mpmath.hyp2f0" title="mpmath.hyp2f0"><tt class="xref py py-func docutils literal"><span class="pre">hyp2f0()</span></tt></a>). For higher degrees, the functions
-is evaluated first by attempting to sum it directly as an asymptotic
-series (this only works for tiny <span class="math">\(|z|\)</span>), and then by evaluating the Borel
-regularized sum using numerical integration. Except for
-special parameter combinations, this can be extremely slow.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="p">[],</span> <span class="mf">0.5</span><span class="p">)</span>          <span class="c"># regularization of 2F0</span>
-<span class="go">(1.340965419580146562086448 + 0.8503366631752726568782447j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="mf">0.5</span><span class="p">)</span>     <span class="c"># regularization of 4F1</span>
-<span class="go">(1.108287213689475145830699 + 0.5327107430640678181200491j)</span>
-</pre></div>
-</div>
-<p>With the following magnitude of argument, the asymptotic series for <span class="math">\(\,_3F_1\)</span>
-gives only a few digits. Using Borel summation, <tt class="docutils literal"><span class="pre">hyper</span></tt> can produce
-a value with full accuracy:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mf">5.25</span><span class="p">],</span> <span class="s">&#39;0.08&#39;</span><span class="p">,</span> <span class="n">force_series</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-  <span class="c">...</span>
-<span class="gr">NoConvergence</span>: <span class="n">Hypergeometric series converges too slowly. Try increasing maxterms.</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mf">5.25</span><span class="p">],</span> <span class="s">&#39;0.08&#39;</span><span class="p">,</span> <span class="n">asymp_tol</span><span class="o">=</span><span class="mf">1e-4</span><span class="p">)</span>
-<span class="go">1.0725535790737</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mf">5.25</span><span class="p">],</span> <span class="s">&#39;0.08&#39;</span><span class="p">)</span>
-<span class="go">(1.07269542893559 + 5.54668863216891e-5j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mf">5.25</span><span class="p">],</span> <span class="s">&#39;-0.08&#39;</span><span class="p">,</span> <span class="n">asymp_tol</span><span class="o">=</span><span class="mf">1e-4</span><span class="p">)</span>
-<span class="go">0.946344925484879</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mf">5.25</span><span class="p">],</span> <span class="s">&#39;-0.08&#39;</span><span class="p">)</span>
-<span class="go">0.946312503737771</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mf">5.25</span><span class="p">],</span> <span class="s">&#39;-0.08&#39;</span><span class="p">)</span>
-<span class="go">0.9463125037377662296700858</span>
-</pre></div>
-</div>
-<p>Note that with the positive <span class="math">\(z\)</span> value, there is a complex part in the
-correct result, which falls below the tolerance of the asymptotic series.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="hypercomb">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">hypercomb()</span></tt><a class="headerlink" href="#hypercomb" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.hypercomb">
-<tt class="descclassname">mpmath.</tt><tt class="descname">hypercomb</tt><big>(</big><em>ctx</em>, <em>function</em>, <em>params=</em><span class="optional">[</span><span class="optional">]</span>, <em>discard_known_zeros=True</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.hypercomb" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes a weighted combination of hypergeometric functions</p>
-<div class="math">
-\[\sum_{r=1}^N \left[ \prod_{k=1}^{l_r} {w_{r,k}}^{c_{r,k}}
-\frac{\prod_{k=1}^{m_r} \Gamma(\alpha_{r,k})}{\prod_{k=1}^{n_r}
-\Gamma(\beta_{r,k})}
-\,_{p_r}F_{q_r}(a_{r,1},\ldots,a_{r,p}; b_{r,1},
-\ldots, b_{r,q}; z_r)\right].\]</div>
-<p>Typically the parameters are linear combinations of a small set of base
-parameters; <a class="reference internal" href="#mpmath.hypercomb" title="mpmath.hypercomb"><tt class="xref py py-func docutils literal"><span class="pre">hypercomb()</span></tt></a> permits computing a correct value in
-the case that some of the <span class="math">\(\alpha\)</span>, <span class="math">\(\beta\)</span>, <span class="math">\(b\)</span> turn out to be
-nonpositive integers, or if division by zero occurs for some <span class="math">\(w^c\)</span>,
-assuming that there are opposing singularities that cancel out.
-The limit is computed by evaluating the function with the base
-parameters perturbed, at a higher working precision.</p>
-<p>The first argument should be a function that takes the perturbable
-base parameters <tt class="docutils literal"><span class="pre">params</span></tt> as input and returns <span class="math">\(N\)</span> tuples
-<tt class="docutils literal"><span class="pre">(w,</span> <span class="pre">c,</span> <span class="pre">alpha,</span> <span class="pre">beta,</span> <span class="pre">a,</span> <span class="pre">b,</span> <span class="pre">z)</span></tt>, where the coefficients <tt class="docutils literal"><span class="pre">w</span></tt>, <tt class="docutils literal"><span class="pre">c</span></tt>,
-gamma factors <tt class="docutils literal"><span class="pre">alpha</span></tt>, <tt class="docutils literal"><span class="pre">beta</span></tt>, and hypergeometric coefficients
-<tt class="docutils literal"><span class="pre">a</span></tt>, <tt class="docutils literal"><span class="pre">b</span></tt> each should be lists of numbers, and <tt class="docutils literal"><span class="pre">z</span></tt> should be a single
-number.</p>
-<p><strong>Examples</strong></p>
-<p>The following evaluates</p>
-<div class="math">
-\[(a-1) \frac{\Gamma(a-3)}{\Gamma(a-4)} \,_1F_1(a,a-1,z) = e^z(a-4)(a+z-1)\]</div>
-<p>with <span class="math">\(a=1, z=3\)</span>. There is a zero factor, two gamma function poles, and
-the 1F1 function is singular; all singularities cancel out to give a finite
-value:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hypercomb</span><span class="p">(</span><span class="k">lambda</span> <span class="n">a</span><span class="p">:</span> <span class="p">[([</span><span class="n">a</span><span class="o">-</span><span class="mi">1</span><span class="p">],[</span><span class="mi">1</span><span class="p">],[</span><span class="n">a</span><span class="o">-</span><span class="mi">3</span><span class="p">],[</span><span class="n">a</span><span class="o">-</span><span class="mi">4</span><span class="p">],[</span><span class="n">a</span><span class="p">],[</span><span class="n">a</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span><span class="mi">3</span><span class="p">)],</span> <span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">-180.769832308689</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="mi">9</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">-180.769832308689</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="meijer-g-function">
-<h2>Meijer G-function<a class="headerlink" href="#meijer-g-function" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="meijerg">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">meijerg()</span></tt><a class="headerlink" href="#meijerg" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.meijerg">
-<tt class="descclassname">mpmath.</tt><tt class="descname">meijerg</tt><big>(</big><em>a_s</em>, <em>b_s</em>, <em>z</em>, <em>r=1</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.meijerg" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the Meijer G-function, defined as</p>
-<div class="math">
-\[\begin{split}G^{m,n}_{p,q} \left( \left. \begin{matrix}
-     a_1, \dots, a_n ; a_{n+1} \dots a_p \\
-     b_1, \dots, b_m ; b_{m+1} \dots b_q
-\end{matrix}\; \right| \; z ; r \right) =
-\frac{1}{2 \pi i} \int_L
-\frac{\prod_{j=1}^m \Gamma(b_j+s) \prod_{j=1}^n\Gamma(1-a_j-s)}
-     {\prod_{j=n+1}^{p}\Gamma(a_j+s) \prod_{j=m+1}^q \Gamma(1-b_j-s)}
-     z^{-s/r} ds\end{split}\]</div>
-<p>for an appropriate choice of the contour <span class="math">\(L\)</span> (see references).</p>
-<p>There are <span class="math">\(p\)</span> elements <span class="math">\(a_j\)</span>.
-The argument <em>a_s</em> should be a pair of lists, the first containing the
-<span class="math">\(n\)</span> elements <span class="math">\(a_1, \ldots, a_n\)</span> and the second containing
-the <span class="math">\(p-n\)</span> elements <span class="math">\(a_{n+1}, \ldots a_p\)</span>.</p>
-<p>There are <span class="math">\(q\)</span> elements <span class="math">\(b_j\)</span>.
-The argument <em>b_s</em> should be a pair of lists, the first containing the
-<span class="math">\(m\)</span> elements <span class="math">\(b_1, \ldots, b_m\)</span> and the second containing
-the <span class="math">\(q-m\)</span> elements <span class="math">\(b_{m+1}, \ldots b_q\)</span>.</p>
-<p>The implicit tuple <span class="math">\((m, n, p, q)\)</span> constitutes the order or degree of the
-Meijer G-function, and is determined by the lengths of the coefficient
-vectors. Confusingly, the indices in this tuple appear in a different order
-from the coefficients, but this notation is standard. The many examples
-given below should hopefully clear up any potential confusion.</p>
-<p><strong>Algorithm</strong></p>
-<p>The Meijer G-function is evaluated as a combination of hypergeometric series.
-There are two versions of the function, which can be selected with
-the optional <em>series</em> argument.</p>
-<p><em>series=1</em> uses a sum of <span class="math">\(m\)</span> <span class="math">\(\,_pF_{q-1}\)</span> functions of <span class="math">\(z\)</span></p>
-<p><em>series=2</em> uses a sum of <span class="math">\(n\)</span> <span class="math">\(\,_qF_{p-1}\)</span> functions of <span class="math">\(1/z\)</span></p>
-<p>The default series is chosen based on the degree and <span class="math">\(|z|\)</span> in order
-to be consistent with Mathematica&#8217;s. This definition of the Meijer G-function
-has a discontinuity at <span class="math">\(|z| = 1\)</span> for some orders, which can
-be avoided by explicitly specifying a series.</p>
-<p>Keyword arguments are forwarded to <a class="reference internal" href="#mpmath.hypercomb" title="mpmath.hypercomb"><tt class="xref py py-func docutils literal"><span class="pre">hypercomb()</span></tt></a>.</p>
-<p><strong>Examples</strong></p>
-<p>Many standard functions are special cases of the Meijer G-function
-(possibly rescaled and/or with branch cut corrections). We define
-some test parameters:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">0.75</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">1.5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">2.25</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>The exponential function:
-<span class="math">\(e^z = G^{1,0}_{0,1} \left( \left. \begin{matrix} - \\ 0 \end{matrix} \;
-\right| \; -z \right)\)</span></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">meijerg</span><span class="p">([[],[]],</span> <span class="p">[[</span><span class="mi">0</span><span class="p">],[]],</span> <span class="o">-</span><span class="n">z</span><span class="p">)</span>
-<span class="go">9.487735836358525720550369</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">9.487735836358525720550369</span>
-</pre></div>
-</div>
-<p>The natural logarithm:
-<span class="math">\(\log(1+z) = G^{1,2}_{2,2} \left( \left. \begin{matrix} 1, 1 \\ 1, 0
-\end{matrix} \; \right| \; -z \right)\)</span></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">meijerg</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],[]],</span> <span class="p">[[</span><span class="mi">1</span><span class="p">],[</span><span class="mi">0</span><span class="p">]],</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">1.178654996341646117219023</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">z</span><span class="p">)</span>
-<span class="go">1.178654996341646117219023</span>
-</pre></div>
-</div>
-<p>A rational function:
-<span class="math">\(\frac{z}{z+1} = G^{1,2}_{2,2} \left( \left. \begin{matrix} 1, 1 \\ 1, 1
-\end{matrix} \; \right| \; z \right)\)</span></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">meijerg</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],[]],</span> <span class="p">[[</span><span class="mi">1</span><span class="p">],[</span><span class="mi">1</span><span class="p">]],</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">0.6923076923076923076923077</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span><span class="o">/</span><span class="p">(</span><span class="n">z</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.6923076923076923076923077</span>
-</pre></div>
-</div>
-<p>The sine and cosine functions:</p>
-<p><span class="math">\(\frac{1}{\sqrt \pi} \sin(2 \sqrt z) = G^{1,0}_{0,2} \left( \left. \begin{matrix}
-- \\ \frac{1}{2}, 0 \end{matrix} \; \right| \; z \right)\)</span></p>
-<p><span class="math">\(\frac{1}{\sqrt \pi} \cos(2 \sqrt z) = G^{1,0}_{0,2} \left( \left. \begin{matrix}
-- \\ 0, \frac{1}{2} \end{matrix} \; \right| \; z \right)\)</span></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">meijerg</span><span class="p">([[],[]],</span> <span class="p">[[</span><span class="mf">0.5</span><span class="p">],[</span><span class="mi">0</span><span class="p">]],</span> <span class="p">(</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.4389807929218676682296453</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">0.4389807929218676682296453</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">meijerg</span><span class="p">([[],[]],</span> <span class="p">[[</span><span class="mi">0</span><span class="p">],[</span><span class="mf">0.5</span><span class="p">]],</span> <span class="p">(</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">-0.3544090145996275423331762</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">-0.3544090145996275423331762</span>
-</pre></div>
-</div>
-<p>Bessel functions:</p>
-<p><span class="math">\(J_a(2 \sqrt z) = G^{1,0}_{0,2} \left( \left.
-\begin{matrix} - \\ \frac{a}{2}, -\frac{a}{2}
-\end{matrix} \; \right| \; z \right)\)</span></p>
-<p><span class="math">\(Y_a(2 \sqrt z) = G^{2,0}_{1,3} \left( \left.
-\begin{matrix} \frac{-a-1}{2} \\ \frac{a}{2}, -\frac{a}{2}, \frac{-a-1}{2}
-\end{matrix} \; \right| \; z \right)\)</span></p>
-<p><span class="math">\((-z)^{a/2} z^{-a/2} I_a(2 \sqrt z) = G^{1,0}_{0,2} \left( \left.
-\begin{matrix} - \\ \frac{a}{2}, -\frac{a}{2}
-\end{matrix} \; \right| \; -z \right)\)</span></p>
-<p><span class="math">\(2 K_a(2 \sqrt z) = G^{2,0}_{0,2} \left( \left.
-\begin{matrix} - \\ \frac{a}{2}, -\frac{a}{2}
-\end{matrix} \; \right| \; z \right)\)</span></p>
-<p>As the example with the Bessel <em>I</em> function shows, a branch
-factor is required for some arguments when inverting the square root.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">meijerg</span><span class="p">([[],[]],</span> <span class="p">[[</span><span class="n">a</span><span class="o">/</span><span class="mi">2</span><span class="p">],[</span><span class="o">-</span><span class="n">a</span><span class="o">/</span><span class="mi">2</span><span class="p">]],</span> <span class="p">(</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.5059425789597154858527264</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselj</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.5059425789597154858527264</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">meijerg</span><span class="p">([[],[(</span><span class="o">-</span><span class="n">a</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">]],</span> <span class="p">[[</span><span class="n">a</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="n">a</span><span class="o">/</span><span class="mi">2</span><span class="p">],[(</span><span class="o">-</span><span class="n">a</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">]],</span> <span class="p">(</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.1853868950066556941442559</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bessely</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">0.1853868950066556941442559</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">meijerg</span><span class="p">([[],[]],</span> <span class="p">[[</span><span class="n">a</span><span class="o">/</span><span class="mi">2</span><span class="p">],[</span><span class="o">-</span><span class="n">a</span><span class="o">/</span><span class="mi">2</span><span class="p">]],</span> <span class="o">-</span><span class="p">(</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(0.8685913322427653875717476 + 2.096964974460199200551738j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">a</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">a</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">besseli</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">(0.8685913322427653875717476 + 2.096964974460199200551738j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mf">0.5</span><span class="o">*</span><span class="n">meijerg</span><span class="p">([[],[]],</span> <span class="p">[[</span><span class="n">a</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="n">a</span><span class="o">/</span><span class="mi">2</span><span class="p">],[]],</span> <span class="p">(</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.09334163695597828403796071</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselk</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.09334163695597828403796071</span>
-</pre></div>
-</div>
-<p>Error functions:</p>
-<p><span class="math">\(\sqrt{\pi} z^{2(a-1)} \mathrm{erfc}(z) = G^{2,0}_{1,2} \left( \left.
-\begin{matrix} a \\ a-1, a-\frac{1}{2}
-\end{matrix} \; \right| \; z, \frac{1}{2} \right)\)</span></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">meijerg</span><span class="p">([[],[</span><span class="n">a</span><span class="p">]],</span> <span class="p">[[</span><span class="n">a</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="n">a</span><span class="o">-</span><span class="mf">0.5</span><span class="p">],[]],</span> <span class="n">z</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
-<span class="go">0.00172839843123091957468712</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span> <span class="o">*</span> <span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">erfc</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="go">0.00172839843123091957468712</span>
-</pre></div>
-</div>
-<p>A Meijer G-function of higher degree, (1,1,2,3):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">meijerg</span><span class="p">([[</span><span class="n">a</span><span class="p">],[</span><span class="n">b</span><span class="p">]],</span> <span class="p">[[</span><span class="n">a</span><span class="p">],[</span><span class="n">b</span><span class="p">,</span><span class="n">a</span><span class="o">-</span><span class="mi">1</span><span class="p">]],</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">1.55984467443050210115617</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">((</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">/</span><span class="n">pi</span><span class="o">*</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="p">(</span><span class="n">a</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1.55984467443050210115617</span>
-</pre></div>
-</div>
-<p>A Meijer G-function of still higher degree, (4,1,2,4), that can
-be expanded as a messy combination of exponential integrals:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">meijerg</span><span class="p">([[</span><span class="n">a</span><span class="p">],[</span><span class="mi">2</span><span class="o">*</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="p">]],</span> <span class="p">[[</span><span class="n">b</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="o">-</span><span class="n">a</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">b</span><span class="p">],[]],</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">0.3323667133658557271898061</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="mi">4</span><span class="o">**</span><span class="p">(</span><span class="n">a</span><span class="o">-</span><span class="n">b</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">b</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="n">a</span><span class="o">*</span>\
-<span class="gp">... </span>    <span class="n">expint</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">b</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">))</span><span class="o">*</span><span class="n">expint</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">b</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="o">-</span><span class="n">z</span><span class="p">)))</span>
-<span class="go">0.3323667133658557271898061</span>
-</pre></div>
-</div>
-<p>In the following case, different series give different values:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">meijerg</span><span class="p">([[</span><span class="mi">1</span><span class="p">],[</span><span class="mf">0.25</span><span class="p">]],[[</span><span class="mi">3</span><span class="p">],[</span><span class="mf">0.5</span><span class="p">]],</span><span class="o">-</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">-0.06417628097442437076207337</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">meijerg</span><span class="p">([[</span><span class="mi">1</span><span class="p">],[</span><span class="mf">0.25</span><span class="p">]],[[</span><span class="mi">3</span><span class="p">],[</span><span class="mf">0.5</span><span class="p">]],</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="n">series</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.1428699426155117511873047</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">meijerg</span><span class="p">([[</span><span class="mi">1</span><span class="p">],[</span><span class="mf">0.25</span><span class="p">]],[[</span><span class="mi">3</span><span class="p">],[</span><span class="mf">0.5</span><span class="p">]],</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="n">series</span><span class="o">=</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">-0.06417628097442437076207337</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Meijer_G-function">http://en.wikipedia.org/wiki/Meijer_G-function</a></li>
-<li><a class="reference external" href="http://mathworld.wolfram.com/MeijerG-Function.html">http://mathworld.wolfram.com/MeijerG-Function.html</a></li>
-<li><a class="reference external" href="http://functions.wolfram.com/HypergeometricFunctions/MeijerG/">http://functions.wolfram.com/HypergeometricFunctions/MeijerG/</a></li>
-<li><a class="reference external" href="http://functions.wolfram.com/HypergeometricFunctions/MeijerG1/">http://functions.wolfram.com/HypergeometricFunctions/MeijerG1/</a></li>
-</ol>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="bilateral-hypergeometric-series">
-<h2>Bilateral hypergeometric series<a class="headerlink" href="#bilateral-hypergeometric-series" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="bihyper">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">bihyper()</span></tt><a class="headerlink" href="#bihyper" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.bihyper">
-<tt class="descclassname">mpmath.</tt><tt class="descname">bihyper</tt><big>(</big><em>a_s</em>, <em>b_s</em>, <em>z</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.bihyper" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the bilateral hypergeometric series</p>
-<div class="math">
-\[\,_AH_B(a_1, \ldots, a_k; b_1, \ldots, b_B; z) =
-    \sum_{n=-\infty}^{\infty}
-    \frac{(a_1)_n \ldots (a_A)_n}
-         {(b_1)_n \ldots (b_B)_n} \, z^n\]</div>
-<p>where, for direct convergence, <span class="math">\(A = B\)</span> and <span class="math">\(|z| = 1\)</span>, although a
-regularized sum exists more generally by considering the
-bilateral series as a sum of two ordinary hypergeometric
-functions. In order for the series to make sense, none of the
-parameters may be integers.</p>
-<p><strong>Examples</strong></p>
-<p>The value of <span class="math">\(\,_2H_2\)</span> at <span class="math">\(z = 1\)</span> is given by Dougall&#8217;s formula:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">c</span><span class="p">,</span><span class="n">d</span> <span class="o">=</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mf">2.25</span><span class="p">,</span> <span class="mf">3.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bihyper</span><span class="p">([</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">],[</span><span class="n">c</span><span class="p">,</span><span class="n">d</span><span class="p">],</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-14.49118026212345786148847</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gammaprod</span><span class="p">([</span><span class="n">c</span><span class="p">,</span><span class="n">d</span><span class="p">,</span><span class="mi">1</span><span class="o">-</span><span class="n">a</span><span class="p">,</span><span class="mi">1</span><span class="o">-</span><span class="n">b</span><span class="p">,</span><span class="n">c</span><span class="o">+</span><span class="n">d</span><span class="o">-</span><span class="n">a</span><span class="o">-</span><span class="n">b</span><span class="o">-</span><span class="mi">1</span><span class="p">],[</span><span class="n">c</span><span class="o">-</span><span class="n">a</span><span class="p">,</span><span class="n">d</span><span class="o">-</span><span class="n">a</span><span class="p">,</span><span class="n">c</span><span class="o">-</span><span class="n">b</span><span class="p">,</span><span class="n">d</span><span class="o">-</span><span class="n">b</span><span class="p">])</span>
-<span class="go">-14.49118026212345786148847</span>
-</pre></div>
-</div>
-<p>The regularized function <span class="math">\(\,_1H_0\)</span> can be expressed as the
-sum of one <span class="math">\(\,_2F_0\)</span> function and one <span class="math">\(\,_1F_1\)</span> function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">0.25</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">0.75</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bihyper</span><span class="p">([</span><span class="n">a</span><span class="p">],</span> <span class="p">[],</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">(0.2454393389657273841385582 + 0.2454393389657273841385582j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="n">a</span><span class="p">,</span><span class="mi">1</span><span class="p">],[],</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">hyper</span><span class="p">([</span><span class="mi">1</span><span class="p">],[</span><span class="mi">1</span><span class="o">-</span><span class="n">a</span><span class="p">],</span><span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(0.2454393389657273841385582 + 0.2454393389657273841385582j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper</span><span class="p">([</span><span class="n">a</span><span class="p">,</span><span class="mi">1</span><span class="p">],[],</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="n">hyper</span><span class="p">([</span><span class="mi">1</span><span class="p">],[</span><span class="mi">2</span><span class="o">-</span><span class="n">a</span><span class="p">],</span><span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="n">z</span><span class="o">/</span><span class="p">(</span><span class="n">a</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(0.2454393389657273841385582 + 0.2454393389657273841385582j)</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../../mpmath/references.html#slater">[Slater]</a> (chapter 6: &#8220;Bilateral Series&#8221;, pp. 180-189)</li>
-<li><a class="reference internal" href="../../mpmath/references.html#wikipedia">[Wikipedia]</a> <a class="reference external" href="http://en.wikipedia.org/wiki/Bilateral_hypergeometric_series">http://en.wikipedia.org/wiki/Bilateral_hypergeometric_series</a></li>
-</ol>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="hypergeometric-functions-of-two-variables">
-<h2>Hypergeometric functions of two variables<a class="headerlink" href="#hypergeometric-functions-of-two-variables" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="hyper2d">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">hyper2d()</span></tt><a class="headerlink" href="#hyper2d" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.hyper2d">
-<tt class="descclassname">mpmath.</tt><tt class="descname">hyper2d</tt><big>(</big><em>a</em>, <em>b</em>, <em>x</em>, <em>y</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.hyper2d" title="Permalink to this definition">¶</a></dt>
-<dd><p>Sums the generalized 2D hypergeometric series</p>
-<div class="math">
-\[\sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
-    \frac{P((a),m,n)}{Q((b),m,n)}
-    \frac{x^m y^n} {m! n!}\]</div>
-<p>where <span class="math">\((a) = (a_1,\ldots,a_r)\)</span>, <span class="math">\((b) = (b_1,\ldots,b_s)\)</span> and where
-<span class="math">\(P\)</span> and <span class="math">\(Q\)</span> are products of rising factorials such as <span class="math">\((a_j)_n\)</span> or
-<span class="math">\((a_j)_{m+n}\)</span>. <span class="math">\(P\)</span> and <span class="math">\(Q\)</span> are specified in the form of dicts, with
-the <span class="math">\(m\)</span> and <span class="math">\(n\)</span> dependence as keys and parameter lists as values.
-The supported rising factorials are given in the following table
-(note that only a few are supported in <span class="math">\(Q\)</span>):</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="31%" />
-<col width="49%" />
-<col width="21%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head">Key</th>
-<th class="head">Rising factorial</th>
-<th class="head"><span class="math">\(Q\)</span></th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-even"><td><tt class="docutils literal"><span class="pre">'m'</span></tt></td>
-<td><span class="math">\((a_j)_m\)</span></td>
-<td>Yes</td>
-</tr>
-<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">'n'</span></tt></td>
-<td><span class="math">\((a_j)_n\)</span></td>
-<td>Yes</td>
-</tr>
-<tr class="row-even"><td><tt class="docutils literal"><span class="pre">'m+n'</span></tt></td>
-<td><span class="math">\((a_j)_{m+n}\)</span></td>
-<td>Yes</td>
-</tr>
-<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">'m-n'</span></tt></td>
-<td><span class="math">\((a_j)_{m-n}\)</span></td>
-<td>No</td>
-</tr>
-<tr class="row-even"><td><tt class="docutils literal"><span class="pre">'n-m'</span></tt></td>
-<td><span class="math">\((a_j)_{n-m}\)</span></td>
-<td>No</td>
-</tr>
-<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">'2m+n'</span></tt></td>
-<td><span class="math">\((a_j)_{2m+n}\)</span></td>
-<td>No</td>
-</tr>
-<tr class="row-even"><td><tt class="docutils literal"><span class="pre">'2m-n'</span></tt></td>
-<td><span class="math">\((a_j)_{2m-n}\)</span></td>
-<td>No</td>
-</tr>
-<tr class="row-odd"><td><tt class="docutils literal"><span class="pre">'2n-m'</span></tt></td>
-<td><span class="math">\((a_j)_{2n-m}\)</span></td>
-<td>No</td>
-</tr>
-</tbody>
-</table>
-<p>For example, the Appell F1 and F4 functions</p>
-<div class="math">
-\[F_1 = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
-      \frac{(a)_{m+n} (b)_m (c)_n}{(d)_{m+n}}
-      \frac{x^m y^n}{m! n!}\]\[F_4 = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
-      \frac{(a)_{m+n} (b)_{m+n}}{(c)_m (d)_{n}}
-      \frac{x^m y^n}{m! n!}\]</div>
-<p>can be represented respectively as</p>
-<blockquote>
-<div><p><tt class="docutils literal"><span class="pre">hyper2d({'m+n':[a],</span> <span class="pre">'m':[b],</span> <span class="pre">'n':[c]},</span> <span class="pre">{'m+n':[d]},</span> <span class="pre">x,</span> <span class="pre">y)</span></tt></p>
-<p><tt class="docutils literal"><span class="pre">hyper2d({'m+n':[a,b]},</span> <span class="pre">{'m':[c],</span> <span class="pre">'n':[d]},</span> <span class="pre">x,</span> <span class="pre">y)</span></tt></p>
-</div></blockquote>
-<p>More generally, <a class="reference internal" href="#mpmath.hyper2d" title="mpmath.hyper2d"><tt class="xref py py-func docutils literal"><span class="pre">hyper2d()</span></tt></a> can evaluate any of the 34 distinct
-convergent second-order (generalized Gaussian) hypergeometric
-series enumerated by Horn, as well as the Kampe de Feriet
-function.</p>
-<p>The series is computed by rewriting it so that the inner
-series (i.e. the series containing <span class="math">\(n\)</span> and <span class="math">\(y\)</span>) has the form of an
-ordinary generalized hypergeometric series and thereby can be
-evaluated efficiently using <a class="reference internal" href="#mpmath.hyper" title="mpmath.hyper"><tt class="xref py py-func docutils literal"><span class="pre">hyper()</span></tt></a>. If possible,
-manually swapping <span class="math">\(x\)</span> and <span class="math">\(y\)</span> and the corresponding parameters
-can sometimes give better results.</p>
-<p><strong>Examples</strong></p>
-<p>Two separable cases: a product of two geometric series, and a
-product of two Gaussian hypergeometric functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">0.25</span><span class="p">),</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper2d</span><span class="p">({</span><span class="s">&#39;m&#39;</span><span class="p">:</span><span class="mi">1</span><span class="p">,</span><span class="s">&#39;n&#39;</span><span class="p">:</span><span class="mi">1</span><span class="p">},</span> <span class="p">{},</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>
-<span class="go">2.666666666666666666666667</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">y</span><span class="p">)</span>
-<span class="go">2.666666666666666666666667</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper2d</span><span class="p">({</span><span class="s">&#39;m&#39;</span><span class="p">:[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span><span class="s">&#39;n&#39;</span><span class="p">:[</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">]},</span> <span class="p">{</span><span class="s">&#39;m&#39;</span><span class="p">:[</span><span class="mi">5</span><span class="p">],</span><span class="s">&#39;n&#39;</span><span class="p">:[</span><span class="mi">6</span><span class="p">]},</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>
-<span class="go">4.164358531238938319669856</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f1</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">hyp2f1</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>
-<span class="go">4.164358531238938319669856</span>
-</pre></div>
-</div>
-<p>Some more series that can be done in closed form:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">hyper2d</span><span class="p">({</span><span class="s">&#39;m&#39;</span><span class="p">:</span><span class="mi">1</span><span class="p">,</span><span class="s">&#39;n&#39;</span><span class="p">:</span><span class="mi">1</span><span class="p">},{</span><span class="s">&#39;m+n&#39;</span><span class="p">:</span><span class="mi">1</span><span class="p">},</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>
-<span class="go">2.013417124712514809623881</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="o">-</span><span class="n">exp</span><span class="p">(</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="n">y</span><span class="p">)</span>
-<span class="go">2.013417124712514809623881</span>
-</pre></div>
-</div>
-<p>Six of the 34 Horn functions, G1-G3 and H1-H3:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">10</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="mf">0.0625</span><span class="p">,</span> <span class="mf">0.125</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a1</span><span class="p">,</span><span class="n">a2</span><span class="p">,</span><span class="n">b1</span><span class="p">,</span><span class="n">b2</span><span class="p">,</span><span class="n">c1</span><span class="p">,</span><span class="n">c2</span><span class="p">,</span><span class="n">d</span> <span class="o">=</span> <span class="mf">1.1</span><span class="p">,</span><span class="o">-</span><span class="mf">1.2</span><span class="p">,</span><span class="o">-</span><span class="mf">1.3</span><span class="p">,</span><span class="o">-</span><span class="mf">1.4</span><span class="p">,</span><span class="mf">1.5</span><span class="p">,</span><span class="o">-</span><span class="mf">1.6</span><span class="p">,</span><span class="mf">1.7</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper2d</span><span class="p">({</span><span class="s">&#39;m+n&#39;</span><span class="p">:</span><span class="n">a1</span><span class="p">,</span><span class="s">&#39;n-m&#39;</span><span class="p">:</span><span class="n">b1</span><span class="p">,</span><span class="s">&#39;m-n&#39;</span><span class="p">:</span><span class="n">b2</span><span class="p">},{},</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>  <span class="c"># G1</span>
-<span class="go">1.139090746</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">m</span><span class="p">,</span><span class="n">n</span><span class="p">:</span> <span class="n">rf</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span><span class="n">m</span><span class="o">+</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">rf</span><span class="p">(</span><span class="n">b1</span><span class="p">,</span><span class="n">n</span><span class="o">-</span><span class="n">m</span><span class="p">)</span><span class="o">*</span><span class="n">rf</span><span class="p">(</span><span class="n">b2</span><span class="p">,</span><span class="n">m</span><span class="o">-</span><span class="n">n</span><span class="p">)</span><span class="o">*</span>\
-<span class="gp">... </span>    <span class="n">x</span><span class="o">**</span><span class="n">m</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">m</span><span class="p">)</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">1.139090746</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper2d</span><span class="p">({</span><span class="s">&#39;m&#39;</span><span class="p">:</span><span class="n">a1</span><span class="p">,</span><span class="s">&#39;n&#39;</span><span class="p">:</span><span class="n">a2</span><span class="p">,</span><span class="s">&#39;n-m&#39;</span><span class="p">:</span><span class="n">b1</span><span class="p">,</span><span class="s">&#39;m-n&#39;</span><span class="p">:</span><span class="n">b2</span><span class="p">},{},</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>  <span class="c"># G2</span>
-<span class="go">0.9503682696</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">m</span><span class="p">,</span><span class="n">n</span><span class="p">:</span> <span class="n">rf</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span><span class="n">m</span><span class="p">)</span><span class="o">*</span><span class="n">rf</span><span class="p">(</span><span class="n">a2</span><span class="p">,</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">rf</span><span class="p">(</span><span class="n">b1</span><span class="p">,</span><span class="n">n</span><span class="o">-</span><span class="n">m</span><span class="p">)</span><span class="o">*</span><span class="n">rf</span><span class="p">(</span><span class="n">b2</span><span class="p">,</span><span class="n">m</span><span class="o">-</span><span class="n">n</span><span class="p">)</span><span class="o">*</span>\
-<span class="gp">... </span>    <span class="n">x</span><span class="o">**</span><span class="n">m</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">m</span><span class="p">)</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">0.9503682696</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper2d</span><span class="p">({</span><span class="s">&#39;2n-m&#39;</span><span class="p">:</span><span class="n">a1</span><span class="p">,</span><span class="s">&#39;2m-n&#39;</span><span class="p">:</span><span class="n">a2</span><span class="p">},{},</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>  <span class="c"># G3</span>
-<span class="go">1.029372029</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">m</span><span class="p">,</span><span class="n">n</span><span class="p">:</span> <span class="n">rf</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">-</span><span class="n">m</span><span class="p">)</span><span class="o">*</span><span class="n">rf</span><span class="p">(</span><span class="n">a2</span><span class="p">,</span><span class="mi">2</span><span class="o">*</span><span class="n">m</span><span class="o">-</span><span class="n">n</span><span class="p">)</span><span class="o">*</span>\
-<span class="gp">... </span>    <span class="n">x</span><span class="o">**</span><span class="n">m</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">m</span><span class="p">)</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">1.029372029</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper2d</span><span class="p">({</span><span class="s">&#39;m-n&#39;</span><span class="p">:</span><span class="n">a1</span><span class="p">,</span><span class="s">&#39;m+n&#39;</span><span class="p">:</span><span class="n">b1</span><span class="p">,</span><span class="s">&#39;n&#39;</span><span class="p">:</span><span class="n">c1</span><span class="p">},{</span><span class="s">&#39;m&#39;</span><span class="p">:</span><span class="n">d</span><span class="p">},</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>  <span class="c"># H1</span>
-<span class="go">-1.605331256</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">m</span><span class="p">,</span><span class="n">n</span><span class="p">:</span> <span class="n">rf</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span><span class="n">m</span><span class="o">-</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">rf</span><span class="p">(</span><span class="n">b1</span><span class="p">,</span><span class="n">m</span><span class="o">+</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">rf</span><span class="p">(</span><span class="n">c1</span><span class="p">,</span><span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">rf</span><span class="p">(</span><span class="n">d</span><span class="p">,</span><span class="n">m</span><span class="p">)</span><span class="o">*</span>\
-<span class="gp">... </span>    <span class="n">x</span><span class="o">**</span><span class="n">m</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">m</span><span class="p">)</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">-1.605331256</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper2d</span><span class="p">({</span><span class="s">&#39;m-n&#39;</span><span class="p">:</span><span class="n">a1</span><span class="p">,</span><span class="s">&#39;m&#39;</span><span class="p">:</span><span class="n">b1</span><span class="p">,</span><span class="s">&#39;n&#39;</span><span class="p">:[</span><span class="n">c1</span><span class="p">,</span><span class="n">c2</span><span class="p">]},{</span><span class="s">&#39;m&#39;</span><span class="p">:</span><span class="n">d</span><span class="p">},</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>  <span class="c"># H2</span>
-<span class="go">-2.35405404</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">m</span><span class="p">,</span><span class="n">n</span><span class="p">:</span> <span class="n">rf</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span><span class="n">m</span><span class="o">-</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">rf</span><span class="p">(</span><span class="n">b1</span><span class="p">,</span><span class="n">m</span><span class="p">)</span><span class="o">*</span><span class="n">rf</span><span class="p">(</span><span class="n">c1</span><span class="p">,</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">rf</span><span class="p">(</span><span class="n">c2</span><span class="p">,</span><span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">rf</span><span class="p">(</span><span class="n">d</span><span class="p">,</span><span class="n">m</span><span class="p">)</span><span class="o">*</span>\
-<span class="gp">... </span>    <span class="n">x</span><span class="o">**</span><span class="n">m</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">m</span><span class="p">)</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">-2.35405404</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyper2d</span><span class="p">({</span><span class="s">&#39;2m+n&#39;</span><span class="p">:</span><span class="n">a1</span><span class="p">,</span><span class="s">&#39;n&#39;</span><span class="p">:</span><span class="n">b1</span><span class="p">},{</span><span class="s">&#39;m+n&#39;</span><span class="p">:</span><span class="n">c1</span><span class="p">},</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>  <span class="c"># H3</span>
-<span class="go">0.974479074</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">m</span><span class="p">,</span><span class="n">n</span><span class="p">:</span> <span class="n">rf</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span><span class="mi">2</span><span class="o">*</span><span class="n">m</span><span class="o">+</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">rf</span><span class="p">(</span><span class="n">b1</span><span class="p">,</span><span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">rf</span><span class="p">(</span><span class="n">c1</span><span class="p">,</span><span class="n">m</span><span class="o">+</span><span class="n">n</span><span class="p">)</span><span class="o">*</span>\
-<span class="gp">... </span>    <span class="n">x</span><span class="o">**</span><span class="n">m</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">m</span><span class="p">)</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">0.974479074</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../../mpmath/references.html#srivastavakarlsson">[SrivastavaKarlsson]</a></li>
-<li><a class="reference internal" href="../../mpmath/references.html#weisstein">[Weisstein]</a> <a class="reference external" href="http://mathworld.wolfram.com/HornFunction.html">http://mathworld.wolfram.com/HornFunction.html</a></li>
-<li><a class="reference internal" href="../../mpmath/references.html#weisstein">[Weisstein]</a> <a class="reference external" href="http://mathworld.wolfram.com/AppellHypergeometricFunction.html">http://mathworld.wolfram.com/AppellHypergeometricFunction.html</a></li>
-</ol>
-</dd></dl>
-
-</div>
-<div class="section" id="appellf1">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">appellf1()</span></tt><a class="headerlink" href="#appellf1" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.appellf1">
-<tt class="descclassname">mpmath.</tt><tt class="descname">appellf1</tt><big>(</big><em>a</em>, <em>b1</em>, <em>b2</em>, <em>c</em>, <em>x</em>, <em>y</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.appellf1" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the Appell F1 hypergeometric function of two variables,</p>
-<div class="math">
-\[F_1(a,b_1,b_2,c,x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
-    \frac{(a)_{m+n} (b_1)_m (b_2)_n}{(c)_{m+n}}
-    \frac{x^m y^n}{m! n!}.\]</div>
-<p>This series is only generally convergent when <span class="math">\(|x| &lt; 1\)</span> and <span class="math">\(|y| &lt; 1\)</span>,
-although <a class="reference internal" href="#mpmath.appellf1" title="mpmath.appellf1"><tt class="xref py py-func docutils literal"><span class="pre">appellf1()</span></tt></a> can evaluate an analytic continuation
-with respecto to either variable, and sometimes both.</p>
-<p><strong>Examples</strong></p>
-<p>Evaluation is supported for real and complex parameters:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">appellf1</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.25</span><span class="p">)</span>
-<span class="go">1.154700538379251529018298</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">appellf1</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.5j</span><span class="p">)</span>
-<span class="go">(1.138403860350148085179415 + 1.510544741058517621110615j)</span>
-</pre></div>
-</div>
-<p>For some integer parameters, the F1 series reduces to a polynomial:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">appellf1</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">-816.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">appellf1</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">-20528.0</span>
-</pre></div>
-</div>
-<p>The analytic continuation with respect to either <span class="math">\(x\)</span> or <span class="math">\(y\)</span>,
-and sometimes with respect to both, can be evaluated:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">appellf1</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">100</span><span class="p">,</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="go">(0.0006231042714165329279738662 + 0.0000005769149277148425774499857j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">appellf1</span><span class="p">(</span><span class="s">&#39;1.1&#39;</span><span class="p">,</span> <span class="s">&#39;0.3&#39;</span><span class="p">,</span> <span class="s">&#39;0.2+2j&#39;</span><span class="p">,</span> <span class="s">&#39;0.4&#39;</span><span class="p">,</span> <span class="s">&#39;0.2&#39;</span><span class="p">,</span> <span class="mf">1.5</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(-0.1782604566893954897128702 + 0.002472407104546216117161499j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">appellf1</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="mi">12</span><span class="p">)</span>
-<span class="go">-0.07122993830066776374929313</span>
-</pre></div>
-</div>
-<p>For certain arguments, F1 reduces to an ordinary hypergeometric function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">appellf1</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.25</span><span class="p">)</span>
-<span class="go">1.547902270302684019335555</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">4</span><span class="o">*</span><span class="n">hyp2f1</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="s">&#39;1/3&#39;</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span>
-<span class="go">1.547902270302684019335555</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">appellf1</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mf">1.5</span><span class="p">)</span>
-<span class="go">(-1.717202506168937502740238 - 2.792526803190927323077905j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f1</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mf">1.5</span><span class="p">)</span>
-<span class="go">(-1.717202506168937502740238 - 2.792526803190927323077905j)</span>
-</pre></div>
-</div>
-<p>The F1 function satisfies a system of partial differential equations:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span><span class="n">b1</span><span class="p">,</span><span class="n">b2</span><span class="p">,</span><span class="n">c</span><span class="p">,</span><span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="nb">map</span><span class="p">(</span><span class="n">mpf</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.25</span><span class="p">,</span><span class="mf">1.125</span><span class="p">,</span><span class="mf">0.25</span><span class="p">,</span><span class="o">-</span><span class="mf">0.25</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">:</span> <span class="n">appellf1</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b1</span><span class="p">,</span><span class="n">b2</span><span class="p">,</span><span class="n">c</span><span class="p">,</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">))</span> <span class="o">+</span>
-<span class="gp">... </span>     <span class="n">y</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span> <span class="o">+</span>
-<span class="gp">... </span>     <span class="p">(</span><span class="n">c</span><span class="o">-</span><span class="p">(</span><span class="n">a</span><span class="o">+</span><span class="n">b1</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">))</span> <span class="o">-</span>
-<span class="gp">... </span>     <span class="n">b1</span><span class="o">*</span><span class="n">y</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span> <span class="o">-</span>
-<span class="gp">... </span>     <span class="n">a</span><span class="o">*</span><span class="n">b1</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">))</span>
-<span class="go">0.0</span>
-<span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">y</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span> <span class="o">+</span>
-<span class="gp">... </span>     <span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span> <span class="o">+</span>
-<span class="gp">... </span>     <span class="p">(</span><span class="n">c</span><span class="o">-</span><span class="p">(</span><span class="n">a</span><span class="o">+</span><span class="n">b2</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span> <span class="o">-</span>
-<span class="gp">... </span>     <span class="n">b2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">))</span> <span class="o">-</span>
-<span class="gp">... </span>     <span class="n">a</span><span class="o">*</span><span class="n">b2</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">))</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>The Appell F1 function allows for closed-form evaluation of various
-integrals, such as any integral of the form
-<span class="math">\(\int x^r (x+a)^p (x+b)^q dx\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">integral</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">q</span><span class="p">,</span><span class="n">r</span><span class="p">,</span><span class="n">x1</span><span class="p">,</span><span class="n">x2</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">q</span><span class="p">,</span><span class="n">r</span><span class="p">,</span><span class="n">x1</span><span class="p">,</span><span class="n">x2</span> <span class="o">=</span> <span class="nb">map</span><span class="p">(</span><span class="n">mpmathify</span><span class="p">,</span> <span class="p">[</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">q</span><span class="p">,</span><span class="n">r</span><span class="p">,</span><span class="n">x1</span><span class="p">,</span><span class="n">x2</span><span class="p">])</span>
-<span class="gp">... </span>    <span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">**</span><span class="n">r</span> <span class="o">*</span> <span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">a</span><span class="p">)</span><span class="o">**</span><span class="n">p</span> <span class="o">*</span> <span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="n">q</span>
-<span class="gp">... </span>    <span class="k">def</span> <span class="nf">F</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-<span class="gp">... </span>        <span class="n">v</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">r</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">r</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">a</span><span class="o">+</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="n">p</span> <span class="o">*</span> <span class="p">(</span><span class="n">b</span><span class="o">+</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="n">q</span>
-<span class="gp">... </span>        <span class="n">v</span> <span class="o">*=</span> <span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">x</span><span class="o">/</span><span class="n">a</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">p</span><span class="p">)</span>
-<span class="gp">... </span>        <span class="n">v</span> <span class="o">*=</span> <span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">x</span><span class="o">/</span><span class="n">b</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">q</span><span class="p">)</span>
-<span class="gp">... </span>        <span class="n">v</span> <span class="o">*=</span> <span class="n">appellf1</span><span class="p">(</span><span class="n">r</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="n">p</span><span class="p">,</span><span class="o">-</span><span class="n">q</span><span class="p">,</span><span class="mi">2</span><span class="o">+</span><span class="n">r</span><span class="p">,</span><span class="o">-</span><span class="n">x</span><span class="o">/</span><span class="n">a</span><span class="p">,</span><span class="o">-</span><span class="n">x</span><span class="o">/</span><span class="n">b</span><span class="p">)</span>
-<span class="gp">... </span>        <span class="k">return</span> <span class="n">v</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&quot;Num. quad: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="n">x1</span><span class="p">,</span><span class="n">x2</span><span class="p">]))</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&quot;Appell F1: </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">F</span><span class="p">(</span><span class="n">x2</span><span class="p">)</span><span class="o">-</span><span class="n">F</span><span class="p">(</span><span class="n">x1</span><span class="p">)))</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integral</span><span class="p">(</span><span class="s">&#39;1/5&#39;</span><span class="p">,</span><span class="s">&#39;4/3&#39;</span><span class="p">,</span><span class="s">&#39;-2&#39;</span><span class="p">,</span><span class="s">&#39;3&#39;</span><span class="p">,</span><span class="s">&#39;1/2&#39;</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">Num. quad: 9.073335358785776206576981</span>
-<span class="go">Appell F1: 9.073335358785776206576981</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integral</span><span class="p">(</span><span class="s">&#39;3/2&#39;</span><span class="p">,</span><span class="s">&#39;4/3&#39;</span><span class="p">,</span><span class="s">&#39;-2&#39;</span><span class="p">,</span><span class="s">&#39;3&#39;</span><span class="p">,</span><span class="s">&#39;1/2&#39;</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">Num. quad: 1.092829171999626454344678</span>
-<span class="go">Appell F1: 1.092829171999626454344678</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integral</span><span class="p">(</span><span class="s">&#39;3/2&#39;</span><span class="p">,</span><span class="s">&#39;4/3&#39;</span><span class="p">,</span><span class="s">&#39;-2&#39;</span><span class="p">,</span><span class="s">&#39;3&#39;</span><span class="p">,</span><span class="s">&#39;1/2&#39;</span><span class="p">,</span><span class="mi">12</span><span class="p">,</span><span class="mi">25</span><span class="p">)</span>
-<span class="go">Num. quad: 1106.323225040235116498927</span>
-<span class="go">Appell F1: 1106.323225040235116498927</span>
-</pre></div>
-</div>
-<p>Also incomplete elliptic integrals fall into this category [1]:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">E</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">if</span> <span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">ae</span><span class="p">(</span><span class="n">z</span><span class="p">):</span>
-<span class="gp">... </span>        <span class="k">return</span> <span class="n">ellipe</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-<span class="gp">... </span>    <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="nb">round</span><span class="p">(</span><span class="n">re</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">ellipe</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="o">+</span> <span class="n">mpf</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="nb">round</span><span class="p">(</span><span class="n">re</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span>\
-<span class="gp">... </span>        <span class="n">sin</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">appellf1</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">1.5</span><span class="p">,</span><span class="n">sin</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span><span class="n">m</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="n">m</span><span class="p">);</span> <span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">m</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span><span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span><span class="n">z</span><span class="p">])</span>
-<span class="go">0.9273298836244400669659042</span>
-<span class="go">0.9273298836244400669659042</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="n">m</span><span class="p">);</span> <span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">m</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span><span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span><span class="n">z</span><span class="p">])</span>
-<span class="go">(1.057495752337234229715836 + 1.198140234735592207439922j)</span>
-<span class="go">(1.057495752337234229715836 + 1.198140234735592207439922j)</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../../mpmath/references.html#wolframfunctions">[WolframFunctions]</a> <a class="reference external" href="http://functions.wolfram.com/EllipticIntegrals/EllipticE2/26/01/">http://functions.wolfram.com/EllipticIntegrals/EllipticE2/26/01/</a></li>
-<li><a class="reference internal" href="../../mpmath/references.html#srivastavakarlsson">[SrivastavaKarlsson]</a></li>
-<li><a class="reference internal" href="../../mpmath/references.html#cabralrosetti">[CabralRosetti]</a></li>
-<li><a class="reference internal" href="../../mpmath/references.html#vidunas">[Vidunas]</a></li>
-<li><a class="reference internal" href="../../mpmath/references.html#slater">[Slater]</a></li>
-</ol>
-</dd></dl>
-
-</div>
-<div class="section" id="appellf2">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">appellf2()</span></tt><a class="headerlink" href="#appellf2" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.appellf2">
-<tt class="descclassname">mpmath.</tt><tt class="descname">appellf2</tt><big>(</big><em>a</em>, <em>b1</em>, <em>b2</em>, <em>c1</em>, <em>c2</em>, <em>x</em>, <em>y</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.appellf2" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the Appell F2 hypergeometric function of two variables</p>
-<div class="math">
-\[F_2(a,b_1,b_2,c_1,c_2,x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
-    \frac{(a)_{m+n} (b_1)_m (b_2)_n}{(c_1)_m (c_2)_n}
-    \frac{x^m y^n}{m! n!}.\]</div>
-<p>The series is generally absolutely convergent for <span class="math">\(|x| + |y| &lt; 1\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>Evaluation for real and complex arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">appellf2</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mf">0.25</span><span class="p">,</span><span class="mf">0.125</span><span class="p">)</span>
-<span class="go">1.257417193533135344785602</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">appellf2</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">-42.8</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">appellf2</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.25</span><span class="p">,</span><span class="o">-</span><span class="mf">0.25</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mf">0.25j</span><span class="p">,</span><span class="mf">0.25</span><span class="p">)</span>
-<span class="go">(0.9880539519421899867041719 + 0.01497616165031102661476978j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">appellf2</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span><span class="mi">1</span><span class="o">-</span><span class="n">j</span><span class="p">,</span><span class="mi">3j</span><span class="p">,</span><span class="o">-</span><span class="mi">3j</span><span class="p">,</span><span class="mf">0.25</span><span class="p">,</span><span class="mf">0.25</span><span class="p">))</span>
-<span class="go">1.201311219287411337955192</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">appellf2</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mf">0.125</span><span class="p">,</span><span class="mi">16</span><span class="p">)</span>
-<span class="go">(-0.09455532250274744282125152 - 0.7647282253046207836769297j)</span>
-</pre></div>
-</div>
-<p>A transformation formula:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span><span class="n">b1</span><span class="p">,</span><span class="n">b2</span><span class="p">,</span><span class="n">c1</span><span class="p">,</span><span class="n">c2</span><span class="p">,</span><span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="nb">map</span><span class="p">(</span><span class="n">mpf</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.25</span><span class="p">,</span><span class="mf">1.625</span><span class="p">,</span><span class="o">-</span><span class="mf">0.125</span><span class="p">,</span><span class="mf">0.125</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">appellf2</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b1</span><span class="p">,</span><span class="n">b2</span><span class="p">,</span><span class="n">c1</span><span class="p">,</span><span class="n">c2</span><span class="p">,</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>
-<span class="go">0.2299211717841180783309688</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">appellf2</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">c1</span><span class="o">-</span><span class="n">b1</span><span class="p">,</span><span class="n">b2</span><span class="p">,</span><span class="n">c1</span><span class="p">,</span><span class="n">c2</span><span class="p">,</span><span class="n">x</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span><span class="n">y</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="p">))</span>
-<span class="go">0.2299211717841180783309688</span>
-</pre></div>
-</div>
-<p>A system of partial differential equations satisfied by F2:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span><span class="n">b1</span><span class="p">,</span><span class="n">b2</span><span class="p">,</span><span class="n">c1</span><span class="p">,</span><span class="n">c2</span><span class="p">,</span><span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="nb">map</span><span class="p">(</span><span class="n">mpf</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.25</span><span class="p">,</span><span class="mf">1.125</span><span class="p">,</span><span class="mf">1.5</span><span class="p">,</span><span class="mf">0.0625</span><span class="p">,</span><span class="o">-</span><span class="mf">0.0625</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">:</span> <span class="n">appellf2</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b1</span><span class="p">,</span><span class="n">b2</span><span class="p">,</span><span class="n">c1</span><span class="p">,</span><span class="n">c2</span><span class="p">,</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">))</span> <span class="o">-</span>
-<span class="gp">... </span>     <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span> <span class="o">+</span>
-<span class="gp">... </span>     <span class="p">(</span><span class="n">c1</span><span class="o">-</span><span class="p">(</span><span class="n">a</span><span class="o">+</span><span class="n">b1</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">))</span> <span class="o">-</span>
-<span class="gp">... </span>     <span class="n">b1</span><span class="o">*</span><span class="n">y</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span> <span class="o">-</span>
-<span class="gp">... </span>     <span class="n">a</span><span class="o">*</span><span class="n">b1</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">))</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">y</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span> <span class="o">-</span>
-<span class="gp">... </span>     <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span> <span class="o">+</span>
-<span class="gp">... </span>     <span class="p">(</span><span class="n">c2</span><span class="o">-</span><span class="p">(</span><span class="n">a</span><span class="o">+</span><span class="n">b2</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span> <span class="o">-</span>
-<span class="gp">... </span>     <span class="n">b2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">))</span> <span class="o">-</span>
-<span class="gp">... </span>     <span class="n">a</span><span class="o">*</span><span class="n">b2</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">))</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<p>See references for <a class="reference internal" href="#mpmath.appellf1" title="mpmath.appellf1"><tt class="xref py py-func docutils literal"><span class="pre">appellf1()</span></tt></a>.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="appellf3">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">appellf3()</span></tt><a class="headerlink" href="#appellf3" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.appellf3">
-<tt class="descclassname">mpmath.</tt><tt class="descname">appellf3</tt><big>(</big><em>a1</em>, <em>a2</em>, <em>b1</em>, <em>b2</em>, <em>c</em>, <em>x</em>, <em>y</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.appellf3" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the Appell F3 hypergeometric function of two variables</p>
-<div class="math">
-\[F_3(a_1,a_2,b_1,b_2,c,x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
-    \frac{(a_1)_m (a_2)_n (b_1)_m (b_2)_n}{(c)_{m+n}}
-    \frac{x^m y^n}{m! n!}.\]</div>
-<p>The series is generally absolutely convergent for <span class="math">\(|x| &lt; 1, |y| &lt; 1\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>Evaluation for various parameters and variables:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">appellf3</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.25</span><span class="p">)</span>
-<span class="go">2.221557778107438938158705</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">appellf3</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">0</span><span class="p">);</span> <span class="n">hyp2f1</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">)</span>
-<span class="go">(-0.5189554589089861284537389 - 0.1454441043328607980769742j)</span>
-<span class="go">(-0.5189554589089861284537389 - 0.1454441043328607980769742j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">appellf3</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">6</span><span class="p">)</span>
-<span class="go">-17.4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">appellf3</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">6</span><span class="p">)</span>
-<span class="go">(17.7876136773677356641825 + 19.54768762233649126154534j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">appellf3</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">(85.02054175067929402953645 + 148.4402528821177305173599j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">appellf3</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="o">-</span><span class="n">j</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mf">0.25</span><span class="p">,</span><span class="mf">0.25</span><span class="p">))</span>
-<span class="go">1.719992169545200286696007</span>
-</pre></div>
-</div>
-<p>Many transformations and evaluations for special combinations
-of the parameters are possible, e.g.:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">c</span><span class="p">,</span><span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="nb">map</span><span class="p">(</span><span class="n">mpf</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.25</span><span class="p">,</span><span class="mf">0.125</span><span class="p">,</span><span class="mf">0.125</span><span class="p">,</span><span class="o">-</span><span class="mf">0.125</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">appellf3</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">c</span><span class="o">-</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">c</span><span class="o">-</span><span class="n">b</span><span class="p">,</span><span class="n">c</span><span class="p">,</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>
-<span class="go">1.093432340896087107444363</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="o">-</span><span class="n">c</span><span class="p">)</span><span class="o">*</span><span class="n">hyp2f1</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">c</span><span class="p">,</span><span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="o">-</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
-<span class="go">1.093432340896087107444363</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">appellf3</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="n">x</span><span class="p">,</span><span class="o">-</span><span class="n">x</span><span class="p">)</span>
-<span class="go">0.01568646277445385390945083</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.01568646277445385390945083</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a1</span><span class="p">,</span><span class="n">a2</span><span class="p">,</span><span class="n">b1</span><span class="p">,</span><span class="n">b2</span><span class="p">,</span><span class="n">c</span><span class="p">,</span><span class="n">x</span> <span class="o">=</span> <span class="nb">map</span><span class="p">(</span><span class="n">mpf</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.25</span><span class="p">,</span><span class="mf">0.125</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">4.25</span><span class="p">,</span><span class="mf">0.125</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">appellf3</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span><span class="n">a2</span><span class="p">,</span><span class="n">b1</span><span class="p">,</span><span class="n">b2</span><span class="p">,</span><span class="n">c</span><span class="p">,</span><span class="n">x</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1.03947361709111140096947</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gammaprod</span><span class="p">([</span><span class="n">c</span><span class="p">,</span><span class="n">c</span><span class="o">-</span><span class="n">a2</span><span class="o">-</span><span class="n">b2</span><span class="p">],[</span><span class="n">c</span><span class="o">-</span><span class="n">a2</span><span class="p">,</span><span class="n">c</span><span class="o">-</span><span class="n">b2</span><span class="p">])</span><span class="o">*</span><span class="n">hyp3f2</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span><span class="n">b1</span><span class="p">,</span><span class="n">c</span><span class="o">-</span><span class="n">a2</span><span class="o">-</span><span class="n">b2</span><span class="p">,</span><span class="n">c</span><span class="o">-</span><span class="n">a2</span><span class="p">,</span><span class="n">c</span><span class="o">-</span><span class="n">b2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1.03947361709111140096947</span>
-</pre></div>
-</div>
-<p>The Appell F3 function satisfies a pair of partial
-differential equations:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a1</span><span class="p">,</span><span class="n">a2</span><span class="p">,</span><span class="n">b1</span><span class="p">,</span><span class="n">b2</span><span class="p">,</span><span class="n">c</span><span class="p">,</span><span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="nb">map</span><span class="p">(</span><span class="n">mpf</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.25</span><span class="p">,</span><span class="mf">0.125</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.625</span><span class="p">,</span><span class="mf">0.0625</span><span class="p">,</span><span class="o">-</span><span class="mf">0.0625</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">:</span> <span class="n">appellf3</span><span class="p">(</span><span class="n">a1</span><span class="p">,</span><span class="n">a2</span><span class="p">,</span><span class="n">b1</span><span class="p">,</span><span class="n">b2</span><span class="p">,</span><span class="n">c</span><span class="p">,</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">))</span> <span class="o">+</span>
-<span class="gp">... </span>     <span class="n">y</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span> <span class="o">+</span>
-<span class="gp">... </span>    <span class="p">(</span><span class="n">c</span><span class="o">-</span><span class="p">(</span><span class="n">a1</span><span class="o">+</span><span class="n">b1</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">))</span> <span class="o">-</span>
-<span class="gp">... </span>    <span class="n">a1</span><span class="o">*</span><span class="n">b1</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">))</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">y</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span> <span class="o">+</span>
-<span class="gp">... </span>    <span class="n">x</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span> <span class="o">+</span>
-<span class="gp">... </span>    <span class="p">(</span><span class="n">c</span><span class="o">-</span><span class="p">(</span><span class="n">a2</span><span class="o">+</span><span class="n">b2</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span> <span class="o">-</span>
-<span class="gp">... </span>    <span class="n">a2</span><span class="o">*</span><span class="n">b2</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">))</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<p>See references for <a class="reference internal" href="#mpmath.appellf1" title="mpmath.appellf1"><tt class="xref py py-func docutils literal"><span class="pre">appellf1()</span></tt></a>.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="appellf4">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">appellf4()</span></tt><a class="headerlink" href="#appellf4" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.appellf4">
-<tt class="descclassname">mpmath.</tt><tt class="descname">appellf4</tt><big>(</big><em>a</em>, <em>b</em>, <em>c1</em>, <em>c2</em>, <em>x</em>, <em>y</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.appellf4" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the Appell F4 hypergeometric function of two variables</p>
-<div class="math">
-\[F_4(a,b,c_1,c_2,x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
-    \frac{(a)_{m+n} (b)_{m+n}}{(c_1)_m (c_2)_n}
-    \frac{x^m y^n}{m! n!}.\]</div>
-<p>The series is generally absolutely convergent for
-<span class="math">\(\sqrt{|x|} + \sqrt{|y|} &lt; 1\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>Evaluation for various parameters and arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">appellf4</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mf">0.25</span><span class="p">,</span><span class="mf">0.125</span><span class="p">)</span>
-<span class="go">1.286182069079718313546608</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">appellf4</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">34.8</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">appellf4</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mf">0.25j</span><span class="p">,</span><span class="o">-</span><span class="mf">0.125j</span><span class="p">)</span>
-<span class="go">(-0.2585967215437846642163352 + 2.436102233553582711818743j)</span>
-</pre></div>
-</div>
-<p>Reduction to <span class="math">\(\,_2F_1\)</span> in a special case:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">c</span><span class="p">,</span><span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="nb">map</span><span class="p">(</span><span class="n">mpf</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.25</span><span class="p">,</span><span class="mf">0.125</span><span class="p">,</span><span class="mf">0.125</span><span class="p">,</span><span class="o">-</span><span class="mf">0.125</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">appellf4</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">c</span><span class="p">,</span><span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="o">-</span><span class="n">c</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">y</span><span class="p">),</span><span class="n">y</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="p">))</span>
-<span class="go">1.129143488466850868248364</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyp2f1</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">c</span><span class="p">,</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">hyp2f1</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="o">-</span><span class="n">c</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>
-<span class="go">1.129143488466850868248364</span>
-</pre></div>
-</div>
-<p>A system of partial differential equations satisfied by F4:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">c1</span><span class="p">,</span><span class="n">c2</span><span class="p">,</span><span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="nb">map</span><span class="p">(</span><span class="n">mpf</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.25</span><span class="p">,</span><span class="mf">1.125</span><span class="p">,</span><span class="mf">0.0625</span><span class="p">,</span><span class="o">-</span><span class="mf">0.0625</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">:</span> <span class="n">appellf4</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">c1</span><span class="p">,</span><span class="n">c2</span><span class="p">,</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">))</span> <span class="o">-</span>
-<span class="gp">... </span>     <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span> <span class="o">-</span>
-<span class="gp">... </span>     <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span> <span class="o">+</span>
-<span class="gp">... </span>     <span class="p">(</span><span class="n">c1</span><span class="o">-</span><span class="p">(</span><span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">))</span> <span class="o">-</span>
-<span class="gp">... </span>     <span class="p">((</span><span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span> <span class="o">-</span>
-<span class="gp">... </span>     <span class="n">a</span><span class="o">*</span><span class="n">b</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">))</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">y</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span> <span class="o">-</span>
-<span class="gp">... </span>     <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">))</span> <span class="o">-</span>
-<span class="gp">... </span>     <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span> <span class="o">+</span>
-<span class="gp">... </span>     <span class="p">(</span><span class="n">c2</span><span class="o">-</span><span class="p">(</span><span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span> <span class="o">-</span>
-<span class="gp">... </span>     <span class="p">((</span><span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">F</span><span class="p">,(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">))</span> <span class="o">-</span>
-<span class="gp">... </span>     <span class="n">a</span><span class="o">*</span><span class="n">b</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">))</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<p>See references for <a class="reference internal" href="#mpmath.appellf1" title="mpmath.appellf1"><tt class="xref py py-func docutils literal"><span class="pre">appellf1()</span></tt></a>.</p>
-</dd></dl>
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-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Hypergeometric functions</a><ul>
-<li><a class="reference internal" href="#common-hypergeometric-series">Common hypergeometric series</a><ul>
-<li><a class="reference internal" href="#hyp0f1"><tt class="docutils literal"><span class="pre">hyp0f1()</span></tt></a></li>
-<li><a class="reference internal" href="#hyp1f1"><tt class="docutils literal"><span class="pre">hyp1f1()</span></tt></a></li>
-<li><a class="reference internal" href="#hyp1f2"><tt class="docutils literal"><span class="pre">hyp1f2()</span></tt></a></li>
-<li><a class="reference internal" href="#hyp2f0"><tt class="docutils literal"><span class="pre">hyp2f0()</span></tt></a></li>
-<li><a class="reference internal" href="#hyp2f1"><tt class="docutils literal"><span class="pre">hyp2f1()</span></tt></a></li>
-<li><a class="reference internal" href="#hyp2f2"><tt class="docutils literal"><span class="pre">hyp2f2()</span></tt></a></li>
-<li><a class="reference internal" href="#hyp2f3"><tt class="docutils literal"><span class="pre">hyp2f3()</span></tt></a></li>
-<li><a class="reference internal" href="#hyp3f2"><tt class="docutils literal"><span class="pre">hyp3f2()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#generalized-hypergeometric-functions">Generalized hypergeometric functions</a><ul>
-<li><a class="reference internal" href="#hyper"><tt class="docutils literal"><span class="pre">hyper()</span></tt></a></li>
-<li><a class="reference internal" href="#hypercomb"><tt class="docutils literal"><span class="pre">hypercomb()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#meijer-g-function">Meijer G-function</a><ul>
-<li><a class="reference internal" href="#meijerg"><tt class="docutils literal"><span class="pre">meijerg()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#bilateral-hypergeometric-series">Bilateral hypergeometric series</a><ul>
-<li><a class="reference internal" href="#bihyper"><tt class="docutils literal"><span class="pre">bihyper()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#hypergeometric-functions-of-two-variables">Hypergeometric functions of two variables</a><ul>
-<li><a class="reference internal" href="#hyper2d"><tt class="docutils literal"><span class="pre">hyper2d()</span></tt></a></li>
-<li><a class="reference internal" href="#appellf1"><tt class="docutils literal"><span class="pre">appellf1()</span></tt></a></li>
-<li><a class="reference internal" href="#appellf2"><tt class="docutils literal"><span class="pre">appellf2()</span></tt></a></li>
-<li><a class="reference internal" href="#appellf3"><tt class="docutils literal"><span class="pre">appellf3()</span></tt></a></li>
-<li><a class="reference internal" href="#appellf4"><tt class="docutils literal"><span class="pre">appellf4()</span></tt></a></li>
-</ul>
-</li>
-</ul>
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-  <div class="section" id="mathematical-functions">
-<h1>Mathematical functions<a class="headerlink" href="#mathematical-functions" title="Permalink to this headline">¶</a></h1>
-<p>Mpmath implements the standard functions from Python&#8217;s <tt class="docutils literal"><span class="pre">math</span></tt> and <tt class="docutils literal"><span class="pre">cmath</span></tt> modules, for both real and complex numbers and with arbitrary precision. Many other functions are also available in mpmath, including commonly-used variants of standard functions (such as the alternative trigonometric functions sec, csc, cot), but also a large number of &#8220;special functions&#8221; such as the gamma function, the Riemann zeta function, error functions, Bessel functions, etc.</p>
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-<li class="toctree-l2"><a class="reference internal" href="../functions/constants.html#exact-constants">Exact constants</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../functions/constants.html#pi-pi">Pi (<tt class="docutils literal"><span class="pre">pi</span></tt>)</a></li>
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-  <div class="section" id="number-theoretical-combinatorial-and-integer-functions">
-<h1>Number-theoretical, combinatorial and integer functions<a class="headerlink" href="#number-theoretical-combinatorial-and-integer-functions" title="Permalink to this headline">¶</a></h1>
-<p>For factorial-type functions, including binomial coefficients,
-double factorials, etc., see the separate
-section <a class="reference internal" href="../functions/gamma.html"><em>Factorials and gamma functions</em></a>.</p>
-<div class="section" id="fibonacci-numbers">
-<h2>Fibonacci numbers<a class="headerlink" href="#fibonacci-numbers" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="fibonacci-fib">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">fibonacci()</span></tt>/<tt class="xref py py-func docutils literal"><span class="pre">fib()</span></tt><a class="headerlink" href="#fibonacci-fib" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.fibonacci">
-<tt class="descclassname">mpmath.</tt><tt class="descname">fibonacci</tt><big>(</big><em>n</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.fibonacci" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">fibonacci(n)</span></tt> computes the <span class="math">\(n\)</span>-th Fibonacci number, <span class="math">\(F(n)\)</span>. The
-Fibonacci numbers are defined by the recurrence <span class="math">\(F(n) = F(n-1) + F(n-2)\)</span>
-with the initial values <span class="math">\(F(0) = 0\)</span>, <span class="math">\(F(1) = 1\)</span>. <a class="reference internal" href="#mpmath.fibonacci" title="mpmath.fibonacci"><tt class="xref py py-func docutils literal"><span class="pre">fibonacci()</span></tt></a>
-extends this definition to arbitrary real and complex arguments
-using the formula</p>
-<div class="math">
-\[F(z) = \frac{\phi^z - \cos(\pi z) \phi^{-z}}{\sqrt 5}\]</div>
-<p>where <span class="math">\(\phi\)</span> is the golden ratio. <a class="reference internal" href="#mpmath.fibonacci" title="mpmath.fibonacci"><tt class="xref py py-func docutils literal"><span class="pre">fibonacci()</span></tt></a> also uses this
-continuous formula to compute <span class="math">\(F(n)\)</span> for extremely large <span class="math">\(n\)</span>, where
-calculating the exact integer would be wasteful.</p>
-<p>For convenience, <tt class="xref py py-func docutils literal"><span class="pre">fib()</span></tt> is available as an alias for
-<a class="reference internal" href="#mpmath.fibonacci" title="mpmath.fibonacci"><tt class="xref py py-func docutils literal"><span class="pre">fibonacci()</span></tt></a>.</p>
-<p><strong>Basic examples</strong></p>
-<p>Some small Fibonacci numbers are:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="n">fibonacci</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">0.0</span>
-<span class="go">1.0</span>
-<span class="go">1.0</span>
-<span class="go">2.0</span>
-<span class="go">3.0</span>
-<span class="go">5.0</span>
-<span class="go">8.0</span>
-<span class="go">13.0</span>
-<span class="go">21.0</span>
-<span class="go">34.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fibonacci</span><span class="p">(</span><span class="mi">50</span><span class="p">)</span>
-<span class="go">12586269025.0</span>
-</pre></div>
-</div>
-<p>The recurrence for <span class="math">\(F(n)\)</span> extends backwards to negative <span class="math">\(n\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="n">fibonacci</span><span class="p">(</span><span class="o">-</span><span class="n">i</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">0.0</span>
-<span class="go">1.0</span>
-<span class="go">-1.0</span>
-<span class="go">2.0</span>
-<span class="go">-3.0</span>
-<span class="go">5.0</span>
-<span class="go">-8.0</span>
-<span class="go">13.0</span>
-<span class="go">-21.0</span>
-<span class="go">34.0</span>
-</pre></div>
-</div>
-<p>Large Fibonacci numbers will be computed approximately unless
-the precision is set high enough:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fib</span><span class="p">(</span><span class="mi">200</span><span class="p">)</span>
-<span class="go">2.8057117299251e+41</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">45</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fib</span><span class="p">(</span><span class="mi">200</span><span class="p">)</span>
-<span class="go">280571172992510140037611932413038677189525.0</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.fibonacci" title="mpmath.fibonacci"><tt class="xref py py-func docutils literal"><span class="pre">fibonacci()</span></tt></a> can compute approximate Fibonacci numbers
-of stupendous size:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fibonacci</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">25</span><span class="p">)</span>
-<span class="go">3.49052338550226e+2089876402499787337692720</span>
-</pre></div>
-</div>
-<p><strong>Real and complex arguments</strong></p>
-<p>The extended Fibonacci function is an analytic function. The
-property <span class="math">\(F(z) = F(z-1) + F(z-2)\)</span> holds for arbitrary <span class="math">\(z\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fib</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">2.1170270579161</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fib</span><span class="p">(</span><span class="n">pi</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">fib</span><span class="p">(</span><span class="n">pi</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">2.1170270579161</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fib</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(-5248.51130728372 - 14195.962288353j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fib</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span> <span class="o">+</span> <span class="n">fib</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(-5248.51130728372 - 14195.962288353j)</span>
-</pre></div>
-</div>
-<p>The Fibonacci function has infinitely many roots on the
-negative half-real axis. The first root is at 0, the second is
-close to -0.18, and then there are infinitely many roots that
-asymptotically approach <span class="math">\(-n+1/2\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">fib</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">)</span>
-<span class="go">-0.183802359692956</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">fib</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">-1.57077646820395</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">fib</span><span class="p">,</span> <span class="o">-</span><span class="mi">17</span><span class="p">)</span>
-<span class="go">-16.4999999596115</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">fib</span><span class="p">,</span> <span class="o">-</span><span class="mi">24</span><span class="p">)</span>
-<span class="go">-23.5000000000479</span>
-</pre></div>
-</div>
-<p><strong>Mathematical relationships</strong></p>
-<p>For large <span class="math">\(n\)</span>, <span class="math">\(F(n+1)/F(n)\)</span> approaches the golden ratio:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fibonacci</span><span class="p">(</span><span class="mi">101</span><span class="p">)</span><span class="o">/</span><span class="n">fibonacci</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">1.6180339887498948482045868343656381177203127439638</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">+</span><span class="n">phi</span>
-<span class="go">1.6180339887498948482045868343656381177203091798058</span>
-</pre></div>
-</div>
-<p>The sum of reciprocal Fibonacci numbers converges to an irrational
-number for which no closed form expression is known:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">fib</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">3.35988566624318</span>
-</pre></div>
-</div>
-<p>Amazingly, however, the sum of odd-index reciprocal Fibonacci
-numbers can be expressed in terms of a Jacobi theta function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">fib</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">1.82451515740692</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span><span class="o">*</span><span class="n">jtheta</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">,(</span><span class="mi">3</span><span class="o">-</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">4</span>
-<span class="go">1.82451515740692</span>
-</pre></div>
-</div>
-<p>Some related sums can be done in closed form:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">fib</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">1.11803398874989</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">phi</span> <span class="o">-</span> <span class="mf">0.5</span>
-<span class="go">1.11803398874989</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">k</span><span class="p">:(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="nb">sum</span><span class="p">(</span><span class="n">fib</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="nb">int</span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">0.618033988749895</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">phi</span><span class="o">-</span><span class="mi">1</span>
-<span class="go">0.618033988749895</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference external" href="http://mathworld.wolfram.com/FibonacciNumber.html">http://mathworld.wolfram.com/FibonacciNumber.html</a></li>
-</ol>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="bernoulli-numbers-and-polynomials">
-<h2>Bernoulli numbers and polynomials<a class="headerlink" href="#bernoulli-numbers-and-polynomials" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="bernoulli">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">bernoulli()</span></tt><a class="headerlink" href="#bernoulli" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.bernoulli">
-<tt class="descclassname">mpmath.</tt><tt class="descname">bernoulli</tt><big>(</big><em>n</em><big>)</big><a class="headerlink" href="#mpmath.bernoulli" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="bernfrac">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">bernfrac()</span></tt><a class="headerlink" href="#bernfrac" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.bernfrac">
-<tt class="descclassname">mpmath.</tt><tt class="descname">bernfrac</tt><big>(</big><em>n</em><big>)</big><a class="headerlink" href="#mpmath.bernfrac" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a tuple of integers <span class="math">\((p, q)\)</span> such that <span class="math">\(p/q = B_n\)</span> exactly,
-where <span class="math">\(B_n\)</span> denotes the <span class="math">\(n\)</span>-th Bernoulli number. The fraction is
-always reduced to lowest terms. Note that for <span class="math">\(n &gt; 1\)</span> and <span class="math">\(n\)</span> odd,
-<span class="math">\(B_n = 0\)</span>, and <span class="math">\((0, 1)\)</span> is returned.</p>
-<p><strong>Examples</strong></p>
-<p>The first few Bernoulli numbers are exactly:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">15</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">bernfrac</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">/</span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">0 1/1</span>
-<span class="go">1 -1/2</span>
-<span class="go">2 1/6</span>
-<span class="go">3 0/1</span>
-<span class="go">4 -1/30</span>
-<span class="go">5 0/1</span>
-<span class="go">6 1/42</span>
-<span class="go">7 0/1</span>
-<span class="go">8 -1/30</span>
-<span class="go">9 0/1</span>
-<span class="go">10 5/66</span>
-<span class="go">11 0/1</span>
-<span class="go">12 -691/2730</span>
-<span class="go">13 0/1</span>
-<span class="go">14 7/6</span>
-</pre></div>
-</div>
-<p>This function works for arbitrarily large <span class="math">\(n\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">bernfrac</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-<span class="go">2338224387510</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">p</span><span class="p">)))</span>
-<span class="go">27692</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="o">/</span> <span class="n">q</span><span class="p">)</span>
-<span class="go">-9.04942396360948e+27677</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">bernoulli</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">4</span><span class="p">))</span>
-<span class="go">-9.04942396360948e+27677</span>
-</pre></div>
-</div>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last"><a class="reference internal" href="#mpmath.bernoulli" title="mpmath.bernoulli"><tt class="xref py py-func docutils literal"><span class="pre">bernoulli()</span></tt></a> computes a floating-point approximation
-directly, without computing the exact fraction first.
-This is much faster for large <span class="math">\(n\)</span>.</p>
-</div>
-<p><strong>Algorithm</strong></p>
-<p><a class="reference internal" href="#mpmath.bernfrac" title="mpmath.bernfrac"><tt class="xref py py-func docutils literal"><span class="pre">bernfrac()</span></tt></a> works by computing the value of <span class="math">\(B_n\)</span> numerically
-and then using the von Staudt-Clausen theorem [1] to reconstruct
-the exact fraction. For large <span class="math">\(n\)</span>, this is significantly faster than
-computing <span class="math">\(B_1, B_2, \ldots, B_2\)</span> recursively with exact arithmetic.
-The implementation has been tested for <span class="math">\(n = 10^m\)</span> up to <span class="math">\(m = 6\)</span>.</p>
-<p>In practice, <a class="reference internal" href="#mpmath.bernfrac" title="mpmath.bernfrac"><tt class="xref py py-func docutils literal"><span class="pre">bernfrac()</span></tt></a> appears to be about three times
-slower than the specialized program calcbn.exe [2]</p>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li>MathWorld, von Staudt-Clausen Theorem:
-<a class="reference external" href="http://mathworld.wolfram.com/vonStaudt-ClausenTheorem.html">http://mathworld.wolfram.com/vonStaudt-ClausenTheorem.html</a></li>
-<li>The Bernoulli Number Page:
-<a class="reference external" href="http://www.bernoulli.org/">http://www.bernoulli.org/</a></li>
-</ol>
-</dd></dl>
-
-</div>
-<div class="section" id="bernpoly">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">bernpoly()</span></tt><a class="headerlink" href="#bernpoly" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.bernpoly">
-<tt class="descclassname">mpmath.</tt><tt class="descname">bernpoly</tt><big>(</big><em>n</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.bernpoly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the Bernoulli polynomial <span class="math">\(B_n(z)\)</span>.</p>
-<p>The first few Bernoulli polynomials are:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">6</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">bernpoly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">)))</span>
-<span class="gp">...</span>
-<span class="go">[1.0]</span>
-<span class="go">[-0.5, 1.0]</span>
-<span class="go">[0.166667, -1.0, 1.0]</span>
-<span class="go">[0.0, 0.5, -1.5, 1.0]</span>
-<span class="go">[-0.0333333, 0.0, 1.0, -2.0, 1.0]</span>
-<span class="go">[0.0, -0.166667, 0.0, 1.66667, -2.5, 1.0]</span>
-</pre></div>
-</div>
-<p>At <span class="math">\(z = 0\)</span>, the Bernoulli polynomial evaluates to a
-Bernoulli number (see <a class="reference internal" href="#mpmath.bernoulli" title="mpmath.bernoulli"><tt class="xref py py-func docutils literal"><span class="pre">bernoulli()</span></tt></a>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">bernpoly</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">bernoulli</span><span class="p">(</span><span class="mi">12</span><span class="p">)</span>
-<span class="go">(-0.253113553113553, -0.253113553113553)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bernpoly</span><span class="p">(</span><span class="mi">13</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">bernoulli</span><span class="p">(</span><span class="mi">13</span><span class="p">)</span>
-<span class="go">(0.0, 0.0)</span>
-</pre></div>
-</div>
-<p>Evaluation is accurate for large <span class="math">\(n\)</span> and small <span class="math">\(z\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bernpoly</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
-<span class="go">2.838224957069370695926416e+78</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bernpoly</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span> <span class="mf">10.5</span><span class="p">)</span>
-<span class="go">5.318704469415522036482914e+1769</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="euler-numbers-and-polynomials">
-<h2>Euler numbers and polynomials<a class="headerlink" href="#euler-numbers-and-polynomials" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="eulernum">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">eulernum()</span></tt><a class="headerlink" href="#eulernum" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.eulernum">
-<tt class="descclassname">mpmath.</tt><tt class="descname">eulernum</tt><big>(</big><em>n</em><big>)</big><a class="headerlink" href="#mpmath.eulernum" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the <span class="math">\(n\)</span>-th Euler number, defined as the <span class="math">\(n\)</span>-th derivative of
-<span class="math">\(\mathrm{sech}(t) = 1/\cosh(t)\)</span> evaluated at <span class="math">\(t = 0\)</span>. Equivalently, the
-Euler numbers give the coefficients of the Taylor series</p>
-<div class="math">
-\[\mathrm{sech}(t) = \sum_{n=0}^{\infty} \frac{E_n}{n!} t^n.\]</div>
-<p>The Euler numbers are closely related to Bernoulli numbers
-and Bernoulli polynomials. They can also be evaluated in terms of
-Euler polynomials (see <a class="reference internal" href="#mpmath.eulerpoly" title="mpmath.eulerpoly"><tt class="xref py py-func docutils literal"><span class="pre">eulerpoly()</span></tt></a>) as <span class="math">\(E_n = 2^n E_n(1/2)\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>Computing the first few Euler numbers and verifying that they
-agree with the Taylor series:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">eulernum</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">11</span><span class="p">)]</span>
-<span class="go">[1.0, 0.0, -1.0, 0.0, 5.0, 0.0, -61.0, 0.0, 1385.0, 0.0, -50521.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">diffs</span><span class="p">(</span><span class="n">sech</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
-<span class="go">[1.0, 0.0, -1.0, 0.0, 5.0, 0.0, -61.0, 0.0, 1385.0, 0.0, -50521.0]</span>
-</pre></div>
-</div>
-<p>Euler numbers grow very rapidly. <a class="reference internal" href="#mpmath.eulernum" title="mpmath.eulernum"><tt class="xref py py-func docutils literal"><span class="pre">eulernum()</span></tt></a> efficiently
-computes numerical approximations for large indices:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">eulernum</span><span class="p">(</span><span class="mi">50</span><span class="p">)</span>
-<span class="go">-6.053285248188621896314384e+54</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eulernum</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span>
-<span class="go">3.887561841253070615257336e+2371</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eulernum</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">20</span><span class="p">)</span>
-<span class="go">4.346791453661149089338186e+1936958564106659551331</span>
-</pre></div>
-</div>
-<p>Comparing with an asymptotic formula for the Euler numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="mi">10</span><span class="o">**</span><span class="mi">5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="o">//</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="mi">8</span> <span class="o">*</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">/</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">e</span><span class="p">))</span><span class="o">**</span><span class="n">n</span>
-<span class="go">3.69919063017432362805663e+436961</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eulernum</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">3.699193712834466537941283e+436961</span>
-</pre></div>
-</div>
-<p>Pass <tt class="docutils literal"><span class="pre">exact=True</span></tt> to obtain exact values of Euler numbers as integers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">eulernum</span><span class="p">(</span><span class="mi">50</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">-6053285248188621896314383785111649088103498225146815121</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">eulernum</span><span class="p">(</span><span class="mi">200</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="o">%</span> <span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">1925859625</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eulernum</span><span class="p">(</span><span class="mi">1001</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="eulerpoly">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">eulerpoly()</span></tt><a class="headerlink" href="#eulerpoly" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.eulerpoly">
-<tt class="descclassname">mpmath.</tt><tt class="descname">eulerpoly</tt><big>(</big><em>n</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.eulerpoly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the Euler polynomial <span class="math">\(E_n(z)\)</span>, defined by the generating function
-representation</p>
-<div class="math">
-\[\frac{2e^{zt}}{e^t+1} = \sum_{n=0}^\infty E_n(z) \frac{t^n}{n!}.\]</div>
-<p>The Euler polynomials may also be represented in terms of
-Bernoulli polynomials (see <a class="reference internal" href="#mpmath.bernpoly" title="mpmath.bernpoly"><tt class="xref py py-func docutils literal"><span class="pre">bernpoly()</span></tt></a>) using various formulas, for
-example</p>
-<div class="math">
-\[E_n(z) = \frac{2}{n+1} \left(
-    B_n(z)-2^{n+1}B_n\left(\frac{z}{2}\right)
-\right).\]</div>
-<p>Special values include the Euler numbers <span class="math">\(E_n = 2^n E_n(1/2)\)</span> (see
-<a class="reference internal" href="#mpmath.eulernum" title="mpmath.eulernum"><tt class="xref py py-func docutils literal"><span class="pre">eulernum()</span></tt></a>).</p>
-<p><strong>Examples</strong></p>
-<p>Computing the coefficients of the first few Euler polynomials:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">6</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">eulerpoly</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">z</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">[1.0]</span>
-<span class="go">[-0.5, 1.0]</span>
-<span class="go">[0.0, -1.0, 1.0]</span>
-<span class="go">[0.25, 0.0, -1.5, 1.0]</span>
-<span class="go">[0.0, 1.0, 0.0, -2.0, 1.0]</span>
-<span class="go">[-0.5, 0.0, 2.5, 0.0, -2.5, 1.0]</span>
-</pre></div>
-</div>
-<p>Evaluation for arbitrary <span class="math">\(z\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">eulerpoly</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">6.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eulerpoly</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">423.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eulerpoly</span><span class="p">(</span><span class="mi">35</span><span class="p">,</span> <span class="mi">11111111112</span><span class="p">)</span>
-<span class="go">3.994957561486776072734601e+351</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eulerpoly</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">10</span><span class="o">+</span><span class="mi">20j</span><span class="p">)</span>
-<span class="go">(-47990.0 - 235980.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eulerpoly</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="s">&#39;-3.5e-5&#39;</span><span class="p">)</span>
-<span class="go">0.000035001225</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eulerpoly</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eulerpoly</span><span class="p">(</span><span class="mi">55</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="o">**</span><span class="mi">80</span><span class="p">)</span>
-<span class="go">-1.0e+4400</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eulerpoly</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="o">-</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">-inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eulerpoly</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="o">-</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">+inf</span>
-</pre></div>
-</div>
-<p>Computing Euler numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span><span class="o">**</span><span class="mi">26</span> <span class="o">*</span> <span class="n">eulerpoly</span><span class="p">(</span><span class="mi">26</span><span class="p">,</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="go">-4087072509293123892361.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eulernum</span><span class="p">(</span><span class="mi">26</span><span class="p">)</span>
-<span class="go">-4087072509293123892361.0</span>
-</pre></div>
-</div>
-<p>Evaluation is accurate for large <span class="math">\(n\)</span> and small <span class="math">\(z\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">eulerpoly</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
-<span class="go">2.29047999988194114177943e+108</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eulerpoly</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span> <span class="mf">10.5</span><span class="p">)</span>
-<span class="go">3.628120031122876847764566e+2070</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eulerpoly</span><span class="p">(</span><span class="mi">10000</span><span class="p">,</span> <span class="mf">10.5</span><span class="p">)</span>
-<span class="go">1.149364285543783412210773e+30688</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="bell-numbers-and-polynomials">
-<h2>Bell numbers and polynomials<a class="headerlink" href="#bell-numbers-and-polynomials" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="bell">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">bell()</span></tt><a class="headerlink" href="#bell" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.bell">
-<tt class="descclassname">mpmath.</tt><tt class="descname">bell</tt><big>(</big><em>n</em>, <em>x</em><big>)</big><a class="headerlink" href="#mpmath.bell" title="Permalink to this definition">¶</a></dt>
-<dd><p>For <span class="math">\(n\)</span> a nonnegative integer, <tt class="docutils literal"><span class="pre">bell(n,x)</span></tt> evaluates the Bell
-polynomial <span class="math">\(B_n(x)\)</span>, the first few of which are</p>
-<div class="math">
-\[B_0(x) = 1\]\[B_1(x) = x\]\[B_2(x) = x^2+x\]\[B_3(x) = x^3+3x^2+x\]</div>
-<p>If <span class="math">\(x = 1\)</span> or <a class="reference internal" href="#mpmath.bell" title="mpmath.bell"><tt class="xref py py-func docutils literal"><span class="pre">bell()</span></tt></a> is called with only one argument, it
-gives the <span class="math">\(n\)</span>-th Bell number <span class="math">\(B_n\)</span>, which is the number of
-partitions of a set with <span class="math">\(n\)</span> elements. By setting the precision to
-at least <span class="math">\(\log_{10} B_n\)</span> digits, <a class="reference internal" href="#mpmath.bell" title="mpmath.bell"><tt class="xref py py-func docutils literal"><span class="pre">bell()</span></tt></a> provides fast
-calculation of exact Bell numbers.</p>
-<p>In general, <a class="reference internal" href="#mpmath.bell" title="mpmath.bell"><tt class="xref py py-func docutils literal"><span class="pre">bell()</span></tt></a> computes</p>
-<div class="math">
-\[B_n(x) = e^{-x} \left(\mathrm{sinc}(\pi n) + E_n(x)\right)\]</div>
-<p>where <span class="math">\(E_n(x)\)</span> is the generalized exponential function implemented
-by <a class="reference internal" href="../functions/zeta.html#mpmath.polyexp" title="mpmath.polyexp"><tt class="xref py py-func docutils literal"><span class="pre">polyexp()</span></tt></a>. This is an extension of Dobinski&#8217;s formula [1],
-where the modification is the sinc term ensuring that <span class="math">\(B_n(x)\)</span> is
-continuous in <span class="math">\(n\)</span>; <a class="reference internal" href="#mpmath.bell" title="mpmath.bell"><tt class="xref py py-func docutils literal"><span class="pre">bell()</span></tt></a> can thus be evaluated,
-differentiated, etc for arbitrary complex arguments.</p>
-<p><strong>Examples</strong></p>
-<p>Simple evaluations:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bell</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bell</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">)</span>
-<span class="go">2.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bell</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">)</span>
-<span class="go">8.75</span>
-</pre></div>
-</div>
-<p>Evaluation for arbitrary complex arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">bell</span><span class="p">(</span><span class="mf">5.75</span><span class="o">+</span><span class="mi">1j</span><span class="p">,</span> <span class="mi">2</span><span class="o">-</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(-10767.71345136587098445143 - 15449.55065599872579097221j)</span>
-</pre></div>
-</div>
-<p>The first few Bell polynomials:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">7</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">nprint</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">bell</span><span class="p">(</span><span class="n">k</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">k</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">[1.0]</span>
-<span class="go">[0.0, 1.0]</span>
-<span class="go">[0.0, 1.0, 1.0]</span>
-<span class="go">[0.0, 1.0, 3.0, 1.0]</span>
-<span class="go">[0.0, 1.0, 7.0, 6.0, 1.0]</span>
-<span class="go">[0.0, 1.0, 15.0, 25.0, 10.0, 1.0]</span>
-<span class="go">[0.0, 1.0, 31.0, 90.0, 65.0, 15.0, 1.0]</span>
-</pre></div>
-</div>
-<p>The first few Bell numbers and complementary Bell numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">bell</span><span class="p">(</span><span class="n">k</span><span class="p">))</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">)]</span>
-<span class="go">[1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="nb">int</span><span class="p">(</span><span class="n">bell</span><span class="p">(</span><span class="n">k</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">)]</span>
-<span class="go">[1, -1, 0, 1, 1, -2, -9, -9, 50, 267]</span>
-</pre></div>
-</div>
-<p>Large Bell numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bell</span><span class="p">(</span><span class="mi">50</span><span class="p">)</span>
-<span class="go">185724268771078270438257767181908917499221852770.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bell</span><span class="p">(</span><span class="mi">50</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-29113173035759403920216141265491160286912.0</span>
-</pre></div>
-</div>
-<p>Some even larger values:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bell</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-1.237132026969293954162816e+1869</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bell</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span>
-<span class="go">2.989901335682408421480422e+1927</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bell</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">6.591553486811969380442171e+1987</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bell</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span><span class="mf">100.5</span><span class="p">)</span>
-<span class="go">9.101014101401543575679639e+2529</span>
-</pre></div>
-</div>
-<p>A determinant identity satisfied by Bell numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="mi">8</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">det</span><span class="p">([[</span><span class="n">bell</span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="n">j</span><span class="p">)</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">)]</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">)])</span>
-<span class="go">125411328000.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">superfac</span><span class="p">(</span><span class="n">N</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">125411328000.0</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference external" href="http://mathworld.wolfram.com/DobinskisFormula.html">http://mathworld.wolfram.com/DobinskisFormula.html</a></li>
-</ol>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="prime-counting-functions">
-<h2>Prime counting functions<a class="headerlink" href="#prime-counting-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="primepi">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">primepi()</span></tt><a class="headerlink" href="#primepi" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.primepi">
-<tt class="descclassname">mpmath.</tt><tt class="descname">primepi</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.primepi" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="primepi2">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">primepi2()</span></tt><a class="headerlink" href="#primepi2" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.primepi2">
-<tt class="descclassname">mpmath.</tt><tt class="descname">primepi2</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.primepi2" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns an interval (as an <tt class="docutils literal"><span class="pre">mpi</span></tt> instance) providing bounds
-for the value of the prime counting function <span class="math">\(\pi(x)\)</span>. For small
-<span class="math">\(x\)</span>, <a class="reference internal" href="#mpmath.primepi2" title="mpmath.primepi2"><tt class="xref py py-func docutils literal"><span class="pre">primepi2()</span></tt></a> returns an exact interval based on
-the output of <a class="reference internal" href="#mpmath.primepi" title="mpmath.primepi"><tt class="xref py py-func docutils literal"><span class="pre">primepi()</span></tt></a>. For <span class="math">\(x &gt; 2656\)</span>, a loose interval
-based on Schoenfeld&#8217;s inequality</p>
-<div class="math">
-\[\begin{split}|\pi(x) - \mathrm{li}(x)| &lt; \frac{\sqrt x \log x}{8 \pi}\end{split}\]</div>
-<p>is returned. This estimate is rigorous assuming the truth of
-the Riemann hypothesis, and can be computed very quickly.</p>
-<p><strong>Examples</strong></p>
-<p>Exact values of the prime counting function for small <span class="math">\(x\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">iv</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primepi2</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">[4.0, 4.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primepi2</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">[25.0, 25.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primepi2</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span>
-<span class="go">[168.0, 168.0]</span>
-</pre></div>
-</div>
-<p>Loose intervals are generated for moderately large <span class="math">\(x\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">primepi2</span><span class="p">(</span><span class="mi">10000</span><span class="p">),</span> <span class="n">primepi</span><span class="p">(</span><span class="mi">10000</span><span class="p">)</span>
-<span class="go">([1209.0, 1283.0], 1229)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primepi2</span><span class="p">(</span><span class="mi">50000</span><span class="p">),</span> <span class="n">primepi</span><span class="p">(</span><span class="mi">50000</span><span class="p">)</span>
-<span class="go">([5070.0, 5263.0], 5133)</span>
-</pre></div>
-</div>
-<p>As <span class="math">\(x\)</span> increases, the absolute error gets worse while the relative
-error improves. The exact value of <span class="math">\(\pi(10^{23})\)</span> is
-1925320391606803968923, and <a class="reference internal" href="#mpmath.primepi2" title="mpmath.primepi2"><tt class="xref py py-func docutils literal"><span class="pre">primepi2()</span></tt></a> gives 9 significant
-digits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">primepi2</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">23</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span>
-<span class="go">[1.9253203909477020467e+21, 1.925320392280406229e+21]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpf</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">delta</span><span class="p">)</span> <span class="o">/</span> <span class="n">mpf</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">a</span><span class="p">)</span>
-<span class="go">6.9219865355293e-10</span>
-</pre></div>
-</div>
-<p>A more precise, nonrigorous estimate for <span class="math">\(\pi(x)\)</span> can be
-obtained using the Riemann R function (<a class="reference internal" href="#mpmath.riemannr" title="mpmath.riemannr"><tt class="xref py py-func docutils literal"><span class="pre">riemannr()</span></tt></a>).
-For large enough <span class="math">\(x\)</span>, the value returned by <a class="reference internal" href="#mpmath.primepi2" title="mpmath.primepi2"><tt class="xref py py-func docutils literal"><span class="pre">primepi2()</span></tt></a>
-essentially amounts to a small perturbation of the value returned by
-<a class="reference internal" href="#mpmath.riemannr" title="mpmath.riemannr"><tt class="xref py py-func docutils literal"><span class="pre">riemannr()</span></tt></a>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">primepi2</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">[4.3619719871407024816e+97, 4.3619719871407032404e+97]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">riemannr</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">4.3619719871407e+97</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="riemannr">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">riemannr()</span></tt><a class="headerlink" href="#riemannr" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.riemannr">
-<tt class="descclassname">mpmath.</tt><tt class="descname">riemannr</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.riemannr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the Riemann R function, a smooth approximation of the
-prime counting function <span class="math">\(\pi(x)\)</span> (see <a class="reference internal" href="#mpmath.primepi" title="mpmath.primepi"><tt class="xref py py-func docutils literal"><span class="pre">primepi()</span></tt></a>). The Riemann
-R function gives a fast numerical approximation useful e.g. to
-roughly estimate the number of primes in a given interval.</p>
-<p>The Riemann R function is computed using the rapidly convergent Gram
-series,</p>
-<div class="math">
-\[R(x) = 1 + \sum_{k=1}^{\infty}
-    \frac{\log^k x}{k k! \zeta(k+1)}.\]</div>
-<p>From the Gram series, one sees that the Riemann R function is a
-well-defined analytic function (except for a branch cut along
-the negative real half-axis); it can be evaluated for arbitrary
-real or complex arguments.</p>
-<p>The Riemann R function gives a very accurate approximation
-of the prime counting function. For example, it is wrong by at
-most 2 for <span class="math">\(x &lt; 1000\)</span>, and for <span class="math">\(x = 10^9\)</span> differs from the exact
-value of <span class="math">\(\pi(x)\)</span> by 79, or less than two parts in a million.
-It is about 10 times more accurate than the logarithmic integral
-estimate (see <a class="reference internal" href="../functions/expintegrals.html#mpmath.li" title="mpmath.li"><tt class="xref py py-func docutils literal"><span class="pre">li()</span></tt></a>), which however is even faster to evaluate.
-It is orders of magnitude more accurate than the extremely
-fast <span class="math">\(x/\log x\)</span> estimate.</p>
-<p><strong>Examples</strong></p>
-<p>For small arguments, the Riemann R function almost exactly
-gives the prime counting function if rounded to the nearest
-integer:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primepi</span><span class="p">(</span><span class="mi">50</span><span class="p">),</span> <span class="n">riemannr</span><span class="p">(</span><span class="mi">50</span><span class="p">)</span>
-<span class="go">(15, 14.9757023241462)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">max</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">primepi</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">-</span><span class="nb">int</span><span class="p">(</span><span class="nb">round</span><span class="p">(</span><span class="n">riemannr</span><span class="p">(</span><span class="n">n</span><span class="p">))))</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">100</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">max</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">primepi</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">-</span><span class="nb">int</span><span class="p">(</span><span class="nb">round</span><span class="p">(</span><span class="n">riemannr</span><span class="p">(</span><span class="n">n</span><span class="p">))))</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">300</span><span class="p">))</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-<p>The Riemann R function can be evaluated for arguments far too large
-for exact determination of <span class="math">\(\pi(x)\)</span> to be computationally
-feasible with any presently known algorithm:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">riemannr</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">30</span><span class="p">)</span>
-<span class="go">1.46923988977204e+28</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">riemannr</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">4.3619719871407e+97</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">riemannr</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">1000</span><span class="p">)</span>
-<span class="go">4.3448325764012e+996</span>
-</pre></div>
-</div>
-<p>A comparison of the Riemann R function and logarithmic integral estimates
-for <span class="math">\(\pi(x)\)</span> using exact values of <span class="math">\(\pi(10^n)\)</span> up to <span class="math">\(n = 9\)</span>.
-The fractional error is shown in parentheses:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">exact</span> <span class="o">=</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span><span class="mi">25</span><span class="p">,</span><span class="mi">168</span><span class="p">,</span><span class="mi">1229</span><span class="p">,</span><span class="mi">9592</span><span class="p">,</span><span class="mi">78498</span><span class="p">,</span><span class="mi">664579</span><span class="p">,</span><span class="mi">5761455</span><span class="p">,</span><span class="mi">50847534</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">exact</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">n</span> <span class="o">+=</span> <span class="mi">1</span>
-<span class="gp">... </span>    <span class="n">r</span><span class="p">,</span> <span class="n">l</span> <span class="o">=</span> <span class="n">riemannr</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="n">n</span><span class="p">),</span> <span class="n">li</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="n">n</span><span class="p">)</span>
-<span class="gp">... </span>    <span class="n">rerr</span><span class="p">,</span> <span class="n">lerr</span> <span class="o">=</span> <span class="n">nstr</span><span class="p">((</span><span class="n">r</span><span class="o">-</span><span class="n">p</span><span class="p">)</span><span class="o">/</span><span class="n">p</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="n">nstr</span><span class="p">((</span><span class="n">l</span><span class="o">-</span><span class="n">p</span><span class="p">)</span><span class="o">/</span><span class="n">p</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%i</span><span class="s"> </span><span class="si">%i</span><span class="s"> </span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">) </span><span class="si">%s</span><span class="s">(</span><span class="si">%s</span><span class="s">)&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">rerr</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="n">lerr</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">1 4 4.56458314100509(0.141) 6.1655995047873(0.541)</span>
-<span class="go">2 25 25.6616332669242(0.0265) 30.1261415840796(0.205)</span>
-<span class="go">3 168 168.359446281167(0.00214) 177.609657990152(0.0572)</span>
-<span class="go">4 1229 1226.93121834343(-0.00168) 1246.13721589939(0.0139)</span>
-<span class="go">5 9592 9587.43173884197(-0.000476) 9629.8090010508(0.00394)</span>
-<span class="go">6 78498 78527.3994291277(0.000375) 78627.5491594622(0.00165)</span>
-<span class="go">7 664579 664667.447564748(0.000133) 664918.405048569(0.000511)</span>
-<span class="go">8 5761455 5761551.86732017(1.68e-5) 5762209.37544803(0.000131)</span>
-<span class="go">9 50847534 50847455.4277214(-1.55e-6) 50849234.9570018(3.35e-5)</span>
-</pre></div>
-</div>
-<p>The derivative of the Riemann R function gives the approximate
-probability for a number of magnitude <span class="math">\(x\)</span> to be prime:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">riemannr</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
-<span class="go">0.141903028110784</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpf</span><span class="p">(</span><span class="n">primepi</span><span class="p">(</span><span class="mi">1050</span><span class="p">)</span> <span class="o">-</span> <span class="n">primepi</span><span class="p">(</span><span class="mi">950</span><span class="p">))</span> <span class="o">/</span> <span class="mi">100</span>
-<span class="go">0.15</span>
-</pre></div>
-</div>
-<p>Evaluation is supported for arbitrary arguments and at arbitrary
-precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">30</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">riemannr</span><span class="p">(</span><span class="mf">7.5</span><span class="p">)</span>
-<span class="go">3.72934743264966261918857135136</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">riemannr</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="o">+</span><span class="mi">2j</span><span class="p">)</span>
-<span class="go">(-0.551002208155486427591793957644 + 2.16966398138119450043195899746j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="cyclotomic-polynomials">
-<h2>Cyclotomic polynomials<a class="headerlink" href="#cyclotomic-polynomials" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="cyclotomic">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">cyclotomic()</span></tt><a class="headerlink" href="#cyclotomic" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.cyclotomic">
-<tt class="descclassname">mpmath.</tt><tt class="descname">cyclotomic</tt><big>(</big><em>n</em>, <em>x</em><big>)</big><a class="headerlink" href="#mpmath.cyclotomic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the cyclotomic polynomial <span class="math">\(\Phi_n(x)\)</span>, defined by</p>
-<div class="math">
-\[\Phi_n(x) = \prod_{\zeta} (x - \zeta)\]</div>
-<p>where <span class="math">\(\zeta\)</span> ranges over all primitive <span class="math">\(n\)</span>-th roots of unity
-(see <a class="reference internal" href="../functions/powers.html#mpmath.unitroots" title="mpmath.unitroots"><tt class="xref py py-func docutils literal"><span class="pre">unitroots()</span></tt></a>). An equivalent representation, used
-for computation, is</p>
-<div class="math">
-\[\Phi_n(x) = \prod_{d\mid n}(x^d-1)^{\mu(n/d)} = \Phi_n(x)\]</div>
-<p>where <span class="math">\(\mu(m)\)</span> denotes the Moebius function. The cyclotomic
-polynomials are integer polynomials, the first of which can be
-written explicitly as</p>
-<div class="math">
-\[\Phi_0(x) = 1\]\[\Phi_1(x) = x - 1\]\[\Phi_2(x) = x + 1\]\[\Phi_3(x) = x^3 + x^2 + 1\]\[\Phi_4(x) = x^2 + 1\]\[\Phi_5(x) = x^4 + x^3 + x^2 + x + 1\]\[\Phi_6(x) = x^2 - x + 1\]</div>
-<p><strong>Examples</strong></p>
-<p>The coefficients of low-order cyclotomic polynomials can be recovered
-using Taylor expansion:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">9</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">p</span> <span class="o">=</span> <span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">cyclotomic</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">nstr</span><span class="p">(</span><span class="n">p</span><span class="p">[:</span><span class="mi">10</span><span class="o">+</span><span class="mi">1</span><span class="o">-</span><span class="n">p</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="mi">1</span><span class="p">)])))</span>
-<span class="gp">...</span>
-<span class="go">0 [1.0]</span>
-<span class="go">1 [-1.0, 1.0]</span>
-<span class="go">2 [1.0, 1.0]</span>
-<span class="go">3 [1.0, 1.0, 1.0]</span>
-<span class="go">4 [1.0, 0.0, 1.0]</span>
-<span class="go">5 [1.0, 1.0, 1.0, 1.0, 1.0]</span>
-<span class="go">6 [1.0, -1.0, 1.0]</span>
-<span class="go">7 [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]</span>
-<span class="go">8 [1.0, 0.0, 0.0, 0.0, 1.0]</span>
-</pre></div>
-</div>
-<p>The definition as a product over primitive roots may be checked
-by computing the product explicitly (for a real argument, this
-method will generally introduce numerical noise in the imaginary
-part):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mi">3</span><span class="o">+</span><span class="mi">4j</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cyclotomic</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">(-419.0 - 360.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fprod</span><span class="p">(</span><span class="n">z</span><span class="o">-</span><span class="n">r</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">unitroots</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="n">primitive</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">(-419.0 - 360.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cyclotomic</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">61.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fprod</span><span class="p">(</span><span class="n">z</span><span class="o">-</span><span class="n">r</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">unitroots</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="n">primitive</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">(61.0 - 3.146045605088568607055454e-25j)</span>
-</pre></div>
-</div>
-<p>Up to permutation, the roots of a given cyclotomic polynomial
-can be checked to agree with the list of primitive roots:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">cyclotomic</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">)[:</span><span class="mi">3</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">polyroots</span><span class="p">(</span><span class="n">p</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]):</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="go">(0.5 - 0.8660254037844386467637232j)</span>
-<span class="go">(0.5 + 0.8660254037844386467637232j)</span>
-<span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">unitroots</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="n">primitive</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="go">(0.5 + 0.8660254037844386467637232j)</span>
-<span class="go">(0.5 - 0.8660254037844386467637232j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="arithmetic-functions">
-<h2>Arithmetic functions<a class="headerlink" href="#arithmetic-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="mangoldt">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">mangoldt()</span></tt><a class="headerlink" href="#mangoldt" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.mangoldt">
-<tt class="descclassname">mpmath.</tt><tt class="descname">mangoldt</tt><big>(</big><em>n</em><big>)</big><a class="headerlink" href="#mpmath.mangoldt" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the von Mangoldt function <span class="math">\(\Lambda(n) = \log p\)</span>
-if <span class="math">\(n = p^k\)</span> a power of a prime, and <span class="math">\(\Lambda(n) = 0\)</span> otherwise.</p>
-<p><strong>Examples</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">mangoldt</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)]</span>
-<span class="go">[0.0, 0.0, 0.0, 0.0, 0.6931471805599453094172321]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mangoldt</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mangoldt</span><span class="p">(</span><span class="mi">7</span><span class="p">)</span>
-<span class="go">1.945910149055313305105353</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mangoldt</span><span class="p">(</span><span class="mi">8</span><span class="p">)</span>
-<span class="go">0.6931471805599453094172321</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fsum</span><span class="p">(</span><span class="n">mangoldt</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">101</span><span class="p">))</span>
-<span class="go">94.04531122935739224600493</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fsum</span><span class="p">(</span><span class="n">mangoldt</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10001</span><span class="p">))</span>
-<span class="go">10013.39669326311478372032</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-</div>
-
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-        </div>
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-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Number-theoretical, combinatorial and integer functions</a><ul>
-<li><a class="reference internal" href="#fibonacci-numbers">Fibonacci numbers</a><ul>
-<li><a class="reference internal" href="#fibonacci-fib"><tt class="docutils literal"><span class="pre">fibonacci()</span></tt>/<tt class="docutils literal"><span class="pre">fib()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#bernoulli-numbers-and-polynomials">Bernoulli numbers and polynomials</a><ul>
-<li><a class="reference internal" href="#bernoulli"><tt class="docutils literal"><span class="pre">bernoulli()</span></tt></a></li>
-<li><a class="reference internal" href="#bernfrac"><tt class="docutils literal"><span class="pre">bernfrac()</span></tt></a></li>
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-</ul>
-</li>
-<li><a class="reference internal" href="#euler-numbers-and-polynomials">Euler numbers and polynomials</a><ul>
-<li><a class="reference internal" href="#eulernum"><tt class="docutils literal"><span class="pre">eulernum()</span></tt></a></li>
-<li><a class="reference internal" href="#eulerpoly"><tt class="docutils literal"><span class="pre">eulerpoly()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#bell-numbers-and-polynomials">Bell numbers and polynomials</a><ul>
-<li><a class="reference internal" href="#bell"><tt class="docutils literal"><span class="pre">bell()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#prime-counting-functions">Prime counting functions</a><ul>
-<li><a class="reference internal" href="#primepi"><tt class="docutils literal"><span class="pre">primepi()</span></tt></a></li>
-<li><a class="reference internal" href="#primepi2"><tt class="docutils literal"><span class="pre">primepi2()</span></tt></a></li>
-<li><a class="reference internal" href="#riemannr"><tt class="docutils literal"><span class="pre">riemannr()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#cyclotomic-polynomials">Cyclotomic polynomials</a><ul>
-<li><a class="reference internal" href="#cyclotomic"><tt class="docutils literal"><span class="pre">cyclotomic()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#arithmetic-functions">Arithmetic functions</a><ul>
-<li><a class="reference internal" href="#mangoldt"><tt class="docutils literal"><span class="pre">mangoldt()</span></tt></a></li>
-</ul>
-</li>
-</ul>
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-</ul>
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-  <div class="section" id="orthogonal-polynomials">
-<h1>Orthogonal polynomials<a class="headerlink" href="#orthogonal-polynomials" title="Permalink to this headline">¶</a></h1>
-<p>An orthogonal polynomial sequence is a sequence of polynomials <span class="math">\(P_0(x), P_1(x), \ldots\)</span> of degree <span class="math">\(0, 1, \ldots\)</span>, which are mutually orthogonal in the sense that</p>
-<div class="math">
-\[\begin{split}\int_S P_n(x) P_m(x) w(x) dx =
-\begin{cases}
-c_n \ne 0 &amp; \text{if $m = n$} \\
-0         &amp; \text{if $m \ne n$}
-\end{cases}\end{split}\]</div>
-<p>where <span class="math">\(S\)</span> is some domain (e.g. an interval <span class="math">\([a,b] \in \mathbb{R}\)</span>) and <span class="math">\(w(x)\)</span> is a fixed <em>weight function</em>. A sequence of orthogonal polynomials is determined completely by <span class="math">\(w\)</span>, <span class="math">\(S\)</span>, and a normalization convention (e.g. <span class="math">\(c_n = 1\)</span>). Applications of orthogonal polynomials include function approximation and solution of differential equations.</p>
-<p>Orthogonal polynomials are sometimes defined using the differential equations they satisfy (as functions of <span class="math">\(x\)</span>) or the recurrence relations they satisfy with respect to the order <span class="math">\(n\)</span>. Other ways of defining orthogonal polynomials include differentiation formulas and generating functions. The standard orthogonal polynomials can also be represented as hypergeometric series (see <a class="reference internal" href="../functions/hypergeometric.html"><em>Hypergeometric functions</em></a>), more specifically using the Gauss hypergeometric function <span class="math">\(\,_2F_1\)</span> in most cases. The following functions are generally implemented using hypergeometric functions since this is computationally efficient and easily generalizes.</p>
-<p>For more information, see the <a class="reference external" href="http://en.wikipedia.org/wiki/Orthogonal_polynomials">Wikipedia article on orthogonal polynomials</a>.</p>
-<div class="section" id="legendre-functions">
-<h2>Legendre functions<a class="headerlink" href="#legendre-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="legendre">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">legendre()</span></tt><a class="headerlink" href="#legendre" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.legendre">
-<tt class="descclassname">mpmath.</tt><tt class="descname">legendre</tt><big>(</big><em>n</em>, <em>x</em><big>)</big><a class="headerlink" href="#mpmath.legendre" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">legendre(n,</span> <span class="pre">x)</span></tt> evaluates the Legendre polynomial <span class="math">\(P_n(x)\)</span>.
-The Legendre polynomials are given by the formula</p>
-<div class="math">
-\[P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 -1)^n.\]</div>
-<p>Alternatively, they can be computed recursively using</p>
-<div class="math">
-\[P_0(x) = 1\]\[P_1(x) = x\]\[(n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x).\]</div>
-<p>A third definition is in terms of the hypergeometric function
-<span class="math">\(\,_2F_1\)</span>, whereby they can be generalized to arbitrary <span class="math">\(n\)</span>:</p>
-<div class="math">
-\[P_n(x) = \,_2F_1\left(-n, n+1, 1, \frac{1-x}{2}\right)\]</div>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Legendre polynomials P_n(x) on [-1,1] for n=0,1,2,3,4</span>
-<span class="n">f0</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">legendre</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">legendre</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">legendre</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f4</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">legendre</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">f0</span><span class="p">,</span><span class="n">f1</span><span class="p">,</span><span class="n">f2</span><span class="p">,</span><span class="n">f3</span><span class="p">,</span><span class="n">f4</span><span class="p">],[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/legendre.png" src="../../../_images/legendre.png" />
-<p><strong>Basic evaluation</strong></p>
-<p>The Legendre polynomials assume fixed values at the points
-<span class="math">\(x = -1\)</span> and <span class="math">\(x = 1\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">([</span><span class="n">legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">6</span><span class="p">)])</span>
-<span class="go">[1.0, 1.0, 1.0, 1.0, 1.0, 1.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">([</span><span class="n">legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">6</span><span class="p">)])</span>
-<span class="go">[1.0, -1.0, 1.0, -1.0, 1.0, -1.0]</span>
-</pre></div>
-</div>
-<p>The coefficients of Legendre polynomials can be recovered
-using degree-<span class="math">\(n\)</span> Taylor expansion:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">)))</span>
-<span class="gp">...</span>
-<span class="go">[1.0]</span>
-<span class="go">[0.0, 1.0]</span>
-<span class="go">[-0.5, 0.0, 1.5]</span>
-<span class="go">[0.0, -1.5, 0.0, 2.5]</span>
-<span class="go">[0.375, 0.0, -3.75, 0.0, 4.375]</span>
-</pre></div>
-</div>
-<p>The roots of Legendre polynomials are located symmetrically
-on the interval <span class="math">\([-1, 1]\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">nprint</span><span class="p">(</span><span class="n">polyroots</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">)[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]))</span>
-<span class="gp">...</span>
-<span class="go">[]</span>
-<span class="go">[0.0]</span>
-<span class="go">[-0.57735, 0.57735]</span>
-<span class="go">[-0.774597, 0.0, 0.774597]</span>
-<span class="go">[-0.861136, -0.339981, 0.339981, 0.861136]</span>
-</pre></div>
-</div>
-<p>An example of an evaluation for arbitrary <span class="math">\(n\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="mf">0.75</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(1.94952805264875 + 2.1071073099422j)</span>
-</pre></div>
-</div>
-<p><strong>Orthogonality</strong></p>
-<p>The Legendre polynomials are orthogonal on <span class="math">\([-1, 1]\)</span> with respect
-to the trivial weight <span class="math">\(w(x) = 1\)</span>. That is, <span class="math">\(P_m(x) P_n(x)\)</span>
-integrates to zero if <span class="math">\(m \ne n\)</span> and to <span class="math">\(2/(2n+1)\)</span> if <span class="math">\(m = n\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">legendre</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">legendre</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">legendre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">0.222222222222222</span>
-</pre></div>
-</div>
-<p><strong>Differential equation</strong></p>
-<p>The Legendre polynomials satisfy the differential equation</p>
-<div class="math">
-\[((1-x^2) y')' + n(n+1) y' = 0.\]</div>
-<p>We can verify this numerically:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="mf">3.6</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="mf">0.73</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">legendre</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">t</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">u</span><span class="p">:</span> <span class="n">P</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">u</span><span class="p">),</span> <span class="n">t</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">P</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">A</span><span class="o">+</span><span class="n">B</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">9.0e-16</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="legenp">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">legenp()</span></tt><a class="headerlink" href="#legenp" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.legenp">
-<tt class="descclassname">mpmath.</tt><tt class="descname">legenp</tt><big>(</big><em>n</em>, <em>m</em>, <em>z</em>, <em>type=2</em><big>)</big><a class="headerlink" href="#mpmath.legenp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculates the (associated) Legendre function of the first kind of
-degree <em>n</em> and order <em>m</em>, <span class="math">\(P_n^m(z)\)</span>. Taking <span class="math">\(m = 0\)</span> gives the ordinary
-Legendre function of the first kind, <span class="math">\(P_n(z)\)</span>. The parameters may be
-complex numbers.</p>
-<p>In terms of the Gauss hypergeometric function, the (associated) Legendre
-function is defined as</p>
-<div class="math">
-\[P_n^m(z) = \frac{1}{\Gamma(1-m)} \frac{(1+z)^{m/2}}{(1-z)^{m/2}}
-    \,_2F_1\left(-n, n+1, 1-m, \frac{1-z}{2}\right).\]</div>
-<p>With <em>type=3</em> instead of <em>type=2</em>, the alternative
-definition</p>
-<div class="math">
-\[\hat{P}_n^m(z) = \frac{1}{\Gamma(1-m)} \frac{(z+1)^{m/2}}{(z-1)^{m/2}}
-    \,_2F_1\left(-n, n+1, 1-m, \frac{1-z}{2}\right).\]</div>
-<p>is used. These functions correspond respectively to <tt class="docutils literal"><span class="pre">LegendreP[n,m,2,z]</span></tt>
-and <tt class="docutils literal"><span class="pre">LegendreP[n,m,3,z]</span></tt> in Mathematica.</p>
-<p>The general solution of the (associated) Legendre differential equation</p>
-<div class="math">
-\[(1-z^2) f''(z) - 2zf'(z) + \left(n(n+1)-\frac{m^2}{1-z^2}\right)f(z) = 0\]</div>
-<p>is given by <span class="math">\(C_1 P_n^m(z) + C_2 Q_n^m(z)\)</span> for arbitrary constants
-<span class="math">\(C_1\)</span>, <span class="math">\(C_2\)</span>, where <span class="math">\(Q_n^m(z)\)</span> is a Legendre function of the
-second kind as implemented by <a class="reference internal" href="#mpmath.legenq" title="mpmath.legenq"><tt class="xref py py-func docutils literal"><span class="pre">legenq()</span></tt></a>.</p>
-<p><strong>Examples</strong></p>
-<p>Evaluation for arbitrary parameters and arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legenp</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">);</span> <span class="n">legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">149.5</span>
-<span class="go">149.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legenp</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">)</span>
-<span class="go">(1.972260393822275434196053 - 1.972260393822275434196053j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legenp</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span> <span class="mi">1</span><span class="o">-</span><span class="n">j</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.5</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(-3.335677248386698208736542 - 5.663270217461022307645625j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">legenp</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.5</span><span class="p">,</span> <span class="nb">type</span><span class="o">=</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">28.125</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">legenp</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.5</span><span class="p">,</span> <span class="nb">type</span><span class="o">=</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">-28.125</span>
-</pre></div>
-</div>
-<p>Verifying the associated Legendre differential equation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C1</span><span class="p">,</span> <span class="n">C2</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">C1</span><span class="o">*</span><span class="n">legenp</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">m</span><span class="p">,</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="n">C2</span><span class="o">*</span><span class="n">legenq</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">m</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">deq</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> \
-<span class="gp">... </span>    <span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">-</span><span class="n">m</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="o">+</span><span class="mi">2j</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="n">chop</span><span class="p">(</span><span class="n">deq</span><span class="p">(</span><span class="n">mpmathify</span><span class="p">(</span><span class="n">z</span><span class="p">)))</span>
-<span class="gp">...</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="legenq">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">legenq()</span></tt><a class="headerlink" href="#legenq" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.legenq">
-<tt class="descclassname">mpmath.</tt><tt class="descname">legenq</tt><big>(</big><em>n</em>, <em>m</em>, <em>z</em>, <em>type=2</em><big>)</big><a class="headerlink" href="#mpmath.legenq" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculates the (associated) Legendre function of the second kind of
-degree <em>n</em> and order <em>m</em>, <span class="math">\(Q_n^m(z)\)</span>. Taking <span class="math">\(m = 0\)</span> gives the ordinary
-Legendre function of the second kind, <span class="math">\(Q_n(z)\)</span>. The parameters may
-complex numbers.</p>
-<p>The Legendre functions of the second kind give a second set of
-solutions to the (associated) Legendre differential equation.
-(See <a class="reference internal" href="#mpmath.legenp" title="mpmath.legenp"><tt class="xref py py-func docutils literal"><span class="pre">legenp()</span></tt></a>.)
-Unlike the Legendre functions of the first kind, they are not
-polynomials of <span class="math">\(z\)</span> for integer <span class="math">\(n\)</span>, <span class="math">\(m\)</span> but rational or logarithmic
-functions with poles at <span class="math">\(z = \pm 1\)</span>.</p>
-<p>There are various ways to define Legendre functions of
-the second kind, giving rise to different complex structure.
-A version can be selected using the <em>type</em> keyword argument.
-The <em>type=2</em> and <em>type=3</em> functions are given respectively by</p>
-<div class="math">
-\[Q_n^m(z) = \frac{\pi}{2 \sin(\pi m)}
-    \left( \cos(\pi m) P_n^m(z) -
-    \frac{\Gamma(1+m+n)}{\Gamma(1-m+n)} P_n^{-m}(z)\right)\]\[\hat{Q}_n^m(z) = \frac{\pi}{2 \sin(\pi m)} e^{\pi i m}
-    \left( \hat{P}_n^m(z) -
-    \frac{\Gamma(1+m+n)}{\Gamma(1-m+n)} \hat{P}_n^{-m}(z)\right)\]</div>
-<p>where <span class="math">\(P\)</span> and <span class="math">\(\hat{P}\)</span> are the <em>type=2</em> and <em>type=3</em> Legendre functions
-of the first kind. The formulas above should be understood as limits
-when <span class="math">\(m\)</span> is an integer.</p>
-<p>These functions correspond to <tt class="docutils literal"><span class="pre">LegendreQ[n,m,2,z]</span></tt> (or <tt class="docutils literal"><span class="pre">LegendreQ[n,m,z]</span></tt>)
-and <tt class="docutils literal"><span class="pre">LegendreQ[n,m,3,z]</span></tt> in Mathematica. The <em>type=3</em> function
-is essentially the same as the function defined in
-Abramowitz &amp; Stegun (eq. 8.1.3) but with <span class="math">\((z+1)^{m/2}(z-1)^{m/2}\)</span> instead
-of <span class="math">\((z^2-1)^{m/2}\)</span>, giving slightly different branches.</p>
-<p><strong>Examples</strong></p>
-<p>Evaluation for arbitrary parameters and arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legenq</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
-<span class="go">-0.8186632680417568557122028</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legenq</span><span class="p">(</span><span class="o">-</span><span class="mf">1.5</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">)</span>
-<span class="go">(0.6655964618250228714288277 + 0.3937692045497259717762649j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legenq</span><span class="p">(</span><span class="mi">2</span><span class="o">-</span><span class="n">j</span><span class="p">,</span> <span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">,</span> <span class="o">-</span><span class="mi">6</span><span class="o">+</span><span class="mi">5j</span><span class="p">)</span>
-<span class="go">(-10001.95256487468541686564 - 6011.691337610097577791134j)</span>
-</pre></div>
-</div>
-<p>Different versions of the function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">legenq</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
-<span class="go">0.7298060598018049369381857</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legenq</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">)</span>
-<span class="go">(-7.902916572420817192300921 + 0.1998650072605976600724502j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legenq</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="nb">type</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">(2.040524284763495081918338 - 0.7298060598018049369381857j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">legenq</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="nb">type</span><span class="o">=</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">-0.1998650072605976600724502</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="chebyshev-polynomials">
-<h2>Chebyshev polynomials<a class="headerlink" href="#chebyshev-polynomials" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="chebyt">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">chebyt()</span></tt><a class="headerlink" href="#chebyt" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.chebyt">
-<tt class="descclassname">mpmath.</tt><tt class="descname">chebyt</tt><big>(</big><em>n</em>, <em>x</em><big>)</big><a class="headerlink" href="#mpmath.chebyt" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">chebyt(n,</span> <span class="pre">x)</span></tt> evaluates the Chebyshev polynomial of the first
-kind <span class="math">\(T_n(x)\)</span>, defined by the identity</p>
-<div class="math">
-\[T_n(\cos x) = \cos(n x).\]</div>
-<p>The Chebyshev polynomials of the first kind are a special
-case of the Jacobi polynomials, and by extension of the
-hypergeometric function <span class="math">\(\,_2F_1\)</span>. They can thus also be
-evaluated for nonintegral <span class="math">\(n\)</span>.</p>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Chebyshev polynomials T_n(x) on [-1,1] for n=0,1,2,3,4</span>
-<span class="n">f0</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyt</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyt</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyt</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyt</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f4</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyt</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">f0</span><span class="p">,</span><span class="n">f1</span><span class="p">,</span><span class="n">f2</span><span class="p">,</span><span class="n">f3</span><span class="p">,</span><span class="n">f4</span><span class="p">],[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/chebyt.png" src="../../../_images/chebyt.png" />
-<p><strong>Basic evaluation</strong></p>
-<p>The coefficients of the <span class="math">\(n\)</span>-th polynomial can be recovered
-using using degree-<span class="math">\(n\)</span> Taylor expansion:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyt</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">)))</span>
-<span class="gp">...</span>
-<span class="go">[1.0]</span>
-<span class="go">[0.0, 1.0]</span>
-<span class="go">[-1.0, 0.0, 2.0]</span>
-<span class="go">[0.0, -3.0, 0.0, 4.0]</span>
-<span class="go">[1.0, 0.0, -8.0, 0.0, 8.0]</span>
-</pre></div>
-</div>
-<p><strong>Orthogonality</strong></p>
-<p>The Chebyshev polynomials of the first kind are orthogonal
-on the interval <span class="math">\([-1, 1]\)</span> with respect to the weight
-function <span class="math">\(w(x) = 1/\sqrt{1-x^2}\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyt</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">chebyt</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]),</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">1.57079632596448</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="chebyu">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">chebyu()</span></tt><a class="headerlink" href="#chebyu" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.chebyu">
-<tt class="descclassname">mpmath.</tt><tt class="descname">chebyu</tt><big>(</big><em>n</em>, <em>x</em><big>)</big><a class="headerlink" href="#mpmath.chebyu" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">chebyu(n,</span> <span class="pre">x)</span></tt> evaluates the Chebyshev polynomial of the second
-kind <span class="math">\(U_n(x)\)</span>, defined by the identity</p>
-<div class="math">
-\[U_n(\cos x) = \frac{\sin((n+1)x)}{\sin(x)}.\]</div>
-<p>The Chebyshev polynomials of the second kind are a special
-case of the Jacobi polynomials, and by extension of the
-hypergeometric function <span class="math">\(\,_2F_1\)</span>. They can thus also be
-evaluated for nonintegral <span class="math">\(n\)</span>.</p>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Chebyshev polynomials U_n(x) on [-1,1] for n=0,1,2,3,4</span>
-<span class="n">f0</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyu</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyu</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyu</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyu</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f4</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyu</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">f0</span><span class="p">,</span><span class="n">f1</span><span class="p">,</span><span class="n">f2</span><span class="p">,</span><span class="n">f3</span><span class="p">,</span><span class="n">f4</span><span class="p">],[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/chebyu.png" src="../../../_images/chebyu.png" />
-<p><strong>Basic evaluation</strong></p>
-<p>The coefficients of the <span class="math">\(n\)</span>-th polynomial can be recovered
-using using degree-<span class="math">\(n\)</span> Taylor expansion:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyu</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">)))</span>
-<span class="gp">...</span>
-<span class="go">[1.0]</span>
-<span class="go">[0.0, 2.0]</span>
-<span class="go">[-1.0, 0.0, 4.0]</span>
-<span class="go">[0.0, -4.0, 0.0, 8.0]</span>
-<span class="go">[1.0, 0.0, -12.0, 0.0, 16.0]</span>
-</pre></div>
-</div>
-<p><strong>Orthogonality</strong></p>
-<p>The Chebyshev polynomials of the second kind are orthogonal
-on the interval <span class="math">\([-1, 1]\)</span> with respect to the weight
-function <span class="math">\(w(x) = \sqrt{1-x^2}\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">chebyu</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">chebyu</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">1.5707963267949</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="jacobi-polynomials">
-<h2>Jacobi polynomials<a class="headerlink" href="#jacobi-polynomials" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="jacobi">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">jacobi()</span></tt><a class="headerlink" href="#jacobi" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.jacobi">
-<tt class="descclassname">mpmath.</tt><tt class="descname">jacobi</tt><big>(</big><em>n</em>, <em>a</em>, <em>b</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.jacobi" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">jacobi(n,</span> <span class="pre">a,</span> <span class="pre">b,</span> <span class="pre">x)</span></tt> evaluates the Jacobi polynomial
-<span class="math">\(P_n^{(a,b)}(x)\)</span>. The Jacobi polynomials are a special
-case of the hypergeometric function <span class="math">\(\,_2F_1\)</span> given by:</p>
-<div class="math">
-\[P_n^{(a,b)}(x) = {n+a \choose n}
-  \,_2F_1\left(-n,1+a+b+n,a+1,\frac{1-x}{2}\right).\]</div>
-<p>Note that this definition generalizes to nonintegral values
-of <span class="math">\(n\)</span>. When <span class="math">\(n\)</span> is an integer, the hypergeometric series
-terminates after a finite number of terms, giving
-a polynomial in <span class="math">\(x\)</span>.</p>
-<p><strong>Evaluation of Jacobi polynomials</strong></p>
-<p>A special evaluation is <span class="math">\(P_n^{(a,b)}(1) = {n+a \choose n}\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jacobi</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">2.4609375</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">binomial</span><span class="p">(</span><span class="mi">4</span><span class="o">+</span><span class="mf">0.5</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">2.4609375</span>
-</pre></div>
-</div>
-<p>A Jacobi polynomial of degree <span class="math">\(n\)</span> is equal to its
-Taylor polynomial of degree <span class="math">\(n\)</span>. The explicit
-coefficients of Jacobi polynomials can therefore
-be recovered easily using <a class="reference internal" href="../../mpmath/calculus/approximation.html#mpmath.taylor" title="mpmath.taylor"><tt class="xref py py-func docutils literal"><span class="pre">taylor()</span></tt></a>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">nprint</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">jacobi</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">[1.0]</span>
-<span class="go">[-0.5, 2.5]</span>
-<span class="go">[-0.75, -1.5, 5.25]</span>
-<span class="go">[0.5, -3.5, -3.5, 10.5]</span>
-<span class="go">[0.625, 2.5, -11.25, -7.5, 20.625]</span>
-</pre></div>
-</div>
-<p>For nonintegral <span class="math">\(n\)</span>, the Jacobi &#8220;polynomial&#8221; is no longer
-a polynomial:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">jacobi</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
-<span class="go">[0.309983, 1.84119, -1.26933, 1.26699, -1.34808]</span>
-</pre></div>
-</div>
-<p><strong>Orthogonality</strong></p>
-<p>The Jacobi polynomials are orthogonal on the interval
-<span class="math">\([-1, 1]\)</span> with respect to the weight function
-<span class="math">\(w(x) = (1-x)^a (1+x)^b\)</span>. That is,
-<span class="math">\(w(x) P_n^{(a,b)}(x) P_m^{(a,b)}(x)\)</span> integrates to
-zero if <span class="math">\(m \ne n\)</span> and to a nonzero number if <span class="math">\(m = n\)</span>.</p>
-<p>The orthogonality is easy to verify using numerical
-quadrature:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">jacobi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="n">a</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="n">b</span> <span class="o">*</span> <span class="n">P</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="n">P</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]),</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">1.9047619047619</span>
-</pre></div>
-</div>
-<p><strong>Differential equation</strong></p>
-<p>The Jacobi polynomials are solutions of the differential
-equation</p>
-<div class="math">
-\[(1-x^2) y'' + (b-a-(a+b+2)x) y' + n (n+a+b+1) y = 0.\]</div>
-<p>We can verify that <a class="reference internal" href="#mpmath.jacobi" title="mpmath.jacobi"><tt class="xref py py-func docutils literal"><span class="pre">jacobi()</span></tt></a> approximately satisfies
-this equation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="mf">2.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="mi">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">jacobi</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A0</span> <span class="o">=</span> <span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">y</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A1</span> <span class="o">=</span> <span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="n">a</span><span class="o">-</span><span class="p">(</span><span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="o">+</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">y</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A2</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">y</span><span class="p">,</span><span class="n">x</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">A2</span> <span class="o">+</span> <span class="n">A1</span> <span class="o">+</span> <span class="n">A0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">4.0e-12</span>
-</pre></div>
-</div>
-<p>The difference of order <span class="math">\(10^{-12}\)</span> is as close to zero as
-it could be at 15-digit working precision, since the terms
-are large:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A0</span><span class="p">,</span> <span class="n">A1</span><span class="p">,</span> <span class="n">A2</span>
-<span class="go">(26560.2328981879, -21503.7641037294, -5056.46879445852)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="gegenbauer-polynomials">
-<h2>Gegenbauer polynomials<a class="headerlink" href="#gegenbauer-polynomials" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="gegenbauer">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">gegenbauer()</span></tt><a class="headerlink" href="#gegenbauer" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.gegenbauer">
-<tt class="descclassname">mpmath.</tt><tt class="descname">gegenbauer</tt><big>(</big><em>n</em>, <em>a</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.gegenbauer" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the Gegenbauer polynomial, or ultraspherical polynomial,</p>
-<div class="math">
-\[C_n^{(a)}(z) = {n+2a-1 \choose n} \,_2F_1\left(-n, n+2a;
-    a+\frac{1}{2}; \frac{1}{2}(1-z)\right).\]</div>
-<p>When <span class="math">\(n\)</span> is a nonnegative integer, this formula gives a polynomial
-in <span class="math">\(z\)</span> of degree <span class="math">\(n\)</span>, but all parameters are permitted to be
-complex numbers. With <span class="math">\(a = 1/2\)</span>, the Gegenbauer polynomial
-reduces to a Legendre polynomial.</p>
-<p><strong>Examples</strong></p>
-<p>Evaluation for arbitrary arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gegenbauer</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">-2485.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gegenbauer</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
-<span class="go">3.012757178975667428359374e+2322</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gegenbauer</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.75</span><span class="p">,</span> <span class="o">-</span><span class="mi">1000j</span><span class="p">)</span>
-<span class="go">(-5038991.358609026523401901 + 9414549.285447104177860806j)</span>
-</pre></div>
-</div>
-<p>Evaluation at negative integer orders:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gegenbauer</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mf">1.75</span><span class="p">)</span>
-<span class="go">-1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gegenbauer</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mf">1.75</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gegenbauer</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2j</span><span class="p">,</span> <span class="mf">1.75</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gegenbauer</span><span class="p">(</span><span class="o">-</span><span class="mi">7</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">8989.0</span>
-</pre></div>
-</div>
-<p>The Gegenbauer polynomials solve the differential equation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span> <span class="n">a</span> <span class="o">=</span> <span class="mf">4.5</span><span class="p">,</span> <span class="mi">1</span><span class="o">+</span><span class="mi">2j</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">gegenbauer</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.75</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.5j</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="n">chop</span><span class="p">((</span><span class="mi">1</span><span class="o">-</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>The Gegenbauer polynomials have generating function
-<span class="math">\((1-2zt+t^2)^{-a}\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="o">*</span><span class="n">t</span><span class="o">+</span><span class="n">t</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">a</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">[1.0, 5.0, 15.0, 35.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">gegenbauer</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">z</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">)]</span>
-<span class="go">[1.0, 5.0, 15.0, 35.0]</span>
-</pre></div>
-</div>
-<p>The Gegenbauer polynomials are orthogonal on <span class="math">\([-1, 1]\)</span> with respect
-to the weight <span class="math">\((1-z^2)^{a-\frac{1}{2}}\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Cn</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">gegenbauer</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">zeroprec</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Cm</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">gegenbauer</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">zeroprec</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">Cn</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">Cm</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">a</span><span class="o">-</span><span class="mf">0.5</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]))</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="hermite-polynomials">
-<h2>Hermite polynomials<a class="headerlink" href="#hermite-polynomials" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="hermite">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">hermite()</span></tt><a class="headerlink" href="#hermite" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.hermite">
-<tt class="descclassname">mpmath.</tt><tt class="descname">hermite</tt><big>(</big><em>n</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.hermite" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the Hermite polynomial <span class="math">\(H_n(z)\)</span>, which may be defined using
-the recurrence</p>
-<div class="math">
-\[H_0(z) = 1\]\[H_1(z) = 2z\]\[H_{n+1} = 2z H_n(z) - 2n H_{n-1}(z).\]</div>
-<p>The Hermite polynomials are orthogonal on <span class="math">\((-\infty, \infty)\)</span> with
-respect to the weight <span class="math">\(e^{-z^2}\)</span>. More generally, allowing arbitrary complex
-values of <span class="math">\(n\)</span>, the Hermite function <span class="math">\(H_n(z)\)</span> is defined as</p>
-<div class="math">
-\[H_n(z) = (2z)^n \,_2F_0\left(-\frac{n}{2}, \frac{1-n}{2},
-    -\frac{1}{z^2}\right)\]</div>
-<p>for <span class="math">\(\Re{z} &gt; 0\)</span>, or generally</p>
-<div class="math">
-\[H_n(z) = 2^n \sqrt{\pi} \left(
-    \frac{1}{\Gamma\left(\frac{1-n}{2}\right)}
-    \,_1F_1\left(-\frac{n}{2}, \frac{1}{2}, z^2\right) -
-    \frac{2z}{\Gamma\left(-\frac{n}{2}\right)}
-    \,_1F_1\left(\frac{1-n}{2}, \frac{3}{2}, z^2\right)
-\right).\]</div>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Hermite polynomials H_n(x) on the real line for n=0,1,2,3,4</span>
-<span class="n">f0</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">hermite</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">hermite</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">hermite</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">hermite</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f4</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">hermite</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">f0</span><span class="p">,</span><span class="n">f1</span><span class="p">,</span><span class="n">f2</span><span class="p">,</span><span class="n">f3</span><span class="p">,</span><span class="n">f4</span><span class="p">],[</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">],[</span><span class="o">-</span><span class="mi">25</span><span class="p">,</span><span class="mi">25</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/hermite.png" src="../../../_images/hermite.png" />
-<p><strong>Examples</strong></p>
-<p>Evaluation for arbitrary arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hermite</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hermite</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">);</span> <span class="n">hermite</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">20.0</span>
-<span class="go">398.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hermite</span><span class="p">(</span><span class="mi">10000</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">4.950440066552087387515653e+19334</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hermite</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="o">**</span><span class="mi">8</span><span class="p">)</span>
-<span class="go">-7999999999999998800000000.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hermite</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="o">**</span><span class="mi">8</span><span class="p">)</span>
-<span class="go">1.675159751729877682920301e+4342944819032534</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hermite</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="o">+</span><span class="mi">2j</span><span class="p">)</span>
-<span class="go">(-0.07652130602993513389421901 - 0.1084662449961914580276007j)</span>
-</pre></div>
-</div>
-<p>Coefficients of the first few Hermite polynomials are:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">7</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">hermite</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">[1.0]</span>
-<span class="go">[0.0, 2.0]</span>
-<span class="go">[-2.0, 0.0, 4.0]</span>
-<span class="go">[0.0, -12.0, 0.0, 8.0]</span>
-<span class="go">[12.0, 0.0, -48.0, 0.0, 16.0]</span>
-<span class="go">[0.0, 120.0, 0.0, -160.0, 0.0, 32.0]</span>
-<span class="go">[-120.0, 0.0, 720.0, 0.0, -480.0, 0.0, 64.0]</span>
-</pre></div>
-</div>
-<p>Values at <span class="math">\(z = 0\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">9</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">hermite</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="go">0.02769459142039868792653387</span>
-<span class="go">0.08333333333333333333333333</span>
-<span class="go">0.2215567313631895034122709</span>
-<span class="go">0.5</span>
-<span class="go">0.8862269254527580136490837</span>
-<span class="go">1.0</span>
-<span class="go">0.0</span>
-<span class="go">-2.0</span>
-<span class="go">0.0</span>
-<span class="go">12.0</span>
-<span class="go">0.0</span>
-<span class="go">-120.0</span>
-<span class="go">0.0</span>
-<span class="go">1680.0</span>
-</pre></div>
-</div>
-<p>Hermite functions satisfy the differential equation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="mi">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">hermite</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mf">1.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">z</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>Verifying orthogonality:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">hermite</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">t</span><span class="p">)</span><span class="o">*</span><span class="n">hermite</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="n">t</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">t</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span><span class="n">inf</span><span class="p">]))</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="laguerre-polynomials">
-<h2>Laguerre polynomials<a class="headerlink" href="#laguerre-polynomials" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="laguerre">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">laguerre()</span></tt><a class="headerlink" href="#laguerre" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.laguerre">
-<tt class="descclassname">mpmath.</tt><tt class="descname">laguerre</tt><big>(</big><em>n</em>, <em>a</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.laguerre" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the generalized (associated) Laguerre polynomial, defined by</p>
-<div class="math">
-\[L_n^a(z) = \frac{\Gamma(n+b+1)}{\Gamma(b+1) \Gamma(n+1)}
-    \,_1F_1(-n, a+1, z).\]</div>
-<p>With <span class="math">\(a = 0\)</span> and <span class="math">\(n\)</span> a nonnegative integer, this reduces to an ordinary
-Laguerre polynomial, the sequence of which begins
-<span class="math">\(L_0(z) = 1, L_1(z) = 1-z, L_2(z) = z^2-2z+1, \ldots\)</span>.</p>
-<p>The Laguerre polynomials are orthogonal with respect to the weight
-<span class="math">\(z^a e^{-z}\)</span> on <span class="math">\([0, \infty)\)</span>.</p>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Hermite polynomials L_n(x) on the real line for n=0,1,2,3,4</span>
-<span class="n">f0</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">laguerre</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">laguerre</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">laguerre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f3</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">laguerre</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">f4</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">laguerre</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">f0</span><span class="p">,</span><span class="n">f1</span><span class="p">,</span><span class="n">f2</span><span class="p">,</span><span class="n">f3</span><span class="p">,</span><span class="n">f4</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">10</span><span class="p">],[</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">])</span>
-</pre></div>
-</div>
-<img alt="../../../_images/laguerre.png" src="../../../_images/laguerre.png" />
-<p><strong>Examples</strong></p>
-<p>Evaluation for arbitrary arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">laguerre</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">)</span>
-<span class="go">0.03726399739583333333333333</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">laguerre</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(4.474921610704496808379097 - 11.02058050372068958069241j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">laguerre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10000</span><span class="p">)</span>
-<span class="go">49980001.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">laguerre</span><span class="p">(</span><span class="mf">2.5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10000</span><span class="p">)</span>
-<span class="go">-9.327764910194842158583189e+4328</span>
-</pre></div>
-</div>
-<p>The first few Laguerre polynomials, normalized to have integer
-coefficients:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">7</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">fac</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">*</span><span class="n">laguerre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">[1.0]</span>
-<span class="go">[1.0, -1.0]</span>
-<span class="go">[2.0, -4.0, 1.0]</span>
-<span class="go">[6.0, -18.0, 9.0, -1.0]</span>
-<span class="go">[24.0, -96.0, 72.0, -16.0, 1.0]</span>
-<span class="go">[120.0, -600.0, 600.0, -200.0, 25.0, -1.0]</span>
-<span class="go">[720.0, -4320.0, 5400.0, -2400.0, 450.0, -36.0, 1.0]</span>
-</pre></div>
-</div>
-<p>Verifying orthogonality:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Lm</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">laguerre</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ln</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">laguerre</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">t</span><span class="p">)</span><span class="o">*</span><span class="n">t</span><span class="o">**</span><span class="n">a</span><span class="o">*</span><span class="n">Lm</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">*</span><span class="n">Ln</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">]))</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="spherical-harmonics">
-<h2>Spherical harmonics<a class="headerlink" href="#spherical-harmonics" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="spherharm">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">spherharm()</span></tt><a class="headerlink" href="#spherharm" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.spherharm">
-<tt class="descclassname">mpmath.</tt><tt class="descname">spherharm</tt><big>(</big><em>l</em>, <em>m</em>, <em>theta</em>, <em>phi</em><big>)</big><a class="headerlink" href="#mpmath.spherharm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the spherical harmonic <span class="math">\(Y_l^m(\theta,\phi)\)</span>,</p>
-<div class="math">
-\[Y_l^m(\theta,\phi) = \sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}}
-    P_l^m(\cos \theta) e^{i m \phi}\]</div>
-<p>where <span class="math">\(P_l^m\)</span> is an associated Legendre function (see <a class="reference internal" href="#mpmath.legenp" title="mpmath.legenp"><tt class="xref py py-func docutils literal"><span class="pre">legenp()</span></tt></a>).</p>
-<p>Here <span class="math">\(\theta \in [0, \pi]\)</span> denotes the polar coordinate (ranging
-from the north pole to the south pole) and <span class="math">\(\phi \in [0, 2 \pi]\)</span> denotes the
-azimuthal coordinate on a sphere. Care should be used since many different
-conventions for spherical coordinate variables are used.</p>
-<p>Usually spherical harmonics are considered for <span class="math">\(l \in \mathbb{N}\)</span>,
-<span class="math">\(m \in \mathbb{Z}\)</span>, <span class="math">\(|m| \le l\)</span>. More generally, <span class="math">\(l,m,\theta,\phi\)</span>
-are permitted to be complex numbers.</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last"><a class="reference internal" href="#mpmath.spherharm" title="mpmath.spherharm"><tt class="xref py py-func docutils literal"><span class="pre">spherharm()</span></tt></a> returns a complex number, even the value is
-purely real.</p>
-</div>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Real part of spherical harmonic Y_(4,0)(theta,phi)</span>
-<span class="k">def</span> <span class="nf">Y</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="n">m</span><span class="p">):</span>
-    <span class="k">def</span> <span class="nf">g</span><span class="p">(</span><span class="n">theta</span><span class="p">,</span><span class="n">phi</span><span class="p">):</span>
-        <span class="n">R</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">fp</span><span class="o">.</span><span class="n">re</span><span class="p">(</span><span class="n">fp</span><span class="o">.</span><span class="n">spherharm</span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="n">m</span><span class="p">,</span><span class="n">theta</span><span class="p">,</span><span class="n">phi</span><span class="p">)))</span>
-        <span class="n">x</span> <span class="o">=</span> <span class="n">R</span><span class="o">*</span><span class="n">fp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">phi</span><span class="p">)</span><span class="o">*</span><span class="n">fp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
-        <span class="n">y</span> <span class="o">=</span> <span class="n">R</span><span class="o">*</span><span class="n">fp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">phi</span><span class="p">)</span><span class="o">*</span><span class="n">fp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
-        <span class="n">z</span> <span class="o">=</span> <span class="n">R</span><span class="o">*</span><span class="n">fp</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
-        <span class="k">return</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span><span class="p">]</span>
-    <span class="k">return</span> <span class="n">g</span>
-
-<span class="n">fp</span><span class="o">.</span><span class="n">splot</span><span class="p">(</span><span class="n">Y</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">fp</span><span class="o">.</span><span class="n">pi</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="o">*</span><span class="n">fp</span><span class="o">.</span><span class="n">pi</span><span class="p">],</span> <span class="n">points</span><span class="o">=</span><span class="mi">300</span><span class="p">)</span>
-<span class="c"># fp.splot(Y(4,0), [0,fp.pi], [0,2*fp.pi], points=300)</span>
-<span class="c"># fp.splot(Y(4,1), [0,fp.pi], [0,2*fp.pi], points=300)</span>
-<span class="c"># fp.splot(Y(4,2), [0,fp.pi], [0,2*fp.pi], points=300)</span>
-<span class="c"># fp.splot(Y(4,3), [0,fp.pi], [0,2*fp.pi], points=300)</span>
-</pre></div>
-</div>
-<p><span class="math">\(Y_{4,0}\)</span>:</p>
-<img alt="../../../_images/spherharm40.png" src="../../../_images/spherharm40.png" />
-<p><span class="math">\(Y_{4,1}\)</span>:</p>
-<img alt="../../../_images/spherharm41.png" src="../../../_images/spherharm41.png" />
-<p><span class="math">\(Y_{4,2}\)</span>:</p>
-<img alt="../../../_images/spherharm42.png" src="../../../_images/spherharm42.png" />
-<p><span class="math">\(Y_{4,3}\)</span>:</p>
-<img alt="../../../_images/spherharm43.png" src="../../../_images/spherharm43.png" />
-<p><span class="math">\(Y_{4,4}\)</span>:</p>
-<img alt="../../../_images/spherharm44.png" src="../../../_images/spherharm44.png" />
-<p><strong>Examples</strong></p>
-<p>Some low-order spherical harmonics with reference values:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">theta</span> <span class="o">=</span> <span class="n">pi</span><span class="o">/</span><span class="mi">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">phi</span> <span class="o">=</span> <span class="n">pi</span><span class="o">/</span><span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">spherharm</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">theta</span><span class="p">,</span><span class="n">phi</span><span class="p">);</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">expj</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">(0.2820947917738781434740397 + 0.0j)</span>
-<span class="go">(0.2820947917738781434740397 + 0.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">spherharm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="n">theta</span><span class="p">,</span><span class="n">phi</span><span class="p">);</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span><span class="o">*</span><span class="n">expj</span><span class="p">(</span><span class="o">-</span><span class="n">phi</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
-<span class="go">(0.1221506279757299803965962 - 0.2115710938304086076055298j)</span>
-<span class="go">(0.1221506279757299803965962 - 0.2115710938304086076055298j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">spherharm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">theta</span><span class="p">,</span><span class="n">phi</span><span class="p">);</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="o">/</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span><span class="o">*</span><span class="n">expj</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">(0.3454941494713354792652446 + 0.0j)</span>
-<span class="go">(0.3454941494713354792652446 + 0.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">spherharm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">theta</span><span class="p">,</span><span class="n">phi</span><span class="p">);</span> <span class="o">-</span><span class="mf">0.5</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span><span class="o">*</span><span class="n">expj</span><span class="p">(</span><span class="n">phi</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)</span>
-<span class="go">(-0.1221506279757299803965962 - 0.2115710938304086076055298j)</span>
-<span class="go">(-0.1221506279757299803965962 - 0.2115710938304086076055298j)</span>
-</pre></div>
-</div>
-<p>With the normalization convention used, the spherical harmonics are orthonormal
-on the unit sphere:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sphere</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">pi</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dS</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">:</span> <span class="n">fp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>   <span class="c"># differential element</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">:</span> <span class="n">fp</span><span class="o">.</span><span class="n">spherharm</span><span class="p">(</span><span class="n">l1</span><span class="p">,</span><span class="n">m1</span><span class="p">,</span><span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">:</span> <span class="n">fp</span><span class="o">.</span><span class="n">conj</span><span class="p">(</span><span class="n">fp</span><span class="o">.</span><span class="n">spherharm</span><span class="p">(</span><span class="n">l2</span><span class="p">,</span><span class="n">m2</span><span class="p">,</span><span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l1</span> <span class="o">=</span> <span class="n">l2</span> <span class="o">=</span> <span class="mi">3</span><span class="p">;</span> <span class="n">m1</span> <span class="o">=</span> <span class="n">m2</span> <span class="o">=</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">fp</span><span class="o">.</span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">:</span> <span class="n">Y1</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="n">Y2</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="n">dS</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">),</span> <span class="o">*</span><span class="n">sphere</span><span class="p">))</span>
-<span class="go">(1+0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m2</span> <span class="o">=</span> <span class="mi">1</span>    <span class="c"># m1 != m2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">fp</span><span class="o">.</span><span class="n">chop</span><span class="p">(</span><span class="n">fp</span><span class="o">.</span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">:</span> <span class="n">Y1</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="n">Y2</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="n">dS</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">),</span> <span class="o">*</span><span class="n">sphere</span><span class="p">)))</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>Evaluation is accurate for large orders:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">spherharm</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span><span class="mi">750</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.25</span><span class="p">)</span>
-<span class="go">(3.776445785304252879026585e-102 - 5.82441278771834794493484e-102j)</span>
-</pre></div>
-</div>
-<p>Evaluation works with complex parameter values:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">spherharm</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mi">2j</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.5j</span><span class="p">)</span>
-<span class="go">(64.44922331113759992154992 + 1981.693919841408089681743j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-</div>
-
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-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Orthogonal polynomials</a><ul>
-<li><a class="reference internal" href="#legendre-functions">Legendre functions</a><ul>
-<li><a class="reference internal" href="#legendre"><tt class="docutils literal"><span class="pre">legendre()</span></tt></a></li>
-<li><a class="reference internal" href="#legenp"><tt class="docutils literal"><span class="pre">legenp()</span></tt></a></li>
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-</ul>
-</li>
-<li><a class="reference internal" href="#chebyshev-polynomials">Chebyshev polynomials</a><ul>
-<li><a class="reference internal" href="#chebyt"><tt class="docutils literal"><span class="pre">chebyt()</span></tt></a></li>
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-</ul>
-</li>
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-<li><a class="reference internal" href="#jacobi"><tt class="docutils literal"><span class="pre">jacobi()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#gegenbauer-polynomials">Gegenbauer polynomials</a><ul>
-<li><a class="reference internal" href="#gegenbauer"><tt class="docutils literal"><span class="pre">gegenbauer()</span></tt></a></li>
-</ul>
-</li>
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-<li><a class="reference internal" href="#hermite"><tt class="docutils literal"><span class="pre">hermite()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#laguerre-polynomials">Laguerre polynomials</a><ul>
-<li><a class="reference internal" href="#laguerre"><tt class="docutils literal"><span class="pre">laguerre()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#spherical-harmonics">Spherical harmonics</a><ul>
-<li><a class="reference internal" href="#spherharm"><tt class="docutils literal"><span class="pre">spherharm()</span></tt></a></li>
-</ul>
-</li>
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-            
-  <div class="section" id="powers-and-logarithms">
-<h1>Powers and logarithms<a class="headerlink" href="#powers-and-logarithms" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="nth-roots">
-<h2>Nth roots<a class="headerlink" href="#nth-roots" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="sqrt">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">sqrt()</span></tt><a class="headerlink" href="#sqrt" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.sqrt">
-<tt class="descclassname">mpmath.</tt><tt class="descname">sqrt</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.sqrt" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">sqrt(x)</span></tt> gives the principal square root of <span class="math">\(x\)</span>, <span class="math">\(\sqrt x\)</span>.
-For positive real numbers, the principal root is simply the
-positive square root. For arbitrary complex numbers, the principal
-square root is defined to satisfy <span class="math">\(\sqrt x = \exp(\log(x)/2)\)</span>.
-The function thus has a branch cut along the negative half real axis.</p>
-<p>For all mpmath numbers <tt class="docutils literal"><span class="pre">x</span></tt>, calling <tt class="docutils literal"><span class="pre">sqrt(x)</span></tt> is equivalent to
-performing <tt class="docutils literal"><span class="pre">x**0.5</span></tt>.</p>
-<p><strong>Examples</strong></p>
-<p>Basic examples and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">3.16227766016838</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">10.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">(0.0 + 2.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mi">1j</span><span class="p">)</span>
-<span class="go">(1.09868411346781 + 0.455089860562227j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">+inf</span>
-</pre></div>
-</div>
-<p>Square root evaluation is fast at huge precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">str</span><span class="p">(</span><span class="n">a</span><span class="p">)[</span><span class="o">-</span><span class="mi">10</span><span class="p">:]</span>
-<span class="go">&#39;9329332814&#39;</span>
-</pre></div>
-</div>
-<p><tt class="xref py py-func docutils literal"><span class="pre">mpmath.iv.sqrt()</span></tt> supports interval arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">iv</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">sqrt</span><span class="p">([</span><span class="mi">16</span><span class="p">,</span><span class="mi">100</span><span class="p">])</span>
-<span class="go">[4.0, 10.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">[1.4142135623730949234, 1.4142135623730951455]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span>
-<span class="go">[1.9999999999999995559, 2.0000000000000004441]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="hypot">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">hypot()</span></tt><a class="headerlink" href="#hypot" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.hypot">
-<tt class="descclassname">mpmath.</tt><tt class="descname">hypot</tt><big>(</big><em>x</em>, <em>y</em><big>)</big><a class="headerlink" href="#mpmath.hypot" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the Euclidean norm of the vector <span class="math">\((x, y)\)</span>, equal
-to <span class="math">\(\sqrt{x^2 + y^2}\)</span>. Both <span class="math">\(x\)</span> and <span class="math">\(y\)</span> must be real.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="cbrt">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">cbrt()</span></tt><a class="headerlink" href="#cbrt" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.cbrt">
-<tt class="descclassname">mpmath.</tt><tt class="descname">cbrt</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.cbrt" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">cbrt(x)</span></tt> computes the cube root of <span class="math">\(x\)</span>, <span class="math">\(x^{1/3}\)</span>. This
-function is faster and more accurate than raising to a floating-point
-fraction:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">125</span><span class="o">**</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">mpf(&#39;4.9999999999999991&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cbrt</span><span class="p">(</span><span class="mi">125</span><span class="p">)</span>
-<span class="go">mpf(&#39;5.0&#39;)</span>
-</pre></div>
-</div>
-<p>Every nonzero complex number has three cube roots. This function
-returns the cube root defined by <span class="math">\(\exp(\log(x)/3)\)</span> where the
-principal branch of the natural logarithm is used. Note that this
-does not give a real cube root for negative real numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cbrt</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(0.5 + 0.866025403784439j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="root">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">root()</span></tt><a class="headerlink" href="#root" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.root">
-<tt class="descclassname">mpmath.</tt><tt class="descname">root</tt><big>(</big><em>z</em>, <em>n</em>, <em>k=0</em><big>)</big><a class="headerlink" href="#mpmath.root" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">root(z,</span> <span class="pre">n,</span> <span class="pre">k=0)</span></tt> computes an <span class="math">\(n\)</span>-th root of <span class="math">\(z\)</span>, i.e. returns a number
-<span class="math">\(r\)</span> that (up to possible approximation error) satisfies <span class="math">\(r^n = z\)</span>.
-(<tt class="docutils literal"><span class="pre">nthroot</span></tt> is available as an alias for <tt class="docutils literal"><span class="pre">root</span></tt>.)</p>
-<p>Every complex number <span class="math">\(z \ne 0\)</span> has <span class="math">\(n\)</span> distinct <span class="math">\(n\)</span>-th roots, which are
-equidistant points on a circle with radius <span class="math">\(|z|^{1/n}\)</span>, centered around the
-origin. A specific root may be selected using the optional index
-<span class="math">\(k\)</span>. The roots are indexed counterclockwise, starting with <span class="math">\(k = 0\)</span> for the root
-closest to the positive real half-axis.</p>
-<p>The <span class="math">\(k = 0\)</span> root is the so-called principal <span class="math">\(n\)</span>-th root, often denoted by
-<span class="math">\(\sqrt[n]{z}\)</span> or <span class="math">\(z^{1/n}\)</span>, and also given by <span class="math">\(\exp(\log(z) / n)\)</span>. If <span class="math">\(z\)</span> is
-a positive real number, the principal root is just the unique positive
-<span class="math">\(n\)</span>-th root of <span class="math">\(z\)</span>. Under some circumstances, non-principal real roots exist:
-for positive real <span class="math">\(z\)</span>, <span class="math">\(n\)</span> even, there is a negative root given by <span class="math">\(k = n/2\)</span>;
-for negative real <span class="math">\(z\)</span>, <span class="math">\(n\)</span> odd, there is a negative root given by <span class="math">\(k = (n-1)/2\)</span>.</p>
-<p>To obtain all roots with a simple expression, use
-<tt class="docutils literal"><span class="pre">[root(z,n,k)</span> <span class="pre">for</span> <span class="pre">k</span> <span class="pre">in</span> <span class="pre">range(n)]</span></tt>.</p>
-<p>An important special case, <tt class="docutils literal"><span class="pre">root(1,</span> <span class="pre">n,</span> <span class="pre">k)</span></tt> returns the <span class="math">\(k\)</span>-th <span class="math">\(n\)</span>-th root of
-unity, <span class="math">\(\zeta_k = e^{2 \pi i k / n}\)</span>. Alternatively, <a class="reference internal" href="#mpmath.unitroots" title="mpmath.unitroots"><tt class="xref py py-func docutils literal"><span class="pre">unitroots()</span></tt></a>
-provides a slightly more convenient way to obtain the roots of unity,
-including the option to compute only the primitive roots of unity.</p>
-<p>Both <span class="math">\(k\)</span> and <span class="math">\(n\)</span> should be integers; <span class="math">\(k\)</span> outside of <tt class="docutils literal"><span class="pre">range(n)</span></tt> will be
-reduced modulo <span class="math">\(n\)</span>. If <span class="math">\(n\)</span> is negative, <span class="math">\(x^{-1/n} = 1/x^{1/n}\)</span> (or
-the equivalent reciprocal for a non-principal root with <span class="math">\(k \ne 0\)</span>) is computed.</p>
-<p><a class="reference internal" href="#mpmath.root" title="mpmath.root"><tt class="xref py py-func docutils literal"><span class="pre">root()</span></tt></a> is implemented to use Newton&#8217;s method for small
-<span class="math">\(n\)</span>. At high precision, this makes <span class="math">\(x^{1/n}\)</span> not much more
-expensive than the regular exponentiation, <span class="math">\(x^n\)</span>. For very large
-<span class="math">\(n\)</span>, <tt class="xref py py-func docutils literal"><span class="pre">nthroot()</span></tt> falls back to use the exponential function.</p>
-<p><strong>Examples</strong></p>
-<p><tt class="xref py py-func docutils literal"><span class="pre">nthroot()</span></tt>/<a class="reference internal" href="#mpmath.root" title="mpmath.root"><tt class="xref py py-func docutils literal"><span class="pre">root()</span></tt></a> is faster and more accurate than raising to a
-floating-point fraction:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">16807</span> <span class="o">**</span> <span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">mpf(&#39;7.0000000000000009&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">root</span><span class="p">(</span><span class="mi">16807</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">mpf(&#39;7.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nthroot</span><span class="p">(</span><span class="mi">16807</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>    <span class="c"># Alias</span>
-<span class="go">mpf(&#39;7.0&#39;)</span>
-</pre></div>
-</div>
-<p>A high-precision root:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nthroot</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">1.584893192461113485202101373391507013269442133825</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nthroot</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span> <span class="o">**</span> <span class="mi">5</span>
-<span class="go">10.0</span>
-</pre></div>
-</div>
-<p>Computing principal and non-principal square and cube roots:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">root</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">3.16227766016838</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">root</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">-3.16227766016838</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">root</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">(1.07721734501594 + 1.86579517236206j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">root</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">-2.15443469003188</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">root</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">(1.07721734501594 - 1.86579517236206j)</span>
-</pre></div>
-</div>
-<p>All the 7th roots of a complex number:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="p">[</span><span class="n">root</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">7</span><span class="p">)]:</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">r</span><span class="o">**</span><span class="mi">7</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">(1.24747270589553 + 0.166227124177353j) (3.0 + 4.0j)</span>
-<span class="go">(0.647824911301003 + 1.07895435170559j) (3.0 + 4.0j)</span>
-<span class="go">(-0.439648254723098 + 1.17920694574172j) (3.0 + 4.0j)</span>
-<span class="go">(-1.19605731775069 + 0.391492658196305j) (3.0 + 4.0j)</span>
-<span class="go">(-1.05181082538903 - 0.691023585965793j) (3.0 + 4.0j)</span>
-<span class="go">(-0.115529328478668 - 1.25318497558335j) (3.0 + 4.0j)</span>
-<span class="go">(0.907748109144957 - 0.871672518271819j) (3.0 + 4.0j)</span>
-</pre></div>
-</div>
-<p>Cube roots of unity:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">):</span> <span class="k">print</span><span class="p">(</span><span class="n">root</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">k</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">1.0</span>
-<span class="go">(-0.5 + 0.866025403784439j)</span>
-<span class="go">(-0.5 - 0.866025403784439j)</span>
-</pre></div>
-</div>
-<p>Some exact high order roots:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">root</span><span class="p">(</span><span class="mi">75</span><span class="o">**</span><span class="mi">210</span><span class="p">,</span> <span class="mi">105</span><span class="p">)</span>
-<span class="go">5625.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">root</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">128</span><span class="p">,</span> <span class="mi">96</span><span class="p">)</span>
-<span class="go">(0.0 - 1.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">root</span><span class="p">(</span><span class="mi">4</span><span class="o">**</span><span class="mi">128</span><span class="p">,</span> <span class="mi">128</span><span class="p">,</span> <span class="mi">96</span><span class="p">)</span>
-<span class="go">(0.0 - 4.0j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="unitroots">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">unitroots()</span></tt><a class="headerlink" href="#unitroots" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.unitroots">
-<tt class="descclassname">mpmath.</tt><tt class="descname">unitroots</tt><big>(</big><em>n</em>, <em>primitive=False</em><big>)</big><a class="headerlink" href="#mpmath.unitroots" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">unitroots(n)</span></tt> returns <span class="math">\(\zeta_0, \zeta_1, \ldots, \zeta_{n-1}\)</span>,
-all the distinct <span class="math">\(n\)</span>-th roots of unity, as a list. If the option
-<em>primitive=True</em> is passed, only the primitive roots are returned.</p>
-<p>Every <span class="math">\(n\)</span>-th root of unity satisfies <span class="math">\((\zeta_k)^n = 1\)</span>. There are <span class="math">\(n\)</span> distinct
-roots for each <span class="math">\(n\)</span> (<span class="math">\(\zeta_k\)</span> and <span class="math">\(\zeta_j\)</span> are the same when
-<span class="math">\(k = j \pmod n\)</span>), which form a regular polygon with vertices on the unit
-circle. They are ordered counterclockwise with increasing <span class="math">\(k\)</span>, starting
-with <span class="math">\(\zeta_0 = 1\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>The roots of unity up to <span class="math">\(n = 4\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">unitroots</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span>
-<span class="go">[1.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">unitroots</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">[1.0, -1.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">unitroots</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">[1.0, (-0.5 + 0.866025j), (-0.5 - 0.866025j)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">unitroots</span><span class="p">(</span><span class="mi">4</span><span class="p">))</span>
-<span class="go">[1.0, (0.0 + 1.0j), -1.0, (0.0 - 1.0j)]</span>
-</pre></div>
-</div>
-<p>Roots of unity form a geometric series that sums to 0:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">fsum</span><span class="p">(</span><span class="n">unitroots</span><span class="p">(</span><span class="mi">25</span><span class="p">)))</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>Primitive roots up to <span class="math">\(n = 4\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">unitroots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">primitive</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">[1.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">unitroots</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">primitive</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">[-1.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">unitroots</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">primitive</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">[(-0.5 + 0.866025j), (-0.5 - 0.866025j)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">unitroots</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">primitive</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">[(0.0 + 1.0j), (0.0 - 1.0j)]</span>
-</pre></div>
-</div>
-<p>There are only four primitive 12th roots:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">unitroots</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="n">primitive</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">[(0.866025 + 0.5j), (-0.866025 + 0.5j), (-0.866025 - 0.5j), (0.866025 - 0.5j)]</span>
-</pre></div>
-</div>
-<p>The <span class="math">\(n\)</span>-th roots of unity form a group, the cyclic group of order <span class="math">\(n\)</span>.
-Any primitive root <span class="math">\(r\)</span> is a generator for this group, meaning that
-<span class="math">\(r^0, r^1, \ldots, r^{n-1}\)</span> gives the whole set of unit roots (in
-some permuted order):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">unitroots</span><span class="p">(</span><span class="mi">6</span><span class="p">):</span> <span class="k">print</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="go">1.0</span>
-<span class="go">(0.5 + 0.866025403784439j)</span>
-<span class="go">(-0.5 + 0.866025403784439j)</span>
-<span class="go">-1.0</span>
-<span class="go">(-0.5 - 0.866025403784439j)</span>
-<span class="go">(0.5 - 0.866025403784439j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span> <span class="o">=</span> <span class="n">unitroots</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="n">primitive</span><span class="o">=</span><span class="bp">True</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">6</span><span class="p">):</span> <span class="k">print</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">r</span><span class="o">**</span><span class="n">k</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">1.0</span>
-<span class="go">(0.5 - 0.866025403784439j)</span>
-<span class="go">(-0.5 - 0.866025403784439j)</span>
-<span class="go">-1.0</span>
-<span class="go">(-0.5 + 0.866025403784438j)</span>
-<span class="go">(0.5 + 0.866025403784438j)</span>
-</pre></div>
-</div>
-<p>The number of primitive roots equals the Euler totient function <span class="math">\(\phi(n)\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="n">unitroots</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">primitive</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">20</span><span class="p">)]</span>
-<span class="go">[1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="exponentiation">
-<h2>Exponentiation<a class="headerlink" href="#exponentiation" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="exp">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">exp()</span></tt><a class="headerlink" href="#exp" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.exp">
-<tt class="descclassname">mpmath.</tt><tt class="descname">exp</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.exp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the exponential function,</p>
-<div class="math">
-\[\exp(x) = e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}.\]</div>
-<p>For complex numbers, the exponential function also satisfies</p>
-<div class="math">
-\[\exp(x+yi) = e^x (\cos y + i \sin y).\]</div>
-<p><strong>Basic examples</strong></p>
-<p>Some values of the exponential function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">2.718281828459045235360287</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.3678794411714423215955238</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>Arguments can be arbitrarily large:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="mi">10000</span><span class="p">)</span>
-<span class="go">8.806818225662921587261496e+4342</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="mi">10000</span><span class="p">)</span>
-<span class="go">1.135483865314736098540939e-4343</span>
-</pre></div>
-</div>
-<p>Evaluation is supported for interval arguments via
-<tt class="xref py py-func docutils literal"><span class="pre">mpmath.iv.exp()</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">iv</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">exp</span><span class="p">([</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span>
-<span class="go">[0.0, 1.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">exp</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">[1.0, 2.71828182845904523536028749558]</span>
-</pre></div>
-</div>
-<p>The exponential function can be evaluated efficiently to arbitrary
-precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">10000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>  
-<span class="go">23.140692632779269005729...8984304016040616</span>
-</pre></div>
-</div>
-<p><strong>Functional properties</strong></p>
-<p>Numerical verification of Euler&#8217;s identity for the complex
-exponential function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">j</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">+</span><span class="mi">1</span>
-<span class="go">(0.0 + 1.22464679914735e-16j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">j</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>This recovers the coefficients (reciprocal factorials) in the
-Maclaurin series expansion of exp:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">exp</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
-<span class="go">[1.0, 1.0, 0.5, 0.166667, 0.0416667, 0.00833333]</span>
-</pre></div>
-</div>
-<p>The exponential function is its own derivative and antiderivative:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">23.1406926327793</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">exp</span><span class="p">,</span> <span class="n">pi</span><span class="p">)</span>
-<span class="go">23.1406926327793</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">exp</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span> <span class="n">pi</span><span class="p">])</span>
-<span class="go">23.1406926327793</span>
-</pre></div>
-</div>
-<p>The exponential function can be evaluated using various methods,
-including direct summation of the series, limits, and solving
-the defining differential equation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="n">pi</span><span class="o">**</span><span class="n">k</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">k</span><span class="p">),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">23.1406926327793</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">pi</span><span class="o">/</span><span class="n">k</span><span class="p">)</span><span class="o">**</span><span class="n">k</span><span class="p">,</span> <span class="n">inf</span><span class="p">)</span>
-<span class="go">23.1406926327793</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">odefun</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">,</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">23.1406926327793</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="power">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">power()</span></tt><a class="headerlink" href="#power" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.power">
-<tt class="descclassname">mpmath.</tt><tt class="descname">power</tt><big>(</big><em>x</em>, <em>y</em><big>)</big><a class="headerlink" href="#mpmath.power" title="Permalink to this definition">¶</a></dt>
-<dd><p>Converts <span class="math">\(x\)</span> and <span class="math">\(y\)</span> to mpmath numbers and evaluates
-<span class="math">\(x^y = \exp(y \log(x))\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">30</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">power</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
-<span class="go">1.41421356237309504880168872421</span>
-</pre></div>
-</div>
-<p>This shows the leading few digits of a large Mersenne prime
-(performing the exact calculation <tt class="docutils literal"><span class="pre">2**43112609-1</span></tt> and
-displaying the result in Python would be very slow):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">power</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">43112609</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span>
-<span class="go">3.16470269330255923143453723949e+12978188</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="expj">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">expj()</span></tt><a class="headerlink" href="#expj" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.expj">
-<tt class="descclassname">mpmath.</tt><tt class="descname">expj</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.expj" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convenience function for computing <span class="math">\(e^{ix}\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expj</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">(1.0 + 0.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expj</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(0.5403023058681397174009366 - 0.8414709848078965066525023j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expj</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-<span class="go">(0.3678794411714423215955238 + 0.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expj</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">)</span>
-<span class="go">(0.1987661103464129406288032 + 0.3095598756531121984439128j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="expjpi">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">expjpi()</span></tt><a class="headerlink" href="#expjpi" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.expjpi">
-<tt class="descclassname">mpmath.</tt><tt class="descname">expjpi</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.expjpi" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convenience function for computing <span class="math">\(e^{i \pi x}\)</span>.
-Evaluation is accurate near zeros (see also <a class="reference internal" href="../functions/trigonometric.html#mpmath.cospi" title="mpmath.cospi"><tt class="xref py py-func docutils literal"><span class="pre">cospi()</span></tt></a>,
-<a class="reference internal" href="../functions/trigonometric.html#mpmath.sinpi" title="mpmath.sinpi"><tt class="xref py py-func docutils literal"><span class="pre">sinpi()</span></tt></a>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expjpi</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">(1.0 + 0.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expjpi</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(-1.0 + 0.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expjpi</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="go">(0.0 + 1.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expjpi</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(-1.0 + 0.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expjpi</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-<span class="go">(0.04321391826377224977441774 + 0.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expjpi</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">)</span>
-<span class="go">(-0.04321391826377224977441774 + 0.0j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="expm1">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">expm1()</span></tt><a class="headerlink" href="#expm1" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.expm1">
-<tt class="descclassname">mpmath.</tt><tt class="descname">expm1</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.expm1" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes <span class="math">\(e^x - 1\)</span>, accurately for small <span class="math">\(x\)</span>.</p>
-<p>Unlike the expression <tt class="docutils literal"><span class="pre">exp(x)</span> <span class="pre">-</span> <span class="pre">1</span></tt>, <tt class="docutils literal"><span class="pre">expm1(x)</span></tt> does not suffer from
-potentially catastrophic cancellation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="mf">1e-10</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span> <span class="k">print</span><span class="p">(</span><span class="n">expm1</span><span class="p">(</span><span class="mf">1e-10</span><span class="p">))</span>
-<span class="go">1.00000008274037e-10</span>
-<span class="go">1.00000000005e-10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="mf">1e-20</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span> <span class="k">print</span><span class="p">(</span><span class="n">expm1</span><span class="p">(</span><span class="mf">1e-20</span><span class="p">))</span>
-<span class="go">0.0</span>
-<span class="go">1.0e-20</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="mf">1e-20</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-  <span class="c">...</span>
-<span class="gr">ZeroDivisionError</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="n">expm1</span><span class="p">(</span><span class="mf">1e-20</span><span class="p">)</span>
-<span class="go">1.0e+20</span>
-</pre></div>
-</div>
-<p>Evaluation works for extremely tiny values:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expm1</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expm1</span><span class="p">(</span><span class="s">&#39;1e-10000000&#39;</span><span class="p">)</span>
-<span class="go">1.0e-10000000</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="powm1">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">powm1()</span></tt><a class="headerlink" href="#powm1" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.powm1">
-<tt class="descclassname">mpmath.</tt><tt class="descname">powm1</tt><big>(</big><em>x</em>, <em>y</em><big>)</big><a class="headerlink" href="#mpmath.powm1" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes <span class="math">\(x^y - 1\)</span>, accurately when <span class="math">\(x^y\)</span> is very close to 1.</p>
-<p>This avoids potentially catastrophic cancellation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">power</span><span class="p">(</span><span class="mf">0.99999995</span><span class="p">,</span> <span class="mf">1e-10</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powm1</span><span class="p">(</span><span class="mf">0.99999995</span><span class="p">,</span> <span class="mf">1e-10</span><span class="p">)</span>
-<span class="go">-5.00000012791934e-18</span>
-</pre></div>
-</div>
-<p>Powers exactly equal to 1, and only those powers, yield 0 exactly:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">powm1</span><span class="p">(</span><span class="o">-</span><span class="n">j</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">(0.0 + 0.0j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powm1</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powm1</span><span class="p">(</span><span class="n">fadd</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mf">1e-100</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">),</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">-4.0e-100</span>
-</pre></div>
-</div>
-<p>Evaluation works for extremely tiny <span class="math">\(y\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">powm1</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="s">&#39;1e-100000&#39;</span><span class="p">)</span>
-<span class="go">6.93147180559945e-100001</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powm1</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="s">&#39;1e-1000&#39;</span><span class="p">)</span>
-<span class="go">(-1.23370055013617e-2000 + 1.5707963267949e-1000j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="logarithms">
-<h2>Logarithms<a class="headerlink" href="#logarithms" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="log">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">log()</span></tt><a class="headerlink" href="#log" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.log">
-<tt class="descclassname">mpmath.</tt><tt class="descname">log</tt><big>(</big><em>x</em>, <em>b=None</em><big>)</big><a class="headerlink" href="#mpmath.log" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the base-<span class="math">\(b\)</span> logarithm of <span class="math">\(x\)</span>, <span class="math">\(\log_b(x)\)</span>. If <span class="math">\(b\)</span> is
-unspecified, <a class="reference internal" href="#mpmath.log" title="mpmath.log"><tt class="xref py py-func docutils literal"><span class="pre">log()</span></tt></a> computes the natural (base <span class="math">\(e\)</span>) logarithm
-and is equivalent to <a class="reference internal" href="#mpmath.ln" title="mpmath.ln"><tt class="xref py py-func docutils literal"><span class="pre">ln()</span></tt></a>. In general, the base <span class="math">\(b\)</span> logarithm
-is defined in terms of the natural logarithm as
-<span class="math">\(\log_b(x) = \ln(x)/\ln(b)\)</span>.</p>
-<p>By convention, we take <span class="math">\(\log(0) = -\infty\)</span>.</p>
-<p>The natural logarithm is real if <span class="math">\(x &gt; 0\)</span> and complex if <span class="math">\(x &lt; 0\)</span> or if
-<span class="math">\(x\)</span> is complex. The principal branch of the complex logarithm is
-used, meaning that <span class="math">\(\Im(\ln(x)) = -\pi &lt; \arg(x) \le \pi\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>Some basic values and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.693147180559945</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">3.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>
-<span class="go">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-<span class="go">(0.0 + 1.5707963267949j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(0.0 + 3.14159265358979j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">-inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">+inf</span>
-</pre></div>
-</div>
-<p>The natural logarithm is the antiderivative of <span class="math">\(1/x\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
-<span class="go">1.6094379124341</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">1.6094379124341</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">log</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">0.1</span>
-</pre></div>
-</div>
-<p>The Taylor series expansion of the natural logarithm around
-<span class="math">\(x = 1\)</span> has coefficients <span class="math">\((-1)^{n+1}/n\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">log</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">))</span>
-<span class="go">[0.0, 1.0, -0.5, 0.333333, -0.25, 0.2, -0.166667, 0.142857]</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.log" title="mpmath.log"><tt class="xref py py-func docutils literal"><span class="pre">log()</span></tt></a> supports arbitrary precision evaluation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">1.1447298858494001741434273513530587116472948129153</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="n">pi</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.33333333333333333333333333333333333333333333333333</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(1.609437912434100374600759 + 0.9272952180016122324285125j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="ln">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">ln()</span></tt><a class="headerlink" href="#ln" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.ln">
-<tt class="descclassname">mpmath.</tt><tt class="descname">ln</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.ln" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the base-<span class="math">\(b\)</span> logarithm of <span class="math">\(x\)</span>, <span class="math">\(\log_b(x)\)</span>. If <span class="math">\(b\)</span> is
-unspecified, <a class="reference internal" href="#mpmath.log" title="mpmath.log"><tt class="xref py py-func docutils literal"><span class="pre">log()</span></tt></a> computes the natural (base <span class="math">\(e\)</span>) logarithm
-and is equivalent to <a class="reference internal" href="#mpmath.ln" title="mpmath.ln"><tt class="xref py py-func docutils literal"><span class="pre">ln()</span></tt></a>. In general, the base <span class="math">\(b\)</span> logarithm
-is defined in terms of the natural logarithm as
-<span class="math">\(\log_b(x) = \ln(x)/\ln(b)\)</span>.</p>
-<p>By convention, we take <span class="math">\(\log(0) = -\infty\)</span>.</p>
-<p>The natural logarithm is real if <span class="math">\(x &gt; 0\)</span> and complex if <span class="math">\(x &lt; 0\)</span> or if
-<span class="math">\(x\)</span> is complex. The principal branch of the complex logarithm is
-used, meaning that <span class="math">\(\Im(\ln(x)) = -\pi &lt; \arg(x) \le \pi\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>Some basic values and limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.693147180559945</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">3.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">16</span><span class="p">)</span>
-<span class="go">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="n">j</span><span class="p">)</span>
-<span class="go">(0.0 + 1.5707963267949j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(0.0 + 3.14159265358979j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">-inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">+inf</span>
-</pre></div>
-</div>
-<p>The natural logarithm is the antiderivative of <span class="math">\(1/x\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
-<span class="go">1.6094379124341</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">1.6094379124341</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">log</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">0.1</span>
-</pre></div>
-</div>
-<p>The Taylor series expansion of the natural logarithm around
-<span class="math">\(x = 1\)</span> has coefficients <span class="math">\((-1)^{n+1}/n\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">log</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">))</span>
-<span class="go">[0.0, 1.0, -0.5, 0.333333, -0.25, 0.2, -0.166667, 0.142857]</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.log" title="mpmath.log"><tt class="xref py py-func docutils literal"><span class="pre">log()</span></tt></a> supports arbitrary precision evaluation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">1.1447298858494001741434273513530587116472948129153</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="n">pi</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.33333333333333333333333333333333333333333333333333</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(1.609437912434100374600759 + 0.9272952180016122324285125j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="log10">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">log10()</span></tt><a class="headerlink" href="#log10" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.log10">
-<tt class="descclassname">mpmath.</tt><tt class="descname">log10</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.log10" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the base-10 logarithm of <span class="math">\(x\)</span>, <span class="math">\(\log_{10}(x)\)</span>. <tt class="docutils literal"><span class="pre">log10(x)</span></tt>
-is equivalent to <tt class="docutils literal"><span class="pre">log(x,</span> <span class="pre">10)</span></tt>.</p>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="lambert-w-function">
-<h2>Lambert W function<a class="headerlink" href="#lambert-w-function" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="lambertw">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">lambertw()</span></tt><a class="headerlink" href="#lambertw" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.lambertw">
-<tt class="descclassname">mpmath.</tt><tt class="descname">lambertw</tt><big>(</big><em>z</em>, <em>k=0</em><big>)</big><a class="headerlink" href="#mpmath.lambertw" title="Permalink to this definition">¶</a></dt>
-<dd><p>The Lambert W function <span class="math">\(W(z)\)</span> is defined as the inverse function
-of <span class="math">\(w \exp(w)\)</span>. In other words, the value of <span class="math">\(W(z)\)</span> is such that
-<span class="math">\(z = W(z) \exp(W(z))\)</span> for any complex number <span class="math">\(z\)</span>.</p>
-<p>The Lambert W function is a multivalued function with infinitely
-many branches <span class="math">\(W_k(z)\)</span>, indexed by <span class="math">\(k \in \mathbb{Z}\)</span>. Each branch
-gives a different solution <span class="math">\(w\)</span> of the equation <span class="math">\(z = w \exp(w)\)</span>.
-All branches are supported by <a class="reference internal" href="#mpmath.lambertw" title="mpmath.lambertw"><tt class="xref py py-func docutils literal"><span class="pre">lambertw()</span></tt></a>:</p>
-<ul class="simple">
-<li><tt class="docutils literal"><span class="pre">lambertw(z)</span></tt> gives the principal solution (branch 0)</li>
-<li><tt class="docutils literal"><span class="pre">lambertw(z,</span> <span class="pre">k)</span></tt> gives the solution on branch <span class="math">\(k\)</span></li>
-</ul>
-<p>The Lambert W function has two partially real branches: the
-principal branch (<span class="math">\(k = 0\)</span>) is real for real <span class="math">\(z &gt; -1/e\)</span>, and the
-<span class="math">\(k = -1\)</span> branch is real for <span class="math">\(-1/e &lt; z &lt; 0\)</span>. All branches except
-<span class="math">\(k = 0\)</span> have a logarithmic singularity at <span class="math">\(z = 0\)</span>.</p>
-<p>The definition, implementation and choice of branches
-is based on <a class="reference internal" href="../../mpmath/references.html#corless">[Corless]</a>.</p>
-<p><strong>Plots</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Branches 0 and -1 of the Lambert W function</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">lambertw</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">lambertw</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)],</span> <span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span><span class="mi">2</span><span class="p">],</span> <span class="n">points</span><span class="o">=</span><span class="mi">2000</span><span class="p">)</span>
-</pre></div>
-</div>
-<img alt="../../../_images/lambertw.png" src="../../../_images/lambertw.png" />
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Principal branch of the Lambert W function W(z)</span>
-<span class="n">cplot</span><span class="p">(</span><span class="n">lambertw</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="n">points</span><span class="o">=</span><span class="mi">50000</span><span class="p">)</span>
-</pre></div>
-</div>
-<img alt="../../../_images/lambertw_c.png" src="../../../_images/lambertw_c.png" />
-<p><strong>Basic examples</strong></p>
-<p>The Lambert W function is the inverse of <span class="math">\(w \exp(w)\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">w</span> <span class="o">=</span> <span class="n">lambertw</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">w</span>
-<span class="go">0.5671432904097838729999687</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">w</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">w</span><span class="p">)</span>
-<span class="go">1.0</span>
-</pre></div>
-</div>
-<p>Any branch gives a valid inverse:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">w</span> <span class="o">=</span> <span class="n">lambertw</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">w</span>
-<span class="go">(-2.853581755409037807206819 + 17.11353553941214591260783j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">w</span> <span class="o">=</span> <span class="n">lambertw</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">25</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">w</span>
-<span class="go">(-5.047020464221569709378686 + 155.4763860949415867162066j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">w</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">w</span><span class="p">))</span>
-<span class="go">1.0</span>
-</pre></div>
-</div>
-<p><strong>Applications to equation-solving</strong></p>
-<p>The Lambert W function may be used to solve various kinds of
-equations, such as finding the value of the infinite power
-tower <span class="math">\(z^{z^{z^{\ldots}}}\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">tower</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
-<span class="gp">... </span>        <span class="k">return</span> <span class="n">z</span>
-<span class="gp">... </span>    <span class="k">return</span> <span class="n">z</span> <span class="o">**</span> <span class="n">tower</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tower</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="mf">0.5</span><span class="p">),</span> <span class="mi">100</span><span class="p">)</span>
-<span class="go">0.6411857445049859844862005</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="n">lambertw</span><span class="p">(</span><span class="o">-</span><span class="n">log</span><span class="p">(</span><span class="mf">0.5</span><span class="p">))</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="go">0.6411857445049859844862005</span>
-</pre></div>
-</div>
-<p><strong>Properties</strong></p>
-<p>The Lambert W function grows roughly like the natural logarithm
-for large arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">lambertw</span><span class="p">(</span><span class="mi">1000</span><span class="p">);</span> <span class="n">log</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span>
-<span class="go">5.249602852401596227126056</span>
-<span class="go">6.907755278982137052053974</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lambertw</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">100</span><span class="p">);</span> <span class="n">log</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">224.8431064451185015393731</span>
-<span class="go">230.2585092994045684017991</span>
-</pre></div>
-</div>
-<p>The principal branch of the Lambert W function has a rational
-Taylor series expansion around <span class="math">\(z = 0\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">lambertw</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">[0.0, 1.0, -1.0, 1.5, -2.666666667, 5.208333333, -10.8]</span>
-</pre></div>
-</div>
-<p>Some special values and limits are:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">lambertw</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lambertw</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.5671432904097838729999687</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lambertw</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lambertw</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lambertw</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">k</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lambertw</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">-inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lambertw</span><span class="p">(</span><span class="n">inf</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(+inf + 12.56637061435917295385057j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lambertw</span><span class="p">(</span><span class="n">inf</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">(+inf + 18.84955592153875943077586j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lambertw</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">(+inf + 21.9911485751285526692385j)</span>
-</pre></div>
-</div>
-<p>The <span class="math">\(k = 0\)</span> and <span class="math">\(k = -1\)</span> branches join at <span class="math">\(z = -1/e\)</span> where
-<span class="math">\(W(z) = -1\)</span> for both branches. Since <span class="math">\(-1/e\)</span> can only be represented
-approximately with binary floating-point numbers, evaluating the
-Lambert W function at this point only gives <span class="math">\(-1\)</span> approximately:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">lambertw</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">e</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">-0.9999999999998371330228251</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lambertw</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">e</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-1.000000000000162866977175</span>
-</pre></div>
-</div>
-<p>If <span class="math">\(-1/e\)</span> happens to round in the negative direction, there might be
-a small imaginary part:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lambertw</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">e</span><span class="p">)</span>
-<span class="go">(-1.0 + 8.22007971483662e-9j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lambertw</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="n">e</span><span class="o">+</span><span class="n">eps</span><span class="p">)</span>
-<span class="go">-0.999999966242188</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../../mpmath/references.html#corless">[Corless]</a></li>
-</ol>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="arithmetic-geometric-mean">
-<h2>Arithmetic-geometric mean<a class="headerlink" href="#arithmetic-geometric-mean" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="agm">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">agm()</span></tt><a class="headerlink" href="#agm" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.agm">
-<tt class="descclassname">mpmath.</tt><tt class="descname">agm</tt><big>(</big><em>a</em>, <em>b=1</em><big>)</big><a class="headerlink" href="#mpmath.agm" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">agm(a,</span> <span class="pre">b)</span></tt> computes the arithmetic-geometric mean of <span class="math">\(a\)</span> and
-<span class="math">\(b\)</span>, defined as the limit of the following iteration:</p>
-<div class="math">
-\[a_0 = a\]\[b_0 = b\]\[a_{n+1} = \frac{a_n+b_n}{2}\]\[b_{n+1} = \sqrt{a_n b_n}\]</div>
-<p>This function can be called with a single argument, computing
-<span class="math">\(\mathrm{agm}(a,1) = \mathrm{agm}(1,a)\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>It is a well-known theorem that the geometric mean of
-two distinct positive numbers is less than the arithmetic
-mean. It follows that the arithmetic-geometric mean lies
-between the two means:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">b</span><span class="p">)</span>
-<span class="go">3.46410161513775</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">agm</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">)</span>
-<span class="go">3.48202767635957</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-<span class="go">3.5</span>
-</pre></div>
-</div>
-<p>The arithmetic-geometric mean is scale-invariant:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">agm</span><span class="p">(</span><span class="mi">10</span><span class="o">*</span><span class="n">e</span><span class="p">,</span> <span class="mi">10</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">29.261085515723</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">10</span><span class="o">*</span><span class="n">agm</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">pi</span><span class="p">)</span>
-<span class="go">29.261085515723</span>
-</pre></div>
-</div>
-<p>As an order-of-magnitude estimate, <span class="math">\(\mathrm{agm}(1,x) \approx x\)</span>
-for large <span class="math">\(x\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">agm</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">643448704.760133</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">agm</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">50</span><span class="p">)</span>
-<span class="go">1.34814309345871e+48</span>
-</pre></div>
-</div>
-<p>For tiny <span class="math">\(x\)</span>, <span class="math">\(\mathrm{agm}(1,x) \approx -\pi/(2 \log(x/4))\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">agm</span><span class="p">(</span><span class="s">&#39;0.01&#39;</span><span class="p">)</span>
-<span class="go">0.262166887202249</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="s">&#39;0.0025&#39;</span><span class="p">)</span>
-<span class="go">0.262172347753122</span>
-</pre></div>
-</div>
-<p>The arithmetic-geometric mean can also be computed for complex
-numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">agm</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="n">j</span><span class="p">)</span>
-<span class="go">(2.51055133276184 + 0.547394054060638j)</span>
-</pre></div>
-</div>
-<p>The AGM iteration converges very quickly (each step doubles
-the number of correct digits), so <a class="reference internal" href="#mpmath.agm" title="mpmath.agm"><tt class="xref py py-func docutils literal"><span class="pre">agm()</span></tt></a> supports efficient
-high-precision evaluation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">10000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">agm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">str</span><span class="p">(</span><span class="n">a</span><span class="p">)[</span><span class="o">-</span><span class="mi">10</span><span class="p">:]</span>
-<span class="go">&#39;1679581912&#39;</span>
-</pre></div>
-</div>
-<p><strong>Mathematical relations</strong></p>
-<p>The arithmetic-geometric mean may be used to evaluate the
-following two parametric definite integrals:</p>
-<div class="math">
-\[I_1 = \int_0^{\infty}
-  \frac{1}{\sqrt{(x^2+a^2)(x^2+b^2)}} \,dx\]\[I_2 = \int_0^{\pi/2}
-  \frac{1}{\sqrt{a^2 \cos^2(x) + b^2 \sin^2(x)}} \,dx\]</div>
-<p>We have:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="mi">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f1</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="p">((</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">a</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">b</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span><span class="o">**-</span><span class="mf">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="p">((</span><span class="n">a</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="p">(</span><span class="n">b</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">**-</span><span class="mf">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">f1</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">0.451115405388492</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">f2</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">])</span>
-<span class="go">0.451115405388492</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">agm</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">))</span>
-<span class="go">0.451115405388492</span>
-</pre></div>
-</div>
-<p>A formula for <span class="math">\(\Gamma(1/4)\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="mf">0.25</span><span class="p">)</span>
-<span class="go">3.62560990822191</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="n">agm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)))</span>
-<span class="go">3.62560990822191</span>
-</pre></div>
-</div>
-<p><strong>Possible issues</strong></p>
-<p>The branch cut chosen for complex <span class="math">\(a\)</span> and <span class="math">\(b\)</span> is somewhat
-arbitrary.</p>
-</dd></dl>
-
-</div>
-</div>
-</div>
-
-
-          </div>
-        </div>
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-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Powers and logarithms</a><ul>
-<li><a class="reference internal" href="#nth-roots">Nth roots</a><ul>
-<li><a class="reference internal" href="#sqrt"><tt class="docutils literal"><span class="pre">sqrt()</span></tt></a></li>
-<li><a class="reference internal" href="#hypot"><tt class="docutils literal"><span class="pre">hypot()</span></tt></a></li>
-<li><a class="reference internal" href="#cbrt"><tt class="docutils literal"><span class="pre">cbrt()</span></tt></a></li>
-<li><a class="reference internal" href="#root"><tt class="docutils literal"><span class="pre">root()</span></tt></a></li>
-<li><a class="reference internal" href="#unitroots"><tt class="docutils literal"><span class="pre">unitroots()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#exponentiation">Exponentiation</a><ul>
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-  <div class="section" id="q-functions">
-<h1>q-functions<a class="headerlink" href="#q-functions" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="q-pochhammer-symbol">
-<h2>q-Pochhammer symbol<a class="headerlink" href="#q-pochhammer-symbol" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="qp">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">qp()</span></tt><a class="headerlink" href="#qp" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.qp">
-<tt class="descclassname">mpmath.</tt><tt class="descname">qp</tt><big>(</big><em>a</em>, <em>q=None</em>, <em>n=None</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.qp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the q-Pochhammer symbol (or q-rising factorial)</p>
-<div class="math">
-\[(a; q)_n = \prod_{k=0}^{n-1} (1-a q^k)\]</div>
-<p>where <span class="math">\(n = \infty\)</span> is permitted if <span class="math">\(|q| &lt; 1\)</span>. Called with two arguments,
-<tt class="docutils literal"><span class="pre">qp(a,q)</span></tt> computes <span class="math">\((a;q)_{\infty}\)</span>; with a single argument, <tt class="docutils literal"><span class="pre">qp(q)</span></tt>
-computes <span class="math">\((q;q)_{\infty}\)</span>. The special case</p>
-<div class="math">
-\[\phi(q) = (q; q)_{\infty} = \prod_{k=1}^{\infty} (1-q^k) =
-    \sum_{k=-\infty}^{\infty} (-1)^k q^{(3k^2-k)/2}\]</div>
-<p>is also known as the Euler function, or (up to a factor <span class="math">\(q^{-1/24}\)</span>)
-the Dirichlet eta function.</p>
-<p><strong>Examples</strong></p>
-<p>If <span class="math">\(n\)</span> is a positive integer, the function amounts to a finite product:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qp</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">-725305.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fprod</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="mi">3</span><span class="o">**</span><span class="n">k</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
-<span class="go">-725305.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qp</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">1.0</span>
-</pre></div>
-</div>
-<p>Complex arguments are allowed:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">qp</span><span class="p">(</span><span class="mi">2</span><span class="o">-</span><span class="mi">1j</span><span class="p">,</span> <span class="mf">0.75j</span><span class="p">)</span>
-<span class="go">(0.4628842231660149089976379 + 4.481821753552703090628793j)</span>
-</pre></div>
-</div>
-<p>The regular Pochhammer symbol <span class="math">\((a)_n\)</span> is obtained in the
-following limit as <span class="math">\(q \to 1\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">7</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="k">lambda</span> <span class="n">q</span><span class="p">:</span> <span class="n">qp</span><span class="p">(</span><span class="n">q</span><span class="o">**</span><span class="n">a</span><span class="p">,</span><span class="n">q</span><span class="p">,</span><span class="n">n</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">q</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">604800.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rf</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">n</span><span class="p">)</span>
-<span class="go">604800.0</span>
-</pre></div>
-</div>
-<p>The Taylor series of the reciprocal Euler function gives
-the partition function <span class="math">\(P(n)\)</span>, i.e. the number of ways of writing
-<span class="math">\(n\)</span> as a sum of positive integers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">q</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="n">qp</span><span class="p">(</span><span class="n">q</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">[1.0, 1.0, 2.0, 3.0, 5.0, 7.0, 11.0, 15.0, 22.0, 30.0, 42.0]</span>
-</pre></div>
-</div>
-<p>Special values include:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">qp</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">diffun</span><span class="p">(</span><span class="n">qp</span><span class="p">),</span> <span class="o">-</span><span class="mf">0.4</span><span class="p">)</span>   <span class="c"># location of maximum</span>
-<span class="go">-0.4112484791779547734440257</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qp</span><span class="p">(</span><span class="n">_</span><span class="p">)</span>
-<span class="go">1.228348867038575112586878</span>
-</pre></div>
-</div>
-<p>The q-Pochhammer symbol is related to the Jacobi theta functions.
-For example, the following identity holds:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>    <span class="c"># arbitrary</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qp</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-<span class="go">0.2887880950866024212788997</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">root</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">root</span><span class="p">(</span><span class="n">q</span><span class="p">,</span><span class="o">-</span><span class="mi">24</span><span class="p">)</span><span class="o">*</span><span class="n">jtheta</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">pi</span><span class="o">/</span><span class="mi">6</span><span class="p">,</span><span class="n">root</span><span class="p">(</span><span class="n">q</span><span class="p">,</span><span class="mi">6</span><span class="p">))</span>
-<span class="go">0.2887880950866024212788997</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="q-gamma-and-factorial">
-<h2>q-gamma and factorial<a class="headerlink" href="#q-gamma-and-factorial" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="qgamma">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">qgamma()</span></tt><a class="headerlink" href="#qgamma" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.qgamma">
-<tt class="descclassname">mpmath.</tt><tt class="descname">qgamma</tt><big>(</big><em>z</em>, <em>q</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.qgamma" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the q-gamma function</p>
-<div class="math">
-\[\Gamma_q(z) = \frac{(q; q)_{\infty}}{(q^z; q)_{\infty}} (1-q)^{1-z}.\]</div>
-<p><strong>Examples</strong></p>
-<p>Evaluation for real and complex arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qgamma</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="mf">0.75</span><span class="p">)</span>
-<span class="go">4.046875</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qgamma</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span><span class="mi">6</span><span class="p">)</span>
-<span class="go">121226245.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qgamma</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">,</span> <span class="mf">0.5j</span><span class="p">)</span>
-<span class="go">(0.1663082382255199834630088 + 0.01952474576025952984418217j)</span>
-</pre></div>
-</div>
-<p>The q-gamma function satisfies a functional equation similar
-to that of the ordinary gamma function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">0.25</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mf">2.5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qgamma</span><span class="p">(</span><span class="n">z</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="n">q</span><span class="p">)</span>
-<span class="go">1.428277424823760954685912</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">q</span><span class="o">**</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">q</span><span class="p">)</span><span class="o">*</span><span class="n">qgamma</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="n">q</span><span class="p">)</span>
-<span class="go">1.428277424823760954685912</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="qfac">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">qfac()</span></tt><a class="headerlink" href="#qfac" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.qfac">
-<tt class="descclassname">mpmath.</tt><tt class="descname">qfac</tt><big>(</big><em>z</em>, <em>q</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.qfac" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the q-factorial,</p>
-<div class="math">
-\[[n]_q! = (1+q)(1+q+q^2)\cdots(1+q+\cdots+q^{n-1})\]</div>
-<p>or more generally</p>
-<div class="math">
-\[[z]_q! = \frac{(q;q)_z}{(1-q)^z}.\]</div>
-<p><strong>Examples</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qfac</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qfac</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">2080.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qfac</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">)</span>
-<span class="go">121226245.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qfac</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mi">1j</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">1j</span><span class="p">)</span>
-<span class="go">(0.4370556551322672478613695 + 0.2609739839216039203708921j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="hypergeometric-q-series">
-<h2>Hypergeometric q-series<a class="headerlink" href="#hypergeometric-q-series" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="qhyper">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">qhyper()</span></tt><a class="headerlink" href="#qhyper" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.qhyper">
-<tt class="descclassname">mpmath.</tt><tt class="descname">qhyper</tt><big>(</big><em>a_s</em>, <em>b_s</em>, <em>q</em>, <em>z</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.qhyper" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the basic hypergeometric series or hypergeometric q-series</p>
-<div class="math">
-\[\begin{split}\,_r\phi_s \left[\begin{matrix}
-    a_1 &amp; a_2 &amp; \ldots &amp; a_r \\
-    b_1 &amp; b_2 &amp; \ldots &amp; b_s
-\end{matrix} ; q,z \right] =
-\sum_{n=0}^\infty
-\frac{(a_1;q)_n, \ldots, (a_r;q)_n}
-     {(b_1;q)_n, \ldots, (b_s;q)_n}
-\left((-1)^n q^{n\choose 2}\right)^{1+s-r}
-\frac{z^n}{(q;q)_n}\end{split}\]</div>
-<p>where <span class="math">\((a;q)_n\)</span> denotes the q-Pochhammer symbol (see <a class="reference internal" href="#mpmath.qp" title="mpmath.qp"><tt class="xref py py-func docutils literal"><span class="pre">qp()</span></tt></a>).</p>
-<p><strong>Examples</strong></p>
-<p>Evaluation works for real and complex arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qhyper</span><span class="p">([</span><span class="mf">0.5</span><span class="p">],</span> <span class="p">[</span><span class="mf">2.25</span><span class="p">],</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">-0.1975849091263356009534385</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qhyper</span><span class="p">([</span><span class="mf">0.5</span><span class="p">],</span> <span class="p">[</span><span class="mf">2.25</span><span class="p">],</span> <span class="mf">0.25</span><span class="o">-</span><span class="mf">0.25j</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">(2.806330244925716649839237 + 3.568997623337943121769938j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qhyper</span><span class="p">([</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="o">+</span><span class="mf">0.5j</span><span class="p">],</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(9.112885171773400017270226 - 1.272756997166375050700388j)</span>
-</pre></div>
-</div>
-<p>Comparing with a summation of the defining series, using
-<a class="reference internal" href="../../mpmath/calculus/sums_limits.html#mpmath.nsum" title="mpmath.nsum"><tt class="xref py py-func docutils literal"><span class="pre">nsum()</span></tt></a>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mf">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qhyper</span><span class="p">([],</span> <span class="p">[</span><span class="n">b</span><span class="p">],</span> <span class="n">q</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">0.6221136748254495583228324</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">n</span><span class="p">:</span> <span class="n">z</span><span class="o">**</span><span class="n">n</span> <span class="o">/</span> <span class="n">qp</span><span class="p">(</span><span class="n">q</span><span class="p">,</span><span class="n">q</span><span class="p">,</span><span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">qp</span><span class="p">(</span><span class="n">b</span><span class="p">,</span><span class="n">q</span><span class="p">,</span><span class="n">n</span><span class="p">)</span> <span class="o">*</span> <span class="n">q</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">0.6221136748254495583228324</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-</div>
-
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-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">q-functions</a><ul>
-<li><a class="reference internal" href="#q-pochhammer-symbol">q-Pochhammer symbol</a><ul>
-<li><a class="reference internal" href="#qp"><tt class="docutils literal"><span class="pre">qp()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#q-gamma-and-factorial">q-gamma and factorial</a><ul>
-<li><a class="reference internal" href="#qgamma"><tt class="docutils literal"><span class="pre">qgamma()</span></tt></a></li>
-<li><a class="reference internal" href="#qfac"><tt class="docutils literal"><span class="pre">qfac()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#hypergeometric-q-series">Hypergeometric q-series</a><ul>
-<li><a class="reference internal" href="#qhyper"><tt class="docutils literal"><span class="pre">qhyper()</span></tt></a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-
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-  <p class="topless"><a href="../functions/numtheory.html"
-                        title="previous chapter">Number-theoretical, combinatorial and integer functions</a></p>
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-            
-  <div class="section" id="trigonometric-functions">
-<h1>Trigonometric functions<a class="headerlink" href="#trigonometric-functions" title="Permalink to this headline">¶</a></h1>
-<p>Except where otherwise noted, the trigonometric functions
-take a radian angle as input and the inverse trigonometric
-functions return radian angles.</p>
-<p>The ordinary trigonometric functions are single-valued
-functions defined everywhere in the complex plane
-(except at the poles of tan, sec, csc, and cot).
-They are defined generally via the exponential function,
-e.g.</p>
-<div class="math">
-\[\cos(x) = \frac{e^{ix} + e^{-ix}}{2}.\]</div>
-<p>The inverse trigonometric functions are multivalued,
-thus requiring branch cuts, and are generally real-valued
-only on a part of the real line. Definitions and branch cuts
-are given in the documentation of each function.
-The branch cut conventions used by mpmath are essentially
-the same as those found in most standard mathematical software,
-such as Mathematica and Python&#8217;s own <tt class="docutils literal"><span class="pre">cmath</span></tt> libary (as of Python 2.6;
-earlier Python versions implement some functions
-erroneously).</p>
-<div class="section" id="degree-radian-conversion">
-<h2>Degree-radian conversion<a class="headerlink" href="#degree-radian-conversion" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="degrees">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">degrees()</span></tt><a class="headerlink" href="#degrees" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.degrees">
-<tt class="descclassname">mpmath.</tt><tt class="descname">degrees</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.degrees" title="Permalink to this definition">¶</a></dt>
-<dd><p>Converts the radian angle <span class="math">\(x\)</span> to a degree angle:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">degrees</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">60.0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="radians">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">radians()</span></tt><a class="headerlink" href="#radians" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.radians">
-<tt class="descclassname">mpmath.</tt><tt class="descname">radians</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.radians" title="Permalink to this definition">¶</a></dt>
-<dd><p>Converts the degree angle <span class="math">\(x\)</span> to radians:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">radians</span><span class="p">(</span><span class="mi">60</span><span class="p">)</span>
-<span class="go">1.0471975511966</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="id1">
-<h2>Trigonometric functions<a class="headerlink" href="#id1" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="cos">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">cos()</span></tt><a class="headerlink" href="#cos" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.cos">
-<tt class="descclassname">mpmath.</tt><tt class="descname">cos</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.cos" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the cosine of <span class="math">\(x\)</span>, <span class="math">\(\cos(x)\)</span>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="mi">100000001</span><span class="p">)</span>
-<span class="go">-0.9802850113244713353133243</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(-4.189625690968807230132555 - 9.109227893755336597979197j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">nan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">cos</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">)))</span>
-<span class="go">[1.0, 0.0, -0.5, 0.0, 0.0416667, 0.0, -0.00138889]</span>
-</pre></div>
-</div>
-<p>Intervals are supported via <tt class="xref py py-func docutils literal"><span class="pre">mpmath.iv.cos()</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">iv</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">cos</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">[0.540302305868139717400936602301, 1.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">cos</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
-<span class="go">[-0.41614683654714238699756823214, 1.0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="sin">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">sin()</span></tt><a class="headerlink" href="#sin" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.sin">
-<tt class="descclassname">mpmath.</tt><tt class="descname">sin</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.sin" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the sine of <span class="math">\(x\)</span>, <span class="math">\(\sin(x)\)</span>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.8660254037844386467637232</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="mi">100000001</span><span class="p">)</span>
-<span class="go">0.1975887055794968911438743</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(9.1544991469114295734673 - 4.168906959966564350754813j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">nan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">)))</span>
-<span class="go">[0.0, 1.0, 0.0, -0.166667, 0.0, 0.00833333, 0.0]</span>
-</pre></div>
-</div>
-<p>Intervals are supported via <tt class="xref py py-func docutils literal"><span class="pre">mpmath.iv.sin()</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">iv</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">sin</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">[0.0, 0.841470984807896506652502331201]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">sin</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
-<span class="go">[0.0, 1.0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="tan">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">tan()</span></tt><a class="headerlink" href="#tan" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.tan">
-<tt class="descclassname">mpmath.</tt><tt class="descname">tan</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.tan" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the tangent of <span class="math">\(x\)</span>, <span class="math">\(\tan(x) = \frac{\sin(x)}{\cos(x)}\)</span>.
-The tangent function is singular at <span class="math">\(x = (n+1/2)\pi\)</span>, but
-<tt class="docutils literal"><span class="pre">tan(x)</span></tt> always returns a finite result since <span class="math">\((n+1/2)\pi\)</span>
-cannot be represented exactly using floating-point arithmetic.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tan</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">1.732050807568877293527446</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tan</span><span class="p">(</span><span class="mi">100000001</span><span class="p">)</span>
-<span class="go">-0.2015625081449864533091058</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tan</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(-0.003764025641504248292751221 + 1.003238627353609801446359j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tan</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">nan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">tan</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">)))</span>
-<span class="go">[0.0, 1.0, 0.0, 0.333333, 0.0, 0.133333, 0.0]</span>
-</pre></div>
-</div>
-<p>Intervals are supported via <tt class="xref py py-func docutils literal"><span class="pre">mpmath.iv.tan()</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">iv</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">tan</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">[0.0, 1.55740772465490223050697482944]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">tan</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>  <span class="c"># Interval includes a singularity</span>
-<span class="go">[-inf, +inf]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="sec">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">sec()</span></tt><a class="headerlink" href="#sec" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.sec">
-<tt class="descclassname">mpmath.</tt><tt class="descname">sec</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.sec" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the secant of <span class="math">\(x\)</span>, <span class="math">\(\mathrm{sec}(x) = \frac{1}{\cos(x)}\)</span>.
-The secant function is singular at <span class="math">\(x = (n+1/2)\pi\)</span>, but
-<tt class="docutils literal"><span class="pre">sec(x)</span></tt> always returns a finite result since <span class="math">\((n+1/2)\pi\)</span>
-cannot be represented exactly using floating-point arithmetic.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sec</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">2.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sec</span><span class="p">(</span><span class="mi">10000001</span><span class="p">)</span>
-<span class="go">-1.184723164360392819100265</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sec</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(-0.04167496441114427004834991 + 0.0906111371962375965296612j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sec</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">nan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">sec</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">)))</span>
-<span class="go">[1.0, 0.0, 0.5, 0.0, 0.208333, 0.0, 0.0847222]</span>
-</pre></div>
-</div>
-<p>Intervals are supported via <tt class="xref py py-func docutils literal"><span class="pre">mpmath.iv.sec()</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">iv</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">sec</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">[1.0, 1.85081571768092561791175326276]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">sec</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>  <span class="c"># Interval includes a singularity</span>
-<span class="go">[-inf, +inf]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="csc">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">csc()</span></tt><a class="headerlink" href="#csc" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.csc">
-<tt class="descclassname">mpmath.</tt><tt class="descname">csc</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.csc" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the cosecant of <span class="math">\(x\)</span>, <span class="math">\(\mathrm{csc}(x) = \frac{1}{\sin(x)}\)</span>.
-This cosecant function is singular at <span class="math">\(x = n \pi\)</span>, but with the
-exception of the point <span class="math">\(x = 0\)</span>, <tt class="docutils literal"><span class="pre">csc(x)</span></tt> returns a finite result
-since <span class="math">\(n \pi\)</span> cannot be represented exactly using floating-point
-arithmetic.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">csc</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">1.154700538379251529018298</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">csc</span><span class="p">(</span><span class="mi">10000001</span><span class="p">)</span>
-<span class="go">-1.864910497503629858938891</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">csc</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(0.09047320975320743980579048 + 0.04120098628857412646300981j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">csc</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">nan</span>
-</pre></div>
-</div>
-<p>Intervals are supported via <tt class="xref py py-func docutils literal"><span class="pre">mpmath.iv.csc()</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">iv</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">csc</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>  <span class="c"># Interval includes a singularity</span>
-<span class="go">[1.18839510577812121626159943988, +inf]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">csc</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
-<span class="go">[1.0, +inf]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="cot">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">cot()</span></tt><a class="headerlink" href="#cot" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.cot">
-<tt class="descclassname">mpmath.</tt><tt class="descname">cot</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.cot" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the cotangent of <span class="math">\(x\)</span>,
-<span class="math">\(\mathrm{cot}(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)}\)</span>.
-This cotangent function is singular at <span class="math">\(x = n \pi\)</span>, but with the
-exception of the point <span class="math">\(x = 0\)</span>, <tt class="docutils literal"><span class="pre">cot(x)</span></tt> returns a finite result
-since <span class="math">\(n \pi\)</span> cannot be represented exactly using floating-point
-arithmetic.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cot</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.5773502691896257645091488</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cot</span><span class="p">(</span><span class="mi">10000001</span><span class="p">)</span>
-<span class="go">1.574131876209625656003562</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cot</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">(-0.003739710376336956660117409 - 0.9967577965693583104609688j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cot</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">nan</span>
-</pre></div>
-</div>
-<p>Intervals are supported via <tt class="xref py py-func docutils literal"><span class="pre">mpmath.iv.cot()</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">iv</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">cot</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>  <span class="c"># Interval includes a singularity</span>
-<span class="go">[0.642092615934330703006419974862, +inf]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iv</span><span class="o">.</span><span class="n">cot</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">])</span>
-<span class="go">[-inf, +inf]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="trigonometric-functions-with-modified-argument">
-<h2>Trigonometric functions with modified argument<a class="headerlink" href="#trigonometric-functions-with-modified-argument" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="cospi">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">cospi()</span></tt><a class="headerlink" href="#cospi" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.cospi">
-<tt class="descclassname">mpmath.</tt><tt class="descname">cospi</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.cospi" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the &lt;no doc&gt; of x</p>
-</dd></dl>
-
-</div>
-<div class="section" id="sinpi">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">sinpi()</span></tt><a class="headerlink" href="#sinpi" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.sinpi">
-<tt class="descclassname">mpmath.</tt><tt class="descname">sinpi</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.sinpi" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the &lt;no doc&gt; of x</p>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="inverse-trigonometric-functions">
-<h2>Inverse trigonometric functions<a class="headerlink" href="#inverse-trigonometric-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="acos">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">acos()</span></tt><a class="headerlink" href="#acos" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.acos">
-<tt class="descclassname">mpmath.</tt><tt class="descname">acos</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.acos" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the inverse cosine or arccosine of <span class="math">\(x\)</span>, <span class="math">\(\cos^{-1}(x)\)</span>.
-Since <span class="math">\(-1 \le \cos(x) \le 1\)</span> for real <span class="math">\(x\)</span>, the inverse
-cosine is real-valued only for <span class="math">\(-1 \le x \le 1\)</span>. On this interval,
-<a class="reference internal" href="#mpmath.acos" title="mpmath.acos"><tt class="xref py py-func docutils literal"><span class="pre">acos()</span></tt></a> is defined to be a monotonically decreasing
-function assuming values between <span class="math">\(+\pi\)</span> and <span class="math">\(0\)</span>.</p>
-<p>Basic values are:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">acos</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">3.141592653589793238462643</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">acos</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">1.570796326794896619231322</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">acos</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">acos</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">)))</span>
-<span class="go">[1.5708, -1.0, 0.0, -0.166667, 0.0, -0.075, 0.0]</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.acos" title="mpmath.acos"><tt class="xref py py-func docutils literal"><span class="pre">acos()</span></tt></a> is defined so as to be a proper inverse function of
-<span class="math">\(\cos(\theta)\)</span> for <span class="math">\(0 \le \theta &lt; \pi\)</span>.
-We have <span class="math">\(\cos(\cos^{-1}(x)) = x\)</span> for all <span class="math">\(x\)</span>, but
-<span class="math">\(\cos^{-1}(\cos(x)) = x\)</span> only for <span class="math">\(0 \le \Re[x] &lt; \pi\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span> <span class="mi">10</span><span class="o">+</span><span class="mi">3j</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">acos</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="n">acos</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))))</span>
-<span class="gp">...</span>
-<span class="go">1.0 1.0</span>
-<span class="go">(10.0 + 0.0j) 2.566370614359172953850574</span>
-<span class="go">-1.0 1.0</span>
-<span class="go">(2.0 + 3.0j) (2.0 + 3.0j)</span>
-<span class="go">(10.0 + 3.0j) (2.566370614359172953850574 - 3.0j)</span>
-</pre></div>
-</div>
-<p>The inverse cosine has two branch points: <span class="math">\(x = \pm 1\)</span>. <a class="reference internal" href="#mpmath.acos" title="mpmath.acos"><tt class="xref py py-func docutils literal"><span class="pre">acos()</span></tt></a>
-places the branch cuts along the line segments <span class="math">\((-\infty, -1)\)</span> and
-<span class="math">\((+1, +\infty)\)</span>. In general,</p>
-<div class="math">
-\[\cos^{-1}(x) = \frac{\pi}{2} + i \log\left(ix + \sqrt{1-x^2} \right)\]</div>
-<p>where the principal-branch log and square root are implied.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="asin">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">asin()</span></tt><a class="headerlink" href="#asin" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.asin">
-<tt class="descclassname">mpmath.</tt><tt class="descname">asin</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.asin" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the inverse sine or arcsine of <span class="math">\(x\)</span>, <span class="math">\(\sin^{-1}(x)\)</span>.
-Since <span class="math">\(-1 \le \sin(x) \le 1\)</span> for real <span class="math">\(x\)</span>, the inverse
-sine is real-valued only for <span class="math">\(-1 \le x \le 1\)</span>.
-On this interval, it is defined to be a monotonically increasing
-function assuming values between <span class="math">\(-\pi/2\)</span> and <span class="math">\(\pi/2\)</span>.</p>
-<p>Basic values are:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asin</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-1.570796326794896619231322</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asin</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asin</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1.570796326794896619231322</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">asin</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">)))</span>
-<span class="go">[0.0, 1.0, 0.0, 0.166667, 0.0, 0.075, 0.0]</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.asin" title="mpmath.asin"><tt class="xref py py-func docutils literal"><span class="pre">asin()</span></tt></a> is defined so as to be a proper inverse function of
-<span class="math">\(\sin(\theta)\)</span> for <span class="math">\(-\pi/2 &lt; \theta &lt; \pi/2\)</span>.
-We have <span class="math">\(\sin(\sin^{-1}(x)) = x\)</span> for all <span class="math">\(x\)</span>, but
-<span class="math">\(\sin^{-1}(\sin(x)) = x\)</span> only for <span class="math">\(-\pi/2 &lt; \Re[x] &lt; \pi/2\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">asin</span><span class="p">(</span><span class="n">x</span><span class="p">))),</span> <span class="n">asin</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">))))</span>
-<span class="gp">...</span>
-<span class="go">1.0 1.0</span>
-<span class="go">10.0 -0.5752220392306202846120698</span>
-<span class="go">-1.0 -1.0</span>
-<span class="go">(1.0 + 3.0j) (1.0 + 3.0j)</span>
-<span class="go">(-2.0 + 3.0j) (-1.141592653589793238462643 - 3.0j)</span>
-</pre></div>
-</div>
-<p>The inverse sine has two branch points: <span class="math">\(x = \pm 1\)</span>. <a class="reference internal" href="#mpmath.asin" title="mpmath.asin"><tt class="xref py py-func docutils literal"><span class="pre">asin()</span></tt></a>
-places the branch cuts along the line segments <span class="math">\((-\infty, -1)\)</span> and
-<span class="math">\((+1, +\infty)\)</span>. In general,</p>
-<div class="math">
-\[\sin^{-1}(x) = -i \log\left(ix + \sqrt{1-x^2} \right)\]</div>
-<p>where the principal-branch log and square root are implied.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="atan">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">atan()</span></tt><a class="headerlink" href="#atan" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.atan">
-<tt class="descclassname">mpmath.</tt><tt class="descname">atan</tt><big>(</big><em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.atan" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the inverse tangent or arctangent of <span class="math">\(x\)</span>, <span class="math">\(\tan^{-1}(x)\)</span>.
-This is a real-valued function for all real <span class="math">\(x\)</span>, with range
-<span class="math">\((-\pi/2, \pi/2)\)</span>.</p>
-<p>Basic values are:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">atan</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">-1.570796326794896619231322</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">atan</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-0.7853981633974483096156609</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">atan</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">atan</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.7853981633974483096156609</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">atan</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">1.570796326794896619231322</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">atan</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">)))</span>
-<span class="go">[0.0, 1.0, 0.0, -0.333333, 0.0, 0.2, 0.0]</span>
-</pre></div>
-</div>
-<p>The inverse tangent is often used to compute angles. However,
-the atan2 function is often better for this as it preserves sign
-(see <a class="reference internal" href="#mpmath.atan2" title="mpmath.atan2"><tt class="xref py py-func docutils literal"><span class="pre">atan2()</span></tt></a>).</p>
-<p><a class="reference internal" href="#mpmath.atan" title="mpmath.atan"><tt class="xref py py-func docutils literal"><span class="pre">atan()</span></tt></a> is defined so as to be a proper inverse function of
-<span class="math">\(\tan(\theta)\)</span> for <span class="math">\(-\pi/2 &lt; \theta &lt; \pi/2\)</span>.
-We have <span class="math">\(\tan(\tan^{-1}(x)) = x\)</span> for all <span class="math">\(x\)</span>, but
-<span class="math">\(\tan^{-1}(\tan(x)) = x\)</span> only for <span class="math">\(-\pi/2 &lt; \Re[x] &lt; \pi/2\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> </span><span class="si">%s</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">tan</span><span class="p">(</span><span class="n">atan</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="n">atan</span><span class="p">(</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">))))</span>
-<span class="gp">...</span>
-<span class="go">1.0 1.0</span>
-<span class="go">10.0 0.5752220392306202846120698</span>
-<span class="go">-1.0 -1.0</span>
-<span class="go">(1.0 + 3.0j) (1.000000000000000000000001 + 3.0j)</span>
-<span class="go">(-2.0 + 3.0j) (1.141592653589793238462644 + 3.0j)</span>
-</pre></div>
-</div>
-<p>The inverse tangent has two branch points: <span class="math">\(x = \pm i\)</span>. <a class="reference internal" href="#mpmath.atan" title="mpmath.atan"><tt class="xref py py-func docutils literal"><span class="pre">atan()</span></tt></a>
-places the branch cuts along the line segments <span class="math">\((-i \infty, -i)\)</span> and
-<span class="math">\((+i, +i \infty)\)</span>. In general,</p>
-<div class="math">
-\[\tan^{-1}(x) = \frac{i}{2}\left(\log(1-ix)-\log(1+ix)\right)\]</div>
-<p>where the principal-branch log is implied.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="atan2">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">atan2()</span></tt><a class="headerlink" href="#atan2" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.atan2">
-<tt class="descclassname">mpmath.</tt><tt class="descname">atan2</tt><big>(</big><em>y</em>, <em>x</em><big>)</big><a class="headerlink" href="#mpmath.atan2" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="asec">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">asec()</span></tt><a class="headerlink" href="#asec" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.asec">
-<tt class="descclassname">mpmath.</tt><tt class="descname">asec</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.asec" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the inverse secant of <span class="math">\(x\)</span>,
-<span class="math">\(\mathrm{sec}^{-1}(x) = \cos^{-1}(1/x)\)</span>.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="acsc">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">acsc()</span></tt><a class="headerlink" href="#acsc" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.acsc">
-<tt class="descclassname">mpmath.</tt><tt class="descname">acsc</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.acsc" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the inverse cosecant of <span class="math">\(x\)</span>,
-<span class="math">\(\mathrm{csc}^{-1}(x) = \sin^{-1}(1/x)\)</span>.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="acot">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">acot()</span></tt><a class="headerlink" href="#acot" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.acot">
-<tt class="descclassname">mpmath.</tt><tt class="descname">acot</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.acot" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the inverse cotangent of <span class="math">\(x\)</span>,
-<span class="math">\(\mathrm{cot}^{-1}(x) = \tan^{-1}(1/x)\)</span>.</p>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="sinc-function">
-<h2>Sinc function<a class="headerlink" href="#sinc-function" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="sinc">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">sinc()</span></tt><a class="headerlink" href="#sinc" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.sinc">
-<tt class="descclassname">mpmath.</tt><tt class="descname">sinc</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.sinc" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">sinc(x)</span></tt> computes the unnormalized sinc function, defined as</p>
-<div class="math">
-\[\begin{split}\mathrm{sinc}(x) = \begin{cases}
-    \sin(x)/x, &amp; \mbox{if } x \ne 0 \\
-    1,         &amp; \mbox{if } x = 0.
-\end{cases}\end{split}\]</div>
-<p>See <a class="reference internal" href="#mpmath.sincpi" title="mpmath.sincpi"><tt class="xref py py-func docutils literal"><span class="pre">sincpi()</span></tt></a> for the normalized sinc function.</p>
-<p>Simple values and limits include:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinc</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinc</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.841470984807897</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinc</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>The integral of the sinc function is the sine integral Si:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="n">sinc</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">0.946083070367183</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">si</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.946083070367183</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="sincpi">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">sincpi()</span></tt><a class="headerlink" href="#sincpi" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.sincpi">
-<tt class="descclassname">mpmath.</tt><tt class="descname">sincpi</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.sincpi" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">sincpi(x)</span></tt> computes the normalized sinc function, defined as</p>
-<div class="math">
-\[\begin{split}\mathrm{sinc}_{\pi}(x) = \begin{cases}
-    \sin(\pi x)/(\pi x), &amp; \mbox{if } x \ne 0 \\
-    1,                   &amp; \mbox{if } x = 0.
-\end{cases}\end{split}\]</div>
-<p>Equivalently, we have
-<span class="math">\(\mathrm{sinc}_{\pi}(x) = \mathrm{sinc}(\pi x)\)</span>.</p>
-<p>The normalization entails that the function integrates
-to unity over the entire real line:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quadosc</span><span class="p">(</span><span class="n">sincpi</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="n">inf</span><span class="p">,</span> <span class="n">inf</span><span class="p">],</span> <span class="n">period</span><span class="o">=</span><span class="mf">2.0</span><span class="p">)</span>
-<span class="go">1.0</span>
-</pre></div>
-</div>
-<p>Like, <a class="reference internal" href="#mpmath.sinpi" title="mpmath.sinpi"><tt class="xref py py-func docutils literal"><span class="pre">sinpi()</span></tt></a>, <a class="reference internal" href="#mpmath.sincpi" title="mpmath.sincpi"><tt class="xref py py-func docutils literal"><span class="pre">sincpi()</span></tt></a> is evaluated accurately
-at its roots:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sincpi</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
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-</div>
-
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-  <ul>
-<li><a class="reference internal" href="#">Trigonometric functions</a><ul>
-<li><a class="reference internal" href="#degree-radian-conversion">Degree-radian conversion</a><ul>
-<li><a class="reference internal" href="#degrees"><tt class="docutils literal"><span class="pre">degrees()</span></tt></a></li>
-<li><a class="reference internal" href="#radians"><tt class="docutils literal"><span class="pre">radians()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#id1">Trigonometric functions</a><ul>
-<li><a class="reference internal" href="#cos"><tt class="docutils literal"><span class="pre">cos()</span></tt></a></li>
-<li><a class="reference internal" href="#sin"><tt class="docutils literal"><span class="pre">sin()</span></tt></a></li>
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-</ul>
-</li>
-<li><a class="reference internal" href="#trigonometric-functions-with-modified-argument">Trigonometric functions with modified argument</a><ul>
-<li><a class="reference internal" href="#cospi"><tt class="docutils literal"><span class="pre">cospi()</span></tt></a></li>
-<li><a class="reference internal" href="#sinpi"><tt class="docutils literal"><span class="pre">sinpi()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#inverse-trigonometric-functions">Inverse trigonometric functions</a><ul>
-<li><a class="reference internal" href="#acos"><tt class="docutils literal"><span class="pre">acos()</span></tt></a></li>
-<li><a class="reference internal" href="#asin"><tt class="docutils literal"><span class="pre">asin()</span></tt></a></li>
-<li><a class="reference internal" href="#atan"><tt class="docutils literal"><span class="pre">atan()</span></tt></a></li>
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-<li><a class="reference internal" href="#asec"><tt class="docutils literal"><span class="pre">asec()</span></tt></a></li>
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-</ul>
-</li>
-<li><a class="reference internal" href="#sinc-function">Sinc function</a><ul>
-<li><a class="reference internal" href="#sinc"><tt class="docutils literal"><span class="pre">sinc()</span></tt></a></li>
-<li><a class="reference internal" href="#sincpi"><tt class="docutils literal"><span class="pre">sincpi()</span></tt></a></li>
-</ul>
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-  <div class="section" id="zeta-functions-l-series-and-polylogarithms">
-<h1>Zeta functions, L-series and polylogarithms<a class="headerlink" href="#zeta-functions-l-series-and-polylogarithms" title="Permalink to this headline">¶</a></h1>
-<p>This section includes the Riemann zeta functions
-and associated functions pertaining to analytic number theory.</p>
-<div class="section" id="riemann-and-hurwitz-zeta-functions">
-<h2>Riemann and Hurwitz zeta functions<a class="headerlink" href="#riemann-and-hurwitz-zeta-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="zeta">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">zeta()</span></tt><a class="headerlink" href="#zeta" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.zeta">
-<tt class="descclassname">mpmath.</tt><tt class="descname">zeta</tt><big>(</big><em>s</em>, <em>a=1</em>, <em>derivative=0</em><big>)</big><a class="headerlink" href="#mpmath.zeta" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the Riemann zeta function</p>
-<div class="math">
-\[\zeta(s) = 1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\ldots\]</div>
-<p>or, with <span class="math">\(a \ne 1\)</span>, the more general Hurwitz zeta function</p>
-<div class="math">
-\[\zeta(s,a) = \sum_{k=0}^\infty \frac{1}{(a+k)^s}.\]</div>
-<p>Optionally, <tt class="docutils literal"><span class="pre">zeta(s,</span> <span class="pre">a,</span> <span class="pre">n)</span></tt> computes the <span class="math">\(n\)</span>-th derivative with
-respect to <span class="math">\(s\)</span>,</p>
-<div class="math">
-\[\zeta^{(n)}(s,a) = (-1)^n \sum_{k=0}^\infty \frac{\log^n(a+k)}{(a+k)^s}.\]</div>
-<p>Although these series only converge for <span class="math">\(\Re(s) &gt; 1\)</span>, the Riemann and Hurwitz
-zeta functions are defined through analytic continuation for arbitrary
-complex <span class="math">\(s \ne 1\)</span> (<span class="math">\(s = 1\)</span> is a pole).</p>
-<p>The implementation uses three algorithms: the Borwein algorithm for
-the Riemann zeta function when <span class="math">\(s\)</span> is close to the real line;
-the Riemann-Siegel formula for the Riemann zeta function when <span class="math">\(s\)</span> is
-large imaginary, and Euler-Maclaurin summation in all other cases.
-The reflection formula for <span class="math">\(\Re(s) &lt; 0\)</span> is implemented in some cases.
-The algorithm can be chosen with <tt class="docutils literal"><span class="pre">method</span> <span class="pre">=</span> <span class="pre">'borwein'</span></tt>,
-<tt class="docutils literal"><span class="pre">method='riemann-siegel'</span></tt> or <tt class="docutils literal"><span class="pre">method</span> <span class="pre">=</span> <span class="pre">'euler-maclaurin'</span></tt>.</p>
-<p>The parameter <span class="math">\(a\)</span> is usually a rational number <span class="math">\(a = p/q\)</span>, and may be specified
-as such by passing an integer tuple <span class="math">\((p, q)\)</span>. Evaluation is supported for
-arbitrary complex <span class="math">\(a\)</span>, but may be slow and/or inaccurate when <span class="math">\(\Re(s) &lt; 0\)</span> for
-nonrational <span class="math">\(a\)</span> or when computing derivatives.</p>
-<p><strong>Examples</strong></p>
-<p>Some values of the Riemann zeta function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span> <span class="n">pi</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span> <span class="mi">6</span>
-<span class="go">1.644934066848226436472415</span>
-<span class="go">1.644934066848226436472415</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">-0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-0.08333333333333333333333333</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>For large positive <span class="math">\(s\)</span>, <span class="math">\(\zeta(s)\)</span> rapidly approaches 1:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">50</span><span class="p">)</span>
-<span class="go">1.000000000000000888178421</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">-</span><span class="nb">sum</span><span class="p">((</span><span class="n">zeta</span><span class="p">(</span><span class="n">k</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">k</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">85</span><span class="p">));</span> <span class="o">+</span><span class="n">euler</span>
-<span class="go">0.5772156649015328606065121</span>
-<span class="go">0.5772156649015328606065121</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="n">zeta</span><span class="p">(</span><span class="n">k</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="n">inf</span><span class="p">])</span>
-<span class="go">1.0</span>
-</pre></div>
-</div>
-<p>Evaluation is supported for complex <span class="math">\(s\)</span> and <span class="math">\(a\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(-0.03373057338827757067584698 + 0.2774499251557093745297677j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">3j</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">)</span>
-<span class="go">(389.6841230140842816370741 + 295.2674610150305334025962j)</span>
-</pre></div>
-</div>
-<p>The Riemann zeta function has so-called nontrivial zeros on
-the critical line <span class="math">\(s = 1/2 + it\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">zeta</span><span class="p">,</span> <span class="mf">0.5</span><span class="o">+</span><span class="mi">14j</span><span class="p">);</span> <span class="n">zetazero</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(0.5 + 14.13472514173469379045725j)</span>
-<span class="go">(0.5 + 14.13472514173469379045725j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">zeta</span><span class="p">,</span> <span class="mf">0.5</span><span class="o">+</span><span class="mi">21j</span><span class="p">);</span> <span class="n">zetazero</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(0.5 + 21.02203963877155499262848j)</span>
-<span class="go">(0.5 + 21.02203963877155499262848j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">zeta</span><span class="p">,</span> <span class="mf">0.5</span><span class="o">+</span><span class="mi">25j</span><span class="p">);</span> <span class="n">zetazero</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">(0.5 + 25.01085758014568876321379j)</span>
-<span class="go">(0.5 + 25.01085758014568876321379j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">zeta</span><span class="p">(</span><span class="n">zetazero</span><span class="p">(</span><span class="mi">10</span><span class="p">)))</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>Evaluation on and near the critical line is supported for large
-heights <span class="math">\(t\)</span> by means of the Riemann-Siegel formula (currently
-for <span class="math">\(a = 1\)</span>, <span class="math">\(n \le 4\)</span>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mf">0.5</span><span class="o">+</span><span class="mi">100000j</span><span class="p">)</span>
-<span class="go">(1.073032014857753132114076 + 5.780848544363503984261041j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mf">0.75</span><span class="o">+</span><span class="mi">1000000j</span><span class="p">)</span>
-<span class="go">(0.9535316058375145020351559 + 0.9525945894834273060175651j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mf">0.5</span><span class="o">+</span><span class="mi">10000000j</span><span class="p">)</span>
-<span class="go">(11.45804061057709254500227 - 8.643437226836021723818215j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mf">0.5</span><span class="o">+</span><span class="mi">100000000j</span><span class="p">,</span> <span class="n">derivative</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(51.12433106710194942681869 + 43.87221167872304520599418j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mf">0.5</span><span class="o">+</span><span class="mi">100000000j</span><span class="p">,</span> <span class="n">derivative</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(-444.2760822795430400549229 - 896.3789978119185981665403j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mf">0.5</span><span class="o">+</span><span class="mi">100000000j</span><span class="p">,</span> <span class="n">derivative</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">(3230.72682687670422215339 + 14374.36950073615897616781j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mf">0.5</span><span class="o">+</span><span class="mi">100000000j</span><span class="p">,</span> <span class="n">derivative</span><span class="o">=</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">(-11967.35573095046402130602 - 218945.7817789262839266148j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mi">10000000j</span><span class="p">)</span>    <span class="c"># off the line</span>
-<span class="go">(2.859846483332530337008882 + 0.491808047480981808903986j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mi">10000000j</span><span class="p">,</span> <span class="n">derivative</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(-4.333835494679647915673205 - 0.08405337962602933636096103j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mi">10000000j</span><span class="p">,</span> <span class="n">derivative</span><span class="o">=</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">(453.2764822702057701894278 - 581.963625832768189140995j)</span>
-</pre></div>
-</div>
-<p>For investigation of the zeta function zeros, the Riemann-Siegel
-Z-function is often more convenient than working with the Riemann
-zeta function directly (see <a class="reference internal" href="#mpmath.siegelz" title="mpmath.siegelz"><tt class="xref py py-func docutils literal"><span class="pre">siegelz()</span></tt></a>).</p>
-<p>Some values of the Hurwitz zeta function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">);</span> <span class="o">-</span><span class="mf">5.</span><span class="o">/</span><span class="mi">4</span> <span class="o">+</span> <span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">6</span>
-<span class="go">0.3949340668482264364724152</span>
-<span class="go">0.3949340668482264364724152</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">));</span> <span class="n">pi</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">8</span><span class="o">*</span><span class="n">catalan</span>
-<span class="go">2.541879647671606498397663</span>
-<span class="go">2.541879647671606498397663</span>
-</pre></div>
-</div>
-<p>For positive integer values of <span class="math">\(s\)</span>, the Hurwitz zeta function is
-equivalent to a polygamma function (except for a normalizing factor):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">5</span><span class="p">));</span> <span class="n">psi</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="s">&#39;1/5&#39;</span><span class="p">)</span><span class="o">/</span><span class="mi">6</span>
-<span class="go">625.5408324774542966919938</span>
-<span class="go">625.5408324774542966919938</span>
-</pre></div>
-</div>
-<p>Evaluation of derivatives:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">,</span> <span class="mi">1</span><span class="p">);</span> <span class="n">loggamma</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span> <span class="o">-</span> <span class="n">ln</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-<span class="go">(-2.675565317808456852310934 + 4.742664438034657928194889j)</span>
-<span class="go">(-2.675565317808456852310934 + 4.742664438034657928194889j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">20</span><span class="p">)</span>
-<span class="go">2432902008176640000.000242</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">,</span> <span class="mf">5.5</span><span class="o">+</span><span class="mi">2j</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">(-0.140075548947797130681075 - 0.3109263360275413251313634j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mf">0.5</span><span class="o">+</span><span class="mi">100000j</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">(-10407.16081931495861539236 + 13777.78669862804508537384j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="o">-</span><span class="mi">100</span><span class="o">+</span><span class="mf">0.5j</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="n">derivative</span><span class="o">=</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">(4.007180821099823942702249e+79 + 4.916117957092593868321778e+78j)</span>
-</pre></div>
-</div>
-<p>Generating a Taylor series at <span class="math">\(s = 2\)</span> using derivatives:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">11</span><span class="p">):</span> <span class="k">print</span><span class="p">(</span><span class="s">&quot;</span><span class="si">%s</span><span class="s"> * (s-2)^</span><span class="si">%i</span><span class="s">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">zeta</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">k</span><span class="p">)</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">k</span><span class="p">),</span> <span class="n">k</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">1.644934066848226436472415 * (s-2)^0</span>
-<span class="go">-0.9375482543158437537025741 * (s-2)^1</span>
-<span class="go">0.9946401171494505117104293 * (s-2)^2</span>
-<span class="go">-1.000024300473840810940657 * (s-2)^3</span>
-<span class="go">1.000061933072352565457512 * (s-2)^4</span>
-<span class="go">-1.000006869443931806408941 * (s-2)^5</span>
-<span class="go">1.000000173233769531820592 * (s-2)^6</span>
-<span class="go">-0.9999999569989868493432399 * (s-2)^7</span>
-<span class="go">0.9999999937218844508684206 * (s-2)^8</span>
-<span class="go">-0.9999999996355013916608284 * (s-2)^9</span>
-<span class="go">1.000000000004610645020747 * (s-2)^10</span>
-</pre></div>
-</div>
-<p>Evaluation at zero and for negative integer <span class="math">\(s\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">-9.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">));</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">81</span>
-<span class="go">0.01234567901234567901234568</span>
-<span class="go">0.01234567901234567901234568</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">,</span> <span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
-<span class="go">(0.2899236037682695182085988 + 0.06561206166091757973112783j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="o">-</span><span class="mf">3.25</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">-0.0005117269627574430494396877</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="o">-</span><span class="mf">3.5</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">11.156360390440003294709</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="o">-</span><span class="mf">100.5</span><span class="p">,</span> <span class="p">(</span><span class="mi">8</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">-4.68162300487989766727122e+77</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="o">-</span><span class="mf">10.5</span><span class="p">,</span> <span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">(-0.01521913704446246609237979 + 29907.72510874248161608216j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="o">-</span><span class="mf">1000.5</span><span class="p">,</span> <span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">(1.031911949062334538202567e+1770 + 1.519555750556794218804724e+426j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(-16.32988355630802510888631 - 22.17706465801374033261383j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">(32.48985276392056641594055 - 51.11604466157397267043655j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">s</span><span class="p">:</span> <span class="n">zeta</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">),</span> <span class="o">-</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">(32.48985276392056641594055 - 51.11604466157397267043655j)</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference external" href="http://mathworld.wolfram.com/RiemannZetaFunction.html">http://mathworld.wolfram.com/RiemannZetaFunction.html</a></li>
-<li><a class="reference external" href="http://mathworld.wolfram.com/HurwitzZetaFunction.html">http://mathworld.wolfram.com/HurwitzZetaFunction.html</a></li>
-<li><a class="reference external" href="http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf">http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf</a></li>
-</ol>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="dirichlet-l-series">
-<h2>Dirichlet L-series<a class="headerlink" href="#dirichlet-l-series" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="altzeta">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">altzeta()</span></tt><a class="headerlink" href="#altzeta" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.altzeta">
-<tt class="descclassname">mpmath.</tt><tt class="descname">altzeta</tt><big>(</big><em>s</em><big>)</big><a class="headerlink" href="#mpmath.altzeta" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the Dirichlet eta function, <span class="math">\(\eta(s)\)</span>, also known as the
-alternating zeta function. This function is defined in analogy
-with the Riemann zeta function as providing the sum of the
-alternating series</p>
-<div class="math">
-\[\eta(s) = \sum_{k=0}^{\infty} \frac{(-1)^k}{k^s}
-    = 1-\frac{1}{2^s}+\frac{1}{3^s}-\frac{1}{4^s}+\ldots\]</div>
-<p>The eta function, unlike the Riemann zeta function, is an entire
-function, having a finite value for all complex <span class="math">\(s\)</span>. The special case
-<span class="math">\(\eta(1) = \log(2)\)</span> gives the value of the alternating harmonic series.</p>
-<p>The alternating zeta function may expressed using the Riemann zeta function
-as <span class="math">\(\eta(s) = (1 - 2^{1-s}) \zeta(s)\)</span>. It can also be expressed
-in terms of the Hurwitz zeta function, for example using
-<a class="reference internal" href="#mpmath.dirichlet" title="mpmath.dirichlet"><tt class="xref py py-func docutils literal"><span class="pre">dirichlet()</span></tt></a> (see documentation for that function).</p>
-<p><strong>Examples</strong></p>
-<p>Some special values are:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">altzeta</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.693147180559945</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">altzeta</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">altzeta</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0.25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">altzeta</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>An example of a sum that can be computed more accurately and
-efficiently via <a class="reference internal" href="#mpmath.altzeta" title="mpmath.altzeta"><tt class="xref py py-func docutils literal"><span class="pre">altzeta()</span></tt></a> than via numerical summation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">sum</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">n</span> <span class="o">/</span> <span class="n">n</span><span class="o">**</span><span class="mf">2.5</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">100</span><span class="p">))</span>
-<span class="go">0.86720495150398402</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">altzeta</span><span class="p">(</span><span class="mf">2.5</span><span class="p">)</span>
-<span class="go">0.867199889012184</span>
-</pre></div>
-</div>
-<p>At positive even integers, the Dirichlet eta function
-evaluates to a rational multiple of a power of <span class="math">\(\pi\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">altzeta</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.822467033424113</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">12</span>
-<span class="go">0.822467033424113</span>
-</pre></div>
-</div>
-<p>Like the Riemann zeta function, <span class="math">\(\eta(s)\)</span>, approaches 1
-as <span class="math">\(s\)</span> approaches positive infinity, although it does
-so from below rather than from above:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">altzeta</span><span class="p">(</span><span class="mi">30</span><span class="p">)</span>
-<span class="go">0.999999999068682</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">altzeta</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">altzeta</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span> <span class="n">rounding</span><span class="o">=</span><span class="s">&#39;d&#39;</span><span class="p">)</span>
-<span class="go">mpf(&#39;0.99999999999999989&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">altzeta</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span> <span class="n">rounding</span><span class="o">=</span><span class="s">&#39;u&#39;</span><span class="p">)</span>
-<span class="go">mpf(&#39;1.0&#39;)</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference external" href="http://mathworld.wolfram.com/DirichletEtaFunction.html">http://mathworld.wolfram.com/DirichletEtaFunction.html</a></li>
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Dirichlet_eta_function">http://en.wikipedia.org/wiki/Dirichlet_eta_function</a></li>
-</ol>
-</dd></dl>
-
-</div>
-<div class="section" id="dirichlet">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">dirichlet()</span></tt><a class="headerlink" href="#dirichlet" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.dirichlet">
-<tt class="descclassname">mpmath.</tt><tt class="descname">dirichlet</tt><big>(</big><em>s</em>, <em>chi</em>, <em>derivative=0</em><big>)</big><a class="headerlink" href="#mpmath.dirichlet" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the Dirichlet L-function</p>
-<div class="math">
-\[L(s,\chi) = \sum_{k=1}^\infty \frac{\chi(k)}{k^s}.\]</div>
-<p>where <span class="math">\(\chi\)</span> is a periodic sequence of length <span class="math">\(q\)</span> which should be supplied
-in the form of a list <span class="math">\([\chi(0), \chi(1), \ldots, \chi(q-1)]\)</span>.
-Strictly, <span class="math">\(\chi\)</span> should be a Dirichlet character, but any periodic
-sequence will work.</p>
-<p>For example, <tt class="docutils literal"><span class="pre">dirichlet(s,</span> <span class="pre">[1])</span></tt> gives the ordinary
-Riemann zeta function and <tt class="docutils literal"><span class="pre">dirichlet(s,</span> <span class="pre">[-1,1])</span></tt> gives
-the alternating zeta function (Dirichlet eta function).</p>
-<p>Also the derivative with respect to <span class="math">\(s\)</span> (currently only a first
-derivative) can be evaluated.</p>
-<p><strong>Examples</strong></p>
-<p>The ordinary Riemann zeta function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dirichlet</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">]);</span> <span class="n">zeta</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">1.202056903159594285399738</span>
-<span class="go">1.202056903159594285399738</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dirichlet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">+inf</span>
-</pre></div>
-</div>
-<p>The alternating zeta function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dirichlet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">]);</span> <span class="n">ln</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.6931471805599453094172321</span>
-<span class="go">0.6931471805599453094172321</span>
-</pre></div>
-</div>
-<p>The following defines the Dirichlet beta function
-<span class="math">\(\beta(s) = \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)^s}\)</span> and verifies
-several values of this function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">s</span><span class="p">,</span> <span class="n">d</span><span class="o">=</span><span class="mi">0</span><span class="p">:</span> <span class="n">dirichlet</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="n">d</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span> <span class="mf">1.</span><span class="o">/</span><span class="mi">2</span>
-<span class="go">0.5</span>
-<span class="go">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span> <span class="n">pi</span><span class="o">/</span><span class="mi">4</span>
-<span class="go">0.7853981633974483096156609</span>
-<span class="go">0.7853981633974483096156609</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span> <span class="o">+</span><span class="n">catalan</span>
-<span class="go">0.9159655941772190150546035</span>
-<span class="go">0.9159655941772190150546035</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">diff</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">0.08158073611659279510291217</span>
-<span class="go">0.08158073611659279510291217</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="mi">2</span><span class="o">*</span><span class="n">catalan</span><span class="o">/</span><span class="n">pi</span>
-<span class="go">0.5831218080616375602767689</span>
-<span class="go">0.5831218080616375602767689</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">log</span><span class="p">(</span><span class="n">gamma</span><span class="p">(</span><span class="mf">0.25</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)))</span>
-<span class="go">0.3915943927068367764719453</span>
-<span class="go">0.3915943927068367764719454</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="mf">0.25</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="p">(</span><span class="n">euler</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">ln2</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">ln</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">-</span><span class="mi">4</span><span class="o">*</span><span class="n">ln</span><span class="p">(</span><span class="n">gamma</span><span class="p">(</span><span class="mf">0.25</span><span class="p">)))</span>
-<span class="go">0.1929013167969124293631898</span>
-<span class="go">0.1929013167969124293631898</span>
-</pre></div>
-</div>
-<p>A custom L-series of period 3:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dirichlet</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">0.7059715047839078092146831</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span><span class="o">*</span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">k</span><span class="p">)</span><span class="o">**-</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span> <span class="o">+</span> \
-<span class="gp">... </span>  <span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">k</span><span class="o">+</span><span class="mi">2</span><span class="p">)</span><span class="o">**-</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">0.7059715047839078092146831</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="stieltjes-constants">
-<h2>Stieltjes constants<a class="headerlink" href="#stieltjes-constants" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="stieltjes">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">stieltjes()</span></tt><a class="headerlink" href="#stieltjes" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.stieltjes">
-<tt class="descclassname">mpmath.</tt><tt class="descname">stieltjes</tt><big>(</big><em>n</em>, <em>a=1</em><big>)</big><a class="headerlink" href="#mpmath.stieltjes" title="Permalink to this definition">¶</a></dt>
-<dd><p>For a nonnegative integer <span class="math">\(n\)</span>, <tt class="docutils literal"><span class="pre">stieltjes(n)</span></tt> computes the
-<span class="math">\(n\)</span>-th Stieltjes constant <span class="math">\(\gamma_n\)</span>, defined as the
-<span class="math">\(n\)</span>-th coefficient in the Laurent series expansion of the
-Riemann zeta function around the pole at <span class="math">\(s = 1\)</span>. That is,
-we have:</p>
-<div class="math">
-\[\zeta(s) = \frac{1}{s-1} \sum_{n=0}^{\infty}
-    \frac{(-1)^n}{n!} \gamma_n (s-1)^n\]</div>
-<p>More generally, <tt class="docutils literal"><span class="pre">stieltjes(n,</span> <span class="pre">a)</span></tt> gives the corresponding
-coefficient <span class="math">\(\gamma_n(a)\)</span> for the Hurwitz zeta function
-<span class="math">\(\zeta(s,a)\)</span> (with <span class="math">\(\gamma_n = \gamma_n(1)\)</span>).</p>
-<p><strong>Examples</strong></p>
-<p>The zeroth Stieltjes constant is just Euler&#8217;s constant <span class="math">\(\gamma\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">stieltjes</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.577215664901533</span>
-</pre></div>
-</div>
-<p>Some more values are:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">stieltjes</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-0.0728158454836767</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">stieltjes</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">0.000205332814909065</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">stieltjes</span><span class="p">(</span><span class="mi">30</span><span class="p">)</span>
-<span class="go">0.00355772885557316</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">stieltjes</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span>
-<span class="go">-1.57095384420474e+486</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">stieltjes</span><span class="p">(</span><span class="mi">2000</span><span class="p">)</span>
-<span class="go">2.680424678918e+1109</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">stieltjes</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">)</span>
-<span class="go">-0.23747539175716</span>
-</pre></div>
-</div>
-<p>An alternative way to compute <span class="math">\(\gamma_1\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">extradps</span><span class="p">(</span><span class="mi">15</span><span class="p">)(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">zeta</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">-0.0728158454836767</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.stieltjes" title="mpmath.stieltjes"><tt class="xref py py-func docutils literal"><span class="pre">stieltjes()</span></tt></a> supports arbitrary precision evaluation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">stieltjes</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">-0.0096903631928723184845303860352125293590658061013408</span>
-</pre></div>
-</div>
-<p><strong>Algorithm</strong></p>
-<p><a class="reference internal" href="#mpmath.stieltjes" title="mpmath.stieltjes"><tt class="xref py py-func docutils literal"><span class="pre">stieltjes()</span></tt></a> numerically evaluates the integral in
-the following representation due to Ainsworth, Howell and
-Coffey [1], [2]:</p>
-<div class="math">
-\[\gamma_n(a) = \frac{\log^n a}{2a} - \frac{\log^{n+1}(a)}{n+1} +
-    \frac{2}{a} \Re \int_0^{\infty}
-    \frac{(x/a-i)\log^n(a-ix)}{(1+x^2/a^2)(e^{2\pi x}-1)} dx.\]</div>
-<p>For some reference values with <span class="math">\(a = 1\)</span>, see e.g. [4].</p>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li>O. R. Ainsworth &amp; L. W. Howell, &#8220;An integral representation of
-the generalized Euler-Mascheroni constants&#8221;, NASA Technical
-Paper 2456 (1985),
-<a class="reference external" href="http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19850014994_1985014994.pdf">http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19850014994_1985014994.pdf</a></li>
-<li>M. W. Coffey, &#8220;The Stieltjes constants, their relation to the
-<span class="math">\(\eta_j\)</span> coefficients, and representation of the Hurwitz
-zeta function&#8221;,      arXiv:0706.0343v1 http://arxiv.org/abs/0706.0343</li>
-<li><a class="reference external" href="http://mathworld.wolfram.com/StieltjesConstants.html">http://mathworld.wolfram.com/StieltjesConstants.html</a></li>
-<li><a class="reference external" href="http://pi.lacim.uqam.ca/piDATA/stieltjesgamma.txt">http://pi.lacim.uqam.ca/piDATA/stieltjesgamma.txt</a></li>
-</ol>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="zeta-function-zeros">
-<h2>Zeta function zeros<a class="headerlink" href="#zeta-function-zeros" title="Permalink to this headline">¶</a></h2>
-<p>These functions are used for the study of the Riemann zeta function
-in the critical strip.</p>
-<div class="section" id="zetazero">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">zetazero()</span></tt><a class="headerlink" href="#zetazero" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.zetazero">
-<tt class="descclassname">mpmath.</tt><tt class="descname">zetazero</tt><big>(</big><em>n</em>, <em>verbose=False</em><big>)</big><a class="headerlink" href="#mpmath.zetazero" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the <span class="math">\(n\)</span>-th nontrivial zero of <span class="math">\(\zeta(s)\)</span> on the critical line,
-i.e. returns an approximation of the <span class="math">\(n\)</span>-th largest complex number
-<span class="math">\(s = \frac{1}{2} + ti\)</span> for which <span class="math">\(\zeta(s) = 0\)</span>. Equivalently, the
-imaginary part <span class="math">\(t\)</span> is a zero of the Z-function (<a class="reference internal" href="#mpmath.siegelz" title="mpmath.siegelz"><tt class="xref py py-func docutils literal"><span class="pre">siegelz()</span></tt></a>).</p>
-<p><strong>Examples</strong></p>
-<p>The first few zeros:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zetazero</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(0.5 + 14.13472514173469379045725j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zetazero</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(0.5 + 21.02203963877155499262848j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zetazero</span><span class="p">(</span><span class="mi">20</span><span class="p">)</span>
-<span class="go">(0.5 + 77.14484006887480537268266j)</span>
-</pre></div>
-</div>
-<p>Verifying that the values are zeros:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">5</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">s</span> <span class="o">=</span> <span class="n">zetazero</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="gp">... </span>    <span class="n">chop</span><span class="p">(</span><span class="n">zeta</span><span class="p">(</span><span class="n">s</span><span class="p">)),</span> <span class="n">chop</span><span class="p">(</span><span class="n">siegelz</span><span class="p">(</span><span class="n">s</span><span class="o">.</span><span class="n">imag</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">(0.0, 0.0)</span>
-<span class="go">(0.0, 0.0)</span>
-<span class="go">(0.0, 0.0)</span>
-<span class="go">(0.0, 0.0)</span>
-</pre></div>
-</div>
-<p>Negative indices give the conjugate zeros (<span class="math">\(n = 0\)</span> is undefined):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">zetazero</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(0.5 - 14.13472514173469379045725j)</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.zetazero" title="mpmath.zetazero"><tt class="xref py py-func docutils literal"><span class="pre">zetazero()</span></tt></a> supports arbitrarily large <span class="math">\(n\)</span> and arbitrary precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zetazero</span><span class="p">(</span><span class="mi">1234567</span><span class="p">)</span>
-<span class="go">(0.5 + 727690.906948208j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zetazero</span><span class="p">(</span><span class="mi">1234567</span><span class="p">)</span>
-<span class="go">(0.5 + 727690.9069482075392389420041147142092708393819935j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">zeta</span><span class="p">(</span><span class="n">_</span><span class="p">)</span><span class="o">/</span><span class="n">_</span><span class="p">)</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>with <em>info=True</em>, <a class="reference internal" href="#mpmath.zetazero" title="mpmath.zetazero"><tt class="xref py py-func docutils literal"><span class="pre">zetazero()</span></tt></a> gives additional information:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zetazero</span><span class="p">(</span><span class="mi">542964976</span><span class="p">,</span><span class="n">info</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">((0.5 + 209039046.578535j), [542964969, 542964978], 6, &#39;(013111110)&#39;)</span>
-</pre></div>
-</div>
-<p>This means that the zero is between Gram points 542964969 and 542964978;
-it is the 6-th zero between them. Finally (01311110) is the pattern
-of zeros in this interval. The numbers indicate the number of zeros
-in each Gram interval (Rosser blocks between parenthesis). In this case
-there is only one Rosser block of length nine.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="nzeros">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">nzeros()</span></tt><a class="headerlink" href="#nzeros" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.nzeros">
-<tt class="descclassname">mpmath.</tt><tt class="descname">nzeros</tt><big>(</big><em>t</em><big>)</big><a class="headerlink" href="#mpmath.nzeros" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the number of zeros of the Riemann zeta function in
-<span class="math">\((0,1) \times (0,t]\)</span>, usually denoted by <span class="math">\(N(t)\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>The first zero has imaginary part between 14 and 15:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nzeros</span><span class="p">(</span><span class="mi">14</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nzeros</span><span class="p">(</span><span class="mi">15</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zetazero</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(0.5 + 14.1347251417347j)</span>
-</pre></div>
-</div>
-<p>Some closely spaced zeros:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nzeros</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">7</span><span class="p">)</span>
-<span class="go">21136125</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zetazero</span><span class="p">(</span><span class="mi">21136125</span><span class="p">)</span>
-<span class="go">(0.5 + 9999999.32718175j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zetazero</span><span class="p">(</span><span class="mi">21136126</span><span class="p">)</span>
-<span class="go">(0.5 + 10000000.2400236j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nzeros</span><span class="p">(</span><span class="mf">545439823.215</span><span class="p">)</span>
-<span class="go">1500000001</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zetazero</span><span class="p">(</span><span class="mi">1500000001</span><span class="p">)</span>
-<span class="go">(0.5 + 545439823.201985j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zetazero</span><span class="p">(</span><span class="mi">1500000002</span><span class="p">)</span>
-<span class="go">(0.5 + 545439823.325697j)</span>
-</pre></div>
-</div>
-<p>This confirms the data given by J. van de Lune,
-H. J. J. te Riele and D. T. Winter in 1986.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="siegelz">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">siegelz()</span></tt><a class="headerlink" href="#siegelz" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.siegelz">
-<tt class="descclassname">mpmath.</tt><tt class="descname">siegelz</tt><big>(</big><em>t</em><big>)</big><a class="headerlink" href="#mpmath.siegelz" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the Z-function, also known as the Riemann-Siegel Z function,</p>
-<div class="math">
-\[Z(t) = e^{i \theta(t)} \zeta(1/2+it)\]</div>
-<p>where <span class="math">\(\zeta(s)\)</span> is the Riemann zeta function (<a class="reference internal" href="#mpmath.zeta" title="mpmath.zeta"><tt class="xref py py-func docutils literal"><span class="pre">zeta()</span></tt></a>)
-and where <span class="math">\(\theta(t)\)</span> denotes the Riemann-Siegel theta function
-(see <a class="reference internal" href="#mpmath.siegeltheta" title="mpmath.siegeltheta"><tt class="xref py py-func docutils literal"><span class="pre">siegeltheta()</span></tt></a>).</p>
-<p>Evaluation is supported for real and complex arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">siegelz</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-0.7363054628673177346778998</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">siegelz</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(-0.1852895764366314976003936 - 0.2773099198055652246992479j)</span>
-</pre></div>
-</div>
-<p>The first four derivatives are supported, using the
-optional <em>derivative</em> keyword argument:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">siegelz</span><span class="p">(</span><span class="mi">1234567</span><span class="p">,</span> <span class="n">derivative</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">56.89689348495089294249178</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">siegelz</span><span class="p">,</span> <span class="mi">1234567</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">56.89689348495089294249178</span>
-</pre></div>
-</div>
-<p>The Z-function has a Maclaurin expansion:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">siegelz</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">)))</span>
-<span class="go">[-1.46035, 0.0, 2.73588, 0.0, -8.39357]</span>
-</pre></div>
-</div>
-<p>The Z-function <span class="math">\(Z(t)\)</span> is equal to <span class="math">\(\pm |\zeta(s)|\)</span> on the
-critical line <span class="math">\(s = 1/2+it\)</span> (i.e. for real arguments <span class="math">\(t\)</span>
-to <span class="math">\(Z\)</span>).  Its zeros coincide with those of the Riemann zeta
-function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">siegelz</span><span class="p">,</span> <span class="mi">14</span><span class="p">)</span>
-<span class="go">14.13472514173469379045725</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">siegelz</span><span class="p">,</span> <span class="mi">20</span><span class="p">)</span>
-<span class="go">21.02203963877155499262848</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">zeta</span><span class="p">,</span> <span class="mf">0.5</span><span class="o">+</span><span class="mi">14j</span><span class="p">)</span>
-<span class="go">(0.5 + 14.13472514173469379045725j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">zeta</span><span class="p">,</span> <span class="mf">0.5</span><span class="o">+</span><span class="mi">20j</span><span class="p">)</span>
-<span class="go">(0.5 + 21.02203963877155499262848j)</span>
-</pre></div>
-</div>
-<p>Since the Z-function is real-valued on the critical line
-(and unlike <span class="math">\(|\zeta(s)|\)</span> analytic), it is useful for
-investigating the zeros of the Riemann zeta function.
-For example, one can use a root-finding algorithm based
-on sign changes:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">siegelz</span><span class="p">,</span> <span class="p">[</span><span class="mi">100</span><span class="p">,</span> <span class="mi">200</span><span class="p">],</span> <span class="n">solver</span><span class="o">=</span><span class="s">&#39;bisect&#39;</span><span class="p">)</span>
-<span class="go">176.4414342977104188888926</span>
-</pre></div>
-</div>
-<p>To locate roots, Gram points <span class="math">\(g_n\)</span> which can be computed
-by <a class="reference internal" href="#mpmath.grampoint" title="mpmath.grampoint"><tt class="xref py py-func docutils literal"><span class="pre">grampoint()</span></tt></a> are useful. If <span class="math">\((-1)^n Z(g_n)\)</span> is
-positive for two consecutive <span class="math">\(n\)</span>, then <span class="math">\(Z(t)\)</span> must have
-a zero between those points:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">g10</span> <span class="o">=</span> <span class="n">grampoint</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g11</span> <span class="o">=</span> <span class="n">grampoint</span><span class="p">(</span><span class="mi">11</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">10</span> <span class="o">*</span> <span class="n">siegelz</span><span class="p">(</span><span class="n">g10</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">11</span> <span class="o">*</span> <span class="n">siegelz</span><span class="p">(</span><span class="n">g11</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">siegelz</span><span class="p">,</span> <span class="p">[</span><span class="n">g10</span><span class="p">,</span> <span class="n">g11</span><span class="p">],</span> <span class="n">solver</span><span class="o">=</span><span class="s">&#39;bisect&#39;</span><span class="p">)</span>
-<span class="go">56.44624769706339480436776</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g10</span><span class="p">,</span> <span class="n">g11</span>
-<span class="go">(54.67523744685325626632663, 57.54516517954725443703014)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="siegeltheta">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">siegeltheta()</span></tt><a class="headerlink" href="#siegeltheta" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.siegeltheta">
-<tt class="descclassname">mpmath.</tt><tt class="descname">siegeltheta</tt><big>(</big><em>t</em><big>)</big><a class="headerlink" href="#mpmath.siegeltheta" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the Riemann-Siegel theta function,</p>
-<div class="math">
-\[\theta(t) = \frac{
-\log\Gamma\left(\frac{1+2it}{4}\right) -
-\log\Gamma\left(\frac{1-2it}{4}\right)
-}{2i} - \frac{\log \pi}{2} t.\]</div>
-<p>The Riemann-Siegel theta function is important in
-providing the phase factor for the Z-function
-(see <a class="reference internal" href="#mpmath.siegelz" title="mpmath.siegelz"><tt class="xref py py-func docutils literal"><span class="pre">siegelz()</span></tt></a>). Evaluation is supported for real and
-complex arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">siegeltheta</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">siegeltheta</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">siegeltheta</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">-inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">siegeltheta</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-1.767547952812290388302216</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">siegeltheta</span><span class="p">(</span><span class="mi">10</span><span class="o">+</span><span class="mf">0.25j</span><span class="p">)</span>
-<span class="go">(-3.068638039426838572528867 + 0.05804937947429712998395177j)</span>
-</pre></div>
-</div>
-<p>Arbitrary derivatives may be computed with derivative = k</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">siegeltheta</span><span class="p">(</span><span class="mi">1234</span><span class="p">,</span> <span class="n">derivative</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.0004051864079114053109473741</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">siegeltheta</span><span class="p">,</span> <span class="mi">1234</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.0004051864079114053109473741</span>
-</pre></div>
-</div>
-<p>The Riemann-Siegel theta function has odd symmetry around <span class="math">\(t = 0\)</span>,
-two local extreme points and three real roots including 0 (located
-symmetrically):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="n">siegeltheta</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)))</span>
-<span class="go">[0.0, -2.68609, 0.0, 2.69433, 0.0, -6.40218]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">diffun</span><span class="p">(</span><span class="n">siegeltheta</span><span class="p">),</span> <span class="mi">7</span><span class="p">)</span>
-<span class="go">6.28983598883690277966509</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">siegeltheta</span><span class="p">,</span> <span class="mi">20</span><span class="p">)</span>
-<span class="go">17.84559954041086081682634</span>
-</pre></div>
-</div>
-<p>For large <span class="math">\(t\)</span>, there is a famous asymptotic formula
-for <span class="math">\(\theta(t)\)</span>, to first order given by:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">siegeltheta</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-<span class="go">5488816.353078403444882823</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="n">t</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="n">t</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="o">-</span><span class="n">t</span><span class="o">/</span><span class="mi">2</span>
-<span class="go">5488816.745777464310273645</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="grampoint">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">grampoint()</span></tt><a class="headerlink" href="#grampoint" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.grampoint">
-<tt class="descclassname">mpmath.</tt><tt class="descname">grampoint</tt><big>(</big><em>n</em><big>)</big><a class="headerlink" href="#mpmath.grampoint" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the <span class="math">\(n\)</span>-th Gram point <span class="math">\(g_n\)</span>, defined as the solution
-to the equation <span class="math">\(\theta(g_n) = \pi n\)</span> where <span class="math">\(\theta(t)\)</span>
-is the Riemann-Siegel theta function (<a class="reference internal" href="#mpmath.siegeltheta" title="mpmath.siegeltheta"><tt class="xref py py-func docutils literal"><span class="pre">siegeltheta()</span></tt></a>).</p>
-<p>The first few Gram points are:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">grampoint</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">17.84559954041086081682634</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">grampoint</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">23.17028270124630927899664</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">grampoint</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">27.67018221781633796093849</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">grampoint</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">31.71797995476405317955149</span>
-</pre></div>
-</div>
-<p>Checking the definition:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">siegeltheta</span><span class="p">(</span><span class="n">grampoint</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">9.42477796076937971538793</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">3</span><span class="o">*</span><span class="n">pi</span>
-<span class="go">9.42477796076937971538793</span>
-</pre></div>
-</div>
-<p>A large Gram point:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">grampoint</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">3293531632.728335454561153</span>
-</pre></div>
-</div>
-<p>Gram points are useful when studying the Z-function
-(<a class="reference internal" href="#mpmath.siegelz" title="mpmath.siegelz"><tt class="xref py py-func docutils literal"><span class="pre">siegelz()</span></tt></a>). See the documentation of that function
-for additional examples.</p>
-<p><a class="reference internal" href="#mpmath.grampoint" title="mpmath.grampoint"><tt class="xref py py-func docutils literal"><span class="pre">grampoint()</span></tt></a> can solve the defining equation for
-nonintegral <span class="math">\(n\)</span>. There is a fixed point where <span class="math">\(g(x) = x\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">grampoint</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">x</span><span class="p">,</span> <span class="mi">10000</span><span class="p">)</span>
-<span class="go">9146.698193171459265866198</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference external" href="http://mathworld.wolfram.com/GramPoint.html">http://mathworld.wolfram.com/GramPoint.html</a></li>
-</ol>
-</dd></dl>
-
-</div>
-<div class="section" id="backlunds">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">backlunds()</span></tt><a class="headerlink" href="#backlunds" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.backlunds">
-<tt class="descclassname">mpmath.</tt><tt class="descname">backlunds</tt><big>(</big><em>t</em><big>)</big><a class="headerlink" href="#mpmath.backlunds" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the function
-<span class="math">\(S(t) = \operatorname{arg} \zeta(\frac{1}{2} + it) / \pi\)</span>.</p>
-<p>See Titchmarsh Section 9.3 for details of the definition.</p>
-<p><strong>Examples</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">backlunds</span><span class="p">(</span><span class="mf">217.3</span><span class="p">)</span>
-<span class="go">0.16302205431184</span>
-</pre></div>
-</div>
-<p>Generally, the value is a small number. At Gram points it is an integer,
-frequently equal to 0:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">backlunds</span><span class="p">(</span><span class="n">grampoint</span><span class="p">(</span><span class="mi">200</span><span class="p">)))</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">backlunds</span><span class="p">(</span><span class="n">extraprec</span><span class="p">(</span><span class="mi">10</span><span class="p">)(</span><span class="n">grampoint</span><span class="p">)(</span><span class="mi">211</span><span class="p">))</span>
-<span class="go">1.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">backlunds</span><span class="p">(</span><span class="n">extraprec</span><span class="p">(</span><span class="mi">10</span><span class="p">)(</span><span class="n">grampoint</span><span class="p">)(</span><span class="mi">232</span><span class="p">))</span>
-<span class="go">-1.0</span>
-</pre></div>
-</div>
-<p>The number of zeros of the Riemann zeta function up to height <span class="math">\(t\)</span>
-satisfies <span class="math">\(N(t) = \theta(t)/\pi + 1 + S(t)\)</span> (see :func:nzeros` and
-<tt class="xref py py-func docutils literal"><span class="pre">siegeltheta()</span></tt>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="mf">1234.55</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nzeros</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-<span class="go">842</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">siegeltheta</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">/</span><span class="n">pi</span><span class="o">+</span><span class="mi">1</span><span class="o">+</span><span class="n">backlunds</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
-<span class="go">842.0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="lerch-transcendent">
-<h2>Lerch transcendent<a class="headerlink" href="#lerch-transcendent" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="lerchphi">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">lerchphi()</span></tt><a class="headerlink" href="#lerchphi" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.lerchphi">
-<tt class="descclassname">mpmath.</tt><tt class="descname">lerchphi</tt><big>(</big><em>z</em>, <em>s</em>, <em>a</em><big>)</big><a class="headerlink" href="#mpmath.lerchphi" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the Lerch transcendent, defined for <span class="math">\(|z| &lt; 1\)</span> and
-<span class="math">\(\Re{a} &gt; 0\)</span> by</p>
-<div class="math">
-\[\Phi(z,s,a) = \sum_{k=0}^{\infty} \frac{z^k}{(a+k)^s}\]</div>
-<p>and generally by the recurrence <span class="math">\(\Phi(z,s,a) = z \Phi(z,s,a+1) + a^{-s}\)</span>
-along with the integral representation valid for <span class="math">\(\Re{a} &gt; 0\)</span></p>
-<div class="math">
-\[\Phi(z,s,a) = \frac{1}{2 a^s} +
-        \int_0^{\infty} \frac{z^t}{(a+t)^s} dt -
-        2 \int_0^{\infty} \frac{\sin(t \log z - s
-            \operatorname{arctan}(t/a)}{(a^2 + t^2)^{s/2}
-            (e^{2 \pi t}-1)} dt.\]</div>
-<p>The Lerch transcendent generalizes the Hurwitz zeta function <tt class="xref py py-func docutils literal"><span class="pre">zeta()</span></tt>
-(<span class="math">\(z = 1\)</span>) and the polylogarithm <tt class="xref py py-func docutils literal"><span class="pre">polylog()</span></tt> (<span class="math">\(a = 1\)</span>).</p>
-<p><strong>Examples</strong></p>
-<p>Several evaluations in terms of simpler functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lerchphi</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mf">0.5</span><span class="p">);</span> <span class="mi">4</span><span class="o">*</span><span class="n">catalan</span>
-<span class="go">3.663862376708876060218414</span>
-<span class="go">3.663862376708876060218414</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">lerchphi</span><span class="p">,</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">));</span> <span class="mi">7</span><span class="o">*</span><span class="n">zeta</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.2131391994087528954617607</span>
-<span class="go">0.2131391994087528954617607</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lerchphi</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">log</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span>
-<span class="go">0.4023594781085250936501898</span>
-<span class="go">0.4023594781085250936501898</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lerchphi</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="o">+</span><span class="mi">2j</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mf">0.5</span><span class="p">);</span> <span class="mi">2</span><span class="o">*</span><span class="n">atanh</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="o">+</span><span class="mi">2j</span><span class="p">))</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="o">+</span><span class="mi">2j</span><span class="p">)</span>
-<span class="go">(1.142423447120257137774002 + 0.2118232380980201350495795j)</span>
-<span class="go">(1.142423447120257137774002 + 0.2118232380980201350495795j)</span>
-</pre></div>
-</div>
-<p>Evaluation works for complex arguments and <span class="math">\(|z| \ge 1\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">lerchphi</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mi">2j</span><span class="p">,</span> <span class="mi">3</span><span class="o">-</span><span class="n">j</span><span class="p">,</span> <span class="mi">4</span><span class="o">+</span><span class="mi">2j</span><span class="p">)</span>
-<span class="go">(0.002025009957009908600539469 + 0.003327897536813558807438089j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lerchphi</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="mf">2.5</span><span class="p">)</span>
-<span class="go">-12.28676272353094275265944</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lerchphi</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">(-4.462130727102185701817349e-11 + 1.575172198981096218823481e-12j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lerchphi</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="o">-</span><span class="mf">10.5</span><span class="p">)</span>
-<span class="go">(112658784011940.5605789002 + 498113185.5756221777743631j)</span>
-</pre></div>
-</div>
-<p>Some degenerate cases:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">lerchphi</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lerchphi</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">-0.5</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../../mpmath/references.html#dlmf">[DLMF]</a> section 25.14</li>
-</ol>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="polylogarithms-and-clausen-functions">
-<h2>Polylogarithms and Clausen functions<a class="headerlink" href="#polylogarithms-and-clausen-functions" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="polylog">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">polylog()</span></tt><a class="headerlink" href="#polylog" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.polylog">
-<tt class="descclassname">mpmath.</tt><tt class="descname">polylog</tt><big>(</big><em>s</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.polylog" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the polylogarithm, defined by the sum</p>
-<div class="math">
-\[\mathrm{Li}_s(z) = \sum_{k=1}^{\infty} \frac{z^k}{k^s}.\]</div>
-<p>This series is convergent only for <span class="math">\(|z| &lt; 1\)</span>, so elsewhere
-the analytic continuation is implied.</p>
-<p>The polylogarithm should not be confused with the logarithmic
-integral (also denoted by Li or li), which is implemented
-as <a class="reference internal" href="../functions/expintegrals.html#mpmath.li" title="mpmath.li"><tt class="xref py py-func docutils literal"><span class="pre">li()</span></tt></a>.</p>
-<p><strong>Examples</strong></p>
-<p>The polylogarithm satisfies a huge number of functional identities.
-A sample of polylogarithm evaluations is shown below:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mf">0.5</span><span class="p">),</span> <span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(0.693147180559945, 0.693147180559945)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mf">0.5</span><span class="p">),</span> <span class="p">(</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="mi">6</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">12</span>
-<span class="go">(0.582240526465012, 0.582240526465012)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="n">phi</span><span class="p">),</span> <span class="o">-</span><span class="n">log</span><span class="p">(</span><span class="n">phi</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">10</span>
-<span class="go">(-1.21852526068613, -1.21852526068613)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mf">0.5</span><span class="p">),</span> <span class="mi">7</span><span class="o">*</span><span class="n">zeta</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">8</span><span class="o">-</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">12</span><span class="o">+</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">3</span><span class="o">/</span><span class="mi">6</span>
-<span class="go">(0.53721319360804, 0.53721319360804)</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.polylog" title="mpmath.polylog"><tt class="xref py py-func docutils literal"><span class="pre">polylog()</span></tt></a> can evaluate the analytic continuation of the
-polylogarithm when <span class="math">\(s\)</span> is an integer:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">(0.536301287357863 - 7.23378441241546j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">-4.1982778868581</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">10j</span><span class="p">)</span>
-<span class="go">(-3.05968879432873 + 3.71678149306807j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">-0.150891632373114</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">0.067618332081142</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">10j</span><span class="p">)</span>
-<span class="go">(0.0384353698579347 + 0.0912451798066779j)</span>
-</pre></div>
-</div>
-<p>Some more examples, with arguments on the unit circle (note that
-the series definition cannot be used for computation here):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">j</span><span class="p">)</span>
-<span class="go">(-0.205616758356028 + 0.915965594177219j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">j</span><span class="o">*</span><span class="n">catalan</span><span class="o">-</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">48</span>
-<span class="go">(-0.205616758356028 + 0.915965594177219j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">j</span><span class="o">/</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">(-0.534247512515375 + 0.765587078525922j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="mi">4</span><span class="o">*</span><span class="n">zeta</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">9</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">j</span><span class="o">*</span><span class="n">pi</span><span class="o">**</span><span class="mi">3</span><span class="o">/</span><span class="mi">81</span>
-<span class="go">(-0.534247512515375 + 0.765587078525921j)</span>
-</pre></div>
-</div>
-<p>Polylogarithms of different order are related by integration
-and differentiation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="mf">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="n">s</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">0.517479061673899</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">polylog</span><span class="p">(</span><span class="n">s</span><span class="p">,</span><span class="n">t</span><span class="p">)</span><span class="o">/</span><span class="n">t</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">z</span><span class="p">])</span>
-<span class="go">0.517479061673899</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">polylog</span><span class="p">(</span><span class="n">s</span><span class="o">+</span><span class="mi">2</span><span class="p">,</span><span class="n">t</span><span class="p">),</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">0.517479061673899</span>
-</pre></div>
-</div>
-<p>Taylor series expansions around <span class="math">\(z = 0\)</span> are:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">nprint</span><span class="p">(</span><span class="n">taylor</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">polylog</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
-<span class="gp">...</span>
-<span class="go">[0.0, 1.0, 8.0, 27.0, 64.0, 125.0]</span>
-<span class="go">[0.0, 1.0, 4.0, 9.0, 16.0, 25.0]</span>
-<span class="go">[0.0, 1.0, 2.0, 3.0, 4.0, 5.0]</span>
-<span class="go">[0.0, 1.0, 1.0, 1.0, 1.0, 1.0]</span>
-<span class="go">[0.0, 1.0, 0.5, 0.333333, 0.25, 0.2]</span>
-<span class="go">[0.0, 1.0, 0.25, 0.111111, 0.0625, 0.04]</span>
-<span class="go">[0.0, 1.0, 0.125, 0.037037, 0.015625, 0.008]</span>
-</pre></div>
-</div>
-<p>The series defining the polylogarithm is simultaneously
-a Taylor series and an L-series. For certain values of <span class="math">\(z\)</span>, the
-polylogarithm reduces to a pure zeta function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">zeta</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">(1.17624173838258, 1.17624173838258)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="o">-</span><span class="n">altzeta</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">(-0.909670702980385, -0.909670702980385)</span>
-</pre></div>
-</div>
-<p>Evaluation for arbitrary, nonintegral <span class="math">\(s\)</span> is supported
-for <span class="math">\(z\)</span> within the unit circle:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">)</span>
-<span class="go">(0.24258605789446 - 0.00222938275488344j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="mf">0.25</span><span class="o">**</span><span class="n">k</span> <span class="o">/</span> <span class="n">k</span><span class="o">**</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">(0.24258605789446 - 0.00222938275488344j)</span>
-</pre></div>
-</div>
-<p>It is also currently supported outside of the unit circle for <span class="math">\(z\)</span>
-not too large in magnitude:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mi">20</span><span class="o">+</span><span class="mi">40j</span><span class="p">)</span>
-<span class="go">(-7.1421172179728 - 3.92726697721369j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polylog</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mi">200</span><span class="o">+</span><span class="mi">400j</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-  <span class="c">...</span>
-<span class="gr">NotImplementedError</span>: <span class="n">polylog for arbitrary s and z</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li>Richard Crandall, &#8220;Note on fast polylogarithm computation&#8221;
-<a class="reference external" href="http://people.reed.edu/~crandall/papers/Polylog.pdf">http://people.reed.edu/~crandall/papers/Polylog.pdf</a></li>
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Polylogarithm">http://en.wikipedia.org/wiki/Polylogarithm</a></li>
-<li><a class="reference external" href="http://mathworld.wolfram.com/Polylogarithm.html">http://mathworld.wolfram.com/Polylogarithm.html</a></li>
-</ol>
-</dd></dl>
-
-</div>
-<div class="section" id="clsin">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">clsin()</span></tt><a class="headerlink" href="#clsin" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.clsin">
-<tt class="descclassname">mpmath.</tt><tt class="descname">clsin</tt><big>(</big><em>s</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.clsin" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the Clausen sine function, defined formally by the series</p>
-<div class="math">
-\[\mathrm{Cl}_s(z) = \sum_{k=1}^{\infty} \frac{\sin(kz)}{k^s}.\]</div>
-<p>The special case <span class="math">\(\mathrm{Cl}_2(z)\)</span> (i.e. <tt class="docutils literal"><span class="pre">clsin(2,z)</span></tt>) is the classical
-&#8220;Clausen function&#8221;. More generally, the Clausen function is defined for
-complex <span class="math">\(s\)</span> and <span class="math">\(z\)</span>, even when the series does not converge. The
-Clausen function is related to the polylogarithm (<a class="reference internal" href="#mpmath.polylog" title="mpmath.polylog"><tt class="xref py py-func docutils literal"><span class="pre">polylog()</span></tt></a>) as</p>
-<div class="math">
-\[\mathrm{Cl}_s(z) = \frac{1}{2i}\left(\mathrm{Li}_s\left(e^{iz}\right) -
-                   \mathrm{Li}_s\left(e^{-iz}\right)\right)\]\[= \mathrm{Im}\left[\mathrm{Li}_s(e^{iz})\right] \quad (s, z \in \mathbb{R}),\]</div>
-<p>and this representation can be taken to provide the analytic continuation of the
-series. The complementary function <a class="reference internal" href="#mpmath.clcos" title="mpmath.clcos"><tt class="xref py py-func docutils literal"><span class="pre">clcos()</span></tt></a> gives the corresponding
-cosine sum.</p>
-<p><strong>Examples</strong></p>
-<p>Evaluation for arbitrarily chosen <span class="math">\(s\)</span> and <span class="math">\(z\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">clsin</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">z</span><span class="p">);</span> <span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="n">sin</span><span class="p">(</span><span class="n">z</span><span class="o">*</span><span class="n">k</span><span class="p">)</span><span class="o">/</span><span class="n">k</span><span class="o">**</span><span class="n">s</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">-0.6533010136329338746275795</span>
-<span class="go">-0.6533010136329338746275795</span>
-</pre></div>
-</div>
-<p>Using <span class="math">\(z + \pi\)</span> instead of <span class="math">\(z\)</span> gives an alternating series:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">clsin</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">z</span><span class="o">+</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">0.8860032351260589402871624</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">k</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">z</span><span class="o">*</span><span class="n">k</span><span class="p">)</span><span class="o">/</span><span class="n">k</span><span class="o">**</span><span class="n">s</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">0.8860032351260589402871624</span>
-</pre></div>
-</div>
-<p>With <span class="math">\(s = 1\)</span>, the sum can be expressed in closed form
-using elementary functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">clsin</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">0.2047709230104579724675985</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">((</span><span class="n">log</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">j</span><span class="o">*</span><span class="n">z</span><span class="p">))</span> <span class="o">-</span> <span class="n">log</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">exp</span><span class="p">(</span><span class="n">j</span><span class="o">*</span><span class="n">z</span><span class="p">)))</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">j</span><span class="p">))</span>
-<span class="go">0.2047709230104579724675985</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="n">sin</span><span class="p">(</span><span class="n">k</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="n">k</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">0.2047709230104579724675985</span>
-</pre></div>
-</div>
-<p>The classical Clausen function <span class="math">\(\mathrm{Cl}_2(\theta)\)</span> gives the
-value of the integral <span class="math">\(\int_0^{\theta} -\ln(2\sin(x/2)) dx\)</span> for
-<span class="math">\(0 &lt; \theta &lt; 2 \pi\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">cl2</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">clsin</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">t</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cl2</span><span class="p">(</span><span class="mf">3.5</span><span class="p">)</span>
-<span class="go">-0.2465045302347694216534255</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="n">quad</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">ln</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="mf">0.5</span><span class="o">*</span><span class="n">x</span><span class="p">)),</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">])</span>
-<span class="go">-0.2465045302347694216534255</span>
-</pre></div>
-</div>
-<p>This function is symmetric about <span class="math">\(\theta = \pi\)</span> with zeros and extreme
-points:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">cl2</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span> <span class="n">cl2</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">);</span> <span class="n">chop</span><span class="p">(</span><span class="n">cl2</span><span class="p">(</span><span class="n">pi</span><span class="p">));</span> <span class="n">cl2</span><span class="p">(</span><span class="mi">5</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">);</span> <span class="n">chop</span><span class="p">(</span><span class="n">cl2</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">))</span>
-<span class="go">0.0</span>
-<span class="go">1.014941606409653625021203</span>
-<span class="go">0.0</span>
-<span class="go">-1.014941606409653625021203</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>Catalan&#8217;s constant is a special value:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">cl2</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.9159655941772190150546035</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">+</span><span class="n">catalan</span>
-<span class="go">0.9159655941772190150546035</span>
-</pre></div>
-</div>
-<p>The Clausen sine function can be expressed in closed form when
-<span class="math">\(s\)</span> is an odd integer (becoming zero when <span class="math">\(s\)</span> &lt; 0):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">clsin</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">);</span> <span class="p">(</span><span class="n">pi</span><span class="o">-</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-<span class="go">0.3636895456083490948304773</span>
-<span class="go">0.3636895456083490948304773</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">clsin</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">z</span><span class="p">);</span> <span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">6</span><span class="o">*</span><span class="n">z</span> <span class="o">-</span> <span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">4</span> <span class="o">+</span> <span class="n">z</span><span class="o">**</span><span class="mi">3</span><span class="o">/</span><span class="mi">12</span>
-<span class="go">0.5661751584451144991707161</span>
-<span class="go">0.5661751584451144991707161</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">clsin</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">clsin</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>It can also be expressed in closed form for even integer <span class="math">\(s \le 0\)</span>,
-providing a finite sum for series such as
-<span class="math">\(\sin(z) + \sin(2z) + \sin(3z) + \ldots\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">clsin</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">0.1903105029507513881275865</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cot</span><span class="p">(</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
-<span class="go">0.1903105029507513881275865</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">clsin</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">-0.1089406163841548817581392</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="n">cot</span><span class="p">(</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">csc</span><span class="p">(</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">4</span>
-<span class="go">-0.1089406163841548817581392</span>
-</pre></div>
-</div>
-<p>Call with <tt class="docutils literal"><span class="pre">pi=True</span></tt> to multiply <span class="math">\(z\)</span> by <span class="math">\(\pi\)</span> exactly:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">clsin</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">-8.892316224968072424732898e-26</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">clsin</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">pi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>Evaluation for complex <span class="math">\(s\)</span>, <span class="math">\(z\)</span> in a nonconvergent case:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span><span class="o">-</span><span class="n">j</span><span class="p">,</span> <span class="mi">1</span><span class="o">+</span><span class="mi">2j</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">clsin</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">(-0.593079480117379002516034 + 0.9038644233367868273362446j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">extraprec</span><span class="p">(</span><span class="mi">20</span><span class="p">)(</span><span class="n">nsum</span><span class="p">)(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="n">sin</span><span class="p">(</span><span class="n">k</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="n">k</span><span class="o">**</span><span class="n">s</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">(-0.593079480117379002516034 + 0.9038644233367868273362446j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="clcos">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">clcos()</span></tt><a class="headerlink" href="#clcos" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.clcos">
-<tt class="descclassname">mpmath.</tt><tt class="descname">clcos</tt><big>(</big><em>s</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.clcos" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the Clausen cosine function, defined formally by the series</p>
-<div class="math">
-\[\mathrm{\widetilde{Cl}}_s(z) = \sum_{k=1}^{\infty} \frac{\cos(kz)}{k^s}.\]</div>
-<p>This function is complementary to the Clausen sine function
-<a class="reference internal" href="#mpmath.clsin" title="mpmath.clsin"><tt class="xref py py-func docutils literal"><span class="pre">clsin()</span></tt></a>. In terms of the polylogarithm,</p>
-<div class="math">
-\[\mathrm{\widetilde{Cl}}_s(z) =
-    \frac{1}{2}\left(\mathrm{Li}_s\left(e^{iz}\right) +
-    \mathrm{Li}_s\left(e^{-iz}\right)\right)\]\[= \mathrm{Re}\left[\mathrm{Li}_s(e^{iz})\right] \quad (s, z \in \mathbb{R}).\]</div>
-<p><strong>Examples</strong></p>
-<p>Evaluation for arbitrarily chosen <span class="math">\(s\)</span> and <span class="math">\(z\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">clcos</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">z</span><span class="p">);</span> <span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="n">cos</span><span class="p">(</span><span class="n">z</span><span class="o">*</span><span class="n">k</span><span class="p">)</span><span class="o">/</span><span class="n">k</span><span class="o">**</span><span class="n">s</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">-0.6518926267198991308332759</span>
-<span class="go">-0.6518926267198991308332759</span>
-</pre></div>
-</div>
-<p>Using <span class="math">\(z + \pi\)</span> instead of <span class="math">\(z\)</span> gives an alternating series:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="mf">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">clcos</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">z</span><span class="o">+</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">-0.8155530586502260817855618</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="n">k</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">z</span><span class="o">*</span><span class="n">k</span><span class="p">)</span><span class="o">/</span><span class="n">k</span><span class="o">**</span><span class="n">s</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">-0.8155530586502260817855618</span>
-</pre></div>
-</div>
-<p>With <span class="math">\(s = 1\)</span>, the sum can be expressed in closed form
-using elementary functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">clcos</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">-0.6720334373369714849797918</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="o">-</span><span class="mf">0.5</span><span class="o">*</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">exp</span><span class="p">(</span><span class="n">j</span><span class="o">*</span><span class="n">z</span><span class="p">))</span><span class="o">+</span><span class="n">log</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">j</span><span class="o">*</span><span class="n">z</span><span class="p">))))</span>
-<span class="go">-0.6720334373369714849797918</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="n">log</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="mf">0.5</span><span class="o">*</span><span class="n">z</span><span class="p">)))</span>    <span class="c"># Equivalent to above when z is real</span>
-<span class="go">-0.6720334373369714849797918</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="n">cos</span><span class="p">(</span><span class="n">k</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="n">k</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">-0.6720334373369714849797918</span>
-</pre></div>
-</div>
-<p>It can also be expressed in closed form when <span class="math">\(s\)</span> is an even integer.
-For example,</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">clcos</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">z</span><span class="p">)</span>
-<span class="go">-0.7805359025135583118863007</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">6</span> <span class="o">-</span> <span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">4</span>
-<span class="go">-0.7805359025135583118863007</span>
-</pre></div>
-</div>
-<p>The case <span class="math">\(s = 0\)</span> gives the renormalized sum of
-<span class="math">\(\cos(z) + \cos(2z) + \cos(3z) + \ldots\)</span> (which happens to be the same for
-any value of <span class="math">\(z\)</span>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">clcos</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">-0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="n">cos</span><span class="p">(</span><span class="n">k</span><span class="o">*</span><span class="n">z</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">-0.5</span>
-</pre></div>
-</div>
-<p>Also the sums</p>
-<div class="math">
-\[\cos(z) + 2\cos(2z) + 3\cos(3z) + \ldots\]</div>
-<p>and</p>
-<div class="math">
-\[\cos(z) + 2^n \cos(2z) + 3^n \cos(3z) + \ldots\]</div>
-<p>for higher integer powers <span class="math">\(n = -s\)</span> can be done in closed form. They are zero
-when <span class="math">\(n\)</span> is positive and even (<span class="math">\(s\)</span> negative and even):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">clcos</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">);</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">-0.2607829375240542480694126</span>
-<span class="go">-0.2607829375240542480694126</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">clcos</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="n">z</span><span class="p">);</span> <span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="n">cos</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">*</span><span class="n">csc</span><span class="p">(</span><span class="n">z</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">4</span><span class="o">/</span><span class="mi">8</span>
-<span class="go">0.1472635054979944390848006</span>
-<span class="go">0.1472635054979944390848006</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">clcos</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="n">z</span><span class="p">);</span> <span class="n">clcos</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="n">z</span><span class="p">);</span> <span class="n">clcos</span><span class="p">(</span><span class="o">-</span><span class="mi">6</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>With <span class="math">\(z = \pi\)</span>, the series reduces to that of the Riemann zeta function
-(more generally, if <span class="math">\(z = p \pi/q\)</span>, it is a finite sum over Hurwitz zeta
-function values):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">clcos</span><span class="p">(</span><span class="mf">2.5</span><span class="p">,</span> <span class="mi">0</span><span class="p">);</span> <span class="n">zeta</span><span class="p">(</span><span class="mf">2.5</span><span class="p">)</span>
-<span class="go">1.34148725725091717975677</span>
-<span class="go">1.34148725725091717975677</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">clcos</span><span class="p">(</span><span class="mf">2.5</span><span class="p">,</span> <span class="n">pi</span><span class="p">);</span> <span class="o">-</span><span class="n">altzeta</span><span class="p">(</span><span class="mf">2.5</span><span class="p">)</span>
-<span class="go">-0.8671998890121841381913472</span>
-<span class="go">-0.8671998890121841381913472</span>
-</pre></div>
-</div>
-<p>Call with <tt class="docutils literal"><span class="pre">pi=True</span></tt> to multiply <span class="math">\(z\)</span> by <span class="math">\(\pi\)</span> exactly:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">clcos</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">2.997921055881167659267063e+102</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">clcos</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">pi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">0.008333333333333333333333333</span>
-</pre></div>
-</div>
-<p>Evaluation for complex <span class="math">\(s\)</span>, <span class="math">\(z\)</span> in a nonconvergent case:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span><span class="o">-</span><span class="n">j</span><span class="p">,</span> <span class="mi">1</span><span class="o">+</span><span class="mi">2j</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">clcos</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">(0.9407430121562251476136807 + 0.715826296033590204557054j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">extraprec</span><span class="p">(</span><span class="mi">20</span><span class="p">)(</span><span class="n">nsum</span><span class="p">)(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="n">cos</span><span class="p">(</span><span class="n">k</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">/</span><span class="n">k</span><span class="o">**</span><span class="n">s</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">(0.9407430121562251476136807 + 0.715826296033590204557054j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="polyexp">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">polyexp()</span></tt><a class="headerlink" href="#polyexp" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.polyexp">
-<tt class="descclassname">mpmath.</tt><tt class="descname">polyexp</tt><big>(</big><em>s</em>, <em>z</em><big>)</big><a class="headerlink" href="#mpmath.polyexp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the polyexponential function, defined for arbitrary
-complex <span class="math">\(s\)</span>, <span class="math">\(z\)</span> by the series</p>
-<div class="math">
-\[E_s(z) = \sum_{k=1}^{\infty} \frac{k^s}{k!} z^k.\]</div>
-<p><span class="math">\(E_s(z)\)</span> is constructed from the exponential function analogously
-to how the polylogarithm is constructed from the ordinary
-logarithm; as a function of <span class="math">\(s\)</span> (with <span class="math">\(z\)</span> fixed), <span class="math">\(E_s\)</span> is an L-series
-It is an entire function of both <span class="math">\(s\)</span> and <span class="math">\(z\)</span>.</p>
-<p>The polyexponential function provides a generalization of the
-Bell polynomials <span class="math">\(B_n(x)\)</span> (see <a class="reference internal" href="../functions/numtheory.html#mpmath.bell" title="mpmath.bell"><tt class="xref py py-func docutils literal"><span class="pre">bell()</span></tt></a>) to noninteger orders <span class="math">\(n\)</span>.
-In terms of the Bell polynomials,</p>
-<div class="math">
-\[E_s(z) = e^z B_s(z) - \mathrm{sinc}(\pi s).\]</div>
-<p>Note that <span class="math">\(B_n(x)\)</span> and <span class="math">\(e^{-x} E_n(x)\)</span> are identical if <span class="math">\(n\)</span>
-is a nonzero integer, but not otherwise. In particular, they differ
-at <span class="math">\(n = 0\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>Evaluating a series:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsum</span><span class="p">(</span><span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">k</span><span class="p">)</span><span class="o">/</span><span class="n">fac</span><span class="p">(</span><span class="n">k</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="n">inf</span><span class="p">])</span>
-<span class="go">2.101755547733791780315904</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polyexp</span><span class="p">(</span><span class="mf">0.5</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">2.101755547733791780315904</span>
-</pre></div>
-</div>
-<p>Evaluation for arbitrary arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">polyexp</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="o">-</span><span class="mi">4j</span><span class="p">,</span> <span class="mf">2.5</span><span class="o">+</span><span class="mi">2j</span><span class="p">)</span>
-<span class="go">(2.351660261190434618268706 + 1.202966666673054671364215j)</span>
-</pre></div>
-</div>
-<p>Evaluation is accurate for tiny function values:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">polyexp</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="o">-</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">3.499471750566824369520223e-36</span>
-</pre></div>
-</div>
-<p>If <span class="math">\(n\)</span> is a nonpositive integer, <span class="math">\(E_n\)</span> reduces to a special
-instance of the hypergeometric function <span class="math">\(\,_pF_q\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polyexp</span><span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">,</span><span class="n">x</span><span class="p">)</span>
-<span class="go">4.042192318847986561771779</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">*</span><span class="n">hyper</span><span class="p">([</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">),</span> <span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">4.042192318847986561771779</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="zeta-function-variants">
-<h2>Zeta function variants<a class="headerlink" href="#zeta-function-variants" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="primezeta">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">primezeta()</span></tt><a class="headerlink" href="#primezeta" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.primezeta">
-<tt class="descclassname">mpmath.</tt><tt class="descname">primezeta</tt><big>(</big><em>s</em><big>)</big><a class="headerlink" href="#mpmath.primezeta" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the prime zeta function, which is defined
-in analogy with the Riemann zeta function (<a class="reference internal" href="#mpmath.zeta" title="mpmath.zeta"><tt class="xref py py-func docutils literal"><span class="pre">zeta()</span></tt></a>)
-as</p>
-<div class="math">
-\[P(s) = \sum_p \frac{1}{p^s}\]</div>
-<p>where the sum is taken over all prime numbers <span class="math">\(p\)</span>. Although
-this sum only converges for <span class="math">\(\mathrm{Re}(s) &gt; 1\)</span>, the
-function is defined by analytic continuation in the
-half-plane <span class="math">\(\mathrm{Re}(s) &gt; 0\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>Arbitrary-precision evaluation for real and complex arguments is
-supported:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">30</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primezeta</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.452247420041065498506543364832</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primezeta</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">0.15483752698840284272036497397</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primezeta</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.17476263929944353642311331466570670097541212192615</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">20</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primezeta</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(-0.12085382601645763295 - 0.013370403397787023602j)</span>
-</pre></div>
-</div>
-<p>The prime zeta function has a logarithmic pole at <span class="math">\(s = 1\)</span>,
-with residue equal to the difference of the Mertens and
-Euler constants:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">primezeta</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">+inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">extradps</span><span class="p">(</span><span class="mi">25</span><span class="p">)(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">primezeta</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">x</span><span class="p">)</span><span class="o">+</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">))(</span><span class="o">+</span><span class="n">eps</span><span class="p">)</span>
-<span class="go">-0.31571845205389007685</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mertens</span><span class="o">-</span><span class="n">euler</span>
-<span class="go">-0.31571845205389007685</span>
-</pre></div>
-</div>
-<p>The analytic continuation to <span class="math">\(0 &lt; \mathrm{Re}(s) \le 1\)</span>
-is implemented. In this strip the function exhibits
-very complex behavior; on the unit interval, it has poles at
-<span class="math">\(1/n\)</span> for every squarefree integer <span class="math">\(n\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">primezeta</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>         <span class="c"># Pole at s = 1/2</span>
-<span class="go">(-inf + 3.1415926535897932385j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primezeta</span><span class="p">(</span><span class="mf">0.25</span><span class="p">)</span>
-<span class="go">(-1.0416106801757269036 + 0.52359877559829887308j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primezeta</span><span class="p">(</span><span class="mf">0.5</span><span class="o">+</span><span class="mi">10j</span><span class="p">)</span>
-<span class="go">(0.54892423556409790529 + 0.45626803423487934264j)</span>
-</pre></div>
-</div>
-<p>Although evaluation works in principle for any <span class="math">\(\mathrm{Re}(s) &gt; 0\)</span>,
-it should be noted that the evaluation time increases exponentially
-as <span class="math">\(s\)</span> approaches the imaginary axis.</p>
-<p>For large <span class="math">\(\mathrm{Re}(s)\)</span>, <span class="math">\(P(s)\)</span> is asymptotic to <span class="math">\(2^{-s}\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">primezeta</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primezeta</span><span class="p">(</span><span class="mi">10</span><span class="p">),</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**-</span><span class="mi">10</span>
-<span class="go">(0.00099360357443698021786, 0.0009765625)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primezeta</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span>
-<span class="go">9.3326361850321887899e-302</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primezeta</span><span class="p">(</span><span class="mi">1000</span><span class="o">+</span><span class="mi">1000j</span><span class="p">)</span>
-<span class="go">(-3.8565440833654995949e-302 - 8.4985390447553234305e-302j)</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<p>Carl-Erik Froberg, &#8220;On the prime zeta function&#8221;,
-BIT 8 (1968), pp. 187-202.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="secondzeta">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">secondzeta()</span></tt><a class="headerlink" href="#secondzeta" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.secondzeta">
-<tt class="descclassname">mpmath.</tt><tt class="descname">secondzeta</tt><big>(</big><em>s</em>, <em>a=0.015</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.secondzeta" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the secondary zeta function <span class="math">\(Z(s)\)</span>, defined for
-<span class="math">\(\mathrm{Re}(s)&gt;1\)</span> by</p>
-<div class="math">
-\[Z(s) = \sum_{n=1}^{\infty} \frac{1}{\tau_n^s}\]</div>
-<p>where <span class="math">\(\frac12+i\tau_n\)</span> runs through the zeros of <span class="math">\(\zeta(s)\)</span> with
-imaginary part positive.</p>
-<p><span class="math">\(Z(s)\)</span> extends to a meromorphic function on <span class="math">\(\mathbb{C}\)</span>  with a
-double pole at <span class="math">\(s=1\)</span> and  simple poles at the points <span class="math">\(-2n\)</span> for
-<span class="math">\(n=0\)</span>,  1, 2, ...</p>
-<p><strong>Examples</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">secondzeta</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.023104993115419</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">xi</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">s</span><span class="p">:</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">s</span><span class="o">*</span><span class="p">(</span><span class="n">s</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">pi</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="mf">0.5</span><span class="o">*</span><span class="n">s</span><span class="p">)</span><span class="o">*</span><span class="n">gamma</span><span class="p">(</span><span class="mf">0.5</span><span class="o">*</span><span class="n">s</span><span class="p">)</span><span class="o">*</span><span class="n">zeta</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Xi</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">xi</span><span class="p">(</span><span class="mf">0.5</span><span class="o">+</span><span class="n">t</span><span class="o">*</span><span class="n">j</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="mf">0.5</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">Xi</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">n</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">Xi</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">(0.023104993115419 + 0.0j)</span>
-</pre></div>
-</div>
-<p>We may ask for an approximate error value:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">secondzeta</span><span class="p">(</span><span class="mf">0.5</span><span class="o">+</span><span class="mi">100j</span><span class="p">,</span> <span class="n">error</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">((-0.216272011276718 - 0.844952708937228j), 2.22044604925031e-16)</span>
-</pre></div>
-</div>
-<p>The function has poles at the negative odd integers,
-and dyadic rational values at the negative even integers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">30</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">secondzeta</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">)</span>
-<span class="go">-0.67236328125</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">secondzeta</span><span class="p">(</span><span class="o">-</span><span class="mi">7</span><span class="p">)</span>
-<span class="go">+inf</span>
-</pre></div>
-</div>
-<p><strong>Implementation notes</strong></p>
-<p>The function is computed as sum of four terms <span class="math">\(Z(s)=A(s)-P(s)+E(s)-S(s)\)</span>
-respectively main, prime, exponential and singular terms.
-The main term <span class="math">\(A(s)\)</span> is computed from the zeros of zeta.
-The prime term depends on the von Mangoldt function.
-The singular term is responsible for the poles of the function.</p>
-<p>The four terms depends on a small parameter <span class="math">\(a\)</span>. We may change the
-value of <span class="math">\(a\)</span>. Theoretically this has no effect on the sum of the four
-terms, but in practice may be important.</p>
-<p>A smaller value of the parameter <span class="math">\(a\)</span> makes <span class="math">\(A(s)\)</span> depend on
-a smaller number of zeros of zeta, but <span class="math">\(P(s)\)</span>  uses more values of
-von Mangoldt function.</p>
-<p>We may also add a verbose option to obtain data about the
-values of the four terms.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">secondzeta</span><span class="p">(</span><span class="mf">0.5</span> <span class="o">+</span> <span class="mi">40j</span><span class="p">,</span> <span class="n">error</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">main term = (-30190318549.138656312556 - 13964804384.624622876523j)</span>
-<span class="go">    computed using 19 zeros of zeta</span>
-<span class="go">prime term = (132717176.89212754625045 + 188980555.17563978290601j)</span>
-<span class="go">    computed using 9 values of the von Mangoldt function</span>
-<span class="go">exponential term = (542447428666.07179812536 + 362434922978.80192435203j)</span>
-<span class="go">singular term = (512124392939.98154322355 + 348281138038.65531023921j)</span>
-<span class="go">((0.059471043 + 0.3463514534j), 1.455191523e-11)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">secondzeta</span><span class="p">(</span><span class="mf">0.5</span> <span class="o">+</span> <span class="mi">40j</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="mf">0.04</span><span class="p">,</span> <span class="n">error</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">main term = (-151962888.19606243907725 - 217930683.90210294051982j)</span>
-<span class="go">    computed using 9 zeros of zeta</span>
-<span class="go">prime term = (2476659342.3038722372461 + 28711581821.921627163136j)</span>
-<span class="go">    computed using 37 values of the von Mangoldt function</span>
-<span class="go">exponential term = (178506047114.7838188264 + 819674143244.45677330576j)</span>
-<span class="go">singular term = (175877424884.22441310708 + 790744630738.28669174871j)</span>
-<span class="go">((0.059471043 + 0.3463514534j), 1.455191523e-11)</span>
-</pre></div>
-</div>
-<p>Notice the great cancellation between the four terms. Changing <span class="math">\(a\)</span>, the
-four terms are very different numbers but the cancellation gives
-the good value of Z(s).</p>
-<p><strong>References</strong></p>
-<p>A. Voros, Zeta functions for the Riemann zeros, Ann. Institute Fourier,
-53, (2003) 665&#8211;699.</p>
-<p>A. Voros, Zeta functions over Zeros of Zeta Functions, Lecture Notes
-of the Unione Matematica Italiana, Springer, 2009.</p>
-</dd></dl>
-
-</div>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Zeta functions, L-series and polylogarithms</a><ul>
-<li><a class="reference internal" href="#riemann-and-hurwitz-zeta-functions">Riemann and Hurwitz zeta functions</a><ul>
-<li><a class="reference internal" href="#zeta"><tt class="docutils literal"><span class="pre">zeta()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#dirichlet-l-series">Dirichlet L-series</a><ul>
-<li><a class="reference internal" href="#altzeta"><tt class="docutils literal"><span class="pre">altzeta()</span></tt></a></li>
-<li><a class="reference internal" href="#dirichlet"><tt class="docutils literal"><span class="pre">dirichlet()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#stieltjes-constants">Stieltjes constants</a><ul>
-<li><a class="reference internal" href="#stieltjes"><tt class="docutils literal"><span class="pre">stieltjes()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#zeta-function-zeros">Zeta function zeros</a><ul>
-<li><a class="reference internal" href="#zetazero"><tt class="docutils literal"><span class="pre">zetazero()</span></tt></a></li>
-<li><a class="reference internal" href="#nzeros"><tt class="docutils literal"><span class="pre">nzeros()</span></tt></a></li>
-<li><a class="reference internal" href="#siegelz"><tt class="docutils literal"><span class="pre">siegelz()</span></tt></a></li>
-<li><a class="reference internal" href="#siegeltheta"><tt class="docutils literal"><span class="pre">siegeltheta()</span></tt></a></li>
-<li><a class="reference internal" href="#grampoint"><tt class="docutils literal"><span class="pre">grampoint()</span></tt></a></li>
-<li><a class="reference internal" href="#backlunds"><tt class="docutils literal"><span class="pre">backlunds()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#lerch-transcendent">Lerch transcendent</a><ul>
-<li><a class="reference internal" href="#lerchphi"><tt class="docutils literal"><span class="pre">lerchphi()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#polylogarithms-and-clausen-functions">Polylogarithms and Clausen functions</a><ul>
-<li><a class="reference internal" href="#polylog"><tt class="docutils literal"><span class="pre">polylog()</span></tt></a></li>
-<li><a class="reference internal" href="#clsin"><tt class="docutils literal"><span class="pre">clsin()</span></tt></a></li>
-<li><a class="reference internal" href="#clcos"><tt class="docutils literal"><span class="pre">clcos()</span></tt></a></li>
-<li><a class="reference internal" href="#polyexp"><tt class="docutils literal"><span class="pre">polyexp()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#zeta-function-variants">Zeta function variants</a><ul>
-<li><a class="reference internal" href="#primezeta"><tt class="docutils literal"><span class="pre">primezeta()</span></tt></a></li>
-<li><a class="reference internal" href="#secondzeta"><tt class="docutils literal"><span class="pre">secondzeta()</span></tt></a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
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-            
-  <div class="section" id="utility-functions">
-<h1>Utility functions<a class="headerlink" href="#utility-functions" title="Permalink to this headline">¶</a></h1>
-<p>This page lists functions that perform basic operations
-on numbers or aid general programming.</p>
-<div class="section" id="conversion-and-printing">
-<h2>Conversion and printing<a class="headerlink" href="#conversion-and-printing" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="mpmathify-convert">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">mpmathify()</span></tt> / <tt class="xref py py-func docutils literal"><span class="pre">convert()</span></tt><a class="headerlink" href="#mpmathify-convert" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.mpmathify">
-<tt class="descclassname">mpmath.</tt><tt class="descname">mpmathify</tt><big>(</big><em>x</em>, <em>strings=True</em><big>)</big><a class="headerlink" href="#mpmath.mpmathify" title="Permalink to this definition">¶</a></dt>
-<dd><p>Converts <em>x</em> to an <tt class="docutils literal"><span class="pre">mpf</span></tt> or <tt class="docutils literal"><span class="pre">mpc</span></tt>. If <em>x</em> is of type <tt class="docutils literal"><span class="pre">mpf</span></tt>,
-<tt class="docutils literal"><span class="pre">mpc</span></tt>, <tt class="docutils literal"><span class="pre">int</span></tt>, <tt class="docutils literal"><span class="pre">float</span></tt>, <tt class="docutils literal"><span class="pre">complex</span></tt>, the conversion
-will be performed losslessly.</p>
-<p>If <em>x</em> is a string, the result will be rounded to the present
-working precision. Strings representing fractions or complex
-numbers are permitted.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpmathify</span><span class="p">(</span><span class="mf">3.5</span><span class="p">)</span>
-<span class="go">mpf(&#39;3.5&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpmathify</span><span class="p">(</span><span class="s">&#39;2.1&#39;</span><span class="p">)</span>
-<span class="go">mpf(&#39;2.1000000000000001&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpmathify</span><span class="p">(</span><span class="s">&#39;3/4&#39;</span><span class="p">)</span>
-<span class="go">mpf(&#39;0.75&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpmathify</span><span class="p">(</span><span class="s">&#39;2+3j&#39;</span><span class="p">)</span>
-<span class="go">mpc(real=&#39;2.0&#39;, imag=&#39;3.0&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="nstr">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">nstr()</span></tt><a class="headerlink" href="#nstr" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.nstr">
-<tt class="descclassname">mpmath.</tt><tt class="descname">nstr</tt><big>(</big><em>x</em>, <em>n=6</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.nstr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert an <tt class="docutils literal"><span class="pre">mpf</span></tt> or <tt class="docutils literal"><span class="pre">mpc</span></tt> to a decimal string literal with <em>n</em>
-significant digits. The small default value for <em>n</em> is chosen to
-make this function useful for printing collections of numbers
-(lists, matrices, etc).</p>
-<p>If <em>x</em> is a list or tuple, <a class="reference internal" href="#mpmath.nstr" title="mpmath.nstr"><tt class="xref py py-func docutils literal"><span class="pre">nstr()</span></tt></a> is applied recursively
-to each element. For unrecognized classes, <a class="reference internal" href="#mpmath.nstr" title="mpmath.nstr"><tt class="xref py py-func docutils literal"><span class="pre">nstr()</span></tt></a>
-simply returns <tt class="docutils literal"><span class="pre">str(x)</span></tt>.</p>
-<p>The companion function <a class="reference internal" href="#mpmath.nprint" title="mpmath.nprint"><tt class="xref py py-func docutils literal"><span class="pre">nprint()</span></tt></a> prints the result
-instead of returning it.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nstr</span><span class="p">([</span><span class="o">+</span><span class="n">pi</span><span class="p">,</span> <span class="n">ldexp</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">500</span><span class="p">)])</span>
-<span class="go">&#39;[3.14159, 3.05494e-151]&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">([</span><span class="o">+</span><span class="n">pi</span><span class="p">,</span> <span class="n">ldexp</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">500</span><span class="p">)])</span>
-<span class="go">[3.14159, 3.05494e-151]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="nprint">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">nprint()</span></tt><a class="headerlink" href="#nprint" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.nprint">
-<tt class="descclassname">mpmath.</tt><tt class="descname">nprint</tt><big>(</big><em>x</em>, <em>n=6</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.nprint" title="Permalink to this definition">¶</a></dt>
-<dd><p>Equivalent to <tt class="docutils literal"><span class="pre">print(nstr(x,</span> <span class="pre">n))</span></tt>.</p>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="arithmetic-operations">
-<h2>Arithmetic operations<a class="headerlink" href="#arithmetic-operations" title="Permalink to this headline">¶</a></h2>
-<p>See also <a class="reference internal" href="../mpmath/functions/powers.html#mpmath.sqrt" title="mpmath.sqrt"><tt class="xref py py-func docutils literal"><span class="pre">mpmath.sqrt()</span></tt></a>, <a class="reference internal" href="../mpmath/functions/powers.html#mpmath.exp" title="mpmath.exp"><tt class="xref py py-func docutils literal"><span class="pre">mpmath.exp()</span></tt></a> etc., listed
-in <a class="reference internal" href="../mpmath/functions/powers.html"><em>Powers and logarithms</em></a></p>
-<div class="section" id="fadd">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">fadd()</span></tt><a class="headerlink" href="#fadd" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.fadd">
-<tt class="descclassname">mpmath.</tt><tt class="descname">fadd</tt><big>(</big><em>ctx</em>, <em>x</em>, <em>y</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.fadd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Adds the numbers <em>x</em> and <em>y</em>, giving a floating-point result,
-optionally using a custom precision and rounding mode.</p>
-<p>The default precision is the working precision of the context.
-You can specify a custom precision in bits by passing the <em>prec</em> keyword
-argument, or by providing an equivalent decimal precision with the <em>dps</em>
-keyword argument. If the precision is set to <tt class="docutils literal"><span class="pre">+inf</span></tt>, or if the flag
-<em>exact=True</em> is passed, an exact addition with no rounding is performed.</p>
-<p>When the precision is finite, the optional <em>rounding</em> keyword argument
-specifies the direction of rounding. Valid options are <tt class="docutils literal"><span class="pre">'n'</span></tt> for
-nearest (default), <tt class="docutils literal"><span class="pre">'f'</span></tt> for floor, <tt class="docutils literal"><span class="pre">'c'</span></tt> for ceiling, <tt class="docutils literal"><span class="pre">'d'</span></tt>
-for down, <tt class="docutils literal"><span class="pre">'u'</span></tt> for up.</p>
-<p><strong>Examples</strong></p>
-<p>Using <a class="reference internal" href="#mpmath.fadd" title="mpmath.fadd"><tt class="xref py py-func docutils literal"><span class="pre">fadd()</span></tt></a> with precision and rounding control:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fadd</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">)</span>
-<span class="go">mpf(&#39;2.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fadd</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">,</span> <span class="n">rounding</span><span class="o">=</span><span class="s">&#39;u&#39;</span><span class="p">)</span>
-<span class="go">mpf(&#39;2.0000000000000004&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">fadd</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="mi">100</span><span class="p">),</span> <span class="mi">25</span><span class="p">)</span>
-<span class="go">2.00000000000000000001</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">fadd</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">,</span> <span class="n">dps</span><span class="o">=</span><span class="mi">15</span><span class="p">),</span> <span class="mi">25</span><span class="p">)</span>
-<span class="go">2.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">fadd</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">,</span> <span class="n">dps</span><span class="o">=</span><span class="mi">25</span><span class="p">),</span> <span class="mi">25</span><span class="p">)</span>
-<span class="go">2.00000000000000000001</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">fadd</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">),</span> <span class="mi">25</span><span class="p">)</span>
-<span class="go">2.00000000000000000001</span>
-</pre></div>
-</div>
-<p>Exact addition avoids cancellation errors, enforcing familiar laws
-of numbers such as <span class="math">\(x+y-x = y\)</span>, which don&#8217;t hold in floating-point
-arithmetic with finite precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">mpf</span><span class="p">(</span><span class="s">&#39;1e-1000&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">fadd</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="n">inf</span><span class="p">)</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">1.0e-1000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">fadd</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">1.0e-1000</span>
-</pre></div>
-</div>
-<p>Exact addition can be inefficient and may be impossible to perform
-with large magnitude differences:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fadd</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="s">&#39;1e-100000000000000000000&#39;</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="n">inf</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-  <span class="c">...</span>
-<span class="gr">OverflowError</span>: <span class="n">the exact result does not fit in memory</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="fsub">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">fsub()</span></tt><a class="headerlink" href="#fsub" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.fsub">
-<tt class="descclassname">mpmath.</tt><tt class="descname">fsub</tt><big>(</big><em>ctx</em>, <em>x</em>, <em>y</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.fsub" title="Permalink to this definition">¶</a></dt>
-<dd><p>Subtracts the numbers <em>x</em> and <em>y</em>, giving a floating-point result,
-optionally using a custom precision and rounding mode.</p>
-<p>See the documentation of <a class="reference internal" href="#mpmath.fadd" title="mpmath.fadd"><tt class="xref py py-func docutils literal"><span class="pre">fadd()</span></tt></a> for a detailed description
-of how to specify precision and rounding.</p>
-<p><strong>Examples</strong></p>
-<p>Using <a class="reference internal" href="#mpmath.fsub" title="mpmath.fsub"><tt class="xref py py-func docutils literal"><span class="pre">fsub()</span></tt></a> with precision and rounding control:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fsub</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">)</span>
-<span class="go">mpf(&#39;2.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fsub</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">,</span> <span class="n">rounding</span><span class="o">=</span><span class="s">&#39;d&#39;</span><span class="p">)</span>
-<span class="go">mpf(&#39;1.9999999999999998&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">fsub</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="mi">100</span><span class="p">),</span> <span class="mi">25</span><span class="p">)</span>
-<span class="go">1.99999999999999999999</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">fsub</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">,</span> <span class="n">dps</span><span class="o">=</span><span class="mi">15</span><span class="p">),</span> <span class="mi">25</span><span class="p">)</span>
-<span class="go">2.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">fsub</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">,</span> <span class="n">dps</span><span class="o">=</span><span class="mi">25</span><span class="p">),</span> <span class="mi">25</span><span class="p">)</span>
-<span class="go">1.99999999999999999999</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">fsub</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">),</span> <span class="mi">25</span><span class="p">)</span>
-<span class="go">1.99999999999999999999</span>
-</pre></div>
-</div>
-<p>Exact subtraction avoids cancellation errors, enforcing familiar laws
-of numbers such as <span class="math">\(x-y+y = x\)</span>, which don&#8217;t hold in floating-point
-arithmetic with finite precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">mpf</span><span class="p">(</span><span class="s">&#39;1e1000&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">y</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">fsub</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="n">inf</span><span class="p">)</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">2.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">fsub</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">2.0</span>
-</pre></div>
-</div>
-<p>Exact addition can be inefficient and may be impossible to perform
-with large magnitude differences:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fsub</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="s">&#39;1e-100000000000000000000&#39;</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="n">inf</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-  <span class="c">...</span>
-<span class="gr">OverflowError</span>: <span class="n">the exact result does not fit in memory</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="fneg">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">fneg()</span></tt><a class="headerlink" href="#fneg" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.fneg">
-<tt class="descclassname">mpmath.</tt><tt class="descname">fneg</tt><big>(</big><em>ctx</em>, <em>x</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.fneg" title="Permalink to this definition">¶</a></dt>
-<dd><p>Negates the number <em>x</em>, giving a floating-point result, optionally
-using a custom precision and rounding mode.</p>
-<p>See the documentation of <a class="reference internal" href="#mpmath.fadd" title="mpmath.fadd"><tt class="xref py py-func docutils literal"><span class="pre">fadd()</span></tt></a> for a detailed description
-of how to specify precision and rounding.</p>
-<p><strong>Examples</strong></p>
-<p>An mpmath number is returned:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fneg</span><span class="p">(</span><span class="mf">2.5</span><span class="p">)</span>
-<span class="go">mpf(&#39;-2.5&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fneg</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="o">+</span><span class="mi">2j</span><span class="p">)</span>
-<span class="go">mpc(real=&#39;5.0&#39;, imag=&#39;-2.0&#39;)</span>
-</pre></div>
-</div>
-<p>Precise control over rounding is possible:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">fadd</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1e-100</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fneg</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">mpf(&#39;-2.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fneg</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">rounding</span><span class="o">=</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="go">mpf(&#39;-2.0000000000000004&#39;)</span>
-</pre></div>
-</div>
-<p>Negating with and without roundoff:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="mi">200000000000000000000001</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="o">-</span><span class="n">mpf</span><span class="p">(</span><span class="n">n</span><span class="p">)))</span>
-<span class="go">-200000000000000016777216</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">fneg</span><span class="p">(</span><span class="n">n</span><span class="p">)))</span>
-<span class="go">-200000000000000016777216</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">fneg</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="n">log</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span><span class="o">+</span><span class="mi">1</span><span class="p">)))</span>
-<span class="go">-200000000000000000000001</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">fneg</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">dps</span><span class="o">=</span><span class="n">log</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">10</span><span class="p">)</span><span class="o">+</span><span class="mi">1</span><span class="p">)))</span>
-<span class="go">-200000000000000000000001</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">fneg</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="n">inf</span><span class="p">)))</span>
-<span class="go">-200000000000000000000001</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">fneg</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">dps</span><span class="o">=</span><span class="n">inf</span><span class="p">)))</span>
-<span class="go">-200000000000000000000001</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">fneg</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">)))</span>
-<span class="go">-200000000000000000000001</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="fmul">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">fmul()</span></tt><a class="headerlink" href="#fmul" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.fmul">
-<tt class="descclassname">mpmath.</tt><tt class="descname">fmul</tt><big>(</big><em>ctx</em>, <em>x</em>, <em>y</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.fmul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Multiplies the numbers <em>x</em> and <em>y</em>, giving a floating-point result,
-optionally using a custom precision and rounding mode.</p>
-<p>See the documentation of <a class="reference internal" href="#mpmath.fadd" title="mpmath.fadd"><tt class="xref py py-func docutils literal"><span class="pre">fadd()</span></tt></a> for a detailed description
-of how to specify precision and rounding.</p>
-<p><strong>Examples</strong></p>
-<p>The result is an mpmath number:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fmul</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mf">5.0</span><span class="p">)</span>
-<span class="go">mpf(&#39;10.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fmul</span><span class="p">(</span><span class="mf">0.5j</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
-<span class="go">mpc(real=&#39;0.0&#39;, imag=&#39;0.25&#39;)</span>
-</pre></div>
-</div>
-<p>Avoiding roundoff:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="mi">10</span><span class="o">**</span><span class="mi">10</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="o">**</span><span class="mi">15</span><span class="o">+</span><span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
-<span class="go">10000000001000010000000001</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="n">mpf</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>
-<span class="go">1.0000000001e+25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="n">mpf</span><span class="p">(</span><span class="n">y</span><span class="p">)))</span>
-<span class="go">10000000001000011026399232</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">fmul</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)))</span>
-<span class="go">10000000001000011026399232</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">fmul</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">dps</span><span class="o">=</span><span class="mi">25</span><span class="p">)))</span>
-<span class="go">10000000001000010000000001</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">fmul</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">)))</span>
-<span class="go">10000000001000010000000001</span>
-</pre></div>
-</div>
-<p>Exact multiplication with complex numbers can be inefficient and may
-be impossible to perform with large magnitude differences between
-real and imaginary parts:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="mi">1</span><span class="o">+</span><span class="mi">2j</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">mpc</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="s">&#39;1e-100000000000000000000&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fmul</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">mpc(real=&#39;2.0&#39;, imag=&#39;4.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fmul</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">rounding</span><span class="o">=</span><span class="s">&#39;u&#39;</span><span class="p">)</span>
-<span class="go">mpc(real=&#39;2.0&#39;, imag=&#39;4.0000000000000009&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fmul</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-  <span class="c">...</span>
-<span class="gr">OverflowError</span>: <span class="n">the exact result does not fit in memory</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="fdiv">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">fdiv()</span></tt><a class="headerlink" href="#fdiv" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.fdiv">
-<tt class="descclassname">mpmath.</tt><tt class="descname">fdiv</tt><big>(</big><em>ctx</em>, <em>x</em>, <em>y</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.fdiv" title="Permalink to this definition">¶</a></dt>
-<dd><p>Divides the numbers <em>x</em> and <em>y</em>, giving a floating-point result,
-optionally using a custom precision and rounding mode.</p>
-<p>See the documentation of <a class="reference internal" href="#mpmath.fadd" title="mpmath.fadd"><tt class="xref py py-func docutils literal"><span class="pre">fadd()</span></tt></a> for a detailed description
-of how to specify precision and rounding.</p>
-<p><strong>Examples</strong></p>
-<p>The result is an mpmath number:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fdiv</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">mpf(&#39;1.5&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fdiv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">mpf(&#39;0.66666666666666663&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fdiv</span><span class="p">(</span><span class="mi">2</span><span class="o">+</span><span class="mi">4j</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
-<span class="go">mpc(real=&#39;4.0&#39;, imag=&#39;8.0&#39;)</span>
-</pre></div>
-</div>
-<p>The rounding direction and precision can be controlled:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fdiv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">dps</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>    <span class="c"># Should be accurate to at least 3 digits</span>
-<span class="go">mpf(&#39;0.6666259765625&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fdiv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">rounding</span><span class="o">=</span><span class="s">&#39;d&#39;</span><span class="p">)</span>
-<span class="go">mpf(&#39;0.66666666666666663&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fdiv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="mi">60</span><span class="p">)</span>
-<span class="go">mpf(&#39;0.66666666666666667&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fdiv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">rounding</span><span class="o">=</span><span class="s">&#39;u&#39;</span><span class="p">)</span>
-<span class="go">mpf(&#39;0.66666666666666674&#39;)</span>
-</pre></div>
-</div>
-<p>Checking the error of a division by performing it at higher precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fdiv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="n">fdiv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
-<span class="go">mpf(&#39;-3.7007434154172148e-17&#39;)</span>
-</pre></div>
-</div>
-<p>Unlike <a class="reference internal" href="#mpmath.fadd" title="mpmath.fadd"><tt class="xref py py-func docutils literal"><span class="pre">fadd()</span></tt></a>, <a class="reference internal" href="#mpmath.fmul" title="mpmath.fmul"><tt class="xref py py-func docutils literal"><span class="pre">fmul()</span></tt></a>, etc., exact division is not
-allowed since the quotient of two floating-point numbers generally
-does not have an exact floating-point representation. (In the
-future this might be changed to allow the case where the division
-is actually exact.)</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fdiv</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-  <span class="c">...</span>
-<span class="gr">ValueError</span>: <span class="n">division is not an exact operation</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="fmod">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">fmod()</span></tt><a class="headerlink" href="#fmod" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.fmod">
-<tt class="descclassname">mpmath.</tt><tt class="descname">fmod</tt><big>(</big><em>x</em>, <em>y</em><big>)</big><a class="headerlink" href="#mpmath.fmod" title="Permalink to this definition">¶</a></dt>
-<dd><p>Converts <span class="math">\(x\)</span> and <span class="math">\(y\)</span> to mpmath numbers and returns <span class="math">\(x \mod y\)</span>.
-For mpmath numbers, this is equivalent to <tt class="docutils literal"><span class="pre">x</span> <span class="pre">%</span> <span class="pre">y</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fmod</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span> <span class="n">pi</span><span class="p">)</span>
-<span class="go">2.61062773871641</span>
-</pre></div>
-</div>
-<p>You can use <a class="reference internal" href="#mpmath.fmod" title="mpmath.fmod"><tt class="xref py py-func docutils literal"><span class="pre">fmod()</span></tt></a> to compute fractional parts of numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fmod</span><span class="p">(</span><span class="mf">10.25</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">0.25</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="fsum">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">fsum()</span></tt><a class="headerlink" href="#fsum" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.fsum">
-<tt class="descclassname">mpmath.</tt><tt class="descname">fsum</tt><big>(</big><em>terms</em>, <em>absolute=False</em>, <em>squared=False</em><big>)</big><a class="headerlink" href="#mpmath.fsum" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculates a sum containing a finite number of terms (for infinite
-series, see <a class="reference internal" href="../mpmath/calculus/sums_limits.html#mpmath.nsum" title="mpmath.nsum"><tt class="xref py py-func docutils literal"><span class="pre">nsum()</span></tt></a>). The terms will be converted to
-mpmath numbers. For len(terms) &gt; 2, this function is generally
-faster and produces more accurate results than the builtin
-Python function <tt class="xref py py-func docutils literal"><span class="pre">sum()</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fsum</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mi">7</span><span class="p">])</span>
-<span class="go">mpf(&#39;10.5&#39;)</span>
-</pre></div>
-</div>
-<p>With squared=True each term is squared, and with absolute=True
-the absolute value of each term is used.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="fprod">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">fprod()</span></tt><a class="headerlink" href="#fprod" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.fprod">
-<tt class="descclassname">mpmath.</tt><tt class="descname">fprod</tt><big>(</big><em>factors</em><big>)</big><a class="headerlink" href="#mpmath.fprod" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculates a product containing a finite number of factors (for
-infinite products, see <a class="reference internal" href="../mpmath/calculus/sums_limits.html#mpmath.nprod" title="mpmath.nprod"><tt class="xref py py-func docutils literal"><span class="pre">nprod()</span></tt></a>). The factors will be
-converted to mpmath numbers.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fprod</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mi">7</span><span class="p">])</span>
-<span class="go">mpf(&#39;7.0&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="fdot">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">fdot()</span></tt><a class="headerlink" href="#fdot" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.fdot">
-<tt class="descclassname">mpmath.</tt><tt class="descname">fdot</tt><big>(</big><em>A</em>, <em>B=None</em>, <em>conjugate=False</em><big>)</big><a class="headerlink" href="#mpmath.fdot" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the dot product of the iterables <span class="math">\(A\)</span> and <span class="math">\(B\)</span>,</p>
-<div class="math">
-\[\sum_{k=0} A_k B_k.\]</div>
-<p>Alternatively, <a class="reference internal" href="#mpmath.fdot" title="mpmath.fdot"><tt class="xref py py-func docutils literal"><span class="pre">fdot()</span></tt></a> accepts a single iterable of pairs.
-In other words, <tt class="docutils literal"><span class="pre">fdot(A,B)</span></tt> and <tt class="docutils literal"><span class="pre">fdot(zip(A,B))</span></tt> are equivalent.
-The elements are automatically converted to mpmath numbers.</p>
-<p>With <tt class="docutils literal"><span class="pre">conjugate=True</span></tt>, the elements in the second vector
-will be conjugated:</p>
-<div class="math">
-\[\sum_{k=0} A_k \overline{B_k}\]</div>
-<p><strong>Examples</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fdot</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-<span class="go">mpf(&#39;6.5&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">))</span>
-<span class="go">[(2, 1), (1.5, -1), (3, 2)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fdot</span><span class="p">(</span><span class="n">_</span><span class="p">)</span>
-<span class="go">mpf(&#39;6.5&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mi">3j</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="o">-</span><span class="n">j</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fdot</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-<span class="go">mpc(real=&#39;9.5&#39;, imag=&#39;-1.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fdot</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">conjugate</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">mpc(real=&#39;3.5&#39;, imag=&#39;-5.0&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="complex-components">
-<h2>Complex components<a class="headerlink" href="#complex-components" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="fabs">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">fabs()</span></tt><a class="headerlink" href="#fabs" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.fabs">
-<tt class="descclassname">mpmath.</tt><tt class="descname">fabs</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.fabs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the absolute value of <span class="math">\(x\)</span>, <span class="math">\(|x|\)</span>. Unlike <tt class="xref py py-func docutils literal"><span class="pre">abs()</span></tt>,
-<a class="reference internal" href="#mpmath.fabs" title="mpmath.fabs"><tt class="xref py py-func docutils literal"><span class="pre">fabs()</span></tt></a> converts non-mpmath numbers (such as <tt class="docutils literal"><span class="pre">int</span></tt>)
-into mpmath numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fabs</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">mpf(&#39;3.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fabs</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">mpf(&#39;3.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fabs</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">mpf(&#39;5.0&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="sign">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">sign()</span></tt><a class="headerlink" href="#sign" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.sign">
-<tt class="descclassname">mpmath.</tt><tt class="descname">sign</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.sign" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the sign of <span class="math">\(x\)</span>, defined as <span class="math">\(\mathrm{sign}(x) = x / |x|\)</span>
-(with the special case <span class="math">\(\mathrm{sign}(0) = 0\)</span>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sign</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">mpf(&#39;1.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sign</span><span class="p">(</span><span class="o">-</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">mpf(&#39;-1.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sign</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">mpf(&#39;0.0&#39;)</span>
-</pre></div>
-</div>
-<p>Note that the sign function is also defined for complex numbers,
-for which it gives the projection onto the unit circle:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sign</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">)</span>
-<span class="go">(0.707106781186547 + 0.707106781186547j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="re">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">re()</span></tt><a class="headerlink" href="#re" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.re">
-<tt class="descclassname">mpmath.</tt><tt class="descname">re</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.re" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the real part of <span class="math">\(x\)</span>, <span class="math">\(\Re(x)\)</span>. Unlike <tt class="docutils literal"><span class="pre">x.real</span></tt>,
-<a class="reference internal" href="#mpmath.re" title="mpmath.re"><tt class="xref py py-func docutils literal"><span class="pre">re()</span></tt></a> converts <span class="math">\(x\)</span> to a mpmath number:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">re</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">mpf(&#39;3.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">re</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">mpf(&#39;-1.0&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="im">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">im()</span></tt><a class="headerlink" href="#im" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.im">
-<tt class="descclassname">mpmath.</tt><tt class="descname">im</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.im" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the imaginary part of <span class="math">\(x\)</span>, <span class="math">\(\Im(x)\)</span>. Unlike <tt class="docutils literal"><span class="pre">x.imag</span></tt>,
-<a class="reference internal" href="#mpmath.im" title="mpmath.im"><tt class="xref py py-func docutils literal"><span class="pre">im()</span></tt></a> converts <span class="math">\(x\)</span> to a mpmath number:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">im</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">mpf(&#39;0.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">im</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">mpf(&#39;4.0&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="arg">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">arg()</span></tt><a class="headerlink" href="#arg" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.arg">
-<tt class="descclassname">mpmath.</tt><tt class="descname">arg</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.arg" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the complex argument (phase) of <span class="math">\(x\)</span>, defined as the
-signed angle between the positive real axis and <span class="math">\(x\)</span> in the
-complex plane:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">arg</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">arg</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">0.785398163397448</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">arg</span><span class="p">(</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">1.5707963267949</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">arg</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">3.14159265358979</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">arg</span><span class="p">(</span><span class="o">-</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">-1.5707963267949</span>
-</pre></div>
-</div>
-<p>The angle is defined to satisfy <span class="math">\(-\pi &lt; \arg(x) \le \pi\)</span> and
-with the sign convention that a nonnegative imaginary part
-results in a nonnegative argument.</p>
-<p>The value returned by <a class="reference internal" href="#mpmath.arg" title="mpmath.arg"><tt class="xref py py-func docutils literal"><span class="pre">arg()</span></tt></a> is an <tt class="docutils literal"><span class="pre">mpf</span></tt> instance.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="conj">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">conj()</span></tt><a class="headerlink" href="#conj" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.conj">
-<tt class="descclassname">mpmath.</tt><tt class="descname">conj</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.conj" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the complex conjugate of <span class="math">\(x\)</span>, <span class="math">\(\overline{x}\)</span>. Unlike
-<tt class="docutils literal"><span class="pre">x.conjugate()</span></tt>, <a class="reference internal" href="#mpmath.im" title="mpmath.im"><tt class="xref py py-func docutils literal"><span class="pre">im()</span></tt></a> converts <span class="math">\(x\)</span> to a mpmath number:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">conj</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">mpf(&#39;3.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">conj</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">mpc(real=&#39;-1.0&#39;, imag=&#39;-4.0&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="polar">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">polar()</span></tt><a class="headerlink" href="#polar" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.polar">
-<tt class="descclassname">mpmath.</tt><tt class="descname">polar</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.polar" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the polar representation of the complex number <span class="math">\(z\)</span>
-as a pair <span class="math">\((r, \phi)\)</span> such that <span class="math">\(z = r e^{i \phi}\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polar</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(2.0, 3.14159265358979)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">polar</span><span class="p">(</span><span class="mi">3</span><span class="o">-</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">(5.0, -0.927295218001612)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="rect">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">rect()</span></tt><a class="headerlink" href="#rect" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.rect">
-<tt class="descclassname">mpmath.</tt><tt class="descname">rect</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.rect" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the complex number represented by polar
-coordinates <span class="math">\((r, \phi)\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">rect</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">pi</span><span class="p">))</span>
-<span class="go">-2.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rect</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="o">-</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">(1.0 - 1.0j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="integer-and-fractional-parts">
-<h2>Integer and fractional parts<a class="headerlink" href="#integer-and-fractional-parts" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="floor">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">floor()</span></tt><a class="headerlink" href="#floor" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.floor">
-<tt class="descclassname">mpmath.</tt><tt class="descname">floor</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.floor" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the floor of <span class="math">\(x\)</span>, <span class="math">\(\lfloor x \rfloor\)</span>, defined as
-the largest integer less than or equal to <span class="math">\(x\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">floor</span><span class="p">(</span><span class="mf">3.5</span><span class="p">)</span>
-<span class="go">mpf(&#39;3.0&#39;)</span>
-</pre></div>
-</div>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last"><a class="reference internal" href="#mpmath.floor" title="mpmath.floor"><tt class="xref py py-func docutils literal"><span class="pre">floor()</span></tt></a>, <a class="reference internal" href="#mpmath.ceil" title="mpmath.ceil"><tt class="xref py py-func docutils literal"><span class="pre">ceil()</span></tt></a> and <a class="reference internal" href="#mpmath.nint" title="mpmath.nint"><tt class="xref py py-func docutils literal"><span class="pre">nint()</span></tt></a> return a
-floating-point number, not a Python <tt class="docutils literal"><span class="pre">int</span></tt>. If <span class="math">\(\lfloor x \rfloor\)</span> is
-too large to be represented exactly at the present working precision,
-the result will be rounded, not necessarily in the direction
-implied by the mathematical definition of the function.</p>
-</div>
-<p>To avoid rounding, use <em>prec=0</em>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">floor</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">30</span><span class="o">+</span><span class="mi">1</span><span class="p">)))</span>
-<span class="go">1000000000000000019884624838656</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">floor</span><span class="p">(</span><span class="mi">10</span><span class="o">**</span><span class="mi">30</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">prec</span><span class="o">=</span><span class="mi">0</span><span class="p">)))</span>
-<span class="go">1000000000000000000000000000001</span>
-</pre></div>
-</div>
-<p>The floor function is defined for complex numbers and
-acts on the real and imaginary parts separately:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">floor</span><span class="p">(</span><span class="mf">3.25</span><span class="o">+</span><span class="mf">4.75j</span><span class="p">)</span>
-<span class="go">mpc(real=&#39;3.0&#39;, imag=&#39;4.0&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="ceil">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">ceil()</span></tt><a class="headerlink" href="#ceil" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.ceil">
-<tt class="descclassname">mpmath.</tt><tt class="descname">ceil</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.ceil" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the ceiling of <span class="math">\(x\)</span>, <span class="math">\(\lceil x \rceil\)</span>, defined as
-the smallest integer greater than or equal to <span class="math">\(x\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ceil</span><span class="p">(</span><span class="mf">3.5</span><span class="p">)</span>
-<span class="go">mpf(&#39;4.0&#39;)</span>
-</pre></div>
-</div>
-<p>The ceiling function is defined for complex numbers and
-acts on the real and imaginary parts separately:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ceil</span><span class="p">(</span><span class="mf">3.25</span><span class="o">+</span><span class="mf">4.75j</span><span class="p">)</span>
-<span class="go">mpc(real=&#39;4.0&#39;, imag=&#39;5.0&#39;)</span>
-</pre></div>
-</div>
-<p>See notes about rounding for <a class="reference internal" href="#mpmath.floor" title="mpmath.floor"><tt class="xref py py-func docutils literal"><span class="pre">floor()</span></tt></a>.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="nint">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">nint()</span></tt><a class="headerlink" href="#nint" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.nint">
-<tt class="descclassname">mpmath.</tt><tt class="descname">nint</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.nint" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the nearest integer function, <span class="math">\(\mathrm{nint}(x)\)</span>.
-This gives the nearest integer to <span class="math">\(x\)</span>; on a tie, it
-gives the nearest even integer:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nint</span><span class="p">(</span><span class="mf">3.2</span><span class="p">)</span>
-<span class="go">mpf(&#39;3.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nint</span><span class="p">(</span><span class="mf">3.8</span><span class="p">)</span>
-<span class="go">mpf(&#39;4.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nint</span><span class="p">(</span><span class="mf">3.5</span><span class="p">)</span>
-<span class="go">mpf(&#39;4.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nint</span><span class="p">(</span><span class="mf">4.5</span><span class="p">)</span>
-<span class="go">mpf(&#39;4.0&#39;)</span>
-</pre></div>
-</div>
-<p>The nearest integer function is defined for complex numbers and
-acts on the real and imaginary parts separately:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nint</span><span class="p">(</span><span class="mf">3.25</span><span class="o">+</span><span class="mf">4.75j</span><span class="p">)</span>
-<span class="go">mpc(real=&#39;3.0&#39;, imag=&#39;5.0&#39;)</span>
-</pre></div>
-</div>
-<p>See notes about rounding for <a class="reference internal" href="#mpmath.floor" title="mpmath.floor"><tt class="xref py py-func docutils literal"><span class="pre">floor()</span></tt></a>.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="frac">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">frac()</span></tt><a class="headerlink" href="#frac" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.frac">
-<tt class="descclassname">mpmath.</tt><tt class="descname">frac</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.frac" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the fractional part of <span class="math">\(x\)</span>, defined as
-<span class="math">\(\mathrm{frac}(x) = x - \lfloor x \rfloor\)</span> (see <a class="reference internal" href="#mpmath.floor" title="mpmath.floor"><tt class="xref py py-func docutils literal"><span class="pre">floor()</span></tt></a>).
-In effect, this computes <span class="math">\(x\)</span> modulo 1, or <span class="math">\(x+n\)</span> where
-<span class="math">\(n \in \mathbb{Z}\)</span> is such that <span class="math">\(x+n \in [0,1)\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">frac</span><span class="p">(</span><span class="mf">1.25</span><span class="p">)</span>
-<span class="go">mpf(&#39;0.25&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">frac</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">mpf(&#39;0.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">frac</span><span class="p">(</span><span class="o">-</span><span class="mf">1.25</span><span class="p">)</span>
-<span class="go">mpf(&#39;0.75&#39;)</span>
-</pre></div>
-</div>
-<p>For a complex number, the fractional part function applies to
-the real and imaginary parts separately:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">frac</span><span class="p">(</span><span class="mf">2.25</span><span class="o">+</span><span class="mf">3.75j</span><span class="p">)</span>
-<span class="go">mpc(real=&#39;0.25&#39;, imag=&#39;0.75&#39;)</span>
-</pre></div>
-</div>
-<p>Plotted, the fractional part function gives a sawtooth
-wave. The Fourier series coefficients have a simple
-form:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">fourier</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">frac</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="mi">4</span><span class="p">))</span>
-<span class="go">([0.0, 0.0, 0.0, 0.0, 0.0], [0.0, -0.31831, -0.159155, -0.106103, -0.0795775])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">5</span><span class="p">)])</span>
-<span class="go">[-0.31831, -0.159155, -0.106103, -0.0795775]</span>
-</pre></div>
-</div>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">The fractional part is sometimes defined as a symmetric
-function, i.e. returning <span class="math">\(-\mathrm{frac}(-x)\)</span> if <span class="math">\(x &lt; 0\)</span>.
-This convention is used, for instance, by Mathematica&#8217;s
-<tt class="docutils literal"><span class="pre">FractionalPart</span></tt>.</p>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="tolerances-and-approximate-comparisons">
-<h2>Tolerances and approximate comparisons<a class="headerlink" href="#tolerances-and-approximate-comparisons" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="chop">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">chop()</span></tt><a class="headerlink" href="#chop" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.chop">
-<tt class="descclassname">mpmath.</tt><tt class="descname">chop</tt><big>(</big><em>x</em>, <em>tol=None</em><big>)</big><a class="headerlink" href="#mpmath.chop" title="Permalink to this definition">¶</a></dt>
-<dd><p>Chops off small real or imaginary parts, or converts
-numbers close to zero to exact zeros. The input can be a
-single number or an iterable:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="mi">5</span><span class="o">+</span><span class="mf">1e-10j</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-9</span><span class="p">)</span>
-<span class="go">mpf(&#39;5.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">chop</span><span class="p">([</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">1e-20</span><span class="p">,</span> <span class="mi">3</span><span class="o">+</span><span class="mf">1e-18j</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">]))</span>
-<span class="go">[1.0, 0.0, 3.0, -4.0, 2.0]</span>
-</pre></div>
-</div>
-<p>The tolerance defaults to <tt class="docutils literal"><span class="pre">100*eps</span></tt>.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="almosteq">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">almosteq()</span></tt><a class="headerlink" href="#almosteq" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.almosteq">
-<tt class="descclassname">mpmath.</tt><tt class="descname">almosteq</tt><big>(</big><em>s</em>, <em>t</em>, <em>rel_eps=None</em>, <em>abs_eps=None</em><big>)</big><a class="headerlink" href="#mpmath.almosteq" title="Permalink to this definition">¶</a></dt>
-<dd><p>Determine whether the difference between <span class="math">\(s\)</span> and <span class="math">\(t\)</span> is smaller
-than a given epsilon, either relatively or absolutely.</p>
-<p>Both a maximum relative difference and a maximum difference
-(&#8216;epsilons&#8217;) may be specified. The absolute difference is
-defined as <span class="math">\(|s-t|\)</span> and the relative difference is defined
-as <span class="math">\(|s-t|/\max(|s|, |t|)\)</span>.</p>
-<p>If only one epsilon is given, both are set to the same value.
-If none is given, both epsilons are set to <span class="math">\(2^{-p+m}\)</span> where
-<span class="math">\(p\)</span> is the current working precision and <span class="math">\(m\)</span> is a small
-integer. The default setting typically allows <a class="reference internal" href="#mpmath.almosteq" title="mpmath.almosteq"><tt class="xref py py-func docutils literal"><span class="pre">almosteq()</span></tt></a>
-to be used to check for mathematical equality
-in the presence of small rounding errors.</p>
-<p><strong>Examples</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">almosteq</span><span class="p">(</span><span class="mf">3.141592653589793</span><span class="p">,</span> <span class="mf">3.141592653589790</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">almosteq</span><span class="p">(</span><span class="mf">3.141592653589793</span><span class="p">,</span> <span class="mf">3.141592653589700</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">almosteq</span><span class="p">(</span><span class="mf">3.141592653589793</span><span class="p">,</span> <span class="mf">3.141592653589700</span><span class="p">,</span> <span class="mf">1e-10</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">almosteq</span><span class="p">(</span><span class="mf">1e-20</span><span class="p">,</span> <span class="mf">2e-20</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">almosteq</span><span class="p">(</span><span class="mf">1e-20</span><span class="p">,</span> <span class="mf">2e-20</span><span class="p">,</span> <span class="n">rel_eps</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">abs_eps</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="properties-of-numbers">
-<h2>Properties of numbers<a class="headerlink" href="#properties-of-numbers" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="isinf">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">isinf()</span></tt><a class="headerlink" href="#isinf" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.isinf">
-<tt class="descclassname">mpmath.</tt><tt class="descname">isinf</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.isinf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return <em>True</em> if the absolute value of <em>x</em> is infinite;
-otherwise return <em>False</em>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isinf</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isinf</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isinf</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isinf</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">4j</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isinf</span><span class="p">(</span><span class="n">mpc</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="n">inf</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isinf</span><span class="p">(</span><span class="n">mpc</span><span class="p">(</span><span class="n">inf</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="isnan">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">isnan()</span></tt><a class="headerlink" href="#isnan" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.isnan">
-<tt class="descclassname">mpmath.</tt><tt class="descname">isnan</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.isnan" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return <em>True</em> if <em>x</em> is a NaN (not-a-number), or for a complex
-number, whether either the real or complex part is NaN;
-otherwise return <em>False</em>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isnan</span><span class="p">(</span><span class="mf">3.14</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isnan</span><span class="p">(</span><span class="n">nan</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isnan</span><span class="p">(</span><span class="n">mpc</span><span class="p">(</span><span class="mf">3.14</span><span class="p">,</span><span class="mf">2.72</span><span class="p">))</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isnan</span><span class="p">(</span><span class="n">mpc</span><span class="p">(</span><span class="mf">3.14</span><span class="p">,</span><span class="n">nan</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="isnormal">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">isnormal()</span></tt><a class="headerlink" href="#isnormal" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.isnormal">
-<tt class="descclassname">mpmath.</tt><tt class="descname">isnormal</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.isnormal" title="Permalink to this definition">¶</a></dt>
-<dd><p>Determine whether <em>x</em> is &#8220;normal&#8221; in the sense of floating-point
-representation; that is, return <em>False</em> if <em>x</em> is zero, an
-infinity or NaN; otherwise return <em>True</em>. By extension, a
-complex number <em>x</em> is considered &#8220;normal&#8221; if its magnitude is
-normal:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isnormal</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isnormal</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isnormal</span><span class="p">(</span><span class="n">inf</span><span class="p">);</span> <span class="n">isnormal</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">);</span> <span class="n">isnormal</span><span class="p">(</span><span class="n">nan</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="go">False</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isnormal</span><span class="p">(</span><span class="mi">0</span><span class="o">+</span><span class="mi">0j</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isnormal</span><span class="p">(</span><span class="mi">0</span><span class="o">+</span><span class="mi">3j</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isnormal</span><span class="p">(</span><span class="n">mpc</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="n">nan</span><span class="p">))</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="isint">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">isint()</span></tt><a class="headerlink" href="#isint" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.isint">
-<tt class="descclassname">mpmath.</tt><tt class="descname">isint</tt><big>(</big><em>x</em>, <em>gaussian=False</em><big>)</big><a class="headerlink" href="#mpmath.isint" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return <em>True</em> if <em>x</em> is integer-valued; otherwise return
-<em>False</em>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isint</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isint</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isint</span><span class="p">(</span><span class="mf">3.2</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isint</span><span class="p">(</span><span class="n">inf</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>Optionally, Gaussian integers can be checked for:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">isint</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">0j</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isint</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">2j</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isint</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="mi">2j</span><span class="p">,</span> <span class="n">gaussian</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="ldexp">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">ldexp()</span></tt><a class="headerlink" href="#ldexp" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.ldexp">
-<tt class="descclassname">mpmath.</tt><tt class="descname">ldexp</tt><big>(</big><em>x</em>, <em>n</em><big>)</big><a class="headerlink" href="#mpmath.ldexp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes <span class="math">\(x 2^n\)</span> efficiently. No rounding is performed.
-The argument <span class="math">\(x\)</span> must be a real floating-point number (or
-possible to convert into one) and <span class="math">\(n\)</span> must be a Python <tt class="docutils literal"><span class="pre">int</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ldexp</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">mpf(&#39;1024.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ldexp</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">mpf(&#39;0.125&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="frexp">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">frexp()</span></tt><a class="headerlink" href="#frexp" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.frexp">
-<tt class="descclassname">mpmath.</tt><tt class="descname">frexp</tt><big>(</big><em>x</em>, <em>n</em><big>)</big><a class="headerlink" href="#mpmath.frexp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Given a real number <span class="math">\(x\)</span>, returns <span class="math">\((y, n)\)</span> with <span class="math">\(y \in [0.5, 1)\)</span>,
-<span class="math">\(n\)</span> a Python integer, and such that <span class="math">\(x = y 2^n\)</span>. No rounding is
-performed.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">frexp</span><span class="p">(</span><span class="mf">7.5</span><span class="p">)</span>
-<span class="go">(mpf(&#39;0.9375&#39;), 3)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="mag">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">mag()</span></tt><a class="headerlink" href="#mag" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.mag">
-<tt class="descclassname">mpmath.</tt><tt class="descname">mag</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.mag" title="Permalink to this definition">¶</a></dt>
-<dd><p>Quick logarithmic magnitude estimate of a number. Returns an
-integer or infinity <span class="math">\(m\)</span> such that <span class="math">\(|x| &lt;= 2^m\)</span>. It is not
-guaranteed that <span class="math">\(m\)</span> is an optimal bound, but it will never
-be too large by more than 2 (and probably not more than 1).</p>
-<p><strong>Examples</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mag</span><span class="p">(</span><span class="mi">10</span><span class="p">),</span> <span class="n">mag</span><span class="p">(</span><span class="mf">10.0</span><span class="p">),</span> <span class="n">mag</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="mi">10</span><span class="p">)),</span> <span class="nb">int</span><span class="p">(</span><span class="n">ceil</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">2</span><span class="p">)))</span>
-<span class="go">(4, 4, 4, 4)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mag</span><span class="p">(</span><span class="mi">10j</span><span class="p">),</span> <span class="n">mag</span><span class="p">(</span><span class="mi">10</span><span class="o">+</span><span class="mi">10j</span><span class="p">)</span>
-<span class="go">(4, 5)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mag</span><span class="p">(</span><span class="mf">0.01</span><span class="p">),</span> <span class="nb">int</span><span class="p">(</span><span class="n">ceil</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="mf">0.01</span><span class="p">,</span><span class="mi">2</span><span class="p">)))</span>
-<span class="go">(-6, -6)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mag</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">mag</span><span class="p">(</span><span class="n">inf</span><span class="p">),</span> <span class="n">mag</span><span class="p">(</span><span class="o">-</span><span class="n">inf</span><span class="p">),</span> <span class="n">mag</span><span class="p">(</span><span class="n">nan</span><span class="p">)</span>
-<span class="go">(-inf, +inf, +inf, nan)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="nint-distance">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">nint_distance()</span></tt><a class="headerlink" href="#nint-distance" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.nint_distance">
-<tt class="descclassname">mpmath.</tt><tt class="descname">nint_distance</tt><big>(</big><em>x</em><big>)</big><a class="headerlink" href="#mpmath.nint_distance" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return <span class="math">\((n,d)\)</span> where <span class="math">\(n\)</span> is the nearest integer to <span class="math">\(x\)</span> and <span class="math">\(d\)</span> is
-an estimate of <span class="math">\(\log_2(|x-n|)\)</span>. If <span class="math">\(d &lt; 0\)</span>, <span class="math">\(-d\)</span> gives the precision
-(measured in bits) lost to cancellation when computing <span class="math">\(x-n\)</span>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">nint_distance</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">n</span><span class="p">);</span> <span class="k">print</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-<span class="go">5</span>
-<span class="go">-inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">nint_distance</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">n</span><span class="p">);</span> <span class="k">print</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-<span class="go">5</span>
-<span class="go">-inf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">nint_distance</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="mf">5.00000001</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">n</span><span class="p">);</span> <span class="k">print</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-<span class="go">5</span>
-<span class="go">-26</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">nint_distance</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="mf">4.99999999</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">n</span><span class="p">);</span> <span class="k">print</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-<span class="go">5</span>
-<span class="go">-26</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">nint_distance</span><span class="p">(</span><span class="n">mpc</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">n</span><span class="p">);</span> <span class="k">print</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-<span class="go">5</span>
-<span class="go">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span> <span class="n">d</span> <span class="o">=</span> <span class="n">nint_distance</span><span class="p">(</span><span class="n">mpc</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mf">0.000001</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">n</span><span class="p">);</span> <span class="k">print</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-<span class="go">5</span>
-<span class="go">-19</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="number-generation">
-<h2>Number generation<a class="headerlink" href="#number-generation" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="fraction">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">fraction()</span></tt><a class="headerlink" href="#fraction" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.fraction">
-<tt class="descclassname">mpmath.</tt><tt class="descname">fraction</tt><big>(</big><em>p</em>, <em>q</em><big>)</big><a class="headerlink" href="#mpmath.fraction" title="Permalink to this definition">¶</a></dt>
-<dd><p>Given Python integers <span class="math">\((p, q)\)</span>, returns a lazy <tt class="docutils literal"><span class="pre">mpf</span></tt> representing
-the fraction <span class="math">\(p/q\)</span>. The value is updated with the precision.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">fraction</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">100</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">100</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">a</span><span class="p">);</span> <span class="k">print</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-<span class="go">0.01</span>
-<span class="go">0.01</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">30</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">a</span><span class="p">);</span> <span class="k">print</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>      <span class="c"># a will be accurate</span>
-<span class="go">0.01</span>
-<span class="go">0.0100000000000000002081668171172</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="rand">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">rand()</span></tt><a class="headerlink" href="#rand" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.rand">
-<tt class="descclassname">mpmath.</tt><tt class="descname">rand</tt><big>(</big><big>)</big><a class="headerlink" href="#mpmath.rand" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns an <tt class="docutils literal"><span class="pre">mpf</span></tt> with value chosen randomly from <span class="math">\([0, 1)\)</span>.
-The number of randomly generated bits in the mantissa is equal
-to the working precision.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="arange">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">arange()</span></tt><a class="headerlink" href="#arange" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.arange">
-<tt class="descclassname">mpmath.</tt><tt class="descname">arange</tt><big>(</big><em>*args</em><big>)</big><a class="headerlink" href="#mpmath.arange" title="Permalink to this definition">¶</a></dt>
-<dd><p>This is a generalized version of Python&#8217;s <tt class="xref py py-func docutils literal"><span class="pre">range()</span></tt> function
-that accepts fractional endpoints and step sizes and
-returns a list of <tt class="docutils literal"><span class="pre">mpf</span></tt> instances. Like <tt class="xref py py-func docutils literal"><span class="pre">range()</span></tt>,
-<a class="reference internal" href="#mpmath.arange" title="mpmath.arange"><tt class="xref py py-func docutils literal"><span class="pre">arange()</span></tt></a> can be called with 1, 2 or 3 arguments:</p>
-<dl class="docutils">
-<dt><tt class="docutils literal"><span class="pre">arange(b)</span></tt></dt>
-<dd><span class="math">\([0, 1, 2, \ldots, x]\)</span></dd>
-<dt><tt class="docutils literal"><span class="pre">arange(a,</span> <span class="pre">b)</span></tt></dt>
-<dd><span class="math">\([a, a+1, a+2, \ldots, x]\)</span></dd>
-<dt><tt class="docutils literal"><span class="pre">arange(a,</span> <span class="pre">b,</span> <span class="pre">h)</span></tt></dt>
-<dd><span class="math">\([a, a+h, a+h, \ldots, x]\)</span></dd>
-</dl>
-<p>where <span class="math">\(b-1 \le x &lt; b\)</span> (in the third case, <span class="math">\(b-h \le x &lt; b\)</span>).</p>
-<p>Like Python&#8217;s <tt class="xref py py-func docutils literal"><span class="pre">range()</span></tt>, the endpoint is not included. To
-produce ranges where the endpoint is included, <a class="reference internal" href="#mpmath.linspace" title="mpmath.linspace"><tt class="xref py py-func docutils literal"><span class="pre">linspace()</span></tt></a>
-is more convenient.</p>
-<p><strong>Examples</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">arange</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">[mpf(&#39;0.0&#39;), mpf(&#39;1.0&#39;), mpf(&#39;2.0&#39;), mpf(&#39;3.0&#39;)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">)</span>
-<span class="go">[mpf(&#39;1.0&#39;), mpf(&#39;1.25&#39;), mpf(&#39;1.5&#39;), mpf(&#39;1.75&#39;)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.75</span><span class="p">)</span>
-<span class="go">[mpf(&#39;1.0&#39;), mpf(&#39;0.25&#39;), mpf(&#39;-0.5&#39;)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="linspace">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">linspace()</span></tt><a class="headerlink" href="#linspace" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.linspace">
-<tt class="descclassname">mpmath.</tt><tt class="descname">linspace</tt><big>(</big><em>*args</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.linspace" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">linspace(a,</span> <span class="pre">b,</span> <span class="pre">n)</span></tt> returns a list of <span class="math">\(n\)</span> evenly spaced
-samples from <span class="math">\(a\)</span> to <span class="math">\(b\)</span>. The syntax <tt class="docutils literal"><span class="pre">linspace(mpi(a,b),</span> <span class="pre">n)</span></tt>
-is also valid.</p>
-<p>This function is often more convenient than <a class="reference internal" href="#mpmath.arange" title="mpmath.arange"><tt class="xref py py-func docutils literal"><span class="pre">arange()</span></tt></a>
-for partitioning an interval into subintervals, since
-the endpoint is included:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">linspace</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">[mpf(&#39;1.0&#39;), mpf(&#39;2.0&#39;), mpf(&#39;3.0&#39;), mpf(&#39;4.0&#39;)]</span>
-</pre></div>
-</div>
-<p>You may also provide the keyword argument <tt class="docutils literal"><span class="pre">endpoint=False</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">linspace</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="n">endpoint</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">[mpf(&#39;1.0&#39;), mpf(&#39;1.75&#39;), mpf(&#39;2.5&#39;), mpf(&#39;3.25&#39;)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="precision-management">
-<h2>Precision management<a class="headerlink" href="#precision-management" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="autoprec">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">autoprec()</span></tt><a class="headerlink" href="#autoprec" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.autoprec">
-<tt class="descclassname">mpmath.</tt><tt class="descname">autoprec</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>maxprec=None</em>, <em>catch=()</em>, <em>verbose=False</em><big>)</big><a class="headerlink" href="#mpmath.autoprec" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a wrapped copy of <em>f</em> that repeatedly evaluates <em>f</em>
-with increasing precision until the result converges to the
-full precision used at the point of the call.</p>
-<p>This heuristically protects against rounding errors, at the cost of
-roughly a 2x slowdown compared to manually setting the optimal
-precision. This method can, however, easily be fooled if the results
-from <em>f</em> depend &#8220;discontinuously&#8221; on the precision, for instance
-if catastrophic cancellation can occur. Therefore, <a class="reference internal" href="#mpmath.autoprec" title="mpmath.autoprec"><tt class="xref py py-func docutils literal"><span class="pre">autoprec()</span></tt></a>
-should be used judiciously.</p>
-<p><strong>Examples</strong></p>
-<p>Many functions are sensitive to perturbations of the input arguments.
-If the arguments are decimal numbers, they may have to be converted
-to binary at a much higher precision. If the amount of required
-extra precision is unknown, <a class="reference internal" href="#mpmath.autoprec" title="mpmath.autoprec"><tt class="xref py py-func docutils literal"><span class="pre">autoprec()</span></tt></a> is convenient:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselj</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">125</span> <span class="o">*</span> <span class="mi">10</span><span class="o">**</span><span class="mi">28</span><span class="p">)</span>    <span class="c"># Exact input</span>
-<span class="go">-8.03284785591801e-17</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselj</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="s">&#39;1.25e30&#39;</span><span class="p">)</span>   <span class="c"># Bad</span>
-<span class="go">7.12954868316652e-16</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">autoprec</span><span class="p">(</span><span class="n">besselj</span><span class="p">)(</span><span class="mi">5</span><span class="p">,</span> <span class="s">&#39;1.25e30&#39;</span><span class="p">)</span>   <span class="c"># Good</span>
-<span class="go">-8.03284785591801e-17</span>
-</pre></div>
-</div>
-<p>The following fails to converge because <span class="math">\(\sin(\pi) = 0\)</span> whereas all
-finite-precision approximations of <span class="math">\(\pi\)</span> give nonzero values:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">autoprec</span><span class="p">(</span><span class="n">sin</span><span class="p">)(</span><span class="n">pi</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-  <span class="c">...</span>
-<span class="gr">NoConvergence: autoprec</span>: <span class="n">prec increased to 2910 without convergence</span>
-</pre></div>
-</div>
-<p>As the following example shows, <a class="reference internal" href="#mpmath.autoprec" title="mpmath.autoprec"><tt class="xref py py-func docutils literal"><span class="pre">autoprec()</span></tt></a> can protect against
-cancellation, but is fooled by too severe cancellation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="mf">1e-10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span> <span class="n">expm1</span><span class="p">(</span><span class="n">x</span><span class="p">);</span> <span class="n">autoprec</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">)(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1.00000008274037e-10</span>
-<span class="go">1.00000000005e-10</span>
-<span class="go">1.00000000005e-10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="mf">1e-50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span> <span class="n">expm1</span><span class="p">(</span><span class="n">x</span><span class="p">);</span> <span class="n">autoprec</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="n">t</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">)(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="go">1.0e-50</span>
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>With <em>catch</em>, an exception or list of exceptions to intercept
-may be specified. The raised exception is interpreted
-as signaling insufficient precision. This permits, for example,
-evaluating a function where a too low precision results in a
-division by zero:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="mf">1e-30</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-  <span class="c">...</span>
-<span class="gr">ZeroDivisionError</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">autoprec</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">catch</span><span class="o">=</span><span class="ne">ZeroDivisionError</span><span class="p">)(</span><span class="mf">1e-30</span><span class="p">)</span>
-<span class="go">1.0e+30</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="workprec">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">workprec()</span></tt><a class="headerlink" href="#workprec" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.workprec">
-<tt class="descclassname">mpmath.</tt><tt class="descname">workprec</tt><big>(</big><em>ctx</em>, <em>n</em>, <em>normalize_output=False</em><big>)</big><a class="headerlink" href="#mpmath.workprec" title="Permalink to this definition">¶</a></dt>
-<dd><p>The block</p>
-<blockquote>
-<div><dl class="docutils">
-<dt>with workprec(n):</dt>
-<dd>&lt;code&gt;</dd>
-</dl>
-</div></blockquote>
-<p>sets the precision to n bits, executes &lt;code&gt;, and then restores
-the precision.</p>
-<p>workprec(n)(f) returns a decorated version of the function f
-that sets the precision to n bits before execution,
-and restores the precision afterwards. With normalize_output=True,
-it rounds the return value to the parent precision.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="workdps">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">workdps()</span></tt><a class="headerlink" href="#workdps" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.workdps">
-<tt class="descclassname">mpmath.</tt><tt class="descname">workdps</tt><big>(</big><em>ctx</em>, <em>n</em>, <em>normalize_output=False</em><big>)</big><a class="headerlink" href="#mpmath.workdps" title="Permalink to this definition">¶</a></dt>
-<dd><p>This function is analogous to workprec (see documentation)
-but changes the decimal precision instead of the number of bits.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="extraprec">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">extraprec()</span></tt><a class="headerlink" href="#extraprec" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.extraprec">
-<tt class="descclassname">mpmath.</tt><tt class="descname">extraprec</tt><big>(</big><em>ctx</em>, <em>n</em>, <em>normalize_output=False</em><big>)</big><a class="headerlink" href="#mpmath.extraprec" title="Permalink to this definition">¶</a></dt>
-<dd><p>The block</p>
-<blockquote>
-<div><dl class="docutils">
-<dt>with extraprec(n):</dt>
-<dd>&lt;code&gt;</dd>
-</dl>
-</div></blockquote>
-<p>increases the precision n bits, executes &lt;code&gt;, and then
-restores the precision.</p>
-<p>extraprec(n)(f) returns a decorated version of the function f
-that increases the working precision by n bits before execution,
-and restores the parent precision afterwards. With
-normalize_output=True, it rounds the return value to the parent
-precision.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="extradps">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">extradps()</span></tt><a class="headerlink" href="#extradps" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.extradps">
-<tt class="descclassname">mpmath.</tt><tt class="descname">extradps</tt><big>(</big><em>ctx</em>, <em>n</em>, <em>normalize_output=False</em><big>)</big><a class="headerlink" href="#mpmath.extradps" title="Permalink to this definition">¶</a></dt>
-<dd><p>This function is analogous to extraprec (see documentation)
-but changes the decimal precision instead of the number of bits.</p>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="performance-and-debugging">
-<h2>Performance and debugging<a class="headerlink" href="#performance-and-debugging" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="memoize">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">memoize()</span></tt><a class="headerlink" href="#memoize" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.memoize">
-<tt class="descclassname">mpmath.</tt><tt class="descname">memoize</tt><big>(</big><em>ctx</em>, <em>f</em><big>)</big><a class="headerlink" href="#mpmath.memoize" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a wrapped copy of <em>f</em> that caches computed values, i.e.
-a memoized copy of <em>f</em>. Values are only reused if the cached precision
-is equal to or higher than the working precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">memoize</span><span class="p">(</span><span class="n">maxcalls</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.909297426825682</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.909297426825682</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-  <span class="c">...</span>
-<span class="gr">NoConvergence: maxcalls</span>: <span class="n">function evaluated 1 times</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="maxcalls">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">maxcalls()</span></tt><a class="headerlink" href="#maxcalls" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.maxcalls">
-<tt class="descclassname">mpmath.</tt><tt class="descname">maxcalls</tt><big>(</big><em>ctx</em>, <em>f</em>, <em>N</em><big>)</big><a class="headerlink" href="#mpmath.maxcalls" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a wrapped copy of <em>f</em> that raises <tt class="docutils literal"><span class="pre">NoConvergence</span></tt> when <em>f</em>
-has been called more than <em>N</em> times:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">maxcalls</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">sum</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">)))</span>
-<span class="go">1.95520948210738</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-  <span class="c">...</span>
-<span class="gr">NoConvergence: maxcalls</span>: <span class="n">function evaluated 10 times</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="monitor">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">monitor()</span></tt><a class="headerlink" href="#monitor" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.monitor">
-<tt class="descclassname">mpmath.</tt><tt class="descname">monitor</tt><big>(</big><em>f</em>, <em>input='print'</em>, <em>output='print'</em><big>)</big><a class="headerlink" href="#mpmath.monitor" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a wrapped copy of <em>f</em> that monitors evaluation by calling
-<em>input</em> with every input (<em>args</em>, <em>kwargs</em>) passed to <em>f</em> and
-<em>output</em> with every value returned from <em>f</em>. The default action
-(specify using the special string value <tt class="docutils literal"><span class="pre">'print'</span></tt>) is to print
-inputs and outputs to stdout, along with the total evaluation
-count:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">5</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">monitor</span><span class="p">(</span><span class="n">exp</span><span class="p">),</span> <span class="mi">1</span><span class="p">)</span>   <span class="c"># diff will eval f(x-h) and f(x+h)</span>
-<span class="go">in  0 (mpf(&#39;0.99999999906867742538452148&#39;),) {}</span>
-<span class="go">out 0 mpf(&#39;2.7182818259274480055282064&#39;)</span>
-<span class="go">in  1 (mpf(&#39;1.0000000009313225746154785&#39;),) {}</span>
-<span class="go">out 1 mpf(&#39;2.7182818309906424675501024&#39;)</span>
-<span class="go">mpf(&#39;2.7182808&#39;)</span>
-</pre></div>
-</div>
-<p>To disable either the input or the output handler, you may
-pass <em>None</em> as argument.</p>
-<p>Custom input and output handlers may be used e.g. to store
-results for later analysis:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">input</span> <span class="o">=</span> <span class="p">[]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">output</span> <span class="o">=</span> <span class="p">[]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="n">monitor</span><span class="p">(</span><span class="n">sin</span><span class="p">,</span> <span class="nb">input</span><span class="o">.</span><span class="n">append</span><span class="p">,</span> <span class="n">output</span><span class="o">.</span><span class="n">append</span><span class="p">),</span> <span class="mf">3.0</span><span class="p">)</span>
-<span class="go">mpf(&#39;3.1415926535897932&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">len</span><span class="p">(</span><span class="nb">input</span><span class="p">)</span>  <span class="c"># Count number of evaluations</span>
-<span class="go">9</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">input</span><span class="p">[</span><span class="mi">3</span><span class="p">]);</span> <span class="k">print</span><span class="p">(</span><span class="n">output</span><span class="p">[</span><span class="mi">3</span><span class="p">])</span>
-<span class="go">((mpf(&#39;3.1415076583334066&#39;),), {})</span>
-<span class="go">8.49952562843408e-5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">input</span><span class="p">[</span><span class="mi">4</span><span class="p">]);</span> <span class="k">print</span><span class="p">(</span><span class="n">output</span><span class="p">[</span><span class="mi">4</span><span class="p">])</span>
-<span class="go">((mpf(&#39;3.1415928201669122&#39;),), {})</span>
-<span class="go">-1.66577118985331e-7</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="timing">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">timing()</span></tt><a class="headerlink" href="#timing" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.timing">
-<tt class="descclassname">mpmath.</tt><tt class="descname">timing</tt><big>(</big><em>f</em>, <em>*args</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.timing" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns time elapsed for evaluating <tt class="docutils literal"><span class="pre">f()</span></tt>. Optionally arguments
-may be passed to time the execution of <tt class="docutils literal"><span class="pre">f(*args,</span> <span class="pre">**kwargs)</span></tt>.</p>
-<p>If the first call is very quick, <tt class="docutils literal"><span class="pre">f</span></tt> is called
-repeatedly and the best time is returned.</p>
-</dd></dl>
-
-</div>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../index.html">
-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Utility functions</a><ul>
-<li><a class="reference internal" href="#conversion-and-printing">Conversion and printing</a><ul>
-<li><a class="reference internal" href="#mpmathify-convert"><tt class="docutils literal"><span class="pre">mpmathify()</span></tt> / <tt class="docutils literal"><span class="pre">convert()</span></tt></a></li>
-<li><a class="reference internal" href="#nstr"><tt class="docutils literal"><span class="pre">nstr()</span></tt></a></li>
-<li><a class="reference internal" href="#nprint"><tt class="docutils literal"><span class="pre">nprint()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#arithmetic-operations">Arithmetic operations</a><ul>
-<li><a class="reference internal" href="#fadd"><tt class="docutils literal"><span class="pre">fadd()</span></tt></a></li>
-<li><a class="reference internal" href="#fsub"><tt class="docutils literal"><span class="pre">fsub()</span></tt></a></li>
-<li><a class="reference internal" href="#fneg"><tt class="docutils literal"><span class="pre">fneg()</span></tt></a></li>
-<li><a class="reference internal" href="#fmul"><tt class="docutils literal"><span class="pre">fmul()</span></tt></a></li>
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-  <div class="section" id="number-identification">
-<h1>Number identification<a class="headerlink" href="#number-identification" title="Permalink to this headline">¶</a></h1>
-<p>Most function in mpmath are concerned with producing approximations from exact mathematical formulas. It is also useful to consider the inverse problem: given only a decimal approximation for a number, such as 0.7320508075688772935274463, is it possible to find an exact formula?</p>
-<p>Subject to certain restrictions, such &#8220;reverse engineering&#8221; is indeed possible thanks to the existence of <em>integer relation algorithms</em>. Mpmath implements the PSLQ algorithm (developed by H. Ferguson), which is one such algorithm.</p>
-<p>Automated number recognition based on PSLQ is not a silver bullet. Any occurring transcendental constants (<span class="math">\(\pi\)</span>, <span class="math">\(e\)</span>, etc) must be guessed by the user, and the relation between those constants in the formula must be linear (such as <span class="math">\(x = 3 \pi + 4 e\)</span>). More complex formulas can be found by combining PSLQ with functional transformations; however, this is only feasible to a limited extent since the computation time grows exponentially with the number of operations that need to be combined.</p>
-<p>The number identification facilities in mpmath are inspired by the <a class="reference external" href="http://oldweb.cecm.sfu.ca/projects/ISC/ISCmain.html">Inverse Symbolic Calculator</a> (ISC). The ISC is more powerful than mpmath, as it uses a lookup table of millions of precomputed constants (thereby mitigating the problem with exponential complexity).</p>
-<div class="section" id="constant-recognition">
-<h2>Constant recognition<a class="headerlink" href="#constant-recognition" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="identify">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">identify()</span></tt><a class="headerlink" href="#identify" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.identify">
-<tt class="descclassname">mpmath.</tt><tt class="descname">identify</tt><big>(</big><em>ctx</em>, <em>x</em>, <em>constants=</em><span class="optional">[</span><span class="optional">]</span>, <em>tol=None</em>, <em>maxcoeff=1000</em>, <em>full=False</em>, <em>verbose=False</em><big>)</big><a class="headerlink" href="#mpmath.identify" title="Permalink to this definition">¶</a></dt>
-<dd><p>Given a real number <span class="math">\(x\)</span>, <tt class="docutils literal"><span class="pre">identify(x)</span></tt> attempts to find an exact
-formula for <span class="math">\(x\)</span>. This formula is returned as a string. If no match
-is found, <tt class="docutils literal"><span class="pre">None</span></tt> is returned. With <tt class="docutils literal"><span class="pre">full=True</span></tt>, a list of
-matching formulas is returned.</p>
-<p>As a simple example, <a class="reference internal" href="#mpmath.identify" title="mpmath.identify"><tt class="xref py py-func docutils literal"><span class="pre">identify()</span></tt></a> will find an algebraic
-formula for the golden ratio:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="n">phi</span><span class="p">)</span>
-<span class="go">&#39;((1+sqrt(5))/2)&#39;</span>
-</pre></div>
-</div>
-<p><a class="reference internal" href="#mpmath.identify" title="mpmath.identify"><tt class="xref py py-func docutils literal"><span class="pre">identify()</span></tt></a> can identify simple algebraic numbers and simple
-combinations of given base constants, as well as certain basic
-transformations thereof. More specifically, <a class="reference internal" href="#mpmath.identify" title="mpmath.identify"><tt class="xref py py-func docutils literal"><span class="pre">identify()</span></tt></a>
-looks for the following:</p>
-<blockquote>
-<div><ol class="arabic simple">
-<li>Fractions</li>
-<li>Quadratic algebraic numbers</li>
-<li>Rational linear combinations of the base constants</li>
-<li>Any of the above after first transforming <span class="math">\(x\)</span> into <span class="math">\(f(x)\)</span> where
-<span class="math">\(f(x)\)</span> is <span class="math">\(1/x\)</span>, <span class="math">\(\sqrt x\)</span>, <span class="math">\(x^2\)</span>, <span class="math">\(\log x\)</span> or <span class="math">\(\exp x\)</span>, either
-directly or with <span class="math">\(x\)</span> or <span class="math">\(f(x)\)</span> multiplied or divided by one of
-the base constants</li>
-<li>Products of fractional powers of the base constants and
-small integers</li>
-</ol>
-</div></blockquote>
-<p>Base constants can be given as a list of strings representing mpmath
-expressions (<a class="reference internal" href="#mpmath.identify" title="mpmath.identify"><tt class="xref py py-func docutils literal"><span class="pre">identify()</span></tt></a> will <tt class="docutils literal"><span class="pre">eval</span></tt> the strings to numerical
-values and use the original strings for the output), or as a dict of
-formula:value pairs.</p>
-<p>In order not to produce spurious results, <a class="reference internal" href="#mpmath.identify" title="mpmath.identify"><tt class="xref py py-func docutils literal"><span class="pre">identify()</span></tt></a> should
-be used with high precision; preferably 50 digits or more.</p>
-<p><strong>Examples</strong></p>
-<p>Simple identifications can be performed safely at standard
-precision. Here the default recognition of rational, algebraic,
-and exp/log of algebraic numbers is demonstrated:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="mf">0.22222222222222222</span><span class="p">)</span>
-<span class="go">&#39;(2/9)&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="mf">1.9662210973805663</span><span class="p">)</span>
-<span class="go">&#39;sqrt(((24+sqrt(48))/8))&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="mf">4.1132503787829275</span><span class="p">)</span>
-<span class="go">&#39;exp((sqrt(8)/2))&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="mf">0.881373587019543</span><span class="p">)</span>
-<span class="go">&#39;log(((2+sqrt(8))/2))&#39;</span>
-</pre></div>
-</div>
-<p>By default, <a class="reference internal" href="#mpmath.identify" title="mpmath.identify"><tt class="xref py py-func docutils literal"><span class="pre">identify()</span></tt></a> does not recognize <span class="math">\(\pi\)</span>. At standard
-precision it finds a not too useful approximation. At slightly
-increased precision, this approximation is no longer accurate
-enough and <a class="reference internal" href="#mpmath.identify" title="mpmath.identify"><tt class="xref py py-func docutils literal"><span class="pre">identify()</span></tt></a> more correctly returns <tt class="docutils literal"><span class="pre">None</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">&#39;(2**(176/117)*3**(20/117)*5**(35/39))/(7**(92/117))&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">30</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">&gt;&gt;&gt;</span>
-</pre></div>
-</div>
-<p>Numbers such as <span class="math">\(\pi\)</span>, and simple combinations of user-defined
-constants, can be identified if they are provided explicitly:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">e</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;pi&#39;</span><span class="p">,</span> <span class="s">&#39;e&#39;</span><span class="p">])</span>
-<span class="go">&#39;(3*pi + (-2)*e)&#39;</span>
-</pre></div>
-</div>
-<p>Here is an example using a dict of constants. Note that the
-constants need not be &#8220;atomic&#8221;; <a class="reference internal" href="#mpmath.identify" title="mpmath.identify"><tt class="xref py py-func docutils literal"><span class="pre">identify()</span></tt></a> can just
-as well express the given number in terms of expressions
-given by formulas:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="n">pi</span><span class="o">+</span><span class="n">e</span><span class="p">,</span> <span class="p">{</span><span class="s">&#39;a&#39;</span><span class="p">:</span><span class="n">pi</span><span class="o">+</span><span class="mi">2</span><span class="p">,</span> <span class="s">&#39;b&#39;</span><span class="p">:</span><span class="mi">2</span><span class="o">*</span><span class="n">e</span><span class="p">})</span>
-<span class="go">&#39;((-2) + 1*a + (1/2)*b)&#39;</span>
-</pre></div>
-</div>
-<p>Next, we attempt some identifications with a set of base constants.
-It is necessary to increase the precision a bit.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">base</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#39;sqrt(2)&#39;</span><span class="p">,</span><span class="s">&#39;pi&#39;</span><span class="p">,</span><span class="s">&#39;log(2)&#39;</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="mf">0.25</span><span class="p">,</span> <span class="n">base</span><span class="p">)</span>
-<span class="go">&#39;(1/4)&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">pi</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mi">5</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">7</span><span class="p">,</span> <span class="n">base</span><span class="p">)</span>
-<span class="go">&#39;(2*sqrt(2) + 3*pi + (5/7)*log(2))&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">pi</span><span class="o">+</span><span class="mi">2</span><span class="p">),</span> <span class="n">base</span><span class="p">)</span>
-<span class="go">&#39;exp((2 + 1*pi))&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)),</span> <span class="n">base</span><span class="p">)</span>
-<span class="go">&#39;((3/7) + (-1/7)*sqrt(2))&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">+</span><span class="mi">4</span><span class="p">),</span> <span class="n">base</span><span class="p">)</span>
-<span class="go">&#39;sqrt(2)/(4 + 3*pi)&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="mi">5</span><span class="o">**</span><span class="p">(</span><span class="n">mpf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">base</span><span class="p">)</span>
-<span class="go">&#39;5**(1/3)*pi*log(2)**2&#39;</span>
-</pre></div>
-</div>
-<p>An example of an erroneous solution being found when too low
-precision is used:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">-</span><span class="mi">4</span><span class="o">*</span><span class="n">e</span><span class="o">+</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">8</span><span class="p">)),</span> <span class="p">[</span><span class="s">&#39;pi&#39;</span><span class="p">,</span> <span class="s">&#39;e&#39;</span><span class="p">,</span> <span class="s">&#39;sqrt(2)&#39;</span><span class="p">])</span>
-<span class="go">&#39;((11/25) + (-158/75)*pi + (76/75)*e + (44/15)*sqrt(2))&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">-</span><span class="mi">4</span><span class="o">*</span><span class="n">e</span><span class="o">+</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">8</span><span class="p">)),</span> <span class="p">[</span><span class="s">&#39;pi&#39;</span><span class="p">,</span> <span class="s">&#39;e&#39;</span><span class="p">,</span> <span class="s">&#39;sqrt(2)&#39;</span><span class="p">])</span>
-<span class="go">&#39;1/(3*pi + (-4)*e + 2*sqrt(2))&#39;</span>
-</pre></div>
-</div>
-<p><strong>Finding approximate solutions</strong></p>
-<p>The tolerance <tt class="docutils literal"><span class="pre">tol</span></tt> defaults to 3/4 of the working precision.
-Lowering the tolerance is useful for finding approximate matches.
-We can for example try to generate approximations for pi:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-2</span><span class="p">)</span>
-<span class="go">&#39;(22/7)&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">)</span>
-<span class="go">&#39;(355/113)&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-10</span><span class="p">)</span>
-<span class="go">&#39;(5**(339/269))/(2**(64/269)*3**(13/269)*7**(92/269))&#39;</span>
-</pre></div>
-</div>
-<p>With <tt class="docutils literal"><span class="pre">full=True</span></tt>, and by supplying a few base constants,
-<tt class="docutils literal"><span class="pre">identify</span></tt> can generate almost endless lists of approximations
-for any number (the output below has been truncated to show only
-the first few):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">identify</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;e&#39;</span><span class="p">,</span> <span class="s">&#39;catalan&#39;</span><span class="p">],</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-5</span><span class="p">,</span> <span class="n">full</span><span class="o">=</span><span class="bp">True</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-<span class="gp">... </span> 
-<span class="go">e/log((6 + (-4/3)*e))</span>
-<span class="go">(3**3*5*e*catalan**2)/(2*7**2)</span>
-<span class="go">sqrt(((-13) + 1*e + 22*catalan))</span>
-<span class="go">log(((-6) + 24*e + 4*catalan)/e)</span>
-<span class="go">exp(catalan*((-1/5) + (8/15)*e))</span>
-<span class="go">catalan*(6 + (-6)*e + 15*catalan)</span>
-<span class="go">sqrt((5 + 26*e + (-3)*catalan))/e</span>
-<span class="go">e*sqrt(((-27) + 2*e + 25*catalan))</span>
-<span class="go">log(((-1) + (-11)*e + 59*catalan))</span>
-<span class="go">((3/20) + (21/20)*e + (3/20)*catalan)</span>
-<span class="gp">...</span>
-</pre></div>
-</div>
-<p>The numerical values are roughly as close to <span class="math">\(\pi\)</span> as permitted by the
-specified tolerance:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="mi">6</span><span class="o">-</span><span class="mi">4</span><span class="o">*</span><span class="n">e</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">3.14157719846001</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">135</span><span class="o">*</span><span class="n">e</span><span class="o">*</span><span class="n">catalan</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">98</span>
-<span class="go">3.14166950419369</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrt</span><span class="p">(</span><span class="n">e</span><span class="o">-</span><span class="mi">13</span><span class="o">+</span><span class="mi">22</span><span class="o">*</span><span class="n">catalan</span><span class="p">)</span>
-<span class="go">3.14158000062992</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="mi">24</span><span class="o">*</span><span class="n">e</span><span class="o">-</span><span class="mi">6</span><span class="o">+</span><span class="mi">4</span><span class="o">*</span><span class="n">catalan</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span>
-<span class="go">3.14158791577159</span>
-</pre></div>
-</div>
-<p><strong>Symbolic processing</strong></p>
-<p>The output formula can be evaluated as a Python expression.
-Note however that if fractions (like &#8216;2/3&#8217;) are present in
-the formula, Python&#8217;s <tt class="xref py py-func docutils literal"><span class="pre">eval()</span></tt> may erroneously perform
-integer division. Note also that the output is not necessarily
-in the algebraically simplest form:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">identify</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">&#39;(sqrt(8)/2)&#39;</span>
-</pre></div>
-</div>
-<p>As a solution to both problems, consider using SymPy&#8217;s
-<tt class="xref py py-func docutils literal"><span class="pre">sympify()</span></tt> to convert the formula into a symbolic expression.
-SymPy can be used to pretty-print or further simplify the formula
-symbolically:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sympify</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sympify</span><span class="p">(</span><span class="n">identify</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)))</span>
-<span class="go">sqrt(2)</span>
-</pre></div>
-</div>
-<p>Sometimes <a class="reference internal" href="#mpmath.identify" title="mpmath.identify"><tt class="xref py py-func docutils literal"><span class="pre">identify()</span></tt></a> can simplify an expression further than
-a symbolic algorithm:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">simplify</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="s">&#39;-1/(-3/2+(1/2)*sqrt(5))*sqrt(3/2-1/2*sqrt(5))&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span>
-<span class="go">1/sqrt(3/2 - sqrt(5)/2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span>
-<span class="go">2/sqrt(6 - 2*sqrt(5))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">30</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">sympify</span><span class="p">(</span><span class="n">identify</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">30</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span>
-<span class="go">1/2 + sqrt(5)/2</span>
-</pre></div>
-</div>
-<p>(In fact, this functionality is available directly in SymPy as the
-function <tt class="xref py py-func docutils literal"><span class="pre">nsimplify()</span></tt>, which is essentially a wrapper for
-<a class="reference internal" href="#mpmath.identify" title="mpmath.identify"><tt class="xref py py-func docutils literal"><span class="pre">identify()</span></tt></a>.)</p>
-<p><strong>Miscellaneous issues and limitations</strong></p>
-<p>The input <span class="math">\(x\)</span> must be a real number. All base constants must be
-positive real numbers and must not be rationals or rational linear
-combinations of each other.</p>
-<p>The worst-case computation time grows quickly with the number of
-base constants. Already with 3 or 4 base constants,
-<a class="reference internal" href="#mpmath.identify" title="mpmath.identify"><tt class="xref py py-func docutils literal"><span class="pre">identify()</span></tt></a> may require several seconds to finish. To search
-for relations among a large number of constants, you should
-consider using <a class="reference internal" href="#mpmath.pslq" title="mpmath.pslq"><tt class="xref py py-func docutils literal"><span class="pre">pslq()</span></tt></a> directly.</p>
-<p>The extended transformations are applied to x, not the constants
-separately. As a result, <tt class="docutils literal"><span class="pre">identify</span></tt> will for example be able to
-recognize <tt class="docutils literal"><span class="pre">exp(2*pi+3)</span></tt> with <tt class="docutils literal"><span class="pre">pi</span></tt> given as a base constant, but
-not <tt class="docutils literal"><span class="pre">2*exp(pi)+3</span></tt>. It will be able to recognize the latter if
-<tt class="docutils literal"><span class="pre">exp(pi)</span></tt> is given explicitly as a base constant.</p>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="algebraic-identification">
-<h2>Algebraic identification<a class="headerlink" href="#algebraic-identification" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="findpoly">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">findpoly()</span></tt><a class="headerlink" href="#findpoly" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.findpoly">
-<tt class="descclassname">mpmath.</tt><tt class="descname">findpoly</tt><big>(</big><em>ctx</em>, <em>x</em>, <em>n=1</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#mpmath.findpoly" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">findpoly(x,</span> <span class="pre">n)</span></tt> returns the coefficients of an integer
-polynomial <span class="math">\(P\)</span> of degree at most <span class="math">\(n\)</span> such that <span class="math">\(P(x) \approx 0\)</span>.
-If no polynomial having <span class="math">\(x\)</span> as a root can be found,
-<a class="reference internal" href="#mpmath.findpoly" title="mpmath.findpoly"><tt class="xref py py-func docutils literal"><span class="pre">findpoly()</span></tt></a> returns <tt class="docutils literal"><span class="pre">None</span></tt>.</p>
-<p><a class="reference internal" href="#mpmath.findpoly" title="mpmath.findpoly"><tt class="xref py py-func docutils literal"><span class="pre">findpoly()</span></tt></a> works by successively calling <a class="reference internal" href="#mpmath.pslq" title="mpmath.pslq"><tt class="xref py py-func docutils literal"><span class="pre">pslq()</span></tt></a> with
-the vectors <span class="math">\([1, x]\)</span>, <span class="math">\([1, x, x^2]\)</span>, <span class="math">\([1, x, x^2, x^3]\)</span>, ...,
-<span class="math">\([1, x, x^2, .., x^n]\)</span> as input. Keyword arguments given to
-<a class="reference internal" href="#mpmath.findpoly" title="mpmath.findpoly"><tt class="xref py py-func docutils literal"><span class="pre">findpoly()</span></tt></a> are forwarded verbatim to <a class="reference internal" href="#mpmath.pslq" title="mpmath.pslq"><tt class="xref py py-func docutils literal"><span class="pre">pslq()</span></tt></a>. In
-particular, you can specify a tolerance for <span class="math">\(P(x)\)</span> with <tt class="docutils literal"><span class="pre">tol</span></tt>
-and a maximum permitted coefficient size with <tt class="docutils literal"><span class="pre">maxcoeff</span></tt>.</p>
-<p>For large values of <span class="math">\(n\)</span>, it is recommended to run <a class="reference internal" href="#mpmath.findpoly" title="mpmath.findpoly"><tt class="xref py py-func docutils literal"><span class="pre">findpoly()</span></tt></a>
-at high precision; preferably 50 digits or more.</p>
-<p><strong>Examples</strong></p>
-<p>By default (degree <span class="math">\(n = 1\)</span>), <a class="reference internal" href="#mpmath.findpoly" title="mpmath.findpoly"><tt class="xref py py-func docutils literal"><span class="pre">findpoly()</span></tt></a> simply finds a linear
-polynomial with a rational root:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findpoly</span><span class="p">(</span><span class="mf">0.7</span><span class="p">)</span>
-<span class="go">[-10, 7]</span>
-</pre></div>
-</div>
-<p>The generated coefficient list is valid input to <tt class="docutils literal"><span class="pre">polyval</span></tt> and
-<tt class="docutils literal"><span class="pre">polyroots</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">polyval</span><span class="p">(</span><span class="n">findpoly</span><span class="p">(</span><span class="n">phi</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">phi</span><span class="p">),</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">-2.0e-16</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">polyroots</span><span class="p">(</span><span class="n">findpoly</span><span class="p">(</span><span class="n">phi</span><span class="p">,</span> <span class="mi">2</span><span class="p">)):</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="go">-0.618033988749895</span>
-<span class="go">1.61803398874989</span>
-</pre></div>
-</div>
-<p>Numbers of the form <span class="math">\(m + n \sqrt p\)</span> for integers <span class="math">\((m, n, p)\)</span> are
-solutions to quadratic equations. As we find here, <span class="math">\(1+\sqrt 2\)</span>
-is a root of the polynomial <span class="math">\(x^2 - 2x - 1\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">findpoly</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">[1, -2, -1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findroot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">2.4142135623731</span>
-</pre></div>
-</div>
-<p>Despite only containing square roots, the following number results
-in a polynomial of degree 4:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">findpoly</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">+</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">[1, 0, -10, 0, 1]</span>
-</pre></div>
-</div>
-<p>In fact, <span class="math">\(x^4 - 10x^2 + 1\)</span> is the <em>minimal polynomial</em> of
-<span class="math">\(r = \sqrt 2 + \sqrt 3\)</span>, meaning that a rational polynomial of
-lower degree having <span class="math">\(r\)</span> as a root does not exist. Given sufficient
-precision, <a class="reference internal" href="#mpmath.findpoly" title="mpmath.findpoly"><tt class="xref py py-func docutils literal"><span class="pre">findpoly()</span></tt></a> will usually find the correct
-minimal polynomial of a given algebraic number.</p>
-<p><strong>Non-algebraic numbers</strong></p>
-<p>If <a class="reference internal" href="#mpmath.findpoly" title="mpmath.findpoly"><tt class="xref py py-func docutils literal"><span class="pre">findpoly()</span></tt></a> fails to find a polynomial with given
-coefficient size and tolerance constraints, that means no such
-polynomial exists.</p>
-<p>We can verify that <span class="math">\(\pi\)</span> is not an algebraic number of degree 3 with
-coefficients less than 1000:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findpoly</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">&gt;&gt;&gt;</span>
-</pre></div>
-</div>
-<p>It is always possible to find an algebraic approximation of a number
-using one (or several) of the following methods:</p>
-<blockquote>
-<div><ol class="arabic simple">
-<li>Increasing the permitted degree</li>
-<li>Allowing larger coefficients</li>
-<li>Reducing the tolerance</li>
-</ol>
-</div></blockquote>
-<p>One example of each method is shown below:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findpoly</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">[95, -545, 863, -183, -298]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findpoly</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">maxcoeff</span><span class="o">=</span><span class="mi">10000</span><span class="p">)</span>
-<span class="go">[836, -1734, -2658, -457]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findpoly</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-7</span><span class="p">)</span>
-<span class="go">[-4, 22, -29, -2]</span>
-</pre></div>
-</div>
-<p>It is unknown whether Euler&#8217;s constant is transcendental (or even
-irrational). We can use <a class="reference internal" href="#mpmath.findpoly" title="mpmath.findpoly"><tt class="xref py py-func docutils literal"><span class="pre">findpoly()</span></tt></a> to check that if is
-an algebraic number, its minimal polynomial must have degree
-at least 7 and a coefficient of magnitude at least 1000000:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">200</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">findpoly</span><span class="p">(</span><span class="n">euler</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="n">maxcoeff</span><span class="o">=</span><span class="mi">10</span><span class="o">**</span><span class="mi">6</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-100</span><span class="p">,</span> <span class="n">maxsteps</span><span class="o">=</span><span class="mi">1000</span><span class="p">)</span>
-<span class="go">&gt;&gt;&gt;</span>
-</pre></div>
-</div>
-<p>Note that the high precision and strict tolerance is necessary
-for such high-degree runs, since otherwise unwanted low-accuracy
-approximations will be detected. It may also be necessary to set
-maxsteps high to prevent a premature exit (before the coefficient
-bound has been reached). Running with <tt class="docutils literal"><span class="pre">verbose=True</span></tt> to get an
-idea what is happening can be useful.</p>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="integer-relations-pslq">
-<h2>Integer relations (PSLQ)<a class="headerlink" href="#integer-relations-pslq" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="pslq">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">pslq()</span></tt><a class="headerlink" href="#pslq" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.pslq">
-<tt class="descclassname">mpmath.</tt><tt class="descname">pslq</tt><big>(</big><em>ctx</em>, <em>x</em>, <em>tol=None</em>, <em>maxcoeff=1000</em>, <em>maxsteps=100</em>, <em>verbose=False</em><big>)</big><a class="headerlink" href="#mpmath.pslq" title="Permalink to this definition">¶</a></dt>
-<dd><p>Given a vector of real numbers <span class="math">\(x = [x_0, x_1, ..., x_n]\)</span>, <tt class="docutils literal"><span class="pre">pslq(x)</span></tt>
-uses the PSLQ algorithm to find a list of integers
-<span class="math">\([c_0, c_1, ..., c_n]\)</span> such that</p>
-<div class="math">
-\[\begin{split}|c_1 x_1 + c_2 x_2 + ... + c_n x_n| &lt; \mathrm{tol}\end{split}\]</div>
-<p>and such that <span class="math">\(\max |c_k| &lt; \mathrm{maxcoeff}\)</span>. If no such vector
-exists, <a class="reference internal" href="#mpmath.pslq" title="mpmath.pslq"><tt class="xref py py-func docutils literal"><span class="pre">pslq()</span></tt></a> returns <tt class="docutils literal"><span class="pre">None</span></tt>. The tolerance defaults to
-3/4 of the working precision.</p>
-<p><strong>Examples</strong></p>
-<p>Find rational approximations for <span class="math">\(\pi\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pslq</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">pi</span><span class="p">],</span> <span class="n">tol</span><span class="o">=</span><span class="mf">0.01</span><span class="p">)</span>
-<span class="go">[22, 7]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pslq</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">pi</span><span class="p">],</span> <span class="n">tol</span><span class="o">=</span><span class="mf">0.001</span><span class="p">)</span>
-<span class="go">[355, 113]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpf</span><span class="p">(</span><span class="mi">22</span><span class="p">)</span><span class="o">/</span><span class="mi">7</span><span class="p">;</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">355</span><span class="p">)</span><span class="o">/</span><span class="mi">113</span><span class="p">;</span> <span class="o">+</span><span class="n">pi</span>
-<span class="go">3.14285714285714</span>
-<span class="go">3.14159292035398</span>
-<span class="go">3.14159265358979</span>
-</pre></div>
-</div>
-<p>Pi is not a rational number with denominator less than 1000:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pslq</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">pi</span><span class="p">])</span>
-<span class="go">&gt;&gt;&gt;</span>
-</pre></div>
-</div>
-<p>To within the standard precision, it can however be approximated
-by at least one rational number with denominator less than <span class="math">\(10^{12}\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">pslq</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">pi</span><span class="p">],</span> <span class="n">maxcoeff</span><span class="o">=</span><span class="mi">10</span><span class="o">**</span><span class="mi">12</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">p</span><span class="p">);</span> <span class="k">print</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-<span class="go">238410049439</span>
-<span class="go">75888275702</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpf</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">/</span><span class="n">q</span>
-<span class="go">3.14159265358979</span>
-</pre></div>
-</div>
-<p>The PSLQ algorithm can be applied to long vectors. For example,
-we can investigate the rational (in)dependence of integer square
-roots:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">30</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pslq</span><span class="p">([</span><span class="n">sqrt</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="o">+</span><span class="mi">1</span><span class="p">)])</span>
-<span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pslq</span><span class="p">([</span><span class="n">sqrt</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">6</span><span class="o">+</span><span class="mi">1</span><span class="p">)])</span>
-<span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pslq</span><span class="p">([</span><span class="n">sqrt</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">8</span><span class="o">+</span><span class="mi">1</span><span class="p">)])</span>
-<span class="go">[2, 0, 0, 0, 0, 0, -1]</span>
-</pre></div>
-</div>
-<p><strong>Machin formulas</strong></p>
-<p>A famous formula for <span class="math">\(\pi\)</span> is Machin&#8217;s,</p>
-<div class="math">
-\[\frac{\pi}{4} = 4 \operatorname{acot} 5 - \operatorname{acot} 239\]</div>
-<p>There are actually infinitely many formulas of this type. Two
-others are</p>
-<div class="math">
-\[\frac{\pi}{4} = \operatorname{acot} 1\]\[\frac{\pi}{4} = 12 \operatorname{acot} 49 + 32 \operatorname{acot} 57
-    + 5 \operatorname{acot} 239 + 12 \operatorname{acot} 110443\]</div>
-<p>We can easily verify the formulas using the PSLQ algorithm:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">30</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pslq</span><span class="p">([</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span> <span class="n">acot</span><span class="p">(</span><span class="mi">1</span><span class="p">)])</span>
-<span class="go">[1, -1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pslq</span><span class="p">([</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span> <span class="n">acot</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">acot</span><span class="p">(</span><span class="mi">239</span><span class="p">)])</span>
-<span class="go">[1, -4, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pslq</span><span class="p">([</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">,</span> <span class="n">acot</span><span class="p">(</span><span class="mi">49</span><span class="p">),</span> <span class="n">acot</span><span class="p">(</span><span class="mi">57</span><span class="p">),</span> <span class="n">acot</span><span class="p">(</span><span class="mi">239</span><span class="p">),</span> <span class="n">acot</span><span class="p">(</span><span class="mi">110443</span><span class="p">)])</span>
-<span class="go">[1, -12, -32, 5, -12]</span>
-</pre></div>
-</div>
-<p>We could try to generate a custom Machin-like formula by running
-the PSLQ algorithm with a few inverse cotangent values, for example
-acot(2), acot(3) ... acot(10). Unfortunately, there is a linear
-dependence among these values, resulting in only that dependence
-being detected, with a zero coefficient for <span class="math">\(\pi\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pslq</span><span class="p">([</span><span class="n">pi</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">acot</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">11</span><span class="p">)])</span>
-<span class="go">[0, 1, -1, 0, 0, 0, -1, 0, 0, 0]</span>
-</pre></div>
-</div>
-<p>We get better luck by removing linearly dependent terms:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pslq</span><span class="p">([</span><span class="n">pi</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">acot</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">11</span><span class="p">)</span> <span class="k">if</span> <span class="n">n</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">)])</span>
-<span class="go">[1, -8, 0, 0, 4, 0, 0, 0]</span>
-</pre></div>
-</div>
-<p>In other words, we found the following formula:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">8</span><span class="o">*</span><span class="n">acot</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="mi">4</span><span class="o">*</span><span class="n">acot</span><span class="p">(</span><span class="mi">7</span><span class="p">)</span>
-<span class="go">3.14159265358979323846264338328</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">+</span><span class="n">pi</span>
-<span class="go">3.14159265358979323846264338328</span>
-</pre></div>
-</div>
-<p><strong>Algorithm</strong></p>
-<p>This is a fairly direct translation to Python of the pseudocode given by
-David Bailey, &#8220;The PSLQ Integer Relation Algorithm&#8221;:
-<a class="reference external" href="http://www.cecm.sfu.ca/organics/papers/bailey/paper/html/node3.html">http://www.cecm.sfu.ca/organics/papers/bailey/paper/html/node3.html</a></p>
-<p>The present implementation uses fixed-point instead of floating-point
-arithmetic, since this is significantly (about 7x) faster.</p>
-</dd></dl>
-
-</div>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../index.html">
-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Number identification</a><ul>
-<li><a class="reference internal" href="#constant-recognition">Constant recognition</a><ul>
-<li><a class="reference internal" href="#identify"><tt class="docutils literal"><span class="pre">identify()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#algebraic-identification">Algebraic identification</a><ul>
-<li><a class="reference internal" href="#findpoly"><tt class="docutils literal"><span class="pre">findpoly()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#integer-relations-pslq">Integer relations (PSLQ)</a><ul>
-<li><a class="reference internal" href="#pslq"><tt class="docutils literal"><span class="pre">pslq()</span></tt></a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
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-                        title="previous chapter">Matrices</a></p>
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-                        title="next chapter">Precision and representation issues</a></p>
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-<h1>Welcome to mpmath&#8217;s documentation!<a class="headerlink" href="#welcome-to-mpmath-s-documentation" title="Permalink to this headline">¶</a></h1>
-<p>Mpmath is a Python library for arbitrary-precision floating-point arithmetic.
-For general information about mpmath, see the project website <a class="reference external" href="http://code.google.com/p/mpmath/">http://code.google.com/p/mpmath/</a></p>
-<p>These documentation pages include general information as well as docstring listing with extensive use of examples that can be run in the interactive Python interpreter. For quick access to the docstrings of individual functions, use the <a class="reference internal" href="../../genindex.html"><em>Index</em></a>, or type <tt class="docutils literal"><span class="pre">help(mpmath.function_name)</span></tt> in the Python interactive prompt.</p>
-<div class="section" id="introduction">
-<h2>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline">¶</a></h2>
-<div class="toctree-wrapper compound">
-<ul>
-<li class="toctree-l1"><a class="reference internal" href="../mpmath/setup.html">Setting up mpmath</a><ul>
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-<li class="toctree-l2"><a class="reference internal" href="../mpmath/setup.html#using-gmpy-optional">Using gmpy (optional)</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/setup.html#running-tests">Running tests</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/setup.html#compiling-the-documentation">Compiling the documentation</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/setup.html#mpmath-under-sympy">Mpmath under SymPy</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/setup.html#mpmath-under-sage">Mpmath under Sage</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../mpmath/basics.html">Basic usage</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/basics.html#number-types">Number types</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/basics.html#setting-the-precision">Setting the precision</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/basics.html#providing-correct-input">Providing correct input</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/basics.html#printing">Printing</a></li>
-</ul>
-</li>
-</ul>
-</div>
-</div>
-<div class="section" id="basic-features">
-<h2>Basic features<a class="headerlink" href="#basic-features" title="Permalink to this headline">¶</a></h2>
-<div class="toctree-wrapper compound">
-<ul>
-<li class="toctree-l1"><a class="reference internal" href="../mpmath/contexts.html">Contexts</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/contexts.html#common-interface">Common interface</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/contexts.html#arbitrary-precision-floating-point-mp">Arbitrary-precision floating-point (<tt class="docutils literal"><span class="pre">mp</span></tt>)</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/contexts.html#arbitrary-precision-interval-arithmetic-iv">Arbitrary-precision interval arithmetic (<tt class="docutils literal"><span class="pre">iv</span></tt>)</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/contexts.html#fast-low-precision-arithmetic-fp">Fast low-precision arithmetic (<tt class="docutils literal"><span class="pre">fp</span></tt>)</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../mpmath/general.html">Utility functions</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/general.html#conversion-and-printing">Conversion and printing</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/general.html#arithmetic-operations">Arithmetic operations</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/general.html#complex-components">Complex components</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/general.html#integer-and-fractional-parts">Integer and fractional parts</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/general.html#tolerances-and-approximate-comparisons">Tolerances and approximate comparisons</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/general.html#properties-of-numbers">Properties of numbers</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/general.html#number-generation">Number generation</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/general.html#precision-management">Precision management</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/general.html#performance-and-debugging">Performance and debugging</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../mpmath/plotting.html">Plotting</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/plotting.html#function-curve-plots">Function curve plots</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/plotting.html#complex-function-plots">Complex function plots</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/plotting.html#d-surface-plots">3D surface plots</a></li>
-</ul>
-</li>
-</ul>
-</div>
-</div>
-<div class="section" id="advanced-mathematics">
-<h2>Advanced mathematics<a class="headerlink" href="#advanced-mathematics" title="Permalink to this headline">¶</a></h2>
-<p>On top of its support for arbitrary-precision arithmetic, mpmath
-provides extensive support for transcendental functions, evaluation of sums, integrals, limits, roots, and so on.</p>
-<div class="toctree-wrapper compound">
-<ul>
-<li class="toctree-l1"><a class="reference internal" href="../mpmath/functions/index.html">Mathematical functions</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/functions/constants.html">Mathematical constants</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/functions/powers.html">Powers and logarithms</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/functions/trigonometric.html">Trigonometric functions</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/functions/hyperbolic.html">Hyperbolic functions</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/functions/gamma.html">Factorials and gamma functions</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/functions/expintegrals.html">Exponential integrals and error functions</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/functions/bessel.html">Bessel functions and related functions</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/functions/orthogonal.html">Orthogonal polynomials</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/functions/hypergeometric.html">Hypergeometric functions</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/functions/elliptic.html">Elliptic functions</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/functions/zeta.html">Zeta functions, L-series and polylogarithms</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/functions/numtheory.html">Number-theoretical, combinatorial and integer functions</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/functions/qfunctions.html">q-functions</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../mpmath/calculus/index.html">Numerical calculus</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/calculus/polynomials.html">Polynomials</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/calculus/optimization.html">Root-finding and optimization</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/calculus/sums_limits.html">Sums, products, limits and extrapolation</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/calculus/differentiation.html">Differentiation</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/calculus/integration.html">Numerical integration (quadrature)</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/calculus/odes.html">Ordinary differential equations</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/calculus/approximation.html">Function approximation</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../mpmath/matrices.html">Matrices</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/matrices.html#creating-matrices">Creating matrices</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/matrices.html#matrix-operations">Matrix operations</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/matrices.html#linear-algebra">Linear algebra</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/matrices.html#interval-and-double-precision-matrices">Interval and double-precision matrices</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/matrices.html#matrix-functions">Matrix functions</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../mpmath/identification.html">Number identification</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/identification.html#constant-recognition">Constant recognition</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/identification.html#algebraic-identification">Algebraic identification</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/identification.html#integer-relations-pslq">Integer relations (PSLQ)</a></li>
-</ul>
-</li>
-</ul>
-</div>
-</div>
-<div class="section" id="end-matter">
-<h2>End matter<a class="headerlink" href="#end-matter" title="Permalink to this headline">¶</a></h2>
-<div class="toctree-wrapper compound">
-<ul>
-<li class="toctree-l1"><a class="reference internal" href="../mpmath/technical.html">Precision and representation issues</a><ul>
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-<li class="toctree-l2"><a class="reference internal" href="../mpmath/technical.html#representation-of-numbers">Representation of numbers</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/technical.html#decimal-issues">Decimal issues</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/technical.html#correctness-guarantees">Correctness guarantees</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/technical.html#double-precision-emulation">Double precision emulation</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mpmath/technical.html#further-reading">Further reading</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../mpmath/references.html">References</a></li>
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-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Welcome to mpmath&#8217;s documentation!</a><ul>
-<li><a class="reference internal" href="#introduction">Introduction</a><ul>
-</ul>
-</li>
-<li><a class="reference internal" href="#basic-features">Basic features</a><ul>
-</ul>
-</li>
-<li><a class="reference internal" href="#advanced-mathematics">Advanced mathematics</a><ul>
-</ul>
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-  <div class="section" id="matrices">
-<h1>Matrices<a class="headerlink" href="#matrices" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="creating-matrices">
-<h2>Creating matrices<a class="headerlink" href="#creating-matrices" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="basic-methods">
-<h3>Basic methods<a class="headerlink" href="#basic-methods" title="Permalink to this headline">¶</a></h3>
-<p>Matrices in mpmath are implemented using dictionaries. Only non-zero values are
-stored, so it is cheap to represent sparse matrices.</p>
-<p>The most basic way to create one is to use the <tt class="docutils literal"><span class="pre">matrix</span></tt> class directly. You
-can create an empty matrix specifying the dimensions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;0.0&#39;, &#39;0.0&#39;],</span>
-<span class="go"> [&#39;0.0&#39;, &#39;0.0&#39;]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;0.0&#39;, &#39;0.0&#39;, &#39;0.0&#39;],</span>
-<span class="go"> [&#39;0.0&#39;, &#39;0.0&#39;, &#39;0.0&#39;]])</span>
-</pre></div>
-</div>
-<p>Calling <tt class="docutils literal"><span class="pre">matrix</span></tt> with one dimension will create a square matrix.</p>
-<p>To access the dimensions of a matrix, use the <tt class="docutils literal"><span class="pre">rows</span></tt> or <tt class="docutils literal"><span class="pre">cols</span></tt> keyword:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;0.0&#39;, &#39;0.0&#39;],</span>
-<span class="go"> [&#39;0.0&#39;, &#39;0.0&#39;],</span>
-<span class="go"> [&#39;0.0&#39;, &#39;0.0&#39;]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">rows</span>
-<span class="go">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">cols</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-<p>You can also change the dimension of an existing matrix. This will set the
-new elements to 0. If the new dimension is smaller than before, the
-concerning elements are discarded:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">rows</span> <span class="o">=</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;0.0&#39;, &#39;0.0&#39;],</span>
-<span class="go"> [&#39;0.0&#39;, &#39;0.0&#39;]])</span>
-</pre></div>
-</div>
-<p>Internally <tt class="docutils literal"><span class="pre">convert</span></tt> is applied every time an element is set. This is
-done using the syntax A[row,column], counting from 0:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">+</span> <span class="mi">1j</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">A</span>
-<span class="go">[0.0           0.0]</span>
-<span class="go">[0.0  (1.0 + 1.0j)]</span>
-</pre></div>
-</div>
-<p>A more comfortable way to create a matrix lets you use nested lists:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;1.0&#39;, &#39;2.0&#39;],</span>
-<span class="go"> [&#39;3.0&#39;, &#39;4.0&#39;]])</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="advanced-methods">
-<h3>Advanced methods<a class="headerlink" href="#advanced-methods" title="Permalink to this headline">¶</a></h3>
-<p>Convenient functions are available for creating various standard matrices:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">zeros</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;0.0&#39;, &#39;0.0&#39;],</span>
-<span class="go"> [&#39;0.0&#39;, &#39;0.0&#39;]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ones</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;1.0&#39;, &#39;1.0&#39;],</span>
-<span class="go"> [&#39;1.0&#39;, &#39;1.0&#39;]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diag</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span> <span class="c"># diagonal matrix</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;1.0&#39;, &#39;0.0&#39;, &#39;0.0&#39;],</span>
-<span class="go"> [&#39;0.0&#39;, &#39;2.0&#39;, &#39;0.0&#39;],</span>
-<span class="go"> [&#39;0.0&#39;, &#39;0.0&#39;, &#39;3.0&#39;]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eye</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="c"># identity matrix</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;1.0&#39;, &#39;0.0&#39;],</span>
-<span class="go"> [&#39;0.0&#39;, &#39;1.0&#39;]])</span>
-</pre></div>
-</div>
-<p>You can even create random matrices:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">randmatrix</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> 
-<span class="go">matrix(</span>
-<span class="go">[[&#39;0.53491598236191806&#39;, &#39;0.57195669543302752&#39;],</span>
-<span class="go"> [&#39;0.85589992269513615&#39;, &#39;0.82444367501382143&#39;]])</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="vectors">
-<h3>Vectors<a class="headerlink" href="#vectors" title="Permalink to this headline">¶</a></h3>
-<p>Vectors may also be represented by the <tt class="docutils literal"><span class="pre">matrix</span></tt> class (with rows = 1 or cols = 1).
-For vectors there are some things which make life easier. A column vector can
-be created using a flat list, a row vectors using an almost flat nested list:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">matrix</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;1.0&#39;],</span>
-<span class="go"> [&#39;2.0&#39;],</span>
-<span class="go"> [&#39;3.0&#39;]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;1.0&#39;, &#39;2.0&#39;, &#39;3.0&#39;]])</span>
-</pre></div>
-</div>
-<p>Optionally vectors can be accessed like lists, using only a single index:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-<span class="go">mpf(&#39;2.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">mpf(&#39;2.0&#39;)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="other">
-<h3>Other<a class="headerlink" href="#other" title="Permalink to this headline">¶</a></h3>
-<p>Like you probably expected, matrices can be printed:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">randmatrix</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> 
-<span class="go">[ 0.782963853573023  0.802057689719883  0.427895717335467]</span>
-<span class="go">[0.0541876859348597  0.708243266653103  0.615134039977379]</span>
-<span class="go">[ 0.856151514955773  0.544759264818486  0.686210904770947]</span>
-</pre></div>
-</div>
-<p>Use <tt class="docutils literal"><span class="pre">nstr</span></tt> or <tt class="docutils literal"><span class="pre">nprint</span></tt> to specify the number of digits to print:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">randmatrix</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="mi">3</span><span class="p">)</span> 
-<span class="go">[2.07e-1  1.66e-1  5.06e-1  1.89e-1  8.29e-1]</span>
-<span class="go">[6.62e-1  6.55e-1  4.47e-1  4.82e-1  2.06e-2]</span>
-<span class="go">[4.33e-1  7.75e-1  6.93e-2  2.86e-1  5.71e-1]</span>
-<span class="go">[1.01e-1  2.53e-1  6.13e-1  3.32e-1  2.59e-1]</span>
-<span class="go">[1.56e-1  7.27e-2  6.05e-1  6.67e-2  2.79e-1]</span>
-</pre></div>
-</div>
-<p>As matrices are mutable, you will need to copy them sometimes:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;0.0&#39;, &#39;0.0&#39;],</span>
-<span class="go"> [&#39;0.0&#39;, &#39;0.0&#39;]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;1.0&#39;, &#39;0.0&#39;],</span>
-<span class="go"> [&#39;0.0&#39;, &#39;0.0&#39;]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;0.0&#39;, &#39;0.0&#39;],</span>
-<span class="go"> [&#39;0.0&#39;, &#39;0.0&#39;]])</span>
-</pre></div>
-</div>
-<p>Finally, it is possible to convert a matrix to a nested list. This is very useful,
-as most Python libraries involving matrices or arrays (namely NumPy or SymPy)
-support this format:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">tolist</span><span class="p">()</span>
-<span class="go">[[mpf(&#39;1.0&#39;), mpf(&#39;0.0&#39;)], [mpf(&#39;0.0&#39;), mpf(&#39;0.0&#39;)]]</span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="section" id="matrix-operations">
-<h2>Matrix operations<a class="headerlink" href="#matrix-operations" title="Permalink to this headline">¶</a></h2>
-<p>You can add and substract matrices of compatible dimensions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">([[</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">9</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">+</span> <span class="n">B</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;-1.0&#39;, &#39;6.0&#39;],</span>
-<span class="go"> [&#39;8.0&#39;, &#39;13.0&#39;]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">-</span> <span class="n">B</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;3.0&#39;, &#39;-2.0&#39;],</span>
-<span class="go"> [&#39;-2.0&#39;, &#39;-5.0&#39;]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">+</span> <span class="n">ones</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> 
-<span class="gt">Traceback (most recent call last):</span>
-  File <span class="nb">&quot;&lt;stdin&gt;&quot;</span>, line <span class="m">1</span>, in <span class="n">&lt;module&gt;</span>
-  File <span class="nb">&quot;...&quot;</span>, line <span class="m">238</span>, in <span class="n">__add__</span>
-    <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s">&#39;incompatible dimensions for addition&#39;</span><span class="p">)</span>
-<span class="gr">ValueError</span>: <span class="n">incompatible dimensions for addition</span>
-</pre></div>
-</div>
-<p>It is possible to multiply or add matrices and scalars. In the latter case the
-operation will be done element-wise:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">*</span> <span class="mi">2</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;2.0&#39;, &#39;4.0&#39;],</span>
-<span class="go"> [&#39;6.0&#39;, &#39;8.0&#39;]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">/</span> <span class="mi">4</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;0.25&#39;, &#39;0.5&#39;],</span>
-<span class="go"> [&#39;0.75&#39;, &#39;1.0&#39;]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">-</span> <span class="mi">1</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;0.0&#39;, &#39;1.0&#39;],</span>
-<span class="go"> [&#39;2.0&#39;, &#39;3.0&#39;]])</span>
-</pre></div>
-</div>
-<p>Of course you can perform matrix multiplication, if the dimensions are
-compatible:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">*</span> <span class="n">B</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;8.0&#39;, &#39;22.0&#39;],</span>
-<span class="go"> [&#39;14.0&#39;, &#39;48.0&#39;]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span> <span class="o">*</span> <span class="n">matrix</span><span class="p">([[</span><span class="o">-</span><span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">7</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]])</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;2.0&#39;]])</span>
-</pre></div>
-</div>
-<p>You can raise powers of square matrices:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;7.0&#39;, &#39;10.0&#39;],</span>
-<span class="go"> [&#39;15.0&#39;, &#39;22.0&#39;]])</span>
-</pre></div>
-</div>
-<p>Negative powers will calculate the inverse:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">**-</span><span class="mi">1</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;-2.0&#39;, &#39;1.0&#39;],</span>
-<span class="go"> [&#39;1.5&#39;, &#39;-0.5&#39;]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">A</span> <span class="o">*</span> <span class="n">A</span><span class="o">**-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">[      1.0  1.08e-19]</span>
-<span class="go">[-2.17e-19       1.0]</span>
-</pre></div>
-</div>
-<p>Matrix transposition is straightforward:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">ones</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;1.0&#39;, &#39;1.0&#39;, &#39;1.0&#39;],</span>
-<span class="go"> [&#39;1.0&#39;, &#39;1.0&#39;, &#39;1.0&#39;]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">T</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;1.0&#39;, &#39;1.0&#39;],</span>
-<span class="go"> [&#39;1.0&#39;, &#39;1.0&#39;],</span>
-<span class="go"> [&#39;1.0&#39;, &#39;1.0&#39;]])</span>
-</pre></div>
-</div>
-<div class="section" id="norms">
-<h3>Norms<a class="headerlink" href="#norms" title="Permalink to this headline">¶</a></h3>
-<p>Sometimes you need to know how &#8220;large&#8221; a matrix or vector is. Due to their
-multidimensional nature it&#8217;s not possible to compare them, but there are
-several functions to map a matrix or a vector to a positive real number, the
-so called norms.</p>
-<dl class="function">
-<dt id="mpmath.norm">
-<tt class="descclassname">mpmath.</tt><tt class="descname">norm</tt><big>(</big><em>ctx</em>, <em>x</em>, <em>p=2</em><big>)</big><a class="headerlink" href="#mpmath.norm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the entrywise <span class="math">\(p\)</span>-norm of an iterable <em>x</em>, i.e. the vector norm
-<span class="math">\(\left(\sum_k |x_k|^p\right)^{1/p}\)</span>, for any given <span class="math">\(1 \le p \le \infty\)</span>.</p>
-<p>Special cases:</p>
-<p>If <em>x</em> is not iterable, this just returns <tt class="docutils literal"><span class="pre">absmax(x)</span></tt>.</p>
-<p><tt class="docutils literal"><span class="pre">p=1</span></tt> gives the sum of absolute values.</p>
-<p><tt class="docutils literal"><span class="pre">p=2</span></tt> is the standard Euclidean vector norm.</p>
-<p><tt class="docutils literal"><span class="pre">p=inf</span></tt> gives the magnitude of the largest element.</p>
-<p>For <em>x</em> a matrix, <tt class="docutils literal"><span class="pre">p=2</span></tt> is the Frobenius norm.
-For operator matrix norms, use <a class="reference internal" href="#mpmath.mnorm" title="mpmath.mnorm"><tt class="xref py py-func docutils literal"><span class="pre">mnorm()</span></tt></a> instead.</p>
-<p>You can use the string &#8216;inf&#8217; as well as float(&#8216;inf&#8217;) or mpf(&#8216;inf&#8217;)
-to specify the infinity norm.</p>
-<p><strong>Examples</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">([</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">100</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">norm</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">mpf(&#39;112.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">norm</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">mpf(&#39;100.5186549850325&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">norm</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">inf</span><span class="p">)</span>
-<span class="go">mpf(&#39;100.0&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="mpmath.mnorm">
-<tt class="descclassname">mpmath.</tt><tt class="descname">mnorm</tt><big>(</big><em>ctx</em>, <em>A</em>, <em>p=1</em><big>)</big><a class="headerlink" href="#mpmath.mnorm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the matrix (operator) <span class="math">\(p\)</span>-norm of A. Currently <tt class="docutils literal"><span class="pre">p=1</span></tt> and <tt class="docutils literal"><span class="pre">p=inf</span></tt>
-are supported:</p>
-<p><tt class="docutils literal"><span class="pre">p=1</span></tt> gives the 1-norm (maximal column sum)</p>
-<p><tt class="docutils literal"><span class="pre">p=inf</span></tt> gives the <span class="math">\(\infty\)</span>-norm (maximal row sum).
-You can use the string &#8216;inf&#8217; as well as float(&#8216;inf&#8217;) or mpf(&#8216;inf&#8217;)</p>
-<p><tt class="docutils literal"><span class="pre">p=2</span></tt> (not implemented) for a square matrix is the usual spectral
-matrix norm, i.e. the largest singular value.</p>
-<p><tt class="docutils literal"><span class="pre">p='f'</span></tt> (or &#8216;F&#8217;, &#8216;fro&#8217;, &#8216;Frobenius, &#8216;frobenius&#8217;) gives the
-Frobenius norm, which is the elementwise 2-norm. The Frobenius norm is an
-approximation of the spectral norm and satisfies</p>
-<div class="math">
-\[\frac{1}{\sqrt{\mathrm{rank}(A)}} \|A\|_F \le \|A\|_2 \le \|A\|_F\]</div>
-<p>The Frobenius norm lacks some mathematical properties that might
-be expected of a norm.</p>
-<p>For general elementwise <span class="math">\(p\)</span>-norms, use <a class="reference internal" href="#mpmath.norm" title="mpmath.norm"><tt class="xref py py-func docutils literal"><span class="pre">norm()</span></tt></a> instead.</p>
-<p><strong>Examples</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1000</span><span class="p">],</span> <span class="p">[</span><span class="mi">100</span><span class="p">,</span> <span class="mi">50</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mnorm</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">mpf(&#39;1050.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mnorm</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">inf</span><span class="p">)</span>
-<span class="go">mpf(&#39;1001.0&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mnorm</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="s">&#39;F&#39;</span><span class="p">)</span>
-<span class="go">mpf(&#39;1006.2310867787777&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="linear-algebra">
-<h2>Linear algebra<a class="headerlink" href="#linear-algebra" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="decompositions">
-<h3>Decompositions<a class="headerlink" href="#decompositions" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="mpmath.cholesky">
-<tt class="descclassname">mpmath.</tt><tt class="descname">cholesky</tt><big>(</big><em>ctx</em>, <em>A</em>, <em>tol=None</em><big>)</big><a class="headerlink" href="#mpmath.cholesky" title="Permalink to this definition">¶</a></dt>
-<dd><p>Cholesky decomposition of a symmetric positive-definite matrix <span class="math">\(A\)</span>.
-Returns a lower triangular matrix <span class="math">\(L\)</span> such that <span class="math">\(A = L \times L^T\)</span>.
-More generally, for a complex Hermitian positive-definite matrix,
-a Cholesky decomposition satisfying <span class="math">\(A = L \times L^H\)</span> is returned.</p>
-<p>The Cholesky decomposition can be used to solve linear equation
-systems twice as efficiently as LU decomposition, or to
-test whether <span class="math">\(A\)</span> is positive-definite.</p>
-<p>The optional parameter <tt class="docutils literal"><span class="pre">tol</span></tt> determines the tolerance for
-verifying positive-definiteness.</p>
-<p><strong>Examples</strong></p>
-<p>Cholesky decomposition of a positive-definite symmetric matrix:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">25</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">+</span> <span class="n">hilbert</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-<span class="go">[     2.0      0.5  0.333333]</span>
-<span class="go">[     0.5  1.33333      0.25]</span>
-<span class="go">[0.333333     0.25       1.2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span> <span class="o">=</span> <span class="n">cholesky</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">L</span><span class="p">)</span>
-<span class="go">[ 1.41421      0.0      0.0]</span>
-<span class="go">[0.353553  1.09924      0.0]</span>
-<span class="go">[0.235702  0.15162  1.05899]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">A</span> <span class="o">-</span> <span class="n">L</span><span class="o">*</span><span class="n">L</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
-<span class="go">[0.0  0.0  0.0]</span>
-<span class="go">[0.0  0.0  0.0]</span>
-<span class="go">[0.0  0.0  0.0]</span>
-</pre></div>
-</div>
-<p>Cholesky decomposition of a Hermitian matrix:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">+</span> <span class="n">matrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span><span class="mf">0.25j</span><span class="p">,</span><span class="o">-</span><span class="mf">0.5j</span><span class="p">],[</span><span class="o">-</span><span class="mf">0.25j</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">],[</span><span class="mf">0.5j</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span> <span class="o">=</span> <span class="n">cholesky</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">L</span><span class="p">)</span>
-<span class="go">[          1.0                0.0                0.0]</span>
-<span class="go">[(0.0 - 0.25j)  (0.968246 + 0.0j)                0.0]</span>
-<span class="go">[ (0.0 + 0.5j)  (0.129099 + 0.0j)  (0.856349 + 0.0j)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">A</span> <span class="o">-</span> <span class="n">L</span><span class="o">*</span><span class="n">L</span><span class="o">.</span><span class="n">H</span><span class="p">)</span>
-<span class="go">[0.0  0.0  0.0]</span>
-<span class="go">[0.0  0.0  0.0]</span>
-<span class="go">[0.0  0.0  0.0]</span>
-</pre></div>
-</div>
-<p>Attempted Cholesky decomposition of a matrix that is not positive
-definite:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="o">-</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">+</span> <span class="n">hilbert</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span> <span class="o">=</span> <span class="n">cholesky</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-  <span class="c">...</span>
-<span class="gr">ValueError</span>: <span class="n">matrix is not positive-definite</span>
-</pre></div>
-</div>
-<p><strong>References</strong></p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../mpmath/references.html#wikipedia">[Wikipedia]</a> <a class="reference external" href="http://en.wikipedia.org/wiki/Cholesky_decomposition">http://en.wikipedia.org/wiki/Cholesky_decomposition</a></li>
-</ol>
-</dd></dl>
-
-</div>
-<div class="section" id="linear-equations">
-<h3>Linear equations<a class="headerlink" href="#linear-equations" title="Permalink to this headline">¶</a></h3>
-<p>Basic linear algebra is implemented; you can for example solve the linear
-equation system:</p>
-<div class="highlight-python"><pre>  x + 2*y = -10
-3*x + 4*y =  10</pre>
-</div>
-<p>using <tt class="docutils literal"><span class="pre">lu_solve</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">([</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">lu_solve</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;30.0&#39;],</span>
-<span class="go"> [&#39;-20.0&#39;]])</span>
-</pre></div>
-</div>
-<p>If you don&#8217;t trust the result, use <tt class="docutils literal"><span class="pre">residual</span></tt> to calculate the residual ||A*x-b||:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">residual</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;3.46944695195361e-18&#39;],</span>
-<span class="go"> [&#39;3.46944695195361e-18&#39;]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">str</span><span class="p">(</span><span class="n">eps</span><span class="p">)</span>
-<span class="go">&#39;2.22044604925031e-16&#39;</span>
-</pre></div>
-</div>
-<p>As you can see, the solution is quite accurate. The error is caused by the
-inaccuracy of the internal floating point arithmetic. Though, it&#8217;s even smaller
-than the current machine epsilon, which basically means you can trust the
-result.</p>
-<p>If you need more speed, use NumPy, or use <tt class="docutils literal"><span class="pre">fp</span></tt> instead <tt class="docutils literal"><span class="pre">mp</span></tt> matrices
-and methods:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">fp</span><span class="o">.</span><span class="n">matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">fp</span><span class="o">.</span><span class="n">matrix</span><span class="p">([</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fp</span><span class="o">.</span><span class="n">lu_solve</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-<span class="go">matrix(</span>
-<span class="go">[[&#39;30.0&#39;],</span>
-<span class="go"> [&#39;-20.0&#39;]])</span>
-</pre></div>
-</div>
-<p><tt class="docutils literal"><span class="pre">lu_solve</span></tt> accepts overdetermined systems. It is usually not possible to solve
-such systems, so the residual is minimized instead. Internally this is done
-using Cholesky decomposition to compute a least squares approximation. This means
-that that <tt class="docutils literal"><span class="pre">lu_solve</span></tt> will square the errors. If you can&#8217;t afford this, use
-<tt class="docutils literal"><span class="pre">qr_solve</span></tt> instead. It is twice as slow but more accurate, and it calculates
-the residual automatically.</p>
-</div>
-<div class="section" id="matrix-factorization">
-<h3>Matrix factorization<a class="headerlink" href="#matrix-factorization" title="Permalink to this headline">¶</a></h3>
-<p>The function <tt class="docutils literal"><span class="pre">lu</span></tt> computes an explicit LU factorization of a matrix:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="p">,</span> <span class="n">L</span><span class="p">,</span> <span class="n">U</span> <span class="o">=</span> <span class="n">lu</span><span class="p">(</span><span class="n">matrix</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">],[</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">],[</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">9</span><span class="p">]]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">P</span>
-<span class="go">[0.0  0.0  1.0]</span>
-<span class="go">[1.0  0.0  0.0]</span>
-<span class="go">[0.0  1.0  0.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">L</span>
-<span class="go">[              1.0                0.0  0.0]</span>
-<span class="go">[              0.0                1.0  0.0]</span>
-<span class="go">[0.571428571428571  0.214285714285714  1.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">U</span>
-<span class="go">[7.0  8.0                9.0]</span>
-<span class="go">[0.0  2.0                3.0]</span>
-<span class="go">[0.0  0.0  0.214285714285714]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">P</span><span class="o">.</span><span class="n">T</span><span class="o">*</span><span class="n">L</span><span class="o">*</span><span class="n">U</span>
-<span class="go">[0.0  2.0  3.0]</span>
-<span class="go">[4.0  5.0  6.0]</span>
-<span class="go">[7.0  8.0  9.0]</span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="section" id="interval-and-double-precision-matrices">
-<h2>Interval and double-precision matrices<a class="headerlink" href="#interval-and-double-precision-matrices" title="Permalink to this headline">¶</a></h2>
-<p>The <tt class="docutils literal"><span class="pre">iv.matrix</span></tt> and <tt class="docutils literal"><span class="pre">fp.matrix</span></tt> classes convert inputs
-to intervals and Python floating-point numbers respectively.</p>
-<p>Interval matrices can be used to perform linear algebra operations
-with rigorous error tracking:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">iv</span><span class="o">.</span><span class="n">matrix</span><span class="p">([[</span><span class="s">&#39;0.1&#39;</span><span class="p">,</span><span class="s">&#39;0.3&#39;</span><span class="p">,</span><span class="s">&#39;1.0&#39;</span><span class="p">],</span>
-<span class="gp">... </span>               <span class="p">[</span><span class="s">&#39;7.1&#39;</span><span class="p">,</span><span class="s">&#39;5.5&#39;</span><span class="p">,</span><span class="s">&#39;4.8&#39;</span><span class="p">],</span>
-<span class="gp">... </span>               <span class="p">[</span><span class="s">&#39;3.2&#39;</span><span class="p">,</span><span class="s">&#39;4.4&#39;</span><span class="p">,</span><span class="s">&#39;5.6&#39;</span><span class="p">]])</span>
-<span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">iv</span><span class="o">.</span><span class="n">matrix</span><span class="p">([</span><span class="s">&#39;4&#39;</span><span class="p">,</span><span class="s">&#39;0.6&#39;</span><span class="p">,</span><span class="s">&#39;0.5&#39;</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">iv</span><span class="o">.</span><span class="n">lu_solve</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">c</span>
-<span class="go">[  [5.2582327113062393041, 5.2582327113062749951]]</span>
-<span class="go">[[-13.155049396267856583, -13.155049396267821167]]</span>
-<span class="go">[  [7.4206915477497212555, 7.4206915477497310922]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">a</span><span class="o">*</span><span class="n">c</span>
-<span class="go">[  [3.9999999999999866773, 4.0000000000000133227]]</span>
-<span class="go">[[0.59999999999972430942, 0.60000000000027142733]]</span>
-<span class="go">[[0.49999999999982236432, 0.50000000000018474111]]</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="matrix-functions">
-<h2>Matrix functions<a class="headerlink" href="#matrix-functions" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="mpmath.expm">
-<tt class="descclassname">mpmath.</tt><tt class="descname">expm</tt><big>(</big><em>ctx</em>, <em>A</em>, <em>method='taylor'</em><big>)</big><a class="headerlink" href="#mpmath.expm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the matrix exponential of a square matrix <span class="math">\(A\)</span>, which is defined
-by the power series</p>
-<div class="math">
-\[\exp(A) = I + A + \frac{A^2}{2!} + \frac{A^3}{3!} + \ldots\]</div>
-<p>With method=&#8217;taylor&#8217;, the matrix exponential is computed
-using the Taylor series. With method=&#8217;pade&#8217;, Pade approximants
-are used instead.</p>
-<p><strong>Examples</strong></p>
-<p>Basic examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expm</span><span class="p">(</span><span class="n">zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">[1.0  0.0  0.0]</span>
-<span class="go">[0.0  1.0  0.0]</span>
-<span class="go">[0.0  0.0  1.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expm</span><span class="p">(</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">[2.71828182845905               0.0               0.0]</span>
-<span class="go">[             0.0  2.71828182845905               0.0]</span>
-<span class="go">[             0.0               0.0  2.71828182845905]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expm</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]])</span>
-<span class="go">[ 3.86814500615414  2.26812870852145  0.841130841230196]</span>
-<span class="go">[ 2.26812870852145  2.44114713886289   1.42699786729125]</span>
-<span class="go">[0.841130841230196  1.42699786729125    1.6000162976327]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expm</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]],</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;pade&#39;</span><span class="p">)</span>
-<span class="go">[ 3.86814500615414  2.26812870852145  0.841130841230196]</span>
-<span class="go">[ 2.26812870852145  2.44114713886289   1.42699786729125]</span>
-<span class="go">[0.841130841230196  1.42699786729125    1.6000162976327]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expm</span><span class="p">([[</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span><span class="mi">1</span><span class="p">]])</span>
-<span class="go">[(1.46869393991589 + 2.28735528717884j)                        0.0]</span>
-<span class="go">[  (1.03776739863568 + 3.536943175722j)  (2.71828182845905 + 0.0j)]</span>
-</pre></div>
-</div>
-<p>Matrices with large entries are allowed:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expm</span><span class="p">(</span><span class="n">matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">],[</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">]])</span><span class="o">**</span><span class="mi">25</span><span class="p">)</span>
-<span class="go">[5.65024064048415e+2050488462815550  9.14228140091932e+2050488462815550]</span>
-<span class="go">[9.14228140091932e+2050488462815550  1.47925220414035e+2050488462815551]</span>
-</pre></div>
-</div>
-<p>The identity <span class="math">\(\exp(A+B) = \exp(A) \exp(B)\)</span> does not hold for
-noncommuting matrices:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">hilbert</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">A</span> <span class="o">+</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">mnorm</span><span class="p">(</span><span class="n">A</span><span class="o">*</span><span class="n">B</span> <span class="o">-</span> <span class="n">B</span><span class="o">*</span><span class="n">A</span><span class="p">))</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">mnorm</span><span class="p">(</span><span class="n">expm</span><span class="p">(</span><span class="n">A</span><span class="o">+</span><span class="n">B</span><span class="p">)</span> <span class="o">-</span> <span class="n">expm</span><span class="p">(</span><span class="n">A</span><span class="p">)</span><span class="o">*</span><span class="n">expm</span><span class="p">(</span><span class="n">B</span><span class="p">)))</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">A</span> <span class="o">+</span> <span class="n">ones</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mnorm</span><span class="p">(</span><span class="n">A</span><span class="o">*</span><span class="n">B</span> <span class="o">-</span> <span class="n">B</span><span class="o">*</span><span class="n">A</span><span class="p">)</span>
-<span class="go">1.8</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mnorm</span><span class="p">(</span><span class="n">expm</span><span class="p">(</span><span class="n">A</span><span class="o">+</span><span class="n">B</span><span class="p">)</span> <span class="o">-</span> <span class="n">expm</span><span class="p">(</span><span class="n">A</span><span class="p">)</span><span class="o">*</span><span class="n">expm</span><span class="p">(</span><span class="n">B</span><span class="p">))</span>
-<span class="go">42.0927851137247</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="mpmath.cosm">
-<tt class="descclassname">mpmath.</tt><tt class="descname">cosm</tt><big>(</big><em>ctx</em>, <em>A</em><big>)</big><a class="headerlink" href="#mpmath.cosm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the cosine of a square matrix <span class="math">\(A\)</span>, defined in analogy
-with the matrix exponential.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cosm</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">[0.54030230586814               0.0               0.0]</span>
-<span class="go">[             0.0  0.54030230586814               0.0]</span>
-<span class="go">[             0.0               0.0  0.54030230586814]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">hilbert</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cosm</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">[ 0.424403834569555  -0.316643413047167  -0.221474945949293]</span>
-<span class="go">[-0.316643413047167   0.820646708837824  -0.127183694770039]</span>
-<span class="go">[-0.221474945949293  -0.127183694770039   0.909236687217541]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">([[</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="n">j</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cosm</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">[(0.833730025131149 - 0.988897705762865j)  (1.07485840848393 - 0.17192140544213j)]</span>
-<span class="go">[                                     0.0               (1.54308063481524 + 0.0j)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="mpmath.sinm">
-<tt class="descclassname">mpmath.</tt><tt class="descname">sinm</tt><big>(</big><em>ctx</em>, <em>A</em><big>)</big><a class="headerlink" href="#mpmath.sinm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives the sine of a square matrix <span class="math">\(A\)</span>, defined in analogy
-with the matrix exponential.</p>
-<p>Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinm</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">[0.841470984807897                0.0                0.0]</span>
-<span class="go">[              0.0  0.841470984807897                0.0]</span>
-<span class="go">[              0.0                0.0  0.841470984807897]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">hilbert</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinm</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">[0.711608512150994  0.339783913247439  0.220742837314741]</span>
-<span class="go">[0.339783913247439  0.244113865695532  0.187231271174372]</span>
-<span class="go">[0.220742837314741  0.187231271174372  0.155816730769635]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">([[</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="n">j</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinm</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">[(1.29845758141598 + 0.634963914784736j)  (-1.96751511930922 + 0.314700021761367j)]</span>
-<span class="go">[                                    0.0                  (0.0 - 1.1752011936438j)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="mpmath.sqrtm">
-<tt class="descclassname">mpmath.</tt><tt class="descname">sqrtm</tt><big>(</big><em>ctx</em>, <em>A</em>, <em>_may_rotate=2</em><big>)</big><a class="headerlink" href="#mpmath.sqrtm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes a square root of the square matrix <span class="math">\(A\)</span>, i.e. returns
-a matrix <span class="math">\(B = A^{1/2}\)</span> such that <span class="math">\(B^2 = A\)</span>. The square root
-of a matrix, if it exists, is not unique.</p>
-<p><strong>Examples</strong></p>
-<p>Square roots of some simple matrices:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrtm</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]])</span>
-<span class="go">[1.0  0.0]</span>
-<span class="go">[0.0  1.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrtm</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]])</span>
-<span class="go">[0.0  0.0]</span>
-<span class="go">[0.0  0.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrtm</span><span class="p">([[</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]])</span>
-<span class="go">[1.4142135623731  0.0]</span>
-<span class="go">[            0.0  1.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrtm</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]])</span>
-<span class="go">[ (0.920442065259926 - 0.21728689675164j)  (0.568864481005783 + 0.351577584254143j)]</span>
-<span class="go">[(0.568864481005783 + 0.351577584254143j)  (0.351577584254143 - 0.568864481005783j)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrtm</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]])</span>
-<span class="go">[1.0  0.0]</span>
-<span class="go">[0.0  1.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrtm</span><span class="p">([[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]])</span>
-<span class="go">[(0.0 - 1.0j)           0.0]</span>
-<span class="go">[         0.0  (1.0 + 0.0j)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrtm</span><span class="p">([[</span><span class="n">j</span><span class="p">,</span><span class="mi">0</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="n">j</span><span class="p">]])</span>
-<span class="go">[(0.707106781186547 + 0.707106781186547j)                                       0.0]</span>
-<span class="go">[                                     0.0  (0.707106781186547 + 0.707106781186547j)]</span>
-</pre></div>
-</div>
-<p>A square root of a rotation matrix, giving the corresponding
-half-angle rotation matrix:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">t1</span> <span class="o">=</span> <span class="mf">0.75</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t2</span> <span class="o">=</span> <span class="n">t1</span> <span class="o">*</span> <span class="mf">0.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A1</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">([[</span><span class="n">cos</span><span class="p">(</span><span class="n">t1</span><span class="p">),</span> <span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">t1</span><span class="p">)],</span> <span class="p">[</span><span class="n">sin</span><span class="p">(</span><span class="n">t1</span><span class="p">),</span> <span class="n">cos</span><span class="p">(</span><span class="n">t1</span><span class="p">)]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A2</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">([[</span><span class="n">cos</span><span class="p">(</span><span class="n">t2</span><span class="p">),</span> <span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">t2</span><span class="p">)],</span> <span class="p">[</span><span class="n">sin</span><span class="p">(</span><span class="n">t2</span><span class="p">),</span> <span class="n">cos</span><span class="p">(</span><span class="n">t2</span><span class="p">)]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrtm</span><span class="p">(</span><span class="n">A1</span><span class="p">)</span>
-<span class="go">[0.930507621912314  -0.366272529086048]</span>
-<span class="go">[0.366272529086048   0.930507621912314]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A2</span>
-<span class="go">[0.930507621912314  -0.366272529086048]</span>
-<span class="go">[0.366272529086048   0.930507621912314]</span>
-</pre></div>
-</div>
-<p>The identity <span class="math">\((A^2)^{1/2} = A\)</span> does not necessarily hold:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">([[</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">],[</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">9</span><span class="p">],[</span><span class="mi">10</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">11</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrtm</span><span class="p">(</span><span class="n">A</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">[ 4.0  1.0   4.0]</span>
-<span class="go">[ 7.0  8.0   9.0]</span>
-<span class="go">[10.0  2.0  11.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrtm</span><span class="p">(</span><span class="n">A</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">[ 4.0  1.0   4.0]</span>
-<span class="go">[ 7.0  8.0   9.0]</span>
-<span class="go">[10.0  2.0  11.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">([[</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">],[</span><span class="mi">7</span><span class="p">,</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span><span class="mi">9</span><span class="p">],[</span><span class="mi">10</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">11</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrtm</span><span class="p">(</span><span class="n">A</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">[  7.43715112194995  -0.324127569985474   1.8481718827526]</span>
-<span class="go">[-0.251549715716942    9.32699765900402  2.48221180985147]</span>
-<span class="go">[  4.11609388833616   0.775751877098258   13.017955697342]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">sqrtm</span><span class="p">(</span><span class="n">A</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">[-4.0   1.0   4.0]</span>
-<span class="go">[ 7.0  -8.0   9.0]</span>
-<span class="go">[10.0   2.0  11.0]</span>
-</pre></div>
-</div>
-<p>For some matrices, a square root does not exist:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sqrtm</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]])</span>
-<span class="gt">Traceback (most recent call last):</span>
-  <span class="c">...</span>
-<span class="gr">ZeroDivisionError</span>: <span class="n">matrix is numerically singular</span>
-</pre></div>
-</div>
-<p>Two examples from the documentation for Matlab&#8217;s <tt class="docutils literal"><span class="pre">sqrtm</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrtm</span><span class="p">([[</span><span class="mi">7</span><span class="p">,</span><span class="mi">10</span><span class="p">],[</span><span class="mi">15</span><span class="p">,</span><span class="mi">22</span><span class="p">]])</span>
-<span class="go">[1.56669890360128  1.74077655955698]</span>
-<span class="go">[2.61116483933547  4.17786374293675]</span>
-<span class="go">&gt;&gt;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">(</span>\
-<span class="gp">... </span>  <span class="p">[[</span><span class="mi">5</span><span class="p">,</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span>
-<span class="gp">... </span>  <span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span>
-<span class="gp">... </span>  <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
-<span class="gp">... </span>  <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="o">-</span><span class="mi">4</span><span class="p">],</span>
-<span class="gp">... </span>  <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">(</span>\
-<span class="gp">... </span>  <span class="p">[[</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">0</span><span class="p">],</span>
-<span class="gp">... </span>  <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">0</span><span class="p">],</span>
-<span class="gp">... </span>  <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span>
-<span class="gp">... </span>  <span class="p">[</span><span class="o">-</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span>
-<span class="gp">... </span>  <span class="p">[</span><span class="o">-</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mnorm</span><span class="p">(</span><span class="n">sqrtm</span><span class="p">(</span><span class="n">X</span><span class="p">)</span> <span class="o">-</span> <span class="n">Y</span><span class="p">)</span>
-<span class="go">4.53155328326114e-19</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="mpmath.logm">
-<tt class="descclassname">mpmath.</tt><tt class="descname">logm</tt><big>(</big><em>ctx</em>, <em>A</em><big>)</big><a class="headerlink" href="#mpmath.logm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes a logarithm of the square matrix <span class="math">\(A\)</span>, i.e. returns
-a matrix <span class="math">\(B = \log(A)\)</span> such that <span class="math">\(\exp(B) = A\)</span>. The logarithm
-of a matrix, if it exists, is not unique.</p>
-<p><strong>Examples</strong></p>
-<p>Logarithms of some simple matrices:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">logm</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">[0.0  0.0  0.0]</span>
-<span class="go">[0.0  0.0  0.0]</span>
-<span class="go">[0.0  0.0  0.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">logm</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">X</span><span class="p">)</span>
-<span class="go">[0.693147180559945                0.0                0.0]</span>
-<span class="go">[              0.0  0.693147180559945                0.0]</span>
-<span class="go">[              0.0                0.0  0.693147180559945]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">logm</span><span class="p">(</span><span class="n">expm</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
-<span class="go">[1.0  0.0  0.0]</span>
-<span class="go">[0.0  1.0  0.0]</span>
-<span class="go">[0.0  0.0  1.0]</span>
-</pre></div>
-</div>
-<p>A logarithm of a complex matrix:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">([[</span><span class="mi">2</span><span class="o">+</span><span class="n">j</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="o">-</span><span class="n">j</span><span class="p">,</span> <span class="mi">1</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">j</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="n">j</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">logm</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nprint</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
-<span class="go">[ (0.808757 + 0.107759j)    (2.20752 + 0.202762j)   (1.07376 - 0.773874j)]</span>
-<span class="go">[ (0.905709 - 0.107795j)  (0.0287395 - 0.824993j)  (0.111619 + 0.514272j)]</span>
-<span class="go">[(-0.930151 + 0.399512j)   (-2.06266 - 0.674397j)  (0.791552 + 0.519839j)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">expm</span><span class="p">(</span><span class="n">B</span><span class="p">))</span>
-<span class="go">[(2.0 + 1.0j)           1.0           3.0]</span>
-<span class="go">[(1.0 - 1.0j)  (1.0 - 2.0j)           1.0]</span>
-<span class="go">[        -4.0          -5.0  (0.0 + 1.0j)]</span>
-</pre></div>
-</div>
-<p>A matrix <span class="math">\(X\)</span> close to the identity matrix, for which
-<span class="math">\(\log(\exp(X)) = \exp(\log(X)) = X\)</span> holds:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">+</span> <span class="n">hilbert</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span>
-<span class="go">[              1.25             0.125  0.0833333333333333]</span>
-<span class="go">[             0.125  1.08333333333333              0.0625]</span>
-<span class="go">[0.0833333333333333            0.0625                1.05]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">logm</span><span class="p">(</span><span class="n">expm</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
-<span class="go">[              1.25             0.125  0.0833333333333333]</span>
-<span class="go">[             0.125  1.08333333333333              0.0625]</span>
-<span class="go">[0.0833333333333333            0.0625                1.05]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expm</span><span class="p">(</span><span class="n">logm</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
-<span class="go">[              1.25             0.125  0.0833333333333333]</span>
-<span class="go">[             0.125  1.08333333333333              0.0625]</span>
-<span class="go">[0.0833333333333333            0.0625                1.05]</span>
-</pre></div>
-</div>
-<p>A logarithm of a rotation matrix, giving back the angle of
-the rotation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="mf">3.7</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">([[</span><span class="n">cos</span><span class="p">(</span><span class="n">t</span><span class="p">),</span><span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">)],[</span><span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="p">),</span><span class="n">cos</span><span class="p">(</span><span class="n">t</span><span class="p">)]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">logm</span><span class="p">(</span><span class="n">A</span><span class="p">))</span>
-<span class="go">[             0.0  -2.58318530717959]</span>
-<span class="go">[2.58318530717959                0.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">-</span><span class="n">t</span><span class="p">)</span>
-<span class="go">2.58318530717959</span>
-</pre></div>
-</div>
-<p>For some matrices, a logarithm does not exist:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">logm</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]])</span>
-<span class="gt">Traceback (most recent call last):</span>
-  <span class="c">...</span>
-<span class="gr">ZeroDivisionError</span>: <span class="n">matrix is numerically singular</span>
-</pre></div>
-</div>
-<p>Logarithm of a matrix with large entries:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">logm</span><span class="p">(</span><span class="n">hilbert</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">*</span> <span class="mi">10</span><span class="o">**</span><span class="mi">20</span><span class="p">)</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">re</span><span class="p">)</span>
-<span class="go">[ 45.5597513593433  1.27721006042799  0.317662687717978]</span>
-<span class="go">[ 1.27721006042799  42.5222778973542   2.24003708791604]</span>
-<span class="go">[0.317662687717978  2.24003708791604    42.395212822267]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="mpmath.powm">
-<tt class="descclassname">mpmath.</tt><tt class="descname">powm</tt><big>(</big><em>ctx</em>, <em>A</em>, <em>r</em><big>)</big><a class="headerlink" href="#mpmath.powm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes <span class="math">\(A^r = \exp(A \log r)\)</span> for a matrix <span class="math">\(A\)</span> and complex
-number <span class="math">\(r\)</span>.</p>
-<p><strong>Examples</strong></p>
-<p>Powers and inverse powers of a matrix:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span><span class="p">;</span> <span class="n">mp</span><span class="o">.</span><span class="n">pretty</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">matrix</span><span class="p">([[</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">],[</span><span class="mi">7</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">9</span><span class="p">],[</span><span class="mi">10</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">11</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powm</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">[ 63.0  20.0   69.0]</span>
-<span class="go">[174.0  89.0  199.0]</span>
-<span class="go">[164.0  48.0  179.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">powm</span><span class="p">(</span><span class="n">powm</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="mi">1</span><span class="o">/</span><span class="mf">4.</span><span class="p">))</span>
-<span class="go">[ 4.0  1.0   4.0]</span>
-<span class="go">[ 7.0  8.0   9.0]</span>
-<span class="go">[10.0  2.0  11.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powm</span><span class="p">(</span><span class="n">extraprec</span><span class="p">(</span><span class="mi">20</span><span class="p">)(</span><span class="n">powm</span><span class="p">)(</span><span class="n">A</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="p">),</span> <span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="mf">4.</span><span class="p">)</span>
-<span class="go">[ 4.0  1.0   4.0]</span>
-<span class="go">[ 7.0  8.0   9.0]</span>
-<span class="go">[10.0  2.0  11.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chop</span><span class="p">(</span><span class="n">powm</span><span class="p">(</span><span class="n">powm</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="mi">1</span><span class="o">+</span><span class="mf">0.5j</span><span class="p">),</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="mf">0.5j</span><span class="p">)))</span>
-<span class="go">[ 4.0  1.0   4.0]</span>
-<span class="go">[ 7.0  8.0   9.0]</span>
-<span class="go">[10.0  2.0  11.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powm</span><span class="p">(</span><span class="n">extraprec</span><span class="p">(</span><span class="mi">5</span><span class="p">)(</span><span class="n">powm</span><span class="p">)(</span><span class="n">A</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.5</span><span class="p">),</span> <span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mf">1.5</span><span class="p">))</span>
-<span class="go">[ 4.0  1.0   4.0]</span>
-<span class="go">[ 7.0  8.0   9.0]</span>
-<span class="go">[10.0  2.0  11.0]</span>
-</pre></div>
-</div>
-<p>A Fibonacci-generating matrix:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">powm</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]],</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">[89.0  55.0]</span>
-<span class="go">[55.0  34.0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fib</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">55.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powm</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]],</span> <span class="mf">6.5</span><span class="p">)</span>
-<span class="go">[(16.5166626964253 - 0.0121089837381789j)  (10.2078589271083 + 0.0195927472575932j)]</span>
-<span class="go">[(10.2078589271083 + 0.0195927472575932j)  (6.30880376931698 - 0.0317017309957721j)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">phi</span><span class="o">**</span><span class="mf">6.5</span> <span class="o">-</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">phi</span><span class="p">)</span><span class="o">**</span><span class="mf">6.5</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">(10.2078589271083 - 0.0195927472575932j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powm</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]],</span> <span class="mf">6.2</span><span class="p">)</span>
-<span class="go">[ (14.3076953002666 - 0.008222855781077j)  (8.81733464837593 + 0.0133048601383712j)]</span>
-<span class="go">[(8.81733464837593 + 0.0133048601383712j)  (5.49036065189071 - 0.0215277159194482j)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">phi</span><span class="o">**</span><span class="mf">6.2</span> <span class="o">-</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">phi</span><span class="p">)</span><span class="o">**</span><span class="mf">6.2</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">(8.81733464837593 - 0.0133048601383712j)</span>
-</pre></div>
-</div>
-</dd></dl>
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-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Matrices</a><ul>
-<li><a class="reference internal" href="#creating-matrices">Creating matrices</a><ul>
-<li><a class="reference internal" href="#basic-methods">Basic methods</a></li>
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-</ul>
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-<li><a class="reference internal" href="#norms">Norms</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#linear-algebra">Linear algebra</a><ul>
-<li><a class="reference internal" href="#decompositions">Decompositions</a></li>
-<li><a class="reference internal" href="#linear-equations">Linear equations</a></li>
-<li><a class="reference internal" href="#matrix-factorization">Matrix factorization</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#interval-and-double-precision-matrices">Interval and double-precision matrices</a></li>
-<li><a class="reference internal" href="#matrix-functions">Matrix functions</a></li>
-</ul>
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-</ul>
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-            
-  <div class="section" id="plotting">
-<h1>Plotting<a class="headerlink" href="#plotting" title="Permalink to this headline">¶</a></h1>
-<p>If <a class="reference external" href="http://matplotlib.sourceforge.net/">matplotlib</a> is available, the functions <tt class="docutils literal"><span class="pre">plot</span></tt> and <tt class="docutils literal"><span class="pre">cplot</span></tt> in mpmath can be used to plot functions respectively as x-y graphs and in the complex plane. Also, <tt class="docutils literal"><span class="pre">splot</span></tt> can be used to produce 3D surface plots.</p>
-<div class="section" id="function-curve-plots">
-<h2>Function curve plots<a class="headerlink" href="#function-curve-plots" title="Permalink to this headline">¶</a></h2>
-<img alt="../../_images/plot.png" src="../../_images/plot.png" />
-<p>Output of <tt class="docutils literal"><span class="pre">plot([cos,</span> <span class="pre">sin],</span> <span class="pre">[-4,</span> <span class="pre">4])</span></tt></p>
-<dl class="function">
-<dt id="mpmath.plot">
-<tt class="descclassname">mpmath.</tt><tt class="descname">plot</tt><big>(</big><em>ctx, f, xlim=[-5, 5], ylim=None, points=200, file=None, dpi=None, singularities=[], axes=None</em><big>)</big><a class="headerlink" href="#mpmath.plot" title="Permalink to this definition">¶</a></dt>
-<dd><p>Shows a simple 2D plot of a function <span class="math">\(f(x)\)</span> or list of functions
-<span class="math">\([f_0(x), f_1(x), \ldots, f_n(x)]\)</span> over a given interval
-specified by <em>xlim</em>. Some examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">plot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">li</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">cos</span><span class="p">,</span> <span class="n">sin</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">fresnels</span><span class="p">,</span> <span class="n">fresnelc</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">sqrt</span><span class="p">,</span> <span class="n">cbrt</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
-<span class="n">plot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">zeta</span><span class="p">(</span><span class="mf">0.5</span><span class="o">+</span><span class="n">t</span><span class="o">*</span><span class="n">j</span><span class="p">),</span> <span class="p">[</span><span class="o">-</span><span class="mi">20</span><span class="p">,</span> <span class="mi">20</span><span class="p">])</span>
-<span class="n">plot</span><span class="p">([</span><span class="n">floor</span><span class="p">,</span> <span class="n">ceil</span><span class="p">,</span> <span class="nb">abs</span><span class="p">,</span> <span class="n">sign</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
-</pre></div>
-</div>
-<p>Points where the function raises a numerical exception or
-returns an infinite value are removed from the graph.
-Singularities can also be excluded explicitly
-as follows (useful for removing erroneous vertical lines):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">plot</span><span class="p">(</span><span class="n">cot</span><span class="p">,</span> <span class="n">ylim</span><span class="o">=</span><span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>   <span class="c"># bad</span>
-<span class="n">plot</span><span class="p">(</span><span class="n">cot</span><span class="p">,</span> <span class="n">ylim</span><span class="o">=</span><span class="p">[</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">],</span> <span class="n">singularities</span><span class="o">=</span><span class="p">[</span><span class="o">-</span><span class="n">pi</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="p">])</span>  <span class="c"># good</span>
-</pre></div>
-</div>
-<p>For parts where the function assumes complex values, the
-real part is plotted with dashes and the imaginary part
-is plotted with dots.</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">This function requires matplotlib (pylab).</p>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="complex-function-plots">
-<h2>Complex function plots<a class="headerlink" href="#complex-function-plots" title="Permalink to this headline">¶</a></h2>
-<img alt="../../_images/cplot.png" src="../../_images/cplot.png" />
-<p>Output of <tt class="docutils literal"><span class="pre">fp.cplot(fp.gamma,</span> <span class="pre">points=100000)</span></tt></p>
-<dl class="function">
-<dt id="mpmath.cplot">
-<tt class="descclassname">mpmath.</tt><tt class="descname">cplot</tt><big>(</big><em>ctx, f, re=[-5, 5], im=[-5, 5], points=2000, color=None, verbose=False, file=None, dpi=None, axes=None</em><big>)</big><a class="headerlink" href="#mpmath.cplot" title="Permalink to this definition">¶</a></dt>
-<dd><p>Plots the given complex-valued function <em>f</em> over a rectangular part
-of the complex plane specified by the pairs of intervals <em>re</em> and <em>im</em>.
-For example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">cplot</span><span class="p">(</span><span class="k">lambda</span> <span class="n">z</span><span class="p">:</span> <span class="n">z</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">])</span>
-<span class="n">cplot</span><span class="p">(</span><span class="n">exp</span><span class="p">)</span>
-<span class="n">cplot</span><span class="p">(</span><span class="n">zeta</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">50</span><span class="p">])</span>
-</pre></div>
-</div>
-<p>By default, the complex argument (phase) is shown as color (hue) and
-the magnitude is show as brightness. You can also supply a
-custom color function (<em>color</em>). This function should take a
-complex number as input and return an RGB 3-tuple containing
-floats in the range 0.0-1.0.</p>
-<p>To obtain a sharp image, the number of points may need to be
-increased to 100,000 or thereabout. Since evaluating the
-function that many times is likely to be slow, the &#8216;verbose&#8217;
-option is useful to display progress.</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">This function requires matplotlib (pylab).</p>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="d-surface-plots">
-<h2>3D surface plots<a class="headerlink" href="#d-surface-plots" title="Permalink to this headline">¶</a></h2>
-<img alt="../../_images/splot.png" src="../../_images/splot.png" />
-<p>Output of <tt class="docutils literal"><span class="pre">splot</span></tt> for the donut example.</p>
-<dl class="function">
-<dt id="mpmath.splot">
-<tt class="descclassname">mpmath.</tt><tt class="descname">splot</tt><big>(</big><em>ctx, f, u=[-5, 5], v=[-5, 5], points=100, keep_aspect=True, wireframe=False, file=None, dpi=None, axes=None</em><big>)</big><a class="headerlink" href="#mpmath.splot" title="Permalink to this definition">¶</a></dt>
-<dd><p>Plots the surface defined by <span class="math">\(f\)</span>.</p>
-<p>If <span class="math">\(f\)</span> returns a single component, then this plots the surface
-defined by <span class="math">\(z = f(x,y)\)</span> over the rectangular domain with
-<span class="math">\(x = u\)</span> and <span class="math">\(y = v\)</span>.</p>
-<p>If <span class="math">\(f\)</span> returns three components, then this plots the parametric
-surface <span class="math">\(x, y, z = f(u,v)\)</span> over the pairs of intervals <span class="math">\(u\)</span> and <span class="math">\(v\)</span>.</p>
-<p>For example, to plot a simple function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">splot</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="n">pi</span><span class="p">,</span><span class="n">pi</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="n">pi</span><span class="p">,</span><span class="n">pi</span><span class="p">])</span>    
-</pre></div>
-</div>
-<p>Plotting a donut:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="p">,</span> <span class="n">R</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">2.5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">:</span> <span class="p">[</span><span class="n">r</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="p">(</span><span class="n">R</span><span class="o">+</span><span class="n">r</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">u</span><span class="p">))</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="p">(</span><span class="n">R</span><span class="o">+</span><span class="n">r</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">u</span><span class="p">))</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">v</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">splot</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="p">])</span>    
-</pre></div>
-</div>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">This function requires matplotlib (pylab) 0.98.5.3 or higher.</p>
-</div>
-</dd></dl>
-
-</div>
-</div>
-
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-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Plotting</a><ul>
-<li><a class="reference internal" href="#function-curve-plots">Function curve plots</a></li>
-<li><a class="reference internal" href="#complex-function-plots">Complex function plots</a></li>
-<li><a class="reference internal" href="#d-surface-plots">3D surface plots</a></li>
-</ul>
-</li>
-</ul>
-
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-            
-  <div class="section" id="references">
-<h1>References<a class="headerlink" href="#references" title="Permalink to this headline">¶</a></h1>
-<p>The following is a non-comprehensive list of works used in the development of mpmath
-or cited for examples or mathematical definitions used in this documentation.
-References not listed here can be found in the source code.</p>
-<table class="docutils citation" frame="void" id="abramowitzstegun" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[AbramowitzStegun]</td><td>M Abramowitz &amp; I Stegun. <em>Handbook of Mathematical Functions, 9th Ed.</em>, Tenth Printing, December 1972, with corrections</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="bailey" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Bailey]</td><td>D H Bailey. &#8220;Tanh-Sinh High-Precision Quadrature&#8221;, <a class="reference external" href="http://crd.lbl.gov/~dhbailey/dhbpapers/dhb-tanh-sinh.pdf">http://crd.lbl.gov/~dhbailey/dhbpapers/dhb-tanh-sinh.pdf</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="benderorszag" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[BenderOrszag]</td><td>C M Bender &amp; S A Orszag. <em>Advanced Mathematical Methods for
-Scientists and Engineers</em>, Springer 1999</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="borweinbailey" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[BorweinBailey]</td><td>J Borwein, D H Bailey &amp; R Girgensohn. <em>Experimentation in Mathematics - Computational Paths to Discovery</em>, A K Peters, 2003</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="borweinborwein" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[BorweinBorwein]</td><td>J Borwein &amp; P B Borwein. <em>Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity</em>, Wiley 1987</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="borweinzeta" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[BorweinZeta]</td><td>P Borwein. &#8220;An Efficient Algorithm for the Riemann Zeta Function&#8221;, <a class="reference external" href="http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P115.pdf">http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P115.pdf</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="cabralrosetti" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[CabralRosetti]</td><td>L G Cabral-Rosetti &amp; M A Sanchis-Lozano. &#8220;Appell Functions and the Scalar One-Loop Three-point Integrals in Feynman Diagrams&#8221;. <a class="reference external" href="http://arxiv.org/abs/hep-ph/0206081">http://arxiv.org/abs/hep-ph/0206081</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="carlson" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Carlson]</td><td>B C Carlson. &#8220;Numerical computation of real or complex elliptic integrals&#8221;. <a class="reference external" href="http://arxiv.org/abs/math/9409227v1">http://arxiv.org/abs/math/9409227v1</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="corless" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Corless]</td><td>R M Corless et al. &#8220;On the Lambert W function&#8221;, Adv. Comp. Math. 5 (1996) 329-359. <a class="reference external" href="http://www.apmaths.uwo.ca/~djeffrey/Offprints/W-adv-cm.pdf">http://www.apmaths.uwo.ca/~djeffrey/Offprints/W-adv-cm.pdf</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="dlmf" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[DLMF]</td><td>NIST Digital Library of Mathematical Functions. <a class="reference external" href="http://dlmf.nist.gov/">http://dlmf.nist.gov/</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="gradshteynryzhik" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[GradshteynRyzhik]</td><td>I S Gradshteyn &amp; I M Ryzhik, A Jeffrey &amp; D Zwillinger (eds.), <em>Table of Integrals, Series and Products</em>, Seventh edition (2007), Elsevier</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="gravesmorris" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[GravesMorris]</td><td>P R Graves-Morris, D E Roberts &amp; A Salam. &#8220;The epsilon algorithm and related topics&#8221;, <em>Journal of Computational and Applied Mathematics</em>, Volume 122, Issue 1-2  (October 2000)</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="mpfr" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[MPFR]</td><td>The MPFR team. &#8220;The MPFR Library: Algorithms and Proofs&#8221;, <a class="reference external" href="http://www.mpfr.org/algorithms.pdf">http://www.mpfr.org/algorithms.pdf</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="slater" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Slater]</td><td>L J Slater. <em>Generalized Hypergeometric Functions</em>. Cambridge University Press, 1966</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="spouge" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Spouge]</td><td>J L Spouge. &#8220;Computation of the gamma, digamma, and trigamma functions&#8221;, SIAM J. Numer. Anal. Vol. 31, No. 3, pp. 931-944, June 1994.</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="srivastavakarlsson" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[SrivastavaKarlsson]</td><td>H M Srivastava &amp; P W Karlsson. <em>Multiple Gaussian Hypergeometric Series</em>. Ellis Horwood, 1985.</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="vidunas" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Vidunas]</td><td>R Vidunas. &#8220;Identities between Appell&#8217;s and hypergeometric functions&#8221;. <a class="reference external" href="http://arxiv.org/abs/0804.0655">http://arxiv.org/abs/0804.0655</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="weisstein" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Weisstein]</td><td>E W Weisstein. <em>MathWorld</em>. <a class="reference external" href="http://mathworld.wolfram.com/">http://mathworld.wolfram.com/</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="whittakerwatson" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[WhittakerWatson]</td><td>E T Whittaker &amp; G N Watson. <em>A Course of Modern Analysis</em>. 4th Ed. 1946
-Cambridge University Press</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="wikipedia" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Wikipedia]</td><td><em>Wikipedia, the free encyclopedia</em>. <a class="reference external" href="http://en.wikipedia.org">http://en.wikipedia.org</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="wolframfunctions" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[WolframFunctions]</td><td>Wolfram Research, Inc. <em>The Wolfram Functions Site</em>. <a class="reference external" href="http://functions.wolfram.com/">http://functions.wolfram.com/</a></td></tr>
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-  <div class="section" id="setting-up-mpmath">
-<h1>Setting up mpmath<a class="headerlink" href="#setting-up-mpmath" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="download-and-installation">
-<h2>Download and installation<a class="headerlink" href="#download-and-installation" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="installer">
-<h3>Installer<a class="headerlink" href="#installer" title="Permalink to this headline">¶</a></h3>
-<p>The mpmath setup files can be downloaded from the <a class="reference external" href="http://code.google.com/p/mpmath/downloads/list">mpmath download page</a> or the <a class="reference external" href="http://pypi.python.org/pypi/mpmath/">Python Package Index</a>. Download the source package (available as both .zip and .tar.gz), extract it, open the extracted directory, and run</p>
-<blockquote>
-<div><tt class="docutils literal"><span class="pre">python</span> <span class="pre">setup.py</span> <span class="pre">install</span></tt></div></blockquote>
-<p>If you are using Windows, you can download the binary installer</p>
-<blockquote>
-<div><tt class="docutils literal"><span class="pre">mpmath-(version).win32.exe</span></tt></div></blockquote>
-<p>from the mpmath website or the Python Package Index. Run the installer and follow the instructions.</p>
-</div>
-<div class="section" id="using-setuptools">
-<h3>Using setuptools<a class="headerlink" href="#using-setuptools" title="Permalink to this headline">¶</a></h3>
-<p>If you have <a class="reference external" href="http://pypi.python.org/pypi/setuptools">setuptools</a> installed, you can download and install mpmath in one step by running:</p>
-<blockquote>
-<div><tt class="docutils literal"><span class="pre">easy_install</span> <span class="pre">mpmath</span></tt></div></blockquote>
-<p>or</p>
-<blockquote>
-<div><tt class="docutils literal"><span class="pre">python</span> <span class="pre">-m</span> <span class="pre">easy_install</span> <span class="pre">mpmath</span></tt></div></blockquote>
-<p>If you have an old version of mpmath installed already, you may have to pass <tt class="docutils literal"><span class="pre">easy_install</span></tt> the <tt class="docutils literal"><span class="pre">-U</span></tt> flag to force an upgrade.</p>
-</div>
-<div class="section" id="debian-ubuntu">
-<h3>Debian/Ubuntu<a class="headerlink" href="#debian-ubuntu" title="Permalink to this headline">¶</a></h3>
-<p>Debian and Ubuntu users can install mpmath with</p>
-<blockquote>
-<div><tt class="docutils literal"><span class="pre">sudo</span> <span class="pre">apt-get</span> <span class="pre">install</span> <span class="pre">python-mpmath</span></tt></div></blockquote>
-<p>See <a class="reference external" href="http://packages.debian.org/python-mpmath">debian</a> and <a class="reference external" href="https://launchpad.net/ubuntu/+source/mpmath">ubuntu</a> package information; please verify that you are getting the latest version.</p>
-</div>
-<div class="section" id="opensuse">
-<h3>OpenSUSE<a class="headerlink" href="#opensuse" title="Permalink to this headline">¶</a></h3>
-<p>Mpmath is provided in the &#8220;Science&#8221; repository for all recent versions of <a class="reference external" href="http://www.opensuse.org/">openSUSE</a>. To add this repository to the YAST software management tool, see <a class="reference external" href="http://en.opensuse.org/SDB:Add_package_repositories">http://en.opensuse.org/SDB:Add_package_repositories</a></p>
-<p>Look up <a class="reference external" href="http://download.opensuse.org/repositories/science/">http://download.opensuse.org/repositories/science/</a> for a list
-of supported OpenSUSE versions and use <a class="reference external" href="http://download.opensuse.org/repositories/science/openSUSE_12.2/">http://download.opensuse.org/repositories/science/openSUSE_12.2/</a>
-(or accordingly for your OpenSUSE version) as the repository URL for YAST.</p>
-</div>
-<div class="section" id="current-development-version">
-<h3>Current development version<a class="headerlink" href="#current-development-version" title="Permalink to this headline">¶</a></h3>
-<p>See <a class="reference external" href="http://code.google.com/p/mpmath/source/checkout">http://code.google.com/p/mpmath/source/checkout</a> for instructions on how to check out the mpmath Subversion repository. The source code can also be browsed online from the Google Code page.</p>
-</div>
-<div class="section" id="checking-that-it-works">
-<h3>Checking that it works<a class="headerlink" href="#checking-that-it-works" title="Permalink to this headline">¶</a></h3>
-<p>After the setup has completed, you should be able to fire up the interactive Python interpreter and do the following:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">50</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="n">mpf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">**</span> <span class="n">mpf</span><span class="p">(</span><span class="s">&#39;0.5&#39;</span><span class="p">)</span>
-<span class="go">1.4142135623730950488016887242096980785696718753769</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span>
-<span class="go">6.2831853071795864769252867665590057683943387987502</span>
-</pre></div>
-</div>
-<p><em>Note: if you have are upgrading mpmath from an earlier version, you may have to manually uninstall the old version or remove the old files.</em></p>
-</div>
-</div>
-<div class="section" id="using-gmpy-optional">
-<h2>Using gmpy (optional)<a class="headerlink" href="#using-gmpy-optional" title="Permalink to this headline">¶</a></h2>
-<p>By default, mpmath uses Python integers internally. If <a class="reference external" href="http://code.google.com/p/gmpy/">gmpy</a> version 1.03 or later is installed on your system, mpmath will automatically detect it and transparently use gmpy integers intead. This makes mpmath much faster, especially at high precision (approximately above 100 digits).</p>
-<p>To verify that mpmath uses gmpy, check the internal variable <tt class="docutils literal"><span class="pre">BACKEND</span></tt> is not equal to &#8216;python&#8217;:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">mpmath.libmp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mpmath</span><span class="o">.</span><span class="n">libmp</span><span class="o">.</span><span class="n">BACKEND</span>
-<span class="go">&#39;gmpy&#39;</span>
-</pre></div>
-</div>
-<p>The gmpy mode can be disabled by setting the MPMATH_NOGMPY environment variable. Note that the mode cannot be switched during runtime; mpmath must be re-imported for this change to take effect.</p>
-</div>
-<div class="section" id="running-tests">
-<h2>Running tests<a class="headerlink" href="#running-tests" title="Permalink to this headline">¶</a></h2>
-<p>It is recommended that you run mpmath&#8217;s full set of unit tests to make sure everything works. The tests are located in the <tt class="docutils literal"><span class="pre">tests</span></tt> subdirectory of the main mpmath directory. They can be run in the interactive interpreter using the <tt class="docutils literal"><span class="pre">runtests()</span></tt> function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="kn">import</span> <span class="nn">mpmath</span>
-<span class="n">mpmath</span><span class="o">.</span><span class="n">runtests</span><span class="p">()</span>
-</pre></div>
-</div>
-<p>Alternatively, they can be run from the <tt class="docutils literal"><span class="pre">tests</span></tt> directory via</p>
-<blockquote>
-<div><tt class="docutils literal"><span class="pre">python</span> <span class="pre">runtests.py</span></tt></div></blockquote>
-<p>The tests should finish in about a minute. If you have <a class="reference external" href="http://psyco.sourceforge.net/">psyco</a> installed, the tests can also be run with</p>
-<blockquote>
-<div><tt class="docutils literal"><span class="pre">python</span> <span class="pre">runtests.py</span> <span class="pre">-psyco</span></tt></div></blockquote>
-<p>which will cut the running time in half.</p>
-<p>If any test fails, please send a detailed bug report to the <a class="reference external" href="http://code.google.com/p/sympy/issues/list">mpmath issue tracker</a>. The tests can also be run with <a class="reference external" href="http://codespeak.net/py/dist/">py.test</a>. This will sometimes generate more useful information in case of a failure.</p>
-<p>To run the tests with support for gmpy disabled, use</p>
-<blockquote>
-<div><tt class="docutils literal"><span class="pre">python</span> <span class="pre">runtests.py</span> <span class="pre">-nogmpy</span></tt></div></blockquote>
-<p>To enable extra diagnostics, use</p>
-<blockquote>
-<div><tt class="docutils literal"><span class="pre">python</span> <span class="pre">runtests.py</span> <span class="pre">-strict</span></tt></div></blockquote>
-</div>
-<div class="section" id="compiling-the-documentation">
-<h2>Compiling the documentation<a class="headerlink" href="#compiling-the-documentation" title="Permalink to this headline">¶</a></h2>
-<p>If you downloaded the source package, the text source for these documentation pages is included in the <tt class="docutils literal"><span class="pre">doc</span></tt> directory. The documentation can be compiled to pretty HTML using <a class="reference external" href="http://sphinx.pocoo.org/">Sphinx</a>. Go to the <tt class="docutils literal"><span class="pre">doc</span></tt> directory and run</p>
-<blockquote>
-<div><tt class="docutils literal"><span class="pre">python</span> <span class="pre">build.py</span></tt></div></blockquote>
-<p>You can also test that all the interactive examples in the documentation work by running</p>
-<blockquote>
-<div><tt class="docutils literal"><span class="pre">python</span> <span class="pre">run_doctest.py</span></tt></div></blockquote>
-<p>and by running the individual <tt class="docutils literal"><span class="pre">.py</span></tt> files in the mpmath source.</p>
-<p>(The doctests may take several minutes.)</p>
-<p>Finally, some additional demo scripts are available in the <tt class="docutils literal"><span class="pre">demo</span></tt> directory included in the source package.</p>
-</div>
-<div class="section" id="mpmath-under-sympy">
-<h2>Mpmath under SymPy<a class="headerlink" href="#mpmath-under-sympy" title="Permalink to this headline">¶</a></h2>
-<p>Mpmath is available as a subpackage of <a class="reference external" href="http://sympy.org">SymPy</a>. With SymPy installed, you can just do</p>
-<blockquote>
-<div><tt class="docutils literal"><span class="pre">import</span> <span class="pre">sympy.mpmath</span> <span class="pre">as</span> <span class="pre">mpmath</span></tt></div></blockquote>
-<p>instead of <tt class="docutils literal"><span class="pre">import</span> <span class="pre">mpmath</span></tt>. Note that the SymPy version of mpmath might not be the most recent. You can make a separate mpmath installation even if SymPy is installed; the two mpmath packages will not interfere with each other.</p>
-</div>
-<div class="section" id="mpmath-under-sage">
-<h2>Mpmath under Sage<a class="headerlink" href="#mpmath-under-sage" title="Permalink to this headline">¶</a></h2>
-<p>Mpmath is a standard package in <a class="reference external" href="http://sagemath.org/">Sage</a>, in version 4.1 or later of Sage.
-Mpmath is preinstalled a regular Python module, and can be imported as usual within Sage:</p>
-<div class="highlight-python"><pre>----------------------------------------------------------------------
-| Sage Version 4.1, Release Date: 2009-07-09                         |
-| Type notebook() for the GUI, and license() for information.        |
-----------------------------------------------------------------------
-sage: import mpmath
-sage: mpmath.mp.dps = 50
-sage: print mpmath.mpf(2) ** 0.5
-1.4142135623730950488016887242096980785696718753769</pre>
-</div>
-<p>The mpmath installation under Sage automatically use Sage integers for asymptotically fast arithmetic,
-so there is no need to install GMPY:</p>
-<div class="highlight-python"><pre>sage: mpmath.libmp.BACKEND
-&#x27;sage&#x27;</pre>
-</div>
-<p>In Sage, mpmath can alternatively be imported via the interface library
-<tt class="docutils literal"><span class="pre">sage.libs.mpmath.all</span></tt>. For example:</p>
-<div class="highlight-python"><pre>sage: import sage.libs.mpmath.all as mpmath</pre>
-</div>
-<p>This module provides a few extra conversion functions, including <tt class="xref py py-func docutils literal"><span class="pre">call()</span></tt>
-which permits calling any mpmath function with Sage numbers as input, and getting
-Sage <tt class="docutils literal"><span class="pre">RealNumber</span></tt> or <tt class="docutils literal"><span class="pre">ComplexNumber</span></tt> instances
-with the appropriate precision back:</p>
-<div class="highlight-python"><pre>sage: w = mpmath.call(mpmath.erf, 2+3*I, prec=100)
-sage: w
--20.829461427614568389103088452 + 8.6873182714701631444280787545*I
-sage: type(w)
-&lt;type &#x27;sage.rings.complex_number.ComplexNumber&#x27;&gt;
-sage: w.prec()
-100</pre>
-</div>
-<p>See the help for <tt class="docutils literal"><span class="pre">sage.libs.mpmath.all</span></tt> for further information.</p>
-</div>
-</div>
-
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-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Setting up mpmath</a><ul>
-<li><a class="reference internal" href="#download-and-installation">Download and installation</a><ul>
-<li><a class="reference internal" href="#installer">Installer</a></li>
-<li><a class="reference internal" href="#using-setuptools">Using setuptools</a></li>
-<li><a class="reference internal" href="#debian-ubuntu">Debian/Ubuntu</a></li>
-<li><a class="reference internal" href="#opensuse">OpenSUSE</a></li>
-<li><a class="reference internal" href="#current-development-version">Current development version</a></li>
-<li><a class="reference internal" href="#checking-that-it-works">Checking that it works</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#using-gmpy-optional">Using gmpy (optional)</a></li>
-<li><a class="reference internal" href="#running-tests">Running tests</a></li>
-<li><a class="reference internal" href="#compiling-the-documentation">Compiling the documentation</a></li>
-<li><a class="reference internal" href="#mpmath-under-sympy">Mpmath under SymPy</a></li>
-<li><a class="reference internal" href="#mpmath-under-sage">Mpmath under Sage</a></li>
-</ul>
-</li>
-</ul>
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-  <div class="section" id="precision-and-representation-issues">
-<h1>Precision and representation issues<a class="headerlink" href="#precision-and-representation-issues" title="Permalink to this headline">¶</a></h1>
-<p>Most of the time, using mpmath is simply a matter of setting the desired precision and entering a formula. For verification purposes, a quite (but not always!) reliable technique is to calculate the same thing a second time at a higher precision and verifying that the results agree.</p>
-<p>To perform more advanced calculations, it is important to have some understanding of how mpmath works internally and what the possible sources of error are. This section gives an overview of arbitrary-precision binary floating-point arithmetic and some concepts from numerical analysis.</p>
-<p>The following concepts are important to understand:</p>
-<ul class="simple">
-<li>The main sources of numerical errors are rounding and cancellation, which are due to the use of finite-precision arithmetic, and truncation or approximation errors, which are due to approximating infinite sequences or continuous functions by a finite number of samples.</li>
-<li>Errors propagate between calculations. A small error in the input may result in a large error in the output.</li>
-<li>Most numerical algorithms for complex problems (e.g. integrals, derivatives) give wrong answers for sufficiently ill-behaved input. Sometimes virtually the only way to get a wrong answer is to design some very contrived input, but at other times the chance of accidentally obtaining a wrong result even for reasonable-looking input is quite high.</li>
-<li>Like any complex numerical software, mpmath has implementation bugs. You should be reasonably suspicious about any results computed by mpmath, even those it claims to be able to compute correctly! If possible, verify results analytically, try different algorithms, and cross-compare with other software.</li>
-</ul>
-<div class="section" id="precision-error-and-tolerance">
-<h2>Precision, error and tolerance<a class="headerlink" href="#precision-error-and-tolerance" title="Permalink to this headline">¶</a></h2>
-<p>The following terms are common in this documentation:</p>
-<ul class="simple">
-<li><em>Precision</em> (or <em>working precision</em>) is the precision at which floating-point arithmetic operations are performed.</li>
-<li><em>Error</em> is the difference between a computed approximation and the exact result.</li>
-<li><em>Accuracy</em> is the inverse of error.</li>
-<li><em>Tolerance</em> is the maximum error (or minimum accuracy) desired in a result.</li>
-</ul>
-<p>Error and accuracy can be measured either directly, or logarithmically in bits or digits. Specifically, if a <span class="math">\(\hat y\)</span> is an approximation for <span class="math">\(y\)</span>, then</p>
-<ul class="simple">
-<li>(Direct) absolute error = <span class="math">\(|\hat y - y|\)</span></li>
-<li>(Direct) relative error = <span class="math">\(|\hat y - y| |y|^{-1}\)</span></li>
-<li>(Direct) absolute accuracy = <span class="math">\(|\hat y - y|^{-1}\)</span></li>
-<li>(Direct) relative accuracy = <span class="math">\(|\hat y - y|^{-1} |y|\)</span></li>
-<li>(Logarithmic) absolute error = <span class="math">\(\log_b |\hat y - y|\)</span></li>
-<li>(Logarithmic) relative error = <span class="math">\(\log_b |\hat y - y| - \log_b |y|\)</span></li>
-<li>(Logarithmic) absolute accuracy = <span class="math">\(-\log_b |\hat y - y|\)</span></li>
-<li>(Logarithmic) relative accuracy = <span class="math">\(-\log_b |\hat y - y| + \log_b |y|\)</span></li>
-</ul>
-<p>where <span class="math">\(b = 2\)</span> and <span class="math">\(b = 10\)</span> for bits and digits respectively. Note that:</p>
-<ul class="simple">
-<li>The logarithmic error roughly equals the position of the first incorrect bit or digit</li>
-<li>The logarithmic accuracy roughly equals the number of correct bits or digits in the result</li>
-</ul>
-<p>These definitions also hold for complex numbers, using <span class="math">\(|a+bi| = \sqrt{a^2+b^2}\)</span>.</p>
-<p><em>Full accuracy</em> means that the accuracy of a result at least equals <em>prec</em>-1, i.e. it is correct except possibly for the last bit.</p>
-</div>
-<div class="section" id="representation-of-numbers">
-<h2>Representation of numbers<a class="headerlink" href="#representation-of-numbers" title="Permalink to this headline">¶</a></h2>
-<p>Mpmath uses binary arithmetic. A binary floating-point number is a number of the form <span class="math">\(man \times 2^{exp}\)</span> where both <em>man</em> (the <em>mantissa</em>) and <em>exp</em> (the <em>exponent</em>) are integers. Some examples of floating-point numbers are given in the following table.</p>
-<blockquote>
-<div><table border="1" class="docutils">
-<colgroup>
-<col width="29%" />
-<col width="36%" />
-<col width="36%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head">Number</th>
-<th class="head">Mantissa</th>
-<th class="head">Exponent</th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-even"><td>3</td>
-<td>3</td>
-<td>0</td>
-</tr>
-<tr class="row-odd"><td>10</td>
-<td>5</td>
-<td>1</td>
-</tr>
-<tr class="row-even"><td>-16</td>
-<td>-1</td>
-<td>4</td>
-</tr>
-<tr class="row-odd"><td>1.25</td>
-<td>5</td>
-<td>-2</td>
-</tr>
-</tbody>
-</table>
-</div></blockquote>
-<p>The representation as defined so far is not unique; one can always multiply the mantissa by 2 and subtract 1 from the exponent with no change in the numerical value. In mpmath, numbers are always normalized so that <em>man</em> is an odd number, with the exception of zero which is always taken to have <em>man = exp = 0</em>. With these conventions, every representable number has a unique representation. (Mpmath does not currently distinguish between positive and negative zero.)</p>
-<p>Simple mathematical operations are now easy to define. Due to uniqueness, equality testing of two numbers simply amounts to separately checking equality of the mantissas and the exponents. Multiplication of nonzero numbers is straightforward: <span class="math">\((m 2^e) \times (n 2^f) = (m n) \times 2^{e+f}\)</span>. Addition is a bit more involved: we first need to multiply the mantissa of one of the operands by a suitable power of 2 to obtain equal exponents.</p>
-<p>More technically, mpmath represents a floating-point number as a 4-tuple <em>(sign, man, exp, bc)</em> where <em>sign</em> is 0 or 1 (indicating positive vs negative) and the mantissa is nonnegative; <em>bc</em> (<em>bitcount</em>) is the size of the absolute value of the mantissa as measured in bits. Though redundant, keeping a separate sign field and explicitly keeping track of the bitcount significantly speeds up arithmetic (the bitcount, especially, is frequently needed but slow to compute from scratch due to the lack of a Python built-in function for the purpose).</p>
-<p>Contrary to popular belief, floating-point <em>numbers</em> do not come with an inherent &#8220;small uncertainty&#8221;, although floating-point <em>arithmetic</em> generally is inexact. Every binary floating-point number is an exact rational number. With arbitrary-precision integers used for the mantissa and exponent, floating-point numbers can be added, subtracted and multiplied <em>exactly</em>. In particular, integers and integer multiples of 1/2, 1/4, 1/8, 1/16, etc. can be represented, added and multiplied exactly in binary floating-point arithmetic.</p>
-<p>Floating-point arithmetic is generally approximate because the size of the mantissa must be limited for efficiency reasons. The maximum allowed width (bitcount) of the mantissa is called the precision or <em>prec</em> for short. Sums and products of floating-point numbers are exact as long as the absolute value of the mantissa is smaller than <span class="math">\(2^{prec}\)</span>. As soon as the mantissa becomes larger than this, it is truncated to contain at most <em>prec</em> bits (the exponent is incremented accordingly to preserve the magnitude of the number), and this operation introduces a rounding error. Division is also generally inexact; although we can add and multiply exactly by setting the precision high enough, no precision is high enough to represent for example 1/3 exactly (the same obviously applies for roots, trigonometric functions, etc).</p>
-<p>The special numbers <tt class="docutils literal"><span class="pre">+inf</span></tt>, <tt class="docutils literal"><span class="pre">-inf</span></tt> and <tt class="docutils literal"><span class="pre">nan</span></tt> are represented internally by a zero mantissa and a nonzero exponent.</p>
-<p>Mpmath uses arbitrary precision integers for both the mantissa and the exponent, so numbers can be as large in magnitude as permitted by the computer&#8217;s memory. Some care may be necessary when working with extremely large numbers. Although standard arithmetic operators are safe, it is for example futile to attempt to compute the exponential function of of <span class="math">\(10^{100000}\)</span>. Mpmath does not complain when asked to perform such a calculation, but instead chugs away on the problem to the best of its ability, assuming that computer resources are infinite. In the worst case, this will be slow and allocate a huge amount of memory; if entirely impossible Python will at some point raise <tt class="docutils literal"><span class="pre">OverflowError:</span> <span class="pre">long</span> <span class="pre">int</span> <span class="pre">too</span> <span class="pre">large</span> <span class="pre">to</span> <span class="pre">convert</span> <span class="pre">to</span> <span class="pre">int</span></tt>.</p>
-<p>For further details on how the arithmetic is implemented, refer to the mpmath source code. The basic arithmetic operations are found in the <tt class="docutils literal"><span class="pre">libmp</span></tt> directory; many functions there are commented extensively.</p>
-</div>
-<div class="section" id="decimal-issues">
-<h2>Decimal issues<a class="headerlink" href="#decimal-issues" title="Permalink to this headline">¶</a></h2>
-<p>Mpmath uses binary arithmetic internally, while most interaction with the user is done via the decimal number system. Translating between binary and decimal numbers is a somewhat subtle matter; many Python novices run into the following &#8220;bug&#8221; (addressed in the <a class="reference external" href="http://www.python.org/doc/faq/general/#why-are-floating-point-calculations-so-inaccurate">General Python FAQ</a>):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mf">0.1</span>
-<span class="go">0.10000000000000001</span>
-</pre></div>
-</div>
-<p>Decimal fractions fall into the category of numbers that generally cannot be represented exactly in binary floating-point form. For example, none of the numbers 0.1, 0.01, 0.001 has an exact representation as a binary floating-point number. Although mpmath can approximate decimal fractions with any accuracy, it does not solve this problem for all uses; users who need <em>exact</em> decimal fractions should look at the <tt class="docutils literal"><span class="pre">decimal</span></tt> module in Python&#8217;s standard library (or perhaps use fractions, which are much faster).</p>
-<p>With <em>prec</em> bits of precision, an arbitrary number can be approximated relatively to within <span class="math">\(2^{-prec}\)</span>, or within <span class="math">\(10^{-dps}\)</span> for <em>dps</em> decimal digits. The equivalent values for <em>prec</em> and <em>dps</em> are therefore related proportionally via the factor <span class="math">\(C = \log(10)/\log(2)\)</span>, or roughly 3.32. For example, the standard (binary) precision in mpmath is 53 bits, which corresponds to a decimal precision of 15.95 digits.</p>
-<p>More precisely, mpmath uses the following formulas to translate between <em>prec</em> and <em>dps</em>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">dps</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="nb">round</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">prec</span><span class="p">)</span> <span class="o">/</span> <span class="n">C</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)))</span>
-
-<span class="n">prec</span><span class="p">(</span><span class="n">dps</span><span class="p">)</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="nb">round</span><span class="p">((</span><span class="nb">int</span><span class="p">(</span><span class="n">dps</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="p">)))</span>
-</pre></div>
-</div>
-<p>Note that the dps is set 1 decimal digit lower than the corresponding binary precision. This is done to hide minor rounding errors and artifacts resulting from binary-decimal conversion. As a result, mpmath interprets 53 bits as giving 15 digits of decimal precision, not 16.</p>
-<p>The <em>dps</em> value controls the number of digits to display when printing numbers with <tt class="xref py py-func docutils literal"><span class="pre">str()</span></tt>, while the decimal precision used by <tt class="xref py py-func docutils literal"><span class="pre">repr()</span></tt> is set two or three digits higher. For example, with 15 dps we have:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">mpmath</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">str</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">&#39;3.14159265358979&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">repr</span><span class="p">(</span><span class="o">+</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">&quot;mpf(&#39;3.1415926535897931&#39;)&quot;</span>
-</pre></div>
-</div>
-<p>The extra digits in the output from <tt class="docutils literal"><span class="pre">repr</span></tt> ensure that <tt class="docutils literal"><span class="pre">x</span> <span class="pre">==</span> <span class="pre">eval(repr(x))</span></tt> holds, i.e. that numbers can be converted to strings and back losslessly.</p>
-<p>It should be noted that precision and accuracy do not always correlate when translating between binary and decimal. As a simple example, the number 0.1 has a decimal precision of 1 digit but is an infinitely accurate representation of 1/10. Conversely, the number <span class="math">\(2^{-50}\)</span> has a binary representation with 1 bit of precision that is infinitely accurate; the same number can actually be represented exactly as a decimal, but doing so requires 35 significant digits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="mf">0.00000000000000088817841970012523233890533447265625</span>
-</pre></div>
-</div>
-<p>All binary floating-point numbers can be represented exactly as decimals (possibly requiring many digits), but the converse is false.</p>
-</div>
-<div class="section" id="correctness-guarantees">
-<h2>Correctness guarantees<a class="headerlink" href="#correctness-guarantees" title="Permalink to this headline">¶</a></h2>
-<p>Basic arithmetic operations (with the <tt class="docutils literal"><span class="pre">mp</span></tt> context) are always performed with correct rounding. Results that can be represented exactly are guranteed to be exact, and results from single inexact operations are guaranteed to be the best possible rounded values. For higher-level operations, mpmath does not generally guarantee correct rounding. In general, mpmath only guarantees that it will use at least the user-set precision to perform a given calculation. <em>The user may have to manually set the working precision higher than the desired accuracy for the result, possibly much higher.</em></p>
-<p>Functions for evaluation of transcendental functions, linear algebra operations, numerical integration, etc., usually automatically increase the working precision and use a stricter tolerance to give a correctly rounded result with high probability: for example, at 50 bits the temporary precision might be set to 70 bits and the tolerance might be set to 60 bits. It can often be assumed that such functions return values that have full accuracy, given inputs that are exact (or sufficiently precise approximations of exact values), but the user must exercise judgement about whether to trust mpmath.</p>
-<p>The level of rigor in mpmath covers the entire spectrum from &#8220;always correct by design&#8221; through &#8220;nearly always correct&#8221; and &#8220;handling the most common errors&#8221; to &#8220;just computing blindly and hoping for the best&#8221;. Of course, a long-term development goal is to successively increase the rigor where possible. The following list might give an idea of the current state.</p>
-<p>Operations that are correctly rounded:</p>
-<ul class="simple">
-<li>Addition, subtraction and multiplication of real and complex numbers.</li>
-<li>Division and square roots of real numbers.</li>
-<li>Powers of real numbers, assuming sufficiently small integer exponents (huge powers are rounded in the right direction, but possibly farther than necessary).</li>
-<li>Conversion from decimal to binary, for reasonably sized numbers (roughly between <span class="math">\(10^{-100}\)</span> and <span class="math">\(10^{100}\)</span>).</li>
-<li>Typically, transcendental functions for exact input-output pairs.</li>
-</ul>
-<p>Operations that should be fully accurate (however, the current implementation may be based on a heuristic error analysis):</p>
-<ul class="simple">
-<li>Radix conversion (large or small numbers).</li>
-<li>Mathematical constants like <span class="math">\(\pi\)</span>.</li>
-<li>Both real and imaginary parts of exp, cos, sin, cosh, sinh, log.</li>
-<li>Other elementary functions (the largest of the real and imaginary part).</li>
-<li>The gamma and log-gamma functions (the largest of the real and the imaginary part; both, when close to real axis).</li>
-<li>Some functions based on hypergeometric series (the largest of the real and imaginary part).</li>
-</ul>
-<p>Correctness of root-finding, numerical integration, etc. largely depends on the well-behavedness of the input functions. Specific limitations are sometimes noted in the respective sections of the documentation.</p>
-</div>
-<div class="section" id="double-precision-emulation">
-<h2>Double precision emulation<a class="headerlink" href="#double-precision-emulation" title="Permalink to this headline">¶</a></h2>
-<p>On most systems, Python&#8217;s <tt class="docutils literal"><span class="pre">float</span></tt> type represents an IEEE 754 <em>double precision</em> number, with a precision of 53 bits and rounding-to-nearest. With default precision (<tt class="docutils literal"><span class="pre">mp.prec</span> <span class="pre">=</span> <span class="pre">53</span></tt>), the mpmath <tt class="docutils literal"><span class="pre">mpf</span></tt> type roughly emulates the behavior of the <tt class="docutils literal"><span class="pre">float</span></tt> type. Sources of incompatibility include the following:</p>
-<ul class="simple">
-<li>In hardware floating-point arithmetic, the size of the exponent is restricted to a fixed range: regular Python floats have a range between roughly <span class="math">\(10^{-300}\)</span> and <span class="math">\(10^{300}\)</span>. Mpmath does not emulate overflow or underflow when exponents fall outside this range.</li>
-<li>On some systems, Python uses 80-bit (extended double) registers for floating-point operations. Due to double rounding, this makes the <tt class="docutils literal"><span class="pre">float</span></tt> type less accurate. This problem is only known to occur with Python versions compiled with GCC on 32-bit systems.</li>
-<li>Machine floats very close to the exponent limit round subnormally, meaning that they lose accuracy (Python may raise an exception instead of rounding a <tt class="docutils literal"><span class="pre">float</span></tt> subnormally).</li>
-<li>Mpmath is able to produce more accurate results for transcendental functions.</li>
-</ul>
-</div>
-<div class="section" id="further-reading">
-<h2>Further reading<a class="headerlink" href="#further-reading" title="Permalink to this headline">¶</a></h2>
-<p>There are many excellent textbooks on numerical analysis and floating-point arithmetic. Some good web resources are:</p>
-<ul class="simple">
-<li><a class="reference external" href="http://docs.sun.com/source/806-3568/ncg_goldberg.html">David Goldberg, What Every Computer Scientist Should Know About Floating-Point Arithmetic</a></li>
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Numerical_analysis">The Wikipedia article about numerical analysis</a></li>
-</ul>
-</div>
-</div>
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-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Precision and representation issues</a><ul>
-<li><a class="reference internal" href="#precision-error-and-tolerance">Precision, error and tolerance</a></li>
-<li><a class="reference internal" href="#representation-of-numbers">Representation of numbers</a></li>
-<li><a class="reference internal" href="#decimal-issues">Decimal issues</a></li>
-<li><a class="reference internal" href="#correctness-guarantees">Correctness guarantees</a></li>
-<li><a class="reference internal" href="#double-precision-emulation">Double precision emulation</a></li>
-<li><a class="reference internal" href="#further-reading">Further reading</a></li>
-</ul>
-</li>
-</ul>
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-            
-  <div class="section" id="module-sympy.ntheory.generate">
-<span id="number-theory"></span><h1>Number Theory<a class="headerlink" href="#module-sympy.ntheory.generate" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="ntheory-class-reference">
-<h2>Ntheory Class Reference<a class="headerlink" href="#ntheory-class-reference" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.ntheory.generate.Sieve">
-<em class="property">class </em><tt class="descclassname">sympy.ntheory.generate.</tt><tt class="descname">Sieve</tt><a class="reference internal" href="../_modules/sympy/ntheory/generate.html#Sieve"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.generate.Sieve" title="Permalink to this definition">¶</a></dt>
-<dd><p>An infinite list of prime numbers, implemented as a dynamically
-growing sieve of Eratosthenes. When a lookup is requested involving
-an odd number that has not been sieved, the sieve is automatically
-extended up to that number.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sieve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">array</span> <span class="kn">import</span> <span class="n">array</span> <span class="c"># this line and next for doctest only</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sieve</span><span class="o">.</span><span class="n">_list</span> <span class="o">=</span> <span class="n">array</span><span class="p">(</span><span class="s">&#39;l&#39;</span><span class="p">,</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">13</span><span class="p">])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">25</span> <span class="ow">in</span> <span class="n">sieve</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sieve</span><span class="o">.</span><span class="n">_list</span>
-<span class="go">array(&#39;l&#39;, [2, 3, 5, 7, 11, 13, 17, 19, 23])</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.ntheory.generate.Sieve.extend">
-<tt class="descname">extend</tt><big>(</big><em>self</em>, <em>n</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/generate.html#Sieve.extend"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.generate.Sieve.extend" title="Permalink to this definition">¶</a></dt>
-<dd><p>Grow the sieve to cover all primes &lt;= n (a real number).</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sieve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">array</span> <span class="kn">import</span> <span class="n">array</span> <span class="c"># this line and next for doctest only</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sieve</span><span class="o">.</span><span class="n">_list</span> <span class="o">=</span> <span class="n">array</span><span class="p">(</span><span class="s">&#39;l&#39;</span><span class="p">,</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">13</span><span class="p">])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sieve</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="mi">30</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sieve</span><span class="p">[</span><span class="mi">10</span><span class="p">]</span> <span class="o">==</span> <span class="mi">29</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.generate.Sieve.extend_to_no">
-<tt class="descname">extend_to_no</tt><big>(</big><em>self</em>, <em>i</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/generate.html#Sieve.extend_to_no"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.generate.Sieve.extend_to_no" title="Permalink to this definition">¶</a></dt>
-<dd><p>Extend to include the ith prime number.</p>
-<p>i must be an integer.</p>
-<p>The list is extended by 50% if it is too short, so it is
-likely that it will be longer than requested.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sieve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">array</span> <span class="kn">import</span> <span class="n">array</span> <span class="c"># this line and next for doctest only</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sieve</span><span class="o">.</span><span class="n">_list</span> <span class="o">=</span> <span class="n">array</span><span class="p">(</span><span class="s">&#39;l&#39;</span><span class="p">,</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">13</span><span class="p">])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sieve</span><span class="o">.</span><span class="n">extend_to_no</span><span class="p">(</span><span class="mi">9</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sieve</span><span class="o">.</span><span class="n">_list</span>
-<span class="go">array(&#39;l&#39;, [2, 3, 5, 7, 11, 13, 17, 19, 23])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.generate.Sieve.primerange">
-<tt class="descname">primerange</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/generate.html#Sieve.primerange"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.generate.Sieve.primerange" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generate all prime numbers in the range [a, b).</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sieve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">([</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">sieve</span><span class="o">.</span><span class="n">primerange</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="mi">18</span><span class="p">)])</span>
-<span class="go">[7, 11, 13, 17]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.generate.Sieve.search">
-<tt class="descname">search</tt><big>(</big><em>self</em>, <em>n</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/generate.html#Sieve.search"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.generate.Sieve.search" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the indices i, j of the primes that bound n.</p>
-<p>If n is prime then i == j.</p>
-<p>Although n can be an expression, if ceiling cannot convert
-it to an integer then an n error will be raised.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sieve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sieve</span><span class="o">.</span><span class="n">search</span><span class="p">(</span><span class="mi">25</span><span class="p">)</span>
-<span class="go">(9, 10)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sieve</span><span class="o">.</span><span class="n">search</span><span class="p">(</span><span class="mi">23</span><span class="p">)</span>
-<span class="go">(9, 9)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="ntheory-functions-reference">
-<h2>Ntheory Functions Reference<a class="headerlink" href="#ntheory-functions-reference" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.ntheory.generate.prime">
-<tt class="descclassname">sympy.ntheory.generate.</tt><tt class="descname">prime</tt><big>(</big><em>nth</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/generate.html#prime"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.generate.prime" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the nth prime, with the primes indexed as prime(1) = 2,
-prime(2) = 3, etc.... The nth prime is approximately n*log(n) and
-can never be larger than 2**n.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.ntheory.primetest.isprime" title="sympy.ntheory.primetest.isprime"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.ntheory.primetest.isprime</span></tt></a></dt>
-<dd>Test if n is prime</dd>
-<dt><a class="reference internal" href="#sympy.ntheory.generate.primerange" title="sympy.ntheory.generate.primerange"><tt class="xref py py-obj docutils literal"><span class="pre">primerange</span></tt></a></dt>
-<dd>Generate all primes in a given range</dd>
-<dt><a class="reference internal" href="#sympy.ntheory.generate.primepi" title="sympy.ntheory.generate.primepi"><tt class="xref py py-obj docutils literal"><span class="pre">primepi</span></tt></a></dt>
-<dd>Return the number of primes less than or equal to n</dd>
-</dl>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://primes.utm.edu/glossary/xpage/BertrandsPostulate.html">http://primes.utm.edu/glossary/xpage/BertrandsPostulate.html</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">prime</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">prime</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">29</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">prime</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.generate.primepi">
-<tt class="descclassname">sympy.ntheory.generate.</tt><tt class="descname">primepi</tt><big>(</big><em>n</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/generate.html#primepi"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.generate.primepi" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the value of the prime counting function pi(n) = the number
-of prime numbers less than or equal to n.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.ntheory.primetest.isprime" title="sympy.ntheory.primetest.isprime"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.ntheory.primetest.isprime</span></tt></a></dt>
-<dd>Test if n is prime</dd>
-<dt><a class="reference internal" href="#sympy.ntheory.generate.primerange" title="sympy.ntheory.generate.primerange"><tt class="xref py py-obj docutils literal"><span class="pre">primerange</span></tt></a></dt>
-<dd>Generate all primes in a given range</dd>
-<dt><a class="reference internal" href="#sympy.ntheory.generate.prime" title="sympy.ntheory.generate.prime"><tt class="xref py py-obj docutils literal"><span class="pre">prime</span></tt></a></dt>
-<dd>Return the nth prime</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">primepi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primepi</span><span class="p">(</span><span class="mi">25</span><span class="p">)</span>
-<span class="go">9</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.generate.nextprime">
-<tt class="descclassname">sympy.ntheory.generate.</tt><tt class="descname">nextprime</tt><big>(</big><em>n</em>, <em>ith=1</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/generate.html#nextprime"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.generate.nextprime" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the ith prime greater than n.</p>
-<p>i must be an integer.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.ntheory.generate.prevprime" title="sympy.ntheory.generate.prevprime"><tt class="xref py py-obj docutils literal"><span class="pre">prevprime</span></tt></a></dt>
-<dd>Return the largest prime smaller than n</dd>
-<dt><a class="reference internal" href="#sympy.ntheory.generate.primerange" title="sympy.ntheory.generate.primerange"><tt class="xref py py-obj docutils literal"><span class="pre">primerange</span></tt></a></dt>
-<dd>Generate all primes in a given range</dd>
-</dl>
-</div>
-<p class="rubric">Notes</p>
-<p>Potential primes are located at 6*j +/- 1. This
-property is used during searching.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">nextprime</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="n">nextprime</span><span class="p">(</span><span class="n">i</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">15</span><span class="p">)]</span>
-<span class="go">[(10, 11), (11, 13), (12, 13), (13, 17), (14, 17)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nextprime</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">ith</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span> <span class="c"># the 2nd prime after 2</span>
-<span class="go">5</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.generate.prevprime">
-<tt class="descclassname">sympy.ntheory.generate.</tt><tt class="descname">prevprime</tt><big>(</big><em>n</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/generate.html#prevprime"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.generate.prevprime" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the largest prime smaller than n.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.ntheory.generate.nextprime" title="sympy.ntheory.generate.nextprime"><tt class="xref py py-obj docutils literal"><span class="pre">nextprime</span></tt></a></dt>
-<dd>Return the ith prime greater than n</dd>
-<dt><a class="reference internal" href="#sympy.ntheory.generate.primerange" title="sympy.ntheory.generate.primerange"><tt class="xref py py-obj docutils literal"><span class="pre">primerange</span></tt></a></dt>
-<dd>Generates all primes in a given range</dd>
-</dl>
-</div>
-<p class="rubric">Notes</p>
-<p>Potential primes are located at 6*j +/- 1. This
-property is used during searching.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">prevprime</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[(</span><span class="n">i</span><span class="p">,</span> <span class="n">prevprime</span><span class="p">(</span><span class="n">i</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">15</span><span class="p">)]</span>
-<span class="go">[(10, 7), (11, 7), (12, 11), (13, 11), (14, 13)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.generate.primerange">
-<tt class="descclassname">sympy.ntheory.generate.</tt><tt class="descname">primerange</tt><big>(</big><em>a</em>, <em>b</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/generate.html#primerange"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.generate.primerange" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generate a list of all prime numbers in the range [a, b).</p>
-<p>If the range exists in the default sieve, the values will
-be returned from there; otherwise values will be returned
-but will not modifiy the sieve.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.ntheory.generate.nextprime" title="sympy.ntheory.generate.nextprime"><tt class="xref py py-obj docutils literal"><span class="pre">nextprime</span></tt></a></dt>
-<dd>Return the ith prime greater than n</dd>
-<dt><a class="reference internal" href="#sympy.ntheory.generate.prevprime" title="sympy.ntheory.generate.prevprime"><tt class="xref py py-obj docutils literal"><span class="pre">prevprime</span></tt></a></dt>
-<dd>Return the largest prime smaller than n</dd>
-<dt><a class="reference internal" href="#sympy.ntheory.generate.randprime" title="sympy.ntheory.generate.randprime"><tt class="xref py py-obj docutils literal"><span class="pre">randprime</span></tt></a></dt>
-<dd>Returns a random prime in a given range</dd>
-<dt><a class="reference internal" href="#sympy.ntheory.generate.primorial" title="sympy.ntheory.generate.primorial"><tt class="xref py py-obj docutils literal"><span class="pre">primorial</span></tt></a></dt>
-<dd>Returns the product of primes based on condition</dd>
-<dt><a class="reference internal" href="#sympy.ntheory.generate.Sieve.primerange" title="sympy.ntheory.generate.Sieve.primerange"><tt class="xref py py-obj docutils literal"><span class="pre">Sieve.primerange</span></tt></a></dt>
-<dd>return range from already computed primes or extend the sieve to contain the requested range.</dd>
-</dl>
-</div>
-<p class="rubric">Notes</p>
-<p>Some famous conjectures about the occurence of primes in a given
-range are [1]:</p>
-<ul>
-<li><dl class="first docutils">
-<dt>Twin primes: though often not, the following will give 2 primes</dt>
-<dd><dl class="first last docutils">
-<dt>an infinite number of times:</dt>
-<dd><p class="first last">primerange(6*n - 1, 6*n + 2)</p>
-</dd>
-</dl>
-</dd>
-</dl>
-</li>
-<li><dl class="first docutils">
-<dt>Legendre&#8217;s: the following always yields at least one prime</dt>
-<dd><p class="first last">primerange(n**2, (n+1)**2+1)</p>
-</dd>
-</dl>
-</li>
-<li><dl class="first docutils">
-<dt>Bertrand&#8217;s (proven): there is always a prime in the range</dt>
-<dd><p class="first last">primerange(n, 2*n)</p>
-</dd>
-</dl>
-</li>
-<li><dl class="first docutils">
-<dt>Brocard&#8217;s: there are at least four primes in the range</dt>
-<dd><p class="first last">primerange(prime(n)**2, prime(n+1)**2)</p>
-</dd>
-</dl>
-</li>
-</ul>
-<p>The average gap between primes is log(n) [2]; the gap between
-primes can be arbitrarily large since sequences of composite
-numbers are arbitrarily large, e.g. the numbers in the sequence
-n! + 2, n! + 3 ... n! + n are all composite.</p>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Prime_number">http://en.wikipedia.org/wiki/Prime_number</a></li>
-<li><a class="reference external" href="http://primes.utm.edu/notes/gaps.html">http://primes.utm.edu/notes/gaps.html</a></li>
-</ol>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">primerange</span><span class="p">,</span> <span class="n">sieve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">([</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">primerange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">30</span><span class="p">)])</span>
-<span class="go">[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]</span>
-</pre></div>
-</div>
-<p>The Sieve method, primerange, is generally faster but it will
-occupy more memory as the sieve stores values. The default
-instance of Sieve, named sieve, can be used:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">sieve</span><span class="o">.</span><span class="n">primerange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">30</span><span class="p">))</span>
-<span class="go">[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.generate.randprime">
-<tt class="descclassname">sympy.ntheory.generate.</tt><tt class="descname">randprime</tt><big>(</big><em>a</em>, <em>b</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/generate.html#randprime"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.generate.randprime" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a random prime number in the range [a, b).</p>
-<p>Bertrand&#8217;s postulate assures that
-randprime(a, 2*a) will always succeed for a &gt; 1.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.ntheory.generate.primerange" title="sympy.ntheory.generate.primerange"><tt class="xref py py-obj docutils literal"><span class="pre">primerange</span></tt></a></dt>
-<dd>Generate all primes in a given range</dd>
-</dl>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Bertrand's_postulate">http://en.wikipedia.org/wiki/Bertrand&#8217;s_postulate</a></li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">randprime</span><span class="p">,</span> <span class="n">isprime</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">randprime</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">30</span><span class="p">)</span> 
-<span class="go">13</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isprime</span><span class="p">(</span><span class="n">randprime</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">30</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.generate.primorial">
-<tt class="descclassname">sympy.ntheory.generate.</tt><tt class="descname">primorial</tt><big>(</big><em>n</em>, <em>nth=True</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/generate.html#primorial"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.generate.primorial" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the product of the first n primes (default) or
-the primes less than or equal to n (when <tt class="docutils literal"><span class="pre">nth=False</span></tt>).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory.generate</span> <span class="kn">import</span> <span class="n">primorial</span><span class="p">,</span> <span class="n">randprime</span><span class="p">,</span> <span class="n">primerange</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorint</span><span class="p">,</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">primefactors</span><span class="p">,</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primorial</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span> <span class="c"># the first 4 primes are 2, 3, 5, 7</span>
-<span class="go">210</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primorial</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">nth</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span> <span class="c"># primes &lt;= 4 are 2 and 3</span>
-<span class="go">6</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primorial</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primorial</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">nth</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primorial</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">101</span><span class="p">),</span> <span class="n">nth</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">210</span>
-</pre></div>
-</div>
-<p>One can argue that the primes are infinite since if you take
-a set of primes and multiply them together (e.g. the primorial) and
-then add or subtract 1, the result cannot be divided by any of the
-original factors, hence either 1 or more new primes must divide this
-product of primes.</p>
-<p>In this case, the number itself is a new prime:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factorint</span><span class="p">(</span><span class="n">primorial</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">{211: 1}</span>
-</pre></div>
-</div>
-<p>In this case two new primes are the factors:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factorint</span><span class="p">(</span><span class="n">primorial</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">{11: 1, 19: 1}</span>
-</pre></div>
-</div>
-<p>Here, some primes smaller and larger than the primes multiplied together
-are obtained:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">primerange</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">20</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">sorted</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">primefactors</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="o">*</span><span class="n">p</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">difference</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">p</span><span class="p">)))</span>
-<span class="go">[2, 5, 31, 149]</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.ntheory.generate.primerange" title="sympy.ntheory.generate.primerange"><tt class="xref py py-obj docutils literal"><span class="pre">primerange</span></tt></a></dt>
-<dd>Generate all primes in a given range</dd>
-</dl>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.generate.cycle_length">
-<tt class="descclassname">sympy.ntheory.generate.</tt><tt class="descname">cycle_length</tt><big>(</big><em>f</em>, <em>x0</em>, <em>nmax=None</em>, <em>values=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/generate.html#cycle_length"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.generate.cycle_length" title="Permalink to this definition">¶</a></dt>
-<dd><p>For a given iterated sequence, return a generator that gives
-the length of the iterated cycle (lambda) and the length of terms
-before the cycle begins (mu); if <tt class="docutils literal"><span class="pre">values</span></tt> is True then the
-terms of the sequence will be returned instead. The sequence is
-started with value <tt class="docutils literal"><span class="pre">x0</span></tt>.</p>
-<p>Note: more than the first lambda + mu terms may be returned and this
-is the cost of cycle detection with Brent&#8217;s method; there are, however,
-generally less terms calculated than would have been calculated if the
-proper ending point were determined, e.g. by using Floyd&#8217;s method.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory.generate</span> <span class="kn">import</span> <span class="n">cycle_length</span>
-</pre></div>
-</div>
-<p>This will yield successive values of i &lt;&#8211; func(i):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">iter</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">while</span> <span class="mi">1</span><span class="p">:</span>
-<span class="gp">... </span>        <span class="n">ii</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-<span class="gp">... </span>        <span class="k">yield</span> <span class="n">ii</span>
-<span class="gp">... </span>        <span class="n">i</span> <span class="o">=</span> <span class="n">ii</span>
-<span class="gp">...</span>
-</pre></div>
-</div>
-<p>A function is defined:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">func</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">i</span><span class="p">:</span> <span class="p">(</span><span class="n">i</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="mi">51</span>
-</pre></div>
-</div>
-<p>and given a seed of 4 and the mu and lambda terms calculated:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">next</span><span class="p">(</span><span class="n">cycle_length</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
-<span class="go">(6, 2)</span>
-</pre></div>
-</div>
-<p>We can see what is meant by looking at the output:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">cycle_length</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="n">values</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">ni</span> <span class="k">for</span> <span class="n">ni</span> <span class="ow">in</span> <span class="n">n</span><span class="p">)</span>
-<span class="go">[17, 35, 2, 5, 26, 14, 44, 50, 2, 5, 26, 14]</span>
-</pre></div>
-</div>
-<p>There are 6 repeating values after the first 2.</p>
-<p>If a sequence is suspected of being longer than you might wish, <tt class="docutils literal"><span class="pre">nmax</span></tt>
-can be used to exit early (and mu will be returned as None):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">next</span><span class="p">(</span><span class="n">cycle_length</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="n">nmax</span> <span class="o">=</span> <span class="mi">4</span><span class="p">))</span>
-<span class="go">(4, None)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">ni</span> <span class="k">for</span> <span class="n">ni</span> <span class="ow">in</span> <span class="n">cycle_length</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="n">nmax</span> <span class="o">=</span> <span class="mi">4</span><span class="p">,</span> <span class="n">values</span><span class="o">=</span><span class="bp">True</span><span class="p">)]</span>
-<span class="go">[17, 35, 2, 5]</span>
-</pre></div>
-</div>
-<dl class="docutils">
-<dt>Code modified from:</dt>
-<dd><a class="reference external" href="http://en.wikipedia.org/wiki/Cycle_detection">http://en.wikipedia.org/wiki/Cycle_detection</a>.</dd>
-</dl>
-</dd></dl>
-
-<span class="target" id="module-sympy.ntheory.factor_"></span><dl class="function">
-<dt id="sympy.ntheory.factor_.smoothness">
-<tt class="descclassname">sympy.ntheory.factor_.</tt><tt class="descname">smoothness</tt><big>(</big><em>n</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/factor_.html#smoothness"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.factor_.smoothness" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the B-smooth and B-power smooth values of n.</p>
-<p>The smoothness of n is the largest prime factor of n; the power-
-smoothness is the largest divisor raised to its multiplicity.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory.factor_</span> <span class="kn">import</span> <span class="n">smoothness</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">smoothness</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="mi">7</span><span class="o">*</span><span class="mi">3</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(3, 128)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">smoothness</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="mi">4</span><span class="o">*</span><span class="mi">13</span><span class="p">)</span>
-<span class="go">(13, 16)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">smoothness</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(2, 2)</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.ntheory.factor_.factorint" title="sympy.ntheory.factor_.factorint"><tt class="xref py py-obj docutils literal"><span class="pre">factorint</span></tt></a>, <a class="reference internal" href="#sympy.ntheory.factor_.smoothness_p" title="sympy.ntheory.factor_.smoothness_p"><tt class="xref py py-obj docutils literal"><span class="pre">smoothness_p</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.factor_.smoothness_p">
-<tt class="descclassname">sympy.ntheory.factor_.</tt><tt class="descname">smoothness_p</tt><big>(</big><em>n</em>, <em>m=-1</em>, <em>power=0</em>, <em>visual=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/factor_.html#smoothness_p"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.factor_.smoothness_p" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a list of [m, (p, (M, sm(p + m), psm(p + m)))...]
-where:</p>
-<ol class="arabic simple">
-<li>p**M is the base-p divisor of n</li>
-<li>sm(p + m) is the smoothness of p + m (m = -1 by default)</li>
-<li>psm(p + n) is the power smoothness of p + m</li>
-</ol>
-<p>The list is sorted according to smoothness (default) or by power smoothness
-if power=1.</p>
-<p>The smoothness of the numbers to the left (m = -1) or right (m = 1) of a
-factor govern the results that are obtained from the p +/- 1 type factoring
-methods.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory.factor_</span> <span class="kn">import</span> <span class="n">smoothness_p</span><span class="p">,</span> <span class="n">factorint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">smoothness_p</span><span class="p">(</span><span class="mi">10431</span><span class="p">,</span> <span class="n">m</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(1, [(3, (2, 2, 4)), (19, (1, 5, 5)), (61, (1, 31, 31))])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">smoothness_p</span><span class="p">(</span><span class="mi">10431</span><span class="p">)</span>
-<span class="go">(-1, [(3, (2, 2, 2)), (19, (1, 3, 9)), (61, (1, 5, 5))])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">smoothness_p</span><span class="p">(</span><span class="mi">10431</span><span class="p">,</span> <span class="n">power</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(-1, [(3, (2, 2, 2)), (61, (1, 5, 5)), (19, (1, 3, 9))])</span>
-</pre></div>
-</div>
-<p>If visual=True then an annotated string will be returned:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">smoothness_p</span><span class="p">(</span><span class="mi">21477639576571</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span>
-<span class="go">p**i=4410317**1 has p-1 B=1787, B-pow=1787</span>
-<span class="go">p**i=4869863**1 has p-1 B=2434931, B-pow=2434931</span>
-</pre></div>
-</div>
-<p>This string can also be generated directly from a factorization dictionary
-and vice versa:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factorint</span><span class="p">(</span><span class="mi">17</span><span class="o">*</span><span class="mi">9</span><span class="p">)</span>
-<span class="go">{3: 2, 17: 1}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">smoothness_p</span><span class="p">(</span><span class="n">_</span><span class="p">)</span>
-<span class="go">&#39;p**i=3**2 has p-1 B=2, B-pow=2\np**i=17**1 has p-1 B=2, B-pow=16&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">smoothness_p</span><span class="p">(</span><span class="n">_</span><span class="p">)</span>
-<span class="go">{3: 2, 17: 1}</span>
-</pre></div>
-</div>
-<p>The table of the output logic is:</p>
-<blockquote>
-<div><table border="1" class="docutils">
-<colgroup>
-<col width="23%" />
-<col width="23%" />
-<col width="27%" />
-<col width="27%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head"><div class="first last line-block">
-<div class="line"><br /></div>
-</div>
-</th>
-<th class="head" colspan="3">Visual</th>
-</tr>
-<tr class="row-even"><th class="head">Input</th>
-<th class="head">True</th>
-<th class="head">False</th>
-<th class="head">other</th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-odd"><td>dict</td>
-<td>str</td>
-<td>tuple</td>
-<td>str</td>
-</tr>
-<tr class="row-even"><td>str</td>
-<td>str</td>
-<td>tuple</td>
-<td>dict</td>
-</tr>
-<tr class="row-odd"><td>tuple</td>
-<td>str</td>
-<td>tuple</td>
-<td>str</td>
-</tr>
-<tr class="row-even"><td>n</td>
-<td>str</td>
-<td>tuple</td>
-<td>tuple</td>
-</tr>
-<tr class="row-odd"><td>mul</td>
-<td>str</td>
-<td>tuple</td>
-<td>tuple</td>
-</tr>
-</tbody>
-</table>
-</div></blockquote>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.ntheory.factor_.factorint" title="sympy.ntheory.factor_.factorint"><tt class="xref py py-obj docutils literal"><span class="pre">factorint</span></tt></a>, <a class="reference internal" href="#sympy.ntheory.factor_.smoothness" title="sympy.ntheory.factor_.smoothness"><tt class="xref py py-obj docutils literal"><span class="pre">smoothness</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.factor_.trailing">
-<tt class="descclassname">sympy.ntheory.factor_.</tt><tt class="descname">trailing</tt><big>(</big><em>n</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/factor_.html#trailing"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.factor_.trailing" title="Permalink to this definition">¶</a></dt>
-<dd><p>Count the number of trailing zero digits in the binary
-representation of n, i.e. determine the largest power of 2
-that divides n.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">trailing</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">trailing</span><span class="p">(</span><span class="mi">128</span><span class="p">)</span>
-<span class="go">7</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">trailing</span><span class="p">(</span><span class="mi">63</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.factor_.multiplicity">
-<tt class="descclassname">sympy.ntheory.factor_.</tt><tt class="descname">multiplicity</tt><big>(</big><em>p</em>, <em>n</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/factor_.html#multiplicity"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.factor_.multiplicity" title="Permalink to this definition">¶</a></dt>
-<dd><p>Find the greatest integer m such that p**m divides n.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">multiplicity</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">multiplicity</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">25</span><span class="p">,</span> <span class="mi">125</span><span class="p">,</span> <span class="mi">250</span><span class="p">]]</span>
-<span class="go">[0, 1, 2, 3, 3]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.factor_.perfect_power">
-<tt class="descclassname">sympy.ntheory.factor_.</tt><tt class="descname">perfect_power</tt><big>(</big><em>n</em>, <em>candidates=None</em>, <em>big=True</em>, <em>factor=True</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/factor_.html#perfect_power"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.factor_.perfect_power" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return <tt class="docutils literal"><span class="pre">(b,</span> <span class="pre">e)</span></tt> such that <tt class="docutils literal"><span class="pre">n</span></tt> == <tt class="docutils literal"><span class="pre">b**e</span></tt> if <tt class="docutils literal"><span class="pre">n</span></tt> is a
-perfect power; otherwise return <tt class="docutils literal"><span class="pre">False</span></tt>.</p>
-<p>By default, the base is recursively decomposed and the exponents
-collected so the largest possible <tt class="docutils literal"><span class="pre">e</span></tt> is sought. If <tt class="docutils literal"><span class="pre">big=False</span></tt>
-then the smallest possible <tt class="docutils literal"><span class="pre">e</span></tt> (thus prime) will be chosen.</p>
-<p>If <tt class="docutils literal"><span class="pre">candidates</span></tt> for exponents are given, they are assumed to be sorted
-and the first one that is larger than the computed maximum will signal
-failure for the routine.</p>
-<p>If <tt class="docutils literal"><span class="pre">factor=True</span></tt> then simultaneous factorization of n is attempted
-since finding a factor indicates the only possible root for n. This
-is True by default since only a few small factors will be tested in
-the course of searching for the perfect power.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">perfect_power</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">perfect_power</span><span class="p">(</span><span class="mi">16</span><span class="p">)</span>
-<span class="go">(2, 4)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">perfect_power</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="n">big</span> <span class="o">=</span> <span class="bp">False</span><span class="p">)</span>
-<span class="go">(4, 2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.factor_.pollard_rho">
-<tt class="descclassname">sympy.ntheory.factor_.</tt><tt class="descname">pollard_rho</tt><big>(</big><em>n</em>, <em>s=2</em>, <em>a=1</em>, <em>retries=5</em>, <em>seed=1234</em>, <em>max_steps=None</em>, <em>F=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/factor_.html#pollard_rho"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.factor_.pollard_rho" title="Permalink to this definition">¶</a></dt>
-<dd><p>Use Pollard&#8217;s rho method to try to extract a nontrivial factor
-of <tt class="docutils literal"><span class="pre">n</span></tt>. The returned factor may be a composite number. If no
-factor is found, <tt class="docutils literal"><span class="pre">None</span></tt> is returned.</p>
-<p>The algorithm generates pseudo-random values of x with a generator
-function, replacing x with F(x). If F is not supplied then the
-function x**2 + <tt class="docutils literal"><span class="pre">a</span></tt> is used. The first value supplied to F(x) is <tt class="docutils literal"><span class="pre">s</span></tt>.
-Upon failure (if <tt class="docutils literal"><span class="pre">retries</span></tt> is &gt; 0) a new <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">s</span></tt> will be
-supplied; the <tt class="docutils literal"><span class="pre">a</span></tt> will be ignored if F was supplied.</p>
-<p>The sequence of numbers generated by such functions generally have a
-a lead-up to some number and then loop around back to that number and
-begin to repeat the sequence, e.g. 1, 2, 3, 4, 5, 3, 4, 5 &#8211; this leader
-and loop look a bit like the Greek letter rho, and thus the name, &#8216;rho&#8217;.</p>
-<p>For a given function, very different leader-loop values can be obtained
-so it is a good idea to allow for retries:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory.generate</span> <span class="kn">import</span> <span class="n">cycle_length</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="mi">16843009</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:(</span><span class="mi">2048</span><span class="o">*</span><span class="nb">pow</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">+</span> <span class="mi">32767</span><span class="p">)</span> <span class="o">%</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&#39;loop length = </span><span class="si">%4i</span><span class="s">; leader length = </span><span class="si">%3i</span><span class="s">&#39;</span> <span class="o">%</span> <span class="nb">next</span><span class="p">(</span><span class="n">cycle_length</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">s</span><span class="p">)))</span>
-<span class="gp">...</span>
-<span class="go">loop length = 2489; leader length =  42</span>
-<span class="go">loop length =   78; leader length = 120</span>
-<span class="go">loop length = 1482; leader length =  99</span>
-<span class="go">loop length = 1482; leader length = 285</span>
-<span class="go">loop length = 1482; leader length = 100</span>
-</pre></div>
-</div>
-<p>Here is an explicit example where there is a two element leadup to
-a sequence of 3 numbers (11, 14, 4) that then repeat:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">=</span><span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">9</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">x</span><span class="o">=</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="mi">12</span><span class="p">)</span><span class="o">%</span><span class="mi">17</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">end</span><span class="o">=</span><span class="s">&#39; &#39;</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="go">16 13 11 14 4 11 14 4 11</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">next</span><span class="p">(</span><span class="n">cycle_length</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="mi">12</span><span class="p">)</span><span class="o">%</span><span class="mi">17</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">(3, 2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">cycle_length</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="mi">12</span><span class="p">)</span><span class="o">%</span><span class="mi">17</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">values</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">[16, 13, 11, 14, 4]</span>
-</pre></div>
-</div>
-<p>Instead of checking the differences of all generated values for a gcd
-with n, only the kth and 2*kth numbers are checked, e.g. 1st and 2nd,
-2nd and 4th, 3rd and 6th until it has been detected that the loop has been
-traversed. Loops may be many thousands of steps long before rho finds a
-factor or reports failure. If <tt class="docutils literal"><span class="pre">max_steps</span></tt> is specified, the iteration
-is cancelled with a failure after the specified number of steps.</p>
-<p class="rubric">References</p>
-<ul class="simple">
-<li>Richard Crandall &amp; Carl Pomerance (2005), &#8220;Prime Numbers:
-A Computational Perspective&#8221;, Springer, 2nd edition, 229-231</li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pollard_rho</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="o">=</span><span class="mi">16843009</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="o">=</span><span class="k">lambda</span> <span class="n">x</span><span class="p">:(</span><span class="mi">2048</span><span class="o">*</span><span class="nb">pow</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="n">n</span><span class="p">)</span> <span class="o">+</span> <span class="mi">32767</span><span class="p">)</span> <span class="o">%</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pollard_rho</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span><span class="n">F</span><span class="p">)</span>
-<span class="go">257</span>
-</pre></div>
-</div>
-<p>Use the default setting with a bad value of <tt class="docutils literal"><span class="pre">a</span></tt> and no retries:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pollard_rho</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="n">n</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="n">retries</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>If retries is &gt; 0 then perhaps the problem will correct itself when
-new values are generated for a:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pollard_rho</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="n">n</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="n">retries</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">257</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.factor_.pollard_pm1">
-<tt class="descclassname">sympy.ntheory.factor_.</tt><tt class="descname">pollard_pm1</tt><big>(</big><em>n</em>, <em>B=10</em>, <em>a=2</em>, <em>retries=0</em>, <em>seed=1234</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/factor_.html#pollard_pm1"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.factor_.pollard_pm1" title="Permalink to this definition">¶</a></dt>
-<dd><p>Use Pollard&#8217;s p-1 method to try to extract a nontrivial factor
-of <tt class="docutils literal"><span class="pre">n</span></tt>. Either a divisor (perhaps composite) or <tt class="docutils literal"><span class="pre">None</span></tt> is returned.</p>
-<p>The value of <tt class="docutils literal"><span class="pre">a</span></tt> is the base that is used in the test gcd(a**M - 1, n).
-The default is 2.  If <tt class="docutils literal"><span class="pre">retries</span></tt> &gt; 0 then if no factor is found after the
-first attempt, a new <tt class="docutils literal"><span class="pre">a</span></tt> will be generated randomly (using the <tt class="docutils literal"><span class="pre">seed</span></tt>)
-and the process repeated.</p>
-<p>Note: the value of M is lcm(1..B) = reduce(ilcm, range(2, B + 1)).</p>
-<p>A search is made for factors next to even numbers having a power smoothness
-less than <tt class="docutils literal"><span class="pre">B</span></tt>. Choosing a larger B increases the likelihood of finding a
-larger factor but takes longer. Whether a factor of n is found or not
-depends on <tt class="docutils literal"><span class="pre">a</span></tt> and the power smoothness of the even mumber just less than
-the factor p (hence the name p - 1).</p>
-<p>Although some discussion of what constitutes a good <tt class="docutils literal"><span class="pre">a</span></tt> some
-descriptions are hard to interpret. At the modular.math site referenced
-below it is stated that if gcd(a**M - 1, n) = N then a**M % q**r is 1
-for every prime power divisor of N. But consider the following:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory.factor_</span> <span class="kn">import</span> <span class="n">smoothness_p</span><span class="p">,</span> <span class="n">pollard_pm1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="o">=</span><span class="mi">257</span><span class="o">*</span><span class="mi">1009</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">smoothness_p</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">(-1, [(257, (1, 2, 256)), (1009, (1, 7, 16))])</span>
-</pre></div>
-</div>
-<p>So we should (and can) find a root with B=16:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pollard_pm1</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">B</span><span class="o">=</span><span class="mi">16</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">1009</span>
-</pre></div>
-</div>
-<p>If we attempt to increase B to 256 we find that it doesn&#8217;t work:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pollard_pm1</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">B</span><span class="o">=</span><span class="mi">256</span><span class="p">)</span>
-<span class="go">&gt;&gt;&gt;</span>
-</pre></div>
-</div>
-<p>But if the value of <tt class="docutils literal"><span class="pre">a</span></tt> is changed we find that only multiples of
-257 work, e.g.:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pollard_pm1</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">B</span><span class="o">=</span><span class="mi">256</span><span class="p">,</span> <span class="n">a</span><span class="o">=</span><span class="mi">257</span><span class="p">)</span>
-<span class="go">1009</span>
-</pre></div>
-</div>
-<p>Checking different <tt class="docutils literal"><span class="pre">a</span></tt> values shows that all the ones that didn&#8217;t
-work had a gcd value not equal to <tt class="docutils literal"><span class="pre">n</span></tt> but equal to one of the
-factors:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core.numbers</span> <span class="kn">import</span> <span class="n">ilcm</span><span class="p">,</span> <span class="n">igcd</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorint</span><span class="p">,</span> <span class="n">Pow</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">256</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">M</span> <span class="o">=</span> <span class="n">ilcm</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">set</span><span class="p">([</span><span class="n">igcd</span><span class="p">(</span><span class="nb">pow</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">256</span><span class="p">)</span> <span class="k">if</span>
-<span class="gp">... </span>     <span class="n">igcd</span><span class="p">(</span><span class="nb">pow</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">!=</span> <span class="n">n</span><span class="p">])</span>
-<span class="go">set([1009])</span>
-</pre></div>
-</div>
-<p>But does aM % d for every divisor of n give 1?</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">aM</span> <span class="o">=</span> <span class="nb">pow</span><span class="p">(</span><span class="mi">255</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[(</span><span class="n">d</span><span class="p">,</span> <span class="n">aM</span><span class="o">%</span><span class="n">Pow</span><span class="p">(</span><span class="o">*</span><span class="n">d</span><span class="o">.</span><span class="n">args</span><span class="p">))</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">factorint</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span><span class="o">.</span><span class="n">args</span><span class="p">]</span>
-<span class="go">[(257**1, 1), (1009**1, 1)]</span>
-</pre></div>
-</div>
-<p>No, only one of them. So perhaps the principle is that a root will
-be found for a given value of B provided that:</p>
-<ol class="arabic simple">
-<li>the power smoothness of the p - 1 value next to the root
-does not exceed B</li>
-<li>a**M % p != 1 for any of the divisors of n.</li>
-</ol>
-<p>By trying more than one <tt class="docutils literal"><span class="pre">a</span></tt> it is possible that one of them
-will yield a factor.</p>
-<p class="rubric">References</p>
-<ul>
-<li><p class="first">Richard Crandall &amp; Carl Pomerance (2005), &#8220;Prime Numbers:
-A Computational Perspective&#8221;, Springer, 2nd edition, 236-238</p>
-</li>
-<li><dl class="first docutils">
-<dt><a class="reference external" href="http://modular.math.washington.edu/edu/2007/spring/ent/ent-html/">http://modular.math.washington.edu/edu/2007/spring/ent/ent-html/</a></dt>
-<dd><p class="first last">node81.html</p>
-</dd>
-</dl>
-</li>
-<li><p class="first"><a class="reference external" href="http://www.cs.toronto.edu/~yuvalf/Factorization.pdf">http://www.cs.toronto.edu/~yuvalf/Factorization.pdf</a></p>
-</li>
-</ul>
-<p class="rubric">Examples</p>
-<p>With the default smoothness bound, this number can&#8217;t be cracked:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">pollard_pm1</span><span class="p">,</span> <span class="n">primefactors</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pollard_pm1</span><span class="p">(</span><span class="mi">21477639576571</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Increasing the smoothness bound helps:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pollard_pm1</span><span class="p">(</span><span class="mi">21477639576571</span><span class="p">,</span> <span class="n">B</span><span class="o">=</span><span class="mi">2000</span><span class="p">)</span>
-<span class="go">4410317</span>
-</pre></div>
-</div>
-<p>Looking at the smoothness of the factors of this number we find:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">flatten</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory.factor_</span> <span class="kn">import</span> <span class="n">smoothness_p</span><span class="p">,</span> <span class="n">factorint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">smoothness_p</span><span class="p">(</span><span class="mi">21477639576571</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span>
-<span class="go">p**i=4410317**1 has p-1 B=1787, B-pow=1787</span>
-<span class="go">p**i=4869863**1 has p-1 B=2434931, B-pow=2434931</span>
-</pre></div>
-</div>
-<p>The B and B-pow are the same for the p - 1 factorizations of the divisors
-because those factorizations had a very large prime factor:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factorint</span><span class="p">(</span><span class="mi">4410317</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">{2: 2, 617: 1, 1787: 1}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorint</span><span class="p">(</span><span class="mi">4869863</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">{2: 1, 2434931: 1}</span>
-</pre></div>
-</div>
-<p>Note that until B reaches the B-pow value of 1787, the number is not cracked;</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pollard_pm1</span><span class="p">(</span><span class="mi">21477639576571</span><span class="p">,</span> <span class="n">B</span><span class="o">=</span><span class="mi">1786</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pollard_pm1</span><span class="p">(</span><span class="mi">21477639576571</span><span class="p">,</span> <span class="n">B</span><span class="o">=</span><span class="mi">1787</span><span class="p">)</span>
-<span class="go">4410317</span>
-</pre></div>
-</div>
-<p>The B value has to do with the factors of the number next to the divisor,
-not the divisors themselves. A worst case scenario is that the number next
-to the factor p has a large prime divisisor or is a perfect power. If these
-conditions apply then the power-smoothness will be about p/2 or p. The more
-realistic is that there will be a large prime factor next to p requiring
-a B value on the order of p/2. Although primes may have been searched for
-up to this level, the p/2 is a factor of p - 1, something that we don&#8217;t
-know. The modular.math reference below states that 15% of numbers in the
-range of 10**15 to 15**15 + 10**4 are 10**6 power smooth so a B of 10**6
-will fail 85% of the time in that range. From 10**8 to 10**8 + 10**3 the
-percentages are nearly reversed...but in that range the simple trial
-division is quite fast.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.factor_.factorint">
-<tt class="descclassname">sympy.ntheory.factor_.</tt><tt class="descname">factorint</tt><big>(</big><em>n</em>, <em>limit=None</em>, <em>use_trial=True</em>, <em>use_rho=True</em>, <em>use_pm1=True</em>, <em>verbose=False</em>, <em>visual=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/factor_.html#factorint"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.factor_.factorint" title="Permalink to this definition">¶</a></dt>
-<dd><p>Given a positive integer <tt class="docutils literal"><span class="pre">n</span></tt>, <tt class="docutils literal"><span class="pre">factorint(n)</span></tt> returns a dict containing
-the prime factors of <tt class="docutils literal"><span class="pre">n</span></tt> as keys and their respective multiplicities
-as values. For example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">factorint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorint</span><span class="p">(</span><span class="mi">2000</span><span class="p">)</span>    <span class="c"># 2000 = (2**4) * (5**3)</span>
-<span class="go">{2: 4, 5: 3}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorint</span><span class="p">(</span><span class="mi">65537</span><span class="p">)</span>   <span class="c"># This number is prime</span>
-<span class="go">{65537: 1}</span>
-</pre></div>
-</div>
-<p>For input less than 2, factorint behaves as follows:</p>
-<blockquote>
-<div><ul class="simple">
-<li><tt class="docutils literal"><span class="pre">factorint(1)</span></tt> returns the empty factorization, <tt class="docutils literal"><span class="pre">{}</span></tt></li>
-<li><tt class="docutils literal"><span class="pre">factorint(0)</span></tt> returns <tt class="docutils literal"><span class="pre">{0:1}</span></tt></li>
-<li><tt class="docutils literal"><span class="pre">factorint(-n)</span></tt> adds <tt class="docutils literal"><span class="pre">-1:1</span></tt> to the factors and then factors <tt class="docutils literal"><span class="pre">n</span></tt></li>
-</ul>
-</div></blockquote>
-<p>Partial Factorization:</p>
-<p>If <tt class="docutils literal"><span class="pre">limit</span></tt> (&gt; 3) is specified, the search is stopped after performing
-trial division up to (and including) the limit (or taking a
-corresponding number of rho/p-1 steps). This is useful if one has
-a large number and only is interested in finding small factors (if
-any). Note that setting a limit does not prevent larger factors
-from being found early; it simply means that the largest factor may
-be composite. Since checking for perfect power is relatively cheap, it is
-done regardless of the limit setting.</p>
-<p>This number, for example, has two small factors and a huge
-semi-prime factor that cannot be reduced easily:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">isprime</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="mi">1407633717262338957430697921446883</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">factorint</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">limit</span><span class="o">=</span><span class="mi">10000</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">==</span> <span class="p">{</span><span class="mi">991</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">202916782076162456022877024859</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">:</span> <span class="mi">1</span><span class="p">}</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isprime</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="n">f</span><span class="p">))</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>This number has a small factor and a residual perfect power whose
-base is greater than the limit:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factorint</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="mi">101</span><span class="o">**</span><span class="mi">7</span><span class="p">,</span> <span class="n">limit</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">{3: 1, 101: 7}</span>
-</pre></div>
-</div>
-<p>Visual Factorization:</p>
-<p>If <tt class="docutils literal"><span class="pre">visual</span></tt> is set to <tt class="docutils literal"><span class="pre">True</span></tt>, then it will return a visual
-factorization of the integer.  For example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">factorint</span><span class="p">(</span><span class="mi">4200</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go"> 3  1  2  1</span>
-<span class="go">2 *3 *5 *7</span>
-</pre></div>
-</div>
-<p>Note that this is achieved by using the evaluate=False flag in Mul
-and Pow. If you do other manipulations with an expression where
-evaluate=False, it may evaluate.  Therefore, you should use the
-visual option only for visualization, and use the normal dictionary
-returned by visual=False if you want to perform operations on the
-factors.</p>
-<p>You can easily switch between the two forms by sending them back to
-factorint:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">Pow</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">regular</span> <span class="o">=</span> <span class="n">factorint</span><span class="p">(</span><span class="mi">1764</span><span class="p">);</span> <span class="n">regular</span>
-<span class="go">{2: 2, 3: 2, 7: 2}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">factorint</span><span class="p">(</span><span class="n">regular</span><span class="p">))</span>
-<span class="go"> 2  2  2</span>
-<span class="go">2 *3 *7</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">visual</span> <span class="o">=</span> <span class="n">factorint</span><span class="p">(</span><span class="mi">1764</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="bp">True</span><span class="p">);</span> <span class="n">pprint</span><span class="p">(</span><span class="n">visual</span><span class="p">)</span>
-<span class="go"> 2  2  2</span>
-<span class="go">2 *3 *7</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">factorint</span><span class="p">(</span><span class="n">visual</span><span class="p">))</span>
-<span class="go">{2: 2, 3: 2, 7: 2}</span>
-</pre></div>
-</div>
-<p>If you want to send a number to be factored in a partially factored form
-you can do so with a dictionary or unevaluated expression:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factorint</span><span class="p">(</span><span class="n">factorint</span><span class="p">({</span><span class="mi">4</span><span class="p">:</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">12</span><span class="p">:</span> <span class="mi">3</span><span class="p">}))</span> <span class="c"># twice to toggle to dict form</span>
-<span class="go">{2: 10, 3: 3}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorint</span><span class="p">(</span><span class="n">Mul</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="o">**</span><span class="nb">dict</span><span class="p">(</span><span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)))</span>
-<span class="go">{2: 4, 3: 1}</span>
-</pre></div>
-</div>
-<p>The table of the output logic is:</p>
-<blockquote>
-<div><table border="1" class="docutils">
-<colgroup>
-<col width="23%" />
-<col width="23%" />
-<col width="27%" />
-<col width="27%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head">&nbsp;</th>
-<th class="head" colspan="3">&nbsp;</th>
-</tr>
-<tr class="row-even"><th class="head">Input</th>
-<th class="head">True</th>
-<th class="head">False</th>
-<th class="head">other</th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-odd"><td>dict</td>
-<td>mul</td>
-<td>dict</td>
-<td>mul</td>
-</tr>
-<tr class="row-even"><td>n</td>
-<td>mul</td>
-<td>dict</td>
-<td>dict</td>
-</tr>
-<tr class="row-odd"><td>mul</td>
-<td>mul</td>
-<td>dict</td>
-<td>dict</td>
-</tr>
-</tbody>
-</table>
-</div></blockquote>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.ntheory.factor_.smoothness" title="sympy.ntheory.factor_.smoothness"><tt class="xref py py-obj docutils literal"><span class="pre">smoothness</span></tt></a>, <a class="reference internal" href="#sympy.ntheory.factor_.smoothness_p" title="sympy.ntheory.factor_.smoothness_p"><tt class="xref py py-obj docutils literal"><span class="pre">smoothness_p</span></tt></a>, <a class="reference internal" href="#sympy.ntheory.factor_.divisors" title="sympy.ntheory.factor_.divisors"><tt class="xref py py-obj docutils literal"><span class="pre">divisors</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>Algorithm:</p>
-<p>The function switches between multiple algorithms. Trial division
-quickly finds small factors (of the order 1-5 digits), and finds
-all large factors if given enough time. The Pollard rho and p-1
-algorithms are used to find large factors ahead of time; they
-will often find factors of the order of 10 digits within a few
-seconds:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factors</span> <span class="o">=</span> <span class="n">factorint</span><span class="p">(</span><span class="mi">12345678910111213141516</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">base</span><span class="p">,</span> <span class="n">exp</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">factors</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="n">base</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="go">2 2</span>
-<span class="go">2507191691 1</span>
-<span class="go">1231026625769 1</span>
-</pre></div>
-</div>
-<p>Any of these methods can optionally be disabled with the following
-boolean parameters:</p>
-<blockquote>
-<div><ul class="simple">
-<li><tt class="docutils literal"><span class="pre">use_trial</span></tt>: Toggle use of trial division</li>
-<li><tt class="docutils literal"><span class="pre">use_rho</span></tt>: Toggle use of Pollard&#8217;s rho method</li>
-<li><tt class="docutils literal"><span class="pre">use_pm1</span></tt>: Toggle use of Pollard&#8217;s p-1 method</li>
-</ul>
-</div></blockquote>
-<p><tt class="docutils literal"><span class="pre">factorint</span></tt> also periodically checks if the remaining part is
-a prime number or a perfect power, and in those cases stops.</p>
-<p>If <tt class="docutils literal"><span class="pre">verbose</span></tt> is set to <tt class="docutils literal"><span class="pre">True</span></tt>, detailed progress is printed.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.factor_.primefactors">
-<tt class="descclassname">sympy.ntheory.factor_.</tt><tt class="descname">primefactors</tt><big>(</big><em>n</em>, <em>limit=None</em>, <em>verbose=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/factor_.html#primefactors"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.factor_.primefactors" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a sorted list of n&#8217;s prime factors, ignoring multiplicity
-and any composite factor that remains if the limit was set too low
-for complete factorization. Unlike factorint(), primefactors() does
-not return -1 or 0.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.ntheory.factor_.divisors" title="sympy.ntheory.factor_.divisors"><tt class="xref py py-obj docutils literal"><span class="pre">divisors</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">primefactors</span><span class="p">,</span> <span class="n">factorint</span><span class="p">,</span> <span class="n">isprime</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primefactors</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span>
-<span class="go">[2, 3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primefactors</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">[5]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">sorted</span><span class="p">(</span><span class="n">factorint</span><span class="p">(</span><span class="mi">123456</span><span class="p">)</span><span class="o">.</span><span class="n">items</span><span class="p">())</span>
-<span class="go">[(2, 6), (3, 1), (643, 1)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primefactors</span><span class="p">(</span><span class="mi">123456</span><span class="p">)</span>
-<span class="go">[2, 3, 643]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">sorted</span><span class="p">(</span><span class="n">factorint</span><span class="p">(</span><span class="mi">10000000001</span><span class="p">,</span> <span class="n">limit</span><span class="o">=</span><span class="mi">200</span><span class="p">)</span><span class="o">.</span><span class="n">items</span><span class="p">())</span>
-<span class="go">[(101, 1), (99009901, 1)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isprime</span><span class="p">(</span><span class="mi">99009901</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primefactors</span><span class="p">(</span><span class="mi">10000000001</span><span class="p">,</span> <span class="n">limit</span><span class="o">=</span><span class="mi">300</span><span class="p">)</span>
-<span class="go">[101]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.factor_.divisors">
-<tt class="descclassname">sympy.ntheory.factor_.</tt><tt class="descname">divisors</tt><big>(</big><em>n</em>, <em>generator=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/factor_.html#divisors"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.factor_.divisors" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return all divisors of n sorted from 1..n by default.
-If generator is True an unordered generator is returned.</p>
-<p>The number of divisors of n can be quite large if there are many
-prime factors (counting repeated factors). If only the number of
-factors is desired use divisor_count(n).</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.ntheory.factor_.primefactors" title="sympy.ntheory.factor_.primefactors"><tt class="xref py py-obj docutils literal"><span class="pre">primefactors</span></tt></a>, <a class="reference internal" href="#sympy.ntheory.factor_.factorint" title="sympy.ntheory.factor_.factorint"><tt class="xref py py-obj docutils literal"><span class="pre">factorint</span></tt></a>, <a class="reference internal" href="#sympy.ntheory.factor_.divisor_count" title="sympy.ntheory.factor_.divisor_count"><tt class="xref py py-obj docutils literal"><span class="pre">divisor_count</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">divisors</span><span class="p">,</span> <span class="n">divisor_count</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">divisors</span><span class="p">(</span><span class="mi">24</span><span class="p">)</span>
-<span class="go">[1, 2, 3, 4, 6, 8, 12, 24]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">divisor_count</span><span class="p">(</span><span class="mi">24</span><span class="p">)</span>
-<span class="go">8</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">divisors</span><span class="p">(</span><span class="mi">120</span><span class="p">,</span> <span class="n">generator</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">[1, 2, 4, 8, 3, 6, 12, 24, 5, 10, 20, 40, 15, 30, 60, 120]</span>
-</pre></div>
-</div>
-<p>This is a slightly modified version of Tim Peters referenced at:
-<a class="reference external" href="http://stackoverflow.com/questions/1010381/python-factorization">http://stackoverflow.com/questions/1010381/python-factorization</a></p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.factor_.divisor_count">
-<tt class="descclassname">sympy.ntheory.factor_.</tt><tt class="descname">divisor_count</tt><big>(</big><em>n</em>, <em>modulus=1</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/factor_.html#divisor_count"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.factor_.divisor_count" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the number of divisors of <tt class="docutils literal"><span class="pre">n</span></tt>. If <tt class="docutils literal"><span class="pre">modulus</span></tt> is not 1 then only
-those that are divisible by <tt class="docutils literal"><span class="pre">modulus</span></tt> are counted.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.ntheory.factor_.factorint" title="sympy.ntheory.factor_.factorint"><tt class="xref py py-obj docutils literal"><span class="pre">factorint</span></tt></a>, <a class="reference internal" href="#sympy.ntheory.factor_.divisors" title="sympy.ntheory.factor_.divisors"><tt class="xref py py-obj docutils literal"><span class="pre">divisors</span></tt></a>, <a class="reference internal" href="#sympy.ntheory.factor_.totient" title="sympy.ntheory.factor_.totient"><tt class="xref py py-obj docutils literal"><span class="pre">totient</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://www.mayer.dial.pipex.com/maths/formulae.htm">http://www.mayer.dial.pipex.com/maths/formulae.htm</a></li>
-</ul>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">divisor_count</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">divisor_count</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span>
-<span class="go">4</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.factor_.totient">
-<tt class="descclassname">sympy.ntheory.factor_.</tt><tt class="descname">totient</tt><big>(</big><em>n</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/factor_.html#totient"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.factor_.totient" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculate the Euler totient function phi(n)</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">totient</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">totient</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">totient</span><span class="p">(</span><span class="mi">25</span><span class="p">)</span>
-<span class="go">20</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.ntheory.factor_.divisor_count" title="sympy.ntheory.factor_.divisor_count"><tt class="xref py py-obj docutils literal"><span class="pre">divisor_count</span></tt></a></p>
-</div>
-</dd></dl>
-
-<span class="target" id="module-sympy.ntheory.modular"></span><dl class="function">
-<dt id="sympy.ntheory.modular.symmetric_residue">
-<tt class="descclassname">sympy.ntheory.modular.</tt><tt class="descname">symmetric_residue</tt><big>(</big><em>a</em>, <em>m</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/modular.html#symmetric_residue"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.modular.symmetric_residue" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the residual mod m such that it is within half of the modulus.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory.modular</span> <span class="kn">import</span> <span class="n">symmetric_residue</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">symmetric_residue</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">symmetric_residue</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
-<span class="go">-2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.modular.crt">
-<tt class="descclassname">sympy.ntheory.modular.</tt><tt class="descname">crt</tt><big>(</big><em>m</em>, <em>v</em>, <em>symmetric=False</em>, <em>check=True</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/modular.html#crt"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.modular.crt" title="Permalink to this definition">¶</a></dt>
-<dd><p>Chinese Remainder Theorem.</p>
-<p>The moduli in m are assumed to be pairwise coprime.  The output
-is then an integer f, such that f = v_i mod m_i for each pair out
-of v and m. If <tt class="docutils literal"><span class="pre">symmetric</span></tt> is False a positive integer will be
-returned, else |f| will be less than or equal to the LCM of the
-moduli, and thus f may be negative.</p>
-<p>If the moduli are not co-prime the correct result will be returned
-if/when the test of the result is found to be incorrect. This result
-will be None if there is no solution.</p>
-<p>The keyword <tt class="docutils literal"><span class="pre">check</span></tt> can be set to False if it is known that the moduli
-are coprime.</p>
-<p>As an example consider a set of residues <tt class="docutils literal"><span class="pre">U</span> <span class="pre">=</span> <span class="pre">[49,</span> <span class="pre">76,</span> <span class="pre">65]</span></tt>
-and a set of moduli <tt class="docutils literal"><span class="pre">M</span> <span class="pre">=</span> <span class="pre">[99,</span> <span class="pre">97,</span> <span class="pre">95]</span></tt>. Then we have:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory.modular</span> <span class="kn">import</span> <span class="n">crt</span><span class="p">,</span> <span class="n">solve_congruence</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">crt</span><span class="p">([</span><span class="mi">99</span><span class="p">,</span> <span class="mi">97</span><span class="p">,</span> <span class="mi">95</span><span class="p">],</span> <span class="p">[</span><span class="mi">49</span><span class="p">,</span> <span class="mi">76</span><span class="p">,</span> <span class="mi">65</span><span class="p">])</span>
-<span class="go">(639985, 912285)</span>
-</pre></div>
-</div>
-<p>This is the correct result because:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="mi">639985</span> <span class="o">%</span> <span class="n">m</span> <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">99</span><span class="p">,</span> <span class="mi">97</span><span class="p">,</span> <span class="mi">95</span><span class="p">]]</span>
-<span class="go">[49, 76, 65]</span>
-</pre></div>
-</div>
-<p>If the moduli are not co-prime, you may receive an incorrect result
-if you use <tt class="docutils literal"><span class="pre">check=False</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">crt</span><span class="p">([</span><span class="mi">12</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">17</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="n">check</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">(954, 1224)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="mi">954</span> <span class="o">%</span> <span class="n">m</span> <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">12</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">17</span><span class="p">]]</span>
-<span class="go">[6, 0, 2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">crt</span><span class="p">([</span><span class="mi">12</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">17</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span> <span class="ow">is</span> <span class="bp">None</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">crt</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
-<span class="go">(5, 6)</span>
-</pre></div>
-</div>
-<p>Note: the order of gf_crt&#8217;s arguments is reversed relative to crt,
-and that solve_congruence takes residue, modulus pairs.</p>
-<p>Programmer&#8217;s note: rather than checking that all pairs of moduli share
-no GCD (an O(n**2) test) and rather than factoring all moduli and seeing
-that there is no factor in common, a check that the result gives the
-indicated residuals is performed &#8211; an O(n) operation.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p><a class="reference internal" href="#sympy.ntheory.modular.solve_congruence" title="sympy.ntheory.modular.solve_congruence"><tt class="xref py py-obj docutils literal"><span class="pre">solve_congruence</span></tt></a></p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="polys/internals.html#sympy.polys.galoistools.gf_crt" title="sympy.polys.galoistools.gf_crt"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.polys.galoistools.gf_crt</span></tt></a></dt>
-<dd>low level crt routine used by this routine</dd>
-</dl>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.modular.crt1">
-<tt class="descclassname">sympy.ntheory.modular.</tt><tt class="descname">crt1</tt><big>(</big><em>m</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/modular.html#crt1"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.modular.crt1" title="Permalink to this definition">¶</a></dt>
-<dd><p>First part of Chinese Remainder Theorem, for multiple application.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory.modular</span> <span class="kn">import</span> <span class="n">crt1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">crt1</span><span class="p">([</span><span class="mi">18</span><span class="p">,</span> <span class="mi">42</span><span class="p">,</span> <span class="mi">6</span><span class="p">])</span>
-<span class="go">(4536, [252, 108, 756], [0, 2, 0])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.modular.crt2">
-<tt class="descclassname">sympy.ntheory.modular.</tt><tt class="descname">crt2</tt><big>(</big><em>m</em>, <em>v</em>, <em>mm</em>, <em>e</em>, <em>s</em>, <em>symmetric=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/modular.html#crt2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.modular.crt2" title="Permalink to this definition">¶</a></dt>
-<dd><p>Second part of Chinese Remainder Theorem, for multiple application.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory.modular</span> <span class="kn">import</span> <span class="n">crt1</span><span class="p">,</span> <span class="n">crt2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mm</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">s</span> <span class="o">=</span> <span class="n">crt1</span><span class="p">([</span><span class="mi">18</span><span class="p">,</span> <span class="mi">42</span><span class="p">,</span> <span class="mi">6</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">crt2</span><span class="p">([</span><span class="mi">18</span><span class="p">,</span> <span class="mi">42</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">mm</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span>
-<span class="go">(0, 4536)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.modular.solve_congruence">
-<tt class="descclassname">sympy.ntheory.modular.</tt><tt class="descname">solve_congruence</tt><big>(</big><em>*remainder_modulus_pairs</em>, <em>**hint</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/modular.html#solve_congruence"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.modular.solve_congruence" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the integer <tt class="docutils literal"><span class="pre">n</span></tt> that has the residual <tt class="docutils literal"><span class="pre">ai</span></tt> when it is
-divided by <tt class="docutils literal"><span class="pre">mi</span></tt> where the <tt class="docutils literal"><span class="pre">ai</span></tt> and <tt class="docutils literal"><span class="pre">mi</span></tt> are given as pairs to
-this function: ((a1, m1), (a2, m2), ...). If there is no solution,
-return None. Otherwise return <tt class="docutils literal"><span class="pre">n</span></tt> and its modulus.</p>
-<p>The <tt class="docutils literal"><span class="pre">mi</span></tt> values need not be co-prime. If it is known that the moduli are
-not co-prime then the hint <tt class="docutils literal"><span class="pre">check</span></tt> can be set to False (default=True) and
-the check for a quicker solution via crt() (valid when the moduli are
-co-prime) will be skipped.</p>
-<p>If the hint <tt class="docutils literal"><span class="pre">symmetric</span></tt> is True (default is False), the value of <tt class="docutils literal"><span class="pre">n</span></tt>
-will be within 1/2 of the modulus, possibly negative.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.ntheory.modular.crt" title="sympy.ntheory.modular.crt"><tt class="xref py py-obj docutils literal"><span class="pre">crt</span></tt></a></dt>
-<dd>high level routine implementing the Chinese Remainder Theorem</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory.modular</span> <span class="kn">import</span> <span class="n">solve_congruence</span>
-</pre></div>
-</div>
-<p>What number is 2 mod 3, 3 mod 5 and 2 mod 7?</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve_congruence</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">7</span><span class="p">))</span>
-<span class="go">(23, 105)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="mi">23</span> <span class="o">%</span> <span class="n">m</span> <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">]]</span>
-<span class="go">[2, 3, 2]</span>
-</pre></div>
-</div>
-<p>If you prefer to work with all remainder in one list and
-all moduli in another, send the arguments like this:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve_congruence</span><span class="p">(</span><span class="o">*</span><span class="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">))))</span>
-<span class="go">(23, 105)</span>
-</pre></div>
-</div>
-<p>The moduli need not be co-prime; in this case there may or
-may not be a solution:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve_congruence</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span> <span class="ow">is</span> <span class="bp">None</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve_congruence</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
-<span class="go">(5, 6)</span>
-</pre></div>
-</div>
-<p>The symmetric flag will make the result be within 1/2 of the modulus:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve_congruence</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">symmetric</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(-1, 6)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<span class="target" id="module-sympy.ntheory.multinomial"></span><dl class="function">
-<dt id="sympy.ntheory.multinomial.binomial_coefficients">
-<tt class="descclassname">sympy.ntheory.multinomial.</tt><tt class="descname">binomial_coefficients</tt><big>(</big><em>n</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/multinomial.html#binomial_coefficients"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.multinomial.binomial_coefficients" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a dictionary containing pairs <span class="math">\({(k1,k2) : C_kn}\)</span> where
-<span class="math">\(C_kn\)</span> are binomial coefficients and <span class="math">\(n=k1+k2\)</span>.
-Examples
-========</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">binomial_coefficients</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">binomial_coefficients</span><span class="p">(</span><span class="mi">9</span><span class="p">)</span>
-<span class="go">{(0, 9): 1, (1, 8): 9, (2, 7): 36, (3, 6): 84,</span>
-<span class="go"> (4, 5): 126, (5, 4): 126, (6, 3): 84, (7, 2): 36, (8, 1): 9, (9, 0): 1}</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.ntheory.multinomial.binomial_coefficients_list" title="sympy.ntheory.multinomial.binomial_coefficients_list"><tt class="xref py py-obj docutils literal"><span class="pre">binomial_coefficients_list</span></tt></a>, <a class="reference internal" href="#sympy.ntheory.multinomial.multinomial_coefficients" title="sympy.ntheory.multinomial.multinomial_coefficients"><tt class="xref py py-obj docutils literal"><span class="pre">multinomial_coefficients</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.multinomial.binomial_coefficients_list">
-<tt class="descclassname">sympy.ntheory.multinomial.</tt><tt class="descname">binomial_coefficients_list</tt><big>(</big><em>n</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/multinomial.html#binomial_coefficients_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.multinomial.binomial_coefficients_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a list of binomial coefficients as rows of the Pascal&#8217;s
-triangle.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.ntheory.multinomial.binomial_coefficients" title="sympy.ntheory.multinomial.binomial_coefficients"><tt class="xref py py-obj docutils literal"><span class="pre">binomial_coefficients</span></tt></a>, <a class="reference internal" href="#sympy.ntheory.multinomial.multinomial_coefficients" title="sympy.ntheory.multinomial.multinomial_coefficients"><tt class="xref py py-obj docutils literal"><span class="pre">multinomial_coefficients</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">binomial_coefficients_list</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">binomial_coefficients_list</span><span class="p">(</span><span class="mi">9</span><span class="p">)</span>
-<span class="go">[1, 9, 36, 84, 126, 126, 84, 36, 9, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.multinomial.multinomial_coefficients">
-<tt class="descclassname">sympy.ntheory.multinomial.</tt><tt class="descname">multinomial_coefficients</tt><big>(</big><em>m</em>, <em>n</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/multinomial.html#multinomial_coefficients"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.multinomial.multinomial_coefficients" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a dictionary containing pairs <tt class="docutils literal"><span class="pre">{(k1,k2,..,km)</span> <span class="pre">:</span> <span class="pre">C_kn}</span></tt>
-where <tt class="docutils literal"><span class="pre">C_kn</span></tt> are multinomial coefficients such that
-<tt class="docutils literal"><span class="pre">n=k1+k2+..+km</span></tt>.</p>
-<p>For example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">multinomial_coefficients</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">multinomial_coefficients</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span> <span class="c"># indirect doctest</span>
-<span class="go">{(0, 5): 1, (1, 4): 5, (2, 3): 10, (3, 2): 10, (4, 1): 5, (5, 0): 1}</span>
-</pre></div>
-</div>
-<p>The algorithm is based on the following result:</p>
-<div class="math">
-\[\binom{n}{k_1, \ldots, k_m} =
-\frac{k_1 + 1}{n - k_1} \sum_{i=2}^m \binom{n}{k_1 + 1, \ldots, k_i - 1, \ldots}\]</div>
-<p>Code contributed to Sage by Yann Laigle-Chapuy, copied with permission
-of the author.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.ntheory.multinomial.binomial_coefficients_list" title="sympy.ntheory.multinomial.binomial_coefficients_list"><tt class="xref py py-obj docutils literal"><span class="pre">binomial_coefficients_list</span></tt></a>, <a class="reference internal" href="#sympy.ntheory.multinomial.binomial_coefficients" title="sympy.ntheory.multinomial.binomial_coefficients"><tt class="xref py py-obj docutils literal"><span class="pre">binomial_coefficients</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.multinomial.multinomial_coefficients_iterator">
-<tt class="descclassname">sympy.ntheory.multinomial.</tt><tt class="descname">multinomial_coefficients_iterator</tt><big>(</big><em>m</em>, <em>n</em>, <em>_tuple=&lt;class 'tuple'&gt;</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/multinomial.html#multinomial_coefficients_iterator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.multinomial.multinomial_coefficients_iterator" title="Permalink to this definition">¶</a></dt>
-<dd><p>multinomial coefficient iterator</p>
-<p>This routine has been optimized for <span class="math">\(m\)</span> large with respect to <span class="math">\(n\)</span> by taking
-advantage of the fact that when the monomial tuples <span class="math">\(t\)</span> are stripped of
-zeros, their coefficient is the same as that of the monomial tuples from
-<tt class="docutils literal"><span class="pre">multinomial_coefficients(n,</span> <span class="pre">n)</span></tt>. Therefore, the latter coefficients are
-precomputed to save memory and time.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory.multinomial</span> <span class="kn">import</span> <span class="n">multinomial_coefficients</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m53</span><span class="p">,</span> <span class="n">m33</span> <span class="o">=</span> <span class="n">multinomial_coefficients</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="n">multinomial_coefficients</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m53</span><span class="p">[(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)]</span> <span class="o">==</span> <span class="n">m53</span><span class="p">[(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">)]</span> <span class="o">==</span> <span class="n">m53</span><span class="p">[(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">)]</span> <span class="o">==</span> <span class="n">m33</span><span class="p">[(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)]</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory.multinomial</span> <span class="kn">import</span> <span class="n">multinomial_coefficients_iterator</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">it</span> <span class="o">=</span> <span class="n">multinomial_coefficients_iterator</span><span class="p">(</span><span class="mi">20</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">next</span><span class="p">(</span><span class="n">it</span><span class="p">)</span>
-<span class="go">((3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<span class="target" id="module-sympy.ntheory.partitions_"></span><dl class="function">
-<dt id="sympy.ntheory.partitions_.npartitions">
-<tt class="descclassname">sympy.ntheory.partitions_.</tt><tt class="descname">npartitions</tt><big>(</big><em>n</em>, <em>verbose=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/partitions_.html#npartitions"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.partitions_.npartitions" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculate the partition function P(n), i.e. the number of ways that
-n can be written as a sum of positive integers.</p>
-<p>P(n) is computed using the Hardy-Ramanujan-Rademacher formula,
-described e.g. at <a class="reference external" href="http://mathworld.wolfram.com/PartitionFunctionP.html">http://mathworld.wolfram.com/PartitionFunctionP.html</a></p>
-<p>The correctness of this implementation has been tested for 10**n
-up to n = 8.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">npartitions</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">npartitions</span><span class="p">(</span><span class="mi">25</span><span class="p">)</span>
-<span class="go">1958</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<span class="target" id="module-sympy.ntheory.primetest"></span><dl class="function">
-<dt id="sympy.ntheory.primetest.mr">
-<tt class="descclassname">sympy.ntheory.primetest.</tt><tt class="descname">mr</tt><big>(</big><em>n</em>, <em>bases</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/primetest.html#mr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.primetest.mr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Perform a Miller-Rabin strong pseudoprime test on n using a
-given list of bases/witnesses.</p>
-<p class="rubric">References</p>
-<ul class="simple">
-<li>Richard Crandall &amp; Carl Pomerance (2005), &#8220;Prime Numbers:
-A Computational Perspective&#8221;, Springer, 2nd edition, 135-138</li>
-</ul>
-<p>A list of thresholds and the bases they require are here:
-<a class="reference external" href="http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test#Deterministic_variants_of_the_test">http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test#Deterministic_variants_of_the_test</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory.primetest</span> <span class="kn">import</span> <span class="n">mr</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mr</span><span class="p">(</span><span class="mi">1373651</span><span class="p">,</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mr</span><span class="p">(</span><span class="mi">479001599</span><span class="p">,</span> <span class="p">[</span><span class="mi">31</span><span class="p">,</span> <span class="mi">73</span><span class="p">])</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.primetest.isprime">
-<tt class="descclassname">sympy.ntheory.primetest.</tt><tt class="descname">isprime</tt><big>(</big><em>n</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/primetest.html#isprime"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.primetest.isprime" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test if n is a prime number (True) or not (False). For n &lt; 10**16 the
-answer is accurate; greater n values have a small probability of actually
-being pseudoprimes.</p>
-<p>Negative primes (e.g. -2) are not considered prime.</p>
-<p>The function first looks for trivial factors, and if none is found,
-performs a safe Miller-Rabin strong pseudoprime test with bases
-that are known to prove a number prime. Finally, a general Miller-Rabin
-test is done with the first k bases which, which will report a
-pseudoprime as a prime with an error of about 4**-k. The current value
-of k is 46 so the error is about 2 x 10**-28.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.ntheory.generate.primerange" title="sympy.ntheory.generate.primerange"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.ntheory.generate.primerange</span></tt></a></dt>
-<dd>Generates all primes in a given range</dd>
-<dt><a class="reference internal" href="#sympy.ntheory.generate.primepi" title="sympy.ntheory.generate.primepi"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.ntheory.generate.primepi</span></tt></a></dt>
-<dd>Return the number of primes less than or equal to n</dd>
-<dt><a class="reference internal" href="#sympy.ntheory.generate.prime" title="sympy.ntheory.generate.prime"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.ntheory.generate.prime</span></tt></a></dt>
-<dd>Return the nth prime</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">isprime</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isprime</span><span class="p">(</span><span class="mi">13</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">isprime</span><span class="p">(</span><span class="mi">15</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<span class="target" id="module-sympy.ntheory.residue_ntheory"></span><dl class="function">
-<dt id="sympy.ntheory.residue_ntheory.n_order">
-<tt class="descclassname">sympy.ntheory.residue_ntheory.</tt><tt class="descname">n_order</tt><big>(</big><em>a</em>, <em>n</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/residue_ntheory.html#n_order"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.residue_ntheory.n_order" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the order of <tt class="docutils literal"><span class="pre">a</span></tt> modulo <tt class="docutils literal"><span class="pre">n</span></tt>.</p>
-<p>The order of <tt class="docutils literal"><span class="pre">a</span></tt> modulo <tt class="docutils literal"><span class="pre">n</span></tt> is the smallest integer
-<tt class="docutils literal"><span class="pre">k</span></tt> such that <tt class="docutils literal"><span class="pre">a**k</span></tt> leaves a remainder of 1 with <tt class="docutils literal"><span class="pre">n</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">n_order</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n_order</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
-<span class="go">6</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n_order</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.residue_ntheory.is_primitive_root">
-<tt class="descclassname">sympy.ntheory.residue_ntheory.</tt><tt class="descname">is_primitive_root</tt><big>(</big><em>a</em>, <em>p</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/residue_ntheory.html#is_primitive_root"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.residue_ntheory.is_primitive_root" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">a</span></tt> is a primitive root of <tt class="docutils literal"><span class="pre">p</span></tt></p>
-<p><tt class="docutils literal"><span class="pre">a</span></tt> is said to be the primitive root of <tt class="docutils literal"><span class="pre">p</span></tt> if gcd(a, p) == 1 and
-totient(p) is the smallest positive number s.t.</p>
-<blockquote>
-<div>a**totient(p) cong 1 mod(p)</div></blockquote>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">is_primitive_root</span><span class="p">,</span> <span class="n">n_order</span><span class="p">,</span> <span class="n">totient</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">is_primitive_root</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">is_primitive_root</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n_order</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span> <span class="o">==</span> <span class="n">totient</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n_order</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span> <span class="o">==</span> <span class="n">totient</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.residue_ntheory.is_quad_residue">
-<tt class="descclassname">sympy.ntheory.residue_ntheory.</tt><tt class="descname">is_quad_residue</tt><big>(</big><em>a</em>, <em>p</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/residue_ntheory.html#is_quad_residue"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.residue_ntheory.is_quad_residue" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">a</span></tt> (mod <tt class="docutils literal"><span class="pre">p</span></tt>) is in the set of squares mod <tt class="docutils literal"><span class="pre">p</span></tt>,
-i.e a % p in set([i**2 % p for i in range(p)]). If <tt class="docutils literal"><span class="pre">p</span></tt> is an odd
-prime, an iterative method is used to make the determination:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">is_quad_residue</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="nb">set</span><span class="p">([</span><span class="n">i</span><span class="o">**</span><span class="mi">2</span> <span class="o">%</span> <span class="mi">7</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">7</span><span class="p">)]))</span>
-<span class="go">[0, 1, 2, 4]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">j</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">7</span><span class="p">)</span> <span class="k">if</span> <span class="n">is_quad_residue</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="mi">7</span><span class="p">)]</span>
-<span class="go">[0, 1, 2, 4]</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.ntheory.residue_ntheory.legendre_symbol" title="sympy.ntheory.residue_ntheory.legendre_symbol"><tt class="xref py py-obj docutils literal"><span class="pre">legendre_symbol</span></tt></a>, <a class="reference internal" href="#sympy.ntheory.residue_ntheory.jacobi_symbol" title="sympy.ntheory.residue_ntheory.jacobi_symbol"><tt class="xref py py-obj docutils literal"><span class="pre">jacobi_symbol</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.residue_ntheory.legendre_symbol">
-<tt class="descclassname">sympy.ntheory.residue_ntheory.</tt><tt class="descname">legendre_symbol</tt><big>(</big><em>a</em>, <em>p</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/residue_ntheory.html#legendre_symbol"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.residue_ntheory.legendre_symbol" title="Permalink to this definition">¶</a></dt>
-<dd><table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>1. 0 if a is multiple of p</strong> :</p>
-<p><strong>2. 1 if a is a quadratic residue of p</strong> :</p>
-<p><strong>3. -1 otherwise</strong> :</p>
-<p class="last"><strong>p should be an odd prime by definition</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.ntheory.residue_ntheory.is_quad_residue" title="sympy.ntheory.residue_ntheory.is_quad_residue"><tt class="xref py py-obj docutils literal"><span class="pre">is_quad_residue</span></tt></a>, <a class="reference internal" href="#sympy.ntheory.residue_ntheory.jacobi_symbol" title="sympy.ntheory.residue_ntheory.jacobi_symbol"><tt class="xref py py-obj docutils literal"><span class="pre">jacobi_symbol</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">legendre_symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">legendre_symbol</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">7</span><span class="p">)]</span>
-<span class="go">[0, 1, 1, -1, 1, -1, -1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="nb">set</span><span class="p">([</span><span class="n">i</span><span class="o">**</span><span class="mi">2</span> <span class="o">%</span> <span class="mi">7</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">7</span><span class="p">)]))</span>
-<span class="go">[0, 1, 2, 4]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.ntheory.residue_ntheory.jacobi_symbol">
-<tt class="descclassname">sympy.ntheory.residue_ntheory.</tt><tt class="descname">jacobi_symbol</tt><big>(</big><em>m</em>, <em>n</em><big>)</big><a class="reference internal" href="../_modules/sympy/ntheory/residue_ntheory.html#jacobi_symbol"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.ntheory.residue_ntheory.jacobi_symbol" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the product of the legendre_symbol(m, p)
-for all the prime factors, p, of n.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>1. 0 if m cong 0 mod(n)</strong> :</p>
-<p><strong>2. 1 if x**2 cong m mod(n) has a solution</strong> :</p>
-<p class="last"><strong>3. -1 otherwise</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.ntheory.residue_ntheory.is_quad_residue" title="sympy.ntheory.residue_ntheory.is_quad_residue"><tt class="xref py py-obj docutils literal"><span class="pre">is_quad_residue</span></tt></a>, <a class="reference internal" href="#sympy.ntheory.residue_ntheory.legendre_symbol" title="sympy.ntheory.residue_ntheory.legendre_symbol"><tt class="xref py py-obj docutils literal"><span class="pre">legendre_symbol</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory</span> <span class="kn">import</span> <span class="n">jacobi_symbol</span><span class="p">,</span> <span class="n">legendre_symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Mul</span><span class="p">,</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jacobi_symbol</span><span class="p">(</span><span class="mi">45</span><span class="p">,</span> <span class="mi">77</span><span class="p">)</span>
-<span class="go">-1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jacobi_symbol</span><span class="p">(</span><span class="mi">60</span><span class="p">,</span> <span class="mi">121</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>The relationship between the jacobi_symbol and legendre_symbol can
-be demonstrated as follows:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">L</span> <span class="o">=</span> <span class="n">legendre_symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="p">(</span><span class="mi">45</span><span class="p">)</span><span class="o">.</span><span class="n">factors</span><span class="p">()</span>
-<span class="go">{3: 2, 5: 1}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">jacobi_symbol</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="mi">45</span><span class="p">)</span> <span class="o">==</span> <span class="n">L</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">L</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span><span class="o">**</span><span class="mi">1</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
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-            <p class="logo"><a href="../index.html">
-              <img class="logo" src="../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Number Theory</a><ul>
-<li><a class="reference internal" href="#ntheory-class-reference">Ntheory Class Reference</a></li>
-<li><a class="reference internal" href="#ntheory-functions-reference">Ntheory Functions Reference</a></li>
-</ul>
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-</ul>
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diff --git a/dev-py3k/modules/parsing.html b/dev-py3k/modules/parsing.html
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+++ /dev/null
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-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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-  <div class="section" id="parsing-input">
-<h1>Parsing input<a class="headerlink" href="#parsing-input" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="parsing-functions-reference">
-<h2>Parsing Functions Reference<a class="headerlink" href="#parsing-functions-reference" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.parsing.sympy_parser.parse_expr">
-<tt class="descclassname">sympy.parsing.sympy_parser.</tt><tt class="descname">parse_expr</tt><big>(</big><em>s</em>, <em>local_dict=None</em>, <em>transformations=(&lt;function auto_symbol at 0x112ca5290&gt;</em>, <em>&lt;function auto_number at 0x112ca5440&gt;</em>, <em>&lt;function factorial_notation at 0x112ca5320&gt;)</em><big>)</big><a class="reference internal" href="../_modules/sympy/parsing/sympy_parser.html#parse_expr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.parsing.sympy_parser.parse_expr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Converts the string <tt class="docutils literal"><span class="pre">s</span></tt> to a SymPy expression, in <tt class="docutils literal"><span class="pre">local_dict</span></tt></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.parsing.sympy_parser</span> <span class="kn">import</span> <span class="n">parse_expr</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">parse_expr</span><span class="p">(</span><span class="s">&quot;1/2&quot;</span><span class="p">)</span>
-<span class="go">1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="n">_</span><span class="p">)</span>
-<span class="go">&lt;class &#39;sympy.core.numbers.Half&#39;&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.parsing.sympy_tokenize.printtoken">
-<tt class="descclassname">sympy.parsing.sympy_tokenize.</tt><tt class="descname">printtoken</tt><big>(</big><em>type</em>, <em>token</em>, <em>srow_scol</em>, <em>erow_ecol</em>, <em>line</em><big>)</big><a class="reference internal" href="../_modules/sympy/parsing/sympy_tokenize.html#printtoken"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.parsing.sympy_tokenize.printtoken" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="sympy.parsing.sympy_tokenize.tokenize">
-<tt class="descclassname">sympy.parsing.sympy_tokenize.</tt><tt class="descname">tokenize</tt><big>(</big><em>readline</em>, <em>tokeneater=&lt;function printtoken at 0x112c978c0&gt;</em><big>)</big><a class="reference internal" href="../_modules/sympy/parsing/sympy_tokenize.html#tokenize"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.parsing.sympy_tokenize.tokenize" title="Permalink to this definition">¶</a></dt>
-<dd><p>The tokenize() function accepts two parameters: one representing the
-input stream, and one providing an output mechanism for tokenize().</p>
-<p>The first parameter, readline, must be a callable object which provides
-the same interface as the readline() method of built-in file objects.
-Each call to the function should return one line of input as a string.</p>
-<p>The second parameter, tokeneater, must also be a callable object. It is
-called once for each token, with five arguments, corresponding to the
-tuples generated by generate_tokens().</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.parsing.sympy_tokenize.untokenize">
-<tt class="descclassname">sympy.parsing.sympy_tokenize.</tt><tt class="descname">untokenize</tt><big>(</big><em>iterable</em><big>)</big><a class="reference internal" href="../_modules/sympy/parsing/sympy_tokenize.html#untokenize"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.parsing.sympy_tokenize.untokenize" title="Permalink to this definition">¶</a></dt>
-<dd><p>Transform tokens back into Python source code.</p>
-<p>Each element returned by the iterable must be a token sequence
-with at least two elements, a token number and token value.  If
-only two tokens are passed, the resulting output is poor.</p>
-<dl class="docutils">
-<dt>Round-trip invariant for full input:</dt>
-<dd>Untokenized source will match input source exactly</dd>
-</dl>
-<p>Round-trip invariant for limited intput:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="c"># Output text will tokenize the back to the input</span>
-<span class="n">t1</span> <span class="o">=</span> <span class="p">[</span><span class="n">tok</span><span class="p">[:</span><span class="mi">2</span><span class="p">]</span> <span class="k">for</span> <span class="n">tok</span> <span class="ow">in</span> <span class="n">generate_tokens</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">readline</span><span class="p">)]</span>
-<span class="n">newcode</span> <span class="o">=</span> <span class="n">untokenize</span><span class="p">(</span><span class="n">t1</span><span class="p">)</span>
-<span class="n">readline</span> <span class="o">=</span> <span class="nb">iter</span><span class="p">(</span><span class="n">newcode</span><span class="o">.</span><span class="n">splitlines</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">next</span>
-<span class="n">t2</span> <span class="o">=</span> <span class="p">[</span><span class="n">tok</span><span class="p">[:</span><span class="mi">2</span><span class="p">]</span> <span class="k">for</span> <span class="n">tok</span> <span class="ow">in</span> <span class="n">generate_tokens</span><span class="p">(</span><span class="n">readline</span><span class="p">)]</span>
-<span class="k">assert</span> <span class="n">t1</span> <span class="o">==</span> <span class="n">t2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.parsing.sympy_tokenize.generate_tokens">
-<tt class="descclassname">sympy.parsing.sympy_tokenize.</tt><tt class="descname">generate_tokens</tt><big>(</big><em>readline</em><big>)</big><a class="reference internal" href="../_modules/sympy/parsing/sympy_tokenize.html#generate_tokens"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.parsing.sympy_tokenize.generate_tokens" title="Permalink to this definition">¶</a></dt>
-<dd><p>The generate_tokens() generator requires one argment, readline, which
-must be a callable object which provides the same interface as the
-readline() method of built-in file objects. Each call to the function
-should return one line of input as a string.  Alternately, readline
-can be a callable function terminating with StopIteration:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">readline</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="n">myfile</span><span class="p">)</span><span class="o">.</span><span class="n">next</span>    <span class="c"># Example of alternate readline</span>
-</pre></div>
-</div>
-<p>The generator produces 5-tuples with these members: the token type; the
-token string; a 2-tuple (srow, scol) of ints specifying the row and
-column where the token begins in the source; a 2-tuple (erow, ecol) of
-ints specifying the row and column where the token ends in the source;
-and the line on which the token was found. The line passed is the
-logical line; continuation lines are included.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.parsing.sympy_tokenize.group">
-<tt class="descclassname">sympy.parsing.sympy_tokenize.</tt><tt class="descname">group</tt><big>(</big><em>*choices</em><big>)</big><a class="reference internal" href="../_modules/sympy/parsing/sympy_tokenize.html#group"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.parsing.sympy_tokenize.group" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="sympy.parsing.sympy_tokenize.any">
-<tt class="descclassname">sympy.parsing.sympy_tokenize.</tt><tt class="descname">any</tt><big>(</big><em>*choices</em><big>)</big><a class="reference internal" href="../_modules/sympy/parsing/sympy_tokenize.html#any"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.parsing.sympy_tokenize.any" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="sympy.parsing.sympy_tokenize.maybe">
-<tt class="descclassname">sympy.parsing.sympy_tokenize.</tt><tt class="descname">maybe</tt><big>(</big><em>*choices</em><big>)</big><a class="reference internal" href="../_modules/sympy/parsing/sympy_tokenize.html#maybe"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.parsing.sympy_tokenize.maybe" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="sympy.parsing.maxima.parse_maxima">
-<tt class="descclassname">sympy.parsing.maxima.</tt><tt class="descname">parse_maxima</tt><big>(</big><em>str</em>, <em>globals=None</em>, <em>name_dict={}</em><big>)</big><a class="reference internal" href="../_modules/sympy/parsing/maxima.html#parse_maxima"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.parsing.maxima.parse_maxima" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="sympy.parsing.mathematica.mathematica">
-<tt class="descclassname">sympy.parsing.mathematica.</tt><tt class="descname">mathematica</tt><big>(</big><em>s</em><big>)</big><a class="reference internal" href="../_modules/sympy/parsing/mathematica.html#mathematica"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.parsing.mathematica.mathematica" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="parsing-exceptions-reference">
-<h2>Parsing Exceptions Reference<a class="headerlink" href="#parsing-exceptions-reference" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.parsing.sympy_tokenize.TokenError">
-<em class="property">class </em><tt class="descclassname">sympy.parsing.sympy_tokenize.</tt><tt class="descname">TokenError</tt><a class="reference internal" href="../_modules/sympy/parsing/sympy_tokenize.html#TokenError"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.parsing.sympy_tokenize.TokenError" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.parsing.sympy_tokenize.StopTokenizing">
-<em class="property">class </em><tt class="descclassname">sympy.parsing.sympy_tokenize.</tt><tt class="descname">StopTokenizing</tt><a class="reference internal" href="../_modules/sympy/parsing/sympy_tokenize.html#StopTokenizing"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.parsing.sympy_tokenize.StopTokenizing" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../index.html">
-              <img class="logo" src="../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Parsing input</a><ul>
-<li><a class="reference internal" href="#parsing-functions-reference">Parsing Functions Reference</a></li>
-<li><a class="reference internal" href="#parsing-exceptions-reference">Parsing Exceptions Reference</a></li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="utilities/timeutils.html"
-                        title="previous chapter">Timing Utilities</a></p>
-  <h4>Next topic</h4>
-  <p class="topless"><a href="physics/index.html"
-                        title="next chapter">Physics Module</a></p>
-  <h3>This Page</h3>
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diff --git a/dev-py3k/modules/physics/gaussopt.html b/dev-py3k/modules/physics/gaussopt.html
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-            
-  <div class="section" id="module-sympy.physics.gaussopt">
-<span id="gaussian-optics"></span><h1>Gaussian Optics<a class="headerlink" href="#module-sympy.physics.gaussopt" title="Permalink to this headline">¶</a></h1>
-<p>Gaussian optics.</p>
-<p>The module implements:</p>
-<ul>
-<li><p class="first">Ray transfer matrices for geometrical and gaussian optics.</p>
-<p>See RayTransferMatrix, GeometricRay and BeamParameter</p>
-</li>
-<li><p class="first">Conjugation relations for geometrical and gaussian optics.</p>
-<p>See geometric_conj*, gauss_conj and conjugate_gauss_beams</p>
-</li>
-</ul>
-<p>The conventions for the distances are as follows:</p>
-<dl class="docutils">
-<dt>focal distance</dt>
-<dd>positive for convergent lenses</dd>
-<dt>object distance</dt>
-<dd>positive for real objects</dd>
-<dt>image distance</dt>
-<dd>positive for real images</dd>
-</dl>
-<dl class="class">
-<dt id="sympy.physics.gaussopt.BeamParameter">
-<em class="property">class </em><tt class="descclassname">sympy.physics.gaussopt.</tt><tt class="descname">BeamParameter</tt><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#BeamParameter"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.BeamParameter" title="Permalink to this definition">¶</a></dt>
-<dd><p>Representation for a gaussian ray in the Ray Transfer Matrix formalism.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>wavelen</strong> : the wavelength,</p>
-<p><strong>z</strong> : the distance to waist, and</p>
-<p><strong>w</strong> : the waist, or</p>
-<p class="last"><strong>z_r</strong> : the rayleigh range</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.physics.gaussopt.RayTransferMatrix" title="sympy.physics.gaussopt.RayTransferMatrix"><tt class="xref py py-obj docutils literal"><span class="pre">RayTransferMatrix</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Complex_beam_parameter">http://en.wikipedia.org/wiki/Complex_beam_parameter</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">BeamParameter</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">BeamParameter</span><span class="p">(</span><span class="mf">530e-9</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">w</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">q</span>
-<span class="go">1 + 1.88679245283019*I*pi</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">q</span><span class="o">.</span><span class="n">n</span><span class="p">()</span>
-<span class="go">1.0 + 5.92753330865999*I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">w_0</span><span class="o">.</span><span class="n">n</span><span class="p">()</span>
-<span class="go">0.00100000000000000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">z_r</span><span class="o">.</span><span class="n">n</span><span class="p">()</span>
-<span class="go">5.92753330865999</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">FreeSpace</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fs</span> <span class="o">=</span> <span class="n">FreeSpace</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">fs</span><span class="o">*</span><span class="n">p</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">w</span><span class="o">.</span><span class="n">n</span><span class="p">()</span>
-<span class="go">0.00101413072159615</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="o">.</span><span class="n">w</span><span class="o">.</span><span class="n">n</span><span class="p">()</span>
-<span class="go">0.00210803120913829</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.physics.gaussopt.BeamParameter.divergence">
-<tt class="descname">divergence</tt><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#BeamParameter.divergence"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.BeamParameter.divergence" title="Permalink to this definition">¶</a></dt>
-<dd><p>Half of the total angular spread.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">BeamParameter</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">BeamParameter</span><span class="p">(</span><span class="mf">530e-9</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">w</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">divergence</span>
-<span class="go">0.00053/pi</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.gaussopt.BeamParameter.gouy">
-<tt class="descname">gouy</tt><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#BeamParameter.gouy"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.BeamParameter.gouy" title="Permalink to this definition">¶</a></dt>
-<dd><p>The Gouy phase.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">BeamParameter</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">BeamParameter</span><span class="p">(</span><span class="mf">530e-9</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">w</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">gouy</span>
-<span class="go">atan(0.53/pi)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.gaussopt.BeamParameter.q">
-<tt class="descname">q</tt><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#BeamParameter.q"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.BeamParameter.q" title="Permalink to this definition">¶</a></dt>
-<dd><p>The complex parameter representing the beam.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">BeamParameter</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">BeamParameter</span><span class="p">(</span><span class="mf">530e-9</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">w</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">q</span>
-<span class="go">1 + 1.88679245283019*I*pi</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.gaussopt.BeamParameter.radius">
-<tt class="descname">radius</tt><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#BeamParameter.radius"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.BeamParameter.radius" title="Permalink to this definition">¶</a></dt>
-<dd><p>The radius of curvature of the phase front.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">BeamParameter</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">BeamParameter</span><span class="p">(</span><span class="mf">530e-9</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">w</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">radius</span>
-<span class="go">0.2809/pi**2 + 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.gaussopt.BeamParameter.w">
-<tt class="descname">w</tt><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#BeamParameter.w"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.BeamParameter.w" title="Permalink to this definition">¶</a></dt>
-<dd><p>The beam radius at <span class="math">\(1/e^2\)</span> intensity.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.gaussopt.BeamParameter.w_0" title="sympy.physics.gaussopt.BeamParameter.w_0"><tt class="xref py py-obj docutils literal"><span class="pre">w_0</span></tt></a></dt>
-<dd>the minimal radius of beam</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">BeamParameter</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">BeamParameter</span><span class="p">(</span><span class="mf">530e-9</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">w</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">w</span>
-<span class="go">0.001*sqrt(0.2809/pi**2 + 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.gaussopt.BeamParameter.w_0">
-<tt class="descname">w_0</tt><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#BeamParameter.w_0"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.BeamParameter.w_0" title="Permalink to this definition">¶</a></dt>
-<dd><p>The beam waist (minimal radius).</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.gaussopt.BeamParameter.w" title="sympy.physics.gaussopt.BeamParameter.w"><tt class="xref py py-obj docutils literal"><span class="pre">w</span></tt></a></dt>
-<dd>the beam radius at <span class="math">\(1/e^2\)</span> intensity</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">BeamParameter</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">BeamParameter</span><span class="p">(</span><span class="mf">530e-9</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">w</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">w_0</span>
-<span class="go">0.00100000000000000</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.gaussopt.BeamParameter.waist_approximation_limit">
-<tt class="descname">waist_approximation_limit</tt><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#BeamParameter.waist_approximation_limit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.BeamParameter.waist_approximation_limit" title="Permalink to this definition">¶</a></dt>
-<dd><p>The minimal waist for which the gauss beam approximation is valid.</p>
-<p>The gauss beam is a solution to the paraxial equation. For curvatures
-that are too great it is not a valid approximation.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">BeamParameter</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">BeamParameter</span><span class="p">(</span><span class="mf">530e-9</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">w</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">waist_approximation_limit</span>
-<span class="go">1.06e-6/pi</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.gaussopt.CurvedMirror">
-<em class="property">class </em><tt class="descclassname">sympy.physics.gaussopt.</tt><tt class="descname">CurvedMirror</tt><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#CurvedMirror"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.CurvedMirror" title="Permalink to this definition">¶</a></dt>
-<dd><p>Ray Transfer Matrix for reflection from curved surface.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>R</strong> : radius of curvature (positive for concave)</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.physics.gaussopt.RayTransferMatrix" title="sympy.physics.gaussopt.RayTransferMatrix"><tt class="xref py py-obj docutils literal"><span class="pre">RayTransferMatrix</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">CurvedMirror</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;R&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">CurvedMirror</span><span class="p">(</span><span class="n">R</span><span class="p">)</span>
-<span class="go">[   1, 0]</span>
-<span class="go">[-2/R, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.gaussopt.CurvedRefraction">
-<em class="property">class </em><tt class="descclassname">sympy.physics.gaussopt.</tt><tt class="descname">CurvedRefraction</tt><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#CurvedRefraction"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.CurvedRefraction" title="Permalink to this definition">¶</a></dt>
-<dd><p>Ray Transfer Matrix for refraction on curved interface.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>R</strong> : radius of curvature (positive for concave)</p>
-<p><strong>n1</strong> : refractive index of one medium</p>
-<p class="last"><strong>n2</strong> : refractive index of other medium</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.physics.gaussopt.RayTransferMatrix" title="sympy.physics.gaussopt.RayTransferMatrix"><tt class="xref py py-obj docutils literal"><span class="pre">RayTransferMatrix</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">CurvedRefraction</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="p">,</span> <span class="n">n1</span><span class="p">,</span> <span class="n">n2</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;R n1 n2&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">CurvedRefraction</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">)</span>
-<span class="go">[               1,     0]</span>
-<span class="go">[(n1 - n2)/(R*n2), n1/n2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.gaussopt.FlatMirror">
-<em class="property">class </em><tt class="descclassname">sympy.physics.gaussopt.</tt><tt class="descname">FlatMirror</tt><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#FlatMirror"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.FlatMirror" title="Permalink to this definition">¶</a></dt>
-<dd><p>Ray Transfer Matrix for reflection.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.physics.gaussopt.RayTransferMatrix" title="sympy.physics.gaussopt.RayTransferMatrix"><tt class="xref py py-obj docutils literal"><span class="pre">RayTransferMatrix</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">FlatMirror</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">FlatMirror</span><span class="p">()</span>
-<span class="go">[1, 0]</span>
-<span class="go">[0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.gaussopt.FlatRefraction">
-<em class="property">class </em><tt class="descclassname">sympy.physics.gaussopt.</tt><tt class="descname">FlatRefraction</tt><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#FlatRefraction"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.FlatRefraction" title="Permalink to this definition">¶</a></dt>
-<dd><p>Ray Transfer Matrix for refraction.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>n1</strong> : refractive index of one medium</p>
-<p class="last"><strong>n2</strong> : refractive index of other medium</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.physics.gaussopt.RayTransferMatrix" title="sympy.physics.gaussopt.RayTransferMatrix"><tt class="xref py py-obj docutils literal"><span class="pre">RayTransferMatrix</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">FlatRefraction</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n1</span><span class="p">,</span> <span class="n">n2</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;n1 n2&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">FlatRefraction</span><span class="p">(</span><span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">)</span>
-<span class="go">[1,     0]</span>
-<span class="go">[0, n1/n2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.gaussopt.FreeSpace">
-<em class="property">class </em><tt class="descclassname">sympy.physics.gaussopt.</tt><tt class="descname">FreeSpace</tt><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#FreeSpace"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.FreeSpace" title="Permalink to this definition">¶</a></dt>
-<dd><p>Ray Transfer Matrix for free space.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>distance</strong> :</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.physics.gaussopt.RayTransferMatrix" title="sympy.physics.gaussopt.RayTransferMatrix"><tt class="xref py py-obj docutils literal"><span class="pre">RayTransferMatrix</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">FreeSpace</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;d&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">FreeSpace</span><span class="p">(</span><span class="n">d</span><span class="p">)</span>
-<span class="go">[1, d]</span>
-<span class="go">[0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.gaussopt.GeometricRay">
-<em class="property">class </em><tt class="descclassname">sympy.physics.gaussopt.</tt><tt class="descname">GeometricRay</tt><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#GeometricRay"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.GeometricRay" title="Permalink to this definition">¶</a></dt>
-<dd><p>Representation for a geometric ray in the Ray Transfer Matrix formalism.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>h</strong> : height, and</p>
-<p><strong>angle</strong> : angle, or</p>
-<p><strong>matrix</strong> : a 2x1 matrix (Matrix(2, 1, [height, angle]))</p>
-<p><strong>Examples</strong> :</p>
-<p><strong>=======</strong> :</p>
-<p><strong>&gt;&gt;&gt; from sympy.physics.gaussopt import GeometricRay, FreeSpace</strong> :</p>
-<p><strong>&gt;&gt;&gt; from sympy import symbols, Matrix</strong> :</p>
-<p><strong>&gt;&gt;&gt; d, h, angle = symbols(&#8216;d, h, angle&#8217;)</strong> :</p>
-<p><strong>&gt;&gt;&gt; GeometricRay(h, angle)</strong> :</p>
-<p><strong>[    h]</strong> :</p>
-<p><strong>[angle]</strong> :</p>
-<p><strong>&gt;&gt;&gt; FreeSpace(d)*GeometricRay(h, angle)</strong> :</p>
-<p><strong>[angle*d + h]</strong> :</p>
-<p><strong>[      angle]</strong> :</p>
-<p><strong>&gt;&gt;&gt; GeometricRay( Matrix( ((h,), (angle,)) ) )</strong> :</p>
-<p><strong>[    h]</strong> :</p>
-<p class="last"><strong>[angle]</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.physics.gaussopt.RayTransferMatrix" title="sympy.physics.gaussopt.RayTransferMatrix"><tt class="xref py py-obj docutils literal"><span class="pre">RayTransferMatrix</span></tt></a></p>
-</div>
-<dl class="attribute">
-<dt id="sympy.physics.gaussopt.GeometricRay.angle">
-<tt class="descname">angle</tt><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#GeometricRay.angle"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.GeometricRay.angle" title="Permalink to this definition">¶</a></dt>
-<dd><p>The angle with the optical axis.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">GeometricRay</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span><span class="p">,</span> <span class="n">angle</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;h, angle&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gRay</span> <span class="o">=</span> <span class="n">GeometricRay</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">angle</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gRay</span><span class="o">.</span><span class="n">angle</span>
-<span class="go">angle</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.gaussopt.GeometricRay.height">
-<tt class="descname">height</tt><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#GeometricRay.height"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.GeometricRay.height" title="Permalink to this definition">¶</a></dt>
-<dd><p>The distance from the optical axis.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">GeometricRay</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span><span class="p">,</span> <span class="n">angle</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;h, angle&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gRay</span> <span class="o">=</span> <span class="n">GeometricRay</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">angle</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gRay</span><span class="o">.</span><span class="n">height</span>
-<span class="go">h</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.gaussopt.RayTransferMatrix">
-<em class="property">class </em><tt class="descclassname">sympy.physics.gaussopt.</tt><tt class="descname">RayTransferMatrix</tt><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#RayTransferMatrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.RayTransferMatrix" title="Permalink to this definition">¶</a></dt>
-<dd><p>Base class for a Ray Transfer Matrix.</p>
-<p>It should be used if there isn&#8217;t already a more specific subclass mentioned
-in See Also.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>parameters</strong> : A, B, C and D or 2x2 matrix (Matrix(2, 2, [A, B, C, D]))</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.physics.gaussopt.GeometricRay" title="sympy.physics.gaussopt.GeometricRay"><tt class="xref py py-obj docutils literal"><span class="pre">GeometricRay</span></tt></a>, <a class="reference internal" href="#sympy.physics.gaussopt.BeamParameter" title="sympy.physics.gaussopt.BeamParameter"><tt class="xref py py-obj docutils literal"><span class="pre">BeamParameter</span></tt></a>, <a class="reference internal" href="#sympy.physics.gaussopt.FreeSpace" title="sympy.physics.gaussopt.FreeSpace"><tt class="xref py py-obj docutils literal"><span class="pre">FreeSpace</span></tt></a>, <a class="reference internal" href="#sympy.physics.gaussopt.FlatRefraction" title="sympy.physics.gaussopt.FlatRefraction"><tt class="xref py py-obj docutils literal"><span class="pre">FlatRefraction</span></tt></a>, <a class="reference internal" href="#sympy.physics.gaussopt.CurvedRefraction" title="sympy.physics.gaussopt.CurvedRefraction"><tt class="xref py py-obj docutils literal"><span class="pre">CurvedRefraction</span></tt></a>, <a class="reference internal" href="#sympy.physics.gaussopt.FlatMirror" title="sympy.physics.gaussopt.FlatMirror"><tt class="xref py py-obj docutils literal"><span class="pre">FlatMirror</span></tt></a>, <a class="reference internal" href="#sympy.physics.gaussopt.CurvedMirror" title="sympy.physics.gaussopt.CurvedMirror"><tt class="xref py py-obj docutils literal"><span class="pre">CurvedMirror</span></tt></a>, <a class="reference internal" href="#sympy.physics.gaussopt.ThinLens" title="sympy.physics.gaussopt.ThinLens"><tt class="xref py py-obj docutils literal"><span class="pre">ThinLens</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Ray_transfer_matrix_analysis">http://en.wikipedia.org/wiki/Ray_transfer_matrix_analysis</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">RayTransferMatrix</span><span class="p">,</span> <span class="n">ThinLens</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Matrix</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mat</span> <span class="o">=</span> <span class="n">RayTransferMatrix</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mat</span>
-<span class="go">[1,  2]</span>
-<span class="go">[3,  4]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">RayTransferMatrix</span><span class="p">(</span><span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">]]))</span>
-<span class="go">[1,  2]</span>
-<span class="go">[3,  4]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mat</span><span class="o">.</span><span class="n">A</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lens</span> <span class="o">=</span> <span class="n">ThinLens</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lens</span>
-<span class="go">[   1, 0]</span>
-<span class="go">[-1/f, 1]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">lens</span><span class="o">.</span><span class="n">C</span>
-<span class="go">-1/f</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.physics.gaussopt.RayTransferMatrix.A">
-<tt class="descname">A</tt><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#RayTransferMatrix.A"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.RayTransferMatrix.A" title="Permalink to this definition">¶</a></dt>
-<dd><p>The A parameter of the Matrix.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">RayTransferMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mat</span> <span class="o">=</span> <span class="n">RayTransferMatrix</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mat</span><span class="o">.</span><span class="n">A</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.gaussopt.RayTransferMatrix.B">
-<tt class="descname">B</tt><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#RayTransferMatrix.B"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.RayTransferMatrix.B" title="Permalink to this definition">¶</a></dt>
-<dd><p>The B parameter of the Matrix.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">RayTransferMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mat</span> <span class="o">=</span> <span class="n">RayTransferMatrix</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mat</span><span class="o">.</span><span class="n">B</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.gaussopt.RayTransferMatrix.C">
-<tt class="descname">C</tt><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#RayTransferMatrix.C"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.RayTransferMatrix.C" title="Permalink to this definition">¶</a></dt>
-<dd><p>The C parameter of the Matrix.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">RayTransferMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mat</span> <span class="o">=</span> <span class="n">RayTransferMatrix</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mat</span><span class="o">.</span><span class="n">C</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.gaussopt.RayTransferMatrix.D">
-<tt class="descname">D</tt><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#RayTransferMatrix.D"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.RayTransferMatrix.D" title="Permalink to this definition">¶</a></dt>
-<dd><p>The D parameter of the Matrix.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">RayTransferMatrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mat</span> <span class="o">=</span> <span class="n">RayTransferMatrix</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mat</span><span class="o">.</span><span class="n">D</span>
-<span class="go">4</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.gaussopt.ThinLens">
-<em class="property">class </em><tt class="descclassname">sympy.physics.gaussopt.</tt><tt class="descname">ThinLens</tt><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#ThinLens"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.ThinLens" title="Permalink to this definition">¶</a></dt>
-<dd><p>Ray Transfer Matrix for a thin lens.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>f</strong> : the focal distance</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.physics.gaussopt.RayTransferMatrix" title="sympy.physics.gaussopt.RayTransferMatrix"><tt class="xref py py-obj docutils literal"><span class="pre">RayTransferMatrix</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">ThinLens</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ThinLens</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-<span class="go">[   1, 0]</span>
-<span class="go">[-1/f, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.gaussopt.conjugate_gauss_beams">
-<tt class="descclassname">sympy.physics.gaussopt.</tt><tt class="descname">conjugate_gauss_beams</tt><big>(</big><em>wavelen</em>, <em>waist_in</em>, <em>waist_out</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#conjugate_gauss_beams"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.conjugate_gauss_beams" title="Permalink to this definition">¶</a></dt>
-<dd><p>Find the optical setup conjugating the object/image waists.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>wavelen</strong> : the wavelength of the beam</p>
-<p><strong>waist_in and waist_out</strong> : the waists to be conjugated</p>
-<p><strong>f</strong> : the focal distance of the element used in the conjugation</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>a tuple containing (s_in, s_out, f)</strong> :</p>
-<p><strong>s_in</strong> : the distance before the optical element</p>
-<p><strong>s_out</strong> : the distance after the optical element</p>
-<p class="last"><strong>f</strong> : the focal distance of the optical element</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">conjugate_gauss_beams</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">factor</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l</span><span class="p">,</span> <span class="n">w_i</span><span class="p">,</span> <span class="n">w_o</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;l w_i w_o f&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">conjugate_gauss_beams</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">w_i</span><span class="p">,</span> <span class="n">w_o</span><span class="p">,</span> <span class="n">f</span><span class="o">=</span><span class="n">f</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">f*(-sqrt(w_i**2/w_o**2 - pi**2*w_i**4/(f**2*l**2)) + 1)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">conjugate_gauss_beams</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">w_i</span><span class="p">,</span> <span class="n">w_o</span><span class="p">,</span> <span class="n">f</span><span class="o">=</span><span class="n">f</span><span class="p">)[</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">f*w_o**2*(w_i**2/w_o**2 - sqrt(w_i**2/w_o**2 -</span>
-<span class="go">          pi**2*w_i**4/(f**2*l**2)))/w_i**2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">conjugate_gauss_beams</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">w_i</span><span class="p">,</span> <span class="n">w_o</span><span class="p">,</span> <span class="n">f</span><span class="o">=</span><span class="n">f</span><span class="p">)[</span><span class="mi">2</span><span class="p">]</span>
-<span class="go">f</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.gaussopt.gaussian_conj">
-<tt class="descclassname">sympy.physics.gaussopt.</tt><tt class="descname">gaussian_conj</tt><big>(</big><em>s_in</em>, <em>z_r_in</em>, <em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#gaussian_conj"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.gaussian_conj" title="Permalink to this definition">¶</a></dt>
-<dd><p>Conjugation relation for gaussian beams.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>s_in</strong> : the distance to optical element from the waist</p>
-<p><strong>z_r_in</strong> : the rayleigh range of the incident beam</p>
-<p><strong>f</strong> : the focal length of the optical element</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>a tuple containing (s_out, z_r_out, m)</strong> :</p>
-<p><strong>s_out</strong> : the distance between the new waist and the optical element</p>
-<p><strong>z_r_out</strong> : the rayleigh range of the emergent beam</p>
-<p class="last"><strong>m</strong> : the ration between the new and the old waists</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">gaussian_conj</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s_in</span><span class="p">,</span> <span class="n">z_r_in</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;s_in z_r_in f&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gaussian_conj</span><span class="p">(</span><span class="n">s_in</span><span class="p">,</span> <span class="n">z_r_in</span><span class="p">,</span> <span class="n">f</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">1/(-1/(s_in + z_r_in**2/(-f + s_in)) + 1/f)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gaussian_conj</span><span class="p">(</span><span class="n">s_in</span><span class="p">,</span> <span class="n">z_r_in</span><span class="p">,</span> <span class="n">f</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
-<span class="go">z_r_in/(1 - s_in**2/f**2 + z_r_in**2/f**2)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gaussian_conj</span><span class="p">(</span><span class="n">s_in</span><span class="p">,</span> <span class="n">z_r_in</span><span class="p">,</span> <span class="n">f</span><span class="p">)[</span><span class="mi">2</span><span class="p">]</span>
-<span class="go">1/sqrt(1 - s_in**2/f**2 + z_r_in**2/f**2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.gaussopt.geometric_conj_ab">
-<tt class="descclassname">sympy.physics.gaussopt.</tt><tt class="descname">geometric_conj_ab</tt><big>(</big><em>a</em>, <em>b</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#geometric_conj_ab"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.geometric_conj_ab" title="Permalink to this definition">¶</a></dt>
-<dd><p>Conjugation relation for geometrical beams under paraxial conditions.</p>
-<p>Takes the distances to the optical element and returns the needed
-focal distance.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.physics.gaussopt.geometric_conj_af" title="sympy.physics.gaussopt.geometric_conj_af"><tt class="xref py py-obj docutils literal"><span class="pre">geometric_conj_af</span></tt></a>, <a class="reference internal" href="#sympy.physics.gaussopt.geometric_conj_bf" title="sympy.physics.gaussopt.geometric_conj_bf"><tt class="xref py py-obj docutils literal"><span class="pre">geometric_conj_bf</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">geometric_conj_ab</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a b&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">geometric_conj_ab</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-<span class="go">a*b/(a + b)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.gaussopt.geometric_conj_af">
-<tt class="descclassname">sympy.physics.gaussopt.</tt><tt class="descname">geometric_conj_af</tt><big>(</big><em>a</em>, <em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#geometric_conj_af"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.geometric_conj_af" title="Permalink to this definition">¶</a></dt>
-<dd><p>Conjugation relation for geometrical beams under paraxial conditions.</p>
-<p>Takes the object distance (for geometric_conj_af) or the image distance
-(for geometric_conj_bf) to the optical element and the focal distance.
-Then it returns the other distance needed for conjugation.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.physics.gaussopt.geometric_conj_ab" title="sympy.physics.gaussopt.geometric_conj_ab"><tt class="xref py py-obj docutils literal"><span class="pre">geometric_conj_ab</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">geometric_conj_af</span><span class="p">,</span> <span class="n">geometric_conj_bf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a b f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">geometric_conj_af</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
-<span class="go">a*f/(a - f)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">geometric_conj_bf</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
-<span class="go">b*f/(b - f)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.gaussopt.geometric_conj_bf">
-<tt class="descclassname">sympy.physics.gaussopt.</tt><tt class="descname">geometric_conj_bf</tt><big>(</big><em>a</em>, <em>f</em><big>)</big><a class="headerlink" href="#sympy.physics.gaussopt.geometric_conj_bf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Conjugation relation for geometrical beams under paraxial conditions.</p>
-<p>Takes the object distance (for geometric_conj_af) or the image distance
-(for geometric_conj_bf) to the optical element and the focal distance.
-Then it returns the other distance needed for conjugation.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.physics.gaussopt.geometric_conj_ab" title="sympy.physics.gaussopt.geometric_conj_ab"><tt class="xref py py-obj docutils literal"><span class="pre">geometric_conj_ab</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">geometric_conj_af</span><span class="p">,</span> <span class="n">geometric_conj_bf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a b f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">geometric_conj_af</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
-<span class="go">a*f/(a - f)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">geometric_conj_bf</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
-<span class="go">b*f/(b - f)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.gaussopt.rayleigh2waist">
-<tt class="descclassname">sympy.physics.gaussopt.</tt><tt class="descname">rayleigh2waist</tt><big>(</big><em>z_r</em>, <em>wavelen</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#rayleigh2waist"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.rayleigh2waist" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculate the waist from the rayleigh range of a gaussian beam.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.physics.gaussopt.waist2rayleigh" title="sympy.physics.gaussopt.waist2rayleigh"><tt class="xref py py-obj docutils literal"><span class="pre">waist2rayleigh</span></tt></a>, <a class="reference internal" href="#sympy.physics.gaussopt.BeamParameter" title="sympy.physics.gaussopt.BeamParameter"><tt class="xref py py-obj docutils literal"><span class="pre">BeamParameter</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">rayleigh2waist</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z_r</span><span class="p">,</span> <span class="n">wavelen</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;z_r wavelen&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rayleigh2waist</span><span class="p">(</span><span class="n">z_r</span><span class="p">,</span> <span class="n">wavelen</span><span class="p">)</span>
-<span class="go">sqrt(wavelen*z_r)/sqrt(pi)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.gaussopt.waist2rayleigh">
-<tt class="descclassname">sympy.physics.gaussopt.</tt><tt class="descname">waist2rayleigh</tt><big>(</big><em>w</em>, <em>wavelen</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/gaussopt.html#waist2rayleigh"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.gaussopt.waist2rayleigh" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculate the rayleigh range from the waist of a gaussian beam.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.physics.gaussopt.rayleigh2waist" title="sympy.physics.gaussopt.rayleigh2waist"><tt class="xref py py-obj docutils literal"><span class="pre">rayleigh2waist</span></tt></a>, <a class="reference internal" href="#sympy.physics.gaussopt.BeamParameter" title="sympy.physics.gaussopt.BeamParameter"><tt class="xref py py-obj docutils literal"><span class="pre">BeamParameter</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.gaussopt</span> <span class="kn">import</span> <span class="n">waist2rayleigh</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">w</span><span class="p">,</span> <span class="n">wavelen</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;w wavelen&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">waist2rayleigh</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">wavelen</span><span class="p">)</span>
-<span class="go">pi*w**2/wavelen</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-
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-  <div class="section" id="module-sympy.physics.hydrogen">
-<span id="hydrogen-wavefunctions"></span><h1>Hydrogen Wavefunctions<a class="headerlink" href="#module-sympy.physics.hydrogen" title="Permalink to this headline">¶</a></h1>
-<dl class="function">
-<dt id="sympy.physics.hydrogen.E_nl">
-<tt class="descclassname">sympy.physics.hydrogen.</tt><tt class="descname">E_nl</tt><big>(</big><em>n</em>, <em>Z=1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/hydrogen.html#E_nl"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.hydrogen.E_nl" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the energy of the state (n, l) in Hartree atomic units.</p>
-<p>The energy doesn&#8217;t depend on &#8220;l&#8221;.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">var</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.hydrogen</span> <span class="kn">import</span> <span class="n">E_nl</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&quot;n Z&quot;</span><span class="p">)</span>
-<span class="go">(n, Z)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E_nl</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">Z</span><span class="p">)</span>
-<span class="go">-Z**2/(2*n**2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E_nl</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E_nl</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">-1/8</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E_nl</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">-1/18</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E_nl</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">47</span><span class="p">)</span>
-<span class="go">-2209/18</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.hydrogen.E_nl_dirac">
-<tt class="descclassname">sympy.physics.hydrogen.</tt><tt class="descname">E_nl_dirac</tt><big>(</big><em>n</em>, <em>l</em>, <em>spin_up=True</em>, <em>Z=1</em>, <em>c=137.035999037000</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/hydrogen.html#E_nl_dirac"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.hydrogen.E_nl_dirac" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the relativistic energy of the state (n, l, spin) in Hartree atomic
-units.</p>
-<p>The energy is calculated from the Dirac equation. The rest mass energy is
-<em>not</em> included.</p>
-<dl class="docutils">
-<dt>n, l</dt>
-<dd>quantum numbers &#8216;n&#8217; and &#8216;l&#8217;</dd>
-<dt>spin_up</dt>
-<dd>True if the electron spin is up (default), otherwise down</dd>
-<dt>Z</dt>
-<dd>atomic number (1 for Hydrogen, 2 for Helium, ...)</dd>
-<dt>c</dt>
-<dd>speed of light in atomic units. Default value is 137.035999037,
-taken from: <a class="reference external" href="http://arxiv.org/abs/1012.3627">http://arxiv.org/abs/1012.3627</a></dd>
-</dl>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.hydrogen</span> <span class="kn">import</span> <span class="n">E_nl_dirac</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E_nl_dirac</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">-0.500006656595360</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">E_nl_dirac</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">-0.125002080189006</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E_nl_dirac</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">-0.125000416028342</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E_nl_dirac</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-<span class="go">-0.125002080189006</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">E_nl_dirac</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">-0.0555562951740285</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E_nl_dirac</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">-0.0555558020932949</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E_nl_dirac</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-<span class="go">-0.0555562951740285</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E_nl_dirac</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">-0.0555556377366884</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E_nl_dirac</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="bp">False</span><span class="p">)</span>
-<span class="go">-0.0555558020932949</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.hydrogen.R_nl">
-<tt class="descclassname">sympy.physics.hydrogen.</tt><tt class="descname">R_nl</tt><big>(</big><em>n</em>, <em>l</em>, <em>r</em>, <em>Z=1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/hydrogen.html#R_nl"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.hydrogen.R_nl" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the Hydrogen radial wavefunction R_{nl}.</p>
-<dl class="docutils">
-<dt>n, l</dt>
-<dd>quantum numbers &#8216;n&#8217; and &#8216;l&#8217;</dd>
-<dt>r</dt>
-<dd>radial coordinate</dd>
-<dt>Z</dt>
-<dd>atomic number (1 for Hydrogen, 2 for Helium, ...)</dd>
-</dl>
-<p>Everything is in Hartree atomic units.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.hydrogen</span> <span class="kn">import</span> <span class="n">R_nl</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">var</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&quot;r Z&quot;</span><span class="p">)</span>
-<span class="go">(r, Z)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R_nl</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">Z</span><span class="p">)</span>
-<span class="go">2*sqrt(Z**3)*exp(-Z*r)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R_nl</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">Z</span><span class="p">)</span>
-<span class="go">sqrt(2)*(-Z*r + 2)*sqrt(Z**3)*exp(-Z*r/2)/4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R_nl</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">Z</span><span class="p">)</span>
-<span class="go">sqrt(6)*Z*r*sqrt(Z**3)*exp(-Z*r/2)/12</span>
-</pre></div>
-</div>
-<p>For Hydrogen atom, you can just use the default value of Z=1:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">R_nl</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span>
-<span class="go">2*exp(-r)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R_nl</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span>
-<span class="go">sqrt(2)*(-r + 2)*exp(-r/2)/4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R_nl</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span>
-<span class="go">2*sqrt(3)*(2*r**2/9 - 2*r + 3)*exp(-r/3)/27</span>
-</pre></div>
-</div>
-<p>For Silver atom, you would use Z=47:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">R_nl</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">Z</span><span class="o">=</span><span class="mi">47</span><span class="p">)</span>
-<span class="go">94*sqrt(47)*exp(-47*r)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R_nl</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">Z</span><span class="o">=</span><span class="mi">47</span><span class="p">)</span>
-<span class="go">47*sqrt(94)*(-47*r + 2)*exp(-47*r/2)/4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R_nl</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">Z</span><span class="o">=</span><span class="mi">47</span><span class="p">)</span>
-<span class="go">94*sqrt(141)*(4418*r**2/9 - 94*r + 3)*exp(-47*r/3)/27</span>
-</pre></div>
-</div>
-<p>The normalization of the radial wavefunction is:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">integrate</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">R_nl</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">R_nl</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">R_nl</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>It holds for any atomic number:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">R_nl</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">Z</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">R_nl</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">Z</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">R_nl</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">Z</span><span class="o">=</span><span class="mi">4</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
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-  <div class="section" id="module-sympy.physics">
-<span id="physics-module"></span><span id="physics-docs"></span><h1>Physics Module<a class="headerlink" href="#module-sympy.physics" title="Permalink to this headline">¶</a></h1>
-<p>A module that helps solving problems in physics</p>
-<div class="section" id="contents">
-<h2>Contents<a class="headerlink" href="#contents" title="Permalink to this headline">¶</a></h2>
-<div class="toctree-wrapper compound">
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-  <div class="section" id="module-sympy.physics.matrices">
-<span id="matrices"></span><h1>Matrices<a class="headerlink" href="#module-sympy.physics.matrices" title="Permalink to this headline">¶</a></h1>
-<p>Known matrices related to physics</p>
-<dl class="function">
-<dt id="sympy.physics.matrices.mgamma">
-<tt class="descclassname">sympy.physics.matrices.</tt><tt class="descname">mgamma</tt><big>(</big><em>mu</em>, <em>lower=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/matrices.html#mgamma"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.matrices.mgamma" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a Dirac gamma matrix gamma^mu in the standard
-(Dirac) representation.</p>
-<p>If you want gamma_mu, use gamma(mu, True).</p>
-<p>We use a convention:</p>
-<p>gamma^5 = I * gamma^0 * gamma^1 * gamma^2 * gamma^3
-gamma_5 = I * gamma_0 * gamma_1 * gamma_2 * gamma_3 = - gamma^5</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">http</span></tt></dt>
-<dd>//en.wikipedia.org/wiki/Gamma_matrices</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.matrices</span> <span class="kn">import</span> <span class="n">mgamma</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mgamma</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">[ 0,  0, 0, 1]</span>
-<span class="go">[ 0,  0, 1, 0]</span>
-<span class="go">[ 0, -1, 0, 0]</span>
-<span class="go">[-1,  0, 0, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.matrices.msigma">
-<tt class="descclassname">sympy.physics.matrices.</tt><tt class="descname">msigma</tt><big>(</big><em>i</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/matrices.html#msigma"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.matrices.msigma" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a Pauli matrix sigma_i. i=1,2,3</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">http</span></tt></dt>
-<dd>//en.wikipedia.org/wiki/Pauli_matrices</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.matrices</span> <span class="kn">import</span> <span class="n">msigma</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">msigma</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">[0, 1]</span>
-<span class="go">[1, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.matrices.pat_matrix">
-<tt class="descclassname">sympy.physics.matrices.</tt><tt class="descname">pat_matrix</tt><big>(</big><em>m</em>, <em>dx</em>, <em>dy</em>, <em>dz</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/matrices.html#pat_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.matrices.pat_matrix" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the Parallel Axis Theorem matrix to translate the inertia
-matrix a distance of (dx, dy, dz) for a body of mass m.</p>
-<p class="rubric">Examples</p>
-<p>If the point we want the inertia about is a distance of 2 units of
-length and 1 unit along the x-axis we get:
-&gt;&gt;&gt; from sympy.physics.matrices import pat_matrix
-&gt;&gt;&gt; pat_matrix(2,1,0,0)
-[0, 0, 0]
-[0, 2, 0]
-[0, 0, 2]</p>
-<p>In case we want to find the inertia along a vector of (1,1,1):
-&gt;&gt;&gt; pat_matrix(2,1,1,1)
-[ 4, -2, -2]
-[-2,  4, -2]
-[-2, -2,  4]</p>
-</dd></dl>
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-  <div class="section" id="potential-issues-advanced-topics-future-features-in-physics-mechanics-module">
-<h1>Potential Issues/Advanced Topics/Future Features in Physics/Mechanics Module<a class="headerlink" href="#potential-issues-advanced-topics-future-features-in-physics-mechanics-module" title="Permalink to this headline">¶</a></h1>
-<p>This document will describe some of the more advanced functionality that this
-module offers but which is not part of the &#8220;official&#8221; interface. Here, some of
-the features that will be implemented in the future will also be covered, along
-with unanswered questions about proper functionality. Also, common problems
-will be discussed, along with some solutions.</p>
-<div class="section" id="common-issues">
-<h2>Common Issues<a class="headerlink" href="#common-issues" title="Permalink to this headline">¶</a></h2>
-<p>Here issues with numerically integrating code, choice of <span class="math">\(dynamicsymbols\)</span> for
-coordinate and speed representation, printing, differentiating, and
-substitution will occur.</p>
-<div class="section" id="numerically-integrating-code">
-<h3>Numerically Integrating Code<a class="headerlink" href="#numerically-integrating-code" title="Permalink to this headline">¶</a></h3>
-<p>See Future Features: Code Output</p>
-</div>
-<div class="section" id="choice-of-coordinates-and-speeds">
-<h3>Choice of Coordinates and Speeds<a class="headerlink" href="#choice-of-coordinates-and-speeds" title="Permalink to this headline">¶</a></h3>
-<p>The Kane object is set up with the assumption that the generalized speeds are
-not the same symbol as the time derivatives of the generalized coordinates.
-This isn&#8217;t to say that they can&#8217;t be the same, just that they have to have a
-different symbol. If you did this:</p>
-<div class="highlight-python"><pre>&gt;&gt; KM.coords([q1, q2, q3])
-&gt;&gt; KM.speeds([q1d, q2d, q3d])</pre>
-</div>
-<p>Your code would not work. Currently, kinematic differential equations are
-required to be provided. It is at this point that we hope the user will
-discover they should not attempt the behavior shown in the code above.</p>
-<p>This behavior might not be true for other methods of forming the equations of
-motion though.</p>
-</div>
-<div class="section" id="printing">
-<h3>Printing<a class="headerlink" href="#printing" title="Permalink to this headline">¶</a></h3>
-<p>The default printing options are to use sorting for <tt class="docutils literal"><span class="pre">Vector</span></tt> and <tt class="docutils literal"><span class="pre">Dyad</span></tt>
-measure numbers, and have unsorted output from the <tt class="docutils literal"><span class="pre">mprint</span></tt>, <tt class="docutils literal"><span class="pre">mpprint</span></tt>, and
-<tt class="docutils literal"><span class="pre">mlatex</span></tt> functions. If you are printing something large, please use one of
-those functions, as the sorting can increase printing time from seconds to
-minutes.</p>
-</div>
-<div class="section" id="differentiating">
-<h3>Differentiating<a class="headerlink" href="#differentiating" title="Permalink to this headline">¶</a></h3>
-<p>Differentiation of very large expressions can take some time in SymPy; it is
-possible for large expressions to take minutes for the derivative to be
-evaluated. This will most commonly come up in linearization.</p>
-</div>
-<div class="section" id="substitution">
-<h3>Substitution<a class="headerlink" href="#substitution" title="Permalink to this headline">¶</a></h3>
-<p>Substitution into large expressions can be slow, and take a few minutes.</p>
-</div>
-<div class="section" id="linearization">
-<h3>Linearization<a class="headerlink" href="#linearization" title="Permalink to this headline">¶</a></h3>
-<p>Currently, the <tt class="docutils literal"><span class="pre">Kane</span></tt> object&#8217;s <tt class="docutils literal"><span class="pre">linearize</span></tt> method doesn&#8217;t support cases
-where there are non-coordinate, non-speed dynamic symbols outside of the
-&#8220;dynamic equations&#8221;. It also does not support cases where time derivatives of
-these types of dynamic symbols show up. This means if you have kinematic
-differential equations which have a non-coordinate, non-speed dynamic symbol,
-it will not work. It also means if you have defined a system parameter (say a
-length or distance or mass) as a dynamic symbol, its time derivative is likely
-to show up in the dynamic equations, and this will prevent linearization.</p>
-</div>
-<div class="section" id="acceleration-of-points">
-<h3>Acceleration of Points<a class="headerlink" href="#acceleration-of-points" title="Permalink to this headline">¶</a></h3>
-<p>At a minimum, points need to have their velocities defined, as the acceleration
-can be calculated by taking the time derivative of the velocity in the same
-frame. If the 1 point or 2 point theorems were used to compute the velocity,
-the time derivative of the velocity expression will most likely be more complex
-than if you were to use the acceleration level 1 point and 2 point theorems.
-Using the acceleration level methods can result in shorted expressions at this
-point, which will result in shorter expressions later (such as when forming
-Kane&#8217;s equations).</p>
-</div>
-</div>
-<div class="section" id="advanced-interfaces">
-<h2>Advanced Interfaces<a class="headerlink" href="#advanced-interfaces" title="Permalink to this headline">¶</a></h2>
-<p>Here we will cover advanced options in: <tt class="docutils literal"><span class="pre">ReferenceFrame</span></tt>, <tt class="docutils literal"><span class="pre">dynamicsymbols</span></tt>,
-and some associated functionality.</p>
-<div class="section" id="referenceframe">
-<h3>ReferenceFrame<a class="headerlink" href="#referenceframe" title="Permalink to this headline">¶</a></h3>
-<p><tt class="docutils literal"><span class="pre">ReferenceFrame</span></tt> is shown as having a <tt class="docutils literal"><span class="pre">.name</span></tt> attribute and <tt class="docutils literal"><span class="pre">.x</span></tt>, <tt class="docutils literal"><span class="pre">.y</span></tt>,
-and <tt class="docutils literal"><span class="pre">.z</span></tt> attributes for accessing the basis vectors, as well as a fairly
-rigidly defined print output. If you wish to have a different set of indices
-defined, there is an option for this. This will also require a different
-interface for accessing the basis vectors.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">mprint</span><span class="p">,</span> <span class="n">mpprint</span><span class="p">,</span> <span class="n">mlatex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">,</span> <span class="n">indices</span><span class="o">=</span><span class="p">[</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="s">&#39;j&#39;</span><span class="p">,</span> <span class="s">&#39;k&#39;</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">[</span><span class="s">&#39;i&#39;</span><span class="p">]</span>
-<span class="go">N[&#39;i&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">x</span>
-<span class="go">N[&#39;i&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mlatex</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="go">&#39;\\mathbf{\\hat{n}_{i}}&#39;</span>
-</pre></div>
-</div>
-<p>Also, the latex output can have custom strings; rather than just indices
-though, the entirety of each basis vector can be specified. The custom latex
-strings can occur without custom indices, and also overwrites the latex string
-that would be used if there were custom indices.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">mlatex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">,</span> <span class="n">latexs</span><span class="o">=</span><span class="p">[</span><span class="s">&#39;n1&#39;</span><span class="p">,</span><span class="s">&#39;\mathbf{n}_2&#39;</span><span class="p">,</span><span class="s">&#39;cat&#39;</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mlatex</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="go">&#39;n1&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mlatex</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="go">&#39;\\mathbf{n}_2&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mlatex</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="go">&#39;cat&#39;</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="dynamicsymbols">
-<h3>dynamicsymbols<a class="headerlink" href="#dynamicsymbols" title="Permalink to this headline">¶</a></h3>
-<p>The <tt class="docutils literal"><span class="pre">dynamicsymbols</span></tt> function also has &#8216;hidden&#8217; functionality; the variable
-which is associated with time can be changed, as well as the notation for
-printing derivatives.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">dynamicsymbols</span><span class="p">,</span> <span class="n">mprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span>
-<span class="go">q1(t)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;T&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q2</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q2&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q2</span>
-<span class="go">q2(T)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span>
-<span class="go">q1(t)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1d</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mprint</span><span class="p">(</span><span class="n">q1d</span><span class="p">)</span>
-<span class="go">q1&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_str</span> <span class="o">=</span> <span class="s">&#39;d&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mprint</span><span class="p">(</span><span class="n">q1d</span><span class="p">)</span>
-<span class="go">q1d</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_str</span> <span class="o">=</span> <span class="s">&#39;</span><span class="se">\&#39;</span><span class="s">&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dynamicsymbols</span><span class="o">.</span><span class="n">_t</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Note that only dynamic symbols created after the change are different. The same
-is not true for the <span class="math">\(._str\)</span> attribute; this affects the printing output only,
-so dynamic symbols created before or after will print the same way.</p>
-<p>Also note that <tt class="docutils literal"><span class="pre">Vector</span></tt>&#8216;s <tt class="docutils literal"><span class="pre">.dt</span></tt> method uses the <tt class="docutils literal"><span class="pre">._t</span></tt> attribute of
-<tt class="docutils literal"><span class="pre">dynamicsymbols</span></tt>, along with a number of other important functions and
-methods. Don&#8217;t mix and match symbols representing time.</p>
-</div>
-<div class="section" id="advanced-functionality">
-<h3>Advanced Functionality<a class="headerlink" href="#advanced-functionality" title="Permalink to this headline">¶</a></h3>
-<p>Remember that the <tt class="docutils literal"><span class="pre">Kane</span></tt> object supports bodies which have time-varying
-masses and inertias, although this functionality isn&#8217;t completely compatible
-with the linearization method.</p>
-<p>Operators were discussed earlier as a potential way to do mathematical
-operations on <tt class="docutils literal"><span class="pre">Vector</span></tt> and <tt class="docutils literal"><span class="pre">Dyad</span></tt> objects. The majority of the code in this
-module is actually coded with them, as it can (subjectively) result in cleaner,
-shorter, more readable code. If using this interface in your code, remember to
-take care and use parentheses; the default order of operations in Python
-results in addition occurring before some of the vector products, so use
-parentheses liberally.</p>
-</div>
-</div>
-<div class="section" id="future-features">
-<h2>Future Features<a class="headerlink" href="#future-features" title="Permalink to this headline">¶</a></h2>
-<p>This will cover the planned features to be added to this submodule.</p>
-<div class="section" id="code-output">
-<h3>Code Output<a class="headerlink" href="#code-output" title="Permalink to this headline">¶</a></h3>
-<p>A function for generating code output for numerical integration is the highest
-priority feature to implement next. There are a number of considerations here.</p>
-<p>Code output for C (using the GSL libraries), Fortran 90 (using LSODA), MATLAB,
-and SciPy is the goal. Things to be considered include: use of <tt class="docutils literal"><span class="pre">cse</span></tt> on large
-expressions for MATLAB and SciPy, which are interpretive. It is currently unclear
-whether compiled languages will benefit from common subexpression elimination,
-especially considering that it is a common part of compiler optimization, and
-there can be a significant time penalty when calling <tt class="docutils literal"><span class="pre">cse</span></tt>.</p>
-<p>Care needs to be taken when constructing the strings for these expressions, as
-well as handling of input parameters, and other dynamic symbols. How to deal
-with output quantities when integrating also needs to be decided, with the
-potential for multiple options being considered.</p>
-</div>
-<div class="section" id="additional-options-on-initialization-of-kane-rigidbody-and-particle">
-<h3>Additional Options on Initialization of Kane, RigidBody, and Particle<a class="headerlink" href="#additional-options-on-initialization-of-kane-rigidbody-and-particle" title="Permalink to this headline">¶</a></h3>
-<p>This would allow a user to specify all relevant information using keyword
-arguments when creating these objects. This is fairly clear for <tt class="docutils literal"><span class="pre">RigidBody</span></tt>
-and <tt class="docutils literal"><span class="pre">Point</span></tt>. For <tt class="docutils literal"><span class="pre">Kane</span></tt>, everything but the force and body lists will be
-able to be entered, as computation of Fr and Fr* can take a while, and produce
-an output.</p>
-</div>
-<div class="section" id="additional-methods-for-rigidbody-and-particle">
-<h3>Additional Methods for RigidBody and Particle<a class="headerlink" href="#additional-methods-for-rigidbody-and-particle" title="Permalink to this headline">¶</a></h3>
-<p>For <tt class="docutils literal"><span class="pre">RigidBody</span></tt> and <tt class="docutils literal"><span class="pre">Particle</span></tt> (not all methods for <tt class="docutils literal"><span class="pre">Particle</span></tt> though),
-add methods for getting: momentum, angular momentum, and kinetic energy.
-Additionally, adding a attribute and method for defining potential energy would
-allow for a total energy method/property.</p>
-<p>Also possible is including the method which creates a transformation matrix for
-3D animations; this would require a &#8220;reference orientation&#8221; for a camera as
-well as a &#8220;reference point&#8221; for distance to the camera. Development of this
-could also be tied into code output.</p>
-</div>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Potential Issues/Advanced Topics/Future Features in Physics/Mechanics Module</a><ul>
-<li><a class="reference internal" href="#common-issues">Common Issues</a><ul>
-<li><a class="reference internal" href="#numerically-integrating-code">Numerically Integrating Code</a></li>
-<li><a class="reference internal" href="#choice-of-coordinates-and-speeds">Choice of Coordinates and Speeds</a></li>
-<li><a class="reference internal" href="#printing">Printing</a></li>
-<li><a class="reference internal" href="#differentiating">Differentiating</a></li>
-<li><a class="reference internal" href="#substitution">Substitution</a></li>
-<li><a class="reference internal" href="#linearization">Linearization</a></li>
-<li><a class="reference internal" href="#acceleration-of-points">Acceleration of Points</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#advanced-interfaces">Advanced Interfaces</a><ul>
-<li><a class="reference internal" href="#referenceframe">ReferenceFrame</a></li>
-<li><a class="reference internal" href="#dynamicsymbols">dynamicsymbols</a></li>
-<li><a class="reference internal" href="#advanced-functionality">Advanced Functionality</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#future-features">Future Features</a><ul>
-<li><a class="reference internal" href="#code-output">Code Output</a></li>
-<li><a class="reference internal" href="#additional-options-on-initialization-of-kane-rigidbody-and-particle">Additional Options on Initialization of Kane, RigidBody, and Particle</a></li>
-<li><a class="reference internal" href="#additional-methods-for-rigidbody-and-particle">Additional Methods for RigidBody and Particle</a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../mechanics/examples.html"
-                        title="previous chapter">Examples for Physics.Mechanics</a></p>
-  <h4>Next topic</h4>
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-  <div class="section" id="essential-components-docstrings">
-<h1>Essential Components (Docstrings)<a class="headerlink" href="#essential-components-docstrings" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="referenceframe">
-<h2>ReferenceFrame<a class="headerlink" href="#referenceframe" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.physics.mechanics.essential.ReferenceFrame">
-<em class="property">class </em><tt class="descclassname">sympy.physics.mechanics.essential.</tt><tt class="descname">ReferenceFrame</tt><big>(</big><em>name</em>, <em>indices=None</em>, <em>latexs=None</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#ReferenceFrame"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.ReferenceFrame" title="Permalink to this definition">¶</a></dt>
-<dd><p>A reference frame in classical mechanics.</p>
-<p>ReferenceFrame is a class used to represent a reference frame in classical
-mechanics. It has a standard basis of three unit vectors in the frame&#8217;s
-x, y, and z directions.</p>
-<p>It also can have a rotation relative to a parent frame; this rotation is
-defined by a direction cosine matrix relating this frame&#8217;s basis vectors to
-the parent frame&#8217;s basis vectors.  It can also have an angular velocity
-vector, defined in another frame.</p>
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.ReferenceFrame.ang_acc_in">
-<tt class="descname">ang_acc_in</tt><big>(</big><em>self</em>, <em>otherframe</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#ReferenceFrame.ang_acc_in"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.ReferenceFrame.ang_acc_in" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the angular acceleration Vector of the ReferenceFrame.</p>
-<p>Effectively returns the Vector:
-^N alpha ^B
-which represent the angular acceleration of B in N, where B is self, and
-N is otherframe.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>otherframe</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The ReferenceFrame which the angular acceleration is returned in.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">Vector</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">V</span> <span class="o">=</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">set_ang_acc</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">V</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">ang_acc_in</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">10*N.x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.ReferenceFrame.ang_vel_in">
-<tt class="descname">ang_vel_in</tt><big>(</big><em>self</em>, <em>otherframe</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#ReferenceFrame.ang_vel_in"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.ReferenceFrame.ang_vel_in" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the angular velocity Vector of the ReferenceFrame.</p>
-<p>Effectively returns the Vector:
-^N omega ^B
-which represent the angular velocity of B in N, where B is self, and
-N is otherframe.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>otherframe</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The ReferenceFrame which the angular velocity is returned in.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">Vector</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">V</span> <span class="o">=</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">V</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">10*N.x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.ReferenceFrame.dcm">
-<tt class="descname">dcm</tt><big>(</big><em>self</em>, <em>otherframe</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#ReferenceFrame.dcm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.ReferenceFrame.dcm" title="Permalink to this definition">¶</a></dt>
-<dd><p>The direction cosine matrix between frames.</p>
-<p>This gives the DCM between this frame and the otherframe.
-The format is N.xyz = N.dcm(B) * B.xyz
-A SymPy Matrix is returned.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>otherframe</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The otherframe which the DCM is generated to.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">Vector</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;q1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">dcm</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-<span class="go">[1,       0,        0]</span>
-<span class="go">[0, cos(q1), -sin(q1)]</span>
-<span class="go">[0, sin(q1),  cos(q1)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.ReferenceFrame.orient">
-<tt class="descname">orient</tt><big>(</big><em>self</em>, <em>parent</em>, <em>rot_type</em>, <em>amounts</em>, <em>rot_order=''</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#ReferenceFrame.orient"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.ReferenceFrame.orient" title="Permalink to this definition">¶</a></dt>
-<dd><p>Defines the orientation of this frame relative to a parent frame.</p>
-<p>Supported orientation types are Body, Space, Quaternion, Axis.
-Examples show correct usage.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>parent</strong> : ReferenceFrame</p>
-<blockquote>
-<div><p>The frame that this ReferenceFrame will have its orientation matrix
-defined in relation to.</p>
-</div></blockquote>
-<p><strong>rot_type</strong> : str</p>
-<blockquote>
-<div><p>The type of orientation matrix that is being created.</p>
-</div></blockquote>
-<p><strong>amounts</strong> : list OR value</p>
-<blockquote>
-<div><p>The quantities that the orientation matrix will be defined by.</p>
-</div></blockquote>
-<p><strong>rot_order</strong> : str</p>
-<blockquote class="last">
-<div><p>If applicable, the order of a series of rotations.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">Vector</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q0</span><span class="p">,</span> <span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span><span class="p">,</span> <span class="n">q4</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;q0 q1 q2 q3 q4&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Now we have a choice of how to implement the orientation. First is
-Body. Body orientation takes this reference frame through three
-successive simple rotations. Acceptable rotation orders are of length
-3, expressed in XYZ or 123, and cannot have a rotation about about an
-axis twice in a row.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">orient</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="s">&#39;Body&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span><span class="p">],</span> <span class="s">&#39;123&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">orient</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="s">&#39;Body&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="s">&#39;ZXZ&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">orient</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="s">&#39;Body&#39;</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="s">&#39;XYX&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Next is Space. Space is like Body, but the rotations are applied in the
-opposite order.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">orient</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="s">&#39;Space&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span><span class="p">],</span> <span class="s">&#39;312&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Next is Quaternion. This orients the new ReferenceFrame with
-Quaternions, defined as a finite rotation about lambda, a unit vector,
-by some amount theta.
-This orientation is described by four parameters:
-q0 = cos(theta/2)
-q1 = lambda_x sin(theta/2)
-q2 = lambda_y sin(theta/2)
-q3 = lambda_z sin(theta/2)
-Quaternion does not take in a rotation order.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">orient</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="s">&#39;Quaternion&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q0</span><span class="p">,</span> <span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span><span class="p">])</span>
-</pre></div>
-</div>
-<p>Last is Axis. This is a rotation about an arbitrary, non-time-varying
-axis by some angle. The axis is supplied as a Vector. This is how
-simple rotations are defined.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">orient</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.ReferenceFrame.orientnew">
-<tt class="descname">orientnew</tt><big>(</big><em>self</em>, <em>newname</em>, <em>rot_type</em>, <em>amounts</em>, <em>rot_order=''</em>, <em>indices=None</em>, <em>latexs=None</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#ReferenceFrame.orientnew"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.ReferenceFrame.orientnew" title="Permalink to this definition">¶</a></dt>
-<dd><p>Creates a new ReferenceFrame oriented with respect to this Frame.</p>
-<p>See ReferenceFrame.orient() for acceptable rotation types, amounts,
-and orders. Parent is going to be self.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>parent</strong> : ReferenceFrame</p>
-<blockquote>
-<div><p>The frame that this ReferenceFrame will have its orientation matrix
-defined in relation to.</p>
-</div></blockquote>
-<p><strong>rot_type</strong> : str</p>
-<blockquote>
-<div><p>The type of orientation matrix that is being created.</p>
-</div></blockquote>
-<p><strong>amounts</strong> : list OR value</p>
-<blockquote>
-<div><p>The quantities that the orientation matrix will be defined by.</p>
-</div></blockquote>
-<p><strong>rot_order</strong> : str</p>
-<blockquote class="last">
-<div><p>If applicable, the order of a series of rotations.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">Vector</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q0</span><span class="p">,</span> <span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span><span class="p">,</span> <span class="n">q4</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;q0 q1 q2 q3 q4&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Now we have a choice of how to implement the orientation. First is
-Body. Body orientation takes this reference frame through three
-successive simple rotations. Acceptable rotation orders are of length
-3, expressed in XYZ or 123, and cannot have a rotation about about an
-axis twice in a row.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">orient</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="s">&#39;Body&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span><span class="p">],</span> <span class="s">&#39;123&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">orient</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="s">&#39;Body&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="s">&#39;ZXZ&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">orient</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="s">&#39;Body&#39;</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="s">&#39;XYX&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Next is Space. Space is like Body, but the rotations are applied in the
-opposite order.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">orient</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="s">&#39;Space&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span><span class="p">],</span> <span class="s">&#39;312&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Next is Quaternion. This orients the new ReferenceFrame with
-Quaternions, defined as a finite rotation about lambda, a unit vector,
-by some amount theta.
-This orientation is described by four parameters:
-q0 = cos(theta/2)
-q1 = lambda_x sin(theta/2)
-q2 = lambda_y sin(theta/2)
-q3 = lambda_z sin(theta/2)
-Quaternion does not take in a rotation order.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">orient</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="s">&#39;Quaternion&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q0</span><span class="p">,</span> <span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span><span class="p">])</span>
-</pre></div>
-</div>
-<p>Last is Axis. This is a rotation about an arbitrary, non-time-varying
-axis by some angle. The axis is supplied as a Vector. This is how
-simple rotations are defined.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">orient</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.ReferenceFrame.set_ang_acc">
-<tt class="descname">set_ang_acc</tt><big>(</big><em>self</em>, <em>otherframe</em>, <em>value</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#ReferenceFrame.set_ang_acc"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.ReferenceFrame.set_ang_acc" title="Permalink to this definition">¶</a></dt>
-<dd><p>Define the angular acceleration Vector in a ReferenceFrame.</p>
-<p>Defines the angular acceleration of this ReferenceFrame, in another.
-Angular acceleration can be defined with respect to multiple different
-ReferenceFrames. Care must be taken to not create loops which are
-inconsistent.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>otherframe</strong> : ReferenceFrame</p>
-<blockquote>
-<div><p>A ReferenceFrame to define the angular acceleration in</p>
-</div></blockquote>
-<p><strong>value</strong> : Vector</p>
-<blockquote class="last">
-<div><p>The Vector representing angular acceleration</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">Vector</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">V</span> <span class="o">=</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">set_ang_acc</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">V</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">ang_acc_in</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">10*N.x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.ReferenceFrame.set_ang_vel">
-<tt class="descname">set_ang_vel</tt><big>(</big><em>self</em>, <em>otherframe</em>, <em>value</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#ReferenceFrame.set_ang_vel"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.ReferenceFrame.set_ang_vel" title="Permalink to this definition">¶</a></dt>
-<dd><p>Define the angular velocity vector in a ReferenceFrame.</p>
-<p>Defines the angular velocity of this ReferenceFrame, in another.
-Angular velocity can be defined with respect to multiple different
-ReferenceFrames. Care must be taken to not create loops which are
-inconsistent.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>otherframe</strong> : ReferenceFrame</p>
-<blockquote>
-<div><p>A ReferenceFrame to define the angular velocity in</p>
-</div></blockquote>
-<p><strong>value</strong> : Vector</p>
-<blockquote class="last">
-<div><p>The Vector representing angular velocity</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">Vector</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">V</span> <span class="o">=</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">V</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">10*N.x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.mechanics.essential.ReferenceFrame.x">
-<tt class="descname">x</tt><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#ReferenceFrame.x"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.ReferenceFrame.x" title="Permalink to this definition">¶</a></dt>
-<dd><p>The basis Vector for the ReferenceFrame, in the x direction.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.mechanics.essential.ReferenceFrame.y">
-<tt class="descname">y</tt><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#ReferenceFrame.y"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.ReferenceFrame.y" title="Permalink to this definition">¶</a></dt>
-<dd><p>The basis Vector for the ReferenceFrame, in the y direction.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.mechanics.essential.ReferenceFrame.z">
-<tt class="descname">z</tt><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#ReferenceFrame.z"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.ReferenceFrame.z" title="Permalink to this definition">¶</a></dt>
-<dd><p>The basis Vector for the ReferenceFrame, in the z direction.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="vector">
-<h2>Vector<a class="headerlink" href="#vector" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.physics.mechanics.essential.Vector">
-<em class="property">class </em><tt class="descclassname">sympy.physics.mechanics.essential.</tt><tt class="descname">Vector</tt><big>(</big><em>inlist</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#Vector"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.Vector" title="Permalink to this definition">¶</a></dt>
-<dd><p>The class used to define vectors.</p>
-<p>It along with ReferenceFrame are the building blocks of describing a
-classical mechanics system in PyDy.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="7%" />
-<col width="12%" />
-<col width="82%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>simp</td>
-<td>Boolean</td>
-<td>Let certain methods use trigsimp on their outputs</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.Vector.cross">
-<tt class="descname">cross</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#Vector.cross"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.Vector.cross" title="Permalink to this definition">¶</a></dt>
-<dd><p>The cross product operator for two Vectors.</p>
-<p>Returns a Vector, expressed in the same ReferenceFrames as self.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>other</strong> : Vector</p>
-<blockquote class="last">
-<div><p>The Vector which we are crossing with</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">Vector</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;q1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">x</span> <span class="o">^</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span>
-<span class="go">N.z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">x</span> <span class="o">^</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span>
-<span class="go">N.z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">y</span> <span class="o">^</span> <span class="n">A</span><span class="o">.</span><span class="n">x</span>
-<span class="go">- sin(q1)*A.y - cos(q1)*A.z</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.Vector.diff">
-<tt class="descname">diff</tt><big>(</big><em>self</em>, <em>wrt</em>, <em>otherframe</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#Vector.diff"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.Vector.diff" title="Permalink to this definition">¶</a></dt>
-<dd><p>Takes the partial derivative, with respect to a value, in a frame.</p>
-<p>Returns a Vector.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>wrt</strong> : Symbol</p>
-<blockquote>
-<div><p>What the partial derivative is taken with respect to.</p>
-</div></blockquote>
-<p><strong>otherframe</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The ReferenceFrame that the partial derivative is taken in.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">Vector</span><span class="p">,</span> <span class="n">dynamicsymbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Vector</span><span class="o">.</span><span class="n">simp</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
-<span class="go">- q1&#39;*A.z</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.Vector.doit">
-<tt class="descname">doit</tt><big>(</big><em>self</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#Vector.doit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.Vector.doit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calls .doit() on each term in the Vector</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.Vector.dot">
-<tt class="descname">dot</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#Vector.dot"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.Vector.dot" title="Permalink to this definition">¶</a></dt>
-<dd><p>Dot product of two vectors.</p>
-<p>Returns a scalar, the dot product of the two Vectors</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>other</strong> : Vector</p>
-<blockquote class="last">
-<div><p>The Vector which we are dotting with</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">Vector</span><span class="p">,</span> <span class="n">dot</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;q1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dot</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dot</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dot</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">,</span> <span class="n">A</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="go">cos(q1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.Vector.dt">
-<tt class="descname">dt</tt><big>(</big><em>self</em>, <em>otherframe</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#Vector.dt"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.Vector.dt" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the time derivative of the Vector in a ReferenceFrame.</p>
-<p>Returns a Vector which is the time derivative of the self Vector, taken
-in frame otherframe.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>otherframe</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The ReferenceFrame that the partial derivative is taken in.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">Vector</span><span class="p">,</span> <span class="n">dynamicsymbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;q1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u1</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;u1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v</span> <span class="o">=</span> <span class="n">u1</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">10</span><span class="o">*</span><span class="n">A</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">dt</span><span class="p">(</span><span class="n">N</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v</span><span class="o">.</span><span class="n">dt</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">u1&#39;*N.x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.Vector.express">
-<tt class="descname">express</tt><big>(</big><em>self</em>, <em>otherframe</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#Vector.express"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.Vector.express" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a vector, expressed in the other frame.</p>
-<p>A new Vector is returned, equalivalent to this Vector, but its
-components are all defined in only the otherframe.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>otherframe</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The frame for this Vector to be described in</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">Vector</span><span class="p">,</span> <span class="n">dynamicsymbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">express</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">cos(q1)*N.x - sin(q1)*N.z</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.Vector.magnitude">
-<tt class="descname">magnitude</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#Vector.magnitude"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.Vector.magnitude" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the magnitude (Euclidean norm) of self.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.Vector.normalize">
-<tt class="descname">normalize</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#Vector.normalize"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.Vector.normalize" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a Vector of magnitude 1, codirectional with self.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.Vector.outer">
-<tt class="descname">outer</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#Vector.outer"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.Vector.outer" title="Permalink to this definition">¶</a></dt>
-<dd><p>Outer product between two Vectors.</p>
-<p>A rank increasing operation, which returns a Dyadic from two Vectors</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>other</strong> : Vector</p>
-<blockquote class="last">
-<div><p>The Vector to take the outer product with</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">outer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">outer</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(N.x|N.x)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.Vector.simplify">
-<tt class="descname">simplify</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#Vector.simplify"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.Vector.simplify" title="Permalink to this definition">¶</a></dt>
-<dd><p>Simplify the elements in the Vector in place.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.Vector.subs">
-<tt class="descname">subs</tt><big>(</big><em>self</em>, <em>*args</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#Vector.subs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.Vector.subs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Substituion on the Vector.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;s&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span> <span class="o">*</span> <span class="n">s</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">subs</span><span class="p">({</span><span class="n">s</span><span class="p">:</span> <span class="mi">2</span><span class="p">})</span>
-<span class="go">2*N.x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="dyadic">
-<h2>Dyadic<a class="headerlink" href="#dyadic" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.physics.mechanics.essential.Dyadic">
-<em class="property">class </em><tt class="descclassname">sympy.physics.mechanics.essential.</tt><tt class="descname">Dyadic</tt><big>(</big><em>inlist</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#Dyadic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.Dyadic" title="Permalink to this definition">¶</a></dt>
-<dd><p>A Dyadic object.</p>
-<p>See:
-<a class="reference external" href="http://en.wikipedia.org/wiki/Dyadic_tensor">http://en.wikipedia.org/wiki/Dyadic_tensor</a>
-Kane, T., Levinson, D. Dynamics Theory and Applications. 1985 McGraw-Hill</p>
-<p>A more powerful way to represent a rigid body&#8217;s inertia. While it is more
-complex, by choosing Dyadic components to be in body fixed basis vectors,
-the resulting matrix is equivalent to the inertia tensor.</p>
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.Dyadic.cross">
-<tt class="descname">cross</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="headerlink" href="#sympy.physics.mechanics.essential.Dyadic.cross" title="Permalink to this definition">¶</a></dt>
-<dd><p>For a cross product in the form: Dyadic x Vector.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>other</strong> : Vector</p>
-<blockquote class="last">
-<div><p>The Vector that we are crossing this Dyadic with</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">outer</span><span class="p">,</span> <span class="n">cross</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">outer</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cross</span><span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="go">(N.x|N.z)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.Dyadic.doit">
-<tt class="descname">doit</tt><big>(</big><em>self</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#Dyadic.doit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.Dyadic.doit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calls .doit() on each term in the Dyadic</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.Dyadic.dot">
-<tt class="descname">dot</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="headerlink" href="#sympy.physics.mechanics.essential.Dyadic.dot" title="Permalink to this definition">¶</a></dt>
-<dd><p>The inner product operator for a Dyadic and a Dyadic or Vector.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>other</strong> : Dyadic or Vector</p>
-<blockquote class="last">
-<div><p>The other Dyadic or Vector to take the inner product with</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">outer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D1</span> <span class="o">=</span> <span class="n">outer</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D2</span> <span class="o">=</span> <span class="n">outer</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D1</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">D2</span><span class="p">)</span>
-<span class="go">(N.x|N.y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D1</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="go">N.x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.Dyadic.dt">
-<tt class="descname">dt</tt><big>(</big><em>self</em>, <em>frame</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#Dyadic.dt"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.Dyadic.dt" title="Permalink to this definition">¶</a></dt>
-<dd><p>Take the time derivative of this Dyadic in a frame.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>frame</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The frame to take the time derivative in</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">outer</span><span class="p">,</span> <span class="n">dynamicsymbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">outer</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="o">.</span><span class="n">dt</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
-<span class="go">- q&#39;*(N.y|N.x) - q&#39;*(N.x|N.y)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.Dyadic.express">
-<tt class="descname">express</tt><big>(</big><em>self</em>, <em>frame1</em>, <em>frame2=None</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#Dyadic.express"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.Dyadic.express" title="Permalink to this definition">¶</a></dt>
-<dd><p>Expresses this Dyadic in alternate frame(s)</p>
-<p>The first frame is the list side expression, the second frame is the
-right side; if Dyadic is in form A.x|B.y, you can express it in two
-different frames. If no second frame is given, the Dyadic is expressed in
-only one frame.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>frame1</strong> : ReferenceFrame</p>
-<blockquote>
-<div><p>The frame to express the left side of the Dyadic in</p>
-</div></blockquote>
-<p><strong>frame2</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>If provided, the frame to express the right side of the Dyadic in</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">outer</span><span class="p">,</span> <span class="n">dynamicsymbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">outer</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="o">.</span><span class="n">express</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
-<span class="go">cos(q)*(B.x|N.x) - sin(q)*(B.y|N.x)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.Dyadic.simplify">
-<tt class="descname">simplify</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#Dyadic.simplify"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.Dyadic.simplify" title="Permalink to this definition">¶</a></dt>
-<dd><p>Simplify the elements in the Dyadic in-place.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.Dyadic.subs">
-<tt class="descname">subs</tt><big>(</big><em>self</em>, <em>*args</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#Dyadic.subs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.Dyadic.subs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Substituion on the Dyadic.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;s&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">s</span> <span class="o">*</span> <span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="o">|</span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">subs</span><span class="p">({</span><span class="n">s</span><span class="p">:</span> <span class="mi">2</span><span class="p">})</span>
-<span class="go">2*(N.x|N.x)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-</div>
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-            </a></p>
-  <h3><a href="../../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Essential Components (Docstrings)</a><ul>
-<li><a class="reference internal" href="#referenceframe">ReferenceFrame</a></li>
-<li><a class="reference internal" href="#vector">Vector</a></li>
-<li><a class="reference internal" href="#dyadic">Dyadic</a></li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../../mechanics/reference.html"
-                        title="previous chapter">References for Physics/Mechanics</a></p>
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-  <div class="section" id="related-essential-functions-docstrings">
-<h1>Related Essential Functions (Docstrings)<a class="headerlink" href="#related-essential-functions-docstrings" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="dynamicsymbols">
-<h2>dynamicsymbols<a class="headerlink" href="#dynamicsymbols" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.physics.mechanics.essential.dynamicsymbols">
-<tt class="descclassname">sympy.physics.mechanics.essential.</tt><tt class="descname">dynamicsymbols</tt><big>(</big><em>names</em>, <em>level=0</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/essential.html#dynamicsymbols"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.essential.dynamicsymbols" title="Permalink to this definition">¶</a></dt>
-<dd><p>Uses symbols and Function for functions of time.</p>
-<p>Creates a SymPy UndefinedFunction, which is then initialized as a function
-of a variable, the default being Symbol(&#8216;t&#8217;).</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>names</strong> : str</p>
-<blockquote>
-<div><p>Names of the dynamic symbols you want to create; works the same way as
-inputs to symbols</p>
-</div></blockquote>
-<p><strong>level</strong> : int</p>
-<blockquote>
-<div><p>Level of differentiation of the returned function; d/dt once of t,
-twice of t, etc.</p>
-</div></blockquote>
-<p><strong>Examples</strong> :</p>
-<p><strong>=======</strong> :</p>
-<p><strong>&gt;&gt;&gt; from sympy.physics.mechanics import dynamicsymbols</strong> :</p>
-<p><strong>&gt;&gt;&gt; from sympy import diff, Symbol</strong> :</p>
-<p><strong>&gt;&gt;&gt; q1 = dynamicsymbols(&#8216;q1&#8217;)</strong> :</p>
-<p><strong>&gt;&gt;&gt; q1</strong> :</p>
-<p><strong>q1(t)</strong> :</p>
-<p><strong>&gt;&gt;&gt; diff(q1, Symbol(&#8216;t&#8217;))</strong> :</p>
-<p class="last"><strong>Derivative(q1(t), t)</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</div>
-<div class="section" id="dot">
-<h2>dot<a class="headerlink" href="#dot" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.physics.mechanics.functions.dot">
-<tt class="descclassname">sympy.physics.mechanics.functions.</tt><tt class="descname">dot</tt><big>(</big><em>vec1</em>, <em>vec2</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/functions.html#dot"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.functions.dot" title="Permalink to this definition">¶</a></dt>
-<dd><p>Dot product convenience wrapper for Vector.dot(): 
-Dot product of two vectors.</p>
-<p>Returns a scalar, the dot product of the two Vectors</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>other</strong> : Vector</p>
-<blockquote class="last">
-<div><p>The Vector which we are dotting with</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">Vector</span><span class="p">,</span> <span class="n">dot</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;q1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dot</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dot</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dot</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">,</span> <span class="n">A</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="go">cos(q1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="cross">
-<h2>cross<a class="headerlink" href="#cross" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.physics.mechanics.functions.cross">
-<tt class="descclassname">sympy.physics.mechanics.functions.</tt><tt class="descname">cross</tt><big>(</big><em>vec1</em>, <em>vec2</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/functions.html#cross"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.functions.cross" title="Permalink to this definition">¶</a></dt>
-<dd><p>Cross product convenience wrapper for Vector.cross(): 
-The cross product operator for two Vectors.</p>
-<p>Returns a Vector, expressed in the same ReferenceFrames as self.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>other</strong> : Vector</p>
-<blockquote class="last">
-<div><p>The Vector which we are crossing with</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">Vector</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;q1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">x</span> <span class="o">^</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span>
-<span class="go">N.z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">x</span> <span class="o">^</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span>
-<span class="go">N.z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">y</span> <span class="o">^</span> <span class="n">A</span><span class="o">.</span><span class="n">x</span>
-<span class="go">- sin(q1)*A.y - cos(q1)*A.z</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="outer">
-<h2>outer<a class="headerlink" href="#outer" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.physics.mechanics.functions.outer">
-<tt class="descclassname">sympy.physics.mechanics.functions.</tt><tt class="descname">outer</tt><big>(</big><em>vec1</em>, <em>vec2</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/functions.html#outer"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.functions.outer" title="Permalink to this definition">¶</a></dt>
-<dd><p>Outer prodcut convenience wrapper for Vector.outer():
-Returns a vector, expressed in the other frame.</p>
-<p>A new Vector is returned, equalivalent to this Vector, but its
-components are all defined in only the otherframe.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>otherframe</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The frame for this Vector to be described in</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">Vector</span><span class="p">,</span> <span class="n">dynamicsymbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">express</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">cos(q1)*N.x - sin(q1)*N.z</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="express">
-<h2>express<a class="headerlink" href="#express" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.physics.mechanics.functions.express">
-<tt class="descclassname">sympy.physics.mechanics.functions.</tt><tt class="descname">express</tt><big>(</big><em>vec</em>, <em>frame</em>, <em>frame2=None</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/functions.html#express"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.functions.express" title="Permalink to this definition">¶</a></dt>
-<dd><p>Express convenience wrapper for Vector.express(): 
-Returns a vector, expressed in the other frame.</p>
-<p>A new Vector is returned, equalivalent to this Vector, but its
-components are all defined in only the otherframe.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>otherframe</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The frame for this Vector to be described in</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">Vector</span><span class="p">,</span> <span class="n">dynamicsymbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">express</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">cos(q1)*N.x - sin(q1)*N.z</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-
-
-          </div>
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-              <img class="logo" src="../../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Related Essential Functions (Docstrings)</a><ul>
-<li><a class="reference internal" href="#dynamicsymbols">dynamicsymbols</a></li>
-<li><a class="reference internal" href="#dot">dot</a></li>
-<li><a class="reference internal" href="#cross">cross</a></li>
-<li><a class="reference internal" href="#outer">outer</a></li>
-<li><a class="reference internal" href="#express">express</a></li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../../mechanics/api/essential.html"
-                        title="previous chapter">Essential Components (Docstrings)</a></p>
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-  <div class="section" id="kane-s-method-lagrange-s-method-associated-functions-docstrings">
-<h1>Kane&#8217;s Method, Lagrange&#8217;s Method &amp; Associated Functions (Docstrings)<a class="headerlink" href="#kane-s-method-lagrange-s-method-associated-functions-docstrings" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="module-sympy.physics.mechanics.kane">
-<span id="kane"></span><h2>Kane<a class="headerlink" href="#module-sympy.physics.mechanics.kane" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.physics.mechanics.kane.KanesMethod">
-<em class="property">class </em><tt class="descclassname">sympy.physics.mechanics.kane.</tt><tt class="descname">KanesMethod</tt><big>(</big><em>frame</em>, <em>q_ind</em>, <em>u_ind</em>, <em>kd_eqs=None</em>, <em>q_dependent=</em><span class="optional">[</span><span class="optional">]</span>, <em>configuration_constraints=</em><span class="optional">[</span><span class="optional">]</span>, <em>u_dependent=</em><span class="optional">[</span><span class="optional">]</span>, <em>velocity_constraints=</em><span class="optional">[</span><span class="optional">]</span>, <em>acceleration_constraints=None</em>, <em>u_auxiliary=</em><span class="optional">[</span><span class="optional">]</span><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/kane.html#KanesMethod"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.kane.KanesMethod" title="Permalink to this definition">¶</a></dt>
-<dd><p>Kane&#8217;s method object.</p>
-<p>This object is used to do the &#8220;book-keeping&#8221; as you go through and form
-equations of motion in the way Kane presents in:
-Kane, T., Levinson, D. Dynamics Theory and Applications. 1985 McGraw-Hill</p>
-<p>The attributes are for equations in the form [M] udot = forcing.</p>
-<p class="rubric">Examples</p>
-<p>This is a simple example for a one degree of freedom translational
-spring-mass-damper.</p>
-<p>In this example, we first need to do the kinematics.
-This involves creating generalized speeds and coordinates and their
-derivatives.
-Then we create a point and set its velocity in a frame:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">dynamicsymbols</span><span class="p">,</span> <span class="n">ReferenceFrame</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Particle</span><span class="p">,</span> <span class="n">KanesMethod</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="p">,</span> <span class="n">u</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q u&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qd</span><span class="p">,</span> <span class="n">ud</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q u&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;m c k&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">u</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Next we need to arrange/store information in the way that KanesMethod
-requires.  The kinematic differential equations need to be stored in a
-dict.  A list of forces/torques must be constructed, where each entry in
-the list is a (Point, Vector) or (ReferenceFrame, Vector) tuple, where the
-Vectors represent the Force or Torque.
-Next a particle needs to be created, and it needs to have a point and mass
-assigned to it.
-Finally, a list of all bodies and particles needs to be created:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">kd</span> <span class="o">=</span> <span class="p">[</span><span class="n">qd</span> <span class="o">-</span> <span class="n">u</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">FL</span> <span class="o">=</span> <span class="p">[(</span><span class="n">P</span><span class="p">,</span> <span class="p">(</span><span class="o">-</span><span class="n">k</span> <span class="o">*</span> <span class="n">q</span> <span class="o">-</span> <span class="n">c</span> <span class="o">*</span> <span class="n">u</span><span class="p">)</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pa</span> <span class="o">=</span> <span class="n">Particle</span><span class="p">(</span><span class="s">&#39;pa&#39;</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">BL</span> <span class="o">=</span> <span class="p">[</span><span class="n">pa</span><span class="p">]</span>
-</pre></div>
-</div>
-<p>Finally we can generate the equations of motion.
-First we create the KanesMethod object and supply an inertial frame,
-coordinates, generalized speeds, and the kinematic differential equations.
-Additional quantities such as configuration and motion constraints,
-dependent coordinates and speeds, and auxiliary speeds are also supplied
-here (see the online documentation).
-Next we form FR* and FR to complete: Fr + Fr* = 0.
-We have the equations of motion at this point.
-It makes sense to rearrnge them though, so we calculate the mass matrix and
-the forcing terms, for E.o.M. in the form: [MM] udot = forcing, where MM is
-the mass matrix, udot is a vector of the time derivatives of the
-generalized speeds, and forcing is a vector representing &#8220;forcing&#8221; terms:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span> <span class="o">=</span> <span class="n">KanesMethod</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">q_ind</span><span class="o">=</span><span class="p">[</span><span class="n">q</span><span class="p">],</span> <span class="n">u_ind</span><span class="o">=</span><span class="p">[</span><span class="n">u</span><span class="p">],</span> <span class="n">kd_eqs</span><span class="o">=</span><span class="n">kd</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">fr</span><span class="p">,</span> <span class="n">frstar</span><span class="p">)</span> <span class="o">=</span> <span class="n">KM</span><span class="o">.</span><span class="n">kanes_equations</span><span class="p">(</span><span class="n">FL</span><span class="p">,</span> <span class="n">BL</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">MM</span> <span class="o">=</span> <span class="n">KM</span><span class="o">.</span><span class="n">mass_matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">forcing</span> <span class="o">=</span> <span class="n">KM</span><span class="o">.</span><span class="n">forcing</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rhs</span> <span class="o">=</span> <span class="n">MM</span><span class="o">.</span><span class="n">inv</span><span class="p">()</span> <span class="o">*</span> <span class="n">forcing</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rhs</span>
-<span class="go">[(-c*u(t) - k*q(t))/m]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span><span class="o">.</span><span class="n">linearize</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">[ 0,  1]</span>
-<span class="go">[-k, -c]</span>
-</pre></div>
-</div>
-<p>Please look at the documentation pages for more information on how to
-perform linearization and how to deal with dependent coordinates &amp; speeds,
-and how do deal with bringing non-contributing forces into evidence.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="24%" />
-<col width="9%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>auxiliary</td>
-<td>Matrix</td>
-<td>If applicable, the set of auxiliary Kane&#8217;s
-equations used to solve for non-contributing
-forces.</td>
-</tr>
-<tr class="row-even"><td>mass_matrix</td>
-<td>Matrix</td>
-<td>The system&#8217;s mass matrix</td>
-</tr>
-<tr class="row-odd"><td>forcing</td>
-<td>Matrix</td>
-<td>The system&#8217;s forcing vector</td>
-</tr>
-<tr class="row-even"><td>mass_matrix_full</td>
-<td>Matrix</td>
-<td>The &#8220;mass matrix&#8221; for the u&#8217;s and q&#8217;s</td>
-</tr>
-<tr class="row-odd"><td>forcing_full</td>
-<td>Matrix</td>
-<td>The &#8220;forcing vector&#8221; for the u&#8217;s and q&#8217;s</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.physics.mechanics.kane.KanesMethod.kanes_equations">
-<tt class="descname">kanes_equations</tt><big>(</big><em>self</em>, <em>FL</em>, <em>BL</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/kane.html#KanesMethod.kanes_equations"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.kane.KanesMethod.kanes_equations" title="Permalink to this definition">¶</a></dt>
-<dd><p>Method to form Kane&#8217;s equations, Fr + Fr* = 0.</p>
-<p>Returns (Fr, Fr*). In the case where auxiliary generalized speeds are
-present (say, s auxiliary speeds, o generalized speeds, and m motion
-constraints) the length of the returned vectors will be o - m + s in
-length. The first o - m equations will be the constrained Kane&#8217;s
-equations, then the s auxiliary Kane&#8217;s equations. These auxiliary
-equations can be accessed with the auxiliary_eqs().</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>FL</strong> : list</p>
-<blockquote>
-<div><p>Takes in a list of (Point, Vector) or (ReferenceFrame, Vector)
-tuples which represent the force at a point or torque on a frame.</p>
-</div></blockquote>
-<p><strong>BL</strong> : list</p>
-<blockquote class="last">
-<div><p>A list of all RigidBody&#8217;s and Particle&#8217;s in the system.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.kane.KanesMethod.kindiffdict">
-<tt class="descname">kindiffdict</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/kane.html#KanesMethod.kindiffdict"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.kane.KanesMethod.kindiffdict" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the qdot&#8217;s in a dictionary.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.kane.KanesMethod.linearize">
-<tt class="descname">linearize</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/kane.html#KanesMethod.linearize"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.kane.KanesMethod.linearize" title="Permalink to this definition">¶</a></dt>
-<dd><p>Method used to generate linearized equations.</p>
-<p>Note that for linearization, it is assumed that time is not perturbed,
-but only coordinates and positions. The &#8220;forcing&#8221; vector&#8217;s jacobian is
-computed with respect to the state vector in the form [Qi, Qd, Ui, Ud].
-This is the &#8220;f_lin_A&#8221; matrix.</p>
-<p>It also finds any non-state dynamicsymbols and computes the jacobian of
-the &#8220;forcing&#8221; vector with respect to them. This is the &#8220;f_lin_B&#8221;
-matrix; if this is empty, an empty matrix is created.</p>
-<p>Consider the following:
-If our equations are: [M]qudot = f, where [M] is the full mass matrix,
-qudot is a vector of the deriatives of the coordinates and speeds, and
-f in the full forcing vector, the linearization process is as follows:
-[M]qudot = [f_lin_A]qu + [f_lin_B]y, where qu is the state vector,
-f_lin_A is the jacobian of the full forcing vector with respect to the
-state vector, f_lin_B is the jacobian of the full forcing vector with
-respect to any non-speed/coordinate dynamicsymbols which show up in the
-full forcing vector, and y is a vector of those dynamic symbols (each
-column in f_lin_B corresponds to a row of the y vector, each of which
-is a non-speed/coordinate dynamicsymbol).</p>
-<p>To get the traditional state-space A and B matrix, you need to multiply
-the f_lin_A and f_lin_B matrices by the inverse of the mass matrix.
-Caution needs to be taken when inverting large symbolic matrices;
-substituting in numerical values before inverting will work better.</p>
-<p>A tuple of (f_lin_A, f_lin_B, other_dynamicsymbols) is returned.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.kane.KanesMethod.rhs">
-<tt class="descname">rhs</tt><big>(</big><em>self</em>, <em>inv_method=None</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/kane.html#KanesMethod.rhs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.kane.KanesMethod.rhs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the system&#8217;s equations of motion in first order form.</p>
-<p>The output of this will be the right hand side of:</p>
-<p>[qdot, udot].T = f(q, u, t)</p>
-<p>Or, the equations of motion in first order form.  The right hand side
-is what is needed by most numerical ODE integrators.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>inv_method</strong> : str</p>
-<blockquote class="last">
-<div><p>The specific sympy inverse matrix calculation method to use.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.physics.mechanics.functions">
-<span id="partial-velocity"></span><h2>partial_velocity<a class="headerlink" href="#module-sympy.physics.mechanics.functions" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.physics.mechanics.functions.partial_velocity">
-<tt class="descclassname">sympy.physics.mechanics.functions.</tt><tt class="descname">partial_velocity</tt><big>(</big><em>vel_list</em>, <em>u_list</em>, <em>frame</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/functions.html#partial_velocity"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.functions.partial_velocity" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a list of partial velocities.</p>
-<p>For a list of velocity or angular velocity vectors the partial derivatives
-with respect to the supplied generalized speeds are computed, in the
-specified ReferenceFrame.</p>
-<p>The output is a list of lists. The outer list has a number of elements
-equal to the number of supplied velocity vectors. The inner lists are, for
-each velocity vector, the partial derivatives of that velocity vector with
-respect to the generalized speeds supplied.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>vel_list</strong> : list</p>
-<blockquote>
-<div><p>List of velocities of Point&#8217;s and angular velocities of ReferenceFrame&#8217;s</p>
-</div></blockquote>
-<p><strong>u_list</strong> : list</p>
-<blockquote>
-<div><p>List of independent generalized speeds.</p>
-</div></blockquote>
-<p><strong>frame</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The ReferenceFrame the partial derivatives are going to be taken in.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">ReferenceFrame</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">dynamicsymbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">partial_velocity</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;u&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">u</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">vel_list</span> <span class="o">=</span> <span class="p">[</span><span class="n">P</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">N</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u_list</span> <span class="o">=</span> <span class="p">[</span><span class="n">u</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">partial_velocity</span><span class="p">(</span><span class="n">vel_list</span><span class="p">,</span> <span class="n">u_list</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
-<span class="go">[[N.x]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.physics.mechanics.lagrange">
-<span id="lagrangesmethod"></span><h2>LagrangesMethod<a class="headerlink" href="#module-sympy.physics.mechanics.lagrange" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.physics.mechanics.lagrange.LagrangesMethod">
-<em class="property">class </em><tt class="descclassname">sympy.physics.mechanics.lagrange.</tt><tt class="descname">LagrangesMethod</tt><big>(</big><em>Lagrangian</em>, <em>q_list</em>, <em>coneqs=None</em>, <em>forcelist=None</em>, <em>frame=None</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/lagrange.html#LagrangesMethod"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.lagrange.LagrangesMethod" title="Permalink to this definition">¶</a></dt>
-<dd><p>Lagrange&#8217;s method object.</p>
-<p>This object generates the equations of motion in a two step procedure. The
-first step involves the initialization of LagrangesMethod by supplying the
-Lagrangian and a list of the generalized coordinates, at the bare minimum.
-If there are any constraint equations, they can be supplied as keyword
-arguments. The Lagrangian multipliers are automatically generated and are
-equal in number to the constraint equations.Similarly any non-conservative
-forces can be supplied in a list (as described below and also shown in the
-example) along with a ReferenceFrame. This is also discussed further in the
-__init__ method.</p>
-<p class="rubric">Examples</p>
-<p>This is a simple example for a one degree of freedom translational
-spring-mass-damper.</p>
-<p>In this example, we first need to do the kinematics.$
-This involves creating generalized coordinates and its derivative.
-Then we create a point and set its velocity in a frame:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">LagrangesMethod</span><span class="p">,</span> <span class="n">Lagrangian</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">Particle</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">dynamicsymbols</span><span class="p">,</span> <span class="n">kinetic_energy</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qd</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;m k b&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">qd</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>We need to then prepare the information as required by LagrangesMethod to
-generate equations of motion.
-First we create the Particle, which has a point attached to it.
-Following this the lagrangian is created from the kinetic and potential
-energies.
-Then, a list of nonconservative forces/torques must be constructed, where
-each entry in is a (Point, Vector) or (ReferenceFrame, Vector) tuple, where
-the Vectors represent the nonconservative force or torque.</p>
-<blockquote>
-<div><div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Pa</span> <span class="o">=</span> <span class="n">Particle</span><span class="p">(</span><span class="s">&#39;Pa&#39;</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Pa</span><span class="o">.</span><span class="n">set_potential_energy</span><span class="p">(</span><span class="n">k</span> <span class="o">*</span> <span class="n">q</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span> <span class="mf">2.0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span> <span class="o">=</span> <span class="n">Lagrangian</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">Pa</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fl</span> <span class="o">=</span> <span class="p">[(</span><span class="n">P</span><span class="p">,</span> <span class="o">-</span><span class="n">b</span> <span class="o">*</span> <span class="n">qd</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)]</span>
-</pre></div>
-</div>
-<p>Finally we can generate the equations of motion.
-First we create the LagrangesMethod object.To do this one must supply an
-the Lagrangian, the list of generalized coordinates. Also supplied are the
-constraint equations, the forcelist and the inertial frame, if relevant.
-Next we generate Lagrange&#8217;s equations of motion, such that:
-Lagrange&#8217;s equations of motion = 0.
-We have the equations of motion at this point.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">l</span> <span class="o">=</span> <span class="n">LagrangesMethod</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="p">[</span><span class="n">q</span><span class="p">],</span> <span class="n">forcelist</span> <span class="o">=</span> <span class="n">fl</span><span class="p">,</span> <span class="n">frame</span> <span class="o">=</span> <span class="n">N</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">l</span><span class="o">.</span><span class="n">form_lagranges_equations</span><span class="p">())</span>
-<span class="go">[b*Derivative(q(t), t) + 1.0*k*q(t) + m*Derivative(q(t), t, t)]</span>
-</pre></div>
-</div>
-</div></blockquote>
-<p>We can also solve for the states using the &#8216;rhs&#8217; method.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">l</span><span class="o">.</span><span class="n">rhs</span><span class="p">())</span>
-<span class="go">[                    Derivative(q(t), t)]</span>
-<span class="go">[(-b*Derivative(q(t), t) - 1.0*k*q(t))/m]</span>
-</pre></div>
-</div>
-<p>Please refer to the docstrings on each method for more details.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="21%" />
-<col width="8%" />
-<col width="71%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>mass_matrix</td>
-<td>Matrix</td>
-<td>The system&#8217;s mass matrix</td>
-</tr>
-<tr class="row-even"><td>forcing</td>
-<td>Matrix</td>
-<td>The system&#8217;s forcing vector</td>
-</tr>
-<tr class="row-odd"><td>mass_matrix_full</td>
-<td>Matrix</td>
-<td>The &#8220;mass matrix&#8221; for the qdot&#8217;s, qdoubledot&#8217;s, and the
-lagrange multipliers (lam)</td>
-</tr>
-<tr class="row-even"><td>forcing_full</td>
-<td>Matrix</td>
-<td>The forcing vector for the qdot&#8217;s, qdoubledot&#8217;s and
-lagrange multipliers (lam)</td>
-</tr>
-</tbody>
-</table>
-<dl class="attribute">
-<dt id="sympy.physics.mechanics.lagrange.LagrangesMethod.forcing">
-<tt class="descname">forcing</tt><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/lagrange.html#LagrangesMethod.forcing"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.lagrange.LagrangesMethod.forcing" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the forcing vector from &#8216;lagranges_equations&#8217; method.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.mechanics.lagrange.LagrangesMethod.forcing_full">
-<tt class="descname">forcing_full</tt><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/lagrange.html#LagrangesMethod.forcing_full"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.lagrange.LagrangesMethod.forcing_full" title="Permalink to this definition">¶</a></dt>
-<dd><p>Augments qdots to the forcing vector above.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.lagrange.LagrangesMethod.form_lagranges_equations">
-<tt class="descname">form_lagranges_equations</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/lagrange.html#LagrangesMethod.form_lagranges_equations"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.lagrange.LagrangesMethod.form_lagranges_equations" title="Permalink to this definition">¶</a></dt>
-<dd><p>Method to form Lagrange&#8217;s equations of motion.</p>
-<p>Returns a vector of equations of motion using Lagrange&#8217;s equations of
-the second kind.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.mechanics.lagrange.LagrangesMethod.mass_matrix">
-<tt class="descname">mass_matrix</tt><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/lagrange.html#LagrangesMethod.mass_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.lagrange.LagrangesMethod.mass_matrix" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the mass matrix, which is augmented by the Lagrange
-multipliers, if necessary.</p>
-<p>If the system is described by &#8216;n&#8217; generalized coordinates and there are
-no constraint equations then an n X n matrix is returned.</p>
-<p>If there are &#8216;n&#8217; generalized coordinates and &#8216;m&#8217; constraint equations
-have been supplied during initialization then an n X (n+m) matrix is
-returned. The (n + m - 1)th and (n + m)th columns contain the
-coefficients of the Lagrange multipliers.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.mechanics.lagrange.LagrangesMethod.mass_matrix_full">
-<tt class="descname">mass_matrix_full</tt><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/lagrange.html#LagrangesMethod.mass_matrix_full"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.lagrange.LagrangesMethod.mass_matrix_full" title="Permalink to this definition">¶</a></dt>
-<dd><p>Augments the coefficients of qdots to the mass_matrix.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.lagrange.LagrangesMethod.rhs">
-<tt class="descname">rhs</tt><big>(</big><em>self</em>, <em>method='GE'</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/lagrange.html#LagrangesMethod.rhs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.lagrange.LagrangesMethod.rhs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns equations that can be solved numerically</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>method</strong> : string</p>
-<blockquote class="last">
-<div><p>The method by which matrix inversion of mass_matrix_full must be
-performed such as Gauss Elimination or LU decomposition.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-</div>
-
-
-          </div>
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-            </a></p>
-  <h3><a href="../../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Kane&#8217;s Method, Lagrange&#8217;s Method &amp; Associated Functions (Docstrings)</a><ul>
-<li><a class="reference internal" href="#module-sympy.physics.mechanics.kane">Kane</a></li>
-<li><a class="reference internal" href="#module-sympy.physics.mechanics.functions">partial_velocity</a></li>
-<li><a class="reference internal" href="#module-sympy.physics.mechanics.lagrange">LagrangesMethod</a></li>
-</ul>
-</li>
-</ul>
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-  <p class="topless"><a href="../../mechanics/api/part_bod.html"
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-  <div class="section" id="kinematics-docstrings">
-<h1>Kinematics (Docstrings)<a class="headerlink" href="#kinematics-docstrings" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="module-sympy.physics.mechanics.point">
-<span id="point"></span><h2>Point<a class="headerlink" href="#module-sympy.physics.mechanics.point" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.physics.mechanics.point.Point">
-<em class="property">class </em><tt class="descclassname">sympy.physics.mechanics.point.</tt><tt class="descname">Point</tt><big>(</big><em>name</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/point.html#Point"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.point.Point" title="Permalink to this definition">¶</a></dt>
-<dd><p>This object represents a point in a dynamic system.</p>
-<p>It stores the: position, velocity, and acceleration of a point.
-The position is a vector defined as the vector distance from a parent
-point to this point.</p>
-<dl class="function">
-<dt id="sympy.physics.mechanics.point.Point.a1pt_theory">
-<tt class="descname">a1pt_theory</tt><big>(</big><em>self</em>, <em>otherpoint</em>, <em>outframe</em>, <em>interframe</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/point.html#Point.a1pt_theory"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.point.Point.a1pt_theory" title="Permalink to this definition">¶</a></dt>
-<dd><p>Sets the acceleration of this point with the 1-point theory.</p>
-<p>The 1-point theory for point acceleration looks like this:</p>
-<p>^N a^P = ^B a^P + ^N a^O + ^N alpha^B x r^OP + ^N omega^B x (^N omega^B
-x r^OP) + 2 ^N omega^B x ^B v^P</p>
-<p>where O is a point fixed in B, P is a point moving in B, and B is
-rotating in frame N.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>otherpoint</strong> : Point</p>
-<blockquote>
-<div><p>The first point of the 1-point theory (O)</p>
-</div></blockquote>
-<p><strong>outframe</strong> : ReferenceFrame</p>
-<blockquote>
-<div><p>The frame we want this point&#8217;s acceleration defined in (N)</p>
-</div></blockquote>
-<p><strong>fixedframe</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The intermediate frame in this calculation (B)</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">dynamicsymbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q2</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q2&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qd</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q2d</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q2&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">5</span> <span class="o">*</span> <span class="n">B</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;O&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">O</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">,</span> <span class="n">q</span> <span class="o">*</span> <span class="n">B</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">qd</span> <span class="o">*</span> <span class="n">B</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">q2d</span> <span class="o">*</span> <span class="n">B</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">a1pt_theory</span><span class="p">(</span><span class="n">O</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-<span class="go">(-25*q + q&#39;&#39;)*B.x + q2&#39;&#39;*B.y - 10*q&#39;*B.z</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.point.Point.a2pt_theory">
-<tt class="descname">a2pt_theory</tt><big>(</big><em>self</em>, <em>otherpoint</em>, <em>outframe</em>, <em>fixedframe</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/point.html#Point.a2pt_theory"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.point.Point.a2pt_theory" title="Permalink to this definition">¶</a></dt>
-<dd><p>Sets the acceleration of this point with the 2-point theory.</p>
-<p>The 2-point theory for point acceleration looks like this:</p>
-<p>^N a^P = ^N a^O + ^N alpha^B x r^OP + ^N omega^B x (^N omega^B x r^OP)</p>
-<p>where O and P are both points fixed in frame B, which is rotating in
-frame N.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>otherpoint</strong> : Point</p>
-<blockquote>
-<div><p>The first point of the 2-point theory (O)</p>
-</div></blockquote>
-<p><strong>outframe</strong> : ReferenceFrame</p>
-<blockquote>
-<div><p>The frame we want this point&#8217;s acceleration defined in (N)</p>
-</div></blockquote>
-<p><strong>fixedframe</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The frame in which both points are fixed (B)</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">dynamicsymbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qd</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;O&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">O</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">,</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">B</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">5</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">a2pt_theory</span><span class="p">(</span><span class="n">O</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-<span class="go">- 10*q&#39;**2*B.x + 10*q&#39;&#39;*B.y</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.point.Point.acc">
-<tt class="descname">acc</tt><big>(</big><em>self</em>, <em>frame</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/point.html#Point.acc"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.point.Point.acc" title="Permalink to this definition">¶</a></dt>
-<dd><p>The acceleration Vector of this Point in a ReferenceFrame.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>frame</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The frame in which the returned acceleration vector will be defined in</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">ReferenceFrame</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;p1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="o">.</span><span class="n">set_acc</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="o">.</span><span class="n">acc</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">10*N.x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.point.Point.locatenew">
-<tt class="descname">locatenew</tt><big>(</big><em>self</em>, <em>name</em>, <em>value</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/point.html#Point.locatenew"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.point.Point.locatenew" title="Permalink to this definition">¶</a></dt>
-<dd><p>Creates a new point with a position defined from this point.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>name</strong> : str</p>
-<blockquote>
-<div><p>The name for the new point</p>
-</div></blockquote>
-<p><strong>value</strong> : Vector</p>
-<blockquote class="last">
-<div><p>The position of the new point relative to this point</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;P1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P2</span> <span class="o">=</span> <span class="n">P1</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;P2&#39;</span><span class="p">,</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.point.Point.pos_from">
-<tt class="descname">pos_from</tt><big>(</big><em>self</em>, <em>otherpoint</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/point.html#Point.pos_from"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.point.Point.pos_from" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a Vector distance between this Point and the other Point.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>otherpoint</strong> : Point</p>
-<blockquote class="last">
-<div><p>The otherpoint we are locating this one relative to</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">ReferenceFrame</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;p1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;p2&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="o">.</span><span class="n">set_pos</span><span class="p">(</span><span class="n">p2</span><span class="p">,</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="o">.</span><span class="n">pos_from</span><span class="p">(</span><span class="n">p2</span><span class="p">)</span>
-<span class="go">10*N.x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.point.Point.set_acc">
-<tt class="descname">set_acc</tt><big>(</big><em>self</em>, <em>frame</em>, <em>value</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/point.html#Point.set_acc"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.point.Point.set_acc" title="Permalink to this definition">¶</a></dt>
-<dd><p>Used to set the acceleration of this Point in a ReferenceFrame.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>value</strong> : Vector</p>
-<blockquote>
-<div><p>The vector value of this point&#8217;s acceleration in the frame</p>
-</div></blockquote>
-<p><strong>frame</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The frame in which this point&#8217;s acceleration is defined</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">ReferenceFrame</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;p1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="o">.</span><span class="n">set_acc</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="o">.</span><span class="n">acc</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">10*N.x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.point.Point.set_pos">
-<tt class="descname">set_pos</tt><big>(</big><em>self</em>, <em>otherpoint</em>, <em>value</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/point.html#Point.set_pos"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.point.Point.set_pos" title="Permalink to this definition">¶</a></dt>
-<dd><p>Used to set the position of this point w.r.t. another point.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>value</strong> : Vector</p>
-<blockquote>
-<div><p>The vector which defines the location of this point</p>
-</div></blockquote>
-<p><strong>point</strong> : Point</p>
-<blockquote class="last">
-<div><p>The other point which this point&#8217;s location is defined relative to</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">ReferenceFrame</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;p1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p2</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;p2&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="o">.</span><span class="n">set_pos</span><span class="p">(</span><span class="n">p2</span><span class="p">,</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="o">.</span><span class="n">pos_from</span><span class="p">(</span><span class="n">p2</span><span class="p">)</span>
-<span class="go">10*N.x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.point.Point.set_vel">
-<tt class="descname">set_vel</tt><big>(</big><em>self</em>, <em>frame</em>, <em>value</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/point.html#Point.set_vel"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.point.Point.set_vel" title="Permalink to this definition">¶</a></dt>
-<dd><p>Sets the velocity Vector of this Point in a ReferenceFrame.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>value</strong> : Vector</p>
-<blockquote>
-<div><p>The vector value of this point&#8217;s velocity in the frame</p>
-</div></blockquote>
-<p><strong>frame</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The frame in which this point&#8217;s velocity is defined</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">ReferenceFrame</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;p1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">10*N.x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.point.Point.v1pt_theory">
-<tt class="descname">v1pt_theory</tt><big>(</big><em>self</em>, <em>otherpoint</em>, <em>outframe</em>, <em>interframe</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/point.html#Point.v1pt_theory"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.point.Point.v1pt_theory" title="Permalink to this definition">¶</a></dt>
-<dd><p>Sets the velocity of this point with the 1-point theory.</p>
-<p>The 1-point theory for point velocity looks like this:</p>
-<p>^N v^P = ^B v^P + ^N v^O + ^N omega^B x r^OP</p>
-<p>where O is a point fixed in B, P is a point moving in B, and B is
-rotating in frame N.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>otherpoint</strong> : Point</p>
-<blockquote>
-<div><p>The first point of the 2-point theory (O)</p>
-</div></blockquote>
-<p><strong>outframe</strong> : ReferenceFrame</p>
-<blockquote>
-<div><p>The frame we want this point&#8217;s velocity defined in (N)</p>
-</div></blockquote>
-<p><strong>interframe</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The intermediate frame in this calculation (B)</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">dynamicsymbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q2</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q2&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qd</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q2d</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q2&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">5</span> <span class="o">*</span> <span class="n">B</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;O&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">O</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">,</span> <span class="n">q</span> <span class="o">*</span> <span class="n">B</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">qd</span> <span class="o">*</span> <span class="n">B</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">q2d</span> <span class="o">*</span> <span class="n">B</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">v1pt_theory</span><span class="p">(</span><span class="n">O</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-<span class="go">q&#39;*B.x + q2&#39;*B.y - 5*q*B.z</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.point.Point.v2pt_theory">
-<tt class="descname">v2pt_theory</tt><big>(</big><em>self</em>, <em>otherpoint</em>, <em>outframe</em>, <em>fixedframe</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/point.html#Point.v2pt_theory"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.point.Point.v2pt_theory" title="Permalink to this definition">¶</a></dt>
-<dd><p>Sets the velocity of this point with the 2-point theory.</p>
-<p>The 2-point theory for point velocity looks like this:</p>
-<p>^N v^P = ^N v^O + ^N omega^B x r^OP</p>
-<p>where O and P are both points fixed in frame B, which is rotating in
-frame N.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>otherpoint</strong> : Point</p>
-<blockquote>
-<div><p>The first point of the 2-point theory (O)</p>
-</div></blockquote>
-<p><strong>outframe</strong> : ReferenceFrame</p>
-<blockquote>
-<div><p>The frame we want this point&#8217;s velocity defined in (N)</p>
-</div></blockquote>
-<p><strong>fixedframe</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The frame in which both points are fixed (B)</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">dynamicsymbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qd</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;O&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">O</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">,</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">B</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">5</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">v2pt_theory</span><span class="p">(</span><span class="n">O</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-<span class="go">5*N.x + 10*q&#39;*B.y</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.point.Point.vel">
-<tt class="descname">vel</tt><big>(</big><em>self</em>, <em>frame</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/point.html#Point.vel"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.point.Point.vel" title="Permalink to this definition">¶</a></dt>
-<dd><p>The velocity Vector of this Point in the ReferenceFrame.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>frame</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The frame in which the returned velocity vector will be defined in</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">ReferenceFrame</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;p1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">10*N.x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.physics.mechanics.functions">
-<span id="kinematic-equations"></span><h2>kinematic_equations<a class="headerlink" href="#module-sympy.physics.mechanics.functions" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.physics.mechanics.functions.kinematic_equations">
-<tt class="descclassname">sympy.physics.mechanics.functions.</tt><tt class="descname">kinematic_equations</tt><big>(</big><em>speeds</em>, <em>coords</em>, <em>rot_type</em>, <em>rot_order=''</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/functions.html#kinematic_equations"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.functions.kinematic_equations" title="Permalink to this definition">¶</a></dt>
-<dd><p>Gives equations relating the qdot&#8217;s to u&#8217;s for a rotation type.</p>
-<p>Supply rotation type and order as in orient. Speeds are assumed to be
-body-fixed; if we are defining the orientation of B in A using by rot_type,
-the angular velocity of B in A is assumed to be in the form: speed[0]*B.x +
-speed[1]*B.y + speed[2]*B.z</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>speeds</strong> : list of length 3</p>
-<blockquote>
-<div><p>The body fixed angular velocity measure numbers.</p>
-</div></blockquote>
-<p><strong>coords</strong> : list of length 3 or 4</p>
-<blockquote>
-<div><p>The coordinates used to define the orientation of the two frames.</p>
-</div></blockquote>
-<p><strong>rot_type</strong> : str</p>
-<blockquote>
-<div><p>The type of rotation used to create the equations. Body, Space, or
-Quaternion only</p>
-</div></blockquote>
-<p><strong>rot_order</strong> : str</p>
-<blockquote class="last">
-<div><p>If applicable, the order of a series of rotations.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">dynamicsymbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">kinematic_equations</span><span class="p">,</span> <span class="n">mprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u1</span><span class="p">,</span> <span class="n">u2</span><span class="p">,</span> <span class="n">u3</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;u1 u2 u3&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1 q2 q3&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mprint</span><span class="p">(</span><span class="n">kinematic_equations</span><span class="p">([</span><span class="n">u1</span><span class="p">,</span><span class="n">u2</span><span class="p">,</span><span class="n">u3</span><span class="p">],</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span><span class="n">q2</span><span class="p">,</span><span class="n">q3</span><span class="p">],</span> <span class="s">&#39;body&#39;</span><span class="p">,</span> <span class="s">&#39;313&#39;</span><span class="p">),</span>
-<span class="gp">... </span>    <span class="n">order</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span>
-<span class="go">[-(u1*sin(q3) + u2*cos(q3))/sin(q2) + q1&#39;, -u1*cos(q3) + u2*sin(q3) + q2&#39;, (u1*sin(q3) + u2*cos(q3))*cos(q2)/sin(q2) - u3 + q3&#39;]</span>
-</pre></div>
-</div>
-</dd></dl>
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-  <h3><a href="../../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Kinematics (Docstrings)</a><ul>
-<li><a class="reference internal" href="#module-sympy.physics.mechanics.point">Point</a></li>
-<li><a class="reference internal" href="#module-sympy.physics.mechanics.functions">kinematic_equations</a></li>
-</ul>
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-</ul>
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-<h1>Masses, Inertias &amp; Particles, RigidBodys (Docstrings)<a class="headerlink" href="#masses-inertias-particles-rigidbodys-docstrings" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="module-sympy.physics.mechanics.particle">
-<span id="particle"></span><h2>Particle<a class="headerlink" href="#module-sympy.physics.mechanics.particle" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.physics.mechanics.particle.Particle">
-<em class="property">class </em><tt class="descclassname">sympy.physics.mechanics.particle.</tt><tt class="descname">Particle</tt><big>(</big><em>name</em>, <em>point</em>, <em>mass</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/particle.html#Particle"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.particle.Particle" title="Permalink to this definition">¶</a></dt>
-<dd><p>A particle.</p>
-<p>Particles have a non-zero mass and lack spatial extension; they take up no
-space.</p>
-<p>Values need to be supplied on initialization, but can be changed later.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>name</strong> : str</p>
-<blockquote>
-<div><p>Name of particle</p>
-</div></blockquote>
-<p><strong>point</strong> : Point</p>
-<blockquote>
-<div><p>A physics/mechanics Point which represents the position, velocity, and
-acceleration of this Particle</p>
-</div></blockquote>
-<p><strong>mass</strong> : sympifyable</p>
-<blockquote class="last">
-<div><p>A SymPy expression representing the Particle&#8217;s mass</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Particle</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">po</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;po&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;m&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pa</span> <span class="o">=</span> <span class="n">Particle</span><span class="p">(</span><span class="s">&#39;pa&#39;</span><span class="p">,</span> <span class="n">po</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># Or you could change these later</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pa</span><span class="o">.</span><span class="n">mass</span> <span class="o">=</span> <span class="n">m</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pa</span><span class="o">.</span><span class="n">point</span> <span class="o">=</span> <span class="n">po</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.physics.mechanics.particle.Particle.angular_momentum">
-<tt class="descname">angular_momentum</tt><big>(</big><em>self</em>, <em>point</em>, <em>frame</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/particle.html#Particle.angular_momentum"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.particle.Particle.angular_momentum" title="Permalink to this definition">¶</a></dt>
-<dd><p>Angular momentum of the particle about the point.</p>
-<p>The angular momentum H, about some point O of a particle, P, is given
-by:</p>
-<p>H = r x m * v</p>
-<p>where r is the position vector from point O to the particle P, m is
-the mass of the particle, and v is the velocity of the particle in
-the inertial frame, N.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>point</strong> : Point</p>
-<blockquote>
-<div><p>The point about which angular momentum of the particle is desired.</p>
-</div></blockquote>
-<p><strong>frame</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The frame in which angular momentum is desired.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Particle</span><span class="p">,</span> <span class="n">Point</span><span class="p">,</span> <span class="n">ReferenceFrame</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">dynamicsymbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;m v r&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;O&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">O</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">r</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">Particle</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">point</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">v</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">angular_momentum</span><span class="p">(</span><span class="n">O</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
-<span class="go">m*r*v*N.z</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.particle.Particle.get_mass">
-<tt class="descname">get_mass</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/particle.html#Particle.get_mass"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.particle.Particle.get_mass" title="Permalink to this definition">¶</a></dt>
-<dd><p>Mass of the particle.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.particle.Particle.get_point">
-<tt class="descname">get_point</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/particle.html#Particle.get_point"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.particle.Particle.get_point" title="Permalink to this definition">¶</a></dt>
-<dd><p>Point of the particle.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.particle.Particle.kinetic_energy">
-<tt class="descname">kinetic_energy</tt><big>(</big><em>self</em>, <em>frame</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/particle.html#Particle.kinetic_energy"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.particle.Particle.kinetic_energy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Kinetic energy of the particle</p>
-<p>The kinetic energy, T, of a particle, P, is given by</p>
-<p>&#8216;T = 1/2 m v^2&#8217;</p>
-<p>where m is the mass of particle P, and v is the velocity of the
-particle in the supplied ReferenceFrame.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>frame</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The Particle&#8217;s velocity is typically defined with respect to
-an inertial frame but any relevant frame in which the velocity is
-known can be supplied.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Particle</span><span class="p">,</span> <span class="n">Point</span><span class="p">,</span> <span class="n">ReferenceFrame</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;m v r&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;O&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">Particle</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">point</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">v</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">kinetic_energy</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">m*v**2/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.particle.Particle.linear_momentum">
-<tt class="descname">linear_momentum</tt><big>(</big><em>self</em>, <em>frame</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/particle.html#Particle.linear_momentum"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.particle.Particle.linear_momentum" title="Permalink to this definition">¶</a></dt>
-<dd><p>Linear momentum of the particle.</p>
-<p>The linear momentum L, of a particle P, with respect to frame N is
-given by</p>
-<p>L = m * v</p>
-<p>where m is the mass of the particle, and v is the velocity of the
-particle in the frame N.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>frame</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The frame in which linear momentum is desired.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Particle</span><span class="p">,</span> <span class="n">Point</span><span class="p">,</span> <span class="n">ReferenceFrame</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">dynamicsymbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;m v&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Particle</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">v</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">linear_momentum</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">m*v*N.x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.mechanics.particle.Particle.mass">
-<tt class="descname">mass</tt><a class="headerlink" href="#sympy.physics.mechanics.particle.Particle.mass" title="Permalink to this definition">¶</a></dt>
-<dd><p>Mass of the particle.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.mechanics.particle.Particle.point">
-<tt class="descname">point</tt><a class="headerlink" href="#sympy.physics.mechanics.particle.Particle.point" title="Permalink to this definition">¶</a></dt>
-<dd><p>Point of the particle.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.mechanics.particle.Particle.potential_energy">
-<tt class="descname">potential_energy</tt><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/particle.html#Particle.potential_energy"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.particle.Particle.potential_energy" title="Permalink to this definition">¶</a></dt>
-<dd><p>The potential energy of the Particle.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Particle</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;m g h&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;O&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">Particle</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">set_potential_energy</span><span class="p">(</span><span class="n">m</span> <span class="o">*</span> <span class="n">g</span> <span class="o">*</span> <span class="n">h</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">potential_energy</span>
-<span class="go">g*h*m</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.particle.Particle.set_potential_energy">
-<tt class="descname">set_potential_energy</tt><big>(</big><em>self</em>, <em>scalar</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/particle.html#Particle.set_potential_energy"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.particle.Particle.set_potential_energy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Used to set the potential energy of the Particle.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>scalar</strong> : Sympifyable</p>
-<blockquote class="last">
-<div><p>The potential energy (a scalar) of the Particle.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Particle</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;m g h&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;O&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">Particle</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">,</span> <span class="n">O</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">set_potential_energy</span><span class="p">(</span><span class="n">m</span> <span class="o">*</span> <span class="n">g</span> <span class="o">*</span> <span class="n">h</span><span class="p">)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.physics.mechanics.rigidbody">
-<span id="rigidbody"></span><h2>RigidBody<a class="headerlink" href="#module-sympy.physics.mechanics.rigidbody" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.physics.mechanics.rigidbody.RigidBody">
-<em class="property">class </em><tt class="descclassname">sympy.physics.mechanics.rigidbody.</tt><tt class="descname">RigidBody</tt><big>(</big><em>name</em>, <em>masscenter</em>, <em>frame</em>, <em>mass</em>, <em>inertia</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/rigidbody.html#RigidBody"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.rigidbody.RigidBody" title="Permalink to this definition">¶</a></dt>
-<dd><p>An idealized rigid body.</p>
-<p>This is essentially a container which holds the various components which
-describe a rigid body: a name, mass, center of mass, reference frame, and
-inertia.</p>
-<p>All of these need to be supplied on creation, but can be changed
-afterwards.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">Point</span><span class="p">,</span> <span class="n">RigidBody</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">outer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;m&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">outer</span> <span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">A</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">inertia_tuple</span> <span class="o">=</span> <span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">P</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">RigidBody</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">inertia_tuple</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># Or you could change them afterwards</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m2</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;m2&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">mass</span> <span class="o">=</span> <span class="n">m2</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="11%" />
-<col width="16%" />
-<col width="73%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>name</td>
-<td>string</td>
-<td>The body&#8217;s name.</td>
-</tr>
-<tr class="row-even"><td>masscenter</td>
-<td>Point</td>
-<td>The point which represents the center of mass of the rigid body.</td>
-</tr>
-<tr class="row-odd"><td>frame</td>
-<td>ReferenceFrame</td>
-<td>The ReferenceFrame which the rigid body is fixed in.</td>
-</tr>
-<tr class="row-even"><td>mass</td>
-<td>Sympifyable</td>
-<td>The body&#8217;s mass.</td>
-</tr>
-<tr class="row-odd"><td>inertia</td>
-<td>(Dyadic, Point)</td>
-<td>The body&#8217;s inertia about a point; stored in a tuple as shown above.</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.physics.mechanics.rigidbody.RigidBody.angular_momentum">
-<tt class="descname">angular_momentum</tt><big>(</big><em>self</em>, <em>point</em>, <em>frame</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/rigidbody.html#RigidBody.angular_momentum"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.rigidbody.RigidBody.angular_momentum" title="Permalink to this definition">¶</a></dt>
-<dd><p>Angular momentum of the rigid body.</p>
-<p>The angular momentum H, about some point O, of a rigid body B, in a
-frame N is given by</p>
-<p>H = I* . omega + r* x (M * v)</p>
-<p>where I* is the central inertia dyadic of B, omega is the angular
-velocity of body B in the frame, N, r* is the position vector from
-point O to the mass center of B, and v is the velocity of point O in
-the frame, N.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>point</strong> : Point</p>
-<blockquote>
-<div><p>The point about which angular momentum is desired.</p>
-</div></blockquote>
-<p><strong>frame</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The frame in which angular momentum is desired.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">outer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">RigidBody</span><span class="p">,</span> <span class="n">dynamicsymbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">omega</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;M v r omega&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;b&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">omega</span> <span class="o">*</span> <span class="n">b</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">1</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">outer</span> <span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Inertia_tuple</span> <span class="o">=</span> <span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">P</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">RigidBody</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">Inertia_tuple</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">angular_momentum</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
-<span class="go">omega*b.x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.rigidbody.RigidBody.kinetic_energy">
-<tt class="descname">kinetic_energy</tt><big>(</big><em>self</em>, <em>frame</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/rigidbody.html#RigidBody.kinetic_energy"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.rigidbody.RigidBody.kinetic_energy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Kinetic energy of the rigid body</p>
-<p>The kinetic energy, T, of a rigid body, B, is given by</p>
-<p>&#8216;T = 1/2 (I omega^2 + m v^2)&#8217;</p>
-<p>where I and m are the central inertia dyadic and mass of rigid body B,
-respectively, omega is the body&#8217;s angular velocity and v is the
-velocity of the body&#8217;s mass center in the supplied ReferenceFrame.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>frame</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The RigidBody&#8217;s angular velocity and the velocity of it&#8217;s mass
-center is typically defined with respect to an inertial frame but
-any relevant frame in which the velocity is known can be supplied.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">outer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">RigidBody</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">omega</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;M v r omega&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;b&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">omega</span> <span class="o">*</span> <span class="n">b</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">v</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">outer</span> <span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">inertia_tuple</span> <span class="o">=</span> <span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">P</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">RigidBody</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">inertia_tuple</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">kinetic_energy</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">M*v**2/2 + omega**2/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.rigidbody.RigidBody.linear_momentum">
-<tt class="descname">linear_momentum</tt><big>(</big><em>self</em>, <em>frame</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/rigidbody.html#RigidBody.linear_momentum"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.rigidbody.RigidBody.linear_momentum" title="Permalink to this definition">¶</a></dt>
-<dd><p>Linear momentum of the rigid body.</p>
-<p>The linear momentum L, of a rigid body B, with respect to frame N is
-given by</p>
-<p>L = M * v*</p>
-<p>where M is the mass of the rigid body and v* is the velocity of
-the mass center of B in the frame, N.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>frame</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>The frame in which linear momentum is desired.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">outer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">RigidBody</span><span class="p">,</span> <span class="n">dynamicsymbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;M v&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">v</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">outer</span> <span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Inertia_tuple</span> <span class="o">=</span> <span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">P</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">RigidBody</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">Inertia_tuple</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">linear_momentum</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">M*v*N.x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.mechanics.rigidbody.RigidBody.potential_energy">
-<tt class="descname">potential_energy</tt><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/rigidbody.html#RigidBody.potential_energy"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.rigidbody.RigidBody.potential_energy" title="Permalink to this definition">¶</a></dt>
-<dd><p>The potential energy of the RigidBody.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">RigidBody</span><span class="p">,</span> <span class="n">Point</span><span class="p">,</span> <span class="n">outer</span><span class="p">,</span> <span class="n">ReferenceFrame</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;M g h&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;b&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">outer</span> <span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Inertia_tuple</span> <span class="o">=</span> <span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">P</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">RigidBody</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">Inertia_tuple</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">set_potential_energy</span><span class="p">(</span><span class="n">M</span> <span class="o">*</span> <span class="n">g</span> <span class="o">*</span> <span class="n">h</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">potential_energy</span>
-<span class="go">M*g*h</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.mechanics.rigidbody.RigidBody.set_potential_energy">
-<tt class="descname">set_potential_energy</tt><big>(</big><em>self</em>, <em>scalar</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/rigidbody.html#RigidBody.set_potential_energy"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.rigidbody.RigidBody.set_potential_energy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Used to set the potential energy of this RigidBody.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>scalar: Sympifyable</strong> :</p>
-<blockquote class="last">
-<div><p>The potential energy (a scalar) of the RigidBody.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Particle</span><span class="p">,</span> <span class="n">Point</span><span class="p">,</span> <span class="n">outer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">RigidBody</span><span class="p">,</span> <span class="n">ReferenceFrame</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;b&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;M g h&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">outer</span> <span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">b</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Inertia_tuple</span> <span class="o">=</span> <span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">P</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">RigidBody</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">Inertia_tuple</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">set_potential_energy</span><span class="p">(</span><span class="n">M</span> <span class="o">*</span> <span class="n">g</span> <span class="o">*</span> <span class="n">h</span><span class="p">)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="inertia">
-<h2>inertia<a class="headerlink" href="#inertia" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.physics.mechanics.functions.inertia">
-<tt class="descclassname">sympy.physics.mechanics.functions.</tt><tt class="descname">inertia</tt><big>(</big><em>frame</em>, <em>ixx</em>, <em>iyy</em>, <em>izz</em>, <em>ixy=0</em>, <em>iyz=0</em>, <em>izx=0</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/functions.html#inertia"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.functions.inertia" title="Permalink to this definition">¶</a></dt>
-<dd><p>Simple way to create inertia Dyadic object.</p>
-<p>If you don&#8217;t know what a Dyadic is, just treat this like the inertia
-tensor.  Then, do the easy thing and define it in a body-fixed frame.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>frame</strong> : ReferenceFrame</p>
-<blockquote>
-<div><p>The frame the inertia is defined in</p>
-</div></blockquote>
-<p><strong>ixx</strong> : Sympifyable</p>
-<blockquote>
-<div><p>the xx element in the inertia dyadic</p>
-</div></blockquote>
-<p><strong>iyy</strong> : Sympifyable</p>
-<blockquote>
-<div><p>the yy element in the inertia dyadic</p>
-</div></blockquote>
-<p><strong>izz</strong> : Sympifyable</p>
-<blockquote>
-<div><p>the zz element in the inertia dyadic</p>
-</div></blockquote>
-<p><strong>ixy</strong> : Sympifyable</p>
-<blockquote>
-<div><p>the xy element in the inertia dyadic</p>
-</div></blockquote>
-<p><strong>iyz</strong> : Sympifyable</p>
-<blockquote>
-<div><p>the yz element in the inertia dyadic</p>
-</div></blockquote>
-<p><strong>izx</strong> : Sympifyable</p>
-<blockquote class="last">
-<div><p>the zx element in the inertia dyadic</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">inertia</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">inertia</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">(N.x|N.x) + 2*(N.y|N.y) + 3*(N.z|N.z)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="inertia-of-point-mass">
-<h2>inertia_of_point_mass<a class="headerlink" href="#inertia-of-point-mass" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.physics.mechanics.functions.inertia_of_point_mass">
-<tt class="descclassname">sympy.physics.mechanics.functions.</tt><tt class="descname">inertia_of_point_mass</tt><big>(</big><em>mass</em>, <em>pos_vec</em>, <em>frame</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/functions.html#inertia_of_point_mass"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.functions.inertia_of_point_mass" title="Permalink to this definition">¶</a></dt>
-<dd><p>Inertia dyadic of a point mass realtive to point O.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>mass</strong> : Sympifyable</p>
-<blockquote>
-<div><p>Mass of the point mass</p>
-</div></blockquote>
-<p><strong>pos_vec</strong> : Vector</p>
-<blockquote>
-<div><p>Position from point O to point mass</p>
-</div></blockquote>
-<p><strong>frame</strong> : ReferenceFrame</p>
-<blockquote class="last">
-<div><p>Reference frame to express the dyadic in</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">inertia_of_point_mass</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;r m&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">px</span> <span class="o">=</span> <span class="n">r</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">inertia_of_point_mass</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">px</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
-<span class="go">m*r**2*(N.y|N.y) + m*r**2*(N.z|N.z)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="linear-momentum">
-<h2>linear_momentum<a class="headerlink" href="#linear-momentum" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.physics.mechanics.functions.linear_momentum">
-<tt class="descclassname">sympy.physics.mechanics.functions.</tt><tt class="descname">linear_momentum</tt><big>(</big><em>frame</em>, <em>*body</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/functions.html#linear_momentum"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.functions.linear_momentum" title="Permalink to this definition">¶</a></dt>
-<dd><p>Linear momentum of the system.</p>
-<p>This function returns the linear momentum of a system of Particle&#8217;s and/or
-RigidBody&#8217;s. The linear momentum of a system is equal to the vector sum of
-the linear momentum of its constituents. Consider a system, S, comprised of
-a rigid body, A, and a particle, P. The linear momentum of the system, L,
-is equal to the vector sum of the linear momentum of the particle, L1, and
-the linear momentum of the rigid body, L2, i.e-</p>
-<p>L = L1 + L2</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>frame</strong> : ReferenceFrame</p>
-<blockquote>
-<div><p>The frame in which linear momentum is desired.</p>
-</div></blockquote>
-<p><strong>body1, body2, body3...</strong> : Particle and/or RigidBody</p>
-<blockquote class="last">
-<div><p>The body (or bodies) whose kinetic energy is required.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Particle</span><span class="p">,</span> <span class="n">ReferenceFrame</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">RigidBody</span><span class="p">,</span> <span class="n">outer</span><span class="p">,</span> <span class="n">linear_momentum</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Pa</span> <span class="o">=</span> <span class="n">Particle</span><span class="p">(</span><span class="s">&#39;Pa&#39;</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ac</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;Ac&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ac</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">25</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">outer</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">RigidBody</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">Ac</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">Ac</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">linear_momentum</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">Pa</span><span class="p">)</span>
-<span class="go">10*N.x + 500*N.y</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="angular-momentum">
-<h2>angular_momentum<a class="headerlink" href="#angular-momentum" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.physics.mechanics.functions.angular_momentum">
-<tt class="descclassname">sympy.physics.mechanics.functions.</tt><tt class="descname">angular_momentum</tt><big>(</big><em>point</em>, <em>frame</em>, <em>*body</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/functions.html#angular_momentum"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.functions.angular_momentum" title="Permalink to this definition">¶</a></dt>
-<dd><p>Angular momentum of a system</p>
-<p>This function returns the angular momentum of a system of Particle&#8217;s and/or
-RigidBody&#8217;s. The angular momentum of such a system is equal to the vector
-sum of the angular momentum of its constituents. Consider a system, S,
-comprised of a rigid body, A, and a particle, P. The angular momentum of
-the system, H, is equal to the vector sum of the linear momentum of the
-particle, H1, and the linear momentum of the rigid body, H2, i.e-</p>
-<p>H = H1 + H2</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>point</strong> : Point</p>
-<blockquote>
-<div><p>The point about which angular momentum of the system is desired.</p>
-</div></blockquote>
-<p><strong>frame</strong> : ReferenceFrame</p>
-<blockquote>
-<div><p>The frame in which angular momentum is desired.</p>
-</div></blockquote>
-<p><strong>body1, body2, body3...</strong> : Particle and/or RigidBody</p>
-<blockquote class="last">
-<div><p>The body (or bodies) whose kinetic energy is required.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Particle</span><span class="p">,</span> <span class="n">ReferenceFrame</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">RigidBody</span><span class="p">,</span> <span class="n">outer</span><span class="p">,</span> <span class="n">angular_momentum</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;O&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">0</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">O</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">,</span> <span class="mi">1</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Pa</span> <span class="o">=</span> <span class="n">Particle</span><span class="p">(</span><span class="s">&#39;Pa&#39;</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ac</span> <span class="o">=</span> <span class="n">O</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;Ac&#39;</span><span class="p">,</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ac</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">5</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">outer</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">RigidBody</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">Ac</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">Ac</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">angular_momentum</span><span class="p">(</span><span class="n">O</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">Pa</span><span class="p">,</span> <span class="n">A</span><span class="p">)</span>
-<span class="go">10*N.z</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="kinetic-energy">
-<h2>kinetic_energy<a class="headerlink" href="#kinetic-energy" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.physics.mechanics.functions.kinetic_energy">
-<tt class="descclassname">sympy.physics.mechanics.functions.</tt><tt class="descname">kinetic_energy</tt><big>(</big><em>frame</em>, <em>*body</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/functions.html#kinetic_energy"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.functions.kinetic_energy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Kinetic energy of a multibody system.</p>
-<p>This function returns the kinetic energy of a system of Particle&#8217;s and/or
-RigidBody&#8217;s. The kinetic energy of such a system is equal to the sum of
-the kinetic energies of its constituents. Consider a system, S, comprising
-a rigid body, A, and a particle, P. The kinetic energy of the system, T,
-is equal to the vector sum of the kinetic energy of the particle, T1, and
-the kinetic energy of the rigid body, T2, i.e.</p>
-<p>T = T1 + T2</p>
-<p>Kinetic energy is a scalar.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>frame</strong> : ReferenceFrame</p>
-<blockquote>
-<div><p>The frame in which the velocity or angular velocity of the body is
-defined.</p>
-</div></blockquote>
-<p><strong>body1, body2, body3...</strong> : Particle and/or RigidBody</p>
-<blockquote class="last">
-<div><p>The body (or bodies) whose kinetic energy is required.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Particle</span><span class="p">,</span> <span class="n">ReferenceFrame</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">RigidBody</span><span class="p">,</span> <span class="n">outer</span><span class="p">,</span> <span class="n">kinetic_energy</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;O&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">0</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">O</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">,</span> <span class="mi">1</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Pa</span> <span class="o">=</span> <span class="n">Particle</span><span class="p">(</span><span class="s">&#39;Pa&#39;</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ac</span> <span class="o">=</span> <span class="n">O</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;Ac&#39;</span><span class="p">,</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ac</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">5</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">outer</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">RigidBody</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">Ac</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">Ac</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">kinetic_energy</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">Pa</span><span class="p">,</span> <span class="n">A</span><span class="p">)</span>
-<span class="go">350</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="potential-energy">
-<h2>potential_energy<a class="headerlink" href="#potential-energy" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.physics.mechanics.functions.potential_energy">
-<tt class="descclassname">sympy.physics.mechanics.functions.</tt><tt class="descname">potential_energy</tt><big>(</big><em>*body</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/functions.html#potential_energy"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.functions.potential_energy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Potential energy of a multibody system.</p>
-<p>This function returns the potential energy of a system of Particle&#8217;s and/or
-RigidBody&#8217;s. The potential energy of such a system is equal to the sum of
-the potential energy of its constituents. Consider a system, S, comprising
-a rigid body, A, and a particle, P. The potential energy of the system, V,
-is equal to the vector sum of the potential energy of the particle, V1, and
-the potential energy of the rigid body, V2, i.e.</p>
-<p>V = V1 + V2</p>
-<p>Potential energy is a scalar.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>body1, body2, body3...</strong> : Particle and/or RigidBody</p>
-<blockquote class="last">
-<div><p>The body (or bodies) whose potential energy is required.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Particle</span><span class="p">,</span> <span class="n">ReferenceFrame</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">RigidBody</span><span class="p">,</span> <span class="n">outer</span><span class="p">,</span> <span class="n">potential_energy</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;M m g h&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;O&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">0</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">O</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">,</span> <span class="mi">1</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Pa</span> <span class="o">=</span> <span class="n">Particle</span><span class="p">(</span><span class="s">&#39;Pa&#39;</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ac</span> <span class="o">=</span> <span class="n">O</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;Ac&#39;</span><span class="p">,</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">outer</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">RigidBody</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">Ac</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">Ac</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Pa</span><span class="o">.</span><span class="n">set_potential_energy</span><span class="p">(</span><span class="n">m</span> <span class="o">*</span> <span class="n">g</span> <span class="o">*</span> <span class="n">h</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">set_potential_energy</span><span class="p">(</span><span class="n">M</span> <span class="o">*</span> <span class="n">g</span> <span class="o">*</span> <span class="n">h</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">potential_energy</span><span class="p">(</span><span class="n">Pa</span><span class="p">,</span> <span class="n">A</span><span class="p">)</span>
-<span class="go">M*g*h + g*h*m</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="lagrangian">
-<h2>Lagrangian<a class="headerlink" href="#lagrangian" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.physics.mechanics.functions.Lagrangian">
-<tt class="descclassname">sympy.physics.mechanics.functions.</tt><tt class="descname">Lagrangian</tt><big>(</big><em>frame</em>, <em>*body</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/functions.html#Lagrangian"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.functions.Lagrangian" title="Permalink to this definition">¶</a></dt>
-<dd><p>Lagrangian of a multibody system.</p>
-<p>This function returns the Lagrangian of a system of Particle&#8217;s and/or
-RigidBody&#8217;s. The Lagrangian of such a system is equal to the difference
-between the kinetic energies and potential energies of its constituents. If
-T and V are the kinetic and potential energies of a system then it&#8217;s
-Lagrangian, L, is defined as</p>
-<p>L = T - V</p>
-<p>The Lagrangian is a scalar.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>frame</strong> : ReferenceFrame</p>
-<blockquote>
-<div><p>The frame in which the velocity or angular velocity of the body is
-defined to determine the kinetic energy.</p>
-</div></blockquote>
-<p><strong>body1, body2, body3...</strong> : Particle and/or RigidBody</p>
-<blockquote class="last">
-<div><p>The body (or bodies) whose kinetic energy is required.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Point</span><span class="p">,</span> <span class="n">Particle</span><span class="p">,</span> <span class="n">ReferenceFrame</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">RigidBody</span><span class="p">,</span> <span class="n">outer</span><span class="p">,</span> <span class="n">Lagrangian</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;M m g h&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;O&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">0</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">O</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">,</span> <span class="mi">1</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Pa</span> <span class="o">=</span> <span class="n">Particle</span><span class="p">(</span><span class="s">&#39;Pa&#39;</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ac</span> <span class="o">=</span> <span class="n">O</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;Ac&#39;</span><span class="p">,</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ac</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">5</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">10</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">outer</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">RigidBody</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">Ac</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">Ac</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Pa</span><span class="o">.</span><span class="n">set_potential_energy</span><span class="p">(</span><span class="n">m</span> <span class="o">*</span> <span class="n">g</span> <span class="o">*</span> <span class="n">h</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">set_potential_energy</span><span class="p">(</span><span class="n">M</span> <span class="o">*</span> <span class="n">g</span> <span class="o">*</span> <span class="n">h</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Lagrangian</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">Pa</span><span class="p">,</span> <span class="n">A</span><span class="p">)</span>
-<span class="go">-M*g*h - g*h*m + 350</span>
-</pre></div>
-</div>
-</dd></dl>
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-  <ul>
-<li><a class="reference internal" href="#">Masses, Inertias &amp; Particles, RigidBodys (Docstrings)</a><ul>
-<li><a class="reference internal" href="#module-sympy.physics.mechanics.particle">Particle</a></li>
-<li><a class="reference internal" href="#module-sympy.physics.mechanics.rigidbody">RigidBody</a></li>
-<li><a class="reference internal" href="#inertia">inertia</a></li>
-<li><a class="reference internal" href="#inertia-of-point-mass">inertia_of_point_mass</a></li>
-<li><a class="reference internal" href="#linear-momentum">linear_momentum</a></li>
-<li><a class="reference internal" href="#angular-momentum">angular_momentum</a></li>
-<li><a class="reference internal" href="#kinetic-energy">kinetic_energy</a></li>
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-<li><a class="reference internal" href="#lagrangian">Lagrangian</a></li>
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-            
-  <div class="section" id="printing-docstrings">
-<h1>Printing (Docstrings)<a class="headerlink" href="#printing-docstrings" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="mechanics-printing">
-<h2>mechanics_printing<a class="headerlink" href="#mechanics-printing" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.physics.mechanics.functions.mechanics_printing">
-<tt class="descclassname">sympy.physics.mechanics.functions.</tt><tt class="descname">mechanics_printing</tt><big>(</big><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/functions.html#mechanics_printing"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.functions.mechanics_printing" title="Permalink to this definition">¶</a></dt>
-<dd><p>Sets up interactive printing for mechanics&#8217; derivatives.</p>
-<p>The main benefit of this is for printing of time derivatives;
-instead of displaying as Derivative(f(t),t), it will display f&#8217;
-This is only actually needed for when derivatives are present and are not
-in a physics.mechanics object.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="c"># 2 lines below are for tests to function properly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">sys</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sys</span><span class="o">.</span><span class="n">displayhook</span> <span class="o">=</span> <span class="n">sys</span><span class="o">.</span><span class="n">__displayhook__</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">diff</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">mechanics_printing</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="n">t</span><span class="p">)</span>
-<span class="go">Derivative(f(t), t)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mechanics_printing</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="n">t</span><span class="p">)</span>
-<span class="go">f&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">Derivative(f(x), x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># 2 lines below are for tests to function properly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">sys</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sys</span><span class="o">.</span><span class="n">displayhook</span> <span class="o">=</span> <span class="n">sys</span><span class="o">.</span><span class="n">__displayhook__</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="mprint">
-<h2>mprint<a class="headerlink" href="#mprint" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.physics.mechanics.functions.mprint">
-<tt class="descclassname">sympy.physics.mechanics.functions.</tt><tt class="descname">mprint</tt><big>(</big><em>expr</em>, <em>**settings</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/functions.html#mprint"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.functions.mprint" title="Permalink to this definition">¶</a></dt>
-<dd><p>Function for printing of expressions generated in mechanics.</p>
-<p>Extends SymPy&#8217;s StrPrinter; mprint is equivalent to:
-print sstr()
-mprint takes the same options as sstr.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>expr</strong> : valid sympy object</p>
-<blockquote>
-<div><p>SymPy expression to print</p>
-</div></blockquote>
-<p><strong>settings</strong> : args</p>
-<blockquote class="last">
-<div><p>Same as print for SymPy</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">mprint</span><span class="p">,</span> <span class="n">dynamicsymbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u1</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;u1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">u1</span><span class="p">)</span>
-<span class="go">u1(t)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mprint</span><span class="p">(</span><span class="n">u1</span><span class="p">)</span>
-<span class="go">u1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="mpprint">
-<h2>mpprint<a class="headerlink" href="#mpprint" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.physics.mechanics.functions.mpprint">
-<tt class="descclassname">sympy.physics.mechanics.functions.</tt><tt class="descname">mpprint</tt><big>(</big><em>expr</em>, <em>**settings</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/functions.html#mpprint"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.functions.mpprint" title="Permalink to this definition">¶</a></dt>
-<dd><p>Function for pretty printing of expressions generated in mechanics.</p>
-<p>Mainly used for expressions not inside a vector; the output of running
-scripts and generating equations of motion. Takes the same options as
-SymPy&#8217;s pretty_print(); see that function for more information.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>expr</strong> : valid sympy object</p>
-<blockquote>
-<div><p>SymPy expression to pretty print</p>
-</div></blockquote>
-<p><strong>settings</strong> : args</p>
-<blockquote class="last">
-<div><p>Same as pretty print</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Use in the same way as pprint</p>
-</dd></dl>
-
-</div>
-<div class="section" id="mlatex">
-<h2>mlatex<a class="headerlink" href="#mlatex" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.physics.mechanics.functions.mlatex">
-<tt class="descclassname">sympy.physics.mechanics.functions.</tt><tt class="descname">mlatex</tt><big>(</big><em>expr</em>, <em>**settings</em><big>)</big><a class="reference internal" href="../../../../_modules/sympy/physics/mechanics/functions.html#mlatex"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.mechanics.functions.mlatex" title="Permalink to this definition">¶</a></dt>
-<dd><p>Function for printing latex representation of mechanics objects.</p>
-<p>For latex representation of Vectors, Dyadics, and dynamicsymbols. Takes the
-same options as SymPy&#8217;s latex(); see that function for more information;</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>expr</strong> : valid sympy object</p>
-<blockquote>
-<div><p>SymPy expression to represent in LaTeX form</p>
-</div></blockquote>
-<p><strong>settings</strong> : args</p>
-<blockquote class="last">
-<div><p>Same as latex()</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">mlatex</span><span class="p">,</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">dynamicsymbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1 q2&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1d</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1 q2&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1dd</span><span class="p">,</span> <span class="n">q2dd</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1 q2&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mlatex</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="go">&#39;\\mathbf{\\hat{n}_x} + \\mathbf{\\hat{n}_y}&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mlatex</span><span class="p">(</span><span class="n">q1</span> <span class="o">+</span> <span class="n">q2</span><span class="p">)</span>
-<span class="go">&#39;q_{1} + q_{2}&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mlatex</span><span class="p">(</span><span class="n">q1d</span><span class="p">)</span>
-<span class="go">&#39;\\dot{q}_{1}&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mlatex</span><span class="p">(</span><span class="n">q1</span> <span class="o">*</span> <span class="n">q2d</span><span class="p">)</span>
-<span class="go">&#39;q_{1} \\dot{q}_{2}&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mlatex</span><span class="p">(</span><span class="n">q1dd</span> <span class="o">*</span> <span class="n">q1</span> <span class="o">/</span> <span class="n">q1d</span><span class="p">)</span>
-<span class="go">&#39;\\frac{q_{1} \\ddot{q}_{1}}{\\dot{q}_{1}}&#39;</span>
-</pre></div>
-</div>
-</dd></dl>
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-  <div class="section" id="examples-for-physics-mechanics">
-<h1>Examples for Physics.Mechanics<a class="headerlink" href="#examples-for-physics-mechanics" title="Permalink to this headline">¶</a></h1>
-<p>This module has been developed to aid in formulation of equations of motion;
-here, examples follow to help display how to use this module.</p>
-<div class="section" id="the-rolling-disc">
-<h2>The Rolling Disc<a class="headerlink" href="#the-rolling-disc" title="Permalink to this headline">¶</a></h2>
-<p>The first example is that of a rolling disc. The disc is assumed to be
-infinitely thin, in contact with the ground at only 1 point, and it is rolling
-without slip on the ground. See the image below.</p>
-<div align="center" class="align-center"><img height="550" src="../../../_images/ex_rd.svg" width="550" /></div>
-<p>Here the definition of the rolling disc&#8217;s kinematics is formed from the contact
-point up, removing the need to introduce generalized speeds. Only 3
-configuration and three speed variables are need to describe this system, along
-with the disc&#8217;s mass and radius, and the local gravity (note that mass will
-drop out).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">tan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span><span class="p">,</span> <span class="n">u1</span><span class="p">,</span> <span class="n">u2</span><span class="p">,</span> <span class="n">u3</span>  <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1 q2 q3 u1 u2 u3&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1d</span><span class="p">,</span> <span class="n">q2d</span><span class="p">,</span> <span class="n">q3d</span><span class="p">,</span> <span class="n">u1d</span><span class="p">,</span> <span class="n">u2d</span><span class="p">,</span> <span class="n">u3d</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1 q2 q3 u1 u2 u3&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;r m g&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mechanics_printing</span><span class="p">()</span>
-</pre></div>
-</div>
-<p>The kinematics are formed by a series of simple rotations. Each simple rotation
-creates a new frame, and the next rotation is defined by the new frame&#8217;s basis
-vectors. This example uses a 3-1-2 series of rotations, or Z, X, Y series of
-rotations. Angular velocity for this is defined using the second frame&#8217;s basis
-(the lean frame); it is for this reason that we defined intermediate frames,
-rather than using a body-three orientation.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;Y&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span> <span class="o">=</span> <span class="n">Y</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;L&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q2</span><span class="p">,</span> <span class="n">Y</span><span class="o">.</span><span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">L</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;R&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q3</span><span class="p">,</span> <span class="n">L</span><span class="o">.</span><span class="n">y</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">u1</span> <span class="o">*</span> <span class="n">L</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">u2</span> <span class="o">*</span> <span class="n">L</span><span class="o">.</span><span class="n">y</span> <span class="o">+</span> <span class="n">u3</span> <span class="o">*</span> <span class="n">L</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="o">.</span><span class="n">set_ang_acc</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">R</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">N</span><span class="p">)</span><span class="o">.</span><span class="n">dt</span><span class="p">(</span><span class="n">R</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">R</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">N</span><span class="p">)</span> <span class="o">^</span> <span class="n">R</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">N</span><span class="p">)))</span>
-</pre></div>
-</div>
-<p>This is the translational kinematics. We create a point with no velocity
-in N; this is the contact point between the disc and ground. Next we form
-the position vector from the contact point to the disc&#8217;s center of mass.
-Finally we form the velocity and acceleration of the disc.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;C&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dmc</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;Dmc&#39;</span><span class="p">,</span> <span class="n">r</span> <span class="o">*</span> <span class="n">L</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dmc</span><span class="o">.</span><span class="n">v2pt_theory</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span>
-<span class="go">r*u2*L.x - r*u1*L.y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dmc</span><span class="o">.</span><span class="n">a2pt_theory</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span>
-<span class="go">(r*u1*u3 + r*u2&#39;)*L.x + (-r*(-u3*q3&#39; + u1&#39;) + r*u2*u3)*L.y + (-r*u1**2 - r*u2**2)*L.z</span>
-</pre></div>
-</div>
-<p>This is a simple way to form the inertia dyadic. The inertia of the disc does
-not change within the lean frame as the disc rolls; this will make for simpler
-equations in the end.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">inertia</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="n">m</span> <span class="o">/</span> <span class="mi">4</span> <span class="o">*</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">m</span> <span class="o">/</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">m</span> <span class="o">/</span> <span class="mi">4</span> <span class="o">*</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mprint</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
-<span class="go">m*r**2/4*(L.x|L.x) + m*r**2/2*(L.y|L.y) + m*r**2/4*(L.z|L.z)</span>
-</pre></div>
-</div>
-<p>Kinematic differential equations; how the generalized coordinate time
-derivatives relate to generalized speeds. Here these were computed by hand.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">kd</span> <span class="o">=</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="n">u3</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">q2</span><span class="p">),</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">u1</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="n">u2</span> <span class="o">+</span> <span class="n">u3</span> <span class="o">*</span> <span class="n">tan</span><span class="p">(</span><span class="n">q2</span><span class="p">)]</span>
-</pre></div>
-</div>
-<p>Creation of the force list; it is the gravitational force at the center of mass of
-the disc. Then we create the disc by assigning a Point to the center of mass
-attribute, a ReferenceFrame to the frame attribute, and mass and inertia. Then
-we form the body list.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ForceList</span> <span class="o">=</span> <span class="p">[(</span><span class="n">Dmc</span><span class="p">,</span> <span class="o">-</span> <span class="n">m</span> <span class="o">*</span> <span class="n">g</span> <span class="o">*</span> <span class="n">Y</span><span class="o">.</span><span class="n">z</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">BodyD</span> <span class="o">=</span> <span class="n">RigidBody</span><span class="p">(</span><span class="s">&#39;BodyD&#39;</span><span class="p">,</span> <span class="n">Dmc</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">Dmc</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">BodyList</span> <span class="o">=</span> <span class="p">[</span><span class="n">BodyD</span><span class="p">]</span>
-</pre></div>
-</div>
-<p>Finally we form the equations of motion, using the same steps we did before.
-Specify inertial frame, supply generalized coordinates and speeds, supply
-kinematic differential equation dictionary, compute Fr from the force list and
-Fr* from the body list, compute the mass matrix and forcing terms, then solve
-for the u dots (time derivatives of the generalized speeds).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span> <span class="o">=</span> <span class="n">KanesMethod</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">q_ind</span><span class="o">=</span><span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span><span class="p">],</span> <span class="n">u_ind</span><span class="o">=</span><span class="p">[</span><span class="n">u1</span><span class="p">,</span> <span class="n">u2</span><span class="p">,</span> <span class="n">u3</span><span class="p">],</span> <span class="n">kd_eqs</span><span class="o">=</span><span class="n">kd</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">fr</span><span class="p">,</span> <span class="n">frstar</span><span class="p">)</span> <span class="o">=</span> <span class="n">KM</span><span class="o">.</span><span class="n">kanes_equations</span><span class="p">(</span><span class="n">ForceList</span><span class="p">,</span> <span class="n">BodyList</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">MM</span> <span class="o">=</span> <span class="n">KM</span><span class="o">.</span><span class="n">mass_matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">forcing</span> <span class="o">=</span> <span class="n">KM</span><span class="o">.</span><span class="n">forcing</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rhs</span> <span class="o">=</span> <span class="n">MM</span><span class="o">.</span><span class="n">inv</span><span class="p">()</span> <span class="o">*</span> <span class="n">forcing</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">kdd</span> <span class="o">=</span> <span class="n">KM</span><span class="o">.</span><span class="n">kindiffdict</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rhs</span> <span class="o">=</span> <span class="n">rhs</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">kdd</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rhs</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mprint</span><span class="p">(</span><span class="n">rhs</span><span class="p">)</span>
-<span class="go">[(4*g*sin(q2)/5 + 2*r*u2*u3 - r*u3**2*tan(q2))/r]</span>
-<span class="go">[                                     -2*u1*u3/3]</span>
-<span class="go">[                        (-2*u2 + u3*tan(q2))*u1]</span>
-</pre></div>
-</div>
-<p>This concludes the rolling disc example.</p>
-</div>
-<div class="section" id="the-rolling-disc-again">
-<h2>The Rolling Disc, again<a class="headerlink" href="#the-rolling-disc-again" title="Permalink to this headline">¶</a></h2>
-<p>We will now revisit the rolling disc example, except this time we are bringing
-the non-contributing (constraint) forces into evidence. See <a class="reference internal" href="../mechanics/reference.html#kane1985">[Kane1985]</a> for a
-more thorough explanation of this. Here, we will turn on the automatic
-simplifcation done when doing vector operations. It makes the outputs nicer for
-small problems, but can cause larger vector operations to hang.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mechanics_printing</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Vector</span><span class="o">.</span><span class="n">simp</span> <span class="o">=</span> <span class="bp">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span><span class="p">,</span> <span class="n">u1</span><span class="p">,</span> <span class="n">u2</span><span class="p">,</span> <span class="n">u3</span>  <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1 q2 q3 u1 u2 u3&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1d</span><span class="p">,</span> <span class="n">q2d</span><span class="p">,</span> <span class="n">q3d</span><span class="p">,</span> <span class="n">u1d</span><span class="p">,</span> <span class="n">u2d</span><span class="p">,</span> <span class="n">u3d</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1 q2 q3 u1 u2 u3&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;r m g&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>These two lines introduce the extra quantities needed to find the constraint
-forces.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">u4</span><span class="p">,</span> <span class="n">u5</span><span class="p">,</span> <span class="n">u6</span><span class="p">,</span> <span class="n">f1</span><span class="p">,</span> <span class="n">f2</span><span class="p">,</span> <span class="n">f3</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;u4 u5 u6 f1 f2 f3&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Most of the main code is the same as before.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;Y&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span> <span class="o">=</span> <span class="n">Y</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;L&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q2</span><span class="p">,</span> <span class="n">Y</span><span class="o">.</span><span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">L</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;R&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q3</span><span class="p">,</span> <span class="n">L</span><span class="o">.</span><span class="n">y</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">u1</span> <span class="o">*</span> <span class="n">L</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">u2</span> <span class="o">*</span> <span class="n">L</span><span class="o">.</span><span class="n">y</span> <span class="o">+</span> <span class="n">u3</span> <span class="o">*</span> <span class="n">L</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="o">.</span><span class="n">set_ang_acc</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">R</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">N</span><span class="p">)</span><span class="o">.</span><span class="n">dt</span><span class="p">(</span><span class="n">R</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">R</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">N</span><span class="p">)</span> <span class="o">^</span> <span class="n">R</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">N</span><span class="p">)))</span>
-</pre></div>
-</div>
-<p>The definition of rolling without slip necessitates that the velocity of the
-contact point is zero; as part of bringing the constraint forces into evidence,
-we have to introduce speeds at this point, which will by definition always be
-zero. They are normal to the ground, along the path which the disc is rolling,
-and along the ground in an perpendicular direction.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;C&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">u4</span> <span class="o">*</span> <span class="n">L</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">u5</span> <span class="o">*</span> <span class="p">(</span><span class="n">Y</span><span class="o">.</span><span class="n">z</span> <span class="o">^</span> <span class="n">L</span><span class="o">.</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">u6</span> <span class="o">*</span> <span class="n">Y</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dmc</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;Dmc&#39;</span><span class="p">,</span> <span class="n">r</span> <span class="o">*</span> <span class="n">L</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">vel</span> <span class="o">=</span> <span class="n">Dmc</span><span class="o">.</span><span class="n">v2pt_theory</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">acc</span> <span class="o">=</span> <span class="n">Dmc</span><span class="o">.</span><span class="n">a2pt_theory</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">inertia</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="n">m</span> <span class="o">/</span> <span class="mi">4</span> <span class="o">*</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">m</span> <span class="o">/</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">m</span> <span class="o">/</span> <span class="mi">4</span> <span class="o">*</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">kd</span> <span class="o">=</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="n">u3</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">q2</span><span class="p">),</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">u1</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="n">u2</span> <span class="o">+</span> <span class="n">u3</span> <span class="o">*</span> <span class="n">tan</span><span class="p">(</span><span class="n">q2</span><span class="p">)]</span>
-</pre></div>
-</div>
-<p>Just as we previously introduced three speeds as part of this process, we also
-introduce three forces; they are in the same direction as the speeds, and
-represent the constraint forces in those directions.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ForceList</span> <span class="o">=</span> <span class="p">[(</span><span class="n">Dmc</span><span class="p">,</span> <span class="o">-</span> <span class="n">m</span> <span class="o">*</span> <span class="n">g</span> <span class="o">*</span> <span class="n">Y</span><span class="o">.</span><span class="n">z</span><span class="p">),</span> <span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">f1</span> <span class="o">*</span> <span class="n">L</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">f2</span> <span class="o">*</span> <span class="p">(</span><span class="n">Y</span><span class="o">.</span><span class="n">z</span> <span class="o">^</span> <span class="n">L</span><span class="o">.</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f3</span> <span class="o">*</span> <span class="n">Y</span><span class="o">.</span><span class="n">z</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">BodyD</span> <span class="o">=</span> <span class="n">RigidBody</span><span class="p">(</span><span class="s">&#39;BodyD&#39;</span><span class="p">,</span> <span class="n">Dmc</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">Dmc</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">BodyList</span> <span class="o">=</span> <span class="p">[</span><span class="n">BodyD</span><span class="p">]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span> <span class="o">=</span> <span class="n">KanesMethod</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">q_ind</span><span class="o">=</span><span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span><span class="p">],</span> <span class="n">u_ind</span><span class="o">=</span><span class="p">[</span><span class="n">u1</span><span class="p">,</span> <span class="n">u2</span><span class="p">,</span> <span class="n">u3</span><span class="p">],</span> <span class="n">kd_eqs</span><span class="o">=</span><span class="n">kd</span><span class="p">,</span>
-<span class="gp">... </span>          <span class="n">u_auxiliary</span><span class="o">=</span><span class="p">[</span><span class="n">u4</span><span class="p">,</span> <span class="n">u5</span><span class="p">,</span> <span class="n">u6</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">fr</span><span class="p">,</span> <span class="n">frstar</span><span class="p">)</span> <span class="o">=</span> <span class="n">KM</span><span class="o">.</span><span class="n">kanes_equations</span><span class="p">(</span><span class="n">ForceList</span><span class="p">,</span> <span class="n">BodyList</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">MM</span> <span class="o">=</span> <span class="n">KM</span><span class="o">.</span><span class="n">mass_matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">forcing</span> <span class="o">=</span> <span class="n">KM</span><span class="o">.</span><span class="n">forcing</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rhs</span> <span class="o">=</span> <span class="n">MM</span><span class="o">.</span><span class="n">inv</span><span class="p">()</span> <span class="o">*</span> <span class="n">forcing</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">kdd</span> <span class="o">=</span> <span class="n">KM</span><span class="o">.</span><span class="n">kindiffdict</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rhs</span> <span class="o">=</span> <span class="n">rhs</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">kdd</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rhs</span><span class="o">.</span><span class="n">simplify</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mprint</span><span class="p">(</span><span class="n">rhs</span><span class="p">)</span>
-<span class="go">[(4*g*sin(q2)/5 + 2*r*u2*u3 - r*u3**2*tan(q2))/r]</span>
-<span class="go">[                                     -2*u1*u3/3]</span>
-<span class="go">[                        (-2*u2 + u3*tan(q2))*u1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">signsimp</span><span class="p">,</span> <span class="n">factor_terms</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mprint</span><span class="p">(</span><span class="n">KM</span><span class="o">.</span><span class="n">auxiliary_eqs</span><span class="o">.</span><span class="n">applyfunc</span><span class="p">(</span><span class="k">lambda</span> <span class="n">w</span><span class="p">:</span> <span class="n">factor_terms</span><span class="p">(</span><span class="n">signsimp</span><span class="p">(</span><span class="n">w</span><span class="p">))))</span>
-<span class="go">[                                                   m*r*(u1*u3 + u2&#39;) - f1]</span>
-<span class="go">[      m*r*((u1**2 + u2**2)*sin(q2) + (u2*u3 + u3*q3&#39; - u1&#39;)*cos(q2)) - f2]</span>
-<span class="go">[g*m - m*r*((u1**2 + u2**2)*cos(q2) - (u2*u3 + u3*q3&#39; - u1&#39;)*sin(q2)) - f3]</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="the-bicycle">
-<h2>The Bicycle<a class="headerlink" href="#the-bicycle" title="Permalink to this headline">¶</a></h2>
-<p>The bicycle is an interesting system in that it has multiple rigid bodies,
-non-holonomic constraints, and a holonomic constraint. The linearized equations
-of motion are presented in <a class="reference internal" href="../mechanics/reference.html#meijaard2007">[Meijaard2007]</a>. This example will go through
-construction of the equations of motion in <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;Calculation of Linearized Bicycle </span><span class="se">\&quot;</span><span class="s">A</span><span class="se">\&quot;</span><span class="s"> Matrix, &#39;</span>
-<span class="gp">... </span>      <span class="s">&#39;with States: Roll, Steer, Roll Rate, Steer Rate&#39;</span><span class="p">)</span>
-<span class="go">Calculation of Linearized Bicycle &quot;A&quot; Matrix, with States: Roll, Steer, Roll Rate, Steer Rate</span>
-</pre></div>
-</div>
-<p>Note that this code has been crudely ported from Autolev, which is the reason
-for some of the unusual naming conventions. It was purposefully as similar as
-possible in order to aid initial porting &amp; debugging. We also turn off
-Vector.simp (turned on in the last example) to avoid hangups when doing
-computations in this problem.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Vector</span><span class="o">.</span><span class="n">simp</span> <span class="o">=</span> <span class="bp">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mechanics_printing</span><span class="p">()</span>
-</pre></div>
-</div>
-<p>Declaration of Coordinates &amp; Speeds:
-A Simple definitions for qdots: (qd = u) is used in this code.  Speeds are: yaw
-frame ang. rate, roll frame ang. rate, rear wheel frame ang.  rate (spinning
-motion), frame ang. rate (pitching motion), steering frame ang. rate, and front
-wheel ang. rate (spinning motion).  Wheel positions are ignorable coordinates,
-so they are not introduced.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q4</span><span class="p">,</span> <span class="n">q5</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1 q2 q4 q5&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1d</span><span class="p">,</span> <span class="n">q2d</span><span class="p">,</span> <span class="n">q4d</span><span class="p">,</span> <span class="n">q5d</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1 q2 q4 q5&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u1</span><span class="p">,</span> <span class="n">u2</span><span class="p">,</span> <span class="n">u3</span><span class="p">,</span> <span class="n">u4</span><span class="p">,</span> <span class="n">u5</span><span class="p">,</span> <span class="n">u6</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;u1 u2 u3 u4 u5 u6&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u1d</span><span class="p">,</span> <span class="n">u2d</span><span class="p">,</span> <span class="n">u3d</span><span class="p">,</span> <span class="n">u4d</span><span class="p">,</span> <span class="n">u5d</span><span class="p">,</span> <span class="n">u6d</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;u1 u2 u3 u4 u5 u6&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Declaration of System&#8217;s Parameters:
-The below symbols should be fairly self-explanatory.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">WFrad</span><span class="p">,</span> <span class="n">WRrad</span><span class="p">,</span> <span class="n">htangle</span><span class="p">,</span> <span class="n">forkoffset</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;WFrad WRrad htangle forkoffset&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">forklength</span><span class="p">,</span> <span class="n">framelength</span><span class="p">,</span> <span class="n">forkcg1</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;forklength framelength forkcg1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">forkcg3</span><span class="p">,</span> <span class="n">framecg1</span><span class="p">,</span> <span class="n">framecg3</span><span class="p">,</span> <span class="n">Iwr11</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;forkcg3 framecg1 framecg3 Iwr11&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Iwr22</span><span class="p">,</span> <span class="n">Iwf11</span><span class="p">,</span> <span class="n">Iwf22</span><span class="p">,</span> <span class="n">Iframe11</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;Iwr22 Iwf11 Iwf22 Iframe11&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Iframe22</span><span class="p">,</span> <span class="n">Iframe33</span><span class="p">,</span> <span class="n">Iframe31</span><span class="p">,</span> <span class="n">Ifork11</span> <span class="o">=</span> \
-<span class="gp">... </span>    <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;Iframe22 Iframe33 Iframe31 Ifork11&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ifork22</span><span class="p">,</span> <span class="n">Ifork33</span><span class="p">,</span> <span class="n">Ifork31</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;Ifork22 Ifork33 Ifork31 g&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mframe</span><span class="p">,</span> <span class="n">mfork</span><span class="p">,</span> <span class="n">mwf</span><span class="p">,</span> <span class="n">mwr</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;mframe mfork mwf mwr&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Set up reference frames for the system:
-N - inertial
-Y - yaw
-R - roll
-WR - rear wheel, rotation angle is ignorable coordinate so not oriented
-Frame - bicycle frame
-TempFrame - statically rotated frame for easier reference inertia definition
-Fork - bicycle fork
-TempFork - statically rotated frame for easier reference inertia definition
-WF - front wheel, again posses a ignorable coordinate</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;Y&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">Y</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;R&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q2</span><span class="p">,</span> <span class="n">Y</span><span class="o">.</span><span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Frame</span> <span class="o">=</span> <span class="n">R</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;Frame&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q4</span> <span class="o">+</span> <span class="n">htangle</span><span class="p">,</span> <span class="n">R</span><span class="o">.</span><span class="n">y</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">WR</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;WR&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">TempFrame</span> <span class="o">=</span> <span class="n">Frame</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;TempFrame&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="n">htangle</span><span class="p">,</span> <span class="n">Frame</span><span class="o">.</span><span class="n">y</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Fork</span> <span class="o">=</span> <span class="n">Frame</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;Fork&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q5</span><span class="p">,</span> <span class="n">Frame</span><span class="o">.</span><span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">TempFork</span> <span class="o">=</span> <span class="n">Fork</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;TempFork&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="n">htangle</span><span class="p">,</span> <span class="n">Fork</span><span class="o">.</span><span class="n">y</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">WF</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;WF&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Kinematics of the Bicycle:
-First block of code is forming the positions of the relevant points rear wheel
-contact -&gt; rear wheel&#8217;s center of mass -&gt; frame&#8217;s center of mass + frame/fork connection
--&gt; fork&#8217;s center of mass + front wheel&#8217;s center of mass -&gt; front wheel contact point.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">WR_cont</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;WR_cont&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">WR_mc</span> <span class="o">=</span> <span class="n">WR_cont</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;WR_mc&#39;</span><span class="p">,</span> <span class="n">WRrad</span> <span class="o">*</span> <span class="n">R</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Steer</span> <span class="o">=</span> <span class="n">WR_mc</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;Steer&#39;</span><span class="p">,</span> <span class="n">framelength</span> <span class="o">*</span> <span class="n">Frame</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Frame_mc</span> <span class="o">=</span> <span class="n">WR_mc</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;Frame_mc&#39;</span><span class="p">,</span> <span class="o">-</span><span class="n">framecg1</span> <span class="o">*</span> <span class="n">Frame</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">framecg3</span> <span class="o">*</span> <span class="n">Frame</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Fork_mc</span> <span class="o">=</span> <span class="n">Steer</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;Fork_mc&#39;</span><span class="p">,</span> <span class="o">-</span><span class="n">forkcg1</span> <span class="o">*</span> <span class="n">Fork</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">forkcg3</span> <span class="o">*</span> <span class="n">Fork</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">WF_mc</span> <span class="o">=</span> <span class="n">Steer</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;WF_mc&#39;</span><span class="p">,</span> <span class="n">forklength</span> <span class="o">*</span> <span class="n">Fork</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">forkoffset</span> <span class="o">*</span> <span class="n">Fork</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">WF_cont</span> <span class="o">=</span> <span class="n">WF_mc</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;WF_cont&#39;</span><span class="p">,</span> <span class="n">WFrad</span><span class="o">*</span><span class="p">(</span><span class="n">dot</span><span class="p">(</span><span class="n">Fork</span><span class="o">.</span><span class="n">y</span><span class="p">,</span> <span class="n">Y</span><span class="o">.</span><span class="n">z</span><span class="p">)</span><span class="o">*</span><span class="n">Fork</span><span class="o">.</span><span class="n">y</span> <span class="o">-</span> \
-<span class="gp">... </span>                                            <span class="n">Y</span><span class="o">.</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">normalize</span><span class="p">())</span>
-</pre></div>
-</div>
-<p>Set the angular velocity of each frame:
-Angular accelerations end up being calculated automatically by differentiating
-the angular velocities when first needed. ::
-u1 is yaw rate
-u2 is roll rate
-u3 is rear wheel rate
-u4 is frame pitch rate
-u5 is fork steer rate
-u6 is front wheel rate</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">u1</span> <span class="o">*</span> <span class="n">Y</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">Y</span><span class="p">,</span> <span class="n">u2</span> <span class="o">*</span> <span class="n">R</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">WR</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">Frame</span><span class="p">,</span> <span class="n">u3</span> <span class="o">*</span> <span class="n">Frame</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Frame</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">u4</span> <span class="o">*</span> <span class="n">Frame</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Fork</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">Frame</span><span class="p">,</span> <span class="n">u5</span> <span class="o">*</span> <span class="n">Fork</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">WF</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">Fork</span><span class="p">,</span> <span class="n">u6</span> <span class="o">*</span> <span class="n">Fork</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Form the velocities of the points, using the 2-point theorem.  Accelerations
-again are calculated automatically when first needed.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">WR_cont</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">WR_mc</span><span class="o">.</span><span class="n">v2pt_theory</span><span class="p">(</span><span class="n">WR_cont</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">WR</span><span class="p">)</span>
-<span class="go">WRrad*(u1*sin(q2) + u3 + u4)*R.x - WRrad*u2*R.y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Steer</span><span class="o">.</span><span class="n">v2pt_theory</span><span class="p">(</span><span class="n">WR_mc</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">Frame</span><span class="p">)</span>
-<span class="go">WRrad*(u1*sin(q2) + u3 + u4)*R.x - WRrad*u2*R.y + framelength*(u1*sin(q2) + u4)*Frame.x - framelength*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4))*Frame.y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Frame_mc</span><span class="o">.</span><span class="n">v2pt_theory</span><span class="p">(</span><span class="n">WR_mc</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">Frame</span><span class="p">)</span>
-<span class="go">WRrad*(u1*sin(q2) + u3 + u4)*R.x - WRrad*u2*R.y + framecg3*(u1*sin(q2) + u4)*Frame.x + (-framecg1*(u1*cos(htangle + q4)*cos(q2) + u2*sin(htangle + q4)) - framecg3*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4)))*Frame.y + framecg1*(u1*sin(q2) + u4)*Frame.z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Fork_mc</span><span class="o">.</span><span class="n">v2pt_theory</span><span class="p">(</span><span class="n">Steer</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">Fork</span><span class="p">)</span>
-<span class="go">WRrad*(u1*sin(q2) + u3 + u4)*R.x - WRrad*u2*R.y + framelength*(u1*sin(q2) + u4)*Frame.x - framelength*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4))*Frame.y + forkcg3*((sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*u1 + u2*sin(htangle + q4)*sin(q5) + u4*cos(q5))*Fork.x + (-forkcg1*((-sin(q2)*sin(q5) + cos(htangle + q4)*cos(q2)*cos(q5))*u1 + u2*sin(htangle + q4)*cos(q5) - u4*sin(q5)) - forkcg3*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4) + u5))*Fork.y + forkcg1*((sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*u1 + u2*sin(htangle + q4)*sin(q5) + u4*cos(q5))*Fork.z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">WF_mc</span><span class="o">.</span><span class="n">v2pt_theory</span><span class="p">(</span><span class="n">Steer</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">Fork</span><span class="p">)</span>
-<span class="go">WRrad*(u1*sin(q2) + u3 + u4)*R.x - WRrad*u2*R.y + framelength*(u1*sin(q2) + u4)*Frame.x - framelength*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4))*Frame.y + forkoffset*((sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*u1 + u2*sin(htangle + q4)*sin(q5) + u4*cos(q5))*Fork.x + (forklength*((-sin(q2)*sin(q5) + cos(htangle + q4)*cos(q2)*cos(q5))*u1 + u2*sin(htangle + q4)*cos(q5) - u4*sin(q5)) - forkoffset*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4) + u5))*Fork.y - forklength*((sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*u1 + u2*sin(htangle + q4)*sin(q5) + u4*cos(q5))*Fork.z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">WF_cont</span><span class="o">.</span><span class="n">v2pt_theory</span><span class="p">(</span><span class="n">WF_mc</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">WF</span><span class="p">)</span>
-<span class="go">WRrad*(u1*sin(q2) + u3 + u4)*R.x - WRrad*u2*R.y + framelength*(u1*sin(q2) + u4)*Frame.x - framelength*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4))*Frame.y + (-WFrad*(sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*((-sin(q2)*sin(q5) + cos(htangle + q4)*cos(q2)*cos(q5))*u1 + u2*sin(htangle + q4)*cos(q5) - u4*sin(q5))/sqrt((-sin(q2)*cos(q5) - sin(q5)*cos(htangle + q4)*cos(q2))*(sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2)) + 1) + forkoffset*((sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*u1 + u2*sin(htangle + q4)*sin(q5) + u4*cos(q5)))*Fork.x + (forklength*((-sin(q2)*sin(q5) + cos(htangle + q4)*cos(q2)*cos(q5))*u1 + u2*sin(htangle + q4)*cos(q5) - u4*sin(q5)) - forkoffset*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4) + u5))*Fork.y + (WFrad*(sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*(-u1*sin(htangle + q4)*cos(q2) + u2*cos(htangle + q4) + u5)/sqrt((-sin(q2)*cos(q5) - sin(q5)*cos(htangle + q4)*cos(q2))*(sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2)) + 1) - forklength*((sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2))*u1 + u2*sin(htangle + q4)*sin(q5) + u4*cos(q5)))*Fork.z - WFrad*((-sin(q2)*sin(q5)*cos(htangle + q4) + cos(q2)*cos(q5))*u6 + u4*cos(q2) + u5*sin(htangle + q4)*sin(q2))/sqrt((-sin(q2)*cos(q5) - sin(q5)*cos(htangle + q4)*cos(q2))*(sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2)) + 1)*Y.x + WFrad*(u2 + u5*cos(htangle + q4) + u6*sin(htangle + q4)*sin(q5))/sqrt((-sin(q2)*cos(q5) - sin(q5)*cos(htangle + q4)*cos(q2))*(sin(q2)*cos(q5) + sin(q5)*cos(htangle + q4)*cos(q2)) + 1)*Y.y</span>
-</pre></div>
-</div>
-<p>Sets the inertias of each body. Uses the inertia frame to construct the inertia
-dyadics. Wheel inertias are only defined by principal moments of inertia, and
-are in fact constant in the frame and fork reference frames; it is for this
-reason that the orientations of the wheels does not need to be defined. The
-frame and fork inertias are defined in the &#8216;Temp&#8217; frames which are fixed to the
-appropriate body frames; this is to allow easier input of the reference values
-of the benchmark paper. Note that due to slightly different orientations, the
-products of inertia need to have their signs flipped; this is done later when
-entering the numerical value.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Frame_I</span> <span class="o">=</span> <span class="p">(</span><span class="n">inertia</span><span class="p">(</span><span class="n">TempFrame</span><span class="p">,</span> <span class="n">Iframe11</span><span class="p">,</span> <span class="n">Iframe22</span><span class="p">,</span> <span class="n">Iframe33</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span>
-<span class="gp">... </span>                                                  <span class="n">Iframe31</span><span class="p">),</span> <span class="n">Frame_mc</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Fork_I</span> <span class="o">=</span> <span class="p">(</span><span class="n">inertia</span><span class="p">(</span><span class="n">TempFork</span><span class="p">,</span> <span class="n">Ifork11</span><span class="p">,</span> <span class="n">Ifork22</span><span class="p">,</span> <span class="n">Ifork33</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">Ifork31</span><span class="p">),</span> <span class="n">Fork_mc</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">WR_I</span> <span class="o">=</span> <span class="p">(</span><span class="n">inertia</span><span class="p">(</span><span class="n">Frame</span><span class="p">,</span> <span class="n">Iwr11</span><span class="p">,</span> <span class="n">Iwr22</span><span class="p">,</span> <span class="n">Iwr11</span><span class="p">),</span> <span class="n">WR_mc</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">WF_I</span> <span class="o">=</span> <span class="p">(</span><span class="n">inertia</span><span class="p">(</span><span class="n">Fork</span><span class="p">,</span> <span class="n">Iwf11</span><span class="p">,</span> <span class="n">Iwf22</span><span class="p">,</span> <span class="n">Iwf11</span><span class="p">),</span> <span class="n">WF_mc</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Declaration of the RigidBody containers.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">BodyFrame</span> <span class="o">=</span> <span class="n">RigidBody</span><span class="p">(</span><span class="s">&#39;BodyFrame&#39;</span><span class="p">,</span> <span class="n">Frame_mc</span><span class="p">,</span> <span class="n">Frame</span><span class="p">,</span> <span class="n">mframe</span><span class="p">,</span> <span class="n">Frame_I</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">BodyFork</span> <span class="o">=</span> <span class="n">RigidBody</span><span class="p">(</span><span class="s">&#39;BodyFork&#39;</span><span class="p">,</span> <span class="n">Fork_mc</span><span class="p">,</span> <span class="n">Fork</span><span class="p">,</span> <span class="n">mfork</span><span class="p">,</span> <span class="n">Fork_I</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">BodyWR</span> <span class="o">=</span> <span class="n">RigidBody</span><span class="p">(</span><span class="s">&#39;BodyWR&#39;</span><span class="p">,</span> <span class="n">WR_mc</span><span class="p">,</span> <span class="n">WR</span><span class="p">,</span> <span class="n">mwr</span><span class="p">,</span> <span class="n">WR_I</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">BodyWF</span> <span class="o">=</span> <span class="n">RigidBody</span><span class="p">(</span><span class="s">&#39;BodyWF&#39;</span><span class="p">,</span> <span class="n">WF_mc</span><span class="p">,</span> <span class="n">WF</span><span class="p">,</span> <span class="n">mwf</span><span class="p">,</span> <span class="n">WF_I</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;Before Forming the List of Nonholonomic Constraints.&#39;</span><span class="p">)</span>
-<span class="go">Before Forming the List of Nonholonomic Constraints.</span>
-</pre></div>
-</div>
-<p>The kinematic differential equations; they are defined quite simply. Each entry
-in this list is equal to zero.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">kd</span> <span class="o">=</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="n">u1</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">u2</span><span class="p">,</span> <span class="n">q4d</span> <span class="o">-</span> <span class="n">u4</span><span class="p">,</span> <span class="n">q5d</span> <span class="o">-</span> <span class="n">u5</span><span class="p">]</span>
-</pre></div>
-</div>
-<p>The nonholonomic constraints are the velocity of the front wheel contact point
-dotted into the X, Y, and Z directions; the yaw frame is used as it is &#8220;closer&#8221;
-to the front wheel (1 less DCM connecting them). These constraints force the
-velocity of the front wheel contact point to be 0 in the inertial frame; the X
-and Y direction constraints enforce a &#8220;no-slip&#8221; condition, and the Z direction
-constraint forces the front wheel contact point to not move away from the
-ground frame, essentially replicating the holonomic constraint which does not
-allow the frame pitch to change in an invalid fashion.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">conlist_speed</span> <span class="o">=</span> <span class="p">[</span><span class="n">WF_cont</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">N</span><span class="p">)</span> <span class="o">&amp;</span> <span class="n">Y</span><span class="o">.</span><span class="n">x</span><span class="p">,</span>
-<span class="gp">... </span>                 <span class="n">WF_cont</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">N</span><span class="p">)</span> <span class="o">&amp;</span> <span class="n">Y</span><span class="o">.</span><span class="n">y</span><span class="p">,</span>
-<span class="gp">... </span>                 <span class="n">WF_cont</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">N</span><span class="p">)</span> <span class="o">&amp;</span> <span class="n">Y</span><span class="o">.</span><span class="n">z</span><span class="p">]</span>
-</pre></div>
-</div>
-<p>The holonomic constraint is that the position from the rear wheel contact point
-to the front wheel contact point when dotted into the normal-to-ground plane
-direction must be zero; effectively that the front and rear wheel contact
-points are always touching the ground plane. This is actually not part of the
-dynamic equations, but instead is necessary for the linearization process.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">conlist_coord</span> <span class="o">=</span> <span class="p">[</span><span class="n">WF_cont</span><span class="o">.</span><span class="n">pos_from</span><span class="p">(</span><span class="n">WR_cont</span><span class="p">)</span> <span class="o">&amp;</span> <span class="n">Y</span><span class="o">.</span><span class="n">z</span><span class="p">]</span>
-</pre></div>
-</div>
-<p>The force list; each body has the appropriate gravitational force applied at
-its center of mass.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">FL</span> <span class="o">=</span> <span class="p">[(</span><span class="n">Frame_mc</span><span class="p">,</span> <span class="o">-</span><span class="n">mframe</span> <span class="o">*</span> <span class="n">g</span> <span class="o">*</span> <span class="n">Y</span><span class="o">.</span><span class="n">z</span><span class="p">),</span> <span class="p">(</span><span class="n">Fork_mc</span><span class="p">,</span> <span class="o">-</span><span class="n">mfork</span> <span class="o">*</span> <span class="n">g</span> <span class="o">*</span> <span class="n">Y</span><span class="o">.</span><span class="n">z</span><span class="p">),</span>
-<span class="gp">... </span>      <span class="p">(</span><span class="n">WF_mc</span><span class="p">,</span> <span class="o">-</span><span class="n">mwf</span> <span class="o">*</span> <span class="n">g</span> <span class="o">*</span> <span class="n">Y</span><span class="o">.</span><span class="n">z</span><span class="p">),</span> <span class="p">(</span><span class="n">WR_mc</span><span class="p">,</span> <span class="o">-</span><span class="n">mwr</span> <span class="o">*</span> <span class="n">g</span> <span class="o">*</span> <span class="n">Y</span><span class="o">.</span><span class="n">z</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">BL</span> <span class="o">=</span> <span class="p">[</span><span class="n">BodyFrame</span><span class="p">,</span> <span class="n">BodyFork</span><span class="p">,</span> <span class="n">BodyWR</span><span class="p">,</span> <span class="n">BodyWF</span><span class="p">]</span>
-</pre></div>
-</div>
-<p>The N frame is the inertial frame, coordinates are supplied in the order of
-independent, dependent coordinates. The kinematic differential equations are
-also entered here. Here the independent speeds are specified, followed by the
-dependent speeds, along with the non-holonomic constraints. The dependent
-coordinate is also provided, with the holonomic constraint. Again, this is only
-comes into play in the linearization process, but is necessary for the
-linearization to correctly work.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span> <span class="o">=</span> <span class="n">KanesMethod</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">q_ind</span><span class="o">=</span><span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span><span class="p">],</span>
-<span class="gp">... </span>          <span class="n">q_dependent</span><span class="o">=</span><span class="p">[</span><span class="n">q4</span><span class="p">],</span> <span class="n">configuration_constraints</span><span class="o">=</span><span class="n">conlist_coord</span><span class="p">,</span>
-<span class="gp">... </span>          <span class="n">u_ind</span><span class="o">=</span><span class="p">[</span><span class="n">u2</span><span class="p">,</span> <span class="n">u3</span><span class="p">,</span> <span class="n">u5</span><span class="p">],</span>
-<span class="gp">... </span>          <span class="n">u_dependent</span><span class="o">=</span><span class="p">[</span><span class="n">u1</span><span class="p">,</span> <span class="n">u4</span><span class="p">,</span> <span class="n">u6</span><span class="p">],</span> <span class="n">velocity_constraints</span><span class="o">=</span><span class="n">conlist_speed</span><span class="p">,</span>
-<span class="gp">... </span>          <span class="n">kd_eqs</span><span class="o">=</span><span class="n">kd</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;Before Forming Generalized Active and Inertia Forces, Fr and Fr*&#39;</span><span class="p">)</span>
-<span class="go">Before Forming Generalized Active and Inertia Forces, Fr and Fr*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">fr</span><span class="p">,</span> <span class="n">frstar</span><span class="p">)</span> <span class="o">=</span> <span class="n">KM</span><span class="o">.</span><span class="n">kanes_equations</span><span class="p">(</span><span class="n">FL</span><span class="p">,</span> <span class="n">BL</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;Base Equations of Motion Computed&#39;</span><span class="p">)</span>
-<span class="go">Base Equations of Motion Computed</span>
-</pre></div>
-</div>
-<p>This is the start of entering in the numerical values from the benchmark paper
-to validate the eigenvalues of the linearized equations from this model to the
-reference eigenvalues. Look at the aforementioned paper for more information.
-Some of these are intermediate values, used to transform values from the paper
-into the coordinate systems used in this model.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">PaperRadRear</span>  <span class="o">=</span>  <span class="mf">0.3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">PaperRadFront</span> <span class="o">=</span>  <span class="mf">0.35</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">HTA</span>           <span class="o">=</span>  <span class="n">evalf</span><span class="o">.</span><span class="n">N</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="o">-</span><span class="n">pi</span><span class="o">/</span><span class="mi">10</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">TrailPaper</span>    <span class="o">=</span>  <span class="mf">0.08</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rake</span>          <span class="o">=</span>  <span class="n">evalf</span><span class="o">.</span><span class="n">N</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">TrailPaper</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">HTA</span><span class="p">)</span><span class="o">-</span><span class="p">(</span><span class="n">PaperRadFront</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">HTA</span><span class="p">))))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">PaperWb</span>       <span class="o">=</span>  <span class="mf">1.02</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">PaperFrameCgX</span> <span class="o">=</span>  <span class="mf">0.3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">PaperFrameCgZ</span> <span class="o">=</span>  <span class="mf">0.9</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">PaperForkCgX</span>  <span class="o">=</span>  <span class="mf">0.9</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">PaperForkCgZ</span>  <span class="o">=</span>  <span class="mf">0.7</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">FrameLength</span>   <span class="o">=</span>  <span class="n">evalf</span><span class="o">.</span><span class="n">N</span><span class="p">(</span><span class="n">PaperWb</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">HTA</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="n">rake</span> <span class="o">-</span> \
-<span class="gp">... </span>                        <span class="p">(</span><span class="n">PaperRadFront</span> <span class="o">-</span> <span class="n">PaperRadRear</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">HTA</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">FrameCGNorm</span>   <span class="o">=</span>  <span class="n">evalf</span><span class="o">.</span><span class="n">N</span><span class="p">((</span><span class="n">PaperFrameCgZ</span> <span class="o">-</span> <span class="n">PaperRadRear</span> <span class="o">-</span> \
-<span class="gp">... </span>                         <span class="p">(</span><span class="n">PaperFrameCgX</span><span class="o">/</span><span class="n">sin</span><span class="p">(</span><span class="n">HTA</span><span class="p">))</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">HTA</span><span class="p">))</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">HTA</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">FrameCGPar</span>    <span class="o">=</span>  <span class="n">evalf</span><span class="o">.</span><span class="n">N</span><span class="p">((</span><span class="n">PaperFrameCgX</span> <span class="o">/</span> <span class="n">sin</span><span class="p">(</span><span class="n">HTA</span><span class="p">)</span> <span class="o">+</span> \
-<span class="gp">... </span>                         <span class="p">(</span><span class="n">PaperFrameCgZ</span> <span class="o">-</span> <span class="n">PaperRadRear</span> <span class="o">-</span> \
-<span class="gp">... </span>                          <span class="n">PaperFrameCgX</span> <span class="o">/</span> <span class="n">sin</span><span class="p">(</span><span class="n">HTA</span><span class="p">)</span> <span class="o">*</span> <span class="n">cos</span><span class="p">(</span><span class="n">HTA</span><span class="p">))</span> <span class="o">*</span> <span class="n">cos</span><span class="p">(</span><span class="n">HTA</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tempa</span>         <span class="o">=</span>  <span class="n">evalf</span><span class="o">.</span><span class="n">N</span><span class="p">((</span><span class="n">PaperForkCgZ</span> <span class="o">-</span> <span class="n">PaperRadFront</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tempb</span>         <span class="o">=</span>  <span class="n">evalf</span><span class="o">.</span><span class="n">N</span><span class="p">((</span><span class="n">PaperWb</span><span class="o">-</span><span class="n">PaperForkCgX</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tempc</span>         <span class="o">=</span>  <span class="n">evalf</span><span class="o">.</span><span class="n">N</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">tempa</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">tempb</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">PaperForkL</span>    <span class="o">=</span>  <span class="n">evalf</span><span class="o">.</span><span class="n">N</span><span class="p">((</span><span class="n">PaperWb</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">HTA</span><span class="p">)</span> <span class="o">-</span> \
-<span class="gp">... </span>                         <span class="p">(</span><span class="n">PaperRadFront</span> <span class="o">-</span> <span class="n">PaperRadRear</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">HTA</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ForkCGNorm</span>    <span class="o">=</span>  <span class="n">evalf</span><span class="o">.</span><span class="n">N</span><span class="p">(</span><span class="n">rake</span> <span class="o">+</span> <span class="p">(</span><span class="n">tempc</span> <span class="o">*</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> \
-<span class="gp">... </span>                         <span class="n">HTA</span> <span class="o">-</span> <span class="n">acos</span><span class="p">(</span><span class="n">tempa</span><span class="o">/</span><span class="n">tempc</span><span class="p">))))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ForkCGPar</span>     <span class="o">=</span>  <span class="n">evalf</span><span class="o">.</span><span class="n">N</span><span class="p">(</span><span class="n">tempc</span> <span class="o">*</span> <span class="n">cos</span><span class="p">((</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span> <span class="o">-</span> <span class="n">HTA</span><span class="p">)</span> <span class="o">-</span> \
-<span class="gp">... </span>                         <span class="n">acos</span><span class="p">(</span><span class="n">tempa</span><span class="o">/</span><span class="n">tempc</span><span class="p">))</span> <span class="o">-</span> <span class="n">PaperForkL</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Here is the final assembly of the numerical values. The symbol &#8216;v&#8217; is the
-forward speed of the bicycle (a concept which only makes sense in the upright,
-static equilibrium case?). These are in a dictionary which will later be
-substituted in. Again the sign on the <em>product</em> of inertia values is flipped
-here, due to different orientations of coordinate systems.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">v</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;v&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">val_dict</span> <span class="o">=</span> <span class="p">{</span>
-<span class="gp">... </span>      <span class="n">WFrad</span><span class="p">:</span> <span class="n">PaperRadFront</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">WRrad</span><span class="p">:</span> <span class="n">PaperRadRear</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">htangle</span><span class="p">:</span> <span class="n">HTA</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">forkoffset</span><span class="p">:</span> <span class="n">rake</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">forklength</span><span class="p">:</span> <span class="n">PaperForkL</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">framelength</span><span class="p">:</span> <span class="n">FrameLength</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">forkcg1</span><span class="p">:</span> <span class="n">ForkCGPar</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">forkcg3</span><span class="p">:</span> <span class="n">ForkCGNorm</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">framecg1</span><span class="p">:</span> <span class="n">FrameCGNorm</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">framecg3</span><span class="p">:</span> <span class="n">FrameCGPar</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">Iwr11</span><span class="p">:</span> <span class="mf">0.0603</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">Iwr22</span><span class="p">:</span> <span class="mf">0.12</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">Iwf11</span><span class="p">:</span> <span class="mf">0.1405</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">Iwf22</span><span class="p">:</span> <span class="mf">0.28</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">Ifork11</span><span class="p">:</span> <span class="mf">0.05892</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">Ifork22</span><span class="p">:</span> <span class="mf">0.06</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">Ifork33</span><span class="p">:</span> <span class="mf">0.00708</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">Ifork31</span><span class="p">:</span> <span class="mf">0.00756</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">Iframe11</span><span class="p">:</span> <span class="mf">9.2</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">Iframe22</span><span class="p">:</span> <span class="mi">11</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">Iframe33</span><span class="p">:</span> <span class="mf">2.8</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">Iframe31</span><span class="p">:</span> <span class="o">-</span><span class="mf">2.4</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">mfork</span><span class="p">:</span> <span class="mi">4</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">mframe</span><span class="p">:</span> <span class="mi">85</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">mwf</span><span class="p">:</span> <span class="mi">3</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">mwr</span><span class="p">:</span> <span class="mi">2</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">g</span><span class="p">:</span> <span class="mf">9.81</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">q1</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">q2</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">q4</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">q5</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">u1</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">u2</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">u3</span><span class="p">:</span> <span class="n">v</span><span class="o">/</span><span class="n">PaperRadRear</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">u4</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">u5</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span>
-<span class="gp">... </span>      <span class="n">u6</span><span class="p">:</span> <span class="n">v</span><span class="o">/</span><span class="n">PaperRadFront</span><span class="p">}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">kdd</span> <span class="o">=</span> <span class="n">KM</span><span class="o">.</span><span class="n">kindiffdict</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;Before Linearization of the </span><span class="se">\&quot;</span><span class="s">Forcing</span><span class="se">\&quot;</span><span class="s"> Term&#39;</span><span class="p">)</span>
-<span class="go">Before Linearization of the &quot;Forcing&quot; Term</span>
-</pre></div>
-</div>
-<p>Linearizes the forcing vector; the equations are set up as MM udot = forcing,
-where MM is the mass matrix, udot is the vector representing the time
-derivatives of the generalized speeds, and forcing is a vector which contains
-both external forcing terms and internal forcing terms, such as centripetal or
-Coriolis forces.  This actually returns a matrix with as many rows as <em>total</em>
-coordinates and speeds, but only as many columns as independent coordinates and
-speeds. (Note that below this is commented out, as it takes a few minutes to
-run, which is not good when performing the doctests)</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="c"># forcing_lin = KM.linearize()[0].subs(sub_dict)</span>
-</pre></div>
-</div>
-<p>As mentioned above, the size of the linearized forcing terms is expanded to
-include both q&#8217;s and u&#8217;s, so the mass matrix must have this done as well.  This
-will likely be changed to be part of the linearized process, for future
-reference.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">MM_full</span> <span class="o">=</span> <span class="p">(</span><span class="n">KM</span><span class="o">.</span><span class="n">_k_kqdot</span><span class="p">)</span><span class="o">.</span><span class="n">row_join</span><span class="p">(</span><span class="n">zeros</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span><span class="o">.</span><span class="n">col_join</span><span class="p">(</span>
-<span class="gp">... </span>          <span class="p">(</span><span class="n">zeros</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span><span class="o">.</span><span class="n">row_join</span><span class="p">(</span><span class="n">KM</span><span class="o">.</span><span class="n">mass_matrix</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="s">&#39;Before Substitution of Numerical Values&#39;</span><span class="p">)</span>
-<span class="go">Before Substitution of Numerical Values</span>
-</pre></div>
-</div>
-<p>I think this is pretty self explanatory. It takes a really long time though.
-I&#8217;ve experimented with using evalf with substitution, this failed due to
-maximum recursion depth being exceeded; I also tried lambdifying this, and it
-is also not successful. (again commented out due to speed)</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="c"># MM_full = MM_full.subs(val_dict)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># forcing_lin = forcing_lin.subs(val_dict)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># print(&#39;Before .evalf() call&#39;)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># MM_full = MM_full.evalf()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># forcing_lin = forcing_lin.evalf()</span>
-</pre></div>
-</div>
-<p>Finally, we construct an &#8220;A&#8221; matrix for the form xdot = A x (x being the state
-vector, although in this case, the sizes are a little off). The following line
-extracts only the minimum entries required for eigenvalue analysis, which
-correspond to rows and columns for lean, steer, lean rate, and steer rate.
-(this is all commented out due to being dependent on the above code, which is
-also commented out):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="c"># Amat = MM_full.inv() * forcing_lin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># A = Amat.extract([1,2,4,6],[1,2,3,5])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># print(A)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># print(&#39;v = 1&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># print(A.subs(v, 1).eigenvals())</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># print(&#39;v = 2&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># print(A.subs(v, 2).eigenvals())</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># print(&#39;v = 3&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># print(A.subs(v, 3).eigenvals())</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># print(&#39;v = 4&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># print(A.subs(v, 4).eigenvals())</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># print(&#39;v = 5&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># print(A.subs(v, 5).eigenvals())</span>
-</pre></div>
-</div>
-<p>Upon running the above code yourself, enabling the commented out lines, compare
-the computed eigenvalues to those is the referenced paper. This concludes the
-bicycle example.</p>
-</div>
-<div class="section" id="the-rolling-disc-example-using-lagranges-method">
-<h2>The Rolling Disc Example using Lagranges Method<a class="headerlink" href="#the-rolling-disc-example-using-lagranges-method" title="Permalink to this headline">¶</a></h2>
-<p>Here the rolling disc is formed from the contact point up, removing the
-need to introduce generalized speeds. Only 3 configuration and 3
-speed variables are needed to describe this system, along with the
-disc&#8217;s mass and radius, and the local gravity.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mechanics_printing</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1 q2 q3&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1d</span><span class="p">,</span> <span class="n">q2d</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1 q2 q3&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;r m g&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>The kinematics are formed by a series of simple rotations. Each simple
-rotation creates a new frame, and the next rotation is defined by the new
-frame&#8217;s basis vectors. This example uses a 3-1-2 series of rotations, or
-Z, X, Y series of rotations. Angular velocity for this is defined using
-the second frame&#8217;s basis (the lean frame).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;Y&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span> <span class="o">=</span> <span class="n">Y</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;L&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q2</span><span class="p">,</span> <span class="n">Y</span><span class="o">.</span><span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">L</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;R&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q3</span><span class="p">,</span> <span class="n">L</span><span class="o">.</span><span class="n">y</span><span class="p">])</span>
-</pre></div>
-</div>
-<p>This is the translational kinematics. We create a point with no velocity
-in N; this is the contact point between the disc and ground. Next we form
-the position vector from the contact point to the disc&#8217;s center of mass.
-Finally we form the velocity and acceleration of the disc.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;C&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dmc</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;Dmc&#39;</span><span class="p">,</span> <span class="n">r</span> <span class="o">*</span> <span class="n">L</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dmc</span><span class="o">.</span><span class="n">v2pt_theory</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span>
-<span class="go">r*(sin(q2)*q1&#39; + q3&#39;)*L.x - r*q2&#39;*L.y</span>
-</pre></div>
-</div>
-<p>Forming the inertia dyadic.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">inertia</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="n">m</span> <span class="o">/</span> <span class="mi">4</span> <span class="o">*</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">m</span> <span class="o">/</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">m</span> <span class="o">/</span> <span class="mi">4</span> <span class="o">*</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mprint</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
-<span class="go">m*r**2/4*(L.x|L.x) + m*r**2/2*(L.y|L.y) + m*r**2/4*(L.z|L.z)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">BodyD</span> <span class="o">=</span> <span class="n">RigidBody</span><span class="p">(</span><span class="s">&#39;BodyD&#39;</span><span class="p">,</span> <span class="n">Dmc</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">Dmc</span><span class="p">))</span>
-</pre></div>
-</div>
-<p>We then set the potential energy and determine the Lagrangian of the rolling
-disc.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">BodyD</span><span class="o">.</span><span class="n">set_potential_energy</span><span class="p">(</span><span class="o">-</span> <span class="n">m</span> <span class="o">*</span> <span class="n">g</span> <span class="o">*</span> <span class="n">r</span> <span class="o">*</span> <span class="n">cos</span><span class="p">(</span><span class="n">q2</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Lag</span> <span class="o">=</span> <span class="n">Lagrangian</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">BodyD</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Then the equations of motion are generated by initializing the
-<tt class="docutils literal"><span class="pre">LagrangesMethod</span></tt> object. Finally we solve for the generalized
-accelerations(q double dots) with the <tt class="docutils literal"><span class="pre">rhs</span></tt> method.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l</span> <span class="o">=</span> <span class="n">LagrangesMethod</span><span class="p">(</span><span class="n">Lag</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l</span><span class="o">.</span><span class="n">form_lagranges_equations</span><span class="p">()</span>
-<span class="go">[5*m*r**2*(sin(q2)*q1&#39; + q3&#39;)*cos(q2)*q2&#39;/4 + 5*m*r**2*(sin(q2)*q1&#39;&#39; + cos(q2)*q1&#39;*q2&#39; + q3&#39;&#39;)*sin(q2)/4 + m*r**2*sin(q2)*q3&#39;&#39;/4 + m*r**2*cos(q2)**2*q1&#39;&#39;/8 + m*r**2*cos(q2)*q2&#39;*q3&#39;/4 + (m*r**2*sin(q2)**2/4 + m*r**2*cos(q2)**2/8)*q1&#39;&#39;]</span>
-<span class="go">[                                                                                                                                 g*m*r*sin(q2) - 5*m*r**2*(sin(q2)*q1&#39; + q3&#39;)*cos(q2)*q1&#39;/4 - m*r**2*cos(q2)*q1&#39;*q3&#39;/4 + 5*m*r**2*q2&#39;&#39;/4]</span>
-<span class="go">[                                                            m*r**2*(sin(q2)*q1&#39;&#39; + cos(q2)*q1&#39;*q2&#39; + q3&#39;&#39;)/4 + m*r**2*(2*sin(q2)*q1&#39;&#39; + 2*cos(q2)*q1&#39;*q2&#39; + 2*q3&#39;&#39;)/2 + m*r**2*sin(q2)*q1&#39;&#39;/4 + m*r**2*cos(q2)*q1&#39;*q2&#39;/4 + m*r**2*q3&#39;&#39;/4]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l</span><span class="o">.</span><span class="n">rhs</span><span class="p">()</span>
-<span class="go">[                                                                                                                                                                                                                             q1&#39;]</span>
-<span class="go">[                                                                                                                                                                                                                             q2&#39;]</span>
-<span class="go">[                                                                                                                                                                                                                             q3&#39;]</span>
-<span class="go">[(24*sin(q2)**2/(m*r**2*(5*sin(q2)**2 + 1)*cos(q2)**2) + 4/(m*r**2*(5*sin(q2)**2 + 1)))*(-5*m*r**2*(sin(q2)*q1&#39; + q3&#39;)*cos(q2)*q2&#39;/4 - 5*m*r**2*sin(q2)*cos(q2)*q1&#39;*q2&#39;/4 - m*r**2*cos(q2)*q2&#39;*q3&#39;/4) + 6*sin(q2)*q1&#39;*q2&#39;/cos(q2)]</span>
-<span class="go">[                                                                                                                           4*(-g*m*r*sin(q2) + 5*m*r**2*(sin(q2)*q1&#39; + q3&#39;)*cos(q2)*q1&#39;/4 + m*r**2*cos(q2)*q1&#39;*q3&#39;/4)/(5*m*r**2)]</span>
-<span class="go">[                                               -(5*sin(q2)**2 + 1)*q1&#39;*q2&#39;/cos(q2) - 4*(-5*m*r**2*(sin(q2)*q1&#39; + q3&#39;)*cos(q2)*q2&#39;/4 - 5*m*r**2*sin(q2)*cos(q2)*q1&#39;*q2&#39;/4 - m*r**2*cos(q2)*q2&#39;*q3&#39;/4)*sin(q2)/(m*r**2*cos(q2)**2)]</span>
-</pre></div>
-</div>
-</div>
-</div>
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-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Examples for Physics.Mechanics</a><ul>
-<li><a class="reference internal" href="#the-rolling-disc">The Rolling Disc</a></li>
-<li><a class="reference internal" href="#the-rolling-disc-again">The Rolling Disc, again</a></li>
-<li><a class="reference internal" href="#the-bicycle">The Bicycle</a></li>
-<li><a class="reference internal" href="#the-rolling-disc-example-using-lagranges-method">The Rolling Disc Example using Lagranges Method</a></li>
-</ul>
-</li>
-</ul>
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-            
-  <div class="section" id="classical-mechanics">
-<h1>Classical Mechanics<a class="headerlink" href="#classical-mechanics" title="Permalink to this headline">¶</a></h1>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Authors:</th><td class="field-body">Gilbert Gede, Luke Peterson, Angadh Nanjangud</td>
-</tr>
-</tbody>
-</table>
-<div class="topic">
-<p class="topic-title first">Abstract</p>
-<p>In this documentation many components of the physics/mechanics module will
-be discussed. <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt> has been written to allow for creation of
-symbolic equations of motion for complicated multibody systems.</p>
-</div>
-<div class="section" id="mechanics">
-<h2>Mechanics<a class="headerlink" href="#mechanics" title="Permalink to this headline">¶</a></h2>
-<p>In physics, mechanics describes conditions of rest or motion; statics or
-dynamics. First, an idealized representation of a system is described. Next, we
-use physical laws to generate equations that define the system&#8217;s behaviors.
-Then, we solve these equations, in multiple possible ways. Finally, we extract
-information from these equations and solutions. Mechanics is currently set up
-to create equations of motion, for dynamics. It is also limited to rigid bodies
-and particles.</p>
-</div>
-<div class="section" id="guide-to-mechanics">
-<h2>Guide to Mechanics<a class="headerlink" href="#guide-to-mechanics" title="Permalink to this headline">¶</a></h2>
-<div class="toctree-wrapper compound">
-<ul>
-<li class="toctree-l1"><a class="reference internal" href="../mechanics/vectors.html">Mechanics: Vector &amp; ReferenceFrame</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/vectors.html#vector">Vector</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/vectors.html#vector-algebra">Vector Algebra</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/vectors.html#vector-calculus">Vector Calculus</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/vectors.html#using-vectors-and-reference-frames-in-sympy-s-mechanics">Using Vectors and Reference Frames in SymPy&#8217;s Mechanics</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../mechanics/kinematics.html">Mechanics: Kinematics</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/kinematics.html#introduction-to-kinematics">Introduction to Kinematics</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/kinematics.html#kinematics-in-sympy-s-mechanics">Kinematics in SymPy&#8217;s Mechanics</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../mechanics/masses.html">Mechanics: Masses, Inertias, Particles and Rigid Bodies</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/masses.html#mass">Mass</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/masses.html#particle">Particle</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/masses.html#inertia">Inertia</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/masses.html#rigid-body">Rigid Body</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/masses.html#dyadic">Dyadic</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/masses.html#linear-momentum">Linear Momentum</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/masses.html#angular-momentum">Angular Momentum</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/masses.html#using-momenta-functions-in-mechanics">Using momenta functions in Mechanics</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/masses.html#kinetic-energy">Kinetic Energy</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/masses.html#potential-energy">Potential Energy</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/masses.html#lagrangian">Lagrangian</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/masses.html#using-energy-functions-in-mechanics">Using energy functions in Mechanics</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../mechanics/kane.html">Kane&#8217;s Method in Physics/Mechanics</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/kane.html#structure-of-equations">Structure of Equations</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/kane.html#id2">Kane&#8217;s Method in Physics/Mechanics</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../mechanics/kane.html#lagrange-s-method-in-physics-mechanics">Lagrange&#8217;s Method in Physics/Mechanics</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/kane.html#id3">Structure of Equations</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/kane.html#id4">Lagrange&#8217;s Method in Physics/Mechanics</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../mechanics/examples.html">Examples for Physics.Mechanics</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/examples.html#the-rolling-disc">The Rolling Disc</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/examples.html#the-rolling-disc-again">The Rolling Disc, again</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/examples.html#the-bicycle">The Bicycle</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/examples.html#the-rolling-disc-example-using-lagranges-method">The Rolling Disc Example using Lagranges Method</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../mechanics/advanced.html">Potential Issues/Advanced Topics/Future Features in Physics/Mechanics Module</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/advanced.html#common-issues">Common Issues</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/advanced.html#advanced-interfaces">Advanced Interfaces</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/advanced.html#future-features">Future Features</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../mechanics/reference.html">References for Physics/Mechanics</a></li>
-</ul>
-</div>
-</div>
-<div class="section" id="mechanics-api">
-<h2>Mechanics API<a class="headerlink" href="#mechanics-api" title="Permalink to this headline">¶</a></h2>
-<div class="toctree-wrapper compound">
-<ul>
-<li class="toctree-l1"><a class="reference internal" href="../mechanics/api/essential.html">Essential Components (Docstrings)</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/api/essential.html#referenceframe">ReferenceFrame</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/api/essential.html#vector">Vector</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/api/essential.html#dyadic">Dyadic</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../mechanics/api/functions.html">Related Essential Functions (Docstrings)</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/api/functions.html#dynamicsymbols">dynamicsymbols</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/api/functions.html#dot">dot</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/api/functions.html#cross">cross</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/api/functions.html#outer">outer</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/api/functions.html#express">express</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../mechanics/api/kinematics.html">Kinematics (Docstrings)</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/api/kinematics.html#module-sympy.physics.mechanics.point">Point</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/api/kinematics.html#module-sympy.physics.mechanics.functions">kinematic_equations</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../mechanics/api/part_bod.html">Masses, Inertias &amp; Particles, RigidBodys (Docstrings)</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/api/part_bod.html#module-sympy.physics.mechanics.particle">Particle</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/api/part_bod.html#module-sympy.physics.mechanics.rigidbody">RigidBody</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/api/part_bod.html#inertia">inertia</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../mechanics/api/part_bod.html#inertia-of-point-mass">inertia_of_point_mass</a></li>
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-<li class="toctree-l1"><a class="reference internal" href="../mechanics/api/kane.html">Kane&#8217;s Method, Lagrange&#8217;s Method &amp; Associated Functions (Docstrings)</a><ul>
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-</ul>
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-  <div class="section" id="kane-s-method-in-physics-mechanics">
-<h1>Kane&#8217;s Method in Physics/Mechanics<a class="headerlink" href="#kane-s-method-in-physics-mechanics" title="Permalink to this headline">¶</a></h1>
-<p><tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt> has been written for use with Kane&#8217;s method of forming
-equations of motion <a class="reference internal" href="../mechanics/reference.html#kane1985">[Kane1985]</a>. This document will describe Kane&#8217;s Method
-as used in this module, but not how the equations are actually derived.</p>
-<div class="section" id="structure-of-equations">
-<h2>Structure of Equations<a class="headerlink" href="#structure-of-equations" title="Permalink to this headline">¶</a></h2>
-<p>In <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt> we are assuming there are 5 basic sets of equations needed
-to describe a system. They are: holonomic constraints, non-holonomic
-constraints, kinematic differential equations, dynamic equations, and
-differentiated non-holonomic equations.</p>
-<div class="math">
-\[\begin{split}\mathbf{f_h}(q, t) &amp;= 0\\
-\mathbf{k_{nh}}(q, t) u + \mathbf{f_{nh}}(q, t) &amp;= 0\\
-\mathbf{k_{k\dot{q}}}(q, t) \dot{q} + \mathbf{k_{ku}}(q, t) u +
-\mathbf{f_k}(q, t) &amp;= 0\\
-\mathbf{k_d}(q, t) \dot{u} + \mathbf{f_d}(q, \dot{q}, u, t) &amp;= 0\\
-\mathbf{k_{dnh}}(q, t) \dot{u} + \mathbf{f_{dnh}}(q, \dot{q}, u, t) &amp;= 0\\\end{split}\]</div>
-<p>In <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt> holonomic constraints are only used for the linearization
-process; it is assumed that they will be too complicated to solve for the
-dependent coordinate(s).  If you are able to easily solve a holonomic
-constraint, you should consider redefining your problem in terms of a smaller
-set of coordinates. Alternatively, the time-differentiated holonomic
-constraints can be supplied.</p>
-<p>Kane&#8217;s method forms two expressions, <span class="math">\(F_r\)</span> and <span class="math">\(F_r^*\)</span>, whose sum
-is zero. In this module, these expressions are rearranged into the following
-form:</p>
-<blockquote>
-<div><span class="math">\(\mathbf{M}(q, t) \dot{u} = \mathbf{f}(q, \dot{q}, u, t)\)</span></div></blockquote>
-<p>For a non-holonomic system with <span class="math">\(o\)</span> total speeds and <span class="math">\(m\)</span> motion constraints, we
-will get o - m equations. The mass-matrix/forcing equations are then augmented
-in the following fashion:</p>
-<div class="math">
-\[\begin{split}\mathbf{M}(q, t) &amp;= \begin{bmatrix} \mathbf{k_d}(q, t) \\
-\mathbf{k_{dnh}}(q, t) \end{bmatrix}\\
-\mathbf{_{(forcing)}}(q, \dot{q}, u, t) &amp;= \begin{bmatrix}
-- \mathbf{f_d}(q, \dot{q}, u, t) \\ - \mathbf{f_{dnh}}(q, \dot{q}, u, t)
-\end{bmatrix}\\\end{split}\]</div>
-</div>
-<div class="section" id="id2">
-<h2>Kane&#8217;s Method in Physics/Mechanics<a class="headerlink" href="#id2" title="Permalink to this headline">¶</a></h2>
-<p>The formulation of the equations of motion in <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt> starts with
-creation of a <tt class="docutils literal"><span class="pre">KanesMethod</span></tt> object. Upon initialization of the
-<tt class="docutils literal"><span class="pre">KanesMethod</span></tt> object, an inertial reference frame needs to be supplied. along
-with some basic system information, suchs as coordinates and speeds</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">u1</span><span class="p">,</span> <span class="n">u2</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1 q2 u1 u2&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1d</span><span class="p">,</span> <span class="n">q2d</span><span class="p">,</span> <span class="n">u1d</span><span class="p">,</span> <span class="n">u2d</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1 q2 u1 u2&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span> <span class="o">=</span> <span class="n">KanesMethod</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">],</span> <span class="p">[</span><span class="n">u1</span><span class="p">,</span> <span class="n">u2</span><span class="p">])</span>
-</pre></div>
-</div>
-<p>It is also important to supply the order of coordinates and speeds properly if
-there are dependent coordinates and speeds. They must be supplied after
-independent coordinates and speeds or as a keyword argument; this is shown
-later.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span><span class="p">,</span> <span class="n">q4</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1 q2 q3 q4&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u1</span><span class="p">,</span> <span class="n">u2</span><span class="p">,</span> <span class="n">u3</span><span class="p">,</span> <span class="n">u4</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;u1 u2 u3 u4&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># Here we will assume q2 is dependent, and u2 and u3 are dependent</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># We need the constraint equations to enter them though</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span> <span class="o">=</span> <span class="n">KanesMethod</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">q3</span><span class="p">,</span> <span class="n">q4</span><span class="p">],</span> <span class="p">[</span><span class="n">u1</span><span class="p">,</span> <span class="n">u4</span><span class="p">])</span>
-</pre></div>
-</div>
-<p>Additionally, if there are auxiliary speeds, they need to be identified here.
-See the examples for more information on this. In this example u4 is the
-auxiliary speed.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span> <span class="o">=</span> <span class="n">KanesMethod</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">q3</span><span class="p">,</span> <span class="n">q4</span><span class="p">],</span> <span class="p">[</span><span class="n">u1</span><span class="p">,</span> <span class="n">u2</span><span class="p">,</span> <span class="n">u3</span><span class="p">],</span> <span class="n">u_auxiliary</span><span class="o">=</span><span class="p">[</span><span class="n">u4</span><span class="p">])</span>
-</pre></div>
-</div>
-<p>Kinematic differential equations must also be supplied; there are to be
-provided as a list of expressions which are each equal to zero. A trivial
-example follows:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">kd</span> <span class="o">=</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="n">u1</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">u2</span><span class="p">]</span>
-</pre></div>
-</div>
-<p>Turning on <tt class="docutils literal"><span class="pre">mechanics_printing()</span></tt> makes the expressions significantly
-shorter and is recommended. Alternatively, the <tt class="docutils literal"><span class="pre">mprint</span></tt> and <tt class="docutils literal"><span class="pre">mpprint</span></tt>
-commands can be used.</p>
-<p>If there are non-holonomic constraints, dependent speeds need to be specified
-(and so do dependent coordinates, but they only come into play when linearizing
-the system). The constraints need to be supplied in a list of expressions which
-are equal to zero, trivial motion and configuration constraints are shown
-below:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span><span class="p">,</span> <span class="n">q4</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1 q2 q3 q4&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1d</span><span class="p">,</span> <span class="n">q2d</span><span class="p">,</span> <span class="n">q3d</span><span class="p">,</span> <span class="n">q4d</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1 q2 q3 q4&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u1</span><span class="p">,</span> <span class="n">u2</span><span class="p">,</span> <span class="n">u3</span><span class="p">,</span> <span class="n">u4</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;u1 u2 u3 u4&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c">#Here we will assume q2 is dependent, and u2 and u3 are dependent</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">speed_cons</span> <span class="o">=</span> <span class="p">[</span><span class="n">u2</span> <span class="o">-</span> <span class="n">u1</span><span class="p">,</span> <span class="n">u3</span> <span class="o">-</span> <span class="n">u1</span> <span class="o">-</span> <span class="n">u4</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">coord_cons</span> <span class="o">=</span> <span class="p">[</span><span class="n">q2</span> <span class="o">-</span> <span class="n">q1</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q_ind</span> <span class="o">=</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">q3</span><span class="p">,</span> <span class="n">q4</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q_dep</span> <span class="o">=</span> <span class="p">[</span><span class="n">q2</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u_ind</span> <span class="o">=</span> <span class="p">[</span><span class="n">u1</span><span class="p">,</span> <span class="n">u4</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u_dep</span> <span class="o">=</span> <span class="p">[</span><span class="n">u2</span><span class="p">,</span> <span class="n">u3</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">kd</span> <span class="o">=</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="n">u1</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">-</span> <span class="n">u2</span><span class="p">,</span> <span class="n">q3d</span> <span class="o">-</span> <span class="n">u3</span><span class="p">,</span> <span class="n">q4d</span> <span class="o">-</span> <span class="n">u4</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span> <span class="o">=</span> <span class="n">KanesMethod</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">q_ind</span><span class="p">,</span> <span class="n">u_ind</span><span class="p">,</span> <span class="n">kd</span><span class="p">,</span>
-<span class="gp">... </span>          <span class="n">q_dependent</span><span class="o">=</span><span class="n">q_dep</span><span class="p">,</span>
-<span class="gp">... </span>          <span class="n">configuration_constraints</span><span class="o">=</span><span class="n">coord_cons</span><span class="p">,</span>
-<span class="gp">... </span>          <span class="n">u_dependent</span><span class="o">=</span><span class="n">u_dep</span><span class="p">,</span>
-<span class="gp">... </span>          <span class="n">velocity_constraints</span><span class="o">=</span><span class="n">speed_cons</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>A dictionary returning the solved <span class="math">\(\dot{q}\)</span>&#8216;s can also be solved for:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mechanics_printing</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span><span class="o">.</span><span class="n">kindiffdict</span><span class="p">()</span>
-<span class="go">{q1&#39;: u1, q2&#39;: u2, q3&#39;: u3, q4&#39;: u4}</span>
-</pre></div>
-</div>
-<p>The final step in forming the equations of motion is supplying a list of
-bodies and particles, and a list of 2-tuples of the form <tt class="docutils literal"><span class="pre">(Point,</span> <span class="pre">Vector)</span></tt>
-or <tt class="docutils literal"><span class="pre">(ReferenceFrame,</span> <span class="pre">Vector)</span></tt> to represent applied forces and torques.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="p">,</span> <span class="n">u</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q u&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qd</span><span class="p">,</span> <span class="n">ud</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q u&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">u</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Pa</span> <span class="o">=</span> <span class="n">Particle</span><span class="p">(</span><span class="s">&#39;Pa&#39;</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">BL</span> <span class="o">=</span> <span class="p">[</span><span class="n">Pa</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">FL</span> <span class="o">=</span> <span class="p">[(</span><span class="n">P</span><span class="p">,</span> <span class="mi">7</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span> <span class="o">=</span> <span class="n">KanesMethod</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="p">[</span><span class="n">q</span><span class="p">],</span> <span class="p">[</span><span class="n">u</span><span class="p">],</span> <span class="p">[</span><span class="n">qd</span> <span class="o">-</span> <span class="n">u</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">fr</span><span class="p">,</span> <span class="n">frstar</span><span class="p">)</span> <span class="o">=</span> <span class="n">KM</span><span class="o">.</span><span class="n">kanes_equations</span><span class="p">(</span><span class="n">FL</span><span class="p">,</span> <span class="n">BL</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span><span class="o">.</span><span class="n">mass_matrix</span>
-<span class="go">[5]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span><span class="o">.</span><span class="n">forcing</span>
-<span class="go">[7]</span>
-</pre></div>
-</div>
-<p>When there are motion constraints, the mass matrix is augmented by the
-<span class="math">\(k_{dnh}(q, t)\)</span> matrix, and the forcing vector by the <span class="math">\(f_{dnh}(q,
-\dot{q}, u, t)\)</span> vector.</p>
-<p>There are also the &#8220;full&#8221; mass matrix and &#8220;full&#8221; forcing vector terms, these
-include the kinematic differential equations; the mass matrix is of size (n +
-o) x (n + o), or square and the size of all coordinates and speeds.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span><span class="o">.</span><span class="n">mass_matrix_full</span>
-<span class="go">[1, 0]</span>
-<span class="go">[0, 5]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span><span class="o">.</span><span class="n">forcing_full</span>
-<span class="go">[u]</span>
-<span class="go">[7]</span>
-</pre></div>
-</div>
-<p>The forcing vector can be linearized as well; its Jacobian is taken only with
-respect to the independent coordinates and speeds. The linearized forcing
-vector is of size (n + o) x (n - l + o - m), where l is the number of
-configuration constraints and m is the number of motion constraints. Two
-matrices are returned; the first is an &#8220;A&#8221; matrix, or the Jacobian with respect
-to the independent states, the second is a &#8220;B&#8221; matrix, or the Jacobian with
-respect to &#8216;forces&#8217;; this can be an empty matrix if there are no &#8216;forces&#8217;.
-Forces here are undefined functions of time (dynamic symbols); they are only
-allowed to be in the forcing vector and their derivatives are not allowed to be
-present. If dynamic symbols appear in the mass matrix or kinematic differential
-equations, an error with be raised.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">KM</span><span class="o">.</span><span class="n">linearize</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">[0, 1]</span>
-<span class="go">[0, 0]</span>
-</pre></div>
-</div>
-<p>Exploration of the provided examples is encouraged in order to gain more
-understanding of the <tt class="docutils literal"><span class="pre">KanesMethod</span></tt> object.</p>
-</div>
-</div>
-<div class="section" id="lagrange-s-method-in-physics-mechanics">
-<h1>Lagrange&#8217;s Method in Physics/Mechanics<a class="headerlink" href="#lagrange-s-method-in-physics-mechanics" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="id3">
-<h2>Structure of Equations<a class="headerlink" href="#id3" title="Permalink to this headline">¶</a></h2>
-<p>In <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt> we are assuming there are 3 basic sets of equations needed
-to describe a system; the constraint equations, the time differentiated
-constraint equations and the dynamic equations.</p>
-<div class="math">
-\[\begin{split}\mathbf{m_{c}}(q, t) \dot{q} + \mathbf{f_{c}}(q, t) &amp;= 0\\
-\mathbf{m_{dc}}(\dot{q}, q, t) \ddot{q} + \mathbf{f_{dc}}(\dot{q}, q, t) &amp;= 0\\
-\mathbf{m_d}(\dot{q}, q, t) \ddot{q} + \mathbf{\Lambda_c}(q, t)
-\lambda + \mathbf{f_d}(\dot{q}, q, t) &amp;= 0\\\end{split}\]</div>
-<p>In this module, the expressions formed by using Lagrange&#8217;s equations of the
-second kind are rearranged into the following form:</p>
-<blockquote>
-<div><span class="math">\(\mathbf{M}(q, t) x = \mathbf{f}(q, \dot{q}, t)\)</span></div></blockquote>
-<p>where in the case of a system without constraints:</p>
-<blockquote>
-<div><span class="math">\(x = \ddot{q}\)</span></div></blockquote>
-<p>For a constrained system with <span class="math">\(n\)</span> generalized speeds and <span class="math">\(m\)</span> constraints, we
-will get n - m equations. The mass-matrix/forcing equations are then augmented
-in the following fashion:</p>
-<div class="math">
-\[\begin{split}x = \begin{bmatrix} \ddot{q} \\ \lambda \end{bmatrix} \\
-\mathbf{M}(q, t) &amp;= \begin{bmatrix} \mathbf{m_d}(q, t) &amp;
-\mathbf{\Lambda_c}(q, t) \end{bmatrix}\\
-\mathbf{F}(\dot{q}, q, t) &amp;= \begin{bmatrix} \mathbf{f_d}(q, \dot{q}, t)
-\end{bmatrix}\\\end{split}\]</div>
-</div>
-<div class="section" id="id4">
-<h2>Lagrange&#8217;s Method in Physics/Mechanics<a class="headerlink" href="#id4" title="Permalink to this headline">¶</a></h2>
-<p>The formulation of the equations of motion in <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt> using
-Lagrange&#8217;s Method starts with the creation of generalized coordinates and a
-Lagrangian. The Lagrangian can either be created with the <tt class="docutils literal"><span class="pre">Lagrangian</span></tt>
-function or can be a user supplied function. In this case we will supply the
-Lagrangian.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1 q2&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1d</span><span class="p">,</span> <span class="n">q2d</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1 q2&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span> <span class="o">=</span> <span class="n">q1d</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">q2d</span><span class="o">**</span><span class="mi">2</span>
-</pre></div>
-</div>
-<p>To formulate the equations of motion we create a <tt class="docutils literal"><span class="pre">LagrangesMethod</span></tt>
-object. The Lagrangian and generalized coordinates need to be supplied upon
-initialization.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">LM</span> <span class="o">=</span> <span class="n">LagrangesMethod</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">])</span>
-</pre></div>
-</div>
-<p>With that the equations of motion can be formed.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mechanics_printing</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">LM</span><span class="o">.</span><span class="n">form_lagranges_equations</span><span class="p">()</span>
-<span class="go">[2*q1&#39;&#39;]</span>
-<span class="go">[2*q2&#39;&#39;]</span>
-</pre></div>
-</div>
-<p>It is possible to obtain the mass matrix and the forcing vector.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">LM</span><span class="o">.</span><span class="n">mass_matrix</span>
-<span class="go">[2, 0]</span>
-<span class="go">[0, 2]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">LM</span><span class="o">.</span><span class="n">forcing</span>
-<span class="go">[0]</span>
-<span class="go">[0]</span>
-</pre></div>
-</div>
-<p>If there are any holonomic or non-holonomic constraints, they must be supplied
-as keyword arguments in a list of expressions which are equal to zero. It
-should be noted that <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt> requires that the holonomic constraint
-equations must be supplied as velocity level constraint equations i.e. the
-holonomic constraint equations must be supplied after they have been
-differentiated with respect to time. Modifying the example above, the equations
-of motion can then be generated:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">LM</span> <span class="o">=</span> <span class="n">LagrangesMethod</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">],</span> <span class="n">coneqs</span> <span class="o">=</span> <span class="p">[</span><span class="n">q1d</span> <span class="o">-</span> <span class="n">q2d</span><span class="p">])</span>
-</pre></div>
-</div>
-<p>When the equations of motion are generated in this case, the Lagrange
-multipliers are introduced; they are represented by <tt class="docutils literal"><span class="pre">lam1</span></tt> in this case. In
-general, there will be as many multipliers as there are constraint equations.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">LM</span><span class="o">.</span><span class="n">form_lagranges_equations</span><span class="p">()</span>
-<span class="go">[ lam1 + 2*q1&#39;&#39;]</span>
-<span class="go">[-lam1 + 2*q2&#39;&#39;]</span>
-</pre></div>
-</div>
-<p>Also in the case of systems with constraints, the &#8216;full&#8217; mass matrix is
-augmented by the <span class="math">\(k_{dc}(q, t)\)</span> matrix, and the forcing vector by the
-<span class="math">\(f_{dc}(q, \dot{q}, t)\)</span> vector. The &#8216;full&#8217; mass matrix is of size
-(2n + o) x (2n + o), i.e. it&#8217;s a square matrix.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">LM</span><span class="o">.</span><span class="n">mass_matrix_full</span>
-<span class="go">[1, 0, 0,  0,  0]</span>
-<span class="go">[0, 1, 0,  0,  0]</span>
-<span class="go">[0, 0, 2,  0, -1]</span>
-<span class="go">[0, 0, 0,  2,  1]</span>
-<span class="go">[0, 0, 1, -1,  0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">LM</span><span class="o">.</span><span class="n">forcing_full</span>
-<span class="go">[q1&#39;]</span>
-<span class="go">[q2&#39;]</span>
-<span class="go">[  0]</span>
-<span class="go">[  0]</span>
-<span class="go">[  0]</span>
-</pre></div>
-</div>
-<p>If there are any non-conservative forces or moments acting on the system,
-they must also be supplied as keyword arguments in a list of 2-tuples of the
-form <tt class="docutils literal"><span class="pre">(Point,</span> <span class="pre">Vector)</span></tt> or <tt class="docutils literal"><span class="pre">(ReferenceFrame,</span> <span class="pre">Vector)</span></tt> where the <tt class="docutils literal"><span class="pre">Vector</span></tt>
-represents the non-conservative forces and torques. Along with this 2-tuple,
-the inertial frame must also be specified as a keyword argument. This is shown
-below by modifying the example above:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">q1d</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">FL</span> <span class="o">=</span> <span class="p">[(</span><span class="n">P</span><span class="p">,</span> <span class="mi">7</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">LM</span> <span class="o">=</span> <span class="n">LagrangesMethod</span><span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">],</span> <span class="n">forcelist</span> <span class="o">=</span> <span class="n">FL</span><span class="p">,</span> <span class="n">frame</span> <span class="o">=</span> <span class="n">N</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">LM</span><span class="o">.</span><span class="n">form_lagranges_equations</span><span class="p">()</span>
-<span class="go">[2*q1&#39;&#39; - 7]</span>
-<span class="go">[    2*q2&#39;&#39;]</span>
-</pre></div>
-</div>
-<p>Exploration of the provided examples is encouraged in order to gain more
-understanding of the <tt class="docutils literal"><span class="pre">LagrangesMethod</span></tt> object.</p>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Kane&#8217;s Method in Physics/Mechanics</a><ul>
-<li><a class="reference internal" href="#structure-of-equations">Structure of Equations</a></li>
-<li><a class="reference internal" href="#id2">Kane&#8217;s Method in Physics/Mechanics</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#lagrange-s-method-in-physics-mechanics">Lagrange&#8217;s Method in Physics/Mechanics</a><ul>
-<li><a class="reference internal" href="#id3">Structure of Equations</a></li>
-<li><a class="reference internal" href="#id4">Lagrange&#8217;s Method in Physics/Mechanics</a></li>
-</ul>
-</li>
-</ul>
-
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-  <div class="section" id="mechanics-kinematics">
-<h1>Mechanics: Kinematics<a class="headerlink" href="#mechanics-kinematics" title="Permalink to this headline">¶</a></h1>
-<p>In mechanics, kinematics refers to the motion of bodies (or particles).
-It is the &#8216;a&#8217; in &#8216;f=ma&#8217;. This document will give some mathematical background
-to describing a system&#8217;s kinematics as well as how to represent the kinematics
-in <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt>.</p>
-<div class="section" id="introduction-to-kinematics">
-<h2>Introduction to Kinematics<a class="headerlink" href="#introduction-to-kinematics" title="Permalink to this headline">¶</a></h2>
-<p>The first topic is rigid motion kinematics. A rigid body is an idealized
-representation of a physical object which has mass and rotational inertia.
-Rigid bodies are obviously not flexible. We can break down rigid body motion
-into translational motion, and rotational motion (when dealing with particles, we
-only have translational motion). Rotational motion can further be broken down
-into simple rotations and general rotations.</p>
-<p>Translation of a rigid body is defined as a motion where the orientation of the
-body does not change during the motion; or during the motion any line segment
-would be parallel to itself at the start of the motion.</p>
-<p>Simple rotations are rotations in which the orientation of the body may change,
-but there is always one line which remains parallel to itself at the start of
-the motion.</p>
-<p>General rotations are rotations which there is not always one line parallel to
-itself at the start of the motion.</p>
-<div class="section" id="angular-velocity">
-<h3>Angular Velocity<a class="headerlink" href="#angular-velocity" title="Permalink to this headline">¶</a></h3>
-<p>The angular velocity of a rigid body refers to the rate of change of its
-orientation. The angular velocity of a body is written down as:
-<span class="math">\(^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}}\)</span>, or the angular velocity of
-<span class="math">\(\mathbf{B}\)</span> in <span class="math">\(\mathbf{N}\)</span>, which is a vector. Note that here,
-the term rigid body was used, but reference frames can also have angular
-velocities. Further discussion of the distinction between a rigid body and a
-reference frame will occur later when describing the code representation.</p>
-<p>Angular velocity is defined as being positive in the direction which causes the
-orientation angles to increase (for simple rotations, or series of simple
-rotations).</p>
-<div align="center" class="align-center"><img height="350" src="../../../_images/kin_angvel1.svg" width="250" /></div>
-<p>The angular velocity vector represents the time derivative of the orientation.
-As a time derivative vector quantity, like those covered in the Vector &amp;
-ReferenceFrame documentation, this quantity (angular velocity) needs to be
-defined in a reference frame. That is what the <span class="math">\(\mathbf{N}\)</span> is in the
-above definition of angular velocity; the frame in which the angular velocity
-is defined in.</p>
-<p>The angular velocity of <span class="math">\(\mathbf{B}\)</span> in <span class="math">\(\mathbf{N}\)</span> can also be
-defined by:</p>
-<div class="math">
-\[^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}} =
-(\frac{^{\mathbf{N}}d \mathbf{\hat{b}_y}}{dt}\cdot\mathbf{\hat{b}_z}
-)\mathbf{\hat{b}_x} + (\frac{^{\mathbf{N}}d \mathbf{\hat{b}_z}}{dt}\cdot
-\mathbf{\hat{b}_x})\mathbf{\hat{b}_y} + (\frac{^{\mathbf{N}}d
-\mathbf{\hat{b}_x}}{dt}\cdot\mathbf{\hat{b}_y})\mathbf{\hat{b}_z}\]</div>
-<p>It is also common for a body&#8217;s angular velocity to be written as:</p>
-<div class="math">
-\[^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}} = w_x \mathbf{\hat{b}_x} +
-w_y \mathbf{\hat{b}_y} + w_z \mathbf{\hat{b}_z}\]</div>
-<p>There are a few additional important points relating to angular velocity. The
-first is the addition theorem for angular velocities, a way of relating the
-angular velocities of multiple bodies and frames. The theorem follows:</p>
-<div class="math">
-\[^{\mathbf{N}}\mathbf{\omega}^{\mathbf{D}} =
-^{\mathbf{N}}\mathbf{\omega}^{\mathbf{A}} +
-^{\mathbf{A}}\mathbf{\omega}^{\mathbf{B}} +
-^{\mathbf{B}}\mathbf{\omega}^{\mathbf{C}} +
-^{\mathbf{C}}\mathbf{\omega}^{\mathbf{D}}\]</div>
-<p>This is also shown in the following example:</p>
-<div align="center" class="align-center"><img height="300" src="../../../_images/kin_angvel2.svg" width="450" /></div>
-<div class="math">
-\[\begin{split}^{\mathbf{N}}\mathbf{\omega}^{\mathbf{A}} &amp;= 0\\
-^{\mathbf{A}}\mathbf{\omega}^{\mathbf{B}} &amp;= \dot{q_1} \mathbf{\hat{a}_x}\\
-^{\mathbf{B}}\mathbf{\omega}^{\mathbf{C}} &amp;= - \dot{q_2} \mathbf{\hat{b}_z}\\
-^{\mathbf{C}}\mathbf{\omega}^{\mathbf{D}} &amp;= \dot{q_3} \mathbf{\hat{c}_y}\\
-^{\mathbf{N}}\mathbf{\omega}^{\mathbf{D}} &amp;= \dot{q_1} \mathbf{\hat{a}_x}
-- \dot{q_2} \mathbf{\hat{b}_z} + \dot{q_3} \mathbf{\hat{c}_y}\\\end{split}\]</div>
-<p>Note the signs used in the angular velocity definitions, which are related to
-how the displacement angle is defined in this case.</p>
-<p>This theorem makes defining angular velocities of multibody systems much
-easier, as the angular velocity of a body in a chain needs to only be defined
-to the previous body in order to be fully defined (and the first body needs
-to be defined in the desired reference frame). The following figure shows an
-example of when using this theorem can make things easier.</p>
-<div align="center" class="align-center"><img height="250" src="../../../_images/kin_angvel3.svg" width="400" /></div>
-<p>Here we can easily write the angular velocity of the body
-<span class="math">\(\mathbf{D}\)</span> in the reference frame of the first body <span class="math">\(\mathbf{A}\)</span>:</p>
-<div class="math">
-\[\begin{split}^\mathbf{A}\mathbf{\omega}^\mathbf{D} = w_1 \mathbf{\hat{p_1}} +
-w_2 \mathbf{\hat{p_2}} + w_3 \mathbf{\hat{p_3}}\\\end{split}\]</div>
-<p>It is very important to remember to only use this with angular velocities; you
-cannot use this theorem with the velocities of points.</p>
-<p>There is another theorem commonly used: the derivative theorem. It provides an
-alternative method (which can be easier) to calculate the time derivative of a
-vector in a reference frame:</p>
-<div class="math">
-\[\frac{^{\mathbf{N}} d \mathbf{v}}{dt} = \frac{^{\mathbf{B}} d \mathbf{v}}{dt}
-+ ^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}} \times \mathbf{v}\]</div>
-<p>The vector <span class="math">\(\mathbf{v}\)</span> can be any vector quantity: a position vector,
-a velocity vector, angular velocity vector, etc. Instead of taking the time
-derivative of the vector in <span class="math">\(\mathbf{N}\)</span>, we take it in
-<span class="math">\(\mathbf{B}\)</span>, where <span class="math">\(\mathbf{B}\)</span> can be any reference frame or
-body, usually one in which it is easy to take the derivative on
-<span class="math">\(\mathbf{v}\)</span> in (<span class="math">\(\mathbf{v}\)</span> is usually composed only of the basis
-vector set belonging to <span class="math">\(\mathbf{B}\)</span>). Then we add the cross product of
-the angular velocity of our newer frame,
-<span class="math">\(^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}}\)</span> and our vector quantity
-<span class="math">\(\mathbf{v}\)</span>. Again, you can choose any alternative frame for this.
-Examples follow:</p>
-</div>
-<div class="section" id="angular-acceleration">
-<h3>Angular Acceleration<a class="headerlink" href="#angular-acceleration" title="Permalink to this headline">¶</a></h3>
-<p>Angular acceleration refers to the time rate of change of the angular velocity
-vector. Just as the angular velocity vector is for a body and is specified in a
-frame, the angular acceleration vector is for a body and is specified in a
-frame: <span class="math">\(^{\mathbf{N}}\mathbf{\alpha}^{\mathbf{B}}\)</span>, or the angular
-acceleration of <span class="math">\(\mathbf{B}\)</span> in <span class="math">\(\mathbf{N}\)</span>, which is a vector.</p>
-<p>Calculating the angular acceleration is relatively straight forward:</p>
-<div class="math">
-\[^{\mathbf{N}}\mathbf{\alpha}^{\mathbf{B}} =
-\frac{^{\mathbf{N}} d ^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}}}{dt}\]</div>
-<p>Note that this can be calculated with the derivative theorem, and when the
-angular velocity is defined in a body fixed frame, becomes quite simple:</p>
-<div class="math">
-\[\begin{split}^{\mathbf{N}}\mathbf{\alpha}^{\mathbf{B}} &amp;=
-\frac{^{\mathbf{N}} d ^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}}}{dt}\\\end{split}\]\[\begin{split}^{\mathbf{N}}\mathbf{\alpha}^{\mathbf{B}} &amp;=
-\frac{^{\mathbf{B}} d ^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}}}{dt}
-+ ^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}} \times
-^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}}\\\end{split}\]\[\begin{split}\textrm{if } ^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}} &amp;=
-w_x \mathbf{\hat{b}_x} + w_y \mathbf{\hat{b}_y} + w_z \mathbf{\hat{b}_z}\\\end{split}\]\[\begin{split}\textrm{then } ^{\mathbf{N}}\mathbf{\alpha}^{\mathbf{B}} &amp;=
-\frac{^{\mathbf{B}} d ^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}}}{dt}
-+ \underbrace{^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}} \times
-^{\mathbf{N}}\mathbf{\omega}^{\mathbf{B}}}_{
-\textrm{this is 0 by definition}}\\\end{split}\]\[\begin{split}^{\mathbf{N}}\mathbf{\alpha}^{\mathbf{B}}&amp;=\frac{d w_x}{dt}\mathbf{\hat{b}_x}
-+ \frac{d w_y}{dt}\mathbf{\hat{b}_y} + \frac{d w_z}{dt}\mathbf{\hat{b}_z}\\\end{split}\]\[\begin{split}^{\mathbf{N}}\mathbf{\alpha}^{\mathbf{B}}&amp;= \dot{w_x}\mathbf{\hat{b}_x} +
-\dot{w_y}\mathbf{\hat{b}_y} + \dot{w_z}\mathbf{\hat{b}_z}\\\end{split}\]</div>
-<p>Again, this is only for the case in which the angular velocity of the body is
-defined in body fixed components.</p>
-</div>
-<div class="section" id="point-velocity-acceleration">
-<h3>Point Velocity &amp; Acceleration<a class="headerlink" href="#point-velocity-acceleration" title="Permalink to this headline">¶</a></h3>
-<p>Consider a point, <span class="math">\(P\)</span>: we can define some characteristics of the point.
-First, we can define a position vector from some other point to <span class="math">\(P\)</span>.
-Second, we can define the velocity vector of <span class="math">\(P\)</span> in a reference frame of
-our choice. Third, we can define the acceleration vector of <span class="math">\(P\)</span> in a
-reference frame of our choice.</p>
-<p>These three quantities are read as:</p>
-<div class="math">
-\[\begin{split}\mathbf{r}^{OP} \textrm{, the position vector from } O
-\textrm{ to }P\\
-^{\mathbf{N}}\mathbf{v}^P \textrm{, the velocity of } P
-\textrm{ in the reference frame } \mathbf{N}\\
-^{\mathbf{N}}\mathbf{a}^P \textrm{, the acceleration of } P
-\textrm{ in the reference frame } \mathbf{N}\\\end{split}\]</div>
-<p>Note that the position vector does not have a frame associated with it; this is
-because there is no time derivative involved, unlike the velocity and
-acceleration vectors.</p>
-<p>We can find these quantities for a simple example easily:</p>
-<div align="center" class="align-center"><img height="300" src="../../../_images/kin_1.svg" width="300" /></div>
-<div class="math">
-\[\begin{split}\textrm{Let's define: }
-\mathbf{r}^{OP} &amp;= q_x \mathbf{\hat{n}_x} + q_y \mathbf{\hat{n}_y}\\
-^{\mathbf{N}}\mathbf{v}^P &amp;= \frac{^{\mathbf{N}} d \mathbf{r}^{OP}}{dt}\\
-\textrm{then we can calculate: }
-^{\mathbf{N}}\mathbf{v}^P &amp;= \dot{q}_x\mathbf{\hat{n}_x} +
-\dot{q}_y\mathbf{\hat{n}_y}\\
-\textrm{and :}
-^{\mathbf{N}}\mathbf{a}^P &amp;= \frac{^{\mathbf{N}} d
-^{\mathbf{N}}\mathbf{v}^P}{dt}\\
-^{\mathbf{N}}\mathbf{a}^P &amp;= \ddot{q}_x\mathbf{\hat{n}_x} +
-\ddot{q}_y\mathbf{\hat{n}_y}\\\end{split}\]</div>
-<p>It is critical to understand in the above example that the point <span class="math">\(O\)</span> is
-fixed in the reference frame <span class="math">\(\mathbf{N}\)</span>. There is no addition theorem
-for translational velocities; alternatives will be discussed later though.
-Also note that the position of every point might not
-always need to be defined to form the dynamic equations of motion.
-When you don&#8217;t want to define the position vector of a point, you can start by
-just defining the velocity vector. For the above example:</p>
-<div class="math">
-\[\begin{split}\textrm{Let us instead define the velocity vector as: }
-^{\mathbf{N}}\mathbf{v}^P &amp;= u_x \mathbf{\hat{n}_x} +
-u_y \mathbf{\hat{n}_y}\\
-\textrm{then acceleration can be written as: }
-^{\mathbf{N}}\mathbf{a}^P &amp;= \dot{u}_x \mathbf{\hat{n}_x} +
-\dot{u}_y \mathbf{\hat{n}_y}\\\end{split}\]</div>
-<p>There will often be cases when the velocity of a point is desired and a related
-point&#8217;s velocity is known. For the cases in which we have two points fixed on a
-rigid body, we use the 2-Point Theorem:</p>
-<div align="center" class="align-center"><img height="300" src="../../../_images/kin_2pt.svg" width="300" /></div>
-<p>Let&#8217;s say we know the velocity of the point <span class="math">\(S\)</span> and the angular
-velocity of the body <span class="math">\(\mathbf{B}\)</span>, both defined in the reference frame
-<span class="math">\(\mathbf{N}\)</span>. We can calculate the velocity and acceleration
-of the point <span class="math">\(P\)</span> in <span class="math">\(\mathbf{N}\)</span> as follows:</p>
-<div class="math">
-\[\begin{split}^{\mathbf{N}}\mathbf{v}^P &amp;= ^\mathbf{N}\mathbf{v}^S +
-^\mathbf{N}\mathbf{\omega}^\mathbf{B} \times \mathbf{r}^{SP}\\
-^{\mathbf{N}}\mathbf{a}^P &amp;= ^\mathbf{N}\mathbf{a}^S +
-^\mathbf{N}\mathbf{\alpha}^\mathbf{B} \times \mathbf{r}^{SP} +
-^\mathbf{N}\mathbf{\omega}^\mathbf{B} \times
-(^\mathbf{N}\mathbf{\omega}^\mathbf{B} \times \mathbf{r}^{SP})\\\end{split}\]</div>
-<p>When only one of the two points is fixed on a body, the 1 point theorem is used
-instead.</p>
-<div align="center" class="align-center"><img height="400" src="../../../_images/kin_1pt.svg" width="400" /></div>
-<p>Here, the velocity of point <span class="math">\(S\)</span> is known in the frame <span class="math">\(\mathbf{N}\)</span>,
-the angular velocity of <span class="math">\(\mathbf{B}\)</span> is known in <span class="math">\(\mathbf{N}\)</span>, and
-the velocity of the point <span class="math">\(P\)</span> is known in the frame associated with body
-<span class="math">\(\mathbf{B}\)</span>. We can then write the velocity and acceleration of
-<span class="math">\(P\)</span> in <span class="math">\(\mathbf{N}\)</span> as:</p>
-<div class="math">
-\[\begin{split}^{\mathbf{N}}\mathbf{v}^P &amp;= ^\mathbf{B}\mathbf{v}^P +
-^\mathbf{N}\mathbf{v}^S + ^\mathbf{N}\mathbf{\omega}^\mathbf{B} \times
-\mathbf{r}^{SP}\\\end{split}\]\[\begin{split}^{\mathbf{N}}\mathbf{a}^P &amp;= ^\mathbf{B}\mathbf{a}^S +
-^\mathbf{N}\mathbf{a}^O + ^\mathbf{N}\mathbf{\alpha}^\mathbf{B}
-\times \mathbf{r}^{SP} + ^\mathbf{N}\mathbf{\omega}^\mathbf{B} \times
-(^\mathbf{N}\mathbf{\omega}^\mathbf{B} \times \mathbf{r}^{SP}) +
-2 ^\mathbf{N}\mathbf{\omega}^\mathbf{B} \times ^\mathbf{B} \mathbf{v}^P \\\end{split}\]</div>
-<p>Examples of applications of the 1 point and 2 point theorem follow.</p>
-<div align="center" class="align-center"><img height="300" src="../../../_images/kin_2.svg" width="400" /></div>
-<p>This example has a disc translating and rotating in a plane. We can easily
-define the angular velocity of the body <span class="math">\(\mathbf{B}\)</span> and velocity of the
-point <span class="math">\(O\)</span>:</p>
-<div class="math">
-\[\begin{split}^\mathbf{N}\mathbf{\omega}^\mathbf{B} &amp;= u_3 \mathbf{\hat{n}_z} = u_3
-\mathbf{\hat{b}_z}\\
-^\mathbf{N}\mathbf{v}^O &amp;= u_1 \mathbf{\hat{n}_x} + u_2 \mathbf{\hat{n}_y}\\\end{split}\]</div>
-<p>and accelerations can be written as:</p>
-<div class="math">
-\[\begin{split}^\mathbf{N}\mathbf{\alpha}^\mathbf{B} &amp;= \dot{u_3} \mathbf{\hat{n}_z} =
-\dot{u_3} \mathbf{\hat{b}_z}\\
-^\mathbf{N}\mathbf{a}^O &amp;= \dot{u_1} \mathbf{\hat{n}_x} + \dot{u_2}
-\mathbf{\hat{n}_y}\\\end{split}\]</div>
-<p>We can use the 2 point theorem to calculate the velocity and acceleration of
-point <span class="math">\(P\)</span> now.</p>
-<div class="math">
-\[\begin{split}\mathbf{r}^{OP} &amp;= R \mathbf{\hat{b}_x}\\
-^\mathbf{N}\mathbf{v}^P &amp;= ^\mathbf{N}\mathbf{v}^O +
-^\mathbf{N}\mathbf{\omega}^\mathbf{B} \times \mathbf{r}^{OP}\\
-^\mathbf{N}\mathbf{v}^P &amp;= u_1 \mathbf{\hat{n}_x} + u_2 \mathbf{\hat{n}_y}
-+ u_3 \mathbf{\hat{b}_z} \times R \mathbf{\hat{b}_x} = u_1
-\mathbf{\hat{n}_x} + u_2 \mathbf{\hat{n}_y} + u_3 R \mathbf{\hat{b}_y}\\
-^{\mathbf{N}}\mathbf{a}^P &amp;= ^\mathbf{N}\mathbf{a}^O +
-^\mathbf{N}\mathbf{\alpha}^\mathbf{B} \times \mathbf{r}^{OP} +
-^\mathbf{N}\mathbf{\omega}^\mathbf{B} \times
-(^\mathbf{N}\mathbf{\omega}^\mathbf{B} \times \mathbf{r}^{OP})\\
-^{\mathbf{N}}\mathbf{a}^P &amp;= \dot{u_1} \mathbf{\hat{n}_x} + \dot{u_2}
-\mathbf{\hat{n}_y} + \dot{u_3}\mathbf{\hat{b}_z}\times R \mathbf{\hat{b}_x}
-+u_3\mathbf{\hat{b}_z}\times(u_3\mathbf{\hat{b}_z}\times
-R\mathbf{\hat{b}_x})\\
-^{\mathbf{N}}\mathbf{a}^P &amp;= \dot{u_1} \mathbf{\hat{n}_x} + \dot{u_2}
-\mathbf{\hat{n}_y} + R\dot{u_3}\mathbf{\hat{b}_y} - R u_3^2
-\mathbf{\hat{b}_x}\\\end{split}\]</div>
-<div align="center" class="align-center"><img height="200" src="../../../_images/kin_3.svg" width="200" /></div>
-<p>In this example we have a double pendulum. We can use the two point theorem
-twice here in order to find the velocity of points <span class="math">\(Q\)</span> and <span class="math">\(P\)</span>;
-point <span class="math">\(O\)</span>&#8216;s velocity is zero in <span class="math">\(\mathbf{N}\)</span>.</p>
-<div class="math">
-\[\begin{split}\mathbf{r}^{OQ} &amp;= l \mathbf{\hat{b}_x}\\
-\mathbf{r}^{QP} &amp;= l \mathbf{\hat{c}_x}\\
-^\mathbf{N}\mathbf{\omega}^\mathbf{B} &amp;= u_1 \mathbf{\hat{b}_z}\\
-^\mathbf{N}\mathbf{\omega}^\mathbf{C} &amp;= u_2 \mathbf{\hat{c}_z}\\
-^\mathbf{N}\mathbf{v}^Q &amp;= ^\mathbf{N}\mathbf{v}^O +
-^\mathbf{N}\mathbf{\omega}^\mathbf{B} \times \mathbf{r}^{OQ}\\
-^\mathbf{N}\mathbf{v}^Q &amp;= u_1 l \mathbf{\hat{b}_y}\\
-^\mathbf{N}\mathbf{v}^P &amp;= ^\mathbf{N}\mathbf{v}^Q +
-^\mathbf{N}\mathbf{\omega}^\mathbf{C} \times \mathbf{r}^{QP}\\
-^\mathbf{N}\mathbf{v}^Q &amp;= u_1 l \mathbf{\hat{b}_y} +u_2 \mathbf{\hat{c}_z}
-\times l \mathbf{\hat{c}_x}\\
-^\mathbf{N}\mathbf{v}^Q &amp;= u_1 l\mathbf{\hat{b}_y}+u_2 l\mathbf{\hat{c}_y}\\\end{split}\]</div>
-<div align="center" class="align-center"><img height="400" src="../../../_images/kin_4.svg" width="300" /></div>
-<p>In this example we have a particle moving on a ring; the ring is supported by a
-rod which can rotate about the <span class="math">\(\mathbf{\hat{n}_x}\)</span> axis. First we use
-the two point theorem to find the velocity of the center point of the ring,
-<span class="math">\(Q\)</span>, then use the 1 point theorem to find the velocity of the particle on
-the ring.</p>
-<div class="math">
-\[\begin{split}^\mathbf{N}\mathbf{\omega}^\mathbf{C} &amp;= u_1 \mathbf{\hat{n}_x}\\
-\mathbf{r}^{OQ} &amp;= -l \mathbf{\hat{c}_z}\\
-^\mathbf{N}\mathbf{v}^Q &amp;= u_1 l \mathbf{\hat{c}_y}\\
-\mathbf{r}^{QP} &amp;= R(cos(q_2) \mathbf{\hat{c}_x}
-+ sin(q_2) \mathbf{\hat{c}_y} )\\
-^\mathbf{C}\mathbf{v}^P &amp;= R u_2 (-sin(q_2) \mathbf{\hat{c}_x}
-+ cos(q_2) \mathbf{\hat{c}_y} )\\
-^\mathbf{N}\mathbf{v}^P &amp;= ^\mathbf{C}\mathbf{v}^P +^\mathbf{N}\mathbf{v}^Q
-+ ^\mathbf{N}\mathbf{\omega}^\mathbf{C} \times \mathbf{r}^{QP}\\
-^\mathbf{N}\mathbf{v}^P &amp;= R u_2 (-sin(q_2) \mathbf{\hat{c}_x}
-+ cos(q_2) \mathbf{\hat{c}_y} ) + u_1 l \mathbf{\hat{c}_y} +
-u_1 \mathbf{\hat{c}_x} \times R(cos(q_2) \mathbf{\hat{c}_x}
-+ sin(q_2) \mathbf{\hat{c}_y}\\
-^\mathbf{N}\mathbf{v}^P &amp;= - R u_2 sin(q_2) \mathbf{\hat{c}_x}
-+ (R u_2 cos(q_2)+u_1 l)\mathbf{\hat{c}_y} + R u_1 sin(q_2)
-\mathbf{\hat{c}_z}\\\end{split}\]</div>
-<p>A final topic in the description of velocities of points is that of rolling, or
-rather, rolling without slip. Two bodies are said to be rolling without slip if
-and only if the point of contact on each body has the same velocity in another
-frame. See the following figure:</p>
-<div align="center" class="align-center"><img height="250" src="../../../_images/kin_rolling.svg" width="450" /></div>
-<p>This is commonly used to form the velocity of a point on one object rolling on
-another fixed object, such as in the following example:</p>
-</div>
-</div>
-<div class="section" id="kinematics-in-sympy-s-mechanics">
-<h2>Kinematics in SymPy&#8217;s Mechanics<a class="headerlink" href="#kinematics-in-sympy-s-mechanics" title="Permalink to this headline">¶</a></h2>
-<p>It should be clear by now that the topic of kinematics here has been mostly
-describing the correct way to manipulate vectors into representing the
-velocities of points. Within <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt> there are convenient methods for
-storing these velocities associated with frames and points. We&#8217;ll now revisit
-the above examples and show how to represent them in <a class="reference internal" href="../../matrices/immutablematrices.html#module-sympy" title="sympy"><tt class="xref py py-mod docutils literal"><span class="pre">sympy</span></tt></a>.</p>
-<p>The topic of reference frame creation has already been covered. When a
-<tt class="docutils literal"><span class="pre">ReferenceFrame</span></tt> is created though, it automatically calculates the angular
-velocity of the frame using the time derivative of the DCM and the angular
-velocity definition.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mechanics_printing</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">q1&#39;*N.x</span>
-</pre></div>
-</div>
-<p>Note that the angular velocity can be defined in an alternate way:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u1</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;u1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">u1</span> <span class="o">*</span> <span class="n">B</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">u1*B.y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
-<span class="go">- u1*B.y</span>
-</pre></div>
-</div>
-<p>Both upon frame creation during <tt class="docutils literal"><span class="pre">orientnew</span></tt> and when calling <tt class="docutils literal"><span class="pre">set_ang_vel</span></tt>,
-the angular velocity is set in both frames involved, as seen above.</p>
-<div align="center" class="align-center"><img height="300" src="../../../_images/kin_angvel2.svg" width="450" /></div>
-<p>Here we have multiple bodies with angular velocities defined relative to each
-other. This is coded as:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;C&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;D&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u1</span><span class="p">,</span> <span class="n">u2</span><span class="p">,</span> <span class="n">u3</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;u1 u2 u3&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">u1</span> <span class="o">*</span> <span class="n">A</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="o">-</span><span class="n">u2</span> <span class="o">*</span> <span class="n">B</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">u3</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">u3*C.y - u2*B.z + u1*A.x</span>
-</pre></div>
-</div>
-<p>In <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt> the shortest path between two frames is used when finding
-the angular velocity. That would mean if we went back and set:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span><span class="o">.</span><span class="n">ang_vel_in</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>The path that was just defined is what is used.
-This can cause problems though, as now the angular
-velocity definitions are inconsistent. It is recommended that you avoid
-doing this.</p>
-<p>Points are a translational analog to the rotational <tt class="docutils literal"><span class="pre">ReferenceFrame</span></tt>.
-Creating a <tt class="docutils literal"><span class="pre">Point</span></tt> can be done in two ways, like <tt class="docutils literal"><span class="pre">ReferenceFrame</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">O</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;O&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">O</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">,</span> <span class="mi">3</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">pos_from</span><span class="p">(</span><span class="n">O</span><span class="p">)</span>
-<span class="go">3*N.x + N.y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;Q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span><span class="o">.</span><span class="n">set_pos</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span><span class="o">.</span><span class="n">pos_from</span><span class="p">(</span><span class="n">P</span><span class="p">)</span>
-<span class="go">N.z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span><span class="o">.</span><span class="n">pos_from</span><span class="p">(</span><span class="n">O</span><span class="p">)</span>
-<span class="go">3*N.x + N.y + N.z</span>
-</pre></div>
-</div>
-<p>Similar to <tt class="docutils literal"><span class="pre">ReferenceFrame</span></tt>, the position vector between two points is found
-by the shortest path (number of intermediate points) between them. Unlike
-rotational motion, there is no addition theorem for the velocity of points. In
-order to have the velocity of a <tt class="docutils literal"><span class="pre">Point</span></tt> in a <tt class="docutils literal"><span class="pre">ReferenceFrame</span></tt>, you have to
-set the value.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">O</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;O&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">u1</span><span class="o">*</span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span><span class="o">.</span><span class="n">vel</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">u1*N.x</span>
-</pre></div>
-</div>
-<p>For both translational and rotational accelerations, the value is computed by
-taking the time derivative of the appropriate velocity, unless the user sets it
-otherwise.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">O</span><span class="o">.</span><span class="n">acc</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">u1&#39;*N.x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span><span class="o">.</span><span class="n">set_acc</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">u2</span><span class="o">*</span><span class="n">u1</span><span class="o">*</span><span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span><span class="o">.</span><span class="n">acc</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">u1*u2*N.y</span>
-</pre></div>
-</div>
-<p>Next is a description of the 2 point and 1 point theorems, as used in
-<tt class="docutils literal"><span class="pre">sympy</span></tt>.</p>
-<div align="center" class="align-center"><img height="300" src="../../../_images/kin_2.svg" width="400" /></div>
-<p>First is the translating, rotating disc.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u1</span><span class="p">,</span> <span class="n">u2</span><span class="p">,</span> <span class="n">u3</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;u1 u2 u3&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;R&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;O&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">u1</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">u2</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">O</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">,</span> <span class="n">R</span> <span class="o">*</span> <span class="n">B</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">u3</span> <span class="o">*</span> <span class="n">B</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">v2pt_theory</span><span class="p">(</span><span class="n">O</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-<span class="go">u1*N.x + u2*N.y + R*u3*B.y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">a2pt_theory</span><span class="p">(</span><span class="n">O</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-<span class="go">u1&#39;*N.x + u2&#39;*N.y - R*u3**2*B.x + R*u3&#39;*B.y</span>
-</pre></div>
-</div>
-<p>We will also cover implementation of the 1 point theorem.</p>
-<div align="center" class="align-center"><img height="400" src="../../../_images/kin_4.svg" width="300" /></div>
-<p>This is the particle moving on a ring, again.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u1</span><span class="p">,</span> <span class="n">u2</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;u1 u2&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1 q2&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;l&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;R&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;C&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span><span class="o">.</span><span class="n">set_ang_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">u1</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;O&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span> <span class="o">=</span> <span class="n">O</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;Q&#39;</span><span class="p">,</span> <span class="o">-</span><span class="n">l</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">Q</span><span class="o">.</span><span class="n">locatenew</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">,</span> <span class="n">R</span> <span class="o">*</span> <span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">q2</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">sin</span><span class="p">(</span><span class="n">q2</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">y</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">set_vel</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">R</span> <span class="o">*</span> <span class="n">u2</span> <span class="o">*</span> <span class="p">(</span><span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="n">q2</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">cos</span><span class="p">(</span><span class="n">q2</span><span class="p">)</span> <span class="o">*</span> <span class="n">C</span><span class="o">.</span><span class="n">y</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span><span class="o">.</span><span class="n">v2pt_theory</span><span class="p">(</span><span class="n">O</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">C</span><span class="p">)</span>
-<span class="go">l*u1*C.y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="o">.</span><span class="n">v1pt_theory</span><span class="p">(</span><span class="n">Q</span><span class="p">,</span> <span class="n">N</span><span class="p">,</span> <span class="n">C</span><span class="p">)</span>
-<span class="go">- R*u2*sin(q2)*C.x + (R*u2*cos(q2) + l*u1)*C.y + R*u1*sin(q2)*C.z</span>
-</pre></div>
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-  <div class="section" id="mechanics-masses-inertias-particles-and-rigid-bodies">
-<h1>Mechanics: Masses, Inertias, Particles and Rigid Bodies<a class="headerlink" href="#mechanics-masses-inertias-particles-and-rigid-bodies" title="Permalink to this headline">¶</a></h1>
-<p>This document will describe how to represent masses and inertias in
-<tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt> and use of the <tt class="docutils literal"><span class="pre">RigidBody</span></tt> and <tt class="docutils literal"><span class="pre">Particle</span></tt> classes.</p>
-<p>It is assumed that the reader is familiar with the basics of these topics, such
-as finding the center of mass for a system of particles, how to manipulate an
-inertia tensor, and the definition of a particle and rigid body. Any advanced
-dynamics text can provide a reference for these details.</p>
-<div class="section" id="mass">
-<h2>Mass<a class="headerlink" href="#mass" title="Permalink to this headline">¶</a></h2>
-<p>The only requirement for a mass is that it needs to be a <tt class="docutils literal"><span class="pre">sympify</span></tt>-able
-expression. Keep in mind that masses can be time varying.</p>
-</div>
-<div class="section" id="particle">
-<h2>Particle<a class="headerlink" href="#particle" title="Permalink to this headline">¶</a></h2>
-<p>Particles are created with the class <tt class="docutils literal"><span class="pre">Particle</span></tt> in <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt>.
-A <tt class="docutils literal"><span class="pre">Particle</span></tt> object has an associated point and an associated mass which are
-the only two attributes of the object.:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">Particle</span><span class="p">,</span> <span class="n">Point</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;m&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">po</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;po&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># create a particle container</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pa</span> <span class="o">=</span> <span class="n">Particle</span><span class="p">(</span><span class="s">&#39;pa&#39;</span><span class="p">,</span> <span class="n">po</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>The associated point contains the position, velocity and acceleration of the
-particle. <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt> allows one to perform kinematic analysis of points
-separate from their association with masses.</p>
-</div>
-<div class="section" id="inertia">
-<h2>Inertia<a class="headerlink" href="#inertia" title="Permalink to this headline">¶</a></h2>
-<p>Dyadics are used to define the inertia of bodies within <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt>.
-Inertia dyadics can be defined explicitly but the <tt class="docutils literal"><span class="pre">inertia</span></tt> function is
-typically much more convenient for the user:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">inertia</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-
-<span class="go">Supply a reference frame and the moments of inertia if the object</span>
-<span class="go">is symmetrical:</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">inertia</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">(N.x|N.x) + 2*(N.y|N.y) + 3*(N.z|N.z)</span>
-
-<span class="go">Supply a reference frame along with the products and moments of inertia</span>
-<span class="go">for a general object:</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">inertia</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
-<span class="go">(N.x|N.x) + 4*(N.x|N.y) + 6*(N.x|N.z) + 4*(N.y|N.x) + 2*(N.y|N.y) +</span>
-<span class="go">            5*(N.y|N.z) + 6*(N.z|N.x) + 5*(N.z|N.y) + 3*(N.z|N.z)</span>
-</pre></div>
-</div>
-<p>Notice that the <tt class="docutils literal"><span class="pre">inertia</span></tt> function returns a dyadic with each component
-represented as two unit vectors separated by a <tt class="docutils literal"><span class="pre">|</span></tt>. Refer to the
-<a class="reference internal" href="#dyadic"><em>Dyadic</em></a> section for more information about dyadics.</p>
-</div>
-<div class="section" id="rigid-body">
-<h2>Rigid Body<a class="headerlink" href="#rigid-body" title="Permalink to this headline">¶</a></h2>
-<p>Rigid bodies are created in a similar fashion as particles. The <tt class="docutils literal"><span class="pre">RigidBody</span></tt>
-class generates objects with four attributes: mass, center of mass, a reference
-frame, and an inertia tuple:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">ReferenceFrame</span><span class="p">,</span> <span class="n">Point</span><span class="p">,</span> <span class="n">RigidBody</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="n">outer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;m&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="s">&#39;P&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">outer</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">A</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># create a rigid body</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">RigidBody</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">P</span><span class="p">))</span>
-</pre></div>
-</div>
-<p>The mass is specified exactly as is in a particle. Similar to the
-<tt class="docutils literal"><span class="pre">Particle</span></tt>&#8216;s <tt class="docutils literal"><span class="pre">.point</span></tt>, the <tt class="docutils literal"><span class="pre">RigidBody</span></tt>&#8216;s center of mass, <tt class="docutils literal"><span class="pre">.masscenter</span></tt>
-must be specified. The reference frame is stored in an analogous fashion and
-holds information about the body&#8217;s orientation and angular velocity. Finally,
-the inertia for a rigid body needs to be specified about a point. In
-<tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt>, you are allowed to specify any point for this. The most
-common is the center of mass, as shown in the above code. If a point is selected
-which is not the center of mass, ensure that the position between the point and
-the center of mass has been defined. The inertia is specified as a tuple of length
-two with the first entry being a <tt class="docutils literal"><span class="pre">Dyadic</span></tt> and the second entry being a
-<tt class="docutils literal"><span class="pre">Point</span></tt> of which the inertia dyadic is defined about.</p>
-</div>
-<div class="section" id="dyadic">
-<span id="id1"></span><h2>Dyadic<a class="headerlink" href="#dyadic" title="Permalink to this headline">¶</a></h2>
-<p>In <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt>, dyadics are used to represent inertia (<a class="reference internal" href="../mechanics/reference.html#kane1985">[Kane1985]</a>,
-<a class="reference internal" href="../mechanics/reference.html#wikidyadics">[WikiDyadics]</a>, <a class="reference internal" href="../mechanics/reference.html#wikidyadicproducts">[WikiDyadicProducts]</a>). A dyadic is a linear polynomial of
-component unit dyadics, similar to a vector being a linear polynomial of
-component unit vectors. A dyadic is the outer product between two vectors which
-returns a new quantity representing the juxtaposition of these two vectors. For
-example:</p>
-<div class="math">
-\[\begin{split}\mathbf{\hat{a}_x} \otimes \mathbf{\hat{a}_x} &amp;= \mathbf{\hat{a}_x}
-\mathbf{\hat{a}_x}\\
-\mathbf{\hat{a}_x} \otimes \mathbf{\hat{a}_y} &amp;= \mathbf{\hat{a}_x}
-\mathbf{\hat{a}_y}\\\end{split}\]</div>
-<p>Where <span class="math">\(\mathbf{\hat{a}_x}\mathbf{\hat{a}_x}\)</span> and
-<span class="math">\(\mathbf{\hat{a}_x}\mathbf{\hat{a}_y}\)</span> are the outer products obtained by
-multiplying the left side as a column vector by the right side as a row vector.
-Note that the order is significant.</p>
-<p>Some additional properties of a dyadic are:</p>
-<div class="math">
-\[\begin{split}(x \mathbf{v}) \otimes \mathbf{w} &amp;= \mathbf{v} \otimes (x \mathbf{w}) = x
-(\mathbf{v} \otimes \mathbf{w})\\
-\mathbf{v} \otimes (\mathbf{w} + \mathbf{u}) &amp;= \mathbf{v} \otimes \mathbf{w}
-+ \mathbf{v} \otimes \mathbf{u}\\
-(\mathbf{v} + \mathbf{w}) \otimes \mathbf{u} &amp;= \mathbf{v} \otimes \mathbf{u}
-+ \mathbf{w} \otimes \mathbf{u}\\\end{split}\]</div>
-<p>A vector in a reference frame can be represented as
-<span class="math">\(\begin{bmatrix}a\\b\\c\end{bmatrix}\)</span> or <span class="math">\(a \mathbf{\hat{i}} + b
-\mathbf{\hat{j}} + c \mathbf{\hat{k}}\)</span>. Similarly, a dyadic can be represented
-in tensor form:</p>
-<div class="math">
-\[\begin{split}\begin{bmatrix}
-a_{11} &amp; a_{12} &amp; a_{13} \\
-a_{21} &amp; a_{22} &amp; a_{23} \\
-a_{31} &amp; a_{32} &amp; a_{33}
-\end{bmatrix}\\\end{split}\]</div>
-<p>or in dyadic form:</p>
-<div class="math">
-\[\begin{split}a_{11} \mathbf{\hat{a}_x}\mathbf{\hat{a}_x} +
-a_{12} \mathbf{\hat{a}_x}\mathbf{\hat{a}_y} +
-a_{13} \mathbf{\hat{a}_x}\mathbf{\hat{a}_z} +
-a_{21} \mathbf{\hat{a}_y}\mathbf{\hat{a}_x} +
-a_{22} \mathbf{\hat{a}_y}\mathbf{\hat{a}_y} +
-a_{23} \mathbf{\hat{a}_y}\mathbf{\hat{a}_z} +
-a_{31} \mathbf{\hat{a}_z}\mathbf{\hat{a}_x} +
-a_{32} \mathbf{\hat{a}_z}\mathbf{\hat{a}_y} +
-a_{33} \mathbf{\hat{a}_z}\mathbf{\hat{a}_z}\\\end{split}\]</div>
-<p>Just as with vectors, the later representation makes it possible to keep track
-of which frames the dyadic is defined with respect to. Also, the two
-components of each term in the dyadic need not be in the same frame. The
-following is valid:</p>
-<div class="math">
-\[\mathbf{\hat{a}_x} \otimes \mathbf{\hat{b}_y} = \mathbf{\hat{a}_x}
-\mathbf{\hat{b}_y}\]</div>
-<p>Dyadics can also be crossed and dotted with vectors; again, order matters:</p>
-<div class="math">
-\[\begin{split}\mathbf{\hat{a}_x}\mathbf{\hat{a}_x} \cdot \mathbf{\hat{a}_x} &amp;=
-\mathbf{\hat{a}_x}\\
-\mathbf{\hat{a}_y}\mathbf{\hat{a}_x} \cdot \mathbf{\hat{a}_x} &amp;=
-\mathbf{\hat{a}_y}\\
-\mathbf{\hat{a}_x}\mathbf{\hat{a}_y} \cdot \mathbf{\hat{a}_x} &amp;= 0\\
-\mathbf{\hat{a}_x} \cdot \mathbf{\hat{a}_x}\mathbf{\hat{a}_x} &amp;=
-\mathbf{\hat{a}_x}\\
-\mathbf{\hat{a}_x} \cdot \mathbf{\hat{a}_x}\mathbf{\hat{a}_y} &amp;=
-\mathbf{\hat{a}_y}\\
-\mathbf{\hat{a}_x} \cdot \mathbf{\hat{a}_y}\mathbf{\hat{a}_x} &amp;= 0\\
-\mathbf{\hat{a}_x} \times \mathbf{\hat{a}_y}\mathbf{\hat{a}_x} &amp;=
-\mathbf{\hat{a}_z}\mathbf{\hat{a}_x}\\
-\mathbf{\hat{a}_x} \times \mathbf{\hat{a}_x}\mathbf{\hat{a}_x} &amp;= 0\\
-\mathbf{\hat{a}_y}\mathbf{\hat{a}_x} \times \mathbf{\hat{a}_z} &amp;=
-- \mathbf{\hat{a}_y}\mathbf{\hat{a}_y}\\\end{split}\]</div>
-<p>One can also take the time derivative of dyadics or express them in different
-frames, just like with vectors.</p>
-</div>
-<div class="section" id="linear-momentum">
-<h2>Linear Momentum<a class="headerlink" href="#linear-momentum" title="Permalink to this headline">¶</a></h2>
-<p>The linear momentum of a particle P is defined as:</p>
-<div class="math">
-\[L_P = m\mathbf{v}\]</div>
-<p>where <span class="math">\(m\)</span> is the mass of the particle P and <span class="math">\(\mathbf{v}\)</span> is the
-velocity of the particle in the inertial frame.[Likins1973]_.</p>
-<p>Similarly the linear momentum of a rigid body is defined as:</p>
-<div class="math">
-\[L_B = m\mathbf{v^*}\]</div>
-<p>where <span class="math">\(m\)</span> is the mass of the rigid body, B, and <span class="math">\(\mathbf{v^*}\)</span> is
-the velocity of the mass center of B in the inertial frame.</p>
-</div>
-<div class="section" id="angular-momentum">
-<h2>Angular Momentum<a class="headerlink" href="#angular-momentum" title="Permalink to this headline">¶</a></h2>
-<p>The angular momentum of a particle P about an arbitrary point O in an inertial
-frame N is defined as:</p>
-<div class="math">
-\[^N \mathbf{H} ^ {P/O} = \mathbf{r} \times m\mathbf{v}\]</div>
-<p>where <span class="math">\(\mathbf{r}\)</span> is a position vector from point O to the particle of
-mass <span class="math">\(m\)</span> and <span class="math">\(\mathbf{v}\)</span> is the velocity of the particle in the
-inertial frame.</p>
-<p>Similarly the angular momentum of a rigid body B about a point O in an inertial
-frame N is defined as:</p>
-<div class="math">
-\[^N \mathbf{H} ^ {B/O} = ^N \mathbf{H} ^ {B/B^*} + ^N \mathbf{H} ^ {B^*/O}\]</div>
-<p>where the angular momentum of the body about it&#8217;s mass center is:</p>
-<div class="math">
-\[^N \mathbf{H} ^ {B/B^*} = \mathbf{I^*} \cdot \omega\]</div>
-<p>and the angular momentum of the mass center about O is:</p>
-<div class="math">
-\[^N \mathbf{H} ^ {B^*/O} = \mathbf{r^*} \times m \mathbf{v^*}\]</div>
-<p>where <span class="math">\(\mathbf{I^*}\)</span> is the central inertia dyadic of rigid body B,
-<span class="math">\(\omega\)</span> is the inertial angular velocity of B, <span class="math">\(\mathbf{r^*}\)</span> is a
-position vector from point O to the mass center of B, <span class="math">\(m\)</span> is the mass of
-B and <span class="math">\(\mathbf{v^*}\)</span> is the velocity of the mass center in the inertial
-frame.</p>
-</div>
-<div class="section" id="using-momenta-functions-in-mechanics">
-<h2>Using momenta functions in Mechanics<a class="headerlink" href="#using-momenta-functions-in-mechanics" title="Permalink to this headline">¶</a></h2>
-<p>The following example shows how to use the momenta functions in
-<tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt>.</p>
-<p>One begins by creating the requisite symbols to describe the system. Then
-the reference frame is created and the kinematics are done.</p>
-<div class="highlight-python"><pre>&gt;&gt; from sympy import symbols
-&gt;&gt; from sympy.physics.mechanics import dynamicsymbols, ReferenceFrame
-&gt;&gt; from sympy.physics.mechanics import RigidBody, Particle, Point, outer
-&gt;&gt; from symp.physics.mechanics import linear_momentum, angular_momentum
-&gt;&gt; m, M, l1 = symbols(&#x27;m M l1&#x27;)
-&gt;&gt; q1d = dynamicsymbols(&#x27;q1d&#x27;)
-&gt;&gt; N = ReferenceFrame(&#x27;N&#x27;)
-&gt;&gt; O = Point(&#x27;O&#x27;)
-&gt;&gt; O.set_vel(N, 0 * N.x)
-&gt;&gt; Ac = O.locatenew(&#x27;Ac&#x27;, l1 * N.x)
-&gt;&gt; P = Ac.locatenew(&#x27;P&#x27;, l1 * N.x)
-&gt;&gt; a = ReferenceFrame(&#x27;a&#x27;)
-&gt;&gt; a.set_ang_vel(N, q1d * N.z)
-&gt;&gt; Ac.v2pt_theory(O, N, a)
-&gt;&gt; P.v2pt_theory(O, N, a)</pre>
-</div>
-<p>Finally, the bodies that make up the system are created. In this case the
-system consists of a particle Pa and a RigidBody A.</p>
-<div class="highlight-python"><pre>&gt;&gt; Pa = Particle(&#x27;Pa&#x27;, P, m)
-&gt;&gt; I = outer(N.z, N.z)
-&gt;&gt; A = RigidBody(&#x27;A&#x27;, Ac, a, M, (I, Ac))</pre>
-</div>
-<p>Then one can either choose to evaluate the the momenta of individual components
-of the system or of the entire system itself.</p>
-<div class="highlight-python"><pre>&gt;&gt; linear_momentum(N,A)
-M*l1*q1d*N.y
-&gt;&gt; angular_momentum(O, N, Pa)
-4*l1**2*m*q1d*N.z
-&gt;&gt; linear_momentum(N, A, Pa)
-(M*l1*q1d + 2*l1*m*q1d)*N.y
-&gt;&gt; angular_momentum(O, N, A, Pa)
-(4*l1**2*m*q1d + q1d)*N.z</pre>
-</div>
-<p>It should be noted that the user can determine either momenta in any frame
-in <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt> as the user is allowed to specify the reference frame when
-calling the function. In other words the user is not limited to determining
-just inertial linear and angular momenta. Please refer to the docstrings on
-each function to learn more about how each function works precisely.</p>
-</div>
-<div class="section" id="kinetic-energy">
-<h2>Kinetic Energy<a class="headerlink" href="#kinetic-energy" title="Permalink to this headline">¶</a></h2>
-<p>The kinetic energy of a particle P is defined as</p>
-<div class="math">
-\[T_P = \frac{1}{2} m \mathbf{v^2}\]</div>
-<p>where <span class="math">\(m\)</span> is the mass of the particle P and <span class="math">\(\mathbf{v}\)</span>
-is the velocity of the particle in the inertial frame.</p>
-<p>Similarly the kinetic energy of a rigid body B is defined as</p>
-<div class="math">
-\[T_B = T_t + T_r\]</div>
-<p>where the translational kinetic energy is given by:</p>
-<div class="math">
-\[T_t = \frac{1}{2} m \mathbf{v^*} \cdot \mathbf{v^*}\]</div>
-<p>and the rotational kinetic energy is given by:</p>
-<div class="math">
-\[T_r = \frac{1}{2} \omega \cdot \mathbf{I^*} \cdot \omega\]</div>
-<p>where <span class="math">\(m\)</span> is the mass of the rigid body, <span class="math">\(\mathbf{v^*}\)</span> is the
-velocity of the mass center in the inertial frame, <span class="math">\(\omega\)</span> is the
-inertial angular velocity of the body and <span class="math">\(\mathbf{I^*}\)</span> is the central
-inertia dyadic.</p>
-</div>
-<div class="section" id="potential-energy">
-<h2>Potential Energy<a class="headerlink" href="#potential-energy" title="Permalink to this headline">¶</a></h2>
-<p>Potential energy is defined as the energy possessed by a body or system by
-virtue of its position or arrangement.</p>
-<p>Since there are a variety of definitions for potential energy, this is not
-discussed further here. One can learn more about this in any elementary text
-book on dynamics.</p>
-</div>
-<div class="section" id="lagrangian">
-<h2>Lagrangian<a class="headerlink" href="#lagrangian" title="Permalink to this headline">¶</a></h2>
-<p>The Lagrangian of a body or a system of bodies is defined as:</p>
-<div class="math">
-\[\mathcal{L} = T - V\]</div>
-<p>where <span class="math">\(T\)</span> and <span class="math">\(V\)</span> are the kinetic and potential energies
-respectively.</p>
-</div>
-<div class="section" id="using-energy-functions-in-mechanics">
-<h2>Using energy functions in Mechanics<a class="headerlink" href="#using-energy-functions-in-mechanics" title="Permalink to this headline">¶</a></h2>
-<p>The following example shows how to use the energy functions in
-<tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt>.</p>
-<p>As was discussed above in the momenta functions, one first creates the system
-by going through an identical procedure.</p>
-<div class="highlight-python"><pre>&gt;&gt; from sympy import symbols
-&gt;&gt; from sympy.physics.mechanics import dynamicsymbols, ReferenceFrame, outer
-&gt;&gt; from sympy.physics.mechanics import RigidBody, Particle, mechanics_printing
-&gt;&gt; from symp.physics.mechanics import kinetic_energy, potential_energy, Point
-&gt;&gt; mechanics_printing()
-&gt;&gt; m, M, l1, g, h, H = symbols(&#x27;m M l1 g h H&#x27;)
-&gt;&gt; omega = dynamicsymbols(&#x27;omega&#x27;)
-&gt;&gt; N = ReferenceFrame(&#x27;N&#x27;)
-&gt;&gt; O = Point(&#x27;O&#x27;)
-&gt;&gt; O.set_vel(N, 0 * N.x)
-&gt;&gt; Ac = O.locatenew(&#x27;Ac&#x27;, l1 * N.x)
-&gt;&gt; P = Ac.locatenew(&#x27;P&#x27;, l1 * N.x)
-&gt;&gt; a = ReferenceFrame(&#x27;a&#x27;)
-&gt;&gt; a.set_ang_vel(N, omega * N.z)
-&gt;&gt; Ac.v2pt_theory(O, N, a)
-&gt;&gt; P.v2pt_theory(O, N, a)
-&gt;&gt; Pa = Particle(&#x27;Pa&#x27;, P, m)
-&gt;&gt; I = outer(N.z, N.z)
-&gt;&gt; A = RigidBody(&#x27;A&#x27;, Ac, a, M, (I, Ac))</pre>
-</div>
-<p>The user can then determine the kinetic energy of any number of entities of the
-system:</p>
-<div class="highlight-python"><pre>&gt;&gt; kinetic_energy(N, Pa)
-2*l1**2*m*q1d**2
-&gt;&gt; kinetic_energy(N, Pa, A)
-M*l1**2*q1d**2/2 + 2*l1**2*m*q1d**2 + q1d**2/2</pre>
-</div>
-<p>It should be noted that the user can determine either kinetic energy relative
-to any frame in <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt> as the user is allowed to specify the
-reference frame when calling the function. In other words the user is not
-limited to determining just inertial kinetic energy.</p>
-<p>For potential energies, the user must first specify the potential energy of
-every entity of the system using the <tt class="xref py py-mod docutils literal"><span class="pre">set_potential_energy</span></tt> method. The
-potential energy of any number of entities comprising the system can then be
-determined:</p>
-<div class="highlight-python"><pre>&gt;&gt; Pa.set_potential_energy(m * g * h)
-&gt;&gt; A.set_potential_energy(M * g * H)
-&gt;&gt; potential_energy(A, Pa)
-H*M*g + g*h*m</pre>
-</div>
-<p>One can also determine the Lagrangian for this system:</p>
-<div class="highlight-python"><pre>&gt;&gt; Lagrangian(Pa, A)
--H*M*g + M*l1**2*q1d**2/2 - g*h*m + 2*l1**2*m*q1d**2 + q1d**2/2</pre>
-</div>
-<p>Please refer to the docstrings to learn more about each function.</p>
-</div>
-</div>
-
-
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-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Mechanics: Masses, Inertias, Particles and Rigid Bodies</a><ul>
-<li><a class="reference internal" href="#mass">Mass</a></li>
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-<li><a class="reference internal" href="#linear-momentum">Linear Momentum</a></li>
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-<li><a class="reference internal" href="#using-momenta-functions-in-mechanics">Using momenta functions in Mechanics</a></li>
-<li><a class="reference internal" href="#kinetic-energy">Kinetic Energy</a></li>
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-<li><a class="reference internal" href="#lagrangian">Lagrangian</a></li>
-<li><a class="reference internal" href="#using-energy-functions-in-mechanics">Using energy functions in Mechanics</a></li>
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-</li>
-</ul>
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-  <div class="section" id="references-for-physics-mechanics">
-<h1>References for Physics/Mechanics<a class="headerlink" href="#references-for-physics-mechanics" title="Permalink to this headline">¶</a></h1>
-<table class="docutils citation" frame="void" id="kane1983" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Kane1983]</td><td>Kane, Thomas R., Peter W. Likins, and David A. Levinson.
-Spacecraft Dynamics. New York: McGraw-Hill Book, 1983. Print.</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="kane1985" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Kane1985]</td><td>Kane, Thomas R., and David A. Levinson. Dynamics, Theory and
-Applications. New York: McGraw-Hill, 1985. Print.</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="meijaard2007" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Meijaard2007]</td><td>Meijaard, J.P., Jim M. Papadopoulos, Andy Ruina, and A.L.
-Schwab. &#8220;Linearized Dynamics Equations for the Balance and Steer of a Bicycle:
-a Benchmark and Review.&#8221; Proceedings of the Royal Society A: Mathematical,
-Physical and Engineering Sciences 463.2084 (2007): 1955-982. Print.</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="wikidyadics" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[WikiDyadics]</td><td>&#8220;Dyadics.&#8221; Wikipedia, the Free Encyclopedia. Web. 05 Aug.
-2011. &lt;<a class="reference external" href="http://en.wikipedia.org/wiki/Dyadics">http://en.wikipedia.org/wiki/Dyadics</a>&gt;.</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="wikidyadicproducts" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[WikiDyadicProducts]</td><td>&#8220;Dyadic Product.&#8221; Wikipedia, the Free Encyclopedia.
-Web. 05 Aug. 2011. &lt;<a class="reference external" href="http://en.wikipedia.org/wiki/Dyadic_product">http://en.wikipedia.org/wiki/Dyadic_product</a>&gt;.</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="likins1973" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Likins1973]</td><td>Likins, Peter W. Elements of Engineering Mechanics.
-McGraw-Hill, Inc. 1973. Print.</td></tr>
-</tbody>
-</table>
-</div>
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-  <div class="section" id="mechanics-vector-referenceframe">
-<h1>Mechanics: Vector &amp; ReferenceFrame<a class="headerlink" href="#mechanics-vector-referenceframe" title="Permalink to this headline">¶</a></h1>
-<p>In <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt>, vectors and reference frames are the &#8220;building blocks&#8221; of
-dynamic systems. This document will describe these mathematically and describe
-how to use them with this module&#8217;s code.</p>
-<div class="section" id="vector">
-<h2>Vector<a class="headerlink" href="#vector" title="Permalink to this headline">¶</a></h2>
-<p>A vector is a geometric object that has a magnitude (or length) and a
-direction.  Vectors in 3-space are often represented on paper as:</p>
-<div align="center" class="align-center"><img height="175" src="../../../_images/vec_rep.svg" width="350" /></div>
-</div>
-<div class="section" id="vector-algebra">
-<h2>Vector Algebra<a class="headerlink" href="#vector-algebra" title="Permalink to this headline">¶</a></h2>
-<p>Vector algebra is the first topic to be discussed.</p>
-<p>Two vectors are said to be equal if and only if (iff) they have the same same
-magnitude and orientation.</p>
-<div class="section" id="vector-operations">
-<h3>Vector Operations<a class="headerlink" href="#vector-operations" title="Permalink to this headline">¶</a></h3>
-<p>Multiple algebraic operations can be done with vectors: addition between
-vectors, scalar multiplication, and vector multiplication.</p>
-<p>Vector addition as based on the parallelogram law.</p>
-<div align="center" class="align-center"><img height="200" src="../../../_images/vec_add.svg" width="200" /></div>
-<p>Vector addition is also commutative:</p>
-<div class="math">
-\[\begin{split}\mathbf{a} + \mathbf{b} &amp;= \mathbf{b} + \mathbf{a} \\
-(\mathbf{a} + \mathbf{b}) + \mathbf{c} &amp;= \mathbf{a} + (\mathbf{b} +
-\mathbf{c})\end{split}\]</div>
-<p>Scalar multiplication is the product of a vector and a scalar; the result is a
-vector with the same orientation but whose magnitude is scaled by the scalar.
-Note that multiplication by -1 is equivalent to rotating the vector by 180
-degrees about an arbitrary axis in the plane perpendicular to the vector.</p>
-<div align="center" class="align-center"><img height="150" src="../../../_images/vec_mul.svg" width="200" /></div>
-<p>A unit vector is simply a vector whose magnitude is equal to 1.  Given any
-vector <span class="math">\(\mathbf{v}\)</span> we can define a unit vector as:</p>
-<div class="math">
-\[\mathbf{\hat{n}_v} = \frac{\mathbf{v}}{\Vert \mathbf{v} \Vert}\]</div>
-<p>Note that every vector can be written as the product of a scalar and unit
-vector.</p>
-<p>Three vector products are implemented in <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt>: the dot product, the
-cross product, and the outer product.</p>
-<p>The dot product operation maps two vectors to a scalar.  It is defined as:</p>
-<div class="math">
-\[\begin{split}\mathbf{a} \cdot \mathbf{b} = \Vert \mathbf{a} \Vert \Vert \mathbf{b}
-\Vert \cos(\theta)\\\end{split}\]</div>
-<p>where <span class="math">\(\theta\)</span> is the angle between <span class="math">\(\mathbf{a}\)</span> and
-<span class="math">\(\mathbf{b}\)</span>.</p>
-<p>The dot product of two unit vectors represent the magnitude of the common
-direction; for other vectors, it is the product of the magnitude of the common
-direction and the two vectors&#8217; magnitudes. The dot product of two perpendicular
-is zero. The figure below shows some examples:</p>
-<div align="center" class="align-center"><img height="250" src="../../../_images/vec_dot.svg" width="450" /></div>
-<p>The dot product is commutative:</p>
-<div class="math">
-\[\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}\]</div>
-<p>The cross product vector multiplication operation of two vectors returns a
-vector:</p>
-<div class="math">
-\[\mathbf{a} \times \mathbf{b} = \mathbf{c}\]</div>
-<p>The vector <span class="math">\(\mathbf{c}\)</span> has the following properties: it&#8217;s orientation is
-perpendicular to both <span class="math">\(\mathbf{a}\)</span> and <span class="math">\(\mathbf{b}\)</span>, it&#8217;s magnitude
-is defined as <span class="math">\(\Vert \mathbf{c} \Vert = \Vert \mathbf{a} \Vert \Vert
-\mathbf{b} \Vert \sin(\theta)\)</span> (where <span class="math">\(\theta\)</span> is the angle between
-<span class="math">\(\mathbf{a}\)</span> and <span class="math">\(\mathbf{b}\)</span>), and has a sense defined by using
-the right hand rule between <span class="math">\(\Vert \mathbf{a} \Vert \Vert \mathbf{b}
-\Vert\)</span>. The figure below shows this:</p>
-<div align="center" class="align-center"><img height="350" src="../../../_images/vec_cross.svg" width="700" /></div>
-<p>The cross product has the following properties:</p>
-<p>It is not commutative:</p>
-<div class="math">
-\[\begin{split}\mathbf{a} \times \mathbf{b} &amp;\neq \mathbf{b} \times \mathbf{a} \\
-\mathbf{a} \times \mathbf{b} &amp;= - \mathbf{b} \times \mathbf{a}\end{split}\]</div>
-<p>and not associative:</p>
-<div class="math">
-\[(\mathbf{a} \times \mathbf{b} ) \times \mathbf{c} \neq \mathbf{a} \times
-(\mathbf{b} \times \mathbf{c})\]</div>
-<p>Two parallel vectors will have a zero cross product.</p>
-<p>The outer product between two vectors will not be not be discussed here, but
-instead in the inertia section (that is where it is used). Other useful vector
-properties and relationships are:</p>
-<div class="math">
-\[\begin{split}\alpha (\mathbf{a} + \mathbf{b}) &amp;= \alpha \mathbf{a} + \alpha \mathbf{b}\\
-\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) &amp;= \mathbf{a} \cdot \mathbf{b} +
-\mathbf{a} \cdot \mathbf{c}\\
-\mathbf{a} \times (\mathbf{b} + \mathbf{c}) &amp;= \mathbf{a} \times \mathbf{b} +
-\mathbf{a} \times \mathbf{b}\\
-(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} &amp; \textrm{ gives the scalar
-triple product.}\\
-\mathbf{a} \times (\mathbf{b} \cdot \mathbf{c}) &amp; \textrm{ does not work,
-as you cannot cross a vector and a scalar.}\\
-(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} &amp;= \mathbf{a} \cdot
-(\mathbf{b} \times \mathbf{c})\\
-(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} &amp;= (\mathbf{b} \times
-\mathbf{c}) \cdot \mathbf{a} = (\mathbf{c} \times \mathbf{a}) \cdot
-\mathbf{b}\\
-(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} &amp;= \mathbf{b}(\mathbf{a}
-\cdot \mathbf{c}) - \mathbf{a}(\mathbf{b} \cdot \mathbf{c})\\
-\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) &amp;= \mathbf{b}(\mathbf{a}
-\cdot \mathbf{c}) - \mathbf{c}(\mathbf{a} \cdot \mathbf{b})\\\end{split}\]</div>
-</div>
-<div class="section" id="alternative-representation">
-<h3>Alternative Representation<a class="headerlink" href="#alternative-representation" title="Permalink to this headline">¶</a></h3>
-<p>If we have three non-coplanar unit vectors
-<span class="math">\(\mathbf{\hat{n}_x},\mathbf{\hat{n}_y},\mathbf{\hat{n}_z}\)</span>,
-we can represent any vector
-<span class="math">\(\mathbf{a}\)</span> as <span class="math">\(\mathbf{a} = a_x \mathbf{\hat{n}_x} + a_y
-\mathbf{\hat{n}_y} + a_z \mathbf{\hat{n}_z}\)</span>. In this situation
-<span class="math">\(\mathbf{\hat{n}_x},\mathbf{\hat{n}_y},\mathbf{\hat{n}_z}\)</span>
-are referred to as a basis.  <span class="math">\(a_x, a_y, a_z\)</span>
-are called the measure numbers.
-Usually the unit vectors are mutually perpendicular, in which case we can refer
-to them as an orthonormal basis, and they are usually right-handed.</p>
-<p>To test equality between two vectors, now we can do the following. With
-vectors:</p>
-<div class="math">
-\[\begin{split}\mathbf{a} &amp;= a_x \mathbf{\hat{n}_x} + a_y \mathbf{\hat{n}_y} + a_z
-\mathbf{\hat{n}_z}\\
-\mathbf{b} &amp;= b_x \mathbf{\hat{n}_x} + b_y \mathbf{\hat{n}_y} + b_z
-\mathbf{\hat{n}_z}\\\end{split}\]</div>
-<p>We can claim equality if: <span class="math">\(a_x = b_x, a_y = b_y, a_z = b_z\)</span>.</p>
-<p>Vector addition is then represented, for the same two vectors, as:</p>
-<div class="math">
-\[\mathbf{a} + \mathbf{b} = (a_x + b_x)\mathbf{\hat{n}_x} + (a_y + b_y)
-\mathbf{\hat{n}_y} + (a_z + b_z) \mathbf{\hat{n}_z}\]</div>
-<p>Multiplication operations are now defined as:</p>
-<div class="math">
-\[\begin{split}\alpha \mathbf{b} &amp;= \alpha b_x \mathbf{\hat{n}_x} + \alpha b_y
-\mathbf{\hat{n}_y} + \alpha b_z \mathbf{\hat{n}_z}\\
-\mathbf{a} \cdot \mathbf{b} &amp;= a_x b_x + a_y b_y + a_z b_z\\
-\mathbf{a} \times \mathbf{b} &amp;=
-\textrm{det }\begin{bmatrix} \mathbf{\hat{n}_x} &amp; \mathbf{\hat{n}_y} &amp;
-\mathbf{\hat{n}_z} \\ a_x &amp; a_y &amp; a_z \\ a_x &amp; a_y &amp; a_z \end{bmatrix}\\
-(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} &amp;=
-\textrm{det }\begin{bmatrix} a_x &amp; a_y &amp; a_z \\ b_x &amp; b_y &amp; b_z \\ c_x &amp; c_y
-&amp; c_z \end{bmatrix}\\\end{split}\]</div>
-<p>To write a vector in a given basis, we can do the follow:</p>
-<div class="math">
-\[\begin{split}\mathbf{a} = (\mathbf{a}\cdot\mathbf{\hat{n}_x})\mathbf{\hat{n}_x} +
-(\mathbf{a}\cdot\mathbf{\hat{n}_y})\mathbf{\hat{n}_y} +
-(\mathbf{a}\cdot\mathbf{\hat{n}_z})\mathbf{\hat{n}_z}\\\end{split}\]</div>
-</div>
-<div class="section" id="examples">
-<h3>Examples<a class="headerlink" href="#examples" title="Permalink to this headline">¶</a></h3>
-<p>Some numeric examples of these operations follow:</p>
-<div class="math">
-\[\begin{split}\mathbf{a} &amp;= \mathbf{\hat{n}_x} + 5 \mathbf{\hat{n}_y}\\
-\mathbf{b} &amp;= \mathbf{\hat{n}_y} + \alpha \mathbf{\hat{n}_z}\\
-\mathbf{a} + \mathbf{b} &amp;= \mathbf{\hat{n}_x} + 6 \mathbf{\hat{n}_y} + \alpha
-\mathbf{\hat{n}_z}\\
-\mathbf{a} \cdot \mathbf{b} &amp;= 5\\
-\mathbf{a} \cdot \mathbf{\hat{n}_y} &amp;= 5\\
-\mathbf{a} \cdot \mathbf{\hat{n}_z} &amp;= 0\\
-\mathbf{a} \times \mathbf{b} &amp;= 5 \alpha \mathbf{\hat{n}_x} - \alpha
-\mathbf{\hat{n}_y} + \mathbf{\hat{n}_z}\\
-\mathbf{b} \times \mathbf{a} &amp;= -5 \alpha \mathbf{\hat{n}_x} + \alpha
-\mathbf{\hat{n}_y} - \mathbf{\hat{n}_z}\\\end{split}\]</div>
-</div>
-</div>
-<div class="section" id="vector-calculus">
-<h2>Vector Calculus<a class="headerlink" href="#vector-calculus" title="Permalink to this headline">¶</a></h2>
-<p>To deal with the calculus of vectors with moving object, we have to introduce
-the concept of a reference frame. A classic example is a train moving along its
-tracks, with you and a friend inside. If both you and your friend are sitting,
-the relative velocity between the two of you is zero. From an observer outside
-the train, you will both have velocity though.</p>
-<p>We will now apply more rigor to this definition. A reference frame is a virtual
-&#8220;platform&#8221; which we choose to observe vector quantities from. If we have a
-reference frame <span class="math">\(\mathbf{N}\)</span>, vector <span class="math">\(\mathbf{a}\)</span> is said to be
-fixed in the frame <span class="math">\(\mathbf{N}\)</span> if none of its properties ever change
-when observed from <span class="math">\(\mathbf{N}\)</span>. We will typically assign a fixed
-orthonormal basis vector set with each reference frame; <span class="math">\(\mathbf{N}\)</span> will
-have <span class="math">\(\mathbf{\hat{n}_x}, \mathbf{\hat{n}_y},\mathbf{\hat{n}_z}\)</span> as its
-basis vectors.</p>
-<div class="section" id="derivatives-of-vectors">
-<h3>Derivatives of Vectors<a class="headerlink" href="#derivatives-of-vectors" title="Permalink to this headline">¶</a></h3>
-<p>A vector which is not fixed in a reference frame therefore has changing
-properties when observed from that frame. Calculus is the study of change, and
-in order to deal with the peculiarities of vectors fixed and not fixed in
-different reference frames, we need to be more explicit in our definitions.</p>
-<div align="center" class="align-center"><img height="300" src="../../../_images/vec_fix_notfix.svg" width="450" /></div>
-<p>In the above figure, we have vectors <span class="math">\(\mathbf{c,d,e,f}\)</span>. If one were to
-take the derivative of <span class="math">\(\mathbf{e}\)</span> with respect to <span class="math">\(\theta\)</span>:</p>
-<div class="math">
-\[\frac{d \mathbf{e}}{d \theta}\]</div>
-<p>it is not clear what the derivative is. If you are observing from frame
-<span class="math">\(\mathbf{A}\)</span>, it is clearly non-zero. If you are observing from frame
-<span class="math">\(\mathbf{B}\)</span>, the derivative is zero. We will therefore introduce the
-frame as part of the derivative notation:</p>
-<div class="math">
-\[\begin{split}\frac{^{\mathbf{A}} d \mathbf{e}}{d \theta} &amp;\neq 0 \textrm{,
-the derivative of } \mathbf{e} \textrm{ with respect to } \theta
-\textrm{ in the reference frame } \mathbf{A}\\
-\frac{^{\mathbf{B}} d \mathbf{e}}{d \theta} &amp;= 0 \textrm{,
- the derivative of } \mathbf{e} \textrm{ with respect to } \theta
-\textrm{ in the reference frame } \mathbf{B}\\
-\frac{^{\mathbf{A}} d \mathbf{c}}{d \theta} &amp;= 0 \textrm{,
- the derivative of } \mathbf{c} \textrm{ with respect to } \theta
-\textrm{ in the reference frame } \mathbf{A}\\
-\frac{^{\mathbf{B}} d \mathbf{c}}{d \theta} &amp;\neq 0 \textrm{,
- the derivative of } \mathbf{c} \textrm{ with respect to } \theta
-\textrm{ in the reference frame } \mathbf{B}\\\end{split}\]</div>
-<p>Here are some additional properties of derivatives of vectors in specific
-frames:</p>
-<div class="math">
-\[\begin{split}\frac{^{\mathbf{A}} d}{dt}(\mathbf{a} + \mathbf{b}) &amp;= \frac{^{\mathbf{A}}
-d\mathbf{a}}{dt} + \frac{^{\mathbf{A}} d\mathbf{b}}{dt}\\
-\frac{^{\mathbf{A}} d}{dt}\gamma \mathbf{a} &amp;= \frac{ d \gamma}{dt}\mathbf{a}
-+ \gamma\frac{^{\mathbf{A}} d\mathbf{a}}{dt}\\
-\frac{^{\mathbf{A}} d}{dt}(\mathbf{a} \times \mathbf{b}) &amp;=
-\frac{^{\mathbf{A}} d\mathbf{a}}{dt} \times \mathbf{b} +
-\mathbf{a} \times \frac{^{\mathbf{A}} d\mathbf{b}}{dt}\\\end{split}\]</div>
-</div>
-<div class="section" id="relating-sets-of-basis-vectors">
-<h3>Relating Sets of Basis Vectors<a class="headerlink" href="#relating-sets-of-basis-vectors" title="Permalink to this headline">¶</a></h3>
-<p>We need to now define the relationship between two different reference frames;
-or how to relate the basis vectors of one frame to another. We can do this
-using a direction cosine matrix (DCM). The direction cosine matrix relates
-the basis vectors of one frame to another, in the following fashion:</p>
-<div class="math">
-\[\begin{split}\begin{bmatrix}
-\mathbf{\hat{a}_x} \\ \mathbf{\hat{a}_y} \\ \mathbf{\hat{a}_z} \\
-\end{bmatrix}  =
-\begin{bmatrix} ^{\mathbf{A}} \mathbf{C}^{\mathbf{B}} \end{bmatrix}
-\begin{bmatrix}
-\mathbf{\hat{b}_x} \\ \mathbf{\hat{b}_y} \\ \mathbf{\hat{b}_z} \\
-\end{bmatrix}\end{split}\]</div>
-<p>When two frames (say, <span class="math">\(\mathbf{A}\)</span> &amp; <span class="math">\(\mathbf{B}\)</span>) are initially
-aligned, then one frame has all of its basis vectors rotated around an axis
-which is aligned with a basis vector, we say the frames are related by a simple
-rotation. The figure below shows this:</p>
-<div align="center" class="align-center"><img height="250" src="../../../_images/simp_rot.svg" width="250" /></div>
-<p>The above rotation is a simple rotation about the Z axis by an angle
-<span class="math">\(\theta\)</span>. Note that after the rotation, the basis vectors
-<span class="math">\(\mathbf{\hat{a}_z}\)</span> and <span class="math">\(\mathbf{\hat{b}_z}\)</span> are still aligned.</p>
-<p>This rotation can be characterized by the following direction cosine matrix:</p>
-<div class="math">
-\[\begin{split}^{\mathbf{A}}\mathbf{C}^{\mathbf{B}} =
-\begin{bmatrix}
-\cos(\theta) &amp; - \sin(\theta) &amp; 0\\
-\sin(\theta) &amp; \cos(\theta) &amp; 0\\
-0 &amp; 0 &amp; 1\\
-\end{bmatrix}\end{split}\]</div>
-<p>Simple rotations about the X and Y axes are defined by:</p>
-<div class="math">
-\[\begin{split}\textrm{DCM for x-axis rotation: }
-\begin{bmatrix}
-1 &amp; 0 &amp; 0\\
-0 &amp; \cos(\theta) &amp; -\sin(\theta)\\
-0 &amp; \sin(\theta) &amp; \cos(\theta)
-\end{bmatrix}\end{split}\]\[\begin{split}\textrm{DCM for y-axis rotation: }
-\begin{bmatrix}
-\cos(\theta) &amp; 0 &amp; \sin(\theta)\\
-0 &amp; 1 &amp; 0\\
--\sin(\theta) &amp; 0 &amp; \cos(\theta)\\
-\end{bmatrix}\end{split}\]</div>
-<p>Rotation in the positive direction here will be defined by using the right-hand
-rule.</p>
-<p>The direction cosine matrix is also involved with the definition of the dot
-product between sets of basis vectors. If we have two reference frames with
-associated basis vectors, their direction cosine matrix can be defined as:</p>
-<div class="math">
-\[\begin{split}\begin{bmatrix}
-C_{xx} &amp; C_{xy} &amp; C_{xz}\\
-C_{yx} &amp; C_{yy} &amp; C_{yz}\\
-C_{zx} &amp; C_{zy} &amp; C_{zz}\\
-\end{bmatrix} =
-\begin{bmatrix}
-\mathbf{\hat{a}_x}\cdot\mathbf{\hat{b}_x} &amp;
-\mathbf{\hat{a}_x}\cdot\mathbf{\hat{b}_y} &amp;
-\mathbf{\hat{a}_x}\cdot\mathbf{\hat{b}_z}\\
-\mathbf{\hat{a}_y}\cdot\mathbf{\hat{b}_x} &amp;
-\mathbf{\hat{a}_y}\cdot\mathbf{\hat{b}_y} &amp;
-\mathbf{\hat{a}_y}\cdot\mathbf{\hat{b}_z}\\
-\mathbf{\hat{a}_z}\cdot\mathbf{\hat{b}_x} &amp;
-\mathbf{\hat{a}_z}\cdot\mathbf{\hat{b}_y} &amp;
-\mathbf{\hat{a}_z}\cdot\mathbf{\hat{b}_z}\\
-\end{bmatrix}\end{split}\]</div>
-<p>Additionally, the direction cosine matrix is orthogonal, in that:</p>
-<div class="math">
-\[\begin{split}^{\mathbf{A}}\mathbf{C}^{\mathbf{B}} =
-(^{\mathbf{B}}\mathbf{C}^{\mathbf{A}})^{-1}\\ =
-(^{\mathbf{B}}\mathbf{C}^{\mathbf{A}})^T\\\end{split}\]</div>
-<p>If we have reference frames <span class="math">\(\mathbf{A}\)</span> and <span class="math">\(\mathbf{B}\)</span>, which in
-this example have undergone a simple z-axis rotation by an amount
-<span class="math">\(\theta\)</span>, we will have two sets of basis vectors. We can then define two
-vectors: <span class="math">\(\mathbf{a} = \mathbf{\hat{a}_x} + \mathbf{\hat{a}_y} +
-\mathbf{\hat{a}_z}\)</span> and <span class="math">\(\mathbf{b} = \mathbf{\hat{b}_x} +
-\mathbf{\hat{b}_y} + \mathbf{\hat{b}_z}\)</span>. If we wish to express
-<span class="math">\(\mathbf{b}\)</span> in the <span class="math">\(\mathbf{A}\)</span> frame, we do the following:</p>
-<div class="math">
-\[\begin{split}\mathbf{b} &amp;= \mathbf{\hat{b}_x} + \mathbf{\hat{b}_y} + \mathbf{\hat{b}_z}\\
-\mathbf{b} &amp;= \begin{bmatrix}\mathbf{\hat{a}_x}\cdot (\mathbf{\hat{b}_x} +
-\mathbf{\hat{b}_y} + \mathbf{\hat{b}_z})\end{bmatrix} \mathbf{\hat{a}_x} +
-\begin{bmatrix}\mathbf{\hat{a}_y}\cdot (\mathbf{\hat{b}_x} + \mathbf{\hat{b}_y}
-+ \mathbf{\hat{b}_z})\end{bmatrix} \mathbf{\hat{a}_y} +
-\begin{bmatrix}\mathbf{\hat{a}_z}\cdot (\mathbf{\hat{b}_x} + \mathbf{\hat{b}_y}
-+ \mathbf{\hat{b}_z})\end{bmatrix} \mathbf{\hat{a}_z}\\ \mathbf{b} &amp;=
-(\cos(\theta) - \sin(\theta))\mathbf{\hat{a}_x} +
-(\sin(\theta) + \cos(\theta))\mathbf{\hat{a}_y} + \mathbf{\hat{a}_z}\end{split}\]</div>
-<p>And if we wish to express <span class="math">\(\mathbf{a}\)</span> in the <span class="math">\(\mathbf{B}\)</span>, we do:</p>
-<div class="math">
-\[\begin{split}\mathbf{a} &amp;= \mathbf{\hat{a}_x} + \mathbf{\hat{a}_y} + \mathbf{\hat{a}_z}\\
-\mathbf{a} &amp;= \begin{bmatrix}\mathbf{\hat{b}_x}\cdot (\mathbf{\hat{a}_x} +
-\mathbf{\hat{a}_y} + \mathbf{\hat{a}_z})\end{bmatrix} \mathbf{\hat{b}_x} +
-\begin{bmatrix}\mathbf{\hat{b}_y}\cdot (\mathbf{\hat{a}_x} +
-\mathbf{\hat{a}_y} + \mathbf{\hat{a}_z})\end{bmatrix} \mathbf{\hat{b}_y} +
-\begin{bmatrix}\mathbf{\hat{b}_z}\cdot (\mathbf{\hat{a}_x} +
-\mathbf{\hat{a}_y} + \mathbf{\hat{a}_z})\end{bmatrix} \mathbf{\hat{b}_z}\\
-\mathbf{a} &amp;= (\cos(\theta) + \sin(\theta))\mathbf{\hat{b}_x} +
-(-\sin(\theta)+\cos(\theta))\mathbf{\hat{b}_y} + \mathbf{\hat{b}_z}\end{split}\]</div>
-</div>
-<div class="section" id="derivatives-with-multiple-frames">
-<h3>Derivatives with Multiple Frames<a class="headerlink" href="#derivatives-with-multiple-frames" title="Permalink to this headline">¶</a></h3>
-<p>If we have reference frames <span class="math">\(\mathbf{A}\)</span> and <span class="math">\(\mathbf{B}\)</span>
-we will have two sets of basis vectors. We can then define two
-vectors: <span class="math">\(\mathbf{a} = a_x\mathbf{\hat{a}_x} + a_y\mathbf{\hat{a}_y} +
-a_z\mathbf{\hat{a}_z}\)</span> and <span class="math">\(\mathbf{b} = b_x\mathbf{\hat{b}_x} +
-b_y\mathbf{\hat{b}_y} + b_z\mathbf{\hat{b}_z}\)</span>. If we want to take the
-derivative of <span class="math">\(\mathbf{b}\)</span> in the reference frame <span class="math">\(\mathbf{A}\)</span>, we
-must first express it in <span class="math">\(\mathbf{A}\)</span>, and the take the derivatives of
-the measure numbers:</p>
-<div class="math">
-\[\frac{^{\mathbf{A}} d\mathbf{b}}{dx} = \frac{d (\mathbf{b}\cdot
-\mathbf{\hat{a}_x} )}{dx} \mathbf{\hat{a}_x} + \frac{d (\mathbf{b}\cdot
-\mathbf{\hat{a}_y} )}{dx} \mathbf{\hat{a}_y} + \frac{d (\mathbf{b}\cdot
-\mathbf{\hat{a}_z} )}{dx} \mathbf{\hat{a}_z} +\]</div>
-</div>
-<div class="section" id="id1">
-<h3>Examples<a class="headerlink" href="#id1" title="Permalink to this headline">¶</a></h3>
-<p>An example of vector calculus:</p>
-<div align="center" class="align-center"><img height="500" src="../../../_images/vec_simp_der.svg" width="350" /></div>
-<p>In this example we have two bodies, each with an attached reference frame.
-We will say that <span class="math">\(\theta\)</span> and <span class="math">\(x\)</span> are functions of time.
-We wish to know the time derivative of vector <span class="math">\(\mathbf{c}\)</span> in both the
-<span class="math">\(\mathbf{A}\)</span> and <span class="math">\(\mathbf{B}\)</span> frames.</p>
-<p>First, we need to define <span class="math">\(\mathbf{c}\)</span>;
-<span class="math">\(\mathbf{c}=x\mathbf{\hat{b}_x}+l\mathbf{\hat{b}_y}\)</span>. This provides a
-definition in the <span class="math">\(\mathbf{B}\)</span> frame. We can now do the following:</p>
-<div class="math">
-\[\begin{split}\frac{^{\mathbf{B}} d \mathbf{c}}{dt} &amp;= \frac{dx}{dt} \mathbf{\hat{b}_x} +
-\frac{dl}{dt} \mathbf{\hat{b}_y}\\
-&amp;= \dot{x} \mathbf{\hat{b}_x}\end{split}\]</div>
-<p>To take the derivative in the <span class="math">\(\mathbf{A}\)</span> frame, we have to first relate
-the two frames:</p>
-<div class="math">
-\[\begin{split}^{\mathbf{A}} \mathbf{C} ^{\mathbf{B}} =
-\begin{bmatrix}
-\cos(\theta) &amp; 0 &amp; \sin(\theta)\\
-0 &amp; 1 &amp; 0\\
--\sin(\theta) &amp; 0 &amp; \cos(\theta)\\
-\end{bmatrix}\end{split}\]</div>
-<p>Now we can do the following:</p>
-<div class="math">
-\[\begin{split}\frac{^{\mathbf{A}} d \mathbf{c}}{dt} &amp;= \frac{d (\mathbf{c} \cdot
-\mathbf{\hat{a}_x})}{dt} \mathbf{\hat{a}_x} + \frac{d (\mathbf{c} \cdot
-\mathbf{\hat{a}_y})}{dt} \mathbf{\hat{a}_y} + \frac{d (\mathbf{c} \cdot
-\mathbf{\hat{a}_z})}{dt} \mathbf{\hat{a}_z}\\
-&amp;= \frac{d (\cos(\theta) x)}{dt} \mathbf{\hat{a}_x} +
-\frac{d (l)}{dt} \mathbf{\hat{a}_y} +
-\frac{d (-\sin(\theta) x)}{dt} \mathbf{\hat{a}_z}\\
-&amp;= (-\dot{\theta}\sin(\theta)x + \cos(\theta)\dot{x}) \mathbf{\hat{a}_x} +
-(\dot{\theta}\cos(\theta)x + \sin(\theta)\dot{x}) \mathbf{\hat{a}_z}\end{split}\]</div>
-<p>Note that this is the time derivative of <span class="math">\(\mathbf{c}\)</span> in
-<span class="math">\(\mathbf{A}\)</span>, and is expressed in the <span class="math">\(\mathbf{A}\)</span> frame. We can
-express it in the <span class="math">\(\mathbf{B}\)</span> frame however, and the expression will
-still be valid:</p>
-<div class="math">
-\[\begin{split}\frac{^{\mathbf{A}} d \mathbf{c}}{dt} &amp;= (-\dot{\theta}\sin(\theta)x +
-\cos(\theta)\dot{x}) \mathbf{\hat{a}_x} + (\dot{\theta}\cos(\theta)x +
-\sin(\theta)\dot{x}) \mathbf{\hat{a}_z}\\
-&amp;= \dot{x}\mathbf{\hat{b}_x} - \theta x \mathbf{\hat{b}_z}\\\end{split}\]</div>
-<p>Note the difference in expression complexity between the two forms. They are
-equivalent, but one is much simpler. This is an extremely important concept, as
-defining vectors in the more complex forms can vastly slow down formulation of
-the equations of motion and increase their length, sometimes to a point where
-they cannot be shown on screen.</p>
-</div>
-</div>
-<div class="section" id="using-vectors-and-reference-frames-in-sympy-s-mechanics">
-<h2>Using Vectors and Reference Frames in SymPy&#8217;s Mechanics<a class="headerlink" href="#using-vectors-and-reference-frames-in-sympy-s-mechanics" title="Permalink to this headline">¶</a></h2>
-<p>We have waited until after all of the relevant mathematical relationships have
-been defined for vectors and reference frames to introduce code. This is due to
-how vectors are formed. When starting any problem in <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt>, one of
-the first steps is defining a reference frame (remember to import
-sympy.physics.mechanics first):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.mechanics</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Now we have created a reference frame, <span class="math">\(\mathbf{N}\)</span>. To have access to
-any basis vectors, first a reference frame needs to be created. Now that we
-have made and object representing <span class="math">\(\mathbf{N}\)</span>, we can access its basis
-vectors:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">x</span>
-<span class="go">N.x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">y</span>
-<span class="go">N.y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">z</span>
-<span class="go">N.z</span>
-</pre></div>
-</div>
-<div class="section" id="vector-algebra-in-mechanics">
-<h3>Vector Algebra, in Mechanics<a class="headerlink" href="#vector-algebra-in-mechanics" title="Permalink to this headline">¶</a></h3>
-<p>We can now do basic algebraic operations on these vectors.:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">x</span> <span class="o">==</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">x</span> <span class="o">==</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span>
-<span class="go">N.x + N.y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span>
-<span class="go">2*N.x + N.y</span>
-</pre></div>
-</div>
-<p>Remember, don&#8217;t add a scalar quantity to a vector (<tt class="docutils literal"><span class="pre">N.x</span> <span class="pre">+</span> <span class="pre">5</span></tt>); this will
-raise an error. At this point, we&#8217;ll use SymPy&#8217;s Symbol in our vectors.
-Remember to refer to SymPy&#8217;s Gotchas and Pitfalls when dealing with symbols.:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span>
-<span class="go">x*N.x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="go">x*N.x + x*N.y</span>
-</pre></div>
-</div>
-<p>In <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt> multiple interfaces to vector multiplication have been
-implemented, at the operator level, method level, and function level. The
-vector dot product can work as follows:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">x</span> <span class="o">&amp;</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">x</span> <span class="o">&amp;</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dot</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dot</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>The &#8220;official&#8221; interface is the function interface; this is what will be used
-in all examples. This is to avoid confusion with the attribute and methods
-being next to each other, and in the case of the operator operation priority.
-The operators used in <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt> for vector multiplication do not posses
-the correct order of operations; this can lead to errors. Care with parentheses
-is needed when using operators to represent vector multiplication.</p>
-<p>The cross product is the other vector multiplication which will be discussed
-here. It offers similar interfaces to the dot product, and comes with the same
-warnings.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">x</span> <span class="o">^</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">x</span> <span class="o">^</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span>
-<span class="go">N.z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">cross</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="o">.</span><span class="n">cross</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="go">- N.y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cross</span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
-<span class="go">N.z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">x</span> <span class="o">^</span> <span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">y</span> <span class="o">+</span> <span class="n">N</span><span class="o">.</span><span class="n">z</span><span class="p">)</span>
-<span class="go">- N.y + N.z</span>
-</pre></div>
-</div>
-<p>Two additional operations can be done with vectors: normalizing the vector to
-length 1, and getting its magnitude. These are done as follows:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">normalize</span><span class="p">()</span>
-<span class="go">sqrt(2)/2*N.x + sqrt(2)/2*N.y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">N</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">magnitude</span><span class="p">()</span>
-<span class="go">sqrt(2)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="vector-calculus-in-mechanics">
-<h3>Vector Calculus, in Mechanics<a class="headerlink" href="#vector-calculus-in-mechanics" title="Permalink to this headline">¶</a></h3>
-<p>We have already introduced our first reference frame. We can take the
-derivative in that frame right now, if we desire:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">*</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
-<span class="go">N.x</span>
-</pre></div>
-</div>
-<p>SymPy has a <tt class="docutils literal"><span class="pre">diff</span></tt> function, but it does not currently work with
-<tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt> Vectors, so please use <tt class="docutils literal"><span class="pre">Vector</span></tt>&#8216;s <tt class="docutils literal"><span class="pre">diff</span></tt> method.  The
-reason for this is that when differentiating a <tt class="docutils literal"><span class="pre">Vector</span></tt>, the frame of
-reference must be specified in addition to what you are taking the derivative
-with respect to; SymPy&#8217;s <tt class="docutils literal"><span class="pre">diff</span></tt> function doesn&#8217;t fit this mold.</p>
-<p>The more interesting case arise with multiple reference frames. If we introduce
-a second reference frame, <span class="math">\(\mathbf{A}\)</span>, we now have two frames. Note that
-at this point we can add components of <span class="math">\(\mathbf{N}\)</span> and
-<span class="math">\(\mathbf{A}\)</span> together, but cannot perform vector multiplication, as no
-relationship between the two frames has been defined.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">x</span> <span class="o">+</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span>
-<span class="go">A.x + N.x</span>
-</pre></div>
-</div>
-<p>If we want to do vector multiplication, first we have to define and
-orientation. The <tt class="docutils literal"><span class="pre">orient</span></tt> method of <tt class="docutils literal"><span class="pre">ReferenceFrame</span></tt> provides that
-functionality.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">orient</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">y</span><span class="p">])</span>
-</pre></div>
-</div>
-<p>If we desire, we can view the DCM between these two frames at any time. This
-can be calculated with the <tt class="docutils literal"><span class="pre">dcm</span></tt> method. This code: <tt class="docutils literal"><span class="pre">N.dcm(A)</span></tt> gives the
-dcm <span class="math">\(^{\mathbf{N}} \mathbf{C} ^{\mathbf{A}}\)</span>.</p>
-<p>This orients the <span class="math">\(\mathbf{A}\)</span> frame relative to the <span class="math">\(\mathbf{N}\)</span>
-frame by a simple rotation around the Y axis, by an amount x. Other, more
-complicated rotation types include Body rotations, Space rotations,
-quaternions, and arbitrary axis rotations. Body and space rotations are
-equivalent to doing 3 simple rotations in a row, each about a basis vector in
-the new frame. An example follows:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Bp</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;Bp&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Bpp</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;Bpp&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span><span class="p">,</span><span class="n">q2</span><span class="p">,</span><span class="n">q3</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;q1 q2 q3&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Bpp</span><span class="o">.</span><span class="n">orient</span><span class="p">(</span><span class="n">N</span><span class="p">,</span><span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Bp</span><span class="o">.</span><span class="n">orient</span><span class="p">(</span><span class="n">Bpp</span><span class="p">,</span><span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q2</span><span class="p">,</span> <span class="n">Bpp</span><span class="o">.</span><span class="n">y</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">orient</span><span class="p">(</span><span class="n">Bp</span><span class="p">,</span><span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q3</span><span class="p">,</span> <span class="n">Bp</span><span class="o">.</span><span class="n">z</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">dcm</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
-<span class="go">[                          cos(q2)*cos(q3),                           -sin(q3)*cos(q2),          sin(q2)]</span>
-<span class="go">[sin(q1)*sin(q2)*cos(q3) + sin(q3)*cos(q1), -sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3), -sin(q1)*cos(q2)]</span>
-<span class="go">[sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3),  sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1),  cos(q1)*cos(q2)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">orient</span><span class="p">(</span><span class="n">N</span><span class="p">,</span><span class="s">&#39;Body&#39;</span><span class="p">,[</span><span class="n">q1</span><span class="p">,</span><span class="n">q2</span><span class="p">,</span><span class="n">q3</span><span class="p">],</span><span class="s">&#39;XYZ&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">dcm</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
-<span class="go">[                          cos(q2)*cos(q3),                           -sin(q3)*cos(q2),          sin(q2)]</span>
-<span class="go">[sin(q1)*sin(q2)*cos(q3) + sin(q3)*cos(q1), -sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3), -sin(q1)*cos(q2)]</span>
-<span class="go">[sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3),  sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1),  cos(q1)*cos(q2)]</span>
-</pre></div>
-</div>
-<p>Space orientations are similar to body orientation, but applied from the frame
-to body. Body and space rotations can involve either two or three axes: &#8216;XYZ&#8217;
-works, as does &#8216;YZX&#8217;, &#8216;ZXZ&#8217;, &#8216;YXY&#8217;, etc. What is key is that each simple
-rotation is about a different axis than the previous one; &#8216;ZZX&#8217; does not
-completely orient a set of basis vectors in 3 space.</p>
-<p>Sometimes it will be more convenient to create a new reference frame and orient
-relative to an existing one in one step. The <tt class="docutils literal"><span class="pre">orientnew</span></tt> method allows for
-this functionality, and essentially wraps the <tt class="docutils literal"><span class="pre">orient</span></tt> method. All of the
-things you can do in <tt class="docutils literal"><span class="pre">orient</span></tt>, you can do in <tt class="docutils literal"><span class="pre">orientnew</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;C&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">])</span>
-</pre></div>
-</div>
-<p>Quaternions (or Euler Parameters) use 4 value to characterize the orientation
-of the frame. This and arbitrary axis rotations are described in the <tt class="docutils literal"><span class="pre">orient</span></tt>
-and <tt class="docutils literal"><span class="pre">orientnew</span></tt> method help, or in the references <a class="reference internal" href="../mechanics/reference.html#kane1983">[Kane1983]</a>.</p>
-<p>Finally, before starting multiframe calculus operations, we will introduce
-another <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt> tool: <tt class="docutils literal"><span class="pre">dynamicsymbols</span></tt>. <tt class="docutils literal"><span class="pre">dynamicsymbols</span></tt> is
-a shortcut function to create undefined functions of time within SymPy. The
-derivative of such a &#8216;dynamicsymbol&#8217; is shown below.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">diff</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span><span class="p">,</span> <span class="n">q2</span><span class="p">,</span> <span class="n">q3</span> <span class="o">=</span> <span class="n">dynamicsymbols</span><span class="p">(</span><span class="s">&#39;q1 q2 q3&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">q1</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">))</span>
-<span class="go">Derivative(q1(t), t)</span>
-</pre></div>
-</div>
-<p>The &#8216;dynamicsymbol&#8217; printing is not very clear above; we will also introduce a
-few other tools here. We can use <tt class="docutils literal"><span class="pre">mprint</span></tt> instead of print for
-non-interactive sessions.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span>
-<span class="go">q1(t)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1d</span> <span class="o">=</span> <span class="n">diff</span><span class="p">(</span><span class="n">q1</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mprint</span><span class="p">(</span><span class="n">q1</span><span class="p">)</span>
-<span class="go">q1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">mprint</span><span class="p">(</span><span class="n">q1d</span><span class="p">)</span>
-<span class="go">q1&#39;</span>
-</pre></div>
-</div>
-<p>For interactive sessions use <tt class="docutils literal"><span class="pre">mechanics_printing</span></tt>. There also exist analogs
-for SymPy&#8217;s <tt class="docutils literal"><span class="pre">pprint</span></tt>, <tt class="docutils literal"><span class="pre">mpprint</span></tt>, and <tt class="docutils literal"><span class="pre">latex</span></tt>, <tt class="docutils literal"><span class="pre">mlatex</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mechanics_printing</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1</span>
-<span class="go">q1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q1d</span>
-<span class="go">q1&#39;</span>
-</pre></div>
-</div>
-<p>A &#8216;dynamicsymbol&#8217; should be used to represent any time varying quantity in
-<tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt>, whether it is a coordinate, varying position, or force.  The
-primary use of a &#8216;dynamicsymbol&#8217; is for speeds and coordinates (of which there
-will be more discussion in the Kinematics Section of the documentation).</p>
-<p>Now we will define the orientation of our new frames with a &#8216;dynamicsymbol&#8217;,
-and can take derivatives and time derivatives with ease. Some examples follow.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">ReferenceFrame</span><span class="p">(</span><span class="s">&#39;N&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">N</span><span class="o">.</span><span class="n">orientnew</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="s">&#39;Axis&#39;</span><span class="p">,</span> <span class="p">[</span><span class="n">q1</span><span class="p">,</span> <span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">B</span><span class="o">.</span><span class="n">y</span><span class="o">*</span><span class="n">q2</span> <span class="o">+</span> <span class="n">B</span><span class="o">.</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">q2</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
-<span class="go">B.y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">B</span><span class="o">.</span><span class="n">y</span><span class="o">*</span><span class="n">q2</span> <span class="o">+</span> <span class="n">B</span><span class="o">.</span><span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">dt</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">(-q1&#39; + q2&#39;)*B.y + q2*q1&#39;*B.z</span>
-</pre></div>
-</div>
-<p>Note that the output vectors are kept in the same frames that they were
-provided in. This remains true for vectors with components made of basis
-vectors from multiple frames:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">B</span><span class="o">.</span><span class="n">y</span><span class="o">*</span><span class="n">q2</span> <span class="o">+</span> <span class="n">B</span><span class="o">.</span><span class="n">z</span> <span class="o">+</span> <span class="n">q2</span><span class="o">*</span><span class="n">N</span><span class="o">.</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">q2</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
-<span class="go">B.y + N.x</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="how-vectors-are-coded-in-mechanics">
-<h3>How Vectors are Coded in Mechanics<a class="headerlink" href="#how-vectors-are-coded-in-mechanics" title="Permalink to this headline">¶</a></h3>
-<p>What follows is a short description of how vectors are defined by the code in
-<tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt>. It is provided for those who want to learn more about how
-this part of <tt class="xref py py-mod docutils literal"><span class="pre">mechanics</span></tt> works, and does not need to be read to use this
-module; don&#8217;t read it unless you want to learn how this module was implemented.</p>
-<p>Every <tt class="docutils literal"><span class="pre">Vector</span></tt>&#8216;s main information is stored in the <tt class="docutils literal"><span class="pre">args</span></tt> attribute, which
-stores the three measure numbers for each basis vector in a frame, for every
-relevant frame. A vector does not exist in code until a <tt class="docutils literal"><span class="pre">ReferenceFrame</span></tt>
-is created. At this point, the <tt class="docutils literal"><span class="pre">x</span></tt>, <tt class="docutils literal"><span class="pre">y</span></tt>, and <tt class="docutils literal"><span class="pre">z</span></tt> attributes of the
-reference frame are immutable <tt class="docutils literal"><span class="pre">Vector</span></tt>&#8216;s which have measure numbers of
-[1,0,0], [0,1,0], and [0,0,1] associated with that <tt class="docutils literal"><span class="pre">ReferenceFrame</span></tt>. Once
-these vectors are accessible, new vectors can be created by doing algebraic
-operations with the basis vectors. A vector can have components from multiple
-frames though. That is why <tt class="docutils literal"><span class="pre">args</span></tt> is a list; it has as many elements in the
-list as there are unique <tt class="docutils literal"><span class="pre">ReferenceFrames</span></tt> in its components, i.e. if there
-are <tt class="docutils literal"><span class="pre">A</span></tt> and <tt class="docutils literal"><span class="pre">B</span></tt> frame basis vectors in our new vector, <tt class="docutils literal"><span class="pre">args</span></tt> is of
-length 2; if it has <tt class="docutils literal"><span class="pre">A</span></tt>, <tt class="docutils literal"><span class="pre">B</span></tt>, and <tt class="docutils literal"><span class="pre">C</span></tt> frame basis vector, <tt class="docutils literal"><span class="pre">args</span></tt> is of
-length three.</p>
-<p>Each element in the <tt class="docutils literal"><span class="pre">args</span></tt> list is a 2-tuple; the first element is a SymPy
-<tt class="docutils literal"><span class="pre">Matrix</span></tt> (this is where the measure numbers for each set of basis vectors are
-stored) and the second element is a <tt class="docutils literal"><span class="pre">ReferenceFrame</span></tt> to associate those
-measure numbers with.</p>
-<p><tt class="docutils literal"><span class="pre">ReferenceFrame</span></tt> stores a few things. First, it stores the name you supply it
-on creation (<tt class="docutils literal"><span class="pre">name</span></tt> attribute). It also stores the direction cosine matrices,
-defined upon creation with the <tt class="docutils literal"><span class="pre">orientnew</span></tt> method, or calling the <tt class="docutils literal"><span class="pre">orient</span></tt>
-method after creation. The direction cosine matrices are represented by SymPy&#8217;s
-<tt class="docutils literal"><span class="pre">Matrix</span></tt>, and are part of a dictionary where the keys are the
-<tt class="docutils literal"><span class="pre">ReferenceFrame</span></tt> and the value the <tt class="docutils literal"><span class="pre">Matrix</span></tt>; these are set
-bi-directionally; in that when you orient <tt class="docutils literal"><span class="pre">A</span></tt> to <tt class="docutils literal"><span class="pre">N</span></tt> you are setting <tt class="docutils literal"><span class="pre">A</span></tt>&#8216;s
-orientation dictionary to include <tt class="docutils literal"><span class="pre">N</span></tt> and its <tt class="docutils literal"><span class="pre">Matrix</span></tt>, but also you also
-are setting <tt class="docutils literal"><span class="pre">N</span></tt>&#8216;s orientation dictionary to include <tt class="docutils literal"><span class="pre">A</span></tt> and its <tt class="docutils literal"><span class="pre">Matrix</span></tt>
-(that DCM being the transpose of the other).</p>
-</div>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Mechanics: Vector &amp; ReferenceFrame</a><ul>
-<li><a class="reference internal" href="#vector">Vector</a></li>
-<li><a class="reference internal" href="#vector-algebra">Vector Algebra</a><ul>
-<li><a class="reference internal" href="#vector-operations">Vector Operations</a></li>
-<li><a class="reference internal" href="#alternative-representation">Alternative Representation</a></li>
-<li><a class="reference internal" href="#examples">Examples</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#vector-calculus">Vector Calculus</a><ul>
-<li><a class="reference internal" href="#derivatives-of-vectors">Derivatives of Vectors</a></li>
-<li><a class="reference internal" href="#relating-sets-of-basis-vectors">Relating Sets of Basis Vectors</a></li>
-<li><a class="reference internal" href="#derivatives-with-multiple-frames">Derivatives with Multiple Frames</a></li>
-<li><a class="reference internal" href="#id1">Examples</a></li>
-</ul>
-</li>
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-  <div class="section" id="module-sympy.physics.paulialgebra">
-<span id="pauli-algebra"></span><h1>Pauli Algebra<a class="headerlink" href="#module-sympy.physics.paulialgebra" title="Permalink to this headline">¶</a></h1>
-<p>This module implements Pauli algebra by subclassing Symbol. Only algebraic
-properties of Pauli matrices are used (we don&#8217;t use the Matrix class).</p>
-<p>See the documentation to the class Pauli for examples.</p>
-<dl class="docutils">
-<dt>See also:</dt>
-<dd><a class="reference external" href="http://en.wikipedia.org/wiki/Pauli_matrices">http://en.wikipedia.org/wiki/Pauli_matrices</a></dd>
-</dl>
-<dl class="function">
-<dt id="sympy.physics.paulialgebra.evaluate_pauli_product">
-<tt class="descclassname">sympy.physics.paulialgebra.</tt><tt class="descname">evaluate_pauli_product</tt><big>(</big><em>arg</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/paulialgebra.html#evaluate_pauli_product"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.paulialgebra.evaluate_pauli_product" title="Permalink to this definition">¶</a></dt>
-<dd><p>Help function to evaluate Pauli matrices product
-with symbolic objects</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>arg: symbolic expression that contains Paulimatrices</strong> :</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.paulialgebra</span> <span class="kn">import</span> <span class="n">Pauli</span><span class="p">,</span> <span class="n">evaluate_pauli_product</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">evaluate_pauli_product</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">Pauli</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">Pauli</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">-sigma3</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">evaluate_pauli_product</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">Pauli</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">Pauli</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span>
-<span class="go">-I*x**2*sigma3</span>
-</pre></div>
-</div>
-</dd></dl>
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-  <div class="section" id="module-sympy.physics.qho_1d">
-<span id="quantum-harmonic-oscillator-in-1-d"></span><h1>Quantum Harmonic Oscillator in 1-D<a class="headerlink" href="#module-sympy.physics.qho_1d" title="Permalink to this headline">¶</a></h1>
-<dl class="function">
-<dt id="sympy.physics.qho_1d.E_n">
-<tt class="descclassname">sympy.physics.qho_1d.</tt><tt class="descname">E_n</tt><big>(</big><em>n</em>, <em>omega</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/qho_1d.html#E_n"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.qho_1d.E_n" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the Energy of the One-dimensional harmonic oscillator</p>
-<dl class="docutils">
-<dt><tt class="docutils literal"><span class="pre">n</span></tt></dt>
-<dd>the &#8220;nodal&#8221; quantum number</dd>
-<dt><tt class="docutils literal"><span class="pre">omega</span></tt></dt>
-<dd>the harmonic oscillator angular frequency</dd>
-</dl>
-<p>The unit of the returned value matches the unit of hw, since the energy is
-calculated as:</p>
-<blockquote>
-<div>E_n = hbar * omega*(n + 1/2)</div></blockquote>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.qho_1d</span> <span class="kn">import</span> <span class="n">E_n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">var</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&quot;x omega&quot;</span><span class="p">)</span>
-<span class="go">(x, omega)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E_n</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">omega</span><span class="p">)</span>
-<span class="go">hbar*omega*(x + 1/2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.qho_1d.psi_n">
-<tt class="descclassname">sympy.physics.qho_1d.</tt><tt class="descname">psi_n</tt><big>(</big><em>n</em>, <em>x</em>, <em>m</em>, <em>omega</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/qho_1d.html#psi_n"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.qho_1d.psi_n" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the wavefunction psi_{n} for the One-dimensional harmonic oscillator.</p>
-<dl class="docutils">
-<dt><tt class="docutils literal"><span class="pre">n</span></tt></dt>
-<dd>the &#8220;nodal&#8221; quantum number.  Corresponds to the number of nodes in the
-wavefunction.  n &gt;= 0</dd>
-<dt><tt class="docutils literal"><span class="pre">x</span></tt></dt>
-<dd>x coordinate</dd>
-<dt><tt class="docutils literal"><span class="pre">m</span></tt></dt>
-<dd>mass of the particle</dd>
-<dt><tt class="docutils literal"><span class="pre">omega</span></tt></dt>
-<dd>angular frequency of the oscillator</dd>
-</dl>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.qho_1d</span> <span class="kn">import</span> <span class="n">psi_n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">var</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&quot;x m omega&quot;</span><span class="p">)</span>
-<span class="go">(x, m, omega)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">psi_n</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">omega</span><span class="p">)</span>
-<span class="go">(m*omega)**(1/4)*exp(-m*omega*x**2/(2*hbar))/(hbar**(1/4)*pi**(1/4))</span>
-</pre></div>
-</div>
-</dd></dl>
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-  <h4>Previous topic</h4>
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-                        title="previous chapter">Pauli Algebra</a></p>
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-                        title="next chapter">Quantum Harmonic Oscillator in 3-D</a></p>
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-  <div class="section" id="module-sympy.physics.quantum.anticommutator">
-<span id="anticommutator"></span><h1>Anticommutator<a class="headerlink" href="#module-sympy.physics.quantum.anticommutator" title="Permalink to this headline">¶</a></h1>
-<p>The anti-commutator: <tt class="docutils literal"><span class="pre">{A,B}</span> <span class="pre">=</span> <span class="pre">A*B</span> <span class="pre">+</span> <span class="pre">B*A</span></tt>.</p>
-<dl class="class">
-<dt id="sympy.physics.quantum.anticommutator.AntiCommutator">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.anticommutator.</tt><tt class="descname">AntiCommutator</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/anticommutator.html#AntiCommutator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.anticommutator.AntiCommutator" title="Permalink to this definition">¶</a></dt>
-<dd><p>The standard anticommutator, in an unevaluated state.</p>
-<p>Evaluating an anticommutator is defined <a class="reference internal" href="#r74">[R74]</a> as: <tt class="docutils literal"><span class="pre">{A,</span> <span class="pre">B}</span> <span class="pre">=</span> <span class="pre">A*B</span> <span class="pre">+</span> <span class="pre">B*A</span></tt>.
-This class returns the anticommutator in an unevaluated form.  To evaluate
-the anticommutator, use the <tt class="docutils literal"><span class="pre">.doit()</span></tt> method.</p>
-<p>Cannonical ordering of an anticommutator is <tt class="docutils literal"><span class="pre">{A,</span> <span class="pre">B}</span></tt> for <tt class="docutils literal"><span class="pre">A</span> <span class="pre">&lt;</span> <span class="pre">B</span></tt>. The
-arguments of the anticommutator are put into canonical order using
-<tt class="docutils literal"><span class="pre">__cmp__</span></tt>. If <tt class="docutils literal"><span class="pre">B</span> <span class="pre">&lt;</span> <span class="pre">A</span></tt>, then <tt class="docutils literal"><span class="pre">{A,</span> <span class="pre">B}</span></tt> is returned as <tt class="docutils literal"><span class="pre">{B,</span> <span class="pre">A}</span></tt>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>A</strong> : Expr</p>
-<blockquote>
-<div><p>The first argument of the anticommutator {A,B}.</p>
-</div></blockquote>
-<p><strong>B</strong> : Expr</p>
-<blockquote class="last">
-<div><p>The second argument of the anticommutator {A,B}.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r74" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[R74]</td><td><em>(<a class="fn-backref" href="#id1">1</a>, <a class="fn-backref" href="#id2">2</a>)</em> <a class="reference external" href="http://en.wikipedia.org/wiki/Commutator">http://en.wikipedia.org/wiki/Commutator</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum</span> <span class="kn">import</span> <span class="n">AntiCommutator</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum</span> <span class="kn">import</span> <span class="n">Operator</span><span class="p">,</span> <span class="n">Dagger</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Operator</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Operator</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Create an anticommutator and use <tt class="docutils literal"><span class="pre">doit()</span></tt> to multiply them out.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ac</span> <span class="o">=</span> <span class="n">AntiCommutator</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">B</span><span class="p">);</span> <span class="n">ac</span>
-<span class="go">{A,B}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ac</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-<span class="go">A*B + B*A</span>
-</pre></div>
-</div>
-<p>The commutator orders it arguments in canonical order:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ac</span> <span class="o">=</span> <span class="n">AntiCommutator</span><span class="p">(</span><span class="n">B</span><span class="p">,</span><span class="n">A</span><span class="p">);</span> <span class="n">ac</span>
-<span class="go">{A,B}</span>
-</pre></div>
-</div>
-<p>Commutative constants are factored out:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">AntiCommutator</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">A</span><span class="p">,</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">*</span><span class="n">B</span><span class="p">)</span>
-<span class="go">3*x**2*y*{A,B}</span>
-</pre></div>
-</div>
-<p>Adjoint operations applied to the anticommutator are properly applied to
-the arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="n">AntiCommutator</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">B</span><span class="p">))</span>
-<span class="go">{Dagger(A),Dagger(B)}</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.physics.quantum.anticommutator.AntiCommutator.doit">
-<tt class="descname">doit</tt><big>(</big><em>self</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/anticommutator.html#AntiCommutator.doit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.anticommutator.AntiCommutator.doit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluate anticommutator</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-
-
-          </div>
-        </div>
-      </div>
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-  <div class="section" id="module-sympy.physics.quantum.cartesian">
-<span id="cartesian-operators-and-states"></span><h1>Cartesian Operators and States<a class="headerlink" href="#module-sympy.physics.quantum.cartesian" title="Permalink to this headline">¶</a></h1>
-<p>Operators and states for 1D cartesian position and momentum.</p>
-<p>TODO:</p>
-<ul class="simple">
-<li>Add 3D classes to mappings in operatorset.py</li>
-</ul>
-<dl class="class">
-<dt id="sympy.physics.quantum.cartesian.XOp">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.cartesian.</tt><tt class="descname">XOp</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cartesian.html#XOp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cartesian.XOp" title="Permalink to this definition">¶</a></dt>
-<dd><p>1D cartesian position operator.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.cartesian.YOp">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.cartesian.</tt><tt class="descname">YOp</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cartesian.html#YOp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cartesian.YOp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Y cartesian coordinate operator (for 2D or 3D systems)</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.cartesian.ZOp">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.cartesian.</tt><tt class="descname">ZOp</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cartesian.html#ZOp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cartesian.ZOp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Z cartesian coordinate operator (for 3D systems)</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.cartesian.PxOp">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.cartesian.</tt><tt class="descname">PxOp</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cartesian.html#PxOp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cartesian.PxOp" title="Permalink to this definition">¶</a></dt>
-<dd><p>1D cartesian momentum operator.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.cartesian.XKet">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.cartesian.</tt><tt class="descname">XKet</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cartesian.html#XKet"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cartesian.XKet" title="Permalink to this definition">¶</a></dt>
-<dd><p>1D cartesian position eigenket.</p>
-<dl class="attribute">
-<dt id="sympy.physics.quantum.cartesian.XKet.position">
-<tt class="descname">position</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cartesian.html#XKet.position"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cartesian.XKet.position" title="Permalink to this definition">¶</a></dt>
-<dd><p>The position of the state.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.cartesian.XBra">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.cartesian.</tt><tt class="descname">XBra</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cartesian.html#XBra"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cartesian.XBra" title="Permalink to this definition">¶</a></dt>
-<dd><p>1D cartesian position eigenbra.</p>
-<dl class="attribute">
-<dt id="sympy.physics.quantum.cartesian.XBra.position">
-<tt class="descname">position</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cartesian.html#XBra.position"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cartesian.XBra.position" title="Permalink to this definition">¶</a></dt>
-<dd><p>The position of the state.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.cartesian.PxKet">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.cartesian.</tt><tt class="descname">PxKet</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cartesian.html#PxKet"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cartesian.PxKet" title="Permalink to this definition">¶</a></dt>
-<dd><p>1D cartesian momentum eigenket.</p>
-<dl class="attribute">
-<dt id="sympy.physics.quantum.cartesian.PxKet.momentum">
-<tt class="descname">momentum</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cartesian.html#PxKet.momentum"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cartesian.PxKet.momentum" title="Permalink to this definition">¶</a></dt>
-<dd><p>The momentum of the state.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.cartesian.PxBra">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.cartesian.</tt><tt class="descname">PxBra</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cartesian.html#PxBra"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cartesian.PxBra" title="Permalink to this definition">¶</a></dt>
-<dd><p>1D cartesian momentum eigenbra.</p>
-<dl class="attribute">
-<dt id="sympy.physics.quantum.cartesian.PxBra.momentum">
-<tt class="descname">momentum</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cartesian.html#PxBra.momentum"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cartesian.PxBra.momentum" title="Permalink to this definition">¶</a></dt>
-<dd><p>The momentum of the state.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.cartesian.PositionState3D">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.cartesian.</tt><tt class="descname">PositionState3D</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cartesian.html#PositionState3D"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cartesian.PositionState3D" title="Permalink to this definition">¶</a></dt>
-<dd><p>Base class for 3D cartesian position eigenstates</p>
-<dl class="attribute">
-<dt id="sympy.physics.quantum.cartesian.PositionState3D.position_x">
-<tt class="descname">position_x</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cartesian.html#PositionState3D.position_x"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cartesian.PositionState3D.position_x" title="Permalink to this definition">¶</a></dt>
-<dd><p>The x coordinate of the state</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.cartesian.PositionState3D.position_y">
-<tt class="descname">position_y</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cartesian.html#PositionState3D.position_y"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cartesian.PositionState3D.position_y" title="Permalink to this definition">¶</a></dt>
-<dd><p>The y coordinate of the state</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.cartesian.PositionState3D.position_z">
-<tt class="descname">position_z</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cartesian.html#PositionState3D.position_z"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cartesian.PositionState3D.position_z" title="Permalink to this definition">¶</a></dt>
-<dd><p>The z coordinate of the state</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.cartesian.PositionKet3D">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.cartesian.</tt><tt class="descname">PositionKet3D</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cartesian.html#PositionKet3D"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cartesian.PositionKet3D" title="Permalink to this definition">¶</a></dt>
-<dd><p>3D cartesian position eigenket</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.cartesian.PositionBra3D">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.cartesian.</tt><tt class="descname">PositionBra3D</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cartesian.html#PositionBra3D"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cartesian.PositionBra3D" title="Permalink to this definition">¶</a></dt>
-<dd><p>3D cartesian position eigenbra</p>
-</dd></dl>
-
-</div>
-
-
-          </div>
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-  <div class="section" id="module-sympy.physics.quantum.cg">
-<span id="clebsch-gordan-coefficients"></span><h1>Clebsch-Gordan Coefficients<a class="headerlink" href="#module-sympy.physics.quantum.cg" title="Permalink to this headline">¶</a></h1>
-<p>Clebsch-Gordon Coefficients.</p>
-<dl class="class">
-<dt id="sympy.physics.quantum.cg.CG">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.cg.</tt><tt class="descname">CG</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cg.html#CG"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cg.CG" title="Permalink to this definition">¶</a></dt>
-<dd><p>Class for Clebsch-Gordan coefficient</p>
-<p>Clebsch-Gordan coefficients describe the angular momentum coupling between
-two systems. The coefficients give the expansion of a coupled total angular
-momentum state and an uncoupled tensor product state. The Clebsch-Gordan
-coefficients are defined as <a class="reference internal" href="#r75">[R75]</a>:</p>
-<div class="math">
-\[C^{j_1,m_1}_{j_2,m_2,j_3,m_3} = \langle j_1,m_1;j_2,m_2 | j_3,m_3\rangle\]</div>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>j1, m1, j2, m2, j3, m3</strong> : Number, Symbol</p>
-<blockquote class="last">
-<div><p>Terms determining the angular momentum of coupled angular momentum
-systems.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.quantum.cg.Wigner3j" title="sympy.physics.quantum.cg.Wigner3j"><tt class="xref py py-obj docutils literal"><span class="pre">Wigner3j</span></tt></a></dt>
-<dd>Wigner-3j symbols</dd>
-</dl>
-</div>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r75" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[R75]</td><td><em>(<a class="fn-backref" href="#id1">1</a>, <a class="fn-backref" href="#id2">2</a>)</em> Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.</td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Define a Clebsch-Gordan coefficient and evaluate its value</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.cg</span> <span class="kn">import</span> <span class="n">CG</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cg</span> <span class="o">=</span> <span class="n">CG</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cg</span>
-<span class="go">CG(3/2, 3/2, 1/2, -1/2, 1, 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cg</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-<span class="go">sqrt(3)/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.cg.Wigner3j">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.cg.</tt><tt class="descname">Wigner3j</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cg.html#Wigner3j"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cg.Wigner3j" title="Permalink to this definition">¶</a></dt>
-<dd><p>Class for the Wigner-3j symbols</p>
-<p>Wigner 3j-symbols are coefficients determined by the coupling of
-two angular momenta. When created, they are expressed as symbolic
-quantities that, for numerical parameters, can be evaluated using the
-<tt class="docutils literal"><span class="pre">.doit()</span></tt> method <a class="reference internal" href="#r76">[R76]</a>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>j1, m1, j2, m2, j3, m3</strong> : Number, Symbol</p>
-<blockquote class="last">
-<div><p>Terms determining the angular momentum of coupled angular momentum
-systems.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.quantum.cg.CG" title="sympy.physics.quantum.cg.CG"><tt class="xref py py-obj docutils literal"><span class="pre">CG</span></tt></a></dt>
-<dd>Clebsch-Gordan coefficients</dd>
-</dl>
-</div>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r76" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[R76]</td><td><em>(<a class="fn-backref" href="#id3">1</a>, <a class="fn-backref" href="#id4">2</a>)</em> Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.</td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Declare a Wigner-3j coefficient and calcualte its value</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.cg</span> <span class="kn">import</span> <span class="n">Wigner3j</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">w3j</span> <span class="o">=</span> <span class="n">Wigner3j</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">w3j</span>
-<span class="go">Wigner3j(6, 0, 4, 0, 2, 0)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">w3j</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-<span class="go">sqrt(715)/143</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.cg.Wigner6j">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.cg.</tt><tt class="descname">Wigner6j</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cg.html#Wigner6j"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cg.Wigner6j" title="Permalink to this definition">¶</a></dt>
-<dd><p>Class for the Wigner-6j symbols</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.quantum.cg.Wigner3j" title="sympy.physics.quantum.cg.Wigner3j"><tt class="xref py py-obj docutils literal"><span class="pre">Wigner3j</span></tt></a></dt>
-<dd>Wigner-3j symbols</dd>
-</dl>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.cg.Wigner9j">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.cg.</tt><tt class="descname">Wigner9j</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cg.html#Wigner9j"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cg.Wigner9j" title="Permalink to this definition">¶</a></dt>
-<dd><p>Class for the Wigner-9j symbols</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.quantum.cg.Wigner3j" title="sympy.physics.quantum.cg.Wigner3j"><tt class="xref py py-obj docutils literal"><span class="pre">Wigner3j</span></tt></a></dt>
-<dd>Wigner-3j symbols</dd>
-</dl>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.cg.cg_simp">
-<tt class="descclassname">sympy.physics.quantum.cg.</tt><tt class="descname">cg_simp</tt><big>(</big><em>e</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/cg.html#cg_simp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.cg.cg_simp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Simplify and combine CG coefficients</p>
-<p>This function uses various symmetry and properties of sums and
-products of Clebsch-Gordan coefficients to simplify statements
-involving these terms <a class="reference internal" href="#r77">[R77]</a>.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.quantum.cg.CG" title="sympy.physics.quantum.cg.CG"><tt class="xref py py-obj docutils literal"><span class="pre">CG</span></tt></a></dt>
-<dd>Clebsh-Gordan coefficients</dd>
-</dl>
-</div>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r77" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[R77]</td><td><em>(<a class="fn-backref" href="#id5">1</a>, <a class="fn-backref" href="#id6">2</a>)</em> Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.</td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Simplify the sum over CG(a,alpha,0,0,a,alpha) for all alpha to
-2*a+1</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.cg</span> <span class="kn">import</span> <span class="n">CG</span><span class="p">,</span> <span class="n">cg_simp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">CG</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">CG</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">CG</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cg_simp</span><span class="p">(</span><span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="o">+</span><span class="n">c</span><span class="p">)</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-</dd></dl>
-
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-  <div class="section" id="module-sympy.physics.quantum.circuitplot">
-<span id="circuit-plot"></span><h1>Circuit Plot<a class="headerlink" href="#module-sympy.physics.quantum.circuitplot" title="Permalink to this headline">¶</a></h1>
-<p>Matplotlib based plotting of quantum circuits.</p>
-<p>Todo:</p>
-<ul class="simple">
-<li>Optimize printing of large circuits.</li>
-<li>Get this to work with single gates.</li>
-<li>Do a better job checking the form of circuits to make sure it is a Mul of
-Gates.</li>
-<li>Get multi-target gates plotting.</li>
-<li>Get initial and final states to plot.</li>
-<li>Get measurements to plot. Might need to rethink measurement as a gate
-issue.</li>
-<li>Get scale and figsize to be handled in a better way.</li>
-<li>Write some tests/examples!</li>
-</ul>
-</div>
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-  <div class="section" id="module-sympy.physics.quantum.commutator">
-<span id="commutator"></span><h1>Commutator<a class="headerlink" href="#module-sympy.physics.quantum.commutator" title="Permalink to this headline">¶</a></h1>
-<p>The commutator: [A,B] = A*B - B*A.</p>
-<dl class="class">
-<dt id="sympy.physics.quantum.commutator.Commutator">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.commutator.</tt><tt class="descname">Commutator</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/commutator.html#Commutator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.commutator.Commutator" title="Permalink to this definition">¶</a></dt>
-<dd><p>The standard commutator, in an unevaluated state.</p>
-<p>Evaluating a commutator is defined <a class="reference internal" href="#r78">[R78]</a> as: <tt class="docutils literal"><span class="pre">[A,</span> <span class="pre">B]</span> <span class="pre">=</span> <span class="pre">A*B</span> <span class="pre">-</span> <span class="pre">B*A</span></tt>. This
-class returns the commutator in an unevaluated form. To evaluate the
-commutator, use the <tt class="docutils literal"><span class="pre">.doit()</span></tt> method.</p>
-<p>Cannonical ordering of a commutator is <tt class="docutils literal"><span class="pre">[A,</span> <span class="pre">B]</span></tt> for <tt class="docutils literal"><span class="pre">A</span> <span class="pre">&lt;</span> <span class="pre">B</span></tt>. The
-arguments of the commutator are put into canonical order using <tt class="docutils literal"><span class="pre">__cmp__</span></tt>.
-If <tt class="docutils literal"><span class="pre">B</span> <span class="pre">&lt;</span> <span class="pre">A</span></tt>, then <tt class="docutils literal"><span class="pre">[B,</span> <span class="pre">A]</span></tt> is returned as <tt class="docutils literal"><span class="pre">-[A,</span> <span class="pre">B]</span></tt>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>A</strong> : Expr</p>
-<blockquote>
-<div><p>The first argument of the commutator [A,B].</p>
-</div></blockquote>
-<p><strong>B</strong> : Expr</p>
-<blockquote class="last">
-<div><p>The second argument of the commutator [A,B].</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r78" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[R78]</td><td><em>(<a class="fn-backref" href="#id1">1</a>, <a class="fn-backref" href="#id2">2</a>)</em> <a class="reference external" href="http://en.wikipedia.org/wiki/Commutator">http://en.wikipedia.org/wiki/Commutator</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum</span> <span class="kn">import</span> <span class="n">Commutator</span><span class="p">,</span> <span class="n">Dagger</span><span class="p">,</span> <span class="n">Operator</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Operator</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Operator</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Operator</span><span class="p">(</span><span class="s">&#39;C&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Create a commutator and use <tt class="docutils literal"><span class="pre">.doit()</span></tt> to evaluate it:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">comm</span> <span class="o">=</span> <span class="n">Commutator</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">comm</span>
-<span class="go">[A,B]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">comm</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-<span class="go">A*B - B*A</span>
-</pre></div>
-</div>
-<p>The commutator orders it arguments in canonical order:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">comm</span> <span class="o">=</span> <span class="n">Commutator</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">A</span><span class="p">);</span> <span class="n">comm</span>
-<span class="go">-[A,B]</span>
-</pre></div>
-</div>
-<p>Commutative constants are factored out:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Commutator</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">A</span><span class="p">,</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">*</span><span class="n">B</span><span class="p">)</span>
-<span class="go">3*x**2*y*[A,B]</span>
-</pre></div>
-</div>
-<p>Using <tt class="docutils literal"><span class="pre">.expand(commutator=True)</span></tt>, the standard commutator expansion rules
-can be applied:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Commutator</span><span class="p">(</span><span class="n">A</span><span class="o">+</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">commutator</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">[A,C] + [B,C]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Commutator</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="o">+</span><span class="n">C</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">commutator</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">[A,B] + [A,C]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Commutator</span><span class="p">(</span><span class="n">A</span><span class="o">*</span><span class="n">B</span><span class="p">,</span> <span class="n">C</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">commutator</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">[A,C]*B + A*[B,C]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Commutator</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="o">*</span><span class="n">C</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">commutator</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">[A,B]*C + B*[A,C]</span>
-</pre></div>
-</div>
-<p>Adjoint operations applied to the commutator are properly applied to the
-arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="n">Commutator</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">))</span>
-<span class="go">-[Dagger(A),Dagger(B)]</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.physics.quantum.commutator.Commutator.doit">
-<tt class="descname">doit</tt><big>(</big><em>self</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/commutator.html#Commutator.doit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.commutator.Commutator.doit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluate commutator</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
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-  <div class="section" id="module-sympy.physics.quantum.constants">
-<span id="constants"></span><h1>Constants<a class="headerlink" href="#module-sympy.physics.quantum.constants" title="Permalink to this headline">¶</a></h1>
-<p>Constants (like hbar) related to quantum mechanics.</p>
-</div>
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-  <div class="section" id="module-sympy.physics.quantum.dagger">
-<span id="dagger"></span><h1>Dagger<a class="headerlink" href="#module-sympy.physics.quantum.dagger" title="Permalink to this headline">¶</a></h1>
-<p>Hermitian conjugation.</p>
-<dl class="class">
-<dt id="sympy.physics.quantum.dagger.Dagger">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.dagger.</tt><tt class="descname">Dagger</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/dagger.html#Dagger"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.dagger.Dagger" title="Permalink to this definition">¶</a></dt>
-<dd><p>General Hermitian conjugate operation.</p>
-<p>Take the Hermetian conjugate of an argument <a class="reference internal" href="#r79">[R79]</a>. For matrices this
-operation is equivalent to transpose and complex conjugate <a class="reference internal" href="#r80">[R80]</a>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>arg</strong> : Expr</p>
-<blockquote class="last">
-<div><p>The sympy expression that we want to take the dagger of.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r79" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[R79]</td><td><em>(<a class="fn-backref" href="#id1">1</a>, <a class="fn-backref" href="#id3">2</a>)</em> <a class="reference external" href="http://en.wikipedia.org/wiki/Hermitian_adjoint">http://en.wikipedia.org/wiki/Hermitian_adjoint</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r80" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[R80]</td><td><em>(<a class="fn-backref" href="#id2">1</a>, <a class="fn-backref" href="#id4">2</a>)</em> <a class="reference external" href="http://en.wikipedia.org/wiki/Hermitian_transpose">http://en.wikipedia.org/wiki/Hermitian_transpose</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Daggering various quantum objects:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.dagger</span> <span class="kn">import</span> <span class="n">Dagger</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.state</span> <span class="kn">import</span> <span class="n">Ket</span><span class="p">,</span> <span class="n">Bra</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.operator</span> <span class="kn">import</span> <span class="n">Operator</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="n">Ket</span><span class="p">(</span><span class="s">&#39;psi&#39;</span><span class="p">))</span>
-<span class="go">&lt;psi|</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="n">Bra</span><span class="p">(</span><span class="s">&#39;phi&#39;</span><span class="p">))</span>
-<span class="go">|phi&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="n">Operator</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">))</span>
-<span class="go">Dagger(A)</span>
-</pre></div>
-</div>
-<p>Inner and outer products:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum</span> <span class="kn">import</span> <span class="n">InnerProduct</span><span class="p">,</span> <span class="n">OuterProduct</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="n">InnerProduct</span><span class="p">(</span><span class="n">Bra</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">),</span> <span class="n">Ket</span><span class="p">(</span><span class="s">&#39;b&#39;</span><span class="p">)))</span>
-<span class="go">&lt;b|a&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="n">OuterProduct</span><span class="p">(</span><span class="n">Ket</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">),</span> <span class="n">Bra</span><span class="p">(</span><span class="s">&#39;b&#39;</span><span class="p">)))</span>
-<span class="go">|b&gt;&lt;a|</span>
-</pre></div>
-</div>
-<p>Powers, sums and products:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Operator</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Operator</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="n">A</span><span class="o">*</span><span class="n">B</span><span class="p">)</span>
-<span class="go">Dagger(B)*Dagger(A)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="n">A</span><span class="o">+</span><span class="n">B</span><span class="p">)</span>
-<span class="go">Dagger(A) + Dagger(B)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="n">A</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">Dagger(A)**2</span>
-</pre></div>
-</div>
-<p>Dagger also seamlessly handles complex numbers and matrices:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="n">I</span><span class="p">],[</span><span class="mi">2</span><span class="p">,</span><span class="n">I</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span>
-<span class="go">[1, I]</span>
-<span class="go">[2, I]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-<span class="go">[ 1,  2]</span>
-<span class="go">[-I, -I]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-
-
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-        </div>
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-            
-  <div class="section" id="module-sympy.physics.quantum.gate">
-<span id="gates"></span><h1>Gates<a class="headerlink" href="#module-sympy.physics.quantum.gate" title="Permalink to this headline">¶</a></h1>
-<p>An implementation of gates that act on qubits.</p>
-<p>Gates are unitary operators that act on the space of qubits.</p>
-<p>Medium Term Todo:</p>
-<ul class="simple">
-<li>Optimize Gate._apply_operators_Qubit to remove the creation of many
-intermediate Qubit objects.</li>
-<li>Add commutation relationships to all operators and use this in gate_sort.</li>
-<li>Fix gate_sort and gate_simp.</li>
-<li>Get multi-target UGates plotting properly.</li>
-<li>Get UGate to work with either sympy/numpy matrices and output either
-format. This should also use the matrix slots.</li>
-</ul>
-<dl class="class">
-<dt id="sympy.physics.quantum.gate.Gate">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">Gate</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#Gate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.Gate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Non-controlled unitary gate operator that acts on qubits.</p>
-<p>This is a general abstract gate that needs to be subclassed to do anything
-useful.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>label</strong> : tuple, int</p>
-<blockquote class="last">
-<div><p>A list of the target qubits (as ints) that the gate will apply to.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.physics.quantum.gate.Gate.get_target_matrix">
-<tt class="descname">get_target_matrix</tt><big>(</big><em>self</em>, <em>format='sympy'</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#Gate.get_target_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.Gate.get_target_matrix" title="Permalink to this definition">¶</a></dt>
-<dd><p>The matrix rep. of the target part of the gate.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>format</strong> : str</p>
-<blockquote class="last">
-<div><p>The format string (&#8216;sympy&#8217;,&#8217;numpy&#8217;, etc.)</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.gate.Gate.min_qubits">
-<tt class="descname">min_qubits</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#Gate.min_qubits"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.Gate.min_qubits" title="Permalink to this definition">¶</a></dt>
-<dd><p>The minimum number of qubits this gate needs to act on.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.gate.Gate.nqubits">
-<tt class="descname">nqubits</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#Gate.nqubits"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.Gate.nqubits" title="Permalink to this definition">¶</a></dt>
-<dd><p>The total number of qubits this gate acts on.</p>
-<p>For controlled gate subclasses this includes both target and control
-qubits, so that, for examples the CNOT gate acts on 2 qubits.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.gate.Gate.targets">
-<tt class="descname">targets</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#Gate.targets"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.Gate.targets" title="Permalink to this definition">¶</a></dt>
-<dd><p>A tuple of target qubits.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.gate.CGate">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">CGate</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#CGate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.CGate" title="Permalink to this definition">¶</a></dt>
-<dd><p>A general unitary gate with control qubits.</p>
-<p>A general control gate applies a target gate to a set of targets if all
-of the control qubits have a particular values (set by
-<tt class="docutils literal"><span class="pre">CGate.control_value</span></tt>).</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>label</strong> : tuple</p>
-<blockquote class="last">
-<div><p>The label in this case has the form (controls, gate), where controls
-is a tuple/list of control qubits (as ints) and gate is a <tt class="docutils literal"><span class="pre">Gate</span></tt>
-instance that is the target operator.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<dl class="attribute">
-<dt id="sympy.physics.quantum.gate.CGate.controls">
-<tt class="descname">controls</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#CGate.controls"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.CGate.controls" title="Permalink to this definition">¶</a></dt>
-<dd><p>A tuple of control qubits.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.gate.CGate.decompose">
-<tt class="descname">decompose</tt><big>(</big><em>self</em>, <em>**options</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#CGate.decompose"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.CGate.decompose" title="Permalink to this definition">¶</a></dt>
-<dd><p>Decompose the controlled gate into CNOT and single qubits gates.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.gate.CGate.eval_controls">
-<tt class="descname">eval_controls</tt><big>(</big><em>self</em>, <em>qubit</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#CGate.eval_controls"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.CGate.eval_controls" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True/False to indicate if the controls are satisfied.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.gate.CGate.gate">
-<tt class="descname">gate</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#CGate.gate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.CGate.gate" title="Permalink to this definition">¶</a></dt>
-<dd><p>The non-controlled gate that will be applied to the targets.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.gate.CGate.min_qubits">
-<tt class="descname">min_qubits</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#CGate.min_qubits"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.CGate.min_qubits" title="Permalink to this definition">¶</a></dt>
-<dd><p>The minimum number of qubits this gate needs to act on.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.gate.CGate.nqubits">
-<tt class="descname">nqubits</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#CGate.nqubits"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.CGate.nqubits" title="Permalink to this definition">¶</a></dt>
-<dd><p>The total number of qubits this gate acts on.</p>
-<p>For controlled gate subclasses this includes both target and control
-qubits, so that, for examples the CNOT gate acts on 2 qubits.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.gate.CGate.targets">
-<tt class="descname">targets</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#CGate.targets"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.CGate.targets" title="Permalink to this definition">¶</a></dt>
-<dd><p>A tuple of target qubits.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.gate.UGate">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">UGate</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#UGate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.UGate" title="Permalink to this definition">¶</a></dt>
-<dd><p>General gate specified by a set of targets and a target matrix.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>label</strong> : tuple</p>
-<blockquote class="last">
-<div><p>A tuple of the form (targets, U), where targets is a tuple of the
-target qubits and U is a unitary matrix with dimension of
-len(targets).</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.physics.quantum.gate.UGate.get_target_matrix">
-<tt class="descname">get_target_matrix</tt><big>(</big><em>self</em>, <em>format='sympy'</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#UGate.get_target_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.UGate.get_target_matrix" title="Permalink to this definition">¶</a></dt>
-<dd><p>The matrix rep. of the target part of the gate.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>format</strong> : str</p>
-<blockquote class="last">
-<div><p>The format string (&#8216;sympy&#8217;,&#8217;numpy&#8217;, etc.)</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.gate.UGate.targets">
-<tt class="descname">targets</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#UGate.targets"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.UGate.targets" title="Permalink to this definition">¶</a></dt>
-<dd><p>A tuple of target qubits.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.gate.OneQubitGate">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">OneQubitGate</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#OneQubitGate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.OneQubitGate" title="Permalink to this definition">¶</a></dt>
-<dd><p>A single qubit unitary gate base class.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.gate.TwoQubitGate">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">TwoQubitGate</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#TwoQubitGate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.TwoQubitGate" title="Permalink to this definition">¶</a></dt>
-<dd><p>A two qubit unitary gate base class.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.gate.IdentityGate">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">IdentityGate</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#IdentityGate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.IdentityGate" title="Permalink to this definition">¶</a></dt>
-<dd><p>The single qubit identity gate.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>target</strong> : int</p>
-<blockquote class="last">
-<div><p>The target qubit this gate will apply to.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.gate.HadamardGate">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">HadamardGate</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#HadamardGate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.HadamardGate" title="Permalink to this definition">¶</a></dt>
-<dd><p>The single qubit Hadamard gate.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>target</strong> : int</p>
-<blockquote class="last">
-<div><p>The target qubit this gate will apply to.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.qubit</span> <span class="kn">import</span> <span class="n">Qubit</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.gate</span> <span class="kn">import</span> <span class="n">HadamardGate</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.qapply</span> <span class="kn">import</span> <span class="n">qapply</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qapply</span><span class="p">(</span><span class="n">HadamardGate</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="o">*</span><span class="n">Qubit</span><span class="p">(</span><span class="s">&#39;1&#39;</span><span class="p">))</span>
-<span class="go">sqrt(2)*|0&gt;/2 - sqrt(2)*|1&gt;/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># Hadamard on bell state, applied on 2 qubits.</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">psi</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">Qubit</span><span class="p">(</span><span class="s">&#39;00&#39;</span><span class="p">)</span><span class="o">+</span><span class="n">Qubit</span><span class="p">(</span><span class="s">&#39;11&#39;</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qapply</span><span class="p">(</span><span class="n">HadamardGate</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="o">*</span><span class="n">HadamardGate</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">psi</span><span class="p">)</span>
-<span class="go">sqrt(2)*|00&gt;/2 + sqrt(2)*|11&gt;/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.gate.XGate">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">XGate</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#XGate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.XGate" title="Permalink to this definition">¶</a></dt>
-<dd><p>The single qubit X, or NOT, gate.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>target</strong> : int</p>
-<blockquote class="last">
-<div><p>The target qubit this gate will apply to.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.gate.YGate">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">YGate</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#YGate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.YGate" title="Permalink to this definition">¶</a></dt>
-<dd><p>The single qubit Y gate.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>target</strong> : int</p>
-<blockquote class="last">
-<div><p>The target qubit this gate will apply to.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.gate.ZGate">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">ZGate</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#ZGate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.ZGate" title="Permalink to this definition">¶</a></dt>
-<dd><p>The single qubit Z gate.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>target</strong> : int</p>
-<blockquote class="last">
-<div><p>The target qubit this gate will apply to.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.gate.TGate">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">TGate</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#TGate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.TGate" title="Permalink to this definition">¶</a></dt>
-<dd><p>The single qubit pi/8 gate.</p>
-<p>This gate rotates the phase of the state by pi/4 if the state is <tt class="docutils literal"><span class="pre">|1&gt;</span></tt> and
-does nothing if the state is <tt class="docutils literal"><span class="pre">|0&gt;</span></tt>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>target</strong> : int</p>
-<blockquote class="last">
-<div><p>The target qubit this gate will apply to.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.gate.PhaseGate">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">PhaseGate</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#PhaseGate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.PhaseGate" title="Permalink to this definition">¶</a></dt>
-<dd><p>The single qubit phase, or S, gate.</p>
-<p>This gate rotates the phase of the state by pi/2 if the state is <tt class="docutils literal"><span class="pre">|1&gt;</span></tt> and
-does nothing if the state is <tt class="docutils literal"><span class="pre">|0&gt;</span></tt>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>target</strong> : int</p>
-<blockquote class="last">
-<div><p>The target qubit this gate will apply to.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.gate.SwapGate">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">SwapGate</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#SwapGate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.SwapGate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Two qubit SWAP gate.</p>
-<p>This gate swap the values of the two qubits.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>label</strong> : tuple</p>
-<blockquote class="last">
-<div><p>A tuple of the form (target1, target2).</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.physics.quantum.gate.SwapGate.decompose">
-<tt class="descname">decompose</tt><big>(</big><em>self</em>, <em>**options</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#SwapGate.decompose"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.SwapGate.decompose" title="Permalink to this definition">¶</a></dt>
-<dd><p>Decompose the SWAP gate into CNOT gates.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.gate.CNotGate">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">CNotGate</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#CNotGate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.CNotGate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Two qubit controlled-NOT.</p>
-<p>This gate performs the NOT or X gate on the target qubit if the control
-qubits all have the value 1.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>label</strong> : tuple</p>
-<blockquote class="last">
-<div><p>A tuple of the form (control, target).</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.gate</span> <span class="kn">import</span> <span class="n">CNOT</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.qapply</span> <span class="kn">import</span> <span class="n">qapply</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.qubit</span> <span class="kn">import</span> <span class="n">Qubit</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">CNOT</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qapply</span><span class="p">(</span><span class="n">c</span><span class="o">*</span><span class="n">Qubit</span><span class="p">(</span><span class="s">&#39;10&#39;</span><span class="p">))</span> <span class="c"># note that qubits are indexed from right to left</span>
-<span class="go">|11&gt;</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.physics.quantum.gate.CNotGate.controls">
-<tt class="descname">controls</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#CNotGate.controls"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.CNotGate.controls" title="Permalink to this definition">¶</a></dt>
-<dd><p>A tuple of control qubits.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.gate.CNotGate.gate">
-<tt class="descname">gate</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#CNotGate.gate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.CNotGate.gate" title="Permalink to this definition">¶</a></dt>
-<dd><p>The non-controlled gate that will be applied to the targets.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.gate.CNotGate.min_qubits">
-<tt class="descname">min_qubits</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#CNotGate.min_qubits"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.CNotGate.min_qubits" title="Permalink to this definition">¶</a></dt>
-<dd><p>The minimum number of qubits this gate needs to act on.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.gate.CNotGate.targets">
-<tt class="descname">targets</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#CNotGate.targets"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.CNotGate.targets" title="Permalink to this definition">¶</a></dt>
-<dd><p>A tuple of target qubits.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.gate.CNOT">
-<tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">CNOT</tt><a class="headerlink" href="#sympy.physics.quantum.gate.CNOT" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <a class="reference internal" href="#sympy.physics.quantum.gate.CNotGate" title="sympy.physics.quantum.gate.CNotGate"><tt class="xref py py-class docutils literal"><span class="pre">CNotGate</span></tt></a></p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.gate.SWAP">
-<tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">SWAP</tt><a class="headerlink" href="#sympy.physics.quantum.gate.SWAP" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <a class="reference internal" href="#sympy.physics.quantum.gate.SwapGate" title="sympy.physics.quantum.gate.SwapGate"><tt class="xref py py-class docutils literal"><span class="pre">SwapGate</span></tt></a></p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.gate.H">
-<tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">H</tt><a class="headerlink" href="#sympy.physics.quantum.gate.H" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <a class="reference internal" href="#sympy.physics.quantum.gate.HadamardGate" title="sympy.physics.quantum.gate.HadamardGate"><tt class="xref py py-class docutils literal"><span class="pre">HadamardGate</span></tt></a></p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.gate.X">
-<tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">X</tt><a class="headerlink" href="#sympy.physics.quantum.gate.X" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <a class="reference internal" href="#sympy.physics.quantum.gate.XGate" title="sympy.physics.quantum.gate.XGate"><tt class="xref py py-class docutils literal"><span class="pre">XGate</span></tt></a></p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.gate.Y">
-<tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">Y</tt><a class="headerlink" href="#sympy.physics.quantum.gate.Y" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <a class="reference internal" href="#sympy.physics.quantum.gate.YGate" title="sympy.physics.quantum.gate.YGate"><tt class="xref py py-class docutils literal"><span class="pre">YGate</span></tt></a></p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.gate.Z">
-<tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">Z</tt><a class="headerlink" href="#sympy.physics.quantum.gate.Z" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <a class="reference internal" href="#sympy.physics.quantum.gate.ZGate" title="sympy.physics.quantum.gate.ZGate"><tt class="xref py py-class docutils literal"><span class="pre">ZGate</span></tt></a></p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.gate.T">
-<tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">T</tt><a class="headerlink" href="#sympy.physics.quantum.gate.T" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <a class="reference internal" href="#sympy.physics.quantum.gate.TGate" title="sympy.physics.quantum.gate.TGate"><tt class="xref py py-class docutils literal"><span class="pre">TGate</span></tt></a></p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.gate.S">
-<tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">S</tt><a class="headerlink" href="#sympy.physics.quantum.gate.S" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <a class="reference internal" href="#sympy.physics.quantum.gate.PhaseGate" title="sympy.physics.quantum.gate.PhaseGate"><tt class="xref py py-class docutils literal"><span class="pre">PhaseGate</span></tt></a></p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.gate.Phase">
-<tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">Phase</tt><a class="headerlink" href="#sympy.physics.quantum.gate.Phase" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <a class="reference internal" href="#sympy.physics.quantum.gate.PhaseGate" title="sympy.physics.quantum.gate.PhaseGate"><tt class="xref py py-class docutils literal"><span class="pre">PhaseGate</span></tt></a></p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.gate.normalized">
-<tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">normalized</tt><big>(</big><em>normalize</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#normalized"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.normalized" title="Permalink to this definition">¶</a></dt>
-<dd><p>Should Hadamard gates be normalized by a 1/sqrt(2).</p>
-<p>This is a global setting that can be used to simplify the look of various
-expressions, by leaving of the leading 1/sqrt(2) of the Hadamard gate.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>normalize</strong> : bool</p>
-<blockquote class="last">
-<div><p>Should the Hadamard gate include the 1/sqrt(2) normalization factor?
-When True, the Hadamard gate will have the 1/sqrt(2). When False, the
-Hadamard gate will not have this factor.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.gate.gate_sort">
-<tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">gate_sort</tt><big>(</big><em>circuit</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#gate_sort"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.gate_sort" title="Permalink to this definition">¶</a></dt>
-<dd><p>Sorts the gates while keeping track of commutation relations</p>
-<p>This function uses a bubble sort to rearrange the order of gate
-application. Keeps track of Quantum computations special commutation
-relations (e.g. things that apply to the same Qubit do not commute with
-each other)</p>
-<p>circuit is the Mul of gates that are to be sorted.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.gate.gate_simp">
-<tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">gate_simp</tt><big>(</big><em>circuit</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#gate_simp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.gate_simp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Simplifies gates symbolically</p>
-<p>It first sorts gates using gate_sort. It then applies basic
-simplification rules to the circuit, e.g., XGate**2 = Identity</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.gate.random_circuit">
-<tt class="descclassname">sympy.physics.quantum.gate.</tt><tt class="descname">random_circuit</tt><big>(</big><em>ngates</em>, <em>nqubits</em>, <em>gate_space=(&lt;class 'sympy.physics.quantum.gate.XGate'&gt;</em>, <em>&lt;class 'sympy.physics.quantum.gate.YGate'&gt;</em>, <em>&lt;class 'sympy.physics.quantum.gate.ZGate'&gt;</em>, <em>&lt;class 'sympy.physics.quantum.gate.PhaseGate'&gt;</em>, <em>&lt;class 'sympy.physics.quantum.gate.TGate'&gt;</em>, <em>&lt;class 'sympy.physics.quantum.gate.HadamardGate'&gt;</em>, <em>&lt;class 'sympy.physics.quantum.gate.CNotGate'&gt;</em>, <em>&lt;class 'sympy.physics.quantum.gate.SwapGate'&gt;)</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/gate.html#random_circuit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.gate.random_circuit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a random circuit of ngates and nqubits.</p>
-<p>This uses an equally weighted sample of (X, Y, Z, S, T, H, CNOT, SWAP)
-gates.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>ngates</strong> : int</p>
-<blockquote>
-<div><p>The number of gates in the circuit.</p>
-</div></blockquote>
-<p><strong>nqubits</strong> : int</p>
-<blockquote>
-<div><p>The number of qubits in the circuit.</p>
-</div></blockquote>
-<p><strong>gate_space</strong> : tuple</p>
-<blockquote class="last">
-<div><p>A tuple of the gate classes that will be used in the circuit.
-Repeating gate classes multiple times in this tuple will increase
-the frequency they appear in the random circuit.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</div>
-
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-            
-  <div class="section" id="module-sympy.physics.quantum.grover">
-<span id="grover-s-algorithm"></span><h1>Grover&#8217;s Algorithm<a class="headerlink" href="#module-sympy.physics.quantum.grover" title="Permalink to this headline">¶</a></h1>
-<p>Grover&#8217;s algorithm and helper functions.</p>
-<p>Todo:</p>
-<ul class="simple">
-<li>W gate construction (or perhaps -W gate based on Mermin&#8217;s book)</li>
-<li>Generalize the algorithm for an unknown function that returns 1 on multiple
-qubit states, not just one.</li>
-<li>Implement _represent_ZGate in OracleGate</li>
-</ul>
-<dl class="class">
-<dt id="sympy.physics.quantum.grover.OracleGate">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.grover.</tt><tt class="descname">OracleGate</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/grover.html#OracleGate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.grover.OracleGate" title="Permalink to this definition">¶</a></dt>
-<dd><p>A black box gate.</p>
-<p>The gate marks the desired qubits of an unknown function by flipping
-the sign of the qubits.  The unknown function returns true when it
-finds its desired qubits and false otherwise.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>qubits</strong> : int</p>
-<blockquote>
-<div><p>Number of qubits.</p>
-</div></blockquote>
-<p><strong>oracle</strong> : callable</p>
-<blockquote class="last">
-<div><p>A callable function that returns a boolean on a computational basis.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Apply an Oracle gate that flips the sign of <tt class="docutils literal"><span class="pre">|2&gt;</span></tt> on different qubits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.qubit</span> <span class="kn">import</span> <span class="n">IntQubit</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.qapply</span> <span class="kn">import</span> <span class="n">qapply</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.grover</span> <span class="kn">import</span> <span class="n">OracleGate</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">qubits</span><span class="p">:</span> <span class="n">qubits</span> <span class="o">==</span> <span class="n">IntQubit</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v</span> <span class="o">=</span> <span class="n">OracleGate</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qapply</span><span class="p">(</span><span class="n">v</span><span class="o">*</span><span class="n">IntQubit</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">-|2&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qapply</span><span class="p">(</span><span class="n">v</span><span class="o">*</span><span class="n">IntQubit</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">|3&gt;</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.physics.quantum.grover.OracleGate.search_function">
-<tt class="descname">search_function</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/grover.html#OracleGate.search_function"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.grover.OracleGate.search_function" title="Permalink to this definition">¶</a></dt>
-<dd><p>The unknown function that helps find the sought after qubits.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.grover.OracleGate.targets">
-<tt class="descname">targets</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/grover.html#OracleGate.targets"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.grover.OracleGate.targets" title="Permalink to this definition">¶</a></dt>
-<dd><p>A tuple of target qubits.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.grover.WGate">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.grover.</tt><tt class="descname">WGate</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/grover.html#WGate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.grover.WGate" title="Permalink to this definition">¶</a></dt>
-<dd><p>General n qubit W Gate in Grover&#8217;s algorithm.</p>
-<p>The gate performs the operation <tt class="docutils literal"><span class="pre">2|phi&gt;&lt;phi|</span> <span class="pre">-</span> <span class="pre">1</span></tt> on some qubits.
-<tt class="docutils literal"><span class="pre">|phi&gt;</span> <span class="pre">=</span> <span class="pre">(tensor</span> <span class="pre">product</span> <span class="pre">of</span> <span class="pre">n</span> <span class="pre">Hadamards)*(|0&gt;</span> <span class="pre">with</span> <span class="pre">n</span> <span class="pre">qubits)</span></tt></p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>nqubits</strong> : int</p>
-<blockquote class="last">
-<div><p>The number of qubits to operate on</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.grover.superposition_basis">
-<tt class="descclassname">sympy.physics.quantum.grover.</tt><tt class="descname">superposition_basis</tt><big>(</big><em>nqubits</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/grover.html#superposition_basis"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.grover.superposition_basis" title="Permalink to this definition">¶</a></dt>
-<dd><p>Creates an equal superposition of the computational basis.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>nqubits</strong> : int</p>
-<blockquote>
-<div><p>The number of qubits.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>state</strong> : Qubit</p>
-<blockquote class="last">
-<div><p>An equal superposition of the computational basis with nqubits.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Create an equal superposition of 2 qubits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.grover</span> <span class="kn">import</span> <span class="n">superposition_basis</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">superposition_basis</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">|0&gt;/2 + |1&gt;/2 + |2&gt;/2 + |3&gt;/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.grover.grover_iteration">
-<tt class="descclassname">sympy.physics.quantum.grover.</tt><tt class="descname">grover_iteration</tt><big>(</big><em>qstate</em>, <em>oracle</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/grover.html#grover_iteration"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.grover.grover_iteration" title="Permalink to this definition">¶</a></dt>
-<dd><p>Applies one application of the Oracle and W Gate, WV.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>qstate</strong> : Qubit</p>
-<blockquote>
-<div><p>A superposition of qubits.</p>
-</div></blockquote>
-<p><strong>oracle</strong> : OracleGate</p>
-<blockquote>
-<div><p>The black box operator that flips the sign of the desired basis qubits.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>Qubit</strong> : The qubits after applying the Oracle and W gate.</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Perform one iteration of grover&#8217;s algorithm to see a phase change:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.qapply</span> <span class="kn">import</span> <span class="n">qapply</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.qubit</span> <span class="kn">import</span> <span class="n">IntQubit</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.grover</span> <span class="kn">import</span> <span class="n">OracleGate</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.grover</span> <span class="kn">import</span> <span class="n">superposition_basis</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.grover</span> <span class="kn">import</span> <span class="n">grover_iteration</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">numqubits</span> <span class="o">=</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">basis_states</span> <span class="o">=</span> <span class="n">superposition_basis</span><span class="p">(</span><span class="n">numqubits</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">qubits</span><span class="p">:</span> <span class="n">qubits</span> <span class="o">==</span> <span class="n">IntQubit</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">v</span> <span class="o">=</span> <span class="n">OracleGate</span><span class="p">(</span><span class="n">numqubits</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qapply</span><span class="p">(</span><span class="n">grover_iteration</span><span class="p">(</span><span class="n">basis_states</span><span class="p">,</span> <span class="n">v</span><span class="p">))</span>
-<span class="go">|2&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.grover.apply_grover">
-<tt class="descclassname">sympy.physics.quantum.grover.</tt><tt class="descname">apply_grover</tt><big>(</big><em>oracle</em>, <em>nqubits</em>, <em>iterations=None</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/grover.html#apply_grover"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.grover.apply_grover" title="Permalink to this definition">¶</a></dt>
-<dd><p>Applies grover&#8217;s algorithm.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>oracle</strong> : callable</p>
-<blockquote>
-<div><p>The unknown callable function that returns true when applied to the
-desired qubits and false otherwise.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>state</strong> : Expr</p>
-<blockquote class="last">
-<div><p>The resulting state after Grover&#8217;s algorithm has been iterated.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Apply grover&#8217;s algorithm to an even superposition of 2 qubits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.qapply</span> <span class="kn">import</span> <span class="n">qapply</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.qubit</span> <span class="kn">import</span> <span class="n">IntQubit</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.grover</span> <span class="kn">import</span> <span class="n">apply_grover</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">qubits</span><span class="p">:</span> <span class="n">qubits</span> <span class="o">==</span> <span class="n">IntQubit</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qapply</span><span class="p">(</span><span class="n">apply_grover</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">|2&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
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-  <div class="section" id="module-sympy.physics.quantum.hilbert">
-<span id="hilbert-space"></span><h1>Hilbert Space<a class="headerlink" href="#module-sympy.physics.quantum.hilbert" title="Permalink to this headline">¶</a></h1>
-<p>Hilbert spaces for quantum mechanics.</p>
-<p>Authors:
-* Brian Granger
-* Matt Curry</p>
-<dl class="class">
-<dt id="sympy.physics.quantum.hilbert.HilbertSpace">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.hilbert.</tt><tt class="descname">HilbertSpace</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/hilbert.html#HilbertSpace"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.hilbert.HilbertSpace" title="Permalink to this definition">¶</a></dt>
-<dd><p>An abstract Hilbert space for quantum mechanics.</p>
-<p>In short, a Hilbert space is an abstract vector space that is complete
-with inner products defined <a class="reference internal" href="#r81">[R81]</a>.</p>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r81" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[R81]</td><td><em>(<a class="fn-backref" href="#id1">1</a>, <a class="fn-backref" href="#id2">2</a>)</em> <a class="reference external" href="http://en.wikipedia.org/wiki/Hilbert_space">http://en.wikipedia.org/wiki/Hilbert_space</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.hilbert</span> <span class="kn">import</span> <span class="n">HilbertSpace</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hs</span> <span class="o">=</span> <span class="n">HilbertSpace</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hs</span>
-<span class="go">H</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.physics.quantum.hilbert.HilbertSpace.dimension">
-<tt class="descname">dimension</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/hilbert.html#HilbertSpace.dimension"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.hilbert.HilbertSpace.dimension" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the Hilbert dimension of the space.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.hilbert.ComplexSpace">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.hilbert.</tt><tt class="descname">ComplexSpace</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/hilbert.html#ComplexSpace"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.hilbert.ComplexSpace" title="Permalink to this definition">¶</a></dt>
-<dd><p>Finite dimensional Hilbert space of complex vectors.</p>
-<p>The elements of this Hilbert space are n-dimensional complex valued
-vectors with the usual inner product that takes the complex conjugate
-of the vector on the right.</p>
-<p>A classic example of this type of Hilbert space is spin-1/2, which is
-<tt class="docutils literal"><span class="pre">ComplexSpace(2)</span></tt>. Generalizing to spin-s, the space is
-<tt class="docutils literal"><span class="pre">ComplexSpace(2*s+1)</span></tt>.  Quantum computing with N qubits is done with the
-direct product space <tt class="docutils literal"><span class="pre">ComplexSpace(2)**N</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.hilbert</span> <span class="kn">import</span> <span class="n">ComplexSpace</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span> <span class="o">=</span> <span class="n">ComplexSpace</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span>
-<span class="go">C(2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span><span class="o">.</span><span class="n">dimension</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c2</span> <span class="o">=</span> <span class="n">ComplexSpace</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c2</span>
-<span class="go">C(n)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c2</span><span class="o">.</span><span class="n">dimension</span>
-<span class="go">n</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.hilbert.L2">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.hilbert.</tt><tt class="descname">L2</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/hilbert.html#L2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.hilbert.L2" title="Permalink to this definition">¶</a></dt>
-<dd><p>The Hilbert space of square integrable functions on an interval.</p>
-<p>An L2 object takes in a single sympy Interval argument which represents
-the interval its functions (vectors) are defined on.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Interval</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.hilbert</span> <span class="kn">import</span> <span class="n">L2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hs</span> <span class="o">=</span> <span class="n">L2</span><span class="p">(</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">oo</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hs</span>
-<span class="go">L2([0, oo))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hs</span><span class="o">.</span><span class="n">dimension</span>
-<span class="go">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hs</span><span class="o">.</span><span class="n">interval</span>
-<span class="go">[0, oo)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.hilbert.FockSpace">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.hilbert.</tt><tt class="descname">FockSpace</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/hilbert.html#FockSpace"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.hilbert.FockSpace" title="Permalink to this definition">¶</a></dt>
-<dd><p>The Hilbert space for second quantization.</p>
-<p>Technically, this Hilbert space is a infinite direct sum of direct
-products of single particle Hilbert spaces <a class="reference internal" href="#r82">[R82]</a>. This is a mess, so we have
-a class to represent it directly.</p>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r82" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[R82]</td><td><em>(<a class="fn-backref" href="#id3">1</a>, <a class="fn-backref" href="#id4">2</a>)</em> <a class="reference external" href="http://en.wikipedia.org/wiki/Fock_space">http://en.wikipedia.org/wiki/Fock_space</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.hilbert</span> <span class="kn">import</span> <span class="n">FockSpace</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hs</span> <span class="o">=</span> <span class="n">FockSpace</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hs</span>
-<span class="go">F</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hs</span><span class="o">.</span><span class="n">dimension</span>
-<span class="go">oo</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-
-
-          </div>
-        </div>
-      </div>
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-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../quantum/cartesian.html"
-                        title="previous chapter">Cartesian Operators and States</a></p>
-  <h4>Next topic</h4>
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-  <div class="section" id="module-sympy.physics.quantum.innerproduct">
-<span id="inner-product"></span><h1>Inner Product<a class="headerlink" href="#module-sympy.physics.quantum.innerproduct" title="Permalink to this headline">¶</a></h1>
-<p>Symbolic inner product.</p>
-<dl class="class">
-<dt id="sympy.physics.quantum.innerproduct.InnerProduct">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.innerproduct.</tt><tt class="descname">InnerProduct</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/innerproduct.html#InnerProduct"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.innerproduct.InnerProduct" title="Permalink to this definition">¶</a></dt>
-<dd><p>An unevaluated inner product between a Bra and a Ket [1].</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>bra</strong> : BraBase or subclass</p>
-<blockquote>
-<div><p>The bra on the left side of the inner product.</p>
-</div></blockquote>
-<p><strong>ket</strong> : KetBase or subclass</p>
-<blockquote class="last">
-<div><p>The ket on the right side of the inner product.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r83" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id1">[R83]</a></td><td><a class="reference external" href="http://en.wikipedia.org/wiki/Inner_product">http://en.wikipedia.org/wiki/Inner_product</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Create an InnerProduct and check its properties:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum</span> <span class="kn">import</span> <span class="n">Bra</span><span class="p">,</span> <span class="n">Ket</span><span class="p">,</span> <span class="n">InnerProduct</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Bra</span><span class="p">(</span><span class="s">&#39;b&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="n">Ket</span><span class="p">(</span><span class="s">&#39;k&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ip</span> <span class="o">=</span> <span class="n">b</span><span class="o">*</span><span class="n">k</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ip</span>
-<span class="go">&lt;b|k&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ip</span><span class="o">.</span><span class="n">bra</span>
-<span class="go">&lt;b|</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ip</span><span class="o">.</span><span class="n">ket</span>
-<span class="go">|k&gt;</span>
-</pre></div>
-</div>
-<p>In simple products of kets and bras inner products will be automatically
-identified and created:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">*</span><span class="n">k</span>
-<span class="go">&lt;b|k&gt;</span>
-</pre></div>
-</div>
-<p>But in more complex expressions, there is ambiguity in whether inner or
-outer products should be created:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">k</span><span class="o">*</span><span class="n">b</span><span class="o">*</span><span class="n">k</span><span class="o">*</span><span class="n">b</span>
-<span class="go">|k&gt;&lt;b|*|k&gt;*&lt;b|</span>
-</pre></div>
-</div>
-<p>A user can force the creation of a inner products in a complex expression
-by using parentheses to group the bra and ket:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">k</span><span class="o">*</span><span class="p">(</span><span class="n">b</span><span class="o">*</span><span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="n">b</span>
-<span class="go">&lt;b|k&gt;*|k&gt;*&lt;b|</span>
-</pre></div>
-</div>
-<p>Notice how the inner product &lt;b|k&gt; moved to the left of the expression
-because inner products are commutative complex numbers.</p>
-</dd></dl>
-
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-  <div class="section" id="module-sympy.physics.quantum.operator">
-<span id="operator"></span><h1>Operator<a class="headerlink" href="#module-sympy.physics.quantum.operator" title="Permalink to this headline">¶</a></h1>
-<p>Quantum mechanical operators.</p>
-<p>TODO:</p>
-<ul class="simple">
-<li>Fix early 0 in apply_operators.</li>
-<li>Debug and test apply_operators.</li>
-<li>Get cse working with classes in this file.</li>
-<li>Doctests and documentation of special methods for InnerProduct, Commutator,
-AntiCommutator, represent, apply_operators.</li>
-</ul>
-<dl class="class">
-<dt id="sympy.physics.quantum.operator.Operator">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.operator.</tt><tt class="descname">Operator</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/operator.html#Operator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.operator.Operator" title="Permalink to this definition">¶</a></dt>
-<dd><p>Base class for non-commuting quantum operators.</p>
-<p>An operator maps between quantum states <a class="reference internal" href="#r84">[R84]</a>. In quantum mechanics,
-observables (including, but not limited to, measured physical values) are
-represented as Hermitian operators <a class="reference internal" href="#r85">[R85]</a>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>args</strong> : tuple</p>
-<blockquote class="last">
-<div><p>The list of numbers or parameters that uniquely specify the
-operator. For time-dependent operators, this will include the time.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r84" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[R84]</td><td><em>(<a class="fn-backref" href="#id1">1</a>, <a class="fn-backref" href="#id3">2</a>)</em> <a class="reference external" href="http://en.wikipedia.org/wiki/Operator_%28physics%29">http://en.wikipedia.org/wiki/Operator_%28physics%29</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="r85" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[R85]</td><td><em>(<a class="fn-backref" href="#id2">1</a>, <a class="fn-backref" href="#id4">2</a>)</em> <a class="reference external" href="http://en.wikipedia.org/wiki/Observable">http://en.wikipedia.org/wiki/Observable</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Create an operator and examine its attributes:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum</span> <span class="kn">import</span> <span class="n">Operator</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Operator</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span>
-<span class="go">A</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">hilbert_space</span>
-<span class="go">H</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">label</span>
-<span class="go">(A,)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">is_commutative</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>Create another operator and do some arithmetic operations:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Operator</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">A</span><span class="o">*</span><span class="n">A</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">B</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span>
-<span class="go">2*A**2 + I*B</span>
-</pre></div>
-</div>
-<p>Operators don&#8217;t commute:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">is_commutative</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="o">.</span><span class="n">is_commutative</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">*</span><span class="n">B</span> <span class="o">==</span> <span class="n">B</span><span class="o">*</span><span class="n">A</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>Polymonials of operators respect the commutation properties:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="p">(</span><span class="n">A</span><span class="o">+</span><span class="n">B</span><span class="p">)</span><span class="o">**</span><span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go">A*B*A + A*B**2 + A**2*B + A**3 + B*A*B + B*A**2 + B**2*A + B**3</span>
-</pre></div>
-</div>
-<p>Operator inverses are handle symbolically:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">inv</span><span class="p">()</span>
-<span class="go">A**(-1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">*</span><span class="n">A</span><span class="o">.</span><span class="n">inv</span><span class="p">()</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.operator.HermitianOperator">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.operator.</tt><tt class="descname">HermitianOperator</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/operator.html#HermitianOperator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.operator.HermitianOperator" title="Permalink to this definition">¶</a></dt>
-<dd><p>A Hermitian operator that satisfies H == Dagger(H).</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>args</strong> : tuple</p>
-<blockquote class="last">
-<div><p>The list of numbers or parameters that uniquely specify the
-operator. For time-dependent operators, this will include the time.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum</span> <span class="kn">import</span> <span class="n">Dagger</span><span class="p">,</span> <span class="n">HermitianOperator</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">H</span> <span class="o">=</span> <span class="n">HermitianOperator</span><span class="p">(</span><span class="s">&#39;H&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="n">H</span><span class="p">)</span>
-<span class="go">H</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.operator.UnitaryOperator">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.operator.</tt><tt class="descname">UnitaryOperator</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/operator.html#UnitaryOperator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.operator.UnitaryOperator" title="Permalink to this definition">¶</a></dt>
-<dd><p>A unitary operator that satisfies U*Dagger(U) == 1.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>args</strong> : tuple</p>
-<blockquote class="last">
-<div><p>The list of numbers or parameters that uniquely specify the
-operator. For time-dependent operators, this will include the time.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum</span> <span class="kn">import</span> <span class="n">Dagger</span><span class="p">,</span> <span class="n">UnitaryOperator</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">U</span> <span class="o">=</span> <span class="n">UnitaryOperator</span><span class="p">(</span><span class="s">&#39;U&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">U</span><span class="o">*</span><span class="n">Dagger</span><span class="p">(</span><span class="n">U</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.operator.OuterProduct">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.operator.</tt><tt class="descname">OuterProduct</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/operator.html#OuterProduct"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.operator.OuterProduct" title="Permalink to this definition">¶</a></dt>
-<dd><p>An unevaluated outer product between a ket and bra.</p>
-<p>This constructs an outer product between any subclass of <tt class="docutils literal"><span class="pre">KetBase</span></tt> and
-<tt class="docutils literal"><span class="pre">BraBase</span></tt> as <tt class="docutils literal"><span class="pre">|a&gt;&lt;b|</span></tt>. An <tt class="docutils literal"><span class="pre">OuterProduct</span></tt> inherits from Operator as they act as
-operators in quantum expressions.  For reference see <a class="reference internal" href="#r86">[R86]</a>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>ket</strong> : KetBase</p>
-<blockquote>
-<div><p>The ket on the left side of the outer product.</p>
-</div></blockquote>
-<p><strong>bar</strong> : BraBase</p>
-<blockquote class="last">
-<div><p>The bra on the right side of the outer product.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r86" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[R86]</td><td><em>(<a class="fn-backref" href="#id5">1</a>, <a class="fn-backref" href="#id6">2</a>)</em> <a class="reference external" href="http://en.wikipedia.org/wiki/Outer_product">http://en.wikipedia.org/wiki/Outer_product</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Create a simple outer product by hand and take its dagger:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum</span> <span class="kn">import</span> <span class="n">Ket</span><span class="p">,</span> <span class="n">Bra</span><span class="p">,</span> <span class="n">OuterProduct</span><span class="p">,</span> <span class="n">Dagger</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum</span> <span class="kn">import</span> <span class="n">Operator</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="n">Ket</span><span class="p">(</span><span class="s">&#39;k&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Bra</span><span class="p">(</span><span class="s">&#39;b&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">op</span> <span class="o">=</span> <span class="n">OuterProduct</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">op</span>
-<span class="go">|k&gt;&lt;b|</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">op</span><span class="o">.</span><span class="n">hilbert_space</span>
-<span class="go">H</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">op</span><span class="o">.</span><span class="n">ket</span>
-<span class="go">|k&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">op</span><span class="o">.</span><span class="n">bra</span>
-<span class="go">&lt;b|</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="n">op</span><span class="p">)</span>
-<span class="go">|b&gt;&lt;k|</span>
-</pre></div>
-</div>
-<p>In simple products of kets and bras outer products will be automatically
-identified and created:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">k</span><span class="o">*</span><span class="n">b</span>
-<span class="go">|k&gt;&lt;b|</span>
-</pre></div>
-</div>
-<p>But in more complex expressions, outer products are not automatically
-created:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Operator</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">*</span><span class="n">k</span><span class="o">*</span><span class="n">b</span>
-<span class="go">A*|k&gt;*&lt;b|</span>
-</pre></div>
-</div>
-<p>A user can force the creation of an outer product in a complex expression
-by using parentheses to group the ket and bra:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">*</span><span class="p">(</span><span class="n">k</span><span class="o">*</span><span class="n">b</span><span class="p">)</span>
-<span class="go">A*|k&gt;&lt;b|</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.physics.quantum.operator.OuterProduct.bra">
-<tt class="descname">bra</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/operator.html#OuterProduct.bra"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.operator.OuterProduct.bra" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the bra on the right side of the outer product.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.operator.OuterProduct.ket">
-<tt class="descname">ket</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/operator.html#OuterProduct.ket"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.operator.OuterProduct.ket" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the ket on the left side of the outer product.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.operator.DifferentialOperator">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.operator.</tt><tt class="descname">DifferentialOperator</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/operator.html#DifferentialOperator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.operator.DifferentialOperator" title="Permalink to this definition">¶</a></dt>
-<dd><p>An operator for representing the differential operator, i.e. d/dx</p>
-<p>It is initialized by passing two arguments. The first is an arbitrary
-expression that involves a function, such as <tt class="docutils literal"><span class="pre">Derivative(f(x),</span> <span class="pre">x)</span></tt>. The
-second is the function (e.g. <tt class="docutils literal"><span class="pre">f(x)</span></tt>) which we are to replace with the
-<tt class="docutils literal"><span class="pre">Wavefunction</span></tt> that this <tt class="docutils literal"><span class="pre">DifferentialOperator</span></tt> is applied to.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>expr</strong> : Expr</p>
-<blockquote>
-<div><p>The arbitrary expression which the appropriate Wavefunction is to be
-substituted into</p>
-</div></blockquote>
-<p><strong>func</strong> : Expr</p>
-<blockquote class="last">
-<div><p>A function (e.g. f(x)) which is to be replaced with the appropriate
-Wavefunction when this DifferentialOperator is applied</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>You can define a completely arbitrary expression and specify where the
-Wavefunction is to be substituted</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Derivative</span><span class="p">,</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.operator</span> <span class="kn">import</span> <span class="n">DifferentialOperator</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.state</span> <span class="kn">import</span> <span class="n">Wavefunction</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.qapply</span> <span class="kn">import</span> <span class="n">qapply</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">DifferentialOperator</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="o">*</span><span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">w</span> <span class="o">=</span> <span class="n">Wavefunction</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="o">.</span><span class="n">function</span>
-<span class="go">f(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="o">.</span><span class="n">variables</span>
-<span class="go">(x,)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">qapply</span><span class="p">(</span><span class="n">d</span><span class="o">*</span><span class="n">w</span><span class="p">)</span>
-<span class="go">Wavefunction(2, x)</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.physics.quantum.operator.DifferentialOperator.expr">
-<tt class="descname">expr</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/operator.html#DifferentialOperator.expr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.operator.DifferentialOperator.expr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the arbitary expression which is to have the Wavefunction
-substituted into it</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.operator</span> <span class="kn">import</span> <span class="n">DifferentialOperator</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Derivative</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">DifferentialOperator</span><span class="p">(</span><span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="o">.</span><span class="n">expr</span>
-<span class="go">Derivative(f(x), x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">DifferentialOperator</span><span class="p">(</span><span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span>
-<span class="gp">... </span>                         <span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="n">y</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="o">.</span><span class="n">expr</span>
-<span class="go">Derivative(f(x, y), x) + Derivative(f(x, y), y)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.operator.DifferentialOperator.free_symbols">
-<tt class="descname">free_symbols</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/operator.html#DifferentialOperator.free_symbols"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.operator.DifferentialOperator.free_symbols" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the free symbols of the expression.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.operator.DifferentialOperator.function">
-<tt class="descname">function</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/operator.html#DifferentialOperator.function"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.operator.DifferentialOperator.function" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the function which is to be replaced with the Wavefunction</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.operator</span> <span class="kn">import</span> <span class="n">DifferentialOperator</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Derivative</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">DifferentialOperator</span><span class="p">(</span><span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="o">.</span><span class="n">function</span>
-<span class="go">f(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">DifferentialOperator</span><span class="p">(</span><span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span>
-<span class="gp">... </span>                         <span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="n">y</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="o">.</span><span class="n">function</span>
-<span class="go">f(x, y)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.operator.DifferentialOperator.variables">
-<tt class="descname">variables</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/operator.html#DifferentialOperator.variables"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.operator.DifferentialOperator.variables" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the variables with which the function in the specified
-arbitrary expression is evaluated</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.operator</span> <span class="kn">import</span> <span class="n">DifferentialOperator</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Derivative</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">DifferentialOperator</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="o">*</span><span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="o">.</span><span class="n">variables</span>
-<span class="go">(x,)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">DifferentialOperator</span><span class="p">(</span><span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span>
-<span class="gp">... </span>                         <span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="n">y</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="o">.</span><span class="n">variables</span>
-<span class="go">(x, y)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
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-  <div class="section" id="module-sympy.physics.quantum.operatorset">
-<span id="operator-state-helper-functions"></span><h1>Operator/State Helper Functions<a class="headerlink" href="#module-sympy.physics.quantum.operatorset" title="Permalink to this headline">¶</a></h1>
-<p>A module for mapping operators to their corresponding eigenstates
-and vice versa</p>
-<p>It contains a global dictionary with eigenstate-operator pairings.
-If a new state-operator pair is created, this dictionary should be
-updated as well.</p>
-<p>It also contains functions operators_to_state and state_to_operators
-for mapping between the two. These can handle both classes and
-instances of operators and states. See the individual function
-descriptions for details.</p>
-<p>TODO List:
-- Update the dictionary with a complete list of state-operator pairs</p>
-<dl class="function">
-<dt id="sympy.physics.quantum.operatorset.operators_to_state">
-<tt class="descclassname">sympy.physics.quantum.operatorset.</tt><tt class="descname">operators_to_state</tt><big>(</big><em>operators</em>, <em>**options</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/operatorset.html#operators_to_state"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.operatorset.operators_to_state" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the eigenstate of the given operator or set of operators</p>
-<p>A global function for mapping operator classes to their associated
-states. It takes either an Operator or a set of operators and
-returns the state associated with these.</p>
-<p>This function can handle both instances of a given operator or
-just the class itself (i.e. both XOp() and XOp)</p>
-<p>There are multiple use cases to consider:</p>
-<p>1) A class or set of classes is passed: First, we try to
-instantiate default instances for these operators. If this fails,
-then the class is simply returned. If we succeed in instantiating
-default instances, then we try to call state._operators_to_state
-on the operator instances. If this fails, the class is returned.
-Otherwise, the instance returned by _operators_to_state is returned.</p>
-<p>2) An instance or set of instances is passed: In this case,
-state._operators_to_state is called on the instances passed. If
-this fails, a state class is returned. If the method returns an
-instance, that instance is returned.</p>
-<p>In both cases, if the operator class or set does not exist in the
-state_mapping dictionary, None is returned.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>arg: Operator or set</strong> :</p>
-<blockquote class="last">
-<div><p>The class or instance of the operator or set of operators
-to be mapped to a state</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.cartesian</span> <span class="kn">import</span> <span class="n">XOp</span><span class="p">,</span> <span class="n">PxOp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.operatorset</span> <span class="kn">import</span> <span class="n">operators_to_state</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.operator</span> <span class="kn">import</span> <span class="n">Operator</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">operators_to_state</span><span class="p">(</span><span class="n">XOp</span><span class="p">)</span>
-<span class="go">|x&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">operators_to_state</span><span class="p">(</span><span class="n">XOp</span><span class="p">())</span>
-<span class="go">|x&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">operators_to_state</span><span class="p">(</span><span class="n">PxOp</span><span class="p">)</span>
-<span class="go">|px&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">operators_to_state</span><span class="p">(</span><span class="n">PxOp</span><span class="p">())</span>
-<span class="go">|px&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">operators_to_state</span><span class="p">(</span><span class="n">Operator</span><span class="p">)</span>
-<span class="go">|psi&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">operators_to_state</span><span class="p">(</span><span class="n">Operator</span><span class="p">())</span>
-<span class="go">|psi&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.operatorset.state_to_operators">
-<tt class="descclassname">sympy.physics.quantum.operatorset.</tt><tt class="descname">state_to_operators</tt><big>(</big><em>state</em>, <em>**options</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/operatorset.html#state_to_operators"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.operatorset.state_to_operators" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the operator or set of operators corresponding to the
-given eigenstate</p>
-<p>A global function for mapping state classes to their associated
-operators or sets of operators. It takes either a state class
-or instance.</p>
-<p>This function can handle both instances of a given state or just
-the class itself (i.e. both XKet() and XKet)</p>
-<p>There are multiple use cases to consider:</p>
-<p>1) A state class is passed: In this case, we first try
-instantiating a default instance of the class. If this succeeds,
-then we try to call state._state_to_operators on that instance.
-If the creation of the default instance or if the calling of
-_state_to_operators fails, then either an operator class or set of
-operator classes is returned. Otherwise, the appropriate
-operator instances are returned.</p>
-<p>2) A state instance is returned: Here, state._state_to_operators
-is called for the instance. If this fails, then a class or set of
-operator classes is returned. Otherwise, the instances are returned.</p>
-<p>In either case, if the state&#8217;s class does not exist in
-state_mapping, None is returned.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>arg: StateBase class or instance (or subclasses)</strong> :</p>
-<blockquote class="last">
-<div><p>The class or instance of the state to be mapped to an
-operator or set of operators</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.cartesian</span> <span class="kn">import</span> <span class="n">XKet</span><span class="p">,</span> <span class="n">PxKet</span><span class="p">,</span> <span class="n">XBra</span><span class="p">,</span> <span class="n">PxBra</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.operatorset</span> <span class="kn">import</span> <span class="n">state_to_operators</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.state</span> <span class="kn">import</span> <span class="n">Ket</span><span class="p">,</span> <span class="n">Bra</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">state_to_operators</span><span class="p">(</span><span class="n">XKet</span><span class="p">)</span>
-<span class="go">X</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">state_to_operators</span><span class="p">(</span><span class="n">XKet</span><span class="p">())</span>
-<span class="go">X</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">state_to_operators</span><span class="p">(</span><span class="n">PxKet</span><span class="p">)</span>
-<span class="go">Px</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">state_to_operators</span><span class="p">(</span><span class="n">PxKet</span><span class="p">())</span>
-<span class="go">Px</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">state_to_operators</span><span class="p">(</span><span class="n">PxBra</span><span class="p">)</span>
-<span class="go">Px</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">state_to_operators</span><span class="p">(</span><span class="n">XBra</span><span class="p">)</span>
-<span class="go">X</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">state_to_operators</span><span class="p">(</span><span class="n">Ket</span><span class="p">)</span>
-<span class="go">O</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">state_to_operators</span><span class="p">(</span><span class="n">Bra</span><span class="p">)</span>
-<span class="go">O</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
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-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../quantum/operator.html"
-                        title="previous chapter">Operator</a></p>
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diff --git a/dev-py3k/modules/physics/quantum/piab.html b/dev-py3k/modules/physics/quantum/piab.html
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-  <div class="section" id="module-sympy.physics.quantum.piab">
-<span id="particle-in-a-box"></span><h1>Particle in a Box<a class="headerlink" href="#module-sympy.physics.quantum.piab" title="Permalink to this headline">¶</a></h1>
-<p>1D quantum particle in a box.</p>
-<dl class="class">
-<dt id="sympy.physics.quantum.piab.PIABHamiltonian">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.piab.</tt><tt class="descname">PIABHamiltonian</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/piab.html#PIABHamiltonian"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.piab.PIABHamiltonian" title="Permalink to this definition">¶</a></dt>
-<dd><p>Particle in a box Hamiltonian operator.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.piab.PIABKet">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.piab.</tt><tt class="descname">PIABKet</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/piab.html#PIABKet"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.piab.PIABKet" title="Permalink to this definition">¶</a></dt>
-<dd><p>Particle in a box eigenket.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.piab.PIABBra">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.piab.</tt><tt class="descname">PIABBra</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/piab.html#PIABBra"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.piab.PIABBra" title="Permalink to this definition">¶</a></dt>
-<dd><p>Particle in a box eigenbra.</p>
-</dd></dl>
-
-</div>
-
-
-          </div>
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-  <div class="section" id="module-sympy.physics.quantum.qapply">
-<span id="qapply"></span><h1>Qapply<a class="headerlink" href="#module-sympy.physics.quantum.qapply" title="Permalink to this headline">¶</a></h1>
-<p>Logic for applying operators to states.</p>
-<p>Todo:
-* Sometimes the final result needs to be expanded, we should do this by hand.</p>
-<dl class="function">
-<dt id="sympy.physics.quantum.qapply.qapply">
-<tt class="descclassname">sympy.physics.quantum.qapply.</tt><tt class="descname">qapply</tt><big>(</big><em>e</em>, <em>**options</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/qapply.html#qapply"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.qapply.qapply" title="Permalink to this definition">¶</a></dt>
-<dd><p>Apply operators to states in a quantum expression.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>e</strong> : Expr</p>
-<blockquote>
-<div><p>The expression containing operators and states. This expression tree
-will be walked to find operators acting on states symbolically.</p>
-</div></blockquote>
-<p><strong>options</strong> : dict</p>
-<blockquote>
-<div><p>A dict of key/value pairs that determine how the operator actions
-are carried out.</p>
-<p>The following options are valid:</p>
-<ul class="simple">
-<li><tt class="docutils literal"><span class="pre">dagger</span></tt>: try to apply Dagger operators to the left
-(default: False).</li>
-<li><tt class="docutils literal"><span class="pre">ip_doit</span></tt>: call <tt class="docutils literal"><span class="pre">.doit()</span></tt> in inner products when they are
-encountered (default: True).</li>
-</ul>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>e</strong> : Expr</p>
-<blockquote class="last">
-<div><p>The original expression, but with the operators applied to states.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</div>
-
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-                        title="previous chapter">Operator/State Helper Functions</a></p>
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-  <div class="section" id="module-sympy.physics.quantum.qft">
-<span id="qft"></span><h1>QFT<a class="headerlink" href="#module-sympy.physics.quantum.qft" title="Permalink to this headline">¶</a></h1>
-<p>An implementation of qubits and gates acting on them.</p>
-<p>Todo:</p>
-<ul class="simple">
-<li>Update docstrings.</li>
-<li>Update tests.</li>
-<li>Implement apply using decompose.</li>
-<li>Implement represent using decompose or something smarter. For this to
-work we first have to implement represent for SWAP.</li>
-<li>Decide if we want upper index to be inclusive in the constructor.</li>
-<li>Fix the printing of Rk gates in plotting.</li>
-</ul>
-<dl class="class">
-<dt id="sympy.physics.quantum.qft.QFT">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.qft.</tt><tt class="descname">QFT</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/qft.html#QFT"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.qft.QFT" title="Permalink to this definition">¶</a></dt>
-<dd><p>The forward quantum Fourier transform.</p>
-<dl class="function">
-<dt id="sympy.physics.quantum.qft.QFT.decompose">
-<tt class="descname">decompose</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/qft.html#QFT.decompose"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.qft.QFT.decompose" title="Permalink to this definition">¶</a></dt>
-<dd><p>Decomposes QFT into elementary gates.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.qft.IQFT">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.qft.</tt><tt class="descname">IQFT</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/qft.html#IQFT"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.qft.IQFT" title="Permalink to this definition">¶</a></dt>
-<dd><p>The inverse quantum Fourier transform.</p>
-<dl class="function">
-<dt id="sympy.physics.quantum.qft.IQFT.decompose">
-<tt class="descname">decompose</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/qft.html#IQFT.decompose"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.qft.IQFT.decompose" title="Permalink to this definition">¶</a></dt>
-<dd><p>Decomposes IQFT into elementary gates.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.qft.RkGate">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.qft.</tt><tt class="descname">RkGate</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/qft.html#RkGate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.qft.RkGate" title="Permalink to this definition">¶</a></dt>
-<dd><p>This is the R_k gate of the QTF.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.qft.Rk">
-<tt class="descclassname">sympy.physics.quantum.qft.</tt><tt class="descname">Rk</tt><a class="headerlink" href="#sympy.physics.quantum.qft.Rk" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <a class="reference internal" href="#sympy.physics.quantum.qft.RkGate" title="sympy.physics.quantum.qft.RkGate"><tt class="xref py py-class docutils literal"><span class="pre">RkGate</span></tt></a></p>
-</dd></dl>
-
-</div>
-
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-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../../index.html">
-              <img class="logo" src="../../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../quantum/grover.html"
-                        title="previous chapter">Grover&#8217;s Algorithm</a></p>
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-                        title="next chapter">Qubit</a></p>
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-  <div class="section" id="module-sympy.physics.quantum.qubit">
-<span id="qubit"></span><h1>Qubit<a class="headerlink" href="#module-sympy.physics.quantum.qubit" title="Permalink to this headline">¶</a></h1>
-<p>Qubits for quantum computing.</p>
-<p>Todo:
-* Finish implementing measurement logic. This should include POVM.
-* Update docstrings.
-* Update tests.</p>
-<dl class="class">
-<dt id="sympy.physics.quantum.qubit.Qubit">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.qubit.</tt><tt class="descname">Qubit</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/qubit.html#Qubit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.qubit.Qubit" title="Permalink to this definition">¶</a></dt>
-<dd><p>A multi-qubit ket in the computational (z) basis.</p>
-<p>We use the normal convention that the least significant qubit is on the
-right, so <tt class="docutils literal"><span class="pre">|00001&gt;</span></tt> has a 1 in the least significant qubit.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>values</strong> : list, str</p>
-<blockquote class="last">
-<div><p>The qubit values as a list of ints ([0,0,0,1,1,]) or a string (&#8216;011&#8217;).</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Create a qubit in a couple of different ways and look at their attributes:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.qubit</span> <span class="kn">import</span> <span class="n">Qubit</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Qubit</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">|000&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Qubit</span><span class="p">(</span><span class="s">&#39;0101&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span>
-<span class="go">|0101&gt;</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="o">.</span><span class="n">nqubits</span>
-<span class="go">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">len</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-<span class="go">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="o">.</span><span class="n">dimension</span>
-<span class="go">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="o">.</span><span class="n">qubit_values</span>
-<span class="go">(0, 1, 0, 1)</span>
-</pre></div>
-</div>
-<p>We can flip the value of an individual qubit:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="o">.</span><span class="n">flip</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">|0111&gt;</span>
-</pre></div>
-</div>
-<p>We can take the dagger of a Qubit to get a bra:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.dagger</span> <span class="kn">import</span> <span class="n">Dagger</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-<span class="go">&lt;0101|</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="n">Dagger</span><span class="p">(</span><span class="n">q</span><span class="p">))</span>
-<span class="go">&lt;class &#39;sympy.physics.quantum.qubit.QubitBra&#39;&gt;</span>
-</pre></div>
-</div>
-<p>Inner products work as expected:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ip</span> <span class="o">=</span> <span class="n">Dagger</span><span class="p">(</span><span class="n">q</span><span class="p">)</span><span class="o">*</span><span class="n">q</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ip</span>
-<span class="go">&lt;0101|0101&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ip</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.qubit.QubitBra">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.qubit.</tt><tt class="descname">QubitBra</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/qubit.html#QubitBra"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.qubit.QubitBra" title="Permalink to this definition">¶</a></dt>
-<dd><p>A multi-qubit bra in the computational (z) basis.</p>
-<p>We use the normal convention that the least significant qubit is on the
-right, so <tt class="docutils literal"><span class="pre">|00001&gt;</span></tt> has a 1 in the least significant qubit.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>values</strong> : list, str</p>
-<blockquote class="last">
-<div><p>The qubit values as a list of ints ([0,0,0,1,1,]) or a string (&#8216;011&#8217;).</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.quantum.qubit.Qubit" title="sympy.physics.quantum.qubit.Qubit"><tt class="xref py py-obj docutils literal"><span class="pre">Qubit</span></tt></a></dt>
-<dd>Examples using qubits</dd>
-</dl>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.qubit.IntQubit">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.qubit.</tt><tt class="descname">IntQubit</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/qubit.html#IntQubit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.qubit.IntQubit" title="Permalink to this definition">¶</a></dt>
-<dd><p>A qubit ket that store integers as binary numbers in qubit values.</p>
-<p>The differences between this class and <tt class="docutils literal"><span class="pre">Qubit</span></tt> are:</p>
-<ul class="simple">
-<li>The form of the constructor.</li>
-<li>The qubit values are printed as their corresponding integer, rather
-than the raw qubit values. The internal storage format of the qubit
-values in the same as <tt class="docutils literal"><span class="pre">Qubit</span></tt>.</li>
-</ul>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>values</strong> : int, tuple</p>
-<blockquote class="last">
-<div><p>If a single argument, the integer we want to represent in the qubit
-values. This integer will be represented using the fewest possible
-number of qubits. If a pair of integers, the first integer gives the
-integer to represent in binary form and the second integer gives
-the number of qubits to use.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Create a qubit for the integer 5:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.qubit</span> <span class="kn">import</span> <span class="n">IntQubit</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.qubit</span> <span class="kn">import</span> <span class="n">Qubit</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">IntQubit</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span>
-<span class="go">|5&gt;</span>
-</pre></div>
-</div>
-<p>We can also create an <tt class="docutils literal"><span class="pre">IntQubit</span></tt> by passing a <tt class="docutils literal"><span class="pre">Qubit</span></tt> instance.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">IntQubit</span><span class="p">(</span><span class="n">Qubit</span><span class="p">(</span><span class="s">&#39;101&#39;</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span>
-<span class="go">|5&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="o">.</span><span class="n">as_int</span><span class="p">()</span>
-<span class="go">5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="o">.</span><span class="n">nqubits</span>
-<span class="go">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="o">.</span><span class="n">qubit_values</span>
-<span class="go">(1, 0, 1)</span>
-</pre></div>
-</div>
-<p>We can go back to the regular qubit form.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Qubit</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-<span class="go">|101&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.qubit.IntQubitBra">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.qubit.</tt><tt class="descname">IntQubitBra</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/qubit.html#IntQubitBra"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.qubit.IntQubitBra" title="Permalink to this definition">¶</a></dt>
-<dd><p>A qubit bra that store integers as binary numbers in qubit values.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.qubit.qubit_to_matrix">
-<tt class="descclassname">sympy.physics.quantum.qubit.</tt><tt class="descname">qubit_to_matrix</tt><big>(</big><em>qubit</em>, <em>format='sympy'</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/qubit.html#qubit_to_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.qubit.qubit_to_matrix" title="Permalink to this definition">¶</a></dt>
-<dd><p>Coverts an Add/Mul of Qubit objects into it&#8217;s matrix representation</p>
-<p>This function is the inverse of <tt class="docutils literal"><span class="pre">matrix_to_qubit</span></tt> and is a shorthand
-for <tt class="docutils literal"><span class="pre">represent(qubit)</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.qubit.matrix_to_qubit">
-<tt class="descclassname">sympy.physics.quantum.qubit.</tt><tt class="descname">matrix_to_qubit</tt><big>(</big><em>matrix</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/qubit.html#matrix_to_qubit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.qubit.matrix_to_qubit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert from the matrix repr. to a sum of Qubit objects.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>matrix</strong> : Matrix, numpy.matrix, scipy.sparse</p>
-<blockquote class="last">
-<div><p>The matrix to build the Qubit representation of. This works with
-sympy matrices, numpy matrices and scipy.sparse sparse matrices.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Represent a state and then go back to its qubit form:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.qubit</span> <span class="kn">import</span> <span class="n">matrix_to_qubit</span><span class="p">,</span> <span class="n">Qubit</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.gate</span> <span class="kn">import</span> <span class="n">Z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.represent</span> <span class="kn">import</span> <span class="n">represent</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Qubit</span><span class="p">(</span><span class="s">&#39;01&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">matrix_to_qubit</span><span class="p">(</span><span class="n">represent</span><span class="p">(</span><span class="n">q</span><span class="p">))</span>
-<span class="go">|01&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.qubit.matrix_to_density">
-<tt class="descclassname">sympy.physics.quantum.qubit.</tt><tt class="descname">matrix_to_density</tt><big>(</big><em>mat</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/qubit.html#matrix_to_density"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.qubit.matrix_to_density" title="Permalink to this definition">¶</a></dt>
-<dd><p>Works by finding the eigenvectors and eigenvalues of the matrix.
-We know we can decompose rho by doing:
-sum(EigenVal*|Eigenvect&gt;&lt;Eigenvect|)</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.qubit.measure_all">
-<tt class="descclassname">sympy.physics.quantum.qubit.</tt><tt class="descname">measure_all</tt><big>(</big><em>qubit</em>, <em>format='sympy'</em>, <em>normalize=True</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/qubit.html#measure_all"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.qubit.measure_all" title="Permalink to this definition">¶</a></dt>
-<dd><p>Perform an ensemble measurement of all qubits.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>qubit</strong> : Qubit, Add</p>
-<blockquote>
-<div><p>The qubit to measure. This can be any Qubit or a linear combination
-of them.</p>
-</div></blockquote>
-<p><strong>format</strong> : str</p>
-<blockquote>
-<div><p>The format of the intermediate matrices to use. Possible values are
-(&#8216;sympy&#8217;,&#8217;numpy&#8217;,&#8217;scipy.sparse&#8217;). Currently only &#8216;sympy&#8217; is
-implemented.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>result</strong> : list</p>
-<blockquote class="last">
-<div><p>A list that consists of primitive states and their probabilities.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.qubit</span> <span class="kn">import</span> <span class="n">Qubit</span><span class="p">,</span> <span class="n">measure_all</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.gate</span> <span class="kn">import</span> <span class="n">H</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">Z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.qapply</span> <span class="kn">import</span> <span class="n">qapply</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">H</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="o">*</span><span class="n">H</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">Qubit</span><span class="p">(</span><span class="s">&#39;00&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span>
-<span class="go">H(0)*H(1)*|00&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">qapply</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">measure_all</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
-<span class="go">[(|00&gt;, 1/4), (|01&gt;, 1/4), (|10&gt;, 1/4), (|11&gt;, 1/4)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.qubit.measure_partial">
-<tt class="descclassname">sympy.physics.quantum.qubit.</tt><tt class="descname">measure_partial</tt><big>(</big><em>qubit</em>, <em>bits</em>, <em>format='sympy'</em>, <em>normalize=True</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/qubit.html#measure_partial"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.qubit.measure_partial" title="Permalink to this definition">¶</a></dt>
-<dd><p>Perform a partial ensemble measure on the specifed qubits.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>qubits</strong> : Qubit</p>
-<blockquote>
-<div><p>The qubit to measure.  This can be any Qubit or a linear combination
-of them.</p>
-</div></blockquote>
-<p><strong>bits</strong> : tuple</p>
-<blockquote>
-<div><p>The qubits to measure.</p>
-</div></blockquote>
-<p><strong>format</strong> : str</p>
-<blockquote>
-<div><p>The format of the intermediate matrices to use. Possible values are
-(&#8216;sympy&#8217;,&#8217;numpy&#8217;,&#8217;scipy.sparse&#8217;). Currently only &#8216;sympy&#8217; is
-implemented.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>result</strong> : list</p>
-<blockquote class="last">
-<div><p>A list that consists of primitive states and their probabilities.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.qubit</span> <span class="kn">import</span> <span class="n">Qubit</span><span class="p">,</span> <span class="n">measure_partial</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.gate</span> <span class="kn">import</span> <span class="n">H</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">Z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.qapply</span> <span class="kn">import</span> <span class="n">qapply</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">H</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="o">*</span><span class="n">H</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">Qubit</span><span class="p">(</span><span class="s">&#39;00&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span>
-<span class="go">H(0)*H(1)*|00&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">qapply</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">measure_partial</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="p">(</span><span class="mi">0</span><span class="p">,))</span>
-<span class="go">[(sqrt(2)*|00&gt;/2 + sqrt(2)*|10&gt;/2, 1/2), (sqrt(2)*|01&gt;/2 + sqrt(2)*|11&gt;/2, 1/2)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.qubit.measure_partial_oneshot">
-<tt class="descclassname">sympy.physics.quantum.qubit.</tt><tt class="descname">measure_partial_oneshot</tt><big>(</big><em>qubit</em>, <em>bits</em>, <em>format='sympy'</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/qubit.html#measure_partial_oneshot"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.qubit.measure_partial_oneshot" title="Permalink to this definition">¶</a></dt>
-<dd><p>Perform a partial oneshot measurement on the specified qubits.</p>
-<p>A oneshot measurement is equivalent to performing a measurement on a
-quantum system. This type of measurement does not return the probabilities
-like an ensemble measurement does, but rather returns <em>one</em> of the
-possible resulting states. The exact state that is returned is determined
-by picking a state randomly according to the ensemble probabilities.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>qubits</strong> : Qubit</p>
-<blockquote>
-<div><p>The qubit to measure.  This can be any Qubit or a linear combination
-of them.</p>
-</div></blockquote>
-<p><strong>bits</strong> : tuple</p>
-<blockquote>
-<div><p>The qubits to measure.</p>
-</div></blockquote>
-<p><strong>format</strong> : str</p>
-<blockquote>
-<div><p>The format of the intermediate matrices to use. Possible values are
-(&#8216;sympy&#8217;,&#8217;numpy&#8217;,&#8217;scipy.sparse&#8217;). Currently only &#8216;sympy&#8217; is
-implemented.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>result</strong> : Qubit</p>
-<blockquote class="last">
-<div><p>The qubit that the system collapsed to upon measurement.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.qubit.measure_all_oneshot">
-<tt class="descclassname">sympy.physics.quantum.qubit.</tt><tt class="descname">measure_all_oneshot</tt><big>(</big><em>qubit</em>, <em>format='sympy'</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/qubit.html#measure_all_oneshot"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.qubit.measure_all_oneshot" title="Permalink to this definition">¶</a></dt>
-<dd><p>Perform a oneshot ensemble measurement on all qubits.</p>
-<p>A oneshot measurement is equivalent to performing a measurement on a
-quantum system. This type of measurement does not return the probabilities
-like an ensemble measurement does, but rather returns <em>one</em> of the
-possible resulting states. The exact state that is returned is determined
-by picking a state randomly according to the ensemble probabilities.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>qubits</strong> : Qubit</p>
-<blockquote>
-<div><p>The qubit to measure.  This can be any Qubit or a linear combination
-of them.</p>
-</div></blockquote>
-<p><strong>format</strong> : str</p>
-<blockquote>
-<div><p>The format of the intermediate matrices to use. Possible values are
-(&#8216;sympy&#8217;,&#8217;numpy&#8217;,&#8217;scipy.sparse&#8217;). Currently only &#8216;sympy&#8217; is
-implemented.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>result</strong> : Qubit</p>
-<blockquote class="last">
-<div><p>The qubit that the system collapsed to upon measurement.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</div>
-
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-            
-  <div class="section" id="module-sympy.physics.quantum.represent">
-<span id="represent"></span><h1>Represent<a class="headerlink" href="#module-sympy.physics.quantum.represent" title="Permalink to this headline">¶</a></h1>
-<p>Logic for representing operators in state in various bases.</p>
-<p>TODO:</p>
-<ul class="simple">
-<li>Get represent working with continuous hilbert spaces.</li>
-<li>Document default basis functionality.</li>
-</ul>
-<dl class="function">
-<dt id="sympy.physics.quantum.represent.represent">
-<tt class="descclassname">sympy.physics.quantum.represent.</tt><tt class="descname">represent</tt><big>(</big><em>expr</em>, <em>**options</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/represent.html#represent"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.represent.represent" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represent the quantum expression in the given basis.</p>
-<p>In quantum mechanics abstract states and operators can be represented in
-various basis sets. Under this operation the follow transforms happen:</p>
-<ul class="simple">
-<li>Ket -&gt; column vector or function</li>
-<li>Bra -&gt; row vector of function</li>
-<li>Operator -&gt; matrix or differential operator</li>
-</ul>
-<p>This function is the top-level interface for this action.</p>
-<p>This function walks the sympy expression tree looking for <tt class="docutils literal"><span class="pre">QExpr</span></tt>
-instances that have a <tt class="docutils literal"><span class="pre">_represent</span></tt> method. This method is then called
-and the object is replaced by the representation returned by this method.
-By default, the <tt class="docutils literal"><span class="pre">_represent</span></tt> method will dispatch to other methods
-that handle the representation logic for a particular basis set. The
-naming convention for these methods is the following:</p>
-<div class="highlight-python"><pre>def _represent_FooBasis(self, e, basis, **options)</pre>
-</div>
-<p>This function will have the logic for representing instances of its class
-in the basis set having a class named <tt class="docutils literal"><span class="pre">FooBasis</span></tt>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>expr</strong> : Expr</p>
-<blockquote>
-<div><p>The expression to represent.</p>
-</div></blockquote>
-<p><strong>basis</strong> : Operator, basis set</p>
-<blockquote>
-<div><p>An object that contains the information about the basis set. If an
-operator is used, the basis is assumed to be the orthonormal
-eigenvectors of that operator. In general though, the basis argument
-can be any object that contains the basis set information.</p>
-</div></blockquote>
-<p><strong>options</strong> : dict</p>
-<blockquote>
-<div><p>Key/value pairs of options that are passed to the underlying method
-that does finds the representation. These options can be used to
-control how the representation is done. For example, this is where
-the size of the basis set would be set.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>e</strong> : Expr</p>
-<blockquote class="last">
-<div><p>The sympy expression of the represented quantum expression.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Here we subclass <tt class="docutils literal"><span class="pre">Operator</span></tt> and <tt class="docutils literal"><span class="pre">Ket</span></tt> to create the z-spin operator
-and its spin 1/2 up eigenstate. By definining the <tt class="docutils literal"><span class="pre">_represent_SzOp</span></tt>
-method, the ket can be represented in the z-spin basis.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum</span> <span class="kn">import</span> <span class="n">Operator</span><span class="p">,</span> <span class="n">represent</span><span class="p">,</span> <span class="n">Ket</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">SzUpKet</span><span class="p">(</span><span class="n">Ket</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">def</span> <span class="nf">_represent_SzOp</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="o">**</span><span class="n">options</span><span class="p">):</span>
-<span class="gp">... </span>        <span class="k">return</span> <span class="n">Matrix</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">SzOp</span><span class="p">(</span><span class="n">Operator</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">pass</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sz</span> <span class="o">=</span> <span class="n">SzOp</span><span class="p">(</span><span class="s">&#39;Sz&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">up</span> <span class="o">=</span> <span class="n">SzUpKet</span><span class="p">(</span><span class="s">&#39;up&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">represent</span><span class="p">(</span><span class="n">up</span><span class="p">,</span> <span class="n">basis</span><span class="o">=</span><span class="n">sz</span><span class="p">)</span>
-<span class="go">[1]</span>
-<span class="go">[0]</span>
-</pre></div>
-</div>
-<p>Here we see an example of representations in a continuous
-basis. We see that the result of representing various combinations
-of cartesian position operators and kets give us continuous
-expressions involving DiracDelta functions.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.cartesian</span> <span class="kn">import</span> <span class="n">XOp</span><span class="p">,</span> <span class="n">XKet</span><span class="p">,</span> <span class="n">XBra</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">XOp</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">XKet</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">XBra</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">represent</span><span class="p">(</span><span class="n">X</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">x*DiracDelta(x - x_2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">represent</span><span class="p">(</span><span class="n">X</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
-<span class="go">x*DiracDelta(x - x_3)*DiracDelta(x_1 - y)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.represent.rep_innerproduct">
-<tt class="descclassname">sympy.physics.quantum.represent.</tt><tt class="descname">rep_innerproduct</tt><big>(</big><em>expr</em>, <em>**options</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/represent.html#rep_innerproduct"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.represent.rep_innerproduct" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns an innerproduct like representation (e.g. <tt class="docutils literal"><span class="pre">&lt;x'|x&gt;</span></tt>) for the
-given state.</p>
-<p>Attempts to calculate inner product with a bra from the specified
-basis. Should only be passed an instance of KetBase or BraBase</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>expr</strong> : KetBase or BraBase</p>
-<blockquote class="last">
-<div><p>The expression to be represented</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.represent</span> <span class="kn">import</span> <span class="n">rep_innerproduct</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.cartesian</span> <span class="kn">import</span> <span class="n">XOp</span><span class="p">,</span> <span class="n">XKet</span><span class="p">,</span> <span class="n">PxOp</span><span class="p">,</span> <span class="n">PxKet</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rep_innerproduct</span><span class="p">(</span><span class="n">XKet</span><span class="p">())</span>
-<span class="go">DiracDelta(x - x_1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rep_innerproduct</span><span class="p">(</span><span class="n">XKet</span><span class="p">(),</span> <span class="n">basis</span><span class="o">=</span><span class="n">PxOp</span><span class="p">())</span>
-<span class="go">sqrt(2)*exp(-I*px_1*x/hbar)/(2*sqrt(hbar)*sqrt(pi))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rep_innerproduct</span><span class="p">(</span><span class="n">PxKet</span><span class="p">(),</span> <span class="n">basis</span><span class="o">=</span><span class="n">XOp</span><span class="p">())</span>
-<span class="go">sqrt(2)*exp(I*px*x_1/hbar)/(2*sqrt(hbar)*sqrt(pi))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.represent.rep_expectation">
-<tt class="descclassname">sympy.physics.quantum.represent.</tt><tt class="descname">rep_expectation</tt><big>(</big><em>expr</em>, <em>**options</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/represent.html#rep_expectation"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.represent.rep_expectation" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns an <tt class="docutils literal"><span class="pre">&lt;x'|A|x&gt;</span></tt> type representation for the given operator.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>expr</strong> : Operator</p>
-<blockquote class="last">
-<div><p>Operator to be represented in the specified basis</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.cartesian</span> <span class="kn">import</span> <span class="n">XOp</span><span class="p">,</span> <span class="n">XKet</span><span class="p">,</span> <span class="n">PxOp</span><span class="p">,</span> <span class="n">PxKet</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.represent</span> <span class="kn">import</span> <span class="n">rep_expectation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rep_expectation</span><span class="p">(</span><span class="n">XOp</span><span class="p">())</span>
-<span class="go">x_1*DiracDelta(x_1 - x_2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rep_expectation</span><span class="p">(</span><span class="n">XOp</span><span class="p">(),</span> <span class="n">basis</span><span class="o">=</span><span class="n">PxOp</span><span class="p">())</span>
-<span class="go">&lt;px_2|*X*|px_1&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rep_expectation</span><span class="p">(</span><span class="n">XOp</span><span class="p">(),</span> <span class="n">basis</span><span class="o">=</span><span class="n">PxKet</span><span class="p">())</span>
-<span class="go">&lt;px_2|*X*|px_1&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.represent.integrate_result">
-<tt class="descclassname">sympy.physics.quantum.represent.</tt><tt class="descname">integrate_result</tt><big>(</big><em>orig_expr</em>, <em>result</em>, <em>**options</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/represent.html#integrate_result"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.represent.integrate_result" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the result of integrating over any unities <tt class="docutils literal"><span class="pre">(|x&gt;&lt;x|)</span></tt> in
-the given expression. Intended for integrating over the result of
-representations in continuous bases.</p>
-<p>This function integrates over any unities that may have been
-inserted into the quantum expression and returns the result.
-It uses the interval of the Hilbert space of the basis state
-passed to it in order to figure out the limits of integration.
-The unities option must be
-specified for this to work.</p>
-<p>Note: This is mostly used internally by represent(). Examples are
-given merely to show the use cases.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>orig_expr</strong> : quantum expression</p>
-<blockquote>
-<div><p>The original expression which was to be represented</p>
-</div></blockquote>
-<p><strong>result: Expr</strong> :</p>
-<blockquote class="last">
-<div><p>The resulting representation that we wish to integrate over</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">DiracDelta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.represent</span> <span class="kn">import</span> <span class="n">integrate_result</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.cartesian</span> <span class="kn">import</span> <span class="n">XOp</span><span class="p">,</span> <span class="n">XKet</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x_ket</span> <span class="o">=</span> <span class="n">XKet</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X_op</span> <span class="o">=</span> <span class="n">XOp</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">x_1</span><span class="p">,</span> <span class="n">x_2</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x, x_1, x_2&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate_result</span><span class="p">(</span><span class="n">X_op</span><span class="o">*</span><span class="n">x_ket</span><span class="p">,</span> <span class="n">x</span><span class="o">*</span><span class="n">DiracDelta</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="n">x_1</span><span class="p">)</span><span class="o">*</span><span class="n">DiracDelta</span><span class="p">(</span><span class="n">x_1</span><span class="o">-</span><span class="n">x_2</span><span class="p">))</span>
-<span class="go">x*DiracDelta(x - x_1)*DiracDelta(x_1 - x_2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate_result</span><span class="p">(</span><span class="n">X_op</span><span class="o">*</span><span class="n">x_ket</span><span class="p">,</span> <span class="n">x</span><span class="o">*</span><span class="n">DiracDelta</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="n">x_1</span><span class="p">)</span><span class="o">*</span><span class="n">DiracDelta</span><span class="p">(</span><span class="n">x_1</span><span class="o">-</span><span class="n">x_2</span><span class="p">),</span>
-<span class="gp">... </span>    <span class="n">unities</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">x*DiracDelta(x - x_2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.represent.get_basis">
-<tt class="descclassname">sympy.physics.quantum.represent.</tt><tt class="descname">get_basis</tt><big>(</big><em>expr</em>, <em>**options</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/represent.html#get_basis"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.represent.get_basis" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a basis state instance corresponding to the basis specified in
-options=s. If no basis is specified, the function tries to form a default
-basis state of the given expression.</p>
-<p>There are three behaviors:</p>
-<ol class="arabic simple">
-<li>The basis specified in options is already an instance of StateBase. If
-this is the case, it is simply returned. If the class is specified but
-not an instance, a default instance is returned.</li>
-<li>The basis specified is an operator or set of operators. If this
-is the case, the operator_to_state mapping method is used.</li>
-<li>No basis is specified. If expr is a state, then a default instance of
-its class is returned.  If expr is an operator, then it is mapped to the
-corresponding state.  If it is neither, then we cannot obtain the basis
-state.</li>
-</ol>
-<p>If the basis cannot be mapped, then it is not changed.</p>
-<p>This will be called from within represent, and represent will
-only pass QExpr&#8217;s.</p>
-<p>TODO (?): Support for Muls and other types of expressions?</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>expr</strong> : Operator or StateBase</p>
-<blockquote class="last">
-<div><p>Expression whose basis is sought</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.represent</span> <span class="kn">import</span> <span class="n">get_basis</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.cartesian</span> <span class="kn">import</span> <span class="n">XOp</span><span class="p">,</span> <span class="n">XKet</span><span class="p">,</span> <span class="n">PxOp</span><span class="p">,</span> <span class="n">PxKet</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">XKet</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">XOp</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">get_basis</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">|x&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">get_basis</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">|x&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">get_basis</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">basis</span><span class="o">=</span><span class="n">PxOp</span><span class="p">())</span>
-<span class="go">|px&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">get_basis</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">basis</span><span class="o">=</span><span class="n">PxKet</span><span class="p">)</span>
-<span class="go">|px&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.represent.enumerate_states">
-<tt class="descclassname">sympy.physics.quantum.represent.</tt><tt class="descname">enumerate_states</tt><big>(</big><em>*args</em>, <em>**options</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/represent.html#enumerate_states"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.represent.enumerate_states" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns instances of the given state with dummy indices appended</p>
-<p>Operates in two different modes:</p>
-<ol class="arabic simple">
-<li>Two arguments are passed to it. The first is the base state which is to
-be indexed, and the second argument is a list of indices to append.</li>
-<li>Three arguments are passed. The first is again the base state to be
-indexed. The second is the start index for counting.  The final argument
-is the number of kets you wish to receive.</li>
-</ol>
-<p>Tries to call state._enumerate_state. If this fails, returns an empty list</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>args</strong> : list</p>
-<blockquote class="last">
-<div><p>See list of operation modes above for explanation</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.cartesian</span> <span class="kn">import</span> <span class="n">XBra</span><span class="p">,</span> <span class="n">XKet</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.represent</span> <span class="kn">import</span> <span class="n">enumerate_states</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">test</span> <span class="o">=</span> <span class="n">XKet</span><span class="p">(</span><span class="s">&#39;foo&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">enumerate_states</span><span class="p">(</span><span class="n">test</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">[|foo_1&gt;, |foo_2&gt;, |foo_3&gt;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">test2</span> <span class="o">=</span> <span class="n">XBra</span><span class="p">(</span><span class="s">&#39;bar&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">enumerate_states</span><span class="p">(</span><span class="n">test2</span><span class="p">,</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">10</span><span class="p">])</span>
-<span class="go">[&lt;bar_4|, &lt;bar_5|, &lt;bar_10|]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-
-
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-            
-  <div class="section" id="module-sympy.physics.quantum.shor">
-<span id="shor-s-algorithm"></span><h1>Shor&#8217;s Algorithm<a class="headerlink" href="#module-sympy.physics.quantum.shor" title="Permalink to this headline">¶</a></h1>
-<p>Shor&#8217;s algorithm and helper functions.</p>
-<p>Todo:</p>
-<ul class="simple">
-<li>Get the CMod gate working again using the new Gate API.</li>
-<li>Fix everything.</li>
-<li>Update docstrings and reformat.</li>
-<li>Remove print statements. We may want to think about a better API for this.</li>
-</ul>
-<dl class="class">
-<dt id="sympy.physics.quantum.shor.CMod">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.shor.</tt><tt class="descname">CMod</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/shor.html#CMod"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.shor.CMod" title="Permalink to this definition">¶</a></dt>
-<dd><p>A controlled mod gate.</p>
-<p>This is black box controlled Mod function for use by shor&#8217;s algorithm.
-TODO implement a decompose property that returns how to do this in terms
-of elementary gates</p>
-<dl class="attribute">
-<dt id="sympy.physics.quantum.shor.CMod.N">
-<tt class="descname">N</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/shor.html#CMod.N"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.shor.CMod.N" title="Permalink to this definition">¶</a></dt>
-<dd><p>N is the type of modular arithmetic we are doing.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.shor.CMod.a">
-<tt class="descname">a</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/shor.html#CMod.a"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.shor.CMod.a" title="Permalink to this definition">¶</a></dt>
-<dd><p>Base of the controlled mod function.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.shor.CMod.t">
-<tt class="descname">t</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/shor.html#CMod.t"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.shor.CMod.t" title="Permalink to this definition">¶</a></dt>
-<dd><p>Size of 1/2 input register.  First 1/2 holds output.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.shor.continued_fraction">
-<tt class="descclassname">sympy.physics.quantum.shor.</tt><tt class="descname">continued_fraction</tt><big>(</big><em>x</em>, <em>y</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/shor.html#continued_fraction"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.shor.continued_fraction" title="Permalink to this definition">¶</a></dt>
-<dd><p>This applies the continued fraction expansion to two numbers x/y</p>
-<p>x is the numerator and y is the denominator</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.shor</span> <span class="kn">import</span> <span class="n">continued_fraction</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">continued_fraction</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span>
-<span class="go">[0, 2, 1, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.shor.period_find">
-<tt class="descclassname">sympy.physics.quantum.shor.</tt><tt class="descname">period_find</tt><big>(</big><em>a</em>, <em>N</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/shor.html#period_find"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.shor.period_find" title="Permalink to this definition">¶</a></dt>
-<dd><p>Finds the period of a in modulo N arithmetic</p>
-<p>This is quantum part of Shor&#8217;s algorithm.It takes two registers,
-puts first in superposition of states with Hadamards so: <tt class="docutils literal"><span class="pre">|k&gt;|0&gt;</span></tt>
-with k being all possible choices. It then does a controlled mod and
-a QFT to determine the order of a.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.shor.shor">
-<tt class="descclassname">sympy.physics.quantum.shor.</tt><tt class="descname">shor</tt><big>(</big><em>N</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/shor.html#shor"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.shor.shor" title="Permalink to this definition">¶</a></dt>
-<dd><p>This function implements Shor&#8217;s factoring algorithm on the Integer N</p>
-<p>The algorithm starts by picking a random number (a) and seeing if it is
-coprime with N. If it isn&#8217;t, then the gcd of the two numbers is a factor
-and we are done. Otherwise, it begins the period_finding subroutine which
-finds the period of a in modulo N arithmetic. This period, if even, can
-be used to calculate factors by taking a**(r/2)-1 and a**(r/2)+1.
-These values are returned.</p>
-</dd></dl>
-
-</div>
-
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-            
-  <div class="section" id="module-sympy.physics.quantum.spin">
-<span id="spin"></span><h1>Spin<a class="headerlink" href="#module-sympy.physics.quantum.spin" title="Permalink to this headline">¶</a></h1>
-<p>Quantum mechanical angular momemtum.</p>
-<dl class="class">
-<dt id="sympy.physics.quantum.spin.Rotation">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.spin.</tt><tt class="descname">Rotation</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/spin.html#Rotation"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.spin.Rotation" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wigner D operator in terms of Euler angles.</p>
-<p>Defines the rotation operator in terms of the Euler angles defined by
-the z-y-z convention for a passive transformation. That is the coordinate
-axes are rotated first about the z-axis, giving the new x&#8217;-y&#8217;-z&#8217; axes. Then
-this new coordinate system is rotated about the new y&#8217;-axis, giving new
-x&#8217;&#8216;-y&#8217;&#8216;-z&#8217;&#8217; axes. Then this new coordinate system is rotated about the
-z&#8217;&#8216;-axis. Conventions follow those laid out in <a class="reference internal" href="#r87">[R87]</a>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>alpha</strong> : Number, Symbol</p>
-<blockquote>
-<div><p>First Euler Angle</p>
-</div></blockquote>
-<p><strong>beta</strong> : Number, Symbol</p>
-<blockquote>
-<div><p>Second Euler angle</p>
-</div></blockquote>
-<p><strong>gamma</strong> : Number, Symbol</p>
-<blockquote class="last">
-<div><p>Third Euler angle</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.quantum.spin.WignerD" title="sympy.physics.quantum.spin.WignerD"><tt class="xref py py-obj docutils literal"><span class="pre">WignerD</span></tt></a></dt>
-<dd>Symbolic Wigner-D function</dd>
-<dt><a class="reference internal" href="#sympy.physics.quantum.spin.Rotation.D" title="sympy.physics.quantum.spin.Rotation.D"><tt class="xref py py-obj docutils literal"><span class="pre">D</span></tt></a></dt>
-<dd>Wigner-D function</dd>
-<dt><a class="reference internal" href="#sympy.physics.quantum.spin.Rotation.d" title="sympy.physics.quantum.spin.Rotation.d"><tt class="xref py py-obj docutils literal"><span class="pre">d</span></tt></a></dt>
-<dd>Wigner small-d function</dd>
-</dl>
-</div>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r87" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[R87]</td><td><em>(<a class="fn-backref" href="#id1">1</a>, <a class="fn-backref" href="#id2">2</a>)</em> Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.</td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>A simple example rotation operator:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.spin</span> <span class="kn">import</span> <span class="n">Rotation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rotation</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">R(pi,0,pi/2)</span>
-</pre></div>
-</div>
-<p>With symbolic Euler angles and calculating the inverse rotation operator:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a b c&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rotation</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
-<span class="go">R(a,b,c)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rotation</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span><span class="o">.</span><span class="n">inverse</span><span class="p">()</span>
-<span class="go">R(-c,-b,-a)</span>
-</pre></div>
-</div>
-<dl class="classmethod">
-<dt id="sympy.physics.quantum.spin.Rotation.D">
-<em class="property">classmethod </em><tt class="descname">D</tt><big>(</big><em>j</em>, <em>m</em>, <em>mp</em>, <em>alpha</em>, <em>beta</em>, <em>gamma</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/spin.html#Rotation.D"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.spin.Rotation.D" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wigner D-function.</p>
-<p>Returns an instance of the WignerD class corresponding to the Wigner-D
-function specified by the parameters.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>j</strong> : Number</p>
-<blockquote>
-<div><p>Total angular momentum</p>
-</div></blockquote>
-<p><strong>m</strong> : Number</p>
-<blockquote>
-<div><p>Eigenvalue of angular momentum along axis after rotation</p>
-</div></blockquote>
-<p><strong>mp</strong> : Number</p>
-<blockquote>
-<div><p>Eigenvalue of angular momentum along rotated axis</p>
-</div></blockquote>
-<p><strong>alpha</strong> : Number, Symbol</p>
-<blockquote>
-<div><p>First Euler angle of rotation</p>
-</div></blockquote>
-<p><strong>beta</strong> : Number, Symbol</p>
-<blockquote>
-<div><p>Second Euler angle of rotation</p>
-</div></blockquote>
-<p><strong>gamma</strong> : Number, Symbol</p>
-<blockquote class="last">
-<div><p>Third Euler angle of rotation</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.quantum.spin.WignerD" title="sympy.physics.quantum.spin.WignerD"><tt class="xref py py-obj docutils literal"><span class="pre">WignerD</span></tt></a></dt>
-<dd>Symbolic Wigner-D function</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<p>Return the Wigner-D matrix element for a defined rotation, both
-numerical and symbolic:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.spin</span> <span class="kn">import</span> <span class="n">Rotation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">gamma</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;alpha beta gamma&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rotation</span><span class="o">.</span><span class="n">D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span><span class="n">pi</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="n">pi</span><span class="p">)</span>
-<span class="go">WignerD(1, 1, 0, pi, pi/2, -pi)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.physics.quantum.spin.Rotation.d">
-<em class="property">classmethod </em><tt class="descname">d</tt><big>(</big><em>j</em>, <em>m</em>, <em>mp</em>, <em>beta</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/spin.html#Rotation.d"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.spin.Rotation.d" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wigner small-d function.</p>
-<p>Returns an instance of the WignerD class corresponding to the Wigner-D
-function specified by the parameters with the alpha and gamma angles
-given as 0.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>j</strong> : Number</p>
-<blockquote>
-<div><p>Total angular momentum</p>
-</div></blockquote>
-<p><strong>m</strong> : Number</p>
-<blockquote>
-<div><p>Eigenvalue of angular momentum along axis after rotation</p>
-</div></blockquote>
-<p><strong>mp</strong> : Number</p>
-<blockquote>
-<div><p>Eigenvalue of angular momentum along rotated axis</p>
-</div></blockquote>
-<p><strong>beta</strong> : Number, Symbol</p>
-<blockquote class="last">
-<div><p>Second Euler angle of rotation</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.quantum.spin.WignerD" title="sympy.physics.quantum.spin.WignerD"><tt class="xref py py-obj docutils literal"><span class="pre">WignerD</span></tt></a></dt>
-<dd>Symbolic Wigner-D function</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<p>Return the Wigner-D matrix element for a defined rotation, both
-numerical and symbolic:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.spin</span> <span class="kn">import</span> <span class="n">Rotation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">beta</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;beta&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rotation</span><span class="o">.</span><span class="n">d</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">WignerD(1, 1, 0, 0, pi/2, 0)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.spin.WignerD">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.spin.</tt><tt class="descname">WignerD</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/spin.html#WignerD"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.spin.WignerD" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wigner-D function</p>
-<p>The Wigner D-function gives the matrix elements of the rotation
-operator in the jm-representation. For the Euler angles <span class="math">\(\alpha\)</span>,
-<span class="math">\(\beta\)</span>, <span class="math">\(\gamma\)</span>, the D-function is defined such that:</p>
-<div class="math">
-\[\begin{split}&lt;j,m| \mathcal{R}(\alpha, \beta, \gamma ) |j',m'&gt; = \delta_{jj'} D(j, m, m', \alpha, \beta, \gamma)\end{split}\]</div>
-<p>Where the rotation operator is as defined by the Rotation class <a class="reference internal" href="#r88">[R88]</a>.</p>
-<p>The Wigner D-function defined in this way gives:</p>
-<div class="math">
-\[D(j, m, m', \alpha, \beta, \gamma) = e^{-i m \alpha} d(j, m, m', \beta) e^{-i m' \gamma}\]</div>
-<p>Where d is the Wigner small-d function, which is given by Rotation.d.</p>
-<p>The Wigner small-d function gives the component of the Wigner
-D-function that is determined by the second Euler angle. That is the
-Wigner D-function is:</p>
-<div class="math">
-\[D(j, m, m', \alpha, \beta, \gamma) = e^{-i m \alpha} d(j, m, m', \beta) e^{-i m' \gamma}\]</div>
-<p>Where d is the small-d function. The Wigner D-function is given by
-Rotation.D.</p>
-<p>Note that to evaluate the D-function, the j, m and mp parameters must
-be integer or half integer numbers.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>j</strong> : Number</p>
-<blockquote>
-<div><p>Total angular momentum</p>
-</div></blockquote>
-<p><strong>m</strong> : Number</p>
-<blockquote>
-<div><p>Eigenvalue of angular momentum along axis after rotation</p>
-</div></blockquote>
-<p><strong>mp</strong> : Number</p>
-<blockquote>
-<div><p>Eigenvalue of angular momentum along rotated axis</p>
-</div></blockquote>
-<p><strong>alpha</strong> : Number, Symbol</p>
-<blockquote>
-<div><p>First Euler angle of rotation</p>
-</div></blockquote>
-<p><strong>beta</strong> : Number, Symbol</p>
-<blockquote>
-<div><p>Second Euler angle of rotation</p>
-</div></blockquote>
-<p><strong>gamma</strong> : Number, Symbol</p>
-<blockquote class="last">
-<div><p>Third Euler angle of rotation</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.quantum.spin.Rotation" title="sympy.physics.quantum.spin.Rotation"><tt class="xref py py-obj docutils literal"><span class="pre">Rotation</span></tt></a></dt>
-<dd>Rotation operator</dd>
-</dl>
-</div>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r88" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[R88]</td><td><em>(<a class="fn-backref" href="#id3">1</a>, <a class="fn-backref" href="#id4">2</a>)</em> Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.</td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Evaluate the Wigner-D matrix elements of a simple rotation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.spin</span> <span class="kn">import</span> <span class="n">Rotation</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rot</span> <span class="o">=</span> <span class="n">Rotation</span><span class="o">.</span><span class="n">D</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rot</span>
-<span class="go">WignerD(1, 1, 0, pi, pi/2, 0)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rot</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-<span class="go">sqrt(2)/2</span>
-</pre></div>
-</div>
-<p>Evaluate the Wigner-d matrix elements of a simple rotation</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rot</span> <span class="o">=</span> <span class="n">Rotation</span><span class="o">.</span><span class="n">d</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rot</span>
-<span class="go">WignerD(1, 1, 0, 0, pi/2, 0)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rot</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-<span class="go">-sqrt(2)/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.spin.JxKet">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.spin.</tt><tt class="descname">JxKet</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/spin.html#JxKet"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.spin.JxKet" title="Permalink to this definition">¶</a></dt>
-<dd><p>Eigenket of Jx.</p>
-<p>See JzKet for the usage of spin eigenstates.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.quantum.spin.JzKet" title="sympy.physics.quantum.spin.JzKet"><tt class="xref py py-obj docutils literal"><span class="pre">JzKet</span></tt></a></dt>
-<dd>Usage of spin states</dd>
-</dl>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.spin.JxBra">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.spin.</tt><tt class="descname">JxBra</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/spin.html#JxBra"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.spin.JxBra" title="Permalink to this definition">¶</a></dt>
-<dd><p>Eigenbra of Jx.</p>
-<p>See JzKet for the usage of spin eigenstates.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.quantum.spin.JzKet" title="sympy.physics.quantum.spin.JzKet"><tt class="xref py py-obj docutils literal"><span class="pre">JzKet</span></tt></a></dt>
-<dd>Usage of spin states</dd>
-</dl>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.spin.JyKet">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.spin.</tt><tt class="descname">JyKet</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/spin.html#JyKet"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.spin.JyKet" title="Permalink to this definition">¶</a></dt>
-<dd><p>Eigenket of Jy.</p>
-<p>See JzKet for the usage of spin eigenstates.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.quantum.spin.JzKet" title="sympy.physics.quantum.spin.JzKet"><tt class="xref py py-obj docutils literal"><span class="pre">JzKet</span></tt></a></dt>
-<dd>Usage of spin states</dd>
-</dl>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.spin.JyBra">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.spin.</tt><tt class="descname">JyBra</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/spin.html#JyBra"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.spin.JyBra" title="Permalink to this definition">¶</a></dt>
-<dd><p>Eigenbra of Jy.</p>
-<p>See JzKet for the usage of spin eigenstates.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.quantum.spin.JzKet" title="sympy.physics.quantum.spin.JzKet"><tt class="xref py py-obj docutils literal"><span class="pre">JzKet</span></tt></a></dt>
-<dd>Usage of spin states</dd>
-</dl>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.spin.JzKet">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.spin.</tt><tt class="descname">JzKet</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/spin.html#JzKet"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.spin.JzKet" title="Permalink to this definition">¶</a></dt>
-<dd><p>Eigenket of Jz.</p>
-<p>Spin state which is an eigenstate of the Jz operator. Uncoupled states,
-that is states representing the interaction of multiple separate spin
-states, are defined as a tensor product of states.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>j</strong> : Number, Symbol</p>
-<blockquote>
-<div><p>Total spin angular momentum</p>
-</div></blockquote>
-<p><strong>m</strong> : Number, Symbol</p>
-<blockquote class="last">
-<div><p>Eigenvalue of the Jz spin operator</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.quantum.spin.JzKetCoupled" title="sympy.physics.quantum.spin.JzKetCoupled"><tt class="xref py py-obj docutils literal"><span class="pre">JzKetCoupled</span></tt></a></dt>
-<dd>Coupled eigenstates</dd>
-<dt><tt class="xref py py-obj docutils literal"><span class="pre">TensorProduct</span></tt></dt>
-<dd>Used to specify uncoupled states</dd>
-<dt><a class="reference internal" href="#sympy.physics.quantum.spin.uncouple" title="sympy.physics.quantum.spin.uncouple"><tt class="xref py py-obj docutils literal"><span class="pre">uncouple</span></tt></a></dt>
-<dd>Uncouples states given coupling parameters</dd>
-<dt><a class="reference internal" href="#sympy.physics.quantum.spin.couple" title="sympy.physics.quantum.spin.couple"><tt class="xref py py-obj docutils literal"><span class="pre">couple</span></tt></a></dt>
-<dd>Couples uncoupled states</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<p><em>Normal States:</em></p>
-<p>Defining simple spin states, both numerical and symbolic:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.spin</span> <span class="kn">import</span> <span class="n">JzKet</span><span class="p">,</span> <span class="n">JxKet</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">JzKet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">|1,0&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">j</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;j m&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">JzKet</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="go">|j,m&gt;</span>
-</pre></div>
-</div>
-<p>Rewriting the JzKet in terms of eigenkets of the Jx operator:
-Note: that the resulting eigenstates are JxKet&#8217;s</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">JzKet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="s">&quot;Jx&quot;</span><span class="p">)</span>
-<span class="go">|1,-1&gt;/2 - sqrt(2)*|1,0&gt;/2 + |1,1&gt;/2</span>
-</pre></div>
-</div>
-<p>Get the vector representation of a state in terms of the basis elements
-of the Jx operator:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.represent</span> <span class="kn">import</span> <span class="n">represent</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.spin</span> <span class="kn">import</span> <span class="n">Jx</span><span class="p">,</span> <span class="n">Jz</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">represent</span><span class="p">(</span><span class="n">JzKet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="n">basis</span><span class="o">=</span><span class="n">Jx</span><span class="p">)</span>
-<span class="go">[      1/2]</span>
-<span class="go">[sqrt(2)/2]</span>
-<span class="go">[      1/2]</span>
-</pre></div>
-</div>
-<p>Apply innerproducts between states:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.innerproduct</span> <span class="kn">import</span> <span class="n">InnerProduct</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.spin</span> <span class="kn">import</span> <span class="n">JxBra</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">InnerProduct</span><span class="p">(</span><span class="n">JxBra</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span> <span class="n">JzKet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span>
-<span class="go">&lt;1,1|1,1&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-<span class="go">1/2</span>
-</pre></div>
-</div>
-<p><em>Uncoupled States:</em></p>
-<p>Define an uncoupled state as a TensorProduct between two Jz eigenkets:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.tensorproduct</span> <span class="kn">import</span> <span class="n">TensorProduct</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">j1</span><span class="p">,</span><span class="n">m1</span><span class="p">,</span><span class="n">j2</span><span class="p">,</span><span class="n">m2</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;j1 m1 j2 m2&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">TensorProduct</span><span class="p">(</span><span class="n">JzKet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> <span class="n">JzKet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span>
-<span class="go">|1,0&gt;x|1,1&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">TensorProduct</span><span class="p">(</span><span class="n">JzKet</span><span class="p">(</span><span class="n">j1</span><span class="p">,</span><span class="n">m1</span><span class="p">),</span> <span class="n">JzKet</span><span class="p">(</span><span class="n">j2</span><span class="p">,</span><span class="n">m2</span><span class="p">))</span>
-<span class="go">|j1,m1&gt;x|j2,m2&gt;</span>
-</pre></div>
-</div>
-<p>A TensorProduct can be rewritten, in which case the eigenstates that make
-up the tensor product is rewritten to the new basis:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">TensorProduct</span><span class="p">(</span><span class="n">JzKet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span><span class="n">JxKet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="s">&#39;Jz&#39;</span><span class="p">)</span>
-<span class="go">|1,1&gt;x|1,-1&gt;/2 + sqrt(2)*|1,1&gt;x|1,0&gt;/2 + |1,1&gt;x|1,1&gt;/2</span>
-</pre></div>
-</div>
-<p>The represent method for TensorProduct&#8217;s gives the vector representation of
-the state. Note that the state in the product basis is the equivalent of the
-tensor product of the vector representation of the component eigenstates:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">represent</span><span class="p">(</span><span class="n">TensorProduct</span><span class="p">(</span><span class="n">JzKet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span><span class="n">JzKet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)))</span>
-<span class="go">[0]</span>
-<span class="go">[0]</span>
-<span class="go">[0]</span>
-<span class="go">[1]</span>
-<span class="go">[0]</span>
-<span class="go">[0]</span>
-<span class="go">[0]</span>
-<span class="go">[0]</span>
-<span class="go">[0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">represent</span><span class="p">(</span><span class="n">TensorProduct</span><span class="p">(</span><span class="n">JzKet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span><span class="n">JxKet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)),</span> <span class="n">basis</span><span class="o">=</span><span class="n">Jz</span><span class="p">)</span>
-<span class="go">[      1/2]</span>
-<span class="go">[sqrt(2)/2]</span>
-<span class="go">[      1/2]</span>
-<span class="go">[        0]</span>
-<span class="go">[        0]</span>
-<span class="go">[        0]</span>
-<span class="go">[        0]</span>
-<span class="go">[        0]</span>
-<span class="go">[        0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.spin.JzBra">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.spin.</tt><tt class="descname">JzBra</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/spin.html#JzBra"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.spin.JzBra" title="Permalink to this definition">¶</a></dt>
-<dd><p>Eigenbra of Jz.</p>
-<p>See the JzKet for the usage of spin eigenstates.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.quantum.spin.JzKet" title="sympy.physics.quantum.spin.JzKet"><tt class="xref py py-obj docutils literal"><span class="pre">JzKet</span></tt></a></dt>
-<dd>Usage of spin states</dd>
-</dl>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.spin.JxKetCoupled">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.spin.</tt><tt class="descname">JxKetCoupled</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/spin.html#JxKetCoupled"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.spin.JxKetCoupled" title="Permalink to this definition">¶</a></dt>
-<dd><p>Coupled eigenket of Jx.</p>
-<p>See JzKetCoupled for the usage of coupled spin eigenstates.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.quantum.spin.JzKetCoupled" title="sympy.physics.quantum.spin.JzKetCoupled"><tt class="xref py py-obj docutils literal"><span class="pre">JzKetCoupled</span></tt></a></dt>
-<dd>Usage of coupled spin states</dd>
-</dl>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.spin.JxBraCoupled">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.spin.</tt><tt class="descname">JxBraCoupled</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/spin.html#JxBraCoupled"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.spin.JxBraCoupled" title="Permalink to this definition">¶</a></dt>
-<dd><p>Coupled eigenbra of Jx.</p>
-<p>See JzKetCoupled for the usage of coupled spin eigenstates.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.quantum.spin.JzKetCoupled" title="sympy.physics.quantum.spin.JzKetCoupled"><tt class="xref py py-obj docutils literal"><span class="pre">JzKetCoupled</span></tt></a></dt>
-<dd>Usage of coupled spin states</dd>
-</dl>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.spin.JyKetCoupled">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.spin.</tt><tt class="descname">JyKetCoupled</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/spin.html#JyKetCoupled"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.spin.JyKetCoupled" title="Permalink to this definition">¶</a></dt>
-<dd><p>Coupled eigenket of Jy.</p>
-<p>See JzKetCoupled for the usage of coupled spin eigenstates.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.quantum.spin.JzKetCoupled" title="sympy.physics.quantum.spin.JzKetCoupled"><tt class="xref py py-obj docutils literal"><span class="pre">JzKetCoupled</span></tt></a></dt>
-<dd>Usage of coupled spin states</dd>
-</dl>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.spin.JyBraCoupled">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.spin.</tt><tt class="descname">JyBraCoupled</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/spin.html#JyBraCoupled"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.spin.JyBraCoupled" title="Permalink to this definition">¶</a></dt>
-<dd><p>Coupled eigenbra of Jy.</p>
-<p>See JzKetCoupled for the usage of coupled spin eigenstates.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.quantum.spin.JzKetCoupled" title="sympy.physics.quantum.spin.JzKetCoupled"><tt class="xref py py-obj docutils literal"><span class="pre">JzKetCoupled</span></tt></a></dt>
-<dd>Usage of coupled spin states</dd>
-</dl>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.spin.JzKetCoupled">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.spin.</tt><tt class="descname">JzKetCoupled</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/spin.html#JzKetCoupled"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.spin.JzKetCoupled" title="Permalink to this definition">¶</a></dt>
-<dd><p>Coupled eigenket of Jz</p>
-<p>Spin state that is an eigenket of Jz which represents the coupling of
-separate spin spaces.</p>
-<p>The arguments for creating instances of JzKetCoupled are <tt class="docutils literal"><span class="pre">j</span></tt>, <tt class="docutils literal"><span class="pre">m</span></tt>,
-<tt class="docutils literal"><span class="pre">jn</span></tt> and an optional <tt class="docutils literal"><span class="pre">jcoupling</span></tt> argument. The <tt class="docutils literal"><span class="pre">j</span></tt> and <tt class="docutils literal"><span class="pre">m</span></tt> options
-are the total angular momentum quantum numbers, as used for normal states
-(e.g. JzKet).</p>
-<p>The other required parameter in <tt class="docutils literal"><span class="pre">jn</span></tt>, which is a tuple defining the <span class="math">\(j_n\)</span>
-angular momentum quantum numbers of the product spaces. So for example, if
-a state represented the coupling of the product basis state
-<span class="math">\(|j_1,m_1\rangle\times|j_2,m_2\rangle\)</span>, the <tt class="docutils literal"><span class="pre">jn</span></tt> for this state would be
-<tt class="docutils literal"><span class="pre">(j1,j2)</span></tt>.</p>
-<p>The final option is <tt class="docutils literal"><span class="pre">jcoupling</span></tt>, which is used to define how the spaces
-specified by <tt class="docutils literal"><span class="pre">jn</span></tt> are coupled, which includes both the order these spaces
-are coupled together and the quantum numbers that arise from these
-couplings. The <tt class="docutils literal"><span class="pre">jcoupling</span></tt> parameter itself is a list of lists, such that
-each of the sublists defines a single coupling between the spin spaces. If
-there are N coupled angular momentum spaces, that is <tt class="docutils literal"><span class="pre">jn</span></tt> has N elements,
-then there must be N-1 sublists. Each of these sublists making up the
-<tt class="docutils literal"><span class="pre">jcoupling</span></tt> parameter have length 3. The first two elements are the
-indicies of the product spaces that are considered to be coupled together.
-For example, if we want to couple <span class="math">\(j_1\)</span> and <span class="math">\(j_4\)</span>, the indicies would be 1
-and 4. If a state has already been coupled, it is referenced by the
-smallest index that is coupled, so if <span class="math">\(j_2\)</span> and <span class="math">\(j_4\)</span> has already been
-coupled to some <span class="math">\(j_{24}\)</span>, then this value can be coupled by referencing it
-with index 2. The final element of the sublist is the quantum number of the
-coupled state. So putting everything together, into a valid sublist for
-<tt class="docutils literal"><span class="pre">jcoupling</span></tt>, if <span class="math">\(j_1\)</span> and <span class="math">\(j_2\)</span> are coupled to an angular momentum space
-with quantum number <span class="math">\(j_{12}\)</span> with the value <tt class="docutils literal"><span class="pre">j12</span></tt>, the sublist would be
-<tt class="docutils literal"><span class="pre">(1,2,j12)</span></tt>, N-1 of these sublists are used in the list for
-<tt class="docutils literal"><span class="pre">jcoupling</span></tt>.</p>
-<p>Note the <tt class="docutils literal"><span class="pre">jcoupling</span></tt> parameter is optional, if it is not specified, the
-default coupling is taken. This default value is to coupled the spaces in
-order and take the quantum number of the coupling to be the maximum value.
-For example, if the spin spaces are <span class="math">\(j_1\)</span>, <span class="math">\(j_2\)</span>, <span class="math">\(j_3\)</span>, <span class="math">\(j_4\)</span>, then the
-default coupling couples <span class="math">\(j_1\)</span> and <span class="math">\(j_2\)</span> to <span class="math">\(j_{12}=j_1+j_2\)</span>, then,
-<span class="math">\(j_{12}\)</span> and <span class="math">\(j_3\)</span> are coupled to <span class="math">\(j_{123}=j_{12}+j_3\)</span>, and finally
-<span class="math">\(j_{123}\)</span> and <span class="math">\(j_4\)</span> to <span class="math">\(j=j_{123}+j_4\)</span>. The jcoupling value that would
-correspond to this is:</p>
-<blockquote>
-<div><tt class="docutils literal"><span class="pre">((1,2,j1+j2),(1,3,j1+j2+j3))</span></tt></div></blockquote>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>args</strong> : tuple</p>
-<blockquote class="last">
-<div><p>The arguments that must be passed are <tt class="docutils literal"><span class="pre">j</span></tt>, <tt class="docutils literal"><span class="pre">m</span></tt>, <tt class="docutils literal"><span class="pre">jn</span></tt>, and
-<tt class="docutils literal"><span class="pre">jcoupling</span></tt>. The <tt class="docutils literal"><span class="pre">j</span></tt> value is the total angular momentum. The <tt class="docutils literal"><span class="pre">m</span></tt>
-value is the eigenvalue of the Jz spin operator. The <tt class="docutils literal"><span class="pre">jn</span></tt> list are
-the j values of argular momentum spaces coupled together. The
-<tt class="docutils literal"><span class="pre">jcoupling</span></tt> parameter is an optional parameter defining how the spaces
-are coupled together. See the above description for how these coupling
-parameters are defined.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.quantum.spin.JzKet" title="sympy.physics.quantum.spin.JzKet"><tt class="xref py py-obj docutils literal"><span class="pre">JzKet</span></tt></a></dt>
-<dd>Normal spin eigenstates</dd>
-<dt><a class="reference internal" href="#sympy.physics.quantum.spin.uncouple" title="sympy.physics.quantum.spin.uncouple"><tt class="xref py py-obj docutils literal"><span class="pre">uncouple</span></tt></a></dt>
-<dd>Uncoupling of coupling spin states</dd>
-<dt><a class="reference internal" href="#sympy.physics.quantum.spin.couple" title="sympy.physics.quantum.spin.couple"><tt class="xref py py-obj docutils literal"><span class="pre">couple</span></tt></a></dt>
-<dd>Coupling of uncoupled spin states</dd>
-</dl>
-</div>
-<p class="rubric">Examples</p>
-<p>Defining simple spin states, both numerical and symbolic:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.spin</span> <span class="kn">import</span> <span class="n">JzKetCoupled</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">JzKetCoupled</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">|1,0,j1=1,j2=1&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">j1</span><span class="p">,</span> <span class="n">j2</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;j m j1 j2&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">JzKetCoupled</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="p">(</span><span class="n">j1</span><span class="p">,</span> <span class="n">j2</span><span class="p">))</span>
-<span class="go">|j,m,j1=j1,j2=j2&gt;</span>
-</pre></div>
-</div>
-<p>Defining coupled spin states for more than 2 coupled spaces with various
-coupling parameters:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">JzKetCoupled</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">|2,1,j1=1,j2=1,j3=1,j(1,2)=2&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">JzKetCoupled</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">((</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span> <span class="p">)</span>
-<span class="go">|2,1,j1=1,j2=1,j3=1,j(1,2)=2&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">JzKetCoupled</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">((</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span> <span class="p">)</span>
-<span class="go">|2,1,j1=1,j2=1,j3=1,j(2,3)=1&gt;</span>
-</pre></div>
-</div>
-<p>Rewriting the JzKetCoupled in terms of eigenkets of the Jx operator:
-Note: that the resulting eigenstates are JxKetCoupled</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">JzKetCoupled</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="s">&quot;Jx&quot;</span><span class="p">)</span>
-<span class="go">|1,-1,j1=1,j2=1&gt;/2 - sqrt(2)*|1,0,j1=1,j2=1&gt;/2 + |1,1,j1=1,j2=1&gt;/2</span>
-</pre></div>
-</div>
-<p>The rewrite method can be used to convert a coupled state to an uncoupled
-state. This is done by passing coupled=False to the rewrite function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">JzKetCoupled</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">rewrite</span><span class="p">(</span><span class="s">&#39;Jz&#39;</span><span class="p">,</span> <span class="n">coupled</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">-sqrt(2)*|1,-1&gt;x|1,1&gt;/2 + sqrt(2)*|1,1&gt;x|1,-1&gt;/2</span>
-</pre></div>
-</div>
-<p>Get the vector representation of a state in terms of the basis elements
-of the Jx operator:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.represent</span> <span class="kn">import</span> <span class="n">represent</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.spin</span> <span class="kn">import</span> <span class="n">Jx</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">represent</span><span class="p">(</span><span class="n">JzKetCoupled</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)),</span> <span class="n">basis</span><span class="o">=</span><span class="n">Jx</span><span class="p">)</span>
-<span class="go">[        0]</span>
-<span class="go">[      1/2]</span>
-<span class="go">[sqrt(2)/2]</span>
-<span class="go">[      1/2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.spin.JzBraCoupled">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.spin.</tt><tt class="descname">JzBraCoupled</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/spin.html#JzBraCoupled"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.spin.JzBraCoupled" title="Permalink to this definition">¶</a></dt>
-<dd><p>Coupled eigenbra of Jz.</p>
-<p>See the JzKetCoupled for the usage of coupled spin eigenstates.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="last docutils">
-<dt><a class="reference internal" href="#sympy.physics.quantum.spin.JzKetCoupled" title="sympy.physics.quantum.spin.JzKetCoupled"><tt class="xref py py-obj docutils literal"><span class="pre">JzKetCoupled</span></tt></a></dt>
-<dd>Usage of coupled spin states</dd>
-</dl>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.spin.couple">
-<tt class="descclassname">sympy.physics.quantum.spin.</tt><tt class="descname">couple</tt><big>(</big><em>expr</em>, <em>jcoupling_list=None</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/spin.html#couple"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.spin.couple" title="Permalink to this definition">¶</a></dt>
-<dd><p>Couple a tensor product of spin states</p>
-<p>This function can be used to couple an uncoupled tensor product of spin
-states. All of the eigenstates to be coupled must be of the same class. It
-will return a linear combination of eigenstates that are subclasses of
-CoupledSpinState determined by Clebsch-Gordan angular momentum coupling
-coefficients.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>expr</strong> : Expr</p>
-<blockquote>
-<div><p>An expression involving TensorProducts of spin states to be coupled.
-Each state must be a subclass of SpinState and they all must be the
-same class.</p>
-</div></blockquote>
-<p><strong>jcoupling_list</strong> : list or tuple</p>
-<blockquote class="last">
-<div><p>Elements of this list are sub-lists of length 2 specifying the order of
-the coupling of the spin spaces. The length of this must be N-1, where N
-is the number of states in the tensor product to be coupled. The
-elements of this sublist are the same as the first two elements of each
-sublist in the <tt class="docutils literal"><span class="pre">jcoupling</span></tt> parameter defined for JzKetCoupled. If this
-parameter is not specified, the default value is taken, which couples
-the first and second product basis spaces, then couples this new coupled
-space to the third product space, etc</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Couple a tensor product of numerical states for two spaces:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.spin</span> <span class="kn">import</span> <span class="n">JzKet</span><span class="p">,</span> <span class="n">couple</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.tensorproduct</span> <span class="kn">import</span> <span class="n">TensorProduct</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">couple</span><span class="p">(</span><span class="n">TensorProduct</span><span class="p">(</span><span class="n">JzKet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> <span class="n">JzKet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)))</span>
-<span class="go">-sqrt(2)*|1,1,j1=1,j2=1&gt;/2 + sqrt(2)*|2,1,j1=1,j2=1&gt;/2</span>
-</pre></div>
-</div>
-<p>Numerical coupling of three spaces using the default coupling method, i.e.
-first and second spaces couple, then this couples to the third space:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">couple</span><span class="p">(</span><span class="n">TensorProduct</span><span class="p">(</span><span class="n">JzKet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span> <span class="n">JzKet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span> <span class="n">JzKet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">)))</span>
-<span class="go">sqrt(6)*|2,2,j1=1,j2=1,j3=1,j(1,2)=2&gt;/3 + sqrt(3)*|3,2,j1=1,j2=1,j3=1,j(1,2)=2&gt;/3</span>
-</pre></div>
-</div>
-<p>Perform this same coupling, but we define the coupling to first couple
-the first and third spaces:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">couple</span><span class="p">(</span><span class="n">TensorProduct</span><span class="p">(</span><span class="n">JzKet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span> <span class="n">JzKet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span> <span class="n">JzKet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">)),</span> <span class="p">((</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span> <span class="p">)</span>
-<span class="go">sqrt(2)*|2,2,j1=1,j2=1,j3=1,j(1,3)=1&gt;/2 - sqrt(6)*|2,2,j1=1,j2=1,j3=1,j(1,3)=2&gt;/6 + sqrt(3)*|3,2,j1=1,j2=1,j3=1,j(1,3)=2&gt;/3</span>
-</pre></div>
-</div>
-<p>Couple a tensor product of symbolic states:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">j1</span><span class="p">,</span><span class="n">m1</span><span class="p">,</span><span class="n">j2</span><span class="p">,</span><span class="n">m2</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;j1 m1 j2 m2&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">couple</span><span class="p">(</span><span class="n">TensorProduct</span><span class="p">(</span><span class="n">JzKet</span><span class="p">(</span><span class="n">j1</span><span class="p">,</span><span class="n">m1</span><span class="p">),</span> <span class="n">JzKet</span><span class="p">(</span><span class="n">j2</span><span class="p">,</span><span class="n">m2</span><span class="p">)))</span>
-<span class="go">Sum(CG(j1, m1, j2, m2, j, m1 + m2)*|j,m1 + m2,j1=j1,j2=j2&gt;, (j, m1 + m2, j1 + j2))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.spin.uncouple">
-<tt class="descclassname">sympy.physics.quantum.spin.</tt><tt class="descname">uncouple</tt><big>(</big><em>expr</em>, <em>jn=None</em>, <em>jcoupling_list=None</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/spin.html#uncouple"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.spin.uncouple" title="Permalink to this definition">¶</a></dt>
-<dd><p>Uncouple a coupled spin state</p>
-<p>Gives the uncoupled representation of a coupled spin state. Arguments must
-be either a spin state that is a subclass of CoupledSpinState or a spin
-state that is a subclass of SpinState and an array giving the j values
-of the spaces that are to be coupled</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>expr</strong> : Expr</p>
-<blockquote>
-<div><p>The expression containing states that are to be coupled. If the states
-are a subclass of SpinState, the <tt class="docutils literal"><span class="pre">jn</span></tt> and <tt class="docutils literal"><span class="pre">jcoupling</span></tt> parameters
-must be defined. If the states are a subclass of CoupledSpinState,
-<tt class="docutils literal"><span class="pre">jn</span></tt> and <tt class="docutils literal"><span class="pre">jcoupling</span></tt> will be taken from the state.</p>
-</div></blockquote>
-<p><strong>jn</strong> : list or tuple</p>
-<blockquote>
-<div><p>The list of the j-values that are coupled. If state is a
-CoupledSpinState, this parameter is ignored. This must be defined if
-state is not a subclass of CoupledSpinState. The syntax of this
-parameter is the same as the <tt class="docutils literal"><span class="pre">jn</span></tt> parameter of JzKetCoupled.</p>
-</div></blockquote>
-<p><strong>jcoupling_list</strong> : list or tuple</p>
-<blockquote class="last">
-<div><p>The list defining how the j-values are coupled together. If state is a
-CoupledSpinState, this parameter is ignored. This must be defined if
-state is not a subclass of CoupledSpinState. The syntax of this
-parameter is the same as the <tt class="docutils literal"><span class="pre">jcoupling</span></tt> parameter of JzKetCoupled.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Uncouple a numerical state using a CoupledSpinState state:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.spin</span> <span class="kn">import</span> <span class="n">JzKetCoupled</span><span class="p">,</span> <span class="n">uncouple</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">uncouple</span><span class="p">(</span><span class="n">JzKetCoupled</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)))</span>
-<span class="go">sqrt(2)*|1/2,-1/2&gt;x|1/2,1/2&gt;/2 + sqrt(2)*|1/2,1/2&gt;x|1/2,-1/2&gt;/2</span>
-</pre></div>
-</div>
-<p>Perform the same calculation using a SpinState state:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.spin</span> <span class="kn">import</span> <span class="n">JzKet</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">uncouple</span><span class="p">(</span><span class="n">JzKet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">sqrt(2)*|1/2,-1/2&gt;x|1/2,1/2&gt;/2 + sqrt(2)*|1/2,1/2&gt;x|1/2,-1/2&gt;/2</span>
-</pre></div>
-</div>
-<p>Uncouple a numerical state of three coupled spaces using a CoupledSpinState state:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">uncouple</span><span class="p">(</span><span class="n">JzKetCoupled</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">((</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span> <span class="p">))</span>
-<span class="go">|1,-1&gt;x|1,1&gt;x|1,1&gt;/2 - |1,0&gt;x|1,0&gt;x|1,1&gt;/2 + |1,1&gt;x|1,0&gt;x|1,0&gt;/2 - |1,1&gt;x|1,1&gt;x|1,-1&gt;/2</span>
-</pre></div>
-</div>
-<p>Perform the same calculation using a SpinState state:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">uncouple</span><span class="p">(</span><span class="n">JzKet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">((</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span> <span class="p">)</span>
-<span class="go">|1,-1&gt;x|1,1&gt;x|1,1&gt;/2 - |1,0&gt;x|1,0&gt;x|1,1&gt;/2 + |1,1&gt;x|1,0&gt;x|1,0&gt;/2 - |1,1&gt;x|1,1&gt;x|1,-1&gt;/2</span>
-</pre></div>
-</div>
-<p>Uncouple a symbolic state using a CoupledSpinState state:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">j</span><span class="p">,</span><span class="n">m</span><span class="p">,</span><span class="n">j1</span><span class="p">,</span><span class="n">j2</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;j m j1 j2&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">uncouple</span><span class="p">(</span><span class="n">JzKetCoupled</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="p">(</span><span class="n">j1</span><span class="p">,</span> <span class="n">j2</span><span class="p">)))</span>
-<span class="go">Sum(CG(j1, m1, j2, m2, j, m)*|j1,m1&gt;x|j2,m2&gt;, (m1, -j1, j1), (m2, -j2, j2))</span>
-</pre></div>
-</div>
-<p>Perform the same calculation using a SpinState state</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">uncouple</span><span class="p">(</span><span class="n">JzKet</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">m</span><span class="p">),</span> <span class="p">(</span><span class="n">j1</span><span class="p">,</span> <span class="n">j2</span><span class="p">))</span>
-<span class="go">Sum(CG(j1, m1, j2, m2, j, m)*|j1,m1&gt;x|j2,m2&gt;, (m1, -j1, j1), (m2, -j2, j2))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-
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-            
-  <div class="section" id="module-sympy.physics.quantum.state">
-<span id="state"></span><h1>State<a class="headerlink" href="#module-sympy.physics.quantum.state" title="Permalink to this headline">¶</a></h1>
-<p>Dirac notation for states.</p>
-<dl class="class">
-<dt id="sympy.physics.quantum.state.KetBase">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.state.</tt><tt class="descname">KetBase</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#KetBase"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.KetBase" title="Permalink to this definition">¶</a></dt>
-<dd><p>Base class for Kets.</p>
-<p>This class defines the dual property and the brackets for printing. This is
-an abstract base class and you should not instantiate it directly, instead
-use Ket.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.state.BraBase">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.state.</tt><tt class="descname">BraBase</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#BraBase"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.BraBase" title="Permalink to this definition">¶</a></dt>
-<dd><p>Base class for Bras.</p>
-<p>This class defines the dual property and the brackets for printing. This
-is an abstract base class and you should not instantiate it directly,
-instead use Bra.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.state.StateBase">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.state.</tt><tt class="descname">StateBase</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#StateBase"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.StateBase" title="Permalink to this definition">¶</a></dt>
-<dd><p>Abstract base class for general abstract states in quantum mechanics.</p>
-<p>All other state classes defined will need to inherit from this class. It
-carries the basic structure for all other states such as dual, _eval_adjoint
-and label.</p>
-<p>This is an abstract base class and you should not instantiate it directly,
-instead use State.</p>
-<dl class="attribute">
-<dt id="sympy.physics.quantum.state.StateBase.dual">
-<tt class="descname">dual</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#StateBase.dual"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.StateBase.dual" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the dual state of this one.</p>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.physics.quantum.state.StateBase.dual_class">
-<em class="property">classmethod </em><tt class="descname">dual_class</tt><big>(</big><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#StateBase.dual_class"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.StateBase.dual_class" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the class used to construt the dual.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.state.StateBase.operators">
-<tt class="descname">operators</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#StateBase.operators"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.StateBase.operators" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the operator(s) that this state is an eigenstate of</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.state.State">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.state.</tt><tt class="descname">State</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#State"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.State" title="Permalink to this definition">¶</a></dt>
-<dd><p>General abstract quantum state used as a base class for Ket and Bra.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.state.Ket">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.state.</tt><tt class="descname">Ket</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#Ket"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.Ket" title="Permalink to this definition">¶</a></dt>
-<dd><p>A general time-independent Ket in quantum mechanics.</p>
-<p>Inherits from State and KetBase. This class should be used as the base
-class for all physical, time-independent Kets in a system. This class
-and its subclasses will be the main classes that users will use for
-expressing Kets in Dirac notation <a class="reference internal" href="#r89">[R89]</a>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>args</strong> : tuple</p>
-<blockquote class="last">
-<div><p>The list of numbers or parameters that uniquely specify the
-ket. This will usually be its symbol or its quantum numbers. For
-time-dependent state, this will include the time.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r89" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[R89]</td><td><em>(<a class="fn-backref" href="#id1">1</a>, <a class="fn-backref" href="#id2">2</a>)</em> <a class="reference external" href="http://en.wikipedia.org/wiki/Bra-ket_notation">http://en.wikipedia.org/wiki/Bra-ket_notation</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Create a simple Ket and looking at its properties:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum</span> <span class="kn">import</span> <span class="n">Ket</span><span class="p">,</span> <span class="n">Bra</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="n">Ket</span><span class="p">(</span><span class="s">&#39;psi&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span>
-<span class="go">|psi&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span><span class="o">.</span><span class="n">hilbert_space</span>
-<span class="go">H</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span><span class="o">.</span><span class="n">is_commutative</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span><span class="o">.</span><span class="n">label</span>
-<span class="go">(psi,)</span>
-</pre></div>
-</div>
-<p>Ket&#8217;s know about their associated bra:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">k</span><span class="o">.</span><span class="n">dual</span>
-<span class="go">&lt;psi|</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span><span class="o">.</span><span class="n">dual_class</span><span class="p">()</span>
-<span class="go">&lt;class &#39;sympy.physics.quantum.state.Bra&#39;&gt;</span>
-</pre></div>
-</div>
-<p>Take a linear combination of two kets:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">k0</span> <span class="o">=</span> <span class="n">Ket</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k1</span> <span class="o">=</span> <span class="n">Ket</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span><span class="o">*</span><span class="n">I</span><span class="o">*</span><span class="n">k0</span> <span class="o">-</span> <span class="mi">4</span><span class="o">*</span><span class="n">k1</span>
-<span class="go">2*I*|0&gt; - 4*|1&gt;</span>
-</pre></div>
-</div>
-<p>Compound labels are passed as tuples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;n,m&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="n">Ket</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span>
-<span class="go">|nm&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.state.Bra">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.state.</tt><tt class="descname">Bra</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#Bra"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.Bra" title="Permalink to this definition">¶</a></dt>
-<dd><p>A general time-independent Bra in quantum mechanics.</p>
-<p>Inherits from State and BraBase. A Bra is the dual of a Ket <a class="reference internal" href="#r90">[R90]</a>. This
-class and its subclasses will be the main classes that users will use for
-expressing Bras in Dirac notation.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>args</strong> : tuple</p>
-<blockquote class="last">
-<div><p>The list of numbers or parameters that uniquely specify the
-ket. This will usually be its symbol or its quantum numbers. For
-time-dependent state, this will include the time.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r90" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[R90]</td><td><em>(<a class="fn-backref" href="#id3">1</a>, <a class="fn-backref" href="#id4">2</a>)</em> <a class="reference external" href="http://en.wikipedia.org/wiki/Bra-ket_notation">http://en.wikipedia.org/wiki/Bra-ket_notation</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Create a simple Bra and look at its properties:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum</span> <span class="kn">import</span> <span class="n">Ket</span><span class="p">,</span> <span class="n">Bra</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Bra</span><span class="p">(</span><span class="s">&#39;psi&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span>
-<span class="go">&lt;psi|</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">hilbert_space</span>
-<span class="go">H</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">is_commutative</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>Bra&#8217;s know about their dual Ket&#8217;s:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">dual</span>
-<span class="go">|psi&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">dual_class</span><span class="p">()</span>
-<span class="go">&lt;class &#39;sympy.physics.quantum.state.Ket&#39;&gt;</span>
-</pre></div>
-</div>
-<p>Like Kets, Bras can have compound labels and be manipulated in a similar
-manner:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;n,m&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Bra</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">m</span><span class="p">)</span> <span class="o">-</span> <span class="n">I</span><span class="o">*</span><span class="n">Bra</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="n">n</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span>
-<span class="go">-I*&lt;mn| + &lt;nm|</span>
-</pre></div>
-</div>
-<p>Symbols in a Bra can be substituted using <tt class="docutils literal"><span class="pre">.subs</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">m</span><span class="p">)</span>
-<span class="go">&lt;mm| - I*&lt;mm|</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.state.TimeDepState">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.state.</tt><tt class="descname">TimeDepState</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#TimeDepState"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.TimeDepState" title="Permalink to this definition">¶</a></dt>
-<dd><p>Base class for a general time-dependent quantum state.</p>
-<p>This class is used as a base class for any time-dependent state. The main
-difference between this class and the time-independent state is that this
-class takes a second argument that is the time in addition to the usual
-label argument.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>args</strong> : tuple</p>
-<blockquote class="last">
-<div><p>The list of numbers or parameters that uniquely specify the ket. This
-will usually be its symbol or its quantum numbers. For time-dependent
-state, this will include the time as the final argument.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<dl class="attribute">
-<dt id="sympy.physics.quantum.state.TimeDepState.label">
-<tt class="descname">label</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#TimeDepState.label"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.TimeDepState.label" title="Permalink to this definition">¶</a></dt>
-<dd><p>The label of the state.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.state.TimeDepState.time">
-<tt class="descname">time</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#TimeDepState.time"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.TimeDepState.time" title="Permalink to this definition">¶</a></dt>
-<dd><p>The time of the state.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.state.TimeDepBra">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.state.</tt><tt class="descname">TimeDepBra</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#TimeDepBra"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.TimeDepBra" title="Permalink to this definition">¶</a></dt>
-<dd><p>General time-dependent Bra in quantum mechanics.</p>
-<p>This inherits from TimeDepState and BraBase and is the main class that
-should be used for Bras that vary with time. Its dual is a TimeDepBra.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>args</strong> : tuple</p>
-<blockquote class="last">
-<div><p>The list of numbers or parameters that uniquely specify the ket. This
-will usually be its symbol or its quantum numbers. For time-dependent
-state, this will include the time as the final argument.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum</span> <span class="kn">import</span> <span class="n">TimeDepBra</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">TimeDepBra</span><span class="p">(</span><span class="s">&#39;psi&#39;</span><span class="p">,</span> <span class="s">&#39;t&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span>
-<span class="go">&lt;psi;t|</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">time</span>
-<span class="go">t</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">label</span>
-<span class="go">(psi,)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">hilbert_space</span>
-<span class="go">H</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">dual</span>
-<span class="go">|psi;t&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.state.TimeDepKet">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.state.</tt><tt class="descname">TimeDepKet</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#TimeDepKet"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.TimeDepKet" title="Permalink to this definition">¶</a></dt>
-<dd><p>General time-dependent Ket in quantum mechanics.</p>
-<p>This inherits from <tt class="docutils literal"><span class="pre">TimeDepState</span></tt> and <tt class="docutils literal"><span class="pre">KetBase</span></tt> and is the main class
-that should be used for Kets that vary with time. Its dual is a
-<tt class="docutils literal"><span class="pre">TimeDepBra</span></tt>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>args</strong> : tuple</p>
-<blockquote class="last">
-<div><p>The list of numbers or parameters that uniquely specify the ket. This
-will usually be its symbol or its quantum numbers. For time-dependent
-state, this will include the time as the final argument.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Create a TimeDepKet and look at its attributes:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum</span> <span class="kn">import</span> <span class="n">TimeDepKet</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="n">TimeDepKet</span><span class="p">(</span><span class="s">&#39;psi&#39;</span><span class="p">,</span> <span class="s">&#39;t&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span>
-<span class="go">|psi;t&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span><span class="o">.</span><span class="n">time</span>
-<span class="go">t</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span><span class="o">.</span><span class="n">label</span>
-<span class="go">(psi,)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span><span class="o">.</span><span class="n">hilbert_space</span>
-<span class="go">H</span>
-</pre></div>
-</div>
-<p>TimeDepKets know about their dual bra:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">k</span><span class="o">.</span><span class="n">dual</span>
-<span class="go">&lt;psi;t|</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span><span class="o">.</span><span class="n">dual_class</span><span class="p">()</span>
-<span class="go">&lt;class &#39;sympy.physics.quantum.state.TimeDepBra&#39;&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.quantum.state.Wavefunction">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.state.</tt><tt class="descname">Wavefunction</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#Wavefunction"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.Wavefunction" title="Permalink to this definition">¶</a></dt>
-<dd><p>Class for representations in continuous bases</p>
-<p>This class takes an expression and coordinates in its constructor. It can
-be used to easily calculate normalizations and probabilities.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>expr</strong> : Expr</p>
-<blockquote>
-<div><p>The expression representing the functional form of the w.f.</p>
-</div></blockquote>
-<p><strong>coords</strong> : Symbol or tuple</p>
-<blockquote class="last">
-<div><p>The coordinates to be integrated over, and their bounds</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Particle in a box, specifying bounds in the more primitive way of using
-Piecewise:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Piecewise</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">N</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.state</span> <span class="kn">import</span> <span class="n">Wavefunction</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">L</span> <span class="o">=</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">Piecewise</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span> <span class="o">&gt;</span> <span class="n">L</span><span class="p">),</span> <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">//</span><span class="n">L</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="o">/</span><span class="n">L</span><span class="p">),</span> <span class="bp">True</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Wavefunction</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">norm</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">is_normalized</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">prob</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">(</span><span class="n">L</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">(</span><span class="mf">0.85</span><span class="o">*</span><span class="n">L</span><span class="p">)</span>
-<span class="go">2*sin(0.85*pi)**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="p">(</span><span class="n">p</span><span class="p">(</span><span class="mf">0.85</span><span class="o">*</span><span class="n">L</span><span class="p">))</span>
-<span class="go">0.412214747707527</span>
-</pre></div>
-</div>
-<p>Additionally, you can specify the bounds of the function and the indices in
-a more compact way:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">diff</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.state</span> <span class="kn">import</span> <span class="n">Wavefunction</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">L</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,L&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">/</span><span class="n">L</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="o">/</span><span class="n">L</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Wavefunction</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">L</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">norm</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">L</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">L</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">sqrt(2)*sin(pi*n*(L - 1)/L)/sqrt(L)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="mf">0.85</span><span class="p">)</span>
-<span class="go">sqrt(2)*sin(0.85*pi*n/L)/sqrt(L)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="mf">0.85</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">L</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">sqrt(2)*sin(0.85*pi)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">is_commutative</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>All arguments are automatically sympified, so you can define the variables
-as strings rather than symbols:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expr</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Wavefunction</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="n">f</span><span class="o">.</span><span class="n">variables</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-<span class="go">&lt;class &#39;sympy.core.symbol.Symbol&#39;&gt;</span>
-</pre></div>
-</div>
-<p>Derivatives of Wavefunctions will return Wavefunctions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">Wavefunction(2*x, x)</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>nargs</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="attribute">
-<dt id="sympy.physics.quantum.state.Wavefunction.expr">
-<tt class="descname">expr</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#Wavefunction.expr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.Wavefunction.expr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the expression which is the functional form of the Wavefunction</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.state</span> <span class="kn">import</span> <span class="n">Wavefunction</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x, y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Wavefunction</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">expr</span>
-<span class="go">x**2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.state.Wavefunction.is_commutative">
-<tt class="descname">is_commutative</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#Wavefunction.is_commutative"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.Wavefunction.is_commutative" title="Permalink to this definition">¶</a></dt>
-<dd><p>Override Function&#8217;s is_commutative so that order is preserved in
-represented expressions</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.state.Wavefunction.is_normalized">
-<tt class="descname">is_normalized</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#Wavefunction.is_normalized"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.Wavefunction.is_normalized" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns true if the Wavefunction is properly normalized</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.state</span> <span class="kn">import</span> <span class="n">Wavefunction</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">L</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,L&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">/</span><span class="n">L</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="o">/</span><span class="n">L</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Wavefunction</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">L</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">is_normalized</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.state.Wavefunction.limits">
-<tt class="descname">limits</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#Wavefunction.limits"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.Wavefunction.limits" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the limits of the coordinates which the w.f. depends on If no
-limits are specified, defaults to <tt class="docutils literal"><span class="pre">(-oo,</span> <span class="pre">oo)</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.state</span> <span class="kn">import</span> <span class="n">Wavefunction</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x, y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Wavefunction</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">limits</span>
-<span class="go">{x: (0, 1)}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Wavefunction</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">limits</span>
-<span class="go">{x: (-oo, oo)}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Wavefunction</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">limits</span>
-<span class="go">{x: (-oo, oo), y: (-1, 2)}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.state.Wavefunction.norm">
-<tt class="descname">norm</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#Wavefunction.norm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.Wavefunction.norm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the normalization of the specified functional form.</p>
-<p>This function integrates over the coordinates of the Wavefunction, with
-the bounds specified.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.state</span> <span class="kn">import</span> <span class="n">Wavefunction</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">L</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,L&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">/</span><span class="n">L</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="o">/</span><span class="n">L</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Wavefunction</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">L</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">norm</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="o">/</span><span class="n">L</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Wavefunction</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">L</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">norm</span>
-<span class="go">sqrt(2)*sqrt(L)/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.state.Wavefunction.normalize">
-<tt class="descname">normalize</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#Wavefunction.normalize"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.Wavefunction.normalize" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a normalized version of the Wavefunction</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.state</span> <span class="kn">import</span> <span class="n">Wavefunction</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">L</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,L&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="o">/</span><span class="n">L</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Wavefunction</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">L</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">normalize</span><span class="p">()</span>
-<span class="go">Wavefunction(sqrt(2)*sin(pi*n*x/L)/sqrt(L), (x, 0, L))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.state.Wavefunction.prob">
-<tt class="descname">prob</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#Wavefunction.prob"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.Wavefunction.prob" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the absolute magnitude of the w.f., <span class="math">\(|\psi(x)|^2\)</span></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.state</span> <span class="kn">import</span> <span class="n">Wavefunction</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">L</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,L&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="o">/</span><span class="n">L</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Wavefunction</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">L</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">prob</span><span class="p">()</span>
-<span class="go">Wavefunction(sin(pi*n*x/L)**2, x)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.quantum.state.Wavefunction.variables">
-<tt class="descname">variables</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/state.html#Wavefunction.variables"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.state.Wavefunction.variables" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the coordinates which the wavefunction depends on</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum.state</span> <span class="kn">import</span> <span class="n">Wavefunction</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Wavefunction</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">variables</span>
-<span class="go">(x, y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">Wavefunction</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span><span class="o">.</span><span class="n">variables</span>
-<span class="go">(x,)</span>
-</pre></div>
-</div>
-</dd></dl>
-
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-  <div class="section" id="module-sympy.physics.quantum.tensorproduct">
-<span id="tensor-product"></span><h1>Tensor Product<a class="headerlink" href="#module-sympy.physics.quantum.tensorproduct" title="Permalink to this headline">¶</a></h1>
-<p>Abstract tensor product.</p>
-<dl class="class">
-<dt id="sympy.physics.quantum.tensorproduct.TensorProduct">
-<em class="property">class </em><tt class="descclassname">sympy.physics.quantum.tensorproduct.</tt><tt class="descname">TensorProduct</tt><a class="reference internal" href="../../../_modules/sympy/physics/quantum/tensorproduct.html#TensorProduct"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.tensorproduct.TensorProduct" title="Permalink to this definition">¶</a></dt>
-<dd><p>The tensor product of two or more arguments.</p>
-<p>For matrices, this uses <tt class="docutils literal"><span class="pre">matrix_tensor_product</span></tt> to compute the Kronecker
-or tensor product matrix. For other objects a symbolic <tt class="docutils literal"><span class="pre">TensorProduct</span></tt>
-instance is returned. The tensor product is a non-commutative
-multiplication that is used primarily with operators and states in quantum
-mechanics.</p>
-<p>Currently, the tensor product distinguishes between commutative and non-
-commutative arguments.  Commutative arguments are assumed to be scalars and
-are pulled out in front of the <tt class="docutils literal"><span class="pre">TensorProduct</span></tt>. Non-commutative arguments
-remain in the resulting <tt class="docutils literal"><span class="pre">TensorProduct</span></tt>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>args</strong> : tuple</p>
-<blockquote class="last">
-<div><p>A sequence of the objects to take the tensor product of.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Start with a simple tensor product of sympy matrices:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">I</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum</span> <span class="kn">import</span> <span class="n">TensorProduct</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m1</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">],[</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m2</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">TensorProduct</span><span class="p">(</span><span class="n">m1</span><span class="p">,</span> <span class="n">m2</span><span class="p">)</span>
-<span class="go">[1, 0, 2, 0]</span>
-<span class="go">[0, 1, 0, 2]</span>
-<span class="go">[3, 0, 4, 0]</span>
-<span class="go">[0, 3, 0, 4]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">TensorProduct</span><span class="p">(</span><span class="n">m2</span><span class="p">,</span> <span class="n">m1</span><span class="p">)</span>
-<span class="go">[1, 2, 0, 0]</span>
-<span class="go">[3, 4, 0, 0]</span>
-<span class="go">[0, 0, 1, 2]</span>
-<span class="go">[0, 0, 3, 4]</span>
-</pre></div>
-</div>
-<p>We can also construct tensor products of non-commutative symbols:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span><span class="n">commutative</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span><span class="n">commutative</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tp</span> <span class="o">=</span> <span class="n">TensorProduct</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tp</span>
-<span class="go">AxB</span>
-</pre></div>
-</div>
-<p>We can take the dagger of a tensor product (note the order does NOT reverse
-like the dagger of a normal product):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum</span> <span class="kn">import</span> <span class="n">Dagger</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="n">tp</span><span class="p">)</span>
-<span class="go">Dagger(A)xDagger(B)</span>
-</pre></div>
-</div>
-<p>Expand can be used to distribute a tensor product across addition:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;C&#39;</span><span class="p">,</span><span class="n">commutative</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tp</span> <span class="o">=</span> <span class="n">TensorProduct</span><span class="p">(</span><span class="n">A</span><span class="o">+</span><span class="n">B</span><span class="p">,</span><span class="n">C</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tp</span>
-<span class="go">(A + B)xC</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tp</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">tensorproduct</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">AxC + BxC</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.quantum.tensorproduct.tensor_product_simp">
-<tt class="descclassname">sympy.physics.quantum.tensorproduct.</tt><tt class="descname">tensor_product_simp</tt><big>(</big><em>e</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../../_modules/sympy/physics/quantum/tensorproduct.html#tensor_product_simp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.quantum.tensorproduct.tensor_product_simp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Try to simplify and combine TensorProducts.</p>
-<p>In general this will try to pull expressions inside of <tt class="docutils literal"><span class="pre">TensorProducts</span></tt>.
-It currently only works for relatively simple cases where the products have
-only scalars, raw <tt class="docutils literal"><span class="pre">TensorProducts</span></tt>, not <tt class="docutils literal"><span class="pre">Add</span></tt>, <tt class="docutils literal"><span class="pre">Pow</span></tt>, <tt class="docutils literal"><span class="pre">Commutators</span></tt>
-of <tt class="docutils literal"><span class="pre">TensorProducts</span></tt>. It is best to see what it does by showing examples.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum</span> <span class="kn">import</span> <span class="n">tensor_product_simp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.quantum</span> <span class="kn">import</span> <span class="n">TensorProduct</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span><span class="n">commutative</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span><span class="n">commutative</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;C&#39;</span><span class="p">,</span><span class="n">commutative</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;D&#39;</span><span class="p">,</span><span class="n">commutative</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>First see what happens to products of tensor products:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="n">TensorProduct</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">B</span><span class="p">)</span><span class="o">*</span><span class="n">TensorProduct</span><span class="p">(</span><span class="n">C</span><span class="p">,</span><span class="n">D</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span>
-<span class="go">AxB*CxD</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tensor_product_simp</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-<span class="go">(A*C)x(B*D)</span>
-</pre></div>
-</div>
-<p>This is the core logic of this function, and it works inside, powers, sums,
-commutators and anticommutators as well:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">tensor_product_simp</span><span class="p">(</span><span class="n">e</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(A*C)x(B*D)**2</span>
-</pre></div>
-</div>
-</dd></dl>
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-  <div class="section" id="module-sympy.physics.secondquant">
-<span id="second-quantization"></span><h1>Second Quantization<a class="headerlink" href="#module-sympy.physics.secondquant" title="Permalink to this headline">¶</a></h1>
-<p>Second quantization operators and states for bosons.</p>
-<p>This follow the formulation of Fetter and Welecka, &#8220;Quantum Theory
-of Many-Particle Systems.&#8221;</p>
-<dl class="class">
-<dt id="sympy.physics.secondquant.Dagger">
-<em class="property">class </em><tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">Dagger</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#Dagger"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.Dagger" title="Permalink to this definition">¶</a></dt>
-<dd><p>Hermitian conjugate of creation/annihilation operators.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">Dagger</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">Bd</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">I</span><span class="p">)</span>
-<span class="go">-2*I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="n">B</span><span class="p">(</span><span class="mi">0</span><span class="p">))</span>
-<span class="go">CreateBoson(0)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="n">Bd</span><span class="p">(</span><span class="mi">0</span><span class="p">))</span>
-<span class="go">AnnihilateBoson(0)</span>
-</pre></div>
-</div>
-<dl class="classmethod">
-<dt id="sympy.physics.secondquant.Dagger.eval">
-<em class="property">classmethod </em><tt class="descname">eval</tt><big>(</big><em>arg</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#Dagger.eval"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.Dagger.eval" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the Dagger instance.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">Dagger</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">Bd</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">I</span><span class="p">)</span>
-<span class="go">-2*I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="n">B</span><span class="p">(</span><span class="mi">0</span><span class="p">))</span>
-<span class="go">CreateBoson(0)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="n">Bd</span><span class="p">(</span><span class="mi">0</span><span class="p">))</span>
-<span class="go">AnnihilateBoson(0)</span>
-</pre></div>
-</div>
-<p>The eval() method is called automatically.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.secondquant.KroneckerDelta">
-<em class="property">class </em><tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">KroneckerDelta</tt><a class="headerlink" href="#sympy.physics.secondquant.KroneckerDelta" title="Permalink to this definition">¶</a></dt>
-<dd><p>The discrete, or Kronecker, delta function.</p>
-<p>A function that takes in two integers i and j. It returns 0 if i and j are
-not equal or it returns 1 if i and j are equal.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>i</strong> : Number, Symbol</p>
-<blockquote>
-<div><p>The first index of the delta function.</p>
-</div></blockquote>
-<p><strong>j</strong> : Number, Symbol</p>
-<blockquote class="last">
-<div><p>The second index of the delta function.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.physics.secondquant.KroneckerDelta.eval" title="sympy.physics.secondquant.KroneckerDelta.eval"><tt class="xref py py-obj docutils literal"><span class="pre">eval</span></tt></a>, <a class="reference internal" href="../functions/special.html#sympy.functions.special.delta_functions.DiracDelta" title="sympy.functions.special.delta_functions.DiracDelta"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.functions.special.delta_functions.DiracDelta</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<p><a class="reference external" href="http://en.wikipedia.org/wiki/Kronecker_delta">http://en.wikipedia.org/wiki/Kronecker_delta</a></p>
-<p class="rubric">Examples</p>
-<p>A simple example with integer indices:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions.special.tensor_functions</span> <span class="kn">import</span> <span class="n">KroneckerDelta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>Symbolic indices:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">k</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
-<span class="go">KroneckerDelta(i, j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span>
-<span class="go">KroneckerDelta(i, i + k + 1)</span>
-</pre></div>
-</div>
-<dl class="classmethod">
-<dt id="sympy.physics.secondquant.KroneckerDelta.eval">
-<em class="property">classmethod </em><tt class="descname">eval</tt><big>(</big><em>i</em>, <em>j</em><big>)</big><a class="headerlink" href="#sympy.physics.secondquant.KroneckerDelta.eval" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates the discrete delta function.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions.special.tensor_functions</span> <span class="kn">import</span> <span class="n">KroneckerDelta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">k</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
-<span class="go">KroneckerDelta(i, j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">k</span><span class="p">)</span>
-<span class="go">KroneckerDelta(i, i + k + 1)</span>
-</pre></div>
-</div>
-<p># indirect doctest</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.KroneckerDelta.indices_contain_equal_information">
-<tt class="descname">indices_contain_equal_information</tt><a class="headerlink" href="#sympy.physics.secondquant.KroneckerDelta.indices_contain_equal_information" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if indices are either both above or below fermi.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions.special.tensor_functions</span> <span class="kn">import</span> <span class="n">KroneckerDelta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span><span class="o">.</span><span class="n">indices_contain_equal_information</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">indices_contain_equal_information</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">indices_contain_equal_information</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.KroneckerDelta.is_above_fermi">
-<tt class="descname">is_above_fermi</tt><a class="headerlink" href="#sympy.physics.secondquant.KroneckerDelta.is_above_fermi" title="Permalink to this definition">¶</a></dt>
-<dd><p>True if Delta can be non-zero above fermi</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.physics.secondquant.KroneckerDelta.is_below_fermi" title="sympy.physics.secondquant.KroneckerDelta.is_below_fermi"><tt class="xref py py-obj docutils literal"><span class="pre">is_below_fermi</span></tt></a>, <a class="reference internal" href="#sympy.physics.secondquant.KroneckerDelta.is_only_below_fermi" title="sympy.physics.secondquant.KroneckerDelta.is_only_below_fermi"><tt class="xref py py-obj docutils literal"><span class="pre">is_only_below_fermi</span></tt></a>, <a class="reference internal" href="#sympy.physics.secondquant.KroneckerDelta.is_only_above_fermi" title="sympy.physics.secondquant.KroneckerDelta.is_only_above_fermi"><tt class="xref py py-obj docutils literal"><span class="pre">is_only_above_fermi</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions.special.tensor_functions</span> <span class="kn">import</span> <span class="n">KroneckerDelta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">is_above_fermi</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">is_above_fermi</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span><span class="o">.</span><span class="n">is_above_fermi</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.KroneckerDelta.is_below_fermi">
-<tt class="descname">is_below_fermi</tt><a class="headerlink" href="#sympy.physics.secondquant.KroneckerDelta.is_below_fermi" title="Permalink to this definition">¶</a></dt>
-<dd><p>True if Delta can be non-zero below fermi</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.physics.secondquant.KroneckerDelta.is_above_fermi" title="sympy.physics.secondquant.KroneckerDelta.is_above_fermi"><tt class="xref py py-obj docutils literal"><span class="pre">is_above_fermi</span></tt></a>, <a class="reference internal" href="#sympy.physics.secondquant.KroneckerDelta.is_only_above_fermi" title="sympy.physics.secondquant.KroneckerDelta.is_only_above_fermi"><tt class="xref py py-obj docutils literal"><span class="pre">is_only_above_fermi</span></tt></a>, <a class="reference internal" href="#sympy.physics.secondquant.KroneckerDelta.is_only_below_fermi" title="sympy.physics.secondquant.KroneckerDelta.is_only_below_fermi"><tt class="xref py py-obj docutils literal"><span class="pre">is_only_below_fermi</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions.special.tensor_functions</span> <span class="kn">import</span> <span class="n">KroneckerDelta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">is_below_fermi</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">is_below_fermi</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span><span class="o">.</span><span class="n">is_below_fermi</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.KroneckerDelta.is_only_above_fermi">
-<tt class="descname">is_only_above_fermi</tt><a class="headerlink" href="#sympy.physics.secondquant.KroneckerDelta.is_only_above_fermi" title="Permalink to this definition">¶</a></dt>
-<dd><p>True if Delta is restricted to above fermi</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.physics.secondquant.KroneckerDelta.is_above_fermi" title="sympy.physics.secondquant.KroneckerDelta.is_above_fermi"><tt class="xref py py-obj docutils literal"><span class="pre">is_above_fermi</span></tt></a>, <a class="reference internal" href="#sympy.physics.secondquant.KroneckerDelta.is_below_fermi" title="sympy.physics.secondquant.KroneckerDelta.is_below_fermi"><tt class="xref py py-obj docutils literal"><span class="pre">is_below_fermi</span></tt></a>, <a class="reference internal" href="#sympy.physics.secondquant.KroneckerDelta.is_only_below_fermi" title="sympy.physics.secondquant.KroneckerDelta.is_only_below_fermi"><tt class="xref py py-obj docutils literal"><span class="pre">is_only_below_fermi</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions.special.tensor_functions</span> <span class="kn">import</span> <span class="n">KroneckerDelta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_above_fermi</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_above_fermi</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_above_fermi</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.KroneckerDelta.is_only_below_fermi">
-<tt class="descname">is_only_below_fermi</tt><a class="headerlink" href="#sympy.physics.secondquant.KroneckerDelta.is_only_below_fermi" title="Permalink to this definition">¶</a></dt>
-<dd><p>True if Delta is restricted to below fermi</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.physics.secondquant.KroneckerDelta.is_above_fermi" title="sympy.physics.secondquant.KroneckerDelta.is_above_fermi"><tt class="xref py py-obj docutils literal"><span class="pre">is_above_fermi</span></tt></a>, <a class="reference internal" href="#sympy.physics.secondquant.KroneckerDelta.is_below_fermi" title="sympy.physics.secondquant.KroneckerDelta.is_below_fermi"><tt class="xref py py-obj docutils literal"><span class="pre">is_below_fermi</span></tt></a>, <a class="reference internal" href="#sympy.physics.secondquant.KroneckerDelta.is_only_above_fermi" title="sympy.physics.secondquant.KroneckerDelta.is_only_above_fermi"><tt class="xref py py-obj docutils literal"><span class="pre">is_only_above_fermi</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions.special.tensor_functions</span> <span class="kn">import</span> <span class="n">KroneckerDelta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_below_fermi</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_below_fermi</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_below_fermi</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.KroneckerDelta.killable_index">
-<tt class="descname">killable_index</tt><a class="headerlink" href="#sympy.physics.secondquant.KroneckerDelta.killable_index" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the index which is preferred to substitute in the final
-expression.</p>
-<p>The index to substitute is the index with less information regarding
-fermi level.  If indices contain same information, &#8216;a&#8217; is preferred
-before &#8216;b&#8217;.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.physics.secondquant.KroneckerDelta.preferred_index" title="sympy.physics.secondquant.KroneckerDelta.preferred_index"><tt class="xref py py-obj docutils literal"><span class="pre">preferred_index</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions.special.tensor_functions</span> <span class="kn">import</span> <span class="n">KroneckerDelta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">j</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;j&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">killable_index</span>
-<span class="go">p</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">killable_index</span>
-<span class="go">p</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span><span class="o">.</span><span class="n">killable_index</span>
-<span class="go">j</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.KroneckerDelta.preferred_index">
-<tt class="descname">preferred_index</tt><a class="headerlink" href="#sympy.physics.secondquant.KroneckerDelta.preferred_index" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the index which is preferred to keep in the final expression.</p>
-<p>The preferred index is the index with more information regarding fermi
-level.  If indices contain same information, &#8216;a&#8217; is preferred before
-&#8216;b&#8217;.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.physics.secondquant.KroneckerDelta.killable_index" title="sympy.physics.secondquant.KroneckerDelta.killable_index"><tt class="xref py py-obj docutils literal"><span class="pre">killable_index</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions.special.tensor_functions</span> <span class="kn">import</span> <span class="n">KroneckerDelta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">j</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;j&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">preferred_index</span>
-<span class="go">i</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">preferred_index</span>
-<span class="go">a</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span><span class="o">.</span><span class="n">preferred_index</span>
-<span class="go">i</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.secondquant.AnnihilateBoson">
-<em class="property">class </em><tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">AnnihilateBoson</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#AnnihilateBoson"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.AnnihilateBoson" title="Permalink to this definition">¶</a></dt>
-<dd><p>Bosonic annihilation operator.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">B</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">AnnihilateBoson(x)</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.physics.secondquant.AnnihilateBoson.apply_operator">
-<tt class="descname">apply_operator</tt><big>(</big><em>self</em>, <em>state</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#AnnihilateBoson.apply_operator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.AnnihilateBoson.apply_operator" title="Permalink to this definition">¶</a></dt>
-<dd><p>Apply state to self if self is not symbolic and state is a FockStateKet, else
-multiply self by state.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">B</span><span class="p">,</span> <span class="n">BKet</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">apply_operator</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">y*AnnihilateBoson(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">apply_operator</span><span class="p">(</span><span class="n">BKet</span><span class="p">((</span><span class="n">n</span><span class="p">,)))</span>
-<span class="go">sqrt(n)*FockStateBosonKet((n - 1,))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.secondquant.CreateBoson">
-<em class="property">class </em><tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">CreateBoson</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#CreateBoson"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.CreateBoson" title="Permalink to this definition">¶</a></dt>
-<dd><p>Bosonic creation operator.</p>
-<dl class="function">
-<dt id="sympy.physics.secondquant.CreateBoson.apply_operator">
-<tt class="descname">apply_operator</tt><big>(</big><em>self</em>, <em>state</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#CreateBoson.apply_operator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.CreateBoson.apply_operator" title="Permalink to this definition">¶</a></dt>
-<dd><p>Apply state to self if self is not symbolic and state is a FockStateKet, else
-multiply self by state.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">B</span><span class="p">,</span> <span class="n">Dagger</span><span class="p">,</span> <span class="n">BKet</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="n">B</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">apply_operator</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">y*CreateBoson(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">apply_operator</span><span class="p">(</span><span class="n">BKet</span><span class="p">((</span><span class="n">n</span><span class="p">,)))</span>
-<span class="go">sqrt(n)*FockStateBosonKet((n - 1,))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.secondquant.AnnihilateFermion">
-<em class="property">class </em><tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">AnnihilateFermion</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#AnnihilateFermion"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.AnnihilateFermion" title="Permalink to this definition">¶</a></dt>
-<dd><p>Fermionic annihilation operator.</p>
-<dl class="function">
-<dt id="sympy.physics.secondquant.AnnihilateFermion.apply_operator">
-<tt class="descname">apply_operator</tt><big>(</big><em>self</em>, <em>state</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#AnnihilateFermion.apply_operator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.AnnihilateFermion.apply_operator" title="Permalink to this definition">¶</a></dt>
-<dd><p>Apply state to self if self is not symbolic and state is a FockStateKet, else
-multiply self by state.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">B</span><span class="p">,</span> <span class="n">Dagger</span><span class="p">,</span> <span class="n">BKet</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="n">B</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">apply_operator</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">y*CreateBoson(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">apply_operator</span><span class="p">(</span><span class="n">BKet</span><span class="p">((</span><span class="n">n</span><span class="p">,)))</span>
-<span class="go">sqrt(n)*FockStateBosonKet((n - 1,))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.AnnihilateFermion.is_only_q_annihilator">
-<tt class="descname">is_only_q_annihilator</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#AnnihilateFermion.is_only_q_annihilator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.AnnihilateFermion.is_only_q_annihilator" title="Permalink to this definition">¶</a></dt>
-<dd><p>Always destroy a quasi-particle?  (annihilate hole or annihilate particle)</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">F</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_q_annihilator</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_q_annihilator</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_q_annihilator</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.AnnihilateFermion.is_only_q_creator">
-<tt class="descname">is_only_q_creator</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#AnnihilateFermion.is_only_q_creator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.AnnihilateFermion.is_only_q_creator" title="Permalink to this definition">¶</a></dt>
-<dd><p>Always create a quasi-particle?  (create hole or create particle)</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">F</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_q_creator</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_q_creator</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_q_creator</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.AnnihilateFermion.is_q_annihilator">
-<tt class="descname">is_q_annihilator</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#AnnihilateFermion.is_q_annihilator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.AnnihilateFermion.is_q_annihilator" title="Permalink to this definition">¶</a></dt>
-<dd><p>Can we destroy a quasi-particle?  (annihilate hole or annihilate particle)
-If so, would that be above or below the fermi surface?</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">F</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">is_q_annihilator</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">is_q_annihilator</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">is_q_annihilator</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.AnnihilateFermion.is_q_creator">
-<tt class="descname">is_q_creator</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#AnnihilateFermion.is_q_creator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.AnnihilateFermion.is_q_creator" title="Permalink to this definition">¶</a></dt>
-<dd><p>Can we create a quasi-particle?  (create hole or create particle)
-If so, would that be above or below the fermi surface?</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">F</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">is_q_creator</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">is_q_creator</span>
-<span class="go">-1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">is_q_creator</span>
-<span class="go">-1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.secondquant.CreateFermion">
-<em class="property">class </em><tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">CreateFermion</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#CreateFermion"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.CreateFermion" title="Permalink to this definition">¶</a></dt>
-<dd><p>Fermionic creation operator.</p>
-<dl class="function">
-<dt id="sympy.physics.secondquant.CreateFermion.apply_operator">
-<tt class="descname">apply_operator</tt><big>(</big><em>self</em>, <em>state</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#CreateFermion.apply_operator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.CreateFermion.apply_operator" title="Permalink to this definition">¶</a></dt>
-<dd><p>Apply state to self if self is not symbolic and state is a FockStateKet, else
-multiply self by state.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">B</span><span class="p">,</span> <span class="n">Dagger</span><span class="p">,</span> <span class="n">BKet</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Dagger</span><span class="p">(</span><span class="n">B</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">apply_operator</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">y*CreateBoson(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">apply_operator</span><span class="p">(</span><span class="n">BKet</span><span class="p">((</span><span class="n">n</span><span class="p">,)))</span>
-<span class="go">sqrt(n)*FockStateBosonKet((n - 1,))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.CreateFermion.is_only_q_annihilator">
-<tt class="descname">is_only_q_annihilator</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#CreateFermion.is_only_q_annihilator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.CreateFermion.is_only_q_annihilator" title="Permalink to this definition">¶</a></dt>
-<dd><p>Always destroy a quasi-particle?  (annihilate hole or annihilate particle)</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">Fd</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Fd</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_q_annihilator</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Fd</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_q_annihilator</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Fd</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_q_annihilator</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.CreateFermion.is_only_q_creator">
-<tt class="descname">is_only_q_creator</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#CreateFermion.is_only_q_creator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.CreateFermion.is_only_q_creator" title="Permalink to this definition">¶</a></dt>
-<dd><p>Always create a quasi-particle?  (create hole or create particle)</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">Fd</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Fd</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_q_creator</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Fd</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_q_creator</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Fd</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">is_only_q_creator</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.CreateFermion.is_q_annihilator">
-<tt class="descname">is_q_annihilator</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#CreateFermion.is_q_annihilator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.CreateFermion.is_q_annihilator" title="Permalink to this definition">¶</a></dt>
-<dd><p>Can we destroy a quasi-particle?  (annihilate hole or annihilate particle)
-If so, would that be above or below the fermi surface?</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">Fd</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Fd</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">is_q_annihilator</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Fd</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">is_q_annihilator</span>
-<span class="go">-1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Fd</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">is_q_annihilator</span>
-<span class="go">-1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.CreateFermion.is_q_creator">
-<tt class="descname">is_q_creator</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#CreateFermion.is_q_creator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.CreateFermion.is_q_creator" title="Permalink to this definition">¶</a></dt>
-<dd><p>Can we create a quasi-particle?  (create hole or create particle)
-If so, would that be above or below the fermi surface?</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">Fd</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Fd</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">is_q_creator</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Fd</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">is_q_creator</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Fd</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">is_q_creator</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.secondquant.FockState">
-<em class="property">class </em><tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">FockState</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#FockState"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.FockState" title="Permalink to this definition">¶</a></dt>
-<dd><p>Many particle Fock state with a sequence of occupation numbers.</p>
-<p>Anywhere you can have a FockState, you can also have S.Zero.
-All code must check for this!</p>
-<p>Base class to represent FockStates.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.secondquant.FockStateBra">
-<em class="property">class </em><tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">FockStateBra</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#FockStateBra"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.FockStateBra" title="Permalink to this definition">¶</a></dt>
-<dd><p>Representation of a bra.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.secondquant.FockStateKet">
-<em class="property">class </em><tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">FockStateKet</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#FockStateKet"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.FockStateKet" title="Permalink to this definition">¶</a></dt>
-<dd><p>Representation of a ket.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.secondquant.FockStateBosonKet">
-<em class="property">class </em><tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">FockStateBosonKet</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#FockStateBosonKet"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.FockStateBosonKet" title="Permalink to this definition">¶</a></dt>
-<dd><p>Many particle Fock state with a sequence of occupation numbers.</p>
-<p>Occupation numbers can be any integer &gt;= 0.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">BKet</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">BKet</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="go">FockStateBosonKet((1, 2))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.secondquant.FockStateBosonBra">
-<em class="property">class </em><tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">FockStateBosonBra</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#FockStateBosonBra"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.FockStateBosonBra" title="Permalink to this definition">¶</a></dt>
-<dd><p>Describes a collection of BosonBra particles.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">BBra</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">BBra</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="go">FockStateBosonBra((1, 2))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.BBra">
-<tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">BBra</tt><a class="headerlink" href="#sympy.physics.secondquant.BBra" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <a class="reference internal" href="#sympy.physics.secondquant.FockStateBosonBra" title="sympy.physics.secondquant.FockStateBosonBra"><tt class="xref py py-class docutils literal"><span class="pre">FockStateBosonBra</span></tt></a></p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.BKet">
-<tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">BKet</tt><a class="headerlink" href="#sympy.physics.secondquant.BKet" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <a class="reference internal" href="#sympy.physics.secondquant.FockStateBosonKet" title="sympy.physics.secondquant.FockStateBosonKet"><tt class="xref py py-class docutils literal"><span class="pre">FockStateBosonKet</span></tt></a></p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.FBra">
-<tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">FBra</tt><a class="headerlink" href="#sympy.physics.secondquant.FBra" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <tt class="xref py py-class docutils literal"><span class="pre">FockStateFermionBra</span></tt></p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.FKet">
-<tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">FKet</tt><a class="headerlink" href="#sympy.physics.secondquant.FKet" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <tt class="xref py py-class docutils literal"><span class="pre">FockStateFermionKet</span></tt></p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.F">
-<tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">F</tt><a class="headerlink" href="#sympy.physics.secondquant.F" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <a class="reference internal" href="#sympy.physics.secondquant.AnnihilateFermion" title="sympy.physics.secondquant.AnnihilateFermion"><tt class="xref py py-class docutils literal"><span class="pre">AnnihilateFermion</span></tt></a></p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.Fd">
-<tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">Fd</tt><a class="headerlink" href="#sympy.physics.secondquant.Fd" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <a class="reference internal" href="#sympy.physics.secondquant.CreateFermion" title="sympy.physics.secondquant.CreateFermion"><tt class="xref py py-class docutils literal"><span class="pre">CreateFermion</span></tt></a></p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.B">
-<tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">B</tt><a class="headerlink" href="#sympy.physics.secondquant.B" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <a class="reference internal" href="#sympy.physics.secondquant.AnnihilateBoson" title="sympy.physics.secondquant.AnnihilateBoson"><tt class="xref py py-class docutils literal"><span class="pre">AnnihilateBoson</span></tt></a></p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.Bd">
-<tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">Bd</tt><a class="headerlink" href="#sympy.physics.secondquant.Bd" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <a class="reference internal" href="#sympy.physics.secondquant.CreateBoson" title="sympy.physics.secondquant.CreateBoson"><tt class="xref py py-class docutils literal"><span class="pre">CreateBoson</span></tt></a></p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.secondquant.apply_operators">
-<tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">apply_operators</tt><big>(</big><em>e</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#apply_operators"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.apply_operators" title="Permalink to this definition">¶</a></dt>
-<dd><p>Take a sympy expression with operators and states and apply the operators.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">apply_operators</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sympify</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">apply_operators</span><span class="p">(</span><span class="n">sympify</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">+</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">7</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.secondquant.InnerProduct">
-<em class="property">class </em><tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">InnerProduct</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#InnerProduct"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.InnerProduct" title="Permalink to this definition">¶</a></dt>
-<dd><p>An unevaluated inner product between a bra and ket.</p>
-<p>Currently this class just reduces things to a product of
-Kronecker Deltas.  In the future, we could introduce abstract
-states like <tt class="docutils literal"><span class="pre">|a&gt;</span></tt> and <tt class="docutils literal"><span class="pre">|b&gt;</span></tt>, and leave the inner product unevaluated as
-<tt class="docutils literal"><span class="pre">&lt;a|b&gt;</span></tt>.</p>
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.InnerProduct.bra">
-<tt class="descname">bra</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#InnerProduct.bra"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.InnerProduct.bra" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the bra part of the state</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.InnerProduct.ket">
-<tt class="descname">ket</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#InnerProduct.ket"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.InnerProduct.ket" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the ket part of the state</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.secondquant.BosonicBasis">
-<em class="property">class </em><tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">BosonicBasis</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#BosonicBasis"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.BosonicBasis" title="Permalink to this definition">¶</a></dt>
-<dd><p>Base class for a basis set of bosonic Fock states.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.secondquant.VarBosonicBasis">
-<em class="property">class </em><tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">VarBosonicBasis</tt><big>(</big><em>n_max</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#VarBosonicBasis"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.VarBosonicBasis" title="Permalink to this definition">¶</a></dt>
-<dd><p>A single state, variable particle number basis set.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">VarBosonicBasis</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">VarBosonicBasis</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span>
-<span class="go">[FockState((0,)), FockState((1,)), FockState((2,)),</span>
-<span class="go"> FockState((3,)), FockState((4,))]</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.physics.secondquant.VarBosonicBasis.index">
-<tt class="descname">index</tt><big>(</big><em>self</em>, <em>state</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#VarBosonicBasis.index"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.VarBosonicBasis.index" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the index of state in basis.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">VarBosonicBasis</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">VarBosonicBasis</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">state</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">state</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span>
-<span class="go">[FockState((0,)), FockState((1,)), FockState((2,))]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">state</span>
-<span class="go">FockStateBosonKet((1,))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.secondquant.VarBosonicBasis.state">
-<tt class="descname">state</tt><big>(</big><em>self</em>, <em>i</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#VarBosonicBasis.state"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.VarBosonicBasis.state" title="Permalink to this definition">¶</a></dt>
-<dd><p>The state of a single basis.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">VarBosonicBasis</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">VarBosonicBasis</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">state</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">FockStateBosonKet((3,))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.secondquant.FixedBosonicBasis">
-<em class="property">class </em><tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">FixedBosonicBasis</tt><big>(</big><em>n_particles</em>, <em>n_levels</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#FixedBosonicBasis"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.FixedBosonicBasis" title="Permalink to this definition">¶</a></dt>
-<dd><p>Fixed particle number basis set.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">FixedBosonicBasis</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">FixedBosonicBasis</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">state</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">state</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span>
-<span class="go">[FockState((2, 0)), FockState((1, 1)), FockState((0, 2))]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">state</span>
-<span class="go">FockStateBosonKet((1, 1))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.physics.secondquant.FixedBosonicBasis.index">
-<tt class="descname">index</tt><big>(</big><em>self</em>, <em>state</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#FixedBosonicBasis.index"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.FixedBosonicBasis.index" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the index of state in basis.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">FixedBosonicBasis</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">FixedBosonicBasis</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">state</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.secondquant.FixedBosonicBasis.state">
-<tt class="descname">state</tt><big>(</big><em>self</em>, <em>i</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#FixedBosonicBasis.state"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.FixedBosonicBasis.state" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the state that lies at index i of the basis</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">FixedBosonicBasis</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">FixedBosonicBasis</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span><span class="o">.</span><span class="n">state</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">FockStateBosonKet((1, 0, 1))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.secondquant.Commutator">
-<em class="property">class </em><tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">Commutator</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#Commutator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.Commutator" title="Permalink to this definition">¶</a></dt>
-<dd><p>The Commutator:  [A, B] = A*B - B*A</p>
-<p>The arguments are ordered according to .__cmp__()</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">Commutator</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="p">,</span> <span class="n">B</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;A,B&#39;</span><span class="p">,</span> <span class="n">commutative</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Commutator</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="n">A</span><span class="p">)</span>
-<span class="go">-Commutator(A, B)</span>
-</pre></div>
-</div>
-<p>Evaluate the commutator with .doit()</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">comm</span> <span class="o">=</span> <span class="n">Commutator</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="n">B</span><span class="p">);</span> <span class="n">comm</span>
-<span class="go">Commutator(A, B)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">comm</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-<span class="go">A*B - B*A</span>
-</pre></div>
-</div>
-<p>For two second quantization operators the commutator is evaluated
-immediately:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">Fd</span><span class="p">,</span> <span class="n">F</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">,</span><span class="n">q</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;p,q&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Commutator</span><span class="p">(</span><span class="n">Fd</span><span class="p">(</span><span class="n">a</span><span class="p">),</span><span class="n">Fd</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>
-<span class="go">2*NO(CreateFermion(a)*CreateFermion(i))</span>
-</pre></div>
-</div>
-<p>But for more complicated expressions, the evaluation is triggered by
-a call to .doit()</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">comm</span> <span class="o">=</span> <span class="n">Commutator</span><span class="p">(</span><span class="n">Fd</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="n">Fd</span><span class="p">(</span><span class="n">q</span><span class="p">),</span><span class="n">F</span><span class="p">(</span><span class="n">i</span><span class="p">));</span> <span class="n">comm</span>
-<span class="go">Commutator(CreateFermion(p)*CreateFermion(q), AnnihilateFermion(i))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">comm</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="n">wicks</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">-KroneckerDelta(p, i)*CreateFermion(q) +</span>
-<span class="go"> KroneckerDelta(q, i)*CreateFermion(p)</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.physics.secondquant.Commutator.doit">
-<tt class="descname">doit</tt><big>(</big><em>self</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#Commutator.doit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.Commutator.doit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Enables the computation of complex expressions.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">Commutator</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">Fd</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i,j&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a,b&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">Commutator</span><span class="p">(</span><span class="n">Fd</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">i</span><span class="p">),</span><span class="n">Fd</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">j</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span><span class="o">.</span><span class="n">doit</span><span class="p">(</span><span class="n">wicks</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.physics.secondquant.Commutator.eval">
-<em class="property">classmethod </em><tt class="descname">eval</tt><big>(</big><em>a</em>, <em>b</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#Commutator.eval"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.Commutator.eval" title="Permalink to this definition">¶</a></dt>
-<dd><p>The Commutator [A,B] is on canonical form if A &lt; B.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">Commutator</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">Fd</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c1</span> <span class="o">=</span> <span class="n">Commutator</span><span class="p">(</span><span class="n">F</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">Fd</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c2</span> <span class="o">=</span> <span class="n">Commutator</span><span class="p">(</span><span class="n">Fd</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Commutator</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="n">c1</span><span class="p">,</span> <span class="n">c2</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.secondquant.matrix_rep">
-<tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">matrix_rep</tt><big>(</big><em>op</em>, <em>basis</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#matrix_rep"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.matrix_rep" title="Permalink to this definition">¶</a></dt>
-<dd><p>Find the representation of an operator in a basis.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">VarBosonicBasis</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">matrix_rep</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">VarBosonicBasis</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">o</span> <span class="o">=</span> <span class="n">B</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">matrix_rep</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-<span class="go">[0, 1,       0,       0, 0]</span>
-<span class="go">[0, 0, sqrt(2),       0, 0]</span>
-<span class="go">[0, 0,       0, sqrt(3), 0]</span>
-<span class="go">[0, 0,       0,       0, 2]</span>
-<span class="go">[0, 0,       0,       0, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.secondquant.contraction">
-<tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">contraction</tt><big>(</big><em>a</em>, <em>b</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#contraction"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.contraction" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculates contraction of Fermionic operators a and b.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">F</span><span class="p">,</span> <span class="n">Fd</span><span class="p">,</span> <span class="n">contraction</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;p,q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a,b&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i,j&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>A contraction is non-zero only if a quasi-creator is to the right of a
-quasi-annihilator:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">contraction</span><span class="p">(</span><span class="n">F</span><span class="p">(</span><span class="n">a</span><span class="p">),</span><span class="n">Fd</span><span class="p">(</span><span class="n">b</span><span class="p">))</span>
-<span class="go">KroneckerDelta(a, b)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">contraction</span><span class="p">(</span><span class="n">Fd</span><span class="p">(</span><span class="n">i</span><span class="p">),</span><span class="n">F</span><span class="p">(</span><span class="n">j</span><span class="p">))</span>
-<span class="go">KroneckerDelta(i, j)</span>
-</pre></div>
-</div>
-<p>For general indices a non-zero result restricts the indices to below/above
-the fermi surface:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">contraction</span><span class="p">(</span><span class="n">Fd</span><span class="p">(</span><span class="n">p</span><span class="p">),</span><span class="n">F</span><span class="p">(</span><span class="n">q</span><span class="p">))</span>
-<span class="go">KroneckerDelta(p, q)*KroneckerDelta(q, _i)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">contraction</span><span class="p">(</span><span class="n">F</span><span class="p">(</span><span class="n">p</span><span class="p">),</span><span class="n">Fd</span><span class="p">(</span><span class="n">q</span><span class="p">))</span>
-<span class="go">KroneckerDelta(p, q)*KroneckerDelta(q, _a)</span>
-</pre></div>
-</div>
-<p>Two creators or two annihilators always vanishes:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">contraction</span><span class="p">(</span><span class="n">F</span><span class="p">(</span><span class="n">p</span><span class="p">),</span><span class="n">F</span><span class="p">(</span><span class="n">q</span><span class="p">))</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">contraction</span><span class="p">(</span><span class="n">Fd</span><span class="p">(</span><span class="n">p</span><span class="p">),</span><span class="n">Fd</span><span class="p">(</span><span class="n">q</span><span class="p">))</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.secondquant.wicks">
-<tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">wicks</tt><big>(</big><em>e</em>, <em>**kw_args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#wicks"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.wicks" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the normal ordered equivalent of an expression using Wicks Theorem.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Dummy</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">wicks</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">Fd</span><span class="p">,</span> <span class="n">NO</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">,</span><span class="n">q</span><span class="p">,</span><span class="n">r</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;p,q,r&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">wicks</span><span class="p">(</span><span class="n">Fd</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">q</span><span class="p">))</span>  
-<span class="go">d(p, q)*d(q, _i) + NO(CreateFermion(p)*AnnihilateFermion(q))</span>
-</pre></div>
-</div>
-<p>By default, the expression is expanded:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">wicks</span><span class="p">(</span><span class="n">F</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">F</span><span class="p">(</span><span class="n">q</span><span class="p">)</span><span class="o">+</span><span class="n">F</span><span class="p">(</span><span class="n">r</span><span class="p">)))</span> 
-<span class="go">NO(AnnihilateFermion(p)*AnnihilateFermion(q)) + NO(</span>
-<span class="go">    AnnihilateFermion(p)*AnnihilateFermion(r))</span>
-</pre></div>
-</div>
-<p>With the keyword &#8216;keep_only_fully_contracted=True&#8217;, only fully contracted
-terms are returned.</p>
-<dl class="docutils">
-<dt>By request, the result can be simplified in the following order:</dt>
-<dd>&#8211; KroneckerDelta functions are evaluated
-&#8211; Dummy variables are substituted consistently across terms</dd>
-</dl>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;p q r&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Dummy</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">wicks</span><span class="p">(</span><span class="n">Fd</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">F</span><span class="p">(</span><span class="n">q</span><span class="p">)</span><span class="o">+</span><span class="n">F</span><span class="p">(</span><span class="n">r</span><span class="p">)),</span> <span class="n">keep_only_fully_contracted</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> 
-<span class="go">KroneckerDelta(_i, _q)*KroneckerDelta(</span>
-<span class="go">    _p, _q) + KroneckerDelta(_i, _r)*KroneckerDelta(_p, _r)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.secondquant.NO">
-<em class="property">class </em><tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">NO</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#NO"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.NO" title="Permalink to this definition">¶</a></dt>
-<dd><p>This Object is used to represent normal ordering brackets.</p>
-<p>i.e.  {abcd}  sometimes written  :abcd:</p>
-<p>Applying the function NO(arg) to an argument means that all operators in
-the argument will be assumed to anticommute, and have vanishing
-contractions.  This allows an immediate reordering to canonical form
-upon object creation.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">NO</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">Fd</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">,</span><span class="n">q</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;p,q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">NO</span><span class="p">(</span><span class="n">Fd</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">q</span><span class="p">))</span>
-<span class="go">NO(CreateFermion(p)*AnnihilateFermion(q))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">NO</span><span class="p">(</span><span class="n">F</span><span class="p">(</span><span class="n">q</span><span class="p">)</span><span class="o">*</span><span class="n">Fd</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
-<span class="go">-NO(CreateFermion(p)*AnnihilateFermion(q))</span>
-</pre></div>
-</div>
-<p>Note:
-If you want to generate a normal ordered equivalent of an expression, you
-should use the function wicks().  This class only indicates that all
-operators inside the brackets anticommute, and have vanishing contractions.
-Nothing more, nothing less.</p>
-<dl class="function">
-<dt id="sympy.physics.secondquant.NO.doit">
-<tt class="descname">doit</tt><big>(</big><em>self</em>, <em>**kw_args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#NO.doit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.NO.doit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Either removes the brackets or enables complex computations
-in its arguments.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">NO</span><span class="p">,</span> <span class="n">Fd</span><span class="p">,</span> <span class="n">F</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">textwrap</span> <span class="kn">import</span> <span class="n">fill</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">Dummy</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">,</span><span class="n">q</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;p,q&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Dummy</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">fill</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">NO</span><span class="p">(</span><span class="n">Fd</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">q</span><span class="p">))</span><span class="o">.</span><span class="n">doit</span><span class="p">())))</span>
-<span class="go">KroneckerDelta(_a, _p)*KroneckerDelta(_a,</span>
-<span class="go">_q)*CreateFermion(_a)*AnnihilateFermion(_a) + KroneckerDelta(_a,</span>
-<span class="go">_p)*KroneckerDelta(_i, _q)*CreateFermion(_a)*AnnihilateFermion(_i) -</span>
-<span class="go">KroneckerDelta(_a, _q)*KroneckerDelta(_i,</span>
-<span class="go">_p)*AnnihilateFermion(_a)*CreateFermion(_i) - KroneckerDelta(_i,</span>
-<span class="go">_p)*KroneckerDelta(_i, _q)*AnnihilateFermion(_i)*CreateFermion(_i)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.secondquant.NO.get_subNO">
-<tt class="descname">get_subNO</tt><big>(</big><em>self</em>, <em>i</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#NO.get_subNO"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.NO.get_subNO" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a NO() without FermionicOperator at index i.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">F</span><span class="p">,</span> <span class="n">NO</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">,</span><span class="n">q</span><span class="p">,</span><span class="n">r</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;p,q,r&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">NO</span><span class="p">(</span><span class="n">F</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">q</span><span class="p">)</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">r</span><span class="p">))</span><span class="o">.</span><span class="n">get_subNO</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>  
-<span class="go">NO(AnnihilateFermion(p)*AnnihilateFermion(r))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.NO.has_q_annihilators">
-<tt class="descname">has_q_annihilators</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#NO.has_q_annihilators"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.NO.has_q_annihilators" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return 0 if the rightmost argument of the first argument is a not a
-q_annihilator, else 1 if it is above fermi or -1 if it is below fermi.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">NO</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">Fd</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">NO</span><span class="p">(</span><span class="n">Fd</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">Fd</span><span class="p">(</span><span class="n">i</span><span class="p">))</span><span class="o">.</span><span class="n">has_q_annihilators</span>
-<span class="go">-1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">NO</span><span class="p">(</span><span class="n">F</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">a</span><span class="p">))</span><span class="o">.</span><span class="n">has_q_annihilators</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">NO</span><span class="p">(</span><span class="n">Fd</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">i</span><span class="p">))</span><span class="o">.</span><span class="n">has_q_annihilators</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.NO.has_q_creators">
-<tt class="descname">has_q_creators</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#NO.has_q_creators"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.NO.has_q_creators" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return 0 if the leftmost argument of the first argument is a not a
-q_creator, else 1 if it is above fermi or -1 if it is below fermi.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">NO</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">Fd</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">NO</span><span class="p">(</span><span class="n">Fd</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">Fd</span><span class="p">(</span><span class="n">i</span><span class="p">))</span><span class="o">.</span><span class="n">has_q_creators</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">NO</span><span class="p">(</span><span class="n">F</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">a</span><span class="p">))</span><span class="o">.</span><span class="n">has_q_creators</span>
-<span class="go">-1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">NO</span><span class="p">(</span><span class="n">Fd</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">a</span><span class="p">))</span><span class="o">.</span><span class="n">has_q_creators</span>           
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.secondquant.NO.iter_q_annihilators">
-<tt class="descname">iter_q_annihilators</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#NO.iter_q_annihilators"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.NO.iter_q_annihilators" title="Permalink to this definition">¶</a></dt>
-<dd><p>Iterates over the annihilation operators.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i j&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a b&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">NO</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">Fd</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">no</span> <span class="o">=</span> <span class="n">NO</span><span class="p">(</span><span class="n">Fd</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">Fd</span><span class="p">(</span><span class="n">j</span><span class="p">))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">no</span><span class="o">.</span><span class="n">iter_q_creators</span><span class="p">()</span>
-<span class="go">&lt;generator object... at 0x...&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">no</span><span class="o">.</span><span class="n">iter_q_creators</span><span class="p">())</span>
-<span class="go">[0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">no</span><span class="o">.</span><span class="n">iter_q_annihilators</span><span class="p">())</span>
-<span class="go">[3, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.secondquant.NO.iter_q_creators">
-<tt class="descname">iter_q_creators</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#NO.iter_q_creators"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.NO.iter_q_creators" title="Permalink to this definition">¶</a></dt>
-<dd><p>Iterates over the creation operators.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i j&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a b&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">NO</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="n">Fd</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">no</span> <span class="o">=</span> <span class="n">NO</span><span class="p">(</span><span class="n">Fd</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">*</span><span class="n">F</span><span class="p">(</span><span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">Fd</span><span class="p">(</span><span class="n">j</span><span class="p">))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">no</span><span class="o">.</span><span class="n">iter_q_creators</span><span class="p">()</span>
-<span class="go">&lt;generator object... at 0x...&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">no</span><span class="o">.</span><span class="n">iter_q_creators</span><span class="p">())</span>
-<span class="go">[0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">no</span><span class="o">.</span><span class="n">iter_q_annihilators</span><span class="p">())</span>
-<span class="go">[3, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.secondquant.evaluate_deltas">
-<tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">evaluate_deltas</tt><big>(</big><em>e</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#evaluate_deltas"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.evaluate_deltas" title="Permalink to this definition">¶</a></dt>
-<dd><p>We evaluate KroneckerDelta symbols in the expression assuming Einstein summation.</p>
-<p>If one index is repeated it is summed over and in effect substituted with
-the other one. If both indices are repeated we substitute according to what
-is the preferred index.  this is determined by
-KroneckerDelta.preferred_index and KroneckerDelta.killable_index.</p>
-<p>In case there are no possible substitutions or if a substitution would
-imply a loss of information, nothing is done.</p>
-<p>In case an index appears in more than one KroneckerDelta, the resulting
-substitution depends on the order of the factors.  Since the ordering is platform
-dependent, the literal expression resulting from this function may be hard to
-predict.</p>
-<p class="rubric">Examples</p>
-<p>We assume the following:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Dummy</span><span class="p">,</span> <span class="n">KroneckerDelta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">evaluate_deltas</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span><span class="n">j</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i j&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Dummy</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span><span class="n">b</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a b&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Dummy</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">,</span><span class="n">q</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;p q&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Dummy</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">t</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;t&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>The order of preference for these indices according to KroneckerDelta is
-(a, b, i, j, p, q).</p>
-<p>Trivial cases:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">evaluate_deltas</span><span class="p">(</span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>       <span class="c"># d_ij f(i) -&gt; f(j)</span>
-<span class="go">f(_j)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">evaluate_deltas</span><span class="p">(</span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">j</span><span class="p">))</span>       <span class="c"># d_ij f(j) -&gt; f(i)</span>
-<span class="go">f(_i)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">evaluate_deltas</span><span class="p">(</span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>       <span class="c"># d_ip f(p) -&gt; f(i)</span>
-<span class="go">f(_i)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">evaluate_deltas</span><span class="p">(</span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">q</span><span class="p">,</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>       <span class="c"># d_qp f(p) -&gt; f(q)</span>
-<span class="go">f(_q)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">evaluate_deltas</span><span class="p">(</span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">q</span><span class="p">,</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">q</span><span class="p">))</span>       <span class="c"># d_qp f(q) -&gt; f(p)</span>
-<span class="go">f(_p)</span>
-</pre></div>
-</div>
-<p>More interesting cases:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">evaluate_deltas</span><span class="p">(</span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="n">t</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">i</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">p</span><span class="p">,</span><span class="n">q</span><span class="p">))</span>
-<span class="go">f(_i, _q)*t(_a, _i)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">evaluate_deltas</span><span class="p">(</span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="n">t</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">i</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">p</span><span class="p">,</span><span class="n">q</span><span class="p">))</span>
-<span class="go">f(_a, _q)*t(_a, _i)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">evaluate_deltas</span><span class="p">(</span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span><span class="n">q</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">p</span><span class="p">,</span><span class="n">q</span><span class="p">))</span>
-<span class="go">f(_p, _p)</span>
-</pre></div>
-</div>
-<p>Finally, here are some cases where nothing is done, because that would
-imply a loss of information:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">evaluate_deltas</span><span class="p">(</span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">q</span><span class="p">))</span>
-<span class="go">f(_q)*KroneckerDelta(_i, _p)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">evaluate_deltas</span><span class="p">(</span><span class="n">KroneckerDelta</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>
-<span class="go">f(_i)*KroneckerDelta(_i, _p)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.secondquant.AntiSymmetricTensor">
-<em class="property">class </em><tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">AntiSymmetricTensor</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#AntiSymmetricTensor"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.AntiSymmetricTensor" title="Permalink to this definition">¶</a></dt>
-<dd><p>Stores upper and lower indices in separate Tuple&#8217;s.</p>
-<p>Each group of indices is assumed to be antisymmetric.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">AntiSymmetricTensor</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i j&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a b&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">AntiSymmetricTensor</span><span class="p">(</span><span class="s">&#39;v&#39;</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">i</span><span class="p">),</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span>
-<span class="go">AntiSymmetricTensor(v, (a, i), (b, j))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">AntiSymmetricTensor</span><span class="p">(</span><span class="s">&#39;v&#39;</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">a</span><span class="p">),</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span>
-<span class="go">-AntiSymmetricTensor(v, (a, i), (b, j))</span>
-</pre></div>
-</div>
-<p>As you can see, the indices are automatically sorted to a canonical form.</p>
-<dl class="function">
-<dt id="sympy.physics.secondquant.AntiSymmetricTensor.doit">
-<tt class="descname">doit</tt><big>(</big><em>self</em>, <em>**kw_args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#AntiSymmetricTensor.doit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.AntiSymmetricTensor.doit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns self.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">AntiSymmetricTensor</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i,j&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a,b&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">AntiSymmetricTensor</span><span class="p">(</span><span class="s">&#39;v&#39;</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">i</span><span class="p">),</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-<span class="go">AntiSymmetricTensor(v, (a, i), (b, j))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.AntiSymmetricTensor.lower">
-<tt class="descname">lower</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#AntiSymmetricTensor.lower"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.AntiSymmetricTensor.lower" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the lower indices.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">AntiSymmetricTensor</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i,j&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a,b&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">AntiSymmetricTensor</span><span class="p">(</span><span class="s">&#39;v&#39;</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">i</span><span class="p">),</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span>
-<span class="go">AntiSymmetricTensor(v, (a, i), (b, j))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">AntiSymmetricTensor</span><span class="p">(</span><span class="s">&#39;v&#39;</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">i</span><span class="p">),</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span><span class="o">.</span><span class="n">lower</span>
-<span class="go">(b, j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.AntiSymmetricTensor.symbol">
-<tt class="descname">symbol</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#AntiSymmetricTensor.symbol"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.AntiSymmetricTensor.symbol" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the symbol of the tensor.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">AntiSymmetricTensor</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i,j&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a,b&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">AntiSymmetricTensor</span><span class="p">(</span><span class="s">&#39;v&#39;</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">i</span><span class="p">),</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span>
-<span class="go">AntiSymmetricTensor(v, (a, i), (b, j))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">AntiSymmetricTensor</span><span class="p">(</span><span class="s">&#39;v&#39;</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">i</span><span class="p">),</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span><span class="o">.</span><span class="n">symbol</span>
-<span class="go">v</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.physics.secondquant.AntiSymmetricTensor.upper">
-<tt class="descname">upper</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#AntiSymmetricTensor.upper"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.AntiSymmetricTensor.upper" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the upper indices.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">AntiSymmetricTensor</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i,j&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a,b&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">AntiSymmetricTensor</span><span class="p">(</span><span class="s">&#39;v&#39;</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">i</span><span class="p">),</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span>
-<span class="go">AntiSymmetricTensor(v, (a, i), (b, j))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">AntiSymmetricTensor</span><span class="p">(</span><span class="s">&#39;v&#39;</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">i</span><span class="p">),</span> <span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">j</span><span class="p">))</span><span class="o">.</span><span class="n">upper</span>
-<span class="go">(a, i)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.secondquant.substitute_dummies">
-<tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">substitute_dummies</tt><big>(</big><em>expr</em>, <em>new_indices=False</em>, <em>pretty_indices={}</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#substitute_dummies"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.substitute_dummies" title="Permalink to this definition">¶</a></dt>
-<dd><p>Collect terms by substitution of dummy variables.</p>
-<p>This routine allows simplification of Add expressions containing terms
-which differ only due to dummy variables.</p>
-<p>The idea is to substitute all dummy variables consistently depending on
-the structure of the term.  For each term, we obtain a sequence of all
-dummy variables, where the order is determined by the index range, what
-factors the index belongs to and its position in each factor.  See
-_get_ordered_dummies() for more inforation about the sorting of dummies.
-The index sequence is then substituted consistently in each term.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Dummy</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">substitute_dummies</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">c</span><span class="p">,</span><span class="n">d</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a b c d&#39;</span><span class="p">,</span> <span class="n">above_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Dummy</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span><span class="n">j</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i j&#39;</span><span class="p">,</span> <span class="n">below_fermi</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Dummy</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expr</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">c</span><span class="p">,</span><span class="n">d</span><span class="p">);</span> <span class="n">expr</span>
-<span class="go">f(_a, _b) + f(_c, _d)</span>
-</pre></div>
-</div>
-<p>Since a, b, c and d are equivalent summation indices, the expression can be
-simplified to a single term (for which the dummy indices are still summed over)</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">substitute_dummies</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-<span class="go">2*f(_a, _b)</span>
-</pre></div>
-</div>
-<p>Controlling output:</p>
-<p>By default the dummy symbols that are already present in the expression
-will be reused in a different permuation.  However, if new_indices=True,
-new dummies will be generated and inserted.  The keyword &#8216;pretty_indices&#8217;
-can be used to control this generation of new symbols.</p>
-<p>By default the new dummies will be generated on the form i_1, i_2, a_1,
-etc.  If you supply a dictionary with key:value pairs in the form:</p>
-<blockquote>
-<div>{ index_group: string_of_letters }</div></blockquote>
-<p>The letters will be used as labels for the new dummy symbols.  The
-index_groups must be one of &#8216;above&#8217;, &#8216;below&#8217; or &#8216;general&#8217;.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expr</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">my_dummies</span> <span class="o">=</span> <span class="p">{</span> <span class="s">&#39;above&#39;</span><span class="p">:</span><span class="s">&#39;st&#39;</span><span class="p">,</span> <span class="s">&#39;below&#39;</span><span class="p">:</span><span class="s">&#39;uv&#39;</span> <span class="p">}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">substitute_dummies</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">new_indices</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">pretty_indices</span><span class="o">=</span><span class="n">my_dummies</span><span class="p">)</span>
-<span class="go">f(_s, _t, _u, _v)</span>
-</pre></div>
-</div>
-<p>If we run out of letters, or if there is no keyword for some index_group
-the default dummy generator will be used as a fallback:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">,</span><span class="n">q</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;p q&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Dummy</span><span class="p">)</span>  <span class="c"># general indices</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expr</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">p</span><span class="p">,</span><span class="n">q</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">substitute_dummies</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">new_indices</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">pretty_indices</span><span class="o">=</span><span class="n">my_dummies</span><span class="p">)</span>
-<span class="go">f(_p_0, _p_1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.physics.secondquant.PermutationOperator">
-<em class="property">class </em><tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">PermutationOperator</tt><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#PermutationOperator"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.PermutationOperator" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents the index permutation operator P(ij).</p>
-<p>P(ij)*f(i)*g(j) = f(i)*g(j) - f(j)*g(i)</p>
-<dl class="function">
-<dt id="sympy.physics.secondquant.PermutationOperator.get_permuted">
-<tt class="descname">get_permuted</tt><big>(</big><em>self</em>, <em>expr</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#PermutationOperator.get_permuted"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.PermutationOperator.get_permuted" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns -expr with permuted indices.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">Function</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">PermutationOperator</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">,</span><span class="n">q</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;p,q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">PermutationOperator</span><span class="p">(</span><span class="n">p</span><span class="p">,</span><span class="n">q</span><span class="p">)</span><span class="o">.</span><span class="n">get_permuted</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">p</span><span class="p">,</span><span class="n">q</span><span class="p">))</span>
-<span class="go">-f(q, p)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.secondquant.simplify_index_permutations">
-<tt class="descclassname">sympy.physics.secondquant.</tt><tt class="descname">simplify_index_permutations</tt><big>(</big><em>expr</em>, <em>permutation_operators</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/secondquant.html#simplify_index_permutations"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.secondquant.simplify_index_permutations" title="Permalink to this definition">¶</a></dt>
-<dd><p>Performs simplification by introducing PermutationOperators where appropriate.</p>
-<dl class="docutils">
-<dt>Schematically:</dt>
-<dd>[abij] - [abji] - [baij] + [baji] -&gt;  P(ab)*P(ij)*[abij]</dd>
-</dl>
-<p>permutation_operators is a list of PermutationOperators to consider.</p>
-<p>If permutation_operators=[P(ab),P(ij)] we will try to introduce the
-permutation operators P(ij) and P(ab) in the expression.  If there are other
-possible simplifications, we ignore them.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">Function</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">simplify_index_permutations</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.secondquant</span> <span class="kn">import</span> <span class="n">PermutationOperator</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">,</span><span class="n">q</span><span class="p">,</span><span class="n">r</span><span class="p">,</span><span class="n">s</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;p,q,r,s&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;g&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expr</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">p</span><span class="p">)</span><span class="o">*</span><span class="n">g</span><span class="p">(</span><span class="n">q</span><span class="p">)</span> <span class="o">-</span> <span class="n">f</span><span class="p">(</span><span class="n">q</span><span class="p">)</span><span class="o">*</span><span class="n">g</span><span class="p">(</span><span class="n">p</span><span class="p">);</span> <span class="n">expr</span>
-<span class="go">f(p)*g(q) - f(q)*g(p)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">simplify_index_permutations</span><span class="p">(</span><span class="n">expr</span><span class="p">,[</span><span class="n">PermutationOperator</span><span class="p">(</span><span class="n">p</span><span class="p">,</span><span class="n">q</span><span class="p">)])</span>
-<span class="go">f(p)*g(q)*PermutationOperator(p, q)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">PermutList</span> <span class="o">=</span> <span class="p">[</span><span class="n">PermutationOperator</span><span class="p">(</span><span class="n">p</span><span class="p">,</span><span class="n">q</span><span class="p">),</span><span class="n">PermutationOperator</span><span class="p">(</span><span class="n">r</span><span class="p">,</span><span class="n">s</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expr</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">p</span><span class="p">,</span><span class="n">r</span><span class="p">)</span><span class="o">*</span><span class="n">g</span><span class="p">(</span><span class="n">q</span><span class="p">,</span><span class="n">s</span><span class="p">)</span> <span class="o">-</span> <span class="n">f</span><span class="p">(</span><span class="n">q</span><span class="p">,</span><span class="n">r</span><span class="p">)</span><span class="o">*</span><span class="n">g</span><span class="p">(</span><span class="n">p</span><span class="p">,</span><span class="n">s</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">q</span><span class="p">,</span><span class="n">s</span><span class="p">)</span><span class="o">*</span><span class="n">g</span><span class="p">(</span><span class="n">p</span><span class="p">,</span><span class="n">r</span><span class="p">)</span> <span class="o">-</span> <span class="n">f</span><span class="p">(</span><span class="n">p</span><span class="p">,</span><span class="n">s</span><span class="p">)</span><span class="o">*</span><span class="n">g</span><span class="p">(</span><span class="n">q</span><span class="p">,</span><span class="n">r</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">simplify_index_permutations</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span><span class="n">PermutList</span><span class="p">)</span>
-<span class="go">f(p, r)*g(q, s)*PermutationOperator(p, q)*PermutationOperator(r, s)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-
-
-          </div>
-        </div>
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-            <p class="logo"><a href="../../index.html">
-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../physics/sho.html"
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-                        title="next chapter">Wigner Symbols</a></p>
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-  <div class="section" id="module-sympy.physics.sho">
-<span id="quantum-harmonic-oscillator-in-3-d"></span><h1>Quantum Harmonic Oscillator in 3-D<a class="headerlink" href="#module-sympy.physics.sho" title="Permalink to this headline">¶</a></h1>
-<dl class="function">
-<dt id="sympy.physics.sho.E_nl">
-<tt class="descclassname">sympy.physics.sho.</tt><tt class="descname">E_nl</tt><big>(</big><em>n</em>, <em>l</em>, <em>hw</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/sho.html#E_nl"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.sho.E_nl" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the Energy of an isotropic harmonic oscillator</p>
-<dl class="docutils">
-<dt><tt class="docutils literal"><span class="pre">n</span></tt></dt>
-<dd>the &#8220;nodal&#8221; quantum number</dd>
-<dt><tt class="docutils literal"><span class="pre">l</span></tt></dt>
-<dd>the orbital angular momentum</dd>
-<dt><tt class="docutils literal"><span class="pre">hw</span></tt></dt>
-<dd>the harmonic oscillator parameter.</dd>
-</dl>
-<p>The unit of the returned value matches the unit of hw, since the energy is
-calculated as:</p>
-<blockquote>
-<div>E_nl = (2*n + l + 3/2)*hw</div></blockquote>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.sho</span> <span class="kn">import</span> <span class="n">E_nl</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x, y, z&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E_nl</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">z*(2*x + y + 3/2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.sho.R_nl">
-<tt class="descclassname">sympy.physics.sho.</tt><tt class="descname">R_nl</tt><big>(</big><em>n</em>, <em>l</em>, <em>nu</em>, <em>r</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/sho.html#R_nl"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.sho.R_nl" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the radial wavefunction R_{nl} for a 3d isotropic harmonic
-oscillator.</p>
-<dl class="docutils">
-<dt><tt class="docutils literal"><span class="pre">n</span></tt></dt>
-<dd>the &#8220;nodal&#8221; quantum number.  Corresponds to the number of nodes in
-the wavefunction.  n &gt;= 0</dd>
-<dt><tt class="docutils literal"><span class="pre">l</span></tt></dt>
-<dd>the quantum number for orbital angular momentum</dd>
-<dt><tt class="docutils literal"><span class="pre">nu</span></tt></dt>
-<dd>mass-scaled frequency: nu = m*omega/(2*hbar) where <span class="math">\(m' is the mass
-and `omega\)</span> the frequency of the oscillator.
-(in atomic units nu == omega/2)</dd>
-<dt><tt class="docutils literal"><span class="pre">r</span></tt></dt>
-<dd>Radial coordinate</dd>
-</dl>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.sho</span> <span class="kn">import</span> <span class="n">R_nl</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">var</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&quot;r nu l&quot;</span><span class="p">)</span>
-<span class="go">(r, nu, l)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R_nl</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span>
-<span class="go">2*2**(3/4)*exp(-r**2)/pi**(1/4)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R_nl</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span>
-<span class="go">4*2**(1/4)*sqrt(3)*(-2*r**2 + 3/2)*exp(-r**2)/(3*pi**(1/4))</span>
-</pre></div>
-</div>
-<p>l, nu and r may be symbolic:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">R_nl</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">nu</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span>
-<span class="go">2*2**(3/4)*sqrt(nu**(3/2))*exp(-nu*r**2)/pi**(1/4)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R_nl</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span>
-<span class="go">r**l*sqrt(2**(l + 3/2)*2**(l + 2)/(2*l + 1)!!)*exp(-r**2)/pi**(1/4)</span>
-</pre></div>
-</div>
-<p>The normalization of the radial wavefunction is:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">R_nl</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span><span class="o">.</span><span class="n">n</span><span class="p">()</span>
-<span class="go">1.00000000000000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">R_nl</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span><span class="o">.</span><span class="n">n</span><span class="p">()</span>
-<span class="go">1.00000000000000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Integral</span><span class="p">(</span><span class="n">R_nl</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span><span class="o">.</span><span class="n">n</span><span class="p">()</span>
-<span class="go">1.00000000000000</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
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-            
-  <div class="section" id="units">
-<span id="physics-units"></span><h1>Units<a class="headerlink" href="#units" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="introduction">
-<h2>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline">¶</a></h2>
-<p>This module provides around 200 predefined units that are commonly used in the
-sciences.  Additionally, it provides the <tt class="xref py py-class docutils literal"><span class="pre">Unit</span></tt> class which allows you
-to define your own units.</p>
-</div>
-<div class="section" id="examples">
-<h2>Examples<a class="headerlink" href="#examples" title="Permalink to this headline">¶</a></h2>
-<p>All examples in this tutorial are computable, so one can just copy and
-paste them into a Python shell and do something useful with them. All
-computations were done using the following setup:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.units</span> <span class="kn">import</span> <span class="o">*</span>
-</pre></div>
-</div>
-<div class="section" id="dimensionless-quantities">
-<h3>Dimensionless quantities<a class="headerlink" href="#dimensionless-quantities" title="Permalink to this headline">¶</a></h3>
-<p>Provides variables for commonly written dimensionless (unit-less) quantities.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span><span class="o">*</span><span class="n">ten</span>
-<span class="go">20</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">20</span><span class="o">*</span><span class="n">percent</span>
-<span class="go">1/5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">300</span><span class="o">*</span><span class="n">kilo</span><span class="o">*</span><span class="mi">20</span><span class="o">*</span><span class="n">percent</span>
-<span class="go">60000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nano</span><span class="o">*</span><span class="n">deg</span>
-<span class="go">pi/180000000000</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="base-units">
-<h3>Base units<a class="headerlink" href="#base-units" title="Permalink to this headline">¶</a></h3>
-<p>The SI base units are defined variable name that are commonly used in written
-and verbal communication.  The singular abbreviated versions are what is used
-for display purposes, but the plural non-abbreviated versions are defined in
-case it helps readability.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">5</span><span class="o">*</span><span class="n">meters</span>
-<span class="go">5*m</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">milli</span><span class="o">*</span><span class="n">kilogram</span>
-<span class="go">kg/1000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gram</span>
-<span class="go">kg/1000</span>
-</pre></div>
-</div>
-<p>Note that British Imperial and U.S. customary units are not included.
-We strongly urge the use of SI units; only Myanmar (Burma), Liberia, and the
-United States have not officially accepted the SI system.</p>
-</div>
-<div class="section" id="derived-units">
-<h3>Derived units<a class="headerlink" href="#derived-units" title="Permalink to this headline">¶</a></h3>
-<p>Common SI derived units.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">joule</span>
-<span class="go">kg*m**2/s**2</span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="section" id="module-sympy.physics.units">
-<span id="docstring"></span><h2>Docstring<a class="headerlink" href="#module-sympy.physics.units" title="Permalink to this headline">¶</a></h2>
-<p>Physical units and dimensions.</p>
-<p>The base class is Unit, where all here defined units (~200) inherit from.</p>
-<p>The find_unit function can help you find units for a given quantity:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">sympy.physics.units</span> <span class="kn">as</span> <span class="nn">u</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="o">.</span><span class="n">find_unit</span><span class="p">(</span><span class="s">&#39;coul&#39;</span><span class="p">)</span>
-<span class="go">[&#39;coulomb&#39;, &#39;coulombs&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="o">.</span><span class="n">find_unit</span><span class="p">(</span><span class="n">u</span><span class="o">.</span><span class="n">charge</span><span class="p">)</span>
-<span class="go">[&#39;C&#39;, &#39;charge&#39;, &#39;coulomb&#39;, &#39;coulombs&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="o">.</span><span class="n">coulomb</span>
-<span class="go">A*s</span>
-</pre></div>
-</div>
-<p>Units are always given in terms of base units that have a name and
-an abbreviation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="o">.</span><span class="n">A</span><span class="o">.</span><span class="n">name</span>
-<span class="go">&#39;ampere&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="o">.</span><span class="n">ampere</span><span class="o">.</span><span class="n">abbrev</span>
-<span class="go">&#39;A&#39;</span>
-</pre></div>
-</div>
-<p>The generic name for a unit (like &#8216;length&#8217;, &#8216;mass&#8217;, etc...)
-can help you find units:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="o">.</span><span class="n">find_unit</span><span class="p">(</span><span class="s">&#39;magnet&#39;</span><span class="p">)</span>
-<span class="go">[&#39;magnetic_flux&#39;, &#39;magnetic_constant&#39;, &#39;magnetic_flux_density&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="o">.</span><span class="n">find_unit</span><span class="p">(</span><span class="n">u</span><span class="o">.</span><span class="n">magnetic_flux</span><span class="p">)</span>
-<span class="go">[&#39;Wb&#39;, &#39;wb&#39;, &#39;weber&#39;, &#39;webers&#39;, &#39;magnetic_flux&#39;]</span>
-</pre></div>
-</div>
-<p>If, for a given session, you wish to add a unit you may do so:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="o">.</span><span class="n">find_unit</span><span class="p">(</span><span class="s">&#39;gal&#39;</span><span class="p">)</span>
-<span class="go">[]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="o">.</span><span class="n">gal</span> <span class="o">=</span> <span class="mi">4</span><span class="o">*</span><span class="n">u</span><span class="o">.</span><span class="n">quart</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="o">.</span><span class="n">gal</span><span class="o">/</span><span class="n">u</span><span class="o">.</span><span class="n">inch</span><span class="o">**</span><span class="mi">3</span>
-<span class="go">924</span>
-</pre></div>
-</div>
-<p>To see a given quantity in terms of some other unit, divide by the desired
-unit:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mph</span> <span class="o">=</span> <span class="n">u</span><span class="o">.</span><span class="n">miles</span><span class="o">/</span><span class="n">u</span><span class="o">.</span><span class="n">hours</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">u</span><span class="o">.</span><span class="n">m</span><span class="o">/</span><span class="n">u</span><span class="o">.</span><span class="n">s</span><span class="o">/</span><span class="n">mph</span><span class="p">)</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">2.2</span>
-</pre></div>
-</div>
-<p>The units are defined in terms of base units, so when you divide similar
-units you will obtain a pure number. This means, for example, that if you
-divide a real-world mass (like grams) by the atomic mass unit (amu) you
-will obtain Avogadro&#8217;s number. To obtain the answer in moles you
-should divide by the unit <tt class="docutils literal"><span class="pre">avogadro</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="o">.</span><span class="n">grams</span><span class="o">/</span><span class="n">u</span><span class="o">.</span><span class="n">amu</span>
-<span class="go">602214179000000000000000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">/</span><span class="n">u</span><span class="o">.</span><span class="n">avogadro</span>
-<span class="go">mol</span>
-</pre></div>
-</div>
-<p>For chemical calculations the unit <tt class="docutils literal"><span class="pre">mmu</span></tt> (molar mass unit) has been
-defined so this conversion is handled automatically. For example, the
-number of moles in 1 kg of water might be calculated as:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="o">.</span><span class="n">kg</span><span class="o">/</span><span class="p">(</span><span class="mi">18</span><span class="o">*</span><span class="n">u</span><span class="o">.</span><span class="n">mmu</span><span class="p">)</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">55.5*mol</span>
-</pre></div>
-</div>
-<p>If you need the number of atoms in a mol as a pure number you can use
-<tt class="docutils literal"><span class="pre">avogadro_number</span></tt> but if you need it as a dimensional quantity you should use
-<tt class="docutils literal"><span class="pre">avogadro_constant</span></tt>. (<tt class="docutils literal"><span class="pre">avogadro</span></tt> is a shorthand for the dimensional
-quantity.)</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="o">.</span><span class="n">avogadro_number</span>
-<span class="go">602214179000000000000000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="o">.</span><span class="n">avogadro_constant</span>
-<span class="go">602214179000000000000000/mol</span>
-</pre></div>
-</div>
-<dl class="class">
-<dt id="sympy.physics.units.Unit">
-<em class="property">class </em><tt class="descclassname">sympy.physics.units.</tt><tt class="descname">Unit</tt><a class="reference internal" href="../../_modules/sympy/physics/units.html#Unit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.units.Unit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Base class for base unit of physical units.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.units</span> <span class="kn">import</span> <span class="n">Unit</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Unit</span><span class="p">(</span><span class="s">&quot;meter&quot;</span><span class="p">,</span> <span class="s">&quot;m&quot;</span><span class="p">)</span>
-<span class="go">m</span>
-</pre></div>
-</div>
-<p>Other units are derived from base units:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">sympy.physics.units</span> <span class="kn">as</span> <span class="nn">u</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cm</span> <span class="o">=</span> <span class="n">u</span><span class="o">.</span><span class="n">m</span><span class="o">/</span><span class="mi">100</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">100</span><span class="o">*</span><span class="n">u</span><span class="o">.</span><span class="n">cm</span>
-<span class="go">m</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.units.find_unit">
-<tt class="descclassname">sympy.physics.units.</tt><tt class="descname">find_unit</tt><big>(</big><em>quantity</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/units.html#find_unit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.units.find_unit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a list of matching units names.
-if quantity is a string &#8211; units containing the string <span class="math">\(quantity\)</span>
-if quantity is a unit &#8211; units having matching base units</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics</span> <span class="kn">import</span> <span class="n">units</span> <span class="k">as</span> <span class="n">u</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="o">.</span><span class="n">find_unit</span><span class="p">(</span><span class="s">&#39;charge&#39;</span><span class="p">)</span>
-<span class="go">[&#39;charge&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="o">.</span><span class="n">find_unit</span><span class="p">(</span><span class="n">u</span><span class="o">.</span><span class="n">charge</span><span class="p">)</span>
-<span class="go">[&#39;C&#39;, &#39;charge&#39;, &#39;coulomb&#39;, &#39;coulombs&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="o">.</span><span class="n">find_unit</span><span class="p">(</span><span class="s">&#39;volt&#39;</span><span class="p">)</span>
-<span class="go">[&#39;volt&#39;, &#39;volts&#39;, &#39;voltage&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="o">.</span><span class="n">find_unit</span><span class="p">(</span><span class="n">u</span><span class="o">.</span><span class="n">inch</span><span class="o">**</span><span class="mi">3</span><span class="p">)[:</span><span class="mi">5</span><span class="p">]</span>
-<span class="go">[&#39;l&#39;, &#39;cl&#39;, &#39;dl&#39;, &#39;ml&#39;, &#39;liter&#39;]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
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-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Units</a><ul>
-<li><a class="reference internal" href="#introduction">Introduction</a></li>
-<li><a class="reference internal" href="#examples">Examples</a><ul>
-<li><a class="reference internal" href="#dimensionless-quantities">Dimensionless quantities</a></li>
-<li><a class="reference internal" href="#base-units">Base units</a></li>
-<li><a class="reference internal" href="#derived-units">Derived units</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.physics.units">Docstring</a></li>
-</ul>
-</li>
-</ul>
-
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-                        title="previous chapter">Wigner Symbols</a></p>
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-            
-  <div class="section" id="module-sympy.physics.wigner">
-<span id="wigner-symbols"></span><h1>Wigner Symbols<a class="headerlink" href="#module-sympy.physics.wigner" title="Permalink to this headline">¶</a></h1>
-<p>Wigner, Clebsch-Gordan, Racah, and Gaunt coefficients</p>
-<p>Collection of functions for calculating Wigner 3j, 6j, 9j,
-Clebsch-Gordan, Racah as well as Gaunt coefficients exactly, all
-evaluating to a rational number times the square root of a rational
-number <a class="reference internal" href="#rasch03">[Rasch03]</a>.</p>
-<p>Please see the description of the individual functions for further
-details and examples.</p>
-<p>REFERENCES:</p>
-<table class="docutils citation" frame="void" id="rasch03" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Rasch03]</td><td><em>(<a class="fn-backref" href="#id1">1</a>, <a class="fn-backref" href="#id5">2</a>, <a class="fn-backref" href="#id7">3</a>, <a class="fn-backref" href="#id10">4</a>, <a class="fn-backref" href="#id13">5</a>, <a class="fn-backref" href="#id15">6</a>)</em> J. Rasch and A. C. H. Yu, &#8216;Efficient Storage Scheme for
-Pre-calculated Wigner 3j, 6j and Gaunt Coefficients&#8217;, SIAM
-J. Sci. Comput. Volume 25, Issue 4, pp. 1416-1428 (2003)</td></tr>
-</tbody>
-</table>
-<p>This code was taken from Sage with the permission of all authors:</p>
-<p><a class="reference external" href="http://groups.google.com/group/sage-devel/browse_thread/thread/33835976efbb3b7f">http://groups.google.com/group/sage-devel/browse_thread/thread/33835976efbb3b7f</a></p>
-<p>AUTHORS:</p>
-<ul class="simple">
-<li>Jens Rasch (2009-03-24): initial version for Sage</li>
-<li>Jens Rasch (2009-05-31): updated to sage-4.0</li>
-</ul>
-<p>Copyright (C) 2008 Jens Rasch &lt;<a class="reference external" href="mailto:jyr2000&#37;&#52;&#48;gmail&#46;com">jyr2000<span>&#64;</span>gmail<span>&#46;</span>com</a>&gt;</p>
-<dl class="function">
-<dt id="sympy.physics.wigner.clebsch_gordan">
-<tt class="descclassname">sympy.physics.wigner.</tt><tt class="descname">clebsch_gordan</tt><big>(</big><em>j_1</em>, <em>j_2</em>, <em>j_3</em>, <em>m_1</em>, <em>m_2</em>, <em>m_3</em>, <em>prec=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/wigner.html#clebsch_gordan"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.wigner.clebsch_gordan" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculates the Clebsch-Gordan coefficient
-<span class="math">\(\langle j_1 m_1 \; j_2 m_2 | j_3 m_3 \rangle\)</span>.</p>
-<p>The reference for this function is <a class="reference internal" href="#edmonds74">[Edmonds74]</a>.</p>
-<p>INPUT:</p>
-<ul class="simple">
-<li><tt class="docutils literal"><span class="pre">j_1</span></tt>, <tt class="docutils literal"><span class="pre">j_2</span></tt>, <tt class="docutils literal"><span class="pre">j_3</span></tt>, <tt class="docutils literal"><span class="pre">m_1</span></tt>, <tt class="docutils literal"><span class="pre">m_2</span></tt>, <tt class="docutils literal"><span class="pre">m_3</span></tt> - integer or half integer</li>
-<li><tt class="docutils literal"><span class="pre">prec</span></tt> - precision, default: <tt class="docutils literal"><span class="pre">None</span></tt>. Providing a precision can
-drastically speed up the calculation.</li>
-</ul>
-<p>OUTPUT:</p>
-<p>Rational number times the square root of a rational number
-(if <tt class="docutils literal"><span class="pre">prec=None</span></tt>), or real number if a precision is given.</p>
-<p>EXAMPLES:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.wigner</span> <span class="kn">import</span> <span class="n">clebsch_gordan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">clebsch_gordan</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">clebsch_gordan</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">sqrt(3)/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">clebsch_gordan</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">-sqrt(2)/2</span>
-</pre></div>
-</div>
-<p>NOTES:</p>
-<p>The Clebsch-Gordan coefficient will be evaluated via its relation
-to Wigner 3j symbols:</p>
-<div class="math">
-\[\langle j_1 m_1 \; j_2 m_2 | j_3 m_3 \rangle
-=(-1)^{j_1-j_2+m_3} \sqrt{2j_3+1} \;
-Wigner3j(j_1,j_2,j_3,m_1,m_2,-m_3)\]</div>
-<p>See also the documentation on Wigner 3j symbols which exhibit much
-higher symmetry relations than the Clebsch-Gordan coefficient.</p>
-<p>AUTHORS:</p>
-<ul class="simple">
-<li>Jens Rasch (2009-03-24): initial version</li>
-</ul>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.wigner.gaunt">
-<tt class="descclassname">sympy.physics.wigner.</tt><tt class="descname">gaunt</tt><big>(</big><em>l_1</em>, <em>l_2</em>, <em>l_3</em>, <em>m_1</em>, <em>m_2</em>, <em>m_3</em>, <em>prec=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/wigner.html#gaunt"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.wigner.gaunt" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculate the Gaunt coefficient.</p>
-<p>The Gaunt coefficient is defined as the integral over three
-spherical harmonics:</p>
-<div class="math">
-\[Y(j_1,j_2,j_3,m_1,m_2,m_3)
-=\int Y_{l_1,m_1}(\Omega)
- Y_{l_2,m_2}(\Omega) Y_{l_3,m_3}(\Omega) d\Omega
-=\sqrt{(2l_1+1)(2l_2+1)(2l_3+1)/(4\pi)}
- \; Y(j_1,j_2,j_3,0,0,0) \; Y(j_1,j_2,j_3,m_1,m_2,m_3)\]</div>
-<p>INPUT:</p>
-<ul class="simple">
-<li><tt class="docutils literal"><span class="pre">l_1</span></tt>, <tt class="docutils literal"><span class="pre">l_2</span></tt>, <tt class="docutils literal"><span class="pre">l_3</span></tt>, <tt class="docutils literal"><span class="pre">m_1</span></tt>, <tt class="docutils literal"><span class="pre">m_2</span></tt>, <tt class="docutils literal"><span class="pre">m_3</span></tt> - integer</li>
-<li><tt class="docutils literal"><span class="pre">prec</span></tt> - precision, default: <tt class="docutils literal"><span class="pre">None</span></tt>. Providing a precision can
-drastically speed up the calculation.</li>
-</ul>
-<p>OUTPUT:</p>
-<p>Rational number times the square root of a rational number
-(if <tt class="docutils literal"><span class="pre">prec=None</span></tt>), or real number if a precision is given.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.wigner</span> <span class="kn">import</span> <span class="n">gaunt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gaunt</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">-1/(2*sqrt(pi))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gaunt</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span><span class="mi">1000</span><span class="p">,</span><span class="mi">1200</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="o">-</span><span class="mi">12</span><span class="p">)</span><span class="o">.</span><span class="n">n</span><span class="p">(</span><span class="mi">64</span><span class="p">)</span>
-<span class="go">0.00689500421922113448...</span>
-</pre></div>
-</div>
-<p>It is an error to use non-integer values for <span class="math">\(l\)</span> and <span class="math">\(m\)</span>:</p>
-<div class="highlight-python"><pre>sage: gaunt(1.2,0,1.2,0,0,0)
-Traceback (most recent call last):
-...
-ValueError: l values must be integer
-sage: gaunt(1,0,1,1.1,0,-1.1)
-Traceback (most recent call last):
-...
-ValueError: m values must be integer</pre>
-</div>
-<p>NOTES:</p>
-<p>The Gaunt coefficient obeys the following symmetry rules:</p>
-<ul>
-<li><p class="first">invariant under any permutation of the columns</p>
-<div class="math">
-\[Y(j_1,j_2,j_3,m_1,m_2,m_3)
-=Y(j_3,j_1,j_2,m_3,m_1,m_2)
-=Y(j_2,j_3,j_1,m_2,m_3,m_1)
-=Y(j_3,j_2,j_1,m_3,m_2,m_1)
-=Y(j_1,j_3,j_2,m_1,m_3,m_2)
-=Y(j_2,j_1,j_3,m_2,m_1,m_3)\]</div>
-</li>
-<li><p class="first">invariant under space inflection, i.e.</p>
-<div class="math">
-\[Y(j_1,j_2,j_3,m_1,m_2,m_3)
-=Y(j_1,j_2,j_3,-m_1,-m_2,-m_3)\]</div>
-</li>
-<li><p class="first">symmetric with respect to the 72 Regge symmetries as inherited
-for the <span class="math">\(3j\)</span> symbols <a class="reference internal" href="#regge58">[Regge58]</a></p>
-</li>
-<li><p class="first">zero for <span class="math">\(l_1\)</span>, <span class="math">\(l_2\)</span>, <span class="math">\(l_3\)</span> not fulfilling triangle relation</p>
-</li>
-<li><p class="first">zero for violating any one of the conditions: <span class="math">\(l_1 \ge |m_1|\)</span>,
-<span class="math">\(l_2 \ge |m_2|\)</span>, <span class="math">\(l_3 \ge |m_3|\)</span></p>
-</li>
-<li><p class="first">non-zero only for an even sum of the <span class="math">\(l_i\)</span>, i.e.
-<span class="math">\(J = l_1 + l_2 + l_3 = 2n\)</span> for <span class="math">\(n\)</span> in <span class="math">\(\mathbb{N}\)</span></p>
-</li>
-</ul>
-<p>ALGORITHM:</p>
-<p>This function uses the algorithm of <a class="reference internal" href="#liberatodebrito82">[Liberatodebrito82]</a> to
-calculate the value of the Gaunt coefficient exactly. Note that
-the formula contains alternating sums over large factorials and is
-therefore unsuitable for finite precision arithmetic and only
-useful for a computer algebra system <a class="reference internal" href="#rasch03">[Rasch03]</a>.</p>
-<p>REFERENCES:</p>
-<table class="docutils citation" frame="void" id="liberatodebrito82" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id4">[Liberatodebrito82]</a></td><td>&#8216;FORTRAN program for the integral of three
-spherical harmonics&#8217;, A. Liberato de Brito,
-Comput. Phys. Commun., Volume 25, pp. 81-85 (1982)</td></tr>
-</tbody>
-</table>
-<p>AUTHORS:</p>
-<ul class="simple">
-<li>Jens Rasch (2009-03-24): initial version for Sage</li>
-</ul>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.wigner.racah">
-<tt class="descclassname">sympy.physics.wigner.</tt><tt class="descname">racah</tt><big>(</big><em>aa</em>, <em>bb</em>, <em>cc</em>, <em>dd</em>, <em>ee</em>, <em>ff</em>, <em>prec=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/wigner.html#racah"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.wigner.racah" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculate the Racah symbol <span class="math">\(W(a,b,c,d;e,f)\)</span>.</p>
-<p>INPUT:</p>
-<ul class="simple">
-<li><tt class="docutils literal"><span class="pre">a</span></tt>, ..., <tt class="docutils literal"><span class="pre">f</span></tt> - integer or half integer</li>
-<li><tt class="docutils literal"><span class="pre">prec</span></tt> - precision, default: <tt class="docutils literal"><span class="pre">None</span></tt>. Providing a precision can
-drastically speed up the calculation.</li>
-</ul>
-<p>OUTPUT:</p>
-<p>Rational number times the square root of a rational number
-(if <tt class="docutils literal"><span class="pre">prec=None</span></tt>), or real number if a precision is given.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.wigner</span> <span class="kn">import</span> <span class="n">racah</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">racah</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">-1/14</span>
-</pre></div>
-</div>
-<p>NOTES:</p>
-<p>The Racah symbol is related to the Wigner 6j symbol:</p>
-<div class="math">
-\[Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6)
-=(-1)^{j_1+j_2+j_4+j_5} W(j_1,j_2,j_5,j_4,j_3,j_6)\]</div>
-<p>Please see the 6j symbol for its much richer symmetries and for
-additional properties.</p>
-<p>ALGORITHM:</p>
-<p>This function uses the algorithm of <a class="reference internal" href="#edmonds74">[Edmonds74]</a> to calculate the
-value of the 6j symbol exactly. Note that the formula contains
-alternating sums over large factorials and is therefore unsuitable
-for finite precision arithmetic and only useful for a computer
-algebra system <a class="reference internal" href="#rasch03">[Rasch03]</a>.</p>
-<p>AUTHORS:</p>
-<ul class="simple">
-<li>Jens Rasch (2009-03-24): initial version</li>
-</ul>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.wigner.wigner_3j">
-<tt class="descclassname">sympy.physics.wigner.</tt><tt class="descname">wigner_3j</tt><big>(</big><em>j_1</em>, <em>j_2</em>, <em>j_3</em>, <em>m_1</em>, <em>m_2</em>, <em>m_3</em>, <em>prec=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/wigner.html#wigner_3j"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.wigner.wigner_3j" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculate the Wigner 3j symbol <span class="math">\(Wigner3j(j_1,j_2,j_3,m_1,m_2,m_3)\)</span>.</p>
-<p>INPUT:</p>
-<ul class="simple">
-<li><tt class="docutils literal"><span class="pre">j_1</span></tt>, <tt class="docutils literal"><span class="pre">j_2</span></tt>, <tt class="docutils literal"><span class="pre">j_3</span></tt>, <tt class="docutils literal"><span class="pre">m_1</span></tt>, <tt class="docutils literal"><span class="pre">m_2</span></tt>, <tt class="docutils literal"><span class="pre">m_3</span></tt> - integer or half integer</li>
-<li><tt class="docutils literal"><span class="pre">prec</span></tt> - precision, default: <tt class="docutils literal"><span class="pre">None</span></tt>. Providing a precision can
-drastically speed up the calculation.</li>
-</ul>
-<p>OUTPUT:</p>
-<p>Rational number times the square root of a rational number
-(if <tt class="docutils literal"><span class="pre">prec=None</span></tt>), or real number if a precision is given.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.wigner</span> <span class="kn">import</span> <span class="n">wigner_3j</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">wigner_3j</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">sqrt(715)/143</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">wigner_3j</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>It is an error to have arguments that are not integer or half
-integer values:</p>
-<div class="highlight-python"><pre>sage: wigner_3j(2.1, 6, 4, 0, 0, 0)
-Traceback (most recent call last):
-...
-ValueError: j values must be integer or half integer
-sage: wigner_3j(2, 6, 4, 1, 0, -1.1)
-Traceback (most recent call last):
-...
-ValueError: m values must be integer or half integer</pre>
-</div>
-<p>NOTES:</p>
-<p>The Wigner 3j symbol obeys the following symmetry rules:</p>
-<ul>
-<li><p class="first">invariant under any permutation of the columns (with the
-exception of a sign change where <span class="math">\(J:=j_1+j_2+j_3\)</span>):</p>
-<div class="math">
-\[Wigner3j(j_1,j_2,j_3,m_1,m_2,m_3)
- =Wigner3j(j_3,j_1,j_2,m_3,m_1,m_2)
- =Wigner3j(j_2,j_3,j_1,m_2,m_3,m_1)
- =(-1)^J Wigner3j(j_3,j_2,j_1,m_3,m_2,m_1)
- =(-1)^J Wigner3j(j_1,j_3,j_2,m_1,m_3,m_2)
- =(-1)^J Wigner3j(j_2,j_1,j_3,m_2,m_1,m_3)\]</div>
-</li>
-<li><p class="first">invariant under space inflection, i.e.</p>
-<div class="math">
-\[Wigner3j(j_1,j_2,j_3,m_1,m_2,m_3)
-=(-1)^J Wigner3j(j_1,j_2,j_3,-m_1,-m_2,-m_3)\]</div>
-</li>
-<li><p class="first">symmetric with respect to the 72 additional symmetries based on
-the work by <a class="reference internal" href="#regge58">[Regge58]</a></p>
-</li>
-<li><p class="first">zero for <span class="math">\(j_1\)</span>, <span class="math">\(j_2\)</span>, <span class="math">\(j_3\)</span> not fulfilling triangle relation</p>
-</li>
-<li><p class="first">zero for <span class="math">\(m_1 + m_2 + m_3 \neq 0\)</span></p>
-</li>
-<li><p class="first">zero for violating any one of the conditions
-<span class="math">\(j_1 \ge |m_1|\)</span>,  <span class="math">\(j_2 \ge |m_2|\)</span>,  <span class="math">\(j_3 \ge |m_3|\)</span></p>
-</li>
-</ul>
-<p>ALGORITHM:</p>
-<p>This function uses the algorithm of <a class="reference internal" href="#edmonds74">[Edmonds74]</a> to calculate the
-value of the 3j symbol exactly. Note that the formula contains
-alternating sums over large factorials and is therefore unsuitable
-for finite precision arithmetic and only useful for a computer
-algebra system <a class="reference internal" href="#rasch03">[Rasch03]</a>.</p>
-<p>REFERENCES:</p>
-<table class="docutils citation" frame="void" id="regge58" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Regge58]</td><td><em>(<a class="fn-backref" href="#id3">1</a>, <a class="fn-backref" href="#id8">2</a>)</em> &#8216;Symmetry Properties of Clebsch-Gordan Coefficients&#8217;,
-T. Regge, Nuovo Cimento, Volume 10, pp. 544 (1958)</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="edmonds74" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Edmonds74]</td><td><em>(<a class="fn-backref" href="#id2">1</a>, <a class="fn-backref" href="#id6">2</a>, <a class="fn-backref" href="#id9">3</a>, <a class="fn-backref" href="#id12">4</a>, <a class="fn-backref" href="#id14">5</a>)</em> &#8216;Angular Momentum in Quantum Mechanics&#8217;,
-A. R. Edmonds, Princeton University Press (1974)</td></tr>
-</tbody>
-</table>
-<p>AUTHORS:</p>
-<ul class="simple">
-<li>Jens Rasch (2009-03-24): initial version</li>
-</ul>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.wigner.wigner_6j">
-<tt class="descclassname">sympy.physics.wigner.</tt><tt class="descname">wigner_6j</tt><big>(</big><em>j_1</em>, <em>j_2</em>, <em>j_3</em>, <em>j_4</em>, <em>j_5</em>, <em>j_6</em>, <em>prec=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/wigner.html#wigner_6j"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.wigner.wigner_6j" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculate the Wigner 6j symbol <span class="math">\(Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6)\)</span>.</p>
-<p>INPUT:</p>
-<ul class="simple">
-<li><tt class="docutils literal"><span class="pre">j_1</span></tt>, ..., <tt class="docutils literal"><span class="pre">j_6</span></tt> - integer or half integer</li>
-<li><tt class="docutils literal"><span class="pre">prec</span></tt> - precision, default: <tt class="docutils literal"><span class="pre">None</span></tt>. Providing a precision can
-drastically speed up the calculation.</li>
-</ul>
-<p>OUTPUT:</p>
-<p>Rational number times the square root of a rational number
-(if <tt class="docutils literal"><span class="pre">prec=None</span></tt>), or real number if a precision is given.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.wigner</span> <span class="kn">import</span> <span class="n">wigner_6j</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">wigner_6j</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">-1/14</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">wigner_6j</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">1/52</span>
-</pre></div>
-</div>
-<p>It is an error to have arguments that are not integer or half
-integer values or do not fulfill the triangle relation:</p>
-<div class="highlight-python"><pre>sage: wigner_6j(2.5,2.5,2.5,2.5,2.5,2.5)
-Traceback (most recent call last):
-...
-ValueError: j values must be integer or half integer and fulfill the triangle relation
-sage: wigner_6j(0.5,0.5,1.1,0.5,0.5,1.1)
-Traceback (most recent call last):
-...
-ValueError: j values must be integer or half integer and fulfill the triangle relation</pre>
-</div>
-<p>NOTES:</p>
-<p>The Wigner 6j symbol is related to the Racah symbol but exhibits
-more symmetries as detailed below.</p>
-<div class="math">
-\[Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6)
- =(-1)^{j_1+j_2+j_4+j_5} W(j_1,j_2,j_5,j_4,j_3,j_6)\]</div>
-<p>The Wigner 6j symbol obeys the following symmetry rules:</p>
-<ul>
-<li><p class="first">Wigner 6j symbols are left invariant under any permutation of
-the columns:</p>
-<div class="math">
-\[Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6)
- =Wigner6j(j_3,j_1,j_2,j_6,j_4,j_5)
- =Wigner6j(j_2,j_3,j_1,j_5,j_6,j_4)
- =Wigner6j(j_3,j_2,j_1,j_6,j_5,j_4)
- =Wigner6j(j_1,j_3,j_2,j_4,j_6,j_5)
- =Wigner6j(j_2,j_1,j_3,j_5,j_4,j_6)\]</div>
-</li>
-<li><p class="first">They are invariant under the exchange of the upper and lower
-arguments in each of any two columns, i.e.</p>
-<div class="math">
-\[Wigner6j(j_1,j_2,j_3,j_4,j_5,j_6)
- =Wigner6j(j_1,j_5,j_6,j_4,j_2,j_3)
- =Wigner6j(j_4,j_2,j_6,j_1,j_5,j_3)
- =Wigner6j(j_4,j_5,j_3,j_1,j_2,j_6)\]</div>
-</li>
-<li><p class="first">additional 6 symmetries <a class="reference internal" href="#regge59">[Regge59]</a> giving rise to 144 symmetries
-in total</p>
-</li>
-<li><p class="first">only non-zero if any triple of <span class="math">\(j\)</span>&#8216;s fulfill a triangle relation</p>
-</li>
-</ul>
-<p>ALGORITHM:</p>
-<p>This function uses the algorithm of <a class="reference internal" href="#edmonds74">[Edmonds74]</a> to calculate the
-value of the 6j symbol exactly. Note that the formula contains
-alternating sums over large factorials and is therefore unsuitable
-for finite precision arithmetic and only useful for a computer
-algebra system <a class="reference internal" href="#rasch03">[Rasch03]</a>.</p>
-<p>REFERENCES:</p>
-<table class="docutils citation" frame="void" id="regge59" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id11">[Regge59]</a></td><td>&#8216;Symmetry Properties of Racah Coefficients&#8217;,
-T. Regge, Nuovo Cimento, Volume 11, pp. 116 (1959)</td></tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.physics.wigner.wigner_9j">
-<tt class="descclassname">sympy.physics.wigner.</tt><tt class="descname">wigner_9j</tt><big>(</big><em>j_1</em>, <em>j_2</em>, <em>j_3</em>, <em>j_4</em>, <em>j_5</em>, <em>j_6</em>, <em>j_7</em>, <em>j_8</em>, <em>j_9</em>, <em>prec=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/physics/wigner.html#wigner_9j"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.physics.wigner.wigner_9j" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculate the Wigner 9j symbol
-<span class="math">\(Wigner9j(j_1,j_2,j_3,j_4,j_5,j_6,j_7,j_8,j_9)\)</span>.</p>
-<p>INPUT:</p>
-<ul class="simple">
-<li><tt class="docutils literal"><span class="pre">j_1</span></tt>, ..., <tt class="docutils literal"><span class="pre">j_9</span></tt> - integer or half integer</li>
-<li><tt class="docutils literal"><span class="pre">prec</span></tt> - precision, default: <tt class="docutils literal"><span class="pre">None</span></tt>. Providing a precision can
-drastically speed up the calculation.</li>
-</ul>
-<p>OUTPUT:</p>
-<p>Rational number times the square root of a rational number
-(if <tt class="docutils literal"><span class="pre">prec=None</span></tt>), or real number if a precision is given.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.physics.wigner</span> <span class="kn">import</span> <span class="n">wigner_9j</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">wigner_9j</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span> <span class="p">,</span><span class="n">prec</span><span class="o">=</span><span class="mi">64</span><span class="p">)</span> <span class="c"># ==1/18</span>
-<span class="go">0.05555555...</span>
-</pre></div>
-</div>
-<p>It is an error to have arguments that are not integer or half
-integer values or do not fulfill the triangle relation:</p>
-<div class="highlight-python"><pre>sage: wigner_9j(0.5,0.5,0.5, 0.5,0.5,0.5, 0.5,0.5,0.5,prec=64)
-Traceback (most recent call last):
-...
-ValueError: j values must be integer or half integer and fulfill the triangle relation
-sage: wigner_9j(1,1,1, 0.5,1,1.5, 0.5,1,2.5,prec=64)
-Traceback (most recent call last):
-...
-ValueError: j values must be integer or half integer and fulfill the triangle relation</pre>
-</div>
-<p>ALGORITHM:</p>
-<p>This function uses the algorithm of <a class="reference internal" href="#edmonds74">[Edmonds74]</a> to calculate the
-value of the 3j symbol exactly. Note that the formula contains
-alternating sums over large factorials and is therefore unsuitable
-for finite precision arithmetic and only useful for a computer
-algebra system <a class="reference internal" href="#rasch03">[Rasch03]</a>.</p>
-</dd></dl>
-
-</div>
-
-
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-            
-  <div class="section" id="module-sympy.plotting.plot">
-<span id="plotting-module"></span><h1>Plotting Module<a class="headerlink" href="#module-sympy.plotting.plot" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="introduction">
-<h2>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline">¶</a></h2>
-<p>The plotting module allows you to plot 2-dimensional and 3-dimensional plots.
-Presently the plots are rendered using <tt class="docutils literal"><span class="pre">Matplotlib</span></tt> as its backend. It is
-also possible to plot 2-dimensional plots using a <tt class="docutils literal"><span class="pre">TextBackend</span></tt> if you don&#8217;t
-have <tt class="docutils literal"><span class="pre">Matplotlib</span></tt>.</p>
-<p>The plotting module has the following functions:</p>
-<ul class="simple">
-<li>plot: Plots 2D line plots.</li>
-<li>plot_parametric: Plots 2D parametric plots.</li>
-<li>plot_implicit: Plots 2D implicit and region plots.</li>
-<li>plot3d: Plots 3D plots of functions in two variables.</li>
-<li>plot3d_parametric_line: Plots 3D line plots, defined by a parameter.</li>
-<li>plot3d_parametric_surface: Plots 3D parametric surface plots.</li>
-</ul>
-<p>The above functions are only for convenience and ease of use. It is possible to
-plot any plot by passing the corresponding <tt class="docutils literal"><span class="pre">Series</span></tt> class to <tt class="docutils literal"><span class="pre">Plot</span></tt> as
-argument.</p>
-</div>
-<div class="section" id="plot-class">
-<h2>Plot Class<a class="headerlink" href="#plot-class" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.plotting.plot.Plot">
-<em class="property">class </em><tt class="descclassname">sympy.plotting.plot.</tt><tt class="descname">Plot</tt><big>(</big><em>*args</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../_modules/sympy/plotting/plot.html#Plot"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.plotting.plot.Plot" title="Permalink to this definition">¶</a></dt>
-<dd><p>The central class of the plotting module.</p>
-<p>For interactive work the function <tt class="docutils literal"><span class="pre">plot</span></tt> is better suited.</p>
-<p>This class permits the plotting of sympy expressions using numerous
-backends (matplotlib, textplot, the old pyglet module for sympy, Google
-charts api, etc).</p>
-<p>The figure can contain an arbitrary number of plots of sympy expressions,
-lists of coordinates of points, etc. Plot has a private attribute _series that
-contains all data series to be plotted (expressions for lines or surfaces,
-lists of points, etc (all subclasses of BaseSeries)). Those data series are
-instances of classes not imported by <tt class="docutils literal"><span class="pre">from</span> <span class="pre">sympy</span> <span class="pre">import</span> <span class="pre">*</span></tt>.</p>
-<p>The customization of the figure is on two levels. Global options that
-concern the figure as a whole (eg title, xlabel, scale, etc) and
-per-data series options (eg name) and aesthetics (eg. color, point shape,
-line type, etc.).</p>
-<p>The difference between options and aesthetics is that an aesthetic can be
-a function of the coordinates (or parameters in a parametric plot). The
-supported values for an aesthetic are:
-- None (the backend uses default values)
-- a constant
-- a function of one variable (the first coordinate or parameter)
-- a function of two variables (the first and second coordinate or
-parameters)
-- a function of three variables (only in nonparametric 3D plots)
-Their implementation depends on the backend so they may not work in some
-backends.</p>
-<p>If the plot is parametric and the arity of the aesthetic function permits
-it the aesthetic is calculated over parameters and not over coordinates.
-If the arity does not permit calculation over parameters the calculation is
-done over coordinates.</p>
-<p>Only cartesian coordinates are supported for the moment, but you can use
-the parametric plots to plot in polar, spherical and cylindrical
-coordinates.</p>
-<p>The arguments for the constructor Plot must be subclasses of BaseSeries.</p>
-<p>Any global option can be specified as a keyword argument.</p>
-<p>The global options for a figure are:</p>
-<ul class="simple">
-<li>title : str</li>
-<li>xlabel : str</li>
-<li>ylabel : str</li>
-<li>legend : bool</li>
-<li>xscale : {&#8216;linear&#8217;, &#8216;log&#8217;}</li>
-<li>yscale : {&#8216;linear&#8217;, &#8216;log&#8217;}</li>
-<li>axis : bool</li>
-<li>axis_center : tuple of two floats or {&#8216;center&#8217;, &#8216;auto&#8217;}</li>
-<li>xlim : tuple of two floats</li>
-<li>ylim : tuple of two floats</li>
-<li>aspect_ratio : tuple of two floats or {&#8216;auto&#8217;}</li>
-<li>autoscale : bool</li>
-<li>margin : float in [0, 1]</li>
-</ul>
-<p>The per data series options and aesthetics are:
-There are none in the base series. See below for options for subclasses.</p>
-<p>Some data series support additional aesthetics or options:</p>
-<p>ListSeries, LineOver1DRangeSeries, Parametric2DLineSeries,
-Parametric3DLineSeries support the following:</p>
-<p>Aesthetics:</p>
-<ul class="simple">
-<li>line_color : function which returns a float.</li>
-</ul>
-<p>options:</p>
-<ul class="simple">
-<li>label : str</li>
-<li>steps : bool</li>
-<li>integers_only : bool</li>
-</ul>
-<p>SurfaceOver2DRangeSeries, ParametricSurfaceSeries support the following:</p>
-<p>aesthetics:</p>
-<ul class="simple">
-<li>surface_color : function which returns a float.</li>
-</ul>
-<dl class="function">
-<dt id="sympy.plotting.plot.Plot.append">
-<tt class="descname">append</tt><big>(</big><em>self</em>, <em>*args</em><big>)</big><a class="reference internal" href="../_modules/sympy/plotting/plot.html#Plot.append"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.plotting.plot.Plot.append" title="Permalink to this definition">¶</a></dt>
-<dd><p>Adds one more graph to the figure.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.plotting.plot.Plot.extend">
-<tt class="descname">extend</tt><big>(</big><em>self</em>, <em>arg</em><big>)</big><a class="reference internal" href="../_modules/sympy/plotting/plot.html#Plot.extend"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.plotting.plot.Plot.extend" title="Permalink to this definition">¶</a></dt>
-<dd><p>Adds the series from another plot or a list of series.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="plotting-function-reference">
-<h2>Plotting Function Reference<a class="headerlink" href="#plotting-function-reference" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.plotting.plot.plot">
-<tt class="descclassname">sympy.plotting.plot.</tt><tt class="descname">plot</tt><big>(</big><em>*args</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../_modules/sympy/plotting/plot.html#plot"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.plotting.plot.plot" title="Permalink to this definition">¶</a></dt>
-<dd><p>Plots a function of a single variable.</p>
-<p>The plotting uses an adaptive algorithm which samples recursively to
-accurately plot the plot. The adaptive algorithm uses a random point near
-the midpoint of two points that has to be further sampled. Hence the same
-plots can appear slightly different.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.plotting.plot.Plot" title="sympy.plotting.plot.Plot"><tt class="xref py py-obj docutils literal"><span class="pre">Plot</span></tt></a>, <tt class="xref py py-obj docutils literal"><span class="pre">LineOver1DRangeSeries.</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.plotting</span> <span class="kn">import</span> <span class="n">plot</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Single Plot</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
-</pre></div>
-</div>
-<p>Multiple plots with single range.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
-</pre></div>
-</div>
-<p>Multiple plots with different ranges.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">plot</span><span class="p">((</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">6</span><span class="p">,</span> <span class="mi">6</span><span class="p">)),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">)))</span>
-</pre></div>
-</div>
-<p>No adaptive sampling.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">adaptive</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">nb_of_points</span><span class="o">=</span><span class="mi">400</span><span class="p">)</span>
-</pre></div>
-</div>
-<p class="rubric">Usage</p>
-<p>Single Plot</p>
-<p><tt class="docutils literal"><span class="pre">plot(expr,</span> <span class="pre">range,</span> <span class="pre">**kwargs)</span></tt></p>
-<p>If the range is not specified, then a default range of (-10, 10) is used.</p>
-<p>Multiple plots with same range.</p>
-<p><tt class="docutils literal"><span class="pre">plot(expr1,</span> <span class="pre">expr2,</span> <span class="pre">...,</span> <span class="pre">range,</span> <span class="pre">**kwargs)</span></tt></p>
-<p>If the range is not specified, then a default range of (-10, 10) is used.</p>
-<p>Multiple plots with different ranges.</p>
-<p><tt class="docutils literal"><span class="pre">plot((expr1,</span> <span class="pre">range),</span> <span class="pre">(expr2,</span> <span class="pre">range),</span> <span class="pre">...,</span> <span class="pre">**kwargs)</span></tt></p>
-<p>Range has to be specified for every expression.</p>
-<p>Default range may change in the future if a more advanced default range
-detection algorithm is implemented.</p>
-<p class="rubric">Arguments</p>
-<p><tt class="docutils literal"><span class="pre">expr</span></tt> : Expression representing the function of single variable</p>
-<p><tt class="docutils literal"><span class="pre">range</span></tt>: (x, 0, 5), A 3-tuple denoting the range of the free variable.</p>
-<p class="rubric">Keyword Arguments</p>
-<p>Arguments for <tt class="docutils literal"><span class="pre">LineOver1DRangeSeries</span></tt> class:</p>
-<p><tt class="docutils literal"><span class="pre">adaptive</span></tt>: Boolean. The default value is set to True. Set adaptive to False and
-specify <tt class="docutils literal"><span class="pre">nb_of_points</span></tt> if uniform sampling is required.</p>
-<p><tt class="docutils literal"><span class="pre">depth</span></tt>: int Recursion depth of the adaptive algorithm. A depth of value <tt class="docutils literal"><span class="pre">n</span></tt>
-samples a maximum of <span class="math">\(2^{n}\)</span> points.</p>
-<p><tt class="docutils literal"><span class="pre">nb_of_points</span></tt>: int. Used when the <tt class="docutils literal"><span class="pre">adaptive</span></tt> is set to False. The function
-is uniformly sampled at <tt class="docutils literal"><span class="pre">nb_of_points</span></tt> number of points.</p>
-<p>Aesthetics options:</p>
-<p><tt class="docutils literal"><span class="pre">line_color</span></tt>: float. Specifies the color for the plot.
-See <tt class="docutils literal"><span class="pre">Plot</span></tt> to see how to set color for the plots.</p>
-<p>If there are multiple plots, then the same series series are applied to
-all the plots. If you want to set these options separately, you can index
-the <tt class="docutils literal"><span class="pre">Plot</span></tt> object returned and set it.</p>
-<p>Arguments for <tt class="docutils literal"><span class="pre">Plot</span></tt> class:</p>
-<p><tt class="docutils literal"><span class="pre">title</span></tt> : str. Title of the plot. It is set to the latex representation of
-the expression, if the plot has only one expression.</p>
-<p><tt class="docutils literal"><span class="pre">xlabel</span></tt> : str. Label for the x - axis.</p>
-<p><tt class="docutils literal"><span class="pre">ylabel</span></tt> : str. Label for the y - axis.</p>
-<p><tt class="docutils literal"><span class="pre">xscale</span></tt>: {&#8216;linear&#8217;, &#8216;log&#8217;} Sets the scaling of the x - axis.</p>
-<p><tt class="docutils literal"><span class="pre">yscale</span></tt>: {&#8216;linear&#8217;, &#8216;log&#8217;} Sets the scaling if the y - axis.</p>
-<p><tt class="docutils literal"><span class="pre">axis_center</span></tt>: tuple of two floats denoting the coordinates of the center or
-{&#8216;center&#8217;, &#8216;auto&#8217;}</p>
-<p><tt class="docutils literal"><span class="pre">xlim</span></tt> : tuple of two floats, denoting the x - axis limits.</p>
-<p><tt class="docutils literal"><span class="pre">ylim</span></tt> : tuple of two floats, denoting the y - axis limits.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.plotting.plot.plot_parametric">
-<tt class="descclassname">sympy.plotting.plot.</tt><tt class="descname">plot_parametric</tt><big>(</big><em>*args</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../_modules/sympy/plotting/plot.html#plot_parametric"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.plotting.plot.plot_parametric" title="Permalink to this definition">¶</a></dt>
-<dd><p>Plots a 2D parametric plot.</p>
-<p>The plotting uses an adaptive algorithm which samples recursively to
-accurately plot the plot. The adaptive algorithm uses a random point near
-the midpoint of two points that has to be further sampled. Hence the same
-plots can appear slightly different.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.plotting.plot.Plot" title="sympy.plotting.plot.Plot"><tt class="xref py py-obj docutils literal"><span class="pre">Plot</span></tt></a>, <a class="reference internal" href="#sympy.plotting.plot.Parametric2DLineSeries" title="sympy.plotting.plot.Parametric2DLineSeries"><tt class="xref py py-obj docutils literal"><span class="pre">Parametric2DLineSeries</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.plotting</span> <span class="kn">import</span> <span class="n">plot_parametric</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;u&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Single Parametric plot</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">plot_parametric</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">sin</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
-</pre></div>
-</div>
-<p>Multiple parametric plot with single range.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">plot_parametric</span><span class="p">((</span><span class="n">cos</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">sin</span><span class="p">(</span><span class="n">u</span><span class="p">)),</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">cos</span><span class="p">(</span><span class="n">u</span><span class="p">)))</span>  
-</pre></div>
-</div>
-<p>Multiple parametric plots.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">plot_parametric</span><span class="p">((</span><span class="n">cos</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">sin</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">)),</span>
-<span class="gp">... </span>    <span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">)))</span>  
-</pre></div>
-</div>
-<p class="rubric">Usage</p>
-<p>Single plot.</p>
-<p><tt class="docutils literal"><span class="pre">plot_parametric(expr_x,</span> <span class="pre">expr_y,</span> <span class="pre">range,</span> <span class="pre">**kwargs)</span></tt></p>
-<p>If the range is not specified, then a default range of (-10, 10) is used.</p>
-<p>Multiple plots with same range.</p>
-<p><tt class="docutils literal"><span class="pre">plot_parametric((expr1_x,</span> <span class="pre">expr1_y),</span> <span class="pre">(expr2_x,</span> <span class="pre">expr2_y),</span> <span class="pre">range,</span> <span class="pre">**kwargs)</span></tt></p>
-<p>If the range is not specified, then a default range of (-10, 10) is used.</p>
-<p>Multiple plots with different ranges.</p>
-<p><tt class="docutils literal"><span class="pre">plot_parametric((expr_x,</span> <span class="pre">expr_y,</span> <span class="pre">range),</span> <span class="pre">...,</span> <span class="pre">**kwargs)</span></tt></p>
-<p>Range has to be specified for every expression.</p>
-<p>Default range may change in the future if a more advanced default range
-detection algorithm is implemented.</p>
-<p class="rubric">Arguments</p>
-<p><tt class="docutils literal"><span class="pre">expr_x</span></tt> : Expression representing the function along x.</p>
-<p><tt class="docutils literal"><span class="pre">expr_y</span></tt> : Expression representing the function along y.</p>
-<p><tt class="docutils literal"><span class="pre">range</span></tt>: (u, 0, 5), A 3-tuple denoting the range of the parameter
-variable.</p>
-<p class="rubric">Keyword Arguments</p>
-<p>Arguments for <tt class="docutils literal"><span class="pre">Parametric2DLineSeries</span></tt> class:</p>
-<p><tt class="docutils literal"><span class="pre">adaptive</span></tt>: Boolean. The default value is set to True. Set adaptive to
-False and specify <tt class="docutils literal"><span class="pre">nb_of_points</span></tt> if uniform sampling is required.</p>
-<p><tt class="docutils literal"><span class="pre">depth</span></tt>: int Recursion depth of the adaptive algorithm. A depth of
-value <tt class="docutils literal"><span class="pre">n</span></tt> samples a maximum of <span class="math">\(2^{n}\)</span> points.</p>
-<p><tt class="docutils literal"><span class="pre">nb_of_points</span></tt>: int. Used when the <tt class="docutils literal"><span class="pre">adaptive</span></tt> is set to False. The
-function is uniformly sampled at <tt class="docutils literal"><span class="pre">nb_of_points</span></tt> number of points.</p>
-<p class="rubric">Aesthetics</p>
-<p><tt class="docutils literal"><span class="pre">line_color</span></tt>: function which returns a float. Specifies the color for the
-plot. See <tt class="docutils literal"><span class="pre">sympy.plotting.Plot</span></tt> for more details.</p>
-<p>If there are multiple plots, then the same Series arguments are applied to
-all the plots. If you want to set these options separately, you can index
-the returned <tt class="docutils literal"><span class="pre">Plot</span></tt> object and set it.</p>
-<p>Arguments for <tt class="docutils literal"><span class="pre">Plot</span></tt> class:</p>
-<p><tt class="docutils literal"><span class="pre">xlabel</span></tt> : str. Label for the x - axis.</p>
-<p><tt class="docutils literal"><span class="pre">ylabel</span></tt> : str. Label for the y - axis.</p>
-<p><tt class="docutils literal"><span class="pre">xscale</span></tt>: {&#8216;linear&#8217;, &#8216;log&#8217;} Sets the scaling of the x - axis.</p>
-<p><tt class="docutils literal"><span class="pre">yscale</span></tt>: {&#8216;linear&#8217;, &#8216;log&#8217;} Sets the scaling if the y - axis.</p>
-<p><tt class="docutils literal"><span class="pre">axis_center</span></tt>: tuple of two floats denoting the coordinates of the center
-or {&#8216;center&#8217;, &#8216;auto&#8217;}</p>
-<p><tt class="docutils literal"><span class="pre">xlim</span></tt> : tuple of two floats, denoting the x - axis limits.</p>
-<p><tt class="docutils literal"><span class="pre">ylim</span></tt> : tuple of two floats, denoting the y - axis limits.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.plotting.plot.plot3d">
-<tt class="descclassname">sympy.plotting.plot.</tt><tt class="descname">plot3d</tt><big>(</big><em>*args</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../_modules/sympy/plotting/plot.html#plot3d"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.plotting.plot.plot3d" title="Permalink to this definition">¶</a></dt>
-<dd><p>Plots a 3D surface plot.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.plotting.plot.Plot" title="sympy.plotting.plot.Plot"><tt class="xref py py-obj docutils literal"><span class="pre">Plot</span></tt></a>, <a class="reference internal" href="#sympy.plotting.plot.SurfaceOver2DRangeSeries" title="sympy.plotting.plot.SurfaceOver2DRangeSeries"><tt class="xref py py-obj docutils literal"><span class="pre">SurfaceOver2DRangeSeries</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.plotting</span> <span class="kn">import</span> <span class="n">plot3d</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Single plot</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">plot3d</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>  
-</pre></div>
-</div>
-<p>Multiple plots with same range</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">plot3d</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="o">-</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>  
-</pre></div>
-</div>
-<p>Multiple plots with different ranges.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">plot3d</span><span class="p">((</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">)),</span>
-<span class="gp">... </span>    <span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)))</span>  
-</pre></div>
-</div>
-<p class="rubric">Usage</p>
-<p>Single plot</p>
-<p><tt class="docutils literal"><span class="pre">plot3d(expr,</span> <span class="pre">range_x,</span> <span class="pre">range_y,</span> <span class="pre">**kwargs)</span></tt></p>
-<p>If the ranges are not specified, then a default range of (-10, 10) is used.</p>
-<p>Multiple plot with the same range.</p>
-<p><tt class="docutils literal"><span class="pre">plot3d(expr1,</span> <span class="pre">expr2,</span> <span class="pre">range_x,</span> <span class="pre">range_y,</span> <span class="pre">**kwargs)</span></tt></p>
-<p>If the ranges are not specified, then a default range of (-10, 10) is used.</p>
-<p>Multiple plots with different ranges.</p>
-<p><tt class="docutils literal"><span class="pre">plot3d((expr1,</span> <span class="pre">range_x,</span> <span class="pre">range_y),</span> <span class="pre">(expr2,</span> <span class="pre">range_x,</span> <span class="pre">range_y),</span> <span class="pre">...,</span> <span class="pre">**kwargs)</span></tt></p>
-<p>Ranges have to be specified for every expression.</p>
-<p>Default range may change in the future if a more advanced default range
-detection algorithm is implemented.</p>
-<p class="rubric">Arguments</p>
-<p><tt class="docutils literal"><span class="pre">expr</span></tt> : Expression representing the function along x.</p>
-<p><tt class="docutils literal"><span class="pre">range_x</span></tt>: (x, 0, 5), A 3-tuple denoting the range of the x
-variable.</p>
-<dl class="docutils">
-<dt><tt class="docutils literal"><span class="pre">range_y</span></tt>: (y, 0, 5), A 3-tuple denoting the range of the y</dt>
-<dd>variable.</dd>
-</dl>
-<p class="rubric">Keyword Arguments</p>
-<p>Arguments for <tt class="docutils literal"><span class="pre">SurfaceOver2DRangeSeries</span></tt> class:</p>
-<p><tt class="docutils literal"><span class="pre">nb_of_points_x</span></tt>: int. The x range is sampled uniformly at
-<tt class="docutils literal"><span class="pre">nb_of_points_x</span></tt> of points.</p>
-<p><tt class="docutils literal"><span class="pre">nb_of_points_y</span></tt>: int. The y range is sampled uniformly at
-<tt class="docutils literal"><span class="pre">nb_of_points_y</span></tt> of points.</p>
-<p>Aesthetics:</p>
-<p><tt class="docutils literal"><span class="pre">surface_color</span></tt>: Function which returns a float. Specifies the color for
-the surface of the plot. See <tt class="docutils literal"><span class="pre">sympy.plotting.Plot</span></tt> for more details.</p>
-<p>If there are multiple plots, then the same series arguments are applied to
-all the plots. If you want to set these options separately, you can index
-the returned <tt class="docutils literal"><span class="pre">Plot</span></tt> object and set it.</p>
-<p>Arguments for <tt class="docutils literal"><span class="pre">Plot</span></tt> class:</p>
-<p><tt class="docutils literal"><span class="pre">title</span></tt> : str. Title of the plot.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.plotting.plot.plot3d_parametric_line">
-<tt class="descclassname">sympy.plotting.plot.</tt><tt class="descname">plot3d_parametric_line</tt><big>(</big><em>*args</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../_modules/sympy/plotting/plot.html#plot3d_parametric_line"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.plotting.plot.plot3d_parametric_line" title="Permalink to this definition">¶</a></dt>
-<dd><p>Plots a 3D parametric line plot.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.plotting.plot.Plot" title="sympy.plotting.plot.Plot"><tt class="xref py py-obj docutils literal"><span class="pre">Plot</span></tt></a>, <a class="reference internal" href="#sympy.plotting.plot.Parametric3DLineSeries" title="sympy.plotting.plot.Parametric3DLineSeries"><tt class="xref py py-obj docutils literal"><span class="pre">Parametric3DLineSeries</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.plotting</span> <span class="kn">import</span> <span class="n">plot3d_parametric_line</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;u&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Single plot.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">plot3d_parametric_line</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">sin</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>  
-</pre></div>
-</div>
-<p>Multiple plots.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">plot3d_parametric_line</span><span class="p">((</span><span class="n">cos</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">sin</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">u</span><span class="p">,</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">)),</span>
-<span class="gp">... </span>    <span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">u</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">)))</span>  
-</pre></div>
-</div>
-<p class="rubric">Usage</p>
-<p>Single plot:</p>
-<p><tt class="docutils literal"><span class="pre">plot3d_parametric_line(expr_x,</span> <span class="pre">expr_y,</span> <span class="pre">expr_z,</span> <span class="pre">range,</span> <span class="pre">**kwargs)</span></tt></p>
-<p>If the range is not specified, then a default range of (-10, 10) is used.</p>
-<p>Multiple plots.</p>
-<p><tt class="docutils literal"><span class="pre">plot3d_parametric_line((expr_x,</span> <span class="pre">expr_y,</span> <span class="pre">expr_z,</span> <span class="pre">range),</span> <span class="pre">...,</span> <span class="pre">**kwargs)</span></tt></p>
-<p>Ranges have to be specified for every expression.</p>
-<p>Default range may change in the future if a more advanced default range
-detection algorithm is implemented.</p>
-<p class="rubric">Arguments</p>
-<p><tt class="docutils literal"><span class="pre">expr_x</span></tt> : Expression representing the function along x.</p>
-<p><tt class="docutils literal"><span class="pre">expr_y</span></tt> : Expression representing the function along y.</p>
-<p><tt class="docutils literal"><span class="pre">expr_z</span></tt> : Expression representing the function along z.</p>
-<p><tt class="docutils literal"><span class="pre">range</span></tt>: <tt class="docutils literal"><span class="pre">(u,</span> <span class="pre">0,</span> <span class="pre">5)</span></tt>, A 3-tuple denoting the range of the parameter
-variable.</p>
-<p class="rubric">Keyword Arguments</p>
-<p>Arguments for <tt class="docutils literal"><span class="pre">Parametric3DLineSeries</span></tt> class.</p>
-<p><tt class="docutils literal"><span class="pre">nb_of_points</span></tt>: The range is uniformly sampled at <tt class="docutils literal"><span class="pre">nb_of_points</span></tt>
-number of points.</p>
-<p>Aesthetics:</p>
-<p><tt class="docutils literal"><span class="pre">line_color</span></tt>: function which returns a float. Specifies the color for the
-plot. See <tt class="docutils literal"><span class="pre">sympy.plotting.Plot</span></tt> for more details.</p>
-<p>If there are multiple plots, then the same series arguments are applied to
-all the plots. If you want to set these options separately, you can index
-the returned <tt class="docutils literal"><span class="pre">Plot</span></tt> object and set it.</p>
-<p>Arguments for <tt class="docutils literal"><span class="pre">Plot</span></tt> class.</p>
-<p><tt class="docutils literal"><span class="pre">title</span></tt> : str. Title of the plot.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.plotting.plot.plot3d_parametric_surface">
-<tt class="descclassname">sympy.plotting.plot.</tt><tt class="descname">plot3d_parametric_surface</tt><big>(</big><em>*args</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../_modules/sympy/plotting/plot.html#plot3d_parametric_surface"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.plotting.plot.plot3d_parametric_surface" title="Permalink to this definition">¶</a></dt>
-<dd><p>Plots a 3D parametric surface plot.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.plotting.plot.Plot" title="sympy.plotting.plot.Plot"><tt class="xref py py-obj docutils literal"><span class="pre">Plot</span></tt></a>, <a class="reference internal" href="#sympy.plotting.plot.ParametricSurfaceSeries" title="sympy.plotting.plot.ParametricSurfaceSeries"><tt class="xref py py-obj docutils literal"><span class="pre">ParametricSurfaceSeries</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.plotting</span> <span class="kn">import</span> <span class="n">plot3d_parametric_surface</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;u v&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Single plot.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">plot3d_parametric_surface</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">u</span> <span class="o">+</span> <span class="n">v</span><span class="p">),</span> <span class="n">sin</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="n">v</span><span class="p">),</span> <span class="n">u</span> <span class="o">-</span> <span class="n">v</span><span class="p">,</span>
-<span class="gp">... </span>    <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>  
-</pre></div>
-</div>
-<p class="rubric">Usage</p>
-<p>Single plot.</p>
-<p><tt class="docutils literal"><span class="pre">plot3d_parametric_surface(expr_x,</span> <span class="pre">expr_y,</span> <span class="pre">expr_z,</span> <span class="pre">range_u,</span> <span class="pre">range_v,</span> <span class="pre">**kwargs)</span></tt></p>
-<p>If the ranges is not specified, then a default range of (-10, 10) is used.</p>
-<p>Multiple plots.</p>
-<p><tt class="docutils literal"><span class="pre">plot3d_parametric_surface((expr_x,</span> <span class="pre">expr_y,</span> <span class="pre">expr_z,</span> <span class="pre">range_u,</span> <span class="pre">range_v),</span> <span class="pre">...,</span> <span class="pre">**kwargs)</span></tt></p>
-<p>Ranges have to be specified for every expression.</p>
-<p>Default range may change in the future if a more advanced default range
-detection algorithm is implemented.</p>
-<p class="rubric">Arguments</p>
-<p><tt class="docutils literal"><span class="pre">expr_x</span></tt>: Expression representing the function along <tt class="docutils literal"><span class="pre">x</span></tt>.</p>
-<p><tt class="docutils literal"><span class="pre">expr_y</span></tt>: Expression representing the function along <tt class="docutils literal"><span class="pre">y</span></tt>.</p>
-<p><tt class="docutils literal"><span class="pre">expr_z</span></tt>: Expression representing the function along <tt class="docutils literal"><span class="pre">z</span></tt>.</p>
-<p><tt class="docutils literal"><span class="pre">range_u</span></tt>: <tt class="docutils literal"><span class="pre">(u,</span> <span class="pre">0,</span> <span class="pre">5)</span></tt>,  A 3-tuple denoting the range of the <tt class="docutils literal"><span class="pre">u</span></tt>
-variable.</p>
-<p><tt class="docutils literal"><span class="pre">range_v</span></tt>: <tt class="docutils literal"><span class="pre">(v,</span> <span class="pre">0,</span> <span class="pre">5)</span></tt>,  A 3-tuple denoting the range of the v
-variable.</p>
-<p class="rubric">Keyword Arguments</p>
-<p>Arguments for <tt class="docutils literal"><span class="pre">ParametricSurfaceSeries</span></tt> class:</p>
-<p><tt class="docutils literal"><span class="pre">nb_of_points_u</span></tt>: int. The <tt class="docutils literal"><span class="pre">u</span></tt> range is sampled uniformly at
-<tt class="docutils literal"><span class="pre">nb_of_points_v</span></tt> of points</p>
-<p><tt class="docutils literal"><span class="pre">nb_of_points_y</span></tt>: int. The <tt class="docutils literal"><span class="pre">v</span></tt> range is sampled uniformly at
-<tt class="docutils literal"><span class="pre">nb_of_points_y</span></tt> of points</p>
-<p>Aesthetics:</p>
-<p><tt class="docutils literal"><span class="pre">surface_color</span></tt>: Function which returns a float. Specifies the color for
-the surface of the plot. See <tt class="docutils literal"><span class="pre">sympy.plotting.Plot</span></tt> for more details.</p>
-<p>If there are multiple plots, then the same series arguments are applied for
-all the plots. If you want to set these options separately, you can index
-the returned <tt class="docutils literal"><span class="pre">Plot</span></tt> object and set it.</p>
-<p>Arguments for <tt class="docutils literal"><span class="pre">Plot</span></tt> class:</p>
-<p><tt class="docutils literal"><span class="pre">title</span></tt> : str. Title of the plot.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.plotting.plot_implicit.plot_implicit">
-<tt class="descclassname">sympy.plotting.plot_implicit.</tt><tt class="descname">plot_implicit</tt><big>(</big><em>expr</em>, <em>*args</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../_modules/sympy/plotting/plot_implicit.html#plot_implicit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.plotting.plot_implicit.plot_implicit" title="Permalink to this definition">¶</a></dt>
-<dd><p>A plot function to plot implicit equations / inequalities.</p>
-<p class="rubric">Arguments</p>
-<ul class="simple">
-<li><tt class="docutils literal"><span class="pre">expr</span></tt> : The equation / inequality that is to be plotted.</li>
-<li><tt class="docutils literal"><span class="pre">(x,</span> <span class="pre">xmin,</span> <span class="pre">xmax)</span></tt> optional, 3-tuple denoting the range of symbol
-<tt class="docutils literal"><span class="pre">x</span></tt></li>
-<li><tt class="docutils literal"><span class="pre">(y,</span> <span class="pre">ymin,</span> <span class="pre">ymax)</span></tt> optional, 3-tuple denoting the range of symbol
-<tt class="docutils literal"><span class="pre">y</span></tt></li>
-</ul>
-<p>The following arguments can be passed as named parameters.</p>
-<ul>
-<li><dl class="first docutils">
-<dt><tt class="docutils literal"><span class="pre">adaptive</span></tt>. Boolean. The default value is set to True. It has to be</dt>
-<dd><p class="first last">set to False if you want to use a mesh grid.</p>
-</dd>
-</dl>
-</li>
-<li><dl class="first docutils">
-<dt><tt class="docutils literal"><span class="pre">depth</span></tt> integer. The depth of recursion for adaptive mesh grid.</dt>
-<dd><p class="first last">Default value is 0. Takes value in the range (0, 4).</p>
-</dd>
-</dl>
-</li>
-<li><dl class="first docutils">
-<dt><tt class="docutils literal"><span class="pre">points</span></tt> integer. The number of points if adaptive mesh grid is not</dt>
-<dd><p class="first last">used. Default value is 200.</p>
-</dd>
-</dl>
-</li>
-<li><p class="first"><tt class="docutils literal"><span class="pre">title</span></tt> string .The title for the plot.</p>
-</li>
-<li><p class="first"><tt class="docutils literal"><span class="pre">xlabel</span></tt> string. The label for the x - axis</p>
-</li>
-<li><p class="first"><tt class="docutils literal"><span class="pre">ylabel</span></tt> string. The label for the y - axis</p>
-</li>
-</ul>
-<p>plot_implicit, by default, uses interval arithmetic to plot functions. If
-the expression cannot be plotted using interval arithmetic, it defaults to
-a generating a contour using a mesh grid of fixed number of points. By
-setting adaptive to False, you can force plot_implicit to use the mesh
-grid. The mesh grid method can be effective when adaptive plotting using
-interval arithmetic, fails to plot with small line width.</p>
-<p class="rubric">Examples:</p>
-<p>Plot expressions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">plot_implicit</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">Eq</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Without any ranges for the symbols in the expression</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p1</span> <span class="o">=</span> <span class="n">plot_implicit</span><span class="p">(</span><span class="n">Eq</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>  
-</pre></div>
-</div>
-<p>With the range for the symbols</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p2</span> <span class="o">=</span> <span class="n">plot_implicit</span><span class="p">(</span><span class="n">Eq</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span>
-<span class="gp">... </span>        <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>  
-</pre></div>
-</div>
-<p>With depth of recursion as argument.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p3</span> <span class="o">=</span> <span class="n">plot_implicit</span><span class="p">(</span><span class="n">Eq</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span>
-<span class="gp">... </span>        <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">depth</span> <span class="o">=</span> <span class="mi">2</span><span class="p">)</span>  
-</pre></div>
-</div>
-<p>Using mesh grid and not using adaptive meshing.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p4</span> <span class="o">=</span> <span class="n">plot_implicit</span><span class="p">(</span><span class="n">Eq</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span>
-<span class="gp">... </span>        <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">adaptive</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>  
-</pre></div>
-</div>
-<p>Using mesh grid with number of points as input.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p5</span> <span class="o">=</span> <span class="n">plot_implicit</span><span class="p">(</span><span class="n">Eq</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span>
-<span class="gp">... </span>        <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span>
-<span class="gp">... </span>        <span class="n">adaptive</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">points</span><span class="o">=</span><span class="mi">400</span><span class="p">)</span>  
-</pre></div>
-</div>
-<p>Plotting regions.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p6</span> <span class="o">=</span> <span class="n">plot_implicit</span><span class="p">(</span><span class="n">y</span> <span class="o">&gt;</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>  
-</pre></div>
-</div>
-<p>Plotting Using boolean conjunctions.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p7</span> <span class="o">=</span> <span class="n">plot_implicit</span><span class="p">(</span><span class="n">And</span><span class="p">(</span><span class="n">y</span> <span class="o">&gt;</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">&gt;</span> <span class="o">-</span><span class="n">x</span><span class="p">))</span>  
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="series-classes">
-<h2>Series Classes<a class="headerlink" href="#series-classes" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.plotting.plot.BaseSeries">
-<em class="property">class </em><tt class="descclassname">sympy.plotting.plot.</tt><tt class="descname">BaseSeries</tt><a class="reference internal" href="../_modules/sympy/plotting/plot.html#BaseSeries"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.plotting.plot.BaseSeries" title="Permalink to this definition">¶</a></dt>
-<dd><p>Base class for the data objects containing stuff to be plotted.</p>
-<p>The backend should check if it supports the data series that it&#8217;s given.
-(eg TextBackend supports only LineOver1DRange).
-It&#8217;s the backend responsibility to know how to use the class of
-data series that it&#8217;s given.</p>
-<p>Some data series classes are grouped (using a class attribute like is_2Dline)
-according to the api they present (based only on convention). The backend is
-not obliged to use that api (eg. The LineOver1DRange belongs to the
-is_2Dline group and presents the get_points method, but the
-TextBackend does not use the get_points method).</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.plotting.plot.Line2DBaseSeries">
-<em class="property">class </em><tt class="descclassname">sympy.plotting.plot.</tt><tt class="descname">Line2DBaseSeries</tt><a class="reference internal" href="../_modules/sympy/plotting/plot.html#Line2DBaseSeries"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.plotting.plot.Line2DBaseSeries" title="Permalink to this definition">¶</a></dt>
-<dd><p>A base class for 2D lines.</p>
-<ul class="simple">
-<li>adding the label, steps and only_integers options</li>
-<li>making is_2Dline true</li>
-<li>defining get_segments and get_color_array</li>
-</ul>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.plotting.plot.LineOver1DRangeSeries">
-<em class="property">class </em><tt class="descclassname">sympy.plotting.plot.</tt><tt class="descname">LineOver1DRangeSeries</tt><big>(</big><em>expr</em>, <em>var_start_end</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../_modules/sympy/plotting/plot.html#LineOver1DRangeSeries"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.plotting.plot.LineOver1DRangeSeries" title="Permalink to this definition">¶</a></dt>
-<dd><p>Representation for a line consisting of a sympy expression over a range.</p>
-<dl class="function">
-<dt id="sympy.plotting.plot.LineOver1DRangeSeries.get_segments">
-<tt class="descname">get_segments</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/plotting/plot.html#LineOver1DRangeSeries.get_segments"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.plotting.plot.LineOver1DRangeSeries.get_segments" title="Permalink to this definition">¶</a></dt>
-<dd><p>Adaptively gets segments for plotting.</p>
-<p>The adaptive sampling is done by recursively checking if three
-points are almost collinear. If they are not collinear, then more
-points are added between those points.</p>
-<p class="rubric">References</p>
-<dl class="docutils">
-<dt>[1] Adaptive polygonal approximation of parametric curves,</dt>
-<dd>Luiz Henrique de Figueiredo.</dd>
-</dl>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.plotting.plot.Parametric2DLineSeries">
-<em class="property">class </em><tt class="descclassname">sympy.plotting.plot.</tt><tt class="descname">Parametric2DLineSeries</tt><big>(</big><em>expr_x</em>, <em>expr_y</em>, <em>var_start_end</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../_modules/sympy/plotting/plot.html#Parametric2DLineSeries"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.plotting.plot.Parametric2DLineSeries" title="Permalink to this definition">¶</a></dt>
-<dd><p>Representation for a line consisting of two parametric sympy expressions
-over a range.</p>
-<dl class="function">
-<dt id="sympy.plotting.plot.Parametric2DLineSeries.get_segments">
-<tt class="descname">get_segments</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/plotting/plot.html#Parametric2DLineSeries.get_segments"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.plotting.plot.Parametric2DLineSeries.get_segments" title="Permalink to this definition">¶</a></dt>
-<dd><p>Adaptively gets segments for plotting.</p>
-<p>The adaptive sampling is done by recursively checking if three
-points are almost collinear. If they are not collinear, then more
-points are added between those points.</p>
-<p class="rubric">References</p>
-<dl class="docutils">
-<dt>[1] Adaptive polygonal approximation of parametric curves,</dt>
-<dd>Luiz Henrique de Figueiredo.</dd>
-</dl>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.plotting.plot.Line3DBaseSeries">
-<em class="property">class </em><tt class="descclassname">sympy.plotting.plot.</tt><tt class="descname">Line3DBaseSeries</tt><a class="reference internal" href="../_modules/sympy/plotting/plot.html#Line3DBaseSeries"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.plotting.plot.Line3DBaseSeries" title="Permalink to this definition">¶</a></dt>
-<dd><p>A base class for 3D lines.</p>
-<p>Most of the stuff is derived from Line2DBaseSeries.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.plotting.plot.Parametric3DLineSeries">
-<em class="property">class </em><tt class="descclassname">sympy.plotting.plot.</tt><tt class="descname">Parametric3DLineSeries</tt><big>(</big><em>expr_x</em>, <em>expr_y</em>, <em>expr_z</em>, <em>var_start_end</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../_modules/sympy/plotting/plot.html#Parametric3DLineSeries"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.plotting.plot.Parametric3DLineSeries" title="Permalink to this definition">¶</a></dt>
-<dd><p>Representation for a 3D line consisting of two parametric sympy
-expressions and a range.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.plotting.plot.SurfaceBaseSeries">
-<em class="property">class </em><tt class="descclassname">sympy.plotting.plot.</tt><tt class="descname">SurfaceBaseSeries</tt><a class="reference internal" href="../_modules/sympy/plotting/plot.html#SurfaceBaseSeries"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.plotting.plot.SurfaceBaseSeries" title="Permalink to this definition">¶</a></dt>
-<dd><p>A base class for 3D surfaces.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.plotting.plot.SurfaceOver2DRangeSeries">
-<em class="property">class </em><tt class="descclassname">sympy.plotting.plot.</tt><tt class="descname">SurfaceOver2DRangeSeries</tt><big>(</big><em>expr</em>, <em>var_start_end_x</em>, <em>var_start_end_y</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../_modules/sympy/plotting/plot.html#SurfaceOver2DRangeSeries"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.plotting.plot.SurfaceOver2DRangeSeries" title="Permalink to this definition">¶</a></dt>
-<dd><p>Representation for a 3D surface consisting of a sympy expression and 2D
-range.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.plotting.plot.ParametricSurfaceSeries">
-<em class="property">class </em><tt class="descclassname">sympy.plotting.plot.</tt><tt class="descname">ParametricSurfaceSeries</tt><big>(</big><em>expr_x</em>, <em>expr_y</em>, <em>expr_z</em>, <em>var_start_end_u</em>, <em>var_start_end_v</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../_modules/sympy/plotting/plot.html#ParametricSurfaceSeries"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.plotting.plot.ParametricSurfaceSeries" title="Permalink to this definition">¶</a></dt>
-<dd><p>Representation for a 3D surface consisting of three parametric sympy
-expressions and a range.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.plotting.plot_implicit.ImplicitSeries">
-<em class="property">class </em><tt class="descclassname">sympy.plotting.plot_implicit.</tt><tt class="descname">ImplicitSeries</tt><big>(</big><em>expr</em>, <em>var_start_end_x</em>, <em>var_start_end_y</em>, <em>has_equality</em>, <em>use_interval_math</em>, <em>depth</em>, <em>nb_of_points</em><big>)</big><a class="reference internal" href="../_modules/sympy/plotting/plot_implicit.html#ImplicitSeries"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.plotting.plot_implicit.ImplicitSeries" title="Permalink to this definition">¶</a></dt>
-<dd><p>Representation for Implicit plot</p>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.plotting.pygletplot">
-<span id="pyglet-plotting-module"></span><h1>Pyglet Plotting Module<a class="headerlink" href="#module-sympy.plotting.pygletplot" title="Permalink to this headline">¶</a></h1>
-<p>This is the documentation for the old plotting module that uses pyglet.
-This module has some limitations and is not actively developed anymore.
-For an alternative you can look at the new plotting module.</p>
-<p>The pyglet plotting module can do nice 2D and 3D plots that can be
-controlled by console commands as well as keyboard and mouse, with
-the only dependency being <tt class="docutils literal"><span class="pre">pyglet</span></tt>.</p>
-<p>Here is the simplest usage:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">var</span><span class="p">,</span> <span class="n">Plot</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;x y z&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Plot</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">3</span><span class="o">-</span><span class="n">y</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>To see lots of plotting examples, see <tt class="docutils literal"><span class="pre">examples/pyglet_plotting.py</span></tt> and try running
-it in interactive mode (python -i plotting.py):</p>
-<div class="highlight-python"><pre>$ python -i examples/pyglet_plotting.py</pre>
-</div>
-<p>And type for instance <tt class="docutils literal"><span class="pre">example(7)</span></tt> or <tt class="docutils literal"><span class="pre">example(11)</span></tt>.</p>
-<p>See also the <a class="reference external" href="https://github.com/sympy/sympy/wiki/Plotting-Module">Plotting Module</a>
-wiki page for screenshots.</p>
-<div class="section" id="plot-window-controls">
-<h2>Plot Window Controls<a class="headerlink" href="#plot-window-controls" title="Permalink to this headline">¶</a></h2>
-<table border="1" class="docutils">
-<colgroup>
-<col width="35%" />
-<col width="65%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head">Camera</th>
-<th class="head">Keys</th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-even"><td>Sensitivity Modifier</td>
-<td>SHIFT</td>
-</tr>
-<tr class="row-odd"><td>Zoom</td>
-<td>R and F, Page Up and Down, Numpad + and -</td>
-</tr>
-<tr class="row-even"><td>Rotate View X,Y axis</td>
-<td>Arrow Keys, A,S,D,W, Numpad 4,6,8,2</td>
-</tr>
-<tr class="row-odd"><td>Rotate View Z axis</td>
-<td>Q and E, Numpad 7 and 9</td>
-</tr>
-<tr class="row-even"><td>Rotate Ordinate Z axis</td>
-<td>Z and C, Numpad 1 and 3</td>
-</tr>
-<tr class="row-odd"><td>View XY</td>
-<td>F1</td>
-</tr>
-<tr class="row-even"><td>View XZ</td>
-<td>F2</td>
-</tr>
-<tr class="row-odd"><td>View YZ</td>
-<td>F3</td>
-</tr>
-<tr class="row-even"><td>View Perspective</td>
-<td>F4</td>
-</tr>
-<tr class="row-odd"><td>Reset</td>
-<td>X, Numpad 5</td>
-</tr>
-</tbody>
-</table>
-<table border="1" class="docutils">
-<colgroup>
-<col width="73%" />
-<col width="27%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head">Axes</th>
-<th class="head">Keys</th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-even"><td>Toggle Visible</td>
-<td>F5</td>
-</tr>
-<tr class="row-odd"><td>Toggle Colors</td>
-<td>F6</td>
-</tr>
-</tbody>
-</table>
-<table border="1" class="docutils">
-<colgroup>
-<col width="73%" />
-<col width="27%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head">Window</th>
-<th class="head">Keys</th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-even"><td>Close</td>
-<td>ESCAPE</td>
-</tr>
-<tr class="row-odd"><td>Screenshot</td>
-<td>F8</td>
-</tr>
-</tbody>
-</table>
-<p>The mouse can be used to rotate, zoom, and translate by dragging the left, middle, and right mouse buttons respectively.</p>
-</div>
-<div class="section" id="coordinate-modes">
-<h2>Coordinate Modes<a class="headerlink" href="#coordinate-modes" title="Permalink to this headline">¶</a></h2>
-<p>Plot supports several curvilinear coordinate modes, and they are independent
-for each plotted function. You can specify a coordinate mode explicitly with
-the &#8216;mode&#8217; named argument, but it can be automatically determined for cartesian
-or parametric plots, and therefore must only be specified for polar,
-cylindrical, and spherical modes.</p>
-<p>Specifically, Plot(function arguments) and Plot.__setitem__(i, function
-arguments) (accessed using array-index syntax on the Plot instance) will
-interpret your arguments as a cartesian plot if you provide one function and a
-parametric plot if you provide two or three functions. Similarly, the arguments
-will be interpreted as a curve is one variable is used, and a surface if two
-are used.</p>
-<p>Supported mode names by number of variables:</p>
-<ul class="simple">
-<li>1 (curves): parametric, cartesian, polar</li>
-<li>2 (surfaces): parametric, cartesian, cylindrical, spherical</li>
-</ul>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Plot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="s">&#39;mode=spherical; color=zfade4&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Note that function parameters are given as option strings of the form
-&#8220;key1=value1; key2 = value2&#8221; (spaces are truncated). Keyword arguments given
-directly to plot apply to the plot itself.</p>
-</div>
-<div class="section" id="specifying-intervals-for-variables">
-<h2>Specifying Intervals for Variables<a class="headerlink" href="#specifying-intervals-for-variables" title="Permalink to this headline">¶</a></h2>
-<p>The basic format for variable intervals is [var, min, max, steps]. However, the
-syntax is quite flexible, and arguments not specified are taken from the
-defaults for the current coordinate mode:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Plot</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="c"># implies [x,-5,5,100]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Plot</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">[],</span> <span class="p">[])</span> <span class="c"># [x,-1,1,40], [y,-1,1,40]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Plot</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">100</span><span class="p">],</span> <span class="p">[</span><span class="mi">100</span><span class="p">])</span> <span class="c"># [x,-1,1,100], [y,-1,1,100]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Plot</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span><span class="o">-</span><span class="mi">13</span><span class="p">,</span><span class="mi">13</span><span class="p">,</span><span class="mi">100</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Plot</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="o">-</span><span class="mi">13</span><span class="p">,</span><span class="mi">13</span><span class="p">])</span> <span class="c"># [x,-13,13,100]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Plot</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span><span class="o">-</span><span class="mi">13</span><span class="p">,</span><span class="mi">13</span><span class="p">])</span> <span class="c"># [x,-13,13,100]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Plot</span><span class="p">(</span><span class="mi">1</span><span class="o">*</span><span class="n">x</span><span class="p">,</span> <span class="p">[],</span> <span class="p">[</span><span class="n">x</span><span class="p">],</span> <span class="s">&#39;mode=cylindrical&#39;</span><span class="p">)</span> <span class="c"># [unbound_theta,0,2*Pi,40], [x,-1,1,20]</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="using-the-interactive-interface">
-<h2>Using the Interactive Interface<a class="headerlink" href="#using-the-interactive-interface" title="Permalink to this headline">¶</a></h2>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Plot</span><span class="p">(</span><span class="n">visible</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span>
-<span class="go">[1]: x**2, &#39;mode=cartesian&#39;</span>
-<span class="go">[2]: 2*x, &#39;mode=cartesian&#39;</span>
-<span class="go">[3]: 2, &#39;mode=cartesian&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">clear</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span>
-<span class="go">&lt;blank plot&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span>  <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">style</span> <span class="o">=</span> <span class="s">&#39;solid&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">style</span> <span class="o">=</span> <span class="s">&#39;wireframe&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">color</span> <span class="o">=</span> <span class="n">z</span><span class="p">,</span> <span class="p">(</span><span class="mf">0.4</span><span class="p">,</span><span class="mf">0.4</span><span class="p">,</span><span class="mf">0.9</span><span class="p">),</span> <span class="p">(</span><span class="mf">0.9</span><span class="p">,</span><span class="mf">0.4</span><span class="p">,</span><span class="mf">0.4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">style</span> <span class="o">=</span> <span class="s">&#39;both&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">style</span> <span class="o">=</span> <span class="s">&#39;both&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="using-custom-color-functions">
-<h2>Using Custom Color Functions<a class="headerlink" href="#using-custom-color-functions" title="Permalink to this headline">¶</a></h2>
-<p>The following code plots a saddle and color it by the magnitude of its gradient:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fz</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Fx</span><span class="p">,</span> <span class="n">Fy</span><span class="p">,</span> <span class="n">Fz</span> <span class="o">=</span> <span class="n">fz</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">fz</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">y</span><span class="p">),</span> <span class="mi">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">fz</span><span class="p">,</span> <span class="s">&#39;style=solid&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">color</span> <span class="o">=</span> <span class="p">(</span><span class="n">Fx</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">Fy</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">Fz</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>The coloring algorithm works like this:</p>
-<ol class="arabic simple">
-<li>Evaluate the color function(s) across the curve or surface.</li>
-<li>Find the minimum and maximum value of each component.</li>
-<li>Scale each component to the color gradient.</li>
-</ol>
-<p>When not specified explicitly, the default color gradient is
-f(0.0)=(0.4,0.4,0.4) -&gt; f(1.0)=(0.9,0.9,0.9). In our case, everything is
-gray-scale because we have applied the default color gradient uniformly for
-each color component. When defining a color scheme in this way, you might want
-to supply a color gradient as well:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">color</span> <span class="o">=</span> <span class="p">(</span><span class="n">Fx</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">Fy</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">Fz</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mf">0.5</span><span class="p">),</span> <span class="p">(</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">0.9</span><span class="p">),</span> <span class="p">(</span><span class="mf">0.9</span><span class="p">,</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">0.1</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Here&#8217;s a color gradient with four steps:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gradient</span> <span class="o">=</span> <span class="p">[</span> <span class="mf">0.0</span><span class="p">,</span> <span class="p">(</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">0.9</span><span class="p">),</span> <span class="mf">0.3</span><span class="p">,</span> <span class="p">(</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">0.9</span><span class="p">,</span><span class="mf">0.1</span><span class="p">),</span>
-<span class="gp">... </span>             <span class="mf">0.7</span><span class="p">,</span> <span class="p">(</span><span class="mf">0.9</span><span class="p">,</span><span class="mf">0.9</span><span class="p">,</span><span class="mf">0.1</span><span class="p">),</span> <span class="mf">1.0</span><span class="p">,</span> <span class="p">(</span><span class="mf">1.0</span><span class="p">,</span><span class="mf">0.0</span><span class="p">,</span><span class="mf">0.0</span><span class="p">)</span> <span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">color</span> <span class="o">=</span> <span class="p">(</span><span class="n">Fx</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">Fy</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">Fz</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mf">0.5</span><span class="p">),</span> <span class="n">gradient</span>
-</pre></div>
-</div>
-<p>The other way to specify a color scheme is to give a separate function for each
-component r, g, b. With this syntax, the default color scheme is defined:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">color</span> <span class="o">=</span> <span class="n">z</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">x</span><span class="p">,</span> <span class="p">(</span><span class="mf">0.4</span><span class="p">,</span><span class="mf">0.4</span><span class="p">,</span><span class="mf">0.4</span><span class="p">),</span> <span class="p">(</span><span class="mf">0.9</span><span class="p">,</span><span class="mf">0.9</span><span class="p">,</span><span class="mf">0.9</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>This maps z-&gt;red, y-&gt;green, and x-&gt;blue. In some cases, you might prefer to use
-the following alternative syntax:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">color</span> <span class="o">=</span> <span class="n">z</span><span class="p">,(</span><span class="mf">0.4</span><span class="p">,</span><span class="mf">0.9</span><span class="p">),</span> <span class="n">y</span><span class="p">,(</span><span class="mf">0.4</span><span class="p">,</span><span class="mf">0.9</span><span class="p">),</span> <span class="n">x</span><span class="p">,(</span><span class="mf">0.4</span><span class="p">,</span><span class="mf">0.9</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>You can still use multi-step gradients with three-function color schemes.</p>
-</div>
-<div class="section" id="plotting-geometric-entities">
-<h2>Plotting Geometric Entities<a class="headerlink" href="#plotting-geometric-entities" title="Permalink to this headline">¶</a></h2>
-<p>The plotting module is capable of plotting some 2D geometric entities like
-line, circle and ellipse. The following example plots a circle and a tangent
-line at a random point on the ellipse.</p>
-<div class="highlight-python"><pre>In [1]: p = Plot(axes=&#x27;label_axes=True&#x27;)
-
-In [2]: c = Circle(Point(0,0), 1)
-
-In [3]: t = c.tangent_line(c.random_point())
-
-In [4]: p[0] = c
-
-In [5]: p[1] = t</pre>
-</div>
-<p>Plotting polygons (Polygon, RegularPolygon, Triangle) are not supported
-directly. However a polygon can be plotted through a loop as follows.</p>
-<div class="highlight-python"><pre>In [6]: p = Plot(axes=&#x27;label_axes=True&#x27;)
-
-In [7]: t = RegularPolygon(Point(0,0), 1, 5)
-
-In [8]: for i in range(len(t.sides)):
-   ....:    p[i] = t.sides[i]</pre>
-</div>
-</div>
-</div>
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-            </a></p>
-  <h3><a href="../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Plotting Module</a><ul>
-<li><a class="reference internal" href="#introduction">Introduction</a></li>
-<li><a class="reference internal" href="#plot-class">Plot Class</a></li>
-<li><a class="reference internal" href="#plotting-function-reference">Plotting Function Reference</a></li>
-<li><a class="reference internal" href="#series-classes">Series Classes</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.plotting.pygletplot">Pyglet Plotting Module</a><ul>
-<li><a class="reference internal" href="#plot-window-controls">Plot Window Controls</a></li>
-<li><a class="reference internal" href="#coordinate-modes">Coordinate Modes</a></li>
-<li><a class="reference internal" href="#specifying-intervals-for-variables">Specifying Intervals for Variables</a></li>
-<li><a class="reference internal" href="#using-the-interactive-interface">Using the Interactive Interface</a></li>
-<li><a class="reference internal" href="#using-custom-color-functions">Using Custom Color Functions</a></li>
-<li><a class="reference internal" href="#plotting-geometric-entities">Plotting Geometric Entities</a></li>
-</ul>
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-</ul>
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-  <div class="section" id="agca-algebraic-geometry-and-commutative-algebra-module">
-<span id="polys-agca"></span><h1>AGCA - Algebraic Geometry and Commutative Algebra Module<a class="headerlink" href="#agca-algebraic-geometry-and-commutative-algebra-module" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="introduction">
-<h2>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline">¶</a></h2>
-<blockquote>
-<div><p>Algebraic geometry is a mixture of the ideas of two Mediterranean
-cultures. It is the superposition of the Arab science of the lightening
-calculation of the solutions of equations over the Greek art of position
-and shape.
-This tapestry was originally woven on European soil and is still being refined
-under the influence of international fashion. Algebraic geometry studies the
-delicate balance between the geometrically plausible and the algebraically
-possible.  Whenever one side of this mathematical teeter-totter outweighs the
-other, one immediately loses interest and runs off in search of a more exciting
-amusement.</p>
-<blockquote>
-<div>George R. Kempf
-1944 &#8211; 2002</div></blockquote>
-</div></blockquote>
-<p>Algebraic Geometry refers to the study of geometric problems via algebraic
-methods (and sometimes vice versa). While this is a rather old topic,
-algebraic geometry as understood today is very much a 20th century
-development. Building on ideas of e.g. Riemann and Dedekind, it was realized
-that there is an intimate connection between properties of the set of
-solutions of a system of polynomial equations (called an algebraic variety)
-and the behavior of the set of polynomial functions on that variety
-(called the coordinate ring).</p>
-<p>As in many geometric disciplines, we can distinguish between local and global
-questions (and methods). Local investigations in algebraic geometry are
-essentially equivalent to the study of certain rings, their ideals and modules.
-This latter topic is also called commutative algebra. It is the basic local
-toolset of algebraic geometers, in much the same way that differential analysis
-is the local toolset of differential geometers.</p>
-<p>A good conceptual introduction to commutative algebra is <a class="reference internal" href="../polys/literature.html#atiyah69">[Atiyah69]</a>. An
-introduction more geared towards computations, and the work most of the
-algorithms in this module are based on, is <a class="reference internal" href="../polys/literature.html#greuel2008">[Greuel2008]</a>.</p>
-<p>This module aims to eventually allow expression and solution of both
-local and global geometric problems, both in the classical case over a field
-and in the more modern arithmetic cases. So far, however, there is no geometric
-functionality at all. Currently the module only provides tools for computational
-commutative algebra over fields.</p>
-<p>All code examples assume:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y,z&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">init_printing</span><span class="p">(</span><span class="n">use_unicode</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">wrap_line</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">no_global</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="reference">
-<h2>Reference<a class="headerlink" href="#reference" title="Permalink to this headline">¶</a></h2>
-<p>In this section we document the usage of the AGCA module. For convenience of
-the reader, some definitions and examples/explanations are interspersed.</p>
-<div class="section" id="base-rings">
-<h3>Base Rings<a class="headerlink" href="#base-rings" title="Permalink to this headline">¶</a></h3>
-<p>Almost all computations in commutative algebra are relative to a &#8220;base ring&#8221;.
-(For example, when asking questions about an ideal, the base ring is the ring
-the ideal is a subset of.) In principle all polys &#8220;domains&#8221; can be used as base
-rings. However, useful functionality is only implemented for polynomial rings
-over fields, and various localizations and quotients thereof.</p>
-<p>As demonstrated in
-the examples below, the most convenient method to create objects you are
-interested in is to build them up from the ground field, and then use the
-various methods to create new objects from old. For example, in order to
-create the local ring of the nodal cubic <span class="math">\(y^2 = x^3\)</span> at the origin, over
-<span class="math">\(\mathbb{Q}\)</span>, you do:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">lr</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">poly_ring</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s">&quot;ilex&quot;</span><span class="p">)</span> <span class="o">/</span> <span class="p">[</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lr</span>
-<span class="go">ℚ[x, y, order=ilex]</span>
-<span class="go">───────────────────</span>
-<span class="go">    ╱   3    2╲</span>
-<span class="go">    ╲- x  + y ╱</span>
-</pre></div>
-</div>
-<p>Note how the python list notation can be used as a short cut to express ideals.
-You can use the <tt class="docutils literal"><span class="pre">convert</span></tt> method to return ordinary sympy objects into
-objects understood by the AGCA module (although in many cases this will be done
-automatically &#8211; for example the list was automatically turned into an ideal,
-and in the process the symbols <span class="math">\(x\)</span> and <span class="math">\(y\)</span> were automatically converted into
-other representations). For example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="o">=</span> <span class="n">lr</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">lr</span><span class="o">.</span><span class="n">convert</span><span class="p">(</span><span class="n">y</span><span class="p">)</span> <span class="p">;</span> <span class="n">X</span>
-<span class="go">    ╱   3    2╲</span>
-<span class="go">x + ╲- x  + y ╱</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">==</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">False</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span><span class="o">**</span><span class="mi">3</span> <span class="o">==</span> <span class="n">Y</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>When no localisation is needed, a more mathematical notation can be
-used. For example, let us create the coordinate ring of three-dimensional
-affine space <span class="math">\(\mathbb{A}^3\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ar</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">]</span> <span class="p">;</span> <span class="n">ar</span>
-<span class="go">ℚ[x, y, z]</span>
-</pre></div>
-</div>
-<p>For more details, refer to the following class documentation. Note that
-the base rings, being domains, are the main point of overlap between the
-AGCA module and the rest of the polys module. All domains are documented
-in detail in the polys reference, so we show here only an abridged version,
-with the methods most pertinent to the AGCA module.</p>
-<dl class="class">
-<dt>
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">Ring</tt></dt>
-<dd><p>Represents a ring domain.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>alias</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>dtype</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>one</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>rep</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>zero</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt>
-<tt class="descclassname">Ring.</tt><tt class="descname">free_module</tt><big>(</big><em>self</em>, <em>rank</em><big>)</big></dt>
-<dd><p>Generate a free module of rank <tt class="docutils literal"><span class="pre">rank</span></tt> over self.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">QQ[x]**2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt>
-<tt class="descclassname">Ring.</tt><tt class="descname">ideal</tt><big>(</big><em>self</em>, <em>*gens</em><big>)</big></dt>
-<dd><p>Generate an ideal of <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">&lt;x**2&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt>
-<tt class="descclassname">Ring.</tt><tt class="descname">quotient_ring</tt><big>(</big><em>self</em>, <em>e</em><big>)</big></dt>
-<dd><p>Form a quotient ring of <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<p>Here <tt class="docutils literal"><span class="pre">e</span></tt> can be an ideal or an iterable.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">quotient_ring</span><span class="p">(</span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">QQ[x]/&lt;x**2&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">quotient_ring</span><span class="p">([</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">])</span>
-<span class="go">QQ[x]/&lt;x**2&gt;</span>
-</pre></div>
-</div>
-<p>The division operator has been overloaded for this:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">/</span><span class="p">[</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">]</span>
-<span class="go">QQ[x]/&lt;x**2&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="function">
-<dt>
-<tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">PolynomialRing</tt><big>(</big><em>dom</em>, <em>*gens</em>, <em>**opts</em><big>)</big></dt>
-<dd><p>Create a generalized multivariate polynomial ring.</p>
-<p>A generalized polynomial ring is defined by a ground field <span class="math">\(K\)</span>, a set
-of generators (typically <span class="math">\(x_1, \dots, x_n\)</span>) and a monomial order <span class="math">\(&lt;\)</span>.
-The monomial order can be global, local or mixed. In any case it induces
-a total ordering on the monomials, and there exists for every (non-zero)
-polynomial <span class="math">\(f \in K[x_1, \dots, x_n]\)</span> a well-defined &#8220;leading monomial&#8221;
-<span class="math">\(LM(f) = LM(f, &gt;)\)</span>. One can then define a multiplicative subset
-<span class="math">\(S = S_&gt; = \{f \in K[x_1, \dots, x_n] | LM(f) = 1\}\)</span>. The generalized
-polynomial ring corresponding to the monomial order is
-<span class="math">\(R = S^{-1}K[x_1, \dots, x_n]\)</span>.</p>
-<p>If <span class="math">\(&gt;\)</span> is a so-called global order, that is <span class="math">\(1\)</span> is the smallest monomial,
-then we just have <span class="math">\(S = K\)</span> and <span class="math">\(R = K[x_1, \dots, x_n]\)</span>.</p>
-<p class="rubric">Examples</p>
-<p>A few examples may make this clearer.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-</pre></div>
-</div>
-<p>Our first ring uses global lexicographic order.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">R1</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">poly_ring</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="p">((</span><span class="s">&quot;lex&quot;</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),))</span>
-</pre></div>
-</div>
-<p>The second ring uses local lexicographic order. Note that when using a
-single (non-product) order, you can just specify the name and omit the
-variables:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">R2</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">poly_ring</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s">&quot;ilex&quot;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>The third and fourth rings use a mixed orders:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">o1</span> <span class="o">=</span> <span class="p">((</span><span class="s">&quot;ilex&quot;</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="s">&quot;lex&quot;</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">o2</span> <span class="o">=</span> <span class="p">((</span><span class="s">&quot;lex&quot;</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="s">&quot;ilex&quot;</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R3</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">poly_ring</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">o1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R4</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">poly_ring</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">o2</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>We will investigate what elements of <span class="math">\(K(x, y)\)</span> are contained in the various
-rings.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">L</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="p">),</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">y</span><span class="p">),</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">test</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">R</span><span class="p">:</span> <span class="p">[</span><span class="n">f</span> <span class="ow">in</span> <span class="n">R</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">L</span><span class="p">]</span>
-</pre></div>
-</div>
-<p>The first ring is just <span class="math">\(K[x, y]\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">test</span><span class="p">(</span><span class="n">R1</span><span class="p">)</span>
-<span class="go">[True, False, False, False, False]</span>
-</pre></div>
-</div>
-<p>The second ring is R1 localised at the maximal ideal (x, y):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">test</span><span class="p">(</span><span class="n">R2</span><span class="p">)</span>
-<span class="go">[True, False, True, True, True]</span>
-</pre></div>
-</div>
-<p>The third ring is R1 localised at the prime ideal (x):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">test</span><span class="p">(</span><span class="n">R3</span><span class="p">)</span>
-<span class="go">[True, False, True, False, True]</span>
-</pre></div>
-</div>
-<p>Finally the fourth ring is R1 localised at <span class="math">\(S = K[x, y] \setminus yK[y]\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">test</span><span class="p">(</span><span class="n">R4</span><span class="p">)</span>
-<span class="go">[True, False, False, True, False]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt>
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">QuotientRing</tt><big>(</big><em>ring</em>, <em>ideal</em><big>)</big></dt>
-<dd><p>Class representing (commutative) quotient rings.</p>
-<p>You should not usually instatiate this by hand, instead use the constructor
-from the ring you are quotienting out by:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">quotient_ring</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
-<span class="go">QQ[x]/&lt;x**3 + 1&gt;</span>
-</pre></div>
-</div>
-<p>Shorter versions are possible:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">/</span><span class="n">I</span>
-<span class="go">QQ[x]/&lt;x**3 + 1&gt;</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">/</span><span class="p">[</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
-<span class="go">QQ[x]/&lt;x**3 + 1&gt;</span>
-</pre></div>
-</div>
-<p>Attributes:</p>
-<ul class="simple">
-<li>ring - the base ring</li>
-<li>base_ideal - the ideal we are quotienting by</li>
-</ul>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>alias</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>one</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>rep</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>zero</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</div>
-<div class="section" id="modules-ideals-and-their-elementary-properties">
-<h3>Modules, Ideals and their Elementary Properties<a class="headerlink" href="#modules-ideals-and-their-elementary-properties" title="Permalink to this headline">¶</a></h3>
-<p>Let <span class="math">\(A\)</span> be a ring. An <span class="math">\(A\)</span>-module is a set <span class="math">\(M\)</span>, together with two binary
-operations <span class="math">\(+: M \times M \to M\)</span> and <span class="math">\(\times: R \times M \to M\)</span> called
-addition and scalar multiplication. These are required to satisfy certain
-axioms, which can be found in e.g. <a class="reference internal" href="../polys/literature.html#atiyah69">[Atiyah69]</a>. In this way modules are
-a direct generalisation of both vector spaces (<span class="math">\(A\)</span> being a field) and abelian
-groups (<span class="math">\(A = \mathbb{Z}\)</span>). A <em>submodule</em> of the <span class="math">\(A\)</span>-module <span class="math">\(M\)</span> is a subset
-<span class="math">\(N \subset M\)</span>, such that the binary operations restrict to <span class="math">\(N\)</span>, and <span class="math">\(N\)</span> becomes
-an <span class="math">\(A\)</span>-module with these operations.</p>
-<p>The ring <span class="math">\(A\)</span> itself has a natural <span class="math">\(A\)</span>-module structure where addition and
-multiplication in the module coincide with addition and multiplication in
-the ring. This <span class="math">\(A\)</span>-module is also written as <span class="math">\(A\)</span>. An <span class="math">\(A\)</span>-submodule of <span class="math">\(A\)</span>
-is called an <em>ideal</em> of <span class="math">\(A\)</span>. Ideals come up very naturally in algebraic
-geometry. More general modules can be seen as a technically convenient &#8220;elbow
-room&#8221; beyond talking only about ideals.</p>
-<p>If <span class="math">\(M\)</span>, <span class="math">\(N\)</span> are <span class="math">\(A\)</span>-modules,
-then there is a natural (componentwise) <span class="math">\(A\)</span>-module structure on <span class="math">\(M \times N\)</span>.
-Similarly there are <span class="math">\(A\)</span>-module structures on cartesian products of more
-components. (For the categorically inclined:
-the cartesian product of finitely many <span class="math">\(A\)</span>-modules, with this
-<span class="math">\(A\)</span>-module structure, is the finite biproduct in the category of all
-<span class="math">\(A\)</span>-modules. With infinitely many components, it is the direct product
-(but the infinite direct sum has to be constructed differently).) As usual,
-repeated product of the <span class="math">\(A\)</span>-module <span class="math">\(M\)</span> is denoted <span class="math">\(M, M^2, M^3 \dots\)</span>, or
-<span class="math">\(M^I\)</span> for arbitrary index sets <span class="math">\(I\)</span>.</p>
-<p>An <span class="math">\(A\)</span>-module <span class="math">\(M\)</span> is called <em>free</em> if it is isomorphic to the <span class="math">\(A\)</span>-module
-<span class="math">\(A^I\)</span> for some (not necessarily finite) index set <span class="math">\(I\)</span> (refer to the next
-section for a definition of isomorphism). The cardinality of <span class="math">\(I\)</span> is called
-the <em>rank</em> of <span class="math">\(M\)</span>; one may prove this is well-defined.
-In general, the AGCA module only works with free modules of finite rank, and
-other closely related modules. The easiest way to create modules is to use
-member methods of the objects they are made up from. For example, let us create
-a free module of rank 4 over the coordinate ring of <span class="math">\(\mathbb{A}^2\)</span>
-we created above, together with a submodule:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">ar</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span> <span class="p">;</span> <span class="n">F</span>
-<span class="go">          4</span>
-<span class="go">ℚ[x, y, z]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">y</span><span class="p">])</span> <span class="p">;</span> <span class="n">S</span>
-<span class="go">╱⎡       2   3⎤              ╲</span>
-<span class="go">╲⎣1, x, x , x ⎦, [0, 1, 0, y]╱</span>
-</pre></div>
-</div>
-<p>Note how python lists can be used as a short-cut notation for module
-elements (vectors). As usual, the <tt class="docutils literal"><span class="pre">convert</span></tt> method can be used to convert
-sympy/python objects into the internal AGCA representation (see detailed
-reference below).</p>
-<p>Here is the detailed documentation of the classes for modules, free modules,
-and submodules:</p>
-<dl class="class">
-<dt id="sympy.polys.agca.modules.Module">
-<em class="property">class </em><tt class="descclassname">sympy.polys.agca.modules.</tt><tt class="descname">Module</tt><big>(</big><em>ring</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#Module"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.Module" title="Permalink to this definition">¶</a></dt>
-<dd><p>Abstract base class for modules.</p>
-<p>Do not instantiate - use ring explicit constructors instead:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">QQ[x]**2</span>
-</pre></div>
-</div>
-<p>Attributes:</p>
-<ul class="simple">
-<li>dtype - type of elements</li>
-<li>ring - containing ring</li>
-</ul>
-<p>Non-implemented methods:</p>
-<ul class="simple">
-<li>submodule</li>
-<li>quotient_module</li>
-<li>is_zero</li>
-<li>is_submodule</li>
-<li>multiply_ideal</li>
-</ul>
-<p>The method convert likely needs to be changed in subclasses.</p>
-<dl class="function">
-<dt id="sympy.polys.agca.modules.Module.contains">
-<tt class="descname">contains</tt><big>(</big><em>self</em>, <em>elem</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#Module.contains"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.Module.contains" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if <tt class="docutils literal"><span class="pre">elem</span></tt> is an element of this module.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.Module.convert">
-<tt class="descname">convert</tt><big>(</big><em>self</em>, <em>elem</em>, <em>M=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#Module.convert"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.Module.convert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">elem</span></tt> into internal representation of this module.</p>
-<p>If <tt class="docutils literal"><span class="pre">M</span></tt> is not None, it should be a module containing it.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.Module.identity_hom">
-<tt class="descname">identity_hom</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#Module.identity_hom"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.Module.identity_hom" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the identity homomorphism on <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.Module.is_submodule">
-<tt class="descname">is_submodule</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#Module.is_submodule"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.Module.is_submodule" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">other</span></tt> is a submodule of <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.Module.is_zero">
-<tt class="descname">is_zero</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#Module.is_zero"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.Module.is_zero" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">self</span></tt> is a zero module.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.Module.multiply_ideal">
-<tt class="descname">multiply_ideal</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#Module.multiply_ideal"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.Module.multiply_ideal" title="Permalink to this definition">¶</a></dt>
-<dd><p>Multiply <tt class="docutils literal"><span class="pre">self</span></tt> by the ideal <tt class="docutils literal"><span class="pre">other</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.Module.quotient_module">
-<tt class="descname">quotient_module</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#Module.quotient_module"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.Module.quotient_module" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generate a quotient module.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.Module.submodule">
-<tt class="descname">submodule</tt><big>(</big><em>self</em>, <em>*gens</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#Module.submodule"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.Module.submodule" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generate a submodule.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.Module.subset">
-<tt class="descname">subset</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#Module.subset"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.Module.subset" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">other</span></tt> is is a subset of <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="o">.</span><span class="n">subset</span><span class="p">([(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)])</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="o">.</span><span class="n">subset</span><span class="p">([(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)])</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.agca.modules.FreeModule">
-<em class="property">class </em><tt class="descclassname">sympy.polys.agca.modules.</tt><tt class="descname">FreeModule</tt><big>(</big><em>ring</em>, <em>rank</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#FreeModule"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.FreeModule" title="Permalink to this definition">¶</a></dt>
-<dd><p>Abstract base class for free modules.</p>
-<p>Additional attributes:</p>
-<ul class="simple">
-<li>rank - rank of the free module</li>
-</ul>
-<p>Non-implemented methods:</p>
-<ul class="simple">
-<li>submodule</li>
-</ul>
-<dl class="function">
-<dt id="sympy.polys.agca.modules.FreeModule.basis">
-<tt class="descname">basis</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#FreeModule.basis"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.FreeModule.basis" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a set of basis elements.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">basis</span><span class="p">()</span>
-<span class="go">([1, 0, 0], [0, 1, 0], [0, 0, 1])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.FreeModule.convert">
-<tt class="descname">convert</tt><big>(</big><em>self</em>, <em>elem</em>, <em>M=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#FreeModule.convert"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.FreeModule.convert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">elem</span></tt> into the internal representation.</p>
-<p>This method is called implicitly whenever computations involve elements
-not in the internal representation.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="o">.</span><span class="n">convert</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="go">[1, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.agca.modules.FreeModule.dtype">
-<tt class="descname">dtype</tt><a class="headerlink" href="#sympy.polys.agca.modules.FreeModule.dtype" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <tt class="xref py py-class docutils literal"><span class="pre">FreeModuleElement</span></tt></p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.FreeModule.identity_hom">
-<tt class="descname">identity_hom</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#FreeModule.identity_hom"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.FreeModule.identity_hom" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the identity homomorphism on <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">identity_hom</span><span class="p">()</span>
-<span class="go">[1, 0]</span>
-<span class="go">[0, 1] : QQ[x]**2 -&gt; QQ[x]**2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.FreeModule.is_submodule">
-<tt class="descname">is_submodule</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#FreeModule.is_submodule"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.FreeModule.is_submodule" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">other</span></tt> is a submodule of <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="o">.</span><span class="n">is_submodule</span><span class="p">(</span><span class="n">F</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="o">.</span><span class="n">is_submodule</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">is_submodule</span><span class="p">(</span><span class="n">F</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.FreeModule.is_zero">
-<tt class="descname">is_zero</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#FreeModule.is_zero"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.FreeModule.is_zero" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">self</span></tt> is a zero module.</p>
-<p>(If, as this implementation assumes, the coefficient ring is not the
-zero ring, then this is equivalent to the rank being zero.)</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">is_zero</span><span class="p">()</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">is_zero</span><span class="p">()</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.FreeModule.multiply_ideal">
-<tt class="descname">multiply_ideal</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#FreeModule.multiply_ideal"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.FreeModule.multiply_ideal" title="Permalink to this definition">¶</a></dt>
-<dd><p>Multiply <tt class="docutils literal"><span class="pre">self</span></tt> by the ideal <tt class="docutils literal"><span class="pre">other</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="o">.</span><span class="n">multiply_ideal</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
-<span class="go">&lt;[x, 0], [0, x]&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.FreeModule.quotient_module">
-<tt class="descname">quotient_module</tt><big>(</big><em>self</em>, <em>submodule</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#FreeModule.quotient_module"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.FreeModule.quotient_module" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a quotient module.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">quotient_module</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">]))</span>
-<span class="go">QQ[x]**2/&lt;[1, x], [x, 2]&gt;</span>
-</pre></div>
-</div>
-<p>Or more conicisely, using the overloaded division operator:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">]]</span>
-<span class="go">QQ[x]**2/&lt;[1, x], [x, 2]&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.agca.modules.SubModule">
-<em class="property">class </em><tt class="descclassname">sympy.polys.agca.modules.</tt><tt class="descname">SubModule</tt><big>(</big><em>gens</em>, <em>container</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#SubModule"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.SubModule" title="Permalink to this definition">¶</a></dt>
-<dd><p>Base class for submodules.</p>
-<p>Attributes:</p>
-<ul class="simple">
-<li>container - containing module</li>
-<li>gens - generators (subset of containing module)</li>
-<li>rank - rank of containing module</li>
-</ul>
-<p>Non-implemented methods:</p>
-<ul class="simple">
-<li>_contains</li>
-<li>_syzygies</li>
-<li>_in_terms_of_generators</li>
-<li>_intersect</li>
-<li>_module_quotient</li>
-</ul>
-<p>Methods that likely need change in subclasses:</p>
-<ul class="simple">
-<li>reduce_element</li>
-</ul>
-<dl class="function">
-<dt id="sympy.polys.agca.modules.SubModule.convert">
-<tt class="descname">convert</tt><big>(</big><em>self</em>, <em>elem</em>, <em>M=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#SubModule.convert"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.SubModule.convert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">elem</span></tt> into the internal represantition.</p>
-<p>Mostly called implicitly.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">convert</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">])</span>
-<span class="go">[2, 2*x]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.SubModule.identity_hom">
-<tt class="descname">identity_hom</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#SubModule.identity_hom"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.SubModule.identity_hom" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the identity homomorphism on <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span><span class="o">.</span><span class="n">identity_hom</span><span class="p">()</span>
-<span class="go">[1, 0]</span>
-<span class="go">[0, 1] : &lt;[x, x]&gt; -&gt; &lt;[x, x]&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.SubModule.in_terms_of_generators">
-<tt class="descname">in_terms_of_generators</tt><big>(</big><em>self</em>, <em>e</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#SubModule.in_terms_of_generators"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.SubModule.in_terms_of_generators" title="Permalink to this definition">¶</a></dt>
-<dd><p>Express element <tt class="docutils literal"><span class="pre">e</span></tt> of <tt class="docutils literal"><span class="pre">self</span></tt> in terms of the generators.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">in_terms_of_generators</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">])</span>
-<span class="go">[-x**2 + x, x**2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.SubModule.inclusion_hom">
-<tt class="descname">inclusion_hom</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#SubModule.inclusion_hom"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.SubModule.inclusion_hom" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a homomorphism representing the inclusion map of <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<p>That is, the natural map from <tt class="docutils literal"><span class="pre">self</span></tt> to <tt class="docutils literal"><span class="pre">self.container</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span><span class="o">.</span><span class="n">inclusion_hom</span><span class="p">()</span>
-<span class="go">[1, 0]</span>
-<span class="go">[0, 1] : &lt;[x, x]&gt; -&gt; QQ[x]**2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.SubModule.intersect">
-<tt class="descname">intersect</tt><big>(</big><em>self</em>, <em>other</em>, <em>**options</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#SubModule.intersect"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.SubModule.intersect" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the intersection of <tt class="docutils literal"><span class="pre">self</span></tt> with submodule <tt class="docutils literal"><span class="pre">other</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="n">y</span><span class="p">,</span> <span class="n">y</span><span class="p">]))</span>
-<span class="go">&lt;[x*y, x*y]&gt;</span>
-</pre></div>
-</div>
-<p>Some implementation allow further options to be passed. Currently, to
-only one implemented is <tt class="docutils literal"><span class="pre">relations=True</span></tt>, in which case the function
-will return a triple <tt class="docutils literal"><span class="pre">(res,</span> <span class="pre">rela,</span> <span class="pre">relb)</span></tt>, where <tt class="docutils literal"><span class="pre">res</span></tt> is the
-intersection module, and <tt class="docutils literal"><span class="pre">rela</span></tt> and <tt class="docutils literal"><span class="pre">relb</span></tt> are lists of coefficient
-vectors, expressing the generators of <tt class="docutils literal"><span class="pre">res</span></tt> in terms of the
-generators of <tt class="docutils literal"><span class="pre">self</span></tt> (<tt class="docutils literal"><span class="pre">rela</span></tt>) and <tt class="docutils literal"><span class="pre">other</span></tt> (<tt class="docutils literal"><span class="pre">relb</span></tt>).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="n">y</span><span class="p">,</span> <span class="n">y</span><span class="p">]),</span> <span class="n">relations</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(&lt;[x*y, x*y]&gt;, [(y,)], [(x,)])</span>
-</pre></div>
-</div>
-<p>The above result says: the intersection module is generated by the
-single element <span class="math">\((-xy, -xy) = -y (x, x) = -x (y, y)\)</span>, where
-<span class="math">\((x, x)\)</span> and <span class="math">\((y, y)\)</span> respectively are the unique generators of
-the two modules being intersected.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.SubModule.is_full_module">
-<tt class="descname">is_full_module</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#SubModule.is_full_module"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.SubModule.is_full_module" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if <tt class="docutils literal"><span class="pre">self</span></tt> is the entire free module.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">is_full_module</span><span class="p">()</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span><span class="o">.</span><span class="n">is_full_module</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.SubModule.is_submodule">
-<tt class="descname">is_submodule</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#SubModule.is_submodule"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.SubModule.is_submodule" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">other</span></tt> is a submodule of <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">is_submodule</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">is_submodule</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">is_submodule</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.SubModule.is_zero">
-<tt class="descname">is_zero</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#SubModule.is_zero"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.SubModule.is_zero" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if <tt class="docutils literal"><span class="pre">self</span></tt> is a zero module.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">is_zero</span><span class="p">()</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">is_zero</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.SubModule.module_quotient">
-<tt class="descname">module_quotient</tt><big>(</big><em>self</em>, <em>other</em>, <em>**options</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#SubModule.module_quotient"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.SubModule.module_quotient" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the module quotient of <tt class="docutils literal"><span class="pre">self</span></tt> by submodule <tt class="docutils literal"><span class="pre">other</span></tt>.</p>
-<p>That is, if <tt class="docutils literal"><span class="pre">self</span></tt> is the module <span class="math">\(M\)</span> and <tt class="docutils literal"><span class="pre">other</span></tt> is <span class="math">\(N\)</span>, then
-return the ideal <span class="math">\(\{f \in R | fN \subset M\}\)</span>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">T</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">.</span><span class="n">module_quotient</span><span class="p">(</span><span class="n">T</span><span class="p">)</span>
-<span class="go">&lt;y&gt;</span>
-</pre></div>
-</div>
-<p>Some implementations allow further options to be passed. Currently, the
-only one implemented is <tt class="docutils literal"><span class="pre">relations=True</span></tt>, which may only be passed
-if <tt class="docutils literal"><span class="pre">other</span></tt> is prinicipal. In this case the function
-will return a pair <tt class="docutils literal"><span class="pre">(res,</span> <span class="pre">rel)</span></tt> where <tt class="docutils literal"><span class="pre">res</span></tt> is the ideal, and
-<tt class="docutils literal"><span class="pre">rel</span></tt> is a list of coefficient vectors, expressing the generators of
-the ideal, multiplied by the generator of <tt class="docutils literal"><span class="pre">other</span></tt> in terms of
-generators of <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">.</span><span class="n">module_quotient</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">relations</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(&lt;y&gt;, [[1]])</span>
-</pre></div>
-</div>
-<p>This means that the quotient ideal is generated by the single element
-<span class="math">\(y\)</span>, and that <span class="math">\(y (x, x) = 1 (xy, xy)\)</span>, <span class="math">\((x, x)\)</span> and <span class="math">\((xy, xy)\)</span> being
-the generators of <span class="math">\(T\)</span> and <span class="math">\(S\)</span>, respectively.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.SubModule.multiply_ideal">
-<tt class="descname">multiply_ideal</tt><big>(</big><em>self</em>, <em>I</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#SubModule.multiply_ideal"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.SubModule.multiply_ideal" title="Permalink to this definition">¶</a></dt>
-<dd><p>Multiply <tt class="docutils literal"><span class="pre">self</span></tt> by the ideal <tt class="docutils literal"><span class="pre">I</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span><span class="o">*</span><span class="n">M</span>
-<span class="go">&lt;[x**2, x**2]&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.SubModule.quotient_module">
-<tt class="descname">quotient_module</tt><big>(</big><em>self</em>, <em>other</em>, <em>**opts</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#SubModule.quotient_module"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.SubModule.quotient_module" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a quotient module.</p>
-<p>This is the same as taking a submodule of a quotient of the containing
-module.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S1</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S2</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S1</span><span class="o">.</span><span class="n">quotient_module</span><span class="p">(</span><span class="n">S2</span><span class="p">)</span>
-<span class="go">&lt;[x, 1] + &lt;[x**2, x]&gt;&gt;</span>
-</pre></div>
-</div>
-<p>Or more coincisely, using the overloaded division operator:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span> <span class="o">/</span> <span class="p">[(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)]</span>
-<span class="go">&lt;[x, 1] + &lt;[x**2, x]&gt;&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.SubModule.reduce_element">
-<tt class="descname">reduce_element</tt><big>(</big><em>self</em>, <em>x</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#SubModule.reduce_element"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.SubModule.reduce_element" title="Permalink to this definition">¶</a></dt>
-<dd><p>Reduce the element <tt class="docutils literal"><span class="pre">x</span></tt> of our ring modulo the ideal <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<p>Here &#8220;reduce&#8221; has no specific meaning, it could return a unique normal
-form, simplify the expression a bit, or just do nothing.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.SubModule.submodule">
-<tt class="descname">submodule</tt><big>(</big><em>self</em>, <em>*gens</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#SubModule.submodule"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.SubModule.submodule" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generate a submodule.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span>
-<span class="go">&lt;[x**2, x]&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.SubModule.syzygy_module">
-<tt class="descname">syzygy_module</tt><big>(</big><em>self</em>, <em>**opts</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#SubModule.syzygy_module"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.SubModule.syzygy_module" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the syzygy module of the generators of <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<p>Suppose <span class="math">\(M\)</span> is generated by <span class="math">\(f_1, \dots, f_n\)</span> over the ring
-<span class="math">\(R\)</span>. Consider the homomorphism <span class="math">\(\phi: R^n \to M\)</span>, given by
-sending <span class="math">\((r_1, \dots, r_n) \to r_1 f_1 + \dots + r_n f_n\)</span>.
-The syzygy module is defined to be the kernel of <span class="math">\(\phi\)</span>.</p>
-<p>The syzygy module is zero iff the generators generate freely a free
-submodule:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">syzygy_module</span><span class="p">()</span><span class="o">.</span><span class="n">is_zero</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>A slightly more interesting example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">],</span> <span class="p">[</span><span class="n">y</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="n">y</span><span class="p">,</span> <span class="o">-</span><span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">syzygy_module</span><span class="p">()</span> <span class="o">==</span> <span class="n">S</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.SubModule.union">
-<tt class="descname">union</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#SubModule.union"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.SubModule.union" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the module generated by the union of <tt class="docutils literal"><span class="pre">self</span></tt> and <tt class="docutils literal"><span class="pre">other</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="p">])</span> <span class="c"># &lt;x(x+1)&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span> <span class="c"># &lt;(x-1)(x+1)&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="n">N</span><span class="p">)</span> <span class="o">==</span> <span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="n">x</span><span class="o">+</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<p>Ideals are created very similarly to modules. For example, let&#8217;s verify
-that the nodal cubic is indeed singular at the origin:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">lr</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">==</span> <span class="n">lr</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">False</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">==</span> <span class="n">lr</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p>We are using here the fact that a curve is non-singular at a point if and only
-if the maximal ideal of the local ring is principal, and that in this case at
-least one of <span class="math">\(x\)</span> and <span class="math">\(y\)</span> must be generators.</p>
-<p>This is the detailed documentation of the class ideal. Please note that most
-of the methods regarding properties of ideals (primality etc.) are not yet
-implemented.</p>
-<dl class="class">
-<dt id="sympy.polys.agca.ideals.Ideal">
-<em class="property">class </em><tt class="descclassname">sympy.polys.agca.ideals.</tt><tt class="descname">Ideal</tt><big>(</big><em>ring</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/ideals.html#Ideal"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.ideals.Ideal" title="Permalink to this definition">¶</a></dt>
-<dd><p>Abstract base class for ideals.</p>
-<p>Do not instantiate - use explicit constructors in the ring class instead:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">&lt;x + 1&gt;</span>
-</pre></div>
-</div>
-<p>Attributes</p>
-<ul class="simple">
-<li>ring - the ring this ideal belongs to</li>
-</ul>
-<p>Non-implemented methods:</p>
-<ul class="simple">
-<li>_contains_elem</li>
-<li>_contains_ideal</li>
-<li>_quotient</li>
-<li>_intersect</li>
-<li>_union</li>
-<li>_product</li>
-<li>is_whole_ring</li>
-<li>is_zero</li>
-<li>is_prime, is_maximal, is_primary, is_radical</li>
-<li>is_principal</li>
-<li>height, depth</li>
-<li>radical</li>
-</ul>
-<p>Methods that likely should be overridden in subclasses:</p>
-<ul class="simple">
-<li>reduce_element</li>
-</ul>
-<dl class="function">
-<dt id="sympy.polys.agca.ideals.Ideal.contains">
-<tt class="descname">contains</tt><big>(</big><em>self</em>, <em>elem</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/ideals.html#Ideal.contains"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.ideals.Ideal.contains" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if <tt class="docutils literal"><span class="pre">elem</span></tt> is an element of this ideal.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.ideals.Ideal.depth">
-<tt class="descname">depth</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/ideals.html#Ideal.depth"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.ideals.Ideal.depth" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the depth of <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.ideals.Ideal.height">
-<tt class="descname">height</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/ideals.html#Ideal.height"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.ideals.Ideal.height" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the height of <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.ideals.Ideal.intersect">
-<tt class="descname">intersect</tt><big>(</big><em>self</em>, <em>J</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/ideals.html#Ideal.intersect"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.ideals.Ideal.intersect" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the intersection of self with ideal J.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="n">R</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>
-<span class="go">&lt;x*y&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.ideals.Ideal.is_maximal">
-<tt class="descname">is_maximal</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/ideals.html#Ideal.is_maximal"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.ideals.Ideal.is_maximal" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if <tt class="docutils literal"><span class="pre">self</span></tt> is a maximal ideal.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.ideals.Ideal.is_primary">
-<tt class="descname">is_primary</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/ideals.html#Ideal.is_primary"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.ideals.Ideal.is_primary" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if <tt class="docutils literal"><span class="pre">self</span></tt> is a primary ideal.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.ideals.Ideal.is_prime">
-<tt class="descname">is_prime</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/ideals.html#Ideal.is_prime"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.ideals.Ideal.is_prime" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if <tt class="docutils literal"><span class="pre">self</span></tt> is a prime ideal.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.ideals.Ideal.is_principal">
-<tt class="descname">is_principal</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/ideals.html#Ideal.is_principal"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.ideals.Ideal.is_principal" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if <tt class="docutils literal"><span class="pre">self</span></tt> is a principal ideal.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.ideals.Ideal.is_radical">
-<tt class="descname">is_radical</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/ideals.html#Ideal.is_radical"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.ideals.Ideal.is_radical" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if <tt class="docutils literal"><span class="pre">self</span></tt> is a radical ideal.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.ideals.Ideal.is_whole_ring">
-<tt class="descname">is_whole_ring</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/ideals.html#Ideal.is_whole_ring"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.ideals.Ideal.is_whole_ring" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if <tt class="docutils literal"><span class="pre">self</span></tt> is the whole ring.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.ideals.Ideal.is_zero">
-<tt class="descname">is_zero</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/ideals.html#Ideal.is_zero"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.ideals.Ideal.is_zero" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if <tt class="docutils literal"><span class="pre">self</span></tt> is the zero ideal.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.ideals.Ideal.product">
-<tt class="descname">product</tt><big>(</big><em>self</em>, <em>J</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/ideals.html#Ideal.product"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.ideals.Ideal.product" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the ideal product of <tt class="docutils literal"><span class="pre">self</span></tt> and <tt class="docutils literal"><span class="pre">J</span></tt>.</p>
-<p>That is, compute the ideal generated by products <span class="math">\(xy\)</span>, for <span class="math">\(x\)</span> an element
-of <tt class="docutils literal"><span class="pre">self</span></tt> and <span class="math">\(y \in J\)</span>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">]</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">product</span><span class="p">(</span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">]</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>
-<span class="go">&lt;x*y&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.ideals.Ideal.quotient">
-<tt class="descname">quotient</tt><big>(</big><em>self</em>, <em>J</em>, <em>**opts</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/ideals.html#Ideal.quotient"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.ideals.Ideal.quotient" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the ideal quotient of <tt class="docutils literal"><span class="pre">self</span></tt> by <tt class="docutils literal"><span class="pre">J</span></tt>.</p>
-<p>That is, if <tt class="docutils literal"><span class="pre">self</span></tt> is the ideal <span class="math">\(I\)</span>, compute the set
-<span class="math">\(I : J = \{x \in R | xJ \subset I \}\)</span>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">quotient</span><span class="p">(</span><span class="n">R</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">&lt;y&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.ideals.Ideal.radical">
-<tt class="descname">radical</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/ideals.html#Ideal.radical"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.ideals.Ideal.radical" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the radical of <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.ideals.Ideal.reduce_element">
-<tt class="descname">reduce_element</tt><big>(</big><em>self</em>, <em>x</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/ideals.html#Ideal.reduce_element"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.ideals.Ideal.reduce_element" title="Permalink to this definition">¶</a></dt>
-<dd><p>Reduce the element <tt class="docutils literal"><span class="pre">x</span></tt> of our ring modulo the ideal <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<p>Here &#8220;reduce&#8221; has no specific meaning: it could return a unique normal
-form, simplify the expression a bit, or just do nothing.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.ideals.Ideal.saturate">
-<tt class="descname">saturate</tt><big>(</big><em>self</em>, <em>J</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/ideals.html#Ideal.saturate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.ideals.Ideal.saturate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the ideal saturation of <tt class="docutils literal"><span class="pre">self</span></tt> by <tt class="docutils literal"><span class="pre">J</span></tt>.</p>
-<p>That is, if <tt class="docutils literal"><span class="pre">self</span></tt> is the ideal <span class="math">\(I\)</span>, compute the set
-<span class="math">\(I : J^\infty = \{x \in R | xJ^n \subset I \text{ for some } n\}\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.ideals.Ideal.subset">
-<tt class="descname">subset</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/ideals.html#Ideal.subset"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.ideals.Ideal.subset" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">other</span></tt> is is a subset of <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<p>Here <tt class="docutils literal"><span class="pre">other</span></tt> may be an ideal.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span><span class="o">.</span><span class="n">subset</span><span class="p">([</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span><span class="o">.</span><span class="n">subset</span><span class="p">([</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">I</span><span class="o">.</span><span class="n">subset</span><span class="p">(</span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.ideals.Ideal.union">
-<tt class="descname">union</tt><big>(</big><em>self</em>, <em>J</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/ideals.html#Ideal.union"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.ideals.Ideal.union" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the ideal generated by the union of <tt class="docutils literal"><span class="pre">self</span></tt> and <tt class="docutils literal"><span class="pre">J</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">ideal</span><span class="p">((</span><span class="n">x</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span> <span class="o">==</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<p>If <span class="math">\(M\)</span> is an <span class="math">\(A\)</span>-module and <span class="math">\(N\)</span> is an <span class="math">\(A\)</span>-submodule, we can define two elements
-<span class="math">\(x\)</span> and <span class="math">\(y\)</span> of <span class="math">\(M\)</span> to be equivalent if <span class="math">\(x - y \in N\)</span>. The set of equivalence
-classes is written <span class="math">\(M/N\)</span>, and has a natural <span class="math">\(A\)</span>-module structure. This is
-called the quotient module of <span class="math">\(M\)</span> by <span class="math">\(N\)</span>. If <span class="math">\(K\)</span> is a submodule of <span class="math">\(M\)</span>
-containing <span class="math">\(N\)</span>, then <span class="math">\(K/N\)</span> is in a natural way a submodule of <span class="math">\(M/N\)</span>. Such a
-module is called a subquotient. Here is the documentation of quotient and
-subquotient modules:</p>
-<dl class="class">
-<dt id="sympy.polys.agca.modules.QuotientModule">
-<em class="property">class </em><tt class="descclassname">sympy.polys.agca.modules.</tt><tt class="descname">QuotientModule</tt><big>(</big><em>ring</em>, <em>base</em>, <em>submodule</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#QuotientModule"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.QuotientModule" title="Permalink to this definition">¶</a></dt>
-<dd><p>Class for quotient modules.</p>
-<p>Do not instantiate this directly. For subquotients, see the
-SubQuotientModule class.</p>
-<p>Attributes:</p>
-<ul class="simple">
-<li>base - the base module we are a quotient of</li>
-<li>killed_module - the submodule we are quotienting by</li>
-<li>rank of the base</li>
-</ul>
-<dl class="function">
-<dt id="sympy.polys.agca.modules.QuotientModule.convert">
-<tt class="descname">convert</tt><big>(</big><em>self</em>, <em>elem</em>, <em>M=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#QuotientModule.convert"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.QuotientModule.convert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">elem</span></tt> into the internal representation.</p>
-<p>This method is called implicitly whenever computations involve elements
-not in the internal representation.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="p">[(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="o">.</span><span class="n">convert</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="go">[1, 0] + &lt;[1, 2], [1, x]&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.agca.modules.QuotientModule.dtype">
-<tt class="descname">dtype</tt><a class="headerlink" href="#sympy.polys.agca.modules.QuotientModule.dtype" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <tt class="xref py py-class docutils literal"><span class="pre">QuotientModuleElement</span></tt></p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.QuotientModule.identity_hom">
-<tt class="descname">identity_hom</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#QuotientModule.identity_hom"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.QuotientModule.identity_hom" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the identity homomorphism on <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="p">[(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">identity_hom</span><span class="p">()</span>
-<span class="go">[1, 0]</span>
-<span class="go">[0, 1] : QQ[x]**2/&lt;[1, 2], [1, x]&gt; -&gt; QQ[x]**2/&lt;[1, 2], [1, x]&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.QuotientModule.is_submodule">
-<tt class="descname">is_submodule</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#QuotientModule.is_submodule"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.QuotientModule.is_submodule" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if <tt class="docutils literal"><span class="pre">other</span></tt> is a submodule of <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="p">[(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">Q</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span><span class="o">.</span><span class="n">is_submodule</span><span class="p">(</span><span class="n">S</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">.</span><span class="n">is_submodule</span><span class="p">(</span><span class="n">Q</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.QuotientModule.is_zero">
-<tt class="descname">is_zero</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#QuotientModule.is_zero"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.QuotientModule.is_zero" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if <tt class="docutils literal"><span class="pre">self</span></tt> is a zero module.</p>
-<p>This happens if and only if the base module is the same as the
-submodule being killed.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">F</span><span class="o">/</span><span class="p">[(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)])</span><span class="o">.</span><span class="n">is_zero</span><span class="p">()</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">F</span><span class="o">/</span><span class="p">[(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)])</span><span class="o">.</span><span class="n">is_zero</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.QuotientModule.quotient_hom">
-<tt class="descname">quotient_hom</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#QuotientModule.quotient_hom"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.QuotientModule.quotient_hom" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the quotient homomorphism to <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<p>That is, return a homomorphism representing the natural map from
-<tt class="docutils literal"><span class="pre">self.base</span></tt> to <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="p">[(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">quotient_hom</span><span class="p">()</span>
-<span class="go">[1, 0]</span>
-<span class="go">[0, 1] : QQ[x]**2 -&gt; QQ[x]**2/&lt;[1, 2], [1, x]&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.QuotientModule.submodule">
-<tt class="descname">submodule</tt><big>(</big><em>self</em>, <em>*gens</em>, <em>**opts</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#QuotientModule.submodule"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.QuotientModule.submodule" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generate a submodule.</p>
-<p>This is the same as taking a quotient of a submodule of the base
-module.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="p">[(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="go">&lt;[x, 0] + &lt;[x, x]&gt;&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.agca.modules.SubQuotientModule">
-<em class="property">class </em><tt class="descclassname">sympy.polys.agca.modules.</tt><tt class="descname">SubQuotientModule</tt><big>(</big><em>gens</em>, <em>container</em>, <em>**opts</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#SubQuotientModule"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.SubQuotientModule" title="Permalink to this definition">¶</a></dt>
-<dd><p>Submodule of a quotient module.</p>
-<p>Equivalently, quotient module of a submodule.</p>
-<p>Do not instantiate this, instead use the submodule or quotient_module
-constructing methods:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span> <span class="o">=</span> <span class="n">F</span><span class="o">/</span><span class="p">[(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">/</span><span class="p">[(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)]</span> <span class="o">==</span> <span class="n">Q</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="mi">5</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>Attributes:</p>
-<ul class="simple">
-<li>base - base module we are quotient of</li>
-<li>killed_module - submodule we are quotienting by</li>
-</ul>
-<dl class="function">
-<dt id="sympy.polys.agca.modules.SubQuotientModule.is_full_module">
-<tt class="descname">is_full_module</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#SubQuotientModule.is_full_module"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.SubQuotientModule.is_full_module" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if <tt class="docutils literal"><span class="pre">self</span></tt> is the entire free module.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span><span class="o">.</span><span class="n">is_full_module</span><span class="p">()</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span><span class="o">.</span><span class="n">is_full_module</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.modules.SubQuotientModule.quotient_hom">
-<tt class="descname">quotient_hom</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/modules.html#SubQuotientModule.quotient_hom"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.modules.SubQuotientModule.quotient_hom" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the quotient homomorphism to self.</p>
-<p>That is, return the natural map from <tt class="docutils literal"><span class="pre">self.base</span></tt> to <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="p">(</span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="p">[(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)])</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="o">.</span><span class="n">quotient_hom</span><span class="p">()</span>
-<span class="go">[1, 0]</span>
-<span class="go">[0, 1] : &lt;[1, 0], [1, x]&gt; -&gt; &lt;[1, 0] + &lt;[1, x]&gt;, [1, x] + &lt;[1, x]&gt;&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="module-homomorphisms-and-syzygies">
-<h3>Module Homomorphisms and Syzygies<a class="headerlink" href="#module-homomorphisms-and-syzygies" title="Permalink to this headline">¶</a></h3>
-<p>Let <span class="math">\(M\)</span> and <span class="math">\(N\)</span> be <span class="math">\(A\)</span>-modules. A mapping <span class="math">\(f: M \to N\)</span> satisfying various
-obvious properties (see <a class="reference internal" href="../polys/literature.html#atiyah69">[Atiyah69]</a>) is called an <span class="math">\(A\)</span>-module homomorphism.
-In this case <span class="math">\(M\)</span> is called the <em>domain</em> and <em>N</em> the <em>codomain</em>. The
-set <span class="math">\(\{x \in M | f(x) = 0\}\)</span> is called the <em>kernel</em> <span class="math">\(ker(f)\)</span>, whereas the
-set <span class="math">\(\{f(x) | x \in M\}\)</span> is called the <em>image</em> <span class="math">\(im(f)\)</span>.
-The kernel is a submodule of <span class="math">\(M\)</span>, the image is a submodule of <span class="math">\(N\)</span>.
-The homomorphism <span class="math">\(f\)</span> is injective if and only if <span class="math">\(ker(f) = 0\)</span> and surjective
-if and only if <span class="math">\(im(f) = N\)</span>.
-A bijective homomorphism is called an <em>isomorphism</em>. Equivalently, <span class="math">\(ker(f) = 0\)</span>
-and <span class="math">\(im(f) = N\)</span>. (A related notion, which currently has no special name in
-the AGCA module, is that of the <em>cokernel</em>, <span class="math">\(coker(f) = N/im(f)\)</span>.)</p>
-<p>Suppose now <span class="math">\(M\)</span> is an <span class="math">\(A\)</span>-module. <span class="math">\(M\)</span> is called <em>finitely generated</em> if there
-exists a surjective homomorphism <span class="math">\(A^n \to M\)</span> for some <span class="math">\(n\)</span>. If such a morphism
-<span class="math">\(f\)</span> is chosen, the images of the standard basis of <span class="math">\(A^n\)</span> are called the
-<em>generators</em> of <span class="math">\(M\)</span>. The module <span class="math">\(ker(f)\)</span> is called <em>syzygy module</em> with respect
-to the generators. A module is called <em>finitely presented</em> if it is finitely
-generated with a finitely generated syzygy module. The class of finitely
-presented modules is essentially the largest class we can hope to be able to
-meaningfully compute in.</p>
-<p>It is an important theorem that, for all the rings we are considering,
-all submodules of finitely generated modules are finitely generated, and hence
-finitely generated and finitely presented modules are the same.</p>
-<p>The notion of syzygies, while it may first seem rather abstract, is actually
-very computational. This is because there exist (fairly easy) algorithms for
-computing them, and more general questions (kernels, intersections, ...) are
-often reduced to syzygy computation.</p>
-<p>Let us say a few words about the definition of homomorphisms in the AGCA
-module. Suppose first that <span class="math">\(f : M \to N\)</span> is an arbitrary morphism of
-<span class="math">\(A\)</span>-modules. Then if <span class="math">\(K\)</span> is a submodule of <span class="math">\(M\)</span>, <span class="math">\(f\)</span> naturally defines a new
-homomorphism <span class="math">\(g: K \to N\)</span> (via <span class="math">\(g(x) = f(x)\)</span>), called the <em>restriction</em> of
-<span class="math">\(f\)</span> to <span class="math">\(K\)</span>. If now <span class="math">\(K\)</span> contained in the kernel of
-<span class="math">\(f\)</span>, then moreover <span class="math">\(f\)</span> defines in a natural homomorphism <span class="math">\(g: M/K \to N\)</span>
-(same formula as above!), and we say that <span class="math">\(f\)</span> <em>descends</em> to <span class="math">\(M/K\)</span>.
-Similarly, if <span class="math">\(L\)</span> is a submodule of <span class="math">\(N\)</span>, there is a natural homomorphism
-<span class="math">\(g: M \to N/L\)</span>, we say that <span class="math">\(g\)</span> <em>factors</em> through <span class="math">\(f\)</span>. Finally, if now <span class="math">\(L\)</span>
-contains the image of <span class="math">\(f\)</span>, then there is a natural homomorphism <span class="math">\(g: M \to L\)</span>
-(defined, again, by the same formula), and we say <span class="math">\(g\)</span> is obtained from <span class="math">\(f\)</span>
-by restriction of codomain. Observe also that each of these four operations
-is reversible, in the sense that given <span class="math">\(g\)</span>, one can always (non-uniquely)
-find <span class="math">\(f\)</span> such that <span class="math">\(g\)</span> is obtained from <span class="math">\(f\)</span> in the above way.</p>
-<p>Note that all modules implemented in AGCA are obtained from free modules by
-taking a succession of submodules and quotients. Hence, in order to explain
-how to define a homomorphism between arbitrary modules, in light of the above,
-we need only explain how to define homomorphisms of free modules. But,
-essentially by the definition of free module, a homomorphism from a free module
-<span class="math">\(A^n\)</span> to any module <span class="math">\(M\)</span> is precisely the same as giving <span class="math">\(n\)</span> elements of <span class="math">\(M\)</span>
-(the images of the standard basis), and giving an element of a free module
-<span class="math">\(A^m\)</span> is precisely the same as giving <span class="math">\(m\)</span> elements of <span class="math">\(A\)</span>. Hence a
-homomorphism of free modules <span class="math">\(A^n \to A^m\)</span> can be specified via a matrix,
-entirely analogously to the case of vector spaces.</p>
-<p>The functions <tt class="docutils literal"><span class="pre">restrict_domain</span></tt> etc. of the class <tt class="docutils literal"><span class="pre">Homomorphism</span></tt> can be
-used
-to carry out the operations described above, and homomorphisms of free modules
-can in principle be instantiated by hand. Since these operations are so common,
-there is a convenience function <tt class="docutils literal"><span class="pre">homomorphism</span></tt> to define a homomorphism
-between arbitrary modules via the method outlined above. It is essentially
-the only way homomorphisms need ever be created by the user.</p>
-<dl class="function">
-<dt id="sympy.polys.agca.homomorphisms.homomorphism">
-<tt class="descclassname">sympy.polys.agca.homomorphisms.</tt><tt class="descname">homomorphism</tt><big>(</big><em>domain</em>, <em>codomain</em>, <em>matrix</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/homomorphisms.html#homomorphism"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.homomorphisms.homomorphism" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a homomorphism object.</p>
-<p>This function tries to build a homomorphism from <tt class="docutils literal"><span class="pre">domain</span></tt> to <tt class="docutils literal"><span class="pre">codomain</span></tt>
-via the matrix <tt class="docutils literal"><span class="pre">matrix</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span><span class="p">,</span> <span class="n">homomorphism</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">T</span> <span class="o">=</span> <span class="n">R</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>If <tt class="docutils literal"><span class="pre">domain</span></tt> is a free module generated by <span class="math">\(e_1, \dots, e_n\)</span>, then
-<tt class="docutils literal"><span class="pre">matrix</span></tt> should be an n-element iterable <span class="math">\((b_1, \dots, b_n)\)</span> where
-the <span class="math">\(b_i\)</span> are elements of <tt class="docutils literal"><span class="pre">codomain</span></tt>. The constructed homomorphism is the
-unique homomorphism sending <span class="math">\(e_i\)</span> to <span class="math">\(b_i\)</span>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">R</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="n">homomorphism</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span>
-<span class="go">[1, x**2]</span>
-<span class="go">[x,    0] : QQ[x]**2 -&gt; QQ[x]**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="go">[1, x]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">[x**2, 0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">[x**2 + 1, x]</span>
-</pre></div>
-</div>
-<p>If <tt class="docutils literal"><span class="pre">domain</span></tt> is a submodule of a free module, them <tt class="docutils literal"><span class="pre">matrix</span></tt> determines
-a homomoprhism from the containing free module to <tt class="docutils literal"><span class="pre">codomain</span></tt>, and the
-homomorphism returned is obtained by restriction to <tt class="docutils literal"><span class="pre">domain</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">homomorphism</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="go">[1, x**2]</span>
-<span class="go">[x,    0] : &lt;[1, 0], [0, x]&gt; -&gt; QQ[x]**2</span>
-</pre></div>
-</div>
-<p>If <tt class="docutils literal"><span class="pre">domain</span></tt> is a (sub)quotient <span class="math">\(N/K\)</span>, then <tt class="docutils literal"><span class="pre">matrix</span></tt> determines a
-homomorphism from <span class="math">\(N\)</span> to <tt class="docutils literal"><span class="pre">codomain</span></tt>. If the kernel contains <span class="math">\(K\)</span>, this
-homomorphism descends to <tt class="docutils literal"><span class="pre">domain</span></tt> and is returned; otherwise an exception
-is raised.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">homomorphism</span><span class="p">(</span><span class="n">S</span><span class="o">/</span><span class="p">[(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)],</span> <span class="n">T</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">[</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="go">[0, x**2]</span>
-<span class="go">[0,    0] : &lt;[1, 0] + &lt;[1, 0]&gt;, [0, x] + &lt;[1, 0]&gt;, [1, 0] + &lt;[1, 0]&gt;&gt; -&gt; QQ[x]**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">homomorphism</span><span class="p">(</span><span class="n">S</span><span class="o">/</span><span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">)],</span> <span class="n">T</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">[</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">ValueError</span>: <span class="n">kernel &lt;[1, 0], [0, 0]&gt; must contain sm, got &lt;[0,x]&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<p>Finally, here is the detailed reference of the actual homomorphism class:</p>
-<dl class="class">
-<dt id="sympy.polys.agca.homomorphisms.ModuleHomomorphism">
-<em class="property">class </em><tt class="descclassname">sympy.polys.agca.homomorphisms.</tt><tt class="descname">ModuleHomomorphism</tt><big>(</big><em>domain</em>, <em>codomain</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/homomorphisms.html#ModuleHomomorphism"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.homomorphisms.ModuleHomomorphism" title="Permalink to this definition">¶</a></dt>
-<dd><p>Abstract base class for module homomoprhisms. Do not instantiate.</p>
-<p>Instead, use the <tt class="docutils literal"><span class="pre">homomorphism</span></tt> function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span><span class="p">,</span> <span class="n">homomorphism</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">homomorphism</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="go">[1, 0]</span>
-<span class="go">[0, 1] : QQ[x]**2 -&gt; QQ[x]**2</span>
-</pre></div>
-</div>
-<p>Attributes:</p>
-<ul class="simple">
-<li>ring - the ring over which we are considering modules</li>
-<li>domain - the domain module</li>
-<li>codomain - the codomain module</li>
-<li>_ker - cached kernel</li>
-<li>_img - cachd image</li>
-</ul>
-<p>Non-implemented methods:</p>
-<ul class="simple">
-<li>_kernel</li>
-<li>_image</li>
-<li>_restrict_domain</li>
-<li>_restrict_codomain</li>
-<li>_quotient_domain</li>
-<li>_quotient_codomain</li>
-<li>_apply</li>
-<li>_mul_scalar</li>
-<li>_compose</li>
-<li>_add</li>
-</ul>
-<dl class="function">
-<dt id="sympy.polys.agca.homomorphisms.ModuleHomomorphism.image">
-<tt class="descname">image</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/homomorphisms.html#ModuleHomomorphism.image"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.homomorphisms.ModuleHomomorphism.image" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the image of <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<p>That is, if <tt class="docutils literal"><span class="pre">self</span></tt> is the homomorphism <span class="math">\(\phi: M \to N\)</span>, then compute
-<span class="math">\(im(\phi) = \{\phi(x) | x \in M \}\)</span>.  This is a submodule of <span class="math">\(N\)</span>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">homomorphism</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">homomorphism</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span><span class="o">.</span><span class="n">image</span><span class="p">()</span> <span class="o">==</span> <span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.homomorphisms.ModuleHomomorphism.is_injective">
-<tt class="descname">is_injective</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/homomorphisms.html#ModuleHomomorphism.is_injective"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.homomorphisms.ModuleHomomorphism.is_injective" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if <tt class="docutils literal"><span class="pre">self</span></tt> is injective.</p>
-<p>That is, check if the elements of the domain are mapped to the same
-codomain element.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">homomorphism</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="n">homomorphism</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span><span class="o">.</span><span class="n">is_injective</span><span class="p">()</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span><span class="o">.</span><span class="n">quotient_domain</span><span class="p">(</span><span class="n">h</span><span class="o">.</span><span class="n">kernel</span><span class="p">())</span><span class="o">.</span><span class="n">is_injective</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.homomorphisms.ModuleHomomorphism.is_isomorphism">
-<tt class="descname">is_isomorphism</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/homomorphisms.html#ModuleHomomorphism.is_isomorphism"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.homomorphisms.ModuleHomomorphism.is_isomorphism" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if <tt class="docutils literal"><span class="pre">self</span></tt> is an isomorphism.</p>
-<p>That is, check if every element of the codomain has precisely one
-preimage. Equivalently, <tt class="docutils literal"><span class="pre">self</span></tt> is both injective and surjective.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">homomorphism</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="n">homomorphism</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="n">h</span><span class="o">.</span><span class="n">restrict_codomain</span><span class="p">(</span><span class="n">h</span><span class="o">.</span><span class="n">image</span><span class="p">())</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span><span class="o">.</span><span class="n">is_isomorphism</span><span class="p">()</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span><span class="o">.</span><span class="n">quotient_domain</span><span class="p">(</span><span class="n">h</span><span class="o">.</span><span class="n">kernel</span><span class="p">())</span><span class="o">.</span><span class="n">is_isomorphism</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.homomorphisms.ModuleHomomorphism.is_surjective">
-<tt class="descname">is_surjective</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/homomorphisms.html#ModuleHomomorphism.is_surjective"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.homomorphisms.ModuleHomomorphism.is_surjective" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if <tt class="docutils literal"><span class="pre">self</span></tt> is surjective.</p>
-<p>That is, check if every element of the codomain has at least one
-preimage.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">homomorphism</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="n">homomorphism</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span><span class="o">.</span><span class="n">is_surjective</span><span class="p">()</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span><span class="o">.</span><span class="n">restrict_codomain</span><span class="p">(</span><span class="n">h</span><span class="o">.</span><span class="n">image</span><span class="p">())</span><span class="o">.</span><span class="n">is_surjective</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.homomorphisms.ModuleHomomorphism.is_zero">
-<tt class="descname">is_zero</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/homomorphisms.html#ModuleHomomorphism.is_zero"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.homomorphisms.ModuleHomomorphism.is_zero" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if <tt class="docutils literal"><span class="pre">self</span></tt> is a zero morphism.</p>
-<p>That is, check if every element of the domain is mapped to zero
-under self.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">homomorphism</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="n">homomorphism</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span><span class="o">.</span><span class="n">is_zero</span><span class="p">()</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span><span class="o">.</span><span class="n">restrict_domain</span><span class="p">(</span><span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">())</span><span class="o">.</span><span class="n">is_zero</span><span class="p">()</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span><span class="o">.</span><span class="n">quotient_codomain</span><span class="p">(</span><span class="n">h</span><span class="o">.</span><span class="n">image</span><span class="p">())</span><span class="o">.</span><span class="n">is_zero</span><span class="p">()</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.homomorphisms.ModuleHomomorphism.kernel">
-<tt class="descname">kernel</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/homomorphisms.html#ModuleHomomorphism.kernel"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.homomorphisms.ModuleHomomorphism.kernel" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the kernel of <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<p>That is, if <tt class="docutils literal"><span class="pre">self</span></tt> is the homomorphism <span class="math">\(\phi: M \to N\)</span>, then compute
-<span class="math">\(ker(\phi) = \{x \in M | \phi(x) = 0\}\)</span>.  This is a submodule of <span class="math">\(M\)</span>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">homomorphism</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">homomorphism</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span><span class="o">.</span><span class="n">kernel</span><span class="p">()</span>
-<span class="go">&lt;[x, -1]&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.homomorphisms.ModuleHomomorphism.quotient_codomain">
-<tt class="descname">quotient_codomain</tt><big>(</big><em>self</em>, <em>sm</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/homomorphisms.html#ModuleHomomorphism.quotient_codomain"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.homomorphisms.ModuleHomomorphism.quotient_codomain" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return <tt class="docutils literal"><span class="pre">self</span></tt> with codomain replaced by <tt class="docutils literal"><span class="pre">codomain/sm</span></tt>.</p>
-<p>Here <tt class="docutils literal"><span class="pre">sm</span></tt> must be a submodule of <tt class="docutils literal"><span class="pre">self.codomain</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">homomorphism</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="n">homomorphism</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span>
-<span class="go">[1, x]</span>
-<span class="go">[0, 0] : QQ[x]**2 -&gt; QQ[x]**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span><span class="o">.</span><span class="n">quotient_codomain</span><span class="p">(</span><span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]))</span>
-<span class="go">[1, x]</span>
-<span class="go">[0, 0] : QQ[x]**2 -&gt; QQ[x]**2/&lt;[1, 1]&gt;</span>
-</pre></div>
-</div>
-<p>This is the same as composing with the quotient map on the left:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">F</span><span class="o">/</span><span class="p">[(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)])</span><span class="o">.</span><span class="n">quotient_hom</span><span class="p">()</span> <span class="o">*</span> <span class="n">h</span>
-<span class="go">[1, x]</span>
-<span class="go">[0, 0] : QQ[x]**2 -&gt; QQ[x]**2/&lt;[1, 1]&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.homomorphisms.ModuleHomomorphism.quotient_domain">
-<tt class="descname">quotient_domain</tt><big>(</big><em>self</em>, <em>sm</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/homomorphisms.html#ModuleHomomorphism.quotient_domain"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.homomorphisms.ModuleHomomorphism.quotient_domain" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return <tt class="docutils literal"><span class="pre">self</span></tt> with domain replaced by <tt class="docutils literal"><span class="pre">domain/sm</span></tt>.</p>
-<p>Here <tt class="docutils literal"><span class="pre">sm</span></tt> must be a submodule of <tt class="docutils literal"><span class="pre">self.kernel()</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">homomorphism</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="n">homomorphism</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span>
-<span class="go">[1, x]</span>
-<span class="go">[0, 0] : QQ[x]**2 -&gt; QQ[x]**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span><span class="o">.</span><span class="n">quotient_domain</span><span class="p">(</span><span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="o">-</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">]))</span>
-<span class="go">[1, x]</span>
-<span class="go">[0, 0] : QQ[x]**2/&lt;[-x, 1]&gt; -&gt; QQ[x]**2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.homomorphisms.ModuleHomomorphism.restrict_codomain">
-<tt class="descname">restrict_codomain</tt><big>(</big><em>self</em>, <em>sm</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/homomorphisms.html#ModuleHomomorphism.restrict_codomain"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.homomorphisms.ModuleHomomorphism.restrict_codomain" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return <tt class="docutils literal"><span class="pre">self</span></tt>, with codomain restricted to to <tt class="docutils literal"><span class="pre">sm</span></tt>.</p>
-<p>Here <tt class="docutils literal"><span class="pre">sm</span></tt> has to be a submodule of <tt class="docutils literal"><span class="pre">self.codomain</span></tt> containing the
-image.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">homomorphism</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="n">homomorphism</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span>
-<span class="go">[1, x]</span>
-<span class="go">[0, 0] : QQ[x]**2 -&gt; QQ[x]**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span><span class="o">.</span><span class="n">restrict_codomain</span><span class="p">(</span><span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]))</span>
-<span class="go">[1, x]</span>
-<span class="go">[0, 0] : QQ[x]**2 -&gt; &lt;[1, 0]&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.agca.homomorphisms.ModuleHomomorphism.restrict_domain">
-<tt class="descname">restrict_domain</tt><big>(</big><em>self</em>, <em>sm</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/agca/homomorphisms.html#ModuleHomomorphism.restrict_domain"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.agca.homomorphisms.ModuleHomomorphism.restrict_domain" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return <tt class="docutils literal"><span class="pre">self</span></tt>, with the domain restricted to <tt class="docutils literal"><span class="pre">sm</span></tt>.</p>
-<p>Here <tt class="docutils literal"><span class="pre">sm</span></tt> has to be a submodule of <tt class="docutils literal"><span class="pre">self.domain</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">homomorphism</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="n">homomorphism</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">F</span><span class="p">,</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span>
-<span class="go">[1, x]</span>
-<span class="go">[0, 0] : QQ[x]**2 -&gt; QQ[x]**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span><span class="o">.</span><span class="n">restrict_domain</span><span class="p">(</span><span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]))</span>
-<span class="go">[1, x]</span>
-<span class="go">[0, 0] : &lt;[1, 0]&gt; -&gt; QQ[x]**2</span>
-</pre></div>
-</div>
-<p>This is the same as just composing on the right with the submodule
-inclusion:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">*</span> <span class="n">F</span><span class="o">.</span><span class="n">submodule</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">inclusion_hom</span><span class="p">()</span>
-<span class="go">[1, x]</span>
-<span class="go">[0, 0] : &lt;[1, 0]&gt; -&gt; QQ[x]**2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
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-  <div class="section" id="basic-functionality-of-the-module">
-<span id="polys-basics"></span><h1>Basic functionality of the module<a class="headerlink" href="#basic-functionality-of-the-module" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="introduction">
-<h2>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline">¶</a></h2>
-<p>This tutorial tries to give an overview of the functionality concerning
-polynomials within SymPy. All code examples assume:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y,z&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">init_printing</span><span class="p">(</span><span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">wrap_line</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">no_global</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="basic-functionality">
-<h2>Basic functionality<a class="headerlink" href="#basic-functionality" title="Permalink to this headline">¶</a></h2>
-<p>These functions provide different algorithms dealing with polynomials in the
-form of SymPy expression, like symbols, sums etc.</p>
-<div class="section" id="division">
-<h3>Division<a class="headerlink" href="#division" title="Permalink to this headline">¶</a></h3>
-<p>The function <tt class="xref py py-func docutils literal"><span class="pre">div()</span></tt> provides division of polynomials with remainder.
-That is, for polynomials <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>, it computes <tt class="docutils literal"><span class="pre">q</span></tt> and <tt class="docutils literal"><span class="pre">r</span></tt>, such
-that <span class="math">\(f = g \cdot q + r\)</span> and <span class="math">\(\deg(r) &lt; q\)</span>. For polynomials in one variables
-with coefficients in a field, say, the rational numbers, <tt class="docutils literal"><span class="pre">q</span></tt> and <tt class="docutils literal"><span class="pre">r</span></tt> are
-uniquely defined this way:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">10</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="s">&#39;QQ&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span>
-<span class="go">5*x   5</span>
-<span class="go">--- + -</span>
-<span class="go"> 2    2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span>
-<span class="go">-2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">q</span><span class="o">*</span><span class="n">g</span> <span class="o">+</span> <span class="n">r</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go">   2</span>
-<span class="go">5*x  + 10*x + 3</span>
-</pre></div>
-</div>
-<p>As you can see, <tt class="docutils literal"><span class="pre">q</span></tt> has a non-integer coefficient. If you want to do division
-only in the ring of polynomials with integer coefficients, you can specify an
-additional parameter:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="s">&#39;ZZ&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span>
-<span class="go">   2</span>
-<span class="go">5*x  + 10*x + 3</span>
-</pre></div>
-</div>
-<p>But be warned, that this ring is no longer Euclidean and that the degree of the
-remainder doesn&#8217;t need to be smaller than that of <tt class="docutils literal"><span class="pre">f</span></tt>. Since 2 doesn&#8217;t divide 5,
-<span class="math">\(2 x\)</span> doesn&#8217;t divide <span class="math">\(5 x^2\)</span>, even if the degree is smaller. But:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="mi">5</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="s">&#39;ZZ&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span>
-<span class="go">9*x + 3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">q</span><span class="o">*</span><span class="n">g</span> <span class="o">+</span> <span class="n">r</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go">   2</span>
-<span class="go">5*x  + 10*x + 3</span>
-</pre></div>
-</div>
-<p>This also works for polynomials with multiple variables:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="n">y</span><span class="o">*</span><span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">z</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="s">&#39;QQ&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span>
-<span class="go">y</span>
-<span class="go">-</span>
-<span class="go">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>In the last examples, all of the three variables <tt class="docutils literal"><span class="pre">x</span></tt>, <tt class="docutils literal"><span class="pre">y</span></tt> and <tt class="docutils literal"><span class="pre">z</span></tt> are
-assumed to be variables of the polynomials. But if you have some unrelated
-constant as coefficient, you can specify the variables explicitly:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a,b,c&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">a</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">c</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="s">&#39;QQ&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span>
-<span class="go">a*x   2*a   b</span>
-<span class="go">--- - --- + -</span>
-<span class="go"> 3     9    3</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span>
-<span class="go">4*a   2*b</span>
-<span class="go">--- - --- + c</span>
-<span class="go"> 9     3</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="gcd-and-lcm">
-<h3>GCD and LCM<a class="headerlink" href="#gcd-and-lcm" title="Permalink to this headline">¶</a></h3>
-<p>With division, there is also the computation of the greatest common divisor and
-the least common multiple.</p>
-<p>When the polynomials have integer coefficients, the contents&#8217; gcd is also
-considered:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="p">(</span><span class="mi">12</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">12</span><span class="p">)</span><span class="o">*</span><span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="mi">16</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-<span class="go">4*x</span>
-</pre></div>
-</div>
-<p>But if the polynomials have rational coefficients, then the returned polynomial is
-monic:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="mi">9</span><span class="o">*</span><span class="n">x</span><span class="o">/</span><span class="mi">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-<span class="go">x</span>
-</pre></div>
-</div>
-<p>It also works with multiple variables. In this case, the variables are ordered
-alphabetically, be default, which has influence on the leading coefficient:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">6</span><span class="o">*</span><span class="n">y</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-<span class="go">x + 2*y</span>
-</pre></div>
-</div>
-<p>The lcm is connected with the gcd and one can be computed using the other:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-<span class="go">x*y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lcm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-<span class="go"> 3  2    2  3</span>
-<span class="go">x *y  + x *y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="n">g</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go"> 4  3    3  4</span>
-<span class="go">x *y  + x *y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">lcm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go"> 4  3    3  4</span>
-<span class="go">x *y  + x *y</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="square-free-factorization">
-<h3>Square-free factorization<a class="headerlink" href="#square-free-factorization" title="Permalink to this headline">¶</a></h3>
-<p>The square-free factorization of a univariate polynomial is the product of all
-factors (not necessarily irreducible) of degree 1, 2 etc.:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">5</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqf_list</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-<span class="go">(1, [(x + 2, 1), (x, 2), (x + 1, 2)])</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqf</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-<span class="go"> 2        2</span>
-<span class="go">x *(x + 1) *(x + 2)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="factorization">
-<h3>Factorization<a class="headerlink" href="#factorization" title="Permalink to this headline">¶</a></h3>
-<p>This function provides factorization of univariate and multivariate polynomials
-with rational coefficients:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="o">/</span><span class="mi">12</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span>
-<span class="go"> 2</span>
-<span class="go">x *(2*x - 1)*(3*x + 4)</span>
-<span class="go">----------------------</span>
-<span class="go">          12</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">         2</span>
-<span class="go">(x + 2*y)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="groebner-bases">
-<h3>Groebner bases<a class="headerlink" href="#groebner-bases" title="Permalink to this headline">¶</a></h3>
-<p>Buchberger&#8217;s algorithm is implemented, supporting various monomial orders:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">groebner</span><span class="p">([</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">y</span><span class="o">**</span><span class="mi">4</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">],</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s">&#39;lex&#39;</span><span class="p">)</span>
-<span class="go">             /[ 2       4    ]                            \</span>
-<span class="go">GroebnerBasis\[x  + 1, y  - 1], x, y, domain=ZZ, order=lex/</span>
-
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">groebner</span><span class="p">([</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">y</span><span class="o">**</span><span class="mi">4</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">3</span><span class="p">],</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s">&#39;grevlex&#39;</span><span class="p">)</span>
-<span class="go">             /[ 4       3   2    ]                                   \</span>
-<span class="go">GroebnerBasis\[y  - 1, z , x  + 1], x, y, z, domain=ZZ, order=grevlex/</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="solving-equations">
-<h3>Solving Equations<a class="headerlink" href="#solving-equations" title="Permalink to this headline">¶</a></h3>
-<p>We have (incomplete) methods to find the complex or even symbolic roots of
-polynomials and to solve some systems of polynomial equations:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">roots</span><span class="p">,</span> <span class="n">solve_poly_system</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">           ____          ____</span>
-<span class="go">     1   \/ 11 *I  1   \/ 11 *I</span>
-<span class="go">[-1, - - --------, - + --------]</span>
-<span class="go">     2      2      2      2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">sorted</span><span class="p">(</span><span class="n">solve</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">p</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">q</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">          __________           __________</span>
-<span class="go">         /  2                 /  2</span>
-<span class="go">   p   \/  p  - 4*q     p   \/  p  - 4*q</span>
-<span class="go">[- - + -------------, - - - -------------]</span>
-<span class="go">   2         2          2         2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve_poly_system</span><span class="p">([</span><span class="n">y</span> <span class="o">-</span> <span class="n">x</span><span class="p">,</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">5</span><span class="p">],</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">[(5, 5)]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve_poly_system</span><span class="p">([</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">y</span><span class="o">*</span><span class="n">x</span><span class="p">],</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">                                   ___                 ___</span>
-<span class="go">                             1   \/ 3 *I         1   \/ 3 *I</span>
-<span class="go">[(0, I), (0, -I), (1, 0), (- - + -------, 0), (- - - -------, 0)]</span>
-<span class="go">                             2      2            2      2</span>
-</pre></div>
-</div>
-</div>
-</div>
-</div>
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-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Basic functionality of the module</a><ul>
-<li><a class="reference internal" href="#introduction">Introduction</a></li>
-<li><a class="reference internal" href="#basic-functionality">Basic functionality</a><ul>
-<li><a class="reference internal" href="#division">Division</a></li>
-<li><a class="reference internal" href="#gcd-and-lcm">GCD and LCM</a></li>
-<li><a class="reference internal" href="#square-free-factorization">Square-free factorization</a></li>
-<li><a class="reference internal" href="#factorization">Factorization</a></li>
-<li><a class="reference internal" href="#groebner-bases">Groebner bases</a></li>
-<li><a class="reference internal" href="#solving-equations">Solving Equations</a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-
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-  <p class="topless"><a href="../polys/index.html"
-                        title="previous chapter">Polynomials Manipulation Module</a></p>
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-  <div class="section" id="polynomials-manipulation-module">
-<span id="polys-docs"></span><h1>Polynomials Manipulation Module<a class="headerlink" href="#polynomials-manipulation-module" title="Permalink to this headline">¶</a></h1>
-<p>Computations with polynomials are at the core of computer algebra and
-having a fast and robust polynomials manipulation module is a key for
-building a powerful symbolic manipulation system. SymPy has a dedicated
-module <tt class="xref py py-mod docutils literal"><span class="pre">sympy.polys</span></tt> for computing in polynomial algebras over
-various coefficient domains.</p>
-<p>There is a vast number of methods implemented, ranging from simple tools
-like polynomial division, to advanced concepts including Gröbner bases
-and multivariate factorization over algebraic number domains.</p>
-<div class="section" id="contents">
-<h2>Contents<a class="headerlink" href="#contents" title="Permalink to this headline">¶</a></h2>
-<div class="toctree-wrapper compound">
-<ul>
-<li class="toctree-l1"><a class="reference internal" href="../polys/basics.html">Basic functionality of the module</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../polys/basics.html#introduction">Introduction</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../polys/basics.html#basic-functionality">Basic functionality</a><ul>
-<li class="toctree-l3"><a class="reference internal" href="../polys/basics.html#division">Division</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/basics.html#gcd-and-lcm">GCD and LCM</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/basics.html#square-free-factorization">Square-free factorization</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/basics.html#factorization">Factorization</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/basics.html#groebner-bases">Groebner bases</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/basics.html#solving-equations">Solving Equations</a></li>
-</ul>
-</li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../polys/wester.html">Examples from Wester&#8217;s Article</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../polys/wester.html#introduction">Introduction</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../polys/wester.html#examples">Examples</a><ul>
-<li class="toctree-l3"><a class="reference internal" href="../polys/wester.html#simple-univariate-polynomial-factorization">Simple univariate polynomial factorization</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/wester.html#univariate-gcd-resultant-and-factorization">Univariate GCD, resultant and factorization</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/wester.html#multivariate-gcd-and-factorization">Multivariate GCD and factorization</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/wester.html#support-for-symbols-in-exponents">Support for symbols in exponents</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/wester.html#testing-if-polynomials-have-common-zeros">Testing if polynomials have common zeros</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/wester.html#normalizing-simple-rational-functions">Normalizing simple rational functions</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/wester.html#expanding-expressions-and-factoring-back">Expanding expressions and factoring back</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/wester.html#factoring-in-terms-of-cyclotomic-polynomials">Factoring in terms of cyclotomic polynomials</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/wester.html#univariate-factoring-over-gaussian-numbers">Univariate factoring over Gaussian numbers</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/wester.html#computing-with-automatic-field-extensions">Computing with automatic field extensions</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/wester.html#univariate-factoring-over-various-domains">Univariate factoring over various domains</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/wester.html#factoring-polynomials-into-linear-factors">Factoring polynomials into linear factors</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/wester.html#advanced-factoring-over-finite-fields">Advanced factoring over finite fields</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/wester.html#working-with-expressions-as-polynomials">Working with expressions as polynomials</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/wester.html#computing-reduced-grobner-bases">Computing reduced Gröbner bases</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/wester.html#multivariate-factoring-over-algebraic-numbers">Multivariate factoring over algebraic numbers</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/wester.html#partial-fraction-decomposition">Partial fraction decomposition</a></li>
-</ul>
-</li>
-<li class="toctree-l2"><a class="reference internal" href="../polys/wester.html#literature">Literature</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../polys/reference.html">Polynomials Manipulation Module Reference</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../polys/reference.html#basic-polynomial-manipulation-functions">Basic polynomial manipulation functions</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../polys/reference.html#extra-polynomial-manipulation-functions">Extra polynomial manipulation functions</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../polys/reference.html#domain-constructors">Domain constructors</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../polys/reference.html#algebraic-number-fields">Algebraic number fields</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../polys/reference.html#monomials-encoded-as-tuples">Monomials encoded as tuples</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../polys/reference.html#formal-manipulation-of-roots-of-polynomials">Formal manipulation of roots of polynomials</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../polys/reference.html#symbolic-root-finding-algorithms">Symbolic root-finding algorithms</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../polys/reference.html#special-polynomials">Special polynomials</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../polys/reference.html#orthogonal-polynomials">Orthogonal polynomials</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../polys/reference.html#manipulation-of-rational-functions">Manipulation of rational functions</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../polys/reference.html#partial-fraction-decomposition">Partial fraction decomposition</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../polys/agca.html">AGCA - Algebraic Geometry and Commutative Algebra Module</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../polys/agca.html#introduction">Introduction</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../polys/agca.html#reference">Reference</a><ul>
-<li class="toctree-l3"><a class="reference internal" href="../polys/agca.html#base-rings">Base Rings</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/agca.html#modules-ideals-and-their-elementary-properties">Modules, Ideals and their Elementary Properties</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/agca.html#module-homomorphisms-and-syzygies">Module Homomorphisms and Syzygies</a></li>
-</ul>
-</li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../polys/internals.html">Internals of the Polynomial Manipulation Module</a><ul>
-<li class="toctree-l2"><a class="reference internal" href="../polys/internals.html#domains">Domains</a><ul>
-<li class="toctree-l3"><a class="reference internal" href="../polys/internals.html#abstract-domains">Abstract Domains</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/internals.html#concrete-domains">Concrete Domains</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/internals.html#implementation-domains">Implementation Domains</a></li>
-</ul>
-</li>
-<li class="toctree-l2"><a class="reference internal" href="../polys/internals.html#level-one">Level One</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../polys/internals.html#level-zero">Level Zero</a><ul>
-<li class="toctree-l3"><a class="reference internal" href="../polys/internals.html#manipulation-of-dense-multivariate-polynomials">Manipulation of dense, multivariate polynomials</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/internals.html#manipulation-of-dense-univariate-polynomials-with-finite-field-coefficients">Manipulation of dense, univariate polynomials with finite field coefficients</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/internals.html#manipulation-of-sparse-distributed-polynomials-and-vectors">Manipulation of sparse, distributed polynomials and vectors</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/internals.html#polynomial-factorization-algorithms">Polynomial factorization algorithms</a></li>
-<li class="toctree-l3"><a class="reference internal" href="../polys/internals.html#groebner-basis-algorithms">Groebner basis algorithms</a></li>
-</ul>
-</li>
-<li class="toctree-l2"><a class="reference internal" href="../polys/internals.html#exceptions">Exceptions</a></li>
-<li class="toctree-l2"><a class="reference internal" href="../polys/internals.html#undocumented">Undocumented</a></li>
-</ul>
-</li>
-<li class="toctree-l1"><a class="reference internal" href="../polys/literature.html">Literature</a></li>
-</ul>
-</div>
-</div>
-</div>
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-<li><a class="reference internal" href="#">Polynomials Manipulation Module</a><ul>
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-  <div class="section" id="internals-of-the-polynomial-manipulation-module">
-<span id="polys-internals"></span><h1>Internals of the Polynomial Manipulation Module<a class="headerlink" href="#internals-of-the-polynomial-manipulation-module" title="Permalink to this headline">¶</a></h1>
-<p>The implementation of the polynomials module is structured internally in
-&#8220;levels&#8221;. There are four levels, called L0, L1, L2 and L3. The levels three
-and four contain the user-facing functionality and were described in the
-previous section. This section focuses on levels zero and one.</p>
-<p>Level zero provides core polynomial manipulation functionality with C-like,
-low-level interfaces. Level one wraps this low-level functionality into object
-oriented structures. These are <em>not</em> the classes seen by the user, but rather
-classes used internally throughout the polys module.</p>
-<p>There is one additional complication in the implementation. This comes from the
-fact that all polynomial manipulations are relative to a <em>ground domain</em>. For
-example, when factoring a polynomial like <span class="math">\(x^{10} - 1\)</span>, one has to decide what
-ring the coefficients are supposed to belong to, or less trivially, what
-coefficients are allowed to appear in the factorization. This choice of
-coefficients is called a ground domain. Typical choices include the integers
-<span class="math">\(\mathbb{Z}\)</span>, the rational numbers <span class="math">\(\mathbb{Q}\)</span> or various related rings and
-fields. But it is perfectly legitimate (although in this case uninteresting)
-to factorize over polynomial rings such as <span class="math">\(k[Y]\)</span>, where <span class="math">\(k\)</span> is some fixed
-field.</p>
-<p>Thus the polynomial manipulation algorithms (both
-complicated ones like factoring, and simpler ones like addition or
-multiplication) have to rely on other code to manipulate the coefficients.
-In the polynomial manipulation module, such code is encapsulated in so-called
-&#8220;domains&#8221;. A domain is basically a factory object: it takes various
-representations of data, and converts them into objects with unified interface.
-Every object created by a domain has to implement the arithmetic operations
-<span class="math">\(+\)</span>, <span class="math">\(-\)</span> and <span class="math">\(\times\)</span>. Other operations are accessed through the domain, e.g.
-as in <tt class="docutils literal"><span class="pre">ZZ.quo(ZZ(4),</span> <span class="pre">ZZ(2))</span></tt>.</p>
-<p>Note that there is some amount of <em>circularity</em>: the polynomial ring domains
-use the level one classes, the level one classes use the level zero functions,
-and level zero functions use domains. It is possible, in principle, but not in
-the current implementation, to work in rings like <span class="math">\(k[X][Y]\)</span>. This would create
-even more layers. For this reason, working in the isomorphic ring <span class="math">\(k[X, Y]\)</span>
-is preferred.</p>
-<div class="section" id="domains">
-<h2>Domains<a class="headerlink" href="#domains" title="Permalink to this headline">¶</a></h2>
-<p>Here we document the various implemented ground domains. There are three
-types: abstract domains, concrete domains, and &#8220;implementation domains&#8221;.
-Abstract domains cannot be (usefully) instantiated at all, and just collect
-together functionality shared by many other domains. Concrete domains are
-those meant to be instantiated and used in the polynomial manipulation
-algorithms. In some cases, there are various possible ways to implement the
-data type the domain provides. For example, depending on what libraries are
-available on the system, the integers are implemented either using the python
-built-in integers, or using gmpy. Note that various aliases are created
-automatically depending on the libraries available. As such e.g. <tt class="docutils literal"><span class="pre">ZZ</span></tt> always
-refers to the most efficient implementation of the integer ring available.</p>
-<div class="section" id="abstract-domains">
-<h3>Abstract Domains<a class="headerlink" href="#abstract-domains" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.polys.domains.Domain">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">Domain</tt><a class="headerlink" href="#sympy.polys.domains.Domain" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents an abstract domain.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>alias</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>dtype</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>one</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>rep</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>zero</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.abs">
-<tt class="descname">abs</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.abs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Absolute value of <tt class="docutils literal"><span class="pre">a</span></tt>, implies <tt class="docutils literal"><span class="pre">__abs__</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.add">
-<tt class="descname">add</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Sum of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>, implies <tt class="docutils literal"><span class="pre">__add__</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.algebraic_field">
-<tt class="descname">algebraic_field</tt><big>(</big><em>self</em>, <em>*extension</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.algebraic_field" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns an algebraic field, i.e. <span class="math">\(K(\alpha, \dots)\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.characteristic">
-<tt class="descname">characteristic</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.characteristic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the characteristic of this domain.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.cofactors">
-<tt class="descname">cofactors</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.cofactors" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns GCD and cofactors of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.convert">
-<tt class="descname">convert</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0=None</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.convert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert an object <span class="math">\(a\)</span> from <span class="math">\(K_0\)</span> to <span class="math">\(K_1\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.denom">
-<tt class="descname">denom</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.denom" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns denominator of <tt class="docutils literal"><span class="pre">a</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.div">
-<tt class="descname">div</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.div" title="Permalink to this definition">¶</a></dt>
-<dd><p>Division of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>, implies something.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.evalf">
-<tt class="descname">evalf</tt><big>(</big><em>self</em>, <em>a</em>, <em>prec=None</em>, <em>**args</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.evalf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns numerical approximation of <tt class="docutils literal"><span class="pre">a</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.exquo">
-<tt class="descname">exquo</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.exquo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Exact quotient of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>, implies something.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.frac_field">
-<tt class="descname">frac_field</tt><big>(</big><em>self</em>, <em>*gens</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.frac_field" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a fraction field, i.e. <tt class="docutils literal"><span class="pre">K(X)</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.from_AlgebraicField">
-<tt class="descname">from_AlgebraicField</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.from_AlgebraicField" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a <tt class="docutils literal"><span class="pre">ANP</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.from_ExpressionDomain">
-<tt class="descname">from_ExpressionDomain</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.from_ExpressionDomain" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a <tt class="docutils literal"><span class="pre">EX</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.from_FF_gmpy">
-<tt class="descname">from_FF_gmpy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.from_FF_gmpy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">ModularInteger(mpz)</span></tt> to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.from_FF_python">
-<tt class="descname">from_FF_python</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.from_FF_python" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">ModularInteger(int)</span></tt> to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.from_FF_sympy">
-<tt class="descname">from_FF_sympy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.from_FF_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">ModularInteger(Integer)</span></tt> to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.from_FractionField">
-<tt class="descname">from_FractionField</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.from_FractionField" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a <tt class="docutils literal"><span class="pre">DMF</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.from_GlobalPolynomialRing">
-<tt class="descname">from_GlobalPolynomialRing</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.from_GlobalPolynomialRing" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a <tt class="docutils literal"><span class="pre">DMP</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.from_QQ_gmpy">
-<tt class="descname">from_QQ_gmpy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.from_QQ_gmpy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a GMPY <tt class="docutils literal"><span class="pre">mpq</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.from_QQ_python">
-<tt class="descname">from_QQ_python</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.from_QQ_python" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a Python <tt class="docutils literal"><span class="pre">Fraction</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.from_QQ_sympy">
-<tt class="descname">from_QQ_sympy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.from_QQ_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a SymPy <tt class="docutils literal"><span class="pre">Rational</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.from_RR_mpmath">
-<tt class="descname">from_RR_mpmath</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.from_RR_mpmath" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a mpmath <tt class="docutils literal"><span class="pre">mpf</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.from_RR_sympy">
-<tt class="descname">from_RR_sympy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.from_RR_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a SymPy <tt class="docutils literal"><span class="pre">Float</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.from_ZZ_gmpy">
-<tt class="descname">from_ZZ_gmpy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.from_ZZ_gmpy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a GMPY <tt class="docutils literal"><span class="pre">mpz</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.from_ZZ_python">
-<tt class="descname">from_ZZ_python</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.from_ZZ_python" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a Python <tt class="docutils literal"><span class="pre">int</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.from_ZZ_sympy">
-<tt class="descname">from_ZZ_sympy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.from_ZZ_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a SymPy <tt class="docutils literal"><span class="pre">Integer</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.from_sympy">
-<tt class="descname">from_sympy</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.from_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a SymPy object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.gcd">
-<tt class="descname">gcd</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.gcd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns GCD of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.gcdex">
-<tt class="descname">gcdex</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.gcdex" title="Permalink to this definition">¶</a></dt>
-<dd><p>Extended GCD of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.get_exact">
-<tt class="descname">get_exact</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.get_exact" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns an exact domain associated with <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.get_field">
-<tt class="descname">get_field</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.get_field" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a field associated with <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.get_ring">
-<tt class="descname">get_ring</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.get_ring" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a ring associated with <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.half_gcdex">
-<tt class="descname">half_gcdex</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.half_gcdex" title="Permalink to this definition">¶</a></dt>
-<dd><p>Half extended GCD of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.inject">
-<tt class="descname">inject</tt><big>(</big><em>self</em>, <em>*gens</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.inject" title="Permalink to this definition">¶</a></dt>
-<dd><p>Inject generators into this domain.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.invert">
-<tt class="descname">invert</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.invert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns inversion of <tt class="docutils literal"><span class="pre">a</span> <span class="pre">mod</span> <span class="pre">b</span></tt>, implies something.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.is_negative">
-<tt class="descname">is_negative</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.is_negative" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">a</span></tt> is negative.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.is_nonnegative">
-<tt class="descname">is_nonnegative</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.is_nonnegative" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">a</span></tt> is non-negative.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.is_nonpositive">
-<tt class="descname">is_nonpositive</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.is_nonpositive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">a</span></tt> is non-positive.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.is_one">
-<tt class="descname">is_one</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.is_one" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">a</span></tt> is one.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.is_positive">
-<tt class="descname">is_positive</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.is_positive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">a</span></tt> is positive.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.is_zero">
-<tt class="descname">is_zero</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.is_zero" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">a</span></tt> is zero.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.lcm">
-<tt class="descname">lcm</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.lcm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns LCM of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.log">
-<tt class="descname">log</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.log" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns b-base logarithm of <tt class="docutils literal"><span class="pre">a</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.map">
-<tt class="descname">map</tt><big>(</big><em>self</em>, <em>seq</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.map" title="Permalink to this definition">¶</a></dt>
-<dd><p>Rersively apply <tt class="docutils literal"><span class="pre">self</span></tt> to all elements of <tt class="docutils literal"><span class="pre">seq</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.mul">
-<tt class="descname">mul</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Product of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>, implies <tt class="docutils literal"><span class="pre">__mul__</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.n">
-<tt class="descname">n</tt><big>(</big><em>self</em>, <em>a</em>, <em>prec=None</em>, <em>**args</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.n" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns numerical approximation of <tt class="docutils literal"><span class="pre">a</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.neg">
-<tt class="descname">neg</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.neg" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">a</span></tt> negated, implies <tt class="docutils literal"><span class="pre">__neg__</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.numer">
-<tt class="descname">numer</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.numer" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns numerator of <tt class="docutils literal"><span class="pre">a</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.of_type">
-<tt class="descname">of_type</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.of_type" title="Permalink to this definition">¶</a></dt>
-<dd><p>Check if <tt class="docutils literal"><span class="pre">a</span></tt> is of type <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.poly_ring">
-<tt class="descname">poly_ring</tt><big>(</big><em>self</em>, <em>*gens</em>, <em>**opts</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.poly_ring" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a polynomial ring, i.e. <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.pos">
-<tt class="descname">pos</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.pos" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">a</span></tt> positive, implies <tt class="docutils literal"><span class="pre">__pos__</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.pow">
-<tt class="descname">pow</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.pow" title="Permalink to this definition">¶</a></dt>
-<dd><p>Raise <tt class="docutils literal"><span class="pre">a</span></tt> to power <tt class="docutils literal"><span class="pre">b</span></tt>, implies <tt class="docutils literal"><span class="pre">__pow__</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.quo">
-<tt class="descname">quo</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.quo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Quotient of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>, implies something.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.rem">
-<tt class="descname">rem</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.rem" title="Permalink to this definition">¶</a></dt>
-<dd><p>Remainder of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>, implies <tt class="docutils literal"><span class="pre">__mod__</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.revert">
-<tt class="descname">revert</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.revert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">a**(-1)</span></tt> if possible.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.sqrt">
-<tt class="descname">sqrt</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.sqrt" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns square root of <tt class="docutils literal"><span class="pre">a</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.sub">
-<tt class="descname">sub</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.sub" title="Permalink to this definition">¶</a></dt>
-<dd><p>Difference of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>, implies <tt class="docutils literal"><span class="pre">__sub__</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.to_sympy">
-<tt class="descname">to_sympy</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.to_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">a</span></tt> to a SymPy object.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Domain.unify">
-<tt class="descname">unify</tt><big>(</big><em>K0</em>, <em>K1</em>, <em>gens=None</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Domain.unify" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a maximal domain containing <span class="math">\(K_0\)</span> and <span class="math">\(K_1\)</span>.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.domains.Field">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">Field</tt><a class="headerlink" href="#sympy.polys.domains.Field" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents a field domain.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>alias</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>dtype</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>one</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>rep</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>zero</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.polys.domains.Field.div">
-<tt class="descname">div</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Field.div" title="Permalink to this definition">¶</a></dt>
-<dd><p>Division of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>, implies <tt class="docutils literal"><span class="pre">__div__</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Field.exquo">
-<tt class="descname">exquo</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Field.exquo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Exact quotient of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>, implies <tt class="docutils literal"><span class="pre">__div__</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Field.gcd">
-<tt class="descname">gcd</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Field.gcd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns GCD of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>.</p>
-<p>This definition of GCD over fields allows to clear denominators
-in <span class="math">\(primitive()\)</span>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">gcd</span><span class="p">,</span> <span class="n">primitive</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">QQ</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">9</span><span class="p">))</span>
-<span class="go">2/9</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gcd</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span><span class="o">/</span><span class="mi">9</span><span class="p">)</span>
-<span class="go">2/9</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">primitive</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">/</span><span class="mi">3</span> <span class="o">+</span> <span class="n">S</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span><span class="o">/</span><span class="mi">9</span><span class="p">)</span>
-<span class="go">(2/9, 3*x + 2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Field.get_field">
-<tt class="descname">get_field</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Field.get_field" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a field associated with <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Field.get_ring">
-<tt class="descname">get_ring</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Field.get_ring" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a ring associated with <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Field.lcm">
-<tt class="descname">lcm</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Field.lcm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns LCM of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">lcm</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="o">.</span><span class="n">lcm</span><span class="p">(</span><span class="n">QQ</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">9</span><span class="p">))</span>
-<span class="go">4/3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lcm</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span><span class="o">/</span><span class="mi">9</span><span class="p">)</span>
-<span class="go">4/3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Field.quo">
-<tt class="descname">quo</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Field.quo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Quotient of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>, implies <tt class="docutils literal"><span class="pre">__div__</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Field.rem">
-<tt class="descname">rem</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Field.rem" title="Permalink to this definition">¶</a></dt>
-<dd><p>Remainder of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>, implies nothing.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Field.revert">
-<tt class="descname">revert</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Field.revert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">a**(-1)</span></tt> if possible.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.domains.Ring">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">Ring</tt><a class="headerlink" href="#sympy.polys.domains.Ring" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents a ring domain.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>alias</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>dtype</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>one</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>rep</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>zero</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.polys.domains.Ring.denom">
-<tt class="descname">denom</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Ring.denom" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns denominator of <span class="math">\(a\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Ring.div">
-<tt class="descname">div</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Ring.div" title="Permalink to this definition">¶</a></dt>
-<dd><p>Division of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>, implies <tt class="docutils literal"><span class="pre">__divmod__</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Ring.exquo">
-<tt class="descname">exquo</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Ring.exquo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Exact quotient of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>, implies <tt class="docutils literal"><span class="pre">__floordiv__</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Ring.free_module">
-<tt class="descname">free_module</tt><big>(</big><em>self</em>, <em>rank</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Ring.free_module" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generate a free module of rank <tt class="docutils literal"><span class="pre">rank</span></tt> over self.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">free_module</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">QQ[x]**2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Ring.get_ring">
-<tt class="descname">get_ring</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Ring.get_ring" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a ring associated with <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Ring.ideal">
-<tt class="descname">ideal</tt><big>(</big><em>self</em>, <em>*gens</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Ring.ideal" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generate an ideal of <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">&lt;x**2&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Ring.invert">
-<tt class="descname">invert</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Ring.invert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns inversion of <tt class="docutils literal"><span class="pre">a</span> <span class="pre">mod</span> <span class="pre">b</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Ring.numer">
-<tt class="descname">numer</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Ring.numer" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns numerator of <tt class="docutils literal"><span class="pre">a</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Ring.quo">
-<tt class="descname">quo</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Ring.quo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Quotient of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>, implies <tt class="docutils literal"><span class="pre">__floordiv__</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Ring.quotient_ring">
-<tt class="descname">quotient_ring</tt><big>(</big><em>self</em>, <em>e</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Ring.quotient_ring" title="Permalink to this definition">¶</a></dt>
-<dd><p>Form a quotient ring of <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<p>Here <tt class="docutils literal"><span class="pre">e</span></tt> can be an ideal or an iterable.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">quotient_ring</span><span class="p">(</span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">ideal</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">QQ[x]/&lt;x**2&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">.</span><span class="n">quotient_ring</span><span class="p">([</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">])</span>
-<span class="go">QQ[x]/&lt;x**2&gt;</span>
-</pre></div>
-</div>
-<p>The division operator has been overloaded for this:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">QQ</span><span class="p">[</span><span class="n">x</span><span class="p">]</span><span class="o">/</span><span class="p">[</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">]</span>
-<span class="go">QQ[x]/&lt;x**2&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Ring.rem">
-<tt class="descname">rem</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Ring.rem" title="Permalink to this definition">¶</a></dt>
-<dd><p>Remainder of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>, implies <tt class="docutils literal"><span class="pre">__mod__</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.Ring.revert">
-<tt class="descname">revert</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.Ring.revert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">a**(-1)</span></tt> if possible.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.domains.SimpleDomain">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">SimpleDomain</tt><a class="headerlink" href="#sympy.polys.domains.SimpleDomain" title="Permalink to this definition">¶</a></dt>
-<dd><p>Base class for simple domains, e.g. ZZ, QQ.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>alias</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>dtype</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>one</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>rep</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>zero</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.polys.domains.SimpleDomain.inject">
-<tt class="descname">inject</tt><big>(</big><em>self</em>, <em>*gens</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.SimpleDomain.inject" title="Permalink to this definition">¶</a></dt>
-<dd><p>Inject generators into this domain.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.domains.CompositeDomain">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">CompositeDomain</tt><a class="headerlink" href="#sympy.polys.domains.CompositeDomain" title="Permalink to this definition">¶</a></dt>
-<dd><p>Base class for composite domains, e.g. ZZ[x], ZZ(X).</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>alias</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>dtype</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>one</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>rep</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>zero</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.polys.domains.CompositeDomain.inject">
-<tt class="descname">inject</tt><big>(</big><em>self</em>, <em>*gens</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.CompositeDomain.inject" title="Permalink to this definition">¶</a></dt>
-<dd><p>Inject generators into this domain.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="concrete-domains">
-<h3>Concrete Domains<a class="headerlink" href="#concrete-domains" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.polys.domains.FiniteField">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">FiniteField</tt><big>(</big><em>mod</em>, <em>symmetric=True</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FiniteField" title="Permalink to this definition">¶</a></dt>
-<dd><p>General class for finite fields.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>alias</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>dom</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>dtype</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>mod</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>one</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>zero</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.polys.domains.FiniteField.characteristic">
-<tt class="descname">characteristic</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FiniteField.characteristic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the characteristic of this domain.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FiniteField.from_FF_gmpy">
-<tt class="descname">from_FF_gmpy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0=None</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FiniteField.from_FF_gmpy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">ModularInteger(mpz)</span></tt> to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FiniteField.from_FF_python">
-<tt class="descname">from_FF_python</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0=None</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FiniteField.from_FF_python" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">ModularInteger(int)</span></tt> to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FiniteField.from_FF_sympy">
-<tt class="descname">from_FF_sympy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0=None</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FiniteField.from_FF_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">ModularInteger(Integer)</span></tt> to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FiniteField.from_QQ_gmpy">
-<tt class="descname">from_QQ_gmpy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0=None</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FiniteField.from_QQ_gmpy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert GMPY&#8217;s <tt class="docutils literal"><span class="pre">mpq</span></tt> to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FiniteField.from_QQ_python">
-<tt class="descname">from_QQ_python</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0=None</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FiniteField.from_QQ_python" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert Python&#8217;s <tt class="docutils literal"><span class="pre">Fraction</span></tt> to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FiniteField.from_QQ_sympy">
-<tt class="descname">from_QQ_sympy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0=None</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FiniteField.from_QQ_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert SymPy&#8217;s <tt class="docutils literal"><span class="pre">Rational</span></tt> to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FiniteField.from_RR_mpmath">
-<tt class="descname">from_RR_mpmath</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FiniteField.from_RR_mpmath" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert mpmath&#8217;s <tt class="docutils literal"><span class="pre">mpf</span></tt> to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FiniteField.from_RR_sympy">
-<tt class="descname">from_RR_sympy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0=None</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FiniteField.from_RR_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert SymPy&#8217;s <tt class="docutils literal"><span class="pre">Float</span></tt> to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FiniteField.from_ZZ_gmpy">
-<tt class="descname">from_ZZ_gmpy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0=None</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FiniteField.from_ZZ_gmpy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert GMPY&#8217;s <tt class="docutils literal"><span class="pre">mpz</span></tt> to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FiniteField.from_ZZ_python">
-<tt class="descname">from_ZZ_python</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0=None</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FiniteField.from_ZZ_python" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert Python&#8217;s <tt class="docutils literal"><span class="pre">int</span></tt> to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FiniteField.from_ZZ_sympy">
-<tt class="descname">from_ZZ_sympy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0=None</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FiniteField.from_ZZ_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert SymPy&#8217;s <tt class="docutils literal"><span class="pre">Integer</span></tt> to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FiniteField.from_sympy">
-<tt class="descname">from_sympy</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FiniteField.from_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert SymPy&#8217;s Integer to SymPy&#8217;s <tt class="docutils literal"><span class="pre">Integer</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FiniteField.get_field">
-<tt class="descname">get_field</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FiniteField.get_field" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a field associated with <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FiniteField.to_sympy">
-<tt class="descname">to_sympy</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FiniteField.to_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">a</span></tt> to a SymPy object.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.domains.IntegerRing">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">IntegerRing</tt><a class="headerlink" href="#sympy.polys.domains.IntegerRing" title="Permalink to this definition">¶</a></dt>
-<dd><p>General class for integer rings.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>alias</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>dtype</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>one</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>zero</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.polys.domains.IntegerRing.algebraic_field">
-<tt class="descname">algebraic_field</tt><big>(</big><em>self</em>, <em>*extension</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.IntegerRing.algebraic_field" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns an algebraic field, i.e. <span class="math">\(\mathbb{Q}(\alpha, \dots)\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.IntegerRing.from_AlgebraicField">
-<tt class="descname">from_AlgebraicField</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.IntegerRing.from_AlgebraicField" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a <tt class="docutils literal"><span class="pre">ANP</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.IntegerRing.get_field">
-<tt class="descname">get_field</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.IntegerRing.get_field" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a field associated with <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.IntegerRing.log">
-<tt class="descname">log</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.IntegerRing.log" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns b-base logarithm of <tt class="docutils literal"><span class="pre">a</span></tt>.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.domains.PolynomialRing">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">PolynomialRing</tt><a class="headerlink" href="#sympy.polys.domains.PolynomialRing" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a generalized multivariate polynomial ring.</p>
-<p>A generalized polynomial ring is defined by a ground field <span class="math">\(K\)</span>, a set
-of generators (typically <span class="math">\(x_1, \dots, x_n\)</span>) and a monomial order <span class="math">\(&lt;\)</span>.
-The monomial order can be global, local or mixed. In any case it induces
-a total ordering on the monomials, and there exists for every (non-zero)
-polynomial <span class="math">\(f \in K[x_1, \dots, x_n]\)</span> a well-defined &#8220;leading monomial&#8221;
-<span class="math">\(LM(f) = LM(f, &gt;)\)</span>. One can then define a multiplicative subset
-<span class="math">\(S = S_&gt; = \{f \in K[x_1, \dots, x_n] | LM(f) = 1\}\)</span>. The generalized
-polynomial ring corresponding to the monomial order is
-<span class="math">\(R = S^{-1}K[x_1, \dots, x_n]\)</span>.</p>
-<p>If <span class="math">\(&gt;\)</span> is a so-called global order, that is <span class="math">\(1\)</span> is the smallest monomial,
-then we just have <span class="math">\(S = K\)</span> and <span class="math">\(R = K[x_1, \dots, x_n]\)</span>.</p>
-<p class="rubric">Examples</p>
-<p>A few examples may make this clearer.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">QQ</span>
-</pre></div>
-</div>
-<p>Our first ring uses global lexicographic order.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">R1</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">poly_ring</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="p">((</span><span class="s">&quot;lex&quot;</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),))</span>
-</pre></div>
-</div>
-<p>The second ring uses local lexicographic order. Note that when using a
-single (non-product) order, you can just specify the name and omit the
-variables:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">R2</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">poly_ring</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s">&quot;ilex&quot;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>The third and fourth rings use a mixed orders:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">o1</span> <span class="o">=</span> <span class="p">((</span><span class="s">&quot;ilex&quot;</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="s">&quot;lex&quot;</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">o2</span> <span class="o">=</span> <span class="p">((</span><span class="s">&quot;lex&quot;</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="s">&quot;ilex&quot;</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R3</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">poly_ring</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">o1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R4</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">poly_ring</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="n">o2</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>We will investigate what elements of <span class="math">\(K(x, y)\)</span> are contained in the various
-rings.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">L</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="p">),</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">y</span><span class="p">),</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">test</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">R</span><span class="p">:</span> <span class="p">[</span><span class="n">f</span> <span class="ow">in</span> <span class="n">R</span> <span class="k">for</span> <span class="n">f</span> <span class="ow">in</span> <span class="n">L</span><span class="p">]</span>
-</pre></div>
-</div>
-<p>The first ring is just <span class="math">\(K[x, y]\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">test</span><span class="p">(</span><span class="n">R1</span><span class="p">)</span>
-<span class="go">[True, False, False, False, False]</span>
-</pre></div>
-</div>
-<p>The second ring is R1 localised at the maximal ideal (x, y):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">test</span><span class="p">(</span><span class="n">R2</span><span class="p">)</span>
-<span class="go">[True, False, True, True, True]</span>
-</pre></div>
-</div>
-<p>The third ring is R1 localised at the prime ideal (x):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">test</span><span class="p">(</span><span class="n">R3</span><span class="p">)</span>
-<span class="go">[True, False, True, False, True]</span>
-</pre></div>
-</div>
-<p>Finally the fourth ring is R1 localised at <span class="math">\(S = K[x, y] \setminus yK[y]\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">test</span><span class="p">(</span><span class="n">R4</span><span class="p">)</span>
-<span class="go">[True, False, False, True, False]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.domains.RationalField">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">RationalField</tt><a class="headerlink" href="#sympy.polys.domains.RationalField" title="Permalink to this definition">¶</a></dt>
-<dd><p>General class for rational fields.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>alias</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>dtype</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>one</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>zero</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.polys.domains.RationalField.algebraic_field">
-<tt class="descname">algebraic_field</tt><big>(</big><em>self</em>, <em>*extension</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.RationalField.algebraic_field" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns an algebraic field, i.e. <span class="math">\(\mathbb{Q}(\alpha, \dots)\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.RationalField.from_AlgebraicField">
-<tt class="descname">from_AlgebraicField</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.RationalField.from_AlgebraicField" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a <tt class="docutils literal"><span class="pre">ANP</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.RationalField.get_ring">
-<tt class="descname">get_ring</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.RationalField.get_ring" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a ring associated with <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.domains.AlgebraicField">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">AlgebraicField</tt><big>(</big><em>dom</em>, <em>*ext</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.AlgebraicField" title="Permalink to this definition">¶</a></dt>
-<dd><p>A class for representing algebraic number fields.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>alias</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>one</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>rep</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>zero</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.polys.domains.AlgebraicField.algebraic_field">
-<tt class="descname">algebraic_field</tt><big>(</big><em>self</em>, <em>*extension</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.AlgebraicField.algebraic_field" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns an algebraic field, i.e. <span class="math">\(\mathbb{Q}(\alpha, \dots)\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.AlgebraicField.denom">
-<tt class="descname">denom</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.AlgebraicField.denom" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns denominator of <tt class="docutils literal"><span class="pre">a</span></tt>.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.domains.AlgebraicField.dtype">
-<tt class="descname">dtype</tt><a class="headerlink" href="#sympy.polys.domains.AlgebraicField.dtype" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <tt class="xref py py-class docutils literal"><span class="pre">ANP</span></tt></p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.AlgebraicField.from_QQ_gmpy">
-<tt class="descname">from_QQ_gmpy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.AlgebraicField.from_QQ_gmpy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a GMPY <tt class="docutils literal"><span class="pre">mpq</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.AlgebraicField.from_QQ_python">
-<tt class="descname">from_QQ_python</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.AlgebraicField.from_QQ_python" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a Python <tt class="docutils literal"><span class="pre">Fraction</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.AlgebraicField.from_QQ_sympy">
-<tt class="descname">from_QQ_sympy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.AlgebraicField.from_QQ_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a SymPy <tt class="docutils literal"><span class="pre">Rational</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.AlgebraicField.from_RR_mpmath">
-<tt class="descname">from_RR_mpmath</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.AlgebraicField.from_RR_mpmath" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a mpmath <tt class="docutils literal"><span class="pre">mpf</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.AlgebraicField.from_RR_sympy">
-<tt class="descname">from_RR_sympy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.AlgebraicField.from_RR_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a SymPy <tt class="docutils literal"><span class="pre">Float</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.AlgebraicField.from_ZZ_gmpy">
-<tt class="descname">from_ZZ_gmpy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.AlgebraicField.from_ZZ_gmpy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a GMPY <tt class="docutils literal"><span class="pre">mpz</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.AlgebraicField.from_ZZ_python">
-<tt class="descname">from_ZZ_python</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.AlgebraicField.from_ZZ_python" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a Python <tt class="docutils literal"><span class="pre">int</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.AlgebraicField.from_ZZ_sympy">
-<tt class="descname">from_ZZ_sympy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.AlgebraicField.from_ZZ_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a SymPy <tt class="docutils literal"><span class="pre">Integer</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.AlgebraicField.from_sympy">
-<tt class="descname">from_sympy</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.AlgebraicField.from_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert SymPy&#8217;s expression to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.AlgebraicField.get_ring">
-<tt class="descname">get_ring</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.AlgebraicField.get_ring" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a ring associated with <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.AlgebraicField.is_negative">
-<tt class="descname">is_negative</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.AlgebraicField.is_negative" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">a</span></tt> is negative.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.AlgebraicField.is_nonnegative">
-<tt class="descname">is_nonnegative</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.AlgebraicField.is_nonnegative" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">a</span></tt> is non-negative.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.AlgebraicField.is_nonpositive">
-<tt class="descname">is_nonpositive</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.AlgebraicField.is_nonpositive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">a</span></tt> is non-positive.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.AlgebraicField.is_positive">
-<tt class="descname">is_positive</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.AlgebraicField.is_positive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">a</span></tt> is positive.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.AlgebraicField.numer">
-<tt class="descname">numer</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.AlgebraicField.numer" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns numerator of <tt class="docutils literal"><span class="pre">a</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.AlgebraicField.to_sympy">
-<tt class="descname">to_sympy</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.AlgebraicField.to_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">a</span></tt> to a SymPy object.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.domains.FractionField">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">FractionField</tt><big>(</big><em>dom</em>, <em>*gens</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField" title="Permalink to this definition">¶</a></dt>
-<dd><p>A class for representing rational function fields.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>alias</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>one</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>rep</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>zero</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.polys.domains.FractionField.denom">
-<tt class="descname">denom</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField.denom" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns denominator of <tt class="docutils literal"><span class="pre">a</span></tt>.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.domains.FractionField.dtype">
-<tt class="descname">dtype</tt><a class="headerlink" href="#sympy.polys.domains.FractionField.dtype" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <tt class="xref py py-class docutils literal"><span class="pre">DMF</span></tt></p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FractionField.factorial">
-<tt class="descname">factorial</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField.factorial" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns factorial of <tt class="docutils literal"><span class="pre">a</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FractionField.frac_field">
-<tt class="descname">frac_field</tt><big>(</big><em>self</em>, <em>*gens</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField.frac_field" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a fraction field, i.e. <span class="math">\(K(X)\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FractionField.from_FractionField">
-<tt class="descname">from_FractionField</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField.from_FractionField" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a fraction field element to another fraction field.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.polyclasses</span> <span class="kn">import</span> <span class="n">DMF</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">DMF</span><span class="p">(([</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">2</span><span class="p">)],</span> <span class="p">[</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">)]),</span> <span class="n">ZZ</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">QQx</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">frac_field</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ZZx</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">frac_field</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">QQx</span><span class="o">.</span><span class="n">from_FractionField</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">ZZx</span><span class="p">)</span>
-<span class="go">(x + 2)/(x + 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FractionField.from_GlobalPolynomialRing">
-<tt class="descname">from_GlobalPolynomialRing</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField.from_GlobalPolynomialRing" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a <tt class="docutils literal"><span class="pre">DMF</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FractionField.from_QQ_gmpy">
-<tt class="descname">from_QQ_gmpy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField.from_QQ_gmpy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a GMPY <tt class="docutils literal"><span class="pre">mpq</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FractionField.from_QQ_python">
-<tt class="descname">from_QQ_python</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField.from_QQ_python" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a Python <tt class="docutils literal"><span class="pre">Fraction</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FractionField.from_QQ_sympy">
-<tt class="descname">from_QQ_sympy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField.from_QQ_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a SymPy <tt class="docutils literal"><span class="pre">Rational</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FractionField.from_RR_mpmath">
-<tt class="descname">from_RR_mpmath</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField.from_RR_mpmath" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a mpmath <tt class="docutils literal"><span class="pre">mpf</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FractionField.from_RR_sympy">
-<tt class="descname">from_RR_sympy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField.from_RR_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a SymPy <tt class="docutils literal"><span class="pre">Float</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FractionField.from_ZZ_gmpy">
-<tt class="descname">from_ZZ_gmpy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField.from_ZZ_gmpy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a GMPY <tt class="docutils literal"><span class="pre">mpz</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FractionField.from_ZZ_python">
-<tt class="descname">from_ZZ_python</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField.from_ZZ_python" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a Python <tt class="docutils literal"><span class="pre">int</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FractionField.from_ZZ_sympy">
-<tt class="descname">from_ZZ_sympy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField.from_ZZ_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a SymPy <tt class="docutils literal"><span class="pre">Integer</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FractionField.from_sympy">
-<tt class="descname">from_sympy</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField.from_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert SymPy&#8217;s expression to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FractionField.get_ring">
-<tt class="descname">get_ring</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField.get_ring" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a ring associated with <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FractionField.is_negative">
-<tt class="descname">is_negative</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField.is_negative" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">a</span></tt> is negative.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FractionField.is_nonnegative">
-<tt class="descname">is_nonnegative</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField.is_nonnegative" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">a</span></tt> is non-negative.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FractionField.is_nonpositive">
-<tt class="descname">is_nonpositive</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField.is_nonpositive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">a</span></tt> is non-positive.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FractionField.is_positive">
-<tt class="descname">is_positive</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField.is_positive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">a</span></tt> is positive.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FractionField.numer">
-<tt class="descname">numer</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField.numer" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns numerator of <tt class="docutils literal"><span class="pre">a</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FractionField.poly_ring">
-<tt class="descname">poly_ring</tt><big>(</big><em>self</em>, <em>*gens</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField.poly_ring" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a polynomial ring, i.e. <span class="math">\(K[X]\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.FractionField.to_sympy">
-<tt class="descname">to_sympy</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.FractionField.to_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">a</span></tt> to a SymPy object.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.domains.RealDomain">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">RealDomain</tt><a class="headerlink" href="#sympy.polys.domains.RealDomain" title="Permalink to this definition">¶</a></dt>
-<dd><p>Abstract domain for real numbers.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>alias</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>dtype</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>one</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>zero</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.polys.domains.RealDomain.as_integer_ratio">
-<tt class="descname">as_integer_ratio</tt><big>(</big><em>self</em>, <em>a</em>, <em>**args</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.RealDomain.as_integer_ratio" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert real number to a (numer, denom) pair.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.RealDomain.div">
-<tt class="descname">div</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.RealDomain.div" title="Permalink to this definition">¶</a></dt>
-<dd><p>Division of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.RealDomain.exquo">
-<tt class="descname">exquo</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.RealDomain.exquo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Exact quotient of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.RealDomain.from_sympy">
-<tt class="descname">from_sympy</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.RealDomain.from_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert SymPy&#8217;s number to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.RealDomain.gcd">
-<tt class="descname">gcd</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.RealDomain.gcd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns GCD of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.RealDomain.get_exact">
-<tt class="descname">get_exact</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.RealDomain.get_exact" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns an exact domain associated with <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.RealDomain.get_field">
-<tt class="descname">get_field</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.RealDomain.get_field" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a field associated with <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.RealDomain.get_ring">
-<tt class="descname">get_ring</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.RealDomain.get_ring" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a ring associated with <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.RealDomain.lcm">
-<tt class="descname">lcm</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.RealDomain.lcm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns LCM of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.RealDomain.limit_denom">
-<tt class="descname">limit_denom</tt><big>(</big><em>self</em>, <em>n</em>, <em>d</em>, <em>**args</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.RealDomain.limit_denom" title="Permalink to this definition">¶</a></dt>
-<dd><p>Find closest rational to <span class="math">\(n/d\)</span> (up to <tt class="docutils literal"><span class="pre">max_denom</span></tt>).</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.RealDomain.quo">
-<tt class="descname">quo</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.RealDomain.quo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Quotient of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.RealDomain.rem">
-<tt class="descname">rem</tt><big>(</big><em>self</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.RealDomain.rem" title="Permalink to this definition">¶</a></dt>
-<dd><p>Remainder of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.RealDomain.to_sympy">
-<tt class="descname">to_sympy</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.RealDomain.to_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">a</span></tt> to SymPy number.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.domains.ExpressionDomain">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">ExpressionDomain</tt><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain" title="Permalink to this definition">¶</a></dt>
-<dd><p>A class for arbitrary expressions.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>alias</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="class">
-<dt id="sympy.polys.domains.ExpressionDomain.Expression">
-<em class="property">class </em><tt class="descname">Expression</tt><big>(</big><em>ex</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.Expression" title="Permalink to this definition">¶</a></dt>
-<dd><p>An arbitrary expression.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.ExpressionDomain.denom">
-<tt class="descclassname">ExpressionDomain.</tt><tt class="descname">denom</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.denom" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns denominator of <tt class="docutils literal"><span class="pre">a</span></tt>.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.domains.ExpressionDomain.dtype">
-<tt class="descclassname">ExpressionDomain.</tt><tt class="descname">dtype</tt><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.dtype" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <a class="reference internal" href="#sympy.polys.domains.ExpressionDomain.Expression" title="sympy.polys.domains.ExpressionDomain.Expression"><tt class="xref py py-class docutils literal"><span class="pre">Expression</span></tt></a></p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.ExpressionDomain.from_ExpressionDomain">
-<tt class="descclassname">ExpressionDomain.</tt><tt class="descname">from_ExpressionDomain</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.from_ExpressionDomain" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a <tt class="docutils literal"><span class="pre">EX</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.ExpressionDomain.from_FractionField">
-<tt class="descclassname">ExpressionDomain.</tt><tt class="descname">from_FractionField</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.from_FractionField" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a <tt class="docutils literal"><span class="pre">DMF</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.ExpressionDomain.from_GlobalPolynomialRing">
-<tt class="descclassname">ExpressionDomain.</tt><tt class="descname">from_GlobalPolynomialRing</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.from_GlobalPolynomialRing" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a <tt class="docutils literal"><span class="pre">DMP</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.ExpressionDomain.from_QQ_gmpy">
-<tt class="descclassname">ExpressionDomain.</tt><tt class="descname">from_QQ_gmpy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.from_QQ_gmpy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a GMPY <tt class="docutils literal"><span class="pre">mpq</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.ExpressionDomain.from_QQ_python">
-<tt class="descclassname">ExpressionDomain.</tt><tt class="descname">from_QQ_python</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.from_QQ_python" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a Python <tt class="docutils literal"><span class="pre">Fraction</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.ExpressionDomain.from_QQ_sympy">
-<tt class="descclassname">ExpressionDomain.</tt><tt class="descname">from_QQ_sympy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.from_QQ_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a SymPy <tt class="docutils literal"><span class="pre">Rational</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.ExpressionDomain.from_RR_mpmath">
-<tt class="descclassname">ExpressionDomain.</tt><tt class="descname">from_RR_mpmath</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.from_RR_mpmath" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a mpmath <tt class="docutils literal"><span class="pre">mpf</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.ExpressionDomain.from_RR_sympy">
-<tt class="descclassname">ExpressionDomain.</tt><tt class="descname">from_RR_sympy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.from_RR_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a SymPy <tt class="docutils literal"><span class="pre">Float</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.ExpressionDomain.from_ZZ_gmpy">
-<tt class="descclassname">ExpressionDomain.</tt><tt class="descname">from_ZZ_gmpy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.from_ZZ_gmpy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a GMPY <tt class="docutils literal"><span class="pre">mpz</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.ExpressionDomain.from_ZZ_python">
-<tt class="descclassname">ExpressionDomain.</tt><tt class="descname">from_ZZ_python</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.from_ZZ_python" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a Python <tt class="docutils literal"><span class="pre">int</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.ExpressionDomain.from_ZZ_sympy">
-<tt class="descclassname">ExpressionDomain.</tt><tt class="descname">from_ZZ_sympy</tt><big>(</big><em>K1</em>, <em>a</em>, <em>K0</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.from_ZZ_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a SymPy <tt class="docutils literal"><span class="pre">Integer</span></tt> object to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.ExpressionDomain.from_sympy">
-<tt class="descclassname">ExpressionDomain.</tt><tt class="descname">from_sympy</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.from_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert SymPy&#8217;s expression to <tt class="docutils literal"><span class="pre">dtype</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.ExpressionDomain.get_field">
-<tt class="descclassname">ExpressionDomain.</tt><tt class="descname">get_field</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.get_field" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a field associated with <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.ExpressionDomain.get_ring">
-<tt class="descclassname">ExpressionDomain.</tt><tt class="descname">get_ring</tt><big>(</big><em>self</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.get_ring" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a ring associated with <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.ExpressionDomain.is_negative">
-<tt class="descclassname">ExpressionDomain.</tt><tt class="descname">is_negative</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.is_negative" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">a</span></tt> is negative.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.ExpressionDomain.is_nonnegative">
-<tt class="descclassname">ExpressionDomain.</tt><tt class="descname">is_nonnegative</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.is_nonnegative" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">a</span></tt> is non-negative.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.ExpressionDomain.is_nonpositive">
-<tt class="descclassname">ExpressionDomain.</tt><tt class="descname">is_nonpositive</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.is_nonpositive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">a</span></tt> is non-positive.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.ExpressionDomain.is_positive">
-<tt class="descclassname">ExpressionDomain.</tt><tt class="descname">is_positive</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.is_positive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <tt class="docutils literal"><span class="pre">a</span></tt> is positive.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.ExpressionDomain.numer">
-<tt class="descclassname">ExpressionDomain.</tt><tt class="descname">numer</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.numer" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns numerator of <tt class="docutils literal"><span class="pre">a</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.domains.ExpressionDomain.to_sympy">
-<tt class="descclassname">ExpressionDomain.</tt><tt class="descname">to_sympy</tt><big>(</big><em>self</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.ExpressionDomain.to_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">a</span></tt> to a SymPy object.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="implementation-domains">
-<h3>Implementation Domains<a class="headerlink" href="#implementation-domains" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.polys.domains.PythonFiniteField">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">PythonFiniteField</tt><big>(</big><em>mod</em>, <em>symmetric=True</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.PythonFiniteField" title="Permalink to this definition">¶</a></dt>
-<dd><p>Finite field based on Python&#8217;s integers.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>dtype</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>mod</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>one</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>zero</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.domains.SymPyFiniteField">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">SymPyFiniteField</tt><big>(</big><em>mod</em>, <em>symmetric=True</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.SymPyFiniteField" title="Permalink to this definition">¶</a></dt>
-<dd><p>Finite field based on SymPy&#8217;s integers.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>dtype</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>mod</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>one</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>zero</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.domains.GMPYFiniteField">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">GMPYFiniteField</tt><big>(</big><em>mod</em>, <em>symmetric=True</em><big>)</big><a class="headerlink" href="#sympy.polys.domains.GMPYFiniteField" title="Permalink to this definition">¶</a></dt>
-<dd><p>Finite field based on Python&#8217;s integers.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="33%" />
-<col width="67%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>dtype</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>mod</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>one</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>zero</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.domains.PythonIntegerRing">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">PythonIntegerRing</tt><a class="headerlink" href="#sympy.polys.domains.PythonIntegerRing" title="Permalink to this definition">¶</a></dt>
-<dd><p>Integer ring based on Python&#8217;s <tt class="docutils literal"><span class="pre">int</span></tt> type.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.domains.SymPyIntegerRing">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">SymPyIntegerRing</tt><a class="headerlink" href="#sympy.polys.domains.SymPyIntegerRing" title="Permalink to this definition">¶</a></dt>
-<dd><p>Integer ring based on SymPy&#8217;s <tt class="docutils literal"><span class="pre">Integer</span></tt> type.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.domains.GMPYIntegerRing">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">GMPYIntegerRing</tt><a class="headerlink" href="#sympy.polys.domains.GMPYIntegerRing" title="Permalink to this definition">¶</a></dt>
-<dd><p>Integer ring based on GMPY&#8217;s <tt class="docutils literal"><span class="pre">mpz</span></tt> type.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.domains.PythonRationalField">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">PythonRationalField</tt><a class="headerlink" href="#sympy.polys.domains.PythonRationalField" title="Permalink to this definition">¶</a></dt>
-<dd><p>Rational field based on Python rational number type.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.domains.SymPyRationalField">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">SymPyRationalField</tt><a class="headerlink" href="#sympy.polys.domains.SymPyRationalField" title="Permalink to this definition">¶</a></dt>
-<dd><p>Rational field based on SymPy Rational class.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.domains.GMPYRationalField">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">GMPYRationalField</tt><a class="headerlink" href="#sympy.polys.domains.GMPYRationalField" title="Permalink to this definition">¶</a></dt>
-<dd><p>Rational field based on GMPY mpq class.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.domains.MPmathRealDomain">
-<em class="property">class </em><tt class="descclassname">sympy.polys.domains.</tt><tt class="descname">MPmathRealDomain</tt><a class="headerlink" href="#sympy.polys.domains.MPmathRealDomain" title="Permalink to this definition">¶</a></dt>
-<dd><p>Domain for real numbers based on mpmath mpf type.</p>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="level-one">
-<h2>Level One<a class="headerlink" href="#level-one" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.polys.polyclasses.DMP">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyclasses.</tt><tt class="descname">DMP</tt><big>(</big><em>rep</em>, <em>dom</em>, <em>lev=None</em>, <em>ring=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP" title="Permalink to this definition">¶</a></dt>
-<dd><p>Dense Multivariate Polynomials over <span class="math">\(K\)</span>.</p>
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.LC">
-<tt class="descname">LC</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.LC"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.LC" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the leading coefficient of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.TC">
-<tt class="descname">TC</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.TC"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.TC" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the trailing coefficient of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.abs">
-<tt class="descname">abs</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.abs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.abs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Make all coefficients in <tt class="docutils literal"><span class="pre">f</span></tt> positive.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.add">
-<tt class="descname">add</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Add two multivariate polynomials <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.add_ground">
-<tt class="descname">add_ground</tt><big>(</big><em>f</em>, <em>c</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.add_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.add_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Add an element of the ground domain to <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.all_coeffs">
-<tt class="descname">all_coeffs</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.all_coeffs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.all_coeffs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns all coefficients from <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.all_monoms">
-<tt class="descname">all_monoms</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.all_monoms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.all_monoms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns all monomials from <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.all_terms">
-<tt class="descname">all_terms</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.all_terms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.all_terms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns all terms from a <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.cancel">
-<tt class="descname">cancel</tt><big>(</big><em>f</em>, <em>g</em>, <em>include=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.cancel"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.cancel" title="Permalink to this definition">¶</a></dt>
-<dd><p>Cancel common factors in a rational function <tt class="docutils literal"><span class="pre">f/g</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.clear_denoms">
-<tt class="descname">clear_denoms</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.clear_denoms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.clear_denoms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Clear denominators, but keep the ground domain.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.coeffs">
-<tt class="descname">coeffs</tt><big>(</big><em>f</em>, <em>order=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.coeffs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.coeffs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns all non-zero coefficients from <tt class="docutils literal"><span class="pre">f</span></tt> in lex order.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.cofactors">
-<tt class="descname">cofactors</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.cofactors"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.cofactors" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns GCD of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt> and their cofactors.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.compose">
-<tt class="descname">compose</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.compose"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.compose" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes functional composition of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.content">
-<tt class="descname">content</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.content"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.content" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns GCD of polynomial coefficients.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.convert">
-<tt class="descname">convert</tt><big>(</big><em>f</em>, <em>dom</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.convert"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.convert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert the ground domain of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.count_complex_roots">
-<tt class="descname">count_complex_roots</tt><big>(</big><em>f</em>, <em>inf=None</em>, <em>sup=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.count_complex_roots"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.count_complex_roots" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the number of complex roots of <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">[inf,</span> <span class="pre">sup]</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.count_real_roots">
-<tt class="descname">count_real_roots</tt><big>(</big><em>f</em>, <em>inf=None</em>, <em>sup=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.count_real_roots"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.count_real_roots" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the number of real roots of <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">[inf,</span> <span class="pre">sup]</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.decompose">
-<tt class="descname">decompose</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.decompose"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.decompose" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes functional decomposition of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.deflate">
-<tt class="descname">deflate</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.deflate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.deflate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Reduce degree of <span class="math">\(f\)</span> by mapping <span class="math">\(x_i^m\)</span> to <span class="math">\(y_i\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.degree">
-<tt class="descname">degree</tt><big>(</big><em>f</em>, <em>j=0</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.degree"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.degree" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the leading degree of <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">x_j</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.degree_list">
-<tt class="descname">degree_list</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.degree_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.degree_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a list of degrees of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.diff">
-<tt class="descname">diff</tt><big>(</big><em>f</em>, <em>m=1</em>, <em>j=0</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.diff"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.diff" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the <tt class="docutils literal"><span class="pre">m</span></tt>-th order derivative of <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">x_j</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.discriminant">
-<tt class="descname">discriminant</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.discriminant"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.discriminant" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes discriminant of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.div">
-<tt class="descname">div</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.div"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.div" title="Permalink to this definition">¶</a></dt>
-<dd><p>Polynomial division with remainder of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.eject">
-<tt class="descname">eject</tt><big>(</big><em>f</em>, <em>dom</em>, <em>front=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.eject"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.eject" title="Permalink to this definition">¶</a></dt>
-<dd><p>Eject selected generators into the ground domain.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.eval">
-<tt class="descname">eval</tt><big>(</big><em>f</em>, <em>a</em>, <em>j=0</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.eval"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.eval" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates <tt class="docutils literal"><span class="pre">f</span></tt> at the given point <tt class="docutils literal"><span class="pre">a</span></tt> in <tt class="docutils literal"><span class="pre">x_j</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.exclude">
-<tt class="descname">exclude</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.exclude"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.exclude" title="Permalink to this definition">¶</a></dt>
-<dd><p>Remove useless generators from <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p>Returns the removed generators and the new excluded <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.polyclasses</span> <span class="kn">import</span> <span class="n">DMP</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">DMP</span><span class="p">([[[</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">)]],</span> <span class="p">[[</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">)],</span> <span class="p">[</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">2</span><span class="p">)]]],</span> <span class="n">ZZ</span><span class="p">)</span><span class="o">.</span><span class="n">exclude</span><span class="p">()</span>
-<span class="go">([2], DMP([[1], [1, 2]], ZZ, None))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.exquo">
-<tt class="descname">exquo</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.exquo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.exquo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes polynomial exact quotient of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.exquo_ground">
-<tt class="descname">exquo_ground</tt><big>(</big><em>f</em>, <em>c</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.exquo_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.exquo_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Exact quotient of <tt class="docutils literal"><span class="pre">f</span></tt> by a an element of the ground domain.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.factor_list">
-<tt class="descname">factor_list</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.factor_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.factor_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a list of irreducible factors of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.factor_list_include">
-<tt class="descname">factor_list_include</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.factor_list_include"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.factor_list_include" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a list of irreducible factors of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.polys.polyclasses.DMP.from_dict">
-<em class="property">classmethod </em><tt class="descname">from_dict</tt><big>(</big><em>rep</em>, <em>lev</em>, <em>dom</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.from_dict"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.from_dict" title="Permalink to this definition">¶</a></dt>
-<dd><p>Construct and instance of <tt class="docutils literal"><span class="pre">cls</span></tt> from a <tt class="docutils literal"><span class="pre">dict</span></tt> representation.</p>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.polys.polyclasses.DMP.from_list">
-<em class="property">classmethod </em><tt class="descname">from_list</tt><big>(</big><em>rep</em>, <em>lev</em>, <em>dom</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.from_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.from_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create an instance of <tt class="docutils literal"><span class="pre">cls</span></tt> given a list of native coefficients.</p>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.polys.polyclasses.DMP.from_sympy_list">
-<em class="property">classmethod </em><tt class="descname">from_sympy_list</tt><big>(</big><em>rep</em>, <em>lev</em>, <em>dom</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.from_sympy_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.from_sympy_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create an instance of <tt class="docutils literal"><span class="pre">cls</span></tt> given a list of SymPy coefficients.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.gcd">
-<tt class="descname">gcd</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.gcd"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.gcd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns polynomial GCD of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.gcdex">
-<tt class="descname">gcdex</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.gcdex"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.gcdex" title="Permalink to this definition">¶</a></dt>
-<dd><p>Extended Euclidean algorithm, if univariate.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.gff_list">
-<tt class="descname">gff_list</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.gff_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.gff_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes greatest factorial factorization of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.half_gcdex">
-<tt class="descname">half_gcdex</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.half_gcdex"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.half_gcdex" title="Permalink to this definition">¶</a></dt>
-<dd><p>Half extended Euclidean algorithm, if univariate.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.homogeneous_order">
-<tt class="descname">homogeneous_order</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.homogeneous_order"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.homogeneous_order" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the homogeneous order of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.inject">
-<tt class="descname">inject</tt><big>(</big><em>f</em>, <em>front=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.inject"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.inject" title="Permalink to this definition">¶</a></dt>
-<dd><p>Inject ground domain generators into <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.integrate">
-<tt class="descname">integrate</tt><big>(</big><em>f</em>, <em>m=1</em>, <em>j=0</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.integrate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.integrate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the <tt class="docutils literal"><span class="pre">m</span></tt>-th order indefinite integral of <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">x_j</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.intervals">
-<tt class="descname">intervals</tt><big>(</big><em>f</em>, <em>all=False</em>, <em>eps=None</em>, <em>inf=None</em>, <em>sup=None</em>, <em>fast=False</em>, <em>sqf=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.intervals"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.intervals" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute isolating intervals for roots of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.invert">
-<tt class="descname">invert</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.invert"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.invert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Invert <tt class="docutils literal"><span class="pre">f</span></tt> modulo <tt class="docutils literal"><span class="pre">g</span></tt>, if possible.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polyclasses.DMP.is_cyclotomic">
-<tt class="descname">is_cyclotomic</tt><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.is_cyclotomic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.is_cyclotomic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is a cyclotomic polynomial.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polyclasses.DMP.is_ground">
-<tt class="descname">is_ground</tt><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.is_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.is_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is an element of the ground domain.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polyclasses.DMP.is_homogeneous">
-<tt class="descname">is_homogeneous</tt><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.is_homogeneous"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.is_homogeneous" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is a homogeneous polynomial.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polyclasses.DMP.is_irreducible">
-<tt class="descname">is_irreducible</tt><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.is_irreducible"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.is_irreducible" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> has no factors over its domain.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polyclasses.DMP.is_linear">
-<tt class="descname">is_linear</tt><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.is_linear"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.is_linear" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is linear in all its variables.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polyclasses.DMP.is_monic">
-<tt class="descname">is_monic</tt><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.is_monic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.is_monic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if the leading coefficient of <tt class="docutils literal"><span class="pre">f</span></tt> is one.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polyclasses.DMP.is_monomial">
-<tt class="descname">is_monomial</tt><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.is_monomial"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.is_monomial" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is zero or has only one term.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polyclasses.DMP.is_one">
-<tt class="descname">is_one</tt><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.is_one"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.is_one" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is a unit polynomial.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polyclasses.DMP.is_primitive">
-<tt class="descname">is_primitive</tt><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.is_primitive"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.is_primitive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if the GCD of the coefficients of <tt class="docutils literal"><span class="pre">f</span></tt> is one.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polyclasses.DMP.is_quadratic">
-<tt class="descname">is_quadratic</tt><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.is_quadratic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.is_quadratic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is quadratic in all its variables.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polyclasses.DMP.is_sqf">
-<tt class="descname">is_sqf</tt><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.is_sqf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.is_sqf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is a square-free polynomial.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polyclasses.DMP.is_zero">
-<tt class="descname">is_zero</tt><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.is_zero"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.is_zero" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is a zero polynomial.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.l1_norm">
-<tt class="descname">l1_norm</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.l1_norm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.l1_norm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns l1 norm of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.lcm">
-<tt class="descname">lcm</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.lcm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.lcm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns polynomial LCM of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.lift">
-<tt class="descname">lift</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.lift"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.lift" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert algebraic coefficients to rationals.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.max_norm">
-<tt class="descname">max_norm</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.max_norm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.max_norm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns maximum norm of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.monic">
-<tt class="descname">monic</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.monic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.monic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Divides all coefficients by <tt class="docutils literal"><span class="pre">LC(f)</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.monoms">
-<tt class="descname">monoms</tt><big>(</big><em>f</em>, <em>order=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.monoms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.monoms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns all non-zero monomials from <tt class="docutils literal"><span class="pre">f</span></tt> in lex order.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.mul">
-<tt class="descname">mul</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Multiply two multivariate polynomials <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.mul_ground">
-<tt class="descname">mul_ground</tt><big>(</big><em>f</em>, <em>c</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.mul_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.mul_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Multiply <tt class="docutils literal"><span class="pre">f</span></tt> by a an element of the ground domain.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.neg">
-<tt class="descname">neg</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.neg"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.neg" title="Permalink to this definition">¶</a></dt>
-<dd><p>Negate all cefficients in <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.nth">
-<tt class="descname">nth</tt><big>(</big><em>f</em>, <em>*N</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.nth"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.nth" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the <tt class="docutils literal"><span class="pre">n</span></tt>-th coefficient of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.pdiv">
-<tt class="descname">pdiv</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.pdiv"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.pdiv" title="Permalink to this definition">¶</a></dt>
-<dd><p>Polynomial pseudo-division of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.per">
-<tt class="descname">per</tt><big>(</big><em>f</em>, <em>rep</em>, <em>dom=None</em>, <em>kill=False</em>, <em>ring=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.per"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.per" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a DMP out of the given representation.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.permute">
-<tt class="descname">permute</tt><big>(</big><em>f</em>, <em>P</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.permute"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.permute" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a polynomial in <span class="math">\(K[x_{P(1)}, ..., x_{P(n)}]\)</span>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.polyclasses</span> <span class="kn">import</span> <span class="n">DMP</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">DMP</span><span class="p">([[[</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">2</span><span class="p">)],</span> <span class="p">[</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">0</span><span class="p">)]],</span> <span class="p">[[]]],</span> <span class="n">ZZ</span><span class="p">)</span><span class="o">.</span><span class="n">permute</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="go">DMP([[[2], []], [[1, 0], []]], ZZ, None)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">DMP</span><span class="p">([[[</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">2</span><span class="p">)],</span> <span class="p">[</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">0</span><span class="p">)]],</span> <span class="p">[[]]],</span> <span class="n">ZZ</span><span class="p">)</span><span class="o">.</span><span class="n">permute</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="go">DMP([[[1], []], [[2, 0], []]], ZZ, None)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.pexquo">
-<tt class="descname">pexquo</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.pexquo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.pexquo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Polynomial exact pseudo-quotient of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.pow">
-<tt class="descname">pow</tt><big>(</big><em>f</em>, <em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.pow"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.pow" title="Permalink to this definition">¶</a></dt>
-<dd><p>Raise <tt class="docutils literal"><span class="pre">f</span></tt> to a non-negative power <tt class="docutils literal"><span class="pre">n</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.pquo">
-<tt class="descname">pquo</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.pquo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.pquo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Polynomial pseudo-quotient of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.prem">
-<tt class="descname">prem</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.prem"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.prem" title="Permalink to this definition">¶</a></dt>
-<dd><p>Polynomial pseudo-remainder of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.primitive">
-<tt class="descname">primitive</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.primitive"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.primitive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns content and a primitive form of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.quo">
-<tt class="descname">quo</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.quo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.quo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes polynomial quotient of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.quo_ground">
-<tt class="descname">quo_ground</tt><big>(</big><em>f</em>, <em>c</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.quo_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.quo_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Quotient of <tt class="docutils literal"><span class="pre">f</span></tt> by a an element of the ground domain.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.refine_root">
-<tt class="descname">refine_root</tt><big>(</big><em>f</em>, <em>s</em>, <em>t</em>, <em>eps=None</em>, <em>steps=None</em>, <em>fast=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.refine_root"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.refine_root" title="Permalink to this definition">¶</a></dt>
-<dd><p>Refine an isolating interval to the given precision.</p>
-<p><tt class="docutils literal"><span class="pre">eps</span></tt> should be a rational number.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.rem">
-<tt class="descname">rem</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.rem"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.rem" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes polynomial remainder of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.resultant">
-<tt class="descname">resultant</tt><big>(</big><em>f</em>, <em>g</em>, <em>includePRS=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.resultant"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.resultant" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes resultant of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt> via PRS.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.revert">
-<tt class="descname">revert</tt><big>(</big><em>f</em>, <em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.revert"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.revert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute <tt class="docutils literal"><span class="pre">f**(-1)</span></tt> mod <tt class="docutils literal"><span class="pre">x**n</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.shift">
-<tt class="descname">shift</tt><big>(</big><em>f</em>, <em>a</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.shift"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.shift" title="Permalink to this definition">¶</a></dt>
-<dd><p>Efficiently compute Taylor shift <tt class="docutils literal"><span class="pre">f(x</span> <span class="pre">+</span> <span class="pre">a)</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.slice">
-<tt class="descname">slice</tt><big>(</big><em>f</em>, <em>m</em>, <em>n</em>, <em>j=0</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.slice"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.slice" title="Permalink to this definition">¶</a></dt>
-<dd><p>Take a continuous subsequence of terms of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.sqf_list">
-<tt class="descname">sqf_list</tt><big>(</big><em>f</em>, <em>all=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.sqf_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.sqf_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a list of square-free factors of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.sqf_list_include">
-<tt class="descname">sqf_list_include</tt><big>(</big><em>f</em>, <em>all=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.sqf_list_include"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.sqf_list_include" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a list of square-free factors of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.sqf_norm">
-<tt class="descname">sqf_norm</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.sqf_norm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.sqf_norm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes square-free norm of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.sqf_part">
-<tt class="descname">sqf_part</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.sqf_part"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.sqf_part" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes square-free part of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.sqr">
-<tt class="descname">sqr</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.sqr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.sqr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Square a multivariate polynomial <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.sturm">
-<tt class="descname">sturm</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.sturm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.sturm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the Sturm sequence of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.sub">
-<tt class="descname">sub</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.sub"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.sub" title="Permalink to this definition">¶</a></dt>
-<dd><p>Subtract two multivariate polynomials <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.sub_ground">
-<tt class="descname">sub_ground</tt><big>(</big><em>f</em>, <em>c</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.sub_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.sub_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Subtract an element of the ground domain from <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.subresultants">
-<tt class="descname">subresultants</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.subresultants"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.subresultants" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes subresultant PRS sequence of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.terms">
-<tt class="descname">terms</tt><big>(</big><em>f</em>, <em>order=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.terms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.terms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns all non-zero terms from <tt class="docutils literal"><span class="pre">f</span></tt> in lex order.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.terms_gcd">
-<tt class="descname">terms_gcd</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.terms_gcd"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.terms_gcd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Remove GCD of terms from the polynomial <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.to_dict">
-<tt class="descname">to_dict</tt><big>(</big><em>f</em>, <em>zero=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.to_dict"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.to_dict" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">f</span></tt> to a dict representation with native coefficients.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.to_exact">
-<tt class="descname">to_exact</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.to_exact"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.to_exact" title="Permalink to this definition">¶</a></dt>
-<dd><p>Make the ground domain exact.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.to_field">
-<tt class="descname">to_field</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.to_field"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.to_field" title="Permalink to this definition">¶</a></dt>
-<dd><p>Make the ground domain a field.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.to_ring">
-<tt class="descname">to_ring</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.to_ring"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.to_ring" title="Permalink to this definition">¶</a></dt>
-<dd><p>Make the ground domain a ring.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.to_sympy_dict">
-<tt class="descname">to_sympy_dict</tt><big>(</big><em>f</em>, <em>zero=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.to_sympy_dict"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.to_sympy_dict" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">f</span></tt> to a dict representation with SymPy coefficients.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.to_tuple">
-<tt class="descname">to_tuple</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.to_tuple"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.to_tuple" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">f</span></tt> to a tuple representation with native coefficients.</p>
-<p>This is needed for hashing.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.total_degree">
-<tt class="descname">total_degree</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.total_degree"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.total_degree" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the total degree of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.trunc">
-<tt class="descname">trunc</tt><big>(</big><em>f</em>, <em>p</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.trunc"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.trunc" title="Permalink to this definition">¶</a></dt>
-<dd><p>Reduce <tt class="docutils literal"><span class="pre">f</span></tt> modulo a constant <tt class="docutils literal"><span class="pre">p</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMP.unify">
-<tt class="descname">unify</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMP.unify"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMP.unify" title="Permalink to this definition">¶</a></dt>
-<dd><p>Unify representations of two multivariate polynomials.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyclasses.DMF">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyclasses.</tt><tt class="descname">DMF</tt><big>(</big><em>rep</em>, <em>dom</em>, <em>lev=None</em>, <em>ring=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMF"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMF" title="Permalink to this definition">¶</a></dt>
-<dd><p>Dense Multivariate Fractions over <span class="math">\(K\)</span>.</p>
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMF.add">
-<tt class="descname">add</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMF.add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMF.add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Add two multivariate fractions <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMF.cancel">
-<tt class="descname">cancel</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMF.cancel"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMF.cancel" title="Permalink to this definition">¶</a></dt>
-<dd><p>Remove common factors from <tt class="docutils literal"><span class="pre">f.num</span></tt> and <tt class="docutils literal"><span class="pre">f.den</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMF.denom">
-<tt class="descname">denom</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMF.denom"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMF.denom" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the denominator of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMF.exquo">
-<tt class="descname">exquo</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="headerlink" href="#sympy.polys.polyclasses.DMF.exquo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes quotient of fractions <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMF.frac_unify">
-<tt class="descname">frac_unify</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMF.frac_unify"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMF.frac_unify" title="Permalink to this definition">¶</a></dt>
-<dd><p>Unify representations of two multivariate fractions.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMF.half_per">
-<tt class="descname">half_per</tt><big>(</big><em>f</em>, <em>rep</em>, <em>kill=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMF.half_per"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMF.half_per" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a DMP out of the given representation.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMF.invert">
-<tt class="descname">invert</tt><big>(</big><em>f</em>, <em>check=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMF.invert"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMF.invert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes inverse of a fraction <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polyclasses.DMF.is_one">
-<tt class="descname">is_one</tt><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMF.is_one"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMF.is_one" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is a unit fraction.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polyclasses.DMF.is_zero">
-<tt class="descname">is_zero</tt><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMF.is_zero"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMF.is_zero" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is a zero fraction.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMF.mul">
-<tt class="descname">mul</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMF.mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMF.mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Multiply two multivariate fractions <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMF.neg">
-<tt class="descname">neg</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMF.neg"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMF.neg" title="Permalink to this definition">¶</a></dt>
-<dd><p>Negate all cefficients in <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMF.numer">
-<tt class="descname">numer</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMF.numer"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMF.numer" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the numerator of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMF.per">
-<tt class="descname">per</tt><big>(</big><em>f</em>, <em>num</em>, <em>den</em>, <em>cancel=True</em>, <em>kill=False</em>, <em>ring=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMF.per"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMF.per" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a DMF out of the given representation.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMF.poly_unify">
-<tt class="descname">poly_unify</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMF.poly_unify"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMF.poly_unify" title="Permalink to this definition">¶</a></dt>
-<dd><p>Unify a multivariate fraction and a polynomial.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMF.pow">
-<tt class="descname">pow</tt><big>(</big><em>f</em>, <em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMF.pow"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMF.pow" title="Permalink to this definition">¶</a></dt>
-<dd><p>Raise <tt class="docutils literal"><span class="pre">f</span></tt> to a non-negative power <tt class="docutils literal"><span class="pre">n</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMF.quo">
-<tt class="descname">quo</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMF.quo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMF.quo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes quotient of fractions <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.DMF.sub">
-<tt class="descname">sub</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#DMF.sub"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.DMF.sub" title="Permalink to this definition">¶</a></dt>
-<dd><p>Subtract two multivariate fractions <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyclasses.ANP">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyclasses.</tt><tt class="descname">ANP</tt><big>(</big><em>rep</em>, <em>mod</em>, <em>dom</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#ANP"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.ANP" title="Permalink to this definition">¶</a></dt>
-<dd><p>Dense Algebraic Number Polynomials over a field.</p>
-<dl class="function">
-<dt id="sympy.polys.polyclasses.ANP.LC">
-<tt class="descname">LC</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#ANP.LC"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.ANP.LC" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the leading coefficient of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.ANP.TC">
-<tt class="descname">TC</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#ANP.TC"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.ANP.TC" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the trailing coefficient of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polyclasses.ANP.is_ground">
-<tt class="descname">is_ground</tt><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#ANP.is_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.ANP.is_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is an element of the ground domain.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polyclasses.ANP.is_one">
-<tt class="descname">is_one</tt><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#ANP.is_one"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.ANP.is_one" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is a unit algebraic number.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polyclasses.ANP.is_zero">
-<tt class="descname">is_zero</tt><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#ANP.is_zero"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.ANP.is_zero" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is a zero algebraic number.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.ANP.pow">
-<tt class="descname">pow</tt><big>(</big><em>f</em>, <em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#ANP.pow"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.ANP.pow" title="Permalink to this definition">¶</a></dt>
-<dd><p>Raise <tt class="docutils literal"><span class="pre">f</span></tt> to a non-negative power <tt class="docutils literal"><span class="pre">n</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.ANP.to_dict">
-<tt class="descname">to_dict</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#ANP.to_dict"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.ANP.to_dict" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">f</span></tt> to a dict representation with native coefficients.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.ANP.to_list">
-<tt class="descname">to_list</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#ANP.to_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.ANP.to_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">f</span></tt> to a list representation with native coefficients.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.ANP.to_sympy_dict">
-<tt class="descname">to_sympy_dict</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#ANP.to_sympy_dict"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.ANP.to_sympy_dict" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">f</span></tt> to a dict representation with SymPy coefficients.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.ANP.to_sympy_list">
-<tt class="descname">to_sympy_list</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#ANP.to_sympy_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.ANP.to_sympy_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">f</span></tt> to a list representation with SymPy coefficients.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.ANP.to_tuple">
-<tt class="descname">to_tuple</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#ANP.to_tuple"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.ANP.to_tuple" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">f</span></tt> to a tuple representation with native coefficients.</p>
-<p>This is needed for hashing.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyclasses.ANP.unify">
-<tt class="descname">unify</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyclasses.html#ANP.unify"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyclasses.ANP.unify" title="Permalink to this definition">¶</a></dt>
-<dd><p>Unify representations of two algebraic numbers.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="level-zero">
-<h2>Level Zero<a class="headerlink" href="#level-zero" title="Permalink to this headline">¶</a></h2>
-<p>Level zero contains the bulk code of the polynomial manipulation module.</p>
-<div class="section" id="manipulation-of-dense-multivariate-polynomials">
-<h3>Manipulation of dense, multivariate polynomials<a class="headerlink" href="#manipulation-of-dense-multivariate-polynomials" title="Permalink to this headline">¶</a></h3>
-<p>These functions can be used to manipulate polynomials in <span class="math">\(K[X_0, \dots, X_u]\)</span>.
-Functions for manipulating multivariate polynomials in the dense representation
-have the prefix <tt class="docutils literal"><span class="pre">dmp_</span></tt>. Functions which only apply to univariate polynomials
-(i.e. <span class="math">\(u = 0\)</span>)
-have the prefix <tt class="docutils literal"><span class="pre">dup__</span></tt>. The ground domain <span class="math">\(K\)</span> has to be passed explicitly.
-For many multivariate polynomial manipulation functions also the level <span class="math">\(u\)</span>,
-i.e. the number of generators minus one, has to be passed.
-(Note that, in many cases, <tt class="docutils literal"><span class="pre">dup_</span></tt> versions of functions are available, which
-may be slightly more efficient.)</p>
-<p><strong>Basic manipulation:</strong></p>
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_LC">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_LC</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="headerlink" href="#sympy.polys.densebasic.dmp_LC" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return leading coefficient of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">poly_LC</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">poly_LC</span><span class="p">([],</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly_LC</span><span class="p">([</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">3</span><span class="p">)],</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_TC">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_TC</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="headerlink" href="#sympy.polys.densebasic.dmp_TC" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return trailing coefficient of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">poly_TC</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">poly_TC</span><span class="p">([],</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">poly_TC</span><span class="p">([</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">3</span><span class="p">)],</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_ground_LC">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_ground_LC</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_ground_LC"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_ground_LC" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the ground leading coefficient.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_ground_LC</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_ground_LC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_ground_TC">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_ground_TC</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_ground_TC"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_ground_TC" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the ground trailing coefficient.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_ground_TC</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_ground_TC</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_true_LT">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_true_LT</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_true_LT"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_true_LT" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the leading term <tt class="docutils literal"><span class="pre">c</span> <span class="pre">*</span> <span class="pre">x_1**n_1</span> <span class="pre">...</span> <span class="pre">x_k**n_k</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_true_LT</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_true_LT</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">((2, 0), 4)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_degree">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_degree</tt><big>(</big><em>f</em>, <em>u</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_degree"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_degree" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the leading degree of <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">x_0</span></tt> in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_degree</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_degree</span><span class="p">([[[]]],</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">-1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_degree</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_degree_in">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_degree_in</tt><big>(</big><em>f</em>, <em>j</em>, <em>u</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_degree_in"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_degree_in" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the leading degree of <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">x_j</span></tt> in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_degree_in</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_degree_in</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_degree_in</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_degree_list">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_degree_list</tt><big>(</big><em>f</em>, <em>u</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_degree_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_degree_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a list of degrees of <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_degree_list</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_degree_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">(1, 2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_strip">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_strip</tt><big>(</big><em>f</em>, <em>u</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_strip"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_strip" title="Permalink to this definition">¶</a></dt>
-<dd><p>Remove leading zeros from <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_strip</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_strip</span><span class="p">([[],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">]],</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">[[0, 1, 2], [1]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_validate">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_validate</tt><big>(</big><em>f</em>, <em>K=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_validate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_validate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the number of levels in <tt class="docutils literal"><span class="pre">f</span></tt> and recursively strip it.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_validate</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_validate</span><span class="p">([[],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">]])</span>
-<span class="go">([[1, 2], [1]], 1)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_validate</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">ValueError</span>: <span class="n">invalid data structure for a multivariate polynomial</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dup_reverse">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dup_reverse</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dup_reverse"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dup_reverse" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute <tt class="docutils literal"><span class="pre">x**n</span> <span class="pre">*</span> <span class="pre">f(1/x)</span></tt>, i.e.: reverse <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">K[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dup_reverse</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dup_reverse</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-<span class="go">[3, 2, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_copy">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_copy</tt><big>(</big><em>f</em>, <em>u</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_copy"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_copy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a new copy of a polynomial <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_copy</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_copy</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">[[1], [1, 2]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_to_tuple">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_to_tuple</tt><big>(</big><em>f</em>, <em>u</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_to_tuple"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_to_tuple" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <span class="math">\(f\)</span> into a nested tuple of tuples.</p>
-<p>This is needed for hashing.  This is similar to dmp_copy().</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_to_tuple</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_to_tuple</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">((1,), (1, 2))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_normal">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_normal</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_normal"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_normal" title="Permalink to this definition">¶</a></dt>
-<dd><p>Normalize a multivariate polynomial in the given domain.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_normal</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_normal</span><span class="p">([[],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mi">2</span><span class="p">]],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1, 2]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_convert">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_convert</tt><big>(</big><em>f</em>, <em>u</em>, <em>K0</em>, <em>K1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_convert"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_convert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert the ground domain of <tt class="docutils literal"><span class="pre">f</span></tt> from <tt class="docutils literal"><span class="pre">K0</span></tt> to <tt class="docutils literal"><span class="pre">K1</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.polyclasses</span> <span class="kn">import</span> <span class="n">DMP</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_convert</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="p">[[</span><span class="n">DMP</span><span class="p">([</span><span class="mi">1</span><span class="p">],</span> <span class="n">ZZ</span><span class="p">)],</span> <span class="p">[</span><span class="n">DMP</span><span class="p">([</span><span class="mi">2</span><span class="p">],</span> <span class="n">ZZ</span><span class="p">)]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="p">[[</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">)],</span> <span class="p">[</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">2</span><span class="p">)]]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_convert</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">[</span><span class="s">&#39;x&#39;</span><span class="p">],</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1], [2]]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_convert</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">[</span><span class="s">&#39;x&#39;</span><span class="p">])</span>
-<span class="go">[[1], [2]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_from_sympy">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_from_sympy</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_from_sympy"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_from_sympy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert the ground domain of <tt class="docutils literal"><span class="pre">f</span></tt> from SymPy to <tt class="docutils literal"><span class="pre">K</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_from_sympy</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_from_sympy</span><span class="p">([[</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)],</span> <span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">2</span><span class="p">)]],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span> <span class="o">==</span> <span class="p">[[</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">)],</span> <span class="p">[</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">2</span><span class="p">)]]</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_nth">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_nth</tt><big>(</big><em>f</em>, <em>n</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_nth"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_nth" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the <tt class="docutils literal"><span class="pre">n</span></tt>-th coefficient of <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">K[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_nth</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_nth</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_nth</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_ground_nth">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_ground_nth</tt><big>(</big><em>f</em>, <em>N</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_ground_nth"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_ground_nth" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the ground <tt class="docutils literal"><span class="pre">n</span></tt>-th coefficient of <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">K[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_ground_nth</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_ground_nth</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_zero_p">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_zero_p</tt><big>(</big><em>f</em>, <em>u</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_zero_p"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_zero_p" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is zero in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_zero_p</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_zero_p</span><span class="p">([[[[[]]]]],</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_zero_p</span><span class="p">([[[[[</span><span class="mi">1</span><span class="p">]]]]],</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_zero">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_zero</tt><big>(</big><em>u</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_zero"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_zero" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a multivariate zero.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_zero</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_zero</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">[[[[[]]]]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_one_p">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_one_p</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_one_p"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_one_p" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is one in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_one_p</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_one_p</span><span class="p">([[[</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">)]]],</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_one">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_one</tt><big>(</big><em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_one"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_one" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a multivariate one over <tt class="docutils literal"><span class="pre">K</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_one</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_one</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[[1]]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_ground_p">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_ground_p</tt><big>(</big><em>f</em>, <em>c</em>, <em>u</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_ground_p"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_ground_p" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if <tt class="docutils literal"><span class="pre">f</span></tt> is constant in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_ground_p</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_ground_p</span><span class="p">([[[</span><span class="mi">3</span><span class="p">]]],</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_ground_p</span><span class="p">([[[</span><span class="mi">4</span><span class="p">]]],</span> <span class="bp">None</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_ground">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_ground</tt><big>(</big><em>c</em>, <em>u</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a multivariate constant.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_ground</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_ground</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">[[[[[[3]]]]]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_ground</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_zeros">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_zeros</tt><big>(</big><em>n</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_zeros"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_zeros" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a list of multivariate zeros.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_zeros</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[[[]]], [[[]]], [[[]]]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_zeros</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[0, 0, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_grounds">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_grounds</tt><big>(</big><em>c</em>, <em>n</em>, <em>u</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_grounds"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_grounds" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a list of multivariate constants.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_grounds</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_grounds</span><span class="p">(</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">4</span><span class="p">),</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">[[[[4]]], [[[4]]], [[[4]]]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_grounds</span><span class="p">(</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">4</span><span class="p">),</span> <span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">[4, 4, 4]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_negative_p">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_negative_p</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_negative_p"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_negative_p" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">LC(f)</span></tt> is negative.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_negative_p</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_negative_p</span><span class="p">([[</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">)],</span> <span class="p">[</span><span class="o">-</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">)]],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_negative_p</span><span class="p">([[</span><span class="o">-</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">)],</span> <span class="p">[</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">)]],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_positive_p">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_positive_p</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_positive_p"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_positive_p" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">LC(f)</span></tt> is positive.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_positive_p</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_positive_p</span><span class="p">([[</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">)],</span> <span class="p">[</span><span class="o">-</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">)]],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_positive_p</span><span class="p">([[</span><span class="o">-</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">)],</span> <span class="p">[</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">)]],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_from_dict">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_from_dict</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_from_dict"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_from_dict" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a <tt class="docutils literal"><span class="pre">K[X]</span></tt> polynomial from a <tt class="docutils literal"><span class="pre">dict</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_from_dict</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_from_dict</span><span class="p">({(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">):</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">):</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">):</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">)},</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1, 0], [], [2, 3]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_from_dict</span><span class="p">({},</span> <span class="mi">0</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_to_dict">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_to_dict</tt><big>(</big><em>f</em>, <em>u</em>, <em>K=None</em>, <em>zero=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_to_dict"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_to_dict" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a <tt class="docutils literal"><span class="pre">K[X]</span></tt> polynomial to a <tt class="docutils literal"><span class="pre">dict``</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_to_dict</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_to_dict</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]],</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">{(0, 0): 3, (0, 1): 2, (2, 1): 1}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_to_dict</span><span class="p">([],</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">{}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_swap">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_swap</tt><big>(</big><em>f</em>, <em>i</em>, <em>j</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_swap"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_swap" title="Permalink to this definition">¶</a></dt>
-<dd><p>Transform <tt class="docutils literal"><span class="pre">K[..x_i..x_j..]</span></tt> to <tt class="docutils literal"><span class="pre">K[..x_j..x_i..]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_swap</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]],</span> <span class="p">[]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_swap</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[[2], []], [[1, 0], []]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_swap</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[[1], [2, 0]], [[]]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_swap</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[[1, 0]], [[2, 0], []]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_permute">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_permute</tt><big>(</big><em>f</em>, <em>P</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_permute"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_permute" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a polynomial in <tt class="docutils literal"><span class="pre">K[x_{P(1)},..,x_{P(n)}]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_permute</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]],</span> <span class="p">[]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_permute</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[[2], []], [[1, 0], []]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_permute</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[[1], []], [[2, 0], []]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_nest">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_nest</tt><big>(</big><em>f</em>, <em>l</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_nest"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_nest" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a multivariate value nested <tt class="docutils literal"><span class="pre">l</span></tt>-levels.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_nest</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_nest</span><span class="p">([[</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">)]],</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[[[1]]]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_raise">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_raise</tt><big>(</big><em>f</em>, <em>l</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_raise"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_raise" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a multivariate polynomial raised <tt class="docutils literal"><span class="pre">l</span></tt>-levels.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_raise</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_raise</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[[[]]], [[[1]], [[2]]]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_deflate">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_deflate</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_deflate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_deflate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Map <tt class="docutils literal"><span class="pre">x_i**m_i</span></tt> to <tt class="docutils literal"><span class="pre">y_i</span></tt> in a polynomial in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_deflate</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_deflate</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">((2, 3), [[1, 2], [3, 4]])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_multi_deflate">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_multi_deflate</tt><big>(</big><em>polys</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_multi_deflate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_multi_deflate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Map <tt class="docutils literal"><span class="pre">x_i**m_i</span></tt> to <tt class="docutils literal"><span class="pre">y_i</span></tt> in a set of polynomials in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_multi_deflate</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_multi_deflate</span><span class="p">((</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">((2, 1), ([[1, 0, 0, 2], [3, 0, 0, 4]], [[1, 0, 2], [3, 0, 4]]))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_inflate">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_inflate</tt><big>(</big><em>f</em>, <em>M</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_inflate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_inflate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Map <tt class="docutils literal"><span class="pre">y_i</span></tt> to <tt class="docutils literal"><span class="pre">x_i**k_i</span></tt> in a polynomial in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_inflate</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_inflate</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1, 0, 0, 2], [], [3, 0, 0, 4]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_exclude">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_exclude</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_exclude"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_exclude" title="Permalink to this definition">¶</a></dt>
-<dd><p>Exclude useless levels from <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p>Return the levels excluded, the new excluded <tt class="docutils literal"><span class="pre">f</span></tt>, and the new <tt class="docutils literal"><span class="pre">u</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_exclude</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[[</span><span class="mi">1</span><span class="p">]],</span> <span class="p">[[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_exclude</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">([2], [[1], [1, 2]], 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_include">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_include</tt><big>(</big><em>f</em>, <em>J</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_include"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_include" title="Permalink to this definition">¶</a></dt>
-<dd><p>Include useless levels in <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_include</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_include</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[[1]], [[1], [2]]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_inject">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_inject</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em>, <em>front=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_inject"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_inject" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">f</span></tt> from <tt class="docutils literal"><span class="pre">K[X][Y]</span></tt> to <tt class="docutils literal"><span class="pre">K[X,Y]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_inject</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">K</span> <span class="o">=</span> <span class="n">ZZ</span><span class="p">[</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="s">&#39;y&#39;</span><span class="p">]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_inject</span><span class="p">([</span><span class="n">K</span><span class="p">([[</span><span class="mi">1</span><span class="p">]]),</span> <span class="n">K</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]])],</span> <span class="mi">0</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-<span class="go">([[[1]], [[1], [2]]], 2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_inject</span><span class="p">([</span><span class="n">K</span><span class="p">([[</span><span class="mi">1</span><span class="p">]]),</span> <span class="n">K</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]])],</span> <span class="mi">0</span><span class="p">,</span> <span class="n">K</span><span class="p">,</span> <span class="n">front</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">([[[1]], [[1, 2]]], 2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_eject">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_eject</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em>, <em>front=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_eject"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_eject" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">f</span></tt> from <tt class="docutils literal"><span class="pre">K[X,Y]</span></tt> to <tt class="docutils literal"><span class="pre">K[X][Y]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_eject</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">K</span> <span class="o">=</span> <span class="n">ZZ</span><span class="p">[</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="s">&#39;y&#39;</span><span class="p">]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_eject</span><span class="p">([[[</span><span class="mi">1</span><span class="p">]],</span> <span class="p">[[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]]],</span> <span class="mi">2</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-<span class="go">[1, x + 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_terms_gcd">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_terms_gcd</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_terms_gcd"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_terms_gcd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Remove GCD of terms from <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_terms_gcd</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_terms_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">((2, 1), [[1], [1, 0]])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_list_terms">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_list_terms</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em>, <em>order=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_list_terms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_list_terms" title="Permalink to this definition">¶</a></dt>
-<dd><p>List all non-zero terms from <tt class="docutils literal"><span class="pre">f</span></tt> in the given order <tt class="docutils literal"><span class="pre">order</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_list_terms</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_list_terms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[((1, 1), 1), ((1, 0), 1), ((0, 1), 2), ((0, 0), 3)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_list_terms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s">&#39;grevlex&#39;</span><span class="p">)</span>
-<span class="go">[((1, 1), 1), ((1, 0), 1), ((0, 1), 2), ((0, 0), 3)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_apply_pairs">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_apply_pairs</tt><big>(</big><em>f</em>, <em>g</em>, <em>h</em>, <em>args</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_apply_pairs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_apply_pairs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Apply <tt class="docutils literal"><span class="pre">h</span></tt> to pairs of coefficients of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dmp_apply_pairs</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">:</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">-</span> <span class="n">z</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_apply_pairs</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]],</span> <span class="p">[[</span><span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]],</span> <span class="n">h</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[4], [5, 6]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dmp_slice">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dmp_slice</tt><big>(</big><em>f</em>, <em>m</em>, <em>n</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dmp_slice"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dmp_slice" title="Permalink to this definition">¶</a></dt>
-<dd><p>Take a continuous subsequence of terms of <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densebasic.dup_random">
-<tt class="descclassname">sympy.polys.densebasic.</tt><tt class="descname">dup_random</tt><big>(</big><em>n</em>, <em>a</em>, <em>b</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densebasic.html#dup_random"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densebasic.dup_random" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a polynomial of degree <tt class="docutils literal"><span class="pre">n</span></tt> with coefficients in <tt class="docutils literal"><span class="pre">[a,</span> <span class="pre">b]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densebasic</span> <span class="kn">import</span> <span class="n">dup_random</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dup_random</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span> 
-<span class="go">[-2, -8, 9, -4]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<p><strong>Arithmetic operations:</strong></p>
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_add_term">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_add_term</tt><big>(</big><em>f</em>, <em>c</em>, <em>i</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_add_term"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_add_term" title="Permalink to this definition">¶</a></dt>
-<dd><p>Add <tt class="docutils literal"><span class="pre">c(x_2..x_u)*x_0**i</span></tt> to <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_add_term</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_add_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[2], [1, 0], [1]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_sub_term">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_sub_term</tt><big>(</big><em>f</em>, <em>c</em>, <em>i</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_sub_term"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_sub_term" title="Permalink to this definition">¶</a></dt>
-<dd><p>Subtract <tt class="docutils literal"><span class="pre">c(x_2..x_u)*x_0**i</span></tt> from <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_sub_term</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">2</span><span class="p">])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_sub_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1, 0], [1]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_mul_term">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_mul_term</tt><big>(</big><em>f</em>, <em>c</em>, <em>i</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_mul_term"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_mul_term" title="Permalink to this definition">¶</a></dt>
-<dd><p>Multiply <tt class="docutils literal"><span class="pre">f</span></tt> by <tt class="docutils literal"><span class="pre">c(x_2..x_u)*x_0**i</span></tt> in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_mul_term</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_mul_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[3, 0, 0], [3, 0], [], [], []]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_add_ground">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_add_ground</tt><big>(</big><em>f</em>, <em>c</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_add_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_add_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Add an element of the ground domain to <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_add_ground</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_add_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">4</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1], [2], [3], [8]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_sub_ground">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_sub_ground</tt><big>(</big><em>f</em>, <em>c</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_sub_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_sub_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Subtract an element of the ground domain from <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_sub_ground</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_sub_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">4</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1], [2], [3], []]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_mul_ground">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_mul_ground</tt><big>(</big><em>f</em>, <em>c</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_mul_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_mul_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Multiply <tt class="docutils literal"><span class="pre">f</span></tt> by a constant value in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_mul_ground</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_mul_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[6], [6, 0]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_quo_ground">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_quo_ground</tt><big>(</big><em>f</em>, <em>c</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_quo_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_quo_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Quotient by a constant in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_quo_ground</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="p">[]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="p">[]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_quo_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1, 0], [1], []]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_quo_ground</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[[1/1, 0/1], [3/2], []]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_exquo_ground">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_exquo_ground</tt><big>(</big><em>f</em>, <em>c</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_exquo_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_exquo_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Exact quotient by a constant in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_exquo_ground</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_exquo_ground</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[[1/2, 0/1], [1/1], []]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dup_lshift">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dup_lshift</tt><big>(</big><em>f</em>, <em>n</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dup_lshift"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dup_lshift" title="Permalink to this definition">¶</a></dt>
-<dd><p>Efficiently multiply <tt class="docutils literal"><span class="pre">f</span></tt> by <tt class="docutils literal"><span class="pre">x**n</span></tt> in <tt class="docutils literal"><span class="pre">K[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dup_lshift</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dup_lshift</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[1, 0, 1, 0, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dup_rshift">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dup_rshift</tt><big>(</big><em>f</em>, <em>n</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dup_rshift"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dup_rshift" title="Permalink to this definition">¶</a></dt>
-<dd><p>Efficiently divide <tt class="docutils literal"><span class="pre">f</span></tt> by <tt class="docutils literal"><span class="pre">x**n</span></tt> in <tt class="docutils literal"><span class="pre">K[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dup_rshift</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dup_rshift</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[1, 0, 1]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dup_rshift</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[1, 0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_abs">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_abs</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_abs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_abs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Make all coefficients positive in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_abs</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">[]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_abs</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1, 0], [1], []]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_neg">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_neg</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_neg"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_neg" title="Permalink to this definition">¶</a></dt>
-<dd><p>Negate a polynomial in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_neg</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">[]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_neg</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[-1, 0], [1], []]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_add">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_add</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Add dense polynomials in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_add</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_add</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1, 1], [1], [1, 0]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_sub">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_sub</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_sub"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_sub" title="Permalink to this definition">¶</a></dt>
-<dd><p>Subtract dense polynomials in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_sub</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_sub</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[-1, 1], [-1], [1, 0]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_add_mul">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_add_mul</tt><big>(</big><em>f</em>, <em>g</em>, <em>h</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_add_mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_add_mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">f</span> <span class="pre">+</span> <span class="pre">g*h</span></tt> where <tt class="docutils literal"><span class="pre">f,</span> <span class="pre">g,</span> <span class="pre">h</span></tt> are in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_add_mul</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_add_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[2], [2], [1, 0]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_sub_mul">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_sub_mul</tt><big>(</big><em>f</em>, <em>g</em>, <em>h</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_sub_mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_sub_mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">f</span> <span class="pre">-</span> <span class="pre">g*h</span></tt> where <tt class="docutils literal"><span class="pre">f,</span> <span class="pre">g,</span> <span class="pre">h</span></tt> are in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_sub_mul</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_sub_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[-2], [1, 0]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_mul">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_mul</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Multiply dense polynomials in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_mul</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_mul</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1, 0], [1], []]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_sqr">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_sqr</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_sqr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_sqr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Square dense polynomials in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_sqr</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_sqr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1], [2, 0], [3, 0, 0], [2, 0, 0, 0], [1, 0, 0, 0, 0]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_pow">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_pow</tt><big>(</big><em>f</em>, <em>n</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_pow"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_pow" title="Permalink to this definition">¶</a></dt>
-<dd><p>Raise <tt class="docutils literal"><span class="pre">f</span></tt> to the <tt class="docutils literal"><span class="pre">n</span></tt>-th power in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_pow</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_pow</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1, 0, 0, 0], [3, 0, 0], [3, 0], [1]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_pdiv">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_pdiv</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_pdiv"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_pdiv" title="Permalink to this definition">¶</a></dt>
-<dd><p>Polynomial pseudo-division in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_pdiv</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_pdiv</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">([[2], [2, -2]], [[-4, 4]])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_prem">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_prem</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_prem"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_prem" title="Permalink to this definition">¶</a></dt>
-<dd><p>Polynomial pseudo-remainder in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_prem</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_prem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[-4, 4]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_pquo">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_pquo</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_pquo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_pquo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Polynomial exact pseudo-quotient in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_pquo</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_pquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[2], []]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_pquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[2], [2, -2]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_pexquo">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_pexquo</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_pexquo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_pexquo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Polynomial pseudo-quotient in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_pexquo</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_pexquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[2], []]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_pexquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">ExactQuotientFailed</span>: <span class="n">[[2], [2]] does not divide [[1], [1, 0], []]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_rr_div">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_rr_div</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_rr_div"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_rr_div" title="Permalink to this definition">¶</a></dt>
-<dd><p>Multivariate division with remainder over a ring.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_rr_div</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_rr_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">([[]], [[1], [1, 0], []])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_ff_div">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_ff_div</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_ff_div"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_ff_div" title="Permalink to this definition">¶</a></dt>
-<dd><p>Polynomial division with remainder over a field.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_ff_div</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_ff_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">([[1/2], [1/2, -1/2]], [[-1/1, 1/1]])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_div">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_div</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_div"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_div" title="Permalink to this definition">¶</a></dt>
-<dd><p>Polynomial division with remainder in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_div</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">([[]], [[1], [1, 0], []])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_div</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">([[1/2], [1/2, -1/2]], [[-1/1, 1/1]])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_rem">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_rem</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_rem"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_rem" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns polynomial remainder in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_rem</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_rem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1], [1, 0], []]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_rem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[[-1/1, 1/1]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_quo">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_quo</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_quo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_quo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns exact polynomial quotient in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_quo</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[]]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[[1/2], [1/2, -1/2]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_exquo">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_exquo</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_exquo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_exquo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns polynomial quotient in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_exquo</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_exquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1], []]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_exquo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">ExactQuotientFailed</span>: <span class="n">[[2], [2]] does not divide [[1], [1, 0], []]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_max_norm">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_max_norm</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_max_norm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_max_norm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns maximum norm of a polynomial in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_max_norm</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_max_norm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_l1_norm">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_l1_norm</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_l1_norm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_l1_norm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns l1 norm of a polynomial in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_l1_norm</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_l1_norm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">6</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densearith.dmp_expand">
-<tt class="descclassname">sympy.polys.densearith.</tt><tt class="descname">dmp_expand</tt><big>(</big><em>polys</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densearith.html#dmp_expand"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densearith.dmp_expand" title="Permalink to this definition">¶</a></dt>
-<dd><p>Multiply together several polynomials in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densearith</span> <span class="kn">import</span> <span class="n">dmp_expand</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_expand</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1], [1], [1, 0, 0], [1, 0, 0]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<p><strong>Further tools:</strong></p>
-<dl class="function">
-<dt id="sympy.polys.densetools.dmp_integrate">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dmp_integrate</tt><big>(</big><em>f</em>, <em>m</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dmp_integrate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dmp_integrate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the indefinite integral of <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">x_0</span></tt> in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dmp_integrate</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_integrate</span><span class="p">([[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">)],</span> <span class="p">[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">0</span><span class="p">)]],</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[[1/2], [2/1, 0/1], []]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_integrate</span><span class="p">([[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">)],</span> <span class="p">[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">0</span><span class="p">)]],</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[[1/6], [1/1, 0/1], [], []]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dmp_integrate_in">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dmp_integrate_in</tt><big>(</big><em>f</em>, <em>m</em>, <em>j</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dmp_integrate_in"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dmp_integrate_in" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the indefinite integral of <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">x_j</span></tt> in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dmp_integrate_in</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_integrate_in</span><span class="p">([[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">)],</span> <span class="p">[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">0</span><span class="p">)]],</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[[1/2], [2/1, 0/1], []]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_integrate_in</span><span class="p">([[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">)],</span> <span class="p">[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">0</span><span class="p">)]],</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[[1/1, 0/1], [1/1, 0/1, 0/1]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dmp_diff">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dmp_diff</tt><big>(</big><em>f</em>, <em>m</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dmp_diff"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dmp_diff" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">m</span></tt>-th order derivative in <tt class="docutils literal"><span class="pre">x_0</span></tt> of a polynomial in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dmp_diff</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1, 2, 3]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dmp_diff_in">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dmp_diff_in</tt><big>(</big><em>f</em>, <em>m</em>, <em>j</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dmp_diff_in"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dmp_diff_in" title="Permalink to this definition">¶</a></dt>
-<dd><p><tt class="docutils literal"><span class="pre">m</span></tt>-th order derivative in <tt class="docutils literal"><span class="pre">x_j</span></tt> of a polynomial in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dmp_diff_in</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_diff_in</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1, 2, 3]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_diff_in</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[2, 2], [4, 3]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dmp_eval">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dmp_eval</tt><big>(</big><em>f</em>, <em>a</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dmp_eval"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dmp_eval" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluate a polynomial at <tt class="docutils literal"><span class="pre">x_0</span> <span class="pre">=</span> <span class="pre">a</span></tt> in <tt class="docutils literal"><span class="pre">K[X]</span></tt> using the Horner scheme.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dmp_eval</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_eval</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[5, 8]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dmp_eval_in">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dmp_eval_in</tt><big>(</big><em>f</em>, <em>a</em>, <em>j</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dmp_eval_in"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dmp_eval_in" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluate a polynomial at <tt class="docutils literal"><span class="pre">x_j</span> <span class="pre">=</span> <span class="pre">a</span></tt> in <tt class="docutils literal"><span class="pre">K[X]</span></tt> using the Horner scheme.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dmp_eval_in</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_eval_in</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[5, 8]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_eval_in</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[7, 4]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dmp_eval_tail">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dmp_eval_tail</tt><big>(</big><em>f</em>, <em>A</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dmp_eval_tail"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dmp_eval_tail" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluate a polynomial at <tt class="docutils literal"><span class="pre">x_j</span> <span class="pre">=</span> <span class="pre">a_j,</span> <span class="pre">...</span></tt> in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dmp_eval_tail</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_eval_tail</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">18</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_eval_tail</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">(</span><span class="mi">2</span><span class="p">,),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[7, 4]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dmp_diff_eval_in">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dmp_diff_eval_in</tt><big>(</big><em>f</em>, <em>m</em>, <em>a</em>, <em>j</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dmp_diff_eval_in"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dmp_diff_eval_in" title="Permalink to this definition">¶</a></dt>
-<dd><p>Differentiate and evaluate a polynomial in <tt class="docutils literal"><span class="pre">x_j</span></tt> at <tt class="docutils literal"><span class="pre">a</span></tt> in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dmp_diff_eval_in</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_diff_eval_in</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[1, 2, 3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_diff_eval_in</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[6, 11]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dmp_trunc">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dmp_trunc</tt><big>(</big><em>f</em>, <em>p</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dmp_trunc"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dmp_trunc" title="Permalink to this definition">¶</a></dt>
-<dd><p>Reduce a <tt class="docutils literal"><span class="pre">K[X]</span></tt> polynomial modulo a polynomial <tt class="docutils literal"><span class="pre">p</span></tt> in <tt class="docutils literal"><span class="pre">K[Y]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dmp_trunc</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">8</span><span class="p">],</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_trunc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[11], [11], [5]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dmp_ground_trunc">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dmp_ground_trunc</tt><big>(</big><em>f</em>, <em>p</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dmp_ground_trunc"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dmp_ground_trunc" title="Permalink to this definition">¶</a></dt>
-<dd><p>Reduce a <tt class="docutils literal"><span class="pre">K[X]</span></tt> polynomial modulo a constant <tt class="docutils literal"><span class="pre">p</span></tt> in <tt class="docutils literal"><span class="pre">K</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dmp_ground_trunc</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">8</span><span class="p">],</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_ground_trunc</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[-1], [-1, 0], [-1, 0]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dup_monic">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dup_monic</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dup_monic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dup_monic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Divide all coefficients by <tt class="docutils literal"><span class="pre">LC(f)</span></tt> in <tt class="docutils literal"><span class="pre">K[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dup_monic</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dup_monic</span><span class="p">([</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">6</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">9</span><span class="p">)],</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[1, 2, 3]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dup_monic</span><span class="p">([</span><span class="n">QQ</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">4</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">2</span><span class="p">)],</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[1/1, 4/3, 2/3]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dmp_ground_monic">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dmp_ground_monic</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dmp_ground_monic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dmp_ground_monic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Divide all coefficients by <tt class="docutils literal"><span class="pre">LC(f)</span></tt> in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dmp_ground_monic</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">9</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">8</span><span class="p">],</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_ground_monic</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1, 2], [1, 0], [3, 1]]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_ground_monic</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[[1/1, 8/3], [5/3, 2/1], [2/3, 1/1]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dup_content">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dup_content</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dup_content"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dup_content" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the GCD of coefficients of <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">K[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dup_content</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">6</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">12</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">6</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">12</span><span class="p">])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dup_content</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dup_content</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">2/1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dmp_ground_content">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dmp_ground_content</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dmp_ground_content"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dmp_ground_content" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the GCD of coefficients of <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dmp_ground_content</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">12</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">12</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_ground_content</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_ground_content</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">2/1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dup_primitive">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dup_primitive</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dup_primitive"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dup_primitive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute content and the primitive form of <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">K[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dup_primitive</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">6</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">12</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">6</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">12</span><span class="p">])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dup_primitive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">(2, [3, 4, 6])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dup_primitive</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">(2/1, [3/1, 4/1, 6/1])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dmp_ground_primitive">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dmp_ground_primitive</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dmp_ground_primitive"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dmp_ground_primitive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute content and the primitive form of <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dmp_ground_primitive</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">12</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">12</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_ground_primitive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">(2, [[1, 3], [2, 6]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_ground_primitive</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">(2/1, [[1/1, 3/1], [2/1, 6/1]])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dup_extract">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dup_extract</tt><big>(</big><em>f</em>, <em>g</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dup_extract"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dup_extract" title="Permalink to this definition">¶</a></dt>
-<dd><p>Extract common content from a pair of polynomials in <tt class="docutils literal"><span class="pre">K[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dup_extract</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">6</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">18</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">12</span><span class="p">])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dup_extract</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">(2, [3, 6, 9], [2, 4, 6])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dmp_ground_extract">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dmp_ground_extract</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dmp_ground_extract"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dmp_ground_extract" title="Permalink to this definition">¶</a></dt>
-<dd><p>Extract common content from a pair of polynomials in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dmp_ground_extract</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">6</span><span class="p">,</span> <span class="mi">12</span><span class="p">],</span> <span class="p">[</span><span class="mi">18</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">8</span><span class="p">],</span> <span class="p">[</span><span class="mi">12</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_ground_extract</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">(2, [[3, 6], [9]], [[2, 4], [6]])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dup_real_imag">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dup_real_imag</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dup_real_imag"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dup_real_imag" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return bivariate polynomials <tt class="docutils literal"><span class="pre">f1</span></tt> and <tt class="docutils literal"><span class="pre">f2</span></tt>, such that <tt class="docutils literal"><span class="pre">f</span> <span class="pre">=</span> <span class="pre">f1</span> <span class="pre">+</span> <span class="pre">f2*I</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dup_real_imag</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dup_real_imag</span><span class="p">([</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">)],</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">([[1], [1], [-3, 0, 1], [-1, 0, 1]], [[3, 0], [2, 0], [-1, 0, 1, 0]])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dup_mirror">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dup_mirror</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dup_mirror"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dup_mirror" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluate efficiently the composition <tt class="docutils literal"><span class="pre">f(-x)</span></tt> in <tt class="docutils literal"><span class="pre">K[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dup_mirror</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dup_mirror</span><span class="p">([</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="o">-</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">4</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">2</span><span class="p">)],</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[-1, 2, 4, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dup_scale">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dup_scale</tt><big>(</big><em>f</em>, <em>a</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dup_scale"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dup_scale" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluate efficiently composition <tt class="docutils literal"><span class="pre">f(a*x)</span></tt> in <tt class="docutils literal"><span class="pre">K[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dup_scale</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dup_scale</span><span class="p">([</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="o">-</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">)],</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[4, -4, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dup_shift">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dup_shift</tt><big>(</big><em>f</em>, <em>a</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dup_shift"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dup_shift" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluate efficiently Taylor shift <tt class="docutils literal"><span class="pre">f(x</span> <span class="pre">+</span> <span class="pre">a)</span></tt> in <tt class="docutils literal"><span class="pre">K[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dup_shift</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dup_shift</span><span class="p">([</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="o">-</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">)],</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[1, 2, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dup_transform">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dup_transform</tt><big>(</big><em>f</em>, <em>p</em>, <em>q</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dup_transform"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dup_transform" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluate functional transformation <tt class="docutils literal"><span class="pre">q**n</span> <span class="pre">*</span> <span class="pre">f(p/q)</span></tt> in <tt class="docutils literal"><span class="pre">K[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dup_transform</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dup_transform</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[1, -2, 5, -4, 4]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dmp_compose">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dmp_compose</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dmp_compose"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dmp_compose" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluate functional composition <tt class="docutils literal"><span class="pre">f(g)</span></tt> in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dmp_compose</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_compose</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1, 3, 0]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dup_decompose">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dup_decompose</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dup_decompose"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dup_decompose" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes functional decomposition of <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">K[x]</span></tt>.</p>
-<p>Given a univariate polynomial <tt class="docutils literal"><span class="pre">f</span></tt> with coefficients in a field of
-characteristic zero, returns list <tt class="docutils literal"><span class="pre">[f_1,</span> <span class="pre">f_2,</span> <span class="pre">...,</span> <span class="pre">f_n]</span></tt>, where:</p>
-<div class="highlight-python"><pre>f = f_1 o f_2 o ... f_n = f_1(f_2(... f_n))</pre>
-</div>
-<p>and <tt class="docutils literal"><span class="pre">f_2,</span> <span class="pre">...,</span> <span class="pre">f_n</span></tt> are monic and homogeneous polynomials of at
-least second degree.</p>
-<p>Unlike factorization, complete functional decompositions of
-polynomials are not unique, consider examples:</p>
-<ol class="arabic simple">
-<li><tt class="docutils literal"><span class="pre">f</span> <span class="pre">o</span> <span class="pre">g</span> <span class="pre">=</span> <span class="pre">f(x</span> <span class="pre">+</span> <span class="pre">b)</span> <span class="pre">o</span> <span class="pre">(g</span> <span class="pre">-</span> <span class="pre">b)</span></tt></li>
-<li><tt class="docutils literal"><span class="pre">x**n</span> <span class="pre">o</span> <span class="pre">x**m</span> <span class="pre">=</span> <span class="pre">x**m</span> <span class="pre">o</span> <span class="pre">x**n</span></tt></li>
-<li><tt class="docutils literal"><span class="pre">T_n</span> <span class="pre">o</span> <span class="pre">T_m</span> <span class="pre">=</span> <span class="pre">T_m</span> <span class="pre">o</span> <span class="pre">T_n</span></tt></li>
-</ol>
-<p>where <tt class="docutils literal"><span class="pre">T_n</span></tt> and <tt class="docutils literal"><span class="pre">T_m</span></tt> are Chebyshev polynomials.</p>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../polys/literature.html#kozen89">[Kozen89]</a></li>
-</ol>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dup_decompose</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dup_decompose</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1, 0, 0], [1, -1, 0]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dmp_lift">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dmp_lift</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dmp_lift"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dmp_lift" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert algebraic coefficients to integers in <tt class="docutils literal"><span class="pre">K[X]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dmp_lift</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">K</span> <span class="o">=</span> <span class="n">QQ</span><span class="o">.</span><span class="n">algebraic_field</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="p">[</span><span class="n">K</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">K</span><span class="p">([</span><span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">0</span><span class="p">)]),</span> <span class="n">K</span><span class="p">([</span><span class="n">QQ</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">0</span><span class="p">)])]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_lift</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">K</span><span class="p">)</span>
-<span class="go">[1/1, 0/1, 2/1, 0/1, 9/1, 0/1, -8/1, 0/1, 16/1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dup_sign_variations">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dup_sign_variations</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dup_sign_variations"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dup_sign_variations" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the number of sign variations of <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">K[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dup_sign_variations</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dup_sign_variations</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dmp_clear_denoms">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dmp_clear_denoms</tt><big>(</big><em>f</em>, <em>u</em>, <em>K0</em>, <em>K1=None</em>, <em>convert=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dmp_clear_denoms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dmp_clear_denoms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Clear denominators, i.e. transform <tt class="docutils literal"><span class="pre">K_0</span></tt> to <tt class="docutils literal"><span class="pre">K_1</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">QQ</span><span class="p">,</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dmp_clear_denoms</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="p">[[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)],</span> <span class="p">[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">)]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_clear_denoms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">QQ</span><span class="p">,</span> <span class="n">convert</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">(6, [[3/1], [2/1, 6/1]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="p">[[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)],</span> <span class="p">[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">)]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_clear_denoms</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">QQ</span><span class="p">,</span> <span class="n">convert</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(6, [[3], [2, 6]])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.densetools.dmp_revert">
-<tt class="descclassname">sympy.polys.densetools.</tt><tt class="descname">dmp_revert</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/densetools.html#dmp_revert"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.densetools.dmp_revert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute <tt class="docutils literal"><span class="pre">f**(-1)</span></tt> mod <tt class="docutils literal"><span class="pre">x**n</span></tt> using Newton iteration.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.densetools</span> <span class="kn">import</span> <span class="n">dmp_revert</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="manipulation-of-dense-univariate-polynomials-with-finite-field-coefficients">
-<h3>Manipulation of dense, univariate polynomials with finite field coefficients<a class="headerlink" href="#manipulation-of-dense-univariate-polynomials-with-finite-field-coefficients" title="Permalink to this headline">¶</a></h3>
-<p>Functions in this module carry the prefix <tt class="docutils literal"><span class="pre">gf_</span></tt>, referring to the classical
-name &#8220;Galois Fields&#8221; for finite fields. Note that many polynomial
-factorization algorithms work by reduction to the finite field case, so having
-special implementations for this case is justified both by performance, and by
-the necessity of certain methods which do not even make sense over general
-fields.</p>
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_crt">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_crt</tt><big>(</big><em>U</em>, <em>M</em>, <em>K=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_crt"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_crt" title="Permalink to this definition">¶</a></dt>
-<dd><p>Chinese Remainder Theorem.</p>
-<p>Given a set of integer residues <tt class="docutils literal"><span class="pre">u_0,...,u_n</span></tt> and a set of
-co-prime integer moduli <tt class="docutils literal"><span class="pre">m_0,...,m_n</span></tt>, returns an integer
-<tt class="docutils literal"><span class="pre">u</span></tt>, such that <tt class="docutils literal"><span class="pre">u</span> <span class="pre">=</span> <span class="pre">u_i</span> <span class="pre">mod</span> <span class="pre">m_i</span></tt> for <tt class="docutils literal"><span class="pre">i</span> <span class="pre">=</span> <span class="pre">``0,...,n</span></tt>.</p>
-<p>As an example consider a set of residues <tt class="docutils literal"><span class="pre">U</span> <span class="pre">=</span> <span class="pre">[49,</span> <span class="pre">76,</span> <span class="pre">65]</span></tt>
-and a set of moduli <tt class="docutils literal"><span class="pre">M</span> <span class="pre">=</span> <span class="pre">[99,</span> <span class="pre">97,</span> <span class="pre">95]</span></tt>. Then we have:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_crt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.ntheory.modular</span> <span class="kn">import</span> <span class="n">solve_congruence</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gf_crt</span><span class="p">([</span><span class="mi">49</span><span class="p">,</span> <span class="mi">76</span><span class="p">,</span> <span class="mi">65</span><span class="p">],</span> <span class="p">[</span><span class="mi">99</span><span class="p">,</span> <span class="mi">97</span><span class="p">,</span> <span class="mi">95</span><span class="p">],</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">639985</span>
-</pre></div>
-</div>
-<p>This is the correct result because:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="mi">639985</span> <span class="o">%</span> <span class="n">m</span> <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">99</span><span class="p">,</span> <span class="mi">97</span><span class="p">,</span> <span class="mi">95</span><span class="p">]]</span>
-<span class="go">[49, 76, 65]</span>
-</pre></div>
-</div>
-<p>Note: this is a low-level routine with no error checking.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<dl class="docutils">
-<dt><a class="reference internal" href="../ntheory.html#sympy.ntheory.modular.crt" title="sympy.ntheory.modular.crt"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.ntheory.modular.crt</span></tt></a></dt>
-<dd>a higher level crt routine</dd>
-</dl>
-<p class="last"><a class="reference internal" href="../ntheory.html#sympy.ntheory.modular.solve_congruence" title="sympy.ntheory.modular.solve_congruence"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.ntheory.modular.solve_congruence</span></tt></a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_crt1">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_crt1</tt><big>(</big><em>M</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_crt1"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_crt1" title="Permalink to this definition">¶</a></dt>
-<dd><p>First part of the Chinese Remainder Theorem.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_crt1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_crt1</span><span class="p">([</span><span class="mi">99</span><span class="p">,</span> <span class="mi">97</span><span class="p">,</span> <span class="mi">95</span><span class="p">],</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">(912285, [9215, 9405, 9603], [62, 24, 12])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_crt2">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_crt2</tt><big>(</big><em>U</em>, <em>M</em>, <em>p</em>, <em>E</em>, <em>S</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_crt2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_crt2" title="Permalink to this definition">¶</a></dt>
-<dd><p>Second part of the Chinese Remainder Theorem.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_crt2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">U</span> <span class="o">=</span> <span class="p">[</span><span class="mi">49</span><span class="p">,</span> <span class="mi">76</span><span class="p">,</span> <span class="mi">65</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="p">[</span><span class="mi">99</span><span class="p">,</span> <span class="mi">97</span><span class="p">,</span> <span class="mi">95</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="mi">912285</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E</span> <span class="o">=</span> <span class="p">[</span><span class="mi">9215</span><span class="p">,</span> <span class="mi">9405</span><span class="p">,</span> <span class="mi">9603</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="p">[</span><span class="mi">62</span><span class="p">,</span> <span class="mi">24</span><span class="p">,</span> <span class="mi">12</span><span class="p">]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_crt2</span><span class="p">(</span><span class="n">U</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">639985</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_int">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_int</tt><big>(</big><em>a</em>, <em>p</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_int"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_int" title="Permalink to this definition">¶</a></dt>
-<dd><p>Coerce <tt class="docutils literal"><span class="pre">a</span> <span class="pre">mod</span> <span class="pre">p</span></tt> to an integer in the range <tt class="docutils literal"><span class="pre">[-p/2,</span> <span class="pre">p/2]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_int</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_int</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gf_int</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">)</span>
-<span class="go">-2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_degree">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_degree</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_degree"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_degree" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the leading degree of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_degree</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_degree</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
-<span class="go">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gf_degree</span><span class="p">([])</span>
-<span class="go">-1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_LC">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_LC</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_LC"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_LC" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the leading coefficient of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_LC</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_LC</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_TC">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_TC</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_TC"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_TC" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the trailing coefficient of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_TC</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_TC</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_strip">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_strip</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_strip"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_strip" title="Permalink to this definition">¶</a></dt>
-<dd><p>Remove leading zeros from <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_strip</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_strip</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">[3, 0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_trunc">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_trunc</tt><big>(</big><em>f</em>, <em>p</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_trunc"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_trunc" title="Permalink to this definition">¶</a></dt>
-<dd><p>Reduce all coefficients modulo <tt class="docutils literal"><span class="pre">p</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_trunc</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_trunc</span><span class="p">([</span><span class="mi">7</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">[2, 3, 3]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_normal">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_normal</tt><big>(</big><em>f</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_normal"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_normal" title="Permalink to this definition">¶</a></dt>
-<dd><p>Normalize all coefficients in <tt class="docutils literal"><span class="pre">K</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_normal</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_normal</span><span class="p">([</span><span class="mi">5</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">21</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">],</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[1, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_convert">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_convert</tt><big>(</big><em>f</em>, <em>p</em>, <em>K0</em>, <em>K1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_convert"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_convert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Normalize all coefficients in <tt class="docutils literal"><span class="pre">K</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_convert</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_convert</span><span class="p">([</span><span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">4</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">0</span><span class="p">)],</span> <span class="mi">3</span><span class="p">,</span> <span class="n">QQ</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[1, 1, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_from_dict">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_from_dict</tt><big>(</big><em>f</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_from_dict"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_from_dict" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt> polynomial from a dict.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_from_dict</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_from_dict</span><span class="p">({</span><span class="mi">10</span><span class="p">:</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">4</span><span class="p">),</span> <span class="mi">4</span><span class="p">:</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">33</span><span class="p">),</span> <span class="mi">0</span><span class="p">:</span> <span class="n">ZZ</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)},</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[4, 0, 0, 0, 0, 0, 3, 0, 0, 0, 4]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_to_dict">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_to_dict</tt><big>(</big><em>f</em>, <em>p</em>, <em>symmetric=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_to_dict"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_to_dict" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt> polynomial to a dict.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_to_dict</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_to_dict</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">{0: -1, 4: -2, 10: -1}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gf_to_dict</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="mi">5</span><span class="p">,</span> <span class="n">symmetric</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">{0: 4, 4: 3, 10: 4}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_from_int_poly">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_from_int_poly</tt><big>(</big><em>f</em>, <em>p</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_from_int_poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_from_int_poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt> polynomial from <tt class="docutils literal"><span class="pre">Z[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_from_int_poly</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_from_int_poly</span><span class="p">([</span><span class="mi">7</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">[2, 3, 3]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_to_int_poly">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_to_int_poly</tt><big>(</big><em>f</em>, <em>p</em>, <em>symmetric=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_to_int_poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_to_int_poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt> polynomial to <tt class="docutils literal"><span class="pre">Z[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_to_int_poly</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_to_int_poly</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">[2, -2, -2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gf_to_int_poly</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="mi">5</span><span class="p">,</span> <span class="n">symmetric</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">[2, 3, 3]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_neg">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_neg</tt><big>(</big><em>f</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_neg"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_neg" title="Permalink to this definition">¶</a></dt>
-<dd><p>Negate a polynomial in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_neg</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_neg</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[2, 3, 4, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_add_ground">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_add_ground</tt><big>(</big><em>f</em>, <em>a</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_add_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_add_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute <tt class="docutils literal"><span class="pre">f</span> <span class="pre">+</span> <span class="pre">a</span></tt> where <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt> and <tt class="docutils literal"><span class="pre">a</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_add_ground</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_add_ground</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[3, 2, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_sub_ground">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_sub_ground</tt><big>(</big><em>f</em>, <em>a</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_sub_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_sub_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute <tt class="docutils literal"><span class="pre">f</span> <span class="pre">-</span> <span class="pre">a</span></tt> where <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt> and <tt class="docutils literal"><span class="pre">a</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_sub_ground</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_sub_ground</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[3, 2, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_mul_ground">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_mul_ground</tt><big>(</big><em>f</em>, <em>a</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_mul_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_mul_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute <tt class="docutils literal"><span class="pre">f</span> <span class="pre">*</span> <span class="pre">a</span></tt> where <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt> and <tt class="docutils literal"><span class="pre">a</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_mul_ground</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_mul_ground</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[1, 4, 3]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_quo_ground">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_quo_ground</tt><big>(</big><em>f</em>, <em>a</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_quo_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_quo_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute <tt class="docutils literal"><span class="pre">f/a</span></tt> where <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt> and <tt class="docutils literal"><span class="pre">a</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_quo_ground</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_quo_ground</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">]),</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[4, 1, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_add">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_add</tt><big>(</big><em>f</em>, <em>g</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Add polynomials in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_add</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_add</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[4, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_sub">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_sub</tt><big>(</big><em>f</em>, <em>g</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_sub"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_sub" title="Permalink to this definition">¶</a></dt>
-<dd><p>Subtract polynomials in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_sub</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_sub</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[1, 0, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_mul">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_mul</tt><big>(</big><em>f</em>, <em>g</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Multiply polynomials in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_mul</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_mul</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[1, 0, 3, 2, 3]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_sqr">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_sqr</tt><big>(</big><em>f</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_sqr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_sqr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Square polynomials in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_sqr</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_sqr</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[4, 2, 3, 1, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_add_mul">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_add_mul</tt><big>(</big><em>f</em>, <em>g</em>, <em>h</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_add_mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_add_mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">f</span> <span class="pre">+</span> <span class="pre">g*h</span></tt> where <tt class="docutils literal"><span class="pre">f</span></tt>, <tt class="docutils literal"><span class="pre">g</span></tt>, <tt class="docutils literal"><span class="pre">h</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_add_mul</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gf_add_mul</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[2, 3, 2, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_sub_mul">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_sub_mul</tt><big>(</big><em>f</em>, <em>g</em>, <em>h</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_sub_mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_sub_mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute <tt class="docutils literal"><span class="pre">f</span> <span class="pre">-</span> <span class="pre">g*h</span></tt> where <tt class="docutils literal"><span class="pre">f</span></tt>, <tt class="docutils literal"><span class="pre">g</span></tt>, <tt class="docutils literal"><span class="pre">h</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_sub_mul</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_sub_mul</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[3, 3, 2, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_expand">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_expand</tt><big>(</big><em>F</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_expand"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_expand" title="Permalink to this definition">¶</a></dt>
-<dd><p>Expand results of <tt class="xref py py-func docutils literal"><span class="pre">factor()</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_expand</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_expand</span><span class="p">([([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="mi">1</span><span class="p">),</span> <span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="mi">2</span><span class="p">),</span> <span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="mi">3</span><span class="p">)],</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[4, 3, 0, 3, 0, 1, 4, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_div">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_div</tt><big>(</big><em>f</em>, <em>g</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_div"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_div" title="Permalink to this definition">¶</a></dt>
-<dd><p>Division with remainder in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p>Given univariate polynomials <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt> with coefficients in a
-finite field with <tt class="docutils literal"><span class="pre">p</span></tt> elements, returns polynomials <tt class="docutils literal"><span class="pre">q</span></tt> and <tt class="docutils literal"><span class="pre">r</span></tt>
-(quotient and remainder) such that <tt class="docutils literal"><span class="pre">f</span> <span class="pre">=</span> <span class="pre">q*g</span> <span class="pre">+</span> <span class="pre">r</span></tt>.</p>
-<p>Consider polynomials <tt class="docutils literal"><span class="pre">x**3</span> <span class="pre">+</span> <span class="pre">x</span> <span class="pre">+</span> <span class="pre">1</span></tt> and <tt class="docutils literal"><span class="pre">x**2</span> <span class="pre">+</span> <span class="pre">x</span></tt> in GF(2):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_div</span><span class="p">,</span> <span class="n">gf_add_mul</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gf_div</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]),</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]),</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">([1, 1], [1])</span>
-</pre></div>
-</div>
-<p>As result we obtained quotient <tt class="docutils literal"><span class="pre">x</span> <span class="pre">+</span> <span class="pre">1</span></tt> and remainder <tt class="docutils literal"><span class="pre">1</span></tt>, thus:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_add_mul</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">]),</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]),</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]),</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[1, 0, 1, 1]</span>
-</pre></div>
-</div>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../polys/literature.html#monagan93">[Monagan93]</a></li>
-<li><a class="reference internal" href="../polys/literature.html#gathen99">[Gathen99]</a></li>
-</ol>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_rem">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_rem</tt><big>(</big><em>f</em>, <em>g</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_rem"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_rem" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute polynomial remainder in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_rem</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_rem</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]),</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]),</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_quo">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_quo</tt><big>(</big><em>f</em>, <em>g</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_quo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_quo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute exact quotient in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_quo</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_quo</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]),</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]),</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[1, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gf_quo</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]),</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]),</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[3, 2, 4]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_exquo">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_exquo</tt><big>(</big><em>f</em>, <em>g</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_exquo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_exquo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute polynomial quotient in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_exquo</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_exquo</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]),</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]),</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[3, 2, 4]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_exquo</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]),</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]),</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">ExactQuotientFailed</span>: <span class="n">[1, 1, 0] does not divide [1, 0, 1, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_lshift">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_lshift</tt><big>(</big><em>f</em>, <em>n</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_lshift"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_lshift" title="Permalink to this definition">¶</a></dt>
-<dd><p>Efficiently multiply <tt class="docutils literal"><span class="pre">f</span></tt> by <tt class="docutils literal"><span class="pre">x**n</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_lshift</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_lshift</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="mi">4</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[3, 2, 4, 0, 0, 0, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_rshift">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_rshift</tt><big>(</big><em>f</em>, <em>n</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_rshift"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_rshift" title="Permalink to this definition">¶</a></dt>
-<dd><p>Efficiently divide <tt class="docutils literal"><span class="pre">f</span></tt> by <tt class="docutils literal"><span class="pre">x**n</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_rshift</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_rshift</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="mi">3</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">([1, 2], [3, 4, 0])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_pow">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_pow</tt><big>(</big><em>f</em>, <em>n</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_pow"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_pow" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute <tt class="docutils literal"><span class="pre">f**n</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt> using repeated squaring.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_pow</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_pow</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[2, 4, 4, 2, 2, 1, 4]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_pow_mod">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_pow_mod</tt><big>(</big><em>f</em>, <em>n</em>, <em>g</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_pow_mod"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_pow_mod" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute <tt class="docutils literal"><span class="pre">f**n</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)[x]/(g)</span></tt> using repeated squaring.</p>
-<p>Given polynomials <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt> and a non-negative
-integer <tt class="docutils literal"><span class="pre">n</span></tt>, efficiently computes <tt class="docutils literal"><span class="pre">f**n</span> <span class="pre">(mod</span> <span class="pre">g)</span></tt> i.e. the remainder
-of <tt class="docutils literal"><span class="pre">f**n</span></tt> from division by <tt class="docutils literal"><span class="pre">g</span></tt>, using the repeated squaring algorithm.</p>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../polys/literature.html#gathen99">[Gathen99]</a></li>
-</ol>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_pow_mod</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_pow_mod</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">]),</span> <span class="mi">3</span><span class="p">,</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]),</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_gcd">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_gcd</tt><big>(</big><em>f</em>, <em>g</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_gcd"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_gcd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Euclidean Algorithm in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_gcd</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_gcd</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">]),</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]),</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[1, 3]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_lcm">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_lcm</tt><big>(</big><em>f</em>, <em>g</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_lcm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_lcm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute polynomial LCM in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_lcm</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_lcm</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">]),</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]),</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[1, 2, 0, 4]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_cofactors">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_cofactors</tt><big>(</big><em>f</em>, <em>g</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_cofactors"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_cofactors" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute polynomial GCD and cofactors in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_cofactors</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_cofactors</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">]),</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]),</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">([1, 3], [3, 3], [2, 1])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_gcdex">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_gcdex</tt><big>(</big><em>f</em>, <em>g</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_gcdex"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_gcdex" title="Permalink to this definition">¶</a></dt>
-<dd><p>Extended Euclidean Algorithm in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p>Given polynomials <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>, computes polynomials
-<tt class="docutils literal"><span class="pre">s</span></tt>, <tt class="docutils literal"><span class="pre">t</span></tt> and <tt class="docutils literal"><span class="pre">h</span></tt>, such that <tt class="docutils literal"><span class="pre">h</span> <span class="pre">=</span> <span class="pre">gcd(f,</span> <span class="pre">g)</span></tt> and <tt class="docutils literal"><span class="pre">s*f</span> <span class="pre">+</span> <span class="pre">t*g</span> <span class="pre">=</span> <span class="pre">h</span></tt>.
-The typical application of EEA is solving polynomial diophantine equations.</p>
-<p>Consider polynomials <tt class="docutils literal"><span class="pre">f</span> <span class="pre">=</span> <span class="pre">(x</span> <span class="pre">+</span> <span class="pre">7)</span> <span class="pre">(x</span> <span class="pre">+</span> <span class="pre">1)</span></tt>, <tt class="docutils literal"><span class="pre">g</span> <span class="pre">=</span> <span class="pre">(x</span> <span class="pre">+</span> <span class="pre">7)</span> <span class="pre">(x**2</span> <span class="pre">+</span> <span class="pre">1)</span></tt>
-in <tt class="docutils literal"><span class="pre">GF(11)[x]</span></tt>. Application of Extended Euclidean Algorithm gives:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_gcdex</span><span class="p">,</span> <span class="n">gf_mul</span><span class="p">,</span> <span class="n">gf_add</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">gf_gcdex</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">7</span><span class="p">]),</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">]),</span> <span class="mi">11</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">g</span>
-<span class="go">([5, 6], [6], [1, 7])</span>
-</pre></div>
-</div>
-<p>As result we obtained polynomials <tt class="docutils literal"><span class="pre">s</span> <span class="pre">=</span> <span class="pre">5*x</span> <span class="pre">+</span> <span class="pre">6</span></tt> and <tt class="docutils literal"><span class="pre">t</span> <span class="pre">=</span> <span class="pre">6</span></tt>, and
-additionally <tt class="docutils literal"><span class="pre">gcd(f,</span> <span class="pre">g)</span> <span class="pre">=</span> <span class="pre">x</span> <span class="pre">+</span> <span class="pre">7</span></tt>. This is correct because:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">S</span> <span class="o">=</span> <span class="n">gf_mul</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">7</span><span class="p">]),</span> <span class="mi">11</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">T</span> <span class="o">=</span> <span class="n">gf_mul</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">]),</span> <span class="mi">11</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gf_add</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">]</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../polys/literature.html#gathen99">[Gathen99]</a></li>
-</ol>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_monic">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_monic</tt><big>(</big><em>f</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_monic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_monic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute LC and a monic polynomial in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_monic</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_monic</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">]),</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">(3, [1, 4, 3])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_diff">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_diff</tt><big>(</big><em>f</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_diff"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_diff" title="Permalink to this definition">¶</a></dt>
-<dd><p>Differentiate polynomial in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_diff</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_diff</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[1, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_eval">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_eval</tt><big>(</big><em>f</em>, <em>a</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_eval"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_eval" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluate <tt class="docutils literal"><span class="pre">f(a)</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)</span></tt> using Horner scheme.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_eval</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_eval</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_multi_eval">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_multi_eval</tt><big>(</big><em>f</em>, <em>A</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_multi_eval"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_multi_eval" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluate <tt class="docutils literal"><span class="pre">f(a)</span></tt> for <tt class="docutils literal"><span class="pre">a</span></tt> in <tt class="docutils literal"><span class="pre">[a_1,</span> <span class="pre">...,</span> <span class="pre">a_n]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_multi_eval</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_multi_eval</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[4, 4, 0, 2, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_compose">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_compose</tt><big>(</big><em>f</em>, <em>g</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_compose"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_compose" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute polynomial composition <tt class="docutils literal"><span class="pre">f(g)</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_compose</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_compose</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[2, 4, 0, 3, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_compose_mod">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_compose_mod</tt><big>(</big><em>g</em>, <em>h</em>, <em>f</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_compose_mod"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_compose_mod" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute polynomial composition <tt class="docutils literal"><span class="pre">g(h)</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)[x]/(f)</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_compose_mod</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_compose_mod</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">]),</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]),</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">]),</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[4]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_trace_map">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_trace_map</tt><big>(</big><em>a</em>, <em>b</em>, <em>c</em>, <em>n</em>, <em>f</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_trace_map"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_trace_map" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute polynomial trace map in <tt class="docutils literal"><span class="pre">GF(p)[x]/(f)</span></tt>.</p>
-<p>Given a polynomial <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>, polynomials <tt class="docutils literal"><span class="pre">a</span></tt>, <tt class="docutils literal"><span class="pre">b</span></tt>,
-<tt class="docutils literal"><span class="pre">c</span></tt> in the quotient ring <tt class="docutils literal"><span class="pre">GF(p)[x]/(f)</span></tt> such that <tt class="docutils literal"><span class="pre">b</span> <span class="pre">=</span> <span class="pre">c**t</span>
-<span class="pre">(mod</span> <span class="pre">f)</span></tt> for some positive power <tt class="docutils literal"><span class="pre">t</span></tt> of <tt class="docutils literal"><span class="pre">p</span></tt>, and a positive
-integer <tt class="docutils literal"><span class="pre">n</span></tt>, returns a mapping:</p>
-<div class="highlight-python"><pre>a -&gt; a**t**n, a + a**t + a**t**2 + ... + a**t**n (mod f)</pre>
-</div>
-<p>In factorization context, <tt class="docutils literal"><span class="pre">b</span> <span class="pre">=</span> <span class="pre">x**p</span> <span class="pre">mod</span> <span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">c</span> <span class="pre">=</span> <span class="pre">x</span> <span class="pre">mod</span> <span class="pre">f</span></tt>.
-This way we can efficiently compute trace polynomials in equal
-degree factorization routine, much faster than with other methods,
-like iterated Frobenius algorithm, for large degrees.</p>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../polys/literature.html#gathen92">[Gathen92]</a></li>
-</ol>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_trace_map</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_trace_map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="mi">4</span><span class="p">,</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">([1, 3], [1, 3])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_random">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_random</tt><big>(</big><em>n</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_random"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_random" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generate a random polynomial in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt> of degree <tt class="docutils literal"><span class="pre">n</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_random</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gf_random</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span> 
-<span class="go">[1, 2, 3, 2, 1, 1, 1, 2, 0, 4, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_irreducible">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_irreducible</tt><big>(</big><em>n</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_irreducible"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_irreducible" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generate random irreducible polynomial of degree <tt class="docutils literal"><span class="pre">n</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_irreducible</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gf_irreducible</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span> 
-<span class="go">[1, 4, 2, 2, 3, 2, 4, 1, 4, 0, 4]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_irreducible_p">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_irreducible_p</tt><big>(</big><em>f</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_irreducible_p"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_irreducible_p" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test irreducibility of a polynomial <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_irreducible_p</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_irreducible_p</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">]),</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gf_irreducible_p</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">]),</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_sqf_p">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_sqf_p</tt><big>(</big><em>f</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_sqf_p"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_sqf_p" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is square-free in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_sqf_p</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_sqf_p</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">]),</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gf_sqf_p</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">]),</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_sqf_part">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_sqf_part</tt><big>(</big><em>f</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_sqf_part"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_sqf_part" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return square-free part of a <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt> polynomial.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_sqf_part</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_sqf_part</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]),</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[1, 4, 3]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_sqf_list">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_sqf_list</tt><big>(</big><em>f</em>, <em>p</em>, <em>K</em>, <em>all=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_sqf_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_sqf_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the square-free decomposition of a <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt> polynomial.</p>
-<p>Given a polynomial <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>, returns the leading coefficient
-of <tt class="docutils literal"><span class="pre">f</span></tt> and a square-free decomposition <tt class="docutils literal"><span class="pre">f_1**e_1</span> <span class="pre">f_2**e_2</span> <span class="pre">...</span> <span class="pre">f_k**e_k</span></tt>
-such that all <tt class="docutils literal"><span class="pre">f_i</span></tt> are monic polynomials and <tt class="docutils literal"><span class="pre">(f_i,</span> <span class="pre">f_j)</span></tt> for <tt class="docutils literal"><span class="pre">i</span> <span class="pre">!=</span> <span class="pre">j</span></tt>
-are co-prime and <tt class="docutils literal"><span class="pre">e_1</span> <span class="pre">...</span> <span class="pre">e_k</span></tt> are given in increasing order. All trivial
-terms (i.e. <tt class="docutils literal"><span class="pre">f_i</span> <span class="pre">=</span> <span class="pre">1</span></tt>) aren&#8217;t included in the output.</p>
-<p>Consider polynomial <tt class="docutils literal"><span class="pre">f</span> <span class="pre">=</span> <span class="pre">x**11</span> <span class="pre">+</span> <span class="pre">1</span></tt> over <tt class="docutils literal"><span class="pre">GF(11)[x]</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="p">(</span>
-<span class="gp">... </span>    <span class="n">gf_from_dict</span><span class="p">,</span> <span class="n">gf_diff</span><span class="p">,</span> <span class="n">gf_sqf_list</span><span class="p">,</span> <span class="n">gf_pow</span><span class="p">,</span>
-<span class="gp">... </span><span class="p">)</span>
-<span class="gp">... </span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">gf_from_dict</span><span class="p">({</span><span class="mi">11</span><span class="p">:</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="mi">0</span><span class="p">:</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">)},</span> <span class="mi">11</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Note that <tt class="docutils literal"><span class="pre">f'(x)</span> <span class="pre">=</span> <span class="pre">0</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[]</span>
-</pre></div>
-</div>
-<p>This phenomenon doesn&#8217;t happen in characteristic zero. However we can
-still compute square-free decomposition of <tt class="docutils literal"><span class="pre">f</span></tt> using <tt class="docutils literal"><span class="pre">gf_sqf()</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_sqf_list</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">(1, [([1, 1], 11)])</span>
-</pre></div>
-</div>
-<p>We obtained factorization <tt class="docutils literal"><span class="pre">f</span> <span class="pre">=</span> <span class="pre">(x</span> <span class="pre">+</span> <span class="pre">1)**11</span></tt>. This is correct because:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_pow</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="mi">11</span><span class="p">,</span> <span class="mi">11</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span> <span class="o">==</span> <span class="n">f</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../polys/literature.html#geddes92">[Geddes92]</a></li>
-</ol>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_Qmatrix">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_Qmatrix</tt><big>(</big><em>f</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_Qmatrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_Qmatrix" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculate Berlekamp&#8217;s <tt class="docutils literal"><span class="pre">Q</span></tt> matrix.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_Qmatrix</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_Qmatrix</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1, 0],</span>
-<span class="go"> [3, 4]]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_Qmatrix</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1, 0, 0, 0],</span>
-<span class="go"> [0, 4, 0, 0],</span>
-<span class="go"> [0, 0, 1, 0],</span>
-<span class="go"> [0, 0, 0, 4]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_Qbasis">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_Qbasis</tt><big>(</big><em>Q</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_Qbasis"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_Qbasis" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute a basis of the kernel of <tt class="docutils literal"><span class="pre">Q</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_Qmatrix</span><span class="p">,</span> <span class="n">gf_Qbasis</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_Qbasis</span><span class="p">(</span><span class="n">gf_Qmatrix</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">),</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1, 0, 0, 0], [0, 0, 1, 0]]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_Qbasis</span><span class="p">(</span><span class="n">gf_Qmatrix</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">),</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1, 0]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_berlekamp">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_berlekamp</tt><big>(</big><em>f</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_berlekamp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_berlekamp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Factor a square-free <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt> for small <tt class="docutils literal"><span class="pre">p</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_berlekamp</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_berlekamp</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1, 0, 2], [1, 0, 3]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_zassenhaus">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_zassenhaus</tt><big>(</big><em>f</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_zassenhaus"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_zassenhaus" title="Permalink to this definition">¶</a></dt>
-<dd><p>Factor a square-free <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt> for medium <tt class="docutils literal"><span class="pre">p</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_zassenhaus</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_zassenhaus</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">]),</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1, 1], [1, 3]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_shoup">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_shoup</tt><big>(</big><em>f</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_shoup"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_shoup" title="Permalink to this definition">¶</a></dt>
-<dd><p>Factor a square-free <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt> for large <tt class="docutils literal"><span class="pre">p</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_shoup</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_shoup</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">]),</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1, 1], [1, 3]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_factor_sqf">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_factor_sqf</tt><big>(</big><em>f</em>, <em>p</em>, <em>K</em>, <em>method=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_factor_sqf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_factor_sqf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Factor a square-free polynomial <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_factor_sqf</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_factor_sqf</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">]),</span> <span class="mi">5</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">(3, [[1, 1], [1, 3]])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_factor">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_factor</tt><big>(</big><em>f</em>, <em>p</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_factor"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_factor" title="Permalink to this definition">¶</a></dt>
-<dd><p>Factor (non square-free) polynomials in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>.</p>
-<p>Given a possibly non square-free polynomial <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">GF(p)[x]</span></tt>,
-returns its complete factorization into irreducibles:</p>
-<div class="highlight-python"><pre>f_1(x)**e_1 f_2(x)**e_2 ... f_d(x)**e_d</pre>
-</div>
-<p>where each <tt class="docutils literal"><span class="pre">f_i</span></tt> is a monic polynomial and <tt class="docutils literal"><span class="pre">gcd(f_i,</span> <span class="pre">f_j)</span> <span class="pre">==</span> <span class="pre">1</span></tt>,
-for <tt class="docutils literal"><span class="pre">i</span> <span class="pre">!=</span> <span class="pre">j</span></tt>.  The result is given as a tuple consisting of the
-leading coefficient of <tt class="docutils literal"><span class="pre">f</span></tt> and a list of factors of <tt class="docutils literal"><span class="pre">f</span></tt> with
-their multiplicities.</p>
-<p>The algorithm proceeds by first computing square-free decomposition
-of <tt class="docutils literal"><span class="pre">f</span></tt> and then iteratively factoring each of square-free factors.</p>
-<p>Consider a non square-free polynomial <tt class="docutils literal"><span class="pre">f</span> <span class="pre">=</span> <span class="pre">(7*x</span> <span class="pre">+</span> <span class="pre">1)</span> <span class="pre">(x</span> <span class="pre">+</span> <span class="pre">2)**2</span></tt> in
-<tt class="docutils literal"><span class="pre">GF(11)[x]</span></tt>. We obtain its factorization into irreducibles as follows:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_factor</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gf_factor</span><span class="p">(</span><span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">5</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">2</span><span class="p">]),</span> <span class="mi">11</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">(5, [([1, 2], 1), ([1, 8], 2)])</span>
-</pre></div>
-</div>
-<p>We arrived with factorization <tt class="docutils literal"><span class="pre">f</span> <span class="pre">=</span> <span class="pre">5</span> <span class="pre">(x</span> <span class="pre">+</span> <span class="pre">2)</span> <span class="pre">(x</span> <span class="pre">+</span> <span class="pre">8)**2</span></tt>. We didn&#8217;t
-recover the exact form of the input polynomial because we requested to
-get monic factors of <tt class="docutils literal"><span class="pre">f</span></tt> and its leading coefficient separately.</p>
-<p>Square-free factors of <tt class="docutils literal"><span class="pre">f</span></tt> can be factored into irreducibles over
-<tt class="docutils literal"><span class="pre">GF(p)</span></tt> using three very different methods:</p>
-<dl class="docutils">
-<dt>Berlekamp</dt>
-<dd>efficient for very small values of <tt class="docutils literal"><span class="pre">p</span></tt> (usually <tt class="docutils literal"><span class="pre">p</span> <span class="pre">&lt;</span> <span class="pre">25</span></tt>)</dd>
-<dt>Cantor-Zassenhaus</dt>
-<dd>efficient on average input and with &#8220;typical&#8221; <tt class="docutils literal"><span class="pre">p</span></tt></dd>
-<dt>Shoup-Kaltofen-Gathen</dt>
-<dd>efficient with very large inputs and modulus</dd>
-</dl>
-<p>If you want to use a specific factorization method, instead of the default
-one, set <tt class="docutils literal"><span class="pre">GF_FACTOR_METHOD</span></tt> with one of <tt class="docutils literal"><span class="pre">berlekamp</span></tt>, <tt class="docutils literal"><span class="pre">zassenhaus</span></tt> or
-<tt class="docutils literal"><span class="pre">shoup</span></tt> values.</p>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../polys/literature.html#gathen99">[Gathen99]</a></li>
-</ol>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_value">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_value</tt><big>(</big><em>f</em>, <em>a</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_value"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_value" title="Permalink to this definition">¶</a></dt>
-<dd><p>Value of polynomial &#8216;f&#8217; at &#8216;a&#8217; in field R.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_value</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gf_value</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="mi">11</span><span class="p">)</span>
-<span class="go">2204</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.galoistools.gf_csolve">
-<tt class="descclassname">sympy.polys.galoistools.</tt><tt class="descname">gf_csolve</tt><big>(</big><em>f</em>, <em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/galoistools.html#gf_csolve"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.galoistools.gf_csolve" title="Permalink to this definition">¶</a></dt>
-<dd><p>To solve f(x) congruent 0 mod(n).</p>
-<p>n is divided into canonical factors and f(x) cong 0 mod(p**e) will be
-solved for each factor. Applying the Chinese Remainder Theorem to the
-results returns the final answers.</p>
-<p class="rubric">References</p>
-<dl class="docutils">
-<dt>[1] &#8216;An introduction to the Theory of Numbers&#8217; 5th Edition by Ivan Niven,</dt>
-<dd>Zuckerman and Montgomery.</dd>
-</dl>
-<p class="rubric">Examples</p>
-<p>Solve [1, 1, 7] congruent 0 mod(189):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.galoistools</span> <span class="kn">import</span> <span class="n">gf_csolve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gf_csolve</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">],</span> <span class="mi">189</span><span class="p">)</span>
-<span class="go">[13, 49, 76, 112, 139, 175]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="manipulation-of-sparse-distributed-polynomials-and-vectors">
-<h3>Manipulation of sparse, distributed polynomials and vectors<a class="headerlink" href="#manipulation-of-sparse-distributed-polynomials-and-vectors" title="Permalink to this headline">¶</a></h3>
-<p>Dense representations quickly require infeasible amounts of storage and
-computation time if the number of variables increases. For this reason,
-there is code to manipulate polynomials in a <em>sparse</em> representation. These
-functions carry the <tt class="docutils literal"><span class="pre">sdp_</span></tt> prefix. As in the <tt class="docutils literal"><span class="pre">sdm_</span></tt> case, the ground domain
-<span class="math">\(K\)</span> and level <span class="math">\(u\)</span> often have to be passed explicitly. Furthermore, sparse
-representations are relative to a monomial order <span class="math">\(O\)</span>, which also has to be
-passed.</p>
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_LC">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_LC</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_LC"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_LC" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the leading coeffcient of <span class="math">\(f\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_LM">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_LM</tt><big>(</big><em>f</em>, <em>u</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_LM"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_LM" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the leading monomial of <span class="math">\(f\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_LT">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_LT</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_LT"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_LT" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the leading term of <span class="math">\(f\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_del_LT">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_del_LT</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_del_LT"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_del_LT" title="Permalink to this definition">¶</a></dt>
-<dd><p>Removes the leading from <span class="math">\(f\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_monoms">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_monoms</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_monoms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_monoms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a list of monomials in <span class="math">\(f\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_sort">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_sort</tt><big>(</big><em>f</em>, <em>O</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_sort"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_sort" title="Permalink to this definition">¶</a></dt>
-<dd><p>Sort terms in <span class="math">\(f\)</span> using the given monomial order <span class="math">\(O\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_strip">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_strip</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_strip"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_strip" title="Permalink to this definition">¶</a></dt>
-<dd><p>Remove terms with zero coefficients from <span class="math">\(f\)</span> in <span class="math">\(K[X]\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_normal">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_normal</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_normal"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_normal" title="Permalink to this definition">¶</a></dt>
-<dd><p>Normalize distributed polynomial in the given domain.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_from_dict">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_from_dict</tt><big>(</big><em>f</em>, <em>O</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_from_dict"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_from_dict" title="Permalink to this definition">¶</a></dt>
-<dd><p>Make a distributed polynomial from a dictionary.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_to_dict">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_to_dict</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_to_dict"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_to_dict" title="Permalink to this definition">¶</a></dt>
-<dd><p>Make a dictionary from a distributed polynomial.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_indep_p">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_indep_p</tt><big>(</big><em>f</em>, <em>j</em>, <em>u</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_indep_p"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_indep_p" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <span class="math">\(True\)</span> if a polynomial is independent of <span class="math">\(x_j\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_one_p">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_one_p</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_one_p"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_one_p" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <span class="math">\(f\)</span> is a multivariate one in <span class="math">\(K[X]\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_one">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_one</tt><big>(</big><em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_one"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_one" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a multivariate one in <span class="math">\(K[X]\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_term_p">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_term_p</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_term_p"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_term_p" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if <span class="math">\(f\)</span> has a single term or is zero.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_abs">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_abs</tt><big>(</big><em>f</em>, <em>u</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_abs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_abs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Make all coefficients positive in <span class="math">\(K[X]\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_neg">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_neg</tt><big>(</big><em>f</em>, <em>u</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_neg"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_neg" title="Permalink to this definition">¶</a></dt>
-<dd><p>Negate a polynomial in <span class="math">\(K[X]\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_add_term">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_add_term</tt><big>(</big><em>f</em>, <em>term</em>, <em>u</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_add_term"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_add_term" title="Permalink to this definition">¶</a></dt>
-<dd><p>Add a single term using bisection method.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_sub_term">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_sub_term</tt><big>(</big><em>f</em>, <em>term</em>, <em>u</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_sub_term"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_sub_term" title="Permalink to this definition">¶</a></dt>
-<dd><p>Sub a single term using bisection method.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_mul_term">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_mul_term</tt><big>(</big><em>f</em>, <em>term</em>, <em>u</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_mul_term"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_mul_term" title="Permalink to this definition">¶</a></dt>
-<dd><p>Multiply a distributed polynomial by a term.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_add">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_add</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Add distributed polynomials in <span class="math">\(K[X]\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_sub">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_sub</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_sub"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_sub" title="Permalink to this definition">¶</a></dt>
-<dd><p>Subtract distributed polynomials in <span class="math">\(K[X]\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_mul">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_mul</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Multiply distributed polynomials in <span class="math">\(K[X]\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_sqr">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_sqr</tt><big>(</big><em>f</em>, <em>u</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_sqr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_sqr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Square a distributed polynomial in <span class="math">\(K[X]\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_pow">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_pow</tt><big>(</big><em>f</em>, <em>n</em>, <em>u</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_pow"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_pow" title="Permalink to this definition">¶</a></dt>
-<dd><p>Raise <span class="math">\(f\)</span> to the n-th power in <span class="math">\(K[X]\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_monic">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_monic</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_monic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_monic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Divides all coefficients by <span class="math">\(LC(f)\)</span> in <span class="math">\(K[X]\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_content">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_content</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_content"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_content" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns GCD of coefficients in <span class="math">\(K[X]\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_primitive">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_primitive</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_primitive"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_primitive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns content and a primitive polynomial in <span class="math">\(K[X]\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_div">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_div</tt><big>(</big><em>f</em>, <em>G</em>, <em>u</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_div"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_div" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generalized polynomial division with remainder in <span class="math">\(K[X]\)</span>.</p>
-<p>Given polynomial <span class="math">\(f\)</span> and a set of polynomials <span class="math">\(g = (g_1, ..., g_n)\)</span>
-compute a set of quotients <span class="math">\(q = (q_1, ..., q_n)\)</span> and remainder <span class="math">\(r\)</span>
-such that <span class="math">\(f = q_1*f_1 + ... + q_n*f_n + r\)</span>, where <span class="math">\(r = 0\)</span> or <span class="math">\(r\)</span>
-is a completely reduced polynomial with respect to <span class="math">\(g\)</span>.</p>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../polys/literature.html#cox97">[Cox97]</a></li>
-<li><a class="reference internal" href="../polys/literature.html#ajwa95">[Ajwa95]</a></li>
-</ol>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_rem">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_rem</tt><big>(</big><em>f</em>, <em>G</em>, <em>u</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_rem"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_rem" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns polynomial remainder in <span class="math">\(K[X]\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_quo">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_quo</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_quo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_quo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns polynomial quotient in <span class="math">\(K[x]\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_exquo">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_exquo</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_exquo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_exquo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns exact polynomial quotient in <span class="math">\(K[X]\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_lcm">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_lcm</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_lcm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_lcm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes LCM of two polynomials in <span class="math">\(K[X]\)</span>.</p>
-<p>The LCM is computed as the unique generater of the intersection
-of the two ideals generated by <span class="math">\(f\)</span> and <span class="math">\(g\)</span>. The approach is to
-compute a Groebner basis with respect to lexicographic ordering
-of <span class="math">\(t*f\)</span> and <span class="math">\((1 - t)*g\)</span>, where <span class="math">\(t\)</span> is an unrelated variable and
-then filtering out the solution that doesn&#8217;t contain <span class="math">\(t\)</span>.</p>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../polys/literature.html#cox97">[Cox97]</a></li>
-</ol>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedpolys.sdp_gcd">
-<tt class="descclassname">sympy.polys.distributedpolys.</tt><tt class="descname">sdp_gcd</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedpolys.html#sdp_gcd"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedpolys.sdp_gcd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute GCD of two polynomials in <span class="math">\(K[X]\)</span> via LCM.</p>
-</dd></dl>
-
-<p>In commutative algebra, one often studies not only polynomials, but also
-<em>modules</em> over polynomial rings. The polynomial manipulation module provides
-rudimentary low-level support for finitely generated free modules. This is
-mainly used for Groebner basis computations (see there), so manipulation
-functions are only provided to the extend needed. They carry the prefix
-<tt class="docutils literal"><span class="pre">sdm_</span></tt>. Note that in examples, the generators of the free module are called
-<span class="math">\(f_1, f_2, \dots\)</span>.</p>
-<dl class="function">
-<dt id="sympy.polys.distributedmodules.sdm_monomial_mul">
-<tt class="descclassname">sympy.polys.distributedmodules.</tt><tt class="descname">sdm_monomial_mul</tt><big>(</big><em>M</em>, <em>X</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedmodules.html#sdm_monomial_mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedmodules.sdm_monomial_mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Multiply tuple <tt class="docutils literal"><span class="pre">X</span></tt> representing a monomial of <span class="math">\(K[X]\)</span> into the tuple
-<tt class="docutils literal"><span class="pre">M</span></tt> representing a monomial of <span class="math">\(F\)</span>.</p>
-<p class="rubric">Examples</p>
-<p>Multiplying <span class="math">\(xy^3\)</span> into <span class="math">\(x f_1\)</span> yields <span class="math">\(x^2 y^3 f_1\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.distributedmodules</span> <span class="kn">import</span> <span class="n">sdm_monomial_mul</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_monomial_mul</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
-<span class="go">(1, 2, 3)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedmodules.sdm_monomial_deg">
-<tt class="descclassname">sympy.polys.distributedmodules.</tt><tt class="descname">sdm_monomial_deg</tt><big>(</big><em>M</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedmodules.html#sdm_monomial_deg"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedmodules.sdm_monomial_deg" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the total degree of <tt class="docutils literal"><span class="pre">M</span></tt>.</p>
-<p class="rubric">Examples</p>
-<p>For example, the total degree of <span class="math">\(x^2 y f_5\)</span> is 3:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.distributedmodules</span> <span class="kn">import</span> <span class="n">sdm_monomial_deg</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_monomial_deg</span><span class="p">((</span><span class="mi">5</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedmodules.sdm_monomial_divides">
-<tt class="descclassname">sympy.polys.distributedmodules.</tt><tt class="descname">sdm_monomial_divides</tt><big>(</big><em>A</em>, <em>B</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedmodules.html#sdm_monomial_divides"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedmodules.sdm_monomial_divides" title="Permalink to this definition">¶</a></dt>
-<dd><p>Does there exist a (polynomial) monomial X such that XA = B?</p>
-<p class="rubric">Examples</p>
-<p>Positive examples:</p>
-<p>In the following examples, the monomial is given in terms of x, y and the
-generator(s), f_1, f_2 etc. The tuple form of that monomial is used in
-the call to sdm_monomial_divides.
-Note: the generator appears last in the expression but first in the tuple
-and other factors appear in the same order that they appear in the monomial
-expression.</p>
-<p><span class="math">\(A = f_1\)</span> divides <span class="math">\(B = f_1\)</span></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.distributedmodules</span> <span class="kn">import</span> <span class="n">sdm_monomial_divides</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_monomial_divides</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p><span class="math">\(A = f_1\)</span> divides <span class="math">\(B = x^2 y f_1\)</span></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_monomial_divides</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p><span class="math">\(A = xy f_5\)</span> divides <span class="math">\(B = x^2 y f_5\)</span></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_monomial_divides</span><span class="p">((</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>Negative examples:</p>
-<p><span class="math">\(A = f_1\)</span> does not divide <span class="math">\(B = f_2\)</span></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_monomial_divides</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p><span class="math">\(A = x f_1\)</span> does not divide <span class="math">\(B = f_1\)</span></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_monomial_divides</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p><span class="math">\(A = xy^2 f_5\)</span> does not divide <span class="math">\(B = y f_5\)</span></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_monomial_divides</span><span class="p">((</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedmodules.sdm_LC">
-<tt class="descclassname">sympy.polys.distributedmodules.</tt><tt class="descname">sdm_LC</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="headerlink" href="#sympy.polys.distributedmodules.sdm_LC" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the leading coeffcient of <span class="math">\(f\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedmodules.sdm_to_dict">
-<tt class="descclassname">sympy.polys.distributedmodules.</tt><tt class="descname">sdm_to_dict</tt><big>(</big><em>f</em><big>)</big><a class="headerlink" href="#sympy.polys.distributedmodules.sdm_to_dict" title="Permalink to this definition">¶</a></dt>
-<dd><p>Make a dictionary from a distributed polynomial.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedmodules.sdm_from_dict">
-<tt class="descclassname">sympy.polys.distributedmodules.</tt><tt class="descname">sdm_from_dict</tt><big>(</big><em>d</em>, <em>O</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedmodules.html#sdm_from_dict"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedmodules.sdm_from_dict" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create an sdm from a dictionary.</p>
-<p>Here <tt class="docutils literal"><span class="pre">O</span></tt> is the monomial order to use.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.distributedmodules</span> <span class="kn">import</span> <span class="n">sdm_from_dict</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">QQ</span><span class="p">,</span> <span class="n">lex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dic</span> <span class="o">=</span> <span class="p">{(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">):</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">):</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">):</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">0</span><span class="p">)}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_from_dict</span><span class="p">(</span><span class="n">dic</span><span class="p">,</span> <span class="n">lex</span><span class="p">)</span>
-<span class="go">[((1, 1, 0), 1/1), ((1, 0, 0), 2/1)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedmodules.sdm_add">
-<tt class="descclassname">sympy.polys.distributedmodules.</tt><tt class="descname">sdm_add</tt><big>(</big><em>f</em>, <em>g</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedmodules.html#sdm_add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedmodules.sdm_add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Add two module elements <tt class="docutils literal"><span class="pre">f</span></tt>, <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p>Addition is done over the ground field <tt class="docutils literal"><span class="pre">K</span></tt>, monomials are ordered
-according to <tt class="docutils literal"><span class="pre">O</span></tt>.</p>
-<p class="rubric">Examples</p>
-<p>All examples use lexicographic order.</p>
-<p><span class="math">\((xy f_1) + (f_2) = f_2 + xy f_1\)</span></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.distributedmodules</span> <span class="kn">import</span> <span class="n">sdm_add</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">lex</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_add</span><span class="p">([((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">))],</span> <span class="p">[((</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">))],</span> <span class="n">lex</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[((2, 0, 0), 1/1), ((1, 1, 1), 1/1)]</span>
-</pre></div>
-</div>
-<p><span class="math">\((xy f_1) + (-xy f_1)\)</span> = 0`</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_add</span><span class="p">([((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">))],</span> <span class="p">[((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">))],</span> <span class="n">lex</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[]</span>
-</pre></div>
-</div>
-<p><span class="math">\((f_1) + (2f_1) = 3f_1\)</span></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_add</span><span class="p">([((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">))],</span> <span class="p">[((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">2</span><span class="p">))],</span> <span class="n">lex</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[((1, 0, 0), 3/1)]</span>
-</pre></div>
-</div>
-<p><span class="math">\((yf_1) + (xf_1) = xf_1 + yf_1\)</span></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_add</span><span class="p">([((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">))],</span> <span class="p">[((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">))],</span> <span class="n">lex</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[((1, 1, 0), 1/1), ((1, 0, 1), 1/1)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedmodules.sdm_LM">
-<tt class="descclassname">sympy.polys.distributedmodules.</tt><tt class="descname">sdm_LM</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedmodules.html#sdm_LM"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedmodules.sdm_LM" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the leading monomial of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p>Only valid if <span class="math">\(f \ne 0\)</span>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.distributedmodules</span> <span class="kn">import</span> <span class="n">sdm_LM</span><span class="p">,</span> <span class="n">sdm_from_dict</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">QQ</span><span class="p">,</span> <span class="n">lex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dic</span> <span class="o">=</span> <span class="p">{(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">):</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">):</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">):</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">)}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_LM</span><span class="p">(</span><span class="n">sdm_from_dict</span><span class="p">(</span><span class="n">dic</span><span class="p">,</span> <span class="n">lex</span><span class="p">))</span>
-<span class="go">(4, 0, 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedmodules.sdm_LT">
-<tt class="descclassname">sympy.polys.distributedmodules.</tt><tt class="descname">sdm_LT</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedmodules.html#sdm_LT"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedmodules.sdm_LT" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the leading term of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p>Only valid if <span class="math">\(f \ne 0\)</span>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.distributedmodules</span> <span class="kn">import</span> <span class="n">sdm_LT</span><span class="p">,</span> <span class="n">sdm_from_dict</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">QQ</span><span class="p">,</span> <span class="n">lex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dic</span> <span class="o">=</span> <span class="p">{(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">):</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">):</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">):</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">3</span><span class="p">)}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_LT</span><span class="p">(</span><span class="n">sdm_from_dict</span><span class="p">(</span><span class="n">dic</span><span class="p">,</span> <span class="n">lex</span><span class="p">))</span>
-<span class="go">((4, 0, 1), 3/1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedmodules.sdm_mul_term">
-<tt class="descclassname">sympy.polys.distributedmodules.</tt><tt class="descname">sdm_mul_term</tt><big>(</big><em>f</em>, <em>term</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedmodules.html#sdm_mul_term"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedmodules.sdm_mul_term" title="Permalink to this definition">¶</a></dt>
-<dd><p>Multiply a distributed module element <tt class="docutils literal"><span class="pre">f</span></tt> by a (polynomial) term <tt class="docutils literal"><span class="pre">term</span></tt>.</p>
-<p>Multiplication of coefficients is done over the ground field <tt class="docutils literal"><span class="pre">K</span></tt>, and
-monomials are ordered according to <tt class="docutils literal"><span class="pre">O</span></tt>.</p>
-<p class="rubric">Examples</p>
-<p><span class="math">\(0 f_1 = 0\)</span></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.distributedmodules</span> <span class="kn">import</span> <span class="n">sdm_mul_term</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">lex</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_mul_term</span><span class="p">([((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">))],</span> <span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">0</span><span class="p">)),</span> <span class="n">lex</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[]</span>
-</pre></div>
-</div>
-<p><span class="math">\(x 0 = 0\)</span></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_mul_term</span><span class="p">([],</span> <span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">)),</span> <span class="n">lex</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[]</span>
-</pre></div>
-</div>
-<p><span class="math">\((x) (f_1) = xf_1\)</span></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_mul_term</span><span class="p">([((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">))],</span> <span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">)),</span> <span class="n">lex</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[((1, 1, 0), 1/1)]</span>
-</pre></div>
-</div>
-<p><span class="math">\((2xy) (3x f_1 + 4y f_2) = 8xy^2 f_2 + 6x^2y f_1\)</span></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="p">[((</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">4</span><span class="p">)),</span> <span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">3</span><span class="p">))]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_mul_term</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">2</span><span class="p">)),</span> <span class="n">lex</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[((2, 1, 2), 8/1), ((1, 2, 1), 6/1)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedmodules.sdm_zero">
-<tt class="descclassname">sympy.polys.distributedmodules.</tt><tt class="descname">sdm_zero</tt><big>(</big><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedmodules.html#sdm_zero"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedmodules.sdm_zero" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the zero module element.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedmodules.sdm_deg">
-<tt class="descclassname">sympy.polys.distributedmodules.</tt><tt class="descname">sdm_deg</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedmodules.html#sdm_deg"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedmodules.sdm_deg" title="Permalink to this definition">¶</a></dt>
-<dd><p>Degree of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p>This is the maximum of the degrees of all its monomials.
-Invalid if <tt class="docutils literal"><span class="pre">f</span></tt> is zero.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.distributedmodules</span> <span class="kn">import</span> <span class="n">sdm_deg</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_deg</span><span class="p">([((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="mi">1</span><span class="p">),</span> <span class="p">((</span><span class="mi">10</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="mi">1</span><span class="p">),</span> <span class="p">((</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="mi">4</span><span class="p">)])</span>
-<span class="go">7</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedmodules.sdm_from_vector">
-<tt class="descclassname">sympy.polys.distributedmodules.</tt><tt class="descname">sdm_from_vector</tt><big>(</big><em>vec</em>, <em>O</em>, <em>K</em>, <em>**opts</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedmodules.html#sdm_from_vector"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedmodules.sdm_from_vector" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create an sdm from an iterable of expressions.</p>
-<p>Coefficients are created in the ground field <tt class="docutils literal"><span class="pre">K</span></tt>, and terms are ordered
-according to monomial order <tt class="docutils literal"><span class="pre">O</span></tt>. Named arguments are passed on to the
-polys conversion code and can be used to specify for example generators.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.distributedmodules</span> <span class="kn">import</span> <span class="n">sdm_from_vector</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">QQ</span><span class="p">,</span> <span class="n">lex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_from_vector</span><span class="p">([</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="p">],</span> <span class="n">lex</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[((1, 0, 0, 1), 2/1), ((0, 2, 0, 0), 1/1), ((0, 0, 2, 0), 1/1)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedmodules.sdm_to_vector">
-<tt class="descclassname">sympy.polys.distributedmodules.</tt><tt class="descname">sdm_to_vector</tt><big>(</big><em>f</em>, <em>gens</em>, <em>K</em>, <em>n=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedmodules.html#sdm_to_vector"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedmodules.sdm_to_vector" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert sdm <tt class="docutils literal"><span class="pre">f</span></tt> into a list of polynomial expressions.</p>
-<p>The generators for the polynomial ring are specified via <tt class="docutils literal"><span class="pre">gens</span></tt>. The rank
-of the module is guessed, or passed via <tt class="docutils literal"><span class="pre">n</span></tt>. The ground field is assumed
-to be <tt class="docutils literal"><span class="pre">K</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.distributedmodules</span> <span class="kn">import</span> <span class="n">sdm_to_vector</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">QQ</span><span class="p">,</span> <span class="n">lex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="p">[((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">2</span><span class="p">)),</span> <span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">)),</span> <span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">))]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_to_vector</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">],</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[x**2 + y**2, 2*z]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="polynomial-factorization-algorithms">
-<h3>Polynomial factorization algorithms<a class="headerlink" href="#polynomial-factorization-algorithms" title="Permalink to this headline">¶</a></h3>
-<p>Many variants of Euclid&#8217;s algorithm:</p>
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_half_gcdex">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_half_gcdex</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_half_gcdex"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_half_gcdex" title="Permalink to this definition">¶</a></dt>
-<dd><p>Half extended Euclidean algorithm in <span class="math">\(F[X]\)</span>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_half_gcdex</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_gcdex">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_gcdex</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_gcdex"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_gcdex" title="Permalink to this definition">¶</a></dt>
-<dd><p>Extended Euclidean algorithm in <span class="math">\(F[X]\)</span>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_gcdex</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_invert">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_invert</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_invert"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_invert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute multiplicative inverse of <span class="math">\(f\)</span> modulo <span class="math">\(g\)</span> in <span class="math">\(F[X]\)</span>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_invert</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_euclidean_prs">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_euclidean_prs</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_euclidean_prs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_euclidean_prs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Euclidean polynomial remainder sequence (PRS) in <span class="math">\(K[X]\)</span>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_euclidean_prs</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_primitive_prs">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_primitive_prs</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_primitive_prs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_primitive_prs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Primitive polynomial remainder sequence (PRS) in <span class="math">\(K[X]\)</span>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_primitive_prs</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_inner_subresultants">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_inner_subresultants</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_inner_subresultants"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_inner_subresultants" title="Permalink to this definition">¶</a></dt>
-<dd><p>Subresultant PRS algorithm in <span class="math">\(K[X]\)</span>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_inner_subresultants</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">9</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">27</span><span class="p">,</span> <span class="mi">4</span><span class="p">]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="p">[[</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">12</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">54</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">729</span><span class="p">,</span> <span class="o">-</span><span class="mi">216</span><span class="p">,</span> <span class="mi">16</span><span class="p">]]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">9</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_inner_subresultants</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span> <span class="o">==</span> <span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">B</span><span class="p">,</span> <span class="n">D</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_subresultants">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_subresultants</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_subresultants"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_subresultants" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes subresultant PRS of two polynomials in <span class="math">\(K[X]\)</span>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_subresultants</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="p">]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">9</span><span class="p">]]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">27</span><span class="p">,</span> <span class="mi">4</span><span class="p">]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="p">[[</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">12</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">54</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">729</span><span class="p">,</span> <span class="o">-</span><span class="mi">216</span><span class="p">,</span> <span class="mi">16</span><span class="p">]]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_subresultants</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span> <span class="o">==</span> <span class="p">[</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">]</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_prs_resultant">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_prs_resultant</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_prs_resultant"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_prs_resultant" title="Permalink to this definition">¶</a></dt>
-<dd><p>Resultant algorithm in <span class="math">\(K[X]\)</span> using subresultant PRS.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_prs_resultant</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">9</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">27</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">12</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">54</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">729</span><span class="p">,</span> <span class="o">-</span><span class="mi">216</span><span class="p">,</span> <span class="mi">16</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_prs_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span> <span class="o">==</span> <span class="p">(</span><span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">])</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_zz_modular_resultant">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_zz_modular_resultant</tt><big>(</big><em>f</em>, <em>g</em>, <em>p</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_zz_modular_resultant"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_zz_modular_resultant" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute resultant of <span class="math">\(f\)</span> and <span class="math">\(g\)</span> modulo a prime <span class="math">\(p\)</span>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_zz_modular_resultant</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_zz_modular_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[-2, 0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_zz_collins_resultant">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_zz_collins_resultant</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_zz_collins_resultant"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_zz_collins_resultant" title="Permalink to this definition">¶</a></dt>
-<dd><p>Collins&#8217;s modular resultant algorithm in <span class="math">\(Z[X]\)</span>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_zz_collins_resultant</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_zz_collins_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[-2, -5, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_qq_collins_resultant">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_qq_collins_resultant</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K0</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_qq_collins_resultant"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_qq_collins_resultant" title="Permalink to this definition">¶</a></dt>
-<dd><p>Collins&#8217;s modular resultant algorithm in <span class="math">\(Q[X]\)</span>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_qq_collins_resultant</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="p">[[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)],</span> <span class="p">[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="p">[[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">)],</span> <span class="p">[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">3</span><span class="p">)]]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_qq_collins_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[-2/1, -7/3, 5/6]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_resultant">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_resultant</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em>, <em>includePRS=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_resultant"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_resultant" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes resultant of two polynomials in <span class="math">\(K[X]\)</span>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_resultant</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">4</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">9</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_resultant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[-3, 0, 0, -12, 1, 0, -54, 8, 729, -216, 16]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_discriminant">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_discriminant</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_discriminant"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_discriminant" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes discriminant of a polynomial in <span class="math">\(K[X]\)</span>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_discriminant</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[[[</span><span class="mi">1</span><span class="p">]],</span> <span class="p">[[]]],</span> <span class="p">[[[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[]]],</span> <span class="p">[[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_discriminant</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[[-4, 0]], [[1], [], []]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_rr_prs_gcd">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_rr_prs_gcd</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_rr_prs_gcd"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_rr_prs_gcd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes polynomial GCD using subresultants over a ring.</p>
-<p>Returns <tt class="docutils literal"><span class="pre">(h,</span> <span class="pre">cff,</span> <span class="pre">cfg)</span></tt> such that <tt class="docutils literal"><span class="pre">a</span> <span class="pre">=</span> <span class="pre">gcd(f,</span> <span class="pre">g)</span></tt>, <tt class="docutils literal"><span class="pre">cff</span> <span class="pre">=</span> <span class="pre">quo(f,</span> <span class="pre">h)</span></tt>,
-and <tt class="docutils literal"><span class="pre">cfg</span> <span class="pre">=</span> <span class="pre">quo(g,</span> <span class="pre">h)</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_rr_prs_gcd</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_rr_prs_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">([[1], [1, 0]], [[1], [1, 0]], [[1], []])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_ff_prs_gcd">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_ff_prs_gcd</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_ff_prs_gcd"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_ff_prs_gcd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes polynomial GCD using subresultants over a field.</p>
-<p>Returns <tt class="docutils literal"><span class="pre">(h,</span> <span class="pre">cff,</span> <span class="pre">cfg)</span></tt> such that <tt class="docutils literal"><span class="pre">a</span> <span class="pre">=</span> <span class="pre">gcd(f,</span> <span class="pre">g)</span></tt>, <tt class="docutils literal"><span class="pre">cff</span> <span class="pre">=</span> <span class="pre">quo(f,</span> <span class="pre">h)</span></tt>,
-and <tt class="docutils literal"><span class="pre">cfg</span> <span class="pre">=</span> <span class="pre">quo(g,</span> <span class="pre">h)</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_ff_prs_gcd</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="p">[[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)],</span> <span class="p">[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">0</span><span class="p">)],</span> <span class="p">[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">0</span><span class="p">)]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="p">[[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">)],</span> <span class="p">[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">0</span><span class="p">)],</span> <span class="p">[]]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_ff_prs_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">([[1/1], [1/1, 0/1]], [[1/2], [1/2, 0/1]], [[1/1], []])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_zz_heu_gcd">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_zz_heu_gcd</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_zz_heu_gcd"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_zz_heu_gcd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Heuristic polynomial GCD in <span class="math">\(Z[X]\)</span>.</p>
-<p>Given univariate polynomials <span class="math">\(f\)</span> and <span class="math">\(g\)</span> in <span class="math">\(Z[X]\)</span>, returns
-their GCD and cofactors, i.e. polynomials <tt class="docutils literal"><span class="pre">h</span></tt>, <tt class="docutils literal"><span class="pre">cff</span></tt> and <tt class="docutils literal"><span class="pre">cfg</span></tt>
-such that:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">h</span> <span class="o">=</span> <span class="n">gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">),</span> <span class="n">cff</span> <span class="o">=</span> <span class="n">quo</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">)</span> <span class="ow">and</span> <span class="n">cfg</span> <span class="o">=</span> <span class="n">quo</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">h</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>The algorithm is purely heuristic which means it may fail to compute
-the GCD. This will be signaled by raising an exception. In this case
-you will need to switch to another GCD method.</p>
-<p>The algorithm computes the polynomial GCD by evaluating polynomials
-f and g at certain points and computing (fast) integer GCD of those
-evaluations. The polynomial GCD is recovered from the integer image
-by interpolation. The evaluation proces reduces f and g variable by
-variable into a large integer.  The final step is to verify if the
-interpolated polynomial is the correct GCD. This gives cofactors of
-the input polynomials as a side effect.</p>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../polys/literature.html#liao95">[Liao95]</a></li>
-</ol>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_zz_heu_gcd</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_zz_heu_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">([[1], [1, 0]], [[1], [1, 0]], [[1], []])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_qq_heu_gcd">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_qq_heu_gcd</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K0</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_qq_heu_gcd"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_qq_heu_gcd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Heuristic polynomial GCD in <span class="math">\(Q[X]\)</span>.</p>
-<p>Returns <tt class="docutils literal"><span class="pre">(h,</span> <span class="pre">cff,</span> <span class="pre">cfg)</span></tt> such that <tt class="docutils literal"><span class="pre">a</span> <span class="pre">=</span> <span class="pre">gcd(f,</span> <span class="pre">g)</span></tt>,
-<tt class="docutils literal"><span class="pre">cff</span> <span class="pre">=</span> <span class="pre">quo(f,</span> <span class="pre">h)</span></tt>, and <tt class="docutils literal"><span class="pre">cfg</span> <span class="pre">=</span> <span class="pre">quo(g,</span> <span class="pre">h)</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_qq_heu_gcd</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="p">[[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">)],</span> <span class="p">[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">0</span><span class="p">)],</span> <span class="p">[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">0</span><span class="p">)]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="p">[[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)],</span> <span class="p">[</span><span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">0</span><span class="p">)],</span> <span class="p">[]]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_qq_heu_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">([[1/1], [2/1, 0/1]], [[1/4], [1/2, 0/1]], [[1/2], []])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_inner_gcd">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_inner_gcd</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_inner_gcd"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_inner_gcd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes polynomial GCD and cofactors of <span class="math">\(f\)</span> and <span class="math">\(g\)</span> in <span class="math">\(K[X]\)</span>.</p>
-<p>Returns <tt class="docutils literal"><span class="pre">(h,</span> <span class="pre">cff,</span> <span class="pre">cfg)</span></tt> such that <tt class="docutils literal"><span class="pre">a</span> <span class="pre">=</span> <span class="pre">gcd(f,</span> <span class="pre">g)</span></tt>,
-<tt class="docutils literal"><span class="pre">cff</span> <span class="pre">=</span> <span class="pre">quo(f,</span> <span class="pre">h)</span></tt>, and <tt class="docutils literal"><span class="pre">cfg</span> <span class="pre">=</span> <span class="pre">quo(g,</span> <span class="pre">h)</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_inner_gcd</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_inner_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">([[1], [1, 0]], [[1], [1, 0]], [[1], []])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_gcd">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_gcd</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_gcd"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_gcd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes polynomial GCD of <span class="math">\(f\)</span> and <span class="math">\(g\)</span> in <span class="math">\(K[X]\)</span>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_gcd</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1], [1, 0]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_lcm">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_lcm</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_lcm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_lcm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes polynomial LCM of <span class="math">\(f\)</span> and <span class="math">\(g\)</span> in <span class="math">\(K[X]\)</span>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_lcm</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_lcm</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[[1], [2, 0], [1, 0, 0], []]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_content">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_content</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_content"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_content" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns GCD of multivariate coefficients.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_content</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">12</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_content</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">[2, 6]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_primitive">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_primitive</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_primitive"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_primitive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns multivariate content and a primitive polynomial.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_primitive</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">12</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_primitive</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">([2, 6], [[1], [2]])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.euclidtools.dmp_cancel">
-<tt class="descclassname">sympy.polys.euclidtools.</tt><tt class="descname">dmp_cancel</tt><big>(</big><em>f</em>, <em>g</em>, <em>u</em>, <em>K</em>, <em>include=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/euclidtools.html#dmp_cancel"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.euclidtools.dmp_cancel" title="Permalink to this definition">¶</a></dt>
-<dd><p>Cancel common factors in a rational function <span class="math">\(f/g\)</span>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.euclidtools</span> <span class="kn">import</span> <span class="n">dmp_cancel</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">ZZ</span><span class="o">.</span><span class="n">map</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">]])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_cancel</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">([[2], [2]], [[1], [-1]])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<p>Polynomial factorization in characteristic zero:</p>
-<dl class="function">
-<dt id="sympy.polys.factortools.dmp_trial_division">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dmp_trial_division</tt><big>(</big><em>f</em>, <em>factors</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dmp_trial_division"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dmp_trial_division" title="Permalink to this definition">¶</a></dt>
-<dd><p>Determine multiplicities of factors using trial division.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.factortools.dmp_zz_mignotte_bound">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dmp_zz_mignotte_bound</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dmp_zz_mignotte_bound"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dmp_zz_mignotte_bound" title="Permalink to this definition">¶</a></dt>
-<dd><p>Mignotte bound for multivariate polynomials in <span class="math">\(K[X]\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.factortools.dup_zz_hensel_step">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dup_zz_hensel_step</tt><big>(</big><em>m</em>, <em>f</em>, <em>g</em>, <em>h</em>, <em>s</em>, <em>t</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dup_zz_hensel_step"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dup_zz_hensel_step" title="Permalink to this definition">¶</a></dt>
-<dd><p>One step in Hensel lifting in <span class="math">\(Z[x]\)</span>.</p>
-<p>Given positive integer <span class="math">\(m\)</span> and <span class="math">\(Z[x]\)</span> polynomials <span class="math">\(f\)</span>, <span class="math">\(g\)</span>, <span class="math">\(h\)</span>, <span class="math">\(s\)</span>
-and <span class="math">\(t\)</span> such that:</p>
-<div class="highlight-python"><pre>f == g*h (mod m)
-s*g + t*h == 1 (mod m)
-
-lc(f) is not a zero divisor (mod m)
-lc(h) == 1
-
-deg(f) == deg(g) + deg(h)
-deg(s) &lt; deg(h)
-deg(t) &lt; deg(g)</pre>
-</div>
-<p>returns polynomials <span class="math">\(G\)</span>, <span class="math">\(H\)</span>, <span class="math">\(S\)</span> and <span class="math">\(T\)</span>, such that:</p>
-<div class="highlight-python"><pre>f == G*H (mod m**2)
-S*G + T**H == 1 (mod m**2)</pre>
-</div>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../polys/literature.html#gathen99">[Gathen99]</a></li>
-</ol>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.factortools.dup_zz_hensel_lift">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dup_zz_hensel_lift</tt><big>(</big><em>p</em>, <em>f</em>, <em>f_list</em>, <em>l</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dup_zz_hensel_lift"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dup_zz_hensel_lift" title="Permalink to this definition">¶</a></dt>
-<dd><p>Multifactor Hensel lifting in <span class="math">\(Z[x]\)</span>.</p>
-<p>Given a prime <span class="math">\(p\)</span>, polynomial <span class="math">\(f\)</span> over <span class="math">\(Z[x]\)</span> such that <span class="math">\(lc(f)\)</span>
-is a unit modulo <span class="math">\(p\)</span>, monic pair-wise coprime polynomials <span class="math">\(f_i\)</span>
-over <span class="math">\(Z[x]\)</span> satisfying:</p>
-<div class="highlight-python"><pre>f = lc(f) f_1 ... f_r (mod p)</pre>
-</div>
-<p>and a positive integer <span class="math">\(l\)</span>, returns a list of monic polynomials
-<span class="math">\(F_1\)</span>, <span class="math">\(F_2\)</span>, ..., <span class="math">\(F_r\)</span> satisfying:</p>
-<div class="highlight-python"><pre>f = lc(f) F_1 ... F_r (mod p**l)
-
-F_i = f_i (mod p), i = 1..r</pre>
-</div>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../polys/literature.html#gathen99">[Gathen99]</a></li>
-</ol>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.factortools.dup_zz_zassenhaus">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dup_zz_zassenhaus</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dup_zz_zassenhaus"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dup_zz_zassenhaus" title="Permalink to this definition">¶</a></dt>
-<dd><p>Factor primitive square-free polynomials in <span class="math">\(Z[x]\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.factortools.dup_zz_irreducible_p">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dup_zz_irreducible_p</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dup_zz_irreducible_p"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dup_zz_irreducible_p" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test irreducibility using Eisenstein&#8217;s criterion.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.factortools.dup_cyclotomic_p">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dup_cyclotomic_p</tt><big>(</big><em>f</em>, <em>K</em>, <em>irreducible=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dup_cyclotomic_p"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dup_cyclotomic_p" title="Permalink to this definition">¶</a></dt>
-<dd><p>Efficiently test if <tt class="docutils literal"><span class="pre">f</span></tt> is a cyclotomic polnomial.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.factortools</span> <span class="kn">import</span> <span class="n">dup_cyclotomic_p</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dup_cyclotomic_p</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dup_cyclotomic_p</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.factortools.dup_zz_cyclotomic_poly">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dup_zz_cyclotomic_poly</tt><big>(</big><em>n</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dup_zz_cyclotomic_poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dup_zz_cyclotomic_poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Efficiently generate n-th cyclotomic polnomial.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.factortools.dup_zz_cyclotomic_factor">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dup_zz_cyclotomic_factor</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dup_zz_cyclotomic_factor"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dup_zz_cyclotomic_factor" title="Permalink to this definition">¶</a></dt>
-<dd><p>Efficiently factor polynomials <span class="math">\(x**n - 1\)</span> and <span class="math">\(x**n + 1\)</span> in <span class="math">\(Z[x]\)</span>.</p>
-<p>Given a univariate polynomial <span class="math">\(f\)</span> in <span class="math">\(Z[x]\)</span> returns a list of factors
-of <span class="math">\(f\)</span>, provided that <span class="math">\(f\)</span> is in the form <span class="math">\(x**n - 1\)</span> or <span class="math">\(x**n + 1\)</span> for
-<span class="math">\(n &gt;= 1\)</span>. Otherwise returns None.</p>
-<p>Factorization is performed using using cyclotomic decomposition of <span class="math">\(f\)</span>,
-which makes this method much faster that any other direct factorization
-approach (e.g. Zassenhaus&#8217;s).</p>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../polys/literature.html#weisstein09">[Weisstein09]</a></li>
-</ol>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.factortools.dup_zz_factor_sqf">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dup_zz_factor_sqf</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dup_zz_factor_sqf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dup_zz_factor_sqf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Factor square-free (non-primitive) polyomials in <span class="math">\(Z[x]\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.factortools.dup_zz_factor">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dup_zz_factor</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dup_zz_factor"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dup_zz_factor" title="Permalink to this definition">¶</a></dt>
-<dd><p>Factor (non square-free) polynomials in <span class="math">\(Z[x]\)</span>.</p>
-<p>Given a univariate polynomial <span class="math">\(f\)</span> in <span class="math">\(Z[x]\)</span> computes its complete
-factorization <span class="math">\(f_1, ..., f_n\)</span> into irreducibles over integers:</p>
-<div class="highlight-python"><pre>f = content(f) f_1**k_1 ... f_n**k_n</pre>
-</div>
-<p>The factorization is computed by reducing the input polynomial
-into a primitive square-free polynomial and factoring it using
-Zassenhaus algorithm. Trial division is used to recover the
-multiplicities of factors.</p>
-<p>The result is returned as a tuple consisting of:</p>
-<div class="highlight-python"><pre>(content(f), [(f_1, k_1), ..., (f_n, k_n))</pre>
-</div>
-<p>Consider polynomial <span class="math">\(f = 2*x**4 - 2\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.factortools</span> <span class="kn">import</span> <span class="n">dup_zz_factor</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dup_zz_factor</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">(2, [([1, -1], 1), ([1, 1], 1), ([1, 0, 1], 1)])</span>
-</pre></div>
-</div>
-<p>In result we got the following factorization:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">f</span> <span class="o">=</span> <span class="mi">2</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Note that this is a complete factorization over integers,
-however over Gaussian integers we can factor the last term.</p>
-<p>By default, polynomials <span class="math">\(x**n - 1\)</span> and <span class="math">\(x**n + 1\)</span> are factored
-using cyclotomic decomposition to speedup computations. To
-disable this behaviour set cyclotomic=False.</p>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../polys/literature.html#gathen99">[Gathen99]</a></li>
-</ol>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.factortools.dmp_zz_wang_non_divisors">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dmp_zz_wang_non_divisors</tt><big>(</big><em>E</em>, <em>cs</em>, <em>ct</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dmp_zz_wang_non_divisors"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dmp_zz_wang_non_divisors" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wang/EEZ: Compute a set of valid divisors.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.factortools.dmp_zz_wang_test_points">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dmp_zz_wang_test_points</tt><big>(</big><em>f</em>, <em>T</em>, <em>ct</em>, <em>A</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dmp_zz_wang_test_points"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dmp_zz_wang_test_points" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wang/EEZ: Test evaluation points for suitability.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.factortools.dmp_zz_wang_lead_coeffs">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dmp_zz_wang_lead_coeffs</tt><big>(</big><em>f</em>, <em>T</em>, <em>cs</em>, <em>E</em>, <em>H</em>, <em>A</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dmp_zz_wang_lead_coeffs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dmp_zz_wang_lead_coeffs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wang/EEZ: Compute correct leading coefficients.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.factortools.dmp_zz_diophantine">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dmp_zz_diophantine</tt><big>(</big><em>F</em>, <em>c</em>, <em>A</em>, <em>d</em>, <em>p</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dmp_zz_diophantine"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dmp_zz_diophantine" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wang/EEZ: Solve multivariate Diophantine equations.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.factortools.dmp_zz_wang_hensel_lifting">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dmp_zz_wang_hensel_lifting</tt><big>(</big><em>f</em>, <em>H</em>, <em>LC</em>, <em>A</em>, <em>p</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dmp_zz_wang_hensel_lifting"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dmp_zz_wang_hensel_lifting" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wang/EEZ: Parallel Hensel lifting algorithm.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.factortools.dmp_zz_wang">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dmp_zz_wang</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em>, <em>mod=None</em>, <em>seed=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dmp_zz_wang"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dmp_zz_wang" title="Permalink to this definition">¶</a></dt>
-<dd><p>Factor primitive square-free polynomials in <span class="math">\(Z[X]\)</span>.</p>
-<p>Given a multivariate polynomial <span class="math">\(f\)</span> in <span class="math">\(Z[x_1,...,x_n]\)</span>, which is
-primitive and square-free in <span class="math">\(x_1\)</span>, computes factorization of <span class="math">\(f\)</span> into
-irreducibles over integers.</p>
-<p>The procedure is based on Wang&#8217;s Enhanced Extended Zassenhaus
-algorithm. The algorithm works by viewing <span class="math">\(f\)</span> as a univariate polynomial
-in <span class="math">\(Z[x_2,...,x_n][x_1]\)</span>, for which an evaluation mapping is computed:</p>
-<div class="highlight-python"><pre>x_2 -&gt; a_2, ..., x_n -&gt; a_n</pre>
-</div>
-<p>where <span class="math">\(a_i\)</span>, for <span class="math">\(i = 2, ..., n\)</span>, are carefully chosen integers.  The
-mapping is used to transform <span class="math">\(f\)</span> into a univariate polynomial in <span class="math">\(Z[x_1]\)</span>,
-which can be factored efficiently using Zassenhaus algorithm. The last
-step is to lift univariate factors to obtain true multivariate
-factors. For this purpose a parallel Hensel lifting procedure is used.</p>
-<p>The parameter <tt class="docutils literal"><span class="pre">seed</span></tt> is passed to _randint and can be used to seed randint
-(when an integer) or (for testing purposes) can be a sequence of numbers.</p>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../polys/literature.html#wang78">[Wang78]</a></li>
-<li><a class="reference internal" href="../polys/literature.html#geddes92">[Geddes92]</a></li>
-</ol>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.factortools.dmp_zz_factor">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dmp_zz_factor</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dmp_zz_factor"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dmp_zz_factor" title="Permalink to this definition">¶</a></dt>
-<dd><p>Factor (non square-free) polynomials in <span class="math">\(Z[X]\)</span>.</p>
-<p>Given a multivariate polynomial <span class="math">\(f\)</span> in <span class="math">\(Z[x]\)</span> computes its complete
-factorization <span class="math">\(f_1, ..., f_n\)</span> into irreducibles over integers:</p>
-<div class="highlight-python"><pre>f = content(f) f_1**k_1 ... f_n**k_n</pre>
-</div>
-<p>The factorization is computed by reducing the input polynomial
-into a primitive square-free polynomial and factoring it using
-Enhanced Extended Zassenhaus (EEZ) algorithm. Trial division
-is used to recover the multiplicities of factors.</p>
-<p>The result is returned as a tuple consisting of:</p>
-<div class="highlight-python"><pre>(content(f), [(f_1, k_1), ..., (f_n, k_n))</pre>
-</div>
-<p>Consider polynomial <span class="math">\(f = 2*(x**2 - y**2)\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.factortools</span> <span class="kn">import</span> <span class="n">dmp_zz_factor</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.domains</span> <span class="kn">import</span> <span class="n">ZZ</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dmp_zz_factor</span><span class="p">([[</span><span class="mi">2</span><span class="p">],</span> <span class="p">[],</span> <span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">)</span>
-<span class="go">(2, [([[1], [-1, 0]], 1), ([[1], [1, 0]], 1)])</span>
-</pre></div>
-</div>
-<p>In result we got the following factorization:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">f</span> <span class="o">=</span> <span class="mi">2</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span>
-</pre></div>
-</div>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../polys/literature.html#gathen99">[Gathen99]</a></li>
-</ol>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.factortools.dmp_ext_factor">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dmp_ext_factor</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dmp_ext_factor"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dmp_ext_factor" title="Permalink to this definition">¶</a></dt>
-<dd><p>Factor multivariate polynomials over algebraic number fields.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.factortools.dup_gf_factor">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dup_gf_factor</tt><big>(</big><em>f</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dup_gf_factor"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dup_gf_factor" title="Permalink to this definition">¶</a></dt>
-<dd><p>Factor univariate polynomials over finite fields.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.factortools.dmp_factor_list">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dmp_factor_list</tt><big>(</big><em>f</em>, <em>u</em>, <em>K0</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dmp_factor_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dmp_factor_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Factor polynomials into irreducibles in <span class="math">\(K[X]\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.factortools.dmp_factor_list_include">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dmp_factor_list_include</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dmp_factor_list_include"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dmp_factor_list_include" title="Permalink to this definition">¶</a></dt>
-<dd><p>Factor polynomials into irreducibles in <span class="math">\(K[X]\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.factortools.dmp_irreducible_p">
-<tt class="descclassname">sympy.polys.factortools.</tt><tt class="descname">dmp_irreducible_p</tt><big>(</big><em>f</em>, <em>u</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/factortools.html#dmp_irreducible_p"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.factortools.dmp_irreducible_p" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> has no factors over its domain.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="groebner-basis-algorithms">
-<h3>Groebner basis algorithms<a class="headerlink" href="#groebner-basis-algorithms" title="Permalink to this headline">¶</a></h3>
-<p>Groebner bases can be used to answer many problems in computational
-commutative algebra. Their computation in rather complicated, and very
-performance-sensitive. We present here various low-level implementations of
-Groebner basis computation algorithms; please see the previous section of the
-manual for usage.</p>
-<dl class="function">
-<dt id="sympy.polys.groebnertools.sdp_groebner">
-<tt class="descclassname">sympy.polys.groebnertools.</tt><tt class="descname">sdp_groebner</tt><big>(</big><em>f</em>, <em>u</em>, <em>O</em>, <em>K</em>, <em>gens=''</em>, <em>verbose=False</em>, <em>method=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/groebnertools.html#sdp_groebner"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.groebnertools.sdp_groebner" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes Groebner basis for a set of polynomials in <span class="math">\(K[X]\)</span>.</p>
-<p>Wrapper around the (default) improved Buchberger and the other algorithms
-for computing Groebner bases. The choice of algorithm can be changed via
-<tt class="docutils literal"><span class="pre">method</span></tt> argument or <tt class="xref py py-func docutils literal"><span class="pre">setup()</span></tt> from <tt class="xref py py-mod docutils literal"><span class="pre">sympy.polys.polyconfig</span></tt>,
-where <tt class="docutils literal"><span class="pre">method</span></tt> can be either <tt class="docutils literal"><span class="pre">buchberger</span></tt> or <tt class="docutils literal"><span class="pre">f5b</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.groebnertools.buchberger">
-<tt class="descclassname">sympy.polys.groebnertools.</tt><tt class="descname">buchberger</tt><big>(</big><em>f</em>, <em>u</em>, <em>O</em>, <em>K</em>, <em>gens=''</em>, <em>verbose=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/groebnertools.html#buchberger"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.groebnertools.buchberger" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes Groebner basis for a set of polynomials in <span class="math">\(K[X]\)</span>.</p>
-<p>Given a set of multivariate polynomials <span class="math">\(F\)</span>, finds another
-set <span class="math">\(G\)</span>, such that Ideal <span class="math">\(F = Ideal G\)</span> and <span class="math">\(G\)</span> is a reduced
-Groebner basis.</p>
-<p>The resulting basis is unique and has monic generators if the
-ground domains is a field. Otherwise the result is non-unique
-but Groebner bases over e.g. integers can be computed (if the
-input polynomials are monic).</p>
-<p>Groebner bases can be used to choose specific generators for a
-polynomial ideal. Because these bases are unique you can check
-for ideal equality by comparing the Groebner bases.  To see if
-one polynomial lies in an ideal, divide by the elements in the
-base and see if the remainder vanishes.</p>
-<p>They can also be used to solve systems of polynomial equations
-as,  by choosing lexicographic ordering,  you can eliminate one
-variable at a time, provided that the ideal is zero-dimensional
-(finite number of solutions).</p>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../polys/literature.html#bose03">[Bose03]</a></li>
-<li><a class="reference internal" href="../polys/literature.html#giovini91">[Giovini91]</a></li>
-<li><a class="reference internal" href="../polys/literature.html#ajwa95">[Ajwa95]</a></li>
-<li><a class="reference internal" href="../polys/literature.html#cox97">[Cox97]</a></li>
-</ol>
-<p>Algorithm used: an improved version of Buchberger&#8217;s algorithm
-as presented in T. Becker, V. Weispfenning, Groebner Bases: A
-Computational Approach to Commutative Algebra, Springer, 1993,
-page 232.</p>
-<p>Added optional <tt class="docutils literal"><span class="pre">gens</span></tt> argument to apply <tt class="xref py py-func docutils literal"><span class="pre">sdp_str()</span></tt> for
-the purpose of debugging the algorithm.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.groebnertools.sdp_spoly">
-<tt class="descclassname">sympy.polys.groebnertools.</tt><tt class="descname">sdp_spoly</tt><big>(</big><em>p1</em>, <em>p2</em>, <em>u</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/groebnertools.html#sdp_spoly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.groebnertools.sdp_spoly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute LCM(LM(p1), LM(p2))/LM(p1)*p1 - LCM(LM(p1), LM(p2))/LM(p2)*p2
-This is the S-poly provided p1 and p2 are monic</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.groebnertools.f5b">
-<tt class="descclassname">sympy.polys.groebnertools.</tt><tt class="descname">f5b</tt><big>(</big><em>F</em>, <em>u</em>, <em>O</em>, <em>K</em>, <em>gens=''</em>, <em>verbose=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/groebnertools.html#f5b"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.groebnertools.f5b" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes a reduced Groebner basis for the ideal generated by F.</p>
-<p>f5b is an implementation of the F5B algorithm by Yao Sun and
-Dingkang Wang. Similarly to Buchberger&#8217;s algorithm, the algorithm
-proceeds by computing critical pairs, computing the S-polynomial,
-reducing it and adjoining the reduced S-polynomial if it is not 0.</p>
-<p>Unlike Buchberger&#8217;s algorithm, each polynomial contains additional
-information, namely a signature and a number. The signature
-specifies the path of computation (i.e. from which polynomial in
-the original basis was it derived and how), the number says when
-the polynomial was added to the basis.  With this information it
-is (often) possible to decide if an S-polynomial will reduce to
-0 and can be discarded.</p>
-<p>Optimizations include: Reducing the generators before computing
-a Groebner basis, removing redundant critical pairs when a new
-polynomial enters the basis and sorting the critical pairs and
-the current basis.</p>
-<p>Once a Groebner basis has been found, it gets reduced.</p>
-<p>** References **
-Yao Sun, Dingkang Wang: &#8220;A New Proof for the Correctness of F5
-(F5-Like) Algorithm&#8221;, <a class="reference external" href="http://arxiv.org/abs/1004.0084">http://arxiv.org/abs/1004.0084</a> (specifically
-v4)</p>
-<p>Thomas Becker, Volker Weispfenning, Groebner bases: A computational
-approach to commutative algebra, 1993, p. 203, 216</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.groebnertools.red_groebner">
-<tt class="descclassname">sympy.polys.groebnertools.</tt><tt class="descname">red_groebner</tt><big>(</big><em>G</em>, <em>u</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/groebnertools.html#red_groebner"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.groebnertools.red_groebner" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute reduced Groebner basis, from BeckerWeispfenning93, p. 216</p>
-<p>Selects a subset of generators, that already generate the ideal
-and computes a reduced Groebner basis for them.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.groebnertools.is_groebner">
-<tt class="descclassname">sympy.polys.groebnertools.</tt><tt class="descname">is_groebner</tt><big>(</big><em>G</em>, <em>u</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/groebnertools.html#is_groebner"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.groebnertools.is_groebner" title="Permalink to this definition">¶</a></dt>
-<dd><p>Check if G is a Groebner basis.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.groebnertools.is_minimal">
-<tt class="descclassname">sympy.polys.groebnertools.</tt><tt class="descname">is_minimal</tt><big>(</big><em>G</em>, <em>u</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/groebnertools.html#is_minimal"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.groebnertools.is_minimal" title="Permalink to this definition">¶</a></dt>
-<dd><p>Checks if G is a minimal Groebner basis.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.groebnertools.is_reduced">
-<tt class="descclassname">sympy.polys.groebnertools.</tt><tt class="descname">is_reduced</tt><big>(</big><em>G</em>, <em>u</em>, <em>O</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/groebnertools.html#is_reduced"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.groebnertools.is_reduced" title="Permalink to this definition">¶</a></dt>
-<dd><p>Checks if G is a reduced Groebner basis.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.groebnertools.matrix_fglm">
-<tt class="descclassname">sympy.polys.groebnertools.</tt><tt class="descname">matrix_fglm</tt><big>(</big><em>F</em>, <em>u</em>, <em>O_from</em>, <em>O_to</em>, <em>K</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/groebnertools.html#matrix_fglm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.groebnertools.matrix_fglm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Converts the reduced Groebner basis <tt class="docutils literal"><span class="pre">F</span></tt> of a zero-dimensional
-ideal w.r.t. <tt class="docutils literal"><span class="pre">O_from</span></tt> to a reduced Groebner basis
-w.r.t. <tt class="docutils literal"><span class="pre">O_to</span></tt>.</p>
-<p class="rubric">References</p>
-<p>J.C. Faugere, P. Gianni, D. Lazard, T. Mora (1994). Efficient
-Computation of Zero-dimensional Groebner Bases by Change of
-Ordering</p>
-</dd></dl>
-
-<p>Groebner basis algorithms for modules are also provided:</p>
-<dl class="function">
-<dt id="sympy.polys.distributedmodules.sdm_spoly">
-<tt class="descclassname">sympy.polys.distributedmodules.</tt><tt class="descname">sdm_spoly</tt><big>(</big><em>f</em>, <em>g</em>, <em>O</em>, <em>K</em>, <em>phantom=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedmodules.html#sdm_spoly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedmodules.sdm_spoly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the generalized s-polynomial of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p>The ground field is assumed to be <tt class="docutils literal"><span class="pre">K</span></tt>, and monomials ordered according to
-<tt class="docutils literal"><span class="pre">O</span></tt>.</p>
-<p>This is invalid if either of <tt class="docutils literal"><span class="pre">f</span></tt> or <tt class="docutils literal"><span class="pre">g</span></tt> is zero.</p>
-<p>If the leading terms of <span class="math">\(f\)</span> and <span class="math">\(g\)</span> involve different basis elements of
-<span class="math">\(F\)</span>, their s-poly is defined to be zero. Otherwise it is a certain linear
-combination of <span class="math">\(f\)</span> and <span class="math">\(g\)</span> in which the leading terms cancel.
-See [SCA, defn 2.3.6] for details.</p>
-<p>If <tt class="docutils literal"><span class="pre">phantom</span></tt> is not <tt class="docutils literal"><span class="pre">None</span></tt>, it should be a pair of module elements on
-which to perform the same operation(s) as on <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>. The in this
-case both results are returned.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.distributedmodules</span> <span class="kn">import</span> <span class="n">sdm_spoly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys</span> <span class="kn">import</span> <span class="n">QQ</span><span class="p">,</span> <span class="n">lex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="p">[((</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">)),</span> <span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">))]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="p">[((</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">))]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="p">[((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">QQ</span><span class="p">(</span><span class="mi">1</span><span class="p">))]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_spoly</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">lex</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_spoly</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">lex</span><span class="p">,</span> <span class="n">QQ</span><span class="p">)</span>
-<span class="go">[((1, 2, 1), 1/1)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedmodules.sdm_ecart">
-<tt class="descclassname">sympy.polys.distributedmodules.</tt><tt class="descname">sdm_ecart</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedmodules.html#sdm_ecart"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedmodules.sdm_ecart" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the ecart of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p>This is defined to be the difference of the total degree of <span class="math">\(f\)</span> and the
-total degree of the leading monomial of <span class="math">\(f\)</span> [SCA, defn 2.3.7].</p>
-<p>Invalid if f is zero.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.distributedmodules</span> <span class="kn">import</span> <span class="n">sdm_ecart</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_ecart</span><span class="p">([((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="mi">1</span><span class="p">),</span> <span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="mi">1</span><span class="p">)])</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sdm_ecart</span><span class="p">([((</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="mi">1</span><span class="p">),</span> <span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="mi">1</span><span class="p">)])</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedmodules.sdm_nf_mora">
-<tt class="descclassname">sympy.polys.distributedmodules.</tt><tt class="descname">sdm_nf_mora</tt><big>(</big><em>f</em>, <em>G</em>, <em>O</em>, <em>K</em>, <em>phantom=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedmodules.html#sdm_nf_mora"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedmodules.sdm_nf_mora" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute a weak normal form of <tt class="docutils literal"><span class="pre">f</span></tt> with respect to <tt class="docutils literal"><span class="pre">G</span></tt> and order <tt class="docutils literal"><span class="pre">O</span></tt>.</p>
-<p>The ground field is assumed to be <tt class="docutils literal"><span class="pre">K</span></tt>, and monomials ordered according to
-<tt class="docutils literal"><span class="pre">O</span></tt>.</p>
-<p>Weak normal forms are defined in [SCA, defn 2.3.3]. They are not unique.
-This function deterministically computes a weak normal form, depending on
-the order of <span class="math">\(G\)</span>.</p>
-<p>The most important property of a weak normal form is the following: if
-<span class="math">\(R\)</span> is the ring associated with the monomial ordering (if the ordering is
-global, we just have <span class="math">\(R = K[x_1, \dots, x_n]\)</span>, otherwise it is a certain
-localization thereof), <span class="math">\(I\)</span> any ideal of <span class="math">\(R\)</span> and <span class="math">\(G\)</span> a standard basis for
-<span class="math">\(I\)</span>, then for any <span class="math">\(f \in R\)</span>, we have <span class="math">\(f \in I\)</span> if and only if
-<span class="math">\(NF(f | G) = 0\)</span>.</p>
-<p>This is the generalized Mora algorithm for computing weak normal forms with
-respect to arbitrary monomial orders [SCA, algorithm 2.3.9].</p>
-<p>If <tt class="docutils literal"><span class="pre">phantom</span></tt> is not <tt class="docutils literal"><span class="pre">None</span></tt>, it should be a pair of &#8220;phantom&#8221; arguments
-on which to perform the same computations as on <tt class="docutils literal"><span class="pre">f</span></tt>, <tt class="docutils literal"><span class="pre">G</span></tt>, both results
-are then returned.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.distributedmodules.sdm_groebner">
-<tt class="descclassname">sympy.polys.distributedmodules.</tt><tt class="descname">sdm_groebner</tt><big>(</big><em>G</em>, <em>NF</em>, <em>O</em>, <em>K</em>, <em>extended=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/distributedmodules.html#sdm_groebner"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.distributedmodules.sdm_groebner" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute a minimal standard basis of <tt class="docutils literal"><span class="pre">G</span></tt> with respect to order <tt class="docutils literal"><span class="pre">O</span></tt>.</p>
-<p>The algorithm uses a normal form <tt class="docutils literal"><span class="pre">NF</span></tt>, for example <tt class="docutils literal"><span class="pre">sdm_nf_mora</span></tt>.
-The ground field is assumed to be <tt class="docutils literal"><span class="pre">K</span></tt>, and monomials ordered according
-to <tt class="docutils literal"><span class="pre">O</span></tt>.</p>
-<p>Let <span class="math">\(N\)</span> denote the submodule generated by elements of <span class="math">\(G\)</span>. A standard
-basis for <span class="math">\(N\)</span> is a subset <span class="math">\(S\)</span> of <span class="math">\(N\)</span>, such that <span class="math">\(in(S) = in(N)\)</span>, where for
-any subset <span class="math">\(X\)</span> of <span class="math">\(F\)</span>, <span class="math">\(in(X)\)</span> denotes the submodule generated by the
-initial forms of elements of <span class="math">\(X\)</span>. [SCA, defn 2.3.2]</p>
-<p>A standard basis is called minimal if no subset of it is a standard basis.</p>
-<p>One may show that standard bases are always generating sets.</p>
-<p>Minimal standard bases are not unique. This algorithm computes a
-deterministic result, depending on the particular order of <span class="math">\(G\)</span>.</p>
-<p>If <tt class="docutils literal"><span class="pre">extended=True</span></tt>, also compute the transition matrix from the initial
-generators to the groebner basis. That is, return a list of coefficient
-vectors, expressing the elements of the groebner basis in terms of the
-elements of <tt class="docutils literal"><span class="pre">G</span></tt>.</p>
-<p>This functions implements the &#8220;sugar&#8221; strategy, see</p>
-<p>Giovini et al: &#8220;One sugar cube, please&#8221; OR Selection strategies in
-Buchberger algorithm.</p>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="exceptions">
-<h2>Exceptions<a class="headerlink" href="#exceptions" title="Permalink to this headline">¶</a></h2>
-<p>These are exceptions defined by the polynomials module.</p>
-<p>TODO sort and explain</p>
-<dl class="class">
-<dt id="sympy.polys.polyerrors.BasePolynomialError">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">BasePolynomialError</tt><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#BasePolynomialError"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.BasePolynomialError" title="Permalink to this definition">¶</a></dt>
-<dd><p>Base class for polynomial related exceptions.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.ExactQuotientFailed">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">ExactQuotientFailed</tt><big>(</big><em>f</em>, <em>g</em>, <em>dom=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#ExactQuotientFailed"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.ExactQuotientFailed" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.OperationNotSupported">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">OperationNotSupported</tt><big>(</big><em>poly</em>, <em>func</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#OperationNotSupported"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.OperationNotSupported" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.HeuristicGCDFailed">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">HeuristicGCDFailed</tt><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#HeuristicGCDFailed"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.HeuristicGCDFailed" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.HomomorphismFailed">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">HomomorphismFailed</tt><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#HomomorphismFailed"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.HomomorphismFailed" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.IsomorphismFailed">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">IsomorphismFailed</tt><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#IsomorphismFailed"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.IsomorphismFailed" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.ExtraneousFactors">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">ExtraneousFactors</tt><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#ExtraneousFactors"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.ExtraneousFactors" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.EvaluationFailed">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">EvaluationFailed</tt><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#EvaluationFailed"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.EvaluationFailed" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.RefinementFailed">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">RefinementFailed</tt><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#RefinementFailed"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.RefinementFailed" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.CoercionFailed">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">CoercionFailed</tt><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#CoercionFailed"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.CoercionFailed" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.NotInvertible">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">NotInvertible</tt><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#NotInvertible"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.NotInvertible" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.NotReversible">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">NotReversible</tt><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#NotReversible"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.NotReversible" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.NotAlgebraic">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">NotAlgebraic</tt><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#NotAlgebraic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.NotAlgebraic" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.DomainError">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">DomainError</tt><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#DomainError"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.DomainError" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.PolynomialError">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">PolynomialError</tt><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#PolynomialError"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.PolynomialError" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.UnificationFailed">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">UnificationFailed</tt><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#UnificationFailed"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.UnificationFailed" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.GeneratorsNeeded">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">GeneratorsNeeded</tt><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#GeneratorsNeeded"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.GeneratorsNeeded" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.ComputationFailed">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">ComputationFailed</tt><big>(</big><em>func</em>, <em>nargs</em>, <em>exc</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#ComputationFailed"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.ComputationFailed" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.GeneratorsError">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">GeneratorsError</tt><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#GeneratorsError"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.GeneratorsError" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.UnivariatePolynomialError">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">UnivariatePolynomialError</tt><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#UnivariatePolynomialError"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.UnivariatePolynomialError" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.MultivariatePolynomialError">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">MultivariatePolynomialError</tt><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#MultivariatePolynomialError"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.MultivariatePolynomialError" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.PolificationFailed">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">PolificationFailed</tt><big>(</big><em>opt</em>, <em>origs</em>, <em>exprs</em>, <em>seq=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#PolificationFailed"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.PolificationFailed" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.OptionError">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">OptionError</tt><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#OptionError"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.OptionError" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polyerrors.FlagError">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polyerrors.</tt><tt class="descname">FlagError</tt><a class="reference internal" href="../../_modules/sympy/polys/polyerrors.html#FlagError"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyerrors.FlagError" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</div>
-<div class="section" id="undocumented">
-<h2>Undocumented<a class="headerlink" href="#undocumented" title="Permalink to this headline">¶</a></h2>
-<p>Many parts of the polys module are still undocumented, and even where there is
-documentation it is scarce. Please contribute!</p>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../index.html">
-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Internals of the Polynomial Manipulation Module</a><ul>
-<li><a class="reference internal" href="#domains">Domains</a><ul>
-<li><a class="reference internal" href="#abstract-domains">Abstract Domains</a></li>
-<li><a class="reference internal" href="#concrete-domains">Concrete Domains</a></li>
-<li><a class="reference internal" href="#implementation-domains">Implementation Domains</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#level-one">Level One</a></li>
-<li><a class="reference internal" href="#level-zero">Level Zero</a><ul>
-<li><a class="reference internal" href="#manipulation-of-dense-multivariate-polynomials">Manipulation of dense, multivariate polynomials</a></li>
-<li><a class="reference internal" href="#manipulation-of-dense-univariate-polynomials-with-finite-field-coefficients">Manipulation of dense, univariate polynomials with finite field coefficients</a></li>
-<li><a class="reference internal" href="#manipulation-of-sparse-distributed-polynomials-and-vectors">Manipulation of sparse, distributed polynomials and vectors</a></li>
-<li><a class="reference internal" href="#polynomial-factorization-algorithms">Polynomial factorization algorithms</a></li>
-<li><a class="reference internal" href="#groebner-basis-algorithms">Groebner basis algorithms</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#exceptions">Exceptions</a></li>
-<li><a class="reference internal" href="#undocumented">Undocumented</a></li>
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-            
-  <div class="section" id="literature">
-<span id="polys-literature"></span><h1>Literature<a class="headerlink" href="#literature" title="Permalink to this headline">¶</a></h1>
-<p>The following is a non-comprehensive list of publications that were used as
-a theoretical foundation for implementing polynomials manipulation module.</p>
-<table class="docutils citation" frame="void" id="kozen89" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Kozen89]</td><td>D. Kozen, S. Landau, Polynomial decomposition algorithms,
-Journal of Symbolic Computation 7 (1989), pp. 445-456</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="liao95" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Liao95]</td><td>Hsin-Chao Liao,  R. Fateman, Evaluation of the heuristic
-polynomial GCD, International Symposium on Symbolic and Algebraic
-Computation (ISSAC), ACM Press, Montreal, Quebec, Canada, 1995,
-pp. 240&#8211;247</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="gathen99" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Gathen99]</td><td>J. von zur Gathen, J. Gerhard, Modern Computer Algebra,
-First Edition, Cambridge University Press, 1999</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="weisstein09" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Weisstein09]</td><td>Eric W. Weisstein, Cyclotomic Polynomial, From MathWorld - A
-Wolfram Web Resource, <a class="reference external" href="http://mathworld.wolfram.com/CyclotomicPolynomial.html">http://mathworld.wolfram.com/CyclotomicPolynomial.html</a></td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="wang78" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Wang78]</td><td>P. S. Wang, An Improved Multivariate Polynomial Factoring
-Algorithm, Math. of Computation 32, 1978, pp. 1215&#8211;1231</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="geddes92" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Geddes92]</td><td>K. Geddes, S. R. Czapor, G. Labahn, Algorithms for
-Computer Algebra, Springer, 1992</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="monagan93" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Monagan93]</td><td>Michael Monagan, In-place Arithmetic for Polynomials
-over Z_n, Proceedings of DISCO &#8216;92, Springer-Verlag LNCS, 721,
-1993, pp. 22&#8211;34</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="kaltofen98" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Kaltofen98]</td><td>E. Kaltofen, V. Shoup, Subquadratic-time Factoring of
-Polynomials over Finite Fields, Mathematics of Computation, Volume
-67, Issue 223, 1998, pp. 1179&#8211;1197</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="shoup95" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Shoup95]</td><td>V. Shoup, A New Polynomial Factorization Algorithm and
-its Implementation, Journal of Symbolic Computation, Volume 20,
-Issue 4, 1995, pp. 363&#8211;397</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="gathen92" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Gathen92]</td><td>J. von zur Gathen, V. Shoup, Computing Frobenius Maps
-and Factoring Polynomials, ACM Symposium on Theory of Computing,
-1992, pp. 187&#8211;224</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="shoup91" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Shoup91]</td><td>V. Shoup, A Fast Deterministic Algorithm for Factoring
-Polynomials over Finite Fields of Small Characteristic, In Proceedings
-of International Symposium on Symbolic and Algebraic Computation, 1991,
-pp. 14&#8211;21</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="cox97" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Cox97]</td><td>D. Cox, J. Little, D. O&#8217;Shea, Ideals, Varieties and
-Algorithms, Springer, Second Edition, 1997</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="ajwa95" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Ajwa95]</td><td>I.A. Ajwa, Z. Liu, P.S. Wang, Groebner Bases Algorithm,
-<a class="reference external" href="http://citeseer.ist.psu.edu/ajwa95grbner.html">http://citeseer.ist.psu.edu/ajwa95grbner.html</a>, 1995</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="bose03" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Bose03]</td><td>N.K. Bose, B. Buchberger, J.P. Guiver, Multidimensional
-Systems Theory and Applications, Springer, 2003</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="giovini91" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Giovini91]</td><td>A. Giovini, T. Mora, &#8220;One sugar cube, please&#8221; or
-Selection strategies in Buchberger algorithm, ISSAC &#8216;91, ACM</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="bronstein93" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Bronstein93]</td><td>M. Bronstein, B. Salvy, Full partial fraction
-decomposition of rational functions, Proceedings ISSAC &#8216;93,
-ACM Press, Kiev, Ukraine, 1993, pp. 157&#8211;160</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="buchberger01" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Buchberger01]</td><td>B. Buchberger, Groebner Bases: A Short Introduction for
-Systems Theorists,  In: R. Moreno-Diaz,  B. Buchberger,
-J. L. Freire, Proceedings of EUROCAST&#8216;01, February, 2001</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="davenport88" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Davenport88]</td><td>J.H. Davenport, Y. Siret, E. Tournier, Computer Algebra
-Systems and Algorithms for Algebraic Computation, Academic Press, London,
-1988, pp. 124&#8211;128</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="greuel2008" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Greuel2008]</td><td>G.-M. Greuel, Gerhard Pfister, A Singular Introduction to
-Commutative Algebra, Springer, 2008</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="atiyah69" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[Atiyah69]</td><td>M.F. Atiyah, I.G. MacDonald, Introduction to Commutative Algebra,
-Addison-Wesley, 1969</td></tr>
-</tbody>
-</table>
-</div>
-
-
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-<span id="polys-reference"></span><h1>Polynomials Manipulation Module Reference<a class="headerlink" href="#polynomials-manipulation-module-reference" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="basic-polynomial-manipulation-functions">
-<h2>Basic polynomial manipulation functions<a class="headerlink" href="#basic-polynomial-manipulation-functions" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.polys.polytools.poly">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">poly</tt><big>(</big><em>expr</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Efficiently transform an expression into a polynomial.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">poly</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">Poly(x**5 + 2*x**4 - x**3 - 2*x**2 + x, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.poly_from_expr">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">poly_from_expr</tt><big>(</big><em>expr</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#poly_from_expr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.poly_from_expr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Construct a polynomial from an expression.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.parallel_poly_from_expr">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">parallel_poly_from_expr</tt><big>(</big><em>exprs</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#parallel_poly_from_expr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.parallel_poly_from_expr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Construct polynomials from expressions.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.degree">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">degree</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#degree"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.degree" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the degree of <tt class="docutils literal"><span class="pre">f</span></tt> in the given variable.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">degree</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">degree</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">gen</span><span class="o">=</span><span class="n">x</span><span class="p">)</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">degree</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">gen</span><span class="o">=</span><span class="n">y</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.degree_list">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">degree_list</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#degree_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.degree_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a list of degrees of <tt class="docutils literal"><span class="pre">f</span></tt> in all variables.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">degree_list</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">degree_list</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">(2, 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.LC">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">LC</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#LC"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.LC" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the leading coefficient of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">LC</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">LC</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
-<span class="go">4</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.LM">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">LM</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#LM"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.LM" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the leading monomial of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">LM</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">LM</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
-<span class="go">x**2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.LT">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">LT</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#LT"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.LT" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the leading term of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">LT</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">LT</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
-<span class="go">4*x**2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.pdiv">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">pdiv</tt><big>(</big><em>f</em>, <em>g</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#pdiv"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.pdiv" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute polynomial pseudo-division of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pdiv</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pdiv</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">(2*x + 4, 20)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.prem">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">prem</tt><big>(</big><em>f</em>, <em>g</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#prem"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.prem" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute polynomial pseudo-remainder of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">prem</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">prem</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">20</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.pquo">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">pquo</tt><big>(</big><em>f</em>, <em>g</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#pquo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.pquo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute polynomial pseudo-quotient of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pquo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pquo</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">2*x + 4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pquo</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">2*x + 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.pexquo">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">pexquo</tt><big>(</big><em>f</em>, <em>g</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#pexquo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.pexquo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute polynomial exact pseudo-quotient of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pexquo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pexquo</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">2*x + 2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pexquo</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">ExactQuotientFailed</span>: <span class="n">2*x - 4 does not divide x**2 + 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.div">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">div</tt><big>(</big><em>f</em>, <em>g</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#div"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.div" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute polynomial division of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">div</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">div</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="n">ZZ</span><span class="p">)</span>
-<span class="go">(0, x**2 + 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">div</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="n">QQ</span><span class="p">)</span>
-<span class="go">(x/2 + 1, 5)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.rem">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">rem</tt><big>(</big><em>f</em>, <em>g</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#rem"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.rem" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute polynomial remainder of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">rem</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rem</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="n">ZZ</span><span class="p">)</span>
-<span class="go">x**2 + 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rem</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="n">QQ</span><span class="p">)</span>
-<span class="go">5</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.quo">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">quo</tt><big>(</big><em>f</em>, <em>g</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#quo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.quo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute polynomial quotient of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">quo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">quo</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">x/2 + 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">quo</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">x + 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.exquo">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">exquo</tt><big>(</big><em>f</em>, <em>g</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#exquo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.exquo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute polynomial exact quotient of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exquo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">exquo</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">x + 1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">exquo</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">ExactQuotientFailed</span>: <span class="n">2*x - 4 does not divide x**2 + 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.half_gcdex">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">half_gcdex</tt><big>(</big><em>f</em>, <em>g</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#half_gcdex"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.half_gcdex" title="Permalink to this definition">¶</a></dt>
-<dd><p>Half extended Euclidean algorithm of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p>Returns <tt class="docutils literal"><span class="pre">(s,</span> <span class="pre">h)</span></tt> such that <tt class="docutils literal"><span class="pre">h</span> <span class="pre">=</span> <span class="pre">gcd(f,</span> <span class="pre">g)</span></tt> and <tt class="docutils literal"><span class="pre">s*f</span> <span class="pre">=</span> <span class="pre">h</span> <span class="pre">(mod</span> <span class="pre">g)</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">half_gcdex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">half_gcdex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">6</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">12</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">15</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">(-x/5 + 3/5, x + 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.gcdex">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">gcdex</tt><big>(</big><em>f</em>, <em>g</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#gcdex"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.gcdex" title="Permalink to this definition">¶</a></dt>
-<dd><p>Extended Euclidean algorithm of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p>Returns <tt class="docutils literal"><span class="pre">(s,</span> <span class="pre">t,</span> <span class="pre">h)</span></tt> such that <tt class="docutils literal"><span class="pre">h</span> <span class="pre">=</span> <span class="pre">gcd(f,</span> <span class="pre">g)</span></tt> and <tt class="docutils literal"><span class="pre">s*f</span> <span class="pre">+</span> <span class="pre">t*g</span> <span class="pre">=</span> <span class="pre">h</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">gcdex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gcdex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">6</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">12</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">15</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">(-x/5 + 3/5, x**2/5 - 6*x/5 + 2, x + 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.invert">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">invert</tt><big>(</big><em>f</em>, <em>g</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#invert"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.invert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Invert <tt class="docutils literal"><span class="pre">f</span></tt> modulo <tt class="docutils literal"><span class="pre">g</span></tt> when possible.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">invert</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">invert</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">-4/3</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">invert</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">NotInvertible</span>: <span class="n">zero divisor</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.subresultants">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">subresultants</tt><big>(</big><em>f</em>, <em>g</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#subresultants"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.subresultants" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute subresultant PRS of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">subresultants</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">subresultants</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">[x**2 + 1, x**2 - 1, -2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.resultant">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">resultant</tt><big>(</big><em>f</em>, <em>g</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#resultant"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.resultant" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute resultant of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">resultant</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">resultant</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">4</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.discriminant">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">discriminant</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#discriminant"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.discriminant" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute discriminant of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">discriminant</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">discriminant</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">-8</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.terms_gcd">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">terms_gcd</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#terms_gcd"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.terms_gcd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Remove GCD of terms from <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p>If the <tt class="docutils literal"><span class="pre">deep</span></tt> flag is True, then the arguments of <tt class="docutils literal"><span class="pre">f</span></tt> will have
-terms_gcd applied to them.</p>
-<p>If a fraction is factored out of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">f</span></tt> is an Add, then
-an unevaluated Mul will be returned so that automatic simplification
-does not redistribute it. The hint <tt class="docutils literal"><span class="pre">clear</span></tt>, when set to False, can be
-used to prevent such factoring when all coefficients are not fractions.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="../core.html#sympy.core.exprtools.gcd_terms" title="sympy.core.exprtools.gcd_terms"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.core.exprtools.gcd_terms</span></tt></a>, <a class="reference internal" href="../core.html#sympy.core.exprtools.factor_terms" title="sympy.core.exprtools.factor_terms"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.core.exprtools.factor_terms</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">terms_gcd</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">terms_gcd</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">6</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">x**3*y*(x**3*y + 1)</span>
-</pre></div>
-</div>
-<p>The default action of polys routines is to expand the expression
-given to them. terms_gcd follows this behavior:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">terms_gcd</span><span class="p">((</span><span class="mi">3</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">))</span>
-<span class="go">3*x*(x*y + x + y + 1)</span>
-</pre></div>
-</div>
-<p>If this is not desired then the hint <tt class="docutils literal"><span class="pre">expand</span></tt> can be set to False.
-In this case the expression will be treated as though it were comprised
-of one or more terms:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">terms_gcd</span><span class="p">((</span><span class="mi">3</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">),</span> <span class="n">expand</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">(3*x + 3)*(x*y + x)</span>
-</pre></div>
-</div>
-<p>In order to traverse factors of a Mul or the arguments of other
-functions, the <tt class="docutils literal"><span class="pre">deep</span></tt> hint can be used:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">terms_gcd</span><span class="p">((</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">),</span> <span class="n">expand</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">3*x*(x + 1)*(y + 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">terms_gcd</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">),</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">cos(x*(y + 1))</span>
-</pre></div>
-</div>
-<p>Rationals are factored out by default:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">terms_gcd</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(2*x + y)/2</span>
-</pre></div>
-</div>
-<p>Only the y-term had a coefficient that was a fraction; if one
-does not want to factor out the 1/2 in cases like this, the
-flag <tt class="docutils literal"><span class="pre">clear</span></tt> can be set to False:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">terms_gcd</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">clear</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">x + y/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">terms_gcd</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">clear</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">y*(x/2 + y)</span>
-</pre></div>
-</div>
-<p>The <tt class="docutils literal"><span class="pre">clear</span></tt> flag is ignored if all coefficients are fractions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">terms_gcd</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="mi">3</span> <span class="o">+</span> <span class="n">y</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">clear</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">(2*x + 3*y)/6</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.cofactors">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">cofactors</tt><big>(</big><em>f</em>, <em>g</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#cofactors"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.cofactors" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute GCD and cofactors of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p>Returns polynomials <tt class="docutils literal"><span class="pre">(h,</span> <span class="pre">cff,</span> <span class="pre">cfg)</span></tt> such that <tt class="docutils literal"><span class="pre">h</span> <span class="pre">=</span> <span class="pre">gcd(f,</span> <span class="pre">g)</span></tt>, and
-<tt class="docutils literal"><span class="pre">cff</span> <span class="pre">=</span> <span class="pre">quo(f,</span> <span class="pre">h)</span></tt> and <tt class="docutils literal"><span class="pre">cfg</span> <span class="pre">=</span> <span class="pre">quo(g,</span> <span class="pre">h)</span></tt> are, so called, cofactors
-of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">cofactors</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">cofactors</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">(x - 1, x + 1, x - 2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.gcd">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">gcd</tt><big>(</big><em>f</em>, <em>g=None</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#gcd"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.gcd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute GCD of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">gcd</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gcd</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">x - 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.gcd_list">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">gcd_list</tt><big>(</big><em>seq</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#gcd_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.gcd_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute GCD of a list of polynomials.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">gcd_list</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gcd_list</span><span class="p">([</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">])</span>
-<span class="go">x - 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.lcm">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">lcm</tt><big>(</big><em>f</em>, <em>g=None</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#lcm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.lcm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute LCM of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">lcm</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">lcm</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">x**3 - 2*x**2 - x + 2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.lcm_list">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">lcm_list</tt><big>(</big><em>seq</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#lcm_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.lcm_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute LCM of a list of polynomials.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">lcm_list</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">lcm_list</span><span class="p">([</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">])</span>
-<span class="go">x**5 - x**4 - 2*x**3 - x**2 + x + 2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.trunc">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">trunc</tt><big>(</big><em>f</em>, <em>p</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#trunc"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.trunc" title="Permalink to this definition">¶</a></dt>
-<dd><p>Reduce <tt class="docutils literal"><span class="pre">f</span></tt> modulo a constant <tt class="docutils literal"><span class="pre">p</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">trunc</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">trunc</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">5</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">-x**3 - x + 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.monic">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">monic</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#monic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.monic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Divide all coefficients of <tt class="docutils literal"><span class="pre">f</span></tt> by <tt class="docutils literal"><span class="pre">LC(f)</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">monic</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">monic</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">x**2 + 4*x/3 + 2/3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.content">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">content</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#content"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.content" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute GCD of coefficients of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">content</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">content</span><span class="p">(</span><span class="mi">6</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">8</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">12</span><span class="p">)</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.primitive">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">primitive</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#primitive"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.primitive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute content and the primitive form of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.polytools</span> <span class="kn">import</span> <span class="n">primitive</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">primitive</span><span class="p">(</span><span class="mi">6</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">8</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">12</span><span class="p">)</span>
-<span class="go">(2, 3*x**2 + 4*x + 6)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span>
-</pre></div>
-</div>
-<p>Expansion is performed by default:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">primitive</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span>
-<span class="go">(2, x**2 + x + 1)</span>
-</pre></div>
-</div>
-<p>Set <tt class="docutils literal"><span class="pre">expand</span></tt> to False to shut this off. Note that the
-extraction will not be recursive; use the as_content_primitive method
-for recursive, non-destructive Rational extraction.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">primitive</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">expand</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">(1, x*(2*x + 2) + 2)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span><span class="o">.</span><span class="n">as_content_primitive</span><span class="p">()</span>
-<span class="go">(2, x*(x + 1) + 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.compose">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">compose</tt><big>(</big><em>f</em>, <em>g</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#compose"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.compose" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute functional composition <tt class="docutils literal"><span class="pre">f(g)</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">compose</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">compose</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="p">,</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">x**2 - x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.decompose">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">decompose</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#decompose"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.decompose" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute functional decomposition of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">decompose</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">decompose</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">[x**2 - x - 1, x**2 + x]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.sturm">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">sturm</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#sturm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.sturm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute Sturm sequence of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sturm</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sturm</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">[x**3 - 2*x**2 + x - 3, 3*x**2 - 4*x + 1, 2*x/9 + 25/9, -2079/4]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.gff_list">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">gff_list</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#gff_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.gff_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute a list of greatest factorial factors of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">gff_list</span><span class="p">,</span> <span class="n">ff</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">5</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gff_list</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-<span class="go">[(x, 1), (x + 2, 4)]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">ff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">ff</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span> <span class="o">==</span> <span class="n">f</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.gff">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">gff</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#gff"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.gff" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute greatest factorial factorization of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.sqf_norm">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">sqf_norm</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#sqf_norm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.sqf_norm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute square-free norm of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p>Returns <tt class="docutils literal"><span class="pre">s</span></tt>, <tt class="docutils literal"><span class="pre">f</span></tt>, <tt class="docutils literal"><span class="pre">r</span></tt>, such that <tt class="docutils literal"><span class="pre">g(x)</span> <span class="pre">=</span> <span class="pre">f(x-sa)</span></tt> and
-<tt class="docutils literal"><span class="pre">r(x)</span> <span class="pre">=</span> <span class="pre">Norm(g(x))</span></tt> is a square-free polynomial over <tt class="docutils literal"><span class="pre">K</span></tt>,
-where <tt class="docutils literal"><span class="pre">a</span></tt> is the algebraic extension of the ground domain.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqf_norm</span><span class="p">,</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sqf_norm</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">extension</span><span class="o">=</span><span class="p">[</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)])</span>
-<span class="go">(1, x**2 - 2*sqrt(3)*x + 4, x**4 - 4*x**2 + 16)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.sqf_part">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">sqf_part</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#sqf_part"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.sqf_part" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute square-free part of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqf_part</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sqf_part</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">x**2 - x - 2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.sqf_list">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">sqf_list</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#sqf_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.sqf_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute a list of square-free factors of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqf_list</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sqf_list</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">5</span> <span class="o">+</span> <span class="mi">16</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">+</span> <span class="mi">50</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">76</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">56</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">16</span><span class="p">)</span>
-<span class="go">(2, [(x + 1, 2), (x + 2, 3)])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.sqf">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">sqf</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#sqf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.sqf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute square-free factorization of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqf</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sqf</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">5</span> <span class="o">+</span> <span class="mi">16</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">+</span> <span class="mi">50</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">76</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">56</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">16</span><span class="p">)</span>
-<span class="go">2*(x + 1)**2*(x + 2)**3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.factor_list">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">factor_list</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#factor_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.factor_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute a list of irreducible factors of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factor_list</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor_list</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">5</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
-<span class="go">(2, [(x + y, 1), (x**2 + 1, 2)])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.factor">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">factor</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#factor"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.factor" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the factorization of <tt class="docutils literal"><span class="pre">f</span></tt> into irreducibles. (Use factorint to
-factor an integer.)</p>
-<p>There two modes implemented: symbolic and formal. If <tt class="docutils literal"><span class="pre">f</span></tt> is not an
-instance of <a class="reference internal" href="#sympy.polys.polytools.Poly" title="sympy.polys.polytools.Poly"><tt class="xref py py-class docutils literal"><span class="pre">Poly</span></tt></a> and generators are not specified, then the
-former mode is used. Otherwise, the formal mode is used.</p>
-<p>In symbolic mode, <a class="reference internal" href="#sympy.polys.polytools.factor" title="sympy.polys.polytools.factor"><tt class="xref py py-func docutils literal"><span class="pre">factor()</span></tt></a> will traverse the expression tree and
-factor its components without any prior expansion, unless an instance
-of <tt class="xref py py-class docutils literal"><span class="pre">Add</span></tt> is encountered (in this case formal factorization is
-used). This way <a class="reference internal" href="#sympy.polys.polytools.factor" title="sympy.polys.polytools.factor"><tt class="xref py py-func docutils literal"><span class="pre">factor()</span></tt></a> can handle large or symbolic exponents.</p>
-<p>By default, the factorization is computed over the rationals. To factor
-over other domain, e.g. an algebraic or finite field, use appropriate
-options: <tt class="docutils literal"><span class="pre">extension</span></tt>, <tt class="docutils literal"><span class="pre">modulus</span></tt> or <tt class="docutils literal"><span class="pre">domain</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factor</span><span class="p">,</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">5</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
-<span class="go">2*(x + y)*(x**2 + 1)**2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">x**2 + 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">modulus</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(x + 1)**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">gaussian</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(x - I)*(x + I)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">extension</span><span class="o">=</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">(x - sqrt(2))*(x + sqrt(2))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">((</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">4</span><span class="p">))</span>
-<span class="go">(x - 1)*(x + 1)/(x + 2)**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">((</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">4</span><span class="p">)</span><span class="o">**</span><span class="mi">10000000</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">(x + 2)**20000000*(x**2 + 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.intervals">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">intervals</tt><big>(</big><em>F</em>, <em>all=False</em>, <em>eps=None</em>, <em>inf=None</em>, <em>sup=None</em>, <em>strict=False</em>, <em>fast=False</em>, <em>sqf=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#intervals"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.intervals" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute isolating intervals for roots of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">intervals</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">intervals</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">[((-2, -1), 1), ((1, 2), 1)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">intervals</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="mf">1e-2</span><span class="p">)</span>
-<span class="go">[((-26/15, -19/11), 1), ((19/11, 26/15), 1)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.refine_root">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">refine_root</tt><big>(</big><em>f</em>, <em>s</em>, <em>t</em>, <em>eps=None</em>, <em>steps=None</em>, <em>fast=False</em>, <em>check_sqf=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#refine_root"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.refine_root" title="Permalink to this definition">¶</a></dt>
-<dd><p>Refine an isolating interval of a root to the given precision.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">refine_root</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">refine_root</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="mf">1e-2</span><span class="p">)</span>
-<span class="go">(19/11, 26/15)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.count_roots">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">count_roots</tt><big>(</big><em>f</em>, <em>inf=None</em>, <em>sup=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#count_roots"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.count_roots" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the number of roots of <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">[inf,</span> <span class="pre">sup]</span></tt> interval.</p>
-<p>If one of <tt class="docutils literal"><span class="pre">inf</span></tt> or <tt class="docutils literal"><span class="pre">sup</span></tt> is complex, it will return the number of roots
-in the complex rectangle with corners at <tt class="docutils literal"><span class="pre">inf</span></tt> and <tt class="docutils literal"><span class="pre">sup</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">count_roots</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">count_roots</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">count_roots</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">I</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.real_roots">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">real_roots</tt><big>(</big><em>f</em>, <em>multiple=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#real_roots"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.real_roots" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a list of real roots with multiplicities of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">real_roots</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">real_roots</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">7</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">[-1/2, 2, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.nroots">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">nroots</tt><big>(</big><em>f</em>, <em>n=15</em>, <em>maxsteps=50</em>, <em>cleanup=True</em>, <em>error=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#nroots"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.nroots" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute numerical approximations of roots of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">nroots</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nroots</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-<span class="go">[-1.73205080756888, 1.73205080756888]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nroots</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="mi">30</span><span class="p">)</span>
-<span class="go">[-1.73205080756887729352744634151, 1.73205080756887729352744634151]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.ground_roots">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">ground_roots</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#ground_roots"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.ground_roots" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute roots of <tt class="docutils literal"><span class="pre">f</span></tt> by factorization in the ground domain.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">ground_roots</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ground_roots</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">6</span> <span class="o">-</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">{0: 2, 1: 2}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.nth_power_roots_poly">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">nth_power_roots_poly</tt><big>(</big><em>f</em>, <em>n</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#nth_power_roots_poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.nth_power_roots_poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Construct a polynomial with n-th powers of roots of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">nth_power_roots_poly</span><span class="p">,</span> <span class="n">factor</span><span class="p">,</span> <span class="n">roots</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">factor</span><span class="p">(</span><span class="n">nth_power_roots_poly</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">g</span>
-<span class="go">(x**2 - x + 1)**2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">R_f</span> <span class="o">=</span> <span class="p">[</span> <span class="p">(</span><span class="n">r</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span> <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">roots</span><span class="p">(</span><span class="n">f</span><span class="p">)</span> <span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R_g</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">roots</span><span class="p">(</span><span class="n">g</span><span class="p">)</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">set</span><span class="p">(</span><span class="n">R_f</span><span class="p">)</span> <span class="o">==</span> <span class="nb">set</span><span class="p">(</span><span class="n">R_g</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.cancel">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">cancel</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#cancel"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.cancel" title="Permalink to this definition">¶</a></dt>
-<dd><p>Cancel common factors in a rational function <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">cancel</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">cancel</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">(2*x + 2)/(x - 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.reduced">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">reduced</tt><big>(</big><em>f</em>, <em>G</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#reduced"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.reduced" title="Permalink to this definition">¶</a></dt>
-<dd><p>Reduces a polynomial <tt class="docutils literal"><span class="pre">f</span></tt> modulo a set of polynomials <tt class="docutils literal"><span class="pre">G</span></tt>.</p>
-<p>Given a polynomial <tt class="docutils literal"><span class="pre">f</span></tt> and a set of polynomials <tt class="docutils literal"><span class="pre">G</span> <span class="pre">=</span> <span class="pre">(g_1,</span> <span class="pre">...,</span> <span class="pre">g_n)</span></tt>,
-computes a set of quotients <tt class="docutils literal"><span class="pre">q</span> <span class="pre">=</span> <span class="pre">(q_1,</span> <span class="pre">...,</span> <span class="pre">q_n)</span></tt> and the remainder <tt class="docutils literal"><span class="pre">r</span></tt>
-such that <tt class="docutils literal"><span class="pre">f</span> <span class="pre">=</span> <span class="pre">q_1*f_1</span> <span class="pre">+</span> <span class="pre">...</span> <span class="pre">+</span> <span class="pre">q_n*f_n</span> <span class="pre">+</span> <span class="pre">r</span></tt>, where <tt class="docutils literal"><span class="pre">r</span></tt> vanishes or <tt class="docutils literal"><span class="pre">r</span></tt>
-is a completely reduced polynomial with respect to <tt class="docutils literal"><span class="pre">G</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">reduced</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">reduced</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="n">y</span><span class="p">])</span>
-<span class="go">([2*x, 1], x**2 + y**2 + y)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.groebner">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">groebner</tt><big>(</big><em>F</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#groebner"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.groebner" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the reduced Groebner basis for a set of polynomials.</p>
-<p>Use the <tt class="docutils literal"><span class="pre">order</span></tt> argument to set the monomial ordering that will be
-used to compute the basis. Allowed orders are <tt class="docutils literal"><span class="pre">lex</span></tt>, <tt class="docutils literal"><span class="pre">grlex</span></tt> and
-<tt class="docutils literal"><span class="pre">grevlex</span></tt>. If no order is specified, it defaults to <tt class="docutils literal"><span class="pre">lex</span></tt>.</p>
-<p>For more information on Groebner bases, see the references and the docstring
-of <span class="math">\(solve_poly_system()\)</span>.</p>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference internal" href="../polys/literature.html#buchberger01">[Buchberger01]</a></li>
-<li><a class="reference internal" href="../polys/literature.html#cox97">[Cox97]</a></li>
-</ol>
-<p class="rubric">Examples</p>
-<p>Example taken from [1].</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">groebner</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">groebner</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s">&#39;lex&#39;</span><span class="p">)</span>
-<span class="go">GroebnerBasis([x**2 - 2*y**2, x*y - 2*y, y**3 - 2*y], x, y,</span>
-<span class="go">              domain=&#39;ZZ&#39;, order=&#39;lex&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">groebner</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s">&#39;grlex&#39;</span><span class="p">)</span>
-<span class="go">GroebnerBasis([y**3 - 2*y, x**2 - 2*y**2, x*y - 2*y], x, y,</span>
-<span class="go">              domain=&#39;ZZ&#39;, order=&#39;grlex&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">groebner</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s">&#39;grevlex&#39;</span><span class="p">)</span>
-<span class="go">GroebnerBasis([y**3 - 2*y, x**2 - 2*y**2, x*y - 2*y], x, y,</span>
-<span class="go">              domain=&#39;ZZ&#39;, order=&#39;grevlex&#39;)</span>
-</pre></div>
-</div>
-<p>By default, an improved implementation of the Buchberger algorithm is
-used. Optionally, an implementation of the F5B algorithm can be used.
-The algorithm can be set using <tt class="docutils literal"><span class="pre">method</span></tt> flag or with the <tt class="xref py py-func docutils literal"><span class="pre">setup()</span></tt>
-function from <tt class="xref py py-mod docutils literal"><span class="pre">sympy.polys.polyconfig</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">y</span> <span class="o">-</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">10</span> <span class="o">-</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">10</span><span class="p">)]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">groebner</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;buchberger&#39;</span><span class="p">)</span>
-<span class="go">GroebnerBasis([x**2 - x - 1, y - 55], x, y, domain=&#39;ZZ&#39;, order=&#39;lex&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">groebner</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s">&#39;f5b&#39;</span><span class="p">)</span>
-<span class="go">GroebnerBasis([x**2 - x - 1, y - 55], x, y, domain=&#39;ZZ&#39;, order=&#39;lex&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.is_zero_dimensional">
-<tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">is_zero_dimensional</tt><big>(</big><em>F</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#is_zero_dimensional"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.is_zero_dimensional" title="Permalink to this definition">¶</a></dt>
-<dd><p>Checks if the ideal generated by a Groebner basis is zero-dimensional.</p>
-<p>The algorithm checks if the set of monomials not divisible by the
-leading monomial of any element of <tt class="docutils literal"><span class="pre">F</span></tt> is bounded.</p>
-<p class="rubric">References</p>
-<p>David A. Cox, John B. Little, Donal O&#8217;Shea. Ideals, Varieties and
-Algorithms, 3rd edition, p. 230</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polytools.Poly">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">Poly</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generic class for representing polynomial expressions.</p>
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.EC">
-<tt class="descname">EC</tt><big>(</big><em>f</em>, <em>order=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.EC"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.EC" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the last non-zero coefficient of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">EC</span><span class="p">()</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.EM">
-<tt class="descname">EM</tt><big>(</big><em>f</em>, <em>order=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.EM"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.EM" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the last non-zero monomial of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">EM</span><span class="p">()</span>
-<span class="go">x**0*y**1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.ET">
-<tt class="descname">ET</tt><big>(</big><em>f</em>, <em>order=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.ET"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.ET" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the last non-zero term of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">ET</span><span class="p">()</span>
-<span class="go">(x**0*y**1, 3)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.LC">
-<tt class="descname">LC</tt><big>(</big><em>f</em>, <em>order=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.LC"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.LC" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the leading coefficient of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">LC</span><span class="p">()</span>
-<span class="go">4</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.LM">
-<tt class="descname">LM</tt><big>(</big><em>f</em>, <em>order=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.LM"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.LM" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the leading monomial of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">LM</span><span class="p">()</span>
-<span class="go">x**2*y**0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.LT">
-<tt class="descname">LT</tt><big>(</big><em>f</em>, <em>order=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.LT"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.LT" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the leading term of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">LT</span><span class="p">()</span>
-<span class="go">(x**2*y**0, 4)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.TC">
-<tt class="descname">TC</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.TC"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.TC" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the trailing coefficient of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">TC</span><span class="p">()</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.abs">
-<tt class="descname">abs</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.abs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.abs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Make all coefficients in <tt class="docutils literal"><span class="pre">f</span></tt> positive.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">abs</span><span class="p">()</span>
-<span class="go">Poly(x**2 + 1, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.add">
-<tt class="descname">add</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Add two polynomials <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Poly(x**2 + x - 1, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">Poly</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">Poly(x**2 + x - 1, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.add_ground">
-<tt class="descname">add_ground</tt><big>(</big><em>f</em>, <em>coeff</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.add_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.add_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Add an element of the ground domain to <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">add_ground</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">Poly(x + 3, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.all_coeffs">
-<tt class="descname">all_coeffs</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.all_coeffs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.all_coeffs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns all coefficients from a univariate polynomial <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">all_coeffs</span><span class="p">()</span>
-<span class="go">[1, 0, 2, -1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.all_monoms">
-<tt class="descname">all_monoms</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.all_monoms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.all_monoms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns all monomials from a univariate polynomial <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">all_monoms</span><span class="p">()</span>
-<span class="go">[(3,), (2,), (1,), (0,)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.all_roots">
-<tt class="descname">all_roots</tt><big>(</big><em>f</em>, <em>multiple=True</em>, <em>radicals=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.all_roots"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.all_roots" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a list of real and complex roots with multiplicities.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">7</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">all_roots</span><span class="p">()</span>
-<span class="go">[-1/2, 2, 2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">all_roots</span><span class="p">()</span>
-<span class="go">[RootOf(x**3 + x + 1, 0),</span>
-<span class="go"> RootOf(x**3 + x + 1, 1),</span>
-<span class="go"> RootOf(x**3 + x + 1, 2)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.all_terms">
-<tt class="descname">all_terms</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.all_terms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.all_terms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns all terms from a univariate polynomial <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">all_terms</span><span class="p">()</span>
-<span class="go">[((3,), 1), ((2,), 0), ((1,), 2), ((0,), -1)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.Poly.args">
-<tt class="descname">args</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.args"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.args" title="Permalink to this definition">¶</a></dt>
-<dd><p>Don&#8217;t mess up with the core.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">args</span>
-<span class="go">(x**2 + 1,)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.as_dict">
-<tt class="descname">as_dict</tt><big>(</big><em>f</em>, <em>native=False</em>, <em>zero=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.as_dict"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.as_dict" title="Permalink to this definition">¶</a></dt>
-<dd><p>Switch to a <tt class="docutils literal"><span class="pre">dict</span></tt> representation.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">as_dict</span><span class="p">()</span>
-<span class="go">{(0, 1): -1, (1, 2): 2, (2, 0): 1}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.as_expr">
-<tt class="descname">as_expr</tt><big>(</big><em>f</em>, <em>*gens</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.as_expr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.as_expr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a polynomial an expression.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">()</span>
-<span class="go">x**2 + 2*x*y**2 - y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">({</span><span class="n">x</span><span class="p">:</span> <span class="mi">5</span><span class="p">})</span>
-<span class="go">10*y**2 - y + 25</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">as_expr</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
-<span class="go">379</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.as_list">
-<tt class="descname">as_list</tt><big>(</big><em>f</em>, <em>native=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.as_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.as_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Switch to a <tt class="docutils literal"><span class="pre">list</span></tt> representation.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.cancel">
-<tt class="descname">cancel</tt><big>(</big><em>f</em>, <em>g</em>, <em>include=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.cancel"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.cancel" title="Permalink to this definition">¶</a></dt>
-<dd><p>Cancel common factors in a rational function <tt class="docutils literal"><span class="pre">f/g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">cancel</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">(1, Poly(2*x + 2, x, domain=&#39;ZZ&#39;), Poly(x - 1, x, domain=&#39;ZZ&#39;))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">cancel</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">include</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(Poly(2*x + 2, x, domain=&#39;ZZ&#39;), Poly(x - 1, x, domain=&#39;ZZ&#39;))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.clear_denoms">
-<tt class="descname">clear_denoms</tt><big>(</big><em>f</em>, <em>convert=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.clear_denoms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.clear_denoms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Clear denominators, but keep the ground domain.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="n">QQ</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">clear_denoms</span><span class="p">()</span>
-<span class="go">(6, Poly(3*x + 2, x, domain=&#39;QQ&#39;))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">clear_denoms</span><span class="p">(</span><span class="n">convert</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(6, Poly(3*x + 2, x, domain=&#39;ZZ&#39;))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.coeffs">
-<tt class="descname">coeffs</tt><big>(</big><em>f</em>, <em>order=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.coeffs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.coeffs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns all non-zero coefficients from <tt class="docutils literal"><span class="pre">f</span></tt> in lex order.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">coeffs</span><span class="p">()</span>
-<span class="go">[1, 2, 3]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.cofactors">
-<tt class="descname">cofactors</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.cofactors"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.cofactors" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the GCD of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt> and their cofactors.</p>
-<p>Returns polynomials <tt class="docutils literal"><span class="pre">(h,</span> <span class="pre">cff,</span> <span class="pre">cfg)</span></tt> such that <tt class="docutils literal"><span class="pre">h</span> <span class="pre">=</span> <span class="pre">gcd(f,</span> <span class="pre">g)</span></tt>, and
-<tt class="docutils literal"><span class="pre">cff</span> <span class="pre">=</span> <span class="pre">quo(f,</span> <span class="pre">h)</span></tt> and <tt class="docutils literal"><span class="pre">cfg</span> <span class="pre">=</span> <span class="pre">quo(g,</span> <span class="pre">h)</span></tt> are, so called, cofactors
-of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">cofactors</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">(Poly(x - 1, x, domain=&#39;ZZ&#39;),</span>
-<span class="go"> Poly(x + 1, x, domain=&#39;ZZ&#39;),</span>
-<span class="go"> Poly(x - 2, x, domain=&#39;ZZ&#39;))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.compose">
-<tt class="descname">compose</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.compose"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.compose" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the functional composition of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">compose</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Poly(x**2 - x, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.content">
-<tt class="descname">content</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.content"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.content" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the GCD of polynomial coefficients.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">6</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">8</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">12</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">content</span><span class="p">()</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.count_roots">
-<tt class="descname">count_roots</tt><big>(</big><em>f</em>, <em>inf=None</em>, <em>sup=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.count_roots"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.count_roots" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the number of roots of <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">[inf,</span> <span class="pre">sup]</span></tt> interval.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">count_roots</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">count_roots</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">I</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.decompose">
-<tt class="descname">decompose</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.decompose"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.decompose" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes a functional decomposition of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="s">&#39;ZZ&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">decompose</span><span class="p">()</span>
-<span class="go">[Poly(x**2 - x - 1, x, domain=&#39;ZZ&#39;), Poly(x**2 + x, x, domain=&#39;ZZ&#39;)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.deflate">
-<tt class="descname">deflate</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.deflate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.deflate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Reduce degree of <tt class="docutils literal"><span class="pre">f</span></tt> by mapping <tt class="docutils literal"><span class="pre">x_i**m</span></tt> to <tt class="docutils literal"><span class="pre">y_i</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">6</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">deflate</span><span class="p">()</span>
-<span class="go">((3, 2), Poly(x**2*y + x + 1, x, y, domain=&#39;ZZ&#39;))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.degree">
-<tt class="descname">degree</tt><big>(</big><em>f</em>, <em>gen=0</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.degree"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.degree" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns degree of <tt class="docutils literal"><span class="pre">f</span></tt> in <tt class="docutils literal"><span class="pre">x_j</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">degree</span><span class="p">()</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">degree</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.degree_list">
-<tt class="descname">degree_list</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.degree_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.degree_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a list of degrees of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">degree_list</span><span class="p">()</span>
-<span class="go">(2, 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.diff">
-<tt class="descname">diff</tt><big>(</big><em>f</em>, <em>*specs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.diff"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.diff" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes partial derivative of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">()</span>
-<span class="go">Poly(2*x + 2, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">Poly(2*x*y, x, y, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.discriminant">
-<tt class="descname">discriminant</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.discriminant"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.discriminant" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the discriminant of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">discriminant</span><span class="p">()</span>
-<span class="go">-8</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.div">
-<tt class="descname">div</tt><big>(</big><em>f</em>, <em>g</em>, <em>auto=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.div"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.div" title="Permalink to this definition">¶</a></dt>
-<dd><p>Polynomial division with remainder of <tt class="docutils literal"><span class="pre">f</span></tt> by <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">div</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">(Poly(1/2*x + 1, x, domain=&#39;QQ&#39;), Poly(5, x, domain=&#39;QQ&#39;))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">div</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">auto</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">(Poly(0, x, domain=&#39;ZZ&#39;), Poly(x**2 + 1, x, domain=&#39;ZZ&#39;))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.Poly.domain">
-<tt class="descname">domain</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.domain"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.domain" title="Permalink to this definition">¶</a></dt>
-<dd><p>Get the ground domain of <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.eject">
-<tt class="descname">eject</tt><big>(</big><em>f</em>, <em>*gens</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.eject"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.eject" title="Permalink to this definition">¶</a></dt>
-<dd><p>Eject selected generators into the ground domain.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">eject</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">Poly(x*y**3 + (x**2 + x)*y + 1, y, domain=&#39;ZZ[x]&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">eject</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">Poly(y*x**2 + (y**3 + y)*x + 1, x, domain=&#39;ZZ[y]&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.eval">
-<tt class="descname">eval</tt><big>(</big><em>f</em>, <em>x</em>, <em>a=None</em>, <em>auto=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.eval"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.eval" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluate <tt class="docutils literal"><span class="pre">f</span></tt> at <tt class="docutils literal"><span class="pre">a</span></tt> in the given variable.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">11</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">eval</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">Poly(5*y + 8, y, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">eval</span><span class="p">({</span><span class="n">x</span><span class="p">:</span> <span class="mi">2</span><span class="p">})</span>
-<span class="go">Poly(5*y + 2*z + 6, y, z, domain=&#39;ZZ&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">eval</span><span class="p">({</span><span class="n">x</span><span class="p">:</span> <span class="mi">2</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="mi">5</span><span class="p">})</span>
-<span class="go">Poly(2*z + 31, z, domain=&#39;ZZ&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">eval</span><span class="p">({</span><span class="n">x</span><span class="p">:</span> <span class="mi">2</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="mi">5</span><span class="p">,</span> <span class="n">z</span><span class="p">:</span> <span class="mi">7</span><span class="p">})</span>
-<span class="go">45</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">eval</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
-<span class="go">Poly(2*z + 31, z, domain=&#39;ZZ&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">Poly(2*z + 31, z, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.exclude">
-<tt class="descname">exclude</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.exclude"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.exclude" title="Permalink to this definition">¶</a></dt>
-<dd><p>Remove unnecessary generators from <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">x</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">exclude</span><span class="p">()</span>
-<span class="go">Poly(a + x, a, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.exquo">
-<tt class="descname">exquo</tt><big>(</big><em>f</em>, <em>g</em>, <em>auto=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.exquo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.exquo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes polynomial exact quotient of <tt class="docutils literal"><span class="pre">f</span></tt> by <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">exquo</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Poly(x + 1, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">exquo</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">ExactQuotientFailed</span>: <span class="n">2*x - 4 does not divide x**2 + 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.exquo_ground">
-<tt class="descname">exquo_ground</tt><big>(</big><em>f</em>, <em>coeff</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.exquo_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.exquo_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Exact quotient of <tt class="docutils literal"><span class="pre">f</span></tt> by a an element of the ground domain.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">exquo_ground</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">Poly(x + 2, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">exquo_ground</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">ExactQuotientFailed</span>: <span class="n">2 does not divide 3 in ZZ</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.factor_list">
-<tt class="descname">factor_list</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.factor_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.factor_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a list of irreducible factors of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">5</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">factor_list</span><span class="p">()</span>
-<span class="go">(2, [(Poly(x + y, x, y, domain=&#39;ZZ&#39;), 1),</span>
-<span class="go">     (Poly(x**2 + 1, x, y, domain=&#39;ZZ&#39;), 2)])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.factor_list_include">
-<tt class="descname">factor_list_include</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.factor_list_include"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.factor_list_include" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a list of irreducible factors of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">5</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">factor_list_include</span><span class="p">()</span>
-<span class="go">[(Poly(2*x + 2*y, x, y, domain=&#39;ZZ&#39;), 1),</span>
-<span class="go"> (Poly(x**2 + 1, x, y, domain=&#39;ZZ&#39;), 2)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.Poly.free_symbols">
-<tt class="descname">free_symbols</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.free_symbols"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.free_symbols" title="Permalink to this definition">¶</a></dt>
-<dd><p>Free symbols of a polynomial expression.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">free_symbols</span>
-<span class="go">set([x])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">free_symbols</span>
-<span class="go">set([x, y])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">free_symbols</span>
-<span class="go">set([x, y])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.Poly.free_symbols_in_domain">
-<tt class="descname">free_symbols_in_domain</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.free_symbols_in_domain"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.free_symbols_in_domain" title="Permalink to this definition">¶</a></dt>
-<dd><p>Free symbols of the domain of <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">free_symbols_in_domain</span>
-<span class="go">set()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">free_symbols_in_domain</span>
-<span class="go">set()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">free_symbols_in_domain</span>
-<span class="go">set([y])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.polys.polytools.Poly.from_dict">
-<em class="property">classmethod </em><tt class="descname">from_dict</tt><big>(</big><em>rep</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.from_dict"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.from_dict" title="Permalink to this definition">¶</a></dt>
-<dd><p>Construct a polynomial from a <tt class="docutils literal"><span class="pre">dict</span></tt>.</p>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.polys.polytools.Poly.from_expr">
-<em class="property">classmethod </em><tt class="descname">from_expr</tt><big>(</big><em>rep</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.from_expr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.from_expr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Construct a polynomial from an expression.</p>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.polys.polytools.Poly.from_list">
-<em class="property">classmethod </em><tt class="descname">from_list</tt><big>(</big><em>rep</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.from_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.from_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Construct a polynomial from a <tt class="docutils literal"><span class="pre">list</span></tt>.</p>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.polys.polytools.Poly.from_poly">
-<em class="property">classmethod </em><tt class="descname">from_poly</tt><big>(</big><em>rep</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.from_poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.from_poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Construct a polynomial from a polynomial.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.gcd">
-<tt class="descname">gcd</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.gcd"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.gcd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the polynomial GCD of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">gcd</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Poly(x - 1, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.gcdex">
-<tt class="descname">gcdex</tt><big>(</big><em>f</em>, <em>g</em>, <em>auto=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.gcdex"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.gcdex" title="Permalink to this definition">¶</a></dt>
-<dd><p>Extended Euclidean algorithm of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p>Returns <tt class="docutils literal"><span class="pre">(s,</span> <span class="pre">t,</span> <span class="pre">h)</span></tt> such that <tt class="docutils literal"><span class="pre">h</span> <span class="pre">=</span> <span class="pre">gcd(f,</span> <span class="pre">g)</span></tt> and <tt class="docutils literal"><span class="pre">s*f</span> <span class="pre">+</span> <span class="pre">t*g</span> <span class="pre">=</span> <span class="pre">h</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">6</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">12</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">gcdex</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">g</span><span class="p">))</span>
-<span class="go">(Poly(-1/5*x + 3/5, x, domain=&#39;QQ&#39;),</span>
-<span class="go"> Poly(1/5*x**2 - 6/5*x + 2, x, domain=&#39;QQ&#39;),</span>
-<span class="go"> Poly(x + 1, x, domain=&#39;QQ&#39;))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.Poly.gen">
-<tt class="descname">gen</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.gen"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.gen" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the principal generator.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">gen</span>
-<span class="go">x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.get_domain">
-<tt class="descname">get_domain</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.get_domain"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.get_domain" title="Permalink to this definition">¶</a></dt>
-<dd><p>Get the ground domain of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.get_modulus">
-<tt class="descname">get_modulus</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.get_modulus"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.get_modulus" title="Permalink to this definition">¶</a></dt>
-<dd><p>Get the modulus of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">modulus</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">get_modulus</span><span class="p">()</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.gff_list">
-<tt class="descname">gff_list</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.gff_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.gff_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes greatest factorial factorization of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">5</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">gff_list</span><span class="p">()</span>
-<span class="go">[(Poly(x, x, domain=&#39;ZZ&#39;), 1), (Poly(x + 2, x, domain=&#39;ZZ&#39;), 4)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.ground_roots">
-<tt class="descname">ground_roots</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.ground_roots"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.ground_roots" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute roots of <tt class="docutils literal"><span class="pre">f</span></tt> by factorization in the ground domain.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">6</span> <span class="o">-</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">ground_roots</span><span class="p">()</span>
-<span class="go">{0: 2, 1: 2}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.half_gcdex">
-<tt class="descname">half_gcdex</tt><big>(</big><em>f</em>, <em>g</em>, <em>auto=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.half_gcdex"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.half_gcdex" title="Permalink to this definition">¶</a></dt>
-<dd><p>Half extended Euclidean algorithm of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p>Returns <tt class="docutils literal"><span class="pre">(s,</span> <span class="pre">h)</span></tt> such that <tt class="docutils literal"><span class="pre">h</span> <span class="pre">=</span> <span class="pre">gcd(f,</span> <span class="pre">g)</span></tt> and <tt class="docutils literal"><span class="pre">s*f</span> <span class="pre">=</span> <span class="pre">h</span> <span class="pre">(mod</span> <span class="pre">g)</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">6</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">12</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">half_gcdex</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">g</span><span class="p">))</span>
-<span class="go">(Poly(-1/5*x + 3/5, x, domain=&#39;QQ&#39;), Poly(x + 1, x, domain=&#39;QQ&#39;))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.has_only_gens">
-<tt class="descname">has_only_gens</tt><big>(</big><em>f</em>, <em>*gens</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.has_only_gens"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.has_only_gens" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">Poly(f,</span> <span class="pre">*gens)</span></tt> retains ground domain.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">has_only_gens</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="n">z</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">has_only_gens</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.homogeneous_order">
-<tt class="descname">homogeneous_order</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.homogeneous_order"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.homogeneous_order" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the homogeneous order of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p>A homogeneous polynomial is a polynomial whose all monomials with
-non-zero coefficients have the same total degree. This degree is
-the homogeneous order of <tt class="docutils literal"><span class="pre">f</span></tt>. If you only want to check if a
-polynomial is homogeneous, then use <a class="reference internal" href="#sympy.polys.polytools.Poly.is_homogeneous" title="sympy.polys.polytools.Poly.is_homogeneous"><tt class="xref py py-func docutils literal"><span class="pre">Poly.is_homogeneous()</span></tt></a>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">5</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">9</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">homogeneous_order</span><span class="p">()</span>
-<span class="go">5</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.inject">
-<tt class="descname">inject</tt><big>(</big><em>f</em>, <em>front=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.inject"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.inject" title="Permalink to this definition">¶</a></dt>
-<dd><p>Inject ground domain generators into <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">inject</span><span class="p">()</span>
-<span class="go">Poly(x**2*y + x*y**3 + x*y + 1, x, y, domain=&#39;ZZ&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">inject</span><span class="p">(</span><span class="n">front</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">Poly(y**3*x + y*x**2 + y*x + 1, y, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.integrate">
-<tt class="descname">integrate</tt><big>(</big><em>f</em>, <em>*specs</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.integrate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.integrate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes indefinite integral of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">integrate</span><span class="p">()</span>
-<span class="go">Poly(1/3*x**3 + x**2 + x, x, domain=&#39;QQ&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">integrate</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">Poly(1/2*x**2*y**2 + 1/2*x**2, x, y, domain=&#39;QQ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.intervals">
-<tt class="descname">intervals</tt><big>(</big><em>f</em>, <em>all=False</em>, <em>eps=None</em>, <em>inf=None</em>, <em>sup=None</em>, <em>fast=False</em>, <em>sqf=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.intervals"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.intervals" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute isolating intervals for roots of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">intervals</span><span class="p">()</span>
-<span class="go">[((-2, -1), 1), ((1, 2), 1)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">intervals</span><span class="p">(</span><span class="n">eps</span><span class="o">=</span><span class="mf">1e-2</span><span class="p">)</span>
-<span class="go">[((-26/15, -19/11), 1), ((19/11, 26/15), 1)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.invert">
-<tt class="descname">invert</tt><big>(</big><em>f</em>, <em>g</em>, <em>auto=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.invert"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.invert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Invert <tt class="docutils literal"><span class="pre">f</span></tt> modulo <tt class="docutils literal"><span class="pre">g</span></tt> when possible.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">invert</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Poly(-4/3, x, domain=&#39;QQ&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">invert</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">NotInvertible</span>: <span class="n">zero divisor</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.Poly.is_cyclotomic">
-<tt class="descname">is_cyclotomic</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.is_cyclotomic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.is_cyclotomic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is a cyclotomic polnomial.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">16</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">14</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">10</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">8</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">6</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">is_cyclotomic</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">16</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">14</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">10</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">8</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">6</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">g</span><span class="p">)</span><span class="o">.</span><span class="n">is_cyclotomic</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.Poly.is_ground">
-<tt class="descname">is_ground</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.is_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.is_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is an element of the ground domain.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_ground</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_ground</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_ground</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.Poly.is_homogeneous">
-<tt class="descname">is_homogeneous</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.is_homogeneous"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.is_homogeneous" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is a homogeneous polynomial.</p>
-<p>A homogeneous polynomial is a polynomial whose all monomials with
-non-zero coefficients have the same total degree. If you want not
-only to check if a polynomial is homogeneous but also compute its
-homogeneous order, then use <a class="reference internal" href="#sympy.polys.polytools.Poly.homogeneous_order" title="sympy.polys.polytools.Poly.homogeneous_order"><tt class="xref py py-func docutils literal"><span class="pre">Poly.homogeneous_order()</span></tt></a>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">is_homogeneous</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">is_homogeneous</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.Poly.is_irreducible">
-<tt class="descname">is_irreducible</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.is_irreducible"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.is_irreducible" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> has no factors over its domain.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">modulus</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">is_irreducible</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">modulus</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">is_irreducible</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.Poly.is_linear">
-<tt class="descname">is_linear</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.is_linear"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.is_linear" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is linear in all its variables.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">is_linear</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">is_linear</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.Poly.is_monic">
-<tt class="descname">is_monic</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.is_monic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.is_monic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if the leading coefficient of <tt class="docutils literal"><span class="pre">f</span></tt> is one.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_monic</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_monic</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.Poly.is_monomial">
-<tt class="descname">is_monomial</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.is_monomial"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.is_monomial" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is zero or has only one term.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_monomial</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_monomial</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.Poly.is_multivariate">
-<tt class="descname">is_multivariate</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.is_multivariate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.is_multivariate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is a multivariate polynomial.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_multivariate</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">is_multivariate</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_multivariate</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">is_multivariate</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.Poly.is_one">
-<tt class="descname">is_one</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.is_one"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.is_one" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is a unit polynomial.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_one</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_one</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.Poly.is_primitive">
-<tt class="descname">is_primitive</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.is_primitive"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.is_primitive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if GCD of the coefficients of <tt class="docutils literal"><span class="pre">f</span></tt> is one.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">6</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">12</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_primitive</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">6</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_primitive</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.Poly.is_quadratic">
-<tt class="descname">is_quadratic</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.is_quadratic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.is_quadratic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is quadratic in all its variables.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">is_quadratic</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">is_quadratic</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.Poly.is_sqf">
-<tt class="descname">is_sqf</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.is_sqf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.is_sqf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is a square-free polynomial.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_sqf</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_sqf</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.Poly.is_univariate">
-<tt class="descname">is_univariate</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.is_univariate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.is_univariate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is a univariate polynomial.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_univariate</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">is_univariate</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_univariate</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">is_univariate</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.Poly.is_zero">
-<tt class="descname">is_zero</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.is_zero"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.is_zero" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">f</span></tt> is a zero polynomial.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_zero</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">is_zero</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.l1_norm">
-<tt class="descname">l1_norm</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.l1_norm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.l1_norm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns l1 norm of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">l1_norm</span><span class="p">()</span>
-<span class="go">6</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.lcm">
-<tt class="descname">lcm</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.lcm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.lcm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns polynomial LCM of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">lcm</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Poly(x**3 - 2*x**2 - x + 2, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.length">
-<tt class="descname">length</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.length"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.length" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the number of non-zero terms in <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">length</span><span class="p">()</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.lift">
-<tt class="descname">lift</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.lift"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.lift" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert algebraic coefficients to rationals.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">extension</span><span class="o">=</span><span class="n">I</span><span class="p">)</span><span class="o">.</span><span class="n">lift</span><span class="p">()</span>
-<span class="go">Poly(x**4 + 3*x**2 + 1, x, domain=&#39;QQ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.ltrim">
-<tt class="descname">ltrim</tt><big>(</big><em>f</em>, <em>gen</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.ltrim"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.ltrim" title="Permalink to this definition">¶</a></dt>
-<dd><p>Remove dummy generators from the &#8220;left&#8221; of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">ltrim</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
-<span class="go">Poly(y**2 + y*z**2, y, z, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.max_norm">
-<tt class="descname">max_norm</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.max_norm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.max_norm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns maximum norm of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">max_norm</span><span class="p">()</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.monic">
-<tt class="descname">monic</tt><big>(</big><em>f</em>, <em>auto=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.monic"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.monic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Divides all coefficients by <tt class="docutils literal"><span class="pre">LC(f)</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">6</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">9</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="n">ZZ</span><span class="p">)</span><span class="o">.</span><span class="n">monic</span><span class="p">()</span>
-<span class="go">Poly(x**2 + 2*x + 3, x, domain=&#39;QQ&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="n">ZZ</span><span class="p">)</span><span class="o">.</span><span class="n">monic</span><span class="p">()</span>
-<span class="go">Poly(x**2 + 4/3*x + 2/3, x, domain=&#39;QQ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.monoms">
-<tt class="descname">monoms</tt><big>(</big><em>f</em>, <em>order=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.monoms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.monoms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns all non-zero monomials from <tt class="docutils literal"><span class="pre">f</span></tt> in lex order.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">monoms</span><span class="p">()</span>
-<span class="go">[(2, 0), (1, 2), (1, 1), (0, 1)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.mul">
-<tt class="descname">mul</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Multiply two polynomials <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">mul</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Poly(x**3 - 2*x**2 + x - 2, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">Poly(x**3 - 2*x**2 + x - 2, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.mul_ground">
-<tt class="descname">mul_ground</tt><big>(</big><em>f</em>, <em>coeff</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.mul_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.mul_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Multiply <tt class="docutils literal"><span class="pre">f</span></tt> by a an element of the ground domain.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">mul_ground</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">Poly(2*x + 2, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.neg">
-<tt class="descname">neg</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.neg"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.neg" title="Permalink to this definition">¶</a></dt>
-<dd><p>Negate all coefficients in <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">neg</span><span class="p">()</span>
-<span class="go">Poly(-x**2 + 1, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">Poly(-x**2 + 1, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.polys.polytools.Poly.new">
-<em class="property">classmethod </em><tt class="descname">new</tt><big>(</big><em>rep</em>, <em>*gens</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.new"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.new" title="Permalink to this definition">¶</a></dt>
-<dd><p>Construct <a class="reference internal" href="#sympy.polys.polytools.Poly" title="sympy.polys.polytools.Poly"><tt class="xref py py-class docutils literal"><span class="pre">Poly</span></tt></a> instance from raw representation.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.nroots">
-<tt class="descname">nroots</tt><big>(</big><em>f</em>, <em>n=15</em>, <em>maxsteps=50</em>, <em>cleanup=True</em>, <em>error=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.nroots"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.nroots" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute numerical approximations of roots of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">nroots</span><span class="p">(</span><span class="n">n</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
-<span class="go">[-1.73205080756888, 1.73205080756888]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">nroots</span><span class="p">(</span><span class="n">n</span><span class="o">=</span><span class="mi">30</span><span class="p">)</span>
-<span class="go">[-1.73205080756887729352744634151, 1.73205080756887729352744634151]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.nth">
-<tt class="descname">nth</tt><big>(</big><em>f</em>, <em>*N</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.nth"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.nth" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the <tt class="docutils literal"><span class="pre">n</span></tt>-th coefficient of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">nth</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">nth</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.nth_power_roots_poly">
-<tt class="descname">nth_power_roots_poly</tt><big>(</big><em>f</em>, <em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.nth_power_roots_poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.nth_power_roots_poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Construct a polynomial with n-th powers of roots of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">nth_power_roots_poly</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">Poly(x**4 - 2*x**3 + 3*x**2 - 2*x + 1, x, domain=&#39;ZZ&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">nth_power_roots_poly</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">Poly(x**4 + 2*x**2 + 1, x, domain=&#39;ZZ&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">nth_power_roots_poly</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">Poly(x**4 + 2*x**3 + 3*x**2 + 2*x + 1, x, domain=&#39;ZZ&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">nth_power_roots_poly</span><span class="p">(</span><span class="mi">12</span><span class="p">)</span>
-<span class="go">Poly(x**4 - 4*x**3 + 6*x**2 - 4*x + 1, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.Poly.one">
-<tt class="descname">one</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.one"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.one" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return one polynomial with <tt class="docutils literal"><span class="pre">self</span></tt>&#8216;s properties.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.pdiv">
-<tt class="descname">pdiv</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.pdiv"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.pdiv" title="Permalink to this definition">¶</a></dt>
-<dd><p>Polynomial pseudo-division of <tt class="docutils literal"><span class="pre">f</span></tt> by <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">pdiv</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">(Poly(2*x + 4, x, domain=&#39;ZZ&#39;), Poly(20, x, domain=&#39;ZZ&#39;))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.per">
-<tt class="descname">per</tt><big>(</big><em>f</em>, <em>rep</em>, <em>gens=None</em>, <em>remove=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.per"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.per" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a Poly out of the given representation.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.polyclasses</span> <span class="kn">import</span> <span class="n">DMP</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">per</span><span class="p">(</span><span class="n">DMP</span><span class="p">([</span><span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">ZZ</span><span class="p">(</span><span class="mi">1</span><span class="p">)],</span> <span class="n">ZZ</span><span class="p">),</span> <span class="n">gens</span><span class="o">=</span><span class="p">[</span><span class="n">y</span><span class="p">])</span>
-<span class="go">Poly(y + 1, y, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.pexquo">
-<tt class="descname">pexquo</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.pexquo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.pexquo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Polynomial exact pseudo-quotient of <tt class="docutils literal"><span class="pre">f</span></tt> by <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">pexquo</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Poly(2*x + 2, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">pexquo</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">ExactQuotientFailed</span>: <span class="n">2*x - 4 does not divide x**2 + 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.pow">
-<tt class="descname">pow</tt><big>(</big><em>f</em>, <em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.pow"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.pow" title="Permalink to this definition">¶</a></dt>
-<dd><p>Raise <tt class="docutils literal"><span class="pre">f</span></tt> to a non-negative power <tt class="docutils literal"><span class="pre">n</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">Poly(x**3 - 6*x**2 + 12*x - 8, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">3</span>
-<span class="go">Poly(x**3 - 6*x**2 + 12*x - 8, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.pquo">
-<tt class="descname">pquo</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.pquo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.pquo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Polynomial pseudo-quotient of <tt class="docutils literal"><span class="pre">f</span></tt> by <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">pquo</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Poly(2*x + 4, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">pquo</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Poly(2*x + 2, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.prem">
-<tt class="descname">prem</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.prem"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.prem" title="Permalink to this definition">¶</a></dt>
-<dd><p>Polynomial pseudo-remainder of <tt class="docutils literal"><span class="pre">f</span></tt> by <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">prem</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Poly(20, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.primitive">
-<tt class="descname">primitive</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.primitive"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.primitive" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the content and a primitive form of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">8</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">12</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">primitive</span><span class="p">()</span>
-<span class="go">(2, Poly(x**2 + 4*x + 6, x, domain=&#39;ZZ&#39;))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.quo">
-<tt class="descname">quo</tt><big>(</big><em>f</em>, <em>g</em>, <em>auto=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.quo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.quo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes polynomial quotient of <tt class="docutils literal"><span class="pre">f</span></tt> by <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Poly(1/2*x + 1, x, domain=&#39;QQ&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">quo</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Poly(x + 1, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.quo_ground">
-<tt class="descname">quo_ground</tt><big>(</big><em>f</em>, <em>coeff</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.quo_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.quo_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Quotient of <tt class="docutils literal"><span class="pre">f</span></tt> by a an element of the ground domain.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">quo_ground</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">Poly(x + 2, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">quo_ground</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">Poly(x + 1, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.rat_clear_denoms">
-<tt class="descname">rat_clear_denoms</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.rat_clear_denoms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.rat_clear_denoms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Clear denominators in a rational function <tt class="docutils literal"><span class="pre">f/g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">rat_clear_denoms</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span>
-<span class="go">Poly(x**2 + y, x, domain=&#39;ZZ[y]&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span>
-<span class="go">Poly(y*x**3 + y**2, x, domain=&#39;ZZ[y]&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.real_roots">
-<tt class="descname">real_roots</tt><big>(</big><em>f</em>, <em>multiple=True</em>, <em>radicals=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.real_roots"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.real_roots" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a list of real roots with multiplicities.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">7</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">real_roots</span><span class="p">()</span>
-<span class="go">[-1/2, 2, 2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">real_roots</span><span class="p">()</span>
-<span class="go">[RootOf(x**3 + x + 1, 0)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.refine_root">
-<tt class="descname">refine_root</tt><big>(</big><em>f</em>, <em>s</em>, <em>t</em>, <em>eps=None</em>, <em>steps=None</em>, <em>fast=False</em>, <em>check_sqf=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.refine_root"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.refine_root" title="Permalink to this definition">¶</a></dt>
-<dd><p>Refine an isolating interval of a root to the given precision.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">refine_root</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">eps</span><span class="o">=</span><span class="mf">1e-2</span><span class="p">)</span>
-<span class="go">(19/11, 26/15)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.rem">
-<tt class="descname">rem</tt><big>(</big><em>f</em>, <em>g</em>, <em>auto=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.rem"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.rem" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the polynomial remainder of <tt class="docutils literal"><span class="pre">f</span></tt> by <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">ZZ</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">rem</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Poly(5, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">rem</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">auto</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">Poly(x**2 + 1, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.reorder">
-<tt class="descname">reorder</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.reorder"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.reorder" title="Permalink to this definition">¶</a></dt>
-<dd><p>Efficiently apply new order of generators.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">reorder</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">Poly(y**2*x + x**2, y, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.replace">
-<tt class="descname">replace</tt><big>(</big><em>f</em>, <em>x</em>, <em>y=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.replace"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.replace" title="Permalink to this definition">¶</a></dt>
-<dd><p>Replace <tt class="docutils literal"><span class="pre">x</span></tt> with <tt class="docutils literal"><span class="pre">y</span></tt> in generators list.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">Poly(y**2 + 1, y, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.resultant">
-<tt class="descname">resultant</tt><big>(</big><em>f</em>, <em>g</em>, <em>includePRS=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.resultant"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.resultant" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the resultant of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt> via PRS.</p>
-<p>If includePRS=True, it includes the subresultant PRS in the result.
-Because the PRS is used to calculate the resultant, this is more
-efficient than calling <a class="reference internal" href="#sympy.polys.polytools.subresultants" title="sympy.polys.polytools.subresultants"><tt class="xref py py-func docutils literal"><span class="pre">subresultants()</span></tt></a> separately.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">resultant</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">resultant</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">includePRS</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(4, [Poly(x**2 + 1, x, domain=&#39;ZZ&#39;), Poly(x**2 - 1, x, domain=&#39;ZZ&#39;),</span>
-<span class="go">     Poly(-2, x, domain=&#39;ZZ&#39;)])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.retract">
-<tt class="descname">retract</tt><big>(</big><em>f</em>, <em>field=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.retract"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.retract" title="Permalink to this definition">¶</a></dt>
-<dd><p>Recalculate the ground domain of a polynomial.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="s">&#39;QQ[y]&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span>
-<span class="go">Poly(x**2 + 1, x, domain=&#39;QQ[y]&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">retract</span><span class="p">()</span>
-<span class="go">Poly(x**2 + 1, x, domain=&#39;ZZ&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">retract</span><span class="p">(</span><span class="n">field</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">Poly(x**2 + 1, x, domain=&#39;QQ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.revert">
-<tt class="descname">revert</tt><big>(</big><em>f</em>, <em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.revert"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.revert" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute <tt class="docutils literal"><span class="pre">f**(-1)</span></tt> mod <tt class="docutils literal"><span class="pre">x**n</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.root">
-<tt class="descname">root</tt><big>(</big><em>f</em>, <em>index</em>, <em>radicals=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.root"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.root" title="Permalink to this definition">¶</a></dt>
-<dd><p>Get an indexed root of a polynomial.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">7</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">4</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">root</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">-1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">root</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">root</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="o">.</span><span class="n">root</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">IndexError</span>: <span class="n">root index out of [-3, 2] range, got 3</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">5</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">root</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">RootOf(x**3 - x**2 + 1, 0)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.set_domain">
-<tt class="descname">set_domain</tt><big>(</big><em>f</em>, <em>domain</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.set_domain"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.set_domain" title="Permalink to this definition">¶</a></dt>
-<dd><p>Set the ground domain of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.set_modulus">
-<tt class="descname">set_modulus</tt><big>(</big><em>f</em>, <em>modulus</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.set_modulus"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.set_modulus" title="Permalink to this definition">¶</a></dt>
-<dd><p>Set the modulus of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">set_modulus</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">Poly(x**2 + 1, x, modulus=2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.shift">
-<tt class="descname">shift</tt><big>(</big><em>f</em>, <em>a</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.shift"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.shift" title="Permalink to this definition">¶</a></dt>
-<dd><p>Efficiently compute Taylor shift <tt class="docutils literal"><span class="pre">f(x</span> <span class="pre">+</span> <span class="pre">a)</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">shift</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">Poly(x**2 + 2*x + 1, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.slice">
-<tt class="descname">slice</tt><big>(</big><em>f</em>, <em>x</em>, <em>m</em>, <em>n=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.slice"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.slice" title="Permalink to this definition">¶</a></dt>
-<dd><p>Take a continuous subsequence of terms of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.sqf_list">
-<tt class="descname">sqf_list</tt><big>(</big><em>f</em>, <em>all=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.sqf_list"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.sqf_list" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a list of square-free factors of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">5</span> <span class="o">+</span> <span class="mi">16</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">+</span> <span class="mi">50</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">76</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">56</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">16</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">sqf_list</span><span class="p">()</span>
-<span class="go">(2, [(Poly(x + 1, x, domain=&#39;ZZ&#39;), 2),</span>
-<span class="go">     (Poly(x + 2, x, domain=&#39;ZZ&#39;), 3)])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">sqf_list</span><span class="p">(</span><span class="nb">all</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(2, [(Poly(1, x, domain=&#39;ZZ&#39;), 1),</span>
-<span class="go">     (Poly(x + 1, x, domain=&#39;ZZ&#39;), 2),</span>
-<span class="go">     (Poly(x + 2, x, domain=&#39;ZZ&#39;), 3)])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.sqf_list_include">
-<tt class="descname">sqf_list_include</tt><big>(</big><em>f</em>, <em>all=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.sqf_list_include"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.sqf_list_include" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a list of square-free factors of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">expand</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">expand</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span>
-<span class="go">2*x**7 + 6*x**6 + 6*x**5 + 2*x**4</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">sqf_list_include</span><span class="p">()</span>
-<span class="go">[(Poly(2, x, domain=&#39;ZZ&#39;), 1),</span>
-<span class="go"> (Poly(x + 1, x, domain=&#39;ZZ&#39;), 3),</span>
-<span class="go"> (Poly(x, x, domain=&#39;ZZ&#39;), 4)]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">f</span><span class="p">)</span><span class="o">.</span><span class="n">sqf_list_include</span><span class="p">(</span><span class="nb">all</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">[(Poly(2, x, domain=&#39;ZZ&#39;), 1),</span>
-<span class="go"> (Poly(1, x, domain=&#39;ZZ&#39;), 2),</span>
-<span class="go"> (Poly(x + 1, x, domain=&#39;ZZ&#39;), 3),</span>
-<span class="go"> (Poly(x, x, domain=&#39;ZZ&#39;), 4)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.sqf_norm">
-<tt class="descname">sqf_norm</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.sqf_norm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.sqf_norm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes square-free norm of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p>Returns <tt class="docutils literal"><span class="pre">s</span></tt>, <tt class="docutils literal"><span class="pre">f</span></tt>, <tt class="docutils literal"><span class="pre">r</span></tt>, such that <tt class="docutils literal"><span class="pre">g(x)</span> <span class="pre">=</span> <span class="pre">f(x-sa)</span></tt> and
-<tt class="docutils literal"><span class="pre">r(x)</span> <span class="pre">=</span> <span class="pre">Norm(g(x))</span></tt> is a square-free polynomial over <tt class="docutils literal"><span class="pre">K</span></tt>,
-where <tt class="docutils literal"><span class="pre">a</span></tt> is the algebraic extension of the ground domain.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">extension</span><span class="o">=</span><span class="p">[</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)])</span><span class="o">.</span><span class="n">sqf_norm</span><span class="p">()</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">s</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span>
-<span class="go">Poly(x**2 - 2*sqrt(3)*x + 4, x, domain=&#39;QQ&lt;sqrt(3)&gt;&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r</span>
-<span class="go">Poly(x**4 - 4*x**2 + 16, x, domain=&#39;QQ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.sqf_part">
-<tt class="descname">sqf_part</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.sqf_part"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.sqf_part" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes square-free part of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">sqf_part</span><span class="p">()</span>
-<span class="go">Poly(x**2 - x - 2, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.sqr">
-<tt class="descname">sqr</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.sqr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.sqr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Square a polynomial <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">sqr</span><span class="p">()</span>
-<span class="go">Poly(x**2 - 4*x + 4, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">Poly(x**2 - 4*x + 4, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.sturm">
-<tt class="descname">sturm</tt><big>(</big><em>f</em>, <em>auto=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.sturm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.sturm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the Sturm sequence of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">sturm</span><span class="p">()</span>
-<span class="go">[Poly(x**3 - 2*x**2 + x - 3, x, domain=&#39;QQ&#39;),</span>
-<span class="go"> Poly(3*x**2 - 4*x + 1, x, domain=&#39;QQ&#39;),</span>
-<span class="go"> Poly(2/9*x + 25/9, x, domain=&#39;QQ&#39;),</span>
-<span class="go"> Poly(-2079/4, x, domain=&#39;QQ&#39;)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.sub">
-<tt class="descname">sub</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.sub"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.sub" title="Permalink to this definition">¶</a></dt>
-<dd><p>Subtract two polynomials <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">sub</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Poly(x**2 - x + 3, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">Poly</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">Poly(x**2 - x + 3, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.sub_ground">
-<tt class="descname">sub_ground</tt><big>(</big><em>f</em>, <em>coeff</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.sub_ground"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.sub_ground" title="Permalink to this definition">¶</a></dt>
-<dd><p>Subtract an element of the ground domain from <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">sub_ground</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">Poly(x - 1, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.subresultants">
-<tt class="descname">subresultants</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.subresultants"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.subresultants" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the subresultant PRS of <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">subresultants</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">[Poly(x**2 + 1, x, domain=&#39;ZZ&#39;),</span>
-<span class="go"> Poly(x**2 - 1, x, domain=&#39;ZZ&#39;),</span>
-<span class="go"> Poly(-2, x, domain=&#39;ZZ&#39;)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.terms">
-<tt class="descname">terms</tt><big>(</big><em>f</em>, <em>order=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.terms"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.terms" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns all non-zero terms from <tt class="docutils literal"><span class="pre">f</span></tt> in lex order.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">terms</span><span class="p">()</span>
-<span class="go">[((2, 0), 1), ((1, 2), 2), ((1, 1), 1), ((0, 1), 3)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.terms_gcd">
-<tt class="descname">terms_gcd</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.terms_gcd"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.terms_gcd" title="Permalink to this definition">¶</a></dt>
-<dd><p>Remove GCD of terms from the polynomial <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">6</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">terms_gcd</span><span class="p">()</span>
-<span class="go">((3, 1), Poly(x**3*y + 1, x, y, domain=&#39;ZZ&#39;))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.termwise">
-<tt class="descname">termwise</tt><big>(</big><em>f</em>, <em>func</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.termwise"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.termwise" title="Permalink to this definition">¶</a></dt>
-<dd><p>Apply a function to all terms of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">xxx_todo_changeme</span><span class="p">,</span> <span class="n">coeff</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="p">(</span><span class="n">k</span><span class="p">,)</span> <span class="o">=</span> <span class="n">xxx_todo_changeme</span>
-<span class="gp">... </span>    <span class="k">return</span> <span class="n">coeff</span><span class="o">//</span><span class="mi">10</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">-</span><span class="n">k</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">20</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">400</span><span class="p">)</span><span class="o">.</span><span class="n">termwise</span><span class="p">(</span><span class="n">func</span><span class="p">)</span>
-<span class="go">Poly(x**2 + 2*x + 4, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.to_exact">
-<tt class="descname">to_exact</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.to_exact"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.to_exact" title="Permalink to this definition">¶</a></dt>
-<dd><p>Make the ground domain exact.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">RR</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mf">1.0</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="n">RR</span><span class="p">)</span><span class="o">.</span><span class="n">to_exact</span><span class="p">()</span>
-<span class="go">Poly(x**2 + 1, x, domain=&#39;QQ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.to_field">
-<tt class="descname">to_field</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.to_field"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.to_field" title="Permalink to this definition">¶</a></dt>
-<dd><p>Make the ground domain a field.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">ZZ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="n">ZZ</span><span class="p">)</span><span class="o">.</span><span class="n">to_field</span><span class="p">()</span>
-<span class="go">Poly(x**2 + 1, x, domain=&#39;QQ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.to_ring">
-<tt class="descname">to_ring</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.to_ring"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.to_ring" title="Permalink to this definition">¶</a></dt>
-<dd><p>Make the ground domain a ring.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">QQ</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">domain</span><span class="o">=</span><span class="n">QQ</span><span class="p">)</span><span class="o">.</span><span class="n">to_ring</span><span class="p">()</span>
-<span class="go">Poly(x**2 + 1, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.total_degree">
-<tt class="descname">total_degree</tt><big>(</big><em>f</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.total_degree"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.total_degree" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the total degree of <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">total_degree</span><span class="p">()</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">5</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">total_degree</span><span class="p">()</span>
-<span class="go">5</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.trunc">
-<tt class="descname">trunc</tt><big>(</big><em>f</em>, <em>p</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.trunc"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.trunc" title="Permalink to this definition">¶</a></dt>
-<dd><p>Reduce <tt class="docutils literal"><span class="pre">f</span></tt> modulo a constant <tt class="docutils literal"><span class="pre">p</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">5</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">7</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">trunc</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">Poly(-x**3 - x + 1, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.Poly.unify">
-<tt class="descname">unify</tt><big>(</big><em>f</em>, <em>g</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.unify"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.unify" title="Permalink to this definition">¶</a></dt>
-<dd><p>Make <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt> belong to the same domain.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Poly</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span>
-<span class="go">Poly(1/2*x + 1, x, domain=&#39;QQ&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span>
-<span class="go">Poly(2*x + 1, x, domain=&#39;ZZ&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">F</span><span class="p">,</span> <span class="n">G</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">unify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">F</span>
-<span class="go">Poly(1/2*x + 1, x, domain=&#39;QQ&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span>
-<span class="go">Poly(2*x + 1, x, domain=&#39;QQ&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.Poly.unit">
-<tt class="descname">unit</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.unit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.unit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return unit polynomial with <tt class="docutils literal"><span class="pre">self</span></tt>&#8216;s properties.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.Poly.zero">
-<tt class="descname">zero</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#Poly.zero"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.Poly.zero" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return zero polynomial with <tt class="docutils literal"><span class="pre">self</span></tt>&#8216;s properties.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polytools.PurePoly">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">PurePoly</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#PurePoly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.PurePoly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Class for representing pure polynomials.</p>
-<dl class="attribute">
-<dt id="sympy.polys.polytools.PurePoly.free_symbols">
-<tt class="descname">free_symbols</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#PurePoly.free_symbols"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.PurePoly.free_symbols" title="Permalink to this definition">¶</a></dt>
-<dd><p>Free symbols of a polynomial.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">PurePoly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">PurePoly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">free_symbols</span>
-<span class="go">set()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">PurePoly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">free_symbols</span>
-<span class="go">set()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">PurePoly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">free_symbols</span>
-<span class="go">set([y])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.polytools.GroebnerBasis">
-<em class="property">class </em><tt class="descclassname">sympy.polys.polytools.</tt><tt class="descname">GroebnerBasis</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#GroebnerBasis"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.GroebnerBasis" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents a reduced Groebner basis.</p>
-<dl class="function">
-<dt id="sympy.polys.polytools.GroebnerBasis.contains">
-<tt class="descname">contains</tt><big>(</big><em>self</em>, <em>poly</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#GroebnerBasis.contains"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.GroebnerBasis.contains" title="Permalink to this definition">¶</a></dt>
-<dd><p>Check if <tt class="docutils literal"><span class="pre">poly</span></tt> belongs the ideal generated by <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">groebner</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">groebner</span><span class="p">([</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">2</span><span class="p">])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="n">f</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.GroebnerBasis.fglm">
-<tt class="descname">fglm</tt><big>(</big><em>self</em>, <em>order</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#GroebnerBasis.fglm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.GroebnerBasis.fglm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a Groebner basis from one ordering to another.</p>
-<p>The FGLM algorithm converts reduced Groebner bases of zero-dimensional
-ideals from one ordering to another. This method is often used when it
-is infeasible to compute a Groebner basis with respect to a particular
-ordering directly.</p>
-<p class="rubric">References</p>
-<p>J.C. Faugere, P. Gianni, D. Lazard, T. Mora (1994). Efficient
-Computation of Zero-dimensional Groebner Bases by Change of
-Ordering</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">groebner</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">groebner</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s">&#39;grlex&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">fglm</span><span class="p">(</span><span class="s">&#39;lex&#39;</span><span class="p">))</span>
-<span class="go">[2*x - y**2 - y + 1, y**4 + 2*y**3 - 3*y**2 - 16*y + 7]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">groebner</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="s">&#39;lex&#39;</span><span class="p">))</span>
-<span class="go">[2*x - y**2 - y + 1, y**4 + 2*y**3 - 3*y**2 - 16*y + 7]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.polytools.GroebnerBasis.is_zero_dimensional">
-<tt class="descname">is_zero_dimensional</tt><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#GroebnerBasis.is_zero_dimensional"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.GroebnerBasis.is_zero_dimensional" title="Permalink to this definition">¶</a></dt>
-<dd><p>Checks if the ideal generated by a Groebner basis is zero-dimensional.</p>
-<p>The algorithm checks if the set of monomials not divisible by the
-leading monomial of any element of <tt class="docutils literal"><span class="pre">F</span></tt> is bounded.</p>
-<p class="rubric">References</p>
-<p>David A. Cox, John B. Little, Donal O&#8217;Shea. Ideals, Varieties and
-Algorithms, 3rd edition, p. 230</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polytools.GroebnerBasis.reduce">
-<tt class="descname">reduce</tt><big>(</big><em>self</em>, <em>expr</em>, <em>auto=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polytools.html#GroebnerBasis.reduce"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polytools.GroebnerBasis.reduce" title="Permalink to this definition">¶</a></dt>
-<dd><p>Reduces a polynomial modulo a Groebner basis.</p>
-<p>Given a polynomial <tt class="docutils literal"><span class="pre">f</span></tt> and a set of polynomials <tt class="docutils literal"><span class="pre">G</span> <span class="pre">=</span> <span class="pre">(g_1,</span> <span class="pre">...,</span> <span class="pre">g_n)</span></tt>,
-computes a set of quotients <tt class="docutils literal"><span class="pre">q</span> <span class="pre">=</span> <span class="pre">(q_1,</span> <span class="pre">...,</span> <span class="pre">q_n)</span></tt> and the remainder <tt class="docutils literal"><span class="pre">r</span></tt>
-such that <tt class="docutils literal"><span class="pre">f</span> <span class="pre">=</span> <span class="pre">q_1*f_1</span> <span class="pre">+</span> <span class="pre">...</span> <span class="pre">+</span> <span class="pre">q_n*f_n</span> <span class="pre">+</span> <span class="pre">r</span></tt>, where <tt class="docutils literal"><span class="pre">r</span></tt> vanishes or <tt class="docutils literal"><span class="pre">r</span></tt>
-is a completely reduced polynomial with respect to <tt class="docutils literal"><span class="pre">G</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">groebner</span><span class="p">,</span> <span class="n">expand</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">groebner</span><span class="p">([</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="n">y</span><span class="p">])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">reduce</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-<span class="go">([2*x, 1], x**2 + y**2 + y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Q</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">_</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expand</span><span class="p">(</span><span class="nb">sum</span><span class="p">(</span><span class="n">q</span><span class="o">*</span><span class="n">g</span> <span class="k">for</span> <span class="n">q</span><span class="p">,</span> <span class="n">g</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">Q</span><span class="p">,</span> <span class="n">G</span><span class="p">))</span> <span class="o">+</span> <span class="n">r</span><span class="p">)</span>
-<span class="go">2*x**4 - x**2 + y**3 + y**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span> <span class="o">==</span> <span class="n">f</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="extra-polynomial-manipulation-functions">
-<h2>Extra polynomial manipulation functions<a class="headerlink" href="#extra-polynomial-manipulation-functions" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.polys.polyfuncs.symmetrize">
-<tt class="descclassname">sympy.polys.polyfuncs.</tt><tt class="descname">symmetrize</tt><big>(</big><em>F</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyfuncs.html#symmetrize"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyfuncs.symmetrize" title="Permalink to this definition">¶</a></dt>
-<dd><p>Rewrite a polynomial in terms of elementary symmetric polynomials.</p>
-<p>A symmetric polynomial is a multivariate polynomial that remains invariant
-under any variable permutation, i.e., if <tt class="docutils literal"><span class="pre">f</span> <span class="pre">=</span> <span class="pre">f(x_1,</span> <span class="pre">x_2,</span> <span class="pre">...,</span> <span class="pre">x_n)</span></tt>,
-then <tt class="docutils literal"><span class="pre">f</span> <span class="pre">=</span> <span class="pre">f(x_{i_1},</span> <span class="pre">x_{i_2},</span> <span class="pre">...,</span> <span class="pre">x_{i_n})</span></tt>, where
-<tt class="docutils literal"><span class="pre">(i_1,</span> <span class="pre">i_2,</span> <span class="pre">...,</span> <span class="pre">i_n)</span></tt> is a permutation of <tt class="docutils literal"><span class="pre">(1,</span> <span class="pre">2,</span> <span class="pre">...,</span> <span class="pre">n)</span></tt> (an
-element of the group <tt class="docutils literal"><span class="pre">S_n</span></tt>).</p>
-<p>Returns a tuple of symmetric polynomials <tt class="docutils literal"><span class="pre">(f1,</span> <span class="pre">f2,</span> <span class="pre">...,</span> <span class="pre">fn)</span></tt> such that
-<tt class="docutils literal"><span class="pre">f</span> <span class="pre">=</span> <span class="pre">f1</span> <span class="pre">+</span> <span class="pre">f2</span> <span class="pre">+</span> <span class="pre">...</span> <span class="pre">+</span> <span class="pre">fn</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.polyfuncs</span> <span class="kn">import</span> <span class="n">symmetrize</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">symmetrize</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(-2*x*y + (x + y)**2, 0)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">symmetrize</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">formal</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(s1**2 - 2*s2, 0, [(s1, x + y), (s2, x*y)])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">symmetrize</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(-2*x*y + (x + y)**2, -2*y**2)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">symmetrize</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">formal</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(s1**2 - 2*s2, -2*y**2, [(s1, x + y), (s2, x*y)])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyfuncs.horner">
-<tt class="descclassname">sympy.polys.polyfuncs.</tt><tt class="descname">horner</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyfuncs.html#horner"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyfuncs.horner" title="Permalink to this definition">¶</a></dt>
-<dd><p>Rewrite a polynomial in Horner form.</p>
-<p>Among other applications, evaluation of a polynomial at a point is optimal
-when it is applied using the Horner scheme ([1]).</p>
-<p class="rubric">References</p>
-<p>[1] - <a class="reference external" href="http://en.wikipedia.org/wiki/Horner_scheme">http://en.wikipedia.org/wiki/Horner_scheme</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.polyfuncs</span> <span class="kn">import</span> <span class="n">horner</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">e</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">horner</span><span class="p">(</span><span class="mi">9</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">+</span> <span class="mi">8</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">7</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">6</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">x*(x*(x*(9*x + 8) + 7) + 6) + 5</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">horner</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">c</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">d</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">e</span><span class="p">)</span>
-<span class="go">e + x*(d + x*(c + x*(a*x + b)))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">horner</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">wrt</span><span class="o">=</span><span class="n">x</span><span class="p">)</span>
-<span class="go">x*(x*y*(4*y + 2) + y*(2*y + 1))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">horner</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">wrt</span><span class="o">=</span><span class="n">y</span><span class="p">)</span>
-<span class="go">y*(x*y*(4*x + 2) + x*(2*x + 1))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyfuncs.interpolate">
-<tt class="descclassname">sympy.polys.polyfuncs.</tt><tt class="descname">interpolate</tt><big>(</big><em>data</em>, <em>x</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyfuncs.html#interpolate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyfuncs.interpolate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Construct an interpolating polynomial for the data points.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.polyfuncs</span> <span class="kn">import</span> <span class="n">interpolate</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<p>A list is interpreted as though it were paired with a range starting
-from 1:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">interpolate</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">16</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x**2</span>
-</pre></div>
-</div>
-<p>This can be made explicit by giving a list of coordinates:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">interpolate</span><span class="p">([(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">9</span><span class="p">)],</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x**2</span>
-</pre></div>
-</div>
-<p>The (x, y) coordinates can also be given as keys and values of a
-dictionary (and the points need not be equispaced):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">interpolate</span><span class="p">([(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">)],</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x**2 + 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">interpolate</span><span class="p">({</span><span class="o">-</span><span class="mi">1</span><span class="p">:</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">:</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">:</span> <span class="mi">5</span><span class="p">},</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x**2 + 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.polyfuncs.viete">
-<tt class="descclassname">sympy.polys.polyfuncs.</tt><tt class="descname">viete</tt><big>(</big><em>f</em>, <em>roots=None</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyfuncs.html#viete"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyfuncs.viete" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generate Viete&#8217;s formulas for <tt class="docutils literal"><span class="pre">f</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.polyfuncs</span> <span class="kn">import</span> <span class="n">viete</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">r1</span><span class="p">,</span> <span class="n">r2</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,a:c,r1:3&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">viete</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="p">[</span><span class="n">r1</span><span class="p">,</span> <span class="n">r2</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">[(r1 + r2, -b/a), (r1*r2, c/a)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="domain-constructors">
-<h2>Domain constructors<a class="headerlink" href="#domain-constructors" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.polys.constructor.construct_domain">
-<tt class="descclassname">sympy.polys.constructor.</tt><tt class="descname">construct_domain</tt><big>(</big><em>obj</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/constructor.html#construct_domain"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.constructor.construct_domain" title="Permalink to this definition">¶</a></dt>
-<dd><p>Construct a minimal domain for the list of coefficients.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="algebraic-number-fields">
-<h2>Algebraic number fields<a class="headerlink" href="#algebraic-number-fields" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.polys.numberfields.minimal_polynomial">
-<tt class="descclassname">sympy.polys.numberfields.</tt><tt class="descname">minimal_polynomial</tt><big>(</big><em>ex</em>, <em>x=None</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/numberfields.html#minimal_polynomial"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.numberfields.minimal_polynomial" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the minimal polynomial of an algebraic number.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">minimal_polynomial</span><span class="p">,</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">minimal_polynomial</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x**2 - 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">minimal_polynomial</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x**4 - 10*x**2 + 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.numberfields.minpoly">
-<tt class="descclassname">sympy.polys.numberfields.</tt><tt class="descname">minpoly</tt><big>(</big><em>ex</em>, <em>x=None</em>, <em>**args</em><big>)</big><a class="headerlink" href="#sympy.polys.numberfields.minpoly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the minimal polynomial of an algebraic number.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">minimal_polynomial</span><span class="p">,</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">minimal_polynomial</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x**2 - 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">minimal_polynomial</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x**4 - 10*x**2 + 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.numberfields.primitive_element">
-<tt class="descclassname">sympy.polys.numberfields.</tt><tt class="descname">primitive_element</tt><big>(</big><em>extension</em>, <em>x=None</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/numberfields.html#primitive_element"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.numberfields.primitive_element" title="Permalink to this definition">¶</a></dt>
-<dd><p>Construct a common number field for all extensions.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.numberfields.field_isomorphism">
-<tt class="descclassname">sympy.polys.numberfields.</tt><tt class="descname">field_isomorphism</tt><big>(</big><em>a</em>, <em>b</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/numberfields.html#field_isomorphism"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.numberfields.field_isomorphism" title="Permalink to this definition">¶</a></dt>
-<dd><p>Construct an isomorphism between two number fields.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.numberfields.to_number_field">
-<tt class="descclassname">sympy.polys.numberfields.</tt><tt class="descname">to_number_field</tt><big>(</big><em>extension</em>, <em>theta=None</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/numberfields.html#to_number_field"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.numberfields.to_number_field" title="Permalink to this definition">¶</a></dt>
-<dd><p>Express <span class="math">\(extension\)</span> in the field generated by <span class="math">\(theta\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.numberfields.isolate">
-<tt class="descclassname">sympy.polys.numberfields.</tt><tt class="descname">isolate</tt><big>(</big><em>alg</em>, <em>eps=None</em>, <em>fast=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/numberfields.html#isolate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.numberfields.isolate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Give a rational isolating interval for an algebraic number.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.numberfields.AlgebraicNumber">
-<em class="property">class </em><tt class="descclassname">sympy.polys.numberfields.</tt><tt class="descname">AlgebraicNumber</tt><a class="reference internal" href="../../_modules/sympy/polys/numberfields.html#AlgebraicNumber"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.numberfields.AlgebraicNumber" title="Permalink to this definition">¶</a></dt>
-<dd><p>Class for representing algebraic numbers in SymPy.</p>
-<dl class="function">
-<dt id="sympy.polys.numberfields.AlgebraicNumber.as_expr">
-<tt class="descname">as_expr</tt><big>(</big><em>self</em>, <em>x=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/numberfields.html#AlgebraicNumber.as_expr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.numberfields.AlgebraicNumber.as_expr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a Basic expression from <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.numberfields.AlgebraicNumber.as_poly">
-<tt class="descname">as_poly</tt><big>(</big><em>self</em>, <em>x=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/numberfields.html#AlgebraicNumber.as_poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.numberfields.AlgebraicNumber.as_poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a Poly instance from <tt class="docutils literal"><span class="pre">self</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.numberfields.AlgebraicNumber.coeffs">
-<tt class="descname">coeffs</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/numberfields.html#AlgebraicNumber.coeffs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.numberfields.AlgebraicNumber.coeffs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns all SymPy coefficients of an algebraic number.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.polys.numberfields.AlgebraicNumber.is_aliased">
-<tt class="descname">is_aliased</tt><a class="reference internal" href="../../_modules/sympy/polys/numberfields.html#AlgebraicNumber.is_aliased"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.numberfields.AlgebraicNumber.is_aliased" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">alias</span></tt> was set.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.numberfields.AlgebraicNumber.native_coeffs">
-<tt class="descname">native_coeffs</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/numberfields.html#AlgebraicNumber.native_coeffs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.numberfields.AlgebraicNumber.native_coeffs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns all native coefficients of an algebraic number.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.numberfields.AlgebraicNumber.to_algebraic_integer">
-<tt class="descname">to_algebraic_integer</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/numberfields.html#AlgebraicNumber.to_algebraic_integer"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.numberfields.AlgebraicNumber.to_algebraic_integer" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert <tt class="docutils literal"><span class="pre">self</span></tt> to an algebraic integer.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="monomials-encoded-as-tuples">
-<h2>Monomials encoded as tuples<a class="headerlink" href="#monomials-encoded-as-tuples" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.polys.monomialtools.Monomial">
-<em class="property">class </em><tt class="descclassname">sympy.polys.monomialtools.</tt><tt class="descname">Monomial</tt><big>(</big><em>exponents</em>, <em>gens=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/monomialtools.html#Monomial"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.monomialtools.Monomial" title="Permalink to this definition">¶</a></dt>
-<dd><p>Class representing a monomial, i.e. a product of powers.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.monomialtools.monomials">
-<tt class="descclassname">sympy.polys.monomialtools.</tt><tt class="descname">monomials</tt><big>(</big><em>variables</em>, <em>degree</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/monomialtools.html#monomials"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.monomialtools.monomials" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generate a set of monomials of the given total degree or less.</p>
-<p>Given a set of variables <span class="math">\(V\)</span> and a total degree <span class="math">\(N\)</span> generate
-a set of monomials of degree at most <span class="math">\(N\)</span>. The total number of
-monomials is huge and is given by the following formula:</p>
-<div class="math">
-\[\frac{(\#V + N)!}{\#V! N!}\]</div>
-<p>For example if we would like to generate a dense polynomial of
-a total degree <span class="math">\(N = 50\)</span> in 5 variables, assuming that exponents
-and all of coefficients are 32-bit long and stored in an array we
-would need almost 80 GiB of memory! Fortunately most polynomials,
-that we will encounter, are sparse.</p>
-<p class="rubric">Examples</p>
-<p>Consider monomials in variables <span class="math">\(x\)</span> and <span class="math">\(y\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">monomials</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">sorted</span><span class="p">(</span><span class="n">monomials</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">],</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">[1, x, y, x**2, y**2, x*y]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">sorted</span><span class="p">(</span><span class="n">monomials</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">],</span> <span class="mi">3</span><span class="p">))</span>
-<span class="go">[1, x, y, x**2, x**3, y**2, y**3, x*y, x*y**2, x**2*y]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.monomialtools.monomial_count">
-<tt class="descclassname">sympy.polys.monomialtools.</tt><tt class="descname">monomial_count</tt><big>(</big><em>V</em>, <em>N</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/monomialtools.html#monomial_count"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.monomialtools.monomial_count" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the number of monomials.</p>
-<p>The number of monomials is given by the following formula:</p>
-<div class="math">
-\[\frac{(\#V + N)!}{\#V! N!}\]</div>
-<p>where <span class="math">\(N\)</span> is a total degree and <span class="math">\(V\)</span> is a set of variables.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">monomials</span><span class="p">,</span> <span class="n">monomial_count</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">monomial_count</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">6</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">monomials</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">],</span> <span class="mi">2</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">sorted</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
-<span class="go">[1, x, y, x**2, y**2, x*y]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">len</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
-<span class="go">6</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.monomialtools.LexOrder">
-<em class="property">class </em><tt class="descclassname">sympy.polys.monomialtools.</tt><tt class="descname">LexOrder</tt><a class="reference internal" href="../../_modules/sympy/polys/monomialtools.html#LexOrder"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.monomialtools.LexOrder" title="Permalink to this definition">¶</a></dt>
-<dd><p>Lexicographic order of monomials.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.monomialtools.GradedLexOrder">
-<em class="property">class </em><tt class="descclassname">sympy.polys.monomialtools.</tt><tt class="descname">GradedLexOrder</tt><a class="reference internal" href="../../_modules/sympy/polys/monomialtools.html#GradedLexOrder"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.monomialtools.GradedLexOrder" title="Permalink to this definition">¶</a></dt>
-<dd><p>Graded lexicographic order of monomials.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.monomialtools.ReversedGradedLexOrder">
-<em class="property">class </em><tt class="descclassname">sympy.polys.monomialtools.</tt><tt class="descname">ReversedGradedLexOrder</tt><a class="reference internal" href="../../_modules/sympy/polys/monomialtools.html#ReversedGradedLexOrder"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.monomialtools.ReversedGradedLexOrder" title="Permalink to this definition">¶</a></dt>
-<dd><p>Reversed graded lexicographic order of monomials.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="formal-manipulation-of-roots-of-polynomials">
-<h2>Formal manipulation of roots of polynomials<a class="headerlink" href="#formal-manipulation-of-roots-of-polynomials" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.polys.rootoftools.RootOf">
-<em class="property">class </em><tt class="descclassname">sympy.polys.rootoftools.</tt><tt class="descname">RootOf</tt><a class="reference internal" href="../../_modules/sympy/polys/rootoftools.html#RootOf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.rootoftools.RootOf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents <tt class="docutils literal"><span class="pre">k</span></tt>-th root of a univariate polynomial.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.polys.rootoftools.RootSum">
-<em class="property">class </em><tt class="descclassname">sympy.polys.rootoftools.</tt><tt class="descname">RootSum</tt><a class="reference internal" href="../../_modules/sympy/polys/rootoftools.html#RootSum"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.rootoftools.RootSum" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents a sum of all roots of a univariate polynomial.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="symbolic-root-finding-algorithms">
-<h2>Symbolic root-finding algorithms<a class="headerlink" href="#symbolic-root-finding-algorithms" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.polys.polyroots.roots">
-<tt class="descclassname">sympy.polys.polyroots.</tt><tt class="descname">roots</tt><big>(</big><em>f</em>, <em>*gens</em>, <em>**flags</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/polyroots.html#roots"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.polyroots.roots" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes symbolic roots of a univariate polynomial.</p>
-<p>Given a univariate polynomial f with symbolic coefficients (or
-a list of the polynomial&#8217;s coefficients), returns a dictionary
-with its roots and their multiplicities.</p>
-<p>Only roots expressible via radicals will be returned.  To get
-a complete set of roots use RootOf class or numerical methods
-instead. By default cubic and quartic formulas are used in
-the algorithm. To disable them because of unreadable output
-set <tt class="docutils literal"><span class="pre">cubics=False</span></tt> or <tt class="docutils literal"><span class="pre">quartics=False</span></tt> respectively.</p>
-<p>To get roots from a specific domain set the <tt class="docutils literal"><span class="pre">filter</span></tt> flag with
-one of the following specifiers: Z, Q, R, I, C. By default all
-roots are returned (this is equivalent to setting <tt class="docutils literal"><span class="pre">filter='C'</span></tt>).</p>
-<p>By default a dictionary is returned giving a compact result in
-case of multiple roots.  However to get a tuple containing all
-those roots set the <tt class="docutils literal"><span class="pre">multiple</span></tt> flag to True.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">roots</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">roots</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">{-1: 1, 1: 1}</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">roots</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-<span class="go">{-1: 1, 1: 1}</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Poly</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">-</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">roots</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">{-sqrt(y): 1, sqrt(y): 1}</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">roots</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">{-sqrt(y): 1, sqrt(y): 1}</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">roots</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">])</span>
-<span class="go">{-1: 1, 1: 1}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="special-polynomials">
-<h2>Special polynomials<a class="headerlink" href="#special-polynomials" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.polys.specialpolys.swinnerton_dyer_poly">
-<tt class="descclassname">sympy.polys.specialpolys.</tt><tt class="descname">swinnerton_dyer_poly</tt><big>(</big><em>n</em>, <em>x=None</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/specialpolys.html#swinnerton_dyer_poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.specialpolys.swinnerton_dyer_poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates n-th Swinnerton-Dyer polynomial in <span class="math">\(x\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.specialpolys.interpolating_poly">
-<tt class="descclassname">sympy.polys.specialpolys.</tt><tt class="descname">interpolating_poly</tt><big>(</big><em>n</em>, <em>x</em>, <em>X='x'</em>, <em>Y='y'</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/specialpolys.html#interpolating_poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.specialpolys.interpolating_poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Construct Lagrange interpolating polynomial for <tt class="docutils literal"><span class="pre">n</span></tt> data points.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.specialpolys.cyclotomic_poly">
-<tt class="descclassname">sympy.polys.specialpolys.</tt><tt class="descname">cyclotomic_poly</tt><big>(</big><em>n</em>, <em>x=None</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/specialpolys.html#cyclotomic_poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.specialpolys.cyclotomic_poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates cyclotomic polynomial of order <span class="math">\(n\)</span> in <span class="math">\(x\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.specialpolys.symmetric_poly">
-<tt class="descclassname">sympy.polys.specialpolys.</tt><tt class="descname">symmetric_poly</tt><big>(</big><em>n</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/specialpolys.html#symmetric_poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.specialpolys.symmetric_poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates symmetric polynomial of order <span class="math">\(n\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.specialpolys.random_poly">
-<tt class="descclassname">sympy.polys.specialpolys.</tt><tt class="descname">random_poly</tt><big>(</big><em>x</em>, <em>n</em>, <em>inf</em>, <em>sup</em>, <em>domain=ZZ</em>, <em>polys=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/specialpolys.html#random_poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.specialpolys.random_poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a polynomial of degree <tt class="docutils literal"><span class="pre">n</span></tt> with coefficients in <tt class="docutils literal"><span class="pre">[inf,</span> <span class="pre">sup]</span></tt>.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="orthogonal-polynomials">
-<h2>Orthogonal polynomials<a class="headerlink" href="#orthogonal-polynomials" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.polys.orthopolys.chebyshevt_poly">
-<tt class="descclassname">sympy.polys.orthopolys.</tt><tt class="descname">chebyshevt_poly</tt><big>(</big><em>n</em>, <em>x=None</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/orthopolys.html#chebyshevt_poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.orthopolys.chebyshevt_poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates Chebyshev polynomial of the first kind of degree <span class="math">\(n\)</span> in <span class="math">\(x\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.orthopolys.chebyshevu_poly">
-<tt class="descclassname">sympy.polys.orthopolys.</tt><tt class="descname">chebyshevu_poly</tt><big>(</big><em>n</em>, <em>x=None</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/orthopolys.html#chebyshevu_poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.orthopolys.chebyshevu_poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates Chebyshev polynomial of the second kind of degree <span class="math">\(n\)</span> in <span class="math">\(x\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.orthopolys.gegenbauer_poly">
-<tt class="descclassname">sympy.polys.orthopolys.</tt><tt class="descname">gegenbauer_poly</tt><big>(</big><em>n</em>, <em>a</em>, <em>x=None</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/orthopolys.html#gegenbauer_poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.orthopolys.gegenbauer_poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates Gegenbauer polynomial of degree <span class="math">\(n\)</span> in <span class="math">\(x\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.orthopolys.hermite_poly">
-<tt class="descclassname">sympy.polys.orthopolys.</tt><tt class="descname">hermite_poly</tt><big>(</big><em>n</em>, <em>x=None</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/orthopolys.html#hermite_poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.orthopolys.hermite_poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates Hermite polynomial of degree <span class="math">\(n\)</span> in <span class="math">\(x\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.orthopolys.jacobi_poly">
-<tt class="descclassname">sympy.polys.orthopolys.</tt><tt class="descname">jacobi_poly</tt><big>(</big><em>n</em>, <em>a</em>, <em>b</em>, <em>x=None</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/orthopolys.html#jacobi_poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.orthopolys.jacobi_poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates Jacobi polynomial of degree <span class="math">\(n\)</span> in <span class="math">\(x\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.orthopolys.legendre_poly">
-<tt class="descclassname">sympy.polys.orthopolys.</tt><tt class="descname">legendre_poly</tt><big>(</big><em>n</em>, <em>x=None</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/orthopolys.html#legendre_poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.orthopolys.legendre_poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates Legendre polynomial of degree <span class="math">\(n\)</span> in <span class="math">\(x\)</span>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.polys.orthopolys.laguerre_poly">
-<tt class="descclassname">sympy.polys.orthopolys.</tt><tt class="descname">laguerre_poly</tt><big>(</big><em>n</em>, <em>x=None</em>, <em>alpha=None</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/orthopolys.html#laguerre_poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.orthopolys.laguerre_poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates Laguerre polynomial of degree <span class="math">\(n\)</span> in <span class="math">\(x\)</span>.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="manipulation-of-rational-functions">
-<h2>Manipulation of rational functions<a class="headerlink" href="#manipulation-of-rational-functions" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.polys.rationaltools.together">
-<tt class="descclassname">sympy.polys.rationaltools.</tt><tt class="descname">together</tt><big>(</big><em>expr</em>, <em>deep=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/rationaltools.html#together"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.rationaltools.together" title="Permalink to this definition">¶</a></dt>
-<dd><p>Denest and combine rational expressions using symbolic methods.</p>
-<p>This function takes an expression or a container of expressions
-and puts it (them) together by denesting and combining rational
-subexpressions. No heroic measures are taken to minimize degree
-of the resulting numerator and denominator. To obtain completely
-reduced expression use <a class="reference internal" href="../galgebra/GA/GAsympy.html#cancel" title="cancel"><tt class="xref py py-func docutils literal"><span class="pre">cancel()</span></tt></a>. However, <a class="reference internal" href="#sympy.polys.rationaltools.together" title="sympy.polys.rationaltools.together"><tt class="xref py py-func docutils literal"><span class="pre">together()</span></tt></a>
-can preserve as much as possible of the structure of the input
-expression in the output (no expansion is performed).</p>
-<p>A wide variety of objects can be put together including lists,
-tuples, sets, relational objects, integrals and others. It is
-also possible to transform interior of function applications,
-by setting <tt class="docutils literal"><span class="pre">deep</span></tt> flag to <tt class="docutils literal"><span class="pre">True</span></tt>.</p>
-<p>By definition, <a class="reference internal" href="#sympy.polys.rationaltools.together" title="sympy.polys.rationaltools.together"><tt class="xref py py-func docutils literal"><span class="pre">together()</span></tt></a> is a complement to <tt class="xref py py-func docutils literal"><span class="pre">apart()</span></tt>,
-so <tt class="docutils literal"><span class="pre">apart(together(expr))</span></tt> should return expr unchanged. Note
-however, that <a class="reference internal" href="#sympy.polys.rationaltools.together" title="sympy.polys.rationaltools.together"><tt class="xref py py-func docutils literal"><span class="pre">together()</span></tt></a> uses only symbolic methods, so
-it might be necessary to use <a class="reference internal" href="../galgebra/GA/GAsympy.html#cancel" title="cancel"><tt class="xref py py-func docutils literal"><span class="pre">cancel()</span></tt></a> to perform algebraic
-simplification and minimise degree of the numerator and denominator.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">together</span><span class="p">,</span> <span class="n">exp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">y</span><span class="p">)</span>
-<span class="go">(x + y)/(x*y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">)</span>
-<span class="go">(x*y + x*z + y*z)/(x*y*z)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(x + y)/(x*y**2)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">y</span><span class="p">))</span>
-<span class="go">(x*(y + 1) + y*(x + 1))/((x + 1)*(y + 1))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">y</span><span class="p">))</span>
-<span class="go">exp(1/y + 1/x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">y</span><span class="p">),</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">exp((x + y)/(x*y))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
-<span class="go">(x + 1)*exp(-x)/x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="p">)))</span>
-<span class="go">(x*exp(x) + 1)*exp(-3*x)/x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="partial-fraction-decomposition">
-<h2>Partial fraction decomposition<a class="headerlink" href="#partial-fraction-decomposition" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.polys.partfrac.apart">
-<tt class="descclassname">sympy.polys.partfrac.</tt><tt class="descname">apart</tt><big>(</big><em>expr</em>, <em>*args</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/polys/partfrac.html#apart"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.polys.partfrac.apart" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute partial fraction decomposition of a rational function.</p>
-<p>Given a rational function <tt class="docutils literal"><span class="pre">f</span></tt> compute partial fraction decomposition
-of <tt class="docutils literal"><span class="pre">f</span></tt>. Two algorithms are available: one is based on undetermined
-coefficients method and the other is Bronstein&#8217;s full partial fraction
-decomposition algorithm.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.polys.partfrac</span> <span class="kn">import</span> <span class="n">apart</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">apart</span><span class="p">(</span><span class="n">y</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">-y/(x + 2) + y/(x + 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../index.html">
-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Polynomials Manipulation Module Reference</a><ul>
-<li><a class="reference internal" href="#basic-polynomial-manipulation-functions">Basic polynomial manipulation functions</a></li>
-<li><a class="reference internal" href="#extra-polynomial-manipulation-functions">Extra polynomial manipulation functions</a></li>
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-<li><a class="reference internal" href="#special-polynomials">Special polynomials</a></li>
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-  <div class="section" id="examples-from-wester-s-article">
-<span id="polys-wester"></span><h1>Examples from Wester&#8217;s Article<a class="headerlink" href="#examples-from-wester-s-article" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="introduction">
-<h2>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline">¶</a></h2>
-<p>In this tutorial we present examples from Wester&#8217;s article concerning
-comparison and critique of mathematical abilities of several computer
-algebra systems (see <a class="reference internal" href="#wester1999">[Wester1999]</a>). All the examples are related to
-polynomial and algebraic computations and SymPy specific remarks were
-added to all of them.</p>
-</div>
-<div class="section" id="examples">
-<h2>Examples<a class="headerlink" href="#examples" title="Permalink to this headline">¶</a></h2>
-<p>All examples in this tutorial are computable, so one can just copy and
-paste them into a Python shell and do something useful with them. All
-computations were done using the following setup:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">init_printing</span><span class="p">(</span><span class="n">use_unicode</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">wrap_line</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">no_global</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;x,y,z,s,c,n&#39;</span><span class="p">)</span>
-<span class="go">(x, y, z, s, c, n)</span>
-</pre></div>
-</div>
-<div class="section" id="simple-univariate-polynomial-factorization">
-<h3>Simple univariate polynomial factorization<a class="headerlink" href="#simple-univariate-polynomial-factorization" title="Permalink to this headline">¶</a></h3>
-<p>To obtain a factorization of a polynomial use <tt class="xref py py-func docutils literal"><span class="pre">factor()</span></tt> function.
-By default <tt class="xref py py-func docutils literal"><span class="pre">factor()</span></tt> returns the result in unevaluated form, so the
-content of the input polynomial is left unexpanded, as in the following
-example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="mi">6</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">2⋅(3⋅x - 5)</span>
-</pre></div>
-</div>
-<p>To achieve the same effect in a more systematic way use <tt class="xref py py-func docutils literal"><span class="pre">primitive()</span></tt>
-function, which returns the content and the primitive part of the input
-polynomial:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">primitive</span><span class="p">(</span><span class="mi">6</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">(2, 3⋅x - 5)</span>
-</pre></div>
-</div>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">The content and the primitive part can be computed only over a ring. To
-simplify coefficients of a polynomial over a field use <tt class="xref py py-func docutils literal"><span class="pre">monic()</span></tt>.</p>
-</div>
-</div>
-<div class="section" id="univariate-gcd-resultant-and-factorization">
-<h3>Univariate GCD, resultant and factorization<a class="headerlink" href="#univariate-gcd-resultant-and-factorization" title="Permalink to this headline">¶</a></h3>
-<p>Consider univariate polynomials <tt class="docutils literal"><span class="pre">f</span></tt>, <tt class="docutils literal"><span class="pre">g</span></tt> and <tt class="docutils literal"><span class="pre">h</span></tt> over integers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="mi">64</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">34</span> <span class="o">-</span> <span class="mi">21</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">47</span> <span class="o">-</span> <span class="mi">126</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">8</span> <span class="o">-</span> <span class="mi">46</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">5</span> <span class="o">-</span> <span class="mi">16</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">60</span> <span class="o">-</span> <span class="mi">81</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="mi">72</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">60</span> <span class="o">-</span> <span class="mi">25</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">25</span> <span class="o">-</span> <span class="mi">19</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">23</span> <span class="o">-</span> <span class="mi">22</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">39</span> <span class="o">-</span> <span class="mi">83</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">52</span> <span class="o">+</span> <span class="mi">54</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">10</span> <span class="o">+</span> <span class="mi">81</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="mi">34</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">19</span> <span class="o">-</span> <span class="mi">25</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">16</span> <span class="o">+</span> <span class="mi">70</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">7</span> <span class="o">+</span> <span class="mi">20</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">91</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">86</span>
-</pre></div>
-</div>
-<p>We can compute the greatest common divisor (GCD) of two polynomials using
-<tt class="xref py py-func docutils literal"><span class="pre">gcd()</span></tt> function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>We see that <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt> have no common factors. However, <tt class="docutils literal"><span class="pre">f*h</span></tt> and <tt class="docutils literal"><span class="pre">g*h</span></tt>
-have an obvious factor <tt class="docutils literal"><span class="pre">h</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gcd</span><span class="p">(</span><span class="n">expand</span><span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="n">h</span><span class="p">),</span> <span class="n">expand</span><span class="p">(</span><span class="n">g</span><span class="o">*</span><span class="n">h</span><span class="p">))</span> <span class="o">-</span> <span class="n">h</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>The same can be verified using the resultant of univariate polynomials:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">resultant</span><span class="p">(</span><span class="n">expand</span><span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="n">h</span><span class="p">),</span> <span class="n">expand</span><span class="p">(</span><span class="n">g</span><span class="o">*</span><span class="n">h</span><span class="p">))</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>Factorization of large univariate polynomials (of degree 120 in this case) over
-integers is also possible:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">expand</span><span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="n">g</span><span class="p">))</span>
-<span class="go"> ⎛    60       47       34        8       5     ⎞ ⎛    60       52     39       25       23       10     ⎞</span>
-<span class="go">-⎝16⋅x   + 21⋅x   - 64⋅x   + 126⋅x  + 46⋅x  + 81⎠⋅⎝72⋅x   - 83⋅x - 22⋅x   - 25⋅x   - 19⋅x   + 54⋅x   + 81⎠</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="multivariate-gcd-and-factorization">
-<h3>Multivariate GCD and factorization<a class="headerlink" href="#multivariate-gcd-and-factorization" title="Permalink to this headline">¶</a></h3>
-<p>What can be done in univariate case, can be also done for multivariate
-polynomials. Consider the following polynomials <tt class="docutils literal"><span class="pre">f</span></tt>, <tt class="docutils literal"><span class="pre">g</span></tt> and <tt class="docutils literal"><span class="pre">h</span></tt>
-in <span class="math">\(\mathbb{Z}[x,y,z]\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="mi">24</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">19</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">8</span> <span class="o">-</span> <span class="mi">47</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">17</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">5</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">8</span> <span class="o">+</span> <span class="mi">6</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">15</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">9</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">22</span> <span class="o">+</span> <span class="mi">5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="mi">34</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">5</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">8</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">13</span> <span class="o">+</span> <span class="mi">20</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">7</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">7</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">7</span> <span class="o">+</span> <span class="mi">12</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">9</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">16</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">4</span> <span class="o">+</span> <span class="mi">80</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">14</span><span class="o">*</span><span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="mi">11</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">12</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">7</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">13</span> <span class="o">-</span> <span class="mi">23</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">8</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">10</span> <span class="o">+</span> <span class="mi">47</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">17</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">5</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">8</span>
-</pre></div>
-</div>
-<p>As previously, we can verify that <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt> have no common factors:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gcd</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>However, <tt class="docutils literal"><span class="pre">f*h</span></tt> and <tt class="docutils literal"><span class="pre">g*h</span></tt> have an obvious factor <tt class="docutils literal"><span class="pre">h</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gcd</span><span class="p">(</span><span class="n">expand</span><span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="n">h</span><span class="p">),</span> <span class="n">expand</span><span class="p">(</span><span class="n">g</span><span class="o">*</span><span class="n">h</span><span class="p">))</span> <span class="o">-</span> <span class="n">h</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>Multivariate factorization of large polynomials is also possible:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">expand</span><span class="p">(</span><span class="n">f</span><span class="o">*</span><span class="n">g</span><span class="p">))</span>
-<span class="go">    7   ⎛   9  9  3       7  6       5    12       7⎞ ⎛   22       17  5  8      15  9  2         19  8    ⎞</span>
-<span class="go">-2⋅y ⋅z⋅⎝6⋅x ⋅y ⋅z  + 10⋅x ⋅z  + 17⋅x ⋅y⋅z   + 40⋅y ⎠⋅⎝3⋅x   + 47⋅x  ⋅y ⋅z  - 6⋅x  ⋅y ⋅z  - 24⋅x⋅y  ⋅z  - 5⎠</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="support-for-symbols-in-exponents">
-<h3>Support for symbols in exponents<a class="headerlink" href="#support-for-symbols-in-exponents" title="Permalink to this headline">¶</a></h3>
-<p>Polynomial manipulation functions provided by <tt class="xref py py-mod docutils literal"><span class="pre">sympy.polys</span></tt> are mostly
-used with integer exponents. However, it&#8217;s perfectly valid to compute with
-symbolic exponents, e.g.:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">gcd</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">4</span><span class="p">)</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">2</span><span class="p">),</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="p">)</span>
-<span class="go"> n</span>
-<span class="go">x</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="testing-if-polynomials-have-common-zeros">
-<h3>Testing if polynomials have common zeros<a class="headerlink" href="#testing-if-polynomials-have-common-zeros" title="Permalink to this headline">¶</a></h3>
-<p>To test if two polynomials have a root in common we can use <tt class="xref py py-func docutils literal"><span class="pre">resultant()</span></tt>
-function. The theory says that the resultant of two polynomials vanishes if
-there is a common zero of those polynomials. For example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">resultant</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>We can visualize this fact by factoring the polynomials:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">        ⎛   3        ⎞</span>
-<span class="go">(x + 1)⋅⎝3⋅x  + x - 2⎠</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">        ⎛ 2          ⎞</span>
-<span class="go">(x + 1)⋅⎝x  - 4⋅x + 5⎠</span>
-</pre></div>
-</div>
-<p>In both cases we obtained the factor <span class="math">\(x + 1\)</span> which tells us that the common
-root is <span class="math">\(x = -1\)</span>.</p>
-</div>
-<div class="section" id="normalizing-simple-rational-functions">
-<h3>Normalizing simple rational functions<a class="headerlink" href="#normalizing-simple-rational-functions" title="Permalink to this headline">¶</a></h3>
-<p>To remove common factors from the numerator and the denominator of a rational
-function the elegant way, use <a class="reference internal" href="../galgebra/GA/GAsympy.html#cancel" title="cancel"><tt class="xref py py-func docutils literal"><span class="pre">cancel()</span></tt></a> function. For example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">cancel</span><span class="p">((</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">4</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">4</span><span class="p">))</span>
-<span class="go">x - 2</span>
-<span class="go">─────</span>
-<span class="go">x + 2</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="expanding-expressions-and-factoring-back">
-<h3>Expanding expressions and factoring back<a class="headerlink" href="#expanding-expressions-and-factoring-back" title="Permalink to this headline">¶</a></h3>
-<p>One can work easily we expressions in both expanded and factored forms.
-Consider a polynomial <tt class="docutils literal"><span class="pre">f</span></tt> in expanded form. We differentiate it and
-factor the result back:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">expand</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">20</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-<span class="go">          19</span>
-<span class="go">20⋅(x + 1)</span>
-</pre></div>
-</div>
-<p>The same can be achieved in factored form:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">20</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">          19</span>
-<span class="go">20⋅(x + 1)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="factoring-in-terms-of-cyclotomic-polynomials">
-<h3>Factoring in terms of cyclotomic polynomials<a class="headerlink" href="#factoring-in-terms-of-cyclotomic-polynomials" title="Permalink to this headline">¶</a></h3>
-<p>SymPy can very efficiently decompose polynomials of the form <span class="math">\(x^n \pm 1\)</span> in
-terms of cyclotomic polynomials:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">15</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">        ⎛ 2        ⎞ ⎛ 4    3    2        ⎞ ⎛ 8    7    5    4    3       ⎞</span>
-<span class="go">(x - 1)⋅⎝x  + x + 1⎠⋅⎝x  + x  + x  + x + 1⎠⋅⎝x  - x  + x  - x  + x - x + 1⎠</span>
-</pre></div>
-</div>
-<p>The original Wester`s example was <span class="math">\(x^{100} - 1\)</span>, but was truncated for
-readability purpose. Note that this is not a big struggle for <tt class="xref py py-func docutils literal"><span class="pre">factor()</span></tt>
-to decompose polynomials of degree 1000 or greater.</p>
-</div>
-<div class="section" id="univariate-factoring-over-gaussian-numbers">
-<h3>Univariate factoring over Gaussian numbers<a class="headerlink" href="#univariate-factoring-over-gaussian-numbers" title="Permalink to this headline">¶</a></h3>
-<p>Consider a univariate polynomial <tt class="docutils literal"><span class="pre">f</span></tt> with integer coefficients:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">+</span> <span class="mi">8</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">77</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">18</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">153</span>
-</pre></div>
-</div>
-<p>We want to obtain a factorization of <tt class="docutils literal"><span class="pre">f</span></tt> over Gaussian numbers. To do this
-we use <tt class="xref py py-func docutils literal"><span class="pre">factor()</span></tt> as previously, but this time we set <tt class="docutils literal"><span class="pre">gaussian</span></tt> keyword
-to <tt class="docutils literal"><span class="pre">True</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">gaussian</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">  ⎛    3⋅ⅈ⎞ ⎛    3⋅ⅈ⎞</span>
-<span class="go">4⋅⎜x - ───⎟⋅⎜x + ───⎟⋅(x + 1 - 4⋅ⅈ)⋅(x + 1 + 4⋅ⅈ)</span>
-<span class="go">  ⎝     2 ⎠ ⎝     2 ⎠</span>
-</pre></div>
-</div>
-<p>As the result we got a splitting factorization of <tt class="docutils literal"><span class="pre">f</span></tt> with monic factors
-(this is a general rule when computing in a field with SymPy). The <tt class="docutils literal"><span class="pre">gaussian</span></tt>
-keyword is useful for improving code readability, however the same result can
-be computed using more general syntax:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">extension</span><span class="o">=</span><span class="n">I</span><span class="p">)</span>
-<span class="go">  ⎛    3⋅ⅈ⎞ ⎛    3⋅ⅈ⎞</span>
-<span class="go">4⋅⎜x - ───⎟⋅⎜x + ───⎟⋅(x + 1 - 4⋅ⅈ)⋅(x + 1 + 4⋅ⅈ)</span>
-<span class="go">  ⎝     2 ⎠ ⎝     2 ⎠</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="computing-with-automatic-field-extensions">
-<h3>Computing with automatic field extensions<a class="headerlink" href="#computing-with-automatic-field-extensions" title="Permalink to this headline">¶</a></h3>
-<p>Consider two univariate polynomials <tt class="docutils literal"><span class="pre">f</span></tt> and <tt class="docutils literal"><span class="pre">g</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mi">3</span><span class="p">)</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">2</span>
-</pre></div>
-</div>
-<p>We would like to reduce degrees of the numerator and the denominator of a
-rational function <tt class="docutils literal"><span class="pre">f/g</span></tt>. Do do this we employ <a class="reference internal" href="../galgebra/GA/GAsympy.html#cancel" title="cancel"><tt class="xref py py-func docutils literal"><span class="pre">cancel()</span></tt></a> function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">cancel</span><span class="p">(</span><span class="n">f</span><span class="o">/</span><span class="n">g</span><span class="p">)</span>
-<span class="go"> 3      2     ___  2             ___         ___</span>
-<span class="go">x  - 2⋅x  + ╲╱ 2 ⋅x  - 3⋅x - 2⋅╲╱ 2 ⋅x - 3⋅╲╱ 2</span>
-<span class="go">────────────────────────────────────────────────</span>
-<span class="go">                      2</span>
-<span class="go">                     x  - 2</span>
-</pre></div>
-</div>
-<p>Unfortunately nothing interesting happened. This is because by default SymPy
-treats <span class="math">\(\sqrt{2}\)</span> as a generator, obtaining a bivariate polynomial for the
-numerator. To make <a class="reference internal" href="../galgebra/GA/GAsympy.html#cancel" title="cancel"><tt class="xref py py-func docutils literal"><span class="pre">cancel()</span></tt></a> recognize algebraic properties of <span class="math">\(\sqrt{2}\)</span>,
-one needs to use <tt class="docutils literal"><span class="pre">extension</span></tt> keyword:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">cancel</span><span class="p">(</span><span class="n">f</span><span class="o">/</span><span class="n">g</span><span class="p">,</span> <span class="n">extension</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go"> 2</span>
-<span class="go">x  - 2⋅x - 3</span>
-<span class="go">────────────</span>
-<span class="go">       ___</span>
-<span class="go"> x - ╲╱ 2</span>
-</pre></div>
-</div>
-<p>Setting <tt class="docutils literal"><span class="pre">extension=True</span></tt> tells <a class="reference internal" href="../galgebra/GA/GAsympy.html#cancel" title="cancel"><tt class="xref py py-func docutils literal"><span class="pre">cancel()</span></tt></a> to find minimal algebraic
-number domain for the coefficients of <tt class="docutils literal"><span class="pre">f/g</span></tt>. The automatically inferred
-domain is <span class="math">\(\mathbb{Q}(\sqrt{2})\)</span>. If one doesn&#8217;t want to rely on automatic
-inference, the same result can be obtained by setting the <tt class="docutils literal"><span class="pre">extension</span></tt>
-keyword with an explicit algebraic number:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">cancel</span><span class="p">(</span><span class="n">f</span><span class="o">/</span><span class="n">g</span><span class="p">,</span> <span class="n">extension</span><span class="o">=</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go"> 2</span>
-<span class="go">x  - 2⋅x - 3</span>
-<span class="go">────────────</span>
-<span class="go">       ___</span>
-<span class="go"> x - ╲╱ 2</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="univariate-factoring-over-various-domains">
-<h3>Univariate factoring over various domains<a class="headerlink" href="#univariate-factoring-over-various-domains" title="Permalink to this headline">¶</a></h3>
-<p>Consider a univariate polynomial <tt class="docutils literal"><span class="pre">f</span></tt> with integer coefficients:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span>
-</pre></div>
-</div>
-<p>With <tt class="xref py py-mod docutils literal"><span class="pre">sympy.polys</span></tt> we can obtain factorizations of <tt class="docutils literal"><span class="pre">f</span></tt> over different
-domains, which includes:</p>
-<ul>
-<li><p class="first">rationals:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-<span class="go">⎛ 2        ⎞ ⎛ 2        ⎞</span>
-<span class="go">⎝x  - x - 1⎠⋅⎝x  + x - 1⎠</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">finite fields:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">modulus</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
-<span class="go">       2        2</span>
-<span class="go">(x - 2) ⋅(x + 2)</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">algebraic numbers:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">alg</span> <span class="o">=</span> <span class="n">AlgebraicNumber</span><span class="p">((</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">alias</span><span class="o">=</span><span class="s">&#39;alpha&#39;</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">extension</span><span class="o">=</span><span class="n">alg</span><span class="p">)</span>
-<span class="go">(x - 1 - α)⋅(x - α)⋅(x + α)⋅(x + 1 + α)</span>
-</pre></div>
-</div>
-</li>
-</ul>
-</div>
-<div class="section" id="factoring-polynomials-into-linear-factors">
-<h3>Factoring polynomials into linear factors<a class="headerlink" href="#factoring-polynomials-into-linear-factors" title="Permalink to this headline">¶</a></h3>
-<p>Currently SymPy can factor polynomials into irreducibles over various domains,
-which can result in a splitting factorization (into linear factors). However,
-there is currently no systematic way to infer a splitting field (algebraic
-number field) automatically. In future the following syntax will be
-implemented:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">7</span><span class="p">,</span> <span class="n">split</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">NotImplementedError</span>: <span class="n">&#39;split&#39; option is not implemented yet</span>
-</pre></div>
-</div>
-<p>Note this is different from <tt class="docutils literal"><span class="pre">extension=True</span></tt>, because the later only tells how
-expression parsing should be done, not what should be the domain of computation.
-One can simulate the <tt class="docutils literal"><span class="pre">split</span></tt> keyword for several classes of polynomials using
-<tt class="xref py py-func docutils literal"><span class="pre">solve()</span></tt> function.</p>
-</div>
-<div class="section" id="advanced-factoring-over-finite-fields">
-<h3>Advanced factoring over finite fields<a class="headerlink" href="#advanced-factoring-over-finite-fields" title="Permalink to this headline">¶</a></h3>
-<p>Consider a univariate polynomial <tt class="docutils literal"><span class="pre">f</span></tt> with integer coefficients:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">11</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">1</span>
-</pre></div>
-</div>
-<p>We can factor <tt class="docutils literal"><span class="pre">f</span></tt> over a large finite field <span class="math">\(F_{65537}\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">modulus</span><span class="o">=</span><span class="mi">65537</span><span class="p">)</span>
-<span class="go">⎛ 2        ⎞ ⎛ 9    8    6    5    3    2    ⎞</span>
-<span class="go">⎝x  + x + 1⎠⋅⎝x  - x  + x  - x  + x  - x  + 1⎠</span>
-</pre></div>
-</div>
-<p>and expand the resulting factorization back:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expand</span><span class="p">(</span><span class="n">_</span><span class="p">)</span>
-<span class="go"> 11</span>
-<span class="go">x   + x + 1</span>
-</pre></div>
-</div>
-<p>obtaining polynomial <tt class="docutils literal"><span class="pre">f</span></tt>. This was done using symmetric polynomial
-representation over finite fields The same thing can be done using
-non-symmetric representation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">modulus</span><span class="o">=</span><span class="mi">65537</span><span class="p">,</span> <span class="n">symmetric</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">⎛ 2        ⎞ ⎛ 9          8    6          5    3          2    ⎞</span>
-<span class="go">⎝x  + x + 1⎠⋅⎝x  + 65536⋅x  + x  + 65536⋅x  + x  + 65536⋅x  + 1⎠</span>
-</pre></div>
-</div>
-<p>As with symmetric representation we can expand the factorization
-to get the input polynomial back. This time, however, we need to
-truncate coefficients of the expanded polynomial modulo 65537:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">trunc</span><span class="p">(</span><span class="n">expand</span><span class="p">(</span><span class="n">_</span><span class="p">),</span> <span class="mi">65537</span><span class="p">)</span>
-<span class="go"> 11</span>
-<span class="go">x   + x + 1</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="working-with-expressions-as-polynomials">
-<h3>Working with expressions as polynomials<a class="headerlink" href="#working-with-expressions-as-polynomials" title="Permalink to this headline">¶</a></h3>
-<p>Consider a multivariate polynomial <tt class="docutils literal"><span class="pre">f</span></tt> in <span class="math">\(\mathbb{Z}[x,y,z]\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">expand</span><span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span><span class="o">**</span><span class="mi">20</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>We want to compute factorization of <tt class="docutils literal"><span class="pre">f</span></tt>. To do this we use <tt class="docutils literal"><span class="pre">factor</span></tt> as
-usually, however we note that the polynomial in consideration is already
-in expanded form, so we can tell the factorization routine to skip
-expanding <tt class="docutils literal"><span class="pre">f</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">expand</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">                 20</span>
-<span class="go">⎛       2      3⎞</span>
-<span class="go">⎝x - 2⋅y  + 3⋅z ⎠</span>
-</pre></div>
-</div>
-<p>The default in <tt class="xref py py-mod docutils literal"><span class="pre">sympy.polys</span></tt> is to expand all expressions given as
-arguments to polynomial manipulation functions and <tt class="xref py py-class docutils literal"><span class="pre">Poly</span></tt> class.
-If we know that expanding is unnecessary, then by setting <tt class="docutils literal"><span class="pre">expand=False</span></tt>
-we can save quite a lot of time for complicated inputs. This can be really
-important when computing with expressions like:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">expand</span><span class="p">((</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">tan</span><span class="p">(</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span><span class="o">**</span><span class="mi">20</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">expand</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">                                 20</span>
-<span class="go">⎛               2           3   ⎞</span>
-<span class="go">⎝-sin(x) + 2⋅cos (y) - 3⋅tan (z)⎠</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="computing-reduced-grobner-bases">
-<h3>Computing reduced Gröbner bases<a class="headerlink" href="#computing-reduced-grobner-bases" title="Permalink to this headline">¶</a></h3>
-<p>To compute a reduced Gröbner basis for a set of polynomials use
-<tt class="xref py py-func docutils literal"><span class="pre">groebner()</span></tt> function. The function accepts various monomial
-orderings, e.g.: <tt class="docutils literal"><span class="pre">lex</span></tt>, <tt class="docutils literal"><span class="pre">grlex</span></tt> and <tt class="docutils literal"><span class="pre">grevlex</span></tt>, or a user
-defined one, via <tt class="docutils literal"><span class="pre">order</span></tt> keyword. The <tt class="docutils literal"><span class="pre">lex</span></tt> ordering is the
-most interesting because it has elimination property, which means
-that if the system of polynomial equations to <tt class="xref py py-func docutils literal"><span class="pre">groebner()</span></tt> is
-zero-dimensional (has finite number of solutions) the last element
-of the basis is a univariate polynomial. Consider the following example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">expand</span><span class="p">((</span><span class="mi">1</span> <span class="o">-</span> <span class="n">c</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">5</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">s</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">5</span> <span class="o">*</span> <span class="p">(</span><span class="n">c</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">s</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">10</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">groebner</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">c</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">s</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span>
-<span class="go">             ⎛⎡ 2    2       20      18       16       14      12    10⎤                           ⎞</span>
-<span class="go">GroebnerBasis⎝⎣c  + s  - 1, c   - 5⋅c   + 10⋅c   - 10⋅c   + 5⋅c   - c  ⎦, s, c, domain=ℤ, order=lex⎠</span>
-</pre></div>
-</div>
-<p>The result is an ordinary Python list, so we can easily apply a function to
-all its elements, for example we can factor those elements:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">factor</span><span class="p">,</span> <span class="n">_</span><span class="p">))</span>
-<span class="go">⎡ 2    2       10        5        5⎤</span>
-<span class="go">⎣c  + s  - 1, c  ⋅(c - 1) ⋅(c + 1) ⎦</span>
-</pre></div>
-</div>
-<p>From the above we can easily find all solutions of the system of polynomial
-equations. Or we can use <tt class="xref py py-func docutils literal"><span class="pre">solve()</span></tt> to achieve this in a more systematic
-way:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">([</span><span class="n">f</span><span class="p">,</span> <span class="n">s</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span>
-<span class="go">[(-1, 0), (0, -1), (0, 1), (1, 0)]</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="multivariate-factoring-over-algebraic-numbers">
-<h3>Multivariate factoring over algebraic numbers<a class="headerlink" href="#multivariate-factoring-over-algebraic-numbers" title="Permalink to this headline">¶</a></h3>
-<p>Computing with multivariate polynomials over various domains is as simple as
-in univariate case. For example consider the following factorization over
-<span class="math">\(\mathbb{Q}(\sqrt{-3})\)</span>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">factor</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span> <span class="n">extension</span><span class="o">=</span><span class="n">sqrt</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">        ⎛      ⎛        ___  ⎞⎞ ⎛      ⎛        ___  ⎞⎞</span>
-<span class="go">        ⎜      ⎜  1   ╲╱ 3 ⋅ⅈ⎟⎟ ⎜      ⎜  1   ╲╱ 3 ⋅ⅈ⎟⎟</span>
-<span class="go">(x + y)⋅⎜x + y⋅⎜- ─ - ───────⎟⎟⋅⎜x + y⋅⎜- ─ + ───────⎟⎟</span>
-<span class="go">        ⎝      ⎝  2      2   ⎠⎠ ⎝      ⎝  2      2   ⎠⎠</span>
-</pre></div>
-</div>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">Currently multivariate polynomials over finite fields aren&#8217;t supported.</p>
-</div>
-</div>
-<div class="section" id="partial-fraction-decomposition">
-<h3>Partial fraction decomposition<a class="headerlink" href="#partial-fraction-decomposition" title="Permalink to this headline">¶</a></h3>
-<p>Consider a univariate rational function <tt class="docutils literal"><span class="pre">f</span></tt> with integer coefficients:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">5</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>To decompose <tt class="docutils literal"><span class="pre">f</span></tt> into partial fractions use <tt class="xref py py-func docutils literal"><span class="pre">apart()</span></tt> function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">apart</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-<span class="go">  3       2        2</span>
-<span class="go">───── - ───── + ────────</span>
-<span class="go">x + 2   x + 1          2</span>
-<span class="go">                (x + 1)</span>
-</pre></div>
-</div>
-<p>To return from partial fractions to the rational function use
-a composition of <tt class="xref py py-func docutils literal"><span class="pre">together()</span></tt> and <a class="reference internal" href="../galgebra/GA/GAsympy.html#cancel" title="cancel"><tt class="xref py py-func docutils literal"><span class="pre">cancel()</span></tt></a>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">cancel</span><span class="p">(</span><span class="n">together</span><span class="p">(</span><span class="n">_</span><span class="p">))</span>
-<span class="go">     2</span>
-<span class="go">    x  + 2⋅x + 3</span>
-<span class="go">───────────────────</span>
-<span class="go"> 3      2</span>
-<span class="go">x  + 4⋅x  + 5⋅x + 2</span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="section" id="literature">
-<h2>Literature<a class="headerlink" href="#literature" title="Permalink to this headline">¶</a></h2>
-<table class="docutils citation" frame="void" id="wester1999" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id1">[Wester1999]</a></td><td>Michael J. Wester, A Critique of the Mathematical Abilities of
-CA Systems, 1999, <a class="reference external" href="http://www.math.unm.edu/~wester/cas/book/Wester.pdf">http://www.math.unm.edu/~wester/cas/book/Wester.pdf</a></td></tr>
-</tbody>
-</table>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../index.html">
-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Examples from Wester&#8217;s Article</a><ul>
-<li><a class="reference internal" href="#introduction">Introduction</a></li>
-<li><a class="reference internal" href="#examples">Examples</a><ul>
-<li><a class="reference internal" href="#simple-univariate-polynomial-factorization">Simple univariate polynomial factorization</a></li>
-<li><a class="reference internal" href="#univariate-gcd-resultant-and-factorization">Univariate GCD, resultant and factorization</a></li>
-<li><a class="reference internal" href="#multivariate-gcd-and-factorization">Multivariate GCD and factorization</a></li>
-<li><a class="reference internal" href="#support-for-symbols-in-exponents">Support for symbols in exponents</a></li>
-<li><a class="reference internal" href="#testing-if-polynomials-have-common-zeros">Testing if polynomials have common zeros</a></li>
-<li><a class="reference internal" href="#normalizing-simple-rational-functions">Normalizing simple rational functions</a></li>
-<li><a class="reference internal" href="#expanding-expressions-and-factoring-back">Expanding expressions and factoring back</a></li>
-<li><a class="reference internal" href="#factoring-in-terms-of-cyclotomic-polynomials">Factoring in terms of cyclotomic polynomials</a></li>
-<li><a class="reference internal" href="#univariate-factoring-over-gaussian-numbers">Univariate factoring over Gaussian numbers</a></li>
-<li><a class="reference internal" href="#computing-with-automatic-field-extensions">Computing with automatic field extensions</a></li>
-<li><a class="reference internal" href="#univariate-factoring-over-various-domains">Univariate factoring over various domains</a></li>
-<li><a class="reference internal" href="#factoring-polynomials-into-linear-factors">Factoring polynomials into linear factors</a></li>
-<li><a class="reference internal" href="#advanced-factoring-over-finite-fields">Advanced factoring over finite fields</a></li>
-<li><a class="reference internal" href="#working-with-expressions-as-polynomials">Working with expressions as polynomials</a></li>
-<li><a class="reference internal" href="#computing-reduced-grobner-bases">Computing reduced Gröbner bases</a></li>
-<li><a class="reference internal" href="#multivariate-factoring-over-algebraic-numbers">Multivariate factoring over algebraic numbers</a></li>
-<li><a class="reference internal" href="#partial-fraction-decomposition">Partial fraction decomposition</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#literature">Literature</a></li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../polys/basics.html"
-                        title="previous chapter">Basic functionality of the module</a></p>
-  <h4>Next topic</h4>
-  <p class="topless"><a href="../polys/reference.html"
-                        title="next chapter">Polynomials Manipulation Module Reference</a></p>
-  <h3>This Page</h3>
-  <ul class="this-page-menu">
-    <li><a href="../../_sources/modules/polys/wester.txt"
-           rel="nofollow">Show Source</a></li>
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-  <div class="section" id="printing-system">
-<h1>Printing System<a class="headerlink" href="#printing-system" title="Permalink to this headline">¶</a></h1>
-<p>See the <a class="reference internal" href="../tutorial/tutorial.en.html#printing-tutorial"><em>Printing</em></a> section in Tutorial for introduction into
-printing.</p>
-<p>This guide documents the printing system in SymPy and how it works
-internally.</p>
-<div class="section" id="module-sympy.printing.printer">
-<span id="printer-class"></span><h2>Printer Class<a class="headerlink" href="#module-sympy.printing.printer" title="Permalink to this headline">¶</a></h2>
-<p>Printing subsystem driver</p>
-<p>SymPy&#8217;s printing system works the following way: Any expression can be
-passed to a designated Printer who then is responsible to return an
-adequate representation of that expression.</p>
-<dl class="docutils">
-<dt>The basic concept is the following:</dt>
-<dd><ol class="first last arabic simple">
-<li>Let the object print itself if it knows how.</li>
-<li>Take the best fitting method defined in the printer.</li>
-<li>As fall-back use the emptyPrinter method for the printer.</li>
-</ol>
-</dd>
-</dl>
-<p>Some more information how the single concepts work and who should use which:</p>
-<ol class="arabic">
-<li><p class="first">The object prints itself</p>
-<blockquote>
-<div><p>This was the original way of doing printing in sympy. Every class had
-its own latex, mathml, str and repr methods, but it turned out that it
-is hard to produce a high quality printer, if all the methods are spread
-out that far. Therefor all printing code was combined into the different
-printers, which works great for built-in sympy objects, but not that
-good for user defined classes where it is inconvenient to patch the
-printers.</p>
-<p>Nevertheless, to get a fitting representation, the printers look for a
-specific method in every object, that will be called if it&#8217;s available
-and is then responsible for the representation. The name of that method
-depends on the specific printer and is defined under
-Printer.printmethod.</p>
-</div></blockquote>
-</li>
-<li><p class="first">Take the best fitting method defined in the printer.</p>
-<blockquote>
-<div><p>The printer loops through expr classes (class + its bases), and tries
-to dispatch the work to _print_&lt;EXPR_CLASS&gt;</p>
-<p>e.g., suppose we have the following class hierarchy:</p>
-<div class="highlight-python"><pre>    Basic
-    |
-    Atom
-    |
-    Number
-    |
-Rational</pre>
-</div>
-<p>then, for expr=Rational(...), in order to dispatch, we will try
-calling printer methods as shown in the figure below:</p>
-<div class="highlight-python"><pre>p._print(expr)
-|
-|-- p._print_Rational(expr)
-|
-|-- p._print_Number(expr)
-|
-|-- p._print_Atom(expr)
-|
-`-- p._print_Basic(expr)</pre>
-</div>
-<p>if ._print_Rational method exists in the printer, then it is called,
-and the result is returned back.</p>
-<p>otherwise, we proceed with trying Rational bases in the inheritance
-order.</p>
-</div></blockquote>
-</li>
-<li><p class="first">As fall-back use the emptyPrinter method for the printer.</p>
-<blockquote>
-<div><p>As fall-back self.emptyPrinter will be called with the expression. If
-not defined in the Printer subclass this will be the same as str(expr).</p>
-</div></blockquote>
-</li>
-</ol>
-<p>The main class responsible for printing is <tt class="docutils literal"><span class="pre">Printer</span></tt> (see also its
-<a class="reference external" href="https://github.com/sympy/sympy/blob/master/sympy/printing/printer.py">source code</a>):</p>
-<dl class="class">
-<dt id="sympy.printing.printer.Printer">
-<em class="property">class </em><tt class="descclassname">sympy.printing.printer.</tt><tt class="descname">Printer</tt><big>(</big><em>settings=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/printer.html#Printer"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.printer.Printer" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generic printer</p>
-<p>Its job is to provide infrastructure for implementing new printers easily.</p>
-<p>Basically, if you want to implement a printer, all you have to do is:</p>
-<ol class="arabic">
-<li><p class="first">Subclass Printer.</p>
-</li>
-<li><p class="first">Define Printer.printmethod in your subclass.
-If a object has a method with that name, this method will be used
-for printing.</p>
-</li>
-<li><p class="first">In your subclass, define <tt class="docutils literal"><span class="pre">_print_&lt;CLASS&gt;</span></tt> methods</p>
-<p>For each class you want to provide printing to, define an appropriate
-method how to do it. For example if you want a class FOO to be printed in
-its own way, define _print_FOO:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="k">def</span> <span class="nf">_print_FOO</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
-    <span class="o">...</span>
-</pre></div>
-</div>
-<p>this should return how FOO instance e is printed</p>
-<p>Also, if <tt class="docutils literal"><span class="pre">BAR</span></tt> is a subclass of <tt class="docutils literal"><span class="pre">FOO</span></tt>, <tt class="docutils literal"><span class="pre">_print_FOO(bar)</span></tt> will
-be called for instance of <tt class="docutils literal"><span class="pre">BAR</span></tt>, if no <tt class="docutils literal"><span class="pre">_print_BAR</span></tt> is provided.
-Thus, usually, we don&#8217;t need to provide printing routines for every
-class we want to support &#8211; only generic routine has to be provided
-for a set of classes.</p>
-<p>A good example for this are functions - for example <tt class="docutils literal"><span class="pre">PrettyPrinter</span></tt>
-only defines <tt class="docutils literal"><span class="pre">_print_Function</span></tt>, and there is no <tt class="docutils literal"><span class="pre">_print_sin</span></tt>,
-<tt class="docutils literal"><span class="pre">_print_tan</span></tt>, etc...</p>
-<p>On the other hand, a good printer will probably have to define
-separate routines for <tt class="docutils literal"><span class="pre">Symbol</span></tt>, <tt class="docutils literal"><span class="pre">Atom</span></tt>, <tt class="docutils literal"><span class="pre">Number</span></tt>, <tt class="docutils literal"><span class="pre">Integral</span></tt>,
-<tt class="docutils literal"><span class="pre">Limit</span></tt>, etc...</p>
-</li>
-<li><p class="first">If convenient, override <tt class="docutils literal"><span class="pre">self.emptyPrinter</span></tt></p>
-<p>This callable will be called to obtain printing result as a last resort,
-that is when no appropriate print method was found for an expression.</p>
-</li>
-</ol>
-<p>Examples of overloading StrPrinter:</p>
-<div class="highlight-python"><pre>from sympy import Basic, Function, Symbol
-from sympy.printing.str import StrPrinter
-
-class CustomStrPrinter(StrPrinter):
-    &quot;&quot;&quot;
-    Examples of how to customize the StrPrinter for both a SymPy class and a
-    user defined class subclassed from the SymPy Basic class.
-    &quot;&quot;&quot;
-
-    def _print_Derivative(self, expr):
-        &quot;&quot;&quot;
-        Custom printing of the SymPy Derivative class.
-
-        Instead of:
-
-        D(x(t), t) or D(x(t), t, t)
-
-        We will print:
-
-        x&#x27;     or     x&#x27;&#x27;
-
-        In this example, expr.args == (x(t), t), and expr.args[0] == x(t), and
-        expr.args[0].func == x
-        &quot;&quot;&quot;
-        return str(expr.args[0].func) + &quot;&#x27;&quot;*len(expr.args[1:])
-
-    def _print_MyClass(self, expr):
-        &quot;&quot;&quot;
-        Print the characters of MyClass.s alternatively lower case and upper
-        case
-        &quot;&quot;&quot;
-        s = &quot;&quot;
-        i = 0
-        for char in expr.s:
-            if i % 2 == 0:
-                s += char.lower()
-            else:
-                s += char.upper()
-            i += 1
-        return s
-
-# Override the __str__ method of to use CustromStrPrinter
-Basic.__str__ = lambda self: CustomStrPrinter().doprint(self)
-# Demonstration of CustomStrPrinter:
-t = Symbol(&#x27;t&#x27;)
-x = Function(&#x27;x&#x27;)(t)
-dxdt = x.diff(t)            # dxdt is a Derivative instance
-d2xdt2 = dxdt.diff(t)       # dxdt2 is a Derivative instance
-ex = MyClass(&#x27;I like both lowercase and upper case&#x27;)
-
-print dxdt
-print d2xdt2
-print ex</pre>
-</div>
-<p>The output of the above code is:</p>
-<div class="highlight-python"><pre>x&#x27;
-x&#x27;&#x27;
-i lIkE BoTh lOwErCaSe aNd uPpEr cAsE</pre>
-</div>
-<p>By overriding Basic.__str__, we can customize the printing of anything that
-is subclassed from Basic.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="52%" />
-<col width="48%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>printmethod</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="attribute">
-<dt id="sympy.printing.printer.Printer.printmethod">
-<tt class="descname">printmethod</tt><em class="property"> = None</em><a class="headerlink" href="#sympy.printing.printer.Printer.printmethod" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.printer.Printer.doprint">
-<tt class="descname">doprint</tt><big>(</big><em>self</em>, <em>expr</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/printer.html#Printer.doprint"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.printer.Printer.doprint" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns printer&#8217;s representation for expr (as a string)</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.printer.Printer._print">
-<tt class="descname">_print</tt><big>(</big><em>self</em>, <em>expr</em>, <em>*args</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/printer.html#Printer._print"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.printer.Printer._print" title="Permalink to this definition">¶</a></dt>
-<dd><p>Internal dispatcher</p>
-<dl class="docutils">
-<dt>Tries the following concepts to print an expression:</dt>
-<dd><ol class="first last arabic simple">
-<li>Let the object print itself if it knows how.</li>
-<li>Take the best fitting method defined in the printer.</li>
-<li>As fall-back use the emptyPrinter method for the printer.</li>
-</ol>
-</dd>
-</dl>
-</dd></dl>
-
-<dl class="classmethod">
-<dt id="sympy.printing.printer.Printer.set_global_settings">
-<em class="property">classmethod </em><tt class="descname">set_global_settings</tt><big>(</big><em>**settings</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/printer.html#Printer.set_global_settings"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.printer.Printer.set_global_settings" title="Permalink to this definition">¶</a></dt>
-<dd><p>Set system-wide printing settings.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="prettyprinter-class">
-<h2>PrettyPrinter Class<a class="headerlink" href="#prettyprinter-class" title="Permalink to this headline">¶</a></h2>
-<p>Pretty printing subsystem is implemented in <tt class="docutils literal"><span class="pre">sympy.printing.pretty.pretty</span></tt>
-by the <tt class="docutils literal"><span class="pre">PrettyPrinter</span></tt> class deriving from <tt class="docutils literal"><span class="pre">Printer</span></tt>. It relies on
-modules <tt class="docutils literal"><span class="pre">sympy.printing.pretty.stringPict</span></tt>, and
-<tt class="docutils literal"><span class="pre">sympy.printing.pretty.pretty_symbology</span></tt> for rendering nice-looking
-formulas.</p>
-<p>The module <tt class="docutils literal"><span class="pre">stringPict</span></tt> provides a base class <tt class="docutils literal"><span class="pre">stringPict</span></tt> and a derived
-class <tt class="docutils literal"><span class="pre">prettyForm</span></tt> that ease the creation and manipulation of formulas
-that span across multiple lines.</p>
-<p>The module <tt class="docutils literal"><span class="pre">pretty_symbology</span></tt> provides primitives to construct 2D shapes
-(hline, vline, etc) together with a technique to use unicode automatically
-when possible.</p>
-<span class="target" id="module-sympy.printing.pretty.pretty"></span><dl class="class">
-<dt id="sympy.printing.pretty.pretty.PrettyPrinter">
-<em class="property">class </em><tt class="descclassname">sympy.printing.pretty.pretty.</tt><tt class="descname">PrettyPrinter</tt><big>(</big><em>settings=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/pretty.html#PrettyPrinter"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.pretty.PrettyPrinter" title="Permalink to this definition">¶</a></dt>
-<dd><p>Printer, which converts an expression into 2D ASCII-art figure.</p>
-<dl class="attribute">
-<dt id="sympy.printing.pretty.pretty.PrettyPrinter.printmethod">
-<tt class="descname">printmethod</tt><em class="property"> = '_pretty'</em><a class="headerlink" href="#sympy.printing.pretty.pretty.PrettyPrinter.printmethod" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.pretty.pretty">
-<tt class="descclassname">sympy.printing.pretty.pretty.</tt><tt class="descname">pretty</tt><big>(</big><em>expr</em>, <em>**settings</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/pretty.html#pretty"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.pretty.pretty" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a string containing the prettified form of expr.</p>
-<p>For information on keyword arguments see pretty_print function.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.pretty.pretty_print">
-<tt class="descclassname">sympy.printing.pretty.pretty.</tt><tt class="descname">pretty_print</tt><big>(</big><em>expr</em>, <em>**settings</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/pretty.html#pretty_print"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.pretty.pretty_print" title="Permalink to this definition">¶</a></dt>
-<dd><p>Prints expr in pretty form.</p>
-<p>pprint is just a shortcut for this function.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>expr</strong> : expression</p>
-<blockquote>
-<div><p>the expression to print</p>
-</div></blockquote>
-<p><strong>wrap_line</strong> : bool, optional</p>
-<blockquote>
-<div><p>line wrapping enabled/disabled, defaults to True</p>
-</div></blockquote>
-<p><strong>num_columns</strong> : bool, optional</p>
-<blockquote>
-<div><p>number of columns before line breaking (default to None which reads
-the terminal width), useful when using SymPy without terminal.</p>
-</div></blockquote>
-<p><strong>use_unicode</strong> : bool or None, optional</p>
-<blockquote>
-<div><p>use unicode characters, such as the Greek letter pi instead of
-the string pi.</p>
-</div></blockquote>
-<p><strong>full_prec</strong> : bool or string, optional</p>
-<blockquote>
-<div><p>use full precision. Default to &#8220;auto&#8221;</p>
-</div></blockquote>
-<p><strong>order</strong> : bool or string, optional</p>
-<blockquote class="last">
-<div><p>set to &#8216;none&#8217; for long expressions if slow; default is None</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.printing.ccode">
-<span id="ccodeprinter"></span><h2>CCodePrinter<a class="headerlink" href="#module-sympy.printing.ccode" title="Permalink to this headline">¶</a></h2>
-<p>This class implements C code printing (i.e. it converts Python expressions
-to strings of C code).</p>
-<p>Usage:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing</span> <span class="kn">import</span> <span class="n">print_ccode</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">Abs</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">print_ccode</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">pow(sin(x), 2) + pow(cos(x), 2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">print_ccode</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">assign_to</span><span class="o">=</span><span class="s">&quot;result&quot;</span><span class="p">)</span>
-<span class="go">result = 2*x + cos(x);</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">print_ccode</span><span class="p">(</span><span class="n">Abs</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">fabs(pow(x, 2))</span>
-</pre></div>
-</div>
-<dl class="data">
-<dt id="sympy.printing.ccode.known_functions">
-<tt class="descclassname">sympy.printing.ccode.</tt><tt class="descname">known_functions</tt><em class="property"> = {'ceiling': [(&lt;function &lt;lambda&gt; at 0x10c67f290&gt;, 'ceil')], 'Abs': [(&lt;function &lt;lambda&gt; at 0x10c67fb90&gt;, 'fabs')]}</em><a class="headerlink" href="#sympy.printing.ccode.known_functions" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.printing.ccode.CCodePrinter">
-<em class="property">class </em><tt class="descclassname">sympy.printing.ccode.</tt><tt class="descname">CCodePrinter</tt><big>(</big><em>settings={}</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/ccode.html#CCodePrinter"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.ccode.CCodePrinter" title="Permalink to this definition">¶</a></dt>
-<dd><p>A printer to convert python expressions to strings of c code</p>
-<dl class="attribute">
-<dt id="sympy.printing.ccode.CCodePrinter.printmethod">
-<tt class="descname">printmethod</tt><em class="property"> = '_ccode'</em><a class="headerlink" href="#sympy.printing.ccode.CCodePrinter.printmethod" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.ccode.CCodePrinter.doprint">
-<tt class="descname">doprint</tt><big>(</big><em>self</em>, <em>expr</em>, <em>assign_to=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/ccode.html#CCodePrinter.doprint"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.ccode.CCodePrinter.doprint" title="Permalink to this definition">¶</a></dt>
-<dd><p>Actually format the expression as C code.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.ccode.CCodePrinter.indent_code">
-<tt class="descname">indent_code</tt><big>(</big><em>self</em>, <em>code</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/ccode.html#CCodePrinter.indent_code"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.ccode.CCodePrinter.indent_code" title="Permalink to this definition">¶</a></dt>
-<dd><p>Accepts a string of code or a list of code lines</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.ccode.ccode">
-<tt class="descclassname">sympy.printing.ccode.</tt><tt class="descname">ccode</tt><big>(</big><em>expr</em>, <em>assign_to=None</em>, <em>**settings</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/ccode.html#ccode"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.ccode.ccode" title="Permalink to this definition">¶</a></dt>
-<dd><p>Converts an expr to a string of c code</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>expr</strong> : sympy.core.Expr</p>
-<blockquote>
-<div><p>a sympy expression to be converted</p>
-</div></blockquote>
-<p><strong>precision</strong> : optional</p>
-<blockquote>
-<div><p>the precision for numbers such as pi [default=15]</p>
-</div></blockquote>
-<p><strong>user_functions</strong> : optional</p>
-<blockquote>
-<div><p>A dictionary where keys are FunctionClass instances and values
-are there string representations.  Alternatively, the
-dictionary value can be a list of tuples i.e. [(argument_test,
-cfunction_string)].  See below for examples.</p>
-</div></blockquote>
-<p><strong>human</strong> : optional</p>
-<blockquote class="last">
-<div><p>If True, the result is a single string that may contain some
-constant declarations for the number symbols. If False, the
-same information is returned in a more programmer-friendly
-data structure.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">ccode</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">Rational</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">tau</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">([</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="s">&quot;tau&quot;</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ccode</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">tau</span><span class="p">)</span><span class="o">**</span><span class="n">Rational</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">&#39;8*sqrt(2)*pow(tau, 7.0/2.0)&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ccode</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">assign_to</span><span class="o">=</span><span class="s">&quot;s&quot;</span><span class="p">)</span>
-<span class="go">&#39;s = sin(x);&#39;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.ccode.print_ccode">
-<tt class="descclassname">sympy.printing.ccode.</tt><tt class="descname">print_ccode</tt><big>(</big><em>expr</em>, <em>**settings</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/ccode.html#print_ccode"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.ccode.print_ccode" title="Permalink to this definition">¶</a></dt>
-<dd><p>Prints C representation of the given expression.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="fortran-printing">
-<h2>Fortran Printing<a class="headerlink" href="#fortran-printing" title="Permalink to this headline">¶</a></h2>
-<p>The fcode function translates a sympy expression into Fortran code. The main
-purpose is to take away the burden of manually translating long mathematical
-expressions. Therefore the resulting expression should also require no (or
-very little) manual tweaking to make it compilable. The optional arguments
-of fcode can be used to fine-tune the behavior of fcode in such a way that
-manual changes in the result are no longer needed.</p>
-<span class="target" id="module-sympy.printing.fcode"></span><dl class="function">
-<dt id="sympy.printing.fcode.fcode">
-<tt class="descclassname">sympy.printing.fcode.</tt><tt class="descname">fcode</tt><big>(</big><em>expr</em>, <em>**settings</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/fcode.html#fcode"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.fcode.fcode" title="Permalink to this definition">¶</a></dt>
-<dd><p>Converts an expr to a string of Fortran 77 code</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>expr</strong> : sympy.core.Expr</p>
-<blockquote>
-<div><p>a sympy expression to be converted</p>
-</div></blockquote>
-<p><strong>assign_to</strong> : optional</p>
-<blockquote>
-<div><p>When given, the argument is used as the name of the
-variable to which the Fortran expression is assigned.
-(This is helpful in case of line-wrapping.)</p>
-</div></blockquote>
-<p><strong>precision</strong> : optional</p>
-<blockquote>
-<div><p>the precision for numbers such as pi [default=15]</p>
-</div></blockquote>
-<p><strong>user_functions</strong> : optional</p>
-<blockquote>
-<div><p>A dictionary where keys are FunctionClass instances and values
-are there string representations.</p>
-</div></blockquote>
-<p><strong>human</strong> : optional</p>
-<blockquote>
-<div><p>If True, the result is a single string that may contain some
-parameter statements for the number symbols. If False, the same
-information is returned in a more programmer-friendly data
-structure.</p>
-</div></blockquote>
-<p><strong>source_format</strong> : optional</p>
-<blockquote class="last">
-<div><p>The source format can be either &#8216;fixed&#8217; or &#8216;free&#8217;.
-[default=&#8217;fixed&#8217;]</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">fcode</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">Rational</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">tau</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,tau&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fcode</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">tau</span><span class="p">)</span><span class="o">**</span><span class="n">Rational</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">&#39;      8*sqrt(2.0d0)*tau**(7.0d0/2.0d0)&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fcode</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">assign_to</span><span class="o">=</span><span class="s">&quot;s&quot;</span><span class="p">)</span>
-<span class="go">&#39;      s = sin(x)&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">fcode</span><span class="p">(</span><span class="n">pi</span><span class="p">))</span>
-<span class="go">      parameter (pi = 3.14159265358979d0)</span>
-<span class="go">      pi</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.fcode.print_fcode">
-<tt class="descclassname">sympy.printing.fcode.</tt><tt class="descname">print_fcode</tt><big>(</big><em>expr</em>, <em>**settings</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/fcode.html#print_fcode"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.fcode.print_fcode" title="Permalink to this definition">¶</a></dt>
-<dd><p>Prints the Fortran representation of the given expression.</p>
-<p>See fcode for the meaning of the optional arguments.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.printing.fcode.FCodePrinter">
-<em class="property">class </em><tt class="descclassname">sympy.printing.fcode.</tt><tt class="descname">FCodePrinter</tt><big>(</big><em>settings=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/fcode.html#FCodePrinter"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.fcode.FCodePrinter" title="Permalink to this definition">¶</a></dt>
-<dd><p>A printer to convert sympy expressions to strings of Fortran code</p>
-<dl class="attribute">
-<dt id="sympy.printing.fcode.FCodePrinter.printmethod">
-<tt class="descname">printmethod</tt><em class="property"> = '_fcode'</em><a class="headerlink" href="#sympy.printing.fcode.FCodePrinter.printmethod" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.fcode.FCodePrinter.doprint">
-<tt class="descname">doprint</tt><big>(</big><em>self</em>, <em>expr</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/fcode.html#FCodePrinter.doprint"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.fcode.FCodePrinter.doprint" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns Fortran code for expr (as a string)</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.fcode.FCodePrinter.indent_code">
-<tt class="descname">indent_code</tt><big>(</big><em>self</em>, <em>code</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/fcode.html#FCodePrinter.indent_code"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.fcode.FCodePrinter.indent_code" title="Permalink to this definition">¶</a></dt>
-<dd><p>Accepts a string of code or a list of code lines</p>
-</dd></dl>
-
-</dd></dl>
-
-<p>Two basic examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fcode</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">&#39;      sqrt(-x**2 + 1)&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fcode</span><span class="p">((</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">I</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">conjugate</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
-<span class="go">&#39;      (cmplx(3,4))/(-conjg(x) + 1)&#39;</span>
-</pre></div>
-</div>
-<p>An example where line wrapping is required:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expr</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">n</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span><span class="o">.</span><span class="n">removeO</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">fcode</span><span class="p">(</span><span class="n">expr</span><span class="p">))</span>
-<span class="go">      -715*x**18/65536 - 429*x**16/32768 - 33*x**14/2048 - 21*x**12/1024</span>
-<span class="go">     @ - 7*x**10/256 - 5*x**8/128 - x**6/16 - x**4/8 - x**2/2 + 1</span>
-</pre></div>
-</div>
-<p>In case of line wrapping, it is handy to include the assignment so that lines
-are wrapped properly when the assignment part is added.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">fcode</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">assign_to</span><span class="o">=</span><span class="s">&quot;var&quot;</span><span class="p">))</span>
-<span class="go">      var = -715*x**18/65536 - 429*x**16/32768 - 33*x**14/2048 - 21*x**</span>
-<span class="go">     @ 12/1024 - 7*x**10/256 - 5*x**8/128 - x**6/16 - x**4/8 - x**2/2 +</span>
-<span class="go">     @ 1</span>
-</pre></div>
-</div>
-<p>For piecewise functions, the assign_to option is mandatory:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">fcode</span><span class="p">(</span><span class="n">Piecewise</span><span class="p">((</span><span class="n">x</span><span class="p">,</span><span class="n">x</span><span class="o">&lt;</span><span class="mi">1</span><span class="p">),(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span><span class="bp">True</span><span class="p">)),</span> <span class="n">assign_to</span><span class="o">=</span><span class="s">&quot;var&quot;</span><span class="p">))</span>
-<span class="go">      if (x &lt; 1) then</span>
-<span class="go">        var = x</span>
-<span class="go">      else</span>
-<span class="go">        var = x**2</span>
-<span class="go">      end if</span>
-</pre></div>
-</div>
-<p>Note that only top-level piecewise functions are supported due to the lack of
-a conditional operator in Fortran. Nested piecewise functions would require the
-introduction of temporary variables, which is a type of expression manipulation
-that goes beyond the scope of fcode.</p>
-<p>Loops are generated if there are Indexed objects in the expression.  This
-also requires use of the assign_to option.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="p">,</span> <span class="n">B</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">IndexedBase</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="s">&#39;B&#39;</span><span class="p">]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;m&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Idx</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">fcode</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">assign_to</span><span class="o">=</span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span>
-<span class="go">    do i = 1, m</span>
-<span class="go">        A(i) = 2*B(i)</span>
-<span class="go">    end do</span>
-</pre></div>
-</div>
-<p>Repeated indices in an expression with Indexed objects are interpreted as
-summation. For instance, code for the trace of a matrix can be generated
-with</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">fcode</span><span class="p">(</span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">],</span> <span class="n">assign_to</span><span class="o">=</span><span class="n">x</span><span class="p">))</span>
-<span class="go">      x = 0</span>
-<span class="go">      do i = 1, m</span>
-<span class="go">          x = x + A(i, i)</span>
-<span class="go">      end do</span>
-</pre></div>
-</div>
-<p>By default, number symbols such as <tt class="docutils literal"><span class="pre">pi</span></tt> and <tt class="docutils literal"><span class="pre">E</span></tt> are detected and defined as
-Fortran parameters. The precision of the constants can be tuned with the
-precision argument. Parameter definitions are easily avoided using the <tt class="docutils literal"><span class="pre">N</span></tt>
-function.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">fcode</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">pi</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">E</span><span class="p">))</span>
-<span class="go">      parameter (E = 2.71828182845905d0)</span>
-<span class="go">      parameter (pi = 3.14159265358979d0)</span>
-<span class="go">      x - pi**2 - E</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">fcode</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">pi</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">E</span><span class="p">,</span> <span class="n">precision</span><span class="o">=</span><span class="mi">25</span><span class="p">))</span>
-<span class="go">      parameter (E = 2.718281828459045235360287d0)</span>
-<span class="go">      parameter (pi = 3.141592653589793238462643d0)</span>
-<span class="go">      x - pi**2 - E</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">fcode</span><span class="p">(</span><span class="n">N</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="mi">25</span><span class="p">)))</span>
-<span class="go">      x - 9.869604401089358618834491d0</span>
-</pre></div>
-</div>
-<p>When some functions are not part of the Fortran standard, it might be desirable
-to introduce the names of user-defined functions in the Fortran expression.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">fcode</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">gamma</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">user_functions</span><span class="o">=</span><span class="p">{</span><span class="n">gamma</span><span class="p">:</span> <span class="s">&#39;mygamma&#39;</span><span class="p">}))</span>
-<span class="go">      -mygamma(x)**2 + 1</span>
-</pre></div>
-</div>
-<p>However, when the user_functions argument is not provided, fcode attempts to
-use a reasonable default and adds a comment to inform the user of the issue.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">fcode</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">gamma</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">C     Not Fortran:</span>
-<span class="go">C     gamma(x)</span>
-<span class="go">      -gamma(x)**2 + 1</span>
-</pre></div>
-</div>
-<p>By default the output is human readable code, ready for copy and paste. With the
-option <tt class="docutils literal"><span class="pre">human=False</span></tt>, the return value is suitable for post-processing with
-source code generators that write routines with multiple instructions. The
-return value is a three-tuple containing: (i) a set of number symbols that must
-be defined as &#8216;Fortran parameters&#8217;, (ii) a list functions that can not be
-translated in pure Fortran and (iii) a string of Fortran code. A few examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fcode</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">gamma</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">human</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">(set(), set([gamma(x)]), &#39;      -gamma(x)**2 + 1&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fcode</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">human</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">(set(), set(), &#39;      -sin(x)**2 + 1&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fcode</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">human</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">(set([(pi, &#39;3.14159265358979d0&#39;)]), set(), &#39;      x - pi**2&#39;)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="module-sympy.printing.gtk">
-<span id="gtk"></span><h2>Gtk<a class="headerlink" href="#module-sympy.printing.gtk" title="Permalink to this headline">¶</a></h2>
-<p>You can print to a grkmathview widget using the function <tt class="docutils literal"><span class="pre">print_gtk</span></tt>
-located in <tt class="docutils literal"><span class="pre">sympy.printing.gtk</span></tt> (it requires to have installed
-gtkmatmatview and libgtkmathview-bin in some systems).</p>
-<p>GtkMathView accepts MathML, so this rendering depends on the MathML
-representation of the expression.</p>
-<p>Usage:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="n">print_gtk</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">))</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.printing.gtk.print_gtk">
-<tt class="descclassname">sympy.printing.gtk.</tt><tt class="descname">print_gtk</tt><big>(</big><em>x</em>, <em>start_viewer=True</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/gtk.html#print_gtk"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.gtk.print_gtk" title="Permalink to this definition">¶</a></dt>
-<dd><p>Print to Gtkmathview, a gtk widget capable of rendering MathML.</p>
-<p>Needs libgtkmathview-bin</p>
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.printing.lambdarepr">
-<span id="lambdaprinter"></span><h2>LambdaPrinter<a class="headerlink" href="#module-sympy.printing.lambdarepr" title="Permalink to this headline">¶</a></h2>
-<p>This classes implements printing to strings that can be used by the
-<a class="reference internal" href="utilities/lambdify.html#sympy.utilities.lambdify.lambdify" title="sympy.utilities.lambdify.lambdify"><tt class="xref py py-func docutils literal"><span class="pre">sympy.utilities.lambdify.lambdify()</span></tt></a> function.</p>
-<dl class="class">
-<dt id="sympy.printing.lambdarepr.LambdaPrinter">
-<em class="property">class </em><tt class="descclassname">sympy.printing.lambdarepr.</tt><tt class="descname">LambdaPrinter</tt><big>(</big><em>settings=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/lambdarepr.html#LambdaPrinter"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.lambdarepr.LambdaPrinter" title="Permalink to this definition">¶</a></dt>
-<dd><p>This printer converts expressions into strings that can be used by
-lambdify.</p>
-<dl class="attribute">
-<dt id="sympy.printing.lambdarepr.LambdaPrinter.printmethod">
-<tt class="descname">printmethod</tt><em class="property"> = '_sympystr'</em><a class="headerlink" href="#sympy.printing.lambdarepr.LambdaPrinter.printmethod" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.lambdarepr.lambdarepr">
-<tt class="descclassname">sympy.printing.lambdarepr.</tt><tt class="descname">lambdarepr</tt><big>(</big><em>expr</em>, <em>**settings</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/lambdarepr.html#lambdarepr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.lambdarepr.lambdarepr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a string usable for lambdifying.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.printing.latex">
-<span id="latexprinter"></span><h2>LatexPrinter<a class="headerlink" href="#module-sympy.printing.latex" title="Permalink to this headline">¶</a></h2>
-<p>This class implements LaTeX printing. See <tt class="docutils literal"><span class="pre">sympy.printing.latex</span></tt>.</p>
-<p>See also the extended LatexPrinter: <a class="reference internal" href="galgebra/latex_ex/latex_ex.html#extended-latex"><em>Extended LaTeXModule for SymPy</em></a></p>
-<dl class="data">
-<dt id="sympy.printing.latex.accepted_latex_functions">
-<tt class="descclassname">sympy.printing.latex.</tt><tt class="descname">accepted_latex_functions</tt><em class="property"> = ['arcsin', 'arccos', 'arctan', 'sin', 'cos', 'tan', 'theta', 'beta', 'alpha', 'gamma', 'sinh', 'cosh', 'tanh', 'sqrt', 'ln', 'log', 'sec', 'csc', 'cot', 'coth', 're', 'im', 'frac', 'root', 'arg', 'zeta', 'psi']</em><a class="headerlink" href="#sympy.printing.latex.accepted_latex_functions" title="Permalink to this definition">¶</a></dt>
-<dd><p>list() -&gt; new empty list
-list(iterable) -&gt; new list initialized from iterable&#8217;s items</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.printing.latex.LatexPrinter">
-<em class="property">class </em><tt class="descclassname">sympy.printing.latex.</tt><tt class="descname">LatexPrinter</tt><big>(</big><em>settings=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/latex.html#LatexPrinter"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.latex.LatexPrinter" title="Permalink to this definition">¶</a></dt>
-<dd><dl class="attribute">
-<dt id="sympy.printing.latex.LatexPrinter.printmethod">
-<tt class="descname">printmethod</tt><em class="property"> = '_latex'</em><a class="headerlink" href="#sympy.printing.latex.LatexPrinter.printmethod" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.latex.latex">
-<tt class="descclassname">sympy.printing.latex.</tt><tt class="descname">latex</tt><big>(</big><em>expr</em>, <em>**settings</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/latex.html#latex"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.latex.latex" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert the given expression to LaTeX representation.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">latex</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">asin</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">tau</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">tau</span><span class="p">)</span><span class="o">**</span><span class="n">Rational</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">&#39;8 \\sqrt{2} \\tau^{\\frac{7}{2}}&#39;</span>
-</pre></div>
-</div>
-<p>order: Any of the supported monomial orderings (currently &#8220;lex&#8221;, &#8220;grlex&#8221;, or
-&#8220;grevlex&#8221;), &#8220;old&#8221;, and &#8220;none&#8221;. This parameter does nothing for Mul objects.
-Setting order to &#8220;old&#8221; uses the compatibility ordering for Add defined in
-Printer. For very large expressions, set the &#8216;order&#8217; keyword to &#8216;none&#8217; if
-speed is a concern.</p>
-<p>mode: Specifies how the generated code will be delimited. &#8216;mode&#8217; can be one
-of &#8216;plain&#8217;, &#8216;inline&#8217;, &#8216;equation&#8217; or &#8216;equation*&#8217;.  If &#8216;mode&#8217; is set to
-&#8216;plain&#8217;, then the resulting code will not be delimited at all (this is the
-default). If &#8216;mode&#8217; is set to &#8216;inline&#8217; then inline LaTeX $ $ will be used.
-If &#8216;mode&#8217; is set to &#8216;equation&#8217; or &#8216;equation*&#8217;, the resulting code will be
-enclosed in the &#8216;equation&#8217; or &#8216;equation*&#8217; environment (remember to import
-&#8216;amsmath&#8217; for &#8216;equation*&#8217;), unless the &#8216;itex&#8217; option is set. In the latter
-case, the <tt class="docutils literal"><span class="pre">$$</span> <span class="pre">$$</span></tt> syntax is used.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">mu</span><span class="p">)</span><span class="o">**</span><span class="n">Rational</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;plain&#39;</span><span class="p">)</span>
-<span class="go">&#39;8 \\sqrt{2} \\mu^{\\frac{7}{2}}&#39;</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">tau</span><span class="p">)</span><span class="o">**</span><span class="n">Rational</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;inline&#39;</span><span class="p">)</span>
-<span class="go">&#39;$8 \\sqrt{2} \\tau^{\\frac{7}{2}}$&#39;</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">mu</span><span class="p">)</span><span class="o">**</span><span class="n">Rational</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;equation*&#39;</span><span class="p">)</span>
-<span class="go">&#39;\\begin{equation*}8 \\sqrt{2} \\mu^{\\frac{7}{2}}\\end{equation*}&#39;</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">mu</span><span class="p">)</span><span class="o">**</span><span class="n">Rational</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;equation&#39;</span><span class="p">)</span>
-<span class="go">&#39;\\begin{equation}8 \\sqrt{2} \\mu^{\\frac{7}{2}}\\end{equation}&#39;</span>
-</pre></div>
-</div>
-<p>itex: Specifies if itex-specific syntax is used, including emitting <tt class="docutils literal"><span class="pre">$$</span> <span class="pre">$$</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">mu</span><span class="p">)</span><span class="o">**</span><span class="n">Rational</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;equation&#39;</span><span class="p">,</span> <span class="n">itex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">&#39;$$8 \\sqrt{2} \\mu^{\\frac{7}{2}}$$&#39;</span>
-</pre></div>
-</div>
-<p>fold_frac_powers: Emit &#8220;^{p/q}&#8221; instead of &#8220;^{frac{p}{q}}&#8221; for fractional
-powers.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">tau</span><span class="p">)</span><span class="o">**</span><span class="n">Rational</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="n">fold_frac_powers</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">&#39;8 \\sqrt{2} \\tau^{7/2}&#39;</span>
-</pre></div>
-</div>
-<p>fold_func_brackets: Fold function brackets where applicable.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">tau</span><span class="p">)</span><span class="o">**</span><span class="n">sin</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span><span class="mi">2</span><span class="p">)))</span>
-<span class="go">&#39;\\left(2 \\tau\\right)^{\\sin{\\left (\\frac{7}{2} \\right )}}&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">tau</span><span class="p">)</span><span class="o">**</span><span class="n">sin</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span><span class="mi">2</span><span class="p">)),</span> <span class="n">fold_func_brackets</span> <span class="o">=</span> <span class="bp">True</span><span class="p">)</span>
-<span class="go">&#39;\\left(2 \\tau\\right)^{\\sin {\\frac{7}{2}}}&#39;</span>
-</pre></div>
-</div>
-<p>mul_symbol: The symbol to use for multiplication. Can be one of None,
-&#8220;ldot&#8221;, &#8220;dot&#8221;, or &#8220;times&#8221;.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">((</span><span class="mi">2</span><span class="o">*</span><span class="n">tau</span><span class="p">)</span><span class="o">**</span><span class="n">sin</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span><span class="mi">2</span><span class="p">)),</span> <span class="n">mul_symbol</span><span class="o">=</span><span class="s">&quot;times&quot;</span><span class="p">)</span>
-<span class="go">&#39;\\left(2 \\times \\tau\\right)^{\\sin{\\left (\\frac{7}{2} \\right )}}&#39;</span>
-</pre></div>
-</div>
-<p>inv_trig_style: How inverse trig functions should be displayed. Can be one
-of &#8220;abbreviated&#8221;, &#8220;full&#8221;, or &#8220;power&#8221;. Defaults to &#8220;abbreviated&#8221;.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">asin</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span><span class="mi">2</span><span class="p">)))</span>
-<span class="go">&#39;\\operatorname{asin}{\\left (\\frac{7}{2} \\right )}&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">asin</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span><span class="mi">2</span><span class="p">)),</span> <span class="n">inv_trig_style</span><span class="o">=</span><span class="s">&quot;full&quot;</span><span class="p">)</span>
-<span class="go">&#39;\\arcsin{\\left (\\frac{7}{2} \\right )}&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">asin</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span><span class="mi">2</span><span class="p">)),</span> <span class="n">inv_trig_style</span><span class="o">=</span><span class="s">&quot;power&quot;</span><span class="p">)</span>
-<span class="go">&#39;\\sin^{-1}{\\left (\\frac{7}{2} \\right )}&#39;</span>
-</pre></div>
-</div>
-<p>mat_str: Which matrix environment string to emit. &#8220;smallmatrix&#8221;, &#8220;bmatrix&#8221;,
-etc. Defaults to &#8220;smallmatrix&#8221;.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">Matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">]),</span> <span class="n">mat_str</span> <span class="o">=</span> <span class="s">&quot;array&quot;</span><span class="p">)</span>
-<span class="go">&#39;\\left[\\begin{array}x\\\\y\\end{array}\\right]&#39;</span>
-</pre></div>
-</div>
-<p>mat_delim: The delimiter to wrap around matrices. Can be one of &#8220;[&#8221;, &#8220;(&#8221;,
-or the empty string. Defaults to &#8220;[&#8221;.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">Matrix</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">]),</span> <span class="n">mat_delim</span><span class="o">=</span><span class="s">&quot;(&quot;</span><span class="p">)</span>
-<span class="go">&#39;\\left(\\begin{smallmatrix}x\\\\y\\end{smallmatrix}\\right)&#39;</span>
-</pre></div>
-</div>
-<p>symbol_names: Dictionary of symbols and the custom strings they should be
-emitted as.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">symbol_names</span><span class="o">=</span><span class="p">{</span><span class="n">x</span><span class="p">:</span><span class="s">&#39;x_i&#39;</span><span class="p">})</span>
-<span class="go">&#39;x_i^{2}&#39;</span>
-</pre></div>
-</div>
-<p>Besides all Basic based expressions, you can recursively
-convert Python containers (lists, tuples and dicts) and
-also SymPy matrices:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">([</span><span class="mi">2</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">],</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;inline&#39;</span><span class="p">)</span>
-<span class="go">&#39;$\\begin{bmatrix}\\frac{2}{x}, &amp; y\\end{bmatrix}$&#39;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.latex.print_latex">
-<tt class="descclassname">sympy.printing.latex.</tt><tt class="descname">print_latex</tt><big>(</big><em>expr</em>, <em>**settings</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/latex.html#print_latex"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.latex.print_latex" title="Permalink to this definition">¶</a></dt>
-<dd><p>Prints LaTeX representation of the given expression.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.printing.mathml">
-<span id="mathmlprinter"></span><h2>MathMLPrinter<a class="headerlink" href="#module-sympy.printing.mathml" title="Permalink to this headline">¶</a></h2>
-<p>This class is responsible for MathML printing. See <tt class="docutils literal"><span class="pre">sympy.printing.mathml</span></tt>.</p>
-<p>More info on mathml content: <a class="reference external" href="http://www.w3.org/TR/MathML2/chapter4.html">http://www.w3.org/TR/MathML2/chapter4.html</a></p>
-<dl class="class">
-<dt id="sympy.printing.mathml.MathMLPrinter">
-<em class="property">class </em><tt class="descclassname">sympy.printing.mathml.</tt><tt class="descname">MathMLPrinter</tt><big>(</big><em>settings=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/mathml.html#MathMLPrinter"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.mathml.MathMLPrinter" title="Permalink to this definition">¶</a></dt>
-<dd><p>Prints an expression to the MathML markup language</p>
-<p>Whenever possible tries to use Content markup and not Presentation markup.</p>
-<p>References: <a class="reference external" href="http://www.w3.org/TR/MathML2/">http://www.w3.org/TR/MathML2/</a></p>
-<dl class="attribute">
-<dt id="sympy.printing.mathml.MathMLPrinter.printmethod">
-<tt class="descname">printmethod</tt><em class="property"> = '_mathml'</em><a class="headerlink" href="#sympy.printing.mathml.MathMLPrinter.printmethod" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.mathml.MathMLPrinter.doprint">
-<tt class="descname">doprint</tt><big>(</big><em>self</em>, <em>expr</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/mathml.html#MathMLPrinter.doprint"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.mathml.MathMLPrinter.doprint" title="Permalink to this definition">¶</a></dt>
-<dd><p>Prints the expression as MathML.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.mathml.MathMLPrinter.mathml_tag">
-<tt class="descname">mathml_tag</tt><big>(</big><em>self</em>, <em>e</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/mathml.html#MathMLPrinter.mathml_tag"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.mathml.MathMLPrinter.mathml_tag" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the MathML tag for an expression.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.mathml.mathml">
-<tt class="descclassname">sympy.printing.mathml.</tt><tt class="descname">mathml</tt><big>(</big><em>expr</em>, <em>**settings</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/mathml.html#mathml"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.mathml.mathml" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the MathML representation of expr</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.mathml.print_mathml">
-<tt class="descclassname">sympy.printing.mathml.</tt><tt class="descname">print_mathml</tt><big>(</big><em>expr</em>, <em>**settings</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/mathml.html#print_mathml"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.mathml.print_mathml" title="Permalink to this definition">¶</a></dt>
-<dd><p>Prints a pretty representation of the MathML code for expr</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="c">##</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.mathml</span> <span class="kn">import</span> <span class="n">print_mathml</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">print_mathml</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span> 
-<span class="go">&lt;apply&gt;</span>
-<span class="go">    &lt;plus/&gt;</span>
-<span class="go">    &lt;ci&gt;x&lt;/ci&gt;</span>
-<span class="go">    &lt;cn&gt;1&lt;/cn&gt;</span>
-<span class="go">&lt;/apply&gt;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.printing.python">
-<span id="pythonprinter"></span><h2>PythonPrinter<a class="headerlink" href="#module-sympy.printing.python" title="Permalink to this headline">¶</a></h2>
-<p>This class implements Python printing. Usage:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">print_python</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">print_python</span><span class="p">(</span><span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = 5*x**3 + sin(x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="module-sympy.printing.repr">
-<span id="reprprinter"></span><h2>ReprPrinter<a class="headerlink" href="#module-sympy.printing.repr" title="Permalink to this headline">¶</a></h2>
-<p>This printer generates executable code. This code satisfies the identity
-<tt class="docutils literal"><span class="pre">eval(srepr(expr))=expr</span></tt>.</p>
-<dl class="class">
-<dt id="sympy.printing.repr.ReprPrinter">
-<em class="property">class </em><tt class="descclassname">sympy.printing.repr.</tt><tt class="descname">ReprPrinter</tt><big>(</big><em>settings=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/repr.html#ReprPrinter"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.repr.ReprPrinter" title="Permalink to this definition">¶</a></dt>
-<dd><dl class="attribute">
-<dt id="sympy.printing.repr.ReprPrinter.printmethod">
-<tt class="descname">printmethod</tt><em class="property"> = '_sympyrepr'</em><a class="headerlink" href="#sympy.printing.repr.ReprPrinter.printmethod" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.repr.ReprPrinter.emptyPrinter">
-<tt class="descname">emptyPrinter</tt><big>(</big><em>self</em>, <em>expr</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/repr.html#ReprPrinter.emptyPrinter"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.repr.ReprPrinter.emptyPrinter" title="Permalink to this definition">¶</a></dt>
-<dd><p>The fallback printer.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.repr.ReprPrinter.reprify">
-<tt class="descname">reprify</tt><big>(</big><em>self</em>, <em>args</em>, <em>sep</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/repr.html#ReprPrinter.reprify"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.repr.ReprPrinter.reprify" title="Permalink to this definition">¶</a></dt>
-<dd><p>Prints each item in <span class="math">\(args\)</span> and joins them with <span class="math">\(sep\)</span>.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.repr.srepr">
-<tt class="descclassname">sympy.printing.repr.</tt><tt class="descname">srepr</tt><big>(</big><em>expr</em>, <em>**settings</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/repr.html#srepr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.repr.srepr" title="Permalink to this definition">¶</a></dt>
-<dd><p>return expr in repr form</p>
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.printing.str">
-<span id="strprinter"></span><h2>StrPrinter<a class="headerlink" href="#module-sympy.printing.str" title="Permalink to this headline">¶</a></h2>
-<p>This module generates readable representations of SymPy expressions.</p>
-<dl class="class">
-<dt id="sympy.printing.str.StrPrinter">
-<em class="property">class </em><tt class="descclassname">sympy.printing.str.</tt><tt class="descname">StrPrinter</tt><big>(</big><em>settings=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/str.html#StrPrinter"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.str.StrPrinter" title="Permalink to this definition">¶</a></dt>
-<dd><dl class="attribute">
-<dt id="sympy.printing.str.StrPrinter.printmethod">
-<tt class="descname">printmethod</tt><em class="property"> = '_sympystr'</em><a class="headerlink" href="#sympy.printing.str.StrPrinter.printmethod" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.str.sstrrepr">
-<tt class="descclassname">sympy.printing.str.</tt><tt class="descname">sstrrepr</tt><big>(</big><em>expr</em>, <em>**settings</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/str.html#sstrrepr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.str.sstrrepr" title="Permalink to this definition">¶</a></dt>
-<dd><p>return expr in mixed str/repr form</p>
-<p>i.e. strings are returned in repr form with quotes, and everything else
-is returned in str form.</p>
-<p>This function could be useful for hooking into sys.displayhook</p>
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.printing.tree">
-<span id="tree-printing"></span><h2>Tree Printing<a class="headerlink" href="#module-sympy.printing.tree" title="Permalink to this headline">¶</a></h2>
-<p>The functions in this module create a representation of an expression as a
-tree.</p>
-<dl class="function">
-<dt id="sympy.printing.tree.pprint_nodes">
-<tt class="descclassname">sympy.printing.tree.</tt><tt class="descname">pprint_nodes</tt><big>(</big><em>subtrees</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/tree.html#pprint_nodes"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.tree.pprint_nodes" title="Permalink to this definition">¶</a></dt>
-<dd><p>Prettyprints systems of nodes.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.tree</span> <span class="kn">import</span> <span class="n">pprint_nodes</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">pprint_nodes</span><span class="p">([</span><span class="s">&quot;a&quot;</span><span class="p">,</span> <span class="s">&quot;b1</span><span class="se">\n</span><span class="s">b2&quot;</span><span class="p">,</span> <span class="s">&quot;c&quot;</span><span class="p">]))</span>
-<span class="go">+-a</span>
-<span class="go">+-b1</span>
-<span class="go">| b2</span>
-<span class="go">+-c</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.tree.print_node">
-<tt class="descclassname">sympy.printing.tree.</tt><tt class="descname">print_node</tt><big>(</big><em>node</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/tree.html#print_node"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.tree.print_node" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns an information about the &#8220;node&#8221;.</p>
-<p>This includes class name, string representation and assumptions.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.tree.tree">
-<tt class="descclassname">sympy.printing.tree.</tt><tt class="descname">tree</tt><big>(</big><em>node</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/tree.html#tree"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.tree.tree" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a tree representation of &#8220;node&#8221; as a string.</p>
-<p>It uses print_node() together with pprint_nodes() on node.args recursively.</p>
-<p>See also: print_tree()</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.tree.print_tree">
-<tt class="descclassname">sympy.printing.tree.</tt><tt class="descname">print_tree</tt><big>(</big><em>node</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/tree.html#print_tree"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.tree.print_tree" title="Permalink to this definition">¶</a></dt>
-<dd><p>Prints a tree representation of &#8220;node&#8221;.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing</span> <span class="kn">import</span> <span class="n">print_tree</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">print_tree</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> 
-<span class="go">Pow: x**2</span>
-<span class="go">+-Symbol: x</span>
-<span class="go">| comparable: False</span>
-<span class="go">+-Integer: 2</span>
-<span class="go">  real: True</span>
-<span class="go">  nonzero: True</span>
-<span class="go">  comparable: True</span>
-<span class="go">  commutative: True</span>
-<span class="go">  infinitesimal: False</span>
-<span class="go">  unbounded: False</span>
-<span class="go">  noninteger: False</span>
-<span class="go">  zero: False</span>
-<span class="go">  complex: True</span>
-<span class="go">  bounded: True</span>
-<span class="go">  rational: True</span>
-<span class="go">  integer: True</span>
-<span class="go">  imaginary: False</span>
-<span class="go">  finite: True</span>
-<span class="go">  irrational: False</span>
-</pre></div>
-</div>
-<p>See also: tree()</p>
-</dd></dl>
-
-</div>
-<div class="section" id="preview">
-<h2>Preview<a class="headerlink" href="#preview" title="Permalink to this headline">¶</a></h2>
-<p>A useful function is <tt class="docutils literal"><span class="pre">preview</span></tt>:</p>
-<span class="target" id="module-sympy.printing.preview"></span><dl class="function">
-<dt id="sympy.printing.preview.preview">
-<tt class="descclassname">sympy.printing.preview.</tt><tt class="descname">preview</tt><big>(</big><em>expr</em>, <em>output='png'</em>, <em>viewer=None</em>, <em>euler=True</em>, <em>packages=()</em>, <em>**latex_settings</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/preview.html#preview"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.preview.preview" title="Permalink to this definition">¶</a></dt>
-<dd><p>View expression or LaTeX markup in PNG, DVI, PostScript or PDF form.</p>
-<p>If the expr argument is an expression, it will be exported to LaTeX and
-then compiled using available the TeX distribution.  The first argument,
-&#8216;expr&#8217;, may also be a LaTeX string.  The function will then run the
-appropriate viewer for the given output format or use the user defined
-one. By default png output is generated.</p>
-<p>By default pretty Euler fonts are used for typesetting (they were used to
-typeset the well known &#8220;Concrete Mathematics&#8221; book). For that to work, you
-need the &#8216;eulervm.sty&#8217; LaTeX style (in Debian/Ubuntu, install the
-texlive-fonts-extra package). If you prefer default AMS fonts or your
-system lacks &#8216;eulervm&#8217; LaTeX package then unset the &#8216;euler&#8217; keyword
-argument.</p>
-<p>To use viewer auto-detection, lets say for &#8216;png&#8217; output, issue</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">preview</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&quot;x,y&quot;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">preview</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">,</span> <span class="n">output</span><span class="o">=</span><span class="s">&#39;png&#39;</span><span class="p">)</span> 
-</pre></div>
-</div>
-<p>This will choose &#8216;pyglet&#8217; by default. To select a different one, do</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">preview</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">,</span> <span class="n">output</span><span class="o">=</span><span class="s">&#39;png&#39;</span><span class="p">,</span> <span class="n">viewer</span><span class="o">=</span><span class="s">&#39;gimp&#39;</span><span class="p">)</span> 
-</pre></div>
-</div>
-<p>The &#8216;png&#8217; format is considered special. For all other formats the rules
-are slightly different. As an example we will take &#8216;dvi&#8217; output format. If
-you would run</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">preview</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">,</span> <span class="n">output</span><span class="o">=</span><span class="s">&#39;dvi&#39;</span><span class="p">)</span> 
-</pre></div>
-</div>
-<p>then &#8216;view&#8217; will look for available &#8216;dvi&#8217; viewers on your system
-(predefined in the function, so it will try evince, first, then kdvi and
-xdvi). If nothing is found you will need to set the viewer explicitly.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">preview</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">,</span> <span class="n">output</span><span class="o">=</span><span class="s">&#39;dvi&#39;</span><span class="p">,</span> <span class="n">viewer</span><span class="o">=</span><span class="s">&#39;superior-dvi-viewer&#39;</span><span class="p">)</span> 
-</pre></div>
-</div>
-<p>This will skip auto-detection and will run user specified
-&#8216;superior-dvi-viewer&#8217;. If &#8216;view&#8217; fails to find it on your system it will
-gracefully raise an exception. You may also enter &#8216;file&#8217; for the viewer
-argument. Doing so will cause this function to return a file object in
-read-only mode.</p>
-<p>Currently this depends on pexpect, which is not available for windows.</p>
-<p>Additional keyword args will be passed to the latex call, e.g., the
-symbol_names flag.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">phidd</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;phidd&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">preview</span><span class="p">(</span><span class="n">phidd</span><span class="p">,</span> <span class="n">symbol_names</span><span class="o">=</span><span class="p">{</span><span class="n">phidd</span><span class="p">:</span><span class="s">r&#39;\ddot{\varphi}&#39;</span><span class="p">})</span> 
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.printing.conventions">
-<span id="implementation-helper-classes-functions"></span><h2>Implementation - Helper Classes/Functions<a class="headerlink" href="#module-sympy.printing.conventions" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.printing.conventions.split_super_sub">
-<tt class="descclassname">sympy.printing.conventions.</tt><tt class="descname">split_super_sub</tt><big>(</big><em>text</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/conventions.html#split_super_sub"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.conventions.split_super_sub" title="Permalink to this definition">¶</a></dt>
-<dd><p>Split a symbol name into a name, superscripts and subscripts</p>
-<p>The first part of the symbol name is considered to be its actual
-&#8216;name&#8217;, followed by super- and subscripts. Each superscript is
-preceded with a &#8220;^&#8221; character or by &#8220;__&#8221;. Each subscript is preceded
-by a &#8220;_&#8221; character.  The three return values are the actual name, a
-list with superscripts and a list with subscripts.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.conventions</span> <span class="kn">import</span> <span class="n">split_super_sub</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">split_super_sub</span><span class="p">(</span><span class="s">&#39;a_x^1&#39;</span><span class="p">)</span>
-<span class="go">(&#39;a&#39;, [&#39;1&#39;], [&#39;x&#39;])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">split_super_sub</span><span class="p">(</span><span class="s">&#39;var_sub1__sup_sub2&#39;</span><span class="p">)</span>
-<span class="go">(&#39;var&#39;, [&#39;sup&#39;], [&#39;sub1&#39;, &#39;sub2&#39;])</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<div class="section" id="module-sympy.printing.codeprinter">
-<span id="codeprinter"></span><h3>CodePrinter<a class="headerlink" href="#module-sympy.printing.codeprinter" title="Permalink to this headline">¶</a></h3>
-<p>This class is a base class for other classes that implement code-printing
-functionality, and additionally lists a number of functions that cannot be
-easily translated to C or Fortran.</p>
-<dl class="class">
-<dt id="sympy.printing.codeprinter.CodePrinter">
-<em class="property">class </em><tt class="descclassname">sympy.printing.codeprinter.</tt><tt class="descname">CodePrinter</tt><big>(</big><em>settings=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/codeprinter.html#CodePrinter"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.codeprinter.CodePrinter" title="Permalink to this definition">¶</a></dt>
-<dd><p>The base class for code-printing subclasses.</p>
-<dl class="attribute">
-<dt id="sympy.printing.codeprinter.CodePrinter.printmethod">
-<tt class="descname">printmethod</tt><em class="property"> = '_sympystr'</em><a class="headerlink" href="#sympy.printing.codeprinter.CodePrinter.printmethod" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-</dd></dl>
-
-<dl class="exception">
-<dt id="sympy.printing.codeprinter.AssignmentError">
-<em class="property">exception </em><tt class="descclassname">sympy.printing.codeprinter.</tt><tt class="descname">AssignmentError</tt><a class="reference internal" href="../_modules/sympy/printing/codeprinter.html#AssignmentError"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.codeprinter.AssignmentError" title="Permalink to this definition">¶</a></dt>
-<dd><p>Raised if an assignment variable for a loop is missing.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.printing.precedence">
-<span id="precedence"></span><h3>Precedence<a class="headerlink" href="#module-sympy.printing.precedence" title="Permalink to this headline">¶</a></h3>
-<dl class="data">
-<dt id="sympy.printing.precedence.PRECEDENCE">
-<tt class="descclassname">sympy.printing.precedence.</tt><tt class="descname">PRECEDENCE</tt><em class="property"> = {'Relational': 20, 'Pow': 60, 'Mul': 50, 'And': 30, 'Or': 20, 'Add': 40, 'Atom': 1000, 'Not': 100, 'Lambda': 1}</em><a class="headerlink" href="#sympy.printing.precedence.PRECEDENCE" title="Permalink to this definition">¶</a></dt>
-<dd><p>Default precedence values for some basic types.</p>
-</dd></dl>
-
-<dl class="data">
-<dt id="sympy.printing.precedence.PRECEDENCE_VALUES">
-<tt class="descclassname">sympy.printing.precedence.</tt><tt class="descname">PRECEDENCE_VALUES</tt><em class="property"> = {'Pow': 60, 'MatAdd': 40, 'And': 30, 'NegativeInfinity': 40, 'factorial': 60, 'Relational': 20, 'Or': 20, 'Sub': 40, 'MatMul': 50, 'factorial2': 60, 'HadamardProduct': 50, 'Add': 40, 'Not': 100}</em><a class="headerlink" href="#sympy.printing.precedence.PRECEDENCE_VALUES" title="Permalink to this definition">¶</a></dt>
-<dd><p>A dictionary assigning precedence values to certain classes. These values
-are treated like they were inherited, so not every single class has to be
-named here.</p>
-</dd></dl>
-
-<dl class="data">
-<dt id="sympy.printing.precedence.PRECEDENCE_FUNCTIONS">
-<tt class="descclassname">sympy.printing.precedence.</tt><tt class="descname">PRECEDENCE_FUNCTIONS</tt><em class="property"> = {'Float': &lt;function precedence_Float at 0x10c5b7ef0&gt;, 'Mul': &lt;function precedence_Mul at 0x10c5b7d40&gt;, 'Integer': &lt;function precedence_Integer at 0x10c5b7e60&gt;, 'Rational': &lt;function precedence_Rational at 0x10c5b7dd0&gt;}</em><a class="headerlink" href="#sympy.printing.precedence.PRECEDENCE_FUNCTIONS" title="Permalink to this definition">¶</a></dt>
-<dd><p>Sometimes it&#8217;s not enough to assign a fixed precedence value to a
-class. Then a function can be inserted in this dictionary that takes an
-instance of this class as argument and returns the appropriate precedence
-value.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.precedence.precedence">
-<tt class="descclassname">sympy.printing.precedence.</tt><tt class="descname">precedence</tt><big>(</big><em>item</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/precedence.html#precedence"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.precedence.precedence" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the precedence of a given object.</p>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.printing.pretty.pretty_symbology">
-<span id="pretty-printing-implementation-helpers"></span><h2>Pretty-Printing Implementation Helpers<a class="headerlink" href="#module-sympy.printing.pretty.pretty_symbology" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.printing.pretty.pretty_symbology.U">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">U</tt><big>(</big><em>name</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/pretty_symbology.html#U"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.U" title="Permalink to this definition">¶</a></dt>
-<dd><p>unicode character by name or None if not found</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.pretty_symbology.pretty_use_unicode">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">pretty_use_unicode</tt><big>(</big><em>flag=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/pretty_symbology.html#pretty_use_unicode"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.pretty_use_unicode" title="Permalink to this definition">¶</a></dt>
-<dd><p>Set whether pretty-printer should use unicode by default</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.pretty_symbology.pretty_try_use_unicode">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">pretty_try_use_unicode</tt><big>(</big><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/pretty_symbology.html#pretty_try_use_unicode"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.pretty_try_use_unicode" title="Permalink to this definition">¶</a></dt>
-<dd><p>See if unicode output is available and leverage it if possible</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.pretty_symbology.xstr">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">xstr</tt><big>(</big><em>*args</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/pretty_symbology.html#xstr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.xstr" title="Permalink to this definition">¶</a></dt>
-<dd><p>call str or unicode depending on current mode</p>
-</dd></dl>
-
-<p>The following two functions return the Unicode version of the inputted Greek
-letter.</p>
-<dl class="function">
-<dt id="sympy.printing.pretty.pretty_symbology.g">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">g</tt><big>(</big><em>l</em><big>)</big><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.g" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.pretty_symbology.G">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">G</tt><big>(</big><em>l</em><big>)</big><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.G" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="data">
-<dt id="sympy.printing.pretty.pretty_symbology.greek_letters">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">greek_letters</tt><em class="property"> = ['alpha', 'beta', 'gamma', 'delta', 'epsilon', 'zeta', 'eta', 'theta', 'iota', 'kappa', 'lamda', 'mu', 'nu', 'xi', 'omicron', 'pi', 'rho', 'sigma', 'tau', 'upsilon', 'phi', 'chi', 'psi', 'omega']</em><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.greek_letters" title="Permalink to this definition">¶</a></dt>
-<dd><p>list() -&gt; new empty list
-list(iterable) -&gt; new list initialized from iterable&#8217;s items</p>
-</dd></dl>
-
-<dl class="data">
-<dt id="sympy.printing.pretty.pretty_symbology.greek">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">greek</tt><em class="property"> = {'nu': ('ν', 'Ν'), 'pi': ('π', 'Π'), 'omicron': ('ο', 'Ο'), 'alpha': ('α', 'Α'), 'mu': ('μ', 'Μ'), 'iota': ('ι', 'Ι'), 'gamma': ('γ', 'Γ'), 'epsilon': ('ε', 'Ε'), 'rho': ('ρ', 'Ρ'), 'psi': ('ψ', 'Ψ'), 'chi': ('χ', 'Χ'), 'lamda': ('λ', 'Λ'), 'eta': ('η', 'Η'), 'lambda': ('λ', 'Λ'), 'sigma': ('σ', 'Σ'), 'omega': ('ω', 'Ω'), 'beta': ('β', 'Β'), 'zeta': ('ζ', 'Ζ'), 'phi': ('φ', 'Φ'), 'xi': ('ξ', 'Ξ'), 'tau': ('τ', 'Τ'), 'delta': ('δ', 'Δ'), 'kappa': ('κ', 'Κ'), 'theta': ('θ', 'Θ'), 'upsilon': ('υ', 'Υ')}</em><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.greek" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="data">
-<dt id="sympy.printing.pretty.pretty_symbology.digit_2txt">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">digit_2txt</tt><em class="property"> = {'2': 'TWO', '3': 'THREE', '0': 'ZERO', '1': 'ONE', '6': 'SIX', '7': 'SEVEN', '4': 'FOUR', '5': 'FIVE', '8': 'EIGHT', '9': 'NINE'}</em><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.digit_2txt" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="data">
-<dt id="sympy.printing.pretty.pretty_symbology.symb_2txt">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">symb_2txt</tt><em class="property"> = {'{}': 'CURLY BRACKET', '{': 'LEFT CURLY BRACKET', '}': 'RIGHT CURLY BRACKET', 'sum': 'SUMMATION', 'int': 'INTEGRAL', '[': 'LEFT SQUARE BRACKET', '+': 'PLUS SIGN', '(': 'LEFT PARENTHESIS', ')': 'RIGHT PARENTHESIS', '=': 'EQUALS SIGN', ']': 'RIGHT SQUARE BRACKET', '-': 'MINUS'}</em><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.symb_2txt" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<p>The following functions return the Unicode subscript/superscript version of
-the character.</p>
-<dl class="data">
-<dt id="sympy.printing.pretty.pretty_symbology.sub">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">sub</tt><em class="property"> = {'2': '₂', '9': '₉', 'gamma': 'ᵧ', 'rho': 'ᵨ', 'chi': 'ᵪ', ')': '₎', '6': '₆', '8': '₈', 'beta': 'ᵦ', 'r': 'ᵣ', '3': '₃', '0': '₀', '1': '₁', 'v': 'ᵥ', '7': '₇', '4': '₄', 'u': 'ᵤ', 'phi': 'ᵩ', 'x': 'ₓ', '5': '₅', '=': '₌', 'a': 'ₐ', 'e': 'ₑ', '+': '₊', '(': '₍', 'i': 'ᵢ', 'o': 'ₒ', '-': '₋'}</em><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.sub" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="data">
-<dt id="sympy.printing.pretty.pretty_symbology.sup">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">sup</tt><em class="property"> = {'2': '²', '3': '³', '0': '⁰', '1': '¹', '6': '⁶', '7': '⁷', '4': '⁴', '5': '⁵', '8': '⁸', '9': '⁹', '=': '⁼', ')': '⁾', '+': '⁺', '(': '⁽', 'i': 'ⁱ', 'n': 'ⁿ', '-': '⁻'}</em><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.sup" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<p>The following functions return Unicode vertical objects.</p>
-<dl class="function">
-<dt id="sympy.printing.pretty.pretty_symbology.xobj">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">xobj</tt><big>(</big><em>symb</em>, <em>length</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/pretty_symbology.html#xobj"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.xobj" title="Permalink to this definition">¶</a></dt>
-<dd><p>Construct spatial object of given length.</p>
-<p>return: [] of equal-length strings</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.pretty_symbology.vobj">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">vobj</tt><big>(</big><em>symb</em>, <em>height</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/pretty_symbology.html#vobj"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.vobj" title="Permalink to this definition">¶</a></dt>
-<dd><p>Construct vertical object of a given height</p>
-<p>see: xobj</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.pretty_symbology.hobj">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">hobj</tt><big>(</big><em>symb</em>, <em>width</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/pretty_symbology.html#hobj"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.hobj" title="Permalink to this definition">¶</a></dt>
-<dd><p>Construct horizontal object of a given width</p>
-<p>see: xobj</p>
-</dd></dl>
-
-<p>The following constants are for rendering roots and fractions.</p>
-<dl class="data">
-<dt id="sympy.printing.pretty.pretty_symbology.root">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">root</tt><em class="property"> = {2: '√', 3: '∛', 4: '∜'}</em><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.root" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.pretty_symbology.VF">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">VF</tt><big>(</big><em>txt</em><big>)</big><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.VF" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="data">
-<dt id="sympy.printing.pretty.pretty_symbology.frac">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">frac</tt><em class="property"> = {(1, 2): '½', (1, 3): '⅓', (5, 6): '⅚', (7, 8): '⅞', (1, 4): '¼', (3, 8): '⅜', (1, 5): '⅕', (1, 8): '⅛', (2, 3): '⅔', (4, 5): '⅘', (1, 6): '⅙', (2, 5): '⅖', (3, 4): '¾', (5, 8): '⅝', (3, 5): '⅗'}</em><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.frac" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<p>The following constants/functions are for rendering atoms and symbols.</p>
-<dl class="function">
-<dt id="sympy.printing.pretty.pretty_symbology.xsym">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">xsym</tt><big>(</big><em>sym</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/pretty_symbology.html#xsym"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.xsym" title="Permalink to this definition">¶</a></dt>
-<dd><p>get symbology for a &#8216;character&#8217;</p>
-</dd></dl>
-
-<dl class="data">
-<dt id="sympy.printing.pretty.pretty_symbology.atoms_table">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">atoms_table</tt><em class="property"> = {'Pi': 'π', 'Naturals': 'ℕ', 'Integers': 'ℤ', 'Ring': '∘', 'Infinity': '∞', 'Union': '∪', 'Exp1': 'ℯ', 'NegativeInfinity': '-∞', 'Intersection': '∩', 'ImaginaryUnit': 'ⅈ', 'EmptySet': '∅', 'Reals': 'ℝ'}</em><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.atoms_table" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.pretty_symbology.pretty_atom">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">pretty_atom</tt><big>(</big><em>atom_name</em>, <em>default=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/pretty_symbology.html#pretty_atom"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.pretty_atom" title="Permalink to this definition">¶</a></dt>
-<dd><p>return pretty representation of an atom</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.pretty_symbology.pretty_symbol">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">pretty_symbol</tt><big>(</big><em>symb_name</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/pretty_symbology.html#pretty_symbol"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.pretty_symbol" title="Permalink to this definition">¶</a></dt>
-<dd><p>return pretty representation of a symbol</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.pretty_symbology.annotated">
-<tt class="descclassname">sympy.printing.pretty.pretty_symbology.</tt><tt class="descname">annotated</tt><big>(</big><em>letter</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/pretty_symbology.html#annotated"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.pretty_symbology.annotated" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a stylised drawing of the letter <tt class="docutils literal"><span class="pre">letter</span></tt>, together with
-information on how to put annotations (super- and subscripts to the
-left and to the right) on it.</p>
-<p>See pretty.py functions _print_meijerg, _print_hyper on how to use this
-information.</p>
-</dd></dl>
-
-<span class="target" id="module-sympy.printing.pretty.stringpict"></span><p>Prettyprinter by Jurjen Bos.
-(I hate spammers: mail me at pietjepuk314 at the reverse of ku.oc.oohay).
-All objects have a method that create a &#8220;stringPict&#8221;,
-that can be used in the str method for pretty printing.</p>
-<dl class="docutils">
-<dt>Updates by Jason Gedge (email &lt;my last name&gt; at cs mun ca)</dt>
-<dd><ul class="first last simple">
-<li>terminal_string() method</li>
-<li>minor fixes and changes (mostly to prettyForm)</li>
-</ul>
-</dd>
-<dt>TODO:</dt>
-<dd><ul class="first last simple">
-<li>Allow left/center/right alignment options for above/below and
-top/center/bottom alignment options for left/right</li>
-</ul>
-</dd>
-</dl>
-<dl class="class">
-<dt id="sympy.printing.pretty.stringpict.stringPict">
-<em class="property">class </em><tt class="descclassname">sympy.printing.pretty.stringpict.</tt><tt class="descname">stringPict</tt><big>(</big><em>s</em>, <em>baseline=0</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/stringpict.html#stringPict"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.stringpict.stringPict" title="Permalink to this definition">¶</a></dt>
-<dd><p>An ASCII picture.
-The pictures are represented as a list of equal length strings.</p>
-<dl class="function">
-<dt id="sympy.printing.pretty.stringpict.stringPict.above">
-<tt class="descname">above</tt><big>(</big><em>self</em>, <em>*args</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/stringpict.html#stringPict.above"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.stringpict.stringPict.above" title="Permalink to this definition">¶</a></dt>
-<dd><p>Put pictures above this picture.
-Returns string, baseline arguments for stringPict.
-Baseline is baseline of bottom picture.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.stringpict.stringPict.below">
-<tt class="descname">below</tt><big>(</big><em>self</em>, <em>*args</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/stringpict.html#stringPict.below"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.stringpict.stringPict.below" title="Permalink to this definition">¶</a></dt>
-<dd><p>Put pictures under this picture.
-Returns string, baseline arguments for stringPict.
-Baseline is baseline of top picture</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.pretty.stringpict</span> <span class="kn">import</span> <span class="n">stringPict</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">stringPict</span><span class="p">(</span><span class="s">&quot;x+3&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">below</span><span class="p">(</span>
-<span class="gp">... </span>      <span class="n">stringPict</span><span class="o">.</span><span class="n">LINE</span><span class="p">,</span> <span class="s">&#39;3&#39;</span><span class="p">)[</span><span class="mi">0</span><span class="p">])</span> 
-<span class="go">x+3</span>
-<span class="go">---</span>
-<span class="go"> 3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.stringpict.stringPict.height">
-<tt class="descname">height</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/stringpict.html#stringPict.height"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.stringpict.stringPict.height" title="Permalink to this definition">¶</a></dt>
-<dd><p>The height of the picture in characters.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.stringpict.stringPict.left">
-<tt class="descname">left</tt><big>(</big><em>self</em>, <em>*args</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/stringpict.html#stringPict.left"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.stringpict.stringPict.left" title="Permalink to this definition">¶</a></dt>
-<dd><p>Put pictures (left to right) at left.
-Returns string, baseline arguments for stringPict.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.stringpict.stringPict.leftslash">
-<tt class="descname">leftslash</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/stringpict.html#stringPict.leftslash"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.stringpict.stringPict.leftslash" title="Permalink to this definition">¶</a></dt>
-<dd><p>Precede object by a slash of the proper size.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.stringpict.stringPict.next">
-<tt class="descname">next</tt><big>(</big><em>*args</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/stringpict.html#stringPict.next"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.stringpict.stringPict.next" title="Permalink to this definition">¶</a></dt>
-<dd><p>Put a string of stringPicts next to each other.
-Returns string, baseline arguments for stringPict.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.stringpict.stringPict.parens">
-<tt class="descname">parens</tt><big>(</big><em>self</em>, <em>left='('</em>, <em>right=')'</em>, <em>ifascii_nougly=False</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/stringpict.html#stringPict.parens"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.stringpict.stringPict.parens" title="Permalink to this definition">¶</a></dt>
-<dd><p>Put parentheses around self.
-Returns string, baseline arguments for stringPict.</p>
-<p>left or right can be None or empty string which means &#8216;no paren from
-that side&#8217;</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.stringpict.stringPict.render">
-<tt class="descname">render</tt><big>(</big><em>self</em>, <em>*args</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/stringpict.html#stringPict.render"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.stringpict.stringPict.render" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the string form of self.</p>
-<p>Unless the argument line_break is set to False, it will
-break the expression in a form that can be printed
-on the terminal without being broken up.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.stringpict.stringPict.right">
-<tt class="descname">right</tt><big>(</big><em>self</em>, <em>*args</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/stringpict.html#stringPict.right"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.stringpict.stringPict.right" title="Permalink to this definition">¶</a></dt>
-<dd><p>Put pictures next to this one.
-Returns string, baseline arguments for stringPict.
-(Multiline) strings are allowed, and are given a baseline of 0.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.pretty.stringpict</span> <span class="kn">import</span> <span class="n">stringPict</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">stringPict</span><span class="p">(</span><span class="s">&quot;10&quot;</span><span class="p">)</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="s">&quot; + &quot;</span><span class="p">,</span><span class="n">stringPict</span><span class="p">(</span><span class="s">&quot;1</span><span class="se">\r</span><span class="s">-</span><span class="se">\r</span><span class="s">2&quot;</span><span class="p">,</span><span class="mi">1</span><span class="p">))[</span><span class="mi">0</span><span class="p">])</span>
-<span class="go">     1</span>
-<span class="go">10 + -</span>
-<span class="go">     2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.stringpict.stringPict.root">
-<tt class="descname">root</tt><big>(</big><em>self</em>, <em>n=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/stringpict.html#stringPict.root"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.stringpict.stringPict.root" title="Permalink to this definition">¶</a></dt>
-<dd><p>Produce a nice root symbol.
-Produces ugly results for big n inserts.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.stringpict.stringPict.stack">
-<tt class="descname">stack</tt><big>(</big><em>*args</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/stringpict.html#stringPict.stack"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.stringpict.stringPict.stack" title="Permalink to this definition">¶</a></dt>
-<dd><p>Put pictures on top of each other,
-from top to bottom.
-Returns string, baseline arguments for stringPict.
-The baseline is the baseline of the second picture.
-Everything is centered.
-Baseline is the baseline of the second picture.
-Strings are allowed.
-The special value stringPict.LINE is a row of &#8216;-&#8216; extended to the width.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.stringpict.stringPict.terminal_width">
-<tt class="descname">terminal_width</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/stringpict.html#stringPict.terminal_width"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.stringpict.stringPict.terminal_width" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the terminal width if possible, otherwise return 0.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.printing.pretty.stringpict.stringPict.width">
-<tt class="descname">width</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/stringpict.html#stringPict.width"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.stringpict.stringPict.width" title="Permalink to this definition">¶</a></dt>
-<dd><p>The width of the picture in characters.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.printing.pretty.stringpict.prettyForm">
-<em class="property">class </em><tt class="descclassname">sympy.printing.pretty.stringpict.</tt><tt class="descname">prettyForm</tt><big>(</big><em>s</em>, <em>baseline=0</em>, <em>binding=0</em>, <em>str=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/stringpict.html#prettyForm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.stringpict.prettyForm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Extension of the stringPict class that knows about
-basic math applications, optimizing double minus signs.</p>
-<p>&#8220;Binding&#8221; is interpreted as follows:</p>
-<div class="highlight-python"><pre>ATOM this is an atom: never needs to be parenthesized
-FUNC this is a function application: parenthesize if added (?)
-DIV  this is a division: make wider division if divided
-POW  this is a power: only parenthesize if exponent
-MUL  this is a multiplication: parenthesize if powered
-ADD  this is an addition: parenthesize if multiplied or powered
-NEG  this is a negative number: optimize if added, parenthesize if multiplied or powered</pre>
-</div>
-<dl class="function">
-<dt id="sympy.printing.pretty.stringpict.prettyForm.apply">
-<tt class="descname">apply</tt><big>(</big><em>function</em>, <em>*args</em><big>)</big><a class="reference internal" href="../_modules/sympy/printing/pretty/stringpict.html#prettyForm.apply"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.printing.pretty.stringpict.prettyForm.apply" title="Permalink to this definition">¶</a></dt>
-<dd><p>Functions of one or more variables.</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../index.html">
-              <img class="logo" src="../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Printing System</a><ul>
-<li><a class="reference internal" href="#module-sympy.printing.printer">Printer Class</a></li>
-<li><a class="reference internal" href="#prettyprinter-class">PrettyPrinter Class</a></li>
-<li><a class="reference internal" href="#module-sympy.printing.ccode">CCodePrinter</a></li>
-<li><a class="reference internal" href="#fortran-printing">Fortran Printing</a></li>
-<li><a class="reference internal" href="#module-sympy.printing.gtk">Gtk</a></li>
-<li><a class="reference internal" href="#module-sympy.printing.lambdarepr">LambdaPrinter</a></li>
-<li><a class="reference internal" href="#module-sympy.printing.latex">LatexPrinter</a></li>
-<li><a class="reference internal" href="#module-sympy.printing.mathml">MathMLPrinter</a></li>
-<li><a class="reference internal" href="#module-sympy.printing.python">PythonPrinter</a></li>
-<li><a class="reference internal" href="#module-sympy.printing.repr">ReprPrinter</a></li>
-<li><a class="reference internal" href="#module-sympy.printing.str">StrPrinter</a></li>
-<li><a class="reference internal" href="#module-sympy.printing.tree">Tree Printing</a></li>
-<li><a class="reference internal" href="#preview">Preview</a></li>
-<li><a class="reference internal" href="#module-sympy.printing.conventions">Implementation - Helper Classes/Functions</a><ul>
-<li><a class="reference internal" href="#module-sympy.printing.codeprinter">CodePrinter</a></li>
-<li><a class="reference internal" href="#module-sympy.printing.precedence">Precedence</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.printing.pretty.pretty_symbology">Pretty-Printing Implementation Helpers</a></li>
-</ul>
-</li>
-</ul>
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-  <div class="section" id="term-rewriting">
-<h1>Term rewriting<a class="headerlink" href="#term-rewriting" title="Permalink to this headline">¶</a></h1>
-<p>Term rewriting is a very general class of functionalities which are used to
-convert expressions of one type in terms of expressions of different kind. For
-example expanding, combining and converting expressions apply to term
-rewriting, and also simplification routines can be included here. Currently
-!SymPy has several functions and Basic built-in methods for performing various
-types of rewriting.</p>
-<div class="section" id="expanding">
-<h2>Expanding<a class="headerlink" href="#expanding" title="Permalink to this headline">¶</a></h2>
-<p>The simplest rewrite rule is expanding expressions into a _sparse_ form.
-Expanding has several flavors and include expanding complex valued expressions,
-arithmetic expand of products and powers but also expanding functions in terms
-of more general functions is possible. Below are listed all currently available
-expand rules.</p>
-<dl class="docutils">
-<dt>Expanding of arithmetic expressions involving products and powers:</dt>
-<dd><div class="first last highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y,z&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">basic</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">x**2 - y**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">basic</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">x**2 + 2*x*y + 2*x*z + y**2 + 2*y*z + z**2</span>
-</pre></div>
-</div>
-</dd>
-</dl>
-<p>Arithmetic expand is done by default in <tt class="docutils literal"><span class="pre">expand()</span></tt> so the keyword <tt class="docutils literal"><span class="pre">basic</span></tt> can
-be omitted. However you can set <tt class="docutils literal"><span class="pre">basic=False</span></tt> to avoid this type of expand if
-you use rules described below. This give complete control on what is done with
-the expression.</p>
-<p>Another type of expand rule is expanding complex valued expressions and putting
-them into a normal form. For this <tt class="docutils literal"><span class="pre">complex</span></tt> keyword is used. Note that it will
-always perform arithmetic expand to obtain the desired normal form:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">re(x) + I*re(y) + I*im(x) - im(y)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">sin(re(x) - im(y))*cosh(re(y) + im(x)) + I*cos(re(x) - im(y))*sinh(re(y) + im(x))</span>
-</pre></div>
-</div>
-<p>Note also that the same behavior can be obtained by using <tt class="docutils literal"><span class="pre">as_real_imag()</span></tt>
-method. However it will return a tuple containing the real part in the first
-place and the imaginary part in the other. This can be also done in a two step
-process by using <tt class="docutils literal"><span class="pre">collect</span></tt> function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">as_real_imag</span><span class="p">()</span>
-<span class="go">(re(x) - im(y), re(y) + im(x))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">collect</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">I</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">),</span> <span class="n">I</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">{1: re(x) - im(y), I: re(y) + im(x)}</span>
-</pre></div>
-</div>
-<p>There is also possibility for expanding expressions in terms of expressions of
-different kind. This is very general type of expanding and usually you would
-use <tt class="docutils literal"><span class="pre">rewrite()</span></tt> to do specific type of rewrite:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">GoldenRatio</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">func</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">1/2 + sqrt(5)/2</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="module-sympy.simplify.cse_main">
-<span id="common-subexpression-detection-and-collection"></span><h2>Common Subexpression Detection and Collection<a class="headerlink" href="#module-sympy.simplify.cse_main" title="Permalink to this headline">¶</a></h2>
-<p>Before evaluating a large expression, it is often useful to identify common
-subexpressions, collect them and evaluate them at once. This is implemented
-in the <tt class="docutils literal"><span class="pre">cse</span></tt> function. Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">cse</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">cse</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">))),</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">⎛    ⎡  ________⎤⎞</span>
-<span class="go">⎝[], ⎣╲╱ sin(x) ⎦⎠</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">cse</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">+</span><span class="mi">5</span><span class="p">)</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">+</span><span class="mi">4</span><span class="p">)),</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">⎛                ⎡  ________   ________⎤⎞</span>
-<span class="go">⎝[(x₀, sin(x))], ⎣╲╱ x₀ + 4 ⋅╲╱ x₀ + 5 ⎦⎠</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">cse</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="mi">5</span> <span class="o">+</span> <span class="n">cos</span><span class="p">(</span><span class="n">y</span><span class="p">))</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="mi">4</span> <span class="o">+</span> <span class="n">cos</span><span class="p">(</span><span class="n">y</span><span class="p">))),</span>
-<span class="gp">... </span>    <span class="n">use_unicode</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">⎛                             ⎡  ________   ________⎤⎞</span>
-<span class="go">⎝[(x₀, sin(x + 1) + cos(y))], ⎣╲╱ x₀ + 4 ⋅╲╱ x₀ + 5 ⎦⎠</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">cse</span><span class="p">((</span><span class="n">x</span><span class="o">-</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">z</span><span class="o">-</span><span class="n">y</span><span class="p">)</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">((</span><span class="n">x</span><span class="o">-</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">z</span><span class="o">-</span><span class="n">y</span><span class="p">))),</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">⎛                          ⎡  ____     ⎤⎞</span>
-<span class="go">⎝[(x₀, -(x - y)⋅(y - z))], ⎣╲╱ x₀  + x₀⎦⎠</span>
-</pre></div>
-</div>
-<p>More information:</p>
-</div>
-</div>
-
-
-          </div>
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-        <div class="sphinxsidebarwrapper">
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-              <img class="logo" src="../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Term rewriting</a><ul>
-<li><a class="reference internal" href="#expanding">Expanding</a></li>
-<li><a class="reference internal" href="#module-sympy.simplify.cse_main">Common Subexpression Detection and Collection</a></li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="assumptions/handlers/sets.html"
-                        title="previous chapter">Sets</a></p>
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-                        title="next chapter">Series Expansions</a></p>
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-            
-  <div class="section" id="module-sympy.series">
-<span id="series-expansions"></span><h1>Series Expansions<a class="headerlink" href="#module-sympy.series" title="Permalink to this headline">¶</a></h1>
-<p>The series module implements series expansions as a function and many related
-functions.</p>
-<div class="section" id="limits">
-<h2>Limits<a class="headerlink" href="#limits" title="Permalink to this headline">¶</a></h2>
-<p>The main purpose of this module is the computation of limits.</p>
-<dl class="function">
-<dt id="sympy.series.limits.limit">
-<tt class="descclassname">sympy.series.limits.</tt><tt class="descname">limit</tt><big>(</big><em>e</em>, <em>z</em>, <em>z0</em>, <em>dir='+'</em><big>)</big><a class="reference internal" href="../_modules/sympy/series/limits.html#limit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.limits.limit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the limit of e(z) at the point z0.</p>
-<p>z0 can be any expression, including oo and -oo.</p>
-<p>For dir=&#8221;+&#8221; (default) it calculates the limit from the right
-(z-&gt;z0+) and for dir=&#8221;-&#8221; the limit from the left (z-&gt;z0-). For infinite z0
-(oo or -oo), the dir argument doesn&#8217;t matter.</p>
-<p class="rubric">Notes</p>
-<p>First we try some heuristics for easy and frequent cases like &#8220;x&#8221;, &#8220;1/x&#8221;,
-&#8220;x**2&#8221; and similar, so that it&#8217;s fast. For all other cases, we use the
-Gruntz algorithm (see the gruntz() function).</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">limit</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">dir</span><span class="o">=</span><span class="s">&quot;+&quot;</span><span class="p">)</span>
-<span class="go">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">dir</span><span class="o">=</span><span class="s">&quot;-&quot;</span><span class="p">)</span>
-<span class="go">-oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.series.limits.Limit">
-<em class="property">class </em><tt class="descclassname">sympy.series.limits.</tt><tt class="descname">Limit</tt><a class="reference internal" href="../_modules/sympy/series/limits.html#Limit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.limits.Limit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents an unevaluated limit.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Limit</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Limit</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">Limit(sin(x)/x, x, 0)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Limit</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">dir</span><span class="o">=</span><span class="s">&quot;-&quot;</span><span class="p">)</span>
-<span class="go">Limit(1/x, x, 0, dir=&#39;-&#39;)</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.series.limits.Limit.doit">
-<tt class="descname">doit</tt><big>(</big><em>self</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../_modules/sympy/series/limits.html#Limit.doit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.limits.Limit.doit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Evaluates limit</p>
-</dd></dl>
-
-</dd></dl>
-
-<p>As is explained above, the workhorse for limit computations is the
-function gruntz() which implements Gruntz&#8217; algorithm for computing limits.</p>
-<div class="section" id="the-gruntz-algorithm">
-<h3>The Gruntz Algorithm<a class="headerlink" href="#the-gruntz-algorithm" title="Permalink to this headline">¶</a></h3>
-<p>This section explains the basics of the algorithm used for computing limits.
-Most of the time the limit() function should just work. However it is still
-useful to keep in mind how it is implemented in case something does not work
-as expected.</p>
-<p>First we define an ordering on functions. Suppose <span class="math">\(f(x)\)</span> and <span class="math">\(g(x)\)</span> are two
-real-valued functions such that <span class="math">\(\lim_{x \to \infty} f(x) = \infty\)</span> and
-similarly <span class="math">\(\lim_{x \to \infty} g(x) = \infty\)</span>. We shall say that <span class="math">\(f(x)\)</span>
-<em>dominates</em>
-<span class="math">\(g(x)\)</span>, written <span class="math">\(f(x) \succ g(x)\)</span>, if for all <span class="math">\(a, b \in \mathbb{R}_{&gt;0}\)</span> we have
-<span class="math">\(\lim_{x \to \infty} \frac{f(x)^a}{g(x)^b} = \infty\)</span>.
-We also say that <span class="math">\(f(x)\)</span> and
-<span class="math">\(g(x)\)</span> are <em>of the same comparability class</em> if neither <span class="math">\(f(x) \succ g(x)\)</span> nor
-<span class="math">\(g(x) \succ f(x)\)</span> and shall denote it as <span class="math">\(f(x) \asymp g(x)\)</span>.</p>
-<p>Note that whenever <span class="math">\(a, b \in \mathbb{R}_{&gt;0}\)</span> then
-<span class="math">\(a f(x)^b \asymp f(x)\)</span>, and we shall use this to extend the definition of
-<span class="math">\(\succ\)</span> to all functions which tend to <span class="math">\(0\)</span> or <span class="math">\(\pm \infty\)</span> as <span class="math">\(x \to \infty\)</span>.
-Thus we declare that <span class="math">\(f(x) \asymp 1/f(x)\)</span> and <span class="math">\(f(x) \asymp -f(x)\)</span>.</p>
-<p>It is easy to show the following examples:</p>
-<ul class="simple">
-<li><span class="math">\(e^x \succ x^m\)</span></li>
-<li><span class="math">\(e^{x^2} \succ e^{mx}\)</span></li>
-<li><span class="math">\(e^{e^x} \succ e^{x^m}\)</span></li>
-<li><span class="math">\(x^m \asymp x^n\)</span></li>
-<li><span class="math">\(e^{x + \frac{1}{x}} \asymp e^{x + \log{x}} \asymp e^x\)</span>.</li>
-</ul>
-<p>From the above definition, it is possible to prove the following property:</p>
-<blockquote>
-<div><p>Suppose <span class="math">\(\omega\)</span>, <span class="math">\(g_1, g_2, \dots\)</span> are functions of <span class="math">\(x\)</span>,
-<span class="math">\(\lim_{x \to \infty} \omega = 0\)</span> and <span class="math">\(\omega \succ g_i\)</span> for
-all <span class="math">\(i\)</span>. Let <span class="math">\(c_1, c_2, \dots \in \mathbb{R}\)</span> with <span class="math">\(c_1 &lt; c_2 &lt; \dots\)</span>.</p>
-<p>Then <span class="math">\(\lim_{x \to \infty} \sum_i g_i \omega^{c_i} = \lim_{x \to \infty} g_1 \omega^{c_1}\)</span>.</p>
-</div></blockquote>
-<p>For <span class="math">\(g_1 = g\)</span> and <span class="math">\(\omega\)</span> as above we also have the following easy result:</p>
-<blockquote>
-<div><ul class="simple">
-<li><span class="math">\(\lim_{x \to \infty} g \omega^c = 0\)</span> for <span class="math">\(c &gt; 0\)</span></li>
-<li><span class="math">\(\lim_{x \to \infty} g \omega^c = \pm \infty\)</span> for <span class="math">\(c &lt; 0\)</span>,
-where the sign is determined by the (eventual) sign of <span class="math">\(g\)</span></li>
-<li><span class="math">\(\lim_{x \to \infty} g \omega^0 = \lim_{x \to \infty} g\)</span>.</li>
-</ul>
-</div></blockquote>
-<p>Using these results yields the following strategy for computing
-<span class="math">\(\lim_{x \to \infty} f(x)\)</span>:</p>
-<ol class="arabic simple">
-<li>Find the set of <em>most rapidly varying subexpressions</em> (MRV set) of <span class="math">\(f(x)\)</span>.
-That is, from the set of all subexpressions of <span class="math">\(f(x)\)</span>, find the elements that
-are maximal under the relation <span class="math">\(\succ\)</span>.</li>
-<li>Choose a function <span class="math">\(\omega\)</span> that is in the same comparability class as
-the elements in the MRV set, such that <span class="math">\(\lim_{x \to \infty} \omega = 0\)</span>.</li>
-<li>Expand <span class="math">\(f(x)\)</span> as a series in <span class="math">\(\omega\)</span> in such a way that the antecedents of
-the above theorem are satisfied.</li>
-<li>Apply the theorem and conclude the computation of
-<span class="math">\(\lim_{x \to \infty} f(x)\)</span>, possibly by recursively working on <span class="math">\(g_1(x)\)</span>.</li>
-</ol>
-<div class="section" id="notes">
-<h4>Notes<a class="headerlink" href="#notes" title="Permalink to this headline">¶</a></h4>
-<p>This exposition glossed over several details. Many are described in the file
-gruntz.py, and all can be found in Gruntz&#8217; very readable thesis. The most
-important points that have not been explained are:</p>
-<ol class="arabic simple">
-<li>Given f(x) and g(x), how do we determine if <span class="math">\(f(x) \succ g(x)\)</span>,
-<span class="math">\(g(x) \succ f(x)\)</span> or <span class="math">\(g(x) \asymp f(x)\)</span>?</li>
-<li>How do we find the MRV set of an expression?</li>
-<li>How do we compute series expansions?</li>
-<li>Why does the algorithm terminate?</li>
-</ol>
-<p>If you are interested, be sure to take a look at
-<a class="reference external" href="http://www.cybertester.com/data/gruntz.pdf">Gruntz Thesis</a>.</p>
-</div>
-<div class="section" id="reference">
-<h4>Reference<a class="headerlink" href="#reference" title="Permalink to this headline">¶</a></h4>
-<dl class="function">
-<dt id="sympy.series.gruntz.gruntz">
-<tt class="descclassname">sympy.series.gruntz.</tt><tt class="descname">gruntz</tt><big>(</big><em>e</em>, <em>z</em>, <em>z0</em>, <em>dir='+'</em><big>)</big><a class="reference internal" href="../_modules/sympy/series/gruntz.html#gruntz"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.gruntz.gruntz" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the limit of e(z) at the point z0 using the Gruntz algorithm.</p>
-<p>z0 can be any expression, including oo and -oo.</p>
-<p>For dir=&#8221;+&#8221; (default) it calculates the limit from the right
-(z-&gt;z0+) and for dir=&#8221;-&#8221; the limit from the left (z-&gt;z0-). For infinite z0
-(oo or -oo), the dir argument doesn&#8217;t matter.</p>
-<p>This algorithm is fully described in the module docstring in the gruntz.py
-file. It relies heavily on the series expansion. Most frequently, gruntz()
-is only used if the faster limit() function (which uses heuristics) fails.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.series.gruntz.compare">
-<tt class="descclassname">sympy.series.gruntz.</tt><tt class="descname">compare</tt><big>(</big><em>a</em>, <em>b</em>, <em>x</em><big>)</big><a class="reference internal" href="../_modules/sympy/series/gruntz.html#compare"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.gruntz.compare" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns &#8220;&lt;&#8221; if a&lt;b, &#8220;=&#8221; for a == b, &#8220;&gt;&#8221; for a&gt;b</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.series.gruntz.rewrite">
-<tt class="descclassname">sympy.series.gruntz.</tt><tt class="descname">rewrite</tt><big>(</big><em>e</em>, <em>Omega</em>, <em>x</em>, <em>wsym</em><big>)</big><a class="reference internal" href="../_modules/sympy/series/gruntz.html#rewrite"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.gruntz.rewrite" title="Permalink to this definition">¶</a></dt>
-<dd><p>e(x) ... the function
-Omega ... the mrv set
-wsym ... the symbol which is going to be used for w</p>
-<p>Returns the rewritten e in terms of w and log(w). See test_rewrite1()
-for examples and correct results.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.series.gruntz.build_expression_tree">
-<tt class="descclassname">sympy.series.gruntz.</tt><tt class="descname">build_expression_tree</tt><big>(</big><em>Omega</em>, <em>rewrites</em><big>)</big><a class="reference internal" href="../_modules/sympy/series/gruntz.html#build_expression_tree"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.gruntz.build_expression_tree" title="Permalink to this definition">¶</a></dt>
-<dd><p>Helper function for rewrite.</p>
-<p>We need to sort Omega (mrv set) so that we replace an expression before
-we replace any expression in terms of which it has to be rewritten:</p>
-<div class="highlight-python"><pre>e1 ---&gt; e2 ---&gt; e3
-         \
-          -&gt; e4</pre>
-</div>
-<p>Here we can do e1, e2, e3, e4 or e1, e2, e4, e3.
-To do this we assemble the nodes into a tree, and sort them by height.</p>
-<p>This function builds the tree, rewrites then sorts the nodes.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.series.gruntz.mrv_leadterm">
-<tt class="descclassname">sympy.series.gruntz.</tt><tt class="descname">mrv_leadterm</tt><big>(</big><em>*args</em>, <em>**kw_args</em><big>)</big><a class="reference internal" href="../_modules/sympy/series/gruntz.html#mrv_leadterm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.gruntz.mrv_leadterm" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns (c0, e0) for e.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.series.gruntz.calculate_series">
-<tt class="descclassname">sympy.series.gruntz.</tt><tt class="descname">calculate_series</tt><big>(</big><em>e</em>, <em>x</em>, <em>skip_abs=False</em>, <em>logx=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/series/gruntz.html#calculate_series"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.gruntz.calculate_series" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculates at least one term of the series of &#8220;e&#8221; in &#8220;x&#8221;.</p>
-<p>This is a place that fails most often, so it is in its own function.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.series.gruntz.limitinf">
-<tt class="descclassname">sympy.series.gruntz.</tt><tt class="descname">limitinf</tt><big>(</big><em>*args</em>, <em>**kw_args</em><big>)</big><a class="reference internal" href="../_modules/sympy/series/gruntz.html#limitinf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.gruntz.limitinf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Limit e(x) for x-&gt; oo</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.series.gruntz.sign">
-<tt class="descclassname">sympy.series.gruntz.</tt><tt class="descname">sign</tt><big>(</big><em>*args</em>, <em>**kw_args</em><big>)</big><a class="reference internal" href="../_modules/sympy/series/gruntz.html#sign"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.gruntz.sign" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a sign of an expression e(x) for x-&gt;oo.</p>
-<div class="highlight-python"><pre>e &gt;  0 for x sufficiently large ...  1
-e == 0 for x sufficiently large ...  0
-e &lt;  0 for x sufficiently large ... -1</pre>
-</div>
-<p>The result of this function is currently undefined if e changes sign
-arbitarily often for arbitrarily large x (e.g. sin(x)).</p>
-<p>Note that this returns zero only if e is <em>constantly</em> zero
-for x sufficiently large. [If e is constant, of course, this is just
-the same thing as the sign of e.]</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.series.gruntz.mrv">
-<tt class="descclassname">sympy.series.gruntz.</tt><tt class="descname">mrv</tt><big>(</big><em>e</em>, <em>x</em><big>)</big><a class="reference internal" href="../_modules/sympy/series/gruntz.html#mrv"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.gruntz.mrv" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a SubsSet of most rapidly varying (mrv) subexpressions of &#8216;e&#8217;,
-and e rewritten in terms of these</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.series.gruntz.mrv_max1">
-<tt class="descclassname">sympy.series.gruntz.</tt><tt class="descname">mrv_max1</tt><big>(</big><em>f</em>, <em>g</em>, <em>exps</em>, <em>x</em><big>)</big><a class="reference internal" href="../_modules/sympy/series/gruntz.html#mrv_max1"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.gruntz.mrv_max1" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the maximum of two sets of expressions f and g, which
-are in the same comparability class, i.e. mrv_max1() compares (two elements of)
-f and g and returns the set, which is in the higher comparability class
-of the union of both, if they have the same order of variation.
-Also returns exps, with the appropriate substitutions made.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.series.gruntz.mrv_max3">
-<tt class="descclassname">sympy.series.gruntz.</tt><tt class="descname">mrv_max3</tt><big>(</big><em>f</em>, <em>expsf</em>, <em>g</em>, <em>expsg</em>, <em>union</em>, <em>expsboth</em>, <em>x</em><big>)</big><a class="reference internal" href="../_modules/sympy/series/gruntz.html#mrv_max3"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.gruntz.mrv_max3" title="Permalink to this definition">¶</a></dt>
-<dd><p>Computes the maximum of two sets of expressions f and g, which
-are in the same comparability class, i.e. max() compares (two elements of)
-f and g and returns either (f, expsf) [if f is larger], (g, expsg)
-[if g is larger] or (union, expsboth) [if f, g are of the same class].</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.series.gruntz.SubsSet">
-<em class="property">class </em><tt class="descclassname">sympy.series.gruntz.</tt><tt class="descname">SubsSet</tt><a class="reference internal" href="../_modules/sympy/series/gruntz.html#SubsSet"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.gruntz.SubsSet" title="Permalink to this definition">¶</a></dt>
-<dd><p>Stores (expr, dummy) pairs, and how to rewrite expr-s.</p>
-<p>The gruntz algorithm needs to rewrite certain expressions in term of a new
-variable w. We cannot use subs, because it is just too smart for us. For
-example:</p>
-<div class="highlight-python"><pre>&gt; Omega=[exp(exp(_p - exp(-_p))/(1 - 1/_p)), exp(exp(_p))]
-&gt; O2=[exp(-exp(_p) + exp(-exp(-_p))*exp(_p)/(1 - 1/_p))/_w, 1/_w]
-&gt; e = exp(exp(_p - exp(-_p))/(1 - 1/_p)) - exp(exp(_p))
-&gt; e.subs(Omega[0],O2[0]).subs(Omega[1],O2[1])
--1/w + exp(exp(p)*exp(-exp(-p))/(1 - 1/p))</pre>
-</div>
-<p>is really not what we want!</p>
-<p>So we do it the hard way and keep track of all the things we potentially
-want to substitute by dummy variables. Consider the expression:</p>
-<div class="highlight-python"><pre>exp(x - exp(-x)) + exp(x) + x.</pre>
-</div>
-<p>The mrv set is {exp(x), exp(-x), exp(x - exp(-x))}.
-We introduce corresponding dummy variables d1, d2, d3 and rewrite:</p>
-<div class="highlight-python"><pre>d3 + d1 + x.</pre>
-</div>
-<p>This class first of all keeps track of the mapping expr-&gt;variable, i.e.
-will at this stage be a dictionary:</p>
-<div class="highlight-python"><pre>{exp(x): d1, exp(-x): d2, exp(x - exp(-x)): d3}.</pre>
-</div>
-<p>[It turns out to be more convenient this way round.]
-But sometimes expressions in the mrv set have other expressions from the
-mrv set as subexpressions, and we need to keep track of that as well. In
-this case, d3 is really exp(x - d2), so rewrites at this stage is:</p>
-<div class="highlight-python"><pre>{d3: exp(x-d2)}.</pre>
-</div>
-<p>The function rewrite uses all this information to correctly rewrite our
-expression in terms of w. In this case w can be choosen to be exp(-x),
-i.e. d2. The correct rewriting then is:</p>
-<div class="highlight-python"><pre>exp(-w)/w + 1/w + x.</pre>
-</div>
-<dl class="function">
-<dt id="sympy.series.gruntz.SubsSet.meets">
-<tt class="descname">meets</tt><big>(</big><em>self</em>, <em>s2</em><big>)</big><a class="reference internal" href="../_modules/sympy/series/gruntz.html#SubsSet.meets"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.gruntz.SubsSet.meets" title="Permalink to this definition">¶</a></dt>
-<dd><p>Tell whether or not self and s2 have non-empty intersection</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.series.gruntz.SubsSet.union">
-<tt class="descname">union</tt><big>(</big><em>self</em>, <em>s2</em>, <em>exps=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/series/gruntz.html#SubsSet.union"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.gruntz.SubsSet.union" title="Permalink to this definition">¶</a></dt>
-<dd><p>Compute the union of self and s2, adjusting exps</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-</div>
-</div>
-<div class="section" id="more-intuitive-series-expansion">
-<h2>More Intuitive Series Expansion<a class="headerlink" href="#more-intuitive-series-expansion" title="Permalink to this headline">¶</a></h2>
-<p>This is achieved
-by creating a wrapper around Basic.series(). This allows for the use of
-series(x*cos(x),x), which is possibly more intuative than (x*cos(x)).series(x).</p>
-<div class="section" id="examples">
-<h3>Examples<a class="headerlink" href="#examples" title="Permalink to this headline">¶</a></h3>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">series</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">series</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1 - x**2/2 + x**4/24 + O(x**6)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="id1">
-<h3>Reference<a class="headerlink" href="#id1" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.series.series.series">
-<tt class="descclassname">sympy.series.series.</tt><tt class="descname">series</tt><big>(</big><em>expr</em>, <em>x=None</em>, <em>x0=0</em>, <em>n=6</em>, <em>dir='+'</em><big>)</big><a class="reference internal" href="../_modules/sympy/series/series.html#series"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.series.series" title="Permalink to this definition">¶</a></dt>
-<dd><p>Series expansion of expr around point <span class="math">\(x = x0\)</span>.</p>
-<p>See the doctring of Expr.series() for complete details of this wrapper.</p>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="order-terms">
-<h2>Order Terms<a class="headerlink" href="#order-terms" title="Permalink to this headline">¶</a></h2>
-<p>This module also implements automatic keeping track of the order of your
-expansion.</p>
-<div class="section" id="id2">
-<h3>Examples<a class="headerlink" href="#id2" title="Permalink to this headline">¶</a></h3>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Order</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Order</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">O(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Order</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
-<span class="go">1 + O(x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="id3">
-<h3>Reference<a class="headerlink" href="#id3" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.series.order.Order">
-<em class="property">class </em><tt class="descclassname">sympy.series.order.</tt><tt class="descname">Order</tt><a class="reference internal" href="../_modules/sympy/series/order.html#Order"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.order.Order" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents the limiting behavior of some function</p>
-<p>The order of a function characterizes the function based on the limiting
-behavior of the function as it goes to some limit. Only taking the limit
-point to be 0 is currently supported. This is expressed in big O notation
-<a class="reference internal" href="#r91">[R91]</a>.</p>
-<p>The formal definition for the order of a function <span class="math">\(g(x)\)</span> about a point <span class="math">\(a\)</span>
-is such that <span class="math">\(g(x) = O(f(x))\)</span> as <span class="math">\(x \rightarrow a\)</span> if and only if for any
-<span class="math">\(\delta &gt; 0\)</span> there exists a <span class="math">\(M &gt; 0\)</span> such that <span class="math">\(|g(x)| \leq M|f(x)|\)</span> for
-<span class="math">\(|x-a| &lt; \delta\)</span>. This is equivalent to <span class="math">\(\lim_{x \rightarrow a}
-|g(x)/f(x)| &lt; \infty\)</span>.</p>
-<p>Let&#8217;s illustrate it on the following example by taking the expansion of
-<span class="math">\(\sin(x)\)</span> about 0:</p>
-<div class="math">
-\[\sin(x) = x - x^3/3! + O(x^5)\]</div>
-<p>where in this case <span class="math">\(O(x^5) = x^5/5! - x^7/7! + \cdots\)</span>. By the definition
-of <span class="math">\(O\)</span>, for any <span class="math">\(\delta &gt; 0\)</span> there is an <span class="math">\(M\)</span> such that:</p>
-<div class="math">
-\[\begin{split}|x^5/5! - x^7/7! + ....| &lt;= M|x^5| \text{ for } |x| &lt; \delta\end{split}\]</div>
-<p>or by the alternate definition:</p>
-<div class="math">
-\[\begin{split}\lim_{x \rightarrow 0} | (x^5/5! - x^7/7! + ....) / x^5| &lt; \infty\end{split}\]</div>
-<p>which surely is true, because</p>
-<div class="math">
-\[\lim_{x \rightarrow 0} | (x^5/5! - x^7/7! + ....) / x^5| = 1/5!\]</div>
-<p>As it is usually used, the order of a function can be intuitively thought
-of representing all terms of powers greater than the one specified. For
-example, <span class="math">\(O(x^3)\)</span> corresponds to any terms proportional to <span class="math">\(x^3,
-x^4,\ldots\)</span> and any higher power. For a polynomial, this leaves terms
-proportional to <span class="math">\(x^2\)</span>, <span class="math">\(x\)</span> and constants.</p>
-<p class="rubric">Notes</p>
-<p>In <tt class="docutils literal"><span class="pre">O(f(x),</span> <span class="pre">x)</span></tt> the expression <tt class="docutils literal"><span class="pre">f(x)</span></tt> is assumed to have a leading
-term.  <tt class="docutils literal"><span class="pre">O(f(x),</span> <span class="pre">x)</span></tt> is automatically transformed to
-<tt class="docutils literal"><span class="pre">O(f(x).as_leading_term(x),x)</span></tt>.</p>
-<blockquote>
-<div><p><tt class="docutils literal"><span class="pre">O(expr*f(x),</span> <span class="pre">x)</span></tt> is <tt class="docutils literal"><span class="pre">O(f(x),</span> <span class="pre">x)</span></tt></p>
-<p><tt class="docutils literal"><span class="pre">O(expr,</span> <span class="pre">x)</span></tt> is <tt class="docutils literal"><span class="pre">O(1)</span></tt></p>
-<p><tt class="docutils literal"><span class="pre">O(0,</span> <span class="pre">x)</span></tt> is 0.</p>
-</div></blockquote>
-<p>Multivariate O is also supported:</p>
-<blockquote>
-<div><tt class="docutils literal"><span class="pre">O(f(x,</span> <span class="pre">y),</span> <span class="pre">x,</span> <span class="pre">y)</span></tt> is transformed to
-<tt class="docutils literal"><span class="pre">O(f(x,</span> <span class="pre">y).as_leading_term(x,y).as_leading_term(y),</span> <span class="pre">x,</span> <span class="pre">y)</span></tt></div></blockquote>
-<p>In the multivariate case, it is assumed the limits w.r.t. the various
-symbols commute.</p>
-<p>If no symbols are passed then all symbols in the expression are used:</p>
-<p class="rubric">References</p>
-<table class="docutils citation" frame="void" id="r91" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label">[R91]</td><td><em>(<a class="fn-backref" href="#id4">1</a>, <a class="fn-backref" href="#id5">2</a>)</em> <a class="reference external" href="http://en.wikipedia.org/wiki/Big_O_notation">Big O notation</a></td></tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">O</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">O(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">x</span>
-<span class="go">O(x**2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">O</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">-</span><span class="n">O</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">O(x)</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.series.order.Order.contains">
-<tt class="descname">contains</tt><big>(</big><em>*args</em>, <em>**kw_args</em><big>)</big><a class="reference internal" href="../_modules/sympy/series/order.html#Order.contains"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.order.Order.contains" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if expr belongs to Order(self.expr, *self.variables).
-Return False if self belongs to expr.
-Return None if the inclusion relation cannot be determined
-(e.g. when self and expr have different symbols).</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="series-acceleration">
-<h2>Series Acceleration<a class="headerlink" href="#series-acceleration" title="Permalink to this headline">¶</a></h2>
-<p>TODO</p>
-<div class="section" id="id6">
-<h3>Reference<a class="headerlink" href="#id6" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.series.acceleration.richardson">
-<tt class="descclassname">sympy.series.acceleration.</tt><tt class="descname">richardson</tt><big>(</big><em>A</em>, <em>k</em>, <em>n</em>, <em>N</em><big>)</big><a class="reference internal" href="../_modules/sympy/series/acceleration.html#richardson"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.acceleration.richardson" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculate an approximation for lim k-&gt;oo A(k) using Richardson
-extrapolation with the terms A(n), A(n+1), ..., A(n+N+1).
-Choosing N ~= 2*n often gives good results.</p>
-<p>A simple example is to calculate exp(1) using the limit definition.
-This limit converges slowly; n = 100 only produces two accurate
-digits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">n</span><span class="p">)</span><span class="o">**</span><span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">round</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(),</span> <span class="mi">10</span><span class="p">))</span>
-<span class="go">2.7048138294</span>
-</pre></div>
-</div>
-<p>Richardson extrapolation with 11 appropriately chosen terms gives
-a value that is accurate to the indicated precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">E</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.series.acceleration</span> <span class="kn">import</span> <span class="n">richardson</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">round</span><span class="p">(</span><span class="n">richardson</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">20</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(),</span> <span class="mi">10</span><span class="p">))</span>
-<span class="go">2.7182818285</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">round</span><span class="p">(</span><span class="n">E</span><span class="o">.</span><span class="n">evalf</span><span class="p">(),</span> <span class="mi">10</span><span class="p">))</span>
-<span class="go">2.7182818285</span>
-</pre></div>
-</div>
-<p>Another useful application is to speed up convergence of series.
-Computing 100 terms of the zeta(2) series 1/k**2 yields only
-two accurate digits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">k</span><span class="p">,</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Sum</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Sum</span><span class="p">(</span><span class="n">k</span><span class="o">**-</span><span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">round</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(),</span> <span class="mi">10</span><span class="p">))</span>
-<span class="go">1.6349839002</span>
-</pre></div>
-</div>
-<p>Richardson extrapolation performs much better:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">round</span><span class="p">(</span><span class="n">richardson</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">20</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(),</span> <span class="mi">10</span><span class="p">))</span>
-<span class="go">1.6449340668</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">round</span><span class="p">(((</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">6</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(),</span> <span class="mi">10</span><span class="p">))</span>     <span class="c"># Exact value</span>
-<span class="go">1.6449340668</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.series.acceleration.shanks">
-<tt class="descclassname">sympy.series.acceleration.</tt><tt class="descname">shanks</tt><big>(</big><em>A</em>, <em>k</em>, <em>n</em>, <em>m=1</em><big>)</big><a class="reference internal" href="../_modules/sympy/series/acceleration.html#shanks"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.acceleration.shanks" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculate an approximation for lim k-&gt;oo A(k) using the n-term Shanks
-transformation S(A)(n). With m &gt; 1, calculate the m-fold recursive
-Shanks transformation S(S(...S(A)...))(n).</p>
-<p>The Shanks transformation is useful for summing Taylor series that
-converge slowly near a pole or singularity, e.g. for log(2):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">k</span><span class="p">,</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Sum</span><span class="p">,</span> <span class="n">Integer</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.series.acceleration</span> <span class="kn">import</span> <span class="n">shanks</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Sum</span><span class="p">(</span><span class="n">Integer</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="n">k</span><span class="p">,</span> <span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">round</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span><span class="o">.</span><span class="n">evalf</span><span class="p">(),</span> <span class="mi">10</span><span class="p">))</span>
-<span class="go">0.6881721793</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">round</span><span class="p">(</span><span class="n">shanks</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="mi">25</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(),</span> <span class="mi">10</span><span class="p">))</span>
-<span class="go">0.6931396564</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">round</span><span class="p">(</span><span class="n">shanks</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="mi">25</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(),</span> <span class="mi">10</span><span class="p">))</span>
-<span class="go">0.6931471806</span>
-</pre></div>
-</div>
-<p>The correct value is 0.6931471805599453094172321215.</p>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="residues">
-<h2>Residues<a class="headerlink" href="#residues" title="Permalink to this headline">¶</a></h2>
-<p>TODO</p>
-<div class="section" id="id7">
-<h3>Reference<a class="headerlink" href="#id7" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.series.residues.residue">
-<tt class="descclassname">sympy.series.residues.</tt><tt class="descname">residue</tt><big>(</big><em>expr</em>, <em>x</em>, <em>x0</em><big>)</big><a class="reference internal" href="../_modules/sympy/series/residues.html#residue"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.series.residues.residue" title="Permalink to this definition">¶</a></dt>
-<dd><p>Finds the residue of <tt class="docutils literal"><span class="pre">expr</span></tt> at the point x=x0.</p>
-<p>The residue is defined as the coefficient of 1/(x-x0) in the power series
-expansion about x=x0.</p>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Residue_theorem">http://en.wikipedia.org/wiki/Residue_theorem</a></li>
-</ol>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">residue</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">residue</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">residue</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">residue</span><span class="p">(</span><span class="mi">2</span><span class="o">/</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-<p>This function is essential for the Residue Theorem [1].</p>
-</dd></dl>
-
-</div>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../index.html">
-              <img class="logo" src="../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Series Expansions</a><ul>
-<li><a class="reference internal" href="#limits">Limits</a><ul>
-<li><a class="reference internal" href="#the-gruntz-algorithm">The Gruntz Algorithm</a><ul>
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-            
-  <div class="section" id="module-sympy.core.sets">
-<span id="sets"></span><h1>Sets<a class="headerlink" href="#module-sympy.core.sets" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="set">
-<h2>Set<a class="headerlink" href="#set" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.core.sets.Set">
-<em class="property">class </em><tt class="descclassname">sympy.core.sets.</tt><tt class="descname">Set</tt><a class="reference internal" href="../_modules/sympy/core/sets.html#Set"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Set" title="Permalink to this definition">¶</a></dt>
-<dd><p>The base class for any kind of set.</p>
-<p>This is not meant to be used directly as a container of items.
-It does not behave like the builtin set; see FiniteSet for that.</p>
-<p>Real intervals are represented by the Interval class and unions of sets
-by the Union class. The empty set is represented by the EmptySet class
-and available as a singleton as S.EmptySet.</p>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="60%" />
-<col width="40%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>is_EmptySet</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>is_Intersection</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>is_UniversalSet</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="attribute">
-<dt id="sympy.core.sets.Set.complement">
-<tt class="descname">complement</tt><a class="reference internal" href="../_modules/sympy/core/sets.html#Set.complement"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Set.complement" title="Permalink to this definition">¶</a></dt>
-<dd><p>The complement of &#8216;self&#8217;.</p>
-<p>As a shortcut it is possible to use the &#8216;~&#8217; or &#8216;-&#8216; operators:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Interval</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">complement</span>
-<span class="go">(-oo, 0) U (1, oo)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">~</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">(-oo, 0) U (1, oo)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="o">-</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">(-oo, 0) U (1, oo)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.sets.Set.contains">
-<tt class="descname">contains</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/sets.html#Set.contains"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Set.contains" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if &#8216;other&#8217; is contained in &#8216;self&#8217; as an element.</p>
-<p>As a shortcut it is possible to use the &#8216;in&#8217; operator:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Interval</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mf">0.5</span> <span class="ow">in</span> <span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.sets.Set.inf">
-<tt class="descname">inf</tt><a class="reference internal" href="../_modules/sympy/core/sets.html#Set.inf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Set.inf" title="Permalink to this definition">¶</a></dt>
-<dd><p>The infimum of &#8216;self&#8217;</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Interval</span><span class="p">,</span> <span class="n">Union</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">inf</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Union</span><span class="p">(</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Interval</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span><span class="o">.</span><span class="n">inf</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.sets.Set.intersect">
-<tt class="descname">intersect</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/sets.html#Set.intersect"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Set.intersect" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the intersection of &#8216;self&#8217; and &#8216;other&#8217;.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Interval</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="n">Interval</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">[1, 2]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.sets.Set.measure">
-<tt class="descname">measure</tt><a class="reference internal" href="../_modules/sympy/core/sets.html#Set.measure"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Set.measure" title="Permalink to this definition">¶</a></dt>
-<dd><p>The (Lebesgue) measure of &#8216;self&#8217;</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Interval</span><span class="p">,</span> <span class="n">Union</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">measure</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Union</span><span class="p">(</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Interval</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span><span class="o">.</span><span class="n">measure</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.sets.Set.sort_key">
-<tt class="descname">sort_key</tt><big>(</big><em>self</em>, <em>order=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/sets.html#Set.sort_key"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Set.sort_key" title="Permalink to this definition">¶</a></dt>
-<dd><p>Give sort_key of infimum (if possible) else sort_key of the set.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.sets.Set.subset">
-<tt class="descname">subset</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/sets.html#Set.subset"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Set.subset" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if &#8216;other&#8217; is a subset of &#8216;self&#8217;.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Interval</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">left_open</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span><span class="o">.</span><span class="n">contains</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.sets.Set.sup">
-<tt class="descname">sup</tt><a class="reference internal" href="../_modules/sympy/core/sets.html#Set.sup"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Set.sup" title="Permalink to this definition">¶</a></dt>
-<dd><p>The supremum of &#8216;self&#8217;</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Interval</span><span class="p">,</span> <span class="n">Union</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">sup</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Union</span><span class="p">(</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">Interval</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span><span class="o">.</span><span class="n">sup</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.sets.Set.union">
-<tt class="descname">union</tt><big>(</big><em>self</em>, <em>other</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/sets.html#Set.union"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Set.union" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the union of &#8216;self&#8217; and &#8216;other&#8217;.</p>
-<p>As a shortcut it is possible to use the &#8216;+&#8217; operator:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Interval</span><span class="p">,</span> <span class="n">FiniteSet</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">union</span><span class="p">(</span><span class="n">Interval</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
-<span class="go">[0, 1] U [2, 3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">Interval</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">[0, 1] U [2, 3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="bp">True</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span> <span class="o">+</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">(1, 2] U {3}</span>
-</pre></div>
-</div>
-<p>Similarly it is possible to use the &#8216;-&#8216; operator for set differences:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">(1, 2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">[1, 2) U (2, 3]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<div class="section" id="elementary-sets">
-<h3>Elementary Sets<a class="headerlink" href="#elementary-sets" title="Permalink to this headline">¶</a></h3>
-</div>
-</div>
-<div class="section" id="interval">
-<h2>Interval<a class="headerlink" href="#interval" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.core.sets.Interval">
-<em class="property">class </em><tt class="descclassname">sympy.core.sets.</tt><tt class="descname">Interval</tt><a class="reference internal" href="../_modules/sympy/core/sets.html#Interval"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Interval" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents a real interval as a Set.</p>
-<dl class="docutils">
-<dt>Usage:</dt>
-<dd><p class="first">Returns an interval with end points &#8220;start&#8221; and &#8220;end&#8221;.</p>
-<p class="last">For left_open=True (default left_open is False) the interval
-will be open on the left. Similarly, for right_open=True the interval
-will be open on the right.</p>
-</dd>
-</dl>
-<p class="rubric">Notes</p>
-<ul class="simple">
-<li>Only real end points are supported</li>
-<li>Interval(a, b) with a &gt; b will return the empty set</li>
-<li>Use the evalf() method to turn an Interval into an mpmath
-&#8216;mpi&#8217; interval instance</li>
-</ul>
-<p class="rubric">References</p>
-<p>&lt;<a class="reference external" href="http://en.wikipedia.org/wiki/Interval_(mathematics)">http://en.wikipedia.org/wiki/Interval_(mathematics)</a>&gt;</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Interval</span><span class="p">,</span> <span class="n">sets</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">[0, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="bp">False</span><span class="p">,</span> <span class="bp">True</span><span class="p">)</span>
-<span class="go">[0, 1)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-<span class="go">[0, a]</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="60%" />
-<col width="40%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>is_EmptySet</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>is_Intersection</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>is_UniversalSet</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.core.sets.Interval.as_relational">
-<tt class="descname">as_relational</tt><big>(</big><em>self</em>, <em>symbol</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/sets.html#Interval.as_relational"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Interval.as_relational" title="Permalink to this definition">¶</a></dt>
-<dd><p>Rewrite an interval in terms of inequalities and logic operators.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.sets.Interval.end">
-<tt class="descname">end</tt><a class="reference internal" href="../_modules/sympy/core/sets.html#Interval.end"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Interval.end" title="Permalink to this definition">¶</a></dt>
-<dd><p>The right end point of &#8216;self&#8217;.</p>
-<p>This property takes the same value as the &#8216;sup&#8217; property.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Interval</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">end</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.sets.Interval.is_left_unbounded">
-<tt class="descname">is_left_unbounded</tt><a class="reference internal" href="../_modules/sympy/core/sets.html#Interval.is_left_unbounded"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Interval.is_left_unbounded" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return <tt class="docutils literal"><span class="pre">True</span></tt> if the left endpoint is negative infinity.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.sets.Interval.is_right_unbounded">
-<tt class="descname">is_right_unbounded</tt><a class="reference internal" href="../_modules/sympy/core/sets.html#Interval.is_right_unbounded"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Interval.is_right_unbounded" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return <tt class="docutils literal"><span class="pre">True</span></tt> if the right endpoint is positive infinity.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.sets.Interval.left">
-<tt class="descname">left</tt><a class="headerlink" href="#sympy.core.sets.Interval.left" title="Permalink to this definition">¶</a></dt>
-<dd><p>The left end point of &#8216;self&#8217;.</p>
-<p>This property takes the same value as the &#8216;inf&#8217; property.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Interval</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">start</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.sets.Interval.left_open">
-<tt class="descname">left_open</tt><a class="reference internal" href="../_modules/sympy/core/sets.html#Interval.left_open"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Interval.left_open" title="Permalink to this definition">¶</a></dt>
-<dd><p>True if &#8216;self&#8217; is left-open.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Interval</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">left_open</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span><span class="o">.</span><span class="n">left_open</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">left_open</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span><span class="o">.</span><span class="n">left_open</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.sets.Interval.right">
-<tt class="descname">right</tt><a class="headerlink" href="#sympy.core.sets.Interval.right" title="Permalink to this definition">¶</a></dt>
-<dd><p>The right end point of &#8216;self&#8217;.</p>
-<p>This property takes the same value as the &#8216;sup&#8217; property.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Interval</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">end</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.sets.Interval.right_open">
-<tt class="descname">right_open</tt><a class="reference internal" href="../_modules/sympy/core/sets.html#Interval.right_open"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Interval.right_open" title="Permalink to this definition">¶</a></dt>
-<dd><p>True if &#8216;self&#8217; is right-open.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Interval</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">right_open</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span><span class="o">.</span><span class="n">right_open</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">right_open</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span><span class="o">.</span><span class="n">right_open</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.core.sets.Interval.start">
-<tt class="descname">start</tt><a class="reference internal" href="../_modules/sympy/core/sets.html#Interval.start"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Interval.start" title="Permalink to this definition">¶</a></dt>
-<dd><p>The left end point of &#8216;self&#8217;.</p>
-<p>This property takes the same value as the &#8216;inf&#8217; property.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Interval</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">start</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="finiteset">
-<h2>FiniteSet<a class="headerlink" href="#finiteset" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.core.sets.FiniteSet">
-<em class="property">class </em><tt class="descclassname">sympy.core.sets.</tt><tt class="descname">FiniteSet</tt><a class="reference internal" href="../_modules/sympy/core/sets.html#FiniteSet"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.FiniteSet" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents a finite set of discrete numbers</p>
-<p class="rubric">References</p>
-<p><a class="reference external" href="http://en.wikipedia.org/wiki/Finite_set">http://en.wikipedia.org/wiki/Finite_set</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">FiniteSet</span><span class="p">,</span> <span class="n">sets</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">FiniteSet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">{1, 2, 3, 4}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">3</span> <span class="ow">in</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="60%" />
-<col width="40%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>is_EmptySet</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>is_Intersection</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>is_UniversalSet</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.core.sets.FiniteSet.as_relational">
-<tt class="descname">as_relational</tt><big>(</big><em>self</em>, <em>symbol</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/sets.html#FiniteSet.as_relational"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.FiniteSet.as_relational" title="Permalink to this definition">¶</a></dt>
-<dd><p>Rewrite a FiniteSet in terms of equalities and logic operators.</p>
-</dd></dl>
-
-</dd></dl>
-
-<div class="section" id="compound-sets">
-<h3>Compound Sets<a class="headerlink" href="#compound-sets" title="Permalink to this headline">¶</a></h3>
-</div>
-</div>
-<div class="section" id="union">
-<h2>Union<a class="headerlink" href="#union" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.core.sets.Union">
-<em class="property">class </em><tt class="descclassname">sympy.core.sets.</tt><tt class="descname">Union</tt><a class="reference internal" href="../_modules/sympy/core/sets.html#Union"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Union" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents a union of sets as a Set.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.core.sets.Intersection" title="sympy.core.sets.Intersection"><tt class="xref py py-obj docutils literal"><span class="pre">Intersection</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<p>&lt;<a class="reference external" href="http://en.wikipedia.org/wiki/Union_(set_theory)">http://en.wikipedia.org/wiki/Union_(set_theory)</a>&gt;</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Union</span><span class="p">,</span> <span class="n">Interval</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Union</span><span class="p">(</span><span class="n">Interval</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">Interval</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
-<span class="go">[1, 2] U [3, 4]</span>
-</pre></div>
-</div>
-<p>The Union constructor will always try to merge overlapping intervals,
-if possible. For example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Union</span><span class="p">(</span><span class="n">Interval</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">Interval</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
-<span class="go">[1, 3]</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="60%" />
-<col width="40%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>is_EmptySet</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>is_Intersection</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>is_UniversalSet</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.core.sets.Union.as_relational">
-<tt class="descname">as_relational</tt><big>(</big><em>self</em>, <em>symbol</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/sets.html#Union.as_relational"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Union.as_relational" title="Permalink to this definition">¶</a></dt>
-<dd><p>Rewrite a Union in terms of equalities and logic operators.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.sets.Union.reduce">
-<tt class="descname">reduce</tt><big>(</big><em>args</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/sets.html#Union.reduce"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Union.reduce" title="Permalink to this definition">¶</a></dt>
-<dd><p>Simplify a Union using known rules</p>
-<p>We first start with global rules like
-&#8216;Merge all FiniteSets&#8217;</p>
-<p>Then we iterate through all pairs and ask the constituent sets if they
-can simplify themselves with any other constituent</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="intersection">
-<h2>Intersection<a class="headerlink" href="#intersection" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.core.sets.Intersection">
-<em class="property">class </em><tt class="descclassname">sympy.core.sets.</tt><tt class="descname">Intersection</tt><a class="reference internal" href="../_modules/sympy/core/sets.html#Intersection"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Intersection" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents an intersection of sets as a Set.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.core.sets.Union" title="sympy.core.sets.Union"><tt class="xref py py-obj docutils literal"><span class="pre">Union</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<p>&lt;<a class="reference external" href="http://en.wikipedia.org/wiki/Intersection_(set_theory)">http://en.wikipedia.org/wiki/Intersection_(set_theory)</a>&gt;</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Intersection</span><span class="p">,</span> <span class="n">Interval</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Intersection</span><span class="p">(</span><span class="n">Interval</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">Interval</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
-<span class="go">[2, 3]</span>
-</pre></div>
-</div>
-<p>We often use the .intersect method</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="n">Interval</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
-<span class="go">[2, 3]</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="60%" />
-<col width="40%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>is_EmptySet</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>is_UniversalSet</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-<dl class="function">
-<dt id="sympy.core.sets.Intersection.as_relational">
-<tt class="descname">as_relational</tt><big>(</big><em>self</em>, <em>symbol</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/sets.html#Intersection.as_relational"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Intersection.as_relational" title="Permalink to this definition">¶</a></dt>
-<dd><p>Rewrite an Intersection in terms of equalities and logic operators</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.core.sets.Intersection.reduce">
-<tt class="descname">reduce</tt><big>(</big><em>args</em><big>)</big><a class="reference internal" href="../_modules/sympy/core/sets.html#Intersection.reduce"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.Intersection.reduce" title="Permalink to this definition">¶</a></dt>
-<dd><p>Simplify an intersection using known rules</p>
-<p>We first start with global rules like
-&#8216;if any empty sets return empty set&#8217; and &#8216;distribute any unions&#8217;</p>
-<p>Then we iterate through all pairs and ask the constituent sets if they
-can simplify themselves with any other constituent</p>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="productset">
-<h2>ProductSet<a class="headerlink" href="#productset" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.core.sets.ProductSet">
-<em class="property">class </em><tt class="descclassname">sympy.core.sets.</tt><tt class="descname">ProductSet</tt><a class="reference internal" href="../_modules/sympy/core/sets.html#ProductSet"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.ProductSet" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents a Cartesian Product of Sets.</p>
-<p>Returns a Cartesian product given several sets as either an iterable
-or individual arguments.</p>
-<p>Can use &#8216;*&#8217; operator on any sets for convenient shorthand.</p>
-<p class="rubric">Notes</p>
-<ul class="simple">
-<li>Passes most operations down to the argument sets</li>
-<li>Flattens Products of ProductSets</li>
-</ul>
-<p class="rubric">References</p>
-<p><a class="reference external" href="http://en.wikipedia.org/wiki/Cartesian_product">http://en.wikipedia.org/wiki/Cartesian_product</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Interval</span><span class="p">,</span> <span class="n">FiniteSet</span><span class="p">,</span> <span class="n">ProductSet</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">I</span> <span class="o">=</span> <span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">);</span> <span class="n">S</span> <span class="o">=</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ProductSet</span><span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">S</span><span class="p">)</span>
-<span class="go">[0, 5] x {1, 2, 3}</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="ow">in</span> <span class="n">ProductSet</span><span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">S</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="c"># The unit square</span>
-<span class="go">[0, 1] x [0, 1]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">coin</span> <span class="o">=</span> <span class="n">FiniteSet</span><span class="p">(</span><span class="s">&#39;H&#39;</span><span class="p">,</span> <span class="s">&#39;T&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">set</span><span class="p">(</span><span class="n">coin</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">set([(H, H), (H, T), (T, H), (T, T)])</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="60%" />
-<col width="40%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>is_EmptySet</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>is_Intersection</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>is_UniversalSet</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<div class="section" id="singleton-sets">
-<h3>Singleton Sets<a class="headerlink" href="#singleton-sets" title="Permalink to this headline">¶</a></h3>
-</div>
-</div>
-<div class="section" id="emptyset">
-<h2>EmptySet<a class="headerlink" href="#emptyset" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.core.sets.EmptySet">
-<em class="property">class </em><tt class="descclassname">sympy.core.sets.</tt><tt class="descname">EmptySet</tt><a class="reference internal" href="../_modules/sympy/core/sets.html#EmptySet"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.EmptySet" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents the empty set. The empty set is available as a singleton
-as S.EmptySet.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.core.sets.UniversalSet" title="sympy.core.sets.UniversalSet"><tt class="xref py py-obj docutils literal"><span class="pre">UniversalSet</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<p><a class="reference external" href="http://en.wikipedia.org/wiki/Empty_set">http://en.wikipedia.org/wiki/Empty_set</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">Interval</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">.</span><span class="n">EmptySet</span>
-<span class="go">EmptySet()</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">EmptySet</span><span class="p">)</span>
-<span class="go">EmptySet()</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="60%" />
-<col width="40%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>is_Intersection</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>is_UniversalSet</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</div>
-<div class="section" id="universalset">
-<h2>UniversalSet<a class="headerlink" href="#universalset" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.core.sets.UniversalSet">
-<em class="property">class </em><tt class="descclassname">sympy.core.sets.</tt><tt class="descname">UniversalSet</tt><a class="reference internal" href="../_modules/sympy/core/sets.html#UniversalSet"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.core.sets.UniversalSet" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents the set of all things.
-The universal set is available as a singleton as S.UniversalSet</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.core.sets.EmptySet" title="sympy.core.sets.EmptySet"><tt class="xref py py-obj docutils literal"><span class="pre">EmptySet</span></tt></a></p>
-</div>
-<p class="rubric">References</p>
-<p><a class="reference external" href="http://en.wikipedia.org/wiki/Universal_set">http://en.wikipedia.org/wiki/Universal_set</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">Interval</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">S</span><span class="o">.</span><span class="n">UniversalSet</span>
-<span class="go">UniversalSet()</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Interval</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">UniversalSet</span><span class="p">)</span>
-<span class="go">[1, 2]</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="60%" />
-<col width="40%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>is_EmptySet</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>is_Intersection</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<div class="section" id="module-sympy.sets.fancysets">
-<span id="special-sets"></span><h3>Special Sets<a class="headerlink" href="#module-sympy.sets.fancysets" title="Permalink to this headline">¶</a></h3>
-</div>
-</div>
-<div class="section" id="naturals">
-<h2>Naturals<a class="headerlink" href="#naturals" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.sets.fancysets.Naturals">
-<em class="property">class </em><tt class="descclassname">sympy.sets.fancysets.</tt><tt class="descname">Naturals</tt><a class="reference internal" href="../_modules/sympy/sets/fancysets.html#Naturals"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.sets.fancysets.Naturals" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents the Natural Numbers. The Naturals are available as a singleton
-as S.Naturals</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">Interval</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">5</span> <span class="ow">in</span> <span class="n">S</span><span class="o">.</span><span class="n">Naturals</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iterable</span> <span class="o">=</span> <span class="nb">iter</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Naturals</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">next</span><span class="p">(</span><span class="n">iterable</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">next</span><span class="p">(</span><span class="n">iterable</span><span class="p">))</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">next</span><span class="p">(</span><span class="n">iterable</span><span class="p">))</span>
-<span class="go">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Naturals</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="n">Interval</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)))</span>
-<span class="go">{1, 2, ..., 10}</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="60%" />
-<col width="40%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>is_EmptySet</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>is_Intersection</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>is_UniversalSet</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</div>
-<div class="section" id="integers">
-<h2>Integers<a class="headerlink" href="#integers" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.sets.fancysets.Integers">
-<em class="property">class </em><tt class="descclassname">sympy.sets.fancysets.</tt><tt class="descname">Integers</tt><a class="reference internal" href="../_modules/sympy/sets/fancysets.html#Integers"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.sets.fancysets.Integers" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents the Integers. The Integers are available as a singleton
-as S.Integers</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">Interval</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">5</span> <span class="ow">in</span> <span class="n">S</span><span class="o">.</span><span class="n">Naturals</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iterable</span> <span class="o">=</span> <span class="nb">iter</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Integers</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">next</span><span class="p">(</span><span class="n">iterable</span><span class="p">))</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">next</span><span class="p">(</span><span class="n">iterable</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">next</span><span class="p">(</span><span class="n">iterable</span><span class="p">))</span>
-<span class="go">-1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="nb">next</span><span class="p">(</span><span class="n">iterable</span><span class="p">))</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">Integers</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="n">Interval</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)))</span>
-<span class="go">{-4, -3, ..., 4}</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="60%" />
-<col width="40%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>is_EmptySet</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>is_Intersection</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>is_UniversalSet</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</div>
-<div class="section" id="transformationset">
-<h2>TransformationSet<a class="headerlink" href="#transformationset" title="Permalink to this headline">¶</a></h2>
-<dl class="class">
-<dt id="sympy.sets.fancysets.TransformationSet">
-<em class="property">class </em><tt class="descclassname">sympy.sets.fancysets.</tt><tt class="descname">TransformationSet</tt><a class="reference internal" href="../_modules/sympy/sets/fancysets.html#TransformationSet"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.sets.fancysets.TransformationSet" title="Permalink to this definition">¶</a></dt>
-<dd><p>A set that is a transformation of another through some algebraic expression</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">TransformationSet</span><span class="p">,</span> <span class="n">FiniteSet</span><span class="p">,</span> <span class="n">Lambda</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">S</span><span class="o">.</span><span class="n">Naturals</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">squares</span> <span class="o">=</span> <span class="n">TransformationSet</span><span class="p">(</span><span class="n">Lambda</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="n">N</span><span class="p">)</span> <span class="c"># {x**2 for x in N}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">4</span> <span class="ow">in</span> <span class="n">squares</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">5</span> <span class="ow">in</span> <span class="n">squares</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">FiniteSet</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span><span class="o">.</span><span class="n">intersect</span><span class="p">(</span><span class="n">squares</span><span class="p">)</span>
-<span class="go">{1, 4, 9}</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">square_iterable</span> <span class="o">=</span> <span class="nb">iter</span><span class="p">(</span><span class="n">squares</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="nb">next</span><span class="p">(</span><span class="n">square_iterable</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="go">4</span>
-<span class="go">9</span>
-<span class="go">16</span>
-</pre></div>
-</div>
-<p class="rubric">Attributes</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="60%" />
-<col width="40%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>is_EmptySet</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-even"><td>is_Intersection</td>
-<td>&nbsp;</td>
-</tr>
-<tr class="row-odd"><td>is_UniversalSet</td>
-<td>&nbsp;</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
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-  <h3><a href="../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Sets</a><ul>
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-</ul>
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-<li><a class="reference internal" href="#singleton-sets">Singleton Sets</a></li>
-</ul>
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-  <div class="section" id="details-on-the-hypergeometric-function-expansion-module">
-<h1>Details on the Hypergeometric Function Expansion Module<a class="headerlink" href="#details-on-the-hypergeometric-function-expansion-module" title="Permalink to this headline">¶</a></h1>
-<p>This page describes how the function <tt class="xref py py-func docutils literal"><span class="pre">hyperexpand()</span></tt> and related code
-work. For usage, see the documentation of the symplify module.</p>
-<div class="section" id="hypergeometric-function-expansion-algorithm">
-<h2>Hypergeometric Function Expansion Algorithm<a class="headerlink" href="#hypergeometric-function-expansion-algorithm" title="Permalink to this headline">¶</a></h2>
-<p>This section describes the algorithm used to expand hypergeometric functions.
-Most of it is based on the papers <a class="reference internal" href="#roach1996">[Roach1996]</a> and <a class="reference internal" href="#roach1997">[Roach1997]</a>.</p>
-<p>Recall that the hypergeometric function is (initially) defined as</p>
-<div class="math">
-\[\begin{split}{}_pF_q\left(\begin{matrix} a_1, \dots, a_p \\ b_1, \dots, b_q \end{matrix}
-                 \middle| z \right)
-    = \sum_{n=0}^\infty \frac{(a_1)_n \dots (a_p)_n}{(b_1)_n \dots (b_q)_n}
-                        \frac{z^n}{n!}.\end{split}\]</div>
-<p>It turns out that there are certain differential operators that can change the
-<span class="math">\(a_p\)</span> and <span class="math">\(p_q\)</span> parameters by integers. If a sequence of such
-operators is known that converts the set of indices <span class="math">\(a_r^0\)</span> and
-<span class="math">\(b_s^0\)</span> into <span class="math">\(a_p\)</span> and <span class="math">\(b_q\)</span>, then we shall say the pair <span class="math">\(a_p,
-b_q\)</span> is reachable from <span class="math">\(a_r^0, b_s^0\)</span>. Our general strategy is thus as
-follows: given a set <span class="math">\(a_p, b_q\)</span> of parameters, try to look up an origin
-<span class="math">\(a_r^0, b_s^0\)</span> for which we know an expression, and then apply the
-sequence of differential operators to the known expression to find an
-expression for the Hypergeometric function we are interested in.</p>
-<div class="section" id="notation">
-<h3>Notation<a class="headerlink" href="#notation" title="Permalink to this headline">¶</a></h3>
-<p>In the following, the symbol <span class="math">\(a\)</span> will always denote a numerator parameter
-and the symbol <span class="math">\(b\)</span> will always denote a denominator parameter. The
-subscripts <span class="math">\(p, q, r, s\)</span> denote vectors of that length, so e.g.
-<span class="math">\(a_p\)</span> denotes a vector of <span class="math">\(p\)</span> numerator parameters. The subscripts
-<span class="math">\(i\)</span> and <span class="math">\(j\)</span> denote &#8220;running indices&#8221;, so they should usually be used in
-conjuction with a &#8220;for all <span class="math">\(i\)</span>&#8221;. E.g. <span class="math">\(a_i &lt; 4\)</span> for all <span class="math">\(i\)</span>.
-Uppercase subscripts <span class="math">\(I\)</span> and <span class="math">\(J\)</span> denote a chosen, fixed index. So
-for example <span class="math">\(a_I &gt; 0\)</span> is true if the inequality holds for the one index
-<span class="math">\(I\)</span> we are currently interested in.</p>
-</div>
-<div class="section" id="incrementing-and-decrementing-indices">
-<h3>Incrementing and decrementing indices<a class="headerlink" href="#incrementing-and-decrementing-indices" title="Permalink to this headline">¶</a></h3>
-<p>Suppose <span class="math">\(a_i \ne 0\)</span>. Set <span class="math">\(A(a_i) =
-\frac{z}{a_i}\frac{\mathrm{d}}{dz}+1\)</span>. It is then easy to show that
-<span class="math">\(A(a_i) {}_p F_q\left({a_p \atop b_q} \middle| z \right) = {}_p F_q\left({a_p +
-e_i \atop b_q} \middle| z \right)\)</span>, where <span class="math">\(e_i\)</span> is the i-th unit vector.
-Similarly for <span class="math">\(b_j \ne 1\)</span> we set <span class="math">\(B(b_j) = \frac{z}{b_j-1}
-\frac{\mathrm{d}}{dz}+1\)</span> and find <span class="math">\(B(b_j) {}_p F_q\left({a_p \atop b_q}
-\middle| z \right) = {}_p F_q\left({a_p \atop b_q - e_i} \middle| z \right)\)</span>.
-Thus we can increment upper and decrement lower indices at will, as long as we
-don&#8217;t go through zero. The <span class="math">\(A(a_i)\)</span> and <span class="math">\(B(b_j)\)</span> are called shift
-operators.</p>
-<p>It is also easy to show that <span class="math">\(\frac{\mathrm{d}}{dz} {}_p F_q\left({a_p
-\atop b_q} \middle| z \right) = \frac{a_1 \dots a_p}{b_1 \dots b_q} {}_p
-F_q\left({a_p + 1 \atop b_q + 1} \middle| z \right)\)</span>, where <span class="math">\(a_p + 1\)</span> is
-the vector <span class="math">\(a_1 + 1, a_2 + 1, \dots\)</span> and similarly for <span class="math">\(b_q + 1\)</span>.
-Combining this with the shift operators,  we arrive at one form of the
-Hypergeometric differential equation: <span class="math">\(\left[ \frac{\mathrm{d}}{dz}
-\prod_{j=1}^q B(b_j) - \frac{a_1 \dots a_p}{(b_1-1) \dots (b_q-1)}
-\prod_{i=1}^p A(a_i) \right] {}_p F_q\left({a_p \atop b_q} \middle| z \right) =
-0\)</span>. This holds if all shift operators are defined, i.e. if no <span class="math">\(a_i = 0\)</span>
-and no <span class="math">\(b_j = 1\)</span>. Clearing denominators and multiplying through by z we
-arrive at the following equation: <span class="math">\(\left[ z\frac{\mathrm{d}}{dz}
-\prod_{j=1}^q \left(z\frac{\mathrm{d}}{dz} + b_j-1 \right) - z \prod_{i=1}^p
-\left( z\frac{\mathrm{d}}{\mathrm{d}z} + a_i \right) \right] {}_p F_q\left({a_p
-\atop b_q} \middle| z\right) = 0\)</span>. Even though our derivation does not show it,
-it can be checked that this equation holds whenever the <span class="math">\({}_p F_q\)</span> is
-defined.</p>
-<p>Notice that, under suitable conditions on <span class="math">\(a_I, b_J\)</span>, each of the
-operators <span class="math">\(A(a_i)\)</span>, <span class="math">\(B(b_j)\)</span> and
-<span class="math">\(z\frac{\mathrm{d}}{\mathrm{d}z}\)</span> can be expressed in terms of <span class="math">\(A(a_I)\)</span> or
-<span class="math">\(B(b_J)\)</span>. Our next aim is to write the Hypergeometric differential
-equation as follows: <span class="math">\([X A(a_I) - r] {}_p F_q\left({a_p \atop b_q}
-\middle| z\right) = 0\)</span>, for some operator <span class="math">\(X\)</span> and some constant <span class="math">\(r\)</span>
-to be determined. If
-<span class="math">\(r \ne 0\)</span>, then we can write this as <span class="math">\(\frac{-1}{r} X {}_p F_q\left({a_p +
-e_I \atop b_q} \middle| z\right) = {}_p F_q\left({a_p \atop b_q} \middle|
-z\right)\)</span>, and so <span class="math">\(\frac{-1}{r}X\)</span> undoes the shifting of <span class="math">\(A(a_I)\)</span>,
-whence it will be called an inverse-shift operator.</p>
-<p>Now <span class="math">\(A(a_I)\)</span> exists if <span class="math">\(a_I \ne 0\)</span>, and then
-<span class="math">\(z\frac{\mathrm{d}}{\mathrm{d}z} = a_I A(a_I) - a_I\)</span>. Observe also that all the
-operators <span class="math">\(A(a_i)\)</span>, <span class="math">\(B(b_j)\)</span> and
-<span class="math">\(z\frac{\mathrm{d}}{\mathrm{d}z}\)</span> commute. We have <span class="math">\(\prod_{i=1}^p \left(
-z\frac{\mathrm{d}}{\mathrm{d}z} + a_i \right) = \left(\prod_{i=1, i \ne I}^p
-\left( z\frac{\mathrm{d}}{\mathrm{d}z} + a_i \right)\right) a_I A(a_I)\)</span>, so
-this gives us the first half of <span class="math">\(X\)</span>. The other half does not have such a
-nice expression. We find <span class="math">\(z\frac{\mathrm{d}}{dz} \prod_{j=1}^q
-\left(z\frac{\mathrm{d}}{dz} + b_j-1 \right) = \left(a_I A(a_I) - a_I\right)
-\prod_{j=1}^q \left(a_I A(a_I) - a_I + b_j - 1\right)\)</span>. Since the first
-half had no constant term, we infer <span class="math">\(r = -a_I\prod_{j=1}^q(b_j - 1
--a_I)\)</span>.</p>
-<p>This tells us under which conditions we can &#8220;un-shift&#8221; <span class="math">\(A(a_I)\)</span>, namely
-when <span class="math">\(a_I \ne 0\)</span> and <span class="math">\(r \ne 0\)</span>. Substituting <span class="math">\(a_I - 1\)</span> for
-<span class="math">\(a_I\)</span> then tells us under what conditions we can decrement the index
-<span class="math">\(a_I\)</span>. Doing a similar analysis for <span class="math">\(B(a_J)\)</span>, we arrive at the
-following rules:</p>
-<ul class="simple">
-<li>An index <span class="math">\(a_I\)</span> can be decremented if <span class="math">\(a_I \ne 1\)</span> and
-<span class="math">\(a_I \ne b_j\)</span> for all <span class="math">\(b_j\)</span>.</li>
-<li>An index <span class="math">\(b_J\)</span> can be
-incremented if <span class="math">\(b_J \ne -1\)</span> and <span class="math">\(b_J \ne a_i\)</span> for all
-<span class="math">\(a_i\)</span>.</li>
-</ul>
-<p>Combined with the conditions (stated above) for the existence of shift
-operators, we have thus established the rules of the game!</p>
-</div>
-<div class="section" id="reduction-of-order">
-<h3>Reduction of Order<a class="headerlink" href="#reduction-of-order" title="Permalink to this headline">¶</a></h3>
-<p>Notice that, quite trivially, if <span class="math">\(a_I = b_J\)</span>, we have <span class="math">\({}_p
-F_q\left({a_p \atop b_q} \middle| z \right) = {}_{p-1} F_{q-1}\left({a_p^*
-\atop b_q^*} \middle| z \right)\)</span>, where <span class="math">\(a_p^*\)</span> means <span class="math">\(a_p\)</span> with
-<span class="math">\(a_I\)</span> omitted, and similarly for <span class="math">\(b_q^*\)</span>. We call this reduction of
-order.</p>
-<p>In fact, we can do even better. If <span class="math">\(a_I - b_J \in \mathbb{Z}_{&gt;0}\)</span>, then
-it is easy to see that <span class="math">\(\frac{(a_I)_n}{(b_J)_n}\)</span> is actually a polynomial
-in <span class="math">\(n\)</span>. It is also easy to see that
-<span class="math">\((z\frac{\mathrm{d}}{\mathrm{d}z})^k z^n = n^k z^n\)</span>. Combining these two remarks
-we find:</p>
-<blockquote>
-<div>If <span class="math">\(a_I - b_J \in \mathbb{Z}_{&gt;0}\)</span>, then there exists a polynomial
-<span class="math">\(p(n) = p_0 + p_1 n + \dots\)</span> (of degree <span class="math">\(a_I - b_J\)</span>) such
-that <span class="math">\(\frac{(a_I)_n}{(b_J)_n} = p(n)\)</span> and <span class="math">\({}_p F_q\left({a_p
-\atop b_q} \middle| z \right) = \left(p_0 + p_1
-z\frac{\mathrm{d}}{\mathrm{d}z} + p_2
-\left(z\frac{\mathrm{d}}{\mathrm{d}z}\right)^2 + \dots \right) {}_{p-1}
-F_{q-1}\left({a_p^* \atop b_q^*} \middle| z \right)\)</span>.</div></blockquote>
-<p>Thus any set of parameters <span class="math">\(a_p, b_q\)</span> is reachable from a set of
-parameters <span class="math">\(c_r, d_s\)</span> where <span class="math">\(c_i - d_j \in \mathbb{Z}\)</span> implies
-<span class="math">\(c_i &lt; d_j\)</span>. Such a set of parameters <span class="math">\(c_r, d_s\)</span> is called
-suitable. Our database of known formulae should only contain suitable origins.
-The reasons are twofold: firstly, working from suitable origins is easier, and
-secondly, a formula for a non-suitable origin can be deduced from a lower order
-formula, and we should put this one into the database instead.</p>
-</div>
-<div class="section" id="moving-around-in-the-parameter-space">
-<h3>Moving Around in the Parameter Space<a class="headerlink" href="#moving-around-in-the-parameter-space" title="Permalink to this headline">¶</a></h3>
-<p>It remains to investigate the following question: suppose <span class="math">\(a_p, b_q\)</span> and
-<span class="math">\(a_p^0, b_q^0\)</span> are both suitable, and also <span class="math">\(a_i - a_i^0 \in
-\mathbb{Z}\)</span>, <span class="math">\(b_j - b_j^0 \in \mathbb{Z}\)</span>. When is <span class="math">\(a_p, b_q\)</span>
-reachable from <span class="math">\(a_p^0, b_q^0\)</span>? It is clear that we can treat all
-parameters independently that are incongruent mod 1. So assume that <span class="math">\(a_i\)</span>
-and <span class="math">\(b_j\)</span> are congruent to <span class="math">\(r\)</span> mod 1, for all <span class="math">\(i\)</span> and
-<span class="math">\(j\)</span>. The same then follows for <span class="math">\(a_i^0\)</span> and <span class="math">\(b_j^0\)</span>.</p>
-<p>If <span class="math">\(r \ne 0\)</span>, then any such <span class="math">\(a_p, b_q\)</span> is reachable from any
-<span class="math">\(a_p^0, b_q^0\)</span>. To see this notice that there exist constants <span class="math">\(c, c^0\)</span>,
-congruent mod 1, such that <span class="math">\(a_i &lt; c &lt; b_j\)</span> for all <span class="math">\(i\)</span> and
-<span class="math">\(j\)</span>, and similarly <span class="math">\(a_i^0 &lt; c^0 &lt; b_j^0\)</span>. If <span class="math">\(n = c - c^0 &gt; 0\)</span> then
-we first inverse-shift all the <span class="math">\(b_j^0\)</span> <span class="math">\(n\)</span> times up, and then
-similarly shift shift up all the <span class="math">\(a_i^0\)</span> <span class="math">\(n\)</span> times. If <span class="math">\(n &lt;
-0\)</span> then we first inverse-shift down the <span class="math">\(a_i^0\)</span> and then shift down the
-<span class="math">\(b_j^0\)</span>. This reduces to the case <span class="math">\(c = c^0\)</span>. But evidently we can
-now shift or inverse-shift around the <span class="math">\(a_i^0\)</span> arbitrarily so long as we
-keep them less than <span class="math">\(c\)</span>, and similarly for the <span class="math">\(b_j^0\)</span> so long as
-we keep them bigger than <span class="math">\(c\)</span>. Thus <span class="math">\(a_p, b_q\)</span> is reachable from
-<span class="math">\(a_p^0, b_q^0\)</span>.</p>
-<p>If <span class="math">\(r = 0\)</span> then the problem is slightly more involved. WLOG no parameter
-is zero. We now have one additional complication: no parameter can ever move
-through zero. Hence <span class="math">\(a_p, b_q\)</span> is reachable from <span class="math">\(a_p^0, b_q^0\)</span> if
-and only if the number of <span class="math">\(a_i &lt; 0\)</span> equals the number of <span class="math">\(a_i^0 &lt;
-0\)</span>, and similarly for the <span class="math">\(b_i\)</span> and <span class="math">\(b_i^0\)</span>. But in a suitable set
-of parameters, all <span class="math">\(b_j &gt; 0\)</span>! This is because the Hypergeometric function
-is undefined if one of the <span class="math">\(b_j\)</span> is a non-positive integer and all
-<span class="math">\(a_i\)</span> are smaller than the <span class="math">\(b_j\)</span>. Hence the number of <span class="math">\(b_j \le 0\)</span> is
-always zero.</p>
-<p>We can thus associate to every suitable set of parameters <span class="math">\(a_p, b_q\)</span>,
-where no <span class="math">\(a_i = 0\)</span>, the following invariants:</p>
-<blockquote>
-<div><ul class="simple">
-<li>For every <span class="math">\(r \in [0, 1)\)</span> the number <span class="math">\(\alpha_r\)</span> of parameters
-<span class="math">\(a_i \equiv r \pmod{1}\)</span>, and similarly the number <span class="math">\(\beta_r\)</span>
-of parameters <span class="math">\(b_i \equiv r \pmod{1}\)</span>.</li>
-<li>The number <span class="math">\(\gamma\)</span>
-of integers <span class="math">\(a_i\)</span> with <span class="math">\(a_i &lt; 0\)</span>.</li>
-</ul>
-</div></blockquote>
-<p>The above reasoning shows that <span class="math">\(a_p, b_q\)</span> is reachable from <span class="math">\(a_p^0,
-b_q^0\)</span> if and only if the invariants <span class="math">\(\alpha_r, \beta_r, \gamma\)</span> all
-agree. Thus in particular &#8220;being reachable from&#8221; is a symmetric relation on
-suitable parameters without zeros.</p>
-</div>
-<div class="section" id="applying-the-operators">
-<h3>Applying the Operators<a class="headerlink" href="#applying-the-operators" title="Permalink to this headline">¶</a></h3>
-<p>If all goes well then for a given set of parameters we find an origin in our
-database for which we have a nice formula. We now have to apply (potentially)
-many differential operators to it. If we do this blindly then the result will
-be very messy. This is because with Hypergeometric type functions, the
-derivative is usually expressed as a sum of two contiguous functions. Hence if
-we compute <span class="math">\(N\)</span> derivatives, then the answer will involve <span class="math">\(2N\)</span>
-contiguous functions! This is clearly undesirable. In fact we know from the
-Hypergeometric differential equation that we need at most <span class="math">\(\max(p, q+1)\)</span>
-contiguous functions to express all derivatives.</p>
-<p>Hence instead of differentiating blindly, we will work with a
-<span class="math">\(\mathbb{C}(z)\)</span>-module basis: for an origin <span class="math">\(a_r^0, b_s^0\)</span> we either store
-(for particularly pretty answers) or compute a set of <span class="math">\(N\)</span> functions
-(typically <span class="math">\(N = \max(r, s+1)\)</span>) with the property that the derivative of
-any of them is a <span class="math">\(\mathbb{C}(z)\)</span>-linear combination of them. In formulae,
-we store a vector <span class="math">\(B\)</span> of <span class="math">\(N\)</span> functions, a matrix <span class="math">\(M\)</span> and a
-vector <span class="math">\(C\)</span> (the latter two with entries in <span class="math">\(\mathbb{C}(z)\)</span>), with
-the following properties:</p>
-<ul class="simple">
-<li><span class="math">\({}_r F_s\left({a_r^0 \atop b_s^0} \middle| z \right) = C B\)</span></li>
-<li><span class="math">\(z\frac{\mathrm{d}}{\mathrm{d}z} B = M B\)</span>.</li>
-</ul>
-<p>Then we can compute as many derivatives as we want and we will always end up
-with <span class="math">\(\mathbb{C}(z)\)</span>-linear combination of at most <span class="math">\(N\)</span> special
-functions.</p>
-<p>As hinted above, <span class="math">\(B\)</span>, <span class="math">\(M\)</span> and <span class="math">\(C\)</span> can either all be stored
-(for particularly pretty answers) or computed from a single <span class="math">\({}_p F_q\)</span>
-formula.</p>
-</div>
-<div class="section" id="loose-ends">
-<h3>Loose Ends<a class="headerlink" href="#loose-ends" title="Permalink to this headline">¶</a></h3>
-<p>This describes the bulk of the hypergeometric function algorithm. There a few
-further tricks, described in the hyperexpand.py source file. The extension to
-Meijer G-functions is also described there.</p>
-</div>
-</div>
-<div class="section" id="meijer-g-functions-of-finite-confluence">
-<h2>Meijer G-Functions of Finite Confluence<a class="headerlink" href="#meijer-g-functions-of-finite-confluence" title="Permalink to this headline">¶</a></h2>
-<p>Slater&#8217;s theorem essentially evaluates a <span class="math">\(G\)</span>-function as a sum of residues.
-If all poles are simple, the resulting series can be recognised as
-hypergeometric series. Thus a <span class="math">\(G\)</span>-function can be evaluated as a sum of
-Hypergeometric functions.</p>
-<p>If the poles are not simple, the resulting series are not hypergeometric. This
-is known as the &#8220;confluent&#8221; or &#8220;logarithmic&#8221; case (the latter because the
-resulting series tend to contain logarithms). The answer depends in a
-complicated way on the multiplicities of various poles, and there is no
-accepted notation for representing it (as far as I know).
-However if there are only finitely many
-multiple poles, we can evaluate the <span class="math">\(G\)</span> function as a sum of hypergeometric
-functions, plus finitely many extra terms. I could not find any good reference
-for this, which is why I work it out here.</p>
-<p>Recall the general setup. We define</p>
-<div class="math">
-\[G(z) = \frac{1}{2\pi i} \int_L \frac{\prod_{j=1}^m \Gamma(b_j - s)
-  \prod_{j=1}^n \Gamma(1 - a_j + s)}{\prod_{j=m+1}^q \Gamma(1 - b_j + s)
-  \prod_{j=n+1}^p \Gamma(a_j - s)} z^s \mathrm{d}s,\]</div>
-<p>where <span class="math">\(L\)</span> is a contour starting and ending at <span class="math">\(+\infty\)</span>, enclosing all of the
-poles of <span class="math">\(\Gamma(b_j - s)\)</span> for <span class="math">\(j = 1, \dots, n\)</span> once in the negative
-direction, and no other poles. Also the integral is assumed absolutely
-convergent.</p>
-<p>In what follows, for any complex numbers <span class="math">\(a, b\)</span>, we write <span class="math">\(a \equiv b \pmod{1}\)</span> if
-and only if there exists an integer <span class="math">\(k\)</span> such that <span class="math">\(a - b = k\)</span>. Thus there are
-double poles iff <span class="math">\(a_i \equiv a_j \pmod{1}\)</span> for some <span class="math">\(i \ne j \le n\)</span>.</p>
-<p>We now assume that whenever <span class="math">\(b_j \equiv a_i \pmod{1}\)</span> for <span class="math">\(i \le m\)</span>, <span class="math">\(j &gt; n\)</span>
-then <span class="math">\(b_j &lt; a_i\)</span>. This means that no quotient of the relevant gamma functions
-is a polynomial, and can always be achieved by &#8220;reduction of order&#8221;. Fix a
-complex number <span class="math">\(c\)</span> such that <span class="math">\(\{b_i | b_i \equiv c \pmod{1}, i \le  m\}\)</span> is
-not empty. Enumerate this set as <span class="math">\(b, b+k_1, \dots, b+k_u\)</span>, with <span class="math">\(k_i\)</span>
-non-negative integers. Enumerate similarly
-<span class="math">\(\{a_j | a_j \equiv c \pmod{1}, j &gt; n\}\)</span> as <span class="math">\(b + l_1, \dots, b + l_v\)</span>.
-Then <span class="math">\(l_i &gt; k_j\)</span> for all <span class="math">\(i, j\)</span>. For finite confluence, we need to assume
-<span class="math">\(v \ge u\)</span> for all such <span class="math">\(c\)</span>.</p>
-<p>Let <span class="math">\(c_1, \dots, c_w\)</span> be distinct <span class="math">\(\pmod{1}\)</span> and exhaust the congruence classes
-of the <span class="math">\(b_i\)</span>. I claim</p>
-<div class="math">
-\[G(z) = -\sum_{j=1}^w (F_j(z) + R_j(z)),\]</div>
-<p>where <span class="math">\(F_j(z)\)</span> is a hypergeometric function and <span class="math">\(R_j(z)\)</span> is a finite sum, both
-to be specified later. Indeed corresponding to every <span class="math">\(c_j\)</span> there is
-a sequence of poles, at mostly finitely many of them multiple poles. This is where
-the <span class="math">\(j\)</span>-th term comes from.</p>
-<p>Hence fix again <span class="math">\(c\)</span>, enumerate the relevant <span class="math">\(b_i\)</span> as
-<span class="math">\(b, b + k_1, \dots, b + k_u\)</span>. We will look at the <span class="math">\(a_j\)</span> corresponding to
-<span class="math">\(a + l_1, \dots, a + l_u\)</span>. The other <span class="math">\(a_i\)</span> are not treated specially. The
-corresponding gamma functions have poles at (potentially) <span class="math">\(s = b + r\)</span> for
-<span class="math">\(r = 0, 1, \dots\)</span>. For <span class="math">\(r \ge l_u\)</span>, pole of the integrand is simple. We thus set</p>
-<div class="math">
-\[R(z) = \sum_{r=0}^{l_u - 1} res_{s = r + b}.\]</div>
-<p>We finally need to investigate the other poles. Set <span class="math">\(r = l_u + t\)</span>, <span class="math">\(t \ge 0\)</span>.
-A computation shows</p>
-<div class="math">
-\[\frac{\Gamma(k_i - l_u - t)}{\Gamma(l_i - l_u - t)}
-     = \frac{1}{(k_i - l_u - t)_{l_i - k_i}}
-     = \frac{(-1)^{\delta_i}}{(l_u - l_i + 1)_{\delta_i}}
-       \frac{(l_u - l_i + 1)_t}{(l_u - k_i + 1)_t},\]</div>
-<p>where <span class="math">\(\delta_i = l_i - k_i\)</span>.</p>
-<p>Also</p>
-<div class="math">
-\[\begin{split}\Gamma(b_j - l_u - b - t) =
-    \frac{\Gamma(b_j - l_u - b)}{(-1)^t(l_u + b + 1 - b_j)_t}, \\\end{split}\]\[\Gamma(1 - a_j + l_u + b + t) =
-    \Gamma(1 - a_j + l_u + b) (1 - a_j + l_u + b)_t\]</div>
-<p>and</p>
-<div class="math">
-\[res_{s = b + l_u + t} \Gamma(b - s) = -\frac{(-1)^{l_u + t}}{(l_u + t)!}
-          = -\frac{(-1)^{l_u}}{l_u!} \frac{(-1)^t}{(l_u+1)_t}.\]</div>
-<p>Hence</p>
-<div class="math">
-\[\begin{split}res_{s = b + l_u + t} =&amp; -z^{b + l_u}
-   \frac{(-1)^{l_u}}{l_u!}
-   \prod_{i=1}^{u} \frac{(-1)^{\delta_i}}{(l_u - k_i + 1)_{\delta_i}}
-   \frac{\prod_{j=1}^n \Gamma(1 - a_j + l_u + b)
-         \prod_{j=1}^m \Gamma(b_j - l_u - b)^*}
-        {\prod_{j=n+1}^p \Gamma(a_j - l_u - b)^* \prod_{j=m+1}^q
-         \Gamma(1 - b_j + l_u + b)}
-   \\ &amp;\times
-   z^t
-   \frac{(-1)^t}{(l_u+1)_t}
-   \prod_{i=1}^{u} \frac{(l_u - l_i + 1)_t}{(l_u - k_i + 1)_t}
-   \frac{\prod_{j=1}^n (1 - a_j + l_u + b)_t
-         \prod_{j=n+1}^p (-1)^t (l_u + b + 1 - a_j)_t^*}
-        {\prod_{j=1}^m (-1)^t (l_u + b + 1 - b_j)_t^*
-         \prod_{j=m+1}^q (1 - b_j + l_u + b)_t},\end{split}\]</div>
-<p>where the <span class="math">\(*\)</span> means to omit the terms we treated specially.</p>
-<p>We thus arrive at</p>
-<div class="math">
-\[\begin{split}F(z) = C \times {}_{p+1}F_{q}\left(
-    \begin{matrix} 1, (1 + l_u - l_i), (1 + l_u + b - a_i)^* \\
-                   1 + l_u, (1 + l_u - k_i), (1 + l_u + b - b_i)^*
-    \end{matrix} \middle| (-1)^{p-m-n} z\right),\end{split}\]</div>
-<p>where <span class="math">\(C\)</span> designates the factor in the residue independent of <span class="math">\(t\)</span>.
-(This result can also be written in slightly simpler form by converting
-all the <span class="math">\(l_u\)</span> etc back to <span class="math">\(a_* - b_*\)</span>, but doing so is going to require more
-notation still and is not helpful for computation.)</p>
-</div>
-<div class="section" id="extending-the-hypergeometric-tables">
-<h2>Extending The Hypergeometric Tables<a class="headerlink" href="#extending-the-hypergeometric-tables" title="Permalink to this headline">¶</a></h2>
-<p>Adding new formulae to the tables is straightforward. At the top of the file
-<tt class="docutils literal"><span class="pre">sympy/simplify/hyperexpand.py</span></tt>, there is a function called
-<tt class="xref py py-func docutils literal"><span class="pre">add_formulae()</span></tt>. Nested in it are defined two helpers,
-<tt class="docutils literal"><span class="pre">add(ap,</span> <span class="pre">bq,</span> <span class="pre">res)</span></tt> and <tt class="docutils literal"><span class="pre">addb(ap,</span> <span class="pre">bq,</span> <span class="pre">B,</span> <span class="pre">C,</span> <span class="pre">M)</span></tt>, as well as dummys
-<tt class="docutils literal"><span class="pre">a</span></tt>, <tt class="docutils literal"><span class="pre">b</span></tt>, <tt class="docutils literal"><span class="pre">c</span></tt>, and <tt class="docutils literal"><span class="pre">z</span></tt>.</p>
-<p>The first step in adding a new formula is by using <tt class="docutils literal"><span class="pre">add(ap,</span> <span class="pre">bq,</span> <span class="pre">res)</span></tt>. This
-declares <tt class="docutils literal"><span class="pre">hyper(ap,</span> <span class="pre">bq,</span> <span class="pre">z)</span> <span class="pre">==</span> <span class="pre">res</span></tt>. Here <tt class="docutils literal"><span class="pre">ap</span></tt> and <tt class="docutils literal"><span class="pre">bq</span></tt> may use the
-dummys <tt class="docutils literal"><span class="pre">a</span></tt>, <tt class="docutils literal"><span class="pre">b</span></tt>, and <tt class="docutils literal"><span class="pre">c</span></tt> as free symbols. For example the well-known formula
-<span class="math">\(\sum_0^\infty \frac{(-a)_n z^n}{n!} = (1-z)^a\)</span> is declared by the following
-line: <tt class="docutils literal"><span class="pre">add((-a,</span> <span class="pre">),</span> <span class="pre">(),</span> <span class="pre">(1-z)**a)</span></tt>.</p>
-<p>From the information provided, the matrices <span class="math">\(B\)</span>, <span class="math">\(C\)</span> and <span class="math">\(M\)</span> will be computed,
-and the formula is now available when expanding hypergeometric functions.
-Next the test file <tt class="docutils literal"><span class="pre">sympy/simplify/tests/test_hyperexpand.py</span></tt> should be run,
-in particular the test <tt class="xref py py-func docutils literal"><span class="pre">test_formulae()</span></tt>. This will test the newly added
-formula numerically. If it fails, there is (presumably) a typo in what was
-entered.</p>
-<p>Since all newly-added formulae are probably relatively complicated, chances
-are that the automatically computed basis is rather suboptimal (there is no
-good way of testing this, other than observing very messy output). In this
-case the matrices <span class="math">\(B\)</span>, <span class="math">\(C\)</span> and <span class="math">\(M\)</span> should be computed by hand. Then the helper
-<tt class="docutils literal"><span class="pre">addb</span></tt> can be used to declare a hypergeometric formula with hand-computed
-basis.</p>
-<div class="section" id="an-example">
-<h3>An example<a class="headerlink" href="#an-example" title="Permalink to this headline">¶</a></h3>
-<p>Because this explanation so far might be very theoretical and difficult to
-understand, we walk through an explicit example now. We take the Fresnel
-function <span class="math">\(C(z)\)</span> which obeys the following hypergeometric representation:</p>
-<div class="math">
-\[\begin{split}C(z) = z \cdot {}_{1}F_{2}\left.\left(
-    \begin{matrix} \frac{1}{4} \\
-                   \frac{1}{2}, \frac{5}{4}
-    \end{matrix} \right| -\frac{\pi^2 z^4}{16}\right) \,.\end{split}\]</div>
-<p>First we try to add this formula to the lookup table by using the
-(simpler) function <tt class="docutils literal"><span class="pre">add(ap,</span> <span class="pre">bq,</span> <span class="pre">res)</span></tt>. The first two arguments
-are simply the lists containing the parameter sets of <span class="math">\({}_{1}F_{2}\)</span>.
-The <tt class="docutils literal"><span class="pre">res</span></tt> argument is a little bit more complicated. We only know
-<span class="math">\(C(z)\)</span> in terms of <span class="math">\({}_{1}F_{2}(\ldots | f(z))\)</span> with <span class="math">\(f\)</span>
-a function of <span class="math">\(z\)</span>, in our case</p>
-<div class="math">
-\[f(z) = -\frac{\pi^2 z^4}{16} \,.\]</div>
-<p>What we need is a formula where the hypergeometric function has
-only <span class="math">\(z\)</span> as argument <span class="math">\({}_{1}F_{2}(\ldots | z)\)</span>. We
-introduce the new complex symbol <span class="math">\(w\)</span> and search for a function
-<span class="math">\(g(w)\)</span> such that</p>
-<div class="math">
-\[f(g(w)) = w\]</div>
-<p>holds. Then we can replace every <span class="math">\(z\)</span> in <span class="math">\(C(z)\)</span> by <span class="math">\(g(w)\)</span>.
-In the case of our example the function <span class="math">\(g\)</span> could look like</p>
-<div class="math">
-\[g(w) = \frac{2}{\sqrt{\pi}} \exp\left(\frac{i \pi}{4}\right) w^{\frac{1}{4}} \,.\]</div>
-<p>We get these functions mainly by guessing and testing the result. Hence
-we proceed by computing <span class="math">\(f(g(w))\)</span> (and simplifying naively)</p>
-<div class="math">
-\[\begin{split}f(g(w)) &amp;= -\frac{\pi^2 g(w)^4}{16} \\
-        &amp;= -\frac{\pi^2 g\left(\frac{2}{\sqrt{\pi}} \exp\left(\frac{i \pi}{4}\right) w^{\frac{1}{4}}\right)^4}{16} \\
-        &amp;= -\frac{\pi^2 \frac{2^4}{\sqrt{\pi}^4} \exp\left(\frac{i \pi}{4}\right)^4 {w^{\frac{1}{4}}}^4}{16} \\
-        &amp;= -\exp\left(i \pi\right) w \\
-        &amp;= w\end{split}\]</div>
-<p>and indeed get back <span class="math">\(w\)</span>. (In case of branched functions we have to be
-aware of branch cuts. In that case we take <span class="math">\(w\)</span> to be a positive real
-number and check the formula. If what we have found works for positive
-<span class="math">\(w\)</span>, then just replace <tt class="xref py py-func docutils literal"><span class="pre">exp()</span></tt> inside any branched function by
-<tt class="xref py py-func docutils literal"><span class="pre">exp_polar()</span></tt> and what we get is right for <span class="math">\(all\)</span> <span class="math">\(w\)</span>.) Hence
-we can write the formula as</p>
-<div class="math">
-\[\begin{split}C(g(w)) = g(w) \cdot {}_{1}F_{2}\left.\left(
-     \begin{matrix} \frac{1}{4} \\
-                    \frac{1}{2}, \frac{5}{4}
-     \end{matrix} \right| w\right) \,.\end{split}\]</div>
-<p>and trivially</p>
-<div class="math">
-\[\begin{split}{}_{1}F_{2}\left.\left(
-\begin{matrix} \frac{1}{4} \\
-               \frac{1}{2}, \frac{5}{4}
-\end{matrix} \right| w\right)
-= \frac{C(g(w))}{g(w)}
-= \frac{C\left(\frac{2}{\sqrt{\pi}} \exp\left(\frac{i \pi}{4}\right) w^{\frac{1}{4}}\right)}
-       {\frac{2}{\sqrt{\pi}} \exp\left(\frac{i \pi}{4}\right) w^{\frac{1}{4}}}\end{split}\]</div>
-<p>which is exactly what is needed for the third paramenter,
-<tt class="docutils literal"><span class="pre">res</span></tt>, in <tt class="docutils literal"><span class="pre">add</span></tt>. Finally, the whole function call to add
-this rule to the table looks like:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">add</span><span class="p">([</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">],</span>
-    <span class="p">[</span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">],</span>
-    <span class="n">fresnelc</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="n">root</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="mi">2</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">))</span> <span class="o">/</span> <span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="n">root</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="mi">2</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">))</span>
-   <span class="p">)</span>
-</pre></div>
-</div>
-<p>Using this rule we will find that it works but the results are not really nice
-in terms of simplicity and number of special function instances included.
-We can obtain much better results by adding the formula to the lookup table
-in another way. For this we use the (more complicated) function <tt class="docutils literal"><span class="pre">addb(ap,</span> <span class="pre">bq,</span> <span class="pre">B,</span> <span class="pre">C,</span> <span class="pre">M)</span></tt>.
-The first two arguments are again the lists containing the parameter sets of
-<span class="math">\({}_{1}F_{2}\)</span>. The remaining three are the matrices mentioned earlier
-on this page.</p>
-<p>We know that the <span class="math">\(n = \max{\left(p, q+1\right)}\)</span>-th derivative can be
-expressed as a linear combination of lower order derivatives. The matrix
-<span class="math">\(B\)</span> contains the basis <span class="math">\(\{B_0, B_1, \ldots\}\)</span> and is of shape
-<span class="math">\(n \times 1\)</span>. The best way to get <span class="math">\(B_i\)</span> is to take the first
-<span class="math">\(n = \max(p, q+1)\)</span> derivatives of the expression for <span class="math">\({}_p F_q\)</span>
-and take out usefull pieces. In our case we find that
-<span class="math">\(n = \max{\left(1, 2+1\right)} = 3\)</span>. For computing the derivatives,
-we have to use the operator <span class="math">\(z\frac{\mathrm{d}}{\mathrm{d}z}\)</span>. The
-first basis element <span class="math">\(B_0\)</span> is set to the expression for <span class="math">\({}_1 F_2\)</span>
-from above:</p>
-<div class="math">
-\[B_0 = \frac{ \sqrt{\pi} \exp\left(-\frac{\mathbf{\imath}\pi}{4}\right)
-C\left( \frac{2}{\sqrt{\pi}} \exp\left(\frac{\mathbf{\imath}\pi}{4}\right) z^{\frac{1}{4}}\right)}
-{2 z^{\frac{1}{4}}}\]</div>
-<p>Next we compute <span class="math">\(z\frac{\mathrm{d}}{\mathrm{d}z} B_0\)</span>. For this we can
-directly use SymPy!</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">fresnelc</span><span class="p">,</span> <span class="n">root</span><span class="p">,</span> <span class="n">diff</span><span class="p">,</span> <span class="n">expand</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;z&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B0</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="n">fresnelc</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">root</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">))</span><span class="o">/</span>\
-<span class="gp">... </span>         <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">root</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">*</span> <span class="n">diff</span><span class="p">(</span><span class="n">B0</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">z*(cosh(2*sqrt(z))/(4*z) -</span>
-<span class="go">sqrt(pi)*exp(-I*pi/4)*fresnelc(2*z**(1/4)*exp(I*pi/4)/sqrt(pi))/(8*z**(5/4)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand</span><span class="p">(</span><span class="n">_</span><span class="p">)</span>
-<span class="go">cosh(2*sqrt(z))/4 - sqrt(pi)*exp(-I*pi/4)*fresnelc(2*z**(1/4)*exp(I*pi/4)/sqrt(pi))/(8*z**(1/4))</span>
-</pre></div>
-</div>
-<p>Formatting this result nicely we obtain</p>
-<div class="math">
-\[B_1^\prime =
-- \frac{1}{4} \frac{
-  \sqrt{\pi}
-  \exp\left(-\frac{\mathbf{\imath}\pi}{4}\right)
-  C\left( \frac{2}{\sqrt{\pi}} \exp\left(\frac{\mathbf{\imath}\pi}{4}\right) z^{\frac{1}{4}}\right)
-}
-{2 z^{\frac{1}{4}}}
-+ \frac{1}{4} \cosh{\left( 2 \sqrt{z} \right )}\]</div>
-<p>Computing the second derivative we find</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="p">(</span><span class="n">Symbol</span><span class="p">,</span> <span class="n">cosh</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">fresnelc</span><span class="p">,</span> <span class="n">root</span><span class="p">,</span>
-<span class="gp">... </span>                   <span class="n">diff</span><span class="p">,</span> <span class="n">expand</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;z&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B1prime</span> <span class="o">=</span> <span class="n">cosh</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">/</span><span class="mi">4</span> <span class="o">-</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">*</span>\
-<span class="gp">... </span>          <span class="n">fresnelc</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">root</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">))</span><span class="o">/</span><span class="p">(</span><span class="mi">8</span><span class="o">*</span><span class="n">root</span><span class="p">(</span><span class="n">z</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">*</span> <span class="n">diff</span><span class="p">(</span><span class="n">B1prime</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">z*(-cosh(2*sqrt(z))/(16*z) + sinh(2*sqrt(z))/(4*sqrt(z)) + sqrt(pi)*exp(-I*pi/4)*fresnelc(2*z**(1/4)*exp(I*pi/4)/sqrt(pi))/(32*z**(5/4)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand</span><span class="p">(</span><span class="n">_</span><span class="p">)</span>
-<span class="go">sqrt(z)*sinh(2*sqrt(z))/4 - cosh(2*sqrt(z))/16 + sqrt(pi)*exp(-I*pi/4)*fresnelc(2*z**(1/4)*exp(I*pi/4)/sqrt(pi))/(32*z**(1/4))</span>
-</pre></div>
-</div>
-<p>which can be printed as</p>
-<div class="math">
-\[B_2^\prime =
-\frac{1}{16} \frac{
-  \sqrt{\pi}
-  \exp\left(-\frac{\mathbf{\imath}\pi}{4}\right)
-  C\left( \frac{2}{\sqrt{\pi}} \exp\left(\frac{\mathbf{\imath}\pi}{4}\right) z^{\frac{1}{4}}\right)
-}
-{2 z^{\frac{1}{4}}}
-- \frac{1}{16} \cosh{\left(2\sqrt{z}\right)}
-+ \frac{1}{4} \sinh{\left(2\sqrt{z}\right)} \sqrt{z}\]</div>
-<p>We see the common pattern and can collect the pieces. Hence it makes sense to
-choose <span class="math">\(B_1\)</span> and <span class="math">\(B_2\)</span> as follows</p>
-<div class="math">
-\[\begin{split}B =
-\left( \begin{matrix}
-  B_0 \\ B_1 \\ B_2
-\end{matrix} \right)
-=
-\left( \begin{matrix}
-  \frac{
-    \sqrt{\pi}
-    \exp\left(-\frac{\mathbf{\imath}\pi}{4}\right)
-    C\left( \frac{2}{\sqrt{\pi}} \exp\left(\frac{\mathbf{\imath}\pi}{4}\right) z^{\frac{1}{4}}\right)
-  }{2 z^{\frac{1}{4}}} \\
-  \cosh\left(2\sqrt{z}\right) \\
-  \sinh\left(2\sqrt{z}\right) \sqrt{z}
-\end{matrix} \right)\end{split}\]</div>
-<p>(This is in contrast to the basis <span class="math">\(B = \left(B_0, B_1^\prime, B_2^\prime\right)\)</span> that would
-have been computed automatically if we used just <tt class="docutils literal"><span class="pre">add(ap,</span> <span class="pre">bq,</span> <span class="pre">res)</span></tt>.)</p>
-<p>Because it must hold that <span class="math">\({}_p F_q\left(\cdots \middle| z \right) = C B\)</span>
-the entries of <span class="math">\(C\)</span> are obviously</p>
-<div class="math">
-\[\begin{split}C =
-\left( \begin{matrix}
-  1 \\ 0 \\ 0
-\end{matrix} \right)\end{split}\]</div>
-<p>Finally we have to compute the entries of the <span class="math">\(3 \times 3\)</span> matrix <span class="math">\(M\)</span>
-such that <span class="math">\(z\frac{\mathrm{d}}{\mathrm{d}z} B = M B\)</span> holds. This is easy.
-We already computed the first part <span class="math">\(z\frac{\mathrm{d}}{\mathrm{d}z} B_0\)</span>
-above. This gives us the first row of <span class="math">\(M\)</span>. For the second row we have:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">cosh</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">diff</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;z&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B1</span> <span class="o">=</span> <span class="n">cosh</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">*</span> <span class="n">diff</span><span class="p">(</span><span class="n">B1</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">sqrt(z)*sinh(2*sqrt(z))</span>
-</pre></div>
-</div>
-<p>and for the third one</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">expand</span><span class="p">,</span> <span class="n">diff</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;z&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B2</span> <span class="o">=</span> <span class="n">sinh</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">))</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">z</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand</span><span class="p">(</span><span class="n">z</span> <span class="o">*</span> <span class="n">diff</span><span class="p">(</span><span class="n">B2</span><span class="p">,</span> <span class="n">z</span><span class="p">))</span>
-<span class="go">sqrt(z)*sinh(2*sqrt(z))/2 + z*cosh(2*sqrt(z))</span>
-</pre></div>
-</div>
-<p>Now we have computed the entries of this matrix to be</p>
-<div class="math">
-\[\begin{split}M =
-\left( \begin{matrix}
-  -\frac{1}{4} &amp; \frac{1}{4} &amp; 0 \\
-  0            &amp; 0           &amp; 1 \\
-  0            &amp; z           &amp; \frac{1}{2} \\
-\end{matrix} \right)\end{split}\]</div>
-<p>Note that the entries of <span class="math">\(C\)</span> and <span class="math">\(M\)</span> should typically be
-rational functions in <span class="math">\(z\)</span>, with rational coefficients. This is all
-we need to do in order to add a new formula to the lookup table for
-<tt class="docutils literal"><span class="pre">hyperexpand</span></tt>.</p>
-</div>
-</div>
-<div class="section" id="implemented-hypergeometric-formulae">
-<h2>Implemented Hypergeometric Formulae<a class="headerlink" href="#implemented-hypergeometric-formulae" title="Permalink to this headline">¶</a></h2>
-<p>A vital part of the algorithm is a relatively large table of hypergeometric
-function representations. The following automatically generated list contains
-all the representations implemented in SymPy (of course many more are
-derived from them). These formulae are mostly taken from <a class="reference internal" href="#luke1969">[Luke1969]</a> and
-<a class="reference internal" href="#prudnikov1990">[Prudnikov1990]</a>. They are all tested numerically.</p>
-<span class="target" id="module-sympy.simplify.hyperexpand_doc"></span><div class="math">
-\[\begin{split}{{}_{0}F_{0}\left(\begin{matrix}  \\  \end{matrix}\middle| {z} \right)} = e^{z}\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{1}F_{0}\left(\begin{matrix} - a \\  \end{matrix}\middle| {z} \right)} = \left(- z + 1\right)^{a}\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{2}F_{1}\left(\begin{matrix} a, a - \frac{1}{2} \\ 2 a \end{matrix}\middle| {z} \right)} = 2^{2 a - 1} \left(\sqrt{- z + 1} + 1\right)^{- 2 a + 1}\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{2}F_{1}\left(\begin{matrix} 1, 1 \\ 2 \end{matrix}\middle| {z} \right)} = - \frac{\log{\left (- z + 1 \right )}}{z}\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, 1 \\ \frac{3}{2} \end{matrix}\middle| {z} \right)} = \frac{\operatorname{atanh}{\left (\sqrt{z} \right )}}{\sqrt{z}}\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle| {z} \right)} = \frac{\operatorname{asin}{\left (\sqrt{z} \right )}}{\sqrt{z}}\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{2}F_{1}\left(\begin{matrix} - a, - a + \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle| {z} \right)} = \frac{1}{2} \left(- \sqrt{z} + 1\right)^{2 a} + \frac{1}{2} \left(\sqrt{z} + 1\right)^{2 a}\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{2}F_{1}\left(\begin{matrix} a, - a \\ \frac{1}{2} \end{matrix}\middle| {z} \right)} = \cos{\left (2 a \operatorname{asin}{\left (\sqrt{z} \right )} \right )}\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{2}F_{1}\left(\begin{matrix} 1, 1 \\ \frac{3}{2} \end{matrix}\middle| {z} \right)} = \frac{\operatorname{asin}{\left (\sqrt{z} \right )}}{\sqrt{z} \sqrt{- z + 1}}\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{3}F_{2}\left(\begin{matrix} - \frac{1}{2}, 1, 1 \\ \frac{1}{2}, 2 \end{matrix}\middle| {z} \right)} = - \frac{2}{3} \sqrt{z} \operatorname{atanh}{\left (\sqrt{z} \right )} + \frac{2}{3} - \frac{\log{\left (- z + 1 \right )}}{3 z}\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{3}F_{2}\left(\begin{matrix} - \frac{1}{2}, 1, 1 \\ 2, 2 \end{matrix}\middle| {z} \right)} = \left(\frac{4}{9} - \frac{16}{9 z}\right) \sqrt{- z + 1} + \frac{4}{3} \frac{\log{\left (\frac{1}{2} \sqrt{- z + 1} + \frac{1}{2} \right )}}{z} + \frac{16}{9 z}\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{1}F_{1}\left(\begin{matrix} 1 \\ b \end{matrix}\middle| {z} \right)} = z^{- b + 1} \left(b - 1\right) e^{z} \gamma\left(b - 1, z\right)\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{1}F_{1}\left(\begin{matrix} a \\ 2 a \end{matrix}\middle| {z} \right)} = 4^{a - \frac{1}{2}} z^{- a + \frac{1}{2}} e^{\frac{1}{2} z} I_{a - \frac{1}{2}}\left(\frac{1}{2} z\right) \Gamma\left(a + \frac{1}{2}\right)\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{1}F_{1}\left(\begin{matrix} a \\ a + 1 \end{matrix}\middle| {z} \right)} = a \left(z e^{\mathbf{\imath} \pi}\right)^{- a} \gamma\left(a, z e^{\mathbf{\imath} \pi}\right)\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{1}F_{1}\left(\begin{matrix} - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle| {z} \right)} = \sqrt{z} \mathbf{\imath} \sqrt{\pi} \operatorname{erf}{\left (\sqrt{z} \mathbf{\imath} \right )} + e^{z}\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{1}F_{2}\left(\begin{matrix} 1 \\ \frac{3}{4}, \frac{5}{4} \end{matrix}\middle| {z} \right)} = \frac{\sqrt{\pi} \left(\mathbf{\imath} \sinh{\left (2 \sqrt{z} \right )} S\left(2 \frac{\sqrt[4]{z} e^{\frac{1}{4} \mathbf{\imath} \pi}}{\sqrt{\pi}}\right) + \cosh{\left (2 \sqrt{z} \right )} C\left(2 \frac{\sqrt[4]{z} e^{\frac{1}{4} \mathbf{\imath} \pi}}{\sqrt{\pi}}\right)\right) e^{- \frac{1}{4} \mathbf{\imath} \pi}}{2 \sqrt[4]{z}}\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{2}F_{2}\left(\begin{matrix} \frac{1}{2}, a \\ \frac{3}{2}, a + 1 \end{matrix}\middle| {z} \right)} = - \frac{a \mathbf{\imath} \sqrt{\pi} \sqrt{\frac{1}{z}} \operatorname{erf}{\left (\sqrt{z} \mathbf{\imath} \right )}}{2 a - 1} - \frac{a \left(z e^{\mathbf{\imath} \pi}\right)^{- a} \gamma\left(a, z e^{\mathbf{\imath} \pi}\right)}{2 a - 1}\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{2}F_{2}\left(\begin{matrix} 1, 1 \\ 2, 2 \end{matrix}\middle| {z} \right)} = \frac{- \log{\left (z \right )} + \operatorname{Ei}{\left (z \right )}}{z} - \frac{\gamma}{z}\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{0}F_{1}\left(\begin{matrix}  \\ \frac{1}{2} \end{matrix}\middle| {z} \right)} = \cosh{\left (2 \sqrt{z} \right )}\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{0}F_{1}\left(\begin{matrix}  \\ b \end{matrix}\middle| {z} \right)} = z^{- \frac{1}{2} b + \frac{1}{2}} I_{b - 1}\left(2 \sqrt{z}\right) \Gamma\left(b\right)\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{0}F_{3}\left(\begin{matrix}  \\ \frac{1}{2}, a, a + \frac{1}{2} \end{matrix}\middle| {z} \right)} = 2^{- 2 a} z^{- \frac{1}{2} a + \frac{1}{4}} \left(I_{2 a - 1}\left(4 \sqrt[4]{z}\right) + J_{2 a - 1}\left(4 \sqrt[4]{z}\right)\right) \Gamma\left(2 a\right)\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{0}F_{3}\left(\begin{matrix}  \\ a, a + \frac{1}{2}, 2 a \end{matrix}\middle| {z} \right)} = \left(2 \sqrt{z} e^{\frac{1}{2} \mathbf{\imath} \pi}\right)^{- 2 a + 1} I_{2 a - 1}\left(2 \sqrt{2} \sqrt[4]{z} e^{\frac{1}{4} \mathbf{\imath} \pi}\right) J_{2 a - 1}\left(2 \sqrt{2} \sqrt[4]{z} e^{\frac{1}{4} \mathbf{\imath} \pi}\right) \Gamma^{2}\left(2 a\right)\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{1}F_{2}\left(\begin{matrix} a \\ a - \frac{1}{2}, 2 a \end{matrix}\middle| {z} \right)} = 2 \times 4^{a - 1} z^{- a + 1} I_{a - \frac{3}{2}}\left(\sqrt{z}\right) I_{a - \frac{1}{2}}\left(\sqrt{z}\right) \Gamma\left(a - \frac{1}{2}\right) \Gamma\left(a + \frac{1}{2}\right) - 4^{a - \frac{1}{2}} z^{- a + \frac{1}{2}} I^{2}_{a - \frac{1}{2}}\left(\sqrt{z}\right) \Gamma^{2}\left(a + \frac{1}{2}\right)\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{1}F_{2}\left(\begin{matrix} \frac{1}{2} \\ b, - b + 2 \end{matrix}\middle| {z} \right)} = \frac{\pi \left(- b + 1\right) I_{- b + 1}\left(\sqrt{z}\right) I_{b - 1}\left(\sqrt{z}\right)}{\sin{\left (b \pi \right )}}\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{1}F_{2}\left(\begin{matrix} \frac{1}{2} \\ \frac{3}{2}, \frac{3}{2} \end{matrix}\middle| {z} \right)} = \frac{\operatorname{Shi}{\left (2 \sqrt{z} \right )}}{2 \sqrt{z}}\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{1}F_{2}\left(\begin{matrix} \frac{3}{4} \\ \frac{3}{2}, \frac{7}{4} \end{matrix}\middle| {z} \right)} = \frac{3}{4} \frac{\sqrt{\pi} e^{- \frac{3}{4} \mathbf{\imath} \pi} S\left(2 \frac{\sqrt[4]{z} e^{\frac{1}{4} \mathbf{\imath} \pi}}{\sqrt{\pi}}\right)}{z^{\frac{3}{4}}}\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{1}F_{2}\left(\begin{matrix} \frac{1}{4} \\ \frac{1}{2}, \frac{5}{4} \end{matrix}\middle| {z} \right)} = \frac{\sqrt{\pi} e^{- \frac{1}{4} \mathbf{\imath} \pi} C\left(2 \frac{\sqrt[4]{z} e^{\frac{1}{4} \mathbf{\imath} \pi}}{\sqrt{\pi}}\right)}{2 \sqrt[4]{z}}\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{2}F_{3}\left(\begin{matrix} a, a + \frac{1}{2} \\ 2 a, b, 2 a - b + 1 \end{matrix}\middle| {z} \right)} = \left(\frac{1}{2} \sqrt{z}\right)^{- 2 a + 1} I_{2 a - b}\left(\sqrt{z}\right) I_{b - 1}\left(\sqrt{z}\right) \Gamma\left(b\right) \Gamma\left(2 a - b + 1\right)\end{split}\]</div>
-<div class="math">
-\[\begin{split}{{}_{2}F_{3}\left(\begin{matrix} 1, 1 \\ 2, 2, \frac{3}{2} \end{matrix}\middle| {z} \right)} = \frac{- \log{\left (2 \sqrt{z} \right )} + \operatorname{Chi}{\left (2 \sqrt{z} \right )}}{z} - \frac{\gamma}{z}\end{split}\]</div>
-</div>
-<div class="section" id="references">
-<h2>References<a class="headerlink" href="#references" title="Permalink to this headline">¶</a></h2>
-<table class="docutils citation" frame="void" id="roach1996" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id1">[Roach1996]</a></td><td>Kelly B. Roach.  Hypergeometric Function Representations.
-In: Proceedings of the 1996 International Symposium on Symbolic and
-Algebraic Computation, pages 301-308, New York, 1996. ACM.</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="roach1997" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id2">[Roach1997]</a></td><td>Kelly B. Roach.  Meijer G Function Representations.
-In: Proceedings of the 1997 International Symposium on Symbolic and
-Algebraic Computation, pages 205-211, New York, 1997. ACM.</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="luke1969" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id3">[Luke1969]</a></td><td>Luke, Y. L. (1969), The Special Functions and Their
-Approximations, Volume 1.</td></tr>
-</tbody>
-</table>
-<table class="docutils citation" frame="void" id="prudnikov1990" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id4">[Prudnikov1990]</a></td><td>A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev (1990).
-Integrals and Series: More Special Functions, Vol. 3,
-Gordon and Breach Science Publisher.</td></tr>
-</tbody>
-</table>
-</div>
-</div>
-
-
-          </div>
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-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Details on the Hypergeometric Function Expansion Module</a><ul>
-<li><a class="reference internal" href="#hypergeometric-function-expansion-algorithm">Hypergeometric Function Expansion Algorithm</a><ul>
-<li><a class="reference internal" href="#notation">Notation</a></li>
-<li><a class="reference internal" href="#incrementing-and-decrementing-indices">Incrementing and decrementing indices</a></li>
-<li><a class="reference internal" href="#reduction-of-order">Reduction of Order</a></li>
-<li><a class="reference internal" href="#moving-around-in-the-parameter-space">Moving Around in the Parameter Space</a></li>
-<li><a class="reference internal" href="#applying-the-operators">Applying the Operators</a></li>
-<li><a class="reference internal" href="#loose-ends">Loose Ends</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#meijer-g-functions-of-finite-confluence">Meijer G-Functions of Finite Confluence</a></li>
-<li><a class="reference internal" href="#extending-the-hypergeometric-tables">Extending The Hypergeometric Tables</a><ul>
-<li><a class="reference internal" href="#an-example">An example</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#implemented-hypergeometric-formulae">Implemented Hypergeometric Formulae</a></li>
-<li><a class="reference internal" href="#references">References</a></li>
-</ul>
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-</ul>
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-  <div class="section" id="simplify">
-<h1>Simplify<a class="headerlink" href="#simplify" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="id1">
-<h2>simplify<a class="headerlink" href="#id1" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.simplify.simplify.simplify">
-<tt class="descclassname">sympy.simplify.simplify.</tt><tt class="descname">simplify</tt><big>(</big><em>expr</em>, <em>ratio=1.7</em>, <em>measure=&lt;function count_ops at 0x10c198b00&gt;</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/simplify.html#simplify"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.simplify.simplify" title="Permalink to this definition">¶</a></dt>
-<dd><p>Simplifies the given expression.</p>
-<p>Simplification is not a well defined term and the exact strategies
-this function tries can change in the future versions of SymPy. If
-your algorithm relies on &#8220;simplification&#8221; (whatever it is), try to
-determine what you need exactly  -  is it powsimp()?, radsimp()?,
-together()?, logcombine()?, or something else? And use this particular
-function directly, because those are well defined and thus your algorithm
-will be robust.</p>
-<p>Nonetheless, especially for interactive use, or when you don&#8217;t know
-anything about the structure of the expression, simplify() tries to apply
-intelligent heuristics to make the input expression &#8220;simpler&#8221;.  For
-example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
-<span class="go">(x**2 + x)/(x*sin(y)**2 + x*cos(y)**2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="go">x + 1</span>
-</pre></div>
-</div>
-<p>Note that we could have obtained the same result by using specific
-simplification functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">trigsimp</span><span class="p">,</span> <span class="n">cancel</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">trigsimp</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span>
-<span class="go">(x**2 + x)/x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">cancel</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span>
-<span class="go">x + 1</span>
-</pre></div>
-</div>
-<p>In some cases, applying <a class="reference internal" href="../galgebra/GA/GAsympy.html#simplify" title="simplify"><tt class="xref py py-func docutils literal"><span class="pre">simplify()</span></tt></a> may actually result in some more
-complicated expression. The default <tt class="docutils literal"><span class="pre">ratio=1.7</span></tt> prevents more extreme
-cases: if (result length)/(input length) &gt; ratio, then input is returned
-unmodified.  The <tt class="docutils literal"><span class="pre">measure</span></tt> parameter lets you specify the function used
-to determine how complex an expression is.  The function should take a
-single argument as an expression and return a number such that if
-expression <tt class="docutils literal"><span class="pre">a</span></tt> is more complex than expression <tt class="docutils literal"><span class="pre">b</span></tt>, then
-<tt class="docutils literal"><span class="pre">measure(a)</span> <span class="pre">&gt;</span> <span class="pre">measure(b)</span></tt>.  The default measure function is
-<tt class="xref py py-func docutils literal"><span class="pre">count_ops()</span></tt>, which returns the total number of operations in the
-expression.</p>
-<p>For example, if <tt class="docutils literal"><span class="pre">ratio=1</span></tt>, <tt class="docutils literal"><span class="pre">simplify</span></tt> output can&#8217;t be longer
-than input.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">count_ops</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">root</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">+</span><span class="mi">3</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Since <tt class="docutils literal"><span class="pre">simplify(root)</span></tt> would result in a slightly longer expression,
-root is returned unchanged instead:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">root</span><span class="p">,</span> <span class="n">ratio</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="n">root</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>If <tt class="docutils literal"><span class="pre">ratio=oo</span></tt>, simplify will be applied anyway:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">count_ops</span><span class="p">(</span><span class="n">simplify</span><span class="p">(</span><span class="n">root</span><span class="p">,</span> <span class="n">ratio</span><span class="o">=</span><span class="n">oo</span><span class="p">))</span> <span class="o">&gt;</span> <span class="n">count_ops</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>Note that the shortest expression is not necessary the simplest, so
-setting <tt class="docutils literal"><span class="pre">ratio</span></tt> to 1 may not be a good idea.
-Heuristically, the default value <tt class="docutils literal"><span class="pre">ratio=1.7</span></tt> seems like a reasonable
-choice.</p>
-<p>You can easily define your own measure function based on what you feel
-should represent the &#8220;size&#8221; or &#8220;complexity&#8221; of the input expression.  Note
-that some choices, such as <tt class="docutils literal"><span class="pre">lambda</span> <span class="pre">expr:</span> <span class="pre">len(str(expr))</span></tt> may appear to be
-good metrics, but have other problems (in this case, the measure function
-may slow down simplify too much for very large expressions).  If you don&#8217;t
-know what a good metric would be, the default, <tt class="docutils literal"><span class="pre">count_ops</span></tt>, is a good one.</p>
-<p>For example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">log</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a b&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">g</span> <span class="o">=</span> <span class="n">log</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">+</span> <span class="n">log</span><span class="p">(</span><span class="n">b</span><span class="p">)</span> <span class="o">+</span> <span class="n">log</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">b</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">h</span>
-<span class="go">log(a*b**(log(1/a) + 1))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">count_ops</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-<span class="go">8</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">count_ops</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-<span class="go">6</span>
-</pre></div>
-</div>
-<p>So you can see that <tt class="docutils literal"><span class="pre">h</span></tt> is simpler than <tt class="docutils literal"><span class="pre">g</span></tt> using the count_ops metric.
-However, we may not like how <tt class="docutils literal"><span class="pre">simplify</span></tt> (in this case, using
-<tt class="docutils literal"><span class="pre">logcombine</span></tt>) has created the <tt class="docutils literal"><span class="pre">b**(log(1/a)</span> <span class="pre">+</span> <span class="pre">1)</span></tt> term.  A simple way to
-reduce this would be to give more weight to powers as operations in
-<tt class="docutils literal"><span class="pre">count_ops</span></tt>.  We can do this by using the <tt class="docutils literal"><span class="pre">visual=True</span></tt> option:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">count_ops</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">2*ADD + DIV + 4*LOG + MUL</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">count_ops</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">ADD + DIV + 2*LOG + MUL + POW</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">my_measure</span><span class="p">(</span><span class="n">expr</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">POW</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;POW&#39;</span><span class="p">)</span>
-<span class="gp">... </span>    <span class="c"># Discourage powers by giving POW a weight of 10</span>
-<span class="gp">... </span>    <span class="n">count</span> <span class="o">=</span> <span class="n">count_ops</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">visual</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">POW</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="gp">... </span>    <span class="c"># Every other operation gets a weight of 1 (the default)</span>
-<span class="gp">... </span>    <span class="n">count</span> <span class="o">=</span> <span class="n">count</span><span class="o">.</span><span class="n">replace</span><span class="p">(</span><span class="n">Symbol</span><span class="p">,</span> <span class="nb">type</span><span class="p">(</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">))</span>
-<span class="gp">... </span>    <span class="k">return</span> <span class="n">count</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">my_measure</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
-<span class="go">8</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">my_measure</span><span class="p">(</span><span class="n">h</span><span class="p">)</span>
-<span class="go">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mf">15.</span><span class="o">/</span><span class="mi">8</span> <span class="o">&gt;</span> <span class="mf">1.7</span> <span class="c"># 1.7 is the default ratio</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">g</span><span class="p">,</span> <span class="n">measure</span><span class="o">=</span><span class="n">my_measure</span><span class="p">)</span>
-<span class="go">-log(a)*log(b) + log(a) + log(b)</span>
-</pre></div>
-</div>
-<p>Note that because <tt class="docutils literal"><span class="pre">simplify()</span></tt> internally tries many different
-simplification strategies and then compares them using the measure
-function, we get a completely different result that is still different
-from the input expression by doing this.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="collect">
-<h2>collect<a class="headerlink" href="#collect" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.simplify.simplify.collect">
-<tt class="descclassname">sympy.simplify.simplify.</tt><tt class="descname">collect</tt><big>(</big><em>expr</em>, <em>syms</em>, <em>func=None</em>, <em>evaluate=True</em>, <em>exact=False</em>, <em>distribute_order_term=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/simplify.html#collect"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.simplify.collect" title="Permalink to this definition">¶</a></dt>
-<dd><p>Collect additive terms of an expression.</p>
-<p>This function collects additive terms of an expression with respect
-to a list of expression up to powers with rational exponents. By the
-term symbol here are meant arbitrary expressions, which can contain
-powers, products, sums etc. In other words symbol is a pattern which
-will be searched for in the expression&#8217;s terms.</p>
-<p>The input expression is not expanded by <a class="reference internal" href="../galgebra/GA/GAsympy.html#collect" title="collect"><tt class="xref py py-func docutils literal"><span class="pre">collect()</span></tt></a>, so user is
-expected to provide an expression is an appropriate form. This makes
-<a class="reference internal" href="../galgebra/GA/GAsympy.html#collect" title="collect"><tt class="xref py py-func docutils literal"><span class="pre">collect()</span></tt></a> more predictable as there is no magic happening behind the
-scenes. However, it is important to note, that powers of products are
-converted to products of powers using the <tt class="xref py py-func docutils literal"><span class="pre">expand_power_base()</span></tt>
-function.</p>
-<p>There are two possible types of output. First, if <tt class="docutils literal"><span class="pre">evaluate</span></tt> flag is
-set, this function will return an expression with collected terms or
-else it will return a dictionary with expressions up to rational powers
-as keys and collected coefficients as values.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span><span class="p">,</span> <span class="n">collect</span><span class="p">,</span> <span class="n">expand</span><span class="p">,</span> <span class="n">factor</span><span class="p">,</span> <span class="n">Wild</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-</pre></div>
-</div>
-<p>This function can collect symbolic coefficients in polynomials or
-rational expressions. It will manage to find all integer or rational
-powers of collection variable:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">collect</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="n">b</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">c + x**2*(a + b) + x*(a - b)</span>
-</pre></div>
-</div>
-<p>The same result can be achieved in dictionary form:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">collect</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="n">b</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">c</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="p">[</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">]</span>
-<span class="go">a + b</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="p">[</span><span class="n">x</span><span class="p">]</span>
-<span class="go">a - b</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="p">[</span><span class="n">S</span><span class="o">.</span><span class="n">One</span><span class="p">]</span>
-<span class="go">c</span>
-</pre></div>
-</div>
-<p>You can also work with multivariate polynomials. However, remember that
-this function is greedy so it will care only about a single symbol at time,
-in specification order:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">collect</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">])</span>
-<span class="go">x**2*(y + 1) + x*y + y*(a + 1)</span>
-</pre></div>
-</div>
-<p>Also more complicated expressions can be used as patterns:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">log</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">collect</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">))</span>
-<span class="go">(a + b)*sin(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">collect</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="n">x</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">x*(a + b)*log(x)</span>
-</pre></div>
-</div>
-<p>You can use wildcards in the pattern:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">w</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;w1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">collect</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">y</span> <span class="o">-</span> <span class="n">b</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">y</span><span class="p">,</span> <span class="n">w</span><span class="o">**</span><span class="n">y</span><span class="p">)</span>
-<span class="go">x**y*(a - b)</span>
-</pre></div>
-</div>
-<p>It is also possible to work with symbolic powers, although it has more
-complicated behavior, because in this case power&#8217;s base and symbolic part
-of the exponent are treated as a single symbol:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">collect</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">c</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">a*x**c + b*x**c</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">collect</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">c</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">c</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="n">c</span><span class="p">)</span>
-<span class="go">x**c*(a + b)</span>
-</pre></div>
-</div>
-<p>However if you incorporate rationals to the exponents, then you will get
-well known behavior:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">collect</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">c</span><span class="p">)</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">c</span><span class="p">),</span> <span class="n">x</span><span class="o">**</span><span class="n">c</span><span class="p">)</span>
-<span class="go">x**(2*c)*(a + b)</span>
-</pre></div>
-</div>
-<p>Note also that all previously stated facts about <a class="reference internal" href="../galgebra/GA/GAsympy.html#collect" title="collect"><tt class="xref py py-func docutils literal"><span class="pre">collect()</span></tt></a> function
-apply to the exponential function, so you can get:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">collect</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">(a + b)*exp(2*x)</span>
-</pre></div>
-</div>
-<p>If you are interested only in collecting specific powers of some symbols
-then set <tt class="docutils literal"><span class="pre">exact</span></tt> flag in arguments:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">collect</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">7</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">7</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">a*x**7 + b*x**7</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">collect</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">7</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">7</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">7</span><span class="p">,</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">x**7*(a + b)</span>
-</pre></div>
-</div>
-<p>You can also apply this function to differential equations, where derivatives
-of arbitrary order can be collected. Note that if you collect with respect
-to a function or a derivative of a function, all derivatives of that function
-will also be collected. Use <tt class="docutils literal"><span class="pre">exact=True</span></tt> to prevent this from happening:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Derivative</span> <span class="k">as</span> <span class="n">D</span><span class="p">,</span> <span class="n">collect</span><span class="p">,</span> <span class="n">Function</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span> <span class="p">(</span><span class="n">x</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">collect</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">D</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">D</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">x</span><span class="p">))</span>
-<span class="go">(a + b)*Derivative(f(x), x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">collect</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">D</span><span class="p">(</span><span class="n">D</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">x</span><span class="p">),</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">D</span><span class="p">(</span><span class="n">D</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">x</span><span class="p">),</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">)</span>
-<span class="go">(a + b)*Derivative(f(x), x, x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">collect</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">D</span><span class="p">(</span><span class="n">D</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">x</span><span class="p">),</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">D</span><span class="p">(</span><span class="n">D</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">x</span><span class="p">),</span><span class="n">x</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">x</span><span class="p">),</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">a*Derivative(f(x), x, x) + b*Derivative(f(x), x, x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">collect</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">D</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">D</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="n">f</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">f</span><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
-<span class="go">(a + b)*f(x) + (a + b)*Derivative(f(x), x)</span>
-</pre></div>
-</div>
-<p>Or you can even match both derivative order and exponent at the same time:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">collect</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">D</span><span class="p">(</span><span class="n">D</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">x</span><span class="p">),</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">D</span><span class="p">(</span><span class="n">D</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">x</span><span class="p">),</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">D</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">x</span><span class="p">))</span>
-<span class="go">(a + b)*Derivative(f(x), x, x)**2</span>
-</pre></div>
-</div>
-<p>Finally, you can apply a function to each of the collected coefficients.
-For example you can factorize symbolic coefficients of polynomial:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">expand</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">3</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">collect</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">factor</span><span class="p">)</span>
-<span class="go">x**3 + 3*x**2*(a + 1) + 3*x*(a + 1)**2 + (a + 1)**3</span>
-</pre></div>
-</div>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">Arguments are expected to be in expanded form, so you might have
-to call <a class="reference internal" href="../galgebra/GA/GAsympy.html#expand" title="expand"><tt class="xref py py-func docutils literal"><span class="pre">expand()</span></tt></a> prior to calling this function.</p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.simplify.simplify.rcollect">
-<tt class="descclassname">sympy.simplify.simplify.</tt><tt class="descname">rcollect</tt><big>(</big><em>expr</em>, <em>*vars</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/simplify.html#rcollect"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.simplify.rcollect" title="Permalink to this definition">¶</a></dt>
-<dd><p>Recursively collect sums in an expression.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">rcollect</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">expr</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rcollect</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">(x + y*(x**2 + x + 1))/(x + y)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="separate">
-<h2>separate<a class="headerlink" href="#separate" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.simplify.simplify.separate">
-<tt class="descclassname">sympy.simplify.simplify.</tt><tt class="descname">separate</tt><big>(</big><em>expr</em>, <em>deep=False</em>, <em>force=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/simplify.html#separate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.simplify.separate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Deprecated wrapper around <tt class="docutils literal"><span class="pre">expand_power_base()</span></tt>.  Use that function instead.</p>
-</dd></dl>
-
-</div>
-<div class="section" id="separatevars">
-<h2>separatevars<a class="headerlink" href="#separatevars" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.simplify.simplify.separatevars">
-<tt class="descclassname">sympy.simplify.simplify.</tt><tt class="descname">separatevars</tt><big>(</big><em>expr</em>, <em>symbols=</em><span class="optional">[</span><span class="optional">]</span>, <em>dict=False</em>, <em>force=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/simplify.html#separatevars"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.simplify.separatevars" title="Permalink to this definition">¶</a></dt>
-<dd><p>Separates variables in an expression, if possible.  By
-default, it separates with respect to all symbols in an
-expression and collects constant coefficients that are
-independent of symbols.</p>
-<p>If dict=True then the separated terms will be returned
-in a dictionary keyed to their corresponding symbols.
-By default, all symbols in the expression will appear as
-keys; if symbols are provided, then all those symbols will
-be used as keys, and any terms in the expression containing
-other symbols or non-symbols will be returned keyed to the
-string &#8216;coeff&#8217;. (Passing None for symbols will return the
-expression in a dictionary keyed to &#8216;coeff&#8217;.)</p>
-<p>If force=True, then bases of powers will be separated regardless
-of assumptions on the symbols involved.</p>
-<p class="rubric">Notes</p>
-<p>The order of the factors is determined by Mul, so that the
-separated expressions may not necessarily be grouped together.</p>
-<p>Although factoring is necessary to separate variables in some
-expressions, it is not necessary in all cases, so one should not
-count on the returned factors being factored.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">alpha</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">separatevars</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">separatevars</span><span class="p">((</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="n">y</span><span class="p">)</span>
-<span class="go">(x*y)**y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">separatevars</span><span class="p">((</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="n">y</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">x**y*y**y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">y</span><span class="p">)</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">z</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">separatevars</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-<span class="go">2*x**2*z*(sin(y) + 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">separatevars</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="n">symbols</span><span class="o">=</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="nb">dict</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">{&#39;coeff&#39;: 2*z, x: x**2, y: sin(y) + 1}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">separatevars</span><span class="p">(</span><span class="n">e</span><span class="p">,</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">alpha</span><span class="p">],</span> <span class="nb">dict</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">{&#39;coeff&#39;: 2*z, alpha: 1, x: x**2, y: sin(y) + 1}</span>
-</pre></div>
-</div>
-<p>If the expression is not really separable, or is only partially
-separable, separatevars will do the best it can to separate it
-by using factoring.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">separatevars</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">-x*(3*x - y - 1)</span>
-</pre></div>
-</div>
-<p>If the expression is not separable then expr is returned unchanged
-or (if dict=True) then None is returned.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">separatevars</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span> <span class="o">==</span> <span class="n">eq</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">separatevars</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">symbols</span><span class="o">=</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="nb">dict</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="o">==</span> <span class="bp">None</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="radsimp">
-<h2>radsimp<a class="headerlink" href="#radsimp" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.simplify.simplify.radsimp">
-<tt class="descclassname">sympy.simplify.simplify.</tt><tt class="descname">radsimp</tt><big>(</big><em>expr</em>, <em>symbolic=True</em>, <em>max_terms=4</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/simplify.html#radsimp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.simplify.radsimp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Rationalize the denominator by removing square roots.</p>
-<p>Note: the expression returned from radsimp must be used with caution
-since if the denominator contains symbols, it will be possible to make
-substitutions that violate the assumptions of the simplification process:
-that for a denominator matching a + b*sqrt(c), a != +/-b*sqrt(c). (If
-there are no symbols, this assumptions is made valid by collecting terms
-of sqrt(c) so the match variable <tt class="docutils literal"><span class="pre">a</span></tt> does not contain <tt class="docutils literal"><span class="pre">sqrt(c)</span></tt>.) If
-you do not want the simplification to occur for symbolic denominators, set
-<tt class="docutils literal"><span class="pre">symbolic</span></tt> to False.</p>
-<p>If there are more than <tt class="docutils literal"><span class="pre">max_terms</span></tt> radical terms do not simplify.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">radsimp</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">denom</span><span class="p">,</span> <span class="n">pprint</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">radsimp</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">I</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">(1 - I)/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">radsimp</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)))</span>
-<span class="go">(-sqrt(2) + 2)/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Symbol</span><span class="p">,</span> <span class="s">&#39;xy&#39;</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="p">((</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">8</span><span class="p">))</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">radsimp</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-<span class="go">sqrt(2)*(x + y)</span>
-</pre></div>
-</div>
-<p>Terms are collected automatically:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">r2</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">r5</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">radsimp</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">y</span><span class="o">*</span><span class="n">r2</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">r2</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="n">r5</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">r5</span><span class="p">)))</span>
-<span class="go">         ___              ___</span>
-<span class="go">       \/ 5 *(-a - b) + \/ 2 *(x + y)</span>
-<span class="go">--------------------------------------------</span>
-<span class="go">     2               2      2              2</span>
-<span class="go">- 5*a  - 10*a*b - 5*b  + 2*x  + 4*x*y + 2*y</span>
-</pre></div>
-</div>
-<p>If radicals in the denominator cannot be removed, the original expression
-will be returned. If the denominator was 1 then any square roots will also
-be collected:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">radsimp</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">sqrt(2)*(x + 1)</span>
-</pre></div>
-</div>
-<p>Results with symbols will not always be valid for all substitutions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">c</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">c</span><span class="p">))</span>
-<span class="go">1/(2*b*sqrt(c))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">radsimp</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">c</span><span class="p">))</span>
-<span class="go">nan</span>
-</pre></div>
-</div>
-<p>If symbolic=False, symbolic denominators will not be transformed (but
-numeric denominators will still be processed):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">radsimp</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">symbolic</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">1/(a + b*sqrt(c))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="ratsimp">
-<h2>ratsimp<a class="headerlink" href="#ratsimp" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.simplify.simplify.ratsimp">
-<tt class="descclassname">sympy.simplify.simplify.</tt><tt class="descname">ratsimp</tt><big>(</big><em>expr</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/simplify.html#ratsimp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.simplify.ratsimp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Put an expression over a common denominator, cancel and reduce.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">ratsimp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ratsimp</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">y</span><span class="p">)</span>
-<span class="go">(x + y)/(x*y)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="fraction">
-<h2>fraction<a class="headerlink" href="#fraction" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.simplify.simplify.fraction">
-<tt class="descclassname">sympy.simplify.simplify.</tt><tt class="descname">fraction</tt><big>(</big><em>expr</em>, <em>exact=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/simplify.html#fraction"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.simplify.fraction" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a pair with expression&#8217;s numerator and denominator.
-If the given expression is not a fraction then this function
-will return the tuple (expr, 1).</p>
-<p>This function will not make any attempt to simplify nested
-fractions or to do any term rewriting at all.</p>
-<p>If only one of the numerator/denominator pair is needed then
-use numer(expr) or denom(expr) functions respectively.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">fraction</span><span class="p">,</span> <span class="n">Rational</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fraction</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="n">y</span><span class="p">)</span>
-<span class="go">(x, y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fraction</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">(x, 1)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fraction</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(1, y**2)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fraction</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(x*y, 2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fraction</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">(1, 2)</span>
-</pre></div>
-</div>
-<p>This function will also work fine with assumptions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;k&#39;</span><span class="p">,</span> <span class="n">negative</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fraction</span><span class="p">(</span><span class="n">x</span> <span class="o">*</span> <span class="n">y</span><span class="o">**</span><span class="n">k</span><span class="p">)</span>
-<span class="go">(x, y**(-k))</span>
-</pre></div>
-</div>
-<p>If we know nothing about sign of some exponent and &#8216;exact&#8217;
-flag is unset, then structure this exponent&#8217;s structure will
-be analyzed and pretty fraction will be returned:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">fraction</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">y</span><span class="p">))</span>
-<span class="go">(2, x**y)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fraction</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">))</span>
-<span class="go">(1, exp(x))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">fraction</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">),</span> <span class="n">exact</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(exp(-x), 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="trigsimp">
-<h2>trigsimp<a class="headerlink" href="#trigsimp" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.simplify.simplify.trigsimp">
-<tt class="descclassname">sympy.simplify.simplify.</tt><tt class="descname">trigsimp</tt><big>(</big><em>expr</em>, <em>deep=False</em>, <em>recursive=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/simplify.html#trigsimp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.simplify.trigsimp" title="Permalink to this definition">¶</a></dt>
-<dd><p>reduces expression by using known trig identities</p>
-<p class="rubric">Notes</p>
-<p>deep:
-- Apply trigsimp inside all objects with arguments</p>
-<p>recursive:
-- Use common subexpression elimination (cse()) and apply
-trigsimp recursively (this is quite expensive if the
-expression is large)</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">trigsimp</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">cosh</span><span class="p">,</span> <span class="n">sinh</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">trigsimp</span><span class="p">(</span><span class="n">e</span><span class="p">)</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">trigsimp</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">e</span><span class="p">))</span>
-<span class="go">log(2*sin(x)**2 + 2*cos(x)**2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">trigsimp</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">e</span><span class="p">),</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">log(2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="besselsimp">
-<h2>besselsimp<a class="headerlink" href="#besselsimp" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.simplify.simplify.besselsimp">
-<tt class="descclassname">sympy.simplify.simplify.</tt><tt class="descname">besselsimp</tt><big>(</big><em>expr</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/simplify.html#besselsimp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.simplify.besselsimp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Simplify bessel-type functions.</p>
-<p>This routine tries to simplify bessel-type functions. Currently it only
-works on the Bessel J and I functions, however. It works by looking at all
-such functions in turn, and eliminating factors of &#8220;I&#8221; and &#8220;-1&#8221; (actually
-their polar equivalents) in front of the argument. After that, functions of
-half-integer order are rewritten using trigonometric functions.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">besselj</span><span class="p">,</span> <span class="n">besseli</span><span class="p">,</span> <span class="n">besselsimp</span><span class="p">,</span> <span class="n">polar_lift</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">z</span><span class="p">,</span> <span class="n">nu</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselsimp</span><span class="p">(</span><span class="n">besselj</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="o">*</span><span class="n">polar_lift</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)))</span>
-<span class="go">exp(I*pi*nu)*besselj(nu, z)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselsimp</span><span class="p">(</span><span class="n">besseli</span><span class="p">(</span><span class="n">nu</span><span class="p">,</span> <span class="n">z</span><span class="o">*</span><span class="n">polar_lift</span><span class="p">(</span><span class="o">-</span><span class="n">I</span><span class="p">)))</span>
-<span class="go">exp(-I*pi*nu/2)*besselj(nu, z)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">besselsimp</span><span class="p">(</span><span class="n">besseli</span><span class="p">(</span><span class="n">S</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">z</span><span class="p">))</span>
-<span class="go">sqrt(2)*cosh(z)/(sqrt(pi)*sqrt(z))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="powsimp">
-<h2>powsimp<a class="headerlink" href="#powsimp" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.simplify.simplify.powsimp">
-<tt class="descclassname">sympy.simplify.simplify.</tt><tt class="descname">powsimp</tt><big>(</big><em>expr</em>, <em>deep=False</em>, <em>combine='all'</em>, <em>force=False</em>, <em>measure=&lt;function count_ops at 0x10c198b00&gt;</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/simplify.html#powsimp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.simplify.powsimp" title="Permalink to this definition">¶</a></dt>
-<dd><p>reduces expression by combining powers with similar bases and exponents.</p>
-<p class="rubric">Notes</p>
-<p>If deep is True then powsimp() will also simplify arguments of
-functions. By default deep is set to False.</p>
-<p>If force is True then bases will be combined without checking for
-assumptions, e.g. sqrt(x)*sqrt(y) -&gt; sqrt(x*y) which is not true
-if x and y are both negative.</p>
-<p>You can make powsimp() only combine bases or only combine exponents by
-changing combine=&#8217;base&#8217; or combine=&#8217;exp&#8217;.  By default, combine=&#8217;all&#8217;,
-which does both.  combine=&#8217;base&#8217; will only combine:</p>
-<div class="highlight-python"><pre> a   a          a                          2x      x
-x * y  =&gt;  (x*y)   as well as things like 2   =&gt;  4</pre>
-</div>
-<p>and combine=&#8217;exp&#8217; will only combine</p>
-<div class="highlight-python"><pre> a   b      (a + b)
-x * x  =&gt;  x</pre>
-</div>
-<p>combine=&#8217;exp&#8217; will strictly only combine exponents in the way that used
-to be automatic.  Also use deep=True if you need the old behavior.</p>
-<p>When combine=&#8217;all&#8217;, &#8216;exp&#8217; is evaluated first.  Consider the first
-example below for when there could be an ambiguity relating to this.
-This is done so things like the second example can be completely
-combined.  If you want &#8216;base&#8217; combined first, do something like
-powsimp(powsimp(expr, combine=&#8217;base&#8217;), combine=&#8217;exp&#8217;).</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">powsimp</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powsimp</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">y</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">z</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="n">z</span><span class="p">,</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;all&#39;</span><span class="p">)</span>
-<span class="go">x**(y + z)*y**z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powsimp</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">y</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">z</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="n">z</span><span class="p">,</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;exp&#39;</span><span class="p">)</span>
-<span class="go">x**(y + z)*y**z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powsimp</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">y</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">z</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="n">z</span><span class="p">,</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;base&#39;</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">x**y*(x*y)**z</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">powsimp</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">z</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">y</span><span class="o">*</span><span class="n">n</span><span class="o">**</span><span class="n">z</span><span class="o">*</span><span class="n">n</span><span class="o">**</span><span class="n">y</span><span class="p">,</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;all&#39;</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(n*x)**(y + z)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powsimp</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">z</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">y</span><span class="o">*</span><span class="n">n</span><span class="o">**</span><span class="n">z</span><span class="o">*</span><span class="n">n</span><span class="o">**</span><span class="n">y</span><span class="p">,</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;exp&#39;</span><span class="p">)</span>
-<span class="go">n**(y + z)*x**(y + z)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powsimp</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">z</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">y</span><span class="o">*</span><span class="n">n</span><span class="o">**</span><span class="n">z</span><span class="o">*</span><span class="n">n</span><span class="o">**</span><span class="n">y</span><span class="p">,</span> <span class="n">combine</span><span class="o">=</span><span class="s">&#39;base&#39;</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(n*x)**y*(n*x)**z</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x y&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powsimp</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">y</span><span class="p">)))</span>
-<span class="go">log(exp(x)*exp(y))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powsimp</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="n">y</span><span class="p">)),</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">x + y</span>
-</pre></div>
-</div>
-<p>Radicals with Mul bases will be combined if combine=&#8217;exp&#8217;</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">Mul</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Two radicals are automatically joined through Mul:
-&gt;&gt;&gt; a=sqrt(x*sqrt(y))
-&gt;&gt;&gt; a*a**3 == a**4
-True</p>
-<p>But if an integer power of that radical has been
-autoexpanded then Mul does not join the resulting factors:
-&gt;&gt;&gt; a**4 # auto expands to a Mul, no longer a Pow
-x**2*y
-&gt;&gt;&gt; _*a # so Mul doesn&#8217;t combine them
-x**2*y*sqrt(x*sqrt(y))
-&gt;&gt;&gt; powsimp(_) # but powsimp will
-(x*sqrt(y))**(5/2)
-&gt;&gt;&gt; powsimp(x*y*a) # but won&#8217;t when doing so would violate assumptions
-x*y*sqrt(x*sqrt(y))</p>
-</dd></dl>
-
-</div>
-<div class="section" id="combsimp">
-<h2>combsimp<a class="headerlink" href="#combsimp" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.simplify.simplify.combsimp">
-<tt class="descclassname">sympy.simplify.simplify.</tt><tt class="descname">combsimp</tt><big>(</big><em>expr</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/simplify.html#combsimp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.simplify.combsimp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Simplify combinatorial expressions.</p>
-<p>This function takes as input an expression containing factorials,
-binomials, Pochhammer symbol and other &#8220;combinatorial&#8221; functions,
-and tries to minimize the number of those functions and reduce
-the size of their arguments. The result is be given in terms of
-binomials and factorials.</p>
-<p>The algorithm works by rewriting all combinatorial functions as
-expressions involving rising factorials (Pochhammer symbols) and
-applies recurrence relations and other transformations applicable
-to rising factorials, to reduce their arguments, possibly letting
-the resulting rising factorial to cancel. Rising factorials with
-the second argument being an integer are expanded into polynomial
-forms and finally all other rising factorial are rewritten in terms
-more familiar functions. If the initial expression contained any
-combinatorial functions, the result is expressed using binomial
-coefficients and gamma functions. If the initial expression consisted
-of gamma functions alone, the result is expressed in terms of gamma
-functions.</p>
-<p>If the result is expressed using gamma functions, the following three
-additional steps are performed:</p>
-<ol class="arabic simple">
-<li>Reduce the number of gammas by applying the reflection theorem
-gamma(x)*gamma(1-x) == pi/sin(pi*x).</li>
-<li>Reduce the number of gammas by applying the multiplication theorem
-gamma(x)*gamma(x+1/n)*...*gamma(x+(n-1)/n) == C*gamma(n*x).</li>
-<li>Reduce the number of prefactors by absorbing them into gammas, where
-possible.</li>
-</ol>
-<p>All transformation rules can be found (or was derived from) here:</p>
-<ol class="arabic simple">
-<li><a class="reference external" href="http://functions.wolfram.com/GammaBetaErf/Pochhammer/17/01/02/">http://functions.wolfram.com/GammaBetaErf/Pochhammer/17/01/02/</a></li>
-<li><a class="reference external" href="http://functions.wolfram.com/GammaBetaErf/Pochhammer/27/01/0005/">http://functions.wolfram.com/GammaBetaErf/Pochhammer/27/01/0005/</a></li>
-</ol>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.simplify</span> <span class="kn">import</span> <span class="n">combsimp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">binomial</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">n</span><span class="p">,</span> <span class="n">k</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">combsimp</span><span class="p">(</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">3</span><span class="p">))</span>
-<span class="go">n*(n - 2)*(n - 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">combsimp</span><span class="p">(</span><span class="n">binomial</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="n">binomial</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">))</span>
-<span class="go">(n + 1)/(k + 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="hypersimp">
-<h2>hypersimp<a class="headerlink" href="#hypersimp" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.simplify.simplify.hypersimp">
-<tt class="descclassname">sympy.simplify.simplify.</tt><tt class="descname">hypersimp</tt><big>(</big><em>f</em>, <em>k</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/simplify.html#hypersimp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.simplify.hypersimp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Given combinatorial term f(k) simplify its consecutive term ratio
-i.e. f(k+1)/f(k).  The input term can be composed of functions and
-integer sequences which have equivalent representation in terms
-of gamma special function.</p>
-<p>The algorithm performs three basic steps:</p>
-<ol class="arabic simple">
-<li>Rewrite all functions in terms of gamma, if possible.</li>
-<li>Rewrite all occurrences of gamma in terms of products
-of gamma and rising factorial with integer,  absolute
-constant exponent.</li>
-<li>Perform simplification of nested fractions, powers
-and if the resulting expression is a quotient of
-polynomials, reduce their total degree.</li>
-</ol>
-<p>If f(k) is hypergeometric then as result we arrive with a
-quotient of polynomials of minimal degree. Otherwise None
-is returned.</p>
-<p>For more information on the implemented algorithm refer to:</p>
-<ol class="arabic simple">
-<li>W. Koepf, Algorithms for m-fold Hypergeometric Summation,
-Journal of Symbolic Computation (1995) 20, 399-417</li>
-</ol>
-</dd></dl>
-
-</div>
-<div class="section" id="hypersimilar">
-<h2>hypersimilar<a class="headerlink" href="#hypersimilar" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.simplify.simplify.hypersimilar">
-<tt class="descclassname">sympy.simplify.simplify.</tt><tt class="descname">hypersimilar</tt><big>(</big><em>f</em>, <em>g</em>, <em>k</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/simplify.html#hypersimilar"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.simplify.hypersimilar" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns True if &#8216;f&#8217; and &#8216;g&#8217; are hyper-similar.</p>
-<p>Similarity in hypergeometric sense means that a quotient of
-f(k) and g(k) is a rational function in k.  This procedure
-is useful in solving recurrence relations.</p>
-<p>For more information see hypersimp().</p>
-</dd></dl>
-
-</div>
-<div class="section" id="nsimplify">
-<h2>nsimplify<a class="headerlink" href="#nsimplify" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.simplify.simplify.nsimplify">
-<tt class="descclassname">sympy.simplify.simplify.</tt><tt class="descname">nsimplify</tt><big>(</big><em>expr</em>, <em>constants=</em><span class="optional">[</span><span class="optional">]</span>, <em>tolerance=None</em>, <em>full=False</em>, <em>rational=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/simplify.html#nsimplify"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.simplify.nsimplify" title="Permalink to this definition">¶</a></dt>
-<dd><p>Find a simple representation for a number or, if there are free symbols or
-if rational=True, then replace Floats with their Rational equivalents. If
-no change is made and rational is not False then Floats will at least be
-converted to Rationals.</p>
-<p>For numerical expressions, a simple formula that numerically matches the
-given numerical expression is sought (and the input should be possible
-to evalf to a precision of at least 30 digits).</p>
-<p>Optionally, a list of (rationally independent) constants to
-include in the formula may be given.</p>
-<p>A lower tolerance may be set to find less exact matches. If no tolerance
-is given then the least precise value will set the tolerance (e.g. Floats
-default to 15 digits of precision, so would be tolerance=10**-15).</p>
-<p>With full=True, a more extensive search is performed
-(this is useful to find simpler numbers when the tolerance
-is set low).</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="../core.html#sympy.core.function.nfloat" title="sympy.core.function.nfloat"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.core.function.nfloat</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">nsimplify</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">GoldenRatio</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsimplify</span><span class="p">(</span><span class="mi">4</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">)),</span> <span class="p">[</span><span class="n">GoldenRatio</span><span class="p">])</span>
-<span class="go">-2 + 2*GoldenRatio</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsimplify</span><span class="p">((</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">I</span><span class="o">/</span><span class="mi">5</span><span class="p">)</span><span class="o">+</span><span class="mi">1</span><span class="p">)))</span>
-<span class="go">1/2 - I*sqrt(sqrt(5)/10 + 1/4)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsimplify</span><span class="p">(</span><span class="n">I</span><span class="o">**</span><span class="n">I</span><span class="p">,</span> <span class="p">[</span><span class="n">pi</span><span class="p">])</span>
-<span class="go">exp(-pi/2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsimplify</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="n">tolerance</span><span class="o">=</span><span class="mf">0.01</span><span class="p">)</span>
-<span class="go">22/7</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="collect-sqrt">
-<h2>collect_sqrt<a class="headerlink" href="#collect-sqrt" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.simplify.simplify.collect_sqrt">
-<tt class="descclassname">sympy.simplify.simplify.</tt><tt class="descname">collect_sqrt</tt><big>(</big><em>expr</em>, <em>evaluate=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/simplify.html#collect_sqrt"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.simplify.collect_sqrt" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return expr with terms having common square roots collected together.
-If <tt class="docutils literal"><span class="pre">evaluate</span></tt> is False a count indicating the number of sqrt-containing
-terms will be returned and the returned expression will be an unevaluated
-Add with args ordered by default_sort_key.</p>
-<p>Note: since I = sqrt(-1), it is collected, too.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.simplify.simplify</span> <span class="kn">import</span> <span class="n">collect_sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">r2</span><span class="p">,</span> <span class="n">r3</span><span class="p">,</span> <span class="n">r5</span> <span class="o">=</span> <span class="p">[</span><span class="n">sqrt</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">collect_sqrt</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">r2</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">r2</span><span class="p">)</span>
-<span class="go">sqrt(2)*(a + b)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">collect_sqrt</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">r2</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">r2</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="n">r3</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">r3</span><span class="p">)</span>
-<span class="go">sqrt(2)*(a + b) + sqrt(3)*(a + b)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">collect_sqrt</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">r2</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">r2</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="n">r3</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">r5</span><span class="p">)</span>
-<span class="go">sqrt(3)*a + sqrt(5)*b + sqrt(2)*(a + b)</span>
-</pre></div>
-</div>
-<p>If evaluate is False then the arguments will be sorted and
-returned as a list and a count of the number of sqrt-containing
-terms will be returned:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">collect_sqrt</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">r2</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">r2</span> <span class="o">+</span> <span class="n">a</span><span class="o">*</span><span class="n">r3</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="n">r5</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">((sqrt(2)*(a + b), sqrt(3)*a, sqrt(5)*b), 3)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">collect_sqrt</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">b</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">((b, sqrt(2)*a), 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">collect_sqrt</span><span class="p">(</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">,</span> <span class="n">evaluate</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">((a + b,), 0)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="collect-const">
-<h2>collect_const<a class="headerlink" href="#collect-const" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.simplify.simplify.collect_const">
-<tt class="descclassname">sympy.simplify.simplify.</tt><tt class="descname">collect_const</tt><big>(</big><em>expr</em>, <em>*vars</em>, <em>**first</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/simplify.html#collect_const"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.simplify.collect_const" title="Permalink to this definition">¶</a></dt>
-<dd><p>A non-greedy collection of terms with similar number coefficients in
-an Add expr. If <tt class="docutils literal"><span class="pre">vars</span></tt> is given then only those constants will be
-targeted.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">a</span><span class="p">,</span> <span class="n">s</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.simplify.simplify</span> <span class="kn">import</span> <span class="n">collect_const</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">collect_const</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)))</span>
-<span class="go">sqrt(3)*(sqrt(2) + 2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">collect_const</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">*</span><span class="n">s</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">7</span><span class="p">)</span><span class="o">*</span><span class="n">s</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">7</span><span class="p">))</span>
-<span class="go">(sqrt(3) + sqrt(7))*(s + 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">collect_const</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">*</span><span class="n">s</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">7</span><span class="p">)</span><span class="o">*</span><span class="n">s</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">7</span><span class="p">))</span>
-<span class="go">(sqrt(2) + 3)*(sqrt(3) + sqrt(7))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">collect_const</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">*</span><span class="n">s</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">7</span><span class="p">)</span><span class="o">*</span><span class="n">s</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">7</span><span class="p">),</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">sqrt(7) + sqrt(3)*(sqrt(2) + 3) + sqrt(7)*(sqrt(2) + 2)</span>
-</pre></div>
-</div>
-<p>If no constants are provided then a leading Rational might be returned:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">collect_const</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
-<span class="go">2*(2*sqrt(5)*a + sqrt(3))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">collect_const</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">4*sqrt(5)*a + 2*sqrt(3)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="posify">
-<h2>posify<a class="headerlink" href="#posify" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.simplify.simplify.posify">
-<tt class="descclassname">sympy.simplify.simplify.</tt><tt class="descname">posify</tt><big>(</big><em>eq</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/simplify.html#posify"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.simplify.posify" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return eq (with generic symbols made positive) and a restore
-dictionary.</p>
-<p>Any symbol that has positive=None will be replaced with a positive dummy
-symbol having the same name. This replacement will allow more symbolic
-processing of expressions, especially those involving powers and
-logarithms.</p>
-<p>A dictionary that can be sent to subs to restore eq to its original
-symbols is also returned.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">posify</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">log</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">posify</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="o">+</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">,</span> <span class="n">negative</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">(_x + n + p, {_x: x})</span>
-</pre></div>
-</div>
-<p>&gt;&gt; log(1/x).expand() # should be log(1/x) but it comes back as -log(x)
-log(1/x)</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="n">posify</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)[</span><span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span> <span class="c"># take [0] and ignore replacements</span>
-<span class="go">-log(_x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span><span class="p">,</span> <span class="n">rep</span> <span class="o">=</span> <span class="n">posify</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">log</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">rep</span><span class="p">)</span>
-<span class="go">-log(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">posify</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">x</span><span class="p">])</span>
-<span class="go">([_x, _x + 1], {_x: x})</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="powdenest">
-<h2>powdenest<a class="headerlink" href="#powdenest" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.simplify.simplify.powdenest">
-<tt class="descclassname">sympy.simplify.simplify.</tt><tt class="descname">powdenest</tt><big>(</big><em>eq</em>, <em>force=False</em>, <em>polar=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/simplify.html#powdenest"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.simplify.powdenest" title="Permalink to this definition">¶</a></dt>
-<dd><p>Collect exponents on powers as assumptions allow.</p>
-<dl class="docutils">
-<dt>Given <tt class="docutils literal"><span class="pre">(bb**be)**e</span></tt>, this can be simplified as follows:</dt>
-<dd><ul class="first last simple">
-<li>if <tt class="docutils literal"><span class="pre">bb</span></tt> is positive, or</li>
-<li><tt class="docutils literal"><span class="pre">e</span></tt> is an integer, or</li>
-<li><tt class="docutils literal"><span class="pre">|be|</span> <span class="pre">&lt;</span> <span class="pre">1</span></tt> then this simplifies to <tt class="docutils literal"><span class="pre">bb**(be*e)</span></tt></li>
-</ul>
-</dd>
-</dl>
-<p>Given a product of powers raised to a power, <tt class="docutils literal"><span class="pre">(bb1**be1</span> <span class="pre">*</span>
-<span class="pre">bb2**be2...)**e</span></tt>, simplification can be done as follows:</p>
-<ul class="simple">
-<li>if e is positive, the gcd of all bei can be joined with e;</li>
-<li>all non-negative bb can be separated from those that are negative
-and their gcd can be joined with e; autosimplification already
-handles this separation.</li>
-<li>integer factors from powers that have integers in the denominator
-of the exponent can be removed from any term and the gcd of such
-integers can be joined with e</li>
-</ul>
-<p>Setting <tt class="docutils literal"><span class="pre">force</span></tt> to True will make symbols that are not explicitly
-negative behave as though they are positive, resulting in more
-denesting.</p>
-<p>Setting <tt class="docutils literal"><span class="pre">polar</span></tt> to True will do simplifications on the riemann surface of
-the logarithm, also resulting in more denestings.</p>
-<p>When there are sums of logs in exp() then a product of powers may be
-obtained e.g. <tt class="docutils literal"><span class="pre">exp(3*(log(a)</span> <span class="pre">+</span> <span class="pre">2*log(b)))</span></tt> - &gt; <tt class="docutils literal"><span class="pre">a**3*b**6</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">powdenest</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">powdenest</span><span class="p">((</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">/</span><span class="mi">3</span><span class="p">))</span><span class="o">**</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="p">))</span>
-<span class="go">(x**(2*a/3))**(3*x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powdenest</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)))</span>
-<span class="go">2**(3*x)</span>
-</pre></div>
-</div>
-<p>Assumptions may prevent expansion:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">powdenest</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">sqrt(x**2)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powdenest</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">p</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">p</span>
-</pre></div>
-</div>
-<p>No other expansion is done.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i,j&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powdenest</span><span class="p">((</span><span class="n">x</span><span class="o">**</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">))</span> <span class="c"># -X-&gt; (x**x)**i*(x**x)**j</span>
-<span class="go">x**(x*(i + j))</span>
-</pre></div>
-</div>
-<p>But exp() will be denested by moving all non-log terms outside of
-the function; this may result in the collapsing of the exp to a power
-with a different base:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">powdenest</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
-<span class="go">x**(3*y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powdenest</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">y</span><span class="o">*</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">+</span> <span class="n">log</span><span class="p">(</span><span class="n">b</span><span class="p">))))</span>
-<span class="go">(a*b)**y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powdenest</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">+</span> <span class="n">log</span><span class="p">(</span><span class="n">b</span><span class="p">))))</span>
-<span class="go">a**3*b**3</span>
-</pre></div>
-</div>
-<p>If assumptions allow, symbols can also be moved to the outermost exponent:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powdenest</span><span class="p">(((</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span><span class="p">))</span><span class="o">**</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="p">))</span><span class="o">**</span><span class="n">x</span><span class="p">)</span>
-<span class="go">((x**(2*i))**(3*y))**x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powdenest</span><span class="p">(((</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span><span class="p">))</span><span class="o">**</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="p">))</span><span class="o">**</span><span class="n">x</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">x**(6*i*x*y)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">powdenest</span><span class="p">(((</span><span class="n">p</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">))</span><span class="o">**</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="p">))</span><span class="o">**</span><span class="n">x</span><span class="p">)</span>
-<span class="go">p**(6*a*x*y)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">powdenest</span><span class="p">(((</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">/</span><span class="mi">3</span><span class="p">))</span><span class="o">**</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">y</span><span class="o">/</span><span class="n">i</span><span class="p">))</span><span class="o">**</span><span class="n">x</span><span class="p">)</span>
-<span class="go">((x**(2*a/3))**(3*y/i))**x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powdenest</span><span class="p">((</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span><span class="p">)</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">i</span><span class="p">))</span><span class="o">**</span><span class="n">z</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(x*y**2)**(2*i*z)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">,</span> <span class="n">negative</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">powdenest</span><span class="p">((</span><span class="n">x</span><span class="o">**</span><span class="n">i</span><span class="p">)</span><span class="o">**</span><span class="n">y</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">x**(i*y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">powdenest</span><span class="p">((</span><span class="n">n</span><span class="o">**</span><span class="n">i</span><span class="p">)</span><span class="o">**</span><span class="n">x</span><span class="p">,</span> <span class="n">force</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">(n**i)**x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="logcombine">
-<h2>logcombine<a class="headerlink" href="#logcombine" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.simplify.simplify.logcombine">
-<tt class="descclassname">sympy.simplify.simplify.</tt><tt class="descname">logcombine</tt><big>(</big><em>expr</em>, <em>force=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/simplify.html#logcombine"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.simplify.logcombine" title="Permalink to this definition">¶</a></dt>
-<dd><p>Takes logarithms and combines them using the following rules:</p>
-<ul class="simple">
-<li>log(x)+log(y) == log(x*y)</li>
-<li>a*log(x) == log(x**a)</li>
-</ul>
-<p>These identities are only valid if x and y are positive and if a is real,
-so the function will not combine the terms unless the arguments have the
-proper assumptions on them.  Use logcombine(func, force=True) to
-automatically assume that the arguments of logs are positive and that
-coefficients are real.  Note that this will not change any assumptions
-already in place, so if the coefficient is imaginary or the argument
-negative, combine will still not combine the equations.  Change the
-assumptions on the variables to make them combine.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">logcombine</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">logcombine</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">+</span><span class="n">log</span><span class="p">(</span><span class="n">y</span><span class="p">)</span><span class="o">-</span><span class="n">log</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="go">a*log(x) + log(y) - log(z)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">logcombine</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">+</span><span class="n">log</span><span class="p">(</span><span class="n">y</span><span class="p">)</span><span class="o">-</span><span class="n">log</span><span class="p">(</span><span class="n">z</span><span class="p">),</span> <span class="n">force</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">log(x**a*y/z)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y,z&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">logcombine</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">+</span><span class="n">log</span><span class="p">(</span><span class="n">y</span><span class="p">)</span><span class="o">-</span><span class="n">log</span><span class="p">(</span><span class="n">z</span><span class="p">))</span>
-<span class="go">log(x**a*y/z)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.simplify.sqrtdenest">
-<span id="square-root-denest"></span><h2>Square Root Denest<a class="headerlink" href="#module-sympy.simplify.sqrtdenest" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="sqrtdenest">
-<h3>sqrtdenest<a class="headerlink" href="#sqrtdenest" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.simplify.sqrtdenest.sqrtdenest">
-<tt class="descclassname">sympy.simplify.sqrtdenest.</tt><tt class="descname">sqrtdenest</tt><big>(</big><em>expr</em>, <em>max_iter=3</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/sqrtdenest.html#sqrtdenest"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.sqrtdenest.sqrtdenest" title="Permalink to this definition">¶</a></dt>
-<dd><p>Denests sqrts in an expression that contain other square roots
-if possible, otherwise returns the expr unchanged. This is based on the
-algorithms of [1].</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.solvers.solvers.unrad</span></tt></p>
-</div>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://www.almaden.ibm.com/cs/people/fagin/symb85.pdf">http://www.almaden.ibm.com/cs/people/fagin/symb85.pdf</a></p>
-<p>[2] D. J. Jeffrey and A. D. Rich, &#8216;Symplifying Square Roots of Square Roots
-by Denesting&#8217; (available at <a class="reference external" href="http://www.cybertester.com/data/denest.pdf">http://www.cybertester.com/data/denest.pdf</a>)</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.simplify.sqrtdenest</span> <span class="kn">import</span> <span class="n">sqrtdenest</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sqrtdenest</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">5</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">6</span><span class="p">)))</span>
-<span class="go">sqrt(2) + sqrt(3)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.simplify.cse_main">
-<span id="common-subexpresion-elimination"></span><h2>Common Subexpresion Elimination<a class="headerlink" href="#module-sympy.simplify.cse_main" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="cse">
-<h3>cse<a class="headerlink" href="#cse" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.simplify.cse_main.cse">
-<tt class="descclassname">sympy.simplify.cse_main.</tt><tt class="descname">cse</tt><big>(</big><em>exprs</em>, <em>symbols=None</em>, <em>optimizations=None</em>, <em>postprocess=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/cse_main.html#cse"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.cse_main.cse" title="Permalink to this definition">¶</a></dt>
-<dd><p>Perform common subexpression elimination on an expression.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>exprs</strong> : list of sympy expressions, or a single sympy expression</p>
-<blockquote>
-<div><p>The expressions to reduce.</p>
-</div></blockquote>
-<p><strong>symbols</strong> : infinite iterator yielding unique Symbols</p>
-<blockquote>
-<div><p>The symbols used to label the common subexpressions which are pulled
-out. The <tt class="docutils literal"><span class="pre">numbered_symbols</span></tt> generator is useful. The default is a
-stream of symbols of the form &#8220;x0&#8221;, &#8220;x1&#8221;, etc. This must be an infinite
-iterator.</p>
-</div></blockquote>
-<p><strong>optimizations</strong> : list of (callable, callable) pairs, optional</p>
-<blockquote>
-<div><p>The (preprocessor, postprocessor) pairs. If not provided,
-<tt class="docutils literal"><span class="pre">sympy.simplify.cse.cse_optimizations</span></tt> is used.</p>
-</div></blockquote>
-<p><strong>postprocess</strong> : a function which accepts the two return values of cse and</p>
-<blockquote>
-<div><p>returns the desired form of output from cse, e.g. if you want the
-replacements reversed the function might be the following lambda:
-lambda r, e: return reversed(r), e</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>replacements</strong> : list of (Symbol, expression) pairs</p>
-<blockquote>
-<div><p>All of the common subexpressions that were replaced. Subexpressions
-earlier in this list might show up in subexpressions later in this list.</p>
-</div></blockquote>
-<p><strong>reduced_exprs</strong> : list of sympy expressions</p>
-<blockquote class="last">
-<div><p>The reduced expressions with all of the replacements above.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.simplify.hyperexpand">
-<span id="hypergeometric-function-expansion"></span><h2>Hypergeometric Function Expansion<a class="headerlink" href="#module-sympy.simplify.hyperexpand" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="hyperexpand">
-<h3>hyperexpand<a class="headerlink" href="#hyperexpand" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.simplify.hyperexpand.hyperexpand">
-<tt class="descclassname">sympy.simplify.hyperexpand.</tt><tt class="descname">hyperexpand</tt><big>(</big><em>f</em>, <em>allow_hyper=False</em>, <em>rewrite='default'</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/hyperexpand.html#hyperexpand"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.hyperexpand.hyperexpand" title="Permalink to this definition">¶</a></dt>
-<dd><p>Expand hypergeometric functions. If allow_hyper is True, allow partial
-simplification (that is a result different from input,
-but still containing hypergeometric functions).</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.simplify.hyperexpand</span> <span class="kn">import</span> <span class="n">hyperexpand</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.functions</span> <span class="kn">import</span> <span class="n">hyper</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hyperexpand</span><span class="p">(</span><span class="n">hyper</span><span class="p">([],</span> <span class="p">[],</span> <span class="n">z</span><span class="p">))</span>
-<span class="go">exp(z)</span>
-</pre></div>
-</div>
-<p>Non-hyperegeometric parts of the expression and hypergeometric expressions
-that are not recognised are left unchanged:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">hyperexpand</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">hyper</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[],</span> <span class="n">z</span><span class="p">))</span>
-<span class="go">hyper((1, 1, 1), (), z) + 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.simplify.traversaltools">
-<span id="traversal-tools"></span><h2>Traversal Tools<a class="headerlink" href="#module-sympy.simplify.traversaltools" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="use">
-<h3>use<a class="headerlink" href="#use" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.simplify.traversaltools.use">
-<tt class="descclassname">sympy.simplify.traversaltools.</tt><tt class="descname">use</tt><big>(</big><em>expr</em>, <em>func</em>, <em>level=0</em>, <em>args=()</em>, <em>kwargs={}</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/traversaltools.html#use"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.traversaltools.use" title="Permalink to this definition">¶</a></dt>
-<dd><p>Use <tt class="docutils literal"><span class="pre">func</span></tt> to transform <tt class="docutils literal"><span class="pre">expr</span></tt> at the given level.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">use</span><span class="p">,</span> <span class="n">expand</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">use</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">expand</span><span class="p">,</span> <span class="n">level</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x*(x**2 + 2*x*y + y**2) + 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expand</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
-<span class="go">x**3 + 2*x**2*y + x*y**2 + 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.simplify.epathtools">
-<span id="epath-tools"></span><h2>EPath Tools<a class="headerlink" href="#module-sympy.simplify.epathtools" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="epath-class">
-<h3>EPath class<a class="headerlink" href="#epath-class" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.simplify.epathtools.EPath">
-<em class="property">class </em><tt class="descclassname">sympy.simplify.epathtools.</tt><tt class="descname">EPath</tt><a class="reference internal" href="../../_modules/sympy/simplify/epathtools.html#EPath"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.epathtools.EPath" title="Permalink to this definition">¶</a></dt>
-<dd><p>Manipulate expressions using paths.</p>
-<p>EPath grammar in EBNF notation:</p>
-<div class="highlight-python"><pre>literal   ::= /[A-Za-z_][A-Za-z_0-9]*/
-number    ::= /-?\d+/
-type      ::= literal
-attribute ::= literal &quot;?&quot;
-all       ::= &quot;*&quot;
-slice     ::= &quot;[&quot; number? (&quot;:&quot; number? (&quot;:&quot; number?)?)? &quot;]&quot;
-range     ::= all | slice
-query     ::= (type | attribute) (&quot;|&quot; (type | attribute))*
-selector  ::= range | query range?
-path      ::= &quot;/&quot; selector (&quot;/&quot; selector)*</pre>
-</div>
-<p>See the docstring of the epath() function.</p>
-<dl class="function">
-<dt id="sympy.simplify.epathtools.EPath.apply">
-<tt class="descname">apply</tt><big>(</big><em>self</em>, <em>expr</em>, <em>func</em>, <em>args=None</em>, <em>kwargs=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/epathtools.html#EPath.apply"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.epathtools.EPath.apply" title="Permalink to this definition">¶</a></dt>
-<dd><p>Modify parts of an expression selected by a path.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.simplify.epathtools</span> <span class="kn">import</span> <span class="n">EPath</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">E</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">t</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">path</span> <span class="o">=</span> <span class="n">EPath</span><span class="p">(</span><span class="s">&quot;/*/[0]/Symbol&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expr</span> <span class="o">=</span> <span class="p">[((</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="mi">2</span><span class="p">),</span> <span class="p">((</span><span class="mi">3</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="n">z</span><span class="p">)]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">path</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">expr</span><span class="p">:</span> <span class="n">expr</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">[((x**2, 1), 2), ((3, y**2), z)]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">path</span> <span class="o">=</span> <span class="n">EPath</span><span class="p">(</span><span class="s">&quot;/*/*/Symbol&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expr</span> <span class="o">=</span> <span class="n">t</span> <span class="o">+</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">E</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">path</span><span class="o">.</span><span class="n">apply</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">expr</span><span class="p">:</span> <span class="mi">2</span><span class="o">*</span><span class="n">expr</span><span class="p">)</span>
-<span class="go">t + sin(2*x + 1) + cos(2*x + 2*y + E)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.simplify.epathtools.EPath.select">
-<tt class="descname">select</tt><big>(</big><em>self</em>, <em>expr</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/epathtools.html#EPath.select"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.epathtools.EPath.select" title="Permalink to this definition">¶</a></dt>
-<dd><p>Retrieve parts of an expression selected by a path.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.simplify.epathtools</span> <span class="kn">import</span> <span class="n">EPath</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">E</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">t</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">path</span> <span class="o">=</span> <span class="n">EPath</span><span class="p">(</span><span class="s">&quot;/*/[0]/Symbol&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expr</span> <span class="o">=</span> <span class="p">[((</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="mi">2</span><span class="p">),</span> <span class="p">((</span><span class="mi">3</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="n">z</span><span class="p">)]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">path</span><span class="o">.</span><span class="n">select</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-<span class="go">[x, y]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">path</span> <span class="o">=</span> <span class="n">EPath</span><span class="p">(</span><span class="s">&quot;/*/*/Symbol&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expr</span> <span class="o">=</span> <span class="n">t</span> <span class="o">+</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">E</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">path</span><span class="o">.</span><span class="n">select</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>
-<span class="go">[x, x, y]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-<div class="section" id="epath">
-<h3>epath<a class="headerlink" href="#epath" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.simplify.epathtools.epath">
-<tt class="descclassname">sympy.simplify.epathtools.</tt><tt class="descname">epath</tt><big>(</big><em>path</em>, <em>expr=None</em>, <em>func=None</em>, <em>args=None</em>, <em>kwargs=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/simplify/epathtools.html#epath"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.simplify.epathtools.epath" title="Permalink to this definition">¶</a></dt>
-<dd><p>Manipulate parts of an expression selected by a path.</p>
-<p>This function allows to manipulate large nested expressions in single
-line of code, utilizing techniques to those applied in XML processing
-standards (e.g. XPath).</p>
-<p>If <tt class="docutils literal"><span class="pre">func</span></tt> is <tt class="docutils literal"><span class="pre">None</span></tt>, <a class="reference internal" href="#sympy.simplify.epathtools.epath" title="sympy.simplify.epathtools.epath"><tt class="xref py py-func docutils literal"><span class="pre">epath()</span></tt></a> retrieves elements selected by
-the <tt class="docutils literal"><span class="pre">path</span></tt>. Otherwise it applies <tt class="docutils literal"><span class="pre">func</span></tt> to each matching element.</p>
-<p>Note that it is more efficient to create an EPath object and use the select
-and apply methods of that object, since this will compile the path string
-only once.  This function should only be used as a convenient shortcut for
-interactive use.</p>
-<p>This is the supported syntax:</p>
-<ul>
-<li><dl class="first docutils">
-<dt>select all: <tt class="docutils literal"><span class="pre">/*</span></tt></dt>
-<dd><p class="first last">Equivalent of <tt class="docutils literal"><span class="pre">for</span> <span class="pre">arg</span> <span class="pre">in</span> <span class="pre">args:</span></tt>.</p>
-</dd>
-</dl>
-</li>
-<li><dl class="first docutils">
-<dt>select slice: <tt class="docutils literal"><span class="pre">/[0]</span></tt> or <tt class="docutils literal"><span class="pre">/[1:5]</span></tt> or <tt class="docutils literal"><span class="pre">/[1:5:2]</span></tt></dt>
-<dd><p class="first last">Supports standard Python&#8217;s slice syntax.</p>
-</dd>
-</dl>
-</li>
-<li><dl class="first docutils">
-<dt>select by type: <tt class="docutils literal"><span class="pre">/list</span></tt> or <tt class="docutils literal"><span class="pre">/list|tuple</span></tt></dt>
-<dd><p class="first last">Emulates <tt class="xref py py-func docutils literal"><span class="pre">isinstance()</span></tt>.</p>
-</dd>
-</dl>
-</li>
-<li><dl class="first docutils">
-<dt>select by attribute: <tt class="docutils literal"><span class="pre">/__iter__?</span></tt></dt>
-<dd><p class="first last">Emulates <tt class="xref py py-func docutils literal"><span class="pre">hasattr()</span></tt>.</p>
-</dd>
-</dl>
-</li>
-</ul>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>path</strong> : str | EPath</p>
-<blockquote>
-<div><p>A path as a string or a compiled EPath.</p>
-</div></blockquote>
-<p><strong>expr</strong> : Basic | iterable</p>
-<blockquote>
-<div><p>An expression or a container of expressions.</p>
-</div></blockquote>
-<p><strong>func</strong> : callable (optional)</p>
-<blockquote>
-<div><p>A callable that will be applied to matching parts.</p>
-</div></blockquote>
-<p><strong>args</strong> : tuple (optional)</p>
-<blockquote>
-<div><p>Additional positional arguments to <tt class="docutils literal"><span class="pre">func</span></tt>.</p>
-</div></blockquote>
-<p><strong>kwargs</strong> : dict (optional)</p>
-<blockquote class="last">
-<div><p>Additional keyword arguments to <tt class="docutils literal"><span class="pre">func</span></tt>.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.simplify.epathtools</span> <span class="kn">import</span> <span class="n">epath</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">E</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">t</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">path</span> <span class="o">=</span> <span class="s">&quot;/*/[0]/Symbol&quot;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expr</span> <span class="o">=</span> <span class="p">[((</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="mi">2</span><span class="p">),</span> <span class="p">((</span><span class="mi">3</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="n">z</span><span class="p">)]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">epath</span><span class="p">(</span><span class="n">path</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-<span class="go">[x, y]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">epath</span><span class="p">(</span><span class="n">path</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">expr</span><span class="p">:</span> <span class="n">expr</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">[((x**2, 1), 2), ((3, y**2), z)]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">path</span> <span class="o">=</span> <span class="s">&quot;/*/*/Symbol&quot;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expr</span> <span class="o">=</span> <span class="n">t</span> <span class="o">+</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">E</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">epath</span><span class="p">(</span><span class="n">path</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>
-<span class="go">[x, x, y]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">epath</span><span class="p">(</span><span class="n">path</span><span class="p">,</span> <span class="n">expr</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">expr</span><span class="p">:</span> <span class="mi">2</span><span class="o">*</span><span class="n">expr</span><span class="p">)</span>
-<span class="go">t + sin(2*x + 1) + cos(2*x + 2*y + E)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../index.html">
-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Simplify</a><ul>
-<li><a class="reference internal" href="#id1">simplify</a></li>
-<li><a class="reference internal" href="#collect">collect</a></li>
-<li><a class="reference internal" href="#separate">separate</a></li>
-<li><a class="reference internal" href="#separatevars">separatevars</a></li>
-<li><a class="reference internal" href="#radsimp">radsimp</a></li>
-<li><a class="reference internal" href="#ratsimp">ratsimp</a></li>
-<li><a class="reference internal" href="#fraction">fraction</a></li>
-<li><a class="reference internal" href="#trigsimp">trigsimp</a></li>
-<li><a class="reference internal" href="#besselsimp">besselsimp</a></li>
-<li><a class="reference internal" href="#powsimp">powsimp</a></li>
-<li><a class="reference internal" href="#combsimp">combsimp</a></li>
-<li><a class="reference internal" href="#hypersimp">hypersimp</a></li>
-<li><a class="reference internal" href="#hypersimilar">hypersimilar</a></li>
-<li><a class="reference internal" href="#nsimplify">nsimplify</a></li>
-<li><a class="reference internal" href="#collect-sqrt">collect_sqrt</a></li>
-<li><a class="reference internal" href="#collect-const">collect_const</a></li>
-<li><a class="reference internal" href="#posify">posify</a></li>
-<li><a class="reference internal" href="#powdenest">powdenest</a></li>
-<li><a class="reference internal" href="#logcombine">logcombine</a></li>
-<li><a class="reference internal" href="#module-sympy.simplify.sqrtdenest">Square Root Denest</a><ul>
-<li><a class="reference internal" href="#sqrtdenest">sqrtdenest</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.simplify.cse_main">Common Subexpresion Elimination</a><ul>
-<li><a class="reference internal" href="#cse">cse</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.simplify.hyperexpand">Hypergeometric Function Expansion</a><ul>
-<li><a class="reference internal" href="#hyperexpand">hyperexpand</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.simplify.traversaltools">Traversal Tools</a><ul>
-<li><a class="reference internal" href="#use">use</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.simplify.epathtools">EPath Tools</a><ul>
-<li><a class="reference internal" href="#epath-class">EPath class</a></li>
-<li><a class="reference internal" href="#epath">epath</a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../sets.html"
-                        title="previous chapter">Sets</a></p>
-  <h4>Next topic</h4>
-  <p class="topless"><a href="../simplify/hyperexpand.html"
-                        title="next chapter">Details on the Hypergeometric Function Expansion Module</a></p>
-  <h3>This Page</h3>
-  <ul class="this-page-menu">
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diff --git a/dev-py3k/modules/solvers/ode.html b/dev-py3k/modules/solvers/ode.html
deleted file mode 100644
index f1e34f4650a..00000000000
--- a/dev-py3k/modules/solvers/ode.html
+++ /dev/null
@@ -1,1710 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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-             accesskey="N">next</a> |</li>
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-
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-      <div class="documentwrapper">
-        <div class="bodywrapper">
-          <div class="body">
-            
-  <div class="section" id="ode">
-<span id="ode-docs"></span><h1>ODE<a class="headerlink" href="#ode" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="user-functions">
-<h2>User Functions<a class="headerlink" href="#user-functions" title="Permalink to this headline">¶</a></h2>
-<p>These are functions that are imported into the global namespace with <tt class="docutils literal"><span class="pre">from</span> <span class="pre">sympy</span> <span class="pre">import</span> <span class="pre">*</span></tt>.  They are intended for user use.</p>
-<div class="section" id="dsolve">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">dsolve()</span></tt><a class="headerlink" href="#dsolve" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.solvers.ode.dsolve">
-<tt class="descclassname">sympy.solvers.ode.</tt><tt class="descname">dsolve</tt><big>(</big><em>eq</em>, <em>func=None</em>, <em>hint='default'</em>, <em>simplify=True</em>, <em>prep=True</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/ode.html#dsolve"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.ode.dsolve" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solves any (supported) kind of ordinary differential equation.</p>
-<p><strong>Usage</strong></p>
-<blockquote>
-<div>dsolve(eq, f(x), hint) -&gt; Solve ordinary differential equation
-eq for function f(x), using method hint.</div></blockquote>
-<p><strong>Details</strong></p>
-<blockquote>
-<div><dl class="docutils">
-<dt><tt class="docutils literal"><span class="pre">eq</span></tt> can be any supported ordinary differential equation (see</dt>
-<dd>the ode docstring for supported methods).  This can either
-be an Equality, or an expression, which is assumed to be
-equal to 0.</dd>
-<dt><tt class="docutils literal"><span class="pre">f(x)</span></tt> is a function of one variable whose derivatives in that</dt>
-<dd>variable make up the ordinary differential equation eq. In many
-cases it is not necessary to provide this; it will be autodetected
-(and an error raised if it couldn&#8217;t be detected).</dd>
-<dt><tt class="docutils literal"><span class="pre">hint</span></tt> is the solving method that you want dsolve to use.  Use</dt>
-<dd>classify_ode(eq, f(x)) to get all of the possible hints for
-an ODE.  The default hint, &#8216;default&#8217;, will use whatever hint
-is returned first by classify_ode().  See Hints below for
-more options that you can use for hint.</dd>
-<dt><tt class="docutils literal"><span class="pre">simplify</span></tt> enables simplification by odesimp().  See its</dt>
-<dd>docstring for more information.  Turn this off, for example,
-to disable solving of solutions for func or simplification
-of arbitrary constants.  It will still integrate with this
-hint. Note that the solution may contain more arbitrary
-constants than the order of the ODE with this option
-enabled.</dd>
-<dt><tt class="docutils literal"><span class="pre">prep</span></tt>, when False and when <tt class="docutils literal"><span class="pre">func</span></tt> is given, will skip the</dt>
-<dd>preprocessing step where the equation is cleaned up so it
-is ready for solving.</dd>
-</dl>
-</div></blockquote>
-<p><strong>Hints</strong></p>
-<blockquote>
-<div><p>Aside from the various solving methods, there are also some
-meta-hints that you can pass to dsolve():</p>
-<dl class="docutils">
-<dt>&#8220;default&#8221;:</dt>
-<dd>This uses whatever hint is returned first by
-classify_ode(). This is the default argument to
-dsolve().</dd>
-<dt>&#8220;all&#8221;:</dt>
-<dd><p class="first">To make dsolve apply all relevant classification hints,
-use dsolve(ODE, func, hint=&#8221;all&#8221;).  This will return a
-dictionary of hint:solution terms.  If a hint causes
-dsolve to raise the NotImplementedError, value of that
-hint&#8217;s key will be the exception object raised.  The
-dictionary will also include some special keys:</p>
-<ul class="last simple">
-<li>order: The order of the ODE.  See also ode_order().</li>
-<li>best: The simplest hint; what would be returned by
-&#8220;best&#8221; below.</li>
-<li>best_hint: The hint that would produce the solution
-given by &#8216;best&#8217;.  If more than one hint produces the
-best solution, the first one in the tuple returned by
-classify_ode() is chosen.</li>
-<li>default: The solution that would be returned by
-default.  This is the one produced by the hint that
-appears first in the tuple returned by classify_ode().</li>
-</ul>
-</dd>
-<dt>&#8220;all_Integral&#8221;:</dt>
-<dd>This is the same as &#8220;all&#8221;, except if a hint also has a
-corresponding &#8220;_Integral&#8221; hint, it only returns the
-&#8220;_Integral&#8221; hint.  This is useful if &#8220;all&#8221; causes
-dsolve() to hang because of a difficult or impossible
-integral.  This meta-hint will also be much faster than
-&#8220;all&#8221;, because integrate() is an expensive routine.</dd>
-<dt>&#8220;best&#8221;:</dt>
-<dd>To have dsolve() try all methods and return the simplest
-one.  This takes into account whether the solution is
-solvable in the function, whether it contains any
-Integral classes (i.e. unevaluatable integrals), and
-which one is the shortest in size.</dd>
-</dl>
-<p>See also the classify_ode() docstring for more info on hints,
-and the ode docstring for a list of all supported hints.</p>
-</div></blockquote>
-<dl class="docutils">
-<dt><strong>Tips</strong></dt>
-<dd><ul class="first last">
-<li><dl class="first docutils">
-<dt>You can declare the derivative of an unknown function this way:</dt>
-<dd><div class="first last highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Derivative</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span> <span class="c"># x is the independent variable</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&quot;f&quot;</span><span class="p">)(</span><span class="n">x</span><span class="p">)</span> <span class="c"># f is a function of x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># f_ will be the derivative of f with respect to x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f_</span> <span class="o">=</span> <span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-</pre></div>
-</div>
-</dd>
-</dl>
-</li>
-<li><p class="first">See test_ode.py for many tests, which serves also as a set of
-examples for how to use dsolve().</p>
-</li>
-<li><p class="first">dsolve always returns an Equality class (except for the case
-when the hint is &#8220;all&#8221; or &#8220;all_Integral&#8221;).  If possible, it
-solves the solution explicitly for the function being solved
-for. Otherwise, it returns an implicit solution.</p>
-</li>
-<li><p class="first">Arbitrary constants are symbols named C1, C2, and so on.</p>
-</li>
-<li><p class="first">Because all solutions should be mathematically equivalent,
-some hints may return the exact same result for an ODE. Often,
-though, two different hints will return the same solution
-formatted differently.  The two should be equivalent. Also
-note that sometimes the values of the arbitrary constants in
-two different solutions may not be the same, because one
-constant may have &#8220;absorbed&#8221; other constants into it.</p>
-</li>
-<li><p class="first">Do help(ode.ode_hintname) to get help more information on a
-specific hint, where hintname is the name of a hint without
-&#8220;_Integral&#8221;.</p>
-</li>
-</ul>
-</dd>
-</dl>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">dsolve</span><span class="p">,</span> <span class="n">Eq</span><span class="p">,</span> <span class="n">Derivative</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dsolve</span><span class="p">(</span><span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="n">x</span><span class="p">,</span><span class="n">x</span><span class="p">)</span><span class="o">+</span><span class="mi">9</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">f(x) == C1*sin(3*x) + C2*cos(3*x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dsolve</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="o">+</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span>
-<span class="gp">... </span>    <span class="n">hint</span><span class="o">=</span><span class="s">&#39;separable&#39;</span><span class="p">,</span> <span class="n">simplify</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">-log(sin(f(x))**2 - 1)/2 == C1 + log(sin(x)**2 - 1)/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dsolve</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="o">+</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span>
-<span class="gp">... </span>    <span class="n">hint</span><span class="o">=</span><span class="s">&#39;1st_exact&#39;</span><span class="p">)</span>
-<span class="go">f(x) == acos(C1/cos(x))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dsolve</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="o">+</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span>
-<span class="gp">... </span><span class="n">hint</span><span class="o">=</span><span class="s">&#39;best&#39;</span><span class="p">)</span>
-<span class="go">f(x) == acos(C1/cos(x))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># Note that even though separable is the default, 1st_exact produces</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># a simpler result in this case.</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="classify-ode">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">classify_ode()</span></tt><a class="headerlink" href="#classify-ode" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.solvers.ode.classify_ode">
-<tt class="descclassname">sympy.solvers.ode.</tt><tt class="descname">classify_ode</tt><big>(</big><em>eq</em>, <em>func=None</em>, <em>dict=False</em>, <em>prep=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/ode.html#classify_ode"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.ode.classify_ode" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a tuple of possible dsolve() classifications for an ODE.</p>
-<p>The tuple is ordered so that first item is the classification that
-dsolve() uses to solve the ODE by default.  In general,
-classifications at the near the beginning of the list will produce
-better solutions faster than those near the end, thought there are
-always exceptions.  To make dsolve use a different classification,
-use dsolve(ODE, func, hint=&lt;classification&gt;).  See also the dsolve()
-docstring for different meta-hints you can use.</p>
-<p>If <tt class="docutils literal"><span class="pre">dict</span></tt> is true, classify_ode() will return a dictionary of
-hint:match expression terms. This is intended for internal use by
-dsolve().  Note that because dictionaries are ordered arbitrarily,
-this will most likely not be in the same order as the tuple.</p>
-<p>If <tt class="docutils literal"><span class="pre">prep</span></tt> is False or <tt class="docutils literal"><span class="pre">func</span></tt> is None then the equation
-will be preprocessed to put it in standard form for classification.</p>
-<p>You can get help on different hints by doing help(ode.ode_hintname),
-where hintname is the name of the hint without &#8220;_Integral&#8221;.</p>
-<p>See sympy.ode.allhints or the sympy.ode docstring for a list of all
-supported hints that can be returned from classify_ode.</p>
-<p class="rubric">Notes</p>
-<p>These are remarks on hint names.</p>
-<p><em>&#8220;_Integral&#8221;</em></p>
-<blockquote>
-<div><p>If a classification has &#8220;_Integral&#8221; at the end, it will return
-the expression with an unevaluated Integral class in it.  Note
-that a hint may do this anyway if integrate() cannot do the
-integral, though just using an &#8220;_Integral&#8221; will do so much
-faster.  Indeed, an &#8220;_Integral&#8221; hint will always be faster than
-its corresponding hint without &#8220;_Integral&#8221; because integrate()
-is an expensive routine.  If dsolve() hangs, it is probably
-because integrate() is hanging on a tough or impossible
-integral.  Try using an &#8220;_Integral&#8221; hint or &#8220;all_Integral&#8221; to
-get it return something.</p>
-<p>Note that some hints do not have &#8220;_Integral&#8221; counterparts.  This
-is because integrate() is not used in solving the ODE for those
-method. For example, nth order linear homogeneous ODEs with
-constant coefficients do not require integration to solve, so
-there is no &#8220;nth_linear_homogeneous_constant_coeff_Integrate&#8221;
-hint. You can easily evaluate any unevaluated Integrals in an
-expression by doing expr.doit().</p>
-</div></blockquote>
-<p><em>Ordinals</em></p>
-<blockquote>
-<div>Some hints contain an ordinal such as &#8220;1st_linear&#8221;.  This is to
-help differentiate them from other hints, as well as from other
-methods that may not be implemented yet. If a hint has &#8220;nth&#8221; in
-it, such as the &#8220;nth_linear&#8221; hints, this means that the method
-used to applies to ODEs of any order.</div></blockquote>
-<p><em>&#8220;indep&#8221; and &#8220;dep&#8221;</em></p>
-<blockquote>
-<div>Some hints contain the words &#8220;indep&#8221; or &#8220;dep&#8221;.  These reference
-the independent variable and the dependent function,
-respectively. For example, if an ODE is in terms of f(x), then
-&#8220;indep&#8221; will refer to x and &#8220;dep&#8221; will refer to f.</div></blockquote>
-<p><em>&#8220;subs&#8221;</em></p>
-<blockquote>
-<div>If a hints has the word &#8220;subs&#8221; in it, it means the the ODE is
-solved by substituting the expression given after the word
-&#8220;subs&#8221; for a single dummy variable.  This is usually in terms of
-&#8220;indep&#8221; and &#8220;dep&#8221; as above.  The substituted expression will be
-written only in characters allowed for names of Python objects,
-meaning operators will be spelled out.  For example, indep/dep
-will be written as indep_div_dep.</div></blockquote>
-<p><em>&#8220;coeff&#8221;</em></p>
-<blockquote>
-<div>The word &#8220;coeff&#8221; in a hint refers to the coefficients of
-something in the ODE, usually of the derivative terms.  See the
-docstring for the individual methods for more info (help(ode)).
-This is contrast to &#8220;coefficients&#8221;, as in
-&#8220;undetermined_coefficients&#8221;, which refers to the common name of
-a method.</div></blockquote>
-<p><em>&#8220;_best&#8221;</em></p>
-<blockquote>
-<div>Methods that have more than one fundamental way to solve will
-have a hint for each sub-method and a &#8220;_best&#8221;
-meta-classification. This will evaluate all hints and return the
-best, using the same considerations as the normal &#8220;best&#8221;
-meta-hint.</div></blockquote>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">classify_ode</span><span class="p">,</span> <span class="n">Eq</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">classify_ode</span><span class="p">(</span><span class="n">Eq</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">(&#39;separable&#39;, &#39;1st_linear&#39;, &#39;1st_homogeneous_coeff_best&#39;,</span>
-<span class="go">&#39;1st_homogeneous_coeff_subs_indep_div_dep&#39;,</span>
-<span class="go">&#39;1st_homogeneous_coeff_subs_dep_div_indep&#39;,</span>
-<span class="go">&#39;nth_linear_constant_coeff_homogeneous&#39;, &#39;separable_Integral&#39;,</span>
-<span class="go">&#39;1st_linear_Integral&#39;,</span>
-<span class="go">&#39;1st_homogeneous_coeff_subs_indep_div_dep_Integral&#39;,</span>
-<span class="go">&#39;1st_homogeneous_coeff_subs_dep_div_indep_Integral&#39;)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">classify_ode</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">(&#39;nth_linear_constant_coeff_undetermined_coefficients&#39;,</span>
-<span class="go">&#39;nth_linear_constant_coeff_variation_of_parameters&#39;,</span>
-<span class="go">&#39;nth_linear_constant_coeff_variation_of_parameters_Integral&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="ode-order">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">ode_order()</span></tt><a class="headerlink" href="#ode-order" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.solvers.ode.ode_order">
-<tt class="descclassname">sympy.solvers.ode.</tt><tt class="descname">ode_order</tt><big>(</big><em>expr</em>, <em>func</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/ode.html#ode_order"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.ode.ode_order" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the order of a given ODE with respect to func.</p>
-<p>This function is implemented recursively.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">ode_order</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Function</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;f&#39;</span><span class="p">,</span> <span class="s">&#39;g&#39;</span><span class="p">]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ode_order</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span>
-<span class="gp">... </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ode_order</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">g</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ode_order</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">g</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">g</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="checkodesol">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">checkodesol()</span></tt><a class="headerlink" href="#checkodesol" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.solvers.ode.checkodesol">
-<tt class="descclassname">sympy.solvers.ode.</tt><tt class="descname">checkodesol</tt><big>(</big><em>ode</em>, <em>sol</em>, <em>func=None</em>, <em>order='auto'</em>, <em>solve_for_func=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/ode.html#checkodesol"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.ode.checkodesol" title="Permalink to this definition">¶</a></dt>
-<dd><p>Substitutes sol into the ode and checks that the result is 0.</p>
-<p>This only works when func is one function, like f(x).  sol can be a
-single solution or a list of solutions.  Each solution may be an Equality
-that the solution satisfies, e.g. Eq(f(x), C1), Eq(f(x) + C1, 0); or simply
-an Expr, e.g. f(x) - C1. In most cases it will not be necessary to
-explicitly identify the function, but if the function cannot be inferred
-from the original equation it can be supplied through the &#8216;func&#8217; argument.</p>
-<p>If a sequence of solutions is passed, the same sort of container will be used
-to return the result for each solution.</p>
-<p>It tries the following methods, in order, until it finds zero
-equivalence:</p>
-<blockquote>
-<div><ol class="arabic simple">
-<li>Substitute the solution for f in the original equation.  This
-only works if the ode is solved for f.  It will attempt to solve
-it first unless solve_for_func == False</li>
-<li>Take n derivatives of the solution, where n is the order of
-ode, and check to see if that is equal to the solution.  This
-only works on exact odes.</li>
-<li>Take the 1st, 2nd, ..., nth derivatives of the solution, each
-time solving for the derivative of f of that order (this will
-always be possible because f is a linear operator).  Then back
-substitute each derivative into ode in reverse order.</li>
-</ol>
-</div></blockquote>
-<p>This function returns a tuple.  The first item in the tuple is True
-if the substitution results in 0, and False otherwise. The second
-item in the tuple is what the substitution results in.  It should
-always be 0 if the first item is True. Note that sometimes this
-function will False, but with an expression that is identically
-equal to 0, instead of returning True.  This is because simplify()
-cannot reduce the expression to 0.  If an expression returned by
-this function vanishes identically, then sol really is a solution to
-ode.</p>
-<p>If this function seems to hang, it is probably because of a hard
-simplification.</p>
-<p>To use this function to test, test the first item of the tuple.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Eq</span><span class="p">,</span> <span class="n">Function</span><span class="p">,</span> <span class="n">checkodesol</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">C1</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,C1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">checkodesol</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">Eq</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">C1</span><span class="p">))</span>
-<span class="go">(True, 0)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">assert</span> <span class="n">checkodesol</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">C1</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">assert</span> <span class="ow">not</span> <span class="n">checkodesol</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">checkodesol</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(False, 2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="homogeneous-order">
-<h3><tt class="xref py py-func docutils literal"><span class="pre">homogeneous_order()</span></tt><a class="headerlink" href="#homogeneous-order" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.solvers.ode.homogeneous_order">
-<tt class="descclassname">sympy.solvers.ode.</tt><tt class="descname">homogeneous_order</tt><big>(</big><em>eq</em>, <em>*symbols</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/ode.html#homogeneous_order"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.ode.homogeneous_order" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the order n if g is homogeneous and None if it is not
-homogeneous.</p>
-<p>Determines if a function is homogeneous and if so of what order.
-A function f(x,y,...) is homogeneous of order n if
-f(t*x,t*y,t*...) == t**n*f(x,y,...).</p>
-<p>If the function is of two variables, F(x, y), then f being
-homogeneous of any order is equivalent to being able to rewrite
-F(x, y) as G(x/y) or H(y/x).  This fact is used to solve 1st order
-ordinary differential equations whose coefficients are homogeneous
-of the same order (see the docstrings of
-ode.ode_1st_homogeneous_coeff_subs_indep_div_dep() and
-ode.ode_1st_homogeneous_coeff_subs_indep_div_dep()</p>
-<p>Symbols can be functions, but every argument of the function must be
-a symbol, and the arguments of the function that appear in the
-expression must match those given in the list of symbols.  If a
-declared function appears with different arguments than given in the
-list of symbols, None is returned.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">homogeneous_order</span><span class="p">,</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">homogeneous_order</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="ow">is</span> <span class="bp">None</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">homogeneous_order</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">None</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">homogeneous_order</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">homogeneous_order</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">homogeneous_order</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="ow">is</span> <span class="bp">None</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="hint-methods">
-<h2>Hint Methods<a class="headerlink" href="#hint-methods" title="Permalink to this headline">¶</a></h2>
-<p>These functions are intended for internal use by <tt class="xref py py-func docutils literal"><span class="pre">dsolve()</span></tt> and others.  Nonetheless, they contain useful information in their docstrings on the various ODE solving methods.</p>
-<div class="section" id="preprocess">
-<h3><tt class="xref py py-obj docutils literal"><span class="pre">preprocess</span></tt><a class="headerlink" href="#preprocess" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.solvers.ode.preprocess">
-<tt class="descclassname">sympy.solvers.ode.</tt><tt class="descname">preprocess</tt><big>(</big><em>expr</em>, <em>func=None</em>, <em>hint='_Integral'</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/ode.html#preprocess"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.ode.preprocess" title="Permalink to this definition">¶</a></dt>
-<dd><p>Prepare expr for solving by making sure that differentiation
-is done so that only func remains in unevaluated derivatives and
-(if hint doesn&#8217;t end with _Integral) that doit is applied to all
-other derivatives. If hint is None, don&#8217;t do any differentiation.
-(Currently this may cause some simple differential equations to
-fail.)</p>
-<p>In case func is None, an attempt will be made to autodetect the
-function to be solved for.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.solvers.ode</span> <span class="kn">import</span> <span class="n">preprocess</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Derivative</span><span class="p">,</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">,</span> <span class="n">g</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Function</span><span class="p">,</span> <span class="s">&#39;fg&#39;</span><span class="p">))</span>
-</pre></div>
-</div>
-<p>Apply doit to derivatives that contain more than the function
-of interest:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">preprocess</span><span class="p">(</span><span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">(Derivative(f(x), x) + 1, f(x))</span>
-</pre></div>
-</div>
-<p>Do others if the differentiation variable(s) intersect with those
-of the function of interest or contain the function of interest:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">preprocess</span><span class="p">(</span><span class="n">Derivative</span><span class="p">(</span><span class="n">g</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>
-<span class="go">(0, f(y))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">preprocess</span><span class="p">(</span><span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">y</span><span class="p">),</span> <span class="n">z</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>
-<span class="go">(0, f(y))</span>
-</pre></div>
-</div>
-<p>Do others if the hint doesn&#8217;t end in &#8216;_Integral&#8217; (the default
-assumes that it does):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">preprocess</span><span class="p">(</span><span class="n">Derivative</span><span class="p">(</span><span class="n">g</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">y</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">(Derivative(g(x), y), f(x))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">preprocess</span><span class="p">(</span><span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">y</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">hint</span><span class="o">=</span><span class="s">&#39;&#39;</span><span class="p">)</span>
-<span class="go">(0, f(x))</span>
-</pre></div>
-</div>
-<p>Don&#8217;t do any derivatives if hint is None:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">preprocess</span><span class="p">(</span><span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">y</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">hint</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span>
-<span class="go">(Derivative(f(x) + 1, x) + Derivative(f(x), y), f(x))</span>
-</pre></div>
-</div>
-<p>If it&#8217;s not clear what the function of interest is, it must be given:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span> <span class="o">=</span> <span class="n">Derivative</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">g</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">preprocess</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">g</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">(Derivative(f(x), x) + Derivative(g(x), x), g(x))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">try</span><span class="p">:</span> <span class="n">preprocess</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span>
-<span class="gp">... </span><span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span> <span class="k">print</span><span class="p">(</span><span class="s">&quot;A ValueError was raised.&quot;</span><span class="p">)</span>
-<span class="go">A ValueError was raised.</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="odesimp">
-<h3><tt class="xref py py-obj docutils literal"><span class="pre">odesimp</span></tt><a class="headerlink" href="#odesimp" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.solvers.ode.odesimp">
-<tt class="descclassname">sympy.solvers.ode.</tt><tt class="descname">odesimp</tt><big>(</big><em>*args</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/ode.html#odesimp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.ode.odesimp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Simplifies ODEs, including trying to solve for func and running
-constantsimp().</p>
-<p>It may use knowledge of the type of solution that that hint returns
-to apply additional simplifications.</p>
-<p>It also attempts to integrate any Integrals in the expression, if
-the hint is not an &#8220;_Integral&#8221; hint.</p>
-<p>This function should have no effect on expressions returned by
-dsolve(), as dsolve already calls odesimp(), but the individual hint
-functions do not call odesimp (because the dsolve() wrapper does).
-Therefore, this function is designed for mainly internal use.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">dsolve</span><span class="p">,</span> <span class="n">pprint</span><span class="p">,</span> <span class="n">Function</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.solvers.ode</span> <span class="kn">import</span> <span class="n">odesimp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="p">,</span> <span class="n">u2</span><span class="p">,</span> <span class="n">C1</span><span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,u2,C1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span> <span class="o">=</span> <span class="n">dsolve</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">x</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span>
-<span class="gp">... </span><span class="n">hint</span><span class="o">=</span><span class="s">&#39;1st_homogeneous_coeff_subs_indep_div_dep_Integral&#39;</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">simplify</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">eq</span><span class="p">)</span>
-<span class="go">              x</span>
-<span class="go">                 ----</span>
-<span class="go">                 f(x)</span>
-<span class="go">                   /</span>
-<span class="go">                  |</span>
-<span class="go">       /f(x)\     |  /  1         1     \</span>
-<span class="go">    log|----| -   |  |- -- - -----------| d(u2) = 0</span>
-<span class="go">       \ C1 /     |  |  u2     2    /1 \|</span>
-<span class="go">                  |  |       u2 *sin|--||</span>
-<span class="go">                  |  \              \u2//</span>
-<span class="go">                  |</span>
-<span class="go">                 /</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">odesimp</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">hint</span><span class="o">=</span><span class="s">&#39;1st_homogeneous_coeff_subs_indep_div_dep&#39;</span>
-<span class="gp">... </span><span class="p">))</span> 
-<span class="go">    x</span>
-<span class="go">--------- = C1</span>
-<span class="go">   /f(x)\</span>
-<span class="go">tan|----|</span>
-<span class="go">   \2*x /</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="constant-renumber">
-<h3><tt class="xref py py-obj docutils literal"><span class="pre">constant_renumber</span></tt><a class="headerlink" href="#constant-renumber" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.solvers.ode.constant_renumber">
-<tt class="descclassname">sympy.solvers.ode.</tt><tt class="descname">constant_renumber</tt><big>(</big><em>expr</em>, <em>symbolname</em>, <em>startnumber</em>, <em>endnumber</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/ode.html#constant_renumber"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.ode.constant_renumber" title="Permalink to this definition">¶</a></dt>
-<dd><p>Renumber arbitrary constants in expr to have numbers 1 through N
-where N is <tt class="docutils literal"><span class="pre">endnumber</span></tt> - <tt class="docutils literal"><span class="pre">startnumber</span></tt> + 1 at most.</p>
-<p>This is a simple function that goes through and renumbers any Symbol
-with a name in the form symbolname + num where num is in the range
-from startnumber to endnumber.</p>
-<p>Symbols are renumbered based on <tt class="docutils literal"><span class="pre">.sort_key()</span></tt>, so they should be
-numbered roughly in the order that they appear in the final, printed
-expression.  Note that this ordering is based in part on hashes, so
-it can produce different results on different machines.</p>
-<p>The structure of this function is very similar to that of
-constantsimp().</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">Eq</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.solvers.ode</span> <span class="kn">import</span> <span class="n">constant_renumber</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">C0</span><span class="p">,</span> <span class="n">C1</span><span class="p">,</span> <span class="n">C2</span><span class="p">,</span> <span class="n">C3</span><span class="p">,</span> <span class="n">C4</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,C:5&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Only constants in the given range (inclusive) are renumbered;
-the renumbering always starts from 1:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">constant_renumber</span><span class="p">(</span><span class="n">C1</span> <span class="o">+</span> <span class="n">C3</span> <span class="o">+</span> <span class="n">C4</span><span class="p">,</span> <span class="s">&#39;C&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">C1 + C2 + C4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">constant_renumber</span><span class="p">(</span><span class="n">C0</span> <span class="o">+</span> <span class="n">C1</span> <span class="o">+</span> <span class="n">C3</span> <span class="o">+</span> <span class="n">C4</span><span class="p">,</span> <span class="s">&#39;C&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">C0 + 2*C1 + C2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">constant_renumber</span><span class="p">(</span><span class="n">C0</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">C1</span> <span class="o">+</span> <span class="n">C2</span><span class="p">,</span> <span class="s">&#39;C&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">C1 + 3*C2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">C2</span> <span class="o">+</span> <span class="n">C1</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">C3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">                2</span>
-<span class="go">C1*x + C2 + C3*x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">constant_renumber</span><span class="p">(</span><span class="n">C2</span> <span class="o">+</span> <span class="n">C1</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">C3</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="s">&#39;C&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
-<span class="go">                2</span>
-<span class="go">C1 + C2*x + C3*x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="constantsimp">
-<h3><tt class="xref py py-obj docutils literal"><span class="pre">constantsimp</span></tt><a class="headerlink" href="#constantsimp" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.solvers.ode.constantsimp">
-<tt class="descclassname">sympy.solvers.ode.</tt><tt class="descname">constantsimp</tt><big>(</big><em>*args</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/ode.html#constantsimp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.ode.constantsimp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Simplifies an expression with arbitrary constants in it.</p>
-<p>This function is written specifically to work with dsolve(), and is
-not intended for general use.</p>
-<p>Simplification is done by &#8220;absorbing&#8221; the arbitrary constants in to
-other arbitrary constants, numbers, and symbols that they are not
-independent of.</p>
-<p>The symbols must all have the same name with numbers after it, for
-example, C1, C2, C3.  The symbolname here would be &#8216;C&#8217;, the
-startnumber would be 1, and the end number would be 3.  If the
-arbitrary constants are independent of the variable x, then the
-independent symbol would be x.  There is no need to specify the
-dependent function, such as f(x), because it already has the
-independent symbol, x, in it.</p>
-<p>Because terms are &#8220;absorbed&#8221; into arbitrary constants and because
-constants are renumbered after simplifying, the arbitrary constants
-in expr are not necessarily equal to the ones of the same name in
-the returned result.</p>
-<p>If two or more arbitrary constants are added, multiplied, or raised
-to the power of each other, they are first absorbed together into a
-single arbitrary constant.  Then the new constant is combined into
-other terms if necessary.</p>
-<p>Absorption is done with limited assistance: terms of Adds are collected
-to try join constants and powers with exponents that are Adds are expanded
-so (C1*cos(x) + C2*cos(x))*exp(x) will simplify to C1*cos(x)*exp(x) and
-exp(C1 + x) will be simplified to C1*exp(x).</p>
-<p>Use constant_renumber() to renumber constants after simplification or else
-arbitrary numbers on constants may appear, e.g. C1 + C3*x.</p>
-<p>In rare cases, a single constant can be &#8220;simplified&#8221; into two
-constants.  Every differential equation solution should have as many
-arbitrary constants as the order of the differential equation.  The
-result here will be technically correct, but it may, for example,
-have C1 and C2 in an expression, when C1 is actually equal to C2.
-Use your discretion in such situations, and also take advantage of
-the ability to use hints in dsolve().</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.solvers.ode</span> <span class="kn">import</span> <span class="n">constantsimp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C1</span><span class="p">,</span> <span class="n">C2</span><span class="p">,</span> <span class="n">C3</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;C1,C2,C3,x,y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">constantsimp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">C1</span><span class="o">*</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">C1*x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">constantsimp</span><span class="p">(</span><span class="n">C1</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">C1 + x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">constantsimp</span><span class="p">(</span><span class="n">C1</span><span class="o">*</span><span class="n">C2</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">C3</span><span class="o">*</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">C1 + C3*x</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="sol-simplicity">
-<h3><tt class="xref py py-obj docutils literal"><span class="pre">sol_simplicity</span></tt><a class="headerlink" href="#sol-simplicity" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.solvers.ode.ode_sol_simplicity">
-<tt class="descclassname">sympy.solvers.ode.</tt><tt class="descname">ode_sol_simplicity</tt><big>(</big><em>sol</em>, <em>func</em>, <em>trysolving=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/ode.html#ode_sol_simplicity"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.ode.ode_sol_simplicity" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns an extended integer representing how simple a solution to an
-ODE is.</p>
-<p>The following things are considered, in order from most simple to
-least:
-- sol is solved for func.
-- sol is not solved for func, but can be if passed to solve (e.g.,
-a solution returned by dsolve(ode, func, simplify=False)
-- If sol is not solved for func, then base the result on the length
-of sol, as computed by len(str(sol)).
-- If sol has any unevaluated Integrals, this will automatically be
-considered less simple than any of the above.</p>
-<p>This function returns an integer such that if solution A is simpler
-than solution B by above metric, then ode_sol_simplicity(sola, func)
-&lt; ode_sol_simplicity(solb, func).</p>
-<p>Currently, the following are the numbers returned, but if the
-heuristic is ever improved, this may change.  Only the ordering is
-guaranteed.</p>
-<p>sol solved for func                        -2
-sol not solved for func but can be         -1
-sol is not solved or solvable for func     len(str(sol))
-sol contains an Integral                   oo</p>
-<p>oo here means the SymPy infinity, which should compare greater than
-any integer.</p>
-<p>If you already know solve() cannot solve sol, you can use
-trysolving=False to skip that step, which is the only potentially
-slow step.  For example, dsolve with the simplify=False flag should
-do this.</p>
-<p>If sol is a list of solutions, if the worst solution in the list
-returns oo it returns that, otherwise it returns len(str(sol)), that
-is, the length of the string representation of the whole list.</p>
-<p class="rubric">Examples</p>
-<p>This function is designed to be passed to min as the key argument,
-such as min(listofsolutions, key=lambda i: ode_sol_simplicity(i, f(x))).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Eq</span><span class="p">,</span> <span class="n">tan</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.solvers.ode</span> <span class="kn">import</span> <span class="n">ode_sol_simplicity</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">C1</span><span class="p">,</span> <span class="n">C2</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x, C1, C2&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ode_sol_simplicity</span><span class="p">(</span><span class="n">Eq</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">C1</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">-2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ode_sol_simplicity</span><span class="p">(</span><span class="n">Eq</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">C1</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">-1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ode_sol_simplicity</span><span class="p">(</span><span class="n">Eq</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">C1</span><span class="o">*</span><span class="n">Integral</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eq1</span> <span class="o">=</span> <span class="n">Eq</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">tan</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)),</span> <span class="n">C1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eq2</span> <span class="o">=</span> <span class="n">Eq</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">tan</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="n">C2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">ode_sol_simplicity</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="k">for</span> <span class="n">eq</span> <span class="ow">in</span> <span class="p">[</span><span class="n">eq1</span><span class="p">,</span> <span class="n">eq2</span><span class="p">]]</span>
-<span class="go">[26, 33]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">min</span><span class="p">([</span><span class="n">eq1</span><span class="p">,</span> <span class="n">eq2</span><span class="p">],</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">i</span><span class="p">:</span> <span class="n">ode_sol_simplicity</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
-<span class="go">f(x)/tan(f(x)/(2*x)) == C1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="st-exact">
-<h3><tt class="xref py py-obj docutils literal"><span class="pre">1st_exact</span></tt><a class="headerlink" href="#st-exact" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.solvers.ode.ode_1st_exact">
-<tt class="descclassname">sympy.solvers.ode.</tt><tt class="descname">ode_1st_exact</tt><big>(</big><em>eq</em>, <em>func</em>, <em>order</em>, <em>match</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/ode.html#ode_1st_exact"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.ode.ode_1st_exact" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solves 1st order exact ordinary differential equations.</p>
-<p>A 1st order differential equation is called exact if it is the total
-differential of a function. That is, the differential equation
-P(x, y)dx + Q(x, y)dy = 0 is exact if there is some function F(x, y)
-such that P(x, y) = dF/dx and Q(x, y) = dF/dy (d here refers to the
-partial derivative).  It can be shown that a necessary and
-sufficient condition for a first order ODE to be exact is that
-dP/dy = dQ/dx.  Then, the solution will be as given below:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Eq</span><span class="p">,</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">y0</span><span class="p">,</span> <span class="n">C1</span><span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y,t,x0,y0,C1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="p">,</span> <span class="n">Q</span><span class="p">,</span> <span class="n">F</span><span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Function</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;P&#39;</span><span class="p">,</span> <span class="s">&#39;Q&#39;</span><span class="p">,</span> <span class="s">&#39;F&#39;</span><span class="p">]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">Eq</span><span class="p">(</span><span class="n">Eq</span><span class="p">(</span><span class="n">F</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="n">Integral</span><span class="p">(</span><span class="n">P</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span> <span class="o">+</span>
-<span class="gp">... </span><span class="n">Integral</span><span class="p">(</span><span class="n">Q</span><span class="p">(</span><span class="n">x0</span><span class="p">,</span> <span class="n">t</span><span class="p">),</span> <span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">y0</span><span class="p">,</span> <span class="n">y</span><span class="p">))),</span> <span class="n">C1</span><span class="p">))</span>
-<span class="go">            x                y</span>
-<span class="go">            /                /</span>
-<span class="go">           |                |</span>
-<span class="go">F(x, y) =  |  P(t, y) dt +  |  Q(x0, t) dt = C1</span>
-<span class="go">           |                |</span>
-<span class="go">          /                /</span>
-<span class="go">          x0               y0</span>
-</pre></div>
-</div>
-<p>Where the first partials of P and Q exist and are continuous in a
-simply connected region.</p>
-<p>A note: SymPy currently has no way to represent inert substitution on
-an expression, so the hint &#8216;1st_exact_Integral&#8217; will return an integral
-with dy.  This is supposed to represent the function that you are
-solving for.</p>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Exact_differential_equation">http://en.wikipedia.org/wiki/Exact_differential_equation</a></li>
-<li>M. Tenenbaum &amp; H. Pollard, &#8220;Ordinary Differential Equations&#8221;,
-Dover 1963, pp. 73</li>
-</ul>
-<p># indirect doctest</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">dsolve</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dsolve</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="o">-</span> <span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="o">-</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">),</span>
-<span class="gp">... </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">hint</span><span class="o">=</span><span class="s">&#39;1st_exact&#39;</span><span class="p">)</span>
-<span class="go">x*cos(f(x)) + f(x)**3/3 == C1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="st-homogeneous-coeff-best">
-<h3><tt class="xref py py-obj docutils literal"><span class="pre">1st_homogeneous_coeff_best</span></tt><a class="headerlink" href="#st-homogeneous-coeff-best" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.solvers.ode.ode_1st_homogeneous_coeff_best">
-<tt class="descclassname">sympy.solvers.ode.</tt><tt class="descname">ode_1st_homogeneous_coeff_best</tt><big>(</big><em>eq</em>, <em>func</em>, <em>order</em>, <em>match</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/ode.html#ode_1st_homogeneous_coeff_best"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.ode.ode_1st_homogeneous_coeff_best" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the best solution to an ODE from the two hints
-&#8216;1st_homogeneous_coeff_subs_dep_div_indep&#8217; and
-&#8216;1st_homogeneous_coeff_subs_indep_div_dep&#8217;.</p>
-<p>This is as determined by ode_sol_simplicity().</p>
-<p>See the ode_1st_homogeneous_coeff_subs_indep_div_dep() and
-ode_1st_homogeneous_coeff_subs_dep_div_indep() docstrings for more
-information on these hints.  Note that there is no
-&#8216;1st_homogeneous_coeff_best_Integral&#8217; hint.</p>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Homogeneous_differential_equation">http://en.wikipedia.org/wiki/Homogeneous_differential_equation</a></li>
-<li>M. Tenenbaum &amp; H. Pollard, &#8220;Ordinary Differential Equations&#8221;,
-Dover 1963, pp. 59</li>
-</ul>
-<p># indirect doctest</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">dsolve</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">dsolve</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span>
-<span class="gp">... </span><span class="n">hint</span><span class="o">=</span><span class="s">&#39;1st_homogeneous_coeff_best&#39;</span><span class="p">,</span> <span class="n">simplify</span><span class="o">=</span><span class="bp">False</span><span class="p">))</span>
-<span class="go">               /    2    \</span>
-<span class="go">               | 3*x     |</span>
-<span class="go">            log|----- + 1|</span>
-<span class="go">               | 2       |</span>
-<span class="go">   /f(x)\      \f (x)    /</span>
-<span class="go">log|----| + -------------- = 0</span>
-<span class="go">   \ C1 /         3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="st-homogeneous-coeff-subs-dep-div-indep">
-<h3><tt class="xref py py-obj docutils literal"><span class="pre">1st_homogeneous_coeff_subs_dep_div_indep</span></tt><a class="headerlink" href="#st-homogeneous-coeff-subs-dep-div-indep" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.solvers.ode.ode_1st_homogeneous_coeff_subs_dep_div_indep">
-<tt class="descclassname">sympy.solvers.ode.</tt><tt class="descname">ode_1st_homogeneous_coeff_subs_dep_div_indep</tt><big>(</big><em>eq</em>, <em>func</em>, <em>order</em>, <em>match</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/ode.html#ode_1st_homogeneous_coeff_subs_dep_div_indep"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.ode.ode_1st_homogeneous_coeff_subs_dep_div_indep" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solves a 1st order differential equation with homogeneous coefficients
-using the substitution
-u1 = &lt;dependent variable&gt;/&lt;independent variable&gt;.</p>
-<p>This is a differential equation P(x, y) + Q(x, y)dy/dx = 0, that P
-and Q are homogeneous of the same order.  A function F(x, y) is
-homogeneous of order n if F(xt, yt) = t**n*F(x, y).  Equivalently,
-F(x, y) can be rewritten as G(y/x) or H(x/y).  See also the
-docstring of homogeneous_order().</p>
-<p>If the coefficients P and Q in the differential equation above are
-homogeneous functions of the same order, then it can be shown that
-the substitution y = u1*x (u1 = y/x) will turn the differential
-equation into an equation separable in the variables x and u.  If
-h(u1) is the function that results from making the substitution
-u1 = f(x)/x on P(x, f(x)) and g(u2) is the function that results
-from the substitution on Q(x, f(x)) in the differential equation
-P(x, f(x)) + Q(x, f(x))*diff(f(x), x) = 0, then the general solution
-is:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">dsolve</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Function</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;f&#39;</span><span class="p">,</span> <span class="s">&#39;g&#39;</span><span class="p">,</span> <span class="s">&#39;h&#39;</span><span class="p">]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">genform</span> <span class="o">=</span> <span class="n">g</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">h</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">genform</span><span class="p">)</span>
-<span class="go"> /f(x)\    /f(x)\ d</span>
-<span class="go">g|----| + h|----|*--(f(x))</span>
-<span class="go"> \ x  /    \ x  / dx</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">dsolve</span><span class="p">(</span><span class="n">genform</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span>
-<span class="gp">... </span><span class="n">hint</span><span class="o">=</span><span class="s">&#39;1st_homogeneous_coeff_subs_dep_div_indep_Integral&#39;</span><span class="p">))</span>
-<span class="go">             f(x)</span>
-<span class="go">             ----</span>
-<span class="go">              x</span>
-<span class="go">               /</span>
-<span class="go">              |</span>
-<span class="go">              |       -h(u1)</span>
-<span class="go">log(C1*x) -   |  ---------------- d(u1) = 0</span>
-<span class="go">              |  u1*h(u1) + g(u1)</span>
-<span class="go">              |</span>
-<span class="go">             /</span>
-</pre></div>
-</div>
-<p>Where u1*h(u1) + g(u1) != 0 and x != 0.</p>
-<p>See also the docstrings of ode_1st_homogeneous_coeff_best() and
-ode_1st_homogeneous_coeff_subs_indep_div_dep().</p>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Homogeneous_differential_equation">http://en.wikipedia.org/wiki/Homogeneous_differential_equation</a></li>
-<li>M. Tenenbaum &amp; H. Pollard, &#8220;Ordinary Differential Equations&#8221;,
-Dover 1963, pp. 59</li>
-</ul>
-<p># indirect doctest</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">dsolve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">dsolve</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span>
-<span class="gp">... </span><span class="n">hint</span><span class="o">=</span><span class="s">&#39;1st_homogeneous_coeff_subs_dep_div_indep&#39;</span><span class="p">,</span> <span class="n">simplify</span><span class="o">=</span><span class="bp">False</span><span class="p">))</span>
-<span class="go">                 /          3   \</span>
-<span class="go">                 |3*f(x)   f (x)|</span>
-<span class="go">              log|------ + -----|</span>
-<span class="go">                 |  x         3 |</span>
-<span class="go">       /x \      \           x  /</span>
-<span class="go">    log|--| + ------------------- = 0</span>
-<span class="go">       \C1/            3</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="st-homogeneous-coeff-subs-indep-div-dep">
-<h3><tt class="xref py py-obj docutils literal"><span class="pre">1st_homogeneous_coeff_subs_indep_div_dep</span></tt><a class="headerlink" href="#st-homogeneous-coeff-subs-indep-div-dep" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.solvers.ode.ode_1st_homogeneous_coeff_subs_indep_div_dep">
-<tt class="descclassname">sympy.solvers.ode.</tt><tt class="descname">ode_1st_homogeneous_coeff_subs_indep_div_dep</tt><big>(</big><em>eq</em>, <em>func</em>, <em>order</em>, <em>match</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/ode.html#ode_1st_homogeneous_coeff_subs_indep_div_dep"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.ode.ode_1st_homogeneous_coeff_subs_indep_div_dep" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solves a 1st order differential equation with homogeneous coefficients
-using the substitution
-u2 = &lt;independent variable&gt;/&lt;dependent variable&gt;.</p>
-<p>This is a differential equation P(x, y) + Q(x, y)dy/dx = 0, that P
-and Q are homogeneous of the same order.  A function F(x, y) is
-homogeneous of order n if F(xt, yt) = t**n*F(x, y).  Equivalently,
-F(x, y) can be rewritten as G(y/x) or H(x/y).  See also the
-docstring of homogeneous_order().</p>
-<p>If the coefficients P and Q in the differential equation above are
-homogeneous functions of the same order, then it can be shown that
-the substitution x = u2*y (u2 = x/y) will turn the differential
-equation into an equation separable in the variables y and u2.  If
-h(u2) is the function that results from making the substitution
-u2 = x/f(x) on P(x, f(x)) and g(u2) is the function that results
-from the substitution on Q(x, f(x)) in the differential equation
-P(x, f(x)) + Q(x, f(x))*diff(f(x), x) = 0, then the general solution
-is:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">dsolve</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Function</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;f&#39;</span><span class="p">,</span> <span class="s">&#39;g&#39;</span><span class="p">,</span> <span class="s">&#39;h&#39;</span><span class="p">]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">genform</span> <span class="o">=</span> <span class="n">g</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span> <span class="o">+</span> <span class="n">h</span><span class="p">(</span><span class="n">x</span><span class="o">/</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">genform</span><span class="p">)</span>
-<span class="go"> / x  \    / x  \ d</span>
-<span class="go">g|----| + h|----|*--(f(x))</span>
-<span class="go"> \f(x)/    \f(x)/ dx</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">dsolve</span><span class="p">(</span><span class="n">genform</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span>
-<span class="gp">... </span><span class="n">hint</span><span class="o">=</span><span class="s">&#39;1st_homogeneous_coeff_subs_indep_div_dep_Integral&#39;</span><span class="p">))</span>
-<span class="go">             x</span>
-<span class="go">            ----</span>
-<span class="go">            f(x)</span>
-<span class="go">              /</span>
-<span class="go">             |</span>
-<span class="go">             |        g(u2)</span>
-<span class="go">             |  ----------------- d(u2)</span>
-<span class="go">             |  -u2*g(u2) - h(u2)</span>
-<span class="go">             |</span>
-<span class="go">            /</span>
-
-<span class="go">f(x) = C1*e</span>
-</pre></div>
-</div>
-<p>Where u2*g(u2) + h(u2) != 0 and f(x) != 0.</p>
-<p>See also the docstrings of ode_1st_homogeneous_coeff_best() and
-ode_1st_homogeneous_coeff_subs_dep_div_indep().</p>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Homogeneous_differential_equation">http://en.wikipedia.org/wiki/Homogeneous_differential_equation</a></li>
-<li>M. Tenenbaum &amp; H. Pollard, &#8220;Ordinary Differential Equations&#8221;,
-Dover 1963, pp. 59</li>
-</ul>
-<p># indirect doctest</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">dsolve</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span>
-<span class="gp">... </span><span class="n">hint</span><span class="o">=</span><span class="s">&#39;1st_homogeneous_coeff_subs_indep_div_dep&#39;</span><span class="p">))</span>
-<span class="go">      ___________</span>
-<span class="go">     /     2</span>
-<span class="go">    /   3*x</span>
-<span class="go">   /   ----- + 1 *f(x) = C1</span>
-<span class="go">3 /     2</span>
-<span class="go">\/     f (x)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="st-linear">
-<h3><tt class="xref py py-obj docutils literal"><span class="pre">1st_linear</span></tt><a class="headerlink" href="#st-linear" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.solvers.ode.ode_1st_linear">
-<tt class="descclassname">sympy.solvers.ode.</tt><tt class="descname">ode_1st_linear</tt><big>(</big><em>eq</em>, <em>func</em>, <em>order</em>, <em>match</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/ode.html#ode_1st_linear"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.ode.ode_1st_linear" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solves 1st order linear differential equations.</p>
-<p>These are differential equations of the form dy/dx _ P(x)*y = Q(x).
-These kinds of differential equations can be solved in a general
-way.  The integrating factor exp(Integral(P(x), x)) will turn the
-equation into a separable equation.  The general solution is:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">dsolve</span><span class="p">,</span> <span class="n">Eq</span><span class="p">,</span> <span class="n">pprint</span><span class="p">,</span> <span class="n">diff</span><span class="p">,</span> <span class="n">sin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">Q</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Function</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;f&#39;</span><span class="p">,</span> <span class="s">&#39;P&#39;</span><span class="p">,</span> <span class="s">&#39;Q&#39;</span><span class="p">]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">genform</span> <span class="o">=</span> <span class="n">Eq</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">P</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">Q</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">genform</span><span class="p">)</span>
-<span class="go">            d</span>
-<span class="go">P(x)*f(x) + --(f(x)) = Q(x)</span>
-<span class="go">            dx</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">dsolve</span><span class="p">(</span><span class="n">genform</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">hint</span><span class="o">=</span><span class="s">&#39;1st_linear_Integral&#39;</span><span class="p">))</span>
-<span class="go">       /       /                   \</span>
-<span class="go">       |      |                    |</span>
-<span class="go">       |      |         /          |     /</span>
-<span class="go">       |      |        |           |    |</span>
-<span class="go">       |      |        | P(x) dx   |  - | P(x) dx</span>
-<span class="go">       |      |        |           |    |</span>
-<span class="go">       |      |       /            |   /</span>
-<span class="go">f(x) = |C1 +  | Q(x)*e           dx|*e</span>
-<span class="go">       |      |                    |</span>
-<span class="go">       \     /                     /</span>
-</pre></div>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Linear_differential_equation#First_order_equation">http://en.wikipedia.org/wiki/Linear_differential_equation#First_order_equation</a></li>
-<li>M. Tenenbaum &amp; H. Pollard, &#8220;Ordinary Differential Equations&#8221;,
-Dover 1963, pp. 92</li>
-</ul>
-<p># indirect doctest</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">dsolve</span><span class="p">(</span><span class="n">Eq</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span>
-<span class="gp">... </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="s">&#39;1st_linear&#39;</span><span class="p">))</span>
-<span class="go">f(x) = x*(C1 - cos(x))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="bernoulli">
-<h3><tt class="xref py py-obj docutils literal"><span class="pre">Bernoulli</span></tt><a class="headerlink" href="#bernoulli" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.solvers.ode.ode_Bernoulli">
-<tt class="descclassname">sympy.solvers.ode.</tt><tt class="descname">ode_Bernoulli</tt><big>(</big><em>eq</em>, <em>func</em>, <em>order</em>, <em>match</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/ode.html#ode_Bernoulli"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.ode.ode_Bernoulli" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solves Bernoulli differential equations.</p>
-<p>These are equations of the form dy/dx + P(x)*y = Q(x)*y**n, n != 1.
-The substitution w = 1/y**(1-n) will transform an equation of this
-form into one that is linear (see the docstring of
-ode_1st_linear()).  The general solution is:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">dsolve</span><span class="p">,</span> <span class="n">Eq</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">Q</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Function</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;f&#39;</span><span class="p">,</span> <span class="s">&#39;P&#39;</span><span class="p">,</span> <span class="s">&#39;Q&#39;</span><span class="p">]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">genform</span> <span class="o">=</span> <span class="n">Eq</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">P</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">Q</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">genform</span><span class="p">)</span>
-<span class="go">            d                n</span>
-<span class="go">P(x)*f(x) + --(f(x)) = Q(x)*f (x)</span>
-<span class="go">            dx</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">dsolve</span><span class="p">(</span><span class="n">genform</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">hint</span><span class="o">=</span><span class="s">&#39;Bernoulli_Integral&#39;</span><span class="p">))</span> 
-<span class="go">                                                                               1</span>
-<span class="go">                                                                              ----</span>
-<span class="go">                                                                             1 - n</span>
-<span class="go">       //                /                            \                     \</span>
-<span class="go">       ||               |                             |                     |</span>
-<span class="go">       ||               |                  /          |             /       |</span>
-<span class="go">       ||               |                 |           |            |        |</span>
-<span class="go">       ||               |        (1 - n)* | P(x) dx   |  (-1 + n)* | P(x) dx|</span>
-<span class="go">       ||               |                 |           |            |        |</span>
-<span class="go">       ||               |                /            |           /         |</span>
-<span class="go">f(x) = ||C1 + (-1 + n)* | -Q(x)*e                   dx|*e                   |</span>
-<span class="go">       ||               |                             |                     |</span>
-<span class="go">       \\               /                            /                     /</span>
-</pre></div>
-</div>
-<p>Note that when n = 1, then the equation is separable (see the
-docstring of ode_separable()).</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">dsolve</span><span class="p">(</span><span class="n">Eq</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">P</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">Q</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span>
-<span class="gp">... </span><span class="n">hint</span><span class="o">=</span><span class="s">&#39;separable_Integral&#39;</span><span class="p">))</span>
-<span class="go"> f(x)</span>
-<span class="go">   /</span>
-<span class="go">  |                /</span>
-<span class="go">  |  1            |</span>
-<span class="go">  |  - dy = C1 +  | (-P(x) + Q(x)) dx</span>
-<span class="go">  |  y            |</span>
-<span class="go">  |              /</span>
-<span class="go"> /</span>
-</pre></div>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Bernoulli_differential_equation">http://en.wikipedia.org/wiki/Bernoulli_differential_equation</a></li>
-<li>M. Tenenbaum &amp; H. Pollard, &#8220;Ordinary Differential Equations&#8221;,
-Dover 1963, pp. 95</li>
-</ul>
-<p># indirect doctest</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">dsolve</span><span class="p">,</span> <span class="n">Eq</span><span class="p">,</span> <span class="n">pprint</span><span class="p">,</span> <span class="n">log</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">dsolve</span><span class="p">(</span><span class="n">Eq</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span>
-<span class="gp">... </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">hint</span><span class="o">=</span><span class="s">&#39;Bernoulli&#39;</span><span class="p">))</span>
-<span class="go">                1</span>
-<span class="go">f(x) = -------------------</span>
-<span class="go">         /     log(x)   1\</span>
-<span class="go">       x*|C1 + ------ + -|</span>
-<span class="go">         \       x      x/</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="liouville">
-<h3><tt class="xref py py-obj docutils literal"><span class="pre">Liouville</span></tt><a class="headerlink" href="#liouville" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.solvers.ode.ode_Liouville">
-<tt class="descclassname">sympy.solvers.ode.</tt><tt class="descname">ode_Liouville</tt><big>(</big><em>eq</em>, <em>func</em>, <em>order</em>, <em>match</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/ode.html#ode_Liouville"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.ode.ode_Liouville" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solves 2nd order Liouville differential equations.</p>
-<p>The general form of a Liouville ODE is
-d^2y/dx^2 + g(y)*(dy/dx)**2 + h(x)*dy/dx.  The general solution is:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">dsolve</span><span class="p">,</span> <span class="n">Eq</span><span class="p">,</span> <span class="n">pprint</span><span class="p">,</span> <span class="n">diff</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Function</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;f&#39;</span><span class="p">,</span> <span class="s">&#39;g&#39;</span><span class="p">,</span> <span class="s">&#39;h&#39;</span><span class="p">]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">genform</span> <span class="o">=</span> <span class="n">Eq</span><span class="p">(</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="n">x</span><span class="p">,</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">g</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span>
-<span class="gp">... </span><span class="n">h</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="n">x</span><span class="p">),</span> <span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">genform</span><span class="p">)</span>
-<span class="go">                2                    2</span>
-<span class="go">        d                d          d</span>
-<span class="go">g(f(x))*--(f(x))  + h(x)*--(f(x)) + ---(f(x)) = 0</span>
-<span class="go">        dx               dx           2</span>
-<span class="go">                                    dx</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">dsolve</span><span class="p">(</span><span class="n">genform</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">hint</span><span class="o">=</span><span class="s">&#39;Liouville_Integral&#39;</span><span class="p">))</span>
-<span class="go">                                  f(x)</span>
-<span class="go">          /                     /</span>
-<span class="go">         |                     |</span>
-<span class="go">         |     /               |     /</span>
-<span class="go">         |    |                |    |</span>
-<span class="go">         |  - | h(x) dx        |    | g(y) dy</span>
-<span class="go">         |    |                |    |</span>
-<span class="go">         |   /                 |   /</span>
-<span class="go">C1 + C2* | e            dx +   |  e           dy = 0</span>
-<span class="go">         |                     |</span>
-<span class="go">        /                     /</span>
-</pre></div>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li>Goldstein and Braun, &#8220;Advanced Methods for the Solution of
-Differential Equations&#8221;, pp. 98</li>
-<li><a class="reference external" href="http://www.maplesoft.com/support/help/Maple/view.aspx?path=odeadvisor/Liouville">http://www.maplesoft.com/support/help/Maple/view.aspx?path=odeadvisor/Liouville</a></li>
-</ul>
-<p># indirect doctest</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">dsolve</span><span class="p">,</span> <span class="n">Eq</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">dsolve</span><span class="p">(</span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span>
-<span class="gp">... </span><span class="n">diff</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">hint</span><span class="o">=</span><span class="s">&#39;Liouville&#39;</span><span class="p">))</span>
-<span class="go">           ________________           ________________</span>
-<span class="go">[f(x) = -\/ C1 + C2*log(x) , f(x) = \/ C1 + C2*log(x) ]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="riccati-special-minus2">
-<h3><tt class="xref py py-obj docutils literal"><span class="pre">Riccati_special_minus2</span></tt><a class="headerlink" href="#riccati-special-minus2" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.solvers.ode.ode_Riccati_special_minus2">
-<tt class="descclassname">sympy.solvers.ode.</tt><tt class="descname">ode_Riccati_special_minus2</tt><big>(</big><em>eq</em>, <em>func</em>, <em>order</em>, <em>match</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/ode.html#ode_Riccati_special_minus2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.ode.ode_Riccati_special_minus2" title="Permalink to this definition">¶</a></dt>
-<dd><p>The general Riccati equation has the form dy/dx = f(x)*y**2 + g(x)*y + h(x).
-While it does not have a general solution [1], the &#8220;special&#8221; form,
-dy/dx = a*y**2 - b*x**c, does have solutions in many cases [2]. This routine
-returns a solution for a*dy/dx = b*y**2 + c*y/x + d/x**2 that is obtained by
-using a suitable change of variables to reduce it to the special form and is
-valid when neither a nor b are zero and either c or d is zero.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.solvers.ode</span> <span class="kn">import</span> <span class="n">dsolve</span><span class="p">,</span> <span class="n">checkodesol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pprint</span><span class="p">,</span> <span class="n">Function</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">genform</span> <span class="o">=</span> <span class="n">a</span><span class="o">*</span><span class="n">y</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="n">b</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">c</span><span class="o">*</span><span class="n">y</span><span class="o">/</span><span class="n">x</span> <span class="o">+</span> <span class="n">d</span><span class="o">/</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sol</span> <span class="o">=</span> <span class="n">dsolve</span><span class="p">(</span><span class="n">genform</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">sol</span><span class="p">)</span>
-<span class="go">         /                                 /        __________________      \\</span>
-<span class="go">        |           __________________    |       /                2        ||</span>
-<span class="go">        |          /                2     |     \/  4*b*d - (a + c)  *log(x)||</span>
-<span class="go">       -|a + c - \/  4*b*d - (a + c)  *tan|C1 + ----------------------------||</span>
-<span class="go">        \                                 \                 2*a             //</span>
-<span class="go">f(x) = -----------------------------------------------------------------------</span>
-<span class="go">                                        2*b*x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">checkodesol</span><span class="p">(</span><span class="n">genform</span><span class="p">,</span> <span class="n">sol</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="mi">1</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p class="rubric">References</p>
-<ol class="arabic simple">
-<li><a class="reference external" href="http://www.maplesoft.com/support/help/Maple/view.aspx?path=odeadvisor/Riccati">http://www.maplesoft.com/support/help/Maple/view.aspx?path=odeadvisor/Riccati</a></li>
-<li><a class="reference external" href="http://eqworld.ipmnet.ru/en/solutions/ode/ode0106.pdf">http://eqworld.ipmnet.ru/en/solutions/ode/ode0106.pdf</a> -
-<a class="reference external" href="http://eqworld.ipmnet.ru/en/solutions/ode/ode0123.pdf">http://eqworld.ipmnet.ru/en/solutions/ode/ode0123.pdf</a></li>
-</ol>
-</dd></dl>
-
-</div>
-<div class="section" id="nth-linear-constant-coeff-homogeneous">
-<h3><tt class="xref py py-obj docutils literal"><span class="pre">nth_linear_constant_coeff_homogeneous</span></tt><a class="headerlink" href="#nth-linear-constant-coeff-homogeneous" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.solvers.ode.ode_nth_linear_constant_coeff_homogeneous">
-<tt class="descclassname">sympy.solvers.ode.</tt><tt class="descname">ode_nth_linear_constant_coeff_homogeneous</tt><big>(</big><em>eq</em>, <em>func</em>, <em>order</em>, <em>match</em>, <em>returns='sol'</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/ode.html#ode_nth_linear_constant_coeff_homogeneous"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.ode.ode_nth_linear_constant_coeff_homogeneous" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solves an nth order linear homogeneous differential equation with
-constant coefficients.</p>
-<p>This is an equation of the form a_n*f(x)^(n) + a_(n-1)*f(x)^(n-1) +
-... + a1*f&#8217;(x) + a0*f(x) = 0</p>
-<p>These equations can be solved in a general manner, by taking the
-roots of the characteristic equation a_n*m**n + a_(n-1)*m**(n-1) +
-... + a1*m + a0 = 0.  The solution will then be the sum of
-Cn*x**i*exp(r*x) terms, for each where Cn is an arbitrary constant,
-r is a root of the characteristic equation and i is is one of each
-from 0 to the multiplicity of the root - 1 (for example, a root 3 of
-multiplicity 2 would create the terms C1*exp(3*x) + C2*x*exp(3*x)).
-The exponential is usually expanded for complex roots using Euler&#8217;s
-equation exp(I*x) = cos(x) + I*sin(x).  Complex roots always come in
-conjugate pars in polynomials with real coefficients, so the two
-roots will be represented (after simplifying the constants) as
-exp(a*x)*(C1*cos(b*x) + C2*sin(b*x)).</p>
-<p>If SymPy cannot find exact roots to the characteristic equation, a
-RootOf instance will be return in its stead.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">dsolve</span><span class="p">,</span> <span class="n">Eq</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dsolve</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span> <span class="o">+</span> <span class="mi">10</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span>
-<span class="gp">... </span><span class="n">hint</span><span class="o">=</span><span class="s">&#39;nth_linear_constant_coeff_homogeneous&#39;</span><span class="p">)</span>
-<span class="gp">... </span>
-<span class="go">f(x) == C1*exp(x*RootOf(_x**5 + 10*_x - 2, 0)) +</span>
-<span class="go">C2*exp(x*RootOf(_x**5 + 10*_x - 2, 1)) +</span>
-<span class="go">C3*exp(x*RootOf(_x**5 + 10*_x - 2, 2)) +</span>
-<span class="go">C4*exp(x*RootOf(_x**5 + 10*_x - 2, 3)) +</span>
-<span class="go">C5*exp(x*RootOf(_x**5 + 10*_x - 2, 4))</span>
-</pre></div>
-</div>
-<p>Note that because this method does not involve integration, there is
-no &#8216;nth_linear_constant_coeff_homogeneous_Integral&#8217; hint.</p>
-<p>The following is for internal use:</p>
-<ul class="simple">
-<li>returns = &#8216;sol&#8217; returns the solution to the ODE.</li>
-<li>returns = &#8216;list&#8217; returns a list of linearly independent
-solutions, for use with non homogeneous solution methods like
-variation of parameters and undetermined coefficients.  Note that,
-though the solutions should be linearly independent, this function
-does not explicitly check that.  You can do &#8220;assert
-simplify(wronskian(sollist)) != 0&#8221; to check for linear independence.
-Also, &#8220;assert len(sollist) == order&#8221; will need to pass.</li>
-<li>returns = &#8216;both&#8217;, return a dictionary {&#8216;sol&#8217;:solution to ODE,
-&#8216;list&#8217;: list of linearly independent solutions}.</li>
-</ul>
-<p class="rubric">References</p>
-<ul>
-<li><dl class="first docutils">
-<dt><a class="reference external" href="http://en.wikipedia.org/wiki/Linear_differential_equation">http://en.wikipedia.org/wiki/Linear_differential_equation</a></dt>
-<dd><p class="first last">section: Nonhomogeneous_equation_with_constant_coefficients</p>
-</dd>
-</dl>
-</li>
-<li><p class="first">M. Tenenbaum &amp; H. Pollard, &#8220;Ordinary Differential Equations&#8221;,
-Dover 1963, pp. 211</p>
-</li>
-</ul>
-<p># indirect doctest</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">dsolve</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">dsolve</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="o">-</span>
-<span class="gp">... </span><span class="mi">2</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="mi">6</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="mi">5</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span>
-<span class="gp">... </span><span class="n">hint</span><span class="o">=</span><span class="s">&#39;nth_linear_constant_coeff_homogeneous&#39;</span><span class="p">))</span>
-<span class="go">                    x                            -2*x</span>
-<span class="go">f(x) = (C1 + C2*x)*e  + (C3*sin(x) + C4*cos(x))*e</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="nth-linear-constant-coeff-undetermined-coefficients">
-<h3><tt class="xref py py-obj docutils literal"><span class="pre">nth_linear_constant_coeff_undetermined_coefficients</span></tt><a class="headerlink" href="#nth-linear-constant-coeff-undetermined-coefficients" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.solvers.ode.ode_nth_linear_constant_coeff_undetermined_coefficients">
-<tt class="descclassname">sympy.solvers.ode.</tt><tt class="descname">ode_nth_linear_constant_coeff_undetermined_coefficients</tt><big>(</big><em>eq</em>, <em>func</em>, <em>order</em>, <em>match</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/ode.html#ode_nth_linear_constant_coeff_undetermined_coefficients"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.ode.ode_nth_linear_constant_coeff_undetermined_coefficients" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solves an nth order linear differential equation with constant
-coefficients using the method of undetermined coefficients.</p>
-<p>This method works on differential equations of the form a_n*f(x)^(n)
-+ a_(n-1)*f(x)^(n-1) + ... + a1*f&#8217;(x) + a0*f(x) = P(x), where P(x)
-is a function that has a finite number of linearly independent
-derivatives.</p>
-<p>Functions that fit this requirement are finite sums functions of the
-form a*x**i*exp(b*x)*sin(c*x + d) or a*x**i*exp(b*x)*cos(c*x + d),
-where i is a non-negative integer and a, b, c, and d are constants.
-For example any polynomial in x, functions like x**2*exp(2*x),
-x*sin(x), and exp(x)*cos(x) can all be used.  Products of sin&#8217;s and
-cos&#8217;s have a finite number of derivatives, because they can be
-expanded into sin(a*x) and cos(b*x) terms.  However, SymPy currently
-cannot do that expansion, so you will need to manually rewrite the
-expression in terms of the above to use this method.  So, for example,
-you will need to manually convert sin(x)**2 into (1 + cos(2*x))/2 to
-properly apply the method of undetermined coefficients on it.</p>
-<p>This method works by creating a trial function from the expression
-and all of its linear independent derivatives and substituting them
-into the original ODE.  The coefficients for each term will be a
-system of linear equations, which are be solved for and substituted,
-giving the solution.  If any of the trial functions are linearly
-dependent on the solution to the homogeneous equation, they are
-multiplied by sufficient x to make them linearly independent.</p>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Method_of_undetermined_coefficients">http://en.wikipedia.org/wiki/Method_of_undetermined_coefficients</a></li>
-<li>M. Tenenbaum &amp; H. Pollard, &#8220;Ordinary Differential Equations&#8221;,
-Dover 1963, pp. 221</li>
-</ul>
-<p># indirect doctest</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">dsolve</span><span class="p">,</span> <span class="n">pprint</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">dsolve</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span>
-<span class="gp">... </span><span class="mi">4</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span>
-<span class="gp">... </span><span class="n">hint</span><span class="o">=</span><span class="s">&#39;nth_linear_constant_coeff_undetermined_coefficients&#39;</span><span class="p">))</span>
-<span class="go">       /             4\</span>
-<span class="go">       |            x |  -x   4*sin(2*x)   3*cos(2*x)</span>
-<span class="go">f(x) = |C1 + C2*x + --|*e   - ---------- + ----------</span>
-<span class="go">       \            3 /           25           25</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="nth-linear-constant-coeff-variation-of-parameters">
-<h3><tt class="xref py py-obj docutils literal"><span class="pre">nth_linear_constant_coeff_variation_of_parameters</span></tt><a class="headerlink" href="#nth-linear-constant-coeff-variation-of-parameters" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.solvers.ode.ode_nth_linear_constant_coeff_variation_of_parameters">
-<tt class="descclassname">sympy.solvers.ode.</tt><tt class="descname">ode_nth_linear_constant_coeff_variation_of_parameters</tt><big>(</big><em>eq</em>, <em>func</em>, <em>order</em>, <em>match</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/ode.html#ode_nth_linear_constant_coeff_variation_of_parameters"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.ode.ode_nth_linear_constant_coeff_variation_of_parameters" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solves an nth order linear differential equation with constant
-coefficients using the method of variation of parameters.</p>
-<p>This method works on any differential equations of the form
-f(x)^(n) + a_(n-1)*f(x)^(n-1) + ... + a1*f&#8217;(x) + a0*f(x) = P(x).</p>
-<p>This method works by assuming that the particular solution takes the
-form Sum(c_i(x)*y_i(x), (x, 1, n)), where y_i is the ith solution to
-the homogeneous equation.  The solution is then solved using
-Wronskian&#8217;s and Cramer&#8217;s Rule.  The particular solution is given by
-Sum(Integral(W_i(x)/W(x), x)*y_i(x), (x, 1, n)), where W(x) is the
-Wronskian of the fundamental system (the system of n linearly
-independent solutions to the homogeneous equation), and W_i(x) is
-the Wronskian of the fundamental system with the ith column replaced
-with [0, 0, ..., 0, P(x)].</p>
-<p>This method is general enough to solve any nth order inhomogeneous
-linear differential equation with constant coefficients, but
-sometimes SymPy cannot simplify the Wronskian well enough to
-integrate it.  If this method hangs, try using the
-&#8216;nth_linear_constant_coeff_variation_of_parameters_Integral&#8217; hint
-and simplifying the integrals manually.  Also, prefer using
-&#8216;nth_linear_constant_coeff_undetermined_coefficients&#8217; when it
-applies, because it doesn&#8217;t use integration, making it faster and
-more reliable.</p>
-<p>Warning, using simplify=False with
-&#8216;nth_linear_constant_coeff_variation_of_parameters&#8217; in dsolve()
-may cause it to hang, because it will not attempt to simplify
-the Wronskian before integrating.  It is recommended that you only
-use simplify=False with
-&#8216;nth_linear_constant_coeff_variation_of_parameters_Integral&#8217; for
-this method, especially if the solution to the homogeneous
-equation has trigonometric functions in it.</p>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Variation_of_parameters">http://en.wikipedia.org/wiki/Variation_of_parameters</a></li>
-<li><a class="reference external" href="http://planetmath.org/encyclopedia/VariationOfParameters.html">http://planetmath.org/encyclopedia/VariationOfParameters.html</a></li>
-<li>M. Tenenbaum &amp; H. Pollard, &#8220;Ordinary Differential Equations&#8221;,
-Dover 1963, pp. 233</li>
-</ul>
-<p># indirect doctest</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">dsolve</span><span class="p">,</span> <span class="n">pprint</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">log</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">dsolve</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span>
-<span class="gp">... </span><span class="mi">3</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span>
-<span class="gp">... </span><span class="n">hint</span><span class="o">=</span><span class="s">&#39;nth_linear_constant_coeff_variation_of_parameters&#39;</span><span class="p">))</span>
-<span class="go">       /                     3                \</span>
-<span class="go">       |                2   x *(6*log(x) - 11)|  x</span>
-<span class="go">f(x) = |C1 + C2*x + C3*x  + ------------------|*e</span>
-<span class="go">       \                            36        /</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="separable">
-<h3><tt class="xref py py-obj docutils literal"><span class="pre">separable</span></tt><a class="headerlink" href="#separable" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.solvers.ode.ode_separable">
-<tt class="descclassname">sympy.solvers.ode.</tt><tt class="descname">ode_separable</tt><big>(</big><em>eq</em>, <em>func</em>, <em>order</em>, <em>match</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/ode.html#ode_separable"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.ode.ode_separable" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solves separable 1st order differential equations.</p>
-<p>This is any differential equation that can be written as
-P(y)*dy/dx = Q(x). The solution can then just be found by
-rearranging terms and integrating:
-Integral(P(y), y) = Integral(Q(x), x). This hint uses separatevars()
-as its back end, so if a separable equation is not caught by this
-solver, it is most likely the fault of that function. separatevars()
-is smart enough to do most expansion and factoring necessary to
-convert a separable equation F(x, y) into the proper form P(x)*Q(y).
-The general solution is:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">dsolve</span><span class="p">,</span> <span class="n">Eq</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Function</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="s">&#39;b&#39;</span><span class="p">,</span> <span class="s">&#39;c&#39;</span><span class="p">,</span> <span class="s">&#39;d&#39;</span><span class="p">,</span> <span class="s">&#39;f&#39;</span><span class="p">]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">genform</span> <span class="o">=</span> <span class="n">Eq</span><span class="p">(</span><span class="n">a</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">b</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">c</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">d</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">genform</span><span class="p">)</span>
-<span class="go">             d</span>
-<span class="go">a(x)*b(f(x))*--(f(x)) = c(x)*d(f(x))</span>
-<span class="go">             dx</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">dsolve</span><span class="p">(</span><span class="n">genform</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">hint</span><span class="o">=</span><span class="s">&#39;separable_Integral&#39;</span><span class="p">))</span>
-<span class="go">     f(x)</span>
-<span class="go">   /                  /</span>
-<span class="go">  |                  |</span>
-<span class="go">  |  b(y)            | c(x)</span>
-<span class="go">  |  ---- dy = C1 +  | ---- dx</span>
-<span class="go">  |  d(y)            | a(x)</span>
-<span class="go">  |                  |</span>
-<span class="go"> /                  /</span>
-</pre></div>
-</div>
-<p class="rubric">References</p>
-<ul class="simple">
-<li>M. Tenenbaum &amp; H. Pollard, &#8220;Ordinary Differential Equations&#8221;,
-Dover 1963, pp. 52</li>
-</ul>
-<p># indirect doctest</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">dsolve</span><span class="p">,</span> <span class="n">Eq</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">dsolve</span><span class="p">(</span><span class="n">Eq</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span>
-<span class="gp">... </span><span class="n">hint</span><span class="o">=</span><span class="s">&#39;separable&#39;</span><span class="p">,</span> <span class="n">simplify</span><span class="o">=</span><span class="bp">False</span><span class="p">))</span>
-<span class="go">   /   2       \         2</span>
-<span class="go">log\3*f (x) - 1/        x</span>
-<span class="go">---------------- = C1 + --</span>
-<span class="go">       6                2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="module-sympy.solvers.ode">
-<span id="information-on-the-ode-module"></span><h2>Information on the ode module<a class="headerlink" href="#module-sympy.solvers.ode" title="Permalink to this headline">¶</a></h2>
-<p>This module contains dsolve() and different helper functions that it
-uses.</p>
-<p>dsolve() solves ordinary differential equations. See the docstring on
-the various functions for their uses. Note that partial differential
-equations support is in pde.py.  Note that ode_hint() functions have
-docstrings describing their various methods, but they are intended for
-internal use.  Use dsolve(ode, func, hint=hint) to solve an ode using a
-specific hint.  See also the docstring on dsolve().</p>
-<p><strong>Functions in this module</strong></p>
-<blockquote>
-<div><p>These are the user functions in this module:</p>
-<ul class="simple">
-<li>dsolve() - Solves ODEs.</li>
-<li>classify_ode() - Classifies ODEs into possible hints for dsolve().</li>
-<li>checkodesol() - Checks if an equation is the solution to an ODE.</li>
-<li>ode_order() - Returns the order (degree) of an ODE.</li>
-<li>homogeneous_order() - Returns the homogeneous order of an
-expression.</li>
-</ul>
-<p>These are the non-solver helper functions that are for internal use.
-The user should use the various options to dsolve() to obtain the
-functionality provided by these functions:</p>
-<ul class="simple">
-<li>odesimp() - Does all forms of ODE simplification.</li>
-<li>ode_sol_simplicity() - A key function for comparing solutions by
-simplicity.</li>
-<li>constantsimp() - Simplifies arbitrary constants.</li>
-<li>constant_renumber() - Renumber arbitrary constants</li>
-<li>_handle_Integral() - Evaluate unevaluated Integrals.</li>
-<li>preprocess - prepare the equation and detect function to solve for</li>
-</ul>
-<p>See also the docstrings of these functions.</p>
-</div></blockquote>
-<p><strong>Currently implemented solver methods</strong></p>
-<p>The following methods are implemented for solving ordinary differential
-equations.  See the docstrings of the various ode_hint() functions for
-more information on each (run help(ode)):</p>
-<blockquote>
-<div><ul class="simple">
-<li>1st order separable differential equations</li>
-<li>1st order differential equations whose coefficients or dx and dy
-are functions homogeneous of the same order.</li>
-<li>1st order exact differential equations.</li>
-<li>1st order linear differential equations</li>
-<li>1st order Bernoulli differential equations.</li>
-<li>2nd order Liouville differential equations.</li>
-<li>nth order linear homogeneous differential equation with constant
-coefficients.</li>
-<li>nth order linear inhomogeneous differential equation with constant
-coefficients using the method of undetermined coefficients.</li>
-<li>nth order linear inhomogeneous differential equation with constant
-coefficients using the method of variation of parameters.</li>
-</ul>
-</div></blockquote>
-<p><strong>Philosophy behind this module</strong></p>
-<p>This module is designed to make it easy to add new ODE solving methods
-without having to mess with the solving code for other methods.  The
-idea is that there is a classify_ode() function, which takes in an ODE
-and tells you what hints, if any, will solve the ODE.  It does this
-without attempting to solve the ODE, so it is fast.  Each solving method
-is a hint, and it has its own function, named ode_hint.  That function
-takes in the ODE and any match expression gathered by classify_ode and
-returns a solved result.  If this result has any integrals in it, the
-ode_hint function will return an unevaluated Integral class. dsolve(),
-which is the user wrapper function around all of this, will then call
-odesimp() on the result, which, among other things, will attempt to
-solve the equation for the dependent variable (the function we are
-solving for), simplify the arbitrary constants in the expression, and
-evaluate any integrals, if the hint allows it.</p>
-<p><strong>How to add new solution methods</strong></p>
-<p>If you have an ODE that you want dsolve() to be able to solve, try to
-avoid adding special case code here.  Instead, try finding a general
-method that will solve your ODE, as well as others.  This way, the ode
-module will become more robust, and unhindered by special case hacks.
-WolphramAlpha and Maple&#8217;s DETools[odeadvisor] function are two resources
-you can use to classify a specific ODE.  It is also better for a method
-to work with an nth order ODE instead of only with specific orders, if
-possible.</p>
-<p>To add a new method, there are a few things that you need to do.  First,
-you need a hint name for your method.  Try to name your hint so that it
-is unambiguous with all other methods, including ones that may not be
-implemented yet.  If your method uses integrals, also include a
-&#8220;hint_Integral&#8221; hint.  If there is more than one way to solve ODEs with
-your method, include a hint for each one, as well as a &#8220;hint_best&#8221; hint.
-Your ode_hint_best() function should choose the best using min with
-ode_sol_simplicity as the key argument.  See
-ode_1st_homogeneous_coeff_best(), for example. The function that uses
-your method will be called ode_hint(), so the hint must only use
-characters that are allowed in a Python function name (alphanumeric
-characters and the underscore &#8216;_&#8217; character).  Include a function for
-every hint, except for &#8220;_Integral&#8221; hints (dsolve() takes care of those
-automatically).  Hint names should be all lowercase, unless a word is
-commonly capitalized (such as Integral or Bernoulli). If you have a hint
-that you do not want to run with &#8220;all_Integral&#8221; that doesn&#8217;t have an
-&#8220;_Integral&#8221; counterpart (such as a best hint that would defeat the
-purpose of &#8220;all_Integral&#8221;), you will need to remove it manually in the
-dsolve() code.  See also the classify_ode() docstring for guidelines on
-writing a hint name.</p>
-<p>Determine <em>in general</em> how the solutions returned by your method
-compare with other methods that can potentially solve the same ODEs.
-Then, put your hints in the allhints tuple in the order that they should
-be called.  The ordering of this tuple determines which hints are
-default. Note that exceptions are ok, because it is easy for the user to
-choose individual hints with dsolve().  In general, &#8220;_Integral&#8221; variants
-should go at the end of the list, and &#8220;_best&#8221; variants should go before
-the various hints they apply to.  For example, the
-&#8220;undetermined_coefficients&#8221; hint comes before the
-&#8220;variation_of_parameters&#8221; hint because, even though variation of
-parameters is more general than undetermined coefficients, undetermined
-coefficients generally returns cleaner results for the ODEs that it can
-solve than variation of parameters does, and it does not require
-integration, so it is much faster.</p>
-<p>Next, you need to have a match expression or a function that matches the
-type of the ODE, which you should put in classify_ode() (if the match
-function is more than just a few lines, like
-_undetermined_coefficients_match(), it should go outside of
-classify_ode()).  It should match the ODE without solving for it as much
-as possible, so that classify_ode() remains fast and is not hindered by
-bugs in solving code.  Be sure to consider corner cases. For example, if
-your solution method involves dividing by something, make sure you
-exclude the case where that division will be 0.</p>
-<p>In most cases, the matching of the ODE will also give you the various
-parts that you need to solve it. You should put that in a dictionary
-(.match() will do this for you), and add that as matching_hints[&#8216;hint&#8217;]
-= matchdict in the relevant part of classify_ode.  classify_ode will
-then send this to dsolve(), which will send it to your function as the
-match argument. Your function should be named ode_hint(eq, func, order,
-match). If you need to send more information, put it in the match
-dictionary.  For example, if you had to substitute in a dummy variable
-in classify_ode to match the ODE, you will need to pass it to your
-function using the match dict to access it.  You can access the
-independent variable using func.args[0], and the dependent variable (the
-function you are trying to solve for) as func.func.  If, while trying to
-solve the ODE, you find that you cannot, raise NotImplementedError.
-dsolve() will catch this error with the &#8220;all&#8221; meta-hint, rather than
-causing the whole routine to fail.</p>
-<p>Add a docstring to your function that describes the method employed.
-Like with anything else in SymPy, you will need to add a doctest to the
-docstring, in addition to real tests in test_ode.py.  Try to maintain
-consistency with the other hint functions&#8217; docstrings.  Add your method
-to the list at the top of this docstring.  Also, add your method to
-ode.rst in the docs/src directory, so that the Sphinx docs will pull its
-docstring into the main SymPy documentation.  Be sure to make the Sphinx
-documentation by running &#8220;make html&#8221; from within the doc directory to
-verify that the docstring formats correctly.</p>
-<p>If your solution method involves integrating, use C.Integral() instead
-of integrate().  This allows the user to bypass hard/slow integration by
-using the &#8220;_Integral&#8221; variant of your hint.  In most cases, calling
-.doit() will integrate your solution.  If this is not the case, you will
-need to write special code in _handle_Integral().  Arbitrary constants
-should be symbols named C1, C2, and so on.  All solution methods should
-return an equality instance.  If you need an arbitrary number of
-arbitrary constants, you can use constants =
-numbered_symbols(prefix=&#8217;C&#8217;, cls=Symbol, start=1).  If it is
-possible to solve for the dependent function in a general way, do so.
-Otherwise, do as best as you can, but do not call solve in your
-ode_hint() function.  odesimp() will attempt to solve the solution for
-you, so you do not need to do that. Lastly, if your ODE has a common
-simplification that can be applied to your solutions, you can add a
-special case in odesimp() for it.  For example, solutions returned from
-the &#8220;1st_homogeneous_coeff&#8221; hints often have many log() terms, so
-odesimp() calls logcombine() on them (it also helps to write the
-arbitrary constant as log(C1) instead of C1 in this case).  Also
-consider common ways that you can rearrange your solution to have
-constantsimp() take better advantage of it.  It is better to put
-simplification in odesimp() than in your method, because it can then be
-turned off with the simplify flag in dsolve(). If you have any
-extraneous simplification in your function, be sure to only run it using
-&#8220;if match.get(&#8216;simplify&#8217;, True):&#8221;, especially if it can be slow or if it
-can reduce the domain of the solution.</p>
-<p>Finally, as with every contribution to SymPy, your method will need to
-be tested. Add a test for each method in test_ode.py.  Follow the
-conventions there, i.e., test the solver using dsolve(eq, f(x),
-hint=your_hint), and also test the solution using checkodesol (you can
-put these in a separate tests and skip/XFAIL if it runs too slow/doesn&#8217;t
-work).  Be sure to call your hint specifically in dsolve, that way the
-test won&#8217;t be broken simply by the introduction of another matching
-hint. If your method works for higher order (&gt;1) ODEs, you will need to
-run sol = constant_renumber(sol, &#8216;C&#8217;, 1, order), for each solution, where
-order is the order of the ODE. This is because constant_renumber renumbers
-the arbitrary constants by printing order, which is platform dependent.
-Try to test every corner case of your solver, including a range of
-orders if it is a nth order solver, but if your solver is slow, such as
-if it involves hard integration, try to keep the test run time down.</p>
-<p>Feel free to refactor existing hints to avoid duplicating code or
-creating inconsistencies.  If you can show that your method exactly
-duplicates an existing method, including in the simplicity and speed of
-obtaining the solutions, then you can remove the old, less general
-method. The existing code is tested extensively in test_ode.py, so if
-anything is broken, one of those tests will surely fail.</p>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../index.html">
-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">ODE</a><ul>
-<li><a class="reference internal" href="#user-functions">User Functions</a><ul>
-<li><a class="reference internal" href="#dsolve"><tt class="docutils literal"><span class="pre">dsolve()</span></tt></a></li>
-<li><a class="reference internal" href="#classify-ode"><tt class="docutils literal"><span class="pre">classify_ode()</span></tt></a></li>
-<li><a class="reference internal" href="#ode-order"><tt class="docutils literal"><span class="pre">ode_order()</span></tt></a></li>
-<li><a class="reference internal" href="#checkodesol"><tt class="docutils literal"><span class="pre">checkodesol()</span></tt></a></li>
-<li><a class="reference internal" href="#homogeneous-order"><tt class="docutils literal"><span class="pre">homogeneous_order()</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#hint-methods">Hint Methods</a><ul>
-<li><a class="reference internal" href="#preprocess"><tt class="docutils literal"><span class="pre">preprocess</span></tt></a></li>
-<li><a class="reference internal" href="#odesimp"><tt class="docutils literal"><span class="pre">odesimp</span></tt></a></li>
-<li><a class="reference internal" href="#constant-renumber"><tt class="docutils literal"><span class="pre">constant_renumber</span></tt></a></li>
-<li><a class="reference internal" href="#constantsimp"><tt class="docutils literal"><span class="pre">constantsimp</span></tt></a></li>
-<li><a class="reference internal" href="#sol-simplicity"><tt class="docutils literal"><span class="pre">sol_simplicity</span></tt></a></li>
-<li><a class="reference internal" href="#st-exact"><tt class="docutils literal"><span class="pre">1st_exact</span></tt></a></li>
-<li><a class="reference internal" href="#st-homogeneous-coeff-best"><tt class="docutils literal"><span class="pre">1st_homogeneous_coeff_best</span></tt></a></li>
-<li><a class="reference internal" href="#st-homogeneous-coeff-subs-dep-div-indep"><tt class="docutils literal"><span class="pre">1st_homogeneous_coeff_subs_dep_div_indep</span></tt></a></li>
-<li><a class="reference internal" href="#st-homogeneous-coeff-subs-indep-div-dep"><tt class="docutils literal"><span class="pre">1st_homogeneous_coeff_subs_indep_div_dep</span></tt></a></li>
-<li><a class="reference internal" href="#st-linear"><tt class="docutils literal"><span class="pre">1st_linear</span></tt></a></li>
-<li><a class="reference internal" href="#bernoulli"><tt class="docutils literal"><span class="pre">Bernoulli</span></tt></a></li>
-<li><a class="reference internal" href="#liouville"><tt class="docutils literal"><span class="pre">Liouville</span></tt></a></li>
-<li><a class="reference internal" href="#riccati-special-minus2"><tt class="docutils literal"><span class="pre">Riccati_special_minus2</span></tt></a></li>
-<li><a class="reference internal" href="#nth-linear-constant-coeff-homogeneous"><tt class="docutils literal"><span class="pre">nth_linear_constant_coeff_homogeneous</span></tt></a></li>
-<li><a class="reference internal" href="#nth-linear-constant-coeff-undetermined-coefficients"><tt class="docutils literal"><span class="pre">nth_linear_constant_coeff_undetermined_coefficients</span></tt></a></li>
-<li><a class="reference internal" href="#nth-linear-constant-coeff-variation-of-parameters"><tt class="docutils literal"><span class="pre">nth_linear_constant_coeff_variation_of_parameters</span></tt></a></li>
-<li><a class="reference internal" href="#separable"><tt class="docutils literal"><span class="pre">separable</span></tt></a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#module-sympy.solvers.ode">Information on the ode module</a></li>
-</ul>
-</li>
-</ul>
-
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-            
-  <div class="section" id="module-sympy.solvers">
-<span id="solvers"></span><h1>Solvers<a class="headerlink" href="#module-sympy.solvers" title="Permalink to this headline">¶</a></h1>
-<p>The <em>solvers</em> module in SymPy implements methods for solving equations.</p>
-<div class="section" id="algebraic-equations">
-<h2>Algebraic equations<a class="headerlink" href="#algebraic-equations" title="Permalink to this headline">¶</a></h2>
-<p>Use <tt class="xref py py-func docutils literal"><span class="pre">solve()</span></tt> to solve algebraic equations. We suppose all equations are equaled to 0,
-so solving x**2 == 1 translates into the following code:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.solvers</span> <span class="kn">import</span> <span class="n">solve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">[-1, 1]</span>
-</pre></div>
-</div>
-<p>The first argument for <tt class="xref py py-func docutils literal"><span class="pre">solve()</span></tt> is an equation (equaled to zero) and the second argument
-is the symbol that we want to solve the equation for.</p>
-<dl class="function">
-<dt id="sympy.solvers.solvers.solve">
-<tt class="descclassname">sympy.solvers.solvers.</tt><tt class="descname">solve</tt><big>(</big><em>f</em>, <em>*symbols</em>, <em>**flags</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/solvers.html#solve"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.solvers.solve" title="Permalink to this definition">¶</a></dt>
-<dd><p>Algebraically solves equations and systems of equations.</p>
-<dl class="docutils">
-<dt>Currently supported are:</dt>
-<dd><ul class="first last simple">
-<li>univariate polynomial,</li>
-<li>transcendental</li>
-<li>piecewise combinations of the above</li>
-<li>systems of linear and polynomial equations</li>
-<li>sytems containing relational expressions.</li>
-</ul>
-</dd>
-</dl>
-<p>Input is formed as:</p>
-<ul>
-<li><dl class="first docutils">
-<dt>f</dt>
-<dd><ul class="first last simple">
-<li>a single Expr or Poly that must be zero,</li>
-<li>an Equality</li>
-<li>a Relational expression or boolean</li>
-<li>iterable of one or more of the above</li>
-</ul>
-</dd>
-</dl>
-</li>
-<li><dl class="first docutils">
-<dt>symbols (object(s) to solve for) specified as</dt>
-<dd><ul class="first last simple">
-<li>none given (other non-numeric objects will be used)</li>
-<li>single symbol</li>
-<li>denested list of symbols
-e.g. solve(f, x, y)</li>
-<li>ordered iterable of symbols
-e.g. solve(f, [x, y])</li>
-</ul>
-</dd>
-</dl>
-</li>
-<li><dl class="first docutils">
-<dt>flags</dt>
-<dd><dl class="first last docutils">
-<dt>&#8216;dict&#8217;=True (default is False)</dt>
-<dd><p class="first last">return list (perhaps empty) of solution mappings</p>
-</dd>
-<dt>&#8216;set&#8217;=True (default is False)</dt>
-<dd><p class="first last">return list of symbols and set of tuple(s) of solution(s)</p>
-</dd>
-<dt>&#8216;exclude=[] (default)&#8217;</dt>
-<dd><p class="first last">don&#8217;t try to solve for any of the free symbols in exclude;
-if expressions are given, the free symbols in them will
-be extracted automatically.</p>
-</dd>
-<dt>&#8216;check=True (default)&#8217;</dt>
-<dd><p class="first last">If False, don&#8217;t do any testing of solutions. This can be
-useful if one wants to include solutions that make any
-denominator zero.</p>
-</dd>
-<dt>&#8216;numerical=True (default)&#8217;</dt>
-<dd><p class="first last">do a fast numerical check if <tt class="docutils literal"><span class="pre">f</span></tt> has only one symbol.</p>
-</dd>
-<dt>&#8216;minimal=True (default is False)&#8217;</dt>
-<dd><p class="first last">a very fast, minimal testing.</p>
-</dd>
-<dt>&#8216;warning=True (default is False)&#8217;</dt>
-<dd><p class="first last">print a warning if checksol() could not conclude.</p>
-</dd>
-<dt>&#8216;simplify=True (default)&#8217;</dt>
-<dd><p class="first last">simplify all but cubic and quartic solutions before
-returning them and (if check is not False) use the
-general simplify function on the solutions and the
-expression obtained when they are substituted into the
-function which should be zero</p>
-</dd>
-<dt>&#8216;force=True (default is False)&#8217;</dt>
-<dd><p class="first last">make positive all symbols without assumptions regarding sign.</p>
-</dd>
-<dt>&#8216;rational=True (default)&#8217;</dt>
-<dd><p class="first last">recast Floats as Rational; if this option is not used, the
-system containing floats may fail to solve because of issues
-with polys. If rational=None, Floats will be recast as
-rationals but the answer will be recast as Floats. If the
-flag is False then nothing will be done to the Floats.</p>
-</dd>
-<dt>&#8216;manual=True (default is False)&#8217;</dt>
-<dd><p class="first last">do not use the polys/matrix method to solve a system of
-equations, solve them one at a time as you might &#8220;manually&#8221;.</p>
-</dd>
-<dt>&#8216;implicit=True (default is False)&#8217;</dt>
-<dd><p class="first last">allows solve to return a solution for a pattern in terms of
-other functions that contain that pattern; this is only
-needed if the pattern is inside of some invertible function
-like cos, exp, ....</p>
-</dd>
-</dl>
-</dd>
-</dl>
-</li>
-</ul>
-<p class="rubric">Notes</p>
-<p>assumptions aren&#8217;t checked when <span class="math">\(solve()\)</span> input involves
-relationals or bools.</p>
-<p>When the solutions are checked, those that make any denominator zero
-are automatically excluded. If you do not want to exclude such solutions
-then use the check=False option:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">limit</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">[]</span>
-</pre></div>
-</div>
-<p>If check=False then a solution to the numerator being zero is found: x = 0.
-In this case, this is a spurious solution since sin(x)/x has the well known
-limit (without dicontinuity) of 1 at x = 0:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">check</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">[0]</span>
-</pre></div>
-</div>
-<p>In the following case, however, the limit exists and is equal to the the
-value of x = 0 that is excluded when check=True:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span> <span class="o">-</span> <span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">[]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">check</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">[0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&#39;-&#39;</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="s">&#39;+&#39;</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<p class="rubric">Examples</p>
-<p>The output varies according to the input and can be seen by example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">solve</span><span class="p">,</span> <span class="n">Poly</span><span class="p">,</span> <span class="n">Eq</span><span class="p">,</span> <span class="n">Function</span><span class="p">,</span> <span class="n">exp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<ul>
-<li><p class="first">boolean or univariate Relational</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span> <span class="o">&lt;</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">And(im(x) == 0, re(x) &lt; 3)</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">to always get a list of solution mappings, use flag dict=True</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="nb">dict</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">[{x: 3}]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">([</span><span class="n">x</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">y</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="nb">dict</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">[{x: 3, y: 1}]</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">to get a list of symbols and set of solution(s) use flag set=True</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">([</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">y</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="nb">set</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">([x, y], set([(-sqrt(3), 1), (sqrt(3), 1)]))</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">single expression and single symbol that is in the expression</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">[y]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">[3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">Eq</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">[3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">Poly</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">3</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">[3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">set</span><span class="p">(</span><span class="n">solve</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">set([-y, y])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">set</span><span class="p">(</span><span class="n">solve</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">set([-1, 1, -I, I])</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">single expression with no symbol that is in the expression</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">[]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">[]</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">single expression with no symbol given</p>
-<blockquote>
-<div><p>In this case, all free symbols will be selected as potential
-symbols to solve for. If the equation is univariate then a list
-of solutionsis returned; otherwise &#8211; as is the case when symbols are
-given as an iterable of length &gt; 1 &#8211; a list of mappings will be returned.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">[3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> 
-<span class="go">[{x: -y}, {x: y}]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> 
-<span class="go">[{x: -y}, {x: y}]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">[{x: y**2}]</span>
-</pre></div>
-</div>
-</div></blockquote>
-</li>
-<li><p class="first">when an object other than a Symbol is given as a symbol, it is
-isolated algebraically and an implicit solution may be obtained.
-This is mostly provided as a convenience to save one from replacing
-the object with a Symbol and solving for that Symbol. It will only
-work if the specified object can be replaced with a Symbol using the
-subs method.</p>
-<blockquote>
-<div><div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">x</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">[x]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">x</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">[x + f(x)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">-</span> <span class="n">x</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">[-x + Derivative(f(x), x)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">set</span><span class="p">(</span><span class="n">solve</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">)))</span>
-<span class="go">set([-sqrt(-x), sqrt(-x)])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Indexed</span><span class="p">,</span> <span class="n">IndexedBase</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">,</span> <span class="n">sqrt</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">IndexedBase</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">eqs</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="n">A</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">A</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">-</span> <span class="mi">3</span><span class="p">,</span> <span class="n">A</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">A</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">eqs</span><span class="p">,</span> <span class="n">eqs</span><span class="o">.</span><span class="n">atoms</span><span class="p">(</span><span class="n">Indexed</span><span class="p">))</span>
-<span class="go">{A[1]: 1, A[2]: 2}</span>
-</pre></div>
-</div>
-<ul>
-<li><p class="first">To solve for a <em>symbol</em> implicitly, use &#8216;implicit=True&#8217;:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">[-LambertW(1)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">implicit</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">[-exp(x)]</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">It is possible to solve for anything that can be targeted with
-subs:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">[-sqrt(3)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">4</span> <span class="o">+</span> <span class="n">y</span><span class="p">),</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">{y: -2 + sqrt(3), x + 2: -sqrt(3)}</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">Nothing heroic is done in this implicit solving so you may end up
-with a symbol still in the solution:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">eqs</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">4</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">eqs</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">{y: sqrt(3)/(-x - 3), x + 2: (-2*x - 6 + sqrt(3))/(x + 3)}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">eqs</span><span class="p">,</span> <span class="n">y</span><span class="o">*</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">{x: -y - 4, x*y: -3*y - sqrt(3)}</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">if you attempt to solve for a number remember that the number
-you have obtained does not necessarily mean that the value is
-equivalent to the expression obtained:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">[sqrt(2)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>  <span class="c"># /!\ -1 is targeted, too</span>
-<span class="go">[x/(y - 1)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">_</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="n">solve</span><span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span> <span class="mi">1</span><span class="p">)]</span>
-<span class="go">[-x + y]</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">To solve for a function within a derivative, use dsolve.</p>
-</li>
-</ul>
-</div></blockquote>
-</li>
-<li><p class="first">single expression and more than 1 symbol</p>
-<blockquote>
-<div><ul>
-<li><p class="first">when there is a linear solution</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">[{x: y**2}]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">[{y: x**2}]</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">when undetermined coefficients are identified</p>
-<blockquote>
-<div><ul>
-<li><p class="first">that are linear</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">((</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="n">b</span> <span class="o">+</span> <span class="mi">2</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-<span class="go">{a: -2, b: 2}</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">that are nonlinear</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">set</span><span class="p">(</span><span class="n">solve</span><span class="p">((</span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="p">)</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="n">b</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
-<span class="go">set([(-sqrt(2), sqrt(2)), (sqrt(2), -sqrt(2))])</span>
-</pre></div>
-</div>
-</li>
-</ul>
-</div></blockquote>
-</li>
-<li><p class="first">if there is no linear solution then the first successful
-attempt for a nonlinear solution will be returned</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span> 
-<span class="go">[{x: -y}, {x: y}]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">[{x: 2*LambertW(y/2)}]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> 
-<span class="go">[{y: -x*exp(x/2)}, {y: x*exp(x/2)}]</span>
-</pre></div>
-</div>
-</li>
-</ul>
-</div></blockquote>
-</li>
-<li><p class="first">iterable of one or more of the above</p>
-<blockquote>
-<div><ul>
-<li><p class="first">involving relationals or bools</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">([</span><span class="n">x</span> <span class="o">&lt;</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">])</span>
-<span class="go">And(im(x) == 0, re(x) == 2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">([</span><span class="n">x</span> <span class="o">&gt;</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">])</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">when the system is linear</p>
-<blockquote>
-<div><ul>
-<li><p class="first">with a solution</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">([</span><span class="n">x</span> <span class="o">-</span> <span class="mi">3</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">{x: 3}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">5</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">6</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">15</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">{x: -3, y: 1}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">5</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">6</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">15</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">{x: -3, y: 1}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">5</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">6</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="n">z</span><span class="p">),</span> <span class="n">z</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">{x: -5*y + 2, z: 21*y - 6}</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">without a solution</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">([</span><span class="n">x</span> <span class="o">+</span> <span class="mi">3</span><span class="p">,</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">3</span><span class="p">])</span>
-<span class="go">[]</span>
-</pre></div>
-</div>
-</li>
-</ul>
-</div></blockquote>
-</li>
-<li><p class="first">when the system is not linear</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">set</span><span class="p">(</span><span class="n">solve</span><span class="p">([</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">4</span><span class="p">],</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span>
-<span class="go">set([(-2, -2), (0, 2), (2, -2)])</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">if no symbols are given, all free symbols will be selected and a list
-of mappings returned</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">([</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="p">])</span>
-<span class="go">[{x: 2, y: -4}]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">([</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)],</span> <span class="nb">set</span><span class="p">([</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">]))</span>
-<span class="go">[{x: 2, f(x): -4}]</span>
-</pre></div>
-</div>
-</li>
-<li><p class="first">if any equation doesn&#8217;t depend on the symbol(s) given it will be
-eliminated from the equation set and an answer may be given
-implicitly in terms of variables that were not of interest</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">([</span><span class="n">x</span> <span class="o">-</span> <span class="n">y</span><span class="p">,</span> <span class="n">y</span> <span class="o">-</span> <span class="mi">3</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">{x: y}</span>
-</pre></div>
-</div>
-</li>
-</ul>
-</div></blockquote>
-</li>
-</ul>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.solvers.solvers.solve_linear">
-<tt class="descclassname">sympy.solvers.solvers.</tt><tt class="descname">solve_linear</tt><big>(</big><em>lhs</em>, <em>rhs=0</em>, <em>symbols=</em><span class="optional">[</span><span class="optional">]</span>, <em>exclude=</em><span class="optional">[</span><span class="optional">]</span><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/solvers.html#solve_linear"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.solvers.solve_linear" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a tuple derived from f = lhs - rhs that is either:</p>
-<blockquote>
-<div><dl class="docutils">
-<dt>(numerator, denominator) of <tt class="docutils literal"><span class="pre">f</span></tt></dt>
-<dd><p class="first">If this comes back as (0, 1) it means
-that <tt class="docutils literal"><span class="pre">f</span></tt> is independent of the symbols in <tt class="docutils literal"><span class="pre">symbols</span></tt>, e.g:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">y</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">y</span> <span class="o">=</span> <span class="n">y</span><span class="o">*</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="o">=</span> <span class="mi">0</span>
-<span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">=</span> <span class="mi">1</span>
-</pre></div>
-</div>
-<p>If it comes back as (0, 0) there is no solution to the equation
-amongst the symbols given.</p>
-<p class="last">If the numerator is not zero then the function is guaranteed
-to be dependent on a symbol in <tt class="docutils literal"><span class="pre">symbols</span></tt>.</p>
-</dd>
-</dl>
-<p>or</p>
-<p>(symbol, solution) where symbol appears linearly in the numerator of
-<tt class="docutils literal"><span class="pre">f</span></tt>, is in <tt class="docutils literal"><span class="pre">symbols</span></tt> (if given) and is not in <tt class="docutils literal"><span class="pre">exclude</span></tt> (if given).</p>
-<p>No simplification is done to <tt class="docutils literal"><span class="pre">f</span></tt> other than and mul=True expansion,
-so the solution will correspond strictly to a unique solution.</p>
-</div></blockquote>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.solvers.solvers</span> <span class="kn">import</span> <span class="n">solve_linear</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-</pre></div>
-</div>
-<p>These are linear in x and 1/x:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve_linear</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(x, -y**2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve_linear</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span> <span class="o">-</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">(x, y**(-2))</span>
-</pre></div>
-</div>
-<p>When not linear in x or y then the numerator and denominator are returned.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve_linear</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">(x**2 - 3*y**2, y**2)</span>
-</pre></div>
-</div>
-<p>If the numerator is a symbol then (0, 0) is returned if the solution for
-that symbol would have set any denominator to 0:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve_linear</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">(0, 0)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span> <span class="c"># to SymPy, this looks like x ...</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve_linear</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">))</span> <span class="c"># so a solution is given</span>
-<span class="go">(x, 0)</span>
-</pre></div>
-</div>
-<p>If x is allowed to cancel, then this appears linear, but this sort of
-cancellation is not done so the solution will always satisfy the original
-expression without causing a division by zero error.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve_linear</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span> <span class="o">-</span> <span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="n">x</span><span class="p">))</span>
-<span class="go">(x**2*(-z**2 + 1), x)</span>
-</pre></div>
-</div>
-<p>You can give a list of what you prefer for x candidates:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve_linear</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">z</span><span class="p">,</span> <span class="n">symbols</span><span class="o">=</span><span class="p">[</span><span class="n">y</span><span class="p">])</span>
-<span class="go">(y, -x - z)</span>
-</pre></div>
-</div>
-<p>You can also indicate what variables you don&#8217;t want to consider:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve_linear</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">z</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">z</span><span class="p">])</span>
-<span class="go">(y, -x - z)</span>
-</pre></div>
-</div>
-<p>If only x was excluded then a solution for y or z might be obtained.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.solvers.solvers.solve_linear_system">
-<tt class="descclassname">sympy.solvers.solvers.</tt><tt class="descname">solve_linear_system</tt><big>(</big><em>system</em>, <em>*symbols</em>, <em>**flags</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/solvers.html#solve_linear_system"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.solvers.solve_linear_system" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solve system of N linear equations with M variables, which means
-both under- and overdetermined systems are supported. The possible
-number of solutions is zero, one or infinite. Respectively, this
-procedure will return None or a dictionary with solutions. In the
-case of underdetermined systems, all arbitrary parameters are skipped.
-This may cause a situation in which an empty dictionary is returned.
-In that case, all symbols can be assigned arbitrary values.</p>
-<p>Input to this functions is a Nx(M+1) matrix, which means it has
-to be in augmented form. If you prefer to enter N equations and M
-unknowns then use <span class="math">\(solve(Neqs, *Msymbols)\)</span> instead. Note: a local
-copy of the matrix is made by this routine so the matrix that is
-passed will not be modified.</p>
-<p>The algorithm used here is fraction-free Gaussian elimination,
-which results, after elimination, in an upper-triangular matrix.
-Then solutions are found using back-substitution. This approach
-is more efficient and compact than the Gauss-Jordan method.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">solve_linear_system</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<p>Solve the following system:</p>
-<div class="highlight-python"><pre>   x + 4 y ==  2
--2 x +   y == 14</pre>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">system</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">((</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">14</span><span class="p">)))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve_linear_system</span><span class="p">(</span><span class="n">system</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">{x: -6, y: 2}</span>
-</pre></div>
-</div>
-<p>A degenerate system returns an empty dictionary.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">system</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">((</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">)</span> <span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve_linear_system</span><span class="p">(</span><span class="n">system</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">{}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.solvers.solvers.solve_linear_system_LU">
-<tt class="descclassname">sympy.solvers.solvers.</tt><tt class="descname">solve_linear_system_LU</tt><big>(</big><em>matrix</em>, <em>syms</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/solvers.html#solve_linear_system_LU"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.solvers.solve_linear_system_LU" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solves the augmented matrix system using LUsolve and returns a dictionary
-in which solutions are keyed to the symbols of syms <em>as ordered</em>.</p>
-<p>The matrix must be invertible.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.matrices.LUsolve</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.solvers.solvers</span> <span class="kn">import</span> <span class="n">solve_linear_system_LU</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve_linear_system_LU</span><span class="p">(</span><span class="n">Matrix</span><span class="p">([</span>
-<span class="gp">... </span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
-<span class="gp">... </span><span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
-<span class="gp">... </span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]]),</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">])</span>
-<span class="go">{x: 1/2, y: 1/4, z: -1/2}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.solvers.solvers.solve_undetermined_coeffs">
-<tt class="descclassname">sympy.solvers.solvers.</tt><tt class="descname">solve_undetermined_coeffs</tt><big>(</big><em>equ</em>, <em>coeffs</em>, <em>sym</em>, <em>**flags</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/solvers.html#solve_undetermined_coeffs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.solvers.solve_undetermined_coeffs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solve equation of a type p(x; a_1, ..., a_k) == q(x) where both
-p, q are univariate polynomials and f depends on k parameters.
-The result of this functions is a dictionary with symbolic
-values of those parameters with respect to coefficients in q.</p>
-<p>This functions accepts both Equations class instances and ordinary
-SymPy expressions. Specification of parameters and variable is
-obligatory for efficiency and simplicity reason.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Eq</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.solvers</span> <span class="kn">import</span> <span class="n">solve_undetermined_coeffs</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve_undetermined_coeffs</span><span class="p">(</span><span class="n">Eq</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="p">[</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">{a: 1/2, b: -1/2}</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve_undetermined_coeffs</span><span class="p">(</span><span class="n">Eq</span><span class="p">(</span><span class="n">a</span><span class="o">*</span><span class="n">c</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">a</span><span class="o">+</span><span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="p">[</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">{a: 1/c, b: -1/c}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.solvers.solvers.tsolve">
-<tt class="descclassname">sympy.solvers.solvers.</tt><tt class="descname">tsolve</tt><big>(</big><em>eq</em>, <em>sym</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/solvers.html#tsolve"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.solvers.tsolve" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="sympy.solvers.solvers.nsolve">
-<tt class="descclassname">sympy.solvers.solvers.</tt><tt class="descname">nsolve</tt><big>(</big><em>*args</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/solvers.html#nsolve"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.solvers.nsolve" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solve a nonlinear equation system numerically:</p>
-<div class="highlight-python"><pre>nsolve(f, [args,] x0, modules=[&#x27;mpmath&#x27;], **kwargs)</pre>
-</div>
-<p>f is a vector function of symbolic expressions representing the system.
-args are the variables. If there is only one variable, this argument can
-be omitted.
-x0 is a starting vector close to a solution.</p>
-<p>Use the modules keyword to specify which modules should be used to
-evaluate the function and the Jacobian matrix. Make sure to use a module
-that supports matrices. For more information on the syntax, please see the
-docstring of lambdify.</p>
-<p>Overdetermined systems are supported.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">nsolve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">sympy</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sympy</span><span class="o">.</span><span class="n">mpmath</span><span class="o">.</span><span class="n">mp</span><span class="o">.</span><span class="n">dps</span> <span class="o">=</span> <span class="mi">15</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x1</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x1&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x2</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x2&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f1</span> <span class="o">=</span> <span class="mi">3</span> <span class="o">*</span> <span class="n">x1</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">x2</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f2</span> <span class="o">=</span> <span class="n">x1</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">x1</span> <span class="o">+</span> <span class="n">x2</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">x2</span> <span class="o">-</span> <span class="mi">8</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">nsolve</span><span class="p">((</span><span class="n">f1</span><span class="p">,</span> <span class="n">f2</span><span class="p">),</span> <span class="p">(</span><span class="n">x1</span><span class="p">,</span> <span class="n">x2</span><span class="p">),</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)))</span>
-<span class="go">[-1.19287309935246]</span>
-<span class="go">[ 1.27844411169911]</span>
-</pre></div>
-</div>
-<p>For one-dimensional functions the syntax is simplified:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">nsolve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsolve</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">3.14159265358979</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nsolve</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">3.14159265358979</span>
-</pre></div>
-</div>
-<p>mpmath.findroot is used, you can find there more extensive documentation,
-especially concerning keyword parameters and available solvers.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.solvers.solvers.check_assumptions">
-<tt class="descclassname">sympy.solvers.solvers.</tt><tt class="descname">check_assumptions</tt><big>(</big><em>expr</em>, <em>**assumptions</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/solvers.html#check_assumptions"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.solvers.check_assumptions" title="Permalink to this definition">¶</a></dt>
-<dd><p>Checks whether expression <span class="math">\(expr\)</span> satisfies all assumptions.</p>
-<p><span class="math">\(assumptions\)</span> is a dict of assumptions: {&#8216;assumption&#8217;: True|False, ...}.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">I</span><span class="p">,</span> <span class="n">exp</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.solvers.solvers</span> <span class="kn">import</span> <span class="n">check_assumptions</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">check_assumptions</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">check_assumptions</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">check_assumptions</span><span class="p">(</span><span class="n">pi</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">negative</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">check_assumptions</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">pi</span><span class="o">/</span><span class="mi">7</span><span class="p">),</span> <span class="n">real</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">check_assumptions</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">check_assumptions</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">5</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<p><span class="math">\(None\)</span> is returned if check_assumptions() could not conclude.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">check_assumptions</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">z</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;z&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">check_assumptions</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.solvers.solvers.checksol">
-<tt class="descclassname">sympy.solvers.solvers.</tt><tt class="descname">checksol</tt><big>(</big><em>f</em>, <em>symbol</em>, <em>sol=None</em>, <em>**flags</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/solvers.html#checksol"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.solvers.checksol" title="Permalink to this definition">¶</a></dt>
-<dd><p>Checks whether sol is a solution of equation f == 0.</p>
-<p>Input can be either a single symbol and corresponding value
-or a dictionary of symbols and values. <tt class="docutils literal"><span class="pre">f</span></tt> can be a single
-equation or an iterable of equations. A solution must satisfy
-all equations in <tt class="docutils literal"><span class="pre">f</span></tt> to be considered valid; if a solution
-does not satisfy any equation, False is returned; if one or
-more checks are inconclusive (and none are False) then None
-is returned.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.solvers</span> <span class="kn">import</span> <span class="n">checksol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">checksol</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">checksol</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">checksol</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">5</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="p">{</span><span class="n">x</span><span class="p">:</span> <span class="mi">3</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="mi">4</span><span class="p">})</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>To check if an expression is zero using checksol, pass it
-as <tt class="docutils literal"><span class="pre">f</span></tt> and send an empty dictionary for <tt class="docutils literal"><span class="pre">symbol</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">checksol</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span> <span class="o">-</span> <span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">),</span> <span class="p">{})</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>None is returned if checksol() could not conclude.</p>
-<dl class="docutils">
-<dt>flags:</dt>
-<dd><dl class="first last docutils">
-<dt>&#8216;numerical=True (default)&#8217;</dt>
-<dd>do a fast numerical check if <tt class="docutils literal"><span class="pre">f</span></tt> has only one symbol.</dd>
-<dt>&#8216;minimal=True (default is False)&#8217;</dt>
-<dd>a very fast, minimal testing.</dd>
-<dt>&#8216;warn=True (default is False)&#8217;</dt>
-<dd>print a warning if checksol() could not conclude.</dd>
-<dt>&#8216;simplify=True (default)&#8217;</dt>
-<dd>simplify solution before substituting into function and
-simplify the function before trying specific simplifications</dd>
-<dt>&#8216;force=True (default is False)&#8217;</dt>
-<dd>make positive all symbols without assumptions regarding sign.</dd>
-</dl>
-</dd>
-</dl>
-</dd></dl>
-
-</div>
-<div class="section" id="ordinary-differential-equations-odes">
-<h2>Ordinary Differential equations (ODEs)<a class="headerlink" href="#ordinary-differential-equations-odes" title="Permalink to this headline">¶</a></h2>
-<p>See <a class="reference internal" href="../solvers/ode.html#ode-docs"><em>ODE</em></a>.</p>
-</div>
-<div class="section" id="partial-differential-equations-pdes">
-<h2>Partial Differential Equations (PDEs)<a class="headerlink" href="#partial-differential-equations-pdes" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.solvers.pde.pde_separate">
-<tt class="descclassname">sympy.solvers.pde.</tt><tt class="descname">pde_separate</tt><big>(</big><em>eq</em>, <em>fun</em>, <em>sep</em>, <em>strategy='mul'</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/pde.html#pde_separate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.pde.pde_separate" title="Permalink to this definition">¶</a></dt>
-<dd><p>Separate variables in partial differential equation either by additive
-or multiplicative separation approach. It tries to rewrite an equation so
-that one of the specified variables occurs on a different side of the
-equation than the others.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first last simple">
-<li><strong>eq</strong> &#8211; Partial differential equation</li>
-<li><strong>fun</strong> &#8211; Original function F(x, y, z)</li>
-<li><strong>sep</strong> &#8211; List of separated functions [X(x), u(y, z)]</li>
-<li><strong>strategy</strong> &#8211; Separation strategy. You can choose between additive
-separation (&#8216;add&#8217;) and multiplicative separation (&#8216;mul&#8217;) which is
-default.</li>
-</ul>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">pde_separate_add</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">pde_separate_mul</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">E</span><span class="p">,</span> <span class="n">Eq</span><span class="p">,</span> <span class="n">Function</span><span class="p">,</span> <span class="n">pde_separate</span><span class="p">,</span> <span class="n">Derivative</span> <span class="k">as</span> <span class="n">D</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">t</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Function</span><span class="p">,</span> <span class="s">&#39;uXT&#39;</span><span class="p">))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span> <span class="o">=</span> <span class="n">Eq</span><span class="p">(</span><span class="n">D</span><span class="p">(</span><span class="n">u</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">E</span><span class="o">**</span><span class="p">(</span><span class="n">u</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">))</span><span class="o">*</span><span class="n">D</span><span class="p">(</span><span class="n">u</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">),</span> <span class="n">t</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pde_separate</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">u</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">),</span> <span class="p">[</span><span class="n">X</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">T</span><span class="p">(</span><span class="n">t</span><span class="p">)],</span> <span class="n">strategy</span><span class="o">=</span><span class="s">&#39;add&#39;</span><span class="p">)</span>
-<span class="go">[exp(-X(x))*Derivative(X(x), x), exp(T(t))*Derivative(T(t), t)]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span> <span class="o">=</span> <span class="n">Eq</span><span class="p">(</span><span class="n">D</span><span class="p">(</span><span class="n">u</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="n">u</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">),</span> <span class="n">t</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pde_separate</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">u</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">),</span> <span class="p">[</span><span class="n">X</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">T</span><span class="p">(</span><span class="n">t</span><span class="p">)],</span> <span class="n">strategy</span><span class="o">=</span><span class="s">&#39;mul&#39;</span><span class="p">)</span>
-<span class="go">[Derivative(X(x), x, x)/X(x), Derivative(T(t), t, t)/T(t)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.solvers.pde.pde_separate_add">
-<tt class="descclassname">sympy.solvers.pde.</tt><tt class="descname">pde_separate_add</tt><big>(</big><em>eq</em>, <em>fun</em>, <em>sep</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/pde.html#pde_separate_add"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.pde.pde_separate_add" title="Permalink to this definition">¶</a></dt>
-<dd><p>Helper function for searching additive separable solutions.</p>
-<p>Consider an equation of two independent variables x, y and a dependent
-variable w, we look for the product of two functions depending on different
-arguments:</p>
-<p><span class="math">\(w(x, y, z) = X(x) + y(y, z)\)</span></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">E</span><span class="p">,</span> <span class="n">Eq</span><span class="p">,</span> <span class="n">Function</span><span class="p">,</span> <span class="n">pde_separate_add</span><span class="p">,</span> <span class="n">Derivative</span> <span class="k">as</span> <span class="n">D</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">t</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">T</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Function</span><span class="p">,</span> <span class="s">&#39;uXT&#39;</span><span class="p">))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span> <span class="o">=</span> <span class="n">Eq</span><span class="p">(</span><span class="n">D</span><span class="p">(</span><span class="n">u</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">E</span><span class="o">**</span><span class="p">(</span><span class="n">u</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">))</span><span class="o">*</span><span class="n">D</span><span class="p">(</span><span class="n">u</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">),</span> <span class="n">t</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pde_separate_add</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">u</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">t</span><span class="p">),</span> <span class="p">[</span><span class="n">X</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">T</span><span class="p">(</span><span class="n">t</span><span class="p">)])</span>
-<span class="go">[exp(-X(x))*Derivative(X(x), x), exp(T(t))*Derivative(T(t), t)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.solvers.pde.pde_separate_mul">
-<tt class="descclassname">sympy.solvers.pde.</tt><tt class="descname">pde_separate_mul</tt><big>(</big><em>eq</em>, <em>fun</em>, <em>sep</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/pde.html#pde_separate_mul"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.pde.pde_separate_mul" title="Permalink to this definition">¶</a></dt>
-<dd><p>Helper function for searching multiplicative separable solutions.</p>
-<p>Consider an equation of two independent variables x, y and a dependent
-variable w, we look for the product of two functions depending on different
-arguments:</p>
-<p><span class="math">\(w(x, y, z) = X(x)*u(y, z)\)</span></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Eq</span><span class="p">,</span> <span class="n">pde_separate_mul</span><span class="p">,</span> <span class="n">Derivative</span> <span class="k">as</span> <span class="n">D</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Function</span><span class="p">,</span> <span class="s">&#39;uXY&#39;</span><span class="p">))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">eq</span> <span class="o">=</span> <span class="n">Eq</span><span class="p">(</span><span class="n">D</span><span class="p">(</span><span class="n">u</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">D</span><span class="p">(</span><span class="n">u</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="n">y</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pde_separate_mul</span><span class="p">(</span><span class="n">eq</span><span class="p">,</span> <span class="n">u</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="p">[</span><span class="n">X</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">Y</span><span class="p">(</span><span class="n">y</span><span class="p">)])</span>
-<span class="go">[Derivative(X(x), x, x)/X(x), Derivative(Y(y), y, y)/Y(y)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="recurrence-equtions">
-<h2>Recurrence Equtions<a class="headerlink" href="#recurrence-equtions" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.solvers.recurr.rsolve">
-<tt class="descclassname">sympy.solvers.recurr.</tt><tt class="descname">rsolve</tt><big>(</big><em>f</em>, <em>y</em>, <em>init=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/recurr.html#rsolve"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.recurr.rsolve" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solve univariate recurrence with rational coefficients.</p>
-<p>Given k-th order linear recurrence Ly = f, or equivalently:</p>
-<blockquote>
-<div>a_{k}(n) y(n+k) + a_{k-1}(n) y(n+k-1) + ... + a_{0}(n) y(n) = f</div></blockquote>
-<p>where a_{i}(n), for i=0..k, are polynomials or rational functions
-in n, and f is a hypergeometric function or a sum of a fixed number
-of pairwise dissimilar hypergeometric terms in n, finds all solutions
-or returns None, if none were found.</p>
-<p>Initial conditions can be given as a dictionary in two forms:</p>
-<blockquote>
-<div>[1] {   n_0  : v_0,   n_1  : v_1, ...,   n_m  : v_m }
-[2] { y(n_0) : v_0, y(n_1) : v_1, ..., y(n_m) : v_m }</div></blockquote>
-<p>or as a list L of values:</p>
-<blockquote>
-<div>L = [ v_0, v_1, ..., v_m ]</div></blockquote>
-<p>where L[i] = v_i, for i=0..m, maps to y(n_i).</p>
-<p>As an example lets consider the following recurrence:</p>
-<blockquote>
-<div>(n - 1) y(n + 2) - (n**2 + 3 n - 2) y(n + 1) + 2 n (n + 1) y(n) == 0</div></blockquote>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">rsolve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">y</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="n">n</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="mi">3</span><span class="o">*</span><span class="n">n</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">y</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">y</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rsolve</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">y</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
-<span class="go">2**n*C0 + C1*n!</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rsolve</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">y</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="p">{</span> <span class="n">y</span><span class="p">(</span><span class="mi">0</span><span class="p">):</span><span class="mi">0</span><span class="p">,</span> <span class="n">y</span><span class="p">(</span><span class="mi">1</span><span class="p">):</span><span class="mi">3</span> <span class="p">})</span>
-<span class="go">3*2**n - 3*n!</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.solvers.recurr.rsolve_poly">
-<tt class="descclassname">sympy.solvers.recurr.</tt><tt class="descname">rsolve_poly</tt><big>(</big><em>coeffs</em>, <em>f</em>, <em>n</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/recurr.html#rsolve_poly"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.recurr.rsolve_poly" title="Permalink to this definition">¶</a></dt>
-<dd><p>Given linear recurrence operator L of order &#8216;k&#8217; with polynomial
-coefficients and inhomogeneous equation Ly = f, where &#8216;f&#8217; is a
-polynomial, we seek for all polynomial solutions over field K
-of characteristic zero.</p>
-<p>The algorithm performs two basic steps:</p>
-<blockquote>
-<div><ol class="arabic simple">
-<li>Compute degree N of the general polynomial solution.</li>
-<li>Find all polynomials of degree N or less of Ly = f.</li>
-</ol>
-</div></blockquote>
-<p>There are two methods for computing the polynomial solutions.
-If the degree bound is relatively small, i.e. it&#8217;s smaller than
-or equal to the order of the recurrence, then naive method of
-undetermined coefficients is being used. This gives system
-of algebraic equations with N+1 unknowns.</p>
-<p>In the other case, the algorithm performs transformation of the
-initial equation to an equivalent one, for which the system of
-algebraic equations has only &#8216;r&#8217; indeterminates. This method is
-quite sophisticated (in comparison with the naive one) and was
-invented together by Abramov, Bronstein and Petkovsek.</p>
-<p>It is possible to generalize the algorithm implemented here to
-the case of linear q-difference and differential equations.</p>
-<p>Lets say that we would like to compute m-th Bernoulli polynomial
-up to a constant. For this we can use b(n+1) - b(n) == m*n**(m-1)
-recurrence, which has solution b(n) = B_m + C. For example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">rsolve_poly</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;n&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rsolve_poly</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="mi">4</span><span class="o">*</span><span class="n">n</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-<span class="go">C0 + n**4 - 2*n**3 + n**2</span>
-</pre></div>
-</div>
-<p>For more information on implemented algorithms refer to:</p>
-<dl class="docutils">
-<dt>[1] S. A. Abramov, M. Bronstein and M. Petkovsek, On polynomial</dt>
-<dd>solutions of linear operator equations, in: T. Levelt, ed.,
-Proc. ISSAC &#8216;95, ACM Press, New York, 1995, 290-296.</dd>
-<dt>[2] M. Petkovsek, Hypergeometric solutions of linear recurrences</dt>
-<dd>with polynomial coefficients, J. Symbolic Computation,
-14 (1992), 243-264.</dd>
-</dl>
-<p>[3] M. Petkovsek, H. S. Wilf, D. Zeilberger, A = B, 1996.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.solvers.recurr.rsolve_ratio">
-<tt class="descclassname">sympy.solvers.recurr.</tt><tt class="descname">rsolve_ratio</tt><big>(</big><em>coeffs</em>, <em>f</em>, <em>n</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/recurr.html#rsolve_ratio"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.recurr.rsolve_ratio" title="Permalink to this definition">¶</a></dt>
-<dd><p>Given linear recurrence operator L of order &#8216;k&#8217; with polynomial
-coefficients and inhomogeneous equation Ly = f, where &#8216;f&#8217; is a
-polynomial, we seek for all rational solutions over field K of
-characteristic zero.</p>
-<p>This procedure accepts only polynomials, however if you are
-interested in solving recurrence with rational coefficients
-then use rsolve() which will pre-process the given equation
-and run this procedure with polynomial arguments.</p>
-<p>The algorithm performs two basic steps:</p>
-<blockquote>
-<div><ol class="arabic simple">
-<li>Compute polynomial v(n) which can be used as universal
-denominator of any rational solution of equation Ly = f.</li>
-<li>Construct new linear difference equation by substitution
-y(n) = u(n)/v(n) and solve it for u(n) finding all its
-polynomial solutions. Return None if none were found.</li>
-</ol>
-</div></blockquote>
-<p>Algorithm implemented here is a revised version of the original
-Abramov&#8217;s algorithm, developed in 1989. The new approach is much
-simpler to implement and has better overall efficiency. This
-method can be easily adapted to q-difference equations case.</p>
-<p>Besides finding rational solutions alone, this functions is
-an important part of Hyper algorithm were it is used to find
-particular solution of inhomogeneous part of a recurrence.</p>
-<p>For more information on the implemented algorithm refer to:</p>
-<dl class="docutils">
-<dt>[1] S. A. Abramov, Rational solutions of linear difference</dt>
-<dd>and q-difference equations with polynomial coefficients,
-in: T. Levelt, ed., Proc. ISSAC &#8216;95, ACM Press, New York,
-1995, 285-289</dd>
-</dl>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.solvers.recurr</span> <span class="kn">import</span> <span class="n">rsolve_ratio</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rsolve_ratio</span><span class="p">([</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">6</span><span class="o">*</span><span class="n">x</span><span class="p">,</span>
-<span class="gp">... </span><span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">11</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">18</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">13</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">22</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">8</span><span class="p">],</span> <span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">C2*(2*x - 3)/(2*(x**2 - 1))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.solvers.recurr.rsolve_hyper">
-<tt class="descclassname">sympy.solvers.recurr.</tt><tt class="descname">rsolve_hyper</tt><big>(</big><em>coeffs</em>, <em>f</em>, <em>n</em>, <em>**hints</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/recurr.html#rsolve_hyper"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.recurr.rsolve_hyper" title="Permalink to this definition">¶</a></dt>
-<dd><p>Given linear recurrence operator L of order &#8216;k&#8217; with polynomial
-coefficients and inhomogeneous equation Ly = f we seek for all
-hypergeometric solutions over field K of characteristic zero.</p>
-<p>The inhomogeneous part can be either hypergeometric or a sum
-of a fixed number of pairwise dissimilar hypergeometric terms.</p>
-<p>The algorithm performs three basic steps:</p>
-<blockquote>
-<div><ol class="arabic simple">
-<li>Group together similar hypergeometric terms in the
-inhomogeneous part of Ly = f, and find particular
-solution using Abramov&#8217;s algorithm.</li>
-<li>Compute generating set of L and find basis in it,
-so that all solutions are linearly independent.</li>
-<li>Form final solution with the number of arbitrary
-constants equal to dimension of basis of L.</li>
-</ol>
-</div></blockquote>
-<p>Term a(n) is hypergeometric if it is annihilated by first order
-linear difference equations with polynomial coefficients or, in
-simpler words, if consecutive term ratio is a rational function.</p>
-<p>The output of this procedure is a linear combination of fixed
-number of hypergeometric terms. However the underlying method
-can generate larger class of solutions - D&#8217;Alembertian terms.</p>
-<p>Note also that this method not only computes the kernel of the
-inhomogeneous equation, but also reduces in to a basis so that
-solutions generated by this procedure are linearly independent</p>
-<p>For more information on the implemented algorithm refer to:</p>
-<dl class="docutils">
-<dt>[1] M. Petkovsek, Hypergeometric solutions of linear recurrences</dt>
-<dd>with polynomial coefficients, J. Symbolic Computation,
-14 (1992), 243-264.</dd>
-</dl>
-<p>[2] M. Petkovsek, H. S. Wilf, D. Zeilberger, A = B, 1996.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.solvers</span> <span class="kn">import</span> <span class="n">rsolve_hyper</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rsolve_hyper</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="mi">0</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">C0*(1/2 + sqrt(5)/2)**x + C1*(-sqrt(5)/2 + 1/2)**x</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">rsolve_hyper</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="mi">1</span><span class="o">+</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">C0 + x*(x + 1)/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="systems-of-polynomial-equations">
-<h2>Systems of Polynomial Equations<a class="headerlink" href="#systems-of-polynomial-equations" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.solvers.polysys.solve_poly_system">
-<tt class="descclassname">sympy.solvers.polysys.</tt><tt class="descname">solve_poly_system</tt><big>(</big><em>seq</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/polysys.html#solve_poly_system"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.polysys.solve_poly_system" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solve a system of polynomial equations.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">solve_poly_system</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve_poly_system</span><span class="p">([</span><span class="n">x</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">],</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">[(0, 0), (2, -sqrt(2)), (2, sqrt(2))]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.solvers.polysys.solve_triangulated">
-<tt class="descclassname">sympy.solvers.polysys.</tt><tt class="descname">solve_triangulated</tt><big>(</big><em>polys</em>, <em>*gens</em>, <em>**args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/solvers/polysys.html#solve_triangulated"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.solvers.polysys.solve_triangulated" title="Permalink to this definition">¶</a></dt>
-<dd><p>Solve a polynomial system using Gianni-Kalkbrenner algorithm.</p>
-<p>The algorithm proceeds by computing one Groebner basis in the ground
-domain and then by iteratively computing polynomial factorizations in
-appropriately constructed algebraic extensions of the ground domain.</p>
-<p class="rubric">References</p>
-<p>1. Patrizia Gianni, Teo Mora, Algebraic Solution of System of
-Polynomial Equations using Groebner Bases, AAECC-5 on Applied Algebra,
-Algebraic Algorithms and Error-Correcting Codes, LNCS 356 247&#8211;257, 1989</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.solvers.polysys</span> <span class="kn">import</span> <span class="n">solve_triangulated</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">F</span> <span class="o">=</span> <span class="p">[</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">z</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">z</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">z</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">solve_triangulated</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
-<span class="go">[(0, 0, 1), (0, 1, 0), (1, 0, 0)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-
-
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-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Solvers</a><ul>
-<li><a class="reference internal" href="#algebraic-equations">Algebraic equations</a></li>
-<li><a class="reference internal" href="#ordinary-differential-equations-odes">Ordinary Differential equations (ODEs)</a></li>
-<li><a class="reference internal" href="#partial-differential-equations-pdes">Partial Differential Equations (PDEs)</a></li>
-<li><a class="reference internal" href="#recurrence-equtions">Recurrence Equtions</a></li>
-<li><a class="reference internal" href="#systems-of-polynomial-equations">Systems of Polynomial Equations</a></li>
-</ul>
-</li>
-</ul>
-
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-          <li><a href="index.html" accesskey="U">SymPy Modules Reference</a> &raquo;</li> 
-      </ul>
-    </div>  
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-          <div class="body">
-            
-  <div class="section" id="module-sympy.statistics">
-<span id="statistics"></span><h1>Statistics<a class="headerlink" href="#module-sympy.statistics" title="Permalink to this headline">¶</a></h1>
-<p>Note: This module has been deprecated. See the stats module.</p>
-<p>The <em>statistics</em> module in SymPy implements standard probability distributions
-and related tools. Its contents can be imported with the following statement:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.statistics</span> <span class="kn">import</span> <span class="o">*</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">init_printing</span><span class="p">(</span><span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">wrap_line</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">no_global</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="section" id="normal-distributions">
-<h2>Normal distributions<a class="headerlink" href="#normal-distributions" title="Permalink to this headline">¶</a></h2>
-<p><tt class="docutils literal"><span class="pre">Normal(mu,</span> <span class="pre">sigma)</span></tt> creates a normal distribution with mean value <tt class="docutils literal"><span class="pre">mu</span></tt> and
-standard deviation <tt class="docutils literal"><span class="pre">sigma</span></tt>. The <tt class="docutils literal"><span class="pre">Normal</span></tt> class defines several
-useful methods and properties. Various properties can be accessed directly as
-follows:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">Normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">mean</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">median</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">variance</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">stddev</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>You can generate random numbers from the desired distribution with the
-<tt class="docutils literal"><span class="pre">random</span></tt> method:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">Normal</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">random</span><span class="p">()</span> 
-<span class="go">4.914375200829805834246144514</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">random</span><span class="p">()</span> 
-<span class="go">11.84331557474637897087177407</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">random</span><span class="p">()</span> 
-<span class="go">17.22474580071733640806996846</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">random</span><span class="p">()</span> 
-<span class="go">9.864643097429464546621602494</span>
-</pre></div>
-</div>
-<p>The probability density function (pdf) and cumulative distribution function
-(cdf) of a distribution can be computed, either in symbolic form or for
-particular values:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">Normal</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">   ___</span>
-<span class="go"> \/ 2</span>
-<span class="go">--------</span>
-<span class="go">    ____</span>
-<span class="go">2*\/ pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">0.0539909665131880</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">   /  ___        \</span>
-<span class="go">   |\/ 2 *(x - 1)|</span>
-<span class="go">erf|-------------|</span>
-<span class="go">   \      2      /   1</span>
-<span class="go">------------------ + -</span>
-<span class="go">        2            2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="o">-</span><span class="n">oo</span><span class="p">),</span> <span class="n">N</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span> <span class="n">N</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">oo</span><span class="p">)</span>
-<span class="go">(0, 1/2, 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">0.999968328758167</span>
-</pre></div>
-</div>
-<p>The method <tt class="docutils literal"><span class="pre">probability</span></tt> gives the total probability on a given interval (a
-convenient alternative syntax for cdf(b)-cdf(a)):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">Normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">probability</span><span class="p">(</span><span class="o">-</span><span class="n">oo</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">probability</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">   /  ___\</span>
-<span class="go">   |\/ 2 |</span>
-<span class="go">erf|-----|</span>
-<span class="go">   \  2  /</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">probability</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">0.682689492137086</span>
-</pre></div>
-</div>
-<p>You can also generate a symmetric confidence interval from a given desired
-confidence level (given as a fraction 0-1). For the normal distribution, 68%,
-95% and 99.7% confidence levels respectively correspond to approximately 1, 2
-and 3 standard deviations:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">Normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">confidence</span><span class="p">(</span><span class="mf">0.68</span><span class="p">)</span>
-<span class="go">(-0.994457883209753, 0.994457883209753)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">confidence</span><span class="p">(</span><span class="mf">0.95</span><span class="p">)</span>
-<span class="go">(-1.95996398454005, 1.95996398454005)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">confidence</span><span class="p">(</span><span class="mf">0.997</span><span class="p">)</span>
-<span class="go">(-2.96773792534178, 2.96773792534178)</span>
-</pre></div>
-</div>
-<p>Plug the interval back in to see that the value is correct:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">probability</span><span class="p">(</span><span class="o">*</span><span class="n">N</span><span class="o">.</span><span class="n">confidence</span><span class="p">(</span><span class="mf">0.95</span><span class="p">))</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">0.950000000000000</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="other-distributions">
-<h2>Other distributions<a class="headerlink" href="#other-distributions" title="Permalink to this headline">¶</a></h2>
-<p>Besides the normal distribution, uniform continuous distributions are also
-supported. <tt class="docutils literal"><span class="pre">Uniform(a,</span> <span class="pre">b)</span></tt> represents the distribution with uniform
-probability on the interval [a, b] and zero probability everywhere else. The
-<tt class="docutils literal"><span class="pre">Uniform</span></tt> class supports the same methods as the <tt class="docutils literal"><span class="pre">Normal</span></tt> class.</p>
-<p>Additional distributions, including support for arbitrary user-defined distributions, are planned for the future.</p>
-</div>
-<div class="section" id="api-reference">
-<h2>API Reference<a class="headerlink" href="#api-reference" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="sample">
-<h3>Sample<a class="headerlink" href="#sample" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.statistics.distributions.Sample">
-<em class="property">class </em><tt class="descclassname">sympy.statistics.distributions.</tt><tt class="descname">Sample</tt><a class="reference internal" href="../_modules/sympy/statistics/distributions.html#Sample"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.statistics.distributions.Sample" title="Permalink to this definition">¶</a></dt>
-<dd><p>Sample([x1, x2, x3, ...]) represents a collection of samples.
-Sample parameters like mean, variance and stddev can be accessed as
-properties.
-The sample will be sorted.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.statistics.distributions</span> <span class="kn">import</span> <span class="n">Sample</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sample</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="go">Sample([0, 1, 2, 3])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Sample</span><span class="p">([</span><span class="mi">8</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="go">Sample([1, 2, 2, 3, 4, 6, 8, 9])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">Sample</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="o">.</span><span class="n">mean</span>
-<span class="go">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="o">.</span><span class="n">stddev</span>
-<span class="go">sqrt(2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="o">.</span><span class="n">median</span>
-<span class="go">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span><span class="o">.</span><span class="n">variance</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="continuous-probability-distributions">
-<h3>Continuous Probability Distributions<a class="headerlink" href="#continuous-probability-distributions" title="Permalink to this headline">¶</a></h3>
-<dl class="class">
-<dt id="sympy.statistics.distributions.ContinuousProbability">
-<em class="property">class </em><tt class="descclassname">sympy.statistics.distributions.</tt><tt class="descname">ContinuousProbability</tt><a class="reference internal" href="../_modules/sympy/statistics/distributions.html#ContinuousProbability"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.statistics.distributions.ContinuousProbability" title="Permalink to this definition">¶</a></dt>
-<dd><p>Base class for continuous probability distributions</p>
-<dl class="function">
-<dt id="sympy.statistics.distributions.ContinuousProbability.probability">
-<tt class="descname">probability</tt><big>(</big><em>s</em>, <em>a</em>, <em>b</em><big>)</big><a class="reference internal" href="../_modules/sympy/statistics/distributions.html#ContinuousProbability.probability"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.statistics.distributions.ContinuousProbability.probability" title="Permalink to this definition">¶</a></dt>
-<dd><p>Calculate the probability that a random number x generated
-from the distribution satisfies a &lt;= x &lt;= b</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.statistics</span> <span class="kn">import</span> <span class="n">Normal</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">probability</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">erf(sqrt(2)/2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">probability</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">-erf(sqrt(2)/2)/2 + 1/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.statistics.distributions.ContinuousProbability.random">
-<tt class="descname">random</tt><big>(</big><em>s</em>, <em>n=None</em><big>)</big><a class="reference internal" href="../_modules/sympy/statistics/distributions.html#ContinuousProbability.random"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.statistics.distributions.ContinuousProbability.random" title="Permalink to this definition">¶</a></dt>
-<dd><p>random() &#8211; generate a random number from the distribution.
-random(n) &#8211; generate a Sample of n random numbers.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.statistics</span> <span class="kn">import</span> <span class="n">Uniform</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Uniform</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span><span class="o">.</span><span class="n">random</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">&lt;</span> <span class="mi">5</span> <span class="ow">and</span> <span class="n">x</span> <span class="o">&gt;</span> <span class="mi">1</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Uniform</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">random</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">&lt;</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">x</span> <span class="o">&gt;</span> <span class="o">-</span><span class="mi">4</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.statistics.distributions.Normal">
-<em class="property">class </em><tt class="descclassname">sympy.statistics.distributions.</tt><tt class="descname">Normal</tt><big>(</big><em>mu</em>, <em>sigma</em><big>)</big><a class="reference internal" href="../_modules/sympy/statistics/distributions.html#Normal"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.statistics.distributions.Normal" title="Permalink to this definition">¶</a></dt>
-<dd><p>Normal(mu, sigma) represents the normal or Gaussian distribution
-with mean value mu and standard deviation sigma.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.statistics</span> <span class="kn">import</span> <span class="n">Normal</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">Normal</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">mean</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">variance</span>
-<span class="go">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">probability</span><span class="p">(</span><span class="o">-</span><span class="n">oo</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>   <span class="c"># probability on an interval</span>
-<span class="go">1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">probability</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">probability</span><span class="p">(</span><span class="o">-</span><span class="n">oo</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">probability</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">erf(sqrt(2)/2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">0.682689492137086</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.statistics.distributions.Normal.cdf">
-<tt class="descname">cdf</tt><big>(</big><em>s</em>, <em>x</em><big>)</big><a class="reference internal" href="../_modules/sympy/statistics/distributions.html#Normal.cdf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.statistics.distributions.Normal.cdf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the cumulative density function as an expression in x</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.statistics</span> <span class="kn">import</span> <span class="n">Normal</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Normal</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">-erf(sqrt(2)/4)/2 + 1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Normal</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">erf(sqrt(2)*(x - 1)/4)/2 + 1/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.statistics.distributions.Normal.confidence">
-<tt class="descname">confidence</tt><big>(</big><em>s</em>, <em>p</em><big>)</big><a class="reference internal" href="../_modules/sympy/statistics/distributions.html#Normal.confidence"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.statistics.distributions.Normal.confidence" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a symmetric (p*100)% confidence interval. For example,
-p=0.95 gives a 95% confidence interval. Currently this function
-only handles numerical values except in the trivial case p=1.</p>
-<p>For example, one standard deviation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.statistics</span> <span class="kn">import</span> <span class="n">Normal</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">Normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">confidence</span><span class="p">(</span><span class="mf">0.68</span><span class="p">)</span>
-<span class="go">(-0.994457883209753, 0.994457883209753)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">probability</span><span class="p">(</span><span class="o">*</span><span class="n">_</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">0.680000000000000</span>
-</pre></div>
-</div>
-<p>Two standard deviations:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="n">Normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">confidence</span><span class="p">(</span><span class="mf">0.95</span><span class="p">)</span>
-<span class="go">(-1.95996398454005, 1.95996398454005)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span><span class="o">.</span><span class="n">probability</span><span class="p">(</span><span class="o">*</span><span class="n">_</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">0.950000000000000</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.statistics.distributions.Normal.fit">
-<tt class="descname">fit</tt><big>(</big><em>sample</em><big>)</big><a class="reference internal" href="../_modules/sympy/statistics/distributions.html#Normal.fit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.statistics.distributions.Normal.fit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a normal distribution fit to the mean and standard
-deviation of the given distribution or sample.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.statistics</span> <span class="kn">import</span> <span class="n">Normal</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Normal</span><span class="o">.</span><span class="n">fit</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">])</span>
-<span class="go">Normal(3, sqrt(2))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Normal</span><span class="o">.</span><span class="n">fit</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">])</span>
-<span class="go">Normal(x/2 + y/2, sqrt((-x/2 + y/2)**2/2 + (x/2 - y/2)**2/2))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.statistics.distributions.Normal.pdf">
-<tt class="descname">pdf</tt><big>(</big><em>s</em>, <em>x</em><big>)</big><a class="reference internal" href="../_modules/sympy/statistics/distributions.html#Normal.pdf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.statistics.distributions.Normal.pdf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the probability density function as an expression in x</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.statistics</span> <span class="kn">import</span> <span class="n">Normal</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Normal</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">sqrt(2)*exp(-1/8)/(4*sqrt(pi))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Normal</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">sqrt(2)*exp(-(x - 1)**2/8)/(4*sqrt(pi))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.statistics.distributions.Uniform">
-<em class="property">class </em><tt class="descclassname">sympy.statistics.distributions.</tt><tt class="descname">Uniform</tt><big>(</big><em>a</em>, <em>b</em><big>)</big><a class="reference internal" href="../_modules/sympy/statistics/distributions.html#Uniform"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.statistics.distributions.Uniform" title="Permalink to this definition">¶</a></dt>
-<dd><p>Uniform(a, b) represents a probability distribution with uniform
-probability density on the interval [a, b] and zero density
-everywhere else.</p>
-<dl class="function">
-<dt id="sympy.statistics.distributions.Uniform.cdf">
-<tt class="descname">cdf</tt><big>(</big><em>s</em>, <em>x</em><big>)</big><a class="reference internal" href="../_modules/sympy/statistics/distributions.html#Uniform.cdf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.statistics.distributions.Uniform.cdf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the cumulative density function as an expression in x</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.statistics</span> <span class="kn">import</span> <span class="n">Uniform</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Uniform</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">1/4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Uniform</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">3/4</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.statistics.distributions.Uniform.confidence">
-<tt class="descname">confidence</tt><big>(</big><em>s</em>, <em>p</em><big>)</big><a class="reference internal" href="../_modules/sympy/statistics/distributions.html#Uniform.confidence"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.statistics.distributions.Uniform.confidence" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generate a symmetric (p*100)% confidence interval.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.statistics</span> <span class="kn">import</span> <span class="n">Uniform</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">U</span> <span class="o">=</span> <span class="n">Uniform</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">U</span><span class="o">.</span><span class="n">confidence</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">(1, 2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">U</span><span class="o">.</span><span class="n">confidence</span><span class="p">(</span><span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">(5/4, 7/4)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.statistics.distributions.Uniform.fit">
-<tt class="descname">fit</tt><big>(</big><em>sample</em><big>)</big><a class="reference internal" href="../_modules/sympy/statistics/distributions.html#Uniform.fit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.statistics.distributions.Uniform.fit" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a uniform distribution fit to the mean and standard
-deviation of the given distribution or sample.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.statistics</span> <span class="kn">import</span> <span class="n">Uniform</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Uniform</span><span class="o">.</span><span class="n">fit</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
-<span class="go">Uniform(-sqrt(6) + 3, sqrt(6) + 3)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Uniform</span><span class="o">.</span><span class="n">fit</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="go">Uniform(-sqrt(3)/2 + 3/2, sqrt(3)/2 + 3/2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.statistics.distributions.Uniform.pdf">
-<tt class="descname">pdf</tt><big>(</big><em>s</em>, <em>x</em><big>)</big><a class="reference internal" href="../_modules/sympy/statistics/distributions.html#Uniform.pdf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.statistics.distributions.Uniform.pdf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the probability density function as an expression in x</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.statistics</span> <span class="kn">import</span> <span class="n">Uniform</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Uniform</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">1/4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Uniform</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">1/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.statistics.distributions.PDF">
-<em class="property">class </em><tt class="descclassname">sympy.statistics.distributions.</tt><tt class="descname">PDF</tt><big>(</big><em>func</em>, <em>(x</em>, <em>a</em>, <em>b)) represents continuous probability distribution with probability distribution function func(x) on interval (a</em>, <em>b</em><big>)</big><a class="reference internal" href="../_modules/sympy/statistics/distributions.html#PDF"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.statistics.distributions.PDF" title="Permalink to this definition">¶</a></dt>
-<dd><p>If func is not normalized so that integrate(func, (x, a, b)) == 1,
-it can be normalized using PDF.normalize() method</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.statistics.distributions</span> <span class="kn">import</span> <span class="n">PDF</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">exponential</span> <span class="o">=</span> <span class="n">PDF</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">/</span><span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="n">a</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">oo</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exponential</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">exp(-x/a)/a</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exponential</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1 - exp(-x/a)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exponential</span><span class="o">.</span><span class="n">mean</span>
-<span class="go">a</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exponential</span><span class="o">.</span><span class="n">variance</span>
-<span class="go">a**2</span>
-</pre></div>
-</div>
-<dl class="function">
-<dt id="sympy.statistics.distributions.PDF.cdf">
-<tt class="descname">cdf</tt><big>(</big><em>self</em>, <em>x</em><big>)</big><a class="reference internal" href="../_modules/sympy/statistics/distributions.html#PDF.cdf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.statistics.distributions.PDF.cdf" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the cumulative density function as an expression in x</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.statistics.distributions</span> <span class="kn">import</span> <span class="n">PDF</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">exp</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">PDF</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">/</span><span class="n">y</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">oo</span><span class="p">))</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">y - y*exp(-4/y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">PDF</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span><span class="o">.</span><span class="n">cdf</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">-10*y - 100</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.statistics.distributions.PDF.normalize">
-<tt class="descname">normalize</tt><big>(</big><em>self</em><big>)</big><a class="reference internal" href="../_modules/sympy/statistics/distributions.html#PDF.normalize"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.statistics.distributions.PDF.normalize" title="Permalink to this definition">¶</a></dt>
-<dd><p>Normalize the probability distribution function so that
-integrate(self.pdf(x), (x, a, b)) == 1</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.statistics.distributions</span> <span class="kn">import</span> <span class="n">PDF</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">exponential</span> <span class="o">=</span> <span class="n">PDF</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">/</span><span class="n">a</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">oo</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exponential</span><span class="o">.</span><span class="n">normalize</span><span class="p">()</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">exp(-x/a)/a</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.statistics.distributions.PDF.transform">
-<tt class="descname">transform</tt><big>(</big><em>self</em>, <em>func</em>, <em>var</em><big>)</big><a class="reference internal" href="../_modules/sympy/statistics/distributions.html#PDF.transform"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.statistics.distributions.PDF.transform" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a probability distribution of random variable func(x)
-currently only some simple injective functions are supported</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.statistics.distributions</span> <span class="kn">import</span> <span class="n">PDF</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">PDF</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">PDF(0, ((_w,), x, x))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
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-  <h3><a href="../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Statistics</a><ul>
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-<li><a class="reference internal" href="#other-distributions">Other distributions</a></li>
-<li><a class="reference internal" href="#api-reference">API Reference</a><ul>
-<li><a class="reference internal" href="#sample">Sample</a></li>
-<li><a class="reference internal" href="#continuous-probability-distributions">Continuous Probability Distributions</a></li>
-</ul>
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-            
-  <div class="section" id="module-sympy.stats">
-<span id="stats"></span><h1>Stats<a class="headerlink" href="#module-sympy.stats" title="Permalink to this headline">¶</a></h1>
-<p>SymPy statistics module</p>
-<p>Introduces a random variable type into the SymPy language.</p>
-<p>Random variables may be declared using prebuilt functions such as
-Normal, Exponential, Coin, Die, etc...  or built with functions like FiniteRV.</p>
-<p>Queries on random expressions can be made using the functions</p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="42%" />
-<col width="58%" />
-</colgroup>
-<tbody valign="top">
-<tr class="row-odd"><td>Expression</td>
-<td>Meaning</td>
-</tr>
-<tr class="row-even"><td>P(condition)</td>
-<td>Probability</td>
-</tr>
-<tr class="row-odd"><td>E(expression)</td>
-<td>Expectation value</td>
-</tr>
-<tr class="row-even"><td>variance(expression)</td>
-<td>Variance</td>
-</tr>
-<tr class="row-odd"><td>density(expression)</td>
-<td>Probability Density Function</td>
-</tr>
-<tr class="row-even"><td>sample(expression)</td>
-<td>Produce a realization</td>
-</tr>
-<tr class="row-odd"><td>where(condition)</td>
-<td>Where the condition is true</td>
-</tr>
-</tbody>
-</table>
-<div class="section" id="examples">
-<h2>Examples<a class="headerlink" href="#examples" title="Permalink to this headline">¶</a></h2>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">P</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">variance</span><span class="p">,</span> <span class="n">Die</span><span class="p">,</span> <span class="n">Normal</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Eq</span><span class="p">,</span> <span class="n">simplify</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="o">=</span> <span class="n">Die</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">Die</span><span class="p">(</span><span class="s">&#39;Y&#39;</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span> <span class="c"># Define two six sided dice</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Z</span> <span class="o">=</span> <span class="n">Normal</span><span class="p">(</span><span class="s">&#39;Z&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="c"># Declare a Normal random variable with mean 0, std 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="p">(</span><span class="n">X</span><span class="o">&gt;</span><span class="mi">3</span><span class="p">)</span> <span class="c"># Probability X is greater than 3</span>
-<span class="go">1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E</span><span class="p">(</span><span class="n">X</span><span class="o">+</span><span class="n">Y</span><span class="p">)</span> <span class="c"># Expectation of the sum of two dice</span>
-<span class="go">7</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">variance</span><span class="p">(</span><span class="n">X</span><span class="o">+</span><span class="n">Y</span><span class="p">)</span> <span class="c"># Variance of the sum of two dice</span>
-<span class="go">35/6</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">P</span><span class="p">(</span><span class="n">Z</span><span class="o">&gt;</span><span class="mi">1</span><span class="p">))</span> <span class="c"># Probability of Z being greater than 1</span>
-<span class="go">-erf(sqrt(2)/2)/2 + 1/2</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="random-variable-types">
-<h2>Random Variable Types<a class="headerlink" href="#random-variable-types" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="finite-types">
-<h3>Finite Types<a class="headerlink" href="#finite-types" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.stats.DiscreteUniform">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">DiscreteUniform</tt><big>(</big><em>name</em>, <em>items</em><big>)</big><a class="headerlink" href="#sympy.stats.DiscreteUniform" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a Finite Random Variable representing a uniform distribution over
-the input set.</p>
-<p>Returns a RandomSymbol.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">DiscreteUniform</span><span class="p">,</span> <span class="n">density</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">DiscreteUniform</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a b c&#39;</span><span class="p">))</span> <span class="c"># equally likely over a, b, c</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">{a: 1/3, b: 1/3, c: 1/3}</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span> <span class="o">=</span> <span class="n">DiscreteUniform</span><span class="p">(</span><span class="s">&#39;Y&#39;</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">)))</span> <span class="c"># distribution over a range</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span>
-<span class="go">{0: 1/5, 1: 1/5, 2: 1/5, 3: 1/5, 4: 1/5}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Die">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Die</tt><big>(</big><em>name</em>, <em>sides=6</em><big>)</big><a class="headerlink" href="#sympy.stats.Die" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a Finite Random Variable representing a fair die.</p>
-<p>Returns a RandomSymbol.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Die</span><span class="p">,</span> <span class="n">density</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D6</span> <span class="o">=</span> <span class="n">Die</span><span class="p">(</span><span class="s">&#39;D6&#39;</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span> <span class="c"># Six sided Die</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">D6</span><span class="p">)</span>
-<span class="go">{1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6}</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D4</span> <span class="o">=</span> <span class="n">Die</span><span class="p">(</span><span class="s">&#39;D4&#39;</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="c"># Four sided Die</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">D4</span><span class="p">)</span>
-<span class="go">{1: 1/4, 2: 1/4, 3: 1/4, 4: 1/4}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Bernoulli">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Bernoulli</tt><big>(</big><em>name</em>, <em>p</em>, <em>succ=1</em>, <em>fail=0</em><big>)</big><a class="headerlink" href="#sympy.stats.Bernoulli" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a Finite Random Variable representing a Bernoulli process.</p>
-<p>Returns a RandomSymbol</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Bernoulli</span><span class="p">,</span> <span class="n">density</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Bernoulli</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mi">4</span><span class="p">)</span> <span class="c"># 1-0 Bernoulli variable, probability = 3/4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">{0: 1/4, 1: 3/4}</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Bernoulli</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">,</span> <span class="s">&#39;Heads&#39;</span><span class="p">,</span> <span class="s">&#39;Tails&#39;</span><span class="p">)</span> <span class="c"># A fair coin toss</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">{Heads: 1/2, Tails: 1/2}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Coin">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Coin</tt><big>(</big><em>name</em>, <em>p=1/2</em><big>)</big><a class="headerlink" href="#sympy.stats.Coin" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a Finite Random Variable representing a Coin toss.</p>
-<p>Probability p is the chance of gettings &#8220;Heads.&#8221; Half by default</p>
-<p>Returns a RandomSymbol.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Coin</span><span class="p">,</span> <span class="n">density</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Rational</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">Coin</span><span class="p">(</span><span class="s">&#39;C&#39;</span><span class="p">)</span> <span class="c"># A fair coin toss</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">C</span><span class="p">)</span>
-<span class="go">{H: 1/2, T: 1/2}</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">C2</span> <span class="o">=</span> <span class="n">Coin</span><span class="p">(</span><span class="s">&#39;C2&#39;</span><span class="p">,</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span> <span class="c"># An unfair coin</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">C2</span><span class="p">)</span>
-<span class="go">{H: 3/5, T: 2/5}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Binomial">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Binomial</tt><big>(</big><em>name</em>, <em>n</em>, <em>p</em>, <em>succ=1</em>, <em>fail=0</em><big>)</big><a class="headerlink" href="#sympy.stats.Binomial" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a Finite Random Variable representing a binomial distribution.</p>
-<p>Returns a RandomSymbol.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Binomial</span><span class="p">,</span> <span class="n">density</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Binomial</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="n">S</span><span class="o">.</span><span class="n">Half</span><span class="p">)</span> <span class="c"># Four &quot;coin flips&quot;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">{0: 1/16, 1: 1/4, 2: 3/8, 3: 1/4, 4: 1/16}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Hypergeometric">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Hypergeometric</tt><big>(</big><em>name</em>, <em>N</em>, <em>m</em>, <em>n</em><big>)</big><a class="headerlink" href="#sympy.stats.Hypergeometric" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a Finite Random Variable representing a hypergeometric distribution.</p>
-<p>Returns a RandomSymbol.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Hypergeometric</span><span class="p">,</span> <span class="n">density</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">S</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Hypergeometric</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="c"># 10 marbles, 5 white (success), 3 draws</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">{0: 1/12, 1: 5/12, 2: 5/12, 3: 1/12}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.FiniteRV">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">FiniteRV</tt><big>(</big><em>name</em>, <em>density</em><big>)</big><a class="headerlink" href="#sympy.stats.FiniteRV" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a Finite Random Variable given a dict representing the density.</p>
-<p>Returns a RandomSymbol.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">FiniteRV</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">E</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">density</span> <span class="o">=</span> <span class="p">{</span><span class="mi">0</span><span class="p">:</span> <span class="o">.</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">:</span> <span class="o">.</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">:</span> <span class="o">.</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">:</span> <span class="o">.</span><span class="mi">4</span><span class="p">}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">FiniteRV</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="n">density</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">E</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">2.00000000000000</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="p">(</span><span class="n">X</span><span class="o">&gt;=</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">0.700000000000000</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="continuous-types">
-<h3>Continuous Types<a class="headerlink" href="#continuous-types" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.stats.Arcsin">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Arcsin</tt><big>(</big><em>name</em>, <em>a=0</em>, <em>b=1</em><big>)</big><a class="headerlink" href="#sympy.stats.Arcsin" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a Continuous Random Variable with an arcsin distribution.</p>
-<p>The density of the arcsin distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{1}{\pi\sqrt{(x-a)(b-x)}}\]</div>
-<p>with <span class="math">\(x \in [a,b]\)</span>. It must hold that <span class="math">\(-\infty &lt; a &lt; b &lt; \infty\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>a</strong> : Real number, the left interval boundary</p>
-<p><strong>b</strong> : Real number, the right interval boundary</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Arcsine_distribution">http://en.wikipedia.org/wiki/Arcsine_distribution</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Arcsin</span><span class="p">,</span> <span class="n">density</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;a&quot;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;b&quot;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Arcsin</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_x, 1/(pi*sqrt((-_x + b)*(_x - a))))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Benini">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Benini</tt><big>(</big><em>name</em>, <em>alpha</em>, <em>beta</em>, <em>sigma</em><big>)</big><a class="headerlink" href="#sympy.stats.Benini" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a Continuous Random Variable with a Benini distribution.</p>
-<p>The density of the Benini distribution is given by</p>
-<div class="math">
-\[f(x) := e^{-\alpha\log{\frac{x}{\sigma}}
-        -\beta\log\left[{\frac{x}{\sigma}}\right]^2}
-        \left(\frac{\alpha}{x}+\frac{2\beta\log{\frac{x}{\sigma}}}{x}\right)\]</div>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>alpha</strong> : Real number, <span class="math">\(alpha\)</span> &gt; 0 a shape</p>
-<p><strong>beta</strong> : Real number, <span class="math">\(beta\)</span> &gt; 0 a shape</p>
-<p><strong>sigma</strong> : Real number, <span class="math">\(sigma\)</span> &gt; 0 a scale</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Benini_distribution">http://en.wikipedia.org/wiki/Benini_distribution</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Benini</span><span class="p">,</span> <span class="n">density</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">pprint</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">alpha</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;alpha&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">beta</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;beta&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sigma</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;sigma&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Benini</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">sigma</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">      /                                                             2       \</span>
-<span class="go">      |   /                  /  x  \\             /  x  \            /  x  \|</span>
-<span class="go">      |   |        2*beta*log|-----||  - alpha*log|-----| - beta*log |-----||</span>
-<span class="go">      |   |alpha             \sigma/|             \sigma/            \sigma/|</span>
-<span class="go">Lambda|x, |----- + -----------------|*e                                     |</span>
-<span class="go">      \   \  x             x        /                                       /</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Beta">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Beta</tt><big>(</big><em>name</em>, <em>alpha</em>, <em>beta</em><big>)</big><a class="headerlink" href="#sympy.stats.Beta" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a Continuous Random Variable with a Beta distribution.</p>
-<p>The density of the Beta distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{x^{\alpha-1}(1-x)^{\beta-1}} {\mathrm{B}(\alpha,\beta)}\]</div>
-<p>with <span class="math">\(x \in [0,1]\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>alpha</strong> : Real number, <span class="math">\(alpha\)</span> &gt; 0 a shape</p>
-<p><strong>beta</strong> : Real number, <span class="math">\(beta\)</span> &gt; 0 a shape</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Beta_distribution">http://en.wikipedia.org/wiki/Beta_distribution</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/BetaDistribution.html">http://mathworld.wolfram.com/BetaDistribution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Beta</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">variance</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">pprint</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">alpha</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;alpha&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">beta</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;beta&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Beta</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">      /    alpha - 1         beta - 1                    \</span>
-<span class="go">      |   x         *(-x + 1)        *gamma(alpha + beta)|</span>
-<span class="go">Lambda|x, -----------------------------------------------|</span>
-<span class="go">      \               gamma(alpha)*gamma(beta)           /</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">E</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">meijerg</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">alpha/(alpha + beta)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">variance</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">meijerg</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">alpha*beta/((alpha + beta)**2*(alpha + beta + 1))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.BetaPrime">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">BetaPrime</tt><big>(</big><em>name</em>, <em>alpha</em>, <em>beta</em><big>)</big><a class="headerlink" href="#sympy.stats.BetaPrime" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with a Beta prime distribution.</p>
-<p>The density of the Beta prime distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)}\]</div>
-<p>with <span class="math">\(x &gt; 0\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>alpha</strong> : Real number, <span class="math">\(alpha\)</span> &gt; 0 a shape</p>
-<p><strong>beta</strong> : Real number, <span class="math">\(beta\)</span> &gt; 0 a shape</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Beta_prime_distribution">http://en.wikipedia.org/wiki/Beta_prime_distribution</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/BetaPrimeDistribution.html">http://mathworld.wolfram.com/BetaPrimeDistribution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">BetaPrime</span><span class="p">,</span> <span class="n">density</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">pprint</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">alpha</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;alpha&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">beta</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;beta&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">BetaPrime</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">      /    alpha - 1        -alpha - beta                    \</span>
-<span class="go">      |   x         *(x + 1)             *gamma(alpha + beta)|</span>
-<span class="go">Lambda|x, ---------------------------------------------------|</span>
-<span class="go">      \                 gamma(alpha)*gamma(beta)             /</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Cauchy">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Cauchy</tt><big>(</big><em>name</em>, <em>x0</em>, <em>gamma</em><big>)</big><a class="headerlink" href="#sympy.stats.Cauchy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with a Cauchy distribution.</p>
-<p>The density of the Cauchy distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{1}{\pi} \arctan\left(\frac{x-x_0}{\gamma}\right)
-        +\frac{1}{2}\]</div>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>x0</strong> : Real number, the location</p>
-<p><strong>gamma</strong> : Real number, <span class="math">\(gamma\)</span> &gt; 0 the scale</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Cauchy_distribution">http://en.wikipedia.org/wiki/Cauchy_distribution</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/CauchyDistribution.html">http://mathworld.wolfram.com/CauchyDistribution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Cauchy</span><span class="p">,</span> <span class="n">density</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x0</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x0&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;gamma&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Cauchy</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">gamma</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_x, 1/(pi*gamma*(1 + (_x - x0)**2/gamma**2)))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Chi">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Chi</tt><big>(</big><em>name</em>, <em>k</em><big>)</big><a class="headerlink" href="#sympy.stats.Chi" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with a Chi distribution.</p>
-<p>The density of the Chi distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}\]</div>
-<p>with <span class="math">\(x \geq 0\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>k</strong> : Integer, <span class="math">\(k\)</span> &gt; 0 the number of degrees of freedom</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>A RandomSymbol.</strong> :</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Chi_distribution">http://en.wikipedia.org/wiki/Chi_distribution</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/ChiDistribution.html">http://mathworld.wolfram.com/ChiDistribution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Chi</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">std</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Chi</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_x, 2**(-k/2 + 1)*_x**(k - 1)*exp(-_x**2/2)/gamma(k/2))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.ChiNoncentral">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">ChiNoncentral</tt><big>(</big><em>name</em>, <em>k</em>, <em>l</em><big>)</big><a class="headerlink" href="#sympy.stats.ChiNoncentral" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with a non-central Chi distribution.</p>
-<p>The density of the non-central Chi distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{e^{-(x^2+\lambda^2)/2} x^k\lambda}
-        {(\lambda x)^{k/2}} I_{k/2-1}(\lambda x)\]</div>
-<p>with <span class="math">\(x \geq 0\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>k</strong> : <span class="math">\(k\)</span> &gt; 0 the number of degrees of freedom</p>
-<p><strong>l</strong> : Shift parameter</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Noncentral_chi_distribution">http://en.wikipedia.org/wiki/Noncentral_chi_distribution</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">ChiNoncentral</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">std</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;l&quot;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">ChiNoncentral</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">l</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_x, _x**k*l*(_x*l)**(-k/2)*exp(-_x**2/2 - l**2/2)*besseli(k/2 - 1, _x*l))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.ChiSquared">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">ChiSquared</tt><big>(</big><em>name</em>, <em>k</em><big>)</big><a class="headerlink" href="#sympy.stats.ChiSquared" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with a Chi-squared distribution.</p>
-<p>The density of the Chi-squared distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{1}{2^{\frac{k}{2}}\Gamma\left(\frac{k}{2}\right)}
-        x^{\frac{k}{2}-1} e^{-\frac{x}{2}}\]</div>
-<p>with <span class="math">\(x \geq 0\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>k</strong> : Integer, <span class="math">\(k\)</span> &gt; 0 the number of degrees of freedom</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>A RandomSymbol.</strong> :</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Chi_squared_distribution">http://en.wikipedia.org/wiki/Chi_squared_distribution</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/Chi-SquaredDistribution.html">http://mathworld.wolfram.com/Chi-SquaredDistribution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">ChiSquared</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">variance</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">combsimp</span><span class="p">,</span> <span class="n">expand_func</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">ChiSquared</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_x, 2**(-k/2)*_x**(k/2 - 1)*exp(-_x/2)/gamma(k/2))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">combsimp</span><span class="p">(</span><span class="n">E</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
-<span class="go">k</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">expand_func</span><span class="p">(</span><span class="n">variance</span><span class="p">(</span><span class="n">X</span><span class="p">)))</span>
-<span class="go">2*k</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Dagum">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Dagum</tt><big>(</big><em>name</em>, <em>p</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.stats.Dagum" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with a Dagum distribution.</p>
-<p>The density of the Dagum distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{a p}{x} \left( \frac{(\tfrac{x}{b})^{a p}}
-        {\left((\tfrac{x}{b})^a + 1 \right)^{p+1}} \right)\]</div>
-<p>with <span class="math">\(x &gt; 0\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>p</strong> : Real number, <span class="math">\(p\)</span> &gt; 0 a shape</p>
-<p><strong>a</strong> : Real number, <span class="math">\(a\)</span> &gt; 0 a shape</p>
-<p><strong>b</strong> : Real number, <span class="math">\(b\)</span> &gt; 0 a scale</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Dagum_distribution">http://en.wikipedia.org/wiki/Dagum_distribution</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Dagum</span><span class="p">,</span> <span class="n">density</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;p&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;b&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;a&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Dagum</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_x, a*p*(_x/b)**(a*p)*((_x/b)**a + 1)**(-p - 1)/_x)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Erlang">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Erlang</tt><big>(</big><em>name</em>, <em>k</em>, <em>l</em><big>)</big><a class="headerlink" href="#sympy.stats.Erlang" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with an Erlang distribution.</p>
-<p>The density of the Erlang distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{\lambda^k x^{k-1} e^{-\lambda x}}{(k-1)!}\]</div>
-<p>with <span class="math">\(x \in [0,\infty]\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>k</strong> : Integer</p>
-<p><strong>l</strong> : Real number, <span class="math">\(\lambda\)</span> &gt; 0 the rate</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Erlang_distribution">http://en.wikipedia.org/wiki/Erlang_distribution</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/ErlangDistribution.html">http://mathworld.wolfram.com/ErlangDistribution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Erlang</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">cdf</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">variance</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">pprint</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">l</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;l&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Erlang</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">l</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">      /    k - 1  k  -x*l\</span>
-<span class="go">      |   x     *l *e    |</span>
-<span class="go">Lambda|x, ---------------|</span>
-<span class="go">      \       gamma(k)   /</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">cdf</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">meijerg</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">      /   /  k*lowergamma(k, 0)   k*lowergamma(k, z*l)            \</span>
-<span class="go">Lambda|z, |- ------------------ + --------------------  for z &gt;= 0|</span>
-<span class="go">      |   &lt;     gamma(k + 1)          gamma(k + 1)                |</span>
-<span class="go">      |   |                                                       |</span>
-<span class="go">      \   \                     0                       otherwise /</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">E</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
-<span class="go">k/l</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">variance</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
-<span class="go">(gamma(k)*gamma(k + 2) - gamma(k + 1)**2)/(l**2*gamma(k)**2)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Exponential">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Exponential</tt><big>(</big><em>name</em>, <em>rate</em><big>)</big><a class="headerlink" href="#sympy.stats.Exponential" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with an Exponential distribution.</p>
-<p>The density of the exponential distribution is given by</p>
-<div class="math">
-\[f(x) := \lambda \exp(-\lambda x)\]</div>
-<p>with <span class="math">\(x &gt; 0\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>rate</strong> : Real number, <span class="math">\(rate\)</span> &gt; 0 the rate or inverse scale</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>A RandomSymbol.</strong> :</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Exponential_distribution">http://en.wikipedia.org/wiki/Exponential_distribution</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/ExponentialDistribution.html">http://mathworld.wolfram.com/ExponentialDistribution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Exponential</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">cdf</span><span class="p">,</span> <span class="n">E</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">variance</span><span class="p">,</span> <span class="n">std</span><span class="p">,</span> <span class="n">skewness</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">l</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;lambda&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Exponential</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">l</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_x, lambda*exp(-_x*lambda))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">cdf</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_z, Piecewise((1 - exp(-_z*lambda), _z &gt;= 0), (0, True)))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">E</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">1/lambda</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">variance</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">lambda**(-2)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">skewness</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Exponential</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_x, 10*exp(-10*_x))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">E</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">1/10</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">std</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">1/10</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.FDistribution">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">FDistribution</tt><big>(</big><em>name</em>, <em>d1</em>, <em>d2</em><big>)</big><a class="headerlink" href="#sympy.stats.FDistribution" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with a F distribution.</p>
-<p>The density of the F distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{\sqrt{\frac{(d_1 x)^{d_1} d_2^{d_2}}
-        {(d_1 x + d_2)^{d_1 + d_2}}}}
-        {x \mathrm{B} \left(\frac{d_1}{2}, \frac{d_2}{2}\right)}\]</div>
-<p>with <span class="math">\(x &gt; 0\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>d1</strong> : <span class="math">\(d1\)</span> &gt; 0 a parameter</p>
-<p><strong>d2</strong> : <span class="math">\(d2\)</span> &gt; 0 a parameter</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/F-distribution">http://en.wikipedia.org/wiki/F-distribution</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/F-Distribution.html">http://mathworld.wolfram.com/F-Distribution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">FDistribution</span><span class="p">,</span> <span class="n">density</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">pprint</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">d1</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;d1&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d2</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;d2&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">FDistribution</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">d1</span><span class="p">,</span> <span class="n">d2</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">      /     d2                                                 \</span>
-<span class="go">      |     --    ______________________________               |</span>
-<span class="go">      |     2    /       d1            -d1 - d2       /d1   d2\|</span>
-<span class="go">      |   d2  *\/  (x*d1)  *(x*d1 + d2)         *gamma|-- + --||</span>
-<span class="go">      |                                               \2    2 /|</span>
-<span class="go">Lambda|x, -----------------------------------------------------|</span>
-<span class="go">      |                          /d1\      /d2\                |</span>
-<span class="go">      |                   x*gamma|--|*gamma|--|                |</span>
-<span class="go">      \                          \2 /      \2 /                /</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.FisherZ">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">FisherZ</tt><big>(</big><em>name</em>, <em>d1</em>, <em>d2</em><big>)</big><a class="headerlink" href="#sympy.stats.FisherZ" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a Continuous Random Variable with an Fisher&#8217;s Z distribution.</p>
-<p>The density of the Fisher&#8217;s Z distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{2d_1^{d_1/2} d_2^{d_2/2}} {\mathrm{B}(d_1/2, d_2/2)}
-        \frac{e^{d_1z}}{\left(d_1e^{2z}+d_2\right)^{\left(d_1+d_2\right)/2}}\]</div>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>d1</strong> : <span class="math">\(d1\)</span> &gt; 0, degree of freedom</p>
-<p><strong>d2</strong> : <span class="math">\(d2\)</span> &gt; 0, degree of freedom</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Fisher%27s_z-distribution">http://en.wikipedia.org/wiki/Fisher%27s_z-distribution</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/Fishersz-Distribution.html">http://mathworld.wolfram.com/Fishersz-Distribution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">FisherZ</span><span class="p">,</span> <span class="n">density</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">pprint</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">d1</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;d1&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d2</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;d2&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">FisherZ</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">d1</span><span class="p">,</span> <span class="n">d2</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">      /                               d1   d2                     \</span>
-<span class="go">      |       d1   d2               - -- - --                     |</span>
-<span class="go">      |       --   --                 2    2                      |</span>
-<span class="go">      |       2    2  /    2*x     \           x*d1      /d1   d2\|</span>
-<span class="go">      |   2*d1  *d2  *\d1*e    + d2/         *e    *gamma|-- + --||</span>
-<span class="go">      |                                                  \2    2 /|</span>
-<span class="go">Lambda|x, --------------------------------------------------------|</span>
-<span class="go">      |                          /d1\      /d2\                   |</span>
-<span class="go">      |                     gamma|--|*gamma|--|                   |</span>
-<span class="go">      \                          \2 /      \2 /                   /</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Frechet">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Frechet</tt><big>(</big><em>name</em>, <em>a</em>, <em>s=1</em>, <em>m=0</em><big>)</big><a class="headerlink" href="#sympy.stats.Frechet" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with a Frechet distribution.</p>
-<p>The density of the Frechet distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{\alpha}{s} \left(\frac{x-m}{s}\right)^{-1-\alpha}
-         e^{-(\frac{x-m}{s})^{-\alpha}}\]</div>
-<p>with <span class="math">\(x \geq m\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>a</strong> : Real number, <span class="math">\(a \in \left(0, \infty\right)\)</span> the shape</p>
-<p><strong>s</strong> : Real number, <span class="math">\(s \in \left(0, \infty\right)\)</span> the scale</p>
-<p><strong>m</strong> : Real number, <span class="math">\(m \in \left(-\infty, \infty\right)\)</span> the minimum</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution">http://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Frechet</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">std</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;a&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;s&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;m&quot;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Frechet</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_x, a*((_x - m)/s)**(-a - 1)*exp(-a - (_x - m)/s)/s)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Gamma">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Gamma</tt><big>(</big><em>name</em>, <em>k</em>, <em>theta</em><big>)</big><a class="headerlink" href="#sympy.stats.Gamma" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with a Gamma distribution.</p>
-<p>The density of the Gamma distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{1}{\Gamma(k) \theta^k} x^{k - 1} e^{-\frac{x}{\theta}}\]</div>
-<p>with <span class="math">\(x \in [0,1]\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>k</strong> : Real number, <span class="math">\(k\)</span> &gt; 0 a shape</p>
-<p><strong>theta</strong> : Real number, <span class="math">\(theta\)</span> &gt; 0 a scale</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Gamma_distribution">http://en.wikipedia.org/wiki/Gamma_distribution</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/GammaDistribution.html">http://mathworld.wolfram.com/GammaDistribution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Gamma</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">cdf</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">variance</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">pprint</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">theta</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;theta&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Gamma</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">      /                     -x \</span>
-<span class="go">      |                   -----|</span>
-<span class="go">      |    k - 1      -k  theta|</span>
-<span class="go">      |   x     *theta  *e     |</span>
-<span class="go">Lambda|x, ---------------------|</span>
-<span class="go">      \          gamma(k)      /</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">cdf</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">meijerg</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">      /   /                                   /     z  \            \</span>
-<span class="go">      |   |                       k*lowergamma|k, -----|            |</span>
-<span class="go">      |   |  k*lowergamma(k, 0)               \   theta/            |</span>
-<span class="go">Lambda|z, &lt;- ------------------ + ----------------------  for z &gt;= 0|</span>
-<span class="go">      |   |     gamma(k + 1)           gamma(k + 1)                 |</span>
-<span class="go">      |   |                                                         |</span>
-<span class="go">      \   \                      0                        otherwise /</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">E</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">theta*gamma(k + 1)/gamma(k)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">V</span> <span class="o">=</span> <span class="n">variance</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">       2      2                     -k      k + 1</span>
-<span class="go">  theta *gamma (k + 1)   theta*theta  *theta     *gamma(k + 2)</span>
-<span class="go">- -------------------- + -------------------------------------</span>
-<span class="go">            2                           gamma(k)</span>
-<span class="go">       gamma (k)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.GammaInverse">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">GammaInverse</tt><big>(</big><em>name</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.stats.GammaInverse" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with an inverse Gamma distribution.</p>
-<p>The density of the inverse Gamma distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1}
-        \exp\left(\frac{-\beta}{x}\right)\]</div>
-<p>with <span class="math">\(x &gt; 0\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>a</strong> : Real number, <span class="math">\(a\)</span> &gt; 0 a shape</p>
-<p><strong>b</strong> : Real number, <span class="math">\(b\)</span> &gt; 0 a scale</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Inverse-gamma_distribution">http://en.wikipedia.org/wiki/Inverse-gamma_distribution</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">GammaInverse</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">cdf</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">variance</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">pprint</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;a&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;b&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">GammaInverse</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">      /               -b\</span>
-<span class="go">      |               --|</span>
-<span class="go">      |    -a - 1  a  x |</span>
-<span class="go">      |   x      *b *e  |</span>
-<span class="go">Lambda|x, --------------|</span>
-<span class="go">      \      gamma(a)   /</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Kumaraswamy">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Kumaraswamy</tt><big>(</big><em>name</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.stats.Kumaraswamy" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a Continuous Random Variable with a Kumaraswamy distribution.</p>
-<p>The density of the Kumaraswamy distribution is given by</p>
-<div class="math">
-\[f(x) := a b x^{a-1} (1-x^a)^{b-1}\]</div>
-<p>with <span class="math">\(x \in [0,1]\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>a</strong> : Real number, <span class="math">\(a\)</span> &gt; 0 a shape</p>
-<p><strong>b</strong> : Real number, <span class="math">\(b\)</span> &gt; 0 a shape</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Kumaraswamy_distribution">http://en.wikipedia.org/wiki/Kumaraswamy_distribution</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Kumaraswamy</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">variance</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">pprint</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;a&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;b&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Kumaraswamy</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">      /                        b - 1\</span>
-<span class="go">      |    a - 1     /   a    \     |</span>
-<span class="go">Lambda\x, x     *a*b*\- x  + 1/     /</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">E</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">meijerg</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">gamma(1 + 1/a)*gamma(b + 1)/gamma(b + 1 + 1/a)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Laplace">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Laplace</tt><big>(</big><em>name</em>, <em>mu</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.stats.Laplace" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with a Laplace distribution.</p>
-<p>The density of the Laplace distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{1}{2 b} \exp \left(-\frac{|x-\mu|}b \right)\]</div>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>mu</strong> : Real number, the location</p>
-<p><strong>b</strong> : Real number, <span class="math">\(b\)</span> &gt; 0 a scale</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Laplace_distribution">http://en.wikipedia.org/wiki/Laplace_distribution</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/LaplaceDistribution.html">http://mathworld.wolfram.com/LaplaceDistribution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Laplace</span><span class="p">,</span> <span class="n">density</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mu</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;mu&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;b&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Laplace</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_x, exp(-Abs(_x - mu)/b)/(2*b))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Logistic">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Logistic</tt><big>(</big><em>name</em>, <em>mu</em>, <em>s</em><big>)</big><a class="headerlink" href="#sympy.stats.Logistic" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with a logistic distribution.</p>
-<p>The density of the logistic distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{e^{-(x-\mu)/s}} {s\left(1+e^{-(x-\mu)/s}\right)^2}\]</div>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>mu</strong> : Real number, the location</p>
-<p><strong>s</strong> : Real number, <span class="math">\(s\)</span> &gt; 0 a scale</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Logistic_distribution">http://en.wikipedia.org/wiki/Logistic_distribution</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/LogisticDistribution.html">http://mathworld.wolfram.com/LogisticDistribution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Logistic</span><span class="p">,</span> <span class="n">density</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mu</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;mu&quot;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;s&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Logistic</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_x, exp((-_x + mu)/s)/(s*(exp((-_x + mu)/s) + 1)**2))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.LogNormal">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">LogNormal</tt><big>(</big><em>name</em>, <em>mean</em>, <em>std</em><big>)</big><a class="headerlink" href="#sympy.stats.LogNormal" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with a log-normal distribution.</p>
-<p>The density of the log-normal distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{1}{x\sqrt{2\pi\sigma^2}}
-        e^{-\frac{\left(\ln x-\mu\right)^2}{2\sigma^2}}\]</div>
-<p>with <span class="math">\(x \geq 0\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>mu</strong> : Real number, the log-scale</p>
-<p><strong>sigma</strong> : Real number, <span class="math">\(\sigma^2 &gt; 0\)</span> a shape</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Lognormal">http://en.wikipedia.org/wiki/Lognormal</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/LogNormalDistribution.html">http://mathworld.wolfram.com/LogNormalDistribution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">LogNormal</span><span class="p">,</span> <span class="n">density</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">pprint</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mu</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;mu&quot;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sigma</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;sigma&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">LogNormal</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">sigma</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">      /                         2\</span>
-<span class="go">      |          -(-mu + log(x)) |</span>
-<span class="go">      |          ----------------|</span>
-<span class="go">      |                     2    |</span>
-<span class="go">      |     ___      2*sigma     |</span>
-<span class="go">      |   \/ 2 *e                |</span>
-<span class="go">Lambda|x, -----------------------|</span>
-<span class="go">      |             ____         |</span>
-<span class="go">      \       2*x*\/ pi *sigma   /</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">LogNormal</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="c"># Mean 0, standard deviation 1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_x, sqrt(2)*exp(-log(_x)**2/2)/(2*_x*sqrt(pi)))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Maxwell">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Maxwell</tt><big>(</big><em>name</em>, <em>a</em><big>)</big><a class="headerlink" href="#sympy.stats.Maxwell" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with a Maxwell distribution.</p>
-<p>The density of the Maxwell distribution is given by</p>
-<div class="math">
-\[f(x) := \sqrt{\frac{2}{\pi}} \frac{x^2 e^{-x^2/(2a^2)}}{a^3}\]</div>
-<p>with <span class="math">\(x \geq 0\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>a</strong> : Real number, <span class="math">\(a\)</span> &gt; 0</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>A RandomSymbol.</strong> :</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Maxwell_distribution">http://en.wikipedia.org/wiki/Maxwell_distribution</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/MaxwellDistribution.html">http://mathworld.wolfram.com/MaxwellDistribution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Maxwell</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">variance</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;a&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Maxwell</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_x, sqrt(2)*_x**2*exp(-_x**2/(2*a**2))/(sqrt(pi)*a**3))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">E</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">2*sqrt(2)*a/sqrt(pi)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">variance</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
-<span class="go">a**2*(-8 + 3*pi)/pi</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Nakagami">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Nakagami</tt><big>(</big><em>name</em>, <em>mu</em>, <em>omega</em><big>)</big><a class="headerlink" href="#sympy.stats.Nakagami" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with a Nakagami distribution.</p>
-<p>The density of the Nakagami distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{2\mu^\mu}{\Gamma(\mu)\omega^\mu} x^{2\mu-1}
-        \exp\left(-\frac{\mu}{\omega}x^2 \right)\]</div>
-<p>with <span class="math">\(x &gt; 0\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>mu</strong> : Real number, <span class="math">\(mu \geq \frac{1}{2}\)</span> a shape</p>
-<p><strong>omega</strong> : Real number, <span class="math">\(omega\)</span> &gt; 0 the spread</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Nakagami_distribution">http://en.wikipedia.org/wiki/Nakagami_distribution</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Nakagami</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">variance</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">pprint</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mu</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;mu&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">omega</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;omega&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Nakagami</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">omega</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">      /                                2   \</span>
-<span class="go">      |                              -x *mu|</span>
-<span class="go">      |                              ------|</span>
-<span class="go">      |      2*mu - 1   mu      -mu  omega |</span>
-<span class="go">      |   2*x        *mu  *omega   *e      |</span>
-<span class="go">Lambda|x, ---------------------------------|</span>
-<span class="go">      \               gamma(mu)            /</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">E</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">meijerg</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">sqrt(mu)*sqrt(omega)*gamma(mu + 1/2)/gamma(mu + 1)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">V</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">variance</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">meijerg</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">      /                               2          \</span>
-<span class="go">omega*\gamma(mu)*gamma(mu + 1) - gamma (mu + 1/2)/</span>
-<span class="go">--------------------------------------------------</span>
-<span class="go">             gamma(mu)*gamma(mu + 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Normal">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Normal</tt><big>(</big><em>name</em>, <em>mean</em>, <em>std</em><big>)</big><a class="headerlink" href="#sympy.stats.Normal" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with a Normal distribution.</p>
-<p>The density of the Normal distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{(x-\mu)^2}{2\sigma^2} }\]</div>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>mu</strong> : Real number, the mean</p>
-<p><strong>sigma</strong> : Real number, <span class="math">\(\sigma^2 &gt; 0\)</span> the variance</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Normal_distribution">http://en.wikipedia.org/wiki/Normal_distribution</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/NormalDistributionFunction.html">http://mathworld.wolfram.com/NormalDistributionFunction.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Normal</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">std</span><span class="p">,</span> <span class="n">cdf</span><span class="p">,</span> <span class="n">skewness</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">pprint</span><span class="p">,</span> <span class="n">factor</span><span class="p">,</span> <span class="n">together</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mu</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;mu&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sigma</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;sigma&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Normal</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">sigma</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_x, sqrt(2)*exp(-(_x - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">simplify</span><span class="p">(</span><span class="n">cdf</span><span class="p">(</span><span class="n">X</span><span class="p">))</span> <span class="c"># it needs a little more help...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">C</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">func</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">together</span><span class="p">(</span><span class="n">factor</span><span class="p">(</span><span class="n">C</span><span class="o">.</span><span class="n">args</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="n">deep</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">C</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">          /      /  ___         \    \</span>
-<span class="go">          |      |\/ 2 *(z - mu)|    |</span>
-<span class="go">          |   erf|--------------|    |</span>
-<span class="go">          |      \   2*sigma    /   1|</span>
-<span class="go">    Lambda|z, ------------------- + -|</span>
-<span class="go">          \            2            2/</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">skewness</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Normal</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="c"># Mean 0, standard deviation 1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_x, sqrt(2)*exp(-_x**2/2)/(2*sqrt(pi)))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">E</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">X</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">std</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">X</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Pareto">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Pareto</tt><big>(</big><em>name</em>, <em>xm</em>, <em>alpha</em><big>)</big><a class="headerlink" href="#sympy.stats.Pareto" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with the Pareto distribution.</p>
-<p>The density of the Pareto distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{\alpha\,x_\mathrm{m}^\alpha}{x^{\alpha+1}}\]</div>
-<p>with <span class="math">\(x \in [x_m,\infty]\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>xm</strong> : Real number, <span class="math">\(xm\)</span> &gt; 0 a scale</p>
-<p><strong>alpha</strong> : Real number, <span class="math">\(alpha\)</span> &gt; 0 a shape</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Pareto_distribution">http://en.wikipedia.org/wiki/Pareto_distribution</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/ParetoDistribution.html">http://mathworld.wolfram.com/ParetoDistribution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Pareto</span><span class="p">,</span> <span class="n">density</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">xm</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;xm&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">beta</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;beta&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Pareto</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">xm</span><span class="p">,</span> <span class="n">beta</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_x, _x**(-beta - 1)*beta*xm**beta)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.QuadraticU">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">QuadraticU</tt><big>(</big><em>name</em>, <em>a</em>, <em>b</em><big>)</big><a class="headerlink" href="#sympy.stats.QuadraticU" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a Continuous Random Variable with a U-quadratic distribution.</p>
-<p>The density of the U-quadratic distribution is given by</p>
-<div class="math">
-\[f(x) := \alpha (x-\beta)^2\]</div>
-<p>with <span class="math">\(x \in [a,b]\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>a</strong> : Real number</p>
-<p><strong>b</strong> : Real number, <span class="math">\(a &lt; b\)</span></p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/U-quadratic_distribution">http://en.wikipedia.org/wiki/U-quadratic_distribution</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">QuadraticU</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">variance</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">factor</span><span class="p">,</span> <span class="n">pprint</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;a&quot;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;b&quot;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">QuadraticU</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">      /   /              2                         \</span>
-<span class="go">      |   |   /    a   b\                          |</span>
-<span class="go">      |   |12*|x - - - -|                          |</span>
-<span class="go">      |   |   \    2   2/                          |</span>
-<span class="go">Lambda|x, &lt;---------------  for And(x &lt;= b, a &lt;= x)|</span>
-<span class="go">      |   |           3                            |</span>
-<span class="go">      |   |   (-a + b)                             |</span>
-<span class="go">      |   |                                        |</span>
-<span class="go">      \   \       0                otherwise       /</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.RaisedCosine">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">RaisedCosine</tt><big>(</big><em>name</em>, <em>mu</em>, <em>s</em><big>)</big><a class="headerlink" href="#sympy.stats.RaisedCosine" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a Continuous Random Variable with a raised cosine distribution.</p>
-<p>The density of the raised cosine distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{1}{2s}\left(1+\cos\left(\frac{x-\mu}{s}\pi\right)\right)\]</div>
-<p>with <span class="math">\(x \in [\mu-s,\mu+s]\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>mu</strong> : Real number</p>
-<p><strong>s</strong> : Real number, <span class="math">\(s\)</span> &gt; 0</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Raised_cosine_distribution">http://en.wikipedia.org/wiki/Raised_cosine_distribution</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">RaisedCosine</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">variance</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">pprint</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mu</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;mu&quot;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;s&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">RaisedCosine</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">s</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">      /   /   /pi*(x - mu)\                                       \</span>
-<span class="go">      |   |cos|-----------| + 1                                   |</span>
-<span class="go">      |   |   \     s     /                                       |</span>
-<span class="go">Lambda|x, &lt;--------------------  for And(x &lt;= mu + s, mu - s &lt;= x)|</span>
-<span class="go">      |   |        2*s                                            |</span>
-<span class="go">      |   |                                                       |</span>
-<span class="go">      \   \         0                        otherwise            /</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Rayleigh">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Rayleigh</tt><big>(</big><em>name</em>, <em>sigma</em><big>)</big><a class="headerlink" href="#sympy.stats.Rayleigh" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with a Rayleigh distribution.</p>
-<p>The density of the Rayleigh distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{x}{\sigma^2} e^{-x^2/2\sigma^2}\]</div>
-<p>with <span class="math">\(x &gt; 0\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>sigma</strong> : Real number, <span class="math">\(sigma\)</span> &gt; 0</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>A RandomSymbol.</strong> :</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Rayleigh_distribution">http://en.wikipedia.org/wiki/Rayleigh_distribution</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/RayleighDistribution.html">http://mathworld.wolfram.com/RayleighDistribution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Rayleigh</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">variance</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sigma</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;sigma&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Rayleigh</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">sigma</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_x, _x*exp(-_x**2/(2*sigma**2))/sigma**2)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">E</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">sqrt(2)*sqrt(pi)*sigma/2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">variance</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">-pi*sigma**2/2 + 2*sigma**2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.StudentT">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">StudentT</tt><big>(</big><em>name</em>, <em>nu</em><big>)</big><a class="headerlink" href="#sympy.stats.StudentT" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with a student&#8217;s t distribution.</p>
-<p>The density of the student&#8217;s t distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{\Gamma \left(\frac{\nu+1}{2} \right)}
-        {\sqrt{\nu\pi}\Gamma \left(\frac{\nu}{2} \right)}
-        \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}\]</div>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>nu</strong> : Real number, <span class="math">\(nu\)</span> &gt; 0, the degrees of freedom</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>A RandomSymbol.</strong> :</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Student_t-distribution">http://en.wikipedia.org/wiki/Student_t-distribution</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/Studentst-Distribution.html">http://mathworld.wolfram.com/Studentst-Distribution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">StudentT</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">variance</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">pprint</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">nu</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;nu&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">StudentT</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">nu</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">      /             nu   1              \</span>
-<span class="go">      |           - -- - -              |</span>
-<span class="go">      |             2    2              |</span>
-<span class="go">      |   / 2    \                      |</span>
-<span class="go">      |   |x     |              /nu   1\|</span>
-<span class="go">      |   |-- + 1|        *gamma|-- + -||</span>
-<span class="go">      |   \nu    /              \2    2/|</span>
-<span class="go">Lambda|x, ------------------------------|</span>
-<span class="go">      |        ____   ____      /nu\    |</span>
-<span class="go">      |      \/ pi *\/ nu *gamma|--|    |</span>
-<span class="go">      \                         \2 /    /</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Triangular">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Triangular</tt><big>(</big><em>name</em>, <em>a</em>, <em>b</em>, <em>c</em><big>)</big><a class="headerlink" href="#sympy.stats.Triangular" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with a triangular distribution.</p>
-<p>The density of the triangular distribution is given by</p>
-<div class="math">
-\[\begin{split}f(x) := \begin{cases}
-          0 &amp; \mathrm{for\ } x &lt; a, \\
-          \frac{2(x-a)}{(b-a)(c-a)} &amp; \mathrm{for\ } a \le x &lt; c, \\
-          \frac{2}{b-a} &amp; \mathrm{for\ } x = c, \\
-          \frac{2(b-x)}{(b-a)(b-c)} &amp; \mathrm{for\ } c &lt; x \le b, \\
-          0 &amp; \mathrm{for\ } b &lt; x.
-        \end{cases}\end{split}\]</div>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>a</strong> : Real number, <span class="math">\(a \in \left(-\infty, \infty\right)\)</span></p>
-<p><strong>b</strong> : Real number, <span class="math">\(a &lt; b\)</span></p>
-<p><strong>c</strong> : Real number, <span class="math">\(a \leq c \leq b\)</span></p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Triangular_distribution">http://en.wikipedia.org/wiki/Triangular_distribution</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/TriangularDistribution.html">http://mathworld.wolfram.com/TriangularDistribution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Triangular</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">E</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;a&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;b&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">c</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;c&quot;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Triangular</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">c</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_x, Piecewise(((2*_x - 2*a)/((-a + b)*(-a + c)),</span>
-<span class="go">                    And(_x &lt; c, a &lt;= _x)),</span>
-<span class="go">                    (2/(-a + b), _x == c),</span>
-<span class="go">                    ((-2*_x + 2*b)/((-a + b)*(b - c)),</span>
-<span class="go">                    And(_x &lt;= b, c &lt; _x)),</span>
-<span class="go">                    (0, True)))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Uniform">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Uniform</tt><big>(</big><em>name</em>, <em>left</em>, <em>right</em><big>)</big><a class="headerlink" href="#sympy.stats.Uniform" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with a uniform distribution.</p>
-<p>The density of the uniform distribution is given by</p>
-<div class="math">
-\[\begin{split}f(x) := \begin{cases}
-          \frac{1}{b - a} &amp; \text{for } x \in [a,b]  \\
-          0               &amp; \text{otherwise}
-        \end{cases}\end{split}\]</div>
-<p>with <span class="math">\(x \in [a,b]\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>a</strong> : Real number, <span class="math">\(-\infty &lt; a\)</span> the left boundary</p>
-<p><strong>b</strong> : Real number, <span class="math">\(a &lt; b &lt; \infty\)</span> the right boundary</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29">http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/UniformDistribution.html">http://mathworld.wolfram.com/UniformDistribution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Uniform</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">cdf</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">variance</span><span class="p">,</span> <span class="n">skewness</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;a&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;b&quot;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Uniform</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_x, Piecewise((1/(-a + b), And(_x &lt;= b, a &lt;= _x)), (0, True)))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">cdf</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_z, _z/(-a + b) - a/(-a + b))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">E</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
-<span class="go">a/2 + b/2</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">variance</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
-<span class="go">a**2/12 - a*b/6 + b**2/12</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">skewness</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.UniformSum">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">UniformSum</tt><big>(</big><em>name</em>, <em>n</em><big>)</big><a class="headerlink" href="#sympy.stats.UniformSum" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with an Irwin-Hall distribution.</p>
-<p>The probability distribution function depends on a single parameter
-<span class="math">\(n\)</span> which is an integer.</p>
-<p>The density of the Irwin-Hall distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{1}{(n-1)!}\sum_{k=0}^{\lfloor x\rfloor}(-1)^k
-        \binom{n}{k}(x-k)^{n-1}\]</div>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>n</strong> : Integral number, <span class="math">\(n\)</span> &gt; 0</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>A RandomSymbol.</strong> :</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Uniform_sum_distribution">http://en.wikipedia.org/wiki/Uniform_sum_distribution</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/UniformSumDistribution.html">http://mathworld.wolfram.com/UniformSumDistribution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">UniformSum</span><span class="p">,</span> <span class="n">density</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">pprint</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">UniformSum</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">      /   floor(x)                        \</span>
-<span class="go">      |     ___                           |</span>
-<span class="go">      |     \  `                          |</span>
-<span class="go">      |      \         k         n - 1 /n\|</span>
-<span class="go">      |       )    (-1) *(-k + x)     *| ||</span>
-<span class="go">      |      /                         \k/|</span>
-<span class="go">      |     /__,                          |</span>
-<span class="go">      |    k = 0                          |</span>
-<span class="go">Lambda|x, --------------------------------|</span>
-<span class="go">      \               (n - 1)!            /</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.VonMises">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">VonMises</tt><big>(</big><em>name</em>, <em>mu</em>, <em>k</em><big>)</big><a class="headerlink" href="#sympy.stats.VonMises" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a Continuous Random Variable with a von Mises distribution.</p>
-<p>The density of the von Mises distribution is given by</p>
-<div class="math">
-\[f(x) := \frac{e^{\kappa\cos(x-\mu)}}{2\pi I_0(\kappa)}\]</div>
-<p>with <span class="math">\(x \in [0,2\pi]\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>mu</strong> : Real number, measure of location</p>
-<p><strong>k</strong> : Real number, measure of concentration</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Von_Mises_distribution">http://en.wikipedia.org/wiki/Von_Mises_distribution</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/vonMisesDistribution.html">http://mathworld.wolfram.com/vonMisesDistribution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">VonMises</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">variance</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">pprint</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">mu</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;mu&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">VonMises</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">mu</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">D</span><span class="p">,</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">      /      k*cos(x - mu)  \</span>
-<span class="go">      |     e               |</span>
-<span class="go">Lambda|x, ------------------|</span>
-<span class="go">      \   2*pi*besseli(0, k)/</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.Weibull">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Weibull</tt><big>(</big><em>name</em>, <em>alpha</em>, <em>beta</em><big>)</big><a class="headerlink" href="#sympy.stats.Weibull" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with a Weibull distribution.</p>
-<p>The density of the Weibull distribution is given by</p>
-<div class="math">
-\[\begin{split}f(x) := \begin{cases}
-          \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}
-          e^{-(x/\lambda)^{k}} &amp; x\geq0\\
-          0 &amp; x&lt;0
-        \end{cases}\end{split}\]</div>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>lambda</strong> : Real number, <span class="math">\(\lambda &gt; 0\)</span> a scale</p>
-<p><strong>k</strong> : Real number, <span class="math">\(k\)</span> &gt; 0 a shape</p>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first last"><strong>A RandomSymbol.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Weibull_distribution">http://en.wikipedia.org/wiki/Weibull_distribution</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/WeibullDistribution.html">http://mathworld.wolfram.com/WeibullDistribution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Weibull</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">variance</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">l</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;lambda&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">k</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;k&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Weibull</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="n">k</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_x, k*(_x/lambda)**(k - 1)*exp(-(_x/lambda)**k)/lambda)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">E</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
-<span class="go">lambda*gamma(1 + 1/k)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">variance</span><span class="p">(</span><span class="n">X</span><span class="p">))</span>
-<span class="go">lambda**2*(-gamma(1 + 1/k)**2 + gamma(1 + 2/k))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.WignerSemicircle">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">WignerSemicircle</tt><big>(</big><em>name</em>, <em>R</em><big>)</big><a class="headerlink" href="#sympy.stats.WignerSemicircle" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a continuous random variable with a Wigner semicircle distribution.</p>
-<p>The density of the Wigner semicircle distribution is given by</p>
-<div class="math">
-\[f(x) := \frac2{\pi R^2}\,\sqrt{R^2-x^2}\]</div>
-<p>with <span class="math">\(x \in [-R,R]\)</span>.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><strong>R</strong> : Real number, <span class="math">\(R\)</span> &gt; 0 the radius</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><strong>A `RandomSymbol`.</strong> :</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://en.wikipedia.org/wiki/Wigner_semicircle_distribution">http://en.wikipedia.org/wiki/Wigner_semicircle_distribution</a>
-[2] <a class="reference external" href="http://mathworld.wolfram.com/WignersSemicircleLaw.html">http://mathworld.wolfram.com/WignersSemicircleLaw.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">WignerSemicircle</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">E</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">simplify</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;R&quot;</span><span class="p">,</span> <span class="n">positive</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">WignerSemicircle</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_x, 2*sqrt(-_x**2 + R**2)/(pi*R**2))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">E</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.ContinuousRV">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">ContinuousRV</tt><big>(</big><em>symbol</em>, <em>density</em>, <em>set=(-oo</em>, <em>oo)</em><big>)</big><a class="headerlink" href="#sympy.stats.ContinuousRV" title="Permalink to this definition">¶</a></dt>
-<dd><p>Create a Continuous Random Variable given the following:</p>
-<p>&#8211; a symbol
-&#8211; a probability density function
-&#8211; set on which the pdf is valid (defaults to entire real line)</p>
-<p>Returns a RandomSymbol.</p>
-<p>Many common continuous random variable types are already implemented.
-This function should be necessary only very rarely.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">pi</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">ContinuousRV</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">E</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pdf</span> <span class="o">=</span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="n">pi</span><span class="p">))</span> <span class="c"># Normal distribution</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">ContinuousRV</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">pdf</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">E</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="p">(</span><span class="n">X</span><span class="o">&gt;</span><span class="mi">0</span><span class="p">)</span>
-<span class="go">1/2</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-<div class="section" id="interface">
-<h2>Interface<a class="headerlink" href="#interface" title="Permalink to this headline">¶</a></h2>
-<dl class="function">
-<dt id="sympy.stats.P">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">P</tt><big>(</big><em>condition</em>, <em>given_condition=None</em>, <em>numsamples=None</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#sympy.stats.P" title="Permalink to this definition">¶</a></dt>
-<dd><p>Probability that a condition is true, optionally given a second condition</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>expr</strong> : Relational containing RandomSymbols</p>
-<blockquote>
-<div><p>The condition of which you want to compute the probability</p>
-</div></blockquote>
-<p><strong>given_condition</strong> : Relational containing RandomSymbols</p>
-<blockquote>
-<div><p>A conditional expression. P(X&gt;1, X&gt;0) is expectation of X&gt;1 given X&gt;0</p>
-</div></blockquote>
-<p><strong>numsamples</strong> : int</p>
-<blockquote>
-<div><p>Enables sampling and approximates the probability with this many samples</p>
-</div></blockquote>
-<p><strong>evalf</strong> : Bool (defaults to True)</p>
-<blockquote>
-<div><p>If sampling return a number rather than a complex expression</p>
-</div></blockquote>
-<p><strong>evaluate</strong> : Bool (defaults to True)</p>
-<blockquote class="last">
-<div><p>In case of continuous systems return unevaluated integral</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">P</span><span class="p">,</span> <span class="n">Die</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Eq</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="o">=</span> <span class="n">Die</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">Die</span><span class="p">(</span><span class="s">&#39;Y&#39;</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="p">(</span><span class="n">X</span><span class="o">&gt;</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="p">(</span><span class="n">Eq</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> <span class="n">X</span><span class="o">&gt;</span><span class="mi">2</span><span class="p">)</span> <span class="c"># Probability that X == 5 given that X &gt; 2</span>
-<span class="go">1/4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">P</span><span class="p">(</span><span class="n">X</span><span class="o">&gt;</span><span class="n">Y</span><span class="p">)</span>
-<span class="go">5/12</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.E">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">E</tt><big>(</big><em>expr</em>, <em>condition=None</em>, <em>numsamples=None</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#sympy.stats.E" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the expected value of a random expression</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>expr</strong> : Expr containing RandomSymbols</p>
-<blockquote>
-<div><p>The expression of which you want to compute the expectation value</p>
-</div></blockquote>
-<p><strong>given</strong> : Expr containing RandomSymbols</p>
-<blockquote>
-<div><p>A conditional expression. E(X, X&gt;0) is expectation of X given X &gt; 0</p>
-</div></blockquote>
-<p><strong>numsamples</strong> : int</p>
-<blockquote>
-<div><p>Enables sampling and approximates the expectation with this many samples</p>
-</div></blockquote>
-<p><strong>evalf</strong> : Bool (defaults to True)</p>
-<blockquote>
-<div><p>If sampling return a number rather than a complex expression</p>
-</div></blockquote>
-<p><strong>evaluate</strong> : Bool (defaults to True)</p>
-<blockquote class="last">
-<div><p>In case of continuous systems return unevaluated integral</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">E</span><span class="p">,</span> <span class="n">Die</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Die</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">7/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">X</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">8</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">E</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">X</span><span class="o">&gt;</span><span class="mi">3</span><span class="p">)</span> <span class="c"># Expectation of X given that it is above 3</span>
-<span class="go">5</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.density">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">density</tt><big>(</big><em>expr</em>, <em>condition=None</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#sympy.stats.density" title="Permalink to this definition">¶</a></dt>
-<dd><p>Probability density of a random expression</p>
-<p>Optionally given a second condition</p>
-<p>This density will take on different forms for different types of
-probability spaces.
-Discrete variables produce Dicts.
-Continuous variables produce Lambdas.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">density</span><span class="p">,</span> <span class="n">Die</span><span class="p">,</span> <span class="n">Normal</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">Die</span><span class="p">(</span><span class="s">&#39;D&#39;</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Normal</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">D</span><span class="p">)</span>
-<span class="go">{1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">D</span><span class="p">)</span>
-<span class="go">{2: 1/6, 4: 1/6, 6: 1/6, 8: 1/6, 10: 1/6, 12: 1/6}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
-<span class="go">Lambda(_x, sqrt(2)*exp(-_x**2/2)/(2*sqrt(pi)))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.given">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">given</tt><big>(</big><em>expr</em>, <em>condition=None</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#sympy.stats.given" title="Permalink to this definition">¶</a></dt>
-<dd><p>From a random expression and a condition on that expression creates a new
-probability space from the condition and returns the same expression on that
-conditional probability space.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">given</span><span class="p">,</span> <span class="n">density</span><span class="p">,</span> <span class="n">Die</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Die</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Y</span> <span class="o">=</span> <span class="n">given</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">X</span><span class="o">&gt;</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">density</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span>
-<span class="go">{4: 1/3, 5: 1/3, 6: 1/3}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.where">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">where</tt><big>(</big><em>condition</em>, <em>given_condition=None</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#sympy.stats.where" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the domain where a condition is True.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">where</span><span class="p">,</span> <span class="n">Die</span><span class="p">,</span> <span class="n">Normal</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">And</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">D1</span><span class="p">,</span> <span class="n">D2</span> <span class="o">=</span> <span class="n">Die</span><span class="p">(</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">Die</span><span class="p">(</span><span class="s">&#39;b&#39;</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">D1</span><span class="o">.</span><span class="n">symbol</span><span class="p">,</span> <span class="n">D2</span><span class="o">.</span><span class="n">symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Normal</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">where</span><span class="p">(</span><span class="n">X</span><span class="o">**</span><span class="mi">2</span><span class="o">&lt;</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">Domain: And(-1 &lt; x, x &lt; 1)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">where</span><span class="p">(</span><span class="n">X</span><span class="o">**</span><span class="mi">2</span><span class="o">&lt;</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">set</span>
-<span class="go">(-1, 1)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">where</span><span class="p">(</span><span class="n">And</span><span class="p">(</span><span class="n">D1</span><span class="o">&lt;=</span><span class="n">D2</span> <span class="p">,</span> <span class="n">D2</span><span class="o">&lt;</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">Domain: Or(And(a == 1, b == 1), And(a == 1, b == 2), And(a == 2, b == 2))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.variance">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">variance</tt><big>(</big><em>X</em>, <em>condition=None</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#sympy.stats.variance" title="Permalink to this definition">¶</a></dt>
-<dd><p>Variance of a random expression</p>
-<p>Expectation of (X-E(X))**2</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Die</span><span class="p">,</span> <span class="n">E</span><span class="p">,</span> <span class="n">Bernoulli</span><span class="p">,</span> <span class="n">variance</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">simplify</span><span class="p">,</span> <span class="n">Symbol</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Die</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Bernoulli</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">variance</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">X</span><span class="p">)</span>
-<span class="go">35/3</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">simplify</span><span class="p">(</span><span class="n">variance</span><span class="p">(</span><span class="n">B</span><span class="p">))</span>
-<span class="go">p*(-p + 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.std">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">std</tt><big>(</big><em>X</em>, <em>condition=None</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#sympy.stats.std" title="Permalink to this definition">¶</a></dt>
-<dd><p>Standard Deviation of a random expression</p>
-<p>Square root of the Expectation of (X-E(X))**2</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Bernoulli</span><span class="p">,</span> <span class="n">std</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">Bernoulli</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">std</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
-<span class="go">sqrt(-p**2 + p)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.sample">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">sample</tt><big>(</big><em>expr</em>, <em>condition=None</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#sympy.stats.sample" title="Permalink to this definition">¶</a></dt>
-<dd><p>A realization of the random expression</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Die</span><span class="p">,</span> <span class="n">sample</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">Z</span> <span class="o">=</span> <span class="n">Die</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">Die</span><span class="p">(</span><span class="s">&#39;Y&#39;</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">Die</span><span class="p">(</span><span class="s">&#39;Z&#39;</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">die_roll</span> <span class="o">=</span> <span class="n">sample</span><span class="p">(</span><span class="n">X</span><span class="o">+</span><span class="n">Y</span><span class="o">+</span><span class="n">Z</span><span class="p">)</span> <span class="c"># A random realization of three dice</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.sample_iter">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">sample_iter</tt><big>(</big><em>expr</em>, <em>condition=None</em>, <em>numsamples=oo</em>, <em>**kwargs</em><big>)</big><a class="headerlink" href="#sympy.stats.sample_iter" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns an iterator of realizations from the expression given a condition</p>
-<p>expr: Random expression to be realized
-condition: A conditional expression (optional)
-numsamples: Length of the iterator (defaults to infinity)</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">Sample</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">sampling_P</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">sampling_E</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">sample_iter_lambdify</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">sample_iter_subs</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">Normal</span><span class="p">,</span> <span class="n">sample_iter</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Normal</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expr</span> <span class="o">=</span> <span class="n">X</span><span class="o">*</span><span class="n">X</span> <span class="o">+</span> <span class="mi">3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">iterator</span> <span class="o">=</span> <span class="n">sample_iter</span><span class="p">(</span><span class="n">expr</span><span class="p">,</span> <span class="n">numsamples</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">iterator</span><span class="p">)</span> 
-<span class="go">[12, 4, 7]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-<div class="section" id="module-sympy.stats.rv">
-<span id="mechanics"></span><h2>Mechanics<a class="headerlink" href="#module-sympy.stats.rv" title="Permalink to this headline">¶</a></h2>
-<p>SymPy Stats employs a relatively complex class hierarchy.</p>
-<p>RandomDomains are a mapping of variables to possible values. For example we
-might say that the symbol Symbol(&#8216;x&#8217;) can take on the values {1,2,3,4,5,6}.</p>
-<dl class="class">
-<dt id="sympy.stats.rv.RandomDomain">
-<em class="property">class </em><tt class="descclassname">sympy.stats.rv.</tt><tt class="descname">RandomDomain</tt><a class="reference internal" href="../_modules/sympy/stats/rv.html#RandomDomain"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.stats.rv.RandomDomain" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<p>A PSpace, or Probability Space, combines a RandomDomain with a density to
-provide probabilistic information. For example the above domain could be
-enhanced by a finite density {1:1/6, 2:1/6, 3:1/6, 4:1/6, 5:1/6, 6:1/6} to
-fully define the roll of a fair die named &#8216;x&#8217;.</p>
-<dl class="class">
-<dt id="sympy.stats.rv.PSpace">
-<em class="property">class </em><tt class="descclassname">sympy.stats.rv.</tt><tt class="descname">PSpace</tt><a class="reference internal" href="../_modules/sympy/stats/rv.html#PSpace"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.stats.rv.PSpace" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<p>A RandomSymbol represents the PSpace&#8217;s symbol &#8216;x&#8217; inside of SymPy expressions.</p>
-<dl class="class">
-<dt id="sympy.stats.rv.RandomSymbol">
-<em class="property">class </em><tt class="descclassname">sympy.stats.rv.</tt><tt class="descname">RandomSymbol</tt><a class="reference internal" href="../_modules/sympy/stats/rv.html#RandomSymbol"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.stats.rv.RandomSymbol" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<p>The RandomDomain and PSpace classes are almost never directly instantiated.
-Instead they are subclassed for a variety of situations.</p>
-<p>RandomDomains and PSpaces must be sufficiently general to represent domains and
-spaces of several variables with arbitrarily complex densities. This generality
-is often unnecessary. Instead we often build SingleDomains and SinglePSpaces to
-represent single, univariate events and processes such as a single die or a
-single normal variable.</p>
-<dl class="class">
-<dt id="sympy.stats.rv.SinglePSpace">
-<em class="property">class </em><tt class="descclassname">sympy.stats.rv.</tt><tt class="descname">SinglePSpace</tt><a class="reference internal" href="../_modules/sympy/stats/rv.html#SinglePSpace"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.stats.rv.SinglePSpace" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.stats.rv.SingleDomain">
-<em class="property">class </em><tt class="descclassname">sympy.stats.rv.</tt><tt class="descname">SingleDomain</tt><a class="reference internal" href="../_modules/sympy/stats/rv.html#SingleDomain"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.stats.rv.SingleDomain" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<p>Another common case is to collect together a set of such univariate random
-variables. A collection of independent SinglePSpaces or SingleDomains can be
-brought together to form a ProductDomain or ProductPSpace. These objects would
-be useful in representing three dice rolled together for example.</p>
-<dl class="class">
-<dt id="sympy.stats.rv.ProductDomain">
-<em class="property">class </em><tt class="descclassname">sympy.stats.rv.</tt><tt class="descname">ProductDomain</tt><a class="reference internal" href="../_modules/sympy/stats/rv.html#ProductDomain"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.stats.rv.ProductDomain" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.stats.rv.ProductPSpace">
-<em class="property">class </em><tt class="descclassname">sympy.stats.rv.</tt><tt class="descname">ProductPSpace</tt><a class="reference internal" href="../_modules/sympy/stats/rv.html#ProductPSpace"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.stats.rv.ProductPSpace" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<p>The Conditional adjective is added whenever we add a global condition to a
-RandomDomain or PSpace. A common example would be three independent dice where
-we know their sum to be greater than 12.</p>
-<dl class="class">
-<dt id="sympy.stats.rv.ConditionalDomain">
-<em class="property">class </em><tt class="descclassname">sympy.stats.rv.</tt><tt class="descname">ConditionalDomain</tt><a class="reference internal" href="../_modules/sympy/stats/rv.html#ConditionalDomain"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.stats.rv.ConditionalDomain" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<p>We specialize further into Finite and Continuous versions of these classes to
-represent finite (such as dice) and continuous (such as normals) random
-variables.</p>
-<span class="target" id="module-sympy.stats.frv"></span><dl class="class">
-<dt id="sympy.stats.frv.FiniteDomain">
-<em class="property">class </em><tt class="descclassname">sympy.stats.frv.</tt><tt class="descname">FiniteDomain</tt><a class="reference internal" href="../_modules/sympy/stats/frv.html#FiniteDomain"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.stats.frv.FiniteDomain" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.stats.frv.FinitePSpace">
-<em class="property">class </em><tt class="descclassname">sympy.stats.frv.</tt><tt class="descname">FinitePSpace</tt><a class="reference internal" href="../_modules/sympy/stats/frv.html#FinitePSpace"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.stats.frv.FinitePSpace" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<span class="target" id="module-sympy.stats.crv"></span><dl class="class">
-<dt id="sympy.stats.crv.ContinuousDomain">
-<em class="property">class </em><tt class="descclassname">sympy.stats.crv.</tt><tt class="descname">ContinuousDomain</tt><a class="reference internal" href="../_modules/sympy/stats/crv.html#ContinuousDomain"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.stats.crv.ContinuousDomain" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="class">
-<dt id="sympy.stats.crv.ContinuousPSpace">
-<em class="property">class </em><tt class="descclassname">sympy.stats.crv.</tt><tt class="descname">ContinuousPSpace</tt><a class="reference internal" href="../_modules/sympy/stats/crv.html#ContinuousPSpace"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.stats.crv.ContinuousPSpace" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<p>Additionally there are a few specialized classes that implement certain common
-random variable types. There is for example a DiePSpace that implements
-SingleFinitePSpace and a NormalPSpace that implements SingleContinuousPSpace.</p>
-<span class="target" id="module-sympy.stats.frv_types"></span><dl class="class">
-<dt id="sympy.stats.frv_types.DiePSpace">
-<em class="property">class </em><tt class="descclassname">sympy.stats.frv_types.</tt><tt class="descname">DiePSpace</tt><a class="reference internal" href="../_modules/sympy/stats/frv_types.html#DiePSpace"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.stats.frv_types.DiePSpace" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<span class="target" id="module-sympy.stats.crv_types"></span><dl class="class">
-<dt id="sympy.stats.crv_types.NormalPSpace">
-<em class="property">class </em><tt class="descclassname">sympy.stats.crv_types.</tt><tt class="descname">NormalPSpace</tt><a class="reference internal" href="../_modules/sympy/stats/crv_types.html#NormalPSpace"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.stats.crv_types.NormalPSpace" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<p>RandomVariables can be extracted from these objects using the PSpace.values
-method.</p>
-<p>As previously mentioned SymPy Stats employs a relatively complex class
-structure. Inheritance is widely used in the implementation of end-level
-classes. This tactic was chosen to balance between the need to allow SymPy to
-represent arbitrarily defined random variables and optimizing for common cases.
-This complicates the code but is structured to only be important to those
-working on extending SymPy Stats to other random variable types.</p>
-<p>Users will not use this class structure. Instead these mechanics are exposed
-through variable creation functions Die, Coin, FiniteRV, Normal, Exponential,
-etc.... These build the appropriate SinglePSpaces and return the corresponding
-RandomVariable. Conditional and Product spaces are formed in the natural
-construction of SymPy expressions and the use of interface functions E, Given,
-Density, etc....</p>
-<dl class="function">
-<dt id="sympy.stats.crv_types.sympy.stats.Die">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Die</tt><big>(</big><big>)</big><a class="headerlink" href="#sympy.stats.crv_types.sympy.stats.Die" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.crv_types.sympy.stats.Normal">
-<tt class="descclassname">sympy.stats.</tt><tt class="descname">Normal</tt><big>(</big><big>)</big><a class="headerlink" href="#sympy.stats.crv_types.sympy.stats.Normal" title="Permalink to this definition">¶</a></dt>
-<dd></dd></dl>
-
-<p>There are some additional functions that may be useful. They are largely used
-internally.</p>
-<dl class="function">
-<dt id="sympy.stats.rv.random_symbols">
-<tt class="descclassname">sympy.stats.rv.</tt><tt class="descname">random_symbols</tt><big>(</big><em>expr</em><big>)</big><a class="reference internal" href="../_modules/sympy/stats/rv.html#random_symbols"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.stats.rv.random_symbols" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns all RandomSymbols within a SymPy Expression.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.rv.pspace">
-<tt class="descclassname">sympy.stats.rv.</tt><tt class="descname">pspace</tt><big>(</big><em>expr</em><big>)</big><a class="reference internal" href="../_modules/sympy/stats/rv.html#pspace"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.stats.rv.pspace" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the underlying Probability Space of a random expression.</p>
-<p>For internal use.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats</span> <span class="kn">import</span> <span class="n">pspace</span><span class="p">,</span> <span class="n">Normal</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.stats.rv</span> <span class="kn">import</span> <span class="n">ProductPSpace</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">Normal</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pspace</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">X</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">==</span> <span class="n">X</span><span class="o">.</span><span class="n">pspace</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.stats.rv.rs_swap">
-<tt class="descclassname">sympy.stats.rv.</tt><tt class="descname">rs_swap</tt><big>(</big><em>a</em>, <em>b</em><big>)</big><a class="reference internal" href="../_modules/sympy/stats/rv.html#rs_swap"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.stats.rv.rs_swap" title="Permalink to this definition">¶</a></dt>
-<dd><p>Build a dictionary to swap RandomSymbols based on their underlying symbol.</p>
-<p>i.e.
-if    X = (&#8216;x&#8217;, pspace1)
-and   Y = (&#8216;x&#8217;, pspace2)
-then X and Y match and the key, value pair
-{X:Y} will appear in the result</p>
-<p>Inputs: collections a and b of random variables which share common symbols
-Output: dict mapping RVs in a to RVs in b</p>
-</dd></dl>
-
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../index.html">
-              <img class="logo" src="../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Stats</a><ul>
-<li><a class="reference internal" href="#examples">Examples</a></li>
-<li><a class="reference internal" href="#random-variable-types">Random Variable Types</a><ul>
-<li><a class="reference internal" href="#finite-types">Finite Types</a></li>
-<li><a class="reference internal" href="#continuous-types">Continuous Types</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#interface">Interface</a></li>
-<li><a class="reference internal" href="#module-sympy.stats.rv">Mechanics</a></li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
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-                        title="previous chapter">Statistics</a></p>
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-<span id="id1"></span><span id="tensor-module"></span><h1>Tensor Module<a class="headerlink" href="#module-sympy.tensor" title="Permalink to this headline">¶</a></h1>
-<p>A module to manipulate symbolic objects with indices including tensors</p>
-<div class="section" id="contents">
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-  <div class="section" id="module-sympy.tensor.index_methods">
-<span id="methods"></span><h1>Methods<a class="headerlink" href="#module-sympy.tensor.index_methods" title="Permalink to this headline">¶</a></h1>
-<p>Module with functions operating on IndexedBase, Indexed and Idx objects</p>
-<ul class="simple">
-<li>Check shape conformance</li>
-<li>Determine indices in resulting expression</li>
-</ul>
-<p>etc.</p>
-<p>Methods in this module could be implemented by calling methods on Expr
-objects instead.  When things stabilize this could be a useful
-refactoring.</p>
-<dl class="function">
-<dt id="sympy.tensor.index_methods.get_contraction_structure">
-<tt class="descclassname">sympy.tensor.index_methods.</tt><tt class="descname">get_contraction_structure</tt><big>(</big><em>expr</em><big>)</big><a class="reference internal" href="../../_modules/sympy/tensor/index_methods.html#get_contraction_structure"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.tensor.index_methods.get_contraction_structure" title="Permalink to this definition">¶</a></dt>
-<dd><p>Determine dummy indices of <tt class="docutils literal"><span class="pre">expr</span></tt> and describe it&#8217;s structure</p>
-<p>By <em>dummy</em> we mean indices that are summation indices.</p>
-<p>The stucture of the expression is determined and described as follows:</p>
-<ol class="arabic">
-<li><p class="first">A conforming summation of Indexed objects is described with a dict where
-the keys are summation indices and the corresponding values are sets
-containing all terms for which the summation applies.  All Add objects
-in the SymPy expression tree are described like this.</p>
-</li>
-<li><p class="first">For all nodes in the SymPy expression tree that are <em>not</em> of type Add, the
-following applies:</p>
-<p>If a node discovers contractions in one of it&#8217;s arguments, the node
-itself will be stored as a key in the dict.  For that key, the
-corresponding value is a list of dicts, each of which is the result of a
-recursive call to get_contraction_structure().  The list contains only
-dicts for the non-trivial deeper contractions, ommitting dicts with None
-as the one and only key.</p>
-</li>
-</ol>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">The presence of expressions among the dictinary keys indicates
-multiple levels of index contractions.  A nested dict displays nested
-contractions and may itself contain dicts from a deeper level.  In
-practical calculations the summation in the deepest nested level must be
-calculated first so that the outer expression can access the resulting
-indexed object.</p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.tensor.index_methods</span> <span class="kn">import</span> <span class="n">get_contraction_structure</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">default_sort_key</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.tensor</span> <span class="kn">import</span> <span class="n">IndexedBase</span><span class="p">,</span> <span class="n">Idx</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">A</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">IndexedBase</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="s">&#39;y&#39;</span><span class="p">,</span> <span class="s">&#39;A&#39;</span><span class="p">]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">l</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">Idx</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="s">&#39;j&#39;</span><span class="p">,</span> <span class="s">&#39;k&#39;</span><span class="p">,</span> <span class="s">&#39;l&#39;</span><span class="p">]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">get_contraction_structure</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">A</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">])</span>
-<span class="go">{(i,): set([x[i]*y[i]]), (j,): set([A[j, j]])}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">get_contraction_structure</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-<span class="go">{None: set([x[i]*y[j]])}</span>
-</pre></div>
-</div>
-<p>A multiplication of contracted factors results in nested dicts representing
-the internal contractions.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">get_contraction_structure</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">sorted</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">keys</span><span class="p">()),</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-<span class="go">[None, x[i, i]*y[j, j]]</span>
-</pre></div>
-</div>
-<p>In this case, the product has no contractions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="p">[</span><span class="bp">None</span><span class="p">]</span>
-<span class="go">set([x[i, i]*y[j, j]])</span>
-</pre></div>
-</div>
-<p>Factors are contracted &#8220;first&#8221;:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">sorted</span><span class="p">(</span><span class="n">d</span><span class="p">[</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">]],</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-<span class="go">[{(i,): set([x[i, i]])}, {(j,): set([y[j, j]])}]</span>
-</pre></div>
-</div>
-<p>A parenthesized Add object is also returned as a nested dictionary.  The
-term containing the parenthesis is a Mul with a contraction among the
-arguments, so it will be found as a key in the result.  It stores the
-dictionary resulting from a recursive call on the Add expression.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">get_contraction_structure</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">j</span><span class="p">]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">sorted</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">d</span><span class="o">.</span><span class="n">keys</span><span class="p">()),</span> <span class="n">key</span><span class="o">=</span><span class="n">default_sort_key</span><span class="p">)</span>
-<span class="go">[x[i]*(y[i] + A[i, j]*x[j]), (i,)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="p">[(</span><span class="n">i</span><span class="p">,)]</span>
-<span class="go">set([x[i]*(y[i] + A[i, j]*x[j])])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="p">[</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">+</span> <span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">])]</span>
-<span class="go">[{None: set([y[i]]), (j,): set([A[i, j]*x[j]])}]</span>
-</pre></div>
-</div>
-<p>Powers with contractions in either base or exponent will also be found as
-keys in the dictionary, mapping to a list of results from recursive calls:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">d</span> <span class="o">=</span> <span class="n">get_contraction_structure</span><span class="p">(</span><span class="n">A</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">**</span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="p">[</span><span class="bp">None</span><span class="p">]</span>
-<span class="go">set([A[j, j]**A[i, i]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nested_contractions</span> <span class="o">=</span> <span class="n">d</span><span class="p">[</span><span class="n">A</span><span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">**</span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nested_contractions</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-<span class="go">{(j,): set([A[j, j]])}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">nested_contractions</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
-<span class="go">{(i,): set([A[i, i]])}</span>
-</pre></div>
-</div>
-<p>The description of the contraction structure may appear complicated when
-represented with a string in the above examples, but it is easy to iterate
-over:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Expr</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">key</span> <span class="ow">in</span> <span class="n">d</span><span class="p">:</span>
-<span class="gp">... </span>    <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">Expr</span><span class="p">):</span>
-<span class="gp">... </span>        <span class="k">continue</span>
-<span class="gp">... </span>    <span class="k">for</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">d</span><span class="p">[</span><span class="n">key</span><span class="p">]:</span>
-<span class="gp">... </span>        <span class="k">if</span> <span class="n">term</span> <span class="ow">in</span> <span class="n">d</span><span class="p">:</span>
-<span class="gp">... </span>            <span class="c"># treat deepest contraction first</span>
-<span class="gp">... </span>            <span class="k">pass</span>
-<span class="gp">... </span>    <span class="c"># treat outermost contactions here</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.tensor.index_methods.get_indices">
-<tt class="descclassname">sympy.tensor.index_methods.</tt><tt class="descname">get_indices</tt><big>(</big><em>expr</em><big>)</big><a class="reference internal" href="../../_modules/sympy/tensor/index_methods.html#get_indices"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.tensor.index_methods.get_indices" title="Permalink to this definition">¶</a></dt>
-<dd><p>Determine the outer indices of expression <tt class="docutils literal"><span class="pre">expr</span></tt></p>
-<p>By <em>outer</em> we mean indices that are not summation indices.  Returns a set
-and a dict.  The set contains outer indices and the dict contains
-information about index symmetries.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.tensor.index_methods</span> <span class="kn">import</span> <span class="n">get_indices</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.tensor</span> <span class="kn">import</span> <span class="n">IndexedBase</span><span class="p">,</span> <span class="n">Idx</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">A</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">IndexedBase</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="s">&#39;y&#39;</span><span class="p">,</span> <span class="s">&#39;A&#39;</span><span class="p">]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i j a z&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>The indices of the total expression is determined, Repeated indices imply a
-summation, for instance the trace of a matrix A:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">get_indices</span><span class="p">(</span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">i</span><span class="p">])</span>
-<span class="go">(set(), {})</span>
-</pre></div>
-</div>
-<p>In the case of many terms, the terms are required to have identical
-outer indices.  Else an IndexConformanceException is raised.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">get_indices</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">*</span><span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-<span class="go">(set([i]), {})</span>
-</pre></div>
-</div>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Exceptions :</th><td class="field-body"></td>
-</tr>
-</tbody>
-</table>
-<p>An IndexConformanceException means that the terms ar not compatible, e.g.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">get_indices</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>                
-<span class="go">        (...)</span>
-<span class="go">IndexConformanceException: Indices are not consistent: x(i) + y(j)</span>
-</pre></div>
-</div>
-<div class="admonition warning">
-<p class="first admonition-title">Warning</p>
-<p>The concept of <em>outer</em> indices applies recursively, starting on the deepest
-level.  This implies that dummies inside parenthesis are assumed to be
-summed first, so that the following expression is handled gracefully:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">get_indices</span><span class="p">((</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">*</span><span class="n">y</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
-<span class="go">(set([i, j]), {})</span>
-</pre></div>
-</div>
-<p class="last">This is correct and may appear convenient, but you need to be careful
-with this as SymPy will happily .expand() the product, if requested.  The
-resulting expression would mix the outer <tt class="docutils literal"><span class="pre">j</span></tt> with the dummies inside
-the parenthesis, which makes it a different expression.  To be on the
-safe side, it is best to avoid such ambiguities by using unique indices
-for all contractions that should be held separate.</p>
-</div>
-</dd></dl>
-
-</div>
-
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-            
-  <div class="section" id="module-sympy.tensor.indexed">
-<span id="indexed-objects"></span><h1>Indexed Objects<a class="headerlink" href="#module-sympy.tensor.indexed" title="Permalink to this headline">¶</a></h1>
-<p>Module that defines indexed objects</p>
-<p>The classes IndexedBase, Indexed and Idx would represent a matrix element
-M[i, j] as in the following graph:</p>
-<div class="highlight-python"><pre>1) The Indexed class represents the entire indexed object.
-           |
-        ___|___
-       &#x27;       &#x27;
-        M[i, j]
-       /   \__\______
-       |             |
-       |             |
-       |     2) The Idx class represent indices and each Idx can
-       |        optionally contain information about its range.
-       |
- 3) IndexedBase represents the `stem&#x27; of an indexed object, here `M&#x27;.
-    The stem used by itself is usually taken to represent the entire
-    array.</pre>
-</div>
-<p>There can be any number of indices on an Indexed object.  No
-transformation properties are implemented in these Base objects, but
-implicit contraction of repeated indices is supported.</p>
-<p>Note that the support for complicated (i.e. non-atomic) integer
-expressions as indices is limited.  (This should be improved in
-future releases.)</p>
-<div class="section" id="examples">
-<h2>Examples<a class="headerlink" href="#examples" title="Permalink to this headline">¶</a></h2>
-<p>To express the above matrix element example you would write:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.tensor</span> <span class="kn">import</span> <span class="n">IndexedBase</span><span class="p">,</span> <span class="n">Idx</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="n">IndexedBase</span><span class="p">(</span><span class="s">&#39;M&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i j&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Idx</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-<span class="go">M[i, j]</span>
-</pre></div>
-</div>
-<p>Repeated indices in a product implies a summation, so to express a
-matrix-vector product in terms of Indexed objects:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">IndexedBase</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-<span class="go">M[i, j]*x[j]</span>
-</pre></div>
-</div>
-<p>If the indexed objects will be converted to component based arrays, e.g.
-with the code printers or the autowrap framework, you also need to provide
-(symbolic or numerical) dimensions.  This can be done by passing an
-optional shape parameter to IndexedBase upon construction:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">dim1</span><span class="p">,</span> <span class="n">dim2</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;dim1 dim2&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">IndexedBase</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="n">dim1</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">dim1</span><span class="p">,</span> <span class="n">dim2</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">.</span><span class="n">shape</span>
-<span class="go">(dim1, 2*dim1, dim2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span>
-<span class="go">(dim1, 2*dim1, dim2)</span>
-</pre></div>
-</div>
-<p>If an IndexedBase object has no shape information, it is assumed that the
-array is as large as the ranges of it&#8217;s indices:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;n m&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Idx</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">j</span> <span class="o">=</span> <span class="n">Idx</span><span class="p">(</span><span class="s">&#39;j&#39;</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span>
-<span class="go">(m, n)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">M</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">ranges</span>
-<span class="go">[(0, m - 1), (0, n - 1)]</span>
-</pre></div>
-</div>
-<p>The above can be compared with the following:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span>
-<span class="go">(dim1, 2*dim1, dim2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">ranges</span>
-<span class="go">[(0, m - 1), None, (0, n - 1)]</span>
-</pre></div>
-</div>
-<p>To analyze the structure of indexed expressions, you can use the methods
-get_indices() and get_contraction_structure():</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.tensor</span> <span class="kn">import</span> <span class="n">get_indices</span><span class="p">,</span> <span class="n">get_contraction_structure</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">get_indices</span><span class="p">(</span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">])</span>
-<span class="go">(set([i]), {})</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">get_contraction_structure</span><span class="p">(</span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">j</span><span class="p">])</span>
-<span class="go">{(j,): set([A[i, j, j]])}</span>
-</pre></div>
-</div>
-<p>See the appropriate docstrings for a detailed explanation of the output.</p>
-<dl class="class">
-<dt id="sympy.tensor.indexed.Idx">
-<em class="property">class </em><tt class="descclassname">sympy.tensor.indexed.</tt><tt class="descname">Idx</tt><a class="reference internal" href="../../_modules/sympy/tensor/indexed.html#Idx"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.tensor.indexed.Idx" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents an integer index as an Integer or integer expression.</p>
-<p>There are a number of ways to create an Idx object.  The constructor
-takes two arguments:</p>
-<dl class="docutils">
-<dt><tt class="docutils literal"><span class="pre">label</span></tt></dt>
-<dd>An integer or a symbol that labels the index.</dd>
-<dt><tt class="docutils literal"><span class="pre">range</span></tt></dt>
-<dd>Optionally you can specify a range as either</dd>
-</dl>
-<ul class="simple">
-<li>Symbol or integer: This is interpreted as a dimension. Lower and
-upper bounds are set to 0 and range - 1, respectively.</li>
-<li>tuple: The two elements are interpreted as the lower and upper
-bounds of the range, respectively.</li>
-</ul>
-<p>Note: the Idx constructor is rather pedantic in that it only accepts
-integer arguments.  The only exception is that you can use oo and -oo to
-specify an unbounded range.  For all other cases, both label and bounds
-must be declared as integers, e.g. if n is given as an argument then
-n.is_integer must return True.</p>
-<p>For convenience, if the label is given as a string it is automatically
-converted to an integer symbol.  (Note: this conversion is not done for
-range or dimension arguments.)</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.tensor</span> <span class="kn">import</span> <span class="n">IndexedBase</span><span class="p">,</span> <span class="n">Idx</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">L</span><span class="p">,</span> <span class="n">U</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;n i L U&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>If a string is given for the label an integer Symbol is created and the
-bounds are both None:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">idx</span> <span class="o">=</span> <span class="n">Idx</span><span class="p">(</span><span class="s">&#39;qwerty&#39;</span><span class="p">);</span> <span class="n">idx</span>
-<span class="go">qwerty</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">idx</span><span class="o">.</span><span class="n">lower</span><span class="p">,</span> <span class="n">idx</span><span class="o">.</span><span class="n">upper</span>
-<span class="go">(None, None)</span>
-</pre></div>
-</div>
-<p>Both upper and lower bounds can be specified:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">idx</span> <span class="o">=</span> <span class="n">Idx</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">L</span><span class="p">,</span> <span class="n">U</span><span class="p">));</span> <span class="n">idx</span>
-<span class="go">i</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">idx</span><span class="o">.</span><span class="n">lower</span><span class="p">,</span> <span class="n">idx</span><span class="o">.</span><span class="n">upper</span>
-<span class="go">(L, U)</span>
-</pre></div>
-</div>
-<p>When only a single bound is given it is interpreted as the dimension
-and the lower bound defaults to 0:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">idx</span> <span class="o">=</span> <span class="n">Idx</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">n</span><span class="p">);</span> <span class="n">idx</span><span class="o">.</span><span class="n">lower</span><span class="p">,</span> <span class="n">idx</span><span class="o">.</span><span class="n">upper</span>
-<span class="go">(0, n - 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">idx</span> <span class="o">=</span> <span class="n">Idx</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">4</span><span class="p">);</span> <span class="n">idx</span><span class="o">.</span><span class="n">lower</span><span class="p">,</span> <span class="n">idx</span><span class="o">.</span><span class="n">upper</span>
-<span class="go">(0, 3)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">idx</span> <span class="o">=</span> <span class="n">Idx</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">oo</span><span class="p">);</span> <span class="n">idx</span><span class="o">.</span><span class="n">lower</span><span class="p">,</span> <span class="n">idx</span><span class="o">.</span><span class="n">upper</span>
-<span class="go">(0, oo)</span>
-</pre></div>
-</div>
-<p>The label can be a literal integer instead of a string/Symbol:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">idx</span> <span class="o">=</span> <span class="n">Idx</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n</span><span class="p">);</span> <span class="n">idx</span><span class="o">.</span><span class="n">lower</span><span class="p">,</span> <span class="n">idx</span><span class="o">.</span><span class="n">upper</span>
-<span class="go">(0, n - 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">idx</span><span class="o">.</span><span class="n">label</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.tensor.indexed.Idx.label">
-<tt class="descname">label</tt><a class="reference internal" href="../../_modules/sympy/tensor/indexed.html#Idx.label"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.tensor.indexed.Idx.label" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the label (Integer or integer expression) of the Idx object.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Idx</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Idx</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">label</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">j</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;j&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Idx</span><span class="p">(</span><span class="n">j</span><span class="p">)</span><span class="o">.</span><span class="n">label</span>
-<span class="go">j</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Idx</span><span class="p">(</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">label</span>
-<span class="go">j + 1</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.tensor.indexed.Idx.lower">
-<tt class="descname">lower</tt><a class="reference internal" href="../../_modules/sympy/tensor/indexed.html#Idx.lower"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.tensor.indexed.Idx.lower" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the lower bound of the Index.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Idx</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Idx</span><span class="p">(</span><span class="s">&#39;j&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">lower</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Idx</span><span class="p">(</span><span class="s">&#39;j&#39;</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span><span class="o">.</span><span class="n">lower</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Idx</span><span class="p">(</span><span class="s">&#39;j&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">lower</span> <span class="ow">is</span> <span class="bp">None</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.tensor.indexed.Idx.upper">
-<tt class="descname">upper</tt><a class="reference internal" href="../../_modules/sympy/tensor/indexed.html#Idx.upper"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.tensor.indexed.Idx.upper" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the upper bound of the Index.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Idx</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Idx</span><span class="p">(</span><span class="s">&#39;j&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">upper</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Idx</span><span class="p">(</span><span class="s">&#39;j&#39;</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span><span class="o">.</span><span class="n">upper</span>
-<span class="go">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Idx</span><span class="p">(</span><span class="s">&#39;j&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">upper</span> <span class="ow">is</span> <span class="bp">None</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.tensor.indexed.Indexed">
-<em class="property">class </em><tt class="descclassname">sympy.tensor.indexed.</tt><tt class="descname">Indexed</tt><a class="reference internal" href="../../_modules/sympy/tensor/indexed.html#Indexed"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.tensor.indexed.Indexed" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represents a mathematical object with indices.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.tensor</span> <span class="kn">import</span> <span class="n">Indexed</span><span class="p">,</span> <span class="n">IndexedBase</span><span class="p">,</span> <span class="n">Idx</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i j&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Idx</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Indexed</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
-<span class="go">A[i, j]</span>
-</pre></div>
-</div>
-<p>It is recommended that Indexed objects are created via IndexedBase:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">IndexedBase</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Indexed</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="o">==</span> <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.tensor.indexed.Indexed.base">
-<tt class="descname">base</tt><a class="reference internal" href="../../_modules/sympy/tensor/indexed.html#Indexed.base"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.tensor.indexed.Indexed.base" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the IndexedBase of the Indexed object.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.tensor</span> <span class="kn">import</span> <span class="n">Indexed</span><span class="p">,</span> <span class="n">IndexedBase</span><span class="p">,</span> <span class="n">Idx</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i j&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Idx</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Indexed</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span><span class="o">.</span><span class="n">base</span>
-<span class="go">A</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">IndexedBase</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">==</span> <span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">base</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.tensor.indexed.Indexed.indices">
-<tt class="descname">indices</tt><a class="reference internal" href="../../_modules/sympy/tensor/indexed.html#Indexed.indices"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.tensor.indexed.Indexed.indices" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the indices of the Indexed object.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.tensor</span> <span class="kn">import</span> <span class="n">Indexed</span><span class="p">,</span> <span class="n">Idx</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i j&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Idx</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Indexed</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span><span class="o">.</span><span class="n">indices</span>
-<span class="go">(i, j)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.tensor.indexed.Indexed.ranges">
-<tt class="descname">ranges</tt><a class="reference internal" href="../../_modules/sympy/tensor/indexed.html#Indexed.ranges"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.tensor.indexed.Indexed.ranges" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a list of tuples with lower and upper range of each index.</p>
-<p>If an index does not define the data members upper and lower, the
-corresponding slot in the list contains <tt class="docutils literal"><span class="pre">None</span></tt> instead of a tuple.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Indexed</span><span class="p">,</span><span class="n">Idx</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Indexed</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">Idx</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">Idx</span><span class="p">(</span><span class="s">&#39;j&#39;</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">Idx</span><span class="p">(</span><span class="s">&#39;k&#39;</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span><span class="o">.</span><span class="n">ranges</span>
-<span class="go">[(0, 1), (0, 3), (0, 7)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Indexed</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">Idx</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">Idx</span><span class="p">(</span><span class="s">&#39;j&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">Idx</span><span class="p">(</span><span class="s">&#39;k&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span><span class="o">.</span><span class="n">ranges</span>
-<span class="go">[(0, 2), (0, 2), (0, 2)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x y z&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Indexed</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span><span class="o">.</span><span class="n">ranges</span>
-<span class="go">[None, None, None]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.tensor.indexed.Indexed.rank">
-<tt class="descname">rank</tt><a class="reference internal" href="../../_modules/sympy/tensor/indexed.html#Indexed.rank"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.tensor.indexed.Indexed.rank" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the rank of the Indexed object.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.tensor</span> <span class="kn">import</span> <span class="n">Indexed</span><span class="p">,</span> <span class="n">Idx</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i:m&#39;</span><span class="p">,</span> <span class="n">cls</span><span class="o">=</span><span class="n">Idx</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Indexed</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span><span class="o">.</span><span class="n">rank</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Indexed</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">k</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="o">.</span><span class="n">rank</span>
-<span class="go">5</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span><span class="o">.</span><span class="n">rank</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">indices</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.tensor.indexed.Indexed.shape">
-<tt class="descname">shape</tt><a class="reference internal" href="../../_modules/sympy/tensor/indexed.html#Indexed.shape"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.tensor.indexed.Indexed.shape" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a list with dimensions of each index.</p>
-<p>Dimensions is a property of the array, not of the indices.  Still, if
-the IndexedBase does not define a shape attribute, it is assumed that
-the ranges of the indices correspond to the shape of the array.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.tensor.indexed</span> <span class="kn">import</span> <span class="n">IndexedBase</span><span class="p">,</span> <span class="n">Idx</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;n m&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Idx</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">j</span> <span class="o">=</span> <span class="n">Idx</span><span class="p">(</span><span class="s">&#39;j&#39;</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">IndexedBase</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">IndexedBase</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span>
-<span class="go">(n, n)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span>
-<span class="go">(m, m)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.tensor.indexed.IndexedBase">
-<em class="property">class </em><tt class="descclassname">sympy.tensor.indexed.</tt><tt class="descname">IndexedBase</tt><a class="reference internal" href="../../_modules/sympy/tensor/indexed.html#IndexedBase"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.tensor.indexed.IndexedBase" title="Permalink to this definition">¶</a></dt>
-<dd><p>Represent the base or stem of an indexed object</p>
-<p>The IndexedBase class represent an array that contains elements. The main purpose
-of this class is to allow the convenient creation of objects of the Indexed
-class.  The __getitem__ method of IndexedBase returns an instance of
-Indexed.  Alone, without indices, the IndexedBase class can be used as a
-notation for e.g. matrix equations, resembling what you could do with the
-Symbol class.  But, the IndexedBase class adds functionality that is not
-available for Symbol instances:</p>
-<blockquote>
-<div><ul class="simple">
-<li>An IndexedBase object can optionally store shape information.  This can
-be used in to check array conformance and conditions for numpy
-broadcasting.  (TODO)</li>
-<li>An IndexedBase object implements syntactic sugar that allows easy symbolic
-representation of array operations, using implicit summation of
-repeated indices.</li>
-<li>The IndexedBase object symbolizes a mathematical structure equivalent
-to arrays, and is recognized as such for code generation and automatic
-compilation and wrapping.</li>
-</ul>
-</div></blockquote>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.tensor</span> <span class="kn">import</span> <span class="n">IndexedBase</span><span class="p">,</span> <span class="n">Idx</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">IndexedBase</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">);</span> <span class="n">A</span>
-<span class="go">A</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
-<span class="go">&lt;class &#39;sympy.tensor.indexed.IndexedBase&#39;&gt;</span>
-</pre></div>
-</div>
-<p>When an IndexedBase object receives indices, it returns an array with named
-axes, represented by an Indexed object:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i j&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
-<span class="go">A[i, j, 2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="go">&lt;class &#39;sympy.tensor.indexed.Indexed&#39;&gt;</span>
-</pre></div>
-</div>
-<p>The IndexedBase constructor takes an optional shape argument.  If given,
-it overrides any shape information in the indices. (But not the index
-ranges!)</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">o</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;m n o p&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Idx</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">j</span> <span class="o">=</span> <span class="n">Idx</span><span class="p">(</span><span class="s">&#39;j&#39;</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span>
-<span class="go">(m, n)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">IndexedBase</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="n">o</span><span class="p">,</span> <span class="n">p</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span>
-<span class="go">(o, p)</span>
-</pre></div>
-</div>
-<dl class="attribute">
-<dt id="sympy.tensor.indexed.IndexedBase.args">
-<tt class="descname">args</tt><a class="reference internal" href="../../_modules/sympy/tensor/indexed.html#IndexedBase.args"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.tensor.indexed.IndexedBase.args" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the arguments used to create this IndexedBase object.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">IndexedBase</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">IndexedBase</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">args</span>
-<span class="go">(A, (x, y))</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.tensor.indexed.IndexedBase.label">
-<tt class="descname">label</tt><a class="reference internal" href="../../_modules/sympy/tensor/indexed.html#IndexedBase.label"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.tensor.indexed.IndexedBase.label" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the label of the IndexedBase object.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">IndexedBase</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">IndexedBase</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">label</span>
-<span class="go">A</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.tensor.indexed.IndexedBase.shape">
-<tt class="descname">shape</tt><a class="reference internal" href="../../_modules/sympy/tensor/indexed.html#IndexedBase.shape"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.tensor.indexed.IndexedBase.shape" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the shape of the IndexedBase object.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">IndexedBase</span><span class="p">,</span> <span class="n">Idx</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">IndexedBase</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">shape</span>
-<span class="go">(x, y)</span>
-</pre></div>
-</div>
-<p>Note: If the shape of the IndexedBase is specified, it will override
-any shape information given by the indices.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">IndexedBase</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="n">IndexedBase</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Idx</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">j</span> <span class="o">=</span> <span class="n">Idx</span><span class="p">(</span><span class="s">&#39;j&#39;</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span>
-<span class="go">(x, y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">B</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">shape</span>
-<span class="go">(2, 1)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</dd></dl>
-
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../index.html">
-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Indexed Objects</a><ul>
-<li><a class="reference internal" href="#examples">Examples</a></li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../tensor/index.html"
-                        title="previous chapter">Tensor Module</a></p>
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-                        title="next chapter">Methods</a></p>
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-            
-  <div class="section" id="autowrap-module">
-<h1>Autowrap Module<a class="headerlink" href="#autowrap-module" title="Permalink to this headline">¶</a></h1>
-<p>The autowrap module works very well in tandem with the Indexed classes of the
-<a class="reference internal" href="../tensor/index.html#tensor-module"><em>Tensor Module</em></a>.  Here is a simple example that shows how to setup a binary
-routine that calculates a matrix-vector product.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.autowrap</span> <span class="kn">import</span> <span class="n">autowrap</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">IndexedBase</span><span class="p">,</span> <span class="n">Idx</span><span class="p">,</span> <span class="n">Eq</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">IndexedBase</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;A&#39;</span><span class="p">,</span> <span class="s">&#39;x&#39;</span><span class="p">,</span> <span class="s">&#39;y&#39;</span><span class="p">]))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;m n&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span> <span class="o">=</span> <span class="n">Idx</span><span class="p">(</span><span class="s">&#39;i&#39;</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">j</span> <span class="o">=</span> <span class="n">Idx</span><span class="p">(</span><span class="s">&#39;j&#39;</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">instruction</span> <span class="o">=</span> <span class="n">Eq</span><span class="p">(</span><span class="n">y</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">A</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="n">j</span><span class="p">]);</span> <span class="n">instruction</span>
-<span class="go">y[i] == A[i, j]*x[j]</span>
-</pre></div>
-</div>
-<p>Because the code printers treat Indexed objects with repeated indices as a
-summation, the above equality instance will be translated to low-level code for
-a matrix vector product.  This is how you tell SymPy to generate the code,
-compile it and wrap it as a python function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">matvec</span> <span class="o">=</span> <span class="n">autowrap</span><span class="p">(</span><span class="n">instruction</span><span class="p">)</span>                 
-</pre></div>
-</div>
-<p>That&#8217;s it.  Now let&#8217;s test it with some numpy arrays.  The default wrapper
-backend is f2py.  The wrapper function it provides is set up to accept python
-lists, which it will silently convert to numpy arrays.  So we can test the
-matrix vector product like this:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">M</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
-<span class="gp">... </span>     <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">matvec</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>                              
-<span class="go">[ 3.  2.]</span>
-</pre></div>
-</div>
-<div class="section" id="implementation-details">
-<h2>Implementation details<a class="headerlink" href="#implementation-details" title="Permalink to this headline">¶</a></h2>
-<p>The autowrap module is implemented with a backend consisting of CodeWrapper
-objects.  The base class <tt class="docutils literal"><span class="pre">CodeWrapper</span></tt> takes care of details about module
-name, filenames and options.  It also contains the driver routine, which runs
-through all steps in the correct order, and also takes care of setting up and
-removing the temporary working directory.</p>
-<p>The actual compilation and wrapping is done by external resources, such as the
-system installed f2py command. The Cython backend runs a distutils setup script
-in a subprocess. Subclasses of CodeWrapper takes care of these
-backend-dependent details.</p>
-</div>
-<div class="section" id="module-sympy.utilities.autowrap">
-<span id="api-reference"></span><h2>API Reference<a class="headerlink" href="#module-sympy.utilities.autowrap" title="Permalink to this headline">¶</a></h2>
-<p>Module for compiling codegen output, and wrap the binary for use in
-python.</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p>To use the autowrap module it must first be imported</p>
-<div class="last highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.autowrap</span> <span class="kn">import</span> <span class="n">autowrap</span>
-</pre></div>
-</div>
-</div>
-<p>This module provides a common interface for different external backends, such
-as f2py, fwrap, Cython, SWIG(?) etc. (Currently only f2py and Cython are
-implemented) The goal is to provide access to compiled binaries of acceptable
-performance with a one-button user interface, i.e.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expr</span> <span class="o">=</span> <span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">25</span><span class="p">))</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">binary_callable</span> <span class="o">=</span> <span class="n">autowrap</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>           
-<span class="gp">&gt;&gt;&gt; </span><span class="n">binary_callable</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>                      
-<span class="go">-1.0</span>
-</pre></div>
-</div>
-<p>The callable returned from autowrap() is a binary python function, not a
-SymPy object.  If it is desired to use the compiled function in symbolic
-expressions, it is better to use binary_function() which returns a SymPy
-Function object.  The binary callable is attached as the _imp_ attribute and
-invoked when a numerical evaluation is requested with evalf(), or with
-lambdify().</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.autowrap</span> <span class="kn">import</span> <span class="n">binary_function</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">binary_function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>             
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span> <span class="o">+</span> <span class="n">y</span>                              
-<span class="go">y + 2*f(x, y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">subs</span><span class="o">=</span><span class="p">{</span><span class="n">x</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span><span class="mi">2</span><span class="p">})</span> 
-<span class="go">0.0</span>
-</pre></div>
-</div>
-<p>The idea is that a SymPy user will primarily be interested in working with
-mathematical expressions, and should not have to learn details about wrapping
-tools in order to evaluate expressions numerically, even if they are
-computationally expensive.</p>
-<p>When is this useful?</p>
-<blockquote>
-<div><ol class="arabic simple">
-<li>For computations on large arrays, Python iterations may be too slow,
-and depending on the mathematical expression, it may be difficult to
-exploit the advanced index operations provided by NumPy.</li>
-<li>For <em>really</em> long expressions that will be called repeatedly, the
-compiled binary should be significantly faster than SymPy&#8217;s .evalf()</li>
-<li>If you are generating code with the codegen utility in order to use
-it in another project, the automatic python wrappers let you test the
-binaries immediately from within SymPy.</li>
-<li>To create customized ufuncs for use with numpy arrays.
-See <em>ufuncify</em>.</li>
-</ol>
-</div></blockquote>
-<p>When is this module NOT the best approach?</p>
-<blockquote>
-<div><ol class="arabic simple">
-<li>If you are really concerned about speed or memory optimizations,
-you will probably get better results by working directly with the
-wrapper tools and the low level code.  However, the files generated
-by this utility may provide a useful starting point and reference
-code. Temporary files will be left intact if you supply the keyword
-tempdir=&#8221;path/to/files/&#8221;.</li>
-<li>If the array computation can be handled easily by numpy, and you
-don&#8217;t need the binaries for another project.</li>
-</ol>
-</div></blockquote>
-<dl class="class">
-<dt id="sympy.utilities.autowrap.CodeWrapper">
-<em class="property">class </em><tt class="descclassname">sympy.utilities.autowrap.</tt><tt class="descname">CodeWrapper</tt><big>(</big><em>generator</em>, <em>filepath=None</em>, <em>flags=</em><span class="optional">[</span><span class="optional">]</span>, <em>verbose=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/autowrap.html#CodeWrapper"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.autowrap.CodeWrapper" title="Permalink to this definition">¶</a></dt>
-<dd><p>Base Class for code wrappers</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.utilities.autowrap.CythonCodeWrapper">
-<em class="property">class </em><tt class="descclassname">sympy.utilities.autowrap.</tt><tt class="descname">CythonCodeWrapper</tt><big>(</big><em>generator</em>, <em>filepath=None</em>, <em>flags=</em><span class="optional">[</span><span class="optional">]</span>, <em>verbose=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/autowrap.html#CythonCodeWrapper"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.autowrap.CythonCodeWrapper" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wrapper that uses Cython</p>
-<dl class="function">
-<dt id="sympy.utilities.autowrap.CythonCodeWrapper.dump_pyx">
-<tt class="descname">dump_pyx</tt><big>(</big><em>self</em>, <em>routines</em>, <em>f</em>, <em>prefix</em>, <em>header=True</em>, <em>empty=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/autowrap.html#CythonCodeWrapper.dump_pyx"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.autowrap.CythonCodeWrapper.dump_pyx" title="Permalink to this definition">¶</a></dt>
-<dd><p>Write a Cython file with python wrappers</p>
-<p>This file contains all the definitions of the routines in c code and
-refers to the header file.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Arguments :</th><td class="field-body"></td>
-</tr>
-</tbody>
-</table>
-<dl class="docutils">
-<dt>routines</dt>
-<dd>List of Routine instances</dd>
-<dt>f</dt>
-<dd>File-like object to write the file to</dd>
-<dt>prefix</dt>
-<dd>The filename prefix, used to refer to the proper header file.
-Only the basename of the prefix is used.</dd>
-<dt>empty</dt>
-<dd>Optional. When True, empty lines are included to structure the
-source files. [DEFAULT=True]</dd>
-</dl>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.utilities.autowrap.DummyWrapper">
-<em class="property">class </em><tt class="descclassname">sympy.utilities.autowrap.</tt><tt class="descname">DummyWrapper</tt><big>(</big><em>generator</em>, <em>filepath=None</em>, <em>flags=</em><span class="optional">[</span><span class="optional">]</span>, <em>verbose=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/autowrap.html#DummyWrapper"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.autowrap.DummyWrapper" title="Permalink to this definition">¶</a></dt>
-<dd><p>Class used for testing independent of backends</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.utilities.autowrap.F2PyCodeWrapper">
-<em class="property">class </em><tt class="descclassname">sympy.utilities.autowrap.</tt><tt class="descname">F2PyCodeWrapper</tt><big>(</big><em>generator</em>, <em>filepath=None</em>, <em>flags=</em><span class="optional">[</span><span class="optional">]</span>, <em>verbose=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/autowrap.html#F2PyCodeWrapper"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.autowrap.F2PyCodeWrapper" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wrapper that uses f2py</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.autowrap.autowrap">
-<tt class="descclassname">sympy.utilities.autowrap.</tt><tt class="descname">autowrap</tt><big>(</big><em>expr</em>, <em>language='F95'</em>, <em>backend='f2py'</em>, <em>tempdir=None</em>, <em>args=None</em>, <em>flags=</em><span class="optional">[</span><span class="optional">]</span>, <em>verbose=False</em>, <em>helpers=</em><span class="optional">[</span><span class="optional">]</span><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/autowrap.html#autowrap"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.autowrap.autowrap" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates python callable binaries based on the math expression.</p>
-<dl class="docutils">
-<dt>expr</dt>
-<dd>The SymPy expression that should be wrapped as a binary routine</dd>
-</dl>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name" colspan="2">Optional arguments:</th></tr>
-<tr class="field-odd field"><td>&nbsp;</td><td class="field-body"></td>
-</tr>
-</tbody>
-</table>
-<dl class="docutils">
-<dt>language</dt>
-<dd>The programming language to use, currently &#8216;C&#8217; or &#8216;F95&#8217;</dd>
-<dt>backend</dt>
-<dd>The wrapper backend to use, currently f2py or Cython</dd>
-<dt>tempdir</dt>
-<dd>Path to directory for temporary files.  If this argument is supplied,
-the generated code and the wrapper input files are left intact in the
-specified path.</dd>
-<dt>args</dt>
-<dd>Sequence of the formal parameters of the generated code, if ommited the
-function signature is determined by the code generator.</dd>
-<dt>flags</dt>
-<dd>Additional option flags that will be passed to the backend</dd>
-<dt>verbose</dt>
-<dd>If True, autowrap will not mute the command line backends.  This can be
-helpful for debugging.</dd>
-<dt>helpers</dt>
-<dd>Used to define auxillary expressions needed for the main expr.  If the
-main expression need to do call a specialized function it should be put
-in the <tt class="docutils literal"><span class="pre">helpers</span></tt> list.  Autowrap will then make sure that the compiled
-main expression can link to the helper routine.  Items should be tuples
-with (&lt;funtion_name&gt;, &lt;sympy_expression&gt;, &lt;arguments&gt;).  It is
-mandatory to supply an argument sequence to helper routines.</dd>
-</dl>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.autowrap</span> <span class="kn">import</span> <span class="n">autowrap</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expr</span> <span class="o">=</span> <span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="n">y</span> <span class="o">+</span> <span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">13</span><span class="p">))</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">binary_func</span> <span class="o">=</span> <span class="n">autowrap</span><span class="p">(</span><span class="n">expr</span><span class="p">)</span>               
-<span class="gp">&gt;&gt;&gt; </span><span class="n">binary_func</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>                       
-<span class="go">-1.0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.autowrap.binary_function">
-<tt class="descclassname">sympy.utilities.autowrap.</tt><tt class="descname">binary_function</tt><big>(</big><em>symfunc</em>, <em>expr</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/autowrap.html#binary_function"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.autowrap.binary_function" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a sympy function with expr as binary implementation</p>
-<p>This is a convenience function that automates the steps needed to
-autowrap the SymPy expression and attaching it to a Function object
-with implemented_function().</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.autowrap</span> <span class="kn">import</span> <span class="n">binary_function</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">expr</span> <span class="o">=</span> <span class="p">((</span><span class="n">x</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">25</span><span class="p">))</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">binary_function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span>             
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>                                    
-<span class="go">&lt;class &#39;sympy.core.function.FunctionClass&#39;&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">2</span><span class="o">*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>                                  
-<span class="go">2*f(x, y)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">subs</span><span class="o">=</span><span class="p">{</span><span class="n">x</span><span class="p">:</span> <span class="mi">1</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="mi">2</span><span class="p">})</span>        
-<span class="go">-1.0</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.autowrap.ufuncify">
-<tt class="descclassname">sympy.utilities.autowrap.</tt><tt class="descname">ufuncify</tt><big>(</big><em>args</em>, <em>expr</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/autowrap.html#ufuncify"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.autowrap.ufuncify" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates a binary ufunc-like lambda function for numpy arrays</p>
-<dl class="docutils">
-<dt><tt class="docutils literal"><span class="pre">args</span></tt></dt>
-<dd>Either a Symbol or a tuple of symbols. Specifies the argument sequence
-for the ufunc-like function.</dd>
-<dt><tt class="docutils literal"><span class="pre">expr</span></tt></dt>
-<dd>A SymPy expression that defines the element wise operation</dd>
-<dt><tt class="docutils literal"><span class="pre">kwargs</span></tt></dt>
-<dd>Optional keyword arguments are forwarded to autowrap().</dd>
-</dl>
-<p>The returned function can only act on one array at a time, as only the
-first argument accept arrays as input.</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">a <em>proper</em> numpy ufunc is required to support broadcasting, type
-casting and more.  The function returned here, may not qualify for
-numpy&#8217;s definition of a ufunc.  That why we use the term ufunc-like.</p>
-</div>
-<p class="rubric">References</p>
-<p>[1] <a class="reference external" href="http://docs.scipy.org/doc/numpy/reference/ufuncs.html">http://docs.scipy.org/doc/numpy/reference/ufuncs.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.autowrap</span> <span class="kn">import</span> <span class="n">ufuncify</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">ufuncify</span><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">],</span> <span class="n">y</span> <span class="o">+</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>             
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="mi">2</span><span class="p">)</span>                            
-<span class="go">[2.  5.  10.]</span>
-</pre></div>
-</div>
-</dd></dl>
-
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-</div>
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-  <h3><a href="../../index.html">Table Of Contents</a></h3>
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-<li><a class="reference internal" href="#">Autowrap Module</a><ul>
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-            
-  <div class="section" id="codegen">
-<h1>Codegen<a class="headerlink" href="#codegen" title="Permalink to this headline">¶</a></h1>
-<p>This module provides functionality to generate directly compilable code from
-SymPy expressions.  The <tt class="docutils literal"><span class="pre">codegen</span></tt> function is the user interface to the code
-generation functionality in SymPy.  Some details of the implementation is given
-below for advanced users that may want to use the framework directly.</p>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p>The <tt class="docutils literal"><span class="pre">codegen</span></tt> callable is not in the sympy namespace automatically,
-to use it you must first execute</p>
-<div class="last highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.codegen</span> <span class="kn">import</span> <span class="n">codegen</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="impementation-details">
-<h2>Impementation Details<a class="headerlink" href="#impementation-details" title="Permalink to this headline">¶</a></h2>
-<p>Here we present the most important pieces of the internal structure, as
-advanced users may want to use it directly, for instance by subclassing a code
-generator for a specialized application.  <strong>It is very likely that you would
-prefer to use the codegen() function documented above.</strong></p>
-<p>Basic assumptions:</p>
-<ul class="simple">
-<li>A generic Routine data structure describes the routine that must be translated
-into C/Fortran/... code. This data structure covers all features present in
-one or more of the supported languages.</li>
-<li>Descendants from the CodeGen class transform multiple Routine instances into
-compilable code. Each derived class translates into a specific language.</li>
-<li>In many cases, one wants a simple workflow. The friendly functions in the last
-part are a simple api on top of the Routine/CodeGen stuff. They are easier to
-use, but are less powerful.</li>
-</ul>
-</div>
-<div class="section" id="routine">
-<h2>Routine<a class="headerlink" href="#routine" title="Permalink to this headline">¶</a></h2>
-<p>The Routine class is a very important piece of the codegen module. Viewing the
-codegen utility as a translator of mathematical expressions into a set of
-statements in a programming language, the Routine instances are responsible for
-extracting and storing information about how the math can be encapsulated in a
-function call.  Thus, it is the Routine constructor that decides what arguments
-the routine will need and if there should be a return value.</p>
-</div>
-<div class="section" id="module-sympy.utilities.codegen">
-<span id="api-reference"></span><h2>API Reference<a class="headerlink" href="#module-sympy.utilities.codegen" title="Permalink to this headline">¶</a></h2>
-<p>module for generating C, C++, Fortran77, Fortran90 and python routines that
-evaluate sympy expressions. This module is work in progress. Only the
-milestones with a &#8216;+&#8217; character in the list below have been completed.</p>
-<p>&#8212; How is sympy.utilities.codegen different from sympy.printing.ccode? &#8212;</p>
-<p>We considered the idea to extend the printing routines for sympy functions in
-such a way that it prints complete compilable code, but this leads to a few
-unsurmountable issues that can only be tackled with dedicated code generator:</p>
-<ul class="simple">
-<li>For C, one needs both a code and a header file, while the printing routines
-generate just one string. This code generator can be extended to support
-.pyf files for f2py.</li>
-<li>SymPy functions are not concerned with programming-technical issues, such
-as input, output and input-output arguments. Other examples are contiguous
-or non-contiguous arrays, including headers of other libraries such as gsl
-or others.</li>
-<li>It is highly interesting to evaluate several sympy functions in one C
-routine, eventually sharing common intermediate results with the help
-of the cse routine. This is more than just printing.</li>
-<li>From the programming perspective, expressions with constants should be
-evaluated in the code generator as much as possible. This is different
-for printing.</li>
-</ul>
-<p>&#8212; Basic assumptions &#8212;</p>
-<ul class="simple">
-<li>A generic Routine data structure describes the routine that must be
-translated into C/Fortran/... code. This data structure covers all
-features present in one or more of the supported languages.</li>
-<li>Descendants from the CodeGen class transform multiple Routine instances
-into compilable code. Each derived class translates into a specific
-language.</li>
-<li>In many cases, one wants a simple workflow. The friendly functions in the
-last part are a simple api on top of the Routine/CodeGen stuff. They are
-easier to use, but are less powerful.</li>
-</ul>
-<p>&#8212; Milestones &#8212;</p>
-<ul class="simple">
-<li>First working version with scalar input arguments, generating C code,
-tests</li>
-<li>Friendly functions that are easier to use than the rigorous
-Routine/CodeGen workflow.</li>
-<li>Integer and Real numbers as input and output</li>
-<li>Output arguments</li>
-<li>InputOutput arguments</li>
-<li>Sort input/output arguments properly</li>
-<li>Contiguous array arguments (numpy matrices)</li>
-<li>Also generate .pyf code for f2py (in autowrap module)</li>
-<li>Isolate constants and evaluate them beforehand in double precision</li>
-<li>Fortran 90</li>
-</ul>
-<ul class="simple">
-<li>Common Subexpression Elimination</li>
-<li>User defined comments in the generated code</li>
-<li>Optional extra include lines for libraries/objects that can eval special
-functions</li>
-<li>Test other C compilers and libraries: gcc, tcc, libtcc, gcc+gsl, ...</li>
-<li>Contiguous array arguments (sympy matrices)</li>
-<li>Non-contiguous array arguments (sympy matrices)</li>
-<li>ccode must raise an error when it encounters something that can not be
-translated into c. ccode(integrate(sin(x)/x, x)) does not make sense.</li>
-<li>Complex numbers as input and output</li>
-<li>A default complex datatype</li>
-<li>Include extra information in the header: date, user, hostname, sha1
-hash, ...</li>
-<li>Fortran 77</li>
-<li>C++</li>
-<li>Python</li>
-<li>...</li>
-</ul>
-<dl class="class">
-<dt id="sympy.utilities.codegen.Routine">
-<em class="property">class </em><tt class="descclassname">sympy.utilities.codegen.</tt><tt class="descname">Routine</tt><big>(</big><em>name</em>, <em>expr</em>, <em>argument_sequence=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/codegen.html#Routine"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.codegen.Routine" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generic description of an evaluation routine for a set of sympy expressions.</p>
-<p>A CodeGen class can translate instances of this class into C/Fortran/...
-code. The routine specification covers all the features present in these
-languages. The CodeGen part must raise an exception when certain features
-are not present in the target language. For example, multiple return
-values are possible in Python, but not in C or Fortran. Another example:
-Fortran and Python support complex numbers, while C does not.</p>
-<dl class="attribute">
-<dt id="sympy.utilities.codegen.Routine.result_variables">
-<tt class="descname">result_variables</tt><a class="reference internal" href="../../_modules/sympy/utilities/codegen.html#Routine.result_variables"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.codegen.Routine.result_variables" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a list of OutputArgument, InOutArgument and Result.</p>
-<p>If return values are present, they are at the end ot the list.</p>
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.utilities.codegen.Routine.variables">
-<tt class="descname">variables</tt><a class="reference internal" href="../../_modules/sympy/utilities/codegen.html#Routine.variables"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.codegen.Routine.variables" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a set containing all variables possibly used in this routine.</p>
-<p>For routines with unnamed return values, the dummies that may or may
-not be used will be included in the set.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.utilities.codegen.DataType">
-<em class="property">class </em><tt class="descclassname">sympy.utilities.codegen.</tt><tt class="descname">DataType</tt><big>(</big><em>cname</em>, <em>fname</em>, <em>pyname</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/codegen.html#DataType"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.codegen.DataType" title="Permalink to this definition">¶</a></dt>
-<dd><p>Holds strings for a certain datatype in different programming languages.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.codegen.get_default_datatype">
-<tt class="descclassname">sympy.utilities.codegen.</tt><tt class="descname">get_default_datatype</tt><big>(</big><em>expr</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/codegen.html#get_default_datatype"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.codegen.get_default_datatype" title="Permalink to this definition">¶</a></dt>
-<dd><p>Derives a decent data type based on the assumptions on the expression.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.utilities.codegen.Argument">
-<em class="property">class </em><tt class="descclassname">sympy.utilities.codegen.</tt><tt class="descname">Argument</tt><big>(</big><em>name</em>, <em>datatype=None</em>, <em>dimensions=None</em>, <em>precision=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/codegen.html#Argument"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.codegen.Argument" title="Permalink to this definition">¶</a></dt>
-<dd><p>An abstract Argument data structure: a name and a data type.</p>
-<p>This structure is refined in the descendants below.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.utilities.codegen.Result">
-<em class="property">class </em><tt class="descclassname">sympy.utilities.codegen.</tt><tt class="descname">Result</tt><big>(</big><em>expr</em>, <em>datatype=None</em>, <em>precision=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/codegen.html#Result"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.codegen.Result" title="Permalink to this definition">¶</a></dt>
-<dd><p>An expression for a scalar return value.</p>
-<p>The name result is used to avoid conflicts with the reserved word
-&#8216;return&#8217; in the python language. It is also shorter than ReturnValue.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.utilities.codegen.CodeGen">
-<em class="property">class </em><tt class="descclassname">sympy.utilities.codegen.</tt><tt class="descname">CodeGen</tt><big>(</big><em>project='project'</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/codegen.html#CodeGen"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.codegen.CodeGen" title="Permalink to this definition">¶</a></dt>
-<dd><p>Abstract class for the code generators.</p>
-<dl class="function">
-<dt id="sympy.utilities.codegen.CodeGen.dump_code">
-<tt class="descname">dump_code</tt><big>(</big><em>self</em>, <em>routines</em>, <em>f</em>, <em>prefix</em>, <em>header=True</em>, <em>empty=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/codegen.html#CodeGen.dump_code"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.codegen.CodeGen.dump_code" title="Permalink to this definition">¶</a></dt>
-<dd><p>Write the code file by calling language specific methods in correct order</p>
-<p>The generated file contains all the definitions of the routines in
-low-level code and refers to the header file if appropriate.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Arguments :</th><td class="field-body"></td>
-</tr>
-</tbody>
-</table>
-<dl class="docutils">
-<dt>routines</dt>
-<dd>A list of Routine instances</dd>
-<dt>f</dt>
-<dd>A file-like object to write the file to</dd>
-<dt>prefix</dt>
-<dd>The filename prefix, used to refer to the proper header file. Only
-the basename of the prefix is used.</dd>
-</dl>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name" colspan="2">Optional arguments:</th></tr>
-<tr class="field-odd field"><td>&nbsp;</td><td class="field-body"></td>
-</tr>
-</tbody>
-</table>
-<dl class="docutils">
-<dt>header</dt>
-<dd>When True, a header comment is included on top of each source file.
-[DEFAULT=True]</dd>
-<dt>empty</dt>
-<dd>When True, empty lines are included to structure the source files.
-[DEFAULT=True]</dd>
-</dl>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.codegen.CodeGen.write">
-<tt class="descname">write</tt><big>(</big><em>self</em>, <em>routines</em>, <em>prefix</em>, <em>to_files=False</em>, <em>header=True</em>, <em>empty=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/codegen.html#CodeGen.write"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.codegen.CodeGen.write" title="Permalink to this definition">¶</a></dt>
-<dd><p>Writes all the source code files for the given routines.</p>
-<p>The generate source is returned as a list of (filename, contents)
-tuples, or is written to files (see options). Each filename consists
-of the given prefix, appended with an appropriate extension.</p>
-<dl class="docutils">
-<dt><tt class="docutils literal"><span class="pre">routines</span></tt></dt>
-<dd>A list of Routine instances to be written</dd>
-<dt><tt class="docutils literal"><span class="pre">prefix</span></tt></dt>
-<dd>The prefix for the output files</dd>
-<dt><tt class="docutils literal"><span class="pre">to_files</span></tt></dt>
-<dd>When True, the output is effectively written to files.
-[DEFAULT=False] Otherwise, a list of (filename, contents)
-tuples is returned.</dd>
-<dt><tt class="docutils literal"><span class="pre">header</span></tt></dt>
-<dd>When True, a header comment is included on top of each source
-file. [DEFAULT=True]</dd>
-<dt><tt class="docutils literal"><span class="pre">empty</span></tt></dt>
-<dd>When True, empty lines are included to structure the source
-files. [DEFAULT=True]</dd>
-</dl>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.utilities.codegen.CCodeGen">
-<em class="property">class </em><tt class="descclassname">sympy.utilities.codegen.</tt><tt class="descname">CCodeGen</tt><big>(</big><em>project='project'</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/codegen.html#CCodeGen"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.codegen.CCodeGen" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generator for C code</p>
-<p>The .write() method inherited from CodeGen will output a code file and an
-inteface file, &lt;prefix&gt;.c and &lt;prefix&gt;.h respectively.</p>
-<dl class="function">
-<dt id="sympy.utilities.codegen.CCodeGen.dump_c">
-<tt class="descname">dump_c</tt><big>(</big><em>self</em>, <em>routines</em>, <em>f</em>, <em>prefix</em>, <em>header=True</em>, <em>empty=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/codegen.html#CCodeGen.dump_c"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.codegen.CCodeGen.dump_c" title="Permalink to this definition">¶</a></dt>
-<dd><p>Write the code file by calling language specific methods in correct order</p>
-<p>The generated file contains all the definitions of the routines in
-low-level code and refers to the header file if appropriate.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Arguments :</th><td class="field-body"></td>
-</tr>
-</tbody>
-</table>
-<dl class="docutils">
-<dt>routines</dt>
-<dd>A list of Routine instances</dd>
-<dt>f</dt>
-<dd>A file-like object to write the file to</dd>
-<dt>prefix</dt>
-<dd>The filename prefix, used to refer to the proper header file. Only
-the basename of the prefix is used.</dd>
-</dl>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name" colspan="2">Optional arguments:</th></tr>
-<tr class="field-odd field"><td>&nbsp;</td><td class="field-body"></td>
-</tr>
-</tbody>
-</table>
-<dl class="docutils">
-<dt>header</dt>
-<dd>When True, a header comment is included on top of each source file.
-[DEFAULT=True]</dd>
-<dt>empty</dt>
-<dd>When True, empty lines are included to structure the source files.
-[DEFAULT=True]</dd>
-</dl>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.codegen.CCodeGen.dump_h">
-<tt class="descname">dump_h</tt><big>(</big><em>self</em>, <em>routines</em>, <em>f</em>, <em>prefix</em>, <em>header=True</em>, <em>empty=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/codegen.html#CCodeGen.dump_h"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.codegen.CCodeGen.dump_h" title="Permalink to this definition">¶</a></dt>
-<dd><p>Writes the C header file.</p>
-<p>This file contains all the function declarations.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Arguments :</th><td class="field-body"></td>
-</tr>
-</tbody>
-</table>
-<dl class="docutils">
-<dt>routines</dt>
-<dd>A list of Routine instances</dd>
-<dt>f</dt>
-<dd>A file-like object to write the file to</dd>
-<dt>prefix</dt>
-<dd>The filename prefix, used to construct the include guards.</dd>
-</dl>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name" colspan="2">Optional arguments:</th></tr>
-<tr class="field-odd field"><td>&nbsp;</td><td class="field-body"></td>
-</tr>
-</tbody>
-</table>
-<dl class="docutils">
-<dt>header</dt>
-<dd>When True, a header comment is included on top of each source
-file. [DEFAULT=True]</dd>
-<dt>empty</dt>
-<dd>When True, empty lines are included to structure the source
-files. [DEFAULT=True]</dd>
-</dl>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.codegen.CCodeGen.get_prototype">
-<tt class="descname">get_prototype</tt><big>(</big><em>self</em>, <em>routine</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/codegen.html#CCodeGen.get_prototype"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.codegen.CCodeGen.get_prototype" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a string for the function prototype for the given routine.</p>
-<p>If the routine has multiple result objects, an CodeGenError is
-raised.</p>
-<p>See: <a class="reference external" href="http://en.wikipedia.org/wiki/Function_prototype">http://en.wikipedia.org/wiki/Function_prototype</a></p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.utilities.codegen.FCodeGen">
-<em class="property">class </em><tt class="descclassname">sympy.utilities.codegen.</tt><tt class="descname">FCodeGen</tt><big>(</big><em>project='project'</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/codegen.html#FCodeGen"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.codegen.FCodeGen" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generator for Fortran 95 code</p>
-<p>The .write() method inherited from CodeGen will output a code file and an
-inteface file, &lt;prefix&gt;.f90 and &lt;prefix&gt;.h respectively.</p>
-<dl class="function">
-<dt id="sympy.utilities.codegen.FCodeGen.dump_f95">
-<tt class="descname">dump_f95</tt><big>(</big><em>self</em>, <em>routines</em>, <em>f</em>, <em>prefix</em>, <em>header=True</em>, <em>empty=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/codegen.html#FCodeGen.dump_f95"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.codegen.FCodeGen.dump_f95" title="Permalink to this definition">¶</a></dt>
-<dd><p>Write the code file by calling language specific methods in correct order</p>
-<p>The generated file contains all the definitions of the routines in
-low-level code and refers to the header file if appropriate.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Arguments :</th><td class="field-body"></td>
-</tr>
-</tbody>
-</table>
-<dl class="docutils">
-<dt>routines</dt>
-<dd>A list of Routine instances</dd>
-<dt>f</dt>
-<dd>A file-like object to write the file to</dd>
-<dt>prefix</dt>
-<dd>The filename prefix, used to refer to the proper header file. Only
-the basename of the prefix is used.</dd>
-</dl>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name" colspan="2">Optional arguments:</th></tr>
-<tr class="field-odd field"><td>&nbsp;</td><td class="field-body"></td>
-</tr>
-</tbody>
-</table>
-<dl class="docutils">
-<dt>header</dt>
-<dd>When True, a header comment is included on top of each source file.
-[DEFAULT=True]</dd>
-<dt>empty</dt>
-<dd>When True, empty lines are included to structure the source files.
-[DEFAULT=True]</dd>
-</dl>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.codegen.FCodeGen.dump_h">
-<tt class="descname">dump_h</tt><big>(</big><em>self</em>, <em>routines</em>, <em>f</em>, <em>prefix</em>, <em>header=True</em>, <em>empty=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/codegen.html#FCodeGen.dump_h"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.codegen.FCodeGen.dump_h" title="Permalink to this definition">¶</a></dt>
-<dd><p>Writes the interface to a header file.</p>
-<p>This file contains all the function declarations.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Arguments :</th><td class="field-body"></td>
-</tr>
-</tbody>
-</table>
-<dl class="docutils">
-<dt>routines</dt>
-<dd>A list of Routine instances</dd>
-<dt>f</dt>
-<dd>A file-like object to write the file to</dd>
-<dt>prefix</dt>
-<dd>The filename prefix</dd>
-</dl>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name" colspan="2">Optional arguments:</th></tr>
-<tr class="field-odd field"><td>&nbsp;</td><td class="field-body"></td>
-</tr>
-</tbody>
-</table>
-<dl class="docutils">
-<dt>header</dt>
-<dd>When True, a header comment is included on top of each source
-file. [DEFAULT=True]</dd>
-<dt>empty</dt>
-<dd>When True, empty lines are included to structure the source
-files. [DEFAULT=True]</dd>
-</dl>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.codegen.FCodeGen.get_interface">
-<tt class="descname">get_interface</tt><big>(</big><em>self</em>, <em>routine</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/codegen.html#FCodeGen.get_interface"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.codegen.FCodeGen.get_interface" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a string for the function interface for the given routine and
-a single result object, which can be None.</p>
-<p>If the routine has multiple result objects, a CodeGenError is
-raised.</p>
-<p>See: <a class="reference external" href="http://en.wikipedia.org/wiki/Function_prototype">http://en.wikipedia.org/wiki/Function_prototype</a></p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.codegen.codegen">
-<tt class="descclassname">sympy.utilities.codegen.</tt><tt class="descname">codegen</tt><big>(</big><em>name_expr</em>, <em>language</em>, <em>prefix</em>, <em>project='project'</em>, <em>to_files=False</em>, <em>header=True</em>, <em>empty=True</em>, <em>argument_sequence=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/codegen.html#codegen"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.codegen.codegen" title="Permalink to this definition">¶</a></dt>
-<dd><p>Write source code for the given expressions in the given language.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name" colspan="2">Mandatory Arguments:</th></tr>
-<tr class="field-odd field"><td>&nbsp;</td><td class="field-body"></td>
-</tr>
-</tbody>
-</table>
-<dl class="docutils">
-<dt><tt class="docutils literal"><span class="pre">name_expr</span></tt></dt>
-<dd>A single (name, expression) tuple or a list of (name, expression)
-tuples. Each tuple corresponds to a routine.  If the expression is an
-equality (an instance of class Equality) the left hand side is
-considered an output argument.</dd>
-<dt><tt class="docutils literal"><span class="pre">language</span></tt></dt>
-<dd>A string that indicates the source code language. This is case
-insensitive. For the moment, only &#8216;C&#8217; and &#8216;F95&#8217; is supported.</dd>
-<dt><tt class="docutils literal"><span class="pre">prefix</span></tt></dt>
-<dd>A prefix for the names of the files that contain the source code.
-Proper (language dependent) suffixes will be appended.</dd>
-</dl>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name" colspan="2">Optional Arguments:</th></tr>
-<tr class="field-odd field"><td>&nbsp;</td><td class="field-body"></td>
-</tr>
-</tbody>
-</table>
-<dl class="docutils">
-<dt><tt class="docutils literal"><span class="pre">project</span></tt></dt>
-<dd>A project name, used for making unique preprocessor instructions.
-[DEFAULT=&#8221;project&#8221;]</dd>
-<dt><tt class="docutils literal"><span class="pre">to_files</span></tt></dt>
-<dd>When True, the code will be written to one or more files with the given
-prefix, otherwise strings with the names and contents of these files
-are returned. [DEFAULT=False]</dd>
-<dt><tt class="docutils literal"><span class="pre">header</span></tt></dt>
-<dd>When True, a header is written on top of each source file.
-[DEFAULT=True]</dd>
-<dt><tt class="docutils literal"><span class="pre">empty</span></tt></dt>
-<dd>When True, empty lines are used to structure the code.  [DEFAULT=True]</dd>
-<dt><tt class="docutils literal"><span class="pre">argument_sequence</span></tt></dt>
-<dd><p class="first">sequence of arguments for the routine in a preferred order.  A
-CodeGenError is raised if required arguments are missing.  Redundant
-arguments are used without warning.</p>
-<p class="last">If omitted, arguments will be ordered alphabetically, but with all
-input aguments first, and then output or in-out arguments.</p>
-</dd>
-</dl>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.codegen</span> <span class="kn">import</span> <span class="n">codegen</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[(</span><span class="n">c_name</span><span class="p">,</span> <span class="n">c_code</span><span class="p">),</span> <span class="p">(</span><span class="n">h_name</span><span class="p">,</span> <span class="n">c_header</span><span class="p">)]</span> <span class="o">=</span> <span class="n">codegen</span><span class="p">(</span>
-<span class="gp">... </span>    <span class="p">(</span><span class="s">&quot;f&quot;</span><span class="p">,</span> <span class="n">x</span><span class="o">+</span><span class="n">y</span><span class="o">*</span><span class="n">z</span><span class="p">),</span> <span class="s">&quot;C&quot;</span><span class="p">,</span> <span class="s">&quot;test&quot;</span><span class="p">,</span> <span class="n">header</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">empty</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">c_name</span><span class="p">)</span>
-<span class="go">test.c</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">c_code</span><span class="p">,</span> <span class="n">end</span><span class="o">=</span><span class="s">&#39; &#39;</span><span class="p">)</span>
-<span class="go">#include &quot;test.h&quot;</span>
-<span class="go">#include &lt;math.h&gt;</span>
-<span class="go">double f(double x, double y, double z) {</span>
-<span class="go">  return x + y*z;</span>
-<span class="go">}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">h_name</span><span class="p">)</span>
-<span class="go">test.h</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">c_header</span><span class="p">,</span> <span class="n">end</span><span class="o">=</span><span class="s">&#39; &#39;</span><span class="p">)</span>
-<span class="go">#ifndef PROJECT__TEST__H</span>
-<span class="go">#define PROJECT__TEST__H</span>
-<span class="go">double f(double x, double y, double z);</span>
-<span class="go">#endif</span>
-</pre></div>
-</div>
-</dd></dl>
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-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Codegen</a><ul>
-<li><a class="reference internal" href="#impementation-details">Impementation Details</a></li>
-<li><a class="reference internal" href="#routine">Routine</a></li>
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-<span id="decorator"></span><h1>Decorator<a class="headerlink" href="#module-sympy.utilities.decorator" title="Permalink to this headline">¶</a></h1>
-<dl class="function">
-<dt id="sympy.utilities.decorator.conserve_mpmath_dps">
-<tt class="descclassname">sympy.utilities.decorator.</tt><tt class="descname">conserve_mpmath_dps</tt><big>(</big><em>func</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/decorator.html#conserve_mpmath_dps"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.decorator.conserve_mpmath_dps" title="Permalink to this definition">¶</a></dt>
-<dd><p>After the function finishes, resets the value of mpmath.mp.dps to
-the value it had before the function was run.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.decorator.threaded">
-<tt class="descclassname">sympy.utilities.decorator.</tt><tt class="descname">threaded</tt><big>(</big><em>func</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/decorator.html#threaded"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.decorator.threaded" title="Permalink to this definition">¶</a></dt>
-<dd><p>Apply <tt class="docutils literal"><span class="pre">func</span></tt> to sub&#8211;elements of an object, including <tt class="xref py py-class docutils literal"><span class="pre">Add</span></tt>.</p>
-<p>This decorator is intended to make it uniformly possible to apply a
-function to all elements of composite objects, e.g. matrices, lists, tuples
-and other iterable containers, or just expressions.</p>
-<p>This version of <a class="reference internal" href="#sympy.utilities.decorator.threaded" title="sympy.utilities.decorator.threaded"><tt class="xref py py-func docutils literal"><span class="pre">threaded()</span></tt></a> decorator allows threading over
-elements of <tt class="xref py py-class docutils literal"><span class="pre">Add</span></tt> class. If this behavior is not desirable
-use <a class="reference internal" href="#sympy.utilities.decorator.xthreaded" title="sympy.utilities.decorator.xthreaded"><tt class="xref py py-func docutils literal"><span class="pre">xthreaded()</span></tt></a> decorator.</p>
-<p>Functions using this decorator must have the following signature:</p>
-<div class="highlight-python"><pre>@threaded
-def function(expr, *args, **kwargs):</pre>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.decorator.threaded_factory">
-<tt class="descclassname">sympy.utilities.decorator.</tt><tt class="descname">threaded_factory</tt><big>(</big><em>func</em>, <em>use_add</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/decorator.html#threaded_factory"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.decorator.threaded_factory" title="Permalink to this definition">¶</a></dt>
-<dd><p>A factory for <tt class="docutils literal"><span class="pre">threaded</span></tt> decorators.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.decorator.xthreaded">
-<tt class="descclassname">sympy.utilities.decorator.</tt><tt class="descname">xthreaded</tt><big>(</big><em>func</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/decorator.html#xthreaded"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.decorator.xthreaded" title="Permalink to this definition">¶</a></dt>
-<dd><p>Apply <tt class="docutils literal"><span class="pre">func</span></tt> to sub&#8211;elements of an object, excluding <tt class="xref py py-class docutils literal"><span class="pre">Add</span></tt>.</p>
-<p>This decorator is intended to make it uniformly possible to apply a
-function to all elements of composite objects, e.g. matrices, lists, tuples
-and other iterable containers, or just expressions.</p>
-<p>This version of <a class="reference internal" href="#sympy.utilities.decorator.threaded" title="sympy.utilities.decorator.threaded"><tt class="xref py py-func docutils literal"><span class="pre">threaded()</span></tt></a> decorator disallows threading over
-elements of <tt class="xref py py-class docutils literal"><span class="pre">Add</span></tt> class. If this behavior is not desirable
-use <a class="reference internal" href="#sympy.utilities.decorator.threaded" title="sympy.utilities.decorator.threaded"><tt class="xref py py-func docutils literal"><span class="pre">threaded()</span></tt></a> decorator.</p>
-<p>Functions using this decorator must have the following signature:</p>
-<div class="highlight-python"><pre>@xthreaded
-def function(expr, *args, **kwargs):</pre>
-</div>
-</dd></dl>
-
-</div>
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-            
-  <div class="section" id="iterables">
-<h1>Iterables<a class="headerlink" href="#iterables" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="cartes">
-<h2>cartes<a class="headerlink" href="#cartes" title="Permalink to this headline">¶</a></h2>
-<p>Returns the cartesian product of sequences as a generator.</p>
-<dl class="docutils">
-<dt>Examples::</dt>
-<dd><div class="first last highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">cartes</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">cartes</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">],</span> <span class="s">&#39;ab&#39;</span><span class="p">))</span>
-<span class="go">[(1, &#39;a&#39;), (1, &#39;b&#39;), (2, &#39;a&#39;), (2, &#39;b&#39;), (3, &#39;a&#39;), (3, &#39;b&#39;)]</span>
-</pre></div>
-</div>
-</dd>
-</dl>
-</div>
-<div class="section" id="variations">
-<h2>variations<a class="headerlink" href="#variations" title="Permalink to this headline">¶</a></h2>
-<p>variations(seq, n) Returns all the variations of the list of size n.</p>
-<p>Has an optional third argument. Must be a boolean value and makes the method
-return the variations with repetition if set to True, or the variations
-without repetition if set to False.</p>
-<dl class="docutils">
-<dt>Examples::</dt>
-<dd><div class="first last highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">variations</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">variations</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">],</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">[(1, 2), (1, 3), (2, 1), (2, 3), (3, 1), (3, 2)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">variations</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">],</span> <span class="mi">2</span><span class="p">,</span> <span class="bp">True</span><span class="p">))</span>
-<span class="go">[(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)]</span>
-</pre></div>
-</div>
-</dd>
-</dl>
-</div>
-<div class="section" id="partitions">
-<h2>partitions<a class="headerlink" href="#partitions" title="Permalink to this headline">¶</a></h2>
-<p>Although the combinatorics module contains Partition and IntegerPartition
-classes for investigation and manipulation of partitions, there are a few
-functions to generate partitions that can be used as low-level tools for
-routines:  <tt class="docutils literal"><span class="pre">partitions</span></tt> and <tt class="docutils literal"><span class="pre">multiset_partitions</span></tt>. The former gives
-integer partitions, and the latter gives enumerated partitions of elements.
-There is also a routine <tt class="docutils literal"><span class="pre">kbins</span></tt> that will give a variety of permutations
-of partions.</p>
-<p>partitions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">partitions</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">p</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span> <span class="k">for</span> <span class="n">s</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">partitions</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="n">m</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span> <span class="k">if</span> <span class="n">s</span> <span class="o">==</span> <span class="mi">2</span><span class="p">]</span>
-<span class="go">[{1: 1, 6: 1}, {2: 1, 5: 1}, {3: 1, 4: 1}]</span>
-</pre></div>
-</div>
-<p>multiset_partitions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">multiset_partitions</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">p</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">multiset_partitions</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">)]</span>
-<span class="go">[[[0, 1], [2]], [[0, 2], [1]], [[0], [1, 2]]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">p</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">multiset_partitions</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="mi">2</span><span class="p">)]</span>
-<span class="go">[[[1, 1, 1], [2]], [[1, 1, 2], [1]], [[1, 1], [1, 2]]]</span>
-</pre></div>
-</div>
-<p>kbins:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">kbins</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">show</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">rv</span> <span class="o">=</span> <span class="p">[]</span>
-<span class="gp">... </span>    <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">k</span><span class="p">:</span>
-<span class="gp">... </span>        <span class="n">rv</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s">&#39;,&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">j</span><span class="p">)</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">p</span><span class="p">]))</span>
-<span class="gp">... </span>    <span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">rv</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">show</span><span class="p">(</span><span class="n">kbins</span><span class="p">(</span><span class="s">&quot;ABCD&quot;</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">[&#39;A,BCD&#39;, &#39;AB,CD&#39;, &#39;ABC,D&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">show</span><span class="p">(</span><span class="n">kbins</span><span class="p">(</span><span class="s">&quot;ABC&quot;</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">[&#39;A,BC&#39;, &#39;AB,C&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">show</span><span class="p">(</span><span class="n">kbins</span><span class="p">(</span><span class="s">&quot;ABC&quot;</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ordered</span><span class="o">=</span><span class="mo">00</span><span class="p">))</span>  <span class="c"># same as multiset_partitions</span>
-<span class="go">[&#39;A,BC&#39;, &#39;AB,C&#39;, &#39;AC,B&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">show</span><span class="p">(</span><span class="n">kbins</span><span class="p">(</span><span class="s">&quot;ABC&quot;</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ordered</span><span class="o">=</span><span class="mi">0</span><span class="n">o1</span><span class="p">))</span>
-<span class="go">[&#39;A,BC&#39;, &#39;A,CB&#39;,</span>
-<span class="go"> &#39;B,AC&#39;, &#39;B,CA&#39;,</span>
-<span class="go"> &#39;C,AB&#39;, &#39;C,BA&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">show</span><span class="p">(</span><span class="n">kbins</span><span class="p">(</span><span class="s">&quot;ABC&quot;</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ordered</span><span class="o">=</span><span class="mi">10</span><span class="p">))</span>
-<span class="go">[&#39;A,BC&#39;, &#39;AB,C&#39;, &#39;AC,B&#39;,</span>
-<span class="go"> &#39;B,AC&#39;, &#39;BC,A&#39;,</span>
-<span class="go"> &#39;C,AB&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">show</span><span class="p">(</span><span class="n">kbins</span><span class="p">(</span><span class="s">&quot;ABC&quot;</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ordered</span><span class="o">=</span><span class="mi">11</span><span class="p">))</span>
-<span class="go">[&#39;A,BC&#39;, &#39;A,CB&#39;, &#39;AB,C&#39;, &#39;AC,B&#39;,</span>
-<span class="go"> &#39;B,AC&#39;, &#39;B,CA&#39;, &#39;BA,C&#39;, &#39;BC,A&#39;,</span>
-<span class="go"> &#39;C,AB&#39;, &#39;C,BA&#39;, &#39;CA,B&#39;, &#39;CB,A&#39;]</span>
-</pre></div>
-</div>
-<div class="section" id="module-sympy.utilities.iterables">
-<span id="docstring"></span><h3>Docstring<a class="headerlink" href="#module-sympy.utilities.iterables" title="Permalink to this headline">¶</a></h3>
-<dl class="function">
-<dt id="sympy.utilities.iterables.binary_partitions">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">binary_partitions</tt><big>(</big><em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#binary_partitions"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.binary_partitions" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates the binary partition of n.</p>
-<p>A binary partition consists only of numbers that are
-powers of two. Each step reduces a 2**(k+1) to 2**k and
-2**k. Thus 16 is converted to 8 and 8.</p>
-<p>Reference: TAOCP 4, section 7.2.1.5, problem 64</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">binary_partitions</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">binary_partitions</span><span class="p">(</span><span class="mi">5</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="go">[4, 1]</span>
-<span class="go">[2, 2, 1]</span>
-<span class="go">[2, 1, 1, 1]</span>
-<span class="go">[1, 1, 1, 1, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.bracelets">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">bracelets</tt><big>(</big><em>n</em>, <em>k</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#bracelets"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.bracelets" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wrapper to necklaces to return a free (unrestricted) necklace.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.capture">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">capture</tt><big>(</big><em>func</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#capture"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.capture" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the printed output of func().</p>
-<p><span class="math">\(func\)</span> should be a function without arguments that produces output with
-print statements.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">capture</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">foo</span><span class="p">():</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&#39;hello world!&#39;</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="s">&#39;hello&#39;</span> <span class="ow">in</span> <span class="n">capture</span><span class="p">(</span><span class="n">foo</span><span class="p">)</span> <span class="c"># foo, not foo()</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">capture</span><span class="p">(</span><span class="k">lambda</span><span class="p">:</span> <span class="n">pprint</span><span class="p">(</span><span class="mi">2</span><span class="o">/</span><span class="n">x</span><span class="p">))</span>
-<span class="go">&#39;2\n-\nx\n&#39;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.common_prefix">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">common_prefix</tt><big>(</big><em>*seqs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#common_prefix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.common_prefix" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the subsequence that is a common start of sequences in <tt class="docutils literal"><span class="pre">seqs</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">common_prefix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">common_prefix</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">)))</span>
-<span class="go">[0, 1, 2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">common_prefix</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">)),</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">)))</span>
-<span class="go">[0, 1, 2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">common_prefix</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
-<span class="go">[1, 2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">common_prefix</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
-<span class="go">[1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.common_suffix">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">common_suffix</tt><big>(</big><em>*seqs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#common_suffix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.common_suffix" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the subsequence that is a common ending of sequences in <tt class="docutils literal"><span class="pre">seqs</span></tt>.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">common_suffix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">common_suffix</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">)))</span>
-<span class="go">[0, 1, 2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">common_suffix</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">)),</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">)))</span>
-<span class="go">[]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">common_suffix</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">9</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="go">[2, 3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">common_suffix</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">9</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="go">[3]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.dict_merge">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">dict_merge</tt><big>(</big><em>*dicts</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#dict_merge"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.dict_merge" title="Permalink to this definition">¶</a></dt>
-<dd><p>Merge dictionaries into a single dictionary.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.flatten">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">flatten</tt><big>(</big><em>iterable</em>, <em>levels=None</em>, <em>cls=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#flatten"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.flatten" title="Permalink to this definition">¶</a></dt>
-<dd><p>Recursively denest iterable containers.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">flatten</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">flatten</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="go">[1, 2, 3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">flatten</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">3</span><span class="p">]])</span>
-<span class="go">[1, 2, 3]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">flatten</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">]])</span>
-<span class="go">[1, 2, 3, 4, 5]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">flatten</span><span class="p">([</span><span class="mf">1.0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="bp">None</span><span class="p">)])</span>
-<span class="go">[1.0, 2, 1, None]</span>
-</pre></div>
-</div>
-<p>If you want to denest only a specified number of levels of
-nested containers, then set <tt class="docutils literal"><span class="pre">levels</span></tt> flag to the desired
-number of levels:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">ls</span> <span class="o">=</span> <span class="p">[[(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)],</span> <span class="p">[(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)]]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">flatten</span><span class="p">(</span><span class="n">ls</span><span class="p">,</span> <span class="n">levels</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-<span class="go">[(-2, -1), (1, 2), (0, 0)]</span>
-</pre></div>
-</div>
-<p>If cls argument is specified, it will only flatten instances of that
-class, for example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core</span> <span class="kn">import</span> <span class="n">Basic</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">class</span> <span class="nc">MyOp</span><span class="p">(</span><span class="n">Basic</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">pass</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">flatten</span><span class="p">([</span><span class="n">MyOp</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">MyOp</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))],</span> <span class="n">cls</span><span class="o">=</span><span class="n">MyOp</span><span class="p">)</span>
-<span class="go">[1, 2, 3]</span>
-</pre></div>
-</div>
-<p>adapted from <a class="reference external" href="http://kogs-www.informatik.uni-hamburg.de/~meine/python_tricks">http://kogs-www.informatik.uni-hamburg.de/~meine/python_tricks</a></p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.generate_bell">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">generate_bell</tt><big>(</big><em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#generate_bell"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.generate_bell" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates the bell permutations of length <tt class="docutils literal"><span class="pre">n</span></tt>.</p>
-<p class="rubric">References</p>
-<ul class="simple">
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/Method_ringing">http://en.wikipedia.org/wiki/Method_ringing</a></li>
-<li><a class="reference external" href="http://stackoverflow.com/questions/4856615/recursive-permutation/4857018">http://stackoverflow.com/questions/4856615/recursive-permutation/4857018</a></li>
-<li><a class="reference external" href="http://programminggeeks.com/bell-algorithm-for-permutation/">http://programminggeeks.com/bell-algorithm-for-permutation/</a></li>
-<li><a class="reference external" href="http://en.wikipedia.org/wiki/">http://en.wikipedia.org/wiki/</a>
-Steinhaus%E2%80%93Johnson%E2%80%93Trotter_algorithm</li>
-<li>Generating involutions, derangements, and relatives by ECO
-Vincent Vajnovszki, DMTCS vol 1 issue 12, 2010</li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">generate_bell</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">zeros</span><span class="p">,</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">permutations</span>
-</pre></div>
-</div>
-<p>This is the sort of permutation used in the ringing of physical bells,
-and does not produce permutations in lexicographical order. Rather, the
-permutations differ from each other by exactly one inversion, and the
-position at which the swapping occurs varies periodically in a simple
-fashion. Consider the first few permutations of 4 elements generated
-by <tt class="docutils literal"><span class="pre">permutations</span></tt> and <tt class="docutils literal"><span class="pre">generate_bell</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">permutations</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">))))[:</span><span class="mi">5</span><span class="p">]</span>
-<span class="go">[(0, 1, 2, 3), (0, 1, 3, 2), (0, 2, 1, 3), (0, 2, 3, 1), (0, 3, 1, 2)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">generate_bell</span><span class="p">(</span><span class="mi">4</span><span class="p">))[:</span><span class="mi">5</span><span class="p">]</span>
-<span class="go">[(0, 1, 2, 3), (1, 0, 2, 3), (1, 2, 0, 3), (1, 2, 3, 0), (2, 1, 3, 0)]</span>
-</pre></div>
-</div>
-<p>Notice how the 2nd and 3rd lexicographical permutations have 3 elements
-out of place whereas each bell permutations always has only two
-elements out of place relative to the previous permutation.</p>
-<p>How the position of inversion varies across the elements can be seen
-by tracing out where the 0 appears in the permutations:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">m</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">24</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">generate_bell</span><span class="p">(</span><span class="mi">4</span><span class="p">)):</span>
-<span class="gp">... </span>    <span class="n">m</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">m</span><span class="o">.</span><span class="n">print_nonzero</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">)</span>
-<span class="go">[ XXXXXX  XXXXXX  XXXXXX ]</span>
-<span class="go">[X XXXX XX XXXX XX XXXX X]</span>
-<span class="go">[XX XX XXXX XX XXXX XX XX]</span>
-<span class="go">[XXX  XXXXXX  XXXXXX  XXX]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.generate_derangements">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">generate_derangements</tt><big>(</big><em>perm</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#generate_derangements"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.generate_derangements" title="Permalink to this definition">¶</a></dt>
-<dd><p>Routine to generate unique derangements.</p>
-<p>TODO: This will be rewritten to use the
-ECO operator approach once the permutations
-branch is in master.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.functions.combinatorial.factorials.subfactorial</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">generate_derangements</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">generate_derangements</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]))</span>
-<span class="go">[[1, 2, 0], [2, 0, 1]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">generate_derangements</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]))</span>
-<span class="go">[[1, 0, 3, 2], [1, 2, 3, 0], [1, 3, 0, 2], [2, 0, 3, 1],     [2, 3, 0, 1], [2, 3, 1, 0], [3, 0, 1, 2], [3, 2, 0, 1],     [3, 2, 1, 0]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">generate_derangements</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]))</span>
-<span class="go">[]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.generate_involutions">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">generate_involutions</tt><big>(</big><em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#generate_involutions"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.generate_involutions" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates involutions.</p>
-<p>An involution is a permutation that when multiplied
-by itself equals the identity permutation. In this
-implementation the involutions are generated using
-Fixed Points.</p>
-<p>Alternatively, an involution can be considered as
-a permutation that does not contain any cycles with
-a length that is greater than two.</p>
-<p>Reference:
-<a class="reference external" href="http://mathworld.wolfram.com/PermutationInvolution.html">http://mathworld.wolfram.com/PermutationInvolution.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">generate_involutions</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">generate_involutions</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">[(0, 1, 2), (0, 2, 1), (1, 0, 2), (2, 1, 0)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">len</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">generate_involutions</span><span class="p">(</span><span class="mi">4</span><span class="p">)))</span>
-<span class="go">10</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.generate_oriented_forest">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">generate_oriented_forest</tt><big>(</big><em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#generate_oriented_forest"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.generate_oriented_forest" title="Permalink to this definition">¶</a></dt>
-<dd><p>This algorithm generates oriented forests.</p>
-<p>An oriented graph is a directed graph having no symmetric pair of directed
-edges. A forest is an acyclic graph, i.e., it has no cycles. A forest can
-also be described as a disjoint union of trees, which are graphs in which
-any two vertices are connected by exactly one simple path.</p>
-<p>Reference:
-[1] T. Beyer and S.M. Hedetniemi: constant time generation of         rooted trees, SIAM J. Computing Vol. 9, No. 4, November 1980
-[2] <a class="reference external" href="http://stackoverflow.com/questions/1633833/oriented-forest-taocp-algorithm-in-python">http://stackoverflow.com/questions/1633833/oriented-forest-taocp-algorithm-in-python</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">generate_oriented_forest</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">generate_oriented_forest</span><span class="p">(</span><span class="mi">4</span><span class="p">))</span>
-<span class="go">[[0, 1, 2, 3], [0, 1, 2, 2], [0, 1, 2, 1], [0, 1, 2, 0],     [0, 1, 1, 1], [0, 1, 1, 0], [0, 1, 0, 1], [0, 1, 0, 0], [0, 0, 0, 0]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.group">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">group</tt><big>(</big><em>seq</em>, <em>multiple=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#group"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.group" title="Permalink to this definition">¶</a></dt>
-<dd><p>Splits a sequence into a list of lists of equal, adjacent elements.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.utilities.iterables.multiset" title="sympy.utilities.iterables.multiset"><tt class="xref py py-obj docutils literal"><span class="pre">multiset</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">group</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">group</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>
-<span class="go">[[1, 1, 1], [2, 2], [3]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">group</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">[(1, 3), (2, 2), (3, 1)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">group</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">multiple</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">[(1, 2), (3, 1), (2, 2), (1, 1)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.has_dups">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">has_dups</tt><big>(</big><em>seq</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#has_dups"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.has_dups" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if there are any duplicate elements in <tt class="docutils literal"><span class="pre">seq</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">has_dups</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Dict</span><span class="p">,</span> <span class="n">Set</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">has_dups</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">has_dups</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">)))</span>
-<span class="go">False</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">all</span><span class="p">(</span><span class="n">has_dups</span><span class="p">(</span><span class="n">c</span><span class="p">)</span> <span class="ow">is</span> <span class="bp">False</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="p">(</span><span class="nb">set</span><span class="p">(),</span> <span class="n">Set</span><span class="p">(),</span> <span class="nb">dict</span><span class="p">(),</span> <span class="n">Dict</span><span class="p">()))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.has_variety">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">has_variety</tt><big>(</big><em>seq</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#has_variety"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.has_variety" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if there are any different elements in <tt class="docutils literal"><span class="pre">seq</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">has_variety</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Dict</span><span class="p">,</span> <span class="n">Set</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">has_variety</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">has_variety</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.ibin">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">ibin</tt><big>(</big><em>n</em>, <em>bits=0</em>, <em>str=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#ibin"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.ibin" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a list of length <tt class="docutils literal"><span class="pre">bits</span></tt> corresponding to the binary value
-of <tt class="docutils literal"><span class="pre">n</span></tt> with small bits to the right (last). If bits is omitted, the
-length will be the number required to represent <tt class="docutils literal"><span class="pre">n</span></tt>. If the bits are
-desired in reversed order, use the [::-1] slice of the returned list.</p>
-<p>If a sequence of all bits-length lists starting from [0, 0,..., 0]
-through [1, 1, ..., 1] are desired, pass a non-integer for bits, e.g.
-&#8216;all&#8217;.</p>
-<p>If the bit <em>string</em> is desired pass <tt class="docutils literal"><span class="pre">str=True</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">ibin</span><span class="p">,</span> <span class="n">variations</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ibin</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">[1, 0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ibin</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
-<span class="go">[0, 0, 1, 0]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ibin</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-<span class="go">[0, 1, 0, 0]</span>
-</pre></div>
-</div>
-<p>If all lists corresponding to 0 to 2**n - 1, pass a non-integer
-for bits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">bits</span> <span class="o">=</span> <span class="mi">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">ibin</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="s">&#39;all&#39;</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-<span class="go">(0, 0)</span>
-<span class="go">(0, 1)</span>
-<span class="go">(1, 0)</span>
-<span class="go">(1, 1)</span>
-</pre></div>
-</div>
-<p>If a bit string is desired of a given length, use str=True:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="mi">123</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">bits</span> <span class="o">=</span> <span class="mi">10</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ibin</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">bits</span><span class="p">,</span> <span class="nb">str</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">&#39;0001111011&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ibin</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">bits</span><span class="p">,</span> <span class="nb">str</span><span class="o">=</span><span class="bp">True</span><span class="p">)[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>  <span class="c"># small bits left</span>
-<span class="go">&#39;1101111000&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">ibin</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="s">&#39;all&#39;</span><span class="p">,</span> <span class="nb">str</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">[&#39;000&#39;, &#39;001&#39;, &#39;010&#39;, &#39;011&#39;, &#39;100&#39;, &#39;101&#39;, &#39;110&#39;, &#39;111&#39;]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.interactive_traversal">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">interactive_traversal</tt><big>(</big><em>expr</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#interactive_traversal"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.interactive_traversal" title="Permalink to this definition">¶</a></dt>
-<dd><p>Traverse a tree asking a user which branch to choose.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.kbins">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">kbins</tt><big>(</big><em>l</em>, <em>k</em>, <em>ordered=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#kbins"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.kbins" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return sequence <tt class="docutils literal"><span class="pre">l</span></tt> partitioned into <tt class="docutils literal"><span class="pre">k</span></tt> bins.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.utilities.iterables.partitions" title="sympy.utilities.iterables.partitions"><tt class="xref py py-obj docutils literal"><span class="pre">partitions</span></tt></a>, <a class="reference internal" href="#sympy.utilities.iterables.multiset_partitions" title="sympy.utilities.iterables.multiset_partitions"><tt class="xref py py-obj docutils literal"><span class="pre">multiset_partitions</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">kbins</span>
-</pre></div>
-</div>
-<p>The default is to give the items in the same order, but grouped
-into k partitions without any reordering:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">kbins</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">)),</span> <span class="mi">2</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="go">[[0], [1, 2, 3, 4]]</span>
-<span class="go">[[0, 1], [2, 3, 4]]</span>
-<span class="go">[[0, 1, 2], [3, 4]]</span>
-<span class="go">[[0, 1, 2, 3], [4]]</span>
-</pre></div>
-</div>
-<p>The <tt class="docutils literal"><span class="pre">ordered</span></tt> flag which is either None (to give the simple partition
-of the the elements) or is a 2 digit integer indicating whether the order of
-the bins and the order of the items in the bins matters. Given:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">A</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]]</span>
-<span class="n">B</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">]]</span>
-<span class="n">C</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">]]</span>
-<span class="n">D</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]]</span>
-</pre></div>
-</div>
-<p>the following values for <tt class="docutils literal"><span class="pre">ordered</span></tt> have the shown meanings:</p>
-<div class="highlight-python"><pre>00 means A == B == C == D
-01 means A == B
-10 means A == D
-11 means A == A</pre>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">ordered</span> <span class="ow">in</span> <span class="p">[</span><span class="bp">None</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">11</span><span class="p">]:</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="s">&#39;ordered =&#39;</span><span class="p">,</span> <span class="n">ordered</span><span class="p">)</span>
-<span class="gp">... </span>    <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">kbins</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">)),</span> <span class="mi">2</span><span class="p">,</span> <span class="n">ordered</span><span class="o">=</span><span class="n">ordered</span><span class="p">):</span>
-<span class="gp">... </span>        <span class="k">print</span><span class="p">(</span><span class="s">&#39;    &#39;</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="go">ordered = None</span>
-<span class="go">     [[0], [1, 2]]</span>
-<span class="go">     [[0, 1], [2]]</span>
-<span class="go">ordered = 0</span>
-<span class="go">     [[0, 1], [2]]</span>
-<span class="go">     [[0, 2], [1]]</span>
-<span class="go">     [[0], [1, 2]]</span>
-<span class="go">ordered = 1</span>
-<span class="go">     [[0], [1, 2]]</span>
-<span class="go">     [[0], [2, 1]]</span>
-<span class="go">     [[1], [0, 2]]</span>
-<span class="go">     [[1], [2, 0]]</span>
-<span class="go">     [[2], [0, 1]]</span>
-<span class="go">     [[2], [1, 0]]</span>
-<span class="go">ordered = 10</span>
-<span class="go">     [[0, 1], [2]]</span>
-<span class="go">     [[2], [0, 1]]</span>
-<span class="go">     [[0, 2], [1]]</span>
-<span class="go">     [[1], [0, 2]]</span>
-<span class="go">     [[0], [1, 2]]</span>
-<span class="go">     [[1, 2], [0]]</span>
-<span class="go">ordered = 11</span>
-<span class="go">     [[0], [1, 2]]</span>
-<span class="go">     [[0, 1], [2]]</span>
-<span class="go">     [[0], [2, 1]]</span>
-<span class="go">     [[0, 2], [1]]</span>
-<span class="go">     [[1], [0, 2]]</span>
-<span class="go">     [[1, 0], [2]]</span>
-<span class="go">     [[1], [2, 0]]</span>
-<span class="go">     [[1, 2], [0]]</span>
-<span class="go">     [[2], [0, 1]]</span>
-<span class="go">     [[2, 0], [1]]</span>
-<span class="go">     [[2], [1, 0]]</span>
-<span class="go">     [[2, 1], [0]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.minlex">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">minlex</tt><big>(</big><em>seq</em>, <em>directed=True</em>, <em>is_set=False</em>, <em>small=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#minlex"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.minlex" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a tuple where the smallest element appears first; if
-<tt class="docutils literal"><span class="pre">directed</span></tt> is True (default) then the order is preserved, otherwise
-the sequence will be reversed if that gives a smaller ordering.</p>
-<p>If every element appears only once then is_set can be set to True
-for more efficient processing.</p>
-<p>If the smallest element is known at the time of calling, it can be
-passed and the calculation of the smallest element will be omitted.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.polyhedron</span> <span class="kn">import</span> <span class="n">minlex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">minlex</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
-<span class="go">(0, 1, 2)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">minlex</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">(0, 2, 1)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">minlex</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">directed</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">(0, 1, 2)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">minlex</span><span class="p">(</span><span class="s">&#39;11010011000&#39;</span><span class="p">,</span> <span class="n">directed</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">&#39;00011010011&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">minlex</span><span class="p">(</span><span class="s">&#39;11010011000&#39;</span><span class="p">,</span> <span class="n">directed</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>
-<span class="go">&#39;00011001011&#39;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.multiset">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">multiset</tt><big>(</big><em>seq</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#multiset"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.multiset" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the hashable sequence in multiset form with values being the
-multiplicity of the item in the sequence.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.utilities.iterables.group" title="sympy.utilities.iterables.group"><tt class="xref py py-obj docutils literal"><span class="pre">group</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">multiset</span><span class="p">,</span> <span class="n">group</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">multiset</span><span class="p">(</span><span class="s">&#39;mississippi&#39;</span><span class="p">)</span>
-<span class="go">{&#39;i&#39;: 4, &#39;m&#39;: 1, &#39;p&#39;: 2, &#39;s&#39;: 4}</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.multiset_combinations">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">multiset_combinations</tt><big>(</big><em>m</em>, <em>n</em>, <em>g=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#multiset_combinations"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.multiset_combinations" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the unique combinations of size <tt class="docutils literal"><span class="pre">n</span></tt> from multiset <tt class="docutils literal"><span class="pre">m</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">multiset_combinations</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.core.compatibility</span> <span class="kn">import</span> <span class="n">combinations</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span>  <span class="n">multiset_combinations</span><span class="p">(</span><span class="s">&#39;baby&#39;</span><span class="p">,</span> <span class="mi">3</span><span class="p">)]</span>
-<span class="go">[&#39;abb&#39;, &#39;aby&#39;, &#39;bby&#39;]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">count</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">s</span><span class="p">):</span> <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="mi">3</span><span class="p">)))</span>
-</pre></div>
-</div>
-<p>The number of combinations depends on the number of letters; the
-number of unique combinations depends on how the letters are
-repeated.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">s1</span> <span class="o">=</span> <span class="s">&#39;abracadabra&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s2</span> <span class="o">=</span> <span class="s">&#39;banana tree&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">count</span><span class="p">(</span><span class="n">combinations</span><span class="p">,</span> <span class="n">s1</span><span class="p">),</span> <span class="n">count</span><span class="p">(</span><span class="n">multiset_combinations</span><span class="p">,</span> <span class="n">s1</span><span class="p">)</span>
-<span class="go">(165, 23)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">count</span><span class="p">(</span><span class="n">combinations</span><span class="p">,</span> <span class="n">s2</span><span class="p">),</span> <span class="n">count</span><span class="p">(</span><span class="n">multiset_combinations</span><span class="p">,</span> <span class="n">s2</span><span class="p">)</span>
-<span class="go">(165, 54)</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.multiset_partitions">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">multiset_partitions</tt><big>(</big><em>multiset</em>, <em>m=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#multiset_partitions"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.multiset_partitions" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return unique partitions of the given multiset (in list form).
-If <tt class="docutils literal"><span class="pre">m</span></tt> is None, all multisets will be returned, otherwise only
-partitions with <tt class="docutils literal"><span class="pre">m</span></tt> parts will be returned.</p>
-<p>If <tt class="docutils literal"><span class="pre">multiset</span></tt> is an integer, a range [0, 1, ..., multiset - 1]
-will be supplied.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="#sympy.utilities.iterables.partitions" title="sympy.utilities.iterables.partitions"><tt class="xref py py-obj docutils literal"><span class="pre">partitions</span></tt></a>, <a class="reference internal" href="../combinatorics/partitions.html#sympy.combinatorics.partitions.Partition" title="sympy.combinatorics.partitions.Partition"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.combinatorics.partitions.Partition</span></tt></a>, <a class="reference internal" href="../combinatorics/partitions.html#sympy.combinatorics.partitions.IntegerPartition" title="sympy.combinatorics.partitions.IntegerPartition"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.combinatorics.partitions.IntegerPartition</span></tt></a></p>
-</div>
-<p class="rubric">Notes</p>
-<p>When all the elements are the same in the multiset, the order
-of the returned partitions is determined by the <tt class="docutils literal"><span class="pre">partitions</span></tt>
-routine.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">multiset_partitions</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">multiset_partitions</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">[[[1, 2, 3], [4]], [[1, 2, 4], [3]], [[1, 2], [3, 4]],</span>
-<span class="go">[[1, 3, 4], [2]], [[1, 3], [2, 4]], [[1, 4], [2, 3]],</span>
-<span class="go">[[1], [2, 3, 4]]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">multiset_partitions</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">[[[1, 2, 3, 4]]]</span>
-</pre></div>
-</div>
-<p>Only unique partitions are returned and these will be returned in a
-canonical order regardless of the order of the input:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">ans</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">multiset_partitions</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">multiset_partitions</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span> <span class="o">==</span> <span class="n">ans</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">multiset_partitions</span><span class="p">(</span><span class="n">a</span><span class="p">))</span> <span class="o">==</span>
-<span class="gp">... </span> <span class="nb">list</span><span class="p">(</span><span class="n">multiset_partitions</span><span class="p">(</span><span class="nb">sorted</span><span class="p">(</span><span class="n">a</span><span class="p">))))</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>If m is omitted then all partitions will be returned:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">multiset_partitions</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]))</span>
-<span class="go">[[[1, 1, 2]], [[1, 1], [2]], [[1, 2], [1]], [[1], [1], [2]]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">multiset_partitions</span><span class="p">([</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="mi">3</span><span class="p">))</span>
-<span class="go">[[[1, 1, 1]], [[1], [1, 1]], [[1], [1], [1]]]</span>
-</pre></div>
-</div>
-<p class="rubric">Counting</p>
-<p>The number of partitions of a set is given by the bell number:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">bell</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">len</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">multiset_partitions</span><span class="p">(</span><span class="mi">5</span><span class="p">)))</span> <span class="o">==</span> <span class="n">bell</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span> <span class="o">==</span> <span class="mi">52</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>The number of partitions of length k from a set of size n is given by the
-Stirling Number of the 2nd kind:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">S2</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Dummy</span><span class="p">,</span> <span class="n">binomial</span><span class="p">,</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">Sum</span>
-<span class="gp">... </span>    <span class="k">if</span> <span class="n">k</span> <span class="o">&gt;</span> <span class="n">n</span><span class="p">:</span>
-<span class="gp">... </span>        <span class="k">return</span> <span class="mi">0</span>
-<span class="gp">... </span>    <span class="n">j</span> <span class="o">=</span> <span class="n">Dummy</span><span class="p">()</span>
-<span class="gp">... </span>    <span class="n">arg</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">k</span><span class="o">-</span><span class="n">j</span><span class="p">)</span><span class="o">*</span><span class="n">j</span><span class="o">**</span><span class="n">n</span><span class="o">*</span><span class="n">binomial</span><span class="p">(</span><span class="n">k</span><span class="p">,</span><span class="n">j</span><span class="p">)</span>
-<span class="gp">... </span>    <span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span><span class="n">k</span><span class="p">)</span><span class="o">*</span><span class="n">Sum</span><span class="p">(</span><span class="n">arg</span><span class="p">,(</span><span class="n">j</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">k</span><span class="p">))</span><span class="o">.</span><span class="n">doit</span><span class="p">()</span>
-<span class="gp">...</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">S2</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">multiset_partitions</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">2</span><span class="p">)))</span> <span class="o">==</span> <span class="mi">15</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-<p>These comments on counting apply to <em>sets</em>, not multisets.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.multiset_permutations">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">multiset_permutations</tt><big>(</big><em>m</em>, <em>k=None</em>, <em>g=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#multiset_permutations"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.multiset_permutations" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return the unique permutations of multiset <tt class="docutils literal"><span class="pre">m</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">multiset_permutations</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">multiset_permutations</span><span class="p">(</span><span class="s">&#39;aab&#39;</span><span class="p">)]</span>
-<span class="go">[&#39;aab&#39;, &#39;aba&#39;, &#39;baa&#39;]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="s">&#39;banana&#39;</span><span class="p">))</span>
-<span class="go">720</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">len</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">multiset_permutations</span><span class="p">(</span><span class="s">&#39;banana&#39;</span><span class="p">)))</span>
-<span class="go">60</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.necklaces">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">necklaces</tt><big>(</big><em>n</em>, <em>k</em>, <em>free=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#necklaces"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.necklaces" title="Permalink to this definition">¶</a></dt>
-<dd><p>A routine to generate necklaces that may (free=True) or may not
-(free=False) be turned over to be viewed. The &#8220;necklaces&#8221; returned
-are comprised of <tt class="docutils literal"><span class="pre">n</span></tt> integers (beads) with <tt class="docutils literal"><span class="pre">k</span></tt> different
-values (colors). Only unique necklaces are returned.</p>
-<p class="rubric">References</p>
-<p><a class="reference external" href="http://mathworld.wolfram.com/Necklace.html">http://mathworld.wolfram.com/Necklace.html</a></p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">necklaces</span><span class="p">,</span> <span class="n">bracelets</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">show</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">i</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">return</span> <span class="s">&#39;&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="n">i</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>The &#8220;unrestricted necklace&#8221; is sometimes also referred to as a
-&#8220;bracelet&#8221; (an object that can be turned over, a sequence that can
-be reversed) and the term &#8220;necklace&#8221; is used to imply a sequence
-that cannot be reversed. So ACB == ABC for a bracelet (rotate and
-reverse) while the two are different for a necklace since rotation
-alone cannot make the two sequences the same.</p>
-<p>(mnemonic: Bracelets can be viewed Backwards, but Not Necklaces.)</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">B</span> <span class="o">=</span> <span class="p">[</span><span class="n">show</span><span class="p">(</span><span class="s">&#39;ABC&#39;</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">bracelets</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">N</span> <span class="o">=</span> <span class="p">[</span><span class="n">show</span><span class="p">(</span><span class="s">&#39;ABC&#39;</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">necklaces</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">set</span><span class="p">(</span><span class="n">N</span><span class="p">)</span> <span class="o">-</span> <span class="nb">set</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
-<span class="go">set([&#39;ACB&#39;])</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">necklaces</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">[(0, 0, 0, 0), (0, 0, 0, 1), (0, 0, 1, 1),</span>
-<span class="go"> (0, 1, 0, 1), (0, 1, 1, 1), (1, 1, 1, 1)]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">show</span><span class="p">(</span><span class="s">&#39;.o&#39;</span><span class="p">,</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">bracelets</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">)]</span>
-<span class="go">[&#39;....&#39;, &#39;...o&#39;, &#39;..oo&#39;, &#39;.o.o&#39;, &#39;.ooo&#39;, &#39;oooo&#39;]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.numbered_symbols">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">numbered_symbols</tt><big>(</big><em>prefix='x'</em>, <em>cls=None</em>, <em>start=0</em>, <em>*args</em>, <em>**assumptions</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#numbered_symbols"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.numbered_symbols" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generate an infinite stream of Symbols consisting of a prefix and
-increasing subscripts.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>prefix</strong> : str, optional</p>
-<blockquote>
-<div><p>The prefix to use. By default, this function will generate symbols of
-the form &#8220;x0&#8221;, &#8220;x1&#8221;, etc.</p>
-</div></blockquote>
-<p><strong>cls</strong> : class, optional</p>
-<blockquote>
-<div><p>The class to use. By default, it uses Symbol, but you can also use Wild or Dummy.</p>
-</div></blockquote>
-<p><strong>start</strong> : int, optional</p>
-<blockquote>
-<div><p>The start number.  By default, it is 0.</p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>sym</strong> : Symbol</p>
-<blockquote class="last">
-<div><p>The subscripted symbols.</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.partitions">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">partitions</tt><big>(</big><em>n</em>, <em>m=None</em>, <em>k=None</em>, <em>size=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#partitions"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.partitions" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generate all partitions of integer n (&gt;= 0).</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>``m``</strong> : integer (default gives partitions of all sizes)</p>
-<blockquote>
-<div><p>limits number of parts in parition (mnemonic: m, maximum parts)</p>
-</div></blockquote>
-<p><strong>``k``</strong> : integer (default gives partitions number from 1 through n)</p>
-<blockquote>
-<div><p>limits the numbers that are kept in the partition (mnemonic: k, keys)</p>
-</div></blockquote>
-<p><strong>``size``</strong> : bool (default False, only partition is returned)</p>
-<blockquote>
-<div><p>when <tt class="docutils literal"><span class="pre">True</span></tt> then (M, P) is returned where M is the sum of the
-multiplicities and P is the generated partition.</p>
-</div></blockquote>
-<p><strong>Each partition is represented as a dictionary, mapping an integer</strong> :</p>
-<p><strong>to the number of copies of that integer in the partition.  For example,</strong> :</p>
-<p class="last"><strong>the first partition of 4 returned is {4: 1}, &#8220;4: one of them&#8221;.</strong> :</p>
-</td>
-</tr>
-</tbody>
-</table>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference internal" href="../combinatorics/partitions.html#sympy.combinatorics.partitions.Partition" title="sympy.combinatorics.partitions.Partition"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.combinatorics.partitions.Partition</span></tt></a>, <a class="reference internal" href="../combinatorics/partitions.html#sympy.combinatorics.partitions.IntegerPartition" title="sympy.combinatorics.partitions.IntegerPartition"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.combinatorics.partitions.IntegerPartition</span></tt></a></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">partitions</span>
-</pre></div>
-</div>
-<p>The numbers appearing in the partition (the key of the returned dict)
-are limited with k:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">partitions</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">2</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-<span class="go">{2: 3}</span>
-<span class="go">{1: 2, 2: 2}</span>
-<span class="go">{1: 4, 2: 1}</span>
-<span class="go">{1: 6}</span>
-</pre></div>
-</div>
-<p>The maximum number of parts in the partion (the sum of the values in
-the returned dict) are limited with m:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">partitions</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="n">m</span><span class="o">=</span><span class="mi">2</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="k">print</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
-<span class="gp">...</span>
-<span class="go">{6: 1}</span>
-<span class="go">{1: 1, 5: 1}</span>
-<span class="go">{2: 1, 4: 1}</span>
-<span class="go">{3: 2}</span>
-</pre></div>
-</div>
-<p>Note that the _same_ dictionary object is returned each time.
-This is for speed:  generating each partition goes quickly,
-taking constant time, independent of n.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">p</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">partitions</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">2</span><span class="p">)]</span>
-<span class="go">[{1: 6}, {1: 6}, {1: 6}, {1: 6}]</span>
-</pre></div>
-</div>
-<p>If you want to build a list of the returned dictionaries then
-make a copy of them:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">p</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">partitions</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">2</span><span class="p">)]</span>
-<span class="go">[{2: 3}, {1: 2, 2: 2}, {1: 4, 2: 1}, {1: 6}]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">[(</span><span class="n">M</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">copy</span><span class="p">())</span> <span class="k">for</span> <span class="n">M</span><span class="p">,</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">partitions</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="bp">True</span><span class="p">)]</span>
-<span class="go">[(3, {2: 3}), (4, {1: 2, 2: 2}), (5, {1: 4, 2: 1}), (6, {1: 6})]</span>
-</pre></div>
-</div>
-<dl class="docutils">
-<dt>Reference:</dt>
-<dd>modified from Tim Peter&#8217;s version to allow for k and m values:
-code.activestate.com/recipes/218332-generator-for-integer-partitions/</dd>
-</dl>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.postfixes">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">postfixes</tt><big>(</big><em>seq</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#postfixes"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.postfixes" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generate all postfixes of a sequence.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">postfixes</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">postfixes</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">]))</span>
-<span class="go">[[4], [3, 4], [2, 3, 4], [1, 2, 3, 4]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.postorder_traversal">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">postorder_traversal</tt><big>(</big><em>node</em>, <em>keys=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#postorder_traversal"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.postorder_traversal" title="Permalink to this definition">¶</a></dt>
-<dd><p>Do a postorder traversal of a tree.</p>
-<p>This generator recursively yields nodes that it has visited in a postorder
-fashion. That is, it descends through the tree depth-first to yield all of
-a node&#8217;s children&#8217;s postorder traversal before yielding the node itself.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>node</strong> : sympy expression</p>
-<blockquote>
-<div><p>The expression to traverse.</p>
-</div></blockquote>
-<p><strong>keys</strong> : (default None) sort key(s)</p>
-<blockquote class="last">
-<div><p>The key(s) used to sort args of Basic objects. When None, args of Basic
-objects are processed in arbitrary order. If key is defined, it will
-be passed along to ordered() as the only key(s) to use to sort the
-arguments; if <tt class="docutils literal"><span class="pre">key</span></tt> is simply True then the default keys of
-<tt class="docutils literal"><span class="pre">ordered</span></tt> will be used (node count and default_sort_key).</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">postorder_traversal</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">w</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-</pre></div>
-</div>
-<p>The nodes are returned in the order that they are encountered unless key
-is given; simply passing key=True will guarantee that the traversal is
-unique.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">postorder_traversal</span><span class="p">(</span><span class="n">w</span> <span class="o">+</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">))</span> 
-<span class="go">[z, y, x, x + y, z*(x + y), w, w + z*(x + y)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">postorder_traversal</span><span class="p">(</span><span class="n">w</span> <span class="o">+</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">z</span><span class="p">,</span> <span class="n">keys</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">[w, z, x, y, x + y, z*(x + y), w + z*(x + y)]</span>
-</pre></div>
-</div>
-<p class="rubric">Yields</p>
-<dl class="docutils">
-<dt>subtree <span class="classifier-delimiter">:</span> <span class="classifier">sympy expression</span></dt>
-<dd>All of the subtrees in the tree.</dd>
-</dl>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.prefixes">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">prefixes</tt><big>(</big><em>seq</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#prefixes"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.prefixes" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generate all prefixes of a sequence.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">prefixes</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">prefixes</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">]))</span>
-<span class="go">[[1], [1, 2], [1, 2, 3], [1, 2, 3, 4]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.reshape">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">reshape</tt><big>(</big><em>seq</em>, <em>how</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#reshape"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.reshape" title="Permalink to this definition">¶</a></dt>
-<dd><p>Reshape the sequence according to the template in <tt class="docutils literal"><span class="pre">how</span></tt>.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">reshape</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">seq</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">9</span><span class="p">))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">reshape</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="p">[</span><span class="mi">4</span><span class="p">])</span> <span class="c"># lists of 4</span>
-<span class="go">[[1, 2, 3, 4], [5, 6, 7, 8]]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">reshape</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="p">(</span><span class="mi">4</span><span class="p">,))</span> <span class="c"># tuples of 4</span>
-<span class="go">[(1, 2, 3, 4), (5, 6, 7, 8)]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">reshape</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span> <span class="c"># tuples of 4</span>
-<span class="go">[(1, 2, 3, 4), (5, 6, 7, 8)]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">reshape</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">2</span><span class="p">]))</span> <span class="c"># (i, i, [i, i])</span>
-<span class="go">[(1, 2, [3, 4]), (5, 6, [7, 8])]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">reshape</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="p">((</span><span class="mi">2</span><span class="p">,),</span> <span class="p">[</span><span class="mi">2</span><span class="p">]))</span> <span class="c"># etc....</span>
-<span class="go">[((1, 2), [3, 4]), ((5, 6), [7, 8])]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">reshape</span><span class="p">(</span><span class="n">seq</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">[(1, [2, 3], 4), (5, [6, 7], 8)]</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">reshape</span><span class="p">(</span><span class="nb">tuple</span><span class="p">(</span><span class="n">seq</span><span class="p">),</span> <span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="mi">2</span><span class="p">,)],))</span>
-<span class="go">(([[1], 2, (3, 4)],), ([[5], 6, (7, 8)],))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">reshape</span><span class="p">(</span><span class="nb">tuple</span><span class="p">(</span><span class="n">seq</span><span class="p">),</span> <span class="p">([</span><span class="mi">1</span><span class="p">],</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="mi">2</span><span class="p">,)))</span>
-<span class="go">(([1], 2, (3, 4)), ([5], 6, (7, 8)))</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">reshape</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">12</span><span class="p">)),</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="nb">set</span><span class="p">([</span><span class="mi">2</span><span class="p">]),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="mi">3</span><span class="p">,),</span> <span class="mi">1</span><span class="p">)])</span>
-<span class="go">[[0, 1, [2, 3, 4], set([5, 6]), (7, (8, 9, 10), 11)]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.rotate_left">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">rotate_left</tt><big>(</big><em>x</em>, <em>y</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#rotate_left"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.rotate_left" title="Permalink to this definition">¶</a></dt>
-<dd><p>Left rotates a list x by the number of steps specified
-in y.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">rotate_left</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rotate_left</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">[1, 2, 0]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.rotate_right">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">rotate_right</tt><big>(</big><em>x</em>, <em>y</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#rotate_right"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.rotate_right" title="Permalink to this definition">¶</a></dt>
-<dd><p>Left rotates a list x by the number of steps specified
-in y.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">rotate_right</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">rotate_right</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">[2, 0, 1]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.runs">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">runs</tt><big>(</big><em>seq</em>, <em>op=&lt;built-in function gt&gt;</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#runs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.runs" title="Permalink to this definition">¶</a></dt>
-<dd><p>Group the sequence into lists in which successive elements
-all compare the same with the comparison operator, <tt class="docutils literal"><span class="pre">op</span></tt>:
-op(seq[i + 1], seq[i]) is True from all elements in a run.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">runs</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">operator</span> <span class="kn">import</span> <span class="n">ge</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">runs</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
-<span class="go">[[0, 1, 2], [2], [1, 4], [3], [2], [2]]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">runs</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="n">op</span><span class="o">=</span><span class="n">ge</span><span class="p">)</span>
-<span class="go">[[0, 1, 2, 2], [1, 4], [3], [2, 2]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.sift">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">sift</tt><big>(</big><em>seq</em>, <em>keyfunc</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#sift"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.sift" title="Permalink to this definition">¶</a></dt>
-<dd><p>Sift the sequence, <tt class="docutils literal"><span class="pre">seq</span></tt> into a dictionary according to keyfunc.</p>
-<p>OUTPUT: each element in expr is stored in a list keyed to the value
-of keyfunc for the element.</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">ordered</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities</span> <span class="kn">import</span> <span class="n">sift</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">exp</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sift</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">)),</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span> <span class="o">%</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">{0: [0, 2, 4], 1: [1, 3]}</span>
-</pre></div>
-</div>
-<p>sift() returns a defaultdict() object, so any key that has no matches will
-give [].</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sift</span><span class="p">([</span><span class="n">x</span><span class="p">],</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">is_commutative</span><span class="p">)</span>
-<span class="go">{True: [x]}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="p">[</span><span class="bp">False</span><span class="p">]</span>
-<span class="go">[]</span>
-</pre></div>
-</div>
-<p>Sometimes you won&#8217;t know how many keys you will get:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sift</span><span class="p">([</span><span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">exp</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">y</span><span class="o">**</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">],</span>
-<span class="gp">... </span>     <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">.</span><span class="n">as_base_exp</span><span class="p">()[</span><span class="mi">0</span><span class="p">])</span>
-<span class="go">{E: [exp(x)], x: [sqrt(x)], y: [y**(2*x)]}</span>
-</pre></div>
-</div>
-<p>If you need to sort the sifted items it might be better to use
-<tt class="docutils literal"><span class="pre">ordered</span></tt> which can economically apply multiple sort keys
-to a squence while sorting.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.subsets">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">subsets</tt><big>(</big><em>seq</em>, <em>k=None</em>, <em>repetition=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#subsets"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.subsets" title="Permalink to this definition">¶</a></dt>
-<dd><p>Generates all k-subsets (combinations) from an n-element set, seq.</p>
-<p>A k-subset of an n-element set is any subset of length exactly k. The
-number of k-subsets of an n-element set is given by binomial(n, k),
-whereas there are 2**n subsets all together. If k is None then all
-2**n subsets will be returned from shortest to longest.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">subsets</span>
-</pre></div>
-</div>
-<p>subsets(seq, k) will return the n!/k!/(n - k)! k-subsets (combinations)
-without repetition, i.e. once an item has been removed, it can no
-longer be &#8220;taken&#8221;:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">subsets</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">[(1, 2)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">subsets</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]))</span>
-<span class="go">[(), (1,), (2,), (1, 2)]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">subsets</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">[(1, 2), (1, 3), (2, 3)]</span>
-</pre></div>
-</div>
-<p>subsets(seq, k, repetition=True) will return the (n - 1 + k)!/k!/(n - 1)!
-combinations <em>with</em> repetition:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">subsets</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="mi">2</span><span class="p">,</span> <span class="n">repetition</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">[(1, 1), (1, 2), (2, 2)]</span>
-</pre></div>
-</div>
-<p>If you ask for more items than are in the set you get the empty set unless
-you allow repetitions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">subsets</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="mi">3</span><span class="p">,</span> <span class="n">repetition</span><span class="o">=</span><span class="bp">False</span><span class="p">))</span>
-<span class="go">[]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">subsets</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="mi">3</span><span class="p">,</span> <span class="n">repetition</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">[(0, 0, 0), (0, 0, 1), (0, 1, 1), (1, 1, 1)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.take">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">take</tt><big>(</big><em>iter</em>, <em>n</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#take"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.take" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return <tt class="docutils literal"><span class="pre">n</span></tt> items from <tt class="docutils literal"><span class="pre">iter</span></tt> iterator.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.topological_sort">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">topological_sort</tt><big>(</big><em>graph</em>, <em>key=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#topological_sort"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.topological_sort" title="Permalink to this definition">¶</a></dt>
-<dd><p>Topological sort of graph&#8217;s vertices.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>``graph``</strong> : <tt class="docutils literal"><span class="pre">tuple[list,</span> <span class="pre">list[tuple[T,</span> <span class="pre">T]]</span></tt></p>
-<blockquote>
-<div><p>A tuple consisting of a list of vertices and a list of edges of
-a graph to be sorted topologically.</p>
-</div></blockquote>
-<p><strong>``key``</strong> : <tt class="docutils literal"><span class="pre">callable[T]</span></tt> (optional)</p>
-<blockquote class="last">
-<div><p>Ordering key for vertices on the same level. By default the natural
-(e.g. lexicographic) ordering is used (in this case the base type
-must implement ordering relations).</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<p>Consider a graph:</p>
-<div class="highlight-python"><pre>+---+     +---+     +---+
-| 7 |\    | 5 |     | 3 |
-+---+ \   +---+     +---+
-  |   _\___/ ____   _/ |
-  |  /  \___/    \ /   |
-  V  V           V V   |
- +----+         +---+  |
- | 11 |         | 8 |  |
- +----+         +---+  |
-  | | \____   ___/ _   |
-  | \      \ /    / \  |
-  V  \     V V   /  V  V
-+---+ \   +---+ |  +----+
-| 2 |  |  | 9 | |  | 10 |
-+---+  |  +---+ |  +----+
-       \________/</pre>
-</div>
-<p>where vertices are integers. This graph can be encoded using
-elementary Python&#8217;s data structures as follows:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">V</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">11</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">E</span> <span class="o">=</span> <span class="p">[(</span><span class="mi">7</span><span class="p">,</span> <span class="mi">11</span><span class="p">),</span> <span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">),</span> <span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">11</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">8</span><span class="p">),</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">10</span><span class="p">),</span>
-<span class="gp">... </span>     <span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="mi">9</span><span class="p">),</span> <span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="mi">10</span><span class="p">),</span> <span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">)]</span>
-</pre></div>
-</div>
-<p>To compute a topological sort for graph <tt class="docutils literal"><span class="pre">(V,</span> <span class="pre">E)</span></tt> issue:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">topological_sort</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">topological_sort</span><span class="p">((</span><span class="n">V</span><span class="p">,</span> <span class="n">E</span><span class="p">))</span>
-<span class="go">[3, 5, 7, 8, 11, 2, 9, 10]</span>
-</pre></div>
-</div>
-<p>If specific tie breaking approach is needed, use <tt class="docutils literal"><span class="pre">key</span></tt> parameter:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">topological_sort</span><span class="p">((</span><span class="n">V</span><span class="p">,</span> <span class="n">E</span><span class="p">),</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">v</span><span class="p">:</span> <span class="o">-</span><span class="n">v</span><span class="p">)</span>
-<span class="go">[7, 5, 11, 3, 10, 8, 9, 2]</span>
-</pre></div>
-</div>
-<p>Only acyclic graphs can be sorted. If the input graph has a cycle,
-then <tt class="xref py py-exc docutils literal"><span class="pre">ValueError</span></tt> will be raised:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">topological_sort</span><span class="p">((</span><span class="n">V</span><span class="p">,</span> <span class="n">E</span> <span class="o">+</span> <span class="p">[(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">7</span><span class="p">)]))</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">ValueError</span>: <span class="n">cycle detected</span>
-</pre></div>
-</div>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><a class="reference external" href="http://en.wikipedia.org/wiki/Topological_sorting">http://en.wikipedia.org/wiki/Topological_sorting</a></p>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.unflatten">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">unflatten</tt><big>(</big><em>iter</em>, <em>n=2</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#unflatten"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.unflatten" title="Permalink to this definition">¶</a></dt>
-<dd><p>Group <tt class="docutils literal"><span class="pre">iter</span></tt> into tuples of length <tt class="docutils literal"><span class="pre">n</span></tt>. Raise an error if
-the length of <tt class="docutils literal"><span class="pre">iter</span></tt> is not a multiple of <tt class="docutils literal"><span class="pre">n</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.uniq">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">uniq</tt><big>(</big><em>seq</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#uniq"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.uniq" title="Permalink to this definition">¶</a></dt>
-<dd><p>Yield unique elements from <tt class="docutils literal"><span class="pre">seq</span></tt> as an iterator.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">uniq</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dat</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">type</span><span class="p">(</span><span class="n">uniq</span><span class="p">(</span><span class="n">dat</span><span class="p">))</span> <span class="ow">in</span> <span class="p">(</span><span class="nb">list</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">)</span>
-<span class="go">False</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">uniq</span><span class="p">(</span><span class="n">dat</span><span class="p">))</span>
-<span class="go">[1, 4, 5, 2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">uniq</span><span class="p">(</span><span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">dat</span><span class="p">))</span>
-<span class="go">[1, 4, 5, 2]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">uniq</span><span class="p">([[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">]]))</span>
-<span class="go">[[1], [2, 1], [1]]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.unrestricted_necklace">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">unrestricted_necklace</tt><big>(</big><em>*args</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#unrestricted_necklace"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.unrestricted_necklace" title="Permalink to this definition">¶</a></dt>
-<dd><p>Wrapper to necklaces to return a free (unrestricted) necklace.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.iterables.variations">
-<tt class="descclassname">sympy.utilities.iterables.</tt><tt class="descname">variations</tt><big>(</big><em>seq</em>, <em>n</em>, <em>repetition=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/iterables.html#variations"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.iterables.variations" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a generator of the n-sized variations of <tt class="docutils literal"><span class="pre">seq</span></tt> (size N).
-<tt class="docutils literal"><span class="pre">repetition</span></tt> controls whether items in <tt class="docutils literal"><span class="pre">seq</span></tt> can appear more than once;</p>
-<div class="admonition-see-also admonition seealso">
-<p class="first admonition-title">See also</p>
-<p class="last"><tt class="xref py py-obj docutils literal"><span class="pre">sympy.core.compatibility.permutations</span></tt>, <tt class="xref py py-obj docutils literal"><span class="pre">sympy.core.compatibility.product</span></tt></p>
-</div>
-<p class="rubric">Examples</p>
-<p>variations(seq, n) will return N! / (N - n)! permutations without
-repetition of seq&#8217;s elements:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.iterables</span> <span class="kn">import</span> <span class="n">variations</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">variations</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="mi">2</span><span class="p">))</span>
-<span class="go">[(1, 2), (2, 1)]</span>
-</pre></div>
-</div>
-<p>variations(seq, n, True) will return the N**n permutations obtained
-by allowing repetition of elements:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">variations</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="mi">2</span><span class="p">,</span> <span class="n">repetition</span><span class="o">=</span><span class="bp">True</span><span class="p">))</span>
-<span class="go">[(1, 1), (1, 2), (2, 1), (2, 2)]</span>
-</pre></div>
-</div>
-<p>If you ask for more items than are in the set you get the empty set unless
-you allow repetitions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">variations</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="mi">3</span><span class="p">,</span> <span class="n">repetition</span><span class="o">=</span><span class="bp">False</span><span class="p">))</span>
-<span class="go">[]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">variations</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="mi">3</span><span class="p">,</span> <span class="n">repetition</span><span class="o">=</span><span class="bp">True</span><span class="p">))[:</span><span class="mi">4</span><span class="p">]</span>
-<span class="go">[(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1)]</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
-</div>
-</div>
-
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-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../../index.html">
-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Iterables</a><ul>
-<li><a class="reference internal" href="#cartes">cartes</a></li>
-<li><a class="reference internal" href="#variations">variations</a></li>
-<li><a class="reference internal" href="#partitions">partitions</a><ul>
-<li><a class="reference internal" href="#module-sympy.utilities.iterables">Docstring</a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../utilities/decorator.html"
-                        title="previous chapter">Decorator</a></p>
-  <h4>Next topic</h4>
-  <p class="topless"><a href="../utilities/lambdify.html"
-                        title="next chapter">Lambdify</a></p>
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-            
-  <div class="section" id="module-sympy.utilities.lambdify">
-<span id="lambdify"></span><h1>Lambdify<a class="headerlink" href="#module-sympy.utilities.lambdify" title="Permalink to this headline">¶</a></h1>
-<p>This module provides convenient functions to transform sympy expressions to
-lambda functions which can be used to calculate numerical values very fast.</p>
-<dl class="function">
-<dt id="sympy.utilities.lambdify.implemented_function">
-<tt class="descclassname">sympy.utilities.lambdify.</tt><tt class="descname">implemented_function</tt><big>(</big><em>symfunc</em>, <em>implementation</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/lambdify.html#implemented_function"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.lambdify.implemented_function" title="Permalink to this definition">¶</a></dt>
-<dd><p>Add numerical <tt class="docutils literal"><span class="pre">implementation</span></tt> to function <tt class="docutils literal"><span class="pre">symfunc</span></tt>.</p>
-<p><tt class="docutils literal"><span class="pre">symfunc</span></tt> can be an <tt class="docutils literal"><span class="pre">UndefinedFunction</span></tt> instance, or a name string.
-In the latter case we create an <tt class="docutils literal"><span class="pre">UndefinedFunction</span></tt> instance with that
-name.</p>
-<p>Be aware that this is a quick workaround, not a general method to create
-special symbolic functions. If you want to create a symbolic function to be
-used by all the machinery of sympy you should subclass the <tt class="docutils literal"><span class="pre">Function</span></tt>
-class.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>symfunc</strong> : <tt class="docutils literal"><span class="pre">str</span></tt> or <tt class="docutils literal"><span class="pre">UndefinedFunction</span></tt> instance</p>
-<blockquote>
-<div><p>If <tt class="docutils literal"><span class="pre">str</span></tt>, then create new <tt class="docutils literal"><span class="pre">UndefinedFunction</span></tt> with this as
-name.  If <span class="math">\(symfunc\)</span> is a sympy function, attach implementation to it.</p>
-</div></blockquote>
-<p><strong>implementation</strong> : callable</p>
-<blockquote>
-<div><p>numerical implementation to be called by <tt class="docutils literal"><span class="pre">evalf()</span></tt> or <tt class="docutils literal"><span class="pre">lambdify</span></tt></p>
-</div></blockquote>
-</td>
-</tr>
-<tr class="field-even field"><th class="field-name">Returns :</th><td class="field-body"><p class="first"><strong>afunc</strong> : sympy.FunctionClass instance</p>
-<blockquote class="last">
-<div><p>function with attached implementation</p>
-</div></blockquote>
-</td>
-</tr>
-</tbody>
-</table>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.lambdify</span> <span class="kn">import</span> <span class="n">lambdify</span><span class="p">,</span> <span class="n">implemented_function</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">implemented_function</span><span class="p">(</span><span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">),</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lam_f</span> <span class="o">=</span> <span class="n">lambdify</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lam_f</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">5</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.lambdify.lambdastr">
-<tt class="descclassname">sympy.utilities.lambdify.</tt><tt class="descname">lambdastr</tt><big>(</big><em>args</em>, <em>expr</em>, <em>printer=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/lambdify.html#lambdastr"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.lambdify.lambdastr" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a string that can be evaluated to a lambda function.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.lambdify</span> <span class="kn">import</span> <span class="n">lambdastr</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lambdastr</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">&#39;lambda x: (x**2)&#39;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">lambdastr</span><span class="p">((</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span><span class="p">),</span> <span class="p">[</span><span class="n">z</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">x</span><span class="p">])</span>
-<span class="go">&#39;lambda x,y,z: ([z, y, x])&#39;</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.lambdify.lambdify">
-<tt class="descclassname">sympy.utilities.lambdify.</tt><tt class="descname">lambdify</tt><big>(</big><em>args</em>, <em>expr</em>, <em>modules=None</em>, <em>printer=None</em>, <em>use_imps=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/lambdify.html#lambdify"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.lambdify.lambdify" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns a lambda function for fast calculation of numerical values.</p>
-<p>If not specified differently by the user, SymPy functions are replaced as
-far as possible by either python-math, numpy (if available) or mpmath
-functions - exactly in this order. To change this behavior, the &#8220;modules&#8221;
-argument can be used. It accepts:</p>
-<blockquote>
-<div><ul class="simple">
-<li>the strings &#8220;math&#8221;, &#8220;mpmath&#8221;, &#8220;numpy&#8221;, &#8220;sympy&#8221;</li>
-<li>any modules (e.g. math)</li>
-<li>dictionaries that map names of sympy functions to arbitrary functions</li>
-<li>lists that contain a mix of the arguments above, with higher priority
-given to entries appearing first.</li>
-</ul>
-</div></blockquote>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.lambdify</span> <span class="kn">import</span> <span class="n">implemented_function</span><span class="p">,</span> <span class="n">lambdify</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sqrt</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">Matrix</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">lambdify</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">4</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">lambdify</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">),</span> <span class="p">[</span><span class="n">z</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-<span class="go">[3, 2, 1]</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">lambdify</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">2.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">lambdify</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-<span class="go">0.0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">lambdify</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">),</span> <span class="n">Matrix</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">))</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">modules</span><span class="o">=</span><span class="s">&#39;sympy&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">[1, 3]</span>
-</pre></div>
-</div>
-<p>Functions present in <span class="math">\(expr\)</span> can also carry their own numerical
-implementations, in a callable attached to the <tt class="docutils literal"><span class="pre">_imp_</span></tt>
-attribute.  Usually you attach this using the
-<tt class="docutils literal"><span class="pre">implemented_function</span></tt> factory:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">implemented_function</span><span class="p">(</span><span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">),</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">func</span> <span class="o">=</span> <span class="n">lambdify</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">func</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<span class="go">5</span>
-</pre></div>
-</div>
-<p><tt class="docutils literal"><span class="pre">lambdify</span></tt> always prefers <tt class="docutils literal"><span class="pre">_imp_</span></tt> implementations to implementations
-in other namespaces, unless the <tt class="docutils literal"><span class="pre">use_imps</span></tt> input parameter is False.</p>
-<p class="rubric">Usage</p>
-<ol class="arabic">
-<li><p class="first">Use one of the provided modules:</p>
-<p>&gt;&gt; f = lambdify(x, sin(x), &#8220;math&#8221;)</p>
-<dl class="docutils">
-<dt>Attention: Functions that are not in the math module will throw a name</dt>
-<dd><p class="first last">error when the lambda function is evaluated! So this would
-be better:</p>
-</dd>
-</dl>
-<p>&gt;&gt; f = lambdify(x, sin(x)*gamma(x), (&#8220;math&#8221;, &#8220;mpmath&#8221;, &#8220;sympy&#8221;))</p>
-</li>
-<li><p class="first">Use some other module:</p>
-<p>&gt;&gt; import numpy
-&gt;&gt; f = lambdify((x,y), tan(x*y), numpy)</p>
-<dl class="docutils">
-<dt>Attention: There are naming differences between numpy and sympy. So if</dt>
-<dd><p class="first last">you simply take the numpy module, e.g. sympy.atan will not be
-translated to numpy.arctan. Use the modified module instead
-by passing the string &#8220;numpy&#8221;:</p>
-</dd>
-</dl>
-<p>&gt;&gt; f = lambdify((x,y), tan(x*y), &#8220;numpy&#8221;)
-&gt;&gt; f(1, 2)
--2.18503986326
-&gt;&gt; from numpy import array
-&gt;&gt; f(array([1, 2, 3]), array([2, 3, 5]))
-[-2.18503986 -0.29100619 -0.8559934 ]</p>
-</li>
-<li><p class="first">Use own dictionaries:</p>
-<p>&gt;&gt; def my_cool_function(x): ...
-&gt;&gt; dic = {&#8220;sin&#8221; : my_cool_function}
-&gt;&gt; f = lambdify(x, sin(x), dic)</p>
-<p>Now f would look like:</p>
-<p>&gt;&gt; lambda x: my_cool_function(x)</p>
-</li>
-</ol>
-</dd></dl>
-
-</div>
-
-
-          </div>
-        </div>
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-            <p class="logo"><a href="../../index.html">
-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../utilities/iterables.html"
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-            
-  <div class="section" id="module-sympy.utilities.memoization">
-<span id="memoization"></span><h1>Memoization<a class="headerlink" href="#module-sympy.utilities.memoization" title="Permalink to this headline">¶</a></h1>
-<dl class="function">
-<dt id="sympy.utilities.memoization.assoc_recurrence_memo">
-<tt class="descclassname">sympy.utilities.memoization.</tt><tt class="descname">assoc_recurrence_memo</tt><big>(</big><em>base_seq</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/memoization.html#assoc_recurrence_memo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.memoization.assoc_recurrence_memo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Memo decorator for associated sequences defined by recurrence starting from base</p>
-<p>base_seq(n) &#8211; callable to get base sequence elements</p>
-<p>XXX works only for Pn0 = base_seq(0) cases
-XXX works only for m &lt;= n cases</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.memoization.recurrence_memo">
-<tt class="descclassname">sympy.utilities.memoization.</tt><tt class="descname">recurrence_memo</tt><big>(</big><em>initial</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/memoization.html#recurrence_memo"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.memoization.recurrence_memo" title="Permalink to this definition">¶</a></dt>
-<dd><p>Memo decorator for sequences defined by recurrence</p>
-<p>See usage examples e.g. in the specfun/combinatorial module</p>
-</dd></dl>
-
-</div>
-
-
-          </div>
-        </div>
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-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../utilities/lambdify.html"
-                        title="previous chapter">Lambdify</a></p>
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-                        title="next chapter">Miscellaneous</a></p>
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-  <div class="section" id="module-sympy.utilities.misc">
-<span id="miscellaneous"></span><h1>Miscellaneous<a class="headerlink" href="#module-sympy.utilities.misc" title="Permalink to this headline">¶</a></h1>
-<p>Miscellaneous stuff that doesn&#8217;t really fit anywhere else.</p>
-<dl class="function">
-<dt id="sympy.utilities.misc.debug">
-<tt class="descclassname">sympy.utilities.misc.</tt><tt class="descname">debug</tt><big>(</big><em>*args</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/misc.html#debug"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.misc.debug" title="Permalink to this definition">¶</a></dt>
-<dd><p>Print <tt class="docutils literal"><span class="pre">*args</span></tt> if SYMPY_DEBUG is True, else do nothing.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.misc.rawlines">
-<tt class="descclassname">sympy.utilities.misc.</tt><tt class="descname">rawlines</tt><big>(</big><em>s</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/misc.html#rawlines"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.misc.rawlines" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a cut-and-pastable string that, when printed, is equivalent
-to the input. The string returned is formatted so it can be indented
-nicely within tests; in some cases it is wrapped in the dedent
-function which has to be imported from textwrap.</p>
-<p class="rubric">Examples</p>
-<p>Note: because there are characters in the examples below that need
-to be escaped because they are themselves within a triple quoted
-docstring, expressions below look more complicated than they would
-be if they were printed in an interpreter window.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.misc</span> <span class="kn">import</span> <span class="n">rawlines</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">TableForm</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">TableForm</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">]],</span> <span class="n">headings</span><span class="o">=</span><span class="p">(</span><span class="bp">None</span><span class="p">,</span> <span class="p">[</span><span class="s">&#39;a&#39;</span><span class="p">,</span> <span class="s">&#39;bee&#39;</span><span class="p">])))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">rawlines</span><span class="p">(</span><span class="n">s</span><span class="p">))</span> <span class="c"># the \ appears as \ when printed</span>
-<span class="go">(</span>
-<span class="go">    &#39;a bee\n&#39;</span>
-<span class="go">    &#39;-----\n&#39;</span>
-<span class="go">    &#39;1 10 &#39;</span>
-<span class="go">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">rawlines</span><span class="p">(</span><span class="s">&#39;&#39;&#39;this</span>
-<span class="gp">... </span><span class="s">that&#39;&#39;&#39;</span><span class="p">))</span>
-<span class="go">dedent(&#39;&#39;&#39;\</span>
-<span class="go">    this</span>
-<span class="go">    that&#39;&#39;&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">rawlines</span><span class="p">(</span><span class="s">&#39;&#39;&#39;this</span>
-<span class="gp">... </span><span class="s">that</span>
-<span class="gp">... </span><span class="s">&#39;&#39;&#39;</span><span class="p">))</span>
-<span class="go">dedent(&#39;&#39;&#39;\</span>
-<span class="go">    this</span>
-<span class="go">    that</span>
-<span class="go">    &#39;&#39;&#39;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="s">&quot;&quot;&quot;this</span>
-<span class="gp">... </span><span class="s">is a triple &#39;&#39;&#39;</span>
-<span class="gp">... </span><span class="s">&quot;&quot;&quot;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">rawlines</span><span class="p">(</span><span class="n">s</span><span class="p">))</span>
-<span class="go">dedent(&quot;&quot;&quot;\</span>
-<span class="go">    this</span>
-<span class="go">    is a triple &#39;&#39;&#39;</span>
-<span class="go">    &quot;&quot;&quot;)</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">rawlines</span><span class="p">(</span><span class="s">&#39;&#39;&#39;this</span>
-<span class="gp">... </span><span class="s">that</span>
-<span class="gp">... </span><span class="s">    &#39;&#39;&#39;</span><span class="p">))</span>
-<span class="go">(</span>
-<span class="go">    &#39;this\n&#39;</span>
-<span class="go">    &#39;that\n&#39;</span>
-<span class="go">    &#39;    &#39;</span>
-<span class="go">)</span>
-</pre></div>
-</div>
-</dd></dl>
-
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diff --git a/dev-py3k/modules/utilities/pkgdata.html b/dev-py3k/modules/utilities/pkgdata.html
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-
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-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
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-            
-  <div class="section" id="module-sympy.utilities.pkgdata">
-<span id="pkgdata"></span><h1>PKGDATA<a class="headerlink" href="#module-sympy.utilities.pkgdata" title="Permalink to this headline">¶</a></h1>
-<p>pkgdata is a simple, extensible way for a package to acquire data file
-resources.</p>
-<p>The getResource function is equivalent to the standard idioms, such as
-the following minimal implementation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="kn">import</span> <span class="nn">sys</span><span class="o">,</span> <span class="nn">os</span>
-
-<span class="k">def</span> <span class="nf">getResource</span><span class="p">(</span><span class="n">identifier</span><span class="p">,</span> <span class="n">pkgname</span><span class="o">=</span><span class="n">__name__</span><span class="p">):</span>
-    <span class="n">pkgpath</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">dirname</span><span class="p">(</span><span class="n">sys</span><span class="o">.</span><span class="n">modules</span><span class="p">[</span><span class="n">pkgname</span><span class="p">]</span><span class="o">.</span><span class="n">__file__</span><span class="p">)</span>
-    <span class="n">path</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">pkgpath</span><span class="p">,</span> <span class="n">identifier</span><span class="p">)</span>
-    <span class="k">return</span> <span class="nb">file</span><span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">normpath</span><span class="p">(</span><span class="n">path</span><span class="p">),</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;rb&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>When a __loader__ is present on the module given by __name__, it will defer
-getResource to its get_data implementation and return it as a file-like
-object (such as StringIO).</p>
-<dl class="function">
-<dt id="sympy.utilities.pkgdata.get_resource">
-<tt class="descclassname">sympy.utilities.pkgdata.</tt><tt class="descname">get_resource</tt><big>(</big><em>identifier</em>, <em>pkgname='sympy.utilities.pkgdata'</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/pkgdata.html#get_resource"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.pkgdata.get_resource" title="Permalink to this definition">¶</a></dt>
-<dd><p>Acquire a readable object for a given package name and identifier.
-An IOError will be raised if the resource can not be found.</p>
-<p>For example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">mydata</span> <span class="o">=</span> <span class="n">get_esource</span><span class="p">(</span><span class="s">&#39;mypkgdata.jpg&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">read</span><span class="p">()</span>
-</pre></div>
-</div>
-<p>Note that the package name must be fully qualified, if given, such
-that it would be found in sys.modules.</p>
-<p>In some cases, getResource will return a real file object.  In that
-case, it may be useful to use its name attribute to get the path
-rather than use it as a file-like object.  For example, you may
-be handing data off to a C API.</p>
-</dd></dl>
-
-</div>
-
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-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../utilities/misc.html"
-                        title="previous chapter">Miscellaneous</a></p>
-  <h4>Next topic</h4>
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-                        title="next chapter">pytest</a></p>
-  <h3>This Page</h3>
-  <ul class="this-page-menu">
-    <li><a href="../../_sources/modules/utilities/pkgdata.txt"
-           rel="nofollow">Show Source</a></li>
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-  <h3>Quick search</h3>
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-            
-  <div class="section" id="module-sympy.utilities.pytest">
-<span id="pytest"></span><h1>pytest<a class="headerlink" href="#module-sympy.utilities.pytest" title="Permalink to this headline">¶</a></h1>
-<p>py.test hacks to support XFAIL/XPASS</p>
-<dl class="function">
-<dt id="sympy.utilities.pytest.SKIP">
-<tt class="descclassname">sympy.utilities.pytest.</tt><tt class="descname">SKIP</tt><big>(</big><em>reason</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/pytest.html#SKIP"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.pytest.SKIP" title="Permalink to this definition">¶</a></dt>
-<dd><p>Similar to <tt class="xref py py-func docutils literal"><span class="pre">skip()</span></tt>, but this is a decorator.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.pytest.raises">
-<tt class="descclassname">sympy.utilities.pytest.</tt><tt class="descname">raises</tt><big>(</big><em>expectedException</em>, <em>code=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/pytest.html#raises"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.pytest.raises" title="Permalink to this definition">¶</a></dt>
-<dd><p>Tests that <tt class="docutils literal"><span class="pre">code</span></tt> raises the exception <tt class="docutils literal"><span class="pre">expectedException</span></tt>.</p>
-<p><tt class="docutils literal"><span class="pre">code</span></tt> may be a callable, such as a lambda expression or function
-name.</p>
-<p>If <tt class="docutils literal"><span class="pre">code</span></tt> is not given or None, <tt class="docutils literal"><span class="pre">raises</span></tt> will return a context
-manager for use in <tt class="docutils literal"><span class="pre">with</span></tt> statements; the code to execute then
-comes from the scope of the <tt class="docutils literal"><span class="pre">with</span></tt>. The calling module must have
-<tt class="docutils literal"><span class="pre">from</span> <span class="pre">__future__</span> <span class="pre">import</span> <span class="pre">with_statement</span></tt> as its first code line
-for compatibility with Python 2.5 in that case.</p>
-<p>raises does nothing if the callable raises the expected exception,
-otherwise it raises an AssertionError.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.pytest</span> <span class="kn">import</span> <span class="n">raises</span>
-</pre></div>
-</div>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">raises</span><span class="p">(</span><span class="ne">ZeroDivisionError</span><span class="p">,</span> <span class="k">lambda</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="mi">0</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">raises</span><span class="p">(</span><span class="ne">ZeroDivisionError</span><span class="p">,</span> <span class="k">lambda</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">AssertionError</span>: <span class="n">DID NOT RAISE</span>
-</pre></div>
-</div>
-<p>(Python 2.5&#8217;s doctest cannot support with statements due to a bug in
-its __import__ handling. That&#8217;s why the following examples are
-excluded from doctesting; the doctest: annotations can go once SymPy
-stops Python 2.5 support.)</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">with</span> <span class="n">raises</span><span class="p">(</span><span class="ne">ZeroDivisionError</span><span class="p">):</span> 
-<span class="gp">... </span>    <span class="n">n</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="mi">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">with</span> <span class="n">raises</span><span class="p">(</span><span class="ne">ZeroDivisionError</span><span class="p">):</span> 
-<span class="gp">... </span>    <span class="n">n</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="mi">2</span>
-<span class="gt">Traceback (most recent call last):</span>
-<span class="c">...</span>
-<span class="gr">AssertionError</span>: <span class="n">DID NOT RAISE</span>
-</pre></div>
-</div>
-<p>Note that you cannot test multiple statements via
-<tt class="docutils literal"><span class="pre">with</span> <span class="pre">raises</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">with</span> <span class="n">raises</span><span class="p">(</span><span class="ne">ZeroDivisionError</span><span class="p">):</span> 
-<span class="gp">... </span>    <span class="n">n</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="mi">0</span>    <span class="c"># will execute and raise, aborting the ``with``</span>
-<span class="gp">... </span>    <span class="n">n</span> <span class="o">=</span> <span class="mi">9999</span><span class="o">/</span><span class="mi">0</span> <span class="c"># never executed</span>
-</pre></div>
-</div>
-<p>This is just what <tt class="docutils literal"><span class="pre">with</span></tt> is supposed to do: abort the
-contained statement sequence at the first exception and let
-the context manager deal with the exception.</p>
-<p>To test multiple statements, you&#8217;ll need a separate <tt class="docutils literal"><span class="pre">with</span></tt>
-for each:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">with</span> <span class="n">raises</span><span class="p">(</span><span class="ne">ZeroDivisionError</span><span class="p">):</span> 
-<span class="gp">... </span>    <span class="n">n</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="mi">0</span>    <span class="c"># will execute and raise</span>
-<span class="gp">... </span><span class="k">with</span> <span class="n">raises</span><span class="p">(</span><span class="ne">ZeroDivisionError</span><span class="p">):</span>
-<span class="gp">... </span>    <span class="n">n</span> <span class="o">=</span> <span class="mi">9999</span><span class="o">/</span><span class="mi">0</span> <span class="c"># will also execute and raise</span>
-</pre></div>
-</div>
-</dd></dl>
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-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../utilities/pkgdata.html"
-                        title="previous chapter">PKGDATA</a></p>
-  <h4>Next topic</h4>
-  <p class="topless"><a href="../utilities/randtest.html"
-                        title="next chapter">Randomised Testing</a></p>
-  <h3>This Page</h3>
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diff --git a/dev-py3k/modules/utilities/randtest.html b/dev-py3k/modules/utilities/randtest.html
deleted file mode 100644
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-
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-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
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-             >modules</a> |</li>
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-            
-  <div class="section" id="module-sympy.utilities.randtest">
-<span id="randomised-testing"></span><h1>Randomised Testing<a class="headerlink" href="#module-sympy.utilities.randtest" title="Permalink to this headline">¶</a></h1>
-<p>Helpers for randomized testing</p>
-<dl class="function">
-<dt id="sympy.utilities.randtest.comp">
-<tt class="descclassname">sympy.utilities.randtest.</tt><tt class="descname">comp</tt><big>(</big><em>z1</em>, <em>z2</em>, <em>tol</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/randtest.html#comp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.randtest.comp" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a bool indicating whether the error between z1 and z2 is &lt;= tol.</p>
-<p>If z2 is non-zero and <tt class="docutils literal"><span class="pre">|z1|</span> <span class="pre">&gt;</span> <span class="pre">1</span></tt> the error is normalized by <tt class="docutils literal"><span class="pre">|z1|</span></tt>, so
-if you want the absolute error, call this as <tt class="docutils literal"><span class="pre">comp(z1</span> <span class="pre">-</span> <span class="pre">z2,</span> <span class="pre">0,</span> <span class="pre">tol)</span></tt>.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.randtest.random_complex_number">
-<tt class="descclassname">sympy.utilities.randtest.</tt><tt class="descname">random_complex_number</tt><big>(</big><em>a=2</em>, <em>b=-1</em>, <em>c=3</em>, <em>d=1</em>, <em>rational=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/randtest.html#random_complex_number"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.randtest.random_complex_number" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return a random complex number.</p>
-<p>To reduce chance of hitting branch cuts or anything, we guarantee
-b &lt;= Im z &lt;= d, a &lt;= Re z &lt;= c</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.randtest.test_derivative_numerically">
-<tt class="descclassname">sympy.utilities.randtest.</tt><tt class="descname">test_derivative_numerically</tt><big>(</big><em>f</em>, <em>z</em>, <em>tol=1e-06</em>, <em>a=2</em>, <em>b=-1</em>, <em>c=3</em>, <em>d=1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/randtest.html#test_derivative_numerically"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.randtest.test_derivative_numerically" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test numerically that the symbolically computed derivative of f
-with respect to z is correct.</p>
-<p>This routine does not test whether there are Floats present with
-precision higher than 15 digits so if there are, your results may
-not be what you expect due to round-off errors.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.randtest</span> <span class="kn">import</span> <span class="n">test_derivative_numerically</span> <span class="k">as</span> <span class="n">td</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">td</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.randtest.test_numerically">
-<tt class="descclassname">sympy.utilities.randtest.</tt><tt class="descname">test_numerically</tt><big>(</big><em>f</em>, <em>g</em>, <em>z=None</em>, <em>tol=1e-06</em>, <em>a=2</em>, <em>b=-1</em>, <em>c=3</em>, <em>d=1</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/randtest.html#test_numerically"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.randtest.test_numerically" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test numerically that f and g agree when evaluated in the argument z.</p>
-<p>If z is None, all symbols will be tested. This routine does not test
-whether there are Floats present with precision higher than 15 digits
-so if there are, your results may not be what you expect due to round-
-off errors.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">S</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.randtest</span> <span class="kn">import</span> <span class="n">test_numerically</span> <span class="k">as</span> <span class="n">tn</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">tn</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">True</span>
-</pre></div>
-</div>
-</dd></dl>
-
-</div>
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-            
-  <div class="section" id="module-sympy.utilities.runtests">
-<span id="run-tests"></span><h1>Run Tests<a class="headerlink" href="#module-sympy.utilities.runtests" title="Permalink to this headline">¶</a></h1>
-<p>This is our testing framework.</p>
-<p>Goals:</p>
-<ul class="simple">
-<li>it should be compatible with py.test and operate very similarly
-(or identically)</li>
-<li>doesn&#8217;t require any external dependencies</li>
-<li>preferably all the functionality should be in this file only</li>
-<li>no magic, just import the test file and execute the test functions, that&#8217;s it</li>
-<li>portable</li>
-</ul>
-<dl class="class">
-<dt id="sympy.utilities.runtests.PyTestReporter">
-<em class="property">class </em><tt class="descclassname">sympy.utilities.runtests.</tt><tt class="descname">PyTestReporter</tt><big>(</big><em>verbose=False</em>, <em>tb='short'</em>, <em>colors=True</em>, <em>force_colors=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/runtests.html#PyTestReporter"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.runtests.PyTestReporter" title="Permalink to this definition">¶</a></dt>
-<dd><p>Py.test like reporter. Should produce output identical to py.test.</p>
-<dl class="function">
-<dt id="sympy.utilities.runtests.PyTestReporter.write">
-<tt class="descname">write</tt><big>(</big><em>self</em>, <em>text</em>, <em>color=''</em>, <em>align='left'</em>, <em>width=None</em>, <em>force_colors=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/runtests.html#PyTestReporter.write"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.runtests.PyTestReporter.write" title="Permalink to this definition">¶</a></dt>
-<dd><p>Prints a text on the screen.</p>
-<p>It uses sys.stdout.write(), so no readline library is necessary.</p>
-<table class="docutils field-list" frame="void" rules="none">
-<col class="field-name" />
-<col class="field-body" />
-<tbody valign="top">
-<tr class="field-odd field"><th class="field-name">Parameters :</th><td class="field-body"><p class="first"><strong>color</strong> : choose from the colors below, &#8220;&#8221; means default color</p>
-<p><strong>align</strong> : &#8220;left&#8221;/&#8221;right&#8221;, &#8220;left&#8221; is a normal print, &#8220;right&#8221; is aligned on</p>
-<blockquote>
-<div><p>the right-hand side of the screen, filled with spaces if
-necessary</p>
-</div></blockquote>
-<p class="last"><strong>width</strong> : the screen width</p>
-</td>
-</tr>
-</tbody>
-</table>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.utilities.runtests.Reporter">
-<em class="property">class </em><tt class="descclassname">sympy.utilities.runtests.</tt><tt class="descname">Reporter</tt><a class="reference internal" href="../../_modules/sympy/utilities/runtests.html#Reporter"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.runtests.Reporter" title="Permalink to this definition">¶</a></dt>
-<dd><p>Parent class for all reporters.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.utilities.runtests.SymPyDocTestFinder">
-<em class="property">class </em><tt class="descclassname">sympy.utilities.runtests.</tt><tt class="descname">SymPyDocTestFinder</tt><big>(</big><em>verbose=False</em>, <em>parser=&lt;doctest.DocTestParser object at 0x10b695610&gt;</em>, <em>recurse=True</em>, <em>exclude_empty=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/runtests.html#SymPyDocTestFinder"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.runtests.SymPyDocTestFinder" title="Permalink to this definition">¶</a></dt>
-<dd><p>A class used to extract the DocTests that are relevant to a given
-object, from its docstring and the docstrings of its contained
-objects.  Doctests can currently be extracted from the following
-object types: modules, functions, classes, methods, staticmethods,
-classmethods, and properties.</p>
-<p>Modified from doctest&#8217;s version by looking harder for code in the
-case that it looks like the the code comes from a different module.
-In the case of decorated functions (e.g. &#64;vectorize) they appear
-to come from a different module (e.g. multidemensional) even though
-their code is not there.</p>
-</dd></dl>
-
-<dl class="class">
-<dt id="sympy.utilities.runtests.SymPyDocTestRunner">
-<em class="property">class </em><tt class="descclassname">sympy.utilities.runtests.</tt><tt class="descname">SymPyDocTestRunner</tt><big>(</big><em>checker=None</em>, <em>verbose=None</em>, <em>optionflags=0</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/runtests.html#SymPyDocTestRunner"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.runtests.SymPyDocTestRunner" title="Permalink to this definition">¶</a></dt>
-<dd><p>A class used to run DocTest test cases, and accumulate statistics.
-The <tt class="docutils literal"><span class="pre">run</span></tt> method is used to process a single DocTest case.  It
-returns a tuple <tt class="docutils literal"><span class="pre">(f,</span> <span class="pre">t)</span></tt>, where <tt class="docutils literal"><span class="pre">t</span></tt> is the number of test cases
-tried, and <tt class="docutils literal"><span class="pre">f</span></tt> is the number of test cases that failed.</p>
-<p>Modified from the doctest version to not reset the sys.displayhook (see
-issue 2041).</p>
-<p>See the docstring of the original DocTestRunner for more information.</p>
-<dl class="function">
-<dt id="sympy.utilities.runtests.SymPyDocTestRunner.run">
-<tt class="descname">run</tt><big>(</big><em>self</em>, <em>test</em>, <em>compileflags=None</em>, <em>out=None</em>, <em>clear_globs=True</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/runtests.html#SymPyDocTestRunner.run"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.runtests.SymPyDocTestRunner.run" title="Permalink to this definition">¶</a></dt>
-<dd><p>Run the examples in <tt class="docutils literal"><span class="pre">test</span></tt>, and display the results using the
-writer function <tt class="docutils literal"><span class="pre">out</span></tt>.</p>
-<p>The examples are run in the namespace <tt class="docutils literal"><span class="pre">test.globs</span></tt>.  If
-<tt class="docutils literal"><span class="pre">clear_globs</span></tt> is true (the default), then this namespace will
-be cleared after the test runs, to help with garbage
-collection.  If you would like to examine the namespace after
-the test completes, then use <tt class="docutils literal"><span class="pre">clear_globs=False</span></tt>.</p>
-<p><tt class="docutils literal"><span class="pre">compileflags</span></tt> gives the set of flags that should be used by
-the Python compiler when running the examples.  If not
-specified, then it will default to the set of future-import
-flags that apply to <tt class="docutils literal"><span class="pre">globs</span></tt>.</p>
-<p>The output of each example is checked using
-<tt class="docutils literal"><span class="pre">SymPyDocTestRunner.check_output</span></tt>, and the results are
-formatted by the <tt class="docutils literal"><span class="pre">SymPyDocTestRunner.report_*</span></tt> methods.</p>
-</dd></dl>
-
-</dd></dl>
-
-<dl class="attribute">
-<dt id="sympy.utilities.runtests.SymPyTestResults">
-<tt class="descclassname">sympy.utilities.runtests.</tt><tt class="descname">SymPyTestResults</tt><a class="headerlink" href="#sympy.utilities.runtests.SymPyTestResults" title="Permalink to this definition">¶</a></dt>
-<dd><p>alias of <tt class="xref py py-class docutils literal"><span class="pre">TestResults</span></tt></p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.runtests.convert_to_native_paths">
-<tt class="descclassname">sympy.utilities.runtests.</tt><tt class="descname">convert_to_native_paths</tt><big>(</big><em>lst</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/runtests.html#convert_to_native_paths"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.runtests.convert_to_native_paths" title="Permalink to this definition">¶</a></dt>
-<dd><p>Converts a list of &#8216;/&#8217; separated paths into a list of
-native (os.sep separated) paths and converts to lowercase
-if the system is case insensitive.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.runtests.doctest">
-<tt class="descclassname">sympy.utilities.runtests.</tt><tt class="descname">doctest</tt><big>(</big><em>*paths</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/runtests.html#doctest"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.runtests.doctest" title="Permalink to this definition">¶</a></dt>
-<dd><p>Runs doctests in all *.py files in the sympy directory which match
-any of the given strings in <tt class="docutils literal"><span class="pre">paths</span></tt> or all tests if paths=[].</p>
-<p>Notes:</p>
-<ul class="simple">
-<li>Paths can be entered in native system format or in unix,
-forward-slash format.</li>
-<li>Files that are on the blacklist can be tested by providing
-their path; they are only excluded if no paths are given.</li>
-</ul>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">sympy</span>
-</pre></div>
-</div>
-<p>Run all tests:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sympy</span><span class="o">.</span><span class="n">doctest</span><span class="p">()</span> 
-</pre></div>
-</div>
-<p>Run one file:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sympy</span><span class="o">.</span><span class="n">doctest</span><span class="p">(</span><span class="s">&quot;sympy/core/basic.py&quot;</span><span class="p">)</span> 
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sympy</span><span class="o">.</span><span class="n">doctest</span><span class="p">(</span><span class="s">&quot;polynomial.rst&quot;</span><span class="p">)</span> 
-</pre></div>
-</div>
-<p>Run all tests in sympy/functions/ and some particular file:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sympy</span><span class="o">.</span><span class="n">doctest</span><span class="p">(</span><span class="s">&quot;/functions&quot;</span><span class="p">,</span> <span class="s">&quot;basic.py&quot;</span><span class="p">)</span> 
-</pre></div>
-</div>
-<p>Run any file having polynomial in its name, doc/src/modules/polynomial.rst,
-sympy/functions/special/polynomials.py, and sympy/polys/polynomial.py:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sympy</span><span class="o">.</span><span class="n">doctest</span><span class="p">(</span><span class="s">&quot;polynomial&quot;</span><span class="p">)</span> 
-</pre></div>
-</div>
-<p>The <tt class="docutils literal"><span class="pre">subprocess</span></tt> and <tt class="docutils literal"><span class="pre">verbose</span></tt> options are the same as with the function
-<tt class="docutils literal"><span class="pre">test()</span></tt>.  See the docstring of that function for more information.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.runtests.get_sympy_dir">
-<tt class="descclassname">sympy.utilities.runtests.</tt><tt class="descname">get_sympy_dir</tt><big>(</big><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/runtests.html#get_sympy_dir"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.runtests.get_sympy_dir" title="Permalink to this definition">¶</a></dt>
-<dd><p>Returns the root sympy directory and set the global value
-indicating whether the system is case sensitive or not.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.runtests.isgeneratorfunction">
-<tt class="descclassname">sympy.utilities.runtests.</tt><tt class="descname">isgeneratorfunction</tt><big>(</big><em>object</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/runtests.html#isgeneratorfunction"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.runtests.isgeneratorfunction" title="Permalink to this definition">¶</a></dt>
-<dd><p>Return True if the object is a user-defined generator function.</p>
-<p>Generator function objects provides same attributes as functions.</p>
-<p>See isfunction.__doc__ for attributes listing.</p>
-<p>Adapted from Python 2.6.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.runtests.run_all_tests">
-<tt class="descclassname">sympy.utilities.runtests.</tt><tt class="descname">run_all_tests</tt><big>(</big><em>test_args=()</em>, <em>test_kwargs={}</em>, <em>doctest_args=()</em>, <em>doctest_kwargs={}</em>, <em>examples_args=()</em>, <em>examples_kwargs={'quiet': True}</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/runtests.html#run_all_tests"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.runtests.run_all_tests" title="Permalink to this definition">¶</a></dt>
-<dd><p>Run all tests.</p>
-<p>Right now, this runs the regular tests (bin/test), the doctests
-(bin/doctest), the examples (examples/all.py), and the sage tests (see
-sympy/external/tests/test_sage.py).</p>
-<p>This is what <tt class="docutils literal"><span class="pre">setup.py</span> <span class="pre">test</span></tt> uses.</p>
-<p>You can pass arguments and keyword arguments to the test functions that
-support them (for now, test,  doctest, and the examples). See the
-docstrings of those functions for a description of the available options.</p>
-<p>For example, to run the solvers tests with colors turned off:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.runtests</span> <span class="kn">import</span> <span class="n">run_all_tests</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">run_all_tests</span><span class="p">(</span><span class="n">test_args</span><span class="o">=</span><span class="p">(</span><span class="s">&quot;solvers&quot;</span><span class="p">,),</span>
-<span class="gp">... </span><span class="n">test_kwargs</span><span class="o">=</span><span class="p">{</span><span class="s">&quot;colors:False&quot;</span><span class="p">})</span> 
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.runtests.run_in_subprocess_with_hash_randomization">
-<tt class="descclassname">sympy.utilities.runtests.</tt><tt class="descname">run_in_subprocess_with_hash_randomization</tt><big>(</big><em>function</em>, <em>function_args=()</em>, <em>function_kwargs={}</em>, <em>command='/sw/bin/python3.3'</em>, <em>module='sympy.utilities.runtests'</em>, <em>force=False</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/runtests.html#run_in_subprocess_with_hash_randomization"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.runtests.run_in_subprocess_with_hash_randomization" title="Permalink to this definition">¶</a></dt>
-<dd><p>Run a function in a Python subprocess with hash randomization enabled.</p>
-<p>If hash randomization is not supported by the version of Python given, it
-returns False.  Otherwise, it returns the exit value of the command.  The
-function is passed to sys.exit(), so the return value of the function will
-be the return value.</p>
-<p>The environment variable PYTHONHASHSEED is used to seed Python&#8217;s hash
-randomization.  If it is set, this function will return False, because
-starting a new subprocess is unnecessary in that case.  If it is not set,
-one is set at random, and the tests are run.  Note that if this
-environment variable is set when Python starts, hash randomization is
-automatically enabled.  To force a subprocess to be created even if
-PYTHONHASHSEED is set, pass <tt class="docutils literal"><span class="pre">force=True</span></tt>.  This flag will not force a
-subprocess in Python versions that do not support hash randomization (see
-below), because those versions of Python do not support the <tt class="docutils literal"><span class="pre">-R</span></tt> flag.</p>
-<p><tt class="docutils literal"><span class="pre">function</span></tt> should be a string name of a function that is importable from
-the module <tt class="docutils literal"><span class="pre">module</span></tt>, like &#8220;_test&#8221;.  The default for <tt class="docutils literal"><span class="pre">module</span></tt> is
-&#8220;sympy.utilities.runtests&#8221;.  <tt class="docutils literal"><span class="pre">function_args</span></tt> and <tt class="docutils literal"><span class="pre">function_kwargs</span></tt>
-should be a repr-able tuple and dict, respectively.  The default Python
-command is sys.executable, which is the currently running Python command.</p>
-<p>This function is necessary because the seed for hash randomization must be
-set by the environment variable before Python starts.  Hence, in order to
-use a predetermined seed for tests, we must start Python in a separate
-subprocess.</p>
-<p>Hash randomization was added in the minor Python versions 2.6.8, 2.7.3,
-3.1.5, and 3.2.3, and is enabled by default in all Python versions after
-and including 3.3.0.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.runtests</span> <span class="kn">import</span> <span class="p">(</span>
-<span class="gp">... </span><span class="n">run_in_subprocess_with_hash_randomization</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="c"># run the core tests in verbose mode</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">run_in_subprocess_with_hash_randomization</span><span class="p">(</span><span class="s">&quot;_test&quot;</span><span class="p">,</span>
-<span class="gp">... </span><span class="n">function_args</span><span class="o">=</span><span class="p">(</span><span class="s">&quot;core&quot;</span><span class="p">,),</span>
-<span class="gp">... </span><span class="n">function_kwargs</span><span class="o">=</span><span class="p">{</span><span class="s">&#39;verbose&#39;</span><span class="p">:</span> <span class="bp">True</span><span class="p">})</span> 
-<span class="go"># Will return 0 if sys.executable supports hash randomization and tests</span>
-<span class="go"># pass, 1 if they fail, and False if it does not support hash</span>
-<span class="go"># randomization.</span>
-</pre></div>
-</div>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.runtests.sympytestfile">
-<tt class="descclassname">sympy.utilities.runtests.</tt><tt class="descname">sympytestfile</tt><big>(</big><em>filename</em>, <em>module_relative=True</em>, <em>name=None</em>, <em>package=None</em>, <em>globs=None</em>, <em>verbose=None</em>, <em>report=True</em>, <em>optionflags=0</em>, <em>extraglobs=None</em>, <em>raise_on_error=False</em>, <em>parser=&lt;doctest.DocTestParser object at 0x10b5ff290&gt;</em>, <em>encoding=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/runtests.html#sympytestfile"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.runtests.sympytestfile" title="Permalink to this definition">¶</a></dt>
-<dd><p>Test examples in the given file.  Return (#failures, #tests).</p>
-<p>Optional keyword arg <tt class="docutils literal"><span class="pre">module_relative</span></tt> specifies how filenames
-should be interpreted:</p>
-<ul class="simple">
-<li>If <tt class="docutils literal"><span class="pre">module_relative</span></tt> is True (the default), then <tt class="docutils literal"><span class="pre">filename</span></tt>
-specifies a module-relative path.  By default, this path is
-relative to the calling module&#8217;s directory; but if the
-<tt class="docutils literal"><span class="pre">package</span></tt> argument is specified, then it is relative to that
-package.  To ensure os-independence, <tt class="docutils literal"><span class="pre">filename</span></tt> should use
-&#8220;/&#8221; characters to separate path segments, and should not
-be an absolute path (i.e., it may not begin with &#8220;/&#8221;).</li>
-<li>If <tt class="docutils literal"><span class="pre">module_relative</span></tt> is False, then <tt class="docutils literal"><span class="pre">filename</span></tt> specifies an
-os-specific path.  The path may be absolute or relative (to
-the current working directory).</li>
-</ul>
-<p>Optional keyword arg <tt class="docutils literal"><span class="pre">name</span></tt> gives the name of the test; by default
-use the file&#8217;s basename.</p>
-<p>Optional keyword argument <tt class="docutils literal"><span class="pre">package</span></tt> is a Python package or the
-name of a Python package whose directory should be used as the
-base directory for a module relative filename.  If no package is
-specified, then the calling module&#8217;s directory is used as the base
-directory for module relative filenames.  It is an error to
-specify <tt class="docutils literal"><span class="pre">package</span></tt> if <tt class="docutils literal"><span class="pre">module_relative</span></tt> is False.</p>
-<p>Optional keyword arg <tt class="docutils literal"><span class="pre">globs</span></tt> gives a dict to be used as the globals
-when executing examples; by default, use {}.  A copy of this dict
-is actually used for each docstring, so that each docstring&#8217;s
-examples start with a clean slate.</p>
-<p>Optional keyword arg <tt class="docutils literal"><span class="pre">extraglobs</span></tt> gives a dictionary that should be
-merged into the globals that are used to execute examples.  By
-default, no extra globals are used.</p>
-<p>Optional keyword arg <tt class="docutils literal"><span class="pre">verbose</span></tt> prints lots of stuff if true, prints
-only failures if false; by default, it&#8217;s true iff &#8220;-v&#8221; is in sys.argv.</p>
-<p>Optional keyword arg <tt class="docutils literal"><span class="pre">report</span></tt> prints a summary at the end when true,
-else prints nothing at the end.  In verbose mode, the summary is
-detailed, else very brief (in fact, empty if all tests passed).</p>
-<p>Optional keyword arg <tt class="docutils literal"><span class="pre">optionflags</span></tt> or&#8217;s together module constants,
-and defaults to 0.  Possible values (see the docs for details):</p>
-<ul class="simple">
-<li>DONT_ACCEPT_TRUE_FOR_1</li>
-<li>DONT_ACCEPT_BLANKLINE</li>
-<li>NORMALIZE_WHITESPACE</li>
-<li>ELLIPSIS</li>
-<li>SKIP</li>
-<li>IGNORE_EXCEPTION_DETAIL</li>
-<li>REPORT_UDIFF</li>
-<li>REPORT_CDIFF</li>
-<li>REPORT_NDIFF</li>
-<li>REPORT_ONLY_FIRST_FAILURE</li>
-</ul>
-<p>Optional keyword arg <tt class="docutils literal"><span class="pre">raise_on_error</span></tt> raises an exception on the
-first unexpected exception or failure. This allows failures to be
-post-mortem debugged.</p>
-<p>Optional keyword arg <tt class="docutils literal"><span class="pre">parser</span></tt> specifies a DocTestParser (or
-subclass) that should be used to extract tests from the files.</p>
-<p>Optional keyword arg <tt class="docutils literal"><span class="pre">encoding</span></tt> specifies an encoding that should
-be used to convert the file to unicode.</p>
-<p>Advanced tomfoolery:  testmod runs methods of a local instance of
-class doctest.Tester, then merges the results into (or creates)
-global Tester instance doctest.master.  Methods of doctest.master
-can be called directly too, if you want to do something unusual.
-Passing report=0 to testmod is especially useful then, to delay
-displaying a summary.  Invoke doctest.master.summarize(verbose)
-when you&#8217;re done fiddling.</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.runtests.test">
-<tt class="descclassname">sympy.utilities.runtests.</tt><tt class="descname">test</tt><big>(</big><em>*paths</em>, <em>**kwargs</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/runtests.html#test"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.runtests.test" title="Permalink to this definition">¶</a></dt>
-<dd><p>Run tests in the specified test_*.py files.</p>
-<p>Tests in a particular test_*.py file are run if any of the given strings
-in <tt class="docutils literal"><span class="pre">paths</span></tt> matches a part of the test file&#8217;s path. If <tt class="docutils literal"><span class="pre">paths=[]</span></tt>,
-tests in all test_*.py files are run.</p>
-<p>Notes:</p>
-<ul class="simple">
-<li>If sort=False, tests are run in random order (not default).</li>
-<li>Paths can be entered in native system format or in unix,
-forward-slash format.</li>
-</ul>
-<p><strong>Explanation of test results</strong></p>
-<table border="1" class="docutils">
-<colgroup>
-<col width="9%" />
-<col width="91%" />
-</colgroup>
-<thead valign="bottom">
-<tr class="row-odd"><th class="head">Output</th>
-<th class="head">Meaning</th>
-</tr>
-</thead>
-<tbody valign="top">
-<tr class="row-even"><td>.</td>
-<td>passed</td>
-</tr>
-<tr class="row-odd"><td>F</td>
-<td>failed</td>
-</tr>
-<tr class="row-even"><td>X</td>
-<td>XPassed (expected to fail but passed)</td>
-</tr>
-<tr class="row-odd"><td>f</td>
-<td>XFAILed (expected to fail and indeed failed)</td>
-</tr>
-<tr class="row-even"><td>s</td>
-<td>skipped</td>
-</tr>
-<tr class="row-odd"><td>w</td>
-<td>slow</td>
-</tr>
-<tr class="row-even"><td>T</td>
-<td>timeout (e.g., when <tt class="docutils literal"><span class="pre">--timeout</span></tt> is used)</td>
-</tr>
-<tr class="row-odd"><td>K</td>
-<td>KeyboardInterrupt (when running the slow tests with <tt class="docutils literal"><span class="pre">--slow</span></tt>,
-you can interrupt one of them without killing the test runner)</td>
-</tr>
-</tbody>
-</table>
-<p>Colors have no additional meaning and are used just to facilitate
-interpreting the output.</p>
-<p class="rubric">Examples</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">sympy</span>
-</pre></div>
-</div>
-<p>Run all tests:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sympy</span><span class="o">.</span><span class="n">test</span><span class="p">()</span>    
-</pre></div>
-</div>
-<p>Run one file:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sympy</span><span class="o">.</span><span class="n">test</span><span class="p">(</span><span class="s">&quot;sympy/core/tests/test_basic.py&quot;</span><span class="p">)</span>    
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sympy</span><span class="o">.</span><span class="n">test</span><span class="p">(</span><span class="s">&quot;_basic&quot;</span><span class="p">)</span>    
-</pre></div>
-</div>
-<p>Run all tests in sympy/functions/ and some particular file:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sympy</span><span class="o">.</span><span class="n">test</span><span class="p">(</span><span class="s">&quot;sympy/core/tests/test_basic.py&quot;</span><span class="p">,</span>
-<span class="gp">... </span>       <span class="s">&quot;sympy/functions&quot;</span><span class="p">)</span>    
-</pre></div>
-</div>
-<p>Run all tests in sympy/core and sympy/utilities:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sympy</span><span class="o">.</span><span class="n">test</span><span class="p">(</span><span class="s">&quot;/core&quot;</span><span class="p">,</span> <span class="s">&quot;/util&quot;</span><span class="p">)</span>    
-</pre></div>
-</div>
-<p>Run specific test from a file:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sympy</span><span class="o">.</span><span class="n">test</span><span class="p">(</span><span class="s">&quot;sympy/core/tests/test_basic.py&quot;</span><span class="p">,</span>
-<span class="gp">... </span>       <span class="n">kw</span><span class="o">=</span><span class="s">&quot;test_equality&quot;</span><span class="p">)</span>    
-</pre></div>
-</div>
-<p>Run specific test from any file:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sympy</span><span class="o">.</span><span class="n">test</span><span class="p">(</span><span class="n">kw</span><span class="o">=</span><span class="s">&quot;subs&quot;</span><span class="p">)</span>    
-</pre></div>
-</div>
-<p>Run the tests with verbose mode on:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sympy</span><span class="o">.</span><span class="n">test</span><span class="p">(</span><span class="n">verbose</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>    
-</pre></div>
-</div>
-<p>Don&#8217;t sort the test output:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sympy</span><span class="o">.</span><span class="n">test</span><span class="p">(</span><span class="n">sort</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>    
-</pre></div>
-</div>
-<p>Turn on post-mortem pdb:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sympy</span><span class="o">.</span><span class="n">test</span><span class="p">(</span><span class="n">pdb</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>    
-</pre></div>
-</div>
-<p>Turn off colors:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sympy</span><span class="o">.</span><span class="n">test</span><span class="p">(</span><span class="n">colors</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>    
-</pre></div>
-</div>
-<p>Force colors, even when the output is not to a terminal (this is useful,
-e.g., if you are piping to <tt class="docutils literal"><span class="pre">less</span> <span class="pre">-r</span></tt> and you still want colors)</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sympy</span><span class="o">.</span><span class="n">test</span><span class="p">(</span><span class="n">force_colors</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>    
-</pre></div>
-</div>
-<p>The traceback verboseness can be set to &#8220;short&#8221; or &#8220;no&#8221; (default is
-&#8220;short&#8221;)</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sympy</span><span class="o">.</span><span class="n">test</span><span class="p">(</span><span class="n">tb</span><span class="o">=</span><span class="s">&#39;no&#39;</span><span class="p">)</span>    
-</pre></div>
-</div>
-<p>You can disable running the tests in a separate subprocess using
-<tt class="docutils literal"><span class="pre">subprocess=False</span></tt>.  This is done to support seeding hash randomization,
-which is enabled by default in the Python versions where it is supported.
-If subprocess=False, hash randomization is enabled/disabled according to
-whether it has been enabled or not in the calling Python process.
-However, even if it is enabled, the seed cannot be printed unless it is
-called from a new Python process.</p>
-<p>Hash randomization was added in the minor Python versions 2.6.8, 2.7.3,
-3.1.5, and 3.2.3, and is enabled by default in all Python versions after
-and including 3.3.0.</p>
-<p>If hash randomization is not supported <tt class="docutils literal"><span class="pre">subprocess=False</span></tt> is used
-automatically.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">sympy</span><span class="o">.</span><span class="n">test</span><span class="p">(</span><span class="n">subprocess</span><span class="o">=</span><span class="bp">False</span><span class="p">)</span>     
-</pre></div>
-</div>
-<p>To set the hash randomization seed, set the environment variable
-<tt class="docutils literal"><span class="pre">PYTHONHASHSEED</span></tt> before running the tests.  This can be done from within
-Python using</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span> <span class="nn">os</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">os</span><span class="o">.</span><span class="n">environ</span><span class="p">[</span><span class="s">&#39;PYTHONHASHSEED&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="mi">42</span> 
-</pre></div>
-</div>
-<p>Or from the command line using</p>
-<p>$ PYTHONHASHSEED=42 ./bin/test</p>
-<p>If the seed is not set, a random seed will be chosen.</p>
-<p>Note that to reproduce the same hash values, you must use both the same as
-well as the same architecture (32-bit vs. 64-bit).</p>
-</dd></dl>
-
-</div>
-
-
-          </div>
-        </div>
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-              <img class="logo" src="../../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../utilities/randtest.html"
-                        title="previous chapter">Randomised Testing</a></p>
-  <h4>Next topic</h4>
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-                        title="next chapter">Source Code Inspection</a></p>
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-<span id="source-code-inspection"></span><h1>Source Code Inspection<a class="headerlink" href="#module-sympy.utilities.source" title="Permalink to this headline">¶</a></h1>
-<p>This module adds several functions for interactive source code inspection.</p>
-<dl class="function">
-<dt id="sympy.utilities.source.get_class">
-<tt class="descclassname">sympy.utilities.source.</tt><tt class="descname">get_class</tt><big>(</big><em>lookup_view</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/source.html#get_class"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.source.get_class" title="Permalink to this definition">¶</a></dt>
-<dd><p>Convert a string version of a class name to the object.</p>
-<p>For example, get_class(&#8216;sympy.core.Basic&#8217;) will return
-class Basic located in module sympy.core</p>
-</dd></dl>
-
-<dl class="function">
-<dt id="sympy.utilities.source.get_mod_func">
-<tt class="descclassname">sympy.utilities.source.</tt><tt class="descname">get_mod_func</tt><big>(</big><em>callback</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/source.html#get_mod_func"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.source.get_mod_func" title="Permalink to this definition">¶</a></dt>
-<dd><p>splits the string path to a class into a string path to the module
-and the name of the class. For example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.utilities.source</span> <span class="kn">import</span> <span class="n">get_mod_func</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">get_mod_func</span><span class="p">(</span><span class="s">&#39;sympy.core.basic.Basic&#39;</span><span class="p">)</span>
-<span class="go">(&#39;sympy.core.basic&#39;, &#39;Basic&#39;)</span>
-</pre></div>
-</div>
-</dd></dl>
-
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-<dt id="sympy.utilities.source.source">
-<tt class="descclassname">sympy.utilities.source.</tt><tt class="descname">source</tt><big>(</big><em>object</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/source.html#source"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.source.source" title="Permalink to this definition">¶</a></dt>
-<dd><p>Prints the source code of a given object.</p>
-</dd></dl>
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-<span id="timing-utilities"></span><h1>Timing Utilities<a class="headerlink" href="#module-sympy.utilities.timeutils" title="Permalink to this headline">¶</a></h1>
-<p>Simple tools for timing functions execution, when IPython is not
-available.</p>
-<dl class="function">
-<dt id="sympy.utilities.timeutils.timed">
-<tt class="descclassname">sympy.utilities.timeutils.</tt><tt class="descname">timed</tt><big>(</big><em>func</em>, <em>setup=None</em><big>)</big><a class="reference internal" href="../../_modules/sympy/utilities/timeutils.html#timed"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#sympy.utilities.timeutils.timed" title="Permalink to this definition">¶</a></dt>
-<dd><p>Adaptively measure execution time of a function.</p>
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diff --git a/dev-py3k/objects.inv b/dev-py3k/objects.inv
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-<h1>SymPy Papers<a class="headerlink" href="#sympy-papers" title="Permalink to this headline">¶</a></h1>
-<p>A list of papers which either use or mention SymPy, as well as presentations
-given about SymPy at conferences can be seen at <a class="reference external" href="https://github.com/sympy/sympy/wiki/SymPy-Papers">SymPy Papers</a>.</p>
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-<h1>Planet SymPy<a class="headerlink" href="#planet-sympy" title="Permalink to this headline">¶</a></h1>
-<p>We have a blog agregator at <a class="reference external" href="http://planet.sympy.org">http://planet.sympy.org</a></p>
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-<h1>SymPy logos<a class="headerlink" href="#sympy-logos" title="Permalink to this headline">¶</a></h1>
-<p>SymPy has a collection of official logos, which can be accessed here:</p>
-<p><a class="reference external" href="https://github.com/sympy/sympy/tree/master/doc/logo">https://github.com/sympy/sympy/tree/master/doc/logo</a></p>
-<p>The license of all the logos is the same as SymPy: BSD. See the LICENSE file in
-the trunk for more information.</p>
-</div>
-<div class="section" id="projects-using-sympy">
-<h1>Projects using SymPy<a class="headerlink" href="#projects-using-sympy" title="Permalink to this headline">¶</a></h1>
-<p>This is an (incomplete) list of projects that use SymPy. If you use SymPy in
-your project, please let us know on our <a class="reference external" href="http://groups.google.com/group/sympy">mailinglist</a>, so that we can add your
-project here as well.</p>
-<ul class="simple">
-<li><a class="reference external" href="http://sfepy.org/">SfePy</a> (simple finite elements in Python)</li>
-<li><a class="reference external" href="http://quameon.sourceforge.net/">Quameon</a> (Quantum Monte Carlo in Python)</li>
-</ul>
-</div>
-<div class="section" id="blogs-news-magazines">
-<h1>Blogs, News, Magazines<a class="headerlink" href="#blogs-news-magazines" title="Permalink to this headline">¶</a></h1>
-<p>You can see a list of blogs/news/magazines mentioning SymPy (that we know of) on
-our <a class="reference external" href="https://github.com/sympy/sympy/wiki/SymPy-in-the-news">wiki page</a>.</p>
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-  <ul>
-<li><a class="reference internal" href="#">SymPy Papers</a></li>
-<li><a class="reference internal" href="#planet-sympy">Planet SymPy</a></li>
-<li><a class="reference internal" href="#sympy-logos">SymPy logos</a></li>
-<li><a class="reference internal" href="#projects-using-sympy">Projects using SymPy</a></li>
-<li><a class="reference internal" href="#blogs-news-magazines">Blogs, News, Magazines</a></li>
-</ul>
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-            
-  <div class="section" id="development-tips-comparisons-in-python">
-<h1>Development Tips: Comparisons in Python<a class="headerlink" href="#development-tips-comparisons-in-python" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="introduction">
-<h2>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline">¶</a></h2>
-<p>When debugging comparisons and hashes in SymPy, it is necessary to understand
-when exactly Python calls each method.
-Unfortunately, the official Python documentation for this is
-not very detailed (see the docs for <a class="reference external" href="http://docs.python.org/dev/reference/datamodel.html#object.__lt__">rich comparison</a>,
-<a class="reference external" href="http://docs.python.org/dev/reference/datamodel.html#object.__cmp__">__cmp__()</a>
-and <a class="reference external" href="http://docs.python.org/dev/reference/datamodel.html#object.__hash__">__hash__()</a>
-methods).</p>
-<p>We wrote this guide to fill in the missing gaps. After reading it, you should
-be able to understand which methods do (and do not) get called and the order in
-which they are called.</p>
-</div>
-<div class="section" id="hashing">
-<h2>Hashing<a class="headerlink" href="#hashing" title="Permalink to this headline">¶</a></h2>
-<p>Every Python class has a <tt class="docutils literal"><span class="pre">__hash__()</span></tt> method, the default
-implementation of which is:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-    <span class="k">return</span> <span class="nb">id</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>You can reimplement it to return a different integer that you compute on your own.
-<tt class="docutils literal"><span class="pre">hash(x)</span></tt> just calls <tt class="docutils literal"><span class="pre">x.__hash__()</span></tt>. Python builtin classes usually redefine
-the <tt class="docutils literal"><span class="pre">__hash__()</span></tt> method. For example, an <tt class="docutils literal"><span class="pre">int</span></tt> has something like this:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-    <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>and a <tt class="docutils literal"><span class="pre">list</span></tt> does something like this:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-    <span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s">&quot;list objects are unhashable&quot;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>The general
-idea about hashes is that if two objects have a different hash, they are not
-equal, but if they have the same hash, they <em>might</em> be equal. (This is usually
-called a &#8220;hash collision&#8221; and you need to use the methods described in the
-next section to determine if the objects really are equal).</p>
-<p>The only requirement from the Python side is
-that the hash value mustn&#8217;t change after it is returned by the
-<tt class="docutils literal"><span class="pre">__hash__()</span></tt> method.</p>
-<p>Please be aware that hashing is <em>platform-dependent</em>. This means that you can
-get different hashes for the same SymPy object on different platforms. This
-affects for instance sorting of sympy expressions. You can also get SymPy
-objects printed in different order.</p>
-<p>When developing, you have to be careful about this, especially when writing
-tests. It is possible that your test runs on a 32-bit platform, but not on
-64-bit. An example:</p>
-<div class="highlight-python"><pre>&gt;&gt; from sympy import *
-&gt;&gt; x = Symbol(&#x27;x&#x27;)
-&gt;&gt; r = rootfinding.roots_quartic(Poly(x**4 - 6*x**3 + 17*x**2 - 26*x + 20, x))
-&gt;&gt; [i.evalf(2) for i in r]
-[1.0 + 1.7*I, 2.0 - 1.0*I, 2.0 + I, 1.0 - 1.7*I]</pre>
-</div>
-<p>If you get this order of solutions, you are probably running 32-bit system.
-On a 64-bit system you would get the following:</p>
-<div class="highlight-python"><pre>&gt;&gt; [i.evalf(2) for i in r]
-[1.0 - 1.7*I, 1.0 + 1.7*I, 2.0 + I, 2.0 - 1.0*I</pre>
-</div>
-<p>When you now write a test like this:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">r</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">r</span><span class="p">]</span>
-<span class="k">assert</span> <span class="n">r</span> <span class="o">==</span> <span class="p">[</span><span class="mf">1.0</span> <span class="o">+</span> <span class="mf">1.7</span><span class="o">*</span><span class="n">I</span><span class="p">,</span> <span class="mf">2.0</span> <span class="o">-</span> <span class="mf">1.0</span><span class="o">*</span><span class="n">I</span><span class="p">,</span> <span class="mf">2.0</span> <span class="o">+</span> <span class="n">I</span><span class="p">,</span> <span class="mf">1.0</span> <span class="o">-</span> <span class="mf">1.7</span><span class="o">*</span><span class="n">I</span><span class="p">]</span>
-</pre></div>
-</div>
-<p>it will fail on a 64-bit platforms, even if it works for your 32-bit system. You can
-avoid this by using the <tt class="docutils literal"><span class="pre">sorted()</span></tt> or <tt class="docutils literal"><span class="pre">set()</span></tt> Python built-in:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">r</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span><span class="o">.</span><span class="n">evalf</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">r</span><span class="p">]</span>
-<span class="k">assert</span> <span class="nb">set</span><span class="p">(</span><span class="n">r</span><span class="p">)</span> <span class="o">==</span> <span class="nb">set</span><span class="p">([</span><span class="mf">1.0</span> <span class="o">+</span> <span class="mf">1.7</span><span class="o">*</span><span class="n">I</span><span class="p">,</span> <span class="mf">2.0</span> <span class="o">-</span> <span class="mf">1.0</span><span class="o">*</span><span class="n">I</span><span class="p">,</span> <span class="mf">2.0</span> <span class="o">+</span> <span class="n">I</span><span class="p">,</span> <span class="mf">1.0</span> <span class="o">-</span> <span class="mf">1.7</span><span class="o">*</span><span class="n">I</span><span class="p">])</span>
-</pre></div>
-</div>
-<p>This approach does not work for doctests since they always compare strings that would
-be printed after a prompt. In that case you could make your test print results using
-a combination of <tt class="docutils literal"><span class="pre">str()</span></tt> and <tt class="docutils literal"><span class="pre">sorted()</span></tt>:</p>
-<div class="highlight-python"><pre>&gt;&gt; sorted([str(i.evalf(2)) for i in r])
-[&#x27;1.0 + 1.7*I&#x27;, &#x27;1.0 - 1.7*I&#x27;, &#x27;2.0 + I&#x27;, &#x27;2.0 - 1.0*I&#x27;]</pre>
-</div>
-<p>or, if you don&#8217;t want to show the values as strings, then sympify the results or the
-sorted list:</p>
-<div class="highlight-python"><pre>&gt;&gt; [S(s) for s in sorted([str(i.evalf(2)) for i in r])]
-[1.0 + 1.7*I, 1.0 - 1.7*I, 2.0 + I, 2.0 - I]</pre>
-</div>
-<p>The printing of SymPy expressions might be also affected, so be careful
-with doctests. If you get the following on a 32-bit system:</p>
-<div class="highlight-python"><pre>&gt;&gt; print dsolve(f(x).diff(x, 2) + 2*f(x).diff(x) - f(x), f(x))
-f(x) == C1*exp(-x + x*sqrt(2)) + C2*exp(-x - x*sqrt(2))</pre>
-</div>
-<p>you might get the following on a 64-bit platform:</p>
-<div class="highlight-python"><pre>&gt;&gt; print dsolve(f(x).diff(x, 2) + 2*f(x).diff(x) - f(x), f(x))
-f(x) == C1*exp(-x - x*sqrt(2)) + C2*exp(-x + x*sqrt(2))</pre>
-</div>
-</div>
-<div class="section" id="method-resolution">
-<h2>Method Resolution<a class="headerlink" href="#method-resolution" title="Permalink to this headline">¶</a></h2>
-<p>Let <tt class="docutils literal"><span class="pre">a</span></tt>, <tt class="docutils literal"><span class="pre">b</span></tt> and <tt class="docutils literal"><span class="pre">c</span></tt> be instances of any one of the Python classes.
-As can be easily checked by the <a class="reference internal" href="#python-script">python script</a> at the end of this guide,
-if you write:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">a</span> <span class="o">==</span> <span class="n">b</span>
-</pre></div>
-</div>
-<p>Python calls the following &#8211; in this order:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">a</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-<span class="n">b</span><span class="o">.</span><span class="n">__eq__</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="n">a</span><span class="o">.</span><span class="n">__cmp__</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-<span class="n">b</span><span class="o">.</span><span class="n">__cmp__</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="nb">id</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">==</span> <span class="nb">id</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>If a particular method is not implemented (or a method
-returns <tt class="docutils literal"><span class="pre">NotImplemented</span></tt> <a class="footnote-reference" href="#id2" id="id1">[1]</a>) Python skips it
-and tries the next one until it succeeds (i.e. until the method returns something
-meaningful). The last line is a catch-all method that always succeeds.</p>
-<p>If you write:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">a</span> <span class="o">!=</span> <span class="n">b</span>
-</pre></div>
-</div>
-<p>Python tries to call:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">a</span><span class="o">.</span><span class="n">__ne__</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-<span class="n">b</span><span class="o">.</span><span class="n">__ne__</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="n">a</span><span class="o">.</span><span class="n">__cmp__</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-<span class="n">b</span><span class="o">.</span><span class="n">__cmp__</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="nb">id</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">==</span> <span class="nb">id</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>If you write:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">a</span> <span class="o">&lt;</span> <span class="n">b</span>
-</pre></div>
-</div>
-<p>Python tries to call:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">a</span><span class="o">.</span><span class="n">__lt__</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-<span class="n">b</span><span class="o">.</span><span class="n">__gt__</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="n">a</span><span class="o">.</span><span class="n">__cmp__</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-<span class="n">b</span><span class="o">.</span><span class="n">__cmp__</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="nb">id</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">&lt;</span> <span class="nb">id</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>If you write:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">a</span> <span class="o">&lt;=</span> <span class="n">b</span>
-</pre></div>
-</div>
-<p>Python tries to call:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">a</span><span class="o">.</span><span class="n">__le__</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-<span class="n">b</span><span class="o">.</span><span class="n">__ge__</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="n">a</span><span class="o">.</span><span class="n">__cmp__</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-<span class="n">b</span><span class="o">.</span><span class="n">__cmp__</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
-<span class="nb">id</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="nb">id</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>And similarly for <tt class="docutils literal"><span class="pre">a</span> <span class="pre">&gt;</span> <span class="pre">b</span></tt> and <tt class="docutils literal"><span class="pre">a</span> <span class="pre">&gt;=</span> <span class="pre">b</span></tt>.</p>
-<p>If you write:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="nb">sorted</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">])</span>
-</pre></div>
-</div>
-<p>Python calls the same chain of methods as for the <tt class="docutils literal"><span class="pre">b</span> <span class="pre">&lt;</span> <span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">c</span> <span class="pre">&lt;</span> <span class="pre">b</span></tt>
-comparisons.</p>
-<p>If you write any of the following:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">a</span> <span class="ow">in</span> <span class="p">{</span><span class="n">d</span><span class="p">:</span> <span class="mi">5</span><span class="p">}</span>
-<span class="n">a</span> <span class="ow">in</span> <span class="nb">set</span><span class="p">([</span><span class="n">d</span><span class="p">,</span> <span class="n">d</span><span class="p">,</span> <span class="n">d</span><span class="p">])</span>
-<span class="nb">set</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">])</span> <span class="o">==</span> <span class="nb">set</span><span class="p">([</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">])</span>
-</pre></div>
-</div>
-<p>Python first compares hashes, e.g.:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="n">a</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-<span class="n">d</span><span class="o">.</span><span class="n">__hash__</span><span class="p">()</span>
-</pre></div>
-</div>
-<p>If <tt class="docutils literal"><span class="pre">hash(a)</span> <span class="pre">!=</span> <span class="pre">hash(d)</span></tt> then the result of the statement <tt class="docutils literal"><span class="pre">a</span> <span class="pre">in</span> <span class="pre">{d:</span> <span class="pre">5}</span></tt> is
-immediately <tt class="docutils literal"><span class="pre">False</span></tt> (remember how hashes work in general). If
-<tt class="docutils literal"><span class="pre">hash(a)</span> <span class="pre">==</span> <span class="pre">hash(d)</span></tt>) Python goes through the method resolution of the
-<tt class="docutils literal"><span class="pre">==</span></tt> operator as shown above.</p>
-</div>
-<div class="section" id="general-notes-and-caveats">
-<h2>General Notes and Caveats<a class="headerlink" href="#general-notes-and-caveats" title="Permalink to this headline">¶</a></h2>
-<p>In the method resolution for <tt class="docutils literal"><span class="pre">&lt;</span></tt>, <tt class="docutils literal"><span class="pre">&lt;=</span></tt>, <tt class="docutils literal"><span class="pre">==</span></tt>, <tt class="docutils literal"><span class="pre">!=</span></tt>, <tt class="docutils literal"><span class="pre">&gt;=</span></tt>, <tt class="docutils literal"><span class="pre">&gt;</span></tt> and
-<tt class="docutils literal"><span class="pre">sorted([a,</span> <span class="pre">b,</span> <span class="pre">c])</span></tt> operators the <tt class="docutils literal"><span class="pre">__hash__()</span></tt> method is <em>not</em> called, so
-in these cases it doesn&#8217;t matter what it returns. The <tt class="docutils literal"><span class="pre">__hash__()</span></tt> method is
-only called for sets and dictionaries.</p>
-<p>In the official Python documentation you can read about <a class="reference external" href="http://docs.python.org/dev/glossary.html#term-hashable">hashable and
-non-hashable</a> objects.
-In reality, you don&#8217;t have to think about it, you just follow the method
-resolution described here. E.g. if you try to use lists as dictionary keys, the
-list&#8217;s <tt class="docutils literal"><span class="pre">__hash__()</span></tt> method will be called and it returns an exception.</p>
-<p>In SymPy, every instance of any subclass of <tt class="docutils literal"><span class="pre">Basic</span></tt> is
-immutable.  Technically this means, that it&#8217;s behavior through all the methods
-above mustn&#8217;t change once the instance is created. Especially, the hash value
-mustn&#8217;t change (as already stated above) or else objects will get mixed up in
-dictionaries and wrong values will be returned for a given key, etc....</p>
-</div>
-<div class="section" id="script-to-verify-this-guide">
-<span id="python-script"></span><h2>Script To Verify This Guide<a class="headerlink" href="#script-to-verify-this-guide" title="Permalink to this headline">¶</a></h2>
-<p>The above method resolution can be verified using the following program:</p>
-<div class="highlight-python"><pre>class A(object):
-
-    def __init__(self, a, hash):
-        self.a = a
-        self._hash = hash
-
-    def __lt__(self, o):
-        print &quot;%s.__lt__(%s)&quot; % (self.a, o.a)
-        return NotImplemented
-
-    def __le__(self, o):
-        print &quot;%s.__le__(%s)&quot; % (self.a, o.a)
-        return NotImplemented
-
-    def __gt__(self, o):
-        print &quot;%s.__gt__(%s)&quot; % (self.a, o.a)
-        return NotImplemented
-
-    def __ge__(self, o):
-        print &quot;%s.__ge__(%s)&quot; % (self.a, o.a)
-        return NotImplemented
-
-    def __cmp__(self, o):
-        print &quot;%s.__cmp__(%s)&quot; % (self.a, o.a)
-        #return cmp(self._hash, o._hash)
-        return NotImplemented
-
-    def __eq__(self, o):
-        print &quot;%s.__eq__(%s)&quot; % (self.a, o.a)
-        return NotImplemented
-
-    def __ne__(self, o):
-        print &quot;%s.__ne__(%s)&quot; % (self.a, o.a)
-        return NotImplemented
-
-    def __hash__(self):
-        print &quot;%s.__hash__()&quot; % (self.a)
-        return self._hash
-
-def show(s):
-    print &quot;--- %s &quot; % s + &quot;-&quot;*40
-    eval(s)
-
-a = A(&quot;a&quot;, 1)
-b = A(&quot;b&quot;, 2)
-c = A(&quot;c&quot;, 3)
-d = A(&quot;d&quot;, 1)
-
-show(&quot;a == b&quot;)
-show(&quot;a != b&quot;)
-show(&quot;a &lt; b&quot;)
-show(&quot;a &lt;= b&quot;)
-show(&quot;a &gt; b&quot;)
-show(&quot;a &gt;= b&quot;)
-show(&quot;sorted([a, b, c])&quot;)
-show(&quot;{d: 5}&quot;)
-show(&quot;a in {d: 5}&quot;)
-show(&quot;set([d, d, d])&quot;)
-show(&quot;a in set([d, d, d])&quot;)
-show(&quot;set([a, b])&quot;)
-
-print &quot;--- x = set([a, b]); y = set([a, b]); ---&quot;
-x = set([a, b])
-y = set([a, b])
-print &quot;               x == y :&quot;
-x == y
-
-print &quot;--- x = set([a, b]); y = set([b, d]); ---&quot;
-x = set([a, b])
-y = set([b, d])
-print &quot;               x == y :&quot;
-x == y</pre>
-</div>
-<p>and its output:</p>
-<div class="highlight-python"><pre>--- a == b ----------------------------------------
-a.__eq__(b)
-b.__eq__(a)
-a.__cmp__(b)
-b.__cmp__(a)
---- a != b ----------------------------------------
-a.__ne__(b)
-b.__ne__(a)
-a.__cmp__(b)
-b.__cmp__(a)
---- a &lt; b ----------------------------------------
-a.__lt__(b)
-b.__gt__(a)
-a.__cmp__(b)
-b.__cmp__(a)
---- a &lt;= b ----------------------------------------
-a.__le__(b)
-b.__ge__(a)
-a.__cmp__(b)
-b.__cmp__(a)
---- a &gt; b ----------------------------------------
-a.__gt__(b)
-b.__lt__(a)
-a.__cmp__(b)
-b.__cmp__(a)
---- a &gt;= b ----------------------------------------
-a.__ge__(b)
-b.__le__(a)
-a.__cmp__(b)
-b.__cmp__(a)
---- sorted([a, b, c]) ----------------------------------------
-b.__lt__(a)
-a.__gt__(b)
-b.__cmp__(a)
-a.__cmp__(b)
-c.__lt__(b)
-b.__gt__(c)
-c.__cmp__(b)
-b.__cmp__(c)
---- {d: 5} ----------------------------------------
-d.__hash__()
---- a in {d: 5} ----------------------------------------
-d.__hash__()
-a.__hash__()
-d.__eq__(a)
-a.__eq__(d)
-d.__cmp__(a)
-a.__cmp__(d)
---- set([d, d, d]) ----------------------------------------
-d.__hash__()
-d.__hash__()
-d.__hash__()
---- a in set([d, d, d]) ----------------------------------------
-d.__hash__()
-d.__hash__()
-d.__hash__()
-a.__hash__()
-d.__eq__(a)
-a.__eq__(d)
-d.__cmp__(a)
-a.__cmp__(d)
---- set([a, b]) ----------------------------------------
-a.__hash__()
-b.__hash__()
---- x = set([a, b]); y = set([a, b]); ---
-a.__hash__()
-b.__hash__()
-a.__hash__()
-b.__hash__()
-               x == y :
---- x = set([a, b]); y = set([b, d]); ---
-a.__hash__()
-b.__hash__()
-b.__hash__()
-d.__hash__()
-               x == y :
-d.__eq__(a)
-a.__eq__(d)
-d.__cmp__(a)
-a.__cmp__(d)</pre>
-</div>
-<hr class="docutils" />
-<table class="docutils footnote" frame="void" id="id2" rules="none">
-<colgroup><col class="label" /><col /></colgroup>
-<tbody valign="top">
-<tr><td class="label"><a class="fn-backref" href="#id1">[1]</a></td><td><p class="first">There is also the similar <tt class="docutils literal"><span class="pre">NotImplementedError</span></tt> exception, which one may
-be tempted to raise to obtain the same effect as returning
-<tt class="docutils literal"><span class="pre">NotImplemented</span></tt>.</p>
-<p>But these are <strong>not</strong> the same, and Python will completely ignore
-<tt class="docutils literal"><span class="pre">NotImplementedError</span></tt> with respect to choosing appropriate comparison
-method, and will just propagate this exception upwards, to the caller.</p>
-<p class="last">So <tt class="docutils literal"><span class="pre">return</span> <span class="pre">NotImplemented</span></tt> is not the same as <tt class="docutils literal"><span class="pre">raise</span> <span class="pre">NotImplementedError</span></tt>.</p>
-</td></tr>
-</tbody>
-</table>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="index.html">
-              <img class="logo" src="_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Development Tips: Comparisons in Python</a><ul>
-<li><a class="reference internal" href="#introduction">Introduction</a></li>
-<li><a class="reference internal" href="#hashing">Hashing</a></li>
-<li><a class="reference internal" href="#method-resolution">Method Resolution</a></li>
-<li><a class="reference internal" href="#general-notes-and-caveats">General Notes and Caveats</a></li>
-<li><a class="reference internal" href="#script-to-verify-this-guide">Script To Verify This Guide</a></li>
-</ul>
-</li>
-</ul>
-
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f:99,platon:3,three:[1,40,133,42,108,45,3,4,53,9,109,12,57,118,105,18,106,60,61,146,23,67,25,161,15,163,54,30,90,164,113,123,124,169,82,38,39],anal:30,polish:[140,119],classof:43,prettyform:164,kluwer:108,digamma:[67,163,30],a2pt_theori:[129,12,42],"\u2082":164,kellei:140,voss:140,"101124983878j":67,cyclotom:[53,21,66,6,28,38,172],inverse_cosine_transform:112,straw:140,jxketcoupl:169,tab:[77,161],j0zero:47,tag:164,expand_func:[163,106,54],unhash:95,tensor_gen:77,tan:[40,140,12,54,14,152,112,45,164,128,69,172,83,149],"\u03bc":164,whole:[163,124,152,64,90,113,50,18,15],disco:21,tar:[131,24,45],coeffic:108,memoiz:[5,98,147,121],tau:[40,164,23,161],set:[0,1,2,3,4,8,9,11,12,13,18,19,22,23,25,5,90,33,38,42,43,44,45,47,53,54,55,56,57,59,60,61,62,64,65,66,67,70,72,75,76,78,80,86,87,89,31,68,95,96,99,100,102,105,106,131,108,109,110,111,112,113,115,117,15,124,129,73,85,133,134,136,137,140,142,144,145,147,148,149,152,157,158,159,161,162,163,164,167,168,172],ls2:162,sdp_ab:38,betaprimedistribut:106,kanesmethod:[0,68,12],mayer:134,setupegg:24,hadamard:[32,43,74,137],ubuntu:[131,164],meta:[152,54],gf_monic:38,densetool:38,dup_real_imag:38,as_coeff_expon:54,fast:[85,14,43,47,4,6,53,11,54,13,97,57,111,19,131,134,152,21,22,66,117,15,119,64,123,35,38],sigmoid:69,pick:[18,74,122],transposit:[73,43,33,37,86],axiom:[9,108,124],symmetri:[40,123,43,146,171,33,76,163,67,3,84,117,97,77],cachit:140,lusolv:[43,57],von:[53,123,21,106],vol:[97,30,55,43,90,163,7,86],dmp_prem:38,regardless:[134,133,55,112,4,86,119],spheric:[163,146,108,45,113,78,28,83],labeledtre:160,ode_1st_homogeneous_coeff_subs_dep_div_indep:152,"0447701735209082j":47,along:[0,85,43,45,47,4,50,53,9,68,54,55,56,18,148,60,109,58,169,67,156,113,5,97,117,15,163,12,40,123,128,37,39],omicron:[164,161],mti:43,start_view:164,vec:[132,43,38],upward:95,referencefram:[0,132,68,109,12,42,58,56,87,89,158,127,129,39],libtcc:144,rotate_right:55,winter:123,"______":80,gammaprod:[67,136],lambda:[40,85,133,14,86,43,31,4,136,53,9,140,11,12,13,117,171,164,17,106,134,152,65,154,67,5,97,96,15,119,75,82,54,47,32,39,123,124,78,37,38,55],aritrari:161,fault:152,joseph:140,signifi:[9,54],manipul:[133,42,108,6,9,142,147,43,60,58,21,66,115,27,70,29,164,134,35,38,55,172],confus:[123,54,109,171,67,167,136],contagi:43,"_xi_1":54,slowest:112,ration:[133,43,45,90,4,136,6,51,140,54,117,171,57,118,103,18,21,149,107,108,60,61,62,110,59,112,23,66,67,157,161,72,15,155,75,163,64,106,123,164,167,78,38,97,146,172],unfair:106,hilbert_spac:[116,138,29],nakagami_distribut:106,omega:[9,139,164,58,171,111,87,106,47,129,161,49,39],cannot:[1,85,133,42,86,43,111,45,31,136,142,109,54,171,99,17,105,18,64,19,131,134,61,62,152,59,112,113,159,161,162,164,39,123,128,38,55],awar:[54,90,14,62,161,95],andrej:140,nome:40,orthon:108,unabl:[18,4],mail:[164,24],gianni:[38,66,57],main:[43,45,47,109,12,152,105,111,131,108,61,110,64,65,158,161,29,54,164,123,124,80],necessarili:[75,112,133,54,152,108,57,124,5,73,118],linear:[0,86,43,87,45,89,47,136,6,68,12,142,56,57,152,18,147,148,153,108,61,62,58,59,22,66,113,140,115,73,29,75,163,90,122,35,169,112,38,172],"_3mm":115,"20000000j":136,dirac_eq:108,conjug:[9,163,141,164,152,97,43,44,123,31,25,159,102,103,116,5,83,149],workflow:144,luca:[167,140,83],edit:[134,30,21,112,66,45,67,140,38],safe:[75,61,142,134,76,148],awai:[164,97,61,12,115],andrew:[18,140],distributedmodul:38,against:[5,43,54],factor:[40,85,133,86,43,90,136,6,137,9,10,74,54,13,97,57,102,103,59,108,61,62,152,21,154,66,5,118,119,163,134,76,106,123,124,167,79,73,38,171,172],expint:[171,163,4,136],pic:115,weibul:106,val_dict:12,unrank_trotterjohnson:86,jha:140,acquir:88,"1e300j":67,rapid:117,matchdict:152,ted:140,xbra:[1,48,145],regula:119,"58778525229247312917j":25,dmp_eject:38,gammainc:[13,171],"711508683721355528322567e":97,"844952708937228j":123,"49j":136,creation:[0,109,12,42,164,161,64,43,87,89,102,36,145,80,106,137,138],realiz:[124,64,161,106,150],varianc:[155,171,106],libari:128,symp:58,sdp_lt:38,cookbook:45,"0215277159194482j":73,only_integ:113,signatur:[65,54,120,86,108,157,38,148],ashwini:140,variant:[123,152,28,38,119],symb:[164,43],basis_st:32,"168906959966564350754813j":128,calculu:[41,109,54,135,2,110,142,22,45,89,47,108,167,112,119],get_seg:113,screen:[99,64,109],randomis:[98,147,51],repr:[61,164,122,99,148,19],distanc:[7,86,108,56,156,115,103,129,18,97],laplace_transform:112,riemannr:53,polificationfail:38,inradiu:18,is_q_creat:102,cauchy_distribut:106,put:[133,43,90,10,11,54,79,100,19,149,108,152,64,65,66,24,77,161,162,74,164,169],york:[90,153,163,57],master:[99,24,55,143],sift:[9,55,168],elbenfuerst:140,g10:123,g11:123,author:[134,146,171,108,89,140,116,105],auxiliari:[0,112,68],gamma_distribut:106,dmp_pexquo:38,pareto_distribut:106,nanjangud:[89,140],tempa:12,tempb:12,tempc:12,staticmethod:[99,107,108,110],absorpt:152,wouldn:43,meaning:[95,107,115,110],who:[164,109,61],volker:38,smith:140,travers:[133,54,55,134,66,147],engin:[75,153,30,161],r61:163,str_format:23,createfermion:102,n_adj:86,mantissa:[5,61],nasa:123,remark:[107,152,90,110,117,172],atiyah69:[124,21],everyon:64,colin:86,plana:4,html:[86,134,47,4,136,7,53,54,55,18,106,149,131,112,152,21,65,67,160,97,75,163,3,164,123,167,171],conflict:[144,149],plane:[40,112,60,12,42,171,108,123,163,109,136,128,5,82,97],"\u015bwirski":140,basicmeth:64,nikla:140,bitlist_from_subset:104,must:[0,1,133,85,86,43,88,131,90,4,40,144,9,152,68,11,12,13,55,57,99,164,102,110,17,105,18,147,106,126,107,108,60,61,58,64,65,67,25,140,109,160,161,5,117,15,119,75,163,54,120,113,76,148,123,124,77,167,168,169,80,112,134,39,146],"539991603850462856j":136,get_permut:102,hyperexpand:[90,163,60,54,133],finitedomain:106,wigner3j:84,prinicip:124,texliv:164,rcollect:133,correspond:[1,133,126,86,43,111,45,31,4,136,49,144,9,68,54,55,57,145,104,18,106,19,108,60,61,139,152,59,155,113,77,159,115,117,12,90,134,76,122,123,124,78,169,80,73,38,97,160],johansson:140,jbase:108,set_global_set:164,split_super_sub:164,"24851410674842e":67,chebyshev_polynomi:163,firstli:[90,105],realist:134,mutual:[60,109,78],nonconverg:123,aim:[124,90,60,45],popov:140,dmp_terms_gcd:38,gf_compos:38,sfu:[75,163,30,171,123,136,97],"4763860949415867162066j":15,zetazero:123,ruskei:86,redund:[9,38,144,61,168],topolog:[50,55,60],forc:[0,131,68,138,54,12,133,62,149,112,56,57,99,109,113,4,36,119,60],issu:[40,45,89,4,144,140,54,55,56,57,99,18,21,131,112,61,146,59,22,24,161,117,15,75,30,64,164,166,127],"0d0":164,compil:[107,131,22,133,61,164,65,56,99,140,80,105,144],"2464022641351420167819697j":97,set_modulu:66,normalpspac:106,prateek:140,docherti:140,bodyfram:12,scheme:[62,146,86,66,47,113,38],delta__j_k:105,rect_0:50,"_print_foo":164,definit:[40,85,42,43,45,4,136,50,144,53,9,109,12,171,162,18,111,19,149,108,60,61,58,64,65,66,67,161,5,73,15,163,54,30,164,134,123,124,167,78,128,112,38],your:[0,131,93,109,54,95,152,64,43,56,45,24,113,143,161,105,51,8,111,164,133],symarrai:[43,140],unsuccess:[9,168,45],definin:1,run_all_test:99,"60784095465201e":67,"\u1d64":164,covarderivativeop:50,"521291677592745527851168e":40,"189190365227034385898282e":136,x_00:37,x_01:37,is_full_modul:124,compileflag:[99,161],"275758229007902299823821e":136,below:[0,133,43,45,47,4,136,144,9,68,11,12,99,100,102,16,152,18,148,134,60,109,146,64,112,23,24,113,158,115,161,75,163,54,164,123,124,83],paprocki:140,ode_ord:[152,140],"0x112ca5440":126,immutablematrix:[85,43,31,35,159,37],slowli:[40,62,97,47,4,136,117,111,119],mesh:113,includ:[0,40,133,86,43,45,90,136,108,5,6,50,137,138,53,93,109,12,169,171,56,57,99,100,144,152,18,148,126,131,172,65,61,62,110,111,22,23,66,67,140,26,161,112,27,28,117,15,29,155,119,54,120,134,154,122,123,164,77,35,78,128,69,37,38,97,115],unrecogn:5,componentwis:124,latexdemo:105,monitor:5,deltafunct:[112,163],handbook:[9,163,168,30],test_formula:90,mess:[116,152,66],factori:[133,14,43,45,136,53,54,55,62,146,59,22,154,66,67,28,15,163,120,38,164,167,83],inclus:[92,111,152,124],accurate_smal:136,bspline_basis_set:163,uniformli:[120,113,18],kitouni:140,"return":[0,1,14,3,4,7,8,9,10,11,12,13,15,16,17,18,19,20,23,25,26,5,29,32,33,34,37,38,39,40,41,43,44,45,47,49,50,51,53,54,55,57,59,61,62,58,64,65,66,67,68,70,96,74,75,76,78,80,82,83,86,87,88,31,93,95,97,99,100,102,103,104,105,106,107,108,109,110,111,112,113,115,116,117,118,119,122,123,124,126,128,129,73,85,132,133,134,136,137,138,139,144,145,148,149,150,151,152,155,156,157,77,159,160,161,163,164,167,168,169,170,79,172],factor_:134,tech:43,howev:[133,43,45,90,4,50,53,9,140,109,54,13,171,57,99,100,102,152,111,149,107,134,60,61,110,59,65,66,113,77,161,5,119,75,163,47,164,148,124,167,128,38,97,172],"135483865314736098540939e":15,frac_unifi:38,subsequ:[55,38,66,54,161],tempdir:65,readabl:[93,164,111,56,88,115,172],occup:102,occur:[61,75,134,60,54,42,62,171,43,56,57,4,86,161,136,5,83,55,133],what:[133,42,86,43,140,87,45,90,136,144,93,109,54,95,57,99,100,102,17,18,147,111,19,108,60,61,151,152,64,112,68,160,161,75,163,134,51,124,167,80,38,39,172],"013370403397787023602j":123,"585621643835618941044855j":97,"82752982813540993502j":136,cot:[82,54,164,112,123,128,28,83,149],"_eval_conjug":64,liao:21,deltaintegr:112,cox:[21,66],init_print:[43,62,112,45,155,124,118,172],is_cyclotom:[38,66],stabil:[62,9,77,168,76],my_funct:64,"6141851416j":136,dup_random:38,hermite_polynomi:163,col:[31,73,43,85],com:[85,133,134,45,47,4,136,53,9,54,97,143,152,18,106,149,131,112,60,62,146,21,22,67,24,160,163,30,3,123,167,55,171],productdomain:106,eberspäch:140,integ:[40,131,133,86,43,111,44,45,3,4,136,106,8,91,144,53,9,152,97,118,74,54,13,72,171,57,102,146,167,65,147,64,148,107,20,60,61,62,110,59,22,154,66,67,24,113,140,160,161,5,28,96,105,15,29,75,163,90,164,134,157,76,122,123,95,77,108,78,169,80,149,38,55,172],"_state_to_oper":145,inverse_hankel_transform:112,read:[60,61,42,95,64,22,164,45,109,4,88,147,86],convent:[1,40,108,9,54,169,97,103,105,18,61,152,112,156,113,77,5,15,12,164,122,78,128],"71130938718796e":171,conveni:[40,132,133,42,14,86,45,4,136,53,11,54,57,148,65,109,58,111,112,155,113,161,5,96,15,163,164,76,123,124,167,80,73],theori:[40,84,86,108,44,47,9,140,142,102,147,20,21,153,67,68,115,30,134,123,167,168,169,129,172,38,39,171],densiti:[155,171,140,106],wendi:86,twoqubitg:137,subpackag:131,biject:[124,86,160],express:[0,1,14,6,8,15,16,17,18,126,20,23,29,90,35,36,37,38,39,40,41,43,45,50,53,54,55,56,57,59,60,62,58,64,65,66,67,70,72,75,76,80,84,86,87,89,31,91,95,96,97,100,101,102,105,106,107,108,109,110,111,112,113,114,118,120,123,124,132,133,134,136,137,138,140,141,171,144,147,149,151,152,155,157,158,159,161,163,164,167,169,170,142,172],furthermor:[38,54],"2660553j":97,digit_2txt:164,orient:[109,12,42,58,55,56,129,18,38,39],futur:[99,12,155,60,54,172,147,64,108,56,89,113,102,127,161,80,5,18,117,133],"59e":73,i_1:[157,66,102],dvi:[164,105,23],topological_sort:55,said:[134,109,42],mangoldt:[53,123],notinvert:[38,66],"078258353474186729184421e":171,forward:[92,75,12,42,65,99,136],to_cnf:70,terms_gcd:[38,66,54],"17e":73,ode_1st_homogeneous_coeff_best:152,lineover1drangeseri:113,base1:77,continued_fract:74,bateman:163,construt:29,php:9,reduce_bas:157,"15j":97,stefan:140,die_rol:106,phy:[146,97],unrestrict:[55,44],averag:[134,18,38,140],longer:[75,11,164,55,134,78,18,118,133],"\u2156":164,elist:157,quart:93,confirm:[123,9],pydi:39,decompress:24,"25e6":148,dmp_max_norm:38,implement:[157,133,150,42,14,86,43,111,163,88,46,47,4,136,6,137,53,9,152,97,11,54,172,95,171,98,56,57,100,101,118,103,104,18,147,64,19,149,107,106,60,61,92,110,21,65,23,66,67,25,140,109,115,55,148,70,28,167,73,15,74,155,75,162,164,90,32,83,134,113,76,122,123,124,108,78,128,80,112,37,38,39,144],"337048736538615712436929e":97,placehold:108,two:[0,1,4,9,11,12,18,19,25,5,28,29,90,38,39,40,42,43,44,45,47,50,53,54,55,57,59,60,61,62,58,64,66,74,75,80,83,84,86,31,68,95,97,100,102,104,105,106,131,108,109,146,111,112,113,115,117,15,119,123,124,126,128,129,73,85,132,133,134,136,137,138,171,145,149,150,151,152,154,155,77,160,161,162,163,164,167,169,172],unpack:45,nbar:108,good:[133,43,90,4,140,54,171,18,148,19,134,60,61,62,111,5,117,119,12,47,164,123,124,108,82],jzbra:169,heavili:111,"012757178975667428359374e":78,dhbailei:[47,30],dmp_nth:38,domainerror:38,infrom:168,gear:124,"22568799288165021578758j":171,is_right_unbound:96,lectur:123,largest:[134,61,86,43,123,124,163,4,160,5,73,149],wignerd:169,pars:[54,86,147,142,126,172],sdp_pow:38,random_integer_partit:44,matrix_multiply_elementwis:43,jborwein:47,unequ:[54,161],ann:123,wave:[5,97,28,47],yuvalf:134,formul:[0,108,109,102,12],astr:115,nvar:108,orthocent:18,facil:[75,45],jzketcoupl:169,charpoli:43,arc:40,denomin:[75,163,74,54,172,62,66,57,67,90,45,38,133],converg:[40,53,163,11,90,62,171,112,123,47,25,4,136,103,5,119,117,111,15,60],implemented_funct:[14,65],compositemorph:115,absenc:77,aris:[140,60,109,97,167,169],ludecompositionff:43,sdp_monom:38,dicontinu:57,absent:7,convert:[40,85,133,86,43,46,90,7,54,169,55,99,100,18,19,108,61,62,152,65,23,66,157,77,161,5,70,15,75,163,164,134,122,124,34,126,128,80,73,38],qfac:154,andrea:140,convers:[131,93,61,54,64,22,108,128,80,5,28,38,19],coplanar:109,"593897639029319267398803e":171,"24570818422368e":67,grade2:108,expectedexcept:17,base_vector:50,"4342944819025j":97,systemat:172,mul:[1,66,41,147,133,54,164,72,64,43,57,76,122,166,137,110,112,37,38,134],kinetic_energi:[89,58,68,87],r40:163,natalia:140,marshall2389:140,cost:[117,5,134,4],"7432082105196321803869551j":4,testmod:99,f_diff:97,symmetric_group:33,lower_triangular_solv:43,viet:66,boilerpl:50,have:[0,1,3,4,6,7,9,12,13,15,17,18,19,23,24,25,5,29,38,39,41,42,43,45,47,50,54,55,56,57,59,60,61,62,63,64,65,66,67,68,69,75,76,78,86,87,90,92,93,95,97,99,102,105,131,108,109,110,111,112,113,115,116,117,118,119,120,123,124,128,133,134,137,138,171,143,144,146,147,148,149,151,152,77,161,163,164,168,169,172],"30566371247811e":97,lazenbi:[157,108],thumb:117,ratint:112,t_n:[38,78],"55065599872579097221j":53,intput:126,column:[1,9,152,68,12,85,164,58,43,31,4,115,146,73,126],wish:[1,93,60,54,62,108,23,56,109,4,161,105,134],reader:[58,115,54,124],shorter:[0,54,56,124,18,144],auto_symbol:126,successfulli:18,nsum:[53,123,171,154,67,47,4,15,5,19,136],print_tre:164,memori:[5,65,134,66,61],measure_partial_oneshot:122,diff:[40,132,43,45,90,4,136,53,140,54,95,171,57,119,149,108,109,152,66,67,24,113,158,5,97,117,15,29,163,135,164,38,123,13,167,78,83,39,172],wise:[85,73,65,38],from_ff_gmpi:38,derang:55,thereof:[124,75,38,136],behind:[152,64,133],quotientmodul:124,fdistribut:106,greaterthan:54,assumptions0:54,dtype:[124,38],"202762j":73,arcsin:[164,161,106,128],patch:[164,50,140,64],partit:[53,9,134,55,98,154,44,5,130],free_symbol:[149,54,59,112,66,18,138],gudchenko:140,is_alt_sym:[9,168],a0dota1:108,associatedlaguerrepolynomi:163,influenc:[124,118],refer:[0,3,4,6,7,9,10,11,12,13,18,126,22,24,25,29,30,90,33,35,36,37,38,39,40,42,43,44,45,47,50,53,54,55,56,57,59,60,61,58,64,65,66,67,96,78,79,84,86,87,89,31,68,97,98,102,103,106,109,146,111,112,113,115,116,15,123,124,127,73,85,133,134,136,144,138,141,171,147,149,152,153,155,157,159,160,161,163,164,167,168,169,142],no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ocal:[18,103],lseri:54,guru:140,cox97:[21,38,66],role:50,textbook:61,flat:73,constantli:111,bbp:140,dump_cod:144,rank_nonlex:86,morphism:[124,115],bby:55,xml:133,materi:140,"219637303741238195688575j":67,is_reduc:38,comfort:73,index_group:102,conycl:18,chebyshevt:[163,45],chebyshevu:163,"53e":73,quadt:47,sousa:140,sup:[164,96,38,66],symb_2txt:164,alexand:140,randtest:51,barnett:97,adv:30,almosteq:5,sub:[85,133,43,111,45,54,57,99,18,59,149,108,65,152,64,112,23,66,161,70,29,163,12,120,164,124,169,37,38,39],s0020:86,row_op:[31,85],add:[85,41,133,42,14,43,77,44,90,48,4,8,137,93,109,54,56,57,151,143,102,152,18,147,106,131,134,61,62,110,64,23,66,113,140,114,161,5,72,120,76,122,123,164,34,80,73,37,38],twinprim:[46,28],sum:[0,84,133,86,43,111,87,44,45,47,4,136,53,140,118,54,141,171,57,151,102,152,147,64,19,108,60,61,62,146,59,22,154,66,67,116,5,73,15,163,135,90,164,134,76,122,123,124,167,106,169,112,37,38,55],sun:38,pcfu:97,pcfv:97,"\u2089":164,floyd:134,mora:[21,38,66,57],kroneck:[151,108,102,163],confusingli:[163,136],usag:[41,14,43,90,8,54,18,147,148,19,149,108,152,22,113,161,96,164,121,124,169,38,39],consequenti:140,tricki:60,is_squar:43,precedence_mul:164,academ:[117,21,7,108,115],"86579517236206j":15,noncommut:[73,72,54],psf:140,expr_z:113,off:[123,12,54,124,152,64,43,66,88,99,47,4,5,105,51,60],seress:9,v_i:[134,57],term:[0,40,84,133,42,43,111,47,4,136,50,6,137,142,53,9,93,74,54,13,72,171,57,100,102,152,147,64,19,148,108,117,60,61,62,58,59,65,66,67,68,161,5,97,96,15,119,75,163,12,90,83,134,76,39,123,124,167,78,169,80,112,149,38,55,172],stinson:86,qualiti:[164,140],boson:102,"\u2157":164,revisit:[12,42],brychkov:90,summer:140,built:[54,61,95,55,164,45,100,161,148,149,38,106,126,119],nonholonom:12,hessenberg:43,multiply_elementwis:43,gci:140,arcco:[164,161],askintegerhandl:72,infunct:54,powdenest:[147,133,149],qfrom:40,sdm_monomial_divid:38,mnemon:[55,161],previous:[133,12,59,45,106,172],jezreel:140,fiddl:99,zgate:137,maxdegre:47,akshai:140,from_zz_gmpi:38,postprocessor:133,mygamma:164,taocp:55,polylog:[123,171,163,4,136],reimplement:[95,140],bracelet:55,eight:164,behaved:61,a_0_0_1:43,ldldecomposit:[85,43],mv_fmt:[105,23],termwis:66,pxket:[1,48,145],dmp_qq_heu_gcd:38,regge58:146,favorit:24,rmul_with_af:86,gcd:[140,74,54,21,134,66,6,172,38,118,133],mayb:[64,126],pborwein:[123,30],hope:[9,56,61,124],toggl:[113,134],dmp_mul_term:38,dmp_subresult:38,set_str_format:108,analyitc:60,quadraticu:106,oooo:55,constant_renumb:152,from_rr_sympi:38,pytestreport:99,"2453596101351438273844725j":97,timedepbra:29,ill:[40,117,11,61],gf_degre:38,hintnam:152,nonstandard:117,notimpl:95,"8952500412050989295774e":97,mean:[40,133,42,86,43,45,47,4,50,93,11,54,95,171,56,57,99,102,106,108,60,61,152,64,112,67,159,161,28,15,155,75,90,164,134,76,123,124,73,38,55,172],parent:[54,99,129,5,50,39],fox:60,exhaust:[90,54],"__sub__":38,from_ff_sympi:38,arrow_formatt:115,stat:[140,142,155,7,147,106],rearrang:[0,152,137],ode_hintnam:152,waksman:140,biholomorph:60,unitari:[112,137,138],collin:38,consum:163,edmonds74:146,postiv:163,hundr:47,counter:18,libmp:[131,61],operators_to_st:145,stupend:53,positionbra3d:48,siegelz:123,as_term:54,strong:[9,134,47,77,4,168,162],"003327897536813558807438089j":123,reset:[9,120,99,3,113,148],issac:[21,112,57],suppli:[0,86,134,87,47,9,68,54,55,102,152,18,109,58,65,113,115,161,75,12,123,129,82,39],stephen:157,bulk:[90,38],runtest:[99,131],bailei:[47,75,30],heroic:[66,57],riccati:152,maxim:[9,111,38,73,124],less:[133,86,43,46,90,136,144,53,93,11,12,171,57,99,102,152,18,19,149,108,61,62,110,59,66,114,5,15,75,163,54,134,38],symarri:140,berkowitz:43,universalset:[147,96],polyexp:[53,123],auto_numb:126,ctimesd:161,fermion:102,mpmathifi:[5,78,136],per:[11,108,66,47,113,38,119],pep:54,sarda:140,same_as_bas:114,is_group:[9,33],scipi:[119,65,56,163,122],anywai:[152,64,60,133],quantiti:[40,93,109,12,42,58,86,108,56,163,84,127,161,68,18,39],"999977909503001j":171,higher:[85,14,43,45,4,136,11,54,152,18,108,61,62,146,111,161,5,163,51,123,82,38],pedant:80,col_del:[31,43,85],"366197716545672122983857e":97,generate_schreier_sim:[9,33],bill:140,unifi:[38,66,54],compound:[96,54,29],makelov:140,pat_matrix:156,priit:140,primetest:134,deri:68,prodcut:132,extradp:[40,5,123,136,148],gert:140,"988897705762865j":73,"4f3":136,"4f1":136,guarante:[51,9,43,61,54,152,55,22,57,5,18,86,19],rframe:108,otherwis:[41,133,42,86,43,47,144,53,9,140,74,54,55,57,99,145,152,17,105,18,147,106,134,60,62,110,59,112,23,66,77,26,5,119,75,163,164,32,123,124,128,38,115],base_f:77,dmp_from_sympi:38,enpoint:18,collis:95,"742664438034657928194889j":[67,123],position_i:48,transfer:103,"365499358305579597618785e":136,"\u215a":164,"32e":73,milton:163,"3619719871407024816e":53,isomorphismfail:38,manifold:50,decai:[47,97,4],is_simpl:163,cbrt:[40,82,15],ioerror:88,dure:[131,68,42,64,43,4,161,105,134,148],vector_fct:[157,108],perfect:[47,134,4],testutil:[9,162,168],piecewise_fold:149,xsym:164,paperradfront:12,xalpha:105,"0000005769149277148425774499857j":136,ever:[90,152,109,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-  <div class="section" id="tutorial">
-<span id="id1"></span><h1>Урок<a class="headerlink" href="#tutorial" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="introduction">
-<h2>Въведение<a class="headerlink" href="#introduction" title="Permalink to this headline">¶</a></h2>
-<p>SymPy е библиотека на Python за символна математика. Тя цели да стане пълнофункционална алгебрична система (CAS - computer algebra system), като поддържа кода възможно най-опростен с цел да бъде разбираем и лесен за разширяване.SymPy е изцяло написана на Python и не изисква външни библиотеки.</p>
-<p>Този урок е въведение към SymPy. Прочетете го, за да научите какво и как може SymPy, и ако искате да научите повече, прочетете <a class="reference external" href="../guide.html">SymPy User&#8217;s Guide</a>, <a class="reference external" href="../modules/index.html">SymPy Modules Reference</a>, или  <a class="reference external" href="https://github.com/sympy/sympy/">сорс код</a> директно.</p>
-</div>
-<div class="section" id="first-steps-with-sympy">
-<h2>Първи стъпки със SymPy<a class="headerlink" href="#first-steps-with-sympy" title="Permalink to this headline">¶</a></h2>
-<p>Най-лесният начин да свалите SymPy е да отидете на <a class="reference external" href="http://code.google.com/p/sympy/">http://code.google.com/p/sympy/</a>  и да изтеглите последния архив.</p>
-<img alt="../_images/featured-downloads.png" src="../_images/featured-downloads.png" />
-<p>Разархивирайте го:</p>
-<pre class="literal-block">
-$ <strong class="input">tar xzf sympy-0.5.12.tar.gz</strong>
-</pre>
-<p>и го изпробвайте от интерпретатор на Python:</p>
-<pre class="literal-block">
-$ <strong class="input">cd sympy-0.5.12</strong>
-$ <strong class="input">python</strong>
-Python 2.4.4 (#2, Jan  3 2008, 13:36:28)
-[GCC 4.2.3 20071123 (prerelease) (Debian 4.2.2-4)] on linux2
-Type &quot;help&quot;, &quot;copyright&quot;, &quot;credits&quot; or &quot;license&quot; for more information.
-&gt;&gt;&gt; from sympy import Symbol, cos
-&gt;&gt;&gt; x = Symbol(&quot;x&quot;)
-&gt;&gt;&gt; (1/cos(x)).series(x, 0, 10)
-1 + x**2/2 + 5*x**4/24 + 61*x**6/720 + 277*x**8/8064 + O(x**10)
-</pre>
-<p>Може да използвате SymPy както е показано отгоре, което всъщност е препоръчителният начин, ако го използвате във вашата програма. Също така може да го инсталирате, използвайки ./setup.py install, както който и да е модул на Python,или просто инсталирайте пакет във вашата любима Linux дистрибуция, например:</p>
-<div class="topic">
-<p class="topic-title first">Installing SymPy in Debian</p>
-<pre class="literal-block">
-$ <strong class="input">sudo apt-get install python-sympy</strong>
-Reading package lists... Done
-Building dependency tree
-Reading state information... Done
-The following NEW packages will be installed:
-  python-sympy
-0 upgraded, 1 newly installed, 0 to remove and 18 not upgraded.
-Need to get 991kB of archives.
-After this operation, 5976kB of additional disk space will be used.
-Get:1 http://ftp.cz.debian.org unstable/main python-sympy 0.5.12-1 [991kB]
-Fetched 991kB in 2s (361kB/s)
-Selecting previously deselected package python-sympy.
-(Reading database ... 232619 files and directories currently installed.)
-Unpacking python-sympy (from .../python-sympy_0.5.12-1_all.deb) ...
-Setting up python-sympy (0.5.12-1) ...
-</pre>
-</div>
-<p>For other means how to install SymPy, consult the wiki page
-<a class="reference external" href="https://github.com/sympy/sympy/wiki/Download-Installation">Download and Installation</a>.</p>
-<div class="section" id="isympy-console">
-<h3>isympy Конзола<a class="headerlink" href="#isympy-console" title="Permalink to this headline">¶</a></h3>
-<p>За да експериментирате с нови функционалности или когато се опитвате да разберете как да направите нещата, може да използвате нашата специална обвивка около IPython, наречена isympy (намираща се в bin/isympy,  ако сте компилирали през сорс директорията), която е стандартна конзола на Python и която включва съответните SymPy модули и дефинираните символи  x, y, z, както и някои други неща:</p>
-<pre class="literal-block">
-$ <strong class="input">cd sympy</strong>
-$ <strong class="input">./bin/isympy</strong>
-IPython console for SymPy 0.7.2-git (Python 2.7.1) (ground types: gmpy)
-
-These commands were executed:
-&gt;&gt;&gt;
-&gt;&gt;&gt; from sympy import *
-&gt;&gt;&gt; x, y, z, t = symbols('x y z t')
-&gt;&gt;&gt; k, m, n = symbols('k m n', integer=True)
-&gt;&gt;&gt; f, g, h = symbols('f g h', cls=Function)
-
-Documentation can be found at <a class="reference external" href="http://www.sympy.org">http://www.sympy.org</a>
-
-In [1]: <strong class="input">(1/cos(x)).series(x, 0, 10)</strong>
-Out[1]:
-     2      4       6        8
-    x    5*x    61*x    277*x     / 10\
-1 + -- + ---- + ----- + ------ + O\x  /
-    2     24     720     8064
-</pre>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">Командите, въведени от вас, са удебелени. Така, това което направихме в 3 линии на обикновен Python, може да се направи с 1 линия на isympy.</p>
-</div>
-</div>
-<div class="section" id="using-sympy-as-a-calculator">
-<h3>Използване на SymPy като калкулатор<a class="headerlink" href="#using-sympy-as-a-calculator" title="Permalink to this headline">¶</a></h3>
-<p>SymPy има 3 вградени типове числа: Цели, с плаваща запетая и рационални.</p>
-<p>Рационалният клас представлява рационална дроб като двойка от цели числа: числителя и знаменателя, така че Rational(1, 2) представлява 1/2, Rational(5, 2) 5/2 и така нататък.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
-<span class="go">1/2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">*</span><span class="mi">2</span>
-<span class="go">1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">50</span><span class="o">/</span><span class="n">Rational</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span><span class="o">**</span><span class="mi">50</span>
-<span class="go">1/88817841970012523233890533447265625</span>
-</pre></div>
-</div>
-<p>Бъдете предпазливи, когато работите с цели и с плаваща запетая числа в Python особено когато делите, защото може да създадете Python число, а не SymPy число. От съотношението на 2 цели числа в Python може да се получи число с плаваща запетая &#8211; &#8220;истинското деление&#8221;, което е стандарт в Python 3, и подразбиращото се поведение на <tt class="docutils literal"><span class="pre">isympy</span></tt>, което въвежда делението от __future__:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span> 
-<span class="go">0.5</span>
-</pre></div>
-</div>
-<p>Но в по-ранните версии на Python, където &#8220;истинското делението&#8221; не бе въведено, като резултат ще бъде закръглено число:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span> 
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>Все пак и в двата случая не използваме SymPy числа, защото Python създава свои собствени. През повечето време най-вероятно ще работите с рационални числа, така че се уверете, че използвате Rational, за да получите очаквания резултат. Може да сметнете за удобно да използвате <tt class="docutils literal"><span class="pre">R</span></tt> като Rational:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="c"># R(1) is a SymPy Integer and Integer/int gives a Rational</span>
-<span class="go">1/2</span>
-</pre></div>
-</div>
-<p>Ние също така имаме някои специални константи, като e и pi, които се третират като символи (1 + pi няма да се изчисли като някое число, а ще си остане 1 + pi) и имат арбитрарна точност:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">E</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">pi**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">3.14159265358979</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">pi</span> <span class="o">+</span> <span class="n">E</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">5.85987448204884</span>
-</pre></div>
-</div>
-<p>Както виждате, evalf изчислява израза до число с плаваща запетая.</p>
-<p>Също така има и клас, представляващ математическа безкрайност, наречен  <tt class="docutils literal"><span class="pre">oo</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">oo</span> <span class="o">&gt;</span> <span class="mi">99999</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">oo</span> <span class="o">+</span> <span class="mi">1</span>
-<span class="go">oo</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="symbols">
-<h3>Символи<a class="headerlink" href="#symbols" title="Permalink to this headline">¶</a></h3>
-<p>За разлика от други компютърни алгебрични системи (CAS), в SymPy вие трябва изрично да декларирате символните променливи:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Отляво е променливата, на която задаваме като стойност инстанция на SymPy класа Symbol. Инстанциите на този клас могат да се комбинират и да направят израз:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">theta</span>
-</pre></div>
-</div>
-<p>Symbols can also be created with the <tt class="docutils literal"><span class="pre">symbols</span></tt> or <tt class="docutils literal"><span class="pre">var</span></tt> functions, the
-latter automatically adding the created symbols to the namespace, and both
-accepting a range notation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">var</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a,b,c&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;d:f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;g:h&#39;</span><span class="p">)</span>
-<span class="go">(g, h)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;g:2&#39;</span><span class="p">)</span>
-<span class="go">(g0, g1)</span>
-</pre></div>
-</div>
-<p>Инстанциите на този клас могат да се комбинират и да направят израз:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">x</span> <span class="o">-</span> <span class="n">y</span>
-<span class="go">2*x</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">(x + y)**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go">x**2 + 2*x*y + y**2</span>
-</pre></div>
-</div>
-<p>И да ги замествате с други символи или числа като използвате <tt class="docutils literal"><span class="pre">subs(old,</span> <span class="pre">new)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">(y + 1)**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">4*y**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>До края на този урок предполагаме, че сте изпълнили следния код:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">init_printing</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">init_printing</span><span class="p">(</span><span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">wrap_line</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">no_global</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Това ще направи нещата да изглеждат по-добре, когато се принтират (pretty printing). Погледнете секцията за принтиране по-нататък. Ако имате инсталиран някой unicode шрифт, може да подадете use_unicode=True за доста по-красив изход.</p>
-</div>
-</div>
-<div class="section" id="algebra">
-<h2>Алгебра<a class="headerlink" href="#algebra" title="Permalink to this headline">¶</a></h2>
-<p>За да разложите непълни дроби, използвайте <tt class="docutils literal"><span class="pre">apart(expr,</span> <span class="pre">x)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">apart</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">)</span>
-<span class="go">       1</span>
-<span class="go">---------------</span>
-<span class="go">(x + 1)*(x + 2)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">apart</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">    1       1</span>
-<span class="go">- ----- + -----</span>
-<span class="go">  x + 2   x + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">x + 1</span>
-<span class="go">-----</span>
-<span class="go">x - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">apart</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">      2</span>
-<span class="go">1 + -----</span>
-<span class="go">    x - 1</span>
-</pre></div>
-</div>
-<p>За да комбинирате нещата отново заедно, използвайте <tt class="docutils literal"><span class="pre">together(expr,</span> <span class="pre">x)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">together</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">)</span>
-<span class="go">x*y + x*z + y*z</span>
-<span class="go">---------------</span>
-<span class="go">     x*y*z</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="n">apart</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x + 1</span>
-<span class="go">-----</span>
-<span class="go">x - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="n">apart</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">       1</span>
-<span class="go">---------------</span>
-<span class="go">(x + 1)*(x + 2)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="calculus">
-<span id="index-0"></span><h2>Висша математика<a class="headerlink" href="#calculus" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="limits">
-<span id="index-1"></span><h3>Граници<a class="headerlink" href="#limits" title="Permalink to this headline">¶</a></h3>
-<p>Границите са изключително лесни за използване в SymPy. Те имат следния синтаксис limit(function, variable, point), така че за да изчислите границата на f(x), където  x -&gt; 0 бихте използвали <tt class="docutils literal"><span class="pre">limit(f,</span> <span class="pre">x,</span> <span class="pre">0)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">limit</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>също така може да изчислите граница до безкрайност:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">oo</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">0</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>За някои не толкова тривиални примери, може да прегледате тестовия файл <a class="reference external" href="https://github.com/sympy/sympy/blob/master/sympy/series/tests/test_demidovich.py">test_demidovich.py</a></p>
-</div>
-<div class="section" id="differentiation">
-<span id="index-2"></span><h3>Диференциално смятане<a class="headerlink" href="#differentiation" title="Permalink to this headline">¶</a></h3>
-<p>Може да изчислите производните на който и да е израз в SymPy, като използвате <tt class="docutils literal"><span class="pre">diff(func,</span> <span class="pre">var)</span></tt>. Примери:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">diff</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">tan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">cos(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">2*cos(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">tan (x) + 1</span>
-</pre></div>
-</div>
-<p>Можете да проверите, че е правилно, като:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">limit</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">delta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">((</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">delta</span><span class="p">)</span> <span class="o">-</span> <span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">/</span><span class="n">delta</span><span class="p">,</span> <span class="n">delta</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">tan (x) + 1</span>
-</pre></div>
-</div>
-<p>Производни от по-висок ред могат да бъдат пресметнати чрез използването на метода <tt class="docutils literal"><span class="pre">diff(func,</span> <span class="pre">var,</span> <span class="pre">n)</span></tt> :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">2*cos(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">-4*sin(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">-8*cos(2*x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="series-expansion">
-<span id="index-3"></span><h3>Разлагане в ред<a class="headerlink" href="#series-expansion" title="Permalink to this headline">¶</a></h3>
-<p>Използвайте метода <tt class="docutils literal"><span class="pre">.series(var,</span> <span class="pre">point,</span> <span class="pre">order)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     2    4     6      8</span>
-<span class="go">    x    x     x      x      / 10\</span>
-<span class="go">1 - -- + -- - --- + ----- + O\x  /</span>
-<span class="go">    2    24   720   40320</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     2      4       6        8</span>
-<span class="go">    x    5*x    61*x    277*x     / 10\</span>
-<span class="go">1 + -- + ---- + ----- + ------ + O\x  /</span>
-<span class="go">    2     24     720     8064</span>
-</pre></div>
-</div>
-<p>Друг прост пример:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">pprint</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;y&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="go">1/y - x/y**2 + x**2/y**3 - x**3/y**4 + x**4/y**5 + O(x**5)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="go">          2    3    4</span>
-<span class="go">1   x    x    x    x     / 5\</span>
-<span class="go">- - -- + -- - -- + -- + O\x /</span>
-<span class="go">y    2    3    4    5</span>
-<span class="go">    y    y    y    y</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="summation">
-<span id="index-4"></span><h3>Summation<a class="headerlink" href="#summation" title="Permalink to this headline">¶</a></h3>
-<p>Compute the summation of f with respect to the given summation variable over the given limits.</p>
-<p>summation(f, (i, a, b)) computes the sum of f with respect to i from a to b,
-i.e.,</p>
-<div class="highlight-python"><pre>                            b
-                          ____
-                          \   `
-summation(f, (i, a, b)) =  )    f
-                          /___,
-                          i = a</pre>
-</div>
-<p>If it cannot compute the sum, it prints the corresponding summation formula.
-Repeated sums can be computed by introducing additional limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">summation</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">log</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i n m&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-<span class="go"> 2</span>
-<span class="go">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="o">**</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">  oo</span>
-<span class="go"> ___</span>
-<span class="go"> \  `</span>
-<span class="go">  \     -n</span>
-<span class="go">  /   log (n)</span>
-<span class="go"> /__,</span>
-<span class="go">n = 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">),</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">m</span><span class="p">))</span>
-<span class="go">      3    2</span>
-<span class="go">m    m    m</span>
-<span class="go">-- + -- + -</span>
-<span class="go">6    2    3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go"> x</span>
-<span class="go">e</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="integration">
-<span id="index-5"></span><h3>Интегриране<a class="headerlink" href="#integration" title="Permalink to this headline">¶</a></h3>
-<p>SymPy поддръжа определени и неопределени интеграли на елементарни и специални трансцедентни функции с помощта на integrate(), който използва мощния разширен алгоритъм на Risch-Norman и няколко еврестики и сравнения с шаблони:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">integrate</span><span class="p">,</span> <span class="n">erf</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Можете да декларирате елементарни функции:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="mi">6</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">5</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go"> 6</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">-cos(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x*log(x) - x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">sinh</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go"> 2</span>
-<span class="go">x  + cosh(x)</span>
-</pre></div>
-</div>
-<p>Лесно можете да се справите и със специалните функции:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">erf</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">  ____    2</span>
-<span class="go">\/ pi *erf (x)</span>
-<span class="go">--------------</span>
-<span class="go">      4</span>
-</pre></div>
-</div>
-<p>Възможно е да изчислите даден интеграл:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-<p>Също така се поддържат и неопределени интеграли:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">-1</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="complex-numbers">
-<span id="index-6"></span><h3>Комплексни числа<a class="headerlink" href="#complex-numbers" title="Permalink to this headline">¶</a></h3>
-<p>Besides the imaginary unit, I, which is imaginary, symbols can be created with
-attributes (e.g. real, positive, complex, etc...) and this will affect how
-they behave:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span> <span class="c"># a plain x with no attributes</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go"> I*x</span>
-<span class="go">e</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">   -im(x)               -im(x)</span>
-<span class="go">I*e      *sin(re(x)) + e      *cos(re(x))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">I*sin(x) + cos(x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="functions">
-<h3>Функции<a class="headerlink" href="#functions" title="Permalink to this headline">¶</a></h3>
-<p><strong>тригонометрични</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">asin</span><span class="p">,</span> <span class="n">asinh</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">sin(x)*cos(y) + sin(y)*cos(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">-sin(x)*sin(y) + cos(x)*cos(y)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*sinh(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinh</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*sin(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
-<span class="go">I*pi</span>
-<span class="go">----</span>
-<span class="go"> 2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*asin(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3     5     7       9</span>
-<span class="go">    x     x     x       x       / 10\</span>
-<span class="go">x - -- + --- - ---- + ------ + O\x  /</span>
-<span class="go">    6    120   5040   362880</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3     5     7       9</span>
-<span class="go">    x     x     x       x       / 10\</span>
-<span class="go">x + -- + --- + ---- + ------ + O\x  /</span>
-<span class="go">    6    120   5040   362880</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3      5      7       9</span>
-<span class="go">    x    3*x    5*x    35*x     / 10\</span>
-<span class="go">x + -- + ---- + ---- + ----- + O\x  /</span>
-<span class="go">    6     40    112     1152</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3      5      7       9</span>
-<span class="go">    x    3*x    5*x    35*x     / 10\</span>
-<span class="go">x - -- + ---- - ---- + ----- + O\x  /</span>
-<span class="go">    6     40    112     1152</span>
-</pre></div>
-</div>
-<p><strong>сферични хармонични</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Ylm</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">  ___</span>
-<span class="go">\/ 3 *cos(theta)</span>
-<span class="go">----------------</span>
-<span class="go">        ____</span>
-<span class="go">    2*\/ pi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">   ___  I*phi</span>
-<span class="go">-\/ 6 *e     *sin(theta)</span>
-<span class="go">------------------------</span>
-<span class="go">            ____</span>
-<span class="go">        4*\/ pi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">   ____  I*phi</span>
-<span class="go">-\/ 30 *e     *sin(theta)*cos(theta)</span>
-<span class="go">------------------------------------</span>
-<span class="go">                  ____</span>
-<span class="go">              4*\/ pi</span>
-</pre></div>
-</div>
-<p><strong>факториел и гама функции</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">x!</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">n!</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="c"># i.e. factorial(x)</span>
-<span class="go">                      /          2     2\</span>
-<span class="go">                    2 |EulerGamma    pi |    / 3\</span>
-<span class="go">1 - EulerGamma*x + x *|----------- + ---| + O\x /</span>
-<span class="go">                      \     2         12/</span>
-</pre></div>
-</div>
-<p><strong>дзета функции</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">zeta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">zeta(4, x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">  4</span>
-<span class="go">pi</span>
-<span class="go">---</span>
-<span class="go"> 90</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">       4</span>
-<span class="go">     pi</span>
-<span class="go">-1 + ---</span>
-<span class="go">      90</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">         4</span>
-<span class="go">  17   pi</span>
-<span class="go">- -- + ---</span>
-<span class="go">  16    90</span>
-</pre></div>
-</div>
-<p><strong>полиноми</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">assoc_legendre</span><span class="p">,</span> <span class="n">chebyshevt</span><span class="p">,</span> <span class="n">legendre</span><span class="p">,</span> <span class="n">hermite</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">2*x  - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   4      2</span>
-<span class="go">8*x  - 8*x  + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">3*x    1</span>
-<span class="go">---- - -</span>
-<span class="go"> 2     2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">      8         6         4        2</span>
-<span class="go">6435*x    3003*x    3465*x    315*x     35</span>
-<span class="go">------- - ------- + ------- - ------ + ---</span>
-<span class="go">  128        32        64       32     128</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">        __________</span>
-<span class="go">       /    2</span>
-<span class="go">-3*x*\/  - x  + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">     2</span>
-<span class="go">- 3*x  + 3</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hermite</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   3</span>
-<span class="go">8*x  - 12*x</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="differential-equations">
-<span id="index-7"></span><h3>Диференциални уравнения<a class="headerlink" href="#differential-equations" title="Permalink to this headline">¶</a></h3>
-<p>В <tt class="docutils literal"><span class="pre">isympy</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">dsolve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">        2</span>
-<span class="go">       d</span>
-<span class="go">f(x) + ---(f(x))</span>
-<span class="go">         2</span>
-<span class="go">       dx</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dsolve</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">f(x) = C1*sin(x) + C2*cos(x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="algebraic-equations">
-<span id="index-8"></span><h3>Алгебрични уравнения<a class="headerlink" href="#algebraic-equations" title="Permalink to this headline">¶</a></h3>
-<p>В <tt class="docutils literal"><span class="pre">isympy</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">solve</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">[-1, 1, -I, I]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">([</span><span class="n">x</span> <span class="o">+</span> <span class="mi">5</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">6</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">15</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">])</span>
-<span class="go">{x: -3, y: 1}</span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="section" id="linear-algebra">
-<span id="index-9"></span><h2>Линейна алгебра<a class="headerlink" href="#linear-algebra" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="matrices">
-<span id="index-10"></span><h3>Матрици<a class="headerlink" href="#matrices" title="Permalink to this headline">¶</a></h3>
-<p>Матриците се създават като инстанции на Matrix класа:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="go">[1  0]</span>
-<span class="go">[    ]</span>
-<span class="go">[0  1]</span>
-</pre></div>
-</div>
-<p>също така може да слагате символи в тях:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">],</span> <span class="p">[</span><span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span>
-<span class="go">[1  x]</span>
-<span class="go">[    ]</span>
-<span class="go">[y  1]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">[x*y + 1    2*x  ]</span>
-<span class="go">[                ]</span>
-<span class="go">[  2*y    x*y + 1]</span>
-</pre></div>
-</div>
-<p>За повече информация и примери с матрици вижте Linear Algebra tutorial.</p>
-</div>
-</div>
-<div class="section" id="pattern-matching">
-<span id="index-11"></span><h2>Сравняване на шаблони<a class="headerlink" href="#pattern-matching" title="Permalink to this headline">¶</a></h2>
-<p>Използвайте <tt class="docutils literal"><span class="pre">.match()</span></tt> метода заедно с класа Wild за да сравнявате изрази с даден шаблон. Методът ще върне речник с изисканите замествания, както следва:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Wild</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">{p: 5}</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">q</span><span class="p">)</span>
-<span class="go">{p: 1, q: 2}</span>
-</pre></div>
-</div>
-<p>Ако съвпадението е неуспешно, методът връща <tt class="docutils literal"><span class="pre">None</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">**</span><span class="n">x</span><span class="p">))</span>
-<span class="go">None</span>
-</pre></div>
-</div>
-<p>Можете да използвате и втория незадължителен параметър exclude, за да се уверите, че някои неща не се появяват в резултата:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">p</span><span class="p">))</span> <span class="c"># 1 is excluded</span>
-<span class="go">None</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span> <span class="c"># x is excluded</span>
-<span class="go">None</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">p</span><span class="p">))</span> <span class="c"># -1 is not excluded</span>
-<span class="go">{p_: -1}</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="printing">
-<span id="printing-tutorial"></span><h2>Принтиране<a class="headerlink" href="#printing" title="Permalink to this headline">¶</a></h2>
-<p>Има много начини, по които изразите може да се отпечатат.</p>
-<p><strong>Стандартен начин</strong></p>
-<p>Това е резултатът от <tt class="docutils literal"><span class="pre">str(expression)</span></tt> и изглежда така:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1/x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Integral(x**2, x)</span>
-</pre></div>
-</div>
-<p><strong>Красиво отпечатване (pretty printing)</strong></p>
-<p>Това е хубаво отпечатване тип ascii-art, направено от <tt class="docutils literal"><span class="pre">pprint</span></tt> функция:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go"> 2</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="go">-</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">  /</span>
-<span class="go"> |</span>
-<span class="go"> |  2</span>
-<span class="go"> | x  dx</span>
-<span class="go"> |</span>
-<span class="go">/</span>
-</pre></div>
-</div>
-<p>Ако имате инсталиран unicode шрифт, то той би трябвало да бъде използван по подразбиране. Може смените това поведение, като използвате опцията <tt class="docutils literal"><span class="pre">use_unicode</span></tt>.:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">⌠</span>
-<span class="go">⎮  2</span>
-<span class="go">⎮ x  dx</span>
-<span class="go">⌡</span>
-</pre></div>
-</div>
-<p>Също така вижте уикито <a class="reference external" href="https://github.com/sympy/sympy/wiki/Pretty-Printing">Pretty Printing</a> за повече примери за добро unicode принтиране.</p>
-<p>Съвет:За да направите красивото отпечатване(pretty printing) по подразбиране в интерпретатора на Python, използвайте:</p>
-<div class="highlight-python"><pre>$ python
-Python 2.5.2 (r252:60911, Jun 25 2008, 17:58:32)
-[GCC 4.3.1] on linux2
-Type &quot;help&quot;, &quot;copyright&quot;, &quot;credits&quot; or &quot;license&quot; for more information.
-&gt;&gt;&gt; from sympy import init_printing, var, Integral
-&gt;&gt;&gt; init_printing(use_unicode=False, wrap_line=False, no_global=True)
-&gt;&gt;&gt; var(&quot;x&quot;)
-x
-&gt;&gt;&gt; x**3/3
- 3
-x
---
-3
-&gt;&gt;&gt; Integral(x**2, x) #doctest: +NORMALIZE_WHITESPACE
-  /
- |
- |  2
- | x  dx
- |
-/</pre>
-</div>
-<p><strong>Python отпечатване</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.python</span> <span class="kn">import</span> <span class="n">python</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = x**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = 1/x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = Integral(x**2, x)</span>
-</pre></div>
-</div>
-<p><strong>LaTeX отпечатване</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">latex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x^{2}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;inline&#39;</span><span class="p">)</span>
-<span class="go">$x^{2}$</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;equation&#39;</span><span class="p">)</span>
-<span class="go">\begin{equation}x^{2}\end{equation}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;equation*&#39;</span><span class="p">)</span>
-<span class="go">\begin{equation*}x^{2}\end{equation*}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">\frac{1}{x}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">\int x^{2}\, dx</span>
-</pre></div>
-</div>
-<p><em>MathML*</em></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.mathml</span> <span class="kn">import</span> <span class="n">mathml</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">latex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">mathml</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">&lt;apply&gt;&lt;power/&gt;&lt;ci&gt;x&lt;/ci&gt;&lt;cn&gt;2&lt;/cn&gt;&lt;/apply&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">mathml</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">))</span>
-<span class="go">&lt;apply&gt;&lt;power/&gt;&lt;ci&gt;x&lt;/ci&gt;&lt;cn&gt;-1&lt;/cn&gt;&lt;/apply&gt;</span>
-</pre></div>
-</div>
-<p><strong>Pyglet</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">preview</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">preview</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span> 
-</pre></div>
-</div>
-<p>И pyglet прозорец с LaTeX рендиран израз ще се появи:</p>
-<img alt="../_images/pngview1.png" src="../_images/pngview1.png" />
-<div class="section" id="notes">
-<h3>Бележки<a class="headerlink" href="#notes" title="Permalink to this headline">¶</a></h3>
-<p><tt class="docutils literal"><span class="pre">isympy</span></tt> извиква <tt class="docutils literal"><span class="pre">pprint</span></tt> автоматично, поради тази причина виждате красивото отпечатване (pretty printing) по подразбиране.</p>
-<p>Забележете, че също така има модул за принтиране, <tt class="docutils literal"><span class="pre">sympy.printing</span></tt>. Други методи за принтиране, налични чрез този модул, са:</p>
-<ul class="simple">
-<li><tt class="docutils literal"><span class="pre">pretty(expr)</span></tt>, <tt class="docutils literal"><span class="pre">pretty_print(expr)</span></tt>, <tt class="docutils literal"><span class="pre">pprint(expr)</span></tt>: Връща или принтира, съответно, красива репрезентация на <tt class="docutils literal"><span class="pre">expr</span></tt>. Това е еквивалентно на второто ниво на репрезентация показано по-горе.</li>
-<li><tt class="docutils literal"><span class="pre">latex(expr)</span></tt>, <tt class="docutils literal"><span class="pre">print_latex(expr)</span></tt>: Връща или принтира, съответно, <a class="reference external" href="http://www.latex-project.org/">LaTeX</a>  репрезентация на <tt class="docutils literal"><span class="pre">expr</span></tt></li>
-<li><tt class="docutils literal"><span class="pre">mathml(expr)</span></tt>, <tt class="docutils literal"><span class="pre">print_mathml(expr)</span></tt>: Връща или принтира, съответно, <a class="reference external" href="http://www.w3.org/Math/">MathML</a> репрезентация на <tt class="docutils literal"><span class="pre">expr</span></tt>.</li>
-<li><tt class="docutils literal"><span class="pre">print_gtk(expr)</span></tt>: Print <tt class="docutils literal"><span class="pre">expr</span></tt> to <a class="reference external" href="http://helm.cs.unibo.it/mml-widget/">Gtkmathview</a>, a GTK widget that displays MathML code. The <a class="reference external" href="http://helm.cs.unibo.it/mml-widget/">Gtkmathview</a> program is required.</li>
-</ul>
-</div>
-</div>
-<div class="section" id="further-documentation">
-<h2>Допълнителна документация<a class="headerlink" href="#further-documentation" title="Permalink to this headline">¶</a></h2>
-<p>Време е да научите повече за SymPy. Прегледайте <a class="reference external" href="../guide.html">SymPy User&#8217;s Guide</a> и <a class="reference external" href="../modules/index.html">SymPy Modules Reference</a>.</p>
-<p>Не пропускайте и да прегледате нашето публично уики – <a class="reference external" href="http://wiki.sympy.org/">wiki.sympy.org</a>, което съдържа много полезни примери, уроци и наръчници, за които ние и нашите потребители допринасяме и Ви окуражаваме да редактирате и подобрите.</p>
-<div class="section" id="translations">
-<h3>Translations<a class="headerlink" href="#translations" title="Permalink to this headline">¶</a></h3>
-<p>This tutorial is also available in other languages:</p>
-</div>
-<ul class="simple">
-<li><a class="reference external" href="tutorial.bg.html">Български</a></li>
-<li><a class="reference external" href="tutorial.cs.html">Česky</a></li>
-<li><a class="reference external" href="tutorial.de.html">Deutsch</a></li>
-<li><a class="reference external" href="tutorial.en.html">English</a></li>
-<li><a class="reference external" href="tutorial.fr.html">Français</a></li>
-<li><a class="reference external" href="tutorial.pl.html">Polski</a></li>
-<li><a class="reference external" href="tutorial.ru.html">Русский</a></li>
-<li><a class="reference external" href="tutorial.sr.html">Српски</a></li>
-<li><a class="reference external" href="tutorial.zh.html">中文</a></li>
-</ul>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="tutorial.html">
-              <img class="logo" src="../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="tutorial.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Урок</a><ul>
-<li><a class="reference internal" href="#introduction">Въведение</a></li>
-<li><a class="reference internal" href="#first-steps-with-sympy">Първи стъпки със SymPy</a><ul>
-<li><a class="reference internal" href="#isympy-console">isympy Конзола</a></li>
-<li><a class="reference internal" href="#using-sympy-as-a-calculator">Използване на SymPy като калкулатор</a></li>
-<li><a class="reference internal" href="#symbols">Символи</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#algebra">Алгебра</a></li>
-<li><a class="reference internal" href="#calculus">Висша математика</a><ul>
-<li><a class="reference internal" href="#limits">Граници</a></li>
-<li><a class="reference internal" href="#differentiation">Диференциално смятане</a></li>
-<li><a class="reference internal" href="#series-expansion">Разлагане в ред</a></li>
-<li><a class="reference internal" href="#summation">Summation</a></li>
-<li><a class="reference internal" href="#integration">Интегриране</a></li>
-<li><a class="reference internal" href="#complex-numbers">Комплексни числа</a></li>
-<li><a class="reference internal" href="#functions">Функции</a></li>
-<li><a class="reference internal" href="#differential-equations">Диференциални уравнения</a></li>
-<li><a class="reference internal" href="#algebraic-equations">Алгебрични уравнения</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#linear-algebra">Линейна алгебра</a><ul>
-<li><a class="reference internal" href="#matrices">Матрици</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#pattern-matching">Сравняване на шаблони</a></li>
-<li><a class="reference internal" href="#printing">Принтиране</a><ul>
-<li><a class="reference internal" href="#notes">Бележки</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#further-documentation">Допълнителна документация</a><ul>
-<li><a class="reference internal" href="#translations">Translations</a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-
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-<div id="searchbox" style="display: none">
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-  <div class="section" id="tutorial">
-<span id="id1"></span><h1>Tutoriál<a class="headerlink" href="#tutorial" title="Trvalý odkaz na tento nadpis">¶</a></h1>
-<div class="section" id="introduction">
-<h2>Úvod<a class="headerlink" href="#introduction" title="Trvalý odkaz na tento nadpis">¶</a></h2>
-<p>SymPy je knihovna jazyka Python pro symbolickou matematiku. Má za cíl stát se kompletním počítačovým algebraickým systémem (CAS, Computer algebra system) a zároveň být co nejjednodušší, aby byla pochopitelná a jednoduše rozšiřitelná. SymPy je napsaná kompletně v jazyce Python a nepotřebuje žádné externí knihovny.</p>
-<p>Tento tutoriál je úvodem do SymPy a zároveň ukazuje, co SymPy umí. Přečtěte si ho, aby jste zjistili, jak pro vás může být SymPy užitečná a pokud budete chtít vědět víc, přečtěte si <a class="reference external" href="../guide.html">Uživatelskou příručku SymPy</a>, <a class="reference external" href="../modules/index.html">Přehled modulů SymPy</a>, nebo přímo <a class="reference external" href="https://github.com/sympy/sympy/">zdrojové kódy</a>.</p>
-</div>
-<div class="section" id="first-steps-with-sympy">
-<h2>První kroky se SymPy<a class="headerlink" href="#first-steps-with-sympy" title="Trvalý odkaz na tento nadpis">¶</a></h2>
-<p>Nejjednodušší způsob, jak stáhnout SymPy je jít na <a class="reference external" href="http://code.google.com/p/sympy/">http://code.google.com/p/sympy/</a> a stáhnout nejnovější verzi z doporučených zdrojů:</p>
-<img alt="../_images/featured-downloads.png" src="../_images/featured-downloads.png" />
-<p>Rozbalit ji:</p>
-<pre class="literal-block">
-$ <strong class="input">tar xzf sympy-0.5.12.tar.gz</strong>
-</pre>
-<p>a vyzkoušet jí v interpretru Pythonu:</p>
-<pre class="literal-block">
-$ <strong class="input">cd sympy-0.5.12</strong>
-$ <strong class="input">python</strong>
-Python 2.4.4 (#2, Jan  3 2008, 13:36:28)
-[GCC 4.2.3 20071123 (prerelease) (Debian 4.2.2-4)] on linux2
-Type &quot;help&quot;, &quot;copyright&quot;, &quot;credits&quot; or &quot;license&quot; for more information.
-&gt;&gt;&gt; from sympy import Symbol, cos
-&gt;&gt;&gt; x = Symbol(&quot;x&quot;)
-&gt;&gt;&gt; (1/cos(x)).series(x, 0, 10)
-1 + x**2/2 + 5*x**4/24 + 61*x**6/720 + 277*x**8/8064 + O(x**10)
-</pre>
-<p>Můžete používat SymPy, jak vidíte výše, což je doporučený způsob pokud jí používate ve svém programu. Můžete si jí také nainstalovat pomocí <tt class="docutils literal"><span class="pre">./setup.py</span> <span class="pre">install</span></tt> jako každý modul Pythonu nebo si jednoduše nainstalovat balíček do vaší oblíbené distribuce Linuxu. Například:</p>
-<div class="topic">
-<p class="topic-title first">Installing SymPy in Debian</p>
-<pre class="literal-block">
-$ <strong class="input">sudo apt-get install python-sympy</strong>
-Reading package lists... Done
-Building dependency tree
-Reading state information... Done
-The following NEW packages will be installed:
-  python-sympy
-0 upgraded, 1 newly installed, 0 to remove and 18 not upgraded.
-Need to get 991kB of archives.
-After this operation, 5976kB of additional disk space will be used.
-Get:1 http://ftp.cz.debian.org unstable/main python-sympy 0.5.12-1 [991kB]
-Fetched 991kB in 2s (361kB/s)
-Selecting previously deselected package python-sympy.
-(Reading database ... 232619 files and directories currently installed.)
-Unpacking python-sympy (from .../python-sympy_0.5.12-1_all.deb) ...
-Setting up python-sympy (0.5.12-1) ...
-</pre>
-</div>
-<p>For other means how to install SymPy, consult the wiki page
-<a class="reference external" href="https://github.com/sympy/sympy/wiki/Download-Installation">Download and Installation</a>.</p>
-<div class="section" id="isympy-console">
-<h3>isympy konzole<a class="headerlink" href="#isympy-console" title="Trvalý odkaz na tento nadpis">¶</a></h3>
-<p>For experimenting with new features, or when figuring out how to do things, you
-can use our special wrapper around IPython called <tt class="docutils literal"><span class="pre">isympy</span></tt> (located in
-<tt class="docutils literal"><span class="pre">bin/isympy</span></tt> if you are running from the source directory) which is just a
-standard Python shell that has already imported the relevant SymPy modules and
-defined the symbols x, y, z and some other things:</p>
-<pre class="literal-block">
-$ <strong class="input">cd sympy</strong>
-$ <strong class="input">./bin/isympy</strong>
-IPython console for SymPy 0.7.2-git (Python 2.7.1) (ground types: gmpy)
-
-These commands were executed:
-&gt;&gt;&gt;
-&gt;&gt;&gt; from sympy import *
-&gt;&gt;&gt; x, y, z, t = symbols('x y z t')
-&gt;&gt;&gt; k, m, n = symbols('k m n', integer=True)
-&gt;&gt;&gt; f, g, h = symbols('f g h', cls=Function)
-
-Documentation can be found at <a class="reference external" href="http://www.sympy.org">http://www.sympy.org</a>
-
-In [1]: <strong class="input">(1/cos(x)).series(x, 0, 10)</strong>
-Out[1]:
-     2      4       6        8
-    x    5*x    61*x    277*x     / 10\
-1 + -- + ---- + ----- + ------ + O\x  /
-    2     24     720     8064
-</pre>
-<div class="admonition note">
-<p class="first admonition-title">Poznámka</p>
-<p class="last">Vámi zadané příkazy jsou tučně. Vidíme, že to, co bychom udělali ve 3 řádcích v běžném interpretru Pythonu, lze udělat v isympy v 1 řádku.</p>
-</div>
-</div>
-<div class="section" id="using-sympy-as-a-calculator">
-<h3>SymPy jako kalkulačka<a class="headerlink" href="#using-sympy-as-a-calculator" title="Trvalý odkaz na tento nadpis">¶</a></h3>
-<p>SymPy has three built-in numeric types: Float, Rational and Integer.</p>
-<p>Třída Rational reprezentuje racionální čísla jako dvojici Integerů: čitatel a jmenovatel Takže Rational(1, 2) reprezentuje 1/2, Rational(5, 2) reprezentuje 5/2, a tak dále.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
-<span class="go">1/2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">*</span><span class="mi">2</span>
-<span class="go">1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">50</span><span class="o">/</span><span class="n">Rational</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span><span class="o">**</span><span class="mi">50</span>
-<span class="go">1/88817841970012523233890533447265625</span>
-</pre></div>
-</div>
-<p>Postupujte opatrně, když pracujete s Inty a Floaty v Pythonu, obzvlášť při dělení, jelikož můžete vytvořit číslo Pythonu místo SymPy. Podíl dvou Intů v Pythonu může vytvořit Float &#8211; &#8220;pravé dělení&#8221;, standard Pythonu 3 a defaultní chování <tt class="docutils literal"><span class="pre">isympy</span></tt>, které vkládá dělení z __future__:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span> 
-<span class="go">0.5</span>
-</pre></div>
-</div>
-<p>Ale v dřívějších verzích Pythonu, kde nebylo dělení zakomponováno, vyšel oříznutý int:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span> 
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>V obou případech se nejedná o číslo SymPy, protože Python vytvořil své vlastní číslo. Většinu času budete pravděpodobně pracovat s čísly Rational, takže dávejte pozor, abyste používali Rational a dostali tak výsledek SymPy. Někomu může připadat příhodné používat místo dlouhého Rational krátké <tt class="docutils literal"><span class="pre">R</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="c"># R(1) is a SymPy Integer and Integer/int gives a Rational</span>
-<span class="go">1/2</span>
-</pre></div>
-</div>
-<p>Také máme k dispozici několik konstant, jak e nebo pí, se kterými se pracuje jako se symboly (1 + pi se nevypočítá jako numerická hodnota, ale ponechá se 1 + pi) a mají libovolnou přesnost:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">E</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">pi**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">3.14159265358979</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">pi</span> <span class="o">+</span> <span class="n">E</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">5.85987448204884</span>
-</pre></div>
-</div>
-<p>jak vidíte, evalf vyhodnotí výraz na číslo s pohyblivou desetinnou čárkou.</p>
-<p>Symbol <tt class="docutils literal"><span class="pre">oo</span></tt> se používá pro třídu definující matematické nekonečno:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">oo</span> <span class="o">&gt;</span> <span class="mi">99999</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">oo</span> <span class="o">+</span> <span class="mi">1</span>
-<span class="go">oo</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="symbols">
-<h3>Symboly<a class="headerlink" href="#symbols" title="Trvalý odkaz na tento nadpis">¶</a></h3>
-<p>Narozdíl od jiných počítačových algebraických systémů, v SymPy je třeba explicitně deklarovat symbolické proměnné:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>On the left is the normal Python variable which has been assigned to the
-SymPy Symbol class. Predefined symbols (including those for symbols with
-Greek names) are available for import from abc:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">theta</span>
-</pre></div>
-</div>
-<p>Symboly mohou být vytvořeny pomocí funkcí <tt class="docutils literal"><span class="pre">symbols</span></tt> nebo <tt class="docutils literal"><span class="pre">var</span></tt>, kde funkce <tt class="docutils literal"><span class="pre">var</span></tt> automaticky přídá vytvořené symboly do namespace, a jak <tt class="docutils literal"><span class="pre">symbols</span></tt>, tak <tt class="docutils literal"><span class="pre">var</span></tt> podporují range notaci:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">var</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a,b,c&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;d:f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;g:h&#39;</span><span class="p">)</span>
-<span class="go">(g, h)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;g:2&#39;</span><span class="p">)</span>
-<span class="go">(g0, g1)</span>
-</pre></div>
-</div>
-<p>Instances of the Symbol class &#8220;play well together&#8221; and are the building blocks
-of expresions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">x</span> <span class="o">-</span> <span class="n">y</span>
-<span class="go">2*x</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">(x + y)**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go">x**2 + 2*x*y + y**2</span>
-</pre></div>
-</div>
-<p>Mohou být nahrazeny jinými čísly, symboly nebo výrazy použitím <tt class="docutils literal"><span class="pre">subs(stary,</span> <span class="pre">novy)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">(y + 1)**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">4*y**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>Pro celý zbytek tutoriálu předpokládejme, že jsme zadali:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">init_printing</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">init_printing</span><span class="p">(</span><span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">wrap_line</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">no_global</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>To nám zajistí hezčí výstup. Koukněte do sekce níže <a class="reference internal" href="#printing-tutorial"><em>Výpis</em></a>. Pokud máte nainstalovaný font unicode, volbou <tt class="docutils literal"><span class="pre">use_unicode=True</span></tt> dostanete o něco hezčí výstup.</p>
-</div>
-</div>
-<div class="section" id="algebra">
-<h2>Algebra<a class="headerlink" href="#algebra" title="Trvalý odkaz na tento nadpis">¶</a></h2>
-<p>Pro rozklad na parciální zlomky, použijte <tt class="docutils literal"><span class="pre">apart(expr,</span> <span class="pre">x)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">apart</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">)</span>
-<span class="go">       1</span>
-<span class="go">---------------</span>
-<span class="go">(x + 1)*(x + 2)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">apart</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">    1       1</span>
-<span class="go">- ----- + -----</span>
-<span class="go">  x + 2   x + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">x + 1</span>
-<span class="go">-----</span>
-<span class="go">x - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">apart</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">      2</span>
-<span class="go">1 + -----</span>
-<span class="go">    x - 1</span>
-</pre></div>
-</div>
-<p>Pro zpětné sdružení věcí dohromady, použijte <tt class="docutils literal"><span class="pre">together(expr,</span> <span class="pre">x)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">together</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">)</span>
-<span class="go">x*y + x*z + y*z</span>
-<span class="go">---------------</span>
-<span class="go">     x*y*z</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="n">apart</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x + 1</span>
-<span class="go">-----</span>
-<span class="go">x - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="n">apart</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">       1</span>
-<span class="go">---------------</span>
-<span class="go">(x + 1)*(x + 2)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="calculus">
-<span id="index-0"></span><h2>Analýza<a class="headerlink" href="#calculus" title="Trvalý odkaz na tento nadpis">¶</a></h2>
-<div class="section" id="limits">
-<span id="index-1"></span><h3>Limity<a class="headerlink" href="#limits" title="Trvalý odkaz na tento nadpis">¶</a></h3>
-<p>Limits are easy to use in SymPy, they follow the syntax <tt class="docutils literal"><span class="pre">limit(function,</span>
-<span class="pre">variable,</span> <span class="pre">point)</span></tt>, so to compute the limit of f(x) as x -&gt; 0, you would issue
-<tt class="docutils literal"><span class="pre">limit(f,</span> <span class="pre">x,</span> <span class="pre">0)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">limit</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>také můžete spočítat limitu v nekonečnu:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">oo</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">0</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>pro některé netriviální příklady limit si můžete přečíst testovací soubor  <a class="reference external" href="https://github.com/sympy/sympy/blob/master/sympy/series/tests/test_demidovich.py">test_demidovich.py</a></p>
-</div>
-<div class="section" id="differentiation">
-<span id="index-2"></span><h3>Derivace<a class="headerlink" href="#differentiation" title="Trvalý odkaz na tento nadpis">¶</a></h3>
-<p>Můžete derivovat libovolný výraz v SymPy pomocí <tt class="docutils literal"><span class="pre">diff(func,</span> <span class="pre">var)</span></tt>. Například:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">diff</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">tan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">cos(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">2*cos(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">tan (x) + 1</span>
-</pre></div>
-</div>
-<p>Výsledek si můžete zkontrolovat:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">limit</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">delta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">((</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">delta</span><span class="p">)</span> <span class="o">-</span> <span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">/</span><span class="n">delta</span><span class="p">,</span> <span class="n">delta</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">tan (x) + 1</span>
-</pre></div>
-</div>
-<p>Vyšší derivace můžete spočítat pomocí <tt class="docutils literal"><span class="pre">diff(func,</span> <span class="pre">var,</span> <span class="pre">n)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">2*cos(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">-4*sin(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">-8*cos(2*x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="series-expansion">
-<span id="index-3"></span><h3>Rozvoj v řady<a class="headerlink" href="#series-expansion" title="Trvalý odkaz na tento nadpis">¶</a></h3>
-<p>Použijte <tt class="docutils literal"><span class="pre">.series(var,</span> <span class="pre">point,</span> <span class="pre">order)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     2    4     6      8</span>
-<span class="go">    x    x     x      x      / 10\</span>
-<span class="go">1 - -- + -- - --- + ----- + O\x  /</span>
-<span class="go">    2    24   720   40320</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     2      4       6        8</span>
-<span class="go">    x    5*x    61*x    277*x     / 10\</span>
-<span class="go">1 + -- + ---- + ----- + ------ + O\x  /</span>
-<span class="go">    2     24     720     8064</span>
-</pre></div>
-</div>
-<p>Další jednoduchý příklad:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">pprint</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;y&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="go">1/y - x/y**2 + x**2/y**3 - x**3/y**4 + x**4/y**5 + O(x**5)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="go">          2    3    4</span>
-<span class="go">1   x    x    x    x     / 5\</span>
-<span class="go">- - -- + -- - -- + -- + O\x /</span>
-<span class="go">y    2    3    4    5</span>
-<span class="go">    y    y    y    y</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="summation">
-<span id="index-4"></span><h3>Summation<a class="headerlink" href="#summation" title="Trvalý odkaz na tento nadpis">¶</a></h3>
-<p>Compute the summation of f with respect to the given summation variable over the given limits.</p>
-<p>summation(f, (i, a, b)) computes the sum of f with respect to i from a to b,
-i.e.,</p>
-<div class="highlight-python"><pre>                            b
-                          ____
-                          \   `
-summation(f, (i, a, b)) =  )    f
-                          /___,
-                          i = a</pre>
-</div>
-<p>If it cannot compute the sum, it prints the corresponding summation formula.
-Repeated sums can be computed by introducing additional limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">summation</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">log</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i n m&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-<span class="go"> 2</span>
-<span class="go">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="o">**</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">  oo</span>
-<span class="go"> ___</span>
-<span class="go"> \  `</span>
-<span class="go">  \     -n</span>
-<span class="go">  /   log (n)</span>
-<span class="go"> /__,</span>
-<span class="go">n = 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">),</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">m</span><span class="p">))</span>
-<span class="go">      3    2</span>
-<span class="go">m    m    m</span>
-<span class="go">-- + -- + -</span>
-<span class="go">6    2    3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go"> x</span>
-<span class="go">e</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="integration">
-<span id="index-5"></span><h3>Integrály<a class="headerlink" href="#integration" title="Trvalý odkaz na tento nadpis">¶</a></h3>
-<p>SymPy umí počítat neurčité a určité integrály elementárních a speciálních transcendentních funkcí pomocí fuknce <tt class="docutils literal"><span class="pre">integrate()</span></tt>, která využívá silného rozšířeného Risch-Normanova algoritmu, některých heuristik a porovnávání vzorků:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">integrate</span><span class="p">,</span> <span class="n">erf</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Můžete integrovat elementární funkce:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="mi">6</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">5</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go"> 6</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">-cos(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x*log(x) - x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">sinh</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go"> 2</span>
-<span class="go">x  + cosh(x)</span>
-</pre></div>
-</div>
-<p>Také speciální funkce lze jednoduše integrovat:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">erf</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">  ____    2</span>
-<span class="go">\/ pi *erf (x)</span>
-<span class="go">--------------</span>
-<span class="go">      4</span>
-</pre></div>
-</div>
-<p>It is possible to compute definite integrals:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-<p>Also, improper integrals are supported as well:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">-1</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="complex-numbers">
-<span id="index-6"></span><h3>Komplexní čísla<a class="headerlink" href="#complex-numbers" title="Trvalý odkaz na tento nadpis">¶</a></h3>
-<p>Kromě imaginárního čísla, I, symboly mohou být vytvořeny s atributy (například real, positive, complex, atd...), a to ovlivní jak se budou chovat:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span> <span class="c"># a plain x with no attributes</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go"> I*x</span>
-<span class="go">e</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">   -im(x)               -im(x)</span>
-<span class="go">I*e      *sin(re(x)) + e      *cos(re(x))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">I*sin(x) + cos(x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="functions">
-<h3>Funkce<a class="headerlink" href="#functions" title="Trvalý odkaz na tento nadpis">¶</a></h3>
-<p><strong>trigonometrické</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">asin</span><span class="p">,</span> <span class="n">asinh</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">sin(x)*cos(y) + sin(y)*cos(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">-sin(x)*sin(y) + cos(x)*cos(y)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*sinh(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinh</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*sin(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
-<span class="go">I*pi</span>
-<span class="go">----</span>
-<span class="go"> 2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*asin(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3     5     7       9</span>
-<span class="go">    x     x     x       x       / 10\</span>
-<span class="go">x - -- + --- - ---- + ------ + O\x  /</span>
-<span class="go">    6    120   5040   362880</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3     5     7       9</span>
-<span class="go">    x     x     x       x       / 10\</span>
-<span class="go">x + -- + --- + ---- + ------ + O\x  /</span>
-<span class="go">    6    120   5040   362880</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3      5      7       9</span>
-<span class="go">    x    3*x    5*x    35*x     / 10\</span>
-<span class="go">x + -- + ---- + ---- + ----- + O\x  /</span>
-<span class="go">    6     40    112     1152</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3      5      7       9</span>
-<span class="go">    x    3*x    5*x    35*x     / 10\</span>
-<span class="go">x - -- + ---- - ---- + ----- + O\x  /</span>
-<span class="go">    6     40    112     1152</span>
-</pre></div>
-</div>
-<p><strong>Sférické funkce</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Ylm</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">  ___</span>
-<span class="go">\/ 3 *cos(theta)</span>
-<span class="go">----------------</span>
-<span class="go">        ____</span>
-<span class="go">    2*\/ pi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">   ___  I*phi</span>
-<span class="go">-\/ 6 *e     *sin(theta)</span>
-<span class="go">------------------------</span>
-<span class="go">            ____</span>
-<span class="go">        4*\/ pi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">   ____  I*phi</span>
-<span class="go">-\/ 30 *e     *sin(theta)*cos(theta)</span>
-<span class="go">------------------------------------</span>
-<span class="go">                  ____</span>
-<span class="go">              4*\/ pi</span>
-</pre></div>
-</div>
-<p><strong>Faktoriály a gamma funkce</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">x!</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">n!</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="c"># i.e. factorial(x)</span>
-<span class="go">                      /          2     2\</span>
-<span class="go">                    2 |EulerGamma    pi |    / 3\</span>
-<span class="go">1 - EulerGamma*x + x *|----------- + ---| + O\x /</span>
-<span class="go">                      \     2         12/</span>
-</pre></div>
-</div>
-<p><strong>Zeta funkce</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">zeta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">zeta(4, x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">  4</span>
-<span class="go">pi</span>
-<span class="go">---</span>
-<span class="go"> 90</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">       4</span>
-<span class="go">     pi</span>
-<span class="go">-1 + ---</span>
-<span class="go">      90</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">         4</span>
-<span class="go">  17   pi</span>
-<span class="go">- -- + ---</span>
-<span class="go">  16    90</span>
-</pre></div>
-</div>
-<p><strong>Polynomy</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">assoc_legendre</span><span class="p">,</span> <span class="n">chebyshevt</span><span class="p">,</span> <span class="n">legendre</span><span class="p">,</span> <span class="n">hermite</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">2*x  - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   4      2</span>
-<span class="go">8*x  - 8*x  + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">3*x    1</span>
-<span class="go">---- - -</span>
-<span class="go"> 2     2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">      8         6         4        2</span>
-<span class="go">6435*x    3003*x    3465*x    315*x     35</span>
-<span class="go">------- - ------- + ------- - ------ + ---</span>
-<span class="go">  128        32        64       32     128</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">        __________</span>
-<span class="go">       /    2</span>
-<span class="go">-3*x*\/  - x  + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">     2</span>
-<span class="go">- 3*x  + 3</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hermite</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   3</span>
-<span class="go">8*x  - 12*x</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="differential-equations">
-<span id="index-7"></span><h3>Diferenciální rovnice<a class="headerlink" href="#differential-equations" title="Trvalý odkaz na tento nadpis">¶</a></h3>
-<p>V <tt class="docutils literal"><span class="pre">isympy</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">dsolve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">        2</span>
-<span class="go">       d</span>
-<span class="go">f(x) + ---(f(x))</span>
-<span class="go">         2</span>
-<span class="go">       dx</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dsolve</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">f(x) = C1*sin(x) + C2*cos(x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="algebraic-equations">
-<span id="index-8"></span><h3>Algebraické rovnice<a class="headerlink" href="#algebraic-equations" title="Trvalý odkaz na tento nadpis">¶</a></h3>
-<p>V <tt class="docutils literal"><span class="pre">isympy</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">solve</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">[-1, 1, -I, I]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">([</span><span class="n">x</span> <span class="o">+</span> <span class="mi">5</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">6</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">15</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">])</span>
-<span class="go">{x: -3, y: 1}</span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="section" id="linear-algebra">
-<span id="index-9"></span><h2>Lineární algebra<a class="headerlink" href="#linear-algebra" title="Trvalý odkaz na tento nadpis">¶</a></h2>
-<div class="section" id="matrices">
-<span id="index-10"></span><h3>Matice<a class="headerlink" href="#matrices" title="Trvalý odkaz na tento nadpis">¶</a></h3>
-<p>Matice jsou vytvořeny jako instance třídy Matrix:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="go">[1  0]</span>
-<span class="go">[    ]</span>
-<span class="go">[0  1]</span>
-</pre></div>
-</div>
-<p>They can also contain symbols:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">],</span> <span class="p">[</span><span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span>
-<span class="go">[1  x]</span>
-<span class="go">[    ]</span>
-<span class="go">[y  1]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">[x*y + 1    2*x  ]</span>
-<span class="go">[                ]</span>
-<span class="go">[  2*y    x*y + 1]</span>
-</pre></div>
-</div>
-<p>For more about Matrices, see the Linear Algebra tutorial.</p>
-</div>
-</div>
-<div class="section" id="pattern-matching">
-<span id="index-11"></span><h2>Porovnávání vzorků<a class="headerlink" href="#pattern-matching" title="Trvalý odkaz na tento nadpis">¶</a></h2>
-<p>Použijte metodu <tt class="docutils literal"><span class="pre">.match()</span></tt> spolu s třídou <tt class="docutils literal"><span class="pre">Wild</span></tt> pro porovnávání výrazů. Metoda vrátí slovník s požadovanými substitucemi, jako vidíme zde:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Wild</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">{p: 5}</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">q</span><span class="p">)</span>
-<span class="go">{p: 1, q: 2}</span>
-</pre></div>
-</div>
-<p>Pokud je porovnání neúspěšné, vrátí <tt class="docutils literal"><span class="pre">None</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">**</span><span class="n">x</span><span class="p">))</span>
-<span class="go">None</span>
-</pre></div>
-</div>
-<p>Lze také použít parametr <tt class="docutils literal"><span class="pre">exclude</span></tt> třídy <tt class="docutils literal"><span class="pre">Wild</span></tt>, aby se určité věci nezobrazily ve výsledku:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">p</span><span class="p">))</span> <span class="c"># 1 is excluded</span>
-<span class="go">None</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span> <span class="c"># x is excluded</span>
-<span class="go">None</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">p</span><span class="p">))</span> <span class="c"># -1 is not excluded</span>
-<span class="go">{p_: -1}</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="printing">
-<span id="printing-tutorial"></span><h2>Výpis<a class="headerlink" href="#printing" title="Trvalý odkaz na tento nadpis">¶</a></h2>
-<p>Je spousta způsobů, jak mohou být výrazy vypisovány.</p>
-<p><strong>Standardní</strong></p>
-<p>To je to, co vrátí <tt class="docutils literal"><span class="pre">str(expression)</span></tt> a vypadá to takto:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1/x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Integral(x**2, x)</span>
-</pre></div>
-</div>
-<p><strong>Pěkný výpis</strong></p>
-<p>Nice ascii-art printing is produced by the <tt class="docutils literal"><span class="pre">pprint</span></tt> function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go"> 2</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="go">-</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">  /</span>
-<span class="go"> |</span>
-<span class="go"> |  2</span>
-<span class="go"> | x  dx</span>
-<span class="go"> |</span>
-<span class="go">/</span>
-</pre></div>
-</div>
-<p>If you have a unicode font installed, the <tt class="docutils literal"><span class="pre">pprint</span></tt> function will use it by
-default. You can override this using the <tt class="docutils literal"><span class="pre">use_unicode</span></tt> option.:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">⌠</span>
-<span class="go">⎮  2</span>
-<span class="go">⎮ x  dx</span>
-<span class="go">⌡</span>
-</pre></div>
-</div>
-<p>Pro více příkladů pěkného unicode výstupu koukněte na wiki na <a class="reference external" href="https://github.com/sympy/sympy/wiki/Pretty-Printing">Pretty Printing</a></p>
-<p>Tip: To make pretty printing the default in the Python interpreter, use:</p>
-<div class="highlight-python"><pre>$ python
-Python 2.5.2 (r252:60911, Jun 25 2008, 17:58:32)
-[GCC 4.3.1] on linux2
-Type &quot;help&quot;, &quot;copyright&quot;, &quot;credits&quot; or &quot;license&quot; for more information.
-&gt;&gt;&gt; from sympy import init_printing, var, Integral
-&gt;&gt;&gt; init_printing(use_unicode=False, wrap_line=False, no_global=True)
-&gt;&gt;&gt; var(&quot;x&quot;)
-x
-&gt;&gt;&gt; x**3/3
- 3
-x
---
-3
-&gt;&gt;&gt; Integral(x**2, x) #doctest: +NORMALIZE_WHITESPACE
-  /
- |
- |  2
- | x  dx
- |
-/</pre>
-</div>
-<p><strong>Python výstup</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.python</span> <span class="kn">import</span> <span class="n">python</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = x**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = 1/x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = Integral(x**2, x)</span>
-</pre></div>
-</div>
-<p><strong>LaTeX výstup</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">latex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x^{2}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;inline&#39;</span><span class="p">)</span>
-<span class="go">$x^{2}$</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;equation&#39;</span><span class="p">)</span>
-<span class="go">\begin{equation}x^{2}\end{equation}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;equation*&#39;</span><span class="p">)</span>
-<span class="go">\begin{equation*}x^{2}\end{equation*}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">\frac{1}{x}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">\int x^{2}\, dx</span>
-</pre></div>
-</div>
-<p><strong>MathML</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.mathml</span> <span class="kn">import</span> <span class="n">mathml</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">latex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">mathml</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">&lt;apply&gt;&lt;power/&gt;&lt;ci&gt;x&lt;/ci&gt;&lt;cn&gt;2&lt;/cn&gt;&lt;/apply&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">mathml</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">))</span>
-<span class="go">&lt;apply&gt;&lt;power/&gt;&lt;ci&gt;x&lt;/ci&gt;&lt;cn&gt;-1&lt;/cn&gt;&lt;/apply&gt;</span>
-</pre></div>
-</div>
-<p><strong>Pyglet</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">preview</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">preview</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span> 
-</pre></div>
-</div>
-<p>If pyglet is installed, a pyglet window will open containing the LaTeX
-rendered expression:</p>
-<img alt="../_images/pngview1.png" src="../_images/pngview1.png" />
-<div class="section" id="notes">
-<h3>Poznámky<a class="headerlink" href="#notes" title="Trvalý odkaz na tento nadpis">¶</a></h3>
-<p><tt class="docutils literal"><span class="pre">isympy</span></tt> volá <tt class="docutils literal"><span class="pre">pprint</span></tt> automaticky, proto standardně vidíte pěkný výpis.</p>
-<p>Všimněte si, že je přístupný modul pro výpisy, <tt class="docutils literal"><span class="pre">sympy.printing</span></tt>. Další metody dostupné díky tomuto modulu jsou:</p>
-<ul class="simple">
-<li><tt class="docutils literal"><span class="pre">pretty(expr)</span></tt>, <tt class="docutils literal"><span class="pre">pretty_print(expr)</span></tt>, <tt class="docutils literal"><span class="pre">pprint(expr)</span></tt>: Return or print, respectively, a pretty representation of <tt class="docutils literal"><span class="pre">expr</span></tt>. This is the same as the second level of representation described above.</li>
-<li><tt class="docutils literal"><span class="pre">latex(expr)</span></tt>, <tt class="docutils literal"><span class="pre">print_latex(expr)</span></tt>: Vrací, respektive vypisuje, <a class="reference external" href="http://www.latex-project.org/">LaTeX</a>  reprezentaci <tt class="docutils literal"><span class="pre">expr</span></tt></li>
-<li><tt class="docutils literal"><span class="pre">mathml(expr)</span></tt>, <tt class="docutils literal"><span class="pre">print_mathml(expr)</span></tt>: Return or print, respectively, a <a class="reference external" href="http://www.w3.org/Math/">MathML</a> representation of <tt class="docutils literal"><span class="pre">expr</span></tt>.</li>
-<li><tt class="docutils literal"><span class="pre">print_gtk(expr)</span></tt>: Print <tt class="docutils literal"><span class="pre">expr</span></tt> to <a class="reference external" href="http://helm.cs.unibo.it/mml-widget/">Gtkmathview</a>, a GTK widget that displays MathML code. The <a class="reference external" href="http://helm.cs.unibo.it/mml-widget/">Gtkmathview</a> program is required.</li>
-</ul>
-</div>
-</div>
-<div class="section" id="further-documentation">
-<h2>Další dokumentace<a class="headerlink" href="#further-documentation" title="Trvalý odkaz na tento nadpis">¶</a></h2>
-<p>Nyní je čas dozvědět se více o SymPy. Pročtěte si <a class="reference external" href="../guide.html">Uživatelskou příručku SymPy</a> a <a class="reference external" href="../modules/index.html">Přehled modulů SymPy</a>.</p>
-<p>Be sure to also browse our public <a class="reference external" href="http://wiki.sympy.org/">wiki.sympy.org</a>,
-that contains a lot of useful examples, tutorials, cookbooks that we and our
-users contributed, and feel free to edit it.</p>
-<div class="section" id="translations">
-<h3>Překlady<a class="headerlink" href="#translations" title="Trvalý odkaz na tento nadpis">¶</a></h3>
-<p>Tento tutoriál je také k dispozici v těchto jazycích:</p>
-</div>
-<ul class="simple">
-<li><a class="reference external" href="tutorial.bg.html">Български</a></li>
-<li><a class="reference external" href="tutorial.cs.html">Česky</a></li>
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-<li><a class="reference external" href="tutorial.ru.html">Русский</a></li>
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-<li><a class="reference external" href="tutorial.zh.html">中文</a></li>
-</ul>
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-
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-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="tutorial.html">
-              <img class="logo" src="../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="tutorial.html">Obsah</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Tutoriál</a><ul>
-<li><a class="reference internal" href="#introduction">Úvod</a></li>
-<li><a class="reference internal" href="#first-steps-with-sympy">První kroky se SymPy</a><ul>
-<li><a class="reference internal" href="#isympy-console">isympy konzole</a></li>
-<li><a class="reference internal" href="#using-sympy-as-a-calculator">SymPy jako kalkulačka</a></li>
-<li><a class="reference internal" href="#symbols">Symboly</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#algebra">Algebra</a></li>
-<li><a class="reference internal" href="#calculus">Analýza</a><ul>
-<li><a class="reference internal" href="#limits">Limity</a></li>
-<li><a class="reference internal" href="#differentiation">Derivace</a></li>
-<li><a class="reference internal" href="#series-expansion">Rozvoj v řady</a></li>
-<li><a class="reference internal" href="#summation">Summation</a></li>
-<li><a class="reference internal" href="#integration">Integrály</a></li>
-<li><a class="reference internal" href="#complex-numbers">Komplexní čísla</a></li>
-<li><a class="reference internal" href="#functions">Funkce</a></li>
-<li><a class="reference internal" href="#differential-equations">Diferenciální rovnice</a></li>
-<li><a class="reference internal" href="#algebraic-equations">Algebraické rovnice</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#linear-algebra">Lineární algebra</a><ul>
-<li><a class="reference internal" href="#matrices">Matice</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#pattern-matching">Porovnávání vzorků</a></li>
-<li><a class="reference internal" href="#printing">Výpis</a><ul>
-<li><a class="reference internal" href="#notes">Poznámky</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#further-documentation">Další dokumentace</a><ul>
-<li><a class="reference internal" href="#translations">Překlady</a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-
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diff --git a/dev-py3k/tutorial/tutorial.de.html b/dev-py3k/tutorial/tutorial.de.html
deleted file mode 100644
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+++ /dev/null
@@ -1,974 +0,0 @@
-
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-            
-  <div class="section" id="tutorial">
-<span id="id1"></span><h1>Tutorial<a class="headerlink" href="#tutorial" title="Permalink zu dieser Überschrift">¶</a></h1>
-<div class="section" id="introduction">
-<h2>Einführung<a class="headerlink" href="#introduction" title="Permalink zu dieser Überschrift">¶</a></h2>
-<p>SymPy ist eine Python-Bibliothek für symbolische Mathematik. SymPy soll ein vollständiges Computer-Algebra-System (CAS) bereitstellen und dabei den Programmcode so einfach halten, dass er nachvollziehbar und einfach erweiterbar bleibt. SymPy ist vollständig in Python geschrieben und benötigt keine weiteren Bibliotheken.</p>
-<p>Dieses Tutorial schafft einen Überblick und gibt eine Einführung in SymPy. Wenn du wissen möchtest, was SymPy tun kann (und wie), lies diese Seite, und wenn du mehr erfahren möchtest, lies den <a class="reference external" href="../guide.html">SymPy User&#8217;s Guide</a>, die <a class="reference external" href="../modules/index.html">SymPy Modules Reference</a>, oder den <a class="reference external" href="https://github.com/sympy/sympy/">Quellcode</a> selbst.</p>
-</div>
-<div class="section" id="first-steps-with-sympy">
-<h2>Erste Schritte mit SymPy<a class="headerlink" href="#first-steps-with-sympy" title="Permalink zu dieser Überschrift">¶</a></h2>
-<p>Am einfachsten kannst du SymPy mit der aktuellsten .tar-Datei aus den &#8220;Featured Downloads&#8221; von <a class="reference external" href="http://code.google.com/p/sympy/">http://code.google.com/p/sympy/</a> installieren:</p>
-<img alt="../_images/featured-downloads.png" src="../_images/featured-downloads.png" />
-<p>Dann musst du die Datei auspacken:</p>
-<pre class="literal-block">
-$ <strong class="input">tar xzf sympy-0.5.12.tar.gz</strong>
-</pre>
-<p>und kannst sie mit einem Python-Interpreter ausprobieren.</p>
-<pre class="literal-block">
-$ <strong class="input">cd sympy-0.5.12</strong>
-$ <strong class="input">python</strong>
-Python 2.4.4 (#2, Jan  3 2008, 13:36:28)
-[GCC 4.2.3 20071123 (prerelease) (Debian 4.2.2-4)] on linux2
-Type &quot;help&quot;, &quot;copyright&quot;, &quot;credits&quot; or &quot;license&quot; for more information.
-&gt;&gt;&gt; from sympy import Symbol, cos
-&gt;&gt;&gt; x = Symbol(&quot;x&quot;)
-&gt;&gt;&gt; (1/cos(x)).series(x, 0, 10)
-1 + x**2/2 + 5*x**4/24 + 61*x**6/720 + 277*x**8/8064 + O(x**10)
-</pre>
-<p>SymPy kann wie im Beispiel gezeigt benutzt werden, und dies ist auch der übliche Weg, es von einem anderen Programm aus zu verwenden. Du kannst es ebenfalls wie jedes andere Modul mithilfe von <tt class="docutils literal"><span class="pre">./setup.py</span> <span class="pre">install</span></tt> installieren. Auch ist es natürlich möglich, einfach ein Paket deiner Linux-Distribution zu verwenden, zum Beispiel:</p>
-<div class="topic">
-<p class="topic-title first">Installing SymPy in Debian</p>
-<pre class="literal-block">
-$ <strong class="input">sudo apt-get install python-sympy</strong>
-Reading package lists... Done
-Building dependency tree
-Reading state information... Done
-The following NEW packages will be installed:
-  python-sympy
-0 upgraded, 1 newly installed, 0 to remove and 18 not upgraded.
-Need to get 991kB of archives.
-After this operation, 5976kB of additional disk space will be used.
-Get:1 http://ftp.cz.debian.org unstable/main python-sympy 0.5.12-1 [991kB]
-Fetched 991kB in 2s (361kB/s)
-Selecting previously deselected package python-sympy.
-(Reading database ... 232619 files and directories currently installed.)
-Unpacking python-sympy (from .../python-sympy_0.5.12-1_all.deb) ...
-Setting up python-sympy (0.5.12-1) ...
-</pre>
-</div>
-<p>Weitere Installationsmöglichkeiten findest du unter <a class="reference external" href="https://github.com/sympy/sympy/wiki/Download-Installation">Download and Installation (auf Englisch)</a> auf der SymPy-Wiki.</p>
-<div class="section" id="isympy-console">
-<h3>isympy-Konsole<a class="headerlink" href="#isympy-console" title="Permalink zu dieser Überschrift">¶</a></h3>
-<p>Für Experimente mit neuen Funktionen und zum Ausprobieren kann die um IPython aufgebaute Umgebung namens <tt class="docutils literal"><span class="pre">isympy</span></tt> (im Quellverzeichnis unter <tt class="docutils literal"><span class="pre">bin/isympy</span></tt>) benutzt werden. Es handelt sich hierbei nur um eine Standard-Python-Konsole, die allerdings bereits die wichtigen sympy-Module importiert hat und unter anderem die Symbole x, y und z bereits definiert hat:</p>
-<pre class="literal-block">
-$ <strong class="input">cd sympy</strong>
-$ <strong class="input">./bin/isympy</strong>
-IPython console for SymPy 0.7.2-git (Python 2.7.1) (ground types: gmpy)
-
-These commands were executed:
-&gt;&gt;&gt;
-&gt;&gt;&gt; from sympy import *
-&gt;&gt;&gt; x, y, z, t = symbols('x y z t')
-&gt;&gt;&gt; k, m, n = symbols('k m n', integer=True)
-&gt;&gt;&gt; f, g, h = symbols('f g h', cls=Function)
-
-Documentation can be found at <a class="reference external" href="http://www.sympy.org">http://www.sympy.org</a>
-
-In [1]: <strong class="input">(1/cos(x)).series(x, 0, 10)</strong>
-Out[1]:
-     2      4       6        8
-    x    5*x    61*x    277*x     / 10\
-1 + -- + ---- + ----- + ------ + O\x  /
-    2     24     720     8064
-</pre>
-<div class="admonition note">
-<p class="first admonition-title">Bemerkung</p>
-<p class="last">Benutzereingaben sind fett dargestellt. Was in einem normalen Python-Interpreter drei Zeilen gebraucht hätte, kann mit isympy in nur einer Zeile ausgedrückt werden.</p>
-</div>
-</div>
-<div class="section" id="using-sympy-as-a-calculator">
-<h3>Mit SymPy rechnen<a class="headerlink" href="#using-sympy-as-a-calculator" title="Permalink zu dieser Überschrift">¶</a></h3>
-<p>SymPy hat drei eingebaute Zahlentypen: Float, Rational und Integer.</p>
-<p>Die Klasse Rational stellt eine rationale Zahl als Paar von zwei Ganzzahlen dar: dem Zähler und dem Nenner. Rational(1, 2) repräsentiert also 1/2, Rational(5, 2) repräsentiert 5/2 und so weiter.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
-<span class="go">1/2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">*</span><span class="mi">2</span>
-<span class="go">1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">50</span><span class="o">/</span><span class="n">Rational</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span><span class="o">**</span><span class="mi">50</span>
-<span class="go">1/88817841970012523233890533447265625</span>
-</pre></div>
-</div>
-<p>Beim Arbeiten mit Pythons Ganzzahlen und Fließkommazahlen ist Vorsicht geboten, besonders bei Division, da eine Python-Zahl statt einer SymPy-Zahl das Ergebnis sein kann. Eine Division von zwei Python-Ganzzahlen hat eine Fließkommazahl zum Ergebnis &#8211; die &#8220;echte Divison&#8221; von Python 3 und das Standardverhalten von <tt class="docutils literal"><span class="pre">isympy</span></tt>, welches division aus __future__ importiert:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span> 
-<span class="go">0.5</span>
-</pre></div>
-</div>
-<p>In älteren Python-Versionen ohne den Import von division ist das Ergebnis hingegen eine abgerundete Ganzzahl:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span> 
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>In beiden Fällen handelt es sich jedoch nicht um eine SymPy-Zahl, weil Python seine eigenen Zahl zurückgegeben hat. Meistens wirst du aber mit Zahlen des Typs Rational arbeiten wollen, also achte darauf, auch solche zu erhalten. Oft wird zur besseren Lesbarkeit <tt class="docutils literal"><span class="pre">R</span></tt> mit Rational gleichgesetzt:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="c"># R(1) is a SymPy Integer and Integer/int gives a Rational</span>
-<span class="go">1/2</span>
-</pre></div>
-</div>
-<p>Es gibt auch einige spezielle Konstanten wie e oder pi, die als Symbole behandelt werden (1 + pi wird nicht als Zahl ausgerechnet, sondern bleibt 1 + pi) und in beliebiger Präzision verfügbar sind:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">E</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">pi**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">3.14159265358979</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">pi</span> <span class="o">+</span> <span class="n">E</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">5.85987448204884</span>
-</pre></div>
-</div>
-<p>Wie man sieht, berechnet evalf aus dem Ausdruck eine Fließkommazahl.</p>
-<p>Das Symnol <tt class="docutils literal"><span class="pre">oo</span></tt> wird für eine Klasse benutzt, die mathematische Unendlichkeit darstellt:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">oo</span> <span class="o">&gt;</span> <span class="mi">99999</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">oo</span> <span class="o">+</span> <span class="mi">1</span>
-<span class="go">oo</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="symbols">
-<h3>Symbole<a class="headerlink" href="#symbols" title="Permalink zu dieser Überschrift">¶</a></h3>
-<p>Im Gegensatz zu anderen Computer-Algebra-Systemen müssen in SymPy symbolische Variablen ausdrücklich deklariert werden:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Links steht eine normale Python-Variable, der ein SymPy-Symbol-Objekt zugewiesen wird. Vordefiniert Symbole (inklusiv Symbole mit griechischen Namen) sind von Import von abc verfügbar:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">theta</span>
-</pre></div>
-</div>
-<p>Symbole können auch mit den <tt class="docutils literal"><span class="pre">symbols</span></tt> oder <tt class="docutils literal"><span class="pre">var</span></tt> Funktionen erstellt werden. Die letztere fügt automatisch die Symbole in den Namespace ein, und beide Funktionen akzeptieren eine Reihennotation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">var</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a,b,c&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;d:f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;g:h&#39;</span><span class="p">)</span>
-<span class="go">(g, h)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;g:2&#39;</span><span class="p">)</span>
-<span class="go">(g0, g1)</span>
-</pre></div>
-</div>
-<p>Aus Symbol-Objekten kann man bequem Ausdrücke zusammensetzen:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">x</span> <span class="o">-</span> <span class="n">y</span>
-<span class="go">2*x</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">(x + y)**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go">x**2 + 2*x*y + y**2</span>
-</pre></div>
-</div>
-<p>Sie können mit <tt class="docutils literal"><span class="pre">subs(old,</span> <span class="pre">new)</span></tt> durch Zahlen, andere Symbole oder Ausdrücke ersetzt werden:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">(y + 1)**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">4*y**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>Für den Rest des Tutorials gehen wir davon aus, dass folgende Zeile ausgeführt wurde:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">init_printing</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">init_printing</span><span class="p">(</span><span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">wrap_line</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">no_global</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Dies sorgt dafür, dass Ausdrücke bei der Ausgabe besser aussehen (siehe <a class="reference internal" href="#printing-tutorial"><em>Ausgabe</em></a> weiter unten). Wenn eine Unicode-Schrift installiert ist, erreichst du mit use_unicode=True eine noch hübschere Ausgabe.</p>
-</div>
-</div>
-<div class="section" id="algebra">
-<h2>Algebra<a class="headerlink" href="#algebra" title="Permalink zu dieser Überschrift">¶</a></h2>
-<p>Für die Partialbruchzerlegung kannst du <tt class="docutils literal"><span class="pre">apart(expr,</span> <span class="pre">x)</span></tt> benutzen:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">apart</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">)</span>
-<span class="go">       1</span>
-<span class="go">---------------</span>
-<span class="go">(x + 1)*(x + 2)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">apart</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">    1       1</span>
-<span class="go">- ----- + -----</span>
-<span class="go">  x + 2   x + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">x + 1</span>
-<span class="go">-----</span>
-<span class="go">x - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">apart</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">      2</span>
-<span class="go">1 + -----</span>
-<span class="go">    x - 1</span>
-</pre></div>
-</div>
-<p>Zum Kombinieren gibt es die Funktion <tt class="docutils literal"><span class="pre">together(expr,</span> <span class="pre">x)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">together</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">)</span>
-<span class="go">x*y + x*z + y*z</span>
-<span class="go">---------------</span>
-<span class="go">     x*y*z</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="n">apart</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x + 1</span>
-<span class="go">-----</span>
-<span class="go">x - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="n">apart</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">       1</span>
-<span class="go">---------------</span>
-<span class="go">(x + 1)*(x + 2)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="calculus">
-<span id="index-0"></span><h2>Infinitesimalrechnung<a class="headerlink" href="#calculus" title="Permalink zu dieser Überschrift">¶</a></h2>
-<div class="section" id="limits">
-<span id="index-1"></span><h3>Limes<a class="headerlink" href="#limits" title="Permalink zu dieser Überschrift">¶</a></h3>
-<p>Grenzwerte sind in SymPy einfach zu benutzen, sie folgen der Syntax <tt class="docutils literal"><span class="pre">limit(function,</span> <span class="pre">variable,</span> <span class="pre">point)</span></tt>, um also den Grenzwert von f(x) bei x -&gt; 0 zu berechnen, kann <tt class="docutils literal"><span class="pre">limit(f,</span> <span class="pre">x,</span> <span class="pre">0)</span></tt> benutzt werden:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">limit</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>Analog kann der Limes für x gegen unendlich berechnet werden:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">oo</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">0</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>Einige nicht-triviale Beispiele zu Grenzwerten finden sich in der Datei <a class="reference external" href="https://github.com/sympy/sympy/blob/master/sympy/series/tests/test_demidovich.py">test_demidovich.py</a></p>
-</div>
-<div class="section" id="differentiation">
-<span id="index-2"></span><h3>Differentialrechnung<a class="headerlink" href="#differentiation" title="Permalink zu dieser Überschrift">¶</a></h3>
-<p>Mithilfe von <tt class="docutils literal"><span class="pre">diff(func,</span> <span class="pre">var)</span></tt> kann jeder SymPy-Ausdruck differenziert werden. Beispiele:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">diff</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">tan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">cos(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">2*cos(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">tan (x) + 1</span>
-</pre></div>
-</div>
-<p>Folgendermaßen kann überprüft werden, ob dies korrekt ist:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">limit</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">delta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">((</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">delta</span><span class="p">)</span> <span class="o">-</span> <span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">/</span><span class="n">delta</span><span class="p">,</span> <span class="n">delta</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">tan (x) + 1</span>
-</pre></div>
-</div>
-<p>Höhere Ableitungen können mithilfe von <tt class="docutils literal"><span class="pre">diff(func,</span> <span class="pre">var,</span> <span class="pre">n)</span></tt> berechnet werden:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">2*cos(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">-4*sin(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">-8*cos(2*x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="series-expansion">
-<span id="index-3"></span><h3>Reihenentwicklung<a class="headerlink" href="#series-expansion" title="Permalink zu dieser Überschrift">¶</a></h3>
-<p>Benutze <tt class="docutils literal"><span class="pre">.series(var,</span> <span class="pre">point,</span> <span class="pre">order)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     2    4     6      8</span>
-<span class="go">    x    x     x      x      / 10\</span>
-<span class="go">1 - -- + -- - --- + ----- + O\x  /</span>
-<span class="go">    2    24   720   40320</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     2      4       6        8</span>
-<span class="go">    x    5*x    61*x    277*x     / 10\</span>
-<span class="go">1 + -- + ---- + ----- + ------ + O\x  /</span>
-<span class="go">    2     24     720     8064</span>
-</pre></div>
-</div>
-<p>Ein weiteres einfaches Beispiel:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">pprint</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;y&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="go">1/y - x/y**2 + x**2/y**3 - x**3/y**4 + x**4/y**5 + O(x**5)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="go">          2    3    4</span>
-<span class="go">1   x    x    x    x     / 5\</span>
-<span class="go">- - -- + -- - -- + -- + O\x /</span>
-<span class="go">y    2    3    4    5</span>
-<span class="go">    y    y    y    y</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="summation">
-<span id="index-4"></span><h3>Summation<a class="headerlink" href="#summation" title="Permalink zu dieser Überschrift">¶</a></h3>
-<p>Compute the summation of f with respect to the given summation variable over the given limits.</p>
-<p>summation(f, (i, a, b)) computes the sum of f with respect to i from a to b,
-i.e.,</p>
-<div class="highlight-python"><pre>                            b
-                          ____
-                          \   `
-summation(f, (i, a, b)) =  )    f
-                          /___,
-                          i = a</pre>
-</div>
-<p>If it cannot compute the sum, it prints the corresponding summation formula.
-Repeated sums can be computed by introducing additional limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">summation</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">log</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i n m&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-<span class="go"> 2</span>
-<span class="go">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="o">**</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">  oo</span>
-<span class="go"> ___</span>
-<span class="go"> \  `</span>
-<span class="go">  \     -n</span>
-<span class="go">  /   log (n)</span>
-<span class="go"> /__,</span>
-<span class="go">n = 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">),</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">m</span><span class="p">))</span>
-<span class="go">      3    2</span>
-<span class="go">m    m    m</span>
-<span class="go">-- + -- + -</span>
-<span class="go">6    2    3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go"> x</span>
-<span class="go">e</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="integration">
-<span id="index-5"></span><h3>Integralrechnung<a class="headerlink" href="#integration" title="Permalink zu dieser Überschrift">¶</a></h3>
-<p>SymPy unterstützt unendliche und endliche Integration transzendenter elementarer und spezieller Funktionen durch <tt class="docutils literal"><span class="pre">integrate()</span></tt>, welches den starken Risch-Norman-Algorithmus nutzt, sowie einige Heuristiken und Mustererkennungen:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">integrate</span><span class="p">,</span> <span class="n">erf</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Es können elementare Funktionen integriert werden:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="mi">6</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">5</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go"> 6</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">-cos(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x*log(x) - x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">sinh</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go"> 2</span>
-<span class="go">x  + cosh(x)</span>
-</pre></div>
-</div>
-<p>Aber auch mit speziellen Funktionen kann einfach umgegangen werden:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">erf</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">  ____    2</span>
-<span class="go">\/ pi *erf (x)</span>
-<span class="go">--------------</span>
-<span class="go">      4</span>
-</pre></div>
-</div>
-<p>Es ist möglich, ein endliches Integral zu berechnen:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-<p>Auch uneigentliche Integrale werden unterstützt:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">-1</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="complex-numbers">
-<span id="index-6"></span><h3>Komplexe Zahlen<a class="headerlink" href="#complex-numbers" title="Permalink zu dieser Überschrift">¶</a></h3>
-<p>Mit Ausnahme der imaginären Einheit, I, die rein imaginär ist, können Symbole mit Attributen (z.B. reale, positive, komplex, usw.) erstellt werden, und dies hat Auswirkungen darauf, wie sie sich verhalten:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span> <span class="c"># a plain x with no attributes</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go"> I*x</span>
-<span class="go">e</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">   -im(x)               -im(x)</span>
-<span class="go">I*e      *sin(re(x)) + e      *cos(re(x))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">I*sin(x) + cos(x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="functions">
-<h3>Funktionen<a class="headerlink" href="#functions" title="Permalink zu dieser Überschrift">¶</a></h3>
-<p><strong>trigonometrische</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">asin</span><span class="p">,</span> <span class="n">asinh</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">sin(x)*cos(y) + sin(y)*cos(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">-sin(x)*sin(y) + cos(x)*cos(y)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*sinh(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinh</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*sin(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
-<span class="go">I*pi</span>
-<span class="go">----</span>
-<span class="go"> 2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*asin(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3     5     7       9</span>
-<span class="go">    x     x     x       x       / 10\</span>
-<span class="go">x - -- + --- - ---- + ------ + O\x  /</span>
-<span class="go">    6    120   5040   362880</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3     5     7       9</span>
-<span class="go">    x     x     x       x       / 10\</span>
-<span class="go">x + -- + --- + ---- + ------ + O\x  /</span>
-<span class="go">    6    120   5040   362880</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3      5      7       9</span>
-<span class="go">    x    3*x    5*x    35*x     / 10\</span>
-<span class="go">x + -- + ---- + ---- + ----- + O\x  /</span>
-<span class="go">    6     40    112     1152</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3      5      7       9</span>
-<span class="go">    x    3*x    5*x    35*x     / 10\</span>
-<span class="go">x - -- + ---- - ---- + ----- + O\x  /</span>
-<span class="go">    6     40    112     1152</span>
-</pre></div>
-</div>
-<p><strong>Kugelflächen</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Ylm</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">  ___</span>
-<span class="go">\/ 3 *cos(theta)</span>
-<span class="go">----------------</span>
-<span class="go">        ____</span>
-<span class="go">    2*\/ pi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">   ___  I*phi</span>
-<span class="go">-\/ 6 *e     *sin(theta)</span>
-<span class="go">------------------------</span>
-<span class="go">            ____</span>
-<span class="go">        4*\/ pi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">   ____  I*phi</span>
-<span class="go">-\/ 30 *e     *sin(theta)*cos(theta)</span>
-<span class="go">------------------------------------</span>
-<span class="go">                  ____</span>
-<span class="go">              4*\/ pi</span>
-</pre></div>
-</div>
-<p><strong>Fakultät und Gamma-Funktion</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">x!</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">n!</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="c"># i.e. factorial(x)</span>
-<span class="go">                      /          2     2\</span>
-<span class="go">                    2 |EulerGamma    pi |    / 3\</span>
-<span class="go">1 - EulerGamma*x + x *|----------- + ---| + O\x /</span>
-<span class="go">                      \     2         12/</span>
-</pre></div>
-</div>
-<p><strong>Zeta-Funktion</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">zeta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">zeta(4, x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">  4</span>
-<span class="go">pi</span>
-<span class="go">---</span>
-<span class="go"> 90</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">       4</span>
-<span class="go">     pi</span>
-<span class="go">-1 + ---</span>
-<span class="go">      90</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">         4</span>
-<span class="go">  17   pi</span>
-<span class="go">- -- + ---</span>
-<span class="go">  16    90</span>
-</pre></div>
-</div>
-<p><strong>Polynome</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">assoc_legendre</span><span class="p">,</span> <span class="n">chebyshevt</span><span class="p">,</span> <span class="n">legendre</span><span class="p">,</span> <span class="n">hermite</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">2*x  - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   4      2</span>
-<span class="go">8*x  - 8*x  + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">3*x    1</span>
-<span class="go">---- - -</span>
-<span class="go"> 2     2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">      8         6         4        2</span>
-<span class="go">6435*x    3003*x    3465*x    315*x     35</span>
-<span class="go">------- - ------- + ------- - ------ + ---</span>
-<span class="go">  128        32        64       32     128</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">        __________</span>
-<span class="go">       /    2</span>
-<span class="go">-3*x*\/  - x  + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">     2</span>
-<span class="go">- 3*x  + 3</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hermite</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   3</span>
-<span class="go">8*x  - 12*x</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="differential-equations">
-<span id="index-7"></span><h3>Differenzialgleichungen<a class="headerlink" href="#differential-equations" title="Permalink zu dieser Überschrift">¶</a></h3>
-<p>In <tt class="docutils literal"><span class="pre">isympy</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">dsolve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">        2</span>
-<span class="go">       d</span>
-<span class="go">f(x) + ---(f(x))</span>
-<span class="go">         2</span>
-<span class="go">       dx</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dsolve</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">f(x) = C1*sin(x) + C2*cos(x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="algebraic-equations">
-<span id="index-8"></span><h3>Algebraische Gleichungen<a class="headerlink" href="#algebraic-equations" title="Permalink zu dieser Überschrift">¶</a></h3>
-<p>In <tt class="docutils literal"><span class="pre">isympy</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">solve</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">[-1, 1, -I, I]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">([</span><span class="n">x</span> <span class="o">+</span> <span class="mi">5</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">6</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">15</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">])</span>
-<span class="go">{x: -3, y: 1}</span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="section" id="linear-algebra">
-<span id="index-9"></span><h2>Lineare Algebra<a class="headerlink" href="#linear-algebra" title="Permalink zu dieser Überschrift">¶</a></h2>
-<div class="section" id="matrices">
-<span id="index-10"></span><h3>Matrizen<a class="headerlink" href="#matrices" title="Permalink zu dieser Überschrift">¶</a></h3>
-<p>Matrizen werden als Instanzen der Matrix-Klasse erzeugt:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="go">[1  0]</span>
-<span class="go">[    ]</span>
-<span class="go">[0  1]</span>
-</pre></div>
-</div>
-<p>Sie können auch Symbole enthalten:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">],</span> <span class="p">[</span><span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span>
-<span class="go">[1  x]</span>
-<span class="go">[    ]</span>
-<span class="go">[y  1]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">[x*y + 1    2*x  ]</span>
-<span class="go">[                ]</span>
-<span class="go">[  2*y    x*y + 1]</span>
-</pre></div>
-</div>
-<p>Mehr Informationen und Beispiele zu Matrizen finden sich im LinearAlgebraTutorial.</p>
-</div>
-</div>
-<div class="section" id="pattern-matching">
-<span id="index-11"></span><h2>Musterabgleich<a class="headerlink" href="#pattern-matching" title="Permalink zu dieser Überschrift">¶</a></h2>
-<p>Die Methode <tt class="docutils literal"><span class="pre">.match()</span></tt> kann gemeinsam mit der Klasse <tt class="docutils literal"><span class="pre">Wild</span></tt> Ausdrücke auf Muster überprüfen. Die Methode gibt ein dictionary mit den nötigen Ersetzungen zurück, wie im folgenden Beispiel ersichtlich:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Wild</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">{p: 5}</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">q</span><span class="p">)</span>
-<span class="go">{p: 1, q: 2}</span>
-</pre></div>
-</div>
-<p>Wenn der Musterabgleich fehlschlägt, wird <tt class="docutils literal"><span class="pre">None</span></tt> zurückgegeben:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">**</span><span class="n">x</span><span class="p">))</span>
-<span class="go">None</span>
-</pre></div>
-</div>
-<p>Über den Parameter <tt class="docutils literal"><span class="pre">exclude</span></tt> kann man manches aus dem Ergebnis ausschließen:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">p</span><span class="p">))</span> <span class="c"># 1 is excluded</span>
-<span class="go">None</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span> <span class="c"># x is excluded</span>
-<span class="go">None</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">p</span><span class="p">))</span> <span class="c"># -1 is not excluded</span>
-<span class="go">{p_: -1}</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="printing">
-<span id="printing-tutorial"></span><h2>Ausgabe<a class="headerlink" href="#printing" title="Permalink zu dieser Überschrift">¶</a></h2>
-<p>Es existieren mehrere Wege, Ausdrücke auszugeben:</p>
-<p><strong>Standard</strong></p>
-<p>Dies ist die Ausgabe von <tt class="docutils literal"><span class="pre">str(expression)</span></tt> und sieht so aus:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1/x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Integral(x**2, x)</span>
-</pre></div>
-</div>
-<p><strong>ASCII-Art-Ausgabe</strong></p>
-<p>Eine <tt class="docutils literal"><span class="pre">pprint</span></tt>-Funktion erzeugt diese hübschere ASCII-Art-Ausgabe.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go"> 2</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="go">-</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">  /</span>
-<span class="go"> |</span>
-<span class="go"> |  2</span>
-<span class="go"> | x  dx</span>
-<span class="go"> |</span>
-<span class="go">/</span>
-</pre></div>
-</div>
-<p>Wenn eine Unicode-Schriftart installiert ist, sollte die ASCII-Art-Ausgabe standardmäßig die Unicode-Fassung verwenden. Dies kann mit dem Parameter <tt class="docutils literal"><span class="pre">use_unicode</span></tt> erzwungen oder abgeschaltet werden.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">⌠</span>
-<span class="go">⎮  2</span>
-<span class="go">⎮ x  dx</span>
-<span class="go">⌡</span>
-</pre></div>
-</div>
-<p>Siehe auch die Wiki-Seite <a class="reference external" href="https://github.com/sympy/sympy/wiki/Pretty-Printing">Pretty Printing</a> für mehr Beispiele von hübschen Unicode-Ausgaben.</p>
-<p>Tipp: Die ASCII-Art-Ausgabe kann auch als Standard-Methode gesetzt werden:</p>
-<div class="highlight-python"><pre>$ python
-Python 2.5.2 (r252:60911, Jun 25 2008, 17:58:32)
-[GCC 4.3.1] on linux2
-Type &quot;help&quot;, &quot;copyright&quot;, &quot;credits&quot; or &quot;license&quot; for more information.
-&gt;&gt;&gt; from sympy import init_printing, var, Integral
-&gt;&gt;&gt; init_printing(use_unicode=False, wrap_line=False, no_global=True)
-&gt;&gt;&gt; var(&quot;x&quot;)
-x
-&gt;&gt;&gt; x**3/3
- 3
-x
---
-3
-&gt;&gt;&gt; Integral(x**2, x) #doctest: +NORMALIZE_WHITESPACE
-  /
- |
- |  2
- | x  dx
- |
-/</pre>
-</div>
-<p><strong>Python-Ausgabe</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.python</span> <span class="kn">import</span> <span class="n">python</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = x**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = 1/x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = Integral(x**2, x)</span>
-</pre></div>
-</div>
-<p><strong>LaTeX-Ausgabe</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">latex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x^{2}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;inline&#39;</span><span class="p">)</span>
-<span class="go">$x^{2}$</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;equation&#39;</span><span class="p">)</span>
-<span class="go">\begin{equation}x^{2}\end{equation}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;equation*&#39;</span><span class="p">)</span>
-<span class="go">\begin{equation*}x^{2}\end{equation*}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">\frac{1}{x}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">\int x^{2}\, dx</span>
-</pre></div>
-</div>
-<p><strong>MathML</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.mathml</span> <span class="kn">import</span> <span class="n">mathml</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">latex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">mathml</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">&lt;apply&gt;&lt;power/&gt;&lt;ci&gt;x&lt;/ci&gt;&lt;cn&gt;2&lt;/cn&gt;&lt;/apply&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">mathml</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">))</span>
-<span class="go">&lt;apply&gt;&lt;power/&gt;&lt;ci&gt;x&lt;/ci&gt;&lt;cn&gt;-1&lt;/cn&gt;&lt;/apply&gt;</span>
-</pre></div>
-</div>
-<p><strong>Pyglet</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">preview</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">preview</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span> 
-</pre></div>
-</div>
-<p>Dies öffnet ein pyglet-Fenster mit dem in LaTeX gerenderten Ausdruck, wenn pyglet installiert ist:</p>
-<img alt="../_images/pngview1.png" src="../_images/pngview1.png" />
-<div class="section" id="notes">
-<h3>Hinweise<a class="headerlink" href="#notes" title="Permalink zu dieser Überschrift">¶</a></h3>
-<p><tt class="docutils literal"><span class="pre">isympy</span></tt> ruft <tt class="docutils literal"><span class="pre">pprint</span></tt> automatisch auf &#8211; deswegen sind die Ausgaben standardmäßig hübsch.</p>
-<p>Es ist gibt auch ein Ausgabemodul <tt class="docutils literal"><span class="pre">sympy.printing</span></tt>. Andere Ausgabemethoden, die durch dieses Modul erreichbar sind:</p>
-<ul class="simple">
-<li><tt class="docutils literal"><span class="pre">pretty(expr)</span></tt>, <tt class="docutils literal"><span class="pre">pretty_print(expr)</span></tt>, <tt class="docutils literal"><span class="pre">pprint(expr)</span></tt>: Return or print, respectively, a pretty representation of <tt class="docutils literal"><span class="pre">expr</span></tt>. This is the same as the second level of representation described above.</li>
-<li><tt class="docutils literal"><span class="pre">latex(expr)</span></tt>, <tt class="docutils literal"><span class="pre">print_latex(expr)</span></tt>: Return or print, respectively, a <a class="reference external" href="http://www.latex-project.org/">LaTeX</a>  representation of <tt class="docutils literal"><span class="pre">expr</span></tt></li>
-<li><tt class="docutils literal"><span class="pre">mathml(expr)</span></tt>, <tt class="docutils literal"><span class="pre">print_mathml(expr)</span></tt>: Return or print, respectively, a <a class="reference external" href="http://www.w3.org/Math/">MathML</a> representation of <tt class="docutils literal"><span class="pre">expr</span></tt>.</li>
-<li><tt class="docutils literal"><span class="pre">print_gtk(expr)</span></tt>: Print <tt class="docutils literal"><span class="pre">expr</span></tt> to <a class="reference external" href="http://helm.cs.unibo.it/mml-widget/">Gtkmathview</a>, a GTK widget that displays MathML code. The <a class="reference external" href="http://helm.cs.unibo.it/mml-widget/">Gtkmathview</a> program is required.</li>
-</ul>
-</div>
-</div>
-<div class="section" id="further-documentation">
-<h2>Weitere Dokumentation<a class="headerlink" href="#further-documentation" title="Permalink zu dieser Überschrift">¶</a></h2>
-<p>Nun ist Zeit, mehr über SymPy zu lernen. Lies den <a class="reference external" href="../guide.html">SymPy User&#8217;s Guide</a> und die <a class="reference external" href="../modules/index.html">SymPy Modules Reference</a>.</p>
-<p>Unser öffentliches Wiki unter <a class="reference external" href="http://wiki.sympy.org/">wiki.sympy.org</a>, enthält einen Haufen nützlicher Beispiele und Anleitungen von uns und unseren Nutzern. (Fühle dich frei, dazu beizutragen und Dinge zu verändern!)</p>
-<div class="section" id="translations">
-<h3>Translations<a class="headerlink" href="#translations" title="Permalink zu dieser Überschrift">¶</a></h3>
-<p>This tutorial is also available in other languages:</p>
-</div>
-<ul class="simple">
-<li><a class="reference external" href="tutorial.bg.html">Български</a></li>
-<li><a class="reference external" href="tutorial.cs.html">Česky</a></li>
-<li><a class="reference external" href="tutorial.de.html">Deutsch</a></li>
-<li><a class="reference external" href="tutorial.en.html">English</a></li>
-<li><a class="reference external" href="tutorial.fr.html">Français</a></li>
-<li><a class="reference external" href="tutorial.pl.html">Polski</a></li>
-<li><a class="reference external" href="tutorial.ru.html">Русский</a></li>
-<li><a class="reference external" href="tutorial.sr.html">Српски</a></li>
-<li><a class="reference external" href="tutorial.zh.html">中文</a></li>
-</ul>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="tutorial.html">
-              <img class="logo" src="../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="tutorial.html">Inhalt</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Tutorial</a><ul>
-<li><a class="reference internal" href="#introduction">Einführung</a></li>
-<li><a class="reference internal" href="#first-steps-with-sympy">Erste Schritte mit SymPy</a><ul>
-<li><a class="reference internal" href="#isympy-console">isympy-Konsole</a></li>
-<li><a class="reference internal" href="#using-sympy-as-a-calculator">Mit SymPy rechnen</a></li>
-<li><a class="reference internal" href="#symbols">Symbole</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#algebra">Algebra</a></li>
-<li><a class="reference internal" href="#calculus">Infinitesimalrechnung</a><ul>
-<li><a class="reference internal" href="#limits">Limes</a></li>
-<li><a class="reference internal" href="#differentiation">Differentialrechnung</a></li>
-<li><a class="reference internal" href="#series-expansion">Reihenentwicklung</a></li>
-<li><a class="reference internal" href="#summation">Summation</a></li>
-<li><a class="reference internal" href="#integration">Integralrechnung</a></li>
-<li><a class="reference internal" href="#complex-numbers">Komplexe Zahlen</a></li>
-<li><a class="reference internal" href="#functions">Funktionen</a></li>
-<li><a class="reference internal" href="#differential-equations">Differenzialgleichungen</a></li>
-<li><a class="reference internal" href="#algebraic-equations">Algebraische Gleichungen</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#linear-algebra">Lineare Algebra</a><ul>
-<li><a class="reference internal" href="#matrices">Matrizen</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#pattern-matching">Musterabgleich</a></li>
-<li><a class="reference internal" href="#printing">Ausgabe</a><ul>
-<li><a class="reference internal" href="#notes">Hinweise</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#further-documentation">Weitere Dokumentation</a><ul>
-<li><a class="reference internal" href="#translations">Translations</a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-
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-            
-  <div class="section" id="tutorial">
-<span id="id1"></span><h1>Tutorial<a class="headerlink" href="#tutorial" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="introduction">
-<h2>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline">¶</a></h2>
-<p>SymPy is a Python library for symbolic mathematics. It aims to become a
-full-featured computer algebra system (CAS) while keeping the code as simple as
-possible in order to be comprehensible and easily extensible.  SymPy is written
-entirely in Python and does not require any external libraries.</p>
-<p>This tutorial gives an overview and introduction to SymPy.
-Read this to have an idea what SymPy can do for you (and how) and if you want
-to know more, read the <a class="reference internal" href="../guide.html#guide"><em>SymPy User&#8217;s Guide</em></a>, the
-<a class="reference internal" href="../modules/index.html#module-docs"><em>SymPy Modules Reference</em></a>, or the
-<a class="reference external" href="https://github.com/sympy/sympy/">sources</a> directly.</p>
-</div>
-<div class="section" id="first-steps-with-sympy">
-<h2>First Steps with SymPy<a class="headerlink" href="#first-steps-with-sympy" title="Permalink to this headline">¶</a></h2>
-<p>The easiest way to download it is to go to
-<a class="reference external" href="http://code.google.com/p/sympy/">http://code.google.com/p/sympy/</a> and
-download the latest tarball from the Featured Downloads:</p>
-<img alt="../_images/featured-downloads.png" src="../_images/featured-downloads.png" />
-<p>Unpack it:</p>
-<pre class="literal-block">
-$ <strong class="input">tar xzf sympy-0.5.12.tar.gz</strong>
-</pre>
-<p>and try it from a Python interpreter:</p>
-<pre class="literal-block">
-$ <strong class="input">cd sympy-0.5.12</strong>
-$ <strong class="input">python</strong>
-Python 2.4.4 (#2, Jan  3 2008, 13:36:28)
-[GCC 4.2.3 20071123 (prerelease) (Debian 4.2.2-4)] on linux2
-Type &quot;help&quot;, &quot;copyright&quot;, &quot;credits&quot; or &quot;license&quot; for more information.
-&gt;&gt;&gt; from sympy import Symbol, cos
-&gt;&gt;&gt; x = Symbol(&quot;x&quot;)
-&gt;&gt;&gt; (1/cos(x)).series(x, 0, 10)
-1 + x**2/2 + 5*x**4/24 + 61*x**6/720 + 277*x**8/8064 + O(x**10)
-</pre>
-<p>You can use SymPy as shown above and this is indeed the recommended way if you
-use it in your program. You can also install it using <tt class="docutils literal"><span class="pre">./setup.py</span> <span class="pre">install</span></tt> as
-any other Python module, or just install a package in your favourite Linux
-distribution, e.g.:</p>
-<div class="topic">
-<p class="topic-title first">Installing SymPy in Debian</p>
-<pre class="literal-block">
-$ <strong class="input">sudo apt-get install python-sympy</strong>
-Reading package lists... Done
-Building dependency tree
-Reading state information... Done
-The following NEW packages will be installed:
-  python-sympy
-0 upgraded, 1 newly installed, 0 to remove and 18 not upgraded.
-Need to get 991kB of archives.
-After this operation, 5976kB of additional disk space will be used.
-Get:1 http://ftp.cz.debian.org unstable/main python-sympy 0.5.12-1 [991kB]
-Fetched 991kB in 2s (361kB/s)
-Selecting previously deselected package python-sympy.
-(Reading database ... 232619 files and directories currently installed.)
-Unpacking python-sympy (from .../python-sympy_0.5.12-1_all.deb) ...
-Setting up python-sympy (0.5.12-1) ...
-</pre>
-</div>
-<p>For other means how to install SymPy, consult the wiki page
-<a class="reference external" href="https://github.com/sympy/sympy/wiki/Download-Installation">Download and Installation</a>.</p>
-<div class="section" id="isympy-console">
-<h3>isympy Console<a class="headerlink" href="#isympy-console" title="Permalink to this headline">¶</a></h3>
-<p>For experimenting with new features, or when figuring out how to do things, you
-can use our special wrapper around IPython called <tt class="docutils literal"><span class="pre">isympy</span></tt> (located in
-<tt class="docutils literal"><span class="pre">bin/isympy</span></tt> if you are running from the source directory) which is just a
-standard Python shell that has already imported the relevant SymPy modules and
-defined the symbols x, y, z and some other things:</p>
-<pre class="literal-block">
-$ <strong class="input">cd sympy</strong>
-$ <strong class="input">./bin/isympy</strong>
-IPython console for SymPy 0.7.2-git (Python 2.7.1) (ground types: gmpy)
-
-These commands were executed:
-&gt;&gt;&gt;
-&gt;&gt;&gt; from sympy import *
-&gt;&gt;&gt; x, y, z, t = symbols('x y z t')
-&gt;&gt;&gt; k, m, n = symbols('k m n', integer=True)
-&gt;&gt;&gt; f, g, h = symbols('f g h', cls=Function)
-
-Documentation can be found at <a class="reference external" href="http://www.sympy.org">http://www.sympy.org</a>
-
-In [1]: <strong class="input">(1/cos(x)).series(x, 0, 10)</strong>
-Out[1]:
-     2      4       6        8
-    x    5*x    61*x    277*x     / 10\
-1 + -- + ---- + ----- + ------ + O\x  /
-    2     24     720     8064
-</pre>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">Commands entered by you are bold. Thus what we did in 3 lines in a regular
-Python interpreter can be done in 1 line in isympy.</p>
-</div>
-</div>
-<div class="section" id="using-sympy-as-a-calculator">
-<h3>Using SymPy as a calculator<a class="headerlink" href="#using-sympy-as-a-calculator" title="Permalink to this headline">¶</a></h3>
-<p>SymPy has three built-in numeric types: Float, Rational and Integer.</p>
-<p>The Rational class represents a rational number as a pair of two Integers:
-the numerator and the denominator. So Rational(1, 2) represents 1/2,
-Rational(5, 2) represents 5/2, and so on.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
-<span class="go">1/2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">*</span><span class="mi">2</span>
-<span class="go">1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">50</span><span class="o">/</span><span class="n">Rational</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span><span class="o">**</span><span class="mi">50</span>
-<span class="go">1/88817841970012523233890533447265625</span>
-</pre></div>
-</div>
-<p>Proceed with caution while working with Python int&#8217;s and floating
-point numbers, especially in division, since you may create a
-Python number, not a SymPy number. A ratio of two Python ints may
-create a float &#8211; the &#8220;true division&#8221; standard of Python 3
-and the default behavior of <tt class="docutils literal"><span class="pre">isympy</span></tt> which imports division
-from __future__:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span> 
-<span class="go">0.5</span>
-</pre></div>
-</div>
-<p>But in earlier Python versions where division has not been imported, a
-truncated int will result:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span> 
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>In both cases, however, you are not dealing with a SymPy Number because
-Python created its own number. Most of the time you will probably be
-working with Rational numbers, so make sure to use Rational to get
-the SymPy result. One might find it convenient to equate <tt class="docutils literal"><span class="pre">R</span></tt> and
-Rational:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="c"># R(1) is a SymPy Integer and Integer/int gives a Rational</span>
-<span class="go">1/2</span>
-</pre></div>
-</div>
-<p>We also have some special constants, like e and pi, that are treated as symbols
-(1 + pi won&#8217;t evaluate to something numeric, rather it will remain as 1 + pi), and
-have arbitrary precision:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">E</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">pi**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">3.14159265358979</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">pi</span> <span class="o">+</span> <span class="n">E</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">5.85987448204884</span>
-</pre></div>
-</div>
-<p>as you see, evalf evaluates the expression to a floating-point number</p>
-<p>The symbol <tt class="docutils literal"><span class="pre">oo</span></tt> is used for a class defining mathematical infinity:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">oo</span> <span class="o">&gt;</span> <span class="mi">99999</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">oo</span> <span class="o">+</span> <span class="mi">1</span>
-<span class="go">oo</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="symbols">
-<h3>Symbols<a class="headerlink" href="#symbols" title="Permalink to this headline">¶</a></h3>
-<p>In contrast to other Computer Algebra Systems, in SymPy you have to declare
-symbolic variables explicitly:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>On the left is the normal Python variable which has been assigned to the
-SymPy Symbol class. Predefined symbols (including those for symbols with
-Greek names) are available for import from abc:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">theta</span>
-</pre></div>
-</div>
-<p>Symbols can also be created with the <tt class="docutils literal"><span class="pre">symbols</span></tt> or <tt class="docutils literal"><span class="pre">var</span></tt> functions, the
-latter automatically adding the created symbols to the namespace, and both
-accepting a range notation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">var</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a,b,c&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;d:f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;g:h&#39;</span><span class="p">)</span>
-<span class="go">(g, h)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;g:2&#39;</span><span class="p">)</span>
-<span class="go">(g0, g1)</span>
-</pre></div>
-</div>
-<p>Instances of the Symbol class &#8220;play well together&#8221; and are the building blocks
-of expresions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">x</span> <span class="o">-</span> <span class="n">y</span>
-<span class="go">2*x</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">(x + y)**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go">x**2 + 2*x*y + y**2</span>
-</pre></div>
-</div>
-<p>They can be substituted with other numbers, symbols or expressions using <tt class="docutils literal"><span class="pre">subs(old,</span> <span class="pre">new)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">(y + 1)**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">4*y**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>For the remainder of the tutorial, we assume that we have run:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">init_printing</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">init_printing</span><span class="p">(</span><span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">wrap_line</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">no_global</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>This will make things look better when printed. See the <a class="reference internal" href="#printing-tutorial"><em>Printing</em></a>
-section below. If you have a unicode font installed, you can pass
-use_unicode=True for a slightly nicer output.</p>
-</div>
-</div>
-<div class="section" id="algebra">
-<h2>Algebra<a class="headerlink" href="#algebra" title="Permalink to this headline">¶</a></h2>
-<p>For partial fraction decomposition, use <tt class="docutils literal"><span class="pre">apart(expr,</span> <span class="pre">x)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">apart</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">)</span>
-<span class="go">       1</span>
-<span class="go">---------------</span>
-<span class="go">(x + 1)*(x + 2)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">apart</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">    1       1</span>
-<span class="go">- ----- + -----</span>
-<span class="go">  x + 2   x + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">x + 1</span>
-<span class="go">-----</span>
-<span class="go">x - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">apart</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">      2</span>
-<span class="go">1 + -----</span>
-<span class="go">    x - 1</span>
-</pre></div>
-</div>
-<p>To combine things back together, use <tt class="docutils literal"><span class="pre">together(expr,</span> <span class="pre">x)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">together</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">)</span>
-<span class="go">x*y + x*z + y*z</span>
-<span class="go">---------------</span>
-<span class="go">     x*y*z</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="n">apart</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x + 1</span>
-<span class="go">-----</span>
-<span class="go">x - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="n">apart</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">       1</span>
-<span class="go">---------------</span>
-<span class="go">(x + 1)*(x + 2)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="calculus">
-<span id="index-0"></span><h2>Calculus<a class="headerlink" href="#calculus" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="limits">
-<span id="index-1"></span><h3>Limits<a class="headerlink" href="#limits" title="Permalink to this headline">¶</a></h3>
-<p>Limits are easy to use in SymPy, they follow the syntax <tt class="docutils literal"><span class="pre">limit(function,</span>
-<span class="pre">variable,</span> <span class="pre">point)</span></tt>, so to compute the limit of f(x) as x -&gt; 0, you would issue
-<tt class="docutils literal"><span class="pre">limit(f,</span> <span class="pre">x,</span> <span class="pre">0)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">limit</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>you can also calculate the limit at infinity:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">oo</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">0</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>for some non-trivial examples on limits, you can read the test file
-<a class="reference external" href="https://github.com/sympy/sympy/blob/master/sympy/series/tests/test_demidovich.py">test_demidovich.py</a></p>
-</div>
-<div class="section" id="differentiation">
-<span id="index-2"></span><h3>Differentiation<a class="headerlink" href="#differentiation" title="Permalink to this headline">¶</a></h3>
-<p>You can differentiate any SymPy expression using <tt class="docutils literal"><span class="pre">diff(func,</span> <span class="pre">var)</span></tt>. Examples:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">diff</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">tan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">cos(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">2*cos(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">tan (x) + 1</span>
-</pre></div>
-</div>
-<p>You can check, that it is correct by:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">limit</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">delta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">((</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">delta</span><span class="p">)</span> <span class="o">-</span> <span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">/</span><span class="n">delta</span><span class="p">,</span> <span class="n">delta</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">tan (x) + 1</span>
-</pre></div>
-</div>
-<p>Higher derivatives can be calculated using the <tt class="docutils literal"><span class="pre">diff(func,</span> <span class="pre">var,</span> <span class="pre">n)</span></tt> method:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">2*cos(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">-4*sin(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">-8*cos(2*x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="series-expansion">
-<span id="index-3"></span><h3>Series expansion<a class="headerlink" href="#series-expansion" title="Permalink to this headline">¶</a></h3>
-<p>Use <tt class="docutils literal"><span class="pre">.series(var,</span> <span class="pre">point,</span> <span class="pre">order)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     2    4     6      8</span>
-<span class="go">    x    x     x      x      / 10\</span>
-<span class="go">1 - -- + -- - --- + ----- + O\x  /</span>
-<span class="go">    2    24   720   40320</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     2      4       6        8</span>
-<span class="go">    x    5*x    61*x    277*x     / 10\</span>
-<span class="go">1 + -- + ---- + ----- + ------ + O\x  /</span>
-<span class="go">    2     24     720     8064</span>
-</pre></div>
-</div>
-<p>Another simple example:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">pprint</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;y&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="go">1/y - x/y**2 + x**2/y**3 - x**3/y**4 + x**4/y**5 + O(x**5)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="go">          2    3    4</span>
-<span class="go">1   x    x    x    x     / 5\</span>
-<span class="go">- - -- + -- - -- + -- + O\x /</span>
-<span class="go">y    2    3    4    5</span>
-<span class="go">    y    y    y    y</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="summation">
-<span id="index-4"></span><h3>Summation<a class="headerlink" href="#summation" title="Permalink to this headline">¶</a></h3>
-<p>Compute the summation of f with respect to the given summation variable over the given limits.</p>
-<p>summation(f, (i, a, b)) computes the sum of f with respect to i from a to b,
-i.e.,</p>
-<div class="highlight-python"><pre>                            b
-                          ____
-                          \   `
-summation(f, (i, a, b)) =  )    f
-                          /___,
-                          i = a</pre>
-</div>
-<p>If it cannot compute the sum, it prints the corresponding summation formula.
-Repeated sums can be computed by introducing additional limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">summation</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">log</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i n m&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-<span class="go"> 2</span>
-<span class="go">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="o">**</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">  oo</span>
-<span class="go"> ___</span>
-<span class="go"> \  `</span>
-<span class="go">  \     -n</span>
-<span class="go">  /   log (n)</span>
-<span class="go"> /__,</span>
-<span class="go">n = 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">),</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">m</span><span class="p">))</span>
-<span class="go">      3    2</span>
-<span class="go">m    m    m</span>
-<span class="go">-- + -- + -</span>
-<span class="go">6    2    3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go"> x</span>
-<span class="go">e</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="integration">
-<span id="index-5"></span><h3>Integration<a class="headerlink" href="#integration" title="Permalink to this headline">¶</a></h3>
-<p>SymPy has support for indefinite and definite integration of transcendental
-elementary and special functions via <tt class="docutils literal"><span class="pre">integrate()</span></tt> facility, which uses
-powerful extended Risch-Norman algorithm and some heuristics and pattern
-matching:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">integrate</span><span class="p">,</span> <span class="n">erf</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>You can integrate elementary functions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="mi">6</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">5</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go"> 6</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">-cos(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x*log(x) - x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">sinh</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go"> 2</span>
-<span class="go">x  + cosh(x)</span>
-</pre></div>
-</div>
-<p>Also special functions are handled easily:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">erf</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">  ____    2</span>
-<span class="go">\/ pi *erf (x)</span>
-<span class="go">--------------</span>
-<span class="go">      4</span>
-</pre></div>
-</div>
-<p>It is possible to compute definite integrals:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-<p>Also, improper integrals are supported as well:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">-1</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="complex-numbers">
-<span id="index-6"></span><h3>Complex numbers<a class="headerlink" href="#complex-numbers" title="Permalink to this headline">¶</a></h3>
-<p>Besides the imaginary unit, I, which is imaginary, symbols can be created with
-attributes (e.g. real, positive, complex, etc...) and this will affect how
-they behave:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span> <span class="c"># a plain x with no attributes</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go"> I*x</span>
-<span class="go">e</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">   -im(x)               -im(x)</span>
-<span class="go">I*e      *sin(re(x)) + e      *cos(re(x))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">I*sin(x) + cos(x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="functions">
-<h3>Functions<a class="headerlink" href="#functions" title="Permalink to this headline">¶</a></h3>
-<p><strong>trigonometric</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">asin</span><span class="p">,</span> <span class="n">asinh</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">sin(x)*cos(y) + sin(y)*cos(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">-sin(x)*sin(y) + cos(x)*cos(y)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*sinh(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinh</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*sin(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
-<span class="go">I*pi</span>
-<span class="go">----</span>
-<span class="go"> 2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*asin(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3     5     7       9</span>
-<span class="go">    x     x     x       x       / 10\</span>
-<span class="go">x - -- + --- - ---- + ------ + O\x  /</span>
-<span class="go">    6    120   5040   362880</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3     5     7       9</span>
-<span class="go">    x     x     x       x       / 10\</span>
-<span class="go">x + -- + --- + ---- + ------ + O\x  /</span>
-<span class="go">    6    120   5040   362880</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3      5      7       9</span>
-<span class="go">    x    3*x    5*x    35*x     / 10\</span>
-<span class="go">x + -- + ---- + ---- + ----- + O\x  /</span>
-<span class="go">    6     40    112     1152</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3      5      7       9</span>
-<span class="go">    x    3*x    5*x    35*x     / 10\</span>
-<span class="go">x - -- + ---- - ---- + ----- + O\x  /</span>
-<span class="go">    6     40    112     1152</span>
-</pre></div>
-</div>
-<p><strong>spherical harmonics</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Ylm</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">  ___</span>
-<span class="go">\/ 3 *cos(theta)</span>
-<span class="go">----------------</span>
-<span class="go">        ____</span>
-<span class="go">    2*\/ pi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">   ___  I*phi</span>
-<span class="go">-\/ 6 *e     *sin(theta)</span>
-<span class="go">------------------------</span>
-<span class="go">            ____</span>
-<span class="go">        4*\/ pi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">   ____  I*phi</span>
-<span class="go">-\/ 30 *e     *sin(theta)*cos(theta)</span>
-<span class="go">------------------------------------</span>
-<span class="go">                  ____</span>
-<span class="go">              4*\/ pi</span>
-</pre></div>
-</div>
-<p><strong>factorials and gamma function</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">x!</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">n!</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="c"># i.e. factorial(x)</span>
-<span class="go">                      /          2     2\</span>
-<span class="go">                    2 |EulerGamma    pi |    / 3\</span>
-<span class="go">1 - EulerGamma*x + x *|----------- + ---| + O\x /</span>
-<span class="go">                      \     2         12/</span>
-</pre></div>
-</div>
-<p><strong>zeta function</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">zeta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">zeta(4, x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">  4</span>
-<span class="go">pi</span>
-<span class="go">---</span>
-<span class="go"> 90</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">       4</span>
-<span class="go">     pi</span>
-<span class="go">-1 + ---</span>
-<span class="go">      90</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">         4</span>
-<span class="go">  17   pi</span>
-<span class="go">- -- + ---</span>
-<span class="go">  16    90</span>
-</pre></div>
-</div>
-<p><strong>polynomials</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">assoc_legendre</span><span class="p">,</span> <span class="n">chebyshevt</span><span class="p">,</span> <span class="n">legendre</span><span class="p">,</span> <span class="n">hermite</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">2*x  - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   4      2</span>
-<span class="go">8*x  - 8*x  + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">3*x    1</span>
-<span class="go">---- - -</span>
-<span class="go"> 2     2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">      8         6         4        2</span>
-<span class="go">6435*x    3003*x    3465*x    315*x     35</span>
-<span class="go">------- - ------- + ------- - ------ + ---</span>
-<span class="go">  128        32        64       32     128</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">        __________</span>
-<span class="go">       /    2</span>
-<span class="go">-3*x*\/  - x  + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">     2</span>
-<span class="go">- 3*x  + 3</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hermite</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   3</span>
-<span class="go">8*x  - 12*x</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="differential-equations">
-<span id="index-7"></span><h3>Differential Equations<a class="headerlink" href="#differential-equations" title="Permalink to this headline">¶</a></h3>
-<p>In <tt class="docutils literal"><span class="pre">isympy</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">dsolve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">        2</span>
-<span class="go">       d</span>
-<span class="go">f(x) + ---(f(x))</span>
-<span class="go">         2</span>
-<span class="go">       dx</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dsolve</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">f(x) = C1*sin(x) + C2*cos(x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="algebraic-equations">
-<span id="index-8"></span><h3>Algebraic equations<a class="headerlink" href="#algebraic-equations" title="Permalink to this headline">¶</a></h3>
-<p>In <tt class="docutils literal"><span class="pre">isympy</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">solve</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">[-1, 1, -I, I]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">([</span><span class="n">x</span> <span class="o">+</span> <span class="mi">5</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">6</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">15</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">])</span>
-<span class="go">{x: -3, y: 1}</span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="section" id="linear-algebra">
-<span id="index-9"></span><h2>Linear Algebra<a class="headerlink" href="#linear-algebra" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="matrices">
-<span id="index-10"></span><h3>Matrices<a class="headerlink" href="#matrices" title="Permalink to this headline">¶</a></h3>
-<p>Matrices are created as instances from the Matrix class:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="go">[1  0]</span>
-<span class="go">[    ]</span>
-<span class="go">[0  1]</span>
-</pre></div>
-</div>
-<p>They can also contain symbols:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">],</span> <span class="p">[</span><span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span>
-<span class="go">[1  x]</span>
-<span class="go">[    ]</span>
-<span class="go">[y  1]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">[x*y + 1    2*x  ]</span>
-<span class="go">[                ]</span>
-<span class="go">[  2*y    x*y + 1]</span>
-</pre></div>
-</div>
-<p>For more about Matrices, see the Linear Algebra tutorial.</p>
-</div>
-</div>
-<div class="section" id="pattern-matching">
-<span id="index-11"></span><h2>Pattern matching<a class="headerlink" href="#pattern-matching" title="Permalink to this headline">¶</a></h2>
-<p>Use the <tt class="docutils literal"><span class="pre">.match()</span></tt> method, along with the <tt class="docutils literal"><span class="pre">Wild</span></tt> class, to perform pattern
-matching on expressions. The method will return a dictionary with the required
-substitutions, as follows:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Wild</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">{p: 5}</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">q</span><span class="p">)</span>
-<span class="go">{p: 1, q: 2}</span>
-</pre></div>
-</div>
-<p>If the match is unsuccessful, it returns <tt class="docutils literal"><span class="pre">None</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">**</span><span class="n">x</span><span class="p">))</span>
-<span class="go">None</span>
-</pre></div>
-</div>
-<p>One can also use the exclude parameter of the <tt class="docutils literal"><span class="pre">Wild</span></tt> class to ensure that
-certain things do not show up in the result:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">p</span><span class="p">))</span> <span class="c"># 1 is excluded</span>
-<span class="go">None</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span> <span class="c"># x is excluded</span>
-<span class="go">None</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">p</span><span class="p">))</span> <span class="c"># -1 is not excluded</span>
-<span class="go">{p_: -1}</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="printing">
-<span id="printing-tutorial"></span><h2>Printing<a class="headerlink" href="#printing" title="Permalink to this headline">¶</a></h2>
-<p>There are many ways to print expressions.</p>
-<p><strong>Standard</strong></p>
-<p>This is what <tt class="docutils literal"><span class="pre">str(expression)</span></tt> returns and it looks like this:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1/x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Integral(x**2, x)</span>
-</pre></div>
-</div>
-<p><strong>Pretty printing</strong></p>
-<p>Nice ascii-art printing is produced by the <tt class="docutils literal"><span class="pre">pprint</span></tt> function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go"> 2</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="go">-</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">  /</span>
-<span class="go"> |</span>
-<span class="go"> |  2</span>
-<span class="go"> | x  dx</span>
-<span class="go"> |</span>
-<span class="go">/</span>
-</pre></div>
-</div>
-<p>If you have a unicode font installed, the <tt class="docutils literal"><span class="pre">pprint</span></tt> function will use it by
-default. You can override this using the <tt class="docutils literal"><span class="pre">use_unicode</span></tt> option.:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">⌠</span>
-<span class="go">⎮  2</span>
-<span class="go">⎮ x  dx</span>
-<span class="go">⌡</span>
-</pre></div>
-</div>
-<p>See also the wiki <a class="reference external" href="https://github.com/sympy/sympy/wiki/Pretty-Printing">Pretty Printing</a> for more examples of a nice
-unicode printing.</p>
-<p>Tip: To make pretty printing the default in the Python interpreter, use:</p>
-<div class="highlight-python"><pre>$ python
-Python 2.5.2 (r252:60911, Jun 25 2008, 17:58:32)
-[GCC 4.3.1] on linux2
-Type &quot;help&quot;, &quot;copyright&quot;, &quot;credits&quot; or &quot;license&quot; for more information.
-&gt;&gt;&gt; from sympy import init_printing, var, Integral
-&gt;&gt;&gt; init_printing(use_unicode=False, wrap_line=False, no_global=True)
-&gt;&gt;&gt; var(&quot;x&quot;)
-x
-&gt;&gt;&gt; x**3/3
- 3
-x
---
-3
-&gt;&gt;&gt; Integral(x**2, x) #doctest: +NORMALIZE_WHITESPACE
-  /
- |
- |  2
- | x  dx
- |
-/</pre>
-</div>
-<p><strong>Python printing</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.python</span> <span class="kn">import</span> <span class="n">python</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = x**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = 1/x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = Integral(x**2, x)</span>
-</pre></div>
-</div>
-<p><strong>LaTeX printing</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">latex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x^{2}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;inline&#39;</span><span class="p">)</span>
-<span class="go">$x^{2}$</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;equation&#39;</span><span class="p">)</span>
-<span class="go">\begin{equation}x^{2}\end{equation}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;equation*&#39;</span><span class="p">)</span>
-<span class="go">\begin{equation*}x^{2}\end{equation*}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">\frac{1}{x}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">\int x^{2}\, dx</span>
-</pre></div>
-</div>
-<p><strong>MathML</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.mathml</span> <span class="kn">import</span> <span class="n">mathml</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">latex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">mathml</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">&lt;apply&gt;&lt;power/&gt;&lt;ci&gt;x&lt;/ci&gt;&lt;cn&gt;2&lt;/cn&gt;&lt;/apply&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">mathml</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">))</span>
-<span class="go">&lt;apply&gt;&lt;power/&gt;&lt;ci&gt;x&lt;/ci&gt;&lt;cn&gt;-1&lt;/cn&gt;&lt;/apply&gt;</span>
-</pre></div>
-</div>
-<p><strong>Pyglet</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">preview</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">preview</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span> 
-</pre></div>
-</div>
-<p>If pyglet is installed, a pyglet window will open containing the LaTeX
-rendered expression:</p>
-<img alt="../_images/pngview1.png" src="../_images/pngview1.png" />
-<div class="section" id="notes">
-<h3>Notes<a class="headerlink" href="#notes" title="Permalink to this headline">¶</a></h3>
-<p><tt class="docutils literal"><span class="pre">isympy</span></tt> calls <tt class="docutils literal"><span class="pre">pprint</span></tt> automatically, so that&#8217;s why you see pretty
-printing by default.</p>
-<p>Note that there is also a printing module available, <tt class="docutils literal"><span class="pre">sympy.printing</span></tt>.  Other
-printing methods available through this module are:</p>
-<ul class="simple">
-<li><tt class="docutils literal"><span class="pre">pretty(expr)</span></tt>, <tt class="docutils literal"><span class="pre">pretty_print(expr)</span></tt>, <tt class="docutils literal"><span class="pre">pprint(expr)</span></tt>: Return or print, respectively, a pretty representation of <tt class="docutils literal"><span class="pre">expr</span></tt>. This is the same as the second level of representation described above.</li>
-<li><tt class="docutils literal"><span class="pre">latex(expr)</span></tt>, <tt class="docutils literal"><span class="pre">print_latex(expr)</span></tt>: Return or print, respectively, a <a class="reference external" href="http://www.latex-project.org/">LaTeX</a>  representation of <tt class="docutils literal"><span class="pre">expr</span></tt></li>
-<li><tt class="docutils literal"><span class="pre">mathml(expr)</span></tt>, <tt class="docutils literal"><span class="pre">print_mathml(expr)</span></tt>: Return or print, respectively, a <a class="reference external" href="http://www.w3.org/Math/">MathML</a> representation of <tt class="docutils literal"><span class="pre">expr</span></tt>.</li>
-<li><tt class="docutils literal"><span class="pre">print_gtk(expr)</span></tt>: Print <tt class="docutils literal"><span class="pre">expr</span></tt> to <a class="reference external" href="http://helm.cs.unibo.it/mml-widget/">Gtkmathview</a>, a GTK widget that displays MathML code. The <a class="reference external" href="http://helm.cs.unibo.it/mml-widget/">Gtkmathview</a> program is required.</li>
-</ul>
-</div>
-</div>
-<div class="section" id="further-documentation">
-<h2>Further documentation<a class="headerlink" href="#further-documentation" title="Permalink to this headline">¶</a></h2>
-<p>Now it&#8217;s time to learn more about SymPy. Go through the
-<a class="reference internal" href="../guide.html#guide"><em>SymPy User&#8217;s Guide</em></a> and
-<a class="reference internal" href="../modules/index.html#module-docs"><em>SymPy Modules Reference</em></a>.</p>
-<p>Be sure to also browse our public <a class="reference external" href="http://wiki.sympy.org/">wiki.sympy.org</a>,
-that contains a lot of useful examples, tutorials, cookbooks that we and our
-users contributed, and feel free to edit it.</p>
-<div class="section" id="translations">
-<h3>Translations<a class="headerlink" href="#translations" title="Permalink to this headline">¶</a></h3>
-<p>This tutorial is also available in other languages:</p>
-</div>
-<ul class="simple">
-<li><a class="reference external" href="tutorial.bg.html">Български</a></li>
-<li><a class="reference external" href="tutorial.cs.html">Česky</a></li>
-<li><a class="reference external" href="tutorial.de.html">Deutsch</a></li>
-<li><a class="reference external" href="tutorial.en.html">English</a></li>
-<li><a class="reference external" href="tutorial.fr.html">Français</a></li>
-<li><a class="reference external" href="tutorial.pl.html">Polski</a></li>
-<li><a class="reference external" href="tutorial.ru.html">Русский</a></li>
-<li><a class="reference external" href="tutorial.sr.html">Српски</a></li>
-<li><a class="reference external" href="tutorial.zh.html">中文</a></li>
-</ul>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="../index.html">
-              <img class="logo" src="../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="../index.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Tutorial</a><ul>
-<li><a class="reference internal" href="#introduction">Introduction</a></li>
-<li><a class="reference internal" href="#first-steps-with-sympy">First Steps with SymPy</a><ul>
-<li><a class="reference internal" href="#isympy-console">isympy Console</a></li>
-<li><a class="reference internal" href="#using-sympy-as-a-calculator">Using SymPy as a calculator</a></li>
-<li><a class="reference internal" href="#symbols">Symbols</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#algebra">Algebra</a></li>
-<li><a class="reference internal" href="#calculus">Calculus</a><ul>
-<li><a class="reference internal" href="#limits">Limits</a></li>
-<li><a class="reference internal" href="#differentiation">Differentiation</a></li>
-<li><a class="reference internal" href="#series-expansion">Series expansion</a></li>
-<li><a class="reference internal" href="#summation">Summation</a></li>
-<li><a class="reference internal" href="#integration">Integration</a></li>
-<li><a class="reference internal" href="#complex-numbers">Complex numbers</a></li>
-<li><a class="reference internal" href="#functions">Functions</a></li>
-<li><a class="reference internal" href="#differential-equations">Differential Equations</a></li>
-<li><a class="reference internal" href="#algebraic-equations">Algebraic equations</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#linear-algebra">Linear Algebra</a><ul>
-<li><a class="reference internal" href="#matrices">Matrices</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#pattern-matching">Pattern matching</a></li>
-<li><a class="reference internal" href="#printing">Printing</a><ul>
-<li><a class="reference internal" href="#notes">Notes</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#further-documentation">Further documentation</a><ul>
-<li><a class="reference internal" href="#translations">Translations</a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-
-  <h4>Previous topic</h4>
-  <p class="topless"><a href="../install.html"
-                        title="previous chapter">Installation</a></p>
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-                        title="next chapter">Gotchas and Pitfalls</a></p>
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diff --git a/dev-py3k/tutorial/tutorial.fr.html b/dev-py3k/tutorial/tutorial.fr.html
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+++ /dev/null
@@ -1,972 +0,0 @@
-
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-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
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-  <div class="section" id="tutorial">
-<span id="id1"></span><h1>Tutoriel<a class="headerlink" href="#tutorial" title="Lien permanent vers ce titre">¶</a></h1>
-<div class="section" id="introduction">
-<h2>Introduction<a class="headerlink" href="#introduction" title="Lien permanent vers ce titre">¶</a></h2>
-<p>SymPy est une bibliothèque Python pour les mathématiques symboliques. SymPy prévoit devenir un système complet de calcul formel (&#8220;CAS&#8221; en anglais : &#8220;Computer Algebra System&#8221;) tout en gardant le code aussi simple que possible afin qu&#8217;il soit compréhensible et facilement extensible. SymPy est entièrement écrit en Python et ne nécessite aucune bibliothèque externe.</p>
-<p>Ce tutoriel donne un aperçu et une introduction à SymPy. Lisez-le pour vous faire une idée de ce que SymPy peut faire pour vous (et comment). Pour en savoir plus, consultez également le <a class="reference external" href="../guide.html">SymPy User&#8217;s Guide</a>, la <a class="reference external" href="../modules/index.html">SymPy Modules Reference</a>, ou le <a class="reference external" href="https://github.com/sympy/sympy/">code source</a> directement.</p>
-</div>
-<div class="section" id="first-steps-with-sympy">
-<h2>Premiers pas avec SymPy<a class="headerlink" href="#first-steps-with-sympy" title="Lien permanent vers ce titre">¶</a></h2>
-<p>La façon la plus simple de télécharger SymPy est de se rendre sur <a class="reference external" href="http://code.google.com/p/sympy/">http://code.google.com/p/sympy/</a> et de télécharger la dernière archive des Téléchargements Recommandés (&#8220;Featured Downloads&#8221; en anglais).</p>
-<img alt="../_images/featured-downloads.png" src="../_images/featured-downloads.png" />
-<p>Décompressez-la :</p>
-<pre class="literal-block">
-$ <strong class="input">tar xzf sympy-0.5.12.tar.gz</strong>
-</pre>
-<p>et essayez-la dans un interpréteur Python :</p>
-<pre class="literal-block">
-$ <strong class="input">cd sympy-0.5.12</strong>
-$ <strong class="input">python</strong>
-Python 2.4.4 (#2, Jan  3 2008, 13:36:28)
-[GCC 4.2.3 20071123 (prerelease) (Debian 4.2.2-4)] on linux2
-Type &quot;help&quot;, &quot;copyright&quot;, &quot;credits&quot; or &quot;license&quot; for more information.
-&gt;&gt;&gt; from sympy import Symbol, cos
-&gt;&gt;&gt; x = Symbol(&quot;x&quot;)
-&gt;&gt;&gt; (1/cos(x)).series(x, 0, 10)
-1 + x**2/2 + 5*x**4/24 + 61*x**6/720 + 277*x**8/8064 + O(x**10)
-</pre>
-<p>Vous pouvez utiliser SymPy comme illustré ci-dessus et c&#8217;est en effet la méthode recommandée si vous voulez l&#8217;utiliser dans votre programme. Vous pouvez aussi l&#8217;installer en utilisant <tt class="docutils literal"><span class="pre">./setup.py</span> <span class="pre">install</span></tt> comme tout autre module Python, ou simplement installer un paquet dans votre distribution Linux préférée, par exemple :</p>
-<div class="topic">
-<p class="topic-title first">Installing SymPy in Debian</p>
-<pre class="literal-block">
-$ <strong class="input">sudo apt-get install python-sympy</strong>
-Reading package lists... Done
-Building dependency tree
-Reading state information... Done
-The following NEW packages will be installed:
-  python-sympy
-0 upgraded, 1 newly installed, 0 to remove and 18 not upgraded.
-Need to get 991kB of archives.
-After this operation, 5976kB of additional disk space will be used.
-Get:1 http://ftp.cz.debian.org unstable/main python-sympy 0.5.12-1 [991kB]
-Fetched 991kB in 2s (361kB/s)
-Selecting previously deselected package python-sympy.
-(Reading database ... 232619 files and directories currently installed.)
-Unpacking python-sympy (from .../python-sympy_0.5.12-1_all.deb) ...
-Setting up python-sympy (0.5.12-1) ...
-</pre>
-</div>
-<p>Pour d&#8217;autres moyens d&#8217;installer SymPy, consultez <a class="reference external" href="https://github.com/sympy/sympy/wiki/Download-Installation">Download and Installation (en anglais)</a> sur le site wiki de SymPy.</p>
-<div class="section" id="isympy-console">
-<h3>La console isympy<a class="headerlink" href="#isympy-console" title="Lien permanent vers ce titre">¶</a></h3>
-<p>Pour tester de nouvelles fonctionnalités, ou si vous cherchez à savoir comment accomplir certaines choses, vous pouvez utiliser notre emballage (wrapper) spécial enrobant IPython appelé <tt class="docutils literal"><span class="pre">isympy</span></tt> (situé dans <tt class="docutils literal"><span class="pre">bin/isympy</span></tt> si vous vous trouvez dans le répertoire source). Cette commande lance un shell Python standard qui importe les modules SymPy les plus courants, les symboles définis x, y, z et d&#8217;autres choses à l&#8217;avance :</p>
-<pre class="literal-block">
-$ <strong class="input">cd sympy</strong>
-$ <strong class="input">./bin/isympy</strong>
-IPython console for SymPy 0.7.2-git (Python 2.7.1) (ground types: gmpy)
-
-These commands were executed:
-&gt;&gt;&gt;
-&gt;&gt;&gt; from sympy import *
-&gt;&gt;&gt; x, y, z, t = symbols('x y z t')
-&gt;&gt;&gt; k, m, n = symbols('k m n', integer=True)
-&gt;&gt;&gt; f, g, h = symbols('f g h', cls=Function)
-
-Documentation can be found at <a class="reference external" href="http://www.sympy.org">http://www.sympy.org</a>
-
-In [1]: <strong class="input">(1/cos(x)).series(x, 0, 10)</strong>
-Out[1]:
-     2      4       6        8
-    x    5*x    61*x    277*x     / 10\
-1 + -- + ---- + ----- + ------ + O\x  /
-    2     24     720     8064
-</pre>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">Les commandes que vous entrez sont en gras. Ainsi ce que nous avons fait en 3 lignes dans un interpréteur Python habituel peut être fait en une ligne avec isympy.</p>
-</div>
-</div>
-<div class="section" id="using-sympy-as-a-calculator">
-<h3>Utiliser SymPy comme une calculatrice<a class="headerlink" href="#using-sympy-as-a-calculator" title="Lien permanent vers ce titre">¶</a></h3>
-<p>SymPy possède trois types numériques natifs : Float (Flottant), Rational (Rationnel) et Integer (Entier).</p>
-<p>La classe Rational représente un nombre rationnel en tant que paire de deux Integers : le numérateur et le dénominateur. Donc Rational(1, 2) représente 1/2, Rational(5, 2) représente 5/2, et ainsi de suite.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
-<span class="go">1/2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">*</span><span class="mi">2</span>
-<span class="go">1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">50</span><span class="o">/</span><span class="n">Rational</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span><span class="o">**</span><span class="mi">50</span>
-<span class="go">1/88817841970012523233890533447265625</span>
-</pre></div>
-</div>
-<p>Faites attention lorsque vous travaillez avec les nombres entiers et à virgule flottante de Python, surtout dans des divisions, puisque vous pourriez créer un nombre Python, et non pas un nombre SymPy. Un ratio de deux entiers Python pourrait créer un flottant &#8211; la &#8220;vraie division&#8221;, standard sous Python 3 et le comportement par défaut de <tt class="docutils literal"><span class="pre">isympy</span></tt> qui importe l&#8217;opération division du module __future__ :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span> 
-<span class="go">0.5</span>
-</pre></div>
-</div>
-<p>Cependant, dans les versions antérieures à Python 3, l&#8217;opération division résulte par défaut en une divison euclidienne et entraînera :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span> 
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>Dans les deux cas, toutefois, vous n&#8217;avez pas affaire à un nombre SymPy parce que Python a créé son propre nombre. La plupart du temps vous travaillerez fort probablement avec des nombres rationels, alors prenez garde à utiliser Rational pour obtenir le résultat en terme d&#8217;objets SymPy. On pourrait trouver pratique d&#8217;assigner Rational à <tt class="docutils literal"><span class="pre">R</span></tt> :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="c"># R(1) is a SymPy Integer and Integer/int gives a Rational</span>
-<span class="go">1/2</span>
-</pre></div>
-</div>
-<p>SymPy offre également quelques constantes spéciales, comme e et pi, qui sont traitées comme des symboles (1 + pi ne sera pas évalué sous forme numérique, mais restera plutôt sous la forme 1 + pi), et qui ont une précision arbitraire :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">E</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">pi**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">3.14159265358979</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">pi</span> <span class="o">+</span> <span class="n">E</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">5.85987448204884</span>
-</pre></div>
-</div>
-<p>Comme vous le voyez, evalf convertit l&#8217;expression en un nombre à virgule flottante.</p>
-<p>Le symbole <tt class="docutils literal"><span class="pre">oo</span></tt> est utilisé pour une classe définissant l&#8217;infini mathématique :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">oo</span> <span class="o">&gt;</span> <span class="mi">99999</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">oo</span> <span class="o">+</span> <span class="mi">1</span>
-<span class="go">oo</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="symbols">
-<h3>Symboles<a class="headerlink" href="#symbols" title="Lien permanent vers ce titre">¶</a></h3>
-<p>Contrairement à d&#8217;autres systèmes de calcul formel, SymPy vous oblige à déclarer les variables symboliques explicitement :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>À gauche se trouve la variable Python normale à laquelle est assignée une instance de la classe Symbol de SymPy. Certains symboles prédefinis (dont les symboles portant un nom grec) sont disponibles après importation :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">theta</span>
-</pre></div>
-</div>
-<p>Plusieurs symboles peuvent également être déclarés en utilisant une seule commande avec les fonctions <tt class="docutils literal"><span class="pre">symbols</span></tt> ou <tt class="docutils literal"><span class="pre">var</span></tt>. Ces fonctions acceptent une chaîne de texte contenant une liste ou un interval de symboles à déclarer, la variante <tt class="docutils literal"><span class="pre">var</span></tt> ajoutant également les symboles à la liste de noms courante (namespace) :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">var</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a,b,c&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;d:f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;g:h&#39;</span><span class="p">)</span>
-<span class="go">(g, h)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;g:2&#39;</span><span class="p">)</span>
-<span class="go">(g0, g1)</span>
-</pre></div>
-</div>
-<p>Les instances de la classe Symbol travaillent ensemble et sont les fondements nécessaires pour construire des expressions :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">x</span> <span class="o">-</span> <span class="n">y</span>
-<span class="go">2*x</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">(x + y)**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go">x**2 + 2*x*y + y**2</span>
-</pre></div>
-</div>
-<p>Elles peuvent être remplacées par d&#8217;autres nombres, symboles ou expressions grâce à l&#8217;utilisation de <tt class="docutils literal"><span class="pre">subs(old,</span> <span class="pre">new)</span></tt> :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">(y + 1)**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">4*y**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>Pour le reste du tutoriel, il est présumé que nous avons exécuté :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">init_printing</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">init_printing</span><span class="p">(</span><span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">wrap_line</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">no_global</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Ceci donnera un meilleur aspect aux résultats affichées. Pour en savoir plus sur l&#8217;affichage, consultez la section <a class="reference internal" href="#printing-tutorial"><em>Affichage</em></a> ci-dessous. Si vous avez une police unicode installée, vous pouvez passer use_unicode=True pour un affichage un peu plus agréable.</p>
-</div>
-</div>
-<div class="section" id="algebra">
-<h2>Algèbre<a class="headerlink" href="#algebra" title="Lien permanent vers ce titre">¶</a></h2>
-<p>Pour la décomposition en éléments simples des fractions, utilisez <tt class="docutils literal"><span class="pre">apart(expr,</span> <span class="pre">x)</span></tt> :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">apart</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">)</span>
-<span class="go">       1</span>
-<span class="go">---------------</span>
-<span class="go">(x + 1)*(x + 2)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">apart</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">    1       1</span>
-<span class="go">- ----- + -----</span>
-<span class="go">  x + 2   x + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">x + 1</span>
-<span class="go">-----</span>
-<span class="go">x - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">apart</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">      2</span>
-<span class="go">1 + -----</span>
-<span class="go">    x - 1</span>
-</pre></div>
-</div>
-<p>Pour remettre tout ensemble, utilisez <tt class="docutils literal"><span class="pre">together(expr,</span> <span class="pre">x)</span></tt> :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">together</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">)</span>
-<span class="go">x*y + x*z + y*z</span>
-<span class="go">---------------</span>
-<span class="go">     x*y*z</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="n">apart</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x + 1</span>
-<span class="go">-----</span>
-<span class="go">x - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="n">apart</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">       1</span>
-<span class="go">---------------</span>
-<span class="go">(x + 1)*(x + 2)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="calculus">
-<span id="index-0"></span><h2>Calcul<a class="headerlink" href="#calculus" title="Lien permanent vers ce titre">¶</a></h2>
-<div class="section" id="limits">
-<span id="index-1"></span><h3>Limites<a class="headerlink" href="#limits" title="Lien permanent vers ce titre">¶</a></h3>
-<p>Les limites sont simples à utiliser dans SymPy, elles suivent la syntaxe <tt class="docutils literal"><span class="pre">limit(function,</span> <span class="pre">variable,</span> <span class="pre">point)</span></tt>, donc pour calculer la limite de f(x) pour x tendant vers 0, il suffit d&#8217;entrer <tt class="docutils literal"><span class="pre">limit(f,</span> <span class="pre">x,</span> <span class="pre">0)</span></tt> :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">limit</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>Vous pouvez aussi calculer la limite tendant vers l&#8217;infini :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">oo</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">0</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>Pour des exemples non-triviaux de limites, vous pouvez lire le fichier de test <a class="reference external" href="https://github.com/sympy/sympy/blob/master/sympy/series/tests/test_demidovich.py">test_demidovich.py</a></p>
-</div>
-<div class="section" id="differentiation">
-<span id="index-2"></span><h3>Dérivation<a class="headerlink" href="#differentiation" title="Lien permanent vers ce titre">¶</a></h3>
-<p>Vous pouvez dériver n&#8217;importe quelle expression SymPy en utilisant <tt class="docutils literal"><span class="pre">diff(func,</span> <span class="pre">var)</span></tt>. Exemples :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">diff</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">tan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">cos(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">2*cos(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">tan (x) + 1</span>
-</pre></div>
-</div>
-<p>Vous pouvez vérifier que c&#8217;est correct avec :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">limit</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">delta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">((</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">delta</span><span class="p">)</span> <span class="o">-</span> <span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">/</span><span class="n">delta</span><span class="p">,</span> <span class="n">delta</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">tan (x) + 1</span>
-</pre></div>
-</div>
-<p>Des dérivées d&#8217;ordre supérieur peuvent être calculées en utilisant la méthode <tt class="docutils literal"><span class="pre">diff(func,</span> <span class="pre">var,</span> <span class="pre">n)</span></tt> :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">2*cos(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">-4*sin(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">-8*cos(2*x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="series-expansion">
-<span id="index-3"></span><h3>Développement en série<a class="headerlink" href="#series-expansion" title="Lien permanent vers ce titre">¶</a></h3>
-<p>Utilisez <tt class="docutils literal"><span class="pre">.series(var,</span> <span class="pre">point,</span> <span class="pre">order)</span></tt> :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     2    4     6      8</span>
-<span class="go">    x    x     x      x      / 10\</span>
-<span class="go">1 - -- + -- - --- + ----- + O\x  /</span>
-<span class="go">    2    24   720   40320</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     2      4       6        8</span>
-<span class="go">    x    5*x    61*x    277*x     / 10\</span>
-<span class="go">1 + -- + ---- + ----- + ------ + O\x  /</span>
-<span class="go">    2     24     720     8064</span>
-</pre></div>
-</div>
-<p>Un autre exemple simple :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">pprint</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;y&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="go">1/y - x/y**2 + x**2/y**3 - x**3/y**4 + x**4/y**5 + O(x**5)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="go">          2    3    4</span>
-<span class="go">1   x    x    x    x     / 5\</span>
-<span class="go">- - -- + -- - -- + -- + O\x /</span>
-<span class="go">y    2    3    4    5</span>
-<span class="go">    y    y    y    y</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="summation">
-<span id="index-4"></span><h3>Sommes<a class="headerlink" href="#summation" title="Lien permanent vers ce titre">¶</a></h3>
-<p>Calcule la somme de f indexée sur la variable muette donnée entre les limites données.</p>
-<p>summation(f, (i, a, b)) calcule la somme de f sur la variable muette i entre les limites a et b, c&#8217;est-à-dire :</p>
-<div class="highlight-python"><pre>                            b
-                          ____
-                          \   `
-summation(f, (i, a, b)) =  )    f
-                          /___,
-                          i = a</pre>
-</div>
-<p>Si la somme ne peut pas être résolue, la formule de sommation correspondante est imprimée. Les sommes répétées peuvent être calculées en introduisant des limites additionnelles.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">summation</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">log</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i n m&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-<span class="go"> 2</span>
-<span class="go">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="o">**</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">  oo</span>
-<span class="go"> ___</span>
-<span class="go"> \  `</span>
-<span class="go">  \     -n</span>
-<span class="go">  /   log (n)</span>
-<span class="go"> /__,</span>
-<span class="go">n = 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">),</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">m</span><span class="p">))</span>
-<span class="go">      3    2</span>
-<span class="go">m    m    m</span>
-<span class="go">-- + -- + -</span>
-<span class="go">6    2    3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go"> x</span>
-<span class="go">e</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="integration">
-<span id="index-5"></span><h3>Intégration<a class="headerlink" href="#integration" title="Lien permanent vers ce titre">¶</a></h3>
-<p>SymPy supporte l&#8217;intégration définie et indéfinie de fonctions élémentaires et spéciales avec <tt class="docutils literal"><span class="pre">integrate()</span></tt>, qui utilise le puissant algorithme de Risch-Norman étendu et quelques heuristiques ainsi que le filtrage par motif :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">integrate</span><span class="p">,</span> <span class="n">erf</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Vous pouvez intégrer des fonctions élémentaires :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="mi">6</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">5</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go"> 6</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">-cos(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x*log(x) - x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">sinh</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go"> 2</span>
-<span class="go">x  + cosh(x)</span>
-</pre></div>
-</div>
-<p>Les fonctions spéciales sont aussi facilement gérées :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">erf</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">  ____    2</span>
-<span class="go">\/ pi *erf (x)</span>
-<span class="go">--------------</span>
-<span class="go">      4</span>
-</pre></div>
-</div>
-<p>Il est possible de calculer l&#8217;intégrale définie :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-<p>Les intégrales impropres sont aussi supportées :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">-1</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="complex-numbers">
-<span id="index-6"></span><h3>Nombres complexes<a class="headerlink" href="#complex-numbers" title="Lien permanent vers ce titre">¶</a></h3>
-<p>Outre l&#8217;unité imaginaire, I, qui est imaginaire, les symboles peuvent être créés avec certains attributs (par exemple réel, positif, complexe, etc.) affectant leur comportement :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span> <span class="c"># a plain x with no attributes</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go"> I*x</span>
-<span class="go">e</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">   -im(x)               -im(x)</span>
-<span class="go">I*e      *sin(re(x)) + e      *cos(re(x))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">I*sin(x) + cos(x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="functions">
-<h3>Fonctions<a class="headerlink" href="#functions" title="Lien permanent vers ce titre">¶</a></h3>
-<p><strong>trigonométrie</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">asin</span><span class="p">,</span> <span class="n">asinh</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">sin(x)*cos(y) + sin(y)*cos(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">-sin(x)*sin(y) + cos(x)*cos(y)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*sinh(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinh</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*sin(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
-<span class="go">I*pi</span>
-<span class="go">----</span>
-<span class="go"> 2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*asin(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3     5     7       9</span>
-<span class="go">    x     x     x       x       / 10\</span>
-<span class="go">x - -- + --- - ---- + ------ + O\x  /</span>
-<span class="go">    6    120   5040   362880</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3     5     7       9</span>
-<span class="go">    x     x     x       x       / 10\</span>
-<span class="go">x + -- + --- + ---- + ------ + O\x  /</span>
-<span class="go">    6    120   5040   362880</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3      5      7       9</span>
-<span class="go">    x    3*x    5*x    35*x     / 10\</span>
-<span class="go">x + -- + ---- + ---- + ----- + O\x  /</span>
-<span class="go">    6     40    112     1152</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3      5      7       9</span>
-<span class="go">    x    3*x    5*x    35*x     / 10\</span>
-<span class="go">x - -- + ---- - ---- + ----- + O\x  /</span>
-<span class="go">    6     40    112     1152</span>
-</pre></div>
-</div>
-<p><strong>harmoniques sphériques</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Ylm</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">  ___</span>
-<span class="go">\/ 3 *cos(theta)</span>
-<span class="go">----------------</span>
-<span class="go">        ____</span>
-<span class="go">    2*\/ pi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">   ___  I*phi</span>
-<span class="go">-\/ 6 *e     *sin(theta)</span>
-<span class="go">------------------------</span>
-<span class="go">            ____</span>
-<span class="go">        4*\/ pi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">   ____  I*phi</span>
-<span class="go">-\/ 30 *e     *sin(theta)*cos(theta)</span>
-<span class="go">------------------------------------</span>
-<span class="go">                  ____</span>
-<span class="go">              4*\/ pi</span>
-</pre></div>
-</div>
-<p><strong>factorielles et fonction gamma</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">x!</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">n!</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="c"># i.e. factorial(x)</span>
-<span class="go">                      /          2     2\</span>
-<span class="go">                    2 |EulerGamma    pi |    / 3\</span>
-<span class="go">1 - EulerGamma*x + x *|----------- + ---| + O\x /</span>
-<span class="go">                      \     2         12/</span>
-</pre></div>
-</div>
-<p><strong>fonction zeta</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">zeta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">zeta(4, x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">  4</span>
-<span class="go">pi</span>
-<span class="go">---</span>
-<span class="go"> 90</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">       4</span>
-<span class="go">     pi</span>
-<span class="go">-1 + ---</span>
-<span class="go">      90</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">         4</span>
-<span class="go">  17   pi</span>
-<span class="go">- -- + ---</span>
-<span class="go">  16    90</span>
-</pre></div>
-</div>
-<p><strong>polynômes</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">assoc_legendre</span><span class="p">,</span> <span class="n">chebyshevt</span><span class="p">,</span> <span class="n">legendre</span><span class="p">,</span> <span class="n">hermite</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">2*x  - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   4      2</span>
-<span class="go">8*x  - 8*x  + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">3*x    1</span>
-<span class="go">---- - -</span>
-<span class="go"> 2     2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">      8         6         4        2</span>
-<span class="go">6435*x    3003*x    3465*x    315*x     35</span>
-<span class="go">------- - ------- + ------- - ------ + ---</span>
-<span class="go">  128        32        64       32     128</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">        __________</span>
-<span class="go">       /    2</span>
-<span class="go">-3*x*\/  - x  + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">     2</span>
-<span class="go">- 3*x  + 3</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hermite</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   3</span>
-<span class="go">8*x  - 12*x</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="differential-equations">
-<span id="index-7"></span><h3>Équations différentielles<a class="headerlink" href="#differential-equations" title="Lien permanent vers ce titre">¶</a></h3>
-<p>Dans <tt class="docutils literal"><span class="pre">isympy</span></tt> :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">dsolve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">        2</span>
-<span class="go">       d</span>
-<span class="go">f(x) + ---(f(x))</span>
-<span class="go">         2</span>
-<span class="go">       dx</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dsolve</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">f(x) = C1*sin(x) + C2*cos(x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="algebraic-equations">
-<span id="index-8"></span><h3>Équations algébriques<a class="headerlink" href="#algebraic-equations" title="Lien permanent vers ce titre">¶</a></h3>
-<p>Dans <tt class="docutils literal"><span class="pre">isympy</span></tt> :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">solve</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">[-1, 1, -I, I]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">([</span><span class="n">x</span> <span class="o">+</span> <span class="mi">5</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">6</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">15</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">])</span>
-<span class="go">{x: -3, y: 1}</span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="section" id="linear-algebra">
-<span id="index-9"></span><h2>Algèbre linéaire<a class="headerlink" href="#linear-algebra" title="Lien permanent vers ce titre">¶</a></h2>
-<div class="section" id="matrices">
-<span id="index-10"></span><h3>Matrices<a class="headerlink" href="#matrices" title="Lien permanent vers ce titre">¶</a></h3>
-<p>Les matrices sont créées en instanciant la classe Matrix :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="go">[1  0]</span>
-<span class="go">[    ]</span>
-<span class="go">[0  1]</span>
-</pre></div>
-</div>
-<p>Elles peuvent aussi contenir des symboles :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">],</span> <span class="p">[</span><span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span>
-<span class="go">[1  x]</span>
-<span class="go">[    ]</span>
-<span class="go">[y  1]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">[x*y + 1    2*x  ]</span>
-<span class="go">[                ]</span>
-<span class="go">[  2*y    x*y + 1]</span>
-</pre></div>
-</div>
-<p>Pour plus d&#8217;informations et d&#8217;exemples avec les Matrices, voyez le Tutoriel Algèbre Linéaire.</p>
-</div>
-</div>
-<div class="section" id="pattern-matching">
-<span id="index-11"></span><h2>Filtrage par motif<a class="headerlink" href="#pattern-matching" title="Lien permanent vers ce titre">¶</a></h2>
-<p>Utilisez la méthode <tt class="docutils literal"><span class="pre">.match()</span></tt>, accompagnée de la classe <tt class="docutils literal"><span class="pre">Wild</span></tt>, pour effectuer un filtrage par motif sur des expressions. La méthode renvoit un dictionnaire avec les substitutions requises, comme suit :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Wild</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">{p: 5}</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">q</span><span class="p">)</span>
-<span class="go">{p: 1, q: 2}</span>
-</pre></div>
-</div>
-<p>Si le filtrage est infructueux, <tt class="docutils literal"><span class="pre">None</span></tt> est renvoyé :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">**</span><span class="n">x</span><span class="p">))</span>
-<span class="go">None</span>
-</pre></div>
-</div>
-<p>On peut aussi utiliser le paramètre <tt class="docutils literal"><span class="pre">exclude</span></tt> de la classe <tt class="docutils literal"><span class="pre">Wild</span></tt> pour s&#8217;assurer que certaines choses ne figurent pas dans le résultat :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">p</span><span class="p">))</span> <span class="c"># 1 is excluded</span>
-<span class="go">None</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span> <span class="c"># x is excluded</span>
-<span class="go">None</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">p</span><span class="p">))</span> <span class="c"># -1 is not excluded</span>
-<span class="go">{p_: -1}</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="printing">
-<span id="printing-tutorial"></span><h2>Affichage<a class="headerlink" href="#printing" title="Lien permanent vers ce titre">¶</a></h2>
-<p>Les expressions peuvent être affichées de plusieures façons.</p>
-<p><strong>Affichage standard</strong></p>
-<p>C&#8217;est ce que <tt class="docutils literal"><span class="pre">str(expression)</span></tt> renvoit et ça ressemble à ça :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1/x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Integral(x**2, x)</span>
-</pre></div>
-</div>
-<p><strong>Affichage amélioré</strong></p>
-<p>Ceci est un affichage amélioré en art ASCII produit par la fonction <tt class="docutils literal"><span class="pre">pprint</span></tt> :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go"> 2</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="go">-</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">  /</span>
-<span class="go"> |</span>
-<span class="go"> |  2</span>
-<span class="go"> | x  dx</span>
-<span class="go"> |</span>
-<span class="go">/</span>
-</pre></div>
-</div>
-<p>Si vous avez une police unicode installée, l&#8217;affichage amélioré l&#8217;utilisera par défaut. Vous pouvez outrepasser ce comportement en utilisant l&#8217;option <tt class="docutils literal"><span class="pre">use_unicode</span></tt> :</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">⌠</span>
-<span class="go">⎮  2</span>
-<span class="go">⎮ x  dx</span>
-<span class="go">⌡</span>
-</pre></div>
-</div>
-<p>Voyez aussi la page wiki <a class="reference external" href="https://github.com/sympy/sympy/wiki/Pretty-Printing">Pretty Printing</a> pour plus d&#8217;exemples concernant l&#8217;affichage amélioré et l&#8217;option unicode.</p>
-<p>Astuce : pour activer l&#8217;affichage amélioré par défaut dans l&#8217;interpréteur Python, utilisez :</p>
-<div class="highlight-python"><pre>$ python
-Python 2.5.2 (r252:60911, Jun 25 2008, 17:58:32)
-[GCC 4.3.1] on linux2
-Type &quot;help&quot;, &quot;copyright&quot;, &quot;credits&quot; or &quot;license&quot; for more information.
-&gt;&gt;&gt; from sympy import init_printing, var, Integral
-&gt;&gt;&gt; init_printing(use_unicode=False, wrap_line=False, no_global=True)
-&gt;&gt;&gt; var(&quot;x&quot;)
-x
-&gt;&gt;&gt; x**3/3
- 3
-x
---
-3
-&gt;&gt;&gt; Integral(x**2, x) #doctest: +NORMALIZE_WHITESPACE
-  /
- |
- |  2
- | x  dx
- |
-/</pre>
-</div>
-<p><strong>Python</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.python</span> <span class="kn">import</span> <span class="n">python</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = x**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = 1/x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = Integral(x**2, x)</span>
-</pre></div>
-</div>
-<p><strong>LaTeX</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">latex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x^{2}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;inline&#39;</span><span class="p">)</span>
-<span class="go">$x^{2}$</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;equation&#39;</span><span class="p">)</span>
-<span class="go">\begin{equation}x^{2}\end{equation}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;equation*&#39;</span><span class="p">)</span>
-<span class="go">\begin{equation*}x^{2}\end{equation*}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">\frac{1}{x}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">\int x^{2}\, dx</span>
-</pre></div>
-</div>
-<p><strong>MathML</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.mathml</span> <span class="kn">import</span> <span class="n">mathml</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">latex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">mathml</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">&lt;apply&gt;&lt;power/&gt;&lt;ci&gt;x&lt;/ci&gt;&lt;cn&gt;2&lt;/cn&gt;&lt;/apply&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">mathml</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">))</span>
-<span class="go">&lt;apply&gt;&lt;power/&gt;&lt;ci&gt;x&lt;/ci&gt;&lt;cn&gt;-1&lt;/cn&gt;&lt;/apply&gt;</span>
-</pre></div>
-</div>
-<p><strong>Pyglet</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">preview</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">preview</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span> 
-</pre></div>
-</div>
-<p>Si pyglet est installé, une fenêtre pyglet apparaîtra avec l&#8217;expression rendue par LaTeX :</p>
-<img alt="../_images/pngview1.png" src="../_images/pngview1.png" />
-<div class="section" id="notes">
-<h3>Notes<a class="headerlink" href="#notes" title="Lien permanent vers ce titre">¶</a></h3>
-<p><tt class="docutils literal"><span class="pre">isympy</span></tt> appelle <tt class="docutils literal"><span class="pre">pprint</span></tt> automatiquement, c&#8217;est donc pourquoi vous voyez un affichage amélioré par défaut.</p>
-<p>Notez qu&#8217;il y a aussi un module d&#8217;affichage disponible, <tt class="docutils literal"><span class="pre">sympy.printing</span></tt>. D&#8217;autres méthodes d&#8217;affichage disponibles dans ce module sont :</p>
-<ul class="simple">
-<li><tt class="docutils literal"><span class="pre">pretty(expr)</span></tt>, <tt class="docutils literal"><span class="pre">pretty_print(expr)</span></tt>, <tt class="docutils literal"><span class="pre">pprint(expr)</span></tt>: Renvoit ou affiche, respectivement, une représentation améliorée de <tt class="docutils literal"><span class="pre">expr</span></tt>. C&#8217;est la même chose que le second niveau de représentation décrit ci-dessus.</li>
-<li><tt class="docutils literal"><span class="pre">latex(expr)</span></tt>, <tt class="docutils literal"><span class="pre">print_latex(expr)</span></tt>: Renvoit ou affiche, respectivement, une représentation <a class="reference external" href="http://www.latex-project.org/">LaTeX</a> de <tt class="docutils literal"><span class="pre">expr</span></tt></li>
-<li><tt class="docutils literal"><span class="pre">mathml(expr)</span></tt>, <tt class="docutils literal"><span class="pre">print_mathml(expr)</span></tt>: Renvoit ou affiche, respectivement, une représentation <a class="reference external" href="http://www.w3.org/Math/">MathML</a> de <tt class="docutils literal"><span class="pre">expr</span></tt>.</li>
-<li><tt class="docutils literal"><span class="pre">print_gtk(expr)</span></tt>: Print <tt class="docutils literal"><span class="pre">expr</span></tt> to <a class="reference external" href="http://helm.cs.unibo.it/mml-widget/">Gtkmathview</a>, a GTK widget that displays MathML code. The <a class="reference external" href="http://helm.cs.unibo.it/mml-widget/">Gtkmathview</a> program is required.</li>
-</ul>
-</div>
-</div>
-<div class="section" id="further-documentation">
-<h2>Documentation<a class="headerlink" href="#further-documentation" title="Lien permanent vers ce titre">¶</a></h2>
-<p>Il est maintenant temps d&#8217;en apprendre plus sur SymPy. Consultez le <a class="reference external" href="../guide.html">SymPy User&#8217;s Guide</a> et la <a class="reference external" href="../modules/index.html">SymPy Modules Reference</a>.</p>
-<p>Veillez également à consulter notre <a class="reference external" href="http://wiki.sympy.org/">wiki.sympy.org</a> public, qui contient beaucoup d&#8217;exemples utiles, des tutoriels, des livres de recettes auxquels nous et nos utilisateurs ont contribué et que nous vous encourageons à modifier.</p>
-<div class="section" id="translations">
-<h3>Traductions<a class="headerlink" href="#translations" title="Lien permanent vers ce titre">¶</a></h3>
-<p>Ce tutoriel est aussi disponible dans les langues suivantes :</p>
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-            <p class="logo"><a href="tutorial.html">
-              <img class="logo" src="../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="tutorial.html">Table des matières</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Tutoriel</a><ul>
-<li><a class="reference internal" href="#introduction">Introduction</a></li>
-<li><a class="reference internal" href="#first-steps-with-sympy">Premiers pas avec SymPy</a><ul>
-<li><a class="reference internal" href="#isympy-console">La console isympy</a></li>
-<li><a class="reference internal" href="#using-sympy-as-a-calculator">Utiliser SymPy comme une calculatrice</a></li>
-<li><a class="reference internal" href="#symbols">Symboles</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#algebra">Algèbre</a></li>
-<li><a class="reference internal" href="#calculus">Calcul</a><ul>
-<li><a class="reference internal" href="#limits">Limites</a></li>
-<li><a class="reference internal" href="#differentiation">Dérivation</a></li>
-<li><a class="reference internal" href="#series-expansion">Développement en série</a></li>
-<li><a class="reference internal" href="#summation">Sommes</a></li>
-<li><a class="reference internal" href="#integration">Intégration</a></li>
-<li><a class="reference internal" href="#complex-numbers">Nombres complexes</a></li>
-<li><a class="reference internal" href="#functions">Fonctions</a></li>
-<li><a class="reference internal" href="#differential-equations">Équations différentielles</a></li>
-<li><a class="reference internal" href="#algebraic-equations">Équations algébriques</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#linear-algebra">Algèbre linéaire</a><ul>
-<li><a class="reference internal" href="#matrices">Matrices</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#pattern-matching">Filtrage par motif</a></li>
-<li><a class="reference internal" href="#printing">Affichage</a><ul>
-<li><a class="reference internal" href="#notes">Notes</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#further-documentation">Documentation</a><ul>
-<li><a class="reference internal" href="#translations">Traductions</a></li>
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-  <div class="section" id="tutorial">
-<span id="id1"></span><h1>Tutorial<a class="headerlink" href="#tutorial" title="Stały odnośnik do tego nagłówka">¶</a></h1>
-<div class="section" id="introduction">
-<h2>Wprowadzenie<a class="headerlink" href="#introduction" title="Stały odnośnik do tego nagłówka">¶</a></h2>
-<p>SymPy jest biblioteką Pythona służącą do wykonywania obliczeń symbolicznych. Za cel stawia sobie stanie się zaawansowanym, w pełni funkcjonalnym systemem algebry komputerowej (ang. Computer Algebra System, w skrócie CAS). Przy tym dąży do zachowania kodu tak prostego, jak to tylko możliwe, by był on zrozumiały i elastyczny. SymPy został napisany całkowicie w Pythonie i nie wymaga żadnych zewnętrznych bibliotek.</p>
-<p>Ten tutorial przedstawia ogólny zarys i przegląd wybranych funkcji SymPy. Dzięki niemu dowiesz się, co SymPy może dla Ciebie zrobić (i jak). Jeśli będziesz chciał dowiedzieć się więcej, przeczytaj <a class="reference external" href="../guide.html">Przewodnik użytkownika</a>, <a class="reference external" href="../modules/index.html">Opis modułów</a>, lub też bezpośrednio <a class="reference external" href="https://github.com/sympy/sympy/">źródła</a>.</p>
-</div>
-<div class="section" id="first-steps-with-sympy">
-<h2>Pierwsze kroki z SymPy<a class="headerlink" href="#first-steps-with-sympy" title="Stały odnośnik do tego nagłówka">¶</a></h2>
-<p>Żeby ściągnąć SymPy udaj się pod adres <a class="reference external" href="http://code.google.com/p/sympy/">http://code.google.com/p/sympy/</a> i pobierz ostatniego tarballa z sekcji Featured Downloads:</p>
-<img alt="../_images/featured-downloads.png" src="../_images/featured-downloads.png" />
-<p>Rozpakuj:</p>
-<pre class="literal-block">
-$ <strong class="input">tar xzf sympy-0.5.12.tar.gz</strong>
-</pre>
-<p>i sprawdź pod interpreterem Pythona:</p>
-<pre class="literal-block">
-$ <strong class="input">cd sympy-0.5.12</strong>
-$ <strong class="input">python</strong>
-Python 2.4.4 (#2, Jan  3 2008, 13:36:28)
-[GCC 4.2.3 20071123 (prerelease) (Debian 4.2.2-4)] on linux2
-Type &quot;help&quot;, &quot;copyright&quot;, &quot;credits&quot; or &quot;license&quot; for more information.
-&gt;&gt;&gt; from sympy import Symbol, cos
-&gt;&gt;&gt; x = Symbol(&quot;x&quot;)
-&gt;&gt;&gt; (1/cos(x)).series(x, 0, 10)
-1 + x**2/2 + 5*x**4/24 + 61*x**6/720 + 277*x**8/8064 + O(x**10)
-</pre>
-<p>Możesz używać SymPy tak jak pokazano powyżej - ten sposób jest zalecany, jeśli używasz SymPy w swoim własnym programie. Możesz też zainstalować SymPy użwając <tt class="docutils literal"><span class="pre">./setup.py</span> <span class="pre">install</span></tt> (jak w przypadku każdego modułu Pythona), albo po prostu zainstalować odpowiednią paczkę w używanej przez siebie dystrybucji Linuksa, np.:</p>
-<div class="topic">
-<p class="topic-title first">Installing SymPy in Debian</p>
-<pre class="literal-block">
-$ <strong class="input">sudo apt-get install python-sympy</strong>
-Reading package lists... Done
-Building dependency tree
-Reading state information... Done
-The following NEW packages will be installed:
-  python-sympy
-0 upgraded, 1 newly installed, 0 to remove and 18 not upgraded.
-Need to get 991kB of archives.
-After this operation, 5976kB of additional disk space will be used.
-Get:1 http://ftp.cz.debian.org unstable/main python-sympy 0.5.12-1 [991kB]
-Fetched 991kB in 2s (361kB/s)
-Selecting previously deselected package python-sympy.
-(Reading database ... 232619 files and directories currently installed.)
-Unpacking python-sympy (from .../python-sympy_0.5.12-1_all.deb) ...
-Setting up python-sympy (0.5.12-1) ...
-</pre>
-</div>
-<p>For other means how to install SymPy, consult the wiki page
-<a class="reference external" href="https://github.com/sympy/sympy/wiki/Download-Installation">Download and Installation</a>.</p>
-<div class="section" id="isympy-console">
-<h3>isympy<a class="headerlink" href="#isympy-console" title="Stały odnośnik do tego nagłówka">¶</a></h3>
-<p>For experimenting with new features, or when figuring out how to do things, you
-can use our special wrapper around IPython called <tt class="docutils literal"><span class="pre">isympy</span></tt> (located in
-<tt class="docutils literal"><span class="pre">bin/isympy</span></tt> if you are running from the source directory) which is just a
-standard Python shell that has already imported the relevant SymPy modules and
-defined the symbols x, y, z and some other things:</p>
-<pre class="literal-block">
-$ <strong class="input">cd sympy</strong>
-$ <strong class="input">./bin/isympy</strong>
-IPython console for SymPy 0.7.2-git (Python 2.7.1) (ground types: gmpy)
-
-These commands were executed:
-&gt;&gt;&gt;
-&gt;&gt;&gt; from sympy import *
-&gt;&gt;&gt; x, y, z, t = symbols('x y z t')
-&gt;&gt;&gt; k, m, n = symbols('k m n', integer=True)
-&gt;&gt;&gt; f, g, h = symbols('f g h', cls=Function)
-
-Documentation can be found at <a class="reference external" href="http://www.sympy.org">http://www.sympy.org</a>
-
-In [1]: <strong class="input">(1/cos(x)).series(x, 0, 10)</strong>
-Out[1]:
-     2      4       6        8
-    x    5*x    61*x    277*x     / 10\
-1 + -- + ---- + ----- + ------ + O\x  /
-    2     24     720     8064
-</pre>
-<div class="admonition note">
-<p class="first admonition-title">Uwaga</p>
-<p class="last">Polecenia wprowadzane przez Ciebie są pogrubione. Tak więc to, co zrobiliśmy w trzech linijkach w zwykłym interpreterze Pythona, może być zrobione w jednej linijce w isympy.</p>
-</div>
-</div>
-<div class="section" id="using-sympy-as-a-calculator">
-<h3>SymPy jako kalkulator<a class="headerlink" href="#using-sympy-as-a-calculator" title="Stały odnośnik do tego nagłówka">¶</a></h3>
-<p>SymPy has three built-in numeric types: Float, Rational and Integer.</p>
-<p>Klasa Rational reprezentuje liczbę wymierną jako parę dwóch liczb całkowitych - licznik i mianownik. Rational(1, 2) reprezentuje więc 1/2, Rational(5, 2) reprezentuje 5/2 itd.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
-<span class="go">1/2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">*</span><span class="mi">2</span>
-<span class="go">1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">50</span><span class="o">/</span><span class="n">Rational</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span><span class="o">**</span><span class="mi">50</span>
-<span class="go">1/88817841970012523233890533447265625</span>
-</pre></div>
-</div>
-<p>Należy zachować ostrożność, pracując z liczbami całkowitymi i zmiennoprzecinkowymi w Pythonie, szczególnie podczas dzielenia. Można bowiem przypadkowo utworzyć liczbę obsługiwaną przez samego Pythona, a nie przez SymPy. Dzielenie przez siebie dwóch Pythonowych liczb całkowitych może dać w wyniku liczbę zmiennoprzecinkową &#8211; jest to standard &#8220;prawidłowego dzielenia&#8221; z Pythona 3 i domyślne zachowanie <tt class="docutils literal"><span class="pre">isympy</span></tt>, który importuje dzielenie z __future__:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span> 
-<span class="go">0.5</span>
-</pre></div>
-</div>
-<p>We wcześniejszych wersjach Pythona, gdzie takie dzielenie nie było zaimportowane, wynik był ucinany do liczby całkowitej:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span> 
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>Jednakże, w obu przypadkach, nie operowaliśmy na liczbach obsługiwanych przez SymPy, ponieważ Python stworzył swoją własną zmienną. Prawdopodobnie większość czasu będziesz operował na liczbach wymiernych &#8211; dlatego upewnij się, że na pewno używasz klasy Rational, aby uzyskać prawidłowy wynik obsługiwany przez SymPy. Dla wygody możesz na przykład przyrównać <tt class="docutils literal"><span class="pre">R</span></tt> i Rational:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="c"># R(1) is a SymPy Integer and Integer/int gives a Rational</span>
-<span class="go">1/2</span>
-</pre></div>
-</div>
-<p>Możemy skorzystać też z kilku specjalnych stałych, takich jak e czy pi. Są one traktowane jako symbole (np. w przypadku 1 + pi nie zostanie obliczona przybliżona wartość tego wyrażenia, lecz pozostanie ono jako 1 + pi), i mają arbitralną precyzję:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">E</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">pi**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">3.14159265358979</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">pi</span> <span class="o">+</span> <span class="n">E</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">5.85987448204884</span>
-</pre></div>
-</div>
-<p>Jak widzisz, evalf wylicza wartość wyrażenia jako liczbę zmiennoprzecinkową.</p>
-<p>Symbol <tt class="docutils literal"><span class="pre">oo</span></tt> jest używany jako klasa definiująca matematyczną nieskończoność:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">oo</span> <span class="o">&gt;</span> <span class="mi">99999</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">oo</span> <span class="o">+</span> <span class="mi">1</span>
-<span class="go">oo</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="symbols">
-<h3>Symbole<a class="headerlink" href="#symbols" title="Stały odnośnik do tego nagłówka">¶</a></h3>
-<p>W przeciwieństwie do innych systemów algebry komputerowej (CAS), w SymPy sam musisz zadeklarować zmienne symboliczne:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>On the left is the normal Python variable which has been assigned to the
-SymPy Symbol class. Predefined symbols (including those for symbols with
-Greek names) are available for import from abc:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">theta</span>
-</pre></div>
-</div>
-<p>Symbols can also be created with the <tt class="docutils literal"><span class="pre">symbols</span></tt> or <tt class="docutils literal"><span class="pre">var</span></tt> functions, the
-latter automatically adding the created symbols to the namespace, and both
-accepting a range notation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">var</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a,b,c&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;d:f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;g:h&#39;</span><span class="p">)</span>
-<span class="go">(g, h)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;g:2&#39;</span><span class="p">)</span>
-<span class="go">(g0, g1)</span>
-</pre></div>
-</div>
-<p>Instances of the Symbol class &#8220;play well together&#8221; and are the building blocks
-of expresions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">x</span> <span class="o">-</span> <span class="n">y</span>
-<span class="go">2*x</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">(x + y)**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go">x**2 + 2*x*y + y**2</span>
-</pre></div>
-</div>
-<p>Można podstawiać w ich miejsce inne symbole i liczby używając <tt class="docutils literal"><span class="pre">subs(stary,</span> <span class="pre">nowy)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">(y + 1)**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">4*y**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>W pozostałej części tutoriala zakładamy, że wykonaliśmy następujące polecenie:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">init_printing</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">init_printing</span><span class="p">(</span><span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">wrap_line</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">no_global</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Dzięki temu wszystko będzie się ładnie wyświetlało (zobacz sekcję <a class="reference internal" href="#printing-tutorial"><em>Wyświetlanie</em></a> poniżej). Jeśli masz zainstalowaną czcionkę unicode, możesz zamienić use_unicode=False na use_unicode=True, dzięki czemu uzyskasz nieco ładniejsze wyjście.</p>
-</div>
-</div>
-<div class="section" id="algebra">
-<h2>Algebra<a class="headerlink" href="#algebra" title="Stały odnośnik do tego nagłówka">¶</a></h2>
-<p>Żeby dokonać rozkładu na ułamki proste, użyj <tt class="docutils literal"><span class="pre">apart(wyrażenie,</span> <span class="pre">x)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">apart</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">)</span>
-<span class="go">       1</span>
-<span class="go">---------------</span>
-<span class="go">(x + 1)*(x + 2)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">apart</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">    1       1</span>
-<span class="go">- ----- + -----</span>
-<span class="go">  x + 2   x + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">x + 1</span>
-<span class="go">-----</span>
-<span class="go">x - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">apart</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">      2</span>
-<span class="go">1 + -----</span>
-<span class="go">    x - 1</span>
-</pre></div>
-</div>
-<p>Aby wszystko znowu ze sobą połączyć, użyj <tt class="docutils literal"><span class="pre">together(wyrażenie,</span> <span class="pre">x)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">together</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">)</span>
-<span class="go">x*y + x*z + y*z</span>
-<span class="go">---------------</span>
-<span class="go">     x*y*z</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="n">apart</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x + 1</span>
-<span class="go">-----</span>
-<span class="go">x - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="n">apart</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">       1</span>
-<span class="go">---------------</span>
-<span class="go">(x + 1)*(x + 2)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="calculus">
-<span id="index-0"></span><h2>Rachunek różniczkowy i całkowy<a class="headerlink" href="#calculus" title="Stały odnośnik do tego nagłówka">¶</a></h2>
-<div class="section" id="limits">
-<span id="index-1"></span><h3>Granice<a class="headerlink" href="#limits" title="Stały odnośnik do tego nagłówka">¶</a></h3>
-<p>Limits are easy to use in SymPy, they follow the syntax <tt class="docutils literal"><span class="pre">limit(function,</span>
-<span class="pre">variable,</span> <span class="pre">point)</span></tt>, so to compute the limit of f(x) as x -&gt; 0, you would issue
-<tt class="docutils literal"><span class="pre">limit(f,</span> <span class="pre">x,</span> <span class="pre">0)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">limit</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>Można również obliczyć granicę w nieskończoności:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">oo</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">0</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>Kilka nietrywialnych przykładów granic znajdziesz w pliku testowym <a class="reference external" href="https://github.com/sympy/sympy/blob/master/sympy/series/tests/test_demidovich.py">test_demidovich.py</a></p>
-</div>
-<div class="section" id="differentiation">
-<span id="index-2"></span><h3>Różniczkowanie<a class="headerlink" href="#differentiation" title="Stały odnośnik do tego nagłówka">¶</a></h3>
-<p>Możesz różniczkować dowolne wyrażenie w SymPy używając <tt class="docutils literal"><span class="pre">diff(funkcja,</span> <span class="pre">zmienna)</span></tt>. Przykłady:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">diff</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">tan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">cos(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">2*cos(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">tan (x) + 1</span>
-</pre></div>
-</div>
-<p>Możesz też sprawdzić poprawność:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">limit</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">delta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">((</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">delta</span><span class="p">)</span> <span class="o">-</span> <span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">/</span><span class="n">delta</span><span class="p">,</span> <span class="n">delta</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">tan (x) + 1</span>
-</pre></div>
-</div>
-<p>Pochodne wyższych rzędów mogą być obliczone przy użyciu <tt class="docutils literal"><span class="pre">diff(funkcja,</span> <span class="pre">zmienna,</span> <span class="pre">n)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">2*cos(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">-4*sin(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">-8*cos(2*x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="series-expansion">
-<span id="index-3"></span><h3>Rozwinięcie w szereg<a class="headerlink" href="#series-expansion" title="Stały odnośnik do tego nagłówka">¶</a></h3>
-<p>Użyj <tt class="docutils literal"><span class="pre">.series(var,</span> <span class="pre">point,</span> <span class="pre">order)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     2    4     6      8</span>
-<span class="go">    x    x     x      x      / 10\</span>
-<span class="go">1 - -- + -- - --- + ----- + O\x  /</span>
-<span class="go">    2    24   720   40320</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     2      4       6        8</span>
-<span class="go">    x    5*x    61*x    277*x     / 10\</span>
-<span class="go">1 + -- + ---- + ----- + ------ + O\x  /</span>
-<span class="go">    2     24     720     8064</span>
-</pre></div>
-</div>
-<p>Inny prosty przykład:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">pprint</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;y&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="go">1/y - x/y**2 + x**2/y**3 - x**3/y**4 + x**4/y**5 + O(x**5)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="go">          2    3    4</span>
-<span class="go">1   x    x    x    x     / 5\</span>
-<span class="go">- - -- + -- - -- + -- + O\x /</span>
-<span class="go">y    2    3    4    5</span>
-<span class="go">    y    y    y    y</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="summation">
-<span id="index-4"></span><h3>Summation<a class="headerlink" href="#summation" title="Stały odnośnik do tego nagłówka">¶</a></h3>
-<p>Compute the summation of f with respect to the given summation variable over the given limits.</p>
-<p>summation(f, (i, a, b)) computes the sum of f with respect to i from a to b,
-i.e.,</p>
-<div class="highlight-python"><pre>                            b
-                          ____
-                          \   `
-summation(f, (i, a, b)) =  )    f
-                          /___,
-                          i = a</pre>
-</div>
-<p>If it cannot compute the sum, it prints the corresponding summation formula.
-Repeated sums can be computed by introducing additional limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">summation</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">log</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i n m&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-<span class="go"> 2</span>
-<span class="go">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="o">**</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">  oo</span>
-<span class="go"> ___</span>
-<span class="go"> \  `</span>
-<span class="go">  \     -n</span>
-<span class="go">  /   log (n)</span>
-<span class="go"> /__,</span>
-<span class="go">n = 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">),</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">m</span><span class="p">))</span>
-<span class="go">      3    2</span>
-<span class="go">m    m    m</span>
-<span class="go">-- + -- + -</span>
-<span class="go">6    2    3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go"> x</span>
-<span class="go">e</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="integration">
-<span id="index-5"></span><h3>Całkowanie<a class="headerlink" href="#integration" title="Stały odnośnik do tego nagłówka">¶</a></h3>
-<p>SymPy potrafi obliczyć całki oznaczone i nieoznaczone dla niektórych funkcji poprzez <tt class="docutils literal"><span class="pre">integrate()</span></tt>. Używany jest do tego rozszerzony algorytm Rischa-Normana, kilka heurystyk i wyszukiwanie wzorca:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">integrate</span><span class="p">,</span> <span class="n">erf</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Możesz całkować podstawowe funkcje:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="mi">6</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">5</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go"> 6</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">-cos(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x*log(x) - x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">sinh</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go"> 2</span>
-<span class="go">x  + cosh(x)</span>
-</pre></div>
-</div>
-<p>Bardziej skomplikowane funkcje również są obsługiwane:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">erf</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">  ____    2</span>
-<span class="go">\/ pi *erf (x)</span>
-<span class="go">--------------</span>
-<span class="go">      4</span>
-</pre></div>
-</div>
-<p>It is possible to compute definite integrals:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-<p>Also, improper integrals are supported as well:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">-1</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="complex-numbers">
-<span id="index-6"></span><h3>Liczby zespolone<a class="headerlink" href="#complex-numbers" title="Stały odnośnik do tego nagłówka">¶</a></h3>
-<p>Besides the imaginary unit, I, which is imaginary, symbols can be created with
-attributes (e.g. real, positive, complex, etc...) and this will affect how
-they behave:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span> <span class="c"># a plain x with no attributes</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go"> I*x</span>
-<span class="go">e</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">   -im(x)               -im(x)</span>
-<span class="go">I*e      *sin(re(x)) + e      *cos(re(x))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">I*sin(x) + cos(x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="functions">
-<h3>Funkcje<a class="headerlink" href="#functions" title="Stały odnośnik do tego nagłówka">¶</a></h3>
-<p><strong>trygonometria</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">asin</span><span class="p">,</span> <span class="n">asinh</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">sin(x)*cos(y) + sin(y)*cos(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">-sin(x)*sin(y) + cos(x)*cos(y)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*sinh(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinh</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*sin(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
-<span class="go">I*pi</span>
-<span class="go">----</span>
-<span class="go"> 2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*asin(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3     5     7       9</span>
-<span class="go">    x     x     x       x       / 10\</span>
-<span class="go">x - -- + --- - ---- + ------ + O\x  /</span>
-<span class="go">    6    120   5040   362880</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3     5     7       9</span>
-<span class="go">    x     x     x       x       / 10\</span>
-<span class="go">x + -- + --- + ---- + ------ + O\x  /</span>
-<span class="go">    6    120   5040   362880</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3      5      7       9</span>
-<span class="go">    x    3*x    5*x    35*x     / 10\</span>
-<span class="go">x + -- + ---- + ---- + ----- + O\x  /</span>
-<span class="go">    6     40    112     1152</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3      5      7       9</span>
-<span class="go">    x    3*x    5*x    35*x     / 10\</span>
-<span class="go">x - -- + ---- - ---- + ----- + O\x  /</span>
-<span class="go">    6     40    112     1152</span>
-</pre></div>
-</div>
-<p><strong>harmoniki sferyczne</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Ylm</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">  ___</span>
-<span class="go">\/ 3 *cos(theta)</span>
-<span class="go">----------------</span>
-<span class="go">        ____</span>
-<span class="go">    2*\/ pi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">   ___  I*phi</span>
-<span class="go">-\/ 6 *e     *sin(theta)</span>
-<span class="go">------------------------</span>
-<span class="go">            ____</span>
-<span class="go">        4*\/ pi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">   ____  I*phi</span>
-<span class="go">-\/ 30 *e     *sin(theta)*cos(theta)</span>
-<span class="go">------------------------------------</span>
-<span class="go">                  ____</span>
-<span class="go">              4*\/ pi</span>
-</pre></div>
-</div>
-<p><strong>silnia i funkcja gamma</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">x!</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">n!</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="c"># i.e. factorial(x)</span>
-<span class="go">                      /          2     2\</span>
-<span class="go">                    2 |EulerGamma    pi |    / 3\</span>
-<span class="go">1 - EulerGamma*x + x *|----------- + ---| + O\x /</span>
-<span class="go">                      \     2         12/</span>
-</pre></div>
-</div>
-<p><strong>funkcja zeta</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">zeta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">zeta(4, x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">  4</span>
-<span class="go">pi</span>
-<span class="go">---</span>
-<span class="go"> 90</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">       4</span>
-<span class="go">     pi</span>
-<span class="go">-1 + ---</span>
-<span class="go">      90</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">         4</span>
-<span class="go">  17   pi</span>
-<span class="go">- -- + ---</span>
-<span class="go">  16    90</span>
-</pre></div>
-</div>
-<p><strong>wielomiany</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">assoc_legendre</span><span class="p">,</span> <span class="n">chebyshevt</span><span class="p">,</span> <span class="n">legendre</span><span class="p">,</span> <span class="n">hermite</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">2*x  - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   4      2</span>
-<span class="go">8*x  - 8*x  + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">3*x    1</span>
-<span class="go">---- - -</span>
-<span class="go"> 2     2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">      8         6         4        2</span>
-<span class="go">6435*x    3003*x    3465*x    315*x     35</span>
-<span class="go">------- - ------- + ------- - ------ + ---</span>
-<span class="go">  128        32        64       32     128</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">        __________</span>
-<span class="go">       /    2</span>
-<span class="go">-3*x*\/  - x  + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">     2</span>
-<span class="go">- 3*x  + 3</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hermite</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   3</span>
-<span class="go">8*x  - 12*x</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="differential-equations">
-<span id="index-7"></span><h3>Równania różniczkowe<a class="headerlink" href="#differential-equations" title="Stały odnośnik do tego nagłówka">¶</a></h3>
-<p>W <tt class="docutils literal"><span class="pre">isympy</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">dsolve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">        2</span>
-<span class="go">       d</span>
-<span class="go">f(x) + ---(f(x))</span>
-<span class="go">         2</span>
-<span class="go">       dx</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dsolve</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">f(x) = C1*sin(x) + C2*cos(x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="algebraic-equations">
-<span id="index-8"></span><h3>Równania algebraiczne<a class="headerlink" href="#algebraic-equations" title="Stały odnośnik do tego nagłówka">¶</a></h3>
-<p>W <tt class="docutils literal"><span class="pre">isympy</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">solve</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">[-1, 1, -I, I]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">([</span><span class="n">x</span> <span class="o">+</span> <span class="mi">5</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">6</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">15</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">])</span>
-<span class="go">{x: -3, y: 1}</span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="section" id="linear-algebra">
-<span id="index-9"></span><h2>Algebra liniowa<a class="headerlink" href="#linear-algebra" title="Stały odnośnik do tego nagłówka">¶</a></h2>
-<div class="section" id="matrices">
-<span id="index-10"></span><h3>Macierze<a class="headerlink" href="#matrices" title="Stały odnośnik do tego nagłówka">¶</a></h3>
-<p>Macierze są tworzone jako obiekty klasy Matrix:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="go">[1  0]</span>
-<span class="go">[    ]</span>
-<span class="go">[0  1]</span>
-</pre></div>
-</div>
-<p>They can also contain symbols:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">],</span> <span class="p">[</span><span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span>
-<span class="go">[1  x]</span>
-<span class="go">[    ]</span>
-<span class="go">[y  1]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">[x*y + 1    2*x  ]</span>
-<span class="go">[                ]</span>
-<span class="go">[  2*y    x*y + 1]</span>
-</pre></div>
-</div>
-<p>For more about Matrices, see the Linear Algebra tutorial.</p>
-</div>
-</div>
-<div class="section" id="pattern-matching">
-<span id="index-11"></span><h2>Wyszukiwanie wzorca<a class="headerlink" href="#pattern-matching" title="Stały odnośnik do tego nagłówka">¶</a></h2>
-<p>Użyj metody <tt class="docutils literal"><span class="pre">.match()</span></tt> z klasy <tt class="docutils literal"><span class="pre">Wild</span></tt>, by wyszukać wzorzec w wyrażeniu (porównać wyrażenia). Metoda ta zwróci zbiór odpowiadających sobie współczynników:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Wild</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">{p: 5}</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">q</span><span class="p">)</span>
-<span class="go">{p: 1, q: 2}</span>
-</pre></div>
-</div>
-<p>Jeśli wyszukiwanie zakończyło się niepowodzeniem, zwracane jest <tt class="docutils literal"><span class="pre">None</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">**</span><span class="n">x</span><span class="p">))</span>
-<span class="go">None</span>
-</pre></div>
-</div>
-<p>Można również użyć parametru wykluczania (exclude) klasy <tt class="docutils literal"><span class="pre">Wild</span></tt>, by zagwarantować, że w wyniku nie zostaną pokazane niektóre rzeczy:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">p</span><span class="p">))</span> <span class="c"># 1 is excluded</span>
-<span class="go">None</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span> <span class="c"># x is excluded</span>
-<span class="go">None</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">p</span><span class="p">))</span> <span class="c"># -1 is not excluded</span>
-<span class="go">{p_: -1}</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="printing">
-<span id="printing-tutorial"></span><h2>Wyświetlanie<a class="headerlink" href="#printing" title="Stały odnośnik do tego nagłówka">¶</a></h2>
-<p>There are many ways to print expressions.</p>
-<p><strong>Standardowe</strong></p>
-<p>Jest to to, co zwraca <tt class="docutils literal"><span class="pre">str(wyrażenie)</span></tt> i wygląda tak:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1/x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Integral(x**2, x)</span>
-</pre></div>
-</div>
-<p><strong>Ładne wyświetlanie</strong></p>
-<p>Nice ascii-art printing is produced by the <tt class="docutils literal"><span class="pre">pprint</span></tt> function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go"> 2</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="go">-</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">  /</span>
-<span class="go"> |</span>
-<span class="go"> |  2</span>
-<span class="go"> | x  dx</span>
-<span class="go"> |</span>
-<span class="go">/</span>
-</pre></div>
-</div>
-<p>If you have a unicode font installed, the <tt class="docutils literal"><span class="pre">pprint</span></tt> function will use it by
-default. You can override this using the <tt class="docutils literal"><span class="pre">use_unicode</span></tt> option.:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">⌠</span>
-<span class="go">⎮  2</span>
-<span class="go">⎮ x  dx</span>
-<span class="go">⌡</span>
-</pre></div>
-</div>
-<p>Zobacz również wiki <a class="reference external" href="https://github.com/sympy/sympy/wiki/Pretty-Printing">Pretty Printing</a>, gdzie znajdziesz więcej przykładów ładnego wyświetlania w unicode.</p>
-<p>Tip: To make pretty printing the default in the Python interpreter, use:</p>
-<div class="highlight-python"><pre>$ python
-Python 2.5.2 (r252:60911, Jun 25 2008, 17:58:32)
-[GCC 4.3.1] on linux2
-Type &quot;help&quot;, &quot;copyright&quot;, &quot;credits&quot; or &quot;license&quot; for more information.
-&gt;&gt;&gt; from sympy import init_printing, var, Integral
-&gt;&gt;&gt; init_printing(use_unicode=False, wrap_line=False, no_global=True)
-&gt;&gt;&gt; var(&quot;x&quot;)
-x
-&gt;&gt;&gt; x**3/3
- 3
-x
---
-3
-&gt;&gt;&gt; Integral(x**2, x) #doctest: +NORMALIZE_WHITESPACE
-  /
- |
- |  2
- | x  dx
- |
-/</pre>
-</div>
-<p><strong>Wyświetlanie w Pythonie</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.python</span> <span class="kn">import</span> <span class="n">python</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = x**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = 1/x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = Integral(x**2, x)</span>
-</pre></div>
-</div>
-<p><strong>Wyświetlanie w LaTeX-u</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">latex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x^{2}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;inline&#39;</span><span class="p">)</span>
-<span class="go">$x^{2}$</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;equation&#39;</span><span class="p">)</span>
-<span class="go">\begin{equation}x^{2}\end{equation}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;equation*&#39;</span><span class="p">)</span>
-<span class="go">\begin{equation*}x^{2}\end{equation*}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">\frac{1}{x}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">\int x^{2}\, dx</span>
-</pre></div>
-</div>
-<p><strong>MathML</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.mathml</span> <span class="kn">import</span> <span class="n">mathml</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">latex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">mathml</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">&lt;apply&gt;&lt;power/&gt;&lt;ci&gt;x&lt;/ci&gt;&lt;cn&gt;2&lt;/cn&gt;&lt;/apply&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">mathml</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">))</span>
-<span class="go">&lt;apply&gt;&lt;power/&gt;&lt;ci&gt;x&lt;/ci&gt;&lt;cn&gt;-1&lt;/cn&gt;&lt;/apply&gt;</span>
-</pre></div>
-</div>
-<p><strong>Pyglet</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">preview</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">preview</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span> 
-</pre></div>
-</div>
-<p>If pyglet is installed, a pyglet window will open containing the LaTeX
-rendered expression:</p>
-<img alt="../_images/pngview1.png" src="../_images/pngview1.png" />
-<div class="section" id="notes">
-<h3>Uwagi<a class="headerlink" href="#notes" title="Stały odnośnik do tego nagłówka">¶</a></h3>
-<p><tt class="docutils literal"><span class="pre">isympy</span></tt> wywołuje <tt class="docutils literal"><span class="pre">pprint</span></tt> automatycznie, dlatego domyślnie jest tam używane ładne wyświetlanie.</p>
-<p>Warto zauważyć, że jest dostępny także moduł wyświetlania, <tt class="docutils literal"><span class="pre">sympy.printing</span></tt>. Dzięki niemu można skorzystać z innych sposobów wyświetlania:</p>
-<ul class="simple">
-<li><tt class="docutils literal"><span class="pre">pretty(expr)</span></tt>, <tt class="docutils literal"><span class="pre">pretty_print(expr)</span></tt>, <tt class="docutils literal"><span class="pre">pprint(expr)</span></tt>: odpowiednio zwraca lub wyświetla ładną reprezentację wyrażenia <tt class="docutils literal"><span class="pre">expr</span></tt>. Jest to to samo, co drugi sposób wyświetlania opisany powyżej.</li>
-<li><tt class="docutils literal"><span class="pre">latex(expr)</span></tt>, <tt class="docutils literal"><span class="pre">print_latex(expr)</span></tt>: odpowiednio zwraca lub wyświetla reprezentację wyrażenia <tt class="docutils literal"><span class="pre">expr</span></tt> w <a class="reference external" href="http://www.latex-project.org/">LaTeX</a>-u.</li>
-<li><tt class="docutils literal"><span class="pre">mathml(expr)</span></tt>, <tt class="docutils literal"><span class="pre">print_mathml(expr)</span></tt>: odpowiednio zwraca lub wyświetla reprezentację wyrażenia <tt class="docutils literal"><span class="pre">expr</span></tt> w <a class="reference external" href="http://www.w3.org/Math/">MathML</a>-u.</li>
-<li><tt class="docutils literal"><span class="pre">print_gtk(expr)</span></tt>: Print <tt class="docutils literal"><span class="pre">expr</span></tt> to <a class="reference external" href="http://helm.cs.unibo.it/mml-widget/">Gtkmathview</a>, a GTK widget that displays MathML code. The <a class="reference external" href="http://helm.cs.unibo.it/mml-widget/">Gtkmathview</a> program is required.</li>
-</ul>
-</div>
-</div>
-<div class="section" id="further-documentation">
-<h2>Dalsza dokumentacja<a class="headerlink" href="#further-documentation" title="Stały odnośnik do tego nagłówka">¶</a></h2>
-<p>Teraz nadszedł czas, by dowiedzieć się więcej na temat SymPy. Przejdź do <a class="reference external" href="../guide.html">Przewodniku użytkownika</a> oraz <a class="reference external" href="../modules/index.html">Opisu modułów SymPy</a>.</p>
-<p>Be sure to also browse our public <a class="reference external" href="http://wiki.sympy.org/">wiki.sympy.org</a>,
-that contains a lot of useful examples, tutorials, cookbooks that we and our
-users contributed, and feel free to edit it.</p>
-<div class="section" id="translations">
-<h3>Translations<a class="headerlink" href="#translations" title="Stały odnośnik do tego nagłówka">¶</a></h3>
-<p>This tutorial is also available in other languages:</p>
-</div>
-<ul class="simple">
-<li><a class="reference external" href="tutorial.bg.html">Български</a></li>
-<li><a class="reference external" href="tutorial.cs.html">Česky</a></li>
-<li><a class="reference external" href="tutorial.de.html">Deutsch</a></li>
-<li><a class="reference external" href="tutorial.en.html">English</a></li>
-<li><a class="reference external" href="tutorial.fr.html">Français</a></li>
-<li><a class="reference external" href="tutorial.pl.html">Polski</a></li>
-<li><a class="reference external" href="tutorial.ru.html">Русский</a></li>
-<li><a class="reference external" href="tutorial.sr.html">Српски</a></li>
-<li><a class="reference external" href="tutorial.zh.html">中文</a></li>
-</ul>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="tutorial.html">
-              <img class="logo" src="../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="tutorial.html">Spis treści</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Tutorial</a><ul>
-<li><a class="reference internal" href="#introduction">Wprowadzenie</a></li>
-<li><a class="reference internal" href="#first-steps-with-sympy">Pierwsze kroki z SymPy</a><ul>
-<li><a class="reference internal" href="#isympy-console">isympy</a></li>
-<li><a class="reference internal" href="#using-sympy-as-a-calculator">SymPy jako kalkulator</a></li>
-<li><a class="reference internal" href="#symbols">Symbole</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#algebra">Algebra</a></li>
-<li><a class="reference internal" href="#calculus">Rachunek różniczkowy i całkowy</a><ul>
-<li><a class="reference internal" href="#limits">Granice</a></li>
-<li><a class="reference internal" href="#differentiation">Różniczkowanie</a></li>
-<li><a class="reference internal" href="#series-expansion">Rozwinięcie w szereg</a></li>
-<li><a class="reference internal" href="#summation">Summation</a></li>
-<li><a class="reference internal" href="#integration">Całkowanie</a></li>
-<li><a class="reference internal" href="#complex-numbers">Liczby zespolone</a></li>
-<li><a class="reference internal" href="#functions">Funkcje</a></li>
-<li><a class="reference internal" href="#differential-equations">Równania różniczkowe</a></li>
-<li><a class="reference internal" href="#algebraic-equations">Równania algebraiczne</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#linear-algebra">Algebra liniowa</a><ul>
-<li><a class="reference internal" href="#matrices">Macierze</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#pattern-matching">Wyszukiwanie wzorca</a></li>
-<li><a class="reference internal" href="#printing">Wyświetlanie</a><ul>
-<li><a class="reference internal" href="#notes">Uwagi</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#further-documentation">Dalsza dokumentacja</a><ul>
-<li><a class="reference internal" href="#translations">Translations</a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-
-  <h3>Ta strona</h3>
-  <ul class="this-page-menu">
-    <li><a href="../_sources/tutorial/tutorial.txt"
-           rel="nofollow">Pokaż źródło</a></li>
-  </ul>
-<div id="searchbox" style="display: none">
-  <h3>Szybkie wyszukiwanie</h3>
-    <form class="search" action="../search.html" method="get">
-      <input type="text" name="q" />
-      <input type="submit" value="Szukaj" />
-      <input type="hidden" name="check_keywords" value="yes" />
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-      <ul>
-        <li class="right" style="margin-right: 10px">
-          <a href="../genindex.html" title="Indeks ogólny"
-             >indeks</a></li>
-        <li><a href="tutorial.html">SymPy 0.7.2-git documentation</a> &raquo;</li> 
-      </ul>
-    </div>
-    <div class="footer">
-        &copy; Copyright 2013 SymPy Development Team.
-      Ostatnia modyfikacja Jan 20, 2013.
-      Utworzone przy pomocy <a href="http://sphinx.pocoo.org/">Sphinx</a>'a 1.1.3.
-    </div>
-  </body>
-</html>
diff --git a/dev-py3k/tutorial/tutorial.ru.html b/dev-py3k/tutorial/tutorial.ru.html
deleted file mode 100644
index e2009e2444d..00000000000
--- a/dev-py3k/tutorial/tutorial.ru.html
+++ /dev/null
@@ -1,975 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
-
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-  <head>
-    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
-    
-    <title>Краткое руководство &mdash; SymPy 0.7.2-git documentation</title>
-    
-    <link rel="stylesheet" href="../_static/default.css" type="text/css" />
-    <link rel="stylesheet" href="../_static/pygments.css" type="text/css" />
-    <link rel="stylesheet" href="http://live.sympy.org/static/live-core.css" type="text/css" />
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-    
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-        VERSION:     '0.7.2-git',
-        COLLAPSE_INDEX: false,
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-        HAS_SOURCE:  true
-      };
-    </script>
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-      <ul>
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-          <a href="../genindex.html" title="Словарь-указатель"
-             accesskey="I">словарь</a></li>
-        <li><a href="tutorial.html">SymPy 0.7.2-git documentation</a> &raquo;</li> 
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-            
-  <div class="section" id="tutorial">
-<span id="id1"></span><h1>Краткое руководство<a class="headerlink" href="#tutorial" title="Ссылка на этот заголовок">¶</a></h1>
-<div class="section" id="introduction">
-<h2>Введение<a class="headerlink" href="#introduction" title="Ссылка на этот заголовок">¶</a></h2>
-<p>SymPy представляет собой открытую библиотеку символьных вычислений на языке Python. Цель SymPy - стать полнофункциональной системой компьютерной алгебры (CAS), при этом сохраняя код максимально понятным и легко расширяемым. SymPy полностью написан на Python и не требует сторонних библиотек.</p>
-<p>Данное руководство представляет из себя введение в SymPy. В нем вы узнаете об основных возможностях SymPy и каким образом использовать эту программу. Если же вы желаете прочитать более подробное руководство, то обратитесь к <a class="reference external" href="../guide.html">Руководству пользователя SymPy</a>, <a class="reference external" href="../modules/index.html">Описанию модулей SymPy</a>, Можно также обратиться и к <a class="reference external" href="https://github.com/sympy/sympy/">исходному коду</a> библиотеки.</p>
-</div>
-<div class="section" id="first-steps-with-sympy">
-<h2>Первые шаги с SymPy<a class="headerlink" href="#first-steps-with-sympy" title="Ссылка на этот заголовок">¶</a></h2>
-<p>Скачать SymPy проще всего с <a class="reference external" href="http://code.google.com/p/sympy/">http://code.google.com/p/sympy/</a>. В разделе Featured, Downloads нужно найти и загрузить последнюю версию дистрибутива:</p>
-<img alt="../_images/featured-downloads.png" src="../_images/featured-downloads.png" />
-<p>Для Windows-систем, нужно скачать и запустить установочный .exe файл. В POSIX-совместимых системах нужно скачать .tar.gz файл и распаковать его:</p>
-<pre class="literal-block">
-$ <strong class="input">tar xzf sympy-0.5.12.tar.gz</strong>
-</pre>
-<p>и запустить из интерпретатора Python:</p>
-<pre class="literal-block">
-$ <strong class="input">cd sympy-0.5.12</strong>
-$ <strong class="input">python</strong>
-Python 2.4.4 (#2, Jan  3 2008, 13:36:28)
-[GCC 4.2.3 20071123 (prerelease) (Debian 4.2.2-4)] on linux2
-Type &quot;help&quot;, &quot;copyright&quot;, &quot;credits&quot; or &quot;license&quot; for more information.
-&gt;&gt;&gt; from sympy import Symbol, cos
-&gt;&gt;&gt; x = Symbol(&quot;x&quot;)
-&gt;&gt;&gt; (1/cos(x)).series(x, 0, 10)
-1 + x**2/2 + 5*x**4/24 + 61*x**6/720 + 277*x**8/8064 + O(x**10)
-</pre>
-<p>Если вы собираетесь использовать SymPy в вашей программе как модуль, рекомендуется обращаться к нему таким же способом, как показано выше. Установить модуль в систему можно или из скаченных исходников, используя команду ./setup.py install. Или, если вы работаете в Linux, можно установить пакет <tt class="docutils literal"><span class="pre">python-sympy</span></tt> с помощью системы установки программ:</p>
-<div class="topic">
-<p class="topic-title first">Installing SymPy in Debian</p>
-<pre class="literal-block">
-$ <strong class="input">sudo apt-get install python-sympy</strong>
-Reading package lists... Done
-Building dependency tree
-Reading state information... Done
-The following NEW packages will be installed:
-  python-sympy
-0 upgraded, 1 newly installed, 0 to remove and 18 not upgraded.
-Need to get 991kB of archives.
-After this operation, 5976kB of additional disk space will be used.
-Get:1 http://ftp.cz.debian.org unstable/main python-sympy 0.5.12-1 [991kB]
-Fetched 991kB in 2s (361kB/s)
-Selecting previously deselected package python-sympy.
-(Reading database ... 232619 files and directories currently installed.)
-Unpacking python-sympy (from .../python-sympy_0.5.12-1_all.deb) ...
-Setting up python-sympy (0.5.12-1) ...
-</pre>
-</div>
-<p>For other means how to install SymPy, consult the wiki page
-<a class="reference external" href="https://github.com/sympy/sympy/wiki/Download-Installation">Download and Installation</a>.</p>
-<div class="section" id="isympy-console">
-<h3>Консоль isympy<a class="headerlink" href="#isympy-console" title="Ссылка на этот заголовок">¶</a></h3>
-<p>SymPy можно использовать не только как модуль, но и как отдельную программу <tt class="docutils literal"><span class="pre">isympy</span></tt>, которая расположена в папке <tt class="docutils literal"><span class="pre">bin</span></tt> относительно директории с исходным кодом. Программа удобна для экспериментов с новыми функциями или для обучения SymPy. Она использует стандартный терминал IPython, но с уже включенными в нее важными модулями SymPy и определенными переменными x, y, z:</p>
-<pre class="literal-block">
-$ <strong class="input">cd sympy</strong>
-$ <strong class="input">./bin/isympy</strong>
-IPython console for SymPy 0.7.2-git (Python 2.7.1) (ground types: gmpy)
-
-These commands were executed:
-&gt;&gt;&gt;
-&gt;&gt;&gt; from sympy import *
-&gt;&gt;&gt; x, y, z, t = symbols('x y z t')
-&gt;&gt;&gt; k, m, n = symbols('k m n', integer=True)
-&gt;&gt;&gt; f, g, h = symbols('f g h', cls=Function)
-
-Documentation can be found at <a class="reference external" href="http://www.sympy.org">http://www.sympy.org</a>
-
-In [1]: <strong class="input">(1/cos(x)).series(x, 0, 10)</strong>
-Out[1]:
-     2      4       6        8
-    x    5*x    61*x    277*x     / 10\
-1 + -- + ---- + ----- + ------ + O\x  /
-    2     24     720     8064
-</pre>
-<div class="admonition note">
-<p class="first admonition-title">Примечание</p>
-<p class="last">Команды введенные вами, обозначены жирным шрифтом. То, что мы делали тремя строчками в стандартном терминале Python, мы можем сделать одной строчкой в isympy. Кроме того программа поддерживает различные способы отображения результатов, в том числе графические.</p>
-</div>
-</div>
-<div class="section" id="using-sympy-as-a-calculator">
-<h3>Использование SymPy в качестве калькулятора<a class="headerlink" href="#using-sympy-as-a-calculator" title="Ссылка на этот заголовок">¶</a></h3>
-<p>SymPy поддерживает три типа численных данных: Float, Rational и Integer.</p>
-<p>Rational представляет собой обыкновенную дробь, которая задается с помощью двух целых чисел: числителя и знаменателя. Например, Rational(1, 2) представляет дробь 1/2, Rational(5, 2) представляет дробь 5/2, и так далее.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
-<span class="go">1/2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">*</span><span class="mi">2</span>
-<span class="go">1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">50</span><span class="o">/</span><span class="n">Rational</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span><span class="o">**</span><span class="mi">50</span>
-<span class="go">1/88817841970012523233890533447265625</span>
-</pre></div>
-</div>
-<p>Важная особенность Python-интерпретатора, о которой нужно сказать отдельно, связана с делением целых чисел При делении двух питоновских чисел типа int с помощью оператора &#8220;/&#8221; в старых версиях Python в результате получается число питоновского типа int. В Python 3 этот стандарт изменен на &#8220;true division&#8221;, и в результате получается питоновский тип float.И этот же стандарт &#8220;true division&#8221; по умолчанию включен и в isympy:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span> 
-<span class="go">0.5</span>
-</pre></div>
-</div>
-<p>В более ранних версиях Python этого не получится, и результатом будет целочисленное деление:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span> 
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>Обратите внимание, что и в том и в другом случае вы имеете дело не с объектом Number из библиотеки SymPy, который представляет число в SymPy, а с питоновскими числами, которые создаются самим интерпретатором Python. Скорее всего, вам нужно будете работать с дробными числами из библиотеки SymPy, поэтому для того чтобы получать результат в виде объектов SymPy убедитесь, что вы используете класс Rational. Кому-то может показаться удобным обозначать Rational как <tt class="docutils literal"><span class="pre">R</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="c"># R(1) is a SymPy Integer and Integer/int gives a Rational</span>
-<span class="go">1/2</span>
-</pre></div>
-</div>
-<p>В модуле Sympy имеются особые константы, такие как e и pi, которые ведут себя как переменные (то есть выражение 1 + pi не преобразуется сразу в число, а так и останется 1 + pi):</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">E</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">pi**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">3.14159265358979</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">pi</span> <span class="o">+</span> <span class="n">E</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">5.85987448204884</span>
-</pre></div>
-</div>
-<p>как вы видите, функция evalf переводит исходное выражение в число с плавающей точкой. Вычисления можно проводить с большей точностью. Для этого нужно передать в качестве аргумента этой функции требуемое число десятичных знаков.</p>
-<p>Для работы с математической бесконечностью используется символ <tt class="docutils literal"><span class="pre">oo</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">oo</span> <span class="o">&gt;</span> <span class="mi">99999</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">oo</span> <span class="o">+</span> <span class="mi">1</span>
-<span class="go">oo</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="symbols">
-<h3>Переменные<a class="headerlink" href="#symbols" title="Ссылка на этот заголовок">¶</a></h3>
-<p>В отличие от многих других систем компьютерной алгебры, вам нужно явно декларировать символьные переменные:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>В левой части этого выражения находится переменная Python, которая питоновским присваиванием соотносится с объектом класса Symbol из SymPy.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">theta</span>
-</pre></div>
-</div>
-<p>Символьные переменные могут также задаваться и с помощью функций <tt class="docutils literal"><span class="pre">symbols</span></tt> или <tt class="docutils literal"><span class="pre">var</span></tt>. Они допускают указание диапазона. Их отличие состоит в том, что <tt class="docutils literal"><span class="pre">var</span></tt> добавляет созданные переменные в текущее пространство имен:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">var</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a,b,c&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;d:f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;g:h&#39;</span><span class="p">)</span>
-<span class="go">(g, h)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;g:2&#39;</span><span class="p">)</span>
-<span class="go">(g0, g1)</span>
-</pre></div>
-</div>
-<p>Экземпляры класса Symbol взаимодействуют друг с другом. Таким образом, с помощью них конструируются алгебраические выражения:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">x</span> <span class="o">-</span> <span class="n">y</span>
-<span class="go">2*x</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">(x + y)**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go">x**2 + 2*x*y + y**2</span>
-</pre></div>
-</div>
-<p>Переменные могут быть заменены на другие переменные, числа или выражения с помощью функции подстановки <tt class="docutils literal"><span class="pre">subs(old,</span> <span class="pre">new)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">(y + 1)**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">4*y**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>Теперь, с этого момента, для всех написанных ниже примеров мы будем предполагать, что запустили следующую команду по настройке системы отображения результатов:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">init_printing</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">init_printing</span><span class="p">(</span><span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">wrap_line</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">no_global</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Она придаст более качественное отображение выражений. Подробнее по системе отображения и печати написано в разделе <a class="reference internal" href="#printing-tutorial"><em>Печать</em></a>. Если же у вас установлен шрифт с юникодом, вы можете использовать опцию use_unicode=True для еще более красивого вывода.</p>
-</div>
-</div>
-<div class="section" id="algebra">
-<h2>Алгебра<a class="headerlink" href="#algebra" title="Ссылка на этот заголовок">¶</a></h2>
-<p>Чтобы разложить выражение на простейшие дроби используется функция <tt class="docutils literal"><span class="pre">apart(expr,</span> <span class="pre">x)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">apart</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">)</span>
-<span class="go">       1</span>
-<span class="go">---------------</span>
-<span class="go">(x + 1)*(x + 2)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">apart</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">    1       1</span>
-<span class="go">- ----- + -----</span>
-<span class="go">  x + 2   x + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">x + 1</span>
-<span class="go">-----</span>
-<span class="go">x - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">apart</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">      2</span>
-<span class="go">1 + -----</span>
-<span class="go">    x - 1</span>
-</pre></div>
-</div>
-<p>Чтобы снова привести дробь к общему знаменателю используется функция <tt class="docutils literal"><span class="pre">together(expr,</span> <span class="pre">x)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">together</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">)</span>
-<span class="go">x*y + x*z + y*z</span>
-<span class="go">---------------</span>
-<span class="go">     x*y*z</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="n">apart</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x + 1</span>
-<span class="go">-----</span>
-<span class="go">x - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="n">apart</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">       1</span>
-<span class="go">---------------</span>
-<span class="go">(x + 1)*(x + 2)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="calculus">
-<span id="index-0"></span><h2>Вычисления<a class="headerlink" href="#calculus" title="Ссылка на этот заголовок">¶</a></h2>
-<div class="section" id="limits">
-<span id="index-1"></span><h3>Пределы<a class="headerlink" href="#limits" title="Ссылка на этот заголовок">¶</a></h3>
-<p>Пределы считаются в SymPy очень легко. Чтобы вычислить предел функции, используйте функцию <tt class="docutils literal"><span class="pre">limit(function,</span> <span class="pre">variable,</span> <span class="pre">point)</span></tt>. Например, чтобы вычислить предел f(x) при x -&gt; 0, вам нужно ввести <tt class="docutils literal"><span class="pre">limit(f,</span> <span class="pre">x,</span> <span class="pre">0)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">limit</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>также вы можете вычислять пределы при x, стремящемся к бесконечности:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">oo</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">0</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>Для более сложных примеров вычисления пределов, вы можете обратится к файлу с тестами <a class="reference external" href="https://github.com/sympy/sympy/blob/master/sympy/series/tests/test_demidovich.py">test_demidovich.py</a></p>
-</div>
-<div class="section" id="differentiation">
-<span id="index-2"></span><h3>Дифференцирование<a class="headerlink" href="#differentiation" title="Ссылка на этот заголовок">¶</a></h3>
-<p>Продифференцировать любое выражение SymPy, можно используя <tt class="docutils literal"><span class="pre">diff(func,</span> <span class="pre">var)</span></tt>. Примеры:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">diff</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">tan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">cos(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">2*cos(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">tan (x) + 1</span>
-</pre></div>
-</div>
-<p>Можно, кстати, через пределы проверить правильность вычислений производной:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">limit</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">delta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">((</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">delta</span><span class="p">)</span> <span class="o">-</span> <span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">/</span><span class="n">delta</span><span class="p">,</span> <span class="n">delta</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">tan (x) + 1</span>
-</pre></div>
-</div>
-<p>Производные более высших порядков можно вычислить, используя дополнительный параметр этой же функции <tt class="docutils literal"><span class="pre">diff(func,</span> <span class="pre">var,</span> <span class="pre">n)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">2*cos(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">-4*sin(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">-8*cos(2*x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="series-expansion">
-<span id="index-3"></span><h3>Разложение в ряд<a class="headerlink" href="#series-expansion" title="Ссылка на этот заголовок">¶</a></h3>
-<p>Для разложения в ряд используйте метод <tt class="docutils literal"><span class="pre">.series(var,</span> <span class="pre">point,</span> <span class="pre">order)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     2    4     6      8</span>
-<span class="go">    x    x     x      x      / 10\</span>
-<span class="go">1 - -- + -- - --- + ----- + O\x  /</span>
-<span class="go">    2    24   720   40320</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     2      4       6        8</span>
-<span class="go">    x    5*x    61*x    277*x     / 10\</span>
-<span class="go">1 + -- + ---- + ----- + ------ + O\x  /</span>
-<span class="go">    2     24     720     8064</span>
-</pre></div>
-</div>
-<p>Еще один простой пример:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">pprint</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;y&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="go">1/y - x/y**2 + x**2/y**3 - x**3/y**4 + x**4/y**5 + O(x**5)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="go">          2    3    4</span>
-<span class="go">1   x    x    x    x     / 5\</span>
-<span class="go">- - -- + -- - -- + -- + O\x /</span>
-<span class="go">y    2    3    4    5</span>
-<span class="go">    y    y    y    y</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="summation">
-<span id="index-4"></span><h3>Summation<a class="headerlink" href="#summation" title="Ссылка на этот заголовок">¶</a></h3>
-<p>Compute the summation of f with respect to the given summation variable over the given limits.</p>
-<p>summation(f, (i, a, b)) computes the sum of f with respect to i from a to b,
-i.e.,</p>
-<div class="highlight-python"><pre>                            b
-                          ____
-                          \   `
-summation(f, (i, a, b)) =  )    f
-                          /___,
-                          i = a</pre>
-</div>
-<p>If it cannot compute the sum, it prints the corresponding summation formula.
-Repeated sums can be computed by introducing additional limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">summation</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">log</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i n m&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-<span class="go"> 2</span>
-<span class="go">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="o">**</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">  oo</span>
-<span class="go"> ___</span>
-<span class="go"> \  `</span>
-<span class="go">  \     -n</span>
-<span class="go">  /   log (n)</span>
-<span class="go"> /__,</span>
-<span class="go">n = 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">),</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">m</span><span class="p">))</span>
-<span class="go">      3    2</span>
-<span class="go">m    m    m</span>
-<span class="go">-- + -- + -</span>
-<span class="go">6    2    3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go"> x</span>
-<span class="go">e</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="integration">
-<span id="index-5"></span><h3>Интегрирование<a class="headerlink" href="#integration" title="Ссылка на этот заголовок">¶</a></h3>
-<p>SymPy поддерживает вычисление определенных и неопределенных интегралов с помощью функции <tt class="docutils literal"><span class="pre">integrate()</span></tt>. Она использует расширенный алгоритм Риша-Нормана и некоторые шаблоны и эвристики. Можно вычислять интегралы трансцендентных, простых и специальных функций:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">integrate</span><span class="p">,</span> <span class="n">erf</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Вы можете интегрировать простейшие функции:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="mi">6</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">5</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go"> 6</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">-cos(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x*log(x) - x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">sinh</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go"> 2</span>
-<span class="go">x  + cosh(x)</span>
-</pre></div>
-</div>
-<p>Примеры интегрирования некоторых специальных функций:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">erf</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">  ____    2</span>
-<span class="go">\/ pi *erf (x)</span>
-<span class="go">--------------</span>
-<span class="go">      4</span>
-</pre></div>
-</div>
-<p>Возможно также вычислить определенный интеграл:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-<p>Поддерживаются и несобственные интегралы:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">-1</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="complex-numbers">
-<span id="index-6"></span><h3>Комплексные числа<a class="headerlink" href="#complex-numbers" title="Ссылка на этот заголовок">¶</a></h3>
-<p>Помимо мнимой единицы I, которое является мнимым числом, символы тоже могут иметь специальные атрибуты (real, positive, complex и т.д), которые определяют поведение этих символов при вычислении символьных выражений:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span> <span class="c"># a plain x with no attributes</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go"> I*x</span>
-<span class="go">e</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">   -im(x)               -im(x)</span>
-<span class="go">I*e      *sin(re(x)) + e      *cos(re(x))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">I*sin(x) + cos(x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="functions">
-<h3>Функции<a class="headerlink" href="#functions" title="Ссылка на этот заголовок">¶</a></h3>
-<p><strong>тригонометрические</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">asin</span><span class="p">,</span> <span class="n">asinh</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">sin(x)*cos(y) + sin(y)*cos(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">-sin(x)*sin(y) + cos(x)*cos(y)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*sinh(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinh</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*sin(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
-<span class="go">I*pi</span>
-<span class="go">----</span>
-<span class="go"> 2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*asin(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3     5     7       9</span>
-<span class="go">    x     x     x       x       / 10\</span>
-<span class="go">x - -- + --- - ---- + ------ + O\x  /</span>
-<span class="go">    6    120   5040   362880</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3     5     7       9</span>
-<span class="go">    x     x     x       x       / 10\</span>
-<span class="go">x + -- + --- + ---- + ------ + O\x  /</span>
-<span class="go">    6    120   5040   362880</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3      5      7       9</span>
-<span class="go">    x    3*x    5*x    35*x     / 10\</span>
-<span class="go">x + -- + ---- + ---- + ----- + O\x  /</span>
-<span class="go">    6     40    112     1152</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3      5      7       9</span>
-<span class="go">    x    3*x    5*x    35*x     / 10\</span>
-<span class="go">x - -- + ---- - ---- + ----- + O\x  /</span>
-<span class="go">    6     40    112     1152</span>
-</pre></div>
-</div>
-<p><strong>сферические</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Ylm</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">  ___</span>
-<span class="go">\/ 3 *cos(theta)</span>
-<span class="go">----------------</span>
-<span class="go">        ____</span>
-<span class="go">    2*\/ pi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">   ___  I*phi</span>
-<span class="go">-\/ 6 *e     *sin(theta)</span>
-<span class="go">------------------------</span>
-<span class="go">            ____</span>
-<span class="go">        4*\/ pi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">   ____  I*phi</span>
-<span class="go">-\/ 30 *e     *sin(theta)*cos(theta)</span>
-<span class="go">------------------------------------</span>
-<span class="go">                  ____</span>
-<span class="go">              4*\/ pi</span>
-</pre></div>
-</div>
-<p><strong>факториалы и гамма-функции</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">x!</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">n!</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="c"># i.e. factorial(x)</span>
-<span class="go">                      /          2     2\</span>
-<span class="go">                    2 |EulerGamma    pi |    / 3\</span>
-<span class="go">1 - EulerGamma*x + x *|----------- + ---| + O\x /</span>
-<span class="go">                      \     2         12/</span>
-</pre></div>
-</div>
-<p><strong>дзета-функции</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">zeta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">zeta(4, x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">  4</span>
-<span class="go">pi</span>
-<span class="go">---</span>
-<span class="go"> 90</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">       4</span>
-<span class="go">     pi</span>
-<span class="go">-1 + ---</span>
-<span class="go">      90</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">         4</span>
-<span class="go">  17   pi</span>
-<span class="go">- -- + ---</span>
-<span class="go">  16    90</span>
-</pre></div>
-</div>
-<p><strong>многочлены</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">assoc_legendre</span><span class="p">,</span> <span class="n">chebyshevt</span><span class="p">,</span> <span class="n">legendre</span><span class="p">,</span> <span class="n">hermite</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">2*x  - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   4      2</span>
-<span class="go">8*x  - 8*x  + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">3*x    1</span>
-<span class="go">---- - -</span>
-<span class="go"> 2     2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">      8         6         4        2</span>
-<span class="go">6435*x    3003*x    3465*x    315*x     35</span>
-<span class="go">------- - ------- + ------- - ------ + ---</span>
-<span class="go">  128        32        64       32     128</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">        __________</span>
-<span class="go">       /    2</span>
-<span class="go">-3*x*\/  - x  + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">     2</span>
-<span class="go">- 3*x  + 3</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hermite</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   3</span>
-<span class="go">8*x  - 12*x</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="differential-equations">
-<span id="index-7"></span><h3>Дифференциальные уравнения<a class="headerlink" href="#differential-equations" title="Ссылка на этот заголовок">¶</a></h3>
-<p>В <tt class="docutils literal"><span class="pre">isympy</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">dsolve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">        2</span>
-<span class="go">       d</span>
-<span class="go">f(x) + ---(f(x))</span>
-<span class="go">         2</span>
-<span class="go">       dx</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dsolve</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">f(x) = C1*sin(x) + C2*cos(x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="algebraic-equations">
-<span id="index-8"></span><h3>Алгебраические уравнения<a class="headerlink" href="#algebraic-equations" title="Ссылка на этот заголовок">¶</a></h3>
-<p>В <tt class="docutils literal"><span class="pre">isympy</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">solve</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">[-1, 1, -I, I]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">([</span><span class="n">x</span> <span class="o">+</span> <span class="mi">5</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">6</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">15</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">])</span>
-<span class="go">{x: -3, y: 1}</span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="section" id="linear-algebra">
-<span id="index-9"></span><h2>Линейная алгебра<a class="headerlink" href="#linear-algebra" title="Ссылка на этот заголовок">¶</a></h2>
-<div class="section" id="matrices">
-<span id="index-10"></span><h3>Матрицы<a class="headerlink" href="#matrices" title="Ссылка на этот заголовок">¶</a></h3>
-<p>Матрицы задаются с помощью конструктора Matrix:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="go">[1  0]</span>
-<span class="go">[    ]</span>
-<span class="go">[0  1]</span>
-</pre></div>
-</div>
-<p>В матрицах вы также можете использовать символьные переменные:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">],</span> <span class="p">[</span><span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span>
-<span class="go">[1  x]</span>
-<span class="go">[    ]</span>
-<span class="go">[y  1]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">[x*y + 1    2*x  ]</span>
-<span class="go">[                ]</span>
-<span class="go">[  2*y    x*y + 1]</span>
-</pre></div>
-</div>
-<p>Для того, чтобы узнать о матрицах подробнее, прочитайте, пожалуйста, Руководство по Линейной Алгебре.</p>
-</div>
-</div>
-<div class="section" id="pattern-matching">
-<span id="index-11"></span><h2>Сопоставление с образцом<a class="headerlink" href="#pattern-matching" title="Ссылка на этот заголовок">¶</a></h2>
-<p>Чтобы сопоставить выражения с образцами, используйте функцию <tt class="docutils literal"><span class="pre">.match()</span></tt> вместе со вспомогательным классом <tt class="docutils literal"><span class="pre">Wild</span></tt>. Эта функция вернет словарь с необходимыми заменами, например:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Wild</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">{p: 5}</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">q</span><span class="p">)</span>
-<span class="go">{p: 1, q: 2}</span>
-</pre></div>
-</div>
-<p>Если же сопоставление не удалось, функция вернет``None``:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">**</span><span class="n">x</span><span class="p">))</span>
-<span class="go">None</span>
-</pre></div>
-</div>
-<p>Также можно использовать параметр exclude для исключения некоторых значений из результата:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">p</span><span class="p">))</span> <span class="c"># 1 is excluded</span>
-<span class="go">None</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span> <span class="c"># x is excluded</span>
-<span class="go">None</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">p</span><span class="p">))</span> <span class="c"># -1 is not excluded</span>
-<span class="go">{p_: -1}</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="printing">
-<span id="printing-tutorial"></span><h2>Печать<a class="headerlink" href="#printing" title="Ссылка на этот заголовок">¶</a></h2>
-<p>Реализовано несколько способов вывода выражений на экран.</p>
-<p><strong>Стандартный</strong></p>
-<p>Стандартный способ представлен функцией <tt class="docutils literal"><span class="pre">str(expression)</span></tt>, которая работает следующим образом:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1/x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Integral(x**2, x)</span>
-</pre></div>
-</div>
-<p><strong>Красивая печать</strong></p>
-<p>Этот способ печати выражений основан на ascii-графике и реализован через функцию <tt class="docutils literal"><span class="pre">pprint</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go"> 2</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="go">-</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">  /</span>
-<span class="go"> |</span>
-<span class="go"> |  2</span>
-<span class="go"> | x  dx</span>
-<span class="go"> |</span>
-<span class="go">/</span>
-</pre></div>
-</div>
-<p>Если у вас установлен шрифт с юникодом, он будет использовать Pretty-print с юникодом по умолчанию. Эту настройку можно отключить, используя <tt class="docutils literal"><span class="pre">use_unicode</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">⌠</span>
-<span class="go">⎮  2</span>
-<span class="go">⎮ x  dx</span>
-<span class="go">⌡</span>
-</pre></div>
-</div>
-<p>Для изучения подробных примеров работы Pretty-print с юникодом вы можете обратится к статье <a class="reference external" href="https://github.com/sympy/sympy/wiki/Pretty-Printing">Pretty Printing</a> из нашего Вики.</p>
-<p>Совет: Чтобы активировать Pretty-print по умолчанию в интерпретаторе Python, используйте:</p>
-<div class="highlight-python"><pre>$ python
-Python 2.5.2 (r252:60911, Jun 25 2008, 17:58:32)
-[GCC 4.3.1] on linux2
-Type &quot;help&quot;, &quot;copyright&quot;, &quot;credits&quot; or &quot;license&quot; for more information.
-&gt;&gt;&gt; from sympy import init_printing, var, Integral
-&gt;&gt;&gt; init_printing(use_unicode=False, wrap_line=False, no_global=True)
-&gt;&gt;&gt; var(&quot;x&quot;)
-x
-&gt;&gt;&gt; x**3/3
- 3
-x
---
-3
-&gt;&gt;&gt; Integral(x**2, x) #doctest: +NORMALIZE_WHITESPACE
-  /
- |
- |  2
- | x  dx
- |
-/</pre>
-</div>
-<p><strong>Печать объектов Python</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.python</span> <span class="kn">import</span> <span class="n">python</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = x**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = 1/x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = Integral(x**2, x)</span>
-</pre></div>
-</div>
-<p><strong>Печать в формате LaTeX</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">latex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x^{2}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;inline&#39;</span><span class="p">)</span>
-<span class="go">$x^{2}$</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;equation&#39;</span><span class="p">)</span>
-<span class="go">\begin{equation}x^{2}\end{equation}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;equation*&#39;</span><span class="p">)</span>
-<span class="go">\begin{equation*}x^{2}\end{equation*}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">\frac{1}{x}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">\int x^{2}\, dx</span>
-</pre></div>
-</div>
-<p><strong>MathML</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.mathml</span> <span class="kn">import</span> <span class="n">mathml</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">latex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">mathml</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">&lt;apply&gt;&lt;power/&gt;&lt;ci&gt;x&lt;/ci&gt;&lt;cn&gt;2&lt;/cn&gt;&lt;/apply&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">mathml</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">))</span>
-<span class="go">&lt;apply&gt;&lt;power/&gt;&lt;ci&gt;x&lt;/ci&gt;&lt;cn&gt;-1&lt;/cn&gt;&lt;/apply&gt;</span>
-</pre></div>
-</div>
-<p><strong>Pyglet</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">preview</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">preview</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span> 
-</pre></div>
-</div>
-<p>Появится окно pyglet с отрисованным выражением LaTeX:</p>
-<img alt="../_images/pngview1.png" src="../_images/pngview1.png" />
-<div class="section" id="notes">
-<h3>Примечания<a class="headerlink" href="#notes" title="Ссылка на этот заголовок">¶</a></h3>
-<p><tt class="docutils literal"><span class="pre">isympy</span></tt> вызывает <tt class="docutils literal"><span class="pre">pprint</span></tt> автоматически, по этой причине Pretty-print будет включен в <tt class="docutils literal"><span class="pre">isympy</span></tt> по умолчанию.</p>
-<p>Также доступен модуль печати - <tt class="docutils literal"><span class="pre">sympy.printing</span></tt>. Через этот модуль доступны следующий функции печати:</p>
-<ul class="simple">
-<li><tt class="docutils literal"><span class="pre">pretty(expr)</span></tt>, <tt class="docutils literal"><span class="pre">pretty_print(expr)</span></tt>, <tt class="docutils literal"><span class="pre">pprint(expr)</span></tt>: Возвращает или выводит на экран, соответственно, &#8220;Красивое&#8221; представление выражения <tt class="docutils literal"><span class="pre">expr</span></tt>.</li>
-<li><tt class="docutils literal"><span class="pre">latex(expr)</span></tt>, <tt class="docutils literal"><span class="pre">print_latex(expr)</span></tt>: Возвращает или выводит на экран, соответственно, <a class="reference external" href="http://www.latex-project.org/">LaTeX</a> -представление <tt class="docutils literal"><span class="pre">expr</span></tt></li>
-<li><tt class="docutils literal"><span class="pre">mathml(expr)</span></tt>, <tt class="docutils literal"><span class="pre">print_mathml(expr)</span></tt>: Возвращает или выводит на экран, соответственно, <a class="reference external" href="http://www.w3.org/Math/">MathML</a> -представление <tt class="docutils literal"><span class="pre">expr</span></tt>.</li>
-<li><tt class="docutils literal"><span class="pre">print_gtk(expr)</span></tt>: Print <tt class="docutils literal"><span class="pre">expr</span></tt> to <a class="reference external" href="http://helm.cs.unibo.it/mml-widget/">Gtkmathview</a>, a GTK widget that displays MathML code. The <a class="reference external" href="http://helm.cs.unibo.it/mml-widget/">Gtkmathview</a> program is required.</li>
-</ul>
-</div>
-</div>
-<div class="section" id="further-documentation">
-<h2>Другие справочники<a class="headerlink" href="#further-documentation" title="Ссылка на этот заголовок">¶</a></h2>
-<p>Чтобы узнать о SymPy подробнее, обратитесь <a class="reference external" href="../guide.html">Руководство пользователя SymPy</a> и <a class="reference external" href="../modules/index.html">Описание модулей SymPy</a>.</p>
-<p>Также можно обратится на <a class="reference external" href="http://wiki.sympy.org/">wiki.sympy.org</a> - сайт, который содержит множество полезных примеров, руководств и советов. Они созданны нами и нашим сообществом. Мы будем рады, если и вы внесете в него свой весомый вклад.</p>
-<div class="section" id="translations">
-<h3>Переводы<a class="headerlink" href="#translations" title="Ссылка на этот заголовок">¶</a></h3>
-<p>Этот текст доступен на других языках:</p>
-</div>
-<ul class="simple">
-<li><a class="reference external" href="tutorial.bg.html">Български</a></li>
-<li><a class="reference external" href="tutorial.cs.html">Česky</a></li>
-<li><a class="reference external" href="tutorial.de.html">Deutsch</a></li>
-<li><a class="reference external" href="tutorial.en.html">English</a></li>
-<li><a class="reference external" href="tutorial.fr.html">Français</a></li>
-<li><a class="reference external" href="tutorial.pl.html">Polski</a></li>
-<li><a class="reference external" href="tutorial.ru.html">Русский</a></li>
-<li><a class="reference external" href="tutorial.sr.html">Српски</a></li>
-<li><a class="reference external" href="tutorial.zh.html">中文</a></li>
-</ul>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="tutorial.html">
-              <img class="logo" src="../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="tutorial.html">Содержание</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Краткое руководство</a><ul>
-<li><a class="reference internal" href="#introduction">Введение</a></li>
-<li><a class="reference internal" href="#first-steps-with-sympy">Первые шаги с SymPy</a><ul>
-<li><a class="reference internal" href="#isympy-console">Консоль isympy</a></li>
-<li><a class="reference internal" href="#using-sympy-as-a-calculator">Использование SymPy в качестве калькулятора</a></li>
-<li><a class="reference internal" href="#symbols">Переменные</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#algebra">Алгебра</a></li>
-<li><a class="reference internal" href="#calculus">Вычисления</a><ul>
-<li><a class="reference internal" href="#limits">Пределы</a></li>
-<li><a class="reference internal" href="#differentiation">Дифференцирование</a></li>
-<li><a class="reference internal" href="#series-expansion">Разложение в ряд</a></li>
-<li><a class="reference internal" href="#summation">Summation</a></li>
-<li><a class="reference internal" href="#integration">Интегрирование</a></li>
-<li><a class="reference internal" href="#complex-numbers">Комплексные числа</a></li>
-<li><a class="reference internal" href="#functions">Функции</a></li>
-<li><a class="reference internal" href="#differential-equations">Дифференциальные уравнения</a></li>
-<li><a class="reference internal" href="#algebraic-equations">Алгебраические уравнения</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#linear-algebra">Линейная алгебра</a><ul>
-<li><a class="reference internal" href="#matrices">Матрицы</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#pattern-matching">Сопоставление с образцом</a></li>
-<li><a class="reference internal" href="#printing">Печать</a><ul>
-<li><a class="reference internal" href="#notes">Примечания</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#further-documentation">Другие справочники</a><ul>
-<li><a class="reference internal" href="#translations">Переводы</a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-
-  <h3>На этой странице</h3>
-  <ul class="this-page-menu">
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-           rel="nofollow">Показать исходный текст</a></li>
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-<div id="searchbox" style="display: none">
-  <h3>Быстрый поиск</h3>
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-      <ul>
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-          <a href="../genindex.html" title="Словарь-указатель"
-             >словарь</a></li>
-        <li><a href="tutorial.html">SymPy 0.7.2-git documentation</a> &raquo;</li> 
-      </ul>
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diff --git a/dev-py3k/tutorial/tutorial.sr.html b/dev-py3k/tutorial/tutorial.sr.html
deleted file mode 100644
index 0bbc0c98f36..00000000000
--- a/dev-py3k/tutorial/tutorial.sr.html
+++ /dev/null
@@ -1,991 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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-    
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-    <link rel="stylesheet" href="../_static/pygments.css" type="text/css" />
-    <link rel="stylesheet" href="http://live.sympy.org/static/live-core.css" type="text/css" />
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-    
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-        VERSION:     '0.7.2-git',
-        COLLAPSE_INDEX: false,
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-        HAS_SOURCE:  true
-      };
-    </script>
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-    <script type="text/javascript" src="http://live.sympy.org/static/utilities.js"></script>
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-             accesskey="I">index</a></li>
-        <li><a href="tutorial.html">SymPy 0.7.2-git documentation</a> &raquo;</li> 
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-            
-  <div class="section" id="tutorial">
-<span id="id1"></span><h1>Туторијал<a class="headerlink" href="#tutorial" title="Permalink to this headline">¶</a></h1>
-<div class="section" id="introduction">
-<h2>Увод<a class="headerlink" href="#introduction" title="Permalink to this headline">¶</a></h2>
-<p>SymPy je Python библиотека за симболичку математику. Циља да постане потпуно садржани алгебарски систем (енглески CAS: &#8220;Computer Algebra System&#8221;), чувајући код једноставнијим колико је могуће да би била разумљива и лакше проширива. SymPy је комплетно написан у програмском језику Python и не захтева никакве екстерне библиотеке.</p>
-<p>Овај туторијал даје вам преглед и увод у SymPy. Прочитајте га да бисте имали идеју шта SymPy можете урадити (и како) а ако хоћете да сазнате више, прочитајте <a class="reference external" href="../guide.html">SymPy кориснички водич</a>, <a class="reference external" href="../modules/index.html">SymPy референтни модул</a>, или <a class="reference external" href="https://github.com/sympy/sympy/">изворни код</a> директно.</p>
-</div>
-<div class="section" id="first-steps-with-sympy">
-<h2>Први кораци<a class="headerlink" href="#first-steps-with-sympy" title="Permalink to this headline">¶</a></h2>
-<p>Најлакши начин да се преузме је да одете на <a class="reference external" href="http://code.google.com/p/sympy/">http://code.google.com/p/sympy/</a> и преузмeте последњу архиву из истакнутих преузимања:</p>
-<img alt="../_images/featured-downloads.png" src="../_images/featured-downloads.png" />
-<p>Отпакујте:</p>
-<pre class="literal-block">
-$ <strong class="input">tar xzf sympy-0.5.12.tar.gz</strong>
-</pre>
-<p>и покушајте из Python интерпретера:</p>
-<pre class="literal-block">
-$ <strong class="input">cd sympy-0.5.12</strong>
-$ <strong class="input">python</strong>
-Python 2.4.4 (#2, Jan  3 2008, 13:36:28)
-[GCC 4.2.3 20071123 (prerelease) (Debian 4.2.2-4)] on linux2
-Type &quot;help&quot;, &quot;copyright&quot;, &quot;credits&quot; or &quot;license&quot; for more information.
-&gt;&gt;&gt; from sympy import Symbol, cos
-&gt;&gt;&gt; x = Symbol(&quot;x&quot;)
-&gt;&gt;&gt; (1/cos(x)).series(x, 0, 10)
-1 + x**2/2 + 5*x**4/24 + 61*x**6/720 + 277*x**8/8064 + O(x**10)
-</pre>
-<p>Можете да користите SymPy како је указано горе и то је препоручен начин ако се користи у вашем програму. Такође га можете инсталирати користећи <tt class="docutils literal"><span class="pre">./setup.py</span> <span class="pre">install</span></tt> као и сваки други Python модул, или само инсталирати пакет ваше Линукс дистрибуције:</p>
-<div class="topic">
-<p class="topic-title first">Installing SymPy in Debian</p>
-<pre class="literal-block">
-$ <strong class="input">sudo apt-get install python-sympy</strong>
-Reading package lists... Done
-Building dependency tree
-Reading state information... Done
-The following NEW packages will be installed:
-  python-sympy
-0 upgraded, 1 newly installed, 0 to remove and 18 not upgraded.
-Need to get 991kB of archives.
-After this operation, 5976kB of additional disk space will be used.
-Get:1 http://ftp.cz.debian.org unstable/main python-sympy 0.5.12-1 [991kB]
-Fetched 991kB in 2s (361kB/s)
-Selecting previously deselected package python-sympy.
-(Reading database ... 232619 files and directories currently installed.)
-Unpacking python-sympy (from .../python-sympy_0.5.12-1_all.deb) ...
-Setting up python-sympy (0.5.12-1) ...
-</pre>
-</div>
-<p>For other means how to install SymPy, consult the wiki page
-<a class="reference external" href="https://github.com/sympy/sympy/wiki/Download-Installation">Download and Installation</a>.</p>
-<div class="section" id="isympy-console">
-<h3>isympy конзола<a class="headerlink" href="#isympy-console" title="Permalink to this headline">¶</a></h3>
-<p>For experimenting with new features, or when figuring out how to do things, you
-can use our special wrapper around IPython called <tt class="docutils literal"><span class="pre">isympy</span></tt> (located in
-<tt class="docutils literal"><span class="pre">bin/isympy</span></tt> if you are running from the source directory) which is just a
-standard Python shell that has already imported the relevant SymPy modules and
-defined the symbols x, y, z and some other things:</p>
-<pre class="literal-block">
-$ <strong class="input">cd sympy</strong>
-$ <strong class="input">./bin/isympy</strong>
-IPython console for SymPy 0.7.2-git (Python 2.7.1) (ground types: gmpy)
-
-These commands were executed:
-&gt;&gt;&gt;
-&gt;&gt;&gt; from sympy import *
-&gt;&gt;&gt; x, y, z, t = symbols('x y z t')
-&gt;&gt;&gt; k, m, n = symbols('k m n', integer=True)
-&gt;&gt;&gt; f, g, h = symbols('f g h', cls=Function)
-
-Documentation can be found at <a class="reference external" href="http://www.sympy.org">http://www.sympy.org</a>
-
-In [1]: <strong class="input">(1/cos(x)).series(x, 0, 10)</strong>
-Out[1]:
-     2      4       6        8
-    x    5*x    61*x    277*x     / 10\
-1 + -- + ---- + ----- + ------ + O\x  /
-    2     24     720     8064
-</pre>
-<div class="admonition note">
-<p class="first admonition-title">Note</p>
-<p class="last">Команде су подебљане. То што смо урадили са 3 линије у регуларном Python интерпретеру може бити урађено у једној линији у isympy.</p>
-</div>
-</div>
-<div class="section" id="using-sympy-as-a-calculator">
-<h3>SymPy као дигитрон<a class="headerlink" href="#using-sympy-as-a-calculator" title="Permalink to this headline">¶</a></h3>
-<p>SymPy has three built-in numeric types: Float, Rational and Integer.</p>
-<p>Класа &#8220;Rational&#8221; представља рацоинални број као пар од два цела, именилац и садржилац, тако да рационални број Rational(1, 2) представаља 1/2, Rational(5, 2) представља 5/2 и тако даље.</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
-<span class="go">1/2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">*</span><span class="mi">2</span>
-<span class="go">1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">50</span><span class="o">/</span><span class="n">Rational</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span><span class="o">**</span><span class="mi">50</span>
-<span class="go">1/88817841970012523233890533447265625</span>
-</pre></div>
-</div>
-<p>Наставите са пажњом док радите са целим и променљивим бројевима, посебно у дељењу, јер можете направити Python број, а не SymPy број. Однос два цела броја у Python може направити променљливи &#8211; истинито дељење(true divison) стандардно за Python 3 у подразумевано понашање <tt class="docutils literal"><span class="pre">isympy</span></tt> који увози дељење из __future__:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span> 
-<span class="go">0.5</span>
-</pre></div>
-</div>
-<p>Али у ранијим Python верзијама где дељење није увежено, скраћени цели број ће бити резултат:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span> 
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>У оба случаја, ипак, не радите са SymPy бројевима јер је Python креирао своје посебне бројеве. Већину времена ћете вероватно радити са рационалним бројевима, тако да будите сигурни да користите рационалне бројеве да бисте добили SymPy резултат. Могло би бити згодно изједначити <tt class="docutils literal"><span class="pre">R</span></tt> и Rational:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="c"># R(1) is a SymPy Integer and Integer/int gives a Rational</span>
-<span class="go">1/2</span>
-</pre></div>
-</div>
-<p>Такође имамо неке специјалне константе као e и pi, који се третирају као симболи (1 + pi неће прерачунати у нумерчку форму, већ ће остати као 1 + pi), могу имати произвољну прецизност:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">E</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">pi**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">3.14159265358979</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">pi</span> <span class="o">+</span> <span class="n">E</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">5.85987448204884</span>
-</pre></div>
-</div>
-<p>као што видите, evalf рачуна израз у реалан број</p>
-<p>Симбол <tt class="docutils literal"><span class="pre">oo</span></tt> представља бесконачно:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">oo</span> <span class="o">&gt;</span> <span class="mi">99999</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">oo</span> <span class="o">+</span> <span class="mi">1</span>
-<span class="go">oo</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="symbols">
-<h3>Симболи<a class="headerlink" href="#symbols" title="Permalink to this headline">¶</a></h3>
-<p>За разлику од осталих компјутерских алгебарских система, у SymPy ви морате да декларишете симболичке променљиве експлицитно:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>On the left is the normal Python variable which has been assigned to the
-SymPy Symbol class. Predefined symbols (including those for symbols with
-Greek names) are available for import from abc:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">theta</span>
-</pre></div>
-</div>
-<p>Symbols can also be created with the <tt class="docutils literal"><span class="pre">symbols</span></tt> or <tt class="docutils literal"><span class="pre">var</span></tt> functions, the
-latter automatically adding the created symbols to the namespace, and both
-accepting a range notation:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">var</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a,b,c&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;d:f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;g:h&#39;</span><span class="p">)</span>
-<span class="go">(g, h)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;g:2&#39;</span><span class="p">)</span>
-<span class="go">(g0, g1)</span>
-</pre></div>
-</div>
-<p>Instances of the Symbol class &#8220;play well together&#8221; and are the building blocks
-of expresions:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">x</span> <span class="o">-</span> <span class="n">y</span>
-<span class="go">2*x</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">(x + y)**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go">x**2 + 2*x*y + y**2</span>
-</pre></div>
-</div>
-<p>Они могу бити замењени са другим бројевима, симболима или изразима користећи команду <tt class="docutils literal"><span class="pre">subs(old,</span> <span class="pre">new)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">(y + 1)**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">4*y**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>За остатак овог туторијала, претпостављамо да смо извршили:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">init_printing</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">init_printing</span><span class="p">(</span><span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">wrap_line</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">no_global</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Ово ће лепше штампати резултате. Погледајте <a class="reference internal" href="#printing-tutorial"><em>Штампање</em></a> секцију. Ако имате неки уникод фонт инсталиран, можете користити use_unicode=True за много лепше резултате.</p>
-</div>
-</div>
-<div class="section" id="algebra">
-<h2>Алгебра<a class="headerlink" href="#algebra" title="Permalink to this headline">¶</a></h2>
-<p>За делимични разломак, користите <tt class="docutils literal"><span class="pre">apart(expr,</span> <span class="pre">x)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">apart</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">)</span>
-<span class="go">       1</span>
-<span class="go">---------------</span>
-<span class="go">(x + 1)*(x + 2)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">apart</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">    1       1</span>
-<span class="go">- ----- + -----</span>
-<span class="go">  x + 2   x + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">x + 1</span>
-<span class="go">-----</span>
-<span class="go">x - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">apart</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">      2</span>
-<span class="go">1 + -----</span>
-<span class="go">    x - 1</span>
-</pre></div>
-</div>
-<p>Да бисте комбиновали ствари заједно, користите <tt class="docutils literal"><span class="pre">together(expr,</span> <span class="pre">x)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">together</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">)</span>
-<span class="go">x*y + x*z + y*z</span>
-<span class="go">---------------</span>
-<span class="go">     x*y*z</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="n">apart</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x + 1</span>
-<span class="go">-----</span>
-<span class="go">x - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="n">apart</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">       1</span>
-<span class="go">---------------</span>
-<span class="go">(x + 1)*(x + 2)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="calculus">
-<span id="index-0"></span><h2>Математичка анализа<a class="headerlink" href="#calculus" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="limits">
-<span id="index-1"></span><h3>Лимити<a class="headerlink" href="#limits" title="Permalink to this headline">¶</a></h3>
-<p>Limits are easy to use in SymPy, they follow the syntax <tt class="docutils literal"><span class="pre">limit(function,</span>
-<span class="pre">variable,</span> <span class="pre">point)</span></tt>, so to compute the limit of f(x) as x -&gt; 0, you would issue
-<tt class="docutils literal"><span class="pre">limit(f,</span> <span class="pre">x,</span> <span class="pre">0)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">limit</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>Можете такође израчунати у бесконачности:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">oo</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">0</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>За неке не-тривијалне примере са лимитима, можете прочитати тест датотеку <a class="reference external" href="https://github.com/sympy/sympy/blob/master/sympy/series/tests/test_demidovich.py">test_demidovich.py</a></p>
-</div>
-<div class="section" id="differentiation">
-<span id="index-2"></span><h3>Изводи<a class="headerlink" href="#differentiation" title="Permalink to this headline">¶</a></h3>
-<p>Можете израчунати извод сваког израза користећи <tt class="docutils literal"><span class="pre">diff(func,</span> <span class="pre">var)</span></tt>. Примери:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">diff</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">tan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">cos(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">2*cos(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">tan (x) + 1</span>
-</pre></div>
-</div>
-<p>можете проверити да ли је то тачно са:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">limit</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">delta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">((</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">delta</span><span class="p">)</span> <span class="o">-</span> <span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">/</span><span class="n">delta</span><span class="p">,</span> <span class="n">delta</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">tan (x) + 1</span>
-</pre></div>
-</div>
-<p>Виши изводи се могу израчунати користећи метод <tt class="docutils literal"><span class="pre">diff(func,</span> <span class="pre">var,</span> <span class="pre">n)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">2*cos(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">-4*sin(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">-8*cos(2*x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="series-expansion">
-<span id="index-3"></span><h3>Развој редова<a class="headerlink" href="#series-expansion" title="Permalink to this headline">¶</a></h3>
-<p>Користите <tt class="docutils literal"><span class="pre">.series(var,</span> <span class="pre">point,</span> <span class="pre">order)</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     2    4     6      8</span>
-<span class="go">    x    x     x      x      / 10\</span>
-<span class="go">1 - -- + -- - --- + ----- + O\x  /</span>
-<span class="go">    2    24   720   40320</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     2      4       6        8</span>
-<span class="go">    x    5*x    61*x    277*x     / 10\</span>
-<span class="go">1 + -- + ---- + ----- + ------ + O\x  /</span>
-<span class="go">    2     24     720     8064</span>
-</pre></div>
-</div>
-<p>Још један једноставан пример:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">pprint</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;y&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="go">1/y - x/y**2 + x**2/y**3 - x**3/y**4 + x**4/y**5 + O(x**5)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="go">          2    3    4</span>
-<span class="go">1   x    x    x    x     / 5\</span>
-<span class="go">- - -- + -- - -- + -- + O\x /</span>
-<span class="go">y    2    3    4    5</span>
-<span class="go">    y    y    y    y</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="summation">
-<span id="index-4"></span><h3>Summation<a class="headerlink" href="#summation" title="Permalink to this headline">¶</a></h3>
-<p>Compute the summation of f with respect to the given summation variable over the given limits.</p>
-<p>summation(f, (i, a, b)) computes the sum of f with respect to i from a to b,
-i.e.,</p>
-<div class="highlight-python"><pre>                            b
-                          ____
-                          \   `
-summation(f, (i, a, b)) =  )    f
-                          /___,
-                          i = a</pre>
-</div>
-<p>If it cannot compute the sum, it prints the corresponding summation formula.
-Repeated sums can be computed by introducing additional limits:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">summation</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">log</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i n m&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-<span class="go"> 2</span>
-<span class="go">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="o">**</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">  oo</span>
-<span class="go"> ___</span>
-<span class="go"> \  `</span>
-<span class="go">  \     -n</span>
-<span class="go">  /   log (n)</span>
-<span class="go"> /__,</span>
-<span class="go">n = 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">),</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">m</span><span class="p">))</span>
-<span class="go">      3    2</span>
-<span class="go">m    m    m</span>
-<span class="go">-- + -- + -</span>
-<span class="go">6    2    3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go"> x</span>
-<span class="go">e</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="integration">
-<span id="index-5"></span><h3>Интеграција<a class="headerlink" href="#integration" title="Permalink to this headline">¶</a></h3>
-<p>SymPy има подршку за одређену и неодређену интеграцију трансцендентних, елементарних и специјалних функција преко методе <tt class="docutils literal"><span class="pre">integrate()</span></tt> које користи моћни проширени Risch-Norman алгоритам и неке хеуристике и обрасце поклапања:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">integrate</span><span class="p">,</span> <span class="n">erf</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>Можете интегрирати елементарне функције:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="mi">6</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">5</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go"> 6</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">-cos(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x*log(x) - x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">sinh</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go"> 2</span>
-<span class="go">x  + cosh(x)</span>
-</pre></div>
-</div>
-<p>Можете користити и разне специјалне функције:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">erf</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">  ____    2</span>
-<span class="go">\/ pi *erf (x)</span>
-<span class="go">--------------</span>
-<span class="go">      4</span>
-</pre></div>
-</div>
-<p>It is possible to compute definite integrals:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-<p>Also, improper integrals are supported as well:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">-1</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="complex-numbers">
-<span id="index-6"></span><h3>Комплексни бројеви<a class="headerlink" href="#complex-numbers" title="Permalink to this headline">¶</a></h3>
-<p>Besides the imaginary unit, I, which is imaginary, symbols can be created with
-attributes (e.g. real, positive, complex, etc...) and this will affect how
-they behave:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span> <span class="c"># a plain x with no attributes</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go"> I*x</span>
-<span class="go">e</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">   -im(x)               -im(x)</span>
-<span class="go">I*e      *sin(re(x)) + e      *cos(re(x))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">I*sin(x) + cos(x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="functions">
-<h3>Функције<a class="headerlink" href="#functions" title="Permalink to this headline">¶</a></h3>
-<p><strong>тригонометријске</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">asin</span><span class="p">,</span> <span class="n">asinh</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">sin(x)*cos(y) + sin(y)*cos(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">-sin(x)*sin(y) + cos(x)*cos(y)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*sinh(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinh</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*sin(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
-<span class="go">I*pi</span>
-<span class="go">----</span>
-<span class="go"> 2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*asin(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3     5     7       9</span>
-<span class="go">    x     x     x       x       / 10\</span>
-<span class="go">x - -- + --- - ---- + ------ + O\x  /</span>
-<span class="go">    6    120   5040   362880</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3     5     7       9</span>
-<span class="go">    x     x     x       x       / 10\</span>
-<span class="go">x + -- + --- + ---- + ------ + O\x  /</span>
-<span class="go">    6    120   5040   362880</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3      5      7       9</span>
-<span class="go">    x    3*x    5*x    35*x     / 10\</span>
-<span class="go">x + -- + ---- + ---- + ----- + O\x  /</span>
-<span class="go">    6     40    112     1152</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3      5      7       9</span>
-<span class="go">    x    3*x    5*x    35*x     / 10\</span>
-<span class="go">x - -- + ---- - ---- + ----- + O\x  /</span>
-<span class="go">    6     40    112     1152</span>
-</pre></div>
-</div>
-<p><strong>сферне хармонике</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Ylm</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">  ___</span>
-<span class="go">\/ 3 *cos(theta)</span>
-<span class="go">----------------</span>
-<span class="go">        ____</span>
-<span class="go">    2*\/ pi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">   ___  I*phi</span>
-<span class="go">-\/ 6 *e     *sin(theta)</span>
-<span class="go">------------------------</span>
-<span class="go">            ____</span>
-<span class="go">        4*\/ pi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">   ____  I*phi</span>
-<span class="go">-\/ 30 *e     *sin(theta)*cos(theta)</span>
-<span class="go">------------------------------------</span>
-<span class="go">                  ____</span>
-<span class="go">              4*\/ pi</span>
-</pre></div>
-</div>
-<p><strong>Факторијали и гама функције</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">x!</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">n!</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="c"># i.e. factorial(x)</span>
-<span class="go">                      /          2     2\</span>
-<span class="go">                    2 |EulerGamma    pi |    / 3\</span>
-<span class="go">1 - EulerGamma*x + x *|----------- + ---| + O\x /</span>
-<span class="go">                      \     2         12/</span>
-</pre></div>
-</div>
-<p><strong>Зета функције</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">zeta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">zeta(4, x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">  4</span>
-<span class="go">pi</span>
-<span class="go">---</span>
-<span class="go"> 90</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">       4</span>
-<span class="go">     pi</span>
-<span class="go">-1 + ---</span>
-<span class="go">      90</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">         4</span>
-<span class="go">  17   pi</span>
-<span class="go">- -- + ---</span>
-<span class="go">  16    90</span>
-</pre></div>
-</div>
-<p><strong>Полиноми</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">assoc_legendre</span><span class="p">,</span> <span class="n">chebyshevt</span><span class="p">,</span> <span class="n">legendre</span><span class="p">,</span> <span class="n">hermite</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">2*x  - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   4      2</span>
-<span class="go">8*x  - 8*x  + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">3*x    1</span>
-<span class="go">---- - -</span>
-<span class="go"> 2     2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">      8         6         4        2</span>
-<span class="go">6435*x    3003*x    3465*x    315*x     35</span>
-<span class="go">------- - ------- + ------- - ------ + ---</span>
-<span class="go">  128        32        64       32     128</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">        __________</span>
-<span class="go">       /    2</span>
-<span class="go">-3*x*\/  - x  + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">     2</span>
-<span class="go">- 3*x  + 3</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hermite</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   3</span>
-<span class="go">8*x  - 12*x</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="differential-equations">
-<span id="index-7"></span><h3>Диференцијални рачун<a class="headerlink" href="#differential-equations" title="Permalink to this headline">¶</a></h3>
-<p>Користећи <tt class="docutils literal"><span class="pre">isympy</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">dsolve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">        2</span>
-<span class="go">       d</span>
-<span class="go">f(x) + ---(f(x))</span>
-<span class="go">         2</span>
-<span class="go">       dx</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dsolve</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">f(x) = C1*sin(x) + C2*cos(x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="algebraic-equations">
-<span id="index-8"></span><h3>Алгебарска рачун<a class="headerlink" href="#algebraic-equations" title="Permalink to this headline">¶</a></h3>
-<p>Користећи <tt class="docutils literal"><span class="pre">isympy</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">solve</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">[-1, 1, -I, I]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">([</span><span class="n">x</span> <span class="o">+</span> <span class="mi">5</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">6</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">15</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">])</span>
-<span class="go">{x: -3, y: 1}</span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="section" id="linear-algebra">
-<span id="index-9"></span><h2>Линеарна алгебра<a class="headerlink" href="#linear-algebra" title="Permalink to this headline">¶</a></h2>
-<div class="section" id="matrices">
-<span id="index-10"></span><h3>Матрице<a class="headerlink" href="#matrices" title="Permalink to this headline">¶</a></h3>
-<p>Матрице се праве као инстанце класе Matrix:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="go">[1  0]</span>
-<span class="go">[    ]</span>
-<span class="go">[0  1]</span>
-</pre></div>
-</div>
-<p>They can also contain symbols:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">],</span> <span class="p">[</span><span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span>
-<span class="go">[1  x]</span>
-<span class="go">[    ]</span>
-<span class="go">[y  1]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">[x*y + 1    2*x  ]</span>
-<span class="go">[                ]</span>
-<span class="go">[  2*y    x*y + 1]</span>
-</pre></div>
-</div>
-<p>For more about Matrices, see the Linear Algebra tutorial.</p>
-</div>
-</div>
-<div class="section" id="pattern-matching">
-<span id="index-11"></span><h2>Поклапање образаца<a class="headerlink" href="#pattern-matching" title="Permalink to this headline">¶</a></h2>
-<p>Користите метод <tt class="docutils literal"><span class="pre">.match()</span></tt> са класом <tt class="docutils literal"><span class="pre">Wild</span></tt>, да би извршили образац одговарајуће на изразе. Овај метод ће вратити речник (dictionary) са потребним изменама, као:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Wild</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">{p: 5}</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">q</span><span class="p">)</span>
-<span class="go">{p: 1, q: 2}</span>
-</pre></div>
-</div>
-<p>Ако је претрага неуспешна, метода враћа <tt class="docutils literal"><span class="pre">None</span></tt>:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">**</span><span class="n">x</span><span class="p">))</span>
-<span class="go">None</span>
-</pre></div>
-</div>
-<p>Можете користити параметар exclude класе <tt class="docutils literal"><span class="pre">Wild</span></tt> да би били сигурни да се одређене ствари неће приказати у реѕултату:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">p</span><span class="p">))</span> <span class="c"># 1 is excluded</span>
-<span class="go">None</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span> <span class="c"># x is excluded</span>
-<span class="go">None</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">p</span><span class="p">))</span> <span class="c"># -1 is not excluded</span>
-<span class="go">{p_: -1}</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="printing">
-<span id="printing-tutorial"></span><h2>Штампање<a class="headerlink" href="#printing" title="Permalink to this headline">¶</a></h2>
-<p>There are many ways to print expressions.</p>
-<p><strong>Стандардно</strong></p>
-<p>Ово је резултат методе <tt class="docutils literal"><span class="pre">str(expression)</span></tt> и изгледа као:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1/x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Integral(x**2, x)</span>
-</pre></div>
-</div>
-<p><strong>Лепше штампање</strong></p>
-<p>Nice ascii-art printing is produced by the <tt class="docutils literal"><span class="pre">pprint</span></tt> function:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go"> 2</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="go">-</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">  /</span>
-<span class="go"> |</span>
-<span class="go"> |  2</span>
-<span class="go"> | x  dx</span>
-<span class="go"> |</span>
-<span class="go">/</span>
-</pre></div>
-</div>
-<p>If you have a unicode font installed, the <tt class="docutils literal"><span class="pre">pprint</span></tt> function will use it by
-default. You can override this using the <tt class="docutils literal"><span class="pre">use_unicode</span></tt> option.:</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">⌠</span>
-<span class="go">⎮  2</span>
-<span class="go">⎮ x  dx</span>
-<span class="go">⌡</span>
-</pre></div>
-</div>
-<p>Погледајте и вики <a class="reference external" href="https://github.com/sympy/sympy/wiki/Pretty-Printing">Pretty Printing</a> за више примера лепог уникодног штампања.</p>
-<p>Tip: To make pretty printing the default in the Python interpreter, use:</p>
-<div class="highlight-python"><pre>$ python
-Python 2.5.2 (r252:60911, Jun 25 2008, 17:58:32)
-[GCC 4.3.1] on linux2
-Type &quot;help&quot;, &quot;copyright&quot;, &quot;credits&quot; or &quot;license&quot; for more information.
-&gt;&gt;&gt; from sympy import init_printing, var, Integral
-&gt;&gt;&gt; init_printing(use_unicode=False, wrap_line=False, no_global=True)
-&gt;&gt;&gt; var(&quot;x&quot;)
-x
-&gt;&gt;&gt; x**3/3
- 3
-x
---
-3
-&gt;&gt;&gt; Integral(x**2, x) #doctest: +NORMALIZE_WHITESPACE
-  /
- |
- |  2
- | x  dx
- |
-/</pre>
-</div>
-<p><strong>Python штампање</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.python</span> <span class="kn">import</span> <span class="n">python</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = x**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = 1/x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = Integral(x**2, x)</span>
-</pre></div>
-</div>
-<p><strong>LaTeX штампање</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">latex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x^{2}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;inline&#39;</span><span class="p">)</span>
-<span class="go">$x^{2}$</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;equation&#39;</span><span class="p">)</span>
-<span class="go">\begin{equation}x^{2}\end{equation}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;equation*&#39;</span><span class="p">)</span>
-<span class="go">\begin{equation*}x^{2}\end{equation*}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">\frac{1}{x}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">\int x^{2}\, dx</span>
-</pre></div>
-</div>
-<p><strong>MathML</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.mathml</span> <span class="kn">import</span> <span class="n">mathml</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">latex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">mathml</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">&lt;apply&gt;&lt;power/&gt;&lt;ci&gt;x&lt;/ci&gt;&lt;cn&gt;2&lt;/cn&gt;&lt;/apply&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">mathml</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">))</span>
-<span class="go">&lt;apply&gt;&lt;power/&gt;&lt;ci&gt;x&lt;/ci&gt;&lt;cn&gt;-1&lt;/cn&gt;&lt;/apply&gt;</span>
-</pre></div>
-</div>
-<p><strong>Pyglet</strong></p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">preview</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">preview</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span> 
-</pre></div>
-</div>
-<p>If pyglet is installed, a pyglet window will open containing the LaTeX
-rendered expression:</p>
-<img alt="../_images/pngview1.png" src="../_images/pngview1.png" />
-<div class="section" id="notes">
-<h3>Белешке<a class="headerlink" href="#notes" title="Permalink to this headline">¶</a></h3>
-<p><tt class="docutils literal"><span class="pre">isympy</span></tt> позива <tt class="docutils literal"><span class="pre">pprint</span></tt> аутоматски , зато видите лепо штампање као подразумевано.</p>
-<p>Такође постоји модул за штампање, <tt class="docutils literal"><span class="pre">sympy.printing</span></tt>. Остали доступни методи штампања су:</p>
-<ul class="simple">
-<li><tt class="docutils literal"><span class="pre">pretty(expr)</span></tt>, <tt class="docutils literal"><span class="pre">pretty_print(expr)</span></tt>, <tt class="docutils literal"><span class="pre">pprint(expr)</span></tt>: Враћа или штампа, лепо представљање од <tt class="docutils literal"><span class="pre">expr</span></tt>. Ово је исто као и други ново представљања опсаног изнад.</li>
-<li><tt class="docutils literal"><span class="pre">latex(expr)</span></tt>, <tt class="docutils literal"><span class="pre">print_latex(expr)</span></tt>: Враћа или штампа <a class="reference external" href="http://www.latex-project.org/">LaTeX</a> представљање од <tt class="docutils literal"><span class="pre">expr</span></tt></li>
-<li><tt class="docutils literal"><span class="pre">mathml(expr)</span></tt>, <tt class="docutils literal"><span class="pre">print_mathml(expr)</span></tt>: Враћа или штампа <a class="reference external" href="http://www.w3.org/Math/">MathML</a> представљање од <tt class="docutils literal"><span class="pre">expr</span></tt>.</li>
-<li><tt class="docutils literal"><span class="pre">print_gtk(expr)</span></tt>: Print <tt class="docutils literal"><span class="pre">expr</span></tt> to <a class="reference external" href="http://helm.cs.unibo.it/mml-widget/">Gtkmathview</a>, a GTK widget that displays MathML code. The <a class="reference external" href="http://helm.cs.unibo.it/mml-widget/">Gtkmathview</a> program is required.</li>
-</ul>
-</div>
-</div>
-<div class="section" id="further-documentation">
-<h2>Додатна документација<a class="headerlink" href="#further-documentation" title="Permalink to this headline">¶</a></h2>
-<p>Сада можете учити даље. Погледајте: <a class="reference external" href="../guide.html">SymPy User&#8217;s Guide</a> и <a class="reference external" href="../modules/index.html">SymPy Modules Reference</a>. Напомена: Остатак документације је доступан само на енглеском језику.</p>
-<p>Be sure to also browse our public <a class="reference external" href="http://wiki.sympy.org/">wiki.sympy.org</a>,
-that contains a lot of useful examples, tutorials, cookbooks that we and our
-users contributed, and feel free to edit it.</p>
-<div class="section" id="translations">
-<h3>Translations<a class="headerlink" href="#translations" title="Permalink to this headline">¶</a></h3>
-<p>This tutorial is also available in other languages:</p>
-</div>
-<ul class="simple">
-<li><a class="reference external" href="tutorial.bg.html">Български</a></li>
-<li><a class="reference external" href="tutorial.cs.html">Česky</a></li>
-<li><a class="reference external" href="tutorial.de.html">Deutsch</a></li>
-<li><a class="reference external" href="tutorial.en.html">English</a></li>
-<li><a class="reference external" href="tutorial.fr.html">Français</a></li>
-<li><a class="reference external" href="tutorial.pl.html">Polski</a></li>
-<li><a class="reference external" href="tutorial.ru.html">Русский</a></li>
-<li><a class="reference external" href="tutorial.sr.html">Српски</a></li>
-<li><a class="reference external" href="tutorial.zh.html">中文</a></li>
-</ul>
-</div>
-</div>
-
-
-          </div>
-        </div>
-      </div>
-      <div class="sphinxsidebar">
-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="tutorial.html">
-              <img class="logo" src="../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="tutorial.html">Table Of Contents</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">Туторијал</a><ul>
-<li><a class="reference internal" href="#introduction">Увод</a></li>
-<li><a class="reference internal" href="#first-steps-with-sympy">Први кораци</a><ul>
-<li><a class="reference internal" href="#isympy-console">isympy конзола</a></li>
-<li><a class="reference internal" href="#using-sympy-as-a-calculator">SymPy као дигитрон</a></li>
-<li><a class="reference internal" href="#symbols">Симболи</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#algebra">Алгебра</a></li>
-<li><a class="reference internal" href="#calculus">Математичка анализа</a><ul>
-<li><a class="reference internal" href="#limits">Лимити</a></li>
-<li><a class="reference internal" href="#differentiation">Изводи</a></li>
-<li><a class="reference internal" href="#series-expansion">Развој редова</a></li>
-<li><a class="reference internal" href="#summation">Summation</a></li>
-<li><a class="reference internal" href="#integration">Интеграција</a></li>
-<li><a class="reference internal" href="#complex-numbers">Комплексни бројеви</a></li>
-<li><a class="reference internal" href="#functions">Функције</a></li>
-<li><a class="reference internal" href="#differential-equations">Диференцијални рачун</a></li>
-<li><a class="reference internal" href="#algebraic-equations">Алгебарска рачун</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#linear-algebra">Линеарна алгебра</a><ul>
-<li><a class="reference internal" href="#matrices">Матрице</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#pattern-matching">Поклапање образаца</a></li>
-<li><a class="reference internal" href="#printing">Штампање</a><ul>
-<li><a class="reference internal" href="#notes">Белешке</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#further-documentation">Додатна документација</a><ul>
-<li><a class="reference internal" href="#translations">Translations</a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-
-  <h3>This Page</h3>
-  <ul class="this-page-menu">
-    <li><a href="../_sources/tutorial/tutorial.txt"
-           rel="nofollow">Show Source</a></li>
-  </ul>
-<div id="searchbox" style="display: none">
-  <h3>Quick search</h3>
-    <form class="search" action="../search.html" method="get">
-      <input type="text" name="q" />
-      <input type="submit" value="Go" />
-      <input type="hidden" name="check_keywords" value="yes" />
-      <input type="hidden" name="area" value="default" />
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-             >index</a></li>
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-  <div class="section" id="tutorial">
-<span id="id1"></span><h1>教程<a class="headerlink" href="#tutorial" title="永久链接至标题">¶</a></h1>
-<div class="section" id="introduction">
-<h2>简介<a class="headerlink" href="#introduction" title="永久链接至标题">¶</a></h2>
-<p>SymPy是一个用来处理数学符号的Python库，旨在成为一个多功能但代码尽可能简洁以便于理解和扩展的计算机代数系统(CAS)。同时，SymPy完全是用Python编写的，并且不依赖任何外部的库。</p>
-<p>本教程主要讲述SymPy的基础知识，阅读它有助于你理解能用SymPy来做什么(怎样做)，如果你想要更深入地学习SymPy，请阅读 <a class="reference external" href="../guide.html">SymPy User&#8217;s Guide</a>, <a class="reference external" href="../modules/index.html">SymPy Modules Reference</a>, 或者直接阅读 <a class="reference external" href="https://github.com/sympy/sympy/">源代码</a> 。</p>
-</div>
-<div class="section" id="first-steps-with-sympy">
-<h2>开始学习SymPy<a class="headerlink" href="#first-steps-with-sympy" title="永久链接至标题">¶</a></h2>
-<p>下载SymPy最简单的方法就是从 <a class="reference external" href="http://code.google.com/p/sympy/">http://code.google.com/p/sympy/</a> 下载最新的压缩包：</p>
-<img alt="../_images/featured-downloads.png" src="../_images/featured-downloads.png" />
-<p>解压：</p>
-<pre class="literal-block">
-$ <strong class="input">tar xzf sympy-0.5.12.tar.gz</strong>
-</pre>
-<p>然后在Python解释器中尝试使用：</p>
-<pre class="literal-block">
-$ <strong class="input">cd sympy-0.5.12</strong>
-$ <strong class="input">python</strong>
-Python 2.4.4 (#2, Jan  3 2008, 13:36:28)
-[GCC 4.2.3 20071123 (prerelease) (Debian 4.2.2-4)] on linux2
-Type &quot;help&quot;, &quot;copyright&quot;, &quot;credits&quot; or &quot;license&quot; for more information.
-&gt;&gt;&gt; from sympy import Symbol, cos
-&gt;&gt;&gt; x = Symbol(&quot;x&quot;)
-&gt;&gt;&gt; (1/cos(x)).series(x, 0, 10)
-1 + x**2/2 + 5*x**4/24 + 61*x**6/720 + 277*x**8/8064 + O(x**10)
-</pre>
-<p>我们推荐你在你的程序中按上面的方法使用SymPy。你还可以像安装其它Python模块一样，通过 <tt class="docutils literal"><span class="pre">./setup.py</span> <span class="pre">install</span></tt> 安装SymPy，或者就像安装一个软件包一样，在你喜欢的Linux发行版中安装它：</p>
-<div class="topic">
-<p class="topic-title first">Installing SymPy in Debian</p>
-<pre class="literal-block">
-$ <strong class="input">sudo apt-get install python-sympy</strong>
-Reading package lists... Done
-Building dependency tree
-Reading state information... Done
-The following NEW packages will be installed:
-  python-sympy
-0 upgraded, 1 newly installed, 0 to remove and 18 not upgraded.
-Need to get 991kB of archives.
-After this operation, 5976kB of additional disk space will be used.
-Get:1 http://ftp.cz.debian.org unstable/main python-sympy 0.5.12-1 [991kB]
-Fetched 991kB in 2s (361kB/s)
-Selecting previously deselected package python-sympy.
-(Reading database ... 232619 files and directories currently installed.)
-Unpacking python-sympy (from .../python-sympy_0.5.12-1_all.deb) ...
-Setting up python-sympy (0.5.12-1) ...
-</pre>
-</div>
-<p>想了解其它安装SymPy的方法，参考wiki页面 <a class="reference external" href="https://github.com/sympy/sympy/wiki/Download-Installation">Download and Installation</a> 。</p>
-<div class="section" id="isympy-console">
-<h3>isympy控制台<a class="headerlink" href="#isympy-console" title="永久链接至标题">¶</a></h3>
-<p>为了验证新特性，或者想弄清楚怎样使用这些特性，你可以使用一个对IPython封装好的包 <tt class="docutils literal"><span class="pre">isympy</span></tt> (如果从源代码目录中运行，则为 <tt class="docutils literal"><span class="pre">bin/isympy</span></tt> )，这只是一个已经导入了相关SymPy模块并且定义了符号x，y，z和一些其他东西的标准Python脚本。</p>
-<pre class="literal-block">
-$ <strong class="input">cd sympy</strong>
-$ <strong class="input">./bin/isympy</strong>
-IPython console for SymPy 0.7.2-git (Python 2.7.1) (ground types: gmpy)
-
-These commands were executed:
-&gt;&gt;&gt;
-&gt;&gt;&gt; from sympy import *
-&gt;&gt;&gt; x, y, z, t = symbols('x y z t')
-&gt;&gt;&gt; k, m, n = symbols('k m n', integer=True)
-&gt;&gt;&gt; f, g, h = symbols('f g h', cls=Function)
-
-Documentation can be found at <a class="reference external" href="http://www.sympy.org">http://www.sympy.org</a>
-
-In [1]: <strong class="input">(1/cos(x)).series(x, 0, 10)</strong>
-Out[1]:
-     2      4       6        8
-    x    5*x    61*x    277*x     / 10\
-1 + -- + ---- + ----- + ------ + O\x  /
-    2     24     720     8064
-</pre>
-<div class="admonition note">
-<p class="first admonition-title">注解</p>
-<p class="last">你所输入的命令是很粗糙的，所以我们在常规的Python解释器中要用三条命令才能完成的事，在isympy中用一条语句就可以完成。</p>
-</div>
-</div>
-<div class="section" id="using-sympy-as-a-calculator">
-<h3>使用SymPy做计算器<a class="headerlink" href="#using-sympy-as-a-calculator" title="永久链接至标题">¶</a></h3>
-<p>SymPy有三个内置的数值类型：浮点型，分数型和整型</p>
-<p>分数类通过一对整型数来表示一个分数：分子和分母，所以Rational(1, 2)表示1/2，Rational(5, 2)表示5/2，等等。</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">Rational</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span>
-<span class="go">1/2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="o">*</span><span class="mi">2</span>
-<span class="go">1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Rational</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">50</span><span class="o">/</span><span class="n">Rational</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span><span class="o">**</span><span class="mi">50</span>
-<span class="go">1/88817841970012523233890533447265625</span>
-</pre></div>
-</div>
-<p>因为你可能建立的是一个基于Python数值类型的数据，而不是基于SymPy数值类型的，所以请小心使用Python中的整型数和浮点数，特别是在除法运算中。两个Python中的整型数的比值可能会形成一个浮点数——这是Python 3所使用的&#8221;真正的除法&#8221;的标准，也是从__future__中导入除法的 <tt class="docutils literal"><span class="pre">isympy</span></tt> 的默认操作。</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span> 
-<span class="go">0.5</span>
-</pre></div>
-</div>
-<p>但是在一些早期的Python版本当中，除法还没有被导入，所以会对除法操作的结果取整，截断小数点后面的部分。</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span> 
-<span class="go">0</span>
-</pre></div>
-</div>
-<p>然而，不管在哪种情况下，你都不是在处理一个SymPy的数据，因为是Python创建它自己的数据。在大多数情况下，你可能需要处理分数，所以这时要确保SymPy的输出结果也是分数。当 <tt class="docutils literal"><span class="pre">R</span></tt> 等同于 <tt class="docutils literal"><span class="pre">Rational</span></tt> 时，更便于我们编写语句：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">R</span> <span class="o">=</span> <span class="n">Rational</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">1/2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">R</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span> <span class="c"># R(1) is a SymPy Integer and Integer/int gives a Rational</span>
-<span class="go">1/2</span>
-</pre></div>
-</div>
-<p>SymPy中还有一些特殊的常量，例如e和pi，这些常量在SymPy中被看作是符号(1 + pi不会等于一个数值，它只是等于1 + pi)，并且拥有任意精度：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">E</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">pi**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pi</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">3.14159265358979</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">pi</span> <span class="o">+</span> <span class="n">E</span><span class="p">)</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
-<span class="go">5.85987448204884</span>
-</pre></div>
-</div>
-<p>正如你所看到的，通过evalf函数对以上表达式求值，可以得到一个浮点数。</p>
-<p>符号 <tt class="docutils literal"><span class="pre">oo</span></tt> 表示数学上所说的无穷：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">oo</span> <span class="o">&gt;</span> <span class="mi">99999</span>
-<span class="go">True</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">oo</span> <span class="o">+</span> <span class="mi">1</span>
-<span class="go">oo</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="symbols">
-<h3>符号<a class="headerlink" href="#symbols" title="永久链接至标题">¶</a></h3>
-<p>跟其它计算机代数系统不同，在SymPy中必须明确声明符号变量：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>左边的是常规的Python变量，已经将SymPy符号类分配给这些变量。预定义符号(包括含有希腊字母的符号)可以从abc中导入：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">theta</span>
-</pre></div>
-</div>
-<p>符号还可以通过 <tt class="docutils literal"><span class="pre">symbols</span></tt> 或者 <tt class="docutils literal"><span class="pre">var</span></tt> 函数来创建，其中，使用 <tt class="docutils literal"><span class="pre">var</span></tt> 函数会自动添加新建的符号到命名空间，这两个函数都允许使用范围表示作为参数：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">var</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;a,b,c&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">d</span><span class="p">,</span> <span class="n">e</span><span class="p">,</span> <span class="n">f</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;d:f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;g:h&#39;</span><span class="p">)</span>
-<span class="go">(g, h)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">var</span><span class="p">(</span><span class="s">&#39;g:2&#39;</span><span class="p">)</span>
-<span class="go">(g0, g1)</span>
-</pre></div>
-</div>
-<p>符号类的实例能够&#8221;很好地结合&#8221;，由多个符号可以构建一个表达式：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="o">+</span> <span class="n">x</span> <span class="o">-</span> <span class="n">y</span>
-<span class="go">2*x</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">(x + y)**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go">x**2 + 2*x*y + y**2</span>
-</pre></div>
-</div>
-<p>使用函数 <tt class="docutils literal"><span class="pre">subs(old,</span> <span class="pre">new)</span></tt> 可以将给定的符号代换成其它的数字，符号或者表达式：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">(y + 1)**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">4*y**2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>对于此教程的余下部分，我们假设已经运行以下语句：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">init_printing</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">init_printing</span><span class="p">(</span><span class="n">use_unicode</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">wrap_line</span><span class="o">=</span><span class="bp">False</span><span class="p">,</span> <span class="n">no_global</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>运行这些语句会让输出的结果更雅观，查看以下章节 <a class="reference internal" href="#printing-tutorial"><em>打印</em></a> ，如果你已经安装了unicode字体，可以设置 use_unicode=True，使得输出更加美观。</p>
-</div>
-</div>
-<div class="section" id="algebra">
-<h2>代数<a class="headerlink" href="#algebra" title="永久链接至标题">¶</a></h2>
-<p>实现部分分数分解，使用函数 <tt class="docutils literal"><span class="pre">apart(expr,</span> <span class="pre">x)</span></tt> ：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">apart</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">)</span>
-<span class="go">       1</span>
-<span class="go">---------------</span>
-<span class="go">(x + 1)*(x + 2)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">apart</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">    1       1</span>
-<span class="go">- ----- + -----</span>
-<span class="go">  x + 2   x + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">x + 1</span>
-<span class="go">-----</span>
-<span class="go">x - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">apart</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">      2</span>
-<span class="go">1 + -----</span>
-<span class="go">    x - 1</span>
-</pre></div>
-</div>
-<p>将分数组合在一起，使用函数 <tt class="docutils literal"><span class="pre">together(expr,</span> <span class="pre">x)</span></tt> ：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">together</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">z</span><span class="p">)</span>
-<span class="go">x*y + x*z + y*z</span>
-<span class="go">---------------</span>
-<span class="go">     x*y*z</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="n">apart</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x + 1</span>
-<span class="go">-----</span>
-<span class="go">x - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">together</span><span class="p">(</span><span class="n">apart</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">),</span> <span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">       1</span>
-<span class="go">---------------</span>
-<span class="go">(x + 1)*(x + 2)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="calculus">
-<span id="index-0"></span><h2>微积分<a class="headerlink" href="#calculus" title="永久链接至标题">¶</a></h2>
-<div class="section" id="limits">
-<span id="index-1"></span><h3>极限<a class="headerlink" href="#limits" title="永久链接至标题">¶</a></h3>
-<p>在SymPy中使用极限很简单，使用函数 <tt class="docutils literal"><span class="pre">limit(function,</span> <span class="pre">variable,</span> <span class="pre">point)</span></tt> ，所以为了计算当 x -&gt; 0 时f(x)的极限，只需要输入 <tt class="docutils literal"><span class="pre">limit(f,</span> <span class="pre">x,</span> <span class="pre">0)</span></tt> ：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">limit</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">oo</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>你还可以计算在无穷远处的极限：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">oo</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">oo</span><span class="p">)</span>
-<span class="go">0</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">1</span>
-</pre></div>
-</div>
-<p>对于一些关于极限的非平凡的例子，你可以查阅测试文件 <a class="reference external" href="https://github.com/sympy/sympy/blob/master/sympy/series/tests/test_demidovich.py">test_demidovich.py</a></p>
-</div>
-<div class="section" id="differentiation">
-<span id="index-2"></span><h3>求导<a class="headerlink" href="#differentiation" title="永久链接至标题">¶</a></h3>
-<p>你可以对任何SymPy的表达式进行求导，使用函数 <tt class="docutils literal"><span class="pre">diff(func,</span> <span class="pre">var)</span></tt> ，例如：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">diff</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">tan</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">cos(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">2*cos(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">tan (x) + 1</span>
-</pre></div>
-</div>
-<p>你可以检查，但是这是正确的：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">limit</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">delta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">limit</span><span class="p">((</span><span class="n">tan</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">delta</span><span class="p">)</span> <span class="o">-</span> <span class="n">tan</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">/</span><span class="n">delta</span><span class="p">,</span> <span class="n">delta</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">tan (x) + 1</span>
-</pre></div>
-</div>
-<p>计算更高阶的导数，使用函数 <tt class="docutils literal"><span class="pre">diff(func,</span> <span class="pre">var,</span> <span class="pre">n)</span></tt> ：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">2*cos(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">-4*sin(2*x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">diff</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">-8*cos(2*x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="series-expansion">
-<span id="index-3"></span><h3>级数展开<a class="headerlink" href="#series-expansion" title="永久链接至标题">¶</a></h3>
-<p>使用 <tt class="docutils literal"><span class="pre">.series(var,</span> <span class="pre">point,</span> <span class="pre">order)</span></tt> ：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">cos</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     2    4     6      8</span>
-<span class="go">    x    x     x      x      / 10\</span>
-<span class="go">1 - -- + -- - --- + ----- + O\x  /</span>
-<span class="go">    2    24   720   40320</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">))</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     2      4       6        8</span>
-<span class="go">    x    5*x    61*x    277*x     / 10\</span>
-<span class="go">1 + -- + ---- + ----- + ------ + O\x  /</span>
-<span class="go">    2     24     720     8064</span>
-</pre></div>
-</div>
-<p>另一个简单的例子：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">pprint</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;y&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">e</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">s</span> <span class="o">=</span> <span class="n">e</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="go">1/y - x/y**2 + x**2/y**3 - x**3/y**4 + x**4/y**5 + O(x**5)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
-<span class="go">          2    3    4</span>
-<span class="go">1   x    x    x    x     / 5\</span>
-<span class="go">- - -- + -- - -- + -- + O\x /</span>
-<span class="go">y    2    3    4    5</span>
-<span class="go">    y    y    y    y</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="summation">
-<span id="index-4"></span><h3>求和<a class="headerlink" href="#summation" title="永久链接至标题">¶</a></h3>
-<p>计算求和变量在给定的范围内函数f的和。</p>
-<p>函数summation(f, (i, a, b))计算当i在[a, b]内时f的和是多少，即：</p>
-<div class="highlight-python"><pre>                            b
-                          ____
-                          \   `
-summation(f, (i, a, b)) =  )    f
-                          /___,
-                          i = a</pre>
-</div>
-<p>如果不能求和，会输出相应的求和公式。通过引入变量的额外限制，可以复合求和：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">summation</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">log</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">i</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;i n m&#39;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
-<span class="go"> 2</span>
-<span class="go">n</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span><span class="o">**</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">log</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">**</span><span class="n">n</span><span class="p">,</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">  oo</span>
-<span class="go"> ___</span>
-<span class="go"> \  `</span>
-<span class="go">  \     -n</span>
-<span class="go">  /   log (n)</span>
-<span class="go"> /__,</span>
-<span class="go">n = 2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">),</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">m</span><span class="p">))</span>
-<span class="go">      3    2</span>
-<span class="go">m    m    m</span>
-<span class="go">-- + -- + -</span>
-<span class="go">6    2    3</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">summation</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="n">n</span><span class="o">/</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go"> x</span>
-<span class="go">e</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="integration">
-<span id="index-5"></span><h3>积分<a class="headerlink" href="#integration" title="永久链接至标题">¶</a></h3>
-<p>SymPy支持初等超越函数和特使函数的不定积分和定积分，通过使用函数 <tt class="docutils literal"><span class="pre">integrate()</span></tt> ，这个函数用到了强大的扩展Risch-Norman算法、一些启发式算法和模式匹配：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">integrate</span><span class="p">,</span> <span class="n">erf</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">log</span><span class="p">,</span> <span class="n">oo</span><span class="p">,</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-</pre></div>
-</div>
-<p>求初等函数的积分：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="mi">6</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">5</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go"> 6</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">-cos(x)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">x*log(x) - x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="n">sinh</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go"> 2</span>
-<span class="go">x  + cosh(x)</span>
-</pre></div>
-</div>
-<p>求特殊函数的积分也很容易：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">erf</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">  ____    2</span>
-<span class="go">\/ pi *erf (x)</span>
-<span class="go">--------------</span>
-<span class="go">      4</span>
-</pre></div>
-</div>
-<p>计算定积分：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">0</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">2</span>
-</pre></div>
-</div>
-<p>SymPy还支持广义积分：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">oo</span><span class="p">))</span>
-<span class="go">1</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">integrate</span><span class="p">(</span><span class="n">log</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
-<span class="go">-1</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="complex-numbers">
-<span id="index-6"></span><h3>复数<a class="headerlink" href="#complex-numbers" title="永久链接至标题">¶</a></h3>
-<p>除了虚数单位I，其它符号的创建都可以带有属性(例如实数、正数、复数等等)，而这些属性会影响操作结果：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span> <span class="c"># a plain x with no attributes</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">()</span>
-<span class="go"> I*x</span>
-<span class="go">e</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">   -im(x)               -im(x)</span>
-<span class="go">I*e      *sin(re(x)) + e      *cos(re(x))</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">,</span> <span class="n">real</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">exp</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="nb">complex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">I*sin(x) + cos(x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="functions">
-<h3>函数<a class="headerlink" href="#functions" title="永久链接至标题">¶</a></h3>
-<p>三角函数</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">asin</span><span class="p">,</span> <span class="n">asinh</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">sinh</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">I</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">sin(x)*cos(y) + sin(y)*cos(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">cos</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">y</span><span class="p">)</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">trig</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">-sin(x)*sin(y) + cos(x)*cos(y)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*sinh(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinh</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*sin(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
-<span class="go">I*pi</span>
-<span class="go">----</span>
-<span class="go"> 2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">I</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
-<span class="go">I*asin(x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3     5     7       9</span>
-<span class="go">    x     x     x       x       / 10\</span>
-<span class="go">x - -- + --- - ---- + ------ + O\x  /</span>
-<span class="go">    6    120   5040   362880</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">sinh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3     5     7       9</span>
-<span class="go">    x     x     x       x       / 10\</span>
-<span class="go">x + -- + --- + ---- + ------ + O\x  /</span>
-<span class="go">    6    120   5040   362880</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3      5      7       9</span>
-<span class="go">    x    3*x    5*x    35*x     / 10\</span>
-<span class="go">x + -- + ---- + ---- + ----- + O\x  /</span>
-<span class="go">    6     40    112     1152</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">asinh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
-<span class="go">     3      5      7       9</span>
-<span class="go">    x    3*x    5*x    35*x     / 10\</span>
-<span class="go">x - -- + ---- - ---- + ----- + O\x  /</span>
-<span class="go">    6     40    112     1152</span>
-</pre></div>
-</div>
-<p>球谐函数</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Ylm</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">  ___</span>
-<span class="go">\/ 3 *cos(theta)</span>
-<span class="go">----------------</span>
-<span class="go">        ____</span>
-<span class="go">    2*\/ pi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">   ___  I*phi</span>
-<span class="go">-\/ 6 *e     *sin(theta)</span>
-<span class="go">------------------------</span>
-<span class="go">            ____</span>
-<span class="go">        4*\/ pi</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Ylm</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">theta</span><span class="p">,</span> <span class="n">phi</span><span class="p">)</span>
-<span class="go">   ____  I*phi</span>
-<span class="go">-\/ 30 *e     *sin(theta)*cos(theta)</span>
-<span class="go">------------------------------------</span>
-<span class="go">                  ____</span>
-<span class="go">              4*\/ pi</span>
-</pre></div>
-</div>
-<p>阶乘和gamma函数</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;x&quot;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">n</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&quot;n&quot;</span><span class="p">,</span> <span class="n">integer</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">x!</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<span class="go">n!</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">gamma</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">series</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="c"># i.e. factorial(x)</span>
-<span class="go">                      /          2     2\</span>
-<span class="go">                    2 |EulerGamma    pi |    / 3\</span>
-<span class="go">1 - EulerGamma*x + x *|----------- + ---| + O\x /</span>
-<span class="go">                      \     2         12/</span>
-</pre></div>
-</div>
-<p>zeta函数</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">zeta</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">zeta(4, x)</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
-<span class="go">  4</span>
-<span class="go">pi</span>
-<span class="go">---</span>
-<span class="go"> 90</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
-<span class="go">       4</span>
-<span class="go">     pi</span>
-<span class="go">-1 + ---</span>
-<span class="go">      90</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">zeta</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
-<span class="go">         4</span>
-<span class="go">  17   pi</span>
-<span class="go">- -- + ---</span>
-<span class="go">  16    90</span>
-</pre></div>
-</div>
-<p>多项式</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">assoc_legendre</span><span class="p">,</span> <span class="n">chebyshevt</span><span class="p">,</span> <span class="n">legendre</span><span class="p">,</span> <span class="n">hermite</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">2*x  - 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">chebyshevt</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   4      2</span>
-<span class="go">8*x  - 8*x  + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   2</span>
-<span class="go">3*x    1</span>
-<span class="go">---- - -</span>
-<span class="go"> 2     2</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">legendre</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">      8         6         4        2</span>
-<span class="go">6435*x    3003*x    3465*x    315*x     35</span>
-<span class="go">------- - ------- + ------- - ------ + ---</span>
-<span class="go">  128        32        64       32     128</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">        __________</span>
-<span class="go">       /    2</span>
-<span class="go">-3*x*\/  - x  + 1</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">assoc_legendre</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">     2</span>
-<span class="go">- 3*x  + 3</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">hermite</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">   3</span>
-<span class="go">8*x  - 12*x</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="differential-equations">
-<span id="index-7"></span><h3>微分方程<a class="headerlink" href="#differential-equations" title="永久链接至标题">¶</a></h3>
-<p>在 <tt class="docutils literal"><span class="pre">isympy</span></tt> ：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Function</span><span class="p">,</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">dsolve</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span> <span class="o">=</span> <span class="n">Function</span><span class="p">(</span><span class="s">&#39;f&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
-<span class="go">        2</span>
-<span class="go">       d</span>
-<span class="go">f(x) + ---(f(x))</span>
-<span class="go">         2</span>
-<span class="go">       dx</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">dsolve</span><span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
-<span class="go">f(x) = C1*sin(x) + C2*cos(x)</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="algebraic-equations">
-<span id="index-8"></span><h3>代数方程<a class="headerlink" href="#algebraic-equations" title="永久链接至标题">¶</a></h3>
-<p>在 <tt class="docutils literal"><span class="pre">isympy</span></tt> ：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">solve</span><span class="p">,</span> <span class="n">symbols</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s">&#39;x,y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">4</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
-<span class="go">[-1, 1, -I, I]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">solve</span><span class="p">([</span><span class="n">x</span> <span class="o">+</span> <span class="mi">5</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">6</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">15</span><span class="p">],</span> <span class="p">[</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">])</span>
-<span class="go">{x: -3, y: 1}</span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="section" id="linear-algebra">
-<span id="index-9"></span><h2>线性代数<a class="headerlink" href="#linear-algebra" title="永久链接至标题">¶</a></h2>
-<div class="section" id="matrices">
-<span id="index-10"></span><h3>矩阵<a class="headerlink" href="#matrices" title="永久链接至标题">¶</a></h3>
-<p>新建的矩阵是矩阵类的实例：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">Symbol</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="go">[1  0]</span>
-<span class="go">[    ]</span>
-<span class="go">[0  1]</span>
-</pre></div>
-</div>
-<p>矩阵还可以包含符号：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;y&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">],</span> <span class="p">[</span><span class="n">y</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span>
-<span class="go">[1  x]</span>
-<span class="go">[    ]</span>
-<span class="go">[y  1]</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">A</span><span class="o">**</span><span class="mi">2</span>
-<span class="go">[x*y + 1    2*x  ]</span>
-<span class="go">[                ]</span>
-<span class="go">[  2*y    x*y + 1]</span>
-</pre></div>
-</div>
-<p>想了解更多关于矩阵的内容，请看线性代数教程。</p>
-</div>
-</div>
-<div class="section" id="pattern-matching">
-<span id="index-11"></span><h2>模式匹配<a class="headerlink" href="#pattern-matching" title="永久链接至标题">¶</a></h2>
-<p>使用 <tt class="docutils literal"><span class="pre">.match()</span></tt> 模式和 <tt class="docutils literal"><span class="pre">Wild</span></tt> 类，可以对表达式执行模式匹配，返回结果就是所需要的各个匹配值，以字典的形式呈现，就像：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Symbol</span><span class="p">,</span> <span class="n">Wild</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">Symbol</span><span class="p">(</span><span class="s">&#39;x&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">{p: 5}</span>
-
-<span class="gp">&gt;&gt;&gt; </span><span class="n">q</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;q&#39;</span><span class="p">)</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="n">q</span><span class="p">)</span>
-<span class="go">{p: 1, q: 2}</span>
-</pre></div>
-</div>
-<p>如果匹配不成功，则返回 <tt class="docutils literal"><span class="pre">None</span></tt> ：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span><span class="o">**</span><span class="n">x</span><span class="p">))</span>
-<span class="go">None</span>
-</pre></div>
-</div>
-<p>还可以在 <tt class="docutils literal"><span class="pre">Wild</span></tt> 类中添加排除参数，使得这些被排除的参数不会出现在结果当中：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">p</span> <span class="o">=</span> <span class="n">Wild</span><span class="p">(</span><span class="s">&#39;p&#39;</span><span class="p">,</span> <span class="n">exclude</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="n">p</span><span class="p">))</span> <span class="c"># 1 is excluded</span>
-<span class="go">None</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">p</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span> <span class="c"># x is excluded</span>
-<span class="go">None</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">((</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">match</span><span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">p</span><span class="p">))</span> <span class="c"># -1 is not excluded</span>
-<span class="go">{p_: -1}</span>
-</pre></div>
-</div>
-</div>
-<div class="section" id="printing">
-<span id="printing-tutorial"></span><h2>打印<a class="headerlink" href="#printing" title="永久链接至标题">¶</a></h2>
-<p>打印表达式有很多方式。</p>
-<p>标准打印</p>
-<p>就是 <tt class="docutils literal"><span class="pre">str(expression)</span></tt> 的返回值，它看起来就像：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1/x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">Integral(x**2, x)</span>
-</pre></div>
-</div>
-<p>Pretty打印</p>
-<p>使用 <tt class="docutils literal"><span class="pre">pprint</span></tt> 函数可以打印出好看的ASCII形式：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">pprint</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go"> 2</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">1</span>
-<span class="go">-</span>
-<span class="go">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">  /</span>
-<span class="go"> |</span>
-<span class="go"> |  2</span>
-<span class="go"> | x  dx</span>
-<span class="go"> |</span>
-<span class="go">/</span>
-</pre></div>
-</div>
-<p>如果你安装了unicode字体，那么打印方式会默认使用 <tt class="docutils literal"><span class="pre">pprint</span></tt> 函数，可以使用 <tt class="docutils literal"><span class="pre">use_unicode</span></tt> 选项来替换这种打印方式：</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="n">pprint</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">),</span> <span class="n">use_unicode</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
-<span class="go">⌠</span>
-<span class="go">⎮  2</span>
-<span class="go">⎮ x  dx</span>
-<span class="go">⌡</span>
-</pre></div>
-</div>
-<p>想获得更多unicode打印方式的例子，还可以查看维基百科 <a class="reference external" href="https://github.com/sympy/sympy/wiki/Pretty-Printing">Pretty Printing</a> 。</p>
-<p>提示：通过这种方法可以在Python解释器中默认使用pretty打印：</p>
-<div class="highlight-python"><pre>$ python
-Python 2.5.2 (r252:60911, Jun 25 2008, 17:58:32)
-[GCC 4.3.1] on linux2
-Type &quot;help&quot;, &quot;copyright&quot;, &quot;credits&quot; or &quot;license&quot; for more information.
-&gt;&gt;&gt; from sympy import init_printing, var, Integral
-&gt;&gt;&gt; init_printing(use_unicode=False, wrap_line=False, no_global=True)
-&gt;&gt;&gt; var(&quot;x&quot;)
-x
-&gt;&gt;&gt; x**3/3
- 3
-x
---
-3
-&gt;&gt;&gt; Integral(x**2, x) #doctest: +NORMALIZE_WHITESPACE
-  /
- |
- |  2
- | x  dx
- |
-/</pre>
-</div>
-<p>Python打印</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.python</span> <span class="kn">import</span> <span class="n">python</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = x**2</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = 1/x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">python</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">)))</span>
-<span class="go">x = Symbol(&#39;x&#39;)</span>
-<span class="go">e = Integral(x**2, x)</span>
-</pre></div>
-</div>
-<p>LaTeX打印</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">latex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
-<span class="go">x^{2}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;inline&#39;</span><span class="p">)</span>
-<span class="go">$x^{2}$</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;equation&#39;</span><span class="p">)</span>
-<span class="go">\begin{equation}x^{2}\end{equation}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">mode</span><span class="o">=</span><span class="s">&#39;equation*&#39;</span><span class="p">)</span>
-<span class="go">\begin{equation*}x^{2}\end{equation*}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">)</span>
-<span class="go">\frac{1}{x}</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">latex</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span>
-<span class="go">\int x^{2}\, dx</span>
-</pre></div>
-</div>
-<p>MathML</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.printing.mathml</span> <span class="kn">import</span> <span class="n">mathml</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">latex</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">mathml</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
-<span class="go">&lt;apply&gt;&lt;power/&gt;&lt;ci&gt;x&lt;/ci&gt;&lt;cn&gt;2&lt;/cn&gt;&lt;/apply&gt;</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="k">print</span><span class="p">(</span><span class="n">mathml</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">x</span><span class="p">))</span>
-<span class="go">&lt;apply&gt;&lt;power/&gt;&lt;ci&gt;x&lt;/ci&gt;&lt;cn&gt;-1&lt;/cn&gt;&lt;/apply&gt;</span>
-</pre></div>
-</div>
-<p>Pyglet</p>
-<div class="highlight-python"><div class="highlight"><pre><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">Integral</span><span class="p">,</span> <span class="n">preview</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.abc</span> <span class="kn">import</span> <span class="n">x</span>
-<span class="gp">&gt;&gt;&gt; </span><span class="n">preview</span><span class="p">(</span><span class="n">Integral</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">x</span><span class="p">))</span> 
-</pre></div>
-</div>
-<p>如果安装了pyglet，一个呈现LaTeX表达式的窗口会被打开：</p>
-<img alt="../_images/pngview1.png" src="../_images/pngview1.png" />
-<div class="section" id="notes">
-<h3>注释<a class="headerlink" href="#notes" title="永久链接至标题">¶</a></h3>
-<p><tt class="docutils literal"><span class="pre">isympy</span></tt> 自动调用 <tt class="docutils literal"><span class="pre">pprint</span></tt> ，所以在默认情况下你看到的是pretty打印形式的输出。</p>
-<p>值得注意的是，SymPy还提供一个打印模块 <tt class="docutils literal"><span class="pre">sympy.printing</span></tt> ，这个模块提供的打印方式有：</p>
-<ul class="simple">
-<li><tt class="docutils literal"><span class="pre">pretty(expr)</span></tt>, <tt class="docutils literal"><span class="pre">pretty_print(expr)</span></tt>, <tt class="docutils literal"><span class="pre">pprint(expr)</span></tt> ：这三个函数都返回或打印 <tt class="docutils literal"><span class="pre">expr</span></tt> 的pretty表示形式，这跟前面描述的第二种表示方法一样。</li>
-<li><tt class="docutils literal"><span class="pre">latex(expr)</span></tt>, <tt class="docutils literal"><span class="pre">print_latex(expr)</span></tt> ：这两个函数都返回或打印 <tt class="docutils literal"><span class="pre">expr</span></tt> 的 <a class="reference external" href="http://www.latex-project.org/">LaTeX</a> 表示形式。</li>
-<li><tt class="docutils literal"><span class="pre">mathml(expr)</span></tt>, <tt class="docutils literal"><span class="pre">print_mathml(expr)</span></tt> ：这两个函数都返回或打印 <tt class="docutils literal"><span class="pre">expr</span></tt> 的 <a class="reference external" href="http://www.w3.org/Math/">MathML</a> 表示形式。</li>
-<li><tt class="docutils literal"><span class="pre">print_gtk(expr)</span></tt>: Print <tt class="docutils literal"><span class="pre">expr</span></tt> to <a class="reference external" href="http://helm.cs.unibo.it/mml-widget/">Gtkmathview</a>, a GTK widget that displays MathML code. The <a class="reference external" href="http://helm.cs.unibo.it/mml-widget/">Gtkmathview</a> program is required.</li>
-</ul>
-</div>
-</div>
-<div class="section" id="further-documentation">
-<h2>更详细的文档<a class="headerlink" href="#further-documentation" title="永久链接至标题">¶</a></h2>
-<p>看完以上的教程之后，现在是时候进一步了解SymPy，请阅读 <a class="reference external" href="../guide.html">SymPy User&#8217;s Guide</a> 和 <a class="reference external" href="../modules/index.html">SymPy Modules Reference</a> 。</p>
-<p>还可以浏览我们的公共网页 <a class="reference external" href="http://wiki.sympy.org/">wiki.sympy.org</a> ，上面有很多有用的例子、教程和cookbooks供大家参考，这些资料都是我们和用户编辑的，欢迎您的参与。</p>
-<div class="section" id="translations">
-<h3>翻译<a class="headerlink" href="#translations" title="永久链接至标题">¶</a></h3>
-<p>此教程还有其它的语言版本：</p>
-</div>
-<ul class="simple">
-<li><a class="reference external" href="tutorial.bg.html">Български</a></li>
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-<li><a class="reference external" href="tutorial.ru.html">Русский</a></li>
-<li><a class="reference external" href="tutorial.sr.html">Српски</a></li>
-<li><a class="reference external" href="tutorial.zh.html">中文</a></li>
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-</div>
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-        <div class="sphinxsidebarwrapper">
-            <p class="logo"><a href="tutorial.html">
-              <img class="logo" src="../_static/sympylogo.png" alt="Logo"/>
-            </a></p>
-  <h3><a href="tutorial.html">內容目录</a></h3>
-  <ul>
-<li><a class="reference internal" href="#">教程</a><ul>
-<li><a class="reference internal" href="#introduction">简介</a></li>
-<li><a class="reference internal" href="#first-steps-with-sympy">开始学习SymPy</a><ul>
-<li><a class="reference internal" href="#isympy-console">isympy控制台</a></li>
-<li><a class="reference internal" href="#using-sympy-as-a-calculator">使用SymPy做计算器</a></li>
-<li><a class="reference internal" href="#symbols">符号</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#algebra">代数</a></li>
-<li><a class="reference internal" href="#calculus">微积分</a><ul>
-<li><a class="reference internal" href="#limits">极限</a></li>
-<li><a class="reference internal" href="#differentiation">求导</a></li>
-<li><a class="reference internal" href="#series-expansion">级数展开</a></li>
-<li><a class="reference internal" href="#summation">求和</a></li>
-<li><a class="reference internal" href="#integration">积分</a></li>
-<li><a class="reference internal" href="#complex-numbers">复数</a></li>
-<li><a class="reference internal" href="#functions">函数</a></li>
-<li><a class="reference internal" href="#differential-equations">微分方程</a></li>
-<li><a class="reference internal" href="#algebraic-equations">代数方程</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#linear-algebra">线性代数</a><ul>
-<li><a class="reference internal" href="#matrices">矩阵</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#pattern-matching">模式匹配</a></li>
-<li><a class="reference internal" href="#printing">打印</a><ul>
-<li><a class="reference internal" href="#notes">注释</a></li>
-</ul>
-</li>
-<li><a class="reference internal" href="#further-documentation">更详细的文档</a><ul>
-<li><a class="reference internal" href="#translations">翻译</a></li>
-</ul>
-</li>
-</ul>
-</li>
-</ul>
-
-  <h3>本页</h3>
-  <ul class="this-page-menu">
-    <li><a href="../_sources/tutorial/tutorial.txt"
-           rel="nofollow">显示源代码</a></li>
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-  <h3>快速搜索</h3>
-    <form class="search" action="../search.html" method="get">
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-    输入相关的模块，术语，类或者函数名称进行搜索
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-      <h3>导航</h3>
-      <ul>
-        <li class="right" style="margin-right: 10px">
-          <a href="../genindex.html" title="总目录"
-             >索引</a></li>
-        <li><a href="tutorial.html">SymPy 0.7.2-git documentation</a> &raquo;</li> 
-      </ul>
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-      最后更新日期是 Jan 20, 2013.
-      使用 <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.1.3.
-    </div>
-  </body>
-</html>
diff --git a/dev-py3k/wiki.html b/dev-py3k/wiki.html
deleted file mode 100644
index f1644a09686..00000000000
--- a/dev-py3k/wiki.html
+++ /dev/null
@@ -1,154 +0,0 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
-
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-<html xmlns="http://www.w3.org/1999/xhtml">
-  <head>
-    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
-    
-    <title>Wiki &mdash; SymPy 0.7.2-git documentation</title>
-    
-    <link rel="stylesheet" href="_static/default.css" type="text/css" />
-    <link rel="stylesheet" href="_static/pygments.css" type="text/css" />
-    <link rel="stylesheet" href="http://live.sympy.org/static/live-core.css" type="text/css" />
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-    <link rel="stylesheet" href="http://live.sympy.org/static/live-sphinx.css" type="text/css" />
-    
-    <script type="text/javascript">
-      var DOCUMENTATION_OPTIONS = {
-        URL_ROOT:    '',
-        VERSION:     '0.7.2-git',
-        COLLAPSE_INDEX: false,
-        FILE_SUFFIX: '.html',
-        HAS_SOURCE:  true
-      };
-    </script>
-    <script type="text/javascript" src="_static/jquery.js"></script>
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-    <script type="text/javascript" src="http://live.sympy.org/static/utilities.js"></script>
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-    <link rel="prev" title="Development Tips: Comparisons in Python" href="python-comparisons.html" /> 
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-<h1>Wiki<a class="headerlink" href="#wiki" title="Permalink to this headline">¶</a></h1>
-<p>We have a public wiki: <a class="reference external" href="http://wiki.sympy.org">http://wiki.sympy.org</a></p>
-<p>Anyone please feel free to contribute to it anything you find
-interesting/useful to other users.</p>
-<div class="section" id="faq">
-<h2>FAQ<a class="headerlink" href="#faq" title="Permalink to this headline">¶</a></h2>
-<p><a class="reference external" href="https://github.com/sympy/sympy/wiki/Faq">FAQ</a> is one of the very useful wiki pages.</p>
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diff --git a/dev/index.html b/dev/index.html
index e77f8f83159..403a5acde3f 100644
--- a/dev/index.html
+++ b/dev/index.html
@@ -197,7 +197,6 @@ <h4>Docs for other versions</h4>
             <p class="topless"><a href="../0.7.2-py3k/index.html">SymPy 0.7.2 (Python 3)</a></p>
             <p class="topless"><a href="../0.7.3/index.html">SymPy 0.7.3</a></p>
             <p class="topless"><a href="../dev/index.html">SymPy git</a></p>
-            <p class="topless"><a href="../dev-py3k/index.html">SymPy git (Python 3)</a></p>
             <p class="topless"><a href="../0.7.4/index.html">SymPy 0.7.4</a></p>
             <p class="topless"><a href="../0.7.4.1/index.html">SymPy 0.7.4.1</a></p>
             <p class="topless"><a href="../0.7.5/index.html">SymPy 0.7.5</a></p>
diff --git a/releases.txt b/releases.txt
index 72785038841..3c68c773ab5 100644
--- a/releases.txt
+++ b/releases.txt
@@ -5,7 +5,6 @@
 0.7.2-py3k:SymPy 0.7.2 (Python 3)
 0.7.3:SymPy 0.7.3
 dev:SymPy git
-dev-py3k:SymPy git (Python 3)
 0.7.4:SymPy 0.7.4
 0.7.4.1:SymPy 0.7.4.1
 0.7.5:SymPy 0.7.5

From 8f3585a980688e997c461a4b45bc9296d19ccbf0 Mon Sep 17 00:00:00 2001
From: Aaron Meurer <asmeurer@gmail.com>
Date: Thu, 11 Feb 2016 11:45:38 -0500
Subject: [PATCH 2/2] Remove dev-py3k from latest/index.html

The generate_indexes script didn't change it for some reason.
---
 latest/index.html | 479 +++++++++++++++++++++++-----------------------
 1 file changed, 239 insertions(+), 240 deletions(-)

diff --git a/latest/index.html b/latest/index.html
index 3998731fb2a..11ffbf90127 100644
--- a/latest/index.html
+++ b/latest/index.html
@@ -1,195 +1,195 @@
-
-
-<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
-  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
-
-
-<html xmlns="http://www.w3.org/1999/xhtml">
-  <head>
-    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
-    
-    <title>Welcome to SymPy’s documentation! &mdash; SymPy 0.7.6.1 documentation</title>
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-      var DOCUMENTATION_OPTIONS = {
-        URL_ROOT:    '',
-        VERSION:     '0.7.6.1',
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-<h1>Welcome to SymPy&#8217;s documentation!<a class="headerlink" href="#welcome-to-sympy-s-documentation" title="Permalink to this headline">¶</a></h1>
-<p><a class="reference external" href="http://sympy.org">SymPy</a> is a Python library for symbolic mathematics.
-If you are new to SymPy, start with the <a class="reference internal" href="tutorial/index.html#tutorial"><em>Tutorial</em></a>.</p>
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